Space of modular forms of level 117, weight 8, and Character 117.g

• Label: 117.8.g
• Level: $117$
• Weight: $8$
• Character orbit: 117.g
• Rep. character: $\chi_{117}(55,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $80$
• Newform subspaces: $5$
• Sturm bound: $112$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 117 = 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	117.g (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 13 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(112\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(117, [\chi])\).

	Total	New	Old
Modular forms	204	84	120
Cusp forms	188	80	108
Eisenstein series	16	4	12

Trace form

80 * q - 7 * q^2 - 2497 * q^4 + 392 * q^5 - 530 * q^7 + 222 * q^8 - 1893 * q^10 - 6762 * q^11 - 10276 * q^13 - 29924 * q^14 - 167305 * q^16 + 16482 * q^17 - 19964 * q^19 - 89713 * q^20 - 49458 * q^22 + 71242 * q^23 + 1150220 * q^25 + 18405 * q^26 + 55372 * q^28 - 11316 * q^29 - 198356 * q^31 + 442625 * q^32 + 1617150 * q^34 + 473130 * q^35 + 368902 * q^37 + 6028 * q^38 + 575862 * q^40 - 338006 * q^41 - 1363922 * q^43 + 1634680 * q^44 + 2645814 * q^46 - 2598588 * q^47 - 5593782 * q^49 + 974366 * q^50 + 7062530 * q^52 + 3270140 * q^53 - 2438256 * q^55 + 1428152 * q^56 + 488289 * q^58 + 2234086 * q^59 + 2242108 * q^61 + 1856720 * q^62 + 13228034 * q^64 - 7803832 * q^65 + 8617918 * q^67 + 1054635 * q^68 + 22193304 * q^70 + 8509030 * q^71 + 6007984 * q^73 - 3414317 * q^74 - 8956976 * q^76 - 5825260 * q^77 + 6378124 * q^79 - 4508669 * q^80 + 20701413 * q^82 + 6153456 * q^83 + 850542 * q^85 + 44499348 * q^86 + 9338352 * q^88 + 11996560 * q^89 - 5091230 * q^91 - 69899528 * q^92 - 21188196 * q^94 - 7726892 * q^95 + 6623566 * q^97 - 13670273 * q^98

Decomposition of \(S_{8}^{\mathrm{new}}(117, [\chi])\) 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				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
117.8.g.a	$117$	$8$	117.g	13.c	$3$	$2$	$1$	$36.549$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		✓			117.8.g.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-1763\)	$1$	$\mathrm{U}(1)[D_{3}]$	\(q+2^{7}\zeta_{6}q^{4}-1763\zeta_{6}q^{7}+(3532+5541\zeta_{6})q^{13}+\cdots\)
117.8.g.b	$117$	$8$	117.g	13.c	$3$	$14$	$7$	$36.549$	\(\mathbb{Q}[x]/(x^{14} - \cdots)\)	None					39.8.e.a	$2$	$0$	\(-8\)	\(0\)	\(642\)	\(-688\)	$2^{12}\cdot 3^{3}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1-\beta _{1}+\beta _{4})q^{2}+(-66\beta _{4}+\beta _{10}+\cdots)q^{4}+\cdots\)
117.8.g.c	$117$	$8$	117.g	13.c	$3$	$16$	$8$	$36.549$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None					39.8.e.b	$2$	$0$	\(-8\)	\(0\)	\(134\)	\(325\)	$2^{12}\cdot 3^{3}\cdot 13^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(\beta _{1}-\beta _{3})q^{2}+(-38+3\beta _{2}+38\beta _{3}+\cdots)q^{4}+\cdots\)
117.8.g.d	$117$	$8$	117.g	13.c	$3$	$16$	$8$	$36.549$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None					13.8.c.a	$2$	$0$	\(9\)	\(0\)	\(-384\)	\(196\)	$2^{12}\cdot 3\cdot 13^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1+\beta _{1}+\beta _{2})q^{2}+(72\beta _{2}+2\beta _{3}+\beta _{5}+\cdots)q^{4}+\cdots\)
117.8.g.e	$117$	$8$	117.g	13.c	$3$	$32$	$16$	$36.549$		None		✓	✓		117.8.g.e	$4$	$0$	\(0\)	\(0\)	\(0\)	\(1400\)		$\mathrm{SU}(2)[C_{3}]$

Decomposition of \(S_{8}^{\mathrm{old}}(117, [\chi])\) into lower level spaces

\( S_{8}^{\mathrm{old}}(117, [\chi]) \simeq \) \(S_{8}^{\mathrm{new}}(13, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(39, [\chi])\)\(^{\oplus 2}\)