Space of modular forms of level 42, weight 2, and Character 42.f

• Label: 42.2.f
• Level: $42$
• Weight: $2$
• Character orbit: 42.f
• Rep. character: $\chi_{42}(5,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $4$
• Newform subspaces: $1$
• Sturm bound: $16$
• Trace bound: $0$

Defining parameters

Level:	\( N \)	\(=\)	\( 42 = 2 \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
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:			\(16\)
Trace bound:			\(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(42, [\chi])\).

	Total	New	Old
Modular forms	24	4	20
Cusp forms	8	4	4
Eisenstein series	16	0	16

Trace form

4 * q + 2 * q^4 - 10 * q^7 - 6 * q^9 - 6 * q^10 + 12 * q^15 - 2 * q^16 + 12 * q^19 + 12 * q^22 + 6 * q^24 + 4 * q^25 - 2 * q^28 - 6 * q^31 - 18 * q^33 - 12 * q^36 + 4 * q^37 - 6 * q^40 - 6 * q^42 - 32 * q^43 - 12 * q^46 + 22 * q^49 + 12 * q^51 + 12 * q^52 + 18 * q^54 + 6 * q^58 + 6 * q^60 + 24 * q^63 - 4 * q^64 - 4 * q^67 + 18 * q^70 + 24 * q^73 - 24 * q^78 + 2 * q^79 - 18 * q^81 - 24 * q^82 - 24 * q^85 - 18 * q^87 + 6 * q^88 - 12 * q^91 - 24 * q^94 + 6 * q^96

Decomposition of \(S_{2}^{\mathrm{new}}(42, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
42.2.f.a	$42$	$2$	42.f	21.g	$6$	$4$	$2$	$0.335$	\(\Q(\zeta_{12})\)	None		✓	✓	✓	42.2.f.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-10\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+\zeta_{12}q^{2}+(-\zeta_{12}-\zeta_{12}^{3})q^{3}+\zeta_{12}^{2}q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(42, [\chi])\) into lower level spaces

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