Space of modular forms of level 510, weight 2, and trivial character

• Label: 510.2.a
• Level: $510$
• Weight: $2$
• Character orbit: 510.a
• Rep. character: $\chi_{510}(1,\cdot)$
• Character field: $\Q$
• Dimension: $9$
• Newform subspaces: $8$
• Sturm bound: $216$
• Trace bound: $7$

Defining parameters

Level:	$N$	$=$	$510 = 2 \cdot 3 \cdot 5 \cdot 17$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	510.a (trivial)
Character field:			$\Q$
Newform subspaces:			$8$
Sturm bound:			$216$
Trace bound:			$7$
Distinguishing $T_p$:			$7$, $11$, $13$

Dimensions

The following table gives the dimensions of various subspaces of $M_{2}(\Gamma_0(510))$.

	Total	New	Old
Modular forms	116	9	107
Cusp forms	101	9	92
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.

$2$	$3$	$5$	$17$	Fricke		Total				Cusp				Eisenstein
						All	New	Old		All	New	Old		All	New	Old
$+$	$+$	$+$	$+$	$+$		$3$	$0$	$3$		$3$	$0$	$3$		$0$	$0$	$0$
$+$	$+$	$+$	$-$	$-$		$10$	$1$	$9$		$9$	$1$	$8$		$1$	$0$	$1$
$+$	$+$	$-$	$+$	$-$		$10$	$2$	$8$		$9$	$2$	$7$		$1$	$0$	$1$
$+$	$+$	$-$	$-$	$+$		$6$	$0$	$6$		$5$	$0$	$5$		$1$	$0$	$1$
$+$	$-$	$+$	$+$	$-$		$10$	$0$	$10$		$9$	$0$	$9$		$1$	$0$	$1$
$+$	$-$	$+$	$-$	$+$		$6$	$0$	$6$		$5$	$0$	$5$		$1$	$0$	$1$
$+$	$-$	$-$	$+$	$+$		$6$	$0$	$6$		$5$	$0$	$5$		$1$	$0$	$1$
$+$	$-$	$-$	$-$	$-$		$7$	$1$	$6$		$6$	$1$	$5$		$1$	$0$	$1$
$-$	$+$	$+$	$+$	$-$		$6$	$1$	$5$		$5$	$1$	$4$		$1$	$0$	$1$
$-$	$+$	$+$	$-$	$+$		$8$	$1$	$7$		$7$	$1$	$6$		$1$	$0$	$1$
$-$	$+$	$-$	$+$	$+$		$7$	$0$	$7$		$6$	$0$	$6$		$1$	$0$	$1$
$-$	$+$	$-$	$-$	$-$		$8$	$1$	$7$		$7$	$1$	$6$		$1$	$0$	$1$
$-$	$-$	$+$	$+$	$+$		$10$	$0$	$10$		$9$	$0$	$9$		$1$	$0$	$1$
$-$	$-$	$+$	$-$	$-$		$5$	$1$	$4$		$4$	$1$	$3$		$1$	$0$	$1$
$-$	$-$	$-$	$+$	$-$		$6$	$1$	$5$		$5$	$1$	$4$		$1$	$0$	$1$
$-$	$-$	$-$	$-$	$+$		$8$	$0$	$8$		$7$	$0$	$7$		$1$	$0$	$1$
Plus space				$+$		$54$	$1$	$53$		$47$	$1$	$46$		$7$	$0$	$7$
Minus space				$-$		$62$	$8$	$54$		$54$	$8$	$46$		$8$	$0$	$8$

Trace form

9 * q + q^2 - 3 * q^3 + 9 * q^4 + q^5 + q^6 + q^8 + 9 * q^9 - 3 * q^10 + 4 * q^11 - 3 * q^12 + 6 * q^13 + q^15 + 9 * q^16 + q^17 + q^18 + 4 * q^19 + q^20 + 4 * q^22 + 8 * q^23 + q^24 + 9 * q^25 - 10 * q^26 - 3 * q^27 + 30 * q^29 + q^30 + 8 * q^31 + q^32 + 12 * q^33 + q^34 + 9 * q^36 + 6 * q^37 - 12 * q^38 - 10 * q^39 - 3 * q^40 + 2 * q^41 + 8 * q^42 + 4 * q^43 + 4 * q^44 + q^45 - 3 * q^48 + 17 * q^49 + q^50 + q^51 + 6 * q^52 + 6 * q^53 + q^54 + 12 * q^55 - 12 * q^57 - 10 * q^58 + 4 * q^59 + q^60 + 6 * q^61 + 9 * q^64 - 10 * q^65 - 4 * q^66 - 4 * q^67 + q^68 + 8 * q^69 + 8 * q^70 - 24 * q^71 + q^72 - 46 * q^73 - 18 * q^74 - 3 * q^75 + 4 * q^76 + 6 * q^78 + 8 * q^79 + q^80 + 9 * q^81 - 38 * q^82 - 28 * q^83 - 3 * q^85 - 28 * q^86 - 2 * q^87 + 4 * q^88 - 6 * q^89 - 3 * q^90 - 32 * q^91 + 8 * q^92 - 40 * q^93 + 20 * q^95 + q^96 - 6 * q^97 - 39 * q^98 + 4 * q^99

Decomposition of $S_{2}^{\mathrm{new}}(\Gamma_0(510))$ 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	17
510.2.a.a	$510$	$2$	510.a	1.a	$1$	$1$	$1$	$4.072$	$\Q$	None	✓	✓	✓	✓	510.2.a.a	$1$	$0$	$-1$	$-1$	$-1$	$2$		$+$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-q^{3}+q^{4}-q^{5}+q^{6}+2q^{7}+\cdots$
510.2.a.b	$510$	$2$	510.a	1.a	$1$	$1$	$1$	$4.072$	$\Q$	None	✓	✓	✓	✓	510.2.a.b	$1$	$0$	$-1$	$1$	$1$	$-2$		$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{3}+q^{4}+q^{5}-q^{6}-2q^{7}+\cdots$
510.2.a.c	$510$	$2$	510.a	1.a	$1$	$1$	$1$	$4.072$	$\Q$	None	✓	✓	✓	✓	510.2.a.c	$1$	$1$	$1$	$-1$	$-1$	$-4$		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-q^{3}+q^{4}-q^{5}-q^{6}-4q^{7}+\cdots$
510.2.a.d	$510$	$2$	510.a	1.a	$1$	$1$	$1$	$4.072$	$\Q$	None	✓	✓	✓	✓	510.2.a.d	$1$	$0$	$1$	$-1$	$-1$	$2$		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-q^{3}+q^{4}-q^{5}-q^{6}+2q^{7}+\cdots$
510.2.a.e	$510$	$2$	510.a	1.a	$1$	$1$	$1$	$4.072$	$\Q$	None	✓	✓	✓	✓	510.2.a.e	$1$	$0$	$1$	$-1$	$1$	$0$		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}+q^{8}+\cdots$
510.2.a.f	$510$	$2$	510.a	1.a	$1$	$1$	$1$	$4.072$	$\Q$	None	✓	✓	✓	✓	510.2.a.f	$1$	$0$	$1$	$1$	$-1$	$0$		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}+q^{8}+\cdots$
510.2.a.g	$510$	$2$	510.a	1.a	$1$	$1$	$1$	$4.072$	$\Q$	None	✓	✓	✓	✓	510.2.a.g	$1$	$0$	$1$	$1$	$1$	$2$		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{3}+q^{4}+q^{5}+q^{6}+2q^{7}+\cdots$
510.2.a.h	$510$	$2$	510.a	1.a	$1$	$2$	$2$	$4.072$	$\Q(\sqrt{6})$	None	✓	✓	✓	✓	510.2.a.h	$1$	$0$	$-2$	$-2$	$2$	$0$		$+$	$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	$q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}+\beta q^{7}+\cdots$

Decomposition of $S_{2}^{\mathrm{old}}(\Gamma_0(510))$ into lower level spaces

$S_{2}^{\mathrm{old}}(\Gamma_0(510)) \simeq$ $S_{2}^{\mathrm{new}}(\Gamma_0(15))$$^{\oplus 4}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(17))$$^{\oplus 8}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(30))$$^{\oplus 2}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(34))$$^{\oplus 4}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(51))$$^{\oplus 4}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(85))$$^{\oplus 4}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(102))$$^{\oplus 2}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(170))$$^{\oplus 2}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(255))$$^{\oplus 2}$