# Properties

 Label 108.5.d Level 108 Weight 5 Character orbit d Rep. character $$\chi_{108}(55,\cdot)$$ Character field $$\Q$$ Dimension 32 Newform subspaces 2 Sturm bound 90 Trace bound 4

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## Defining parameters

 Level: $$N$$ $$=$$ $$108 = 2^{2} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 108.d (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$4$$ Character field: $$\Q$$ Newform subspaces: $$2$$ Sturm bound: $$90$$ Trace bound: $$4$$ Distinguishing $$T_p$$: $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{5}(108, [\chi])$$.

Total New Old
Modular forms 78 32 46
Cusp forms 66 32 34
Eisenstein series 12 0 12

## Trace form

 $$32q + 14q^{4} + O(q^{10})$$ $$32q + 14q^{4} - 26q^{10} - 176q^{13} - 118q^{16} + 1122q^{22} + 4368q^{25} + 2154q^{28} - 1016q^{34} + 3280q^{37} - 3542q^{40} - 1788q^{46} - 8800q^{49} - 2948q^{52} - 9092q^{58} - 4400q^{61} + 5738q^{64} + 12522q^{70} + 8320q^{73} - 41472q^{76} - 38396q^{82} - 8032q^{85} + 18774q^{88} + 48084q^{94} + 7936q^{97} + O(q^{100})$$

## Decomposition of $$S_{5}^{\mathrm{new}}(108, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
108.5.d.a $$16$$ $$11.164$$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}+(-1-\beta _{2})q^{4}+(-\beta _{1}-\beta _{7}+\cdots)q^{5}+\cdots$$
108.5.d.b $$16$$ $$11.164$$ $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{7}q^{2}+(2+\beta _{2})q^{4}+(-\beta _{7}+\beta _{9}+\cdots)q^{5}+\cdots$$

## Decomposition of $$S_{5}^{\mathrm{old}}(108, [\chi])$$ into lower level spaces

$$S_{5}^{\mathrm{old}}(108, [\chi]) \cong$$ $$S_{5}^{\mathrm{new}}(4, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{5}^{\mathrm{new}}(12, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{5}^{\mathrm{new}}(36, [\chi])$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ ($$1 + 7 T^{2} + 76 T^{4} + 304 T^{6} + 94720 T^{8} + 77824 T^{10} + 4980736 T^{12} + 117440512 T^{14} + 4294967296 T^{16}$$)($$1 - 14 T^{2} + 76 T^{4} - 992 T^{6} - 2816 T^{8} - 253952 T^{10} + 4980736 T^{12} - 234881024 T^{14} + 4294967296 T^{16}$$)
$3$ 1
$5$ ($$( 1 + 2092 T^{2} + 2149522 T^{4} + 1622274208 T^{6} + 1066877108443 T^{8} + 633700862500000 T^{10} + 327991027832031250 T^{12} +$$$$12\!\cdots\!00$$$$T^{14} +$$$$23\!\cdots\!25$$$$T^{16} )^{2}$$)($$( 1 + 1816 T^{2} + 1373884 T^{4} + 428961832 T^{6} + 50788147846 T^{8} + 167563215625000 T^{10} + 209638061523437500 T^{12} +$$$$10\!\cdots\!00$$$$T^{14} +$$$$23\!\cdots\!25$$$$T^{16} )^{2}$$)
$7$ ($$( 1 - 9380 T^{2} + 36105250 T^{4} - 73168453568 T^{6} + 129372615567787 T^{8} - 421801574297259968 T^{10} +$$$$11\!\cdots\!50$$$$T^{12} -$$$$17\!\cdots\!80$$$$T^{14} +$$$$11\!\cdots\!01$$$$T^{16} )^{2}$$)($$( 1 - 7628 T^{2} + 28548058 T^{4} - 90794922320 T^{6} + 251781425444899 T^{8} - 523414658985258320 T^{10} +$$$$94\!\cdots\!58$$$$T^{12} -$$$$14\!\cdots\!28$$$$T^{14} +$$$$11\!\cdots\!01$$$$T^{16} )^{2}$$)
$11$ ($$( 1 - 48044 T^{2} + 1167571642 T^{4} - 20151290866448 T^{6} + 304446182867375107 T^{8} -$$$$43\!\cdots\!88$$$$T^{10} +$$$$53\!\cdots\!62$$$$T^{12} -$$$$47\!\cdots\!04$$$$T^{14} +$$$$21\!\cdots\!21$$$$T^{16} )^{2}$$)($$( 1 - 66488 T^{2} + 2449218172 T^{4} - 58905964133768 T^{6} + 1015821535494509830 T^{8} -$$$$12\!\cdots\!08$$$$T^{10} +$$$$11\!\cdots\!92$$$$T^{12} -$$$$65\!\cdots\!08$$$$T^{14} +$$$$21\!\cdots\!21$$$$T^{16} )^{2}$$)
$13$ ($$( 1 + 88 T + 63508 T^{2} + 5077384 T^{3} + 2650180198 T^{4} + 145015164424 T^{5} + 51805426629268 T^{6} + 2050231490778328 T^{7} + 665416609183179841 T^{8} )^{4}$$)($$( 1 - 44 T + 67090 T^{2} - 966368 T^{3} + 2192145307 T^{4} - 27600436448 T^{5} + 54727374071890 T^{6} - 1025115745389164 T^{7} + 665416609183179841 T^{8} )^{4}$$)
$17$ ($$( 1 + 239464 T^{2} + 41438060380 T^{4} + 5059782297489112 T^{6} +$$$$47\!\cdots\!02$$$$T^{8} +$$$$35\!\cdots\!92$$$$T^{10} +$$$$20\!\cdots\!80$$$$T^{12} +$$$$81\!\cdots\!44$$$$T^{14} +$$$$23\!\cdots\!61$$$$T^{16} )^{2}$$)($$( 1 + 422296 T^{2} + 87803746876 T^{4} + 11972376766816936 T^{6} +$$$$11\!\cdots\!30$$$$T^{8} +$$$$83\!\cdots\!76$$$$T^{10} +$$$$42\!\cdots\!56$$$$T^{12} +$$$$14\!\cdots\!16$$$$T^{14} +$$$$23\!\cdots\!61$$$$T^{16} )^{2}$$)
$19$ ($$( 1 - 680120 T^{2} + 233557156252 T^{4} - 51394020863660552 T^{6} +$$$$79\!\cdots\!98$$$$T^{8} -$$$$87\!\cdots\!32$$$$T^{10} +$$$$67\!\cdots\!12$$$$T^{12} -$$$$33\!\cdots\!20$$$$T^{14} +$$$$83\!\cdots\!61$$$$T^{16} )^{2}$$)($$( 1 - 298556 T^{2} + 58331731402 T^{4} - 10017587005489136 T^{6} +$$$$14\!\cdots\!39$$$$T^{8} -$$$$17\!\cdots\!76$$$$T^{10} +$$$$16\!\cdots\!62$$$$T^{12} -$$$$14\!\cdots\!76$$$$T^{14} +$$$$83\!\cdots\!61$$$$T^{16} )^{2}$$)
$23$ ($$( 1 - 675608 T^{2} + 314533671388 T^{4} - 115630955446337192 T^{6} +$$$$35\!\cdots\!18$$$$T^{8} -$$$$90\!\cdots\!52$$$$T^{10} +$$$$19\!\cdots\!68$$$$T^{12} -$$$$32\!\cdots\!28$$$$T^{14} +$$$$37\!\cdots\!21$$$$T^{16} )^{2}$$)($$( 1 - 1184696 T^{2} + 509745640444 T^{4} - 84604026939000968 T^{6} +$$$$71\!\cdots\!22$$$$T^{8} -$$$$66\!\cdots\!08$$$$T^{10} +$$$$31\!\cdots\!84$$$$T^{12} -$$$$56\!\cdots\!36$$$$T^{14} +$$$$37\!\cdots\!21$$$$T^{16} )^{2}$$)
$29$ ($$( 1 + 2998744 T^{2} + 4863922457692 T^{4} + 5416371986440228072 T^{6} +$$$$44\!\cdots\!14$$$$T^{8} +$$$$27\!\cdots\!92$$$$T^{10} +$$$$12\!\cdots\!32$$$$T^{12} +$$$$37\!\cdots\!64$$$$T^{14} +$$$$62\!\cdots\!41$$$$T^{16} )^{2}$$)($$( 1 + 1223560 T^{2} + 603752953756 T^{4} - 126384872678073416 T^{6} -$$$$34\!\cdots\!30$$$$T^{8} -$$$$63\!\cdots\!76$$$$T^{10} +$$$$15\!\cdots\!76$$$$T^{12} +$$$$15\!\cdots\!60$$$$T^{14} +$$$$62\!\cdots\!41$$$$T^{16} )^{2}$$)
$31$ ($$( 1 - 1328756 T^{2} + 2683287954034 T^{4} - 2424947225563311392 T^{6} +$$$$29\!\cdots\!59$$$$T^{8} -$$$$20\!\cdots\!72$$$$T^{10} +$$$$19\!\cdots\!54$$$$T^{12} -$$$$82\!\cdots\!76$$$$T^{14} +$$$$52\!\cdots\!61$$$$T^{16} )^{2}$$)($$( 1 - 4495112 T^{2} + 10589743228444 T^{4} - 16346503421024387384 T^{6} +$$$$17\!\cdots\!66$$$$T^{8} -$$$$13\!\cdots\!44$$$$T^{10} +$$$$77\!\cdots\!64$$$$T^{12} -$$$$27\!\cdots\!52$$$$T^{14} +$$$$52\!\cdots\!61$$$$T^{16} )^{2}$$)
$37$ ($$( 1 - 800 T + 1852732 T^{2} - 3784975328 T^{3} + 1716048721222 T^{4} - 7093653145699808 T^{5} + 6507683083621962172 T^{6} -$$$$52\!\cdots\!00$$$$T^{7} +$$$$12\!\cdots\!41$$$$T^{8} )^{4}$$)($$( 1 - 20 T + 5679442 T^{2} - 534101024 T^{3} + 14604995159131 T^{4} - 1000991309240864 T^{5} + 19948923334735992082 T^{6} -$$$$13\!\cdots\!20$$$$T^{7} +$$$$12\!\cdots\!41$$$$T^{8} )^{4}$$)
$41$ ($$( 1 + 14045080 T^{2} + 100482710328412 T^{4} +$$$$46\!\cdots\!84$$$$T^{6} +$$$$15\!\cdots\!54$$$$T^{8} +$$$$37\!\cdots\!64$$$$T^{10} +$$$$64\!\cdots\!92$$$$T^{12} +$$$$71\!\cdots\!80$$$$T^{14} +$$$$40\!\cdots\!81$$$$T^{16} )^{2}$$)($$( 1 + 8498440 T^{2} + 39457953574684 T^{4} +$$$$14\!\cdots\!68$$$$T^{6} +$$$$47\!\cdots\!42$$$$T^{8} +$$$$11\!\cdots\!28$$$$T^{10} +$$$$25\!\cdots\!44$$$$T^{12} +$$$$43\!\cdots\!40$$$$T^{14} +$$$$40\!\cdots\!81$$$$T^{16} )^{2}$$)
$43$ ($$( 1 - 11990648 T^{2} + 69115070774812 T^{4} -$$$$29\!\cdots\!24$$$$T^{6} +$$$$10\!\cdots\!18$$$$T^{8} -$$$$34\!\cdots\!24$$$$T^{10} +$$$$94\!\cdots\!12$$$$T^{12} -$$$$19\!\cdots\!48$$$$T^{14} +$$$$18\!\cdots\!01$$$$T^{16} )^{2}$$)($$( 1 - 16979720 T^{2} + 140301823474588 T^{4} -$$$$77\!\cdots\!28$$$$T^{6} +$$$$30\!\cdots\!14$$$$T^{8} -$$$$90\!\cdots\!28$$$$T^{10} +$$$$19\!\cdots\!88$$$$T^{12} -$$$$27\!\cdots\!20$$$$T^{14} +$$$$18\!\cdots\!01$$$$T^{16} )^{2}$$)
$47$ ($$( 1 - 27701336 T^{2} + 374201000701276 T^{4} -$$$$31\!\cdots\!48$$$$T^{6} +$$$$18\!\cdots\!14$$$$T^{8} -$$$$75\!\cdots\!28$$$$T^{10} +$$$$21\!\cdots\!96$$$$T^{12} -$$$$37\!\cdots\!16$$$$T^{14} +$$$$32\!\cdots\!41$$$$T^{16} )^{2}$$)($$( 1 - 11004728 T^{2} + 98757031754236 T^{4} -$$$$67\!\cdots\!20$$$$T^{6} +$$$$36\!\cdots\!06$$$$T^{8} -$$$$16\!\cdots\!20$$$$T^{10} +$$$$55\!\cdots\!56$$$$T^{12} -$$$$14\!\cdots\!68$$$$T^{14} +$$$$32\!\cdots\!41$$$$T^{16} )^{2}$$)
$53$ ($$( 1 + 35837884 T^{2} + 582188420839618 T^{4} +$$$$60\!\cdots\!16$$$$T^{6} +$$$$50\!\cdots\!79$$$$T^{8} +$$$$37\!\cdots\!76$$$$T^{10} +$$$$22\!\cdots\!78$$$$T^{12} +$$$$86\!\cdots\!04$$$$T^{14} +$$$$15\!\cdots\!41$$$$T^{16} )^{2}$$)($$( 1 + 25978312 T^{2} + 440128249758748 T^{4} +$$$$51\!\cdots\!52$$$$T^{6} +$$$$46\!\cdots\!86$$$$T^{8} +$$$$32\!\cdots\!72$$$$T^{10} +$$$$17\!\cdots\!08$$$$T^{12} +$$$$62\!\cdots\!72$$$$T^{14} +$$$$15\!\cdots\!41$$$$T^{16} )^{2}$$)
$59$ ($$( 1 - 70312136 T^{2} + 2419955263154716 T^{4} -$$$$51\!\cdots\!68$$$$T^{6} +$$$$75\!\cdots\!74$$$$T^{8} -$$$$76\!\cdots\!28$$$$T^{10} +$$$$52\!\cdots\!56$$$$T^{12} -$$$$22\!\cdots\!96$$$$T^{14} +$$$$46\!\cdots\!81$$$$T^{16} )^{2}$$)($$( 1 - 31826744 T^{2} + 538590143907196 T^{4} -$$$$92\!\cdots\!92$$$$T^{6} +$$$$13\!\cdots\!74$$$$T^{8} -$$$$13\!\cdots\!32$$$$T^{10} +$$$$11\!\cdots\!36$$$$T^{12} -$$$$10\!\cdots\!84$$$$T^{14} +$$$$46\!\cdots\!81$$$$T^{16} )^{2}$$)
$61$ ($$( 1 + 688 T + 35342980 T^{2} + 28994874640 T^{3} + 632810106882118 T^{4} + 401458424080372240 T^{5} +$$$$67\!\cdots\!80$$$$T^{6} +$$$$18\!\cdots\!48$$$$T^{7} +$$$$36\!\cdots\!61$$$$T^{8} )^{4}$$)($$( 1 + 412 T + 9607138 T^{2} + 3267599488 T^{3} + 216368249231659 T^{4} + 45242662962529408 T^{5} +$$$$18\!\cdots\!78$$$$T^{6} +$$$$10\!\cdots\!52$$$$T^{7} +$$$$36\!\cdots\!61$$$$T^{8} )^{4}$$)
$67$ ($$( 1 - 29579000 T^{2} + 1274223737389084 T^{4} -$$$$30\!\cdots\!28$$$$T^{6} +$$$$73\!\cdots\!58$$$$T^{8} -$$$$12\!\cdots\!48$$$$T^{10} +$$$$21\!\cdots\!04$$$$T^{12} -$$$$19\!\cdots\!00$$$$T^{14} +$$$$27\!\cdots\!61$$$$T^{16} )^{2}$$)($$( 1 - 38213180 T^{2} + 1702311028453834 T^{4} -$$$$38\!\cdots\!28$$$$T^{6} +$$$$99\!\cdots\!27$$$$T^{8} -$$$$15\!\cdots\!48$$$$T^{10} +$$$$28\!\cdots\!54$$$$T^{12} -$$$$25\!\cdots\!80$$$$T^{14} +$$$$27\!\cdots\!61$$$$T^{16} )^{2}$$)
$71$ ($$( 1 - 86649128 T^{2} + 3567527319446236 T^{4} -$$$$11\!\cdots\!36$$$$T^{6} +$$$$32\!\cdots\!90$$$$T^{8} -$$$$74\!\cdots\!96$$$$T^{10} +$$$$14\!\cdots\!56$$$$T^{12} -$$$$23\!\cdots\!68$$$$T^{14} +$$$$17\!\cdots\!41$$$$T^{16} )^{2}$$)($$( 1 - 102214088 T^{2} + 3903098982427036 T^{4} -$$$$62\!\cdots\!96$$$$T^{6} +$$$$63\!\cdots\!58$$$$T^{8} -$$$$40\!\cdots\!56$$$$T^{10} +$$$$16\!\cdots\!56$$$$T^{12} -$$$$27\!\cdots\!28$$$$T^{14} +$$$$17\!\cdots\!41$$$$T^{16} )^{2}$$)
$73$ ($$( 1 - 2060 T + 55108594 T^{2} - 101224244576 T^{3} + 2192760100160827 T^{4} - 2874590492512190816 T^{5} +$$$$44\!\cdots\!14$$$$T^{6} -$$$$47\!\cdots\!60$$$$T^{7} +$$$$65\!\cdots\!61$$$$T^{8} )^{4}$$)($$( 1 - 20 T + 67167226 T^{2} + 662878096 T^{3} + 2724353301452611 T^{4} + 18824571923829136 T^{5} +$$$$54\!\cdots\!06$$$$T^{6} -$$$$45\!\cdots\!20$$$$T^{7} +$$$$65\!\cdots\!61$$$$T^{8} )^{4}$$)
$79$ ($$( 1 - 162885800 T^{2} + 12409200314583772 T^{4} -$$$$63\!\cdots\!36$$$$T^{6} +$$$$26\!\cdots\!06$$$$T^{8} -$$$$96\!\cdots\!96$$$$T^{10} +$$$$28\!\cdots\!12$$$$T^{12} -$$$$56\!\cdots\!00$$$$T^{14} +$$$$52\!\cdots\!41$$$$T^{16} )^{2}$$)($$( 1 - 164533964 T^{2} + 15212503116962266 T^{4} -$$$$95\!\cdots\!24$$$$T^{6} +$$$$43\!\cdots\!71$$$$T^{8} -$$$$14\!\cdots\!64$$$$T^{10} +$$$$35\!\cdots\!86$$$$T^{12} -$$$$57\!\cdots\!84$$$$T^{14} +$$$$52\!\cdots\!41$$$$T^{16} )^{2}$$)
$83$ ($$( 1 - 140155436 T^{2} + 8287556309932090 T^{4} -$$$$42\!\cdots\!92$$$$T^{6} +$$$$22\!\cdots\!03$$$$T^{8} -$$$$95\!\cdots\!72$$$$T^{10} +$$$$42\!\cdots\!90$$$$T^{12} -$$$$16\!\cdots\!56$$$$T^{14} +$$$$25\!\cdots\!61$$$$T^{16} )^{2}$$)($$( 1 - 185955656 T^{2} + 18958921421058076 T^{4} -$$$$13\!\cdots\!16$$$$T^{6} +$$$$73\!\cdots\!26$$$$T^{8} -$$$$30\!\cdots\!56$$$$T^{10} +$$$$96\!\cdots\!56$$$$T^{12} -$$$$21\!\cdots\!76$$$$T^{14} +$$$$25\!\cdots\!61$$$$T^{16} )^{2}$$)
$89$ ($$( 1 + 257253400 T^{2} + 38619499284608860 T^{4} +$$$$38\!\cdots\!44$$$$T^{6} +$$$$28\!\cdots\!58$$$$T^{8} +$$$$15\!\cdots\!64$$$$T^{10} +$$$$59\!\cdots\!60$$$$T^{12} +$$$$15\!\cdots\!00$$$$T^{14} +$$$$24\!\cdots\!21$$$$T^{16} )^{2}$$)($$( 1 + 103821208 T^{2} + 12140769019811644 T^{4} +$$$$81\!\cdots\!28$$$$T^{6} +$$$$66\!\cdots\!06$$$$T^{8} +$$$$32\!\cdots\!68$$$$T^{10} +$$$$18\!\cdots\!84$$$$T^{12} +$$$$63\!\cdots\!28$$$$T^{14} +$$$$24\!\cdots\!21$$$$T^{16} )^{2}$$)
$97$ ($$( 1 + 1732 T + 297359818 T^{2} + 428249450896 T^{3} + 37002869299122643 T^{4} + 37912615976467685776 T^{5} +$$$$23\!\cdots\!98$$$$T^{6} +$$$$12\!\cdots\!12$$$$T^{7} +$$$$61\!\cdots\!21$$$$T^{8} )^{4}$$)($$( 1 - 3716 T + 259176202 T^{2} - 799969063952 T^{3} + 30273937496959507 T^{4} - 70820686053913578512 T^{5} +$$$$20\!\cdots\!22$$$$T^{6} -$$$$25\!\cdots\!56$$$$T^{7} +$$$$61\!\cdots\!21$$$$T^{8} )^{4}$$)
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