Space of modular forms of level 1680, weight 2, and Character 1680.cz

• Label: 1680.2.cz
• Level: $1680$
• Weight: $2$
• Character orbit: 1680.cz
• Rep. character: $\chi_{1680}(97,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $96$
• Newform subspaces: $6$
• Sturm bound: $768$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1680.cz (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 35 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(768\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1680, [\chi])\).

	Total	New	Old
Modular forms	816	96	720
Cusp forms	720	96	624
Eisenstein series	96	0	96

Trace form

\( 96 q + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1680, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1680.2.cz.a	$1680$	$2$	1680.cz	35.f	$4$	$8$	$4$	$13.415$	8.0.1698758656.6	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-8\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{1}q^{3}-\beta _{2}q^{5}+(-1-\beta _{3}+\beta _{4}+\cdots)q^{7}+\cdots\)
1680.2.cz.b	$1680$	$2$	1680.cz	35.f	$4$	$8$	$4$	$13.415$	8.0.1698758656.6	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{1}q^{3}+\beta _{2}q^{5}+(-\beta _{1}+\beta _{3}-\beta _{5}+\cdots)q^{7}+\cdots\)
1680.2.cz.c	$1680$	$2$	1680.cz	35.f	$4$	$16$	$8$	$13.415$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{7}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{7}q^{3}+(1+\beta _{1}+\beta _{2}-\beta _{3}+\beta _{5}+\cdots)q^{5}+\cdots\)
1680.2.cz.d	$1680$	$2$	1680.cz	35.f	$4$	$16$	$8$	$13.415$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(8\)	$2^{8}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{10}q^{3}+(1+\beta _{1}-\beta _{2}+\beta _{3}+\beta _{4}+\cdots)q^{5}+\cdots\)
1680.2.cz.e	$1680$	$2$	1680.cz	35.f	$4$	$24$	$12$	$13.415$		None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{4}]$
1680.2.cz.f	$1680$	$2$	1680.cz	35.f	$4$	$24$	$12$	$13.415$		None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(4\)		$\mathrm{SU}(2)[C_{4}]$

Decomposition of \(S_{2}^{\mathrm{old}}(1680, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1680, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(70, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(140, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(210, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(280, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(420, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(560, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(840, [\chi])\)\(^{\oplus 2}\)