Properties

Label 363.2.e
Level 363
Weight 2
Character orbit e
Rep. character \(\chi_{363}(124,\cdot)\)
Character field \(\Q(\zeta_{5})\)
Dimension 72
Newform subspaces 14
Sturm bound 88
Trace bound 4

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Defining parameters

Level: \( N \) = \( 363 = 3 \cdot 11^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 363.e (of order \(5\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 11 \)
Character field: \(\Q(\zeta_{5})\)
Newform subspaces: \( 14 \)
Sturm bound: \(88\)
Trace bound: \(4\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(363, [\chi])\).

Total New Old
Modular forms 224 72 152
Cusp forms 128 72 56
Eisenstein series 96 0 96

Trace form

\( 72q + 4q^{2} - 14q^{4} + 4q^{5} + 2q^{6} + 2q^{7} - 8q^{8} - 18q^{9} + O(q^{10}) \) \( 72q + 4q^{2} - 14q^{4} + 4q^{5} + 2q^{6} + 2q^{7} - 8q^{8} - 18q^{9} - 4q^{10} + 8q^{12} + 2q^{13} + 6q^{14} - 2q^{15} - 22q^{16} - 14q^{17} - 6q^{18} + 20q^{19} + 2q^{20} - 16q^{21} - 4q^{23} - 12q^{24} - 20q^{25} + 2q^{26} + 12q^{28} + 4q^{29} + 8q^{31} + 48q^{32} - 56q^{34} - 14q^{36} - 2q^{37} + 14q^{38} + 16q^{39} - 4q^{40} - 20q^{41} + 14q^{42} - 16q^{43} + 4q^{45} - 10q^{46} - 6q^{47} - 8q^{48} - 16q^{49} - 18q^{50} - 10q^{52} - 6q^{53} - 8q^{54} - 108q^{56} + 44q^{58} + 46q^{59} + 12q^{60} + 18q^{61} + 28q^{62} + 2q^{63} + 2q^{64} + 28q^{65} - 44q^{67} + 28q^{68} + 10q^{69} + 34q^{70} + 32q^{71} + 2q^{72} - 20q^{73} + 30q^{74} - 40q^{76} - 64q^{78} + 6q^{79} + 14q^{80} - 18q^{81} + 46q^{82} - 34q^{83} + 2q^{85} + 24q^{86} - 24q^{87} - 60q^{89} - 4q^{90} - 18q^{91} + 22q^{92} - 56q^{93} - 18q^{94} - 30q^{95} + 8q^{96} + 10q^{97} - 40q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(363, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
363.2.e.a \(4\) \(2.899\) \(\Q(\zeta_{10})\) None \(-5\) \(-1\) \(-2\) \(-10\) \(q+(-1-\zeta_{10}+\zeta_{10}^{2}+\zeta_{10}^{3})q^{2}+\cdots\)
363.2.e.b \(4\) \(2.899\) \(\Q(\zeta_{10})\) None \(-4\) \(1\) \(2\) \(-1\) \(q+(-2+\zeta_{10}-\zeta_{10}^{2}+2\zeta_{10}^{3})q^{2}+\cdots\)
363.2.e.c \(4\) \(2.899\) \(\Q(\zeta_{10})\) None \(-2\) \(-1\) \(4\) \(3\) \(q+(-\zeta_{10}+\zeta_{10}^{2})q^{2}+\zeta_{10}^{2}q^{3}+(-1+\cdots)q^{4}+\cdots\)
363.2.e.d \(4\) \(2.899\) \(\Q(\zeta_{10})\) None \(-2\) \(1\) \(-4\) \(1\) \(q+(-2+2\zeta_{10}-2\zeta_{10}^{2}+2\zeta_{10}^{3})q^{2}+\cdots\)
363.2.e.e \(4\) \(2.899\) \(\Q(\zeta_{10})\) None \(-1\) \(1\) \(2\) \(-4\) \(q+(-1+\zeta_{10}-\zeta_{10}^{2}+\zeta_{10}^{3})q^{2}+\cdots\)
363.2.e.f \(4\) \(2.899\) \(\Q(\zeta_{10})\) None \(1\) \(1\) \(-3\) \(-1\) \(q+(-1+2\zeta_{10}-2\zeta_{10}^{2}+\zeta_{10}^{3})q^{2}+\cdots\)
363.2.e.g \(4\) \(2.899\) \(\Q(\zeta_{10})\) None \(1\) \(1\) \(2\) \(4\) \(q+(1-\zeta_{10}+\zeta_{10}^{2}-\zeta_{10}^{3})q^{2}-\zeta_{10}^{2}q^{3}+\cdots\)
363.2.e.h \(4\) \(2.899\) \(\Q(\zeta_{10})\) None \(2\) \(-1\) \(4\) \(-3\) \(q+(\zeta_{10}-\zeta_{10}^{2})q^{2}+\zeta_{10}^{2}q^{3}+(-1+\cdots)q^{4}+\cdots\)
363.2.e.i \(4\) \(2.899\) \(\Q(\zeta_{10})\) None \(2\) \(1\) \(-4\) \(-1\) \(q+(2-2\zeta_{10}+2\zeta_{10}^{2}-2\zeta_{10}^{3})q^{2}+\cdots\)
363.2.e.j \(4\) \(2.899\) \(\Q(\zeta_{10})\) None \(3\) \(-1\) \(-1\) \(3\) \(q+(1-\zeta_{10}^{3})q^{2}+\zeta_{10}^{2}q^{3}+(1-\zeta_{10}+\cdots)q^{4}+\cdots\)
363.2.e.k \(4\) \(2.899\) \(\Q(\zeta_{10})\) None \(4\) \(1\) \(2\) \(1\) \(q+(2-\zeta_{10}+\zeta_{10}^{2}-2\zeta_{10}^{3})q^{2}-\zeta_{10}^{2}q^{3}+\cdots\)
363.2.e.l \(4\) \(2.899\) \(\Q(\zeta_{10})\) None \(5\) \(-1\) \(-2\) \(10\) \(q+(1+\zeta_{10}-\zeta_{10}^{2}-\zeta_{10}^{3})q^{2}+\zeta_{10}^{2}q^{3}+\cdots\)
363.2.e.m \(8\) \(2.899\) 8.0.324000000.3 None \(0\) \(2\) \(6\) \(0\) \(q+\beta _{7}q^{2}-\beta _{6}q^{3}+\beta _{4}q^{4}+(3+3\beta _{2}+\cdots)q^{5}+\cdots\)
363.2.e.n \(16\) \(2.899\) 16.0.\(\cdots\).3 None \(0\) \(-4\) \(-2\) \(0\) \(q-\beta _{12}q^{2}+\beta _{4}q^{3}+(\beta _{3}-\beta _{6}+\beta _{8}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(363, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(363, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(33, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(121, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 5 T + 13 T^{2} + 25 T^{3} + 39 T^{4} + 50 T^{5} + 52 T^{6} + 40 T^{7} + 16 T^{8} \))(\( 1 + 4 T + 4 T^{2} - 7 T^{3} - 21 T^{4} - 14 T^{5} + 16 T^{6} + 32 T^{7} + 16 T^{8} \))(\( 1 + 2 T + 2 T^{2} + 5 T^{3} + 11 T^{4} + 10 T^{5} + 8 T^{6} + 16 T^{7} + 16 T^{8} \))(\( 1 + 2 T + 2 T^{2} - 4 T^{4} + 8 T^{6} + 16 T^{7} + 16 T^{8} \))(\( 1 + T - T^{2} - 3 T^{3} - T^{4} - 6 T^{5} - 4 T^{6} + 8 T^{7} + 16 T^{8} \))(\( 1 - T + 4 T^{2} - 2 T^{3} + 9 T^{4} - 4 T^{5} + 16 T^{6} - 8 T^{7} + 16 T^{8} \))(\( 1 - T - T^{2} + 3 T^{3} - T^{4} + 6 T^{5} - 4 T^{6} - 8 T^{7} + 16 T^{8} \))(\( 1 - 2 T + 2 T^{2} - 5 T^{3} + 11 T^{4} - 10 T^{5} + 8 T^{6} - 16 T^{7} + 16 T^{8} \))(\( 1 - 2 T + 2 T^{2} - 4 T^{4} + 8 T^{6} - 16 T^{7} + 16 T^{8} \))(\( 1 - 3 T + 2 T^{2} + T^{4} + 8 T^{6} - 24 T^{7} + 16 T^{8} \))(\( 1 - 4 T + 4 T^{2} + 7 T^{3} - 21 T^{4} + 14 T^{5} + 16 T^{6} - 32 T^{7} + 16 T^{8} \))(\( 1 - 5 T + 13 T^{2} - 25 T^{3} + 39 T^{4} - 50 T^{5} + 52 T^{6} - 40 T^{7} + 16 T^{8} \))(\( 1 - T^{2} - 3 T^{4} + 7 T^{6} + 5 T^{8} + 28 T^{10} - 48 T^{12} - 64 T^{14} + 256 T^{16} \))(\( 1 - T^{2} + T^{4} - 5 T^{6} - 7 T^{8} - 40 T^{10} + 44 T^{12} + 64 T^{14} + 208 T^{16} + 256 T^{18} + 704 T^{20} - 2560 T^{22} - 1792 T^{24} - 5120 T^{26} + 4096 T^{28} - 16384 T^{30} + 65536 T^{32} \))
$3$ (\( 1 + T + T^{2} + T^{3} + T^{4} \))(\( 1 - T + T^{2} - T^{3} + T^{4} \))(\( 1 + T + T^{2} + T^{3} + T^{4} \))(\( 1 - T + T^{2} - T^{3} + T^{4} \))(\( 1 - T + T^{2} - T^{3} + T^{4} \))(\( 1 - T + T^{2} - T^{3} + T^{4} \))(\( 1 - T + T^{2} - T^{3} + T^{4} \))(\( 1 + T + T^{2} + T^{3} + T^{4} \))(\( 1 - T + T^{2} - T^{3} + T^{4} \))(\( 1 + T + T^{2} + T^{3} + T^{4} \))(\( 1 - T + T^{2} - T^{3} + T^{4} \))(\( 1 + T + T^{2} + T^{3} + T^{4} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \))(\( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \))
$5$ (\( 1 + 2 T - T^{2} - 12 T^{3} - 19 T^{4} - 60 T^{5} - 25 T^{6} + 250 T^{7} + 625 T^{8} \))(\( 1 - 2 T - T^{2} - 8 T^{3} + 41 T^{4} - 40 T^{5} - 25 T^{6} - 250 T^{7} + 625 T^{8} \))(\( 1 - 4 T + T^{2} + 16 T^{3} - 39 T^{4} + 80 T^{5} + 25 T^{6} - 500 T^{7} + 625 T^{8} \))(\( 1 + 4 T + 11 T^{2} + 24 T^{3} + 41 T^{4} + 120 T^{5} + 275 T^{6} + 500 T^{7} + 625 T^{8} \))(\( 1 - 2 T - T^{2} + 12 T^{3} - 19 T^{4} + 60 T^{5} - 25 T^{6} - 250 T^{7} + 625 T^{8} \))(\( 1 + 3 T - T^{2} - 3 T^{3} + 16 T^{4} - 15 T^{5} - 25 T^{6} + 375 T^{7} + 625 T^{8} \))(\( 1 - 2 T - T^{2} + 12 T^{3} - 19 T^{4} + 60 T^{5} - 25 T^{6} - 250 T^{7} + 625 T^{8} \))(\( 1 - 4 T + T^{2} + 16 T^{3} - 39 T^{4} + 80 T^{5} + 25 T^{6} - 500 T^{7} + 625 T^{8} \))(\( 1 + 4 T + 11 T^{2} + 24 T^{3} + 41 T^{4} + 120 T^{5} + 275 T^{6} + 500 T^{7} + 625 T^{8} \))(\( 1 + T + T^{2} + 11 T^{3} + 36 T^{4} + 55 T^{5} + 25 T^{6} + 125 T^{7} + 625 T^{8} \))(\( 1 - 2 T - T^{2} - 8 T^{3} + 41 T^{4} - 40 T^{5} - 25 T^{6} - 250 T^{7} + 625 T^{8} \))(\( 1 + 2 T - T^{2} - 12 T^{3} - 19 T^{4} - 60 T^{5} - 25 T^{6} + 250 T^{7} + 625 T^{8} \))(\( ( 1 - 3 T + 4 T^{2} + 3 T^{3} - 29 T^{4} + 15 T^{5} + 100 T^{6} - 375 T^{7} + 625 T^{8} )^{2} \))(\( ( 1 + T - T^{2} + 2 T^{3} - 16 T^{4} + 89 T^{5} + 156 T^{6} - 152 T^{7} + 381 T^{8} - 760 T^{9} + 3900 T^{10} + 11125 T^{11} - 10000 T^{12} + 6250 T^{13} - 15625 T^{14} + 78125 T^{15} + 390625 T^{16} )^{2} \))
$7$ (\( 1 + 10 T + 53 T^{2} + 200 T^{3} + 589 T^{4} + 1400 T^{5} + 2597 T^{6} + 3430 T^{7} + 2401 T^{8} \))(\( 1 + T - 6 T^{2} - 13 T^{3} + 29 T^{4} - 91 T^{5} - 294 T^{6} + 343 T^{7} + 2401 T^{8} \))(\( 1 - 3 T + 2 T^{2} + 15 T^{3} - 59 T^{4} + 105 T^{5} + 98 T^{6} - 1029 T^{7} + 2401 T^{8} \))(\( 1 - T - 6 T^{2} + 13 T^{3} + 29 T^{4} + 91 T^{5} - 294 T^{6} - 343 T^{7} + 2401 T^{8} \))(\( 1 + 4 T + 9 T^{2} + 8 T^{3} - 31 T^{4} + 56 T^{5} + 441 T^{6} + 1372 T^{7} + 2401 T^{8} \))(\( 1 + T - 6 T^{2} - 13 T^{3} + 29 T^{4} - 91 T^{5} - 294 T^{6} + 343 T^{7} + 2401 T^{8} \))(\( 1 - 4 T + 9 T^{2} - 8 T^{3} - 31 T^{4} - 56 T^{5} + 441 T^{6} - 1372 T^{7} + 2401 T^{8} \))(\( 1 + 3 T + 2 T^{2} - 15 T^{3} - 59 T^{4} - 105 T^{5} + 98 T^{6} + 1029 T^{7} + 2401 T^{8} \))(\( 1 + T - 6 T^{2} - 13 T^{3} + 29 T^{4} - 91 T^{5} - 294 T^{6} + 343 T^{7} + 2401 T^{8} \))(\( 1 - 3 T + 2 T^{2} + 15 T^{3} - 59 T^{4} + 105 T^{5} + 98 T^{6} - 1029 T^{7} + 2401 T^{8} \))(\( 1 - T - 6 T^{2} + 13 T^{3} + 29 T^{4} + 91 T^{5} - 294 T^{6} - 343 T^{7} + 2401 T^{8} \))(\( 1 - 10 T + 53 T^{2} - 200 T^{3} + 589 T^{4} - 1400 T^{5} + 2597 T^{6} - 3430 T^{7} + 2401 T^{8} \))(\( 1 - 2 T^{2} - 45 T^{4} + 188 T^{6} + 1829 T^{8} + 9212 T^{10} - 108045 T^{12} - 235298 T^{14} + 5764801 T^{16} \))(\( 1 - 21 T^{2} + 241 T^{4} - 1890 T^{6} + 10698 T^{8} - 33285 T^{10} - 74446 T^{12} + 1750014 T^{14} - 13296727 T^{16} + 85750686 T^{18} - 178744846 T^{20} - 3915946965 T^{22} + 61671841098 T^{24} - 533878220610 T^{26} + 3335750215441 T^{28} - 14242684529829 T^{30} + 33232930569601 T^{32} \))
$11$ (\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))
$13$ (\( 1 - 13 T^{2} + 169 T^{4} - 2197 T^{6} + 28561 T^{8} \))(\( 1 - 3 T + 6 T^{2} - 59 T^{3} + 339 T^{4} - 767 T^{5} + 1014 T^{6} - 6591 T^{7} + 28561 T^{8} \))(\( 1 + T + 18 T^{2} + 5 T^{3} + 251 T^{4} + 65 T^{5} + 3042 T^{6} + 2197 T^{7} + 28561 T^{8} \))(\( 1 + 2 T - 9 T^{2} - 44 T^{3} + 29 T^{4} - 572 T^{5} - 1521 T^{6} + 4394 T^{7} + 28561 T^{8} \))(\( 1 - 2 T - 9 T^{2} + 44 T^{3} + 29 T^{4} + 572 T^{5} - 1521 T^{6} - 4394 T^{7} + 28561 T^{8} \))(\( 1 + 7 T + 6 T^{2} - 49 T^{3} - 181 T^{4} - 637 T^{5} + 1014 T^{6} + 15379 T^{7} + 28561 T^{8} \))(\( 1 + 2 T - 9 T^{2} - 44 T^{3} + 29 T^{4} - 572 T^{5} - 1521 T^{6} + 4394 T^{7} + 28561 T^{8} \))(\( 1 - T + 18 T^{2} - 5 T^{3} + 251 T^{4} - 65 T^{5} + 3042 T^{6} - 2197 T^{7} + 28561 T^{8} \))(\( 1 - 2 T - 9 T^{2} + 44 T^{3} + 29 T^{4} + 572 T^{5} - 1521 T^{6} - 4394 T^{7} + 28561 T^{8} \))(\( 1 - 9 T + 18 T^{2} + 115 T^{3} - 789 T^{4} + 1495 T^{5} + 3042 T^{6} - 19773 T^{7} + 28561 T^{8} \))(\( 1 + 3 T + 6 T^{2} + 59 T^{3} + 339 T^{4} + 767 T^{5} + 1014 T^{6} + 6591 T^{7} + 28561 T^{8} \))(\( 1 - 13 T^{2} + 169 T^{4} - 2197 T^{6} + 28561 T^{8} \))(\( 1 - 23 T^{2} + 360 T^{4} - 4393 T^{6} + 40199 T^{8} - 742417 T^{10} + 10281960 T^{12} - 111016607 T^{14} + 815730721 T^{16} \))(\( 1 - T^{2} - 263 T^{4} + 358 T^{6} + 40682 T^{8} - 80065 T^{10} - 3208942 T^{12} + 7246006 T^{14} - 308472935 T^{16} + 1224575014 T^{18} - 91650592462 T^{20} - 386458462585 T^{22} + 33185557191722 T^{24} + 49353340081942 T^{26} - 6127396387212503 T^{28} - 3937376385699289 T^{30} + 665416609183179841 T^{32} \))
$17$ (\( 1 + 10 T + 43 T^{2} + 200 T^{3} + 1029 T^{4} + 3400 T^{5} + 12427 T^{6} + 49130 T^{7} + 83521 T^{8} \))(\( 1 - 3 T + 37 T^{2} - 45 T^{3} + 676 T^{4} - 765 T^{5} + 10693 T^{6} - 14739 T^{7} + 83521 T^{8} \))(\( 1 - 3 T - 13 T^{2} + 15 T^{3} + 256 T^{4} + 255 T^{5} - 3757 T^{6} - 14739 T^{7} + 83521 T^{8} \))(\( 1 - 4 T - T^{2} + 72 T^{3} - 271 T^{4} + 1224 T^{5} - 289 T^{6} - 19652 T^{7} + 83521 T^{8} \))(\( 1 - 2 T - 13 T^{2} + 60 T^{3} + 101 T^{4} + 1020 T^{5} - 3757 T^{6} - 9826 T^{7} + 83521 T^{8} \))(\( 1 + 12 T + 37 T^{2} - 180 T^{3} - 1619 T^{4} - 3060 T^{5} + 10693 T^{6} + 58956 T^{7} + 83521 T^{8} \))(\( 1 + 2 T - 13 T^{2} - 60 T^{3} + 101 T^{4} - 1020 T^{5} - 3757 T^{6} + 9826 T^{7} + 83521 T^{8} \))(\( 1 + 3 T - 13 T^{2} - 15 T^{3} + 256 T^{4} - 255 T^{5} - 3757 T^{6} + 14739 T^{7} + 83521 T^{8} \))(\( 1 + 4 T - T^{2} - 72 T^{3} - 271 T^{4} - 1224 T^{5} - 289 T^{6} + 19652 T^{7} + 83521 T^{8} \))(\( 1 + 2 T - 13 T^{2} + 20 T^{3} + 341 T^{4} + 340 T^{5} - 3757 T^{6} + 9826 T^{7} + 83521 T^{8} \))(\( 1 + 3 T + 37 T^{2} + 45 T^{3} + 676 T^{4} + 765 T^{5} + 10693 T^{6} + 14739 T^{7} + 83521 T^{8} \))(\( 1 - 10 T + 43 T^{2} - 200 T^{3} + 1029 T^{4} - 3400 T^{5} + 12427 T^{6} - 49130 T^{7} + 83521 T^{8} \))(\( 1 - 31 T^{2} + 672 T^{4} - 11873 T^{6} + 173855 T^{8} - 3431297 T^{10} + 56126112 T^{12} - 748264639 T^{14} + 6975757441 T^{16} \))(\( 1 - 25 T^{2} + 97 T^{4} + 3550 T^{6} - 42862 T^{8} - 2988025 T^{10} + 63581474 T^{12} + 1318750 T^{14} - 8435629295 T^{16} + 381118750 T^{18} + 5310388289954 T^{20} - 72123659611225 T^{22} - 298994915436142 T^{24} + 7156778346593950 T^{26} + 56514357011286817 T^{28} - 4209445663985023225 T^{30} + 48661191875666868481 T^{32} \))
$19$ (\( 1 + 10 T + 41 T^{2} + 200 T^{3} + 1141 T^{4} + 3800 T^{5} + 14801 T^{6} + 68590 T^{7} + 130321 T^{8} \))(\( 1 + 5 T + 21 T^{2} + 145 T^{3} + 956 T^{4} + 2755 T^{5} + 7581 T^{6} + 34295 T^{7} + 130321 T^{8} \))(\( 1 + 5 T + 21 T^{2} + 145 T^{3} + 956 T^{4} + 2755 T^{5} + 7581 T^{6} + 34295 T^{7} + 130321 T^{8} \))(\( 1 + 3 T - 10 T^{2} - 87 T^{3} - 71 T^{4} - 1653 T^{5} - 3610 T^{6} + 20577 T^{7} + 130321 T^{8} \))(\( 1 - 19 T^{2} + 361 T^{4} - 6859 T^{6} + 130321 T^{8} \))(\( 1 - 10 T + 21 T^{2} + 70 T^{3} - 469 T^{4} + 1330 T^{5} + 7581 T^{6} - 68590 T^{7} + 130321 T^{8} \))(\( 1 - 19 T^{2} + 361 T^{4} - 6859 T^{6} + 130321 T^{8} \))(\( 1 - 5 T + 21 T^{2} - 145 T^{3} + 956 T^{4} - 2755 T^{5} + 7581 T^{6} - 34295 T^{7} + 130321 T^{8} \))(\( 1 - 3 T - 10 T^{2} + 87 T^{3} - 71 T^{4} + 1653 T^{5} - 3610 T^{6} - 20577 T^{7} + 130321 T^{8} \))(\( 1 - 10 T + 21 T^{2} + 70 T^{3} - 469 T^{4} + 1330 T^{5} + 7581 T^{6} - 68590 T^{7} + 130321 T^{8} \))(\( 1 - 5 T + 21 T^{2} - 145 T^{3} + 956 T^{4} - 2755 T^{5} + 7581 T^{6} - 34295 T^{7} + 130321 T^{8} \))(\( 1 - 10 T + 41 T^{2} - 200 T^{3} + 1141 T^{4} - 3800 T^{5} + 14801 T^{6} - 68590 T^{7} + 130321 T^{8} \))(\( 1 + 10 T^{2} - 261 T^{4} - 6220 T^{6} + 32021 T^{8} - 2245420 T^{10} - 34013781 T^{12} + 470458810 T^{14} + 16983563041 T^{16} \))(\( 1 - 57 T^{2} + 1789 T^{4} - 39330 T^{6} + 672438 T^{8} - 6082185 T^{10} - 58925794 T^{12} + 3527528562 T^{14} - 77517140647 T^{16} + 1273437810882 T^{18} - 7679268399874 T^{20} - 286141751729985 T^{22} + 11420393164163958 T^{24} - 241134835919313330 T^{26} + 3959620390209362029 T^{28} - 45543381089624394897 T^{30} + \)\(28\!\cdots\!81\)\( T^{32} \))
$23$ (\( ( 1 + 4 T + 23 T^{2} )^{4} \))(\( ( 1 + 4 T + 45 T^{2} + 92 T^{3} + 529 T^{4} )^{2} \))(\( ( 1 + 2 T + 27 T^{2} + 46 T^{3} + 529 T^{4} )^{2} \))(\( ( 1 - 2 T + 23 T^{2} )^{4} \))(\( ( 1 - 8 T + 23 T^{2} )^{4} \))(\( ( 1 + 4 T + 45 T^{2} + 92 T^{3} + 529 T^{4} )^{2} \))(\( ( 1 - 8 T + 23 T^{2} )^{4} \))(\( ( 1 + 2 T + 27 T^{2} + 46 T^{3} + 529 T^{4} )^{2} \))(\( ( 1 - 2 T + 23 T^{2} )^{4} \))(\( ( 1 + 2 T + 27 T^{2} + 46 T^{3} + 529 T^{4} )^{2} \))(\( ( 1 + 4 T + 45 T^{2} + 92 T^{3} + 529 T^{4} )^{2} \))(\( ( 1 + 4 T + 23 T^{2} )^{4} \))(\( ( 1 + 6 T + 23 T^{2} )^{8} \))(\( ( 1 - 2 T + 23 T^{2} )^{16} \))
$29$ (\( 1 - 10 T + 31 T^{2} - 200 T^{3} + 1821 T^{4} - 5800 T^{5} + 26071 T^{6} - 243890 T^{7} + 707281 T^{8} \))(\( 1 + 6 T + 7 T^{2} - 132 T^{3} - 995 T^{4} - 3828 T^{5} + 5887 T^{6} + 146334 T^{7} + 707281 T^{8} \))(\( 1 + 10 T + 31 T^{2} + 200 T^{3} + 1821 T^{4} + 5800 T^{5} + 26071 T^{6} + 243890 T^{7} + 707281 T^{8} \))(\( 1 - 6 T + 7 T^{2} + 132 T^{3} - 995 T^{4} + 3828 T^{5} + 5887 T^{6} - 146334 T^{7} + 707281 T^{8} \))(\( 1 - 6 T + 7 T^{2} + 132 T^{3} - 995 T^{4} + 3828 T^{5} + 5887 T^{6} - 146334 T^{7} + 707281 T^{8} \))(\( 1 + 6 T + 7 T^{2} - 132 T^{3} - 995 T^{4} - 3828 T^{5} + 5887 T^{6} + 146334 T^{7} + 707281 T^{8} \))(\( 1 + 6 T + 7 T^{2} - 132 T^{3} - 995 T^{4} - 3828 T^{5} + 5887 T^{6} + 146334 T^{7} + 707281 T^{8} \))(\( 1 - 10 T + 31 T^{2} - 200 T^{3} + 1821 T^{4} - 5800 T^{5} + 26071 T^{6} - 243890 T^{7} + 707281 T^{8} \))(\( 1 + 6 T + 7 T^{2} - 132 T^{3} - 995 T^{4} - 3828 T^{5} + 5887 T^{6} + 146334 T^{7} + 707281 T^{8} \))(\( 1 - 10 T + 31 T^{2} - 200 T^{3} + 1821 T^{4} - 5800 T^{5} + 26071 T^{6} - 243890 T^{7} + 707281 T^{8} \))(\( 1 - 6 T + 7 T^{2} + 132 T^{3} - 995 T^{4} + 3828 T^{5} + 5887 T^{6} - 146334 T^{7} + 707281 T^{8} \))(\( 1 + 10 T + 31 T^{2} + 200 T^{3} + 1821 T^{4} + 5800 T^{5} + 26071 T^{6} + 243890 T^{7} + 707281 T^{8} \))(\( 1 - 55 T^{2} + 2184 T^{4} - 73865 T^{6} + 2225831 T^{8} - 62120465 T^{10} + 1544701704 T^{12} - 32715282655 T^{14} + 500246412961 T^{16} \))(\( 1 - 109 T^{2} + 7237 T^{4} - 374306 T^{6} + 16475366 T^{8} - 642740845 T^{10} + 22740816518 T^{12} - 739401320150 T^{14} + 22253089167217 T^{16} - 621836510246150 T^{18} + 16084147447667558 T^{20} - 382317243965246245 T^{22} + 8241742743719618726 T^{24} - \)\(15\!\cdots\!06\)\( T^{26} + \)\(25\!\cdots\!17\)\( T^{28} - \)\(32\!\cdots\!29\)\( T^{30} + \)\(25\!\cdots\!21\)\( T^{32} \))
$31$ (\( 1 - 31 T^{2} + 961 T^{4} - 29791 T^{6} + 923521 T^{8} \))(\( 1 - 13 T + 63 T^{2} - 341 T^{3} + 2480 T^{4} - 10571 T^{5} + 60543 T^{6} - 387283 T^{7} + 923521 T^{8} \))(\( 1 + 7 T + 3 T^{2} + 119 T^{3} + 1640 T^{4} + 3689 T^{5} + 2883 T^{6} + 208537 T^{7} + 923521 T^{8} \))(\( 1 - 5 T - 6 T^{2} + 185 T^{3} - 739 T^{4} + 5735 T^{5} - 5766 T^{6} - 148955 T^{7} + 923521 T^{8} \))(\( 1 - 8 T + 33 T^{2} - 16 T^{3} - 895 T^{4} - 496 T^{5} + 31713 T^{6} - 238328 T^{7} + 923521 T^{8} \))(\( 1 + 12 T + 63 T^{2} + 434 T^{3} + 3255 T^{4} + 13454 T^{5} + 60543 T^{6} + 357492 T^{7} + 923521 T^{8} \))(\( 1 - 8 T + 33 T^{2} - 16 T^{3} - 895 T^{4} - 496 T^{5} + 31713 T^{6} - 238328 T^{7} + 923521 T^{8} \))(\( 1 + 7 T + 3 T^{2} + 119 T^{3} + 1640 T^{4} + 3689 T^{5} + 2883 T^{6} + 208537 T^{7} + 923521 T^{8} \))(\( 1 - 5 T - 6 T^{2} + 185 T^{3} - 739 T^{4} + 5735 T^{5} - 5766 T^{6} - 148955 T^{7} + 923521 T^{8} \))(\( 1 - 8 T + 3 T^{2} - 46 T^{3} + 1175 T^{4} - 1426 T^{5} + 2883 T^{6} - 238328 T^{7} + 923521 T^{8} \))(\( 1 - 13 T + 63 T^{2} - 341 T^{3} + 2480 T^{4} - 10571 T^{5} + 60543 T^{6} - 387283 T^{7} + 923521 T^{8} \))(\( 1 - 31 T^{2} + 961 T^{4} - 29791 T^{6} + 923521 T^{8} \))(\( ( 1 + 4 T - 15 T^{2} - 184 T^{3} - 271 T^{4} - 5704 T^{5} - 14415 T^{6} + 119164 T^{7} + 923521 T^{8} )^{2} \))(\( ( 1 + 9 T + 7 T^{2} - 324 T^{3} - 1884 T^{4} - 1935 T^{5} + 24878 T^{6} + 152820 T^{7} + 805067 T^{8} + 4737420 T^{9} + 23907758 T^{10} - 57645585 T^{11} - 1739913564 T^{12} - 9275844924 T^{13} + 6212525767 T^{14} + 247613526999 T^{15} + 852891037441 T^{16} )^{2} \))
$37$ (\( 1 + 2 T - 33 T^{2} - 140 T^{3} + 941 T^{4} - 5180 T^{5} - 45177 T^{6} + 101306 T^{7} + 1874161 T^{8} \))(\( 1 + T - 6 T^{2} + 197 T^{3} + 1379 T^{4} + 7289 T^{5} - 8214 T^{6} + 50653 T^{7} + 1874161 T^{8} \))(\( 1 - 7 T - 18 T^{2} + 145 T^{3} + 371 T^{4} + 5365 T^{5} - 24642 T^{6} - 354571 T^{7} + 1874161 T^{8} \))(\( 1 + 3 T - 28 T^{2} - 195 T^{3} + 451 T^{4} - 7215 T^{5} - 38332 T^{6} + 151959 T^{7} + 1874161 T^{8} \))(\( 1 + 6 T - T^{2} - 228 T^{3} - 1331 T^{4} - 8436 T^{5} - 1369 T^{6} + 303918 T^{7} + 1874161 T^{8} \))(\( 1 - 9 T - 6 T^{2} + 307 T^{3} - 1581 T^{4} + 11359 T^{5} - 8214 T^{6} - 455877 T^{7} + 1874161 T^{8} \))(\( 1 + 6 T - T^{2} - 228 T^{3} - 1331 T^{4} - 8436 T^{5} - 1369 T^{6} + 303918 T^{7} + 1874161 T^{8} \))(\( 1 - 7 T - 18 T^{2} + 145 T^{3} + 371 T^{4} + 5365 T^{5} - 24642 T^{6} - 354571 T^{7} + 1874161 T^{8} \))(\( 1 + 3 T - 28 T^{2} - 195 T^{3} + 451 T^{4} - 7215 T^{5} - 38332 T^{6} + 151959 T^{7} + 1874161 T^{8} \))(\( 1 + 3 T - 18 T^{2} + 155 T^{3} + 1851 T^{4} + 5735 T^{5} - 24642 T^{6} + 151959 T^{7} + 1874161 T^{8} \))(\( 1 + T - 6 T^{2} + 197 T^{3} + 1379 T^{4} + 7289 T^{5} - 8214 T^{6} + 50653 T^{7} + 1874161 T^{8} \))(\( 1 + 2 T - 33 T^{2} - 140 T^{3} + 941 T^{4} - 5180 T^{5} - 45177 T^{6} + 101306 T^{7} + 1874161 T^{8} \))(\( ( 1 - 11 T + 84 T^{2} - 517 T^{3} + 2579 T^{4} - 19129 T^{5} + 114996 T^{6} - 557183 T^{7} + 1874161 T^{8} )^{2} \))(\( ( 1 + 5 T - 12 T^{2} - 245 T^{3} - 781 T^{4} - 9065 T^{5} - 16428 T^{6} + 253265 T^{7} + 1874161 T^{8} )^{4} \))
$41$ (\( 1 + 10 T + 19 T^{2} + 200 T^{3} + 2901 T^{4} + 8200 T^{5} + 31939 T^{6} + 689210 T^{7} + 2825761 T^{8} \))(\( 1 + 7 T - 22 T^{2} - 161 T^{3} + 575 T^{4} - 6601 T^{5} - 36982 T^{6} + 482447 T^{7} + 2825761 T^{8} \))(\( 1 - 17 T + 208 T^{2} - 1879 T^{3} + 13815 T^{4} - 77039 T^{5} + 349648 T^{6} - 1171657 T^{7} + 2825761 T^{8} \))(\( 1 + 2 T - 37 T^{2} - 156 T^{3} + 1205 T^{4} - 6396 T^{5} - 62197 T^{6} + 137842 T^{7} + 2825761 T^{8} \))(\( 1 - 2 T - 37 T^{2} + 156 T^{3} + 1205 T^{4} + 6396 T^{5} - 62197 T^{6} - 137842 T^{7} + 2825761 T^{8} \))(\( 1 - 3 T - 22 T^{2} - 171 T^{3} + 2215 T^{4} - 7011 T^{5} - 36982 T^{6} - 206763 T^{7} + 2825761 T^{8} \))(\( 1 + 2 T - 37 T^{2} - 156 T^{3} + 1205 T^{4} - 6396 T^{5} - 62197 T^{6} + 137842 T^{7} + 2825761 T^{8} \))(\( 1 + 17 T + 208 T^{2} + 1879 T^{3} + 13815 T^{4} + 77039 T^{5} + 349648 T^{6} + 1171657 T^{7} + 2825761 T^{8} \))(\( 1 - 2 T - 37 T^{2} + 156 T^{3} + 1205 T^{4} + 6396 T^{5} - 62197 T^{6} - 137842 T^{7} + 2825761 T^{8} \))(\( 1 + 23 T + 208 T^{2} + 961 T^{3} + 3975 T^{4} + 39401 T^{5} + 349648 T^{6} + 1585183 T^{7} + 2825761 T^{8} \))(\( 1 - 7 T - 22 T^{2} + 161 T^{3} + 575 T^{4} + 6601 T^{5} - 36982 T^{6} - 482447 T^{7} + 2825761 T^{8} \))(\( 1 - 10 T + 19 T^{2} - 200 T^{3} + 2901 T^{4} - 8200 T^{5} + 31939 T^{6} - 689210 T^{7} + 2825761 T^{8} \))(\( 1 - 79 T^{2} + 4560 T^{4} - 227441 T^{6} + 10302479 T^{8} - 382328321 T^{10} + 12885470160 T^{12} - 375258235039 T^{14} + 7984925229121 T^{16} \))(\( 1 - 13 T^{2} - 1379 T^{4} + 16198 T^{6} - 617554 T^{8} - 145400749 T^{10} + 6388932854 T^{12} + 109747963234 T^{14} - 6394289003543 T^{16} + 184486326196354 T^{18} + 18053597290451894 T^{20} - 690668714469476509 T^{22} - 4931122514944590034 T^{24} + \)\(21\!\cdots\!98\)\( T^{26} - \)\(31\!\cdots\!99\)\( T^{28} - \)\(49\!\cdots\!93\)\( T^{30} + \)\(63\!\cdots\!41\)\( T^{32} \))
$43$ (\( ( 1 + 66 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 + 41 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 + 8 T + 97 T^{2} + 344 T^{3} + 1849 T^{4} )^{2} \))(\( ( 1 + 12 T + 43 T^{2} )^{4} \))(\( ( 1 + 43 T^{2} )^{4} \))(\( ( 1 + 41 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 + 43 T^{2} )^{4} \))(\( ( 1 - 8 T + 97 T^{2} - 344 T^{3} + 1849 T^{4} )^{2} \))(\( ( 1 - 12 T + 43 T^{2} )^{4} \))(\( ( 1 + 8 T + 97 T^{2} + 344 T^{3} + 1849 T^{4} )^{2} \))(\( ( 1 + 41 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 + 66 T^{2} + 1849 T^{4} )^{2} \))(\( ( 1 + 74 T^{2} + 1849 T^{4} )^{4} \))(\( ( 1 + 42 T^{2} + 1849 T^{4} )^{8} \))
$47$ (\( 1 + 8 T + 17 T^{2} - 240 T^{3} - 2719 T^{4} - 11280 T^{5} + 37553 T^{6} + 830584 T^{7} + 4879681 T^{8} \))(\( 1 + 8 T + 67 T^{2} + 610 T^{3} + 6231 T^{4} + 28670 T^{5} + 148003 T^{6} + 830584 T^{7} + 4879681 T^{8} \))(\( 1 - 2 T - 43 T^{2} - 50 T^{3} + 2351 T^{4} - 2350 T^{5} - 94987 T^{6} - 207646 T^{7} + 4879681 T^{8} \))(\( 1 + 2 T - 43 T^{2} - 180 T^{3} + 1661 T^{4} - 8460 T^{5} - 94987 T^{6} + 207646 T^{7} + 4879681 T^{8} \))(\( 1 + 8 T + 17 T^{2} - 240 T^{3} - 2719 T^{4} - 11280 T^{5} + 37553 T^{6} + 830584 T^{7} + 4879681 T^{8} \))(\( 1 - 17 T + 67 T^{2} + 335 T^{3} - 4344 T^{4} + 15745 T^{5} + 148003 T^{6} - 1764991 T^{7} + 4879681 T^{8} \))(\( 1 + 8 T + 17 T^{2} - 240 T^{3} - 2719 T^{4} - 11280 T^{5} + 37553 T^{6} + 830584 T^{7} + 4879681 T^{8} \))(\( 1 - 2 T - 43 T^{2} - 50 T^{3} + 2351 T^{4} - 2350 T^{5} - 94987 T^{6} - 207646 T^{7} + 4879681 T^{8} \))(\( 1 + 2 T - 43 T^{2} - 180 T^{3} + 1661 T^{4} - 8460 T^{5} - 94987 T^{6} + 207646 T^{7} + 4879681 T^{8} \))(\( 1 + 3 T - 43 T^{2} - 45 T^{3} + 2116 T^{4} - 2115 T^{5} - 94987 T^{6} + 311469 T^{7} + 4879681 T^{8} \))(\( 1 + 8 T + 67 T^{2} + 610 T^{3} + 6231 T^{4} + 28670 T^{5} + 148003 T^{6} + 830584 T^{7} + 4879681 T^{8} \))(\( 1 + 8 T + 17 T^{2} - 240 T^{3} - 2719 T^{4} - 11280 T^{5} + 37553 T^{6} + 830584 T^{7} + 4879681 T^{8} \))(\( ( 1 - 47 T^{2} + 2209 T^{4} - 103823 T^{6} + 4879681 T^{8} )^{2} \))(\( ( 1 - 14 T + 86 T^{2} - 322 T^{3} + 2051 T^{4} - 15022 T^{5} + 6600 T^{6} + 921760 T^{7} - 8276823 T^{8} + 43322720 T^{9} + 14579400 T^{10} - 1559629106 T^{11} + 10008225731 T^{12} - 73849092254 T^{13} + 927012518294 T^{14} - 7092723686482 T^{15} + 23811286661761 T^{16} )^{2} \))
$53$ (\( 1 + 6 T - 17 T^{2} - 420 T^{3} - 1619 T^{4} - 22260 T^{5} - 47753 T^{6} + 893262 T^{7} + 7890481 T^{8} \))(\( 1 + T - 47 T^{2} + 155 T^{3} + 2916 T^{4} + 8215 T^{5} - 132023 T^{6} + 148877 T^{7} + 7890481 T^{8} \))(\( 1 - 11 T + 23 T^{2} + 475 T^{3} - 5824 T^{4} + 25175 T^{5} + 64607 T^{6} - 1637647 T^{7} + 7890481 T^{8} \))(\( 1 + 6 T - 17 T^{2} - 420 T^{3} - 1619 T^{4} - 22260 T^{5} - 47753 T^{6} + 893262 T^{7} + 7890481 T^{8} \))(\( 1 + 6 T - 17 T^{2} - 420 T^{3} - 1619 T^{4} - 22260 T^{5} - 47753 T^{6} + 893262 T^{7} + 7890481 T^{8} \))(\( 1 - 4 T - 47 T^{2} + 160 T^{3} + 2121 T^{4} + 8480 T^{5} - 132023 T^{6} - 595508 T^{7} + 7890481 T^{8} \))(\( 1 + 6 T - 17 T^{2} - 420 T^{3} - 1619 T^{4} - 22260 T^{5} - 47753 T^{6} + 893262 T^{7} + 7890481 T^{8} \))(\( 1 - 11 T + 23 T^{2} + 475 T^{3} - 5824 T^{4} + 25175 T^{5} + 64607 T^{6} - 1637647 T^{7} + 7890481 T^{8} \))(\( 1 + 6 T - 17 T^{2} - 420 T^{3} - 1619 T^{4} - 22260 T^{5} - 47753 T^{6} + 893262 T^{7} + 7890481 T^{8} \))(\( 1 - 6 T + 23 T^{2} + 120 T^{3} - 1319 T^{4} + 6360 T^{5} + 64607 T^{6} - 893262 T^{7} + 7890481 T^{8} \))(\( 1 + T - 47 T^{2} + 155 T^{3} + 2916 T^{4} + 8215 T^{5} - 132023 T^{6} + 148877 T^{7} + 7890481 T^{8} \))(\( 1 + 6 T - 17 T^{2} - 420 T^{3} - 1619 T^{4} - 22260 T^{5} - 47753 T^{6} + 893262 T^{7} + 7890481 T^{8} \))(\( ( 1 - 9 T + 28 T^{2} + 225 T^{3} - 3509 T^{4} + 11925 T^{5} + 78652 T^{6} - 1339893 T^{7} + 7890481 T^{8} )^{2} \))(\( ( 1 + 9 T + 29 T^{2} + 270 T^{3} + 2406 T^{4} + 62217 T^{5} + 482170 T^{6} + 1493478 T^{7} + 11287517 T^{8} + 79154334 T^{9} + 1354415530 T^{10} + 9262680309 T^{11} + 18984497286 T^{12} + 112912783110 T^{13} + 642766472741 T^{14} + 10572400258533 T^{15} + 62259690411361 T^{16} )^{2} \))
$59$ (\( 1 - 59 T^{2} + 3481 T^{4} - 205379 T^{6} + 12117361 T^{8} \))(\( 1 + 11 T + 17 T^{2} - 577 T^{3} - 6700 T^{4} - 34043 T^{5} + 59177 T^{6} + 2259169 T^{7} + 12117361 T^{8} \))(\( 1 - 15 T + 131 T^{2} - 1395 T^{3} + 14176 T^{4} - 82305 T^{5} + 456011 T^{6} - 3080685 T^{7} + 12117361 T^{8} \))(\( 1 - 10 T + 41 T^{2} + 180 T^{3} - 4219 T^{4} + 10620 T^{5} + 142721 T^{6} - 2053790 T^{7} + 12117361 T^{8} \))(\( 1 - 4 T - 43 T^{2} + 408 T^{3} + 905 T^{4} + 24072 T^{5} - 149683 T^{6} - 821516 T^{7} + 12117361 T^{8} \))(\( 1 + 6 T + 17 T^{2} - 222 T^{3} - 1685 T^{4} - 13098 T^{5} + 59177 T^{6} + 1232274 T^{7} + 12117361 T^{8} \))(\( 1 - 4 T - 43 T^{2} + 408 T^{3} + 905 T^{4} + 24072 T^{5} - 149683 T^{6} - 821516 T^{7} + 12117361 T^{8} \))(\( 1 - 15 T + 131 T^{2} - 1395 T^{3} + 14176 T^{4} - 82305 T^{5} + 456011 T^{6} - 3080685 T^{7} + 12117361 T^{8} \))(\( 1 - 10 T + 41 T^{2} + 180 T^{3} - 4219 T^{4} + 10620 T^{5} + 142721 T^{6} - 2053790 T^{7} + 12117361 T^{8} \))(\( 1 + 20 T + 131 T^{2} + 530 T^{3} + 3851 T^{4} + 31270 T^{5} + 456011 T^{6} + 4107580 T^{7} + 12117361 T^{8} \))(\( 1 + 11 T + 17 T^{2} - 577 T^{3} - 6700 T^{4} - 34043 T^{5} + 59177 T^{6} + 2259169 T^{7} + 12117361 T^{8} \))(\( 1 - 59 T^{2} + 3481 T^{4} - 205379 T^{6} + 12117361 T^{8} \))(\( ( 1 - 6 T - 23 T^{2} + 492 T^{3} - 1595 T^{4} + 29028 T^{5} - 80063 T^{6} - 1232274 T^{7} + 12117361 T^{8} )^{2} \))(\( ( 1 - 6 T - 23 T^{2} + 492 T^{3} - 1595 T^{4} + 29028 T^{5} - 80063 T^{6} - 1232274 T^{7} + 12117361 T^{8} )^{4} \))
$61$ (\( 1 + 20 T + 179 T^{2} + 1600 T^{3} + 15001 T^{4} + 97600 T^{5} + 666059 T^{6} + 4539620 T^{7} + 13845841 T^{8} \))(\( 1 + 24 T + 245 T^{2} + 1896 T^{3} + 15169 T^{4} + 115656 T^{5} + 911645 T^{6} + 5447544 T^{7} + 13845841 T^{8} \))(\( 1 - 12 T - 7 T^{2} + 576 T^{3} - 3335 T^{4} + 35136 T^{5} - 26047 T^{6} - 2723772 T^{7} + 13845841 T^{8} \))(\( 1 - 3 T - 52 T^{2} + 339 T^{3} + 2155 T^{4} + 20679 T^{5} - 193492 T^{6} - 680943 T^{7} + 13845841 T^{8} \))(\( 1 + 6 T - 25 T^{2} - 516 T^{3} - 1571 T^{4} - 31476 T^{5} - 93025 T^{6} + 1361886 T^{7} + 13845841 T^{8} \))(\( 1 - 21 T + 245 T^{2} - 2559 T^{3} + 23404 T^{4} - 156099 T^{5} + 911645 T^{6} - 4766601 T^{7} + 13845841 T^{8} \))(\( 1 - 6 T - 25 T^{2} + 516 T^{3} - 1571 T^{4} + 31476 T^{5} - 93025 T^{6} - 1361886 T^{7} + 13845841 T^{8} \))(\( 1 + 12 T - 7 T^{2} - 576 T^{3} - 3335 T^{4} - 35136 T^{5} - 26047 T^{6} + 2723772 T^{7} + 13845841 T^{8} \))(\( 1 + 3 T - 52 T^{2} - 339 T^{3} + 2155 T^{4} - 20679 T^{5} - 193492 T^{6} + 680943 T^{7} + 13845841 T^{8} \))(\( 1 + 3 T - 7 T^{2} + 441 T^{3} + 4900 T^{4} + 26901 T^{5} - 26047 T^{6} + 680943 T^{7} + 13845841 T^{8} \))(\( 1 - 24 T + 245 T^{2} - 1896 T^{3} + 15169 T^{4} - 115656 T^{5} + 911645 T^{6} - 5447544 T^{7} + 13845841 T^{8} \))(\( 1 - 20 T + 179 T^{2} - 1600 T^{3} + 15001 T^{4} - 97600 T^{5} + 666059 T^{6} - 4539620 T^{7} + 13845841 T^{8} \))(\( ( 1 - 61 T^{2} + 3721 T^{4} - 226981 T^{6} + 13845841 T^{8} )^{2} \))(\( 1 - 201 T^{2} + 23065 T^{4} - 1899450 T^{6} + 118420890 T^{8} - 6776311593 T^{10} + 410378301938 T^{12} - 27281888695530 T^{14} + 1797850314334025 T^{16} - 101515907836067130 T^{18} + 5682032718483539858 T^{20} - \)\(34\!\cdots\!73\)\( T^{22} + \)\(22\!\cdots\!90\)\( T^{24} - \)\(13\!\cdots\!50\)\( T^{26} + \)\(61\!\cdots\!65\)\( T^{28} - \)\(19\!\cdots\!41\)\( T^{30} + \)\(36\!\cdots\!61\)\( T^{32} \))
$67$ (\( ( 1 + 12 T + 67 T^{2} )^{4} \))(\( ( 1 + 3 T + 125 T^{2} + 201 T^{3} + 4489 T^{4} )^{2} \))(\( ( 1 - T + 33 T^{2} - 67 T^{3} + 4489 T^{4} )^{2} \))(\( ( 1 + T + 67 T^{2} )^{4} \))(\( ( 1 + 4 T + 67 T^{2} )^{4} \))(\( ( 1 + 3 T + 125 T^{2} + 201 T^{3} + 4489 T^{4} )^{2} \))(\( ( 1 + 4 T + 67 T^{2} )^{4} \))(\( ( 1 - T + 33 T^{2} - 67 T^{3} + 4489 T^{4} )^{2} \))(\( ( 1 + T + 67 T^{2} )^{4} \))(\( ( 1 - T + 33 T^{2} - 67 T^{3} + 4489 T^{4} )^{2} \))(\( ( 1 + 3 T + 125 T^{2} + 201 T^{3} + 4489 T^{4} )^{2} \))(\( ( 1 + 12 T + 67 T^{2} )^{4} \))(\( ( 1 + 2 T + 67 T^{2} )^{8} \))(\( ( 1 - 15 T + 116 T^{2} - 1005 T^{3} + 4489 T^{4} )^{8} \))
$71$ (\( 1 - 8 T - 7 T^{2} + 624 T^{3} - 4495 T^{4} + 44304 T^{5} - 35287 T^{6} - 2863288 T^{7} + 25411681 T^{8} \))(\( 1 + 20 T + 119 T^{2} + 470 T^{3} + 4871 T^{4} + 33370 T^{5} + 599879 T^{6} + 7158220 T^{7} + 25411681 T^{8} \))(\( 1 - 18 T + 253 T^{2} - 2826 T^{3} + 28855 T^{4} - 200646 T^{5} + 1275373 T^{6} - 6442398 T^{7} + 25411681 T^{8} \))(\( 1 - 71 T^{2} + 5041 T^{4} - 357911 T^{6} + 25411681 T^{8} \))(\( 1 - 71 T^{2} + 5041 T^{4} - 357911 T^{6} + 25411681 T^{8} \))(\( 1 - 15 T + 119 T^{2} - 1455 T^{3} + 17296 T^{4} - 103305 T^{5} + 599879 T^{6} - 5368665 T^{7} + 25411681 T^{8} \))(\( 1 - 71 T^{2} + 5041 T^{4} - 357911 T^{6} + 25411681 T^{8} \))(\( 1 - 18 T + 253 T^{2} - 2826 T^{3} + 28855 T^{4} - 200646 T^{5} + 1275373 T^{6} - 6442398 T^{7} + 25411681 T^{8} \))(\( 1 - 71 T^{2} + 5041 T^{4} - 357911 T^{6} + 25411681 T^{8} \))(\( 1 + 27 T + 253 T^{2} + 819 T^{3} + 100 T^{4} + 58149 T^{5} + 1275373 T^{6} + 9663597 T^{7} + 25411681 T^{8} \))(\( 1 + 20 T + 119 T^{2} + 470 T^{3} + 4871 T^{4} + 33370 T^{5} + 599879 T^{6} + 7158220 T^{7} + 25411681 T^{8} \))(\( 1 - 8 T - 7 T^{2} + 624 T^{3} - 4495 T^{4} + 44304 T^{5} - 35287 T^{6} - 2863288 T^{7} + 25411681 T^{8} \))(\( ( 1 - 6 T - 35 T^{2} + 636 T^{3} - 1331 T^{4} + 45156 T^{5} - 176435 T^{6} - 2147466 T^{7} + 25411681 T^{8} )^{2} \))(\( ( 1 - 10 T - 34 T^{2} + 970 T^{3} - 3085 T^{4} - 69530 T^{5} + 591384 T^{6} + 703760 T^{7} - 21365271 T^{8} + 49966960 T^{9} + 2981166744 T^{10} - 24885551830 T^{11} - 78395035885 T^{12} + 1750102470470 T^{13} - 4355409653314 T^{14} - 90951201583910 T^{15} + 645753531245761 T^{16} )^{2} \))
$73$ (\( 1 + 20 T + 167 T^{2} + 1600 T^{3} + 17569 T^{4} + 116800 T^{5} + 889943 T^{6} + 7780340 T^{7} + 28398241 T^{8} \))(\( 1 - 16 T + 63 T^{2} - 470 T^{3} + 8141 T^{4} - 34310 T^{5} + 335727 T^{6} - 6224272 T^{7} + 28398241 T^{8} \))(\( 1 - 4 T - 57 T^{2} - 170 T^{3} + 6221 T^{4} - 12410 T^{5} - 303753 T^{6} - 1556068 T^{7} + 28398241 T^{8} \))(\( 1 + 11 T + 48 T^{2} - 275 T^{3} - 6529 T^{4} - 20075 T^{5} + 255792 T^{6} + 4279187 T^{7} + 28398241 T^{8} \))(\( 1 - 14 T + 123 T^{2} - 700 T^{3} + 821 T^{4} - 51100 T^{5} + 655467 T^{6} - 5446238 T^{7} + 28398241 T^{8} \))(\( 1 + 14 T + 63 T^{2} + 850 T^{3} + 12521 T^{4} + 62050 T^{5} + 335727 T^{6} + 5446238 T^{7} + 28398241 T^{8} \))(\( 1 + 14 T + 123 T^{2} + 700 T^{3} + 821 T^{4} + 51100 T^{5} + 655467 T^{6} + 5446238 T^{7} + 28398241 T^{8} \))(\( 1 + 4 T - 57 T^{2} + 170 T^{3} + 6221 T^{4} + 12410 T^{5} - 303753 T^{6} + 1556068 T^{7} + 28398241 T^{8} \))(\( 1 - 11 T + 48 T^{2} + 275 T^{3} - 6529 T^{4} + 20075 T^{5} + 255792 T^{6} - 4279187 T^{7} + 28398241 T^{8} \))(\( 1 + 6 T - 57 T^{2} - 130 T^{3} + 4761 T^{4} - 9490 T^{5} - 303753 T^{6} + 2334102 T^{7} + 28398241 T^{8} \))(\( 1 + 16 T + 63 T^{2} + 470 T^{3} + 8141 T^{4} + 34310 T^{5} + 335727 T^{6} + 6224272 T^{7} + 28398241 T^{8} \))(\( 1 - 20 T + 167 T^{2} - 1600 T^{3} + 17569 T^{4} - 116800 T^{5} + 889943 T^{6} - 7780340 T^{7} + 28398241 T^{8} \))(\( 1 - 98 T^{2} + 4275 T^{4} + 103292 T^{6} - 32904091 T^{8} + 550443068 T^{10} + 121402480275 T^{12} - 14830754176322 T^{14} + 806460091894081 T^{16} \))(\( 1 - 153 T^{2} + 7105 T^{4} + 592110 T^{6} - 110084430 T^{8} + 8204907591 T^{10} - 145073008078 T^{12} - 38635616703330 T^{14} + 4712959112277665 T^{16} - 205889201412045570 T^{18} - 4119818245993990798 T^{20} + \)\(12\!\cdots\!99\)\( T^{22} - \)\(88\!\cdots\!30\)\( T^{24} + \)\(25\!\cdots\!90\)\( T^{26} + \)\(16\!\cdots\!05\)\( T^{28} - \)\(18\!\cdots\!77\)\( T^{30} + \)\(65\!\cdots\!61\)\( T^{32} \))
$79$ (\( 1 + 30 T + 461 T^{2} + 5400 T^{3} + 52861 T^{4} + 426600 T^{5} + 2877101 T^{6} + 14791170 T^{7} + 38950081 T^{8} \))(\( 1 - 11 T + 42 T^{2} + 407 T^{3} - 7795 T^{4} + 32153 T^{5} + 262122 T^{6} - 5423429 T^{7} + 38950081 T^{8} \))(\( 1 - 15 T + 6 T^{2} + 815 T^{3} - 5979 T^{4} + 64385 T^{5} + 37446 T^{6} - 7395585 T^{7} + 38950081 T^{8} \))(\( 1 - 11 T + 42 T^{2} + 407 T^{3} - 7795 T^{4} + 32153 T^{5} + 262122 T^{6} - 5423429 T^{7} + 38950081 T^{8} \))(\( 1 - 4 T - 63 T^{2} + 568 T^{3} + 2705 T^{4} + 44872 T^{5} - 393183 T^{6} - 1972156 T^{7} + 38950081 T^{8} \))(\( 1 - 11 T + 42 T^{2} + 407 T^{3} - 7795 T^{4} + 32153 T^{5} + 262122 T^{6} - 5423429 T^{7} + 38950081 T^{8} \))(\( 1 + 4 T - 63 T^{2} - 568 T^{3} + 2705 T^{4} - 44872 T^{5} - 393183 T^{6} + 1972156 T^{7} + 38950081 T^{8} \))(\( 1 + 15 T + 6 T^{2} - 815 T^{3} - 5979 T^{4} - 64385 T^{5} + 37446 T^{6} + 7395585 T^{7} + 38950081 T^{8} \))(\( 1 + 11 T + 42 T^{2} - 407 T^{3} - 7795 T^{4} - 32153 T^{5} + 262122 T^{6} + 5423429 T^{7} + 38950081 T^{8} \))(\( 1 + 5 T + 6 T^{2} + 715 T^{3} + 9821 T^{4} + 56485 T^{5} + 37446 T^{6} + 2465195 T^{7} + 38950081 T^{8} \))(\( 1 + 11 T + 42 T^{2} - 407 T^{3} - 7795 T^{4} - 32153 T^{5} + 262122 T^{6} + 5423429 T^{7} + 38950081 T^{8} \))(\( 1 - 30 T + 461 T^{2} - 5400 T^{3} + 52861 T^{4} - 426600 T^{5} + 2877101 T^{6} - 14791170 T^{7} + 38950081 T^{8} \))(\( ( 1 - 79 T^{2} + 6241 T^{4} - 493039 T^{6} + 38950081 T^{8} )^{2} \))(\( 1 - 265 T^{2} + 40261 T^{4} - 4382570 T^{6} + 354324590 T^{8} - 29112124345 T^{10} + 2777740858574 T^{12} - 279089215200710 T^{14} + 25072966990928449 T^{16} - 1741795792067631110 T^{18} + \)\(10\!\cdots\!94\)\( T^{20} - \)\(70\!\cdots\!45\)\( T^{22} + \)\(53\!\cdots\!90\)\( T^{24} - \)\(41\!\cdots\!70\)\( T^{26} + \)\(23\!\cdots\!01\)\( T^{28} - \)\(97\!\cdots\!65\)\( T^{30} + \)\(23\!\cdots\!21\)\( T^{32} \))
$83$ (\( 1 - 20 T + 157 T^{2} - 1600 T^{3} + 19929 T^{4} - 132800 T^{5} + 1081573 T^{6} - 11435740 T^{7} + 47458321 T^{8} \))(\( 1 - 7 T - 14 T^{2} - 641 T^{3} + 11409 T^{4} - 53203 T^{5} - 96446 T^{6} - 4002509 T^{7} + 47458321 T^{8} \))(\( 1 - 9 T + 88 T^{2} - 1185 T^{3} + 16681 T^{4} - 98355 T^{5} + 606232 T^{6} - 5146083 T^{7} + 47458321 T^{8} \))(\( 1 - 6 T - 47 T^{2} + 780 T^{3} - 779 T^{4} + 64740 T^{5} - 323783 T^{6} - 3430722 T^{7} + 47458321 T^{8} \))(\( 1 + 12 T + 61 T^{2} - 264 T^{3} - 8231 T^{4} - 21912 T^{5} + 420229 T^{6} + 6861444 T^{7} + 47458321 T^{8} \))(\( 1 + 13 T - 14 T^{2} - 421 T^{3} + 1449 T^{4} - 34943 T^{5} - 96446 T^{6} + 7433231 T^{7} + 47458321 T^{8} \))(\( 1 - 12 T + 61 T^{2} + 264 T^{3} - 8231 T^{4} + 21912 T^{5} + 420229 T^{6} - 6861444 T^{7} + 47458321 T^{8} \))(\( 1 + 9 T + 88 T^{2} + 1185 T^{3} + 16681 T^{4} + 98355 T^{5} + 606232 T^{6} + 5146083 T^{7} + 47458321 T^{8} \))(\( 1 + 6 T - 47 T^{2} - 780 T^{3} - 779 T^{4} - 64740 T^{5} - 323783 T^{6} + 3430722 T^{7} + 47458321 T^{8} \))(\( 1 + 21 T + 88 T^{2} - 915 T^{3} - 13199 T^{4} - 75945 T^{5} + 606232 T^{6} + 12007527 T^{7} + 47458321 T^{8} \))(\( 1 + 7 T - 14 T^{2} + 641 T^{3} + 11409 T^{4} + 53203 T^{5} - 96446 T^{6} + 4002509 T^{7} + 47458321 T^{8} \))(\( 1 + 20 T + 157 T^{2} + 1600 T^{3} + 19929 T^{4} + 132800 T^{5} + 1081573 T^{6} + 11435740 T^{7} + 47458321 T^{8} \))(\( ( 1 - 83 T^{2} + 6889 T^{4} - 571787 T^{6} + 47458321 T^{8} )^{2} \))(\( 1 - 256 T^{2} + 36562 T^{4} - 3706112 T^{6} + 293436467 T^{8} - 14868563200 T^{10} + 105192677348 T^{12} + 62258419495936 T^{14} - 6690049905923675 T^{16} + 428898251907503104 T^{18} + 4992267848430812708 T^{20} - \)\(48\!\cdots\!00\)\( T^{22} + \)\(66\!\cdots\!47\)\( T^{24} - \)\(57\!\cdots\!88\)\( T^{26} + \)\(39\!\cdots\!82\)\( T^{28} - \)\(18\!\cdots\!24\)\( T^{30} + \)\(50\!\cdots\!81\)\( T^{32} \))
$89$ (\( ( 1 + 14 T + 89 T^{2} )^{4} \))(\( ( 1 + 12 T + 209 T^{2} + 1068 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 - 10 T + 183 T^{2} - 890 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 - 12 T + 89 T^{2} )^{4} \))(\( ( 1 + 6 T + 89 T^{2} )^{4} \))(\( ( 1 + 12 T + 209 T^{2} + 1068 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 + 6 T + 89 T^{2} )^{4} \))(\( ( 1 - 10 T + 183 T^{2} - 890 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 - 12 T + 89 T^{2} )^{4} \))(\( ( 1 - 10 T + 183 T^{2} - 890 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 + 12 T + 209 T^{2} + 1068 T^{3} + 7921 T^{4} )^{2} \))(\( ( 1 + 14 T + 89 T^{2} )^{4} \))(\( ( 1 - 9 T + 89 T^{2} )^{8} \))(\( ( 1 + 7 T + 182 T^{2} + 623 T^{3} + 7921 T^{4} )^{8} \))
$97$ (\( 1 + 2 T - 93 T^{2} - 380 T^{3} + 8261 T^{4} - 36860 T^{5} - 875037 T^{6} + 1825346 T^{7} + 88529281 T^{8} \))(\( 1 + 12 T - 43 T^{2} - 900 T^{3} - 1859 T^{4} - 87300 T^{5} - 404587 T^{6} + 10952076 T^{7} + 88529281 T^{8} \))(\( 1 - 32 T + 537 T^{2} - 7060 T^{3} + 77501 T^{4} - 684820 T^{5} + 5052633 T^{6} - 29205536 T^{7} + 88529281 T^{8} \))(\( 1 + 5 T - 72 T^{2} - 845 T^{3} + 2759 T^{4} - 81965 T^{5} - 677448 T^{6} + 4563365 T^{7} + 88529281 T^{8} \))(\( 1 + 2 T - 93 T^{2} - 380 T^{3} + 8261 T^{4} - 36860 T^{5} - 875037 T^{6} + 1825346 T^{7} + 88529281 T^{8} \))(\( 1 - 3 T - 43 T^{2} - 765 T^{3} + 11236 T^{4} - 74205 T^{5} - 404587 T^{6} - 2738019 T^{7} + 88529281 T^{8} \))(\( 1 + 2 T - 93 T^{2} - 380 T^{3} + 8261 T^{4} - 36860 T^{5} - 875037 T^{6} + 1825346 T^{7} + 88529281 T^{8} \))(\( 1 - 32 T + 537 T^{2} - 7060 T^{3} + 77501 T^{4} - 684820 T^{5} + 5052633 T^{6} - 29205536 T^{7} + 88529281 T^{8} \))(\( 1 + 5 T - 72 T^{2} - 845 T^{3} + 2759 T^{4} - 81965 T^{5} - 677448 T^{6} + 4563365 T^{7} + 88529281 T^{8} \))(\( 1 + 33 T + 537 T^{2} + 6655 T^{3} + 71196 T^{4} + 645535 T^{5} + 5052633 T^{6} + 30118209 T^{7} + 88529281 T^{8} \))(\( 1 + 12 T - 43 T^{2} - 900 T^{3} - 1859 T^{4} - 87300 T^{5} - 404587 T^{6} + 10952076 T^{7} + 88529281 T^{8} \))(\( 1 + 2 T - 93 T^{2} - 380 T^{3} + 8261 T^{4} - 36860 T^{5} - 875037 T^{6} + 1825346 T^{7} + 88529281 T^{8} \))(\( ( 1 - 7 T - 48 T^{2} + 1015 T^{3} - 2449 T^{4} + 98455 T^{5} - 451632 T^{6} - 6388711 T^{7} + 88529281 T^{8} )^{2} \))(\( ( 1 - 2 T - 59 T^{2} + 50 T^{3} - 5404 T^{4} - 76196 T^{5} + 1038275 T^{6} + 3301724 T^{7} - 6386513 T^{8} + 320267228 T^{9} + 9769129475 T^{10} - 69542031908 T^{11} - 478412234524 T^{12} + 429367012850 T^{13} - 49145348290811 T^{14} - 161596568956226 T^{15} + 7837433594376961 T^{16} )^{2} \))
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