Space of modular forms of level 90, weight 8, and Character 90.c

• Label: 90.8.c
• Level: $90$
• Weight: $8$
• Character orbit: 90.c
• Rep. character: $\chi_{90}(19,\cdot)$
• Character field: $\Q$
• Dimension: $18$
• Newform subspaces: $4$
• Sturm bound: $144$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	90.c (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(144\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(90, [\chi])\).

	Total	New	Old
Modular forms	134	18	116
Cusp forms	118	18	100
Eisenstein series	16	0	16

Trace form

18 * q - 1152 * q^4 - 224 * q^5 + 1104 * q^10 + 668 * q^11 - 2464 * q^14 + 73728 * q^16 - 69648 * q^19 + 14336 * q^20 - 221682 * q^25 - 266976 * q^26 + 241704 * q^29 - 318288 * q^31 - 257568 * q^34 + 191996 * q^35 - 70656 * q^40 - 1440148 * q^41 - 42752 * q^44 - 1179456 * q^46 - 4486506 * q^49 - 1145024 * q^50 + 1512 * q^55 + 157696 * q^56 + 3334100 * q^59 + 941988 * q^61 - 4718592 * q^64 - 9450108 * q^65 - 335808 * q^70 - 3472632 * q^71 + 10493600 * q^74 + 4457472 * q^76 - 7286400 * q^79 - 917504 * q^80 + 26131212 * q^85 - 14532800 * q^86 + 61574908 * q^89 + 41015256 * q^91 - 20186112 * q^94 - 14199504 * q^95

Decomposition of \(S_{8}^{\mathrm{new}}(90, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
90.8.c.a	$90$	$8$	90.c	5.b	$2$	$2$	$2$	$28.115$	\(\Q(\sqrt{-1}) \)	None					30.8.c.a	$2$	$0$	\(0\)	\(0\)	\(100\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+8 i q^{2}-64 q^{4}+(275 i+50)q^{5}+\cdots\)
90.8.c.b	$90$	$8$	90.c	5.b	$2$	$4$	$4$	$28.115$	\(\Q(i, \sqrt{2641})\)	None					30.8.c.b	$2$	$0$	\(0\)	\(0\)	\(-264\)	\(0\)	$2^{4}\cdot 5$	$\mathrm{SU}(2)[C_{2}]$	\(q+4\beta _{1}q^{2}-2^{6}q^{4}+(-66-44\beta _{1}+\cdots)q^{5}+\cdots\)
90.8.c.c	$90$	$8$	90.c	5.b	$2$	$4$	$4$	$28.115$	\(\Q(i, \sqrt{31})\)	None					10.8.b.a	$2$	$0$	\(0\)	\(0\)	\(-60\)	\(0\)	$2^{9}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}-2^{6}q^{4}+(-15+4\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)
90.8.c.d	$90$	$8$	90.c	5.b	$2$	$8$	$8$	$28.115$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		✓	✓		90.8.c.d	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{28}\cdot 3^{4}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}-2^{6}q^{4}+(-6\beta _{1}+\beta _{3})q^{5}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(90, [\chi])\) into lower level spaces

\( S_{8}^{\mathrm{old}}(90, [\chi]) \simeq \) \(S_{8}^{\mathrm{new}}(5, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 2}\)