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The answer is yes when $k$ is a multiple of $4$. There is a unique form of weight $k$ of the form $f_k=1+a_dq^d+\cdots$. When $k$ is a multiple of $4$ this is the theta series for a putative extremal …

Some vague thoughts, rather than an actual answer. The zeroes of a modular form $f$ of weight $k$ correspond (except at the cusps) to zeros of the modular function $F=f^{12k}/\Delta^k$ so the problem …

You can do this if and only if the trace of $M$ is an integer. By the theory of the rational canonical form if matrices $A$ and $B$ over $\mathbb{Q}$ have the same characteristic polynomial and neithe …

I am looking for a reference for the transformation formulae for the classical theta-functions $$\theta_4(\tau)=\sum_{n=-\infty}^\infty (-1)^n q^{n^2}$$ and $$\theta_2(\tau)=\sum_{n=-\infty}^\infty q^ …

This question is something of a follow-up to Transformation formulae for classical theta functions . How does one recognise whether a subgroup of the modular group $\Gamma=\mathrm{SL}_2(\mathbb{Z})$ …

I'm afraid I know nothing about $B_3$ but here is my favourite construction of $\widetilde{\mathrm{SL}}_2(\mathbb{R})$. The elements of $\widetilde{\mathrm{SL}}_2(\mathbb{R})$ are pairs $(A,f)$ where …

To answer Scott; yes, every modular form of integer weight $k$ for a congruence group can be expressed rationally in terms of $\eta(r\tau)$ for rational $r$. For a start, $g=f/\eta^{2k}$ is a modular …

Here's the standard example. I found it in Lang's Algebraic Number Theory where he attributes it to Artin. Let $K$ be the splitting field of $X^5-X+1$ over $\mathbb{Q}$. Then $K$ has Galois group $S_5 …