Properties

Label 135.1.d
Level 135
Weight 1
Character orbit d
Rep. character \(\chi_{135}(134,\cdot)\)
Character field \(\Q\)
Dimension 2
Newform subspaces 2
Sturm bound 18
Trace bound 2

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Defining parameters

Level: \( N \) \(=\) \( 135 = 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 135.d (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 15 \)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(18\)
Trace bound: \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(135, [\chi])\).

Total New Old
Modular forms 8 2 6
Cusp forms 2 2 0
Eisenstein series 6 0 6

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

\(D_n\) \(A_4\) \(S_4\) \(A_5\)
Dimension 2 0 0 0

Trace form

\( 2q + O(q^{10}) \) \( 2q - 2q^{10} - 2q^{16} - 2q^{19} + 2q^{25} - 2q^{31} + 2q^{34} + 2q^{40} + 2q^{46} + 2q^{49} - 2q^{61} + 2q^{64} - 2q^{79} - 2q^{85} - 4q^{94} + O(q^{100}) \)

Decomposition of \(S_{1}^{\mathrm{new}}(135, [\chi])\) into newform subspaces

Label Dim. \(A\) Field Image CM RM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
135.1.d.a \(1\) \(0.067\) \(\Q\) \(D_{3}\) \(\Q(\sqrt{-15}) \) None \(-1\) \(0\) \(1\) \(0\) \(q-q^{2}+q^{5}+q^{8}-q^{10}-q^{16}-q^{17}+\cdots\)
135.1.d.b \(1\) \(0.067\) \(\Q\) \(D_{3}\) \(\Q(\sqrt{-15}) \) None \(1\) \(0\) \(-1\) \(0\) \(q+q^{2}-q^{5}-q^{8}-q^{10}-q^{16}+q^{17}+\cdots\)

Hecke characteristic polynomials

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