# Properties

 Label 80.10.a Level 80 Weight 10 Character orbit a Rep. character $$\chi_{80}(1,\cdot)$$ Character field $$\Q$$ Dimension 18 Newform subspaces 11 Sturm bound 120 Trace bound 3

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$80 = 2^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 80.a (trivial) Character field: $$\Q$$ Newform subspaces: $$11$$ Sturm bound: $$120$$ Trace bound: $$3$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{10}(\Gamma_0(80))$$.

Total New Old
Modular forms 114 18 96
Cusp forms 102 18 84
Eisenstein series 12 0 12

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.
$$+$$$$+$$$$+$$$$4$$
$$+$$$$-$$$$-$$$$5$$
$$-$$$$+$$$$-$$$$5$$
$$-$$$$-$$$$+$$$$4$$
Plus space$$+$$$$8$$
Minus space$$-$$$$10$$

## Trace form

 $$18q + 162q^{3} - 2750q^{7} + 145838q^{9} + O(q^{10})$$ $$18q + 162q^{3} - 2750q^{7} + 145838q^{9} - 65860q^{11} - 101250q^{15} - 242212q^{17} - 429696q^{19} + 317084q^{21} + 53942q^{23} + 7031250q^{25} + 10576572q^{27} - 2572316q^{29} - 18789876q^{31} + 9585408q^{33} + 9003750q^{35} - 1214776q^{37} - 45636732q^{39} - 5888256q^{41} + 103684082q^{43} - 6902500q^{45} - 103174654q^{47} + 133620662q^{49} + 220753972q^{51} + 54219264q^{53} - 36602500q^{55} + 10596936q^{57} + 158263288q^{59} + 155249152q^{61} - 118675982q^{63} - 87902500q^{65} + 730750038q^{67} - 206565164q^{69} + 87544716q^{71} + 52841052q^{73} + 63281250q^{75} + 160708208q^{77} + 1224570360q^{79} + 1368858734q^{81} - 754085846q^{83} + 214605000q^{85} - 136660980q^{87} - 997034220q^{89} - 2165382476q^{91} + 803195336q^{93} - 651605000q^{95} + 1276631188q^{97} - 5943909556q^{99} + O(q^{100})$$

## Decomposition of $$S_{10}^{\mathrm{new}}(\Gamma_0(80))$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces A-L signs $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$ 2 5
80.10.a.a $$1$$ $$41.203$$ $$\Q$$ None $$0$$ $$-174$$ $$-625$$ $$-4658$$ $$-$$ $$+$$ $$q-174q^{3}-5^{4}q^{5}-4658q^{7}+10593q^{9}+\cdots$$
80.10.a.b $$1$$ $$41.203$$ $$\Q$$ None $$0$$ $$-46$$ $$-625$$ $$10318$$ $$-$$ $$+$$ $$q-46q^{3}-5^{4}q^{5}+10318q^{7}-17567q^{9}+\cdots$$
80.10.a.c $$1$$ $$41.203$$ $$\Q$$ None $$0$$ $$48$$ $$625$$ $$532$$ $$-$$ $$-$$ $$q+48q^{3}+5^{4}q^{5}+532q^{7}-17379q^{9}+\cdots$$
80.10.a.d $$1$$ $$41.203$$ $$\Q$$ None $$0$$ $$114$$ $$-625$$ $$-4242$$ $$-$$ $$+$$ $$q+114q^{3}-5^{4}q^{5}-4242q^{7}-6687q^{9}+\cdots$$
80.10.a.e $$1$$ $$41.203$$ $$\Q$$ None $$0$$ $$204$$ $$625$$ $$-5432$$ $$-$$ $$-$$ $$q+204q^{3}+5^{4}q^{5}-5432q^{7}+21933q^{9}+\cdots$$
80.10.a.f $$2$$ $$41.203$$ $$\Q(\sqrt{1009})$$ None $$0$$ $$-260$$ $$1250$$ $$-1700$$ $$-$$ $$-$$ $$q+(-130-\beta )q^{3}+5^{4}q^{5}+(-850+\cdots)q^{7}+\cdots$$
80.10.a.g $$2$$ $$41.203$$ $$\Q(\sqrt{46})$$ None $$0$$ $$-108$$ $$-1250$$ $$908$$ $$+$$ $$+$$ $$q+(-54+\beta )q^{3}-5^{4}q^{5}+(454+13\beta )q^{7}+\cdots$$
80.10.a.h $$2$$ $$41.203$$ $$\Q(\sqrt{6049})$$ None $$0$$ $$92$$ $$1250$$ $$6908$$ $$+$$ $$-$$ $$q+(46-\beta )q^{3}+5^{4}q^{5}+(3454+37\beta )q^{7}+\cdots$$
80.10.a.i $$2$$ $$41.203$$ $$\Q(\sqrt{22})$$ None $$0$$ $$116$$ $$-1250$$ $$-11284$$ $$+$$ $$+$$ $$q+(58+\beta )q^{3}-5^{4}q^{5}+(-5642-35\beta )q^{7}+\cdots$$
80.10.a.j $$2$$ $$41.203$$ $$\Q(\sqrt{79})$$ None $$0$$ $$260$$ $$-1250$$ $$380$$ $$-$$ $$+$$ $$q+(130+\beta )q^{3}-5^{4}q^{5}+(190+69\beta )q^{7}+\cdots$$
80.10.a.k $$3$$ $$41.203$$ 3.3.7117.1 None $$0$$ $$-84$$ $$1875$$ $$5520$$ $$+$$ $$-$$ $$q+(-28-\beta _{1})q^{3}+5^{4}q^{5}+(1840-17\beta _{1}+\cdots)q^{7}+\cdots$$

## Decomposition of $$S_{10}^{\mathrm{old}}(\Gamma_0(80))$$ into lower level spaces

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ ($$1 + 174 T + 19683 T^{2}$$)($$1 + 46 T + 19683 T^{2}$$)($$1 - 48 T + 19683 T^{2}$$)($$1 - 114 T + 19683 T^{2}$$)($$1 - 204 T + 19683 T^{2}$$)($$1 + 260 T + 52230 T^{2} + 5117580 T^{3} + 387420489 T^{4}$$)($$1 + 108 T + 15786 T^{2} + 2125764 T^{3} + 387420489 T^{4}$$)($$1 - 92 T + 17286 T^{2} - 1810836 T^{3} + 387420489 T^{4}$$)($$1 - 116 T + 7530 T^{2} - 2283228 T^{3} + 387420489 T^{4}$$)($$1 - 260 T + 36042 T^{2} - 5117580 T^{3} + 387420489 T^{4}$$)($$1 + 84 T + 9513 T^{2} + 2615544 T^{3} + 187244379 T^{4} + 32543321076 T^{5} + 7625597484987 T^{6}$$)
$5$ ($$1 + 625 T$$)($$1 + 625 T$$)($$1 - 625 T$$)($$1 + 625 T$$)($$1 - 625 T$$)($$( 1 - 625 T )^{2}$$)($$( 1 + 625 T )^{2}$$)($$( 1 - 625 T )^{2}$$)($$( 1 + 625 T )^{2}$$)($$( 1 + 625 T )^{2}$$)($$( 1 - 625 T )^{3}$$)
$7$ ($$1 + 4658 T + 40353607 T^{2}$$)($$1 - 10318 T + 40353607 T^{2}$$)($$1 - 532 T + 40353607 T^{2}$$)($$1 + 4242 T + 40353607 T^{2}$$)($$1 + 5432 T + 40353607 T^{2}$$)($$1 + 1700 T + 35221550 T^{2} + 68601131900 T^{3} + 1628413597910449 T^{4}$$)($$1 - 908 T + 76435506 T^{2} - 36641075156 T^{3} + 1628413597910449 T^{4}$$)($$1 - 6908 T + 59513006 T^{2} - 278762717156 T^{3} + 1628413597910449 T^{4}$$)($$1 + 11284 T + 69419378 T^{2} + 455350101388 T^{3} + 1628413597910449 T^{4}$$)($$1 - 380 T - 15543150 T^{2} - 15334370660 T^{3} + 1628413597910449 T^{4}$$)($$1 - 5520 T + 62218149 T^{2} - 328164567136 T^{3} + 2510726733013443 T^{4} - 8988843060465678480 T^{5} +$$$$65\!\cdots\!43$$$$T^{6}$$)
$11$ ($$1 + 28992 T + 2357947691 T^{2}$$)($$1 - 5568 T + 2357947691 T^{2}$$)($$1 - 33180 T + 2357947691 T^{2}$$)($$1 - 46208 T + 2357947691 T^{2}$$)($$1 + 73932 T + 2357947691 T^{2}$$)($$1 + 23984 T + 1217213446 T^{2} + 56553017420944 T^{3} + 5559917313492231481 T^{4}$$)($$1 + 25120 T + 4397886806 T^{2} + 59231645997920 T^{3} + 5559917313492231481 T^{4}$$)($$1 - 8080 T + 4065472006 T^{2} - 19052217343280 T^{3} + 5559917313492231481 T^{4}$$)($$1 - 101408 T + 6891283798 T^{2} - 239114759448928 T^{3} + 5559917313492231481 T^{4}$$)($$1 + 102720 T + 7335543382 T^{2} + 242208386819520 T^{3} + 5559917313492231481 T^{4}$$)($$1 + 5556 T + 2594856177 T^{2} + 72877014191416 T^{3} + 6118535131034237307 T^{4} +$$$$30\!\cdots\!36$$$$T^{5} +$$$$13\!\cdots\!71$$$$T^{6}$$)
$13$ ($$1 + 164446 T + 10604499373 T^{2}$$)($$1 - 45986 T + 10604499373 T^{2}$$)($$1 + 99682 T + 10604499373 T^{2}$$)($$1 + 115934 T + 10604499373 T^{2}$$)($$1 + 114514 T + 10604499373 T^{2}$$)($$1 - 115020 T + 22672043710 T^{2} - 1219729517882460 T^{3} +$$$$11\!\cdots\!29$$$$T^{4}$$)($$1 - 146948 T + 18650572638 T^{2} - 1558309973863604 T^{3} +$$$$11\!\cdots\!29$$$$T^{4}$$)($$1 - 111948 T + 22805544638 T^{2} - 1187152495808604 T^{3} +$$$$11\!\cdots\!29$$$$T^{4}$$)($$1 + 21372 T + 21196469342 T^{2} + 226639360599756 T^{3} +$$$$11\!\cdots\!29$$$$T^{4}$$)($$1 - 179140 T + 22798610142 T^{2} - 1899690017679220 T^{3} +$$$$11\!\cdots\!29$$$$T^{4}$$)($$1 + 83094 T + 20077033299 T^{2} + 754871996338436 T^{3} +$$$$21\!\cdots\!27$$$$T^{4} +$$$$93\!\cdots\!26$$$$T^{5} +$$$$11\!\cdots\!17$$$$T^{6}$$)
$17$ ($$1 + 594822 T + 118587876497 T^{2}$$)($$1 + 381318 T + 118587876497 T^{2}$$)($$1 + 443454 T + 118587876497 T^{2}$$)($$1 - 494842 T + 118587876497 T^{2}$$)($$1 - 41682 T + 118587876497 T^{2}$$)($$1 - 412820 T + 113016614470 T^{2} - 48955447175491540 T^{3} +$$$$14\!\cdots\!09$$$$T^{4}$$)($$1 - 169268 T + 244287370694 T^{2} - 20073132678894196 T^{3} +$$$$14\!\cdots\!09$$$$T^{4}$$)($$1 + 327532 T + 241881460294 T^{2} + 38841324364815404 T^{3} +$$$$14\!\cdots\!09$$$$T^{4}$$)($$1 + 296780 T + 144639901894 T^{2} + 35194509986779660 T^{3} +$$$$14\!\cdots\!09$$$$T^{4}$$)($$1 - 316020 T + 259693705798 T^{2} - 37476140730581940 T^{3} +$$$$14\!\cdots\!09$$$$T^{4}$$)($$1 - 367062 T + 268505189631 T^{2} - 69711192496917428 T^{3} +$$$$31\!\cdots\!07$$$$T^{4} -$$$$51\!\cdots\!58$$$$T^{5} +$$$$16\!\cdots\!73$$$$T^{6}$$)
$19$ ($$1 - 295780 T + 322687697779 T^{2}$$)($$1 + 610460 T + 322687697779 T^{2}$$)($$1 - 357244 T + 322687697779 T^{2}$$)($$1 - 1008740 T + 322687697779 T^{2}$$)($$1 + 1057460 T + 322687697779 T^{2}$$)($$1 - 296520 T + 659218232758 T^{2} - 95683356145429080 T^{3} +$$$$10\!\cdots\!41$$$$T^{4}$$)($$1 - 25480 T + 371498689782 T^{2} - 8222082539408920 T^{3} +$$$$10\!\cdots\!41$$$$T^{4}$$)($$1 - 1156680 T + 686155308982 T^{2} - 373246406267013720 T^{3} +$$$$10\!\cdots\!41$$$$T^{4}$$)($$1 + 275832 T + 612347527414 T^{2} + 89007593053777128 T^{3} +$$$$10\!\cdots\!41$$$$T^{4}$$)($$1 + 137272 T + 111610161654 T^{2} + 44295985649518888 T^{3} +$$$$10\!\cdots\!41$$$$T^{4}$$)($$1 + 1489116 T + 1342580105481 T^{2} + 830560424045529832 T^{3} +$$$$43\!\cdots\!99$$$$T^{4} +$$$$15\!\cdots\!56$$$$T^{5} +$$$$33\!\cdots\!39$$$$T^{6}$$)
$23$ ($$1 + 2544534 T + 1801152661463 T^{2}$$)($$1 - 1447914 T + 1801152661463 T^{2}$$)($$1 - 142956 T + 1801152661463 T^{2}$$)($$1 - 532554 T + 1801152661463 T^{2}$$)($$1 + 1599336 T + 1801152661463 T^{2}$$)($$1 - 1049220 T + 3497852029390 T^{2} - 1889805395460208860 T^{3} +$$$$32\!\cdots\!69$$$$T^{4}$$)($$1 + 1782748 T + 3965893132658 T^{2} + 3211001304917840324 T^{3} +$$$$32\!\cdots\!69$$$$T^{4}$$)($$1 - 1057252 T + 1690239470158 T^{2} - 1904272253637079676 T^{3} +$$$$32\!\cdots\!69$$$$T^{4}$$)($$1 - 585284 T + 3430385383090 T^{2} - 1054185834311710492 T^{3} +$$$$32\!\cdots\!69$$$$T^{4}$$)($$1 - 665460 T + 2886450615250 T^{2} - 1198595050097167980 T^{3} +$$$$32\!\cdots\!69$$$$T^{4}$$)($$1 - 499920 T - 191246545323 T^{2} + 4150866877865471776 T^{3} -$$$$34\!\cdots\!49$$$$T^{4} -$$$$16\!\cdots\!80$$$$T^{5} +$$$$58\!\cdots\!47$$$$T^{6}$$)
$29$ ($$1 + 3722970 T + 14507145975869 T^{2}$$)($$1 - 5385510 T + 14507145975869 T^{2}$$)($$1 - 1527966 T + 14507145975869 T^{2}$$)($$1 - 4196390 T + 14507145975869 T^{2}$$)($$1 - 2184510 T + 14507145975869 T^{2}$$)($$1 + 3666980 T + 20832571957438 T^{2} + 53197414150592105620 T^{3} +$$$$21\!\cdots\!61$$$$T^{4}$$)($$1 - 7323340 T + 29495257443614 T^{2} -$$$$10\!\cdots\!60$$$$T^{3} +$$$$21\!\cdots\!61$$$$T^{4}$$)($$1 + 4212260 T + 32604383130814 T^{2} + 61107870708313953940 T^{3} +$$$$21\!\cdots\!61$$$$T^{4}$$)($$1 + 9928756 T + 50878662529822 T^{2} +$$$$14\!\cdots\!64$$$$T^{3} +$$$$21\!\cdots\!61$$$$T^{4}$$)($$1 + 6893748 T + 40195999658014 T^{2} +$$$$10\!\cdots\!12$$$$T^{3} +$$$$21\!\cdots\!61$$$$T^{4}$$)($$1 - 5234682 T + 42736985914563 T^{2} -$$$$14\!\cdots\!08$$$$T^{3} +$$$$61\!\cdots\!47$$$$T^{4} -$$$$11\!\cdots\!02$$$$T^{5} +$$$$30\!\cdots\!09$$$$T^{6}$$)
$31$ ($$1 + 2335772 T + 26439622160671 T^{2}$$)($$1 + 3053852 T + 26439622160671 T^{2}$$)($$1 + 7323416 T + 26439622160671 T^{2}$$)($$1 - 3365028 T + 26439622160671 T^{2}$$)($$1 - 9619648 T + 26439622160671 T^{2}$$)($$1 + 1613144 T + 29382323902526 T^{2} + 42650917850753459624 T^{3} +$$$$69\!\cdots\!41$$$$T^{4}$$)($$1 + 10677272 T + 73017326150862 T^{2} +$$$$28\!\cdots\!12$$$$T^{3} +$$$$69\!\cdots\!41$$$$T^{4}$$)($$1 - 11361128 T + 71055092544062 T^{2} -$$$$30\!\cdots\!88$$$$T^{3} +$$$$69\!\cdots\!41$$$$T^{4}$$)($$1 + 5131480 T + 36749057608142 T^{2} +$$$$13\!\cdots\!80$$$$T^{3} +$$$$69\!\cdots\!41$$$$T^{4}$$)($$1 + 291832 T + 38964935800398 T^{2} + 7715927814392939272 T^{3} +$$$$69\!\cdots\!41$$$$T^{4}$$)($$1 + 12708912 T + 128130963574173 T^{2} +$$$$72\!\cdots\!04$$$$T^{3} +$$$$33\!\cdots\!83$$$$T^{4} +$$$$88\!\cdots\!92$$$$T^{5} +$$$$18\!\cdots\!11$$$$T^{6}$$)
$37$ ($$1 - 10840418 T + 129961739795077 T^{2}$$)($$1 - 12889442 T + 129961739795077 T^{2}$$)($$1 + 2666842 T + 129961739795077 T^{2}$$)($$1 + 14931358 T + 129961739795077 T^{2}$$)($$1 - 4799942 T + 129961739795077 T^{2}$$)($$1 + 21121940 T + 328931801286510 T^{2} +$$$$27\!\cdots\!80$$$$T^{3} +$$$$16\!\cdots\!29$$$$T^{4}$$)($$1 + 5750460 T + 104099098360718 T^{2} +$$$$74\!\cdots\!20$$$$T^{3} +$$$$16\!\cdots\!29$$$$T^{4}$$)($$1 + 7251860 T + 207906306187118 T^{2} +$$$$94\!\cdots\!20$$$$T^{3} +$$$$16\!\cdots\!29$$$$T^{4}$$)($$1 + 11007932 T + 240473943386510 T^{2} +$$$$14\!\cdots\!64$$$$T^{3} +$$$$16\!\cdots\!29$$$$T^{4}$$)($$1 - 11261380 T + 218879982937230 T^{2} -$$$$14\!\cdots\!60$$$$T^{3} +$$$$16\!\cdots\!29$$$$T^{4}$$)($$1 - 21724434 T + 286542561812475 T^{2} -$$$$28\!\cdots\!32$$$$T^{3} +$$$$37\!\cdots\!75$$$$T^{4} -$$$$36\!\cdots\!86$$$$T^{5} +$$$$21\!\cdots\!33$$$$T^{6}$$)
$41$ ($$1 - 21593862 T + 327381934393961 T^{2}$$)($$1 + 33786618 T + 327381934393961 T^{2}$$)($$1 + 7939014 T + 327381934393961 T^{2}$$)($$1 - 11056262 T + 327381934393961 T^{2}$$)($$1 - 9531882 T + 327381934393961 T^{2}$$)($$1 + 26957276 T + 811945448362966 T^{2} +$$$$88\!\cdots\!36$$$$T^{3} +$$$$10\!\cdots\!21$$$$T^{4}$$)($$1 + 7795764 T + 505248245475190 T^{2} +$$$$25\!\cdots\!04$$$$T^{3} +$$$$10\!\cdots\!21$$$$T^{4}$$)($$1 - 13030436 T - 112698304084010 T^{2} -$$$$42\!\cdots\!96$$$$T^{3} +$$$$10\!\cdots\!21$$$$T^{4}$$)($$1 + 41835956 T + 1084325213081206 T^{2} +$$$$13\!\cdots\!16$$$$T^{3} +$$$$10\!\cdots\!21$$$$T^{4}$$)($$1 - 29773452 T + 771012402449398 T^{2} -$$$$97\!\cdots\!72$$$$T^{3} +$$$$10\!\cdots\!21$$$$T^{4}$$)($$1 - 27440478 T + 1215792148252263 T^{2} -$$$$18\!\cdots\!72$$$$T^{3} +$$$$39\!\cdots\!43$$$$T^{4} -$$$$29\!\cdots\!38$$$$T^{5} +$$$$35\!\cdots\!81$$$$T^{6}$$)
$43$ ($$1 + 10832294 T + 502592611936843 T^{2}$$)($$1 - 36886234 T + 502592611936843 T^{2}$$)($$1 - 21174520 T + 502592611936843 T^{2}$$)($$1 - 6396794 T + 502592611936843 T^{2}$$)($$1 - 13464484 T + 502592611936843 T^{2}$$)($$1 + 52889700 T + 1703843788760950 T^{2} +$$$$26\!\cdots\!00$$$$T^{3} +$$$$25\!\cdots\!49$$$$T^{4}$$)($$1 + 16770524 T + 1074543668703834 T^{2} +$$$$84\!\cdots\!32$$$$T^{3} +$$$$25\!\cdots\!49$$$$T^{4}$$)($$1 - 47934076 T + 1577772318092534 T^{2} -$$$$24\!\cdots\!68$$$$T^{3} +$$$$25\!\cdots\!49$$$$T^{4}$$)($$1 - 23394052 T + 925906061281562 T^{2} -$$$$11\!\cdots\!36$$$$T^{3} +$$$$25\!\cdots\!49$$$$T^{4}$$)($$1 - 11708180 T + 838769843899386 T^{2} -$$$$58\!\cdots\!40$$$$T^{3} +$$$$25\!\cdots\!49$$$$T^{4}$$)($$1 - 23218260 T + 251755624778721 T^{2} -$$$$25\!\cdots\!68$$$$T^{3} +$$$$12\!\cdots\!03$$$$T^{4} -$$$$58\!\cdots\!40$$$$T^{5} +$$$$12\!\cdots\!07$$$$T^{6}$$)
$47$ ($$1 + 5172138 T + 1119130473102767 T^{2}$$)($$1 - 44163798 T + 1119130473102767 T^{2}$$)($$1 + 16059636 T + 1119130473102767 T^{2}$$)($$1 - 35559158 T + 1119130473102767 T^{2}$$)($$1 + 11441952 T + 1119130473102767 T^{2}$$)($$1 + 58412180 T + 2814913257457630 T^{2} +$$$$65\!\cdots\!60$$$$T^{3} +$$$$12\!\cdots\!89$$$$T^{4}$$)($$1 + 15393892 T + 1097719682118050 T^{2} +$$$$17\!\cdots\!64$$$$T^{3} +$$$$12\!\cdots\!89$$$$T^{4}$$)($$1 + 30914292 T + 2034610190905950 T^{2} +$$$$34\!\cdots\!64$$$$T^{3} +$$$$12\!\cdots\!89$$$$T^{4}$$)($$1 + 11711748 T - 733120788879390 T^{2} +$$$$13\!\cdots\!16$$$$T^{3} +$$$$12\!\cdots\!89$$$$T^{4}$$)($$1 + 62493300 T + 3177958884734338 T^{2} +$$$$69\!\cdots\!00$$$$T^{3} +$$$$12\!\cdots\!89$$$$T^{4}$$)($$1 - 28701528 T + 2689800471894429 T^{2} -$$$$45\!\cdots\!28$$$$T^{3} +$$$$30\!\cdots\!43$$$$T^{4} -$$$$35\!\cdots\!92$$$$T^{5} +$$$$14\!\cdots\!63$$$$T^{6}$$)
$53$ ($$1 - 98179674 T + 3299763591802133 T^{2}$$)($$1 - 29746266 T + 3299763591802133 T^{2}$$)($$1 + 87822234 T + 3299763591802133 T^{2}$$)($$1 - 39738586 T + 3299763591802133 T^{2}$$)($$1 - 53615766 T + 3299763591802133 T^{2}$$)($$1 + 39035140 T + 5675030678030830 T^{2} +$$$$12\!\cdots\!20$$$$T^{3} +$$$$10\!\cdots\!89$$$$T^{4}$$)($$1 + 58529292 T + 3439533688251982 T^{2} +$$$$19\!\cdots\!36$$$$T^{3} +$$$$10\!\cdots\!89$$$$T^{4}$$)($$1 - 100922108 T + 9120851179993582 T^{2} -$$$$33\!\cdots\!64$$$$T^{3} +$$$$10\!\cdots\!89$$$$T^{4}$$)($$1 + 46384268 T + 6816046668377422 T^{2} +$$$$15\!\cdots\!44$$$$T^{3} +$$$$10\!\cdots\!89$$$$T^{4}$$)($$1 - 9417780 T + 5708185761526990 T^{2} -$$$$31\!\cdots\!40$$$$T^{3} +$$$$10\!\cdots\!89$$$$T^{4}$$)($$1 + 45629982 T + 7541283402476907 T^{2} +$$$$30\!\cdots\!96$$$$T^{3} +$$$$24\!\cdots\!31$$$$T^{4} +$$$$49\!\cdots\!98$$$$T^{5} +$$$$35\!\cdots\!37$$$$T^{6}$$)
$59$ ($$1 + 16162860 T + 8662995818654939 T^{2}$$)($$1 - 65575380 T + 8662995818654939 T^{2}$$)($$1 + 120625212 T + 8662995818654939 T^{2}$$)($$1 - 85185620 T + 8662995818654939 T^{2}$$)($$1 + 81862620 T + 8662995818654939 T^{2}$$)($$1 - 54995560 T + 15674484224932678 T^{2} -$$$$47\!\cdots\!40$$$$T^{3} +$$$$75\!\cdots\!21$$$$T^{4}$$)($$1 + 59618264 T + 11433756193805126 T^{2} +$$$$51\!\cdots\!96$$$$T^{3} +$$$$75\!\cdots\!21$$$$T^{4}$$)($$1 - 47362536 T + 6051611540480326 T^{2} -$$$$41\!\cdots\!04$$$$T^{3} +$$$$75\!\cdots\!21$$$$T^{4}$$)($$1 + 178239576 T + 18856238015031622 T^{2} +$$$$15\!\cdots\!64$$$$T^{3} +$$$$75\!\cdots\!21$$$$T^{4}$$)($$1 - 92930856 T + 16656477955483462 T^{2} -$$$$80\!\cdots\!84$$$$T^{3} +$$$$75\!\cdots\!21$$$$T^{4}$$)($$1 - 268721868 T + 46457702853914817 T^{2} -$$$$50\!\cdots\!40$$$$T^{3} +$$$$40\!\cdots\!63$$$$T^{4} -$$$$20\!\cdots\!28$$$$T^{5} +$$$$65\!\cdots\!19$$$$T^{6}$$)
$61$ ($$1 + 43928158 T + 11694146092834141 T^{2}$$)($$1 - 40183202 T + 11694146092834141 T^{2}$$)($$1 - 93576542 T + 11694146092834141 T^{2}$$)($$1 - 45748642 T + 11694146092834141 T^{2}$$)($$1 + 104691298 T + 11694146092834141 T^{2}$$)($$1 + 274579716 T + 41753623519328446 T^{2} +$$$$32\!\cdots\!56$$$$T^{3} +$$$$13\!\cdots\!81$$$$T^{4}$$)($$1 + 188163772 T + 22324076261167422 T^{2} +$$$$22\!\cdots\!52$$$$T^{3} +$$$$13\!\cdots\!81$$$$T^{4}$$)($$1 - 203634428 T + 30920737906297022 T^{2} -$$$$23\!\cdots\!48$$$$T^{3} +$$$$13\!\cdots\!81$$$$T^{4}$$)($$1 - 31825220 T + 17079149243685182 T^{2} -$$$$37\!\cdots\!20$$$$T^{3} +$$$$13\!\cdots\!81$$$$T^{4}$$)($$1 - 195673924 T + 22434263296171326 T^{2} -$$$$22\!\cdots\!84$$$$T^{3} +$$$$13\!\cdots\!81$$$$T^{4}$$)($$1 - 155970138 T + 27601780143557283 T^{2} -$$$$34\!\cdots\!16$$$$T^{3} +$$$$32\!\cdots\!03$$$$T^{4} -$$$$21\!\cdots\!78$$$$T^{5} +$$$$15\!\cdots\!21$$$$T^{6}$$)
$67$ ($$1 - 81557422 T + 27206534396294947 T^{2}$$)($$1 - 115706158 T + 27206534396294947 T^{2}$$)($$1 + 193621688 T + 27206534396294947 T^{2}$$)($$1 - 45286158 T + 27206534396294947 T^{2}$$)($$1 + 140571092 T + 27206534396294947 T^{2}$$)($$1 - 318580 T + 48520062064444070 T^{2} -$$$$86\!\cdots\!60$$$$T^{3} +$$$$74\!\cdots\!09$$$$T^{4}$$)($$1 - 105998252 T + 21995694557368746 T^{2} -$$$$28\!\cdots\!44$$$$T^{3} +$$$$74\!\cdots\!09$$$$T^{4}$$)($$1 - 58872852 T + 54767076047787046 T^{2} -$$$$16\!\cdots\!44$$$$T^{3} +$$$$74\!\cdots\!09$$$$T^{4}$$)($$1 + 89480628 T + 42050726086531690 T^{2} +$$$$24\!\cdots\!16$$$$T^{3} +$$$$74\!\cdots\!09$$$$T^{4}$$)($$1 - 219767420 T + 65652945987990090 T^{2} -$$$$59\!\cdots\!40$$$$T^{3} +$$$$74\!\cdots\!09$$$$T^{4}$$)($$1 - 526916604 T + 169230786613422441 T^{2} -$$$$33\!\cdots\!48$$$$T^{3} +$$$$46\!\cdots\!27$$$$T^{4} -$$$$39\!\cdots\!36$$$$T^{5} +$$$$20\!\cdots\!23$$$$T^{6}$$)
$71$ ($$1 + 161307732 T + 45848500718449031 T^{2}$$)($$1 - 231681708 T + 45848500718449031 T^{2}$$)($$1 + 417763488 T + 45848500718449031 T^{2}$$)($$1 - 189967468 T + 45848500718449031 T^{2}$$)($$1 + 97098792 T + 45848500718449031 T^{2}$$)($$1 - 7130936 T + 51935375688707086 T^{2} -$$$$32\!\cdots\!16$$$$T^{3} +$$$$21\!\cdots\!61$$$$T^{4}$$)($$1 - 168665592 T + 68474091725761822 T^{2} -$$$$77\!\cdots\!52$$$$T^{3} +$$$$21\!\cdots\!61$$$$T^{4}$$)($$1 - 349900792 T + 118671349360183822 T^{2} -$$$$16\!\cdots\!52$$$$T^{3} +$$$$21\!\cdots\!61$$$$T^{4}$$)($$1 + 112319176 T + 82264334516632606 T^{2} +$$$$51\!\cdots\!56$$$$T^{3} +$$$$21\!\cdots\!61$$$$T^{4}$$)($$1 + 311207016 T + 76405636625293726 T^{2} +$$$$14\!\cdots\!96$$$$T^{3} +$$$$21\!\cdots\!61$$$$T^{4}$$)($$1 - 239894424 T + 117827037562982037 T^{2} -$$$$16\!\cdots\!88$$$$T^{3} +$$$$54\!\cdots\!47$$$$T^{4} -$$$$50\!\cdots\!64$$$$T^{5} +$$$$96\!\cdots\!91$$$$T^{6}$$)
$73$ ($$1 + 247147966 T + 58871586708267913 T^{2}$$)($$1 - 358691906 T + 58871586708267913 T^{2}$$)($$1 + 450372742 T + 58871586708267913 T^{2}$$)($$1 - 412170946 T + 58871586708267913 T^{2}$$)($$1 - 171848906 T + 58871586708267913 T^{2}$$)($$1 - 120858180 T + 42707263689423190 T^{2} -$$$$71\!\cdots\!40$$$$T^{3} +$$$$34\!\cdots\!69$$$$T^{4}$$)($$1 + 390212412 T + 130694746296409238 T^{2} +$$$$22\!\cdots\!56$$$$T^{3} +$$$$34\!\cdots\!69$$$$T^{4}$$)($$1 - 71160388 T - 50883178339169962 T^{2} -$$$$41\!\cdots\!44$$$$T^{3} +$$$$34\!\cdots\!69$$$$T^{4}$$)($$1 + 93294524 T + 109959841455461270 T^{2} +$$$$54\!\cdots\!12$$$$T^{3} +$$$$34\!\cdots\!69$$$$T^{4}$$)($$1 + 99224060 T + 35402447061205782 T^{2} +$$$$58\!\cdots\!80$$$$T^{3} +$$$$34\!\cdots\!69$$$$T^{4}$$)($$1 - 198362430 T + 177547240540777287 T^{2} -$$$$22\!\cdots\!56$$$$T^{3} +$$$$10\!\cdots\!31$$$$T^{4} -$$$$68\!\cdots\!70$$$$T^{5} +$$$$20\!\cdots\!97$$$$T^{6}$$)
$79$ ($$1 - 583345720 T + 119851595982618319 T^{2}$$)($$1 - 486017080 T + 119851595982618319 T^{2}$$)($$1 - 91425472 T + 119851595982618319 T^{2}$$)($$1 + 95040840 T + 119851595982618319 T^{2}$$)($$1 - 117380080 T + 119851595982618319 T^{2}$$)($$1 + 6877520 T - 115982362290712162 T^{2} +$$$$82\!\cdots\!80$$$$T^{3} +$$$$14\!\cdots\!61$$$$T^{4}$$)($$1 + 466946256 T + 276599246367485918 T^{2} +$$$$55\!\cdots\!64$$$$T^{3} +$$$$14\!\cdots\!61$$$$T^{4}$$)($$1 - 452087344 T + 272781342616725918 T^{2} -$$$$54\!\cdots\!36$$$$T^{3} +$$$$14\!\cdots\!61$$$$T^{4}$$)($$1 - 191601328 T - 39905777688415266 T^{2} -$$$$22\!\cdots\!32$$$$T^{3} +$$$$14\!\cdots\!61$$$$T^{4}$$)($$1 + 542261776 T + 313115996157615582 T^{2} +$$$$64\!\cdots\!44$$$$T^{3} +$$$$14\!\cdots\!61$$$$T^{4}$$)($$1 - 413839728 T + 365148310414214253 T^{2} -$$$$92\!\cdots\!64$$$$T^{3} +$$$$43\!\cdots\!07$$$$T^{4} -$$$$59\!\cdots\!08$$$$T^{5} +$$$$17\!\cdots\!59$$$$T^{6}$$)
$83$ ($$1 - 14571786 T + 186940255267540403 T^{2}$$)($$1 + 251168886 T + 186940255267540403 T^{2}$$)($$1 - 652637376 T + 186940255267540403 T^{2}$$)($$1 + 261706326 T + 186940255267540403 T^{2}$$)($$1 + 323637636 T + 186940255267540403 T^{2}$$)($$1 + 1402348740 T + 857904310704391270 T^{2} +$$$$26\!\cdots\!20$$$$T^{3} +$$$$34\!\cdots\!09$$$$T^{4}$$)($$1 + 329535164 T + 370572645121965674 T^{2} +$$$$61\!\cdots\!92$$$$T^{3} +$$$$34\!\cdots\!09$$$$T^{4}$$)($$1 - 244865436 T + 172811505943449574 T^{2} -$$$$45\!\cdots\!08$$$$T^{3} +$$$$34\!\cdots\!09$$$$T^{4}$$)($$1 - 17270436 T + 372491529649343530 T^{2} -$$$$32\!\cdots\!08$$$$T^{3} +$$$$34\!\cdots\!09$$$$T^{4}$$)($$1 - 1256915700 T + 768086791626261130 T^{2} -$$$$23\!\cdots\!00$$$$T^{3} +$$$$34\!\cdots\!09$$$$T^{4}$$)($$1 + 371949828 T + 338245334081122329 T^{2} +$$$$14\!\cdots\!04$$$$T^{3} +$$$$63\!\cdots\!87$$$$T^{4} +$$$$12\!\cdots\!52$$$$T^{5} +$$$$65\!\cdots\!27$$$$T^{6}$$)
$89$ ($$1 - 470133690 T + 350356403707485209 T^{2}$$)($$1 + 526039110 T + 350356403707485209 T^{2}$$)($$1 + 170059206 T + 350356403707485209 T^{2}$$)($$1 + 19938630 T + 350356403707485209 T^{2}$$)($$1 + 894379110 T + 350356403707485209 T^{2}$$)($$1 - 830088660 T + 692293619421117718 T^{2} -$$$$29\!\cdots\!40$$$$T^{3} +$$$$12\!\cdots\!81$$$$T^{4}$$)($$1 + 108334860 T - 479764388871867818 T^{2} +$$$$37\!\cdots\!40$$$$T^{3} +$$$$12\!\cdots\!81$$$$T^{4}$$)($$1 + 256073260 T + 592174274852066582 T^{2} +$$$$89\!\cdots\!40$$$$T^{3} +$$$$12\!\cdots\!81$$$$T^{4}$$)($$1 + 615067148 T + 322694158807723094 T^{2} +$$$$21\!\cdots\!32$$$$T^{3} +$$$$12\!\cdots\!81$$$$T^{4}$$)($$1 + 462291852 T + 159603168035249494 T^{2} +$$$$16\!\cdots\!68$$$$T^{3} +$$$$12\!\cdots\!81$$$$T^{4}$$)($$1 - 754926606 T + 926540806511526711 T^{2} -$$$$52\!\cdots\!12$$$$T^{3} +$$$$32\!\cdots\!99$$$$T^{4} -$$$$92\!\cdots\!86$$$$T^{5} +$$$$43\!\cdots\!29$$$$T^{6}$$)
$97$ ($$1 + 117838462 T + 760231058654565217 T^{2}$$)($$1 + 1075981438 T + 760231058654565217 T^{2}$$)($$1 + 10947022 T + 760231058654565217 T^{2}$$)($$1 + 19503358 T + 760231058654565217 T^{2}$$)($$1 - 232678562 T + 760231058654565217 T^{2}$$)($$1 - 638394580 T + 1615411126351062630 T^{2} -$$$$48\!\cdots\!60$$$$T^{3} +$$$$57\!\cdots\!89$$$$T^{4}$$)($$1 - 2043058628 T + 2509217520410601030 T^{2} -$$$$15\!\cdots\!76$$$$T^{3} +$$$$57\!\cdots\!89$$$$T^{4}$$)($$1 + 184950572 T - 615454564650127770 T^{2} +$$$$14\!\cdots\!24$$$$T^{3} +$$$$57\!\cdots\!89$$$$T^{4}$$)($$1 + 996545468 T + 943405561624881990 T^{2} +$$$$75\!\cdots\!56$$$$T^{3} +$$$$57\!\cdots\!89$$$$T^{4}$$)($$1 - 1671716740 T + 2048690578856969670 T^{2} -$$$$12\!\cdots\!80$$$$T^{3} +$$$$57\!\cdots\!89$$$$T^{4}$$)($$1 + 903451002 T + 2439888940908067119 T^{2} +$$$$13\!\cdots\!72$$$$T^{3} +$$$$18\!\cdots\!23$$$$T^{4} +$$$$52\!\cdots\!78$$$$T^{5} +$$$$43\!\cdots\!13$$$$T^{6}$$)