Space of modular forms of level 42, weight 4, and trivial character

• Label: 42.4.a
• Level: $42$
• Weight: $4$
• Character orbit: 42.a
• Rep. character: $\chi_{42}(1,\cdot)$
• Character field: $\Q$
• Dimension: $2$
• Newform subspaces: $2$
• Sturm bound: $32$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 42 = 2 \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	42.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(32\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(42))\).

	Total	New	Old
Modular forms	28	2	26
Cusp forms	20	2	18
Eisenstein series	8	0	8

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		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(5\)	\(0\)	\(5\)		\(4\)	\(0\)	\(4\)		\(1\)	\(0\)	\(1\)
\(+\)	\(+\)	\(-\)	\(-\)		\(3\)	\(0\)	\(3\)		\(2\)	\(0\)	\(2\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(+\)	\(-\)		\(3\)	\(0\)	\(3\)		\(2\)	\(0\)	\(2\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(-\)	\(+\)		\(3\)	\(0\)	\(3\)		\(2\)	\(0\)	\(2\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(+\)	\(-\)		\(3\)	\(0\)	\(3\)		\(2\)	\(0\)	\(2\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)	\(+\)		\(4\)	\(1\)	\(3\)		\(3\)	\(1\)	\(2\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)	\(+\)		\(3\)	\(1\)	\(2\)		\(2\)	\(1\)	\(1\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(-\)	\(-\)		\(4\)	\(0\)	\(4\)		\(3\)	\(0\)	\(3\)		\(1\)	\(0\)	\(1\)
Plus space			\(+\)		\(15\)	\(2\)	\(13\)		\(11\)	\(2\)	\(9\)		\(4\)	\(0\)	\(4\)
Minus space			\(-\)		\(13\)	\(0\)	\(13\)		\(9\)	\(0\)	\(9\)		\(4\)	\(0\)	\(4\)

Trace form

2 * q + 4 * q^2 + 8 * q^4 + 20 * q^5 + 16 * q^8 + 18 * q^9 + 40 * q^10 - 80 * q^11 - 76 * q^13 - 48 * q^15 + 32 * q^16 + 4 * q^17 + 36 * q^18 - 32 * q^19 + 80 * q^20 - 42 * q^21 - 160 * q^22 - 104 * q^23 + 78 * q^25 - 152 * q^26 + 140 * q^29 - 96 * q^30 - 16 * q^31 + 64 * q^32 + 192 * q^33 + 8 * q^34 + 112 * q^35 + 72 * q^36 + 364 * q^37 - 64 * q^38 - 24 * q^39 + 160 * q^40 + 468 * q^41 - 84 * q^42 + 376 * q^43 - 320 * q^44 + 180 * q^45 - 208 * q^46 - 96 * q^47 + 98 * q^49 + 156 * q^50 - 24 * q^51 - 304 * q^52 + 492 * q^53 - 1312 * q^55 - 648 * q^57 + 280 * q^58 - 1264 * q^59 - 192 * q^60 - 220 * q^61 - 32 * q^62 + 128 * q^64 - 696 * q^65 + 384 * q^66 - 888 * q^67 + 16 * q^68 + 768 * q^69 + 224 * q^70 - 200 * q^71 + 144 * q^72 + 1428 * q^73 + 728 * q^74 - 960 * q^75 - 128 * q^76 - 448 * q^77 - 48 * q^78 + 608 * q^79 + 320 * q^80 + 162 * q^81 + 936 * q^82 + 1264 * q^83 - 168 * q^84 + 104 * q^85 + 752 * q^86 + 1104 * q^87 - 640 * q^88 + 420 * q^89 + 360 * q^90 + 56 * q^91 - 416 * q^92 - 384 * q^93 - 192 * q^94 + 1408 * q^95 - 1260 * q^97 + 196 * q^98 - 720 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(42))\) 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
42.4.a.a	$42$	$4$	42.a	1.a	$1$	$1$	$1$	$2.478$	\(\Q\)	None	✓	✓	✓	✓	42.4.a.a	$1$	$0$	\(2\)	\(-3\)	\(18\)	\(7\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-3q^{3}+4q^{4}+18q^{5}-6q^{6}+\cdots\)
42.4.a.b	$42$	$4$	42.a	1.a	$1$	$1$	$1$	$2.478$	\(\Q\)	None	✓	✓	✓	✓	42.4.a.b	$1$	$0$	\(2\)	\(3\)	\(2\)	\(-7\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+3q^{3}+4q^{4}+2q^{5}+6q^{6}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(42))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(42)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 2}\)