# Properties

 Label 108.6.e Level 108 Weight 6 Character orbit e Rep. character $$\chi_{108}(37,\cdot)$$ Character field $$\Q(\zeta_{3})$$ Dimension 10 Newform subspaces 1 Sturm bound 108 Trace bound 0

# Related objects

## Defining parameters

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

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{6}(108, [\chi])$$.

Total New Old
Modular forms 198 10 188
Cusp forms 162 10 152
Eisenstein series 36 0 36

## Trace form

 $$10q + 21q^{5} + 29q^{7} + O(q^{10})$$ $$10q + 21q^{5} + 29q^{7} - 177q^{11} - 181q^{13} - 2280q^{17} - 832q^{19} - 399q^{23} - 4778q^{25} + 6033q^{29} + 2759q^{31} - 37146q^{35} - 15172q^{37} + 18435q^{41} + 1469q^{43} + 25155q^{47} - 4056q^{49} - 116844q^{53} + 14778q^{55} + 90537q^{59} + 1403q^{61} + 148407q^{65} + 13907q^{67} - 229368q^{71} + 15200q^{73} + 211983q^{77} + 29993q^{79} + 228951q^{83} - 49662q^{85} - 598332q^{89} + 124930q^{91} + 394764q^{95} + 40541q^{97} + O(q^{100})$$

## Decomposition of $$S_{6}^{\mathrm{new}}(108, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
108.6.e.a $$10$$ $$17.321$$ $$\mathbb{Q}[x]/(x^{10} + \cdots)$$ None $$0$$ $$0$$ $$21$$ $$29$$ $$q+(4-4\beta _{1}+\beta _{7})q^{5}+(6\beta _{1}+\beta _{2}+\beta _{7}+\cdots)q^{7}+\cdots$$

## Decomposition of $$S_{6}^{\mathrm{old}}(108, [\chi])$$ into lower level spaces

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 - 21 T - 5203 T^{2} + 519930 T^{3} + 14035794 T^{4} - 2854822770 T^{5} + 76722872007 T^{6} + 9761967315441 T^{7} - 599011867854189 T^{8} - 11924309583255600 T^{9} + 2533145723872694124 T^{10} - 37263467447673750000 T^{11} -$$$$58\!\cdots\!25$$$$T^{12} +$$$$29\!\cdots\!25$$$$T^{13} +$$$$73\!\cdots\!75$$$$T^{14} -$$$$85\!\cdots\!50$$$$T^{15} +$$$$13\!\cdots\!50$$$$T^{16} +$$$$15\!\cdots\!50$$$$T^{17} -$$$$47\!\cdots\!75$$$$T^{18} -$$$$59\!\cdots\!25$$$$T^{19} +$$$$88\!\cdots\!25$$$$T^{20}$$
$7$ $$1 - 29 T - 39569 T^{2} + 3762444 T^{3} + 440397336 T^{4} - 77352503496 T^{5} - 2769093584103 T^{6} - 560172784984473 T^{7} + 238615372451780007 T^{8} + 16031898530170676332 T^{9} -$$$$69\!\cdots\!60$$$$T^{10} +$$$$26\!\cdots\!24$$$$T^{11} +$$$$67\!\cdots\!43$$$$T^{12} -$$$$26\!\cdots\!39$$$$T^{13} -$$$$22\!\cdots\!03$$$$T^{14} -$$$$10\!\cdots\!72$$$$T^{15} +$$$$99\!\cdots\!64$$$$T^{16} +$$$$14\!\cdots\!92$$$$T^{17} -$$$$25\!\cdots\!69$$$$T^{18} -$$$$31\!\cdots\!03$$$$T^{19} +$$$$17\!\cdots\!49$$$$T^{20}$$
$11$ $$1 + 177 T - 396232 T^{2} + 71434269 T^{3} + 104816625882 T^{4} - 33726096455301 T^{5} - 6913987980717606 T^{6} + 8552599160812456257 T^{7} -$$$$10\!\cdots\!51$$$$T^{8} -$$$$50\!\cdots\!82$$$$T^{9} +$$$$47\!\cdots\!16$$$$T^{10} -$$$$81\!\cdots\!82$$$$T^{11} -$$$$27\!\cdots\!51$$$$T^{12} +$$$$35\!\cdots\!07$$$$T^{13} -$$$$46\!\cdots\!06$$$$T^{14} -$$$$36\!\cdots\!51$$$$T^{15} +$$$$18\!\cdots\!82$$$$T^{16} +$$$$20\!\cdots\!19$$$$T^{17} -$$$$17\!\cdots\!32$$$$T^{18} +$$$$12\!\cdots\!27$$$$T^{19} +$$$$11\!\cdots\!01$$$$T^{20}$$
$13$ $$1 + 181 T - 1012331 T^{2} + 14482182 T^{3} + 446454243174 T^{4} - 84043375137762 T^{5} - 192479505557683773 T^{6} - 802846347498861897 T^{7} +$$$$98\!\cdots\!51$$$$T^{8} +$$$$77\!\cdots\!72$$$$T^{9} -$$$$41\!\cdots\!24$$$$T^{10} +$$$$28\!\cdots\!96$$$$T^{11} +$$$$13\!\cdots\!99$$$$T^{12} -$$$$41\!\cdots\!29$$$$T^{13} -$$$$36\!\cdots\!73$$$$T^{14} -$$$$59\!\cdots\!66$$$$T^{15} +$$$$11\!\cdots\!26$$$$T^{16} +$$$$14\!\cdots\!74$$$$T^{17} -$$$$36\!\cdots\!31$$$$T^{18} +$$$$24\!\cdots\!33$$$$T^{19} +$$$$49\!\cdots\!49$$$$T^{20}$$
$17$ $$( 1 + 1140 T + 4980550 T^{2} + 3443850354 T^{3} + 10068870522169 T^{4} + 5069379208548852 T^{5} + 14296356292995309833 T^{6} +$$$$69\!\cdots\!46$$$$T^{7} +$$$$14\!\cdots\!50$$$$T^{8} +$$$$46\!\cdots\!40$$$$T^{9} +$$$$57\!\cdots\!57$$$$T^{10} )^{2}$$
$19$ $$( 1 + 416 T + 5046258 T^{2} + 6215761044 T^{3} + 20272296121125 T^{4} + 15898268281316088 T^{5} + 50196212153221491375 T^{6} +$$$$38\!\cdots\!44$$$$T^{7} +$$$$76\!\cdots\!42$$$$T^{8} +$$$$15\!\cdots\!16$$$$T^{9} +$$$$93\!\cdots\!99$$$$T^{10} )^{2}$$
$23$ $$1 + 399 T - 16077241 T^{2} + 38108825820 T^{3} + 155650662506976 T^{4} - 562944417983120520 T^{5} - 48522958516353490863 T^{6} +$$$$48\!\cdots\!51$$$$T^{7} -$$$$71\!\cdots\!41$$$$T^{8} -$$$$11\!\cdots\!52$$$$T^{9} +$$$$81\!\cdots\!80$$$$T^{10} -$$$$74\!\cdots\!36$$$$T^{11} -$$$$29\!\cdots\!09$$$$T^{12} +$$$$12\!\cdots\!57$$$$T^{13} -$$$$83\!\cdots\!63$$$$T^{14} -$$$$62\!\cdots\!60$$$$T^{15} +$$$$11\!\cdots\!24$$$$T^{16} +$$$$17\!\cdots\!40$$$$T^{17} -$$$$47\!\cdots\!41$$$$T^{18} +$$$$75\!\cdots\!57$$$$T^{19} +$$$$12\!\cdots\!49$$$$T^{20}$$
$29$ $$1 - 6033 T + 3652157 T^{2} - 31641196734 T^{3} + 283528398607854 T^{4} - 668469168127712358 T^{5} +$$$$64\!\cdots\!39$$$$T^{6} -$$$$82\!\cdots\!55$$$$T^{7} -$$$$90\!\cdots\!17$$$$T^{8} -$$$$14\!\cdots\!60$$$$T^{9} +$$$$58\!\cdots\!16$$$$T^{10} -$$$$29\!\cdots\!40$$$$T^{11} -$$$$38\!\cdots\!17$$$$T^{12} -$$$$71\!\cdots\!95$$$$T^{13} +$$$$11\!\cdots\!39$$$$T^{14} -$$$$24\!\cdots\!42$$$$T^{15} +$$$$21\!\cdots\!54$$$$T^{16} -$$$$48\!\cdots\!66$$$$T^{17} +$$$$11\!\cdots\!57$$$$T^{18} -$$$$38\!\cdots\!17$$$$T^{19} +$$$$13\!\cdots\!01$$$$T^{20}$$
$31$ $$1 - 2759 T - 54902477 T^{2} - 189444651072 T^{3} + 2052158291804100 T^{4} + 13274031992302596720 T^{5} -$$$$32\!\cdots\!47$$$$T^{6} -$$$$42\!\cdots\!39$$$$T^{7} -$$$$13\!\cdots\!49$$$$T^{8} +$$$$36\!\cdots\!48$$$$T^{9} +$$$$63\!\cdots\!24$$$$T^{10} +$$$$10\!\cdots\!48$$$$T^{11} -$$$$11\!\cdots\!49$$$$T^{12} -$$$$99\!\cdots\!89$$$$T^{13} -$$$$21\!\cdots\!47$$$$T^{14} +$$$$25\!\cdots\!20$$$$T^{15} +$$$$11\!\cdots\!00$$$$T^{16} -$$$$29\!\cdots\!72$$$$T^{17} -$$$$24\!\cdots\!77$$$$T^{18} -$$$$35\!\cdots\!09$$$$T^{19} +$$$$36\!\cdots\!01$$$$T^{20}$$
$37$ $$( 1 + 7586 T + 201201093 T^{2} + 803146672896 T^{3} + 19241810738464926 T^{4} + 60351714230064941916 T^{5} +$$$$13\!\cdots\!82$$$$T^{6} +$$$$38\!\cdots\!04$$$$T^{7} +$$$$67\!\cdots\!49$$$$T^{8} +$$$$17\!\cdots\!86$$$$T^{9} +$$$$16\!\cdots\!57$$$$T^{10} )^{2}$$
$41$ $$1 - 18435 T - 117679042 T^{2} + 4344492069675 T^{3} - 505249106564622 T^{4} -$$$$52\!\cdots\!97$$$$T^{5} +$$$$16\!\cdots\!40$$$$T^{6} +$$$$39\!\cdots\!43$$$$T^{7} -$$$$31\!\cdots\!03$$$$T^{8} -$$$$10\!\cdots\!42$$$$T^{9} +$$$$29\!\cdots\!40$$$$T^{10} -$$$$11\!\cdots\!42$$$$T^{11} -$$$$41\!\cdots\!03$$$$T^{12} +$$$$62\!\cdots\!43$$$$T^{13} +$$$$30\!\cdots\!40$$$$T^{14} -$$$$10\!\cdots\!97$$$$T^{15} -$$$$12\!\cdots\!22$$$$T^{16} +$$$$12\!\cdots\!75$$$$T^{17} -$$$$38\!\cdots\!42$$$$T^{18} -$$$$69\!\cdots\!35$$$$T^{19} +$$$$43\!\cdots\!01$$$$T^{20}$$
$43$ $$1 - 1469 T - 271863536 T^{2} + 4016430594327 T^{3} + 12129147672135834 T^{4} -$$$$75\!\cdots\!27$$$$T^{5} +$$$$55\!\cdots\!62$$$$T^{6} -$$$$12\!\cdots\!57$$$$T^{7} -$$$$29\!\cdots\!39$$$$T^{8} +$$$$61\!\cdots\!62$$$$T^{9} -$$$$11\!\cdots\!84$$$$T^{10} +$$$$90\!\cdots\!66$$$$T^{11} -$$$$64\!\cdots\!11$$$$T^{12} -$$$$38\!\cdots\!99$$$$T^{13} +$$$$26\!\cdots\!62$$$$T^{14} -$$$$52\!\cdots\!61$$$$T^{15} +$$$$12\!\cdots\!66$$$$T^{16} +$$$$59\!\cdots\!89$$$$T^{17} -$$$$59\!\cdots\!36$$$$T^{18} -$$$$47\!\cdots\!67$$$$T^{19} +$$$$47\!\cdots\!49$$$$T^{20}$$
$47$ $$1 - 25155 T - 401246233 T^{2} + 14349179861244 T^{3} + 97557609874842960 T^{4} -$$$$41\!\cdots\!12$$$$T^{5} -$$$$25\!\cdots\!27$$$$T^{6} +$$$$55\!\cdots\!85$$$$T^{7} +$$$$12\!\cdots\!11$$$$T^{8} -$$$$42\!\cdots\!56$$$$T^{9} -$$$$37\!\cdots\!60$$$$T^{10} -$$$$96\!\cdots\!92$$$$T^{11} +$$$$67\!\cdots\!39$$$$T^{12} +$$$$67\!\cdots\!55$$$$T^{13} -$$$$71\!\cdots\!27$$$$T^{14} -$$$$26\!\cdots\!84$$$$T^{15} +$$$$14\!\cdots\!40$$$$T^{16} +$$$$47\!\cdots\!92$$$$T^{17} -$$$$30\!\cdots\!33$$$$T^{18} -$$$$44\!\cdots\!85$$$$T^{19} +$$$$40\!\cdots\!49$$$$T^{20}$$
$53$ $$( 1 + 58422 T + 3354568213 T^{2} + 110313236959296 T^{3} + 3390725554692289246 T^{4} +$$$$71\!\cdots\!28$$$$T^{5} +$$$$14\!\cdots\!78$$$$T^{6} +$$$$19\!\cdots\!04$$$$T^{7} +$$$$24\!\cdots\!41$$$$T^{8} +$$$$17\!\cdots\!22$$$$T^{9} +$$$$12\!\cdots\!93$$$$T^{10} )^{2}$$
$59$ $$1 - 90537 T + 2831117840 T^{2} - 13805150996349 T^{3} - 966660594685472478 T^{4} +$$$$10\!\cdots\!09$$$$T^{5} +$$$$69\!\cdots\!78$$$$T^{6} -$$$$39\!\cdots\!93$$$$T^{7} +$$$$84\!\cdots\!01$$$$T^{8} +$$$$17\!\cdots\!74$$$$T^{9} -$$$$12\!\cdots\!20$$$$T^{10} +$$$$12\!\cdots\!26$$$$T^{11} +$$$$42\!\cdots\!01$$$$T^{12} -$$$$14\!\cdots\!07$$$$T^{13} +$$$$18\!\cdots\!78$$$$T^{14} +$$$$19\!\cdots\!91$$$$T^{15} -$$$$12\!\cdots\!78$$$$T^{16} -$$$$13\!\cdots\!51$$$$T^{17} +$$$$19\!\cdots\!40$$$$T^{18} -$$$$44\!\cdots\!63$$$$T^{19} +$$$$34\!\cdots\!01$$$$T^{20}$$
$61$ $$1 - 1403 T - 3536905883 T^{2} - 452840008146 T^{3} + 7065863261737144698 T^{4} +$$$$54\!\cdots\!90$$$$T^{5} -$$$$10\!\cdots\!33$$$$T^{6} -$$$$70\!\cdots\!89$$$$T^{7} +$$$$11\!\cdots\!67$$$$T^{8} +$$$$32\!\cdots\!04$$$$T^{9} -$$$$10\!\cdots\!12$$$$T^{10} +$$$$27\!\cdots\!04$$$$T^{11} +$$$$80\!\cdots\!67$$$$T^{12} -$$$$42\!\cdots\!89$$$$T^{13} -$$$$51\!\cdots\!33$$$$T^{14} +$$$$23\!\cdots\!90$$$$T^{15} +$$$$25\!\cdots\!98$$$$T^{16} -$$$$13\!\cdots\!46$$$$T^{17} -$$$$91\!\cdots\!83$$$$T^{18} -$$$$30\!\cdots\!03$$$$T^{19} +$$$$18\!\cdots\!01$$$$T^{20}$$
$67$ $$1 - 13907 T - 3876685544 T^{2} - 77425491657903 T^{3} + 10014688417385231130 T^{4} +$$$$30\!\cdots\!39$$$$T^{5} -$$$$79\!\cdots\!54$$$$T^{6} -$$$$69\!\cdots\!51$$$$T^{7} -$$$$33\!\cdots\!67$$$$T^{8} +$$$$37\!\cdots\!46$$$$T^{9} +$$$$22\!\cdots\!76$$$$T^{10} +$$$$51\!\cdots\!22$$$$T^{11} -$$$$60\!\cdots\!83$$$$T^{12} -$$$$16\!\cdots\!93$$$$T^{13} -$$$$26\!\cdots\!54$$$$T^{14} +$$$$13\!\cdots\!73$$$$T^{15} +$$$$60\!\cdots\!70$$$$T^{16} -$$$$63\!\cdots\!29$$$$T^{17} -$$$$42\!\cdots\!44$$$$T^{18} -$$$$20\!\cdots\!49$$$$T^{19} +$$$$20\!\cdots\!49$$$$T^{20}$$
$71$ $$( 1 + 114684 T + 7758380659 T^{2} + 426246123888336 T^{3} + 19260501229393543450 T^{4} +$$$$77\!\cdots\!40$$$$T^{5} +$$$$34\!\cdots\!50$$$$T^{6} +$$$$13\!\cdots\!36$$$$T^{7} +$$$$45\!\cdots\!09$$$$T^{8} +$$$$12\!\cdots\!84$$$$T^{9} +$$$$19\!\cdots\!51$$$$T^{10} )^{2}$$
$73$ $$( 1 - 7600 T + 3606834246 T^{2} - 31056473559714 T^{3} + 12288417972789256281 T^{4} -$$$$80\!\cdots\!84$$$$T^{5} +$$$$25\!\cdots\!33$$$$T^{6} -$$$$13\!\cdots\!86$$$$T^{7} +$$$$32\!\cdots\!22$$$$T^{8} -$$$$14\!\cdots\!00$$$$T^{9} +$$$$38\!\cdots\!93$$$$T^{10} )^{2}$$
$79$ $$1 - 29993 T - 5352351629 T^{2} - 358913063028768 T^{3} + 26234825811851125236 T^{4} +$$$$21\!\cdots\!52$$$$T^{5} +$$$$27\!\cdots\!85$$$$T^{6} -$$$$84\!\cdots\!45$$$$T^{7} -$$$$35\!\cdots\!45$$$$T^{8} +$$$$81\!\cdots\!80$$$$T^{9} +$$$$16\!\cdots\!00$$$$T^{10} +$$$$25\!\cdots\!20$$$$T^{11} -$$$$33\!\cdots\!45$$$$T^{12} -$$$$24\!\cdots\!55$$$$T^{13} +$$$$24\!\cdots\!85$$$$T^{14} +$$$$60\!\cdots\!48$$$$T^{15} +$$$$22\!\cdots\!36$$$$T^{16} -$$$$93\!\cdots\!32$$$$T^{17} -$$$$43\!\cdots\!29$$$$T^{18} -$$$$74\!\cdots\!07$$$$T^{19} +$$$$76\!\cdots\!01$$$$T^{20}$$
$83$ $$1 - 228951 T + 21403431983 T^{2} - 1202282302650156 T^{3} + 62567029919071222368 T^{4} -$$$$36\!\cdots\!68$$$$T^{5} +$$$$11\!\cdots\!01$$$$T^{6} +$$$$11\!\cdots\!41$$$$T^{7} -$$$$18\!\cdots\!73$$$$T^{8} +$$$$14\!\cdots\!84$$$$T^{9} -$$$$88\!\cdots\!72$$$$T^{10} +$$$$55\!\cdots\!12$$$$T^{11} -$$$$28\!\cdots\!77$$$$T^{12} +$$$$67\!\cdots\!87$$$$T^{13} +$$$$28\!\cdots\!01$$$$T^{14} -$$$$34\!\cdots\!24$$$$T^{15} +$$$$23\!\cdots\!32$$$$T^{16} -$$$$17\!\cdots\!92$$$$T^{17} +$$$$12\!\cdots\!83$$$$T^{18} -$$$$52\!\cdots\!93$$$$T^{19} +$$$$89\!\cdots\!49$$$$T^{20}$$
$89$ $$( 1 + 299166 T + 52616244181 T^{2} + 6660261403977288 T^{3} +$$$$67\!\cdots\!10$$$$T^{4} +$$$$55\!\cdots\!64$$$$T^{5} +$$$$37\!\cdots\!90$$$$T^{6} +$$$$20\!\cdots\!88$$$$T^{7} +$$$$91\!\cdots\!69$$$$T^{8} +$$$$29\!\cdots\!66$$$$T^{9} +$$$$54\!\cdots\!49$$$$T^{10} )^{2}$$
$97$ $$1 - 40541 T - 17893496138 T^{2} + 2263333692661293 T^{3} + 99710157551726941410 T^{4} -$$$$30\!\cdots\!95$$$$T^{5} +$$$$10\!\cdots\!20$$$$T^{6} +$$$$21\!\cdots\!29$$$$T^{7} -$$$$20\!\cdots\!15$$$$T^{8} -$$$$65\!\cdots\!06$$$$T^{9} +$$$$19\!\cdots\!00$$$$T^{10} -$$$$56\!\cdots\!42$$$$T^{11} -$$$$15\!\cdots\!35$$$$T^{12} +$$$$13\!\cdots\!97$$$$T^{13} +$$$$56\!\cdots\!20$$$$T^{14} -$$$$14\!\cdots\!15$$$$T^{15} +$$$$39\!\cdots\!90$$$$T^{16} +$$$$77\!\cdots\!49$$$$T^{17} -$$$$52\!\cdots\!38$$$$T^{18} -$$$$10\!\cdots\!37$$$$T^{19} +$$$$21\!\cdots\!49$$$$T^{20}$$