Space of modular forms of level 117, weight 6, and trivial character

• Label: 117.6.a
• Level: $117$
• Weight: $6$
• Character orbit: 117.a
• Rep. character: $\chi_{117}(1,\cdot)$
• Character field: $\Q$
• Dimension: $25$
• Newform subspaces: $8$
• Sturm bound: $84$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 117 = 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	117.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(84\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(117))\).

	Total	New	Old
Modular forms	74	25	49
Cusp forms	66	25	41
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
\(+\)	\(+\)	\(+\)	\(4\)
\(+\)	\(-\)	\(-\)	\(6\)
\(-\)	\(+\)	\(-\)	\(8\)
\(-\)	\(-\)	\(+\)	\(7\)
Plus space		\(+\)	\(11\)
Minus space		\(-\)	\(14\)

Trace form

25 * q - 6 * q^2 + 358 * q^4 + 30 * q^5 + 56 * q^7 + 168 * q^8 - 222 * q^10 - 1704 * q^11 + 169 * q^13 + 882 * q^14 + 5530 * q^16 + 2358 * q^17 + 4556 * q^19 + 2712 * q^20 - 2688 * q^22 - 4872 * q^23 + 19675 * q^25 - 2028 * q^26 - 10600 * q^28 + 14310 * q^29 - 808 * q^31 + 9540 * q^32 - 2220 * q^34 + 5040 * q^35 - 2602 * q^37 + 36180 * q^38 + 26226 * q^40 - 6414 * q^41 + 6596 * q^43 - 57720 * q^44 + 780 * q^46 - 42204 * q^47 + 101037 * q^49 + 39990 * q^50 - 11492 * q^52 - 14874 * q^53 - 42888 * q^55 + 26118 * q^56 + 75672 * q^58 - 110376 * q^59 - 67042 * q^61 + 33024 * q^62 - 39830 * q^64 - 17238 * q^65 - 25228 * q^67 + 6630 * q^68 - 87924 * q^70 - 62748 * q^71 + 76730 * q^73 + 75198 * q^74 + 30176 * q^76 + 253440 * q^77 - 129280 * q^79 + 412416 * q^80 - 136956 * q^82 - 14136 * q^83 + 51180 * q^85 - 160284 * q^86 - 209640 * q^88 - 191550 * q^89 + 70304 * q^91 - 479124 * q^92 - 210858 * q^94 + 269136 * q^95 - 239734 * q^97 - 461094 * q^98

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(117))\) 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
117.6.a.a	$117$	$6$	117.a	1.a	$1$	$1$	$1$	$18.765$	\(\Q\)	None	✓				39.6.a.a	$1$	$1$	\(-2\)	\(0\)	\(74\)	\(-112\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-28q^{4}+74q^{5}-112q^{7}+\cdots\)
117.6.a.b	$117$	$6$	117.a	1.a	$1$	$2$	$2$	$18.765$	\(\Q(\sqrt{14}) \)	None	✓				39.6.a.b	$1$	$0$	\(4\)	\(0\)	\(48\)	\(72\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(2+\beta )q^{2}+(28+4\beta )q^{4}+(24-5\beta )q^{5}+\cdots\)
117.6.a.c	$117$	$6$	117.a	1.a	$1$	$2$	$2$	$18.765$	\(\Q(\sqrt{17}) \)	None	✓				13.6.a.a	$1$	$0$	\(5\)	\(0\)	\(42\)	\(-36\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(3-\beta )q^{2}+(-19-5\beta )q^{4}+(41-40\beta )q^{5}+\cdots\)
117.6.a.d	$117$	$6$	117.a	1.a	$1$	$3$	$3$	$18.765$	3.3.168897.1	None	✓				13.6.a.b	$1$	$1$	\(-7\)	\(0\)	\(-56\)	\(-60\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-2-\beta _{1})q^{2}+(40+\beta _{1}+4\beta _{2})q^{4}+\cdots\)
117.6.a.e	$117$	$6$	117.a	1.a	$1$	$3$	$3$	$18.765$	3.3.125308.1	None	✓				39.6.a.c	$1$	$1$	\(0\)	\(0\)	\(-54\)	\(84\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(5+2\beta _{1}+\beta _{2})q^{4}+(-18+\cdots)q^{5}+\cdots\)
117.6.a.f	$117$	$6$	117.a	1.a	$1$	$4$	$4$	$18.765$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓		✓		39.6.a.d	$1$	$0$	\(-6\)	\(0\)	\(-24\)	\(72\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-2-\beta _{1})q^{2}+(33-3\beta _{1}+2\beta _{2}+\cdots)q^{4}+\cdots\)
117.6.a.g	$117$	$6$	117.a	1.a	$1$	$4$	$4$	$18.765$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓	✓	✓	117.6.a.g	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-288\)		$+$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(15+\beta _{3})q^{4}+(-4\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)
117.6.a.h	$117$	$6$	117.a	1.a	$1$	$6$	$6$	$18.765$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓	✓	✓	117.6.a.h	$2$	$0$	\(0\)	\(0\)	\(0\)	\(324\)		$+$	$-$	$-$	$2^{5}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(6+\beta _{2})q^{4}+\beta _{3}q^{5}+(54+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(117))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(117)) \simeq \) \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(13))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 2}\)