Space of modular forms of level 1170, weight 2, and trivial character

• Label: 1170.2.a
• Level: $1170$
• Weight: $2$
• Character orbit: 1170.a
• Rep. character: $\chi_{1170}(1,\cdot)$
• Character field: $\Q$
• Dimension: $20$
• Newform subspaces: $17$
• Sturm bound: $504$
• Trace bound: $17$

Defining parameters

Level:	\( N \)	\(=\)	\( 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1170.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 17 \)
Sturm bound:			\(504\)
Trace bound:			\(17\)
Distinguishing \(T_p\):			\(7\), \(11\), \(17\), \(31\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1170))\).

	Total	New	Old
Modular forms	268	20	248
Cusp forms	237	20	217
Eisenstein series	31	0	31

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

\(2\)	\(3\)	\(5\)	\(13\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(-\)	\(-\)	\(2\)
\(+\)	\(+\)	\(-\)	\(+\)	\(-\)	\(1\)
\(+\)	\(+\)	\(-\)	\(-\)	\(+\)	\(1\)
\(+\)	\(-\)	\(+\)	\(+\)	\(-\)	\(2\)
\(+\)	\(-\)	\(+\)	\(-\)	\(+\)	\(2\)
\(+\)	\(-\)	\(-\)	\(+\)	\(+\)	\(2\)
\(+\)	\(-\)	\(-\)	\(-\)	\(-\)	\(1\)
\(-\)	\(+\)	\(+\)	\(+\)	\(-\)	\(1\)
\(-\)	\(+\)	\(+\)	\(-\)	\(+\)	\(1\)
\(-\)	\(+\)	\(-\)	\(-\)	\(-\)	\(2\)
\(-\)	\(-\)	\(+\)	\(+\)	\(+\)	\(1\)
\(-\)	\(-\)	\(+\)	\(-\)	\(-\)	\(2\)
\(-\)	\(-\)	\(-\)	\(+\)	\(-\)	\(2\)
Plus space				\(+\)	\(7\)
Minus space				\(-\)	\(13\)

Trace form

20 * q - 2 * q^2 + 20 * q^4 - 2 * q^5 - 2 * q^8 - 4 * q^11 + 2 * q^13 + 20 * q^16 + 8 * q^17 + 20 * q^19 - 2 * q^20 + 16 * q^22 - 16 * q^23 + 20 * q^25 + 16 * q^29 - 8 * q^31 - 2 * q^32 - 4 * q^34 + 8 * q^35 - 20 * q^37 + 8 * q^38 - 8 * q^41 + 4 * q^43 - 4 * q^44 + 20 * q^46 + 32 * q^47 + 36 * q^49 - 2 * q^50 + 2 * q^52 + 4 * q^53 - 8 * q^55 - 28 * q^58 - 4 * q^59 - 16 * q^61 + 20 * q^62 + 20 * q^64 - 4 * q^65 + 24 * q^67 + 8 * q^68 + 16 * q^70 - 12 * q^73 + 24 * q^74 + 20 * q^76 - 2 * q^80 - 12 * q^82 + 12 * q^85 + 16 * q^86 + 16 * q^88 + 64 * q^89 - 8 * q^91 - 16 * q^92 + 8 * q^94 + 8 * q^95 + 4 * q^97 + 14 * q^98

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1170))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces					A-L signs				Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	3	5	13
1170.2.a.a	$1170$	$2$	1170.a	1.a	$1$	$1$	$1$	$9.342$	\(\Q\)	None	✓				390.2.a.f	$1$	$1$	\(-1\)	\(0\)	\(-1\)	\(0\)		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-q^{5}-q^{8}+q^{10}-4q^{11}+\cdots\)
1170.2.a.b	$1170$	$2$	1170.a	1.a	$1$	$1$	$1$	$9.342$	\(\Q\)	None	✓				130.2.a.b	$1$	$1$	\(-1\)	\(0\)	\(-1\)	\(0\)		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-q^{5}-q^{8}+q^{10}+q^{13}+\cdots\)
1170.2.a.c	$1170$	$2$	1170.a	1.a	$1$	$1$	$1$	$9.342$	\(\Q\)	None	✓	✓	✓	✓	1170.2.a.c	$1$	$1$	\(-1\)	\(0\)	\(1\)	\(-4\)		$+$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+q^{5}-4q^{7}-q^{8}-q^{10}+\cdots\)
1170.2.a.d	$1170$	$2$	1170.a	1.a	$1$	$1$	$1$	$9.342$	\(\Q\)	None	✓				130.2.a.c	$1$	$1$	\(-1\)	\(0\)	\(1\)	\(-4\)		$+$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+q^{5}-4q^{7}-q^{8}-q^{10}+\cdots\)
1170.2.a.e	$1170$	$2$	1170.a	1.a	$1$	$1$	$1$	$9.342$	\(\Q\)	None	✓				390.2.a.e	$1$	$1$	\(-1\)	\(0\)	\(1\)	\(2\)		$+$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+q^{5}+2q^{7}-q^{8}-q^{10}+\cdots\)
1170.2.a.f	$1170$	$2$	1170.a	1.a	$1$	$1$	$1$	$9.342$	\(\Q\)	None	✓	✓	✓	✓	1170.2.a.f	$1$	$0$	\(-1\)	\(0\)	\(1\)	\(2\)		$+$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+q^{5}+2q^{7}-q^{8}-q^{10}+\cdots\)
1170.2.a.g	$1170$	$2$	1170.a	1.a	$1$	$1$	$1$	$9.342$	\(\Q\)	None	✓		✓	✓	390.2.a.g	$1$	$0$	\(-1\)	\(0\)	\(1\)	\(2\)		$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+q^{5}+2q^{7}-q^{8}-q^{10}+\cdots\)
1170.2.a.h	$1170$	$2$	1170.a	1.a	$1$	$1$	$1$	$9.342$	\(\Q\)	None	✓	✓	✓	✓	1170.2.a.c	$1$	$1$	\(1\)	\(0\)	\(-1\)	\(-4\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-q^{5}-4q^{7}+q^{8}-q^{10}+\cdots\)
1170.2.a.i	$1170$	$2$	1170.a	1.a	$1$	$1$	$1$	$9.342$	\(\Q\)	None	✓				130.2.a.a	$1$	$0$	\(1\)	\(0\)	\(-1\)	\(-4\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-q^{5}-4q^{7}+q^{8}-q^{10}+\cdots\)
1170.2.a.j	$1170$	$2$	1170.a	1.a	$1$	$1$	$1$	$9.342$	\(\Q\)	None	✓		✓	✓	390.2.a.b	$1$	$1$	\(1\)	\(0\)	\(-1\)	\(-2\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-q^{5}-2q^{7}+q^{8}-q^{10}+\cdots\)
1170.2.a.k	$1170$	$2$	1170.a	1.a	$1$	$1$	$1$	$9.342$	\(\Q\)	None	✓				390.2.a.d	$1$	$0$	\(1\)	\(0\)	\(-1\)	\(2\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-q^{5}+2q^{7}+q^{8}-q^{10}+\cdots\)
1170.2.a.l	$1170$	$2$	1170.a	1.a	$1$	$1$	$1$	$9.342$	\(\Q\)	None	✓	✓	✓	✓	1170.2.a.f	$1$	$0$	\(1\)	\(0\)	\(-1\)	\(2\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-q^{5}+2q^{7}+q^{8}-q^{10}+\cdots\)
1170.2.a.m	$1170$	$2$	1170.a	1.a	$1$	$1$	$1$	$9.342$	\(\Q\)	None	✓				390.2.a.a	$1$	$0$	\(1\)	\(0\)	\(1\)	\(0\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+q^{5}+q^{8}+q^{10}-q^{13}+\cdots\)
1170.2.a.n	$1170$	$2$	1170.a	1.a	$1$	$1$	$1$	$9.342$	\(\Q\)	None	✓				390.2.a.c	$1$	$0$	\(1\)	\(0\)	\(1\)	\(4\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+q^{5}+4q^{7}+q^{8}+q^{10}+\cdots\)
1170.2.a.o	$1170$	$2$	1170.a	1.a	$1$	$2$	$2$	$9.342$	\(\Q(\sqrt{2}) \)	None	✓		✓	✓	390.2.a.h	$1$	$0$	\(-2\)	\(0\)	\(-2\)	\(0\)		$+$	$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-q^{5}+\beta q^{7}-q^{8}+q^{10}+\cdots\)
1170.2.a.p	$1170$	$2$	1170.a	1.a	$1$	$2$	$2$	$9.342$	\(\Q(\sqrt{13}) \)	None	✓	✓	✓	✓	1170.2.a.p	$1$	$0$	\(-2\)	\(0\)	\(-2\)	\(2\)		$+$	$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-q^{5}+(1+\beta )q^{7}-q^{8}+\cdots\)
1170.2.a.q	$1170$	$2$	1170.a	1.a	$1$	$2$	$2$	$9.342$	\(\Q(\sqrt{13}) \)	None	✓	✓	✓	✓	1170.2.a.p	$1$	$0$	\(2\)	\(0\)	\(2\)	\(2\)		$-$	$+$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+q^{5}+(1+\beta )q^{7}+q^{8}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1170))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1170)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(65))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(78))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(117))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(130))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(195))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(234))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(390))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(585))\)\(^{\oplus 2}\)