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

• Label: 1680.2.di
• Level: $1680$
• Weight: $2$
• Character orbit: 1680.di
• Rep. character: $\chi_{1680}(289,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $96$
• Newform subspaces: $7$
• Sturm bound: $768$
• Trace bound: $11$

Defining parameters

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

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

\( 96 q + 48 q^{9} + O(q^{10}) \)

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.di.a	$1680$	$2$	1680.di	35.j	$6$	$4$	$2$	$13.415$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(-4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-\zeta_{12}+\zeta_{12}^{3})q^{3}+(-2+\zeta_{12}+\cdots)q^{5}+\cdots\)
1680.2.di.b	$1680$	$2$	1680.di	35.j	$6$	$4$	$2$	$13.415$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(-4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q-\zeta_{12}q^{3}+(-\zeta_{12}-2\zeta_{12}^{2}+\zeta_{12}^{3})q^{5}+\cdots\)
1680.2.di.c	$1680$	$2$	1680.di	35.j	$6$	$12$	$6$	$13.415$	12.0.\(\cdots\).1	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q-\beta _{2}q^{3}+(-\beta _{1}+\beta _{4}-\beta _{5}-\beta _{8}+\beta _{10}+\cdots)q^{5}+\cdots\)
1680.2.di.d	$1680$	$2$	1680.di	35.j	$6$	$16$	$8$	$13.415$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(2\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{3}q^{3}+(1-\beta _{2}+\beta _{4}-\beta _{8}-\beta _{11}+\cdots)q^{5}+\cdots\)
1680.2.di.e	$1680$	$2$	1680.di	35.j	$6$	$16$	$8$	$13.415$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(2\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(\beta _{6}+\beta _{7})q^{3}+(\beta _{8}+\beta _{12}+\beta _{14})q^{5}+\cdots\)
1680.2.di.f	$1680$	$2$	1680.di	35.j	$6$	$20$	$10$	$13.415$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(2\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q-\beta _{6}q^{3}+(-\beta _{7}+\beta _{12}-\beta _{17})q^{5}+\cdots\)
1680.2.di.g	$1680$	$2$	1680.di	35.j	$6$	$24$	$12$	$13.415$		None		$4$	$0$	\(0\)	\(0\)	\(2\)	\(0\)		$\mathrm{SU}(2)[C_{6}]$

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

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