# Properties

 Label 108.5.g Level 108 Weight 5 Character orbit g Rep. character $$\chi_{108}(17,\cdot)$$ Character field $$\Q(\zeta_{6})$$ Dimension 8 Newform subspaces 1 Sturm bound 90 Trace bound 0

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$108 = 2^{2} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 108.g (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$9$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$1$$ Sturm bound: $$90$$ Trace bound: $$0$$

## Dimensions

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

Total New Old
Modular forms 162 8 154
Cusp forms 126 8 118
Eisenstein series 36 0 36

## Trace form

 $$8q + 9q^{5} + 13q^{7} + O(q^{10})$$ $$8q + 9q^{5} + 13q^{7} + 18q^{11} - 5q^{13} + 562q^{19} + 1719q^{23} + 353q^{25} - 2115q^{29} + 187q^{31} + 16q^{37} + 7920q^{41} - 68q^{43} - 13689q^{47} - 327q^{49} - 1818q^{55} + 20052q^{59} - 1937q^{61} - 25965q^{65} + 154q^{67} - 7802q^{73} + 25641q^{77} - 2195q^{79} - 37017q^{83} - 3042q^{85} + 15830q^{91} + 37116q^{95} + 7282q^{97} + O(q^{100})$$

## 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.g.a $$8$$ $$11.164$$ $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ None $$0$$ $$0$$ $$9$$ $$13$$ $$q+(2+\beta _{2}+\beta _{4}+\beta _{7})q^{5}+(3-\beta _{1}+3\beta _{2}+\cdots)q^{7}+\cdots$$

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

$$S_{5}^{\mathrm{old}}(108, [\chi]) \cong$$ $$S_{5}^{\mathrm{new}}(9, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{5}^{\mathrm{new}}(18, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{5}^{\mathrm{new}}(27, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{5}^{\mathrm{new}}(36, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{5}^{\mathrm{new}}(54, [\chi])$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 - 9 T + 1114 T^{2} - 9783 T^{3} + 533599 T^{4} - 10528056 T^{5} + 59806456 T^{6} - 10305069192 T^{7} - 6001445444 T^{8} - 6440668245000 T^{9} + 23361896875000 T^{10} - 2570326171875000 T^{11} + 81420745849609375 T^{12} - 932979583740234375 T^{13} + 66399574279785156250 T^{14} -$$$$33\!\cdots\!25$$$$T^{15} +$$$$23\!\cdots\!25$$$$T^{16}$$
$7$ $$1 - 13 T - 4554 T^{2} + 124753 T^{3} + 8962391 T^{4} - 347580324 T^{5} + 2901735784 T^{6} + 447642835484 T^{7} - 30706182623268 T^{8} + 1074790447997084 T^{9} + 16727929349338984 T^{10} - 4810959089900633124 T^{11} +$$$$29\!\cdots\!91$$$$T^{12} +$$$$99\!\cdots\!53$$$$T^{13} -$$$$87\!\cdots\!54$$$$T^{14} -$$$$59\!\cdots\!13$$$$T^{15} +$$$$11\!\cdots\!01$$$$T^{16}$$
$11$ $$1 - 18 T + 33850 T^{2} - 607356 T^{3} + 452208721 T^{4} - 17443950048 T^{5} + 9074477968558 T^{6} - 445095012639474 T^{7} + 191968264536523468 T^{8} - 6516636080054538834 T^{9} +$$$$19\!\cdots\!98$$$$T^{10} -$$$$54\!\cdots\!08$$$$T^{11} +$$$$20\!\cdots\!81$$$$T^{12} -$$$$40\!\cdots\!56$$$$T^{13} +$$$$33\!\cdots\!50$$$$T^{14} -$$$$25\!\cdots\!58$$$$T^{15} +$$$$21\!\cdots\!21$$$$T^{16}$$
$13$ $$1 + 5 T - 42054 T^{2} - 7266665 T^{3} + 352176575 T^{4} + 250092029040 T^{5} + 23191943061904 T^{6} - 3464858176098460 T^{7} - 548440340475170196 T^{8} - 98959814367548116060 T^{9} +$$$$18\!\cdots\!84$$$$T^{10} +$$$$58\!\cdots\!40$$$$T^{11} +$$$$23\!\cdots\!75$$$$T^{12} -$$$$13\!\cdots\!65$$$$T^{13} -$$$$22\!\cdots\!94$$$$T^{14} +$$$$77\!\cdots\!05$$$$T^{15} +$$$$44\!\cdots\!81$$$$T^{16}$$
$17$ $$1 - 288125 T^{2} + 52320681154 T^{4} - 6759733382202755 T^{6} +$$$$63\!\cdots\!86$$$$T^{8} -$$$$47\!\cdots\!55$$$$T^{10} +$$$$25\!\cdots\!74$$$$T^{12} -$$$$97\!\cdots\!25$$$$T^{14} +$$$$23\!\cdots\!61$$$$T^{16}$$
$19$ $$( 1 - 281 T + 110170 T^{2} + 68843041 T^{3} - 21846246566 T^{4} + 8971693946161 T^{5} + 1871079140226970 T^{6} - 621941492257591241 T^{7} +$$$$28\!\cdots\!81$$$$T^{8} )^{2}$$
$23$ $$1 - 1719 T + 1529458 T^{2} - 935945649 T^{3} + 342642958747 T^{4} + 16333135609500 T^{5} - 118516645434237428 T^{6} +$$$$10\!\cdots\!68$$$$T^{7} -$$$$65\!\cdots\!00$$$$T^{8} +$$$$29\!\cdots\!88$$$$T^{9} -$$$$92\!\cdots\!68$$$$T^{10} +$$$$35\!\cdots\!00$$$$T^{11} +$$$$21\!\cdots\!67$$$$T^{12} -$$$$16\!\cdots\!49$$$$T^{13} +$$$$73\!\cdots\!78$$$$T^{14} -$$$$23\!\cdots\!39$$$$T^{15} +$$$$37\!\cdots\!21$$$$T^{16}$$
$29$ $$1 + 2115 T + 4091014 T^{2} + 5498870985 T^{3} + 7048695081595 T^{4} + 8024737206821040 T^{5} + 8297140249856169556 T^{6} +$$$$79\!\cdots\!80$$$$T^{7} +$$$$68\!\cdots\!24$$$$T^{8} +$$$$56\!\cdots\!80$$$$T^{9} +$$$$41\!\cdots\!16$$$$T^{10} +$$$$28\!\cdots\!40$$$$T^{11} +$$$$17\!\cdots\!95$$$$T^{12} +$$$$97\!\cdots\!85$$$$T^{13} +$$$$51\!\cdots\!34$$$$T^{14} +$$$$18\!\cdots\!15$$$$T^{15} +$$$$62\!\cdots\!41$$$$T^{16}$$
$31$ $$1 - 187 T - 2516004 T^{2} + 186847537 T^{3} + 3362431719041 T^{4} - 296059350096 T^{5} - 3460282108678916846 T^{6} - 13487262715981377034 T^{7} +$$$$31\!\cdots\!92$$$$T^{8} -$$$$12\!\cdots\!14$$$$T^{9} -$$$$29\!\cdots\!86$$$$T^{10} -$$$$23\!\cdots\!56$$$$T^{11} +$$$$24\!\cdots\!21$$$$T^{12} +$$$$12\!\cdots\!37$$$$T^{13} -$$$$15\!\cdots\!84$$$$T^{14} -$$$$10\!\cdots\!67$$$$T^{15} +$$$$52\!\cdots\!61$$$$T^{16}$$
$37$ $$( 1 - 8 T + 3611368 T^{2} + 1256575624 T^{3} + 6911619203950 T^{4} + 2355025028051464 T^{5} + 12684855900547773928 T^{6} - 52663616046720282248 T^{7} +$$$$12\!\cdots\!41$$$$T^{8} )^{2}$$
$41$ $$1 - 7920 T + 37687894 T^{2} - 132890424480 T^{3} + 385083705354505 T^{4} - 963185727644706960 T^{5} +$$$$21\!\cdots\!66$$$$T^{6} -$$$$41\!\cdots\!20$$$$T^{7} +$$$$74\!\cdots\!64$$$$T^{8} -$$$$11\!\cdots\!20$$$$T^{9} +$$$$16\!\cdots\!86$$$$T^{10} -$$$$21\!\cdots\!60$$$$T^{11} +$$$$24\!\cdots\!05$$$$T^{12} -$$$$23\!\cdots\!80$$$$T^{13} +$$$$19\!\cdots\!34$$$$T^{14} -$$$$11\!\cdots\!20$$$$T^{15} +$$$$40\!\cdots\!81$$$$T^{16}$$
$43$ $$1 + 68 T - 12950604 T^{2} - 209786648 T^{3} + 102574547445791 T^{4} + 59145034173804 T^{5} -$$$$54\!\cdots\!36$$$$T^{6} +$$$$27\!\cdots\!16$$$$T^{7} +$$$$21\!\cdots\!12$$$$T^{8} +$$$$94\!\cdots\!16$$$$T^{9} -$$$$63\!\cdots\!36$$$$T^{10} +$$$$23\!\cdots\!04$$$$T^{11} +$$$$14\!\cdots\!91$$$$T^{12} -$$$$97\!\cdots\!48$$$$T^{13} -$$$$20\!\cdots\!04$$$$T^{14} +$$$$37\!\cdots\!68$$$$T^{15} +$$$$18\!\cdots\!01$$$$T^{16}$$
$47$ $$1 + 13689 T + 103685338 T^{2} + 564293857959 T^{3} + 2435028217967227 T^{4} + 8736748337842042500 T^{5} +$$$$26\!\cdots\!72$$$$T^{6} +$$$$71\!\cdots\!32$$$$T^{7} +$$$$16\!\cdots\!20$$$$T^{8} +$$$$35\!\cdots\!92$$$$T^{9} +$$$$63\!\cdots\!92$$$$T^{10} +$$$$10\!\cdots\!00$$$$T^{11} +$$$$13\!\cdots\!67$$$$T^{12} +$$$$15\!\cdots\!59$$$$T^{13} +$$$$13\!\cdots\!78$$$$T^{14} +$$$$90\!\cdots\!29$$$$T^{15} +$$$$32\!\cdots\!41$$$$T^{16}$$
$53$ $$1 - 5145920 T^{2} + 115452291970684 T^{4} - 84051566001475463360 T^{6} +$$$$67\!\cdots\!26$$$$T^{8} -$$$$52\!\cdots\!60$$$$T^{10} +$$$$44\!\cdots\!64$$$$T^{12} -$$$$12\!\cdots\!20$$$$T^{14} +$$$$15\!\cdots\!41$$$$T^{16}$$
$59$ $$1 - 20052 T + 216711700 T^{2} - 1657982214864 T^{3} + 9931594296358591 T^{4} - 49579528565018409012 T^{5} +$$$$21\!\cdots\!48$$$$T^{6} -$$$$84\!\cdots\!76$$$$T^{7} +$$$$30\!\cdots\!08$$$$T^{8} -$$$$10\!\cdots\!36$$$$T^{9} +$$$$31\!\cdots\!08$$$$T^{10} -$$$$88\!\cdots\!72$$$$T^{11} +$$$$21\!\cdots\!31$$$$T^{12} -$$$$43\!\cdots\!64$$$$T^{13} +$$$$68\!\cdots\!00$$$$T^{14} -$$$$76\!\cdots\!92$$$$T^{15} +$$$$46\!\cdots\!81$$$$T^{16}$$
$61$ $$1 + 1937 T - 10529634 T^{2} + 149647181023 T^{3} + 416288373490931 T^{4} - 1350680282380662864 T^{5} +$$$$12\!\cdots\!24$$$$T^{6} +$$$$40\!\cdots\!44$$$$T^{7} -$$$$98\!\cdots\!68$$$$T^{8} +$$$$55\!\cdots\!04$$$$T^{9} +$$$$23\!\cdots\!44$$$$T^{10} -$$$$35\!\cdots\!44$$$$T^{11} +$$$$15\!\cdots\!91$$$$T^{12} +$$$$76\!\cdots\!23$$$$T^{13} -$$$$74\!\cdots\!94$$$$T^{14} +$$$$18\!\cdots\!97$$$$T^{15} +$$$$13\!\cdots\!21$$$$T^{16}$$
$67$ $$1 - 154 T - 33835854 T^{2} - 25606229228 T^{3} + 539365905411977 T^{4} + 738160924156362336 T^{5} +$$$$70\!\cdots\!02$$$$T^{6} -$$$$11\!\cdots\!78$$$$T^{7} -$$$$26\!\cdots\!64$$$$T^{8} -$$$$22\!\cdots\!38$$$$T^{9} +$$$$28\!\cdots\!82$$$$T^{10} +$$$$60\!\cdots\!96$$$$T^{11} +$$$$88\!\cdots\!37$$$$T^{12} -$$$$85\!\cdots\!28$$$$T^{13} -$$$$22\!\cdots\!34$$$$T^{14} -$$$$20\!\cdots\!14$$$$T^{15} +$$$$27\!\cdots\!61$$$$T^{16}$$
$71$ $$1 - 68871716 T^{2} + 3244147638477940 T^{4} -$$$$11\!\cdots\!24$$$$T^{6} +$$$$30\!\cdots\!74$$$$T^{8} -$$$$73\!\cdots\!64$$$$T^{10} +$$$$13\!\cdots\!40$$$$T^{12} -$$$$18\!\cdots\!96$$$$T^{14} +$$$$17\!\cdots\!41$$$$T^{16}$$
$73$ $$( 1 + 3901 T + 59309470 T^{2} + 292589317519 T^{3} + 2279602007321194 T^{4} + 8309021952930084079 T^{5} +$$$$47\!\cdots\!70$$$$T^{6} +$$$$89\!\cdots\!21$$$$T^{7} +$$$$65\!\cdots\!61$$$$T^{8} )^{2}$$
$79$ $$1 + 2195 T - 87724914 T^{2} - 187644610415 T^{3} + 3128319215246375 T^{4} + 3711455091635884260 T^{5} -$$$$16\!\cdots\!36$$$$T^{6} +$$$$76\!\cdots\!80$$$$T^{7} +$$$$88\!\cdots\!64$$$$T^{8} +$$$$29\!\cdots\!80$$$$T^{9} -$$$$24\!\cdots\!96$$$$T^{10} +$$$$21\!\cdots\!60$$$$T^{11} +$$$$72\!\cdots\!75$$$$T^{12} -$$$$16\!\cdots\!15$$$$T^{13} -$$$$30\!\cdots\!34$$$$T^{14} +$$$$29\!\cdots\!95$$$$T^{15} +$$$$52\!\cdots\!41$$$$T^{16}$$
$83$ $$1 + 37017 T + 725723290 T^{2} + 9956481997959 T^{3} + 104510585134438411 T^{4} +$$$$87\!\cdots\!72$$$$T^{5} +$$$$62\!\cdots\!28$$$$T^{6} +$$$$40\!\cdots\!76$$$$T^{7} +$$$$26\!\cdots\!48$$$$T^{8} +$$$$19\!\cdots\!96$$$$T^{9} +$$$$14\!\cdots\!48$$$$T^{10} +$$$$93\!\cdots\!92$$$$T^{11} +$$$$53\!\cdots\!91$$$$T^{12} +$$$$23\!\cdots\!59$$$$T^{13} +$$$$82\!\cdots\!90$$$$T^{14} +$$$$20\!\cdots\!97$$$$T^{15} +$$$$25\!\cdots\!61$$$$T^{16}$$
$89$ $$1 - 294759296 T^{2} + 46567064448316540 T^{4} -$$$$48\!\cdots\!04$$$$T^{6} +$$$$35\!\cdots\!14$$$$T^{8} -$$$$19\!\cdots\!24$$$$T^{10} +$$$$72\!\cdots\!40$$$$T^{12} -$$$$17\!\cdots\!36$$$$T^{14} +$$$$24\!\cdots\!21$$$$T^{16}$$
$97$ $$1 - 7282 T - 283226964 T^{2} + 993163976152 T^{3} + 57034963146137471 T^{4} - 97460573991801682656 T^{5} -$$$$75\!\cdots\!16$$$$T^{6} +$$$$33\!\cdots\!26$$$$T^{7} +$$$$75\!\cdots\!52$$$$T^{8} +$$$$29\!\cdots\!06$$$$T^{9} -$$$$58\!\cdots\!76$$$$T^{10} -$$$$67\!\cdots\!96$$$$T^{11} +$$$$35\!\cdots\!91$$$$T^{12} +$$$$54\!\cdots\!52$$$$T^{13} -$$$$13\!\cdots\!84$$$$T^{14} -$$$$31\!\cdots\!02$$$$T^{15} +$$$$37\!\cdots\!41$$$$T^{16}$$