with respect to the field that ranges from 5 to 20 (with a preferred value
$f_{enh}=6.0^{+9.0}_{-3.0}$ with respect to the COSMOS field at comparable
redshifts and stellar mass range), suggesting a significant environmental
effect triggering nuclear activity in the Spiderweb protocluster.

Unfortunately, the decreasing sensitivity of Chandra due to the contamination
layer on the ACIS detector makes future studies of this kind very hard, and we
can realistically foresee that only a handful of new targets will be studied
with comparable detail in the future. Nevertheless, the confirmation of an
enhanced X-ray AGN activity in protoclusters is an important ingredient with
which to plan studies of high-z large-scale structure with future X-ray
facilities. By exploiting the available multiwavelength dataset on the
Spiderweb field, we plan to further explore the properties of the X-ray
protocluster members to investigate the main physical mechanism responsible
for triggering the X-ray emission.

###### Acknowledgements.

This work was carried out during the ongoing COVID-19 pandemic. The authors
would like to acknowledge the health workers all over the world for their role
in fighting in the frontline of this crisis. We thank the anonymous referee
for detailed comments and positive criticism that helped improving the quality
of the paper. We thank Stefano Marchesi for useful discussions. P.T. and R.G.
acknowledge financial contribution from the agreement ASI-INAF n.2017-14-H.0.
M.N. acknowledges INAF-1.05.01.86.20. S.B. acknowledges partial financial
support from the Indark INFN Grant. A.S. is supported by the FARE-MIUR grant
’ClustersXEuclid’ R165SBKTMA and INFN InDark grant. AS, MP, VS, LDM are
supported by the ERC-StG ‘ClustersXCosmo’ grant agreement 716762. HD
acknowledges financial support from the AEI-MCINN under grant with reference
PID2019-105776GB-I00/DOI:10.13039/501100011033, and from the ACIISI,
Consejería de Economía, Conocimiento y Empleo del Gobierno de Canarias and the
European Regional Development Fund (ERDF) under grant with reference
PROID2020010107. We thank Hans Moritz Günther for help with the use of the
MARX software. We thank Malgorzata Sobolewska for assistance during the
Chandra observations. This work is based in part on data collected at Subaru
Telescope and obtained from the SMOKA, which is operated by the Astronomy Data
Center, National Astronomical Observatory of Japan, (Baba et al., 2002). This
research has made use of the services of the ESO Science Archive Facility.

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##  Appendix A Source list

In Table 7 we list the properties of the 107 X-ray sources detected in the FOV
included in a circle of 5 arcmin centered on the Spiderweb Galaxy. Soft (0.5-2
keV) and hard (2-7 keV) net counts are obtained with aperture photometry.
Values have been rounded to the first decimal place except in a few cases.
Negative values are sometimes obtained due to background fluctuations. Soft
and hard (2-10 keV) band energy fluxes are obtained multiplying the net
counts, corrected for vignetting, by the corresponding conversion factors
listed in Table 4. The errors are obtained combining independently the
statistical photometric error with the systematic uncertainty on the spectral
shape (hence, on the conversion factors). In case the $S/N$ in a given band
does not reach our threshold value $S/N>2$, we quote the $2\,\sigma$ upper
limit to the energy flux. The original ASCII file of Table 1 can be retrieved
from the webpage191919http://www.arcetri.inaf.it/spiderweb/.

Table 7: Catalog of the X-ray detected sources within 5 arcmin from the Spiderweb Galaxy.  XID | RA | DEC | Cts/s (soft) | Cts/s (hard) | $F_{S}$ | $F_{H}$
---|---|---|---|---|---|---
|  |  |  |  |  $10^{-16}$ erg/s/cm2 |  $10^{-16}$ erg/s/cm2
1 | 175.18593 | -26.54934 | $26.2\pm 6.6$ | $164.0\pm 13.8$ | $4.3\pm 1.1$ | $71.4\pm 11.2$
2 | 175.27884 | -26.50376 | $174.8\pm 13.9$ | $246.6\pm 17.5$ | $34.7\pm 3.8$ | $121.2\pm 18.1$
3 | 175.24638 | -26.50113 | $6.9\pm 4.0$ | $19.4\pm 6.1$ | $<2.0$ | $7.6\pm 2.6$
4 | 175.22588 | -26.49542 | $220.5\pm 15.2$ | $174.6\pm 13.7$ | $29.3\pm 3.0$ | $69.2\pm 10.6$
5 | 175.20786 | -26.49363 | $29.0\pm 5.9$ | $28.1\pm 6.0$ | $3.7\pm 0.8$ | $11.0\pm 2.8$
6 | 175.22015 | -26.48645 | $241.1\pm 15.8$ | $189.1\pm 14.2$ | $31.9\pm 3.2$ | $75.1\pm 11.4$
7 | 175.19974 | -26.48512 | $132.6\pm 12.6$ | $235.3\pm 16.0$ | $17.1\pm 2.1$ | $92.4\pm 13.7$
8 | 175.28694 | -26.47671 | $19.0\pm 5.0$ | $12.9\pm 5.2$ | $4.4\pm 1.2$ | $7.6\pm 3.2$
9 | 175.23038 | -26.47326 | $23.5\pm 5.4$ | $12.2\pm 4.7$ | $3.1\pm 0.7$ | $4.8\pm 1.9$
10 | 175.25893 | -26.51378 | $18.5\pm 5.2$ | $-1.7\pm 4.3$ | $2.9\pm 0.8$ | $<3.0$
11 | 175.29327 | -26.48855 | $7.3\pm 4.7$ | $18.6\pm 7.1$ | $<3.8$ | $10.4\pm 4.2$
12 | 175.22990 | -26.47804 | $15.2\pm 4.9$ | $9.8\pm 5.2$ | $2.0\pm 0.7$ | $3.8\pm 2.1$
13r3 | 175.28102 | -26.44954 | $8.3\pm 3.5$ | $-4.6\pm 2.2$ | $1.9\pm 0.8$ | $<1.0$
14 | 175.13669 | -26.53742 | $22.0\pm 6.3$ | $6.7\pm 6.1$ | $4.9\pm 1.5$ | $<10.0$
15 | 175.12944 | -26.49295 | $16.6\pm 5.6$ | $3.2\pm 5.5$ | $2.5\pm 0.9$ | $<6.0$
16 | 175.23132 | -26.54720 | $22.4\pm 5.9$ | $38.1\pm 8.2$ | $3.9\pm 1.1$ | $16.7\pm 4.2$
17r1 | 175.18042 | -26.53844 | $3.7\pm 3.1$ | $11.5\pm 4.8$ | $<1.5$ | $4.9\pm 2.1$
18 | 175.21500 | -26.52062 | $5.3\pm 3.7$ | $14.5\pm 5.3$ | $<1.7$ | $5.7\pm 2.2$
19 | 175.17738 | -26.51848 | $-2.4\pm 2.1$ | $13.9\pm 5.4$ | $<0.3$ | $5.6\pm 2.3$
20 | 175.11499 | -26.48742 | $-3.7\pm 3.8$ | $24.8\pm 7.7$ | $<0.8$ | $15.2\pm 5.1$
21 | 175.21033 | -26.55919 | $44.8\pm 7.7$ | $56.6\pm 9.1$ | $10.2\pm 1.9$ | $33.4\pm 7.0$
22 | 175.20374 | -26.55444 | $141.0\pm 12.5$ | $259.1\pm 16.9$ | $25.2\pm 2.9$ | $118.5\pm 17.4$
23 | 175.18608 | -26.54500 | $378.1\pm 19.9$ | $263.3\pm 17.1$ | $62.6\pm 5.7$ | $119.6\pm 17.6$
24 | 175.25270 | -26.53977 | $83.4\pm 9.8$ | $86.2\pm 10.7$ | $15.2\pm 2.1$ | $39.6\pm 7.2$
25 | 175.26349 | -26.53928 | $5.0\pm 4.0$ | $25.6\pm 7.6$ | $<3.0$ | $15.1\pm 4.9$
26r1 | 175.16380 | -26.53818 | $9.9\pm 4.6$ | $1.8\pm 4.5$ | $1.6\pm 0.8$ | $<4.5$
27 | 175.24773 | -26.53220 | $619.8\pm 25.4$ | $574.2\pm 24.6$ | $109.5\pm 9.3$ | $265.3\pm 36.7$
28 | 175.24940 | -26.53080 | $29.9\pm 7.2$ | $32.2\pm 7.5$ | $5.3\pm 1.3$ | $14.8\pm 3.9$
29 | 175.24535 | -26.53001 | $28.6\pm 6.6$ | $14.9\pm 6.1$ | $5.1\pm 1.2$ | $7.1\pm 3.0$
30 | 175.27881 | -26.52859 | $339.7\pm 18.7$ | $337.3\pm 19.1$ | $91.4\pm 8.4$ | $218.5\pm 31.3$
31 | 175.18436 | -26.52488 | $107.7\pm 11.0$ | $277.6\pm 17.0$ | $14.7\pm 1.8$ | $112.4\pm 16.3$
32 | 175.17839 | -26.51327 | $74.8\pm 9.0$ | $98.4\pm 10.5$ | $10.0\pm 1.4$ | $39.7\pm 6.7$
33 | 175.24586 | -26.50403 | $2.7\pm 2.7$ | $7.7\pm 4.2$ | $<1.0$ | $3.0\pm 1.5$
34 | 175.24149 | -26.49342 | $1.1\pm 2.9$ | $16.2\pm 5.5$ | $<1.0$ | $6.4\pm 2.3$
35 | 175.19250 | -26.48165 | $10.3\pm 3.9$ | $5.6\pm 3.8$ | $1.34\pm 0.52$ | $<5.0$
36 | 175.16551 | -26.47922 | $260.8\pm 16.6$ | $175.2\pm 14.0$ | $35.7\pm 3.5$ | $72.9\pm 11.2$
37 | 175.18231 | -26.46532 | $164.2\pm 13.3$ | $359.2\pm 19.5$ | $22.0\pm 2.4$ | $146.5\pm 20.9$
38 | 175.26686 | -26.46100 | $-1.5\pm 2.1$ | $8.1\pm 4.3$ | $<0.5$ | $3.5\pm 1.7$
39 | 175.26224 | -26.45923 | $77.5\pm 9.4$ | $19.0\pm 6.3$ | $12.5\pm 1.8$ | $8.3\pm 2.9$
40 | 175.28162 | -26.45306 | $16.0\pm 5.7$ | $62.8\pm 9.4$ | $3.6\pm 1.3$ | $36.7\pm 7.3$
41 | 175.27260 | -26.44206 | $13.2\pm 5.1$ | $18.3\pm 6.8$ | $3.0\pm 1.2$ | $11.2\pm 4.4$
42 | 175.16692 | -26.43220 | $90.8\pm 10.8$ | $101.6\pm 12.3$ | $13.8\pm 1.9$ | $42.5\pm 7.6$
43 | 175.20650 | -26.42832 | $119.2\pm 12.0$ | $28.0\pm 7.7$ | $17.6\pm 2.2$ | $11.5\pm 3.5$
44 | 175.14221 | -26.54348 | $1.6\pm 4.7$ | $24.5\pm 7.7$ | $<2.4$ | $14.4\pm 4.9$
45 | 175.15338 | -26.53828 | $8.9\pm 5.3$ | $14.9\pm 6.7$ | $<3.2$ | $6.4\pm 3.0$
46 | 175.13422 | -26.51784 | $7.7\pm 5.0$ | $12.6\pm 6.7$ | $<2.8$ | $5.5\pm 2.7$
47 | 175.26782 | -26.45793 | $3.6\pm 4.4$ | $11.6\pm 6.4$ | $<2.0$ | $4.9\pm 2.5$
48 | 175.15282 | -26.45285 | $12.4\pm 5.1$ | $22.5\pm 6.8$ | $1.86\pm 0.77$ | $9.7\pm 3.2$
49 | 175.16095 | -26.44319 | $7.5\pm 4.4$ | $11.0\pm 5.5$ | $<2.3$ | $4.6\pm 2.4$
50 | 175.14491 | -26.44095 | $2.2\pm 5.3$ | $51.5\pm 9.7$ | $<2.0$ | $22.2\pm 5.1$
51 | 175.23083 | -26.43491 | $11.7\pm 4.9$ | $8.7\pm 5.9$ | $1.68\pm 0.70$ | $3.5\pm 2.3$
52 | 175.20361 | -26.43283 | $6.3\pm 4.3$ | $68.0\pm 9.3$ | $<2.0$ | $27.6\pm 5.3$
53 | 175.19453 | -26.42296 | $1.2\pm 4.0$ | $44.7\pm 8.6$ | $<1.4$ | $18.3\pm 4.3$
54 | 175.13251 | -26.46288 | $3.3\pm 4.8$ | $17.4\pm 7.3$ | $1.9$ | $7.2\pm 3.2$
55 | 175.18709 | -26.50135 | $1.4\pm 2.7$ | $38.6\pm 6.8$ | $<0.8$ | $16.0\pm 3.5$
56 | 175.18121 | -26.47711 | $4.9\pm 3.0$ | $34.1\pm 6.6$ | $<1.5$ | $14.5\pm 4.0$
57 | 175.15547 | -26.50483 | $41.2\pm 7.1$ | $26.6\pm 6.4$ | $5.6\pm 1.1$ | $10.8\pm 3.0$
58 | 175.20148 | -26.48577 | $5355.1\pm 73.4$ | $4033.0\pm 63.7$ | $689.6\pm 52.0$ | $1575.0\pm 208.8$
59 | 175.27609 | -26.48332 | $4.0\pm 4.3$ | $20.6\pm 6.9$ | $<2.0$ | $8.9\pm 3.2$
60 | 175.22218 | -26.43210 | $9.9\pm 4.6$ | $17.5\pm 6.4$ | $1.43\pm 0.68$ | $7.1\pm 2.7$
61 | 175.20068 | -26.43743 | $8.2\pm 4.1$ | $23.8\pm 6.7$ | $1.15\pm 0.58$ | $9.7\pm 3.0$
62 | 175.19254 | -26.43634 | $7.9\pm 4.0$ | $7.9\pm 4.6$ | $1.12\pm 0.55$ | $3.2\pm 1.9$
63 | 175.22252 | -26.45065 | $26.4\pm 5.9$ | $49.4\pm 8.0$ | $3.6\pm 0.8$ | $19.7\pm 4.1$
64 | 175.25787 | -26.44586 | $30.0\pm 6.8$ | $14.0\pm 6.7$ | $4.7\pm 1.1$ | $5.8\pm 2.9$
65 | 175.27296 | -26.44957 | $21.9\pm 6.0$ | $22.6\pm 7.2$ | $4.2\pm 1.2$ | $11.2\pm 3.9$
66 | 175.27232 | -26.45238 | $9.7\pm 4.5$ | $23.9\pm 7.8$ | $1.73\pm 0.81$ | $11.1\pm 3.9$
67 | 175.13544 | -26.52808 | $53.3\pm 8.3$ | $53.8\pm 9.2$ | $10.2\pm 1.8$ | $27.9\pm 6.0$
68 | 175.14703 | -26.47453 | $3.7\pm 3.9$ | $18.9\pm 6.4$ | $<1.6$ | $7.7\pm 2.8$
69 | 175.16040 | -26.46140 | $5.3\pm 3.9$ | $29.6\pm 6.8$ | $<1.8$ | $12.0\pm 3.2$
70 | 175.16412 | -26.46660 | $11.2\pm 4.3$ | $13.3\pm 5.0$ | $1.50\pm 0.59$ | $5.4\pm 2.1$
71 | 175.16164 | -26.48617 | $73.7\pm 9.0$ | $17.4\pm 5.3$ | $9.9\pm 1.4$ | $7.1\pm 2.4$
72 | 175.18102 | -26.46208 | $20.4\pm 5.1$ | $11.2\pm 4.6$ | $2.8\pm 0.7$ | $4.6\pm 2.0$
73 | 175.18962 | -26.46955 | $12.5\pm 4.2$ | $24.9\pm 5.7$ | $1.68\pm 0.57$ | $10.3\pm 2.7$
74 | 175.18440 | -26.48541 | $60.7\pm 8.0$ | $39.7\pm 6.8$ | $8.5\pm 1.3$ | $17.0\pm 3.7$
75 | 175.19153 | -26.48805 | $495.0\pm 22.4$ | $334.8\pm 18.7$ | $64.2\pm 5.6$ | $133.0\pm 19.0$
76 | 175.20314 | -26.46463 | $1.1\pm 2.5$ | $11.8\pm 4.5$ | $<0.8$ | $4.8\pm 1.9$
77 | 175.22797 | -26.45959 | $5.0\pm 3.4$ | $42.3\pm 7.4$ | $<1.6$ | $17.2\pm 3.8$
78 | 175.23540 | -26.46467 | $14.4\pm 4.6$ | $9.8\pm 4.7$ | $1.90\pm 0.63$ | $3.8\pm 1.9$
79 | 175.25610 | -26.46475 | $0.5\pm 3.2$ | $24.4\pm 6.6$ | $<1.1$ | $10.9\pm 3.3$
80 | 175.25993 | -26.46278 | $248.4\pm 16.1$ | $314.3\pm 18.6$ | $41.2\pm 4.1$ | $144.2\pm 20.8$
81 | 175.20267 | -26.48068 | $58.9\pm 8.0$ | $43.6\pm 7.2$ | $7.6\pm 1.2$ | $17.1\pm 3.6$
82 | 175.27940 | -26.47438 | $10.1\pm 4.8$ | $29.9\pm 7.7$ | $1.7\pm 0.8$ | $13.4\pm 3.9$
83 | 175.14645 | -26.49300 | $4.4\pm 3.6$ | $38.8\pm 7.5$ | $<1.6$ | $16.0\pm 3.7$
84 | 175.15465 | -26.49537 | $7.1\pm 4.0$ | $15.2\pm 5.4$ | $<2.0$ | $6.2\pm 2.4$
85 | 175.16492 | -26.49530 | $56.2\pm 8.0$ | $100.1\pm 10.5$ | $7.5\pm 1.2$ | $40.7\pm 6.8$
86 | 175.18535 | -26.48916 | $51.6\pm 7.5$ | $52.0\pm 7.8$ | $7.1\pm 1.2$ | $21.8\pm 4.4$
87 | 175.19443 | -26.48622 | $24.2\pm 5.4$ | $6.3\pm 3.8$ | $3.1\pm 0.7$ | $2.5\pm 1.5$
88 | 175.20360 | -26.48595 | $37.7\pm 8.0$ | $23.5\pm 6.2$ | $4.9\pm 1.1$ | $9.2\pm 2.7$
89 | 175.21483 | -26.48368 | $7.6\pm 3.5$ | $42.7\pm 7.2$ | $1.00\pm 0.46$ | $16.9\pm 3.6$
90 | 175.21136 | -26.49230 | $25.2\pm 5.5$ | $23.3\pm 5.6$ | $3.25\pm 0.75$ | $9.0\pm 2.5$
91 | 175.22507 | -26.49026 | $17.8\pm 4.8$ | $20.5\pm 5.4$ | $2.4\pm 0.7$ | $8.1\pm 2.4$
92 | 175.22772 | -26.49112 | $89.5\pm 9.7$ | $14.6\pm 4.9$ | $11.8\pm 1.5$ | $5.7\pm 2.1$
93 | 175.23019 | -26.49082 | $66.7\pm 8.5$ | $34.6\pm 6.7$ | $8.7\pm 1.3$ | $13.5\pm 3.1$
94r1 | 175.23413 | -26.48683 | $5.2\pm 2.9$ | $8.7\pm 3.8$ | $<1.3$ | $3.4\pm 1.5$
95 | 175.24232 | -26.50758 | $47.2\pm 7.5$ | $54.0\pm 8.5$ | $6.3\pm 1.1$ | $21.1\pm 4.3$
96 | 175.26640 | -26.51311 | $116.2\pm 11.3$ | $344.1\pm 19.3$ | $18.4\pm 2.3$ | $146.2\pm 20.9$
97 | 175.26569 | -26.51449 | $13.4\pm 4.4$ | $41.4\pm 7.4$ | $2.1\pm 0.7$ | $17.4\pm 3.9$
98 | 175.24670 | -26.53411 | $77.1\pm 10.0$ | $99.3\pm 11.4$ | $12.9\pm 1.9$ | $43.1\pm 7.5$
99 | 175.24873 | -26.53539 | $53.5\pm 8.5$ | $52.1\pm 8.9$ | $8.8\pm 1.5$ | $21.8\pm 4.7$
100 | 175.22601 | -26.56204 | $40.9\pm 7.2$ | $36.2\pm 7.8$ | $11.3\pm 2.2$ | $25.8\pm 6.5$
101 | 175.18367 | -26.51827 | $14.0\pm 4.7$ | $3.5\pm 3.9$ | $1.9\pm 0.6$ | $<4.0$
102 | 175.14857 | -26.54114 | $163.1\pm 13.8$ | $135.8\pm 13.8$ | $31.4\pm 3.5$ | $70.2\pm 11.7$
103 | 175.16844 | -26.55954 | $6.7\pm 5.1$ | $61.5\pm 9.8$ | $<3.6$ | $35.3\pm 7.3$
104r3 | 175.20998 | -26.56691 | $-1.3\pm 1.8$ | $9.0\pm 4.2$ | $<0.7$ | $6.8\pm 3.3$
105 | 175.13370 | -26.49614 | $-2.1\pm 1.5$ | $7.5\pm 3.6$ | $<0.2$ | $3.2\pm 1.6$
106r3 | 175.26247 | -26.43484 | $8.3\pm 3.5$ | $3.7\pm 3.6$ | $1.61\pm 0.70$ | $<5.5$
107 | 175.26004 | -26.49588 | $2.0\pm 3.5$ | $13.8\pm 5.6$ | $<1.3$ | $5.9\pm 2.5$
202020The RA and DEC positions refer to the centroid of the X-ray detected
emission. If a source does not satisfy $S/N>2$ in one band, we report the
$2\,\sigma$ upper limit. Sources with apex r1 and r3 have an extraction radius
1 and 3 pixels, respectively, smaller than $r_{ext}$ as defined in Section
5.1.

##  Appendix B Spectra of X-ray members

In Figure 23, we show the folded spectra of the eight X-ray members of the
Spiderweb protocluster that have at least 40 net counts in the soft (0.5-2
keV) or hard (2-7 keV) band, along with the best-fit spectral model. Spectra
have been rebinned with a minimum of 20 counts per energy bin for display
purpose. The spectrum of the Spiderweb Galaxy (XID 58) will be discussed in
much greater detail in a companion paper (Tozzi et. al. in preparation). The
spectral model used in all the fits consists in a fixed modelization of the
Galactic absorption with the Xspec model tbabs, with $N_{\rm
H\,gal}=3.18\times 10^{20}$ cm-2, an intrinsic power law with a free slope
$\Gamma$, an intrinsic absorption modeled with zwabs, and a Gaussian line
component that is allowed to vary in a range $1.9-2.1$ keV, approximately
centered on the energy 2.03 keV corresponding to the neutral iron Kα line at a
rest frame energy of 6.4 keV.

For completeness, in Figure 24, we show the folded spectra of the six X-ray
members of the Spiderweb protocluster that do not reach 40 in the soft and in
the hard band. Spectra have been rebinned with a minimum of only ten counts
per energy bin for display purposes. The spectral model used in the fits is
significantly simpler than the previous one, and consists in a fixed
modelization of the Galactic absorption with the Xspec model tbabs, with
$N_{\rm H\,gal}=3.18\times 10^{20}$ cm-2, an intrinsic power law with a slope
frozen to $\Gamma=1.8$, and an intrinsic absorption modeled with zwabs. We
note that for sources XID 34 and XID 87, the spectral slope is left free
instead of the intrinsic absorption, because the simple Compton-thin model is
not able to provide acceptable fits. Best-fit parameters for all the sources
are shown in Table 5.

Figure 23: Folded spectra of the X-ray members of the Spiderweb protocluster
with more than 40 net counts in the soft or in the hard band, including the
Spiderweb Galaxy itself (XID 58).

Figure 23:  \- continued .

Figure 24: Folded spectra of the X-ray members of the Spiderweb protocluster
with less than 40 net counts in the soft and in the hard band.

##  Appendix C Notes on single sources

In this Appendix we briefly discuss specific features of all the X-ray-
detected members of the Spiderweb protocluster, except the Spiderweb Galaxy
(XID 58).

###  C.1 XID 7

This source has been identified as a candidate H$\alpha$ emitter in Kurk et
al. (2004b) (ID=272), and a photometric redshift of $z=2.18^{+0.08}_{-0.05}$
(ID=493) has been measured by Tanaka et al. (2013). Kuiper et al. (2011)
investigated the NIR spectrum with SINFONI (see their ID 12), and found a
spectroscopic redshift of $z=2.1447\pm 0.0023$ from the O[III] line, observed
with a flux of $8.2\pm 1.5\times 10^{-17}$ erg/s/cm2/$\AA$. In the X-ray, this
source shows a Compton thin spectrum with the highest intrinsic absorption in
the sample ($\sim 4\times 10^{23}$ cm-2), and a luminosity in the QSO range
($L_{0.5-10keV}\sim 7.5\times 10^{44}$ erg/s). It is the X-ray source closest
to the Spiderweb Galaxy.

###  C.2 XID 9

The optical counterpart of source XID 9 has been originally selected as a
candidate Ly$\alpha$ emitter. The optical spectrum is shown in Pentericci et
al. (2002) (ID=1612), where the Ly$\alpha$ line has been measured to have a
flux of $3.9\times 10^{-17}$ erg/s/cm2, with an $EW>30$ $\AA$ and a $FWHM=490$
km/s This source has been confirmed also with Keck spectroscopy by Croft et
al. (2005) (ID=L675) and the Ly$\alpha$ has been revised to have $FWHM=200\pm
30$ km/s with a flux of $6.4\times 10^{-17}$ erg/s/cm2. The NV line is also
detected with $FWHM=200\pm 30$ km/s for a flux of $4.1\times 10^{-17}$
erg/s/cm2.

###  C.3 XID 12

The redshift of XID 12 (z=2.1628) has been obtained with Subaru in Shimakawa
et al. (2014) (ID=71). The X-ray spectrum is consistent with negligible
intrinsic absorption, while the total luminosity $L_{0.5-10keV}\sim 2.2\times
10^{43}$ erg/s is the lowest among the X-ray protocluster member, but still in
the Seyfert-like regime. This source has the largest offset from its optical
counterpart, due to the extended nature of its emission as noted in the text.
Possible association of the extended emission with radio jets will be
investigated when the wide-field, high-resolution radio images are available.

###  C.4 XID 34

This source has been identified with APEX LABOCA in Dannerbauer et al. (2014)
(ID=DKB12a) with a flux $S_{870\mu}=5.0\pm 1.4$ mJy for a $S/N=3.6$. It also
has a strong detection at 24$\mu$ with $S_{24\mu}=303.4\pm 5.0$ $\mu$Jy and
significant detection in the range 160 - 500 $\mu$Jy, but it only have an
upper limit at 1.4 GHz with JVLA. There are four H$\alpha$ emitters consistent
with the LABOCA position, however DKB12a is the closest to the X-ray position.
The H$\alpha$ and FIR-SED derived SFR are 240 and 860 $M_{\odot}$/yr,
respectively, suggesting the presence of dust. The total FIR luminosity is
$5.0\times 10^{12}\,L_{\odot}$. Eventually, a redshift of $z=2.1690$ was
measured with Subaru in Shimakawa et al. (2014) (ID 29). Shimakawa et al.
(2018) identified a possible subgroup at the position of this source. The
source was also detected in the CO line in the COALAS survey (Jin et al. 2021,
see their Table 3), with an offset of 2 arcsec between the ATCA and X-ray
centroids. The high FIR luminosity is consistent with the X-ray tentative
classification as a Compton-thick AGN, due to an extremely flat (if not
inverted) power-law spectrum. Despite the very low flux, the estimated
intrinsic luminosity would be $L_{2-10keV}\sim 2.6\times 10^{43}$ erg/s in our
simplistic assumption that the observed (reflected) luminosity is $\sim 10$%
of the intrinsic emission.

###  C.5 XID 36

The optical counterpart of source XID 36 was originally selected as a
Ly$\alpha$ emitter with $B=24.8$. The optical spectrum is shown in Pentericci
et al. (2002), where the source was classified as a QSO (ID=1687). Its
spectrum shows Ly$\alpha$, SiIV and CIV emission lines. The Ly$\alpha$ line
was measured to have a flux of $56\times 10^{-17}$ erg/s/cm2, with an $EW=164$
$\AA$ and a $FWHM=5800$ km/s. The Ly$\alpha$ and CIV lines have a double-
peaked profile due to absorption by neutral, metal-enriched gas. The presence
of dust is suggested by the low Ly$\alpha$/CIV line ratio compared to normal
values for QSOs (Ly$\alpha$/CIV $\sim 2.5$ from composite spectra of QSO BLRs
Osterbrock 1989).

The spectrum (ID=X3/L968d) obtained by Croft et al. (2005) found only moderate
absorption blueward of Ly$\alpha$, and corrected the redshift from $z=2.183$
Pentericci et al. (2002) to $z=2.162$. This AGN shows broad and narrow
Ly$\alpha$, broad and narrow CIV $\lambda$1549, and broad NV. The CIV emission
was fitted with two Gaussian components, and the broad and narrow Ly$\alpha$
and NV were fitted with three Gaussian components. For the Ly$\alpha,$ these
latter authors found $FWHM=4790\pm 40$ km/s and $FWHM=122\pm 20$ km/s for the
broad and narrow component, respectively. For the CIV they found
$FWHM=12180\pm 280$ km/s and $FWHM=1070\pm 50$ km/s for the broad and narrow
component, respectively. The NV line has a measured $FWHM=4041\pm 40$ km/s.
The QSO nature of this source is confirmed by the high X-ray luminosity
($L_{0.5-10keV}\sim 3.1\times 10^{44}$ erg/s). The source shows a moderate
absorption in the X-ray at the level of $\sim 10^{22}$ cm-2, but also shows
hints of a strong Fe line with rest-frame EW $\sim 0.22$ keV, despite with
large uncertainties.

###  C.6 XID 57

The redshift of XID 57 (z=2.1426) was obtained with Subaru in Shimakawa et al.
(2014) (ID=14). The X-ray spectrum is typical of a Type I AGN with typical
Seyfert luminosity $L_{0.5-10keV}\sim 5.0\times 10^{43}$ erg/s.

###  C.7 XID 73

The source was identified with an ERO candidate in Kurk et al. (2004b) with
ID=463. It is the only X-ray-emitting protocluster member in our catalog that
has not been spectroscopically confirmed. In the X-ray, it appears as an
obscured Seyfert galaxy.

###  C.8 XID 74

This source is classified as a quiescent galaxy despite it being detected in
H$\alpha$ probably due to the presence of the AGN that nevertheless does not
contribute significantly to the overall SED. This source is investigated in
detail in Doherty et al. (2010) where they measure it to have $z=2.172$ (ID
456) and to be consistent with being formed in an intense burst of star
formation, with a derived age of $1.6^{+1.1}_{-0.7}$ Gyr. The low current star
formation rate $<1.0\,M_{\odot}$/yr derived from the SED fit confirms that the
galaxy is now in a quiescent phase. The X-ray spectrum is consistent with a
Type I or moderately obscured Seyfert galaxy. Also, the neutral Fe emission
line is detected with $EW\sim 0.5$ keV, despite with significant
uncertainties.

###  C.9 XID 75

This source was selected as an H$\alpha$ emitter candidate in Kurk et al.
(2004b) and eventually spectroscopically confirmed (Kurk et al. 2004a)
(ID=215). Its spectrum shows continuum emission plus a very broad line which
almost covers the full spectral range, and it is identified with H$\alpha$.
For this AGN the N[II] line is blended with the very broad H$\alpha$ line and
impossible to discern. The measured FWHM of the line is $5300\pm 800$ km/s for
a flux of $46.2\pm 8.8\times 10^{-17}$ erg/s/cm2 and an $EW=150\pm 30\AA$. The
H$\alpha$ emission is therefore not considered to be associated to star
formation rate. The X-ray spectrum is typical of an unobscured QSO, with
$L_{0.5-10keV}\sim 5.2\times 10^{44}$ erg/s, consistent with the presence of
the broad optical line.

###  C.10 XID 80

Source XID 80 has been identified with source ID 16 in Pentericci et al.
(2002) as a candidate Ly$\alpha$ emitter encompassed by a very strong and
extended Ly$\alpha$ emission, and therefore classified as an AGN. Eventually
its spectrum was obtained by Croft et al. (2005) at $z=2.149$ (ID X16/L778d).
The spectrum shows a broad Ly$\alpha$ (rest-frame deconvolved FWHM 890 km/s),
and several emission lines such as NV ($\lambda$1240), CIV ($\lambda$1549),
HeII ($\lambda$1640), CIII] ($\lambda$1909),[CII] ($\lambda$2326), and MgII
($\lambda$2798), all with $FWHM$ ranging from 1200 to 2000 km/s (see Table 2
of Croft et al. 2005).

This source was also identified with APEX LABOCA at 870 $\mu m$ with
$S_{870}\mu m=4.2\pm 1.4$ mJy for a $S/N=2.9$ in Dannerbauer et al. (2014)
(ID=DKB16). Despite the faint signal, this source is clearly detected in IR
and radio bands (from $S_{24\mu}=572.1\pm 5.0$ $\mu$Jy to $S_{1.4GHz}=76.2\pm
19.0$ $\mu$Jy with the VLA (see their Table 2). The estimated SFR is $\sim
1140M_{\odot}$/yr and $\sim 830M_{\odot}$/yr from the H$\alpha$ and FIR SED,
respectively, and the total FIR luminosity is measured to be $4.8\times
10^{12}\,L_{\odot}$. The X-ray spectrum shows that this source is an obscured
Type II QSO with with an unobscured luminosity $L_{0.5-10keV}\sim 7.4\times
10^{44}$ erg/s.

Finally, this source has the second-largest offset from its optical
counterpart due to a possible extended nature of the emission as noted in the
text. Possible association of the extended emission with radio jets will be
investigated when the wide-field, high-resolution radio images are available.

###  C.11 XID 86

Source XID 86 is identified in Pentericci et al. (2002) (Source ID=5) with a
faint extended ERO with $R-K=5.8$ ($I-K=5.3$) consistent with having a
redshift close to the Spiderweb Galaxy. Eventually, the source was
spectroscopically confirmed by Croft et al. (2005) at $z=2.162$ (ID 5Xd).
Faint but significant Ly$\alpha$ emission (rest-frame deconvolved $FWHM\sim
400$ km/s) has been detected. It was not possible to classify this source as a
starburst or an AGN, but the X-ray luminosity provides strong evidence for the
presence of an AGN, as already noted in the literature (Pentericci et al.
2002; Croft et al. 2005). In the X-ray, this source 86 shows a flat spectrum
($\Gamma\sim 1$), with hardness ratio $HR\sim 0$. We consider it a candidate
Compton-thick source, with the X-ray spectrum dominated by a reflection
component, and an estimated intrinsic luminosity $L_{2-10keV}\sim 3.4\times
10^{44}$ erg/s. The detection of the neutral Fe emission line with $EW\sim
0.50$ keV strengthen the Compton-thick hypothesis.

###  C.12 XID 87

Source XID 87 was selected as an ERO candidate in Kurk et al. (2004b) with
ID=284 and identified as an H$\alpha$ emitter in Shimakawa et al. (2018)
(ID=48). The redshift was measured thanks to a ALMA CO(3–2) observation
presented in Tadaki et al. (2019). The CO(3-2) line has a redshift of 2.157
with a $FWHM=232$ km/s. This source was also detected in the CO(1-0) line in
the COALAS survey (Jin et al. 2021, see their Table 3). The X-ray spectrum
shows the steepest slope in the sample, and is typical of a Type I AGN, with
$L_{0.5-10keV}\sim 3.3\times 10^{43}$ erg/s.

###  C.13 XID 90

This source was originally selected as an H$\alpha$ emitter candidate in Kurk
et al. (2004b) with ID=154. It was also detected in the CO line in the COALAS
survey (Jin et al. 2021, see their Table 3). We note that the X-ray centroid
is closer to a nearby source that nevertheless has similar color and
preliminary photo-z consistent with being in the protocluster. This is the
only X-ray source in our sample that has two possible counterparts, both
consistent with being in the protocluster, but with only one spectroscopically
confirmed. The X-ray spectrum is typical of Type II AGN, with a Seyfert-like
unabsorbed luminosity of $L_{0.5-10keV}\sim 4.6\times 10^{43}$ erg/s.

##  Appendix D $L_{X}$-$M_{*}$ relation

In Figure 25 we show the $L_{X}-M_{*}$ plane for all the protocluster members
with spectroscopic redshift. We plot 11 sources with X-ray emission, excluding
the spiderweb and excluding also the two Compton-thick candidates, whose
intrinsic luminosity cannot be estimated accurately. In addition, we also show
the average X-ray luminosity per source that we obtain by stacking the sources
without X-ray detection in four stellar mass bins. The mass bins have been
chosen in order to have a similar number of sources ($\geq 10$) in each of
them, except for the first bin ($9.5<{\rm log}(M_{*})<10.0$) where we include
all the sources without a $K_{\rm S}$-band counterpart. Before stacking the
X-ray images at the source position, we removed the sources closer than 12
arcsec to the Spiderweb Galaxy in order to avoid spurious signal due to the
irregular diffuse emission. We have 38 sources in the first bin, then 13
sources in the $10.0<{\rm log}(M_{*})<10.5$ bin, and 11 and 12 sources in the
$10.5<log(M_{*})<10.9$ and $10.9<{\rm log}(M_{*})<11.6$ bin, respectively.
Despite in all the cases we formally have a positive photometry, most of them
are very close to zero, and, in all the cases, consistent with zero whitin 1
or 1.5 $\sigma$. Therefore, we must consider the blue points and the
corresponding error bars as upper limits. The error bars on ${\rm log}(M_{*})$
are $\sim 0.2$ for the X-ray detected sources, while corresponds to the bin
width for the stacked points.

Apart from a very scattered, global trend of having higher intrinsic $L_{X}$
at higher $M_{*}$, we note that the sources not detected in X-ray do not have
luminosity close to our detection limit, but at least an order of magnitude
lower, implying that they are not hosting AGN comparable to those hosted by
the X-ray detected protocluster members. In the right panel of Figure 25 we
also plot, as a reference, the average $L_{X}$-$M_{*}$ relation obtained in
the field at $z\sim 2$ in Mullaney et al. (2012a) for the rest-frame 2-10 keV
band. Apart from the expected scatter at high masses (see Mullaney et al.
2012b), which is entirely expected because the relation holds for average
values of the X-ray luminosity, we notice that the stacked values of the
protocluster galaxies without X-ray detection appear to be lower than the
relation measured in the field. This comparison is preliminary, because only
star forming galaxies should be included. However, this is a hint that, at low
mass, the accretion onto the SMBH may have been suppressed among the
protocluster members, while it is enhanced at high masses. In other words, the
average $L_{X}$–$M_{*}$ relation may be significantly steeper within the
protocluster. This issue will be explored in Pannella et al. (in preparation).

Figure 25: X-ray luminosity and stellar mass relations. Left panel: Red points
show the distribution of 12 protocluster members (including the Spiderweb
Galaxy but excluding the two Compton-thick candidates) in the soft-band,
intrinsic luminosity and stellar mass plane. Blue points are obtained stacking
the protocluster members without X-ray detection in bins of stellar mass. We
note that stacked $L_{X}$ values should be considered upper limits, because,
despite the formal positive detection, all the stacked values are consistent
with noise within 1 or 1.5 $\sigma$. Right panel: Same as the left panel but
in the rest-frame hard band.


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