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

• Label: 370.2.h
• Level: $370$
• Weight: $2$
• Character orbit: 370.h
• Rep. character: $\chi_{370}(117,\cdot)$
• 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.h (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

\( 38 q - 2 q^{2} + 8 q^{3} + 38 q^{4} - 6 q^{5} - 2 q^{8} + O(q^{10}) \)

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	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
370.2.h.a	$370$	$2$	370.h	185.k	$4$	$2$	$1$	$2.954$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(2\)	\(0\)	\(-2\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+q^{2}+q^{4}+(-1+2i)q^{5}+(-2+2i)q^{7}+\cdots\)
370.2.h.b	$370$	$2$	370.h	185.k	$4$	$2$	$1$	$2.954$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(2\)	\(2\)	\(-2\)	\(2\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+q^{2}+(1-i)q^{3}+q^{4}+(-1-2i)q^{5}+\cdots\)
370.2.h.c	$370$	$2$	370.h	185.k	$4$	$4$	$2$	$2.954$	\(\Q(\zeta_{8})\)	None		$2$	$0$	\(4\)	\(0\)	\(0\)	\(8\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+q^{2}+2\zeta_{8}q^{3}+q^{4}+(-\zeta_{8}-2\zeta_{8}^{3})q^{5}+\cdots\)
370.2.h.d	$370$	$2$	370.h	185.k	$4$	$10$	$5$	$2.954$	\(\mathbb{Q}[x]/(x^{10} + \cdots)\)	None		$2$	$0$	\(10\)	\(2\)	\(2\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+q^{2}+(\beta _{1}-\beta _{4})q^{3}+q^{4}+(-\beta _{1}+\beta _{3}+\cdots)q^{5}+\cdots\)
370.2.h.e	$370$	$2$	370.h	185.k	$4$	$20$	$10$	$2.954$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None		$2$	$0$	\(-20\)	\(4\)	\(-4\)	\(-2\)	$2^{6}$	$\mathrm{SU}(2)[C_{4}]$	\(q-q^{2}-\beta _{4}q^{3}+q^{4}+\beta _{13}q^{5}+\beta _{4}q^{6}+\cdots\)

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

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