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

• Label: 810.2.k
• Level: $810$
• Weight: $2$
• Character orbit: 810.k
• Rep. character: $\chi_{810}(91,\cdot)$
• Character field: $\Q(\zeta_{9})$
• Dimension: $72$
• Newform subspaces: $5$
• Sturm bound: $324$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	810.k (of order \(9\) and degree \(6\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 27 \)
Character field:			\(\Q(\zeta_{9})\)
Newform subspaces:			\( 5 \)
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	1044	72	972
Cusp forms	900	72	828
Eisenstein series	144	0	144

Trace form

72 * q - 6 * q^8 - 18 * q^11 - 6 * q^14 - 24 * q^17 + 18 * q^22 + 72 * q^23 + 66 * q^29 + 18 * q^34 - 12 * q^35 + 12 * q^38 - 18 * q^41 + 18 * q^43 + 72 * q^47 + 18 * q^49 + 72 * q^53 - 6 * q^56 - 48 * q^59 + 18 * q^61 - 48 * q^62 - 36 * q^64 + 24 * q^65 + 18 * q^67 + 12 * q^68 - 36 * q^70 + 48 * q^71 + 36 * q^73 + 12 * q^74 - 36 * q^76 - 60 * q^77 - 72 * q^79 + 12 * q^80 + 60 * q^83 - 72 * q^85 + 90 * q^86 - 36 * q^88 + 6 * q^89 + 36 * q^91 - 36 * q^92 - 36 * q^94 + 48 * q^95 + 18 * q^97 - 30 * q^98

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	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
810.2.k.a	$810$	$2$	810.k	27.e	$9$	$6$	$1$	$6.468$	\(\Q(\zeta_{18})\)	None					270.2.k.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(6\)	$1$	$\mathrm{SU}(2)[C_{9}]$	\(q+(\zeta_{18}-\zeta_{18}^{4})q^{2}-\zeta_{18}^{5}q^{4}+(\zeta_{18}^{2}+\cdots)q^{5}+\cdots\)
810.2.k.b	$810$	$2$	810.k	27.e	$9$	$12$	$2$	$6.468$	12.0.\(\cdots\).1	None					270.2.k.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-12\)	$3^{3}$	$\mathrm{SU}(2)[C_{9}]$	\(q-\beta _{7}q^{2}+\beta _{9}q^{4}+(-\beta _{8}+\beta _{9})q^{5}+\cdots\)
810.2.k.c	$810$	$2$	810.k	27.e	$9$	$12$	$2$	$6.468$	12.0.\(\cdots\).1	None					270.2.k.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(6\)	$3$	$\mathrm{SU}(2)[C_{9}]$	\(q+(-\beta _{8}+\beta _{9})q^{2}+\beta _{7}q^{4}-\beta _{10}q^{5}+\cdots\)
810.2.k.d	$810$	$2$	810.k	27.e	$9$	$18$	$3$	$6.468$	\(\mathbb{Q}[x]/(x^{18} - \cdots)\)	None					270.2.k.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-3\)	$3^{5}$	$\mathrm{SU}(2)[C_{9}]$	\(q-\beta _{1}q^{2}-\beta _{2}q^{4}+(\beta _{2}+\beta _{4})q^{5}+(\beta _{1}+\cdots)q^{7}+\cdots\)
810.2.k.e	$810$	$2$	810.k	27.e	$9$	$24$	$4$	$6.468$		None			✓		270.2.k.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(3\)		$\mathrm{SU}(2)[C_{9}]$

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

\( S_{2}^{\mathrm{old}}(810, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(27, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(54, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(81, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(135, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(162, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(270, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(405, [\chi])\)\(^{\oplus 2}\)