Let $k$ be a positive integer and let $\Gamma$ be a finite index subgroup of the modular group $\SL(2,\Z)$.
A (classical) modular form $f$ of weight $k$ on $\Gamma$, is a holomorphic function defined on the upper half plane $\mathcal{H}$, which satisfies the transformation property \[ f(\gamma z) = (cz+d)^k f(z) \] for all $z\in\mathcal{H}$ and $\gamma=\begin{pmatrix}a&b\\c&d\end{pmatrix}\in \Gamma$ and is holomorphic at all the cusps of $\Gamma$.
If $\Gamma$ contains the principal congruence subgroup $\Gamma(N)$ then $f$ is said to be a modular form of level $N$.
For each fixed choice of $k$ and $\Gamma$ the set of modular forms of weight $k$ on $G$ form a finite-dimensional $\mathbb{C}$-vector space denoted $M_k(\Gamma)$.
For the congruence subgroup $\Gamma_1(N)$ the space $M_k(\Gamma_1(N))$ decomposes as a direct sum of subspaces $M_k(N,\chi)$ over the group of Dirichlet characters $\chi$ of modulus $N$, where $M_k(N,\chi)$ is the subspace of forms $f\in M_k(N)$ that satisfy \[ f(\gamma z) = \chi(d)(cz+d)^k f(z) \] for all $\gamma=\begin{pmatrix}a&b\\c&d\end{pmatrix}$ in $\Gamma_0(N)$.
Elements of $M_k(N,\chi)$ are said to be modular forms of weight $k$, level $N$, and character $\chi$.
For trivial character $\chi$ of modulus $N$ we have $M_k(N,\chi)=M_k(\Gamma_0(N))$.
- Review status: reviewed
- Last edited by Alex J. Best on 2018-12-19 06:32:25
- cmf.bad_prime
- cmf.character
- cmf.cm_form
- cmf.cusp_form
- cmf.eisenstein
- cmf.fouriercoefficients
- cmf.hecke_operator
- cmf.level
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- dq.ec.reliability
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- rcs
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- rcs.rigor.cmf
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- lmfdb/classical_modular_forms/__init__.py (line 6)
- lmfdb/classical_modular_forms/main.py (line 210)
- lmfdb/classical_modular_forms/main.py (line 222)
- 2018-12-19 06:32:25 by Alex J. Best (Reviewed)