# Properties

 Label 20.5 Level 20 Weight 5 Dimension 22 Nonzero newspaces 3 Newform subspaces 5 Sturm bound 120 Trace bound 1

## Defining parameters

 Level: $$N$$ = $$20 = 2^{2} \cdot 5$$ Weight: $$k$$ = $$5$$ Nonzero newspaces: $$3$$ Newform subspaces: $$5$$ Sturm bound: $$120$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{5}(\Gamma_1(20))$$.

Total New Old
Modular forms 58 26 32
Cusp forms 38 22 16
Eisenstein series 20 4 16

## Trace form

 $$22q + 6q^{2} - 10q^{3} - 36q^{4} + 4q^{5} + 32q^{6} + 110q^{7} + 216q^{8} - 170q^{9} + O(q^{10})$$ $$22q + 6q^{2} - 10q^{3} - 36q^{4} + 4q^{5} + 32q^{6} + 110q^{7} + 216q^{8} - 170q^{9} - 210q^{10} - 300q^{11} - 200q^{12} - 8q^{13} - 184q^{14} + 542q^{15} - 592q^{16} + 912q^{17} + 286q^{18} + 100q^{20} - 2612q^{21} + 800q^{22} - 810q^{23} + 3216q^{24} + 2066q^{25} + 324q^{26} + 2120q^{27} + 40q^{28} + 1508q^{29} - 920q^{30} - 836q^{31} - 2304q^{32} - 2780q^{33} - 4308q^{34} - 2562q^{35} - 9220q^{36} - 6388q^{37} - 3360q^{38} + 4200q^{40} + 10280q^{41} + 12120q^{42} + 3270q^{43} + 9840q^{44} + 3344q^{45} + 14792q^{46} - 2250q^{47} + 8640q^{48} - 5130q^{49} - 11730q^{50} + 1948q^{51} - 12488q^{52} + 4572q^{53} - 26768q^{54} - 6780q^{55} - 25168q^{56} - 10000q^{57} - 7428q^{58} + 11320q^{60} + 22184q^{61} + 25680q^{62} + 12950q^{63} + 33984q^{64} - 3852q^{65} + 38000q^{66} - 4810q^{67} + 2712q^{68} - 18248q^{69} - 10800q^{70} - 5988q^{71} - 36264q^{72} - 28508q^{73} - 42108q^{74} - 11282q^{75} - 36000q^{76} + 1380q^{77} - 14480q^{78} + 19120q^{80} - 1062q^{81} + 27412q^{82} + 19950q^{83} + 80672q^{84} + 36536q^{85} + 46352q^{86} + 24800q^{87} + 18080q^{88} + 60068q^{89} - 29130q^{90} - 11124q^{91} - 52680q^{92} - 22700q^{93} - 67704q^{94} - 15576q^{95} - 78208q^{96} - 7548q^{97} - 21474q^{98} + O(q^{100})$$

## Decomposition of $$S_{5}^{\mathrm{new}}(\Gamma_1(20))$$

We only show spaces with odd parity, since no modular forms exist when this condition is not satisfied. Within each space $$S_k^{\mathrm{new}}(N, \chi)$$ we list the newforms together with their dimension.

Label $$\chi$$ Newforms Dimension $$\chi$$ degree
20.5.b $$\chi_{20}(11, \cdot)$$ 20.5.b.a 8 1
20.5.d $$\chi_{20}(19, \cdot)$$ 20.5.d.a 1 1
20.5.d.b 1
20.5.d.c 8
20.5.f $$\chi_{20}(13, \cdot)$$ 20.5.f.a 4 2

## Decomposition of $$S_{5}^{\mathrm{old}}(\Gamma_1(20))$$ into lower level spaces

$$S_{5}^{\mathrm{old}}(\Gamma_1(20)) \cong$$ $$S_{5}^{\mathrm{new}}(\Gamma_1(4))$$$$^{\oplus 2}$$$$\oplus$$$$S_{5}^{\mathrm{new}}(\Gamma_1(5))$$$$^{\oplus 3}$$$$\oplus$$$$S_{5}^{\mathrm{new}}(\Gamma_1(10))$$$$^{\oplus 2}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$1 - 6 T + 28 T^{2} - 168 T^{3} + 784 T^{4} - 2688 T^{5} + 7168 T^{6} - 24576 T^{7} + 65536 T^{8}$$)($$1 + 4 T$$)($$1 - 4 T$$)($$1 + 24 T^{2} + 496 T^{4} + 6144 T^{6} + 65536 T^{8}$$)()
$3$ ($$1 - 160 T^{2} + 19676 T^{4} - 2014560 T^{6} + 180027846 T^{8} - 13217528160 T^{10} + 846987282396 T^{12} - 45188725836960 T^{14} + 1853020188851841 T^{16}$$)($$1 - 2 T + 81 T^{2}$$)($$1 + 2 T + 81 T^{2}$$)($$( 1 + 84 T^{2} + 9126 T^{4} + 551124 T^{6} + 43046721 T^{8} )^{2}$$)($$1 + 10 T + 50 T^{2} - 270 T^{3} - 10206 T^{4} - 21870 T^{5} + 328050 T^{6} + 5314410 T^{7} + 43046721 T^{8}$$)
$5$ ($$( 1 - 125 T^{2} )^{4}$$)($$1 - 25 T$$)($$1 - 25 T$$)($$( 1 + 20 T + 710 T^{2} + 12500 T^{3} + 390625 T^{4} )^{2}$$)($$1 + 6 T - 910 T^{2} + 3750 T^{3} + 390625 T^{4}$$)
$7$ ($$1 - 6720 T^{2} + 31323036 T^{4} - 102209887680 T^{6} + 281929138377926 T^{8} - 589219662707551680 T^{10} +$$$$10\!\cdots\!36$$$$T^{12} -$$$$12\!\cdots\!20$$$$T^{14} +$$$$11\!\cdots\!01$$$$T^{16}$$)($$1 - 82 T + 2401 T^{2}$$)($$1 + 82 T + 2401 T^{2}$$)($$( 1 + 6804 T^{2} + 23087206 T^{4} + 39223706004 T^{6} + 33232930569601 T^{8} )^{2}$$)($$1 - 110 T + 6050 T^{2} - 311190 T^{3} + 15823298 T^{4} - 747167190 T^{5} + 34877046050 T^{6} - 1522541592110 T^{7} + 33232930569601 T^{8}$$)
$11$ ($$1 - 77928 T^{2} + 2885754588 T^{4} - 68373804388056 T^{6} + 1162395919152747590 T^{8} -$$$$14\!\cdots\!36$$$$T^{10} +$$$$13\!\cdots\!68$$$$T^{12} -$$$$76\!\cdots\!48$$$$T^{14} +$$$$21\!\cdots\!21$$$$T^{16}$$)($$( 1 - 121 T )( 1 + 121 T )$$)($$( 1 - 121 T )( 1 + 121 T )$$)($$( 1 - 10564 T^{2} + 433024326 T^{4} - 2264487218884 T^{6} + 45949729863572161 T^{8} )^{2}$$)($$( 1 + 150 T + 28882 T^{2} + 2196150 T^{3} + 214358881 T^{4} )^{2}$$)
$13$ ($$( 1 - 176 T + 77628 T^{2} - 12420688 T^{3} + 2794789574 T^{4} - 354747269968 T^{5} + 63323544409788 T^{6} - 4100462981556656 T^{7} + 665416609183179841 T^{8} )^{2}$$)($$( 1 - 169 T )( 1 + 169 T )$$)($$( 1 - 169 T )( 1 + 169 T )$$)($$( 1 - 93316 T^{2} + 3781796166 T^{4} - 76120727960836 T^{6} + 665416609183179841 T^{8} )^{2}$$)($$1 + 360 T + 64800 T^{2} + 14552280 T^{3} + 3127330814 T^{4} + 415627669080 T^{5} + 52859350720800 T^{6} + 8387310644093160 T^{7} + 665416609183179841 T^{8}$$)
$17$ ($$( 1 + 24 T + 157788 T^{2} - 20618328 T^{3} + 13093999814 T^{4} - 1722063372888 T^{5} + 1100690815100508 T^{6} + 13982933693514264 T^{7} + 48661191875666868481 T^{8} )^{2}$$)($$( 1 - 289 T )( 1 + 289 T )$$)($$( 1 - 289 T )( 1 + 289 T )$$)($$( 1 - 307716 T^{2} + 37466348806 T^{4} - 2146552176714756 T^{6} + 48661191875666868481 T^{8} )^{2}$$)($$1 - 960 T + 460800 T^{2} - 168098880 T^{3} + 52934858494 T^{4} - 14039786556480 T^{5} + 3214429028812800 T^{6} - 559317347740570560 T^{7} + 48661191875666868481 T^{8}$$)
$19$ ($$1 - 730888 T^{2} + 257781714588 T^{4} - 57630026124579896 T^{6} +$$$$89\!\cdots\!90$$$$T^{8} -$$$$97\!\cdots\!36$$$$T^{10} +$$$$74\!\cdots\!28$$$$T^{12} -$$$$35\!\cdots\!48$$$$T^{14} +$$$$83\!\cdots\!61$$$$T^{16}$$)($$( 1 - 361 T )( 1 + 361 T )$$)($$( 1 - 361 T )( 1 + 361 T )$$)($$( 1 - 73924 T^{2} - 4326450234 T^{4} - 1255492914242884 T^{6} +$$$$28\!\cdots\!81$$$$T^{8} )^{2}$$)($$1 - 79796 T^{2} + 32223922086 T^{4} - 1355220396419636 T^{6} +$$$$28\!\cdots\!81$$$$T^{8}$$)
$23$ ($$1 - 885760 T^{2} + 416439331356 T^{4} - 136723474364764160 T^{6} +$$$$39\!\cdots\!66$$$$T^{8} -$$$$10\!\cdots\!60$$$$T^{10} +$$$$25\!\cdots\!16$$$$T^{12} -$$$$42\!\cdots\!60$$$$T^{14} +$$$$37\!\cdots\!21$$$$T^{16}$$)($$1 + 878 T + 279841 T^{2}$$)($$1 - 878 T + 279841 T^{2}$$)($$( 1 + 769044 T^{2} + 304476266086 T^{4} + 60224593364441364 T^{6} +$$$$61\!\cdots\!61$$$$T^{8} )^{2}$$)($$1 + 810 T + 328050 T^{2} + 264893490 T^{3} + 211669226338 T^{4} + 74128059135090 T^{5} + 25689918721432050 T^{6} + 17750845789936460010 T^{7} +$$$$61\!\cdots\!61$$$$T^{8}$$)
$29$ ($$( 1 - 600 T + 1936636 T^{2} - 649163880 T^{3} + 1640402898886 T^{4} - 459141278210280 T^{5} + 968795212211139196 T^{6} -$$$$21\!\cdots\!00$$$$T^{7} +$$$$25\!\cdots\!21$$$$T^{8} )^{2}$$)($$1 + 1198 T + 707281 T^{2}$$)($$1 + 1198 T + 707281 T^{2}$$)($$( 1 - 676 T + 1518566 T^{2} - 478121956 T^{3} + 500246412961 T^{4} )^{4}$$)($$1 - 1448516 T^{2} + 1055067026886 T^{4} - 724614933116615876 T^{6} +$$$$25\!\cdots\!21$$$$T^{8}$$)
$31$ ($$1 - 1907688 T^{2} + 1721712272988 T^{4} - 1980674566102865496 T^{6} +$$$$24\!\cdots\!90$$$$T^{8} -$$$$16\!\cdots\!36$$$$T^{10} +$$$$12\!\cdots\!28$$$$T^{12} -$$$$11\!\cdots\!48$$$$T^{14} +$$$$52\!\cdots\!61$$$$T^{16}$$)($$( 1 - 961 T )( 1 + 961 T )$$)($$( 1 - 961 T )( 1 + 961 T )$$)($$( 1 - 2680324 T^{2} + 3246634805766 T^{4} - 2286024317038010884 T^{6} +$$$$72\!\cdots\!81$$$$T^{8} )^{2}$$)($$( 1 + 418 T + 535098 T^{2} + 386031778 T^{3} + 852891037441 T^{4} )^{2}$$)
$37$ ($$( 1 + 2864 T + 5437308 T^{2} + 5448048592 T^{3} + 7051577121734 T^{4} + 10210520197231312 T^{5} + 19098432634640284668 T^{6} +$$$$18\!\cdots\!84$$$$T^{7} +$$$$12\!\cdots\!41$$$$T^{8} )^{2}$$)($$( 1 - 1369 T )( 1 + 1369 T )$$)($$( 1 - 1369 T )( 1 + 1369 T )$$)($$( 1 - 4488196 T^{2} + 9924276565446 T^{4} - 15764696235170416516 T^{6} +$$$$12\!\cdots\!41$$$$T^{8} )^{2}$$)($$1 + 660 T + 217800 T^{2} + 345245340 T^{3} - 1278103400242 T^{4} + 647045351659740 T^{5} + 765018025063993800 T^{6} +$$$$43\!\cdots\!60$$$$T^{7} +$$$$12\!\cdots\!41$$$$T^{8}$$)
$41$ ($$( 1 - 2448 T + 9585788 T^{2} - 17538082416 T^{3} + 39790776226950 T^{4} - 49558429305918576 T^{5} + 76541800442205332348 T^{6} -$$$$55\!\cdots\!88$$$$T^{7} +$$$$63\!\cdots\!41$$$$T^{8} )^{2}$$)($$1 - 482 T + 2825761 T^{2}$$)($$1 - 482 T + 2825761 T^{2}$$)($$( 1 - 364 T + 1690406 T^{2} - 1028577004 T^{3} + 7984925229121 T^{4} )^{4}$$)($$( 1 - 1482 T + 6194578 T^{2} - 4187777802 T^{3} + 7984925229121 T^{4} )^{2}$$)
$43$ ($$1 - 18909280 T^{2} + 176096852184156 T^{4} -$$$$10\!\cdots\!20$$$$T^{6} +$$$$42\!\cdots\!86$$$$T^{8} -$$$$12\!\cdots\!20$$$$T^{10} +$$$$24\!\cdots\!56$$$$T^{12} -$$$$30\!\cdots\!80$$$$T^{14} +$$$$18\!\cdots\!01$$$$T^{16}$$)($$1 + 2078 T + 3418801 T^{2}$$)($$1 - 2078 T + 3418801 T^{2}$$)($$( 1 + 3640404 T^{2} + 1840427380006 T^{4} + 42549771043379790804 T^{6} +$$$$13\!\cdots\!01$$$$T^{8} )^{2}$$)($$1 - 3270 T + 5346450 T^{2} - 11205966270 T^{3} + 23487235317602 T^{4} - 38310968689842270 T^{5} + 62490378374179866450 T^{6} -$$$$13\!\cdots\!70$$$$T^{7} +$$$$13\!\cdots\!01$$$$T^{8}$$)
$47$ ($$1 - 19070080 T^{2} + 224159050635036 T^{4} -$$$$17\!\cdots\!40$$$$T^{6} +$$$$99\!\cdots\!26$$$$T^{8} -$$$$41\!\cdots\!40$$$$T^{10} +$$$$12\!\cdots\!56$$$$T^{12} -$$$$25\!\cdots\!80$$$$T^{14} +$$$$32\!\cdots\!41$$$$T^{16}$$)($$1 - 4402 T + 4879681 T^{2}$$)($$1 + 4402 T + 4879681 T^{2}$$)($$( 1 + 18767124 T^{2} + 135582308428006 T^{4} +$$$$44\!\cdots\!64$$$$T^{6} +$$$$56\!\cdots\!21$$$$T^{8} )^{2}$$)($$1 + 2250 T + 2531250 T^{2} + 7891319250 T^{3} + 22718088596834 T^{4} + 38507120609159250 T^{5} + 60272319362582531250 T^{6} +$$$$26\!\cdots\!50$$$$T^{7} +$$$$56\!\cdots\!21$$$$T^{8}$$)
$53$ ($$( 1 - 1296 T + 16848508 T^{2} - 32857590768 T^{3} + 141852843045574 T^{4} - 259262195660679408 T^{5} +$$$$10\!\cdots\!88$$$$T^{6} -$$$$63\!\cdots\!36$$$$T^{7} +$$$$38\!\cdots\!21$$$$T^{8} )^{2}$$)($$( 1 - 2809 T )( 1 + 2809 T )$$)($$( 1 - 2809 T )( 1 + 2809 T )$$)($$( 1 - 7023876 T^{2} + 6572518202566 T^{4} -$$$$43\!\cdots\!36$$$$T^{6} +$$$$38\!\cdots\!21$$$$T^{8} )^{2}$$)($$1 - 1980 T + 1960200 T^{2} - 16406396820 T^{3} + 137161065548878 T^{4} - 129454362386670420 T^{5} +$$$$12\!\cdots\!00$$$$T^{6} -$$$$97\!\cdots\!80$$$$T^{7} +$$$$38\!\cdots\!21$$$$T^{8}$$)
$59$ ($$1 - 57863048 T^{2} + 1797979000179868 T^{4} -$$$$36\!\cdots\!16$$$$T^{6} +$$$$52\!\cdots\!90$$$$T^{8} -$$$$53\!\cdots\!36$$$$T^{10} +$$$$38\!\cdots\!88$$$$T^{12} -$$$$18\!\cdots\!28$$$$T^{14} +$$$$46\!\cdots\!81$$$$T^{16}$$)($$( 1 - 3481 T )( 1 + 3481 T )$$)($$( 1 - 3481 T )( 1 + 3481 T )$$)($$( 1 - 18192964 T^{2} + 372799944667206 T^{4} -$$$$26\!\cdots\!44$$$$T^{6} +$$$$21\!\cdots\!41$$$$T^{8} )^{2}$$)($$1 + 162508 T^{2} + 44050763806758 T^{4} + 23861120754202997068 T^{6} +$$$$21\!\cdots\!41$$$$T^{8}$$)
$61$ ($$( 1 - 3968 T + 48790428 T^{2} - 144590839936 T^{3} + 984937331366150 T^{4} - 2001981779810306176 T^{5} +$$$$93\!\cdots\!68$$$$T^{6} -$$$$10\!\cdots\!28$$$$T^{7} +$$$$36\!\cdots\!61$$$$T^{8} )^{2}$$)($$1 + 4078 T + 13845841 T^{2}$$)($$1 + 4078 T + 13845841 T^{2}$$)($$( 1 - 1644 T + 17384326 T^{2} - 22762562604 T^{3} + 191707312997281 T^{4} )^{4}$$)($$( 1 - 7914 T + 36788306 T^{2} - 109575985674 T^{3} + 191707312997281 T^{4} )^{2}$$)
$67$ ($$1 - 73661280 T^{2} + 1417160078043996 T^{4} +$$$$25\!\cdots\!60$$$$T^{6} -$$$$13\!\cdots\!74$$$$T^{8} +$$$$10\!\cdots\!60$$$$T^{10} +$$$$23\!\cdots\!76$$$$T^{12} -$$$$49\!\cdots\!80$$$$T^{14} +$$$$27\!\cdots\!61$$$$T^{16}$$)($$1 + 4478 T + 20151121 T^{2}$$)($$1 - 4478 T + 20151121 T^{2}$$)($$( 1 + 69567444 T^{2} + 1999220647242406 T^{4} +$$$$28\!\cdots\!04$$$$T^{6} +$$$$16\!\cdots\!81$$$$T^{8} )^{2}$$)($$1 + 4810 T + 11568050 T^{2} + 108537019890 T^{3} + 1012520446566818 T^{4} + 2187142620782796690 T^{5} +$$$$46\!\cdots\!50$$$$T^{6} +$$$$39\!\cdots\!10$$$$T^{7} +$$$$16\!\cdots\!81$$$$T^{8}$$)
$71$ ($$1 - 93361128 T^{2} + 5034263138166108 T^{4} -$$$$18\!\cdots\!76$$$$T^{6} +$$$$54\!\cdots\!90$$$$T^{8} -$$$$12\!\cdots\!36$$$$T^{10} +$$$$20\!\cdots\!68$$$$T^{12} -$$$$25\!\cdots\!68$$$$T^{14} +$$$$17\!\cdots\!41$$$$T^{16}$$)($$( 1 - 5041 T )( 1 + 5041 T )$$)($$( 1 - 5041 T )( 1 + 5041 T )$$)($$( 1 - 36881284 T^{2} + 596367162990726 T^{4} -$$$$23\!\cdots\!24$$$$T^{6} +$$$$41\!\cdots\!21$$$$T^{8} )^{2}$$)($$( 1 + 2994 T + 49877146 T^{2} + 76082572914 T^{3} + 645753531245761 T^{4} )^{2}$$)
$73$ ($$( 1 + 7224 T + 116019228 T^{2} + 579521463432 T^{3} + 4936809828072134 T^{4} + 16457390183214623112 T^{5} +$$$$93\!\cdots\!68$$$$T^{6} +$$$$16\!\cdots\!04$$$$T^{7} +$$$$65\!\cdots\!61$$$$T^{8} )^{2}$$)($$( 1 - 5329 T )( 1 + 5329 T )$$)($$( 1 - 5329 T )( 1 + 5329 T )$$)($$( 1 - 45412996 T^{2} + 2127886061569926 T^{4} -$$$$36\!\cdots\!76$$$$T^{6} +$$$$65\!\cdots\!61$$$$T^{8} )^{2}$$)($$1 + 14060 T + 98841800 T^{2} + 628627141380 T^{3} + 3731941946353934 T^{4} + 17851905060050312580 T^{5} +$$$$79\!\cdots\!00$$$$T^{6} +$$$$32\!\cdots\!60$$$$T^{7} +$$$$65\!\cdots\!61$$$$T^{8}$$)
$79$ ($$1 - 254373768 T^{2} + 29821166949528348 T^{4} -$$$$21\!\cdots\!76$$$$T^{6} +$$$$99\!\cdots\!90$$$$T^{8} -$$$$32\!\cdots\!36$$$$T^{10} +$$$$68\!\cdots\!08$$$$T^{12} -$$$$88\!\cdots\!08$$$$T^{14} +$$$$52\!\cdots\!41$$$$T^{16}$$)($$( 1 - 6241 T )( 1 + 6241 T )$$)($$( 1 - 6241 T )( 1 + 6241 T )$$)($$( 1 - 128520964 T^{2} + 7130553234151686 T^{4} -$$$$19\!\cdots\!04$$$$T^{6} +$$$$23\!\cdots\!21$$$$T^{8} )^{2}$$)($$1 - 127811332 T^{2} + 7111883058262278 T^{4} -$$$$19\!\cdots\!52$$$$T^{6} +$$$$23\!\cdots\!21$$$$T^{8}$$)
$83$ ($$1 - 315969760 T^{2} + 46108155295297116 T^{4} -$$$$40\!\cdots\!60$$$$T^{6} +$$$$23\!\cdots\!86$$$$T^{8} -$$$$91\!\cdots\!60$$$$T^{10} +$$$$23\!\cdots\!96$$$$T^{12} -$$$$36\!\cdots\!60$$$$T^{14} +$$$$25\!\cdots\!61$$$$T^{16}$$)($$1 - 8002 T + 47458321 T^{2}$$)($$1 + 8002 T + 47458321 T^{2}$$)($$( 1 + 71614164 T^{2} + 3304864070081446 T^{4} +$$$$16\!\cdots\!24$$$$T^{6} +$$$$50\!\cdots\!81$$$$T^{8} )^{2}$$)($$1 - 19950 T + 199001250 T^{2} - 1392338210550 T^{3} + 9242910030806018 T^{4} - 66078033736847486550 T^{5} +$$$$44\!\cdots\!50$$$$T^{6} -$$$$21\!\cdots\!50$$$$T^{7} +$$$$50\!\cdots\!81$$$$T^{8}$$)
$89$ ($$( 1 - 11880 T + 292095196 T^{2} - 2292250543320 T^{3} + 28806792351684166 T^{4} -$$$$14\!\cdots\!20$$$$T^{5} +$$$$11\!\cdots\!76$$$$T^{6} -$$$$29\!\cdots\!80$$$$T^{7} +$$$$15\!\cdots\!61$$$$T^{8} )^{2}$$)($$1 - 4322 T + 62742241 T^{2}$$)($$1 - 4322 T + 62742241 T^{2}$$)($$( 1 - 6916 T + 63458246 T^{2} - 433925338756 T^{3} + 3936588805702081 T^{4} )^{4}$$)($$1 - 213169796 T^{2} + 19233126298796166 T^{4} -$$$$83\!\cdots\!76$$$$T^{6} +$$$$15\!\cdots\!61$$$$T^{8}$$)
$97$ ($$( 1 + 2184 T + 140081788 T^{2} - 468341237448 T^{3} + 7891029582726534 T^{4} - 41461913013921714888 T^{5} +$$$$10\!\cdots\!68$$$$T^{6} +$$$$15\!\cdots\!44$$$$T^{7} +$$$$61\!\cdots\!21$$$$T^{8} )^{2}$$)($$( 1 - 9409 T )( 1 + 9409 T )$$)($$( 1 - 9409 T )( 1 + 9409 T )$$)($$( 1 - 267347716 T^{2} + 33011083228247046 T^{4} -$$$$20\!\cdots\!76$$$$T^{6} +$$$$61\!\cdots\!21$$$$T^{8} )^{2}$$)($$1 + 3180 T + 5056200 T^{2} - 122563753980 T^{3} - 13176145025425522 T^{4} - 10850481016510288380 T^{5} +$$$$39\!\cdots\!00$$$$T^{6} +$$$$22\!\cdots\!80$$$$T^{7} +$$$$61\!\cdots\!21$$$$T^{8}$$)