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

• Label: 810.2.e
• Level: $810$
• Weight: $2$
• Character orbit: 810.e
• Rep. character: $\chi_{810}(271,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $32$
• Newform subspaces: $14$
• Sturm bound: $324$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	810.e (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 9 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 14 \)
Sturm bound:			\(324\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(7\), \(11\), \(13\), \(17\), \(23\)

Dimensions

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

	Total	New	Old
Modular forms	372	32	340
Cusp forms	276	32	244
Eisenstein series	96	0	96

Trace form

32 * q - 16 * q^4 - 20 * q^7 - 20 * q^13 - 16 * q^16 + 16 * q^19 + 12 * q^22 - 16 * q^25 + 40 * q^28 - 20 * q^31 + 12 * q^34 + 40 * q^37 - 8 * q^43 - 24 * q^46 - 24 * q^49 - 20 * q^52 + 16 * q^61 + 32 * q^64 + 16 * q^67 + 12 * q^70 - 32 * q^73 - 8 * q^76 + 4 * q^79 - 24 * q^82 + 24 * q^85 + 12 * q^88 - 16 * q^91 + 12 * q^94 - 8 * q^97

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.e.a	$810$	$2$	810.e	9.c	$3$	$2$	$1$	$6.468$	\(\Q(\sqrt{-3}) \)	None					270.2.a.a	$2$	$0$	\(-1\)	\(0\)	\(-1\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\zeta_{6})q^{2}-\zeta_{6}q^{4}-\zeta_{6}q^{5}+(-2+\cdots)q^{7}+\cdots\)
810.2.e.b	$810$	$2$	810.e	9.c	$3$	$2$	$1$	$6.468$	\(\Q(\sqrt{-3}) \)	None					30.2.a.a	$2$	$0$	\(-1\)	\(0\)	\(-1\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\zeta_{6})q^{2}-\zeta_{6}q^{4}-\zeta_{6}q^{5}+(4+\cdots)q^{7}+\cdots\)
810.2.e.c	$810$	$2$	810.e	9.c	$3$	$2$	$1$	$6.468$	\(\Q(\sqrt{-3}) \)	None		✓			810.2.a.d	$2$	$0$	\(-1\)	\(0\)	\(1\)	\(-5\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\zeta_{6})q^{2}-\zeta_{6}q^{4}+\zeta_{6}q^{5}+(-5+\cdots)q^{7}+\cdots\)
810.2.e.d	$810$	$2$	810.e	9.c	$3$	$2$	$1$	$6.468$	\(\Q(\sqrt{-3}) \)	None					270.2.a.b	$2$	$0$	\(-1\)	\(0\)	\(1\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\zeta_{6})q^{2}-\zeta_{6}q^{4}+\zeta_{6}q^{5}+(-2+\cdots)q^{7}+\cdots\)
810.2.e.e	$810$	$2$	810.e	9.c	$3$	$2$	$1$	$6.468$	\(\Q(\sqrt{-3}) \)	None					90.2.a.a	$2$	$0$	\(-1\)	\(0\)	\(1\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\zeta_{6})q^{2}-\zeta_{6}q^{4}+\zeta_{6}q^{5}+(-2+\cdots)q^{7}+\cdots\)
810.2.e.f	$810$	$2$	810.e	9.c	$3$	$2$	$1$	$6.468$	\(\Q(\sqrt{-3}) \)	None		✓			810.2.a.c	$2$	$0$	\(-1\)	\(0\)	\(1\)	\(1\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\zeta_{6})q^{2}-\zeta_{6}q^{4}+\zeta_{6}q^{5}+(1+\cdots)q^{7}+\cdots\)
810.2.e.g	$810$	$2$	810.e	9.c	$3$	$2$	$1$	$6.468$	\(\Q(\sqrt{-3}) \)	None		✓			810.2.a.d	$2$	$0$	\(1\)	\(0\)	\(-1\)	\(-5\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{2}-\zeta_{6}q^{4}-\zeta_{6}q^{5}+(-5+\cdots)q^{7}+\cdots\)
810.2.e.h	$810$	$2$	810.e	9.c	$3$	$2$	$1$	$6.468$	\(\Q(\sqrt{-3}) \)	None					90.2.a.a	$2$	$0$	\(1\)	\(0\)	\(-1\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{2}-\zeta_{6}q^{4}-\zeta_{6}q^{5}+(-2+\cdots)q^{7}+\cdots\)
810.2.e.i	$810$	$2$	810.e	9.c	$3$	$2$	$1$	$6.468$	\(\Q(\sqrt{-3}) \)	None					270.2.a.b	$2$	$0$	\(1\)	\(0\)	\(-1\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{2}-\zeta_{6}q^{4}-\zeta_{6}q^{5}+(-2+\cdots)q^{7}+\cdots\)
810.2.e.j	$810$	$2$	810.e	9.c	$3$	$2$	$1$	$6.468$	\(\Q(\sqrt{-3}) \)	None		✓			810.2.a.c	$2$	$0$	\(1\)	\(0\)	\(-1\)	\(1\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{2}-\zeta_{6}q^{4}-\zeta_{6}q^{5}+(1-\zeta_{6})q^{7}+\cdots\)
810.2.e.k	$810$	$2$	810.e	9.c	$3$	$2$	$1$	$6.468$	\(\Q(\sqrt{-3}) \)	None					270.2.a.a	$2$	$0$	\(1\)	\(0\)	\(1\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{2}-\zeta_{6}q^{4}+\zeta_{6}q^{5}+(-2+\cdots)q^{7}+\cdots\)
810.2.e.l	$810$	$2$	810.e	9.c	$3$	$2$	$1$	$6.468$	\(\Q(\sqrt{-3}) \)	None					30.2.a.a	$2$	$0$	\(1\)	\(0\)	\(1\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{2}-\zeta_{6}q^{4}+\zeta_{6}q^{5}+(4-4\zeta_{6})q^{7}+\cdots\)
810.2.e.m	$810$	$2$	810.e	9.c	$3$	$4$	$2$	$6.468$	\(\Q(\zeta_{12})\)	None		✓			810.2.a.j	$2$	$0$	\(-2\)	\(0\)	\(-2\)	\(-4\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta_1 q^{2}+(\beta_1-1)q^{4}+(\beta_1-1)q^{5}+\cdots\)
810.2.e.n	$810$	$2$	810.e	9.c	$3$	$4$	$2$	$6.468$	\(\Q(\zeta_{12})\)	None		✓			810.2.a.j	$2$	$0$	\(2\)	\(0\)	\(2\)	\(-4\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta_1 q^{2}+(\beta_1-1)q^{4}+(-\beta_1+1)q^{5}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(810, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(54, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(81, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(90, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(135, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(162, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(270, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(405, [\chi])\)\(^{\oplus 2}\)