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

• Label: 78.3.h
• Level: $78$
• Weight: $3$
• Character orbit: 78.h
• Rep. character: $\chi_{78}(29,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $20$
• 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.h (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 39 \)
Character field:			\(\Q(\zeta_{6})\)
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	20	44
Cusp forms	48	20	28
Eisenstein series	16	0	16

Trace form

\( 20 q + 20 q^{4} + 10 q^{7} + 4 q^{9} + O(q^{10}) \)

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.h.a	$78$	$3$	78.h	39.i	$6$	$8$	$4$	$2.125$	8.0.4857532416.2	None		$4$	$0$	\(0\)	\(-2\)	\(0\)	\(-12\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{2}q^{2}+(-1-\beta _{1}+\beta _{3}-\beta _{4}-\beta _{5}+\cdots)q^{3}+\cdots\)
78.3.h.b	$78$	$3$	78.h	39.i	$6$	$12$	$6$	$2.125$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		$4$	$0$	\(0\)	\(2\)	\(0\)	\(22\)	$3^{4}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{7}q^{2}+(-\beta _{4}+\beta _{9})q^{3}+2\beta _{1}q^{4}+\cdots\)

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

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