Properties

Label 45.2.f
Level 45
Weight 2
Character orbit f
Rep. character \(\chi_{45}(8,\cdot)\)
Character field \(\Q(\zeta_{4})\)
Dimension 4
Newform subspaces 1
Sturm bound 12
Trace bound 0

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Defining parameters

Level: \( N \) = \( 45 = 3^{2} \cdot 5 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 45.f (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 15 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 1 \)
Sturm bound: \(12\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(45, [\chi])\).

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

Trace form

\( 4q - 8q^{7} + O(q^{10}) \) \( 4q - 8q^{7} - 8q^{10} + 4q^{13} + 4q^{16} + 8q^{22} + 16q^{25} + 8q^{28} - 16q^{31} + 4q^{37} - 12q^{40} - 32q^{43} - 16q^{46} + 4q^{52} + 8q^{55} + 12q^{58} + 32q^{61} + 16q^{67} + 24q^{70} + 4q^{73} - 4q^{82} - 32q^{85} - 24q^{88} - 16q^{91} - 44q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(45, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
45.2.f.a \(4\) \(0.359\) \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(-8\) \(q+\zeta_{8}q^{2}-\zeta_{8}^{2}q^{4}+(-\zeta_{8}+2\zeta_{8}^{3})q^{5}+\cdots\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{4} + 16 T^{8} \)
$3$ \( \)
$5$ \( 1 - 8 T^{2} + 25 T^{4} \)
$7$ \( ( 1 + 4 T + 8 T^{2} + 28 T^{3} + 49 T^{4} )^{2} \)
$11$ \( ( 1 - 6 T + 11 T^{2} )^{2}( 1 + 6 T + 11 T^{2} )^{2} \)
$13$ \( ( 1 - 6 T + 13 T^{2} )^{2}( 1 + 4 T + 13 T^{2} )^{2} \)
$17$ \( 1 - 254 T^{4} + 83521 T^{8} \)
$19$ \( ( 1 - 19 T^{2} )^{4} \)
$23$ \( 1 - 158 T^{4} + 279841 T^{8} \)
$29$ \( ( 1 + 40 T^{2} + 841 T^{4} )^{2} \)
$31$ \( ( 1 + 4 T + 31 T^{2} )^{4} \)
$37$ \( ( 1 - 2 T + 2 T^{2} - 74 T^{3} + 1369 T^{4} )^{2} \)
$41$ \( ( 1 - 80 T^{2} + 1681 T^{4} )^{2} \)
$43$ \( ( 1 + 16 T + 128 T^{2} + 688 T^{3} + 1849 T^{4} )^{2} \)
$47$ \( 1 - 3518 T^{4} + 4879681 T^{8} \)
$53$ \( ( 1 - 56 T^{2} + 2809 T^{4} )( 1 + 56 T^{2} + 2809 T^{4} ) \)
$59$ \( ( 1 + 46 T^{2} + 3481 T^{4} )^{2} \)
$61$ \( ( 1 - 8 T + 61 T^{2} )^{4} \)
$67$ \( ( 1 - 8 T + 32 T^{2} - 536 T^{3} + 4489 T^{4} )^{2} \)
$71$ \( ( 1 - 110 T^{2} + 5041 T^{4} )^{2} \)
$73$ \( ( 1 - 2 T + 2 T^{2} - 146 T^{3} + 5329 T^{4} )^{2} \)
$79$ \( ( 1 - 14 T^{2} + 6241 T^{4} )^{2} \)
$83$ \( 1 + 8722 T^{4} + 47458321 T^{8} \)
$89$ \( ( 1 + 16 T^{2} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 + 22 T + 242 T^{2} + 2134 T^{3} + 9409 T^{4} )^{2} \)
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