Space of modular forms of level 117, weight 3, and Character 117.bd

• Label: 117.3.bd
• Level: $117$
• Weight: $3$
• Character orbit: 117.bd
• Rep. character: $\chi_{117}(19,\cdot)$
• Character field: $\Q(\zeta_{12})$
• Dimension: $44$
• Newform subspaces: $5$
• Sturm bound: $42$
• Trace bound: $2$

Defining parameters

Level:	$N$	$=$	$117 = 3^{2} \cdot 13$
Weight:	$k$	$=$	$3$
Character orbit:	$[\chi]$	$=$	117.bd (of order $12$ and degree $4$)
Character conductor:	$\operatorname{cond}(\chi)$	$=$	$13$
Character field:			$\Q(\zeta_{12})$
Newform subspaces:			$5$
Sturm bound:			$42$
Trace bound:			$2$
Distinguishing $T_p$:			$2$

Dimensions

The following table gives the dimensions of various subspaces of $M_{3}(117, [\chi])$.

	Total	New	Old
Modular forms	128	52	76
Cusp forms	96	44	52
Eisenstein series	32	8	24

Trace form

44 * q + 6 * q^2 - 6 * q^4 - 6 * q^5 - 20 * q^7 + 54 * q^8 - 12 * q^11 + 6 * q^13 - 24 * q^14 + 46 * q^16 + 22 * q^19 + 114 * q^20 + 108 * q^22 + 30 * q^23 - 144 * q^26 - 148 * q^28 - 66 * q^29 - 112 * q^31 - 252 * q^32 - 354 * q^34 - 168 * q^35 + 56 * q^37 - 72 * q^40 + 204 * q^41 + 192 * q^43 + 240 * q^44 - 270 * q^46 + 360 * q^47 + 60 * q^49 + 258 * q^50 + 320 * q^52 - 240 * q^53 + 204 * q^55 + 120 * q^56 + 432 * q^58 + 276 * q^59 + 108 * q^61 - 522 * q^62 - 780 * q^65 - 242 * q^67 - 264 * q^68 - 756 * q^70 - 78 * q^71 - 482 * q^73 - 288 * q^74 - 386 * q^76 + 120 * q^79 + 96 * q^80 - 144 * q^82 + 228 * q^83 + 264 * q^85 + 1188 * q^86 + 1092 * q^88 + 714 * q^89 + 1120 * q^91 + 36 * q^92 + 546 * q^94 + 798 * q^95 - 10 * q^97 + 894 * q^98

Decomposition of $S_{3}^{\mathrm{new}}(117, [\chi])$ 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
117.3.bd.a	$117$	$3$	117.bd	13.f	$12$	$4$	$1$	$3.188$	$\Q(\zeta_{12})$	$\Q(\sqrt{-3})$		✓			117.3.bd.a	$4$	$0$	$0$	$0$	$0$	$-22$	$1$	$\mathrm{U}(1)[D_{12}]$	$q-4\zeta_{12}q^{4}+(-8-8\zeta_{12}+5\zeta_{12}^{2}+\cdots)q^{7}+\cdots$
117.3.bd.b	$117$	$3$	117.bd	13.f	$12$	$4$	$1$	$3.188$	$\Q(\zeta_{12})$	None					13.3.f.a	$2$	$0$	$2$	$0$	$14$	$16$	$1$	$\mathrm{SU}(2)[C_{12}]$	$q+(1-\zeta_{12}^{2}+\zeta_{12}^{3})q^{2}+(-1-2\zeta_{12}+\cdots)q^{4}+\cdots$
117.3.bd.c	$117$	$3$	117.bd	13.f	$12$	$8$	$2$	$3.188$	8.0.$\cdots$.10	None					39.3.l.a	$2$	$0$	$2$	$0$	$-16$	$14$	$1$	$\mathrm{SU}(2)[C_{12}]$	$q+(1+\beta _{2}-\beta _{3}+\beta _{4})q^{2}+(2-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots$
117.3.bd.d	$117$	$3$	117.bd	13.f	$12$	$12$	$3$	$3.188$	$\mathbb{Q}[x]/(x^{12} - \cdots)$	None					39.3.l.b	$2$	$0$	$2$	$0$	$-4$	$-32$	$1$	$\mathrm{SU}(2)[C_{12}]$	$q+(\beta _{1}+\beta _{2})q^{2}+(-2-3\beta _{5}-\beta _{6}-\beta _{11})q^{4}+\cdots$
117.3.bd.e	$117$	$3$	117.bd	13.f	$12$	$16$	$4$	$3.188$	$\mathbb{Q}[x]/(x^{16} + \cdots)$	None		✓	✓		117.3.bd.e	$4$	$0$	$0$	$0$	$0$	$4$	$2^{4}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{12}]$	$q+\beta _{12}q^{2}+(-1+2\beta _{1}+\beta _{2}-3\beta _{3}+\cdots)q^{4}+\cdots$

Decomposition of $S_{3}^{\mathrm{old}}(117, [\chi])$ into lower level spaces

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