Space of modular forms of level 117, weight 4, and trivial character

• Label: 117.4.a
• Level: $117$
• Weight: $4$
• Character orbit: 117.a
• Rep. character: $\chi_{117}(1,\cdot)$
• Character field: $\Q$
• Dimension: $15$
• Newform subspaces: $7$
• Sturm bound: $56$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 117 = 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	117.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(56\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(117))\).

	Total	New	Old
Modular forms	46	15	31
Cusp forms	38	15	23
Eisenstein series	8	0	8

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

\(3\)	\(13\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(4\)
\(+\)	\(-\)	\(-\)	\(2\)
\(-\)	\(+\)	\(-\)	\(4\)
\(-\)	\(-\)	\(+\)	\(5\)
Plus space		\(+\)	\(9\)
Minus space		\(-\)	\(6\)

Trace form

15 * q + 82 * q^4 - 6 * q^5 + 2 * q^7 - 54 * q^10 + 42 * q^11 - 13 * q^13 + 78 * q^14 + 218 * q^16 - 36 * q^17 - 162 * q^19 + 192 * q^20 - 16 * q^22 + 36 * q^23 + 635 * q^25 + 78 * q^26 + 888 * q^28 - 294 * q^29 - 730 * q^31 + 180 * q^32 - 184 * q^34 - 366 * q^35 + 78 * q^37 - 732 * q^38 - 1550 * q^40 - 402 * q^41 + 806 * q^43 + 600 * q^44 - 1996 * q^46 - 282 * q^47 - 91 * q^49 + 144 * q^50 - 624 * q^52 + 834 * q^53 + 452 * q^55 + 390 * q^56 - 1172 * q^58 + 1734 * q^59 + 1622 * q^61 + 48 * q^62 + 3626 * q^64 + 468 * q^65 + 2178 * q^67 - 2634 * q^68 - 3388 * q^70 + 1098 * q^71 - 1870 * q^73 - 3210 * q^74 - 1664 * q^76 - 1812 * q^77 - 1096 * q^79 + 1392 * q^80 - 168 * q^82 + 1422 * q^83 - 712 * q^85 - 5004 * q^86 - 456 * q^88 + 534 * q^89 - 676 * q^91 + 204 * q^92 + 5138 * q^94 + 3720 * q^95 + 1758 * q^97 + 5424 * q^98

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(117))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3	13
117.4.a.a	$117$	$4$	117.a	1.a	$1$	$1$	$1$	$6.903$	\(\Q\)	None	✓				39.4.a.a	$1$	$0$	\(0\)	\(0\)	\(12\)	\(2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-8q^{4}+12q^{5}+2q^{7}+6^{2}q^{11}+13q^{13}+\cdots\)
117.4.a.b	$117$	$4$	117.a	1.a	$1$	$1$	$1$	$6.903$	\(\Q\)	None	✓				13.4.a.a	$1$	$0$	\(5\)	\(0\)	\(7\)	\(-13\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+5q^{2}+17q^{4}+7q^{5}-13q^{7}+45q^{8}+\cdots\)
117.4.a.c	$117$	$4$	117.a	1.a	$1$	$2$	$2$	$6.903$	\(\Q(\sqrt{14}) \)	None	✓				39.4.a.b	$1$	$1$	\(-2\)	\(0\)	\(-24\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{2}+(7-2\beta )q^{4}+(-12+\cdots)q^{5}+\cdots\)
117.4.a.d	$117$	$4$	117.a	1.a	$1$	$2$	$2$	$6.903$	\(\Q(\sqrt{17}) \)	None	✓				13.4.a.b	$1$	$1$	\(-1\)	\(0\)	\(3\)	\(-9\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(-4+\beta )q^{4}+(2-\beta )q^{5}+(-10+\cdots)q^{7}+\cdots\)
117.4.a.e	$117$	$4$	117.a	1.a	$1$	$2$	$2$	$6.903$	\(\Q(\sqrt{7}) \)	None	✓	✓	✓	✓	117.4.a.e	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-44\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}-q^{4}-4\beta q^{5}-22q^{7}-9\beta q^{8}+\cdots\)
117.4.a.f	$117$	$4$	117.a	1.a	$1$	$3$	$3$	$6.903$	3.3.3144.1	None	✓		✓		39.4.a.c	$1$	$0$	\(-2\)	\(0\)	\(-4\)	\(30\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(3+\beta _{2})q^{4}+(-2+\cdots)q^{5}+\cdots\)
117.4.a.g	$117$	$4$	117.a	1.a	$1$	$4$	$4$	$6.903$	4.4.1520092.1	None	✓	✓	✓	✓	117.4.a.g	$2$	$0$	\(0\)	\(0\)	\(0\)	\(36\)		$+$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(14+\beta _{2})q^{4}-\beta _{3}q^{5}+(8+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(117))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(117)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(13))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 2}\)