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I realized right after finishing writing this question that Li75 already answered my questions. So the question is closed before it is opened, but I'm leaving this question as a note to myself and for …

This question is about some basic(classical) results on Atkin-Lehner-Li theory of newforms. Let $f$ be a (normalized) newform of level $N$ and character $\epsilon$. Denote the $n$th Fourier coefficien …

Let $\widetilde{M}_k^{\leq \ell}$ be the space of weight $k$ depth $\leq \ell$ quasimodular forms, and $\widetilde{M}_{k,\mathbb R}^{\leq \ell}$ be a subspace of $\widetilde{M}_k^{\leq \ell}$ whose e …

Let $G=\mathrm{SL}_2(\mathbb{R})$, $K=\mathrm{SO}(2)$, and $f$ be a holomorphic modular form of weight $k$ for $\Gamma$ a Fuchsian group of the first kind. In Borel's book, 'automorphic forms on $\mat …

It is known that for $\Gamma_0(N)$, a weakly holomorhpic modular form is a harmonic maass form. Here is the definitions. A weakly modular form $f$ for $\Gamma_0(N)$ is a meromorphic function on th …

In p.312 of 'Rhoades, Robert C., Linear relations among Poincaré series via harmonic weak Maass forms. Ramanujan J. 29 (2012), no. 1-3, 311–320', the author defines the extended principal part at infi …

I am studying the paper M. Ram Murty, V. Kumar Murty: Mean values of derivatives of modular $L$-series, Ann. of Math. (2) 133 (1991), no. 3, 447-475. Let $L(s)=\sum_{n=1}^{\infty} \frac{a(m)}{m^s}$ b …

Let $q=e^{\pi i \tau}$ for $\tau \in \mathbb H$ and $$ P_2(q)=\frac{P(q)+2P(q^2)}{3}, \quad Q_2(q)=\frac{Q(q)+4Q(q^2)}{5}, \\ R_2(q)=\frac{R(q)+8R(q^2)}{9}, $$ a Fourier series of quasi-modular form …

I think the following answers the question. It does not depend on the above comments because I didn't understand them. This is just my approach. (But not fully my idea because it depends on a strong p …

Let $P,Q,R$ be the Fourier series of the Eisenstein series $E_2,E_4,E_6$, that is, $$ P(q)=1-24\sum_{n=1}^{\infty}\sigma_1(n)q^n, $$ $$ Q(q)=1+240\sum_{n=1}^{\infty}\sigma_3(n)q^n, $$ $$ R(q)=1-504 …