Properties

Label 1680.1.dp
Level $1680$
Weight $1$
Character orbit 1680.dp
Rep. character $\chi_{1680}(479,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $8$
Newform subspaces $4$
Sturm bound $384$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1680.dp (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 420 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 4 \)
Sturm bound: \(384\)
Trace bound: \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(1680, [\chi])\).

Total New Old
Modular forms 80 8 72
Cusp forms 32 8 24
Eisenstein series 48 0 48

The following table gives the dimensions of subspaces with specified projective image type.

\(D_n\) \(A_4\) \(S_4\) \(A_5\)
Dimension 8 0 0 0

Trace form

\( 8 q + 2 q^{9} + O(q^{10}) \) \( 8 q + 2 q^{9} + 6 q^{21} - 4 q^{25} - 6 q^{45} - 4 q^{49} + 12 q^{61} + 2 q^{81} + O(q^{100}) \)

Decomposition of \(S_{1}^{\mathrm{new}}(1680, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field Image CM RM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1680.1.dp.a 1680.dp 420.ae $2$ $0.838$ \(\Q(\sqrt{-3}) \) $D_{6}$ \(\Q(\sqrt{-5}) \) None \(0\) \(-2\) \(-1\) \(-1\) \(q-q^{3}-\zeta_{6}q^{5}+\zeta_{6}^{2}q^{7}+q^{9}+\zeta_{6}q^{15}+\cdots\)
1680.1.dp.b 1680.dp 420.ae $2$ $0.838$ \(\Q(\sqrt{-3}) \) $D_{6}$ \(\Q(\sqrt{-5}) \) None \(0\) \(-1\) \(1\) \(-1\) \(q-\zeta_{6}q^{3}+\zeta_{6}q^{5}+\zeta_{6}^{2}q^{7}+\zeta_{6}^{2}q^{9}+\cdots\)
1680.1.dp.c 1680.dp 420.ae $2$ $0.838$ \(\Q(\sqrt{-3}) \) $D_{6}$ \(\Q(\sqrt{-5}) \) None \(0\) \(1\) \(1\) \(1\) \(q+\zeta_{6}q^{3}+\zeta_{6}q^{5}-\zeta_{6}^{2}q^{7}+\zeta_{6}^{2}q^{9}+\cdots\)
1680.1.dp.d 1680.dp 420.ae $2$ $0.838$ \(\Q(\sqrt{-3}) \) $D_{6}$ \(\Q(\sqrt{-5}) \) None \(0\) \(2\) \(-1\) \(1\) \(q+q^{3}-\zeta_{6}q^{5}-\zeta_{6}^{2}q^{7}+q^{9}-\zeta_{6}q^{15}+\cdots\)

Decomposition of \(S_{1}^{\mathrm{old}}(1680, [\chi])\) into lower level spaces

\( S_{1}^{\mathrm{old}}(1680, [\chi]) \cong \) \(S_{1}^{\mathrm{new}}(420, [\chi])\)\(^{\oplus 3}\)