Weight 2, Level 11

The unique newform of weight 2 and level 11 is given by

$$f(z)=\eta(z)^2\eta(11z)^2=q\prod_{n=1}^\infty(1-q^{n})^2(1-q^{11n})^2$$

where $q=e(z)=e^{2\pi i z}$. Thus

$$\begin{eqnarray*}f(z)&=&q-2q^2-q^3+2 q^4+q^5+2q^6-2q^7-2q^9-2q^{10} \\ && \qqua... ...q^{13}+4 q^{14}-q^{15}-4q^{16}\\ &=& \sum_{n=1}^\infty a_n q^n. \end{eqnarray*}$$

This modular form is associated with the elliptic curve $E_{11}: y^2+y=x^3-x^2.$ The $L$-function associated with $f$ is

$$L_{11}(s) =\sum_{n=1}^\infty \frac{a_n}{n^{s+1/2}}.$$

For $d<0$, we have, by the formula of Kohnen and Zagier, as computed by Rodriguez-Villegas using the algorithm of Gross,

$$L_{11}(/12,\chi_d)= \kappa_{11} \frac{c_{11}(\vert d\vert)^2}{\vert d\vert^{1/2}}$$

where $\kappa_{11}=2.917633\dots$ and where

$$g(z)=\sum_{n=1}^\infty c_{11}(n) q^n=(\theta_1(q)-\theta_2(q))/2$$

with

$$\theta_1(q)=\sum_{(x,y,z)\in Z^3\atop x\equiv y \mod 2} q^{x^2+11 y^2 +11 z^2}$$

and

$$\theta_2(q)=\sum_{ x\equiv y \mod 3 \atop y\equiv z \mod 2} q^{(x^2+11 y^2 +33 z^2)/3}$$

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