Space of modular forms of level 42, weight 6, and trivial character

• Label: 42.6.a
• Level: $42$
• Weight: $6$
• Character orbit: 42.a
• Rep. character: $\chi_{42}(1,\cdot)$
• Character field: $\Q$
• Dimension: $6$
• Newform subspaces: $6$
• Sturm bound: $48$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 42 = 2 \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	42.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(48\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(42))\).

	Total	New	Old
Modular forms	44	6	38
Cusp forms	36	6	30
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
\(+\)	\(+\)	\(+\)	\(+\)		\(4\)	\(1\)	\(3\)		\(3\)	\(1\)	\(2\)		\(1\)	\(0\)	\(1\)
\(+\)	\(+\)	\(-\)	\(-\)		\(6\)	\(1\)	\(5\)		\(5\)	\(1\)	\(4\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(+\)	\(-\)		\(6\)	\(1\)	\(5\)		\(5\)	\(1\)	\(4\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(-\)	\(+\)		\(6\)	\(1\)	\(5\)		\(5\)	\(1\)	\(4\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(+\)	\(-\)		\(6\)	\(1\)	\(5\)		\(5\)	\(1\)	\(4\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)	\(+\)		\(5\)	\(0\)	\(5\)		\(4\)	\(0\)	\(4\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)	\(+\)		\(6\)	\(0\)	\(6\)		\(5\)	\(0\)	\(5\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(-\)	\(-\)		\(5\)	\(1\)	\(4\)		\(4\)	\(1\)	\(3\)		\(1\)	\(0\)	\(1\)
Plus space			\(+\)		\(21\)	\(2\)	\(19\)		\(17\)	\(2\)	\(15\)		\(4\)	\(0\)	\(4\)
Minus space			\(-\)		\(23\)	\(4\)	\(19\)		\(19\)	\(4\)	\(15\)		\(4\)	\(0\)	\(4\)

Trace form

6 * q - 8 * q^2 + 96 * q^4 + 44 * q^5 - 128 * q^8 + 486 * q^9 + 624 * q^10 + 712 * q^11 - 36 * q^13 - 792 * q^15 + 1536 * q^16 + 1468 * q^17 - 648 * q^18 - 4080 * q^19 + 704 * q^20 + 882 * q^21 + 2880 * q^22 - 2336 * q^23 - 1686 * q^25 + 7024 * q^26 - 12508 * q^29 - 576 * q^30 + 2976 * q^31 - 2048 * q^32 - 720 * q^33 - 12048 * q^34 - 12152 * q^35 + 7776 * q^36 - 1692 * q^37 - 6016 * q^38 - 11160 * q^39 + 9984 * q^40 + 1548 * q^41 + 3528 * q^42 - 6888 * q^43 + 11392 * q^44 + 3564 * q^45 - 22560 * q^46 + 5904 * q^47 + 14406 * q^49 + 7560 * q^50 - 21888 * q^51 - 576 * q^52 + 59652 * q^53 + 65712 * q^55 + 39096 * q^57 + 24720 * q^58 + 54560 * q^59 - 12672 * q^60 - 55476 * q^61 - 92480 * q^62 + 24576 * q^64 + 39576 * q^65 - 39168 * q^66 + 59928 * q^67 + 23488 * q^68 + 94320 * q^69 + 28224 * q^70 - 128912 * q^71 - 10368 * q^72 - 79716 * q^73 - 29296 * q^74 - 37728 * q^75 - 65280 * q^76 - 47824 * q^77 - 3168 * q^78 - 119760 * q^79 + 11264 * q^80 + 39366 * q^81 - 90576 * q^82 - 56144 * q^83 + 14112 * q^84 + 109416 * q^85 + 78944 * q^86 - 138816 * q^87 + 46080 * q^88 + 154140 * q^89 + 50544 * q^90 + 5880 * q^91 - 37376 * q^92 - 170352 * q^93 - 50304 * q^94 - 369440 * q^95 - 2436 * q^97 - 19208 * q^98 + 57672 * q^99

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(42))\) 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
42.6.a.a	$42$	$6$	42.a	1.a	$1$	$1$	$1$	$6.736$	\(\Q\)	None	✓	✓	✓	✓	42.6.a.a	$1$	$0$	\(-4\)	\(-9\)	\(-54\)	\(49\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}-9q^{3}+2^{4}q^{4}-54q^{5}+6^{2}q^{6}+\cdots\)
42.6.a.b	$42$	$6$	42.a	1.a	$1$	$1$	$1$	$6.736$	\(\Q\)	None	✓	✓	✓	✓	42.6.a.b	$1$	$1$	\(-4\)	\(-9\)	\(44\)	\(-49\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}-9q^{3}+2^{4}q^{4}+44q^{5}+6^{2}q^{6}+\cdots\)
42.6.a.c	$42$	$6$	42.a	1.a	$1$	$1$	$1$	$6.736$	\(\Q\)	None	✓	✓	✓	✓	42.6.a.c	$1$	$1$	\(-4\)	\(9\)	\(-72\)	\(49\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}+9q^{3}+2^{4}q^{4}-72q^{5}-6^{2}q^{6}+\cdots\)
42.6.a.d	$42$	$6$	42.a	1.a	$1$	$1$	$1$	$6.736$	\(\Q\)	None	✓	✓	✓	✓	42.6.a.d	$1$	$0$	\(-4\)	\(9\)	\(26\)	\(-49\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}+9q^{3}+2^{4}q^{4}+26q^{5}-6^{2}q^{6}+\cdots\)
42.6.a.e	$42$	$6$	42.a	1.a	$1$	$1$	$1$	$6.736$	\(\Q\)	None	✓	✓	✓	✓	42.6.a.e	$1$	$0$	\(4\)	\(-9\)	\(76\)	\(-49\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}-9q^{3}+2^{4}q^{4}+76q^{5}-6^{2}q^{6}+\cdots\)
42.6.a.f	$42$	$6$	42.a	1.a	$1$	$1$	$1$	$6.736$	\(\Q\)	None	✓	✓	✓	✓	42.6.a.f	$1$	$0$	\(4\)	\(9\)	\(24\)	\(49\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}+9q^{3}+2^{4}q^{4}+24q^{5}+6^{2}q^{6}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(42))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(42)) \simeq \) \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 2}\)