Space of modular forms of level 810, weight 4, and trivial character

• Label: 810.4.a
• Level: $810$
• Weight: $4$
• Character orbit: 810.a
• Rep. character: $\chi_{810}(1,\cdot)$
• Character field: $\Q$
• Dimension: $48$
• Newform subspaces: $22$
• Sturm bound: $648$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	810.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 22 \)
Sturm bound:			\(648\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(810))\).

	Total	New	Old
Modular forms	510	48	462
Cusp forms	462	48	414
Eisenstein series	48	0	48

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
\(+\)	\(+\)	\(+\)	\(+\)	\(6\)
\(+\)	\(+\)	\(-\)	\(-\)	\(5\)
\(+\)	\(-\)	\(+\)	\(-\)	\(6\)
\(+\)	\(-\)	\(-\)	\(+\)	\(7\)
\(-\)	\(+\)	\(+\)	\(-\)	\(5\)
\(-\)	\(+\)	\(-\)	\(+\)	\(8\)
\(-\)	\(-\)	\(+\)	\(+\)	\(7\)
\(-\)	\(-\)	\(-\)	\(-\)	\(4\)
Plus space			\(+\)	\(28\)
Minus space			\(-\)	\(20\)

Trace form

48 * q + 192 * q^4 - 48 * q^7 + 96 * q^13 + 768 * q^16 + 60 * q^19 - 72 * q^22 + 1200 * q^25 - 192 * q^28 - 120 * q^31 + 360 * q^34 + 672 * q^37 + 852 * q^43 - 504 * q^46 + 2916 * q^49 + 384 * q^52 + 2436 * q^61 + 3072 * q^64 + 3228 * q^67 + 360 * q^70 + 132 * q^73 + 240 * q^76 - 192 * q^79 + 2088 * q^82 - 1440 * q^85 - 288 * q^88 + 1200 * q^91 + 3096 * q^94 + 6756 * q^97

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(810))\) 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					A-L signs			Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	3	5
810.4.a.a	$810$	$4$	810.a	1.a	$1$	$1$	$1$	$47.792$	\(\Q\)	None	✓	✓			810.4.a.a	$1$	$1$	\(-2\)	\(0\)	\(5\)	\(-28\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+5q^{5}-28q^{7}-8q^{8}+\cdots\)
810.4.a.b	$810$	$4$	810.a	1.a	$1$	$1$	$1$	$47.792$	\(\Q\)	None	✓				90.4.e.a	$1$	$0$	\(-2\)	\(0\)	\(5\)	\(-16\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+5q^{5}-2^{4}q^{7}-8q^{8}+\cdots\)
810.4.a.c	$810$	$4$	810.a	1.a	$1$	$1$	$1$	$47.792$	\(\Q\)	None	✓	✓			810.4.a.c	$1$	$1$	\(-2\)	\(0\)	\(5\)	\(2\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+5q^{5}+2q^{7}-8q^{8}+\cdots\)
810.4.a.d	$810$	$4$	810.a	1.a	$1$	$1$	$1$	$47.792$	\(\Q\)	None	✓	✓			810.4.a.a	$1$	$1$	\(2\)	\(0\)	\(-5\)	\(-28\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}-5q^{5}-28q^{7}+8q^{8}+\cdots\)
810.4.a.e	$810$	$4$	810.a	1.a	$1$	$1$	$1$	$47.792$	\(\Q\)	None	✓				90.4.e.a	$1$	$1$	\(2\)	\(0\)	\(-5\)	\(-16\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}-5q^{5}-2^{4}q^{7}+8q^{8}+\cdots\)
810.4.a.f	$810$	$4$	810.a	1.a	$1$	$1$	$1$	$47.792$	\(\Q\)	None	✓	✓			810.4.a.c	$1$	$1$	\(2\)	\(0\)	\(-5\)	\(2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}-5q^{5}+2q^{7}+8q^{8}+\cdots\)
810.4.a.g	$810$	$4$	810.a	1.a	$1$	$2$	$2$	$47.792$	\(\Q(\sqrt{3}) \)	None	✓	✓			810.4.a.g	$1$	$1$	\(-4\)	\(0\)	\(-10\)	\(-26\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}-5q^{5}+(-13+\beta )q^{7}+\cdots\)
810.4.a.h	$810$	$4$	810.a	1.a	$1$	$2$	$2$	$47.792$	\(\Q(\sqrt{3081}) \)	None	✓	✓			810.4.a.h	$1$	$0$	\(-4\)	\(0\)	\(-10\)	\(-5\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}-5q^{5}+(-2-\beta )q^{7}+\cdots\)
810.4.a.i	$810$	$4$	810.a	1.a	$1$	$2$	$2$	$47.792$	\(\Q(\sqrt{6}) \)	None	✓				90.4.e.c	$1$	$0$	\(-4\)	\(0\)	\(-10\)	\(-2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}-5q^{5}+(-1+7\beta )q^{7}+\cdots\)
810.4.a.j	$810$	$4$	810.a	1.a	$1$	$2$	$2$	$47.792$	\(\Q(\sqrt{489}) \)	None	✓	✓			810.4.a.j	$1$	$0$	\(-4\)	\(0\)	\(-10\)	\(7\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}-5q^{5}+(4-\beta )q^{7}-8q^{8}+\cdots\)
810.4.a.k	$810$	$4$	810.a	1.a	$1$	$2$	$2$	$47.792$	\(\Q(\sqrt{15}) \)	None	✓				90.4.e.b	$1$	$0$	\(-4\)	\(0\)	\(10\)	\(16\)		$+$	$-$	$-$	$+$	$3$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+5q^{5}+(8+\beta )q^{7}-8q^{8}+\cdots\)
810.4.a.l	$810$	$4$	810.a	1.a	$1$	$2$	$2$	$47.792$	\(\Q(\sqrt{15}) \)	None	✓		✓		90.4.e.b	$1$	$1$	\(4\)	\(0\)	\(-10\)	\(16\)		$-$	$+$	$+$	$-$	$3$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}-5q^{5}+(8+\beta )q^{7}+8q^{8}+\cdots\)
810.4.a.m	$810$	$4$	810.a	1.a	$1$	$2$	$2$	$47.792$	\(\Q(\sqrt{3}) \)	None	✓	✓			810.4.a.g	$1$	$1$	\(4\)	\(0\)	\(10\)	\(-26\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+5q^{5}+(-13+\beta )q^{7}+\cdots\)
810.4.a.n	$810$	$4$	810.a	1.a	$1$	$2$	$2$	$47.792$	\(\Q(\sqrt{3081}) \)	None	✓	✓			810.4.a.h	$1$	$0$	\(4\)	\(0\)	\(10\)	\(-5\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+5q^{5}+(-2-\beta )q^{7}+\cdots\)
810.4.a.o	$810$	$4$	810.a	1.a	$1$	$2$	$2$	$47.792$	\(\Q(\sqrt{6}) \)	None	✓				90.4.e.c	$1$	$1$	\(4\)	\(0\)	\(10\)	\(-2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+5q^{5}+(-1+7\beta )q^{7}+\cdots\)
810.4.a.p	$810$	$4$	810.a	1.a	$1$	$2$	$2$	$47.792$	\(\Q(\sqrt{489}) \)	None	✓	✓			810.4.a.j	$1$	$0$	\(4\)	\(0\)	\(10\)	\(7\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+5q^{5}+(4-\beta )q^{7}+8q^{8}+\cdots\)
810.4.a.q	$810$	$4$	810.a	1.a	$1$	$3$	$3$	$47.792$	3.3.3732.1	None	✓		✓		90.4.e.d	$1$	$1$	\(-6\)	\(0\)	\(15\)	\(3\)		$+$	$+$	$-$	$-$	$3^{2}$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+5q^{5}+(1-2\beta _{1}+\beta _{2})q^{7}+\cdots\)
810.4.a.r	$810$	$4$	810.a	1.a	$1$	$3$	$3$	$47.792$	3.3.3732.1	None	✓				90.4.e.d	$1$	$0$	\(6\)	\(0\)	\(-15\)	\(3\)		$-$	$-$	$+$	$+$	$3^{2}$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}-5q^{5}+(1-2\beta _{1}+\beta _{2})q^{7}+\cdots\)
810.4.a.s	$810$	$4$	810.a	1.a	$1$	$4$	$4$	$47.792$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓		✓		90.4.e.e	$1$	$1$	\(-8\)	\(0\)	\(-20\)	\(23\)		$+$	$-$	$+$	$-$	$3^{3}$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}-5q^{5}+(6+\beta _{1})q^{7}+\cdots\)
810.4.a.t	$810$	$4$	810.a	1.a	$1$	$4$	$4$	$47.792$	4.4.8657424.2	None	✓	✓	✓		810.4.a.t	$1$	$0$	\(-8\)	\(0\)	\(20\)	\(2\)		$+$	$-$	$-$	$+$	$3^{4}$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+5q^{5}+(1+\beta _{1}+\beta _{3})q^{7}+\cdots\)
810.4.a.u	$810$	$4$	810.a	1.a	$1$	$4$	$4$	$47.792$	4.4.8657424.2	None	✓	✓	✓		810.4.a.t	$1$	$0$	\(8\)	\(0\)	\(-20\)	\(2\)		$-$	$-$	$+$	$+$	$3^{4}$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}-5q^{5}+(1+\beta _{1}+\beta _{3})q^{7}+\cdots\)
810.4.a.v	$810$	$4$	810.a	1.a	$1$	$4$	$4$	$47.792$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓		✓		90.4.e.e	$1$	$0$	\(8\)	\(0\)	\(20\)	\(23\)		$-$	$+$	$-$	$+$	$3^{3}$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+5q^{5}+(6+\beta _{1})q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(810))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(810)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(54))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(81))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(135))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(162))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(270))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(405))\)\(^{\oplus 2}\)