LMFDB - Space of modular forms of level 3135, weight 2, and trivial character

Defining parameters

Level:	$ N $	$=$	$ 3135 = 3 \cdot 5 \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3135.a (trivial)
Character field:			$\Q$
Newform subspaces:			$ 25 $
Sturm bound:			$960$
Trace bound:			$11$
Distinguishing $T_p$:			$2$, $7$, $13$, $17$

Dimensions

The following table gives the dimensions of various subspaces of $M_{2}(\Gamma_0(3135))$.

	Total	New	Old
Modular forms	488	121	367
Cusp forms	473	121	352
Eisenstein series	15	0	15

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

$3$	$5$	$11$	$19$	Fricke	Dim
$+$	$+$	$+$	$+$	$+$	$6$
$+$	$+$	$+$	$-$	$-$	$7$
$+$	$+$	$-$	$+$	$-$	$9$
$+$	$+$	$-$	$-$	$+$	$8$
$+$	$-$	$+$	$+$	$-$	$11$
$+$	$-$	$+$	$-$	$+$	$6$
$+$	$-$	$-$	$+$	$+$	$4$
$+$	$-$	$-$	$-$	$-$	$9$
$-$	$+$	$+$	$+$	$-$	$9$
$-$	$+$	$+$	$-$	$+$	$4$
$-$	$+$	$-$	$+$	$+$	$6$
$-$	$+$	$-$	$-$	$-$	$11$
$-$	$-$	$+$	$+$	$+$	$4$
$-$	$-$	$+$	$-$	$-$	$13$
$-$	$-$	$-$	$+$	$-$	$11$
$-$	$-$	$-$	$-$	$+$	$3$
Plus space				$+$	$41$
Minus space				$-$	$80$

Trace form

$ 121 q + 3 q^{2} + q^{3} + 127 q^{4} + q^{5} + 3 q^{6} + 8 q^{7} + 15 q^{8} + 121 q^{9} + O(q^{10}) $

\( 121 q + 3 q^{2} + q^{3} + 127 q^{4} + q^{5} + 3 q^{6} + 8 q^{7} + 15 q^{8} + 121 q^{9} + 3 q^{10} + q^{11} + 7 q^{12} + 14 q^{13} + 8 q^{14} + q^{15} + 135 q^{16} + 18 q^{17} + 3 q^{18} + q^{19} - 9 q^{20} + 8 q^{21} + 3 q^{22} + 24 q^{23} + 15 q^{24} + 121 q^{25} + 26 q^{26} + q^{27} + 56 q^{28} + 14 q^{29} + 3 q^{30} + 16 q^{31} + 63 q^{32} + q^{33} + 38 q^{34} - 8 q^{35} + 127 q^{36} + 38 q^{37} + 3 q^{38} + 14 q^{39} + 15 q^{40} + 10 q^{41} + 8 q^{42} - 4 q^{43} - 9 q^{44} + q^{45} + 40 q^{46} + 16 q^{47} + 31 q^{48} + 145 q^{49} + 3 q^{50} - 14 q^{51} + 98 q^{52} + 22 q^{53} + 3 q^{54} - 15 q^{55} + 72 q^{56} + q^{57} + 42 q^{58} - 52 q^{59} + 7 q^{60} + 30 q^{61} + 16 q^{62} + 8 q^{63} + 167 q^{64} - 2 q^{65} + 3 q^{66} - 28 q^{67} + 78 q^{68} + 8 q^{69} + 8 q^{70} - 8 q^{71} + 15 q^{72} + 58 q^{73} + 34 q^{74} + q^{75} + 7 q^{76} - 8 q^{77} + 10 q^{78} + 48 q^{79} - q^{80} + 121 q^{81} + 46 q^{82} - 28 q^{83} + 56 q^{84} + 2 q^{85} + 52 q^{86} + 14 q^{87} + 15 q^{88} + 58 q^{89} + 3 q^{90} - 16 q^{91} - 24 q^{92} + 16 q^{93} - 64 q^{94} + q^{95} + 63 q^{96} + 34 q^{97} - 37 q^{98} + q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of $S_{2}^{\mathrm{new}}(\Gamma_0(3135))$ into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces				A-L signs				Fricke sign	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*	$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$	3	5	11	19	Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
3135.2.a.a	$3135$	$2$	3135.a	1.a	$1$	$1$	$1$	$25.033$	$\Q$	None	✓	$1$	$0$	$-2$	$1$	$1$	$-2$	$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-2q^{2}+q^{3}+2q^{4}+q^{5}-2q^{6}-2q^{7}+\cdots$
3135.2.a.b	$3135$	$2$	3135.a	1.a	$1$	$1$	$1$	$25.033$	$\Q$	None	✓	$1$	$1$	$-1$	$-1$	$-1$	$-4$	$+$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-q^{3}-q^{4}-q^{5}+q^{6}-4q^{7}+\cdots$
3135.2.a.c	$3135$	$2$	3135.a	1.a	$1$	$1$	$1$	$25.033$	$\Q$	None	✓	$1$	$1$	$-1$	$-1$	$1$	$0$	$+$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-q^{3}-q^{4}+q^{5}+q^{6}+3q^{8}+\cdots$
3135.2.a.d	$3135$	$2$	3135.a	1.a	$1$	$1$	$1$	$25.033$	$\Q$	None	✓	$1$	$0$	$-1$	$1$	$-1$	$4$	$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{3}-q^{4}-q^{5}-q^{6}+4q^{7}+\cdots$
3135.2.a.e	$3135$	$2$	3135.a	1.a	$1$	$1$	$1$	$25.033$	$\Q$	None	✓	$1$	$1$	$-1$	$1$	$1$	$0$	$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{3}-q^{4}+q^{5}-q^{6}+3q^{8}+\cdots$
3135.2.a.f	$3135$	$2$	3135.a	1.a	$1$	$1$	$1$	$25.033$	$\Q$	None	✓	$1$	$0$	$1$	$1$	$1$	$4$	$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{3}-q^{4}+q^{5}+q^{6}+4q^{7}+\cdots$
3135.2.a.g	$3135$	$2$	3135.a	1.a	$1$	$1$	$1$	$25.033$	$\Q$	None	✓	$1$	$0$	$1$	$1$	$1$	$4$	$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{3}-q^{4}+q^{5}+q^{6}+4q^{7}+\cdots$
3135.2.a.h	$3135$	$2$	3135.a	1.a	$1$	$1$	$1$	$25.033$	$\Q$	None	✓	$1$	$1$	$2$	$1$	$-1$	$-2$	$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+2q^{2}+q^{3}+2q^{4}-q^{5}+2q^{6}-2q^{7}+\cdots$
3135.2.a.i	$3135$	$2$	3135.a	1.a	$1$	$3$	$3$	$25.033$	$\Q(\zeta_{14})^+$	None	✓	$1$	$1$	$-2$	$3$	$-3$	$-3$	$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+(-1-\beta _{2})q^{2}+q^{3}+(\beta _{1}+\beta _{2})q^{4}+\cdots$
3135.2.a.j	$3135$	$2$	3135.a	1.a	$1$	$3$	$3$	$25.033$	$\Q(\zeta_{14})^+$	None	✓	$1$	$1$	$-2$	$3$	$3$	$-5$	$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+(-1-\beta _{2})q^{2}+q^{3}+(\beta _{1}+\beta _{2})q^{4}+\cdots$
3135.2.a.k	$3135$	$2$	3135.a	1.a	$1$	$3$	$3$	$25.033$	$\Q(\zeta_{18})^+$	None	✓	$1$	$1$	$0$	$-3$	$3$	$-3$	$+$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-\beta _{1}q^{2}-q^{3}+\beta _{2}q^{4}+q^{5}+\beta _{1}q^{6}+\cdots$
3135.2.a.l	$3135$	$2$	3135.a	1.a	$1$	$3$	$3$	$25.033$	$\Q(\zeta_{18})^+$	None	✓	$1$	$1$	$0$	$3$	$3$	$-3$	$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-\beta _{1}q^{2}+q^{3}+\beta _{2}q^{4}+q^{5}-\beta _{1}q^{6}+\cdots$
3135.2.a.m	$3135$	$2$	3135.a	1.a	$1$	$4$	$4$	$25.033$	4.4.132889.1	None	✓	$1$	$0$	$-1$	$-4$	$4$	$-5$	$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-\beta _{1}q^{2}-q^{3}+(3+\beta _{2})q^{4}+q^{5}+\beta _{1}q^{6}+\cdots$
3135.2.a.n	$3135$	$2$	3135.a	1.a	$1$	$6$	$6$	$25.033$	6.6.31252925.1	None	✓	$1$	$1$	$-2$	$-6$	$-6$	$5$	$+$	$+$	$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	$q-\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}-q^{5}+\beta _{1}q^{6}+\cdots$
3135.2.a.o	$3135$	$2$	3135.a	1.a	$1$	$6$	$6$	$25.033$	6.6.20413244.1	None	✓	$1$	$1$	$-2$	$6$	$-6$	$-7$	$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-\beta _{1}q^{2}+q^{3}+(\beta _{2}+\beta _{3})q^{4}-q^{5}+\cdots$
3135.2.a.p	$3135$	$2$	3135.a	1.a	$1$	$6$	$6$	$25.033$	6.6.905177.1	None	✓	$1$	$1$	$-1$	$-6$	$6$	$3$	$+$	$-$	$+$	$-$	$+$	$2^{3}$	$\mathrm{SU}(2)$	$q+\beta _{3}q^{2}-q^{3}+(1+\beta _{1})q^{4}+q^{5}-\beta _{3}q^{6}+\cdots$
3135.2.a.q	$3135$	$2$	3135.a	1.a	$1$	$7$	$7$	$25.033$	$\mathbb{Q}[x]/(x^{7} - \cdots)$	None	✓	$1$	$1$	$-3$	$-7$	$-7$	$3$	$+$	$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	$q-\beta _{1}q^{2}-q^{3}+(1+\beta _{1}+\beta _{2})q^{4}-q^{5}+\cdots$
3135.2.a.r	$3135$	$2$	3135.a	1.a	$1$	$7$	$7$	$25.033$	$\mathbb{Q}[x]/(x^{7} - \cdots)$	None	✓	$1$	$0$	$1$	$-7$	$7$	$4$	$+$	$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	$q-\beta _{2}q^{2}-q^{3}+\beta _{6}q^{4}+q^{5}+\beta _{2}q^{6}+\cdots$
3135.2.a.s	$3135$	$2$	3135.a	1.a	$1$	$7$	$7$	$25.033$	$\mathbb{Q}[x]/(x^{7} - \cdots)$	None	✓	$1$	$0$	$3$	$-7$	$-7$	$-3$	$+$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}-q^{5}-\beta _{1}q^{6}+\cdots$
3135.2.a.t	$3135$	$2$	3135.a	1.a	$1$	$9$	$9$	$25.033$	$\mathbb{Q}[x]/(x^{9} - \cdots)$	None	✓	$1$	$0$	$-1$	$9$	$-9$	$7$	$-$	$+$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	$q-\beta _{1}q^{2}+q^{3}+(1+\beta _{2})q^{4}-q^{5}-\beta _{1}q^{6}+\cdots$
3135.2.a.u	$3135$	$2$	3135.a	1.a	$1$	$9$	$9$	$25.033$	$\mathbb{Q}[x]/(x^{9} - \cdots)$	None	✓	$1$	$0$	$2$	$-9$	$9$	$-3$	$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}+q^{5}-\beta _{1}q^{6}+\cdots$
3135.2.a.v	$3135$	$2$	3135.a	1.a	$1$	$9$	$9$	$25.033$	$\mathbb{Q}[x]/(x^{9} - \cdots)$	None	✓	$1$	$0$	$3$	$-9$	$-9$	$3$	$+$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{2}-q^{3}+(1+\beta _{1}+\beta _{2})q^{4}-q^{5}+\cdots$
3135.2.a.w	$3135$	$2$	3135.a	1.a	$1$	$9$	$9$	$25.033$	$\mathbb{Q}[x]/(x^{9} - \cdots)$	None	✓	$1$	$0$	$5$	$9$	$9$	$5$	$-$	$-$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	$q+(1-\beta _{1})q^{2}+q^{3}+(2-\beta _{1}+\beta _{2})q^{4}+\cdots$
3135.2.a.x	$3135$	$2$	3135.a	1.a	$1$	$10$	$10$	$25.033$	$\mathbb{Q}[x]/(x^{10} - \cdots)$	None	✓	$1$	$0$	$4$	$10$	$-10$	$5$	$-$	$+$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{2}+q^{3}+(2+\beta _{2})q^{4}-q^{5}+\beta _{1}q^{6}+\cdots$
3135.2.a.y	$3135$	$2$	3135.a	1.a	$1$	$12$	$12$	$25.033$	$\mathbb{Q}[x]/(x^{12} - \cdots)$	None	✓	$1$	$0$	$1$	$12$	$12$	$1$	$-$	$-$	$+$	$-$	$-$	$2^{5}$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{2}+q^{3}+(2+\beta _{2})q^{4}+q^{5}+\beta _{1}q^{6}+\cdots$

Decomposition of $S_{2}^{\mathrm{old}}(\Gamma_0(3135))$ into lower level spaces

Overview	Random
Universe	Knowledge

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:	\( N \)	\(=\)	\( 3135 = 3 \cdot 5 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3135.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 25 \)
Sturm bound:			\(960\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(2\), \(7\), \(13\), \(17\)

\(3\)	\(5\)	\(11\)	\(19\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(+\)	$+$	\(6\)
\(+\)	\(+\)	\(+\)	\(-\)	$-$	\(7\)
\(+\)	\(+\)	\(-\)	\(+\)	$-$	\(9\)
\(+\)	\(+\)	\(-\)	\(-\)	$+$	\(8\)
\(+\)	\(-\)	\(+\)	\(+\)	$-$	\(11\)
\(+\)	\(-\)	\(+\)	\(-\)	$+$	\(6\)
\(+\)	\(-\)	\(-\)	\(+\)	$+$	\(4\)
\(+\)	\(-\)	\(-\)	\(-\)	$-$	\(9\)
\(-\)	\(+\)	\(+\)	\(+\)	$-$	\(9\)
\(-\)	\(+\)	\(+\)	\(-\)	$+$	\(4\)
\(-\)	\(+\)	\(-\)	\(+\)	$+$	\(6\)
\(-\)	\(+\)	\(-\)	\(-\)	$-$	\(11\)
\(-\)	\(-\)	\(+\)	\(+\)	$+$	\(4\)
\(-\)	\(-\)	\(+\)	\(-\)	$-$	\(13\)
\(-\)	\(-\)	\(-\)	\(+\)	$-$	\(11\)
\(-\)	\(-\)	\(-\)	\(-\)	$+$	\(3\)
Plus space				\(+\)	\(41\)
Minus space				\(-\)	\(80\)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Defining parameters

Dimensions

Trace form

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3135))\) into newform subspaces

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3135))\) 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	3135.2.a

Level	$3135$
Weight	$2$
Character orbit	3135.a
Rep. character	$\chi_{3135}(1,\cdot)$
Character field	$\Q$
Dimension	$121$
Newform subspaces	$25$
Sturm bound	$960$
Trace bound	$11$