Space of modular forms of level 90, weight 5, and Character 90.g

• Label: 90.5.g
• Level: $90$
• Weight: $5$
• Character orbit: 90.g
• Rep. character: $\chi_{90}(37,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $20$
• Newform subspaces: $6$
• Sturm bound: $90$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 5 \)
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:			\(90\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{5}(90, [\chi])\).

	Total	New	Old
Modular forms	160	20	140
Cusp forms	128	20	108
Eisenstein series	32	0	32

Trace form

20 * q - 60 * q^5 - 140 * q^7 + 32 * q^10 - 456 * q^11 - 300 * q^13 - 1280 * q^16 + 780 * q^17 + 192 * q^20 - 1280 * q^22 - 1860 * q^23 - 2876 * q^25 - 1344 * q^26 + 1120 * q^28 + 3656 * q^31 + 5052 * q^35 - 300 * q^37 - 4800 * q^38 + 256 * q^40 - 7512 * q^41 + 5460 * q^43 - 1408 * q^46 + 16620 * q^47 + 7488 * q^50 - 2400 * q^52 + 780 * q^53 + 4584 * q^55 + 2240 * q^58 + 28872 * q^61 + 1920 * q^62 - 996 * q^65 - 9740 * q^67 - 6240 * q^68 - 16576 * q^70 - 26376 * q^71 - 3820 * q^73 + 4352 * q^76 + 9480 * q^77 + 3840 * q^80 + 21760 * q^82 + 11340 * q^83 + 18916 * q^85 + 23232 * q^86 + 10240 * q^88 - 72600 * q^91 - 14880 * q^92 - 35424 * q^95 - 50700 * q^97 - 42240 * q^98

Decomposition of \(S_{5}^{\mathrm{new}}(90, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
90.5.g.a	$90$	$5$	90.g	5.c	$4$	$2$	$1$	$9.303$	\(\Q(\sqrt{-1}) \)	None					10.5.c.b	$2$	$0$	\(-4\)	\(0\)	\(30\)	\(-38\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-2 i-2)q^{2}+8 i q^{4}+(20 i+15)q^{5}+\cdots\)
90.5.g.b	$90$	$5$	90.g	5.c	$4$	$2$	$1$	$9.303$	\(\Q(\sqrt{-1}) \)	None					10.5.c.a	$2$	$0$	\(4\)	\(0\)	\(30\)	\(58\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(2 i+2)q^{2}+8 i q^{4}+(-20 i+15)q^{5}+\cdots\)
90.5.g.c	$90$	$5$	90.g	5.c	$4$	$4$	$2$	$9.303$	\(\Q(i, \sqrt{6})\)	None					30.5.f.b	$2$	$0$	\(-8\)	\(0\)	\(-36\)	\(68\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-2+2\beta _{2})q^{2}-8\beta _{2}q^{4}+(-9-11\beta _{1}+\cdots)q^{5}+\cdots\)
90.5.g.d	$90$	$5$	90.g	5.c	$4$	$4$	$2$	$9.303$	\(\Q(i, \sqrt{26})\)	None		✓			90.5.g.d	$2$	$0$	\(-8\)	\(0\)	\(-24\)	\(-100\)	$3^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-2-2\beta _{2})q^{2}+8\beta _{2}q^{4}+(-6-2\beta _{1}+\cdots)q^{5}+\cdots\)
90.5.g.e	$90$	$5$	90.g	5.c	$4$	$4$	$2$	$9.303$	\(\Q(i, \sqrt{6})\)	None					30.5.f.a	$2$	$0$	\(8\)	\(0\)	\(-84\)	\(-28\)	$3^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(2-2\beta _{2})q^{2}-8\beta _{2}q^{4}+(-21+\beta _{1}+\cdots)q^{5}+\cdots\)
90.5.g.f	$90$	$5$	90.g	5.c	$4$	$4$	$2$	$9.303$	\(\Q(i, \sqrt{26})\)	None		✓			90.5.g.d	$2$	$0$	\(8\)	\(0\)	\(24\)	\(-100\)	$3^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(2+2\beta _{2})q^{2}+8\beta _{2}q^{4}+(6-2\beta _{1}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{5}^{\mathrm{old}}(90, [\chi])\) into lower level spaces

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