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Properties

Label 1680.3.s
Level $1680$
Weight $3$
Character orbit 1680.s
Rep. character $\chi_{1680}(1441,\cdot)$
Character field $\Q$
Dimension $64$
Newform subspaces $4$
Sturm bound $1152$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1680.s (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(1152\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(1680, [\chi])\).

Total New Old
Modular forms 792 64 728
Cusp forms 744 64 680
Eisenstein series 48 0 48

Trace form

\( 64 q - 16 q^{7} - 192 q^{9} + O(q^{10}) \) \( 64 q - 16 q^{7} - 192 q^{9} + 32 q^{11} + 64 q^{23} - 320 q^{25} - 32 q^{29} + 96 q^{37} - 96 q^{39} - 32 q^{43} - 64 q^{49} + 96 q^{53} - 96 q^{57} + 48 q^{63} + 160 q^{65} + 160 q^{67} - 128 q^{71} - 160 q^{77} + 96 q^{79} + 576 q^{81} - 192 q^{91} - 96 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{3}^{\mathrm{new}}(1680, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1680.3.s.a 1680.s 7.b $8$ $45.777$ 8.0.3317760000.3 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{5}q^{3}-\beta _{6}q^{5}+(-2\beta _{1}+\beta _{3}-\beta _{4}+\cdots)q^{7}+\cdots\)
1680.3.s.b 1680.s 7.b $12$ $45.777$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(0\) \(0\) \(-8\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{2}q^{3}-\beta _{3}q^{5}+(-1-\beta _{4})q^{7}-3q^{9}+\cdots\)
1680.3.s.c 1680.s 7.b $12$ $45.777$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(0\) \(0\) \(8\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{3}q^{3}+\beta _{8}q^{5}+(1+\beta _{3}+\beta _{8}-\beta _{10}+\cdots)q^{7}+\cdots\)
1680.3.s.d 1680.s 7.b $32$ $45.777$ None \(0\) \(0\) \(0\) \(-16\) $\mathrm{SU}(2)[C_{2}]$

Decomposition of \(S_{3}^{\mathrm{old}}(1680, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(1680, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 20}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(14, [\chi])\)\(^{\oplus 16}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(56, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(112, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(168, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(336, [\chi])\)\(^{\oplus 2}\)