Properties

Label 72.1
Level 72
Weight 1
Dimension 2
Nonzero newspaces 1
Newform subspaces 1
Sturm bound 288
Trace bound 0

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Defining parameters

Level: \( N \) = \( 72\( 72 = 2^{3} \cdot 3^{2} \) \)
Weight: \( k \) = \( 1 \)
Nonzero newspaces: \( 1 \)
Newform subspaces: \( 1 \)
Sturm bound: \(288\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(\Gamma_1(72))\).

Total New Old
Modular forms 50 11 39
Cusp forms 2 2 0
Eisenstein series 48 9 39

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

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

Decomposition of \(S_{1}^{\mathrm{new}}(\Gamma_1(72))\)

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
72.1.b \(\chi_{72}(19, \cdot)\) None 0 1
72.1.e \(\chi_{72}(17, \cdot)\) None 0 1
72.1.g \(\chi_{72}(55, \cdot)\) None 0 1
72.1.h \(\chi_{72}(53, \cdot)\) None 0 1
72.1.j \(\chi_{72}(5, \cdot)\) None 0 2
72.1.k \(\chi_{72}(7, \cdot)\) None 0 2
72.1.m \(\chi_{72}(41, \cdot)\) None 0 2
72.1.p \(\chi_{72}(43, \cdot)\) 72.1.p.a 2 2

Hecke characteristic polynomials

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