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

\( 15 q + 82 q^{4} - 6 q^{5} + 2 q^{7} + O(q^{10}) \)

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

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

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