Properties

Label 10.8.a
Level 10
Weight 8
Character orbit a
Rep. character \(\chi_{10}(1,\cdot)\)
Character field \(\Q\)
Dimension 1
Newform subspaces 1
Sturm bound 12
Trace bound 0

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Defining parameters

Level: \( N \) \(=\) \( 10 = 2 \cdot 5 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 10.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 1 \)
Sturm bound: \(12\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(10))\).

Total New Old
Modular forms 13 1 12
Cusp forms 9 1 8
Eisenstein series 4 0 4

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

\(2\)\(5\)FrickeDim.
\(-\)\(-\)\(+\)\(1\)
Plus space\(+\)\(1\)
Minus space\(-\)\(0\)

Trace form

\( q + 8q^{2} + 28q^{3} + 64q^{4} + 125q^{5} + 224q^{6} + 104q^{7} + 512q^{8} - 1403q^{9} + O(q^{10}) \) \( q + 8q^{2} + 28q^{3} + 64q^{4} + 125q^{5} + 224q^{6} + 104q^{7} + 512q^{8} - 1403q^{9} + 1000q^{10} - 5148q^{11} + 1792q^{12} - 8602q^{13} + 832q^{14} + 3500q^{15} + 4096q^{16} + 20274q^{17} - 11224q^{18} + 45500q^{19} + 8000q^{20} + 2912q^{21} - 41184q^{22} - 72072q^{23} + 14336q^{24} + 15625q^{25} - 68816q^{26} - 100520q^{27} + 6656q^{28} + 231510q^{29} + 28000q^{30} - 80128q^{31} + 32768q^{32} - 144144q^{33} + 162192q^{34} + 13000q^{35} - 89792q^{36} + 104654q^{37} + 364000q^{38} - 240856q^{39} + 64000q^{40} + 584922q^{41} + 23296q^{42} - 795532q^{43} - 329472q^{44} - 175375q^{45} - 576576q^{46} + 425664q^{47} + 114688q^{48} - 812727q^{49} + 125000q^{50} + 567672q^{51} - 550528q^{52} + 1500798q^{53} - 804160q^{54} - 643500q^{55} + 53248q^{56} + 1274000q^{57} + 1852080q^{58} + 246420q^{59} + 224000q^{60} + 893942q^{61} - 641024q^{62} - 145912q^{63} + 262144q^{64} - 1075250q^{65} - 1153152q^{66} - 2336836q^{67} + 1297536q^{68} - 2018016q^{69} + 104000q^{70} - 203688q^{71} - 718336q^{72} - 3805702q^{73} + 837232q^{74} + 437500q^{75} + 2912000q^{76} - 535392q^{77} - 1926848q^{78} + 5053040q^{79} + 512000q^{80} + 253801q^{81} + 4679376q^{82} - 45492q^{83} + 186368q^{84} + 2534250q^{85} - 6364256q^{86} + 6482280q^{87} - 2635776q^{88} + 980010q^{89} - 1403000q^{90} - 894608q^{91} - 4612608q^{92} - 2243584q^{93} + 3405312q^{94} + 5687500q^{95} + 917504q^{96} - 5247646q^{97} - 6501816q^{98} + 7222644q^{99} + O(q^{100}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(10))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 5
10.8.a.a \(1\) \(3.124\) \(\Q\) None \(8\) \(28\) \(125\) \(104\) \(-\) \(-\) \(q+8q^{2}+28q^{3}+2^{6}q^{4}+5^{3}q^{5}+224q^{6}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(10))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(10)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 8 T \)
$3$ \( 1 - 28 T + 2187 T^{2} \)
$5$ \( 1 - 125 T \)
$7$ \( 1 - 104 T + 823543 T^{2} \)
$11$ \( 1 + 5148 T + 19487171 T^{2} \)
$13$ \( 1 + 8602 T + 62748517 T^{2} \)
$17$ \( 1 - 20274 T + 410338673 T^{2} \)
$19$ \( 1 - 45500 T + 893871739 T^{2} \)
$23$ \( 1 + 72072 T + 3404825447 T^{2} \)
$29$ \( 1 - 231510 T + 17249876309 T^{2} \)
$31$ \( 1 + 80128 T + 27512614111 T^{2} \)
$37$ \( 1 - 104654 T + 94931877133 T^{2} \)
$41$ \( 1 - 584922 T + 194754273881 T^{2} \)
$43$ \( 1 + 795532 T + 271818611107 T^{2} \)
$47$ \( 1 - 425664 T + 506623120463 T^{2} \)
$53$ \( 1 - 1500798 T + 1174711139837 T^{2} \)
$59$ \( 1 - 246420 T + 2488651484819 T^{2} \)
$61$ \( 1 - 893942 T + 3142742836021 T^{2} \)
$67$ \( 1 + 2336836 T + 6060711605323 T^{2} \)
$71$ \( 1 + 203688 T + 9095120158391 T^{2} \)
$73$ \( 1 + 3805702 T + 11047398519097 T^{2} \)
$79$ \( 1 - 5053040 T + 19203908986159 T^{2} \)
$83$ \( 1 + 45492 T + 27136050989627 T^{2} \)
$89$ \( 1 - 980010 T + 44231334895529 T^{2} \)
$97$ \( 1 + 5247646 T + 80798284478113 T^{2} \)
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