# 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

# Related objects

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