Properties

Label 10.8
Level 10
Weight 8
Dimension 5
Nonzero newspaces 2
Newform subspaces 2
Sturm bound 48
Trace bound 1

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Defining parameters

Level: \( N \) = \( 10\( 10 = 2 \cdot 5 \) \)
Weight: \( k \) = \( 8 \)
Nonzero newspaces: \( 2 \)
Newform subspaces: \( 2 \)
Sturm bound: \(48\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_1(10))\).

Total New Old
Modular forms 25 5 20
Cusp forms 17 5 12
Eisenstein series 8 0 8

Trace form

\( 5q + 8q^{2} + 28q^{3} - 192q^{4} + 185q^{5} + 1120q^{6} + 104q^{7} + 512q^{8} - 8191q^{9} + O(q^{10}) \) \( 5q + 8q^{2} + 28q^{3} - 192q^{4} + 185q^{5} + 1120q^{6} + 104q^{7} + 512q^{8} - 8191q^{9} + 360q^{10} + 12660q^{11} + 1792q^{12} - 8602q^{13} - 5696q^{14} - 31460q^{15} + 20480q^{16} + 20274q^{17} - 11224q^{18} + 109100q^{19} + 4160q^{20} - 61040q^{21} - 41184q^{22} - 72072q^{23} - 43008q^{24} + 101725q^{25} - 124880q^{26} - 100520q^{27} + 6656q^{28} + 401070q^{29} + 438240q^{30} - 474240q^{31} + 32768q^{32} - 144144q^{33} - 536176q^{34} - 263720q^{35} + 344640q^{36} + 104654q^{37} + 364000q^{38} + 922568q^{39} + 104960q^{40} + 817410q^{41} + 23296q^{42} - 795532q^{43} - 1469184q^{44} - 3054795q^{45} + 569920q^{46} + 425664q^{47} + 114688q^{48} + 1707381q^{49} + 2486600q^{50} + 928760q^{51} - 550528q^{52} + 1500798q^{53} - 5920320q^{54} - 1169980q^{55} + 471040q^{56} + 1274000q^{57} + 1852080q^{58} + 2339940q^{59} + 2461440q^{60} - 4357490q^{61} - 641024q^{62} - 145912q^{63} - 786432q^{64} + 1686190q^{65} + 1248640q^{66} - 2336836q^{67} + 1297536q^{68} + 4496848q^{69} + 2783680q^{70} - 8036040q^{71} - 718336q^{72} - 3805702q^{73} - 2289296q^{74} - 6960100q^{75} - 1158400q^{76} - 535392q^{77} - 1926848q^{78} + 12780080q^{79} + 757760q^{80} + 16682205q^{81} + 4679376q^{82} - 45492q^{83} + 4279296q^{84} - 5461270q^{85} - 1952480q^{86} + 6482280q^{87} - 2635776q^{88} - 32490030q^{89} - 16982520q^{90} + 4446160q^{91} - 4612608q^{92} - 2243584q^{93} + 18945664q^{94} + 37591900q^{95} + 4587520q^{96} - 5247646q^{97} - 6501816q^{98} - 11887132q^{99} + O(q^{100}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_1(10))\)

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
10.8.a \(\chi_{10}(1, \cdot)\) 10.8.a.a 1 1
10.8.b \(\chi_{10}(9, \cdot)\) 10.8.b.a 4 1

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_1(10))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_1(10)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(5))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 - 8 T \))(\( ( 1 + 64 T^{2} )^{2} \))
$3$ (\( 1 - 28 T + 2187 T^{2} \))(\( 1 - 980 T^{2} + 84438 T^{4} - 4687309620 T^{6} + 22876792454961 T^{8} \))
$5$ (\( 1 - 125 T \))(\( 1 - 60 T - 41250 T^{2} - 4687500 T^{3} + 6103515625 T^{4} \))
$7$ (\( 1 - 104 T + 823543 T^{2} \))(\( 1 - 2907140 T^{2} + 3444026008998 T^{4} - 1971689424002241860 T^{6} + \)\(45\!\cdots\!01\)\( T^{8} \))
$11$ (\( 1 + 5148 T + 19487171 T^{2} \))(\( ( 1 - 8904 T + 58001046 T^{2} - 173513770584 T^{3} + 379749833583241 T^{4} )^{2} \))
$13$ (\( 1 + 8602 T + 62748517 T^{2} \))(\( 1 - 207134260 T^{2} + 18369334894869078 T^{4} - \)\(81\!\cdots\!40\)\( T^{6} + \)\(15\!\cdots\!21\)\( T^{8} \))
$17$ (\( 1 - 20274 T + 410338673 T^{2} \))(\( 1 - 513791940 T^{2} + 236061419749305158 T^{4} - \)\(86\!\cdots\!60\)\( T^{6} + \)\(28\!\cdots\!41\)\( T^{8} \))
$19$ (\( 1 - 45500 T + 893871739 T^{2} \))(\( ( 1 - 31800 T + 833487878 T^{2} - 28425121300200 T^{3} + 799006685782884121 T^{4} )^{2} \))
$23$ (\( 1 + 72072 T + 3404825447 T^{2} \))(\( 1 - 6583044420 T^{2} + 22546653652650262118 T^{4} - \)\(76\!\cdots\!80\)\( T^{6} + \)\(13\!\cdots\!81\)\( T^{8} \))
$29$ (\( 1 - 231510 T + 17249876309 T^{2} \))(\( ( 1 - 84780 T + 15469426318 T^{2} - 1462444513477020 T^{3} + \)\(29\!\cdots\!81\)\( T^{4} )^{2} \))
$31$ (\( 1 + 80128 T + 27512614111 T^{2} \))(\( ( 1 + 197056 T + 59904732606 T^{2} + 5421525686257216 T^{3} + \)\(75\!\cdots\!21\)\( T^{4} )^{2} \))
$37$ (\( 1 - 104654 T + 94931877133 T^{2} \))(\( 1 - 238521132500 T^{2} + \)\(29\!\cdots\!78\)\( T^{4} - \)\(21\!\cdots\!00\)\( T^{6} + \)\(81\!\cdots\!21\)\( T^{8} \))
$41$ (\( 1 - 584922 T + 194754273881 T^{2} \))(\( ( 1 - 116244 T + 205827060246 T^{2} - 22639015813022964 T^{3} + \)\(37\!\cdots\!61\)\( T^{4} )^{2} \))
$43$ (\( 1 + 795532 T + 271818611107 T^{2} \))(\( 1 - 1046615537780 T^{2} + \)\(42\!\cdots\!98\)\( T^{4} - \)\(77\!\cdots\!20\)\( T^{6} + \)\(54\!\cdots\!01\)\( T^{8} \))
$47$ (\( 1 - 425664 T + 506623120463 T^{2} \))(\( 1 + 115388894940 T^{2} - \)\(27\!\cdots\!62\)\( T^{4} + \)\(29\!\cdots\!60\)\( T^{6} + \)\(65\!\cdots\!61\)\( T^{8} \))
$53$ (\( 1 - 1500798 T + 1174711139837 T^{2} \))(\( 1 - 1677817211540 T^{2} + \)\(14\!\cdots\!38\)\( T^{4} - \)\(23\!\cdots\!60\)\( T^{6} + \)\(19\!\cdots\!61\)\( T^{8} \))
$59$ (\( 1 - 246420 T + 2488651484819 T^{2} \))(\( ( 1 - 1046760 T + 4817827968438 T^{2} - 2605020828249136440 T^{3} + \)\(61\!\cdots\!61\)\( T^{4} )^{2} \))
$61$ (\( 1 - 893942 T + 3142742836021 T^{2} \))(\( ( 1 + 2625716 T + 8002437582606 T^{2} + 8251950148425716036 T^{3} + \)\(98\!\cdots\!41\)\( T^{4} )^{2} \))
$67$ (\( 1 + 2336836 T + 6060711605323 T^{2} \))(\( 1 - 16580251270100 T^{2} + \)\(13\!\cdots\!58\)\( T^{4} - \)\(60\!\cdots\!00\)\( T^{6} + \)\(13\!\cdots\!41\)\( T^{8} \))
$71$ (\( 1 + 203688 T + 9095120158391 T^{2} \))(\( ( 1 + 3916176 T + 13255881578926 T^{2} + 35618091281407032816 T^{3} + \)\(82\!\cdots\!81\)\( T^{4} )^{2} \))
$73$ (\( 1 + 3805702 T + 11047398519097 T^{2} \))(\( 1 - 37988250868580 T^{2} + \)\(59\!\cdots\!18\)\( T^{4} - \)\(46\!\cdots\!20\)\( T^{6} + \)\(14\!\cdots\!81\)\( T^{8} \))
$79$ (\( 1 - 5053040 T + 19203908986159 T^{2} \))(\( ( 1 - 3863520 T + 34443978644318 T^{2} - 74194686446205019680 T^{3} + \)\(36\!\cdots\!81\)\( T^{4} )^{2} \))
$83$ (\( 1 + 45492 T + 27136050989627 T^{2} \))(\( 1 - 7805733448980 T^{2} + \)\(31\!\cdots\!58\)\( T^{4} - \)\(57\!\cdots\!20\)\( T^{6} + \)\(54\!\cdots\!41\)\( T^{8} \))
$89$ (\( 1 - 980010 T + 44231334895529 T^{2} \))(\( ( 1 + 16735020 T + 154740910351158 T^{2} + \)\(74\!\cdots\!80\)\( T^{3} + \)\(19\!\cdots\!41\)\( T^{4} )^{2} \))
$97$ (\( 1 + 5247646 T + 80798284478113 T^{2} \))(\( 1 - 161931097215620 T^{2} + \)\(13\!\cdots\!38\)\( T^{4} - \)\(10\!\cdots\!80\)\( T^{6} + \)\(42\!\cdots\!61\)\( T^{8} \))
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