Notation:
\(K\) imaginary quadratic field.
\(\mathscr A\) ideal class of \(K\)
\(D\) discriminant \(K\)
\(\epsilon (n) = \legendre{D}{n}\) associated Dirichlet character.
\(h = \#\Cl_K\text{.}\)
\(w = 2u \) twice number of units.
\(f\in S_{2k}^{new}(\Gamma _0(N))\) for \(k\ge 1\text{.}\) \((N,D) = 1\) write
\begin{equation*}
f(z) = \sum_{n \ge 1} a(n) e^{2\pi i n z}
\end{equation*}
\begin{equation*}
L(f,s) = \sum a_n n^s
\end{equation*}
define
\begin{equation*}
L_{\mathscr A } (f,s) = L^{(N)} (2s-2k +1 , \epsilon ) \cdot \sum_{n=1}^\infty a(n) r_{\mathscr A} (n)n^{-s}
\end{equation*}
\begin{equation*}
L^{(N)} = \sum _{(n,N) = 1} \epsilon (n) n^{-2s +2k -1}\text{.}
\end{equation*}
Subsection 6.10.1 Rankin's method
\begin{equation*}
\theta _{\mathscr A} (z) = \sum_{n=1}^\infty r_{\mathscr A}(n) q^n \in S_1(|D|, \epsilon )
\end{equation*}
\begin{equation*}
\frac{\Gamma (s+2k -1)}{(4\pi )^{s+2k-1}} \cdot \sum_{n=1}^\infty \frac{a(n) r_{\mathscr A}(n)}{n^{s+2k-1}}
\end{equation*}
\begin{equation*}
= \int_0^\infty \sum_{n=1}^\infty a(n)r_{\mathscr A}(n) e^{-4\pi n y} y^{s+2k-2} \diff y
\end{equation*}
\begin{equation*}
= \int_0^\infty \int_0^1 f(x+iy) \overline{\theta _{\mathscr A} (x+iy)} \diff x y^{s+2k -2 } \diff y
\end{equation*}
\begin{equation*}
= \int\int_{\Gamma _\infty \backslash \HH} f(z) \overline{\theta _{\mathscr A} (z)} \diff x \diff y/y^2
\end{equation*}
\begin{equation*}
= ?????
\end{equation*}
Choose \(\mathcal F\) to be a fundamental domain for \(\Gamma _0(M)\) where \(M = N|D|\) consider
\begin{equation*}
\bigcup _{\gamma \in \Gamma _\infty \backslash \Gamma _0(M) } \gamma \mathcal F
\end{equation*}
have
\begin{equation*}
\sum_{\gamma \in \Gamma _\alpha \backslash \Gamma _0(M)} \int \in_{\gamma \mathcal F} f(z) \overline{\theta _{\mathscr A}(z)} y^{s+2k} \frac{\diff x\diff y}{y^2}
\end{equation*}
\begin{equation*}
= \int\int _{ \mathcal F} f(\gamma z) \overline{\theta _{\mathscr A}(\gamma z)} (\Im \gamma z)^{s+2k} \frac{\diff x\diff y}{y^2}
\end{equation*}
\begin{equation*}
\sum_{\gamma =\pm \begin{pmatrix} \bullet \amp \bullet \\\ c \amp d \end{pmatrix} \in \Gamma _\infty \backslash \Gamma _0(M)} \int\int_{\mathcal F} f(z) \overline{\theta _{\mathscr A}(z)}
\end{equation*}
More formulae
\begin{equation*}
\frac{\Gamma (s+2k-1)}{(4\pi )^{s+2k-1}} L_{\mathscr A }(f, s+2k-1)
\end{equation*}
\begin{equation*}
= \int\int_{\mathcal F} f(z) \overline{\theta _{\mathscr A}(z) E_s(z)} y^{2k} \frac{\diff x\diff y}{y^2}
\end{equation*}
\begin{equation*}
= (f, \theta _{\mathscr A} f)_{\Gamma _0(N)}
\end{equation*}
\begin{equation*}
E_s(z) = \frac 12 \sum_{c,d\in \ZZ,\,c \equiv 0 \pmod M} \frac{\epsilon (d)}{(cz+d)^{2k-1}} \frac{y^s}{|cz+d|^{2s}}
\end{equation*}
want to take \((d,M) =1\) to \((d, N)=1\) .
we resolve this by letting
\begin{equation*}
\tr_N^M \colon \widetilde M_{2k}(\Gamma _0(M)) \hookrightarrow \widetilde M_{2k}(\Gamma _0(N))
\end{equation*}
\begin{equation*}
g \mapsto \sum _{\gamma \in \Gamma _0(M)\backslash \Gamma _0(N)} g|_{2k} \gamma
\end{equation*}
\begin{equation*}
(f,g)_{\Gamma _0(M)} = (f,\tr_N^M f)_{\Gamma _0(N)}
\end{equation*}
so
\begin{equation*}
(4\pi )^{-s-2k +1} \Gamma (s+2k-1) L_{\mathscr A}(f, s+2k-1)= (f, \tr_N^M \theta _{\mathscr A}E_s)
\end{equation*}
Proposition 6.10.1.
\(D\) a fundamental discriminant
\begin{equation*}
N \ge 1
\end{equation*}
prime to \(D\text{.}\)
\begin{equation*}
\tilde \phi _s(z) = \tr_N^{DN}
\end{equation*}
Then for \(f\in S_{2k}^{new}(\Gamma _0(N))\)
\begin{equation*}
(4\pi )^{-s-2k +1} N^s\Gamma (s+2k-1) L_{\mathscr A}(f, s+2k-1)= (f, \tilde \phi _s)
\end{equation*}
Computing the trace
\begin{equation*}
\tilde \phi _s = \epsilon _s(Nz) \theta _{\mathscr A}(z) U_{|D|} ???
\end{equation*}
where
\begin{equation*}
\epsilon _s= \frac{\sum_{D= D_1\cdot D_2} \epsilon _{D_1}(N) \chi _{D_1D_2}(\mathscr A) E_s ^{(D_1)} (|D_2|z)}{\kappa (D_1) |D_1|^{s+2k-1}}
\end{equation*}
\(D\) odd \(\equiv 1\pmod 4\) \(D_1,D_2\) fund disc. \(\chi _{D_1,D_2}\) genus character \(\chi_{(a)} = \epsilon _{D_1}(N(a) ) = \epsilon _{D_2}(N(a))\) for ideal prime to \(D\) with \(\kappa = 1 , D_1\gt 0\text{,}\) \(\kappa = i\) \(D_2 \lt 0\text{.}\)
\begin{equation*}
E_s^{(D_1)} (z) =\frac 12 \sum_{m,n \in \ZZ,\,D_2 |m} \frac{\epsilon _1(m)\epsilon _2(n) y^s}{(mz+n)^{2k-1}{|mz+n|}^{2s}}
\end{equation*}
\begin{equation*}
U_n \colon f(z) \mapsto \frac 1n \sum_{j\pmod n} f\left(\frac{z+j}{n}\right)
\end{equation*}
on a function \(f\) of period 1.
Fourier coefficients. Consider
\begin{equation*}
\epsilon _s(z) = \sum_{n\in \ZZ} e_s(n,y) e(nx)
\end{equation*}