Properties

Label 450.2.l
Level 450
Weight 2
Character orbit l
Rep. character \(\chi_{450}(19,\cdot)\)
Character field \(\Q(\zeta_{10})\)
Dimension 48
Newform subspaces 4
Sturm bound 180
Trace bound 5

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Defining parameters

Level: \( N \) = \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 450.l (of order \(10\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 25 \)
Character field: \(\Q(\zeta_{10})\)
Newform subspaces: \( 4 \)
Sturm bound: \(180\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(450, [\chi])\).

Total New Old
Modular forms 392 48 344
Cusp forms 328 48 280
Eisenstein series 64 0 64

Trace form

\( 48q + 12q^{4} + 6q^{5} + O(q^{10}) \) \( 48q + 12q^{4} + 6q^{5} + 2q^{10} - 8q^{11} + 4q^{14} - 12q^{16} + 10q^{17} - 14q^{19} + 4q^{20} - 20q^{22} + 10q^{23} + 44q^{25} + 20q^{26} - 10q^{28} + 22q^{29} + 18q^{31} - 10q^{34} + 14q^{35} - 2q^{40} - 30q^{41} - 2q^{44} + 16q^{46} + 70q^{47} - 32q^{49} + 8q^{50} - 2q^{55} - 4q^{56} + 30q^{61} - 60q^{62} + 12q^{64} - 62q^{65} - 70q^{67} - 52q^{70} + 42q^{71} - 44q^{74} - 16q^{76} - 120q^{77} - 40q^{79} - 4q^{80} - 90q^{83} - 74q^{85} + 2q^{86} + 10q^{88} - 56q^{89} + 36q^{91} + 10q^{92} + 8q^{94} - 40q^{97} - 40q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(450, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
450.2.l.a \(8\) \(3.593\) \(\Q(\zeta_{20})\) None \(0\) \(0\) \(0\) \(0\) \(q+\zeta_{20}q^{2}+\zeta_{20}^{2}q^{4}+(\zeta_{20}+\zeta_{20}^{2}+\cdots)q^{5}+\cdots\)
450.2.l.b \(8\) \(3.593\) \(\Q(\zeta_{20})\) None \(0\) \(0\) \(10\) \(0\) \(q+\zeta_{20}q^{2}+\zeta_{20}^{2}q^{4}+(1-\zeta_{20}-\zeta_{20}^{3}+\cdots)q^{5}+\cdots\)
450.2.l.c \(16\) \(3.593\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(-4\) \(0\) \(q-\beta _{8}q^{2}-\beta _{10}q^{4}+(-1-\beta _{3}+\beta _{5}+\cdots)q^{5}+\cdots\)
450.2.l.d \(16\) \(3.593\) \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{11}q^{2}+(1+\beta _{3}+\beta _{4}+\beta _{5})q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(450, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(450, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(150, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(225, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 - T^{2} + T^{4} - T^{6} + T^{8} \))(\( 1 - T^{2} + T^{4} - T^{6} + T^{8} \))(\( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \))(\( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \))
$3$ 1
$5$ (\( 1 + 5 T^{2} - 10 T^{3} + 5 T^{4} - 50 T^{5} + 125 T^{6} + 625 T^{8} \))(\( 1 - 10 T + 45 T^{2} - 130 T^{3} + 305 T^{4} - 650 T^{5} + 1125 T^{6} - 1250 T^{7} + 625 T^{8} \))(\( 1 + 4 T - 4 T^{2} - 6 T^{3} + 161 T^{4} + 260 T^{5} - 580 T^{6} + 520 T^{7} + 7205 T^{8} + 2600 T^{9} - 14500 T^{10} + 32500 T^{11} + 100625 T^{12} - 18750 T^{13} - 62500 T^{14} + 312500 T^{15} + 390625 T^{16} \))(\( 1 - 10 T^{2} + 35 T^{4} + 100 T^{6} - 1475 T^{8} + 2500 T^{10} + 21875 T^{12} - 156250 T^{14} + 390625 T^{16} \))
$7$ (\( 1 - 32 T^{2} + 500 T^{4} - 5152 T^{6} + 40294 T^{8} - 252448 T^{10} + 1200500 T^{12} - 3764768 T^{14} + 5764801 T^{16} \))(\( 1 - 20 T^{2} + 146 T^{4} - 160 T^{6} - 2789 T^{8} - 7840 T^{10} + 350546 T^{12} - 2352980 T^{14} + 5764801 T^{16} \))(\( 1 - 30 T^{2} + 527 T^{4} - 6940 T^{6} + 74703 T^{8} - 716280 T^{10} + 6275029 T^{12} - 50516750 T^{14} + 372187880 T^{16} - 2475320750 T^{18} + 15066344629 T^{20} - 84269625720 T^{22} + 430647929103 T^{24} - 1960378228060 T^{26} + 7294358354927 T^{28} - 20346692185470 T^{30} + 33232930569601 T^{32} \))(\( ( 1 - 35 T^{2} + 611 T^{4} - 7045 T^{6} + 57976 T^{8} - 345205 T^{10} + 1467011 T^{12} - 4117715 T^{14} + 5764801 T^{16} )^{2} \))
$11$ (\( 1 + 10 T + 28 T^{2} - 60 T^{3} - 617 T^{4} - 1790 T^{5} - 794 T^{6} + 18400 T^{7} + 94725 T^{8} + 202400 T^{9} - 96074 T^{10} - 2382490 T^{11} - 9033497 T^{12} - 9663060 T^{13} + 49603708 T^{14} + 194871710 T^{15} + 214358881 T^{16} \))(\( 1 - 4 T - 25 T^{2} + 80 T^{3} + 260 T^{4} - 752 T^{5} + 1353 T^{6} + 2310 T^{7} - 44765 T^{8} + 25410 T^{9} + 163713 T^{10} - 1000912 T^{11} + 3806660 T^{12} + 12884080 T^{13} - 44289025 T^{14} - 77948684 T^{15} + 214358881 T^{16} \))(\( 1 + 2 T + 11 T^{2} + 50 T^{3} + 270 T^{4} - 764 T^{5} - 168 T^{6} - 4404 T^{7} - 16480 T^{8} - 222470 T^{9} + 461835 T^{10} + 66610 T^{11} + 2053685 T^{12} - 10233320 T^{13} + 85541270 T^{14} - 166971000 T^{15} - 141281840 T^{16} - 1836681000 T^{17} + 10350493670 T^{18} - 13620548920 T^{19} + 30068002085 T^{20} + 10727607110 T^{21} + 818168874435 T^{22} - 4335310932370 T^{23} - 3532634358880 T^{24} - 10384401631164 T^{25} - 4357487332968 T^{26} - 217978116346804 T^{27} + 847375661714670 T^{28} + 1726135607196550 T^{29} + 4177248169415651 T^{30} + 8354496338831302 T^{31} + 45949729863572161 T^{32} \))(\( 1 - 27 T^{2} + 396 T^{4} - 3528 T^{6} + 1916 T^{8} + 265119 T^{10} - 1853541 T^{12} - 18034104 T^{14} + 470113072 T^{16} - 2182126584 T^{18} - 27137693781 T^{20} + 469674480759 T^{22} + 410711615996 T^{24} - 91507233992328 T^{26} + 1242817637181516 T^{28} - 10253245506747507 T^{30} + 45949729863572161 T^{32} \))
$13$ (\( 1 + 20 T + 202 T^{2} + 1420 T^{3} + 8095 T^{4} + 40660 T^{5} + 185692 T^{6} + 772160 T^{7} + 2918789 T^{8} + 10038080 T^{9} + 31381948 T^{10} + 89330020 T^{11} + 231201295 T^{12} + 527236060 T^{13} + 975015418 T^{14} + 1254970340 T^{15} + 815730721 T^{16} \))(\( 1 - 5 T^{2} + 20 T^{3} + 96 T^{4} - 100 T^{5} + 25 T^{6} + 6470 T^{7} - 6509 T^{8} + 84110 T^{9} + 4225 T^{10} - 219700 T^{11} + 2741856 T^{12} + 7425860 T^{13} - 24134045 T^{14} + 815730721 T^{16} \))(\( 1 - 20 T + 240 T^{2} - 2100 T^{3} + 14722 T^{4} - 86440 T^{5} + 437000 T^{6} - 1922660 T^{7} + 7352283 T^{8} - 23888120 T^{9} + 61481440 T^{10} - 91205000 T^{11} - 205909876 T^{12} + 2578228580 T^{13} - 14615222400 T^{14} + 65003714260 T^{15} - 249510986035 T^{16} + 845048285380 T^{17} - 2469972585600 T^{18} + 5664368190260 T^{19} - 5880991968436 T^{20} - 33863778065000 T^{21} + 296759167924960 T^{22} - 1498944103918040 T^{23} + 5997483112586043 T^{24} - 20388846764492180 T^{25} + 60244160938013000 T^{26} - 154914344460558280 T^{27} + 342994409173165282 T^{28} - 636037723843731300 T^{29} + 944970332567829360 T^{30} - 1023717860281815140 T^{31} + 665416609183179841 T^{32} \))(\( ( 1 + 40 T^{2} + 20 T^{3} + 501 T^{4} + 800 T^{5} - 2120 T^{6} + 16580 T^{7} - 105479 T^{8} + 215540 T^{9} - 358280 T^{10} + 1757600 T^{11} + 14309061 T^{12} + 7425860 T^{13} + 193072360 T^{14} + 815730721 T^{16} )^{2} \))
$17$ (\( 1 + 10 T + 78 T^{2} + 590 T^{3} + 3615 T^{4} + 20330 T^{5} + 105768 T^{6} + 488740 T^{7} + 2089449 T^{8} + 8308580 T^{9} + 30566952 T^{10} + 99881290 T^{11} + 301928415 T^{12} + 837715630 T^{13} + 1882730382 T^{14} + 4103386730 T^{15} + 6975757441 T^{16} \))(\( 1 + 10 T + 75 T^{2} + 420 T^{3} + 2171 T^{4} + 9960 T^{5} + 43585 T^{6} + 177290 T^{7} + 711536 T^{8} + 3013930 T^{9} + 12596065 T^{10} + 48933480 T^{11} + 181324091 T^{12} + 596339940 T^{13} + 1810317675 T^{14} + 4103386730 T^{15} + 6975757441 T^{16} \))(\( 1 - 30 T + 500 T^{2} - 5990 T^{3} + 56782 T^{4} - 447460 T^{5} + 3015460 T^{6} - 17669550 T^{7} + 90820363 T^{8} - 409992380 T^{9} + 1611336300 T^{10} - 5357847740 T^{11} + 13820507184 T^{12} - 18014105810 T^{13} - 71676889400 T^{14} + 711604275870 T^{15} - 3544790874535 T^{16} + 12097272689790 T^{17} - 20714621036600 T^{18} - 88503301844530 T^{19} + 1154302580514864 T^{20} - 7607377618573180 T^{21} + 38893741123454700 T^{22} - 168235729149311740 T^{23} + 633540822991571083 T^{24} - 2095394413157566350 T^{25} + 6079148967047941540 T^{26} - 15335302721813462180 T^{27} + 33082455874380289102 T^{28} - 59328422417106562630 T^{29} + 84188913279700464500 T^{30} - 85872691545294473790 T^{31} + 48661191875666868481 T^{32} \))(\( 1 - 4 T^{2} + 514 T^{4} - 2824 T^{6} + 133771 T^{8} + 1671148 T^{10} + 1558516 T^{12} + 1283966848 T^{14} - 7563163843 T^{16} + 371066419072 T^{18} + 130168814836 T^{20} + 40337450159212 T^{22} + 933154048640011 T^{24} - 5693166774867976 T^{26} + 299467829936097154 T^{28} - 673511306237603716 T^{30} + 48661191875666868481 T^{32} \))
$19$ (\( 1 + 8 T - 10 T^{2} - 270 T^{3} - 565 T^{4} + 5224 T^{5} + 28422 T^{6} - 55860 T^{7} - 788595 T^{8} - 1061340 T^{9} + 10260342 T^{10} + 35831416 T^{11} - 73631365 T^{12} - 668546730 T^{13} - 470458810 T^{14} + 7150973912 T^{15} + 16983563041 T^{16} \))(\( 1 - 10 T + 22 T^{2} - 30 T^{3} + 1023 T^{4} - 4210 T^{5} - 4736 T^{6} - 17200 T^{7} + 424105 T^{8} - 326800 T^{9} - 1709696 T^{10} - 28876390 T^{11} + 133318383 T^{12} - 74282970 T^{13} + 1035009382 T^{14} - 8938717390 T^{15} + 16983563041 T^{16} \))(\( 1 + 24 T^{2} - 30 T^{3} + 330 T^{4} + 740 T^{5} + 7630 T^{6} + 18810 T^{7} + 160255 T^{8} - 244020 T^{9} + 7105152 T^{10} - 5384250 T^{11} + 141174008 T^{12} - 174364190 T^{13} + 1779959840 T^{14} + 4165821270 T^{15} + 38170716365 T^{16} + 79150604130 T^{17} + 642565502240 T^{18} - 1195963979210 T^{19} + 18397937896568 T^{20} - 13331936040750 T^{21} + 334268135478912 T^{22} - 218122581750780 T^{23} + 2721700895135455 T^{24} + 6069755595222990 T^{25} + 46780035547021630 T^{26} + 86202791584682060 T^{27} + 730393923291833130 T^{28} - 1261589503867711770 T^{29} + 19176160458789218904 T^{30} + \)\(28\!\cdots\!81\)\( T^{32} \))(\( ( 1 + 8 T + 10 T^{2} - 90 T^{3} - 225 T^{4} + 3464 T^{5} + 17692 T^{6} - 28120 T^{7} - 383015 T^{8} - 534280 T^{9} + 6386812 T^{10} + 23759576 T^{11} - 29322225 T^{12} - 222848910 T^{13} + 470458810 T^{14} + 7150973912 T^{15} + 16983563041 T^{16} )^{2} \))
$23$ (\( 1 - 10 T + 116 T^{2} - 950 T^{3} + 6927 T^{4} - 45910 T^{5} + 267168 T^{6} - 1460600 T^{7} + 7269605 T^{8} - 33593800 T^{9} + 141331872 T^{10} - 558586970 T^{11} + 1938458607 T^{12} - 6114525850 T^{13} + 17172163124 T^{14} - 34048254470 T^{15} + 78310985281 T^{16} \))(\( 1 + 10 T + 105 T^{2} + 790 T^{3} + 5156 T^{4} + 30260 T^{5} + 156735 T^{6} + 798320 T^{7} + 3801671 T^{8} + 18361360 T^{9} + 82912815 T^{10} + 368173420 T^{11} + 1442860196 T^{12} + 5084710970 T^{13} + 15543768345 T^{14} + 34048254470 T^{15} + 78310985281 T^{16} \))(\( 1 - 10 T + 30 T^{2} + 250 T^{3} - 2718 T^{4} + 8680 T^{5} + 13790 T^{6} - 227450 T^{7} + 942143 T^{8} - 3042820 T^{9} + 4039430 T^{10} + 83513580 T^{11} - 909320196 T^{12} + 3918445730 T^{13} - 578679600 T^{14} - 92103435490 T^{15} + 633292773765 T^{16} - 2118379016270 T^{17} - 306121508400 T^{18} + 47675729196910 T^{19} - 254465072968836 T^{20} + 537522046037940 T^{21} + 597980611103270 T^{22} - 10360270966640540 T^{23} + 73780146605597183 T^{24} - 409672172849759350 T^{25} + 571271589636219710 T^{26} + 8270388698692886360 T^{27} - 59563949206231232478 T^{28} + \)\(12\!\cdots\!50\)\( T^{29} + \)\(34\!\cdots\!70\)\( T^{30} - \)\(26\!\cdots\!70\)\( T^{31} + \)\(61\!\cdots\!61\)\( T^{32} \))(\( 1 + 92 T^{2} + 3170 T^{4} + 33160 T^{6} - 518445 T^{8} - 6754564 T^{10} + 15464652 T^{12} - 19324637120 T^{14} - 817634880635 T^{16} - 10222733036480 T^{18} + 4327643680332 T^{20} - 999917886547396 T^{22} - 40599938764008045 T^{24} + 1373703111844600840 T^{26} + 69469359449504417570 T^{28} + \)\(10\!\cdots\!28\)\( T^{30} + \)\(61\!\cdots\!61\)\( T^{32} \))
$29$ (\( 1 - 22 T + 290 T^{2} - 2930 T^{3} + 24395 T^{4} - 175526 T^{5} + 1131792 T^{6} - 6698140 T^{7} + 37055745 T^{8} - 194246060 T^{9} + 951837072 T^{10} - 4280903614 T^{11} + 17254119995 T^{12} - 60097666570 T^{13} + 172498763090 T^{14} - 379497278798 T^{15} + 500246412961 T^{16} \))(\( 1 + 10 T + 47 T^{2} + 470 T^{3} + 4888 T^{4} + 28460 T^{5} + 139689 T^{6} + 999650 T^{7} + 6593475 T^{8} + 28989850 T^{9} + 117478449 T^{10} + 694110940 T^{11} + 3457189528 T^{12} + 9640240030 T^{13} + 27956696087 T^{14} + 172498763090 T^{15} + 500246412961 T^{16} \))(\( 1 - 10 T - 36 T^{2} + 270 T^{3} + 3910 T^{4} - 4460 T^{5} - 185440 T^{6} - 333090 T^{7} + 4522435 T^{8} + 38473780 T^{9} - 69131068 T^{10} - 1395128220 T^{11} - 3409890032 T^{12} + 36344819210 T^{13} + 245632726680 T^{14} - 543098432830 T^{15} - 7660398480495 T^{16} - 15749854552070 T^{17} + 206577123137880 T^{18} + 886413795712690 T^{19} - 2411750431722992 T^{20} - 28615682794524780 T^{21} - 41120771452036828 T^{22} + 663667946139678020 T^{23} + 2262331886599280035 T^{24} - 4832185253102205210 T^{25} - 78015949343189273440 T^{26} - 54414273555047997340 T^{27} + \)\(13\!\cdots\!10\)\( T^{28} + \)\(27\!\cdots\!30\)\( T^{29} - \)\(10\!\cdots\!16\)\( T^{30} - \)\(86\!\cdots\!90\)\( T^{31} + \)\(25\!\cdots\!21\)\( T^{32} \))(\( 1 - 84 T^{2} + 4610 T^{4} - 235420 T^{6} + 11114035 T^{8} - 439686452 T^{10} + 15471460428 T^{12} - 517724582880 T^{14} + 16060864040285 T^{16} - 435406374202080 T^{18} + 10942670002976268 T^{20} - 261535755577347092 T^{22} + 5559756142273007635 T^{24} - 99042896863533319420 T^{26} + \)\(16\!\cdots\!10\)\( T^{28} - \)\(24\!\cdots\!04\)\( T^{30} + \)\(25\!\cdots\!21\)\( T^{32} \))
$31$ (\( 1 - 24 T + 230 T^{2} - 990 T^{3} - 685 T^{4} + 32568 T^{5} - 115202 T^{6} - 726180 T^{7} + 8308045 T^{8} - 22511580 T^{9} - 110709122 T^{10} + 970233288 T^{11} - 632611885 T^{12} - 28342859490 T^{13} + 204125846630 T^{14} - 660302738664 T^{15} + 852891037441 T^{16} \))(\( 1 - 6 T + 5 T^{2} - 110 T^{3} + 1840 T^{4} - 4048 T^{5} - 9417 T^{6} - 113520 T^{7} + 1708895 T^{8} - 3519120 T^{9} - 9049737 T^{10} - 120593968 T^{11} + 1699278640 T^{12} - 3149206610 T^{13} + 4437518405 T^{14} - 165075684666 T^{15} + 852891037441 T^{16} \))(\( 1 + 18 T + 121 T^{2} + 610 T^{3} + 6060 T^{4} + 54724 T^{5} + 309962 T^{6} + 1456784 T^{7} + 9332270 T^{8} + 64865970 T^{9} + 327433735 T^{10} + 1367357590 T^{11} + 7702756335 T^{12} + 47160385620 T^{13} + 227624335970 T^{14} + 1111476930100 T^{15} + 6249005152660 T^{16} + 34455784833100 T^{17} + 218746986867170 T^{18} + 1404955048005420 T^{19} + 7113657233255535 T^{20} + 39146286915106090 T^{21} + 290598645096078535 T^{22} + 1784632401545702670 T^{23} + 7959409441979521070 T^{24} + 38516818529710942064 T^{25} + \)\(25\!\cdots\!62\)\( T^{26} + \)\(13\!\cdots\!44\)\( T^{27} + \)\(47\!\cdots\!60\)\( T^{28} + \)\(14\!\cdots\!10\)\( T^{29} + \)\(91\!\cdots\!41\)\( T^{30} + \)\(42\!\cdots\!18\)\( T^{31} + \)\(72\!\cdots\!81\)\( T^{32} \))(\( ( 1 - 3 T - 34 T^{2} + 148 T^{3} - 554 T^{4} - 1909 T^{5} + 27369 T^{6} - 24516 T^{7} - 77548 T^{8} - 759996 T^{9} + 26301609 T^{10} - 56871019 T^{11} - 511630634 T^{12} + 4237114348 T^{13} - 30175125154 T^{14} - 82537842333 T^{15} + 852891037441 T^{16} )^{2} \))
$37$ (\( 1 + 20 T + 275 T^{2} + 3140 T^{3} + 29831 T^{4} + 251620 T^{5} + 1934025 T^{6} + 13441540 T^{7} + 85392936 T^{8} + 497336980 T^{9} + 2647680225 T^{10} + 12745307860 T^{11} + 55908096791 T^{12} + 217740024980 T^{13} + 705574762475 T^{14} + 1898637542660 T^{15} + 3512479453921 T^{16} \))(\( 1 + 10 T + 65 T^{2} + 230 T^{3} + 1471 T^{4} + 9410 T^{5} + 48855 T^{6} - 117890 T^{7} - 967744 T^{8} - 4361930 T^{9} + 66882495 T^{10} + 476644730 T^{11} + 2756890831 T^{12} + 15949110110 T^{13} + 166772216585 T^{14} + 949318771330 T^{15} + 3512479453921 T^{16} \))(\( 1 - 20 T + 265 T^{2} - 2300 T^{3} + 17532 T^{4} - 115640 T^{5} + 825500 T^{6} - 5506160 T^{7} + 38914148 T^{8} - 253492260 T^{9} + 1794329215 T^{10} - 11470149820 T^{11} + 74232183599 T^{12} - 424648096560 T^{13} + 2693507120800 T^{14} - 16225080394680 T^{15} + 105486161388440 T^{16} - 600327974603160 T^{17} + 3687411248375200 T^{18} - 21509700035053680 T^{19} + 139123063446085439 T^{20} - 795385575901637740 T^{21} + 4603757853365738935 T^{22} - 24064496080486490580 T^{23} + \)\(13\!\cdots\!08\)\( T^{24} - \)\(71\!\cdots\!20\)\( T^{25} + \)\(39\!\cdots\!00\)\( T^{26} - \)\(20\!\cdots\!20\)\( T^{27} + \)\(11\!\cdots\!92\)\( T^{28} - \)\(56\!\cdots\!00\)\( T^{29} + \)\(23\!\cdots\!85\)\( T^{30} - \)\(66\!\cdots\!60\)\( T^{31} + \)\(12\!\cdots\!41\)\( T^{32} \))(\( ( 1 - 5 T + 110 T^{2} - 970 T^{3} + 7756 T^{4} - 73855 T^{5} + 499995 T^{6} - 3335450 T^{7} + 23205836 T^{8} - 123411650 T^{9} + 684493155 T^{10} - 3740977315 T^{11} + 14535992716 T^{12} - 67263638290 T^{13} + 282229904990 T^{14} - 474659385665 T^{15} + 3512479453921 T^{16} )^{2} \))
$41$ (\( 1 + 22 T + 146 T^{2} - 142 T^{3} - 6829 T^{4} - 52634 T^{5} - 236776 T^{6} + 967444 T^{7} + 17720057 T^{8} + 39665204 T^{9} - 398020456 T^{10} - 3627587914 T^{11} - 19297121869 T^{12} - 16451580542 T^{13} + 693515219186 T^{14} + 4284594025382 T^{15} + 7984925229121 T^{16} \))(\( 1 - 14 T + 45 T^{2} - 130 T^{3} + 2680 T^{4} - 7652 T^{5} - 46977 T^{6} + 402900 T^{7} - 2944785 T^{8} + 16518900 T^{9} - 78968337 T^{10} - 527383492 T^{11} + 7573039480 T^{12} - 15061306130 T^{13} + 213754690845 T^{14} - 2726559834334 T^{15} + 7984925229121 T^{16} \))(\( 1 + 22 T + 316 T^{2} + 3710 T^{3} + 39530 T^{4} + 407436 T^{5} + 3918652 T^{6} + 35382126 T^{7} + 301988495 T^{8} + 2475859380 T^{9} + 19707200260 T^{10} + 150750040860 T^{11} + 1111070917960 T^{12} + 7891095298530 T^{13} + 54497096107120 T^{14} + 366799083339050 T^{15} + 2387443151993985 T^{16} + 15038762416901050 T^{17} + 91609618556068720 T^{18} + 543862179069986130 T^{19} + 3139620868205567560 T^{20} + 17465327034634372860 T^{21} + 93611255533262302660 T^{22} + \)\(48\!\cdots\!80\)\( T^{23} + \)\(24\!\cdots\!95\)\( T^{24} + \)\(11\!\cdots\!86\)\( T^{25} + \)\(52\!\cdots\!52\)\( T^{26} + \)\(22\!\cdots\!76\)\( T^{27} + \)\(89\!\cdots\!30\)\( T^{28} + \)\(34\!\cdots\!10\)\( T^{29} + \)\(11\!\cdots\!76\)\( T^{30} + \)\(34\!\cdots\!22\)\( T^{31} + \)\(63\!\cdots\!41\)\( T^{32} \))(\( 1 - 64 T^{2} + 5410 T^{4} - 287220 T^{6} + 19007235 T^{8} - 803484352 T^{10} + 43037275588 T^{12} - 1651130949280 T^{14} + 79326977742685 T^{16} - 2775551125739680 T^{18} + 121613054902822468 T^{20} - 3816634428012336832 T^{22} + \)\(15\!\cdots\!35\)\( T^{24} - \)\(38\!\cdots\!20\)\( T^{26} + \)\(12\!\cdots\!10\)\( T^{28} - \)\(24\!\cdots\!04\)\( T^{30} + \)\(63\!\cdots\!41\)\( T^{32} \))
$43$ (\( 1 - 120 T^{2} + 10856 T^{4} - 676920 T^{6} + 33149566 T^{8} - 1251625080 T^{10} + 37114503656 T^{12} - 758563565880 T^{14} + 11688200277601 T^{16} \))(\( 1 - 120 T^{2} + 10666 T^{4} - 636400 T^{6} + 31214171 T^{8} - 1176703600 T^{10} + 36464931466 T^{12} - 758563565880 T^{14} + 11688200277601 T^{16} \))(\( 1 - 360 T^{2} + 63972 T^{4} - 7536360 T^{6} + 666842308 T^{8} - 47508203160 T^{10} + 2846509586204 T^{12} - 147613243096600 T^{14} + 6742598283189430 T^{16} - 272936886485613400 T^{18} + 9731649819823821404 T^{20} - \)\(30\!\cdots\!40\)\( T^{22} + \)\(77\!\cdots\!08\)\( T^{24} - \)\(16\!\cdots\!40\)\( T^{26} + \)\(25\!\cdots\!72\)\( T^{28} - \)\(26\!\cdots\!40\)\( T^{30} + \)\(13\!\cdots\!01\)\( T^{32} \))(\( ( 1 - 168 T^{2} + 14140 T^{4} - 823768 T^{6} + 38677094 T^{8} - 1523147032 T^{10} + 48341846140 T^{12} - 1061988992232 T^{14} + 11688200277601 T^{16} )^{2} \))
$47$ (\( 1 + 10 T + 164 T^{2} + 1060 T^{3} + 6767 T^{4} + 10550 T^{5} - 301398 T^{6} - 3376520 T^{7} - 34256075 T^{8} - 158696440 T^{9} - 665788182 T^{10} + 1095332650 T^{11} + 33020801327 T^{12} + 243105707420 T^{13} + 1767791313956 T^{14} + 5066231204630 T^{15} + 23811286661761 T^{16} \))(\( 1 - 30 T + 425 T^{2} - 3450 T^{3} + 13416 T^{4} + 40020 T^{5} - 992525 T^{6} + 8396100 T^{7} - 57415609 T^{8} + 394616700 T^{9} - 2192487725 T^{10} + 4154996460 T^{11} + 65465800296 T^{12} - 791240274150 T^{13} + 4581166514825 T^{14} - 15198693613890 T^{15} + 23811286661761 T^{16} \))(\( 1 - 50 T + 1290 T^{2} - 23780 T^{3} + 355482 T^{4} - 4545300 T^{5} + 51239280 T^{6} - 520209820 T^{7} + 4818856003 T^{8} - 41087515780 T^{9} + 324882626890 T^{10} - 2399419588970 T^{11} + 16710816340744 T^{12} - 111327617454640 T^{13} + 724257830956800 T^{14} - 4728319081214120 T^{15} + 31780196573425365 T^{16} - 222230996817063640 T^{17} + 1599885548583571200 T^{18} - 11558367226993088720 T^{19} + 81543452992418022664 T^{20} - \)\(55\!\cdots\!90\)\( T^{21} + \)\(35\!\cdots\!10\)\( T^{22} - \)\(20\!\cdots\!40\)\( T^{23} + \)\(11\!\cdots\!83\)\( T^{24} - \)\(58\!\cdots\!40\)\( T^{25} + \)\(26\!\cdots\!20\)\( T^{26} - \)\(11\!\cdots\!00\)\( T^{27} + \)\(41\!\cdots\!62\)\( T^{28} - \)\(12\!\cdots\!60\)\( T^{29} + \)\(33\!\cdots\!10\)\( T^{30} - \)\(60\!\cdots\!50\)\( T^{31} + \)\(56\!\cdots\!21\)\( T^{32} \))(\( ( 1 + 14 T^{2} + 3387 T^{4} + 127192 T^{6} + 5908805 T^{8} + 280967128 T^{10} + 16527479547 T^{12} + 150909014606 T^{14} + 23811286661761 T^{16} )^{2} \))
$53$ (\( 1 - 30 T + 425 T^{2} - 3090 T^{3} + 5031 T^{4} + 120570 T^{5} - 1120625 T^{6} + 3254310 T^{7} + 1334576 T^{8} + 172478430 T^{9} - 3147835625 T^{10} + 17950099890 T^{11} + 39697009911 T^{12} - 1292224073370 T^{13} + 9419853479825 T^{14} - 35241334195110 T^{15} + 62259690411361 T^{16} \))(\( 1 + 90 T^{2} - 900 T^{3} + 5291 T^{4} - 81000 T^{5} + 628380 T^{6} - 3853800 T^{7} + 44701381 T^{8} - 204251400 T^{9} + 1765119420 T^{10} - 12059037000 T^{11} + 41748534971 T^{12} - 376375943700 T^{13} + 1994792501610 T^{14} + 62259690411361 T^{16} \))(\( 1 + 30 T + 655 T^{2} + 10630 T^{3} + 152332 T^{4} + 1919860 T^{5} + 22723040 T^{6} + 248684220 T^{7} + 2591225028 T^{8} + 25379094630 T^{9} + 238630725505 T^{10} + 2134501287330 T^{11} + 18442776242119 T^{12} + 152410720886420 T^{13} + 1218238235823200 T^{14} + 9336287074370240 T^{15} + 69349166231444240 T^{16} + 494823214941622720 T^{17} + 3422031204427368800 T^{18} + 22690450893407550340 T^{19} + \)\(14\!\cdots\!39\)\( T^{20} + \)\(89\!\cdots\!90\)\( T^{21} + \)\(52\!\cdots\!45\)\( T^{22} + \)\(29\!\cdots\!10\)\( T^{23} + \)\(16\!\cdots\!08\)\( T^{24} + \)\(82\!\cdots\!60\)\( T^{25} + \)\(39\!\cdots\!60\)\( T^{26} + \)\(17\!\cdots\!20\)\( T^{27} + \)\(74\!\cdots\!12\)\( T^{28} + \)\(27\!\cdots\!90\)\( T^{29} + \)\(90\!\cdots\!95\)\( T^{30} + \)\(21\!\cdots\!10\)\( T^{31} + \)\(38\!\cdots\!21\)\( T^{32} \))(\( 1 + 225 T^{2} + 23512 T^{4} + 1434600 T^{6} + 47561688 T^{8} - 356685525 T^{10} - 175945692541 T^{12} - 15108285618000 T^{14} - 907892015735920 T^{16} - 42439174300962000 T^{18} - 1388296144026602221 T^{20} - 7905706785586957725 T^{22} + \)\(29\!\cdots\!68\)\( T^{24} + \)\(25\!\cdots\!00\)\( T^{26} + \)\(11\!\cdots\!92\)\( T^{28} + \)\(31\!\cdots\!25\)\( T^{30} + \)\(38\!\cdots\!21\)\( T^{32} \))
$59$ (\( 1 - 20 T + 242 T^{2} - 2760 T^{3} + 31483 T^{4} - 335220 T^{5} + 3074684 T^{6} - 25371200 T^{7} + 201486805 T^{8} - 1496900800 T^{9} + 10702975004 T^{10} - 68847148380 T^{11} + 381490876363 T^{12} - 1973191065240 T^{13} + 10207689141122 T^{14} - 49773029696380 T^{15} + 146830437604321 T^{16} \))(\( 1 - 118 T^{2} + 3563 T^{4} + 324004 T^{6} - 38801675 T^{8} + 1127857924 T^{10} + 43174157243 T^{12} - 4977302969638 T^{14} + 146830437604321 T^{16} \))(\( 1 + 20 T - 111 T^{2} - 3640 T^{3} + 11660 T^{4} + 317820 T^{5} - 1817980 T^{6} - 11679820 T^{7} + 217773860 T^{8} - 627241760 T^{9} - 13630850673 T^{10} + 133224376740 T^{11} + 186113511463 T^{12} - 9371240514920 T^{13} + 48131664631280 T^{14} + 252534601553160 T^{15} - 4526062111550040 T^{16} + 14899541491636440 T^{17} + 167546324581485680 T^{18} - 1924656005713754680 T^{19} + 2255204605374809143 T^{20} + 95245344150556405260 T^{21} - \)\(57\!\cdots\!93\)\( T^{22} - \)\(15\!\cdots\!40\)\( T^{23} + \)\(31\!\cdots\!60\)\( T^{24} - \)\(10\!\cdots\!80\)\( T^{25} - \)\(92\!\cdots\!80\)\( T^{26} + \)\(95\!\cdots\!80\)\( T^{27} + \)\(20\!\cdots\!60\)\( T^{28} - \)\(38\!\cdots\!60\)\( T^{29} - \)\(68\!\cdots\!71\)\( T^{30} + \)\(73\!\cdots\!80\)\( T^{31} + \)\(21\!\cdots\!41\)\( T^{32} \))(\( 1 - 239 T^{2} + 20720 T^{4} - 262340 T^{6} - 99474320 T^{8} + 8721612103 T^{10} - 165516489857 T^{12} - 25053022225200 T^{14} + 2461592465750520 T^{16} - 87209570365921200 T^{18} - 2005623059050107377 T^{20} + \)\(36\!\cdots\!23\)\( T^{22} - \)\(14\!\cdots\!20\)\( T^{24} - \)\(13\!\cdots\!40\)\( T^{26} + \)\(36\!\cdots\!20\)\( T^{28} - \)\(14\!\cdots\!79\)\( T^{30} + \)\(21\!\cdots\!41\)\( T^{32} \))
$61$ (\( 1 + 118 T^{2} + 10203 T^{4} + 835676 T^{6} + 64821605 T^{8} + 3109550396 T^{10} + 141269115723 T^{12} + 6079404174598 T^{14} + 191707312997281 T^{16} \))(\( 1 + 14 T + 30 T^{2} - 30 T^{3} + 5215 T^{4} + 25262 T^{5} - 186192 T^{6} - 1508020 T^{7} - 4396815 T^{8} - 91989220 T^{9} - 692820432 T^{10} + 5733994022 T^{11} + 72206060815 T^{12} - 25337889030 T^{13} + 1545611230830 T^{14} + 43998399704294 T^{15} + 191707312997281 T^{16} \))(\( 1 - 12 T - 104 T^{2} + 1460 T^{3} + 9310 T^{4} - 110696 T^{5} - 1128308 T^{6} + 9605804 T^{7} + 95707895 T^{8} - 545482080 T^{9} - 8568890440 T^{10} + 31012865040 T^{11} + 647686771260 T^{12} - 1172976299580 T^{13} - 45238788397880 T^{14} + 2951178504100 T^{15} + 3155660575885285 T^{16} + 180021888750100 T^{17} - 168333531628511480 T^{18} - 266243333454967980 T^{19} + 8967768052669329660 T^{20} + 26193351096196217040 T^{21} - \)\(44\!\cdots\!40\)\( T^{22} - \)\(17\!\cdots\!80\)\( T^{23} + \)\(18\!\cdots\!95\)\( T^{24} + \)\(11\!\cdots\!64\)\( T^{25} - \)\(80\!\cdots\!08\)\( T^{26} - \)\(48\!\cdots\!56\)\( T^{27} + \)\(24\!\cdots\!10\)\( T^{28} + \)\(23\!\cdots\!60\)\( T^{29} - \)\(10\!\cdots\!64\)\( T^{30} - \)\(72\!\cdots\!12\)\( T^{31} + \)\(36\!\cdots\!61\)\( T^{32} \))(\( ( 1 - 16 T + 120 T^{2} - 960 T^{3} + 9505 T^{4} - 123568 T^{5} + 1138368 T^{6} - 7181200 T^{7} + 51027345 T^{8} - 438053200 T^{9} + 4235867328 T^{10} - 28047588208 T^{11} + 131604718705 T^{12} - 810812448960 T^{13} + 6182444923320 T^{14} - 50283885376336 T^{15} + 191707312997281 T^{16} )^{2} \))
$67$ (\( 1 - 10 T + 88 T^{2} - 1660 T^{3} + 11615 T^{4} - 86790 T^{5} + 911158 T^{6} - 5793400 T^{7} + 42525189 T^{8} - 388157800 T^{9} + 4090188262 T^{10} - 26103220770 T^{11} + 234055270415 T^{12} - 2241207677620 T^{13} + 7960337630872 T^{14} - 60607116053230 T^{15} + 406067677556641 T^{16} \))(\( 1 - 10 T + 195 T^{2} - 2790 T^{3} + 29716 T^{4} - 356960 T^{5} + 3452465 T^{6} - 30345750 T^{7} + 284471411 T^{8} - 2033165250 T^{9} + 15498115385 T^{10} - 107360360480 T^{11} + 598810711636 T^{12} - 3766849048530 T^{13} + 17639384522955 T^{14} - 60607116053230 T^{15} + 406067677556641 T^{16} \))(\( 1 + 50 T + 1390 T^{2} + 30100 T^{3} + 553682 T^{4} + 8918100 T^{5} + 129754300 T^{6} + 1739003620 T^{7} + 21685053163 T^{8} + 253853201940 T^{9} + 2808527300190 T^{10} + 29485013121570 T^{11} + 294663718701784 T^{12} + 2810511152550240 T^{13} + 25630455288876000 T^{14} + 223725991474439120 T^{15} + 1870714073010261365 T^{16} + 14989641428787421040 T^{17} + \)\(11\!\cdots\!00\)\( T^{18} + \)\(84\!\cdots\!20\)\( T^{19} + \)\(59\!\cdots\!64\)\( T^{20} + \)\(39\!\cdots\!90\)\( T^{21} + \)\(25\!\cdots\!10\)\( T^{22} + \)\(15\!\cdots\!20\)\( T^{23} + \)\(88\!\cdots\!83\)\( T^{24} + \)\(47\!\cdots\!40\)\( T^{25} + \)\(23\!\cdots\!00\)\( T^{26} + \)\(10\!\cdots\!00\)\( T^{27} + \)\(45\!\cdots\!02\)\( T^{28} + \)\(16\!\cdots\!00\)\( T^{29} + \)\(51\!\cdots\!10\)\( T^{30} + \)\(12\!\cdots\!50\)\( T^{31} + \)\(16\!\cdots\!81\)\( T^{32} \))(\( ( 1 + 20 T + 198 T^{2} + 1470 T^{3} + 10135 T^{4} + 65680 T^{5} + 492368 T^{6} + 6337980 T^{7} + 70845689 T^{8} + 424644660 T^{9} + 2210239952 T^{10} + 19754113840 T^{11} + 204231611335 T^{12} + 1984683907290 T^{13} + 17910759669462 T^{14} + 121214232106460 T^{15} + 406067677556641 T^{16} )^{2} \))
$71$ (\( 1 + 20 T + 118 T^{2} - 490 T^{3} - 14397 T^{4} - 149580 T^{5} - 402414 T^{6} + 9696700 T^{7} + 137916525 T^{8} + 688465700 T^{9} - 2028568974 T^{10} - 53536327380 T^{11} - 365851971357 T^{12} - 884072381990 T^{13} + 15115833502678 T^{14} + 181902403167820 T^{15} + 645753531245761 T^{16} \))(\( 1 - 34 T + 425 T^{2} - 2570 T^{3} + 16960 T^{4} - 219512 T^{5} + 2212323 T^{6} - 19311440 T^{7} + 171308535 T^{8} - 1371112240 T^{9} + 11152320243 T^{10} - 78565759432 T^{11} + 430982109760 T^{12} - 4636869432070 T^{13} + 54442620666425 T^{14} - 309234085385294 T^{15} + 645753531245761 T^{16} \))(\( 1 - 28 T + 256 T^{2} - 1150 T^{3} + 20770 T^{4} - 328604 T^{5} + 2351902 T^{6} - 21115094 T^{7} + 321977895 T^{8} - 2983992020 T^{9} + 18685321160 T^{10} - 180970630790 T^{11} + 2127756781760 T^{12} - 17298444226470 T^{13} + 123520240637920 T^{14} - 1239239342309850 T^{15} + 12274938369025885 T^{16} - 87985993303999350 T^{17} + 622665533055754720 T^{18} - 6191303471540104170 T^{19} + 54069876583671738560 T^{20} - \)\(32\!\cdots\!90\)\( T^{21} + \)\(23\!\cdots\!60\)\( T^{22} - \)\(27\!\cdots\!20\)\( T^{23} + \)\(20\!\cdots\!95\)\( T^{24} - \)\(96\!\cdots\!14\)\( T^{25} + \)\(76\!\cdots\!02\)\( T^{26} - \)\(75\!\cdots\!84\)\( T^{27} + \)\(34\!\cdots\!70\)\( T^{28} - \)\(13\!\cdots\!50\)\( T^{29} + \)\(21\!\cdots\!36\)\( T^{30} - \)\(16\!\cdots\!28\)\( T^{31} + \)\(41\!\cdots\!21\)\( T^{32} \))(\( 1 - 276 T^{2} + 34250 T^{4} - 2419780 T^{6} + 80093635 T^{8} + 3842028412 T^{10} - 741573573372 T^{12} + 57460499529840 T^{14} - 3784985104999915 T^{16} + 289658378129923440 T^{18} - 18844631084559358332 T^{20} + \)\(49\!\cdots\!52\)\( T^{22} + \)\(51\!\cdots\!35\)\( T^{24} - \)\(78\!\cdots\!80\)\( T^{26} + \)\(56\!\cdots\!50\)\( T^{28} - \)\(22\!\cdots\!56\)\( T^{30} + \)\(41\!\cdots\!21\)\( T^{32} \))
$73$ (\( 1 + 20 T + 282 T^{2} + 3780 T^{3} + 46315 T^{4} + 505780 T^{5} + 5254892 T^{6} + 48564720 T^{7} + 416447549 T^{8} + 3545224560 T^{9} + 28003319468 T^{10} + 196757018260 T^{11} + 1315264531915 T^{12} + 7836210621540 T^{13} + 42676251813498 T^{14} + 220947970381940 T^{15} + 806460091894081 T^{16} \))(\( 1 + 110 T^{2} + 840 T^{3} + 6291 T^{4} + 92400 T^{5} + 907660 T^{6} + 5561520 T^{7} + 91719461 T^{8} + 405990960 T^{9} + 4836920140 T^{10} + 35945170800 T^{11} + 178653334131 T^{12} + 1741380138120 T^{13} + 16646764891790 T^{14} + 806460091894081 T^{16} \))(\( 1 - 20 T + 380 T^{2} - 7420 T^{3} + 107282 T^{4} - 1509240 T^{5} + 20338260 T^{6} - 240254580 T^{7} + 2794851843 T^{8} - 31155740720 T^{9} + 320911861980 T^{10} - 3274927660920 T^{11} + 32181004976604 T^{12} - 299417039187980 T^{13} + 2775262759264800 T^{14} - 24764197420905260 T^{15} + 211758644546106765 T^{16} - 1807786411726083980 T^{17} + 14789375244122119200 T^{18} - \)\(11\!\cdots\!60\)\( T^{19} + \)\(91\!\cdots\!64\)\( T^{20} - \)\(67\!\cdots\!60\)\( T^{21} + \)\(48\!\cdots\!20\)\( T^{22} - \)\(34\!\cdots\!40\)\( T^{23} + \)\(22\!\cdots\!83\)\( T^{24} - \)\(14\!\cdots\!40\)\( T^{25} + \)\(87\!\cdots\!40\)\( T^{26} - \)\(47\!\cdots\!80\)\( T^{27} + \)\(24\!\cdots\!22\)\( T^{28} - \)\(12\!\cdots\!60\)\( T^{29} + \)\(46\!\cdots\!20\)\( T^{30} - \)\(17\!\cdots\!40\)\( T^{31} + \)\(65\!\cdots\!61\)\( T^{32} \))(\( ( 1 - 34 T^{2} - 300 T^{3} + 1527 T^{4} + 10200 T^{5} + 149668 T^{6} + 870600 T^{7} - 18783595 T^{8} + 63553800 T^{9} + 797580772 T^{10} + 3967973400 T^{11} + 43364114007 T^{12} - 621921477900 T^{13} - 5145363693826 T^{14} + 806460091894081 T^{16} )^{2} \))
$79$ (\( 1 - 16 T + 274 T^{2} - 3476 T^{3} + 48131 T^{4} - 506768 T^{5} + 5479676 T^{6} - 51684448 T^{7} + 496487797 T^{8} - 4083071392 T^{9} + 34198657916 T^{10} - 249856387952 T^{11} + 1874706348611 T^{12} - 10695848042924 T^{13} + 66605962812754 T^{14} - 307262543778544 T^{15} + 1517108809906561 T^{16} \))(\( 1 - 138 T^{2} + 1500 T^{3} + 4403 T^{4} - 205200 T^{5} + 1430844 T^{6} + 7779600 T^{7} - 206126795 T^{8} + 614588400 T^{9} + 8929897404 T^{10} - 101171602800 T^{11} + 171497206643 T^{12} + 4615584598500 T^{13} - 33546068861898 T^{14} + 1517108809906561 T^{16} \))(\( 1 + 20 T + 224 T^{2} + 1780 T^{3} + 6070 T^{4} + 5760 T^{5} + 875320 T^{6} + 20333540 T^{7} + 305719495 T^{8} + 3198483920 T^{9} + 22059896192 T^{10} + 116979470440 T^{11} + 660067500148 T^{12} + 7893347694940 T^{13} + 139701887310360 T^{14} + 1828379420794980 T^{15} + 18018691485968485 T^{16} + 144441974242803420 T^{17} + 871879478703956760 T^{18} + 3891728254165522660 T^{19} + 25709682596232111988 T^{20} + \)\(35\!\cdots\!60\)\( T^{21} + \)\(53\!\cdots\!32\)\( T^{22} + \)\(61\!\cdots\!80\)\( T^{23} + \)\(46\!\cdots\!95\)\( T^{24} + \)\(24\!\cdots\!60\)\( T^{25} + \)\(82\!\cdots\!20\)\( T^{26} + \)\(43\!\cdots\!40\)\( T^{27} + \)\(35\!\cdots\!70\)\( T^{28} + \)\(83\!\cdots\!20\)\( T^{29} + \)\(82\!\cdots\!44\)\( T^{30} + \)\(58\!\cdots\!80\)\( T^{31} + \)\(23\!\cdots\!21\)\( T^{32} \))(\( ( 1 + 18 T + 30 T^{2} - 1440 T^{3} - 11395 T^{4} + 29514 T^{5} + 920082 T^{6} - 1186800 T^{7} - 91344965 T^{8} - 93757200 T^{9} + 5742231762 T^{10} + 14551553046 T^{11} - 443836172995 T^{12} - 4430961214560 T^{13} + 7292623665630 T^{14} + 345670361750862 T^{15} + 1517108809906561 T^{16} )^{2} \))
$83$ (\( 1 + 70 T + 2432 T^{2} + 56730 T^{3} + 1010935 T^{4} + 14768130 T^{5} + 184247072 T^{6} + 2010015520 T^{7} + 19419043189 T^{8} + 166831288160 T^{9} + 1269278079008 T^{10} + 8444224748310 T^{11} + 47977277740135 T^{12} + 223461775677390 T^{13} + 795118988033408 T^{14} + 1899523569273890 T^{15} + 2252292232139041 T^{16} \))(\( 1 + 50 T + 1335 T^{2} + 23750 T^{3} + 305836 T^{4} + 2886800 T^{5} + 19447445 T^{6} + 89310250 T^{7} + 442867371 T^{8} + 7412750750 T^{9} + 133973448605 T^{10} + 1650634711600 T^{11} + 14514463061356 T^{12} + 93552215271250 T^{13} + 436465398447615 T^{14} + 1356802549481350 T^{15} + 2252292232139041 T^{16} \))(\( 1 - 30 T + 535 T^{2} - 5780 T^{3} + 43882 T^{4} - 205010 T^{5} + 590510 T^{6} - 2468290 T^{7} + 123659138 T^{8} - 2415976440 T^{9} + 27231081235 T^{10} - 158815212750 T^{11} + 405339323209 T^{12} - 3244201552980 T^{13} + 126271405899100 T^{14} - 2149314211865960 T^{15} + 23236430083058540 T^{16} - 178393079584874680 T^{17} + 869883715238899900 T^{18} - 1854992273373775260 T^{19} + 19236723714775472089 T^{20} - \)\(62\!\cdots\!50\)\( T^{21} + \)\(89\!\cdots\!15\)\( T^{22} - \)\(65\!\cdots\!80\)\( T^{23} + \)\(27\!\cdots\!58\)\( T^{24} - \)\(46\!\cdots\!70\)\( T^{25} + \)\(91\!\cdots\!90\)\( T^{26} - \)\(26\!\cdots\!70\)\( T^{27} + \)\(46\!\cdots\!02\)\( T^{28} - \)\(51\!\cdots\!40\)\( T^{29} + \)\(39\!\cdots\!15\)\( T^{30} - \)\(18\!\cdots\!10\)\( T^{31} + \)\(50\!\cdots\!81\)\( T^{32} \))(\( 1 + 269 T^{2} + 25224 T^{4} + 550144 T^{6} - 77992844 T^{8} - 7340856293 T^{10} - 150970152289 T^{12} + 37124680615772 T^{14} + 5186035445336672 T^{16} + 255751924762053308 T^{18} - 7164789948750246769 T^{20} - \)\(24\!\cdots\!17\)\( T^{22} - \)\(17\!\cdots\!04\)\( T^{24} + \)\(85\!\cdots\!56\)\( T^{26} + \)\(26\!\cdots\!64\)\( T^{28} + \)\(19\!\cdots\!01\)\( T^{30} + \)\(50\!\cdots\!81\)\( T^{32} \))
$89$ (\( 1 - 34 T + 409 T^{2} - 1014 T^{3} - 27449 T^{4} + 344318 T^{5} - 919769 T^{6} - 22713862 T^{7} + 350578752 T^{8} - 2021533718 T^{9} - 7285490249 T^{10} + 242733516142 T^{11} - 1722211773209 T^{12} - 5662236281286 T^{13} + 203265348003049 T^{14} - 1503865386447986 T^{15} + 3936588805702081 T^{16} \))(\( ( 1 + 10 T + 71 T^{2} + 1290 T^{3} + 19001 T^{4} + 114810 T^{5} + 562391 T^{6} + 7049690 T^{7} + 62742241 T^{8} )^{2} \))(\( 1 + 70 T + 2119 T^{2} + 35330 T^{3} + 343460 T^{4} + 2032360 T^{5} + 10870620 T^{6} + 64684440 T^{7} - 584089940 T^{8} - 17002587830 T^{9} - 99745085123 T^{10} + 366225143690 T^{11} + 5693869232523 T^{12} + 24776539056740 T^{13} + 308063609313680 T^{14} + 3148141734399680 T^{15} + 20769024208268760 T^{16} + 280184614361571520 T^{17} + 2440171849373659280 T^{18} + 17466691962290941060 T^{19} + \)\(35\!\cdots\!43\)\( T^{20} + \)\(20\!\cdots\!10\)\( T^{21} - \)\(49\!\cdots\!03\)\( T^{22} - \)\(75\!\cdots\!70\)\( T^{23} - \)\(22\!\cdots\!40\)\( T^{24} + \)\(22\!\cdots\!60\)\( T^{25} + \)\(33\!\cdots\!20\)\( T^{26} + \)\(56\!\cdots\!40\)\( T^{27} + \)\(84\!\cdots\!60\)\( T^{28} + \)\(77\!\cdots\!70\)\( T^{29} + \)\(41\!\cdots\!79\)\( T^{30} + \)\(12\!\cdots\!30\)\( T^{31} + \)\(15\!\cdots\!61\)\( T^{32} \))(\( 1 + 117 T^{2} + 37096 T^{4} + 2404428 T^{6} + 479330076 T^{8} + 8822390031 T^{10} + 2916006253619 T^{12} - 127003185538956 T^{14} + 15092894209169912 T^{16} - 1005992232654070476 T^{18} + \)\(18\!\cdots\!79\)\( T^{20} + \)\(43\!\cdots\!91\)\( T^{22} + \)\(18\!\cdots\!56\)\( T^{24} + \)\(74\!\cdots\!28\)\( T^{26} + \)\(91\!\cdots\!16\)\( T^{28} + \)\(22\!\cdots\!97\)\( T^{30} + \)\(15\!\cdots\!61\)\( T^{32} \))
$97$ (\( 1 - 60 T + 1990 T^{2} - 47940 T^{3} + 920391 T^{4} - 14754060 T^{5} + 202723580 T^{6} - 2424996600 T^{7} + 25463108381 T^{8} - 235224670200 T^{9} + 1907426164220 T^{10} - 13465632202380 T^{11} + 81481553468871 T^{12} - 411677091920580 T^{13} + 1657614289808710 T^{14} - 4847897068686780 T^{15} + 7837433594376961 T^{16} \))(\( 1 + 20 T + 270 T^{2} + 3780 T^{3} + 51031 T^{4} + 591220 T^{5} + 6636980 T^{6} + 69655800 T^{7} + 673071701 T^{8} + 6756612600 T^{9} + 62447344820 T^{10} + 539590531060 T^{11} + 4517737738711 T^{12} + 32460146171460 T^{13} + 224902441330830 T^{14} + 1615965689562260 T^{15} + 7837433594376961 T^{16} \))(\( 1 + 10 T + 205 T^{2} + 4320 T^{3} + 44952 T^{4} + 667870 T^{5} + 8897460 T^{6} + 68756910 T^{7} + 834554408 T^{8} + 7477618800 T^{9} + 22708693755 T^{10} + 118605463990 T^{11} - 2491804851081 T^{12} - 84263627884560 T^{13} - 895266623177600 T^{14} - 11056342779615180 T^{15} - 138673183182585960 T^{16} - 1072465249622672460 T^{17} - 8423563657478038400 T^{18} - 76905138052285028880 T^{19} - \)\(22\!\cdots\!61\)\( T^{20} + \)\(10\!\cdots\!30\)\( T^{21} + \)\(18\!\cdots\!95\)\( T^{22} + \)\(60\!\cdots\!00\)\( T^{23} + \)\(65\!\cdots\!88\)\( T^{24} + \)\(52\!\cdots\!70\)\( T^{25} + \)\(65\!\cdots\!40\)\( T^{26} + \)\(47\!\cdots\!10\)\( T^{27} + \)\(31\!\cdots\!32\)\( T^{28} + \)\(29\!\cdots\!40\)\( T^{29} + \)\(13\!\cdots\!45\)\( T^{30} + \)\(63\!\cdots\!30\)\( T^{31} + \)\(61\!\cdots\!21\)\( T^{32} \))(\( ( 1 + 35 T + 600 T^{2} + 7860 T^{3} + 99976 T^{4} + 1207285 T^{5} + 13936775 T^{6} + 160561200 T^{7} + 1706485496 T^{8} + 15574436400 T^{9} + 131131115975 T^{10} + 1101856422805 T^{11} + 8850803397256 T^{12} + 67496494420020 T^{13} + 499783202957400 T^{14} + 2827939956733955 T^{15} + 7837433594376961 T^{16} )^{2} \))
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