Space of modular forms of level 90, weight 4, and trivial character

• Label: 90.4.a
• Level: $90$
• Weight: $4$
• Character orbit: 90.a
• Rep. character: $\chi_{90}(1,\cdot)$
• Character field: $\Q$
• Dimension: $5$
• Newform subspaces: $5$
• Sturm bound: $72$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	90.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(72\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(90))\).

	Total	New	Old
Modular forms	62	5	57
Cusp forms	46	5	41
Eisenstein series	16	0	16

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	Dim
\(+\)	\(+\)	\(+\)	$+$	\(1\)
\(+\)	\(-\)	\(+\)	$-$	\(1\)
\(+\)	\(-\)	\(-\)	$+$	\(1\)
\(-\)	\(+\)	\(-\)	$+$	\(1\)
\(-\)	\(-\)	\(+\)	$+$	\(1\)
Plus space			\(+\)	\(4\)
Minus space			\(-\)	\(1\)

Trace form

\( 5 q - 2 q^{2} + 20 q^{4} - 5 q^{5} + 52 q^{7} - 8 q^{8} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(90))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	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
90.4.a.a	$90$	$4$	90.a	1.a	$1$	$1$	$1$	$5.310$	\(\Q\)	None	✓	$1$	$1$	\(-2\)	\(0\)	\(-5\)	\(-4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}-5q^{5}-4q^{7}-8q^{8}+\cdots\)
90.4.a.b	$90$	$4$	90.a	1.a	$1$	$1$	$1$	$5.310$	\(\Q\)	None	✓	$1$	$0$	\(-2\)	\(0\)	\(-5\)	\(14\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}-5q^{5}+14q^{7}-8q^{8}+\cdots\)
90.4.a.c	$90$	$4$	90.a	1.a	$1$	$1$	$1$	$5.310$	\(\Q\)	None	✓	$1$	$0$	\(-2\)	\(0\)	\(5\)	\(-4\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+5q^{5}-4q^{7}-8q^{8}+\cdots\)
90.4.a.d	$90$	$4$	90.a	1.a	$1$	$1$	$1$	$5.310$	\(\Q\)	None	✓	$1$	$0$	\(2\)	\(0\)	\(-5\)	\(32\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}-5q^{5}+2^{5}q^{7}+8q^{8}+\cdots\)
90.4.a.e	$90$	$4$	90.a	1.a	$1$	$1$	$1$	$5.310$	\(\Q\)	None	✓	$1$	$0$	\(2\)	\(0\)	\(5\)	\(14\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+5q^{5}+14q^{7}+8q^{8}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(90))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(90)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 2}\)