Space of modular forms of level 42, weight 6, and Character 42.e

• Label: 42.6.e
• Level: $42$
• Weight: $6$
• Character orbit: 42.e
• Rep. character: $\chi_{42}(25,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $12$
• Newform subspaces: $4$
• Sturm bound: $48$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 42 = 2 \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	42.e (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(48\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(42, [\chi])\).

	Total	New	Old
Modular forms	88	12	76
Cusp forms	72	12	60
Eisenstein series	16	0	16

Trace form

12 * q - 96 * q^4 - 44 * q^5 - 144 * q^6 - 234 * q^7 - 486 * q^9 + 648 * q^10 + 296 * q^11 - 3312 * q^13 - 704 * q^14 - 396 * q^15 - 1536 * q^16 + 2612 * q^17 + 1020 * q^19 + 1408 * q^20 + 1764 * q^21 - 5232 * q^22 + 4964 * q^23 + 1152 * q^24 - 7404 * q^25 + 880 * q^26 + 9504 * q^28 + 18376 * q^29 - 4176 * q^30 - 18834 * q^31 + 6714 * q^33 - 12384 * q^34 + 200 * q^35 + 15552 * q^36 - 1512 * q^37 + 816 * q^38 + 8028 * q^39 + 10368 * q^40 + 53424 * q^41 + 20520 * q^42 + 34032 * q^43 + 4736 * q^44 - 3564 * q^45 - 18000 * q^46 - 99492 * q^47 - 24054 * q^49 - 109376 * q^50 - 16884 * q^51 + 26496 * q^52 - 10404 * q^53 + 5832 * q^54 - 119028 * q^55 + 30976 * q^56 + 98928 * q^57 - 25896 * q^58 + 73096 * q^59 + 3168 * q^60 - 47592 * q^61 - 18080 * q^62 - 29160 * q^63 + 49152 * q^64 + 58116 * q^65 + 34848 * q^66 - 26916 * q^67 + 41792 * q^68 - 197784 * q^69 + 94008 * q^70 - 616 * q^71 - 2544 * q^73 - 183696 * q^74 + 9432 * q^75 - 32640 * q^76 + 188308 * q^77 + 38304 * q^78 - 28434 * q^79 - 11264 * q^80 - 39366 * q^81 + 75264 * q^82 + 446624 * q^83 + 21312 * q^84 + 421656 * q^85 + 32496 * q^86 - 119574 * q^87 + 41856 * q^88 - 127008 * q^89 - 104976 * q^90 + 292812 * q^91 - 158848 * q^92 - 102456 * q^93 + 108192 * q^94 - 442348 * q^95 + 18432 * q^96 - 342372 * q^97 + 152544 * q^98 - 47952 * q^99

Decomposition of \(S_{6}^{\mathrm{new}}(42, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
42.6.e.a	$42$	$6$	42.e	7.c	$3$	$2$	$1$	$6.736$	\(\Q(\sqrt{-3}) \)	None		✓			42.6.e.a	$2$	$0$	\(-4\)	\(-9\)	\(6\)	\(119\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-4+4\zeta_{6})q^{2}-9\zeta_{6}q^{3}-2^{4}\zeta_{6}q^{4}+\cdots\)
42.6.e.b	$42$	$6$	42.e	7.c	$3$	$2$	$1$	$6.736$	\(\Q(\sqrt{-3}) \)	None		✓			42.6.e.b	$2$	$0$	\(4\)	\(9\)	\(-86\)	\(49\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(4-4\zeta_{6})q^{2}+9\zeta_{6}q^{3}-2^{4}\zeta_{6}q^{4}+\cdots\)
42.6.e.c	$42$	$6$	42.e	7.c	$3$	$4$	$2$	$6.736$	\(\Q(\sqrt{-3}, \sqrt{9601})\)	None		✓			42.6.e.c	$2$	$0$	\(-8\)	\(18\)	\(53\)	\(6\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-4\beta _{2}q^{2}+(9-9\beta _{2})q^{3}+(-2^{4}+2^{4}\beta _{2}+\cdots)q^{4}+\cdots\)
42.6.e.d	$42$	$6$	42.e	7.c	$3$	$4$	$2$	$6.736$	\(\Q(\sqrt{-3}, \sqrt{505})\)	None		✓			42.6.e.d	$2$	$0$	\(8\)	\(-18\)	\(-17\)	\(-408\)	$7$	$\mathrm{SU}(2)[C_{3}]$	\(q+(4-4\beta _{1})q^{2}-9\beta _{1}q^{3}-2^{4}\beta _{1}q^{4}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(42, [\chi])\) into lower level spaces

\( S_{6}^{\mathrm{old}}(42, [\chi]) \simeq \) \(S_{6}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(14, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 2}\)