Properties

Label 525.2.t
Level 525
Weight 2
Character orbit t
Rep. character \(\chi_{525}(26,\cdot)\)
Character field \(\Q(\zeta_{6})\)
Dimension 90
Newform subspaces 10
Sturm bound 160
Trace bound 7

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Defining parameters

Level: \( N \) = \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 525.t (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 21 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 10 \)
Sturm bound: \(160\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(2\), \(13\), \(37\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(525, [\chi])\).

Total New Old
Modular forms 184 114 70
Cusp forms 136 90 46
Eisenstein series 48 24 24

Trace form

\( 90q + 3q^{3} + 42q^{4} + 5q^{7} - 3q^{9} + O(q^{10}) \) \( 90q + 3q^{3} + 42q^{4} + 5q^{7} - 3q^{9} + 24q^{12} - 48q^{16} + 2q^{18} - 3q^{19} - 6q^{21} + 56q^{22} + 36q^{24} - 12q^{28} + 15q^{31} - 24q^{33} - 12q^{36} - 5q^{37} - 3q^{39} - 30q^{42} - 26q^{43} + 9q^{49} + 18q^{51} - 42q^{52} - 90q^{54} - 54q^{57} + 32q^{58} - 66q^{61} + 19q^{63} - 144q^{64} - 72q^{66} - 23q^{67} + 10q^{72} - 15q^{73} + 48q^{78} - 3q^{79} + 21q^{81} - 72q^{82} + 36q^{84} + 6q^{87} + 16q^{88} - 93q^{91} - 27q^{93} + 36q^{94} + 162q^{96} + 132q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(525, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
525.2.t.a \(2\) \(4.192\) \(\Q(\sqrt{-3}) \) None \(-3\) \(0\) \(0\) \(5\) \(q+(-2+\zeta_{6})q^{2}+(-1+2\zeta_{6})q^{3}+(1+\cdots)q^{4}+\cdots\)
525.2.t.b \(2\) \(4.192\) \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(-3\) \(0\) \(-4\) \(q+(-1-\zeta_{6})q^{3}+(-2+2\zeta_{6})q^{4}+(-3+\cdots)q^{7}+\cdots\)
525.2.t.c \(2\) \(4.192\) \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(3\) \(0\) \(-1\) \(q+(1+\zeta_{6})q^{3}+(-2+2\zeta_{6})q^{4}+(-2+\cdots)q^{7}+\cdots\)
525.2.t.d \(2\) \(4.192\) \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(3\) \(0\) \(4\) \(q+(1+\zeta_{6})q^{3}+(-2+2\zeta_{6})q^{4}+(3-2\zeta_{6})q^{7}+\cdots\)
525.2.t.e \(2\) \(4.192\) \(\Q(\sqrt{-3}) \) None \(3\) \(3\) \(0\) \(5\) \(q+(2-\zeta_{6})q^{2}+(2-\zeta_{6})q^{3}+(1-\zeta_{6})q^{4}+\cdots\)
525.2.t.f \(8\) \(4.192\) 8.0.856615824.2 None \(-3\) \(-2\) \(0\) \(-2\) \(q+(\beta _{1}+\beta _{3})q^{2}+(-1-\beta _{1}-\beta _{3}-\beta _{7})q^{3}+\cdots\)
525.2.t.g \(8\) \(4.192\) 8.0.856615824.2 None \(3\) \(-1\) \(0\) \(-2\) \(q-\beta _{3}q^{2}+(-1-\beta _{1}-\beta _{5}-\beta _{7})q^{3}+\cdots\)
525.2.t.h \(20\) \(4.192\) \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None \(0\) \(-3\) \(0\) \(0\) \(q+\beta _{3}q^{2}-\beta _{1}q^{3}+(1-\beta _{1}-\beta _{4}+\beta _{11}+\cdots)q^{4}+\cdots\)
525.2.t.i \(20\) \(4.192\) \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None \(0\) \(3\) \(0\) \(0\) \(q-\beta _{2}q^{2}+\beta _{4}q^{3}+(\beta _{5}-\beta _{11})q^{4}+(-\beta _{1}+\cdots)q^{6}+\cdots\)
525.2.t.j \(24\) \(4.192\) None \(0\) \(0\) \(0\) \(0\)

Decomposition of \(S_{2}^{\mathrm{old}}(525, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(525, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 3 T + 5 T^{2} + 6 T^{3} + 4 T^{4} \))(\( 1 + 2 T^{2} + 4 T^{4} \))(\( 1 + 2 T^{2} + 4 T^{4} \))(\( 1 + 2 T^{2} + 4 T^{4} \))(\( 1 - 3 T + 5 T^{2} - 6 T^{3} + 4 T^{4} \))(\( 1 + 3 T + 7 T^{2} + 12 T^{3} + 16 T^{4} + 12 T^{5} - 8 T^{6} - 36 T^{7} - 68 T^{8} - 72 T^{9} - 32 T^{10} + 96 T^{11} + 256 T^{12} + 384 T^{13} + 448 T^{14} + 384 T^{15} + 256 T^{16} \))(\( 1 - 3 T + 7 T^{2} - 12 T^{3} + 16 T^{4} - 12 T^{5} - 8 T^{6} + 36 T^{7} - 68 T^{8} + 72 T^{9} - 32 T^{10} - 96 T^{11} + 256 T^{12} - 384 T^{13} + 448 T^{14} - 384 T^{15} + 256 T^{16} \))(\( 1 + 3 T^{2} + 2 T^{4} + 5 T^{6} + 9 T^{8} - 29 T^{10} + 47 T^{12} + 294 T^{14} + 40 T^{16} - 144 T^{18} + 1017 T^{20} - 576 T^{22} + 640 T^{24} + 18816 T^{26} + 12032 T^{28} - 29696 T^{30} + 36864 T^{32} + 81920 T^{34} + 131072 T^{36} + 786432 T^{38} + 1048576 T^{40} \))(\( 1 + 3 T^{2} + 2 T^{4} + 5 T^{6} + 9 T^{8} - 29 T^{10} + 47 T^{12} + 294 T^{14} + 40 T^{16} - 144 T^{18} + 1017 T^{20} - 576 T^{22} + 640 T^{24} + 18816 T^{26} + 12032 T^{28} - 29696 T^{30} + 36864 T^{32} + 81920 T^{34} + 131072 T^{36} + 786432 T^{38} + 1048576 T^{40} \))
$3$ (\( 1 + 3 T^{2} \))(\( 1 + 3 T + 3 T^{2} \))(\( 1 - 3 T + 3 T^{2} \))(\( 1 - 3 T + 3 T^{2} \))(\( 1 - 3 T + 3 T^{2} \))(\( 1 + 2 T - 4 T^{3} - 11 T^{4} - 12 T^{5} + 54 T^{7} + 81 T^{8} \))(\( 1 + T + 3 T^{2} + 4 T^{3} + 16 T^{4} + 12 T^{5} + 27 T^{6} + 27 T^{7} + 81 T^{8} \))(\( 1 + 3 T + 8 T^{2} + 15 T^{3} + 18 T^{4} + 45 T^{5} + 59 T^{6} + 147 T^{7} + 271 T^{8} + 330 T^{9} + 879 T^{10} + 990 T^{11} + 2439 T^{12} + 3969 T^{13} + 4779 T^{14} + 10935 T^{15} + 13122 T^{16} + 32805 T^{17} + 52488 T^{18} + 59049 T^{19} + 59049 T^{20} \))(\( 1 - 3 T + 8 T^{2} - 15 T^{3} + 18 T^{4} - 45 T^{5} + 59 T^{6} - 147 T^{7} + 271 T^{8} - 330 T^{9} + 879 T^{10} - 990 T^{11} + 2439 T^{12} - 3969 T^{13} + 4779 T^{14} - 10935 T^{15} + 13122 T^{16} - 32805 T^{17} + 52488 T^{18} - 59049 T^{19} + 59049 T^{20} \))
$5$ 1
$7$ (\( 1 - 5 T + 7 T^{2} \))(\( 1 + 4 T + 7 T^{2} \))(\( 1 + T + 7 T^{2} \))(\( 1 - 4 T + 7 T^{2} \))(\( 1 - 5 T + 7 T^{2} \))(\( 1 + 2 T + 4 T^{2} - 10 T^{3} - 41 T^{4} - 70 T^{5} + 196 T^{6} + 686 T^{7} + 2401 T^{8} \))(\( 1 + 2 T + 4 T^{2} - 10 T^{3} - 41 T^{4} - 70 T^{5} + 196 T^{6} + 686 T^{7} + 2401 T^{8} \))(\( ( 1 + T^{2} + 13 T^{3} + 7 T^{4} + 133 T^{5} + 49 T^{6} + 637 T^{7} + 343 T^{8} + 16807 T^{10} )^{2} \))(\( ( 1 + T^{2} - 13 T^{3} + 7 T^{4} - 133 T^{5} + 49 T^{6} - 637 T^{7} + 343 T^{8} + 16807 T^{10} )^{2} \))
$11$ (\( 1 - 6 T + 23 T^{2} - 66 T^{3} + 121 T^{4} \))(\( 1 + 11 T^{2} + 121 T^{4} \))(\( 1 + 11 T^{2} + 121 T^{4} \))(\( 1 + 11 T^{2} + 121 T^{4} \))(\( 1 + 6 T + 23 T^{2} + 66 T^{3} + 121 T^{4} \))(\( 1 + 16 T^{2} - 2 T^{4} - 30 T^{5} + 268 T^{6} - 1548 T^{7} + 21079 T^{8} - 17028 T^{9} + 32428 T^{10} - 39930 T^{11} - 29282 T^{12} + 28344976 T^{14} + 214358881 T^{16} \))(\( 1 + 16 T^{2} - 2 T^{4} + 30 T^{5} + 268 T^{6} + 1548 T^{7} + 21079 T^{8} + 17028 T^{9} + 32428 T^{10} + 39930 T^{11} - 29282 T^{12} + 28344976 T^{14} + 214358881 T^{16} \))(\( 1 + 60 T^{2} + 2114 T^{4} + 53114 T^{6} + 1038747 T^{8} + 16310488 T^{10} + 209475293 T^{12} + 2203457790 T^{14} + 19212402091 T^{16} + 148735009794 T^{18} + 1341299566641 T^{20} + 17996936185074 T^{22} + 281288779014331 T^{24} + 3903559885910190 T^{26} + 44902889404627133 T^{28} + 423052052705515288 T^{30} + 3260033061033808587 T^{32} + 20170032660940262474 T^{34} + 97137728931591548354 T^{36} + \)\(33\!\cdots\!60\)\( T^{38} + \)\(67\!\cdots\!01\)\( T^{40} \))(\( 1 + 60 T^{2} + 2114 T^{4} + 53114 T^{6} + 1038747 T^{8} + 16310488 T^{10} + 209475293 T^{12} + 2203457790 T^{14} + 19212402091 T^{16} + 148735009794 T^{18} + 1341299566641 T^{20} + 17996936185074 T^{22} + 281288779014331 T^{24} + 3903559885910190 T^{26} + 44902889404627133 T^{28} + 423052052705515288 T^{30} + 3260033061033808587 T^{32} + 20170032660940262474 T^{34} + 97137728931591548354 T^{36} + \)\(33\!\cdots\!60\)\( T^{38} + \)\(67\!\cdots\!01\)\( T^{40} \))
$13$ (\( 1 - 14 T^{2} + 169 T^{4} \))(\( ( 1 - 2 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} ) \))(\( ( 1 - 7 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} ) \))(\( ( 1 - 2 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} ) \))(\( 1 - 14 T^{2} + 169 T^{4} \))(\( 1 - 83 T^{2} + 3217 T^{4} - 76058 T^{6} + 1197778 T^{8} - 12853802 T^{10} + 91880737 T^{12} - 400625147 T^{14} + 815730721 T^{16} \))(\( 1 - 83 T^{2} + 3217 T^{4} - 76058 T^{6} + 1197778 T^{8} - 12853802 T^{10} + 91880737 T^{12} - 400625147 T^{14} + 815730721 T^{16} \))(\( ( 1 - 58 T^{2} + 2067 T^{4} - 50385 T^{6} + 943419 T^{8} - 13724229 T^{10} + 159437811 T^{12} - 1439045985 T^{14} + 9977014203 T^{16} - 47312381818 T^{18} + 137858491849 T^{20} )^{2} \))(\( ( 1 - 58 T^{2} + 2067 T^{4} - 50385 T^{6} + 943419 T^{8} - 13724229 T^{10} + 159437811 T^{12} - 1439045985 T^{14} + 9977014203 T^{16} - 47312381818 T^{18} + 137858491849 T^{20} )^{2} \))
$17$ (\( 1 + 6 T + 19 T^{2} + 102 T^{3} + 289 T^{4} \))(\( 1 - 17 T^{2} + 289 T^{4} \))(\( 1 - 17 T^{2} + 289 T^{4} \))(\( 1 - 17 T^{2} + 289 T^{4} \))(\( 1 - 6 T + 19 T^{2} - 102 T^{3} + 289 T^{4} \))(\( 1 + 12 T + 34 T^{2} + 12 T^{3} + 1078 T^{4} + 6882 T^{5} + 8740 T^{6} + 70272 T^{7} + 637627 T^{8} + 1194624 T^{9} + 2525860 T^{10} + 33811266 T^{11} + 90035638 T^{12} + 17038284 T^{13} + 820677346 T^{14} + 4924064076 T^{15} + 6975757441 T^{16} \))(\( 1 - 12 T + 34 T^{2} - 12 T^{3} + 1078 T^{4} - 6882 T^{5} + 8740 T^{6} - 70272 T^{7} + 637627 T^{8} - 1194624 T^{9} + 2525860 T^{10} - 33811266 T^{11} + 90035638 T^{12} - 17038284 T^{13} + 820677346 T^{14} - 4924064076 T^{15} + 6975757441 T^{16} \))(\( 1 - 68 T^{2} + 1606 T^{4} - 19422 T^{6} + 550335 T^{8} - 18001236 T^{10} + 282479529 T^{12} - 4249504470 T^{14} + 113206321539 T^{16} - 2028696191474 T^{18} + 28292432369185 T^{20} - 586293199335986 T^{22} + 9455105181258819 T^{24} - 102572707360433430 T^{26} + 1970508676351925289 T^{28} - 36290381976542954964 T^{30} + \)\(32\!\cdots\!35\)\( T^{32} - \)\(32\!\cdots\!38\)\( T^{34} + \)\(78\!\cdots\!86\)\( T^{36} - \)\(95\!\cdots\!12\)\( T^{38} + \)\(40\!\cdots\!01\)\( T^{40} \))(\( 1 - 68 T^{2} + 1606 T^{4} - 19422 T^{6} + 550335 T^{8} - 18001236 T^{10} + 282479529 T^{12} - 4249504470 T^{14} + 113206321539 T^{16} - 2028696191474 T^{18} + 28292432369185 T^{20} - 586293199335986 T^{22} + 9455105181258819 T^{24} - 102572707360433430 T^{26} + 1970508676351925289 T^{28} - 36290381976542954964 T^{30} + \)\(32\!\cdots\!35\)\( T^{32} - \)\(32\!\cdots\!38\)\( T^{34} + \)\(78\!\cdots\!86\)\( T^{36} - \)\(95\!\cdots\!12\)\( T^{38} + \)\(40\!\cdots\!01\)\( T^{40} \))
$19$ (\( 1 + 12 T + 67 T^{2} + 228 T^{3} + 361 T^{4} \))(\( ( 1 - 7 T + 19 T^{2} )( 1 + T + 19 T^{2} ) \))(\( ( 1 + T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \))(\( ( 1 - 7 T + 19 T^{2} )( 1 + T + 19 T^{2} ) \))(\( 1 + 12 T + 67 T^{2} + 228 T^{3} + 361 T^{4} \))(\( 1 - 9 T + 100 T^{2} - 657 T^{3} + 4723 T^{4} - 26244 T^{5} + 148996 T^{6} - 704196 T^{7} + 3331528 T^{8} - 13379724 T^{9} + 53787556 T^{10} - 180007596 T^{11} + 615506083 T^{12} - 1626797043 T^{13} + 4704588100 T^{14} - 8044845651 T^{15} + 16983563041 T^{16} \))(\( 1 - 9 T + 100 T^{2} - 657 T^{3} + 4723 T^{4} - 26244 T^{5} + 148996 T^{6} - 704196 T^{7} + 3331528 T^{8} - 13379724 T^{9} + 53787556 T^{10} - 180007596 T^{11} + 615506083 T^{12} - 1626797043 T^{13} + 4704588100 T^{14} - 8044845651 T^{15} + 16983563041 T^{16} \))(\( ( 1 + 59 T^{2} + 1650 T^{4} + 38478 T^{6} + 13230 T^{7} + 896922 T^{8} + 780570 T^{9} + 18506640 T^{10} + 14830830 T^{11} + 323788842 T^{12} + 90744570 T^{13} + 5014491438 T^{14} + 77625703650 T^{16} + 1002030219419 T^{18} + 6131066257801 T^{20} )^{2} \))(\( ( 1 + 59 T^{2} + 1650 T^{4} + 38478 T^{6} + 13230 T^{7} + 896922 T^{8} + 780570 T^{9} + 18506640 T^{10} + 14830830 T^{11} + 323788842 T^{12} + 90744570 T^{13} + 5014491438 T^{14} + 77625703650 T^{16} + 1002030219419 T^{18} + 6131066257801 T^{20} )^{2} \))
$23$ (\( 1 - 3 T + 26 T^{2} - 69 T^{3} + 529 T^{4} \))(\( 1 + 23 T^{2} + 529 T^{4} \))(\( 1 + 23 T^{2} + 529 T^{4} \))(\( 1 + 23 T^{2} + 529 T^{4} \))(\( 1 + 3 T + 26 T^{2} + 69 T^{3} + 529 T^{4} \))(\( 1 - 27 T + 391 T^{2} - 3996 T^{3} + 31651 T^{4} - 205875 T^{5} + 1157938 T^{6} - 5917779 T^{7} + 28782226 T^{8} - 136108917 T^{9} + 612549202 T^{10} - 2504881125 T^{11} + 8857247491 T^{12} - 25719626628 T^{13} + 57882032599 T^{14} - 91930287069 T^{15} + 78310985281 T^{16} \))(\( 1 + 27 T + 391 T^{2} + 3996 T^{3} + 31651 T^{4} + 205875 T^{5} + 1157938 T^{6} + 5917779 T^{7} + 28782226 T^{8} + 136108917 T^{9} + 612549202 T^{10} + 2504881125 T^{11} + 8857247491 T^{12} + 25719626628 T^{13} + 57882032599 T^{14} + 91930287069 T^{15} + 78310985281 T^{16} \))(\( 1 + 99 T^{2} + 4250 T^{4} + 102299 T^{6} + 1576890 T^{8} + 17142031 T^{10} + 181616261 T^{12} + 10191303513 T^{14} + 591206119363 T^{16} + 19834960319622 T^{18} + 492118045969095 T^{20} + 10492694009080038 T^{22} + 165443711648661283 T^{24} + 1508678675615778057 T^{26} + 14222548341961254341 T^{28} + \)\(71\!\cdots\!19\)\( T^{30} + \)\(34\!\cdots\!90\)\( T^{32} + \)\(11\!\cdots\!91\)\( T^{34} + \)\(26\!\cdots\!50\)\( T^{36} + \)\(32\!\cdots\!31\)\( T^{38} + \)\(17\!\cdots\!01\)\( T^{40} \))(\( 1 + 99 T^{2} + 4250 T^{4} + 102299 T^{6} + 1576890 T^{8} + 17142031 T^{10} + 181616261 T^{12} + 10191303513 T^{14} + 591206119363 T^{16} + 19834960319622 T^{18} + 492118045969095 T^{20} + 10492694009080038 T^{22} + 165443711648661283 T^{24} + 1508678675615778057 T^{26} + 14222548341961254341 T^{28} + \)\(71\!\cdots\!19\)\( T^{30} + \)\(34\!\cdots\!90\)\( T^{32} + \)\(11\!\cdots\!91\)\( T^{34} + \)\(26\!\cdots\!50\)\( T^{36} + \)\(32\!\cdots\!31\)\( T^{38} + \)\(17\!\cdots\!01\)\( T^{40} \))
$29$ (\( 1 - 55 T^{2} + 841 T^{4} \))(\( ( 1 - 29 T^{2} )^{2} \))(\( ( 1 - 29 T^{2} )^{2} \))(\( ( 1 - 29 T^{2} )^{2} \))(\( 1 - 55 T^{2} + 841 T^{4} \))(\( 1 - 53 T^{2} + 3250 T^{4} - 128951 T^{6} + 4063174 T^{8} - 108447791 T^{10} + 2298663250 T^{12} - 31525636013 T^{14} + 500246412961 T^{16} \))(\( 1 - 53 T^{2} + 3250 T^{4} - 128951 T^{6} + 4063174 T^{8} - 108447791 T^{10} + 2298663250 T^{12} - 31525636013 T^{14} + 500246412961 T^{16} \))(\( ( 1 - 132 T^{2} + 8782 T^{4} - 413831 T^{6} + 16000213 T^{8} - 513486626 T^{10} + 13456179133 T^{12} - 292694803511 T^{14} + 5223738405022 T^{16} - 66032526510852 T^{18} + 420707233300201 T^{20} )^{2} \))(\( ( 1 - 132 T^{2} + 8782 T^{4} - 413831 T^{6} + 16000213 T^{8} - 513486626 T^{10} + 13456179133 T^{12} - 292694803511 T^{14} + 5223738405022 T^{16} - 66032526510852 T^{18} + 420707233300201 T^{20} )^{2} \))
$31$ (\( 1 + 6 T + 43 T^{2} + 186 T^{3} + 961 T^{4} \))(\( ( 1 + 4 T + 31 T^{2} )( 1 + 11 T + 31 T^{2} ) \))(\( ( 1 - 11 T + 31 T^{2} )( 1 - 4 T + 31 T^{2} ) \))(\( ( 1 + 4 T + 31 T^{2} )( 1 + 11 T + 31 T^{2} ) \))(\( 1 + 6 T + 43 T^{2} + 186 T^{3} + 961 T^{4} \))(\( 1 + 21 T + 262 T^{2} + 2415 T^{3} + 17293 T^{4} + 101304 T^{5} + 505090 T^{6} + 2328618 T^{7} + 11769748 T^{8} + 72187158 T^{9} + 485391490 T^{10} + 3017947464 T^{11} + 15970448653 T^{12} + 69139399665 T^{13} + 232525964422 T^{14} + 577764896331 T^{15} + 852891037441 T^{16} \))(\( 1 + 21 T + 262 T^{2} + 2415 T^{3} + 17293 T^{4} + 101304 T^{5} + 505090 T^{6} + 2328618 T^{7} + 11769748 T^{8} + 72187158 T^{9} + 485391490 T^{10} + 3017947464 T^{11} + 15970448653 T^{12} + 69139399665 T^{13} + 232525964422 T^{14} + 577764896331 T^{15} + 852891037441 T^{16} \))(\( ( 1 - 21 T + 326 T^{2} - 3759 T^{3} + 37146 T^{4} - 316851 T^{5} + 2454105 T^{6} - 17262699 T^{7} + 113223339 T^{8} - 689960868 T^{9} + 3971445087 T^{10} - 21388786908 T^{11} + 108807628779 T^{12} - 514273065909 T^{13} + 2266417503705 T^{14} - 9071175123501 T^{15} + 32967211734426 T^{16} - 103419916443249 T^{17} + 278042478205766 T^{18} - 555232065374091 T^{19} + 819628286980801 T^{20} )^{2} \))(\( ( 1 - 21 T + 326 T^{2} - 3759 T^{3} + 37146 T^{4} - 316851 T^{5} + 2454105 T^{6} - 17262699 T^{7} + 113223339 T^{8} - 689960868 T^{9} + 3971445087 T^{10} - 21388786908 T^{11} + 108807628779 T^{12} - 514273065909 T^{13} + 2266417503705 T^{14} - 9071175123501 T^{15} + 32967211734426 T^{16} - 103419916443249 T^{17} + 278042478205766 T^{18} - 555232065374091 T^{19} + 819628286980801 T^{20} )^{2} \))
$37$ (\( 1 - 4 T - 21 T^{2} - 148 T^{3} + 1369 T^{4} \))(\( ( 1 + T + 37 T^{2} )( 1 + 10 T + 37 T^{2} ) \))(\( ( 1 - 11 T + 37 T^{2} )( 1 + 10 T + 37 T^{2} ) \))(\( ( 1 - 10 T + 37 T^{2} )( 1 - T + 37 T^{2} ) \))(\( 1 - 4 T - 21 T^{2} - 148 T^{3} + 1369 T^{4} \))(\( 1 + 7 T - 24 T^{2} - 493 T^{3} - 973 T^{4} + 16188 T^{5} + 118336 T^{6} - 258098 T^{7} - 5756772 T^{8} - 9549626 T^{9} + 162001984 T^{10} + 819970764 T^{11} - 1823558653 T^{12} - 34186570801 T^{13} - 61577433816 T^{14} + 664523139931 T^{15} + 3512479453921 T^{16} \))(\( 1 + 7 T - 24 T^{2} - 493 T^{3} - 973 T^{4} + 16188 T^{5} + 118336 T^{6} - 258098 T^{7} - 5756772 T^{8} - 9549626 T^{9} + 162001984 T^{10} + 819970764 T^{11} - 1823558653 T^{12} - 34186570801 T^{13} - 61577433816 T^{14} + 664523139931 T^{15} + 3512479453921 T^{16} \))(\( ( 1 + 12 T - 49 T^{2} - 490 T^{3} + 7278 T^{4} + 25228 T^{5} - 430924 T^{6} - 718902 T^{7} + 17278462 T^{8} - 5308020 T^{9} - 787415472 T^{10} - 196396740 T^{11} + 23654214478 T^{12} - 36414543006 T^{13} - 807620954764 T^{14} + 1749409347196 T^{15} + 18673356804702 T^{16} - 46516619795170 T^{17} - 172111493242129 T^{18} + 1559540877540924 T^{19} + 4808584372417849 T^{20} )^{2} \))(\( ( 1 - 12 T - 49 T^{2} + 490 T^{3} + 7278 T^{4} - 25228 T^{5} - 430924 T^{6} + 718902 T^{7} + 17278462 T^{8} + 5308020 T^{9} - 787415472 T^{10} + 196396740 T^{11} + 23654214478 T^{12} + 36414543006 T^{13} - 807620954764 T^{14} - 1749409347196 T^{15} + 18673356804702 T^{16} + 46516619795170 T^{17} - 172111493242129 T^{18} - 1559540877540924 T^{19} + 4808584372417849 T^{20} )^{2} \))
$41$ (\( ( 1 - 3 T + 41 T^{2} )^{2} \))(\( ( 1 + 41 T^{2} )^{2} \))(\( ( 1 + 41 T^{2} )^{2} \))(\( ( 1 + 41 T^{2} )^{2} \))(\( ( 1 + 3 T + 41 T^{2} )^{2} \))(\( ( 1 - 15 T + 218 T^{2} - 1791 T^{3} + 14136 T^{4} - 73431 T^{5} + 366458 T^{6} - 1033815 T^{7} + 2825761 T^{8} )^{2} \))(\( ( 1 + 15 T + 218 T^{2} + 1791 T^{3} + 14136 T^{4} + 73431 T^{5} + 366458 T^{6} + 1033815 T^{7} + 2825761 T^{8} )^{2} \))(\( ( 1 + 149 T^{2} + 12123 T^{4} + 648375 T^{6} + 26231484 T^{8} + 1031235516 T^{10} + 44095124604 T^{12} + 1832152788375 T^{14} + 57585513713643 T^{16} + 1189753859139029 T^{18} + 13422659310152401 T^{20} )^{2} \))(\( ( 1 + 149 T^{2} + 12123 T^{4} + 648375 T^{6} + 26231484 T^{8} + 1031235516 T^{10} + 44095124604 T^{12} + 1832152788375 T^{14} + 57585513713643 T^{16} + 1189753859139029 T^{18} + 13422659310152401 T^{20} )^{2} \))
$43$ (\( ( 1 + T + 43 T^{2} )^{2} \))(\( ( 1 - 5 T + 43 T^{2} )^{2} \))(\( ( 1 - 5 T + 43 T^{2} )^{2} \))(\( ( 1 + 5 T + 43 T^{2} )^{2} \))(\( ( 1 + T + 43 T^{2} )^{2} \))(\( ( 1 + 8 T + 184 T^{2} + 1022 T^{3} + 12127 T^{4} + 43946 T^{5} + 340216 T^{6} + 636056 T^{7} + 3418801 T^{8} )^{2} \))(\( ( 1 + 8 T + 184 T^{2} + 1022 T^{3} + 12127 T^{4} + 43946 T^{5} + 340216 T^{6} + 636056 T^{7} + 3418801 T^{8} )^{2} \))(\( ( 1 + 9 T + 103 T^{2} + 589 T^{3} + 6250 T^{4} + 34300 T^{5} + 268750 T^{6} + 1089061 T^{7} + 8189221 T^{8} + 30769209 T^{9} + 147008443 T^{10} )^{4} \))(\( ( 1 - 9 T + 103 T^{2} - 589 T^{3} + 6250 T^{4} - 34300 T^{5} + 268750 T^{6} - 1089061 T^{7} + 8189221 T^{8} - 30769209 T^{9} + 147008443 T^{10} )^{4} \))
$47$ (\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 + 6 T - 116 T^{2} - 252 T^{3} + 10126 T^{4} + 1986 T^{5} - 595736 T^{6} - 157218 T^{7} + 25623007 T^{8} - 7389246 T^{9} - 1315980824 T^{10} + 206192478 T^{11} + 49411649806 T^{12} - 57794941764 T^{13} - 1250388978164 T^{14} + 3039738722778 T^{15} + 23811286661761 T^{16} \))(\( 1 - 6 T - 116 T^{2} + 252 T^{3} + 10126 T^{4} - 1986 T^{5} - 595736 T^{6} + 157218 T^{7} + 25623007 T^{8} + 7389246 T^{9} - 1315980824 T^{10} - 206192478 T^{11} + 49411649806 T^{12} + 57794941764 T^{13} - 1250388978164 T^{14} - 3039738722778 T^{15} + 23811286661761 T^{16} \))(\( 1 - 320 T^{2} + 55258 T^{4} - 6481134 T^{6} + 564305583 T^{8} - 37299364464 T^{10} + 1825870594845 T^{12} - 57025827049134 T^{14} + 130434887196171 T^{16} + 112418780517368770 T^{18} - 7605195418763792351 T^{20} + \)\(24\!\cdots\!30\)\( T^{22} + \)\(63\!\cdots\!51\)\( T^{24} - \)\(61\!\cdots\!86\)\( T^{26} + \)\(43\!\cdots\!45\)\( T^{28} - \)\(19\!\cdots\!36\)\( T^{30} + \)\(65\!\cdots\!03\)\( T^{32} - \)\(16\!\cdots\!46\)\( T^{34} + \)\(31\!\cdots\!18\)\( T^{36} - \)\(40\!\cdots\!80\)\( T^{38} + \)\(27\!\cdots\!01\)\( T^{40} \))(\( 1 - 320 T^{2} + 55258 T^{4} - 6481134 T^{6} + 564305583 T^{8} - 37299364464 T^{10} + 1825870594845 T^{12} - 57025827049134 T^{14} + 130434887196171 T^{16} + 112418780517368770 T^{18} - 7605195418763792351 T^{20} + \)\(24\!\cdots\!30\)\( T^{22} + \)\(63\!\cdots\!51\)\( T^{24} - \)\(61\!\cdots\!86\)\( T^{26} + \)\(43\!\cdots\!45\)\( T^{28} - \)\(19\!\cdots\!36\)\( T^{30} + \)\(65\!\cdots\!03\)\( T^{32} - \)\(16\!\cdots\!46\)\( T^{34} + \)\(31\!\cdots\!18\)\( T^{36} - \)\(40\!\cdots\!80\)\( T^{38} + \)\(27\!\cdots\!01\)\( T^{40} \))
$53$ (\( 1 + 53 T^{2} + 2809 T^{4} \))(\( 1 + 53 T^{2} + 2809 T^{4} \))(\( 1 + 53 T^{2} + 2809 T^{4} \))(\( 1 + 53 T^{2} + 2809 T^{4} \))(\( 1 + 53 T^{2} + 2809 T^{4} \))(\( 1 - 24 T + 340 T^{2} - 3552 T^{3} + 29050 T^{4} - 180120 T^{5} + 750160 T^{6} - 1659096 T^{7} + 1273315 T^{8} - 87932088 T^{9} + 2107199440 T^{10} - 26815725240 T^{11} + 229218473050 T^{12} - 1485430391136 T^{13} + 7535882783860 T^{14} - 28193067356088 T^{15} + 62259690411361 T^{16} \))(\( 1 + 24 T + 340 T^{2} + 3552 T^{3} + 29050 T^{4} + 180120 T^{5} + 750160 T^{6} + 1659096 T^{7} + 1273315 T^{8} + 87932088 T^{9} + 2107199440 T^{10} + 26815725240 T^{11} + 229218473050 T^{12} + 1485430391136 T^{13} + 7535882783860 T^{14} + 28193067356088 T^{15} + 62259690411361 T^{16} \))(\( 1 + 348 T^{2} + 65006 T^{4} + 8168402 T^{6} + 754331535 T^{8} + 52520483476 T^{10} + 2690516083409 T^{12} + 86297805646650 T^{14} - 153773100809741 T^{16} - 235981740156615306 T^{18} - 17360136557008258575 T^{20} - \)\(66\!\cdots\!54\)\( T^{22} - \)\(12\!\cdots\!21\)\( T^{24} + \)\(19\!\cdots\!50\)\( T^{26} + \)\(16\!\cdots\!49\)\( T^{28} + \)\(91\!\cdots\!24\)\( T^{30} + \)\(37\!\cdots\!35\)\( T^{32} + \)\(11\!\cdots\!38\)\( T^{34} + \)\(25\!\cdots\!26\)\( T^{36} + \)\(37\!\cdots\!72\)\( T^{38} + \)\(30\!\cdots\!01\)\( T^{40} \))(\( 1 + 348 T^{2} + 65006 T^{4} + 8168402 T^{6} + 754331535 T^{8} + 52520483476 T^{10} + 2690516083409 T^{12} + 86297805646650 T^{14} - 153773100809741 T^{16} - 235981740156615306 T^{18} - 17360136557008258575 T^{20} - \)\(66\!\cdots\!54\)\( T^{22} - \)\(12\!\cdots\!21\)\( T^{24} + \)\(19\!\cdots\!50\)\( T^{26} + \)\(16\!\cdots\!49\)\( T^{28} + \)\(91\!\cdots\!24\)\( T^{30} + \)\(37\!\cdots\!35\)\( T^{32} + \)\(11\!\cdots\!38\)\( T^{34} + \)\(25\!\cdots\!26\)\( T^{36} + \)\(37\!\cdots\!72\)\( T^{38} + \)\(30\!\cdots\!01\)\( T^{40} \))
$59$ (\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 - 12 T - 80 T^{2} + 1164 T^{3} + 7690 T^{4} - 80082 T^{5} - 434420 T^{6} + 1772232 T^{7} + 28861927 T^{8} + 104561688 T^{9} - 1512216020 T^{10} - 16447161078 T^{11} + 93182506090 T^{12} + 832171884036 T^{13} - 3374442691280 T^{14} - 29863817817828 T^{15} + 146830437604321 T^{16} \))(\( 1 + 12 T - 80 T^{2} - 1164 T^{3} + 7690 T^{4} + 80082 T^{5} - 434420 T^{6} - 1772232 T^{7} + 28861927 T^{8} - 104561688 T^{9} - 1512216020 T^{10} + 16447161078 T^{11} + 93182506090 T^{12} - 832171884036 T^{13} - 3374442691280 T^{14} + 29863817817828 T^{15} + 146830437604321 T^{16} \))(\( 1 - 281 T^{2} + 43822 T^{4} - 4385565 T^{6} + 294312414 T^{8} - 11205257169 T^{10} - 28140925407 T^{12} + 33950080736241 T^{14} - 2072476947516237 T^{16} + 52044476827656370 T^{18} - 492615579799524713 T^{20} + \)\(18\!\cdots\!70\)\( T^{22} - \)\(25\!\cdots\!57\)\( T^{24} + \)\(14\!\cdots\!81\)\( T^{26} - \)\(41\!\cdots\!47\)\( T^{28} - \)\(57\!\cdots\!69\)\( T^{30} + \)\(52\!\cdots\!34\)\( T^{32} - \)\(27\!\cdots\!65\)\( T^{34} + \)\(94\!\cdots\!02\)\( T^{36} - \)\(21\!\cdots\!01\)\( T^{38} + \)\(26\!\cdots\!01\)\( T^{40} \))(\( 1 - 281 T^{2} + 43822 T^{4} - 4385565 T^{6} + 294312414 T^{8} - 11205257169 T^{10} - 28140925407 T^{12} + 33950080736241 T^{14} - 2072476947516237 T^{16} + 52044476827656370 T^{18} - 492615579799524713 T^{20} + \)\(18\!\cdots\!70\)\( T^{22} - \)\(25\!\cdots\!57\)\( T^{24} + \)\(14\!\cdots\!81\)\( T^{26} - \)\(41\!\cdots\!47\)\( T^{28} - \)\(57\!\cdots\!69\)\( T^{30} + \)\(52\!\cdots\!34\)\( T^{32} - \)\(27\!\cdots\!65\)\( T^{34} + \)\(94\!\cdots\!02\)\( T^{36} - \)\(21\!\cdots\!01\)\( T^{38} + \)\(26\!\cdots\!01\)\( T^{40} \))
$61$ (\( 1 + 9 T + 88 T^{2} + 549 T^{3} + 3721 T^{4} \))(\( ( 1 - 14 T + 61 T^{2} )( 1 - 13 T + 61 T^{2} ) \))(\( ( 1 - 13 T + 61 T^{2} )( 1 + T + 61 T^{2} ) \))(\( ( 1 - 14 T + 61 T^{2} )( 1 - 13 T + 61 T^{2} ) \))(\( 1 + 9 T + 88 T^{2} + 549 T^{3} + 3721 T^{4} \))(\( 1 - 15 T + 223 T^{2} - 2220 T^{3} + 19711 T^{4} - 141723 T^{5} + 816310 T^{6} - 5175267 T^{7} + 31433836 T^{8} - 315691287 T^{9} + 3037489510 T^{10} - 32168428263 T^{11} + 272915371951 T^{12} - 1875003788220 T^{13} + 11489043482503 T^{14} - 47141142540315 T^{15} + 191707312997281 T^{16} \))(\( 1 - 15 T + 223 T^{2} - 2220 T^{3} + 19711 T^{4} - 141723 T^{5} + 816310 T^{6} - 5175267 T^{7} + 31433836 T^{8} - 315691287 T^{9} + 3037489510 T^{10} - 32168428263 T^{11} + 272915371951 T^{12} - 1875003788220 T^{13} + 11489043482503 T^{14} - 47141142540315 T^{15} + 191707312997281 T^{16} \))(\( ( 1 + 45 T + 1163 T^{2} + 21960 T^{3} + 334221 T^{4} + 4323609 T^{5} + 49194321 T^{6} + 504078669 T^{7} + 4728952065 T^{8} + 41065652565 T^{9} + 332029517802 T^{10} + 2505004806465 T^{11} + 17596430633865 T^{12} + 114416280368289 T^{13} + 681136746668961 T^{14} + 3651704168370309 T^{15} + 17219191039307781 T^{16} + 69014632679021160 T^{17} + 222955605015837803 T^{18} + 526236574177536345 T^{19} + 713342911662882601 T^{20} )^{2} \))(\( ( 1 + 45 T + 1163 T^{2} + 21960 T^{3} + 334221 T^{4} + 4323609 T^{5} + 49194321 T^{6} + 504078669 T^{7} + 4728952065 T^{8} + 41065652565 T^{9} + 332029517802 T^{10} + 2505004806465 T^{11} + 17596430633865 T^{12} + 114416280368289 T^{13} + 681136746668961 T^{14} + 3651704168370309 T^{15} + 17219191039307781 T^{16} + 69014632679021160 T^{17} + 222955605015837803 T^{18} + 526236574177536345 T^{19} + 713342911662882601 T^{20} )^{2} \))
$67$ (\( 1 + 13 T + 102 T^{2} + 871 T^{3} + 4489 T^{4} \))(\( ( 1 + 5 T + 67 T^{2} )( 1 + 11 T + 67 T^{2} ) \))(\( ( 1 - 16 T + 67 T^{2} )( 1 + 5 T + 67 T^{2} ) \))(\( ( 1 - 11 T + 67 T^{2} )( 1 - 5 T + 67 T^{2} ) \))(\( 1 + 13 T + 102 T^{2} + 871 T^{3} + 4489 T^{4} \))(\( 1 + 4 T - 234 T^{2} - 412 T^{3} + 35255 T^{4} + 28434 T^{5} - 3551522 T^{6} - 717722 T^{7} + 271900824 T^{8} - 48087374 T^{9} - 15942782258 T^{10} + 8551895142 T^{11} + 710427770855 T^{12} - 556251544084 T^{13} - 21167261427546 T^{14} + 24242846421292 T^{15} + 406067677556641 T^{16} \))(\( 1 + 4 T - 234 T^{2} - 412 T^{3} + 35255 T^{4} + 28434 T^{5} - 3551522 T^{6} - 717722 T^{7} + 271900824 T^{8} - 48087374 T^{9} - 15942782258 T^{10} + 8551895142 T^{11} + 710427770855 T^{12} - 556251544084 T^{13} - 21167261427546 T^{14} + 24242846421292 T^{15} + 406067677556641 T^{16} \))(\( ( 1 + 10 T - 194 T^{2} - 1284 T^{3} + 29439 T^{4} + 97824 T^{5} - 3224136 T^{6} - 5037786 T^{7} + 270996189 T^{8} + 90868948 T^{9} - 19829421434 T^{10} + 6088219516 T^{11} + 1216501892421 T^{12} - 1515179630718 T^{13} - 64969954656456 T^{14} + 132074638467168 T^{15} + 2663004312673191 T^{16} - 7781953701234732 T^{17} - 78777129445988354 T^{18} + 272065343962949470 T^{19} + 1822837804551761449 T^{20} )^{2} \))(\( ( 1 - 10 T - 194 T^{2} + 1284 T^{3} + 29439 T^{4} - 97824 T^{5} - 3224136 T^{6} + 5037786 T^{7} + 270996189 T^{8} - 90868948 T^{9} - 19829421434 T^{10} - 6088219516 T^{11} + 1216501892421 T^{12} + 1515179630718 T^{13} - 64969954656456 T^{14} - 132074638467168 T^{15} + 2663004312673191 T^{16} + 7781953701234732 T^{17} - 78777129445988354 T^{18} - 272065343962949470 T^{19} + 1822837804551761449 T^{20} )^{2} \))
$71$ (\( 1 - 94 T^{2} + 5041 T^{4} \))(\( ( 1 - 71 T^{2} )^{2} \))(\( ( 1 - 71 T^{2} )^{2} \))(\( ( 1 - 71 T^{2} )^{2} \))(\( 1 - 94 T^{2} + 5041 T^{4} \))(\( 1 - 464 T^{2} + 99532 T^{4} - 12936548 T^{6} + 1114829374 T^{8} - 65213138468 T^{10} + 2529275433292 T^{12} - 59438531739344 T^{14} + 645753531245761 T^{16} \))(\( 1 - 464 T^{2} + 99532 T^{4} - 12936548 T^{6} + 1114829374 T^{8} - 65213138468 T^{10} + 2529275433292 T^{12} - 59438531739344 T^{14} + 645753531245761 T^{16} \))(\( ( 1 - 417 T^{2} + 83080 T^{4} - 10852019 T^{6} + 1067389843 T^{8} - 84042150656 T^{10} + 5380712198563 T^{12} - 275768045033939 T^{14} + 10642571588156680 T^{16} - 269279222529482337 T^{18} + 3255243551009881201 T^{20} )^{2} \))(\( ( 1 - 417 T^{2} + 83080 T^{4} - 10852019 T^{6} + 1067389843 T^{8} - 84042150656 T^{10} + 5380712198563 T^{12} - 275768045033939 T^{14} + 10642571588156680 T^{16} - 269279222529482337 T^{18} + 3255243551009881201 T^{20} )^{2} \))
$73$ (\( 1 + 6 T + 85 T^{2} + 438 T^{3} + 5329 T^{4} \))(\( ( 1 - 10 T + 73 T^{2} )( 1 + 7 T + 73 T^{2} ) \))(\( ( 1 - 17 T + 73 T^{2} )( 1 - 10 T + 73 T^{2} ) \))(\( ( 1 - 7 T + 73 T^{2} )( 1 + 10 T + 73 T^{2} ) \))(\( 1 + 6 T + 85 T^{2} + 438 T^{3} + 5329 T^{4} \))(\( 1 + 15 T + 280 T^{2} + 3075 T^{3} + 38779 T^{4} + 422928 T^{5} + 4017052 T^{6} + 39506334 T^{7} + 310273396 T^{8} + 2883962382 T^{9} + 21406870108 T^{10} + 164526181776 T^{11} + 1101255387739 T^{12} + 6374695148475 T^{13} + 42373583360920 T^{14} + 165710977786455 T^{15} + 806460091894081 T^{16} \))(\( 1 + 15 T + 280 T^{2} + 3075 T^{3} + 38779 T^{4} + 422928 T^{5} + 4017052 T^{6} + 39506334 T^{7} + 310273396 T^{8} + 2883962382 T^{9} + 21406870108 T^{10} + 164526181776 T^{11} + 1101255387739 T^{12} + 6374695148475 T^{13} + 42373583360920 T^{14} + 165710977786455 T^{15} + 806460091894081 T^{16} \))(\( ( 1 - 24 T + 572 T^{2} - 9120 T^{3} + 139305 T^{4} - 1702236 T^{5} + 20269650 T^{6} - 207027012 T^{7} + 2089522809 T^{8} - 18747463404 T^{9} + 169031552934 T^{10} - 1368564828492 T^{11} + 11135067049161 T^{12} - 80537027127204 T^{13} + 575622405685650 T^{14} - 3528857096181948 T^{15} + 21081614393189145 T^{16} - 100752274494164640 T^{17} + 461295172563414332 T^{18} - 1412918080998429912 T^{19} + 4297625829703557649 T^{20} )^{2} \))(\( ( 1 + 24 T + 572 T^{2} + 9120 T^{3} + 139305 T^{4} + 1702236 T^{5} + 20269650 T^{6} + 207027012 T^{7} + 2089522809 T^{8} + 18747463404 T^{9} + 169031552934 T^{10} + 1368564828492 T^{11} + 11135067049161 T^{12} + 80537027127204 T^{13} + 575622405685650 T^{14} + 3528857096181948 T^{15} + 21081614393189145 T^{16} + 100752274494164640 T^{17} + 461295172563414332 T^{18} + 1412918080998429912 T^{19} + 4297625829703557649 T^{20} )^{2} \))
$79$ (\( 1 - 16 T + 177 T^{2} - 1264 T^{3} + 6241 T^{4} \))(\( ( 1 + 4 T + 79 T^{2} )( 1 + 13 T + 79 T^{2} ) \))(\( ( 1 - 17 T + 79 T^{2} )( 1 + 4 T + 79 T^{2} ) \))(\( ( 1 + 4 T + 79 T^{2} )( 1 + 13 T + 79 T^{2} ) \))(\( 1 - 16 T + 177 T^{2} - 1264 T^{3} + 6241 T^{4} \))(\( 1 + 29 T + 294 T^{2} + 1975 T^{3} + 27377 T^{4} + 260496 T^{5} + 598654 T^{6} + 2403434 T^{7} + 77714340 T^{8} + 189871286 T^{9} + 3736199614 T^{10} + 128434687344 T^{11} + 1066336367537 T^{12} + 6077186388025 T^{13} + 71467711923174 T^{14} + 556913360598611 T^{15} + 1517108809906561 T^{16} \))(\( 1 + 29 T + 294 T^{2} + 1975 T^{3} + 27377 T^{4} + 260496 T^{5} + 598654 T^{6} + 2403434 T^{7} + 77714340 T^{8} + 189871286 T^{9} + 3736199614 T^{10} + 128434687344 T^{11} + 1066336367537 T^{12} + 6077186388025 T^{13} + 71467711923174 T^{14} + 556913360598611 T^{15} + 1517108809906561 T^{16} \))(\( ( 1 - 23 T + 19 T^{2} + 1650 T^{3} + 22269 T^{4} - 308997 T^{5} - 2262141 T^{6} + 18480543 T^{7} + 256079211 T^{8} - 288694607 T^{9} - 26891047898 T^{10} - 22806873953 T^{11} + 1598190355851 T^{12} + 9111628440177 T^{13} - 88110575183421 T^{14} - 950801196121803 T^{15} + 5413314546997149 T^{16} + 31686449827162350 T^{17} + 28825067388224659 T^{18} - 2756586707600221337 T^{19} + 9468276082626847201 T^{20} )^{2} \))(\( ( 1 - 23 T + 19 T^{2} + 1650 T^{3} + 22269 T^{4} - 308997 T^{5} - 2262141 T^{6} + 18480543 T^{7} + 256079211 T^{8} - 288694607 T^{9} - 26891047898 T^{10} - 22806873953 T^{11} + 1598190355851 T^{12} + 9111628440177 T^{13} - 88110575183421 T^{14} - 950801196121803 T^{15} + 5413314546997149 T^{16} + 31686449827162350 T^{17} + 28825067388224659 T^{18} - 2756586707600221337 T^{19} + 9468276082626847201 T^{20} )^{2} \))
$83$ (\( ( 1 + 9 T + 83 T^{2} )^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 - 9 T + 83 T^{2} )^{2} \))(\( ( 1 - 15 T + 380 T^{2} - 3759 T^{3} + 49260 T^{4} - 311997 T^{5} + 2617820 T^{6} - 8576805 T^{7} + 47458321 T^{8} )^{2} \))(\( ( 1 + 15 T + 380 T^{2} + 3759 T^{3} + 49260 T^{4} + 311997 T^{5} + 2617820 T^{6} + 8576805 T^{7} + 47458321 T^{8} )^{2} \))(\( ( 1 + 200 T^{2} + 27258 T^{4} + 1670937 T^{6} + 63680961 T^{8} - 932587014 T^{10} + 438698140329 T^{12} + 79299864516777 T^{14} + 8911740697292202 T^{16} + 450458446427808200 T^{18} + 15516041187205853449 T^{20} )^{2} \))(\( ( 1 + 200 T^{2} + 27258 T^{4} + 1670937 T^{6} + 63680961 T^{8} - 932587014 T^{10} + 438698140329 T^{12} + 79299864516777 T^{14} + 8911740697292202 T^{16} + 450458446427808200 T^{18} + 15516041187205853449 T^{20} )^{2} \))
$89$ (\( 1 - 3 T - 80 T^{2} - 267 T^{3} + 7921 T^{4} \))(\( 1 - 89 T^{2} + 7921 T^{4} \))(\( 1 - 89 T^{2} + 7921 T^{4} \))(\( 1 - 89 T^{2} + 7921 T^{4} \))(\( 1 + 3 T - 80 T^{2} + 267 T^{3} + 7921 T^{4} \))(\( 1 - 3 T - 53 T^{2} - 2820 T^{3} + 14227 T^{4} + 160275 T^{5} + 3467116 T^{6} - 26593569 T^{7} - 193500020 T^{8} - 2366827641 T^{9} + 27463025836 T^{10} + 112988906475 T^{11} + 892633862707 T^{12} - 15747047646180 T^{13} - 26340008420933 T^{14} - 132694004686587 T^{15} + 3936588805702081 T^{16} \))(\( 1 + 3 T - 53 T^{2} + 2820 T^{3} + 14227 T^{4} - 160275 T^{5} + 3467116 T^{6} + 26593569 T^{7} - 193500020 T^{8} + 2366827641 T^{9} + 27463025836 T^{10} - 112988906475 T^{11} + 892633862707 T^{12} + 15747047646180 T^{13} - 26340008420933 T^{14} + 132694004686587 T^{15} + 3936588805702081 T^{16} \))(\( 1 - 548 T^{2} + 149638 T^{4} - 27617742 T^{6} + 3982290303 T^{8} - 496827970068 T^{10} + 57439702492761 T^{12} - 6298039134882534 T^{14} + 650541332844392067 T^{16} - 62975586452546418242 T^{18} + \)\(57\!\cdots\!53\)\( T^{20} - \)\(49\!\cdots\!82\)\( T^{22} + \)\(40\!\cdots\!47\)\( T^{24} - \)\(31\!\cdots\!74\)\( T^{26} + \)\(22\!\cdots\!41\)\( T^{28} - \)\(15\!\cdots\!68\)\( T^{30} + \)\(98\!\cdots\!63\)\( T^{32} - \)\(54\!\cdots\!22\)\( T^{34} + \)\(23\!\cdots\!18\)\( T^{36} - \)\(67\!\cdots\!88\)\( T^{38} + \)\(97\!\cdots\!01\)\( T^{40} \))(\( 1 - 548 T^{2} + 149638 T^{4} - 27617742 T^{6} + 3982290303 T^{8} - 496827970068 T^{10} + 57439702492761 T^{12} - 6298039134882534 T^{14} + 650541332844392067 T^{16} - 62975586452546418242 T^{18} + \)\(57\!\cdots\!53\)\( T^{20} - \)\(49\!\cdots\!82\)\( T^{22} + \)\(40\!\cdots\!47\)\( T^{24} - \)\(31\!\cdots\!74\)\( T^{26} + \)\(22\!\cdots\!41\)\( T^{28} - \)\(15\!\cdots\!68\)\( T^{30} + \)\(98\!\cdots\!63\)\( T^{32} - \)\(54\!\cdots\!22\)\( T^{34} + \)\(23\!\cdots\!18\)\( T^{36} - \)\(67\!\cdots\!88\)\( T^{38} + \)\(97\!\cdots\!01\)\( T^{40} \))
$97$ (\( 1 - 86 T^{2} + 9409 T^{4} \))(\( ( 1 - 19 T + 97 T^{2} )( 1 + 19 T + 97 T^{2} ) \))(\( ( 1 - 14 T + 97 T^{2} )( 1 + 14 T + 97 T^{2} ) \))(\( ( 1 - 19 T + 97 T^{2} )( 1 + 19 T + 97 T^{2} ) \))(\( 1 - 86 T^{2} + 9409 T^{4} \))(\( 1 - 368 T^{2} + 81676 T^{4} - 12257504 T^{6} + 1385094598 T^{8} - 115330855136 T^{10} + 7230717554956 T^{12} - 306533697813872 T^{14} + 7837433594376961 T^{16} \))(\( 1 - 368 T^{2} + 81676 T^{4} - 12257504 T^{6} + 1385094598 T^{8} - 115330855136 T^{10} + 7230717554956 T^{12} - 306533697813872 T^{14} + 7837433594376961 T^{16} \))(\( ( 1 - 673 T^{2} + 225150 T^{4} - 48497952 T^{6} + 7395278745 T^{8} - 830846466435 T^{10} + 69582177711705 T^{12} - 4293488820532512 T^{14} + 187543646909764350 T^{16} - 5274592809015694753 T^{18} + 73742412689492826049 T^{20} )^{2} \))(\( ( 1 - 673 T^{2} + 225150 T^{4} - 48497952 T^{6} + 7395278745 T^{8} - 830846466435 T^{10} + 69582177711705 T^{12} - 4293488820532512 T^{14} + 187543646909764350 T^{16} - 5274592809015694753 T^{18} + 73742412689492826049 T^{20} )^{2} \))
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