Space of modular forms of level 42, weight 4, and Character 42.f

• Label: 42.4.f
• Level: $42$
• Weight: $4$
• Character orbit: 42.f
• Rep. character: $\chi_{42}(5,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $16$
• Newform subspaces: $1$
• Sturm bound: $32$
• Trace bound: $0$

Defining parameters

Level:	\( N \)	\(=\)	\( 42 = 2 \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	42.f (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 21 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 1 \)
Sturm bound:			\(32\)
Trace bound:			\(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(42, [\chi])\).

	Total	New	Old
Modular forms	56	16	40
Cusp forms	40	16	24
Eisenstein series	16	0	16

Trace form

\( 16 q + 32 q^{4} + 80 q^{7} + 18 q^{9} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(42, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
42.4.f.a	$42$	$4$	42.f	21.g	$6$	$16$	$8$	$2.478$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(80\)	$2^{6}\cdot 3^{8}$	$\mathrm{SU}(2)[C_{6}]$	\(q-\beta _{2}q^{2}+\beta _{1}q^{3}+(4+4\beta _{5})q^{4}+(-\beta _{2}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(42, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(42, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 2}\)