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

• Label: 1680.2.t
• Level: $1680$
• Weight: $2$
• Character orbit: 1680.t
• Rep. character: $\chi_{1680}(1009,\cdot)$
• Character field: $\Q$
• Dimension: $36$
• Newform subspaces: $11$
• 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.t (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 11 \)
Sturm bound:			\(768\)
Trace bound:			\(15\)
Distinguishing \(T_p\):			\(11\), \(13\), \(19\)

Dimensions

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

	Total	New	Old
Modular forms	408	36	372
Cusp forms	360	36	324
Eisenstein series	48	0	48

Trace form

\( 36 q - 4 q^{5} - 36 q^{9} + O(q^{10}) \)

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.t.a	$1680$	$2$	1680.t	5.b	$2$	$2$	$2$	$13.415$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-iq^{3}+(-2-i)q^{5}-iq^{7}-q^{9}+\cdots\)
1680.2.t.b	$1680$	$2$	1680.t	5.b	$2$	$2$	$2$	$13.415$	\(\Q(\sqrt{-1}) \)	None		$2$	$1$	\(0\)	\(0\)	\(-2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{3}+(-1+2i)q^{5}+iq^{7}-q^{9}+\cdots\)
1680.2.t.c	$1680$	$2$	1680.t	5.b	$2$	$2$	$2$	$13.415$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{3}+(-1-2i)q^{5}+iq^{7}-q^{9}+\cdots\)
1680.2.t.d	$1680$	$2$	1680.t	5.b	$2$	$2$	$2$	$13.415$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-iq^{3}+(-1+2i)q^{5}-iq^{7}-q^{9}+\cdots\)
1680.2.t.e	$1680$	$2$	1680.t	5.b	$2$	$2$	$2$	$13.415$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-iq^{3}+(1+2i)q^{5}-iq^{7}-q^{9}-2q^{11}+\cdots\)
1680.2.t.f	$1680$	$2$	1680.t	5.b	$2$	$2$	$2$	$13.415$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{3}+(1+2i)q^{5}+iq^{7}-q^{9}+6q^{11}+\cdots\)
1680.2.t.g	$1680$	$2$	1680.t	5.b	$2$	$2$	$2$	$13.415$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-iq^{3}+(2-i)q^{5}-iq^{7}-q^{9}-4q^{11}+\cdots\)
1680.2.t.h	$1680$	$2$	1680.t	5.b	$2$	$4$	$4$	$13.415$	\(\Q(i, \sqrt{5})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{3}-\beta _{3}q^{5}-\beta _{1}q^{7}-q^{9}+2q^{11}+\cdots\)
1680.2.t.i	$1680$	$2$	1680.t	5.b	$2$	$6$	$6$	$13.415$	6.0.350464.1	None		$2$	$0$	\(0\)	\(0\)	\(-2\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}+\beta _{5}q^{5}-\beta _{1}q^{7}-q^{9}+(-\beta _{2}+\cdots)q^{11}+\cdots\)
1680.2.t.j	$1680$	$2$	1680.t	5.b	$2$	$6$	$6$	$13.415$	6.0.350464.1	None		$2$	$0$	\(0\)	\(0\)	\(-2\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{3}+\beta _{5}q^{5}+\beta _{1}q^{7}-q^{9}+(\beta _{2}+\cdots)q^{11}+\cdots\)
1680.2.t.k	$1680$	$2$	1680.t	5.b	$2$	$6$	$6$	$13.415$	6.0.350464.1	None		$2$	$0$	\(0\)	\(0\)	\(2\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{3}+\beta _{2}q^{5}+\beta _{1}q^{7}-q^{9}-2q^{11}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(1680, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(70, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(80, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(140, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(210, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(280, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(420, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(560, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(840, [\chi])\)\(^{\oplus 2}\)