Space of modular forms of level 474, weight 4, and Character 474.e

• Label: 474.4.e
• Level: $474$
• Weight: $4$
• Character orbit: 474.e
• Rep. character: $\chi_{474}(55,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $80$
• Newform subspaces: $4$
• Sturm bound: $320$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 474 = 2 \cdot 3 \cdot 79 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	474.e (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 79 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(320\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(474, [\chi])\).

	Total	New	Old
Modular forms	488	80	408
Cusp forms	472	80	392
Eisenstein series	16	0	16

Trace form

80 * q - 6 * q^3 - 160 * q^4 - 12 * q^6 + 10 * q^7 - 360 * q^9 + 24 * q^10 - 80 * q^11 + 48 * q^12 + 14 * q^13 + 84 * q^15 - 640 * q^16 + 64 * q^17 + 124 * q^19 - 72 * q^21 + 168 * q^22 + 72 * q^23 - 48 * q^24 - 1214 * q^25 + 108 * q^27 + 40 * q^28 + 32 * q^29 - 120 * q^30 - 112 * q^31 + 276 * q^33 - 296 * q^34 - 344 * q^35 - 1440 * q^36 - 414 * q^37 + 1248 * q^38 + 474 * q^39 - 48 * q^40 + 56 * q^41 - 180 * q^42 - 686 * q^43 - 320 * q^44 + 304 * q^46 + 540 * q^47 - 96 * q^48 - 3046 * q^49 - 112 * q^50 - 672 * q^51 - 112 * q^52 + 2148 * q^53 - 108 * q^54 + 246 * q^55 + 2496 * q^57 + 200 * q^58 - 1468 * q^59 - 168 * q^60 + 72 * q^61 - 352 * q^62 + 90 * q^63 + 5120 * q^64 - 3032 * q^65 - 720 * q^66 - 1204 * q^67 - 128 * q^68 - 960 * q^69 + 1524 * q^70 - 368 * q^71 - 1924 * q^73 + 504 * q^74 - 1194 * q^75 + 496 * q^76 - 856 * q^77 - 48 * q^78 - 5714 * q^79 - 3240 * q^81 - 776 * q^82 + 948 * q^83 + 144 * q^84 - 2336 * q^85 - 896 * q^86 + 636 * q^87 - 336 * q^88 + 6832 * q^89 - 108 * q^90 + 3072 * q^91 + 288 * q^92 - 2652 * q^93 - 1264 * q^94 - 3616 * q^95 + 384 * q^96 - 3800 * q^97 - 496 * q^98 - 720 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(474, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
474.4.e.a	$474$	$4$	474.e	79.c	$3$	$18$	$9$	$27.967$	\(\mathbb{Q}[x]/(x^{18} - \cdots)\)	None		✓			474.4.e.a	$2$	$0$	\(-18\)	\(27\)	\(7\)	\(7\)	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q-2\beta _{3}q^{2}+3\beta _{3}q^{3}+(-4+4\beta _{3})q^{4}+\cdots\)
474.4.e.b	$474$	$4$	474.e	79.c	$3$	$20$	$10$	$27.967$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None		✓			474.4.e.b	$2$	$0$	\(20\)	\(-30\)	\(3\)	\(13\)	$2^{2}\cdot 3^{6}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(2+2\beta _{1})q^{2}+(-3-3\beta _{1})q^{3}+4\beta _{1}q^{4}+\cdots\)
474.4.e.c	$474$	$4$	474.e	79.c	$3$	$20$	$10$	$27.967$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None		✓			474.4.e.c	$2$	$0$	\(20\)	\(30\)	\(0\)	\(-8\)	$2^{2}\cdot 3^{3}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(2+2\beta _{2})q^{2}+(3+3\beta _{2})q^{3}+4\beta _{2}q^{4}+\cdots\)
474.4.e.d	$474$	$4$	474.e	79.c	$3$	$22$	$11$	$27.967$		None		✓	✓		474.4.e.d	$2$	$0$	\(-22\)	\(-33\)	\(-10\)	\(-2\)		$\mathrm{SU}(2)[C_{3}]$

Decomposition of \(S_{4}^{\mathrm{old}}(474, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(474, [\chi]) \simeq \) \(S_{4}^{\mathrm{new}}(79, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(158, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(237, [\chi])\)\(^{\oplus 2}\)