Properties

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 $7$

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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

\( 64 q + 4 q^{3} + 4 q^{7} - 32 q^{9} + O(q^{10}) \) \( 64 q + 4 q^{3} + 4 q^{7} - 32 q^{9} - 8 q^{11} - 4 q^{19} - 8 q^{23} - 32 q^{25} - 8 q^{27} - 32 q^{29} - 4 q^{31} + 16 q^{37} - 4 q^{39} - 16 q^{41} + 24 q^{43} + 24 q^{49} + 32 q^{53} + 32 q^{55} + 16 q^{57} + 16 q^{59} + 16 q^{61} + 4 q^{63} + 8 q^{65} - 12 q^{67} - 16 q^{71} - 8 q^{73} + 4 q^{75} - 16 q^{77} - 12 q^{79} - 32 q^{81} - 24 q^{87} - 16 q^{89} - 44 q^{91} - 32 q^{97} + 16 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1680, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1680.2.bg.a 1680.bg 7.c $2$ $13.415$ \(\Q(\sqrt{-3}) \) None \(0\) \(-1\) \(-1\) \(-5\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{6})q^{3}-\zeta_{6}q^{5}+(-2-\zeta_{6})q^{7}+\cdots\)
1680.2.bg.b 1680.bg 7.c $2$ $13.415$ \(\Q(\sqrt{-3}) \) None \(0\) \(-1\) \(-1\) \(-4\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{6})q^{3}-\zeta_{6}q^{5}+(-3+2\zeta_{6})q^{7}+\cdots\)
1680.2.bg.c 1680.bg 7.c $2$ $13.415$ \(\Q(\sqrt{-3}) \) None \(0\) \(-1\) \(-1\) \(1\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{6})q^{3}-\zeta_{6}q^{5}+(2-3\zeta_{6})q^{7}+\cdots\)
1680.2.bg.d 1680.bg 7.c $2$ $13.415$ \(\Q(\sqrt{-3}) \) None \(0\) \(-1\) \(-1\) \(4\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{6})q^{3}-\zeta_{6}q^{5}+(1+2\zeta_{6})q^{7}+\cdots\)
1680.2.bg.e 1680.bg 7.c $2$ $13.415$ \(\Q(\sqrt{-3}) \) None \(0\) \(-1\) \(-1\) \(5\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{6})q^{3}-\zeta_{6}q^{5}+(2+\zeta_{6})q^{7}+\cdots\)
1680.2.bg.f 1680.bg 7.c $2$ $13.415$ \(\Q(\sqrt{-3}) \) None \(0\) \(-1\) \(1\) \(-5\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{6})q^{3}+\zeta_{6}q^{5}+(-2-\zeta_{6})q^{7}+\cdots\)
1680.2.bg.g 1680.bg 7.c $2$ $13.415$ \(\Q(\sqrt{-3}) \) None \(0\) \(-1\) \(1\) \(-4\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{6})q^{3}+\zeta_{6}q^{5}+(-1-2\zeta_{6})q^{7}+\cdots\)
1680.2.bg.h 1680.bg 7.c $2$ $13.415$ \(\Q(\sqrt{-3}) \) None \(0\) \(-1\) \(1\) \(-1\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{6})q^{3}+\zeta_{6}q^{5}+(-2+3\zeta_{6})q^{7}+\cdots\)
1680.2.bg.i 1680.bg 7.c $2$ $13.415$ \(\Q(\sqrt{-3}) \) None \(0\) \(-1\) \(1\) \(4\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{6})q^{3}+\zeta_{6}q^{5}+(3-2\zeta_{6})q^{7}+\cdots\)
1680.2.bg.j 1680.bg 7.c $2$ $13.415$ \(\Q(\sqrt{-3}) \) None \(0\) \(-1\) \(1\) \(5\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{6})q^{3}+\zeta_{6}q^{5}+(2+\zeta_{6})q^{7}+\cdots\)
1680.2.bg.k 1680.bg 7.c $2$ $13.415$ \(\Q(\sqrt{-3}) \) None \(0\) \(1\) \(-1\) \(-4\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{3}-\zeta_{6}q^{5}+(-3+2\zeta_{6})q^{7}+\cdots\)
1680.2.bg.l 1680.bg 7.c $2$ $13.415$ \(\Q(\sqrt{-3}) \) None \(0\) \(1\) \(-1\) \(1\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{3}-\zeta_{6}q^{5}+(2-3\zeta_{6})q^{7}+\cdots\)
1680.2.bg.m 1680.bg 7.c $2$ $13.415$ \(\Q(\sqrt{-3}) \) None \(0\) \(1\) \(1\) \(-5\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{3}+\zeta_{6}q^{5}+(-2-\zeta_{6})q^{7}+\cdots\)
1680.2.bg.n 1680.bg 7.c $2$ $13.415$ \(\Q(\sqrt{-3}) \) None \(0\) \(1\) \(1\) \(4\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{3}+\zeta_{6}q^{5}+(3-2\zeta_{6})q^{7}+\cdots\)
1680.2.bg.o 1680.bg 7.c $4$ $13.415$ \(\Q(\zeta_{12})\) None \(0\) \(-2\) \(-2\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-\zeta_{12}^{2}q^{3}+(-1+\zeta_{12}^{2})q^{5}+(-\zeta_{12}+\cdots)q^{7}+\cdots\)
1680.2.bg.p 1680.bg 7.c $4$ $13.415$ \(\Q(\sqrt{2}, \sqrt{-3})\) None \(0\) \(-2\) \(2\) \(2\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{2}q^{3}+(1+\beta _{2})q^{5}+(1+\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots\)
1680.2.bg.q 1680.bg 7.c $4$ $13.415$ \(\Q(\sqrt{-3}, \sqrt{7})\) None \(0\) \(2\) \(-2\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1+\beta _{2})q^{3}+\beta _{2}q^{5}+(\beta _{1}+\beta _{3})q^{7}+\cdots\)
1680.2.bg.r 1680.bg 7.c $4$ $13.415$ \(\Q(\sqrt{2}, \sqrt{-3})\) None \(0\) \(2\) \(-2\) \(2\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1+\beta _{2})q^{3}+\beta _{2}q^{5}+(-\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots\)
1680.2.bg.s 1680.bg 7.c $4$ $13.415$ \(\Q(\sqrt{-3}, \sqrt{7})\) None \(0\) \(2\) \(2\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1+\beta _{2})q^{3}-\beta _{2}q^{5}+(\beta _{1}+\beta _{3})q^{7}+\cdots\)
1680.2.bg.t 1680.bg 7.c $4$ $13.415$ \(\Q(\sqrt{2}, \sqrt{-3})\) None \(0\) \(2\) \(2\) \(2\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1+\beta _{1})q^{3}-\beta _{1}q^{5}+(-\beta _{1}+\beta _{3})q^{7}+\cdots\)
1680.2.bg.u 1680.bg 7.c $6$ $13.415$ 6.0.38363328.2 None \(0\) \(3\) \(-3\) \(2\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1+\beta _{3})q^{3}+\beta _{3}q^{5}+(\beta _{1}-\beta _{4})q^{7}+\cdots\)
1680.2.bg.v 1680.bg 7.c $6$ $13.415$ 6.0.29428272.1 None \(0\) \(3\) \(3\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{3}q^{3}+(1+\beta _{3})q^{5}+(-\beta _{1}-\beta _{5})q^{7}+\cdots\)

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

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