Space of modular forms of level 90, weight 2, and trivial character

• Label: 90.2.a
• Level: $90$
• Weight: $2$
• Character orbit: 90.a
• Rep. character: $\chi_{90}(1,\cdot)$
• Character field: $\Q$
• Dimension: $3$
• Newform subspaces: $3$
• Sturm bound: $36$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	90.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(36\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(90))\).

	Total	New	Old
Modular forms	26	3	23
Cusp forms	11	3	8
Eisenstein series	15	0	15

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)	\(3\)	\(5\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(2\)	\(0\)	\(2\)		\(1\)	\(0\)	\(1\)		\(1\)	\(0\)	\(1\)
\(+\)	\(+\)	\(-\)	\(-\)		\(4\)	\(1\)	\(3\)		\(2\)	\(1\)	\(1\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(+\)	\(-\)		\(4\)	\(0\)	\(4\)		\(2\)	\(0\)	\(2\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)	\(+\)		\(3\)	\(0\)	\(3\)		\(1\)	\(0\)	\(1\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(+\)	\(-\)		\(3\)	\(1\)	\(2\)		\(1\)	\(1\)	\(0\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)	\(+\)		\(3\)	\(0\)	\(3\)		\(1\)	\(0\)	\(1\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)	\(+\)		\(3\)	\(0\)	\(3\)		\(1\)	\(0\)	\(1\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(-\)	\(-\)		\(4\)	\(1\)	\(3\)		\(2\)	\(1\)	\(1\)		\(2\)	\(0\)	\(2\)
Plus space			\(+\)		\(11\)	\(0\)	\(11\)		\(4\)	\(0\)	\(4\)		\(7\)	\(0\)	\(7\)
Minus space			\(-\)		\(15\)	\(3\)	\(12\)		\(7\)	\(3\)	\(4\)		\(8\)	\(0\)	\(8\)

Trace form

3 * q + q^2 + 3 * q^4 + q^5 + q^8 - q^10 - 6 * q^13 - 4 * q^14 + 3 * q^16 - 6 * q^17 - 12 * q^19 + q^20 - 12 * q^22 + 3 * q^25 + 2 * q^26 + 6 * q^29 + q^32 + 6 * q^34 - 4 * q^35 + 18 * q^37 - 4 * q^38 - q^40 + 6 * q^41 + 12 * q^43 + 3 * q^49 + q^50 - 6 * q^52 + 6 * q^53 + 12 * q^55 - 4 * q^56 + 18 * q^58 - 6 * q^61 + 8 * q^62 + 3 * q^64 + 2 * q^65 - 12 * q^67 - 6 * q^68 - 8 * q^70 - 18 * q^73 + 2 * q^74 - 12 * q^76 + q^80 + 6 * q^82 - 12 * q^83 - 18 * q^85 - 4 * q^86 - 12 * q^88 - 18 * q^89 - 24 * q^91 - 4 * q^95 + 6 * q^97 + 9 * q^98

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(90))\) 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					A-L signs			Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	3	5
90.2.a.a	$90$	$2$	90.a	1.a	$1$	$1$	$1$	$0.719$	\(\Q\)	None	✓	✓	✓	✓	90.2.a.a	$1$	$0$	\(-1\)	\(0\)	\(1\)	\(2\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+q^{5}+2q^{7}-q^{8}-q^{10}+\cdots\)
90.2.a.b	$90$	$2$	90.a	1.a	$1$	$1$	$1$	$0.719$	\(\Q\)	None	✓	✓	✓	✓	90.2.a.a	$1$	$0$	\(1\)	\(0\)	\(-1\)	\(2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-q^{5}+2q^{7}+q^{8}-q^{10}+\cdots\)
90.2.a.c	$90$	$2$	90.a	1.a	$1$	$1$	$1$	$0.719$	\(\Q\)	None	✓		✓	✓	30.2.a.a	$1$	$0$	\(1\)	\(0\)	\(1\)	\(-4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+q^{5}-4q^{7}+q^{8}+q^{10}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(90))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(90)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 2}\)