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Modular forms are used to solve Fermat's last theorem, Mock modular forms are used in black holes theory. I read also in Quanta magazine that Eisenstein series are used to compute what we call Monster …

Let $f$ be a primitive form of an even weight $k\geq 2$ for the full modular group $SL_2(Z)$. Following from , Proposition 2.3 from [Z. Rudnick and P. Sarnak, Zeros of principal $L$-functions and ra …

Let $f$ be a primitive form of an even weight $k\geq 2$ for the full modular group $SL_2(Z)$ and $\lambda_f(n)$ be the $n$-th normalized Fourier coefficient. Define a multiplicative function $g$ by $ …

In their paper 'Sign changes of Hecke eigenvalues', Matomaki and Radziwill established in Lemma 6.2 the following result: There exists absolute positive constants $c$ and $\eta$ such that uniformly in …

Let $f$ be a primitive form of an even weight $k$ for the full modular group and let $L(Sym^rf,s)$ be the symmetric $r$th $(r\geq 2)$ power $L$-function associated to $f.$ I have three questions rela …

When I was reading the paper of Winfried Kohnen, Yuk-Kam Lau and Igore E. Shparlinski (ON THE NUMBER OF SIGN CHANGES OF HECKE EIGENVALUES OF NEWFORMS), I found this result (which is Theorem 2 of the p …

I want to write a survey on the subject 'Sign changes for coefficients of symmetric power $L$-functions'. So, I browse the Web and I got some papers. I read it and I gave special interest to the paper …

Using The Euler product, we can express the sum $$\sum_{\substack{l=1\\\gcd(l,a)=\gcd(l,b)=1}}^{+\infty}\left(\frac{\lambda_f(l)}{l^{3/4}}\right)^3$$ as an infinite product over prime numbers so that …

I did some computations but I am stuck in finding the exression of the sum $$\lambda_f(n^2)+\lambda_f(n)^2 $$ in terms of $\lambda_f(n),$ where $f$ is a modular form for the full modular group. Any he …

Given a modular form $f$ of an even weight $k$ for the full modular group. Let $\lambda_f(n)$ the $n$-th normalized Fourier coefficient of $f.$ For a fixed positive integers $a$ and $b,$ I want to dis …

From the Hecke relation, we get $$ \lambda_f(n)^2 =\sum_{d|n} \lambda_f\left(\frac{n^2}{d^2}\right).$$

Let $f$ be a modular form of an even weight $k$ over the modular group $SL_2(Z).$ Denote $\lambda_f(n)$ the $n$-th normalized Fourier coefficient of $f.$ I am doing some calculations and I am stack in …

Sorry if my question is stupid but it comes to my mind whenever I read about the theory of symmetric power $L$ functions. Out of curiosity, I did a web search and found only the explicit expression of …

I have three questions related to the theory of modular forms and it was frequently asked to me by my collegues and even my invited teacher in our seminars of the number theory at the faculty of scien …

I want to understand how we construct from a modular form $f$ its symmetric power function $Sym^rf.$ I read that there is a particular representation that does this but I am not familiar with this no …