Space of modular forms of level 304, weight 7, and Character 304.e

• Label: 304.7.e
• Level: $304$
• Weight: $7$
• Character orbit: 304.e
• Rep. character: $\chi_{304}(113,\cdot)$
• Character field: $\Q$
• Dimension: $59$
• Newform subspaces: $6$
• Sturm bound: $280$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 304 = 2^{4} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	304.e (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 19 \)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(280\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{7}(304, [\chi])\).

	Total	New	Old
Modular forms	246	61	185
Cusp forms	234	59	175
Eisenstein series	12	2	10

Trace form

\( 59 q - 2 q^{5} + 722 q^{7} - 14773 q^{9} + O(q^{10}) \)

Decomposition of \(S_{7}^{\mathrm{new}}(304, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
304.7.e.a	$304$	$7$	304.e	19.b	$2$	$1$	$1$	$69.936$	\(\Q\)	\(\Q(\sqrt{-19}) \)	✓	$2$	$0$	\(0\)	\(0\)	\(-54\)	\(-610\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q-54q^{5}-610q^{7}+3^{6}q^{9}+1062q^{11}+\cdots\)
304.7.e.b	$304$	$7$	304.e	19.b	$2$	$2$	$2$	$69.936$	\(\Q(\sqrt{57}) \)	\(\Q(\sqrt{-19}) \)	✓	$2$	$0$	\(0\)	\(0\)	\(54\)	\(610\)	$2^{3}$	$\mathrm{U}(1)[D_{2}]$	\(q+(3^{3}+7\beta )q^{5}+(305-9\beta )q^{7}+3^{6}q^{9}+\cdots\)
304.7.e.c	$304$	$7$	304.e	19.b	$2$	$8$	$8$	$69.936$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(2\)	\(-362\)	$2^{8}\cdot 3\cdot 5$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}+\beta _{7}q^{5}+(-46-\beta _{2}-2\beta _{3}+\cdots)q^{7}+\cdots\)
304.7.e.d	$304$	$7$	304.e	19.b	$2$	$8$	$8$	$69.936$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(108\)	\(140\)	$2^{11}\cdot 29$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}+(13-\beta _{5})q^{5}+(18-2\beta _{3}+\cdots)q^{7}+\cdots\)
304.7.e.e	$304$	$7$	304.e	19.b	$2$	$10$	$10$	$69.936$	\(\mathbb{Q}[x]/(x^{10} + \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(-112\)	\(224\)	$2^{13}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{5}q^{3}+(-11-\beta _{1})q^{5}+(23-2\beta _{1}+\cdots)q^{7}+\cdots\)
304.7.e.f	$304$	$7$	304.e	19.b	$2$	$30$	$30$	$69.936$		None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(720\)		$\mathrm{SU}(2)[C_{2}]$

Decomposition of \(S_{7}^{\mathrm{old}}(304, [\chi])\) into lower level spaces

\( S_{7}^{\mathrm{old}}(304, [\chi]) \cong \) \(S_{7}^{\mathrm{new}}(19, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(38, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(76, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(152, [\chi])\)\(^{\oplus 2}\)