Space of modular forms of level 717, weight 2, and trivial character

• Label: 717.2.a
• Level: $717$
• Weight: $2$
• Character orbit: 717.a
• Rep. character: $\chi_{717}(1,\cdot)$
• Character field: $\Q$
• Dimension: $39$
• Newform subspaces: $7$
• Sturm bound: $160$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 717 = 3 \cdot 239 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	717.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(160\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(717))\).

	Total	New	Old
Modular forms	82	39	43
Cusp forms	79	39	40
Eisenstein series	3	0	3

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

\(3\)	\(239\)	Fricke	Dim
\(+\)	\(+\)	$+$	\(8\)
\(+\)	\(-\)	$-$	\(12\)
\(-\)	\(+\)	$-$	\(11\)
\(-\)	\(-\)	$+$	\(8\)
Plus space		\(+\)	\(16\)
Minus space		\(-\)	\(23\)

Trace form

\( 39 q + q^{2} - q^{3} + 37 q^{4} - 2 q^{5} - 3 q^{6} - 3 q^{8} + 39 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(717))\) 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	239
717.2.a.a	$717$	$2$	717.a	1.a	$1$	$2$	$2$	$5.725$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(1\)	\(-2\)	\(-2\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}-q^{3}+(-1+\beta )q^{4}-q^{5}-\beta q^{6}+\cdots\)
717.2.a.b	$717$	$2$	717.a	1.a	$1$	$2$	$2$	$5.725$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(1\)	\(2\)	\(6\)	\(2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+q^{3}+(-1+\beta )q^{4}+3q^{5}+\cdots\)
717.2.a.c	$717$	$2$	717.a	1.a	$1$	$2$	$2$	$5.725$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(3\)	\(2\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+q^{3}+3\beta q^{4}+(1-2\beta )q^{5}+\cdots\)
717.2.a.d	$717$	$2$	717.a	1.a	$1$	$6$	$6$	$5.725$	6.6.1767625.1	None	✓	$1$	$1$	\(-2\)	\(-6\)	\(5\)	\(-9\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{5}q^{2}-q^{3}+(1+\beta _{2}+\beta _{3}-\beta _{4}+\cdots)q^{4}+\cdots\)
717.2.a.e	$717$	$2$	717.a	1.a	$1$	$7$	$7$	$5.725$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(7\)	\(3\)	\(3\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{2}+q^{3}+(1+\beta _{6})q^{4}+\beta _{6}q^{5}+\cdots\)
717.2.a.f	$717$	$2$	717.a	1.a	$1$	$8$	$8$	$5.725$	8.8.2585660609.1	None	✓	$1$	$1$	\(-5\)	\(8\)	\(-13\)	\(-7\)		$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{2}+q^{3}+(1+\beta _{2}+\beta _{3}+\cdots)q^{4}+\cdots\)
717.2.a.g	$717$	$2$	717.a	1.a	$1$	$12$	$12$	$5.725$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	$1$	$0$	\(3\)	\(-12\)	\(-1\)	\(11\)		$+$	$-$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}-\beta _{9}q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(717))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(717)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(239))\)\(^{\oplus 2}\)