Space of modular forms of level 78, weight 2, and Character 78.e

• Label: 78.2.e
• Level: $78$
• Weight: $2$
• Character orbit: 78.e
• Rep. character: $\chi_{78}(55,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $4$
• Newform subspaces: $2$
• Sturm bound: $28$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 78 = 2 \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	78.e (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 13 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(28\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(78, [\chi])\).

	Total	New	Old
Modular forms	36	4	32
Cusp forms	20	4	16
Eisenstein series	16	0	16

Trace form

4 * q + 2 * q^2 - 2 * q^4 + 4 * q^5 - 4 * q^8 - 2 * q^9 + 2 * q^10 - 8 * q^11 - 2 * q^13 - 4 * q^15 - 2 * q^16 - 2 * q^17 - 4 * q^18 - 2 * q^20 + 8 * q^21 + 8 * q^22 + 2 * q^26 + 6 * q^29 + 4 * q^30 - 16 * q^31 + 2 * q^32 - 4 * q^33 - 4 * q^34 - 8 * q^35 - 2 * q^36 + 18 * q^37 + 12 * q^39 - 4 * q^40 - 2 * q^41 + 4 * q^42 + 16 * q^44 - 2 * q^45 + 16 * q^47 + 6 * q^49 - 16 * q^51 + 4 * q^52 + 4 * q^53 - 16 * q^55 + 8 * q^57 - 6 * q^58 + 8 * q^59 + 8 * q^60 + 18 * q^61 - 8 * q^62 + 4 * q^64 - 26 * q^65 - 8 * q^66 + 8 * q^67 - 2 * q^68 + 12 * q^69 - 16 * q^70 + 8 * q^71 + 2 * q^72 - 52 * q^73 - 18 * q^74 - 8 * q^75 + 16 * q^77 - 16 * q^79 - 2 * q^80 - 2 * q^81 + 2 * q^82 - 4 * q^84 + 14 * q^85 - 12 * q^87 + 8 * q^88 - 20 * q^89 - 4 * q^90 + 8 * q^94 - 8 * q^95 - 12 * q^97 - 6 * q^98 + 16 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(78, [\chi])\) 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				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
78.2.e.a	$78$	$2$	78.e	13.c	$3$	$2$	$1$	$0.623$	\(\Q(\sqrt{-3}) \)	None		✓			78.2.e.a	$2$	$0$	\(1\)	\(-1\)	\(6\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{2}+(-1+\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots\)
78.2.e.b	$78$	$2$	78.e	13.c	$3$	$2$	$1$	$0.623$	\(\Q(\sqrt{-3}) \)	None		✓			78.2.e.b	$2$	$0$	\(1\)	\(1\)	\(-2\)	\(2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{2}+(1-\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(78, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(78, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(26, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(39, [\chi])\)\(^{\oplus 2}\)