Space of modular forms of level 810, weight 4, and Character 810.e

• Label: 810.4.e
• Level: $810$
• Weight: $4$
• Character orbit: 810.e
• Rep. character: $\chi_{810}(271,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $96$
• Newform subspaces: $34$
• Sturm bound: $648$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	810.e (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 9 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 34 \)
Sturm bound:			\(648\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(810, [\chi])\).

	Total	New	Old
Modular forms	1020	96	924
Cusp forms	924	96	828
Eisenstein series	96	0	96

Trace form

\( 96 q - 192 q^{4} + 120 q^{7} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(810, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
810.4.e.a	$810$	$4$	810.e	9.c	$3$	$2$	$1$	$47.792$	\(\Q(\sqrt{-3}) \)	None					90.4.a.b	$2$	$0$	\(-2\)	\(0\)	\(-5\)	\(-14\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-2+2\zeta_{6})q^{2}-4\zeta_{6}q^{4}-5\zeta_{6}q^{5}+\cdots\)
810.4.e.b	$810$	$4$	810.e	9.c	$3$	$2$	$1$	$47.792$	\(\Q(\sqrt{-3}) \)	None					270.4.a.b	$2$	$0$	\(-2\)	\(0\)	\(-5\)	\(-8\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-2+2\zeta_{6})q^{2}-4\zeta_{6}q^{4}-5\zeta_{6}q^{5}+\cdots\)
810.4.e.c	$810$	$4$	810.e	9.c	$3$	$2$	$1$	$47.792$	\(\Q(\sqrt{-3}) \)	None					10.4.a.a	$2$	$0$	\(-2\)	\(0\)	\(-5\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-2+2\zeta_{6})q^{2}-4\zeta_{6}q^{4}-5\zeta_{6}q^{5}+\cdots\)
810.4.e.d	$810$	$4$	810.e	9.c	$3$	$2$	$1$	$47.792$	\(\Q(\sqrt{-3}) \)	None					270.4.a.a	$2$	$0$	\(-2\)	\(0\)	\(-5\)	\(34\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-2+2\zeta_{6})q^{2}-4\zeta_{6}q^{4}-5\zeta_{6}q^{5}+\cdots\)
810.4.e.e	$810$	$4$	810.e	9.c	$3$	$2$	$1$	$47.792$	\(\Q(\sqrt{-3}) \)	None					30.4.a.a	$2$	$0$	\(-2\)	\(0\)	\(5\)	\(-32\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-2+2\zeta_{6})q^{2}-4\zeta_{6}q^{4}+5\zeta_{6}q^{5}+\cdots\)
810.4.e.f	$810$	$4$	810.e	9.c	$3$	$2$	$1$	$47.792$	\(\Q(\sqrt{-3}) \)	None					270.4.a.f	$2$	$0$	\(-2\)	\(0\)	\(5\)	\(-14\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-2+2\zeta_{6})q^{2}-4\zeta_{6}q^{4}+5\zeta_{6}q^{5}+\cdots\)
810.4.e.g	$810$	$4$	810.e	9.c	$3$	$2$	$1$	$47.792$	\(\Q(\sqrt{-3}) \)	None		✓			810.4.a.c	$2$	$0$	\(-2\)	\(0\)	\(5\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-2+2\zeta_{6})q^{2}-4\zeta_{6}q^{4}+5\zeta_{6}q^{5}+\cdots\)
810.4.e.h	$810$	$4$	810.e	9.c	$3$	$2$	$1$	$47.792$	\(\Q(\sqrt{-3}) \)	None					270.4.a.e	$2$	$0$	\(-2\)	\(0\)	\(5\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-2+2\zeta_{6})q^{2}-4\zeta_{6}q^{4}+5\zeta_{6}q^{5}+\cdots\)
810.4.e.i	$810$	$4$	810.e	9.c	$3$	$2$	$1$	$47.792$	\(\Q(\sqrt{-3}) \)	None					30.4.a.b	$2$	$0$	\(-2\)	\(0\)	\(5\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-2+2\zeta_{6})q^{2}-4\zeta_{6}q^{4}+5\zeta_{6}q^{5}+\cdots\)
810.4.e.j	$810$	$4$	810.e	9.c	$3$	$2$	$1$	$47.792$	\(\Q(\sqrt{-3}) \)	None					270.4.a.d	$2$	$0$	\(-2\)	\(0\)	\(5\)	\(13\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-2+2\zeta_{6})q^{2}-4\zeta_{6}q^{4}+5\zeta_{6}q^{5}+\cdots\)
810.4.e.k	$810$	$4$	810.e	9.c	$3$	$2$	$1$	$47.792$	\(\Q(\sqrt{-3}) \)	None					270.4.a.c	$2$	$0$	\(-2\)	\(0\)	\(5\)	\(22\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-2+2\zeta_{6})q^{2}-4\zeta_{6}q^{4}+5\zeta_{6}q^{5}+\cdots\)
810.4.e.l	$810$	$4$	810.e	9.c	$3$	$2$	$1$	$47.792$	\(\Q(\sqrt{-3}) \)	None		✓			810.4.a.a	$2$	$0$	\(-2\)	\(0\)	\(5\)	\(28\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-2+2\zeta_{6})q^{2}-4\zeta_{6}q^{4}+5\zeta_{6}q^{5}+\cdots\)
810.4.e.m	$810$	$4$	810.e	9.c	$3$	$2$	$1$	$47.792$	\(\Q(\sqrt{-3}) \)	None					30.4.a.a	$2$	$0$	\(2\)	\(0\)	\(-5\)	\(-32\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(2-2\zeta_{6})q^{2}-4\zeta_{6}q^{4}-5\zeta_{6}q^{5}+\cdots\)
810.4.e.n	$810$	$4$	810.e	9.c	$3$	$2$	$1$	$47.792$	\(\Q(\sqrt{-3}) \)	None					270.4.a.f	$2$	$0$	\(2\)	\(0\)	\(-5\)	\(-14\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(2-2\zeta_{6})q^{2}-4\zeta_{6}q^{4}-5\zeta_{6}q^{5}+\cdots\)
810.4.e.o	$810$	$4$	810.e	9.c	$3$	$2$	$1$	$47.792$	\(\Q(\sqrt{-3}) \)	None		✓			810.4.a.c	$2$	$0$	\(2\)	\(0\)	\(-5\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(2-2\zeta_{6})q^{2}-4\zeta_{6}q^{4}-5\zeta_{6}q^{5}+\cdots\)
810.4.e.p	$810$	$4$	810.e	9.c	$3$	$2$	$1$	$47.792$	\(\Q(\sqrt{-3}) \)	None					30.4.a.b	$2$	$0$	\(2\)	\(0\)	\(-5\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(2-2\zeta_{6})q^{2}-4\zeta_{6}q^{4}-5\zeta_{6}q^{5}+\cdots\)
810.4.e.q	$810$	$4$	810.e	9.c	$3$	$2$	$1$	$47.792$	\(\Q(\sqrt{-3}) \)	None					270.4.a.e	$2$	$0$	\(2\)	\(0\)	\(-5\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(2-2\zeta_{6})q^{2}-4\zeta_{6}q^{4}-5\zeta_{6}q^{5}+\cdots\)
810.4.e.r	$810$	$4$	810.e	9.c	$3$	$2$	$1$	$47.792$	\(\Q(\sqrt{-3}) \)	None					270.4.a.d	$2$	$0$	\(2\)	\(0\)	\(-5\)	\(13\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(2-2\zeta_{6})q^{2}-4\zeta_{6}q^{4}-5\zeta_{6}q^{5}+\cdots\)
810.4.e.s	$810$	$4$	810.e	9.c	$3$	$2$	$1$	$47.792$	\(\Q(\sqrt{-3}) \)	None					270.4.a.c	$2$	$0$	\(2\)	\(0\)	\(-5\)	\(22\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(2-2\zeta_{6})q^{2}-4\zeta_{6}q^{4}-5\zeta_{6}q^{5}+\cdots\)
810.4.e.t	$810$	$4$	810.e	9.c	$3$	$2$	$1$	$47.792$	\(\Q(\sqrt{-3}) \)	None		✓			810.4.a.a	$2$	$0$	\(2\)	\(0\)	\(-5\)	\(28\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(2-2\zeta_{6})q^{2}-4\zeta_{6}q^{4}-5\zeta_{6}q^{5}+\cdots\)
810.4.e.u	$810$	$4$	810.e	9.c	$3$	$2$	$1$	$47.792$	\(\Q(\sqrt{-3}) \)	None					90.4.a.b	$2$	$0$	\(2\)	\(0\)	\(5\)	\(-14\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(2-2\zeta_{6})q^{2}-4\zeta_{6}q^{4}+5\zeta_{6}q^{5}+\cdots\)
810.4.e.v	$810$	$4$	810.e	9.c	$3$	$2$	$1$	$47.792$	\(\Q(\sqrt{-3}) \)	None					270.4.a.b	$2$	$0$	\(2\)	\(0\)	\(5\)	\(-8\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(2-2\zeta_{6})q^{2}-4\zeta_{6}q^{4}+5\zeta_{6}q^{5}+\cdots\)
810.4.e.w	$810$	$4$	810.e	9.c	$3$	$2$	$1$	$47.792$	\(\Q(\sqrt{-3}) \)	None					10.4.a.a	$2$	$0$	\(2\)	\(0\)	\(5\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(2-2\zeta_{6})q^{2}-4\zeta_{6}q^{4}+5\zeta_{6}q^{5}+\cdots\)
810.4.e.x	$810$	$4$	810.e	9.c	$3$	$2$	$1$	$47.792$	\(\Q(\sqrt{-3}) \)	None					270.4.a.a	$2$	$0$	\(2\)	\(0\)	\(5\)	\(34\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(2-2\zeta_{6})q^{2}-4\zeta_{6}q^{4}+5\zeta_{6}q^{5}+\cdots\)
810.4.e.y	$810$	$4$	810.e	9.c	$3$	$4$	$2$	$47.792$	\(\Q(\sqrt{-3}, \sqrt{-163})\)	None		✓			810.4.a.j	$2$	$0$	\(-4\)	\(0\)	\(-10\)	\(-7\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-2-2\beta _{1})q^{2}+4\beta _{1}q^{4}+5\beta _{1}q^{5}+\cdots\)
810.4.e.z	$810$	$4$	810.e	9.c	$3$	$4$	$2$	$47.792$	\(\Q(\sqrt{-3}, \sqrt{401})\)	None					270.4.a.m	$2$	$0$	\(-4\)	\(0\)	\(-10\)	\(-1\)	$3^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q-2\beta _{1}q^{2}+(-4+4\beta _{1})q^{4}+(-5+5\beta _{1}+\cdots)q^{5}+\cdots\)
810.4.e.ba	$810$	$4$	810.e	9.c	$3$	$4$	$2$	$47.792$	\(\Q(\sqrt{-3}, \sqrt{-1027})\)	None		✓			810.4.a.h	$2$	$0$	\(-4\)	\(0\)	\(-10\)	\(5\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-2-2\beta _{1})q^{2}+4\beta _{1}q^{4}+5\beta _{1}q^{5}+\cdots\)
810.4.e.bb	$810$	$4$	810.e	9.c	$3$	$4$	$2$	$47.792$	\(\Q(\zeta_{12})\)	None		✓			810.4.a.g	$2$	$0$	\(-4\)	\(0\)	\(-10\)	\(26\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(2\beta_1-2)q^{2}-4\beta_1 q^{4}-5\beta_1 q^{5}+\cdots\)
810.4.e.bc	$810$	$4$	810.e	9.c	$3$	$4$	$2$	$47.792$	\(\Q(\sqrt{-3}, \sqrt{-163})\)	None		✓			810.4.a.j	$2$	$0$	\(4\)	\(0\)	\(10\)	\(-7\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(2+2\beta _{1})q^{2}+4\beta _{1}q^{4}-5\beta _{1}q^{5}+\cdots\)
810.4.e.bd	$810$	$4$	810.e	9.c	$3$	$4$	$2$	$47.792$	\(\Q(\sqrt{-3}, \sqrt{401})\)	None					270.4.a.m	$2$	$0$	\(4\)	\(0\)	\(10\)	\(-1\)	$3^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+2\beta _{1}q^{2}+(-4+4\beta _{1})q^{4}+(5-5\beta _{1}+\cdots)q^{5}+\cdots\)
810.4.e.be	$810$	$4$	810.e	9.c	$3$	$4$	$2$	$47.792$	\(\Q(\sqrt{-3}, \sqrt{-1027})\)	None		✓			810.4.a.h	$2$	$0$	\(4\)	\(0\)	\(10\)	\(5\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(2+2\beta _{1})q^{2}+4\beta _{1}q^{4}-5\beta _{1}q^{5}+\cdots\)
810.4.e.bf	$810$	$4$	810.e	9.c	$3$	$4$	$2$	$47.792$	\(\Q(\zeta_{12})\)	None		✓			810.4.a.g	$2$	$0$	\(4\)	\(0\)	\(10\)	\(26\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-2\beta_1+2)q^{2}-4\beta_1 q^{4}+5\beta_1 q^{5}+\cdots\)
810.4.e.bg	$810$	$4$	810.e	9.c	$3$	$8$	$4$	$47.792$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None		✓			810.4.a.t	$2$	$0$	\(-8\)	\(0\)	\(20\)	\(-2\)	$3^{10}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-2-2\beta _{1})q^{2}+4\beta _{1}q^{4}-5\beta _{1}q^{5}+\cdots\)
810.4.e.bh	$810$	$4$	810.e	9.c	$3$	$8$	$4$	$47.792$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None		✓			810.4.a.t	$2$	$0$	\(8\)	\(0\)	\(-20\)	\(-2\)	$3^{10}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(2+2\beta _{1})q^{2}+4\beta _{1}q^{4}+5\beta _{1}q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(810, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(810, [\chi]) \simeq \) \(S_{4}^{\mathrm{new}}(9, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(27, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(54, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(81, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(90, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(135, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(162, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(270, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(405, [\chi])\)\(^{\oplus 2}\)