Space of modular forms of level 810, weight 2, and Character 810.f

• Label: 810.2.f
• Level: $810$
• Weight: $2$
• Character orbit: 810.f
• Rep. character: $\chi_{810}(323,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $48$
• Newform subspaces: $4$
• Sturm bound: $324$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	810.f (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 15 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(324\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(810, [\chi])\).

	Total	New	Old
Modular forms	372	48	324
Cusp forms	276	48	228
Eisenstein series	96	0	96

Trace form

\( 48 q + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(810, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
810.2.f.a	$810$	$2$	810.f	15.e	$4$	$8$	$4$	$6.468$	\(\Q(\zeta_{24})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(4\)	$2^{2}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\zeta_{24}^{3}q^{2}-\zeta_{24}^{2}q^{4}+(\zeta_{24}-2\zeta_{24}^{3}+\cdots)q^{5}+\cdots\)
810.2.f.b	$810$	$2$	810.f	15.e	$4$	$8$	$4$	$6.468$	\(\Q(\zeta_{24})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(16\)	$2^{4}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\zeta_{24}q^{2}+\zeta_{24}^{3}q^{4}+(\zeta_{24}+\zeta_{24}^{4}+\cdots)q^{5}+\cdots\)
810.2.f.c	$810$	$2$	810.f	15.e	$4$	$16$	$8$	$6.468$	16.0.\(\cdots\).9	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-16\)	$2^{8}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{12}q^{2}+\beta _{13}q^{4}+(\beta _{2}+\beta _{3}-\beta _{4}+\cdots)q^{5}+\cdots\)
810.2.f.d	$810$	$2$	810.f	15.e	$4$	$16$	$8$	$6.468$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)	$2^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{14}q^{2}-\beta _{6}q^{4}+(-\beta _{9}+\beta _{13})q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(810, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(810, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(90, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(135, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(270, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(405, [\chi])\)\(^{\oplus 2}\)