Properties

Label 6025.2.a
Level 6025
Weight 2
Character orbit a
Rep. character \(\chi_{6025}(1,\cdot)\)
Character field \(\Q\)
Dimension 380
Newform subspaces 17
Sturm bound 1210
Trace bound 3

Related objects

Downloads

Learn more about

Defining parameters

Level: \( N \) \(=\) \( 6025 = 5^{2} \cdot 241 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6025.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 17 \)
Sturm bound: \(1210\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(6025))\).

Total New Old
Modular forms 610 380 230
Cusp forms 599 380 219
Eisenstein series 11 0 11

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

\(5\)\(241\)FrickeDim.
\(+\)\(+\)\(+\)\(87\)
\(+\)\(-\)\(-\)\(93\)
\(-\)\(+\)\(-\)\(106\)
\(-\)\(-\)\(+\)\(94\)
Plus space\(+\)\(181\)
Minus space\(-\)\(199\)

Trace form

\( 380q + 2q^{2} - 2q^{3} + 382q^{4} + 6q^{6} + 6q^{8} + 374q^{9} + O(q^{10}) \) \( 380q + 2q^{2} - 2q^{3} + 382q^{4} + 6q^{6} + 6q^{8} + 374q^{9} - 4q^{11} - 10q^{12} - 12q^{14} + 378q^{16} + 8q^{17} + 2q^{18} - 8q^{19} + 2q^{22} - 2q^{23} - 18q^{24} - 2q^{26} - 14q^{27} - 6q^{28} + 4q^{29} - 10q^{31} + 34q^{32} - 4q^{33} - 10q^{34} + 356q^{36} - 2q^{37} + 10q^{38} - 30q^{39} - 10q^{41} + 42q^{42} - 20q^{43} + 18q^{44} + 46q^{46} - 20q^{47} - 26q^{48} + 370q^{49} - 50q^{51} + 14q^{53} + 8q^{54} - 42q^{56} - 28q^{57} + 42q^{58} - 8q^{59} + 4q^{61} + 16q^{62} + 28q^{63} + 390q^{64} - 42q^{66} - 24q^{67} + 60q^{68} + 12q^{69} + 4q^{71} + 78q^{72} + 10q^{74} - 60q^{76} + 50q^{77} + 26q^{78} + 18q^{79} + 356q^{81} + 22q^{82} - 20q^{83} - 36q^{84} + 22q^{86} - 12q^{87} + 50q^{88} + 18q^{89} - 52q^{91} + 14q^{92} - 18q^{93} + 6q^{96} + 30q^{97} + 56q^{98} - 52q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(6025))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 5 241
6025.2.a.a \(2\) \(48.110\) \(\Q(\sqrt{5}) \) None \(-2\) \(1\) \(0\) \(-2\) \(-\) \(-\) \(q-q^{2}+\beta q^{3}-q^{4}-\beta q^{6}+(-2+2\beta )q^{7}+\cdots\)
6025.2.a.b \(2\) \(48.110\) \(\Q(\sqrt{5}) \) None \(0\) \(-3\) \(0\) \(2\) \(-\) \(-\) \(q+(1-2\beta )q^{2}+(-2+\beta )q^{3}+3q^{4}+\cdots\)
6025.2.a.c \(2\) \(48.110\) \(\Q(\sqrt{5}) \) None \(0\) \(3\) \(0\) \(-2\) \(-\) \(-\) \(q+(1-2\beta )q^{2}+(1+\beta )q^{3}+3q^{4}+(-1+\cdots)q^{6}+\cdots\)
6025.2.a.d \(2\) \(48.110\) \(\Q(\sqrt{5}) \) None \(2\) \(-1\) \(0\) \(2\) \(-\) \(-\) \(q+q^{2}+(-1+\beta )q^{3}-q^{4}+(-1+\beta )q^{6}+\cdots\)
6025.2.a.e \(5\) \(48.110\) 5.5.38569.1 None \(1\) \(5\) \(0\) \(10\) \(+\) \(-\) \(q+(\beta _{1}+\beta _{3})q^{2}+(1-\beta _{1}-\beta _{3}-\beta _{4})q^{3}+\cdots\)
6025.2.a.f \(7\) \(48.110\) 7.7.31056073.1 None \(4\) \(3\) \(0\) \(7\) \(+\) \(+\) \(q+(1-\beta _{1})q^{2}-\beta _{6}q^{3}+(1-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
6025.2.a.g \(11\) \(48.110\) \(\mathbb{Q}[x]/(x^{11} - \cdots)\) None \(4\) \(8\) \(0\) \(9\) \(+\) \(-\) \(q+(1-\beta _{1}+\beta _{2})q^{2}+(1+\beta _{2}+\beta _{3})q^{3}+\cdots\)
6025.2.a.h \(12\) \(48.110\) \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(-3\) \(-1\) \(0\) \(-3\) \(+\) \(-\) \(q-\beta _{1}q^{2}-\beta _{8}q^{3}+(1+\beta _{2})q^{4}+(1-\beta _{3}+\cdots)q^{6}+\cdots\)
6025.2.a.i \(15\) \(48.110\) \(\mathbb{Q}[x]/(x^{15} - \cdots)\) None \(-2\) \(7\) \(0\) \(3\) \(+\) \(+\) \(q-\beta _{1}q^{2}-\beta _{11}q^{3}+\beta _{2}q^{4}+(-\beta _{8}+\cdots)q^{6}+\cdots\)
6025.2.a.j \(25\) \(48.110\) None \(-6\) \(-15\) \(0\) \(-19\) \(+\) \(+\)
6025.2.a.k \(25\) \(48.110\) None \(4\) \(-9\) \(0\) \(-7\) \(+\) \(-\)
6025.2.a.l \(40\) \(48.110\) None \(-11\) \(-8\) \(0\) \(-16\) \(+\) \(+\)
6025.2.a.m \(40\) \(48.110\) None \(-9\) \(-8\) \(0\) \(-20\) \(-\) \(-\)
6025.2.a.n \(40\) \(48.110\) None \(9\) \(8\) \(0\) \(20\) \(+\) \(-\)
6025.2.a.o \(40\) \(48.110\) None \(11\) \(8\) \(0\) \(16\) \(-\) \(+\)
6025.2.a.p \(46\) \(48.110\) None \(0\) \(0\) \(0\) \(0\) \(-\) \(-\)
6025.2.a.q \(66\) \(48.110\) None \(0\) \(0\) \(0\) \(0\) \(-\) \(+\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(6025))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(6025)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(241))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1205))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 + T + 2 T^{2} )^{2} \))(\( 1 - T^{2} + 4 T^{4} \))(\( 1 - T^{2} + 4 T^{4} \))(\( ( 1 - T + 2 T^{2} )^{2} \))(\( 1 - T + 4 T^{2} - 3 T^{3} + 5 T^{4} - 5 T^{5} + 10 T^{6} - 12 T^{7} + 32 T^{8} - 16 T^{9} + 32 T^{10} \))(\( 1 - 4 T + 14 T^{2} - 34 T^{3} + 74 T^{4} - 134 T^{5} + 223 T^{6} - 327 T^{7} + 446 T^{8} - 536 T^{9} + 592 T^{10} - 544 T^{11} + 448 T^{12} - 256 T^{13} + 128 T^{14} \))(\( 1 - 4 T + 16 T^{2} - 44 T^{3} + 119 T^{4} - 265 T^{5} + 569 T^{6} - 1075 T^{7} + 1955 T^{8} - 3212 T^{9} + 5073 T^{10} - 7317 T^{11} + 10146 T^{12} - 12848 T^{13} + 15640 T^{14} - 17200 T^{15} + 18208 T^{16} - 16960 T^{17} + 15232 T^{18} - 11264 T^{19} + 8192 T^{20} - 4096 T^{21} + 2048 T^{22} \))(\( 1 + 3 T + 10 T^{2} + 22 T^{3} + 49 T^{4} + 87 T^{5} + 157 T^{6} + 246 T^{7} + 389 T^{8} + 556 T^{9} + 839 T^{10} + 1158 T^{11} + 1699 T^{12} + 2316 T^{13} + 3356 T^{14} + 4448 T^{15} + 6224 T^{16} + 7872 T^{17} + 10048 T^{18} + 11136 T^{19} + 12544 T^{20} + 11264 T^{21} + 10240 T^{22} + 6144 T^{23} + 4096 T^{24} \))(\( 1 + 2 T + 14 T^{2} + 25 T^{3} + 103 T^{4} + 168 T^{5} + 530 T^{6} + 801 T^{7} + 2121 T^{8} + 2987 T^{9} + 6933 T^{10} + 9114 T^{11} + 19011 T^{12} + 23293 T^{13} + 44373 T^{14} + 50485 T^{15} + 88746 T^{16} + 93172 T^{17} + 152088 T^{18} + 145824 T^{19} + 221856 T^{20} + 191168 T^{21} + 271488 T^{22} + 205056 T^{23} + 271360 T^{24} + 172032 T^{25} + 210944 T^{26} + 102400 T^{27} + 114688 T^{28} + 32768 T^{29} + 32768 T^{30} \))
$3$ (\( 1 - T + 5 T^{2} - 3 T^{3} + 9 T^{4} \))(\( 1 + 3 T + 7 T^{2} + 9 T^{3} + 9 T^{4} \))(\( 1 - 3 T + 7 T^{2} - 9 T^{3} + 9 T^{4} \))(\( 1 + T + 5 T^{2} + 3 T^{3} + 9 T^{4} \))(\( 1 - 5 T + 17 T^{2} - 43 T^{3} + 99 T^{4} - 185 T^{5} + 297 T^{6} - 387 T^{7} + 459 T^{8} - 405 T^{9} + 243 T^{10} \))(\( 1 - 3 T + 16 T^{2} - 35 T^{3} + 110 T^{4} - 191 T^{5} + 467 T^{6} - 679 T^{7} + 1401 T^{8} - 1719 T^{9} + 2970 T^{10} - 2835 T^{11} + 3888 T^{12} - 2187 T^{13} + 2187 T^{14} \))(\( 1 - 8 T + 44 T^{2} - 184 T^{3} + 644 T^{4} - 1954 T^{5} + 5282 T^{6} - 12921 T^{7} + 28973 T^{8} - 59994 T^{9} + 115445 T^{10} - 206986 T^{11} + 346335 T^{12} - 539946 T^{13} + 782271 T^{14} - 1046601 T^{15} + 1283526 T^{16} - 1424466 T^{17} + 1408428 T^{18} - 1207224 T^{19} + 866052 T^{20} - 472392 T^{21} + 177147 T^{22} \))(\( 1 + T + 11 T^{2} + 8 T^{3} + 68 T^{4} + 30 T^{5} + 303 T^{6} + 40 T^{7} + 1099 T^{8} - 195 T^{9} + 3502 T^{10} - 1444 T^{11} + 10528 T^{12} - 4332 T^{13} + 31518 T^{14} - 5265 T^{15} + 89019 T^{16} + 9720 T^{17} + 220887 T^{18} + 65610 T^{19} + 446148 T^{20} + 157464 T^{21} + 649539 T^{22} + 177147 T^{23} + 531441 T^{24} \))(\( 1 - 7 T + 44 T^{2} - 194 T^{3} + 773 T^{4} - 2617 T^{5} + 8164 T^{6} - 22967 T^{7} + 60449 T^{8} - 147310 T^{9} + 339467 T^{10} - 734507 T^{11} + 1513337 T^{12} - 2950637 T^{13} + 5499816 T^{14} - 9736601 T^{15} + 16499448 T^{16} - 26555733 T^{17} + 40860099 T^{18} - 59495067 T^{19} + 82490481 T^{20} - 107388990 T^{21} + 132201963 T^{22} - 150686487 T^{23} + 160692012 T^{24} - 154531233 T^{25} + 136934631 T^{26} - 103099554 T^{27} + 70150212 T^{28} - 33480783 T^{29} + 14348907 T^{30} \))
$5$ 1
$7$ (\( 1 + 2 T + 10 T^{2} + 14 T^{3} + 49 T^{4} \))(\( 1 - 2 T + 10 T^{2} - 14 T^{3} + 49 T^{4} \))(\( 1 + 2 T + 10 T^{2} + 14 T^{3} + 49 T^{4} \))(\( 1 - 2 T + 10 T^{2} - 14 T^{3} + 49 T^{4} \))(\( 1 - 10 T + 63 T^{2} - 277 T^{3} + 980 T^{4} - 2809 T^{5} + 6860 T^{6} - 13573 T^{7} + 21609 T^{8} - 24010 T^{9} + 16807 T^{10} \))(\( 1 - 7 T + 46 T^{2} - 196 T^{3} + 786 T^{4} - 2528 T^{5} + 7897 T^{6} - 21047 T^{7} + 55279 T^{8} - 123872 T^{9} + 269598 T^{10} - 470596 T^{11} + 773122 T^{12} - 823543 T^{13} + 823543 T^{14} \))(\( 1 - 9 T + 77 T^{2} - 443 T^{3} + 2422 T^{4} - 10808 T^{5} + 45823 T^{6} - 168895 T^{7} + 593821 T^{8} - 1863877 T^{9} + 5589939 T^{10} - 15122740 T^{11} + 39129573 T^{12} - 91329973 T^{13} + 203680603 T^{14} - 405516895 T^{15} + 770147161 T^{16} - 1271550392 T^{17} + 1994621146 T^{18} - 2553806843 T^{19} + 3107227739 T^{20} - 2542277241 T^{21} + 1977326743 T^{22} \))(\( 1 + 3 T + 51 T^{2} + 135 T^{3} + 1169 T^{4} + 2891 T^{5} + 16678 T^{6} + 41432 T^{7} + 176813 T^{8} + 460010 T^{9} + 1555215 T^{10} + 4088130 T^{11} + 11751926 T^{12} + 28616910 T^{13} + 76205535 T^{14} + 157783430 T^{15} + 424528013 T^{16} + 696347624 T^{17} + 1962150022 T^{18} + 2380862813 T^{19} + 6739052369 T^{20} + 5447736945 T^{21} + 14406237699 T^{22} + 5931980229 T^{23} + 13841287201 T^{24} \))(\( 1 - 3 T + 69 T^{2} - 184 T^{3} + 2285 T^{4} - 5435 T^{5} + 48514 T^{6} - 103450 T^{7} + 746705 T^{8} - 1437121 T^{9} + 8946471 T^{10} - 15668832 T^{11} + 87418730 T^{12} - 140495069 T^{13} + 717670379 T^{14} - 1064822293 T^{15} + 5023692653 T^{16} - 6884258381 T^{17} + 29984624390 T^{18} - 37620865632 T^{19} + 150363338097 T^{20} - 169075848529 T^{21} + 614943675815 T^{22} - 596368663450 T^{23} + 1957714889998 T^{24} - 1535252978315 T^{25} + 4518191607755 T^{26} - 2546796844984 T^{27} + 6685341718083 T^{28} - 2034669218547 T^{29} + 4747561509943 T^{30} \))
$11$ (\( 1 - 5 T + 27 T^{2} - 55 T^{3} + 121 T^{4} \))(\( 1 + T + 21 T^{2} + 11 T^{3} + 121 T^{4} \))(\( 1 + T + 21 T^{2} + 11 T^{3} + 121 T^{4} \))(\( 1 - 5 T + 27 T^{2} - 55 T^{3} + 121 T^{4} \))(\( 1 + 3 T + 30 T^{2} + 64 T^{3} + 402 T^{4} + 681 T^{5} + 4422 T^{6} + 7744 T^{7} + 39930 T^{8} + 43923 T^{9} + 161051 T^{10} \))(\( 1 + 18 T + 194 T^{2} + 1471 T^{3} + 8839 T^{4} + 43563 T^{5} + 182353 T^{6} + 651389 T^{7} + 2005883 T^{8} + 5271123 T^{9} + 11764709 T^{10} + 21536911 T^{11} + 31243894 T^{12} + 31888098 T^{13} + 19487171 T^{14} \))(\( 1 + 3 T + 62 T^{2} + 154 T^{3} + 1657 T^{4} + 2555 T^{5} + 22142 T^{6} - 7625 T^{7} + 93562 T^{8} - 937944 T^{9} - 1275223 T^{10} - 15528862 T^{11} - 14027453 T^{12} - 113491224 T^{13} + 124531022 T^{14} - 111637625 T^{15} + 3565991242 T^{16} + 4526338355 T^{17} + 32290242347 T^{18} + 33011267674 T^{19} + 146192756842 T^{20} + 77812273803 T^{21} + 285311670611 T^{22} \))(\( 1 - 22 T + 309 T^{2} - 3215 T^{3} + 27241 T^{4} - 195612 T^{5} + 1224926 T^{6} - 6803622 T^{7} + 33951641 T^{8} - 153501494 T^{9} + 632754137 T^{10} - 2387213463 T^{11} + 8263362466 T^{12} - 26259348093 T^{13} + 76563250577 T^{14} - 204310488514 T^{15} + 497085975881 T^{16} - 1095730126722 T^{17} + 2170031129486 T^{18} - 3811924493652 T^{19} + 5839350277321 T^{20} - 7580801826565 T^{21} + 8014664201709 T^{22} - 6276856753442 T^{23} + 3138428376721 T^{24} \))(\( 1 + 10 T + 140 T^{2} + 1028 T^{3} + 8603 T^{4} + 51007 T^{5} + 324685 T^{6} + 1638350 T^{7} + 8671398 T^{8} + 38435840 T^{9} + 176810244 T^{10} + 701777246 T^{11} + 2875783160 T^{12} + 10334097764 T^{13} + 38247516248 T^{14} + 125038989415 T^{15} + 420722678728 T^{16} + 1250425829444 T^{17} + 3827667385960 T^{18} + 10274720658686 T^{19} + 28475466606444 T^{20} + 68091435146240 T^{21} + 168981015635058 T^{22} + 351194872686350 T^{23} + 765590246052335 T^{24} + 1322990216623207 T^{25} + 2454536302266433 T^{26} + 3226304371269188 T^{27} + 4833179700150340 T^{28} + 3797498335832410 T^{29} + 4177248169415651 T^{30} \))
$13$ (\( 1 - 3 T + 27 T^{2} - 39 T^{3} + 169 T^{4} \))(\( 1 - T + 15 T^{2} - 13 T^{3} + 169 T^{4} \))(\( 1 + T + 15 T^{2} + 13 T^{3} + 169 T^{4} \))(\( 1 + 3 T + 27 T^{2} + 39 T^{3} + 169 T^{4} \))(\( 1 - T + 48 T^{2} - 34 T^{3} + 1070 T^{4} - 547 T^{5} + 13910 T^{6} - 5746 T^{7} + 105456 T^{8} - 28561 T^{9} + 371293 T^{10} \))(\( 1 - T + 43 T^{2} - 16 T^{3} + 962 T^{4} - 171 T^{5} + 16575 T^{6} - 3431 T^{7} + 215475 T^{8} - 28899 T^{9} + 2113514 T^{10} - 456976 T^{11} + 15965599 T^{12} - 4826809 T^{13} + 62748517 T^{14} \))(\( 1 - 9 T + 97 T^{2} - 587 T^{3} + 3646 T^{4} - 16217 T^{5} + 71982 T^{6} - 244837 T^{7} + 834242 T^{8} - 2260088 T^{9} + 6932271 T^{10} - 20417578 T^{11} + 90119523 T^{12} - 381954872 T^{13} + 1832829674 T^{14} - 6992789557 T^{15} + 26726412726 T^{16} - 78276361553 T^{17} + 228781092982 T^{18} - 478833933227 T^{19} + 1028636439181 T^{20} - 1240726426641 T^{21} + 1792160394037 T^{22} \))(\( 1 - 5 T + 94 T^{2} - 419 T^{3} + 4519 T^{4} - 18313 T^{5} + 144981 T^{6} - 535786 T^{7} + 3431095 T^{8} - 11548019 T^{9} + 62910413 T^{10} - 191545534 T^{11} + 914969618 T^{12} - 2490091942 T^{13} + 10631859797 T^{14} - 25370997743 T^{15} + 97995504295 T^{16} - 198933591298 T^{17} + 699795595629 T^{18} - 1149113591821 T^{19} + 3686287128199 T^{20} - 4443285237287 T^{21} + 12958698233806 T^{22} - 8960801970185 T^{23} + 23298085122481 T^{24} \))(\( 1 - 8 T + 135 T^{2} - 856 T^{3} + 8138 T^{4} - 42812 T^{5} + 297820 T^{6} - 1347080 T^{7} + 7616486 T^{8} - 30614433 T^{9} + 149729256 T^{10} - 552303058 T^{11} + 2443009380 T^{12} - 8488918786 T^{13} + 35043186281 T^{14} - 116137458321 T^{15} + 455561421653 T^{16} - 1434627274834 T^{17} + 5367291607860 T^{18} - 15774327639538 T^{19} + 55593424648008 T^{20} - 147770020734297 T^{21} + 477923201251262 T^{22} - 1098854539644680 T^{23} + 3158232003266860 T^{24} - 5901997753039388 T^{25} + 14584601286673106 T^{26} - 19943160864843736 T^{27} + 40888139389954155 T^{28} - 31499011085594312 T^{29} + 51185893014090757 T^{30} \))
$17$ (\( 1 - T + 23 T^{2} - 17 T^{3} + 289 T^{4} \))(\( 1 + T + 3 T^{2} + 17 T^{3} + 289 T^{4} \))(\( 1 - T + 3 T^{2} - 17 T^{3} + 289 T^{4} \))(\( 1 + T + 23 T^{2} + 17 T^{3} + 289 T^{4} \))(\( 1 - 5 T + 33 T^{2} - 181 T^{3} + 789 T^{4} - 3625 T^{5} + 13413 T^{6} - 52309 T^{7} + 162129 T^{8} - 417605 T^{9} + 1419857 T^{10} \))(\( 1 - 2 T + 54 T^{2} - 118 T^{3} + 1511 T^{4} - 4204 T^{5} + 32789 T^{6} - 93345 T^{7} + 557413 T^{8} - 1214956 T^{9} + 7423543 T^{10} - 9855478 T^{11} + 76672278 T^{12} - 48275138 T^{13} + 410338673 T^{14} \))(\( 1 - 4 T + 106 T^{2} - 449 T^{3} + 6030 T^{4} - 25167 T^{5} + 232247 T^{6} - 915996 T^{7} + 6617860 T^{8} - 23891579 T^{9} + 144618675 T^{10} - 466504364 T^{11} + 2458517475 T^{12} - 6904666331 T^{13} + 32513546180 T^{14} - 76504901916 T^{15} + 329757528679 T^{16} - 607470199023 T^{17} + 2474342198190 T^{18} - 3132115091009 T^{19} + 12570314908682 T^{20} - 8063975601796 T^{21} + 34271896307633 T^{22} \))(\( 1 - 4 T + 107 T^{2} - 378 T^{3} + 5581 T^{4} - 16942 T^{5} + 191746 T^{6} - 492741 T^{7} + 4974231 T^{8} - 10970245 T^{9} + 105662523 T^{10} - 208156762 T^{11} + 1924983462 T^{12} - 3538664954 T^{13} + 30536469147 T^{14} - 53896813685 T^{15} + 415452747351 T^{16} - 699621758037 T^{17} + 4628282305474 T^{18} - 6951957797966 T^{19} + 38931702278221 T^{20} - 44826217315866 T^{21} + 215711347348043 T^{22} - 137087585230532 T^{23} + 582622237229761 T^{24} \))(\( 1 - T + 126 T^{2} - 269 T^{3} + 8190 T^{4} - 24298 T^{5} + 373120 T^{6} - 1283146 T^{7} + 13229934 T^{8} - 48031043 T^{9} + 380644474 T^{10} - 1384341208 T^{11} + 9095231436 T^{12} - 31895449037 T^{13} + 182878337902 T^{14} - 598744414275 T^{15} + 3108931744334 T^{16} - 9217784771693 T^{17} + 44684872045068 T^{18} - 115621562033368 T^{19} + 540460720920218 T^{20} - 1159352614554467 T^{21} + 5428753561437582 T^{22} - 8950915257389386 T^{23} + 44247508478560640 T^{24} - 48984619793109802 T^{25} + 280686830759514270 T^{26} - 156725381814805709 T^{27} + 1247976832146148062 T^{28} - 168377826559400929 T^{29} + 2862423051509815793 T^{30} \))
$19$ (\( 1 + T + 37 T^{2} + 19 T^{3} + 361 T^{4} \))(\( 1 + 13 T + 79 T^{2} + 247 T^{3} + 361 T^{4} \))(\( 1 + 13 T + 79 T^{2} + 247 T^{3} + 361 T^{4} \))(\( 1 + T + 37 T^{2} + 19 T^{3} + 361 T^{4} \))(\( 1 - 3 T + 72 T^{2} - 170 T^{3} + 2368 T^{4} - 4385 T^{5} + 44992 T^{6} - 61370 T^{7} + 493848 T^{8} - 390963 T^{9} + 2476099 T^{10} \))(\( 1 + 6 T + 77 T^{2} + 408 T^{3} + 3328 T^{4} + 14851 T^{5} + 91875 T^{6} + 346087 T^{7} + 1745625 T^{8} + 5361211 T^{9} + 22826752 T^{10} + 53170968 T^{11} + 190659623 T^{12} + 282275286 T^{13} + 893871739 T^{14} \))(\( 1 + 33 T + 583 T^{2} + 7144 T^{3} + 68237 T^{4} + 542625 T^{5} + 3756768 T^{6} + 23329258 T^{7} + 132524391 T^{8} + 695866631 T^{9} + 3391438531 T^{10} + 15344291290 T^{11} + 64437332089 T^{12} + 251207853791 T^{13} + 908984797869 T^{14} + 3040292231818 T^{15} + 9302129488032 T^{16} + 25528271177625 T^{17} + 60995125854143 T^{18} + 121330574364904 T^{19} + 188126927805157 T^{20} + 202325186507433 T^{21} + 116490258898219 T^{22} \))(\( 1 + 6 T + 142 T^{2} + 730 T^{3} + 10024 T^{4} + 46417 T^{5} + 468739 T^{6} + 1979928 T^{7} + 16137130 T^{8} + 62321901 T^{9} + 430966450 T^{10} + 1510794001 T^{11} + 9148767508 T^{12} + 28705086019 T^{13} + 155578888450 T^{14} + 427465918959 T^{15} + 2103006918730 T^{16} + 4902497740872 T^{17} + 22052239214059 T^{18} + 41490844509163 T^{19} + 170243235922984 T^{20} + 235562019378670 T^{21} + 870611408607742 T^{22} + 698941553389314 T^{23} + 2213314919066161 T^{24} \))(\( 1 + 30 T + 617 T^{2} + 9225 T^{3} + 114598 T^{4} + 1205122 T^{5} + 11195374 T^{6} + 92718239 T^{7} + 698401371 T^{8} + 4807897553 T^{9} + 30589810478 T^{10} + 180290400740 T^{11} + 991023236386 T^{12} + 5083321480560 T^{13} + 24432500370967 T^{14} + 109938656945393 T^{15} + 464217507048373 T^{16} + 1835079054482160 T^{17} + 6797428378371574 T^{18} + 23495625314837540 T^{19} + 75743399134765322 T^{20} + 226191776138629193 T^{21} + 624281248015754169 T^{22} + 1574686057107004799 T^{23} + 3612609461834874346 T^{24} + 7388682830733656722 T^{25} + 13349550689218100962 T^{26} + 20417830128385335225 T^{27} + 25946690796212605403 T^{28} + 23970200573486523630 T^{29} + 15181127029874798299 T^{30} \))
$23$ (\( 1 + 8 T + 42 T^{2} + 184 T^{3} + 529 T^{4} \))(\( 1 + 26 T^{2} + 529 T^{4} \))(\( 1 + 26 T^{2} + 529 T^{4} \))(\( 1 - 8 T + 42 T^{2} - 184 T^{3} + 529 T^{4} \))(\( 1 - 8 T + 105 T^{2} - 615 T^{3} + 4696 T^{4} - 20085 T^{5} + 108008 T^{6} - 325335 T^{7} + 1277535 T^{8} - 2238728 T^{9} + 6436343 T^{10} \))(\( 1 - 22 T + 329 T^{2} - 3499 T^{3} + 30354 T^{4} - 215473 T^{5} + 1308858 T^{6} - 6746533 T^{7} + 30103734 T^{8} - 113985217 T^{9} + 369317118 T^{10} - 979163659 T^{11} + 2117556847 T^{12} - 3256789558 T^{13} + 3404825447 T^{14} \))(\( 1 - 31 T + 645 T^{2} - 9693 T^{3} + 119003 T^{4} - 1220599 T^{5} + 10860307 T^{6} - 84638686 T^{7} + 587470399 T^{8} - 3645456586 T^{9} + 20391619600 T^{10} - 102819393318 T^{11} + 469007250800 T^{12} - 1928446533994 T^{13} + 7147752344633 T^{14} - 23685374528926 T^{15} + 69900660937301 T^{16} - 180692458077511 T^{17} + 405184442669341 T^{18} - 759068380328733 T^{19} + 1161743466643635 T^{20} - 1284221847623119 T^{21} + 952809757913927 T^{22} \))(\( 1 + 32 T + 580 T^{2} + 7469 T^{3} + 76528 T^{4} + 663240 T^{5} + 5077858 T^{6} + 35272611 T^{7} + 225993748 T^{8} + 1346655168 T^{9} + 7505092384 T^{10} + 39288214509 T^{11} + 193919253422 T^{12} + 903628933707 T^{13} + 3970193871136 T^{14} + 16384753429056 T^{15} + 63242316434068 T^{16} + 227026622901573 T^{17} + 751705223245762 T^{18} + 2258216429468280 T^{19} + 5992983081584368 T^{20} + 13452809228467147 T^{21} + 24027376503916420 T^{22} + 30489912253245664 T^{23} + 21914624432020321 T^{24} \))(\( 1 + 19 T + 345 T^{2} + 4344 T^{3} + 48872 T^{4} + 467208 T^{5} + 4038140 T^{6} + 31447888 T^{7} + 225466890 T^{8} + 1495229706 T^{9} + 9276905985 T^{10} + 54168988017 T^{11} + 300138423930 T^{12} + 1586908275369 T^{13} + 8051845774445 T^{14} + 39328563854155 T^{15} + 185192452812235 T^{16} + 839474477670201 T^{17} + 3651784203956310 T^{18} + 15158703775665297 T^{19} + 59709348898212855 T^{20} + 221347658786918634 T^{21} + 767675404527949830 T^{22} + 2462715094286536528 T^{23} + 7273306608360198820 T^{24} + 19354797451106521992 T^{25} + 46565718488769440344 T^{26} + 95197128532696274424 T^{27} + \)\(17\!\cdots\!35\)\( T^{28} + \)\(22\!\cdots\!71\)\( T^{29} + \)\(26\!\cdots\!07\)\( T^{30} \))
$29$ (\( 1 + 3 T + 29 T^{2} + 87 T^{3} + 841 T^{4} \))(\( 1 + 3 T + 29 T^{2} + 87 T^{3} + 841 T^{4} \))(\( 1 + 3 T + 29 T^{2} + 87 T^{3} + 841 T^{4} \))(\( 1 + 3 T + 29 T^{2} + 87 T^{3} + 841 T^{4} \))(\( 1 - 9 T + 138 T^{2} - 782 T^{3} + 7068 T^{4} - 29739 T^{5} + 204972 T^{6} - 657662 T^{7} + 3365682 T^{8} - 6365529 T^{9} + 20511149 T^{10} \))(\( 1 + 16 T + 228 T^{2} + 1945 T^{3} + 15113 T^{4} + 86708 T^{5} + 503802 T^{6} + 2527253 T^{7} + 14610258 T^{8} + 72921428 T^{9} + 368590957 T^{10} + 1375661545 T^{11} + 4676541972 T^{12} + 9517173136 T^{13} + 17249876309 T^{14} \))(\( 1 - T + 165 T^{2} - 481 T^{3} + 13537 T^{4} - 58070 T^{5} + 790904 T^{6} - 3714933 T^{7} + 36720806 T^{8} - 160589230 T^{9} + 1356513220 T^{10} - 5258726656 T^{11} + 39338883380 T^{12} - 135055542430 T^{13} + 895583737534 T^{14} - 2627501527173 T^{15} + 16222349788696 T^{16} - 34541390250470 T^{17} + 233511575594933 T^{18} - 240618524634241 T^{19} + 2393679086018385 T^{20} - 420707233300201 T^{21} + 12200509765705829 T^{22} \))(\( 1 - 6 T + 135 T^{2} - 539 T^{3} + 8952 T^{4} - 35377 T^{5} + 479022 T^{6} - 1973598 T^{7} + 19968664 T^{8} - 81475880 T^{9} + 699787907 T^{10} - 2930882253 T^{11} + 22109454828 T^{12} - 84995585337 T^{13} + 588521629787 T^{14} - 1987115237320 T^{15} + 14123456642584 T^{16} - 40480762644102 T^{17} + 284933456872062 T^{18} - 610248874183493 T^{19} + 4478205888826872 T^{20} - 7819351680993391 T^{21} + 56795476495527135 T^{22} - 73203058594234974 T^{23} + 353814783205469041 T^{24} \))(\( 1 + 12 T + 339 T^{2} + 3188 T^{3} + 50639 T^{4} + 382964 T^{5} + 4444766 T^{6} + 27152996 T^{7} + 256269972 T^{8} + 1242735270 T^{9} + 10314402824 T^{10} + 38009283932 T^{11} + 309955761183 T^{12} + 831254958912 T^{13} + 8114646936272 T^{14} + 18821183700381 T^{15} + 235324761151888 T^{16} + 699085420444992 T^{17} + 7559511059492187 T^{18} + 26883244348708892 T^{19} + 211560253169084776 T^{20} + 739207920425231670 T^{21} + 4420625318710893348 T^{22} + 13583188850144381156 T^{23} + 64480869190579351654 T^{24} + \)\(16\!\cdots\!64\)\( T^{25} + \)\(61\!\cdots\!31\)\( T^{26} + \)\(11\!\cdots\!08\)\( T^{27} + \)\(34\!\cdots\!71\)\( T^{28} + \)\(35\!\cdots\!72\)\( T^{29} + \)\(86\!\cdots\!49\)\( T^{30} \))
$31$ (\( 1 + 12 T + 78 T^{2} + 372 T^{3} + 961 T^{4} \))(\( 1 + 4 T + 46 T^{2} + 124 T^{3} + 961 T^{4} \))(\( 1 + 4 T + 46 T^{2} + 124 T^{3} + 961 T^{4} \))(\( 1 + 12 T + 78 T^{2} + 372 T^{3} + 961 T^{4} \))(\( 1 + 16 T + 230 T^{2} + 2063 T^{3} + 16548 T^{4} + 97141 T^{5} + 512988 T^{6} + 1982543 T^{7} + 6851930 T^{8} + 14776336 T^{9} + 28629151 T^{10} \))(\( 1 + 18 T + 321 T^{2} + 3457 T^{3} + 35295 T^{4} + 269386 T^{5} + 1944797 T^{6} + 11129437 T^{7} + 60288707 T^{8} + 258879946 T^{9} + 1051473345 T^{10} + 3192612097 T^{11} + 9189957471 T^{12} + 15975066258 T^{13} + 27512614111 T^{14} \))(\( 1 - 6 T + 167 T^{2} - 640 T^{3} + 12503 T^{4} - 33462 T^{5} + 660282 T^{6} - 1410885 T^{7} + 28528780 T^{8} - 49501876 T^{9} + 1017551600 T^{10} - 1515235802 T^{11} + 31544099600 T^{12} - 47571302836 T^{13} + 849900884980 T^{14} - 1302981926085 T^{15} + 18903313080582 T^{16} - 29697648173622 T^{17} + 343990214229833 T^{18} - 545850263962240 T^{19} + 4415416900832057 T^{20} - 4917769721884806 T^{21} + 25408476896404831 T^{22} \))(\( 1 - 8 T + 110 T^{2} - 561 T^{3} + 5136 T^{4} - 26697 T^{5} + 222132 T^{6} - 1395873 T^{7} + 10324443 T^{8} - 60423563 T^{9} + 375353534 T^{10} - 1867603862 T^{11} + 11382213016 T^{12} - 57895719722 T^{13} + 360714746174 T^{14} - 1800078365333 T^{15} + 9534839923803 T^{16} - 39962658893823 T^{17} + 197142967667892 T^{18} - 734504258921367 T^{19} + 4380448368296976 T^{20} - 14832628032136431 T^{21} + 90159111567888110 T^{22} - 203267815171238648 T^{23} + 787662783788549761 T^{24} \))(\( 1 + 22 T + 440 T^{2} + 5691 T^{3} + 69881 T^{4} + 688745 T^{5} + 6648869 T^{6} + 55558941 T^{7} + 460139509 T^{8} + 3403260763 T^{9} + 25026167067 T^{10} + 166971287488 T^{11} + 1109762228609 T^{12} + 6751797338530 T^{13} + 40964110917559 T^{14} + 228369831804309 T^{15} + 1269887438444329 T^{16} + 6488477242327330 T^{17} + 33060926552490719 T^{18} + 154201490392205248 T^{19} + 716477915912370117 T^{20} + 3020406454565368603 T^{21} + 12659640748342011499 T^{22} + 47385722828613309981 T^{23} + \)\(17\!\cdots\!99\)\( T^{24} + \)\(56\!\cdots\!45\)\( T^{25} + \)\(17\!\cdots\!11\)\( T^{26} + \)\(44\!\cdots\!51\)\( T^{27} + \)\(10\!\cdots\!40\)\( T^{28} + \)\(16\!\cdots\!62\)\( T^{29} + \)\(23\!\cdots\!51\)\( T^{30} \))
$37$ (\( 1 + 54 T^{2} + 1369 T^{4} \))(\( 1 + 54 T^{2} + 1369 T^{4} \))(\( 1 + 54 T^{2} + 1369 T^{4} \))(\( 1 + 54 T^{2} + 1369 T^{4} \))(\( 1 + 7 T + 136 T^{2} + 703 T^{3} + 8869 T^{4} + 36613 T^{5} + 328153 T^{6} + 962407 T^{7} + 6888808 T^{8} + 13119127 T^{9} + 69343957 T^{10} \))(\( 1 + 8 T + 140 T^{2} + 446 T^{3} + 5906 T^{4} - 5317 T^{5} + 140628 T^{6} - 725991 T^{7} + 5203236 T^{8} - 7278973 T^{9} + 299156618 T^{10} + 835875806 T^{11} + 9708153980 T^{12} + 20525811272 T^{13} + 94931877133 T^{14} \))(\( 1 - 23 T + 452 T^{2} - 6195 T^{3} + 76439 T^{4} - 786870 T^{5} + 7479142 T^{6} - 62919285 T^{7} + 496649263 T^{8} - 3562483986 T^{9} + 24196899884 T^{10} - 151017785580 T^{11} + 895285295708 T^{12} - 4877040576834 T^{13} + 25156775118739 T^{14} - 117920870094885 T^{15} + 518633301244894 T^{16} - 2018893139449830 T^{17} + 7256497756169387 T^{18} - 21759810217040595 T^{19} + 58742706387374804 T^{20} - 110597440565610527 T^{21} + 177917621779460413 T^{22} \))(\( 1 - 8 T + 285 T^{2} - 2328 T^{3} + 41990 T^{4} - 329585 T^{5} + 4122834 T^{6} - 29895592 T^{7} + 295299907 T^{8} - 1931900130 T^{9} + 16051467481 T^{10} - 93483683185 T^{11} + 673998628252 T^{12} - 3458896277845 T^{13} + 21974458981489 T^{14} - 97856537284890 T^{15} + 553439569003027 T^{16} - 2073078646137544 T^{17} + 10578064073723106 T^{18} - 31288122724879805 T^{19} + 147489012270142790 T^{20} - 302550930242939256 T^{21} + 1370446546139086965 T^{22} - 1423340974235683304 T^{23} + 6582952005840035281 T^{24} \))(\( 1 - 12 T + 314 T^{2} - 3412 T^{3} + 50098 T^{4} - 494865 T^{5} + 5446581 T^{6} - 48638912 T^{7} + 448085578 T^{8} - 3616250406 T^{9} + 29279798873 T^{10} - 214642130278 T^{11} + 1561641890826 T^{12} - 10455854594328 T^{13} + 69101820742268 T^{14} - 423482062973499 T^{15} + 2556767367463916 T^{16} - 14314064939635032 T^{17} + 79101846696009378 T^{18} - 402273909523946758 T^{19} + 2030377114017960461 T^{20} - 9278309168231172054 T^{21} + 42537605035765287874 T^{22} - \)\(17\!\cdots\!52\)\( T^{23} + \)\(70\!\cdots\!37\)\( T^{24} - \)\(23\!\cdots\!85\)\( T^{25} + \)\(89\!\cdots\!74\)\( T^{26} - \)\(22\!\cdots\!72\)\( T^{27} + \)\(76\!\cdots\!58\)\( T^{28} - \)\(10\!\cdots\!68\)\( T^{29} + \)\(33\!\cdots\!93\)\( T^{30} \))
$41$ (\( 1 - 3 T + 53 T^{2} - 123 T^{3} + 1681 T^{4} \))(\( 1 - 7 T + 93 T^{2} - 287 T^{3} + 1681 T^{4} \))(\( 1 - 7 T + 93 T^{2} - 287 T^{3} + 1681 T^{4} \))(\( 1 - 3 T + 53 T^{2} - 123 T^{3} + 1681 T^{4} \))(\( 1 - 9 T + 232 T^{2} - 1503 T^{3} + 20127 T^{4} - 92977 T^{5} + 825207 T^{6} - 2526543 T^{7} + 15989672 T^{8} - 25431849 T^{9} + 115856201 T^{10} \))(\( 1 + 15 T + 165 T^{2} + 1716 T^{3} + 15349 T^{4} + 112837 T^{5} + 825907 T^{6} + 5652081 T^{7} + 33862187 T^{8} + 189678997 T^{9} + 1057868429 T^{10} + 4849005876 T^{11} + 19116273165 T^{12} + 71251563615 T^{13} + 194754273881 T^{14} \))(\( 1 - 8 T + 248 T^{2} - 1581 T^{3} + 27398 T^{4} - 146428 T^{5} + 1949997 T^{6} - 9676579 T^{7} + 112075607 T^{8} - 551199103 T^{9} + 5581442406 T^{10} - 25768211940 T^{11} + 228839138646 T^{12} - 926565692143 T^{13} + 7724362910047 T^{14} - 27343699551619 T^{15} + 225919244381397 T^{16} - 695548263801148 T^{17} + 5335877595791638 T^{18} - 12624166787240301 T^{19} + 81190719729702328 T^{20} - 107381274481219208 T^{21} + 550329031716248441 T^{22} \))(\( 1 + T + 230 T^{2} - 257 T^{3} + 24637 T^{4} - 76060 T^{5} + 1796100 T^{6} - 7778661 T^{7} + 108454823 T^{8} - 482085656 T^{9} + 5703268056 T^{10} - 22805439650 T^{11} + 255177914992 T^{12} - 935023025650 T^{13} + 9587193602136 T^{14} - 33225825497176 T^{15} + 306467409095303 T^{16} - 901206112326861 T^{17} + 8531662227260100 T^{18} - 14813010071388860 T^{19} + 196724602869854077 T^{20} - 84137157139247977 T^{21} + 3087211641335052230 T^{22} + 550329031716248441 T^{23} + 22563490300366186081 T^{24} \))(\( 1 + 13 T + 341 T^{2} + 3629 T^{3} + 52543 T^{4} + 466994 T^{5} + 4839600 T^{6} + 36354346 T^{7} + 298170361 T^{8} + 1902608617 T^{9} + 13180820992 T^{10} + 71589476876 T^{11} + 455476595943 T^{12} + 2172102398188 T^{13} + 14836124371336 T^{14} + 73017536046731 T^{15} + 608281099224776 T^{16} + 3651304131354028 T^{17} + 31391902468987503 T^{18} + 202294751766602636 T^{19} + 1527079846194171392 T^{20} + 9037589260574844697 T^{21} + 58069952149390641041 T^{22} + \)\(29\!\cdots\!66\)\( T^{23} + \)\(15\!\cdots\!00\)\( T^{24} + \)\(62\!\cdots\!94\)\( T^{25} + \)\(28\!\cdots\!63\)\( T^{26} + \)\(81\!\cdots\!49\)\( T^{27} + \)\(31\!\cdots\!61\)\( T^{28} + \)\(49\!\cdots\!93\)\( T^{29} + \)\(15\!\cdots\!01\)\( T^{30} \))
$43$ (\( 1 + 4 T + 70 T^{2} + 172 T^{3} + 1849 T^{4} \))(\( 1 + 12 T + 102 T^{2} + 516 T^{3} + 1849 T^{4} \))(\( 1 - 12 T + 102 T^{2} - 516 T^{3} + 1849 T^{4} \))(\( 1 - 4 T + 70 T^{2} - 172 T^{3} + 1849 T^{4} \))(\( 1 - 32 T + 556 T^{2} - 6781 T^{3} + 63014 T^{4} - 461877 T^{5} + 2709602 T^{6} - 12538069 T^{7} + 44205892 T^{8} - 109401632 T^{9} + 147008443 T^{10} \))(\( 1 + 14 T + 160 T^{2} + 1152 T^{3} + 8465 T^{4} + 39218 T^{5} + 236797 T^{6} + 1042279 T^{7} + 10182271 T^{8} + 72514082 T^{9} + 673026755 T^{10} + 3938458752 T^{11} + 23521350880 T^{12} + 88499082686 T^{13} + 271818611107 T^{14} \))(\( 1 - 19 T + 467 T^{2} - 6416 T^{3} + 94986 T^{4} - 1036284 T^{5} + 11574715 T^{6} - 104788953 T^{7} + 953247821 T^{8} - 7321184820 T^{9} + 56082791347 T^{10} - 368610934200 T^{11} + 2411560027921 T^{12} - 13536870732180 T^{13} + 75789874504247 T^{14} - 358252577305353 T^{15} + 1701580830318745 T^{16} - 6550727385869916 T^{17} + 25818962594609502 T^{18} - 74991492981088016 T^{19} + 234710749774505681 T^{20} - 410618163952400731 T^{21} + 929293739471222707 T^{22} \))(\( 1 - 2 T + 279 T^{2} - 920 T^{3} + 38932 T^{4} - 175056 T^{5} + 3671987 T^{6} - 19664643 T^{7} + 264316714 T^{8} - 1510203737 T^{9} + 15303609823 T^{10} - 85368924579 T^{11} + 725763544528 T^{12} - 3670863756897 T^{13} + 28296374562727 T^{14} - 120071768517659 T^{15} + 903646246139914 T^{16} - 2890868549580849 T^{17} + 23211962938208363 T^{18} - 47583478785946992 T^{19} + 455045013207562132 T^{20} - 462385202981895560 T^{21} + 6029603565406305471 T^{22} - 1858587478942445414 T^{23} + 39959630797262576401 T^{24} \))(\( 1 - 25 T + 664 T^{2} - 11070 T^{3} + 178891 T^{4} - 2300836 T^{5} + 28352101 T^{6} - 301544277 T^{7} + 3080085239 T^{8} - 28211205837 T^{9} + 249502562636 T^{10} - 2021210853870 T^{11} + 15897696311733 T^{12} - 115971363872945 T^{13} + 824791787040357 T^{14} - 5473357413730351 T^{15} + 35466046842735351 T^{16} - 214431051801075305 T^{17} + 1263978140656955631 T^{18} - 6910117688421609870 T^{19} + 36678983257628335748 T^{20} - \)\(17\!\cdots\!13\)\( T^{21} + \)\(83\!\cdots\!73\)\( T^{22} - \)\(35\!\cdots\!77\)\( T^{23} + \)\(14\!\cdots\!43\)\( T^{24} - \)\(49\!\cdots\!64\)\( T^{25} + \)\(16\!\cdots\!37\)\( T^{26} - \)\(44\!\cdots\!70\)\( T^{27} + \)\(11\!\cdots\!52\)\( T^{28} - \)\(18\!\cdots\!25\)\( T^{29} + \)\(31\!\cdots\!07\)\( T^{30} \))
$47$ (\( 1 + 9 T + 103 T^{2} + 423 T^{3} + 2209 T^{4} \))(\( 1 - 9 T + 113 T^{2} - 423 T^{3} + 2209 T^{4} \))(\( 1 + 9 T + 113 T^{2} + 423 T^{3} + 2209 T^{4} \))(\( 1 - 9 T + 103 T^{2} - 423 T^{3} + 2209 T^{4} \))(\( 1 - 7 T + 210 T^{2} - 1139 T^{3} + 18579 T^{4} - 76733 T^{5} + 873213 T^{6} - 2516051 T^{7} + 21802830 T^{8} - 34157767 T^{9} + 229345007 T^{10} \))(\( 1 - 10 T + 213 T^{2} - 1823 T^{3} + 23033 T^{4} - 162514 T^{5} + 1573229 T^{6} - 9290969 T^{7} + 73941763 T^{8} - 358993426 T^{9} + 2391355159 T^{10} - 8895658463 T^{11} + 48850486491 T^{12} - 107792153290 T^{13} + 506623120463 T^{14} \))(\( 1 - 35 T + 880 T^{2} - 15376 T^{3} + 224009 T^{4} - 2672909 T^{5} + 28152005 T^{6} - 258452490 T^{7} + 2180451303 T^{8} - 16759642682 T^{9} + 123192609095 T^{10} - 854553467168 T^{11} + 5790052627465 T^{12} - 37022050684538 T^{13} + 226380995631369 T^{14} - 1261165704855690 T^{15} + 6456521783789035 T^{16} - 28811861665822061 T^{17} + 113488138591796167 T^{18} - 366122343711237136 T^{19} + 984834816330434960 T^{20} - 1840969628254051715 T^{21} + 2472159215084012303 T^{22} \))(\( 1 + 34 T + 896 T^{2} + 16727 T^{3} + 267326 T^{4} + 3590905 T^{5} + 43102036 T^{6} + 460340779 T^{7} + 4494844159 T^{8} + 40090808867 T^{9} + 331074847308 T^{10} + 2529205380764 T^{11} + 17993329433412 T^{12} + 118872652895908 T^{13} + 731344337703372 T^{14} + 4162348048998541 T^{15} + 21933405640633279 T^{16} + 105576859182140453 T^{17} + 464606127162309844 T^{18} + 1819235496386189015 T^{19} + 6365376018141921086 T^{20} + 18719695423589983609 T^{21} + 47128822483303723904 T^{22} + 84053413312856418302 T^{23} + \)\(11\!\cdots\!41\)\( T^{24} \))(\( 1 + 16 T + 492 T^{2} + 5699 T^{3} + 99873 T^{4} + 871873 T^{5} + 11279555 T^{6} + 72810887 T^{7} + 783290515 T^{8} + 3220474720 T^{9} + 33378181673 T^{10} + 17288882580 T^{11} + 707170695241 T^{12} - 7584884037603 T^{13} - 4437797603730 T^{14} - 537960786887469 T^{15} - 208576487375310 T^{16} - 16755008839065027 T^{17} + 73420583092006343 T^{18} + 84364231836856980 T^{19} + 7655119309441456711 T^{20} + 34714190468480982880 T^{21} + \)\(39\!\cdots\!45\)\( T^{22} + \)\(17\!\cdots\!07\)\( T^{23} + \)\(12\!\cdots\!85\)\( T^{24} + \)\(45\!\cdots\!77\)\( T^{25} + \)\(24\!\cdots\!19\)\( T^{26} + \)\(66\!\cdots\!59\)\( T^{27} + \)\(26\!\cdots\!84\)\( T^{28} + \)\(41\!\cdots\!04\)\( T^{29} + \)\(12\!\cdots\!43\)\( T^{30} \))
$53$ (\( ( 1 - 4 T + 53 T^{2} )^{2} \))(\( 1 - 12 T + 122 T^{2} - 636 T^{3} + 2809 T^{4} \))(\( 1 + 12 T + 122 T^{2} + 636 T^{3} + 2809 T^{4} \))(\( ( 1 + 4 T + 53 T^{2} )^{2} \))(\( 1 - 32 T + 655 T^{2} - 9013 T^{3} + 95970 T^{4} - 781289 T^{5} + 5086410 T^{6} - 25317517 T^{7} + 97514435 T^{8} - 252495392 T^{9} + 418195493 T^{10} \))(\( 1 + 15 T + 248 T^{2} + 2459 T^{3} + 26359 T^{4} + 226500 T^{5} + 2031262 T^{6} + 14891311 T^{7} + 107656886 T^{8} + 636238500 T^{9} + 3924248843 T^{10} + 19402692779 T^{11} + 103712482264 T^{12} + 332465416935 T^{13} + 1174711139837 T^{14} \))(\( 1 + 14 T + 282 T^{2} + 2766 T^{3} + 33849 T^{4} + 298024 T^{5} + 3100616 T^{6} + 26287378 T^{7} + 243443677 T^{8} + 1835387119 T^{9} + 15238389718 T^{10} + 103865986416 T^{11} + 807634655054 T^{12} + 5155602417271 T^{13} + 36243164300729 T^{14} + 207420056648818 T^{15} + 1296663636723688 T^{16} + 6605511561109096 T^{17} + 39762797372342613 T^{18} + 172210303677824526 T^{19} + 930533332888201506 T^{20} + 2448424585117182686 T^{21} + 9269035929372191597 T^{22} \))(\( 1 + 5 T + 441 T^{2} + 1896 T^{3} + 94214 T^{4} + 351860 T^{5} + 12993530 T^{6} + 42520621 T^{7} + 1294765158 T^{8} + 3740747497 T^{9} + 98490903865 T^{10} + 252621044096 T^{11} + 5872652829352 T^{12} + 13388915337088 T^{13} + 276660948956785 T^{14} + 556911265110869 T^{15} + 10216319878660998 T^{16} + 17781932061761153 T^{17} + 287993291260495370 T^{18} + 413333861663046820 T^{19} + 5865734472415965254 T^{20} + 6256351770056844168 T^{21} + 77125374431191254609 T^{22} + 46345179646860957985 T^{23} + \)\(49\!\cdots\!41\)\( T^{24} \))(\( 1 - 4 T + 464 T^{2} - 1139 T^{3} + 104229 T^{4} - 117368 T^{5} + 15276402 T^{6} + 996280 T^{7} + 1653955482 T^{8} + 1832413844 T^{9} + 141924674600 T^{10} + 273324058399 T^{11} + 10108497538498 T^{12} + 24328445769833 T^{13} + 616167868168061 T^{14} + 1516066725600641 T^{15} + 32656897012907233 T^{16} + 68338604167460897 T^{17} + 1504922788038966746 T^{18} + 2156658289640199919 T^{19} + 59352259263211577800 T^{20} + 40614282176195069876 T^{21} + \)\(19\!\cdots\!34\)\( T^{22} + 62028084363030737080 T^{23} + \)\(50\!\cdots\!66\)\( T^{24} - \)\(20\!\cdots\!32\)\( T^{25} + \)\(96\!\cdots\!13\)\( T^{26} - \)\(55\!\cdots\!99\)\( T^{27} + \)\(12\!\cdots\!72\)\( T^{28} - \)\(55\!\cdots\!76\)\( T^{29} + \)\(73\!\cdots\!57\)\( T^{30} \))
$59$ (\( 1 + 10 T + 138 T^{2} + 590 T^{3} + 3481 T^{4} \))(\( 1 - 2 T + 74 T^{2} - 118 T^{3} + 3481 T^{4} \))(\( 1 - 2 T + 74 T^{2} - 118 T^{3} + 3481 T^{4} \))(\( 1 + 10 T + 138 T^{2} + 590 T^{3} + 3481 T^{4} \))(\( 1 + 8 T + 291 T^{2} + 1707 T^{3} + 33590 T^{4} + 145309 T^{5} + 1981810 T^{6} + 5942067 T^{7} + 59765289 T^{8} + 96938888 T^{9} + 714924299 T^{10} \))(\( 1 + 18 T + 283 T^{2} + 3342 T^{3} + 39787 T^{4} + 377400 T^{5} + 3440044 T^{6} + 26583077 T^{7} + 202962596 T^{8} + 1313729400 T^{9} + 8171414273 T^{10} + 40496220462 T^{11} + 202323576617 T^{12} + 759249605538 T^{13} + 2488651484819 T^{14} \))(\( 1 + 6 T + 348 T^{2} + 1596 T^{3} + 61335 T^{4} + 232670 T^{5} + 7399972 T^{6} + 24581802 T^{7} + 680201535 T^{8} + 2028486741 T^{9} + 49647828250 T^{10} + 133465942558 T^{11} + 2929221866750 T^{12} + 7061162345421 T^{13} + 139699111056765 T^{14} + 297866568864522 T^{15} + 5290419794719628 T^{16} + 9814144762251470 T^{17} + 152641438821373365 T^{18} + 234341378416496316 T^{19} + 3014722544891918772 T^{20} + 3066700519803848406 T^{21} + 30155888444737842659 T^{22} \))(\( 1 - 26 T + 730 T^{2} - 11536 T^{3} + 185282 T^{4} - 2076094 T^{5} + 23928077 T^{6} - 201664335 T^{7} + 1842506089 T^{8} - 12100923663 T^{9} + 98573053649 T^{10} - 562897464460 T^{11} + 5101887162824 T^{12} - 33210950403140 T^{13} + 343132799752169 T^{14} - 2485275600983277 T^{15} + 22326311425111129 T^{16} - 144174733333176165 T^{17} + 1009299056862938357 T^{18} - 5166674415723816986 T^{19} + 27205037140203803522 T^{20} - 99936319764003376304 T^{21} + \)\(37\!\cdots\!30\)\( T^{22} - \)\(78\!\cdots\!34\)\( T^{23} + \)\(17\!\cdots\!81\)\( T^{24} \))(\( 1 + 50 T + 1656 T^{2} + 41091 T^{3} + 846567 T^{4} + 14990210 T^{5} + 235271476 T^{6} + 3323944602 T^{7} + 42862165094 T^{8} + 508599205506 T^{9} + 5594191512530 T^{10} + 57296999928431 T^{11} + 548677398738234 T^{12} + 4924051700103359 T^{13} + 41498895626633429 T^{14} + 328708825551825739 T^{15} + 2448434841971372311 T^{16} + 17140623968059792679 T^{17} + \)\(11\!\cdots\!86\)\( T^{18} + \)\(69\!\cdots\!91\)\( T^{19} + \)\(39\!\cdots\!70\)\( T^{20} + \)\(21\!\cdots\!46\)\( T^{21} + \)\(10\!\cdots\!86\)\( T^{22} + \)\(48\!\cdots\!42\)\( T^{23} + \)\(20\!\cdots\!64\)\( T^{24} + \)\(76\!\cdots\!10\)\( T^{25} + \)\(25\!\cdots\!53\)\( T^{26} + \)\(73\!\cdots\!71\)\( T^{27} + \)\(17\!\cdots\!24\)\( T^{28} + \)\(30\!\cdots\!50\)\( T^{29} + \)\(36\!\cdots\!99\)\( T^{30} \))
$61$ (\( 1 - 3 T + 113 T^{2} - 183 T^{3} + 3721 T^{4} \))(\( 1 - 19 T + 201 T^{2} - 1159 T^{3} + 3721 T^{4} \))(\( 1 - 19 T + 201 T^{2} - 1159 T^{3} + 3721 T^{4} \))(\( 1 - 3 T + 113 T^{2} - 183 T^{3} + 3721 T^{4} \))(\( 1 + 12 T + 226 T^{2} + 1909 T^{3} + 24382 T^{4} + 164073 T^{5} + 1487302 T^{6} + 7103389 T^{7} + 51297706 T^{8} + 166150092 T^{9} + 844596301 T^{10} \))(\( 1 - 4 T + 173 T^{2} + 229 T^{3} + 13815 T^{4} + 53354 T^{5} + 1189845 T^{6} + 3012271 T^{7} + 72580545 T^{8} + 198530234 T^{9} + 3135742515 T^{10} + 3170697589 T^{11} + 146115160073 T^{12} - 206081497444 T^{13} + 3142742836021 T^{14} \))(\( 1 - 9 T + 523 T^{2} - 4167 T^{3} + 130382 T^{4} - 923100 T^{5} + 20475161 T^{6} - 128835705 T^{7} + 2253750195 T^{8} - 12545722717 T^{9} + 182770561509 T^{10} - 890638865574 T^{11} + 11149004252049 T^{12} - 46682634229957 T^{13} + 511558473011295 T^{14} - 1783838686552905 T^{15} + 17293245242979461 T^{16} - 47558457572639100 T^{17} + 409757096446090022 T^{18} - 798844373259669927 T^{19} + 6116038406552255743 T^{20} - 6420086204965943409 T^{21} + 43513917611435838661 T^{22} \))(\( 1 + 26 T + 752 T^{2} + 13491 T^{3} + 235281 T^{4} + 3266857 T^{5} + 43103756 T^{6} + 492402970 T^{7} + 5338943579 T^{8} + 52076341808 T^{9} + 483789691118 T^{10} + 4119791757429 T^{11} + 33509307831364 T^{12} + 251307297203169 T^{13} + 1800181440650078 T^{14} + 11820340139921648 T^{15} + 73922163902804939 T^{16} + 415881727063413970 T^{17} + 2220721645485199916 T^{18} + 10266891433055055997 T^{19} + 45105088309313270961 T^{20} + \)\(15\!\cdots\!31\)\( T^{21} + \)\(53\!\cdots\!52\)\( T^{22} + \)\(11\!\cdots\!86\)\( T^{23} + \)\(26\!\cdots\!21\)\( T^{24} \))(\( 1 + 41 T + 1266 T^{2} + 28143 T^{3} + 537421 T^{4} + 8723591 T^{5} + 127846442 T^{6} + 1684406586 T^{7} + 20552628279 T^{8} + 231552538619 T^{9} + 2451586546111 T^{10} + 24315296310024 T^{11} + 228555938207149 T^{12} + 2028097124596243 T^{13} + 17129570660768330 T^{14} + 137038654279098775 T^{15} + 1044903810306868130 T^{16} + 7546549400622620203 T^{17} + 51877855410196887169 T^{18} + \)\(33\!\cdots\!84\)\( T^{19} + \)\(20\!\cdots\!11\)\( T^{20} + \)\(11\!\cdots\!59\)\( T^{21} + \)\(64\!\cdots\!59\)\( T^{22} + \)\(32\!\cdots\!66\)\( T^{23} + \)\(14\!\cdots\!22\)\( T^{24} + \)\(62\!\cdots\!91\)\( T^{25} + \)\(23\!\cdots\!81\)\( T^{26} + \)\(74\!\cdots\!03\)\( T^{27} + \)\(20\!\cdots\!46\)\( T^{28} + \)\(40\!\cdots\!81\)\( T^{29} + \)\(60\!\cdots\!01\)\( T^{30} \))
$67$ (\( 1 + T - 17 T^{2} + 67 T^{3} + 4489 T^{4} \))(\( 1 + 3 T + 105 T^{2} + 201 T^{3} + 4489 T^{4} \))(\( 1 - 3 T + 105 T^{2} - 201 T^{3} + 4489 T^{4} \))(\( 1 - T - 17 T^{2} - 67 T^{3} + 4489 T^{4} \))(\( 1 - 5 T + 152 T^{2} - 454 T^{3} + 14354 T^{4} - 39039 T^{5} + 961718 T^{6} - 2038006 T^{7} + 45715976 T^{8} - 100755605 T^{9} + 1350125107 T^{10} \))(\( 1 + 18 T + 312 T^{2} + 3657 T^{3} + 46177 T^{4} + 442394 T^{5} + 4388546 T^{6} + 34987571 T^{7} + 294032582 T^{8} + 1985906666 T^{9} + 13888333051 T^{10} + 73692649497 T^{11} + 421239033384 T^{12} + 1628250879042 T^{13} + 6060711605323 T^{14} \))(\( 1 - 54 T + 1793 T^{2} - 43376 T^{3} + 848860 T^{4} - 14000247 T^{5} + 200942236 T^{6} - 2552265686 T^{7} + 29093829595 T^{8} - 299932159848 T^{9} + 2814333707634 T^{10} - 24095922393102 T^{11} + 188560358411478 T^{12} - 1346395465557672 T^{13} + 8750347470480985 T^{14} - 51431014662734006 T^{15} + 271297157880319252 T^{16} - 1266439693586395743 T^{17} + 5144695653294481780 T^{18} - 17613591581696860016 T^{19} + 48781316172556839971 T^{20} - 98433241445795118246 T^{21} + \)\(12\!\cdots\!83\)\( T^{22} \))(\( 1 + 6 T + 375 T^{2} + 1475 T^{3} + 71151 T^{4} + 178925 T^{5} + 9349095 T^{6} + 14634395 T^{7} + 969096135 T^{8} + 986114059 T^{9} + 83407729450 T^{10} + 63325311176 T^{11} + 6068485120114 T^{12} + 4242795848792 T^{13} + 374417297501050 T^{14} + 296586622727017 T^{15} + 19528373477017335 T^{16} + 19758264115255265 T^{17} + 845704008444287055 T^{18} + 1084412823982417775 T^{19} + 28892121325832563791 T^{20} + 40129638234535046825 T^{21} + \)\(68\!\cdots\!75\)\( T^{22} + \)\(73\!\cdots\!98\)\( T^{23} + \)\(81\!\cdots\!61\)\( T^{24} \))(\( 1 - 43 T + 1412 T^{2} - 34612 T^{3} + 725254 T^{4} - 13148340 T^{5} + 213596249 T^{6} - 3140878835 T^{7} + 42386968766 T^{8} - 528391319525 T^{9} + 6126124618550 T^{10} - 66308482555692 T^{11} + 672588399620473 T^{12} - 6405417740114407 T^{13} + 57392366612773961 T^{14} - 484028819246150401 T^{15} + 3845288563055855387 T^{16} - 28753920235373573023 T^{17} + \)\(20\!\cdots\!99\)\( T^{18} - \)\(13\!\cdots\!32\)\( T^{19} + \)\(82\!\cdots\!50\)\( T^{20} - \)\(47\!\cdots\!25\)\( T^{21} + \)\(25\!\cdots\!18\)\( T^{22} - \)\(12\!\cdots\!35\)\( T^{23} + \)\(58\!\cdots\!03\)\( T^{24} - \)\(23\!\cdots\!60\)\( T^{25} + \)\(88\!\cdots\!82\)\( T^{26} - \)\(28\!\cdots\!32\)\( T^{27} + \)\(77\!\cdots\!44\)\( T^{28} - \)\(15\!\cdots\!47\)\( T^{29} + \)\(24\!\cdots\!43\)\( T^{30} \))
$71$ (\( 1 + 5 T - 3 T^{2} + 355 T^{3} + 5041 T^{4} \))(\( 1 + 5 T + 47 T^{2} + 355 T^{3} + 5041 T^{4} \))(\( 1 + 5 T + 47 T^{2} + 355 T^{3} + 5041 T^{4} \))(\( 1 + 5 T - 3 T^{2} + 355 T^{3} + 5041 T^{4} \))(\( 1 + 11 T + 293 T^{2} + 2666 T^{3} + 38033 T^{4} + 267381 T^{5} + 2700343 T^{6} + 13439306 T^{7} + 104867923 T^{8} + 279528491 T^{9} + 1804229351 T^{10} \))(\( 1 + 50 T + 1452 T^{2} + 29886 T^{3} + 479876 T^{4} + 6256313 T^{5} + 67998420 T^{6} + 622620957 T^{7} + 4827887820 T^{8} + 31538073833 T^{9} + 171752899036 T^{10} + 759453498366 T^{11} + 2619741017652 T^{12} + 6405014196050 T^{13} + 9095120158391 T^{14} \))(\( 1 + 5 T + 369 T^{2} + 2726 T^{3} + 77887 T^{4} + 632327 T^{5} + 11819152 T^{6} + 93566238 T^{7} + 1360132324 T^{8} + 10053683216 T^{9} + 121983570726 T^{10} + 816591365980 T^{11} + 8660833521546 T^{12} + 50680617091856 T^{13} + 486806320215164 T^{14} + 2377675392426078 T^{15} + 21324460942330352 T^{16} + 81001268230914167 T^{17} + 708391623776599817 T^{18} + 1760324126175944486 T^{19} + 16918096765107692439 T^{20} + 16276217755049406005 T^{21} + \)\(23\!\cdots\!71\)\( T^{22} \))(\( 1 - 94 T + 4589 T^{2} - 153184 T^{3} + 3904973 T^{4} - 80682013 T^{5} + 1402386991 T^{6} - 21038436416 T^{7} + 277565120451 T^{8} - 3266755261035 T^{9} + 34672343703145 T^{10} - 334475660641729 T^{11} + 2946747864969272 T^{12} - 23747771905562759 T^{13} + 174783284607553945 T^{14} - 1169207642232297885 T^{15} + 7053396297627388131 T^{16} - 37958164480894446016 T^{17} + \)\(17\!\cdots\!11\)\( T^{18} - \)\(73\!\cdots\!83\)\( T^{19} + \)\(25\!\cdots\!53\)\( T^{20} - \)\(70\!\cdots\!04\)\( T^{21} + \)\(14\!\cdots\!89\)\( T^{22} - \)\(21\!\cdots\!74\)\( T^{23} + \)\(16\!\cdots\!41\)\( T^{24} \))(\( 1 + 14 T + 714 T^{2} + 9134 T^{3} + 256359 T^{4} + 2978414 T^{5} + 60611852 T^{6} + 640396335 T^{7} + 10494156375 T^{8} + 101038970433 T^{9} + 1405479787448 T^{10} + 12341749470266 T^{11} + 150201975448272 T^{12} + 1201578068635484 T^{13} + 13032406954284030 T^{14} + 94635638066606051 T^{15} + 925300893754166130 T^{16} + 6057155043991474844 T^{17} + 53758939234666479792 T^{18} + \)\(31\!\cdots\!46\)\( T^{19} + \)\(25\!\cdots\!48\)\( T^{20} + \)\(12\!\cdots\!93\)\( T^{21} + \)\(95\!\cdots\!25\)\( T^{22} + \)\(41\!\cdots\!35\)\( T^{23} + \)\(27\!\cdots\!12\)\( T^{24} + \)\(96\!\cdots\!14\)\( T^{25} + \)\(59\!\cdots\!89\)\( T^{26} + \)\(14\!\cdots\!94\)\( T^{27} + \)\(83\!\cdots\!54\)\( T^{28} + \)\(11\!\cdots\!34\)\( T^{29} + \)\(58\!\cdots\!51\)\( T^{30} \))
$73$ (\( 1 + 9 T + 135 T^{2} + 657 T^{3} + 5329 T^{4} \))(\( 1 - 13 T + 187 T^{2} - 949 T^{3} + 5329 T^{4} \))(\( 1 + 13 T + 187 T^{2} + 949 T^{3} + 5329 T^{4} \))(\( 1 - 9 T + 135 T^{2} - 657 T^{3} + 5329 T^{4} \))(\( 1 - 29 T + 676 T^{2} - 9972 T^{3} + 124570 T^{4} - 1149081 T^{5} + 9093610 T^{6} - 53140788 T^{7} + 262975492 T^{8} - 823548989 T^{9} + 2073071593 T^{10} \))(\( 1 + 133 T^{2} - 1068 T^{3} + 10948 T^{4} - 119175 T^{5} + 1571649 T^{6} - 6028685 T^{7} + 114730377 T^{8} - 635083575 T^{9} + 4258958116 T^{10} - 30329321388 T^{11} + 275718521869 T^{12} + 11047398519097 T^{14} \))(\( 1 + 17 T + 578 T^{2} + 8389 T^{3} + 169543 T^{4} + 2083597 T^{5} + 31678731 T^{6} + 336054025 T^{7} + 4171190164 T^{8} + 38460101207 T^{9} + 405026306682 T^{10} + 3250225364632 T^{11} + 29566920387786 T^{12} + 204953879332103 T^{13} + 1622663884028788 T^{14} + 9543343190970025 T^{15} + 65672277338388483 T^{16} + 315319539893081533 T^{17} + 1873009087123262671 T^{18} + 6765393710899445509 T^{19} + 34027777117378853714 T^{20} + 73059639104960480033 T^{21} + \)\(31\!\cdots\!77\)\( T^{22} \))(\( 1 - 22 T + 668 T^{2} - 9806 T^{3} + 171777 T^{4} - 1817947 T^{5} + 22903099 T^{6} - 169880788 T^{7} + 1651008275 T^{8} - 6379023100 T^{9} + 53269643441 T^{10} + 183530814455 T^{11} + 710939601926 T^{12} + 13397749455215 T^{13} + 283873929897089 T^{14} - 2481548429292700 T^{15} + 46885730886444275 T^{16} - 352175035799255284 T^{17} + 3466022766785369611 T^{18} - 20083584995596833859 T^{19} + \)\(13\!\cdots\!37\)\( T^{20} - \)\(57\!\cdots\!78\)\( T^{21} + \)\(28\!\cdots\!32\)\( T^{22} - \)\(69\!\cdots\!94\)\( T^{23} + \)\(22\!\cdots\!21\)\( T^{24} \))(\( 1 - 10 T + 494 T^{2} - 3751 T^{3} + 122454 T^{4} - 787530 T^{5} + 21586985 T^{6} - 125494338 T^{7} + 3021032393 T^{8} - 16172684699 T^{9} + 348789318111 T^{10} - 1732253844491 T^{11} + 34136454539357 T^{12} - 158366641271012 T^{13} + 2879386341403827 T^{14} - 12464863590107445 T^{15} + 210195202922479371 T^{16} - 843935831333222948 T^{17} + 13279661135537042069 T^{18} - 49192962149031940331 T^{19} + \)\(72\!\cdots\!23\)\( T^{20} - \)\(24\!\cdots\!11\)\( T^{21} + \)\(33\!\cdots\!21\)\( T^{22} - \)\(10\!\cdots\!78\)\( T^{23} + \)\(12\!\cdots\!05\)\( T^{24} - \)\(33\!\cdots\!70\)\( T^{25} + \)\(38\!\cdots\!58\)\( T^{26} - \)\(85\!\cdots\!71\)\( T^{27} + \)\(82\!\cdots\!02\)\( T^{28} - \)\(12\!\cdots\!90\)\( T^{29} + \)\(89\!\cdots\!57\)\( T^{30} \))
$79$ (\( 1 + 2 T - 86 T^{2} + 158 T^{3} + 6241 T^{4} \))(\( 1 - 10 T + 58 T^{2} - 790 T^{3} + 6241 T^{4} \))(\( 1 - 10 T + 58 T^{2} - 790 T^{3} + 6241 T^{4} \))(\( 1 + 2 T - 86 T^{2} + 158 T^{3} + 6241 T^{4} \))(\( 1 - 16 T + 424 T^{2} - 4911 T^{3} + 69310 T^{4} - 576225 T^{5} + 5475490 T^{6} - 30649551 T^{7} + 209048536 T^{8} - 623201296 T^{9} + 3077056399 T^{10} \))(\( 1 + 15 T + 468 T^{2} + 5553 T^{3} + 98197 T^{4} + 946068 T^{5} + 12199714 T^{6} + 95010077 T^{7} + 963777406 T^{8} + 5904410388 T^{9} + 48414950683 T^{10} + 216289799793 T^{11} + 1440062394732 T^{12} + 3646311832815 T^{13} + 19203908986159 T^{14} \))(\( 1 + 16 T + 643 T^{2} + 8618 T^{3} + 185961 T^{4} + 2117687 T^{5} + 32353783 T^{6} + 320137040 T^{7} + 3904045847 T^{8} + 34623954136 T^{9} + 365890566412 T^{10} + 2989968768694 T^{11} + 28905354746548 T^{12} + 216088097762776 T^{13} + 1924846860359033 T^{14} + 12469363639100240 T^{15} + 99554415012007417 T^{16} + 514783144419899927 T^{17} + 3571178118975113799 T^{18} + 13074443723774742698 T^{19} + 77064576216823579117 T^{20} + \)\(15\!\cdots\!16\)\( T^{21} + \)\(74\!\cdots\!79\)\( T^{22} \))(\( 1 - 9 T + 367 T^{2} - 2038 T^{3} + 62223 T^{4} - 141043 T^{5} + 6475151 T^{6} + 14731329 T^{7} + 423885699 T^{8} + 4946425456 T^{9} + 15523331202 T^{10} + 639046178665 T^{11} + 470674369434 T^{12} + 50484648114535 T^{13} + 96881110031682 T^{14} + 2438780660400784 T^{15} + 16510382310791619 T^{16} + 45329130165224271 T^{17} + 1574027980704258671 T^{18} - 2708576935134823837 T^{19} + 94399061478815945103 T^{20} - \)\(24\!\cdots\!22\)\( T^{21} + \)\(34\!\cdots\!67\)\( T^{22} - \)\(67\!\cdots\!11\)\( T^{23} + \)\(59\!\cdots\!41\)\( T^{24} \))(\( 1 + 44 T + 1760 T^{2} + 47139 T^{3} + 1146715 T^{4} + 22849120 T^{5} + 419964256 T^{6} + 6763660759 T^{7} + 101700966064 T^{8} + 1382466434370 T^{9} + 17690936696558 T^{10} + 208066457594283 T^{11} + 2315894010741594 T^{12} + 23901530554084313 T^{13} + 234140016077896982 T^{14} + 2135073976875200373 T^{15} + 18497061270153861578 T^{16} + \)\(14\!\cdots\!33\)\( T^{17} + \)\(11\!\cdots\!66\)\( T^{18} + \)\(81\!\cdots\!23\)\( T^{19} + \)\(54\!\cdots\!42\)\( T^{20} + \)\(33\!\cdots\!70\)\( T^{21} + \)\(19\!\cdots\!76\)\( T^{22} + \)\(10\!\cdots\!99\)\( T^{23} + \)\(50\!\cdots\!64\)\( T^{24} + \)\(21\!\cdots\!20\)\( T^{25} + \)\(85\!\cdots\!85\)\( T^{26} + \)\(27\!\cdots\!99\)\( T^{27} + \)\(82\!\cdots\!40\)\( T^{28} + \)\(16\!\cdots\!64\)\( T^{29} + \)\(29\!\cdots\!99\)\( T^{30} \))
$83$ (\( 1 - 29 T + 375 T^{2} - 2407 T^{3} + 6889 T^{4} \))(\( 1 + 13 T + 177 T^{2} + 1079 T^{3} + 6889 T^{4} \))(\( 1 - 13 T + 177 T^{2} - 1079 T^{3} + 6889 T^{4} \))(\( 1 + 29 T + 375 T^{2} + 2407 T^{3} + 6889 T^{4} \))(\( 1 - 10 T + 298 T^{2} - 2663 T^{3} + 41844 T^{4} - 313927 T^{5} + 3473052 T^{6} - 18345407 T^{7} + 170392526 T^{8} - 474583210 T^{9} + 3939040643 T^{10} \))(\( 1 - 24 T + 705 T^{2} - 11938 T^{3} + 194891 T^{4} - 2473798 T^{5} + 28249166 T^{6} - 273618813 T^{7} + 2344680778 T^{8} - 17041994422 T^{9} + 111436140217 T^{10} - 566557436098 T^{11} + 2777023653315 T^{12} - 7846568960856 T^{13} + 27136050989627 T^{14} \))(\( 1 - 29 T + 828 T^{2} - 15713 T^{3} + 281415 T^{4} - 4136872 T^{5} + 57745871 T^{6} - 709662065 T^{7} + 8324128788 T^{8} - 88511841411 T^{9} + 899942775938 T^{10} - 8382930168216 T^{11} + 74695250402854 T^{12} - 609758075480379 T^{13} + 4759628627304156 T^{14} - 33679370082292865 T^{15} + 227463332834435053 T^{16} - 1352510476259761768 T^{17} + 7636491789245882205 T^{18} - 35390267843600751233 T^{19} + \)\(15\!\cdots\!84\)\( T^{20} - \)\(44\!\cdots\!21\)\( T^{21} + \)\(12\!\cdots\!67\)\( T^{22} \))(\( 1 - 8 T + 448 T^{2} - 2918 T^{3} + 100176 T^{4} - 424192 T^{5} + 13340059 T^{6} - 13375319 T^{7} + 1118067097 T^{8} + 5037020331 T^{9} + 59946946589 T^{10} + 927626542684 T^{11} + 3267800357708 T^{12} + 76993003042772 T^{13} + 412974515051621 T^{14} + 2880102744001497 T^{15} + 53061587188964137 T^{16} - 52685925154090117 T^{17} + 4361403870224488771 T^{18} - 11510895741391856384 T^{19} + \)\(22\!\cdots\!16\)\( T^{20} - \)\(54\!\cdots\!54\)\( T^{21} + \)\(69\!\cdots\!52\)\( T^{22} - \)\(10\!\cdots\!36\)\( T^{23} + \)\(10\!\cdots\!61\)\( T^{24} \))(\( 1 + 7 T + 555 T^{2} + 4899 T^{3} + 155129 T^{4} + 1499352 T^{5} + 29597218 T^{6} + 276408647 T^{7} + 4274594676 T^{8} + 35670610542 T^{9} + 487071844767 T^{10} + 3584592585017 T^{11} + 45880931425585 T^{12} + 310684877413560 T^{13} + 3884329781286276 T^{14} + 25729978545945169 T^{15} + 322399371846760908 T^{16} + 2140308120502014840 T^{17} + 26234120137040970395 T^{18} + \)\(17\!\cdots\!57\)\( T^{19} + \)\(19\!\cdots\!81\)\( T^{20} + \)\(11\!\cdots\!98\)\( T^{21} + \)\(11\!\cdots\!52\)\( T^{22} + \)\(62\!\cdots\!27\)\( T^{23} + \)\(55\!\cdots\!54\)\( T^{24} + \)\(23\!\cdots\!48\)\( T^{25} + \)\(19\!\cdots\!43\)\( T^{26} + \)\(52\!\cdots\!39\)\( T^{27} + \)\(49\!\cdots\!65\)\( T^{28} + \)\(51\!\cdots\!03\)\( T^{29} + \)\(61\!\cdots\!07\)\( T^{30} \))
$89$ (\( 1 - 24 T + 302 T^{2} - 2136 T^{3} + 7921 T^{4} \))(\( 1 + 12 T + 134 T^{2} + 1068 T^{3} + 7921 T^{4} \))(\( 1 + 12 T + 134 T^{2} + 1068 T^{3} + 7921 T^{4} \))(\( 1 - 24 T + 302 T^{2} - 2136 T^{3} + 7921 T^{4} \))(\( 1 - 9 T + 152 T^{2} - 1354 T^{3} + 16110 T^{4} - 77787 T^{5} + 1433790 T^{6} - 10725034 T^{7} + 107155288 T^{8} - 564680169 T^{9} + 5584059449 T^{10} \))(\( 1 + 13 T + 260 T^{2} + 2275 T^{3} + 16962 T^{4} + 2186 T^{5} - 807494 T^{6} - 17411725 T^{7} - 71866966 T^{8} + 17315306 T^{9} + 11957684178 T^{10} + 142738598275 T^{11} + 1451855456740 T^{12} + 6460756782493 T^{13} + 44231334895529 T^{14} \))(\( 1 + 5 T + 195 T^{2} + 1230 T^{3} + 33329 T^{4} + 250880 T^{5} + 4285411 T^{6} + 36248750 T^{7} + 498318379 T^{8} + 4149499905 T^{9} + 49565915508 T^{10} + 406334801638 T^{11} + 4411366480212 T^{12} + 32868188747505 T^{13} + 351299009325251 T^{14} + 2274327808448750 T^{15} + 23929989787398539 T^{16} + 124682666276295680 T^{17} + 1474186160733086041 T^{18} + 4842004231013559630 T^{19} + 68319498722959615755 T^{20} + \)\(15\!\cdots\!05\)\( T^{21} + \)\(27\!\cdots\!89\)\( T^{22} \))(\( 1 + 3 T + 589 T^{2} + 1274 T^{3} + 175478 T^{4} + 280583 T^{5} + 35372418 T^{6} + 43704857 T^{7} + 5387584795 T^{8} + 5415512026 T^{9} + 651937772393 T^{10} + 561070696873 T^{11} + 64130514360812 T^{12} + 49935292021697 T^{13} + 5163999095124953 T^{14} + 3817768097457194 T^{15} + 338029143615825595 T^{16} + 244050519698043793 T^{17} + 17579429962052113698 T^{18} + 12410560638992213407 T^{19} + \)\(69\!\cdots\!18\)\( T^{20} + \)\(44\!\cdots\!66\)\( T^{21} + \)\(18\!\cdots\!89\)\( T^{22} + \)\(83\!\cdots\!67\)\( T^{23} + \)\(24\!\cdots\!21\)\( T^{24} \))(\( 1 - 4 T + 765 T^{2} - 3907 T^{3} + 286678 T^{4} - 1784983 T^{5} + 70653364 T^{6} - 512993615 T^{7} + 12961873059 T^{8} - 104848697801 T^{9} + 1895706212342 T^{10} - 16279343143646 T^{11} + 230359494603497 T^{12} - 1995366970479321 T^{13} + 23824090802589747 T^{14} - 197140730599035623 T^{15} + 2120344081430487483 T^{16} - 15805301773166701641 T^{17} + \)\(16\!\cdots\!93\)\( T^{18} - \)\(10\!\cdots\!86\)\( T^{19} + \)\(10\!\cdots\!58\)\( T^{20} - \)\(52\!\cdots\!61\)\( T^{21} + \)\(57\!\cdots\!11\)\( T^{22} - \)\(20\!\cdots\!15\)\( T^{23} + \)\(24\!\cdots\!76\)\( T^{24} - \)\(55\!\cdots\!83\)\( T^{25} + \)\(79\!\cdots\!42\)\( T^{26} - \)\(96\!\cdots\!47\)\( T^{27} + \)\(16\!\cdots\!85\)\( T^{28} - \)\(78\!\cdots\!64\)\( T^{29} + \)\(17\!\cdots\!49\)\( T^{30} \))
$97$ (\( 1 - 30 T + 414 T^{2} - 2910 T^{3} + 9409 T^{4} \))(\( 1 + 10 T + 214 T^{2} + 970 T^{3} + 9409 T^{4} \))(\( 1 - 10 T + 214 T^{2} - 970 T^{3} + 9409 T^{4} \))(\( 1 + 30 T + 414 T^{2} + 2910 T^{3} + 9409 T^{4} \))(\( 1 - 43 T + 1116 T^{2} - 20334 T^{3} + 284864 T^{4} - 3137841 T^{5} + 27631808 T^{6} - 191322606 T^{7} + 1018543068 T^{8} - 3806759083 T^{9} + 8587340257 T^{10} \))(\( 1 + T + 396 T^{2} - 357 T^{3} + 75395 T^{4} - 180194 T^{5} + 9562600 T^{6} - 26454385 T^{7} + 927572200 T^{8} - 1695445346 T^{9} + 68810980835 T^{10} - 31604953317 T^{11} + 3400586741772 T^{12} + 832972004929 T^{13} + 80798284478113 T^{14} \))(\( 1 + 6 T + 496 T^{2} + 4042 T^{3} + 136231 T^{4} + 1232695 T^{5} + 26746490 T^{6} + 239578498 T^{7} + 4080758420 T^{8} + 33830250458 T^{9} + 495002175583 T^{10} + 3696666740452 T^{11} + 48015211031551 T^{12} + 318308826559322 T^{13} + 3724398029456660 T^{14} + 21209712170999938 T^{15} + 229681210310447930 T^{16} + 1026800425615953655 T^{17} + 11007231092737812103 T^{18} + 31678906588471676362 T^{19} + \)\(37\!\cdots\!32\)\( T^{20} + \)\(44\!\cdots\!94\)\( T^{21} + \)\(71\!\cdots\!53\)\( T^{22} \))(\( 1 - 29 T + 1249 T^{2} - 25366 T^{3} + 632678 T^{4} - 10020564 T^{5} + 185253972 T^{6} - 2425870975 T^{7} + 36546622034 T^{8} - 409964738669 T^{9} + 5276102462169 T^{10} - 51721589331614 T^{11} + 581746423923696 T^{12} - 5016994165166558 T^{13} + 49642848066548121 T^{14} - 374163747935252237 T^{15} + 3235446171648777554 T^{16} - 20831779481905340575 T^{17} + \)\(15\!\cdots\!88\)\( T^{18} - \)\(80\!\cdots\!32\)\( T^{19} + \)\(49\!\cdots\!58\)\( T^{20} - \)\(19\!\cdots\!22\)\( T^{21} + \)\(92\!\cdots\!01\)\( T^{22} - \)\(20\!\cdots\!37\)\( T^{23} + \)\(69\!\cdots\!41\)\( T^{24} \))(\( 1 + 9 T + 911 T^{2} + 8593 T^{3} + 418332 T^{4} + 4024990 T^{5} + 127456591 T^{6} + 1224782348 T^{7} + 28721075133 T^{8} + 270358448786 T^{9} + 5062473091192 T^{10} + 45799947473449 T^{11} + 720603157804507 T^{12} + 6142121503270985 T^{13} + 84329245944514849 T^{14} + 662618094351958005 T^{15} + 8179936856617940353 T^{16} + 57791221224276697865 T^{17} + \)\(65\!\cdots\!11\)\( T^{18} + \)\(40\!\cdots\!69\)\( T^{19} + \)\(43\!\cdots\!44\)\( T^{20} + \)\(22\!\cdots\!94\)\( T^{21} + \)\(23\!\cdots\!29\)\( T^{22} + \)\(95\!\cdots\!28\)\( T^{23} + \)\(96\!\cdots\!47\)\( T^{24} + \)\(29\!\cdots\!10\)\( T^{25} + \)\(29\!\cdots\!96\)\( T^{26} + \)\(59\!\cdots\!13\)\( T^{27} + \)\(61\!\cdots\!47\)\( T^{28} + \)\(58\!\cdots\!21\)\( T^{29} + \)\(63\!\cdots\!93\)\( T^{30} \))
show more
show less