Properties

Label 1005.2.a
Level 1005
Weight 2
Character orbit a
Rep. character \(\chi_{1005}(1,\cdot)\)
Character field \(\Q\)
Dimension 43
Newform subspaces 10
Sturm bound 272
Trace bound 2

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Defining parameters

Level: \( N \) = \( 1005 = 3 \cdot 5 \cdot 67 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1005.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 10 \)
Sturm bound: \(272\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1005))\).

Total New Old
Modular forms 140 43 97
Cusp forms 133 43 90
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.

\(3\)\(5\)\(67\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(4\)
\(+\)\(+\)\(-\)\(-\)\(8\)
\(+\)\(-\)\(+\)\(-\)\(5\)
\(+\)\(-\)\(-\)\(+\)\(5\)
\(-\)\(+\)\(+\)\(-\)\(5\)
\(-\)\(+\)\(-\)\(+\)\(5\)
\(-\)\(-\)\(+\)\(+\)\(4\)
\(-\)\(-\)\(-\)\(-\)\(7\)
Plus space\(+\)\(18\)
Minus space\(-\)\(25\)

Trace form

\( 43q - 3q^{2} - q^{3} + 41q^{4} - q^{5} + q^{6} - 8q^{7} - 15q^{8} + 43q^{9} + O(q^{10}) \) \( 43q - 3q^{2} - q^{3} + 41q^{4} - q^{5} + q^{6} - 8q^{7} - 15q^{8} + 43q^{9} + 5q^{10} - 4q^{11} - 7q^{12} - 14q^{13} + 3q^{15} + 41q^{16} - 14q^{17} - 3q^{18} + 9q^{20} - 8q^{21} - 28q^{22} - 4q^{23} - 3q^{24} + 43q^{25} + 38q^{26} - q^{27} - 16q^{28} - 26q^{29} - 3q^{30} + 17q^{32} + 4q^{33} + 26q^{34} + 41q^{36} - 2q^{37} + 52q^{38} - 14q^{39} + 9q^{40} + 6q^{41} - 8q^{42} - 28q^{43} - 4q^{44} - q^{45} + 32q^{46} - 20q^{47} + q^{48} + 27q^{49} - 3q^{50} - 2q^{51} - 26q^{52} - 30q^{53} + q^{54} + 4q^{55} + 48q^{56} - 20q^{57} + 30q^{58} - 16q^{59} + 5q^{60} - 30q^{61} + 16q^{62} - 8q^{63} + q^{64} + 10q^{65} + 20q^{66} + 7q^{67} - 14q^{68} + 8q^{69} + 16q^{70} + 24q^{71} - 15q^{72} - 6q^{73} - 34q^{74} - q^{75} + 76q^{76} - 16q^{77} - 2q^{78} - 48q^{79} + q^{80} + 43q^{81} + 66q^{82} + 20q^{83} + 16q^{84} - 2q^{85} + 4q^{86} - 22q^{87} + 4q^{88} + 42q^{89} + 5q^{90} - 8q^{91} - 104q^{92} - 24q^{93} - 40q^{94} - 20q^{95} + 5q^{96} + 14q^{97} - 11q^{98} - 4q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1005))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3 5 67
1005.2.a.a \(1\) \(8.025\) \(\Q\) None \(0\) \(1\) \(-1\) \(2\) \(-\) \(+\) \(-\) \(q+q^{3}-2q^{4}-q^{5}+2q^{7}+q^{9}-6q^{11}+\cdots\)
1005.2.a.b \(1\) \(8.025\) \(\Q\) None \(1\) \(-1\) \(1\) \(0\) \(+\) \(-\) \(+\) \(q+q^{2}-q^{3}-q^{4}+q^{5}-q^{6}-3q^{8}+\cdots\)
1005.2.a.c \(4\) \(8.025\) 4.4.1957.1 None \(-4\) \(4\) \(4\) \(-9\) \(-\) \(-\) \(+\) \(q+(-1+\beta _{1})q^{2}+q^{3}+(1-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
1005.2.a.d \(4\) \(8.025\) 4.4.2525.1 None \(-2\) \(4\) \(-4\) \(-1\) \(-\) \(+\) \(-\) \(q-\beta _{1}q^{2}+q^{3}+(1+\beta _{1}+\beta _{2})q^{4}-q^{5}+\cdots\)
1005.2.a.e \(4\) \(8.025\) 4.4.1957.1 None \(0\) \(-4\) \(-4\) \(1\) \(+\) \(+\) \(+\) \(q+\beta _{1}q^{2}-q^{3}+(\beta _{1}+\beta _{2}+\beta _{3})q^{4}+\cdots\)
1005.2.a.f \(4\) \(8.025\) 4.4.9301.1 None \(2\) \(-4\) \(4\) \(11\) \(+\) \(-\) \(+\) \(q+(1+\beta _{2})q^{2}-q^{3}+(2+\beta _{2}-\beta _{3})q^{4}+\cdots\)
1005.2.a.g \(5\) \(8.025\) 5.5.273397.1 None \(-2\) \(-5\) \(5\) \(-9\) \(+\) \(-\) \(-\) \(q-\beta _{1}q^{2}-q^{3}+(1+\beta _{1}+\beta _{2})q^{4}+q^{5}+\cdots\)
1005.2.a.h \(5\) \(8.025\) 5.5.772525.1 None \(1\) \(5\) \(-5\) \(-3\) \(-\) \(+\) \(+\) \(q+\beta _{1}q^{2}+q^{3}+(1+\beta _{2})q^{4}-q^{5}+\beta _{1}q^{6}+\cdots\)
1005.2.a.i \(7\) \(8.025\) \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(4\) \(7\) \(7\) \(3\) \(-\) \(-\) \(-\) \(q+(1-\beta _{1})q^{2}+q^{3}+(2-\beta _{1}+\beta _{2})q^{4}+\cdots\)
1005.2.a.j \(8\) \(8.025\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-3\) \(-8\) \(-8\) \(-3\) \(+\) \(+\) \(-\) \(q-\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}-q^{5}+\beta _{1}q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1005))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1005)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(67))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(201))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(335))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 2 T^{2} \))(\( 1 - T + 2 T^{2} \))(\( 1 + 4 T + 10 T^{2} + 19 T^{3} + 29 T^{4} + 38 T^{5} + 40 T^{6} + 32 T^{7} + 16 T^{8} \))(\( 1 + 2 T + 4 T^{2} + 7 T^{3} + 13 T^{4} + 14 T^{5} + 16 T^{6} + 16 T^{7} + 16 T^{8} \))(\( 1 + 4 T^{2} - T^{3} + 9 T^{4} - 2 T^{5} + 16 T^{6} + 16 T^{8} \))(\( 1 - 2 T + 2 T^{2} - 3 T^{3} + 7 T^{4} - 6 T^{5} + 8 T^{6} - 16 T^{7} + 16 T^{8} \))(\( 1 + 2 T + 4 T^{2} + 7 T^{3} + 13 T^{4} + 16 T^{5} + 26 T^{6} + 28 T^{7} + 32 T^{8} + 32 T^{9} + 32 T^{10} \))(\( 1 - T + 4 T^{2} - 3 T^{3} + 8 T^{4} - 5 T^{5} + 16 T^{6} - 12 T^{7} + 32 T^{8} - 16 T^{9} + 32 T^{10} \))(\( 1 - 4 T + 11 T^{2} - 23 T^{3} + 43 T^{4} - 73 T^{5} + 119 T^{6} - 176 T^{7} + 238 T^{8} - 292 T^{9} + 344 T^{10} - 368 T^{11} + 352 T^{12} - 256 T^{13} + 128 T^{14} \))(\( 1 + 3 T + 7 T^{2} + 12 T^{3} + 18 T^{4} + 20 T^{5} + 18 T^{6} + 11 T^{7} + 16 T^{8} + 22 T^{9} + 72 T^{10} + 160 T^{11} + 288 T^{12} + 384 T^{13} + 448 T^{14} + 384 T^{15} + 256 T^{16} \))
$3$ (\( 1 - T \))(\( 1 + T \))(\( ( 1 - T )^{4} \))(\( ( 1 - T )^{4} \))(\( ( 1 + T )^{4} \))(\( ( 1 + T )^{4} \))(\( ( 1 + T )^{5} \))(\( ( 1 - T )^{5} \))(\( ( 1 - T )^{7} \))(\( ( 1 + T )^{8} \))
$5$ (\( 1 + T \))(\( 1 - T \))(\( ( 1 - T )^{4} \))(\( ( 1 + T )^{4} \))(\( ( 1 + T )^{4} \))(\( ( 1 - T )^{4} \))(\( ( 1 - T )^{5} \))(\( ( 1 + T )^{5} \))(\( ( 1 - T )^{7} \))(\( ( 1 + T )^{8} \))
$7$ (\( 1 - 2 T + 7 T^{2} \))(\( 1 + 7 T^{2} \))(\( 1 + 9 T + 50 T^{2} + 191 T^{3} + 571 T^{4} + 1337 T^{5} + 2450 T^{6} + 3087 T^{7} + 2401 T^{8} \))(\( 1 + T + 12 T^{2} - 19 T^{3} + 45 T^{4} - 133 T^{5} + 588 T^{6} + 343 T^{7} + 2401 T^{8} \))(\( 1 - T + 24 T^{2} - 19 T^{3} + 241 T^{4} - 133 T^{5} + 1176 T^{6} - 343 T^{7} + 2401 T^{8} \))(\( 1 - 11 T + 68 T^{2} - 283 T^{3} + 869 T^{4} - 1981 T^{5} + 3332 T^{6} - 3773 T^{7} + 2401 T^{8} \))(\( 1 + 9 T + 51 T^{2} + 212 T^{3} + 727 T^{4} + 2072 T^{5} + 5089 T^{6} + 10388 T^{7} + 17493 T^{8} + 21609 T^{9} + 16807 T^{10} \))(\( 1 + 3 T + 25 T^{2} + 76 T^{3} + 301 T^{4} + 762 T^{5} + 2107 T^{6} + 3724 T^{7} + 8575 T^{8} + 7203 T^{9} + 16807 T^{10} \))(\( 1 - 3 T + 31 T^{2} - 88 T^{3} + 514 T^{4} - 1223 T^{5} + 5328 T^{6} - 10684 T^{7} + 37296 T^{8} - 59927 T^{9} + 176302 T^{10} - 211288 T^{11} + 521017 T^{12} - 352947 T^{13} + 823543 T^{14} \))(\( 1 + 3 T + 20 T^{2} + 45 T^{3} + 205 T^{4} + 315 T^{5} + 1112 T^{6} + 1129 T^{7} + 6444 T^{8} + 7903 T^{9} + 54488 T^{10} + 108045 T^{11} + 492205 T^{12} + 756315 T^{13} + 2352980 T^{14} + 2470629 T^{15} + 5764801 T^{16} \))
$11$ (\( 1 + 6 T + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 + 5 T + 22 T^{2} + 51 T^{3} + 223 T^{4} + 561 T^{5} + 2662 T^{6} + 6655 T^{7} + 14641 T^{8} \))(\( 1 + 5 T + 42 T^{2} + 135 T^{3} + 653 T^{4} + 1485 T^{5} + 5082 T^{6} + 6655 T^{7} + 14641 T^{8} \))(\( 1 + 3 T + 32 T^{2} + 83 T^{3} + 459 T^{4} + 913 T^{5} + 3872 T^{6} + 3993 T^{7} + 14641 T^{8} \))(\( 1 - T + 30 T^{2} - 5 T^{3} + 409 T^{4} - 55 T^{5} + 3630 T^{6} - 1331 T^{7} + 14641 T^{8} \))(\( 1 + T + 37 T^{2} + 56 T^{3} + 631 T^{4} + 992 T^{5} + 6941 T^{6} + 6776 T^{7} + 49247 T^{8} + 14641 T^{9} + 161051 T^{10} \))(\( 1 - 11 T + 87 T^{2} - 482 T^{3} + 2185 T^{4} - 7926 T^{5} + 24035 T^{6} - 58322 T^{7} + 115797 T^{8} - 161051 T^{9} + 161051 T^{10} \))(\( 1 - 5 T + 55 T^{2} - 260 T^{3} + 1512 T^{4} - 6217 T^{5} + 25450 T^{6} - 87196 T^{7} + 279950 T^{8} - 752257 T^{9} + 2012472 T^{10} - 3806660 T^{11} + 8857805 T^{12} - 8857805 T^{13} + 19487171 T^{14} \))(\( 1 + T + 40 T^{2} + 121 T^{3} + 841 T^{4} + 3559 T^{5} + 14452 T^{6} + 55671 T^{7} + 190980 T^{8} + 612381 T^{9} + 1748692 T^{10} + 4737029 T^{11} + 12313081 T^{12} + 19487171 T^{13} + 70862440 T^{14} + 19487171 T^{15} + 214358881 T^{16} \))
$13$ (\( 1 - 2 T + 13 T^{2} \))(\( 1 - 4 T + 13 T^{2} \))(\( 1 + 5 T + 23 T^{2} + 82 T^{3} + 263 T^{4} + 1066 T^{5} + 3887 T^{6} + 10985 T^{7} + 28561 T^{8} \))(\( 1 + 9 T + 55 T^{2} + 214 T^{3} + 821 T^{4} + 2782 T^{5} + 9295 T^{6} + 19773 T^{7} + 28561 T^{8} \))(\( 1 + 5 T + 51 T^{2} + 192 T^{3} + 989 T^{4} + 2496 T^{5} + 8619 T^{6} + 10985 T^{7} + 28561 T^{8} \))(\( 1 - 7 T + 55 T^{2} - 242 T^{3} + 1085 T^{4} - 3146 T^{5} + 9295 T^{6} - 15379 T^{7} + 28561 T^{8} \))(\( 1 + 7 T + 38 T^{2} + 131 T^{3} + 594 T^{4} + 2038 T^{5} + 7722 T^{6} + 22139 T^{7} + 83486 T^{8} + 199927 T^{9} + 371293 T^{10} \))(\( 1 + T + 24 T^{2} - 11 T^{3} + 398 T^{4} - 56 T^{5} + 5174 T^{6} - 1859 T^{7} + 52728 T^{8} + 28561 T^{9} + 371293 T^{10} \))(\( 1 + T + 74 T^{2} + 67 T^{3} + 2521 T^{4} + 1987 T^{5} + 51128 T^{6} + 33418 T^{7} + 664664 T^{8} + 335803 T^{9} + 5538637 T^{10} + 1913587 T^{11} + 27475682 T^{12} + 4826809 T^{13} + 62748517 T^{14} \))(\( 1 - T + 41 T^{2} - 30 T^{3} + 991 T^{4} - 906 T^{5} + 16883 T^{6} - 16459 T^{7} + 234792 T^{8} - 213967 T^{9} + 2853227 T^{10} - 1990482 T^{11} + 28303951 T^{12} - 11138790 T^{13} + 197899169 T^{14} - 62748517 T^{15} + 815730721 T^{16} \))
$17$ (\( 1 + 3 T + 17 T^{2} \))(\( 1 + 4 T + 17 T^{2} \))(\( 1 + 14 T + 128 T^{2} + 807 T^{3} + 3817 T^{4} + 13719 T^{5} + 36992 T^{6} + 68782 T^{7} + 83521 T^{8} \))(\( 1 + 14 T + 126 T^{2} + 787 T^{3} + 3705 T^{4} + 13379 T^{5} + 36414 T^{6} + 68782 T^{7} + 83521 T^{8} \))(\( 1 - 6 T + 52 T^{2} - 221 T^{3} + 1287 T^{4} - 3757 T^{5} + 15028 T^{6} - 29478 T^{7} + 83521 T^{8} \))(\( 1 + 38 T^{2} + 41 T^{3} + 781 T^{4} + 697 T^{5} + 10982 T^{6} + 83521 T^{8} \))(\( 1 + T + 45 T^{2} - T^{3} + 1206 T^{4} + 155 T^{5} + 20502 T^{6} - 289 T^{7} + 221085 T^{8} + 83521 T^{9} + 1419857 T^{10} \))(\( 1 - 12 T + 127 T^{2} - 855 T^{3} + 5013 T^{4} - 22106 T^{5} + 85221 T^{6} - 247095 T^{7} + 623951 T^{8} - 1002252 T^{9} + 1419857 T^{10} \))(\( 1 - 11 T + 141 T^{2} - 1045 T^{3} + 7693 T^{4} - 42302 T^{5} + 222987 T^{6} - 942348 T^{7} + 3790779 T^{8} - 12225278 T^{9} + 37795709 T^{10} - 87279445 T^{11} + 200199837 T^{12} - 265513259 T^{13} + 410338673 T^{14} \))(\( 1 + 7 T + 96 T^{2} + 442 T^{3} + 3338 T^{4} + 10181 T^{5} + 58600 T^{6} + 121086 T^{7} + 837090 T^{8} + 2058462 T^{9} + 16935400 T^{10} + 50019253 T^{11} + 278793098 T^{12} + 627576794 T^{13} + 2317206624 T^{14} + 2872370711 T^{15} + 6975757441 T^{16} \))
$19$ (\( 1 + T + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( 1 + 15 T + 137 T^{2} + 850 T^{3} + 4183 T^{4} + 16150 T^{5} + 49457 T^{6} + 102885 T^{7} + 130321 T^{8} \))(\( 1 + 11 T + 69 T^{2} + 394 T^{3} + 1979 T^{4} + 7486 T^{5} + 24909 T^{6} + 75449 T^{7} + 130321 T^{8} \))(\( 1 + 15 T + 129 T^{2} + 746 T^{3} + 3563 T^{4} + 14174 T^{5} + 46569 T^{6} + 102885 T^{7} + 130321 T^{8} \))(\( 1 - 5 T + 53 T^{2} - 206 T^{3} + 1423 T^{4} - 3914 T^{5} + 19133 T^{6} - 34295 T^{7} + 130321 T^{8} \))(\( 1 + 12 T + 131 T^{2} + 912 T^{3} + 5592 T^{4} + 26011 T^{5} + 106248 T^{6} + 329232 T^{7} + 898529 T^{8} + 1563852 T^{9} + 2476099 T^{10} \))(\( 1 - 9 T + 116 T^{2} - 697 T^{3} + 4802 T^{4} - 19984 T^{5} + 91238 T^{6} - 251617 T^{7} + 795644 T^{8} - 1172889 T^{9} + 2476099 T^{10} \))(\( 1 - 8 T + 89 T^{2} - 528 T^{3} + 3935 T^{4} - 19229 T^{5} + 109927 T^{6} - 441582 T^{7} + 2088613 T^{8} - 6941669 T^{9} + 26990165 T^{10} - 68809488 T^{11} + 220372811 T^{12} - 376367048 T^{13} + 893871739 T^{14} \))(\( 1 - 28 T + 428 T^{2} - 4572 T^{3} + 37878 T^{4} - 257677 T^{5} + 1498536 T^{6} - 7666133 T^{7} + 35171082 T^{8} - 145656527 T^{9} + 540971496 T^{10} - 1767406543 T^{11} + 4936298838 T^{12} - 11320724628 T^{13} + 20135637068 T^{14} - 25028408692 T^{15} + 16983563041 T^{16} \))
$23$ (\( 1 + 9 T + 23 T^{2} \))(\( 1 - 8 T + 23 T^{2} \))(\( 1 + 10 T + 80 T^{2} + 504 T^{3} + 2953 T^{4} + 11592 T^{5} + 42320 T^{6} + 121670 T^{7} + 279841 T^{8} \))(\( 1 - 6 T + 60 T^{2} - 216 T^{3} + 1461 T^{4} - 4968 T^{5} + 31740 T^{6} - 73002 T^{7} + 279841 T^{8} \))(\( 1 - 8 T + 62 T^{2} - 336 T^{3} + 1983 T^{4} - 7728 T^{5} + 32798 T^{6} - 97336 T^{7} + 279841 T^{8} \))(\( 1 + 2 T + 72 T^{2} + 104 T^{3} + 2289 T^{4} + 2392 T^{5} + 38088 T^{6} + 24334 T^{7} + 279841 T^{8} \))(\( 1 + 9 T + 113 T^{2} + 666 T^{3} + 4817 T^{4} + 20995 T^{5} + 110791 T^{6} + 352314 T^{7} + 1374871 T^{8} + 2518569 T^{9} + 6436343 T^{10} \))(\( 1 - 4 T + 37 T^{2} - 288 T^{3} + 1793 T^{4} - 5636 T^{5} + 41239 T^{6} - 152352 T^{7} + 450179 T^{8} - 1119364 T^{9} + 6436343 T^{10} \))(\( 1 - 11 T + 171 T^{2} - 1380 T^{3} + 12168 T^{4} - 75196 T^{5} + 472258 T^{6} - 2266394 T^{7} + 10861934 T^{8} - 39778684 T^{9} + 148048056 T^{10} - 386180580 T^{11} + 1100614653 T^{12} - 1628394779 T^{13} + 3404825447 T^{14} \))(\( 1 + 11 T + 162 T^{2} + 1093 T^{3} + 9033 T^{4} + 41312 T^{5} + 257590 T^{6} + 902626 T^{7} + 5677020 T^{8} + 20760398 T^{9} + 136265110 T^{10} + 502643104 T^{11} + 2527803753 T^{12} + 7034922899 T^{13} + 23981814018 T^{14} + 37453079917 T^{15} + 78310985281 T^{16} \))
$29$ (\( 1 - 3 T + 29 T^{2} \))(\( 1 + 2 T + 29 T^{2} \))(\( 1 + 17 T + 173 T^{2} + 1164 T^{3} + 6793 T^{4} + 33756 T^{5} + 145493 T^{6} + 414613 T^{7} + 707281 T^{8} \))(\( 1 + 21 T + 269 T^{2} + 2264 T^{3} + 14279 T^{4} + 65656 T^{5} + 226229 T^{6} + 512169 T^{7} + 707281 T^{8} \))(\( 1 - 3 T + 15 T^{2} + 32 T^{3} + 771 T^{4} + 928 T^{5} + 12615 T^{6} - 73167 T^{7} + 707281 T^{8} \))(\( 1 + 11 T + 125 T^{2} + 856 T^{3} + 5657 T^{4} + 24824 T^{5} + 105125 T^{6} + 268279 T^{7} + 707281 T^{8} \))(\( 1 - 6 T + 129 T^{2} - 676 T^{3} + 7068 T^{4} - 29103 T^{5} + 204972 T^{6} - 568516 T^{7} + 3146181 T^{8} - 4243686 T^{9} + 20511149 T^{10} \))(\( 1 - 11 T + 178 T^{2} - 1291 T^{3} + 11238 T^{4} - 56354 T^{5} + 325902 T^{6} - 1085731 T^{7} + 4341242 T^{8} - 7780091 T^{9} + 20511149 T^{10} \))(\( 1 + 153 T^{2} + 82 T^{3} + 11011 T^{4} + 7597 T^{5} + 486831 T^{6} + 302674 T^{7} + 14118099 T^{8} + 6389077 T^{9} + 268547279 T^{10} + 57997042 T^{11} + 3138205797 T^{12} + 17249876309 T^{14} \))(\( 1 - 2 T + 132 T^{2} - 46 T^{3} + 8898 T^{4} + 2309 T^{5} + 416254 T^{6} + 184407 T^{7} + 14023366 T^{8} + 5347803 T^{9} + 350069614 T^{10} + 56314201 T^{11} + 6293386338 T^{12} - 943512854 T^{13} + 78516678372 T^{14} - 34499752618 T^{15} + 500246412961 T^{16} \))
$31$ (\( 1 + 4 T + 31 T^{2} \))(\( 1 - 2 T + 31 T^{2} \))(\( 1 - T + 10 T^{2} - 185 T^{3} + 567 T^{4} - 5735 T^{5} + 9610 T^{6} - 29791 T^{7} + 923521 T^{8} \))(\( 1 + 7 T + 66 T^{2} + 257 T^{3} + 2259 T^{4} + 7967 T^{5} + 63426 T^{6} + 208537 T^{7} + 923521 T^{8} \))(\( 1 + 13 T + 144 T^{2} + 1017 T^{3} + 6677 T^{4} + 31527 T^{5} + 138384 T^{6} + 387283 T^{7} + 923521 T^{8} \))(\( 1 - 9 T + 84 T^{2} - 485 T^{3} + 3101 T^{4} - 15035 T^{5} + 80724 T^{6} - 268119 T^{7} + 923521 T^{8} \))(\( 1 + T + 27 T^{2} - 162 T^{3} - 67 T^{4} - 8298 T^{5} - 2077 T^{6} - 155682 T^{7} + 804357 T^{8} + 923521 T^{9} + 28629151 T^{10} \))(\( 1 - T + 115 T^{2} - 168 T^{3} + 5887 T^{4} - 8492 T^{5} + 182497 T^{6} - 161448 T^{7} + 3425965 T^{8} - 923521 T^{9} + 28629151 T^{10} \))(\( 1 + 3 T + 121 T^{2} + 424 T^{3} + 5638 T^{4} + 26819 T^{5} + 151054 T^{6} + 1026708 T^{7} + 4682674 T^{8} + 25773059 T^{9} + 167961658 T^{10} + 391572904 T^{11} + 3464127271 T^{12} + 2662511043 T^{13} + 27512614111 T^{14} \))(\( 1 - 15 T + 306 T^{2} - 3169 T^{3} + 36843 T^{4} - 288427 T^{5} + 2400478 T^{6} - 14728753 T^{7} + 94404168 T^{8} - 456591343 T^{9} + 2306859358 T^{10} - 8592528757 T^{11} + 34025284203 T^{12} - 90725779519 T^{13} + 271576126386 T^{14} - 412689211665 T^{15} + 852891037441 T^{16} \))
$37$ (\( 1 + 7 T + 37 T^{2} \))(\( 1 - 6 T + 37 T^{2} \))(\( 1 - 4 T + 90 T^{2} - 520 T^{3} + 4011 T^{4} - 19240 T^{5} + 123210 T^{6} - 202612 T^{7} + 1874161 T^{8} \))(\( 1 - 6 T + 92 T^{2} - 756 T^{3} + 4045 T^{4} - 27972 T^{5} + 125948 T^{6} - 303918 T^{7} + 1874161 T^{8} \))(\( 1 + 78 T^{2} + 136 T^{3} + 3223 T^{4} + 5032 T^{5} + 106782 T^{6} + 1874161 T^{8} \))(\( 1 - 12 T + 90 T^{2} - 184 T^{3} + 427 T^{4} - 6808 T^{5} + 123210 T^{6} - 607836 T^{7} + 1874161 T^{8} \))(\( 1 + 9 T + 131 T^{2} + 1062 T^{3} + 8277 T^{4} + 55727 T^{5} + 306249 T^{6} + 1453878 T^{7} + 6635543 T^{8} + 16867449 T^{9} + 69343957 T^{10} \))(\( 1 + 6 T + 147 T^{2} + 696 T^{3} + 9797 T^{4} + 35726 T^{5} + 362489 T^{6} + 952824 T^{7} + 7445991 T^{8} + 11244966 T^{9} + 69343957 T^{10} \))(\( 1 + 5 T + 173 T^{2} + 924 T^{3} + 14388 T^{4} + 77116 T^{5} + 765750 T^{6} + 3682550 T^{7} + 28332750 T^{8} + 105571804 T^{9} + 728795364 T^{10} + 1731724764 T^{11} + 11996504561 T^{12} + 12828632045 T^{13} + 94931877133 T^{14} \))(\( 1 + 3 T + 68 T^{2} + 235 T^{3} + 5309 T^{4} + 17712 T^{5} + 214178 T^{6} + 867814 T^{7} + 10310776 T^{8} + 32109118 T^{9} + 293209682 T^{10} + 897165936 T^{11} + 9949920749 T^{12} + 16295829895 T^{13} + 174469395812 T^{14} + 284795631399 T^{15} + 3512479453921 T^{16} \))
$41$ (\( 1 - 6 T + 41 T^{2} \))(\( 1 - 6 T + 41 T^{2} \))(\( 1 - T + 104 T^{2} - 107 T^{3} + 5313 T^{4} - 4387 T^{5} + 174824 T^{6} - 68921 T^{7} + 2825761 T^{8} \))(\( 1 + 29 T + 402 T^{2} + 3705 T^{3} + 26351 T^{4} + 151905 T^{5} + 675762 T^{6} + 1998709 T^{7} + 2825761 T^{8} \))(\( 1 - 7 T + 74 T^{2} - 95 T^{3} + 1473 T^{4} - 3895 T^{5} + 124394 T^{6} - 482447 T^{7} + 2825761 T^{8} \))(\( 1 + 9 T + 74 T^{2} + 351 T^{3} + 1491 T^{4} + 14391 T^{5} + 124394 T^{6} + 620289 T^{7} + 2825761 T^{8} \))(\( 1 + T + 91 T^{2} + 12 T^{3} + 5241 T^{4} - 1716 T^{5} + 214881 T^{6} + 20172 T^{7} + 6271811 T^{8} + 2825761 T^{9} + 115856201 T^{10} \))(\( 1 - 19 T + 223 T^{2} - 1526 T^{3} + 8361 T^{4} - 41536 T^{5} + 342801 T^{6} - 2565206 T^{7} + 15369383 T^{8} - 53689459 T^{9} + 115856201 T^{10} \))(\( 1 - T + 219 T^{2} - 186 T^{3} + 22068 T^{4} - 16565 T^{5} + 1356280 T^{6} - 870784 T^{7} + 55607480 T^{8} - 27845765 T^{9} + 1520948628 T^{10} - 525591546 T^{11} + 25372508019 T^{12} - 4750104241 T^{13} + 194754273881 T^{14} \))(\( 1 - 5 T + 184 T^{2} - 1129 T^{3} + 19423 T^{4} - 108459 T^{5} + 1361428 T^{6} - 6691655 T^{7} + 65710152 T^{8} - 274357855 T^{9} + 2288560468 T^{10} - 7475102739 T^{11} + 54884755903 T^{12} - 130801650929 T^{13} + 874019180344 T^{14} - 973771369405 T^{15} + 7984925229121 T^{16} \))
$43$ (\( 1 + 10 T + 43 T^{2} \))(\( 1 + 43 T^{2} \))(\( 1 + 19 T + 270 T^{2} + 2407 T^{3} + 18531 T^{4} + 103501 T^{5} + 499230 T^{6} + 1510633 T^{7} + 3418801 T^{8} \))(\( 1 + T + 86 T^{2} - 31 T^{3} + 4693 T^{4} - 1333 T^{5} + 159014 T^{6} + 79507 T^{7} + 3418801 T^{8} \))(\( 1 + 13 T + 186 T^{2} + 1639 T^{3} + 12301 T^{4} + 70477 T^{5} + 343914 T^{6} + 1033591 T^{7} + 3418801 T^{8} \))(\( 1 - 23 T + 338 T^{2} - 3331 T^{3} + 25361 T^{4} - 143233 T^{5} + 624962 T^{6} - 1828661 T^{7} + 3418801 T^{8} \))(\( 1 + 29 T + 527 T^{2} + 6494 T^{3} + 61823 T^{4} + 453268 T^{5} + 2658389 T^{6} + 12007406 T^{7} + 41900189 T^{8} + 99145229 T^{9} + 147008443 T^{10} \))(\( 1 - 9 T + 193 T^{2} - 1250 T^{3} + 15073 T^{4} - 74082 T^{5} + 648139 T^{6} - 2311250 T^{7} + 15344851 T^{8} - 30769209 T^{9} + 147008443 T^{10} \))(\( 1 + 3 T + 77 T^{2} + 92 T^{3} + 5452 T^{4} + 8199 T^{5} + 307840 T^{6} + 455092 T^{7} + 13237120 T^{8} + 15159951 T^{9} + 433472164 T^{10} + 314529692 T^{11} + 11319650111 T^{12} + 18964089147 T^{13} + 271818611107 T^{14} \))(\( 1 - 15 T + 136 T^{2} - 483 T^{3} + 2021 T^{4} - 3433 T^{5} - 2876 T^{6} + 1774083 T^{7} - 14440660 T^{8} + 76285569 T^{9} - 5317724 T^{10} - 272947531 T^{11} + 6909396821 T^{12} - 71005077969 T^{13} + 859705374664 T^{14} - 4077279166605 T^{15} + 11688200277601 T^{16} \))
$47$ (\( 1 - 9 T + 47 T^{2} \))(\( 1 - 12 T + 47 T^{2} \))(\( 1 + 7 T + 101 T^{2} + 830 T^{3} + 5123 T^{4} + 39010 T^{5} + 223109 T^{6} + 726761 T^{7} + 4879681 T^{8} \))(\( 1 + T + 101 T^{2} + 228 T^{3} + 6095 T^{4} + 10716 T^{5} + 223109 T^{6} + 103823 T^{7} + 4879681 T^{8} \))(\( 1 + 7 T + 167 T^{2} + 824 T^{3} + 11217 T^{4} + 38728 T^{5} + 368903 T^{6} + 726761 T^{7} + 4879681 T^{8} \))(\( 1 + 13 T + 175 T^{2} + 1594 T^{3} + 11699 T^{4} + 74918 T^{5} + 386575 T^{6} + 1349699 T^{7} + 4879681 T^{8} \))(\( 1 + 20 T + 217 T^{2} + 1642 T^{3} + 11404 T^{4} + 76615 T^{5} + 535988 T^{6} + 3627178 T^{7} + 22529591 T^{8} + 97593620 T^{9} + 229345007 T^{10} \))(\( 1 - 3 T + 212 T^{2} - 529 T^{3} + 18924 T^{4} - 36456 T^{5} + 889428 T^{6} - 1168561 T^{7} + 22010476 T^{8} - 14639043 T^{9} + 229345007 T^{10} \))(\( 1 - 10 T + 303 T^{2} - 2234 T^{3} + 38297 T^{4} - 220319 T^{5} + 2785015 T^{6} - 12922282 T^{7} + 130895705 T^{8} - 486684671 T^{9} + 3976109431 T^{10} - 10901207354 T^{11} + 69491537121 T^{12} - 107792153290 T^{13} + 506623120463 T^{14} \))(\( 1 + 6 T + 148 T^{2} + 1280 T^{3} + 14942 T^{4} + 113573 T^{5} + 1137996 T^{6} + 7371255 T^{7} + 60993090 T^{8} + 346448985 T^{9} + 2513833164 T^{10} + 11791489579 T^{11} + 72912193502 T^{12} + 293561608960 T^{13} + 1595323868692 T^{14} + 3039738722778 T^{15} + 23811286661761 T^{16} \))
$53$ (\( 1 + 12 T + 53 T^{2} \))(\( 1 + 2 T + 53 T^{2} \))(\( 1 + 11 T + 229 T^{2} + 1640 T^{3} + 18389 T^{4} + 86920 T^{5} + 643261 T^{6} + 1637647 T^{7} + 7890481 T^{8} \))(\( 1 - T + 191 T^{2} - 164 T^{3} + 14653 T^{4} - 8692 T^{5} + 536519 T^{6} - 148877 T^{7} + 7890481 T^{8} \))(\( 1 + 7 T + 151 T^{2} + 906 T^{3} + 10477 T^{4} + 48018 T^{5} + 424159 T^{6} + 1042139 T^{7} + 7890481 T^{8} \))(\( 1 + T + 93 T^{2} + 22 T^{3} + 6555 T^{4} + 1166 T^{5} + 261237 T^{6} + 148877 T^{7} + 7890481 T^{8} \))(\( 1 - 9 T + 286 T^{2} - 1903 T^{3} + 31382 T^{4} - 151140 T^{5} + 1663246 T^{6} - 5345527 T^{7} + 42578822 T^{8} - 71014329 T^{9} + 418195493 T^{10} \))(\( 1 - 9 T + 150 T^{2} - 1289 T^{3} + 13922 T^{4} - 88254 T^{5} + 737866 T^{6} - 3620801 T^{7} + 22331550 T^{8} - 71014329 T^{9} + 418195493 T^{10} \))(\( 1 - 3 T + 116 T^{2} - 5 T^{3} + 8361 T^{4} + 7871 T^{5} + 592234 T^{6} - 24670 T^{7} + 31388402 T^{8} + 22109639 T^{9} + 1244760597 T^{10} - 39452405 T^{11} + 48510677188 T^{12} - 66493083387 T^{13} + 1174711139837 T^{14} \))(\( 1 + 19 T + 373 T^{2} + 5038 T^{3} + 62291 T^{4} + 653034 T^{5} + 6205675 T^{6} + 52255769 T^{7} + 402171640 T^{8} + 2769555757 T^{9} + 17431741075 T^{10} + 97221742818 T^{11} + 491505951971 T^{12} + 2106868893734 T^{13} + 8267306701117 T^{14} + 22319511656903 T^{15} + 62259690411361 T^{16} \))
$59$ (\( 1 + 15 T + 59 T^{2} \))(\( 1 + 6 T + 59 T^{2} \))(\( 1 + 17 T + 235 T^{2} + 2402 T^{3} + 21209 T^{4} + 141718 T^{5} + 818035 T^{6} + 3491443 T^{7} + 12117361 T^{8} \))(\( 1 - T + 45 T^{2} + 258 T^{3} + 3803 T^{4} + 15222 T^{5} + 156645 T^{6} - 205379 T^{7} + 12117361 T^{8} \))(\( 1 + 23 T + 383 T^{2} + 4426 T^{3} + 38523 T^{4} + 261134 T^{5} + 1333223 T^{6} + 4723717 T^{7} + 12117361 T^{8} \))(\( 1 - 17 T + 283 T^{2} - 2798 T^{3} + 26585 T^{4} - 165082 T^{5} + 985123 T^{6} - 3491443 T^{7} + 12117361 T^{8} \))(\( 1 + 8 T + 283 T^{2} + 1788 T^{3} + 32806 T^{4} + 155275 T^{5} + 1935554 T^{6} + 6224028 T^{7} + 58122257 T^{8} + 96938888 T^{9} + 714924299 T^{10} \))(\( 1 - 21 T + 380 T^{2} - 4537 T^{3} + 48516 T^{4} - 392478 T^{5} + 2862444 T^{6} - 15793297 T^{7} + 78044020 T^{8} - 254464581 T^{9} + 714924299 T^{10} \))(\( 1 + 4 T + 235 T^{2} + 898 T^{3} + 26281 T^{4} + 104315 T^{5} + 1971233 T^{6} + 7615250 T^{7} + 116302747 T^{8} + 363120515 T^{9} + 5397565499 T^{10} + 10881390178 T^{11} + 168007210265 T^{12} + 168722134564 T^{13} + 2488651484819 T^{14} \))(\( 1 - 18 T + 340 T^{2} - 4366 T^{3} + 55750 T^{4} - 575249 T^{5} + 5761890 T^{6} - 48961049 T^{7} + 405749798 T^{8} - 2888701891 T^{9} + 20057139090 T^{10} - 118144064371 T^{11} + 675542875750 T^{12} - 3121359489434 T^{13} + 14341381437940 T^{14} - 44795726726742 T^{15} + 146830437604321 T^{16} \))
$61$ (\( 1 - 8 T + 61 T^{2} \))(\( 1 - 14 T + 61 T^{2} \))(\( 1 + 11 T + 169 T^{2} + 1452 T^{3} + 15613 T^{4} + 88572 T^{5} + 628849 T^{6} + 2496791 T^{7} + 13845841 T^{8} \))(\( 1 + 3 T + 113 T^{2} + 146 T^{3} + 7545 T^{4} + 8906 T^{5} + 420473 T^{6} + 680943 T^{7} + 13845841 T^{8} \))(\( 1 + 35 T + 679 T^{2} + 8658 T^{3} + 79435 T^{4} + 528138 T^{5} + 2526559 T^{6} + 7944335 T^{7} + 13845841 T^{8} \))(\( 1 + 5 T + 135 T^{2} + 474 T^{3} + 8631 T^{4} + 28914 T^{5} + 502335 T^{6} + 1134905 T^{7} + 13845841 T^{8} \))(\( 1 - 5 T + 276 T^{2} - 1177 T^{3} + 32142 T^{4} - 106212 T^{5} + 1960662 T^{6} - 4379617 T^{7} + 62646756 T^{8} - 69229205 T^{9} + 844596301 T^{10} \))(\( 1 + 7 T + 132 T^{2} - 117 T^{3} + 1418 T^{4} - 66142 T^{5} + 86498 T^{6} - 435357 T^{7} + 29961492 T^{8} + 96920887 T^{9} + 844596301 T^{10} \))(\( 1 + 9 T + 208 T^{2} + 1433 T^{3} + 21559 T^{4} + 149839 T^{5} + 1726296 T^{6} + 11206958 T^{7} + 105304056 T^{8} + 557550919 T^{9} + 4893483379 T^{10} + 19841090153 T^{11} + 175676030608 T^{12} + 463683369249 T^{13} + 3142742836021 T^{14} \))(\( 1 - 13 T + 249 T^{2} - 2276 T^{3} + 22031 T^{4} - 136368 T^{5} + 786367 T^{6} - 2878763 T^{7} + 17491408 T^{8} - 175604543 T^{9} + 2926071607 T^{10} - 30952945008 T^{11} + 305037723071 T^{12} - 1922301181076 T^{13} + 12828573215889 T^{14} - 40855656868273 T^{15} + 191707312997281 T^{16} \))
$67$ (\( 1 - T \))(\( 1 + T \))(\( ( 1 + T )^{4} \))(\( ( 1 - T )^{4} \))(\( ( 1 + T )^{4} \))(\( ( 1 + T )^{4} \))(\( ( 1 - T )^{5} \))(\( ( 1 + T )^{5} \))(\( ( 1 - T )^{7} \))(\( ( 1 - T )^{8} \))
$71$ (\( 1 + 12 T + 71 T^{2} \))(\( 1 - 14 T + 71 T^{2} \))(\( 1 - 22 T + 358 T^{2} - 3751 T^{3} + 36155 T^{4} - 266321 T^{5} + 1804678 T^{6} - 7874042 T^{7} + 25411681 T^{8} \))(\( 1 - 2 T + 258 T^{2} - 399 T^{3} + 26705 T^{4} - 28329 T^{5} + 1300578 T^{6} - 715822 T^{7} + 25411681 T^{8} \))(\( 1 + 128 T^{2} + 659 T^{3} + 8377 T^{4} + 46789 T^{5} + 645248 T^{6} + 25411681 T^{8} \))(\( 1 + 14 T + 126 T^{2} - 193 T^{3} - 4161 T^{4} - 13703 T^{5} + 635166 T^{6} + 5010754 T^{7} + 25411681 T^{8} \))(\( 1 + 4 T + 315 T^{2} + 1095 T^{3} + 41951 T^{4} + 115214 T^{5} + 2978521 T^{6} + 5519895 T^{7} + 112741965 T^{8} + 101646724 T^{9} + 1804229351 T^{10} \))(\( 1 - 6 T + 255 T^{2} - 1255 T^{3} + 30851 T^{4} - 120132 T^{5} + 2190421 T^{6} - 6326455 T^{7} + 91267305 T^{8} - 152470086 T^{9} + 1804229351 T^{10} \))(\( 1 - 6 T + 373 T^{2} - 1867 T^{3} + 65584 T^{4} - 278000 T^{5} + 7076796 T^{6} - 24924110 T^{7} + 502452516 T^{8} - 1401398000 T^{9} + 23473235024 T^{10} - 47443608427 T^{11} + 672977547923 T^{12} - 768601703526 T^{13} + 9095120158391 T^{14} \))(\( 1 - 4 T + 318 T^{2} - 583 T^{3} + 46377 T^{4} + 13841 T^{5} + 4375642 T^{6} + 7805764 T^{7} + 330545308 T^{8} + 554209244 T^{9} + 22057611322 T^{10} + 4953846151 T^{11} + 1178517529737 T^{12} - 1051865711633 T^{13} + 40735890286878 T^{14} - 36380480633564 T^{15} + 645753531245761 T^{16} \))
$73$ (\( 1 - 11 T + 73 T^{2} \))(\( 1 - 2 T + 73 T^{2} \))(\( 1 - 16 T + 138 T^{2} - 1741 T^{3} + 21079 T^{4} - 127093 T^{5} + 735402 T^{6} - 6224272 T^{7} + 28398241 T^{8} \))(\( 1 + 4 T + 70 T^{2} - 201 T^{3} + 581 T^{4} - 14673 T^{5} + 373030 T^{6} + 1556068 T^{7} + 28398241 T^{8} \))(\( 1 + 6 T + 118 T^{2} + 755 T^{3} + 12739 T^{4} + 55115 T^{5} + 628822 T^{6} + 2334102 T^{7} + 28398241 T^{8} \))(\( 1 + 2 T + 188 T^{2} + 431 T^{3} + 17035 T^{4} + 31463 T^{5} + 1001852 T^{6} + 778034 T^{7} + 28398241 T^{8} \))(\( 1 + 31 T + 709 T^{2} + 10639 T^{3} + 131178 T^{4} + 1223747 T^{5} + 9575994 T^{6} + 56695231 T^{7} + 275813053 T^{8} + 880345471 T^{9} + 2073071593 T^{10} \))(\( 1 - 6 T + 237 T^{2} - 2097 T^{3} + 25013 T^{4} - 242224 T^{5} + 1825949 T^{6} - 11174913 T^{7} + 92197029 T^{8} - 170389446 T^{9} + 2073071593 T^{10} \))(\( 1 + 9 T + 307 T^{2} + 1049 T^{3} + 32123 T^{4} - 49894 T^{5} + 1981485 T^{6} - 10888384 T^{7} + 144648405 T^{8} - 265885126 T^{9} + 12496393091 T^{10} + 29789754809 T^{11} + 636432979051 T^{12} + 1362008036601 T^{13} + 11047398519097 T^{14} \))(\( 1 - 11 T + 292 T^{2} - 2152 T^{3} + 40918 T^{4} - 244691 T^{5} + 4358362 T^{6} - 24400938 T^{7} + 371863318 T^{8} - 1781268474 T^{9} + 23225711098 T^{10} - 95188958747 T^{11} + 1161999225238 T^{12} - 4461250068136 T^{13} + 44189594076388 T^{14} - 121521383710067 T^{15} + 806460091894081 T^{16} \))
$79$ (\( 1 + 4 T + 79 T^{2} \))(\( 1 + 10 T + 79 T^{2} \))(\( 1 - 5 T + 204 T^{2} - 569 T^{3} + 19293 T^{4} - 44951 T^{5} + 1273164 T^{6} - 2465195 T^{7} + 38950081 T^{8} \))(\( 1 + 19 T + 274 T^{2} + 3111 T^{3} + 29969 T^{4} + 245769 T^{5} + 1710034 T^{6} + 9367741 T^{7} + 38950081 T^{8} \))(\( 1 + 19 T + 258 T^{2} + 1701 T^{3} + 15083 T^{4} + 134379 T^{5} + 1610178 T^{6} + 9367741 T^{7} + 38950081 T^{8} \))(\( 1 - 11 T + 330 T^{2} - 2453 T^{3} + 39303 T^{4} - 193787 T^{5} + 2059530 T^{6} - 5423429 T^{7} + 38950081 T^{8} \))(\( 1 + 19 T + 283 T^{2} + 3278 T^{3} + 32683 T^{4} + 279898 T^{5} + 2581957 T^{6} + 20457998 T^{7} + 139530037 T^{8} + 740051539 T^{9} + 3077056399 T^{10} \))(\( 1 + 15 T + 357 T^{2} + 4096 T^{3} + 55935 T^{4} + 457756 T^{5} + 4418865 T^{6} + 25563136 T^{7} + 176014923 T^{8} + 584251215 T^{9} + 3077056399 T^{10} \))(\( 1 + 11 T + 265 T^{2} + 2306 T^{3} + 44836 T^{4} + 361043 T^{5} + 4937068 T^{6} + 32081136 T^{7} + 390028372 T^{8} + 2253269363 T^{9} + 22105896604 T^{10} + 89818886786 T^{11} + 815419945735 T^{12} + 2673962010731 T^{13} + 19203908986159 T^{14} \))(\( 1 - 33 T + 998 T^{2} - 19359 T^{3} + 341613 T^{4} - 4717225 T^{5} + 59741882 T^{6} - 627920035 T^{7} + 6081406052 T^{8} - 49605682765 T^{9} + 372849085562 T^{10} - 2325775896775 T^{11} + 13305854020653 T^{12} - 59568734828241 T^{13} + 242601280609958 T^{14} - 633728996543247 T^{15} + 1517108809906561 T^{16} \))
$83$ (\( 1 + 12 T + 83 T^{2} \))(\( 1 + 16 T + 83 T^{2} \))(\( 1 + 12 T + 266 T^{2} + 2603 T^{3} + 30789 T^{4} + 216049 T^{5} + 1832474 T^{6} + 6861444 T^{7} + 47458321 T^{8} \))(\( 1 + 2 T + 114 T^{2} + 449 T^{3} + 14765 T^{4} + 37267 T^{5} + 785346 T^{6} + 1143574 T^{7} + 47458321 T^{8} \))(\( 1 + 10 T + 242 T^{2} + 1745 T^{3} + 27947 T^{4} + 144835 T^{5} + 1667138 T^{6} + 5717870 T^{7} + 47458321 T^{8} \))(\( 1 - 18 T + 420 T^{2} - 4553 T^{3} + 55785 T^{4} - 377899 T^{5} + 2893380 T^{6} - 10292166 T^{7} + 47458321 T^{8} \))(\( 1 + 20 T + 529 T^{2} + 6723 T^{3} + 96813 T^{4} + 840510 T^{5} + 8035479 T^{6} + 46314747 T^{7} + 302475323 T^{8} + 949166420 T^{9} + 3939040643 T^{10} \))(\( 1 + 303 T^{2} + 113 T^{3} + 43103 T^{4} + 14634 T^{5} + 3577549 T^{6} + 778457 T^{7} + 173251461 T^{8} + 3939040643 T^{10} \))(\( 1 - 30 T + 669 T^{2} - 9803 T^{3} + 126106 T^{4} - 1286038 T^{5} + 12772542 T^{6} - 112810898 T^{7} + 1060120986 T^{8} - 8859515782 T^{9} + 72105771422 T^{10} - 465233920763 T^{11} + 2635218190167 T^{12} - 9808211201070 T^{13} + 27136050989627 T^{14} \))(\( 1 - 44 T + 1228 T^{2} - 23263 T^{3} + 342959 T^{4} - 3986161 T^{5} + 39431064 T^{6} - 347954706 T^{7} + 3109391720 T^{8} - 28880240598 T^{9} + 271640599896 T^{10} - 2279235039707 T^{11} + 16276258311839 T^{12} - 91633902478109 T^{13} + 401482778497132 T^{14} - 1193986243543588 T^{15} + 2252292232139041 T^{16} \))
$89$ (\( 1 - 15 T + 89 T^{2} \))(\( 1 + 18 T + 89 T^{2} \))(\( 1 - 16 T + 236 T^{2} - 2399 T^{3} + 21365 T^{4} - 213511 T^{5} + 1869356 T^{6} - 11279504 T^{7} + 62742241 T^{8} \))(\( 1 + 32 T + 612 T^{2} + 8509 T^{3} + 92139 T^{4} + 757301 T^{5} + 4847652 T^{6} + 22559008 T^{7} + 62742241 T^{8} \))(\( 1 - 2 T + 190 T^{2} - 617 T^{3} + 22787 T^{4} - 54913 T^{5} + 1504990 T^{6} - 1409938 T^{7} + 62742241 T^{8} \))(\( 1 - 16 T + 204 T^{2} - 2513 T^{3} + 31369 T^{4} - 223657 T^{5} + 1615884 T^{6} - 11279504 T^{7} + 62742241 T^{8} \))(\( 1 - 9 T + 89 T^{2} - 689 T^{3} + 12370 T^{4} - 174381 T^{5} + 1100930 T^{6} - 5457569 T^{7} + 62742241 T^{8} - 564680169 T^{9} + 5584059449 T^{10} \))(\( 1 - 16 T + 507 T^{2} - 5589 T^{3} + 94823 T^{4} - 740144 T^{5} + 8439247 T^{6} - 44270469 T^{7} + 357419283 T^{8} - 1003875856 T^{9} + 5584059449 T^{10} \))(\( 1 - 13 T + 443 T^{2} - 5153 T^{3} + 95527 T^{4} - 981724 T^{5} + 12777365 T^{6} - 110629596 T^{7} + 1137185485 T^{8} - 7776235804 T^{9} + 67343573663 T^{10} - 323310767873 T^{11} + 2473738335907 T^{12} - 6460756782493 T^{13} + 44231334895529 T^{14} \))(\( 1 - 5 T + 342 T^{2} + 216 T^{3} + 48954 T^{4} + 284307 T^{5} + 5337544 T^{6} + 43342794 T^{7} + 528885070 T^{8} + 3857508666 T^{9} + 42278686024 T^{10} + 200427621483 T^{11} + 3071483665914 T^{12} + 1206156840984 T^{13} + 169967601508662 T^{14} - 221156674477645 T^{15} + 3936588805702081 T^{16} \))
$97$ (\( 1 - 2 T + 97 T^{2} \))(\( 1 + 8 T + 97 T^{2} \))(\( 1 + 7 T + 243 T^{2} + 1894 T^{3} + 28531 T^{4} + 183718 T^{5} + 2286387 T^{6} + 6388711 T^{7} + 88529281 T^{8} \))(\( 1 - 25 T + 565 T^{2} - 7740 T^{3} + 91193 T^{4} - 750780 T^{5} + 5316085 T^{6} - 22816825 T^{7} + 88529281 T^{8} \))(\( 1 - 5 T + 183 T^{2} - 726 T^{3} + 25533 T^{4} - 70422 T^{5} + 1721847 T^{6} - 4563365 T^{7} + 88529281 T^{8} \))(\( 1 - 9 T + 241 T^{2} - 1676 T^{3} + 28909 T^{4} - 162572 T^{5} + 2267569 T^{6} - 8214057 T^{7} + 88529281 T^{8} \))(\( 1 + 17 T + 340 T^{2} + 4581 T^{3} + 55228 T^{4} + 609166 T^{5} + 5357116 T^{6} + 43102629 T^{7} + 310308820 T^{8} + 1504997777 T^{9} + 8587340257 T^{10} \))(\( 1 - 5 T + 190 T^{2} - 1845 T^{3} + 32260 T^{4} - 179092 T^{5} + 3129220 T^{6} - 17359605 T^{7} + 173407870 T^{8} - 442646405 T^{9} + 8587340257 T^{10} \))(\( 1 + 7 T + 550 T^{2} + 3015 T^{3} + 137689 T^{4} + 606381 T^{5} + 20571332 T^{6} + 73458322 T^{7} + 1995419204 T^{8} + 5705438829 T^{9} + 125665032697 T^{10} + 266915782215 T^{11} + 4723037141350 T^{12} + 5830804034503 T^{13} + 80798284478113 T^{14} \))(\( 1 - 7 T + 325 T^{2} - 2478 T^{3} + 61127 T^{4} - 422922 T^{5} + 8377095 T^{6} - 50847841 T^{7} + 906753496 T^{8} - 4932240577 T^{9} + 78820086855 T^{10} - 385989490506 T^{11} + 5411529359687 T^{12} - 21279429156846 T^{13} + 270715901601925 T^{14} - 565587991346791 T^{15} + 7837433594376961 T^{16} \))
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