Properties

Label 20.8.a
Level 20
Weight 8
Character orbit a
Rep. character \(\chi_{20}(1,\cdot)\)
Character field \(\Q\)
Dimension 3
Newform subspaces 2
Sturm bound 24
Trace bound 1

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Defining parameters

Level: \( N \) \(=\) \( 20 = 2^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 20.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(24\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(20))\).

Total New Old
Modular forms 24 3 21
Cusp forms 18 3 15
Eisenstein series 6 0 6

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.
\(-\)\(+\)\(-\)\(1\)
\(-\)\(-\)\(+\)\(2\)
Plus space\(+\)\(2\)
Minus space\(-\)\(1\)

Trace form

\( 3q - 26q^{3} + 125q^{5} + 954q^{7} + 2707q^{9} + O(q^{10}) \) \( 3q - 26q^{3} + 125q^{5} + 954q^{7} + 2707q^{9} - 240q^{11} + 9126q^{13} - 1750q^{15} + 6318q^{17} - 19428q^{19} - 93652q^{21} + 43518q^{23} + 46875q^{25} - 159452q^{27} + 35562q^{29} - 261492q^{31} + 799920q^{33} + 295750q^{35} - 191946q^{37} - 432628q^{39} - 1042662q^{41} + 2083230q^{43} + 876125q^{45} - 1832862q^{47} + 137199q^{49} - 4286724q^{51} + 3235902q^{53} + 930000q^{55} - 1347304q^{57} - 1362996q^{59} - 4346634q^{61} + 7176506q^{63} + 2154250q^{65} + 290634q^{67} - 662844q^{69} - 805116q^{71} + 3433686q^{73} - 406250q^{75} - 1616880q^{77} + 5934696q^{79} - 229433q^{81} - 5962722q^{83} + 575250q^{85} - 3939564q^{87} - 9031362q^{89} + 16727892q^{91} - 26418136q^{93} - 7689500q^{95} + 18000534q^{97} + 746640q^{99} + O(q^{100}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(20))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 5
20.8.a.a \(1\) \(6.248\) \(\Q\) None \(0\) \(-6\) \(-125\) \(-706\) \(-\) \(+\) \(q-6q^{3}-5^{3}q^{5}-706q^{7}-2151q^{9}+\cdots\)
20.8.a.b \(2\) \(6.248\) \(\Q(\sqrt{1129}) \) None \(0\) \(-20\) \(250\) \(1660\) \(-\) \(-\) \(q+(-10-\beta )q^{3}+5^{3}q^{5}+(830+9\beta )q^{7}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(20))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(20)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( 1 + 6 T + 2187 T^{2} \))(\( 1 + 20 T - 42 T^{2} + 43740 T^{3} + 4782969 T^{4} \))
$5$ (\( 1 + 125 T \))(\( ( 1 - 125 T )^{2} \))
$7$ (\( 1 + 706 T + 823543 T^{2} \))(\( 1 - 1660 T + 1970190 T^{2} - 1367081380 T^{3} + 678223072849 T^{4} \))
$11$ (\( 1 + 3840 T + 19487171 T^{2} \))(\( 1 - 3600 T + 5634742 T^{2} - 70153815600 T^{3} + 379749833583241 T^{4} \))
$13$ (\( 1 + 4054 T + 62748517 T^{2} \))(\( 1 - 13180 T + 163072398 T^{2} - 827025454060 T^{3} + 3937376385699289 T^{4} \))
$17$ (\( 1 - 858 T + 410338673 T^{2} \))(\( 1 - 5460 T - 160982138 T^{2} - 2240449154580 T^{3} + 168377826559400929 T^{4} \))
$19$ (\( 1 - 21044 T + 893871739 T^{2} \))(\( 1 + 40472 T + 2050920774 T^{2} + 36176777020808 T^{3} + 799006685782884121 T^{4} \))
$23$ (\( 1 - 85338 T + 3404825447 T^{2} \))(\( 1 + 41820 T + 7228954990 T^{2} + 142389800193540 T^{3} + 11592836324538749809 T^{4} \))
$29$ (\( 1 + 83106 T + 17249876309 T^{2} \))(\( 1 - 118668 T + 37435002574 T^{2} - 2047008321836412 T^{3} + \)\(29\!\cdots\!81\)\( T^{4} \))
$31$ (\( 1 + 145564 T + 27512614111 T^{2} \))(\( 1 + 115928 T + 13575043518 T^{2} + 3189482328660008 T^{3} + \)\(75\!\cdots\!21\)\( T^{4} \))
$37$ (\( 1 + 498886 T + 94931877133 T^{2} \))(\( 1 - 306940 T + 120498758430 T^{2} - 29138390367203020 T^{3} + \)\(90\!\cdots\!89\)\( T^{4} \))
$41$ (\( 1 + 689514 T + 194754273881 T^{2} \))(\( 1 + 353148 T + 314020811638 T^{2} + 68777082312527388 T^{3} + \)\(37\!\cdots\!61\)\( T^{4} \))
$43$ (\( 1 - 867890 T + 271818611107 T^{2} \))(\( 1 - 1215340 T + 866800610214 T^{2} - 330352030822781380 T^{3} + \)\(73\!\cdots\!49\)\( T^{4} \))
$47$ (\( 1 - 235638 T + 506623120463 T^{2} \))(\( 1 + 2068500 T + 2082813588382 T^{2} + 1047949924677715500 T^{3} + \)\(25\!\cdots\!69\)\( T^{4} \))
$53$ (\( 1 - 1835442 T + 1174711139837 T^{2} \))(\( 1 - 1400460 T + 2025459025630 T^{2} - 1645135962896125020 T^{3} + \)\(13\!\cdots\!69\)\( T^{4} \))
$59$ (\( 1 - 629508 T + 2488651484819 T^{2} \))(\( 1 + 1992504 T + 5477605919542 T^{2} + 4958648038107796776 T^{3} + \)\(61\!\cdots\!61\)\( T^{4} \))
$61$ (\( 1 + 2667958 T + 3142742836021 T^{2} \))(\( 1 + 1678676 T + 900787415886 T^{2} + 5275646973000388196 T^{3} + \)\(98\!\cdots\!41\)\( T^{4} \))
$67$ (\( 1 + 3373306 T + 6060711605323 T^{2} \))(\( 1 - 3663940 T + 12525094986870 T^{2} - 22206083679207152620 T^{3} + \)\(36\!\cdots\!29\)\( T^{4} \))
$71$ (\( 1 + 2600052 T + 9095120158391 T^{2} \))(\( 1 - 1794936 T + 17428289847406 T^{2} - 16325158596621707976 T^{3} + \)\(82\!\cdots\!81\)\( T^{4} \))
$73$ (\( 1 + 1628494 T + 11047398519097 T^{2} \))(\( 1 - 5062180 T + 19516555685238 T^{2} - 55923919835402451460 T^{3} + \)\(12\!\cdots\!09\)\( T^{4} \))
$79$ (\( 1 + 4243528 T + 19203908986159 T^{2} \))(\( 1 - 10178224 T + 57668559431262 T^{2} - \)\(19\!\cdots\!16\)\( T^{3} + \)\(36\!\cdots\!81\)\( T^{4} \))
$83$ (\( 1 - 1251378 T + 27136050989627 T^{2} \))(\( 1 + 7214100 T + 67277011758070 T^{2} + \)\(19\!\cdots\!00\)\( T^{3} + \)\(73\!\cdots\!29\)\( T^{4} \))
$89$ (\( 1 - 6299466 T + 44231334895529 T^{2} \))(\( 1 + 15330828 T + 138366637288054 T^{2} + \)\(67\!\cdots\!12\)\( T^{3} + \)\(19\!\cdots\!41\)\( T^{4} \))
$97$ (\( 1 - 3976514 T + 80798284478113 T^{2} \))(\( 1 - 14024020 T + 210744479822310 T^{2} - \)\(11\!\cdots\!60\)\( T^{3} + \)\(65\!\cdots\!69\)\( T^{4} \))
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