Space of modular forms of level 778, weight 2, and trivial character

• Label: 778.2.a
• Level: $778$
• Weight: $2$
• Character orbit: 778.a
• Rep. character: $\chi_{778}(1,\cdot)$
• Character field: $\Q$
• Dimension: $32$
• Newform subspaces: $4$
• Sturm bound: $195$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 778 = 2 \cdot 389 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	778.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(195\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(778))\).

	Total	New	Old
Modular forms	99	32	67
Cusp forms	96	32	64
Eisenstein series	3	0	3

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

\(2\)	\(389\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(20\)	\(9\)	\(11\)		\(20\)	\(9\)	\(11\)		\(0\)	\(0\)	\(0\)
\(+\)	\(-\)	\(-\)		\(29\)	\(7\)	\(22\)		\(28\)	\(7\)	\(21\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)		\(24\)	\(12\)	\(12\)		\(23\)	\(12\)	\(11\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)		\(26\)	\(4\)	\(22\)		\(25\)	\(4\)	\(21\)		\(1\)	\(0\)	\(1\)
Plus space		\(+\)		\(46\)	\(13\)	\(33\)		\(45\)	\(13\)	\(32\)		\(1\)	\(0\)	\(1\)
Minus space		\(-\)		\(53\)	\(19\)	\(34\)		\(51\)	\(19\)	\(32\)		\(2\)	\(0\)	\(2\)

Trace form

32 * q + 32 * q^4 + 6 * q^5 + 2 * q^6 + 38 * q^9 + 4 * q^10 + 2 * q^11 - 6 * q^13 + 8 * q^15 + 32 * q^16 + 12 * q^17 - 14 * q^19 + 6 * q^20 + 4 * q^21 - 4 * q^22 - 4 * q^23 + 2 * q^24 + 38 * q^25 - 12 * q^27 + 4 * q^29 - 16 * q^30 - 8 * q^31 - 32 * q^33 - 16 * q^35 + 38 * q^36 - 16 * q^37 - 16 * q^38 - 32 * q^39 + 4 * q^40 - 24 * q^41 - 8 * q^43 + 2 * q^44 + 30 * q^45 + 8 * q^46 + 28 * q^49 + 8 * q^50 - 28 * q^51 - 6 * q^52 - 16 * q^54 + 12 * q^55 - 32 * q^57 + 10 * q^58 - 22 * q^59 + 8 * q^60 - 32 * q^61 - 8 * q^62 - 40 * q^63 + 32 * q^64 - 16 * q^65 + 24 * q^66 - 10 * q^67 + 12 * q^68 - 24 * q^69 + 16 * q^70 + 8 * q^71 + 12 * q^73 + 14 * q^74 - 28 * q^75 - 14 * q^76 - 20 * q^77 + 24 * q^78 - 8 * q^79 + 6 * q^80 + 56 * q^81 + 16 * q^82 + 4 * q^84 + 4 * q^85 - 6 * q^86 + 4 * q^87 - 4 * q^88 + 32 * q^90 - 48 * q^91 - 4 * q^92 - 56 * q^93 + 28 * q^94 - 20 * q^95 + 2 * q^96 - 44 * q^97 + 8 * q^98 + 2 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(778))\) 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	389
778.2.a.a	$778$	$2$	778.a	1.a	$1$	$4$	$4$	$6.212$	\(\Q(\zeta_{15})^+\)	None	✓	✓	✓	✓	778.2.a.a	$1$	$1$	\(4\)	\(-3\)	\(-7\)	\(-5\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+(-1-\beta _{2}-\beta _{3})q^{3}+q^{4}+(-2+\cdots)q^{5}+\cdots\)
778.2.a.b	$778$	$2$	778.a	1.a	$1$	$7$	$7$	$6.212$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	✓	✓	✓	778.2.a.b	$1$	$0$	\(-7\)	\(5\)	\(9\)	\(3\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+(1+\beta _{2})q^{3}+q^{4}+(1-\beta _{4})q^{5}+\cdots\)
778.2.a.c	$778$	$2$	778.a	1.a	$1$	$9$	$9$	$6.212$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	✓	✓	✓	778.2.a.c	$1$	$1$	\(-9\)	\(-6\)	\(-8\)	\(-3\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+(-1+\beta _{1})q^{3}+q^{4}+(-1+\beta _{4}+\cdots)q^{5}+\cdots\)
778.2.a.d	$778$	$2$	778.a	1.a	$1$	$12$	$12$	$6.212$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	✓	✓	✓	778.2.a.d	$1$	$0$	\(12\)	\(4\)	\(12\)	\(5\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+\beta _{1}q^{3}+q^{4}+(1+\beta _{6})q^{5}+\beta _{1}q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(778))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(778)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(389))\)\(^{\oplus 2}\)