Properties

Label 1342.2.a
Level 1342
Weight 2
Character orbit a
Rep. character \(\chi_{1342}(1,\cdot)\)
Character field \(\Q\)
Dimension 49
Newform subspaces 14
Sturm bound 372
Trace bound 5

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Defining parameters

Level: \( N \) = \( 1342 = 2 \cdot 11 \cdot 61 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1342.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 14 \)
Sturm bound: \(372\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1342))\).

Total New Old
Modular forms 190 49 141
Cusp forms 183 49 134
Eisenstein series 7 0 7

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

\(2\)\(11\)\(61\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(6\)
\(+\)\(+\)\(-\)\(-\)\(6\)
\(+\)\(-\)\(+\)\(-\)\(9\)
\(+\)\(-\)\(-\)\(+\)\(4\)
\(-\)\(+\)\(+\)\(-\)\(6\)
\(-\)\(+\)\(-\)\(+\)\(6\)
\(-\)\(-\)\(+\)\(+\)\(4\)
\(-\)\(-\)\(-\)\(-\)\(8\)
Plus space\(+\)\(20\)
Minus space\(-\)\(29\)

Trace form

\( 49q - q^{2} + 49q^{4} - 2q^{5} - 4q^{6} - 8q^{7} - q^{8} + 41q^{9} + O(q^{10}) \) \( 49q - q^{2} + 49q^{4} - 2q^{5} - 4q^{6} - 8q^{7} - q^{8} + 41q^{9} - 6q^{10} + q^{11} - 14q^{13} - 8q^{14} + 8q^{15} + 49q^{16} + 14q^{17} - 13q^{18} - 12q^{19} - 2q^{20} - 8q^{21} - q^{22} + 4q^{23} - 4q^{24} + 47q^{25} - 2q^{26} + 24q^{27} - 8q^{28} + 10q^{29} - 4q^{31} - q^{32} + 4q^{33} - 10q^{34} + 32q^{35} + 41q^{36} - 6q^{37} - 6q^{40} + 14q^{41} + 24q^{42} - 28q^{43} + q^{44} - 2q^{45} - 16q^{46} - 12q^{47} + 17q^{49} - 31q^{50} - 16q^{51} - 14q^{52} - 14q^{53} + 8q^{54} + 6q^{55} - 8q^{56} - 32q^{57} - 22q^{58} + 12q^{59} + 8q^{60} - q^{61} + 24q^{62} + 49q^{64} - 12q^{65} - 4q^{66} - 36q^{67} + 14q^{68} + 80q^{69} - 16q^{70} - 4q^{71} - 13q^{72} - 10q^{73} - 14q^{74} + 8q^{75} - 12q^{76} - 8q^{77} + 8q^{78} - 32q^{79} - 2q^{80} + 25q^{81} - 10q^{82} + 52q^{83} - 8q^{84} - 36q^{85} - 20q^{86} - 8q^{87} - q^{88} + 42q^{89} - 6q^{90} - 56q^{91} + 4q^{92} + 24q^{93} + 16q^{94} + 24q^{95} - 4q^{96} - 46q^{97} + 7q^{98} + 13q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1342))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 11 61
1342.2.a.a \(1\) \(10.716\) \(\Q\) None \(-1\) \(1\) \(-2\) \(4\) \(+\) \(+\) \(+\) \(q-q^{2}+q^{3}+q^{4}-2q^{5}-q^{6}+4q^{7}+\cdots\)
1342.2.a.b \(1\) \(10.716\) \(\Q\) None \(1\) \(-1\) \(-4\) \(-2\) \(-\) \(-\) \(-\) \(q+q^{2}-q^{3}+q^{4}-4q^{5}-q^{6}-2q^{7}+\cdots\)
1342.2.a.c \(1\) \(10.716\) \(\Q\) None \(1\) \(-1\) \(0\) \(-2\) \(-\) \(-\) \(+\) \(q+q^{2}-q^{3}+q^{4}-q^{6}-2q^{7}+q^{8}+\cdots\)
1342.2.a.d \(1\) \(10.716\) \(\Q\) None \(1\) \(3\) \(0\) \(2\) \(-\) \(-\) \(-\) \(q+q^{2}+3q^{3}+q^{4}+3q^{6}+2q^{7}+q^{8}+\cdots\)
1342.2.a.e \(2\) \(10.716\) \(\Q(\sqrt{2}) \) None \(-2\) \(2\) \(0\) \(0\) \(+\) \(+\) \(-\) \(q-q^{2}+(1+\beta )q^{3}+q^{4}+\beta q^{5}+(-1+\cdots)q^{6}+\cdots\)
1342.2.a.f \(2\) \(10.716\) \(\Q(\sqrt{5}) \) None \(2\) \(3\) \(2\) \(1\) \(-\) \(-\) \(-\) \(q+q^{2}+(1+\beta )q^{3}+q^{4}+2\beta q^{5}+(1+\cdots)q^{6}+\cdots\)
1342.2.a.g \(3\) \(10.716\) 3.3.148.1 None \(3\) \(-3\) \(-4\) \(-4\) \(-\) \(-\) \(+\) \(q+q^{2}+(-1-\beta _{2})q^{3}+q^{4}+(-1-\beta _{1}+\cdots)q^{5}+\cdots\)
1342.2.a.h \(4\) \(10.716\) 4.4.14656.1 None \(-4\) \(-2\) \(-2\) \(-2\) \(+\) \(-\) \(-\) \(q-q^{2}+(-1+\beta _{1})q^{3}+q^{4}+(-1-\beta _{2}+\cdots)q^{5}+\cdots\)
1342.2.a.i \(4\) \(10.716\) 4.4.48396.1 None \(-4\) \(-2\) \(0\) \(3\) \(+\) \(+\) \(-\) \(q-q^{2}+(-1+\beta _{1})q^{3}+q^{4}+(-\beta _{2}+\cdots)q^{5}+\cdots\)
1342.2.a.j \(4\) \(10.716\) 4.4.2225.1 None \(4\) \(-2\) \(4\) \(1\) \(-\) \(-\) \(-\) \(q+q^{2}+(-1-\beta _{2})q^{3}+q^{4}+(1+\beta _{3})q^{5}+\cdots\)
1342.2.a.k \(5\) \(10.716\) 5.5.3362852.1 None \(-5\) \(-2\) \(0\) \(-3\) \(+\) \(+\) \(+\) \(q-q^{2}-\beta _{1}q^{3}+q^{4}+\beta _{3}q^{5}+\beta _{1}q^{6}+\cdots\)
1342.2.a.l \(6\) \(10.716\) 6.6.8248384.1 None \(6\) \(-3\) \(-4\) \(-9\) \(-\) \(+\) \(-\) \(q+q^{2}+(-1-\beta _{3})q^{3}+q^{4}+(\beta _{3}-\beta _{5})q^{5}+\cdots\)
1342.2.a.m \(6\) \(10.716\) 6.6.136632976.1 None \(6\) \(2\) \(2\) \(5\) \(-\) \(+\) \(+\) \(q+q^{2}+\beta _{1}q^{3}+q^{4}-\beta _{2}q^{5}+\beta _{1}q^{6}+\cdots\)
1342.2.a.n \(9\) \(10.716\) \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(-9\) \(5\) \(6\) \(-2\) \(+\) \(-\) \(+\) \(q-q^{2}+(1-\beta _{1})q^{3}+q^{4}+(1-\beta _{5})q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1342))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1342)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(61))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(122))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(671))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{3} \))(\( ( 1 + T )^{4} \))(\( ( 1 + T )^{4} \))(\( ( 1 - T )^{4} \))(\( ( 1 + T )^{5} \))(\( ( 1 - T )^{6} \))(\( ( 1 - T )^{6} \))(\( ( 1 + T )^{9} \))
$3$ (\( 1 - T + 3 T^{2} \))(\( 1 + T + 3 T^{2} \))(\( 1 + T + 3 T^{2} \))(\( 1 - 3 T + 3 T^{2} \))(\( 1 - 2 T + 5 T^{2} - 6 T^{3} + 9 T^{4} \))(\( 1 - 3 T + 7 T^{2} - 9 T^{3} + 9 T^{4} \))(\( 1 + 3 T + 8 T^{2} + 13 T^{3} + 24 T^{4} + 27 T^{5} + 27 T^{6} \))(\( 1 + 2 T + 8 T^{2} + 12 T^{3} + 31 T^{4} + 36 T^{5} + 72 T^{6} + 54 T^{7} + 81 T^{8} \))(\( 1 + 2 T + 5 T^{2} + 8 T^{3} + 16 T^{4} + 24 T^{5} + 45 T^{6} + 54 T^{7} + 81 T^{8} \))(\( 1 + 2 T + 7 T^{2} + 12 T^{3} + 28 T^{4} + 36 T^{5} + 63 T^{6} + 54 T^{7} + 81 T^{8} \))(\( 1 + 2 T + 8 T^{2} + 13 T^{3} + 35 T^{4} + 43 T^{5} + 105 T^{6} + 117 T^{7} + 216 T^{8} + 162 T^{9} + 243 T^{10} \))(\( 1 + 3 T + 13 T^{2} + 29 T^{3} + 70 T^{4} + 129 T^{5} + 241 T^{6} + 387 T^{7} + 630 T^{8} + 783 T^{9} + 1053 T^{10} + 729 T^{11} + 729 T^{12} \))(\( 1 - 2 T + 8 T^{2} - 16 T^{3} + 48 T^{4} - 76 T^{5} + 162 T^{6} - 228 T^{7} + 432 T^{8} - 432 T^{9} + 648 T^{10} - 486 T^{11} + 729 T^{12} \))(\( 1 - 5 T + 16 T^{2} - 42 T^{3} + 94 T^{4} - 183 T^{5} + 339 T^{6} - 576 T^{7} + 952 T^{8} - 1604 T^{9} + 2856 T^{10} - 5184 T^{11} + 9153 T^{12} - 14823 T^{13} + 22842 T^{14} - 30618 T^{15} + 34992 T^{16} - 32805 T^{17} + 19683 T^{18} \))
$5$ (\( 1 + 2 T + 5 T^{2} \))(\( 1 + 4 T + 5 T^{2} \))(\( 1 + 5 T^{2} \))(\( 1 + 5 T^{2} \))(\( 1 + 8 T^{2} + 25 T^{4} \))(\( 1 - 2 T + 6 T^{2} - 10 T^{3} + 25 T^{4} \))(\( 1 + 4 T + 17 T^{2} + 38 T^{3} + 85 T^{4} + 100 T^{5} + 125 T^{6} \))(\( 1 + 2 T + 12 T^{2} + 22 T^{3} + 88 T^{4} + 110 T^{5} + 300 T^{6} + 250 T^{7} + 625 T^{8} \))(\( 1 - 2 T^{2} + 4 T^{3} + 34 T^{4} + 20 T^{5} - 50 T^{6} + 625 T^{8} \))(\( ( 1 - 2 T + 6 T^{2} - 10 T^{3} + 25 T^{4} )^{2} \))(\( 1 + 11 T^{2} + 6 T^{3} + 76 T^{4} + 52 T^{5} + 380 T^{6} + 150 T^{7} + 1375 T^{8} + 3125 T^{10} \))(\( 1 + 4 T + 18 T^{2} + 28 T^{3} + 33 T^{4} - 132 T^{5} - 328 T^{6} - 660 T^{7} + 825 T^{8} + 3500 T^{9} + 11250 T^{10} + 12500 T^{11} + 15625 T^{12} \))(\( 1 - 2 T + 20 T^{2} - 32 T^{3} + 193 T^{4} - 266 T^{5} + 1188 T^{6} - 1330 T^{7} + 4825 T^{8} - 4000 T^{9} + 12500 T^{10} - 6250 T^{11} + 15625 T^{12} \))(\( 1 - 6 T + 31 T^{2} - 122 T^{3} + 420 T^{4} - 1336 T^{5} + 3840 T^{6} - 10294 T^{7} + 25672 T^{8} - 59076 T^{9} + 128360 T^{10} - 257350 T^{11} + 480000 T^{12} - 835000 T^{13} + 1312500 T^{14} - 1906250 T^{15} + 2421875 T^{16} - 2343750 T^{17} + 1953125 T^{18} \))
$7$ (\( 1 - 4 T + 7 T^{2} \))(\( 1 + 2 T + 7 T^{2} \))(\( 1 + 2 T + 7 T^{2} \))(\( 1 - 2 T + 7 T^{2} \))(\( 1 + 12 T^{2} + 49 T^{4} \))(\( 1 - T + 3 T^{2} - 7 T^{3} + 49 T^{4} \))(\( 1 + 4 T + 13 T^{2} + 22 T^{3} + 91 T^{4} + 196 T^{5} + 343 T^{6} \))(\( 1 + 2 T + 20 T^{2} + 26 T^{3} + 184 T^{4} + 182 T^{5} + 980 T^{6} + 686 T^{7} + 2401 T^{8} \))(\( 1 - 3 T + 17 T^{2} - 51 T^{3} + 150 T^{4} - 357 T^{5} + 833 T^{6} - 1029 T^{7} + 2401 T^{8} \))(\( 1 - T + 23 T^{2} - 19 T^{3} + 228 T^{4} - 133 T^{5} + 1127 T^{6} - 343 T^{7} + 2401 T^{8} \))(\( 1 + 3 T + 30 T^{2} + 72 T^{3} + 391 T^{4} + 724 T^{5} + 2737 T^{6} + 3528 T^{7} + 10290 T^{8} + 7203 T^{9} + 16807 T^{10} \))(\( 1 + 9 T + 57 T^{2} + 265 T^{3} + 1025 T^{4} + 3318 T^{5} + 9468 T^{6} + 23226 T^{7} + 50225 T^{8} + 90895 T^{9} + 136857 T^{10} + 151263 T^{11} + 117649 T^{12} \))(\( 1 - 5 T + 29 T^{2} - 99 T^{3} + 323 T^{4} - 914 T^{5} + 2412 T^{6} - 6398 T^{7} + 15827 T^{8} - 33957 T^{9} + 69629 T^{10} - 84035 T^{11} + 117649 T^{12} \))(\( 1 + 2 T + 20 T^{2} + 76 T^{3} + 208 T^{4} + 996 T^{5} + 2292 T^{6} + 7250 T^{7} + 23077 T^{8} + 46344 T^{9} + 161539 T^{10} + 355250 T^{11} + 786156 T^{12} + 2391396 T^{13} + 3495856 T^{14} + 8941324 T^{15} + 16470860 T^{16} + 11529602 T^{17} + 40353607 T^{18} \))
$11$ (\( 1 + T \))(\( 1 - T \))(\( 1 - T \))(\( 1 - T \))(\( ( 1 + T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{3} \))(\( ( 1 - T )^{4} \))(\( ( 1 + T )^{4} \))(\( ( 1 - T )^{4} \))(\( ( 1 + T )^{5} \))(\( ( 1 + T )^{6} \))(\( ( 1 + T )^{6} \))(\( ( 1 - T )^{9} \))
$13$ (\( 1 + 3 T + 13 T^{2} \))(\( 1 + T + 13 T^{2} \))(\( 1 - T + 13 T^{2} \))(\( 1 + T + 13 T^{2} \))(\( 1 - 6 T + 27 T^{2} - 78 T^{3} + 169 T^{4} \))(\( 1 + 2 T + 22 T^{2} + 26 T^{3} + 169 T^{4} \))(\( 1 + 3 T + 26 T^{2} + 79 T^{3} + 338 T^{4} + 507 T^{5} + 2197 T^{6} \))(\( 1 + 38 T^{2} + 16 T^{3} + 655 T^{4} + 208 T^{5} + 6422 T^{6} + 28561 T^{8} \))(\( 1 + 7 T + 56 T^{2} + 228 T^{3} + 1051 T^{4} + 2964 T^{5} + 9464 T^{6} + 15379 T^{7} + 28561 T^{8} \))(\( 1 - 5 T + 32 T^{2} - 120 T^{3} + 519 T^{4} - 1560 T^{5} + 5408 T^{6} - 10985 T^{7} + 28561 T^{8} \))(\( 1 + T + 26 T^{2} + 19 T^{3} + 531 T^{4} + 432 T^{5} + 6903 T^{6} + 3211 T^{7} + 57122 T^{8} + 28561 T^{9} + 371293 T^{10} \))(\( 1 + 10 T + 60 T^{2} + 262 T^{3} + 940 T^{4} + 3058 T^{5} + 10966 T^{6} + 39754 T^{7} + 158860 T^{8} + 575614 T^{9} + 1713660 T^{10} + 3712930 T^{11} + 4826809 T^{12} \))(\( 1 - 3 T + 65 T^{2} - 179 T^{3} + 1880 T^{4} - 4443 T^{5} + 31303 T^{6} - 57759 T^{7} + 317720 T^{8} - 393263 T^{9} + 1856465 T^{10} - 1113879 T^{11} + 4826809 T^{12} \))(\( 1 + T + 36 T^{2} + 76 T^{3} + 1054 T^{4} + 1859 T^{5} + 20287 T^{6} + 42068 T^{7} + 328246 T^{8} + 556176 T^{9} + 4267198 T^{10} + 7109492 T^{11} + 44570539 T^{12} + 53094899 T^{13} + 391342822 T^{14} + 366837484 T^{15} + 2258946612 T^{16} + 815730721 T^{17} + 10604499373 T^{18} \))
$17$ (\( 1 + 3 T + 17 T^{2} \))(\( 1 - 3 T + 17 T^{2} \))(\( 1 + 5 T + 17 T^{2} \))(\( 1 + 5 T + 17 T^{2} \))(\( ( 1 - 3 T + 17 T^{2} )^{2} \))(\( 1 - 6 T + 38 T^{2} - 102 T^{3} + 289 T^{4} \))(\( 1 + T + 6 T^{2} + 9 T^{3} + 102 T^{4} + 289 T^{5} + 4913 T^{6} \))(\( 1 + 8 T + 62 T^{2} + 352 T^{3} + 1587 T^{4} + 5984 T^{5} + 17918 T^{6} + 39304 T^{7} + 83521 T^{8} \))(\( 1 - 15 T + 138 T^{2} - 852 T^{3} + 4045 T^{4} - 14484 T^{5} + 39882 T^{6} - 73695 T^{7} + 83521 T^{8} \))(\( 1 - 5 T + 34 T^{2} - 160 T^{3} + 957 T^{4} - 2720 T^{5} + 9826 T^{6} - 24565 T^{7} + 83521 T^{8} \))(\( 1 + 9 T + 74 T^{2} + 303 T^{3} + 1431 T^{4} + 4464 T^{5} + 24327 T^{6} + 87567 T^{7} + 363562 T^{8} + 751689 T^{9} + 1419857 T^{10} \))(\( 1 + 10 T + 76 T^{2} + 326 T^{3} + 1120 T^{4} + 1930 T^{5} + 5246 T^{6} + 32810 T^{7} + 323680 T^{8} + 1601638 T^{9} + 6347596 T^{10} + 14198570 T^{11} + 24137569 T^{12} \))(\( 1 - 9 T + 115 T^{2} - 721 T^{3} + 5168 T^{4} - 23853 T^{5} + 119049 T^{6} - 405501 T^{7} + 1493552 T^{8} - 3542273 T^{9} + 9604915 T^{10} - 12778713 T^{11} + 24137569 T^{12} \))(\( 1 - 11 T + 134 T^{2} - 1044 T^{3} + 7840 T^{4} - 47725 T^{5} + 276187 T^{6} - 1381588 T^{7} + 6605858 T^{8} - 27820984 T^{9} + 112299586 T^{10} - 399278932 T^{11} + 1356906731 T^{12} - 3986039725 T^{13} + 11131678880 T^{14} - 25199622036 T^{15} + 54985382182 T^{16} - 76733331851 T^{17} + 118587876497 T^{18} \))
$19$ (\( 1 + 19 T^{2} \))(\( 1 + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( 1 + 19 T^{2} \))(\( ( 1 + 4 T + 19 T^{2} )^{2} \))(\( 1 + 5 T + 43 T^{2} + 95 T^{3} + 361 T^{4} \))(\( 1 + 4 T + 25 T^{2} + 184 T^{3} + 475 T^{4} + 1444 T^{5} + 6859 T^{6} \))(\( 1 - 4 T + 44 T^{2} - 100 T^{3} + 982 T^{4} - 1900 T^{5} + 15884 T^{6} - 27436 T^{7} + 130321 T^{8} \))(\( 1 - T + 27 T^{2} + 111 T^{3} + 160 T^{4} + 2109 T^{5} + 9747 T^{6} - 6859 T^{7} + 130321 T^{8} \))(\( 1 - 9 T + 63 T^{2} - 369 T^{3} + 1928 T^{4} - 7011 T^{5} + 22743 T^{6} - 61731 T^{7} + 130321 T^{8} \))(\( 1 - 3 T + 48 T^{2} - 52 T^{3} + 1139 T^{4} - 674 T^{5} + 21641 T^{6} - 18772 T^{7} + 329232 T^{8} - 390963 T^{9} + 2476099 T^{10} \))(\( 1 + 17 T + 191 T^{2} + 1451 T^{3} + 8995 T^{4} + 45718 T^{5} + 211898 T^{6} + 868642 T^{7} + 3247195 T^{8} + 9952409 T^{9} + 24891311 T^{10} + 42093683 T^{11} + 47045881 T^{12} \))(\( 1 - 7 T + 81 T^{2} - 309 T^{3} + 2003 T^{4} - 4290 T^{5} + 31446 T^{6} - 81510 T^{7} + 723083 T^{8} - 2119431 T^{9} + 10556001 T^{10} - 17332693 T^{11} + 47045881 T^{12} \))(\( 1 + 6 T + 80 T^{2} + 280 T^{3} + 2744 T^{4} + 6348 T^{5} + 68154 T^{6} + 118120 T^{7} + 1465897 T^{8} + 2163292 T^{9} + 27852043 T^{10} + 42641320 T^{11} + 467468286 T^{12} + 827277708 T^{13} + 6794415656 T^{14} + 13172846680 T^{15} + 71509739120 T^{16} + 101901378246 T^{17} + 322687697779 T^{18} \))
$23$ (\( 1 + 4 T + 23 T^{2} \))(\( 1 - 4 T + 23 T^{2} \))(\( 1 + 4 T + 23 T^{2} \))(\( 1 + 4 T + 23 T^{2} \))(\( ( 1 + 2 T + 23 T^{2} )^{2} \))(\( 1 - 13 T + 87 T^{2} - 299 T^{3} + 529 T^{4} \))(\( 1 + 6 T + 53 T^{2} + 280 T^{3} + 1219 T^{4} + 3174 T^{5} + 12167 T^{6} \))(\( 1 + 12 T + 124 T^{2} + 836 T^{3} + 4634 T^{4} + 19228 T^{5} + 65596 T^{6} + 146004 T^{7} + 279841 T^{8} \))(\( 1 - 12 T + 112 T^{2} - 748 T^{3} + 4030 T^{4} - 17204 T^{5} + 59248 T^{6} - 146004 T^{7} + 279841 T^{8} \))(\( ( 1 + 23 T^{2} )^{4} \))(\( 1 + 3 T + 98 T^{2} + 234 T^{3} + 4185 T^{4} + 7738 T^{5} + 96255 T^{6} + 123786 T^{7} + 1192366 T^{8} + 839523 T^{9} + 6436343 T^{10} \))(\( 1 + 19 T + 243 T^{2} + 2141 T^{3} + 15431 T^{4} + 90806 T^{5} + 470734 T^{6} + 2088538 T^{7} + 8162999 T^{8} + 26049547 T^{9} + 68001363 T^{10} + 122290517 T^{11} + 148035889 T^{12} \))(\( 1 - 10 T + 102 T^{2} - 742 T^{3} + 4859 T^{4} - 27980 T^{5} + 139356 T^{6} - 643540 T^{7} + 2570411 T^{8} - 9027914 T^{9} + 28543782 T^{10} - 64363430 T^{11} + 148035889 T^{12} \))(\( 1 - 21 T + 250 T^{2} - 2038 T^{3} + 13007 T^{4} - 72588 T^{5} + 404207 T^{6} - 2352218 T^{7} + 13453793 T^{8} - 69026974 T^{9} + 309437239 T^{10} - 1244323322 T^{11} + 4917986569 T^{12} - 20313098508 T^{13} + 83717513401 T^{14} - 301697141782 T^{15} + 851206361750 T^{16} - 1644530690901 T^{17} + 1801152661463 T^{18} \))
$29$ (\( 1 + 29 T^{2} \))(\( 1 - 10 T + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 + 2 T + 29 T^{2} \))(\( 1 - 12 T + 76 T^{2} - 348 T^{3} + 841 T^{4} \))(\( 1 + 5 T + 63 T^{2} + 145 T^{3} + 841 T^{4} \))(\( 1 + 6 T + 29 T^{2} + 130 T^{3} + 841 T^{4} + 5046 T^{5} + 24389 T^{6} \))(\( 1 + 2 T + 108 T^{2} + 166 T^{3} + 4600 T^{4} + 4814 T^{5} + 90828 T^{6} + 48778 T^{7} + 707281 T^{8} \))(\( 1 - 18 T + 202 T^{2} - 1630 T^{3} + 10042 T^{4} - 47270 T^{5} + 169882 T^{6} - 439002 T^{7} + 707281 T^{8} \))(\( 1 - 6 T + 108 T^{2} - 466 T^{3} + 4598 T^{4} - 13514 T^{5} + 90828 T^{6} - 146334 T^{7} + 707281 T^{8} \))(\( 1 + 7 T + 32 T^{2} - 56 T^{3} + 339 T^{4} + 992 T^{5} + 9831 T^{6} - 47096 T^{7} + 780448 T^{8} + 4950967 T^{9} + 20511149 T^{10} \))(\( 1 + 7 T + 81 T^{2} + 821 T^{3} + 5177 T^{4} + 32834 T^{5} + 220468 T^{6} + 952186 T^{7} + 4353857 T^{8} + 20023369 T^{9} + 57289761 T^{10} + 143578043 T^{11} + 594823321 T^{12} \))(\( 1 - 4 T + 76 T^{2} - 222 T^{3} + 3117 T^{4} - 6302 T^{5} + 100988 T^{6} - 182758 T^{7} + 2621397 T^{8} - 5414358 T^{9} + 53753356 T^{10} - 82044596 T^{11} + 594823321 T^{12} \))(\( 1 + 5 T + 114 T^{2} + 394 T^{3} + 7317 T^{4} + 22572 T^{5} + 355709 T^{6} + 964458 T^{7} + 12898767 T^{8} + 30261078 T^{9} + 374064243 T^{10} + 811109178 T^{11} + 8675386801 T^{12} + 15964746732 T^{13} + 150080077233 T^{14} + 234360388474 T^{15} + 1966485899226 T^{16} + 2501232064805 T^{17} + 14507145975869 T^{18} \))
$31$ (\( 1 + 3 T + 31 T^{2} \))(\( 1 - 7 T + 31 T^{2} \))(\( 1 + 7 T + 31 T^{2} \))(\( 1 + 5 T + 31 T^{2} \))(\( 1 - 6 T + 21 T^{2} - 186 T^{3} + 961 T^{4} \))(\( 1 + T + 31 T^{2} + 31 T^{3} + 961 T^{4} \))(\( 1 + T + 72 T^{2} + 91 T^{3} + 2232 T^{4} + 961 T^{5} + 29791 T^{6} \))(\( 1 - 6 T + 100 T^{2} - 460 T^{3} + 4459 T^{4} - 14260 T^{5} + 96100 T^{6} - 178746 T^{7} + 923521 T^{8} \))(\( 1 + 20 T + 221 T^{2} + 1644 T^{3} + 10000 T^{4} + 50964 T^{5} + 212381 T^{6} + 595820 T^{7} + 923521 T^{8} \))(\( 1 - 8 T + 127 T^{2} - 652 T^{3} + 5788 T^{4} - 20212 T^{5} + 122047 T^{6} - 238328 T^{7} + 923521 T^{8} \))(\( 1 - 4 T + 90 T^{2} - 305 T^{3} + 4583 T^{4} - 12917 T^{5} + 142073 T^{6} - 293105 T^{7} + 2681190 T^{8} - 3694084 T^{9} + 28629151 T^{10} \))(\( 1 - T + 73 T^{2} + 49 T^{3} + 2594 T^{4} + 5909 T^{5} + 76153 T^{6} + 183179 T^{7} + 2492834 T^{8} + 1459759 T^{9} + 67417033 T^{10} - 28629151 T^{11} + 887503681 T^{12} \))(\( 1 - 8 T + 148 T^{2} - 1010 T^{3} + 10316 T^{4} - 56466 T^{5} + 413130 T^{6} - 1750446 T^{7} + 9913676 T^{8} - 30088910 T^{9} + 136681108 T^{10} - 229033208 T^{11} + 887503681 T^{12} \))(\( 1 + 7 T + 114 T^{2} + 884 T^{3} + 7356 T^{4} + 45161 T^{5} + 311079 T^{6} + 1569724 T^{7} + 9268908 T^{8} + 49775280 T^{9} + 287336148 T^{10} + 1508504764 T^{11} + 9267354489 T^{12} + 41707131881 T^{13} + 210596034756 T^{14} + 784553254004 T^{15} + 3136438008654 T^{16} + 5970237262087 T^{17} + 26439622160671 T^{18} \))
$37$ (\( 1 - 3 T + 37 T^{2} \))(\( 1 - 3 T + 37 T^{2} \))(\( 1 - 5 T + 37 T^{2} \))(\( 1 - 3 T + 37 T^{2} \))(\( ( 1 + T + 37 T^{2} )^{2} \))(\( 1 - 11 T + 93 T^{2} - 407 T^{3} + 1369 T^{4} \))(\( 1 + 25 T + 310 T^{2} + 2341 T^{3} + 11470 T^{4} + 34225 T^{5} + 50653 T^{6} \))(\( 1 + 4 T + 78 T^{2} + 352 T^{3} + 3819 T^{4} + 13024 T^{5} + 106782 T^{6} + 202612 T^{7} + 1874161 T^{8} \))(\( 1 + 7 T + 80 T^{2} + 132 T^{3} + 1979 T^{4} + 4884 T^{5} + 109520 T^{6} + 354571 T^{7} + 1874161 T^{8} \))(\( 1 + 5 T + 112 T^{2} + 340 T^{3} + 5359 T^{4} + 12580 T^{5} + 153328 T^{6} + 253265 T^{7} + 1874161 T^{8} \))(\( 1 + 2 T + 168 T^{2} + 299 T^{3} + 11855 T^{4} + 16605 T^{5} + 438635 T^{6} + 409331 T^{7} + 8509704 T^{8} + 3748322 T^{9} + 69343957 T^{10} \))(\( 1 - 3 T + 83 T^{2} - 601 T^{3} + 5486 T^{4} - 29411 T^{5} + 281839 T^{6} - 1088207 T^{7} + 7510334 T^{8} - 30442453 T^{9} + 155555363 T^{10} - 208031871 T^{11} + 2565726409 T^{12} \))(\( 1 + 5 T + 189 T^{2} + 895 T^{3} + 15822 T^{4} + 65169 T^{5} + 754493 T^{6} + 2411253 T^{7} + 21660318 T^{8} + 45334435 T^{9} + 354216429 T^{10} + 346719785 T^{11} + 2565726409 T^{12} \))(\( 1 - 16 T + 232 T^{2} - 2618 T^{3} + 26183 T^{4} - 225854 T^{5} + 1840656 T^{6} - 13292224 T^{7} + 91205811 T^{8} - 572147502 T^{9} + 3374615007 T^{10} - 18197054656 T^{11} + 93234748368 T^{12} - 423286758494 T^{13} + 1815632826131 T^{14} - 6717071738762 T^{15} + 22024195494856 T^{16} - 56199671262736 T^{17} + 129961739795077 T^{18} \))
$41$ (\( 1 + 41 T^{2} \))(\( 1 + 8 T + 41 T^{2} \))(\( 1 + 41 T^{2} \))(\( 1 + 41 T^{2} \))(\( ( 1 - 8 T + 41 T^{2} )^{2} \))(\( 1 - 9 T + 71 T^{2} - 369 T^{3} + 1681 T^{4} \))(\( ( 1 + 2 T + 41 T^{2} )^{3} \))(\( 1 + 124 T^{2} - 104 T^{3} + 6782 T^{4} - 4264 T^{5} + 208444 T^{6} + 2825761 T^{8} \))(\( 1 + 8 T + 52 T^{2} + 344 T^{3} + 1926 T^{4} + 14104 T^{5} + 87412 T^{6} + 551368 T^{7} + 2825761 T^{8} \))(\( ( 1 + 4 T + 66 T^{2} + 164 T^{3} + 1681 T^{4} )^{2} \))(\( 1 + 9 T + 168 T^{2} + 1062 T^{3} + 11991 T^{4} + 58530 T^{5} + 491631 T^{6} + 1785222 T^{7} + 11578728 T^{8} + 25431849 T^{9} + 115856201 T^{10} \))(\( 1 - 13 T + 143 T^{2} - 873 T^{3} + 8067 T^{4} - 59330 T^{5} + 495442 T^{6} - 2432530 T^{7} + 13560627 T^{8} - 60168033 T^{9} + 404083823 T^{10} - 1506130613 T^{11} + 4750104241 T^{12} \))(\( 1 - 2 T + 126 T^{2} + 134 T^{3} + 7751 T^{4} + 17484 T^{5} + 371828 T^{6} + 716844 T^{7} + 13029431 T^{8} + 9235414 T^{9} + 356045886 T^{10} - 231712402 T^{11} + 4750104241 T^{12} \))(\( 1 - 13 T + 218 T^{2} - 2144 T^{3} + 23767 T^{4} - 193876 T^{5} + 1744549 T^{6} - 12347120 T^{7} + 94532801 T^{8} - 585471006 T^{9} + 3875844841 T^{10} - 20755508720 T^{11} + 120236061629 T^{12} - 547847239636 T^{13} + 2753554329167 T^{14} - 10184223492704 T^{15} + 42456431706058 T^{16} - 103804027978573 T^{17} + 327381934393961 T^{18} \))
$43$ (\( 1 - T + 43 T^{2} \))(\( 1 + 11 T + 43 T^{2} \))(\( 1 + 11 T + 43 T^{2} \))(\( 1 + 7 T + 43 T^{2} \))(\( 1 + 6 T + 93 T^{2} + 258 T^{3} + 1849 T^{4} \))(\( 1 - 8 T + 82 T^{2} - 344 T^{3} + 1849 T^{4} \))(\( 1 - 13 T + 72 T^{2} - 247 T^{3} + 3096 T^{4} - 24037 T^{5} + 79507 T^{6} \))(\( 1 + 6 T + 152 T^{2} + 664 T^{3} + 9307 T^{4} + 28552 T^{5} + 281048 T^{6} + 477042 T^{7} + 3418801 T^{8} \))(\( 1 - 10 T + 125 T^{2} - 912 T^{3} + 8008 T^{4} - 39216 T^{5} + 231125 T^{6} - 795070 T^{7} + 3418801 T^{8} \))(\( 1 - 14 T + 187 T^{2} - 1568 T^{3} + 12268 T^{4} - 67424 T^{5} + 345763 T^{6} - 1113098 T^{7} + 3418801 T^{8} \))(\( 1 + 3 T + 122 T^{2} + 411 T^{3} + 7249 T^{4} + 24320 T^{5} + 311707 T^{6} + 759939 T^{7} + 9699854 T^{8} + 10256403 T^{9} + 147008443 T^{10} \))(\( 1 + 38 T + 826 T^{2} + 12348 T^{3} + 140788 T^{4} + 1266758 T^{5} + 9226438 T^{6} + 54470594 T^{7} + 260317012 T^{8} + 981752436 T^{9} + 2823929626 T^{10} + 5586320834 T^{11} + 6321363049 T^{12} \))(\( 1 - 8 T + 178 T^{2} - 1552 T^{3} + 14474 T^{4} - 127142 T^{5} + 745826 T^{6} - 5467106 T^{7} + 26762426 T^{8} - 123394864 T^{9} + 608546578 T^{10} - 1176067544 T^{11} + 6321363049 T^{12} \))(\( 1 + 199 T^{2} - 148 T^{3} + 21623 T^{4} - 24914 T^{5} + 1612305 T^{6} - 2087020 T^{7} + 89865648 T^{8} - 111451004 T^{9} + 3864222864 T^{10} - 3858899980 T^{11} + 128189533635 T^{12} - 85176008114 T^{13} + 3178763562989 T^{14} - 935561731252 T^{15} + 54091903610293 T^{16} + 502592611936843 T^{18} \))
$47$ (\( 1 + 6 T + 47 T^{2} \))(\( 1 + 12 T + 47 T^{2} \))(\( 1 + 8 T + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 - 4 T + 48 T^{2} - 188 T^{3} + 2209 T^{4} \))(\( 1 + 14 T + 138 T^{2} + 658 T^{3} + 2209 T^{4} \))(\( 1 + 8 T + 125 T^{2} + 622 T^{3} + 5875 T^{4} + 17672 T^{5} + 103823 T^{6} \))(\( 1 + 14 T + 220 T^{2} + 1754 T^{3} + 15488 T^{4} + 82438 T^{5} + 485980 T^{6} + 1453522 T^{7} + 4879681 T^{8} \))(\( 1 - 5 T + 175 T^{2} - 621 T^{3} + 11942 T^{4} - 29187 T^{5} + 386575 T^{6} - 519115 T^{7} + 4879681 T^{8} \))(\( 1 - 33 T + 573 T^{2} - 6479 T^{3} + 52208 T^{4} - 304513 T^{5} + 1265757 T^{6} - 3426159 T^{7} + 4879681 T^{8} \))(\( 1 + 12 T + 251 T^{2} + 2134 T^{3} + 24422 T^{4} + 147604 T^{5} + 1147834 T^{6} + 4714006 T^{7} + 26059573 T^{8} + 58556172 T^{9} + 229345007 T^{10} \))(\( 1 + 54 T^{2} + 140 T^{3} + 5673 T^{4} + 9864 T^{5} + 207768 T^{6} + 463608 T^{7} + 12531657 T^{8} + 14535220 T^{9} + 263502774 T^{10} + 10779215329 T^{12} \))(\( 1 - 11 T + 257 T^{2} - 2271 T^{3} + 28915 T^{4} - 199864 T^{5} + 1789288 T^{6} - 9393608 T^{7} + 63873235 T^{8} - 235782033 T^{9} + 1254078017 T^{10} - 2522795077 T^{11} + 10779215329 T^{12} \))(\( 1 - 9 T + 276 T^{2} - 1772 T^{3} + 33469 T^{4} - 167492 T^{5} + 2635771 T^{6} - 11146848 T^{7} + 157728193 T^{8} - 586993486 T^{9} + 7413225071 T^{10} - 24623387232 T^{11} + 273653652533 T^{12} - 817307530052 T^{13} + 7675948039283 T^{14} - 19100769562988 T^{15} + 139827981247788 T^{16} - 214301579955849 T^{17} + 1119130473102767 T^{18} \))
$53$ (\( 1 - T + 53 T^{2} \))(\( 1 - 9 T + 53 T^{2} \))(\( 1 - 7 T + 53 T^{2} \))(\( 1 - 9 T + 53 T^{2} \))(\( ( 1 - 13 T + 53 T^{2} )^{2} \))(\( 1 + 7 T + 117 T^{2} + 371 T^{3} + 2809 T^{4} \))(\( 1 + 31 T + 474 T^{2} + 4335 T^{3} + 25122 T^{4} + 87079 T^{5} + 148877 T^{6} \))(\( 1 + 20 T + 302 T^{2} + 3328 T^{3} + 26475 T^{4} + 176384 T^{5} + 848318 T^{6} + 2977540 T^{7} + 7890481 T^{8} \))(\( 1 - 13 T + 208 T^{2} - 1620 T^{3} + 15287 T^{4} - 85860 T^{5} + 584272 T^{6} - 1935401 T^{7} + 7890481 T^{8} \))(\( 1 - 3 T + 176 T^{2} - 404 T^{3} + 13019 T^{4} - 21412 T^{5} + 494384 T^{6} - 446631 T^{7} + 7890481 T^{8} \))(\( 1 + 18 T + 272 T^{2} + 2865 T^{3} + 27931 T^{4} + 214495 T^{5} + 1480343 T^{6} + 8047785 T^{7} + 40494544 T^{8} + 142028658 T^{9} + 418195493 T^{10} \))(\( 1 + 31 T + 707 T^{2} + 10725 T^{3} + 133278 T^{4} + 1284767 T^{5} + 10462735 T^{6} + 68092651 T^{7} + 374377902 T^{8} + 1596705825 T^{9} + 5578570067 T^{10} + 12964060283 T^{11} + 22164361129 T^{12} \))(\( 1 - 7 T + 189 T^{2} - 1229 T^{3} + 18114 T^{4} - 114035 T^{5} + 1167265 T^{6} - 6043855 T^{7} + 50882226 T^{8} - 182969833 T^{9} + 1491300909 T^{10} - 2927368451 T^{11} + 22164361129 T^{12} \))(\( 1 - 18 T + 422 T^{2} - 5156 T^{3} + 73181 T^{4} - 696380 T^{5} + 7512016 T^{6} - 59349904 T^{7} + 533482159 T^{8} - 3634369218 T^{9} + 28274554427 T^{10} - 166713880336 T^{11} + 1118366406032 T^{12} - 5494773158780 T^{13} + 30603964373233 T^{14} - 114279445981124 T^{15} + 495728101011214 T^{16} - 1120674427404498 T^{17} + 3299763591802133 T^{18} \))
$59$ (\( 1 - 6 T + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( 1 + 4 T + 59 T^{2} \))(\( 1 - 12 T + 59 T^{2} \))(\( 1 + 12 T + 136 T^{2} + 708 T^{3} + 3481 T^{4} \))(\( 1 + 38 T^{2} + 3481 T^{4} \))(\( 1 - 16 T + 61 T^{2} + 90 T^{3} + 3599 T^{4} - 55696 T^{5} + 205379 T^{6} \))(\( 1 + 14 T + 164 T^{2} + 1018 T^{3} + 8952 T^{4} + 60062 T^{5} + 570884 T^{6} + 2875306 T^{7} + 12117361 T^{8} \))(\( 1 - 9 T + 39 T^{2} - 609 T^{3} + 7390 T^{4} - 35931 T^{5} + 135759 T^{6} - 1848411 T^{7} + 12117361 T^{8} \))(\( 1 - 15 T + 225 T^{2} - 2185 T^{3} + 20552 T^{4} - 128915 T^{5} + 783225 T^{6} - 3080685 T^{7} + 12117361 T^{8} \))(\( 1 + 6 T + 187 T^{2} + 718 T^{3} + 16086 T^{4} + 43432 T^{5} + 949074 T^{6} + 2499358 T^{7} + 38405873 T^{8} + 72704166 T^{9} + 714924299 T^{10} \))(\( 1 + 6 T + 262 T^{2} + 1430 T^{3} + 32017 T^{4} + 152568 T^{5} + 2366672 T^{6} + 9001512 T^{7} + 111451177 T^{8} + 293691970 T^{9} + 3174748582 T^{10} + 4289545794 T^{11} + 42180533641 T^{12} \))(\( 1 + 5 T + 189 T^{2} + 893 T^{3} + 21867 T^{4} + 85560 T^{5} + 1531384 T^{6} + 5048040 T^{7} + 76119027 T^{8} + 183403447 T^{9} + 2290181229 T^{10} + 3574621495 T^{11} + 42180533641 T^{12} \))(\( 1 - T + 242 T^{2} + 430 T^{3} + 29891 T^{4} + 113960 T^{5} + 2804109 T^{6} + 11363250 T^{7} + 218245095 T^{8} + 749506226 T^{9} + 12876460605 T^{10} + 39555473250 T^{11} + 575905102311 T^{12} + 1380894459560 T^{13} + 21369802221409 T^{14} + 18137629465630 T^{15} + 602253659326198 T^{16} - 146830437604321 T^{17} + 8662995818654939 T^{18} \))
$61$ (\( 1 + T \))(\( 1 - T \))(\( 1 + T \))(\( 1 - T \))(\( ( 1 - T )^{2} \))(\( ( 1 - T )^{2} \))(\( ( 1 + T )^{3} \))(\( ( 1 - T )^{4} \))(\( ( 1 - T )^{4} \))(\( ( 1 - T )^{4} \))(\( ( 1 + T )^{5} \))(\( ( 1 - T )^{6} \))(\( ( 1 + T )^{6} \))(\( ( 1 + T )^{9} \))
$67$ (\( 1 + 8 T + 67 T^{2} \))(\( 1 - 8 T + 67 T^{2} \))(\( 1 - 16 T + 67 T^{2} \))(\( 1 + 8 T + 67 T^{2} \))(\( 1 + 6 T^{2} + 4489 T^{4} \))(\( 1 + 4 T - 42 T^{2} + 268 T^{3} + 4489 T^{4} \))(\( ( 1 + 12 T + 67 T^{2} )^{3} \))(\( 1 + 8 T + 140 T^{2} + 584 T^{3} + 10294 T^{4} + 39128 T^{5} + 628460 T^{6} + 2406104 T^{7} + 20151121 T^{8} \))(\( 1 + 33 T + 605 T^{2} + 7689 T^{3} + 72924 T^{4} + 515163 T^{5} + 2715845 T^{6} + 9925179 T^{7} + 20151121 T^{8} \))(\( 1 - 17 T + 301 T^{2} - 3249 T^{3} + 31292 T^{4} - 217683 T^{5} + 1351189 T^{6} - 5112971 T^{7} + 20151121 T^{8} \))(\( ( 1 - 4 T + 67 T^{2} )^{5} \))(\( 1 + 12 T + 226 T^{2} + 820 T^{3} + 5319 T^{4} - 127048 T^{5} - 722948 T^{6} - 8512216 T^{7} + 23876991 T^{8} + 246625660 T^{9} + 4554153346 T^{10} + 16201501284 T^{11} + 90458382169 T^{12} \))(\( 1 + 5 T + 247 T^{2} + 1571 T^{3} + 32259 T^{4} + 189082 T^{5} + 2696170 T^{6} + 12668494 T^{7} + 144810651 T^{8} + 472498673 T^{9} + 4977326887 T^{10} + 6750625535 T^{11} + 90458382169 T^{12} \))(\( 1 - 17 T + 460 T^{2} - 5684 T^{3} + 96217 T^{4} - 985716 T^{5} + 12774201 T^{6} - 111024556 T^{7} + 1181157343 T^{8} - 8781931542 T^{9} + 79137541981 T^{10} - 498389231884 T^{11} + 3842007015363 T^{12} - 19863282387636 T^{13} + 129904987420219 T^{14} - 514165444248596 T^{15} + 2787927338448580 T^{16} - 6903150518462897 T^{17} + 27206534396294947 T^{18} \))
$71$ (\( 1 + 13 T + 71 T^{2} \))(\( 1 + 3 T + 71 T^{2} \))(\( 1 + 13 T + 71 T^{2} \))(\( 1 - 9 T + 71 T^{2} \))(\( 1 + 2 T + 141 T^{2} + 142 T^{3} + 5041 T^{4} \))(\( 1 + 11 T + 141 T^{2} + 781 T^{3} + 5041 T^{4} \))(\( 1 - 15 T + 150 T^{2} - 1027 T^{3} + 10650 T^{4} - 75615 T^{5} + 357911 T^{6} \))(\( 1 + 6 T + 136 T^{2} + 748 T^{3} + 10079 T^{4} + 53108 T^{5} + 685576 T^{6} + 2147466 T^{7} + 25411681 T^{8} \))(\( 1 - 8 T + 229 T^{2} - 1272 T^{3} + 22244 T^{4} - 90312 T^{5} + 1154389 T^{6} - 2863288 T^{7} + 25411681 T^{8} \))(\( 1 + 4 T + 231 T^{2} + 688 T^{3} + 23216 T^{4} + 48848 T^{5} + 1164471 T^{6} + 1431644 T^{7} + 25411681 T^{8} \))(\( 1 - 18 T + 204 T^{2} - 2069 T^{3} + 23817 T^{4} - 241345 T^{5} + 1691007 T^{6} - 10429829 T^{7} + 73013844 T^{8} - 457410258 T^{9} + 1804229351 T^{10} \))(\( 1 + 17 T + 315 T^{2} + 3767 T^{3} + 47722 T^{4} + 437907 T^{5} + 4194103 T^{6} + 31091397 T^{7} + 240566602 T^{8} + 1348250737 T^{9} + 8004679515 T^{10} + 30671898967 T^{11} + 128100283921 T^{12} \))(\( 1 - 10 T + 320 T^{2} - 2326 T^{3} + 47076 T^{4} - 278796 T^{5} + 4230246 T^{6} - 19794516 T^{7} + 237310116 T^{8} - 832500986 T^{9} + 8131737920 T^{10} - 18042293510 T^{11} + 128100283921 T^{12} \))(\( 1 - 5 T + 194 T^{2} + 752 T^{3} + 19778 T^{4} + 99249 T^{5} + 3171451 T^{6} + 8320444 T^{7} + 231247366 T^{8} + 1274528688 T^{9} + 16418562986 T^{10} + 41943358204 T^{11} + 1135097198861 T^{12} + 2522083927569 T^{13} + 35684048104078 T^{14} + 96331413508592 T^{15} + 1764453310727854 T^{16} - 3228767656228805 T^{17} + 45848500718449031 T^{18} \))
$73$ (\( 1 + 73 T^{2} \))(\( 1 - 14 T + 73 T^{2} \))(\( 1 + 6 T + 73 T^{2} \))(\( 1 - 2 T + 73 T^{2} \))(\( 1 + 12 T + 132 T^{2} + 876 T^{3} + 5329 T^{4} \))(\( 1 + 7 T + 147 T^{2} + 511 T^{3} + 5329 T^{4} \))(\( 1 - 12 T + 197 T^{2} - 1622 T^{3} + 14381 T^{4} - 63948 T^{5} + 389017 T^{6} \))(\( 1 + 14 T + 348 T^{2} + 3078 T^{3} + 39936 T^{4} + 224694 T^{5} + 1854492 T^{6} + 5446238 T^{7} + 28398241 T^{8} \))(\( 1 - 36 T + 710 T^{2} - 9312 T^{3} + 91410 T^{4} - 679776 T^{5} + 3783590 T^{6} - 14004612 T^{7} + 28398241 T^{8} \))(\( 1 + 16 T + 272 T^{2} + 3072 T^{3} + 29358 T^{4} + 224256 T^{5} + 1449488 T^{6} + 6224272 T^{7} + 28398241 T^{8} \))(\( 1 + 5 T + 282 T^{2} + 1110 T^{3} + 36801 T^{4} + 114200 T^{5} + 2686473 T^{6} + 5915190 T^{7} + 109702794 T^{8} + 141991205 T^{9} + 2073071593 T^{10} \))(\( 1 - 7 T + 157 T^{2} - 769 T^{3} + 15285 T^{4} - 87814 T^{5} + 1451916 T^{6} - 6410422 T^{7} + 81453765 T^{8} - 299154073 T^{9} + 4458523837 T^{10} - 14511501151 T^{11} + 151334226289 T^{12} \))(\( 1 + 16 T + 360 T^{2} + 3658 T^{3} + 48441 T^{4} + 372882 T^{5} + 4022268 T^{6} + 27220386 T^{7} + 258142089 T^{8} + 1423024186 T^{9} + 10223366760 T^{10} + 33169145488 T^{11} + 151334226289 T^{12} \))(\( 1 + 5 T + 50 T^{2} + 622 T^{3} + 11805 T^{4} + 24532 T^{5} + 512033 T^{6} + 2937914 T^{7} + 43894463 T^{8} - 26789682 T^{9} + 3204295799 T^{10} + 15656143706 T^{11} + 199189541561 T^{12} + 696665648212 T^{13} + 24472610155365 T^{14} + 94129888751758 T^{15} + 552369925954850 T^{16} + 4032300459470405 T^{17} + 58871586708267913 T^{18} \))
$79$ (\( 1 + 2 T + 79 T^{2} \))(\( 1 + 79 T^{2} \))(\( 1 - 4 T + 79 T^{2} \))(\( 1 - 12 T + 79 T^{2} \))(\( 1 - 12 T + 192 T^{2} - 948 T^{3} + 6241 T^{4} \))(\( 1 - 20 T + 238 T^{2} - 1580 T^{3} + 6241 T^{4} \))(\( 1 + 22 T + 377 T^{2} + 3750 T^{3} + 29783 T^{4} + 137302 T^{5} + 493039 T^{6} \))(\( 1 - 34 T + 720 T^{2} - 10026 T^{3} + 104640 T^{4} - 792054 T^{5} + 4493520 T^{6} - 16763326 T^{7} + 38950081 T^{8} \))(\( 1 + 7 T + 195 T^{2} + 1215 T^{3} + 17974 T^{4} + 95985 T^{5} + 1216995 T^{6} + 3451273 T^{7} + 38950081 T^{8} \))(\( 1 + 21 T + 361 T^{2} + 3735 T^{3} + 38400 T^{4} + 295065 T^{5} + 2253001 T^{6} + 10353819 T^{7} + 38950081 T^{8} \))(\( 1 + 16 T + 259 T^{2} + 3054 T^{3} + 28970 T^{4} + 284388 T^{5} + 2288630 T^{6} + 19060014 T^{7} + 127697101 T^{8} + 623201296 T^{9} + 3077056399 T^{10} \))(\( 1 + 10 T + 306 T^{2} + 2558 T^{3} + 44825 T^{4} + 319308 T^{5} + 4262792 T^{6} + 25225332 T^{7} + 279752825 T^{8} + 1261193762 T^{9} + 11918724786 T^{10} + 30770563990 T^{11} + 243087455521 T^{12} \))(\( 1 - 3 T + 331 T^{2} + 247 T^{3} + 42987 T^{4} + 161222 T^{5} + 3641184 T^{6} + 12736538 T^{7} + 268281867 T^{8} + 121780633 T^{9} + 12892476811 T^{10} - 9231169197 T^{11} + 243087455521 T^{12} \))(\( 1 + 39 T + 932 T^{2} + 15656 T^{3} + 202741 T^{4} + 2015204 T^{5} + 15170471 T^{6} + 75722000 T^{7} + 142418509 T^{8} - 788540422 T^{9} + 11251062211 T^{10} + 472581002000 T^{11} + 7479633851369 T^{12} + 78492359031524 T^{13} + 623845491389659 T^{14} + 3805777203636776 T^{15} + 17898043175100188 T^{16} + 59167243586355879 T^{17} + 119851595982618319 T^{18} \))
$83$ (\( 1 + 12 T + 83 T^{2} \))(\( 1 - 14 T + 83 T^{2} \))(\( 1 + 2 T + 83 T^{2} \))(\( 1 + 6 T + 83 T^{2} \))(\( 1 + 8 T + 20 T^{2} + 664 T^{3} + 6889 T^{4} \))(\( 1 - 3 T + 167 T^{2} - 249 T^{3} + 6889 T^{4} \))(\( 1 + 10 T + 269 T^{2} + 1634 T^{3} + 22327 T^{4} + 68890 T^{5} + 571787 T^{6} \))(\( 1 + 2 T + 244 T^{2} + 110 T^{3} + 26696 T^{4} + 9130 T^{5} + 1680916 T^{6} + 1143574 T^{7} + 47458321 T^{8} \))(\( 1 - 14 T + 262 T^{2} - 2606 T^{3} + 32562 T^{4} - 216298 T^{5} + 1804918 T^{6} - 8005018 T^{7} + 47458321 T^{8} \))(\( 1 - 6 T + 312 T^{2} - 1462 T^{3} + 38078 T^{4} - 121346 T^{5} + 2149368 T^{6} - 3430722 T^{7} + 47458321 T^{8} \))(\( 1 + 5 T + 32 T^{2} + 1398 T^{3} + 12149 T^{4} + 15608 T^{5} + 1008367 T^{6} + 9630822 T^{7} + 18297184 T^{8} + 237291605 T^{9} + 3939040643 T^{10} \))(\( 1 - 5 T + 337 T^{2} - 1821 T^{3} + 49773 T^{4} - 281198 T^{5} + 4762708 T^{6} - 23339434 T^{7} + 342886197 T^{8} - 1041224127 T^{9} + 15993454177 T^{10} - 19695203215 T^{11} + 326940373369 T^{12} \))(\( 1 - 12 T + 402 T^{2} - 4374 T^{3} + 72229 T^{4} - 678634 T^{5} + 7591408 T^{6} - 56326622 T^{7} + 497585581 T^{8} - 2500996338 T^{9} + 19078245042 T^{10} - 47268487716 T^{11} + 326940373369 T^{12} \))(\( 1 - 43 T + 1172 T^{2} - 23198 T^{3} + 377561 T^{4} - 5208008 T^{5} + 63677991 T^{6} - 700003106 T^{7} + 7105350837 T^{8} - 66849705946 T^{9} + 589744119471 T^{10} - 4822321397234 T^{11} + 36410247439917 T^{12} - 247163315434568 T^{13} + 1487228124211723 T^{14} - 7584362781414062 T^{15} + 31803451759842844 T^{16} - 96848565981978763 T^{17} + 186940255267540403 T^{18} \))
$89$ (\( 1 + 6 T + 89 T^{2} \))(\( 1 + 89 T^{2} \))(\( 1 + 89 T^{2} \))(\( 1 - 12 T + 89 T^{2} \))(\( 1 + 16 T^{2} + 7921 T^{4} \))(\( 1 + 30 T + 398 T^{2} + 2670 T^{3} + 7921 T^{4} \))(\( 1 - 16 T + 337 T^{2} - 2898 T^{3} + 29993 T^{4} - 126736 T^{5} + 704969 T^{6} \))(\( 1 + 26 T + 460 T^{2} + 5534 T^{3} + 57416 T^{4} + 492526 T^{5} + 3643660 T^{6} + 18329194 T^{7} + 62742241 T^{8} \))(\( 1 - 18 T + 330 T^{2} - 4422 T^{3} + 42874 T^{4} - 393558 T^{5} + 2613930 T^{6} - 12689442 T^{7} + 62742241 T^{8} \))(\( 1 - 6 T + 212 T^{2} - 1418 T^{3} + 23990 T^{4} - 126202 T^{5} + 1679252 T^{6} - 4229814 T^{7} + 62742241 T^{8} \))(\( 1 - 16 T + 347 T^{2} - 4494 T^{3} + 55856 T^{4} - 565444 T^{5} + 4971184 T^{6} - 35596974 T^{7} + 244624243 T^{8} - 1003875856 T^{9} + 5584059449 T^{10} \))(\( 1 + 202 T^{2} - 16 T^{3} + 33529 T^{4} - 7316 T^{5} + 3237880 T^{6} - 651124 T^{7} + 265583209 T^{8} - 11279504 T^{9} + 12673932682 T^{10} + 496981290961 T^{12} \))(\( 1 - 4 T + 436 T^{2} - 1422 T^{3} + 85797 T^{4} - 225062 T^{5} + 9773948 T^{6} - 20030518 T^{7} + 679598037 T^{8} - 1002465918 T^{9} + 27355617076 T^{10} - 22336237796 T^{11} + 496981290961 T^{12} \))(\( 1 - 32 T + 971 T^{2} - 19410 T^{3} + 361664 T^{4} - 5389036 T^{5} + 75274460 T^{6} - 894234046 T^{7} + 10000424448 T^{8} - 97151852680 T^{9} + 890037775872 T^{10} - 7083227878366 T^{11} + 53066160791740 T^{12} - 338120195469676 T^{13} + 2019553276563136 T^{14} - 9646406857553010 T^{15} + 42948626183558659 T^{16} - 125970841782466592 T^{17} + 350356403707485209 T^{18} \))
$97$ (\( 1 - 11 T + 97 T^{2} \))(\( 1 + 17 T + 97 T^{2} \))(\( 1 - 7 T + 97 T^{2} \))(\( 1 - 7 T + 97 T^{2} \))(\( 1 + 2 T - 5 T^{2} + 194 T^{3} + 9409 T^{4} \))(\( 1 + 19 T + 273 T^{2} + 1843 T^{3} + 9409 T^{4} \))(\( 1 - 5 T + 86 T^{2} - 657 T^{3} + 8342 T^{4} - 47045 T^{5} + 912673 T^{6} \))(\( 1 + 8 T + 374 T^{2} + 2264 T^{3} + 53839 T^{4} + 219608 T^{5} + 3518966 T^{6} + 7301384 T^{7} + 88529281 T^{8} \))(\( 1 + 11 T + 286 T^{2} + 3260 T^{3} + 37021 T^{4} + 316220 T^{5} + 2690974 T^{6} + 10039403 T^{7} + 88529281 T^{8} \))(\( 1 - 5 T + 294 T^{2} - 1300 T^{3} + 38157 T^{4} - 126100 T^{5} + 2766246 T^{6} - 4563365 T^{7} + 88529281 T^{8} \))(\( 1 + 34 T + 836 T^{2} + 13839 T^{3} + 190379 T^{4} + 2031669 T^{5} + 18466763 T^{6} + 130211151 T^{7} + 762994628 T^{8} + 3009995554 T^{9} + 8587340257 T^{10} \))(\( 1 - 5 T + 311 T^{2} - 1939 T^{3} + 54334 T^{4} - 331097 T^{5} + 6238027 T^{6} - 32116409 T^{7} + 511228606 T^{8} - 1769672947 T^{9} + 27532606391 T^{10} - 42936701285 T^{11} + 832972004929 T^{12} \))(\( 1 + 17 T + 311 T^{2} + 4019 T^{3} + 48666 T^{4} + 543805 T^{5} + 5329703 T^{6} + 52749085 T^{7} + 457898394 T^{8} + 3668032787 T^{9} + 27532606391 T^{10} + 145984784369 T^{11} + 832972004929 T^{12} \))(\( 1 - 22 T + 732 T^{2} - 12614 T^{3} + 237897 T^{4} - 3344796 T^{5} + 46693926 T^{6} - 549960164 T^{7} + 6273125169 T^{8} - 62879230546 T^{9} + 608493141393 T^{10} - 5174575183076 T^{11} + 42616285524198 T^{12} - 296112384971676 T^{13} + 2042902485119529 T^{14} - 10507108870174406 T^{15} + 59144344237978716 T^{16} - 172423539076293142 T^{17} + 760231058654565217 T^{18} \))
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