Space of modular forms of level 115, weight 4, and trivial character

• Label: 115.4.a
• Level: $115$
• Weight: $4$
• Character orbit: 115.a
• Rep. character: $\chi_{115}(1,\cdot)$
• Character field: $\Q$
• Dimension: $22$
• Newform subspaces: $6$
• Sturm bound: $48$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 115 = 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	115.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(48\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(115))\).

	Total	New	Old
Modular forms	38	22	16
Cusp forms	34	22	12
Eisenstein series	4	0	4

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	Dim
\(+\)	\(+\)	\(+\)	\(5\)
\(+\)	\(-\)	\(-\)	\(6\)
\(-\)	\(+\)	\(-\)	\(3\)
\(-\)	\(-\)	\(+\)	\(8\)
Plus space		\(+\)	\(13\)
Minus space		\(-\)	\(9\)

Trace form

22 * q + 4 * q^2 - 4 * q^3 + 88 * q^4 - 10 * q^6 - 40 * q^7 + 42 * q^8 + 290 * q^9 - 92 * q^11 + 46 * q^12 + 60 * q^13 + 284 * q^14 - 40 * q^15 + 408 * q^16 - 220 * q^17 - 402 * q^18 - 196 * q^19 - 40 * q^20 - 376 * q^21 + 464 * q^22 + 138 * q^23 - 12 * q^24 + 550 * q^25 - 242 * q^26 - 268 * q^27 - 280 * q^28 - 432 * q^29 + 80 * q^30 + 460 * q^31 + 40 * q^32 + 200 * q^33 - 596 * q^34 + 300 * q^35 + 1146 * q^36 - 856 * q^37 - 660 * q^38 - 596 * q^39 + 284 * q^41 - 712 * q^42 - 1044 * q^43 - 840 * q^44 + 1396 * q^47 + 398 * q^48 + 1158 * q^49 + 100 * q^50 + 88 * q^51 - 1430 * q^52 + 1416 * q^53 - 1786 * q^54 + 440 * q^55 + 4544 * q^56 - 2704 * q^57 + 2198 * q^58 + 156 * q^59 + 340 * q^60 - 864 * q^61 - 1670 * q^62 + 192 * q^63 + 1118 * q^64 - 640 * q^65 + 1044 * q^66 - 2196 * q^67 - 4640 * q^68 + 120 * q^70 + 204 * q^71 - 5838 * q^72 + 1636 * q^73 - 432 * q^74 - 100 * q^75 + 1092 * q^76 + 784 * q^77 + 1406 * q^78 + 1696 * q^79 - 2640 * q^80 + 3574 * q^81 + 1938 * q^82 - 740 * q^83 - 10392 * q^84 + 780 * q^85 + 3488 * q^86 + 2068 * q^87 + 2372 * q^88 - 396 * q^89 + 2340 * q^90 + 1176 * q^91 + 1104 * q^92 + 3016 * q^93 - 6534 * q^94 + 1480 * q^95 - 5166 * q^96 - 2196 * q^97 - 6772 * q^98 + 2212 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(115))\) 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
115.4.a.a	$115$	$4$	115.a	1.a	$1$	$1$	$1$	$6.785$	\(\Q\)	None	✓	✓			115.4.a.a	$1$	$1$	\(1\)	\(4\)	\(-5\)	\(-32\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+4q^{3}-7q^{4}-5q^{5}+4q^{6}+\cdots\)
115.4.a.b	$115$	$4$	115.a	1.a	$1$	$1$	$1$	$6.785$	\(\Q\)	None	✓	✓			115.4.a.b	$1$	$1$	\(2\)	\(-3\)	\(5\)	\(-2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-3q^{3}-4q^{4}+5q^{5}-6q^{6}+\cdots\)
115.4.a.c	$115$	$4$	115.a	1.a	$1$	$2$	$2$	$6.785$	\(\Q(\sqrt{109}) \)	None	✓	✓	✓		115.4.a.c	$1$	$1$	\(-6\)	\(-3\)	\(10\)	\(1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{2}+(-1-\beta )q^{3}+q^{4}+5q^{5}+\cdots\)
115.4.a.d	$115$	$4$	115.a	1.a	$1$	$5$	$5$	$6.785$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓	✓		115.4.a.d	$1$	$1$	\(-5\)	\(-6\)	\(-25\)	\(-15\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(-1-\beta _{2}-\beta _{3}-\beta _{4})q^{3}+\cdots\)
115.4.a.e	$115$	$4$	115.a	1.a	$1$	$5$	$5$	$6.785$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓	✓	✓	115.4.a.e	$1$	$0$	\(6\)	\(4\)	\(-25\)	\(-3\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{2}+(1+\beta _{1}-\beta _{2}+\beta _{3})q^{3}+\cdots\)
115.4.a.f	$115$	$4$	115.a	1.a	$1$	$8$	$8$	$6.785$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓	✓	✓	115.4.a.f	$1$	$0$	\(6\)	\(0\)	\(40\)	\(11\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+\beta _{6}q^{3}+(5+\beta _{2})q^{4}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(115))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(115)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 2}\)