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

• Label: 370.2.r
• Level: $370$
• Weight: $2$
• Character orbit: 370.r
• Rep. character: $\chi_{370}(23,\cdot)$
• 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.r (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

\( 76 q - 4 q^{3} + 38 q^{4} - 4 q^{5} + O(q^{10}) \)

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

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

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