Space of modular forms of level 304, weight 10, and trivial character

• Label: 304.10.a
• Level: $304$
• Weight: $10$
• Character orbit: 304.a
• Rep. character: $\chi_{304}(1,\cdot)$
• Character field: $\Q$
• Dimension: $81$
• Newform subspaces: $13$
• Sturm bound: $400$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 304 = 2^{4} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	304.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 13 \)
Sturm bound:			\(400\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(304))\).

	Total	New	Old
Modular forms	366	81	285
Cusp forms	354	81	273
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
\(+\)	\(+\)	$+$	\(19\)
\(+\)	\(-\)	$-$	\(22\)
\(-\)	\(+\)	$-$	\(20\)
\(-\)	\(-\)	$+$	\(20\)
Plus space		\(+\)	\(39\)
Minus space		\(-\)	\(42\)

Trace form

\( 81 q + 718 q^{5} - 4802 q^{7} + 531441 q^{9} + O(q^{10}) \)

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(304))\) 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
304.10.a.a	$304$	$10$	304.a	1.a	$1$	$1$	$1$	$156.571$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-102\)	\(-1581\)	\(4865\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-102q^{3}-1581q^{5}+4865q^{7}-9279q^{9}+\cdots\)
304.10.a.b	$304$	$10$	304.a	1.a	$1$	$1$	$1$	$156.571$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(119\)	\(-684\)	\(-9149\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+119q^{3}-684q^{5}-9149q^{7}-5522q^{9}+\cdots\)
304.10.a.c	$304$	$10$	304.a	1.a	$1$	$3$	$3$	$156.571$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-3\)	\(486\)	\(13317\)		$-$	$-$	$+$	$2^{3}\cdot 3$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1})q^{3}+(162+2\beta _{1}+\beta _{2})q^{5}+\cdots\)
304.10.a.d	$304$	$10$	304.a	1.a	$1$	$4$	$4$	$156.571$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-226\)	\(866\)	\(-2670\)		$-$	$-$	$+$	$2^{4}\cdot 3$	$\mathrm{SU}(2)$	\(q+(-57-\beta _{2})q^{3}+(218-\beta _{1}+3\beta _{2}+\cdots)q^{5}+\cdots\)
304.10.a.e	$304$	$10$	304.a	1.a	$1$	$4$	$4$	$156.571$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(-84\)	\(-1395\)	\(-12307\)		$-$	$+$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+(-21-\beta _{1})q^{3}+(-350-5\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)
304.10.a.f	$304$	$10$	304.a	1.a	$1$	$6$	$6$	$156.571$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(155\)	\(-3612\)	\(-4085\)		$-$	$-$	$+$	$2^{6}$	$\mathrm{SU}(2)$	\(q+(26-\beta _{2})q^{3}+(-601-4\beta _{1}-2\beta _{2}+\cdots)q^{5}+\cdots\)
304.10.a.g	$304$	$10$	304.a	1.a	$1$	$6$	$6$	$156.571$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(155\)	\(2061\)	\(4580\)		$-$	$+$	$-$	$2^{9}$	$\mathrm{SU}(2)$	\(q+(26-\beta _{1})q^{3}+(7^{3}+\beta _{1}+\beta _{2})q^{5}+\cdots\)
304.10.a.h	$304$	$10$	304.a	1.a	$1$	$7$	$7$	$156.571$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-7\)	\(-445\)	\(-16242\)		$-$	$-$	$+$	$2^{10}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1})q^{3}+(-2^{6}+\beta _{3})q^{5}+(-2321+\cdots)q^{7}+\cdots\)
304.10.a.i	$304$	$10$	304.a	1.a	$1$	$8$	$8$	$156.571$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(-7\)	\(3894\)	\(7133\)		$-$	$+$	$-$	$2^{16}\cdot 3^{3}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+(487-2\beta _{1}-\beta _{7})q^{5}+\cdots\)
304.10.a.j	$304$	$10$	304.a	1.a	$1$	$9$	$9$	$156.571$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(152\)	\(-1510\)	\(9232\)		$+$	$+$	$+$	$2^{19}\cdot 3$	$\mathrm{SU}(2)$	\(q+(17-\beta _{1})q^{3}+(-168-\beta _{1}+\beta _{3}+\cdots)q^{5}+\cdots\)
304.10.a.k	$304$	$10$	304.a	1.a	$1$	$10$	$10$	$156.571$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-233\)	\(2074\)	\(-6755\)		$+$	$+$	$+$	$2^{26}\cdot 3$	$\mathrm{SU}(2)$	\(q+(-23-\beta _{1})q^{3}+(207+\beta _{1}-\beta _{3}+\cdots)q^{5}+\cdots\)
304.10.a.l	$304$	$10$	304.a	1.a	$1$	$10$	$10$	$156.571$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(10\)	\(1449\)	\(7651\)		$+$	$-$	$-$	$2^{27}\cdot 3^{4}$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{3}+(145+2\beta _{1}+\beta _{2})q^{5}+\cdots\)
304.10.a.m	$304$	$10$	304.a	1.a	$1$	$12$	$12$	$156.571$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(71\)	\(-885\)	\(-372\)		$+$	$-$	$-$	$2^{41}\cdot 3^{4}$	$\mathrm{SU}(2)$	\(q+(6-\beta _{1})q^{3}+(-74-\beta _{2})q^{5}+(-2^{5}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_0(304))\) into lower level spaces

\( S_{10}^{\mathrm{old}}(\Gamma_0(304)) \cong \) \(S_{10}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 8}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 6}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 2}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 5}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(38))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(76))\)\(^{\oplus 3}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(152))\)\(^{\oplus 2}\)