Space of modular forms of level 1710, weight 2, and Character 1710.p

• Label: 1710.2.p
• Level: $1710$
• Weight: $2$
• Character orbit: 1710.p
• Rep. character: $\chi_{1710}(37,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $100$
• Newform subspaces: $5$
• Sturm bound: $720$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 1710 = 2 \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1710.p (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 95 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(720\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1710, [\chi])\).

	Total	New	Old
Modular forms	752	100	652
Cusp forms	688	100	588
Eisenstein series	64	0	64

Trace form

\( 100 q - 4 q^{5} + 4 q^{7} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1710, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1710.2.p.a	$1710$	$2$	1710.p	95.g	$4$	$4$	$2$	$13.654$	\(\Q(\zeta_{8})\)	None		$4$	$0$	\(0\)	\(0\)	\(4\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+\zeta_{8}^{3}q^{2}-\zeta_{8}^{2}q^{4}+(1+2\zeta_{8}^{2})q^{5}+\cdots\)
1710.2.p.b	$1710$	$2$	1710.p	95.g	$4$	$16$	$8$	$13.654$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(8\)	$2^{5}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{9}q^{2}-\beta _{2}q^{4}+(-\beta _{2}-\beta _{4})q^{5}+\cdots\)
1710.2.p.c	$1710$	$2$	1710.p	95.g	$4$	$20$	$10$	$13.654$	\(\mathbb{Q}[x]/(x^{20} + \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(-12\)	\(-4\)	$2^{9}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{7}q^{2}+\beta _{2}q^{4}+(-1-\beta _{6})q^{5}+(-1+\cdots)q^{7}+\cdots\)
1710.2.p.d	$1710$	$2$	1710.p	95.g	$4$	$20$	$10$	$13.654$	\(\mathbb{Q}[x]/(x^{20} + \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(4\)	\(-4\)	$2^{10}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{2}q^{2}+\beta _{4}q^{4}-\beta _{6}q^{5}+(\beta _{6}-\beta _{9}+\cdots)q^{7}+\cdots\)
1710.2.p.e	$1710$	$2$	1710.p	95.g	$4$	$40$	$20$	$13.654$		None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(8\)		$\mathrm{SU}(2)[C_{4}]$

Decomposition of \(S_{2}^{\mathrm{old}}(1710, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1710, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(95, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(190, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(285, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(570, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(855, [\chi])\)\(^{\oplus 2}\)