otherwise we have a positive number under the square-root sign by the constraints on $\nu$. We note that the above definition is invariant under $\nu\to -\nu$, and for future reference we record that $$\label{normwhit2}
\tilde{W}_{\frac{q}{2},
\nu}(y)=\eta_{q,\nu}\frac{W_{\operatorname{sgn}(y)\frac{q}{2},
\nu}(4\pi|y|)}{\Gamma(\frac{1}{2}+\nu+\operatorname{sgn}(y)\frac{q}{2})},\qquad
q\in 2{\mathbb{Z}},\quad \nu\in i{\mathbb{R}},\quad y\in{\mathbb{R}}^\times,$$ where $\eta_{q,\nu}$ is a constant of modulus $1$ depending on $q$ and $\nu$ but not on $y$.[^7]

By [@BrMo Section 4], the functions $\tilde{W}_{q/2, \nu}$ $(q\in{\mathbb{Z}})$ for fixed $\nu$ form an orthonormal basis of the Hilbert space $L^2({\mathbb{R}}^{\times}, d^\times y)$ which justifies our normalization: $$\label{orth}
 L^2({\mathbb{R}}^{\times}, d^\times y) = \bigoplus_{q \in {\mathbb{Z}}} {\mathbb{C}}\tilde{W}_{\frac{q}{2},
 \nu},\qquad
\langle\tilde{W}_{\frac{q}{2},\nu},\tilde{W}_{\frac{q'}{2},\nu}\rangle=\delta_{q,q'}.$$ We review the uniform bounds [@BH (24)–(26)]. For all $\nu$ we have $$\label{whittaker1}
 \tilde{W}_{\frac{q}{2}, \nu}(y) \ll |y|^{1/2} \left(\frac{|y|}{|q| + |\nu| + 1}\right)^{-1-|\Re\nu|} \exp\left(-\frac{|y|}{|q| + |\nu| +
 1}\right).$$ For $\nu\in\frac{1}{2}{\mathbb{Z}}\cup i{\mathbb{R}}$ we have, for any $0 < {\varepsilon}<1/4$, $$\label{whittaker2}
 \tilde{W}_{\frac{q}{2}, \nu}(y) \ll_{\varepsilon}|y|^{1/2-{\varepsilon}} (|q| + |\nu| + 1).$$ For $\nu\in(-\frac{1}{2},\frac{1}{2})$ we have, for any $0 < {\varepsilon}<
1$, $$\label{whittaker3}
 \tilde{W}_{\frac{q}{2}, \nu}(y) \ll_{\varepsilon}|y|^{1/2-|\nu| -{\varepsilon}} (|q| + |\nu| + 1)
^{1+|\nu|}.$$ For $q \in {\mathbb{Z}}^d$ and appropriate $\nu\in{\mathbb{C}}^d$, we define $$\label{convent}
\tilde{W}_{\frac{q}{2}, \nu}(y) := \prod_{j=1}^d \tilde{W}_{\frac{q_j}{2},
\nu_j}(y_j),\qquad y \in K_\infty^\times.$$

Hecke eigenvalues and Fourier expansion
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Let ${\mathfrak{c}}\subseteq {\mathfrak{c}}_\pi$ be a nonzero ideal. There is a character 