Properties

Label 525.4.d
Level 525
Weight 4
Character orbit d
Rep. character \(\chi_{525}(274,\cdot)\)
Character field \(\Q\)
Dimension 56
Newform subspaces 15
Sturm bound 320
Trace bound 11

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Defining parameters

Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 525.d (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q\)
Newform subspaces: \( 15 \)
Sturm bound: \(320\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(2\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(525, [\chi])\).

Total New Old
Modular forms 252 56 196
Cusp forms 228 56 172
Eisenstein series 24 0 24

Trace form

\( 56q - 216q^{4} + 24q^{6} - 504q^{9} + O(q^{10}) \) \( 56q - 216q^{4} + 24q^{6} - 504q^{9} + 1016q^{16} + 96q^{19} + 84q^{21} - 648q^{24} + 184q^{26} + 560q^{29} + 96q^{31} + 160q^{34} + 1944q^{36} - 576q^{39} - 2112q^{41} - 1580q^{44} + 2044q^{46} - 2744q^{49} + 2304q^{51} - 216q^{54} + 84q^{56} + 112q^{59} - 4208q^{61} - 5988q^{64} - 1296q^{66} + 2304q^{69} - 1616q^{71} + 8708q^{74} - 8952q^{76} - 6904q^{79} + 4536q^{81} - 1008q^{84} - 1460q^{86} + 4288q^{89} + 1456q^{91} + 1216q^{94} + 7968q^{96} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(525, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
525.4.d.a \(2\) \(30.976\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q+5iq^{2}+3iq^{3}-17q^{4}-15q^{6}+\cdots\)
525.4.d.b \(2\) \(30.976\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q+4iq^{2}+3iq^{3}-8q^{4}-12q^{6}-7iq^{7}+\cdots\)
525.4.d.c \(2\) \(30.976\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q+3iq^{2}-3iq^{3}-q^{4}+9q^{6}-7iq^{7}+\cdots\)
525.4.d.d \(2\) \(30.976\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q+3iq^{2}-3iq^{3}-q^{4}+9q^{6}-7iq^{7}+\cdots\)
525.4.d.e \(2\) \(30.976\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q+2iq^{2}+3iq^{3}+4q^{4}-6q^{6}+7iq^{7}+\cdots\)
525.4.d.f \(2\) \(30.976\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q-3iq^{3}+8q^{4}-7iq^{7}-9q^{9}+42q^{11}+\cdots\)
525.4.d.g \(4\) \(30.976\) \(\Q(i, \sqrt{57})\) None \(0\) \(0\) \(0\) \(0\) \(q+(\beta _{1}-\beta _{2})q^{2}-3\beta _{2}q^{3}+(-10+3\beta _{3})q^{4}+\cdots\)
525.4.d.h \(4\) \(30.976\) \(\Q(i, \sqrt{65})\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{2}+3\beta _{2}q^{3}+(-9+\beta _{3})q^{4}+\cdots\)
525.4.d.i \(4\) \(30.976\) \(\Q(i, \sqrt{17})\) None \(0\) \(0\) \(0\) \(0\) \(q+(\beta _{1}+4\beta _{2})q^{2}-3\beta _{2}q^{3}+(-5-7\beta _{3})q^{4}+\cdots\)
525.4.d.j \(4\) \(30.976\) \(\Q(i, \sqrt{41})\) None \(0\) \(0\) \(0\) \(0\) \(q+(\beta _{1}-\beta _{2})q^{2}+3\beta _{2}q^{3}+(-6+3\beta _{3})q^{4}+\cdots\)
525.4.d.k \(4\) \(30.976\) \(\Q(i, \sqrt{5})\) None \(0\) \(0\) \(0\) \(0\) \(q+(-2\beta _{1}+\beta _{2})q^{2}-3\beta _{1}q^{3}+(-1+\cdots)q^{4}+\cdots\)
525.4.d.l \(4\) \(30.976\) \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(0\) \(q+(\zeta_{8}+\zeta_{8}^{2})q^{2}-3\zeta_{8}q^{3}+(-1-2\zeta_{8}^{3})q^{4}+\cdots\)
525.4.d.m \(4\) \(30.976\) \(\Q(i, \sqrt{17})\) None \(0\) \(0\) \(0\) \(0\) \(q+(\beta _{1}-\beta _{2})q^{2}-3\beta _{2}q^{3}+3\beta _{3}q^{4}+\cdots\)
525.4.d.n \(8\) \(30.976\) \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{2}-3\beta _{3}q^{3}+(-4+\beta _{2})q^{4}+\cdots\)
525.4.d.o \(8\) \(30.976\) \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q+(\beta _{1}+\beta _{4})q^{2}-3\beta _{4}q^{3}+(-5+\beta _{2}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(525, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(525, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(175, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 9 T^{2} + 64 T^{4} \))(\( ( 1 - 4 T + 8 T^{2} )( 1 + 4 T + 8 T^{2} ) \))(\( 1 - 7 T^{2} + 64 T^{4} \))(\( 1 - 7 T^{2} + 64 T^{4} \))(\( 1 - 12 T^{2} + 64 T^{4} \))(\( ( 1 - 8 T^{2} )^{2} \))(\( 1 + T^{2} + 64 T^{6} + 4096 T^{8} \))(\( 1 + T^{2} + 112 T^{4} + 64 T^{6} + 4096 T^{8} \))(\( 1 + T^{2} - 80 T^{4} + 64 T^{6} + 4096 T^{8} \))(\( 1 - 7 T^{2} + 48 T^{4} - 448 T^{6} + 4096 T^{8} \))(\( 1 - 14 T^{2} + 97 T^{4} - 896 T^{6} + 4096 T^{8} \))(\( 1 - 14 T^{2} + 145 T^{4} - 896 T^{6} + 4096 T^{8} \))(\( 1 - 19 T^{2} + 180 T^{4} - 1216 T^{6} + 4096 T^{8} \))(\( 1 - 16 T^{2} + 156 T^{4} - 1751 T^{6} + 18356 T^{8} - 112064 T^{10} + 638976 T^{12} - 4194304 T^{14} + 16777216 T^{16} \))(\( 1 - 16 T^{2} + 84 T^{4} + 481 T^{6} - 10156 T^{8} + 30784 T^{10} + 344064 T^{12} - 4194304 T^{14} + 16777216 T^{16} \))
$3$ (\( 1 + 9 T^{2} \))(\( 1 + 9 T^{2} \))(\( 1 + 9 T^{2} \))(\( 1 + 9 T^{2} \))(\( 1 + 9 T^{2} \))(\( 1 + 9 T^{2} \))(\( ( 1 + 9 T^{2} )^{2} \))(\( ( 1 + 9 T^{2} )^{2} \))(\( ( 1 + 9 T^{2} )^{2} \))(\( ( 1 + 9 T^{2} )^{2} \))(\( ( 1 + 9 T^{2} )^{2} \))(\( ( 1 + 9 T^{2} )^{2} \))(\( ( 1 + 9 T^{2} )^{2} \))(\( ( 1 + 9 T^{2} )^{4} \))(\( ( 1 + 9 T^{2} )^{4} \))
$5$ 1
$7$ (\( 1 + 49 T^{2} \))(\( 1 + 49 T^{2} \))(\( 1 + 49 T^{2} \))(\( 1 + 49 T^{2} \))(\( 1 + 49 T^{2} \))(\( 1 + 49 T^{2} \))(\( ( 1 + 49 T^{2} )^{2} \))(\( ( 1 + 49 T^{2} )^{2} \))(\( ( 1 + 49 T^{2} )^{2} \))(\( ( 1 + 49 T^{2} )^{2} \))(\( ( 1 + 49 T^{2} )^{2} \))(\( ( 1 + 49 T^{2} )^{2} \))(\( ( 1 + 49 T^{2} )^{2} \))(\( ( 1 + 49 T^{2} )^{4} \))(\( ( 1 + 49 T^{2} )^{4} \))
$11$ (\( ( 1 - 12 T + 1331 T^{2} )^{2} \))(\( ( 1 - 62 T + 1331 T^{2} )^{2} \))(\( ( 1 + 36 T + 1331 T^{2} )^{2} \))(\( ( 1 + 6 T + 1331 T^{2} )^{2} \))(\( ( 1 + 21 T + 1331 T^{2} )^{2} \))(\( ( 1 - 42 T + 1331 T^{2} )^{2} \))(\( ( 1 + 6 T + 1246 T^{2} + 7986 T^{3} + 1771561 T^{4} )^{2} \))(\( ( 1 + 22 T + 2718 T^{2} + 29282 T^{3} + 1771561 T^{4} )^{2} \))(\( ( 1 + 26 T + 2406 T^{2} + 34606 T^{3} + 1771561 T^{4} )^{2} \))(\( ( 1 - 62 T + 3582 T^{2} - 82522 T^{3} + 1771561 T^{4} )^{2} \))(\( ( 1 + 92 T + 4758 T^{2} + 122452 T^{3} + 1771561 T^{4} )^{2} \))(\( ( 1 + 16 T - 474 T^{2} + 21296 T^{3} + 1771561 T^{4} )^{2} \))(\( ( 1 + 31 T + 2796 T^{2} + 41261 T^{3} + 1771561 T^{4} )^{2} \))(\( ( 1 - 21 T + 2681 T^{2} - 22236 T^{3} + 3289690 T^{4} - 29596116 T^{5} + 4749555041 T^{6} - 49516901511 T^{7} + 3138428376721 T^{8} )^{2} \))(\( ( 1 - 57 T + 4001 T^{2} - 182772 T^{3} + 7206874 T^{4} - 243269532 T^{5} + 7088015561 T^{6} - 134403018387 T^{7} + 3138428376721 T^{8} )^{2} \))
$13$ (\( 1 - 3494 T^{2} + 4826809 T^{4} \))(\( 1 - 550 T^{2} + 4826809 T^{4} \))(\( 1 - 3238 T^{2} + 4826809 T^{4} \))(\( 1 - 2713 T^{2} + 4826809 T^{4} \))(\( 1 - 3818 T^{2} + 4826809 T^{4} \))(\( 1 - 3994 T^{2} + 4826809 T^{4} \))(\( 1 - 4556 T^{2} + 14317590 T^{4} - 21990941804 T^{6} + 23298085122481 T^{8} \))(\( 1 - 8416 T^{2} + 27329422 T^{4} - 40622424544 T^{6} + 23298085122481 T^{8} \))(\( 1 - 4576 T^{2} + 14485390 T^{4} - 22087477984 T^{6} + 23298085122481 T^{8} \))(\( 1 - 6720 T^{2} + 20906318 T^{4} - 32436156480 T^{6} + 23298085122481 T^{8} \))(\( 1 + 5684 T^{2} + 17268502 T^{4} + 27435582356 T^{6} + 23298085122481 T^{8} \))(\( 1 - 5836 T^{2} + 17983510 T^{4} - 28169257324 T^{6} + 23298085122481 T^{8} \))(\( 1 - 5571 T^{2} + 15544460 T^{4} - 26890152939 T^{6} + 23298085122481 T^{8} \))(\( 1 - 3671 T^{2} + 2135358 T^{4} - 10866464017 T^{6} + 58692802872482 T^{8} - 52450346315431753 T^{10} + 49749752450970783198 T^{12} - \)\(41\!\cdots\!59\)\( T^{14} + \)\(54\!\cdots\!61\)\( T^{16} \))(\( 1 - 7079 T^{2} + 27317214 T^{4} - 70304963521 T^{6} + 159254918186594 T^{8} - 339348630667834489 T^{10} + \)\(63\!\cdots\!34\)\( T^{12} - \)\(79\!\cdots\!91\)\( T^{14} + \)\(54\!\cdots\!61\)\( T^{16} \))
$17$ (\( 1 + 8130 T^{2} + 24137569 T^{4} \))(\( 1 - 2770 T^{2} + 24137569 T^{4} \))(\( 1 - 8062 T^{2} + 24137569 T^{4} \))(\( 1 - 9097 T^{2} + 24137569 T^{4} \))(\( 1 - 9342 T^{2} + 24137569 T^{4} \))(\( 1 - 5470 T^{2} + 24137569 T^{4} \))(\( 1 - 19520 T^{2} + 143530686 T^{4} - 471165346880 T^{6} + 582622237229761 T^{8} \))(\( 1 - 4604 T^{2} - 2402618 T^{4} - 111129367676 T^{6} + 582622237229761 T^{8} \))(\( 1 - 12860 T^{2} + 88767046 T^{4} - 310409137340 T^{6} + 582622237229761 T^{8} \))(\( 1 - 15900 T^{2} + 109116038 T^{4} - 383787347100 T^{6} + 582622237229761 T^{8} \))(\( 1 + 676 T^{2} + 29648902 T^{4} + 16316996644 T^{6} + 582622237229761 T^{8} \))(\( 1 - 11900 T^{2} + 83185606 T^{4} - 287237071100 T^{6} + 582622237229761 T^{8} \))(\( 1 - 10335 T^{2} + 55642016 T^{4} - 249461775615 T^{6} + 582622237229761 T^{8} \))(\( 1 - 16795 T^{2} + 189484074 T^{4} - 1396068939605 T^{6} + 8048224142687690 T^{8} - 33697710358472520245 T^{10} + \)\(11\!\cdots\!14\)\( T^{12} - \)\(23\!\cdots\!55\)\( T^{14} + \)\(33\!\cdots\!21\)\( T^{16} \))(\( 1 - 17779 T^{2} + 188146650 T^{4} - 1407976054013 T^{6} + 7815459662716778 T^{8} - 33985119154086514397 T^{10} + \)\(10\!\cdots\!50\)\( T^{12} - \)\(25\!\cdots\!11\)\( T^{14} + \)\(33\!\cdots\!21\)\( T^{16} \))
$19$ (\( ( 1 - 92 T + 6859 T^{2} )^{2} \))(\( ( 1 + 100 T + 6859 T^{2} )^{2} \))(\( ( 1 - 124 T + 6859 T^{2} )^{2} \))(\( ( 1 - 4 T + 6859 T^{2} )^{2} \))(\( ( 1 + 16 T + 6859 T^{2} )^{2} \))(\( ( 1 + 38 T + 6859 T^{2} )^{2} \))(\( ( 1 + 64 T + 6534 T^{2} + 438976 T^{3} + 47045881 T^{4} )^{2} \))(\( ( 1 + 102 T + 13134 T^{2} + 699618 T^{3} + 47045881 T^{4} )^{2} \))(\( ( 1 + 174 T + 18414 T^{2} + 1193466 T^{3} + 47045881 T^{4} )^{2} \))(\( ( 1 - 122 T + 17398 T^{2} - 836798 T^{3} + 47045881 T^{4} )^{2} \))(\( ( 1 - 108 T + 13254 T^{2} - 740772 T^{3} + 47045881 T^{4} )^{2} \))(\( ( 1 - 96 T + 13974 T^{2} - 658464 T^{3} + 47045881 T^{4} )^{2} \))(\( ( 1 - 56 T + 1174 T^{2} - 384104 T^{3} + 47045881 T^{4} )^{2} \))(\( ( 1 + 72 T + 25164 T^{2} + 1424592 T^{3} + 252125750 T^{4} + 9771276528 T^{5} + 1183862549484 T^{6} + 23233514240088 T^{7} + 2213314919066161 T^{8} )^{2} \))(\( ( 1 - 12 T - 10212 T^{2} - 154020 T^{3} + 95206646 T^{4} - 1056423180 T^{5} - 480432536772 T^{6} - 3872252373348 T^{7} + 2213314919066161 T^{8} )^{2} \))
$23$ (\( 1 - 11790 T^{2} + 148035889 T^{4} \))(\( 1 - 22570 T^{2} + 148035889 T^{4} \))(\( ( 1 - 12167 T^{2} )^{2} \))(\( 1 - 18709 T^{2} + 148035889 T^{4} \))(\( 1 - 23709 T^{2} + 148035889 T^{4} \))(\( 1 - 24190 T^{2} + 148035889 T^{4} \))(\( 1 - 15704 T^{2} + 357132654 T^{4} - 2324755600856 T^{6} + 21914624432020321 T^{8} \))(\( 1 - 1868 T^{2} - 142455866 T^{4} - 276531040652 T^{6} + 21914624432020321 T^{8} \))(\( 1 - 5084 T^{2} - 148699226 T^{4} - 752614459676 T^{6} + 21914624432020321 T^{8} \))(\( 1 - 1308 T^{2} + 290453798 T^{4} - 193630942812 T^{6} + 21914624432020321 T^{8} \))(\( 1 + 6532 T^{2} + 101938534 T^{4} + 966970426948 T^{6} + 21914624432020321 T^{8} \))(\( 1 - 30044 T^{2} + 519364774 T^{4} - 4447590249116 T^{6} + 21914624432020321 T^{8} \))(\( 1 - 11514 T^{2} + 171279659 T^{4} - 1704485225946 T^{6} + 21914624432020321 T^{8} \))(\( 1 - 66652 T^{2} + 2057185554 T^{4} - 40083083332832 T^{6} + 562578923508162875 T^{8} - \)\(59\!\cdots\!48\)\( T^{10} + \)\(45\!\cdots\!34\)\( T^{12} - \)\(21\!\cdots\!88\)\( T^{14} + \)\(48\!\cdots\!41\)\( T^{16} \))(\( 1 - 31828 T^{2} + 620170842 T^{4} - 8468979898736 T^{6} + 101809268764343459 T^{8} - \)\(12\!\cdots\!04\)\( T^{10} + \)\(13\!\cdots\!82\)\( T^{12} - \)\(10\!\cdots\!32\)\( T^{14} + \)\(48\!\cdots\!41\)\( T^{16} \))
$29$ (\( ( 1 - 58 T + 24389 T^{2} )^{2} \))(\( ( 1 - 10 T + 24389 T^{2} )^{2} \))(\( ( 1 + 102 T + 24389 T^{2} )^{2} \))(\( ( 1 - 123 T + 24389 T^{2} )^{2} \))(\( ( 1 + 167 T + 24389 T^{2} )^{2} \))(\( ( 1 - 258 T + 24389 T^{2} )^{2} \))(\( ( 1 - 252 T + 56446 T^{2} - 6146028 T^{3} + 594823321 T^{4} )^{2} \))(\( ( 1 - 196 T + 20942 T^{2} - 4780244 T^{3} + 594823321 T^{4} )^{2} \))(\( ( 1 - 32 T + 19046 T^{2} - 780448 T^{3} + 594823321 T^{4} )^{2} \))(\( ( 1 + 352 T + 78278 T^{2} + 8584928 T^{3} + 594823321 T^{4} )^{2} \))(\( ( 1 - 236 T + 61982 T^{2} - 5755804 T^{3} + 594823321 T^{4} )^{2} \))(\( ( 1 + 188 T + 34286 T^{2} + 4585132 T^{3} + 594823321 T^{4} )^{2} \))(\( ( 1 - 62 T + 19751 T^{2} - 1512118 T^{3} + 594823321 T^{4} )^{2} \))(\( ( 1 - 240 T + 86430 T^{2} - 14441664 T^{3} + 3156579983 T^{4} - 352217743296 T^{5} + 51410579634030 T^{6} - 3481715034208560 T^{7} + 353814783205469041 T^{8} )^{2} \))(\( ( 1 + 378 T + 143760 T^{2} + 29729988 T^{3} + 5852097929 T^{4} + 725084677332 T^{5} + 85511800626960 T^{6} + 5483701178878482 T^{7} + 353814783205469041 T^{8} )^{2} \))
$31$ (\( ( 1 + 224 T + 29791 T^{2} )^{2} \))(\( ( 1 + 48 T + 29791 T^{2} )^{2} \))(\( ( 1 + 160 T + 29791 T^{2} )^{2} \))(\( ( 1 + 205 T + 29791 T^{2} )^{2} \))(\( ( 1 - 10 T + 29791 T^{2} )^{2} \))(\( ( 1 - 146 T + 29791 T^{2} )^{2} \))(\( ( 1 - 40 T - 13890 T^{2} - 1191640 T^{3} + 887503681 T^{4} )^{2} \))(\( ( 1 - 150 T + 36542 T^{2} - 4468650 T^{3} + 887503681 T^{4} )^{2} \))(\( ( 1 - 330 T + 85430 T^{2} - 9831030 T^{3} + 887503681 T^{4} )^{2} \))(\( ( 1 - 66 T + 45870 T^{2} - 1966206 T^{3} + 887503681 T^{4} )^{2} \))(\( ( 1 + 60 T + 8462 T^{2} + 1787460 T^{3} + 887503681 T^{4} )^{2} \))(\( ( 1 + 120 T + 60590 T^{2} + 3574920 T^{3} + 887503681 T^{4} )^{2} \))(\( ( 1 + 135 T + 63420 T^{2} + 4021785 T^{3} + 887503681 T^{4} )^{2} \))(\( ( 1 - 351 T + 102010 T^{2} - 21726711 T^{3} + 4440563682 T^{4} - 647260447401 T^{5} + 90534250498810 T^{6} - 9280307378395521 T^{7} + 787662783788549761 T^{8} )^{2} \))(\( ( 1 + 93 T + 16330 T^{2} - 936891 T^{3} + 385099458 T^{4} - 27910919781 T^{5} + 14492935110730 T^{6} + 2458884860942403 T^{7} + 787662783788549761 T^{8} )^{2} \))
$37$ (\( 1 - 79990 T^{2} + 2565726409 T^{4} \))(\( 1 - 40790 T^{2} + 2565726409 T^{4} \))(\( 1 + 57098 T^{2} + 2565726409 T^{4} \))(\( 1 - 32662 T^{2} + 2565726409 T^{4} \))(\( 1 - 83617 T^{2} + 2565726409 T^{4} \))(\( 1 + 87050 T^{2} + 2565726409 T^{4} \))(\( 1 - 134924 T^{2} + 8546718390 T^{4} - 346178070007916 T^{6} + 6582952005840035281 T^{8} \))(\( 1 - 155884 T^{2} + 11012319222 T^{4} - 399955695540556 T^{6} + 6582952005840035281 T^{8} \))(\( 1 - 189004 T^{2} + 14019426870 T^{4} - 484932554206636 T^{6} + 6582952005840035281 T^{8} \))(\( 1 - 53740 T^{2} + 3534883318 T^{4} - 137882137219660 T^{6} + 6582952005840035281 T^{8} \))(\( 1 - 176044 T^{2} + 12759471222 T^{4} - 451680739945996 T^{6} + 6582952005840035281 T^{8} \))(\( 1 - 124204 T^{2} + 8380919670 T^{4} - 318673482903436 T^{6} + 6582952005840035281 T^{8} \))(\( 1 - 193159 T^{2} + 14439461800 T^{4} - 495593147436031 T^{6} + 6582952005840035281 T^{8} \))(\( 1 - 87569 T^{2} + 8318550313 T^{4} - 613716143379878 T^{6} + 29309622284809592422 T^{8} - \)\(15\!\cdots\!02\)\( T^{10} + \)\(54\!\cdots\!53\)\( T^{12} - \)\(14\!\cdots\!01\)\( T^{14} + \)\(43\!\cdots\!61\)\( T^{16} \))(\( 1 - 207809 T^{2} + 23288320585 T^{4} - 1818846583640678 T^{6} + \)\(10\!\cdots\!58\)\( T^{8} - \)\(46\!\cdots\!02\)\( T^{10} + \)\(15\!\cdots\!85\)\( T^{12} - \)\(35\!\cdots\!61\)\( T^{14} + \)\(43\!\cdots\!61\)\( T^{16} \))
$41$ (\( ( 1 - 18 T + 68921 T^{2} )^{2} \))(\( ( 1 + 248 T + 68921 T^{2} )^{2} \))(\( ( 1 + 318 T + 68921 T^{2} )^{2} \))(\( ( 1 - 57 T + 68921 T^{2} )^{2} \))(\( ( 1 + 168 T + 68921 T^{2} )^{2} \))(\( ( 1 + 282 T + 68921 T^{2} )^{2} \))(\( ( 1 + 450 T + 175642 T^{2} + 31014450 T^{3} + 4750104241 T^{4} )^{2} \))(\( ( 1 + 176 T - 16914 T^{2} + 12130096 T^{3} + 4750104241 T^{4} )^{2} \))(\( ( 1 - 200 T + 64542 T^{2} - 13784200 T^{3} + 4750104241 T^{4} )^{2} \))(\( ( 1 - 16 T + 18350 T^{2} - 1102736 T^{3} + 4750104241 T^{4} )^{2} \))(\( ( 1 - 44 T + 106326 T^{2} - 3032524 T^{3} + 4750104241 T^{4} )^{2} \))(\( ( 1 - 100 T + 89142 T^{2} - 6892100 T^{3} + 4750104241 T^{4} )^{2} \))(\( ( 1 - 235 T + 143042 T^{2} - 16196435 T^{3} + 4750104241 T^{4} )^{2} \))(\( ( 1 - 381 T + 273692 T^{2} - 71526783 T^{3} + 28134293782 T^{4} - 4929697411143 T^{5} + 1300065529927772 T^{6} - 124732517004099141 T^{7} + 22563490300366186081 T^{8} )^{2} \))(\( ( 1 + 465 T + 105716 T^{2} + 566235 T^{3} - 2050285754 T^{4} + 39025482435 T^{5} + 502162019941556 T^{6} + 152232599493191865 T^{7} + 22563490300366186081 T^{8} )^{2} \))
$43$ (\( 1 - 43414 T^{2} + 6321363049 T^{4} \))(\( 1 - 154390 T^{2} + 6321363049 T^{4} \))(\( 1 - 87190 T^{2} + 6321363049 T^{4} \))(\( 1 + 6635 T^{2} + 6321363049 T^{4} \))(\( 1 - 149605 T^{2} + 6321363049 T^{4} \))(\( 1 - 158614 T^{2} + 6321363049 T^{4} \))(\( 1 - 181676 T^{2} + 16252611510 T^{4} - 1148439953290124 T^{6} + 39959630797262576401 T^{8} \))(\( 1 - 225580 T^{2} + 23395199158 T^{4} - 1425973076593420 T^{6} + 39959630797262576401 T^{8} \))(\( 1 - 212476 T^{2} + 21325427350 T^{4} - 1343137935199324 T^{6} + 39959630797262576401 T^{8} \))(\( 1 - 34620 T^{2} - 3131277802 T^{4} - 218845588756380 T^{6} + 39959630797262576401 T^{8} \))(\( 1 - 144940 T^{2} + 16379434678 T^{4} - 916218360322060 T^{6} + 39959630797262576401 T^{8} \))(\( 1 - 112876 T^{2} + 6993047350 T^{4} - 713530175518924 T^{6} + 39959630797262576401 T^{8} \))(\( 1 + 33774 T^{2} + 3686087315 T^{4} + 213497715616926 T^{6} + 39959630797262576401 T^{8} \))(\( 1 - 271076 T^{2} + 44752340058 T^{4} - 5319114613093072 T^{6} + \)\(48\!\cdots\!07\)\( T^{8} - \)\(33\!\cdots\!28\)\( T^{10} + \)\(17\!\cdots\!58\)\( T^{12} - \)\(68\!\cdots\!24\)\( T^{14} + \)\(15\!\cdots\!01\)\( T^{16} \))(\( 1 - 125228 T^{2} + 15455427282 T^{4} - 1545044296825216 T^{6} + \)\(15\!\cdots\!19\)\( T^{8} - \)\(97\!\cdots\!84\)\( T^{10} + \)\(61\!\cdots\!82\)\( T^{12} - \)\(31\!\cdots\!72\)\( T^{14} + \)\(15\!\cdots\!01\)\( T^{16} \))
$47$ (\( 1 - 164382 T^{2} + 10779215329 T^{4} \))(\( 1 - 102670 T^{2} + 10779215329 T^{4} \))(\( 1 - 150046 T^{2} + 10779215329 T^{4} \))(\( 1 - 204046 T^{2} + 10779215329 T^{4} \))(\( 1 - 47646 T^{2} + 10779215329 T^{4} \))(\( 1 - 202462 T^{2} + 10779215329 T^{4} \))(\( 1 - 283436 T^{2} + 41632933734 T^{4} - 3055217675990444 T^{6} + \)\(11\!\cdots\!41\)\( T^{8} \))(\( 1 - 183612 T^{2} + 18245588294 T^{4} - 1979193284988348 T^{6} + \)\(11\!\cdots\!41\)\( T^{8} \))(\( 1 - 273516 T^{2} + 36034474214 T^{4} - 2948287859926764 T^{6} + \)\(11\!\cdots\!41\)\( T^{8} \))(\( 1 + 4180 T^{2} + 14462189158 T^{4} + 45057120075220 T^{6} + \)\(11\!\cdots\!41\)\( T^{8} \))(\( 1 - 53052 T^{2} - 317140666 T^{4} - 571858931634108 T^{6} + \)\(11\!\cdots\!41\)\( T^{8} \))(\( 1 + 43524 T^{2} + 9878986694 T^{4} + 469154567979396 T^{6} + \)\(11\!\cdots\!41\)\( T^{8} \))(\( 1 - 337916 T^{2} + 49344837574 T^{4} - 3642469327114364 T^{6} + \)\(11\!\cdots\!41\)\( T^{8} \))(\( 1 - 257496 T^{2} + 49366812892 T^{4} - 6931866867101352 T^{6} + \)\(84\!\cdots\!54\)\( T^{8} - \)\(74\!\cdots\!08\)\( T^{10} + \)\(57\!\cdots\!72\)\( T^{12} - \)\(32\!\cdots\!44\)\( T^{14} + \)\(13\!\cdots\!81\)\( T^{16} \))(\( 1 - 623016 T^{2} + 180431963644 T^{4} - 32308674389768664 T^{6} + \)\(39\!\cdots\!54\)\( T^{8} - \)\(34\!\cdots\!56\)\( T^{10} + \)\(20\!\cdots\!04\)\( T^{12} - \)\(78\!\cdots\!24\)\( T^{14} + \)\(13\!\cdots\!81\)\( T^{16} \))
$53$ (\( 1 + 270762 T^{2} + 22164361129 T^{4} \))(\( 1 - 231190 T^{2} + 22164361129 T^{4} \))(\( 1 - 49750 T^{2} + 22164361129 T^{4} \))(\( 1 - 190825 T^{2} + 22164361129 T^{4} \))(\( 1 - 264630 T^{2} + 22164361129 T^{4} \))(\( 1 - 184858 T^{2} + 22164361129 T^{4} \))(\( 1 + 14356 T^{2} + 44102355894 T^{4} + 318191568367924 T^{6} + \)\(49\!\cdots\!41\)\( T^{8} \))(\( 1 - 302000 T^{2} + 54356941198 T^{4} - 6693637060958000 T^{6} + \)\(49\!\cdots\!41\)\( T^{8} \))(\( 1 - 205664 T^{2} + 54678146734 T^{4} - 4558411167234656 T^{6} + \)\(49\!\cdots\!41\)\( T^{8} \))(\( 1 - 44384 T^{2} + 11570351278 T^{4} - 983743004349536 T^{6} + \)\(49\!\cdots\!41\)\( T^{8} \))(\( 1 - 521420 T^{2} + 112288959478 T^{4} - 11556941179883180 T^{6} + \)\(49\!\cdots\!41\)\( T^{8} \))(\( 1 - 139244 T^{2} + 23569194934 T^{4} - 3086254301046476 T^{6} + \)\(49\!\cdots\!41\)\( T^{8} \))(\( 1 - 517079 T^{2} + 110424015304 T^{4} - 11460725688222191 T^{6} + \)\(49\!\cdots\!41\)\( T^{8} \))(\( 1 - 641683 T^{2} + 172519035378 T^{4} - 26517744919429925 T^{6} + \)\(35\!\cdots\!22\)\( T^{8} - \)\(58\!\cdots\!25\)\( T^{10} + \)\(84\!\cdots\!98\)\( T^{12} - \)\(69\!\cdots\!87\)\( T^{14} + \)\(24\!\cdots\!81\)\( T^{16} \))(\( 1 - 135451 T^{2} + 38529689730 T^{4} - 8651841044933597 T^{6} + \)\(85\!\cdots\!18\)\( T^{8} - \)\(19\!\cdots\!13\)\( T^{10} + \)\(18\!\cdots\!30\)\( T^{12} - \)\(14\!\cdots\!39\)\( T^{14} + \)\(24\!\cdots\!81\)\( T^{16} \))
$59$ (\( ( 1 + 380 T + 205379 T^{2} )^{2} \))(\( ( 1 + 120 T + 205379 T^{2} )^{2} \))(\( ( 1 - 132 T + 205379 T^{2} )^{2} \))(\( ( 1 + 33 T + 205379 T^{2} )^{2} \))(\( ( 1 + 488 T + 205379 T^{2} )^{2} \))(\( ( 1 - 360 T + 205379 T^{2} )^{2} \))(\( ( 1 + 804 T + 380614 T^{2} + 165124716 T^{3} + 42180533641 T^{4} )^{2} \))(\( ( 1 - 844 T + 474182 T^{2} - 173339876 T^{3} + 42180533641 T^{4} )^{2} \))(\( ( 1 + 364 T + 438374 T^{2} + 74757956 T^{3} + 42180533641 T^{4} )^{2} \))(\( ( 1 + 516 T + 418118 T^{2} + 105975564 T^{3} + 42180533641 T^{4} )^{2} \))(\( ( 1 - 464 T + 458102 T^{2} - 95295856 T^{3} + 42180533641 T^{4} )^{2} \))(\( ( 1 + 104 T + 80534 T^{2} + 21359416 T^{3} + 42180533641 T^{4} )^{2} \))(\( ( 1 - 441 T + 273734 T^{2} - 90572139 T^{3} + 42180533641 T^{4} )^{2} \))(\( ( 1 - 855 T + 767616 T^{2} - 344685519 T^{3} + 195821792246 T^{4} - 70791167206701 T^{5} + 32378452511369856 T^{6} - 7406861424949972845 T^{7} + \)\(17\!\cdots\!81\)\( T^{8} )^{2} \))(\( ( 1 + 231 T + 701280 T^{2} + 148812447 T^{3} + 204019205798 T^{4} + 30562951552413 T^{5} + 29580364631760480 T^{6} + 2001152034109290909 T^{7} + \)\(17\!\cdots\!81\)\( T^{8} )^{2} \))
$61$ (\( ( 1 - 718 T + 226981 T^{2} )^{2} \))(\( ( 1 - 622 T + 226981 T^{2} )^{2} \))(\( ( 1 - 398 T + 226981 T^{2} )^{2} \))(\( ( 1 + 427 T + 226981 T^{2} )^{2} \))(\( ( 1 - 28 T + 226981 T^{2} )^{2} \))(\( ( 1 + 682 T + 226981 T^{2} )^{2} \))(\( ( 1 + 428 T + 425886 T^{2} + 97147868 T^{3} + 51520374361 T^{4} )^{2} \))(\( ( 1 + 204 T + 455006 T^{2} + 46304124 T^{3} + 51520374361 T^{4} )^{2} \))(\( ( 1 - 792 T + 340886 T^{2} - 179768952 T^{3} + 51520374361 T^{4} )^{2} \))(\( ( 1 + 880 T + 575238 T^{2} + 199743280 T^{3} + 51520374361 T^{4} )^{2} \))(\( ( 1 + 684 T + 535646 T^{2} + 155255004 T^{3} + 51520374361 T^{4} )^{2} \))(\( ( 1 + 468 T + 494606 T^{2} + 106227108 T^{3} + 51520374361 T^{4} )^{2} \))(\( ( 1 + 223 T + 149646 T^{2} + 50616763 T^{3} + 51520374361 T^{4} )^{2} \))(\( ( 1 - 687 T + 715504 T^{2} - 385624593 T^{3} + 235308406830 T^{4} - 87529455743733 T^{5} + 36863033936792944 T^{6} - 8033878365777054867 T^{7} + \)\(26\!\cdots\!21\)\( T^{8} )^{2} \))(\( ( 1 + 1353 T + 1374808 T^{2} + 931878711 T^{3} + 511309944510 T^{4} + 211518761701491 T^{5} + 70830622834497688 T^{6} + 15822179663604592773 T^{7} + \)\(26\!\cdots\!21\)\( T^{8} )^{2} \))
$67$ (\( 1 - 431782 T^{2} + 90458382169 T^{4} \))(\( 1 + 215690 T^{2} + 90458382169 T^{4} \))(\( 1 - 593062 T^{2} + 90458382169 T^{4} \))(\( 1 - 207142 T^{2} + 90458382169 T^{4} \))(\( 1 + 333563 T^{2} + 90458382169 T^{4} \))(\( 1 + 57818 T^{2} + 90458382169 T^{4} \))(\( 1 - 859676 T^{2} + 362036762934 T^{4} - 77764900149517244 T^{6} + \)\(81\!\cdots\!61\)\( T^{8} \))(\( 1 - 1064524 T^{2} + 463499688022 T^{4} - 96295118820072556 T^{6} + \)\(81\!\cdots\!61\)\( T^{8} \))(\( 1 - 885916 T^{2} + 375059568694 T^{4} - 80138528097631804 T^{6} + \)\(81\!\cdots\!61\)\( T^{8} \))(\( 1 - 74012 T^{2} + 114756705078 T^{4} - 6695005781092028 T^{6} + \)\(81\!\cdots\!61\)\( T^{8} \))(\( 1 - 663244 T^{2} + 218042637142 T^{4} - 59995979223296236 T^{6} + \)\(81\!\cdots\!61\)\( T^{8} \))(\( 1 + 244724 T^{2} + 162973601494 T^{4} + 22137337117926356 T^{6} + \)\(81\!\cdots\!61\)\( T^{8} \))(\( 1 - 405131 T^{2} + 135876758064 T^{4} - 36647494826509139 T^{6} + \)\(81\!\cdots\!61\)\( T^{8} \))(\( 1 - 651893 T^{2} + 433059221049 T^{4} - 177336848417309398 T^{6} + \)\(61\!\cdots\!50\)\( T^{8} - \)\(16\!\cdots\!62\)\( T^{10} + \)\(35\!\cdots\!89\)\( T^{12} - \)\(48\!\cdots\!37\)\( T^{14} + \)\(66\!\cdots\!21\)\( T^{16} \))(\( 1 - 995573 T^{2} + 706524334905 T^{4} - 318789658849139158 T^{6} + \)\(11\!\cdots\!14\)\( T^{8} - \)\(28\!\cdots\!02\)\( T^{10} + \)\(57\!\cdots\!05\)\( T^{12} - \)\(73\!\cdots\!57\)\( T^{14} + \)\(66\!\cdots\!21\)\( T^{16} \))
$71$ (\( ( 1 + 960 T + 357911 T^{2} )^{2} \))(\( ( 1 + 678 T + 357911 T^{2} )^{2} \))(\( ( 1 + 720 T + 357911 T^{2} )^{2} \))(\( ( 1 - 300 T + 357911 T^{2} )^{2} \))(\( ( 1 + 285 T + 357911 T^{2} )^{2} \))(\( ( 1 - 810 T + 357911 T^{2} )^{2} \))(\( ( 1 - 954 T + 930526 T^{2} - 341447094 T^{3} + 128100283921 T^{4} )^{2} \))(\( ( 1 - 1670 T + 1384382 T^{2} - 597711370 T^{3} + 128100283921 T^{4} )^{2} \))(\( ( 1 - 454 T + 304526 T^{2} - 162491594 T^{3} + 128100283921 T^{4} )^{2} \))(\( ( 1 - 310 T + 664038 T^{2} - 110952410 T^{3} + 128100283921 T^{4} )^{2} \))(\( ( 1 + 740 T + 757502 T^{2} + 264854140 T^{3} + 128100283921 T^{4} )^{2} \))(\( ( 1 + 136 T + 540446 T^{2} + 48675896 T^{3} + 128100283921 T^{4} )^{2} \))(\( ( 1 - 619 T + 576906 T^{2} - 221546909 T^{3} + 128100283921 T^{4} )^{2} \))(\( ( 1 + 681 T + 1310019 T^{2} + 668258544 T^{3} + 678493576820 T^{4} + 239177083741584 T^{5} + 167813805841904499 T^{6} + 31222828989263790111 T^{7} + \)\(16\!\cdots\!41\)\( T^{8} )^{2} \))(\( ( 1 + 1725 T + 1434819 T^{2} + 706812600 T^{3} + 347044577276 T^{4} + 252976004478600 T^{5} + 183800721275245299 T^{6} + 79088663739324578475 T^{7} + \)\(16\!\cdots\!41\)\( T^{8} )^{2} \))
$73$ (\( 1 + 358322 T^{2} + 151334226289 T^{4} \))(\( 1 - 365870 T^{2} + 151334226289 T^{4} \))(\( 1 - 526030 T^{2} + 151334226289 T^{4} \))(\( 1 - 768430 T^{2} + 151334226289 T^{4} \))(\( 1 - 75790 T^{2} + 151334226289 T^{4} \))(\( 1 - 762658 T^{2} + 151334226289 T^{4} \))(\( 1 - 977372 T^{2} + 539124333606 T^{4} - 147909835416532508 T^{6} + \)\(22\!\cdots\!21\)\( T^{8} \))(\( 1 - 155440 T^{2} + 209914818238 T^{4} - 23523392134362160 T^{6} + \)\(22\!\cdots\!21\)\( T^{8} \))(\( 1 - 984112 T^{2} + 463281832126 T^{4} - 148929828101720368 T^{6} + \)\(22\!\cdots\!21\)\( T^{8} \))(\( 1 - 1473328 T^{2} + 843769310398 T^{4} - 222964952949919792 T^{6} + \)\(22\!\cdots\!21\)\( T^{8} \))(\( 1 - 1317340 T^{2} + 723135691558 T^{4} - 199358629659551260 T^{6} + \)\(22\!\cdots\!21\)\( T^{8} \))(\( 1 + 231908 T^{2} + 102056920486 T^{4} + 35095617750229412 T^{6} + \)\(22\!\cdots\!21\)\( T^{8} \))(\( 1 - 1454332 T^{2} + 829074972646 T^{4} - 220090207987333948 T^{6} + \)\(22\!\cdots\!21\)\( T^{8} \))(\( 1 + 270040 T^{2} + 305634878940 T^{4} + 41673922681088936 T^{6} + \)\(45\!\cdots\!18\)\( T^{8} + \)\(63\!\cdots\!04\)\( T^{10} + \)\(69\!\cdots\!40\)\( T^{12} + \)\(93\!\cdots\!60\)\( T^{14} + \)\(52\!\cdots\!41\)\( T^{16} \))(\( 1 - 1799816 T^{2} + 1700330801820 T^{4} - 1053085527973751032 T^{6} + \)\(47\!\cdots\!58\)\( T^{8} - \)\(15\!\cdots\!48\)\( T^{10} + \)\(38\!\cdots\!20\)\( T^{12} - \)\(62\!\cdots\!04\)\( T^{14} + \)\(52\!\cdots\!41\)\( T^{16} \))
$79$ (\( ( 1 + 896 T + 493039 T^{2} )^{2} \))(\( ( 1 + 740 T + 493039 T^{2} )^{2} \))(\( ( 1 - 1024 T + 493039 T^{2} )^{2} \))(\( ( 1 + 686 T + 493039 T^{2} )^{2} \))(\( ( 1 - 469 T + 493039 T^{2} )^{2} \))(\( ( 1 + 1136 T + 493039 T^{2} )^{2} \))(\( ( 1 - 572 T + 901662 T^{2} - 282018308 T^{3} + 243087455521 T^{4} )^{2} \))(\( ( 1 - 888 T + 1007454 T^{2} - 437818632 T^{3} + 243087455521 T^{4} )^{2} \))(\( ( 1 + 408 T + 994782 T^{2} + 201159912 T^{3} + 243087455521 T^{4} )^{2} \))(\( ( 1 + 1832 T + 1824478 T^{2} + 903247448 T^{3} + 243087455521 T^{4} )^{2} \))(\( ( 1 - 408 T - 143586 T^{2} - 201159912 T^{3} + 243087455521 T^{4} )^{2} \))(\( ( 1 - 432 T + 602142 T^{2} - 212992848 T^{3} + 243087455521 T^{4} )^{2} \))(\( ( 1 - 427 T + 986572 T^{2} - 210527653 T^{3} + 243087455521 T^{4} )^{2} \))(\( ( 1 + 345 T + 1235121 T^{2} + 721148448 T^{3} + 716751093026 T^{4} + 355554309653472 T^{5} + 300242421150553041 T^{6} + 41348800614003320055 T^{7} + \)\(59\!\cdots\!41\)\( T^{8} )^{2} \))(\( ( 1 + 1629 T + 1989837 T^{2} + 1837769472 T^{3} + 1504809477470 T^{4} + 906092022705408 T^{5} + 483704413231540077 T^{6} + \)\(19\!\cdots\!51\)\( T^{7} + \)\(59\!\cdots\!41\)\( T^{8} )^{2} \))
$83$ (\( 1 - 953478 T^{2} + 326940373369 T^{4} \))(\( 1 - 924550 T^{2} + 326940373369 T^{4} \))(\( 1 - 1101958 T^{2} + 326940373369 T^{4} \))(\( 1 + 819227 T^{2} + 326940373369 T^{4} \))(\( 1 - 978738 T^{2} + 326940373369 T^{4} \))(\( 1 - 1119238 T^{2} + 326940373369 T^{4} \))(\( 1 - 134924 T^{2} + 162125495574 T^{4} - 44112102936438956 T^{6} + \)\(10\!\cdots\!61\)\( T^{8} \))(\( 1 - 340236 T^{2} + 29905619222 T^{4} - 111236884873575084 T^{6} + \)\(10\!\cdots\!61\)\( T^{8} \))(\( 1 - 1049484 T^{2} + 546979674134 T^{4} - 343118690804791596 T^{6} + \)\(10\!\cdots\!61\)\( T^{8} \))(\( 1 - 1027340 T^{2} + 679923446038 T^{4} - 335878923176908460 T^{6} + \)\(10\!\cdots\!61\)\( T^{8} \))(\( 1 - 2031756 T^{2} + 1672847111702 T^{4} - 664263065234705964 T^{6} + \)\(10\!\cdots\!61\)\( T^{8} \))(\( 1 - 2003724 T^{2} + 1638356284694 T^{4} - 655098272688426156 T^{6} + \)\(10\!\cdots\!61\)\( T^{8} \))(\( 1 + 25081 T^{2} - 503757220916 T^{4} + 8199991504467889 T^{6} + \)\(10\!\cdots\!61\)\( T^{8} \))(\( 1 - 2098899 T^{2} + 2942941076302 T^{4} - 2619304253314086933 T^{6} + \)\(17\!\cdots\!54\)\( T^{8} - \)\(85\!\cdots\!77\)\( T^{10} + \)\(31\!\cdots\!22\)\( T^{12} - \)\(73\!\cdots\!91\)\( T^{14} + \)\(11\!\cdots\!21\)\( T^{16} \))(\( 1 - 1219827 T^{2} + 504702020110 T^{4} - 464473092130980597 T^{6} + \)\(44\!\cdots\!74\)\( T^{8} - \)\(15\!\cdots\!93\)\( T^{10} + \)\(53\!\cdots\!10\)\( T^{12} - \)\(42\!\cdots\!43\)\( T^{14} + \)\(11\!\cdots\!21\)\( T^{16} \))
$89$ (\( ( 1 - 1038 T + 704969 T^{2} )^{2} \))(\( ( 1 + 200 T + 704969 T^{2} )^{2} \))(\( ( 1 + 354 T + 704969 T^{2} )^{2} \))(\( ( 1 + 714 T + 704969 T^{2} )^{2} \))(\( ( 1 + 324 T + 704969 T^{2} )^{2} \))(\( ( 1 - 1038 T + 704969 T^{2} )^{2} \))(\( ( 1 + 366 T + 1156090 T^{2} + 258018654 T^{3} + 496981290961 T^{4} )^{2} \))(\( ( 1 + 588 T + 1495334 T^{2} + 414521772 T^{3} + 496981290961 T^{4} )^{2} \))(\( ( 1 + 36 T - 842170 T^{2} + 25378884 T^{3} + 496981290961 T^{4} )^{2} \))(\( ( 1 + 796 T + 411158 T^{2} + 561155324 T^{3} + 496981290961 T^{4} )^{2} \))(\( ( 1 - 1332 T + 1790774 T^{2} - 939018708 T^{3} + 496981290961 T^{4} )^{2} \))(\( ( 1 - 1404 T + 1802390 T^{2} - 989776476 T^{3} + 496981290961 T^{4} )^{2} \))(\( ( 1 + 466 T + 306170 T^{2} + 328515554 T^{3} + 496981290961 T^{4} )^{2} \))(\( ( 1 - 198 T + 1184884 T^{2} + 492558966 T^{3} + 650980125750 T^{4} + 347238801702054 T^{5} + 588865179959033524 T^{6} - 69370567934082071382 T^{7} + \)\(24\!\cdots\!21\)\( T^{8} )^{2} \))(\( ( 1 - 978 T + 526228 T^{2} + 159100050 T^{3} - 360762420714 T^{4} + 112160603148450 T^{5} + 261525470779825108 T^{6} - \)\(34\!\cdots\!02\)\( T^{7} + \)\(24\!\cdots\!21\)\( T^{8} )^{2} \))
$97$ (\( 1 - 1332542 T^{2} + 832972004929 T^{4} \))(\( 1 - 222590 T^{2} + 832972004929 T^{4} \))(\( 1 - 1743550 T^{2} + 832972004929 T^{4} \))(\( 1 - 1581310 T^{2} + 832972004929 T^{4} \))(\( 1 - 1812350 T^{2} + 832972004929 T^{4} \))(\( 1 - 366082 T^{2} + 832972004929 T^{4} \))(\( 1 - 1153244 T^{2} + 1289747845830 T^{4} - 960619966852339676 T^{6} + \)\(69\!\cdots\!41\)\( T^{8} \))(\( 1 - 310080 T^{2} + 1253411633918 T^{4} - 258287959288384320 T^{6} + \)\(69\!\cdots\!41\)\( T^{8} \))(\( 1 - 3345504 T^{2} + 4441575836990 T^{4} - 2786711174377989216 T^{6} + \)\(69\!\cdots\!41\)\( T^{8} \))(\( 1 - 1841920 T^{2} + 2158510258558 T^{4} - 1534267795318823680 T^{6} + \)\(69\!\cdots\!41\)\( T^{8} \))(\( 1 - 580380 T^{2} + 1528544052038 T^{4} - 483440292220693020 T^{6} + \)\(69\!\cdots\!41\)\( T^{8} \))(\( 1 - 2911164 T^{2} + 3760771737350 T^{4} - 2424918113757127356 T^{6} + \)\(69\!\cdots\!41\)\( T^{8} \))(\( 1 + 576836 T^{2} + 1686814161670 T^{4} + 480488239435224644 T^{6} + \)\(69\!\cdots\!41\)\( T^{8} \))(\( 1 - 3683064 T^{2} + 6611927715868 T^{4} - 7859364237946659528 T^{6} + \)\(75\!\cdots\!62\)\( T^{8} - \)\(65\!\cdots\!12\)\( T^{10} + \)\(45\!\cdots\!88\)\( T^{12} - \)\(21\!\cdots\!96\)\( T^{14} + \)\(48\!\cdots\!81\)\( T^{16} \))(\( 1 - 3522744 T^{2} + 6104761341340 T^{4} - 7977627762356281608 T^{6} + \)\(83\!\cdots\!78\)\( T^{8} - \)\(66\!\cdots\!32\)\( T^{10} + \)\(42\!\cdots\!40\)\( T^{12} - \)\(20\!\cdots\!16\)\( T^{14} + \)\(48\!\cdots\!81\)\( T^{16} \))
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