Space of modular forms of level 78, weight 2, and trivial character

• Label: 78.2.a
• Level: $78$
• Weight: $2$
• Character orbit: 78.a
• Rep. character: $\chi_{78}(1,\cdot)$
• Character field: $\Q$
• Dimension: $1$
• Newform subspaces: $1$
• Sturm bound: $28$
• Trace bound: $0$

Defining parameters

Level:	\( N \)	\(=\)	\( 78 = 2 \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	78.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 1 \)
Sturm bound:			\(28\)
Trace bound:			\(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(78))\).

	Total	New	Old
Modular forms	18	1	17
Cusp forms	11	1	10
Eisenstein series	7	0	7

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

\(2\)	\(3\)	\(13\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(-\)	\(-\)		\(4\)	\(1\)	\(3\)		\(3\)	\(1\)	\(2\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(+\)	\(-\)		\(3\)	\(0\)	\(3\)		\(2\)	\(0\)	\(2\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(-\)	\(+\)		\(2\)	\(0\)	\(2\)		\(1\)	\(0\)	\(1\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(+\)	\(-\)		\(2\)	\(0\)	\(2\)		\(1\)	\(0\)	\(1\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)	\(+\)		\(2\)	\(0\)	\(2\)		\(1\)	\(0\)	\(1\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)	\(+\)		\(4\)	\(0\)	\(4\)		\(3\)	\(0\)	\(3\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(-\)	\(-\)		\(1\)	\(0\)	\(1\)		\(0\)	\(0\)	\(0\)		\(1\)	\(0\)	\(1\)
Plus space			\(+\)		\(8\)	\(0\)	\(8\)		\(5\)	\(0\)	\(5\)		\(3\)	\(0\)	\(3\)
Minus space			\(-\)		\(10\)	\(1\)	\(9\)		\(6\)	\(1\)	\(5\)		\(4\)	\(0\)	\(4\)

Trace form

q - q^2 - q^3 + q^4 + 2 * q^5 + q^6 + 4 * q^7 - q^8 + q^9 - 2 * q^10 - 4 * q^11 - q^12 + q^13 - 4 * q^14 - 2 * q^15 + q^16 + 2 * q^17 - q^18 - 8 * q^19 + 2 * q^20 - 4 * q^21 + 4 * q^22 + q^24 - q^25 - q^26 - q^27 + 4 * q^28 + 6 * q^29 + 2 * q^30 - 4 * q^31 - q^32 + 4 * q^33 - 2 * q^34 + 8 * q^35 + q^36 - 2 * q^37 + 8 * q^38 - q^39 - 2 * q^40 - 10 * q^41 + 4 * q^42 + 4 * q^43 - 4 * q^44 + 2 * q^45 + 8 * q^47 - q^48 + 9 * q^49 + q^50 - 2 * q^51 + q^52 - 10 * q^53 + q^54 - 8 * q^55 - 4 * q^56 + 8 * q^57 - 6 * q^58 + 4 * q^59 - 2 * q^60 - 2 * q^61 + 4 * q^62 + 4 * q^63 + q^64 + 2 * q^65 - 4 * q^66 - 16 * q^67 + 2 * q^68 - 8 * q^70 - 8 * q^71 - q^72 + 2 * q^73 + 2 * q^74 + q^75 - 8 * q^76 - 16 * q^77 + q^78 + 8 * q^79 + 2 * q^80 + q^81 + 10 * q^82 + 12 * q^83 - 4 * q^84 + 4 * q^85 - 4 * q^86 - 6 * q^87 + 4 * q^88 + 14 * q^89 - 2 * q^90 + 4 * q^91 + 4 * q^93 - 8 * q^94 - 16 * q^95 + q^96 + 10 * q^97 - 9 * q^98 - 4 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(78))\) 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	3	13
78.2.a.a	$78$	$2$	78.a	1.a	$1$	$1$	$1$	$0.623$	\(\Q\)	None	✓	✓	✓	✓	78.2.a.a	$1$	$0$	\(-1\)	\(-1\)	\(2\)	\(4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}+2q^{5}+q^{6}+4q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(78))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(78)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 2}\)