Space of modular forms of level 370, weight 2, and Character 370.q

• Label: 370.2.q
• Level: $370$
• Weight: $2$
• Character orbit: 370.q
• Rep. character: $\chi_{370}(97,\cdot)$
• Character field: $\Q(\zeta_{12})$
• Dimension: $76$
• Newform subspaces: $6$
• Sturm bound: $114$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 370 = 2 \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	370.q (of order \(12\) and degree \(4\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 185 \)
Character field:			\(\Q(\zeta_{12})\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(114\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(370, [\chi])\).

	Total	New	Old
Modular forms	244	76	168
Cusp forms	212	76	136
Eisenstein series	32	0	32

Trace form

76 * q + 2 * q^2 + 4 * q^3 - 38 * q^4 + 6 * q^5 - 4 * q^8 - 2 * q^10 + 4 * q^12 + 8 * q^13 + 8 * q^14 + 8 * q^15 - 38 * q^16 - 12 * q^19 + 8 * q^23 + 2 * q^25 + 8 * q^26 + 64 * q^27 - 18 * q^29 - 24 * q^30 + 16 * q^31 + 2 * q^32 + 12 * q^33 - 16 * q^35 + 14 * q^37 - 64 * q^39 - 14 * q^40 - 18 * q^41 - 84 * q^42 - 16 * q^43 - 12 * q^44 + 8 * q^45 + 8 * q^47 - 8 * q^48 + 72 * q^49 - 20 * q^50 + 8 * q^52 - 42 * q^53 - 12 * q^55 - 4 * q^56 + 24 * q^57 + 6 * q^58 + 20 * q^59 + 8 * q^60 - 4 * q^61 - 20 * q^62 + 76 * q^64 + 12 * q^65 + 8 * q^66 + 68 * q^67 - 80 * q^69 + 8 * q^70 - 8 * q^71 + 4 * q^73 + 52 * q^74 - 120 * q^75 - 24 * q^76 - 44 * q^77 + 20 * q^78 - 16 * q^79 - 6 * q^80 + 62 * q^81 - 8 * q^83 - 8 * q^86 - 36 * q^87 + 58 * q^89 + 40 * q^90 - 20 * q^91 - 4 * q^92 + 72 * q^93 - 36 * q^94 + 68 * q^95 - 24 * q^98

Decomposition of \(S_{2}^{\mathrm{new}}(370, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
370.2.q.a	$370$	$2$	370.q	185.p	$12$	$4$	$1$	$2.954$	\(\Q(\zeta_{12})\)	None		✓			370.2.q.a	$2$	$0$	\(-2\)	\(4\)	\(-4\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q-\zeta_{12}^{2}q^{2}+(1+\zeta_{12}-\zeta_{12}^{3})q^{3}+\cdots\)
370.2.q.b	$370$	$2$	370.q	185.p	$12$	$4$	$1$	$2.954$	\(\Q(\zeta_{12})\)	None		✓			370.2.q.b	$2$	$0$	\(-2\)	\(6\)	\(2\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q-\zeta_{12}^{2}q^{2}+(1+\zeta_{12}+\zeta_{12}^{2}-2\zeta_{12}^{3})q^{3}+\cdots\)
370.2.q.c	$370$	$2$	370.q	185.p	$12$	$8$	$2$	$2.954$	\(\Q(\zeta_{24})\)	None		✓			370.2.q.c	$2$	$0$	\(4\)	\(4\)	\(4\)	\(12\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q+\zeta_{24}^{4}q^{2}+(\zeta_{24}^{4}+\zeta_{24}^{5}+\zeta_{24}^{6}+\cdots)q^{3}+\cdots\)
370.2.q.d	$370$	$2$	370.q	185.p	$12$	$12$	$3$	$2.954$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		✓			370.2.q.d	$2$	$0$	\(-6\)	\(0\)	\(2\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q+(-1-\beta _{7})q^{2}+(1-\beta _{1}-\beta _{3}+2\beta _{4}+\cdots)q^{3}+\cdots\)
370.2.q.e	$370$	$2$	370.q	185.p	$12$	$16$	$4$	$2.954$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		✓			370.2.q.e	$2$	$0$	\(-8\)	\(-8\)	\(-4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q+(-1-\beta _{14})q^{2}+(-1+\beta _{2}+\beta _{13}+\cdots)q^{3}+\cdots\)
370.2.q.f	$370$	$2$	370.q	185.p	$12$	$32$	$8$	$2.954$		None		✓	✓		370.2.q.f	$2$	$0$	\(16\)	\(-2\)	\(6\)	\(-10\)		$\mathrm{SU}(2)[C_{12}]$

Decomposition of \(S_{2}^{\mathrm{old}}(370, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(370, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(185, [\chi])\)\(^{\oplus 2}\)