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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.

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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 $GL(3)$.

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  • $\begingroup$ 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 $S_k(\Gamma_0(N),\chi)$ ? $\endgroup$ – reuns Oct 20 '16 at 3:51
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    $\begingroup$ @user1952009, an obvious complication with non-trivial level is that the functional equation(s) of Eisenstein series is/are more complicated: "scattering matrix". Understanding this is best done on the adele group, I think, since the most-elementary approach still involves, in effect, Fourier transform on adeles. And, also, best to use (adelic) Whittaker models, and choose the "special vector" (corresponding to "newform") at bad primes. From some viewpoint, these are just technicalities, but they're not trivial... $\endgroup$ – paul garrett Oct 20 '16 at 13:36

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