Properties

Label 3600.1.bk
Level $3600$
Weight $1$
Character orbit 3600.bk
Rep. character $\chi_{3600}(143,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $4$
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.bk (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 60 \)
Character field: \(\Q(i)\)
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 240 4 236
Cusp forms 96 4 92
Eisenstein series 144 0 144

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

\(D_n\) \(A_4\) \(S_4\) \(A_5\)
Dimension 4 0 0 0

Trace form

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

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.bk.a \(4\) \(1.797\) \(\Q(\zeta_{8})\) \(D_{4}\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q+(-1-\zeta_{8}^{2})q^{13}-\zeta_{8}q^{17}+(-\zeta_{8}+\cdots)q^{29}+\cdots\)

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

\( S_{1}^{\mathrm{old}}(3600, [\chi]) \cong \) \(S_{1}^{\mathrm{new}}(180, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(300, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(720, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(900, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(1200, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

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