Properties

Label 1680.2.t
Level $1680$
Weight $2$
Character orbit 1680.t
Rep. character $\chi_{1680}(1009,\cdot)$
Character field $\Q$
Dimension $36$
Newform subspaces $11$
Sturm bound $768$
Trace bound $15$

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Defining parameters

Level: \( N \) \(=\) \( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1680.t (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q\)
Newform subspaces: \( 11 \)
Sturm bound: \(768\)
Trace bound: \(15\)
Distinguishing \(T_p\): \(11\), \(13\), \(19\)

Dimensions

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

Total New Old
Modular forms 408 36 372
Cusp forms 360 36 324
Eisenstein series 48 0 48

Trace form

\( 36 q - 4 q^{5} - 36 q^{9} + O(q^{10}) \) \( 36 q - 4 q^{5} - 36 q^{9} - 8 q^{19} - 4 q^{25} + 8 q^{29} + 8 q^{31} + 16 q^{39} + 8 q^{41} + 4 q^{45} - 36 q^{49} + 32 q^{51} - 40 q^{55} - 24 q^{61} + 16 q^{65} + 16 q^{69} - 16 q^{75} + 32 q^{79} + 36 q^{81} - 8 q^{85} - 40 q^{89} + 24 q^{91} - 64 q^{95} + O(q^{100}) \)

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.t.a 1680.t 5.b $2$ $13.415$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(-4\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-iq^{3}+(-2-i)q^{5}-iq^{7}-q^{9}+\cdots\)
1680.2.t.b 1680.t 5.b $2$ $13.415$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(-2\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{3}+(-1+2i)q^{5}+iq^{7}-q^{9}+\cdots\)
1680.2.t.c 1680.t 5.b $2$ $13.415$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(-2\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{3}+(-1-2i)q^{5}+iq^{7}-q^{9}+\cdots\)
1680.2.t.d 1680.t 5.b $2$ $13.415$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(-2\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-iq^{3}+(-1+2i)q^{5}-iq^{7}-q^{9}+\cdots\)
1680.2.t.e 1680.t 5.b $2$ $13.415$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(2\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-iq^{3}+(1+2i)q^{5}-iq^{7}-q^{9}-2q^{11}+\cdots\)
1680.2.t.f 1680.t 5.b $2$ $13.415$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(2\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{3}+(1+2i)q^{5}+iq^{7}-q^{9}+6q^{11}+\cdots\)
1680.2.t.g 1680.t 5.b $2$ $13.415$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(4\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-iq^{3}+(2-i)q^{5}-iq^{7}-q^{9}-4q^{11}+\cdots\)
1680.2.t.h 1680.t 5.b $4$ $13.415$ \(\Q(i, \sqrt{5})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{1}q^{3}-\beta _{3}q^{5}-\beta _{1}q^{7}-q^{9}+2q^{11}+\cdots\)
1680.2.t.i 1680.t 5.b $6$ $13.415$ 6.0.350464.1 None \(0\) \(0\) \(-2\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{3}+\beta _{5}q^{5}-\beta _{1}q^{7}-q^{9}+(-\beta _{2}+\cdots)q^{11}+\cdots\)
1680.2.t.j 1680.t 5.b $6$ $13.415$ 6.0.350464.1 None \(0\) \(0\) \(-2\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{1}q^{3}+\beta _{5}q^{5}+\beta _{1}q^{7}-q^{9}+(\beta _{2}+\cdots)q^{11}+\cdots\)
1680.2.t.k 1680.t 5.b $6$ $13.415$ 6.0.350464.1 None \(0\) \(0\) \(2\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{1}q^{3}+\beta _{2}q^{5}+\beta _{1}q^{7}-q^{9}-2q^{11}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(1680, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(60, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(70, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(80, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(120, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(140, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(210, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(240, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(280, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(420, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(560, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(840, [\chi])\)\(^{\oplus 2}\)