Properties

Label 20.2
Level 20
Weight 2
Dimension 3
Nonzero newspaces 2
Newform subspaces 2
Sturm bound 48
Trace bound 1

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Defining parameters

Level: \( N \) = \( 20\( 20 = 2^{2} \cdot 5 \) \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 2 \)
Newform subspaces: \( 2 \)
Sturm bound: \(48\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(20))\).

Total New Old
Modular forms 22 11 11
Cusp forms 3 3 0
Eisenstein series 19 8 11

Trace form

\( 3q - 2q^{2} - 2q^{3} - 5q^{5} + 2q^{7} + 4q^{8} + q^{9} + O(q^{10}) \) \( 3q - 2q^{2} - 2q^{3} - 5q^{5} + 2q^{7} + 4q^{8} + q^{9} + 6q^{10} + 2q^{15} - 8q^{16} - 6q^{18} - 4q^{19} - 4q^{20} - 4q^{21} + 6q^{23} + 7q^{25} + 4q^{26} + 4q^{27} + 6q^{29} - 4q^{31} + 8q^{32} - 2q^{35} + 12q^{36} - 12q^{37} - 4q^{39} - 4q^{40} - 10q^{41} - 10q^{43} + 5q^{45} - 6q^{47} - 3q^{49} - 14q^{50} + 12q^{51} - 4q^{52} + 12q^{53} + 8q^{57} + 8q^{58} + 12q^{59} + 26q^{61} + 2q^{63} + 2q^{67} - 12q^{68} - 12q^{69} - 12q^{71} - 12q^{72} - 20q^{73} - 2q^{75} + 8q^{79} + 16q^{80} - 29q^{81} + 16q^{82} + 6q^{83} - 12q^{85} - 12q^{87} - 6q^{89} + 6q^{90} + 4q^{91} + 8q^{93} + 4q^{95} + 28q^{97} + 14q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(20))\)

We only show spaces with even 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
20.2.a \(\chi_{20}(1, \cdot)\) 20.2.a.a 1 1
20.2.c \(\chi_{20}(9, \cdot)\) None 0 1
20.2.e \(\chi_{20}(3, \cdot)\) 20.2.e.a 2 2

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 2 T + 2 T^{2} \))
$3$ (\( 1 + 2 T + 3 T^{2} \))(\( 1 + 9 T^{4} \))
$5$ (\( 1 + T \))(\( 1 + 4 T + 5 T^{2} \))
$7$ (\( 1 - 2 T + 7 T^{2} \))(\( 1 + 49 T^{4} \))
$11$ (\( 1 + 11 T^{2} \))(\( ( 1 - 11 T^{2} )^{2} \))
$13$ (\( 1 - 2 T + 13 T^{2} \))(\( ( 1 - 4 T + 13 T^{2} )( 1 + 6 T + 13 T^{2} ) \))
$17$ (\( 1 + 6 T + 17 T^{2} \))(\( ( 1 - 8 T + 17 T^{2} )( 1 + 2 T + 17 T^{2} ) \))
$19$ (\( 1 + 4 T + 19 T^{2} \))(\( ( 1 + 19 T^{2} )^{2} \))
$23$ (\( 1 - 6 T + 23 T^{2} \))(\( 1 + 529 T^{4} \))
$29$ (\( 1 - 6 T + 29 T^{2} \))(\( ( 1 - 10 T + 29 T^{2} )( 1 + 10 T + 29 T^{2} ) \))
$31$ (\( 1 + 4 T + 31 T^{2} \))(\( ( 1 - 31 T^{2} )^{2} \))
$37$ (\( 1 - 2 T + 37 T^{2} \))(\( ( 1 + 2 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} ) \))
$41$ (\( 1 - 6 T + 41 T^{2} \))(\( ( 1 + 8 T + 41 T^{2} )^{2} \))
$43$ (\( 1 + 10 T + 43 T^{2} \))(\( 1 + 1849 T^{4} \))
$47$ (\( 1 + 6 T + 47 T^{2} \))(\( 1 + 2209 T^{4} \))
$53$ (\( 1 + 6 T + 53 T^{2} \))(\( ( 1 - 14 T + 53 T^{2} )( 1 - 4 T + 53 T^{2} ) \))
$59$ (\( 1 - 12 T + 59 T^{2} \))(\( ( 1 + 59 T^{2} )^{2} \))
$61$ (\( 1 - 2 T + 61 T^{2} \))(\( ( 1 - 12 T + 61 T^{2} )^{2} \))
$67$ (\( 1 - 2 T + 67 T^{2} \))(\( 1 + 4489 T^{4} \))
$71$ (\( 1 + 12 T + 71 T^{2} \))(\( ( 1 - 71 T^{2} )^{2} \))
$73$ (\( 1 - 2 T + 73 T^{2} \))(\( ( 1 + 6 T + 73 T^{2} )( 1 + 16 T + 73 T^{2} ) \))
$79$ (\( 1 - 8 T + 79 T^{2} \))(\( ( 1 + 79 T^{2} )^{2} \))
$83$ (\( 1 - 6 T + 83 T^{2} \))(\( 1 + 6889 T^{4} \))
$89$ (\( 1 + 6 T + 89 T^{2} \))(\( ( 1 - 10 T + 89 T^{2} )( 1 + 10 T + 89 T^{2} ) \))
$97$ (\( 1 - 2 T + 97 T^{2} \))(\( ( 1 - 18 T + 97 T^{2} )( 1 - 8 T + 97 T^{2} ) \))
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