Properties

Label 387.3.b
Level 387
Weight 3
Character orbit b
Rep. character \(\chi_{387}(343,\cdot)\)
Character field \(\Q\)
Dimension 35
Newform subspaces 5
Sturm bound 132
Trace bound 1

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Defining parameters

Level: \( N \) \(=\) \( 387 = 3^{2} \cdot 43 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 387.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 43 \)
Character field: \(\Q\)
Newform subspaces: \( 5 \)
Sturm bound: \(132\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(387, [\chi])\).

Total New Old
Modular forms 92 37 55
Cusp forms 84 35 49
Eisenstein series 8 2 6

Trace form

\( 35q - 60q^{4} + O(q^{10}) \) \( 35q - 60q^{4} + 14q^{10} - 3q^{11} - 3q^{13} - 24q^{14} + 100q^{16} + 47q^{17} - 73q^{23} - 103q^{25} + 19q^{31} + 28q^{35} + 2q^{38} - 74q^{40} + 183q^{41} - q^{43} + 76q^{44} - 104q^{47} - 121q^{49} - 84q^{52} - 223q^{53} - 36q^{56} + 38q^{58} + 310q^{59} - 188q^{64} + 349q^{67} - 682q^{68} + 130q^{74} + 140q^{79} + 161q^{83} - 96q^{86} + 718q^{92} - 10q^{95} + 87q^{97} + O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(387, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
387.3.b.a \(1\) \(10.545\) \(\Q\) \(\Q(\sqrt{-43}) \) \(0\) \(0\) \(0\) \(0\) \(q+4q^{4}+21q^{11}-17q^{13}+2^{4}q^{16}+\cdots\)
387.3.b.b \(2\) \(10.545\) \(\Q(\sqrt{43}) \) \(\Q(\sqrt{-43}) \) \(0\) \(0\) \(0\) \(0\) \(q+4q^{4}+\beta q^{11}+17q^{13}+2^{4}q^{16}+\cdots\)
387.3.b.c \(6\) \(10.545\) \(\mathbb{Q}[x]/(x^{6} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{2}+(-2+\beta _{3}+\beta _{4})q^{4}-\beta _{2}q^{5}+\cdots\)
387.3.b.d \(12\) \(10.545\) \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{2}+(-2+\beta _{2})q^{4}+\beta _{4}q^{5}+\beta _{8}q^{7}+\cdots\)
387.3.b.e \(14\) \(10.545\) \(\mathbb{Q}[x]/(x^{14} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{1}q^{2}+(-3+\beta _{2})q^{4}+\beta _{11}q^{5}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(387, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(387, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(43, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(129, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 - 2 T )( 1 + 2 T ) \))(\( ( 1 - 2 T )^{2}( 1 + 2 T )^{2} \))(\( 1 - 4 T^{2} + 41 T^{4} - 114 T^{6} + 656 T^{8} - 1024 T^{10} + 4096 T^{12} \))(\( ( 1 - 7 T^{2} + 41 T^{4} - 147 T^{6} + 656 T^{8} - 1792 T^{10} + 4096 T^{12} )^{2} \))(\( 1 - 10 T^{2} + 43 T^{4} - 148 T^{6} + 603 T^{8} - 2066 T^{10} + 7025 T^{12} - 30360 T^{14} + 112400 T^{16} - 528896 T^{18} + 2469888 T^{20} - 9699328 T^{22} + 45088768 T^{24} - 167772160 T^{26} + 268435456 T^{28} \))
$3$ 1
$5$ (\( ( 1 - 5 T )( 1 + 5 T ) \))(\( ( 1 - 5 T )^{2}( 1 + 5 T )^{2} \))(\( 1 - 33 T^{2} + 1538 T^{4} - 41006 T^{6} + 961250 T^{8} - 12890625 T^{10} + 244140625 T^{12} \))(\( ( 1 - 60 T^{2} + 2522 T^{4} - 76910 T^{6} + 1576250 T^{8} - 23437500 T^{10} + 244140625 T^{12} )^{2} \))(\( 1 - 158 T^{2} + 12245 T^{4} - 646152 T^{6} + 26887244 T^{8} - 946961944 T^{10} + 29003099582 T^{12} - 776091411812 T^{14} + 18126937238750 T^{16} - 369907009375000 T^{18} + 6564268554687500 T^{20} - 98594970703125000 T^{22} + 1167774200439453125 T^{24} - 9417533874511718750 T^{26} + 37252902984619140625 T^{28} \))
$7$ (\( ( 1 - 7 T )( 1 + 7 T ) \))(\( ( 1 - 7 T )^{2}( 1 + 7 T )^{2} \))(\( 1 - 144 T^{2} + 11511 T^{4} - 668464 T^{6} + 27637911 T^{8} - 830131344 T^{10} + 13841287201 T^{12} \))(\( ( 1 - 84 T^{2} + 8232 T^{4} - 386386 T^{6} + 19765032 T^{8} - 484243284 T^{10} + 13841287201 T^{12} )^{2} \))(\( 1 - 338 T^{2} + 58561 T^{4} - 6653660 T^{6} + 554265016 T^{8} - 36391844060 T^{10} + 2025986266086 T^{12} - 102453858581484 T^{14} + 4864393024872486 T^{16} - 209791739028932060 T^{18} + 7671741271922860216 T^{20} - \)\(22\!\cdots\!60\)\( T^{22} + \)\(46\!\cdots\!61\)\( T^{24} - \)\(64\!\cdots\!38\)\( T^{26} + \)\(45\!\cdots\!01\)\( T^{28} \))
$11$ (\( 1 - 21 T + 121 T^{2} \))(\( 1 + 199 T^{2} + 14641 T^{4} \))(\( ( 1 + 19 T + 411 T^{2} + 4354 T^{3} + 49731 T^{4} + 278179 T^{5} + 1771561 T^{6} )^{2} \))(\( ( 1 + 236 T^{2} + 31210 T^{4} + 3995438 T^{6} + 456945610 T^{8} + 50588695916 T^{10} + 3138428376721 T^{12} )^{2} \))(\( ( 1 - 7 T + 543 T^{2} - 3942 T^{3} + 146986 T^{4} - 1007864 T^{5} + 25813058 T^{6} - 153013646 T^{7} + 3123380018 T^{8} - 14756136824 T^{9} + 260394665146 T^{10} - 845002708902 T^{11} + 14084021558343 T^{12} - 21968998637047 T^{13} + 379749833583241 T^{14} )^{2} \))
$13$ (\( 1 + 17 T + 169 T^{2} \))(\( ( 1 - 17 T + 169 T^{2} )^{2} \))(\( ( 1 - 15 T + 257 T^{2} - 1570 T^{3} + 43433 T^{4} - 428415 T^{5} + 4826809 T^{6} )^{2} \))(\( ( 1 + 12 T + 506 T^{2} + 4022 T^{3} + 85514 T^{4} + 342732 T^{5} + 4826809 T^{6} )^{4} \))(\( ( 1 + T + 617 T^{2} + 1916 T^{3} + 199528 T^{4} + 976928 T^{5} + 43815810 T^{6} + 223819278 T^{7} + 7404871890 T^{8} + 27902040608 T^{9} + 963083546152 T^{10} + 1562940061436 T^{11} + 85058689470833 T^{12} + 23298085122481 T^{13} + 3937376385699289 T^{14} )^{2} \))
$17$ (\( 1 - 9 T + 289 T^{2} \))(\( 1 - 497 T^{2} + 83521 T^{4} \))(\( ( 1 - 10 T + 767 T^{2} - 5655 T^{3} + 221663 T^{4} - 835210 T^{5} + 24137569 T^{6} )^{2} \))(\( ( 1 + 978 T^{2} + 505355 T^{4} + 175816052 T^{6} + 42207754955 T^{8} + 6822290777298 T^{10} + 582622237229761 T^{12} )^{2} \))(\( ( 1 - 9 T + 683 T^{2} - 5314 T^{3} + 315413 T^{4} - 2502531 T^{5} + 123338639 T^{6} - 787992708 T^{7} + 35644866671 T^{8} - 209013891651 T^{9} + 7613303050997 T^{10} - 37069175041474 T^{11} + 1376923834006667 T^{12} - 5243600135067849 T^{13} + 168377826559400929 T^{14} )^{2} \))
$19$ (\( ( 1 - 19 T )( 1 + 19 T ) \))(\( ( 1 - 19 T )^{2}( 1 + 19 T )^{2} \))(\( 1 - 691 T^{2} + 378040 T^{4} - 133433020 T^{6} + 49266550840 T^{8} - 11735642061331 T^{10} + 2213314919066161 T^{12} \))(\( ( 1 - 220 T^{2} + 218248 T^{4} - 72222058 T^{6} + 28442297608 T^{8} - 3736383869020 T^{10} + 2213314919066161 T^{12} )^{2} \))(\( 1 - 3106 T^{2} + 4764233 T^{4} - 4822168116 T^{6} + 3619085950600 T^{8} - 2137515640209476 T^{10} + 1025890729835393262 T^{12} - \)\(40\!\cdots\!08\)\( T^{14} + \)\(13\!\cdots\!02\)\( T^{16} - \)\(36\!\cdots\!16\)\( T^{18} + \)\(80\!\cdots\!00\)\( T^{20} - \)\(13\!\cdots\!96\)\( T^{22} + \)\(17\!\cdots\!33\)\( T^{24} - \)\(15\!\cdots\!26\)\( T^{26} + \)\(63\!\cdots\!41\)\( T^{28} \))
$23$ (\( 1 + 3 T + 529 T^{2} \))(\( 1 - 1049 T^{2} + 279841 T^{4} \))(\( ( 1 - 40 T + 1387 T^{2} - 28195 T^{3} + 733723 T^{4} - 11193640 T^{5} + 148035889 T^{6} )^{2} \))(\( ( 1 + 1186 T^{2} + 999595 T^{4} + 548809204 T^{6} + 279727664395 T^{8} + 92876828543266 T^{10} + 21914624432020321 T^{12} )^{2} \))(\( ( 1 + 75 T + 5463 T^{2} + 247242 T^{3} + 10318521 T^{4} + 330847929 T^{5} + 9750023775 T^{6} + 233835070804 T^{7} + 5157762576975 T^{8} + 92584815299289 T^{9} + 1527511429400169 T^{10} + 19361764622845002 T^{11} + 226313030760164487 T^{12} + 1643596832401524075 T^{13} + 11592836324538749809 T^{14} )^{2} \))
$29$ (\( ( 1 - 29 T )( 1 + 29 T ) \))(\( ( 1 - 29 T )^{2}( 1 + 29 T )^{2} \))(\( 1 - 1621 T^{2} + 1946890 T^{4} - 2007466870 T^{6} + 1376998306090 T^{8} - 810899435409781 T^{10} + 353814783205469041 T^{12} \))(\( ( 1 - 4648 T^{2} + 9308386 T^{4} - 10271462278 T^{6} + 6583644558466 T^{8} - 2325145327442728 T^{10} + 353814783205469041 T^{12} )^{2} \))(\( 1 - 5734 T^{2} + 16275365 T^{4} - 31011157008 T^{6} + 45539850410092 T^{8} - 55656176891335376 T^{10} + 58467585466871243502 T^{12} - \)\(52\!\cdots\!56\)\( T^{14} + \)\(41\!\cdots\!62\)\( T^{16} - \)\(27\!\cdots\!36\)\( T^{18} + \)\(16\!\cdots\!72\)\( T^{20} - \)\(77\!\cdots\!68\)\( T^{22} + \)\(28\!\cdots\!65\)\( T^{24} - \)\(71\!\cdots\!54\)\( T^{26} + \)\(88\!\cdots\!61\)\( T^{28} \))
$31$ (\( 1 - 19 T + 961 T^{2} \))(\( ( 1 + 19 T + 961 T^{2} )^{2} \))(\( ( 1 + 56 T + 2591 T^{2} + 96591 T^{3} + 2489951 T^{4} + 51717176 T^{5} + 887503681 T^{6} )^{2} \))(\( ( 1 - 22 T + 1565 T^{2} - 52440 T^{3} + 1503965 T^{4} - 20317462 T^{5} + 887503681 T^{6} )^{4} \))(\( ( 1 - 31 T + 3679 T^{2} - 95614 T^{3} + 6776409 T^{4} - 169871117 T^{5} + 8879136119 T^{6} - 201888437196 T^{7} + 8532849810359 T^{8} - 156879543842957 T^{9} + 6014087931461529 T^{10} - 81548323653883774 T^{11} + 3015412467802366879 T^{12} - 24417546297445042591 T^{13} + \)\(75\!\cdots\!21\)\( T^{14} )^{2} \))
$37$ (\( ( 1 - 37 T )( 1 + 37 T ) \))(\( ( 1 - 37 T )^{2}( 1 + 37 T )^{2} \))(\( 1 - 5259 T^{2} + 14237096 T^{4} - 24083763884 T^{6} + 26682610076456 T^{8} - 18472129448170539 T^{10} + 6582952005840035281 T^{12} \))(\( ( 1 - 5274 T^{2} + 14176787 T^{4} - 23887209716 T^{6} + 26569581300707 T^{8} - 18524816639979354 T^{10} + 6582952005840035281 T^{12} )^{2} \))(\( 1 - 4790 T^{2} + 17457059 T^{4} - 43407304428 T^{6} + 95846122850417 T^{8} - 174019995659920522 T^{10} + \)\(29\!\cdots\!39\)\( T^{12} - \)\(41\!\cdots\!96\)\( T^{14} + \)\(54\!\cdots\!79\)\( T^{16} - \)\(61\!\cdots\!62\)\( T^{18} + \)\(63\!\cdots\!77\)\( T^{20} - \)\(53\!\cdots\!48\)\( T^{22} + \)\(40\!\cdots\!59\)\( T^{24} - \)\(20\!\cdots\!90\)\( T^{26} + \)\(81\!\cdots\!21\)\( T^{28} \))
$41$ (\( 1 + 39 T + 1681 T^{2} \))(\( 1 - 1841 T^{2} + 2825761 T^{4} \))(\( ( 1 - 86 T + 6451 T^{2} - 292121 T^{3} + 10844131 T^{4} - 243015446 T^{5} + 4750104241 T^{6} )^{2} \))(\( ( 1 + 7678 T^{2} + 28068223 T^{4} + 59998308292 T^{6} + 79314089892703 T^{8} + 61308255909191038 T^{10} + 22563490300366186081 T^{12} )^{2} \))(\( ( 1 - 25 T + 6491 T^{2} - 125490 T^{3} + 20823001 T^{4} - 270807191 T^{5} + 44179722027 T^{6} - 449185257596 T^{7} + 74266112727387 T^{8} - 765236398847351 T^{9} + 98911425360447241 T^{10} - 1002028267002394290 T^{11} + 87126481582199234891 T^{12} - \)\(56\!\cdots\!25\)\( T^{13} + \)\(37\!\cdots\!61\)\( T^{14} )^{2} \))
$43$ (\( 1 + 43 T \))(\( ( 1 - 43 T )^{2} \))(\( 1 - 10 T + 147 T^{2} + 135020 T^{3} + 271803 T^{4} - 34188010 T^{5} + 6321363049 T^{6} \))(\( ( 1 + 50 T + 3123 T^{2} + 206228 T^{3} + 5774427 T^{4} + 170940050 T^{5} + 6321363049 T^{6} )^{2} \))(\( 1 - 46 T + 4923 T^{2} - 127548 T^{3} + 8944129 T^{4} - 175732658 T^{5} + 11683156115 T^{6} - 256027169288 T^{7} + 21602155656635 T^{8} - 600794986903058 T^{9} + 56539086566089321 T^{10} - 1490806569007452348 T^{11} + \)\(10\!\cdots\!27\)\( T^{12} - \)\(18\!\cdots\!46\)\( T^{13} + \)\(73\!\cdots\!49\)\( T^{14} \))
$47$ (\( 1 - 78 T + 2209 T^{2} \))(\( 1 + 1666 T^{2} + 4879681 T^{4} \))(\( ( 1 + 15 T + 5052 T^{2} + 79770 T^{3} + 11159868 T^{4} + 73195215 T^{5} + 10779215329 T^{6} )^{2} \))(\( ( 1 + 7584 T^{2} + 28835970 T^{4} + 73363427822 T^{6} + 140710334925570 T^{8} + 180584798042795424 T^{10} + \)\(11\!\cdots\!41\)\( T^{12} )^{2} \))(\( ( 1 + 76 T + 9421 T^{2} + 486558 T^{3} + 37453290 T^{4} + 1772397162 T^{5} + 112478821196 T^{6} + 4844498862752 T^{7} + 248465716021964 T^{8} + 8648732755865322 T^{9} + 403717077689482410 T^{10} + 11585572015573108638 T^{11} + \)\(49\!\cdots\!29\)\( T^{12} + \)\(88\!\cdots\!16\)\( T^{13} + \)\(25\!\cdots\!69\)\( T^{14} )^{2} \))
$53$ (\( 1 + 63 T + 2809 T^{2} \))(\( 1 - 1649 T^{2} + 7890481 T^{4} \))(\( ( 1 - 55 T + 4677 T^{2} - 168490 T^{3} + 13137693 T^{4} - 433976455 T^{5} + 22164361129 T^{6} )^{2} \))(\( ( 1 + 11870 T^{2} + 67173439 T^{4} + 233080504292 T^{6} + 530030744134159 T^{8} + 739022525182855070 T^{10} + \)\(49\!\cdots\!41\)\( T^{12} )^{2} \))(\( ( 1 + 135 T + 19255 T^{2} + 1740194 T^{3} + 149657933 T^{4} + 10173177849 T^{5} + 659267373483 T^{6} + 35658979252380 T^{7} + 1851882052113747 T^{8} + 80271266527155369 T^{9} + 3317072472831686357 T^{10} + \)\(10\!\cdots\!34\)\( T^{11} + \)\(33\!\cdots\!95\)\( T^{12} + \)\(66\!\cdots\!35\)\( T^{13} + \)\(13\!\cdots\!69\)\( T^{14} )^{2} \))
$59$ (\( 1 - 54 T + 3481 T^{2} \))(\( 1 - 4046 T^{2} + 12117361 T^{4} \))(\( ( 1 - 6 T + 4271 T^{2} - 147196 T^{3} + 14867351 T^{4} - 72704166 T^{5} + 42180533641 T^{6} )^{2} \))(\( ( 1 + 8314 T^{2} + 57506075 T^{4} + 216996472356 T^{6} + 696821870468075 T^{8} + 1220748258242324794 T^{10} + \)\(17\!\cdots\!81\)\( T^{12} )^{2} \))(\( ( 1 - 122 T + 17963 T^{2} - 1308828 T^{3} + 126371597 T^{4} - 7649411134 T^{5} + 623982716663 T^{6} - 32425945779992 T^{7} + 2172083836703903 T^{8} - 92690676148097374 T^{9} + 5330421398525394677 T^{10} - \)\(19\!\cdots\!88\)\( T^{11} + \)\(91\!\cdots\!63\)\( T^{12} - \)\(21\!\cdots\!82\)\( T^{13} + \)\(61\!\cdots\!61\)\( T^{14} )^{2} \))
$61$ (\( ( 1 - 61 T )( 1 + 61 T ) \))(\( ( 1 - 61 T )^{2}( 1 + 61 T )^{2} \))(\( 1 - 6176 T^{2} + 53251015 T^{4} - 178029474320 T^{6} + 737305086778615 T^{8} - 1183984365071207456 T^{10} + \)\(26\!\cdots\!21\)\( T^{12} \))(\( ( 1 + 298 T^{2} + 31588111 T^{4} + 19337532460 T^{6} + 437363962396351 T^{8} + 57128779273189738 T^{10} + \)\(26\!\cdots\!21\)\( T^{12} )^{2} \))(\( 1 - 33798 T^{2} + 564994107 T^{4} - 6205944738108 T^{6} + 50113713323620665 T^{8} - \)\(31\!\cdots\!30\)\( T^{10} + \)\(15\!\cdots\!07\)\( T^{12} - \)\(65\!\cdots\!80\)\( T^{14} + \)\(21\!\cdots\!87\)\( T^{16} - \)\(60\!\cdots\!30\)\( T^{18} + \)\(13\!\cdots\!65\)\( T^{20} - \)\(22\!\cdots\!88\)\( T^{22} + \)\(28\!\cdots\!07\)\( T^{24} - \)\(23\!\cdots\!18\)\( T^{26} + \)\(97\!\cdots\!81\)\( T^{28} \))
$67$ (\( 1 - 91 T + 4489 T^{2} \))(\( ( 1 + 91 T + 4489 T^{2} )^{2} \))(\( ( 1 + 35 T + 9017 T^{2} + 350730 T^{3} + 40477313 T^{4} + 705289235 T^{5} + 90458382169 T^{6} )^{2} \))(\( ( 1 - 94 T + 12035 T^{2} - 629964 T^{3} + 54025115 T^{4} - 1894205374 T^{5} + 90458382169 T^{6} )^{4} \))(\( ( 1 - 67 T + 20995 T^{2} - 909910 T^{3} + 195521481 T^{4} - 6071498237 T^{5} + 1189501163939 T^{6} - 30048384588084 T^{7} + 5339670724922171 T^{8} - 122347495625073677 T^{9} + 17686556850546872289 T^{10} - \)\(36\!\cdots\!10\)\( T^{11} + \)\(38\!\cdots\!55\)\( T^{12} - \)\(54\!\cdots\!87\)\( T^{13} + \)\(36\!\cdots\!29\)\( T^{14} )^{2} \))
$71$ (\( ( 1 - 71 T )( 1 + 71 T ) \))(\( ( 1 - 71 T )^{2}( 1 + 71 T )^{2} \))(\( 1 - 22246 T^{2} + 239223215 T^{4} - 1523736510420 T^{6} + 6079064027374415 T^{8} - 14365433056093199206 T^{10} + \)\(16\!\cdots\!41\)\( T^{12} \))(\( ( 1 - 8554 T^{2} + 44826155 T^{4} - 144692605764 T^{6} + 1139107951316555 T^{8} - 5523775706276239594 T^{10} + \)\(16\!\cdots\!41\)\( T^{12} )^{2} \))(\( 1 - 35326 T^{2} + 656197363 T^{4} - 8235510824956 T^{6} + 77924894067415761 T^{8} - \)\(59\!\cdots\!06\)\( T^{10} + \)\(37\!\cdots\!79\)\( T^{12} - \)\(20\!\cdots\!40\)\( T^{14} + \)\(95\!\cdots\!99\)\( T^{16} - \)\(38\!\cdots\!66\)\( T^{18} + \)\(12\!\cdots\!01\)\( T^{20} - \)\(34\!\cdots\!76\)\( T^{22} + \)\(69\!\cdots\!63\)\( T^{24} - \)\(95\!\cdots\!06\)\( T^{26} + \)\(68\!\cdots\!61\)\( T^{28} \))
$73$ (\( ( 1 - 73 T )( 1 + 73 T ) \))(\( ( 1 - 73 T )^{2}( 1 + 73 T )^{2} \))(\( 1 - 24604 T^{2} + 277197791 T^{4} - 1859020745064 T^{6} + 7871929673485631 T^{8} - 19842144100961968924 T^{10} + \)\(22\!\cdots\!21\)\( T^{12} \))(\( ( 1 - 23266 T^{2} + 242784035 T^{4} - 1568732835396 T^{6} + 6894639536882435 T^{8} - 18763100498007688546 T^{10} + \)\(22\!\cdots\!21\)\( T^{12} )^{2} \))(\( 1 - 39502 T^{2} + 752139715 T^{4} - 9087927750076 T^{6} + 78026579614304289 T^{8} - \)\(51\!\cdots\!58\)\( T^{10} + \)\(28\!\cdots\!55\)\( T^{12} - \)\(15\!\cdots\!36\)\( T^{14} + \)\(81\!\cdots\!55\)\( T^{16} - \)\(41\!\cdots\!98\)\( T^{18} + \)\(17\!\cdots\!69\)\( T^{20} - \)\(59\!\cdots\!36\)\( T^{22} + \)\(13\!\cdots\!15\)\( T^{24} - \)\(20\!\cdots\!82\)\( T^{26} + \)\(14\!\cdots\!81\)\( T^{28} \))
$79$ (\( 1 + 14 T + 6241 T^{2} \))(\( ( 1 - 14 T + 6241 T^{2} )^{2} \))(\( ( 1 - 89 T + 20896 T^{2} - 1122134 T^{3} + 130411936 T^{4} - 3466557209 T^{5} + 243087455521 T^{6} )^{2} \))(\( ( 1 - 2 T + 6073 T^{2} - 530216 T^{3} + 37901593 T^{4} - 77900162 T^{5} + 243087455521 T^{6} )^{4} \))(\( ( 1 + 30 T + 15435 T^{2} + 932364 T^{3} + 166994349 T^{4} + 8589161514 T^{5} + 1387372139847 T^{6} + 61448902589272 T^{7} + 8658589524785127 T^{8} + 334548536692382634 T^{9} + 40594231384795850829 T^{10} + \)\(14\!\cdots\!04\)\( T^{11} + \)\(14\!\cdots\!35\)\( T^{12} + \)\(17\!\cdots\!30\)\( T^{13} + \)\(36\!\cdots\!81\)\( T^{14} )^{2} \))
$83$ (\( 1 + 123 T + 6889 T^{2} \))(\( 1 + 1351 T^{2} + 47458321 T^{4} \))(\( ( 1 + 5 T + 14767 T^{2} + 128390 T^{3} + 101729863 T^{4} + 237291605 T^{5} + 326940373369 T^{6} )^{2} \))(\( ( 1 + 29896 T^{2} + 438236338 T^{4} + 3805686578182 T^{6} + 20797960802668498 T^{8} + 67334528572028769736 T^{10} + \)\(10\!\cdots\!61\)\( T^{12} )^{2} \))(\( ( 1 - 147 T + 32295 T^{2} - 3413526 T^{3} + 469160262 T^{4} - 40188269148 T^{5} + 4362001508118 T^{6} - 322962400211822 T^{7} + 30049828389424902 T^{8} - 1907267777660180508 T^{9} + \)\(15\!\cdots\!78\)\( T^{10} - \)\(76\!\cdots\!66\)\( T^{11} + \)\(50\!\cdots\!55\)\( T^{12} - \)\(15\!\cdots\!67\)\( T^{13} + \)\(73\!\cdots\!29\)\( T^{14} )^{2} \))
$89$ (\( ( 1 - 89 T )( 1 + 89 T ) \))(\( ( 1 - 89 T )^{2}( 1 + 89 T )^{2} \))(\( 1 - 38876 T^{2} + 668902015 T^{4} - 6712320081320 T^{6} + 41968411430515615 T^{8} - \)\(15\!\cdots\!56\)\( T^{10} + \)\(24\!\cdots\!21\)\( T^{12} \))(\( ( 1 - 29354 T^{2} + 401491435 T^{4} - 3655307590724 T^{6} + 25190472374205835 T^{8} - \)\(11\!\cdots\!74\)\( T^{10} + \)\(24\!\cdots\!21\)\( T^{12} )^{2} \))(\( 1 - 55518 T^{2} + 1593591987 T^{4} - 31321862328060 T^{6} + 472577821360946961 T^{8} - \)\(57\!\cdots\!22\)\( T^{10} + \)\(58\!\cdots\!91\)\( T^{12} - \)\(50\!\cdots\!52\)\( T^{14} + \)\(36\!\cdots\!31\)\( T^{16} - \)\(22\!\cdots\!82\)\( T^{18} + \)\(11\!\cdots\!81\)\( T^{20} - \)\(48\!\cdots\!60\)\( T^{22} + \)\(15\!\cdots\!87\)\( T^{24} - \)\(33\!\cdots\!38\)\( T^{26} + \)\(38\!\cdots\!81\)\( T^{28} \))
$97$ (\( 1 + 193 T + 9409 T^{2} \))(\( ( 1 + 193 T + 9409 T^{2} )^{2} \))(\( ( 1 + 190 T + 26827 T^{2} + 2529295 T^{3} + 252415243 T^{4} + 16820563390 T^{5} + 832972004929 T^{6} )^{2} \))(\( ( 1 - 128 T + 30130 T^{2} - 2323646 T^{3} + 283493170 T^{4} - 11331747968 T^{5} + 832972004929 T^{6} )^{4} \))(\( ( 1 - 267 T + 83769 T^{2} - 14712948 T^{3} + 2641758456 T^{4} - 339877882320 T^{5} + 43330018949466 T^{6} - 4235858138068682 T^{7} + 407692148295525594 T^{8} - 30089144549592211920 T^{9} + \)\(22\!\cdots\!24\)\( T^{10} - \)\(11\!\cdots\!28\)\( T^{11} + \)\(61\!\cdots\!81\)\( T^{12} - \)\(18\!\cdots\!47\)\( T^{13} + \)\(65\!\cdots\!69\)\( T^{14} )^{2} \))
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