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		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
$+$	$+$	$+$	$+$		$67$	$6$	$61$		$61$	$6$	$55$		$6$	$0$	$6$
$+$	$+$	$-$	$-$		$62$	$5$	$57$		$56$	$5$	$51$		$6$	$0$	$6$
$+$	$-$	$+$	$-$		$61$	$6$	$55$		$55$	$6$	$49$		$6$	$0$	$6$
$+$	$-$	$-$	$+$		$65$	$7$	$58$		$59$	$7$	$52$		$6$	$0$	$6$
$-$	$+$	$+$	$-$		$64$	$5$	$59$		$58$	$5$	$53$		$6$	$0$	$6$
$-$	$+$	$-$	$+$		$65$	$8$	$57$		$59$	$8$	$51$		$6$	$0$	$6$
$-$	$-$	$+$	$+$		$64$	$7$	$57$		$58$	$7$	$51$		$6$	$0$	$6$
$-$	$-$	$-$	$-$		$62$	$4$	$58$		$56$	$4$	$52$		$6$	$0$	$6$
Plus space			$+$		$261$	$28$	$233$		$237$	$28$	$209$		$24$	$0$	$24$
Minus space			$-$		$249$	$20$	$229$		$225$	$20$	$205$		$24$	$0$	$24$

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}$