Properties

Label 20.3
Level 20
Weight 3
Dimension 10
Nonzero newspaces 3
Newform subspaces 5
Sturm bound 72
Trace bound 2

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Defining parameters

Level: \( N \) = \( 20 = 2^{2} \cdot 5 \)
Weight: \( k \) = \( 3 \)
Nonzero newspaces: \( 3 \)
Newform subspaces: \( 5 \)
Sturm bound: \(72\)
Trace bound: \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(\Gamma_1(20))\).

Total New Old
Modular forms 34 14 20
Cusp forms 14 10 4
Eisenstein series 20 4 16

Trace form

\( 10q - 2q^{2} + 2q^{3} - 4q^{4} - 10q^{5} - 16q^{6} - 14q^{7} - 8q^{8} - 8q^{9} + O(q^{10}) \) \( 10q - 2q^{2} + 2q^{3} - 4q^{4} - 10q^{5} - 16q^{6} - 14q^{7} - 8q^{8} - 8q^{9} + 6q^{10} + 20q^{11} + 40q^{12} + 2q^{13} + 56q^{14} + 2q^{15} + 48q^{16} - 22q^{17} + 22q^{18} - 44q^{20} - 20q^{21} - 80q^{22} - 46q^{23} - 144q^{24} + 42q^{25} - 108q^{26} + 32q^{27} - 40q^{28} + 32q^{29} + 40q^{30} - 28q^{31} + 128q^{32} + 100q^{33} + 156q^{34} + 98q^{35} + 92q^{36} + 82q^{37} + 80q^{38} - 24q^{40} - 52q^{41} - 120q^{42} - 30q^{43} - 80q^{44} - 220q^{45} - 136q^{46} - 78q^{47} - 168q^{49} + 86q^{50} + 4q^{51} + 56q^{52} - 190q^{53} + 112q^{54} - 60q^{55} + 144q^{56} - 16q^{57} - 36q^{58} + 40q^{60} + 140q^{61} - 80q^{62} + 98q^{63} - 64q^{64} + 250q^{65} + 80q^{66} - 14q^{67} - 136q^{68} + 472q^{69} - 80q^{70} + 196q^{71} - 72q^{72} + 362q^{73} - 204q^{74} - 62q^{75} + 100q^{77} - 80q^{78} - 144q^{80} - 366q^{81} + 116q^{82} - 126q^{83} + 32q^{84} - 302q^{85} + 224q^{86} - 16q^{87} + 160q^{88} - 528q^{89} - 114q^{90} - 252q^{91} - 120q^{92} - 428q^{93} - 104q^{94} + 64q^{95} - 256q^{96} - 198q^{97} + 102q^{98} + O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(\Gamma_1(20))\)

We only show spaces with odd 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
20.3.b \(\chi_{20}(11, \cdot)\) 20.3.b.a 4 1
20.3.d \(\chi_{20}(19, \cdot)\) 20.3.d.a 1 1
20.3.d.b 1
20.3.d.c 2
20.3.f \(\chi_{20}(13, \cdot)\) 20.3.f.a 2 2

Decomposition of \(S_{3}^{\mathrm{old}}(\Gamma_1(20))\) into lower level spaces

\( S_{3}^{\mathrm{old}}(\Gamma_1(20)) \cong \) \(S_{3}^{\mathrm{new}}(\Gamma_1(10))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 2 T + 4 T^{2} + 8 T^{3} + 16 T^{4} \))(\( 1 + 2 T \))(\( 1 - 2 T \))(\( 1 + 4 T^{2} \))
$3$ (\( 1 - 16 T^{2} + 206 T^{4} - 1296 T^{6} + 6561 T^{8} \))(\( 1 - 4 T + 9 T^{2} \))(\( 1 + 4 T + 9 T^{2} \))(\( ( 1 + 9 T^{2} )^{2} \))(\( 1 - 2 T + 2 T^{2} - 18 T^{3} + 81 T^{4} \))
$5$ (\( ( 1 - 5 T^{2} )^{2} \))(\( 1 + 5 T \))(\( 1 + 5 T \))(\( 1 - 6 T + 25 T^{2} \))(\( 1 + 6 T + 25 T^{2} \))
$7$ (\( 1 - 96 T^{2} + 6606 T^{4} - 230496 T^{6} + 5764801 T^{8} \))(\( 1 + 4 T + 49 T^{2} \))(\( 1 - 4 T + 49 T^{2} \))(\( ( 1 + 49 T^{2} )^{2} \))(\( ( 1 + 7 T )^{2}( 1 + 49 T^{2} ) \))
$11$ (\( 1 - 84 T^{2} - 7674 T^{4} - 1229844 T^{6} + 214358881 T^{8} \))(\( ( 1 - 11 T )( 1 + 11 T ) \))(\( ( 1 - 11 T )( 1 + 11 T ) \))(\( ( 1 - 11 T )^{2}( 1 + 11 T )^{2} \))(\( ( 1 - 10 T + 121 T^{2} )^{2} \))
$13$ (\( ( 1 + 8 T + 334 T^{2} + 1352 T^{3} + 28561 T^{4} )^{2} \))(\( ( 1 - 13 T )( 1 + 13 T ) \))(\( ( 1 - 13 T )( 1 + 13 T ) \))(\( ( 1 - 10 T + 169 T^{2} )( 1 + 10 T + 169 T^{2} ) \))(\( 1 - 18 T + 162 T^{2} - 3042 T^{3} + 28561 T^{4} \))
$17$ (\( ( 1 + 12 T + 294 T^{2} + 3468 T^{3} + 83521 T^{4} )^{2} \))(\( ( 1 - 17 T )( 1 + 17 T ) \))(\( ( 1 - 17 T )( 1 + 17 T ) \))(\( ( 1 - 30 T + 289 T^{2} )( 1 + 30 T + 289 T^{2} ) \))(\( 1 - 2 T + 2 T^{2} - 578 T^{3} + 83521 T^{4} \))
$19$ (\( 1 - 1124 T^{2} + 571366 T^{4} - 146480804 T^{6} + 16983563041 T^{8} \))(\( ( 1 - 19 T )( 1 + 19 T ) \))(\( ( 1 - 19 T )( 1 + 19 T ) \))(\( ( 1 - 19 T )^{2}( 1 + 19 T )^{2} \))(\( 1 - 658 T^{2} + 130321 T^{4} \))
$23$ (\( 1 - 1856 T^{2} + 1404046 T^{4} - 519384896 T^{6} + 78310985281 T^{8} \))(\( 1 - 44 T + 529 T^{2} \))(\( 1 + 44 T + 529 T^{2} \))(\( ( 1 + 529 T^{2} )^{2} \))(\( ( 1 + 23 T )^{2}( 1 + 529 T^{2} ) \))
$29$ (\( ( 1 + 4 T + 1606 T^{2} + 3364 T^{3} + 707281 T^{4} )^{2} \))(\( 1 + 22 T + 841 T^{2} \))(\( 1 + 22 T + 841 T^{2} \))(\( ( 1 - 42 T + 841 T^{2} )^{2} \))(\( 1 - 1618 T^{2} + 707281 T^{4} \))
$31$ (\( 1 - 1524 T^{2} + 1236966 T^{4} - 1407446004 T^{6} + 852891037441 T^{8} \))(\( ( 1 - 31 T )( 1 + 31 T ) \))(\( ( 1 - 31 T )( 1 + 31 T ) \))(\( ( 1 - 31 T )^{2}( 1 + 31 T )^{2} \))(\( ( 1 + 14 T + 961 T^{2} )^{2} \))
$37$ (\( ( 1 - 8 T + 2254 T^{2} - 10952 T^{3} + 1874161 T^{4} )^{2} \))(\( ( 1 - 37 T )( 1 + 37 T ) \))(\( ( 1 - 37 T )( 1 + 37 T ) \))(\( ( 1 - 70 T + 1369 T^{2} )( 1 + 70 T + 1369 T^{2} ) \))(\( 1 - 66 T + 2178 T^{2} - 90354 T^{3} + 1874161 T^{4} \))
$41$ (\( ( 1 + 56 T + 3966 T^{2} + 94136 T^{3} + 2825761 T^{4} )^{2} \))(\( 1 - 62 T + 1681 T^{2} \))(\( 1 - 62 T + 1681 T^{2} \))(\( ( 1 + 18 T + 1681 T^{2} )^{2} \))(\( ( 1 + 14 T + 1681 T^{2} )^{2} \))
$43$ (\( 1 - 6896 T^{2} + 18665806 T^{4} - 23576051696 T^{6} + 11688200277601 T^{8} \))(\( 1 + 76 T + 1849 T^{2} \))(\( 1 - 76 T + 1849 T^{2} \))(\( ( 1 + 1849 T^{2} )^{2} \))(\( 1 + 30 T + 450 T^{2} + 55470 T^{3} + 3418801 T^{4} \))
$47$ (\( 1 - 4736 T^{2} + 14725966 T^{4} - 23110169216 T^{6} + 23811286661761 T^{8} \))(\( 1 + 4 T + 2209 T^{2} \))(\( 1 - 4 T + 2209 T^{2} \))(\( ( 1 + 2209 T^{2} )^{2} \))(\( 1 + 78 T + 3042 T^{2} + 172302 T^{3} + 4879681 T^{4} \))
$53$ (\( ( 1 + 88 T + 7054 T^{2} + 247192 T^{3} + 7890481 T^{4} )^{2} \))(\( ( 1 - 53 T )( 1 + 53 T ) \))(\( ( 1 - 53 T )( 1 + 53 T ) \))(\( ( 1 - 90 T + 2809 T^{2} )( 1 + 90 T + 2809 T^{2} ) \))(\( 1 + 14 T + 98 T^{2} + 39326 T^{3} + 7890481 T^{4} \))
$59$ (\( 1 - 8164 T^{2} + 34261926 T^{4} - 98926135204 T^{6} + 146830437604321 T^{8} \))(\( ( 1 - 59 T )( 1 + 59 T ) \))(\( ( 1 - 59 T )( 1 + 59 T ) \))(\( ( 1 - 59 T )^{2}( 1 + 59 T )^{2} \))(\( 1 - 3826 T^{2} + 12117361 T^{4} \))
$61$ (\( ( 1 - 64 T + 5086 T^{2} - 238144 T^{3} + 13845841 T^{4} )^{2} \))(\( 1 + 58 T + 3721 T^{2} \))(\( 1 + 58 T + 3721 T^{2} \))(\( ( 1 - 22 T + 3721 T^{2} )^{2} \))(\( ( 1 - 42 T + 3721 T^{2} )^{2} \))
$67$ (\( 1 - 7536 T^{2} + 47276046 T^{4} - 151858847856 T^{6} + 406067677556641 T^{8} \))(\( 1 - 116 T + 4489 T^{2} \))(\( 1 + 116 T + 4489 T^{2} \))(\( ( 1 + 4489 T^{2} )^{2} \))(\( 1 + 14 T + 98 T^{2} + 62846 T^{3} + 20151121 T^{4} \))
$71$ (\( 1 - 12084 T^{2} + 81146406 T^{4} - 307074753204 T^{6} + 645753531245761 T^{8} \))(\( ( 1 - 71 T )( 1 + 71 T ) \))(\( ( 1 - 71 T )( 1 + 71 T ) \))(\( ( 1 - 71 T )^{2}( 1 + 71 T )^{2} \))(\( ( 1 - 98 T + 5041 T^{2} )^{2} \))
$73$ (\( ( 1 - 132 T + 9894 T^{2} - 703428 T^{3} + 28398241 T^{4} )^{2} \))(\( ( 1 - 73 T )( 1 + 73 T ) \))(\( ( 1 - 73 T )( 1 + 73 T ) \))(\( ( 1 - 110 T + 5329 T^{2} )( 1 + 110 T + 5329 T^{2} ) \))(\( 1 - 98 T + 4802 T^{2} - 522242 T^{3} + 28398241 T^{4} \))
$79$ (\( 1 - 11844 T^{2} + 72414726 T^{4} - 461324759364 T^{6} + 1517108809906561 T^{8} \))(\( ( 1 - 79 T )( 1 + 79 T ) \))(\( ( 1 - 79 T )( 1 + 79 T ) \))(\( ( 1 - 79 T )^{2}( 1 + 79 T )^{2} \))(\( 1 - 3266 T^{2} + 38950081 T^{4} \))
$83$ (\( 1 - 21296 T^{2} + 201120526 T^{4} - 1010672404016 T^{6} + 2252292232139041 T^{8} \))(\( 1 + 76 T + 6889 T^{2} \))(\( 1 - 76 T + 6889 T^{2} \))(\( ( 1 + 6889 T^{2} )^{2} \))(\( 1 + 126 T + 7938 T^{2} + 868014 T^{3} + 47458321 T^{4} \))
$89$ (\( ( 1 + 44 T + 8326 T^{2} + 348524 T^{3} + 62742241 T^{4} )^{2} \))(\( 1 + 142 T + 7921 T^{2} \))(\( 1 + 142 T + 7921 T^{2} \))(\( ( 1 + 78 T + 7921 T^{2} )^{2} \))(\( 1 - 3298 T^{2} + 62742241 T^{4} \))
$97$ (\( ( 1 + 132 T + 22454 T^{2} + 1241988 T^{3} + 88529281 T^{4} )^{2} \))(\( ( 1 - 97 T )( 1 + 97 T ) \))(\( ( 1 - 97 T )( 1 + 97 T ) \))(\( ( 1 - 130 T + 9409 T^{2} )( 1 + 130 T + 9409 T^{2} ) \))(\( 1 - 66 T + 2178 T^{2} - 620994 T^{3} + 88529281 T^{4} \))
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