Space of modular forms of level 520, weight 2, and trivial character

• Label: 520.2.a
• Level: $520$
• Weight: $2$
• Character orbit: 520.a
• Rep. character: $\chi_{520}(1,\cdot)$
• Character field: $\Q$
• Dimension: $12$
• Newform subspaces: $7$
• Sturm bound: $168$
• Trace bound: $11$

Defining parameters

Level:	$N$	$=$	$520 = 2^{3} \cdot 5 \cdot 13$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	520.a (trivial)
Character field:			$\Q$
Newform subspaces:			$7$
Sturm bound:			$168$
Trace bound:			$11$
Distinguishing $T_p$:			$3$, $11$

Dimensions

The following table gives the dimensions of various subspaces of $M_{2}(\Gamma_0(520))$.

	Total	New	Old
Modular forms	92	12	80
Cusp forms	77	12	65
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.

$2$	$5$	$13$	Fricke	Dim
$+$	$+$	$+$	$+$	$1$
$+$	$+$	$-$	$-$	$2$
$+$	$-$	$+$	$-$	$2$
$-$	$+$	$+$	$-$	$2$
$-$	$+$	$-$	$+$	$2$
$-$	$-$	$+$	$+$	$2$
$-$	$-$	$-$	$-$	$1$
Plus space			$+$	$5$
Minus space			$-$	$7$

Trace form

12 * q - 4 * q^3 - 2 * q^5 + 20 * q^9 - 4 * q^11 - 2 * q^13 + 8 * q^17 - 12 * q^19 + 8 * q^21 + 12 * q^25 - 16 * q^27 + 8 * q^29 + 16 * q^33 - 20 * q^37 - 12 * q^43 + 6 * q^45 - 20 * q^49 - 8 * q^51 + 4 * q^53 + 16 * q^57 - 12 * q^59 - 16 * q^61 - 4 * q^65 - 8 * q^67 - 16 * q^69 + 16 * q^71 + 4 * q^73 - 4 * q^75 - 8 * q^77 - 16 * q^79 + 52 * q^81 + 48 * q^83 - 4 * q^85 + 64 * q^87 - 24 * q^93 - 8 * q^95 - 4 * q^97 + 20 * q^99

Decomposition of $S_{2}^{\mathrm{new}}(\Gamma_0(520))$ 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	5	13
520.2.a.a	$520$	$2$	520.a	1.a	$1$	$1$	$1$	$4.152$	$\Q$	None	✓	✓	✓	✓	520.2.a.a	$1$	$1$	$0$	$0$	$-1$	$0$		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{5}-3q^{9}-4q^{11}-q^{13}-6q^{17}+\cdots$
520.2.a.b	$520$	$2$	520.a	1.a	$1$	$1$	$1$	$4.152$	$\Q$	None	✓	✓	✓	✓	520.2.a.b	$1$	$0$	$0$	$2$	$1$	$0$		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+2q^{3}+q^{5}+q^{9}+2q^{11}+q^{13}+\cdots$
520.2.a.c	$520$	$2$	520.a	1.a	$1$	$2$	$2$	$4.152$	$\Q(\sqrt{2})$	None	✓	✓	✓	✓	520.2.a.c	$1$	$1$	$0$	$-4$	$2$	$-4$		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+(-2+\beta )q^{3}+q^{5}-2q^{7}+(3-4\beta )q^{9}+\cdots$
520.2.a.d	$520$	$2$	520.a	1.a	$1$	$2$	$2$	$4.152$	$\Q(\sqrt{3})$	None	✓	✓	✓	✓	520.2.a.d	$1$	$1$	$0$	$-2$	$-2$	$0$		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+(-1+\beta )q^{3}-q^{5}-2\beta q^{7}+(1-2\beta )q^{9}+\cdots$
520.2.a.e	$520$	$2$	520.a	1.a	$1$	$2$	$2$	$4.152$	$\Q(\sqrt{3})$	None	✓	✓	✓	✓	520.2.a.e	$1$	$0$	$0$	$-2$	$-2$	$0$		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+(-1+\beta )q^{3}-q^{5}+2\beta q^{7}+(1-2\beta )q^{9}+\cdots$
520.2.a.f	$520$	$2$	520.a	1.a	$1$	$2$	$2$	$4.152$	$\Q(\sqrt{6})$	None	✓	✓	✓	✓	520.2.a.f	$1$	$0$	$0$	$0$	$2$	$4$		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+\beta q^{3}+q^{5}+2q^{7}+3q^{9}+(-2+\beta )q^{11}+\cdots$
520.2.a.g	$520$	$2$	520.a	1.a	$1$	$2$	$2$	$4.152$	$\Q(\sqrt{5})$	None	✓	✓	✓	✓	520.2.a.g	$1$	$0$	$0$	$2$	$-2$	$0$		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	$q+(1+\beta )q^{3}-q^{5}+(3+2\beta )q^{9}+(3-\beta )q^{11}+\cdots$

Decomposition of $S_{2}^{\mathrm{old}}(\Gamma_0(520))$ into lower level spaces

$S_{2}^{\mathrm{old}}(\Gamma_0(520)) \simeq$ $S_{2}^{\mathrm{new}}(\Gamma_0(20))$$^{\oplus 4}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(26))$$^{\oplus 6}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(40))$$^{\oplus 2}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(52))$$^{\oplus 4}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(65))$$^{\oplus 4}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(104))$$^{\oplus 2}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(130))$$^{\oplus 3}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(260))$$^{\oplus 2}$