Properties

Label 68.1
Level 68
Weight 1
Dimension 3
Nonzero newspaces 2
Newform subspaces 2
Sturm bound 288
Trace bound 1

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Defining parameters

Level: \( N \) = \( 68 = 2^{2} \cdot 17 \)
Weight: \( k \) = \( 1 \)
Nonzero newspaces: \( 2 \)
Newform subspaces: \( 2 \)
Sturm bound: \(288\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(\Gamma_1(68))\).

Total New Old
Modular forms 43 17 26
Cusp forms 3 3 0
Eisenstein series 40 14 26

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

\(D_n\) \(A_4\) \(S_4\) \(A_5\)
Dimension 3 0 0 0

Trace form

\( 3q - q^{2} - q^{4} - 2q^{5} - q^{8} - q^{9} + O(q^{10}) \) \( 3q - q^{2} - q^{4} - 2q^{5} - q^{8} - q^{9} + 2q^{10} - 2q^{13} + 3q^{16} - q^{17} - q^{18} + 2q^{20} + q^{25} + 2q^{26} + 2q^{29} - q^{32} - q^{34} - q^{36} - 2q^{37} - 2q^{40} + 2q^{41} + 2q^{45} - q^{49} - 3q^{50} - 2q^{52} + 2q^{53} - 2q^{58} - 2q^{61} - q^{64} + 3q^{68} + 3q^{72} + 2q^{73} + 2q^{74} - 2q^{80} - q^{81} + 2q^{82} + 2q^{85} - 2q^{89} + 2q^{90} - 2q^{97} + 3q^{98} + O(q^{100}) \)

Decomposition of \(S_{1}^{\mathrm{new}}(\Gamma_1(68))\)

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
68.1.c \(\chi_{68}(35, \cdot)\) None 0 1
68.1.d \(\chi_{68}(67, \cdot)\) 68.1.d.a 1 1
68.1.f \(\chi_{68}(47, \cdot)\) 68.1.f.a 2 2
68.1.g \(\chi_{68}(15, \cdot)\) None 0 4
68.1.j \(\chi_{68}(5, \cdot)\) None 0 8

Hecke characteristic polynomials

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