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Questions tagged [modular-forms]

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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Asymptotics for average of Fourier coefficients of cusp form

Iwaniec Topics in Classical Automorphic Forms, after introducing the Rankin-Selberg convolution $L$-function $$L(f \otimes \bar{f}, s) = \sum_{n = 1}^\infty \frac{|a(n)|^s}{n^s}$$ of a weight $k$ ...
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Additional components of the family of identities $ \eta $ functions of level 6.

Can these equations be considered as the family of $\eta$ identities functions of level 6? q6_24_64=729*u1^4*u3^20 – 1944*u1^3*u2^5*u3^15*u6 + 1728*u1^2*u2^10*u3^10*u6^2 – 512*u2^20*u6^4; q6_48_108=...
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Prerequisites for reading “Automorphic Forms and Representations”

I am hoping to read (at least the first chapter of) "Automorphic forms and Representations" by D. Bump. As per the introductions, prerequisites are "basic knowledge of algebraic number theory, the ...
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First Fourier coefficient of weight $k$ holomorphic cusp form.

Let $f$ a weight $k$ holomorphic Hecke cusp form with $\|f\|^2=\langle f,f\rangle=1$ with fourier expansion $$f(z)=\sum_{r\geq 1}a_f(r)e(rz)$$ Let $\displaystyle\lambda_f(r)=\frac{a_f(r)r^{(-k+1)/2}}{...
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An integral involving Euler's $\Phi(q)$ function

Let $\phi(q)$ be the Euler's function given by the infinite product : $$\phi(q)=\prod_{n=1}^{\infty}\frac{1}{1-q^{n}}\;\;\;\;|q|<1$$ And let $\mu(k)$ be the $\text{M}\ddot{\text{o}}\text{bius}$ ...
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A modular equation of 29th degree of Dedekind’s $\eta$ function.

Regarding the Post Additional values of Dedekind's $\eta$ function in radical form I wrote the equation that has as root the value $\frac{\eta(29i)}{\eta(i)}$ that is missing. Can someone help ...
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40 views

A modular equation of 23rd degree of Dedekind’s $\eta$ function.

Regarding the Post Additional values of Dedekind's $\eta$ function in radical form I wrote the equation that has as root the value $\frac{\eta(23i)}{\eta(i)}$ that is missing. Can someone help ...
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34 views

A modular equation of 19th degree of Dedekind’s $\eta$ function.

Regarding the Post Additional values of Dedekind's $\eta$ function in radical form I wrote the equation that has as root the value $\frac{\eta(19i)}{\eta(i)}$ that is missing. Can someone help ...
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1answer
53 views

A modular equation of 17th degree of Dedekind’s $\eta$ function.

Regarding the Post Additional values of Dedekind's $\eta$ function in radical form I wrote the equation that has as root the value $\frac{\eta(17i)}{\eta(i)}$ that is missing. Can someone help ...
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1answer
60 views

A modular equation of 13th degree of Dedekind’s $ \eta$ function.

Regarding the Post Additional values of Dedekind's $\eta$ function in radical form I wrote the equation that has as root the value $\frac{\eta(13i)} {\eta(i)}$ that is missing. Can someone help ...
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1answer
71 views

A modular equation of 11th degree of Dedekind's $\eta$ function.

Regarding the Post Additional values of Dedekind's $\eta$ function in radical form I wrote the equation that has as root the value $\frac{\eta(11i)} {\eta(i)}$ that is missing. Can someone help ...
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Which root lattices have a theta series with this property?

Suppose $\Lambda$ is a root lattice (the integral lattice generated by a crystallographic root system). Consider its theta series $$\theta_{\Lambda}(q) = \sum_{a\in \Lambda} q^{(a,a)/2},$$ where $(\...
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1answer
49 views

Why are elliptic points called elliptic?

Points on the upper half plane $\mathbb H := \{ z \in \mathbb C : \Im(z)>0 \}$ are called elliptic with respect to some $\gamma \in \operatorname{SL}_2(\mathbb Z)$ if they are fixpoints of the ...
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Why are some Ramanujan $G_n$ and $g_n$ functions highly factorable?

Given the Dedekind eta function $\eta(\tau)$ with $\tau = \sqrt{-n}$. Define the Ramanujan $G_n$ and $g_n$ functions as, $$G_n = 2^{-1/4}\frac{\eta^2(\tau)}{\eta(\tau/2)\,\eta(2\tau)}$$ $$g_n = 2^{-1/...
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Proving $j$-invariant is surjective

I'm trying to solve exercise $1.1.9$ from Diamond's A First Course in Modular Forms, in which we must prove the $j$-invariant $j:\mathcal{H}\to\mathbb{C}$ with $j(\tau)=1728\frac{g_2(\tau)^3}{\Delta(\...
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Simple question about Hecke operators for the adeles

I have a question about Hecke operators which should not be too difficult. I'm reading these notes by Jerry Shurman on translating modular forms to the adeles and don't understand a certain sentence. ...
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Decomposing the space of modular forms into $\chi$-eigenspaces via representation theory

I'm reading Diamond and Shurman's introductory book on modular forms and, in chapter 4.3, they give a decomposition of $M_k(\Gamma_1(N))$ as a direct sum of eigenspaces defined for Dirichlet ...
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28 views

The index of $\{\pm 1\}$ in $\gamma^{-1}\Gamma_0(N)\gamma$

Let $N$ be a positive integer, and let $\Gamma_0(N)$ be the Hecke subgroup. Let $\gamma\in\mathrm{SL}_2(\mathbb{Z})$. My question is: what is the generator of $$\gamma^{-1}\Gamma_0(N)\gamma/\{\pm 1\}...
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36 views

What are all the subgroups of SL_2(Z)?

I'm learning about modular forms and I dislike the congruence subgroups. They feel inadequately motivated to me, and they beg the question of what other subgroups of SL_2(Z) are there that may be just ...
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Why aren't holomorphic modular forms bounded?

Let $f$ be any non-zero integral weight (holomorphic) modular form with respect to $SL_2(\mathbb{Z})$ and of weight $k, k\geq 4$. Since it is holomorphic at infinity, for given $\epsilon > 0$, it ...
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Cusps of congruence subgroup

Let $\Gamma$ be a congruence subgroup and let $F$ be its fundamental domain. My first question is: Do the cusps of $\Gamma$ can be defined as elements of $\partial F\cap\mathbb{P^1(\mathbb{Q})}$? ($\...
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Test vector for local zeta integral with ramified character

Suppose $\pi$ is an unramified principal series representation of ${\rm GL}_2(F)$, where $F$ is a non-archimedean local field with integers $\mathfrak{o}$. Let $W$ be a function in its Whittaker model....
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Computation of $\Phi(T,[\gamma])$ in a paper of Poonen and Rodriguez-Villegas

I am reading the paper "Lattice Polygons and the Number 12" by Bjorn Poonen and Fernando Rodriguez-Villegas (a copy of this paper can be found, e.g., on the first author's webpage). In it, the authors ...
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Linear relation between theta functions $\theta(z;u) \in S_8(\Gamma_0(4))$?

Using spherical Harmonics, I can modular forms on $\Gamma_0(4)$ as follows. Let $u(\vec{m}) \in S^2$ be a spherical harmonic: $$ \theta(z; u) = \sum_{m \in \mathbb{Z}^4} u(m) e(|m|z) = \sum_{(a,b,c,...
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For a cusp modular form of weight $k$, what is remarkable about $y^{k/2}f$ staying bounded as $y\rightarrow \infty$?

A holomorphic modular form is said to be a cusp form if its constant Fourier coefficients at the cusps are zero. In the case of a modular form of weight $k$ for the full modular group $\mathrm{SL}_2(\...
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39 views

Cusp form becomes a bounded function on $\operatorname{SL}_2(\mathbb R)$

I'm reading Gelbart's notes on the decomposition of $L^2(\operatorname{SL}_2(\mathbb Z) \backslash \operatorname{SL}_2(\mathbb R))$, and am stuck on a small detail about reinterpreting modular forms ...
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62 views

Theta function squared is a weight $1$ modular form

Let $$\vartheta(\tau) = \sum_{n\in\mathbb{Z}}e^{\pi in^2\tau}.$$ I know that $\vartheta$ satisfies the transfromation properties $$\vartheta(\tau + 2) = \vartheta(\tau), \quad \vartheta\left(-\frac{1}{...
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If $f$ is a modular form and $y^{k/2}f$ stays bounded as $y\rightarrow \infty$, its constant Fourier coefficients at the cusps are zero

Let $f(\tau)$ be a modular form of weight $k$, where $\tau = x + iy \in \mathbb H$. Then $f$ is a cusp form if its constant Fourier coefficients at the cusps are zero. By modularity, it suffices to ...
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Generating function for sum of two squares is a modular form

Let $$\theta(\tau) = \sum_{n = -\infty}^{\infty}q^{n^2}, \quad q:=e^{2\pi i\tau}$$ be the classical theta series. The Fourier coefficients $r_2(n)$ of $\theta(\tau)^2$ give the number of ways of ...
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Consider more cusps in modular form?

A holomorphic function on $\mathbb{H}$ is called a modular form of weight $k$ for SL$_2(\mathbb{Z)}$ if it satisfies $f(Mz)= (cz+d)^kf(z) \ \ \ \ \ \ \ \forall M \in$ SL$_2(\mathbb{Z)}$ $f$ is ...
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Showing existence of modular function

I am studying Mock Modular forms in the book Harmonic Maass Forms and Mock Modular Forms: Theory and Applications and i have problems with a theorem. In Theorem 6.4 it says If $Z_E^+ $ has poles in $...
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1answer
71 views

Eisenstein and Weierstrass zeta - series identity

Let $\zeta_\Lambda$ be the Weierstrass $\zeta$-function for lattice $\Lambda$ and $G_2$ the Eisenstein series of weight $2$. The quasiperiod is defined by $\eta_\Lambda(\lambda) := \zeta_\Lambda(z + \...
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Order of modular form $f$ at $p$ is equal to order of $f$ at $g(p)$

Let $f\ $ be a modular form of weight $k$ and $G=\operatorname{SL}_2(\mathbb{Z})/\{\pm 1\}$ be the modular group. In Lattices and Codes from Ebeling, they say that the identity $$f(\tau)=(c\tau +d)^...
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1answer
36 views

Bibliography request for differential equation involving Dedekind Eta Function

I would like to find the reference or bibliographic source of the fact that the following differential equation: $$36(y')^2-24y''y+y''' = 0$$ is satisfied by $y(z) = \frac{\eta'(z)}{\eta(z)}$ ($\eta(...
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Asymptotic's for Fourier coefficients of $GL(3)$ Maass forms

Let $f$ be a $GL(3)$ Hecke-Maass cusp form and $A(m,n)$ denote its Fourier coefficients. (1) Are there any lower bounds known for $\sum_{p\leq x}|A(1,p)|^2$ or $\sum_{n\leq x}|A(1,n)|^2$ ? (we know ...
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Factoring out Kloosterman Sum

I'm reading Iwaniec's book and he says Kloosterman sum factors into $S(n,n;c)=S(n\bar{q},n\bar{q};r)T(n\bar{r},n\bar{r};q)$ where $n$ is square free and $c=rq$ such that $(q,n)=(q,r)=1$(i.e. $q$ is ...
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1answer
36 views

Another algebraic connection from a Ramanujan theorem.

I think I have interpreted a Ramanujan theorem well, but I ask someone for help to confirm it. In the Notebook II of Ramanujan, we read: Berndt (Vol. IV p. 24 Entry 13), algebraically demonstrates ...
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Hecke operator on half integral weight modular form

(Sorry for my poor english skill..) Let $N$ be an odd integer and $k$ be a positive integer. Let $\chi$ be a Dirichlet character modulo $4N$ and $f=\sum_{n=1}^{\infty} a(n)q^n \in S_{k+\frac{1}{2}}(\...
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1answer
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A clarification of Ramanujan's theorem.

I would like to know if my interpretation is right. Who can give me comfort? In Notebook IV, Berndt reports Entry 11 (p. 328) and the proof that from, it's purely algebraic. However, I am sure ...
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Do all hauptmoduln have integral coefficients in their Fourier expansion

Define Klein's modular $j$-invariant as $$j(\tau) = 1728 \frac{g_2^3}{\Delta},$$ where $$ g_2 = 60 \sum_{(m,n)}' \Big( \frac{1}{m + n \tau} \Big)^4$$ and $$ g_3 = 60 \sum_{(m,n)}' \Big( \frac{1}{m + ...
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1answer
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Hypergeometric functions and modular forms

May I please ask if it is possible to write Hypergeometric functions in terms of Jacobi theta functions? I am trying to bring the following Hypergeometric expression (pg.9, eq 4.3) into a known ...
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1answer
45 views

Generalized Jacobi theta functions - Laurent series expansion of H(w,q,S).

Can someone please assist me with the missing steps in the proof of 'proposition 3' in M. Kaneko and D. Zagier paper (https://people.mpim-bonn.mpg.de/zagier/files/progmath/129/165/fulltext.pdf, pg. 4)....
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1answer
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Subgroup of $\text{PSL}(2,\mathbb{Z})$ generated by $S$ and $T^2$

In the group $PSL(2,\mathbb{Z})$ (which acts on the upper half plane of $\mathbb{C}$), suppose $S$ is the inversion and $T$ is the translation by $1$, i.e. $$S= \left( {\begin{array}{cc} 0 & ...
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Find weight 2 newforms with given coefficients using websites or softwares, if exists

Given a finite set $P$ that consists of "small" prime numbers $$P=\{p_1, ..., p_N\}.$$ Further if given integers $a_{p_i}$ for each $p_i \in P$, then are there websites or software that could give us ...
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How to prove the integrality of this Eisenstein Series?

Cohen and Strömberg included in their book Modular Forms: A Classical Approach the chapter "A Brief Introduction to Complex Multiplication" (pp. 199-203). In this chapter (p. 202) we find Proposition ...
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1answer
50 views

How to know the j-invariant of the modular elliptic curve from the modular form?

How do people compute the $j$-invariant of an elliptic curve $E$ over $\mathbb Q$ from the associated modular form $f=\sum_{n=1}^{\infty} a_nq^n$? In other words, how to compute (at least giving some ...
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Does Weil's converse theorem holds for weight 1?

I'm reading Iwaneic's "Topics in classical automorphic forms". Now, I'm reading the proof the theorem that for any Hecke character $\xi$ of a quadratic field $K/\mathbb{Q}$, there exists a $\mathrm{GL}...
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48 views

Fundamental domains of modular groups $\Gamma_0(N)$

For the modular group $\Gamma_0(N)$, where $N\in \mathbb{Z}_+$, there exists a fundamental domain $D_N$ which lies in the strip $-\frac{1}{2} < z < \frac{1}{2}$ of the upper half plane, since ...
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1answer
50 views

$p$-th coefficient of weight $2$ new form with $p | N$ must be $1$ or $0$ or $-1$?

Let $f= \sum_{n=1}^{\infty}a_nq^n \in S_2^{new}(N)$ be a normalized new form of weight $2$ with respect to $\Gamma_0(N)$ and assume $p|N$ is a prime. Then must $a_p=0$ if $p^2|N$ and belongs to $\{-1,...
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85 views

Do general discrete subgroups of $\operatorname{SL}_2(\mathbb R)$ have fundamental domains in the upper half plane?

Let $\Gamma$ be a congruence subgroup of $\operatorname{SL}_2(\mathbb Z)$. The quotient $\Gamma \backslash \mathbb H$ has the structure of a one dimensional complex manifold, such that the quotient ...