Space of modular forms of level 304, weight 2, and trivial character

• Label: 304.2.a
• Level: $304$
• Weight: $2$
• Character orbit: 304.a
• Rep. character: $\chi_{304}(1,\cdot)$
• Character field: $\Q$
• Dimension: $9$
• Newform subspaces: $7$
• Sturm bound: $80$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 304 = 2^{4} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	304.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(80\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(3\), \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(304))\).

	Total	New	Old
Modular forms	46	9	37
Cusp forms	35	9	26
Eisenstein series	11	0	11

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		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(10\)	\(1\)	\(9\)		\(8\)	\(1\)	\(7\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)		\(13\)	\(4\)	\(9\)		\(10\)	\(4\)	\(6\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)		\(13\)	\(2\)	\(11\)		\(10\)	\(2\)	\(8\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)		\(10\)	\(2\)	\(8\)		\(7\)	\(2\)	\(5\)		\(3\)	\(0\)	\(3\)
Plus space		\(+\)		\(20\)	\(3\)	\(17\)		\(15\)	\(3\)	\(12\)		\(5\)	\(0\)	\(5\)
Minus space		\(-\)		\(26\)	\(6\)	\(20\)		\(20\)	\(6\)	\(14\)		\(6\)	\(0\)	\(6\)

Trace form

9 * q - 2 * q^5 - 2 * q^7 + 9 * q^9 + 2 * q^11 - 2 * q^13 + 12 * q^15 + 2 * q^17 + 3 * q^19 - 8 * q^21 - 4 * q^23 + 3 * q^25 - 2 * q^29 - 8 * q^33 + 18 * q^35 - 2 * q^37 + 4 * q^39 + 2 * q^41 - 14 * q^43 - 18 * q^45 + 14 * q^47 + q^49 + 4 * q^51 + 6 * q^53 - 18 * q^55 - 16 * q^59 - 10 * q^61 - 30 * q^63 + 12 * q^65 - 20 * q^67 - 24 * q^69 + 4 * q^71 + 2 * q^73 - 4 * q^75 - 8 * q^77 - 8 * q^79 + 17 * q^81 + 4 * q^83 - 12 * q^85 + 12 * q^87 + 18 * q^89 - 40 * q^91 + 8 * q^93 - 8 * q^95 + 10 * q^97 + 42 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(304))\) 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					A-L signs		Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	19
304.2.a.a	$304$	$2$	304.a	1.a	$1$	$1$	$1$	$2.427$	\(\Q\)	None	✓				76.2.a.a	$1$	$1$	\(0\)	\(-2\)	\(-1\)	\(3\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-q^{5}+3q^{7}+q^{9}-5q^{11}+\cdots\)
304.2.a.b	$304$	$2$	304.a	1.a	$1$	$1$	$1$	$2.427$	\(\Q\)	None	✓		✓	✓	152.2.a.b	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-3\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-3q^{7}-2q^{9}-2q^{11}+q^{13}+\cdots\)
304.2.a.c	$304$	$2$	304.a	1.a	$1$	$1$	$1$	$2.427$	\(\Q\)	None	✓				38.2.a.a	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}-2q^{9}+6q^{11}+5q^{13}+\cdots\)
304.2.a.d	$304$	$2$	304.a	1.a	$1$	$1$	$1$	$2.427$	\(\Q\)	None	✓				38.2.a.b	$1$	$1$	\(0\)	\(1\)	\(-4\)	\(-3\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-4q^{5}-3q^{7}-2q^{9}-2q^{11}+\cdots\)
304.2.a.e	$304$	$2$	304.a	1.a	$1$	$1$	$1$	$2.427$	\(\Q\)	None	✓				152.2.a.a	$1$	$0$	\(0\)	\(2\)	\(-1\)	\(3\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-q^{5}+3q^{7}+q^{9}+3q^{11}+\cdots\)
304.2.a.f	$304$	$2$	304.a	1.a	$1$	$1$	$1$	$2.427$	\(\Q\)	None	✓				19.2.a.a	$1$	$0$	\(0\)	\(2\)	\(3\)	\(1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+3q^{5}+q^{7}+q^{9}-3q^{11}+\cdots\)
304.2.a.g	$304$	$2$	304.a	1.a	$1$	$3$	$3$	$2.427$	3.3.961.1	None	✓		✓		152.2.a.c	$1$	$0$	\(0\)	\(-1\)	\(1\)	\(-4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}-\beta _{2}q^{5}+(-2+\beta _{1}-\beta _{2})q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(304))\) into lower level spaces

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