Properties

Label 1764.1.y
Level $1764$
Weight $1$
Character orbit 1764.y
Rep. character $\chi_{1764}(667,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $14$
Newform subspaces $4$
Sturm bound $336$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1764.y (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 28 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 4 \)
Sturm bound: \(336\)
Trace bound: \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(1764, [\chi])\).

Total New Old
Modular forms 84 22 62
Cusp forms 20 14 6
Eisenstein series 64 8 56

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

\(D_n\) \(A_4\) \(S_4\) \(A_5\)
Dimension 14 0 0 0

Trace form

\( 14 q - q^{2} - 3 q^{4} + 2 q^{8} + O(q^{10}) \) \( 14 q - q^{2} - 3 q^{4} + 2 q^{8} - 7 q^{16} + 8 q^{22} - q^{25} + 4 q^{29} - q^{32} - 2 q^{37} - 4 q^{46} - 2 q^{50} + 2 q^{53} + 6 q^{58} + 6 q^{64} + 2 q^{74} - 16 q^{85} + 4 q^{88} + O(q^{100}) \)

Decomposition of \(S_{1}^{\mathrm{new}}(1764, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field Image CM RM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1764.1.y.a 1764.y 28.g $2$ $0.880$ \(\Q(\sqrt{-3}) \) $D_{2}$ \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-7}) \) \(\Q(\sqrt{7}) \) \(-1\) \(0\) \(0\) \(0\) \(q-\zeta_{6}q^{2}+\zeta_{6}^{2}q^{4}+q^{8}-\zeta_{6}q^{16}+\cdots\)
1764.1.y.b 1764.y 28.g $4$ $0.880$ \(\Q(\sqrt{2}, \sqrt{-3})\) $D_{4}$ \(\Q(\sqrt{-1}) \) None \(-2\) \(0\) \(0\) \(0\) \(q+\beta _{2}q^{2}+(-1-\beta _{2})q^{4}+(-\beta _{1}-\beta _{3})q^{5}+\cdots\)
1764.1.y.c 1764.y 28.g $4$ $0.880$ \(\Q(\zeta_{12})\) $D_{2}$ \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-21}) \) \(\Q(\sqrt{3}) \) \(0\) \(0\) \(0\) \(0\) \(q-\zeta_{12}q^{2}+\zeta_{12}^{2}q^{4}-\zeta_{12}^{3}q^{8}+\zeta_{12}^{5}q^{11}+\cdots\)
1764.1.y.d 1764.y 28.g $4$ $0.880$ \(\Q(\sqrt{2}, \sqrt{-3})\) $D_{4}$ \(\Q(\sqrt{-1}) \) None \(2\) \(0\) \(0\) \(0\) \(q-\beta _{2}q^{2}+(-1-\beta _{2})q^{4}+(-\beta _{1}-\beta _{3})q^{5}+\cdots\)

Decomposition of \(S_{1}^{\mathrm{old}}(1764, [\chi])\) into lower level spaces

\( S_{1}^{\mathrm{old}}(1764, [\chi]) \cong \) \(S_{1}^{\mathrm{new}}(196, [\chi])\)\(^{\oplus 3}\)