LMFDB - Space of modular forms of level 1638, weight 2, and Character 1638.j

Defining parameters

Level:	$ N $	$=$	$ 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1638.j (of order $3$ and degree $2$)
Character conductor:	$\operatorname{cond}(\chi)$	$=$	$ 7 $
Character field:			$\Q(\zeta_{3})$
Newform subspaces:			$ 20 $
Sturm bound:			$672$
Trace bound:			$7$
Distinguishing $T_p$:			$5$, $11$, $17$

Dimensions

The following table gives the dimensions of various subspaces of $M_{2}(1638, [\chi])$.

	Total	New	Old
Modular forms	704	80	624
Cusp forms	640	80	560
Eisenstein series	64	0	64

Trace form

$ 80 q - 40 q^{4} - 8 q^{5} + O(q^{10}) $

$ 80 q - 40 q^{4} - 8 q^{5} + 4 q^{10} - 4 q^{11} - 4 q^{13} + 10 q^{14} - 40 q^{16} + 18 q^{17} + 4 q^{19} + 16 q^{20} + 12 q^{22} + 12 q^{23} - 44 q^{25} - 6 q^{26} - 12 q^{28} + 4 q^{29} - 4 q^{31} - 32 q^{34} + 12 q^{35} - 10 q^{38} + 4 q^{40} - 32 q^{41} + 16 q^{43} - 4 q^{44} - 20 q^{46} - 12 q^{47} + 62 q^{49} - 16 q^{50} + 2 q^{52} - 22 q^{53} + 8 q^{55} - 2 q^{56} + 8 q^{59} - 34 q^{61} - 20 q^{62} + 80 q^{64} + 4 q^{65} - 8 q^{67} + 18 q^{68} + 8 q^{71} - 4 q^{73} - 8 q^{76} - 24 q^{77} - 8 q^{80} - 24 q^{82} + 24 q^{83} + 64 q^{85} + 12 q^{86} - 6 q^{88} - 12 q^{89} + 16 q^{91} - 24 q^{92} + 22 q^{94} + 8 q^{95} - 16 q^{97} + 72 q^{98} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Decomposition of $S_{2}^{\mathrm{new}}(1638, [\chi])$ into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$	Coefficient ring index	Sato-Tate	$q$-expansion
1638.2.j.a	$1638$	$2$	1638.j	7.c	$3$	$2$	$1$	$13.079$	$\Q(\sqrt{-3}) $	None		$2$	$0$	$-1$	$0$	$-2$	$-5$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q-\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}-2\zeta_{6}q^{5}+\cdots$
1638.2.j.b	$1638$	$2$	1638.j	7.c	$3$	$2$	$1$	$13.079$	$\Q(\sqrt{-3}) $	None		$2$	$0$	$-1$	$0$	$-2$	$-1$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q-\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}-2\zeta_{6}q^{5}+\cdots$
1638.2.j.c	$1638$	$2$	1638.j	7.c	$3$	$2$	$1$	$13.079$	$\Q(\sqrt{-3}) $	None		$2$	$0$	$-1$	$0$	$0$	$5$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q-\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+(2+\zeta_{6})q^{7}+\cdots$
1638.2.j.d	$1638$	$2$	1638.j	7.c	$3$	$2$	$1$	$13.079$	$\Q(\sqrt{-3}) $	None		$2$	$1$	$-1$	$0$	$1$	$-5$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q-\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+\zeta_{6}q^{5}+(-3+\cdots)q^{7}+\cdots$
1638.2.j.e	$1638$	$2$	1638.j	7.c	$3$	$2$	$1$	$13.079$	$\Q(\sqrt{-3}) $	None		$2$	$0$	$-1$	$0$	$1$	$-1$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q-\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+\zeta_{6}q^{5}+(1+\cdots)q^{7}+\cdots$
1638.2.j.f	$1638$	$2$	1638.j	7.c	$3$	$2$	$1$	$13.079$	$\Q(\sqrt{-3}) $	None		$2$	$0$	$1$	$0$	$-4$	$-1$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q+\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}-4\zeta_{6}q^{5}+\cdots$
1638.2.j.g	$1638$	$2$	1638.j	7.c	$3$	$2$	$1$	$13.079$	$\Q(\sqrt{-3}) $	None		$2$	$0$	$1$	$0$	$-1$	$-5$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q+\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}-\zeta_{6}q^{5}+(-3+\cdots)q^{7}+\cdots$
1638.2.j.h	$1638$	$2$	1638.j	7.c	$3$	$2$	$1$	$13.079$	$\Q(\sqrt{-3}) $	None		$2$	$0$	$1$	$0$	$-1$	$-1$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q+\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}-\zeta_{6}q^{5}+(1+\cdots)q^{7}+\cdots$
1638.2.j.i	$1638$	$2$	1638.j	7.c	$3$	$2$	$1$	$13.079$	$\Q(\sqrt{-3}) $	None		$2$	$0$	$1$	$0$	$0$	$-5$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q+\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+(-2-\zeta_{6})q^{7}+\cdots$
1638.2.j.j	$1638$	$2$	1638.j	7.c	$3$	$2$	$1$	$13.079$	$\Q(\sqrt{-3}) $	None		$2$	$0$	$1$	$0$	$3$	$-1$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q+\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+3\zeta_{6}q^{5}+\cdots$
1638.2.j.k	$1638$	$2$	1638.j	7.c	$3$	$4$	$2$	$13.079$	$\Q(\sqrt{-3}, \sqrt{7})$	None		$2$	$0$	$-2$	$0$	$-2$	$0$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(-1-\beta _{2})q^{2}+\beta _{2}q^{4}+(-1+\beta _{1}+\cdots)q^{5}+\cdots$
1638.2.j.l	$1638$	$2$	1638.j	7.c	$3$	$4$	$2$	$13.079$	$\Q(\sqrt{-2}, \sqrt{-3})$	None		$2$	$0$	$-2$	$0$	$0$	$10$	$3$	$\mathrm{SU}(2)[C_{3}]$	$q-\beta _{1}q^{2}+(-1+\beta _{1})q^{4}-\beta _{2}q^{5}+(2+\cdots)q^{7}+\cdots$
1638.2.j.m	$1638$	$2$	1638.j	7.c	$3$	$4$	$2$	$13.079$	$\Q(\sqrt{2}, \sqrt{-3})$	None		$2$	$0$	$2$	$0$	$-4$	$-2$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q-\beta _{2}q^{2}+(-1-\beta _{2})q^{4}+(-\beta _{1}+2\beta _{2}+\cdots)q^{5}+\cdots$
1638.2.j.n	$1638$	$2$	1638.j	7.c	$3$	$4$	$2$	$13.079$	$\Q(\sqrt{2}, \sqrt{-3})$	None		$2$	$0$	$2$	$0$	$0$	$2$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(1+\beta _{2})q^{2}+\beta _{2}q^{4}+\beta _{1}q^{5}+(1+\beta _{1}+\cdots)q^{7}+\cdots$
1638.2.j.o	$1638$	$2$	1638.j	7.c	$3$	$4$	$2$	$13.079$	$\Q(\sqrt{-2}, \sqrt{-3})$	None		$2$	$0$	$2$	$0$	$0$	$10$	$3$	$\mathrm{SU}(2)[C_{3}]$	$q+\beta _{1}q^{2}+(-1+\beta _{1})q^{4}-\beta _{2}q^{5}+(2+\cdots)q^{7}+\cdots$
1638.2.j.p	$1638$	$2$	1638.j	7.c	$3$	$6$	$3$	$13.079$	6.0.309123.1	None		$2$	$0$	$-3$	$0$	$-2$	$-4$	$3$	$\mathrm{SU}(2)[C_{3}]$	$q-\beta _{4}q^{2}+(-1+\beta _{4})q^{4}+(-\beta _{1}-\beta _{4}+\cdots)q^{5}+\cdots$
1638.2.j.q	$1638$	$2$	1638.j	7.c	$3$	$6$	$3$	$13.079$	6.0.21870000.1	None		$2$	$0$	$-3$	$0$	$3$	$3$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q+\beta _{2}q^{2}+(-1-\beta _{2})q^{4}+(-\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots$
1638.2.j.r	$1638$	$2$	1638.j	7.c	$3$	$8$	$4$	$13.079$	8.0.8681953329.1	None		$2$	$0$	$4$	$0$	$2$	$7$	$3$	$\mathrm{SU}(2)[C_{3}]$	$q+(1+\beta _{3})q^{2}+\beta _{3}q^{4}+(1+\beta _{1}+\beta _{3}+\cdots)q^{5}+\cdots$
1638.2.j.s	$1638$	$2$	1638.j	7.c	$3$	$10$	$5$	$13.079$	$\mathbb{Q}[x]/(x^{10} + \cdots)$	None		$2$	$0$	$-5$	$0$	$1$	$-3$	$2^{2}\cdot 3$	$\mathrm{SU}(2)[C_{3}]$	$q+(-1-\beta _{2})q^{2}+\beta _{2}q^{4}+(\beta _{6}-\beta _{8}+\cdots)q^{5}+\cdots$
1638.2.j.t	$1638$	$2$	1638.j	7.c	$3$	$10$	$5$	$13.079$	$\mathbb{Q}[x]/(x^{10} + \cdots)$	None		$2$	$0$	$5$	$0$	$-1$	$-3$	$2^{2}\cdot 3$	$\mathrm{SU}(2)[C_{3}]$	$q-\beta _{2}q^{2}+(-1-\beta _{2})q^{4}-\beta _{8}q^{5}+(-1+\cdots)q^{7}+\cdots$

Decomposition of $S_{2}^{\mathrm{old}}(1638, [\chi])$ into lower level spaces

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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1638.j (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 20 \)
Sturm bound:			\(672\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(5\), \(11\), \(17\)

Introduction

L-functions

Modular forms

Varieties

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Groups

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Defining parameters

Dimensions

Trace form

Decomposition of \(S_{2}^{\mathrm{new}}(1638, [\chi])\) into newform subspaces

Decomposition of \(S_{2}^{\mathrm{old}}(1638, [\chi])\) into lower level spaces

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	1638.2.j

Level	$1638$
Weight	$2$
Character orbit	1638.j
Rep. character	$\chi_{1638}(235,\cdot)$
Character field	$\Q(\zeta_{3})$
Dimension	$80$
Newform subspaces	$20$
Sturm bound	$672$
Trace bound	$7$