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

• Label: 1680.2.k
• Level: $1680$
• Weight: $2$
• Character orbit: 1680.k
• Rep. character: $\chi_{1680}(209,\cdot)$
• Character field: $\Q$
• Dimension: $92$
• Newform subspaces: $9$
• Sturm bound: $768$
• Trace bound: $15$

Defining parameters

Level:	\( N \)	\(=\)	\( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1680.k (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 105 \)
Character field:			\(\Q\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(768\)
Trace bound:			\(15\)
Distinguishing \(T_p\):			\(11\), \(13\), \(23\)

Dimensions

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

	Total	New	Old
Modular forms	408	100	308
Cusp forms	360	92	268
Eisenstein series	48	8	40

Trace form

\( 92 q - 4 q^{9} + O(q^{10}) \)

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.k.a	$1680$	$2$	1680.k	105.g	$2$	$4$	$4$	$13.415$	\(\Q(\sqrt{-2}, \sqrt{3})\)	None		$4$	$0$	\(0\)	\(-4\)	\(0\)	\(4\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-1-\beta _{1})q^{3}+(-\beta _{1}+\beta _{2})q^{5}+(1+\cdots)q^{7}+\cdots\)
1680.2.k.b	$1680$	$2$	1680.k	105.g	$2$	$4$	$4$	$13.415$	\(\Q(\sqrt{-5}, \sqrt{7})\)	\(\Q(\sqrt{-35}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+\beta _{3}q^{3}+(-\beta _{1}-\beta _{3})q^{5}+(-\beta _{1}+\beta _{3})q^{7}+\cdots\)
1680.2.k.c	$1680$	$2$	1680.k	105.g	$2$	$4$	$4$	$13.415$	\(\Q(\sqrt{-2}, \sqrt{3})\)	None		$4$	$0$	\(0\)	\(4\)	\(0\)	\(-4\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+(1+\beta _{1})q^{3}+(\beta _{1}+\beta _{2})q^{5}+(-1+\beta _{3})q^{7}+\cdots\)
1680.2.k.d	$1680$	$2$	1680.k	105.g	$2$	$8$	$8$	$13.415$	8.0.\(\cdots\).2	\(\Q(\sqrt{-35}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}\cdot 3^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q+\beta _{6}q^{3}-\beta _{4}q^{5}+(\beta _{1}-\beta _{6})q^{7}-\beta _{5}q^{9}+\cdots\)
1680.2.k.e	$1680$	$2$	1680.k	105.g	$2$	$8$	$8$	$13.415$	8.0.\(\cdots\).11	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{7}q^{3}+(\beta _{1}+\beta _{2}+\beta _{3}-\beta _{5}+\beta _{6}+\cdots)q^{5}+\cdots\)
1680.2.k.f	$1680$	$2$	1680.k	105.g	$2$	$8$	$8$	$13.415$	8.0.\(\cdots\).11	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{7}q^{3}-\beta _{3}q^{5}+(\beta _{2}+\beta _{6})q^{7}+(-\beta _{2}+\cdots)q^{9}+\cdots\)
1680.2.k.g	$1680$	$2$	1680.k	105.g	$2$	$8$	$8$	$13.415$	8.0.40960000.1	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{8}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{6}q^{3}-\beta _{2}q^{5}+(-\beta _{5}-\beta _{6})q^{7}+\cdots\)
1680.2.k.h	$1680$	$2$	1680.k	105.g	$2$	$24$	$24$	$13.415$		None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{2}]$
1680.2.k.i	$1680$	$2$	1680.k	105.g	$2$	$24$	$24$	$13.415$		None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{2}]$

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

\( S_{2}^{\mathrm{old}}(1680, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(210, [\chi])\)\(^{\oplus 4}\)