# On the square coeffecients of a modular form

Let $k\in \mathbb{N}$. Let $f\in M_k(\Gamma_0(N),\chi)$ be a modular form of weight $k$ on $\Gamma_0(N)$ with a Dirichlet character $\chi$. If $f$ has a Fourier expansion of the form $$f(\tau)=\sum_{n\in \mathbb{N}} a(n) q^n \qquad (q=e^{2\pi i }).$$

Set

$$F_f(\tau)=\sum_{\substack{n\in \mathbb{N}\\n=\square}} a(n) q^n$$

Question: is $F_f$ a modular form. If yes, which weight and level?

Background: If $k\in \mathbb{Z}+\frac{1}{2}$, one can associate using Shimura lifting, to each Hecke eigenforms $f$ of weight $k$ a modular form $g$ of weight $2k-1$ such that $$\lambda(f,l^2)=\lambda(g,l)$$ where $\lambda(f,l^2),\lambda(g,l)$ are the Hecke-eigenvalues with respect to the operator $T(l^2),T(l)$.

As we see, for normalized Hecke-eigenform $f$ the coefficient $a_f(l^2)$ are important in this case, since it is give us information about the corresponding coefficients $a_g(l)$.

If $k\in \mathbb{Z}$, the situation not clear for me!.

Bests.

The associated Dirichlet series $\sum c(n^2)/n^{2s}$ is obtained by integrating $f$ against the product of the weight $1/2$ theta series and a half-integral-weight Eisenstein series, giving Shimura's (c. 1975) Rankin-Selberg-style integral representation of the symmetric square $L$-function attached to $f$. Gelbart-Jacquet showed a case of "functoriality", namely, that these Dirichlet series are the standard $L$-functions attached to (rather special) cuspforms on $GL(3)$.
• Hi Paul. There, I have remade for $\Gamma_0(N)$ all your proof of the Rankin-Selberg convolution for $SL_2(\mathbb{Z})$. Now I have a problem with the needed real Eisenstein series for $\Gamma_1(N)$. I also tried with $\Gamma_0(N),\chi$ but I'm not sure of what I obtained. Do you remember the trick for generalizing what I wrote to $f,g \in S_k(\Gamma_1(N))$ or $S_k(\Gamma_0(N),\chi)$ ? Commented Oct 20, 2016 at 3:51