Properties

Label 12.20
Level 12
Weight 20
Dimension 40
Nonzero newspaces 2
Newform subspaces 3
Sturm bound 160
Trace bound 1

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Defining parameters

Level: \( N \) = \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) = \( 20 \)
Nonzero newspaces: \( 2 \)
Newform subspaces: \( 3 \)
Sturm bound: \(160\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{20}(\Gamma_1(12))\).

Total New Old
Modular forms 81 44 37
Cusp forms 71 40 31
Eisenstein series 10 4 6

Trace form

\( 40q - 47880q^{4} - 2642328q^{5} + 21771144q^{6} - 22356880q^{7} + 740208312q^{9} + O(q^{10}) \) \( 40q - 47880q^{4} - 2642328q^{5} + 21771144q^{6} - 22356880q^{7} + 740208312q^{9} - 4162950000q^{10} + 19697749344q^{11} - 2268767880q^{12} + 33702689744q^{13} - 76604031504q^{15} + 791689296672q^{16} - 593676014040q^{17} + 2852271258192q^{18} - 1120481284000q^{19} - 2274601593528q^{21} + 7247870602416q^{22} + 3058170374880q^{23} + 23719435803936q^{24} - 82077069150416q^{25} - 7629081673968q^{28} - 20632695071256q^{29} + 37456727138640q^{30} - 353268036790864q^{31} + 562050181030320q^{33} - 248241675948480q^{34} + 101839235263776q^{35} - 1177548745309416q^{36} + 3492436239515120q^{37} + 151949185567200q^{39} - 40433256181440q^{40} + 481400137188360q^{41} - 2709671988768336q^{42} - 5148001223004640q^{43} - 5389102119946392q^{45} + 17206125364795104q^{46} - 10457602667272800q^{47} + 1561106052682272q^{48} - 33043775752066320q^{49} + 1342847550674016q^{51} + 16558671470242896q^{52} + 30131832529686600q^{53} + 40087412948587896q^{54} - 13367303894026368q^{55} + 83994822579003192q^{57} - 34645536255764880q^{58} - 95254877571624576q^{59} + 50260381053157440q^{60} + 142698311428025072q^{61} - 8661513382114320q^{63} - 139766490424918656q^{64} - 301750589490544080q^{65} - 87772097712685968q^{66} + 420432590712268160q^{67} - 780638608309943808q^{69} + 244663962371376480q^{70} + 147467537428629600q^{71} + 54134736422921280q^{72} - 1085559448828078192q^{73} + 1083175956794066112q^{75} + 2744130151937256048q^{76} - 3446123571889300800q^{77} + 1125465917961999024q^{78} + 249052172722763888q^{79} + 1633676782541038920q^{81} + 1064985418859680800q^{82} - 4459007835083711520q^{83} + 1920709484261873424q^{84} + 3133310833050055056q^{85} + 6329919533648769360q^{87} - 9209792381828477760q^{88} - 5967971715685201560q^{89} - 5194944179528261040q^{90} + 6611734573552861600q^{91} + 17075256530728117896q^{93} - 3835186345170303552q^{94} - 34539126254816774400q^{95} - 11947184361208227456q^{96} + 30758007426443375888q^{97} + 7631311683051909216q^{99} + O(q^{100}) \)

Decomposition of \(S_{20}^{\mathrm{new}}(\Gamma_1(12))\)

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
12.20.a \(\chi_{12}(1, \cdot)\) 12.20.a.a 2 1
12.20.a.b 2
12.20.b \(\chi_{12}(11, \cdot)\) 12.20.b.a 36 1

Decomposition of \(S_{20}^{\mathrm{old}}(\Gamma_1(12))\) into lower level spaces

\( S_{20}^{\mathrm{old}}(\Gamma_1(12)) \cong \) \(S_{20}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 6}\)\(\oplus\)\(S_{20}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 4}\)\(\oplus\)\(S_{20}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 3}\)\(\oplus\)\(S_{20}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 2}\)\(\oplus\)\(S_{20}^{\mathrm{new}}(\Gamma_1(6))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( \))(\( \))
$3$ (\( ( 1 + 19683 T )^{2} \))(\( ( 1 - 19683 T )^{2} \))
$5$ (\( 1 - 624780 T + 23825765979550 T^{2} - 11916732788085937500 T^{3} + \)\(36\!\cdots\!25\)\( T^{4} \))(\( 1 + 3267108 T + 1452068386750 T^{2} + 62315139770507812500 T^{3} + \)\(36\!\cdots\!25\)\( T^{4} \))
$7$ (\( 1 - 4667104 T + 872249874518190 T^{2} - \)\(53\!\cdots\!72\)\( T^{3} + \)\(12\!\cdots\!49\)\( T^{4} \))(\( 1 + 27023984 T + 14123598636960750 T^{2} + \)\(30\!\cdots\!12\)\( T^{3} + \)\(12\!\cdots\!49\)\( T^{4} \))
$11$ (\( 1 - 9015783192 T + 76236163959728858998 T^{2} - \)\(55\!\cdots\!72\)\( T^{3} + \)\(37\!\cdots\!81\)\( T^{4} \))(\( 1 - 10681966152 T + \)\(12\!\cdots\!58\)\( T^{2} - \)\(65\!\cdots\!32\)\( T^{3} + \)\(37\!\cdots\!81\)\( T^{4} \))
$13$ (\( 1 - 25948151500 T + \)\(23\!\cdots\!54\)\( T^{2} - \)\(37\!\cdots\!00\)\( T^{3} + \)\(21\!\cdots\!29\)\( T^{4} \))(\( 1 - 33667969900 T + \)\(31\!\cdots\!54\)\( T^{2} - \)\(49\!\cdots\!00\)\( T^{3} + \)\(21\!\cdots\!29\)\( T^{4} \))
$17$ (\( 1 + 330949868796 T + \)\(16\!\cdots\!10\)\( T^{2} + \)\(79\!\cdots\!88\)\( T^{3} + \)\(57\!\cdots\!09\)\( T^{4} \))(\( 1 + 262726145244 T + \)\(49\!\cdots\!90\)\( T^{2} + \)\(62\!\cdots\!32\)\( T^{3} + \)\(57\!\cdots\!09\)\( T^{4} \))
$19$ (\( 1 + 1555799112920 T + \)\(23\!\cdots\!58\)\( T^{2} + \)\(30\!\cdots\!80\)\( T^{3} + \)\(39\!\cdots\!41\)\( T^{4} \))(\( 1 - 435317828920 T + \)\(90\!\cdots\!58\)\( T^{2} - \)\(86\!\cdots\!80\)\( T^{3} + \)\(39\!\cdots\!41\)\( T^{4} \))
$23$ (\( 1 + 2955794188464 T + \)\(12\!\cdots\!98\)\( T^{2} + \)\(22\!\cdots\!68\)\( T^{3} + \)\(55\!\cdots\!69\)\( T^{4} \))(\( 1 - 6013964563344 T + \)\(49\!\cdots\!58\)\( T^{2} - \)\(44\!\cdots\!28\)\( T^{3} + \)\(55\!\cdots\!69\)\( T^{4} \))
$29$ (\( 1 + 171112962219588 T + \)\(18\!\cdots\!74\)\( T^{2} + \)\(10\!\cdots\!72\)\( T^{3} + \)\(37\!\cdots\!61\)\( T^{4} \))(\( 1 - 150480267148332 T + \)\(15\!\cdots\!94\)\( T^{2} - \)\(91\!\cdots\!08\)\( T^{3} + \)\(37\!\cdots\!61\)\( T^{4} \))
$31$ (\( 1 + 439686204047792 T + \)\(91\!\cdots\!58\)\( T^{2} + \)\(95\!\cdots\!32\)\( T^{3} + \)\(46\!\cdots\!41\)\( T^{4} \))(\( 1 - 86418167256928 T - \)\(50\!\cdots\!62\)\( T^{2} - \)\(18\!\cdots\!88\)\( T^{3} + \)\(46\!\cdots\!41\)\( T^{4} \))
$37$ (\( 1 + 912841755316868 T + \)\(11\!\cdots\!02\)\( T^{2} + \)\(57\!\cdots\!64\)\( T^{3} + \)\(39\!\cdots\!29\)\( T^{4} \))(\( 1 - 2557439394663868 T + \)\(28\!\cdots\!02\)\( T^{2} - \)\(15\!\cdots\!64\)\( T^{3} + \)\(39\!\cdots\!29\)\( T^{4} \))
$41$ (\( 1 + 2039868884446380 T + \)\(80\!\cdots\!22\)\( T^{2} + \)\(89\!\cdots\!80\)\( T^{3} + \)\(19\!\cdots\!21\)\( T^{4} \))(\( 1 - 2521269021634740 T + \)\(63\!\cdots\!22\)\( T^{2} - \)\(11\!\cdots\!40\)\( T^{3} + \)\(19\!\cdots\!21\)\( T^{4} \))
$43$ (\( 1 + 5607123893986664 T + \)\(22\!\cdots\!38\)\( T^{2} + \)\(60\!\cdots\!48\)\( T^{3} + \)\(11\!\cdots\!49\)\( T^{4} \))(\( 1 - 459122670982024 T + \)\(16\!\cdots\!58\)\( T^{2} - \)\(49\!\cdots\!68\)\( T^{3} + \)\(11\!\cdots\!49\)\( T^{4} \))
$47$ (\( 1 + 720086631286464 T + \)\(33\!\cdots\!90\)\( T^{2} + \)\(42\!\cdots\!12\)\( T^{3} + \)\(34\!\cdots\!89\)\( T^{4} \))(\( 1 + 9737516035986336 T + \)\(12\!\cdots\!90\)\( T^{2} + \)\(57\!\cdots\!88\)\( T^{3} + \)\(34\!\cdots\!89\)\( T^{4} \))
$53$ (\( 1 - 56983015881991692 T + \)\(16\!\cdots\!50\)\( T^{2} - \)\(32\!\cdots\!64\)\( T^{3} + \)\(33\!\cdots\!89\)\( T^{4} \))(\( 1 + 26851183352305092 T + \)\(13\!\cdots\!50\)\( T^{2} + \)\(15\!\cdots\!64\)\( T^{3} + \)\(33\!\cdots\!89\)\( T^{4} \))
$59$ (\( 1 - 17959203991017432 T + \)\(86\!\cdots\!34\)\( T^{2} - \)\(79\!\cdots\!48\)\( T^{3} + \)\(19\!\cdots\!21\)\( T^{4} \))(\( 1 + 113214081562642008 T + \)\(11\!\cdots\!94\)\( T^{2} + \)\(50\!\cdots\!12\)\( T^{3} + \)\(19\!\cdots\!21\)\( T^{4} \))
$61$ (\( 1 - 29244471161814604 T + \)\(16\!\cdots\!86\)\( T^{2} - \)\(24\!\cdots\!64\)\( T^{3} + \)\(69\!\cdots\!81\)\( T^{4} \))(\( 1 + 61288509973786676 T + \)\(14\!\cdots\!26\)\( T^{2} + \)\(51\!\cdots\!16\)\( T^{3} + \)\(69\!\cdots\!81\)\( T^{4} \))
$67$ (\( 1 + 37110594569246744 T + \)\(82\!\cdots\!90\)\( T^{2} + \)\(18\!\cdots\!32\)\( T^{3} + \)\(24\!\cdots\!09\)\( T^{4} \))(\( 1 - 457543185281514904 T + \)\(15\!\cdots\!10\)\( T^{2} - \)\(22\!\cdots\!12\)\( T^{3} + \)\(24\!\cdots\!09\)\( T^{4} \))
$71$ (\( 1 + 604502474341905360 T + \)\(29\!\cdots\!62\)\( T^{2} + \)\(90\!\cdots\!60\)\( T^{3} + \)\(22\!\cdots\!61\)\( T^{4} \))(\( 1 - 751970011770534960 T + \)\(37\!\cdots\!62\)\( T^{2} - \)\(11\!\cdots\!60\)\( T^{3} + \)\(22\!\cdots\!61\)\( T^{4} \))
$73$ (\( 1 + 659663813215165772 T + \)\(54\!\cdots\!70\)\( T^{2} + \)\(16\!\cdots\!64\)\( T^{3} + \)\(64\!\cdots\!69\)\( T^{4} \))(\( 1 - 1019226905786569012 T + \)\(62\!\cdots\!10\)\( T^{2} - \)\(25\!\cdots\!44\)\( T^{3} + \)\(64\!\cdots\!69\)\( T^{4} \))
$79$ (\( 1 + 352041739978389776 T + \)\(49\!\cdots\!82\)\( T^{2} + \)\(39\!\cdots\!44\)\( T^{3} + \)\(12\!\cdots\!61\)\( T^{4} \))(\( 1 - 601093912701153664 T + \)\(54\!\cdots\!62\)\( T^{2} - \)\(68\!\cdots\!16\)\( T^{3} + \)\(12\!\cdots\!61\)\( T^{4} \))
$83$ (\( 1 + 2492903589286330584 T + \)\(55\!\cdots\!58\)\( T^{2} + \)\(72\!\cdots\!48\)\( T^{3} + \)\(84\!\cdots\!09\)\( T^{4} \))(\( 1 + 1966104245797380936 T + \)\(60\!\cdots\!18\)\( T^{2} + \)\(57\!\cdots\!92\)\( T^{3} + \)\(84\!\cdots\!09\)\( T^{4} \))
$89$ (\( 1 - 2523728600070118740 T - \)\(42\!\cdots\!82\)\( T^{2} - \)\(27\!\cdots\!60\)\( T^{3} + \)\(11\!\cdots\!81\)\( T^{4} \))(\( 1 + 8491700315755320300 T + \)\(38\!\cdots\!18\)\( T^{2} + \)\(92\!\cdots\!00\)\( T^{3} + \)\(11\!\cdots\!81\)\( T^{4} \))
$97$ (\( 1 - 17246738574828858628 T + \)\(16\!\cdots\!62\)\( T^{2} - \)\(96\!\cdots\!24\)\( T^{3} + \)\(31\!\cdots\!89\)\( T^{4} \))(\( 1 - 746042691667512772 T + \)\(43\!\cdots\!62\)\( T^{2} - \)\(41\!\cdots\!76\)\( T^{3} + \)\(31\!\cdots\!89\)\( T^{4} \))
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