Space of modular forms of level 42, weight 4, and Character 42.e

• Label: 42.4.e
• Level: $42$
• Weight: $4$
• Character orbit: 42.e
• Rep. character: $\chi_{42}(25,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $8$
• Newform subspaces: $3$
• Sturm bound: $32$
• Trace bound: $3$

Defining parameters

Level:	$N$	$=$	$42 = 2 \cdot 3 \cdot 7$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	42.e (of order $3$ and degree $2$)
Character conductor:	$\operatorname{cond}(\chi)$	$=$	$7$
Character field:			$\Q(\zeta_{3})$
Newform subspaces:			$3$
Sturm bound:			$32$
Trace bound:			$3$
Distinguishing $T_p$:			$5$

Dimensions

The following table gives the dimensions of various subspaces of $M_{4}(42, [\chi])$.

	Total	New	Old
Modular forms	56	8	48
Cusp forms	40	8	32
Eisenstein series	16	0	16

Trace form

8 * q - 6 * q^3 - 16 * q^4 + 16 * q^5 + 24 * q^6 + 28 * q^7 - 36 * q^9 - 52 * q^10 - 28 * q^11 - 24 * q^12 + 12 * q^13 - 32 * q^14 + 84 * q^15 - 64 * q^16 + 260 * q^17 - 50 * q^19 - 128 * q^20 - 36 * q^21 + 424 * q^22 - 148 * q^23 - 48 * q^24 - 446 * q^25 - 152 * q^26 + 108 * q^27 - 104 * q^28 - 104 * q^29 + 24 * q^30 + 156 * q^31 - 138 * q^33 + 304 * q^34 + 860 * q^35 + 288 * q^36 + 94 * q^37 - 72 * q^38 - 546 * q^39 - 208 * q^40 - 936 * q^41 + 252 * q^42 - 212 * q^43 - 112 * q^44 + 144 * q^45 - 296 * q^46 - 72 * q^47 + 192 * q^48 + 1220 * q^49 + 1696 * q^50 + 276 * q^51 - 24 * q^52 + 372 * q^53 - 108 * q^54 - 380 * q^55 - 224 * q^56 - 948 * q^57 - 412 * q^58 + 112 * q^59 - 168 * q^60 - 1764 * q^61 - 3536 * q^62 - 18 * q^63 + 512 * q^64 - 432 * q^65 - 528 * q^66 + 2 * q^67 + 1040 * q^68 + 1896 * q^69 + 844 * q^70 + 2432 * q^71 + 746 * q^73 - 216 * q^74 - 738 * q^75 + 400 * q^76 + 532 * q^77 - 1200 * q^78 + 2280 * q^79 + 256 * q^80 - 324 * q^81 + 768 * q^82 - 3256 * q^83 + 648 * q^84 - 2312 * q^85 + 216 * q^86 + 462 * q^87 - 848 * q^88 + 864 * q^89 + 936 * q^90 - 2106 * q^91 + 1184 * q^92 + 1578 * q^93 - 2160 * q^94 + 2888 * q^95 - 192 * q^96 + 4340 * q^97 - 240 * q^98 + 504 * q^99

Decomposition of $S_{4}^{\mathrm{new}}(42, [\chi])$ 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
42.4.e.a	$42$	$4$	42.e	7.c	$3$	$2$	$1$	$2.478$	$\Q(\sqrt{-3})$	None		✓			42.4.e.a	$2$	$0$	$2$	$-3$	$6$	$-7$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(2-2\zeta_{6})q^{2}-3\zeta_{6}q^{3}-4\zeta_{6}q^{4}+\cdots$
42.4.e.b	$42$	$4$	42.e	7.c	$3$	$2$	$1$	$2.478$	$\Q(\sqrt{-3})$	None		✓			42.4.e.b	$2$	$0$	$2$	$3$	$15$	$35$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(2-2\zeta_{6})q^{2}+3\zeta_{6}q^{3}-4\zeta_{6}q^{4}+\cdots$
42.4.e.c	$42$	$4$	42.e	7.c	$3$	$4$	$2$	$2.478$	$\Q(\sqrt{-3}, \sqrt{1345})$	None		✓	✓		42.4.e.c	$2$	$0$	$-4$	$-6$	$-5$	$0$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q-2\beta _{2}q^{2}+(-3+3\beta _{2})q^{3}+(-4+4\beta _{2}+\cdots)q^{4}+\cdots$

Decomposition of $S_{4}^{\mathrm{old}}(42, [\chi])$ into lower level spaces

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