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

• Label: 370.2.g
• Level: $370$
• Weight: $2$
• Character orbit: 370.g
• Rep. character: $\chi_{370}(43,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $38$
• Newform subspaces: $5$
• Sturm bound: $114$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 370 = 2 \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	370.g (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 185 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 5 \)
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	122	38	84
Cusp forms	106	38	68
Eisenstein series	16	0	16

Trace form

38 * q - 8 * q^3 - 38 * q^4 + 4 * q^5 + 2 * q^10 + 8 * q^12 - 4 * q^14 + 38 * q^16 + 4 * q^17 + 10 * q^18 - 12 * q^19 - 4 * q^20 - 16 * q^22 - 10 * q^25 - 8 * q^26 + 40 * q^27 - 18 * q^29 - 16 * q^31 + 8 * q^35 - 12 * q^37 + 8 * q^39 - 2 * q^40 - 56 * q^42 - 6 * q^45 - 8 * q^47 - 8 * q^48 - 8 * q^50 - 6 * q^53 - 12 * q^55 + 4 * q^56 + 18 * q^58 + 20 * q^59 + 22 * q^61 + 40 * q^62 - 38 * q^64 - 24 * q^65 - 8 * q^66 - 16 * q^67 - 4 * q^68 + 88 * q^69 + 16 * q^70 - 16 * q^71 - 10 * q^72 - 2 * q^73 - 26 * q^74 - 72 * q^75 + 12 * q^76 + 40 * q^77 + 20 * q^78 + 32 * q^79 + 4 * q^80 + 10 * q^81 + 48 * q^82 + 20 * q^83 - 16 * q^86 + 24 * q^87 + 16 * q^88 + 46 * q^89 - 40 * q^90 - 40 * q^91 - 36 * q^94 - 40 * q^95 - 112 * q^97 + 34 * 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.g.a	$370$	$2$	370.g	185.f	$4$	$2$	$1$	$2.954$	\(\Q(\sqrt{-1}) \)	None		✓			370.2.g.a	$2$	$0$	\(0\)	\(-2\)	\(4\)	\(2\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+i q^{2}+(-i-1)q^{3}-q^{4}+(i+2)q^{5}+\cdots\)
370.2.g.b	$370$	$2$	370.g	185.f	$4$	$2$	$1$	$2.954$	\(\Q(\sqrt{-1}) \)	None		✓			370.2.g.b	$2$	$0$	\(0\)	\(0\)	\(-4\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+i q^{2}-q^{4}+(i-2)q^{5}+(-2 i-2)q^{7}+\cdots\)
370.2.g.c	$370$	$2$	370.g	185.f	$4$	$4$	$2$	$2.954$	\(\Q(\zeta_{8})\)	None		✓			370.2.g.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(8\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+\zeta_{8}^{2}q^{2}+2\zeta_{8}q^{3}-q^{4}+(2\zeta_{8}-\zeta_{8}^{3})q^{5}+\cdots\)
370.2.g.d	$370$	$2$	370.g	185.f	$4$	$10$	$5$	$2.954$	\(\mathbb{Q}[x]/(x^{10} + \cdots)\)	None		✓			370.2.g.d	$2$	$0$	\(0\)	\(-2\)	\(2\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{6}q^{2}+(-\beta _{2}-\beta _{3}-\beta _{8})q^{3}-q^{4}+\cdots\)
370.2.g.e	$370$	$2$	370.g	185.f	$4$	$20$	$10$	$2.954$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None		✓	✓		370.2.g.e	$2$	$0$	\(0\)	\(-4\)	\(2\)	\(-2\)	$2^{6}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{10}q^{2}-\beta _{1}q^{3}-q^{4}-\beta _{8}q^{5}+\beta _{4}q^{6}+\cdots\)

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

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