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

• Label: 78.2.i
• Level: $78$
• Weight: $2$
• Character orbit: 78.i
• Rep. character: $\chi_{78}(43,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $8$
• 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.i (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 13 \)
Character field:			\(\Q(\zeta_{6})\)
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	8	28
Cusp forms	20	8	12
Eisenstein series	16	0	16

Trace form

8 * q + 4 * q^4 - 4 * q^9 + 4 * q^10 - 4 * q^13 - 16 * q^14 - 12 * q^15 - 4 * q^16 - 8 * q^17 - 8 * q^23 + 8 * q^26 + 8 * q^29 + 4 * q^30 + 12 * q^33 + 16 * q^35 + 4 * q^36 - 12 * q^37 + 4 * q^39 + 8 * q^40 + 48 * q^41 - 4 * q^42 + 4 * q^49 + 24 * q^50 + 16 * q^51 - 8 * q^52 - 8 * q^56 - 12 * q^58 - 48 * q^59 - 12 * q^61 - 16 * q^62 - 8 * q^64 - 8 * q^65 + 24 * q^66 - 48 * q^67 + 8 * q^68 + 4 * q^69 + 24 * q^71 - 8 * q^74 + 8 * q^75 - 48 * q^77 - 16 * q^78 - 32 * q^79 - 4 * q^81 - 4 * q^82 - 12 * q^84 + 36 * q^85 - 4 * q^87 - 24 * q^89 - 8 * q^90 + 32 * q^91 - 16 * q^92 - 24 * q^93 + 24 * q^95 + 24 * q^98

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.i.a	$78$	$2$	78.i	13.e	$6$	$4$	$2$	$0.623$	\(\Q(\zeta_{12})\)	None		✓			78.2.i.a	$2$	$0$	\(0\)	\(-2\)	\(0\)	\(6\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+\zeta_{12}q^{2}+(-1+\zeta_{12}^{2})q^{3}+\zeta_{12}^{2}q^{4}+\cdots\)
78.2.i.b	$78$	$2$	78.i	13.e	$6$	$4$	$2$	$0.623$	\(\Q(\zeta_{12})\)	None		✓			78.2.i.b	$2$	$0$	\(0\)	\(2\)	\(0\)	\(-6\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+\zeta_{12}q^{2}+(1-\zeta_{12}^{2})q^{3}+\zeta_{12}^{2}q^{4}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(78, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(13, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(39, [\chi])\)\(^{\oplus 2}\)