Space of modular forms of level 78, weight 3, and Character 78.f

• Label: 78.3.f
• Level: $78$
• Weight: $3$
• Character orbit: 78.f
• Rep. character: $\chi_{78}(31,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $12$
• Newform subspaces: $2$
• Sturm bound: $42$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 78 = 2 \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	78.f (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 13 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(42\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(78, [\chi])\).

	Total	New	Old
Modular forms	64	12	52
Cusp forms	48	12	36
Eisenstein series	16	0	16

Trace form

\( 12 q - 4 q^{2} + 12 q^{5} + 8 q^{8} + 36 q^{9} + O(q^{10}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(78, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
78.3.f.a	$78$	$3$	78.f	13.d	$4$	$4$	$2$	$2.125$	\(\Q(\zeta_{12})\)	None		$2$	$0$	\(4\)	\(0\)	\(12\)	\(-4\)	$2^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(1+\zeta_{12})q^{2}+\zeta_{12}^{3}q^{3}+2\zeta_{12}q^{4}+\cdots\)
78.3.f.b	$78$	$3$	78.f	13.d	$4$	$8$	$4$	$2.125$	8.0.\(\cdots\).1	None		$2$	$0$	\(-8\)	\(0\)	\(0\)	\(4\)	$2^{6}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-1-\beta _{6})q^{2}+\beta _{1}q^{3}+2\beta _{6}q^{4}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(78, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(78, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(13, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(26, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(39, [\chi])\)\(^{\oplus 2}\)