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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$\left(\frac{\Delta(6z)\Delta(z)}{\Delta(3z)\Delta(2z)}\right)^{1/2}$ generates the field of modular functions on $\Gamma_1(6)$

I have already shown that the function, defined on the upper half plane $\{z\in\mathbb{C}:\Im(z)>0\}$, $$t(z)=\left(\frac{\Delta(6z)\Delta(z)}{\Delta(3z)\Delta(2z)}\right)^{1/2}$$ is a modular ...
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How did Hecke come up with Hecke-operators?

I'm currently studying Hecke-operators and I'm curious how Hecke came up with them. The original definition he gave in his paper is $$\left( f \mid T_n\right) (z) = n^{k - 1} \sum_{ad = n, \, b \mod d,...
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$A(z)$ is a automorphic form $\Rightarrow$ $A(-1/z)z^{-k}$ is a form

I've found this: Let A be an automorphic form (holomorphic) of degree $-k$, where $k\ge 2$. Call $a_0$ the constant term in its expansion at $i\infty$, $q$ the period of the form $A(-1/z)z^{-k}$ ...
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Reference/Literature on Eichler integrals

Has anyone reference or literature advices on Eichler integrals? I want to know what the Eichler integral associated to a modular form is. It should help to understand this, page 65, 4th sentence in ...
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Why is 12 the smallest weight for which a cusp forms exists

On wikipedia (here) I have read the following: Twelve is the smallest weight for which a cusp form exists. [...] This fact is related to a constellation of interesting appearances of the number ...
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Is $\eta^{24}(\tau)\,j(\tau) = {E_4}^3(q)$?

Given the j-function $j(\tau)$, $j(\tau) = 1728J(\tau)$, where $J(\tau)$ is Klein’s absolute invariant, the Dedekind eta function $\eta(\tau)$, and the following Eisenstein series, $\begin{align} ...
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Modular equation for a quotient of eta functions

In this article at page $4$ F. Beukers introduces a function $$y(\tau) = \frac{\eta^{8}(6\tau)\eta^{4}(\tau)}{\eta^{8}(2\tau)\eta^{4}(3\tau)}\tag{1}$$ where $\tau$ is a complex number with positive ...
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Modular transformations of $\eta(\tau)$

Under a modular transformation the Dedekind $\eta$ function transforms as $$\eta(-1/\tau) = \sqrt{-i}\eta(\tau).\tag*{$(*)$}$$Siegel gives a proof in this paper here that uses complex analytic ...
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Values of modular functions on the cusps

I have the meromorphic modular function (weight $0$) on $\Gamma_1(6)$ $$y(z)=\frac{\eta(6z)^8\eta(z)^4}{\eta(2z)^8\eta(3z)^4},$$whereby $\eta$ is the Dedekind eta function. The set of cusps of $\...
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64 views

Does viewing an Eisenstein series as a sum over cusps explain the antagonism between Eisenstein serieses and cusp forms?

I'm trying to understand the relationship between various aspects of the concept of "Eisenstein series" (as discussed for example in Diamond & Shurman's "A First Course in Modular Forms"), in ...
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calculation with modular forms

Let $\Delta(z)=(2\pi)^{12} q\prod_{n=1}^{\infty}(1-q^n)^{24}$ with $q=e^{2\pi i z}$ be the modular discriminant. I found $$\left(\dfrac{\Delta(6z)\Delta(z)}{\Delta(3z)\Delta(2z)}\right)^{1/2}=q\prod_{...
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roots of modular forms in the complex field

For $\begin{pmatrix}a&b\\c&d\end{pmatrix}\in SL_2(\mathbb{Z})$ the modular discriminant $$\Delta(z)=(2\pi)^{12}\eta(z)^{24}\qquad(1)$$ holds $$\Delta\left(\dfrac{az+b}{cz+d}\right)=(cz+d)^{12}\...
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87 views

What are these theta functions appearing in Sloane's database

Looking at Sloane's database, I found a neat formula for the lambda-invariant. Let $q:\tau \mapsto \exp(\pi i \tau)$ on the complex upper-half plane. Then $$\lambda(q) = 16q\;\prod_{k>0} \frac{(1 +...
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What is an L-Function and how can it be used for categorization?

On hackernews, I came across this article, describing the LMFDB, the database of L-functions, modular forms, and related objects which aims at the categorization of mathematical objects where ...
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295 views

Demystifying modular forms

I am really struggling to understand what modular forms are and how I should think of them. Unfortunately I often see others being in the same shoes as me when it comes to modular forms, I imagine ...
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358 views

Nice formulas for the lambda invariant of an elliptic curve

Where can I find some nice formulas for the lambda invariant of an elliptic curve? I vaguely recall there's a nice product formula in terms of $q$, but a Google search didn't give me much. Also, are ...
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27 views

contradiction to $M_k(\Gamma_1(6))=\mathbb{C}$

I just read the fact $M_0(\Gamma)=\mathbb{C}$ (constant functions) where $M_0(\Gamma)$ is the space of modular forms of weight $0$ on the congruence subgroup $\Gamma$. But in this article at page ...
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$f\in M_k (\Gamma_1(N)$ and $g(z)=f(dz)$, then $g\in M_k(\Gamma_1 (dN)$

Let $N,d\in\mathbb{N}$ and $\Gamma_1(N)=\{\begin{pmatrix}a&b\\c&d\end{pmatrix}\in SL_2(\mathbb{Z}):a\equiv d\equiv 1\mod N,~c\equiv 0\mod N\}$. I need a proof (references or your ideas) of: ...
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Is a modular form on $\text{SL}_2(\mathbb{Z})$ also a modular form on congruence subgroups?

Is a modular form $f$ of weight $k$ with respect to $\text{SL}_2(\mathbb{Z})$ always a modular form to a congruence subgroup $\Gamma$ (for example $\Gamma_1(N)$)? If the transformation law $f|_k\...
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Proving $\zeta(3)\in\mathbb{R}\setminus\mathbb{Q}$ using modular forms

I am looking for references proving $\zeta(3)\in\mathbb{R}\setminus\mathbb{Q}$ using modular forms, like this paper written by F. Beukers. Does anybody know some different papers or books? Thanks.
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Geometric interpretation of the width (of cusps)

Given a congruence subgroup $\Gamma$ and a cusp $\alpha$. Choose $\gamma\in SL_2(\mathbb{Z})$ such that $\gamma\infty=\alpha$. The minimal $h$ such that $\begin{pmatrix}1&h\\0&1\end{pmatrix}\...
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two congruence subgroups with the same set of cusps

Are there two congruence subgroups $\Gamma, \Gamma'$ of $SL_2(\mathbb{Z})$ with the same set of cusps? Thanks in advance.
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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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Need of cusps (with respect to Modular Forms)

In every introduction about Modular Forms (on $SL_2(\mathbb{Z})$ and congruence subgroups) one reads the term 'cusps'. A Modular Form should be holomorphic in the cusps. Can anybody explain to me, ...
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Hecke $L$-function of cusp form is entire

Let $f=\sum a(n)q^n\in S_k(N,\chi)$ be a cusp form of integral weight. Can someone give me the proof of the fact that : the Hecke $L$-function $L(f,s)=\sum\frac{a(n)}{n^s}$ is entire. I searched in ...
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Derivative of the Klein j-invariant

To prove that the field of meromorphic functions on $X(1)$ is generated by the Klein j-invariant, we need to show that the derivative of $j(\tau)$ does not vanish. (It only has simple zeroes). But I ...
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Regarding the weight of the given modular form

I am following this course. A question regarding the modular form of weight two that is constructed in lecture 33 emerged. Let me briefly tell you what's going on there. Let $\mathbb{H}$ denote the ...
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Convergence behaviour of Eichler integral

Consiger $g : \mathbb H \to \mathbb C$ a modular form of weight $2-k, k \in \frac{1}{2}\mathbb Z$. Let $z \in \mathbb H$ and consider the following integral: $$\int_{-\overline{z}}^{i\infty}(\omega+z)...
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Generators of the congruence subgroup $\Gamma (5) \subset SL(2,\mathbb Z)$

Recall that $SL(2,\mathbb Z)=\left\{A=\begin{pmatrix} a&b \\ c&d \end{pmatrix}: \det(A)=1; a,b,c,d \in \mathbb Z \right\} $ and $\Gamma(5)=\left\{A=\begin{pmatrix} a&b \\ c&d \end{...
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Modular properties of weight $\frac{3}{2}$ theta functions

Define the theta-functions $g(z) = - \sum_{n \in \mathbf{Z}} \left(n + \frac{1}{6} \right) e^{3 \pi i \left( n + \frac{1}{6} \right)^2 z}$. How can I show that $$g(z + 1) = e^{\frac{1}{24}2 \pi i} g(z)...
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a new continued fraction for $\sqrt{2}$

In a q-continued fraction related to the octahedral group I defined a new q-continued fraction for the square of ramanujan's octic continued fraction which I discovered using certain three term ...
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What is an exponential singularity?

In Ken Ono's Lecture notes on Harmonic Maass forms and Mock Modular Functions (here) the author uses the term "exponential singularity". What does this mean? Thanks a lot!
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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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Discriminant cusp form and generator of an ideal in the Hilbert class field (proof of prop. 14.1 Heegner points on $X_0(N)$, Gross)

Let $K$ be a quadratic imaginary field where the rational prime $N$ splits: $\mathfrak{n}\cdot\bar{\mathfrak{n}}=(N)$ and denote with $H_K$ the Hilbert class field of $K$. At the beginnig of the ...
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$p$ adic modular forms and wide open neighbourhood (e.g. Coleman primitive): is it possible to obtain a holomorphic function?

It is well known that a modular form (of weight $k$ and level $N$) is in particular also a classical modular form; this can be seen both using Serre's definition with $q$-expansion and the Katz's one, ...
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Hauptmodul for $\Gamma_1(3)$

I'm looking for a nice formula for a Hauptmodul for $\Gamma_1(3)$. This is certainly very classical.A reference would be great.
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How to compute the Hecke operator on an Eisenstein series?

In my current course on Modular Forms we are now discussing Hecke operators and we are asked the following: Prove that for any even integer $k \geq 4$ and prime $p$ we have $T_pG_k = \sigma_{k−1}(p)...
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show that $\sum P(x,y) e^{x^2 + y^2}$ is a modular form over $\Gamma_0(4)$ where $P(x,y) = x^4 - 6 x^2 y^2 + y^4$

In these lecture notes of Zagier, I read that generalized theta functions are still modular forms. Let $q = e^{2\pi i z}$ $$\theta(z) = \sum_{(x,y) \in \mathbb{Z}} \Big[ x^4 - 6 x^2 y^2 + y^4 \Big] \...
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How can we show the equality $[SL_2(\mathbb Z): \Gamma_0(N)]=N\prod_{p\mid N}\left(1+\frac1p\right)$?

If you want to read question directly, please go to down. Firstly, let's give the definitions of $\Gamma(N)$, $\Gamma_1(N)$ ve $\Gamma_0(N)$: $$\Gamma(N) : = \left \{\left[ \begin{array}{cc}a & b ...
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Entring in the world of Modular Forms

Following wikipedia, a modular form of weight $k$ for the modular group $\mathrm{SL}_2(\mathbb{Z})$ is a complex-valued function  $f$  on the upper half-plane $H = \{z\in\mathbb{C}\colon Im(z)>0\}$ ...
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Finiteness of Dimension of $M_{k}(\Gamma)$

Let $M_{k}(\Gamma)$ denote the space of weight $k$ modular forms for the congruence subgroup $\Gamma$. Are there any proofs of the finiteness of the dimension of $M_{k}(\Gamma)$ that don't rely on ...
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About regular/irregular cusps and dimension formula for modular forms and cusp forms

I'm reading Diamond and Shurman's book about modular forms, in particular I'm reading the proof of the dimension formula of the vector spaces $\mathcal{M}_{k}(\Gamma)$ and $\mathcal{S}_{k}(\Gamma)$ ...
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Are all Dirichlet coefficients of any element of the Selberg class necessarily algebraic?

The title says it all: do we know at least one element of the Selberg class having at least one transcendental coefficient in its development in a Dirichlet series for $\Re(s)>1$? Or are such ...
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How to visualize the region $\mathbb{H}/\Gamma_0(4)$ and its cusps?

In number theory we learn that $\theta(z) = \sum q^{n^2}$ is a modular form with respect to $\Gamma = \Gamma_0(4)$. This boils down to two properties: $\theta(z)= \theta(z+1)$ this shift symmetry ...
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Zeros of the second derivative of the modular $j$-function

I am interested in the zeros of $j''(z)$, where $j:\mathbb{H}\rightarrow\mathbb{C}$ is the classic modular function. Specifically I am interested in knowing if the zeros of $j''$ are algebraic over $\...
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Proof of $\frac{1}{e^{\pi}+1}+\frac{3}{e^{3\pi}+1}+\frac{5}{e^{5\pi}+1}+\ldots=\frac{1}{24}$

I would like to prove that $\displaystyle\sum_{\substack{n=1\\n\text{ odd}}}^{\infty}\frac{n}{e^{n\pi}+1}=\frac1{24}$. I found a solution by myself 10 hours after I posted it, here it is: $$f(x)=\...
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$p$-depletion of a modular form

Let $p$ a prime and $N$ an integer such that $p\not\mid N$. I will denote with $X_0(m)$ the modular curve with respect to the congruence subgroup $\Gamma_0(m)$. Let $f$ be a modular form with ...
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About period of elliptic points

I'm reading about modular forms and I've seen some results which only make use of elliptic points of period 2 and 3 (and cusps). I would like to know if it's possible to find elliptic points of period ...
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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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Modular Discriminant and Pentagonal Numbers

I am asked to show $$(2\pi)^{-12}\Delta(\tau) = q \cdot \Big (\sum_{n\in \mathbb{Z}} (-1)^n \cdot q^{(3n^2+n)/2} \Big)^{24}$$ where $\Delta:\mathbb{H} \to \mathbb{C}$ is the modular discriminant, $q=e^...