Space of modular forms of level 490, weight 3, and Character 490.f

• Label: 490.3.f
• Level: $490$
• Weight: $3$
• Character orbit: 490.f
• Rep. character: $\chi_{490}(197,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $82$
• Newform subspaces: $17$
• Sturm bound: $252$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 490 = 2 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	490.f (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 17 \)
Sturm bound:			\(252\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(3\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(490, [\chi])\).

	Total	New	Old
Modular forms	368	82	286
Cusp forms	304	82	222
Eisenstein series	64	0	64

Trace form

82 * q - 2 * q^2 - 4 * q^3 - 8 * q^5 - 8 * q^6 + 4 * q^8 + 22 * q^10 + 16 * q^11 - 8 * q^12 + 6 * q^13 - 52 * q^15 - 328 * q^16 - 2 * q^17 - 62 * q^18 + 20 * q^20 - 16 * q^22 + 76 * q^23 + 86 * q^25 + 52 * q^26 + 176 * q^27 - 72 * q^31 + 8 * q^32 + 120 * q^33 + 508 * q^36 - 206 * q^37 - 104 * q^38 - 20 * q^40 - 240 * q^41 - 92 * q^43 - 126 * q^45 - 136 * q^46 + 28 * q^47 + 16 * q^48 - 142 * q^50 + 376 * q^51 + 12 * q^52 + 18 * q^53 + 328 * q^55 - 264 * q^57 - 192 * q^58 + 24 * q^60 + 288 * q^61 + 232 * q^62 - 326 * q^65 - 64 * q^66 + 92 * q^67 + 4 * q^68 - 72 * q^71 - 124 * q^72 + 106 * q^73 + 180 * q^75 - 80 * q^76 + 472 * q^78 + 32 * q^80 - 1010 * q^81 - 240 * q^82 - 564 * q^83 + 6 * q^85 + 440 * q^86 + 408 * q^87 + 32 * q^88 + 54 * q^90 + 152 * q^92 + 376 * q^93 + 240 * q^95 + 32 * q^96 + 514 * q^97

Decomposition of \(S_{3}^{\mathrm{new}}(490, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
490.3.f.a	$490$	$3$	490.f	5.c	$4$	$2$	$1$	$13.352$	\(\Q(\sqrt{-1}) \)	None		✓			490.3.f.a	$2$	$0$	\(-2\)	\(-8\)	\(-6\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-i-1)q^{2}+(4 i-4)q^{3}+2 i q^{4}+\cdots\)
490.3.f.b	$490$	$3$	490.f	5.c	$4$	$2$	$1$	$13.352$	\(\Q(\sqrt{-1}) \)	None					10.3.c.a	$2$	$0$	\(-2\)	\(4\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-i-1)q^{2}+(-2 i+2)q^{3}+2 i q^{4}+\cdots\)
490.3.f.c	$490$	$3$	490.f	5.c	$4$	$2$	$1$	$13.352$	\(\Q(\sqrt{-1}) \)	None		✓			490.3.f.a	$2$	$0$	\(-2\)	\(8\)	\(6\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-i-1)q^{2}+(-4 i+4)q^{3}+2 i q^{4}+\cdots\)
490.3.f.d	$490$	$3$	490.f	5.c	$4$	$4$	$2$	$13.352$	\(\Q(i, \sqrt{14})\)	None					70.3.f.a	$2$	$0$	\(-4\)	\(-4\)	\(-12\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-1-\beta _{2})q^{2}+(-1+\beta _{2}+\beta _{3})q^{3}+\cdots\)
490.3.f.e	$490$	$3$	490.f	5.c	$4$	$4$	$2$	$13.352$	\(\Q(i, \sqrt{30})\)	None					70.3.l.a	$2$	$0$	\(-4\)	\(-4\)	\(12\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-1+\beta _{2})q^{2}+(-1+\beta _{1}-\beta _{2})q^{3}+\cdots\)
490.3.f.f	$490$	$3$	490.f	5.c	$4$	$4$	$2$	$13.352$	\(\Q(i, \sqrt{11})\)	None					70.3.l.b	$2$	$0$	\(-4\)	\(-2\)	\(6\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-1-\beta _{2})q^{2}+(-1+\beta _{1}-\beta _{3})q^{3}+\cdots\)
490.3.f.g	$490$	$3$	490.f	5.c	$4$	$4$	$2$	$13.352$	\(\Q(\zeta_{8})\)	None		✓			490.3.f.g	$4$	$0$	\(-4\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-1+\zeta_{8}^{2})q^{2}+5\zeta_{8}q^{3}-2\zeta_{8}^{2}q^{4}+\cdots\)
490.3.f.h	$490$	$3$	490.f	5.c	$4$	$4$	$2$	$13.352$	\(\Q(\zeta_{8})\)	None		✓			490.3.f.h	$4$	$0$	\(-4\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-1+\zeta_{8}^{2})q^{2}+\zeta_{8}q^{3}-2\zeta_{8}^{2}q^{4}+\cdots\)
490.3.f.i	$490$	$3$	490.f	5.c	$4$	$4$	$2$	$13.352$	\(\Q(\zeta_{8})\)	None		✓			490.3.f.i	$4$	$0$	\(-4\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-1+\zeta_{8}^{2})q^{2}+4\zeta_{8}q^{3}-2\zeta_{8}^{2}q^{4}+\cdots\)
490.3.f.j	$490$	$3$	490.f	5.c	$4$	$4$	$2$	$13.352$	\(\Q(\zeta_{8})\)	None		✓			490.3.f.j	$4$	$0$	\(-4\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-1+\zeta_{8}^{2})q^{2}+\zeta_{8}q^{3}-2\zeta_{8}^{2}q^{4}+\cdots\)
490.3.f.k	$490$	$3$	490.f	5.c	$4$	$4$	$2$	$13.352$	\(\Q(i, \sqrt{11})\)	None					70.3.l.b	$2$	$0$	\(-4\)	\(2\)	\(-6\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-1+\beta _{2})q^{2}+(-\beta _{1}+\beta _{2}-\beta _{3})q^{3}+\cdots\)
490.3.f.l	$490$	$3$	490.f	5.c	$4$	$4$	$2$	$13.352$	\(\Q(i, \sqrt{30})\)	None					70.3.l.a	$2$	$0$	\(-4\)	\(4\)	\(-12\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-1+\beta _{2})q^{2}+(1+\beta _{1}+\beta _{2})q^{3}+\cdots\)
490.3.f.m	$490$	$3$	490.f	5.c	$4$	$4$	$2$	$13.352$	\(\Q(\zeta_{8})\)	None		✓			490.3.f.m	$4$	$0$	\(4\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(1-\zeta_{8}^{2})q^{2}+\zeta_{8}q^{3}-2\zeta_{8}^{2}q^{4}+\cdots\)
490.3.f.n	$490$	$3$	490.f	5.c	$4$	$8$	$4$	$13.352$	8.0.\(\cdots\).20	None					70.3.f.b	$2$	$0$	\(8\)	\(-4\)	\(4\)	\(0\)	$2^{3}\cdot 5$	$\mathrm{SU}(2)[C_{4}]$	\(q+(1-\beta _{1})q^{2}+(-\beta _{1}+\beta _{3})q^{3}-2\beta _{1}q^{4}+\cdots\)
490.3.f.o	$490$	$3$	490.f	5.c	$4$	$8$	$4$	$13.352$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None					70.3.l.c	$2$	$0$	\(8\)	\(-2\)	\(2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(1+\beta _{2})q^{2}-\beta _{1}q^{3}+2\beta _{2}q^{4}+(-\beta _{2}+\cdots)q^{5}+\cdots\)
490.3.f.p	$490$	$3$	490.f	5.c	$4$	$8$	$4$	$13.352$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None					70.3.l.c	$2$	$0$	\(8\)	\(2\)	\(-2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(1-\beta _{2})q^{2}+\beta _{4}q^{3}-2\beta _{2}q^{4}+(-\beta _{2}+\cdots)q^{5}+\cdots\)
490.3.f.q	$490$	$3$	490.f	5.c	$4$	$12$	$6$	$13.352$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		✓	✓		490.3.f.q	$4$	$0$	\(12\)	\(0\)	\(0\)	\(0\)	$2^{10}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(1-\beta _{4})q^{2}+\beta _{7}q^{3}-2\beta _{4}q^{4}+(\beta _{5}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(490, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(490, [\chi]) \simeq \) \(S_{3}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(70, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(245, [\chi])\)\(^{\oplus 2}\)