Space of modular forms of level 90, weight 6, and trivial character

• Label: 90.6.a
• Level: $90$
• Weight: $6$
• Character orbit: 90.a
• Rep. character: $\chi_{90}(1,\cdot)$
• Character field: $\Q$
• Dimension: $7$
• Newform subspaces: $7$
• Sturm bound: $108$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	90.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(108\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(90))\).

	Total	New	Old
Modular forms	98	7	91
Cusp forms	82	7	75
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
\(+\)	\(+\)	\(-\)	$-$	\(1\)
\(+\)	\(-\)	\(+\)	$-$	\(1\)
\(+\)	\(-\)	\(-\)	$+$	\(1\)
\(-\)	\(+\)	\(+\)	$-$	\(1\)
\(-\)	\(-\)	\(+\)	$+$	\(1\)
\(-\)	\(-\)	\(-\)	$-$	\(2\)
Plus space			\(+\)	\(2\)
Minus space			\(-\)	\(5\)

Trace form

\( 7 q + 4 q^{2} + 112 q^{4} + 25 q^{5} + 80 q^{7} + 64 q^{8} + O(q^{10}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(90))\) 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
90.6.a.a	$90$	$6$	90.a	1.a	$1$	$1$	$1$	$14.435$	\(\Q\)	None	✓	$1$	$0$	\(-4\)	\(0\)	\(-25\)	\(32\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}+2^{4}q^{4}-5^{2}q^{5}+2^{5}q^{7}-2^{6}q^{8}+\cdots\)
90.6.a.b	$90$	$6$	90.a	1.a	$1$	$1$	$1$	$14.435$	\(\Q\)	None	✓	$1$	$1$	\(-4\)	\(0\)	\(25\)	\(-118\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}+2^{4}q^{4}+5^{2}q^{5}-118q^{7}+\cdots\)
90.6.a.c	$90$	$6$	90.a	1.a	$1$	$1$	$1$	$14.435$	\(\Q\)	None	✓	$1$	$0$	\(-4\)	\(0\)	\(25\)	\(98\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}+2^{4}q^{4}+5^{2}q^{5}+98q^{7}-2^{6}q^{8}+\cdots\)
90.6.a.d	$90$	$6$	90.a	1.a	$1$	$1$	$1$	$14.435$	\(\Q\)	None	✓	$1$	$1$	\(4\)	\(0\)	\(-25\)	\(-172\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}+2^{4}q^{4}-5^{2}q^{5}-172q^{7}+\cdots\)
90.6.a.e	$90$	$6$	90.a	1.a	$1$	$1$	$1$	$14.435$	\(\Q\)	None	✓	$1$	$0$	\(4\)	\(0\)	\(-25\)	\(98\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}+2^{4}q^{4}-5^{2}q^{5}+98q^{7}+2^{6}q^{8}+\cdots\)
90.6.a.f	$90$	$6$	90.a	1.a	$1$	$1$	$1$	$14.435$	\(\Q\)	None	✓	$1$	$0$	\(4\)	\(0\)	\(25\)	\(-22\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}+2^{4}q^{4}+5^{2}q^{5}-22q^{7}+2^{6}q^{8}+\cdots\)
90.6.a.g	$90$	$6$	90.a	1.a	$1$	$1$	$1$	$14.435$	\(\Q\)	None	✓	$1$	$0$	\(4\)	\(0\)	\(25\)	\(164\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}+2^{4}q^{4}+5^{2}q^{5}+164q^{7}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(90))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(90)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 2}\)