Space of modular forms of level 90, weight 2, and Character 90.e

• Label: 90.2.e
• Level: $90$
• Weight: $2$
• Character orbit: 90.e
• Rep. character: $\chi_{90}(31,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $8$
• Newform subspaces: $3$
• Sturm bound: $36$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	90.e (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 9 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(36\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(90, [\chi])\).

	Total	New	Old
Modular forms	44	8	36
Cusp forms	28	8	20
Eisenstein series	16	0	16

Trace form

8 * q + 2 * q^2 + 2 * q^3 - 4 * q^4 + 2 * q^5 - 2 * q^6 + 4 * q^7 - 4 * q^8 - 4 * q^9 - 6 * q^11 - 4 * q^12 + 4 * q^13 - 2 * q^14 + 4 * q^15 - 4 * q^16 - 12 * q^17 + 4 * q^18 + 4 * q^19 + 2 * q^20 + 4 * q^21 - 6 * q^22 - 12 * q^23 + 4 * q^24 - 4 * q^25 + 16 * q^26 - 16 * q^27 - 8 * q^28 - 6 * q^29 + 2 * q^30 + 4 * q^31 + 2 * q^32 - 6 * q^33 - 6 * q^34 - 8 * q^35 + 2 * q^36 - 8 * q^37 + 10 * q^38 + 28 * q^39 + 12 * q^41 + 20 * q^42 - 2 * q^43 + 12 * q^44 + 4 * q^45 + 12 * q^46 + 12 * q^47 + 2 * q^48 - 6 * q^49 + 2 * q^50 - 18 * q^51 + 4 * q^52 + 24 * q^53 - 8 * q^54 - 2 * q^56 - 26 * q^57 - 6 * q^59 + 4 * q^60 + 22 * q^61 - 8 * q^62 + 40 * q^63 + 8 * q^64 + 4 * q^65 - 36 * q^66 + 22 * q^67 + 6 * q^68 - 42 * q^69 - 6 * q^70 - 24 * q^71 + 10 * q^72 - 44 * q^73 - 20 * q^74 + 2 * q^75 - 2 * q^76 + 36 * q^77 - 4 * q^78 - 8 * q^79 - 4 * q^80 - 4 * q^81 + 12 * q^82 - 12 * q^83 + 10 * q^84 - 12 * q^85 - 14 * q^86 + 36 * q^87 - 6 * q^88 + 60 * q^89 - 16 * q^90 - 40 * q^91 - 12 * q^92 - 20 * q^93 - 6 * q^94 - 8 * q^95 - 2 * q^96 - 2 * q^97 - 36 * q^98 - 24 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(90, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
90.2.e.a	$90$	$2$	90.e	9.c	$3$	$2$	$1$	$0.719$	\(\Q(\sqrt{-3}) \)	None		✓			90.2.e.a	$2$	$0$	\(-1\)	\(3\)	\(1\)	\(1\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\zeta_{6})q^{2}+(2-\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots\)
90.2.e.b	$90$	$2$	90.e	9.c	$3$	$2$	$1$	$0.719$	\(\Q(\sqrt{-3}) \)	None		✓			90.2.e.b	$2$	$0$	\(1\)	\(-3\)	\(-1\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{2}+(-1-\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots\)
90.2.e.c	$90$	$2$	90.e	9.c	$3$	$4$	$2$	$0.719$	\(\Q(\sqrt{-3}, \sqrt{-11})\)	None		✓	✓		90.2.e.c	$2$	$0$	\(2\)	\(2\)	\(2\)	\(-1\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{2}q^{2}+(1-\beta _{1}+\beta _{3})q^{3}+(-1+\beta _{2}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(90, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(90, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 2}\)