Properties

Label 150.2.h
Level 150
Weight 2
Character orbit h
Rep. character \(\chi_{150}(19,\cdot)\)
Character field \(\Q(\zeta_{10})\)
Dimension 24
Newform subspaces 2
Sturm bound 60
Trace bound 1

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Defining parameters

Level: \( N \) = \( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 150.h (of order \(10\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 25 \)
Character field: \(\Q(\zeta_{10})\)
Newform subspaces: \( 2 \)
Sturm bound: \(60\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(150, [\chi])\).

Total New Old
Modular forms 136 24 112
Cusp forms 104 24 80
Eisenstein series 32 0 32

Trace form

\( 24q + 6q^{4} + 4q^{5} - 2q^{6} + 6q^{9} + O(q^{10}) \) \( 24q + 6q^{4} + 4q^{5} - 2q^{6} + 6q^{9} + 2q^{10} + 12q^{11} - 2q^{15} - 6q^{16} - 20q^{17} - 8q^{19} - 4q^{20} - 4q^{21} - 20q^{22} - 20q^{23} - 8q^{24} + 14q^{25} + 8q^{26} - 10q^{28} - 32q^{29} - 16q^{30} + 6q^{31} - 20q^{33} + 20q^{34} - 24q^{35} - 6q^{36} - 2q^{40} + 44q^{41} + 10q^{42} + 8q^{44} - 4q^{45} + 4q^{46} - 40q^{47} - 44q^{49} - 8q^{50} + 16q^{51} + 2q^{54} + 28q^{55} + 12q^{60} + 12q^{61} + 60q^{62} + 20q^{63} + 6q^{64} + 12q^{65} - 8q^{66} - 40q^{67} + 16q^{69} - 22q^{70} - 8q^{71} + 8q^{74} + 8q^{75} + 8q^{76} + 80q^{77} - 4q^{79} + 4q^{80} - 6q^{81} + 40q^{83} + 4q^{84} + 16q^{85} - 24q^{86} - 20q^{87} + 10q^{88} + 36q^{89} - 2q^{90} - 12q^{91} + 32q^{94} - 2q^{96} + 50q^{97} + 80q^{98} + 8q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(150, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
150.2.h.a \(8\) \(1.198\) \(\Q(\zeta_{20})\) None \(0\) \(0\) \(0\) \(0\) \(q+\zeta_{20}q^{2}-\zeta_{20}^{7}q^{3}+\zeta_{20}^{2}q^{4}+(\zeta_{20}+\cdots)q^{5}+\cdots\)
150.2.h.b \(16\) \(1.198\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(4\) \(0\) \(q+\beta _{8}q^{2}+\beta _{6}q^{3}-\beta _{10}q^{4}+(1+\beta _{3}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(150, [\chi])\) into lower level spaces

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 - T^{2} + T^{4} - T^{6} + T^{8} \))(\( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \))
$3$ (\( 1 - T^{2} + T^{4} - T^{6} + T^{8} \))(\( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \))
$5$ (\( 1 + 5 T^{2} + 10 T^{3} + 5 T^{4} + 50 T^{5} + 125 T^{6} + 625 T^{8} \))(\( 1 - 4 T - 4 T^{2} + 6 T^{3} + 161 T^{4} - 260 T^{5} - 580 T^{6} - 520 T^{7} + 7205 T^{8} - 2600 T^{9} - 14500 T^{10} - 32500 T^{11} + 100625 T^{12} + 18750 T^{13} - 62500 T^{14} - 312500 T^{15} + 390625 T^{16} \))
$7$ (\( 1 - 32 T^{2} + 500 T^{4} - 5152 T^{6} + 40294 T^{8} - 252448 T^{10} + 1200500 T^{12} - 3764768 T^{14} + 5764801 T^{16} \))(\( 1 - 30 T^{2} + 527 T^{4} - 6940 T^{6} + 74703 T^{8} - 716280 T^{10} + 6275029 T^{12} - 50516750 T^{14} + 372187880 T^{16} - 2475320750 T^{18} + 15066344629 T^{20} - 84269625720 T^{22} + 430647929103 T^{24} - 1960378228060 T^{26} + 7294358354927 T^{28} - 20346692185470 T^{30} + 33232930569601 T^{32} \))
$11$ (\( 1 - 10 T + 28 T^{2} + 60 T^{3} - 617 T^{4} + 1790 T^{5} - 794 T^{6} - 18400 T^{7} + 94725 T^{8} - 202400 T^{9} - 96074 T^{10} + 2382490 T^{11} - 9033497 T^{12} + 9663060 T^{13} + 49603708 T^{14} - 194871710 T^{15} + 214358881 T^{16} \))(\( 1 - 2 T + 11 T^{2} - 50 T^{3} + 270 T^{4} + 764 T^{5} - 168 T^{6} + 4404 T^{7} - 16480 T^{8} + 222470 T^{9} + 461835 T^{10} - 66610 T^{11} + 2053685 T^{12} + 10233320 T^{13} + 85541270 T^{14} + 166971000 T^{15} - 141281840 T^{16} + 1836681000 T^{17} + 10350493670 T^{18} + 13620548920 T^{19} + 30068002085 T^{20} - 10727607110 T^{21} + 818168874435 T^{22} + 4335310932370 T^{23} - 3532634358880 T^{24} + 10384401631164 T^{25} - 4357487332968 T^{26} + 217978116346804 T^{27} + 847375661714670 T^{28} - 1726135607196550 T^{29} + 4177248169415651 T^{30} - 8354496338831302 T^{31} + 45949729863572161 T^{32} \))
$13$ (\( 1 + 20 T + 202 T^{2} + 1420 T^{3} + 8095 T^{4} + 40660 T^{5} + 185692 T^{6} + 772160 T^{7} + 2918789 T^{8} + 10038080 T^{9} + 31381948 T^{10} + 89330020 T^{11} + 231201295 T^{12} + 527236060 T^{13} + 975015418 T^{14} + 1254970340 T^{15} + 815730721 T^{16} \))(\( 1 - 20 T + 240 T^{2} - 2100 T^{3} + 14722 T^{4} - 86440 T^{5} + 437000 T^{6} - 1922660 T^{7} + 7352283 T^{8} - 23888120 T^{9} + 61481440 T^{10} - 91205000 T^{11} - 205909876 T^{12} + 2578228580 T^{13} - 14615222400 T^{14} + 65003714260 T^{15} - 249510986035 T^{16} + 845048285380 T^{17} - 2469972585600 T^{18} + 5664368190260 T^{19} - 5880991968436 T^{20} - 33863778065000 T^{21} + 296759167924960 T^{22} - 1498944103918040 T^{23} + 5997483112586043 T^{24} - 20388846764492180 T^{25} + 60244160938013000 T^{26} - 154914344460558280 T^{27} + 342994409173165282 T^{28} - 636037723843731300 T^{29} + 944970332567829360 T^{30} - 1023717860281815140 T^{31} + 665416609183179841 T^{32} \))
$17$ (\( 1 - 10 T + 78 T^{2} - 590 T^{3} + 3615 T^{4} - 20330 T^{5} + 105768 T^{6} - 488740 T^{7} + 2089449 T^{8} - 8308580 T^{9} + 30566952 T^{10} - 99881290 T^{11} + 301928415 T^{12} - 837715630 T^{13} + 1882730382 T^{14} - 4103386730 T^{15} + 6975757441 T^{16} \))(\( 1 + 30 T + 500 T^{2} + 5990 T^{3} + 56782 T^{4} + 447460 T^{5} + 3015460 T^{6} + 17669550 T^{7} + 90820363 T^{8} + 409992380 T^{9} + 1611336300 T^{10} + 5357847740 T^{11} + 13820507184 T^{12} + 18014105810 T^{13} - 71676889400 T^{14} - 711604275870 T^{15} - 3544790874535 T^{16} - 12097272689790 T^{17} - 20714621036600 T^{18} + 88503301844530 T^{19} + 1154302580514864 T^{20} + 7607377618573180 T^{21} + 38893741123454700 T^{22} + 168235729149311740 T^{23} + 633540822991571083 T^{24} + 2095394413157566350 T^{25} + 6079148967047941540 T^{26} + 15335302721813462180 T^{27} + 33082455874380289102 T^{28} + 59328422417106562630 T^{29} + 84188913279700464500 T^{30} + 85872691545294473790 T^{31} + 48661191875666868481 T^{32} \))
$19$ (\( 1 + 8 T - 10 T^{2} - 270 T^{3} - 565 T^{4} + 5224 T^{5} + 28422 T^{6} - 55860 T^{7} - 788595 T^{8} - 1061340 T^{9} + 10260342 T^{10} + 35831416 T^{11} - 73631365 T^{12} - 668546730 T^{13} - 470458810 T^{14} + 7150973912 T^{15} + 16983563041 T^{16} \))(\( 1 + 24 T^{2} - 30 T^{3} + 330 T^{4} + 740 T^{5} + 7630 T^{6} + 18810 T^{7} + 160255 T^{8} - 244020 T^{9} + 7105152 T^{10} - 5384250 T^{11} + 141174008 T^{12} - 174364190 T^{13} + 1779959840 T^{14} + 4165821270 T^{15} + 38170716365 T^{16} + 79150604130 T^{17} + 642565502240 T^{18} - 1195963979210 T^{19} + 18397937896568 T^{20} - 13331936040750 T^{21} + 334268135478912 T^{22} - 218122581750780 T^{23} + 2721700895135455 T^{24} + 6069755595222990 T^{25} + 46780035547021630 T^{26} + 86202791584682060 T^{27} + 730393923291833130 T^{28} - 1261589503867711770 T^{29} + 19176160458789218904 T^{30} + \)\(28\!\cdots\!81\)\( T^{32} \))
$23$ (\( 1 + 10 T + 116 T^{2} + 950 T^{3} + 6927 T^{4} + 45910 T^{5} + 267168 T^{6} + 1460600 T^{7} + 7269605 T^{8} + 33593800 T^{9} + 141331872 T^{10} + 558586970 T^{11} + 1938458607 T^{12} + 6114525850 T^{13} + 17172163124 T^{14} + 34048254470 T^{15} + 78310985281 T^{16} \))(\( 1 + 10 T + 30 T^{2} - 250 T^{3} - 2718 T^{4} - 8680 T^{5} + 13790 T^{6} + 227450 T^{7} + 942143 T^{8} + 3042820 T^{9} + 4039430 T^{10} - 83513580 T^{11} - 909320196 T^{12} - 3918445730 T^{13} - 578679600 T^{14} + 92103435490 T^{15} + 633292773765 T^{16} + 2118379016270 T^{17} - 306121508400 T^{18} - 47675729196910 T^{19} - 254465072968836 T^{20} - 537522046037940 T^{21} + 597980611103270 T^{22} + 10360270966640540 T^{23} + 73780146605597183 T^{24} + 409672172849759350 T^{25} + 571271589636219710 T^{26} - 8270388698692886360 T^{27} - 59563949206231232478 T^{28} - \)\(12\!\cdots\!50\)\( T^{29} + \)\(34\!\cdots\!70\)\( T^{30} + \)\(26\!\cdots\!70\)\( T^{31} + \)\(61\!\cdots\!61\)\( T^{32} \))
$29$ (\( 1 + 22 T + 290 T^{2} + 2930 T^{3} + 24395 T^{4} + 175526 T^{5} + 1131792 T^{6} + 6698140 T^{7} + 37055745 T^{8} + 194246060 T^{9} + 951837072 T^{10} + 4280903614 T^{11} + 17254119995 T^{12} + 60097666570 T^{13} + 172498763090 T^{14} + 379497278798 T^{15} + 500246412961 T^{16} \))(\( 1 + 10 T - 36 T^{2} - 270 T^{3} + 3910 T^{4} + 4460 T^{5} - 185440 T^{6} + 333090 T^{7} + 4522435 T^{8} - 38473780 T^{9} - 69131068 T^{10} + 1395128220 T^{11} - 3409890032 T^{12} - 36344819210 T^{13} + 245632726680 T^{14} + 543098432830 T^{15} - 7660398480495 T^{16} + 15749854552070 T^{17} + 206577123137880 T^{18} - 886413795712690 T^{19} - 2411750431722992 T^{20} + 28615682794524780 T^{21} - 41120771452036828 T^{22} - 663667946139678020 T^{23} + 2262331886599280035 T^{24} + 4832185253102205210 T^{25} - 78015949343189273440 T^{26} + 54414273555047997340 T^{27} + \)\(13\!\cdots\!10\)\( T^{28} - \)\(27\!\cdots\!30\)\( T^{29} - \)\(10\!\cdots\!16\)\( T^{30} + \)\(86\!\cdots\!90\)\( T^{31} + \)\(25\!\cdots\!21\)\( T^{32} \))
$31$ (\( 1 - 24 T + 230 T^{2} - 990 T^{3} - 685 T^{4} + 32568 T^{5} - 115202 T^{6} - 726180 T^{7} + 8308045 T^{8} - 22511580 T^{9} - 110709122 T^{10} + 970233288 T^{11} - 632611885 T^{12} - 28342859490 T^{13} + 204125846630 T^{14} - 660302738664 T^{15} + 852891037441 T^{16} \))(\( 1 + 18 T + 121 T^{2} + 610 T^{3} + 6060 T^{4} + 54724 T^{5} + 309962 T^{6} + 1456784 T^{7} + 9332270 T^{8} + 64865970 T^{9} + 327433735 T^{10} + 1367357590 T^{11} + 7702756335 T^{12} + 47160385620 T^{13} + 227624335970 T^{14} + 1111476930100 T^{15} + 6249005152660 T^{16} + 34455784833100 T^{17} + 218746986867170 T^{18} + 1404955048005420 T^{19} + 7113657233255535 T^{20} + 39146286915106090 T^{21} + 290598645096078535 T^{22} + 1784632401545702670 T^{23} + 7959409441979521070 T^{24} + 38516818529710942064 T^{25} + \)\(25\!\cdots\!62\)\( T^{26} + \)\(13\!\cdots\!44\)\( T^{27} + \)\(47\!\cdots\!60\)\( T^{28} + \)\(14\!\cdots\!10\)\( T^{29} + \)\(91\!\cdots\!41\)\( T^{30} + \)\(42\!\cdots\!18\)\( T^{31} + \)\(72\!\cdots\!81\)\( T^{32} \))
$37$ (\( 1 + 20 T + 275 T^{2} + 3140 T^{3} + 29831 T^{4} + 251620 T^{5} + 1934025 T^{6} + 13441540 T^{7} + 85392936 T^{8} + 497336980 T^{9} + 2647680225 T^{10} + 12745307860 T^{11} + 55908096791 T^{12} + 217740024980 T^{13} + 705574762475 T^{14} + 1898637542660 T^{15} + 3512479453921 T^{16} \))(\( 1 - 20 T + 265 T^{2} - 2300 T^{3} + 17532 T^{4} - 115640 T^{5} + 825500 T^{6} - 5506160 T^{7} + 38914148 T^{8} - 253492260 T^{9} + 1794329215 T^{10} - 11470149820 T^{11} + 74232183599 T^{12} - 424648096560 T^{13} + 2693507120800 T^{14} - 16225080394680 T^{15} + 105486161388440 T^{16} - 600327974603160 T^{17} + 3687411248375200 T^{18} - 21509700035053680 T^{19} + 139123063446085439 T^{20} - 795385575901637740 T^{21} + 4603757853365738935 T^{22} - 24064496080486490580 T^{23} + \)\(13\!\cdots\!08\)\( T^{24} - \)\(71\!\cdots\!20\)\( T^{25} + \)\(39\!\cdots\!00\)\( T^{26} - \)\(20\!\cdots\!20\)\( T^{27} + \)\(11\!\cdots\!92\)\( T^{28} - \)\(56\!\cdots\!00\)\( T^{29} + \)\(23\!\cdots\!85\)\( T^{30} - \)\(66\!\cdots\!60\)\( T^{31} + \)\(12\!\cdots\!41\)\( T^{32} \))
$41$ (\( 1 - 22 T + 146 T^{2} + 142 T^{3} - 6829 T^{4} + 52634 T^{5} - 236776 T^{6} - 967444 T^{7} + 17720057 T^{8} - 39665204 T^{9} - 398020456 T^{10} + 3627587914 T^{11} - 19297121869 T^{12} + 16451580542 T^{13} + 693515219186 T^{14} - 4284594025382 T^{15} + 7984925229121 T^{16} \))(\( 1 - 22 T + 316 T^{2} - 3710 T^{3} + 39530 T^{4} - 407436 T^{5} + 3918652 T^{6} - 35382126 T^{7} + 301988495 T^{8} - 2475859380 T^{9} + 19707200260 T^{10} - 150750040860 T^{11} + 1111070917960 T^{12} - 7891095298530 T^{13} + 54497096107120 T^{14} - 366799083339050 T^{15} + 2387443151993985 T^{16} - 15038762416901050 T^{17} + 91609618556068720 T^{18} - 543862179069986130 T^{19} + 3139620868205567560 T^{20} - 17465327034634372860 T^{21} + 93611255533262302660 T^{22} - \)\(48\!\cdots\!80\)\( T^{23} + \)\(24\!\cdots\!95\)\( T^{24} - \)\(11\!\cdots\!86\)\( T^{25} + \)\(52\!\cdots\!52\)\( T^{26} - \)\(22\!\cdots\!76\)\( T^{27} + \)\(89\!\cdots\!30\)\( T^{28} - \)\(34\!\cdots\!10\)\( T^{29} + \)\(11\!\cdots\!76\)\( T^{30} - \)\(34\!\cdots\!22\)\( T^{31} + \)\(63\!\cdots\!41\)\( T^{32} \))
$43$ (\( 1 - 120 T^{2} + 10856 T^{4} - 676920 T^{6} + 33149566 T^{8} - 1251625080 T^{10} + 37114503656 T^{12} - 758563565880 T^{14} + 11688200277601 T^{16} \))(\( 1 - 360 T^{2} + 63972 T^{4} - 7536360 T^{6} + 666842308 T^{8} - 47508203160 T^{10} + 2846509586204 T^{12} - 147613243096600 T^{14} + 6742598283189430 T^{16} - 272936886485613400 T^{18} + 9731649819823821404 T^{20} - \)\(30\!\cdots\!40\)\( T^{22} + \)\(77\!\cdots\!08\)\( T^{24} - \)\(16\!\cdots\!40\)\( T^{26} + \)\(25\!\cdots\!72\)\( T^{28} - \)\(26\!\cdots\!40\)\( T^{30} + \)\(13\!\cdots\!01\)\( T^{32} \))
$47$ (\( 1 - 10 T + 164 T^{2} - 1060 T^{3} + 6767 T^{4} - 10550 T^{5} - 301398 T^{6} + 3376520 T^{7} - 34256075 T^{8} + 158696440 T^{9} - 665788182 T^{10} - 1095332650 T^{11} + 33020801327 T^{12} - 243105707420 T^{13} + 1767791313956 T^{14} - 5066231204630 T^{15} + 23811286661761 T^{16} \))(\( 1 + 50 T + 1290 T^{2} + 23780 T^{3} + 355482 T^{4} + 4545300 T^{5} + 51239280 T^{6} + 520209820 T^{7} + 4818856003 T^{8} + 41087515780 T^{9} + 324882626890 T^{10} + 2399419588970 T^{11} + 16710816340744 T^{12} + 111327617454640 T^{13} + 724257830956800 T^{14} + 4728319081214120 T^{15} + 31780196573425365 T^{16} + 222230996817063640 T^{17} + 1599885548583571200 T^{18} + 11558367226993088720 T^{19} + 81543452992418022664 T^{20} + \)\(55\!\cdots\!90\)\( T^{21} + \)\(35\!\cdots\!10\)\( T^{22} + \)\(20\!\cdots\!40\)\( T^{23} + \)\(11\!\cdots\!83\)\( T^{24} + \)\(58\!\cdots\!40\)\( T^{25} + \)\(26\!\cdots\!20\)\( T^{26} + \)\(11\!\cdots\!00\)\( T^{27} + \)\(41\!\cdots\!62\)\( T^{28} + \)\(12\!\cdots\!60\)\( T^{29} + \)\(33\!\cdots\!10\)\( T^{30} + \)\(60\!\cdots\!50\)\( T^{31} + \)\(56\!\cdots\!21\)\( T^{32} \))
$53$ (\( 1 + 30 T + 425 T^{2} + 3090 T^{3} + 5031 T^{4} - 120570 T^{5} - 1120625 T^{6} - 3254310 T^{7} + 1334576 T^{8} - 172478430 T^{9} - 3147835625 T^{10} - 17950099890 T^{11} + 39697009911 T^{12} + 1292224073370 T^{13} + 9419853479825 T^{14} + 35241334195110 T^{15} + 62259690411361 T^{16} \))(\( 1 - 30 T + 655 T^{2} - 10630 T^{3} + 152332 T^{4} - 1919860 T^{5} + 22723040 T^{6} - 248684220 T^{7} + 2591225028 T^{8} - 25379094630 T^{9} + 238630725505 T^{10} - 2134501287330 T^{11} + 18442776242119 T^{12} - 152410720886420 T^{13} + 1218238235823200 T^{14} - 9336287074370240 T^{15} + 69349166231444240 T^{16} - 494823214941622720 T^{17} + 3422031204427368800 T^{18} - 22690450893407550340 T^{19} + \)\(14\!\cdots\!39\)\( T^{20} - \)\(89\!\cdots\!90\)\( T^{21} + \)\(52\!\cdots\!45\)\( T^{22} - \)\(29\!\cdots\!10\)\( T^{23} + \)\(16\!\cdots\!08\)\( T^{24} - \)\(82\!\cdots\!60\)\( T^{25} + \)\(39\!\cdots\!60\)\( T^{26} - \)\(17\!\cdots\!20\)\( T^{27} + \)\(74\!\cdots\!12\)\( T^{28} - \)\(27\!\cdots\!90\)\( T^{29} + \)\(90\!\cdots\!95\)\( T^{30} - \)\(21\!\cdots\!10\)\( T^{31} + \)\(38\!\cdots\!21\)\( T^{32} \))
$59$ (\( 1 + 20 T + 242 T^{2} + 2760 T^{3} + 31483 T^{4} + 335220 T^{5} + 3074684 T^{6} + 25371200 T^{7} + 201486805 T^{8} + 1496900800 T^{9} + 10702975004 T^{10} + 68847148380 T^{11} + 381490876363 T^{12} + 1973191065240 T^{13} + 10207689141122 T^{14} + 49773029696380 T^{15} + 146830437604321 T^{16} \))(\( 1 - 20 T - 111 T^{2} + 3640 T^{3} + 11660 T^{4} - 317820 T^{5} - 1817980 T^{6} + 11679820 T^{7} + 217773860 T^{8} + 627241760 T^{9} - 13630850673 T^{10} - 133224376740 T^{11} + 186113511463 T^{12} + 9371240514920 T^{13} + 48131664631280 T^{14} - 252534601553160 T^{15} - 4526062111550040 T^{16} - 14899541491636440 T^{17} + 167546324581485680 T^{18} + 1924656005713754680 T^{19} + 2255204605374809143 T^{20} - 95245344150556405260 T^{21} - \)\(57\!\cdots\!93\)\( T^{22} + \)\(15\!\cdots\!40\)\( T^{23} + \)\(31\!\cdots\!60\)\( T^{24} + \)\(10\!\cdots\!80\)\( T^{25} - \)\(92\!\cdots\!80\)\( T^{26} - \)\(95\!\cdots\!80\)\( T^{27} + \)\(20\!\cdots\!60\)\( T^{28} + \)\(38\!\cdots\!60\)\( T^{29} - \)\(68\!\cdots\!71\)\( T^{30} - \)\(73\!\cdots\!80\)\( T^{31} + \)\(21\!\cdots\!41\)\( T^{32} \))
$61$ (\( 1 + 118 T^{2} + 10203 T^{4} + 835676 T^{6} + 64821605 T^{8} + 3109550396 T^{10} + 141269115723 T^{12} + 6079404174598 T^{14} + 191707312997281 T^{16} \))(\( 1 - 12 T - 104 T^{2} + 1460 T^{3} + 9310 T^{4} - 110696 T^{5} - 1128308 T^{6} + 9605804 T^{7} + 95707895 T^{8} - 545482080 T^{9} - 8568890440 T^{10} + 31012865040 T^{11} + 647686771260 T^{12} - 1172976299580 T^{13} - 45238788397880 T^{14} + 2951178504100 T^{15} + 3155660575885285 T^{16} + 180021888750100 T^{17} - 168333531628511480 T^{18} - 266243333454967980 T^{19} + 8967768052669329660 T^{20} + 26193351096196217040 T^{21} - \)\(44\!\cdots\!40\)\( T^{22} - \)\(17\!\cdots\!80\)\( T^{23} + \)\(18\!\cdots\!95\)\( T^{24} + \)\(11\!\cdots\!64\)\( T^{25} - \)\(80\!\cdots\!08\)\( T^{26} - \)\(48\!\cdots\!56\)\( T^{27} + \)\(24\!\cdots\!10\)\( T^{28} + \)\(23\!\cdots\!60\)\( T^{29} - \)\(10\!\cdots\!64\)\( T^{30} - \)\(72\!\cdots\!12\)\( T^{31} + \)\(36\!\cdots\!61\)\( T^{32} \))
$67$ (\( 1 - 10 T + 88 T^{2} - 1660 T^{3} + 11615 T^{4} - 86790 T^{5} + 911158 T^{6} - 5793400 T^{7} + 42525189 T^{8} - 388157800 T^{9} + 4090188262 T^{10} - 26103220770 T^{11} + 234055270415 T^{12} - 2241207677620 T^{13} + 7960337630872 T^{14} - 60607116053230 T^{15} + 406067677556641 T^{16} \))(\( 1 + 50 T + 1390 T^{2} + 30100 T^{3} + 553682 T^{4} + 8918100 T^{5} + 129754300 T^{6} + 1739003620 T^{7} + 21685053163 T^{8} + 253853201940 T^{9} + 2808527300190 T^{10} + 29485013121570 T^{11} + 294663718701784 T^{12} + 2810511152550240 T^{13} + 25630455288876000 T^{14} + 223725991474439120 T^{15} + 1870714073010261365 T^{16} + 14989641428787421040 T^{17} + \)\(11\!\cdots\!00\)\( T^{18} + \)\(84\!\cdots\!20\)\( T^{19} + \)\(59\!\cdots\!64\)\( T^{20} + \)\(39\!\cdots\!90\)\( T^{21} + \)\(25\!\cdots\!10\)\( T^{22} + \)\(15\!\cdots\!20\)\( T^{23} + \)\(88\!\cdots\!83\)\( T^{24} + \)\(47\!\cdots\!40\)\( T^{25} + \)\(23\!\cdots\!00\)\( T^{26} + \)\(10\!\cdots\!00\)\( T^{27} + \)\(45\!\cdots\!02\)\( T^{28} + \)\(16\!\cdots\!00\)\( T^{29} + \)\(51\!\cdots\!10\)\( T^{30} + \)\(12\!\cdots\!50\)\( T^{31} + \)\(16\!\cdots\!81\)\( T^{32} \))
$71$ (\( 1 - 20 T + 118 T^{2} + 490 T^{3} - 14397 T^{4} + 149580 T^{5} - 402414 T^{6} - 9696700 T^{7} + 137916525 T^{8} - 688465700 T^{9} - 2028568974 T^{10} + 53536327380 T^{11} - 365851971357 T^{12} + 884072381990 T^{13} + 15115833502678 T^{14} - 181902403167820 T^{15} + 645753531245761 T^{16} \))(\( 1 + 28 T + 256 T^{2} + 1150 T^{3} + 20770 T^{4} + 328604 T^{5} + 2351902 T^{6} + 21115094 T^{7} + 321977895 T^{8} + 2983992020 T^{9} + 18685321160 T^{10} + 180970630790 T^{11} + 2127756781760 T^{12} + 17298444226470 T^{13} + 123520240637920 T^{14} + 1239239342309850 T^{15} + 12274938369025885 T^{16} + 87985993303999350 T^{17} + 622665533055754720 T^{18} + 6191303471540104170 T^{19} + 54069876583671738560 T^{20} + \)\(32\!\cdots\!90\)\( T^{21} + \)\(23\!\cdots\!60\)\( T^{22} + \)\(27\!\cdots\!20\)\( T^{23} + \)\(20\!\cdots\!95\)\( T^{24} + \)\(96\!\cdots\!14\)\( T^{25} + \)\(76\!\cdots\!02\)\( T^{26} + \)\(75\!\cdots\!84\)\( T^{27} + \)\(34\!\cdots\!70\)\( T^{28} + \)\(13\!\cdots\!50\)\( T^{29} + \)\(21\!\cdots\!36\)\( T^{30} + \)\(16\!\cdots\!28\)\( T^{31} + \)\(41\!\cdots\!21\)\( T^{32} \))
$73$ (\( 1 + 20 T + 282 T^{2} + 3780 T^{3} + 46315 T^{4} + 505780 T^{5} + 5254892 T^{6} + 48564720 T^{7} + 416447549 T^{8} + 3545224560 T^{9} + 28003319468 T^{10} + 196757018260 T^{11} + 1315264531915 T^{12} + 7836210621540 T^{13} + 42676251813498 T^{14} + 220947970381940 T^{15} + 806460091894081 T^{16} \))(\( 1 - 20 T + 380 T^{2} - 7420 T^{3} + 107282 T^{4} - 1509240 T^{5} + 20338260 T^{6} - 240254580 T^{7} + 2794851843 T^{8} - 31155740720 T^{9} + 320911861980 T^{10} - 3274927660920 T^{11} + 32181004976604 T^{12} - 299417039187980 T^{13} + 2775262759264800 T^{14} - 24764197420905260 T^{15} + 211758644546106765 T^{16} - 1807786411726083980 T^{17} + 14789375244122119200 T^{18} - \)\(11\!\cdots\!60\)\( T^{19} + \)\(91\!\cdots\!64\)\( T^{20} - \)\(67\!\cdots\!60\)\( T^{21} + \)\(48\!\cdots\!20\)\( T^{22} - \)\(34\!\cdots\!40\)\( T^{23} + \)\(22\!\cdots\!83\)\( T^{24} - \)\(14\!\cdots\!40\)\( T^{25} + \)\(87\!\cdots\!40\)\( T^{26} - \)\(47\!\cdots\!80\)\( T^{27} + \)\(24\!\cdots\!22\)\( T^{28} - \)\(12\!\cdots\!60\)\( T^{29} + \)\(46\!\cdots\!20\)\( T^{30} - \)\(17\!\cdots\!40\)\( T^{31} + \)\(65\!\cdots\!61\)\( T^{32} \))
$79$ (\( 1 - 16 T + 274 T^{2} - 3476 T^{3} + 48131 T^{4} - 506768 T^{5} + 5479676 T^{6} - 51684448 T^{7} + 496487797 T^{8} - 4083071392 T^{9} + 34198657916 T^{10} - 249856387952 T^{11} + 1874706348611 T^{12} - 10695848042924 T^{13} + 66605962812754 T^{14} - 307262543778544 T^{15} + 1517108809906561 T^{16} \))(\( 1 + 20 T + 224 T^{2} + 1780 T^{3} + 6070 T^{4} + 5760 T^{5} + 875320 T^{6} + 20333540 T^{7} + 305719495 T^{8} + 3198483920 T^{9} + 22059896192 T^{10} + 116979470440 T^{11} + 660067500148 T^{12} + 7893347694940 T^{13} + 139701887310360 T^{14} + 1828379420794980 T^{15} + 18018691485968485 T^{16} + 144441974242803420 T^{17} + 871879478703956760 T^{18} + 3891728254165522660 T^{19} + 25709682596232111988 T^{20} + \)\(35\!\cdots\!60\)\( T^{21} + \)\(53\!\cdots\!32\)\( T^{22} + \)\(61\!\cdots\!80\)\( T^{23} + \)\(46\!\cdots\!95\)\( T^{24} + \)\(24\!\cdots\!60\)\( T^{25} + \)\(82\!\cdots\!20\)\( T^{26} + \)\(43\!\cdots\!40\)\( T^{27} + \)\(35\!\cdots\!70\)\( T^{28} + \)\(83\!\cdots\!20\)\( T^{29} + \)\(82\!\cdots\!44\)\( T^{30} + \)\(58\!\cdots\!80\)\( T^{31} + \)\(23\!\cdots\!21\)\( T^{32} \))
$83$ (\( 1 - 70 T + 2432 T^{2} - 56730 T^{3} + 1010935 T^{4} - 14768130 T^{5} + 184247072 T^{6} - 2010015520 T^{7} + 19419043189 T^{8} - 166831288160 T^{9} + 1269278079008 T^{10} - 8444224748310 T^{11} + 47977277740135 T^{12} - 223461775677390 T^{13} + 795118988033408 T^{14} - 1899523569273890 T^{15} + 2252292232139041 T^{16} \))(\( 1 + 30 T + 535 T^{2} + 5780 T^{3} + 43882 T^{4} + 205010 T^{5} + 590510 T^{6} + 2468290 T^{7} + 123659138 T^{8} + 2415976440 T^{9} + 27231081235 T^{10} + 158815212750 T^{11} + 405339323209 T^{12} + 3244201552980 T^{13} + 126271405899100 T^{14} + 2149314211865960 T^{15} + 23236430083058540 T^{16} + 178393079584874680 T^{17} + 869883715238899900 T^{18} + 1854992273373775260 T^{19} + 19236723714775472089 T^{20} + \)\(62\!\cdots\!50\)\( T^{21} + \)\(89\!\cdots\!15\)\( T^{22} + \)\(65\!\cdots\!80\)\( T^{23} + \)\(27\!\cdots\!58\)\( T^{24} + \)\(46\!\cdots\!70\)\( T^{25} + \)\(91\!\cdots\!90\)\( T^{26} + \)\(26\!\cdots\!70\)\( T^{27} + \)\(46\!\cdots\!02\)\( T^{28} + \)\(51\!\cdots\!40\)\( T^{29} + \)\(39\!\cdots\!15\)\( T^{30} + \)\(18\!\cdots\!10\)\( T^{31} + \)\(50\!\cdots\!81\)\( T^{32} \))
$89$ (\( 1 + 34 T + 409 T^{2} + 1014 T^{3} - 27449 T^{4} - 344318 T^{5} - 919769 T^{6} + 22713862 T^{7} + 350578752 T^{8} + 2021533718 T^{9} - 7285490249 T^{10} - 242733516142 T^{11} - 1722211773209 T^{12} + 5662236281286 T^{13} + 203265348003049 T^{14} + 1503865386447986 T^{15} + 3936588805702081 T^{16} \))(\( 1 - 70 T + 2119 T^{2} - 35330 T^{3} + 343460 T^{4} - 2032360 T^{5} + 10870620 T^{6} - 64684440 T^{7} - 584089940 T^{8} + 17002587830 T^{9} - 99745085123 T^{10} - 366225143690 T^{11} + 5693869232523 T^{12} - 24776539056740 T^{13} + 308063609313680 T^{14} - 3148141734399680 T^{15} + 20769024208268760 T^{16} - 280184614361571520 T^{17} + 2440171849373659280 T^{18} - 17466691962290941060 T^{19} + \)\(35\!\cdots\!43\)\( T^{20} - \)\(20\!\cdots\!10\)\( T^{21} - \)\(49\!\cdots\!03\)\( T^{22} + \)\(75\!\cdots\!70\)\( T^{23} - \)\(22\!\cdots\!40\)\( T^{24} - \)\(22\!\cdots\!60\)\( T^{25} + \)\(33\!\cdots\!20\)\( T^{26} - \)\(56\!\cdots\!40\)\( T^{27} + \)\(84\!\cdots\!60\)\( T^{28} - \)\(77\!\cdots\!70\)\( T^{29} + \)\(41\!\cdots\!79\)\( T^{30} - \)\(12\!\cdots\!30\)\( T^{31} + \)\(15\!\cdots\!61\)\( T^{32} \))
$97$ (\( 1 - 60 T + 1990 T^{2} - 47940 T^{3} + 920391 T^{4} - 14754060 T^{5} + 202723580 T^{6} - 2424996600 T^{7} + 25463108381 T^{8} - 235224670200 T^{9} + 1907426164220 T^{10} - 13465632202380 T^{11} + 81481553468871 T^{12} - 411677091920580 T^{13} + 1657614289808710 T^{14} - 4847897068686780 T^{15} + 7837433594376961 T^{16} \))(\( 1 + 10 T + 205 T^{2} + 4320 T^{3} + 44952 T^{4} + 667870 T^{5} + 8897460 T^{6} + 68756910 T^{7} + 834554408 T^{8} + 7477618800 T^{9} + 22708693755 T^{10} + 118605463990 T^{11} - 2491804851081 T^{12} - 84263627884560 T^{13} - 895266623177600 T^{14} - 11056342779615180 T^{15} - 138673183182585960 T^{16} - 1072465249622672460 T^{17} - 8423563657478038400 T^{18} - 76905138052285028880 T^{19} - \)\(22\!\cdots\!61\)\( T^{20} + \)\(10\!\cdots\!30\)\( T^{21} + \)\(18\!\cdots\!95\)\( T^{22} + \)\(60\!\cdots\!00\)\( T^{23} + \)\(65\!\cdots\!88\)\( T^{24} + \)\(52\!\cdots\!70\)\( T^{25} + \)\(65\!\cdots\!40\)\( T^{26} + \)\(47\!\cdots\!10\)\( T^{27} + \)\(31\!\cdots\!32\)\( T^{28} + \)\(29\!\cdots\!40\)\( T^{29} + \)\(13\!\cdots\!45\)\( T^{30} + \)\(63\!\cdots\!30\)\( T^{31} + \)\(61\!\cdots\!21\)\( T^{32} \))
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