Properties

Label 126.2.m
Level 126
Weight 2
Character orbit m
Rep. character \(\chi_{126}(41,\cdot)\)
Character field \(\Q(\zeta_{6})\)
Dimension 16
Newform subspaces 1
Sturm bound 48
Trace bound 0

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Defining parameters

Level: \( N \) = \( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 126.m (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 63 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 1 \)
Sturm bound: \(48\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(126, [\chi])\).

Total New Old
Modular forms 56 16 40
Cusp forms 40 16 24
Eisenstein series 16 0 16

Trace form

\( 16q + 8q^{4} + 2q^{7} + 12q^{9} + O(q^{10}) \) \( 16q + 8q^{4} + 2q^{7} + 12q^{9} - 12q^{11} - 6q^{14} - 8q^{16} - 12q^{18} + 18q^{21} - 48q^{23} - 8q^{25} + 4q^{28} - 12q^{29} - 24q^{30} + 12q^{36} - 8q^{37} - 36q^{39} - 12q^{42} + 4q^{43} + 24q^{46} - 8q^{49} + 60q^{50} + 12q^{51} - 6q^{56} + 48q^{57} - 12q^{58} + 24q^{60} + 24q^{63} - 16q^{64} + 84q^{65} - 28q^{67} + 36q^{74} + 78q^{77} - 24q^{78} - 4q^{79} + 36q^{81} + 18q^{84} - 12q^{85} - 24q^{86} + 24q^{91} - 48q^{92} - 96q^{93} + 12q^{95} - 72q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(126, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
126.2.m.a \(16\) \(1.006\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(0\) \(2\) \(q+\beta _{10}q^{2}-\beta _{1}q^{3}+\beta _{5}q^{4}+(\beta _{1}-\beta _{6}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(126, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(126, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T^{2} + T^{4} )^{4} \)
$3$ \( 1 - 6 T^{2} + 9 T^{4} + 54 T^{6} - 288 T^{8} + 486 T^{10} + 729 T^{12} - 4374 T^{14} + 6561 T^{16} \)
$5$ \( 1 - 16 T^{2} + 123 T^{4} - 584 T^{6} + 1481 T^{8} + 1416 T^{10} - 59414 T^{12} + 576968 T^{14} - 3477114 T^{16} + 14424200 T^{18} - 37133750 T^{20} + 22125000 T^{22} + 578515625 T^{24} - 5703125000 T^{26} + 30029296875 T^{28} - 97656250000 T^{30} + 152587890625 T^{32} \)
$7$ \( 1 - 2 T + 6 T^{2} + 8 T^{3} - 58 T^{4} + 222 T^{5} - 104 T^{6} - 662 T^{7} + 3483 T^{8} - 4634 T^{9} - 5096 T^{10} + 76146 T^{11} - 139258 T^{12} + 134456 T^{13} + 705894 T^{14} - 1647086 T^{15} + 5764801 T^{16} \)
$11$ \( ( 1 + 6 T + 35 T^{2} + 138 T^{3} + 481 T^{4} + 1512 T^{5} + 3854 T^{6} + 13116 T^{7} + 37618 T^{8} + 144276 T^{9} + 466334 T^{10} + 2012472 T^{11} + 7042321 T^{12} + 22225038 T^{13} + 62004635 T^{14} + 116923026 T^{15} + 214358881 T^{16} )^{2} \)
$13$ \( 1 + 68 T^{2} + 2376 T^{4} + 57352 T^{6} + 1082018 T^{8} + 16951644 T^{10} + 232506496 T^{12} + 2987085740 T^{14} + 38351015667 T^{16} + 504817490060 T^{18} + 6640618032256 T^{20} + 81822347823996 T^{22} + 882635323274978 T^{24} + 7906460224523848 T^{26} + 55356250251014856 T^{28} + 267741594227551652 T^{30} + 665416609183179841 T^{32} \)
$17$ \( ( 1 + 94 T^{2} + 4285 T^{4} + 124198 T^{6} + 2503180 T^{8} + 35893222 T^{10} + 357887485 T^{12} + 2268931486 T^{14} + 6975757441 T^{16} )^{2} \)
$19$ \( ( 1 - 98 T^{2} + 4546 T^{4} - 134984 T^{6} + 2932423 T^{8} - 48729224 T^{10} + 592439266 T^{12} - 4610496338 T^{14} + 16983563041 T^{16} )^{2} \)
$23$ \( ( 1 + 24 T + 317 T^{2} + 3000 T^{3} + 22111 T^{4} + 134028 T^{5} + 704756 T^{6} + 3411156 T^{7} + 16228318 T^{8} + 78456588 T^{9} + 372815924 T^{10} + 1630718676 T^{11} + 6187564351 T^{12} + 19309029000 T^{13} + 46927376813 T^{14} + 81715810728 T^{15} + 78310985281 T^{16} )^{2} \)
$29$ \( ( 1 + 6 T + 98 T^{2} + 516 T^{3} + 4846 T^{4} + 25650 T^{5} + 193448 T^{6} + 972210 T^{7} + 6347347 T^{8} + 28194090 T^{9} + 162689768 T^{10} + 625577850 T^{11} + 3427483726 T^{12} + 10583752884 T^{13} + 58292685458 T^{14} + 103499257854 T^{15} + 500246412961 T^{16} )^{2} \)
$31$ \( 1 + 104 T^{2} + 3888 T^{4} + 96880 T^{6} + 4455362 T^{8} + 148421160 T^{10} + 1870813504 T^{12} + 70884338648 T^{14} + 4079738375235 T^{16} + 68119849440728 T^{18} + 1727735558027584 T^{20} + 131724325838289960 T^{22} + 3799938318355208642 T^{24} + 79405588442700000880 T^{26} + \)\(30\!\cdots\!68\)\( T^{28} + \)\(78\!\cdots\!84\)\( T^{30} + \)\(72\!\cdots\!81\)\( T^{32} \)
$37$ \( ( 1 + 2 T + 46 T^{2} + 38 T^{3} + 2002 T^{4} + 1406 T^{5} + 62974 T^{6} + 101306 T^{7} + 1874161 T^{8} )^{4} \)
$41$ \( 1 - 70 T^{2} - 1569 T^{4} + 150526 T^{6} + 5171081 T^{8} - 280356900 T^{10} - 8583026738 T^{12} + 255096492224 T^{14} + 9422146980954 T^{16} + 428817203428544 T^{18} - 24253582218197618 T^{20} - 1331724499683612900 T^{22} + 41290695138728249801 T^{24} + \)\(20\!\cdots\!26\)\( T^{26} - \)\(35\!\cdots\!89\)\( T^{28} - \)\(26\!\cdots\!70\)\( T^{30} + \)\(63\!\cdots\!41\)\( T^{32} \)
$43$ \( ( 1 - 2 T - 129 T^{2} - 46 T^{3} + 9833 T^{4} + 11184 T^{5} - 521114 T^{6} - 232628 T^{7} + 22298490 T^{8} - 10003004 T^{9} - 963539786 T^{10} + 889206288 T^{11} + 33617070233 T^{12} - 6762388378 T^{13} - 815455833321 T^{14} - 543637222214 T^{15} + 11688200277601 T^{16} )^{2} \)
$47$ \( 1 - 136 T^{2} + 8016 T^{4} - 225584 T^{6} - 1533310 T^{8} + 489880632 T^{10} - 30253322048 T^{12} + 1486359308360 T^{14} - 68731587628605 T^{16} + 3283367712167240 T^{18} - 147626560784506688 T^{20} + 5280528817834607928 T^{22} - 36510083951344758910 T^{24} - \)\(11\!\cdots\!16\)\( T^{26} + \)\(93\!\cdots\!56\)\( T^{28} - \)\(34\!\cdots\!84\)\( T^{30} + \)\(56\!\cdots\!21\)\( T^{32} \)
$53$ \( ( 1 - 53 T^{2} )^{16} \)
$59$ \( 1 - 178 T^{2} + 20643 T^{4} - 1496150 T^{6} + 80456981 T^{8} - 3515943660 T^{10} + 180649052698 T^{12} - 13219132050040 T^{14} + 857467356385554 T^{16} - 46015798666189240 T^{18} + 2188989785849689978 T^{20} - \)\(14\!\cdots\!60\)\( T^{22} + \)\(11\!\cdots\!01\)\( T^{24} - \)\(76\!\cdots\!50\)\( T^{26} + \)\(36\!\cdots\!83\)\( T^{28} - \)\(11\!\cdots\!58\)\( T^{30} + \)\(21\!\cdots\!41\)\( T^{32} \)
$61$ \( 1 + 248 T^{2} + 28191 T^{4} + 2052448 T^{6} + 122525357 T^{8} + 7198089144 T^{10} + 457346362462 T^{12} + 32095759051208 T^{14} + 2131627513941198 T^{16} + 119428319429544968 T^{18} + 6332345016577220542 T^{20} + \)\(37\!\cdots\!84\)\( T^{22} + \)\(23\!\cdots\!17\)\( T^{24} + \)\(14\!\cdots\!48\)\( T^{26} + \)\(74\!\cdots\!11\)\( T^{28} + \)\(24\!\cdots\!68\)\( T^{30} + \)\(36\!\cdots\!61\)\( T^{32} \)
$67$ \( ( 1 + 14 T + 39 T^{2} - 110 T^{3} - 2659 T^{4} - 53760 T^{5} - 92054 T^{6} + 2669060 T^{7} + 22240746 T^{8} + 178827020 T^{9} - 413230406 T^{10} - 16169018880 T^{11} - 53581830739 T^{12} - 148513761770 T^{13} + 3527876904591 T^{14} + 84849962474522 T^{15} + 406067677556641 T^{16} )^{2} \)
$71$ \( ( 1 - 478 T^{2} + 105553 T^{4} - 13986142 T^{6} + 1213269316 T^{8} - 70504141822 T^{10} + 2682279164593 T^{12} - 61231935714238 T^{14} + 645753531245761 T^{16} )^{2} \)
$73$ \( ( 1 - 362 T^{2} + 64045 T^{4} - 7316714 T^{6} + 612211324 T^{8} - 38990768906 T^{10} + 1818765344845 T^{12} - 54782989916618 T^{14} + 806460091894081 T^{16} )^{2} \)
$79$ \( ( 1 + 2 T - 183 T^{2} + 982 T^{3} + 19715 T^{4} - 144312 T^{5} - 491612 T^{6} + 7480148 T^{7} - 8945118 T^{8} + 590931692 T^{9} - 3068150492 T^{10} - 71151444168 T^{11} + 767900846915 T^{12} + 3021669383818 T^{13} - 44485004360343 T^{14} + 38407817972318 T^{15} + 1517108809906561 T^{16} )^{2} \)
$83$ \( 1 + 44 T^{2} - 13368 T^{4} - 595880 T^{6} + 87112226 T^{8} + 3596762100 T^{10} - 160886956928 T^{12} - 11841264313180 T^{14} - 673218489607821 T^{16} - 81574469853497020 T^{18} - 7635424846602197888 T^{20} + \)\(11\!\cdots\!00\)\( T^{22} + \)\(19\!\cdots\!66\)\( T^{24} - \)\(92\!\cdots\!20\)\( T^{26} - \)\(14\!\cdots\!48\)\( T^{28} + \)\(32\!\cdots\!76\)\( T^{30} + \)\(50\!\cdots\!81\)\( T^{32} \)
$89$ \( ( 1 + 496 T^{2} + 119404 T^{4} + 18272464 T^{6} + 1934931814 T^{8} + 144736187344 T^{10} + 7491674544364 T^{12} + 246502720316656 T^{14} + 3936588805702081 T^{16} )^{2} \)
$97$ \( 1 + 74 T^{2} + 11511 T^{4} - 803858 T^{6} - 150210775 T^{8} - 25062425316 T^{10} - 114467134418 T^{12} + 73367970993632 T^{14} + 27832786456667274 T^{16} + 690319239079083488 T^{18} - 10133693108155893458 T^{20} - \)\(20\!\cdots\!64\)\( T^{22} - \)\(11\!\cdots\!75\)\( T^{24} - \)\(59\!\cdots\!42\)\( T^{26} + \)\(79\!\cdots\!51\)\( T^{28} + \)\(48\!\cdots\!06\)\( T^{30} + \)\(61\!\cdots\!21\)\( T^{32} \)
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