# Properties

 Label 160.6.a Level 160 Weight 6 Character orbit a Rep. character $$\chi_{160}(1,\cdot)$$ Character field $$\Q$$ Dimension 20 Newform subspaces 8 Sturm bound 144 Trace bound 9

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$160 = 2^{5} \cdot 5$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 160.a (trivial) Character field: $$\Q$$ Newform subspaces: $$8$$ Sturm bound: $$144$$ Trace bound: $$9$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{6}(\Gamma_0(160))$$.

Total New Old
Modular forms 128 20 108
Cusp forms 112 20 92
Eisenstein series 16 0 16

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

$$2$$$$5$$FrickeDim.
$$+$$$$+$$$$+$$$$5$$
$$+$$$$-$$$$-$$$$6$$
$$-$$$$+$$$$-$$$$5$$
$$-$$$$-$$$$+$$$$4$$
Plus space$$+$$$$9$$
Minus space$$-$$$$11$$

## Trace form

 $$20q + 1532q^{9} + O(q^{10})$$ $$20q + 1532q^{9} - 464q^{13} - 2008q^{17} + 10136q^{21} + 12500q^{25} - 7960q^{29} - 7504q^{33} - 4304q^{37} + 44464q^{41} - 11800q^{45} + 5996q^{49} + 35024q^{53} + 183392q^{57} - 125440q^{61} - 42200q^{65} + 147208q^{69} + 212392q^{73} - 149648q^{77} + 38940q^{81} + 282536q^{89} - 667696q^{93} - 81544q^{97} + O(q^{100})$$

## Decomposition of $$S_{6}^{\mathrm{new}}(\Gamma_0(160))$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces A-L signs $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$ 2 5
160.6.a.a $$2$$ $$25.661$$ $$\Q(\sqrt{70})$$ None $$0$$ $$-8$$ $$50$$ $$-104$$ $$-$$ $$-$$ $$q+(-4+\beta )q^{3}+5^{2}q^{5}+(-52-\beta )q^{7}+\cdots$$
160.6.a.b $$2$$ $$25.661$$ $$\Q(\sqrt{5})$$ None $$0$$ $$0$$ $$-50$$ $$0$$ $$-$$ $$+$$ $$q-3\beta q^{3}-5^{2}q^{5}-31\beta q^{7}-63q^{9}+\cdots$$
160.6.a.c $$2$$ $$25.661$$ $$\Q(\sqrt{85})$$ None $$0$$ $$0$$ $$-50$$ $$0$$ $$+$$ $$+$$ $$q-\beta q^{3}-5^{2}q^{5}+3\beta q^{7}+97q^{9}+26\beta q^{11}+\cdots$$
160.6.a.d $$2$$ $$25.661$$ $$\Q(\sqrt{10})$$ None $$0$$ $$0$$ $$50$$ $$0$$ $$-$$ $$-$$ $$q+\beta q^{3}+5^{2}q^{5}+7\beta q^{7}-203q^{9}+\cdots$$
160.6.a.e $$2$$ $$25.661$$ $$\Q(\sqrt{70})$$ None $$0$$ $$8$$ $$50$$ $$104$$ $$+$$ $$-$$ $$q+(4+\beta )q^{3}+5^{2}q^{5}+(52-\beta )q^{7}+(53+\cdots)q^{9}+\cdots$$
160.6.a.f $$3$$ $$25.661$$ 3.3.39180.1 None $$0$$ $$-10$$ $$-75$$ $$6$$ $$+$$ $$+$$ $$q+(-3-\beta _{1})q^{3}-5^{2}q^{5}+(2+\beta _{1}-\beta _{2})q^{7}+\cdots$$
160.6.a.g $$3$$ $$25.661$$ 3.3.39180.1 None $$0$$ $$10$$ $$-75$$ $$-6$$ $$-$$ $$+$$ $$q+(3+\beta _{1})q^{3}-5^{2}q^{5}+(-2-\beta _{1}+\beta _{2})q^{7}+\cdots$$
160.6.a.h $$4$$ $$25.661$$ 4.4.81998080.1 None $$0$$ $$0$$ $$100$$ $$0$$ $$+$$ $$-$$ $$q-\beta _{1}q^{3}+5^{2}q^{5}+(-6\beta _{1}+\beta _{2})q^{7}+\cdots$$

## Decomposition of $$S_{6}^{\mathrm{old}}(\Gamma_0(160))$$ into lower level spaces

$$S_{6}^{\mathrm{old}}(\Gamma_0(160)) \cong$$ $$S_{6}^{\mathrm{new}}(\Gamma_0(4))$$$$^{\oplus 8}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_0(5))$$$$^{\oplus 6}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_0(8))$$$$^{\oplus 6}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_0(10))$$$$^{\oplus 5}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_0(16))$$$$^{\oplus 4}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_0(20))$$$$^{\oplus 4}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_0(32))$$$$^{\oplus 2}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_0(40))$$$$^{\oplus 3}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_0(80))$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ ($$1 + 8 T + 222 T^{2} + 1944 T^{3} + 59049 T^{4}$$)($$1 + 306 T^{2} + 59049 T^{4}$$)($$1 + 146 T^{2} + 59049 T^{4}$$)($$1 + 446 T^{2} + 59049 T^{4}$$)($$1 - 8 T + 222 T^{2} - 1944 T^{3} + 59049 T^{4}$$)($$1 + 10 T + 181 T^{2} + 5268 T^{3} + 43983 T^{4} + 590490 T^{5} + 14348907 T^{6}$$)($$1 - 10 T + 181 T^{2} - 5268 T^{3} + 43983 T^{4} - 590490 T^{5} + 14348907 T^{6}$$)($$1 + 124 T^{2} + 25686 T^{4} + 7322076 T^{6} + 3486784401 T^{8}$$)
$5$ ($$( 1 - 25 T )^{2}$$)($$( 1 + 25 T )^{2}$$)($$( 1 + 25 T )^{2}$$)($$( 1 - 25 T )^{2}$$)($$( 1 - 25 T )^{2}$$)($$( 1 + 25 T )^{3}$$)($$( 1 + 25 T )^{3}$$)($$( 1 - 25 T )^{4}$$)
$7$ ($$1 + 104 T + 36038 T^{2} + 1747928 T^{3} + 282475249 T^{4}$$)($$1 + 14394 T^{2} + 282475249 T^{4}$$)($$1 + 30554 T^{2} + 282475249 T^{4}$$)($$1 + 31654 T^{2} + 282475249 T^{4}$$)($$1 - 104 T + 36038 T^{2} - 1747928 T^{3} + 282475249 T^{4}$$)($$1 - 6 T + 27153 T^{2} - 1137532 T^{3} + 456360471 T^{4} - 1694851494 T^{5} + 4747561509943 T^{6}$$)($$1 + 6 T + 27153 T^{2} + 1137532 T^{3} + 456360471 T^{4} + 1694851494 T^{5} + 4747561509943 T^{6}$$)($$1 - 27060 T^{2} + 497649542 T^{4} - 7643780237940 T^{6} + 79792266297612001 T^{8}$$)
$11$ ($$1 + 320 T + 319702 T^{2} + 51536320 T^{3} + 25937424601 T^{4}$$)($$1 + 254822 T^{2} + 25937424601 T^{4}$$)($$1 + 92262 T^{2} + 25937424601 T^{4}$$)($$1 - 197738 T^{2} + 25937424601 T^{4}$$)($$1 - 320 T + 319702 T^{2} - 51536320 T^{3} + 25937424601 T^{4}$$)($$1 - 396 T + 327873 T^{2} - 67617992 T^{3} + 52804274523 T^{4} - 10271220141996 T^{5} + 4177248169415651 T^{6}$$)($$1 + 396 T + 327873 T^{2} + 67617992 T^{3} + 52804274523 T^{4} + 10271220141996 T^{5} + 4177248169415651 T^{6}$$)($$1 + 382252 T^{2} + 87516901782 T^{4} + 9914632428581452 T^{6} +$$$$67\!\cdots\!01$$$$T^{8}$$)
$13$ ($$1 + 100 T + 297086 T^{2} + 37129300 T^{3} + 137858491849 T^{4}$$)($$( 1 - 154 T + 371293 T^{2} )^{2}$$)($$( 1 - 506 T + 371293 T^{2} )^{2}$$)($$( 1 + 146 T + 371293 T^{2} )^{2}$$)($$1 + 100 T + 297086 T^{2} + 37129300 T^{3} + 137858491849 T^{4}$$)($$1 + 354 T + 845379 T^{2} + 291738444 T^{3} + 313883305047 T^{4} + 48801906114546 T^{5} + 51185893014090757 T^{6}$$)($$1 + 354 T + 845379 T^{2} + 291738444 T^{3} + 313883305047 T^{4} + 48801906114546 T^{5} + 51185893014090757 T^{6}$$)($$( 1 + 292 T - 102402 T^{2} + 108417556 T^{3} + 137858491849 T^{4} )^{2}$$)
$17$ ($$1 - 580 T + 2475814 T^{2} - 823517060 T^{3} + 2015993900449 T^{4}$$)($$( 1 - 178 T + 1419857 T^{2} )^{2}$$)($$( 1 + 1838 T + 1419857 T^{2} )^{2}$$)($$( 1 + 702 T + 1419857 T^{2} )^{2}$$)($$1 - 580 T + 2475814 T^{2} - 823517060 T^{3} + 2015993900449 T^{4}$$)($$1 - 1158 T + 1914111 T^{2} - 544550612 T^{3} + 2717763902127 T^{4} - 2334520936719942 T^{5} + 2862423051509815793 T^{6}$$)($$1 - 1158 T + 1914111 T^{2} - 544550612 T^{3} + 2717763902127 T^{4} - 2334520936719942 T^{5} + 2862423051509815793 T^{6}$$)($$( 1 + 380 T + 2009510 T^{2} + 539545660 T^{3} + 2015993900449 T^{4} )^{2}$$)
$19$ ($$1 + 720 T + 4633798 T^{2} + 1782791280 T^{3} + 6131066257801 T^{4}$$)($$1 + 4019078 T^{2} + 6131066257801 T^{4}$$)($$1 + 687238 T^{2} + 6131066257801 T^{4}$$)($$1 - 2512762 T^{2} + 6131066257801 T^{4}$$)($$1 - 720 T + 4633798 T^{2} - 1782791280 T^{3} + 6131066257801 T^{4}$$)($$1 + 3192 T + 9330057 T^{2} + 15933739216 T^{3} + 23102144807643 T^{4} + 19570363494900792 T^{5} + 15181127029874798299 T^{6}$$)($$1 - 3192 T + 9330057 T^{2} - 15933739216 T^{3} + 23102144807643 T^{4} - 19570363494900792 T^{5} + 15181127029874798299 T^{6}$$)($$1 + 1633036 T^{2} + 154661897526 T^{4} + 10012251917374313836 T^{6} +$$$$37\!\cdots\!01$$$$T^{8}$$)
$23$ ($$1 + 1688 T + 10950502 T^{2} + 10864546984 T^{3} + 41426511213649 T^{4}$$)($$1 + 5934266 T^{2} + 41426511213649 T^{4}$$)($$1 + 9265626 T^{2} + 41426511213649 T^{4}$$)($$1 - 3871674 T^{2} + 41426511213649 T^{4}$$)($$1 - 1688 T + 10950502 T^{2} - 10864546984 T^{3} + 41426511213649 T^{4}$$)($$1 + 6126 T + 29722593 T^{2} + 84141222540 T^{3} + 191304803397399 T^{4} + 253778807694813774 T^{5} +$$$$26\!\cdots\!07$$$$T^{6}$$)($$1 - 6126 T + 29722593 T^{2} - 84141222540 T^{3} + 191304803397399 T^{4} - 253778807694813774 T^{5} +$$$$26\!\cdots\!07$$$$T^{6}$$)($$1 + 25651084 T^{2} + 247347299659206 T^{4} +$$$$10\!\cdots\!16$$$$T^{6} +$$$$17\!\cdots\!01$$$$T^{8}$$)
$29$ ($$1 - 108 T + 39233214 T^{2} - 2215204092 T^{3} + 420707233300201 T^{4}$$)($$( 1 - 4110 T + 20511149 T^{2} )^{2}$$)($$( 1 + 4530 T + 20511149 T^{2} )^{2}$$)($$( 1 + 4010 T + 20511149 T^{2} )^{2}$$)($$1 - 108 T + 39233214 T^{2} - 2215204092 T^{3} + 420707233300201 T^{4}$$)($$1 - 426 T - 939693 T^{2} + 141773503364 T^{3} - 19274183137257 T^{4} - 179221281385885626 T^{5} +$$$$86\!\cdots\!49$$$$T^{6}$$)($$1 - 426 T - 939693 T^{2} + 141773503364 T^{3} - 19274183137257 T^{4} - 179221281385885626 T^{5} +$$$$86\!\cdots\!49$$$$T^{6}$$)($$( 1 + 84 T + 9837118 T^{2} + 1722936516 T^{3} + 420707233300201 T^{4} )^{2}$$)
$31$ ($$1 + 9840 T + 76732702 T^{2} + 281710845840 T^{3} + 819628286980801 T^{4}$$)($$1 + 47289582 T^{2} + 819628286980801 T^{4}$$)($$1 + 43928942 T^{2} + 819628286980801 T^{4}$$)($$1 + 36406942 T^{2} + 819628286980801 T^{4}$$)($$1 - 9840 T + 76732702 T^{2} - 281710845840 T^{3} + 819628286980801 T^{4}$$)($$1 + 3276 T + 2292813 T^{2} - 41639420248 T^{3} + 65641289591763 T^{4} + 2685102268149104076 T^{5} +$$$$23\!\cdots\!51$$$$T^{6}$$)($$1 - 3276 T + 2292813 T^{2} + 41639420248 T^{3} + 65641289591763 T^{4} - 2685102268149104076 T^{5} +$$$$23\!\cdots\!51$$$$T^{6}$$)($$1 + 24283452 T^{2} + 1637830022018822 T^{4} +$$$$19\!\cdots\!52$$$$T^{6} +$$$$67\!\cdots\!01$$$$T^{8}$$)
$37$ ($$1 - 6540 T + 61572814 T^{2} - 453509478780 T^{3} + 4808584372417849 T^{4}$$)($$( 1 - 7442 T + 69343957 T^{2} )^{2}$$)($$( 1 - 338 T + 69343957 T^{2} )^{2}$$)($$( 1 + 14778 T + 69343957 T^{2} )^{2}$$)($$1 - 6540 T + 61572814 T^{2} - 453509478780 T^{3} + 4808584372417849 T^{4}$$)($$1 + 11562 T + 90623691 T^{2} + 525372493468 T^{3} + 6284205331885287 T^{4} + 55596852513895170138 T^{5} +$$$$33\!\cdots\!93$$$$T^{6}$$)($$1 + 11562 T + 90623691 T^{2} + 525372493468 T^{3} + 6284205331885287 T^{4} + 55596852513895170138 T^{5} +$$$$33\!\cdots\!93$$$$T^{6}$$)($$( 1 - 9868 T + 159567054 T^{2} - 684286167676 T^{3} + 4808584372417849 T^{4} )^{2}$$)
$41$ ($$1 + 10620 T + 77106582 T^{2} + 1230392854620 T^{3} + 13422659310152401 T^{4}$$)($$( 1 - 7270 T + 115856201 T^{2} )^{2}$$)($$( 1 + 6330 T + 115856201 T^{2} )^{2}$$)($$( 1 + 4350 T + 115856201 T^{2} )^{2}$$)($$1 + 10620 T + 77106582 T^{2} + 1230392854620 T^{3} + 13422659310152401 T^{4}$$)($$1 - 12450 T + 384843783 T^{2} - 2886023186300 T^{3} + 44586538676848383 T^{4} -$$$$16\!\cdots\!50$$$$T^{5} +$$$$15\!\cdots\!01$$$$T^{6}$$)($$1 - 12450 T + 384843783 T^{2} - 2886023186300 T^{3} + 44586538676848383 T^{4} -$$$$16\!\cdots\!50$$$$T^{5} +$$$$15\!\cdots\!01$$$$T^{6}$$)($$( 1 - 23812 T + 331016342 T^{2} - 2758767858212 T^{3} + 13422659310152401 T^{4} )^{2}$$)
$43$ ($$1 + 25672 T + 364912302 T^{2} + 3774000748696 T^{3} + 21611482313284249 T^{4}$$)($$1 - 26783614 T^{2} + 21611482313284249 T^{4}$$)($$1 - 35859614 T^{2} + 21611482313284249 T^{4}$$)($$1 + 139567886 T^{2} + 21611482313284249 T^{4}$$)($$1 - 25672 T + 364912302 T^{2} - 3774000748696 T^{3} + 21611482313284249 T^{4}$$)($$1 + 26346 T + 640605069 T^{2} + 8098987233524 T^{3} + 94174353771597567 T^{4} +$$$$56\!\cdots\!54$$$$T^{5} +$$$$31\!\cdots\!07$$$$T^{6}$$)($$1 - 26346 T + 640605069 T^{2} - 8098987233524 T^{3} + 94174353771597567 T^{4} -$$$$56\!\cdots\!54$$$$T^{5} +$$$$31\!\cdots\!07$$$$T^{6}$$)($$1 + 385757980 T^{2} + 71059469286049782 T^{4} +$$$$83\!\cdots\!20$$$$T^{6} +$$$$46\!\cdots\!01$$$$T^{8}$$)
$47$ ($$1 + 28296 T + 617782998 T^{2} + 6489546318072 T^{3} + 52599132235830049 T^{4}$$)($$1 + 403777034 T^{2} + 52599132235830049 T^{4}$$)($$1 + 442383274 T^{2} + 52599132235830049 T^{4}$$)($$1 + 422513974 T^{2} + 52599132235830049 T^{4}$$)($$1 - 28296 T + 617782998 T^{2} - 6489546318072 T^{3} + 52599132235830049 T^{4}$$)($$1 + 36762 T + 1119958377 T^{2} + 18490559326820 T^{3} + 256856861812773639 T^{4} +$$$$19\!\cdots\!38$$$$T^{5} +$$$$12\!\cdots\!43$$$$T^{6}$$)($$1 - 36762 T + 1119958377 T^{2} - 18490559326820 T^{3} + 256856861812773639 T^{4} -$$$$19\!\cdots\!38$$$$T^{5} +$$$$12\!\cdots\!43$$$$T^{6}$$)($$1 + 353765740 T^{2} + 66158288717340582 T^{4} +$$$$18\!\cdots\!60$$$$T^{6} +$$$$27\!\cdots\!01$$$$T^{8}$$)
$53$ ($$1 - 31340 T + 755347886 T^{2} - 13106246750620 T^{3} + 174887470365513049 T^{4}$$)($$( 1 - 32226 T + 418195493 T^{2} )^{2}$$)($$( 1 + 15486 T + 418195493 T^{2} )^{2}$$)($$( 1 + 18154 T + 418195493 T^{2} )^{2}$$)($$1 - 31340 T + 755347886 T^{2} - 13106246750620 T^{3} + 174887470365513049 T^{4}$$)($$1 + 21162 T + 395789499 T^{2} - 881498066468 T^{3} + 165517384658528007 T^{4} +$$$$37\!\cdots\!38$$$$T^{5} +$$$$73\!\cdots\!57$$$$T^{6}$$)($$1 + 21162 T + 395789499 T^{2} - 881498066468 T^{3} + 165517384658528007 T^{4} +$$$$37\!\cdots\!38$$$$T^{5} +$$$$73\!\cdots\!57$$$$T^{6}$$)($$( 1 - 8748 T + 833865262 T^{2} - 3658374172764 T^{3} + 174887470365513049 T^{4} )^{2}$$)
$59$ ($$1 + 30800 T + 1666896598 T^{2} + 22019668409200 T^{3} + 511116753300641401 T^{4}$$)($$1 + 270997718 T^{2} + 511116753300641401 T^{4}$$)($$1 + 1376531158 T^{2} + 511116753300641401 T^{4}$$)($$1 + 1041470358 T^{2} + 511116753300641401 T^{4}$$)($$1 - 30800 T + 1666896598 T^{2} - 22019668409200 T^{3} + 511116753300641401 T^{4}$$)($$1 + 35040 T + 1757220897 T^{2} + 50989349593920 T^{3} + 1256279917975876203 T^{4} +$$$$17\!\cdots\!40$$$$T^{5} +$$$$36\!\cdots\!99$$$$T^{6}$$)($$1 - 35040 T + 1757220897 T^{2} - 50989349593920 T^{3} + 1256279917975876203 T^{4} -$$$$17\!\cdots\!40$$$$T^{5} +$$$$36\!\cdots\!99$$$$T^{6}$$)($$1 + 488747308 T^{2} + 896846029819225302 T^{4} +$$$$24\!\cdots\!08$$$$T^{6} +$$$$26\!\cdots\!01$$$$T^{8}$$)
$61$ ($$1 - 24540 T + 1447225822 T^{2} - 20726393226540 T^{3} + 713342911662882601 T^{4}$$)($$( 1 - 26770 T + 844596301 T^{2} )^{2}$$)($$( 1 + 16750 T + 844596301 T^{2} )^{2}$$)($$( 1 + 42130 T + 844596301 T^{2} )^{2}$$)($$1 - 24540 T + 1447225822 T^{2} - 20726393226540 T^{3} + 713342911662882601 T^{4}$$)($$1 + 24138 T + 393086643 T^{2} - 6904061162564 T^{3} + 331999524650307543 T^{4} +$$$$17\!\cdots\!38$$$$T^{5} +$$$$60\!\cdots\!01$$$$T^{6}$$)($$1 + 24138 T + 393086643 T^{2} - 6904061162564 T^{3} + 331999524650307543 T^{4} +$$$$17\!\cdots\!38$$$$T^{5} +$$$$60\!\cdots\!01$$$$T^{6}$$)($$( 1 + 31012 T + 1250446302 T^{2} + 26192620486612 T^{3} + 713342911662882601 T^{4} )^{2}$$)
$67$ ($$1 + 34584 T + 2930656478 T^{2} + 46692726700488 T^{3} + 1822837804551761449 T^{4}$$)($$1 + 219594834 T^{2} + 1822837804551761449 T^{4}$$)($$1 + 2513562674 T^{2} + 1822837804551761449 T^{4}$$)($$1 + 2438310974 T^{2} + 1822837804551761449 T^{4}$$)($$1 - 34584 T + 2930656478 T^{2} - 46692726700488 T^{3} + 1822837804551761449 T^{4}$$)($$1 - 9570 T + 659335509 T^{2} - 83080838420484 T^{3} + 890185424637524463 T^{4} -$$$$17\!\cdots\!30$$$$T^{5} +$$$$24\!\cdots\!43$$$$T^{6}$$)($$1 + 9570 T + 659335509 T^{2} + 83080838420484 T^{3} + 890185424637524463 T^{4} +$$$$17\!\cdots\!30$$$$T^{5} +$$$$24\!\cdots\!43$$$$T^{6}$$)($$1 + 3336863100 T^{2} + 6361754088145172822 T^{4} +$$$$60\!\cdots\!00$$$$T^{6} +$$$$33\!\cdots\!01$$$$T^{8}$$)
$71$ ($$1 - 12400 T + 1143670702 T^{2} - 22372443952400 T^{3} + 3255243551009881201 T^{4}$$)($$1 + 681226622 T^{2} + 3255243551009881201 T^{4}$$)($$1 + 1737195262 T^{2} + 3255243551009881201 T^{4}$$)($$1 + 1542914862 T^{2} + 3255243551009881201 T^{4}$$)($$1 + 12400 T + 1143670702 T^{2} + 22372443952400 T^{3} + 3255243551009881201 T^{4}$$)($$1 + 88092 T + 7289446053 T^{2} + 329541325840584 T^{3} + 13151832521353701603 T^{4} +$$$$28\!\cdots\!92$$$$T^{5} +$$$$58\!\cdots\!51$$$$T^{6}$$)($$1 - 88092 T + 7289446053 T^{2} - 329541325840584 T^{3} + 13151832521353701603 T^{4} -$$$$28\!\cdots\!92$$$$T^{5} +$$$$58\!\cdots\!51$$$$T^{6}$$)($$1 + 6374750812 T^{2} + 16622748841310481702 T^{4} +$$$$20\!\cdots\!12$$$$T^{6} +$$$$10\!\cdots\!01$$$$T^{8}$$)
$73$ ($$1 + 7180 T + 284279286 T^{2} + 14884654037740 T^{3} + 4297625829703557649 T^{4}$$)($$( 1 + 18534 T + 2073071593 T^{2} )^{2}$$)($$( 1 + 20806 T + 2073071593 T^{2} )^{2}$$)($$( 1 - 26266 T + 2073071593 T^{2} )^{2}$$)($$1 + 7180 T + 284279286 T^{2} + 14884654037740 T^{3} + 4297625829703557649 T^{4}$$)($$1 - 66750 T + 3875077479 T^{2} - 111360074258500 T^{3} + 8033313042388954047 T^{4} -$$$$28\!\cdots\!50$$$$T^{5} +$$$$89\!\cdots\!57$$$$T^{6}$$)($$1 - 66750 T + 3875077479 T^{2} - 111360074258500 T^{3} + 8033313042388954047 T^{4} -$$$$28\!\cdots\!50$$$$T^{5} +$$$$89\!\cdots\!57$$$$T^{6}$$)($$( 1 - 59700 T + 5029368950 T^{2} - 123762374102100 T^{3} + 4297625829703557649 T^{4} )^{2}$$)
$79$ ($$1 + 71840 T + 7430807198 T^{2} + 221055731704160 T^{3} + 9468276082626847201 T^{4}$$)($$1 - 1369983522 T^{2} + 9468276082626847201 T^{4}$$)($$1 + 1275471838 T^{2} + 9468276082626847201 T^{4}$$)($$1 + 6078817438 T^{2} + 9468276082626847201 T^{4}$$)($$1 - 71840 T + 7430807198 T^{2} - 221055731704160 T^{3} + 9468276082626847201 T^{4}$$)($$1 - 92952 T + 11164448877 T^{2} - 573846024396496 T^{3} + 34353638858281213923 T^{4} -$$$$88\!\cdots\!52$$$$T^{5} +$$$$29\!\cdots\!99$$$$T^{6}$$)($$1 + 92952 T + 11164448877 T^{2} + 573846024396496 T^{3} + 34353638858281213923 T^{4} +$$$$88\!\cdots\!52$$$$T^{5} +$$$$29\!\cdots\!99$$$$T^{6}$$)($$1 + 10884258108 T^{2} + 48452581383610561862 T^{4} +$$$$10\!\cdots\!08$$$$T^{6} +$$$$89\!\cdots\!01$$$$T^{8}$$)
$83$ ($$1 - 31928 T + 5910993662 T^{2} - 125765689649704 T^{3} + 15516041187205853449 T^{4}$$)($$1 + 1693436786 T^{2} + 15516041187205853449 T^{4}$$)($$1 - 3421895214 T^{2} + 15516041187205853449 T^{4}$$)($$1 - 1875047714 T^{2} + 15516041187205853449 T^{4}$$)($$1 + 31928 T + 5910993662 T^{2} + 125765689649704 T^{3} + 15516041187205853449 T^{4}$$)($$1 + 30258 T + 4293702405 T^{2} + 426012342532708 T^{3} + 16913068282241846415 T^{4} +$$$$46\!\cdots\!42$$$$T^{5} +$$$$61\!\cdots\!07$$$$T^{6}$$)($$1 - 30258 T + 4293702405 T^{2} - 426012342532708 T^{3} + 16913068282241846415 T^{4} -$$$$46\!\cdots\!42$$$$T^{5} +$$$$61\!\cdots\!07$$$$T^{6}$$)($$1 + 7906443964 T^{2} + 42876570344126662806 T^{4} +$$$$12\!\cdots\!36$$$$T^{6} +$$$$24\!\cdots\!01$$$$T^{8}$$)
$89$ ($$1 + 40748 T + 8570866774 T^{2} + 227539254427852 T^{3} + 31181719929966183601 T^{4}$$)($$( 1 + 107590 T + 5584059449 T^{2} )^{2}$$)($$( 1 + 18310 T + 5584059449 T^{2} )^{2}$$)($$( 1 - 30570 T + 5584059449 T^{2} )^{2}$$)($$1 + 40748 T + 8570866774 T^{2} + 227539254427852 T^{3} + 31181719929966183601 T^{4}$$)($$1 - 172686 T + 26445328791 T^{2} - 2103593815517412 T^{3} +$$$$14\!\cdots\!59$$$$T^{4} -$$$$53\!\cdots\!86$$$$T^{5} +$$$$17\!\cdots\!49$$$$T^{6}$$)($$1 - 172686 T + 26445328791 T^{2} - 2103593815517412 T^{3} +$$$$14\!\cdots\!59$$$$T^{4} -$$$$53\!\cdots\!86$$$$T^{5} +$$$$17\!\cdots\!49$$$$T^{6}$$)($$( 1 - 104660 T + 13126874198 T^{2} - 584427661932340 T^{3} + 31181719929966183601 T^{4} )^{2}$$)
$97$ ($$1 + 190140 T + 19653817414 T^{2} + 1632796876465980 T^{3} + 73742412689492826049 T^{4}$$)($$( 1 + 108838 T + 8587340257 T^{2} )^{2}$$)($$( 1 - 49978 T + 8587340257 T^{2} )^{2}$$)($$( 1 - 66882 T + 8587340257 T^{2} )^{2}$$)($$1 + 190140 T + 19653817414 T^{2} + 1632796876465980 T^{3} + 73742412689492826049 T^{4}$$)($$1 - 170910 T + 30283966671 T^{2} - 2852314667192740 T^{3} +$$$$26\!\cdots\!47$$$$T^{4} -$$$$12\!\cdots\!90$$$$T^{5} +$$$$63\!\cdots\!93$$$$T^{6}$$)($$1 - 170910 T + 30283966671 T^{2} - 2852314667192740 T^{3} +$$$$26\!\cdots\!47$$$$T^{4} -$$$$12\!\cdots\!90$$$$T^{5} +$$$$63\!\cdots\!93$$$$T^{6}$$)($$( 1 + 29564 T + 14772618438 T^{2} + 253876127357948 T^{3} + 73742412689492826049 T^{4} )^{2}$$)