Space of modular forms of level 1690, weight 2, and Character 1690.d

• Label: 1690.2.d
• Level: $1690$
• Weight: $2$
• Character orbit: 1690.d
• Rep. character: $\chi_{1690}(1351,\cdot)$
• Character field: $\Q$
• Dimension: $54$
• Newform subspaces: $12$
• Sturm bound: $546$
• Trace bound: $14$

Defining parameters

Level:	\( N \)	\(=\)	\( 1690 = 2 \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1690.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 13 \)
Character field:			\(\Q\)
Newform subspaces:			\( 12 \)
Sturm bound:			\(546\)
Trace bound:			\(14\)
Distinguishing \(T_p\):			\(3\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1690, [\chi])\).

	Total	New	Old
Modular forms	302	54	248
Cusp forms	246	54	192
Eisenstein series	56	0	56

Trace form

\( 54 q - 4 q^{3} - 54 q^{4} + 54 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1690, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1690.2.d.a	$1690$	$2$	1690.d	13.b	$2$	$2$	$2$	$13.495$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(-4\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{2}-2q^{3}-q^{4}+iq^{5}-2iq^{6}+\cdots\)
1690.2.d.b	$1690$	$2$	1690.d	13.b	$2$	$2$	$2$	$13.495$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(-4\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{2}-2q^{3}-q^{4}-iq^{5}-2iq^{6}+\cdots\)
1690.2.d.c	$1690$	$2$	1690.d	13.b	$2$	$2$	$2$	$13.495$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{2}-q^{4}+iq^{5}+3iq^{7}-iq^{8}+\cdots\)
1690.2.d.d	$1690$	$2$	1690.d	13.b	$2$	$2$	$2$	$13.495$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{2}-q^{4}+iq^{5}-iq^{8}-3q^{9}+\cdots\)
1690.2.d.e	$1690$	$2$	1690.d	13.b	$2$	$2$	$2$	$13.495$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(4\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{2}+2q^{3}-q^{4}-iq^{5}+2iq^{6}+\cdots\)
1690.2.d.f	$1690$	$2$	1690.d	13.b	$2$	$4$	$4$	$13.495$	\(\Q(\zeta_{12})\)	None		$2$	$0$	\(0\)	\(-4\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\zeta_{12}q^{2}+(-1+\zeta_{12}^{3})q^{3}-q^{4}+\cdots\)
1690.2.d.g	$1690$	$2$	1690.d	13.b	$2$	$4$	$4$	$13.495$	\(\Q(i, \sqrt{10})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}-\beta _{3}q^{3}-q^{4}-\beta _{1}q^{5}-\beta _{2}q^{6}+\cdots\)
1690.2.d.h	$1690$	$2$	1690.d	13.b	$2$	$4$	$4$	$13.495$	\(\Q(\zeta_{12})\)	None		$2$	$0$	\(0\)	\(4\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\zeta_{12}q^{2}+(1+\zeta_{12}^{3})q^{3}-q^{4}-\zeta_{12}q^{5}+\cdots\)
1690.2.d.i	$1690$	$2$	1690.d	13.b	$2$	$6$	$6$	$13.495$	6.0.153664.1	None		$2$	$0$	\(0\)	\(-2\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{5}q^{2}-\beta _{2}q^{3}-q^{4}-\beta _{5}q^{5}+(\beta _{1}+\cdots)q^{6}+\cdots\)
1690.2.d.j	$1690$	$2$	1690.d	13.b	$2$	$6$	$6$	$13.495$	6.0.153664.1	None		$2$	$0$	\(0\)	\(2\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{5}q^{2}+\beta _{4}q^{3}-q^{4}+\beta _{5}q^{5}-\beta _{1}q^{6}+\cdots\)
1690.2.d.k	$1690$	$2$	1690.d	13.b	$2$	$8$	$8$	$13.495$	8.0.22581504.2	None		$2$	$0$	\(0\)	\(4\)	\(0\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{5}q^{2}+\beta _{2}q^{3}-q^{4}-\beta _{5}q^{5}+\beta _{3}q^{6}+\cdots\)
1690.2.d.l	$1690$	$2$	1690.d	13.b	$2$	$12$	$12$	$13.495$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		$2$	$0$	\(0\)	\(-4\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{2}+(-1+\beta _{5}+\beta _{8})q^{3}-q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1690, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1690, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(26, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(65, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(130, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(169, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(338, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(845, [\chi])\)\(^{\oplus 2}\)