Properties

Label 1764.1.g
Level $1764$
Weight $1$
Character orbit 1764.g
Rep. character $\chi_{1764}(883,\cdot)$
Character field $\Q$
Dimension $7$
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.g (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 4 \)
Character field: \(\Q\)
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 42 12 30
Cusp forms 10 7 3
Eisenstein series 32 5 27

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

\(D_n\) \(A_4\) \(S_4\) \(A_5\)
Dimension 7 0 0 0

Trace form

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

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.g.a \(1\) \(0.880\) \(\Q\) \(D_{2}\) \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-7}) \) \(\Q(\sqrt{7}) \) \(1\) \(0\) \(0\) \(0\) \(q+q^{2}+q^{4}+q^{8}+q^{16}-q^{25}+2q^{29}+\cdots\)
1764.1.g.b \(2\) \(0.880\) \(\Q(\sqrt{2}) \) \(D_{4}\) \(\Q(\sqrt{-1}) \) None \(-2\) \(0\) \(0\) \(0\) \(q-q^{2}+q^{4}-\beta q^{5}-q^{8}+\beta q^{10}-\beta q^{13}+\cdots\)
1764.1.g.c \(2\) \(0.880\) \(\Q(\sqrt{-1}) \) \(D_{2}\) \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-21}) \) \(\Q(\sqrt{3}) \) \(0\) \(0\) \(0\) \(0\) \(q-iq^{2}-q^{4}+iq^{8}+iq^{11}+q^{16}+\cdots\)
1764.1.g.d \(2\) \(0.880\) \(\Q(\sqrt{2}) \) \(D_{4}\) \(\Q(\sqrt{-1}) \) None \(2\) \(0\) \(0\) \(0\) \(q+q^{2}+q^{4}-\beta q^{5}+q^{8}-\beta q^{10}+\beta q^{13}+\cdots\)

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 - T \))(\( ( 1 + T )^{2} \))(\( 1 + T^{2} \))(\( ( 1 - T )^{2} \))
$3$ 1
$5$ (\( 1 + T^{2} \))(\( 1 + T^{4} \))(\( ( 1 + T^{2} )^{2} \))(\( 1 + T^{4} \))
$7$ 1
$11$ (\( ( 1 - T )( 1 + T ) \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 + T^{2} )^{2} \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))
$13$ (\( 1 + T^{2} \))(\( 1 + T^{4} \))(\( ( 1 + T^{2} )^{2} \))(\( 1 + T^{4} \))
$17$ (\( 1 + T^{2} \))(\( 1 + T^{4} \))(\( ( 1 + T^{2} )^{2} \))(\( 1 + T^{4} \))
$19$ (\( ( 1 - T )( 1 + T ) \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))
$23$ (\( ( 1 - T )( 1 + T ) \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 + T^{2} )^{2} \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))
$29$ (\( ( 1 - T )^{2} \))(\( ( 1 - T )^{4} \))(\( ( 1 + T^{2} )^{2} \))(\( ( 1 + T )^{4} \))
$31$ (\( ( 1 - T )( 1 + T ) \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))
$37$ (\( ( 1 + T )^{2} \))(\( ( 1 + T^{2} )^{2} \))(\( ( 1 - T )^{4} \))(\( ( 1 + T^{2} )^{2} \))
$41$ (\( 1 + T^{2} \))(\( 1 + T^{4} \))(\( ( 1 + T^{2} )^{2} \))(\( 1 + T^{4} \))
$43$ (\( ( 1 - T )( 1 + T ) \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))
$47$ (\( ( 1 - T )( 1 + T ) \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))
$53$ (\( ( 1 + T )^{2} \))(\( ( 1 + T^{2} )^{2} \))(\( ( 1 + T^{2} )^{2} \))(\( ( 1 + T^{2} )^{2} \))
$59$ (\( ( 1 - T )( 1 + T ) \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))
$61$ (\( 1 + T^{2} \))(\( 1 + T^{4} \))(\( ( 1 + T^{2} )^{2} \))(\( 1 + T^{4} \))
$67$ (\( ( 1 - T )( 1 + T ) \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))
$71$ (\( ( 1 - T )( 1 + T ) \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 + T^{2} )^{2} \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))
$73$ (\( 1 + T^{2} \))(\( 1 + T^{4} \))(\( ( 1 + T^{2} )^{2} \))(\( 1 + T^{4} \))
$79$ (\( ( 1 - T )( 1 + T ) \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))
$83$ (\( ( 1 - T )( 1 + T ) \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))
$89$ (\( 1 + T^{2} \))(\( 1 + T^{4} \))(\( ( 1 + T^{2} )^{2} \))(\( 1 + T^{4} \))
$97$ (\( 1 + T^{2} \))(\( 1 + T^{4} \))(\( ( 1 + T^{2} )^{2} \))(\( 1 + T^{4} \))
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