and we construct a form $f_0$ – called the $p$-stabilisation of $f$ – for which $p$ does divide the level, and the $L$-functions remain mostly unchanged. In [@Mercuri] we constructed $f_0$ explicitly in the case $n=1$ which was possible through explicit formulae on the action of the Hecke operators involved on the Fourier coefficients. For general $n$, it is easier to work with abstract Hecke rings, the Satake isomorphism, and certain Hecke polynomials, however at the end of this section, we show how all this abstract Hecke yoga reduces to the explicit form found in [@Mercuri], when $n=1$.

Let $({\mathfrak}{b}^{-1}, {\mathfrak}{bc})\subseteq 2\mathbb{Z}\times 2\mathbb{Z}$ be ideals as before; put $\Gamma = \Gamma[{\mathfrak}{b}^{-1}, {\mathfrak}{bc}]$ and $D = D[{\mathfrak}{b}^{-1}, {\mathfrak}{bc}]$. Then define $$\begin{aligned}
    {\text{{{\fontfamily{pag}\selectfont}O}}}_p :&= GL_n(\mathbb{Z}_p),   &&{\text{{{\fontfamily{pag}\selectfont}O}}}:= \prod_p GL_n(\mathbb{Z}_p), \\
    {\text{{{\fontfamily{pag}\selectfont}X}}}_p :&= M_n(\mathbb{Z}_p)\cap GL_n(\mathbb{Q}_p), &&{\text{{{\fontfamily{pag}\selectfont}X}}}:= GL_n(\mathbb{Q})_{\mathbf{f}}\cap\prod_p {\text{{{\fontfamily{pag}\selectfont}X}}}_p, \\
    Z_0 :&= \{{\operatorname{diag}}[\tilde{q}, q]\mid q\in {\text{{{\fontfamily{pag}\selectfont}X}}}\}, &&Z:= D[2, 2]Z_0D[2, 2].\end{aligned}$$

For appropriate groups, $\Delta$ and $\Xi$, the abstract Hecke ring $\mathcal{R}(\Delta, \Xi)$ denotes the ring of formal finite sums $\sum_{\xi}c_{\xi}\Delta\xi\Delta$ where $c_{\xi}\in\mathbb{C}$ and $\xi\in \Xi$. Consider the Hecke ring $\mathcal{R}(V, W)$ defined in [@Shimurahalf p. 39] and let $\mathcal{R}$ denote the factor ring of $\mathcal{R}(V, W)$ defined in [@Shimurahalf p. 41] or analogously to \[factorring\] 