Properties

Label 1024.4.a
Level 1024
Weight 4
Character orbit a
Rep. character \(\chi_{1024}(1,\cdot)\)
Character field \(\Q\)
Dimension 92
Newform subspaces 16
Sturm bound 512
Trace bound 17

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Defining parameters

Level: \( N \) \(=\) \( 1024 = 2^{10} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1024.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 16 \)
Sturm bound: \(512\)
Trace bound: \(17\)
Distinguishing \(T_p\): \(3\), \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1024))\).

Total New Old
Modular forms 408 100 308
Cusp forms 360 92 268
Eisenstein series 48 8 40

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

\(2\)Dim.
\(+\)\(48\)
\(-\)\(44\)

Trace form

\( 92q + 756q^{9} + O(q^{10}) \) \( 92q + 756q^{9} + 8q^{17} + 1900q^{25} - 8q^{33} + 3340q^{49} + 216q^{57} - 8q^{65} + 4868q^{81} - 8q^{97} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1024))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2
1024.4.a.a \(2\) \(60.418\) \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(-16\) \(-\) \(q+4\beta q^{3}+\beta q^{5}-8q^{7}+5q^{9}+4\beta q^{11}+\cdots\)
1024.4.a.b \(2\) \(60.418\) \(\Q(\sqrt{2}) \) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(0\) \(0\) \(+\) \(q+13\beta q^{5}-3^{3}q^{9}+55\beta q^{13}-104q^{17}+\cdots\)
1024.4.a.c \(2\) \(60.418\) \(\Q(\sqrt{2}) \) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(0\) \(0\) \(-\) \(q-9\beta q^{5}-3^{3}q^{9}+37\beta q^{13}+104q^{17}+\cdots\)
1024.4.a.d \(2\) \(60.418\) \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(16\) \(-\) \(q+4\beta q^{3}-\beta q^{5}+8q^{7}+5q^{9}+4\beta q^{11}+\cdots\)
1024.4.a.e \(4\) \(60.418\) \(\Q(\sqrt{2}, \sqrt{43})\) None \(0\) \(-12\) \(0\) \(0\) \(-\) \(q+(-3+\beta _{3})q^{3}+(-4\beta _{1}+\beta _{2})q^{5}+\cdots\)
1024.4.a.f \(4\) \(60.418\) \(\Q(\sqrt{2}, \sqrt{17})\) None \(0\) \(0\) \(0\) \(-104\) \(+\) \(q+(-3\beta _{1}+\beta _{2})q^{3}+(3\beta _{1}-2\beta _{2})q^{5}+\cdots\)
1024.4.a.g \(4\) \(60.418\) \(\Q(\sqrt{2}, \sqrt{5})\) None \(0\) \(0\) \(0\) \(0\) \(-\) \(q+\beta _{2}q^{3}-11\beta _{1}q^{5}+3\beta _{3}q^{7}+13q^{9}+\cdots\)
1024.4.a.h \(4\) \(60.418\) \(\Q(\sqrt{2}, \sqrt{5})\) None \(0\) \(0\) \(0\) \(0\) \(+\) \(q+\beta _{2}q^{3}-7\beta _{1}q^{5}-\beta _{3}q^{7}+13q^{9}+\cdots\)
1024.4.a.i \(4\) \(60.418\) \(\Q(\sqrt{2}, \sqrt{17})\) None \(0\) \(0\) \(0\) \(104\) \(+\) \(q+(-3\beta _{1}+\beta _{2})q^{3}+(-3\beta _{1}+2\beta _{2}+\cdots)q^{5}+\cdots\)
1024.4.a.j \(4\) \(60.418\) \(\Q(\sqrt{2}, \sqrt{43})\) None \(0\) \(12\) \(0\) \(0\) \(+\) \(q+(3-\beta _{3})q^{3}+(-4\beta _{1}+\beta _{2})q^{5}+(5\beta _{1}+\cdots)q^{7}+\cdots\)
1024.4.a.k \(8\) \(60.418\) 8.8.\(\cdots\).3 None \(0\) \(0\) \(0\) \(0\) \(-\) \(q+\beta _{5}q^{3}+(\beta _{1}-\beta _{3}+\beta _{6})q^{5}+(-2\beta _{1}+\cdots)q^{7}+\cdots\)
1024.4.a.l \(8\) \(60.418\) 8.8.\(\cdots\).3 None \(0\) \(0\) \(0\) \(0\) \(+\) \(q-\beta _{5}q^{3}+(\beta _{1}-\beta _{3}+\beta _{6})q^{5}+(2\beta _{1}+\cdots)q^{7}+\cdots\)
1024.4.a.m \(10\) \(60.418\) \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(0\) \(0\) \(0\) \(-28\) \(-\) \(q+\beta _{7}q^{3}+\beta _{4}q^{5}+(-3-\beta _{2})q^{7}+(6+\cdots)q^{9}+\cdots\)
1024.4.a.n \(10\) \(60.418\) \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(0\) \(0\) \(0\) \(28\) \(+\) \(q+\beta _{7}q^{3}-\beta _{4}q^{5}+(3+\beta _{2})q^{7}+(6+\beta _{2}+\cdots)q^{9}+\cdots\)
1024.4.a.o \(12\) \(60.418\) \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(0\) \(-40\) \(0\) \(-\) \(q-\beta _{3}q^{3}+(-3+\beta _{7})q^{5}+(\beta _{3}-\beta _{9}+\cdots)q^{7}+\cdots\)
1024.4.a.p \(12\) \(60.418\) \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(0\) \(40\) \(0\) \(+\) \(q-\beta _{3}q^{3}+(3-\beta _{7})q^{5}+(-\beta _{3}+\beta _{9}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1024))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1024)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 7}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(128))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(256))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(512))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( 1 + 22 T^{2} + 729 T^{4} \))(\( ( 1 + 27 T^{2} )^{2} \))(\( ( 1 + 27 T^{2} )^{2} \))(\( 1 + 22 T^{2} + 729 T^{4} \))(\( ( 1 + 6 T + 20 T^{2} + 162 T^{3} + 729 T^{4} )^{2} \))(\( 1 + 4 T^{2} - 986 T^{4} + 2916 T^{6} + 531441 T^{8} \))(\( ( 1 + 14 T^{2} + 729 T^{4} )^{2} \))(\( ( 1 + 14 T^{2} + 729 T^{4} )^{2} \))(\( 1 + 4 T^{2} - 986 T^{4} + 2916 T^{6} + 531441 T^{8} \))(\( ( 1 - 6 T + 20 T^{2} - 162 T^{3} + 729 T^{4} )^{2} \))(\( ( 1 + 52 T^{2} - 96 T^{3} + 1642 T^{4} - 2592 T^{5} + 37908 T^{6} + 531441 T^{8} )^{2} \))(\( ( 1 + 52 T^{2} + 96 T^{3} + 1642 T^{4} + 2592 T^{5} + 37908 T^{6} + 531441 T^{8} )^{2} \))(\( 1 + 108 T^{2} + 6317 T^{4} + 275312 T^{6} + 9907770 T^{8} + 295366792 T^{10} + 7222764330 T^{12} + 146312084592 T^{14} + 2447335229013 T^{16} + 30502389939948 T^{18} + 205891132094649 T^{20} \))(\( 1 + 108 T^{2} + 6317 T^{4} + 275312 T^{6} + 9907770 T^{8} + 295366792 T^{10} + 7222764330 T^{12} + 146312084592 T^{14} + 2447335229013 T^{16} + 30502389939948 T^{18} + 205891132094649 T^{20} \))(\( 1 + 108 T^{2} + 6778 T^{4} + 308892 T^{6} + 11259807 T^{8} + 353146136 T^{10} + 9938675756 T^{12} + 257443533144 T^{14} + 5983923091887 T^{16} + 119671089688188 T^{18} + 1914307398268218 T^{20} + 22236242266222092 T^{22} + 150094635296999121 T^{24} \))(\( 1 + 108 T^{2} + 6778 T^{4} + 308892 T^{6} + 11259807 T^{8} + 353146136 T^{10} + 9938675756 T^{12} + 257443533144 T^{14} + 5983923091887 T^{16} + 119671089688188 T^{18} + 1914307398268218 T^{20} + 22236242266222092 T^{22} + 150094635296999121 T^{24} \))
$5$ (\( 1 + 248 T^{2} + 15625 T^{4} \))(\( 1 - 88 T^{2} + 15625 T^{4} \))(\( 1 + 88 T^{2} + 15625 T^{4} \))(\( 1 + 248 T^{2} + 15625 T^{4} \))(\( 1 + 264 T^{2} + 37666 T^{4} + 4125000 T^{6} + 244140625 T^{8} \))(\( 1 + 192 T^{2} + 30674 T^{4} + 3000000 T^{6} + 244140625 T^{8} \))(\( ( 1 + 8 T^{2} + 15625 T^{4} )^{2} \))(\( ( 1 + 152 T^{2} + 15625 T^{4} )^{2} \))(\( 1 + 192 T^{2} + 30674 T^{4} + 3000000 T^{6} + 244140625 T^{8} \))(\( 1 + 264 T^{2} + 37666 T^{4} + 4125000 T^{6} + 244140625 T^{8} \))(\( 1 + 336 T^{2} + 48260 T^{4} + 5642352 T^{6} + 727217574 T^{8} + 88161750000 T^{10} + 11782226562500 T^{12} + 1281738281250000 T^{14} + 59604644775390625 T^{16} \))(\( 1 + 336 T^{2} + 48260 T^{4} + 5642352 T^{6} + 727217574 T^{8} + 88161750000 T^{10} + 11782226562500 T^{12} + 1281738281250000 T^{14} + 59604644775390625 T^{16} \))(\( 1 + 600 T^{2} + 191789 T^{4} + 42513504 T^{6} + 7224666922 T^{8} + 994659832592 T^{10} + 112885420656250 T^{12} + 10379273437500000 T^{14} + 731616973876953125 T^{16} + 35762786865234375000 T^{18} + \)\(93\!\cdots\!25\)\( T^{20} \))(\( 1 + 600 T^{2} + 191789 T^{4} + 42513504 T^{6} + 7224666922 T^{8} + 994659832592 T^{10} + 112885420656250 T^{12} + 10379273437500000 T^{14} + 731616973876953125 T^{16} + 35762786865234375000 T^{18} + \)\(93\!\cdots\!25\)\( T^{20} \))(\( ( 1 + 20 T + 500 T^{2} + 8548 T^{3} + 133483 T^{4} + 1677640 T^{5} + 21559400 T^{6} + 209705000 T^{7} + 2085671875 T^{8} + 16695312500 T^{9} + 122070312500 T^{10} + 610351562500 T^{11} + 3814697265625 T^{12} )^{2} \))(\( ( 1 - 20 T + 500 T^{2} - 8548 T^{3} + 133483 T^{4} - 1677640 T^{5} + 21559400 T^{6} - 209705000 T^{7} + 2085671875 T^{8} - 16695312500 T^{9} + 122070312500 T^{10} - 610351562500 T^{11} + 3814697265625 T^{12} )^{2} \))
$7$ (\( ( 1 + 8 T + 343 T^{2} )^{2} \))(\( ( 1 + 343 T^{2} )^{2} \))(\( ( 1 + 343 T^{2} )^{2} \))(\( ( 1 - 8 T + 343 T^{2} )^{2} \))(\( 1 + 1100 T^{2} + 520598 T^{4} + 129413900 T^{6} + 13841287201 T^{8} \))(\( ( 1 + 52 T + 1294 T^{2} + 17836 T^{3} + 117649 T^{4} )^{2} \))(\( ( 1 - 34 T^{2} + 117649 T^{4} )^{2} \))(\( ( 1 + 606 T^{2} + 117649 T^{4} )^{2} \))(\( ( 1 - 52 T + 1294 T^{2} - 17836 T^{3} + 117649 T^{4} )^{2} \))(\( 1 + 1100 T^{2} + 520598 T^{4} + 129413900 T^{6} + 13841287201 T^{8} \))(\( 1 + 664 T^{2} + 411836 T^{4} + 170193192 T^{6} + 69392575814 T^{8} + 20023058845608 T^{10} + 5700340355711036 T^{12} + 1081266629012538136 T^{14} + \)\(19\!\cdots\!01\)\( T^{16} \))(\( 1 + 664 T^{2} + 411836 T^{4} + 170193192 T^{6} + 69392575814 T^{8} + 20023058845608 T^{10} + 5700340355711036 T^{12} + 1081266629012538136 T^{14} + \)\(19\!\cdots\!01\)\( T^{16} \))(\( ( 1 + 14 T + 979 T^{2} + 5832 T^{3} + 372410 T^{4} + 662580 T^{5} + 127736630 T^{6} + 686128968 T^{7} + 39506181253 T^{8} + 193778020814 T^{9} + 4747561509943 T^{10} )^{2} \))(\( ( 1 - 14 T + 979 T^{2} - 5832 T^{3} + 372410 T^{4} - 662580 T^{5} + 127736630 T^{6} - 686128968 T^{7} + 39506181253 T^{8} - 193778020814 T^{9} + 4747561509943 T^{10} )^{2} \))(\( 1 + 1764 T^{2} + 1831330 T^{4} + 1344165268 T^{6} + 766175763663 T^{8} + 351953388506056 T^{10} + 132841523230184540 T^{12} + 41406964204348982344 T^{14} + \)\(10\!\cdots\!63\)\( T^{16} + \)\(21\!\cdots\!32\)\( T^{18} + \)\(35\!\cdots\!30\)\( T^{20} + \)\(39\!\cdots\!36\)\( T^{22} + \)\(26\!\cdots\!01\)\( T^{24} \))(\( 1 + 1764 T^{2} + 1831330 T^{4} + 1344165268 T^{6} + 766175763663 T^{8} + 351953388506056 T^{10} + 132841523230184540 T^{12} + 41406964204348982344 T^{14} + \)\(10\!\cdots\!63\)\( T^{16} + \)\(21\!\cdots\!32\)\( T^{18} + \)\(35\!\cdots\!30\)\( T^{20} + \)\(39\!\cdots\!36\)\( T^{22} + \)\(26\!\cdots\!01\)\( T^{24} \))
$11$ (\( 1 + 2630 T^{2} + 1771561 T^{4} \))(\( ( 1 + 1331 T^{2} )^{2} \))(\( ( 1 + 1331 T^{2} )^{2} \))(\( 1 + 2630 T^{2} + 1771561 T^{4} \))(\( ( 1 + 2 T + 556 T^{2} + 2662 T^{3} + 1771561 T^{4} )^{2} \))(\( 1 + 1508 T^{2} + 2498950 T^{4} + 2671513988 T^{6} + 3138428376721 T^{8} \))(\( ( 1 + 702 T^{2} + 1771561 T^{4} )^{2} \))(\( ( 1 - 578 T^{2} + 1771561 T^{4} )^{2} \))(\( 1 + 1508 T^{2} + 2498950 T^{4} + 2671513988 T^{6} + 3138428376721 T^{8} \))(\( ( 1 - 2 T + 556 T^{2} - 2662 T^{3} + 1771561 T^{4} )^{2} \))(\( ( 1 + 64 T + 5124 T^{2} + 237984 T^{3} + 10089610 T^{4} + 316756704 T^{5} + 9077478564 T^{6} + 150908652224 T^{7} + 3138428376721 T^{8} )^{2} \))(\( ( 1 - 64 T + 5124 T^{2} - 237984 T^{3} + 10089610 T^{4} - 316756704 T^{5} + 9077478564 T^{6} - 150908652224 T^{7} + 3138428376721 T^{8} )^{2} \))(\( 1 + 7260 T^{2} + 24476093 T^{4} + 51189622576 T^{6} + 78543614693626 T^{8} + 105919424238644520 T^{10} + \)\(13\!\cdots\!86\)\( T^{12} + \)\(16\!\cdots\!96\)\( T^{14} + \)\(13\!\cdots\!33\)\( T^{16} + \)\(71\!\cdots\!60\)\( T^{18} + \)\(17\!\cdots\!01\)\( T^{20} \))(\( 1 + 7260 T^{2} + 24476093 T^{4} + 51189622576 T^{6} + 78543614693626 T^{8} + 105919424238644520 T^{10} + \)\(13\!\cdots\!86\)\( T^{12} + \)\(16\!\cdots\!96\)\( T^{14} + \)\(13\!\cdots\!33\)\( T^{16} + \)\(71\!\cdots\!60\)\( T^{18} + \)\(17\!\cdots\!01\)\( T^{20} \))(\( 1 + 7596 T^{2} + 27544026 T^{4} + 62245432732 T^{6} + 98692206820863 T^{8} + 124484093722000536 T^{10} + \)\(15\!\cdots\!08\)\( T^{12} + \)\(22\!\cdots\!96\)\( T^{14} + \)\(30\!\cdots\!23\)\( T^{16} + \)\(34\!\cdots\!92\)\( T^{18} + \)\(27\!\cdots\!66\)\( T^{20} + \)\(13\!\cdots\!96\)\( T^{22} + \)\(30\!\cdots\!61\)\( T^{24} \))(\( 1 + 7596 T^{2} + 27544026 T^{4} + 62245432732 T^{6} + 98692206820863 T^{8} + 124484093722000536 T^{10} + \)\(15\!\cdots\!08\)\( T^{12} + \)\(22\!\cdots\!96\)\( T^{14} + \)\(30\!\cdots\!23\)\( T^{16} + \)\(34\!\cdots\!92\)\( T^{18} + \)\(27\!\cdots\!66\)\( T^{20} + \)\(13\!\cdots\!96\)\( T^{22} + \)\(30\!\cdots\!61\)\( T^{24} \))
$13$ (\( 1 + 4376 T^{2} + 4826809 T^{4} \))(\( 1 - 1656 T^{2} + 4826809 T^{4} \))(\( 1 + 1656 T^{2} + 4826809 T^{4} \))(\( 1 + 4376 T^{2} + 4826809 T^{4} \))(\( 1 + 6680 T^{2} + 20476226 T^{4} + 32243084120 T^{6} + 23298085122481 T^{8} \))(\( 1 + 6240 T^{2} + 19143218 T^{4} + 30119288160 T^{6} + 23298085122481 T^{8} \))(\( ( 1 + 3336 T^{2} + 4826809 T^{4} )^{2} \))(\( ( 1 + 344 T^{2} + 4826809 T^{4} )^{2} \))(\( 1 + 6240 T^{2} + 19143218 T^{4} + 30119288160 T^{6} + 23298085122481 T^{8} \))(\( 1 + 6680 T^{2} + 20476226 T^{4} + 32243084120 T^{6} + 23298085122481 T^{8} \))(\( 1 + 5488 T^{2} + 20166596 T^{4} + 49928047824 T^{6} + 114772460011238 T^{8} + 240993150589313616 T^{10} + \)\(46\!\cdots\!76\)\( T^{12} + \)\(61\!\cdots\!52\)\( T^{14} + \)\(54\!\cdots\!61\)\( T^{16} \))(\( 1 + 5488 T^{2} + 20166596 T^{4} + 49928047824 T^{6} + 114772460011238 T^{8} + 240993150589313616 T^{10} + \)\(46\!\cdots\!76\)\( T^{12} + \)\(61\!\cdots\!52\)\( T^{14} + \)\(54\!\cdots\!61\)\( T^{16} \))(\( 1 + 12168 T^{2} + 76341149 T^{4} + 324341675296 T^{6} + 1028186938895082 T^{8} + 2540163312339385904 T^{10} + \)\(49\!\cdots\!38\)\( T^{12} + \)\(75\!\cdots\!76\)\( T^{14} + \)\(85\!\cdots\!21\)\( T^{16} + \)\(66\!\cdots\!48\)\( T^{18} + \)\(26\!\cdots\!49\)\( T^{20} \))(\( 1 + 12168 T^{2} + 76341149 T^{4} + 324341675296 T^{6} + 1028186938895082 T^{8} + 2540163312339385904 T^{10} + \)\(49\!\cdots\!38\)\( T^{12} + \)\(75\!\cdots\!76\)\( T^{14} + \)\(85\!\cdots\!21\)\( T^{16} + \)\(66\!\cdots\!48\)\( T^{18} + \)\(26\!\cdots\!49\)\( T^{20} \))(\( ( 1 + 52 T + 6372 T^{2} + 209764 T^{3} + 16561659 T^{4} + 356094344 T^{5} + 31074788296 T^{6} + 782339273768 T^{7} + 79939964716131 T^{8} + 2224442206477972 T^{9} + 148455398400448932 T^{10} + 2661666436732719364 T^{11} + \)\(11\!\cdots\!29\)\( T^{12} )^{2} \))(\( ( 1 - 52 T + 6372 T^{2} - 209764 T^{3} + 16561659 T^{4} - 356094344 T^{5} + 31074788296 T^{6} - 782339273768 T^{7} + 79939964716131 T^{8} - 2224442206477972 T^{9} + 148455398400448932 T^{10} - 2661666436732719364 T^{11} + \)\(11\!\cdots\!29\)\( T^{12} )^{2} \))
$17$ (\( ( 1 + 24 T + 4913 T^{2} )^{2} \))(\( ( 1 + 104 T + 4913 T^{2} )^{2} \))(\( ( 1 - 104 T + 4913 T^{2} )^{2} \))(\( ( 1 + 24 T + 4913 T^{2} )^{2} \))(\( ( 1 + 96 T + 10582 T^{2} + 471648 T^{3} + 24137569 T^{4} )^{2} \))(\( ( 1 - 104 T + 12258 T^{2} - 510952 T^{3} + 24137569 T^{4} )^{2} \))(\( ( 1 + 96 T + 4913 T^{2} )^{4} \))(\( ( 1 - 16 T + 4913 T^{2} )^{4} \))(\( ( 1 - 104 T + 12258 T^{2} - 510952 T^{3} + 24137569 T^{4} )^{2} \))(\( ( 1 + 96 T + 10582 T^{2} + 471648 T^{3} + 24137569 T^{4} )^{2} \))(\( ( 1 - 96 T + 16172 T^{2} - 1303200 T^{3} + 112442022 T^{4} - 6402621600 T^{5} + 390352765868 T^{6} - 11384436143712 T^{7} + 582622237229761 T^{8} )^{2} \))(\( ( 1 - 96 T + 16172 T^{2} - 1303200 T^{3} + 112442022 T^{4} - 6402621600 T^{5} + 390352765868 T^{6} - 11384436143712 T^{7} + 582622237229761 T^{8} )^{2} \))(\( ( 1 - 2 T + 12653 T^{2} - 102520 T^{3} + 98460610 T^{4} - 354493580 T^{5} + 483736976930 T^{6} - 2474583573880 T^{7} + 1500492401316541 T^{8} - 1165244474459522 T^{9} + 2862423051509815793 T^{10} )^{2} \))(\( ( 1 - 2 T + 12653 T^{2} - 102520 T^{3} + 98460610 T^{4} - 354493580 T^{5} + 483736976930 T^{6} - 2474583573880 T^{7} + 1500492401316541 T^{8} - 1165244474459522 T^{9} + 2862423051509815793 T^{10} )^{2} \))(\( ( 1 + 9230 T^{2} + 118784 T^{3} + 60569871 T^{4} + 787214336 T^{5} + 385457234500 T^{6} + 3867584032768 T^{7} + 1462009440583599 T^{8} + 14086342321819648 T^{9} + 5377603249630694030 T^{10} + \)\(14\!\cdots\!09\)\( T^{12} )^{2} \))(\( ( 1 + 9230 T^{2} + 118784 T^{3} + 60569871 T^{4} + 787214336 T^{5} + 385457234500 T^{6} + 3867584032768 T^{7} + 1462009440583599 T^{8} + 14086342321819648 T^{9} + 5377603249630694030 T^{10} + \)\(14\!\cdots\!09\)\( T^{12} )^{2} \))
$19$ (\( 1 - 3210 T^{2} + 47045881 T^{4} \))(\( ( 1 + 6859 T^{2} )^{2} \))(\( ( 1 + 6859 T^{2} )^{2} \))(\( 1 - 3210 T^{2} + 47045881 T^{4} \))(\( ( 1 + 82 T + 15356 T^{2} + 562438 T^{3} + 47045881 T^{4} )^{2} \))(\( 1 + 24004 T^{2} + 237911014 T^{4} + 1129289327524 T^{6} + 2213314919066161 T^{8} \))(\( ( 1 + 2158 T^{2} + 47045881 T^{4} )^{2} \))(\( ( 1 + 2158 T^{2} + 47045881 T^{4} )^{2} \))(\( 1 + 24004 T^{2} + 237911014 T^{4} + 1129289327524 T^{6} + 2213314919066161 T^{8} \))(\( ( 1 - 82 T + 15356 T^{2} - 562438 T^{3} + 47045881 T^{4} )^{2} \))(\( ( 1 + 32 T + 10532 T^{2} - 548928 T^{3} + 33116138 T^{4} - 3765097152 T^{5} + 495487218692 T^{6} + 10326006328928 T^{7} + 2213314919066161 T^{8} )^{2} \))(\( ( 1 - 32 T + 10532 T^{2} + 548928 T^{3} + 33116138 T^{4} + 3765097152 T^{5} + 495487218692 T^{6} - 10326006328928 T^{7} + 2213314919066161 T^{8} )^{2} \))(\( 1 + 33948 T^{2} + 631492045 T^{4} + 8119820572464 T^{6} + 79259035180229178 T^{8} + \)\(60\!\cdots\!76\)\( T^{10} + \)\(37\!\cdots\!18\)\( T^{12} + \)\(17\!\cdots\!04\)\( T^{14} + \)\(65\!\cdots\!45\)\( T^{16} + \)\(16\!\cdots\!08\)\( T^{18} + \)\(23\!\cdots\!01\)\( T^{20} \))(\( 1 + 33948 T^{2} + 631492045 T^{4} + 8119820572464 T^{6} + 79259035180229178 T^{8} + \)\(60\!\cdots\!76\)\( T^{10} + \)\(37\!\cdots\!18\)\( T^{12} + \)\(17\!\cdots\!04\)\( T^{14} + \)\(65\!\cdots\!45\)\( T^{16} + \)\(16\!\cdots\!08\)\( T^{18} + \)\(23\!\cdots\!01\)\( T^{20} \))(\( 1 + 49228 T^{2} + 1209451642 T^{4} + 19832677238652 T^{6} + 241982329014671967 T^{8} + \)\(23\!\cdots\!56\)\( T^{10} + \)\(17\!\cdots\!08\)\( T^{12} + \)\(10\!\cdots\!36\)\( T^{14} + \)\(53\!\cdots\!87\)\( T^{16} + \)\(20\!\cdots\!32\)\( T^{18} + \)\(59\!\cdots\!82\)\( T^{20} + \)\(11\!\cdots\!28\)\( T^{22} + \)\(10\!\cdots\!81\)\( T^{24} \))(\( 1 + 49228 T^{2} + 1209451642 T^{4} + 19832677238652 T^{6} + 241982329014671967 T^{8} + \)\(23\!\cdots\!56\)\( T^{10} + \)\(17\!\cdots\!08\)\( T^{12} + \)\(10\!\cdots\!36\)\( T^{14} + \)\(53\!\cdots\!87\)\( T^{16} + \)\(20\!\cdots\!32\)\( T^{18} + \)\(59\!\cdots\!82\)\( T^{20} + \)\(11\!\cdots\!28\)\( T^{22} + \)\(10\!\cdots\!81\)\( T^{24} \))
$23$ (\( ( 1 - 184 T + 12167 T^{2} )^{2} \))(\( ( 1 + 12167 T^{2} )^{2} \))(\( ( 1 + 12167 T^{2} )^{2} \))(\( ( 1 + 184 T + 12167 T^{2} )^{2} \))(\( 1 + 38892 T^{2} + 672566806 T^{4} + 5757411794988 T^{6} + 21914624432020321 T^{8} \))(\( ( 1 - 12 T + 4718 T^{2} - 146004 T^{3} + 148035889 T^{4} )^{2} \))(\( ( 1 + 10814 T^{2} + 148035889 T^{4} )^{2} \))(\( ( 1 + 24254 T^{2} + 148035889 T^{4} )^{2} \))(\( ( 1 + 12 T + 4718 T^{2} + 146004 T^{3} + 148035889 T^{4} )^{2} \))(\( 1 + 38892 T^{2} + 672566806 T^{4} + 5757411794988 T^{6} + 21914624432020321 T^{8} \))(\( 1 + 30936 T^{2} + 649121468 T^{4} + 10376816461800 T^{6} + 129966114652954566 T^{8} + \)\(15\!\cdots\!00\)\( T^{10} + \)\(14\!\cdots\!28\)\( T^{12} + \)\(10\!\cdots\!84\)\( T^{14} + \)\(48\!\cdots\!41\)\( T^{16} \))(\( 1 + 30936 T^{2} + 649121468 T^{4} + 10376816461800 T^{6} + 129966114652954566 T^{8} + \)\(15\!\cdots\!00\)\( T^{10} + \)\(14\!\cdots\!28\)\( T^{12} + \)\(10\!\cdots\!84\)\( T^{14} + \)\(48\!\cdots\!41\)\( T^{16} \))(\( ( 1 + 138 T + 47715 T^{2} + 4085400 T^{3} + 882303386 T^{4} + 56970028764 T^{5} + 10734985297462 T^{6} + 604785820920600 T^{7} + 85941999241707045 T^{8} + 3024218171618804298 T^{9} + \)\(26\!\cdots\!07\)\( T^{10} )^{2} \))(\( ( 1 - 138 T + 47715 T^{2} - 4085400 T^{3} + 882303386 T^{4} - 56970028764 T^{5} + 10734985297462 T^{6} - 604785820920600 T^{7} + 85941999241707045 T^{8} - 3024218171618804298 T^{9} + \)\(26\!\cdots\!07\)\( T^{10} )^{2} \))(\( 1 + 80484 T^{2} + 3070592098 T^{4} + 76586871969684 T^{6} + 1463035780161687439 T^{8} + \)\(23\!\cdots\!20\)\( T^{10} + \)\(30\!\cdots\!16\)\( T^{12} + \)\(34\!\cdots\!80\)\( T^{14} + \)\(32\!\cdots\!19\)\( T^{16} + \)\(24\!\cdots\!96\)\( T^{18} + \)\(14\!\cdots\!18\)\( T^{20} + \)\(57\!\cdots\!16\)\( T^{22} + \)\(10\!\cdots\!61\)\( T^{24} \))(\( 1 + 80484 T^{2} + 3070592098 T^{4} + 76586871969684 T^{6} + 1463035780161687439 T^{8} + \)\(23\!\cdots\!20\)\( T^{10} + \)\(30\!\cdots\!16\)\( T^{12} + \)\(34\!\cdots\!80\)\( T^{14} + \)\(32\!\cdots\!19\)\( T^{16} + \)\(24\!\cdots\!96\)\( T^{18} + \)\(14\!\cdots\!18\)\( T^{20} + \)\(57\!\cdots\!16\)\( T^{22} + \)\(10\!\cdots\!61\)\( T^{24} \))
$29$ (\( 1 + 39256 T^{2} + 594823321 T^{4} \))(\( 1 - 36920 T^{2} + 594823321 T^{4} \))(\( 1 + 36920 T^{2} + 594823321 T^{4} \))(\( 1 + 39256 T^{2} + 594823321 T^{4} \))(\( 1 + 35304 T^{2} + 1047611074 T^{4} + 20999642524584 T^{6} + 353814783205469041 T^{8} \))(\( 1 + 72128 T^{2} + 2328989938 T^{4} + 42903416497088 T^{6} + 353814783205469041 T^{8} \))(\( ( 1 + 27560 T^{2} + 594823321 T^{4} )^{2} \))(\( ( 1 + 39800 T^{2} + 594823321 T^{4} )^{2} \))(\( 1 + 72128 T^{2} + 2328989938 T^{4} + 42903416497088 T^{6} + 353814783205469041 T^{8} \))(\( 1 + 35304 T^{2} + 1047611074 T^{4} + 20999642524584 T^{6} + 353814783205469041 T^{8} \))(\( 1 + 67344 T^{2} + 3992949188 T^{4} + 136432799019696 T^{6} + 4100907536736032358 T^{8} + \)\(81\!\cdots\!16\)\( T^{10} + \)\(14\!\cdots\!08\)\( T^{12} + \)\(14\!\cdots\!84\)\( T^{14} + \)\(12\!\cdots\!81\)\( T^{16} \))(\( 1 + 67344 T^{2} + 3992949188 T^{4} + 136432799019696 T^{6} + 4100907536736032358 T^{8} + \)\(81\!\cdots\!16\)\( T^{10} + \)\(14\!\cdots\!08\)\( T^{12} + \)\(14\!\cdots\!84\)\( T^{14} + \)\(12\!\cdots\!81\)\( T^{16} \))(\( 1 + 154168 T^{2} + 11039253245 T^{4} + 493373494152672 T^{6} + 16008852482124057194 T^{8} + \)\(42\!\cdots\!28\)\( T^{10} + \)\(95\!\cdots\!74\)\( T^{12} + \)\(17\!\cdots\!52\)\( T^{14} + \)\(23\!\cdots\!45\)\( T^{16} + \)\(19\!\cdots\!08\)\( T^{18} + \)\(74\!\cdots\!01\)\( T^{20} \))(\( 1 + 154168 T^{2} + 11039253245 T^{4} + 493373494152672 T^{6} + 16008852482124057194 T^{8} + \)\(42\!\cdots\!28\)\( T^{10} + \)\(95\!\cdots\!74\)\( T^{12} + \)\(17\!\cdots\!52\)\( T^{14} + \)\(23\!\cdots\!45\)\( T^{16} + \)\(19\!\cdots\!08\)\( T^{18} + \)\(74\!\cdots\!01\)\( T^{20} \))(\( ( 1 + 348 T + 125348 T^{2} + 27181868 T^{3} + 6168807579 T^{4} + 1024210224920 T^{5} + 183146629003336 T^{6} + 24979463175573880 T^{7} + 3669350610750749859 T^{8} + \)\(39\!\cdots\!92\)\( T^{9} + \)\(44\!\cdots\!68\)\( T^{10} + \)\(30\!\cdots\!52\)\( T^{11} + \)\(21\!\cdots\!61\)\( T^{12} )^{2} \))(\( ( 1 - 348 T + 125348 T^{2} - 27181868 T^{3} + 6168807579 T^{4} - 1024210224920 T^{5} + 183146629003336 T^{6} - 24979463175573880 T^{7} + 3669350610750749859 T^{8} - \)\(39\!\cdots\!92\)\( T^{9} + \)\(44\!\cdots\!68\)\( T^{10} - \)\(30\!\cdots\!52\)\( T^{11} + \)\(21\!\cdots\!61\)\( T^{12} )^{2} \))
$31$ (\( ( 1 + 176 T + 29791 T^{2} )^{2} \))(\( ( 1 + 29791 T^{2} )^{2} \))(\( ( 1 + 29791 T^{2} )^{2} \))(\( ( 1 - 176 T + 29791 T^{2} )^{2} \))(\( 1 + 13628 T^{2} - 886706170 T^{4} + 12094900164668 T^{6} + 787662783788549761 T^{8} \))(\( ( 1 + 88 T + 48190 T^{2} + 2621608 T^{3} + 887503681 T^{4} )^{2} \))(\( ( 1 + 20862 T^{2} + 887503681 T^{4} )^{2} \))(\( ( 1 - 32898 T^{2} + 887503681 T^{4} )^{2} \))(\( ( 1 - 88 T + 48190 T^{2} - 2621608 T^{3} + 887503681 T^{4} )^{2} \))(\( 1 + 13628 T^{2} - 886706170 T^{4} + 12094900164668 T^{6} + 787662783788549761 T^{8} \))(\( 1 + 159352 T^{2} + 11411140124 T^{4} + 508100706233160 T^{6} + 16871525430515085638 T^{8} + \)\(45\!\cdots\!60\)\( T^{10} + \)\(89\!\cdots\!64\)\( T^{12} + \)\(11\!\cdots\!32\)\( T^{14} + \)\(62\!\cdots\!21\)\( T^{16} \))(\( 1 + 159352 T^{2} + 11411140124 T^{4} + 508100706233160 T^{6} + 16871525430515085638 T^{8} + \)\(45\!\cdots\!60\)\( T^{10} + \)\(89\!\cdots\!64\)\( T^{12} + \)\(11\!\cdots\!32\)\( T^{14} + \)\(62\!\cdots\!21\)\( T^{16} \))(\( ( 1 + 184 T + 134043 T^{2} + 19809056 T^{3} + 7638677322 T^{4} + 852982867024 T^{5} + 227563836099702 T^{6} + 17580610117135136 T^{7} + 3544046273282822853 T^{8} + \)\(14\!\cdots\!24\)\( T^{9} + \)\(23\!\cdots\!51\)\( T^{10} )^{2} \))(\( ( 1 - 184 T + 134043 T^{2} - 19809056 T^{3} + 7638677322 T^{4} - 852982867024 T^{5} + 227563836099702 T^{6} - 17580610117135136 T^{7} + 3544046273282822853 T^{8} - \)\(14\!\cdots\!24\)\( T^{9} + \)\(23\!\cdots\!51\)\( T^{10} )^{2} \))(\( 1 + 181172 T^{2} + 17301312578 T^{4} + 1145992728921316 T^{6} + 57803385823619913071 T^{8} + \)\(23\!\cdots\!24\)\( T^{10} + \)\(76\!\cdots\!84\)\( T^{12} + \)\(20\!\cdots\!44\)\( T^{14} + \)\(45\!\cdots\!31\)\( T^{16} + \)\(80\!\cdots\!56\)\( T^{18} + \)\(10\!\cdots\!38\)\( T^{20} + \)\(99\!\cdots\!72\)\( T^{22} + \)\(48\!\cdots\!81\)\( T^{24} \))(\( 1 + 181172 T^{2} + 17301312578 T^{4} + 1145992728921316 T^{6} + 57803385823619913071 T^{8} + \)\(23\!\cdots\!24\)\( T^{10} + \)\(76\!\cdots\!84\)\( T^{12} + \)\(20\!\cdots\!44\)\( T^{14} + \)\(45\!\cdots\!31\)\( T^{16} + \)\(80\!\cdots\!56\)\( T^{18} + \)\(10\!\cdots\!38\)\( T^{20} + \)\(99\!\cdots\!72\)\( T^{22} + \)\(48\!\cdots\!81\)\( T^{24} \))
$37$ (\( 1 + 8856 T^{2} + 2565726409 T^{4} \))(\( 1 + 84744 T^{2} + 2565726409 T^{4} \))(\( 1 - 84744 T^{2} + 2565726409 T^{4} \))(\( 1 + 8856 T^{2} + 2565726409 T^{4} \))(\( 1 - 94984 T^{2} + 4939729634 T^{4} - 243702957232456 T^{6} + 6582952005840035281 T^{8} \))(\( 1 + 126368 T^{2} + 7731943506 T^{4} + 324225714852512 T^{6} + 6582952005840035281 T^{8} \))(\( ( 1 + 90056 T^{2} + 2565726409 T^{4} )^{2} \))(\( ( 1 + 101144 T^{2} + 2565726409 T^{4} )^{2} \))(\( 1 + 126368 T^{2} + 7731943506 T^{4} + 324225714852512 T^{6} + 6582952005840035281 T^{8} \))(\( 1 - 94984 T^{2} + 4939729634 T^{4} - 243702957232456 T^{6} + 6582952005840035281 T^{8} \))(\( 1 + 194608 T^{2} + 21091595012 T^{4} + 1589205356208144 T^{6} + 90960508533988116902 T^{8} + \)\(40\!\cdots\!96\)\( T^{10} + \)\(13\!\cdots\!72\)\( T^{12} + \)\(32\!\cdots\!32\)\( T^{14} + \)\(43\!\cdots\!61\)\( T^{16} \))(\( 1 + 194608 T^{2} + 21091595012 T^{4} + 1589205356208144 T^{6} + 90960508533988116902 T^{8} + \)\(40\!\cdots\!96\)\( T^{10} + \)\(13\!\cdots\!72\)\( T^{12} + \)\(32\!\cdots\!32\)\( T^{14} + \)\(43\!\cdots\!61\)\( T^{16} \))(\( 1 + 361064 T^{2} + 63572869101 T^{4} + 7160954906480288 T^{6} + \)\(56\!\cdots\!06\)\( T^{8} + \)\(33\!\cdots\!92\)\( T^{10} + \)\(14\!\cdots\!54\)\( T^{12} + \)\(47\!\cdots\!28\)\( T^{14} + \)\(10\!\cdots\!29\)\( T^{16} + \)\(15\!\cdots\!04\)\( T^{18} + \)\(11\!\cdots\!49\)\( T^{20} \))(\( 1 + 361064 T^{2} + 63572869101 T^{4} + 7160954906480288 T^{6} + \)\(56\!\cdots\!06\)\( T^{8} + \)\(33\!\cdots\!92\)\( T^{10} + \)\(14\!\cdots\!54\)\( T^{12} + \)\(47\!\cdots\!28\)\( T^{14} + \)\(10\!\cdots\!29\)\( T^{16} + \)\(15\!\cdots\!04\)\( T^{18} + \)\(11\!\cdots\!49\)\( T^{20} \))(\( ( 1 + 148 T + 208804 T^{2} + 16537060 T^{3} + 18949959467 T^{4} + 712824327880 T^{5} + 1106210758269384 T^{6} + 36106690680105640 T^{7} + 48620411453961464003 T^{8} + \)\(21\!\cdots\!20\)\( T^{9} + \)\(13\!\cdots\!24\)\( T^{10} + \)\(49\!\cdots\!64\)\( T^{11} + \)\(16\!\cdots\!29\)\( T^{12} )^{2} \))(\( ( 1 - 148 T + 208804 T^{2} - 16537060 T^{3} + 18949959467 T^{4} - 712824327880 T^{5} + 1106210758269384 T^{6} - 36106690680105640 T^{7} + 48620411453961464003 T^{8} - \)\(21\!\cdots\!20\)\( T^{9} + \)\(13\!\cdots\!24\)\( T^{10} - \)\(49\!\cdots\!64\)\( T^{11} + \)\(16\!\cdots\!29\)\( T^{12} )^{2} \))
$41$ (\( ( 1 - 200 T + 68921 T^{2} )^{2} \))(\( ( 1 - 472 T + 68921 T^{2} )^{2} \))(\( ( 1 + 472 T + 68921 T^{2} )^{2} \))(\( ( 1 - 200 T + 68921 T^{2} )^{2} \))(\( ( 1 - 320 T + 162754 T^{2} - 22054720 T^{3} + 4750104241 T^{4} )^{2} \))(\( ( 1 - 400 T + 173490 T^{2} - 27568400 T^{3} + 4750104241 T^{4} )^{2} \))(\( ( 1 + 312 T + 68921 T^{2} )^{4} \))(\( ( 1 + 328 T + 68921 T^{2} )^{4} \))(\( ( 1 - 400 T + 173490 T^{2} - 27568400 T^{3} + 4750104241 T^{4} )^{2} \))(\( ( 1 - 320 T + 162754 T^{2} - 22054720 T^{3} + 4750104241 T^{4} )^{2} \))(\( ( 1 + 320 T + 84996 T^{2} + 10066368 T^{3} + 2463544678 T^{4} + 693784148928 T^{5} + 403739860068036 T^{6} + 104762219006067520 T^{7} + 22563490300366186081 T^{8} )^{2} \))(\( ( 1 + 320 T + 84996 T^{2} + 10066368 T^{3} + 2463544678 T^{4} + 693784148928 T^{5} + 403739860068036 T^{6} + 104762219006067520 T^{7} + 22563490300366186081 T^{8} )^{2} \))(\( ( 1 + 220509 T^{2} + 10715136 T^{3} + 24270968490 T^{4} + 1235215122432 T^{5} + 1672779419299290 T^{6} + 50898012956491776 T^{7} + 72190662971277946149 T^{8} + \)\(15\!\cdots\!01\)\( T^{10} )^{2} \))(\( ( 1 + 220509 T^{2} + 10715136 T^{3} + 24270968490 T^{4} + 1235215122432 T^{5} + 1672779419299290 T^{6} + 50898012956491776 T^{7} + 72190662971277946149 T^{8} + \)\(15\!\cdots\!01\)\( T^{10} )^{2} \))(\( ( 1 - 40 T + 306502 T^{2} - 18831240 T^{3} + 44656110879 T^{4} - 2603821604240 T^{5} + 3906631272972244 T^{6} - 179457988785825040 T^{7} + \)\(21\!\cdots\!39\)\( T^{8} - \)\(61\!\cdots\!40\)\( T^{9} + \)\(69\!\cdots\!62\)\( T^{10} - \)\(62\!\cdots\!40\)\( T^{11} + \)\(10\!\cdots\!21\)\( T^{12} )^{2} \))(\( ( 1 - 40 T + 306502 T^{2} - 18831240 T^{3} + 44656110879 T^{4} - 2603821604240 T^{5} + 3906631272972244 T^{6} - 179457988785825040 T^{7} + \)\(21\!\cdots\!39\)\( T^{8} - \)\(61\!\cdots\!40\)\( T^{9} + \)\(69\!\cdots\!62\)\( T^{10} - \)\(62\!\cdots\!40\)\( T^{11} + \)\(10\!\cdots\!21\)\( T^{12} )^{2} \))
$43$ (\( 1 + 139014 T^{2} + 6321363049 T^{4} \))(\( ( 1 + 79507 T^{2} )^{2} \))(\( ( 1 + 79507 T^{2} )^{2} \))(\( 1 + 139014 T^{2} + 6321363049 T^{4} \))(\( ( 1 - 510 T + 205076 T^{2} - 40548570 T^{3} + 6321363049 T^{4} )^{2} \))(\( 1 + 214116 T^{2} + 24054082310 T^{4} + 1353504970599684 T^{6} + 39959630797262576401 T^{8} \))(\( ( 1 + 29054 T^{2} + 6321363049 T^{4} )^{2} \))(\( ( 1 + 98174 T^{2} + 6321363049 T^{4} )^{2} \))(\( 1 + 214116 T^{2} + 24054082310 T^{4} + 1353504970599684 T^{6} + 39959630797262576401 T^{8} \))(\( ( 1 + 510 T + 205076 T^{2} + 40548570 T^{3} + 6321363049 T^{4} )^{2} \))(\( ( 1 + 768 T + 411220 T^{2} + 158851296 T^{3} + 50848290954 T^{4} + 12629789991072 T^{5} + 2599470913009780 T^{6} + 385991125967495424 T^{7} + 39959630797262576401 T^{8} )^{2} \))(\( ( 1 - 768 T + 411220 T^{2} - 158851296 T^{3} + 50848290954 T^{4} - 12629789991072 T^{5} + 2599470913009780 T^{6} - 385991125967495424 T^{7} + 39959630797262576401 T^{8} )^{2} \))(\( 1 + 551948 T^{2} + 145009588605 T^{4} + 24152295654927088 T^{6} + \)\(28\!\cdots\!30\)\( T^{8} + \)\(25\!\cdots\!12\)\( T^{10} + \)\(18\!\cdots\!70\)\( T^{12} + \)\(96\!\cdots\!88\)\( T^{14} + \)\(36\!\cdots\!45\)\( T^{16} + \)\(88\!\cdots\!48\)\( T^{18} + \)\(10\!\cdots\!49\)\( T^{20} \))(\( 1 + 551948 T^{2} + 145009588605 T^{4} + 24152295654927088 T^{6} + \)\(28\!\cdots\!30\)\( T^{8} + \)\(25\!\cdots\!12\)\( T^{10} + \)\(18\!\cdots\!70\)\( T^{12} + \)\(96\!\cdots\!88\)\( T^{14} + \)\(36\!\cdots\!45\)\( T^{16} + \)\(88\!\cdots\!48\)\( T^{18} + \)\(10\!\cdots\!49\)\( T^{20} \))(\( 1 + 115084 T^{2} + 31655055130 T^{4} + 2908825652771836 T^{6} + \)\(44\!\cdots\!03\)\( T^{8} + \)\(33\!\cdots\!48\)\( T^{10} + \)\(35\!\cdots\!60\)\( T^{12} + \)\(20\!\cdots\!52\)\( T^{14} + \)\(17\!\cdots\!03\)\( T^{16} + \)\(73\!\cdots\!64\)\( T^{18} + \)\(50\!\cdots\!30\)\( T^{20} + \)\(11\!\cdots\!16\)\( T^{22} + \)\(63\!\cdots\!01\)\( T^{24} \))(\( 1 + 115084 T^{2} + 31655055130 T^{4} + 2908825652771836 T^{6} + \)\(44\!\cdots\!03\)\( T^{8} + \)\(33\!\cdots\!48\)\( T^{10} + \)\(35\!\cdots\!60\)\( T^{12} + \)\(20\!\cdots\!52\)\( T^{14} + \)\(17\!\cdots\!03\)\( T^{16} + \)\(73\!\cdots\!64\)\( T^{18} + \)\(50\!\cdots\!30\)\( T^{20} + \)\(11\!\cdots\!16\)\( T^{22} + \)\(63\!\cdots\!01\)\( T^{24} \))
$47$ (\( ( 1 + 208 T + 103823 T^{2} )^{2} \))(\( ( 1 + 103823 T^{2} )^{2} \))(\( ( 1 + 103823 T^{2} )^{2} \))(\( ( 1 - 208 T + 103823 T^{2} )^{2} \))(\( 1 + 276348 T^{2} + 40098147526 T^{4} + 2978814597738492 T^{6} + \)\(11\!\cdots\!41\)\( T^{8} \))(\( ( 1 + 360 T + 178846 T^{2} + 37376280 T^{3} + 10779215329 T^{4} )^{2} \))(\( ( 1 + 199646 T^{2} + 10779215329 T^{4} )^{2} \))(\( ( 1 + 38366 T^{2} + 10779215329 T^{4} )^{2} \))(\( ( 1 - 360 T + 178846 T^{2} - 37376280 T^{3} + 10779215329 T^{4} )^{2} \))(\( 1 + 276348 T^{2} + 40098147526 T^{4} + 2978814597738492 T^{6} + \)\(11\!\cdots\!41\)\( T^{8} \))(\( 1 + 434424 T^{2} + 101469073820 T^{4} + 15759447458513352 T^{6} + \)\(18\!\cdots\!62\)\( T^{8} + \)\(16\!\cdots\!08\)\( T^{10} + \)\(11\!\cdots\!20\)\( T^{12} + \)\(54\!\cdots\!36\)\( T^{14} + \)\(13\!\cdots\!81\)\( T^{16} \))(\( 1 + 434424 T^{2} + 101469073820 T^{4} + 15759447458513352 T^{6} + \)\(18\!\cdots\!62\)\( T^{8} + \)\(16\!\cdots\!08\)\( T^{10} + \)\(11\!\cdots\!20\)\( T^{12} + \)\(54\!\cdots\!36\)\( T^{14} + \)\(13\!\cdots\!81\)\( T^{16} \))(\( ( 1 + 472 T + 462219 T^{2} + 171516064 T^{3} + 90105579914 T^{4} + 25593405310224 T^{5} + 9355031623411222 T^{6} + 1848808586238545056 T^{7} + \)\(51\!\cdots\!73\)\( T^{8} + \)\(54\!\cdots\!52\)\( T^{9} + \)\(12\!\cdots\!43\)\( T^{10} )^{2} \))(\( ( 1 - 472 T + 462219 T^{2} - 171516064 T^{3} + 90105579914 T^{4} - 25593405310224 T^{5} + 9355031623411222 T^{6} - 1848808586238545056 T^{7} + \)\(51\!\cdots\!73\)\( T^{8} - \)\(54\!\cdots\!52\)\( T^{9} + \)\(12\!\cdots\!43\)\( T^{10} )^{2} \))(\( 1 + 313844 T^{2} + 72753002882 T^{4} + 13405470205350052 T^{6} + \)\(20\!\cdots\!87\)\( T^{8} + \)\(25\!\cdots\!12\)\( T^{10} + \)\(28\!\cdots\!04\)\( T^{12} + \)\(27\!\cdots\!48\)\( T^{14} + \)\(23\!\cdots\!67\)\( T^{16} + \)\(16\!\cdots\!28\)\( T^{18} + \)\(98\!\cdots\!42\)\( T^{20} + \)\(45\!\cdots\!56\)\( T^{22} + \)\(15\!\cdots\!21\)\( T^{24} \))(\( 1 + 313844 T^{2} + 72753002882 T^{4} + 13405470205350052 T^{6} + \)\(20\!\cdots\!87\)\( T^{8} + \)\(25\!\cdots\!12\)\( T^{10} + \)\(28\!\cdots\!04\)\( T^{12} + \)\(27\!\cdots\!48\)\( T^{14} + \)\(23\!\cdots\!67\)\( T^{16} + \)\(16\!\cdots\!28\)\( T^{18} + \)\(98\!\cdots\!42\)\( T^{20} + \)\(45\!\cdots\!56\)\( T^{22} + \)\(15\!\cdots\!21\)\( T^{24} \))
$53$ (\( 1 + 159416 T^{2} + 22164361129 T^{4} \))(\( 1 + 296296 T^{2} + 22164361129 T^{4} \))(\( 1 - 296296 T^{2} + 22164361129 T^{4} \))(\( 1 + 159416 T^{2} + 22164361129 T^{4} \))(\( 1 + 456120 T^{2} + 91936266658 T^{4} + 10109608398159480 T^{6} + \)\(49\!\cdots\!41\)\( T^{8} \))(\( 1 + 129504 T^{2} + 13654493714 T^{4} + 2870373423650016 T^{6} + \)\(49\!\cdots\!41\)\( T^{8} \))(\( ( 1 + 86504 T^{2} + 22164361129 T^{4} )^{2} \))(\( ( 1 - 33544 T^{2} + 22164361129 T^{4} )^{2} \))(\( 1 + 129504 T^{2} + 13654493714 T^{4} + 2870373423650016 T^{6} + \)\(49\!\cdots\!41\)\( T^{8} \))(\( 1 + 456120 T^{2} + 91936266658 T^{4} + 10109608398159480 T^{6} + \)\(49\!\cdots\!41\)\( T^{8} \))(\( 1 + 593328 T^{2} + 161091742340 T^{4} + 27386249068104336 T^{6} + \)\(39\!\cdots\!94\)\( T^{8} + \)\(60\!\cdots\!44\)\( T^{10} + \)\(79\!\cdots\!40\)\( T^{12} + \)\(64\!\cdots\!92\)\( T^{14} + \)\(24\!\cdots\!81\)\( T^{16} \))(\( 1 + 593328 T^{2} + 161091742340 T^{4} + 27386249068104336 T^{6} + \)\(39\!\cdots\!94\)\( T^{8} + \)\(60\!\cdots\!44\)\( T^{10} + \)\(79\!\cdots\!40\)\( T^{12} + \)\(64\!\cdots\!92\)\( T^{14} + \)\(24\!\cdots\!81\)\( T^{16} \))(\( 1 + 525160 T^{2} + 112387404877 T^{4} + 12025532417207968 T^{6} + \)\(67\!\cdots\!30\)\( T^{8} + \)\(40\!\cdots\!00\)\( T^{10} + \)\(15\!\cdots\!70\)\( T^{12} + \)\(59\!\cdots\!88\)\( T^{14} + \)\(12\!\cdots\!53\)\( T^{16} + \)\(12\!\cdots\!60\)\( T^{18} + \)\(53\!\cdots\!49\)\( T^{20} \))(\( 1 + 525160 T^{2} + 112387404877 T^{4} + 12025532417207968 T^{6} + \)\(67\!\cdots\!30\)\( T^{8} + \)\(40\!\cdots\!00\)\( T^{10} + \)\(15\!\cdots\!70\)\( T^{12} + \)\(59\!\cdots\!88\)\( T^{14} + \)\(12\!\cdots\!53\)\( T^{16} + \)\(12\!\cdots\!60\)\( T^{18} + \)\(53\!\cdots\!49\)\( T^{20} \))(\( ( 1 + 636 T + 850036 T^{2} + 356826412 T^{3} + 273714006091 T^{4} + 85121646768472 T^{5} + 50261228041394792 T^{6} + 12672655405949805944 T^{7} + \)\(60\!\cdots\!39\)\( T^{8} + \)\(11\!\cdots\!96\)\( T^{9} + \)\(41\!\cdots\!76\)\( T^{10} + \)\(46\!\cdots\!52\)\( T^{11} + \)\(10\!\cdots\!89\)\( T^{12} )^{2} \))(\( ( 1 - 636 T + 850036 T^{2} - 356826412 T^{3} + 273714006091 T^{4} - 85121646768472 T^{5} + 50261228041394792 T^{6} - 12672655405949805944 T^{7} + \)\(60\!\cdots\!39\)\( T^{8} - \)\(11\!\cdots\!96\)\( T^{9} + \)\(41\!\cdots\!76\)\( T^{10} - \)\(46\!\cdots\!52\)\( T^{11} + \)\(10\!\cdots\!89\)\( T^{12} )^{2} \))
$59$ (\( 1 + 387430 T^{2} + 42180533641 T^{4} \))(\( ( 1 + 205379 T^{2} )^{2} \))(\( ( 1 + 205379 T^{2} )^{2} \))(\( 1 + 387430 T^{2} + 42180533641 T^{4} \))(\( ( 1 + 390 T + 383380 T^{2} + 80097810 T^{3} + 42180533641 T^{4} )^{2} \))(\( 1 - 14876 T^{2} + 60233947974 T^{4} - 627477618443516 T^{6} + \)\(17\!\cdots\!81\)\( T^{8} \))(\( ( 1 + 271518 T^{2} + 42180533641 T^{4} )^{2} \))(\( ( 1 - 136802 T^{2} + 42180533641 T^{4} )^{2} \))(\( 1 - 14876 T^{2} + 60233947974 T^{4} - 627477618443516 T^{6} + \)\(17\!\cdots\!81\)\( T^{8} \))(\( ( 1 - 390 T + 383380 T^{2} - 80097810 T^{3} + 42180533641 T^{4} )^{2} \))(\( ( 1 + 672 T + 930932 T^{2} + 407252736 T^{3} + 296175415050 T^{4} + 83641159666944 T^{5} + 39267208543483412 T^{6} + 5821533190136119008 T^{7} + \)\(17\!\cdots\!81\)\( T^{8} )^{2} \))(\( ( 1 - 672 T + 930932 T^{2} - 407252736 T^{3} + 296175415050 T^{4} - 83641159666944 T^{5} + 39267208543483412 T^{6} - 5821533190136119008 T^{7} + \)\(17\!\cdots\!81\)\( T^{8} )^{2} \))(\( 1 + 958284 T^{2} + 532967592733 T^{4} + 207330855222256112 T^{6} + \)\(61\!\cdots\!62\)\( T^{8} + \)\(14\!\cdots\!28\)\( T^{10} + \)\(25\!\cdots\!42\)\( T^{12} + \)\(36\!\cdots\!72\)\( T^{14} + \)\(39\!\cdots\!93\)\( T^{16} + \)\(30\!\cdots\!24\)\( T^{18} + \)\(13\!\cdots\!01\)\( T^{20} \))(\( 1 + 958284 T^{2} + 532967592733 T^{4} + 207330855222256112 T^{6} + \)\(61\!\cdots\!62\)\( T^{8} + \)\(14\!\cdots\!28\)\( T^{10} + \)\(25\!\cdots\!42\)\( T^{12} + \)\(36\!\cdots\!72\)\( T^{14} + \)\(39\!\cdots\!93\)\( T^{16} + \)\(30\!\cdots\!24\)\( T^{18} + \)\(13\!\cdots\!01\)\( T^{20} \))(\( 1 + 890060 T^{2} + 493229275226 T^{4} + 200234076913876924 T^{6} + \)\(64\!\cdots\!55\)\( T^{8} + \)\(17\!\cdots\!12\)\( T^{10} + \)\(38\!\cdots\!84\)\( T^{12} + \)\(72\!\cdots\!92\)\( T^{14} + \)\(11\!\cdots\!55\)\( T^{16} + \)\(15\!\cdots\!04\)\( T^{18} + \)\(15\!\cdots\!86\)\( T^{20} + \)\(11\!\cdots\!60\)\( T^{22} + \)\(56\!\cdots\!41\)\( T^{24} \))(\( 1 + 890060 T^{2} + 493229275226 T^{4} + 200234076913876924 T^{6} + \)\(64\!\cdots\!55\)\( T^{8} + \)\(17\!\cdots\!12\)\( T^{10} + \)\(38\!\cdots\!84\)\( T^{12} + \)\(72\!\cdots\!92\)\( T^{14} + \)\(11\!\cdots\!55\)\( T^{16} + \)\(15\!\cdots\!04\)\( T^{18} + \)\(15\!\cdots\!86\)\( T^{20} + \)\(11\!\cdots\!60\)\( T^{22} + \)\(56\!\cdots\!41\)\( T^{24} \))
$61$ (\( 1 + 224120 T^{2} + 51520374361 T^{4} \))(\( 1 - 388440 T^{2} + 51520374361 T^{4} \))(\( 1 + 388440 T^{2} + 51520374361 T^{4} \))(\( 1 + 224120 T^{2} + 51520374361 T^{4} \))(\( 1 + 100568 T^{2} + 46956514178 T^{4} + 5181301008737048 T^{6} + \)\(26\!\cdots\!21\)\( T^{8} \))(\( 1 + 719648 T^{2} + 231280022898 T^{4} + 37076534368144928 T^{6} + \)\(26\!\cdots\!21\)\( T^{8} \))(\( ( 1 + 450920 T^{2} + 51520374361 T^{4} )^{2} \))(\( ( 1 - 239880 T^{2} + 51520374361 T^{4} )^{2} \))(\( 1 + 719648 T^{2} + 231280022898 T^{4} + 37076534368144928 T^{6} + \)\(26\!\cdots\!21\)\( T^{8} \))(\( 1 + 100568 T^{2} + 46956514178 T^{4} + 5181301008737048 T^{6} + \)\(26\!\cdots\!21\)\( T^{8} \))(\( 1 + 1364464 T^{2} + 846539346500 T^{4} + 322819977038049360 T^{6} + \)\(85\!\cdots\!22\)\( T^{8} + \)\(16\!\cdots\!60\)\( T^{10} + \)\(22\!\cdots\!00\)\( T^{12} + \)\(18\!\cdots\!84\)\( T^{14} + \)\(70\!\cdots\!41\)\( T^{16} \))(\( 1 + 1364464 T^{2} + 846539346500 T^{4} + 322819977038049360 T^{6} + \)\(85\!\cdots\!22\)\( T^{8} + \)\(16\!\cdots\!60\)\( T^{10} + \)\(22\!\cdots\!00\)\( T^{12} + \)\(18\!\cdots\!84\)\( T^{14} + \)\(70\!\cdots\!41\)\( T^{16} \))(\( 1 + 1146824 T^{2} + 509607070653 T^{4} + 80080659050426912 T^{6} - \)\(14\!\cdots\!54\)\( T^{8} - \)\(80\!\cdots\!72\)\( T^{10} - \)\(77\!\cdots\!94\)\( T^{12} + \)\(21\!\cdots\!52\)\( T^{14} + \)\(69\!\cdots\!93\)\( T^{16} + \)\(80\!\cdots\!84\)\( T^{18} + \)\(36\!\cdots\!01\)\( T^{20} \))(\( 1 + 1146824 T^{2} + 509607070653 T^{4} + 80080659050426912 T^{6} - \)\(14\!\cdots\!54\)\( T^{8} - \)\(80\!\cdots\!72\)\( T^{10} - \)\(77\!\cdots\!94\)\( T^{12} + \)\(21\!\cdots\!52\)\( T^{14} + \)\(69\!\cdots\!93\)\( T^{16} + \)\(80\!\cdots\!84\)\( T^{18} + \)\(36\!\cdots\!01\)\( T^{20} \))(\( ( 1 + 732 T + 988244 T^{2} + 639388844 T^{3} + 487192576603 T^{4} + 244846010914712 T^{5} + 142614779059820968 T^{6} + 55575392403432244472 T^{7} + \)\(25\!\cdots\!83\)\( T^{8} + \)\(74\!\cdots\!04\)\( T^{9} + \)\(26\!\cdots\!24\)\( T^{10} + \)\(44\!\cdots\!32\)\( T^{11} + \)\(13\!\cdots\!81\)\( T^{12} )^{2} \))(\( ( 1 - 732 T + 988244 T^{2} - 639388844 T^{3} + 487192576603 T^{4} - 244846010914712 T^{5} + 142614779059820968 T^{6} - 55575392403432244472 T^{7} + \)\(25\!\cdots\!83\)\( T^{8} - \)\(74\!\cdots\!04\)\( T^{9} + \)\(26\!\cdots\!24\)\( T^{10} - \)\(44\!\cdots\!32\)\( T^{11} + \)\(13\!\cdots\!81\)\( T^{12} )^{2} \))
$67$ (\( 1 + 133014 T^{2} + 90458382169 T^{4} \))(\( ( 1 + 300763 T^{2} )^{2} \))(\( ( 1 + 300763 T^{2} )^{2} \))(\( 1 + 133014 T^{2} + 90458382169 T^{4} \))(\( ( 1 + 646 T + 705468 T^{2} + 194292898 T^{3} + 90458382169 T^{4} )^{2} \))(\( 1 - 184316 T^{2} + 92156184870 T^{4} - 16672927167861404 T^{6} + \)\(81\!\cdots\!61\)\( T^{8} \))(\( ( 1 + 240526 T^{2} + 90458382169 T^{4} )^{2} \))(\( ( 1 + 513166 T^{2} + 90458382169 T^{4} )^{2} \))(\( 1 - 184316 T^{2} + 92156184870 T^{4} - 16672927167861404 T^{6} + \)\(81\!\cdots\!61\)\( T^{8} \))(\( ( 1 - 646 T + 705468 T^{2} - 194292898 T^{3} + 90458382169 T^{4} )^{2} \))(\( ( 1 + 512 T + 619396 T^{2} + 267786656 T^{3} + 275161349482 T^{4} + 80540318018528 T^{5} + 56029560081949924 T^{6} + 13929745610903012864 T^{7} + \)\(81\!\cdots\!61\)\( T^{8} )^{2} \))(\( ( 1 - 512 T + 619396 T^{2} - 267786656 T^{3} + 275161349482 T^{4} - 80540318018528 T^{5} + 56029560081949924 T^{6} - 13929745610903012864 T^{7} + \)\(81\!\cdots\!61\)\( T^{8} )^{2} \))(\( 1 + 2130652 T^{2} + 2127009110445 T^{4} + 1334413095892987440 T^{6} + \)\(59\!\cdots\!58\)\( T^{8} + \)\(20\!\cdots\!52\)\( T^{10} + \)\(53\!\cdots\!02\)\( T^{12} + \)\(10\!\cdots\!40\)\( T^{14} + \)\(15\!\cdots\!05\)\( T^{16} + \)\(14\!\cdots\!92\)\( T^{18} + \)\(60\!\cdots\!49\)\( T^{20} \))(\( 1 + 2130652 T^{2} + 2127009110445 T^{4} + 1334413095892987440 T^{6} + \)\(59\!\cdots\!58\)\( T^{8} + \)\(20\!\cdots\!52\)\( T^{10} + \)\(53\!\cdots\!02\)\( T^{12} + \)\(10\!\cdots\!40\)\( T^{14} + \)\(15\!\cdots\!05\)\( T^{16} + \)\(14\!\cdots\!92\)\( T^{18} + \)\(60\!\cdots\!49\)\( T^{20} \))(\( 1 + 1784652 T^{2} + 1652400360506 T^{4} + 1050186812010889468 T^{6} + \)\(51\!\cdots\!71\)\( T^{8} + \)\(20\!\cdots\!64\)\( T^{10} + \)\(67\!\cdots\!32\)\( T^{12} + \)\(18\!\cdots\!16\)\( T^{14} + \)\(42\!\cdots\!31\)\( T^{16} + \)\(77\!\cdots\!12\)\( T^{18} + \)\(11\!\cdots\!26\)\( T^{20} + \)\(10\!\cdots\!48\)\( T^{22} + \)\(54\!\cdots\!81\)\( T^{24} \))(\( 1 + 1784652 T^{2} + 1652400360506 T^{4} + 1050186812010889468 T^{6} + \)\(51\!\cdots\!71\)\( T^{8} + \)\(20\!\cdots\!64\)\( T^{10} + \)\(67\!\cdots\!32\)\( T^{12} + \)\(18\!\cdots\!16\)\( T^{14} + \)\(42\!\cdots\!31\)\( T^{16} + \)\(77\!\cdots\!12\)\( T^{18} + \)\(11\!\cdots\!26\)\( T^{20} + \)\(10\!\cdots\!48\)\( T^{22} + \)\(54\!\cdots\!81\)\( T^{24} \))
$71$ (\( ( 1 - 936 T + 357911 T^{2} )^{2} \))(\( ( 1 + 357911 T^{2} )^{2} \))(\( ( 1 + 357911 T^{2} )^{2} \))(\( ( 1 + 936 T + 357911 T^{2} )^{2} \))(\( 1 + 974892 T^{2} + 473570827606 T^{4} + 124883941992311532 T^{6} + \)\(16\!\cdots\!41\)\( T^{8} \))(\( ( 1 - 356 T + 664206 T^{2} - 127416316 T^{3} + 128100283921 T^{4} )^{2} \))(\( ( 1 + 581342 T^{2} + 128100283921 T^{4} )^{2} \))(\( ( 1 + 241502 T^{2} + 128100283921 T^{4} )^{2} \))(\( ( 1 + 356 T + 664206 T^{2} + 127416316 T^{3} + 128100283921 T^{4} )^{2} \))(\( 1 + 974892 T^{2} + 473570827606 T^{4} + 124883941992311532 T^{6} + \)\(16\!\cdots\!41\)\( T^{8} \))(\( 1 + 1119576 T^{2} + 845240011580 T^{4} + 458427553477745256 T^{6} + \)\(18\!\cdots\!26\)\( T^{8} + \)\(58\!\cdots\!76\)\( T^{10} + \)\(13\!\cdots\!80\)\( T^{12} + \)\(23\!\cdots\!36\)\( T^{14} + \)\(26\!\cdots\!81\)\( T^{16} \))(\( 1 + 1119576 T^{2} + 845240011580 T^{4} + 458427553477745256 T^{6} + \)\(18\!\cdots\!26\)\( T^{8} + \)\(58\!\cdots\!76\)\( T^{10} + \)\(13\!\cdots\!80\)\( T^{12} + \)\(23\!\cdots\!36\)\( T^{14} + \)\(26\!\cdots\!81\)\( T^{16} \))(\( ( 1 + 1734 T + 2753587 T^{2} + 2611066280 T^{3} + 2272726706522 T^{4} + 1414434499369348 T^{5} + 813433888257995542 T^{6} + \)\(33\!\cdots\!80\)\( T^{7} + \)\(12\!\cdots\!97\)\( T^{8} + \)\(28\!\cdots\!94\)\( T^{9} + \)\(58\!\cdots\!51\)\( T^{10} )^{2} \))(\( ( 1 - 1734 T + 2753587 T^{2} - 2611066280 T^{3} + 2272726706522 T^{4} - 1414434499369348 T^{5} + 813433888257995542 T^{6} - \)\(33\!\cdots\!80\)\( T^{7} + \)\(12\!\cdots\!97\)\( T^{8} - \)\(28\!\cdots\!94\)\( T^{9} + \)\(58\!\cdots\!51\)\( T^{10} )^{2} \))(\( 1 + 1669540 T^{2} + 1101330408738 T^{4} + 357143535249264468 T^{6} + \)\(85\!\cdots\!79\)\( T^{8} + \)\(44\!\cdots\!08\)\( T^{10} + \)\(21\!\cdots\!16\)\( T^{12} + \)\(56\!\cdots\!68\)\( T^{14} + \)\(14\!\cdots\!39\)\( T^{16} + \)\(75\!\cdots\!48\)\( T^{18} + \)\(29\!\cdots\!78\)\( T^{20} + \)\(57\!\cdots\!40\)\( T^{22} + \)\(44\!\cdots\!21\)\( T^{24} \))(\( 1 + 1669540 T^{2} + 1101330408738 T^{4} + 357143535249264468 T^{6} + \)\(85\!\cdots\!79\)\( T^{8} + \)\(44\!\cdots\!08\)\( T^{10} + \)\(21\!\cdots\!16\)\( T^{12} + \)\(56\!\cdots\!68\)\( T^{14} + \)\(14\!\cdots\!39\)\( T^{16} + \)\(75\!\cdots\!48\)\( T^{18} + \)\(29\!\cdots\!78\)\( T^{20} + \)\(57\!\cdots\!40\)\( T^{22} + \)\(44\!\cdots\!21\)\( T^{24} \))
$73$ (\( ( 1 - 406 T + 389017 T^{2} )^{2} \))(\( ( 1 + 1098 T + 389017 T^{2} )^{2} \))(\( ( 1 + 1098 T + 389017 T^{2} )^{2} \))(\( ( 1 - 406 T + 389017 T^{2} )^{2} \))(\( ( 1 - 436 T + 185546 T^{2} - 169611412 T^{3} + 151334226289 T^{4} )^{2} \))(\( ( 1 - 900 T + 882342 T^{2} - 350115300 T^{3} + 151334226289 T^{4} )^{2} \))(\( ( 1 + 1138 T + 389017 T^{2} )^{4} \))(\( ( 1 - 782 T + 389017 T^{2} )^{4} \))(\( ( 1 - 900 T + 882342 T^{2} - 350115300 T^{3} + 151334226289 T^{4} )^{2} \))(\( ( 1 - 436 T + 185546 T^{2} - 169611412 T^{3} + 151334226289 T^{4} )^{2} \))(\( ( 1 - 8 T + 984740 T^{2} - 22862616 T^{3} + 463531388774 T^{4} - 8893946288472 T^{5} + 149024865995829860 T^{6} - 470972693666143304 T^{7} + \)\(22\!\cdots\!21\)\( T^{8} )^{2} \))(\( ( 1 - 8 T + 984740 T^{2} - 22862616 T^{3} + 463531388774 T^{4} - 8893946288472 T^{5} + 149024865995829860 T^{6} - 470972693666143304 T^{7} + \)\(22\!\cdots\!21\)\( T^{8} )^{2} \))(\( ( 1 + 148 T + 1578093 T^{2} + 253463696 T^{3} + 1105903969594 T^{4} + 150341055768952 T^{5} + 430215444539549098 T^{6} + 38357732326510304144 T^{7} + \)\(92\!\cdots\!09\)\( T^{8} + \)\(33\!\cdots\!08\)\( T^{9} + \)\(89\!\cdots\!57\)\( T^{10} )^{2} \))(\( ( 1 + 148 T + 1578093 T^{2} + 253463696 T^{3} + 1105903969594 T^{4} + 150341055768952 T^{5} + 430215444539549098 T^{6} + 38357732326510304144 T^{7} + \)\(92\!\cdots\!09\)\( T^{8} + \)\(33\!\cdots\!08\)\( T^{9} + \)\(89\!\cdots\!57\)\( T^{10} )^{2} \))(\( ( 1 + 148 T + 947258 T^{2} - 175820956 T^{3} + 371682340127 T^{4} - 204957420553496 T^{5} + 130376255817064620 T^{6} - 79731920871459353432 T^{7} + \)\(56\!\cdots\!03\)\( T^{8} - \)\(10\!\cdots\!28\)\( T^{9} + \)\(21\!\cdots\!18\)\( T^{10} + \)\(13\!\cdots\!36\)\( T^{11} + \)\(34\!\cdots\!69\)\( T^{12} )^{2} \))(\( ( 1 + 148 T + 947258 T^{2} - 175820956 T^{3} + 371682340127 T^{4} - 204957420553496 T^{5} + 130376255817064620 T^{6} - 79731920871459353432 T^{7} + \)\(56\!\cdots\!03\)\( T^{8} - \)\(10\!\cdots\!28\)\( T^{9} + \)\(21\!\cdots\!18\)\( T^{10} + \)\(13\!\cdots\!36\)\( T^{11} + \)\(34\!\cdots\!69\)\( T^{12} )^{2} \))
$79$ (\( ( 1 + 768 T + 493039 T^{2} )^{2} \))(\( ( 1 + 493039 T^{2} )^{2} \))(\( ( 1 + 493039 T^{2} )^{2} \))(\( ( 1 - 768 T + 493039 T^{2} )^{2} \))(\( 1 + 579260 T^{2} + 173243887814 T^{4} + 140810839485094460 T^{6} + \)\(59\!\cdots\!41\)\( T^{8} \))(\( ( 1 - 1600 T + 1469406 T^{2} - 788862400 T^{3} + 243087455521 T^{4} )^{2} \))(\( ( 1 + 493039 T^{2} )^{4} \))(\( ( 1 + 801758 T^{2} + 243087455521 T^{4} )^{2} \))(\( ( 1 + 1600 T + 1469406 T^{2} + 788862400 T^{3} + 243087455521 T^{4} )^{2} \))(\( 1 + 579260 T^{2} + 173243887814 T^{4} + 140810839485094460 T^{6} + \)\(59\!\cdots\!41\)\( T^{8} \))(\( 1 + 2878840 T^{2} + 3993378831260 T^{4} + 3465442907909496648 T^{6} + \)\(20\!\cdots\!86\)\( T^{8} + \)\(84\!\cdots\!08\)\( T^{10} + \)\(23\!\cdots\!60\)\( T^{12} + \)\(41\!\cdots\!40\)\( T^{14} + \)\(34\!\cdots\!81\)\( T^{16} \))(\( 1 + 2878840 T^{2} + 3993378831260 T^{4} + 3465442907909496648 T^{6} + \)\(20\!\cdots\!86\)\( T^{8} + \)\(84\!\cdots\!08\)\( T^{10} + \)\(23\!\cdots\!60\)\( T^{12} + \)\(41\!\cdots\!40\)\( T^{14} + \)\(34\!\cdots\!81\)\( T^{16} \))(\( ( 1 + 2208 T + 3816107 T^{2} + 4320867712 T^{3} + 4245684014154 T^{4} + 3176789940661184 T^{5} + 2093287800654474006 T^{6} + \)\(10\!\cdots\!52\)\( T^{7} + \)\(45\!\cdots\!33\)\( T^{8} + \)\(13\!\cdots\!28\)\( T^{9} + \)\(29\!\cdots\!99\)\( T^{10} )^{2} \))(\( ( 1 - 2208 T + 3816107 T^{2} - 4320867712 T^{3} + 4245684014154 T^{4} - 3176789940661184 T^{5} + 2093287800654474006 T^{6} - \)\(10\!\cdots\!52\)\( T^{7} + \)\(45\!\cdots\!33\)\( T^{8} - \)\(13\!\cdots\!28\)\( T^{9} + \)\(29\!\cdots\!99\)\( T^{10} )^{2} \))(\( 1 + 1934388 T^{2} + 1804679417986 T^{4} + 836598661037609700 T^{6} - \)\(30\!\cdots\!49\)\( T^{8} - \)\(28\!\cdots\!36\)\( T^{10} - \)\(20\!\cdots\!00\)\( T^{12} - \)\(69\!\cdots\!56\)\( T^{14} - \)\(18\!\cdots\!09\)\( T^{16} + \)\(12\!\cdots\!00\)\( T^{18} + \)\(63\!\cdots\!66\)\( T^{20} + \)\(16\!\cdots\!88\)\( T^{22} + \)\(20\!\cdots\!21\)\( T^{24} \))(\( 1 + 1934388 T^{2} + 1804679417986 T^{4} + 836598661037609700 T^{6} - \)\(30\!\cdots\!49\)\( T^{8} - \)\(28\!\cdots\!36\)\( T^{10} - \)\(20\!\cdots\!00\)\( T^{12} - \)\(69\!\cdots\!56\)\( T^{14} - \)\(18\!\cdots\!09\)\( T^{16} + \)\(12\!\cdots\!00\)\( T^{18} + \)\(63\!\cdots\!66\)\( T^{20} + \)\(16\!\cdots\!88\)\( T^{22} + \)\(20\!\cdots\!21\)\( T^{24} \))
$83$ (\( 1 + 878582 T^{2} + 326940373369 T^{4} \))(\( ( 1 + 571787 T^{2} )^{2} \))(\( ( 1 + 571787 T^{2} )^{2} \))(\( 1 + 878582 T^{2} + 326940373369 T^{4} \))(\( ( 1 + 1490 T + 1505572 T^{2} + 851962630 T^{3} + 326940373369 T^{4} )^{2} \))(\( 1 + 1535492 T^{2} + 1242039742054 T^{4} + 502014327785112548 T^{6} + \)\(10\!\cdots\!61\)\( T^{8} \))(\( ( 1 + 596014 T^{2} + 326940373369 T^{4} )^{2} \))(\( ( 1 + 1142574 T^{2} + 326940373369 T^{4} )^{2} \))(\( 1 + 1535492 T^{2} + 1242039742054 T^{4} + 502014327785112548 T^{6} + \)\(10\!\cdots\!61\)\( T^{8} \))(\( ( 1 - 1490 T + 1505572 T^{2} - 851962630 T^{3} + 326940373369 T^{4} )^{2} \))(\( ( 1 + 2080 T + 3182580 T^{2} + 3073449408 T^{3} + 2675991151978 T^{4} + 1757358416652096 T^{5} + 1040513893476712020 T^{6} + \)\(38\!\cdots\!40\)\( T^{7} + \)\(10\!\cdots\!61\)\( T^{8} )^{2} \))(\( ( 1 - 2080 T + 3182580 T^{2} - 3073449408 T^{3} + 2675991151978 T^{4} - 1757358416652096 T^{5} + 1040513893476712020 T^{6} - \)\(38\!\cdots\!40\)\( T^{7} + \)\(10\!\cdots\!61\)\( T^{8} )^{2} \))(\( 1 + 2541516 T^{2} + 3317268679373 T^{4} + 2806354404035135728 T^{6} + \)\(18\!\cdots\!66\)\( T^{8} + \)\(10\!\cdots\!48\)\( T^{10} + \)\(59\!\cdots\!54\)\( T^{12} + \)\(29\!\cdots\!08\)\( T^{14} + \)\(11\!\cdots\!57\)\( T^{16} + \)\(29\!\cdots\!36\)\( T^{18} + \)\(37\!\cdots\!49\)\( T^{20} \))(\( 1 + 2541516 T^{2} + 3317268679373 T^{4} + 2806354404035135728 T^{6} + \)\(18\!\cdots\!66\)\( T^{8} + \)\(10\!\cdots\!48\)\( T^{10} + \)\(59\!\cdots\!54\)\( T^{12} + \)\(29\!\cdots\!08\)\( T^{14} + \)\(11\!\cdots\!57\)\( T^{16} + \)\(29\!\cdots\!36\)\( T^{18} + \)\(37\!\cdots\!49\)\( T^{20} \))(\( 1 + 2150508 T^{2} + 1900786638394 T^{4} + 815373636219633692 T^{6} + \)\(15\!\cdots\!23\)\( T^{8} + \)\(45\!\cdots\!36\)\( T^{10} + \)\(42\!\cdots\!68\)\( T^{12} + \)\(15\!\cdots\!84\)\( T^{14} + \)\(16\!\cdots\!03\)\( T^{16} + \)\(28\!\cdots\!28\)\( T^{18} + \)\(21\!\cdots\!74\)\( T^{20} + \)\(80\!\cdots\!92\)\( T^{22} + \)\(12\!\cdots\!81\)\( T^{24} \))(\( 1 + 2150508 T^{2} + 1900786638394 T^{4} + 815373636219633692 T^{6} + \)\(15\!\cdots\!23\)\( T^{8} + \)\(45\!\cdots\!36\)\( T^{10} + \)\(42\!\cdots\!68\)\( T^{12} + \)\(15\!\cdots\!84\)\( T^{14} + \)\(16\!\cdots\!03\)\( T^{16} + \)\(28\!\cdots\!28\)\( T^{18} + \)\(21\!\cdots\!74\)\( T^{20} + \)\(80\!\cdots\!92\)\( T^{22} + \)\(12\!\cdots\!81\)\( T^{24} \))
$89$ (\( ( 1 + 1306 T + 704969 T^{2} )^{2} \))(\( ( 1 - 1670 T + 704969 T^{2} )^{2} \))(\( ( 1 - 1670 T + 704969 T^{2} )^{2} \))(\( ( 1 + 1306 T + 704969 T^{2} )^{2} \))(\( ( 1 + 268 T + 1365802 T^{2} + 188931692 T^{3} + 496981290961 T^{4} )^{2} \))(\( ( 1 - 580 T + 1472006 T^{2} - 408882020 T^{3} + 496981290961 T^{4} )^{2} \))(\( ( 1 + 1410 T + 704969 T^{2} )^{4} \))(\( ( 1 - 510 T + 704969 T^{2} )^{4} \))(\( ( 1 - 580 T + 1472006 T^{2} - 408882020 T^{3} + 496981290961 T^{4} )^{2} \))(\( ( 1 + 268 T + 1365802 T^{2} + 188931692 T^{3} + 496981290961 T^{4} )^{2} \))(\( ( 1 - 136 T + 2100324 T^{2} - 253838616 T^{3} + 2087259799462 T^{4} - 178948355282904 T^{5} + 1043821732956371364 T^{6} - 47648470904217988424 T^{7} + \)\(24\!\cdots\!21\)\( T^{8} )^{2} \))(\( ( 1 - 136 T + 2100324 T^{2} - 253838616 T^{3} + 2087259799462 T^{4} - 178948355282904 T^{5} + 1043821732956371364 T^{6} - 47648470904217988424 T^{7} + \)\(24\!\cdots\!21\)\( T^{8} )^{2} \))(\( ( 1 - 44 T + 1822557 T^{2} - 625587056 T^{3} + 1619388326906 T^{4} - 839474353817096 T^{5} + 1141618569430595914 T^{6} - \)\(31\!\cdots\!16\)\( T^{7} + \)\(63\!\cdots\!13\)\( T^{8} - \)\(10\!\cdots\!24\)\( T^{9} + \)\(17\!\cdots\!49\)\( T^{10} )^{2} \))(\( ( 1 - 44 T + 1822557 T^{2} - 625587056 T^{3} + 1619388326906 T^{4} - 839474353817096 T^{5} + 1141618569430595914 T^{6} - \)\(31\!\cdots\!16\)\( T^{7} + \)\(63\!\cdots\!13\)\( T^{8} - \)\(10\!\cdots\!24\)\( T^{9} + \)\(17\!\cdots\!49\)\( T^{10} )^{2} \))(\( ( 1 - 44 T + 1702554 T^{2} - 353288092 T^{3} + 2168072947391 T^{4} - 489761622555544 T^{5} + 1662469057460846188 T^{6} - \)\(34\!\cdots\!36\)\( T^{7} + \)\(10\!\cdots\!51\)\( T^{8} - \)\(12\!\cdots\!28\)\( T^{9} + \)\(42\!\cdots\!34\)\( T^{10} - \)\(76\!\cdots\!56\)\( T^{11} + \)\(12\!\cdots\!81\)\( T^{12} )^{2} \))(\( ( 1 - 44 T + 1702554 T^{2} - 353288092 T^{3} + 2168072947391 T^{4} - 489761622555544 T^{5} + 1662469057460846188 T^{6} - \)\(34\!\cdots\!36\)\( T^{7} + \)\(10\!\cdots\!51\)\( T^{8} - \)\(12\!\cdots\!28\)\( T^{9} + \)\(42\!\cdots\!34\)\( T^{10} - \)\(76\!\cdots\!56\)\( T^{11} + \)\(12\!\cdots\!81\)\( T^{12} )^{2} \))
$97$ (\( ( 1 + 472 T + 912673 T^{2} )^{2} \))(\( ( 1 - 1816 T + 912673 T^{2} )^{2} \))(\( ( 1 + 1816 T + 912673 T^{2} )^{2} \))(\( ( 1 + 472 T + 912673 T^{2} )^{2} \))(\( ( 1 + 1120 T + 1994294 T^{2} + 1022193760 T^{3} + 832972004929 T^{4} )^{2} \))(\( ( 1 - 280 T + 1342018 T^{2} - 255548440 T^{3} + 832972004929 T^{4} )^{2} \))(\( ( 1 - 464 T + 912673 T^{2} )^{4} \))(\( ( 1 - 1216 T + 912673 T^{2} )^{4} \))(\( ( 1 - 280 T + 1342018 T^{2} - 255548440 T^{3} + 832972004929 T^{4} )^{2} \))(\( ( 1 + 1120 T + 1994294 T^{2} + 1022193760 T^{3} + 832972004929 T^{4} )^{2} \))(\( ( 1 - 1120 T + 2386796 T^{2} - 1138826400 T^{3} + 2110524053990 T^{4} - 1039376106967200 T^{5} + 1988134249476517484 T^{6} - \)\(85\!\cdots\!40\)\( T^{7} + \)\(69\!\cdots\!41\)\( T^{8} )^{2} \))(\( ( 1 - 1120 T + 2386796 T^{2} - 1138826400 T^{3} + 2110524053990 T^{4} - 1039376106967200 T^{5} + 1988134249476517484 T^{6} - \)\(85\!\cdots\!40\)\( T^{7} + \)\(69\!\cdots\!41\)\( T^{8} )^{2} \))(\( ( 1 + 2 T + 2565789 T^{2} - 723000584 T^{3} + 3231145430658 T^{4} - 1257303020547316 T^{5} + 2948979193634928834 T^{6} - \)\(60\!\cdots\!36\)\( T^{7} + \)\(19\!\cdots\!13\)\( T^{8} + \)\(13\!\cdots\!82\)\( T^{9} + \)\(63\!\cdots\!93\)\( T^{10} )^{2} \))(\( ( 1 + 2 T + 2565789 T^{2} - 723000584 T^{3} + 3231145430658 T^{4} - 1257303020547316 T^{5} + 2948979193634928834 T^{6} - \)\(60\!\cdots\!36\)\( T^{7} + \)\(19\!\cdots\!13\)\( T^{8} + \)\(13\!\cdots\!82\)\( T^{9} + \)\(63\!\cdots\!93\)\( T^{10} )^{2} \))(\( ( 1 + 1488 T + 3210350 T^{2} + 3035978256 T^{3} + 3528379264303 T^{4} + 2145951493961504 T^{5} + 2614419569567271172 T^{6} + \)\(19\!\cdots\!92\)\( T^{7} + \)\(29\!\cdots\!87\)\( T^{8} + \)\(23\!\cdots\!52\)\( T^{9} + \)\(22\!\cdots\!50\)\( T^{10} + \)\(94\!\cdots\!84\)\( T^{11} + \)\(57\!\cdots\!89\)\( T^{12} )^{2} \))(\( ( 1 + 1488 T + 3210350 T^{2} + 3035978256 T^{3} + 3528379264303 T^{4} + 2145951493961504 T^{5} + 2614419569567271172 T^{6} + \)\(19\!\cdots\!92\)\( T^{7} + \)\(29\!\cdots\!87\)\( T^{8} + \)\(23\!\cdots\!52\)\( T^{9} + \)\(22\!\cdots\!50\)\( T^{10} + \)\(94\!\cdots\!84\)\( T^{11} + \)\(57\!\cdots\!89\)\( T^{12} )^{2} \))
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