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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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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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 ...
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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 ...
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euation of modular functions

In this article at page four: It says that one easily checks the equation $(1)$. But I can't check that. I already tried to use $\eta(-1/z)=(-iz)^{1/2}\eta(z),~z$ in the upper half plane, but this ...
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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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280 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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356 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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38 views

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 ...
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40 views

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 ...
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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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1answer
24 views

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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35 views

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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64 views

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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22 views

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: ...
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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 ...
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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} ...
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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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259 views

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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31 views

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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169 views

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 = ...
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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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335 views

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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24 views

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: ...
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48 views

$p$-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, ...
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35 views

Every point in the upper half plane is congruent to a point in the fundamental region of $\lambda$

In Ahlfors complex analysis book, 2nd edition Pg. 273-274 Chapter 7 Section 3.5 we have the theorem $\textbf{Theorem 8.}$ Every point $\tau$ in the upper half plane is equivalent under the ...
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The j-function of a point on the boundary of the fundamental domain is real valued.

I am trying to prove the following statement: Let $z \in \mathbb{D} $ (the standard fundamental domain for $SL_{2}(\mathbb{Z})$). Prove that if $z$ lies on the boundary of $\mathbb{D}$, or if ...
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Weight 2 Newforms of large level computations.

I am stuck with some weight $2$ newform computations of large level. For example I want to compute newforms of level $11520$. Can anyone suggest me a way to do it? I need it to solve some diophantine ...
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28 views

Residue of Rankin-Selberg Dirichlet series

Let $f\in S_k(N,\chi)$ be a cusp forms, and let $R_f(s)=\sum_{n=1}^{\infty}\frac{a(n)^2}{n^s}$ the Rankin-Selberg Dirichlet series then $R_f(s)$ hase a pole at $s=k.$ Can someone suggest to me a ...
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Good description of orbits of upper half plane under $SL_2 (Z)$

It's known that $SL_2(Z)$ acts on $H=\{z\, |\, Im(z)>0\}$, is there a good description of orbits of $i$ and $w$, other than directly write down $=\{ \frac{ac|z|^2+bc\bar z+adz+bd}{c^2|z|^2+dc\bar z ...
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Eisenstein series of congruence subgroup

Let $P$ be a prime number, and let $k >2$ be a positive even integer. Now I want to compute the Fourier expansion of the Eisenstein series of $\Gamma_0(p)$ at the cusp $0$, i.e., $$ E_{k, 0}(z) = ...
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35 views

Elliptic points of modular group

For $SL_2(Z)$ we know the elliptic point is either equivalent to $i$ or $1/2+\sqrt{3}/2i$. Then for finite index subgroup of $SL_2(Z)$, $\Gamma$, is this still true? There will be fewer points fixed ...