Space of modular forms of level 585, weight 2, and trivial character

• Label: 585.2.a
• Level: $585$
• Weight: $2$
• Character orbit: 585.a
• Rep. character: $\chi_{585}(1,\cdot)$
• Character field: $\Q$
• Dimension: $20$
• Newform subspaces: $14$
• Sturm bound: $168$
• Trace bound: $7$

Defining parameters

Level:	$N$	$=$	$585 = 3^{2} \cdot 5 \cdot 13$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	585.a (trivial)
Character field:			$\Q$
Newform subspaces:			$14$
Sturm bound:			$168$
Trace bound:			$7$
Distinguishing $T_p$:			$2$, $7$, $11$

Dimensions

The following table gives the dimensions of various subspaces of $M_{2}(\Gamma_0(585))$.

	Total	New	Old
Modular forms	92	20	72
Cusp forms	77	20	57
Eisenstein series	15	0	15

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

$3$	$5$	$13$	Fricke	Dim
$+$	$+$	$+$	$+$	$3$
$+$	$+$	$-$	$-$	$1$
$+$	$-$	$+$	$-$	$3$
$+$	$-$	$-$	$+$	$1$
$-$	$+$	$+$	$-$	$3$
$-$	$+$	$-$	$+$	$2$
$-$	$-$	$+$	$+$	$2$
$-$	$-$	$-$	$-$	$5$
Plus space			$+$	$8$
Minus space			$-$	$12$

Trace form

20 * q - 2 * q^2 + 20 * q^4 + 2 * q^5 - 4 * q^7 - 6 * q^8 + 4 * q^10 - 4 * q^11 - 2 * q^13 + 16 * q^14 + 20 * q^16 + 4 * q^17 - 8 * q^19 + 6 * q^20 - 24 * q^22 + 20 * q^25 - 12 * q^28 - 16 * q^29 + 12 * q^31 + 6 * q^32 - 4 * q^35 + 12 * q^38 + 28 * q^41 - 16 * q^43 + 32 * q^44 - 32 * q^46 + 4 * q^47 + 12 * q^49 - 2 * q^50 - 6 * q^52 - 32 * q^53 + 4 * q^55 + 48 * q^56 - 8 * q^58 + 16 * q^61 - 16 * q^62 + 20 * q^64 + 4 * q^65 - 36 * q^67 - 20 * q^68 + 8 * q^70 - 24 * q^71 - 24 * q^73 - 16 * q^74 - 44 * q^76 - 36 * q^77 + 14 * q^80 + 48 * q^82 + 44 * q^83 + 8 * q^85 - 80 * q^86 - 52 * q^88 - 8 * q^89 + 4 * q^91 - 12 * q^92 + 72 * q^94 - 8 * q^95 - 20 * q^97 + 38 * q^98

Decomposition of $S_{2}^{\mathrm{new}}(\Gamma_0(585))$ 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3	5	13
585.2.a.a	$585$	$2$	585.a	1.a	$1$	$1$	$1$	$4.671$	$\Q$	None	✓				195.2.a.d	$1$	$1$	$-2$	$0$	$-1$	$-3$		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-2q^{2}+2q^{4}-q^{5}-3q^{7}+2q^{10}+\cdots$
585.2.a.b	$585$	$2$	585.a	1.a	$1$	$1$	$1$	$4.671$	$\Q$	None	✓				195.2.a.b	$1$	$0$	$-2$	$0$	$-1$	$3$		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-2q^{2}+2q^{4}-q^{5}+3q^{7}+2q^{10}+\cdots$
585.2.a.c	$585$	$2$	585.a	1.a	$1$	$1$	$1$	$4.671$	$\Q$	None	✓				195.2.a.c	$1$	$1$	$-2$	$0$	$1$	$-1$		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-2q^{2}+2q^{4}+q^{5}-q^{7}-2q^{10}+\cdots$
585.2.a.d	$585$	$2$	585.a	1.a	$1$	$1$	$1$	$4.671$	$\Q$	None	✓	✓			585.2.a.d	$1$	$1$	$-1$	$0$	$-1$	$2$		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-q^{4}-q^{5}+2q^{7}+3q^{8}+q^{10}+\cdots$
585.2.a.e	$585$	$2$	585.a	1.a	$1$	$1$	$1$	$4.671$	$\Q$	None	✓	✓	✓	✓	585.2.a.e	$1$	$0$	$0$	$0$	$-1$	$-1$		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-2q^{4}-q^{5}-q^{7}+3q^{11}+q^{13}+\cdots$
585.2.a.f	$585$	$2$	585.a	1.a	$1$	$1$	$1$	$4.671$	$\Q$	None	✓	✓	✓	✓	585.2.a.e	$1$	$1$	$0$	$0$	$1$	$-1$		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-2q^{4}+q^{5}-q^{7}-3q^{11}+q^{13}+\cdots$
585.2.a.g	$585$	$2$	585.a	1.a	$1$	$1$	$1$	$4.671$	$\Q$	None	✓				195.2.a.a	$1$	$1$	$1$	$0$	$-1$	$0$		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-q^{4}-q^{5}-3q^{8}-q^{10}-4q^{11}+\cdots$
585.2.a.h	$585$	$2$	585.a	1.a	$1$	$1$	$1$	$4.671$	$\Q$	None	✓				65.2.a.a	$1$	$1$	$1$	$0$	$1$	$-4$		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-q^{4}+q^{5}-4q^{7}-3q^{8}+q^{10}+\cdots$
585.2.a.i	$585$	$2$	585.a	1.a	$1$	$1$	$1$	$4.671$	$\Q$	None	✓	✓			585.2.a.d	$1$	$0$	$1$	$0$	$1$	$2$		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-q^{4}+q^{5}+2q^{7}-3q^{8}+q^{10}+\cdots$
585.2.a.j	$585$	$2$	585.a	1.a	$1$	$2$	$2$	$4.671$	$\Q(\sqrt{17})$	None	✓	✓	✓		585.2.a.j	$1$	$1$	$-1$	$0$	$-2$	$-5$		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-\beta q^{2}+(2+\beta )q^{4}-q^{5}+(-3+\beta )q^{7}+\cdots$
585.2.a.k	$585$	$2$	585.a	1.a	$1$	$2$	$2$	$4.671$	$\Q(\sqrt{3})$	None	✓				65.2.a.c	$1$	$0$	$0$	$0$	$2$	$4$		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+\beta q^{2}+q^{4}+q^{5}+2q^{7}-\beta q^{8}+\beta q^{10}+\cdots$
585.2.a.l	$585$	$2$	585.a	1.a	$1$	$2$	$2$	$4.671$	$\Q(\sqrt{17})$	None	✓	✓	✓		585.2.a.j	$1$	$0$	$1$	$0$	$2$	$-5$		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+\beta q^{2}+(2+\beta )q^{4}+q^{5}+(-3+\beta )q^{7}+\cdots$
585.2.a.m	$585$	$2$	585.a	1.a	$1$	$2$	$2$	$4.671$	$\Q(\sqrt{2})$	None	✓		✓		65.2.a.b	$1$	$0$	$2$	$0$	$-2$	$4$		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+(1+\beta )q^{2}+(1+2\beta )q^{4}-q^{5}+(2+2\beta )q^{7}+\cdots$
585.2.a.n	$585$	$2$	585.a	1.a	$1$	$3$	$3$	$4.671$	3.3.316.1	None	✓		✓		195.2.a.e	$1$	$0$	$0$	$0$	$3$	$1$		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{2}+(3+\beta _{2})q^{4}+q^{5}-\beta _{2}q^{7}+\cdots$

Decomposition of $S_{2}^{\mathrm{old}}(\Gamma_0(585))$ into lower level spaces

$S_{2}^{\mathrm{old}}(\Gamma_0(585)) \simeq$ $S_{2}^{\mathrm{new}}(\Gamma_0(15))$$^{\oplus 4}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(39))$$^{\oplus 4}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(45))$$^{\oplus 2}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(65))$$^{\oplus 3}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(117))$$^{\oplus 2}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(195))$$^{\oplus 2}$