Space of modular forms of level 490, weight 4, and Character 490.e

• Label: 490.4.e
• Level: $490$
• Weight: $4$
• Character orbit: 490.e
• Rep. character: $\chi_{490}(361,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $80$
• Newform subspaces: $27$
• Sturm bound: $336$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 490 = 2 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	490.e (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 27 \)
Sturm bound:			\(336\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(3\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(490, [\chi])\).

	Total	New	Old
Modular forms	536	80	456
Cusp forms	472	80	392
Eisenstein series	64	0	64

Trace form

80 * q + 12 * q^3 - 160 * q^4 + 10 * q^5 + 8 * q^6 - 282 * q^9 + 20 * q^10 + 2 * q^11 + 48 * q^12 - 16 * q^13 + 160 * q^15 - 640 * q^16 + 132 * q^17 - 112 * q^18 - 250 * q^19 - 80 * q^20 - 112 * q^22 - 168 * q^23 - 16 * q^24 - 1000 * q^25 + 156 * q^26 - 816 * q^27 + 748 * q^29 + 140 * q^30 - 96 * q^31 + 128 * q^33 + 480 * q^34 + 2256 * q^36 - 980 * q^37 + 704 * q^38 - 1816 * q^39 + 80 * q^40 - 1096 * q^41 - 464 * q^43 + 8 * q^44 + 380 * q^45 + 424 * q^46 - 500 * q^47 - 384 * q^48 + 1296 * q^51 + 32 * q^52 + 1716 * q^53 - 556 * q^54 + 480 * q^55 - 1992 * q^57 + 240 * q^58 - 1140 * q^59 - 320 * q^60 - 1146 * q^61 + 160 * q^62 + 5120 * q^64 - 1210 * q^65 - 2880 * q^66 + 1124 * q^67 + 528 * q^68 - 1684 * q^69 + 2752 * q^71 - 448 * q^72 + 972 * q^73 + 1108 * q^74 + 300 * q^75 + 2000 * q^76 - 2176 * q^78 + 6108 * q^79 + 160 * q^80 - 752 * q^81 - 1536 * q^82 - 968 * q^83 - 4000 * q^85 + 1692 * q^86 - 2580 * q^87 + 224 * q^88 + 162 * q^89 - 1800 * q^90 + 1344 * q^92 + 4684 * q^93 - 1084 * q^94 - 600 * q^95 - 64 * q^96 - 3504 * q^97 + 17396 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(490, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
490.4.e.a	$490$	$4$	490.e	7.c	$3$	$2$	$1$	$28.911$	\(\Q(\sqrt{-3}) \)	None					10.4.a.a	$2$	$0$	\(-2\)	\(-8\)	\(5\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-2\zeta_{6}q^{2}+(-8+8\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots\)
490.4.e.b	$490$	$4$	490.e	7.c	$3$	$2$	$1$	$28.911$	\(\Q(\sqrt{-3}) \)	None					70.4.a.f	$2$	$0$	\(-2\)	\(-7\)	\(5\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-2\zeta_{6}q^{2}+(-7+7\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots\)
490.4.e.c	$490$	$4$	490.e	7.c	$3$	$2$	$1$	$28.911$	\(\Q(\sqrt{-3}) \)	None					70.4.a.e	$2$	$0$	\(-2\)	\(-5\)	\(-5\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-2\zeta_{6}q^{2}+(-5+5\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots\)
490.4.e.d	$490$	$4$	490.e	7.c	$3$	$2$	$1$	$28.911$	\(\Q(\sqrt{-3}) \)	None		✓			490.4.a.l	$2$	$0$	\(-2\)	\(-1\)	\(5\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-2\zeta_{6}q^{2}+(-1+\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots\)
490.4.e.e	$490$	$4$	490.e	7.c	$3$	$2$	$1$	$28.911$	\(\Q(\sqrt{-3}) \)	None		✓			490.4.a.l	$2$	$0$	\(-2\)	\(1\)	\(-5\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-2\zeta_{6}q^{2}+(1-\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots\)
490.4.e.f	$490$	$4$	490.e	7.c	$3$	$2$	$1$	$28.911$	\(\Q(\sqrt{-3}) \)	None					70.4.e.a	$2$	$0$	\(-2\)	\(1\)	\(-5\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-2\zeta_{6}q^{2}+(1-\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots\)
490.4.e.g	$490$	$4$	490.e	7.c	$3$	$2$	$1$	$28.911$	\(\Q(\sqrt{-3}) \)	None					70.4.a.e	$2$	$0$	\(-2\)	\(5\)	\(5\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-2\zeta_{6}q^{2}+(5-5\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots\)
490.4.e.h	$490$	$4$	490.e	7.c	$3$	$2$	$1$	$28.911$	\(\Q(\sqrt{-3}) \)	None					70.4.a.f	$2$	$0$	\(-2\)	\(7\)	\(-5\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-2\zeta_{6}q^{2}+(7-7\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots\)
490.4.e.i	$490$	$4$	490.e	7.c	$3$	$2$	$1$	$28.911$	\(\Q(\sqrt{-3}) \)	None					10.4.a.a	$2$	$1$	\(-2\)	\(8\)	\(-5\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-2\zeta_{6}q^{2}+(8-8\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots\)
490.4.e.j	$490$	$4$	490.e	7.c	$3$	$2$	$1$	$28.911$	\(\Q(\sqrt{-3}) \)	None					70.4.a.a	$2$	$0$	\(2\)	\(-8\)	\(-5\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+2\zeta_{6}q^{2}+(-8+8\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots\)
490.4.e.k	$490$	$4$	490.e	7.c	$3$	$2$	$1$	$28.911$	\(\Q(\sqrt{-3}) \)	None					70.4.a.d	$2$	$0$	\(2\)	\(-4\)	\(-5\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+2\zeta_{6}q^{2}+(-4+4\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots\)
490.4.e.l	$490$	$4$	490.e	7.c	$3$	$2$	$1$	$28.911$	\(\Q(\sqrt{-3}) \)	None					70.4.a.b	$2$	$0$	\(2\)	\(-3\)	\(5\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+2\zeta_{6}q^{2}+(-3+3\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots\)
490.4.e.m	$490$	$4$	490.e	7.c	$3$	$2$	$1$	$28.911$	\(\Q(\sqrt{-3}) \)	None					70.4.e.c	$2$	$0$	\(2\)	\(-1\)	\(-5\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+2\zeta_{6}q^{2}+(-1+\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots\)
490.4.e.n	$490$	$4$	490.e	7.c	$3$	$2$	$1$	$28.911$	\(\Q(\sqrt{-3}) \)	None					70.4.a.c	$2$	$0$	\(2\)	\(-1\)	\(-5\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+2\zeta_{6}q^{2}+(-1+\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots\)
490.4.e.o	$490$	$4$	490.e	7.c	$3$	$2$	$1$	$28.911$	\(\Q(\sqrt{-3}) \)	None					70.4.a.c	$2$	$0$	\(2\)	\(1\)	\(5\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+2\zeta_{6}q^{2}+(1-\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots\)
490.4.e.p	$490$	$4$	490.e	7.c	$3$	$2$	$1$	$28.911$	\(\Q(\sqrt{-3}) \)	None					70.4.a.b	$2$	$0$	\(2\)	\(3\)	\(-5\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+2\zeta_{6}q^{2}+(3-3\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots\)
490.4.e.q	$490$	$4$	490.e	7.c	$3$	$2$	$1$	$28.911$	\(\Q(\sqrt{-3}) \)	None					70.4.a.d	$2$	$0$	\(2\)	\(4\)	\(5\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+2\zeta_{6}q^{2}+(4-4\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots\)
490.4.e.r	$490$	$4$	490.e	7.c	$3$	$2$	$1$	$28.911$	\(\Q(\sqrt{-3}) \)	None					70.4.a.a	$2$	$0$	\(2\)	\(8\)	\(5\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+2\zeta_{6}q^{2}+(8-8\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots\)
490.4.e.s	$490$	$4$	490.e	7.c	$3$	$2$	$1$	$28.911$	\(\Q(\sqrt{-3}) \)	None					70.4.e.b	$2$	$0$	\(2\)	\(10\)	\(-5\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+2\zeta_{6}q^{2}+(10-10\zeta_{6})q^{3}+(-4+4\zeta_{6})q^{4}+\cdots\)
490.4.e.t	$490$	$4$	490.e	7.c	$3$	$4$	$2$	$28.911$	\(\Q(\sqrt{-3}, \sqrt{-59})\)	None		✓			490.4.a.p	$2$	$0$	\(4\)	\(-5\)	\(-10\)	\(0\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q-2\beta _{1}q^{2}+(-3-3\beta _{1}-\beta _{3})q^{3}+(-4+\cdots)q^{4}+\cdots\)
490.4.e.u	$490$	$4$	490.e	7.c	$3$	$4$	$2$	$28.911$	\(\Q(\sqrt{-3}, \sqrt{46})\)	None					70.4.e.d	$2$	$0$	\(4\)	\(-2\)	\(10\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-2\beta _{2}q^{2}+(-1+\beta _{1}-\beta _{2})q^{3}+(-4+\cdots)q^{4}+\cdots\)
490.4.e.v	$490$	$4$	490.e	7.c	$3$	$4$	$2$	$28.911$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None		✓			490.4.a.q	$2$	$0$	\(4\)	\(-2\)	\(10\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(2+2\beta _{2})q^{2}+(3\beta _{1}+\beta _{2}+3\beta _{3})q^{3}+\cdots\)
490.4.e.w	$490$	$4$	490.e	7.c	$3$	$4$	$2$	$28.911$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None		✓			490.4.a.q	$2$	$0$	\(4\)	\(2\)	\(-10\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(2+2\beta _{2})q^{2}+(3\beta _{1}-\beta _{2}+3\beta _{3})q^{3}+\cdots\)
490.4.e.x	$490$	$4$	490.e	7.c	$3$	$4$	$2$	$28.911$	\(\Q(\sqrt{-3}, \sqrt{-59})\)	None		✓			490.4.a.p	$2$	$0$	\(4\)	\(5\)	\(10\)	\(0\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q-2\beta _{1}q^{2}+(3+3\beta _{1}+\beta _{3})q^{3}+(-4+\cdots)q^{4}+\cdots\)
490.4.e.y	$490$	$4$	490.e	7.c	$3$	$6$	$3$	$28.911$	6.0.\(\cdots\).2	None					70.4.e.e	$2$	$0$	\(-6\)	\(4\)	\(15\)	\(0\)	$3^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-2-2\beta _{3})q^{2}+(\beta _{1}-\beta _{3})q^{3}+4\beta _{3}q^{4}+\cdots\)
490.4.e.z	$490$	$4$	490.e	7.c	$3$	$8$	$4$	$28.911$	\(\Q(\sqrt{2}, \sqrt{-3}, \sqrt{113})\)	None		✓			490.4.a.x	$2$	$0$	\(-8\)	\(-10\)	\(-20\)	\(0\)	$7^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+2\beta _{1}q^{2}+(-3-3\beta _{1}+\beta _{5})q^{3}+(-4+\cdots)q^{4}+\cdots\)
490.4.e.ba	$490$	$4$	490.e	7.c	$3$	$8$	$4$	$28.911$	\(\Q(\sqrt{2}, \sqrt{-3}, \sqrt{113})\)	None		✓			490.4.a.x	$2$	$0$	\(-8\)	\(10\)	\(20\)	\(0\)	$7^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+2\beta _{1}q^{2}+(3+3\beta _{1}-\beta _{5})q^{3}+(-4+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(490, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(490, [\chi]) \simeq \) \(S_{4}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(14, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(49, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(70, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(98, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(245, [\chi])\)\(^{\oplus 2}\)