Properties

Label 108.3.f
Level 108
Weight 3
Character orbit f
Rep. character \(\chi_{108}(19,\cdot)\)
Character field \(\Q(\zeta_{6})\)
Dimension 20
Newform subspaces 3
Sturm bound 54
Trace bound 2

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Defining parameters

Level: \( N \) \(=\) \( 108 = 2^{2} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 108.f (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 36 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 3 \)
Sturm bound: \(54\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(108, [\chi])\).

Total New Old
Modular forms 84 28 56
Cusp forms 60 20 40
Eisenstein series 24 8 16

Trace form

\( 20q + q^{2} - q^{4} + 2q^{5} + 22q^{8} + O(q^{10}) \) \( 20q + q^{2} - q^{4} + 2q^{5} + 22q^{8} + 4q^{10} - 2q^{13} + 24q^{14} - q^{16} + 32q^{17} - 52q^{20} - 9q^{22} - 12q^{25} - 160q^{26} - 36q^{28} + 26q^{29} - 119q^{32} - 11q^{34} - 8q^{37} + 153q^{38} + 4q^{40} - 58q^{41} + 390q^{44} - 96q^{46} - 16q^{49} + 237q^{50} + 22q^{52} - 136q^{53} - 270q^{56} + 52q^{58} - 2q^{61} - 564q^{62} + 2q^{64} - 146q^{65} - 331q^{68} + 102q^{70} + 16q^{73} + 404q^{74} + 93q^{76} + 246q^{77} + 848q^{80} + 202q^{82} - 52q^{85} + 447q^{86} + 75q^{88} + 392q^{89} - 426q^{92} + 48q^{94} - 62q^{97} - 1022q^{98} + O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(108, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
108.3.f.a \(2\) \(2.943\) \(\Q(\sqrt{-3}) \) None \(-4\) \(0\) \(4\) \(-6\) \(q-2q^{2}+4q^{4}+(4-4\zeta_{6})q^{5}+(-4+\cdots)q^{7}+\cdots\)
108.3.f.b \(2\) \(2.943\) \(\Q(\sqrt{-3}) \) None \(2\) \(0\) \(4\) \(6\) \(q+(2-2\zeta_{6})q^{2}-4\zeta_{6}q^{4}+(4-4\zeta_{6})q^{5}+\cdots\)
108.3.f.c \(16\) \(2.943\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(3\) \(0\) \(-6\) \(0\) \(q+\beta _{1}q^{2}+(-\beta _{2}+\beta _{3})q^{4}+(-\beta _{2}-\beta _{6}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(108, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(108, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 + 2 T )^{2} \))(\( 1 - 2 T + 4 T^{2} \))(\( 1 - 3 T + 7 T^{2} - 30 T^{3} + 76 T^{4} - 144 T^{5} + 424 T^{6} - 912 T^{7} + 1552 T^{8} - 3648 T^{9} + 6784 T^{10} - 9216 T^{11} + 19456 T^{12} - 30720 T^{13} + 28672 T^{14} - 49152 T^{15} + 65536 T^{16} \))
$3$ 1
$5$ (\( 1 - 4 T - 9 T^{2} - 100 T^{3} + 625 T^{4} \))(\( 1 - 4 T - 9 T^{2} - 100 T^{3} + 625 T^{4} \))(\( ( 1 + 3 T - 38 T^{2} + 201 T^{3} + 1495 T^{4} - 6984 T^{5} + 24112 T^{6} + 181380 T^{7} - 866084 T^{8} + 4534500 T^{9} + 15070000 T^{10} - 109125000 T^{11} + 583984375 T^{12} + 1962890625 T^{13} - 9277343750 T^{14} + 18310546875 T^{15} + 152587890625 T^{16} )^{2} \))
$7$ (\( 1 + 6 T + 61 T^{2} + 294 T^{3} + 2401 T^{4} \))(\( 1 - 6 T + 61 T^{2} - 294 T^{3} + 2401 T^{4} \))(\( 1 + 167 T^{2} + 13188 T^{4} + 535927 T^{6} + 4595825 T^{8} - 715412784 T^{10} - 48284089874 T^{12} - 2073288609886 T^{14} - 88693520972328 T^{16} - 4977965952336286 T^{18} - 278348169589725074 T^{20} - 9902233810610977584 T^{22} + \)\(15\!\cdots\!25\)\( T^{24} + \)\(42\!\cdots\!27\)\( T^{26} + \)\(25\!\cdots\!88\)\( T^{28} + \)\(76\!\cdots\!67\)\( T^{30} + \)\(11\!\cdots\!01\)\( T^{32} \))
$11$ (\( 1 - 21 T + 268 T^{2} - 2541 T^{3} + 14641 T^{4} \))(\( 1 + 21 T + 268 T^{2} + 2541 T^{3} + 14641 T^{4} \))(\( 1 + 524 T^{2} + 140094 T^{4} + 24486904 T^{6} + 2972860745 T^{8} + 215973352008 T^{10} - 2384433396482 T^{12} - 3310557806176204 T^{14} - 541778612591523276 T^{16} - 48469876840225802764 T^{18} - \)\(51\!\cdots\!42\)\( T^{20} + \)\(67\!\cdots\!68\)\( T^{22} + \)\(13\!\cdots\!45\)\( T^{24} + \)\(16\!\cdots\!04\)\( T^{26} + \)\(13\!\cdots\!54\)\( T^{28} + \)\(75\!\cdots\!44\)\( T^{30} + \)\(21\!\cdots\!21\)\( T^{32} \))
$13$ (\( ( 1 - 23 T + 169 T^{2} )( 1 + T + 169 T^{2} ) \))(\( ( 1 - 23 T + 169 T^{2} )( 1 + T + 169 T^{2} ) \))(\( ( 1 + 23 T - 276 T^{2} - 5009 T^{3} + 162641 T^{4} + 1572720 T^{5} - 34351730 T^{6} - 33339262 T^{7} + 8566506504 T^{8} - 5634335278 T^{9} - 981119760530 T^{10} + 7591219050480 T^{11} + 132671260194161 T^{12} - 690533185671641 T^{13} - 6430271493804756 T^{14} + 90559656871083647 T^{15} + 665416609183179841 T^{16} )^{2} \))
$17$ (\( ( 1 - 11 T + 289 T^{2} )^{2} \))(\( ( 1 - 11 T + 289 T^{2} )^{2} \))(\( ( 1 + 3 T + 578 T^{2} + 6549 T^{3} + 169242 T^{4} + 1892661 T^{5} + 48275138 T^{6} + 72412707 T^{7} + 6975757441 T^{8} )^{4} \))
$19$ (\( 1 - 479 T^{2} + 130321 T^{4} \))(\( 1 - 479 T^{2} + 130321 T^{4} \))(\( ( 1 - 1673 T^{2} + 1559890 T^{4} - 937716023 T^{6} + 401371470970 T^{8} - 122204089833383 T^{10} + 26492490152025490 T^{12} - 3702875859597687353 T^{14} + \)\(28\!\cdots\!81\)\( T^{16} )^{2} \))
$23$ (\( 1 + 42 T + 1117 T^{2} + 22218 T^{3} + 279841 T^{4} \))(\( 1 - 42 T + 1117 T^{2} - 22218 T^{3} + 279841 T^{4} \))(\( 1 + 2687 T^{2} + 3719652 T^{4} + 3461634655 T^{6} + 2442367009985 T^{8} + 1429403163693456 T^{10} + 759315200173466974 T^{12} + \)\(39\!\cdots\!18\)\( T^{14} + \)\(20\!\cdots\!24\)\( T^{16} + \)\(11\!\cdots\!38\)\( T^{18} + \)\(59\!\cdots\!94\)\( T^{20} + \)\(31\!\cdots\!76\)\( T^{22} + \)\(14\!\cdots\!85\)\( T^{24} + \)\(59\!\cdots\!55\)\( T^{26} + \)\(17\!\cdots\!32\)\( T^{28} + \)\(36\!\cdots\!47\)\( T^{30} + \)\(37\!\cdots\!21\)\( T^{32} \))
$29$ (\( 1 - 34 T + 315 T^{2} - 28594 T^{3} + 707281 T^{4} \))(\( 1 - 34 T + 315 T^{2} - 28594 T^{3} + 707281 T^{4} \))(\( ( 1 + 21 T - 2432 T^{2} - 19167 T^{3} + 4062037 T^{4} + 6623136 T^{5} - 4703569190 T^{6} - 3469324686 T^{7} + 4129311885376 T^{8} - 2917702060926 T^{9} - 3326745120272390 T^{10} + 3939595750954656 T^{11} + 2032019438564861557 T^{12} - 8063695540664952567 T^{13} - \)\(86\!\cdots\!12\)\( T^{14} + \)\(62\!\cdots\!01\)\( T^{15} + \)\(25\!\cdots\!21\)\( T^{16} )^{2} \))
$31$ (\( 1 - 12 T + 1009 T^{2} - 11532 T^{3} + 923521 T^{4} \))(\( 1 + 12 T + 1009 T^{2} + 11532 T^{3} + 923521 T^{4} \))(\( 1 + 4715 T^{2} + 10729584 T^{4} + 17585852527 T^{6} + 25170831884477 T^{8} + 32193274685973408 T^{10} + 37136398859226646834 T^{12} + \)\(40\!\cdots\!98\)\( T^{14} + \)\(41\!\cdots\!28\)\( T^{16} + \)\(37\!\cdots\!58\)\( T^{18} + \)\(31\!\cdots\!94\)\( T^{20} + \)\(25\!\cdots\!88\)\( T^{22} + \)\(18\!\cdots\!37\)\( T^{24} + \)\(11\!\cdots\!27\)\( T^{26} + \)\(66\!\cdots\!64\)\( T^{28} + \)\(27\!\cdots\!15\)\( T^{30} + \)\(52\!\cdots\!61\)\( T^{32} \))
$37$ (\( ( 1 + 16 T + 1369 T^{2} )^{2} \))(\( ( 1 + 16 T + 1369 T^{2} )^{2} \))(\( ( 1 - 14 T + 3040 T^{2} - 66050 T^{3} + 4581118 T^{4} - 90422450 T^{5} + 5697449440 T^{6} - 35920169726 T^{7} + 3512479453921 T^{8} )^{4} \))
$41$ (\( 1 - 13 T - 1512 T^{2} - 21853 T^{3} + 2825761 T^{4} \))(\( 1 - 13 T - 1512 T^{2} - 21853 T^{3} + 2825761 T^{4} \))(\( ( 1 + 42 T - 4424 T^{2} - 125040 T^{3} + 14943835 T^{4} + 232367040 T^{5} - 36025798940 T^{6} - 132403432026 T^{7} + 71285616374608 T^{8} - 222570169235706 T^{9} - 101800297638493340 T^{10} + 1103767662172616640 T^{11} + \)\(11\!\cdots\!35\)\( T^{12} - \)\(16\!\cdots\!40\)\( T^{13} - \)\(99\!\cdots\!44\)\( T^{14} + \)\(15\!\cdots\!62\)\( T^{15} + \)\(63\!\cdots\!41\)\( T^{16} )^{2} \))
$43$ (\( 1 + 87 T + 4372 T^{2} + 160863 T^{3} + 3418801 T^{4} \))(\( 1 - 87 T + 4372 T^{2} - 160863 T^{3} + 3418801 T^{4} \))(\( 1 + 10292 T^{2} + 52819302 T^{4} + 202099368136 T^{6} + 665872758097265 T^{8} + 1877680374529631208 T^{10} + \)\(45\!\cdots\!90\)\( T^{12} + \)\(10\!\cdots\!64\)\( T^{14} + \)\(19\!\cdots\!28\)\( T^{16} + \)\(34\!\cdots\!64\)\( T^{18} + \)\(53\!\cdots\!90\)\( T^{20} + \)\(75\!\cdots\!08\)\( T^{22} + \)\(90\!\cdots\!65\)\( T^{24} + \)\(94\!\cdots\!36\)\( T^{26} + \)\(84\!\cdots\!02\)\( T^{28} + \)\(56\!\cdots\!92\)\( T^{30} + \)\(18\!\cdots\!01\)\( T^{32} \))
$47$ (\( 1 + 6 T + 2221 T^{2} + 13254 T^{3} + 4879681 T^{4} \))(\( 1 - 6 T + 2221 T^{2} - 13254 T^{3} + 4879681 T^{4} \))(\( 1 + 12983 T^{2} + 90223140 T^{4} + 428939892679 T^{6} + 1551988218895697 T^{8} + 4556649670813575888 T^{10} + \)\(11\!\cdots\!54\)\( T^{12} + \)\(26\!\cdots\!06\)\( T^{14} + \)\(58\!\cdots\!04\)\( T^{16} + \)\(12\!\cdots\!86\)\( T^{18} + \)\(27\!\cdots\!94\)\( T^{20} + \)\(52\!\cdots\!08\)\( T^{22} + \)\(87\!\cdots\!37\)\( T^{24} + \)\(11\!\cdots\!79\)\( T^{26} + \)\(12\!\cdots\!40\)\( T^{28} + \)\(85\!\cdots\!63\)\( T^{30} + \)\(32\!\cdots\!41\)\( T^{32} \))
$53$ (\( ( 1 + 52 T + 2809 T^{2} )^{2} \))(\( ( 1 + 52 T + 2809 T^{2} )^{2} \))(\( ( 1 - 18 T + 10016 T^{2} - 126558 T^{3} + 40472766 T^{4} - 355501422 T^{5} + 79031057696 T^{6} - 398958500322 T^{7} + 62259690411361 T^{8} )^{4} \))
$59$ (\( 1 - 93 T + 6364 T^{2} - 323733 T^{3} + 12117361 T^{4} \))(\( 1 + 93 T + 6364 T^{2} + 323733 T^{3} + 12117361 T^{4} \))(\( 1 + 18092 T^{2} + 175903662 T^{4} + 1133035156984 T^{6} + 5240167912552121 T^{8} + 17175194950853605128 T^{10} + \)\(35\!\cdots\!18\)\( T^{12} + \)\(21\!\cdots\!16\)\( T^{14} - \)\(83\!\cdots\!68\)\( T^{16} + \)\(26\!\cdots\!76\)\( T^{18} + \)\(52\!\cdots\!78\)\( T^{20} + \)\(30\!\cdots\!68\)\( T^{22} + \)\(11\!\cdots\!61\)\( T^{24} + \)\(29\!\cdots\!84\)\( T^{26} + \)\(55\!\cdots\!82\)\( T^{28} + \)\(69\!\cdots\!32\)\( T^{30} + \)\(46\!\cdots\!81\)\( T^{32} \))
$61$ (\( 1 - 16 T - 3465 T^{2} - 59536 T^{3} + 13845841 T^{4} \))(\( 1 - 16 T - 3465 T^{2} - 59536 T^{3} + 13845841 T^{4} \))(\( ( 1 + 17 T - 3582 T^{2} - 125549 T^{3} - 18305593 T^{4} - 350178696 T^{5} - 12273736208 T^{6} + 2060229775052 T^{7} + 599123836260396 T^{8} + 7666114992968492 T^{9} - 169940200011910928 T^{10} - 18041337511166813256 T^{11} - \)\(35\!\cdots\!33\)\( T^{12} - \)\(89\!\cdots\!49\)\( T^{13} - \)\(95\!\cdots\!22\)\( T^{14} + \)\(16\!\cdots\!97\)\( T^{15} + \)\(36\!\cdots\!61\)\( T^{16} )^{2} \))
$67$ (\( ( 1 - 67 T )^{2}( 1 - 67 T + 4489 T^{2} ) \))(\( ( 1 + 67 T )^{2}( 1 + 67 T + 4489 T^{2} ) \))(\( 1 + 25148 T^{2} + 330341646 T^{4} + 3017899283800 T^{6} + 21559886998912025 T^{8} + \)\(12\!\cdots\!60\)\( T^{10} + \)\(66\!\cdots\!50\)\( T^{12} + \)\(31\!\cdots\!76\)\( T^{14} + \)\(14\!\cdots\!68\)\( T^{16} + \)\(63\!\cdots\!96\)\( T^{18} + \)\(27\!\cdots\!50\)\( T^{20} + \)\(10\!\cdots\!60\)\( T^{22} + \)\(35\!\cdots\!25\)\( T^{24} + \)\(10\!\cdots\!00\)\( T^{26} + \)\(22\!\cdots\!66\)\( T^{28} + \)\(33\!\cdots\!68\)\( T^{30} + \)\(27\!\cdots\!61\)\( T^{32} \))
$71$ (\( ( 1 - 71 T )^{2}( 1 + 71 T )^{2} \))(\( ( 1 - 71 T )^{2}( 1 + 71 T )^{2} \))(\( ( 1 - 12968 T^{2} + 137492380 T^{4} - 912038324888 T^{6} + 5454839368725190 T^{8} - 23176426971828216728 T^{10} + \)\(88\!\cdots\!80\)\( T^{12} - \)\(21\!\cdots\!88\)\( T^{14} + \)\(41\!\cdots\!21\)\( T^{16} )^{2} \))
$73$ (\( ( 1 + 25 T + 5329 T^{2} )^{2} \))(\( ( 1 + 25 T + 5329 T^{2} )^{2} \))(\( ( 1 - 29 T + 7774 T^{2} - 620747 T^{3} + 46170850 T^{4} - 3307960763 T^{5} + 220767925534 T^{6} - 4388692562381 T^{7} + 806460091894081 T^{8} )^{4} \))
$79$ (\( 1 - 48 T + 7009 T^{2} - 299568 T^{3} + 38950081 T^{4} \))(\( 1 + 48 T + 7009 T^{2} + 299568 T^{3} + 38950081 T^{4} \))(\( 1 + 23147 T^{2} + 328058736 T^{4} + 3136165224559 T^{6} + 21315067551811709 T^{8} + 91269506056145418912 T^{10} + \)\(59\!\cdots\!06\)\( T^{12} - \)\(27\!\cdots\!10\)\( T^{14} - \)\(25\!\cdots\!44\)\( T^{16} - \)\(10\!\cdots\!10\)\( T^{18} + \)\(90\!\cdots\!66\)\( T^{20} + \)\(53\!\cdots\!92\)\( T^{22} + \)\(49\!\cdots\!89\)\( T^{24} + \)\(28\!\cdots\!59\)\( T^{26} + \)\(11\!\cdots\!16\)\( T^{28} + \)\(31\!\cdots\!67\)\( T^{30} + \)\(52\!\cdots\!41\)\( T^{32} \))
$83$ (\( 1 + 60 T + 8089 T^{2} + 413340 T^{3} + 47458321 T^{4} \))(\( 1 - 60 T + 8089 T^{2} - 413340 T^{3} + 47458321 T^{4} \))(\( 1 + 17795 T^{2} + 28452672 T^{4} - 239952438809 T^{6} + 12519030554664557 T^{8} + 90364909803409302048 T^{10} - \)\(32\!\cdots\!46\)\( T^{12} + \)\(11\!\cdots\!22\)\( T^{14} + \)\(52\!\cdots\!80\)\( T^{16} + \)\(55\!\cdots\!62\)\( T^{18} - \)\(73\!\cdots\!86\)\( T^{20} + \)\(96\!\cdots\!28\)\( T^{22} + \)\(63\!\cdots\!17\)\( T^{24} - \)\(57\!\cdots\!09\)\( T^{26} + \)\(32\!\cdots\!12\)\( T^{28} + \)\(96\!\cdots\!95\)\( T^{30} + \)\(25\!\cdots\!61\)\( T^{32} \))
$89$ (\( ( 1 - 2 T + 7921 T^{2} )^{2} \))(\( ( 1 - 2 T + 7921 T^{2} )^{2} \))(\( ( 1 - 96 T + 27716 T^{2} - 1940016 T^{3} + 308634966 T^{4} - 15366866736 T^{5} + 1738963951556 T^{6} - 47710203932256 T^{7} + 3936588805702081 T^{8} )^{4} \))
$97$ (\( 1 - 43 T - 7560 T^{2} - 404587 T^{3} + 88529281 T^{4} \))(\( 1 - 43 T - 7560 T^{2} - 404587 T^{3} + 88529281 T^{4} \))(\( ( 1 + 74 T - 29868 T^{2} - 1540328 T^{3} + 607324823 T^{4} + 20751693936 T^{5} - 8122914917000 T^{6} - 74589814314322 T^{7} + 88320904907559480 T^{8} - 701815562883455698 T^{9} - \)\(71\!\cdots\!00\)\( T^{10} + \)\(17\!\cdots\!44\)\( T^{11} + \)\(47\!\cdots\!03\)\( T^{12} - \)\(11\!\cdots\!72\)\( T^{13} - \)\(20\!\cdots\!88\)\( T^{14} + \)\(48\!\cdots\!06\)\( T^{15} + \)\(61\!\cdots\!21\)\( T^{16} )^{2} \))
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