Properties

Label 30.2
Level 30
Weight 2
Dimension 7
Nonzero newspaces 3
Newform subspaces 3
Sturm bound 96
Trace bound 1

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Defining parameters

Level: \( N \) = \( 30 = 2 \cdot 3 \cdot 5 \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 3 \)
Newform subspaces: \( 3 \)
Sturm bound: \(96\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(30))\).

Total New Old
Modular forms 40 7 33
Cusp forms 9 7 2
Eisenstein series 31 0 31

Trace form

\( 7q - q^{2} - 3q^{3} - q^{4} - 5q^{5} - 3q^{6} - 8q^{7} - q^{8} - q^{9} + O(q^{10}) \) \( 7q - q^{2} - 3q^{3} - q^{4} - 5q^{5} - 3q^{6} - 8q^{7} - q^{8} - q^{9} + 3q^{10} + 4q^{11} + 5q^{12} + 2q^{13} + 8q^{14} + 9q^{15} - q^{16} + 6q^{17} + 7q^{18} - 4q^{19} + 3q^{20} - 4q^{22} - 3q^{24} - 9q^{25} - 14q^{26} - 3q^{27} - 8q^{28} - 6q^{29} - 15q^{30} - 16q^{31} - q^{32} - 4q^{33} - 2q^{34} + 8q^{35} - q^{36} + 26q^{37} + 4q^{38} + 14q^{39} + 11q^{40} - 2q^{41} + 8q^{42} + 20q^{43} - 4q^{44} - 5q^{45} + 24q^{46} + 5q^{48} + 15q^{49} + 7q^{50} - 6q^{51} + 2q^{52} - 6q^{53} - 3q^{54} + 4q^{55} - 20q^{57} - 14q^{58} - 20q^{59} - 7q^{60} - 30q^{61} - 8q^{62} - 8q^{63} - q^{64} - 14q^{65} + 12q^{66} - 20q^{67} + 6q^{68} - 8q^{69} - 8q^{70} + 24q^{71} - 9q^{72} - 18q^{73} + 2q^{74} + 21q^{75} + 12q^{76} - 2q^{78} + 8q^{79} - 5q^{80} + 31q^{81} - 10q^{82} + 12q^{83} + 6q^{85} + 12q^{86} + 14q^{87} - 4q^{88} + 38q^{89} + 11q^{90} + 16q^{91} + 16q^{93} - 16q^{94} + 4q^{95} + 5q^{96} + 14q^{97} - 9q^{98} - 4q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(30))\)

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
30.2.a \(\chi_{30}(1, \cdot)\) 30.2.a.a 1 1
30.2.c \(\chi_{30}(19, \cdot)\) 30.2.c.a 2 1
30.2.e \(\chi_{30}(17, \cdot)\) 30.2.e.a 4 2

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(30))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(30)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(15))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T \))(\( 1 + T^{2} \))(\( 1 + T^{4} \))
$3$ (\( 1 - T \))(\( 1 + T^{2} \))(\( 1 + 4 T + 8 T^{2} + 12 T^{3} + 9 T^{4} \))
$5$ (\( 1 + T \))(\( 1 + 4 T + 5 T^{2} \))(\( 1 + 8 T^{2} + 25 T^{4} \))
$7$ (\( 1 + 4 T + 7 T^{2} \))(\( 1 - 10 T^{2} + 49 T^{4} \))(\( ( 1 + 2 T + 2 T^{2} + 14 T^{3} + 49 T^{4} )^{2} \))
$11$ (\( 1 + 11 T^{2} \))(\( ( 1 - 2 T + 11 T^{2} )^{2} \))(\( ( 1 - 20 T^{2} + 121 T^{4} )^{2} \))
$13$ (\( 1 - 2 T + 13 T^{2} \))(\( ( 1 - 4 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} ) \))(\( ( 1 + 169 T^{4} )^{2} \))
$17$ (\( 1 - 6 T + 17 T^{2} \))(\( ( 1 - 8 T + 17 T^{2} )( 1 + 8 T + 17 T^{2} ) \))(\( ( 1 - 16 T^{2} + 289 T^{4} )( 1 + 16 T^{2} + 289 T^{4} ) \))
$19$ (\( 1 + 4 T + 19 T^{2} \))(\( ( 1 + 19 T^{2} )^{2} \))(\( ( 1 - 22 T^{2} + 361 T^{4} )^{2} \))
$23$ (\( 1 + 23 T^{2} \))(\( 1 - 30 T^{2} + 529 T^{4} \))(\( 1 - 158 T^{4} + 279841 T^{8} \))
$29$ (\( 1 + 6 T + 29 T^{2} \))(\( ( 1 + 29 T^{2} )^{2} \))(\( ( 1 + 8 T^{2} + 841 T^{4} )^{2} \))
$31$ (\( 1 - 8 T + 31 T^{2} \))(\( ( 1 + 8 T + 31 T^{2} )^{2} \))(\( ( 1 + 2 T + 31 T^{2} )^{4} \))
$37$ (\( 1 - 2 T + 37 T^{2} \))(\( ( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} ) \))(\( ( 1 - 12 T + 72 T^{2} - 444 T^{3} + 1369 T^{4} )^{2} \))
$41$ (\( 1 + 6 T + 41 T^{2} \))(\( ( 1 - 2 T + 41 T^{2} )^{2} \))(\( ( 1 - 50 T^{2} + 1681 T^{4} )^{2} \))
$43$ (\( 1 + 4 T + 43 T^{2} \))(\( 1 - 70 T^{2} + 1849 T^{4} \))(\( ( 1 - 12 T + 72 T^{2} - 516 T^{3} + 1849 T^{4} )^{2} \))
$47$ (\( 1 + 47 T^{2} \))(\( 1 - 30 T^{2} + 2209 T^{4} \))(\( ( 1 + 2209 T^{4} )^{2} \))
$53$ (\( 1 + 6 T + 53 T^{2} \))(\( 1 - 70 T^{2} + 2809 T^{4} \))(\( ( 1 - 56 T^{2} + 2809 T^{4} )( 1 + 56 T^{2} + 2809 T^{4} ) \))
$59$ (\( 1 + 59 T^{2} \))(\( ( 1 + 10 T + 59 T^{2} )^{2} \))(\( ( 1 + 20 T^{2} + 3481 T^{4} )^{2} \))
$61$ (\( 1 + 10 T + 61 T^{2} \))(\( ( 1 - 2 T + 61 T^{2} )^{2} \))(\( ( 1 + 6 T + 61 T^{2} )^{4} \))
$67$ (\( 1 + 4 T + 67 T^{2} \))(\( 1 - 70 T^{2} + 4489 T^{4} \))(\( ( 1 + 8 T + 32 T^{2} + 536 T^{3} + 4489 T^{4} )^{2} \))
$71$ (\( 1 + 71 T^{2} \))(\( ( 1 - 12 T + 71 T^{2} )^{2} \))(\( ( 1 + 58 T^{2} + 5041 T^{4} )^{2} \))
$73$ (\( 1 - 2 T + 73 T^{2} \))(\( 1 - 130 T^{2} + 5329 T^{4} \))(\( ( 1 - 6 T + 73 T^{2} )^{2}( 1 + 16 T + 73 T^{2} )^{2} \))
$79$ (\( 1 - 8 T + 79 T^{2} \))(\( ( 1 + 79 T^{2} )^{2} \))(\( ( 1 - 122 T^{2} + 6241 T^{4} )^{2} \))
$83$ (\( 1 - 12 T + 83 T^{2} \))(\( 1 - 150 T^{2} + 6889 T^{4} \))(\( 1 - 13294 T^{4} + 47458321 T^{8} \))
$89$ (\( 1 - 18 T + 89 T^{2} \))(\( ( 1 - 10 T + 89 T^{2} )^{2} \))(\( ( 1 + 170 T^{2} + 7921 T^{4} )^{2} \))
$97$ (\( 1 - 2 T + 97 T^{2} \))(\( ( 1 - 18 T + 97 T^{2} )( 1 + 18 T + 97 T^{2} ) \))(\( ( 1 - 6 T + 18 T^{2} - 582 T^{3} + 9409 T^{4} )^{2} \))
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