Properties

Label 1764.1.d
Level 1764
Weight 1
Character orbit d
Rep. character \(\chi_{1764}(685,\cdot)\)
Character field \(\Q\)
Dimension 2
Newform subspaces 1
Sturm bound 336
Trace bound 0

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Defining parameters

Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1764.d (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q\)
Newform subspaces: \( 1 \)
Sturm bound: \(336\)
Trace bound: \(0\)

Dimensions

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

Total New Old
Modular forms 56 2 54
Cusp forms 8 2 6
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 2 0 0 0

Trace form

\( 2q + O(q^{10}) \) \( 2q + 2q^{25} + 2q^{37} + 2q^{43} - 2q^{67} - 2q^{79} + O(q^{100}) \)

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

Label Dim. \(A\) Field Image CM RM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
1764.1.d.a \(2\) \(0.880\) \(\Q(\sqrt{-3}) \) \(D_{6}\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(0\) \(0\) \(q+(\zeta_{6}+\zeta_{6}^{2})q^{13}+(\zeta_{6}+\zeta_{6}^{2})q^{19}+\cdots\)

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

\( S_{1}^{\mathrm{old}}(1764, [\chi]) \cong \) \(S_{1}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 6}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$7$ 1
$11$ \( ( 1 + T^{2} )^{2} \)
$13$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
$17$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$19$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
$23$ \( ( 1 + T^{2} )^{2} \)
$29$ \( ( 1 + T^{2} )^{2} \)
$31$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
$37$ \( ( 1 - T + T^{2} )^{2} \)
$41$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$43$ \( ( 1 - T + T^{2} )^{2} \)
$47$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$53$ \( ( 1 + T^{2} )^{2} \)
$59$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$61$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$67$ \( ( 1 + T + T^{2} )^{2} \)
$71$ \( ( 1 + T^{2} )^{2} \)
$73$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
$79$ \( ( 1 + T + T^{2} )^{2} \)
$83$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$89$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$97$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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