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

• Label: 1680.2.bg
• Level: $1680$
• Weight: $2$
• Character orbit: 1680.bg
• Rep. character: $\chi_{1680}(961,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $64$
• Newform subspaces: $22$
• 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.bg (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 22 \)
Sturm bound:			\(768\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(11\), \(13\), \(17\)

Dimensions

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

	Total	New	Old
Modular forms	816	64	752
Cusp forms	720	64	656
Eisenstein series	96	0	96

Trace form

\( 64 q + 4 q^{3} + 4 q^{7} - 32 q^{9} + O(q^{10}) \)

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	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.bg.a	$1680$	$2$	1680.bg	7.c	$3$	$2$	$1$	$13.415$	\(\Q(\sqrt{-3}) \)	None					420.2.q.a	$2$	$0$	\(0\)	\(-1\)	\(-1\)	\(-5\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\zeta_{6})q^{3}-\zeta_{6}q^{5}+(-2-\zeta_{6})q^{7}+\cdots\)
1680.2.bg.b	$1680$	$2$	1680.bg	7.c	$3$	$2$	$1$	$13.415$	\(\Q(\sqrt{-3}) \)	None					840.2.bg.c	$2$	$0$	\(0\)	\(-1\)	\(-1\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\zeta_{6})q^{3}-\zeta_{6}q^{5}+(-3+2\zeta_{6})q^{7}+\cdots\)
1680.2.bg.c	$1680$	$2$	1680.bg	7.c	$3$	$2$	$1$	$13.415$	\(\Q(\sqrt{-3}) \)	None					840.2.bg.b	$2$	$1$	\(0\)	\(-1\)	\(-1\)	\(1\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\zeta_{6})q^{3}-\zeta_{6}q^{5}+(2-3\zeta_{6})q^{7}+\cdots\)
1680.2.bg.d	$1680$	$2$	1680.bg	7.c	$3$	$2$	$1$	$13.415$	\(\Q(\sqrt{-3}) \)	None					210.2.i.b	$2$	$0$	\(0\)	\(-1\)	\(-1\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\zeta_{6})q^{3}-\zeta_{6}q^{5}+(1+2\zeta_{6})q^{7}+\cdots\)
1680.2.bg.e	$1680$	$2$	1680.bg	7.c	$3$	$2$	$1$	$13.415$	\(\Q(\sqrt{-3}) \)	None					840.2.bg.a	$2$	$0$	\(0\)	\(-1\)	\(-1\)	\(5\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\zeta_{6})q^{3}-\zeta_{6}q^{5}+(2+\zeta_{6})q^{7}+\cdots\)
1680.2.bg.f	$1680$	$2$	1680.bg	7.c	$3$	$2$	$1$	$13.415$	\(\Q(\sqrt{-3}) \)	None					840.2.bg.f	$2$	$0$	\(0\)	\(-1\)	\(1\)	\(-5\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\zeta_{6})q^{3}+\zeta_{6}q^{5}+(-2-\zeta_{6})q^{7}+\cdots\)
1680.2.bg.g	$1680$	$2$	1680.bg	7.c	$3$	$2$	$1$	$13.415$	\(\Q(\sqrt{-3}) \)	None					210.2.i.d	$2$	$0$	\(0\)	\(-1\)	\(1\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\zeta_{6})q^{3}+\zeta_{6}q^{5}+(-1-2\zeta_{6})q^{7}+\cdots\)
1680.2.bg.h	$1680$	$2$	1680.bg	7.c	$3$	$2$	$1$	$13.415$	\(\Q(\sqrt{-3}) \)	None					840.2.bg.e	$2$	$0$	\(0\)	\(-1\)	\(1\)	\(-1\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\zeta_{6})q^{3}+\zeta_{6}q^{5}+(-2+3\zeta_{6})q^{7}+\cdots\)
1680.2.bg.i	$1680$	$2$	1680.bg	7.c	$3$	$2$	$1$	$13.415$	\(\Q(\sqrt{-3}) \)	None					840.2.bg.d	$2$	$0$	\(0\)	\(-1\)	\(1\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\zeta_{6})q^{3}+\zeta_{6}q^{5}+(3-2\zeta_{6})q^{7}+\cdots\)
1680.2.bg.j	$1680$	$2$	1680.bg	7.c	$3$	$2$	$1$	$13.415$	\(\Q(\sqrt{-3}) \)	None					420.2.q.b	$2$	$0$	\(0\)	\(-1\)	\(1\)	\(5\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\zeta_{6})q^{3}+\zeta_{6}q^{5}+(2+\zeta_{6})q^{7}+\cdots\)
1680.2.bg.k	$1680$	$2$	1680.bg	7.c	$3$	$2$	$1$	$13.415$	\(\Q(\sqrt{-3}) \)	None					210.2.i.a	$2$	$0$	\(0\)	\(1\)	\(-1\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{3}-\zeta_{6}q^{5}+(-3+2\zeta_{6})q^{7}+\cdots\)
1680.2.bg.l	$1680$	$2$	1680.bg	7.c	$3$	$2$	$1$	$13.415$	\(\Q(\sqrt{-3}) \)	None					105.2.i.b	$2$	$0$	\(0\)	\(1\)	\(-1\)	\(1\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{3}-\zeta_{6}q^{5}+(2-3\zeta_{6})q^{7}+\cdots\)
1680.2.bg.m	$1680$	$2$	1680.bg	7.c	$3$	$2$	$1$	$13.415$	\(\Q(\sqrt{-3}) \)	None					105.2.i.a	$2$	$0$	\(0\)	\(1\)	\(1\)	\(-5\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{3}+\zeta_{6}q^{5}+(-2-\zeta_{6})q^{7}+\cdots\)
1680.2.bg.n	$1680$	$2$	1680.bg	7.c	$3$	$2$	$1$	$13.415$	\(\Q(\sqrt{-3}) \)	None					210.2.i.c	$2$	$0$	\(0\)	\(1\)	\(1\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{3}+\zeta_{6}q^{5}+(3-2\zeta_{6})q^{7}+\cdots\)
1680.2.bg.o	$1680$	$2$	1680.bg	7.c	$3$	$4$	$2$	$13.415$	\(\Q(\zeta_{12})\)	None					105.2.i.d	$2$	$0$	\(0\)	\(-2\)	\(-2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-\zeta_{12}^{2}q^{3}+(-1+\zeta_{12}^{2})q^{5}+(-\zeta_{12}+\cdots)q^{7}+\cdots\)
1680.2.bg.p	$1680$	$2$	1680.bg	7.c	$3$	$4$	$2$	$13.415$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None					105.2.i.c	$2$	$0$	\(0\)	\(-2\)	\(2\)	\(2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{2}q^{3}+(1+\beta _{2})q^{5}+(1+\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots\)
1680.2.bg.q	$1680$	$2$	1680.bg	7.c	$3$	$4$	$2$	$13.415$	\(\Q(\sqrt{-3}, \sqrt{7})\)	None					420.2.q.c	$2$	$0$	\(0\)	\(2\)	\(-2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1+\beta _{2})q^{3}+\beta _{2}q^{5}+(\beta _{1}+\beta _{3})q^{7}+\cdots\)
1680.2.bg.r	$1680$	$2$	1680.bg	7.c	$3$	$4$	$2$	$13.415$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None					840.2.bg.g	$2$	$0$	\(0\)	\(2\)	\(-2\)	\(2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1+\beta _{2})q^{3}+\beta _{2}q^{5}+(-\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots\)
1680.2.bg.s	$1680$	$2$	1680.bg	7.c	$3$	$4$	$2$	$13.415$	\(\Q(\sqrt{-3}, \sqrt{7})\)	None					840.2.bg.h	$2$	$0$	\(0\)	\(2\)	\(2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1+\beta _{2})q^{3}-\beta _{2}q^{5}+(\beta _{1}+\beta _{3})q^{7}+\cdots\)
1680.2.bg.t	$1680$	$2$	1680.bg	7.c	$3$	$4$	$2$	$13.415$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None					420.2.q.d	$2$	$0$	\(0\)	\(2\)	\(2\)	\(2\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1+\beta _{1})q^{3}-\beta _{1}q^{5}+(-\beta _{1}+\beta _{3})q^{7}+\cdots\)
1680.2.bg.u	$1680$	$2$	1680.bg	7.c	$3$	$6$	$3$	$13.415$	6.0.38363328.2	None					840.2.bg.i	$2$	$0$	\(0\)	\(3\)	\(-3\)	\(2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1+\beta _{3})q^{3}+\beta _{3}q^{5}+(\beta _{1}-\beta _{4})q^{7}+\cdots\)
1680.2.bg.v	$1680$	$2$	1680.bg	7.c	$3$	$6$	$3$	$13.415$	6.0.29428272.1	None					840.2.bg.j	$2$	$0$	\(0\)	\(3\)	\(3\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta _{3}q^{3}+(1+\beta _{3})q^{5}+(-\beta _{1}-\beta _{5})q^{7}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(1680, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(56, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(70, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(112, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(140, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(168, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(210, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(280, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(336, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(420, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(560, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(840, [\chi])\)\(^{\oplus 2}\)