Properties

Label 6.19
Level 6
Weight 19
Dimension 6
Nonzero newspaces 1
Newform subspaces 1
Sturm bound 38
Trace bound 0

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Defining parameters

Level: \( N \) = \( 6 = 2 \cdot 3 \)
Weight: \( k \) = \( 19 \)
Nonzero newspaces: \( 1 \)
Newform subspaces: \( 1 \)
Sturm bound: \(38\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{19}(\Gamma_1(6))\).

Total New Old
Modular forms 20 6 14
Cusp forms 16 6 10
Eisenstein series 4 0 4

Trace form

\( 6q - 6258q^{3} - 786432q^{4} - 15753216q^{6} + 28233804q^{7} + 695971638q^{9} + O(q^{10}) \) \( 6q - 6258q^{3} - 786432q^{4} - 15753216q^{6} + 28233804q^{7} + 695971638q^{9} - 434257920q^{10} + 820248576q^{12} + 29566196220q^{13} + 119627095680q^{15} + 103079215104q^{16} + 315939373056q^{18} - 438814047012q^{19} + 2876527406172q^{21} - 2844452929536q^{22} + 2064805527552q^{24} - 25696048717290q^{25} + 11197265522814q^{27} - 3700661157888q^{28} + 13072787619840q^{30} + 21775814927148q^{31} - 962560003968q^{33} + 99067611119616q^{34} - 91222394535936q^{36} + 638446564817436q^{37} - 736541155104180q^{39} + 56919054090240q^{40} - 203954622480384q^{42} - 1688313718883652q^{43} + 390590504075520q^{45} + 1979247919104q^{46} - 107511621353472q^{48} - 895767896448270q^{49} + 9636273526722048q^{51} - 3875300470947840q^{52} + 2993011804200960q^{54} - 1259959207783680q^{55} + 4023767318253996q^{57} + 21251172660756480q^{58} - 15679762684968960q^{60} - 16279597277700036q^{61} - 32525159214131028q^{63} - 13510798882111488q^{64} - 60047751762690048q^{66} + 11153724314613276q^{67} + 3612037794746112q^{69} + 161904998736691200q^{70} - 41410805505196032q^{72} - 78910243347781140q^{73} + 348225828845090910q^{75} + 57516234769956864q^{76} - 168903906208235520q^{78} - 476518976428926228q^{79} + 630680446106425062q^{81} + 396536393269149696q^{82} - 377032200181776384q^{84} - 967978669078932480q^{85} + 419256510981966720q^{87} + 372828134380142592q^{88} - 2046848246643179520q^{90} - 1032060167365562760q^{91} + 764294136047005116q^{93} + 2566175766871080960q^{94} - 270638190107295744q^{96} + 834695417243310348q^{97} + 761688154713814272q^{99} + O(q^{100}) \)

Decomposition of \(S_{19}^{\mathrm{new}}(\Gamma_1(6))\)

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
6.19.b \(\chi_{6}(5, \cdot)\) 6.19.b.a 6 1

Decomposition of \(S_{19}^{\mathrm{old}}(\Gamma_1(6))\) into lower level spaces

\( S_{19}^{\mathrm{old}}(\Gamma_1(6)) \cong \) \(S_{19}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + 131072 T^{2} )^{3} \)
$3$ \( 1 + 6258 T - 328404537 T^{2} - 5869270541988 T^{3} - 127230646314358593 T^{4} + \)\(93\!\cdots\!18\)\( T^{5} + \)\(58\!\cdots\!69\)\( T^{6} \)
$5$ \( 1 + 1403932561770 T^{2} + \)\(80\!\cdots\!75\)\( T^{4} - \)\(57\!\cdots\!00\)\( T^{6} + \)\(11\!\cdots\!75\)\( T^{8} + \)\(29\!\cdots\!50\)\( T^{10} + \)\(30\!\cdots\!25\)\( T^{12} \)
$7$ \( ( 1 - 14116902 T + 2766205832016543 T^{2} + \)\(35\!\cdots\!92\)\( T^{3} + \)\(45\!\cdots\!07\)\( T^{4} - \)\(37\!\cdots\!02\)\( T^{5} + \)\(43\!\cdots\!49\)\( T^{6} )^{2} \)
$11$ \( 1 - 17996209479583479510 T^{2} + \)\(13\!\cdots\!83\)\( T^{4} - \)\(72\!\cdots\!20\)\( T^{6} + \)\(41\!\cdots\!63\)\( T^{8} - \)\(17\!\cdots\!10\)\( T^{10} + \)\(29\!\cdots\!81\)\( T^{12} \)
$13$ \( ( 1 - 14783098110 T + \)\(32\!\cdots\!87\)\( T^{2} - \)\(27\!\cdots\!80\)\( T^{3} + \)\(36\!\cdots\!23\)\( T^{4} - \)\(18\!\cdots\!10\)\( T^{5} + \)\(14\!\cdots\!89\)\( T^{6} )^{2} \)
$17$ \( 1 + \)\(97\!\cdots\!70\)\( T^{2} + \)\(55\!\cdots\!23\)\( T^{4} + \)\(35\!\cdots\!40\)\( T^{6} + \)\(11\!\cdots\!63\)\( T^{8} + \)\(38\!\cdots\!70\)\( T^{10} + \)\(77\!\cdots\!41\)\( T^{12} \)
$19$ \( ( 1 + 219407023506 T + \)\(24\!\cdots\!15\)\( T^{2} + \)\(33\!\cdots\!60\)\( T^{3} + \)\(25\!\cdots\!15\)\( T^{4} + \)\(23\!\cdots\!86\)\( T^{5} + \)\(11\!\cdots\!21\)\( T^{6} )^{2} \)
$23$ \( 1 - \)\(10\!\cdots\!50\)\( T^{2} + \)\(61\!\cdots\!63\)\( T^{4} - \)\(23\!\cdots\!00\)\( T^{6} + \)\(65\!\cdots\!43\)\( T^{8} - \)\(11\!\cdots\!50\)\( T^{10} + \)\(11\!\cdots\!81\)\( T^{12} \)
$29$ \( 1 - \)\(94\!\cdots\!86\)\( T^{2} + \)\(42\!\cdots\!95\)\( T^{4} - \)\(11\!\cdots\!40\)\( T^{6} + \)\(18\!\cdots\!95\)\( T^{8} - \)\(18\!\cdots\!26\)\( T^{10} + \)\(86\!\cdots\!61\)\( T^{12} \)
$31$ \( ( 1 - 10887907463574 T + \)\(15\!\cdots\!15\)\( T^{2} - \)\(12\!\cdots\!80\)\( T^{3} + \)\(10\!\cdots\!15\)\( T^{4} - \)\(53\!\cdots\!94\)\( T^{5} + \)\(34\!\cdots\!21\)\( T^{6} )^{2} \)
$37$ \( ( 1 - 319223282408718 T + \)\(69\!\cdots\!43\)\( T^{2} - \)\(11\!\cdots\!32\)\( T^{3} + \)\(11\!\cdots\!47\)\( T^{4} - \)\(91\!\cdots\!38\)\( T^{5} + \)\(48\!\cdots\!89\)\( T^{6} )^{2} \)
$41$ \( 1 - \)\(24\!\cdots\!70\)\( T^{2} + \)\(50\!\cdots\!43\)\( T^{4} - \)\(59\!\cdots\!40\)\( T^{6} + \)\(58\!\cdots\!63\)\( T^{8} - \)\(32\!\cdots\!70\)\( T^{10} + \)\(15\!\cdots\!21\)\( T^{12} \)
$43$ \( ( 1 + 844156859441826 T + \)\(94\!\cdots\!71\)\( T^{2} + \)\(43\!\cdots\!12\)\( T^{3} + \)\(23\!\cdots\!79\)\( T^{4} + \)\(53\!\cdots\!26\)\( T^{5} + \)\(16\!\cdots\!49\)\( T^{6} )^{2} \)
$47$ \( 1 - \)\(18\!\cdots\!34\)\( T^{2} + \)\(24\!\cdots\!15\)\( T^{4} - \)\(37\!\cdots\!80\)\( T^{6} + \)\(38\!\cdots\!15\)\( T^{8} - \)\(45\!\cdots\!94\)\( T^{10} + \)\(38\!\cdots\!61\)\( T^{12} \)
$53$ \( 1 - \)\(38\!\cdots\!30\)\( T^{2} + \)\(67\!\cdots\!43\)\( T^{4} - \)\(81\!\cdots\!60\)\( T^{6} + \)\(80\!\cdots\!03\)\( T^{8} - \)\(54\!\cdots\!30\)\( T^{10} + \)\(16\!\cdots\!61\)\( T^{12} \)
$59$ \( 1 - \)\(40\!\cdots\!70\)\( T^{2} + \)\(71\!\cdots\!03\)\( T^{4} - \)\(69\!\cdots\!40\)\( T^{6} + \)\(40\!\cdots\!23\)\( T^{8} - \)\(12\!\cdots\!70\)\( T^{10} + \)\(17\!\cdots\!21\)\( T^{12} \)
$61$ \( ( 1 + 8139798638850018 T + \)\(35\!\cdots\!71\)\( T^{2} + \)\(22\!\cdots\!52\)\( T^{3} + \)\(48\!\cdots\!51\)\( T^{4} + \)\(15\!\cdots\!98\)\( T^{5} + \)\(25\!\cdots\!41\)\( T^{6} )^{2} \)
$67$ \( ( 1 - 5576862157306638 T + \)\(27\!\cdots\!23\)\( T^{2} + \)\(20\!\cdots\!48\)\( T^{3} + \)\(20\!\cdots\!07\)\( T^{4} - \)\(30\!\cdots\!78\)\( T^{5} + \)\(40\!\cdots\!29\)\( T^{6} )^{2} \)
$71$ \( 1 - \)\(34\!\cdots\!86\)\( T^{2} + \)\(14\!\cdots\!95\)\( T^{4} - \)\(30\!\cdots\!40\)\( T^{6} + \)\(62\!\cdots\!95\)\( T^{8} - \)\(67\!\cdots\!26\)\( T^{10} + \)\(86\!\cdots\!61\)\( T^{12} \)
$73$ \( ( 1 + 39455121673890570 T + \)\(49\!\cdots\!27\)\( T^{2} + \)\(22\!\cdots\!60\)\( T^{3} + \)\(17\!\cdots\!63\)\( T^{4} + \)\(47\!\cdots\!70\)\( T^{5} + \)\(41\!\cdots\!09\)\( T^{6} )^{2} \)
$79$ \( ( 1 + 238259488214463114 T + \)\(52\!\cdots\!95\)\( T^{2} + \)\(66\!\cdots\!20\)\( T^{3} + \)\(75\!\cdots\!95\)\( T^{4} + \)\(49\!\cdots\!94\)\( T^{5} + \)\(29\!\cdots\!81\)\( T^{6} )^{2} \)
$83$ \( 1 - \)\(14\!\cdots\!58\)\( T^{2} + \)\(10\!\cdots\!51\)\( T^{4} - \)\(43\!\cdots\!72\)\( T^{6} + \)\(12\!\cdots\!31\)\( T^{8} - \)\(21\!\cdots\!38\)\( T^{10} + \)\(18\!\cdots\!41\)\( T^{12} \)
$89$ \( 1 - \)\(36\!\cdots\!50\)\( T^{2} + \)\(54\!\cdots\!63\)\( T^{4} - \)\(59\!\cdots\!00\)\( T^{6} + \)\(82\!\cdots\!43\)\( T^{8} - \)\(83\!\cdots\!50\)\( T^{10} + \)\(34\!\cdots\!81\)\( T^{12} \)
$97$ \( ( 1 - 417347708621655174 T + \)\(13\!\cdots\!91\)\( T^{2} - \)\(53\!\cdots\!88\)\( T^{3} + \)\(76\!\cdots\!99\)\( T^{4} - \)\(13\!\cdots\!54\)\( T^{5} + \)\(19\!\cdots\!69\)\( T^{6} )^{2} \)
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