Properties

Label 20.6
Level 20
Weight 6
Dimension 29
Nonzero newspaces 3
Newform subspaces 4
Sturm bound 144
Trace bound 1

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Defining parameters

Level: \( N \) = \( 20\( 20 = 2^{2} \cdot 5 \) \)
Weight: \( k \) = \( 6 \)
Nonzero newspaces: \( 3 \)
Newform subspaces: \( 4 \)
Sturm bound: \(144\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_1(20))\).

Total New Old
Modular forms 70 37 33
Cusp forms 50 29 21
Eisenstein series 20 8 12

Trace form

\( 29q - 2q^{2} + 22q^{3} - 39q^{5} - 184q^{6} + 218q^{7} + 244q^{8} + 479q^{9} + O(q^{10}) \) \( 29q - 2q^{2} + 22q^{3} - 39q^{5} - 184q^{6} + 218q^{7} + 244q^{8} + 479q^{9} + 566q^{10} - 680q^{11} - 1280q^{12} - 504q^{13} - 1790q^{15} + 1976q^{16} + 1192q^{17} - 3246q^{18} + 3284q^{19} - 1364q^{20} + 3700q^{21} - 2440q^{22} - 3186q^{23} - 13319q^{25} + 6836q^{26} - 44q^{27} + 5920q^{28} + 21234q^{29} + 8600q^{30} + 5068q^{31} + 17608q^{32} - 160q^{33} - 19090q^{35} - 27908q^{36} - 3796q^{37} - 22160q^{38} + 2684q^{39} - 31444q^{40} - 38870q^{41} - 39400q^{42} - 370q^{43} + 20671q^{45} + 29416q^{46} + 16146q^{47} + 108160q^{48} + 34323q^{49} + 114186q^{50} + 25916q^{51} + 38476q^{52} - 63356q^{53} + 13000q^{55} - 85984q^{56} + 13832q^{57} - 183672q^{58} - 63692q^{59} - 263840q^{60} + 103238q^{61} - 109400q^{62} + 52538q^{63} + 29192q^{65} + 186000q^{66} - 11542q^{67} + 313988q^{68} - 146228q^{69} + 364240q^{70} + 45684q^{71} + 309828q^{72} + 72996q^{73} + 26150q^{75} - 297600q^{76} - 157920q^{77} - 586200q^{78} - 128248q^{79} - 467824q^{80} - 276315q^{81} - 458744q^{82} - 38466q^{83} + 76892q^{85} + 545416q^{86} + 121572q^{87} + 690080q^{88} + 121166q^{89} + 945366q^{90} + 44452q^{91} + 576800q^{92} + 388072q^{93} + 7660q^{95} - 841984q^{96} + 224084q^{97} - 1036906q^{98} - 139480q^{99} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_1(20))\)

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
20.6.a \(\chi_{20}(1, \cdot)\) 20.6.a.a 1 1
20.6.c \(\chi_{20}(9, \cdot)\) 20.6.c.a 2 1
20.6.e \(\chi_{20}(3, \cdot)\) 20.6.e.a 2 2
20.6.e.b 24

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_1(20))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_1(20)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(5))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(10))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 - 8 T + 32 T^{2} \))
$3$ (\( 1 - 22 T + 243 T^{2} \))(\( 1 - 362 T^{2} + 59049 T^{4} \))(\( 1 + 59049 T^{4} \))
$5$ (\( 1 + 25 T \))(\( 1 + 10 T + 3125 T^{2} \))(\( 1 - 76 T + 3125 T^{2} \))
$7$ (\( 1 - 218 T + 16807 T^{2} \))(\( 1 - 18610 T^{2} + 282475249 T^{4} \))(\( 1 + 282475249 T^{4} \))
$11$ (\( 1 + 480 T + 161051 T^{2} \))(\( ( 1 + 100 T + 161051 T^{2} )^{2} \))(\( ( 1 - 161051 T^{2} )^{2} \))
$13$ (\( 1 + 622 T + 371293 T^{2} \))(\( 1 - 202442 T^{2} + 137858491849 T^{4} \))(\( ( 1 - 1194 T + 371293 T^{2} )( 1 - 244 T + 371293 T^{2} ) \))
$17$ (\( 1 - 186 T + 1419857 T^{2} \))(\( 1 - 1879458 T^{2} + 2015993900449 T^{4} \))(\( ( 1 - 808 T + 1419857 T^{2} )( 1 + 2242 T + 1419857 T^{2} ) \))
$19$ (\( 1 + 1204 T + 2476099 T^{2} \))(\( ( 1 - 2244 T + 2476099 T^{2} )^{2} \))(\( ( 1 + 2476099 T^{2} )^{2} \))
$23$ (\( 1 + 3186 T + 6436343 T^{2} \))(\( 1 - 1185810 T^{2} + 41426511213649 T^{4} \))(\( 1 + 41426511213649 T^{4} \))
$29$ (\( 1 - 5526 T + 20511149 T^{2} \))(\( ( 1 - 7854 T + 20511149 T^{2} )^{2} \))(\( ( 1 - 2950 T + 20511149 T^{2} )( 1 + 2950 T + 20511149 T^{2} ) \))
$31$ (\( 1 - 9356 T + 28629151 T^{2} \))(\( ( 1 + 2144 T + 28629151 T^{2} )^{2} \))(\( ( 1 - 28629151 T^{2} )^{2} \))
$37$ (\( 1 - 5618 T + 69343957 T^{2} \))(\( 1 - 30515770 T^{2} + 4808584372417849 T^{4} \))(\( ( 1 + 11292 T + 69343957 T^{2} )( 1 + 12242 T + 69343957 T^{2} ) \))
$41$ (\( 1 + 14394 T + 115856201 T^{2} \))(\( ( 1 + 7414 T + 115856201 T^{2} )^{2} \))(\( ( 1 - 4952 T + 115856201 T^{2} )^{2} \))
$43$ (\( 1 + 370 T + 147008443 T^{2} \))(\( 1 + 21442214 T^{2} + 21611482313284249 T^{4} \))(\( 1 + 21611482313284249 T^{4} \))
$47$ (\( 1 - 16146 T + 229345007 T^{2} \))(\( 1 - 369731298 T^{2} + 52599132235830049 T^{4} \))(\( 1 + 52599132235830049 T^{4} \))
$53$ (\( 1 + 4374 T + 418195493 T^{2} \))(\( 1 - 248174170 T^{2} + 174887470365513049 T^{4} \))(\( ( 1 - 40244 T + 418195493 T^{2} )( 1 - 7294 T + 418195493 T^{2} ) \))
$59$ (\( 1 + 11748 T + 714924299 T^{2} \))(\( ( 1 + 25972 T + 714924299 T^{2} )^{2} \))(\( ( 1 + 714924299 T^{2} )^{2} \))
$61$ (\( 1 - 13202 T + 844596301 T^{2} \))(\( ( 1 + 3058 T + 844596301 T^{2} )^{2} \))(\( ( 1 + 54948 T + 844596301 T^{2} )^{2} \))
$67$ (\( 1 + 11542 T + 1350125107 T^{2} \))(\( 1 + 755362070 T^{2} + 1822837804551761449 T^{4} \))(\( 1 + 1822837804551761449 T^{4} \))
$71$ (\( 1 + 29532 T + 1804229351 T^{2} \))(\( ( 1 - 37608 T + 1804229351 T^{2} )^{2} \))(\( ( 1 - 1804229351 T^{2} )^{2} \))
$73$ (\( 1 - 33698 T + 2073071593 T^{2} \))(\( 1 - 3569749522 T^{2} + 4297625829703557649 T^{4} \))(\( ( 1 - 20144 T + 2073071593 T^{2} )( 1 + 88806 T + 2073071593 T^{2} ) \))
$79$ (\( 1 - 31208 T + 3077056399 T^{2} \))(\( ( 1 + 79728 T + 3077056399 T^{2} )^{2} \))(\( ( 1 + 3077056399 T^{2} )^{2} \))
$83$ (\( 1 + 38466 T + 3939040643 T^{2} \))(\( 1 - 7612675530 T^{2} + 15516041187205853449 T^{4} \))(\( 1 + 15516041187205853449 T^{4} \))
$89$ (\( 1 - 119514 T + 5584059449 T^{2} \))(\( ( 1 - 826 T + 5584059449 T^{2} )^{2} \))(\( ( 1 - 51050 T + 5584059449 T^{2} )( 1 + 51050 T + 5584059449 T^{2} ) \))
$97$ (\( 1 - 94658 T + 8587340257 T^{2} \))(\( 1 - 15761405890 T^{2} + 73742412689492826049 T^{4} \))(\( ( 1 - 160808 T + 8587340257 T^{2} )( 1 + 92142 T + 8587340257 T^{2} ) \))
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