Properties

Label 2.8.a
Level 2
Weight 8
Character orbit a
Rep. character \(\chi_{2}(1,\cdot)\)
Character field \(\Q\)
Dimension 1
Newform subspaces 1
Sturm bound 2
Trace bound 0

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Defining parameters

Level: \( N \) = \( 2 \)
Weight: \( k \) = \( 8 \)
Character orbit: \([\chi]\) = 2.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 1 \)
Sturm bound: \(2\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(2))\).

Total New Old
Modular forms 3 1 2
Cusp forms 1 1 0
Eisenstein series 2 0 2

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators.

\(2\)Dim.
\(+\)\(1\)

Trace form

\( q - 8q^{2} + 12q^{3} + 64q^{4} - 210q^{5} - 96q^{6} + 1016q^{7} - 512q^{8} - 2043q^{9} + O(q^{10}) \) \( q - 8q^{2} + 12q^{3} + 64q^{4} - 210q^{5} - 96q^{6} + 1016q^{7} - 512q^{8} - 2043q^{9} + 1680q^{10} + 1092q^{11} + 768q^{12} + 1382q^{13} - 8128q^{14} - 2520q^{15} + 4096q^{16} + 14706q^{17} + 16344q^{18} - 39940q^{19} - 13440q^{20} + 12192q^{21} - 8736q^{22} + 68712q^{23} - 6144q^{24} - 34025q^{25} - 11056q^{26} - 50760q^{27} + 65024q^{28} - 102570q^{29} + 20160q^{30} + 227552q^{31} - 32768q^{32} + 13104q^{33} - 117648q^{34} - 213360q^{35} - 130752q^{36} + 160526q^{37} + 319520q^{38} + 16584q^{39} + 107520q^{40} + 10842q^{41} - 97536q^{42} - 630748q^{43} + 69888q^{44} + 429030q^{45} - 549696q^{46} + 472656q^{47} + 49152q^{48} + 208713q^{49} + 272200q^{50} + 176472q^{51} + 88448q^{52} - 1494018q^{53} + 406080q^{54} - 229320q^{55} - 520192q^{56} - 479280q^{57} + 820560q^{58} + 2640660q^{59} - 161280q^{60} + 827702q^{61} - 1820416q^{62} - 2075688q^{63} + 262144q^{64} - 290220q^{65} - 104832q^{66} - 126004q^{67} + 941184q^{68} + 824544q^{69} + 1706880q^{70} - 1414728q^{71} + 1046016q^{72} + 980282q^{73} - 1284208q^{74} - 408300q^{75} - 2556160q^{76} + 1109472q^{77} - 132672q^{78} - 3566800q^{79} - 860160q^{80} + 3858921q^{81} - 86736q^{82} + 5672892q^{83} + 780288q^{84} - 3088260q^{85} + 5045984q^{86} - 1230840q^{87} - 559104q^{88} - 11951190q^{89} - 3432240q^{90} + 1404112q^{91} + 4397568q^{92} + 2730624q^{93} - 3781248q^{94} + 8387400q^{95} - 393216q^{96} + 8682146q^{97} - 1669704q^{98} - 2230956q^{99} + O(q^{100}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(2))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2
2.8.a.a \(1\) \(0.625\) \(\Q\) None \(-8\) \(12\) \(-210\) \(1016\) \(+\) \(q-8q^{2}+12q^{3}+2^{6}q^{4}-210q^{5}+\cdots\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 8 T \)
$3$ \( 1 - 12 T + 2187 T^{2} \)
$5$ \( 1 + 210 T + 78125 T^{2} \)
$7$ \( 1 - 1016 T + 823543 T^{2} \)
$11$ \( 1 - 1092 T + 19487171 T^{2} \)
$13$ \( 1 - 1382 T + 62748517 T^{2} \)
$17$ \( 1 - 14706 T + 410338673 T^{2} \)
$19$ \( 1 + 39940 T + 893871739 T^{2} \)
$23$ \( 1 - 68712 T + 3404825447 T^{2} \)
$29$ \( 1 + 102570 T + 17249876309 T^{2} \)
$31$ \( 1 - 227552 T + 27512614111 T^{2} \)
$37$ \( 1 - 160526 T + 94931877133 T^{2} \)
$41$ \( 1 - 10842 T + 194754273881 T^{2} \)
$43$ \( 1 + 630748 T + 271818611107 T^{2} \)
$47$ \( 1 - 472656 T + 506623120463 T^{2} \)
$53$ \( 1 + 1494018 T + 1174711139837 T^{2} \)
$59$ \( 1 - 2640660 T + 2488651484819 T^{2} \)
$61$ \( 1 - 827702 T + 3142742836021 T^{2} \)
$67$ \( 1 + 126004 T + 6060711605323 T^{2} \)
$71$ \( 1 + 1414728 T + 9095120158391 T^{2} \)
$73$ \( 1 - 980282 T + 11047398519097 T^{2} \)
$79$ \( 1 + 3566800 T + 19203908986159 T^{2} \)
$83$ \( 1 - 5672892 T + 27136050989627 T^{2} \)
$89$ \( 1 + 11951190 T + 44231334895529 T^{2} \)
$97$ \( 1 - 8682146 T + 80798284478113 T^{2} \)
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