Space of modular forms of level 42, weight 2, and trivial character

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

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(42))\).

	Total	New	Old
Modular forms	12	1	11
Cusp forms	5	1	4
Eisenstein series	7	0	7

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)	\(3\)	\(7\)	Fricke	Dim
\(-\)	\(+\)	\(+\)	\(-\)	\(1\)
Plus space			\(+\)	\(0\)
Minus space			\(-\)	\(1\)

Trace form

q + q^2 - q^3 + q^4 - 2 * q^5 - q^6 - q^7 + q^8 + q^9 - 2 * q^10 - 4 * q^11 - q^12 + 6 * q^13 - q^14 + 2 * q^15 + q^16 + 2 * q^17 + q^18 - 4 * q^19 - 2 * q^20 + q^21 - 4 * q^22 + 8 * q^23 - q^24 - q^25 + 6 * q^26 - q^27 - q^28 - 2 * q^29 + 2 * q^30 + q^32 + 4 * q^33 + 2 * q^34 + 2 * q^35 + q^36 - 10 * q^37 - 4 * q^38 - 6 * q^39 - 2 * q^40 - 6 * q^41 + q^42 - 4 * q^43 - 4 * q^44 - 2 * q^45 + 8 * q^46 - q^48 + q^49 - q^50 - 2 * q^51 + 6 * q^52 + 6 * q^53 - q^54 + 8 * q^55 - q^56 + 4 * q^57 - 2 * q^58 + 4 * q^59 + 2 * q^60 + 6 * q^61 - q^63 + q^64 - 12 * q^65 + 4 * q^66 + 4 * q^67 + 2 * q^68 - 8 * q^69 + 2 * q^70 + 8 * q^71 + q^72 + 10 * q^73 - 10 * q^74 + q^75 - 4 * q^76 + 4 * q^77 - 6 * q^78 - 2 * q^80 + q^81 - 6 * q^82 - 4 * q^83 + q^84 - 4 * q^85 - 4 * q^86 + 2 * q^87 - 4 * q^88 - 6 * q^89 - 2 * q^90 - 6 * q^91 + 8 * q^92 + 8 * q^95 - q^96 - 14 * q^97 + q^98 - 4 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(42))\) 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					A-L signs			Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	3	7
42.2.a.a	$42$	$2$	42.a	1.a	$1$	$1$	$1$	$0.335$	\(\Q\)	None	✓	✓	✓	✓	42.2.a.a	$1$	$0$	\(1\)	\(-1\)	\(-2\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}-2q^{5}-q^{6}-q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(42))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(42)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 2}\)