Suppose $f$ is the following modular forms

\begin{equation} f=\eta(q^5)^4 [\eta(q)^4+5\,\eta(q)^3\eta(q^{25})+20 \eta(q)^2\eta(q^{25})^2+25\eta(q)\eta(q^{25})^3+25 \eta(q^{25})^4] \end{equation}

where $\eta$ is the Dedekind's eta function, i.e. \begin{equation} \eta(q)=q^{\frac{1}{24}} \prod_{n=1}^{\infty}(1-q^n) \end{equation}

$f$ is of weight $4$ and level 25 (with group $\Gamma_0(25)$). I want to find whether the twisted $L$ functions, $L(f,\chi,s)$ vanishes at $s=2$ or not, where $\chi$ is the quadratic character $(./5)$. I tried to show this use modular symbols, but have not made any progress. Anyone got any ideas?

In the note, Overconvergent Modular Symbols by Robert Pollack,

http://math.bu.edu/people/rpollack/Papers/Overconvergent_modular_symbols.pdf

In page 3 there is a formula

\begin{equation} L(g,\chi,1)=\frac{\tau(\chi)}{N} \sum_{a\, \text{mod}\, N} \bar{\chi}(a)2 \,\pi \,i \int_{i\,\infty}^{-a/N}g(z)dz \end{equation}

where $g$ is a modular form of weight 1. I am wondering whether there is a formula for weight 4. Then we could transform $L(f,\chi,2)$ into a sum of integrals, which might be helpful.

Explicit approaches to modular abelian varieties$\endgroup$ – François Brunault Jan 26 '17 at 8:39Parabolic points and zeta functions of modular curves(see the first formula in the proof of Thm 3.9) wstein.org/edu/Fall2003/252/references/Manin-Parabolic/… (in this article $f$ has weight 2, but the assertion holds for any weight) This is just the Fourier decomposition of a primitive Dirichlet character, I suggest that you work it out on your own or look at math.stackexchange.com/questions/27489/… $\endgroup$ – François Brunault Feb 4 '17 at 18:125more comments