Properties

Label 136.1.j
Level 136
Weight 1
Character orbit j
Rep. character \(\chi_{136}(115,\cdot)\)
Character field \(\Q(\zeta_{4})\)
Dimension 2
Newform subspaces 1
Sturm bound 18
Trace bound 0

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Defining parameters

Level: \( N \) \(=\) \( 136 = 2^{3} \cdot 17 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 136.j (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 136 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 1 \)
Sturm bound: \(18\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(136, [\chi])\).

Total New Old
Modular forms 6 6 0
Cusp forms 2 2 0
Eisenstein series 4 4 0

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

\( 2q - 2q^{3} - 2q^{4} - 2q^{6} + O(q^{10}) \) \( 2q - 2q^{3} - 2q^{4} - 2q^{6} + 2q^{11} + 2q^{12} + 2q^{16} + 2q^{18} - 2q^{22} + 2q^{24} - 4q^{33} - 2q^{34} + 4q^{38} - 2q^{41} - 2q^{44} - 2q^{48} + 2q^{50} - 2q^{51} + 4q^{57} - 2q^{64} - 2q^{72} - 2q^{73} + 2q^{75} + 2q^{81} + 2q^{82} + 2q^{88} - 2q^{96} + 2q^{97} - 2q^{98} + 2q^{99} + O(q^{100}) \)

Decomposition of \(S_{1}^{\mathrm{new}}(136, [\chi])\) into newform subspaces

Label Dim. \(A\) Field Image CM RM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
136.1.j.a \(2\) \(0.068\) \(\Q(\sqrt{-1}) \) \(D_{4}\) \(\Q(\sqrt{-2}) \) None \(0\) \(-2\) \(0\) \(0\) \(q-iq^{2}+(-1-i)q^{3}-q^{4}+(-1+i+\cdots)q^{6}+\cdots\)

Hecke characteristic polynomials

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