Space of modular forms of level 304, weight 6, and Character 304.i

• Label: 304.6.i
• Level: $304$
• Weight: $6$
• Character orbit: 304.i
• Rep. character: $\chi_{304}(49,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $98$
• Newform subspaces: $6$
• Sturm bound: $240$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 304 = 2^{4} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	304.i (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 19 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(240\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(304, [\chi])\).

	Total	New	Old
Modular forms	412	102	310
Cusp forms	388	98	290
Eisenstein series	24	4	20

Trace form

\( 98 q + 19 q^{3} - q^{5} - 120 q^{7} - 3830 q^{9} + O(q^{10}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(304, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
304.6.i.a	$304$	$6$	304.i	19.c	$3$	$6$	$3$	$48.757$	\(\mathbb{Q}[x]/(x^{6} + \cdots)\)	None		$2$	$0$	\(0\)	\(15\)	\(14\)	\(624\)	$2^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(5-\beta _{1}-\beta _{2}-5\beta _{3})q^{3}+(5-\beta _{1}+\cdots)q^{5}+\cdots\)
304.6.i.b	$304$	$6$	304.i	19.c	$3$	$8$	$4$	$48.757$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		$2$	$0$	\(0\)	\(14\)	\(-36\)	\(-76\)	$2^{2}\cdot 3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(3-3\beta _{2}+\beta _{3})q^{3}+(-9+9\beta _{2}+\beta _{4}+\cdots)q^{5}+\cdots\)
304.6.i.c	$304$	$6$	304.i	19.c	$3$	$16$	$8$	$48.757$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$2$	$0$	\(0\)	\(-28\)	\(10\)	\(-208\)	$2^{12}\cdot 3^{3}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-3\beta _{3}-\beta _{5})q^{3}+(\beta _{2}+\beta _{3}-\beta _{4}+\cdots)q^{5}+\cdots\)
304.6.i.d	$304$	$6$	304.i	19.c	$3$	$18$	$9$	$48.757$	\(\mathbb{Q}[x]/(x^{18} - \cdots)\)	None		$2$	$0$	\(0\)	\(11\)	\(11\)	\(-336\)	$2^{16}\cdot 3^{3}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-\beta _{1}-\beta _{2}+\beta _{3})q^{3}+(-\beta _{1}+\beta _{5}+\cdots)q^{5}+\cdots\)
304.6.i.e	$304$	$6$	304.i	19.c	$3$	$24$	$12$	$48.757$		None		$2$	$0$	\(0\)	\(4\)	\(25\)	\(200\)		$\mathrm{SU}(2)[C_{3}]$
304.6.i.f	$304$	$6$	304.i	19.c	$3$	$26$	$13$	$48.757$		None		$2$	$0$	\(0\)	\(3\)	\(-25\)	\(-324\)		$\mathrm{SU}(2)[C_{3}]$

Decomposition of \(S_{6}^{\mathrm{old}}(304, [\chi])\) into lower level spaces

\( S_{6}^{\mathrm{old}}(304, [\chi]) \cong \) \(S_{6}^{\mathrm{new}}(19, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(38, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(76, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(152, [\chi])\)\(^{\oplus 2}\)