# Properties

 Label 15.10 Level 15 Weight 10 Dimension 46 Nonzero newspaces 3 Newform subspaces 6 Sturm bound 160 Trace bound 1

## Defining parameters

 Level: $$N$$ = $$15 = 3 \cdot 5$$ Weight: $$k$$ = $$10$$ Nonzero newspaces: $$3$$ Newform subspaces: $$6$$ Sturm bound: $$160$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{10}(\Gamma_1(15))$$.

Total New Old
Modular forms 80 54 26
Cusp forms 64 46 18
Eisenstein series 16 8 8

## Trace form

 $$46q + 68q^{2} - 150q^{3} + 344q^{4} - 690q^{5} - 7140q^{6} - 21188q^{7} + 67932q^{8} - 13122q^{9} + O(q^{10})$$ $$46q + 68q^{2} - 150q^{3} + 344q^{4} - 690q^{5} - 7140q^{6} - 21188q^{7} + 67932q^{8} - 13122q^{9} + 18520q^{10} - 159064q^{11} + 233412q^{12} + 170148q^{13} + 531816q^{14} - 61290q^{15} - 1915024q^{16} - 420520q^{17} + 1440948q^{18} + 1510496q^{19} + 1173100q^{20} - 3242652q^{21} - 6848280q^{22} + 3122496q^{23} + 4436208q^{24} + 9847270q^{25} - 6770392q^{26} + 4786830q^{27} - 9172776q^{28} - 7070108q^{29} - 15876960q^{30} + 12093928q^{31} + 17859412q^{32} + 18442884q^{33} - 2414552q^{34} - 21174680q^{35} - 31019304q^{36} - 10493548q^{37} - 2385976q^{38} + 7411824q^{39} + 124068200q^{40} + 117594724q^{41} + 2551536q^{42} - 122810748q^{43} - 177425288q^{44} - 125161290q^{45} - 96583992q^{46} + 51128264q^{47} + 340587828q^{48} + 135923910q^{49} + 94360700q^{50} - 109600476q^{51} - 209781040q^{52} - 114062224q^{53} - 23383404q^{54} + 128495720q^{55} - 136664640q^{56} + 45495684q^{57} - 415626152q^{58} + 417468464q^{59} - 208559160q^{60} + 185675724q^{61} - 31830936q^{62} + 14879472q^{63} + 452073416q^{64} - 172199200q^{65} + 324822624q^{66} - 331042820q^{67} + 717440536q^{68} + 172304496q^{69} + 1164742440q^{70} - 73747472q^{71} - 1088344908q^{72} + 37028316q^{73} - 2296939904q^{74} - 390483390q^{75} - 2207705040q^{76} + 72064224q^{77} - 137852280q^{78} + 589394400q^{79} + 733243180q^{80} + 2853197046q^{81} + 5664251624q^{82} + 804884184q^{83} - 1030301856q^{84} - 2307340580q^{85} + 245612816q^{86} - 2840994132q^{87} - 2764190328q^{88} + 27621876q^{89} + 196282680q^{90} - 5025897832q^{91} - 2807106528q^{92} - 1123759992q^{93} + 2549830744q^{94} + 2062249640q^{95} + 9194942832q^{96} + 5375198660q^{97} + 5935831732q^{98} - 98992368q^{99} + O(q^{100})$$

## Decomposition of $$S_{10}^{\mathrm{new}}(\Gamma_1(15))$$

We only show spaces with even 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
15.10.a $$\chi_{15}(1, \cdot)$$ 15.10.a.a 1 1
15.10.a.b 1
15.10.a.c 2
15.10.a.d 2
15.10.b $$\chi_{15}(4, \cdot)$$ 15.10.b.a 8 1
15.10.e $$\chi_{15}(2, \cdot)$$ 15.10.e.a 32 2

## Decomposition of $$S_{10}^{\mathrm{old}}(\Gamma_1(15))$$ into lower level spaces

$$S_{10}^{\mathrm{old}}(\Gamma_1(15)) \cong$$ $$S_{10}^{\mathrm{new}}(\Gamma_1(3))$$$$^{\oplus 2}$$$$\oplus$$$$S_{10}^{\mathrm{new}}(\Gamma_1(5))$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ ($$1 + 4 T + 512 T^{2}$$)($$1 - 22 T + 512 T^{2}$$)($$1 - 19 T - 68 T^{2} - 9728 T^{3} + 262144 T^{4}$$)($$1 - 31 T + 722 T^{2} - 15872 T^{3} + 262144 T^{4}$$)($$1 - 1451 T^{2} + 1581940 T^{4} - 1146579392 T^{6} + 685942325248 T^{8} - 300568908136448 T^{10} + 108710089027747840 T^{12} - 26138892237258358784 T^{14} +$$$$47\!\cdots\!96$$$$T^{16}$$)
$3$ ($$1 - 81 T$$)($$1 + 81 T$$)($$( 1 - 81 T )^{2}$$)($$( 1 + 81 T )^{2}$$)($$( 1 + 6561 T^{2} )^{4}$$)
$5$ ($$1 - 625 T$$)($$1 + 625 T$$)($$( 1 + 625 T )^{2}$$)($$( 1 - 625 T )^{2}$$)($$1 + 690 T - 609700 T^{2} - 1699931250 T^{3} - 2538785156250 T^{4} - 3320178222656250 T^{5} - 2325820922851562500 T^{6} +$$$$51\!\cdots\!50$$$$T^{7} +$$$$14\!\cdots\!25$$$$T^{8}$$)
$7$ ($$1 + 7680 T + 40353607 T^{2}$$)($$1 + 5988 T + 40353607 T^{2}$$)($$1 + 11872 T + 112235774 T^{2} + 479078022304 T^{3} + 1628413597910449 T^{4}$$)($$1 - 14112 T + 103286414 T^{2} - 569470101984 T^{3} + 1628413597910449 T^{4}$$)($$1 - 187079228 T^{2} + 15253712953314292 T^{4} -$$$$75\!\cdots\!16$$$$T^{6} +$$$$30\!\cdots\!14$$$$T^{8} -$$$$12\!\cdots\!84$$$$T^{10} +$$$$40\!\cdots\!92$$$$T^{12} -$$$$80\!\cdots\!72$$$$T^{14} +$$$$70\!\cdots\!01$$$$T^{16}$$)
$11$ ($$1 + 86404 T + 2357947691 T^{2}$$)($$1 + 14648 T + 2357947691 T^{2}$$)($$1 - 35488 T + 526013014 T^{2} - 83678847658208 T^{3} + 5559917313492231481 T^{4}$$)($$1 + 21512 T + 1790961254 T^{2} + 50724170728792 T^{3} + 5559917313492231481 T^{4}$$)($$( 1 + 35994 T + 5523704672 T^{2} + 186668393765490 T^{3} + 15626722625814770286 T^{4} +$$$$44\!\cdots\!90$$$$T^{5} +$$$$30\!\cdots\!32$$$$T^{6} +$$$$47\!\cdots\!74$$$$T^{7} +$$$$30\!\cdots\!61$$$$T^{8} )^{2}$$)
$13$ ($$1 + 149978 T + 10604499373 T^{2}$$)($$1 - 37906 T + 10604499373 T^{2}$$)($$1 - 143676 T + 24105149134 T^{2} - 1523612051915148 T^{3} +$$$$11\!\cdots\!29$$$$T^{4}$$)($$1 - 24284 T + 5870669214 T^{2} - 257519662773932 T^{3} +$$$$11\!\cdots\!29$$$$T^{4}$$)($$1 - 57204710684 T^{2} +$$$$16\!\cdots\!00$$$$T^{4} -$$$$29\!\cdots\!48$$$$T^{6} +$$$$37\!\cdots\!18$$$$T^{8} -$$$$33\!\cdots\!92$$$$T^{10} +$$$$20\!\cdots\!00$$$$T^{12} -$$$$81\!\cdots\!76$$$$T^{14} +$$$$15\!\cdots\!81$$$$T^{16}$$)
$17$ ($$1 + 207622 T + 118587876497 T^{2}$$)($$1 + 441098 T + 118587876497 T^{2}$$)($$1 - 385156 T + 268539949078 T^{2} - 45674832160078532 T^{3} +$$$$14\!\cdots\!09$$$$T^{4}$$)($$1 + 156956 T + 212670095078 T^{2} + 18613078743463132 T^{3} +$$$$14\!\cdots\!09$$$$T^{4}$$)($$1 - 118674378380 T^{2} +$$$$17\!\cdots\!36$$$$T^{4} -$$$$19\!\cdots\!60$$$$T^{6} +$$$$18\!\cdots\!86$$$$T^{8} -$$$$27\!\cdots\!40$$$$T^{10} +$$$$34\!\cdots\!16$$$$T^{12} -$$$$33\!\cdots\!20$$$$T^{14} +$$$$39\!\cdots\!61$$$$T^{16}$$)
$19$ ($$1 - 716284 T + 322687697779 T^{2}$$)($$1 - 441820 T + 322687697779 T^{2}$$)($$1 + 403296 T + 684929514838 T^{2} + 130138657763479584 T^{3} +$$$$10\!\cdots\!41$$$$T^{4}$$)($$1 + 95896 T + 629883192438 T^{2} + 30944459466214984 T^{3} +$$$$10\!\cdots\!41$$$$T^{4}$$)($$( 1 - 425792 T + 640639588492 T^{2} - 117316391220409664 T^{3} +$$$$17\!\cdots\!54$$$$T^{4} -$$$$37\!\cdots\!56$$$$T^{5} +$$$$66\!\cdots\!72$$$$T^{6} -$$$$14\!\cdots\!88$$$$T^{7} +$$$$10\!\cdots\!81$$$$T^{8} )^{2}$$)
$23$ ($$1 - 1369920 T + 1801152661463 T^{2}$$)($$1 - 2264136 T + 1801152661463 T^{2}$$)($$1 - 223704 T - 1375273107794 T^{2} - 402925054979918952 T^{3} +$$$$32\!\cdots\!69$$$$T^{4}$$)($$1 + 735264 T + 3363187908526 T^{2} + 1324322710477931232 T^{3} +$$$$32\!\cdots\!69$$$$T^{4}$$)($$1 - 10272938895992 T^{2} +$$$$48\!\cdots\!32$$$$T^{4} -$$$$14\!\cdots\!84$$$$T^{6} +$$$$30\!\cdots\!94$$$$T^{8} -$$$$46\!\cdots\!96$$$$T^{10} +$$$$51\!\cdots\!52$$$$T^{12} -$$$$35\!\cdots\!28$$$$T^{14} +$$$$11\!\cdots\!21$$$$T^{16}$$)
$29$ ($$1 + 3194402 T + 14507145975869 T^{2}$$)($$1 + 1049350 T + 14507145975869 T^{2}$$)($$1 + 74572 T + 28833430018078 T^{2} + 1081826889712503068 T^{3} +$$$$21\!\cdots\!61$$$$T^{4}$$)($$1 + 2678212 T + 15397908029438 T^{2} + 38853212438324066228 T^{3} +$$$$21\!\cdots\!61$$$$T^{4}$$)($$( 1 + 36786 T + 11698955610980 T^{2} - 38232367405598309058 T^{3} +$$$$21\!\cdots\!18$$$$T^{4} -$$$$55\!\cdots\!02$$$$T^{5} +$$$$24\!\cdots\!80$$$$T^{6} +$$$$11\!\cdots\!74$$$$T^{7} +$$$$44\!\cdots\!21$$$$T^{8} )^{2}$$)
$31$ ($$1 + 2349000 T + 26439622160671 T^{2}$$)($$1 + 7910568 T + 26439622160671 T^{2}$$)($$1 + 5027128 T + 52415931233342 T^{2} +$$$$13\!\cdots\!88$$$$T^{3} +$$$$69\!\cdots\!41$$$$T^{4}$$)($$1 - 10782432 T + 69294691361342 T^{2} -$$$$28\!\cdots\!72$$$$T^{3} +$$$$69\!\cdots\!41$$$$T^{4}$$)($$( 1 - 237044 T + 50199397014268 T^{2} -$$$$19\!\cdots\!72$$$$T^{3} +$$$$11\!\cdots\!74$$$$T^{4} -$$$$51\!\cdots\!12$$$$T^{5} +$$$$35\!\cdots\!88$$$$T^{6} -$$$$43\!\cdots\!84$$$$T^{7} +$$$$48\!\cdots\!81$$$$T^{8} )^{2}$$)
$37$ ($$1 - 18735710 T + 129961739795077 T^{2}$$)($$1 + 20992558 T + 129961739795077 T^{2}$$)($$1 - 5373628 T + 231061724951934 T^{2} -$$$$69\!\cdots\!56$$$$T^{3} +$$$$16\!\cdots\!29$$$$T^{4}$$)($$1 - 21968332 T + 359210373327534 T^{2} -$$$$28\!\cdots\!64$$$$T^{3} +$$$$16\!\cdots\!29$$$$T^{4}$$)($$1 - 624655700715068 T^{2} +$$$$20\!\cdots\!12$$$$T^{4} -$$$$45\!\cdots\!76$$$$T^{6} +$$$$70\!\cdots\!14$$$$T^{8} -$$$$77\!\cdots\!04$$$$T^{10} +$$$$59\!\cdots\!92$$$$T^{12} -$$$$30\!\cdots\!52$$$$T^{14} +$$$$81\!\cdots\!81$$$$T^{16}$$)
$41$ ($$1 + 29282630 T + 327381934393961 T^{2}$$)($$1 - 13285562 T + 327381934393961 T^{2}$$)($$1 - 14211332 T + 443988635955862 T^{2} -$$$$46\!\cdots\!52$$$$T^{3} +$$$$10\!\cdots\!21$$$$T^{4}$$)($$1 - 26060372 T + 693239183881142 T^{2} -$$$$85\!\cdots\!92$$$$T^{3} +$$$$10\!\cdots\!21$$$$T^{4}$$)($$( 1 - 46660044 T + 1629598509234068 T^{2} -$$$$39\!\cdots\!32$$$$T^{3} +$$$$78\!\cdots\!54$$$$T^{4} -$$$$12\!\cdots\!52$$$$T^{5} +$$$$17\!\cdots\!28$$$$T^{6} -$$$$16\!\cdots\!64$$$$T^{7} +$$$$11\!\cdots\!41$$$$T^{8} )^{2}$$)
$43$ ($$1 + 1516724 T + 502592611936843 T^{2}$$)($$1 + 23130764 T + 502592611936843 T^{2}$$)($$1 - 27748920 T + 1170232974699430 T^{2} -$$$$13\!\cdots\!60$$$$T^{3} +$$$$25\!\cdots\!49$$$$T^{4}$$)($$1 + 7191160 T + 750634586008230 T^{2} +$$$$36\!\cdots\!80$$$$T^{3} +$$$$25\!\cdots\!49$$$$T^{4}$$)($$1 - 1288760154444440 T^{2} +$$$$10\!\cdots\!96$$$$T^{4} -$$$$57\!\cdots\!80$$$$T^{6} +$$$$31\!\cdots\!06$$$$T^{8} -$$$$14\!\cdots\!20$$$$T^{10} +$$$$65\!\cdots\!96$$$$T^{12} -$$$$20\!\cdots\!60$$$$T^{14} +$$$$40\!\cdots\!01$$$$T^{16}$$)
$47$ ($$1 - 615752 T + 1119130473102767 T^{2}$$)($$1 + 13873688 T + 1119130473102767 T^{2}$$)($$1 - 95966440 T + 4320659216802910 T^{2} -$$$$10\!\cdots\!80$$$$T^{3} +$$$$12\!\cdots\!89$$$$T^{4}$$)($$1 + 31580240 T + 382042606129310 T^{2} +$$$$35\!\cdots\!80$$$$T^{3} +$$$$12\!\cdots\!89$$$$T^{4}$$)($$1 - 4836473602067240 T^{2} +$$$$12\!\cdots\!56$$$$T^{4} -$$$$21\!\cdots\!80$$$$T^{6} +$$$$28\!\cdots\!26$$$$T^{8} -$$$$27\!\cdots\!20$$$$T^{10} +$$$$19\!\cdots\!76$$$$T^{12} -$$$$95\!\cdots\!60$$$$T^{14} +$$$$24\!\cdots\!41$$$$T^{16}$$)
$53$ ($$1 - 4747430 T + 3299763591802133 T^{2}$$)($$1 + 57635174 T + 3299763591802133 T^{2}$$)($$1 + 64305596 T + 6611083028543086 T^{2} +$$$$21\!\cdots\!68$$$$T^{3} +$$$$10\!\cdots\!89$$$$T^{4}$$)($$1 - 3131116 T + 6584849973489806 T^{2} -$$$$10\!\cdots\!28$$$$T^{3} +$$$$10\!\cdots\!89$$$$T^{4}$$)($$1 - 4341506689340012 T^{2} +$$$$22\!\cdots\!72$$$$T^{4} -$$$$59\!\cdots\!84$$$$T^{6} +$$$$28\!\cdots\!74$$$$T^{8} -$$$$64\!\cdots\!76$$$$T^{10} +$$$$26\!\cdots\!12$$$$T^{12} -$$$$56\!\cdots\!28$$$$T^{14} +$$$$14\!\cdots\!41$$$$T^{16}$$)
$59$ ($$1 - 60616076 T + 8662995818654939 T^{2}$$)($$1 + 32042120 T + 8662995818654939 T^{2}$$)($$1 - 187863136 T + 23071633420288438 T^{2} -$$$$16\!\cdots\!04$$$$T^{3} +$$$$75\!\cdots\!21$$$$T^{4}$$)($$1 + 35494664 T + 7388006896329158 T^{2} +$$$$30\!\cdots\!96$$$$T^{3} +$$$$75\!\cdots\!21$$$$T^{4}$$)($$( 1 - 118263018 T + 24386862278408432 T^{2} -$$$$17\!\cdots\!86$$$$T^{3} +$$$$26\!\cdots\!54$$$$T^{4} -$$$$15\!\cdots\!54$$$$T^{5} +$$$$18\!\cdots\!72$$$$T^{6} -$$$$76\!\cdots\!42$$$$T^{7} +$$$$56\!\cdots\!41$$$$T^{8} )^{2}$$)
$61$ ($$1 + 126745682 T + 11694146092834141 T^{2}$$)($$1 - 110664022 T + 11694146092834141 T^{2}$$)($$1 - 154080060 T + 23302683905802238 T^{2} -$$$$18\!\cdots\!60$$$$T^{3} +$$$$13\!\cdots\!81$$$$T^{4}$$)($$1 - 341497340 T + 52053805546777278 T^{2} -$$$$39\!\cdots\!40$$$$T^{3} +$$$$13\!\cdots\!81$$$$T^{4}$$)($$( 1 + 178713880 T + 47272829372304076 T^{2} +$$$$59\!\cdots\!20$$$$T^{3} +$$$$82\!\cdots\!06$$$$T^{4} +$$$$69\!\cdots\!20$$$$T^{5} +$$$$64\!\cdots\!56$$$$T^{6} +$$$$28\!\cdots\!80$$$$T^{7} +$$$$18\!\cdots\!61$$$$T^{8} )^{2}$$)
$67$ ($$1 + 111182652 T + 27206534396294947 T^{2}$$)($$1 + 118568268 T + 27206534396294947 T^{2}$$)($$1 - 33592376 T - 10819815556424362 T^{2} -$$$$91\!\cdots\!72$$$$T^{3} +$$$$74\!\cdots\!09$$$$T^{4}$$)($$1 + 288195816 T + 74605839041196758 T^{2} +$$$$78\!\cdots\!52$$$$T^{3} +$$$$74\!\cdots\!09$$$$T^{4}$$)($$1 - 128915336297443400 T^{2} +$$$$88\!\cdots\!36$$$$T^{4} -$$$$39\!\cdots\!00$$$$T^{6} +$$$$12\!\cdots\!86$$$$T^{8} -$$$$29\!\cdots\!00$$$$T^{10} +$$$$48\!\cdots\!16$$$$T^{12} -$$$$52\!\cdots\!00$$$$T^{14} +$$$$30\!\cdots\!61$$$$T^{16}$$)
$71$ ($$1 + 175551608 T + 45848500718449031 T^{2}$$)($$1 - 276679712 T + 45848500718449031 T^{2}$$)($$1 + 228270976 T + 45777616900481806 T^{2} +$$$$10\!\cdots\!56$$$$T^{3} +$$$$21\!\cdots\!61$$$$T^{4}$$)($$1 - 210286064 T + 91549406631588686 T^{2} -$$$$96\!\cdots\!84$$$$T^{3} +$$$$21\!\cdots\!61$$$$T^{4}$$)($$( 1 + 78445332 T + 132742682733898508 T^{2} +$$$$14\!\cdots\!24$$$$T^{3} +$$$$78\!\cdots\!70$$$$T^{4} +$$$$65\!\cdots\!44$$$$T^{5} +$$$$27\!\cdots\!88$$$$T^{6} +$$$$75\!\cdots\!12$$$$T^{7} +$$$$44\!\cdots\!21$$$$T^{8} )^{2}$$)
$73$ ($$1 + 61233350 T + 58871586708267913 T^{2}$$)($$1 + 264023294 T + 58871586708267913 T^{2}$$)($$1 + 33122316 T + 68371107952007926 T^{2} +$$$$19\!\cdots\!08$$$$T^{3} +$$$$34\!\cdots\!69$$$$T^{4}$$)($$1 + 232663084 T + 130536207391012086 T^{2} +$$$$13\!\cdots\!92$$$$T^{3} +$$$$34\!\cdots\!69$$$$T^{4}$$)($$1 - 112790784848235992 T^{2} +$$$$16\!\cdots\!32$$$$T^{4} -$$$$11\!\cdots\!84$$$$T^{6} +$$$$89\!\cdots\!94$$$$T^{8} -$$$$39\!\cdots\!96$$$$T^{10} +$$$$19\!\cdots\!52$$$$T^{12} -$$$$46\!\cdots\!28$$$$T^{14} +$$$$14\!\cdots\!21$$$$T^{16}$$)
$79$ ($$1 - 234431160 T + 119851595982618319 T^{2}$$)($$1 - 448202760 T + 119851595982618319 T^{2}$$)($$1 + 932406760 T + 453226630902929438 T^{2} +$$$$11\!\cdots\!40$$$$T^{3} +$$$$14\!\cdots\!61$$$$T^{4}$$)($$1 + 24755040 T + 43090694479668638 T^{2} +$$$$29\!\cdots\!60$$$$T^{3} +$$$$14\!\cdots\!61$$$$T^{4}$$)($$( 1 - 431961140 T + 452072920533003676 T^{2} -$$$$14\!\cdots\!80$$$$T^{3} +$$$$79\!\cdots\!66$$$$T^{4} -$$$$17\!\cdots\!20$$$$T^{5} +$$$$64\!\cdots\!36$$$$T^{6} -$$$$74\!\cdots\!60$$$$T^{7} +$$$$20\!\cdots\!21$$$$T^{8} )^{2}$$)
$83$ ($$1 - 118910388 T + 186940255267540403 T^{2}$$)($$1 - 851015796 T + 186940255267540403 T^{2}$$)($$1 - 207040152 T + 372783310330485238 T^{2} -$$$$38\!\cdots\!56$$$$T^{3} +$$$$34\!\cdots\!09$$$$T^{4}$$)($$1 + 372082152 T + 379908828789982198 T^{2} +$$$$69\!\cdots\!56$$$$T^{3} +$$$$34\!\cdots\!09$$$$T^{4}$$)($$1 - 981210392397464024 T^{2} +$$$$42\!\cdots\!20$$$$T^{4} -$$$$11\!\cdots\!28$$$$T^{6} +$$$$22\!\cdots\!98$$$$T^{8} -$$$$38\!\cdots\!52$$$$T^{10} +$$$$51\!\cdots\!20$$$$T^{12} -$$$$41\!\cdots\!96$$$$T^{14} +$$$$14\!\cdots\!61$$$$T^{16}$$)
$89$ ($$1 + 316534326 T + 350356403707485209 T^{2}$$)($$1 - 189894930 T + 350356403707485209 T^{2}$$)($$1 - 224518164 T + 610925899926766678 T^{2} -$$$$78\!\cdots\!76$$$$T^{3} +$$$$12\!\cdots\!81$$$$T^{4}$$)($$1 + 427639116 T + 702028302670151638 T^{2} +$$$$14\!\cdots\!44$$$$T^{3} +$$$$12\!\cdots\!81$$$$T^{4}$$)($$( 1 - 178691112 T + 708605882924008892 T^{2} -$$$$39\!\cdots\!44$$$$T^{3} +$$$$23\!\cdots\!94$$$$T^{4} -$$$$13\!\cdots\!96$$$$T^{5} +$$$$86\!\cdots\!52$$$$T^{6} -$$$$76\!\cdots\!48$$$$T^{7} +$$$$15\!\cdots\!61$$$$T^{8} )^{2}$$)
$97$ ($$1 - 242912258 T + 760231058654565217 T^{2}$$)($$1 + 1014149278 T + 760231058654565217 T^{2}$$)($$1 - 387134596 T - 734969029248610362 T^{2} -$$$$29\!\cdots\!32$$$$T^{3} +$$$$57\!\cdots\!89$$$$T^{4}$$)($$1 - 1771658884 T + 2207048700436243398 T^{2} -$$$$13\!\cdots\!28$$$$T^{3} +$$$$57\!\cdots\!89$$$$T^{4}$$)($$1 - 1313769052010852360 T^{2} +$$$$18\!\cdots\!56$$$$T^{4} -$$$$20\!\cdots\!20$$$$T^{6} +$$$$15\!\cdots\!26$$$$T^{8} -$$$$11\!\cdots\!80$$$$T^{10} +$$$$62\!\cdots\!76$$$$T^{12} -$$$$25\!\cdots\!40$$$$T^{14} +$$$$11\!\cdots\!41$$$$T^{16}$$)