Space of modular forms of level 764, weight 2, and trivial character

• Label: 764.2.a
• Level: $764$
• Weight: $2$
• Character orbit: 764.a
• Rep. character: $\chi_{764}(1,\cdot)$
• Character field: $\Q$
• Dimension: $16$
• Newform subspaces: $2$
• Sturm bound: $192$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 764 = 2^{2} \cdot 191 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	764.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(192\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(764))\).

	Total	New	Old
Modular forms	99	16	83
Cusp forms	94	16	78
Eisenstein series	5	0	5

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

\(2\)	\(191\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(12\)	\(0\)	\(12\)		\(11\)	\(0\)	\(11\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(-\)		\(38\)	\(0\)	\(38\)		\(36\)	\(0\)	\(36\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)		\(18\)	\(8\)	\(10\)		\(17\)	\(8\)	\(9\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)		\(31\)	\(8\)	\(23\)		\(30\)	\(8\)	\(22\)		\(1\)	\(0\)	\(1\)
Plus space		\(+\)		\(43\)	\(8\)	\(35\)		\(41\)	\(8\)	\(33\)		\(2\)	\(0\)	\(2\)
Minus space		\(-\)		\(56\)	\(8\)	\(48\)		\(53\)	\(8\)	\(45\)		\(3\)	\(0\)	\(3\)

Trace form

16 * q - 2 * q^3 - 4 * q^5 - 2 * q^7 + 10 * q^9 - 4 * q^11 + 6 * q^15 - 2 * q^17 + 2 * q^19 - 6 * q^21 + 4 * q^23 + 12 * q^25 - 8 * q^27 - 2 * q^29 - 6 * q^31 - 6 * q^33 - 14 * q^35 + 2 * q^37 + 6 * q^39 - 8 * q^41 - 10 * q^43 - 14 * q^45 - 2 * q^47 + 10 * q^49 - 16 * q^51 - 4 * q^53 - 26 * q^55 + 30 * q^57 - 14 * q^59 - 4 * q^61 + 10 * q^63 + 4 * q^65 + 6 * q^67 + 2 * q^69 + 22 * q^71 - 2 * q^73 + 16 * q^77 - 22 * q^79 - 24 * q^81 + 22 * q^83 - 28 * q^85 + 2 * q^87 - 12 * q^89 - 2 * q^91 + 44 * q^93 + 6 * q^95 - 10 * q^97 + 10 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(764))\) 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	191
764.2.a.a	$764$	$2$	764.a	1.a	$1$	$8$	$8$	$6.101$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓	✓	✓	764.2.a.a	$1$	$1$	\(0\)	\(-5\)	\(-4\)	\(-7\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+(-\beta _{2}+\beta _{4}-\beta _{5}+\cdots)q^{5}+\cdots\)
764.2.a.b	$764$	$2$	764.a	1.a	$1$	$8$	$8$	$6.101$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓	✓	✓	764.2.a.b	$1$	$0$	\(0\)	\(3\)	\(0\)	\(5\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+\beta _{6}q^{5}+(1-\beta _{4})q^{7}+(\beta _{1}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(764))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(764)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(191))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(382))\)\(^{\oplus 2}\)