LMFDB - Space of modular forms of level 1690, weight 2, and Character 1690.e

Defining parameters

Level:	$ N $	$=$	$ 1690 = 2 \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1690.e (of order $3$ and degree $2$)
Character conductor:	$\operatorname{cond}(\chi)$	$=$	$ 13 $
Character field:			$\Q(\zeta_{3})$
Newform subspaces:			$ 22 $
Sturm bound:			$546$
Trace bound:			$7$
Distinguishing $T_p$:			$3$, $7$, $11$, $19$, $31$

Dimensions

The following table gives the dimensions of various subspaces of $M_{2}(1690, [\chi])$.

	Total	New	Old
Modular forms	600	100	500
Cusp forms	488	100	388
Eisenstein series	112	0	112

Trace form

$ 100 q - 50 q^{4} - 8 q^{7} - 42 q^{9} + O(q^{10}) $

$ 100 q - 50 q^{4} - 8 q^{7} - 42 q^{9} - 2 q^{10} - 2 q^{11} + 12 q^{14} + 4 q^{15} - 50 q^{16} - 12 q^{17} - 16 q^{18} + 6 q^{19} + 24 q^{21} - 12 q^{22} + 8 q^{23} + 100 q^{25} - 24 q^{27} - 8 q^{28} + 8 q^{29} - 4 q^{30} - 8 q^{31} + 20 q^{33} + 16 q^{34} - 2 q^{35} - 42 q^{36} + 12 q^{37} + 32 q^{38} + 4 q^{40} - 4 q^{41} + 12 q^{42} + 16 q^{43} + 4 q^{44} - 16 q^{45} + 8 q^{46} + 32 q^{47} - 72 q^{49} + 8 q^{51} + 40 q^{53} - 12 q^{54} - 12 q^{55} - 6 q^{56} - 64 q^{57} - 12 q^{58} - 8 q^{60} - 4 q^{61} - 12 q^{62} - 32 q^{63} + 100 q^{64} - 24 q^{66} - 12 q^{68} - 28 q^{69} + 40 q^{71} + 8 q^{72} - 56 q^{73} - 2 q^{74} + 6 q^{76} + 24 q^{77} + 40 q^{79} - 26 q^{81} - 24 q^{82} - 12 q^{84} + 12 q^{85} - 8 q^{86} + 32 q^{87} - 12 q^{88} - 10 q^{89} + 52 q^{90} - 16 q^{92} + 44 q^{93} - 10 q^{94} + 16 q^{95} - 4 q^{97} - 16 q^{98} - 20 q^{99} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Decomposition of $S_{2}^{\mathrm{new}}(1690, [\chi])$ 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
1690.2.e.a	$1690$	$2$	1690.e	13.c	$3$	$2$	$1$	$13.495$	$\Q(\sqrt{-3}) $	None		$2$	$0$	$-1$	$-2$	$-2$	$4$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(-1+\zeta_{6})q^{2}+(-2+2\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots$
1690.2.e.b	$1690$	$2$	1690.e	13.c	$3$	$2$	$1$	$13.495$	$\Q(\sqrt{-3}) $	None		$2$	$1$	$-1$	$-2$	$2$	$0$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(-1+\zeta_{6})q^{2}+(-2+2\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots$
1690.2.e.c	$1690$	$2$	1690.e	13.c	$3$	$2$	$1$	$13.495$	$\Q(\sqrt{-3}) $	None		$2$	$0$	$-1$	$0$	$2$	$0$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(-1+\zeta_{6})q^{2}-\zeta_{6}q^{4}+q^{5}+q^{8}+\cdots$
1690.2.e.d	$1690$	$2$	1690.e	13.c	$3$	$2$	$1$	$13.495$	$\Q(\sqrt{-3}) $	None		$2$	$0$	$-1$	$2$	$-2$	$-4$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(-1+\zeta_{6})q^{2}+(2-2\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots$
1690.2.e.e	$1690$	$2$	1690.e	13.c	$3$	$2$	$1$	$13.495$	$\Q(\sqrt{-3}) $	None		$2$	$0$	$-1$	$2$	$2$	$-1$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(-1+\zeta_{6})q^{2}+(2-2\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots$
1690.2.e.f	$1690$	$2$	1690.e	13.c	$3$	$2$	$1$	$13.495$	$\Q(\sqrt{-3}) $	None		$2$	$0$	$1$	$-2$	$-2$	$0$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(1-\zeta_{6})q^{2}+(-2+2\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots$
1690.2.e.g	$1690$	$2$	1690.e	13.c	$3$	$2$	$1$	$13.495$	$\Q(\sqrt{-3}) $	None		$2$	$0$	$1$	$-2$	$2$	$-4$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(1-\zeta_{6})q^{2}+(-2+2\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots$
1690.2.e.h	$1690$	$2$	1690.e	13.c	$3$	$2$	$1$	$13.495$	$\Q(\sqrt{-3}) $	None		$2$	$0$	$1$	$0$	$-2$	$-3$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(1-\zeta_{6})q^{2}-\zeta_{6}q^{4}-q^{5}-3\zeta_{6}q^{7}+\cdots$
1690.2.e.i	$1690$	$2$	1690.e	13.c	$3$	$2$	$1$	$13.495$	$\Q(\sqrt{-3}) $	None		$2$	$0$	$1$	$0$	$-2$	$0$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(1-\zeta_{6})q^{2}-\zeta_{6}q^{4}-q^{5}-q^{8}+3\zeta_{6}q^{9}+\cdots$
1690.2.e.j	$1690$	$2$	1690.e	13.c	$3$	$2$	$1$	$13.495$	$\Q(\sqrt{-3}) $	None		$2$	$0$	$1$	$2$	$2$	$4$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(1-\zeta_{6})q^{2}+(2-2\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots$
1690.2.e.k	$1690$	$2$	1690.e	13.c	$3$	$4$	$2$	$13.495$	$\Q(\zeta_{12})$	None		$2$	$0$	$-2$	$-2$	$-4$	$-2$	$3$	$\mathrm{SU}(2)[C_{3}]$	$q-\zeta_{12}q^{2}+(-\zeta_{12}+\zeta_{12}^{2})q^{3}+(-1+\cdots)q^{4}+\cdots$
1690.2.e.l	$1690$	$2$	1690.e	13.c	$3$	$4$	$2$	$13.495$	$\Q(\zeta_{12})$	None		$2$	$0$	$-2$	$2$	$-4$	$6$	$3$	$\mathrm{SU}(2)[C_{3}]$	$q-\zeta_{12}q^{2}+(\zeta_{12}-\zeta_{12}^{2})q^{3}+(-1+\cdots)q^{4}+\cdots$
1690.2.e.m	$1690$	$2$	1690.e	13.c	$3$	$4$	$2$	$13.495$	$\Q(\sqrt{-3}, \sqrt{10})$	None		$2$	$0$	$2$	$0$	$4$	$-2$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(1+\beta _{2})q^{2}+\beta _{1}q^{3}+\beta _{2}q^{4}+q^{5}+\cdots$
1690.2.e.n	$1690$	$2$	1690.e	13.c	$3$	$4$	$2$	$13.495$	$\Q(\zeta_{12})$	None		$2$	$0$	$2$	$2$	$4$	$-6$	$3$	$\mathrm{SU}(2)[C_{3}]$	$q+\zeta_{12}q^{2}+(\zeta_{12}-\zeta_{12}^{2})q^{3}+(-1+\cdots)q^{4}+\cdots$
1690.2.e.o	$1690$	$2$	1690.e	13.c	$3$	$6$	$3$	$13.495$	6.0.64827.1	None		$2$	$0$	$-3$	$-1$	$-6$	$-5$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(-1+\beta _{5})q^{2}-\beta _{1}q^{3}-\beta _{5}q^{4}-q^{5}+\cdots$
1690.2.e.p	$1690$	$2$	1690.e	13.c	$3$	$6$	$3$	$13.495$	6.0.64827.1	None		$2$	$0$	$-3$	$1$	$-6$	$-2$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(-1+\beta _{5})q^{2}+\beta _{1}q^{3}-\beta _{5}q^{4}-q^{5}+\cdots$
1690.2.e.q	$1690$	$2$	1690.e	13.c	$3$	$6$	$3$	$13.495$	6.0.64827.1	None		$2$	$0$	$3$	$-1$	$6$	$5$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(1-\beta _{5})q^{2}-\beta _{1}q^{3}-\beta _{5}q^{4}+q^{5}+\cdots$
1690.2.e.r	$1690$	$2$	1690.e	13.c	$3$	$6$	$3$	$13.495$	6.0.64827.1	None		$2$	$0$	$3$	$1$	$6$	$2$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(1-\beta _{5})q^{2}+\beta _{1}q^{3}-\beta _{5}q^{4}+q^{5}+\cdots$
1690.2.e.s	$1690$	$2$	1690.e	13.c	$3$	$8$	$4$	$13.495$	8.0.22581504.2	None		$2$	$0$	$-4$	$-2$	$8$	$0$	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{3}]$	$q-\beta _{4}q^{2}+(-1-\beta _{1}-\beta _{2})q^{3}+(-1+\cdots)q^{4}+\cdots$
1690.2.e.t	$1690$	$2$	1690.e	13.c	$3$	$8$	$4$	$13.495$	8.0.22581504.2	None		$2$	$0$	$4$	$-2$	$-8$	$0$	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{3}]$	$q+\beta _{4}q^{2}+(-1-\beta _{1}-\beta _{2})q^{3}+(-1+\cdots)q^{4}+\cdots$
1690.2.e.u	$1690$	$2$	1690.e	13.c	$3$	$12$	$6$	$13.495$	$\mathbb{Q}[x]/(x^{12} - \cdots)$	None		$2$	$0$	$-6$	$2$	$12$	$-3$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(-1-\beta _{8})q^{2}+(-\beta _{1}-\beta _{5})q^{3}+\beta _{8}q^{4}+\cdots$
1690.2.e.v	$1690$	$2$	1690.e	13.c	$3$	$12$	$6$	$13.495$	$\mathbb{Q}[x]/(x^{12} - \cdots)$	None		$2$	$0$	$6$	$2$	$-12$	$3$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(1+\beta _{8})q^{2}+(-\beta _{1}-\beta _{5})q^{3}+\beta _{8}q^{4}+\cdots$

Decomposition of $S_{2}^{\mathrm{old}}(1690, [\chi])$ into lower level spaces

$ S_{2}^{\mathrm{old}}(1690, [\chi]) \cong $ $S_{2}^{\mathrm{new}}(26, [\chi])$$^{\oplus 4}$$\oplus$$S_{2}^{\mathrm{new}}(169, [\chi])$$^{\oplus 4}$$\oplus$$S_{2}^{\mathrm{new}}(338, [\chi])$$^{\oplus 2}$

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Level:	\( N \)	\(=\)	\( 1690 = 2 \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1690.e (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 13 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 22 \)
Sturm bound:			\(546\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(3\), \(7\), \(11\), \(19\), \(31\)

Introduction

L-functions

Modular forms

Varieties

Fields

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Properties

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Defining parameters

Dimensions

Trace form

Decomposition of \(S_{2}^{\mathrm{new}}(1690, [\chi])\) into newform subspaces

Decomposition of \(S_{2}^{\mathrm{old}}(1690, [\chi])\) 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	1690.2.e

Level	$1690$
Weight	$2$
Character orbit	1690.e
Rep. character	$\chi_{1690}(191,\cdot)$
Character field	$\Q(\zeta_{3})$
Dimension	$100$
Newform subspaces	$22$
Sturm bound	$546$
Trace bound	$7$