Space of modular forms of level 115, weight 2, and trivial character

• Label: 115.2.a
• Level: $115$
• Weight: $2$
• Character orbit: 115.a
• Rep. character: $\chi_{115}(1,\cdot)$
• Character field: $\Q$
• Dimension: $7$
• Newform subspaces: $3$
• Sturm bound: $24$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 115 = 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	115.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(24\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(115))\).

	Total	New	Old
Modular forms	14	7	7
Cusp forms	11	7	4
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.

\(5\)	\(23\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(2\)	\(2\)	\(0\)		\(2\)	\(2\)	\(0\)		\(0\)	\(0\)	\(0\)
\(+\)	\(-\)	\(-\)		\(4\)	\(1\)	\(3\)		\(3\)	\(1\)	\(2\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)		\(5\)	\(4\)	\(1\)		\(4\)	\(4\)	\(0\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)		\(3\)	\(0\)	\(3\)		\(2\)	\(0\)	\(2\)		\(1\)	\(0\)	\(1\)
Plus space		\(+\)		\(5\)	\(2\)	\(3\)		\(4\)	\(2\)	\(2\)		\(1\)	\(0\)	\(1\)
Minus space		\(-\)		\(9\)	\(5\)	\(4\)		\(7\)	\(5\)	\(2\)		\(2\)	\(0\)	\(2\)

Trace form

7 * q + q^2 - 4 * q^3 + 9 * q^4 + q^5 + 2 * q^6 - 4 * q^7 + 3 * q^8 - q^9 + 3 * q^10 + 4 * q^11 - 22 * q^12 - 10 * q^13 - 12 * q^14 + 17 * q^16 - 2 * q^17 + 3 * q^18 - 4 * q^19 - q^20 + 12 * q^21 - 8 * q^22 - 5 * q^23 - 24 * q^24 + 7 * q^25 + 12 * q^26 - 4 * q^27 - 8 * q^28 + 16 * q^29 - 4 * q^30 - 2 * q^31 - 3 * q^32 - 10 * q^34 - 2 * q^35 + 11 * q^36 + 2 * q^37 + 28 * q^38 + 8 * q^39 + 15 * q^40 + 8 * q^41 + 8 * q^42 - 12 * q^43 - 32 * q^44 + 13 * q^45 + 3 * q^46 + 16 * q^47 - 34 * q^48 + q^49 + q^50 - 4 * q^51 - 32 * q^52 + 18 * q^53 - 22 * q^54 + 4 * q^55 - 24 * q^56 + 40 * q^58 + 10 * q^59 - 16 * q^60 - 6 * q^61 - 34 * q^62 - 12 * q^63 + 23 * q^64 + 10 * q^65 + 36 * q^66 + 8 * q^67 + 26 * q^68 + 4 * q^69 - 12 * q^70 - 6 * q^71 + 51 * q^72 - 26 * q^73 + 34 * q^74 - 4 * q^75 - 28 * q^76 + 12 * q^77 + 26 * q^78 + 12 * q^79 - q^80 - 17 * q^81 + 16 * q^82 - 16 * q^83 + 16 * q^84 - 8 * q^86 - 8 * q^87 + 12 * q^88 + 14 * q^89 + 3 * q^90 - 44 * q^91 - 5 * q^92 - 12 * q^93 + 22 * q^94 - 4 * q^95 - 46 * q^96 - 22 * q^97 + 17 * q^98 + 4 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(115))\) 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	23
115.2.a.a	$115$	$2$	115.a	1.a	$1$	$1$	$1$	$0.918$	\(\Q\)	None	✓	✓	✓	✓	115.2.a.a	$1$	$0$	\(2\)	\(0\)	\(-1\)	\(1\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+2q^{4}-q^{5}+q^{7}-3q^{9}-2q^{10}+\cdots\)
115.2.a.b	$115$	$2$	115.a	1.a	$1$	$2$	$2$	$0.918$	\(\Q(\sqrt{5}) \)	None	✓	✓	✓	✓	115.2.a.b	$1$	$1$	\(-3\)	\(-2\)	\(-2\)	\(-2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{2}-q^{3}+3\beta q^{4}-q^{5}+\cdots\)
115.2.a.c	$115$	$2$	115.a	1.a	$1$	$4$	$4$	$0.918$	\(\Q(\sqrt{11 +2 \sqrt{17}})\)	None	✓	✓	✓	✓	115.2.a.c	$1$	$0$	\(2\)	\(-2\)	\(4\)	\(-3\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(-1-\beta _{2})q^{3}+(1+\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(115))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(115)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 2}\)