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

• Label: 1680.2.d
• Level: $1680$
• Weight: $2$
• Character orbit: 1680.d
• Rep. character: $\chi_{1680}(1231,\cdot)$
• Character field: $\Q$
• Dimension: $32$
• Newform subspaces: $4$
• Sturm bound: $768$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1680.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 28 \)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(768\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(11\), \(19\)

Dimensions

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

	Total	New	Old
Modular forms	408	32	376
Cusp forms	360	32	328
Eisenstein series	48	0	48

Trace form

\( 32 q + 32 q^{9} - 8 q^{21} - 32 q^{25} - 16 q^{37} + 32 q^{49} + 96 q^{53} - 16 q^{57} + 48 q^{65} + 48 q^{77} + 32 q^{81} + 16 q^{93}+O(q^{100}) \)

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	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1680.2.d.a	$1680$	$2$	1680.d	28.d	$2$	$4$	$4$	$13.415$	\(\Q(i, \sqrt{6})\)	None		✓			1680.2.d.a	$2$	$0$	\(0\)	\(-4\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-q^{3}+\beta _{1}q^{5}+(-\beta _{1}-\beta _{3})q^{7}+q^{9}+\cdots\)
1680.2.d.b	$1680$	$2$	1680.d	28.d	$2$	$4$	$4$	$13.415$	\(\Q(i, \sqrt{6})\)	None		✓			1680.2.d.a	$2$	$0$	\(0\)	\(4\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+q^{3}-\beta _{1}q^{5}+(-\beta _{1}-\beta _{3})q^{7}+q^{9}+\cdots\)
1680.2.d.c	$1680$	$2$	1680.d	28.d	$2$	$12$	$12$	$13.415$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		✓			1680.2.d.c	$2$	$0$	\(0\)	\(-12\)	\(0\)	\(4\)	$2^{14}$	$\mathrm{SU}(2)[C_{2}]$	\(q-q^{3}-\beta _{1}q^{5}-\beta _{9}q^{7}+q^{9}-\beta _{7}q^{11}+\cdots\)
1680.2.d.d	$1680$	$2$	1680.d	28.d	$2$	$12$	$12$	$13.415$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		✓			1680.2.d.c	$2$	$0$	\(0\)	\(12\)	\(0\)	\(-4\)	$2^{14}$	$\mathrm{SU}(2)[C_{2}]$	\(q+q^{3}+\beta _{1}q^{5}+\beta _{8}q^{7}+q^{9}-\beta _{7}q^{11}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(1680, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(112, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(140, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(336, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(420, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(560, [\chi])\)\(^{\oplus 2}\)