Space of modular forms of level 515, weight 2, and trivial character

• Label: 515.2.a
• Level: $515$
• Weight: $2$
• Character orbit: 515.a
• Rep. character: $\chi_{515}(1,\cdot)$
• Character field: $\Q$
• Dimension: $35$
• Newform subspaces: $4$
• Sturm bound: $104$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 515 = 5 \cdot 103 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	515.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(104\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(515))\).

	Total	New	Old
Modular forms	54	35	19
Cusp forms	51	35	16
Eisenstein series	3	0	3

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

\(5\)	\(103\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(10\)	\(8\)	\(2\)		\(10\)	\(8\)	\(2\)		\(0\)	\(0\)	\(0\)
\(+\)	\(-\)	\(-\)		\(16\)	\(9\)	\(7\)		\(15\)	\(9\)	\(6\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)		\(17\)	\(14\)	\(3\)		\(16\)	\(14\)	\(2\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)		\(11\)	\(4\)	\(7\)		\(10\)	\(4\)	\(6\)		\(1\)	\(0\)	\(1\)
Plus space		\(+\)		\(21\)	\(12\)	\(9\)		\(20\)	\(12\)	\(8\)		\(1\)	\(0\)	\(1\)
Minus space		\(-\)		\(33\)	\(23\)	\(10\)		\(31\)	\(23\)	\(8\)		\(2\)	\(0\)	\(2\)

Trace form

35 * q - 3 * q^2 + 33 * q^4 + q^5 - 8 * q^6 + 4 * q^7 - 15 * q^8 + 35 * q^9 - q^10 + 4 * q^12 + 10 * q^13 + 4 * q^14 + 17 * q^16 - 10 * q^17 - 19 * q^18 - q^20 + 4 * q^21 - 8 * q^22 - 8 * q^23 + 12 * q^24 + 35 * q^25 - 4 * q^26 - 24 * q^27 + 18 * q^28 - 6 * q^29 + 8 * q^30 + 16 * q^31 - 13 * q^32 - 40 * q^33 - 10 * q^34 + 8 * q^35 + 49 * q^36 + 6 * q^37 + 26 * q^38 - 20 * q^39 - 9 * q^40 - 10 * q^41 - 4 * q^42 + 20 * q^43 - 4 * q^44 + 5 * q^45 + 10 * q^46 + 8 * q^47 + 16 * q^48 + 59 * q^49 - 3 * q^50 + 12 * q^51 + 50 * q^52 - 22 * q^53 - 44 * q^54 - 12 * q^55 - 6 * q^56 + 12 * q^57 - 18 * q^58 - 4 * q^59 - 12 * q^60 + 6 * q^61 - 28 * q^62 + 32 * q^63 - 39 * q^64 + 10 * q^65 + 60 * q^66 - 12 * q^68 - 20 * q^69 - 8 * q^70 + 24 * q^71 - 103 * q^72 + 30 * q^73 - 34 * q^74 - 24 * q^76 - 12 * q^77 + 60 * q^78 + 4 * q^79 + 15 * q^80 + 35 * q^81 - 68 * q^82 - 108 * q^84 + 2 * q^85 + 4 * q^86 - 20 * q^87 - 68 * q^88 + 46 * q^89 - 13 * q^90 + 64 * q^91 - 72 * q^92 - 40 * q^93 - 48 * q^94 + 4 * q^95 + 32 * q^96 + 10 * q^97 - 45 * q^98 - 16 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(515))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		5	103
515.2.a.a	$515$	$2$	515.a	1.a	$1$	$4$	$4$	$4.112$	\(\Q(\sqrt{22 +2 \sqrt{5}})\)	None	✓	✓	✓	✓	515.2.a.a	$1$	$1$	\(-1\)	\(-1\)	\(4\)	\(-3\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(\beta _{1}-\beta _{3})q^{3}+(-1+\beta _{1}+\cdots)q^{4}+\cdots\)
515.2.a.b	$515$	$2$	515.a	1.a	$1$	$8$	$8$	$4.112$	8.8.\(\cdots\).1	None	✓	✓	✓	✓	515.2.a.b	$1$	$1$	\(-2\)	\(-3\)	\(-8\)	\(-11\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{3}q^{2}-\beta _{1}q^{3}+(1-\beta _{2})q^{4}-q^{5}+\cdots\)
515.2.a.c	$515$	$2$	515.a	1.a	$1$	$9$	$9$	$4.112$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	✓	✓	✓	515.2.a.c	$1$	$0$	\(1\)	\(3\)	\(-9\)	\(9\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+\beta _{7}q^{3}+(1+\beta _{2})q^{4}-q^{5}+\cdots\)
515.2.a.d	$515$	$2$	515.a	1.a	$1$	$14$	$14$	$4.112$	\(\mathbb{Q}[x]/(x^{14} - \cdots)\)	None	✓	✓	✓	✓	515.2.a.d	$1$	$0$	\(-1\)	\(1\)	\(14\)	\(9\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-\beta _{9}q^{3}+(1+\beta _{2})q^{4}+q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(515))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(515)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(103))\)\(^{\oplus 2}\)