Space of modular forms of level 1170, weight 2, and Character 1170.w

• Label: 1170.2.w
• Level: $1170$
• Weight: $2$
• Character orbit: 1170.w
• Rep. character: $\chi_{1170}(307,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $70$
• Newform subspaces: $9$
• Sturm bound: $504$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1170.w (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 65 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(504\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1170, [\chi])\).

	Total	New	Old
Modular forms	536	70	466
Cusp forms	472	70	402
Eisenstein series	64	0	64

Trace form

70 * q - 70 * q^4 + 2 * q^5 + 4 * q^11 + 6 * q^13 + 70 * q^16 - 2 * q^17 - 20 * q^19 - 2 * q^20 - 8 * q^23 - 6 * q^25 + 24 * q^31 + 18 * q^34 - 12 * q^35 + 24 * q^37 + 10 * q^41 + 8 * q^43 - 4 * q^44 + 4 * q^46 + 16 * q^47 + 78 * q^49 - 16 * q^50 - 6 * q^52 + 30 * q^53 + 40 * q^55 + 20 * q^58 + 12 * q^59 + 32 * q^61 - 12 * q^62 - 70 * q^64 + 50 * q^65 + 2 * q^68 - 20 * q^70 + 16 * q^71 + 20 * q^76 - 56 * q^77 + 2 * q^80 - 26 * q^82 - 40 * q^83 - 22 * q^85 - 6 * q^89 + 8 * q^92 + 24 * q^95

Decomposition of \(S_{2}^{\mathrm{new}}(1170, [\chi])\) 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				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1170.2.w.a	$1170$	$2$	1170.w	65.f	$4$	$2$	$1$	$9.342$	\(\Q(\sqrt{-1}) \)	None					130.2.g.b	$2$	$0$	\(0\)	\(0\)	\(-4\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+i q^{2}-q^{4}+(i-2)q^{5}-2 q^{7}+\cdots\)
1170.2.w.b	$1170$	$2$	1170.w	65.f	$4$	$2$	$1$	$9.342$	\(\Q(\sqrt{-1}) \)	None					130.2.g.a	$2$	$1$	\(0\)	\(0\)	\(2\)	\(-8\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+i q^{2}-q^{4}+(2 i+1)q^{5}-4 q^{7}+\cdots\)
1170.2.w.c	$1170$	$2$	1170.w	65.f	$4$	$2$	$1$	$9.342$	\(\Q(\sqrt{-1}) \)	None					130.2.g.c	$2$	$0$	\(0\)	\(0\)	\(4\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q-i q^{2}-q^{4}+(-i+2)q^{5}-2 q^{7}+\cdots\)
1170.2.w.d	$1170$	$2$	1170.w	65.f	$4$	$4$	$2$	$9.342$	\(\Q(\zeta_{12})\)	None					130.2.g.e	$2$	$0$	\(0\)	\(0\)	\(-4\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q-\zeta_{12}^{3}q^{2}-q^{4}+(-2+\zeta_{12}+2\zeta_{12}^{2}+\cdots)q^{5}+\cdots\)
1170.2.w.e	$1170$	$2$	1170.w	65.f	$4$	$4$	$2$	$9.342$	\(\Q(i, \sqrt{11})\)	None					130.2.g.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(12\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{2}q^{2}-q^{4}+(-\beta _{1}+2\beta _{2})q^{5}+3q^{7}+\cdots\)
1170.2.w.f	$1170$	$2$	1170.w	65.f	$4$	$12$	$6$	$9.342$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None					390.2.j.a	$2$	$0$	\(0\)	\(0\)	\(4\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{4}q^{2}-q^{4}+\beta _{9}q^{5}+(\beta _{2}-\beta _{5}+\beta _{7}+\cdots)q^{7}+\cdots\)
1170.2.w.g	$1170$	$2$	1170.w	65.f	$4$	$14$	$7$	$9.342$	\(\mathbb{Q}[x]/(x^{14} - \cdots)\)	None		✓			1170.2.m.g	$2$	$0$	\(0\)	\(0\)	\(-2\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{3}q^{2}-q^{4}+\beta _{11}q^{5}-\beta _{4}q^{7}+\beta _{3}q^{8}+\cdots\)
1170.2.w.h	$1170$	$2$	1170.w	65.f	$4$	$14$	$7$	$9.342$	\(\mathbb{Q}[x]/(x^{14} - \cdots)\)	None		✓			1170.2.m.g	$2$	$0$	\(0\)	\(0\)	\(2\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{3}q^{2}-q^{4}-\beta _{11}q^{5}-\beta _{4}q^{7}-\beta _{3}q^{8}+\cdots\)
1170.2.w.i	$1170$	$2$	1170.w	65.f	$4$	$16$	$8$	$9.342$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None			✓		390.2.j.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{7}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{6}q^{2}-q^{4}-\beta _{9}q^{5}+(\beta _{1}-\beta _{8}-\beta _{9}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1170, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1170, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(65, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(130, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(195, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(390, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(585, [\chi])\)\(^{\oplus 2}\)