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

• Label: 1710.2.n
• Level: $1710$
• Weight: $2$
• Character orbit: 1710.n
• Rep. character: $\chi_{1710}(647,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $72$
• Newform subspaces: $9$
• Sturm bound: $720$
• Trace bound: $10$

Defining parameters

Level:	\( N \)	\(=\)	\( 1710 = 2 \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1710.n (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 15 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(720\)
Trace bound:			\(10\)
Distinguishing \(T_p\):			\(7\), \(17\)

Dimensions

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

	Total	New	Old
Modular forms	752	72	680
Cusp forms	688	72	616
Eisenstein series	64	0	64

Trace form

72 * q - 16 * q^7 + 16 * q^10 - 8 * q^13 - 72 * q^16 + 16 * q^22 - 16 * q^28 + 32 * q^31 - 8 * q^37 + 8 * q^40 + 16 * q^43 - 32 * q^46 + 8 * q^52 - 16 * q^55 - 8 * q^58 - 64 * q^61 - 64 * q^67 + 48 * q^70 + 72 * q^73 + 56 * q^82 - 96 * q^85 + 16 * q^88 + 64 * q^91 + 88 * q^97

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	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1710.2.n.a	$1710$	$2$	1710.n	15.e	$4$	$4$	$2$	$13.654$	\(\Q(\zeta_{8})\)	None		✓			1710.2.n.a	$4$	$1$	\(0\)	\(0\)	\(0\)	\(-12\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q-\zeta_{8}^{3}q^{2}-\zeta_{8}^{2}q^{4}+(-2\zeta_{8}+\zeta_{8}^{3})q^{5}+\cdots\)
1710.2.n.b	$1710$	$2$	1710.n	15.e	$4$	$4$	$2$	$13.654$	\(\Q(\zeta_{8})\)	None		✓			1710.2.n.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+\zeta_{8}q^{2}+\zeta_{8}^{2}q^{4}+(-2\zeta_{8}-\zeta_{8}^{3})q^{5}+\cdots\)
1710.2.n.c	$1710$	$2$	1710.n	15.e	$4$	$4$	$2$	$13.654$	\(\Q(\zeta_{8})\)	None		✓			1710.2.n.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q-\zeta_{8}q^{2}+\zeta_{8}^{2}q^{4}+(-\zeta_{8}+2\zeta_{8}^{3})q^{5}+\cdots\)
1710.2.n.d	$1710$	$2$	1710.n	15.e	$4$	$4$	$2$	$13.654$	\(\Q(\zeta_{8})\)	None		✓			1710.2.n.d	$4$	$0$	\(0\)	\(0\)	\(0\)	\(8\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q-\zeta_{8}q^{2}+\zeta_{8}^{2}q^{4}+(-\zeta_{8}+2\zeta_{8}^{3})q^{5}+\cdots\)
1710.2.n.e	$1710$	$2$	1710.n	15.e	$4$	$4$	$2$	$13.654$	\(\Q(\zeta_{8})\)	None		✓			1710.2.n.e	$4$	$0$	\(0\)	\(0\)	\(0\)	\(12\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+\zeta_{8}^{3}q^{2}-\zeta_{8}^{2}q^{4}+(-\zeta_{8}+2\zeta_{8}^{3})q^{5}+\cdots\)
1710.2.n.f	$1710$	$2$	1710.n	15.e	$4$	$8$	$4$	$13.654$	8.0.110166016.2	None		✓			1710.2.n.f	$2$	$0$	\(0\)	\(0\)	\(-8\)	\(-8\)	$2^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{2}q^{2}+\beta _{5}q^{4}+(-1+\beta _{3}-\beta _{7})q^{5}+\cdots\)
1710.2.n.g	$1710$	$2$	1710.n	15.e	$4$	$8$	$4$	$13.654$	8.0.110166016.2	None		✓			1710.2.n.f	$2$	$0$	\(0\)	\(0\)	\(8\)	\(-8\)	$2^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{2}q^{2}+\beta _{5}q^{4}+(1-\beta _{2}-\beta _{7})q^{5}+\cdots\)
1710.2.n.h	$1710$	$2$	1710.n	15.e	$4$	$16$	$8$	$13.654$	16.0.\(\cdots\).1	None		✓			1710.2.n.h	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{8}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{5}q^{2}+\beta _{9}q^{4}+(\beta _{1}-\beta _{15})q^{5}+(\beta _{4}+\cdots)q^{7}+\cdots\)
1710.2.n.i	$1710$	$2$	1710.n	15.e	$4$	$20$	$10$	$13.654$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None		✓	✓		1710.2.n.i	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{10}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{7}q^{2}-\beta _{10}q^{4}+\beta _{14}q^{5}+\beta _{17}q^{7}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(1710, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(90, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(285, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(570, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(855, [\chi])\)\(^{\oplus 2}\)