Space of modular forms of level 810, weight 2, and trivial character

• Label: 810.2.a
• Level: $810$
• Weight: $2$
• Character orbit: 810.a
• Rep. character: $\chi_{810}(1,\cdot)$
• Character field: $\Q$
• Dimension: $16$
• Newform subspaces: $12$
• Sturm bound: $324$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	810.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 12 \)
Sturm bound:			\(324\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(7\), \(11\), \(17\), \(23\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(810))\).

	Total	New	Old
Modular forms	186	16	170
Cusp forms	139	16	123
Eisenstein series	47	0	47

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
\(+\)	\(+\)	\(+\)	$+$	\(2\)
\(+\)	\(+\)	\(-\)	$-$	\(3\)
\(+\)	\(-\)	\(+\)	$-$	\(2\)
\(+\)	\(-\)	\(-\)	$+$	\(1\)
\(-\)	\(+\)	\(+\)	$-$	\(3\)
\(-\)	\(-\)	\(+\)	$+$	\(1\)
\(-\)	\(-\)	\(-\)	$-$	\(4\)
Plus space			\(+\)	\(4\)
Minus space			\(-\)	\(12\)

Trace form

\( 16 q + 16 q^{4} + 8 q^{7} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(810))\) 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
810.2.a.a	$810$	$2$	810.a	1.a	$1$	$1$	$1$	$6.468$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(0\)	\(1\)	\(-4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+q^{5}-4q^{7}-q^{8}-q^{10}+\cdots\)
810.2.a.b	$810$	$2$	810.a	1.a	$1$	$1$	$1$	$6.468$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(0\)	\(1\)	\(-1\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+q^{5}-q^{7}-q^{8}-q^{10}+\cdots\)
810.2.a.c	$810$	$2$	810.a	1.a	$1$	$1$	$1$	$6.468$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(0\)	\(1\)	\(-1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+q^{5}-q^{7}-q^{8}-q^{10}+\cdots\)
810.2.a.d	$810$	$2$	810.a	1.a	$1$	$1$	$1$	$6.468$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(0\)	\(1\)	\(5\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+q^{5}+5q^{7}-q^{8}-q^{10}+\cdots\)
810.2.a.e	$810$	$2$	810.a	1.a	$1$	$1$	$1$	$6.468$	\(\Q\)	None	✓	$1$	$1$	\(1\)	\(0\)	\(-1\)	\(-4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-q^{5}-4q^{7}+q^{8}-q^{10}+\cdots\)
810.2.a.f	$810$	$2$	810.a	1.a	$1$	$1$	$1$	$6.468$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(0\)	\(-1\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-q^{5}-q^{7}+q^{8}-q^{10}+\cdots\)
810.2.a.g	$810$	$2$	810.a	1.a	$1$	$1$	$1$	$6.468$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(0\)	\(-1\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-q^{5}-q^{7}+q^{8}-q^{10}+\cdots\)
810.2.a.h	$810$	$2$	810.a	1.a	$1$	$1$	$1$	$6.468$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(0\)	\(-1\)	\(5\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-q^{5}+5q^{7}+q^{8}-q^{10}+\cdots\)
810.2.a.i	$810$	$2$	810.a	1.a	$1$	$2$	$2$	$6.468$	\(\Q(\sqrt{33}) \)	None	✓	$1$	$1$	\(-2\)	\(0\)	\(-2\)	\(1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-q^{5}+\beta q^{7}-q^{8}+q^{10}+\cdots\)
810.2.a.j	$810$	$2$	810.a	1.a	$1$	$2$	$2$	$6.468$	\(\Q(\sqrt{3}) \)	None	✓	$1$	$0$	\(-2\)	\(0\)	\(-2\)	\(4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-q^{5}+(2+\beta )q^{7}-q^{8}+\cdots\)
810.2.a.k	$810$	$2$	810.a	1.a	$1$	$2$	$2$	$6.468$	\(\Q(\sqrt{33}) \)	None	✓	$1$	$0$	\(2\)	\(0\)	\(2\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+q^{5}+\beta q^{7}+q^{8}+q^{10}+\cdots\)
810.2.a.l	$810$	$2$	810.a	1.a	$1$	$2$	$2$	$6.468$	\(\Q(\sqrt{3}) \)	None	✓	$1$	$0$	\(2\)	\(0\)	\(2\)	\(4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+q^{5}+(2+\beta )q^{7}+q^{8}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(810))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(810)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(54))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(81))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(135))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(162))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(270))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(405))\)\(^{\oplus 2}\)