Space of modular forms of level 304, weight 6, and trivial character

• Label: 304.6.a
• Level: $304$
• Weight: $6$
• Character orbit: 304.a
• Rep. character: $\chi_{304}(1,\cdot)$
• Character field: $\Q$
• Dimension: $45$
• Newform subspaces: $15$
• Sturm bound: $240$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 304 = 2^{4} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	304.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 15 \)
Sturm bound:			\(240\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(304))\).

	Total	New	Old
Modular forms	206	45	161
Cusp forms	194	45	149
Eisenstein series	12	0	12

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)	\(19\)	Fricke	Dim
\(+\)	\(+\)	$+$	\(10\)
\(+\)	\(-\)	$-$	\(13\)
\(-\)	\(+\)	$-$	\(11\)
\(-\)	\(-\)	$+$	\(11\)
Plus space		\(+\)	\(21\)
Minus space		\(-\)	\(24\)

Trace form

\( 45 q + 38 q^{5} - 98 q^{7} + 3645 q^{9} + O(q^{10}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(304))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces					A-L signs		Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	19
304.6.a.a	$304$	$6$	304.a	1.a	$1$	$1$	$1$	$48.757$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-4\)	\(54\)	\(-248\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{3}+54q^{5}-248q^{7}-227q^{9}+\cdots\)
304.6.a.b	$304$	$6$	304.a	1.a	$1$	$1$	$1$	$48.757$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(-24\)	\(167\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-24q^{5}+167q^{7}-242q^{9}+\cdots\)
304.6.a.c	$304$	$6$	304.a	1.a	$1$	$1$	$1$	$48.757$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(6\)	\(31\)	\(27\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+6q^{3}+31q^{5}+3^{3}q^{7}-207q^{9}+\cdots\)
304.6.a.d	$304$	$6$	304.a	1.a	$1$	$1$	$1$	$48.757$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(7\)	\(16\)	\(-75\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+7q^{3}+2^{4}q^{5}-75q^{7}-194q^{9}+\cdots\)
304.6.a.e	$304$	$6$	304.a	1.a	$1$	$1$	$1$	$48.757$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(14\)	\(-45\)	\(121\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+14q^{3}-45q^{5}+11^{2}q^{7}-47q^{9}+\cdots\)
304.6.a.f	$304$	$6$	304.a	1.a	$1$	$2$	$2$	$48.757$	\(\Q(\sqrt{1441}) \)	None	✓	$1$	$0$	\(0\)	\(-3\)	\(-45\)	\(-114\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{3}+(-24+3\beta )q^{5}+(-59+\cdots)q^{7}+\cdots\)
304.6.a.g	$304$	$6$	304.a	1.a	$1$	$2$	$2$	$48.757$	\(\Q(\sqrt{177}) \)	None	✓	$1$	$1$	\(0\)	\(7\)	\(-133\)	\(-72\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(2+3\beta )q^{3}+(-69+5\beta )q^{5}+(-29+\cdots)q^{7}+\cdots\)
304.6.a.h	$304$	$6$	304.a	1.a	$1$	$3$	$3$	$48.757$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-13\)	\(81\)	\(-228\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-4-\beta _{1})q^{3}+(3^{3}-\beta _{1}+\beta _{2})q^{5}+\cdots\)
304.6.a.i	$304$	$6$	304.a	1.a	$1$	$3$	$3$	$48.757$	3.3.976277.1	None	✓	$1$	$1$	\(0\)	\(7\)	\(58\)	\(197\)		$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(2-\beta _{1})q^{3}+(20+\beta _{1}+\beta _{2})q^{5}+(67+\cdots)q^{7}+\cdots\)
304.6.a.j	$304$	$6$	304.a	1.a	$1$	$3$	$3$	$48.757$	3.3.272193.1	None	✓	$1$	$0$	\(0\)	\(8\)	\(-9\)	\(13\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(3+\beta _{1}+\beta _{2})q^{3}+(-2+3\beta _{1}+\beta _{2})q^{5}+\cdots\)
304.6.a.k	$304$	$6$	304.a	1.a	$1$	$4$	$4$	$48.757$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-10\)	\(-110\)	\(-30\)		$-$	$-$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-3-\beta _{3})q^{3}+(-26+\beta _{1}+2\beta _{2}+\cdots)q^{5}+\cdots\)
304.6.a.l	$304$	$6$	304.a	1.a	$1$	$4$	$4$	$48.757$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(-6\)	\(90\)	\(190\)		$-$	$+$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+(-3-3\beta _{1})q^{3}+(21-4\beta _{1}+\beta _{2}+\cdots)q^{5}+\cdots\)
304.6.a.m	$304$	$6$	304.a	1.a	$1$	$6$	$6$	$48.757$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-23\)	\(0\)	\(-133\)		$+$	$+$	$+$	$2^{6}$	$\mathrm{SU}(2)$	\(q+(-4+\beta _{1})q^{3}+(-\beta _{1}-\beta _{3})q^{5}+(-23+\cdots)q^{7}+\cdots\)
304.6.a.n	$304$	$6$	304.a	1.a	$1$	$6$	$6$	$48.757$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(4\)	\(-25\)	\(161\)		$+$	$-$	$-$	$2^{6}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(-4-\beta _{3})q^{5}+(26+3\beta _{1}+\cdots)q^{7}+\cdots\)
304.6.a.o	$304$	$6$	304.a	1.a	$1$	$7$	$7$	$48.757$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(5\)	\(99\)	\(-74\)		$+$	$-$	$-$	$2^{9}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(14+\beta _{1}+\beta _{2})q^{5}+(-10+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(304))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(304)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 5}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(38))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(76))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(152))\)\(^{\oplus 2}\)