# 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

# Related objects

## 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

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