Properties

Label 136.1
Level 136
Weight 1
Dimension 7
Nonzero newspaces 3
Newform subspaces 3
Sturm bound 1152
Trace bound 1

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Defining parameters

Level: \( N \) = \( 136 = 2^{3} \cdot 17 \)
Weight: \( k \) = \( 1 \)
Nonzero newspaces: \( 3 \)
Newform subspaces: \( 3 \)
Sturm bound: \(1152\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(\Gamma_1(136))\).

Total New Old
Modular forms 109 37 72
Cusp forms 13 7 6
Eisenstein series 96 30 66

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

\(D_n\) \(A_4\) \(S_4\) \(A_5\)
Dimension 7 0 0 0

Trace form

\( 7q - q^{2} - 2q^{3} - q^{4} - 2q^{6} - q^{8} - 3q^{9} + O(q^{10}) \) \( 7q - q^{2} - 2q^{3} - q^{4} - 2q^{6} - q^{8} - 3q^{9} - 2q^{11} + 6q^{12} - q^{16} - q^{17} - 3q^{18} - 2q^{19} - 2q^{22} + 6q^{24} - q^{25} + 4q^{27} - q^{32} - 4q^{33} - q^{34} - 3q^{36} + 6q^{38} - 2q^{41} + 6q^{43} - 2q^{44} - 2q^{48} - q^{49} + 7q^{50} - 2q^{51} + 4q^{54} + 4q^{57} - 2q^{59} - q^{64} + 4q^{66} - 2q^{67} - q^{68} - 3q^{72} - 2q^{73} - 2q^{75} - 2q^{76} + 3q^{81} - 2q^{82} + 6q^{83} - 2q^{86} - 2q^{88} - 2q^{89} - 2q^{96} - 2q^{97} - q^{98} + 2q^{99} + O(q^{100}) \)

Decomposition of \(S_{1}^{\mathrm{new}}(\Gamma_1(136))\)

We only show spaces with odd parity, since no modular forms exist when this condition is not satisfied. Within each space \( S_k^{\mathrm{new}}(N, \chi) \) we list the newforms together with their dimension.

Label \(\chi\) Newforms Dimension \(\chi\) degree
136.1.d \(\chi_{136}(103, \cdot)\) None 0 1
136.1.e \(\chi_{136}(67, \cdot)\) 136.1.e.a 1 1
136.1.f \(\chi_{136}(35, \cdot)\) None 0 1
136.1.g \(\chi_{136}(135, \cdot)\) None 0 1
136.1.j \(\chi_{136}(115, \cdot)\) 136.1.j.a 2 2
136.1.l \(\chi_{136}(47, \cdot)\) None 0 2
136.1.m \(\chi_{136}(15, \cdot)\) None 0 4
136.1.p \(\chi_{136}(19, \cdot)\) 136.1.p.a 4 4
136.1.q \(\chi_{136}(5, \cdot)\) None 0 8
136.1.t \(\chi_{136}(41, \cdot)\) None 0 8

Decomposition of \(S_{1}^{\mathrm{old}}(\Gamma_1(136))\) into lower level spaces

\( S_{1}^{\mathrm{old}}(\Gamma_1(136)) \cong \) \(S_{1}^{\mathrm{new}}(\Gamma_1(68))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T \))(\( 1 + T^{2} \))(\( 1 + T^{4} \))
$3$ (\( ( 1 - T )( 1 + T ) \))(\( ( 1 + T )^{2}( 1 + T^{2} ) \))(\( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \))
$5$ (\( 1 + T^{2} \))(\( 1 + T^{4} \))(\( 1 + T^{8} \))
$7$ (\( 1 + T^{2} \))(\( 1 + T^{4} \))(\( 1 + T^{8} \))
$11$ (\( ( 1 - T )( 1 + T ) \))(\( ( 1 - T )^{2}( 1 + T^{2} ) \))(\( ( 1 + T )^{4}( 1 + T^{4} ) \))
$13$ (\( ( 1 - T )( 1 + T ) \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 + T^{2} )^{4} \))
$17$ (\( 1 + T \))(\( 1 + T^{2} \))(\( 1 + T^{4} \))
$19$ (\( ( 1 + T )^{2} \))(\( ( 1 + T^{2} )^{2} \))(\( ( 1 + T^{4} )^{2} \))
$23$ (\( 1 + T^{2} \))(\( 1 + T^{4} \))(\( 1 + T^{8} \))
$29$ (\( 1 + T^{2} \))(\( 1 + T^{4} \))(\( 1 + T^{8} \))
$31$ (\( 1 + T^{2} \))(\( 1 + T^{4} \))(\( 1 + T^{8} \))
$37$ (\( 1 + T^{2} \))(\( 1 + T^{4} \))(\( 1 + T^{8} \))
$41$ (\( ( 1 - T )( 1 + T ) \))(\( ( 1 + T )^{2}( 1 + T^{2} ) \))(\( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \))
$43$ (\( ( 1 - T )^{2} \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 - T )^{4}( 1 + T^{2} )^{2} \))
$47$ (\( ( 1 - T )( 1 + T ) \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 + T^{2} )^{4} \))
$53$ (\( ( 1 - T )( 1 + T ) \))(\( ( 1 + T^{2} )^{2} \))(\( ( 1 + T^{4} )^{2} \))
$59$ (\( ( 1 - T )^{2} \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 + T )^{4}( 1 + T^{2} )^{2} \))
$61$ (\( 1 + T^{2} \))(\( 1 + T^{4} \))(\( 1 + T^{8} \))
$67$ (\( ( 1 + T )^{2} \))(\( ( 1 + T^{2} )^{2} \))(\( ( 1 + T^{4} )^{2} \))
$71$ (\( 1 + T^{2} \))(\( 1 + T^{4} \))(\( 1 + T^{8} \))
$73$ (\( ( 1 - T )( 1 + T ) \))(\( ( 1 + T )^{2}( 1 + T^{2} ) \))(\( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \))
$79$ (\( 1 + T^{2} \))(\( 1 + T^{4} \))(\( 1 + T^{8} \))
$83$ (\( ( 1 - T )^{2} \))(\( ( 1 - T )^{2}( 1 + T )^{2} \))(\( ( 1 - T )^{4}( 1 + T^{2} )^{2} \))
$89$ (\( ( 1 + T )^{2} \))(\( ( 1 + T^{2} )^{2} \))(\( ( 1 + T^{4} )^{2} \))
$97$ (\( ( 1 - T )( 1 + T ) \))(\( ( 1 - T )^{2}( 1 + T^{2} ) \))(\( ( 1 + T )^{4}( 1 + T^{4} ) \))
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