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

• Label: 370.2.e
• Level: $370$
• Weight: $2$
• Character orbit: 370.e
• Rep. character: $\chi_{370}(121,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $20$
• 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.e (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 37 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(114\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\), \(7\)

Dimensions

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

	Total	New	Old
Modular forms	120	20	100
Cusp forms	104	20	84
Eisenstein series	16	0	16

Trace form

\( 20 q + 4 q^{3} - 10 q^{4} - 8 q^{7} - 2 q^{9} + O(q^{10}) \)

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.e.a	$370$	$2$	370.e	37.c	$3$	$2$	$1$	$2.954$	\(\Q(\sqrt{-3}) \)	None		✓			370.2.e.a	$2$	$0$	\(1\)	\(-2\)	\(-1\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{2}-2\zeta_{6}q^{3}-\zeta_{6}q^{4}-\zeta_{6}q^{5}+\cdots\)
370.2.e.b	$370$	$2$	370.e	37.c	$3$	$2$	$1$	$2.954$	\(\Q(\sqrt{-3}) \)	None		✓			370.2.e.b	$2$	$0$	\(1\)	\(-1\)	\(1\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{2}-\zeta_{6}q^{3}-\zeta_{6}q^{4}+\zeta_{6}q^{5}+\cdots\)
370.2.e.c	$370$	$2$	370.e	37.c	$3$	$2$	$1$	$2.954$	\(\Q(\sqrt{-3}) \)	None		✓			370.2.e.c	$2$	$0$	\(1\)	\(3\)	\(1\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{2}+3\zeta_{6}q^{3}-\zeta_{6}q^{4}+\zeta_{6}q^{5}+\cdots\)
370.2.e.d	$370$	$2$	370.e	37.c	$3$	$4$	$2$	$2.954$	\(\Q(\zeta_{12})\)	None		✓			370.2.e.d	$2$	$0$	\(-2\)	\(2\)	\(-2\)	\(0\)	$2^{2}\cdot 3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(\beta_1-1)q^{2}+\beta_1 q^{3}-\beta_1 q^{4}+\cdots\)
370.2.e.e	$370$	$2$	370.e	37.c	$3$	$4$	$2$	$2.954$	\(\Q(\sqrt{-3}, \sqrt{10})\)	None		✓			370.2.e.e	$2$	$0$	\(2\)	\(2\)	\(-2\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta _{2}q^{2}+(1+\beta _{2})q^{3}+(-1-\beta _{2})q^{4}+\cdots\)
370.2.e.f	$370$	$2$	370.e	37.c	$3$	$6$	$3$	$2.954$	6.0.2696112.1	None		✓	✓		370.2.e.f	$2$	$0$	\(-3\)	\(0\)	\(3\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta _{4}q^{2}+\beta _{5}q^{3}+(-1+\beta _{4})q^{4}+(1+\cdots)q^{5}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(370, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(37, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(74, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(185, [\chi])\)\(^{\oplus 2}\)