Properties

Label 805.2.a
Level 805
Weight 2
Character orbit a
Rep. character \(\chi_{805}(1,\cdot)\)
Character field \(\Q\)
Dimension 43
Newform subspaces 13
Sturm bound 192
Trace bound 11

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Defining parameters

Level: \( N \) \(=\) \( 805 = 5 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 805.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 13 \)
Sturm bound: \(192\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(2\), \(3\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(805))\).

Total New Old
Modular forms 100 43 57
Cusp forms 93 43 50
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.

\(5\)\(7\)\(23\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(4\)
\(+\)\(+\)\(-\)\(-\)\(9\)
\(+\)\(-\)\(+\)\(-\)\(4\)
\(+\)\(-\)\(-\)\(+\)\(5\)
\(-\)\(+\)\(+\)\(-\)\(4\)
\(-\)\(+\)\(-\)\(+\)\(5\)
\(-\)\(-\)\(+\)\(+\)\(4\)
\(-\)\(-\)\(-\)\(-\)\(8\)
Plus space\(+\)\(18\)
Minus space\(-\)\(25\)

Trace form

\( 43q + q^{2} + 4q^{3} + 37q^{4} - q^{5} + 8q^{6} - q^{7} + 9q^{8} + 59q^{9} + O(q^{10}) \) \( 43q + q^{2} + 4q^{3} + 37q^{4} - q^{5} + 8q^{6} - q^{7} + 9q^{8} + 59q^{9} - 3q^{10} + 8q^{12} - 14q^{13} - 3q^{14} + 13q^{16} + 6q^{17} - 7q^{18} + 4q^{19} - 7q^{20} - 12q^{22} + 11q^{23} + 20q^{24} + 43q^{25} + 2q^{26} + 16q^{27} - 7q^{28} - 26q^{29} + 20q^{30} - 16q^{31} + 29q^{32} + 48q^{33} + 10q^{34} + 7q^{35} + 69q^{36} - 22q^{37} - 12q^{38} - 12q^{39} - 15q^{40} + 22q^{41} - 12q^{42} + 4q^{43} + 28q^{44} - 13q^{45} - 3q^{46} - 40q^{47} - 32q^{48} + 43q^{49} + q^{50} - 20q^{51} - 22q^{52} - 46q^{53} - 20q^{54} - 12q^{55} - 15q^{56} - 46q^{58} - 12q^{59} - 12q^{60} - 38q^{61} - 28q^{62} - 13q^{63} - 47q^{64} - 10q^{65} - 96q^{66} - 28q^{67} - 70q^{68} - 4q^{69} + q^{70} - 40q^{71} - 75q^{72} + 14q^{73} - 50q^{74} + 4q^{75} + 12q^{76} + 4q^{77} - 28q^{78} - 28q^{79} + q^{80} + 131q^{81} + 46q^{82} - 20q^{83} - 12q^{84} + 2q^{85} - 68q^{86} + 24q^{87} + 12q^{88} - 10q^{89} - 39q^{90} - 2q^{91} + 17q^{92} + 40q^{93} - 12q^{94} + 20q^{95} + 40q^{96} + 14q^{97} + q^{98} + 20q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(805))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 5 7 23
805.2.a.a \(1\) \(6.428\) \(\Q\) None \(-1\) \(0\) \(-1\) \(-1\) \(+\) \(+\) \(-\) \(q-q^{2}-q^{4}-q^{5}-q^{7}+3q^{8}-3q^{9}+\cdots\)
805.2.a.b \(1\) \(6.428\) \(\Q\) None \(-1\) \(0\) \(-1\) \(-1\) \(+\) \(+\) \(-\) \(q-q^{2}-q^{4}-q^{5}-q^{7}+3q^{8}-3q^{9}+\cdots\)
805.2.a.c \(1\) \(6.428\) \(\Q\) None \(2\) \(-1\) \(-1\) \(-1\) \(+\) \(+\) \(+\) \(q+2q^{2}-q^{3}+2q^{4}-q^{5}-2q^{6}-q^{7}+\cdots\)
805.2.a.d \(1\) \(6.428\) \(\Q\) None \(2\) \(3\) \(-1\) \(-1\) \(+\) \(+\) \(-\) \(q+2q^{2}+3q^{3}+2q^{4}-q^{5}+6q^{6}+\cdots\)
805.2.a.e \(2\) \(6.428\) \(\Q(\sqrt{5}) \) None \(3\) \(0\) \(-2\) \(-2\) \(+\) \(+\) \(-\) \(q+(1+\beta )q^{2}+(1-2\beta )q^{3}+3\beta q^{4}-q^{5}+\cdots\)
805.2.a.f \(3\) \(6.428\) \(\Q(\zeta_{14})^+\) None \(-2\) \(0\) \(-3\) \(-3\) \(+\) \(+\) \(+\) \(q+(-1-\beta _{2})q^{2}+(-1+2\beta _{1}-\beta _{2})q^{3}+\cdots\)
805.2.a.g \(4\) \(6.428\) 4.4.2777.1 None \(-2\) \(-5\) \(4\) \(4\) \(-\) \(-\) \(+\) \(q+(-1+\beta _{3})q^{2}+(-1-\beta _{1})q^{3}+(1+\cdots)q^{4}+\cdots\)
805.2.a.h \(4\) \(6.428\) 4.4.22545.1 None \(-1\) \(0\) \(-4\) \(-4\) \(+\) \(+\) \(-\) \(q-\beta _{2}q^{2}+(\beta _{1}+\beta _{3})q^{3}+(3+\beta _{3})q^{4}+\cdots\)
805.2.a.i \(4\) \(6.428\) 4.4.7537.1 None \(1\) \(4\) \(-4\) \(4\) \(+\) \(-\) \(+\) \(q+\beta _{1}q^{2}+(1+\beta _{3})q^{3}+(1+\beta _{2})q^{4}+\cdots\)
805.2.a.j \(4\) \(6.428\) 4.4.2777.1 None \(3\) \(6\) \(4\) \(-4\) \(-\) \(+\) \(+\) \(q+(1-\beta _{1})q^{2}+(1+\beta _{3})q^{3}+(1-\beta _{1}+\cdots)q^{4}+\cdots\)
805.2.a.k \(5\) \(6.428\) 5.5.122821.1 None \(-3\) \(-6\) \(5\) \(-5\) \(-\) \(+\) \(-\) \(q+(-1+\beta _{1})q^{2}+(-1-\beta _{3})q^{3}+(\beta _{3}+\cdots)q^{4}+\cdots\)
805.2.a.l \(5\) \(6.428\) 5.5.255877.1 None \(-1\) \(-4\) \(-5\) \(5\) \(+\) \(-\) \(-\) \(q-\beta _{1}q^{2}+(-1-\beta _{4})q^{3}+(1+\beta _{2})q^{4}+\cdots\)
805.2.a.m \(8\) \(6.428\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(1\) \(7\) \(8\) \(8\) \(-\) \(-\) \(-\) \(q+\beta _{1}q^{2}+(1+\beta _{6})q^{3}+(1+\beta _{2})q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(805))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(805)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(115))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(161))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T + 2 T^{2} \))(\( 1 + T + 2 T^{2} \))(\( 1 - 2 T + 2 T^{2} \))(\( 1 - 2 T + 2 T^{2} \))(\( 1 - 3 T + 5 T^{2} - 6 T^{3} + 4 T^{4} \))(\( 1 + 2 T + 5 T^{2} + 7 T^{3} + 10 T^{4} + 8 T^{5} + 8 T^{6} \))(\( 1 + 2 T + 5 T^{2} + 9 T^{3} + 14 T^{4} + 18 T^{5} + 20 T^{6} + 16 T^{7} + 16 T^{8} \))(\( 1 + T - T^{2} - 2 T^{3} + T^{4} - 4 T^{5} - 4 T^{6} + 8 T^{7} + 16 T^{8} \))(\( 1 - T + 3 T^{2} - 2 T^{3} + 7 T^{4} - 4 T^{5} + 12 T^{6} - 8 T^{7} + 16 T^{8} \))(\( 1 - 3 T + 7 T^{2} - 12 T^{3} + 19 T^{4} - 24 T^{5} + 28 T^{6} - 24 T^{7} + 16 T^{8} \))(\( 1 + 3 T + 8 T^{2} + 14 T^{3} + 24 T^{4} + 33 T^{5} + 48 T^{6} + 56 T^{7} + 64 T^{8} + 48 T^{9} + 32 T^{10} \))(\( 1 + T + 4 T^{2} + 2 T^{3} + 10 T^{4} + 5 T^{5} + 20 T^{6} + 8 T^{7} + 32 T^{8} + 16 T^{9} + 32 T^{10} \))(\( 1 - T + 5 T^{2} - 5 T^{3} + 16 T^{4} - 17 T^{5} + 46 T^{6} - 41 T^{7} + 102 T^{8} - 82 T^{9} + 184 T^{10} - 136 T^{11} + 256 T^{12} - 160 T^{13} + 320 T^{14} - 128 T^{15} + 256 T^{16} \))
$3$ (\( 1 + 3 T^{2} \))(\( 1 + 3 T^{2} \))(\( 1 + T + 3 T^{2} \))(\( 1 - 3 T + 3 T^{2} \))(\( 1 + T^{2} + 9 T^{4} \))(\( 1 + 2 T^{2} + 7 T^{3} + 6 T^{4} + 27 T^{6} \))(\( 1 + 5 T + 17 T^{2} + 43 T^{3} + 83 T^{4} + 129 T^{5} + 153 T^{6} + 135 T^{7} + 81 T^{8} \))(\( 1 - 3 T^{2} - 3 T^{3} + 12 T^{4} - 9 T^{5} - 27 T^{6} + 81 T^{8} \))(\( 1 - 4 T + 11 T^{2} - 21 T^{3} + 40 T^{4} - 63 T^{5} + 99 T^{6} - 108 T^{7} + 81 T^{8} \))(\( 1 - 6 T + 21 T^{2} - 53 T^{3} + 104 T^{4} - 159 T^{5} + 189 T^{6} - 162 T^{7} + 81 T^{8} \))(\( 1 + 6 T + 23 T^{2} + 65 T^{3} + 149 T^{4} + 283 T^{5} + 447 T^{6} + 585 T^{7} + 621 T^{8} + 486 T^{9} + 243 T^{10} \))(\( 1 + 4 T + 13 T^{2} + 29 T^{3} + 61 T^{4} + 109 T^{5} + 183 T^{6} + 261 T^{7} + 351 T^{8} + 324 T^{9} + 243 T^{10} \))(\( 1 - 7 T + 32 T^{2} - 112 T^{3} + 328 T^{4} - 830 T^{5} + 1863 T^{6} - 3745 T^{7} + 6824 T^{8} - 11235 T^{9} + 16767 T^{10} - 22410 T^{11} + 26568 T^{12} - 27216 T^{13} + 23328 T^{14} - 15309 T^{15} + 6561 T^{16} \))
$5$ (\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{3} \))(\( ( 1 - T )^{4} \))(\( ( 1 + T )^{4} \))(\( ( 1 + T )^{4} \))(\( ( 1 - T )^{4} \))(\( ( 1 - T )^{5} \))(\( ( 1 + T )^{5} \))(\( ( 1 - T )^{8} \))
$7$ (\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( ( 1 + T )^{2} \))(\( ( 1 + T )^{3} \))(\( ( 1 - T )^{4} \))(\( ( 1 + T )^{4} \))(\( ( 1 - T )^{4} \))(\( ( 1 + T )^{4} \))(\( ( 1 + T )^{5} \))(\( ( 1 - T )^{5} \))(\( ( 1 - T )^{8} \))
$11$ (\( 1 + 4 T + 11 T^{2} \))(\( 1 - 2 T + 11 T^{2} \))(\( 1 + 5 T + 11 T^{2} \))(\( 1 + T + 11 T^{2} \))(\( ( 1 - 6 T + 11 T^{2} )^{2} \))(\( 1 + 7 T + 47 T^{2} + 161 T^{3} + 517 T^{4} + 847 T^{5} + 1331 T^{6} \))(\( 1 + 8 T + 47 T^{2} + 213 T^{3} + 833 T^{4} + 2343 T^{5} + 5687 T^{6} + 10648 T^{7} + 14641 T^{8} \))(\( 1 - 5 T + 44 T^{2} - 146 T^{3} + 724 T^{4} - 1606 T^{5} + 5324 T^{6} - 6655 T^{7} + 14641 T^{8} \))(\( 1 - 11 T + 84 T^{2} - 418 T^{3} + 1630 T^{4} - 4598 T^{5} + 10164 T^{6} - 14641 T^{7} + 14641 T^{8} \))(\( 1 - 7 T + 58 T^{2} - 236 T^{3} + 1030 T^{4} - 2596 T^{5} + 7018 T^{6} - 9317 T^{7} + 14641 T^{8} \))(\( 1 + 11 T + 69 T^{2} + 299 T^{3} + 1060 T^{4} + 3464 T^{5} + 11660 T^{6} + 36179 T^{7} + 91839 T^{8} + 161051 T^{9} + 161051 T^{10} \))(\( 1 + 7 T + 51 T^{2} + 201 T^{3} + 922 T^{4} + 2796 T^{5} + 10142 T^{6} + 24321 T^{7} + 67881 T^{8} + 102487 T^{9} + 161051 T^{10} \))(\( 1 - 6 T + 53 T^{2} - 273 T^{3} + 1395 T^{4} - 6203 T^{5} + 24895 T^{6} - 93532 T^{7} + 320928 T^{8} - 1028852 T^{9} + 3012295 T^{10} - 8256193 T^{11} + 20424195 T^{12} - 43966923 T^{13} + 93892733 T^{14} - 116923026 T^{15} + 214358881 T^{16} \))
$13$ (\( 1 + 2 T + 13 T^{2} \))(\( 1 - 4 T + 13 T^{2} \))(\( 1 - 3 T + 13 T^{2} \))(\( 1 - 7 T + 13 T^{2} \))(\( ( 1 - 3 T + 13 T^{2} )^{2} \))(\( 1 - 5 T + 17 T^{2} - 33 T^{3} + 221 T^{4} - 845 T^{5} + 2197 T^{6} \))(\( 1 + 8 T + 59 T^{2} + 291 T^{3} + 1197 T^{4} + 3783 T^{5} + 9971 T^{6} + 17576 T^{7} + 28561 T^{8} \))(\( 1 + 17 T + 154 T^{2} + 914 T^{3} + 3874 T^{4} + 11882 T^{5} + 26026 T^{6} + 37349 T^{7} + 28561 T^{8} \))(\( 1 + 3 T + 42 T^{2} + 94 T^{3} + 752 T^{4} + 1222 T^{5} + 7098 T^{6} + 6591 T^{7} + 28561 T^{8} \))(\( 1 + 5 T + 52 T^{2} + 190 T^{3} + 1012 T^{4} + 2470 T^{5} + 8788 T^{6} + 10985 T^{7} + 28561 T^{8} \))(\( 1 + 7 T + 50 T^{2} + 218 T^{3} + 1149 T^{4} + 4063 T^{5} + 14937 T^{6} + 36842 T^{7} + 109850 T^{8} + 199927 T^{9} + 371293 T^{10} \))(\( 1 + 5 T + 44 T^{2} + 134 T^{3} + 749 T^{4} + 1789 T^{5} + 9737 T^{6} + 22646 T^{7} + 96668 T^{8} + 142805 T^{9} + 371293 T^{10} \))(\( 1 - 8 T + 106 T^{2} - 647 T^{3} + 4834 T^{4} - 23288 T^{5} + 125401 T^{6} - 484541 T^{7} + 2035168 T^{8} - 6299033 T^{9} + 21192769 T^{10} - 51163736 T^{11} + 138063874 T^{12} - 240226571 T^{13} + 511641754 T^{14} - 501988136 T^{15} + 815730721 T^{16} \))
$17$ (\( 1 - 6 T + 17 T^{2} \))(\( 1 + 6 T + 17 T^{2} \))(\( 1 + 5 T + 17 T^{2} \))(\( 1 - 3 T + 17 T^{2} \))(\( 1 - 2 T + 30 T^{2} - 34 T^{3} + 289 T^{4} \))(\( 1 - 5 T + 43 T^{2} - 129 T^{3} + 731 T^{4} - 1445 T^{5} + 4913 T^{6} \))(\( 1 + 6 T + 51 T^{2} + 293 T^{3} + 1187 T^{4} + 4981 T^{5} + 14739 T^{6} + 29478 T^{7} + 83521 T^{8} \))(\( 1 + 9 T + 62 T^{2} + 294 T^{3} + 1380 T^{4} + 4998 T^{5} + 17918 T^{6} + 44217 T^{7} + 83521 T^{8} \))(\( 1 - 13 T + 126 T^{2} - 764 T^{3} + 3760 T^{4} - 12988 T^{5} + 36414 T^{6} - 63869 T^{7} + 83521 T^{8} \))(\( 1 - 5 T + 56 T^{2} - 216 T^{3} + 1304 T^{4} - 3672 T^{5} + 16184 T^{6} - 24565 T^{7} + 83521 T^{8} \))(\( 1 - T + 25 T^{2} - 83 T^{3} + 704 T^{4} - 800 T^{5} + 11968 T^{6} - 23987 T^{7} + 122825 T^{8} - 83521 T^{9} + 1419857 T^{10} \))(\( 1 + 7 T + 47 T^{2} + 133 T^{3} + 566 T^{4} + 1192 T^{5} + 9622 T^{6} + 38437 T^{7} + 230911 T^{8} + 584647 T^{9} + 1419857 T^{10} \))(\( 1 - 4 T + 51 T^{2} - 209 T^{3} + 1445 T^{4} - 3607 T^{5} + 25137 T^{6} - 43060 T^{7} + 346508 T^{8} - 732020 T^{9} + 7264593 T^{10} - 17721191 T^{11} + 120687845 T^{12} - 296750113 T^{13} + 1231016019 T^{14} - 1641354692 T^{15} + 6975757441 T^{16} \))
$19$ (\( 1 - 4 T + 19 T^{2} \))(\( 1 + 8 T + 19 T^{2} \))(\( 1 + 19 T^{2} \))(\( 1 + 8 T + 19 T^{2} \))(\( 1 - 8 T + 34 T^{2} - 152 T^{3} + 361 T^{4} \))(\( 1 + 6 T + 62 T^{2} + 215 T^{3} + 1178 T^{4} + 2166 T^{5} + 6859 T^{6} \))(\( 1 - 2 T + 19 T^{2} - 83 T^{3} + 4 T^{4} - 1577 T^{5} + 6859 T^{6} - 13718 T^{7} + 130321 T^{8} \))(\( 1 - 4 T + 67 T^{2} - 181 T^{3} + 1792 T^{4} - 3439 T^{5} + 24187 T^{6} - 27436 T^{7} + 130321 T^{8} \))(\( 1 - 8 T + 79 T^{2} - 375 T^{3} + 2172 T^{4} - 7125 T^{5} + 28519 T^{6} - 54872 T^{7} + 130321 T^{8} \))(\( 1 - 8 T + 73 T^{2} - 331 T^{3} + 1840 T^{4} - 6289 T^{5} + 26353 T^{6} - 54872 T^{7} + 130321 T^{8} \))(\( 1 - 2 T + 50 T^{2} + T^{3} + 1137 T^{4} + 1030 T^{5} + 21603 T^{6} + 361 T^{7} + 342950 T^{8} - 260642 T^{9} + 2476099 T^{10} \))(\( 1 + 10 T + 116 T^{2} + 713 T^{3} + 4699 T^{4} + 19998 T^{5} + 89281 T^{6} + 257393 T^{7} + 795644 T^{8} + 1303210 T^{9} + 2476099 T^{10} \))(\( 1 + 89 T^{2} - 25 T^{3} + 3758 T^{4} - 2787 T^{5} + 103839 T^{6} - 112070 T^{7} + 2193378 T^{8} - 2129330 T^{9} + 37485879 T^{10} - 19116033 T^{11} + 489746318 T^{12} - 61902475 T^{13} + 4187083409 T^{14} + 16983563041 T^{16} \))
$23$ (\( 1 - T \))(\( 1 - T \))(\( 1 + T \))(\( 1 - T \))(\( ( 1 - T )^{2} \))(\( ( 1 + T )^{3} \))(\( ( 1 + T )^{4} \))(\( ( 1 - T )^{4} \))(\( ( 1 + T )^{4} \))(\( ( 1 + T )^{4} \))(\( ( 1 - T )^{5} \))(\( ( 1 - T )^{5} \))(\( ( 1 - T )^{8} \))
$29$ (\( 1 + 2 T + 29 T^{2} \))(\( 1 - 10 T + 29 T^{2} \))(\( 1 - 3 T + 29 T^{2} \))(\( 1 + 5 T + 29 T^{2} \))(\( 1 - 6 T + 47 T^{2} - 174 T^{3} + 841 T^{4} \))(\( 1 + 12 T + 107 T^{2} + 592 T^{3} + 3103 T^{4} + 10092 T^{5} + 24389 T^{6} \))(\( 1 + T + 108 T^{2} + 83 T^{3} + 4590 T^{4} + 2407 T^{5} + 90828 T^{6} + 24389 T^{7} + 707281 T^{8} \))(\( 1 - 10 T + 128 T^{2} - 790 T^{3} + 5662 T^{4} - 22910 T^{5} + 107648 T^{6} - 243890 T^{7} + 707281 T^{8} \))(\( 1 + 2 T + 64 T^{2} + 302 T^{3} + 1982 T^{4} + 8758 T^{5} + 53824 T^{6} + 48778 T^{7} + 707281 T^{8} \))(\( 1 + 2 T + 80 T^{2} + 222 T^{3} + 2942 T^{4} + 6438 T^{5} + 67280 T^{6} + 48778 T^{7} + 707281 T^{8} \))(\( 1 + 14 T + 106 T^{2} + 324 T^{3} - 723 T^{4} - 12204 T^{5} - 20967 T^{6} + 272484 T^{7} + 2585234 T^{8} + 9901934 T^{9} + 20511149 T^{10} \))(\( 1 + 14 T + 162 T^{2} + 1292 T^{3} + 9157 T^{4} + 50996 T^{5} + 265553 T^{6} + 1086572 T^{7} + 3951018 T^{8} + 9901934 T^{9} + 20511149 T^{10} \))(\( 1 + 3 T + 131 T^{2} + 494 T^{3} + 8858 T^{4} + 36316 T^{5} + 400317 T^{6} + 1617223 T^{7} + 13313754 T^{8} + 46899467 T^{9} + 336666597 T^{10} + 885710924 T^{11} + 6265095098 T^{12} + 10132507606 T^{13} + 77921855051 T^{14} + 51749628927 T^{15} + 500246412961 T^{16} \))
$31$ (\( 1 - 4 T + 31 T^{2} \))(\( 1 - 10 T + 31 T^{2} \))(\( 1 - 6 T + 31 T^{2} \))(\( 1 + 2 T + 31 T^{2} \))(\( 1 + 2 T + 43 T^{2} + 62 T^{3} + 961 T^{4} \))(\( 1 + 4 T + 96 T^{2} + 247 T^{3} + 2976 T^{4} + 3844 T^{5} + 29791 T^{6} \))(\( 1 + 2 T + 43 T^{2} + 9 T^{3} + 878 T^{4} + 279 T^{5} + 41323 T^{6} + 59582 T^{7} + 923521 T^{8} \))(\( 1 + 2 T + 61 T^{2} + 347 T^{3} + 1834 T^{4} + 10757 T^{5} + 58621 T^{6} + 59582 T^{7} + 923521 T^{8} \))(\( 1 - 8 T + 127 T^{2} - 721 T^{3} + 5960 T^{4} - 22351 T^{5} + 122047 T^{6} - 238328 T^{7} + 923521 T^{8} \))(\( 1 + 75 T^{2} + 117 T^{3} + 2696 T^{4} + 3627 T^{5} + 72075 T^{6} + 923521 T^{8} \))(\( 1 + 6 T + 85 T^{2} + 479 T^{3} + 3931 T^{4} + 20655 T^{5} + 121861 T^{6} + 460319 T^{7} + 2532235 T^{8} + 5541126 T^{9} + 28629151 T^{10} \))(\( 1 + 10 T + 169 T^{2} + 1209 T^{3} + 10855 T^{4} + 55721 T^{5} + 336505 T^{6} + 1161849 T^{7} + 5034679 T^{8} + 9235210 T^{9} + 28629151 T^{10} \))(\( 1 + 16 T + 282 T^{2} + 2915 T^{3} + 31017 T^{4} + 243008 T^{5} + 1914656 T^{6} + 11903421 T^{7} + 73922784 T^{8} + 369006051 T^{9} + 1839984416 T^{10} + 7239451328 T^{11} + 28644850857 T^{12} + 83453975165 T^{13} + 250276038042 T^{14} + 440201825776 T^{15} + 852891037441 T^{16} \))
$37$ (\( 1 + 10 T + 37 T^{2} \))(\( 1 - 8 T + 37 T^{2} \))(\( 1 + 4 T + 37 T^{2} \))(\( 1 + 4 T + 37 T^{2} \))(\( 1 + 10 T + 94 T^{2} + 370 T^{3} + 1369 T^{4} \))(\( 1 - 3 T + 93 T^{2} - 195 T^{3} + 3441 T^{4} - 4107 T^{5} + 50653 T^{6} \))(\( 1 + 7 T + 140 T^{2} + 718 T^{3} + 7578 T^{4} + 26566 T^{5} + 191660 T^{6} + 354571 T^{7} + 1874161 T^{8} \))(\( 1 - 5 T + 124 T^{2} - 386 T^{3} + 6202 T^{4} - 14282 T^{5} + 169756 T^{6} - 253265 T^{7} + 1874161 T^{8} \))(\( 1 + 5 T + 76 T^{2} + 504 T^{3} + 3412 T^{4} + 18648 T^{5} + 104044 T^{6} + 253265 T^{7} + 1874161 T^{8} \))(\( 1 - 13 T + 178 T^{2} - 1292 T^{3} + 9928 T^{4} - 47804 T^{5} + 243682 T^{6} - 658489 T^{7} + 1874161 T^{8} \))(\( 1 + T + 17 T^{2} - 11 T^{3} + 376 T^{4} + 10076 T^{5} + 13912 T^{6} - 15059 T^{7} + 861101 T^{8} + 1874161 T^{9} + 69343957 T^{10} \))(\( 1 + 3 T + 127 T^{2} + 247 T^{3} + 7314 T^{4} + 10492 T^{5} + 270618 T^{6} + 338143 T^{7} + 6432931 T^{8} + 5622483 T^{9} + 69343957 T^{10} \))(\( 1 + 7 T + 108 T^{2} + 228 T^{3} + 3794 T^{4} - 11242 T^{5} + 99020 T^{6} - 722845 T^{7} + 4372490 T^{8} - 26745265 T^{9} + 135558380 T^{10} - 569441026 T^{11} + 7110566834 T^{12} + 15810422196 T^{13} + 277098452172 T^{14} + 664523139931 T^{15} + 3512479453921 T^{16} \))
$41$ (\( 1 - 10 T + 41 T^{2} \))(\( 1 + 2 T + 41 T^{2} \))(\( 1 + 41 T^{2} \))(\( 1 + 8 T + 41 T^{2} \))(\( 1 + 37 T^{2} + 1681 T^{4} \))(\( 1 - T + 58 T^{2} + 87 T^{3} + 2378 T^{4} - 1681 T^{5} + 68921 T^{6} \))(\( 1 + 9 T + 151 T^{2} + 1098 T^{3} + 9024 T^{4} + 45018 T^{5} + 253831 T^{6} + 620289 T^{7} + 2825761 T^{8} \))(\( 1 - 7 T + 89 T^{2} - 574 T^{3} + 5722 T^{4} - 23534 T^{5} + 149609 T^{6} - 482447 T^{7} + 2825761 T^{8} \))(\( 1 - 23 T + 277 T^{2} - 2246 T^{3} + 15230 T^{4} - 92086 T^{5} + 465637 T^{6} - 1585183 T^{7} + 2825761 T^{8} \))(\( 1 - T + 121 T^{2} - 242 T^{3} + 6474 T^{4} - 9922 T^{5} + 203401 T^{6} - 68921 T^{7} + 2825761 T^{8} \))(\( 1 - 7 T + 135 T^{2} - 1002 T^{3} + 9005 T^{4} - 58961 T^{5} + 369205 T^{6} - 1684362 T^{7} + 9304335 T^{8} - 19780327 T^{9} + 115856201 T^{10} \))(\( 1 + 15 T + 215 T^{2} + 2090 T^{3} + 18157 T^{4} + 121401 T^{5} + 744437 T^{6} + 3513290 T^{7} + 14818015 T^{8} + 42386415 T^{9} + 115856201 T^{10} \))(\( 1 - 7 T + 158 T^{2} - 1315 T^{3} + 13133 T^{4} - 106952 T^{5} + 819746 T^{6} - 5500878 T^{7} + 39191124 T^{8} - 225535998 T^{9} + 1377993026 T^{10} - 7371238792 T^{11} + 37110719213 T^{12} - 152350904315 T^{13} + 750516470078 T^{14} - 1363279917167 T^{15} + 7984925229121 T^{16} \))
$43$ (\( 1 - 12 T + 43 T^{2} \))(\( 1 + 43 T^{2} \))(\( 1 + 2 T + 43 T^{2} \))(\( 1 - 6 T + 43 T^{2} \))(\( 1 + 6 T + 50 T^{2} + 258 T^{3} + 1849 T^{4} \))(\( 1 - 2 T + 128 T^{2} - 171 T^{3} + 5504 T^{4} - 3698 T^{5} + 79507 T^{6} \))(\( 1 + 4 T + 97 T^{2} + 393 T^{3} + 5906 T^{4} + 16899 T^{5} + 179353 T^{6} + 318028 T^{7} + 3418801 T^{8} \))(\( 1 + 6 T + 133 T^{2} + 705 T^{3} + 7716 T^{4} + 30315 T^{5} + 245917 T^{6} + 477042 T^{7} + 3418801 T^{8} \))(\( 1 - 2 T + 55 T^{2} - 103 T^{3} + 1724 T^{4} - 4429 T^{5} + 101695 T^{6} - 159014 T^{7} + 3418801 T^{8} \))(\( 1 - 14 T + 161 T^{2} - 1421 T^{3} + 9720 T^{4} - 61103 T^{5} + 297689 T^{6} - 1113098 T^{7} + 3418801 T^{8} \))(\( 1 + 12 T + 138 T^{2} + 1233 T^{3} + 10559 T^{4} + 69898 T^{5} + 454037 T^{6} + 2279817 T^{7} + 10971966 T^{8} + 41025612 T^{9} + 147008443 T^{10} \))(\( 1 - 8 T + 200 T^{2} - 1169 T^{3} + 16157 T^{4} - 70810 T^{5} + 694751 T^{6} - 2161481 T^{7} + 15901400 T^{8} - 27350408 T^{9} + 147008443 T^{10} \))(\( 1 + 10 T + 187 T^{2} + 2015 T^{3} + 20782 T^{4} + 189837 T^{5} + 1590085 T^{6} + 11446780 T^{7} + 82824466 T^{8} + 492211540 T^{9} + 2940067165 T^{10} + 15093370359 T^{11} + 71049522382 T^{12} + 296222012645 T^{13} + 1182094890163 T^{14} + 2718186111070 T^{15} + 11688200277601 T^{16} \))
$47$ (\( 1 + 12 T + 47 T^{2} \))(\( 1 - 12 T + 47 T^{2} \))(\( 1 + 9 T + 47 T^{2} \))(\( 1 - 3 T + 47 T^{2} \))(\( 1 + 4 T + 93 T^{2} + 188 T^{3} + 2209 T^{4} \))(\( 1 - 12 T + 140 T^{2} - 1087 T^{3} + 6580 T^{4} - 26508 T^{5} + 103823 T^{6} \))(\( 1 + 21 T + 345 T^{2} + 3461 T^{3} + 28589 T^{4} + 162667 T^{5} + 762105 T^{6} + 2180283 T^{7} + 4879681 T^{8} \))(\( 1 + 12 T + 227 T^{2} + 1731 T^{3} + 16932 T^{4} + 81357 T^{5} + 501443 T^{6} + 1245876 T^{7} + 4879681 T^{8} \))(\( 1 - 2 T + 117 T^{2} - 73 T^{3} + 6904 T^{4} - 3431 T^{5} + 258453 T^{6} - 207646 T^{7} + 4879681 T^{8} \))(\( 1 - 4 T + 109 T^{2} - 53 T^{3} + 5092 T^{4} - 2491 T^{5} + 240781 T^{6} - 415292 T^{7} + 4879681 T^{8} \))(\( 1 + 24 T + 431 T^{2} + 5125 T^{3} + 50209 T^{4} + 375429 T^{5} + 2359823 T^{6} + 11321125 T^{7} + 44747713 T^{8} + 117112344 T^{9} + 229345007 T^{10} \))(\( 1 + 10 T + 219 T^{2} + 1665 T^{3} + 19993 T^{4} + 112659 T^{5} + 939671 T^{6} + 3677985 T^{7} + 22737237 T^{8} + 48796810 T^{9} + 229345007 T^{10} \))(\( 1 - 19 T + 204 T^{2} - 1818 T^{3} + 15942 T^{4} - 130342 T^{5} + 1055049 T^{6} - 8462151 T^{7} + 62857704 T^{8} - 397721097 T^{9} + 2330603241 T^{10} - 13532497466 T^{11} + 77791874502 T^{12} - 416949222726 T^{13} + 2198959927116 T^{14} - 9625839288797 T^{15} + 23811286661761 T^{16} \))
$53$ (\( 1 - 14 T + 53 T^{2} \))(\( 1 + 4 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 - 2 T + 53 T^{2} \))(\( 1 + 22 T + 222 T^{2} + 1166 T^{3} + 2809 T^{4} \))(\( 1 + 9 T + 11 T^{2} - 419 T^{3} + 583 T^{4} + 25281 T^{5} + 148877 T^{6} \))(\( 1 + 11 T + 102 T^{2} + 318 T^{3} + 2720 T^{4} + 16854 T^{5} + 286518 T^{6} + 1637647 T^{7} + 7890481 T^{8} \))(\( 1 - 23 T + 344 T^{2} - 3536 T^{3} + 29014 T^{4} - 187408 T^{5} + 966296 T^{6} - 3424171 T^{7} + 7890481 T^{8} \))(\( 1 + T + 150 T^{2} + 104 T^{3} + 10648 T^{4} + 5512 T^{5} + 421350 T^{6} + 148877 T^{7} + 7890481 T^{8} \))(\( 1 + T + 202 T^{2} + 146 T^{3} + 15792 T^{4} + 7738 T^{5} + 567418 T^{6} + 148877 T^{7} + 7890481 T^{8} \))(\( 1 + 21 T + 407 T^{2} + 4757 T^{3} + 50566 T^{4} + 385716 T^{5} + 2679998 T^{6} + 13362413 T^{7} + 60592939 T^{8} + 165700101 T^{9} + 418195493 T^{10} \))(\( 1 + 9 T + 135 T^{2} + 103 T^{3} + 846 T^{4} - 46032 T^{5} + 44838 T^{6} + 289327 T^{7} + 20098395 T^{8} + 71014329 T^{9} + 418195493 T^{10} \))(\( 1 + T + 184 T^{2} + 630 T^{3} + 18404 T^{4} + 93240 T^{5} + 1334752 T^{6} + 7959493 T^{7} + 76724166 T^{8} + 421853129 T^{9} + 3749318368 T^{10} + 13881291480 T^{11} + 145216412324 T^{12} + 263463160590 T^{13} + 4078242447736 T^{14} + 1174711139837 T^{15} + 62259690411361 T^{16} \))
$59$ (\( 1 - 8 T + 59 T^{2} \))(\( 1 - 14 T + 59 T^{2} \))(\( 1 + 6 T + 59 T^{2} \))(\( 1 - 2 T + 59 T^{2} \))(\( 1 + 4 T + 102 T^{2} + 236 T^{3} + 3481 T^{4} \))(\( 1 + 21 T + 303 T^{2} + 2681 T^{3} + 17877 T^{4} + 73101 T^{5} + 205379 T^{6} \))(\( 1 + 37 T + 708 T^{2} + 8954 T^{3} + 80808 T^{4} + 528286 T^{5} + 2464548 T^{6} + 7599023 T^{7} + 12117361 T^{8} \))(\( 1 - 23 T + 428 T^{2} - 4760 T^{3} + 44440 T^{4} - 280840 T^{5} + 1489868 T^{6} - 4723717 T^{7} + 12117361 T^{8} \))(\( 1 - 15 T + 286 T^{2} - 2620 T^{3} + 26558 T^{4} - 154580 T^{5} + 995566 T^{6} - 3080685 T^{7} + 12117361 T^{8} \))(\( 1 + 7 T + 162 T^{2} + 752 T^{3} + 12318 T^{4} + 44368 T^{5} + 563922 T^{6} + 1437653 T^{7} + 12117361 T^{8} \))(\( 1 + T + 225 T^{2} + 105 T^{3} + 22792 T^{4} + 5604 T^{5} + 1344728 T^{6} + 365505 T^{7} + 46210275 T^{8} + 12117361 T^{9} + 714924299 T^{10} \))(\( 1 + 19 T + 369 T^{2} + 4255 T^{3} + 47176 T^{4} + 369412 T^{5} + 2783384 T^{6} + 14811655 T^{7} + 75784851 T^{8} + 230229859 T^{9} + 714924299 T^{10} \))(\( 1 - 21 T + 444 T^{2} - 4604 T^{3} + 46566 T^{4} - 171266 T^{5} + 163748 T^{6} + 19348241 T^{7} - 146115662 T^{8} + 1141546219 T^{9} + 570006788 T^{10} - 35174439814 T^{11} + 564257032326 T^{12} - 3291511472596 T^{13} + 18728156936604 T^{14} - 52261681181199 T^{15} + 146830437604321 T^{16} \))
$61$ (\( 1 - 10 T + 61 T^{2} \))(\( 1 + 2 T + 61 T^{2} \))(\( 1 - 10 T + 61 T^{2} \))(\( 1 + 14 T + 61 T^{2} \))(\( 1 - 10 T + 102 T^{2} - 610 T^{3} + 3721 T^{4} \))(\( 1 + 9 T + 161 T^{2} + 887 T^{3} + 9821 T^{4} + 33489 T^{5} + 226981 T^{6} \))(\( 1 - T + 234 T^{2} - 170 T^{3} + 21104 T^{4} - 10370 T^{5} + 870714 T^{6} - 226981 T^{7} + 13845841 T^{8} \))(\( 1 + 17 T + 304 T^{2} + 2996 T^{3} + 29488 T^{4} + 182756 T^{5} + 1131184 T^{6} + 3858677 T^{7} + 13845841 T^{8} \))(\( 1 - T + 76 T^{2} - 168 T^{3} + 6712 T^{4} - 10248 T^{5} + 282796 T^{6} - 226981 T^{7} + 13845841 T^{8} \))(\( 1 + 7 T + 190 T^{2} + 984 T^{3} + 15464 T^{4} + 60024 T^{5} + 706990 T^{6} + 1588867 T^{7} + 13845841 T^{8} \))(\( 1 - 7 T + 259 T^{2} - 1417 T^{3} + 29142 T^{4} - 123032 T^{5} + 1777662 T^{6} - 5272657 T^{7} + 58788079 T^{8} - 96920887 T^{9} + 844596301 T^{10} \))(\( 1 + 21 T + 405 T^{2} + 4851 T^{3} + 53432 T^{4} + 434312 T^{5} + 3259352 T^{6} + 18050571 T^{7} + 91927305 T^{8} + 290762661 T^{9} + 844596301 T^{10} \))(\( 1 + 7 T + 224 T^{2} + 1286 T^{3} + 20480 T^{4} + 103824 T^{5} + 1113080 T^{6} + 6058615 T^{7} + 57316462 T^{8} + 369575515 T^{9} + 4141770680 T^{10} + 23566075344 T^{11} + 283562823680 T^{12} + 1086150843086 T^{13} + 11540563856864 T^{14} + 21999199852147 T^{15} + 191707312997281 T^{16} \))
$67$ (\( 1 + 4 T + 67 T^{2} \))(\( 1 + 4 T + 67 T^{2} \))(\( 1 - 4 T + 67 T^{2} \))(\( 1 + 4 T + 67 T^{2} \))(\( 1 - 2 T + 130 T^{2} - 134 T^{3} + 4489 T^{4} \))(\( 1 - 13 T + 185 T^{2} - 1365 T^{3} + 12395 T^{4} - 58357 T^{5} + 300763 T^{6} \))(\( 1 + 31 T + 578 T^{2} + 7134 T^{3} + 67506 T^{4} + 477978 T^{5} + 2594642 T^{6} + 9323653 T^{7} + 20151121 T^{8} \))(\( 1 - 5 T + 154 T^{2} - 26 T^{3} + 9622 T^{4} - 1742 T^{5} + 691306 T^{6} - 1503815 T^{7} + 20151121 T^{8} \))(\( 1 + 5 T + 264 T^{2} + 984 T^{3} + 26394 T^{4} + 65928 T^{5} + 1185096 T^{6} + 1503815 T^{7} + 20151121 T^{8} \))(\( 1 - 15 T + 182 T^{2} - 1634 T^{3} + 14382 T^{4} - 109478 T^{5} + 816998 T^{6} - 4511445 T^{7} + 20151121 T^{8} \))(\( 1 + 35 T + 785 T^{2} + 11891 T^{3} + 140386 T^{4} + 1278568 T^{5} + 9405862 T^{6} + 53378699 T^{7} + 236098955 T^{8} + 705289235 T^{9} + 1350125107 T^{10} \))(\( 1 - 5 T + 191 T^{2} - 1087 T^{3} + 21248 T^{4} - 94260 T^{5} + 1423616 T^{6} - 4879543 T^{7} + 57445733 T^{8} - 100755605 T^{9} + 1350125107 T^{10} \))(\( 1 - 11 T + 274 T^{2} - 2206 T^{3} + 32906 T^{4} - 190468 T^{5} + 2318198 T^{6} - 10429181 T^{7} + 142198330 T^{8} - 698755127 T^{9} + 10406390822 T^{10} - 57285727084 T^{11} + 663092787626 T^{12} - 2978375986042 T^{13} + 24785596714306 T^{14} - 66667827658553 T^{15} + 406067677556641 T^{16} \))
$71$ (\( 1 - 8 T + 71 T^{2} \))(\( 1 - 8 T + 71 T^{2} \))(\( 1 + 8 T + 71 T^{2} \))(\( 1 - 8 T + 71 T^{2} \))(\( 1 + 4 T + 141 T^{2} + 284 T^{3} + 5041 T^{4} \))(\( 1 + 16 T + 198 T^{2} + 1809 T^{3} + 14058 T^{4} + 80656 T^{5} + 357911 T^{6} \))(\( 1 + 8 T + 281 T^{2} + 1677 T^{3} + 29804 T^{4} + 119067 T^{5} + 1416521 T^{6} + 2863288 T^{7} + 25411681 T^{8} \))(\( 1 - 20 T + 293 T^{2} - 2987 T^{3} + 28828 T^{4} - 212077 T^{5} + 1477013 T^{6} - 7158220 T^{7} + 25411681 T^{8} \))(\( 1 + 8 T - 15 T^{2} - 323 T^{3} + 964 T^{4} - 22933 T^{5} - 75615 T^{6} + 2863288 T^{7} + 25411681 T^{8} \))(\( 1 + 213 T^{2} - 199 T^{3} + 20156 T^{4} - 14129 T^{5} + 1073733 T^{6} + 25411681 T^{8} \))(\( 1 + 32 T + 619 T^{2} + 8265 T^{3} + 88661 T^{4} + 796645 T^{5} + 6294931 T^{6} + 41663865 T^{7} + 221546909 T^{8} + 813173792 T^{9} + 1804229351 T^{10} \))(\( 1 + 16 T + 367 T^{2} + 4221 T^{3} + 52733 T^{4} + 439265 T^{5} + 3744043 T^{6} + 21278061 T^{7} + 131353337 T^{8} + 406586896 T^{9} + 1804229351 T^{10} \))(\( 1 - 8 T + 180 T^{2} - 1131 T^{3} + 24643 T^{4} - 164934 T^{5} + 2608896 T^{6} - 14487455 T^{7} + 197532400 T^{8} - 1028609305 T^{9} + 13151444736 T^{10} - 59031692874 T^{11} + 626220054883 T^{12} - 2040583395981 T^{13} + 23058051105780 T^{14} - 72760961267128 T^{15} + 645753531245761 T^{16} \))
$73$ (\( 1 + 6 T + 73 T^{2} \))(\( 1 + 73 T^{2} \))(\( 1 - 10 T + 73 T^{2} \))(\( 1 + 6 T + 73 T^{2} \))(\( ( 1 - 9 T + 73 T^{2} )^{2} \))(\( 1 + 9 T + 197 T^{2} + 1285 T^{3} + 14381 T^{4} + 47961 T^{5} + 389017 T^{6} \))(\( 1 - 25 T + 446 T^{2} - 5146 T^{3} + 50676 T^{4} - 375658 T^{5} + 2376734 T^{6} - 9725425 T^{7} + 28398241 T^{8} \))(\( 1 + 15 T + 190 T^{2} + 1950 T^{3} + 20478 T^{4} + 142350 T^{5} + 1012510 T^{6} + 5835255 T^{7} + 28398241 T^{8} \))(\( 1 - 3 T + 108 T^{2} - 542 T^{3} + 12212 T^{4} - 39566 T^{5} + 575532 T^{6} - 1167051 T^{7} + 28398241 T^{8} \))(\( 1 - 3 T + 122 T^{2} - 960 T^{3} + 8920 T^{4} - 70080 T^{5} + 650138 T^{6} - 1167051 T^{7} + 28398241 T^{8} \))(\( 1 + 7 T + 276 T^{2} + 984 T^{3} + 30101 T^{4} + 64761 T^{5} + 2197373 T^{6} + 5243736 T^{7} + 107368692 T^{8} + 198787687 T^{9} + 2073071593 T^{10} \))(\( 1 - T + 150 T^{2} - 302 T^{3} + 16641 T^{4} - 10797 T^{5} + 1214793 T^{6} - 1609358 T^{7} + 58352550 T^{8} - 28398241 T^{9} + 2073071593 T^{10} \))(\( 1 + 3 T + 269 T^{2} + 1309 T^{3} + 40920 T^{4} + 215618 T^{5} + 4537865 T^{6} + 22059098 T^{7} + 378794378 T^{8} + 1610314154 T^{9} + 24182282585 T^{10} + 83879067506 T^{11} + 1162056021720 T^{12} + 2713650715237 T^{13} + 40708906871741 T^{14} + 33142195557291 T^{15} + 806460091894081 T^{16} \))
$79$ (\( 1 + 79 T^{2} \))(\( 1 - 6 T + 79 T^{2} \))(\( 1 + 15 T + 79 T^{2} \))(\( 1 + 3 T + 79 T^{2} \))(\( 1 - 22 T^{2} + 6241 T^{4} \))(\( 1 + 18 T + 282 T^{2} + 2493 T^{3} + 22278 T^{4} + 112338 T^{5} + 493039 T^{6} \))(\( 1 + 19 T + 275 T^{2} + 3079 T^{3} + 31569 T^{4} + 243241 T^{5} + 1716275 T^{6} + 9367741 T^{7} + 38950081 T^{8} \))(\( 1 - 12 T + 331 T^{2} - 2637 T^{3} + 39342 T^{4} - 208323 T^{5} + 2065771 T^{6} - 5916468 T^{7} + 38950081 T^{8} \))(\( 1 + 169 T^{2} - 121 T^{3} + 17656 T^{4} - 9559 T^{5} + 1054729 T^{6} + 38950081 T^{8} \))(\( 1 + 14 T + 319 T^{2} + 2683 T^{3} + 35728 T^{4} + 211957 T^{5} + 1990879 T^{6} + 6902546 T^{7} + 38950081 T^{8} \))(\( 1 - 18 T + 376 T^{2} - 4711 T^{3} + 57271 T^{4} - 530898 T^{5} + 4524409 T^{6} - 29401351 T^{7} + 185382664 T^{8} - 701101458 T^{9} + 3077056399 T^{10} \))(\( 1 - 20 T + 426 T^{2} - 5881 T^{3} + 69513 T^{4} - 679554 T^{5} + 5491527 T^{6} - 36703321 T^{7} + 210034614 T^{8} - 779001620 T^{9} + 3077056399 T^{10} \))(\( 1 + 15 T + 229 T^{2} + 1061 T^{3} + 8607 T^{4} + 10105 T^{5} + 1259311 T^{6} + 12848707 T^{7} + 191850408 T^{8} + 1015047853 T^{9} + 7859359951 T^{10} + 4982159095 T^{11} + 335243347167 T^{12} + 3264756839339 T^{13} + 55667027314309 T^{14} + 288058634792385 T^{15} + 1517108809906561 T^{16} \))
$83$ (\( 1 - 12 T + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))(\( 1 - 12 T + 83 T^{2} \))(\( 1 - 12 T + 83 T^{2} \))(\( 1 + 6 T + 130 T^{2} + 498 T^{3} + 6889 T^{4} \))(\( 1 + 29 T + 443 T^{2} + 4603 T^{3} + 36769 T^{4} + 199781 T^{5} + 571787 T^{6} \))(\( 1 + 7 T + 292 T^{2} + 1748 T^{3} + 34698 T^{4} + 145084 T^{5} + 2011588 T^{6} + 4002509 T^{7} + 47458321 T^{8} \))(\( 1 + 3 T + 164 T^{2} + 546 T^{3} + 13890 T^{4} + 45318 T^{5} + 1129796 T^{6} + 1715361 T^{7} + 47458321 T^{8} \))(\( 1 - 5 T + 162 T^{2} - 1438 T^{3} + 15118 T^{4} - 119354 T^{5} + 1116018 T^{6} - 2858935 T^{7} + 47458321 T^{8} \))(\( 1 - 3 T + 180 T^{2} - 1556 T^{3} + 14858 T^{4} - 129148 T^{5} + 1240020 T^{6} - 1715361 T^{7} + 47458321 T^{8} \))(\( 1 + T + 211 T^{2} + 561 T^{3} + 26112 T^{4} + 69144 T^{5} + 2167296 T^{6} + 3864729 T^{7} + 120647057 T^{8} + 47458321 T^{9} + 3939040643 T^{10} \))(\( 1 + 31 T + 617 T^{2} + 7921 T^{3} + 85250 T^{4} + 777204 T^{5} + 7075750 T^{6} + 54567769 T^{7} + 352792579 T^{8} + 1471207951 T^{9} + 3939040643 T^{10} \))(\( 1 - 25 T + 632 T^{2} - 8426 T^{3} + 113366 T^{4} - 877076 T^{5} + 7714544 T^{6} - 29380439 T^{7} + 346252610 T^{8} - 2438576437 T^{9} + 53145493616 T^{10} - 501500654812 T^{11} + 5380160018486 T^{12} - 33190356457918 T^{13} + 206626315969208 T^{14} - 678401274740675 T^{15} + 2252292232139041 T^{16} \))
$89$ (\( 1 + 2 T + 89 T^{2} \))(\( 1 - 10 T + 89 T^{2} \))(\( 1 + 10 T + 89 T^{2} \))(\( 1 + 2 T + 89 T^{2} \))(\( 1 + 14 T + 222 T^{2} + 1246 T^{3} + 7921 T^{4} \))(\( 1 - 7 T + 253 T^{2} - 1155 T^{3} + 22517 T^{4} - 55447 T^{5} + 704969 T^{6} \))(\( 1 - 7 T + 254 T^{2} - 810 T^{3} + 26912 T^{4} - 72090 T^{5} + 2011934 T^{6} - 4934783 T^{7} + 62742241 T^{8} \))(\( 1 + 11 T + 206 T^{2} + 1952 T^{3} + 27748 T^{4} + 173728 T^{5} + 1631726 T^{6} + 7754659 T^{7} + 62742241 T^{8} \))(\( 1 - 35 T + 810 T^{2} - 11932 T^{3} + 133804 T^{4} - 1061948 T^{5} + 6416010 T^{6} - 24673915 T^{7} + 62742241 T^{8} \))(\( 1 + 11 T + 232 T^{2} + 1102 T^{3} + 20340 T^{4} + 98078 T^{5} + 1837672 T^{6} + 7754659 T^{7} + 62742241 T^{8} \))(\( 1 - T + 215 T^{2} + 449 T^{3} + 25370 T^{4} + 79560 T^{5} + 2257930 T^{6} + 3556529 T^{7} + 151568335 T^{8} - 62742241 T^{9} + 5584059449 T^{10} \))(\( 1 + 21 T + 417 T^{2} + 4765 T^{3} + 54724 T^{4} + 484844 T^{5} + 4870436 T^{6} + 37743565 T^{7} + 293972073 T^{8} + 1317587061 T^{9} + 5584059449 T^{10} \))(\( 1 - T + 330 T^{2} - 1456 T^{3} + 57344 T^{4} - 365870 T^{5} + 7551134 T^{6} - 47530569 T^{7} + 773518510 T^{8} - 4230220641 T^{9} + 59812532414 T^{10} - 257927008030 T^{11} + 3597891067904 T^{12} - 8130390557744 T^{13} + 164003826017130 T^{14} - 44231334895529 T^{15} + 3936588805702081 T^{16} \))
$97$ (\( 1 + 2 T + 97 T^{2} \))(\( 1 + 2 T + 97 T^{2} \))(\( 1 - 7 T + 97 T^{2} \))(\( 1 - 7 T + 97 T^{2} \))(\( 1 + 14 T + 198 T^{2} + 1358 T^{3} + 9409 T^{4} \))(\( 1 - T + 142 T^{2} - 277 T^{3} + 13774 T^{4} - 9409 T^{5} + 912673 T^{6} \))(\( 1 + 2 T + 176 T^{2} + 1868 T^{3} + 13257 T^{4} + 181196 T^{5} + 1655984 T^{6} + 1825346 T^{7} + 88529281 T^{8} \))(\( 1 - T + 253 T^{2} + 614 T^{3} + 28606 T^{4} + 59558 T^{5} + 2380477 T^{6} - 912673 T^{7} + 88529281 T^{8} \))(\( 1 + 11 T + 177 T^{2} - 318 T^{3} + 2406 T^{4} - 30846 T^{5} + 1665393 T^{6} + 10039403 T^{7} + 88529281 T^{8} \))(\( 1 - 9 T + 241 T^{2} - 2090 T^{3} + 31298 T^{4} - 202730 T^{5} + 2267569 T^{6} - 8214057 T^{7} + 88529281 T^{8} \))(\( 1 + 9 T + 214 T^{2} + 2015 T^{3} + 19163 T^{4} + 226496 T^{5} + 1858811 T^{6} + 18959135 T^{7} + 195312022 T^{8} + 796763529 T^{9} + 8587340257 T^{10} \))(\( 1 - 19 T + 298 T^{2} - 2293 T^{3} + 25987 T^{4} - 186112 T^{5} + 2520739 T^{6} - 21574837 T^{7} + 271976554 T^{8} - 1682056339 T^{9} + 8587340257 T^{10} \))(\( 1 - 10 T + 574 T^{2} - 4842 T^{3} + 155521 T^{4} - 1122152 T^{5} + 26343286 T^{6} - 162485876 T^{7} + 3054647092 T^{8} - 15761129972 T^{9} + 247863977974 T^{10} - 1024157832296 T^{11} + 13768162310401 T^{12} - 41579901524394 T^{13} + 478125930829246 T^{14} - 807982844781130 T^{15} + 7837433594376961 T^{16} \))
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