A modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group.

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how to prove that the Peterson inner product <Ek, f>=0 where f in Sk, an eigenform for all of the level 1 Hecke operators.

how to prove that the Peterson inner product Ek and f is 0 where f in Sk, an eigenform for all of the level 1 Hecke operators.
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Why does the valence formula imply $M_k=0$ for $k <0$?

I'm studying Modular Forms and in the notes I'm reading the author states the following result, known as the valence formula: "Let $f$ be a non-zero weakly modular meromorphic form of weight $k$ ...
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Finding the value of a sum of two Hecke eigenvalues

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 ...
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Transitive action of $SL_2(\mathbb{R})$ on $\mathbb{H}$

I'm studying Modular Forms and I'm not understanding why the action of $SL_2(\mathbb{R})$ on $\mathbb{H}$ defined by $\begin{pmatrix} a & b \\ c & d \end{pmatrix}z=\frac{az+b}{cd+d}$ is ...
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connection between odd primes and a certain q-series

I posed a conjecture about odd primes and a certain q-serieshere.I thought it would be more appropriate ,if I could ask the converse of the aforementioned problem . Is $p$ an odd prime iff ...
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Question about the sign of a certain sum

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 ...
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conjecture about primes and a certain q-series.

Using wolfram Mathematica ,I observed an interesting and surprising property concerning prime numbers and q-series which I could not prove.Yet there is strong evidence supporting it. I would be happy ...
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A theta function around its natural boundary

Let $q = e^{2\pi i\tau}$, if $$\psi(q^2)=\sum_{n=0}^{\infty} q^{n(n+1)}$$ is one of ramanujan theta functions,is it possible to evaluate the limit $$\lim_{q\rightarrow 1} (1-q){\psi^2(q^2)}$$ In fact ...
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What is the limit $\lim_{q\to 1} \vartheta_{2}^{2}(q)(1-q)$

If $\vartheta_{2}(q)$ is jacobi's theta function, what is the limit $$\lim_{q\to 1} \vartheta_{2}^{2}(q)(1-q)$$ for the nome $q$. I would like to know whether the limit exists or not. If it does, ...
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Ramanujan theta function and its continued fraction

I believe Ramanujan would have loved this kind of identity. After deriving the identity, I wanted to share it with the mathematical community. If it's well known, please inform me and give me some ...
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369 views

a new continued fraction for $\sqrt{2}$

In a q-continued fraction related to the octahedral group I defined a new q-continued fraction of the square of ramanujan's octic continued fraction which I discovered using certain three term ...
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the zeros of theta function?

Recall that the theta function with character $(a,b)\in \mathbb{R}^2$ is defined by $$ \vartheta_{a,b}(z, \tau) :=\sum^\infty_{n=-\infty} e^{\pi i (n + a)^{2} \tau + 2 \pi i(n + a)(z + b)},\quad a,b ...
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Can Eisenstein Series output complex numbers?

The 2nd Eisenstein Series, defined by, $$E_{2}(\tau)=1-24\sum_{n=0}^{\infty} \frac{nq^{2n}}{1-q^{2n}},$$ where $q=e^{i\pi \tau}$ is the nome acts on upper-half plane. Must it always output real ...
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Explicit descriptions of the modular curves $Y(5)$ and $Y_1(5)$

The modular curves $Y(5)$ and $Y_1(5)$ associated to the congruence subgroups $\Gamma(5)$ and $\Gamma_1(5)$ are of genus 0. As such, they are nothing but the Riemann sphere $\mathbb P^1$ minus a ...
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Why are rational numbers required in cusps of congruence subgroups?

While we consider the action of congruence subgroups on $\mathbb{H}$ (the upper half plane), we compactify using an additional point at infinity, that is fine. But why do we add even all rational ...
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Why only congruence subgroups for modular forms?

When we define modular forms, why do we restrict ourselves to congrumence subgroups? Why not any subgroup of finite index? Or, even more generally why not any subgroup? Is it just a matter of ...
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Ramanujan's tau function identity

While studying Ramanujan's tau function, I observed that the function satisfies a beautiful identity that I had not seen previously in the literature. Let $\tau(n)$ be Ramanujan's tau function, such ...
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The ratio of jacobi theta functions

Let $q=e^{2\pi i\tau}$. If $\theta_2$ and $\theta_3$ are jacobi theta functions , is it true that the ratio of the two functions can be expressed as a continued fraction of the form $$ ...
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a q-continued fraction related to the octahedral group

Let $q=e^{2\pi i\tau}$. If $u(\tau)$ is Ramanujan's octic continued fraction, $$u(\tau)=\cfrac{\sqrt{2}\,q^{1/8}}{1+\cfrac{q}{1+q+\cfrac{q^2}{1+q^2+\cfrac{q^3}{1+q^3+\ddots}}}}$$ is it true that ...
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Holomorphicity and modularity of Jacobi forms

I am reading this paper Suerconformal algebras and mock theta functions by T. Eguchi and K. Hikami. In section 3.2, the authors define a function $$J(z;w;\tau) = ...
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Fourier coefficients of eigenforms

How does one prove that the fourier coefficients of a normalized eigenform for Hecke operators $T_p$ on $S_k(N)$ all lie in a fixed number field? If the proof is lengthy, a reference to a book that ...
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What is the idea behind the new/old form theory of modular forms

I am interested in the theory of modular forms, since they have nice transformation laws and a connection to arithmetic using the Euler product which one can form using the theory of Hecke operators. ...
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Representations of $\mathbb{H}^{\times}$ and $\mathbb{H}^{\times}/\mathbb{R}^{\times}$.

In an attempt to recapture Eichler's theta correspondence I have hit a stumbling block. Let $D$ be a quaternion algebra over $\mathbb{Q}$, ramified at $p,\infty$. Also let $V_j = ...
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A new $q$-continued fraction of order $12$

I think I may have discovered a $q$-continued fraction of order $12$ with a form different from that established by Mahadeva Naika. Let $q=e^{2i \pi \tau}=\exp(2i \pi \tau)$, then, $$\begin{aligned} ...
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What does holomorphic at the cusp infinity means

In the usual theory of classical modular forms, the modular forms defined to be "holomorphic at the cusp infinity". I do not know what this should mean? can anyone explain it for me? Thanks
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Finding references about modular forms and symmetric power $L$-functions

I want to write an introduction to my thesis which is about modular forms and symmetric power $L$-functions. Could you give me good references to these two topics as simple as possible to get an ...
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Counting points on the Klein quartic

In Moreno's book "Algebraic Curves over Finite Fields", he mentions the following in passing with no further comments ($K$ denotes the Klein quartic defined by $X^3 Y + Y^3 Z + Z^3 X = 0$): The ...
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On the definition of modular forms

In many books, I see people defining modular forms to be holomorphic/meromorphic functions in the upper half plane such that it is invariant under the $|_k$ action of the group ...
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Quotient of the upper-half plane as a projective line

In the discussion of the Belyi theorem here one pointed out that $\mathbb{H}^{*}/SL(2, \mathbb{Z})$ is isomorphic to the (complex) projective line $\mathbb{P}^1$, where $\mathbb{H}^{*}=\mathbb{H} \cup ...
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Extend a map to a 1-cocycle

Let $\Gamma=PSL(2,\mathbb{Z})$ be the modular group with the usual presentation $\Gamma=\langle S,U,T|\ S^2=U^3=1, T=US\rangle$ where ...
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Modular curves over finite fields

I'm looking for a detailed reference for modular curves over finite fields, such as $X(N)$, $X_1(N)$, and $X_0(N)$. There seems to be a lot of literature dealing with them over $\mathbb{C}$, but I'm ...
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What is the exact value of $\eta(6i)$?

Let $\eta(\tau)$ be the Dedekind eta function. In his Lost Notebook, Ramanujan played around with a related function and came up with some of the nice evaluations, $$\begin{aligned} \eta(i) &= ...
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Multiplicative property for the coefficients of cusp form

This question might seen as a duplicate of this, however my aim is to understand the theory which lies beneath the computations of Sage. Let $\Gamma_0(4)$ be a congruence subgroup of ...
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$\Gamma_0(4)$ has no torsion except $-1$

Let $\Gamma_0(4)$ be a congruence subgroup of $SL(2,\mathbb{Z})$ defined as $$\Gamma_0(4)=\{M=\begin{pmatrix} a &b\\ c& d \end{pmatrix}\in SL(2,\mathbb{Z})\mid c=0\bmod 4\}.$$ How to prove ...
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Coefficient in the Fourier expansion of the cusp form

Ideal of cusp for $\Gamma_{0}(4)$ is principal and generated by $f(z)=η(2z)^{12}=q+\sum a(n)q^n $, this is discussed here. How one can compute the coefficient $a(n)$ when $n$ is rather large ? for ...
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$\Gamma(2)$ and $\Gamma_0(4)$ are conjugate

Show that the groups $\Gamma(2)$ and $\Gamma_0(4)$ are conjugate to each other in $SL(2,\mathbb{R})$, where $\Gamma(2)$ and $\Gamma_0(4)$ is congruence subgroups of the modular group Is it the ...
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Elliptic curve over field $\mathbb{C}(\lambda)$

I have the following problems on my modular forms course final exam: Over field $\mathbb{C}(\lambda)$ equation $y^2 =x(x−1)(x−\lambda)$ defines an elliptic curve $E_{\lambda}$ with a base in ...
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Find motivation for calculating $\int_{2}^{X} A^2(t) A(\alpha t)dt$

I read a thesis of Kong Kar Lun (student of Tsang K.M) about the some mean value theorems for certain errors terms in analytic number theory and in which he gave the asymptotic formulas of the ...
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What is the order of a cusp form at a cusp?

This question is about the definition of order of a section of a bundle at a point, and the related notion of associated divisor. Let us look at a specific example, the discriminant $\Delta(z)$ on ...
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Why has $\Phi(X,Y)$ integral coefficients?

Let $j$ be Klein's $j$-invariant and let $M$ be a the set of integral $2\times 2$ matrices with determinant $n$, $A$ and $-A$ identified. Why has the polynomial $\Phi(X,Y)\in\mathbb C[X,Y]$ with ...
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Modular form $f(z)$ vs $f(Nz)$

Is it a fact that if $f(z)$ is a modular form of weight $k$ for $\mathrm{SL}_2(\mathbb{Z})$ then $f(Nz)$ is a modular form of weight $k$ for $\Gamma_0(N)$? I tried considering this by simply making ...
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Show the Dedekind $\eta(\tau)$ function is a modular form of weight $\frac{1}{2}$ for $\Gamma_0(6)$

Given that $\Gamma_0(6)$ is generated by matrices in the set $$G = \left\{\left(\begin{array}{cc} 1 & 1\\ 0 & 1 \end{array}\right), \left(\begin{array}{cc} 7 & -3\\ 12 & -5 ...
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Action of full modular group on higher level modular functions

Let $\Gamma$ be a finite index subgroup of $\Gamma(1)=SL_2(\mathbf{Z})$ and $f$ a modular function for $\Gamma$. By this I mean a meromorphic function defined on the upper half-plane $f: \mathfrak{h} ...
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Ideal of cusp forms for $\Gamma_0(4)$ is principal

Let $\Gamma_0(4)$ be a congruence subgroup of $SL(2,\mathbb{Z})$ defined as $$\Gamma_0(4)=\{M=\begin{pmatrix} a &b\\ c& d \end{pmatrix}\in SL(2,\mathbb{Z})\mid c=0\bmod 4\}.$$ Dedekind ...
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Question about $q$-expansion principle for modular forms

I am reading Zagier's lectures "The 1-2-3 of Modular Forms". Before formulate the question let me introduce some notations. Let $z$ be the coordinate on the upper-half plane $\mathbb{H}$ and ...
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Formula for $q$-expansion of weight 2 modular forms

Is there a general formula for finding the $q$-expansion of weight 2 modular forms?
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The character of a newform

In "Modular Forms and Fermat's Last Theorem" Chp.1, p.7 Glenn Stevens talks about the character of a weight 2 newform $\epsilon$ in relation to the characteristic polynomial of a Frobenius element. ...
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Example of holomorphic modular form of weight 2?

I read that for holomorphic modular forms of weight 2, $f(q ) = \sum a_n q^n $ Hecke proved $|a_n| < Cn$. Are there any holomorphic modular forms of weight 2? There certainly aren't any for the ...
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Definition of a “modular Galois representation”

I am trying to pin down a definition for a $n$-dimensional modular Galois representation $$\rho : \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \rightarrow \text{GL}_n(A).$$ I am just looking for ...
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The ring of modular forms for $\Gamma_0(11)$

Let $\mathcal M(11) = \oplus \mathcal M_k(11)$ be a graded algebra of modular forms for congruence group $\Gamma_0(11)$. I want to find generators and relations between them. I proved that $\dim ...