Properties

Label 360.1.u
Level 360
Weight 1
Character orbit u
Rep. character \(\chi_{360}(37,\cdot)\)
Character field \(\Q(\zeta_{4})\)
Dimension 4
Newform subspaces 1
Sturm bound 72
Trace bound 0

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Defining parameters

Level: \( N \) \(=\) \( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 360.u (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 40 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 1 \)
Sturm bound: \(72\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(360, [\chi])\).

Total New Old
Modular forms 20 8 12
Cusp forms 4 4 0
Eisenstein series 16 4 12

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

\( 4q - 4q^{7} + O(q^{10}) \) \( 4q - 4q^{7} + 4q^{10} - 4q^{16} + 4q^{22} - 4q^{28} - 4q^{55} - 4q^{58} - 4q^{70} + 4q^{73} + 4q^{88} - 4q^{97} + O(q^{100}) \)

Decomposition of \(S_{1}^{\mathrm{new}}(360, [\chi])\) into newform subspaces

Label Dim. \(A\) Field Image CM RM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
360.1.u.a \(4\) \(0.180\) \(\Q(\zeta_{8})\) \(D_{4}\) \(\Q(\sqrt{-6}) \) None \(0\) \(0\) \(0\) \(-4\) \(q-\zeta_{8}^{3}q^{2}-\zeta_{8}^{2}q^{4}+\zeta_{8}q^{5}+(-1+\cdots)q^{7}+\cdots\)

Hecke characteristic polynomials

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