Properties

Label 1512.2.s
Level 1512
Weight 2
Character orbit s
Rep. character \(\chi_{1512}(865,\cdot)\)
Character field \(\Q(\zeta_{3})\)
Dimension 64
Newform subspaces 16
Sturm bound 576
Trace bound 7

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Defining parameters

Level: \( N \) = \( 1512 = 2^{3} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1512.s (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 16 \)
Sturm bound: \(576\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(5\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1512, [\chi])\).

Total New Old
Modular forms 624 64 560
Cusp forms 528 64 464
Eisenstein series 96 0 96

Trace form

\( 64q + O(q^{10}) \) \( 64q - 4q^{13} + 2q^{19} - 22q^{25} - 18q^{31} + 2q^{37} + 24q^{43} - 16q^{49} - 40q^{55} - 4q^{61} - 2q^{67} - 18q^{73} + 30q^{79} + 32q^{85} + 20q^{91} + 4q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1512, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
1512.2.s.a \(2\) \(12.073\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-4\) \(4\) \(q-4\zeta_{6}q^{5}+(3-2\zeta_{6})q^{7}+(2-2\zeta_{6})q^{11}+\cdots\)
1512.2.s.b \(2\) \(12.073\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-2\) \(-4\) \(q-2\zeta_{6}q^{5}+(-1-2\zeta_{6})q^{7}+3q^{13}+\cdots\)
1512.2.s.c \(2\) \(12.073\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-2\) \(1\) \(q-2\zeta_{6}q^{5}+(-1+3\zeta_{6})q^{7}-2q^{13}+\cdots\)
1512.2.s.d \(2\) \(12.073\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-2\) \(4\) \(q-2\zeta_{6}q^{5}+(3-2\zeta_{6})q^{7}+(4-4\zeta_{6})q^{11}+\cdots\)
1512.2.s.e \(2\) \(12.073\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-2\) \(5\) \(q-2\zeta_{6}q^{5}+(2+\zeta_{6})q^{7}+(-3+3\zeta_{6})q^{11}+\cdots\)
1512.2.s.f \(2\) \(12.073\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(2\) \(-4\) \(q+2\zeta_{6}q^{5}+(-1-2\zeta_{6})q^{7}+3q^{13}+\cdots\)
1512.2.s.g \(2\) \(12.073\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(2\) \(1\) \(q+2\zeta_{6}q^{5}+(-1+3\zeta_{6})q^{7}-2q^{13}+\cdots\)
1512.2.s.h \(2\) \(12.073\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(2\) \(4\) \(q+2\zeta_{6}q^{5}+(3-2\zeta_{6})q^{7}+(-4+4\zeta_{6})q^{11}+\cdots\)
1512.2.s.i \(2\) \(12.073\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(2\) \(5\) \(q+2\zeta_{6}q^{5}+(2+\zeta_{6})q^{7}+(3-3\zeta_{6})q^{11}+\cdots\)
1512.2.s.j \(2\) \(12.073\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(4\) \(4\) \(q+4\zeta_{6}q^{5}+(3-2\zeta_{6})q^{7}+(-2+2\zeta_{6})q^{11}+\cdots\)
1512.2.s.k \(6\) \(12.073\) 6.0.309123.1 None \(0\) \(0\) \(-1\) \(-4\) \(q+(\beta _{1}-\beta _{5})q^{5}+(-1-\beta _{1}+\beta _{2}+\beta _{5})q^{7}+\cdots\)
1512.2.s.l \(6\) \(12.073\) 6.0.309123.1 None \(0\) \(0\) \(1\) \(-4\) \(q+(-\beta _{1}+\beta _{5})q^{5}+(-1-\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots\)
1512.2.s.m \(8\) \(12.073\) 8.0.9391935744.3 None \(0\) \(0\) \(-4\) \(-2\) \(q+(\beta _{3}+\beta _{5}-\beta _{6})q^{5}+(-\beta _{1}-\beta _{3}-\beta _{4}+\cdots)q^{7}+\cdots\)
1512.2.s.n \(8\) \(12.073\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(0\) \(-2\) \(-4\) \(q-\beta _{2}q^{5}+(-\beta _{4}-\beta _{6})q^{7}+(-\beta _{1}+\beta _{4}+\cdots)q^{11}+\cdots\)
1512.2.s.o \(8\) \(12.073\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(0\) \(2\) \(-4\) \(q+\beta _{2}q^{5}+(-\beta _{4}-\beta _{6})q^{7}+(\beta _{1}-\beta _{4}+\cdots)q^{11}+\cdots\)
1512.2.s.p \(8\) \(12.073\) 8.0.9391935744.3 None \(0\) \(0\) \(4\) \(-2\) \(q+(-\beta _{3}-\beta _{5}+\beta _{6})q^{5}+(-\beta _{1}-\beta _{3}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1512, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1512, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(56, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(126, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(168, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(189, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(252, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(378, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(504, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(756, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ (\( 1 + 4 T + 11 T^{2} + 20 T^{3} + 25 T^{4} \))(\( 1 + 2 T - T^{2} + 10 T^{3} + 25 T^{4} \))(\( 1 + 2 T - T^{2} + 10 T^{3} + 25 T^{4} \))(\( 1 + 2 T - T^{2} + 10 T^{3} + 25 T^{4} \))(\( 1 + 2 T - T^{2} + 10 T^{3} + 25 T^{4} \))(\( 1 - 2 T - T^{2} - 10 T^{3} + 25 T^{4} \))(\( 1 - 2 T - T^{2} - 10 T^{3} + 25 T^{4} \))(\( 1 - 2 T - T^{2} - 10 T^{3} + 25 T^{4} \))(\( 1 - 2 T - T^{2} - 10 T^{3} + 25 T^{4} \))(\( 1 - 4 T + 11 T^{2} - 20 T^{3} + 25 T^{4} \))(\( 1 + T - 10 T^{2} - 7 T^{3} + 57 T^{4} + 14 T^{5} - 299 T^{6} + 70 T^{7} + 1425 T^{8} - 875 T^{9} - 6250 T^{10} + 3125 T^{11} + 15625 T^{12} \))(\( 1 - T - 10 T^{2} + 7 T^{3} + 57 T^{4} - 14 T^{5} - 299 T^{6} - 70 T^{7} + 1425 T^{8} + 875 T^{9} - 6250 T^{10} - 3125 T^{11} + 15625 T^{12} \))(\( ( 1 + 2 T - 4 T^{2} - 4 T^{3} + 19 T^{4} - 20 T^{5} - 100 T^{6} + 250 T^{7} + 625 T^{8} )^{2} \))(\( 1 + 2 T - T^{2} + 20 T^{3} + 36 T^{4} - 46 T^{5} + 330 T^{6} + 755 T^{7} - 374 T^{8} + 3775 T^{9} + 8250 T^{10} - 5750 T^{11} + 22500 T^{12} + 62500 T^{13} - 15625 T^{14} + 156250 T^{15} + 390625 T^{16} \))(\( 1 - 2 T - T^{2} - 20 T^{3} + 36 T^{4} + 46 T^{5} + 330 T^{6} - 755 T^{7} - 374 T^{8} - 3775 T^{9} + 8250 T^{10} + 5750 T^{11} + 22500 T^{12} - 62500 T^{13} - 15625 T^{14} - 156250 T^{15} + 390625 T^{16} \))(\( ( 1 - 2 T - 4 T^{2} + 4 T^{3} + 19 T^{4} + 20 T^{5} - 100 T^{6} - 250 T^{7} + 625 T^{8} )^{2} \))
$7$ (\( 1 - 4 T + 7 T^{2} \))(\( 1 + 4 T + 7 T^{2} \))(\( 1 - T + 7 T^{2} \))(\( 1 - 4 T + 7 T^{2} \))(\( 1 - 5 T + 7 T^{2} \))(\( 1 + 4 T + 7 T^{2} \))(\( 1 - T + 7 T^{2} \))(\( 1 - 4 T + 7 T^{2} \))(\( 1 - 5 T + 7 T^{2} \))(\( 1 - 4 T + 7 T^{2} \))(\( 1 + 4 T + 14 T^{2} + 55 T^{3} + 98 T^{4} + 196 T^{5} + 343 T^{6} \))(\( 1 + 4 T + 14 T^{2} + 55 T^{3} + 98 T^{4} + 196 T^{5} + 343 T^{6} \))(\( 1 + 2 T + 5 T^{2} + 18 T^{3} + 8 T^{4} + 126 T^{5} + 245 T^{6} + 686 T^{7} + 2401 T^{8} \))(\( 1 + 4 T + 5 T^{2} - 33 T^{3} - 136 T^{4} - 231 T^{5} + 245 T^{6} + 1372 T^{7} + 2401 T^{8} \))(\( 1 + 4 T + 5 T^{2} - 33 T^{3} - 136 T^{4} - 231 T^{5} + 245 T^{6} + 1372 T^{7} + 2401 T^{8} \))(\( 1 + 2 T + 5 T^{2} + 18 T^{3} + 8 T^{4} + 126 T^{5} + 245 T^{6} + 686 T^{7} + 2401 T^{8} \))
$11$ (\( 1 - 2 T - 7 T^{2} - 22 T^{3} + 121 T^{4} \))(\( 1 - 11 T^{2} + 121 T^{4} \))(\( 1 - 11 T^{2} + 121 T^{4} \))(\( 1 - 4 T + 5 T^{2} - 44 T^{3} + 121 T^{4} \))(\( 1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4} \))(\( 1 - 11 T^{2} + 121 T^{4} \))(\( 1 - 11 T^{2} + 121 T^{4} \))(\( 1 + 4 T + 5 T^{2} + 44 T^{3} + 121 T^{4} \))(\( 1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4} \))(\( 1 + 2 T - 7 T^{2} + 22 T^{3} + 121 T^{4} \))(\( 1 - T - 20 T^{2} + 41 T^{3} + 179 T^{4} - 310 T^{5} - 1349 T^{6} - 3410 T^{7} + 21659 T^{8} + 54571 T^{9} - 292820 T^{10} - 161051 T^{11} + 1771561 T^{12} \))(\( 1 + T - 20 T^{2} - 41 T^{3} + 179 T^{4} + 310 T^{5} - 1349 T^{6} + 3410 T^{7} + 21659 T^{8} - 54571 T^{9} - 292820 T^{10} + 161051 T^{11} + 1771561 T^{12} \))(\( 1 + 2 T - 14 T^{2} - 48 T^{3} - 58 T^{4} + 26 T^{5} - 168 T^{6} + 2902 T^{7} + 26203 T^{8} + 31922 T^{9} - 20328 T^{10} + 34606 T^{11} - 849178 T^{12} - 7730448 T^{13} - 24801854 T^{14} + 38974342 T^{15} + 214358881 T^{16} \))(\( 1 + T - 15 T^{2} + 108 T^{3} + 240 T^{4} - 1366 T^{5} + 5502 T^{6} + 16607 T^{7} - 62387 T^{8} + 182677 T^{9} + 665742 T^{10} - 1818146 T^{11} + 3513840 T^{12} + 17393508 T^{13} - 26573415 T^{14} + 19487171 T^{15} + 214358881 T^{16} \))(\( 1 - T - 15 T^{2} - 108 T^{3} + 240 T^{4} + 1366 T^{5} + 5502 T^{6} - 16607 T^{7} - 62387 T^{8} - 182677 T^{9} + 665742 T^{10} + 1818146 T^{11} + 3513840 T^{12} - 17393508 T^{13} - 26573415 T^{14} - 19487171 T^{15} + 214358881 T^{16} \))(\( 1 - 2 T - 14 T^{2} + 48 T^{3} - 58 T^{4} - 26 T^{5} - 168 T^{6} - 2902 T^{7} + 26203 T^{8} - 31922 T^{9} - 20328 T^{10} - 34606 T^{11} - 849178 T^{12} + 7730448 T^{13} - 24801854 T^{14} - 38974342 T^{15} + 214358881 T^{16} \))
$13$ (\( ( 1 - 5 T + 13 T^{2} )^{2} \))(\( ( 1 - 3 T + 13 T^{2} )^{2} \))(\( ( 1 + 2 T + 13 T^{2} )^{2} \))(\( ( 1 + T + 13 T^{2} )^{2} \))(\( ( 1 - 6 T + 13 T^{2} )^{2} \))(\( ( 1 - 3 T + 13 T^{2} )^{2} \))(\( ( 1 + 2 T + 13 T^{2} )^{2} \))(\( ( 1 + T + 13 T^{2} )^{2} \))(\( ( 1 - 6 T + 13 T^{2} )^{2} \))(\( ( 1 - 5 T + 13 T^{2} )^{2} \))(\( ( 1 - 2 T + 36 T^{2} - 49 T^{3} + 468 T^{4} - 338 T^{5} + 2197 T^{6} )^{2} \))(\( ( 1 - 2 T + 36 T^{2} - 49 T^{3} + 468 T^{4} - 338 T^{5} + 2197 T^{6} )^{2} \))(\( ( 1 + 4 T + 17 T^{2} - 48 T^{3} - 160 T^{4} - 624 T^{5} + 2873 T^{6} + 8788 T^{7} + 28561 T^{8} )^{2} \))(\( ( 1 + 10 T + 73 T^{2} + 389 T^{3} + 1556 T^{4} + 5057 T^{5} + 12337 T^{6} + 21970 T^{7} + 28561 T^{8} )^{2} \))(\( ( 1 + 10 T + 73 T^{2} + 389 T^{3} + 1556 T^{4} + 5057 T^{5} + 12337 T^{6} + 21970 T^{7} + 28561 T^{8} )^{2} \))(\( ( 1 + 4 T + 17 T^{2} - 48 T^{3} - 160 T^{4} - 624 T^{5} + 2873 T^{6} + 8788 T^{7} + 28561 T^{8} )^{2} \))
$17$ (\( 1 - 6 T + 19 T^{2} - 102 T^{3} + 289 T^{4} \))(\( 1 + 4 T - T^{2} + 68 T^{3} + 289 T^{4} \))(\( 1 - 6 T + 19 T^{2} - 102 T^{3} + 289 T^{4} \))(\( 1 - 17 T^{2} + 289 T^{4} \))(\( 1 + 4 T - T^{2} + 68 T^{3} + 289 T^{4} \))(\( 1 - 4 T - T^{2} - 68 T^{3} + 289 T^{4} \))(\( 1 + 6 T + 19 T^{2} + 102 T^{3} + 289 T^{4} \))(\( 1 - 17 T^{2} + 289 T^{4} \))(\( 1 - 4 T - T^{2} - 68 T^{3} + 289 T^{4} \))(\( 1 + 6 T + 19 T^{2} + 102 T^{3} + 289 T^{4} \))(\( 1 - 2 T - 36 T^{2} + 14 T^{3} + 826 T^{4} + 262 T^{5} - 16321 T^{6} + 4454 T^{7} + 238714 T^{8} + 68782 T^{9} - 3006756 T^{10} - 2839714 T^{11} + 24137569 T^{12} \))(\( 1 + 2 T - 36 T^{2} - 14 T^{3} + 826 T^{4} - 262 T^{5} - 16321 T^{6} - 4454 T^{7} + 238714 T^{8} - 68782 T^{9} - 3006756 T^{10} + 2839714 T^{11} + 24137569 T^{12} \))(\( ( 1 - 2 T - 4 T^{2} + 52 T^{3} - 293 T^{4} + 884 T^{5} - 1156 T^{6} - 9826 T^{7} + 83521 T^{8} )^{2} \))(\( 1 + 4 T - 29 T^{2} - 246 T^{3} + 203 T^{4} + 5446 T^{5} + 13245 T^{6} - 47608 T^{7} - 341903 T^{8} - 809336 T^{9} + 3827805 T^{10} + 26756198 T^{11} + 16954763 T^{12} - 349284822 T^{13} - 699989501 T^{14} + 1641354692 T^{15} + 6975757441 T^{16} \))(\( 1 - 4 T - 29 T^{2} + 246 T^{3} + 203 T^{4} - 5446 T^{5} + 13245 T^{6} + 47608 T^{7} - 341903 T^{8} + 809336 T^{9} + 3827805 T^{10} - 26756198 T^{11} + 16954763 T^{12} + 349284822 T^{13} - 699989501 T^{14} - 1641354692 T^{15} + 6975757441 T^{16} \))(\( ( 1 + 2 T - 4 T^{2} - 52 T^{3} - 293 T^{4} - 884 T^{5} - 1156 T^{6} + 9826 T^{7} + 83521 T^{8} )^{2} \))
$19$ (\( 1 - 4 T - 3 T^{2} - 76 T^{3} + 361 T^{4} \))(\( 1 + 4 T - 3 T^{2} + 76 T^{3} + 361 T^{4} \))(\( ( 1 - 8 T + 19 T^{2} )( 1 + 7 T + 19 T^{2} ) \))(\( 1 - 4 T - 3 T^{2} - 76 T^{3} + 361 T^{4} \))(\( 1 + 4 T - 3 T^{2} + 76 T^{3} + 361 T^{4} \))(\( 1 + 4 T - 3 T^{2} + 76 T^{3} + 361 T^{4} \))(\( ( 1 - 8 T + 19 T^{2} )( 1 + 7 T + 19 T^{2} ) \))(\( 1 - 4 T - 3 T^{2} - 76 T^{3} + 361 T^{4} \))(\( 1 + 4 T - 3 T^{2} + 76 T^{3} + 361 T^{4} \))(\( 1 - 4 T - 3 T^{2} - 76 T^{3} + 361 T^{4} \))(\( 1 - 5 T - 34 T^{2} + 63 T^{3} + 1465 T^{4} - 1156 T^{5} - 29405 T^{6} - 21964 T^{7} + 528865 T^{8} + 432117 T^{9} - 4430914 T^{10} - 12380495 T^{11} + 47045881 T^{12} \))(\( 1 - 5 T - 34 T^{2} + 63 T^{3} + 1465 T^{4} - 1156 T^{5} - 29405 T^{6} - 21964 T^{7} + 528865 T^{8} + 432117 T^{9} - 4430914 T^{10} - 12380495 T^{11} + 47045881 T^{12} \))(\( 1 + 6 T - 5 T^{2} + 114 T^{3} + 877 T^{4} - 936 T^{5} + 9706 T^{6} + 54060 T^{7} - 159614 T^{8} + 1027140 T^{9} + 3503866 T^{10} - 6420024 T^{11} + 114291517 T^{12} + 282275286 T^{13} - 235229405 T^{14} + 5363230434 T^{15} + 16983563041 T^{16} \))(\( 1 - T + 3 T^{2} + 94 T^{3} - 40 T^{4} - 786 T^{5} - 206 T^{6} - 9541 T^{7} - 158247 T^{8} - 181279 T^{9} - 74366 T^{10} - 5391174 T^{11} - 5212840 T^{12} + 232753306 T^{13} + 141137643 T^{14} - 893871739 T^{15} + 16983563041 T^{16} \))(\( 1 - T + 3 T^{2} + 94 T^{3} - 40 T^{4} - 786 T^{5} - 206 T^{6} - 9541 T^{7} - 158247 T^{8} - 181279 T^{9} - 74366 T^{10} - 5391174 T^{11} - 5212840 T^{12} + 232753306 T^{13} + 141137643 T^{14} - 893871739 T^{15} + 16983563041 T^{16} \))(\( 1 + 6 T - 5 T^{2} + 114 T^{3} + 877 T^{4} - 936 T^{5} + 9706 T^{6} + 54060 T^{7} - 159614 T^{8} + 1027140 T^{9} + 3503866 T^{10} - 6420024 T^{11} + 114291517 T^{12} + 282275286 T^{13} - 235229405 T^{14} + 5363230434 T^{15} + 16983563041 T^{16} \))
$23$ (\( 1 - 6 T + 13 T^{2} - 138 T^{3} + 529 T^{4} \))(\( 1 + 2 T - 19 T^{2} + 46 T^{3} + 529 T^{4} \))(\( 1 + 2 T - 19 T^{2} + 46 T^{3} + 529 T^{4} \))(\( 1 + 6 T + 13 T^{2} + 138 T^{3} + 529 T^{4} \))(\( 1 - 4 T - 7 T^{2} - 92 T^{3} + 529 T^{4} \))(\( 1 - 2 T - 19 T^{2} - 46 T^{3} + 529 T^{4} \))(\( 1 - 2 T - 19 T^{2} - 46 T^{3} + 529 T^{4} \))(\( 1 - 6 T + 13 T^{2} - 138 T^{3} + 529 T^{4} \))(\( 1 + 4 T - 7 T^{2} + 92 T^{3} + 529 T^{4} \))(\( 1 + 6 T + 13 T^{2} + 138 T^{3} + 529 T^{4} \))(\( 1 + 2 T - 60 T^{2} - 38 T^{3} + 2458 T^{4} + 482 T^{5} - 65101 T^{6} + 11086 T^{7} + 1300282 T^{8} - 462346 T^{9} - 16790460 T^{10} + 12872686 T^{11} + 148035889 T^{12} \))(\( 1 - 2 T - 60 T^{2} + 38 T^{3} + 2458 T^{4} - 482 T^{5} - 65101 T^{6} - 11086 T^{7} + 1300282 T^{8} + 462346 T^{9} - 16790460 T^{10} - 12872686 T^{11} + 148035889 T^{12} \))(\( 1 - 2 T - 10 T^{2} + 72 T^{3} - 922 T^{4} + 2222 T^{5} + 760 T^{6} - 57846 T^{7} + 692635 T^{8} - 1330458 T^{9} + 402040 T^{10} + 27035074 T^{11} - 258013402 T^{12} + 463416696 T^{13} - 1480358890 T^{14} - 6809650894 T^{15} + 78310985281 T^{16} \))(\( 1 - 12 T + 61 T^{2} - 194 T^{3} + 271 T^{4} + 838 T^{5} - 11959 T^{6} + 133924 T^{7} - 856813 T^{8} + 3080252 T^{9} - 6326311 T^{10} + 10195946 T^{11} + 75836911 T^{12} - 1248650542 T^{13} + 9030189229 T^{14} - 40857905364 T^{15} + 78310985281 T^{16} \))(\( 1 + 12 T + 61 T^{2} + 194 T^{3} + 271 T^{4} - 838 T^{5} - 11959 T^{6} - 133924 T^{7} - 856813 T^{8} - 3080252 T^{9} - 6326311 T^{10} - 10195946 T^{11} + 75836911 T^{12} + 1248650542 T^{13} + 9030189229 T^{14} + 40857905364 T^{15} + 78310985281 T^{16} \))(\( 1 + 2 T - 10 T^{2} - 72 T^{3} - 922 T^{4} - 2222 T^{5} + 760 T^{6} + 57846 T^{7} + 692635 T^{8} + 1330458 T^{9} + 402040 T^{10} - 27035074 T^{11} - 258013402 T^{12} - 463416696 T^{13} - 1480358890 T^{14} + 6809650894 T^{15} + 78310985281 T^{16} \))
$29$ (\( ( 1 + 6 T + 29 T^{2} )^{2} \))(\( ( 1 + 4 T + 29 T^{2} )^{2} \))(\( ( 1 - 6 T + 29 T^{2} )^{2} \))(\( ( 1 + 29 T^{2} )^{2} \))(\( ( 1 - 5 T + 29 T^{2} )^{2} \))(\( ( 1 - 4 T + 29 T^{2} )^{2} \))(\( ( 1 + 6 T + 29 T^{2} )^{2} \))(\( ( 1 + 29 T^{2} )^{2} \))(\( ( 1 + 5 T + 29 T^{2} )^{2} \))(\( ( 1 - 6 T + 29 T^{2} )^{2} \))(\( ( 1 + T + 37 T^{2} - 71 T^{3} + 1073 T^{4} + 841 T^{5} + 24389 T^{6} )^{2} \))(\( ( 1 - T + 37 T^{2} + 71 T^{3} + 1073 T^{4} - 841 T^{5} + 24389 T^{6} )^{2} \))(\( ( 1 - 8 T + 36 T^{2} - 24 T^{3} - 182 T^{4} - 696 T^{5} + 30276 T^{6} - 195112 T^{7} + 707281 T^{8} )^{2} \))(\( ( 1 - 6 T + 59 T^{2} - 387 T^{3} + 2091 T^{4} - 11223 T^{5} + 49619 T^{6} - 146334 T^{7} + 707281 T^{8} )^{2} \))(\( ( 1 + 6 T + 59 T^{2} + 387 T^{3} + 2091 T^{4} + 11223 T^{5} + 49619 T^{6} + 146334 T^{7} + 707281 T^{8} )^{2} \))(\( ( 1 + 8 T + 36 T^{2} + 24 T^{3} - 182 T^{4} + 696 T^{5} + 30276 T^{6} + 195112 T^{7} + 707281 T^{8} )^{2} \))
$31$ (\( ( 1 - 11 T + 31 T^{2} )( 1 + 4 T + 31 T^{2} ) \))(\( 1 + T - 30 T^{2} + 31 T^{3} + 961 T^{4} \))(\( 1 + T - 30 T^{2} + 31 T^{3} + 961 T^{4} \))(\( 1 + 5 T - 6 T^{2} + 155 T^{3} + 961 T^{4} \))(\( ( 1 - 4 T + 31 T^{2} )( 1 + 11 T + 31 T^{2} ) \))(\( 1 + T - 30 T^{2} + 31 T^{3} + 961 T^{4} \))(\( 1 + T - 30 T^{2} + 31 T^{3} + 961 T^{4} \))(\( 1 + 5 T - 6 T^{2} + 155 T^{3} + 961 T^{4} \))(\( ( 1 - 4 T + 31 T^{2} )( 1 + 11 T + 31 T^{2} ) \))(\( ( 1 - 11 T + 31 T^{2} )( 1 + 4 T + 31 T^{2} ) \))(\( 1 + 4 T - 28 T^{2} - 402 T^{3} - 584 T^{4} + 5648 T^{5} + 67711 T^{6} + 175088 T^{7} - 561224 T^{8} - 11975982 T^{9} - 25858588 T^{10} + 114516604 T^{11} + 887503681 T^{12} \))(\( 1 + 4 T - 28 T^{2} - 402 T^{3} - 584 T^{4} + 5648 T^{5} + 67711 T^{6} + 175088 T^{7} - 561224 T^{8} - 11975982 T^{9} - 25858588 T^{10} + 114516604 T^{11} + 887503681 T^{12} \))(\( 1 + 6 T - 38 T^{2} - 348 T^{3} + 517 T^{4} + 10584 T^{5} + 34246 T^{6} - 188550 T^{7} - 2230412 T^{8} - 5845050 T^{9} + 32910406 T^{10} + 315307944 T^{11} + 477460357 T^{12} - 9962944548 T^{13} - 33725139878 T^{14} + 165075684666 T^{15} + 852891037441 T^{16} \))(\( 1 - 8 T + 9 T^{2} + 102 T^{3} - 1287 T^{4} + 10998 T^{5} - 5977 T^{6} - 318082 T^{7} + 2302767 T^{8} - 9860542 T^{9} - 5743897 T^{10} + 327641418 T^{11} - 1188571527 T^{12} + 2920173402 T^{13} + 7987533129 T^{14} - 220100912888 T^{15} + 852891037441 T^{16} \))(\( 1 - 8 T + 9 T^{2} + 102 T^{3} - 1287 T^{4} + 10998 T^{5} - 5977 T^{6} - 318082 T^{7} + 2302767 T^{8} - 9860542 T^{9} - 5743897 T^{10} + 327641418 T^{11} - 1188571527 T^{12} + 2920173402 T^{13} + 7987533129 T^{14} - 220100912888 T^{15} + 852891037441 T^{16} \))(\( 1 + 6 T - 38 T^{2} - 348 T^{3} + 517 T^{4} + 10584 T^{5} + 34246 T^{6} - 188550 T^{7} - 2230412 T^{8} - 5845050 T^{9} + 32910406 T^{10} + 315307944 T^{11} + 477460357 T^{12} - 9962944548 T^{13} - 33725139878 T^{14} + 165075684666 T^{15} + 852891037441 T^{16} \))
$37$ (\( 1 + 7 T + 12 T^{2} + 259 T^{3} + 1369 T^{4} \))(\( 1 - 3 T - 28 T^{2} - 111 T^{3} + 1369 T^{4} \))(\( 1 + 2 T - 33 T^{2} + 74 T^{3} + 1369 T^{4} \))(\( ( 1 - 10 T + 37 T^{2} )( 1 + 11 T + 37 T^{2} ) \))(\( 1 - 37 T^{2} + 1369 T^{4} \))(\( 1 - 3 T - 28 T^{2} - 111 T^{3} + 1369 T^{4} \))(\( 1 + 2 T - 33 T^{2} + 74 T^{3} + 1369 T^{4} \))(\( ( 1 - 10 T + 37 T^{2} )( 1 + 11 T + 37 T^{2} ) \))(\( 1 - 37 T^{2} + 1369 T^{4} \))(\( 1 + 7 T + 12 T^{2} + 259 T^{3} + 1369 T^{4} \))(\( 1 - 8 T - 2 T^{2} + 254 T^{3} - 1214 T^{4} + 2314 T^{5} + 5399 T^{6} + 85618 T^{7} - 1661966 T^{8} + 12865862 T^{9} - 3748322 T^{10} - 554751656 T^{11} + 2565726409 T^{12} \))(\( 1 - 8 T - 2 T^{2} + 254 T^{3} - 1214 T^{4} + 2314 T^{5} + 5399 T^{6} + 85618 T^{7} - 1661966 T^{8} + 12865862 T^{9} - 3748322 T^{10} - 554751656 T^{11} + 2565726409 T^{12} \))(\( 1 - 61 T^{2} - 648 T^{3} + 2017 T^{4} + 31752 T^{5} + 168050 T^{6} - 910440 T^{7} - 7948706 T^{8} - 33686280 T^{9} + 230060450 T^{10} + 1608334056 T^{11} + 3780182737 T^{12} - 44934884136 T^{13} - 156509310949 T^{14} + 3512479453921 T^{16} \))(\( 1 - 63 T^{2} - 114 T^{3} + 1781 T^{4} + 5700 T^{5} + 37899 T^{6} - 141018 T^{7} - 2640591 T^{8} - 5217666 T^{9} + 51883731 T^{10} + 288722100 T^{11} + 3337880741 T^{12} - 7905211098 T^{13} - 161640763767 T^{14} + 3512479453921 T^{16} \))(\( 1 - 63 T^{2} - 114 T^{3} + 1781 T^{4} + 5700 T^{5} + 37899 T^{6} - 141018 T^{7} - 2640591 T^{8} - 5217666 T^{9} + 51883731 T^{10} + 288722100 T^{11} + 3337880741 T^{12} - 7905211098 T^{13} - 161640763767 T^{14} + 3512479453921 T^{16} \))(\( 1 - 61 T^{2} - 648 T^{3} + 2017 T^{4} + 31752 T^{5} + 168050 T^{6} - 910440 T^{7} - 7948706 T^{8} - 33686280 T^{9} + 230060450 T^{10} + 1608334056 T^{11} + 3780182737 T^{12} - 44934884136 T^{13} - 156509310949 T^{14} + 3512479453921 T^{16} \))
$41$ (\( ( 1 + 2 T + 41 T^{2} )^{2} \))(\( ( 1 + 8 T + 41 T^{2} )^{2} \))(\( ( 1 - 2 T + 41 T^{2} )^{2} \))(\( ( 1 + 4 T + 41 T^{2} )^{2} \))(\( ( 1 + 2 T + 41 T^{2} )^{2} \))(\( ( 1 - 8 T + 41 T^{2} )^{2} \))(\( ( 1 + 2 T + 41 T^{2} )^{2} \))(\( ( 1 - 4 T + 41 T^{2} )^{2} \))(\( ( 1 - 2 T + 41 T^{2} )^{2} \))(\( ( 1 - 2 T + 41 T^{2} )^{2} \))(\( ( 1 + 66 T^{2} - 137 T^{3} + 2706 T^{4} + 68921 T^{6} )^{2} \))(\( ( 1 + 66 T^{2} + 137 T^{3} + 2706 T^{4} + 68921 T^{6} )^{2} \))(\( ( 1 + 8 T + 60 T^{2} + 456 T^{3} + 2554 T^{4} + 18696 T^{5} + 100860 T^{6} + 551368 T^{7} + 2825761 T^{8} )^{2} \))(\( ( 1 + 6 T + 127 T^{2} + 453 T^{3} + 6650 T^{4} + 18573 T^{5} + 213487 T^{6} + 413526 T^{7} + 2825761 T^{8} )^{2} \))(\( ( 1 - 6 T + 127 T^{2} - 453 T^{3} + 6650 T^{4} - 18573 T^{5} + 213487 T^{6} - 413526 T^{7} + 2825761 T^{8} )^{2} \))(\( ( 1 - 8 T + 60 T^{2} - 456 T^{3} + 2554 T^{4} - 18696 T^{5} + 100860 T^{6} - 551368 T^{7} + 2825761 T^{8} )^{2} \))
$43$ (\( ( 1 + 7 T + 43 T^{2} )^{2} \))(\( ( 1 + T + 43 T^{2} )^{2} \))(\( ( 1 - 9 T + 43 T^{2} )^{2} \))(\( ( 1 + T + 43 T^{2} )^{2} \))(\( ( 1 - 8 T + 43 T^{2} )^{2} \))(\( ( 1 + T + 43 T^{2} )^{2} \))(\( ( 1 - 9 T + 43 T^{2} )^{2} \))(\( ( 1 + T + 43 T^{2} )^{2} \))(\( ( 1 - 8 T + 43 T^{2} )^{2} \))(\( ( 1 + 7 T + 43 T^{2} )^{2} \))(\( ( 1 + 3 T + 9 T^{2} - 143 T^{3} + 387 T^{4} + 5547 T^{5} + 79507 T^{6} )^{2} \))(\( ( 1 + 3 T + 9 T^{2} - 143 T^{3} + 387 T^{4} + 5547 T^{5} + 79507 T^{6} )^{2} \))(\( ( 1 + 14 T^{2} - 72 T^{3} - 1881 T^{4} - 3096 T^{5} + 25886 T^{6} + 3418801 T^{8} )^{2} \))(\( ( 1 - T + 144 T^{2} - 50 T^{3} + 8638 T^{4} - 2150 T^{5} + 266256 T^{6} - 79507 T^{7} + 3418801 T^{8} )^{2} \))(\( ( 1 - T + 144 T^{2} - 50 T^{3} + 8638 T^{4} - 2150 T^{5} + 266256 T^{6} - 79507 T^{7} + 3418801 T^{8} )^{2} \))(\( ( 1 + 14 T^{2} - 72 T^{3} - 1881 T^{4} - 3096 T^{5} + 25886 T^{6} + 3418801 T^{8} )^{2} \))
$47$ (\( 1 + 2 T - 43 T^{2} + 94 T^{3} + 2209 T^{4} \))(\( 1 + 8 T + 17 T^{2} + 376 T^{3} + 2209 T^{4} \))(\( 1 - 2 T - 43 T^{2} - 94 T^{3} + 2209 T^{4} \))(\( 1 + 4 T - 31 T^{2} + 188 T^{3} + 2209 T^{4} \))(\( 1 + 2 T - 43 T^{2} + 94 T^{3} + 2209 T^{4} \))(\( 1 - 8 T + 17 T^{2} - 376 T^{3} + 2209 T^{4} \))(\( 1 + 2 T - 43 T^{2} + 94 T^{3} + 2209 T^{4} \))(\( 1 - 4 T - 31 T^{2} - 188 T^{3} + 2209 T^{4} \))(\( 1 - 2 T - 43 T^{2} - 94 T^{3} + 2209 T^{4} \))(\( 1 - 2 T - 43 T^{2} - 94 T^{3} + 2209 T^{4} \))(\( 1 - 3 T - 18 T^{2} + 1225 T^{3} - 2499 T^{4} - 16644 T^{5} + 579911 T^{6} - 782268 T^{7} - 5520291 T^{8} + 127183175 T^{9} - 87834258 T^{10} - 688035021 T^{11} + 10779215329 T^{12} \))(\( 1 + 3 T - 18 T^{2} - 1225 T^{3} - 2499 T^{4} + 16644 T^{5} + 579911 T^{6} + 782268 T^{7} - 5520291 T^{8} - 127183175 T^{9} - 87834258 T^{10} + 688035021 T^{11} + 10779215329 T^{12} \))(\( 1 + 20 T + 196 T^{2} + 1464 T^{3} + 7286 T^{4} + 1508 T^{5} - 284064 T^{6} - 3306668 T^{7} - 27412733 T^{8} - 155413396 T^{9} - 627497376 T^{10} + 156565084 T^{11} + 35553355766 T^{12} + 335761090248 T^{13} + 2112726204484 T^{14} + 10132462409260 T^{15} + 23811286661761 T^{16} \))(\( 1 + 9 T - 67 T^{2} - 660 T^{3} + 2564 T^{4} + 7452 T^{5} - 311076 T^{6} + 354939 T^{7} + 24590333 T^{8} + 16682133 T^{9} - 687166884 T^{10} + 773688996 T^{11} + 12511502084 T^{12} - 151367704620 T^{13} - 722207427043 T^{14} + 4559608084167 T^{15} + 23811286661761 T^{16} \))(\( 1 - 9 T - 67 T^{2} + 660 T^{3} + 2564 T^{4} - 7452 T^{5} - 311076 T^{6} - 354939 T^{7} + 24590333 T^{8} - 16682133 T^{9} - 687166884 T^{10} - 773688996 T^{11} + 12511502084 T^{12} + 151367704620 T^{13} - 722207427043 T^{14} - 4559608084167 T^{15} + 23811286661761 T^{16} \))(\( 1 - 20 T + 196 T^{2} - 1464 T^{3} + 7286 T^{4} - 1508 T^{5} - 284064 T^{6} + 3306668 T^{7} - 27412733 T^{8} + 155413396 T^{9} - 627497376 T^{10} - 156565084 T^{11} + 35553355766 T^{12} - 335761090248 T^{13} + 2112726204484 T^{14} - 10132462409260 T^{15} + 23811286661761 T^{16} \))
$53$ (\( 1 - 6 T - 17 T^{2} - 318 T^{3} + 2809 T^{4} \))(\( 1 + 14 T + 143 T^{2} + 742 T^{3} + 2809 T^{4} \))(\( 1 - 6 T - 17 T^{2} - 318 T^{3} + 2809 T^{4} \))(\( 1 - 6 T - 17 T^{2} - 318 T^{3} + 2809 T^{4} \))(\( 1 - 10 T + 47 T^{2} - 530 T^{3} + 2809 T^{4} \))(\( 1 - 14 T + 143 T^{2} - 742 T^{3} + 2809 T^{4} \))(\( 1 + 6 T - 17 T^{2} + 318 T^{3} + 2809 T^{4} \))(\( 1 + 6 T - 17 T^{2} + 318 T^{3} + 2809 T^{4} \))(\( 1 + 10 T + 47 T^{2} + 530 T^{3} + 2809 T^{4} \))(\( 1 + 6 T - 17 T^{2} + 318 T^{3} + 2809 T^{4} \))(\( 1 + 8 T - 90 T^{2} - 278 T^{3} + 9514 T^{4} + 8150 T^{5} - 568945 T^{6} + 431950 T^{7} + 26724826 T^{8} - 41387806 T^{9} - 710143290 T^{10} + 3345563944 T^{11} + 22164361129 T^{12} \))(\( 1 - 8 T - 90 T^{2} + 278 T^{3} + 9514 T^{4} - 8150 T^{5} - 568945 T^{6} - 431950 T^{7} + 26724826 T^{8} + 41387806 T^{9} - 710143290 T^{10} - 3345563944 T^{11} + 22164361129 T^{12} \))(\( 1 + 10 T - 34 T^{2} - 720 T^{3} - 538 T^{4} + 28730 T^{5} + 169816 T^{6} - 937470 T^{7} - 17639549 T^{8} - 49685910 T^{9} + 477013144 T^{10} + 4277236210 T^{11} - 4245078778 T^{12} - 301100754960 T^{13} - 753588278386 T^{14} + 11747111398370 T^{15} + 62259690411361 T^{16} \))(\( 1 - 7 T - 158 T^{2} + 729 T^{3} + 18893 T^{4} - 54478 T^{5} - 1436103 T^{6} + 995821 T^{7} + 90249700 T^{8} + 52778513 T^{9} - 4034013327 T^{10} - 8110521206 T^{11} + 149074857533 T^{12} + 304864514397 T^{13} - 3501969058382 T^{14} - 8222977978859 T^{15} + 62259690411361 T^{16} \))(\( 1 + 7 T - 158 T^{2} - 729 T^{3} + 18893 T^{4} + 54478 T^{5} - 1436103 T^{6} - 995821 T^{7} + 90249700 T^{8} - 52778513 T^{9} - 4034013327 T^{10} + 8110521206 T^{11} + 149074857533 T^{12} - 304864514397 T^{13} - 3501969058382 T^{14} + 8222977978859 T^{15} + 62259690411361 T^{16} \))(\( 1 - 10 T - 34 T^{2} + 720 T^{3} - 538 T^{4} - 28730 T^{5} + 169816 T^{6} + 937470 T^{7} - 17639549 T^{8} + 49685910 T^{9} + 477013144 T^{10} - 4277236210 T^{11} - 4245078778 T^{12} + 301100754960 T^{13} - 753588278386 T^{14} - 11747111398370 T^{15} + 62259690411361 T^{16} \))
$59$ (\( 1 - 6 T - 23 T^{2} - 354 T^{3} + 3481 T^{4} \))(\( 1 + 12 T + 85 T^{2} + 708 T^{3} + 3481 T^{4} \))(\( 1 - 8 T + 5 T^{2} - 472 T^{3} + 3481 T^{4} \))(\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 + 9 T + 22 T^{2} + 531 T^{3} + 3481 T^{4} \))(\( 1 - 12 T + 85 T^{2} - 708 T^{3} + 3481 T^{4} \))(\( 1 + 8 T + 5 T^{2} + 472 T^{3} + 3481 T^{4} \))(\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 - 9 T + 22 T^{2} - 531 T^{3} + 3481 T^{4} \))(\( 1 + 6 T - 23 T^{2} + 354 T^{3} + 3481 T^{4} \))(\( 1 + 9 T - 54 T^{2} - 171 T^{3} + 4023 T^{4} - 18486 T^{5} - 466229 T^{6} - 1090674 T^{7} + 14004063 T^{8} - 35119809 T^{9} - 654337494 T^{10} + 6434318691 T^{11} + 42180533641 T^{12} \))(\( 1 - 9 T - 54 T^{2} + 171 T^{3} + 4023 T^{4} + 18486 T^{5} - 466229 T^{6} + 1090674 T^{7} + 14004063 T^{8} + 35119809 T^{9} - 654337494 T^{10} - 6434318691 T^{11} + 42180533641 T^{12} \))(\( 1 - 22 T + 106 T^{2} - 48 T^{3} + 14486 T^{4} - 158758 T^{5} + 199560 T^{6} - 3891698 T^{7} + 83505403 T^{8} - 229610182 T^{9} + 694668360 T^{10} - 32605559282 T^{11} + 175532091446 T^{12} - 34316366352 T^{13} + 4471136565946 T^{14} - 54750332666018 T^{15} + 146830437604321 T^{16} \))(\( 1 + 4 T - 21 T^{2} - 738 T^{3} - 6108 T^{4} - 3112 T^{5} + 138210 T^{6} + 1929881 T^{7} + 10657612 T^{8} + 113862979 T^{9} + 481109010 T^{10} - 639139448 T^{11} - 74012840988 T^{12} - 527614132662 T^{13} - 885791206461 T^{14} + 9954605939276 T^{15} + 146830437604321 T^{16} \))(\( 1 - 4 T - 21 T^{2} + 738 T^{3} - 6108 T^{4} + 3112 T^{5} + 138210 T^{6} - 1929881 T^{7} + 10657612 T^{8} - 113862979 T^{9} + 481109010 T^{10} + 639139448 T^{11} - 74012840988 T^{12} + 527614132662 T^{13} - 885791206461 T^{14} - 9954605939276 T^{15} + 146830437604321 T^{16} \))(\( 1 + 22 T + 106 T^{2} + 48 T^{3} + 14486 T^{4} + 158758 T^{5} + 199560 T^{6} + 3891698 T^{7} + 83505403 T^{8} + 229610182 T^{9} + 694668360 T^{10} + 32605559282 T^{11} + 175532091446 T^{12} + 34316366352 T^{13} + 4471136565946 T^{14} + 54750332666018 T^{15} + 146830437604321 T^{16} \))
$61$ (\( 1 - 9 T + 20 T^{2} - 549 T^{3} + 3721 T^{4} \))(\( ( 1 - 13 T + 61 T^{2} )( 1 + 14 T + 61 T^{2} ) \))(\( 1 + 11 T + 60 T^{2} + 671 T^{3} + 3721 T^{4} \))(\( 1 - 3 T - 52 T^{2} - 183 T^{3} + 3721 T^{4} \))(\( 1 - 8 T + 3 T^{2} - 488 T^{3} + 3721 T^{4} \))(\( ( 1 - 13 T + 61 T^{2} )( 1 + 14 T + 61 T^{2} ) \))(\( 1 + 11 T + 60 T^{2} + 671 T^{3} + 3721 T^{4} \))(\( 1 - 3 T - 52 T^{2} - 183 T^{3} + 3721 T^{4} \))(\( 1 - 8 T + 3 T^{2} - 488 T^{3} + 3721 T^{4} \))(\( 1 - 9 T + 20 T^{2} - 549 T^{3} + 3721 T^{4} \))(\( 1 - 13 T - 45 T^{2} + 252 T^{3} + 13021 T^{4} - 33607 T^{5} - 667154 T^{6} - 2050027 T^{7} + 48451141 T^{8} + 57199212 T^{9} - 623062845 T^{10} - 10979751913 T^{11} + 51520374361 T^{12} \))(\( 1 - 13 T - 45 T^{2} + 252 T^{3} + 13021 T^{4} - 33607 T^{5} - 667154 T^{6} - 2050027 T^{7} + 48451141 T^{8} + 57199212 T^{9} - 623062845 T^{10} - 10979751913 T^{11} + 51520374361 T^{12} \))(\( 1 - 2 T - 90 T^{2} + 116 T^{3} + 4577 T^{4} - 7740 T^{5} + 332338 T^{6} - 11534 T^{7} - 30676068 T^{8} - 703574 T^{9} + 1236629698 T^{10} - 1756832940 T^{11} + 63372414257 T^{12} + 97973170916 T^{13} - 4636833692490 T^{14} - 6285485672042 T^{15} + 191707312997281 T^{16} \))(\( 1 + 25 T + 196 T^{2} + 1097 T^{3} + 19941 T^{4} + 171360 T^{5} + 257114 T^{6} + 4239618 T^{7} + 85475992 T^{8} + 258616698 T^{9} + 956721194 T^{10} + 38895464160 T^{11} + 276099915381 T^{12} + 926522142197 T^{13} + 10097993374756 T^{14} + 78568570900525 T^{15} + 191707312997281 T^{16} \))(\( 1 + 25 T + 196 T^{2} + 1097 T^{3} + 19941 T^{4} + 171360 T^{5} + 257114 T^{6} + 4239618 T^{7} + 85475992 T^{8} + 258616698 T^{9} + 956721194 T^{10} + 38895464160 T^{11} + 276099915381 T^{12} + 926522142197 T^{13} + 10097993374756 T^{14} + 78568570900525 T^{15} + 191707312997281 T^{16} \))(\( 1 - 2 T - 90 T^{2} + 116 T^{3} + 4577 T^{4} - 7740 T^{5} + 332338 T^{6} - 11534 T^{7} - 30676068 T^{8} - 703574 T^{9} + 1236629698 T^{10} - 1756832940 T^{11} + 63372414257 T^{12} + 97973170916 T^{13} - 4636833692490 T^{14} - 6285485672042 T^{15} + 191707312997281 T^{16} \))
$67$ (\( 1 - 7 T - 18 T^{2} - 469 T^{3} + 4489 T^{4} \))(\( 1 + 3 T - 58 T^{2} + 201 T^{3} + 4489 T^{4} \))(\( 1 - 12 T + 77 T^{2} - 804 T^{3} + 4489 T^{4} \))(\( ( 1 - 5 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \))(\( 1 - 6 T - 31 T^{2} - 402 T^{3} + 4489 T^{4} \))(\( 1 + 3 T - 58 T^{2} + 201 T^{3} + 4489 T^{4} \))(\( 1 - 12 T + 77 T^{2} - 804 T^{3} + 4489 T^{4} \))(\( ( 1 - 5 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \))(\( 1 - 6 T - 31 T^{2} - 402 T^{3} + 4489 T^{4} \))(\( 1 - 7 T - 18 T^{2} - 469 T^{3} + 4489 T^{4} \))(\( 1 - 16 T + 34 T^{2} + 562 T^{3} + 1498 T^{4} - 52510 T^{5} + 381563 T^{6} - 3518170 T^{7} + 6724522 T^{8} + 169028806 T^{9} + 685138114 T^{10} - 21602001712 T^{11} + 90458382169 T^{12} \))(\( 1 - 16 T + 34 T^{2} + 562 T^{3} + 1498 T^{4} - 52510 T^{5} + 381563 T^{6} - 3518170 T^{7} + 6724522 T^{8} + 169028806 T^{9} + 685138114 T^{10} - 21602001712 T^{11} + 90458382169 T^{12} \))(\( 1 - 2 T - 145 T^{2} - 310 T^{3} + 10229 T^{4} + 49232 T^{5} - 447054 T^{6} - 1987140 T^{7} + 26208538 T^{8} - 133138380 T^{9} - 2006825406 T^{10} + 14807164016 T^{11} + 206125816709 T^{12} - 418538783170 T^{13} - 13116465414505 T^{14} - 12121423210646 T^{15} + 406067677556641 T^{16} \))(\( 1 + 30 T + 335 T^{2} + 2816 T^{3} + 39697 T^{4} + 415264 T^{5} + 2662719 T^{6} + 27218576 T^{7} + 300754815 T^{8} + 1823644592 T^{9} + 11952945591 T^{10} + 124896046432 T^{11} + 799939050337 T^{12} + 3801952301312 T^{13} + 30303558026615 T^{14} + 181821348159690 T^{15} + 406067677556641 T^{16} \))(\( 1 + 30 T + 335 T^{2} + 2816 T^{3} + 39697 T^{4} + 415264 T^{5} + 2662719 T^{6} + 27218576 T^{7} + 300754815 T^{8} + 1823644592 T^{9} + 11952945591 T^{10} + 124896046432 T^{11} + 799939050337 T^{12} + 3801952301312 T^{13} + 30303558026615 T^{14} + 181821348159690 T^{15} + 406067677556641 T^{16} \))(\( 1 - 2 T - 145 T^{2} - 310 T^{3} + 10229 T^{4} + 49232 T^{5} - 447054 T^{6} - 1987140 T^{7} + 26208538 T^{8} - 133138380 T^{9} - 2006825406 T^{10} + 14807164016 T^{11} + 206125816709 T^{12} - 418538783170 T^{13} - 13116465414505 T^{14} - 12121423210646 T^{15} + 406067677556641 T^{16} \))
$71$ (\( ( 1 - 8 T + 71 T^{2} )^{2} \))(\( ( 1 + 6 T + 71 T^{2} )^{2} \))(\( ( 1 - 4 T + 71 T^{2} )^{2} \))(\( ( 1 + 14 T + 71 T^{2} )^{2} \))(\( ( 1 - 12 T + 71 T^{2} )^{2} \))(\( ( 1 - 6 T + 71 T^{2} )^{2} \))(\( ( 1 + 4 T + 71 T^{2} )^{2} \))(\( ( 1 - 14 T + 71 T^{2} )^{2} \))(\( ( 1 + 12 T + 71 T^{2} )^{2} \))(\( ( 1 + 8 T + 71 T^{2} )^{2} \))(\( ( 1 - T + 129 T^{2} - 235 T^{3} + 9159 T^{4} - 5041 T^{5} + 357911 T^{6} )^{2} \))(\( ( 1 + T + 129 T^{2} + 235 T^{3} + 9159 T^{4} + 5041 T^{5} + 357911 T^{6} )^{2} \))(\( ( 1 - 22 T + 402 T^{2} - 4638 T^{3} + 45946 T^{4} - 329298 T^{5} + 2026482 T^{6} - 7874042 T^{7} + 25411681 T^{8} )^{2} \))(\( ( 1 + 11 T + 194 T^{2} + 1324 T^{3} + 16844 T^{4} + 94004 T^{5} + 977954 T^{6} + 3937021 T^{7} + 25411681 T^{8} )^{2} \))(\( ( 1 - 11 T + 194 T^{2} - 1324 T^{3} + 16844 T^{4} - 94004 T^{5} + 977954 T^{6} - 3937021 T^{7} + 25411681 T^{8} )^{2} \))(\( ( 1 + 22 T + 402 T^{2} + 4638 T^{3} + 45946 T^{4} + 329298 T^{5} + 2026482 T^{6} + 7874042 T^{7} + 25411681 T^{8} )^{2} \))
$73$ (\( ( 1 - 7 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} ) \))(\( ( 1 - 7 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} ) \))(\( 1 + 5 T - 48 T^{2} + 365 T^{3} + 5329 T^{4} \))(\( 1 - 14 T + 123 T^{2} - 1022 T^{3} + 5329 T^{4} \))(\( 1 - 11 T + 48 T^{2} - 803 T^{3} + 5329 T^{4} \))(\( ( 1 - 7 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} ) \))(\( 1 + 5 T - 48 T^{2} + 365 T^{3} + 5329 T^{4} \))(\( 1 - 14 T + 123 T^{2} - 1022 T^{3} + 5329 T^{4} \))(\( 1 - 11 T + 48 T^{2} - 803 T^{3} + 5329 T^{4} \))(\( ( 1 - 7 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} ) \))(\( 1 + 3 T - 132 T^{2} - 347 T^{3} + 8433 T^{4} + 8514 T^{5} - 573015 T^{6} + 621522 T^{7} + 44939457 T^{8} - 134988899 T^{9} - 3748567812 T^{10} + 6219214779 T^{11} + 151334226289 T^{12} \))(\( 1 + 3 T - 132 T^{2} - 347 T^{3} + 8433 T^{4} + 8514 T^{5} - 573015 T^{6} + 621522 T^{7} + 44939457 T^{8} - 134988899 T^{9} - 3748567812 T^{10} + 6219214779 T^{11} + 151334226289 T^{12} \))(\( 1 + 10 T - 141 T^{2} - 1714 T^{3} + 12677 T^{4} + 151788 T^{5} - 657782 T^{6} - 4715864 T^{7} + 43753554 T^{8} - 344258072 T^{9} - 3505320278 T^{10} + 59048112396 T^{11} + 360004501157 T^{12} - 3553244710402 T^{13} - 21338125906749 T^{14} + 110473985190970 T^{15} + 806460091894081 T^{16} \))(\( 1 - 4 T - 177 T^{2} + 378 T^{3} + 16890 T^{4} - 8478 T^{5} - 1265668 T^{6} + 148369 T^{7} + 87542238 T^{8} + 10830937 T^{9} - 6744744772 T^{10} - 3298086126 T^{11} + 479646290490 T^{12} + 783621062154 T^{13} - 26786158053153 T^{14} - 44189594076388 T^{15} + 806460091894081 T^{16} \))(\( 1 - 4 T - 177 T^{2} + 378 T^{3} + 16890 T^{4} - 8478 T^{5} - 1265668 T^{6} + 148369 T^{7} + 87542238 T^{8} + 10830937 T^{9} - 6744744772 T^{10} - 3298086126 T^{11} + 479646290490 T^{12} + 783621062154 T^{13} - 26786158053153 T^{14} - 44189594076388 T^{15} + 806460091894081 T^{16} \))(\( 1 + 10 T - 141 T^{2} - 1714 T^{3} + 12677 T^{4} + 151788 T^{5} - 657782 T^{6} - 4715864 T^{7} + 43753554 T^{8} - 344258072 T^{9} - 3505320278 T^{10} + 59048112396 T^{11} + 360004501157 T^{12} - 3553244710402 T^{13} - 21338125906749 T^{14} + 110473985190970 T^{15} + 806460091894081 T^{16} \))
$79$ (\( 1 + T - 78 T^{2} + 79 T^{3} + 6241 T^{4} \))(\( 1 + T - 78 T^{2} + 79 T^{3} + 6241 T^{4} \))(\( ( 1 - 17 T + 79 T^{2} )( 1 + 13 T + 79 T^{2} ) \))(\( ( 1 - 4 T + 79 T^{2} )( 1 + 17 T + 79 T^{2} ) \))(\( 1 + T - 78 T^{2} + 79 T^{3} + 6241 T^{4} \))(\( 1 + T - 78 T^{2} + 79 T^{3} + 6241 T^{4} \))(\( ( 1 - 17 T + 79 T^{2} )( 1 + 13 T + 79 T^{2} ) \))(\( ( 1 - 4 T + 79 T^{2} )( 1 + 17 T + 79 T^{2} ) \))(\( 1 + T - 78 T^{2} + 79 T^{3} + 6241 T^{4} \))(\( 1 + T - 78 T^{2} + 79 T^{3} + 6241 T^{4} \))(\( 1 - 12 T - 84 T^{2} + 454 T^{3} + 14832 T^{4} - 6264 T^{5} - 1475337 T^{6} - 494856 T^{7} + 92566512 T^{8} + 223839706 T^{9} - 3271806804 T^{10} - 36924676788 T^{11} + 243087455521 T^{12} \))(\( 1 - 12 T - 84 T^{2} + 454 T^{3} + 14832 T^{4} - 6264 T^{5} - 1475337 T^{6} - 494856 T^{7} + 92566512 T^{8} + 223839706 T^{9} - 3271806804 T^{10} - 36924676788 T^{11} + 243087455521 T^{12} \))(\( 1 - 8 T - 201 T^{2} + 920 T^{3} + 28853 T^{4} - 55488 T^{5} - 3083690 T^{6} + 2064928 T^{7} + 256704030 T^{8} + 163129312 T^{9} - 19245309290 T^{10} - 27357748032 T^{11} + 1123826687093 T^{12} + 2830891887080 T^{13} - 48860578559721 T^{14} - 153631271889272 T^{15} + 1517108809906561 T^{16} \))(\( 1 - 7 T + 6 T^{2} + 609 T^{3} - 7809 T^{4} + 35112 T^{5} + 213797 T^{6} - 5183759 T^{7} + 23372886 T^{8} - 409516961 T^{9} + 1334307077 T^{10} + 17311585368 T^{11} - 304161182529 T^{12} + 1873927346991 T^{13} + 1458524733126 T^{14} - 134427362903113 T^{15} + 1517108809906561 T^{16} \))(\( 1 - 7 T + 6 T^{2} + 609 T^{3} - 7809 T^{4} + 35112 T^{5} + 213797 T^{6} - 5183759 T^{7} + 23372886 T^{8} - 409516961 T^{9} + 1334307077 T^{10} + 17311585368 T^{11} - 304161182529 T^{12} + 1873927346991 T^{13} + 1458524733126 T^{14} - 134427362903113 T^{15} + 1517108809906561 T^{16} \))(\( 1 - 8 T - 201 T^{2} + 920 T^{3} + 28853 T^{4} - 55488 T^{5} - 3083690 T^{6} + 2064928 T^{7} + 256704030 T^{8} + 163129312 T^{9} - 19245309290 T^{10} - 27357748032 T^{11} + 1123826687093 T^{12} + 2830891887080 T^{13} - 48860578559721 T^{14} - 153631271889272 T^{15} + 1517108809906561 T^{16} \))
$83$ (\( ( 1 + 14 T + 83 T^{2} )^{2} \))(\( ( 1 + 6 T + 83 T^{2} )^{2} \))(\( ( 1 - 4 T + 83 T^{2} )^{2} \))(\( ( 1 - 14 T + 83 T^{2} )^{2} \))(\( ( 1 - 15 T + 83 T^{2} )^{2} \))(\( ( 1 - 6 T + 83 T^{2} )^{2} \))(\( ( 1 + 4 T + 83 T^{2} )^{2} \))(\( ( 1 + 14 T + 83 T^{2} )^{2} \))(\( ( 1 + 15 T + 83 T^{2} )^{2} \))(\( ( 1 - 14 T + 83 T^{2} )^{2} \))(\( ( 1 + T + 129 T^{2} + 313 T^{3} + 10707 T^{4} + 6889 T^{5} + 571787 T^{6} )^{2} \))(\( ( 1 - T + 129 T^{2} - 313 T^{3} + 10707 T^{4} - 6889 T^{5} + 571787 T^{6} )^{2} \))(\( ( 1 + 20 T + 360 T^{2} + 3684 T^{3} + 40318 T^{4} + 305772 T^{5} + 2480040 T^{6} + 11435740 T^{7} + 47458321 T^{8} )^{2} \))(\( ( 1 - 29 T + 560 T^{2} - 7232 T^{3} + 75754 T^{4} - 600256 T^{5} + 3857840 T^{6} - 16581823 T^{7} + 47458321 T^{8} )^{2} \))(\( ( 1 + 29 T + 560 T^{2} + 7232 T^{3} + 75754 T^{4} + 600256 T^{5} + 3857840 T^{6} + 16581823 T^{7} + 47458321 T^{8} )^{2} \))(\( ( 1 - 20 T + 360 T^{2} - 3684 T^{3} + 40318 T^{4} - 305772 T^{5} + 2480040 T^{6} - 11435740 T^{7} + 47458321 T^{8} )^{2} \))
$89$ (\( 1 - 12 T + 55 T^{2} - 1068 T^{3} + 7921 T^{4} \))(\( 1 - 2 T - 85 T^{2} - 178 T^{3} + 7921 T^{4} \))(\( 1 + 18 T + 235 T^{2} + 1602 T^{3} + 7921 T^{4} \))(\( 1 + 6 T - 53 T^{2} + 534 T^{3} + 7921 T^{4} \))(\( 1 + 10 T + 11 T^{2} + 890 T^{3} + 7921 T^{4} \))(\( 1 + 2 T - 85 T^{2} + 178 T^{3} + 7921 T^{4} \))(\( 1 - 18 T + 235 T^{2} - 1602 T^{3} + 7921 T^{4} \))(\( 1 - 6 T - 53 T^{2} - 534 T^{3} + 7921 T^{4} \))(\( 1 - 10 T + 11 T^{2} - 890 T^{3} + 7921 T^{4} \))(\( 1 + 12 T + 55 T^{2} + 1068 T^{3} + 7921 T^{4} \))(\( 1 - 17 T - 57 T^{2} + 344 T^{3} + 36001 T^{4} - 164759 T^{5} - 1843186 T^{6} - 14663551 T^{7} + 285163921 T^{8} + 242509336 T^{9} - 3576307737 T^{10} - 94929010633 T^{11} + 496981290961 T^{12} \))(\( 1 + 17 T - 57 T^{2} - 344 T^{3} + 36001 T^{4} + 164759 T^{5} - 1843186 T^{6} + 14663551 T^{7} + 285163921 T^{8} - 242509336 T^{9} - 3576307737 T^{10} + 94929010633 T^{11} + 496981290961 T^{12} \))(\( 1 + 16 T - 92 T^{2} - 1248 T^{3} + 22646 T^{4} + 83536 T^{5} - 3013152 T^{6} - 6008080 T^{7} + 235094707 T^{8} - 534719120 T^{9} - 23867176992 T^{10} + 58890290384 T^{11} + 1420860789686 T^{12} - 6968906192352 T^{13} - 45722278768412 T^{14} + 707701358328464 T^{15} + 3936588805702081 T^{16} \))(\( 1 - 9 T - 128 T^{2} - 549 T^{3} + 20701 T^{4} + 107280 T^{5} - 329006 T^{6} - 7780266 T^{7} - 45486752 T^{8} - 692443674 T^{9} - 2606056526 T^{10} + 75629074320 T^{11} + 1298827130941 T^{12} - 3065648637501 T^{13} - 63613605243008 T^{14} - 398082014059761 T^{15} + 3936588805702081 T^{16} \))(\( 1 + 9 T - 128 T^{2} + 549 T^{3} + 20701 T^{4} - 107280 T^{5} - 329006 T^{6} + 7780266 T^{7} - 45486752 T^{8} + 692443674 T^{9} - 2606056526 T^{10} - 75629074320 T^{11} + 1298827130941 T^{12} + 3065648637501 T^{13} - 63613605243008 T^{14} + 398082014059761 T^{15} + 3936588805702081 T^{16} \))(\( 1 - 16 T - 92 T^{2} + 1248 T^{3} + 22646 T^{4} - 83536 T^{5} - 3013152 T^{6} + 6008080 T^{7} + 235094707 T^{8} + 534719120 T^{9} - 23867176992 T^{10} - 58890290384 T^{11} + 1420860789686 T^{12} + 6968906192352 T^{13} - 45722278768412 T^{14} - 707701358328464 T^{15} + 3936588805702081 T^{16} \))
$97$ (\( ( 1 + 15 T + 97 T^{2} )^{2} \))(\( ( 1 - T + 97 T^{2} )^{2} \))(\( ( 1 - T + 97 T^{2} )^{2} \))(\( ( 1 - 9 T + 97 T^{2} )^{2} \))(\( ( 1 + 5 T + 97 T^{2} )^{2} \))(\( ( 1 - T + 97 T^{2} )^{2} \))(\( ( 1 - T + 97 T^{2} )^{2} \))(\( ( 1 - 9 T + 97 T^{2} )^{2} \))(\( ( 1 + 5 T + 97 T^{2} )^{2} \))(\( ( 1 + 15 T + 97 T^{2} )^{2} \))(\( ( 1 - 14 T + 331 T^{2} - 2740 T^{3} + 32107 T^{4} - 131726 T^{5} + 912673 T^{6} )^{2} \))(\( ( 1 - 14 T + 331 T^{2} - 2740 T^{3} + 32107 T^{4} - 131726 T^{5} + 912673 T^{6} )^{2} \))(\( ( 1 + 230 T^{2} + 72 T^{3} + 26415 T^{4} + 6984 T^{5} + 2164070 T^{6} + 88529281 T^{8} )^{2} \))(\( ( 1 + 4 T + 192 T^{2} - 412 T^{3} + 15886 T^{4} - 39964 T^{5} + 1806528 T^{6} + 3650692 T^{7} + 88529281 T^{8} )^{2} \))(\( ( 1 + 4 T + 192 T^{2} - 412 T^{3} + 15886 T^{4} - 39964 T^{5} + 1806528 T^{6} + 3650692 T^{7} + 88529281 T^{8} )^{2} \))(\( ( 1 + 230 T^{2} + 72 T^{3} + 26415 T^{4} + 6984 T^{5} + 2164070 T^{6} + 88529281 T^{8} )^{2} \))
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