Properties

Label 1050.2.bc
Level 1050
Weight 2
Character orbit bc
Rep. character \(\chi_{1050}(157,\cdot)\)
Character field \(\Q(\zeta_{12})\)
Dimension 96
Newform subspaces 8
Sturm bound 480
Trace bound 19

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Defining parameters

Level: \( N \) = \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1050.bc (of order \(12\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 35 \)
Character field: \(\Q(\zeta_{12})\)
Newform subspaces: \( 8 \)
Sturm bound: \(480\)
Trace bound: \(19\)
Distinguishing \(T_p\): \(11\), \(13\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1050, [\chi])\).

Total New Old
Modular forms 1056 96 960
Cusp forms 864 96 768
Eisenstein series 192 0 192

Trace form

\( 96q + 12q^{7} + O(q^{10}) \) \( 96q + 12q^{7} - 16q^{11} + 48q^{16} + 8q^{21} - 8q^{22} + 8q^{23} + 48q^{26} + 12q^{28} + 96q^{31} - 12q^{33} - 96q^{36} + 16q^{37} + 48q^{38} - 4q^{42} + 48q^{43} + 16q^{46} + 24q^{47} - 32q^{51} - 16q^{53} - 32q^{56} + 16q^{57} - 36q^{58} + 24q^{61} + 48q^{67} + 128q^{71} + 24q^{73} + 32q^{78} + 48q^{81} - 48q^{82} + 32q^{86} - 60q^{87} - 4q^{88} + 8q^{91} - 16q^{92} - 8q^{93} - 16q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1050, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
1050.2.bc.a \(8\) \(8.384\) \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(0\) \(q-\zeta_{24}^{7}q^{2}+(-\zeta_{24}^{3}+\zeta_{24}^{7})q^{3}+\cdots\)
1050.2.bc.b \(8\) \(8.384\) \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(0\) \(q+(\zeta_{24}^{3}-\zeta_{24}^{7})q^{2}+\zeta_{24}^{7}q^{3}+(\zeta_{24}^{2}+\cdots)q^{4}+\cdots\)
1050.2.bc.c \(8\) \(8.384\) \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(0\) \(q-\zeta_{24}^{7}q^{2}+(\zeta_{24}^{3}-\zeta_{24}^{7})q^{3}-\zeta_{24}^{2}q^{4}+\cdots\)
1050.2.bc.d \(8\) \(8.384\) \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(0\) \(q-\zeta_{24}q^{2}-\zeta_{24}^{5}q^{3}+\zeta_{24}^{2}q^{4}+\zeta_{24}^{6}q^{6}+\cdots\)
1050.2.bc.e \(16\) \(8.384\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{2}q^{2}-\beta _{11}q^{3}+(\beta _{8}-\beta _{9})q^{4}-\beta _{9}q^{6}+\cdots\)
1050.2.bc.f \(16\) \(8.384\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{11}q^{2}+\beta _{2}q^{3}-\beta _{8}q^{4}+\beta _{9}q^{6}+\cdots\)
1050.2.bc.g \(16\) \(8.384\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(0\) \(4\) \(q-\beta _{6}q^{2}-\beta _{12}q^{3}+(-\beta _{5}+\beta _{13})q^{4}+\cdots\)
1050.2.bc.h \(16\) \(8.384\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(0\) \(8\) \(q+\beta _{15}q^{2}+\beta _{2}q^{3}+(\beta _{5}-\beta _{13})q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1050, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1050, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(70, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(175, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(210, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(350, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(525, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 - T^{4} + T^{8} \))(\( 1 - T^{4} + T^{8} \))(\( 1 - T^{4} + T^{8} \))(\( 1 - T^{4} + T^{8} \))(\( ( 1 - T^{4} + T^{8} )^{2} \))(\( ( 1 - T^{4} + T^{8} )^{2} \))(\( ( 1 - T^{4} + T^{8} )^{2} \))(\( ( 1 - T^{4} + T^{8} )^{2} \))
$3$ (\( 1 - T^{4} + T^{8} \))(\( 1 - T^{4} + T^{8} \))(\( 1 - T^{4} + T^{8} \))(\( 1 - T^{4} + T^{8} \))(\( ( 1 - T^{4} + T^{8} )^{2} \))(\( ( 1 - T^{4} + T^{8} )^{2} \))(\( ( 1 - T^{4} + T^{8} )^{2} \))(\( ( 1 - T^{4} + T^{8} )^{2} \))
$5$ 1
$7$ (\( 1 - 94 T^{4} + 2401 T^{8} \))(\( 1 + 71 T^{4} + 2401 T^{8} \))(\( 1 + 71 T^{4} + 2401 T^{8} \))(\( 1 - 94 T^{4} + 2401 T^{8} \))(\( ( 1 + 24 T^{2} + 239 T^{4} + 1176 T^{6} + 2401 T^{8} )^{2} \))(\( ( 1 - 24 T^{2} + 239 T^{4} - 1176 T^{6} + 2401 T^{8} )^{2} \))(\( 1 - 4 T + 34 T^{2} - 80 T^{3} + 445 T^{4} - 704 T^{5} + 3926 T^{6} - 5044 T^{7} + 29688 T^{8} - 35308 T^{9} + 192374 T^{10} - 241472 T^{11} + 1068445 T^{12} - 1344560 T^{13} + 4000066 T^{14} - 3294172 T^{15} + 5764801 T^{16} \))(\( 1 - 8 T + 6 T^{2} + 120 T^{3} - 275 T^{4} - 1120 T^{5} + 4770 T^{6} + 3216 T^{7} - 40872 T^{8} + 22512 T^{9} + 233730 T^{10} - 384160 T^{11} - 660275 T^{12} + 2016840 T^{13} + 705894 T^{14} - 6588344 T^{15} + 5764801 T^{16} \))
$11$ (\( ( 1 + T - 10 T^{2} + 11 T^{3} + 121 T^{4} )^{4} \))(\( ( 1 + 2 T - 16 T^{2} - 4 T^{3} + 235 T^{4} - 44 T^{5} - 1936 T^{6} + 2662 T^{7} + 14641 T^{8} )^{2} \))(\( ( 1 + 2 T - 16 T^{2} - 4 T^{3} + 235 T^{4} - 44 T^{5} - 1936 T^{6} + 2662 T^{7} + 14641 T^{8} )^{2} \))(\( ( 1 + T - 10 T^{2} + 11 T^{3} + 121 T^{4} )^{4} \))(\( ( 1 + 2 T - 14 T^{2} - 48 T^{3} - 58 T^{4} + 26 T^{5} - 168 T^{6} + 2902 T^{7} + 26203 T^{8} + 31922 T^{9} - 20328 T^{10} + 34606 T^{11} - 849178 T^{12} - 7730448 T^{13} - 24801854 T^{14} + 38974342 T^{15} + 214358881 T^{16} )^{2} \))(\( ( 1 + 2 T - 14 T^{2} - 48 T^{3} - 58 T^{4} + 26 T^{5} - 168 T^{6} + 2902 T^{7} + 26203 T^{8} + 31922 T^{9} - 20328 T^{10} + 34606 T^{11} - 849178 T^{12} - 7730448 T^{13} - 24801854 T^{14} + 38974342 T^{15} + 214358881 T^{16} )^{2} \))(\( 1 - 4 T - 42 T^{2} + 72 T^{3} + 1354 T^{4} - 52 T^{5} - 25972 T^{6} - 39092 T^{7} + 336125 T^{8} + 1014028 T^{9} - 2401652 T^{10} - 15137644 T^{11} - 1718882 T^{12} + 141984000 T^{13} + 309093362 T^{14} - 600162532 T^{15} - 4715588236 T^{16} - 6601787852 T^{17} + 37400296802 T^{18} + 188980704000 T^{19} - 25166151362 T^{20} - 2437932703844 T^{21} - 4254673018772 T^{22} + 19760537034788 T^{23} + 72051378876125 T^{24} - 92176891136572 T^{25} - 673646791737172 T^{26} - 14836206871772 T^{27} + 4249432022080234 T^{28} + 2485635274363032 T^{29} - 15949493010496122 T^{30} - 16708992677662604 T^{31} + 45949729863572161 T^{32} \))(\( 1 - 4 T - 42 T^{2} + 72 T^{3} + 1354 T^{4} - 52 T^{5} - 25972 T^{6} - 39092 T^{7} + 336125 T^{8} + 1014028 T^{9} - 2401652 T^{10} - 15137644 T^{11} - 1718882 T^{12} + 141984000 T^{13} + 309093362 T^{14} - 600162532 T^{15} - 4715588236 T^{16} - 6601787852 T^{17} + 37400296802 T^{18} + 188980704000 T^{19} - 25166151362 T^{20} - 2437932703844 T^{21} - 4254673018772 T^{22} + 19760537034788 T^{23} + 72051378876125 T^{24} - 92176891136572 T^{25} - 673646791737172 T^{26} - 14836206871772 T^{27} + 4249432022080234 T^{28} + 2485635274363032 T^{29} - 15949493010496122 T^{30} - 16708992677662604 T^{31} + 45949729863572161 T^{32} \))
$13$ (\( 1 - 194 T^{4} + 28899 T^{8} - 5540834 T^{12} + 815730721 T^{16} \))(\( 1 - 92 T^{4} - 17562 T^{8} - 2627612 T^{12} + 815730721 T^{16} \))(\( 1 - 92 T^{4} - 17562 T^{8} - 2627612 T^{12} + 815730721 T^{16} \))(\( 1 - 194 T^{4} + 28899 T^{8} - 5540834 T^{12} + 815730721 T^{16} \))(\( 1 + 50 T^{4} + 39793 T^{8} + 11442962 T^{12} + 523271620 T^{16} + 326822437682 T^{20} + 32460372580753 T^{24} + 1164904256124050 T^{28} + 665416609183179841 T^{32} \))(\( 1 + 50 T^{4} + 39793 T^{8} + 11442962 T^{12} + 523271620 T^{16} + 326822437682 T^{20} + 32460372580753 T^{24} + 1164904256124050 T^{28} + 665416609183179841 T^{32} \))(\( 1 + 16 T + 128 T^{2} + 744 T^{3} + 3314 T^{4} + 10768 T^{5} + 24864 T^{6} + 28864 T^{7} - 4655 T^{8} + 572216 T^{9} + 6801728 T^{10} + 43610328 T^{11} + 198012786 T^{12} + 641713232 T^{13} + 1520861600 T^{14} + 2393411960 T^{15} + 3297590084 T^{16} + 31114355480 T^{17} + 257025610400 T^{18} + 1409843970704 T^{19} + 5655443180946 T^{20} + 16192209514104 T^{21} + 32830641925952 T^{22} + 35905705403672 T^{23} - 3797226506255 T^{24} + 306088269902272 T^{25} + 3427713541333536 T^{26} + 19297983122990416 T^{27} + 77209854095902034 T^{28} + 225339079304636232 T^{29} + 503984177369508992 T^{30} + 818974288225452112 T^{31} + 665416609183179841 T^{32} \))(\( 1 - 16 T + 128 T^{2} - 744 T^{3} + 3314 T^{4} - 10768 T^{5} + 24864 T^{6} - 28864 T^{7} - 4655 T^{8} - 572216 T^{9} + 6801728 T^{10} - 43610328 T^{11} + 198012786 T^{12} - 641713232 T^{13} + 1520861600 T^{14} - 2393411960 T^{15} + 3297590084 T^{16} - 31114355480 T^{17} + 257025610400 T^{18} - 1409843970704 T^{19} + 5655443180946 T^{20} - 16192209514104 T^{21} + 32830641925952 T^{22} - 35905705403672 T^{23} - 3797226506255 T^{24} - 306088269902272 T^{25} + 3427713541333536 T^{26} - 19297983122990416 T^{27} + 77209854095902034 T^{28} - 225339079304636232 T^{29} + 503984177369508992 T^{30} - 818974288225452112 T^{31} + 665416609183179841 T^{32} \))
$17$ (\( 1 - 96 T^{2} + 4382 T^{4} - 125760 T^{6} + 2520387 T^{8} - 36344640 T^{10} + 365989022 T^{12} - 2317206624 T^{14} + 6975757441 T^{16} \))(\( ( 1 + 16 T^{2} + 289 T^{4} )^{2}( 1 + 16 T^{2} - 33 T^{4} + 4624 T^{6} + 83521 T^{8} ) \))(\( ( 1 - 16 T^{2} + 289 T^{4} )^{2}( 1 - 16 T^{2} - 33 T^{4} - 4624 T^{6} + 83521 T^{8} ) \))(\( 1 + 96 T^{2} + 4382 T^{4} + 125760 T^{6} + 2520387 T^{8} + 36344640 T^{10} + 365989022 T^{12} + 2317206624 T^{14} + 6975757441 T^{16} \))(\( 1 + 96 T^{2} + 4335 T^{4} + 121248 T^{6} + 2296193 T^{8} + 27677184 T^{10} + 58458882 T^{12} - 6525249024 T^{14} - 167337014754 T^{16} - 1885796967936 T^{18} + 4882544283522 T^{20} + 668059938525696 T^{22} + 16017685405722113 T^{24} + 244435228441640352 T^{26} + 2525667398391013935 T^{28} + 16164271349702489184 T^{30} + 48661191875666868481 T^{32} \))(\( 1 - 96 T^{2} + 4335 T^{4} - 121248 T^{6} + 2296193 T^{8} - 27677184 T^{10} + 58458882 T^{12} + 6525249024 T^{14} - 167337014754 T^{16} + 1885796967936 T^{18} + 4882544283522 T^{20} - 668059938525696 T^{22} + 16017685405722113 T^{24} - 244435228441640352 T^{26} + 2525667398391013935 T^{28} - 16164271349702489184 T^{30} + 48661191875666868481 T^{32} \))(\( 1 + 12 T - 760 T^{3} - 4000 T^{4} + 9644 T^{5} + 164912 T^{6} + 417780 T^{7} - 2033646 T^{8} - 15406336 T^{9} - 21694928 T^{10} + 141136884 T^{11} + 768607008 T^{12} + 1405601268 T^{13} - 1667112976 T^{14} - 24205669888 T^{15} - 123458993805 T^{16} - 411496388096 T^{17} - 481795650064 T^{18} + 6905719029684 T^{19} + 64194825915168 T^{20} + 200394192705588 T^{21} - 523662821550032 T^{22} - 6321815470032128 T^{23} - 14186221216859886 T^{24} + 49543643042916660 T^{25} + 332461586110845488 T^{26} + 330518167990812652 T^{27} - 2330488948919044000 T^{28} - 7527479305008512120 T^{29} + 34349076618117789516 T^{31} + 48661191875666868481 T^{32} \))(\( 1 - 12 T + 144 T^{2} - 968 T^{3} + 6368 T^{4} - 26396 T^{5} + 107888 T^{6} - 145332 T^{7} - 233838 T^{8} + 8508352 T^{9} - 40945088 T^{10} + 232534236 T^{11} - 555402720 T^{12} + 1622229852 T^{13} + 5913536576 T^{14} - 33762615104 T^{15} + 261999162867 T^{16} - 573964456768 T^{17} + 1709012070464 T^{18} + 7970015262876 T^{19} - 46387790577120 T^{20} + 330165362724252 T^{21} - 988314886811072 T^{22} + 3491305869096896 T^{23} - 1631197168488558 T^{24} - 17234613267062004 T^{25} + 217501549931641712 T^{26} - 904640974936280668 T^{27} + 3710138406679118048 T^{28} - 9587631535852947016 T^{29} + 24246407024553733776 T^{30} - 34349076618117789516 T^{31} + 48661191875666868481 T^{32} \))
$19$ (\( ( 1 + 8 T + 13 T^{2} + 104 T^{3} + 1024 T^{4} + 1976 T^{5} + 4693 T^{6} + 54872 T^{7} + 130321 T^{8} )^{2} \))(\( ( 1 + 2 T - 8 T^{2} - 52 T^{3} - 293 T^{4} - 988 T^{5} - 2888 T^{6} + 13718 T^{7} + 130321 T^{8} )^{2} \))(\( ( 1 - 2 T - 8 T^{2} + 52 T^{3} - 293 T^{4} + 988 T^{5} - 2888 T^{6} - 13718 T^{7} + 130321 T^{8} )^{2} \))(\( ( 1 - 8 T + 13 T^{2} - 104 T^{3} + 1024 T^{4} - 1976 T^{5} + 4693 T^{6} - 54872 T^{7} + 130321 T^{8} )^{2} \))(\( ( 1 - 2 T - 43 T^{2} - 70 T^{3} + 1187 T^{4} + 3200 T^{5} - 13284 T^{6} - 38196 T^{7} + 120370 T^{8} - 725724 T^{9} - 4795524 T^{10} + 21948800 T^{11} + 154691027 T^{12} - 173326930 T^{13} - 2022972883 T^{14} - 1787743478 T^{15} + 16983563041 T^{16} )^{2} \))(\( ( 1 + 2 T - 43 T^{2} + 70 T^{3} + 1187 T^{4} - 3200 T^{5} - 13284 T^{6} + 38196 T^{7} + 120370 T^{8} + 725724 T^{9} - 4795524 T^{10} - 21948800 T^{11} + 154691027 T^{12} + 173326930 T^{13} - 2022972883 T^{14} + 1787743478 T^{15} + 16983563041 T^{16} )^{2} \))(\( 1 - 8 T - 22 T^{2} - 144 T^{3} + 3371 T^{4} + 4000 T^{5} - 7722 T^{6} - 652040 T^{7} - 370487 T^{8} + 2996624 T^{9} + 89400308 T^{10} + 59920656 T^{11} - 371517522 T^{12} - 10373644960 T^{13} - 8723924296 T^{14} + 39539357168 T^{15} + 977996314130 T^{16} + 751247786192 T^{17} - 3149336670856 T^{18} - 71152830780640 T^{19} - 48416534984562 T^{20} + 148369476400944 T^{21} + 4205916251531348 T^{22} + 2678597506009136 T^{23} - 6292189320370967 T^{24} - 210405286459819160 T^{25} - 47344093642739322 T^{26} + 465961035592876000 T^{27} + 7461084592172028731 T^{28} - 6055629618565016496 T^{29} - 17578147087223450662 T^{30} - \)\(12\!\cdots\!92\)\( T^{31} + \)\(28\!\cdots\!81\)\( T^{32} \))(\( 1 + 8 T - 22 T^{2} + 144 T^{3} + 3371 T^{4} - 4000 T^{5} - 7722 T^{6} + 652040 T^{7} - 370487 T^{8} - 2996624 T^{9} + 89400308 T^{10} - 59920656 T^{11} - 371517522 T^{12} + 10373644960 T^{13} - 8723924296 T^{14} - 39539357168 T^{15} + 977996314130 T^{16} - 751247786192 T^{17} - 3149336670856 T^{18} + 71152830780640 T^{19} - 48416534984562 T^{20} - 148369476400944 T^{21} + 4205916251531348 T^{22} - 2678597506009136 T^{23} - 6292189320370967 T^{24} + 210405286459819160 T^{25} - 47344093642739322 T^{26} - 465961035592876000 T^{27} + 7461084592172028731 T^{28} + 6055629618565016496 T^{29} - 17578147087223450662 T^{30} + \)\(12\!\cdots\!92\)\( T^{31} + \)\(28\!\cdots\!81\)\( T^{32} \))
$23$ (\( 1 - 48 T^{2} + 2009 T^{4} - 59568 T^{6} + 1666512 T^{8} - 31511472 T^{10} + 562200569 T^{12} - 7105722672 T^{14} + 78310985281 T^{16} \))(\( 1 + 24 T^{2} - 223 T^{4} - 9960 T^{6} - 6048 T^{8} - 5268840 T^{10} - 62404543 T^{12} + 3552861336 T^{14} + 78310985281 T^{16} \))(\( 1 - 24 T^{2} - 223 T^{4} + 9960 T^{6} - 6048 T^{8} + 5268840 T^{10} - 62404543 T^{12} - 3552861336 T^{14} + 78310985281 T^{16} \))(\( 1 + 48 T^{2} + 2009 T^{4} + 59568 T^{6} + 1666512 T^{8} + 31511472 T^{10} + 562200569 T^{12} + 7105722672 T^{14} + 78310985281 T^{16} \))(\( 1 - 96 T^{2} + 4119 T^{4} - 100512 T^{6} + 1547777 T^{8} - 13526016 T^{10} - 398667390 T^{12} + 32311398144 T^{14} - 1026908911218 T^{16} + 17092729618176 T^{18} - 111563481084990 T^{20} - 2002335803188224 T^{22} + 121207941865270337 T^{24} - 4163861495106288288 T^{26} + 90266338035491702199 T^{28} - \)\(11\!\cdots\!64\)\( T^{30} + \)\(61\!\cdots\!61\)\( T^{32} \))(\( 1 + 96 T^{2} + 4119 T^{4} + 100512 T^{6} + 1547777 T^{8} + 13526016 T^{10} - 398667390 T^{12} - 32311398144 T^{14} - 1026908911218 T^{16} - 17092729618176 T^{18} - 111563481084990 T^{20} + 2002335803188224 T^{22} + 121207941865270337 T^{24} + 4163861495106288288 T^{26} + 90266338035491702199 T^{28} + \)\(11\!\cdots\!64\)\( T^{30} + \)\(61\!\cdots\!61\)\( T^{32} \))(\( 1 - 40 T + 776 T^{2} - 9784 T^{3} + 91799 T^{4} - 705368 T^{5} + 4787160 T^{6} - 29946728 T^{7} + 172895985 T^{8} - 902021104 T^{9} + 4178360768 T^{10} - 17091640336 T^{11} + 61289409778 T^{12} - 182133009408 T^{13} + 342869112800 T^{14} + 372347426016 T^{15} - 5577275049586 T^{16} + 8563990798368 T^{17} + 181377760671200 T^{18} - 2216012325467136 T^{19} + 17151289721685298 T^{20} - 110007659635131248 T^{21} + 618547350853602752 T^{22} - 3071224408630233488 T^{23} + 13539654936478996785 T^{24} - 53938628839308543064 T^{25} + \)\(19\!\cdots\!40\)\( T^{26} - \)\(67\!\cdots\!36\)\( T^{27} + \)\(20\!\cdots\!79\)\( T^{28} - \)\(49\!\cdots\!72\)\( T^{29} + \)\(89\!\cdots\!84\)\( T^{30} - \)\(10\!\cdots\!80\)\( T^{31} + \)\(61\!\cdots\!61\)\( T^{32} \))(\( 1 + 32 T + 536 T^{2} + 6152 T^{3} + 54743 T^{4} + 405160 T^{5} + 2601288 T^{6} + 14731984 T^{7} + 73350129 T^{8} + 313225808 T^{9} + 1063218752 T^{10} + 1942762160 T^{11} - 9374200334 T^{12} - 140082082224 T^{13} - 1076056625056 T^{14} - 6480812802528 T^{15} - 33308810016754 T^{16} - 149058694458144 T^{17} - 569233954654624 T^{18} - 1704378694419408 T^{19} - 2623285595666894 T^{20} + 12504283629180880 T^{21} + 157394533153790528 T^{22} + 1066479201735536176 T^{23} + 5744120872478451249 T^{24} + 26534552190230332592 T^{25} + \)\(10\!\cdots\!12\)\( T^{26} + \)\(38\!\cdots\!20\)\( T^{27} + \)\(11\!\cdots\!03\)\( T^{28} + \)\(31\!\cdots\!16\)\( T^{29} + \)\(62\!\cdots\!24\)\( T^{30} + \)\(85\!\cdots\!24\)\( T^{31} + \)\(61\!\cdots\!61\)\( T^{32} \))
$29$ (\( ( 1 - 10 T^{2} + 841 T^{4} )^{4} \))(\( ( 1 - 44 T^{2} + 1194 T^{4} - 37004 T^{6} + 707281 T^{8} )^{2} \))(\( ( 1 - 44 T^{2} + 1194 T^{4} - 37004 T^{6} + 707281 T^{8} )^{2} \))(\( ( 1 - 10 T^{2} + 841 T^{4} )^{4} \))(\( ( 1 - 64 T^{2} + 3856 T^{4} - 135376 T^{6} + 4752862 T^{8} - 113851216 T^{10} + 2727275536 T^{12} - 38068692544 T^{14} + 500246412961 T^{16} )^{2} \))(\( ( 1 - 64 T^{2} + 3856 T^{4} - 135376 T^{6} + 4752862 T^{8} - 113851216 T^{10} + 2727275536 T^{12} - 38068692544 T^{14} + 500246412961 T^{16} )^{2} \))(\( 1 - 248 T^{2} + 31418 T^{4} - 2677416 T^{6} + 171374865 T^{8} - 8752284864 T^{10} + 370628380738 T^{12} - 13351487849248 T^{14} + 415501762604420 T^{16} - 11228601281217568 T^{18} + 262138411756753378 T^{20} - 5206063149142513344 T^{22} + 85729661487925625265 T^{24} - \)\(11\!\cdots\!16\)\( T^{26} + \)\(11\!\cdots\!38\)\( T^{28} - \)\(73\!\cdots\!88\)\( T^{30} + \)\(25\!\cdots\!21\)\( T^{32} \))(\( 1 - 248 T^{2} + 31418 T^{4} - 2677416 T^{6} + 171374865 T^{8} - 8752284864 T^{10} + 370628380738 T^{12} - 13351487849248 T^{14} + 415501762604420 T^{16} - 11228601281217568 T^{18} + 262138411756753378 T^{20} - 5206063149142513344 T^{22} + 85729661487925625265 T^{24} - \)\(11\!\cdots\!16\)\( T^{26} + \)\(11\!\cdots\!38\)\( T^{28} - \)\(73\!\cdots\!88\)\( T^{30} + \)\(25\!\cdots\!21\)\( T^{32} \))
$31$ (\( ( 1 - 12 T + 106 T^{2} - 696 T^{3} + 3891 T^{4} - 21576 T^{5} + 101866 T^{6} - 357492 T^{7} + 923521 T^{8} )^{2} \))(\( ( 1 - 24 T + 301 T^{2} - 2616 T^{3} + 16872 T^{4} - 81096 T^{5} + 289261 T^{6} - 714984 T^{7} + 923521 T^{8} )^{2} \))(\( ( 1 - 24 T + 301 T^{2} - 2616 T^{3} + 16872 T^{4} - 81096 T^{5} + 289261 T^{6} - 714984 T^{7} + 923521 T^{8} )^{2} \))(\( ( 1 - 12 T + 106 T^{2} - 696 T^{3} + 3891 T^{4} - 21576 T^{5} + 101866 T^{6} - 357492 T^{7} + 923521 T^{8} )^{2} \))(\( ( 1 + 99 T^{2} + 5549 T^{4} + 2448 T^{5} + 231102 T^{6} + 153072 T^{7} + 7750614 T^{8} + 4745232 T^{9} + 222089022 T^{10} + 72928368 T^{11} + 5124618029 T^{12} + 87862864419 T^{14} + 852891037441 T^{16} )^{2} \))(\( ( 1 + 99 T^{2} + 5549 T^{4} + 2448 T^{5} + 231102 T^{6} + 153072 T^{7} + 7750614 T^{8} + 4745232 T^{9} + 222089022 T^{10} + 72928368 T^{11} + 5124618029 T^{12} + 87862864419 T^{14} + 852891037441 T^{16} )^{2} \))(\( 1 + 24 T + 334 T^{2} + 3408 T^{3} + 27899 T^{4} + 199680 T^{5} + 1273586 T^{6} + 7589688 T^{7} + 43727785 T^{8} + 250780944 T^{9} + 1474786764 T^{10} + 8526426768 T^{11} + 48339918958 T^{12} + 259935770400 T^{13} + 1349761682520 T^{14} + 7075762017840 T^{15} + 37942723845666 T^{16} + 219348622553040 T^{17} + 1297120976901720 T^{18} + 7743746535986400 T^{19} + 44642930296011118 T^{20} + 244104359431513968 T^{21} + 1308878681740078284 T^{22} + 6899639338664300784 T^{23} + 37295035913646998185 T^{24} + \)\(20\!\cdots\!48\)\( T^{25} + \)\(10\!\cdots\!86\)\( T^{26} + \)\(50\!\cdots\!80\)\( T^{27} + \)\(21\!\cdots\!39\)\( T^{28} + \)\(83\!\cdots\!28\)\( T^{29} + \)\(25\!\cdots\!14\)\( T^{30} + \)\(56\!\cdots\!24\)\( T^{31} + \)\(72\!\cdots\!81\)\( T^{32} \))(\( 1 + 24 T + 334 T^{2} + 3408 T^{3} + 27899 T^{4} + 199680 T^{5} + 1273586 T^{6} + 7589688 T^{7} + 43727785 T^{8} + 250780944 T^{9} + 1474786764 T^{10} + 8526426768 T^{11} + 48339918958 T^{12} + 259935770400 T^{13} + 1349761682520 T^{14} + 7075762017840 T^{15} + 37942723845666 T^{16} + 219348622553040 T^{17} + 1297120976901720 T^{18} + 7743746535986400 T^{19} + 44642930296011118 T^{20} + 244104359431513968 T^{21} + 1308878681740078284 T^{22} + 6899639338664300784 T^{23} + 37295035913646998185 T^{24} + \)\(20\!\cdots\!48\)\( T^{25} + \)\(10\!\cdots\!86\)\( T^{26} + \)\(50\!\cdots\!80\)\( T^{27} + \)\(21\!\cdots\!39\)\( T^{28} + \)\(83\!\cdots\!28\)\( T^{29} + \)\(25\!\cdots\!14\)\( T^{30} + \)\(56\!\cdots\!24\)\( T^{31} + \)\(72\!\cdots\!81\)\( T^{32} \))
$37$ (\( 1 + 144 T^{2} + 11089 T^{4} + 601488 T^{6} + 25035696 T^{8} + 823437072 T^{10} + 20782571329 T^{12} + 369464602896 T^{14} + 3512479453921 T^{16} \))(\( 1 - 36 T^{2} - 566 T^{4} + 35928 T^{6} - 286749 T^{8} + 49185432 T^{10} - 1060775126 T^{12} - 92366150724 T^{14} + 3512479453921 T^{16} \))(\( 1 + 36 T^{2} - 566 T^{4} - 35928 T^{6} - 286749 T^{8} - 49185432 T^{10} - 1060775126 T^{12} + 92366150724 T^{14} + 3512479453921 T^{16} \))(\( 1 - 144 T^{2} + 11089 T^{4} - 601488 T^{6} + 25035696 T^{8} - 823437072 T^{10} + 20782571329 T^{12} - 369464602896 T^{14} + 3512479453921 T^{16} \))(\( 1 + 36 T^{2} + 4499 T^{4} + 146412 T^{6} + 10243837 T^{8} + 329458176 T^{10} + 19288003070 T^{12} + 581067044496 T^{14} + 30324317162902 T^{16} + 795480783915024 T^{18} + 36148823121674270 T^{20} + 845299542824169984 T^{22} + 35981266991815734877 T^{24} + \)\(70\!\cdots\!88\)\( T^{26} + \)\(29\!\cdots\!19\)\( T^{28} + \)\(32\!\cdots\!04\)\( T^{30} + \)\(12\!\cdots\!41\)\( T^{32} \))(\( 1 - 36 T^{2} + 4499 T^{4} - 146412 T^{6} + 10243837 T^{8} - 329458176 T^{10} + 19288003070 T^{12} - 581067044496 T^{14} + 30324317162902 T^{16} - 795480783915024 T^{18} + 36148823121674270 T^{20} - 845299542824169984 T^{22} + 35981266991815734877 T^{24} - \)\(70\!\cdots\!88\)\( T^{26} + \)\(29\!\cdots\!19\)\( T^{28} - \)\(32\!\cdots\!04\)\( T^{30} + \)\(12\!\cdots\!41\)\( T^{32} \))(\( 1 - 8 T + 248 T^{2} - 1752 T^{3} + 30199 T^{4} - 206976 T^{5} + 2540200 T^{6} - 17745904 T^{7} + 169033089 T^{8} - 1209867128 T^{9} + 9485737632 T^{10} - 68571970520 T^{11} + 465855584354 T^{12} - 3322795052152 T^{13} + 20345020109280 T^{14} - 139940970795928 T^{15} + 794780046531614 T^{16} - 5177815919449336 T^{17} + 27852332529604320 T^{18} - 168309537776655256 T^{19} + 873088367828476994 T^{20} - 4755051775144147640 T^{21} + 24337807551267523488 T^{22} - \)\(11\!\cdots\!24\)\( T^{23} + \)\(59\!\cdots\!69\)\( T^{24} - \)\(23\!\cdots\!08\)\( T^{25} + \)\(12\!\cdots\!00\)\( T^{26} - \)\(36\!\cdots\!88\)\( T^{27} + \)\(19\!\cdots\!19\)\( T^{28} - \)\(42\!\cdots\!44\)\( T^{29} + \)\(22\!\cdots\!72\)\( T^{30} - \)\(26\!\cdots\!44\)\( T^{31} + \)\(12\!\cdots\!41\)\( T^{32} \))(\( 1 - 8 T - 184 T^{2} + 1704 T^{3} + 16375 T^{4} - 186288 T^{5} - 883880 T^{6} + 13446752 T^{7} + 28690305 T^{8} - 694378184 T^{9} - 355698912 T^{10} + 26449712536 T^{11} - 17643340702 T^{12} - 717223895656 T^{13} + 1384593306720 T^{14} + 9602300166392 T^{15} - 59171912839138 T^{16} + 355285106156504 T^{17} + 1895508236899680 T^{18} - 36329541986663368 T^{19} - 33066461053401022 T^{20} + 1834127728758744952 T^{21} - 912626092170967008 T^{22} - 65918624447323666472 T^{23} + \)\(10\!\cdots\!05\)\( T^{24} + \)\(17\!\cdots\!04\)\( T^{25} - \)\(42\!\cdots\!20\)\( T^{26} - \)\(33\!\cdots\!44\)\( T^{27} + \)\(10\!\cdots\!75\)\( T^{28} + \)\(41\!\cdots\!88\)\( T^{29} - \)\(16\!\cdots\!76\)\( T^{30} - \)\(26\!\cdots\!44\)\( T^{31} + \)\(12\!\cdots\!41\)\( T^{32} \))
$41$ (\( ( 1 - 126 T^{2} + 7139 T^{4} - 211806 T^{6} + 2825761 T^{8} )^{2} \))(\( ( 1 - 138 T^{2} + 8075 T^{4} - 231978 T^{6} + 2825761 T^{8} )^{2} \))(\( ( 1 - 138 T^{2} + 8075 T^{4} - 231978 T^{6} + 2825761 T^{8} )^{2} \))(\( ( 1 - 126 T^{2} + 7139 T^{4} - 211806 T^{6} + 2825761 T^{8} )^{2} \))(\( ( 1 - 158 T^{2} + 12905 T^{4} - 739782 T^{6} + 33449684 T^{8} - 1243573542 T^{10} + 36466445705 T^{12} - 750516470078 T^{14} + 7984925229121 T^{16} )^{2} \))(\( ( 1 - 158 T^{2} + 12905 T^{4} - 739782 T^{6} + 33449684 T^{8} - 1243573542 T^{10} + 36466445705 T^{12} - 750516470078 T^{14} + 7984925229121 T^{16} )^{2} \))(\( 1 - 172 T^{2} + 13974 T^{4} - 791464 T^{6} + 43993793 T^{8} - 2592908552 T^{10} + 134337801798 T^{12} - 5711936189732 T^{14} + 227906401150852 T^{16} - 9601764734939492 T^{18} + 379606521146518278 T^{20} - 12316585909380369032 T^{22} + \)\(35\!\cdots\!53\)\( T^{24} - \)\(10\!\cdots\!64\)\( T^{26} + \)\(31\!\cdots\!94\)\( T^{28} - \)\(65\!\cdots\!92\)\( T^{30} + \)\(63\!\cdots\!41\)\( T^{32} \))(\( 1 - 172 T^{2} + 13974 T^{4} - 791464 T^{6} + 43993793 T^{8} - 2592908552 T^{10} + 134337801798 T^{12} - 5711936189732 T^{14} + 227906401150852 T^{16} - 9601764734939492 T^{18} + 379606521146518278 T^{20} - 12316585909380369032 T^{22} + \)\(35\!\cdots\!53\)\( T^{24} - \)\(10\!\cdots\!64\)\( T^{26} + \)\(31\!\cdots\!94\)\( T^{28} - \)\(65\!\cdots\!92\)\( T^{30} + \)\(63\!\cdots\!41\)\( T^{32} \))
$43$ (\( 1 - 3548 T^{4} + 6433446 T^{8} - 12129905948 T^{12} + 11688200277601 T^{16} \))(\( 1 + 316 T^{4} - 5622234 T^{8} + 1080341116 T^{12} + 11688200277601 T^{16} \))(\( 1 + 316 T^{4} - 5622234 T^{8} + 1080341116 T^{12} + 11688200277601 T^{16} \))(\( 1 - 3548 T^{4} + 6433446 T^{8} - 12129905948 T^{12} + 11688200277601 T^{16} \))(\( 1 - 1504 T^{4} - 258116 T^{8} - 4620736288 T^{12} + 14886307959622 T^{16} - 15797377842150688 T^{20} - 3016911502853259716 T^{24} - \)\(60\!\cdots\!04\)\( T^{28} + \)\(13\!\cdots\!01\)\( T^{32} \))(\( 1 - 1504 T^{4} - 258116 T^{8} - 4620736288 T^{12} + 14886307959622 T^{16} - 15797377842150688 T^{20} - 3016911502853259716 T^{24} - \)\(60\!\cdots\!04\)\( T^{28} + \)\(13\!\cdots\!01\)\( T^{32} \))(\( 1 - 24 T + 288 T^{2} - 2456 T^{3} + 18144 T^{4} - 133752 T^{5} + 1000544 T^{6} - 6828856 T^{7} + 39963644 T^{8} - 227672888 T^{9} + 1517218592 T^{10} - 11117129912 T^{11} + 81274307360 T^{12} - 589957036568 T^{13} + 4310182316384 T^{14} - 31623166948824 T^{15} + 218467749390534 T^{16} - 1359796178799432 T^{17} + 7969527102994016 T^{18} - 46905714106411976 T^{19} + 277860683276675360 T^{20} - 1634311958991847016 T^{21} + 9590889544724607008 T^{22} - 61885728202879567016 T^{23} + \)\(46\!\cdots\!44\)\( T^{24} - \)\(34\!\cdots\!08\)\( T^{25} + \)\(21\!\cdots\!56\)\( T^{26} - \)\(12\!\cdots\!64\)\( T^{27} + \)\(72\!\cdots\!44\)\( T^{28} - \)\(42\!\cdots\!08\)\( T^{29} + \)\(21\!\cdots\!12\)\( T^{30} - \)\(76\!\cdots\!68\)\( T^{31} + \)\(13\!\cdots\!01\)\( T^{32} \))(\( 1 - 24 T + 288 T^{2} - 2456 T^{3} + 18144 T^{4} - 133752 T^{5} + 1000544 T^{6} - 6828856 T^{7} + 39963644 T^{8} - 227672888 T^{9} + 1517218592 T^{10} - 11117129912 T^{11} + 81274307360 T^{12} - 589957036568 T^{13} + 4310182316384 T^{14} - 31623166948824 T^{15} + 218467749390534 T^{16} - 1359796178799432 T^{17} + 7969527102994016 T^{18} - 46905714106411976 T^{19} + 277860683276675360 T^{20} - 1634311958991847016 T^{21} + 9590889544724607008 T^{22} - 61885728202879567016 T^{23} + \)\(46\!\cdots\!44\)\( T^{24} - \)\(34\!\cdots\!08\)\( T^{25} + \)\(21\!\cdots\!56\)\( T^{26} - \)\(12\!\cdots\!64\)\( T^{27} + \)\(72\!\cdots\!44\)\( T^{28} - \)\(42\!\cdots\!08\)\( T^{29} + \)\(21\!\cdots\!12\)\( T^{30} - \)\(76\!\cdots\!68\)\( T^{31} + \)\(13\!\cdots\!01\)\( T^{32} \))
$47$ (\( 1 - 144 T^{2} + 10457 T^{4} - 510480 T^{6} + 22955952 T^{8} - 1127650320 T^{10} + 51026824217 T^{12} - 1552207007376 T^{14} + 23811286661761 T^{16} \))(\( 1 + 3689 T^{4} + 8729040 T^{8} + 18001143209 T^{12} + 23811286661761 T^{16} \))(\( 1 + 3689 T^{4} + 8729040 T^{8} + 18001143209 T^{12} + 23811286661761 T^{16} \))(\( 1 + 144 T^{2} + 10457 T^{4} + 510480 T^{6} + 22955952 T^{8} + 1127650320 T^{10} + 51026824217 T^{12} + 1552207007376 T^{14} + 23811286661761 T^{16} \))(\( 1 + 72 T^{2} - 2097 T^{4} - 275400 T^{6} - 1208767 T^{8} + 434448 T^{10} - 23245686270 T^{12} + 566620331904 T^{14} + 118374567124638 T^{16} + 1251664313175936 T^{18} - 113431533623679870 T^{20} + 4683008541253392 T^{22} - 28782297544276858687 T^{24} - \)\(14\!\cdots\!00\)\( T^{26} - \)\(24\!\cdots\!77\)\( T^{28} + \)\(18\!\cdots\!68\)\( T^{30} + \)\(56\!\cdots\!21\)\( T^{32} \))(\( 1 - 72 T^{2} - 2097 T^{4} + 275400 T^{6} - 1208767 T^{8} - 434448 T^{10} - 23245686270 T^{12} - 566620331904 T^{14} + 118374567124638 T^{16} - 1251664313175936 T^{18} - 113431533623679870 T^{20} - 4683008541253392 T^{22} - 28782297544276858687 T^{24} + \)\(14\!\cdots\!00\)\( T^{26} - \)\(24\!\cdots\!77\)\( T^{28} - \)\(18\!\cdots\!68\)\( T^{30} + \)\(56\!\cdots\!21\)\( T^{32} \))(\( 1 + 72 T^{2} - 616 T^{3} + 3239 T^{4} - 50872 T^{5} + 298520 T^{6} - 2523744 T^{7} + 15099153 T^{8} - 113209408 T^{9} + 814540096 T^{10} - 3210267840 T^{11} + 24971489874 T^{12} - 159883078560 T^{13} + 379015320032 T^{14} - 3041257505584 T^{15} + 39450787699278 T^{16} - 142939102762448 T^{17} + 837244841950688 T^{18} - 16599540865334880 T^{19} + 121852904679850194 T^{20} - 736258900236674880 T^{21} + 8780103088888331584 T^{22} - 57354503546728915904 T^{23} + \)\(35\!\cdots\!33\)\( T^{24} - \)\(28\!\cdots\!48\)\( T^{25} + \)\(15\!\cdots\!80\)\( T^{26} - \)\(12\!\cdots\!16\)\( T^{27} + \)\(37\!\cdots\!99\)\( T^{28} - \)\(33\!\cdots\!32\)\( T^{29} + \)\(18\!\cdots\!68\)\( T^{30} + \)\(56\!\cdots\!21\)\( T^{32} \))(\( 1 - 24 T + 216 T^{2} + 40 T^{3} - 18457 T^{4} + 150040 T^{5} + 44552 T^{6} - 9386184 T^{7} + 61817361 T^{8} + 96146848 T^{9} - 3977203520 T^{10} + 21778214496 T^{11} + 50868959058 T^{12} - 1343760123600 T^{13} + 6047140835168 T^{14} + 23428375060624 T^{15} - 394415861681970 T^{16} + 1101133627849328 T^{17} + 13358134104886112 T^{18} - 139513207312522800 T^{19} + 248224293005100498 T^{20} + 4994724756032621472 T^{21} - 42871133149336758080 T^{22} + 48710216156441750624 T^{23} + \)\(14\!\cdots\!21\)\( T^{24} - \)\(10\!\cdots\!28\)\( T^{25} + \)\(23\!\cdots\!48\)\( T^{26} + \)\(37\!\cdots\!20\)\( T^{27} - \)\(21\!\cdots\!37\)\( T^{28} + \)\(21\!\cdots\!80\)\( T^{29} + \)\(55\!\cdots\!04\)\( T^{30} - \)\(28\!\cdots\!32\)\( T^{31} + \)\(56\!\cdots\!21\)\( T^{32} \))
$53$ (\( 1 - 5407 T^{4} + 21345168 T^{8} - 42663830767 T^{12} + 62259690411361 T^{16} \))(\( 1 + 84 T^{2} + 5642 T^{4} + 276360 T^{6} + 9540387 T^{8} + 776295240 T^{10} + 44518093802 T^{12} + 1861806334836 T^{14} + 62259690411361 T^{16} \))(\( 1 - 84 T^{2} + 5642 T^{4} - 276360 T^{6} + 9540387 T^{8} - 776295240 T^{10} + 44518093802 T^{12} - 1861806334836 T^{14} + 62259690411361 T^{16} \))(\( 1 - 5407 T^{4} + 21345168 T^{8} - 42663830767 T^{12} + 62259690411361 T^{16} \))(\( 1 - 60 T^{2} + 2040 T^{4} - 50400 T^{6} + 8790530 T^{8} + 35607396 T^{10} - 20650486848 T^{12} + 1867209163140 T^{14} - 61066879873293 T^{16} + 5244990539260260 T^{18} - 162942274114893888 T^{20} + 789215183807310084 T^{22} + \)\(54\!\cdots\!30\)\( T^{24} - \)\(88\!\cdots\!00\)\( T^{26} + \)\(10\!\cdots\!40\)\( T^{28} - \)\(82\!\cdots\!40\)\( T^{30} + \)\(38\!\cdots\!21\)\( T^{32} \))(\( 1 + 60 T^{2} + 2040 T^{4} + 50400 T^{6} + 8790530 T^{8} - 35607396 T^{10} - 20650486848 T^{12} - 1867209163140 T^{14} - 61066879873293 T^{16} - 5244990539260260 T^{18} - 162942274114893888 T^{20} - 789215183807310084 T^{22} + \)\(54\!\cdots\!30\)\( T^{24} + \)\(88\!\cdots\!00\)\( T^{26} + \)\(10\!\cdots\!40\)\( T^{28} + \)\(82\!\cdots\!40\)\( T^{30} + \)\(38\!\cdots\!21\)\( T^{32} \))(\( 1 - 28 T + 344 T^{2} - 2912 T^{3} + 25298 T^{4} - 232644 T^{5} + 1917568 T^{6} - 14158268 T^{7} + 104375121 T^{8} - 770687892 T^{9} + 5270701040 T^{10} - 39047642852 T^{11} + 318443399442 T^{12} - 2209044273928 T^{13} + 14320519756920 T^{14} - 111315852463604 T^{15} + 873920983850628 T^{16} - 5899740180571012 T^{17} + 40226339997188280 T^{18} - 328875884369578856 T^{19} + 2512671592872511602 T^{20} - 16329548252980066036 T^{21} + \)\(11\!\cdots\!60\)\( T^{22} - \)\(90\!\cdots\!04\)\( T^{23} + \)\(64\!\cdots\!81\)\( T^{24} - \)\(46\!\cdots\!44\)\( T^{25} + \)\(33\!\cdots\!32\)\( T^{26} - \)\(21\!\cdots\!68\)\( T^{27} + \)\(12\!\cdots\!18\)\( T^{28} - \)\(75\!\cdots\!76\)\( T^{29} + \)\(47\!\cdots\!36\)\( T^{30} - \)\(20\!\cdots\!96\)\( T^{31} + \)\(38\!\cdots\!21\)\( T^{32} \))(\( 1 + 44 T + 1016 T^{2} + 15712 T^{3} + 178130 T^{4} + 1540980 T^{5} + 10390336 T^{6} + 58117132 T^{7} + 357423825 T^{8} + 3614994372 T^{9} + 45716027504 T^{10} + 505985826004 T^{11} + 4503741642258 T^{12} + 32039845594664 T^{13} + 184020453561240 T^{14} + 916952178658276 T^{15} + 5330817007952772 T^{16} + 48598465468888628 T^{17} + 516913454053523160 T^{18} + 4769996092596792328 T^{19} + 35536687857145546098 T^{20} + \)\(21\!\cdots\!72\)\( T^{21} + \)\(10\!\cdots\!16\)\( T^{22} + \)\(42\!\cdots\!64\)\( T^{23} + \)\(22\!\cdots\!25\)\( T^{24} + \)\(19\!\cdots\!56\)\( T^{25} + \)\(18\!\cdots\!64\)\( T^{26} + \)\(14\!\cdots\!60\)\( T^{27} + \)\(87\!\cdots\!30\)\( T^{28} + \)\(40\!\cdots\!76\)\( T^{29} + \)\(14\!\cdots\!04\)\( T^{30} + \)\(32\!\cdots\!08\)\( T^{31} + \)\(38\!\cdots\!21\)\( T^{32} \))
$59$ (\( ( 1 + 16 T + 86 T^{2} + 832 T^{3} + 10315 T^{4} + 49088 T^{5} + 299366 T^{6} + 3286064 T^{7} + 12117361 T^{8} )^{2} \))(\( ( 1 - 8 T - 58 T^{2} - 32 T^{3} + 7627 T^{4} - 1888 T^{5} - 201898 T^{6} - 1643032 T^{7} + 12117361 T^{8} )^{2} \))(\( ( 1 + 8 T - 58 T^{2} + 32 T^{3} + 7627 T^{4} + 1888 T^{5} - 201898 T^{6} + 1643032 T^{7} + 12117361 T^{8} )^{2} \))(\( ( 1 - 16 T + 86 T^{2} - 832 T^{3} + 10315 T^{4} - 49088 T^{5} + 299366 T^{6} - 3286064 T^{7} + 12117361 T^{8} )^{2} \))(\( ( 1 + 20 T + 124 T^{2} - 312 T^{3} - 7894 T^{4} - 53140 T^{5} - 123312 T^{6} + 3441412 T^{7} + 48399379 T^{8} + 203043308 T^{9} - 429249072 T^{10} - 10913840060 T^{11} - 95654447734 T^{12} - 223056381288 T^{13} + 5230386171484 T^{14} + 49773029696380 T^{15} + 146830437604321 T^{16} )^{2} \))(\( ( 1 - 20 T + 124 T^{2} + 312 T^{3} - 7894 T^{4} + 53140 T^{5} - 123312 T^{6} - 3441412 T^{7} + 48399379 T^{8} - 203043308 T^{9} - 429249072 T^{10} + 10913840060 T^{11} - 95654447734 T^{12} + 223056381288 T^{13} + 5230386171484 T^{14} - 49773029696380 T^{15} + 146830437604321 T^{16} )^{2} \))(\( 1 + 8 T - 236 T^{2} - 1632 T^{3} + 30563 T^{4} + 163648 T^{5} - 2682716 T^{6} - 9836488 T^{7} + 178469613 T^{8} + 332320192 T^{9} - 10946443072 T^{10} - 3817987168 T^{11} + 756189977838 T^{12} - 175777223888 T^{13} - 56160273686696 T^{14} + 6297587928448 T^{15} + 3675723794639094 T^{16} + 371557687778432 T^{17} - 195493912703388776 T^{18} - 36100950464893552 T^{19} + 9163026946045045518 T^{20} - 2729571799673395232 T^{21} - \)\(46\!\cdots\!52\)\( T^{22} + \)\(82\!\cdots\!48\)\( T^{23} + \)\(26\!\cdots\!73\)\( T^{24} - \)\(85\!\cdots\!32\)\( T^{25} - \)\(13\!\cdots\!16\)\( T^{26} + \)\(49\!\cdots\!32\)\( T^{27} + \)\(54\!\cdots\!03\)\( T^{28} - \)\(17\!\cdots\!28\)\( T^{29} - \)\(14\!\cdots\!96\)\( T^{30} + \)\(29\!\cdots\!92\)\( T^{31} + \)\(21\!\cdots\!41\)\( T^{32} \))(\( 1 - 8 T - 236 T^{2} + 1632 T^{3} + 30563 T^{4} - 163648 T^{5} - 2682716 T^{6} + 9836488 T^{7} + 178469613 T^{8} - 332320192 T^{9} - 10946443072 T^{10} + 3817987168 T^{11} + 756189977838 T^{12} + 175777223888 T^{13} - 56160273686696 T^{14} - 6297587928448 T^{15} + 3675723794639094 T^{16} - 371557687778432 T^{17} - 195493912703388776 T^{18} + 36100950464893552 T^{19} + 9163026946045045518 T^{20} + 2729571799673395232 T^{21} - \)\(46\!\cdots\!52\)\( T^{22} - \)\(82\!\cdots\!48\)\( T^{23} + \)\(26\!\cdots\!73\)\( T^{24} + \)\(85\!\cdots\!32\)\( T^{25} - \)\(13\!\cdots\!16\)\( T^{26} - \)\(49\!\cdots\!32\)\( T^{27} + \)\(54\!\cdots\!03\)\( T^{28} + \)\(17\!\cdots\!28\)\( T^{29} - \)\(14\!\cdots\!96\)\( T^{30} - \)\(29\!\cdots\!92\)\( T^{31} + \)\(21\!\cdots\!41\)\( T^{32} \))
$61$ (\( ( 1 + 12 T + 118 T^{2} + 840 T^{3} + 4107 T^{4} + 51240 T^{5} + 439078 T^{6} + 2723772 T^{7} + 13845841 T^{8} )^{2} \))(\( ( 1 - 18 T + 208 T^{2} - 1800 T^{3} + 12867 T^{4} - 109800 T^{5} + 773968 T^{6} - 4085658 T^{7} + 13845841 T^{8} )^{2} \))(\( ( 1 - 18 T + 208 T^{2} - 1800 T^{3} + 12867 T^{4} - 109800 T^{5} + 773968 T^{6} - 4085658 T^{7} + 13845841 T^{8} )^{2} \))(\( ( 1 + 12 T + 118 T^{2} + 840 T^{3} + 4107 T^{4} + 51240 T^{5} + 439078 T^{6} + 2723772 T^{7} + 13845841 T^{8} )^{2} \))(\( ( 1 + 12 T + 225 T^{2} + 2124 T^{3} + 22907 T^{4} + 153816 T^{5} + 1374816 T^{6} + 7573500 T^{7} + 71943462 T^{8} + 461983500 T^{9} + 5115690336 T^{10} + 34913309496 T^{11} + 317166679787 T^{12} + 1793922543324 T^{13} + 11592084231225 T^{14} + 37712914032252 T^{15} + 191707312997281 T^{16} )^{2} \))(\( ( 1 + 12 T + 225 T^{2} + 2124 T^{3} + 22907 T^{4} + 153816 T^{5} + 1374816 T^{6} + 7573500 T^{7} + 71943462 T^{8} + 461983500 T^{9} + 5115690336 T^{10} + 34913309496 T^{11} + 317166679787 T^{12} + 1793922543324 T^{13} + 11592084231225 T^{14} + 37712914032252 T^{15} + 191707312997281 T^{16} )^{2} \))(\( 1 - 24 T + 468 T^{2} - 6624 T^{3} + 79480 T^{4} - 829512 T^{5} + 7576872 T^{6} - 61686168 T^{7} + 432435906 T^{8} - 2460066384 T^{9} + 8493304812 T^{10} + 36652670904 T^{11} - 1139116635712 T^{12} + 15272951044440 T^{13} - 160222534806948 T^{14} + 1465410452960880 T^{15} - 12014486118540493 T^{16} + 89390037630613680 T^{17} - 596188052016653508 T^{18} + 3466669701018035640 T^{19} - 15772027818523273792 T^{20} + 30956710267288726104 T^{21} + \)\(43\!\cdots\!32\)\( T^{22} - \)\(77\!\cdots\!64\)\( T^{23} + \)\(82\!\cdots\!86\)\( T^{24} - \)\(72\!\cdots\!88\)\( T^{25} + \)\(54\!\cdots\!72\)\( T^{26} - \)\(36\!\cdots\!32\)\( T^{27} + \)\(21\!\cdots\!80\)\( T^{28} - \)\(10\!\cdots\!44\)\( T^{29} + \)\(46\!\cdots\!88\)\( T^{30} - \)\(14\!\cdots\!24\)\( T^{31} + \)\(36\!\cdots\!61\)\( T^{32} \))(\( 1 - 24 T + 468 T^{2} - 6624 T^{3} + 79480 T^{4} - 829512 T^{5} + 7576872 T^{6} - 61686168 T^{7} + 432435906 T^{8} - 2460066384 T^{9} + 8493304812 T^{10} + 36652670904 T^{11} - 1139116635712 T^{12} + 15272951044440 T^{13} - 160222534806948 T^{14} + 1465410452960880 T^{15} - 12014486118540493 T^{16} + 89390037630613680 T^{17} - 596188052016653508 T^{18} + 3466669701018035640 T^{19} - 15772027818523273792 T^{20} + 30956710267288726104 T^{21} + \)\(43\!\cdots\!32\)\( T^{22} - \)\(77\!\cdots\!64\)\( T^{23} + \)\(82\!\cdots\!86\)\( T^{24} - \)\(72\!\cdots\!88\)\( T^{25} + \)\(54\!\cdots\!72\)\( T^{26} - \)\(36\!\cdots\!32\)\( T^{27} + \)\(21\!\cdots\!80\)\( T^{28} - \)\(10\!\cdots\!44\)\( T^{29} + \)\(46\!\cdots\!88\)\( T^{30} - \)\(14\!\cdots\!24\)\( T^{31} + \)\(36\!\cdots\!61\)\( T^{32} \))
$67$ (\( 1 - 7922 T^{4} + 42606963 T^{8} - 159637180562 T^{12} + 406067677556641 T^{16} \))(\( 1 - 4946 T^{4} + 4311795 T^{8} - 99667444466 T^{12} + 406067677556641 T^{16} \))(\( 1 - 4946 T^{4} + 4311795 T^{8} - 99667444466 T^{12} + 406067677556641 T^{16} \))(\( 1 - 7922 T^{4} + 42606963 T^{8} - 159637180562 T^{12} + 406067677556641 T^{16} \))(\( 1 + 144 T^{2} + 15479 T^{4} + 1233648 T^{6} + 82019857 T^{8} + 5129471520 T^{10} + 354430386290 T^{12} + 27207762471936 T^{14} + 1909973041583038 T^{16} + 122135645736520704 T^{18} + 7142169600206531090 T^{20} + \)\(46\!\cdots\!80\)\( T^{22} + \)\(33\!\cdots\!37\)\( T^{24} + \)\(22\!\cdots\!52\)\( T^{26} + \)\(12\!\cdots\!19\)\( T^{28} + \)\(52\!\cdots\!76\)\( T^{30} + \)\(16\!\cdots\!81\)\( T^{32} \))(\( 1 - 144 T^{2} + 15479 T^{4} - 1233648 T^{6} + 82019857 T^{8} - 5129471520 T^{10} + 354430386290 T^{12} - 27207762471936 T^{14} + 1909973041583038 T^{16} - 122135645736520704 T^{18} + 7142169600206531090 T^{20} - \)\(46\!\cdots\!80\)\( T^{22} + \)\(33\!\cdots\!37\)\( T^{24} - \)\(22\!\cdots\!52\)\( T^{26} + \)\(12\!\cdots\!19\)\( T^{28} - \)\(52\!\cdots\!76\)\( T^{30} + \)\(16\!\cdots\!81\)\( T^{32} \))(\( 1 - 84 T + 3216 T^{2} - 71776 T^{3} + 957024 T^{4} - 5741436 T^{5} - 40997200 T^{6} + 1158667636 T^{7} - 7828301806 T^{8} - 48210145144 T^{9} + 1314996417536 T^{10} - 8000602781236 T^{11} - 45952509042976 T^{12} + 1085658642853028 T^{13} - 5561139754106368 T^{14} - 34677137718483048 T^{15} + 641528682100762995 T^{16} - 2323368227138364216 T^{17} - 24963956356183485952 T^{18} + \)\(32\!\cdots\!64\)\( T^{19} - \)\(92\!\cdots\!96\)\( T^{20} - \)\(10\!\cdots\!52\)\( T^{21} + \)\(11\!\cdots\!84\)\( T^{22} - \)\(29\!\cdots\!12\)\( T^{23} - \)\(31\!\cdots\!46\)\( T^{24} + \)\(31\!\cdots\!92\)\( T^{25} - \)\(74\!\cdots\!00\)\( T^{26} - \)\(70\!\cdots\!88\)\( T^{27} + \)\(78\!\cdots\!64\)\( T^{28} - \)\(39\!\cdots\!12\)\( T^{29} + \)\(11\!\cdots\!64\)\( T^{30} - \)\(20\!\cdots\!12\)\( T^{31} + \)\(16\!\cdots\!81\)\( T^{32} \))(\( 1 + 36 T + 960 T^{2} + 19808 T^{3} + 351456 T^{4} + 5525148 T^{5} + 79071728 T^{6} + 1045655452 T^{7} + 12919151378 T^{8} + 150131218232 T^{9} + 1651372846640 T^{10} + 17248158507668 T^{11} + 171756254361824 T^{12} + 1633413949285436 T^{13} + 14868335084466800 T^{14} + 129646495999196328 T^{15} + 1083758100261010803 T^{16} + 8686315231946153976 T^{17} + 66743956194171465200 T^{18} + \)\(49\!\cdots\!68\)\( T^{19} + \)\(34\!\cdots\!04\)\( T^{20} + \)\(23\!\cdots\!76\)\( T^{21} + \)\(14\!\cdots\!60\)\( T^{22} + \)\(90\!\cdots\!36\)\( T^{23} + \)\(52\!\cdots\!98\)\( T^{24} + \)\(28\!\cdots\!44\)\( T^{25} + \)\(14\!\cdots\!72\)\( T^{26} + \)\(67\!\cdots\!84\)\( T^{27} + \)\(28\!\cdots\!16\)\( T^{28} + \)\(10\!\cdots\!96\)\( T^{29} + \)\(35\!\cdots\!40\)\( T^{30} + \)\(88\!\cdots\!48\)\( T^{31} + \)\(16\!\cdots\!81\)\( T^{32} \))
$71$ (\( ( 1 + 4 T + 98 T^{2} + 284 T^{3} + 5041 T^{4} )^{4} \))(\( ( 1 - 2 T - 49 T^{2} - 142 T^{3} + 5041 T^{4} )^{4} \))(\( ( 1 - 2 T - 49 T^{2} - 142 T^{3} + 5041 T^{4} )^{4} \))(\( ( 1 + 4 T + 98 T^{2} + 284 T^{3} + 5041 T^{4} )^{4} \))(\( ( 1 - 26 T + 513 T^{2} - 6318 T^{3} + 63556 T^{4} - 448578 T^{5} + 2586033 T^{6} - 9305686 T^{7} + 25411681 T^{8} )^{4} \))(\( ( 1 - 26 T + 513 T^{2} - 6318 T^{3} + 63556 T^{4} - 448578 T^{5} + 2586033 T^{6} - 9305686 T^{7} + 25411681 T^{8} )^{4} \))(\( ( 1 + 16 T + 392 T^{2} + 4432 T^{3} + 72412 T^{4} + 697872 T^{5} + 8835896 T^{6} + 71051216 T^{7} + 737387398 T^{8} + 5044636336 T^{9} + 44541751736 T^{10} + 249776065392 T^{11} + 1840110644572 T^{12} + 7996344483632 T^{13} + 50215311297032 T^{14} + 145521922534256 T^{15} + 645753531245761 T^{16} )^{2} \))(\( ( 1 + 16 T + 392 T^{2} + 4432 T^{3} + 72412 T^{4} + 697872 T^{5} + 8835896 T^{6} + 71051216 T^{7} + 737387398 T^{8} + 5044636336 T^{9} + 44541751736 T^{10} + 249776065392 T^{11} + 1840110644572 T^{12} + 7996344483632 T^{13} + 50215311297032 T^{14} + 145521922534256 T^{15} + 645753531245761 T^{16} )^{2} \))
$73$ (\( 1 - 6242 T^{4} + 10564323 T^{8} - 177261820322 T^{12} + 806460091894081 T^{16} \))(\( 1 + 3934 T^{4} - 12921885 T^{8} + 111718680094 T^{12} + 806460091894081 T^{16} \))(\( 1 + 3934 T^{4} - 12921885 T^{8} + 111718680094 T^{12} + 806460091894081 T^{16} \))(\( 1 - 6242 T^{4} + 10564323 T^{8} - 177261820322 T^{12} + 806460091894081 T^{16} \))(\( 1 - 576 T^{2} + 168527 T^{4} - 33370560 T^{6} + 5044720033 T^{8} - 621838271232 T^{10} + 64869439165922 T^{12} - 5836679136073728 T^{14} + 456532218503037886 T^{16} - 31103663116136896512 T^{18} + \)\(18\!\cdots\!02\)\( T^{20} - \)\(94\!\cdots\!48\)\( T^{22} + \)\(40\!\cdots\!73\)\( T^{24} - \)\(14\!\cdots\!40\)\( T^{26} + \)\(38\!\cdots\!67\)\( T^{28} - \)\(70\!\cdots\!84\)\( T^{30} + \)\(65\!\cdots\!61\)\( T^{32} \))(\( 1 + 576 T^{2} + 168527 T^{4} + 33370560 T^{6} + 5044720033 T^{8} + 621838271232 T^{10} + 64869439165922 T^{12} + 5836679136073728 T^{14} + 456532218503037886 T^{16} + 31103663116136896512 T^{18} + \)\(18\!\cdots\!02\)\( T^{20} + \)\(94\!\cdots\!48\)\( T^{22} + \)\(40\!\cdots\!73\)\( T^{24} + \)\(14\!\cdots\!40\)\( T^{26} + \)\(38\!\cdots\!67\)\( T^{28} + \)\(70\!\cdots\!84\)\( T^{30} + \)\(65\!\cdots\!61\)\( T^{32} \))(\( 1 + 16 T + 416 T^{2} + 4440 T^{3} + 68404 T^{4} + 584824 T^{5} + 7229216 T^{6} + 53644504 T^{7} + 544685226 T^{8} + 2736110216 T^{9} + 18336160640 T^{10} - 72833715600 T^{11} - 1347659375280 T^{12} - 28574375923760 T^{13} - 284065502536896 T^{14} - 3320804755103832 T^{15} - 26931854637922413 T^{16} - 242418747122579736 T^{17} - 1513785063019118784 T^{18} - 11115917998733343920 T^{19} - 38271155725110882480 T^{20} - \)\(15\!\cdots\!00\)\( T^{21} + \)\(27\!\cdots\!60\)\( T^{22} + \)\(30\!\cdots\!52\)\( T^{23} + \)\(43\!\cdots\!06\)\( T^{24} + \)\(31\!\cdots\!52\)\( T^{25} + \)\(31\!\cdots\!84\)\( T^{26} + \)\(18\!\cdots\!48\)\( T^{27} + \)\(15\!\cdots\!84\)\( T^{28} + \)\(74\!\cdots\!20\)\( T^{29} + \)\(50\!\cdots\!44\)\( T^{30} + \)\(14\!\cdots\!12\)\( T^{31} + \)\(65\!\cdots\!61\)\( T^{32} \))(\( 1 - 40 T + 512 T^{2} + 1704 T^{3} - 115340 T^{4} + 995264 T^{5} + 4224032 T^{6} - 134702512 T^{7} + 706330218 T^{8} + 4851409768 T^{9} - 89219066272 T^{10} + 434676875784 T^{11} + 2356262737872 T^{12} - 56320191577336 T^{13} + 325034384414112 T^{14} + 1853571681237960 T^{15} - 38795165571506349 T^{16} + 135310732730371080 T^{17} + 1732108234542802848 T^{18} - 21909511966840518712 T^{19} + 66913717089408883152 T^{20} + \)\(90\!\cdots\!12\)\( T^{21} - \)\(13\!\cdots\!08\)\( T^{22} + \)\(53\!\cdots\!96\)\( T^{23} + \)\(56\!\cdots\!58\)\( T^{24} - \)\(79\!\cdots\!56\)\( T^{25} + \)\(18\!\cdots\!68\)\( T^{26} + \)\(31\!\cdots\!28\)\( T^{27} - \)\(26\!\cdots\!40\)\( T^{28} + \)\(28\!\cdots\!32\)\( T^{29} + \)\(62\!\cdots\!08\)\( T^{30} - \)\(35\!\cdots\!80\)\( T^{31} + \)\(65\!\cdots\!61\)\( T^{32} \))
$79$ (\( ( 1 + 24 T + 298 T^{2} + 2544 T^{3} + 20163 T^{4} + 200976 T^{5} + 1859818 T^{6} + 11832936 T^{7} + 38950081 T^{8} )^{2} \))(\( ( 1 + 42 T + 877 T^{2} + 12138 T^{3} + 123732 T^{4} + 958902 T^{5} + 5473357 T^{6} + 20707638 T^{7} + 38950081 T^{8} )^{2} \))(\( ( 1 - 42 T + 877 T^{2} - 12138 T^{3} + 123732 T^{4} - 958902 T^{5} + 5473357 T^{6} - 20707638 T^{7} + 38950081 T^{8} )^{2} \))(\( ( 1 - 24 T + 298 T^{2} - 2544 T^{3} + 20163 T^{4} - 200976 T^{5} + 1859818 T^{6} - 11832936 T^{7} + 38950081 T^{8} )^{2} \))(\( ( 1 - 6 T + 194 T^{2} - 1092 T^{3} + 17173 T^{4} - 108768 T^{5} + 1220606 T^{6} - 9301122 T^{7} + 98127316 T^{8} - 734788638 T^{9} + 7617802046 T^{10} - 53626865952 T^{11} + 668889741013 T^{12} - 3360145587708 T^{13} + 47158966371074 T^{14} - 115223453916954 T^{15} + 1517108809906561 T^{16} )^{2} \))(\( ( 1 + 6 T + 194 T^{2} + 1092 T^{3} + 17173 T^{4} + 108768 T^{5} + 1220606 T^{6} + 9301122 T^{7} + 98127316 T^{8} + 734788638 T^{9} + 7617802046 T^{10} + 53626865952 T^{11} + 668889741013 T^{12} + 3360145587708 T^{13} + 47158966371074 T^{14} + 115223453916954 T^{15} + 1517108809906561 T^{16} )^{2} \))(\( 1 + 12 T + 518 T^{2} + 5640 T^{3} + 137639 T^{4} + 1484376 T^{5} + 26147402 T^{6} + 279843540 T^{7} + 3950719429 T^{8} + 41115674928 T^{9} + 499791767516 T^{10} + 4982987471808 T^{11} + 54520483633690 T^{12} + 515559074594184 T^{13} + 5198103978036912 T^{14} + 46390749479828928 T^{15} + 436708099187124858 T^{16} + 3664869208906485312 T^{17} + 32441366926928367792 T^{18} + \)\(25\!\cdots\!76\)\( T^{19} + \)\(21\!\cdots\!90\)\( T^{20} + \)\(15\!\cdots\!92\)\( T^{21} + \)\(12\!\cdots\!36\)\( T^{22} + \)\(78\!\cdots\!52\)\( T^{23} + \)\(59\!\cdots\!69\)\( T^{24} + \)\(33\!\cdots\!60\)\( T^{25} + \)\(24\!\cdots\!02\)\( T^{26} + \)\(11\!\cdots\!04\)\( T^{27} + \)\(81\!\cdots\!99\)\( T^{28} + \)\(26\!\cdots\!60\)\( T^{29} + \)\(19\!\cdots\!58\)\( T^{30} + \)\(34\!\cdots\!88\)\( T^{31} + \)\(23\!\cdots\!21\)\( T^{32} \))(\( 1 - 12 T + 518 T^{2} - 5640 T^{3} + 137639 T^{4} - 1484376 T^{5} + 26147402 T^{6} - 279843540 T^{7} + 3950719429 T^{8} - 41115674928 T^{9} + 499791767516 T^{10} - 4982987471808 T^{11} + 54520483633690 T^{12} - 515559074594184 T^{13} + 5198103978036912 T^{14} - 46390749479828928 T^{15} + 436708099187124858 T^{16} - 3664869208906485312 T^{17} + 32441366926928367792 T^{18} - \)\(25\!\cdots\!76\)\( T^{19} + \)\(21\!\cdots\!90\)\( T^{20} - \)\(15\!\cdots\!92\)\( T^{21} + \)\(12\!\cdots\!36\)\( T^{22} - \)\(78\!\cdots\!52\)\( T^{23} + \)\(59\!\cdots\!69\)\( T^{24} - \)\(33\!\cdots\!60\)\( T^{25} + \)\(24\!\cdots\!02\)\( T^{26} - \)\(11\!\cdots\!04\)\( T^{27} + \)\(81\!\cdots\!99\)\( T^{28} - \)\(26\!\cdots\!60\)\( T^{29} + \)\(19\!\cdots\!58\)\( T^{30} - \)\(34\!\cdots\!88\)\( T^{31} + \)\(23\!\cdots\!21\)\( T^{32} \))
$83$ (\( ( 1 - 10414 T^{4} + 47458321 T^{8} )^{2} \))(\( 1 - 8708 T^{4} + 41254758 T^{8} - 413267059268 T^{12} + 2252292232139041 T^{16} \))(\( 1 - 8708 T^{4} + 41254758 T^{8} - 413267059268 T^{12} + 2252292232139041 T^{16} \))(\( ( 1 - 10414 T^{4} + 47458321 T^{8} )^{2} \))(\( 1 + 2720 T^{4} - 86504132 T^{8} - 3906190240 T^{12} + 5664126472953286 T^{16} - 185381230296987040 T^{20} - \)\(19\!\cdots\!12\)\( T^{24} + \)\(29\!\cdots\!20\)\( T^{28} + \)\(50\!\cdots\!81\)\( T^{32} \))(\( 1 + 2720 T^{4} - 86504132 T^{8} - 3906190240 T^{12} + 5664126472953286 T^{16} - 185381230296987040 T^{20} - \)\(19\!\cdots\!12\)\( T^{24} + \)\(29\!\cdots\!20\)\( T^{28} + \)\(50\!\cdots\!81\)\( T^{32} \))(\( 1 + 16 T + 128 T^{2} + 48 T^{3} - 16022 T^{4} - 55056 T^{5} + 1171072 T^{6} + 29664624 T^{7} + 191234977 T^{8} - 1519529248 T^{9} - 27088145920 T^{10} - 210706506976 T^{11} + 1209312464722 T^{12} + 24316714882016 T^{13} + 89859490185984 T^{14} - 1324590918805248 T^{15} - 28693493560058396 T^{16} - 109941046260835584 T^{17} + 619042027891243776 T^{18} + 13903981452243282592 T^{19} + 57391939140077851762 T^{20} - \)\(82\!\cdots\!68\)\( T^{21} - \)\(88\!\cdots\!80\)\( T^{22} - \)\(41\!\cdots\!96\)\( T^{23} + \)\(43\!\cdots\!57\)\( T^{24} + \)\(55\!\cdots\!72\)\( T^{25} + \)\(18\!\cdots\!28\)\( T^{26} - \)\(70\!\cdots\!52\)\( T^{27} - \)\(17\!\cdots\!42\)\( T^{28} + \)\(42\!\cdots\!24\)\( T^{29} + \)\(94\!\cdots\!12\)\( T^{30} + \)\(97\!\cdots\!12\)\( T^{31} + \)\(50\!\cdots\!81\)\( T^{32} \))(\( 1 - 16 T + 128 T^{2} - 48 T^{3} - 16022 T^{4} + 55056 T^{5} + 1171072 T^{6} - 29664624 T^{7} + 191234977 T^{8} + 1519529248 T^{9} - 27088145920 T^{10} + 210706506976 T^{11} + 1209312464722 T^{12} - 24316714882016 T^{13} + 89859490185984 T^{14} + 1324590918805248 T^{15} - 28693493560058396 T^{16} + 109941046260835584 T^{17} + 619042027891243776 T^{18} - 13903981452243282592 T^{19} + 57391939140077851762 T^{20} + \)\(82\!\cdots\!68\)\( T^{21} - \)\(88\!\cdots\!80\)\( T^{22} + \)\(41\!\cdots\!96\)\( T^{23} + \)\(43\!\cdots\!57\)\( T^{24} - \)\(55\!\cdots\!72\)\( T^{25} + \)\(18\!\cdots\!28\)\( T^{26} + \)\(70\!\cdots\!52\)\( T^{27} - \)\(17\!\cdots\!42\)\( T^{28} - \)\(42\!\cdots\!24\)\( T^{29} + \)\(94\!\cdots\!12\)\( T^{30} - \)\(97\!\cdots\!12\)\( T^{31} + \)\(50\!\cdots\!81\)\( T^{32} \))
$89$ (\( ( 1 - 130 T^{2} + 8979 T^{4} - 1029730 T^{6} + 62742241 T^{8} )^{2} \))(\( ( 1 - 6 T - 139 T^{2} + 18 T^{3} + 19500 T^{4} + 1602 T^{5} - 1101019 T^{6} - 4229814 T^{7} + 62742241 T^{8} )^{2} \))(\( ( 1 + 6 T - 139 T^{2} - 18 T^{3} + 19500 T^{4} - 1602 T^{5} - 1101019 T^{6} + 4229814 T^{7} + 62742241 T^{8} )^{2} \))(\( ( 1 - 130 T^{2} + 8979 T^{4} - 1029730 T^{6} + 62742241 T^{8} )^{2} \))(\( ( 1 + 30 T + 439 T^{2} + 3930 T^{3} + 17353 T^{4} - 77580 T^{5} - 2924786 T^{6} - 44484000 T^{7} - 479997158 T^{8} - 3959076000 T^{9} - 23167229906 T^{10} - 54691495020 T^{11} + 1088766108073 T^{12} + 21945353634570 T^{13} + 218174786731879 T^{14} + 1326940046865870 T^{15} + 3936588805702081 T^{16} )^{2} \))(\( ( 1 - 30 T + 439 T^{2} - 3930 T^{3} + 17353 T^{4} + 77580 T^{5} - 2924786 T^{6} + 44484000 T^{7} - 479997158 T^{8} + 3959076000 T^{9} - 23167229906 T^{10} + 54691495020 T^{11} + 1088766108073 T^{12} - 21945353634570 T^{13} + 218174786731879 T^{14} - 1326940046865870 T^{15} + 3936588805702081 T^{16} )^{2} \))(\( 1 - 16 T - 440 T^{2} + 6624 T^{3} + 131636 T^{4} - 1629968 T^{5} - 29449232 T^{6} + 280328720 T^{7} + 5329291050 T^{8} - 36595065152 T^{9} - 800017753480 T^{10} + 3635275214288 T^{11} + 102057968083920 T^{12} - 258547916350832 T^{13} - 11209896420041672 T^{14} + 8808353629704256 T^{15} + 1069443829226298003 T^{16} + 783943473043678784 T^{17} - 88793589543150083912 T^{18} - \)\(18\!\cdots\!08\)\( T^{19} + \)\(64\!\cdots\!20\)\( T^{20} + \)\(20\!\cdots\!12\)\( T^{21} - \)\(39\!\cdots\!80\)\( T^{22} - \)\(16\!\cdots\!08\)\( T^{23} + \)\(20\!\cdots\!50\)\( T^{24} + \)\(98\!\cdots\!80\)\( T^{25} - \)\(91\!\cdots\!32\)\( T^{26} - \)\(45\!\cdots\!52\)\( T^{27} + \)\(32\!\cdots\!56\)\( T^{28} + \)\(14\!\cdots\!56\)\( T^{29} - \)\(86\!\cdots\!40\)\( T^{30} - \)\(27\!\cdots\!84\)\( T^{31} + \)\(15\!\cdots\!61\)\( T^{32} \))(\( 1 + 16 T - 440 T^{2} - 6624 T^{3} + 131636 T^{4} + 1629968 T^{5} - 29449232 T^{6} - 280328720 T^{7} + 5329291050 T^{8} + 36595065152 T^{9} - 800017753480 T^{10} - 3635275214288 T^{11} + 102057968083920 T^{12} + 258547916350832 T^{13} - 11209896420041672 T^{14} - 8808353629704256 T^{15} + 1069443829226298003 T^{16} - 783943473043678784 T^{17} - 88793589543150083912 T^{18} + \)\(18\!\cdots\!08\)\( T^{19} + \)\(64\!\cdots\!20\)\( T^{20} - \)\(20\!\cdots\!12\)\( T^{21} - \)\(39\!\cdots\!80\)\( T^{22} + \)\(16\!\cdots\!08\)\( T^{23} + \)\(20\!\cdots\!50\)\( T^{24} - \)\(98\!\cdots\!80\)\( T^{25} - \)\(91\!\cdots\!32\)\( T^{26} + \)\(45\!\cdots\!52\)\( T^{27} + \)\(32\!\cdots\!56\)\( T^{28} - \)\(14\!\cdots\!56\)\( T^{29} - \)\(86\!\cdots\!40\)\( T^{30} + \)\(27\!\cdots\!84\)\( T^{31} + \)\(15\!\cdots\!61\)\( T^{32} \))
$97$ (\( 1 + 2308 T^{4} - 29276922 T^{8} + 204325580548 T^{12} + 7837433594376961 T^{16} \))(\( ( 1 + 2207 T^{4} + 88529281 T^{8} )^{2} \))(\( ( 1 + 2207 T^{4} + 88529281 T^{8} )^{2} \))(\( 1 + 2308 T^{4} - 29276922 T^{8} + 204325580548 T^{12} + 7837433594376961 T^{16} \))(\( 1 + 4700 T^{4} - 3632918 T^{8} + 688067520560 T^{12} + 9198193610860915 T^{16} + 60914122874629517360 T^{20} - \)\(28\!\cdots\!98\)\( T^{24} + \)\(32\!\cdots\!00\)\( T^{28} + \)\(61\!\cdots\!21\)\( T^{32} \))(\( 1 + 4700 T^{4} - 3632918 T^{8} + 688067520560 T^{12} + 9198193610860915 T^{16} + 60914122874629517360 T^{20} - \)\(28\!\cdots\!98\)\( T^{24} + \)\(32\!\cdots\!00\)\( T^{28} + \)\(61\!\cdots\!21\)\( T^{32} \))(\( 1 - 44 T + 968 T^{2} - 15000 T^{3} + 231654 T^{4} - 3860388 T^{5} + 58116000 T^{6} - 743419588 T^{7} + 9111376513 T^{8} - 116138978204 T^{9} + 1430090287664 T^{10} - 15964265880436 T^{11} + 171574947065342 T^{12} - 1888109943974960 T^{13} + 20331243756190952 T^{14} - 203613182755137748 T^{15} + 1985416913924716484 T^{16} - 19750478727248361556 T^{17} + \)\(19\!\cdots\!68\)\( T^{18} - \)\(17\!\cdots\!80\)\( T^{19} + \)\(15\!\cdots\!02\)\( T^{20} - \)\(13\!\cdots\!52\)\( T^{21} + \)\(11\!\cdots\!56\)\( T^{22} - \)\(93\!\cdots\!52\)\( T^{23} + \)\(71\!\cdots\!93\)\( T^{24} - \)\(56\!\cdots\!96\)\( T^{25} + \)\(42\!\cdots\!00\)\( T^{26} - \)\(27\!\cdots\!64\)\( T^{27} + \)\(16\!\cdots\!14\)\( T^{28} - \)\(10\!\cdots\!00\)\( T^{29} + \)\(63\!\cdots\!92\)\( T^{30} - \)\(27\!\cdots\!92\)\( T^{31} + \)\(61\!\cdots\!21\)\( T^{32} \))(\( 1 + 44 T + 968 T^{2} + 15000 T^{3} + 231654 T^{4} + 3860388 T^{5} + 58116000 T^{6} + 743419588 T^{7} + 9111376513 T^{8} + 116138978204 T^{9} + 1430090287664 T^{10} + 15964265880436 T^{11} + 171574947065342 T^{12} + 1888109943974960 T^{13} + 20331243756190952 T^{14} + 203613182755137748 T^{15} + 1985416913924716484 T^{16} + 19750478727248361556 T^{17} + \)\(19\!\cdots\!68\)\( T^{18} + \)\(17\!\cdots\!80\)\( T^{19} + \)\(15\!\cdots\!02\)\( T^{20} + \)\(13\!\cdots\!52\)\( T^{21} + \)\(11\!\cdots\!56\)\( T^{22} + \)\(93\!\cdots\!52\)\( T^{23} + \)\(71\!\cdots\!93\)\( T^{24} + \)\(56\!\cdots\!96\)\( T^{25} + \)\(42\!\cdots\!00\)\( T^{26} + \)\(27\!\cdots\!64\)\( T^{27} + \)\(16\!\cdots\!14\)\( T^{28} + \)\(10\!\cdots\!00\)\( T^{29} + \)\(63\!\cdots\!92\)\( T^{30} + \)\(27\!\cdots\!92\)\( T^{31} + \)\(61\!\cdots\!21\)\( T^{32} \))
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