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

• Label: 1680.2.bl
• Level: $1680$
• Weight: $2$
• Character orbit: 1680.bl
• Rep. character: $\chi_{1680}(127,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $72$
• Newform subspaces: $5$
• Sturm bound: $768$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1680.bl (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 20 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(768\)
Trace bound:			\(13\)
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	72	744
Cusp forms	720	72	648
Eisenstein series	96	0	96

Trace form

\( 72 q - 24 q^{13} - 24 q^{17} - 24 q^{25} + 24 q^{37} + 24 q^{45} + 24 q^{53} + 24 q^{65} + 24 q^{73} - 72 q^{81} - 24 q^{85} - 48 q^{93} + 24 q^{97}+O(q^{100}) \)

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.bl.a	$1680$	$2$	1680.bl	20.e	$4$	$4$	$2$	$13.415$	\(\Q(\zeta_{8})\)	None		✓			1680.2.bl.a	$4$	$0$	\(0\)	\(0\)	\(-8\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q-\zeta_{8}q^{3}+(-2+\zeta_{8}^{2})q^{5}+\zeta_{8}^{3}q^{7}+\cdots\)
1680.2.bl.b	$1680$	$2$	1680.bl	20.e	$4$	$8$	$4$	$13.415$	\(\Q(\zeta_{24})\)	None		✓			1680.2.bl.b	$4$	$0$	\(0\)	\(0\)	\(8\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta_1 q^{3}+(-\beta_{3}-\beta_{2}+1)q^{5}+\cdots\)
1680.2.bl.c	$1680$	$2$	1680.bl	20.e	$4$	$12$	$6$	$13.415$	12.0.\(\cdots\).1	None		✓			1680.2.bl.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{6}q^{3}+\beta _{5}q^{5}+\beta _{1}q^{7}-\beta _{7}q^{9}+\cdots\)
1680.2.bl.d	$1680$	$2$	1680.bl	20.e	$4$	$24$	$12$	$13.415$		None		✓			1680.2.bl.d	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{4}]$
1680.2.bl.e	$1680$	$2$	1680.bl	20.e	$4$	$24$	$12$	$13.415$		None		✓			1680.2.bl.e	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{4}]$

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

\( S_{2}^{\mathrm{old}}(1680, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(20, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(60, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(80, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(140, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(240, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(420, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(560, [\chi])\)\(^{\oplus 2}\)