Space of modular forms of level 90, weight 7, and Character 90.g

• Label: 90.7.g
• Level: $90$
• Weight: $7$
• Character orbit: 90.g
• Rep. character: $\chi_{90}(37,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $30$
• Newform subspaces: $6$
• Sturm bound: $126$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	90.g (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(126\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{7}(90, [\chi])\).

	Total	New	Old
Modular forms	232	30	202
Cusp forms	200	30	170
Eisenstein series	32	0	32

Trace form

30 * q + 8 * q^2 + 180 * q^5 + 120 * q^7 - 256 * q^8 - 408 * q^10 + 2776 * q^11 + 3786 * q^13 - 30720 * q^16 - 6638 * q^17 + 8512 * q^20 + 20544 * q^22 - 51680 * q^23 + 23634 * q^25 + 49584 * q^26 - 3840 * q^28 + 5904 * q^31 - 8192 * q^32 - 11288 * q^35 + 197094 * q^37 + 80224 * q^38 - 9984 * q^40 + 181912 * q^41 - 238992 * q^43 + 78048 * q^46 - 188160 * q^47 - 393032 * q^50 + 121152 * q^52 + 836654 * q^53 + 259224 * q^55 + 128000 * q^56 + 90912 * q^58 - 1065912 * q^61 - 764896 * q^62 - 1134426 * q^65 + 702624 * q^67 + 212416 * q^68 + 351264 * q^70 + 2875856 * q^71 - 392202 * q^73 - 790272 * q^76 - 4789240 * q^77 - 184320 * q^80 + 36768 * q^82 + 766392 * q^83 - 293454 * q^85 + 3095328 * q^86 - 657408 * q^88 - 1953600 * q^91 - 1653760 * q^92 - 2580184 * q^95 + 6872598 * q^97 + 3320776 * q^98

Decomposition of \(S_{7}^{\mathrm{new}}(90, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
90.7.g.a	$90$	$7$	90.g	5.c	$4$	$2$	$1$	$20.705$	\(\Q(\sqrt{-1}) \)	None					10.7.c.a	$2$	$0$	\(8\)	\(0\)	\(150\)	\(-494\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(4 i+4)q^{2}+32 i q^{4}+(-100 i+75)q^{5}+\cdots\)
90.7.g.b	$90$	$7$	90.g	5.c	$4$	$4$	$2$	$20.705$	\(\Q(i, \sqrt{129})\)	None					10.7.c.b	$2$	$0$	\(-16\)	\(0\)	\(-330\)	\(-202\)	$2\cdot 5^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-4-4\beta _{1})q^{2}+2^{5}\beta _{1}q^{4}+(-82+\cdots)q^{5}+\cdots\)
90.7.g.c	$90$	$7$	90.g	5.c	$4$	$4$	$2$	$20.705$	\(\Q(i, \sqrt{6})\)	None					30.7.f.a	$2$	$0$	\(-16\)	\(0\)	\(420\)	\(596\)	$3^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-4-4\beta _{2})q^{2}+2^{5}\beta _{2}q^{4}+(105+\cdots)q^{5}+\cdots\)
90.7.g.d	$90$	$7$	90.g	5.c	$4$	$6$	$3$	$20.705$	\(\mathbb{Q}[x]/(x^{6} + \cdots)\)	None		✓			90.7.g.d	$2$	$0$	\(-24\)	\(0\)	\(6\)	\(-84\)	$2\cdot 3^{2}\cdot 5$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-4+4\beta _{1})q^{2}-2^{5}\beta _{1}q^{4}+(1-12\beta _{1}+\cdots)q^{5}+\cdots\)
90.7.g.e	$90$	$7$	90.g	5.c	$4$	$6$	$3$	$20.705$	\(\mathbb{Q}[x]/(x^{6} + \cdots)\)	None		✓			90.7.g.d	$2$	$0$	\(24\)	\(0\)	\(-6\)	\(-84\)	$2\cdot 3^{2}\cdot 5$	$\mathrm{SU}(2)[C_{4}]$	\(q+(4-4\beta _{1})q^{2}-2^{5}\beta _{1}q^{4}+(-1+12\beta _{1}+\cdots)q^{5}+\cdots\)
90.7.g.f	$90$	$7$	90.g	5.c	$4$	$8$	$4$	$20.705$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None			✓		30.7.f.b	$2$	$0$	\(32\)	\(0\)	\(-60\)	\(388\)	$2^{5}\cdot 3^{8}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(4+4\beta _{1})q^{2}+2^{5}\beta _{1}q^{4}+(-7+23\beta _{1}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{7}^{\mathrm{old}}(90, [\chi])\) into lower level spaces

\( S_{7}^{\mathrm{old}}(90, [\chi]) \simeq \) \(S_{7}^{\mathrm{new}}(5, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 2}\)