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

• Label: 1680.2.f
• Level: $1680$
• Weight: $2$
• Character orbit: 1680.f
• Rep. character: $\chi_{1680}(881,\cdot)$
• Character field: $\Q$
• Dimension: $64$
• Newform subspaces: $12$
• Sturm bound: $768$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1680.f (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 21 \)
Character field:			\(\Q\)
Newform subspaces:			\( 12 \)
Sturm bound:			\(768\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(11\), \(17\), \(41\)

Dimensions

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

	Total	New	Old
Modular forms	408	64	344
Cusp forms	360	64	296
Eisenstein series	48	0	48

Trace form

\( 64 q - 4 q^{7} + 4 q^{21} + 64 q^{25} + 12 q^{39} - 56 q^{43} + 8 q^{49} + 28 q^{51} + 20 q^{63} + 40 q^{67} - 24 q^{79} + 8 q^{81} - 32 q^{93} + 84 q^{99}+O(q^{100}) \)

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.f.a	$1680$	$2$	1680.f	21.c	$2$	$2$	$2$	$13.415$	\(\Q(\sqrt{-3}) \)	None					420.2.d.a	$2$	$1$	\(0\)	\(-3\)	\(2\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-2+\zeta_{6})q^{3}+q^{5}+(-1-2\zeta_{6})q^{7}+\cdots\)
1680.2.f.b	$1680$	$2$	1680.f	21.c	$2$	$2$	$2$	$13.415$	\(\Q(\sqrt{-3}) \)	None					105.2.b.a	$2$	$0$	\(0\)	\(0\)	\(-2\)	\(-4\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{3}-q^{5}+(-\beta-2)q^{7}-3 q^{9}+\cdots\)
1680.2.f.c	$1680$	$2$	1680.f	21.c	$2$	$2$	$2$	$13.415$	\(\Q(\sqrt{-3}) \)	None					105.2.b.a	$2$	$1$	\(0\)	\(0\)	\(2\)	\(-4\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta q^{3}+q^{5}+(\beta-2)q^{7}-3 q^{9}+\cdots\)
1680.2.f.d	$1680$	$2$	1680.f	21.c	$2$	$2$	$2$	$13.415$	\(\Q(\sqrt{-3}) \)	None					420.2.d.a	$2$	$0$	\(0\)	\(3\)	\(-2\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(2-\zeta_{6})q^{3}-q^{5}+(-3+2\zeta_{6})q^{7}+\cdots\)
1680.2.f.e	$1680$	$2$	1680.f	21.c	$2$	$4$	$4$	$13.415$	\(\Q(i, \sqrt{5})\)	None					210.2.b.a	$2$	$0$	\(0\)	\(-2\)	\(-4\)	\(-6\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{1}+\beta _{2})q^{3}-q^{5}+(-1+\beta _{1}+2\beta _{2}+\cdots)q^{7}+\cdots\)
1680.2.f.f	$1680$	$2$	1680.f	21.c	$2$	$4$	$4$	$13.415$	\(\Q(i, \sqrt{5})\)	None					420.2.d.c	$2$	$0$	\(0\)	\(-2\)	\(-4\)	\(6\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-1-\beta _{1})q^{3}-q^{5}+(1-\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots\)
1680.2.f.g	$1680$	$2$	1680.f	21.c	$2$	$4$	$4$	$13.415$	\(\Q(\sqrt{-3}, \sqrt{-11})\)	None					105.2.b.c	$2$	$0$	\(0\)	\(-1\)	\(4\)	\(8\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{3}+q^{5}+(2+\beta _{2})q^{7}+(1+\beta _{2}+\cdots)q^{9}+\cdots\)
1680.2.f.h	$1680$	$2$	1680.f	21.c	$2$	$4$	$4$	$13.415$	\(\Q(\sqrt{-3}, \sqrt{-11})\)	None					105.2.b.c	$2$	$0$	\(0\)	\(1\)	\(-4\)	\(8\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}-q^{5}+(2-\beta _{2})q^{7}+(1+\beta _{2}+\cdots)q^{9}+\cdots\)
1680.2.f.i	$1680$	$2$	1680.f	21.c	$2$	$4$	$4$	$13.415$	\(\Q(i, \sqrt{5})\)	None					210.2.b.a	$2$	$0$	\(0\)	\(2\)	\(4\)	\(-6\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-\beta _{1}-\beta _{2})q^{3}+q^{5}+(-1-2\beta _{2}+\cdots)q^{7}+\cdots\)
1680.2.f.j	$1680$	$2$	1680.f	21.c	$2$	$4$	$4$	$13.415$	\(\Q(i, \sqrt{5})\)	None					420.2.d.c	$2$	$0$	\(0\)	\(2\)	\(4\)	\(6\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(1+\beta _{1})q^{3}+q^{5}+(1+\beta _{2}+\beta _{3})q^{7}+\cdots\)
1680.2.f.k	$1680$	$2$	1680.f	21.c	$2$	$16$	$16$	$13.415$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None					840.2.f.a	$2$	$0$	\(0\)	\(0\)	\(-16\)	\(-2\)	$2^{16}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{3}q^{3}-q^{5}-\beta _{9}q^{7}+\beta _{7}q^{9}+\beta _{8}q^{11}+\cdots\)
1680.2.f.l	$1680$	$2$	1680.f	21.c	$2$	$16$	$16$	$13.415$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None					840.2.f.a	$2$	$0$	\(0\)	\(0\)	\(16\)	\(-2\)	$2^{16}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{3}q^{3}+q^{5}-\beta _{4}q^{7}+\beta _{7}q^{9}+\beta _{8}q^{11}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(1680, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(168, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(210, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(336, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(420, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(840, [\chi])\)\(^{\oplus 2}\)