Properties

Label 20.3.b
Level 20
Weight 3
Character orbit b
Rep. character \(\chi_{20}(11,\cdot)\)
Character field \(\Q\)
Dimension 4
Newform subspaces 1
Sturm bound 9
Trace bound 0

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Defining parameters

Level: \( N \) = \( 20 = 2^{2} \cdot 5 \)
Weight: \( k \) = \( 3 \)
Character orbit: \([\chi]\) = 20.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 4 \)
Character field: \(\Q\)
Newform subspaces: \( 1 \)
Sturm bound: \(9\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(20, [\chi])\).

Total New Old
Modular forms 8 4 4
Cusp forms 4 4 0
Eisenstein series 4 0 4

Trace form

\( 4q - 2q^{2} - 4q^{4} - 8q^{8} - 4q^{9} + O(q^{10}) \) \( 4q - 2q^{2} - 4q^{4} - 8q^{8} - 4q^{9} - 10q^{10} + 40q^{12} - 16q^{13} + 40q^{14} - 16q^{16} - 24q^{17} + 22q^{18} + 20q^{20} + 40q^{21} - 80q^{22} - 80q^{24} + 20q^{25} - 12q^{26} - 40q^{28} - 8q^{29} - 40q^{30} + 128q^{32} + 80q^{33} + 92q^{34} - 36q^{36} + 16q^{37} + 80q^{38} + 40q^{40} - 112q^{41} - 120q^{42} - 80q^{44} - 40q^{45} + 40q^{46} - 4q^{49} - 10q^{50} + 56q^{52} - 176q^{53} + 80q^{54} + 80q^{56} - 36q^{58} + 40q^{60} + 128q^{61} - 80q^{62} - 64q^{64} + 40q^{65} + 80q^{66} - 136q^{68} + 120q^{69} - 72q^{72} + 264q^{73} - 108q^{74} + 240q^{77} - 80q^{78} - 80q^{80} - 276q^{81} + 116q^{82} + 160q^{84} - 160q^{85} - 80q^{86} + 160q^{88} - 88q^{89} + 30q^{90} - 120q^{92} - 400q^{93} - 120q^{94} - 264q^{97} + 102q^{98} + O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(20, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
20.3.b.a \(4\) \(0.545\) \(\Q(\zeta_{10})\) None \(-2\) \(0\) \(0\) \(0\) \(q-\zeta_{10}^{2}q^{2}+\zeta_{10}^{3}q^{3}+(-2-\zeta_{10}+\cdots)q^{4}+\cdots\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T + 4 T^{2} + 8 T^{3} + 16 T^{4} \)
$3$ \( 1 - 16 T^{2} + 206 T^{4} - 1296 T^{6} + 6561 T^{8} \)
$5$ \( ( 1 - 5 T^{2} )^{2} \)
$7$ \( 1 - 96 T^{2} + 6606 T^{4} - 230496 T^{6} + 5764801 T^{8} \)
$11$ \( 1 - 84 T^{2} - 7674 T^{4} - 1229844 T^{6} + 214358881 T^{8} \)
$13$ \( ( 1 + 8 T + 334 T^{2} + 1352 T^{3} + 28561 T^{4} )^{2} \)
$17$ \( ( 1 + 12 T + 294 T^{2} + 3468 T^{3} + 83521 T^{4} )^{2} \)
$19$ \( 1 - 1124 T^{2} + 571366 T^{4} - 146480804 T^{6} + 16983563041 T^{8} \)
$23$ \( 1 - 1856 T^{2} + 1404046 T^{4} - 519384896 T^{6} + 78310985281 T^{8} \)
$29$ \( ( 1 + 4 T + 1606 T^{2} + 3364 T^{3} + 707281 T^{4} )^{2} \)
$31$ \( 1 - 1524 T^{2} + 1236966 T^{4} - 1407446004 T^{6} + 852891037441 T^{8} \)
$37$ \( ( 1 - 8 T + 2254 T^{2} - 10952 T^{3} + 1874161 T^{4} )^{2} \)
$41$ \( ( 1 + 56 T + 3966 T^{2} + 94136 T^{3} + 2825761 T^{4} )^{2} \)
$43$ \( 1 - 6896 T^{2} + 18665806 T^{4} - 23576051696 T^{6} + 11688200277601 T^{8} \)
$47$ \( 1 - 4736 T^{2} + 14725966 T^{4} - 23110169216 T^{6} + 23811286661761 T^{8} \)
$53$ \( ( 1 + 88 T + 7054 T^{2} + 247192 T^{3} + 7890481 T^{4} )^{2} \)
$59$ \( 1 - 8164 T^{2} + 34261926 T^{4} - 98926135204 T^{6} + 146830437604321 T^{8} \)
$61$ \( ( 1 - 64 T + 5086 T^{2} - 238144 T^{3} + 13845841 T^{4} )^{2} \)
$67$ \( 1 - 7536 T^{2} + 47276046 T^{4} - 151858847856 T^{6} + 406067677556641 T^{8} \)
$71$ \( 1 - 12084 T^{2} + 81146406 T^{4} - 307074753204 T^{6} + 645753531245761 T^{8} \)
$73$ \( ( 1 - 132 T + 9894 T^{2} - 703428 T^{3} + 28398241 T^{4} )^{2} \)
$79$ \( 1 - 11844 T^{2} + 72414726 T^{4} - 461324759364 T^{6} + 1517108809906561 T^{8} \)
$83$ \( 1 - 21296 T^{2} + 201120526 T^{4} - 1010672404016 T^{6} + 2252292232139041 T^{8} \)
$89$ \( ( 1 + 44 T + 8326 T^{2} + 348524 T^{3} + 62742241 T^{4} )^{2} \)
$97$ \( ( 1 + 132 T + 22454 T^{2} + 1241988 T^{3} + 88529281 T^{4} )^{2} \)
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