Properties

Label 3600.1.dx
Level 3600
Weight 1
Character orbit dx
Rep. character \(\chi_{3600}(287,\cdot)\)
Character field \(\Q(\zeta_{20})\)
Dimension 16
Newform subspaces 1
Sturm bound 720
Trace bound 0

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Defining parameters

Level: \( N \) \(=\) \( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3600.dx (of order \(20\) and degree \(8\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 300 \)
Character field: \(\Q(\zeta_{20})\)
Newform subspaces: \( 1 \)
Sturm bound: \(720\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(3600, [\chi])\).

Total New Old
Modular forms 256 16 240
Cusp forms 64 16 48
Eisenstein series 192 0 192

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

\(D_n\) \(A_4\) \(S_4\) \(A_5\)
Dimension 16 0 0 0

Trace form

\( 16q + O(q^{10}) \) \( 16q - 4q^{13} + 4q^{37} + 4q^{73} - 12q^{85} + 4q^{97} + O(q^{100}) \)

Decomposition of \(S_{1}^{\mathrm{new}}(3600, [\chi])\) into newform subspaces

Label Dim. \(A\) Field Image CM RM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
3600.1.dx.a \(16\) \(1.797\) \(\Q(\zeta_{40})\) \(D_{20}\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q-\zeta_{40}^{7}q^{5}+(-\zeta_{40}^{4}+\zeta_{40}^{18})q^{13}+\cdots\)

Decomposition of \(S_{1}^{\mathrm{old}}(3600, [\chi])\) into lower level spaces

\( S_{1}^{\mathrm{old}}(3600, [\chi]) \cong \) \(S_{1}^{\mathrm{new}}(900, [\chi])\)\(^{\oplus 3}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 - T^{4} + T^{8} - T^{12} + T^{16} \)
$7$ \( ( 1 + T^{4} )^{8} \)
$11$ \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
$13$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
$17$ \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
$19$ \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
$23$ \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
$29$ \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
$31$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
$37$ \( ( 1 + T^{2} )^{8}( 1 - T + T^{2} - T^{3} + T^{4} )^{4} \)
$41$ \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
$43$ \( ( 1 + T^{4} )^{8} \)
$47$ \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
$53$ \( ( 1 + T^{4} )^{4}( 1 - T^{4} + T^{8} - T^{12} + T^{16} ) \)
$59$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \)
$61$ \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
$67$ \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
$71$ \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
$73$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
$79$ \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{4} \)
$83$ \( ( 1 - T^{4} + T^{8} - T^{12} + T^{16} )^{2} \)
$89$ \( ( 1 + T^{4} )^{4}( 1 - T^{4} + T^{8} - T^{12} + T^{16} ) \)
$97$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \)
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