Properties

Label 64.2
Level 64
Weight 2
Dimension 61
Nonzero newspaces 4
Newform subspaces 4
Sturm bound 512
Trace bound 3

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Defining parameters

Level: \( N \) = \( 64 = 2^{6} \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 4 \)
Newform subspaces: \( 4 \)
Sturm bound: \(512\)
Trace bound: \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(64))\).

Total New Old
Modular forms 164 83 81
Cusp forms 93 61 32
Eisenstein series 71 22 49

Trace form

\( 61q - 8q^{2} - 6q^{3} - 8q^{4} - 8q^{5} - 8q^{6} - 8q^{7} - 8q^{8} - 13q^{9} + O(q^{10}) \) \( 61q - 8q^{2} - 6q^{3} - 8q^{4} - 8q^{5} - 8q^{6} - 8q^{7} - 8q^{8} - 13q^{9} - 8q^{10} - 10q^{11} - 8q^{12} - 16q^{13} - 8q^{14} - 12q^{15} - 8q^{16} - 22q^{17} - 8q^{18} - 14q^{19} - 8q^{20} - 4q^{21} - 8q^{23} + 32q^{24} + q^{25} + 32q^{26} + 32q^{28} + 8q^{29} + 72q^{30} + 16q^{31} + 32q^{32} + 20q^{33} + 32q^{34} - 4q^{35} + 72q^{36} + 32q^{38} - 8q^{39} + 32q^{40} - 10q^{41} + 32q^{42} - 18q^{43} - 12q^{45} - 8q^{46} - 24q^{47} - 8q^{48} - 23q^{49} - 32q^{50} + 20q^{51} - 56q^{52} - 32q^{53} - 72q^{54} + 56q^{55} - 64q^{56} - 16q^{57} - 80q^{58} + 62q^{59} - 104q^{60} - 16q^{61} - 40q^{62} + 60q^{63} - 104q^{64} - 24q^{65} - 88q^{66} + 82q^{67} - 56q^{68} - 20q^{69} - 104q^{70} + 56q^{71} - 80q^{72} - 10q^{73} - 64q^{74} + 50q^{75} - 72q^{76} - 4q^{77} - 32q^{78} + 24q^{79} + 32q^{80} - 11q^{81} + 72q^{82} - 6q^{83} + 104q^{84} + 96q^{86} - 8q^{87} + 72q^{88} + 38q^{89} + 136q^{90} - 12q^{91} + 144q^{92} + 32q^{93} + 88q^{94} + 12q^{95} + 128q^{96} + 34q^{97} + 128q^{98} + 18q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(64))\)

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
64.2.a \(\chi_{64}(1, \cdot)\) 64.2.a.a 1 1
64.2.b \(\chi_{64}(33, \cdot)\) 64.2.b.a 2 1
64.2.e \(\chi_{64}(17, \cdot)\) 64.2.e.a 2 2
64.2.g \(\chi_{64}(9, \cdot)\) None 0 4
64.2.i \(\chi_{64}(5, \cdot)\) 64.2.i.a 56 8

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(64))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(64)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(16))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(32))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( 1 + 3 T^{2} \))(\( 1 - 2 T^{2} + 9 T^{4} \))(\( 1 - 2 T + 2 T^{2} - 6 T^{3} + 9 T^{4} \))
$5$ (\( 1 - 2 T + 5 T^{2} \))(\( ( 1 - 5 T^{2} )^{2} \))(\( ( 1 - 2 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} ) \))
$7$ (\( 1 + 7 T^{2} \))(\( ( 1 + 7 T^{2} )^{2} \))(\( 1 - 10 T^{2} + 49 T^{4} \))
$11$ (\( 1 + 11 T^{2} \))(\( 1 + 14 T^{2} + 121 T^{4} \))(\( 1 + 2 T + 2 T^{2} + 22 T^{3} + 121 T^{4} \))
$13$ (\( 1 + 6 T + 13 T^{2} \))(\( ( 1 - 13 T^{2} )^{2} \))(\( ( 1 - 4 T + 13 T^{2} )( 1 + 6 T + 13 T^{2} ) \))
$17$ (\( 1 - 2 T + 17 T^{2} \))(\( ( 1 + 6 T + 17 T^{2} )^{2} \))(\( ( 1 + 2 T + 17 T^{2} )^{2} \))
$19$ (\( 1 + 19 T^{2} \))(\( 1 - 34 T^{2} + 361 T^{4} \))(\( 1 + 6 T + 18 T^{2} + 114 T^{3} + 361 T^{4} \))
$23$ (\( 1 + 23 T^{2} \))(\( ( 1 + 23 T^{2} )^{2} \))(\( 1 - 10 T^{2} + 529 T^{4} \))
$29$ (\( 1 - 10 T + 29 T^{2} \))(\( ( 1 - 29 T^{2} )^{2} \))(\( ( 1 - 10 T + 29 T^{2} )( 1 + 4 T + 29 T^{2} ) \))
$31$ (\( 1 + 31 T^{2} \))(\( ( 1 + 31 T^{2} )^{2} \))(\( ( 1 - 8 T + 31 T^{2} )^{2} \))
$37$ (\( 1 - 2 T + 37 T^{2} \))(\( ( 1 - 37 T^{2} )^{2} \))(\( 1 - 6 T + 18 T^{2} - 222 T^{3} + 1369 T^{4} \))
$41$ (\( 1 - 10 T + 41 T^{2} \))(\( ( 1 + 6 T + 41 T^{2} )^{2} \))(\( ( 1 - 41 T^{2} )^{2} \))
$43$ (\( 1 + 43 T^{2} \))(\( 1 + 14 T^{2} + 1849 T^{4} \))(\( 1 + 10 T + 50 T^{2} + 430 T^{3} + 1849 T^{4} \))
$47$ (\( 1 + 47 T^{2} \))(\( ( 1 + 47 T^{2} )^{2} \))(\( ( 1 + 8 T + 47 T^{2} )^{2} \))
$53$ (\( 1 + 14 T + 53 T^{2} \))(\( ( 1 - 53 T^{2} )^{2} \))(\( ( 1 - 4 T + 53 T^{2} )( 1 + 14 T + 53 T^{2} ) \))
$59$ (\( 1 + 59 T^{2} \))(\( 1 - 82 T^{2} + 3481 T^{4} \))(\( 1 - 6 T + 18 T^{2} - 354 T^{3} + 3481 T^{4} \))
$61$ (\( 1 - 10 T + 61 T^{2} \))(\( ( 1 - 61 T^{2} )^{2} \))(\( 1 + 18 T + 162 T^{2} + 1098 T^{3} + 3721 T^{4} \))
$67$ (\( 1 + 67 T^{2} \))(\( 1 + 62 T^{2} + 4489 T^{4} \))(\( 1 - 10 T + 50 T^{2} - 670 T^{3} + 4489 T^{4} \))
$71$ (\( 1 + 71 T^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( 1 - 42 T^{2} + 5041 T^{4} \))
$73$ (\( 1 + 6 T + 73 T^{2} \))(\( ( 1 - 2 T + 73 T^{2} )^{2} \))(\( 1 - 130 T^{2} + 5329 T^{4} \))
$79$ (\( 1 + 79 T^{2} \))(\( ( 1 + 79 T^{2} )^{2} \))(\( ( 1 + 79 T^{2} )^{2} \))
$83$ (\( 1 + 83 T^{2} \))(\( 1 + 158 T^{2} + 6889 T^{4} \))(\( 1 - 2 T + 2 T^{2} - 166 T^{3} + 6889 T^{4} \))
$89$ (\( 1 - 10 T + 89 T^{2} \))(\( ( 1 - 18 T + 89 T^{2} )^{2} \))(\( 1 - 162 T^{2} + 7921 T^{4} \))
$97$ (\( 1 - 18 T + 97 T^{2} \))(\( ( 1 - 10 T + 97 T^{2} )^{2} \))(\( ( 1 + 2 T + 97 T^{2} )^{2} \))
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