Properties

Label 96.2
Level 96
Weight 2
Dimension 98
Nonzero newspaces 6
Newform subspaces 7
Sturm bound 1024
Trace bound 5

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Defining parameters

Level: \( N \) = \( 96 = 2^{5} \cdot 3 \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 6 \)
Newform subspaces: \( 7 \)
Sturm bound: \(1024\)
Trace bound: \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(96))\).

Total New Old
Modular forms 320 118 202
Cusp forms 193 98 95
Eisenstein series 127 20 107

Trace form

\( 98q - 2q^{3} - 8q^{4} + 4q^{5} - 4q^{6} - 4q^{7} - 2q^{9} + O(q^{10}) \) \( 98q - 2q^{3} - 8q^{4} + 4q^{5} - 4q^{6} - 4q^{7} - 2q^{9} - 24q^{10} - 20q^{12} - 20q^{13} - 32q^{14} - 12q^{15} - 48q^{16} - 16q^{17} - 12q^{18} - 12q^{19} - 32q^{20} - 20q^{21} - 32q^{22} - 24q^{23} + 8q^{24} - 30q^{25} + 40q^{26} - 38q^{27} + 32q^{28} + 4q^{29} + 44q^{30} - 52q^{31} + 40q^{32} - 8q^{33} + 16q^{34} - 48q^{35} + 48q^{36} + 12q^{37} + 40q^{38} - 20q^{39} + 32q^{40} + 8q^{41} + 56q^{42} - 4q^{43} + 8q^{44} + 32q^{45} - 8q^{46} + 24q^{47} + 48q^{48} + 38q^{49} + 24q^{50} + 16q^{51} + 40q^{52} - 12q^{53} + 48q^{54} + 56q^{55} + 56q^{56} + 16q^{57} + 64q^{58} + 64q^{59} + 56q^{60} - 68q^{61} + 48q^{62} + 12q^{63} + 64q^{64} + 8q^{65} + 20q^{66} + 44q^{67} - 8q^{68} - 68q^{69} + 16q^{70} + 40q^{71} - 64q^{72} - 52q^{73} - 32q^{74} + 38q^{75} - 8q^{76} - 64q^{77} - 68q^{78} - 4q^{79} - 56q^{80} - 38q^{81} - 88q^{82} - 136q^{84} - 72q^{85} - 64q^{86} + 64q^{87} - 96q^{88} - 136q^{90} - 8q^{91} - 80q^{92} + 8q^{93} - 96q^{94} + 16q^{95} - 144q^{96} - 28q^{97} - 80q^{98} + 76q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(96))\)

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
96.2.a \(\chi_{96}(1, \cdot)\) 96.2.a.a 1 1
96.2.a.b 1
96.2.c \(\chi_{96}(95, \cdot)\) 96.2.c.a 4 1
96.2.d \(\chi_{96}(49, \cdot)\) 96.2.d.a 2 1
96.2.f \(\chi_{96}(47, \cdot)\) 96.2.f.a 2 1
96.2.j \(\chi_{96}(25, \cdot)\) None 0 2
96.2.k \(\chi_{96}(23, \cdot)\) None 0 2
96.2.n \(\chi_{96}(13, \cdot)\) 96.2.n.a 32 4
96.2.o \(\chi_{96}(11, \cdot)\) 96.2.o.a 56 4

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(96))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(96)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(16))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(24))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(32))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(48))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( 1 + T \))(\( 1 - T \))(\( 1 - 2 T^{2} + 9 T^{4} \))(\( 1 + T^{2} \))(\( 1 - 2 T + 3 T^{2} \))
$5$ (\( 1 - 2 T + 5 T^{2} \))(\( 1 - 2 T + 5 T^{2} \))(\( ( 1 - 2 T^{2} + 25 T^{4} )^{2} \))(\( ( 1 - 4 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} ) \))(\( ( 1 + 5 T^{2} )^{2} \))
$7$ (\( 1 - 4 T + 7 T^{2} \))(\( 1 + 4 T + 7 T^{2} \))(\( ( 1 - 10 T^{2} + 49 T^{4} )^{2} \))(\( ( 1 - 2 T + 7 T^{2} )^{2} \))(\( ( 1 - 7 T^{2} )^{2} \))
$11$ (\( 1 + 4 T + 11 T^{2} \))(\( 1 - 4 T + 11 T^{2} \))(\( ( 1 + 14 T^{2} + 121 T^{4} )^{2} \))(\( ( 1 - 11 T^{2} )^{2} \))(\( ( 1 - 6 T + 11 T^{2} )( 1 + 6 T + 11 T^{2} ) \))
$13$ (\( 1 + 2 T + 13 T^{2} \))(\( 1 + 2 T + 13 T^{2} \))(\( ( 1 + 2 T + 13 T^{2} )^{4} \))(\( ( 1 - 6 T + 13 T^{2} )( 1 + 6 T + 13 T^{2} ) \))(\( ( 1 - 13 T^{2} )^{2} \))
$17$ (\( 1 + 6 T + 17 T^{2} \))(\( 1 + 6 T + 17 T^{2} \))(\( ( 1 - 17 T^{2} )^{4} \))(\( ( 1 + 2 T + 17 T^{2} )^{2} \))(\( ( 1 - 6 T + 17 T^{2} )( 1 + 6 T + 17 T^{2} ) \))
$19$ (\( 1 - 4 T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))(\( ( 1 - 2 T^{2} + 361 T^{4} )^{2} \))(\( 1 - 22 T^{2} + 361 T^{4} \))(\( ( 1 + 2 T + 19 T^{2} )^{2} \))
$23$ (\( 1 + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( ( 1 + 14 T^{2} + 529 T^{4} )^{2} \))(\( ( 1 + 4 T + 23 T^{2} )^{2} \))(\( ( 1 + 23 T^{2} )^{2} \))
$29$ (\( 1 - 2 T + 29 T^{2} \))(\( 1 - 2 T + 29 T^{2} \))(\( ( 1 - 50 T^{2} + 841 T^{4} )^{2} \))(\( 1 - 22 T^{2} + 841 T^{4} \))(\( ( 1 + 29 T^{2} )^{2} \))
$31$ (\( 1 + 4 T + 31 T^{2} \))(\( 1 - 4 T + 31 T^{2} \))(\( ( 1 - 58 T^{2} + 961 T^{4} )^{2} \))(\( ( 1 + 2 T + 31 T^{2} )^{2} \))(\( ( 1 - 31 T^{2} )^{2} \))
$37$ (\( 1 + 2 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( ( 1 - 6 T + 37 T^{2} )^{4} \))(\( 1 - 10 T^{2} + 1369 T^{4} \))(\( ( 1 - 37 T^{2} )^{2} \))
$41$ (\( 1 - 2 T + 41 T^{2} \))(\( 1 - 2 T + 41 T^{2} \))(\( ( 1 - 50 T^{2} + 1681 T^{4} )^{2} \))(\( ( 1 - 2 T + 41 T^{2} )^{2} \))(\( ( 1 - 6 T + 41 T^{2} )( 1 + 6 T + 41 T^{2} ) \))
$43$ (\( 1 + 4 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( ( 1 - 82 T^{2} + 1849 T^{4} )^{2} \))(\( 1 - 70 T^{2} + 1849 T^{4} \))(\( ( 1 - 10 T + 43 T^{2} )^{2} \))
$47$ (\( 1 + 8 T + 47 T^{2} \))(\( 1 - 8 T + 47 T^{2} \))(\( ( 1 - 34 T^{2} + 2209 T^{4} )^{2} \))(\( ( 1 - 12 T + 47 T^{2} )^{2} \))(\( ( 1 + 47 T^{2} )^{2} \))
$53$ (\( 1 - 10 T + 53 T^{2} \))(\( 1 - 10 T + 53 T^{2} \))(\( ( 1 - 34 T^{2} + 2809 T^{4} )^{2} \))(\( 1 - 70 T^{2} + 2809 T^{4} \))(\( ( 1 + 53 T^{2} )^{2} \))
$59$ (\( 1 - 4 T + 59 T^{2} \))(\( 1 + 4 T + 59 T^{2} \))(\( ( 1 + 110 T^{2} + 3481 T^{4} )^{2} \))(\( 1 - 102 T^{2} + 3481 T^{4} \))(\( ( 1 - 6 T + 59 T^{2} )( 1 + 6 T + 59 T^{2} ) \))
$61$ (\( 1 - 6 T + 61 T^{2} \))(\( 1 - 6 T + 61 T^{2} \))(\( ( 1 + 2 T + 61 T^{2} )^{4} \))(\( ( 1 - 61 T^{2} )^{2} \))(\( ( 1 - 61 T^{2} )^{2} \))
$67$ (\( 1 + 4 T + 67 T^{2} \))(\( 1 - 4 T + 67 T^{2} \))(\( ( 1 - 130 T^{2} + 4489 T^{4} )^{2} \))(\( 1 + 10 T^{2} + 4489 T^{4} \))(\( ( 1 + 14 T + 67 T^{2} )^{2} \))
$71$ (\( 1 - 16 T + 71 T^{2} \))(\( 1 + 16 T + 71 T^{2} \))(\( ( 1 + 110 T^{2} + 5041 T^{4} )^{2} \))(\( ( 1 + 12 T + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))
$73$ (\( 1 + 6 T + 73 T^{2} \))(\( 1 + 6 T + 73 T^{2} \))(\( ( 1 + 6 T + 73 T^{2} )^{4} \))(\( ( 1 + 6 T + 73 T^{2} )^{2} \))(\( ( 1 - 2 T + 73 T^{2} )^{2} \))
$79$ (\( 1 + 4 T + 79 T^{2} \))(\( 1 - 4 T + 79 T^{2} \))(\( ( 1 + 38 T^{2} + 6241 T^{4} )^{2} \))(\( ( 1 + 10 T + 79 T^{2} )^{2} \))(\( ( 1 - 79 T^{2} )^{2} \))
$83$ (\( 1 + 12 T + 83 T^{2} \))(\( 1 - 12 T + 83 T^{2} \))(\( ( 1 + 158 T^{2} + 6889 T^{4} )^{2} \))(\( 1 + 90 T^{2} + 6889 T^{4} \))(\( ( 1 - 18 T + 83 T^{2} )( 1 + 18 T + 83 T^{2} ) \))
$89$ (\( 1 - 10 T + 89 T^{2} \))(\( 1 - 10 T + 89 T^{2} \))(\( ( 1 + 110 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 + 10 T + 89 T^{2} )^{2} \))(\( ( 1 - 18 T + 89 T^{2} )( 1 + 18 T + 89 T^{2} ) \))
$97$ (\( 1 + 14 T + 97 T^{2} \))(\( 1 + 14 T + 97 T^{2} \))(\( ( 1 - 10 T + 97 T^{2} )^{4} \))(\( ( 1 + 2 T + 97 T^{2} )^{2} \))(\( ( 1 + 10 T + 97 T^{2} )^{2} \))
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