Properties

Label 81.4.c
Level $81$
Weight $4$
Character orbit 81.c
Rep. character $\chi_{81}(28,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $22$
Newform subspaces $7$
Sturm bound $36$
Trace bound $4$

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Defining parameters

Level: \( N \) \(=\) \( 81 = 3^{4} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 81.c (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 9 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 7 \)
Sturm bound: \(36\)
Trace bound: \(4\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(81, [\chi])\).

Total New Old
Modular forms 66 26 40
Cusp forms 42 22 20
Eisenstein series 24 4 20

Trace form

\( 22q - 38q^{4} + 32q^{7} + O(q^{10}) \) \( 22q - 38q^{4} + 32q^{7} - 36q^{10} - 58q^{13} - 110q^{16} + 476q^{19} - 18q^{22} - 317q^{25} - 1252q^{28} + 266q^{31} + 1188q^{34} + 692q^{37} + 846q^{40} - 76q^{43} + 1152q^{46} - 1239q^{49} - 1732q^{52} - 4356q^{55} + 684q^{58} - 22q^{61} + 4180q^{64} + 1346q^{67} + 1494q^{70} + 2528q^{73} - 6124q^{76} - 2614q^{79} - 4464q^{82} - 54q^{85} + 5958q^{88} + 6172q^{91} + 6732q^{94} + 86q^{97} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(81, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
81.4.c.a \(2\) \(4.779\) \(\Q(\sqrt{-3}) \) None \(-3\) \(0\) \(-15\) \(25\) \(q+(-3+3\zeta_{6})q^{2}-\zeta_{6}q^{4}-15\zeta_{6}q^{5}+\cdots\)
81.4.c.b \(2\) \(4.779\) \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(-20\) \(q+8\zeta_{6}q^{4}+(-20+20\zeta_{6})q^{7}+70\zeta_{6}q^{13}+\cdots\)
81.4.c.c \(2\) \(4.779\) \(\Q(\sqrt{-3}) \) None \(3\) \(0\) \(15\) \(25\) \(q+(3-3\zeta_{6})q^{2}-\zeta_{6}q^{4}+15\zeta_{6}q^{5}+\cdots\)
81.4.c.d \(4\) \(4.779\) \(\Q(\sqrt{-3}, \sqrt{-19})\) None \(-3\) \(0\) \(12\) \(-10\) \(q+(-1-\beta _{1}-\beta _{3})q^{2}+(7\beta _{1}-3\beta _{2}+\cdots)q^{4}+\cdots\)
81.4.c.e \(4\) \(4.779\) \(\Q(\sqrt{2}, \sqrt{-3})\) None \(0\) \(0\) \(0\) \(-22\) \(q+\beta _{1}q^{2}+10\beta _{2}q^{4}+(4\beta _{1}+4\beta _{3})q^{5}+\cdots\)
81.4.c.f \(4\) \(4.779\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(44\) \(q-\zeta_{12}^{2}q^{2}+(5-5\zeta_{12})q^{4}+(-7\zeta_{12}^{2}+\cdots)q^{5}+\cdots\)
81.4.c.g \(4\) \(4.779\) \(\Q(\sqrt{-3}, \sqrt{-19})\) None \(3\) \(0\) \(-12\) \(-10\) \(q+(1+\beta _{1}+\beta _{3})q^{2}+(7\beta _{1}-3\beta _{2}+3\beta _{3})q^{4}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(81, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(81, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(9, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(27, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 3 T + T^{2} + 24 T^{3} + 64 T^{4} \))(\( 1 - 8 T^{2} + 64 T^{4} \))(\( 1 - 3 T + T^{2} - 24 T^{3} + 64 T^{4} \))(\( 1 + 3 T + 5 T^{2} - 36 T^{3} - 120 T^{4} - 288 T^{5} + 320 T^{6} + 1536 T^{7} + 4096 T^{8} \))(\( 1 + 2 T^{2} - 60 T^{4} + 128 T^{6} + 4096 T^{8} \))(\( 1 - 13 T^{2} + 105 T^{4} - 832 T^{6} + 4096 T^{8} \))(\( 1 - 3 T + 5 T^{2} + 36 T^{3} - 120 T^{4} + 288 T^{5} + 320 T^{6} - 1536 T^{7} + 4096 T^{8} \))
$3$ 1
$5$ (\( 1 + 15 T + 100 T^{2} + 1875 T^{3} + 15625 T^{4} \))(\( 1 - 125 T^{2} + 15625 T^{4} \))(\( 1 - 15 T + 100 T^{2} - 1875 T^{3} + 15625 T^{4} \))(\( 1 - 12 T - 85 T^{2} + 252 T^{3} + 18816 T^{4} + 31500 T^{5} - 1328125 T^{6} - 23437500 T^{7} + 244140625 T^{8} \))(\( 1 + 38 T^{2} - 14181 T^{4} + 593750 T^{6} + 244140625 T^{8} \))(\( 1 - 103 T^{2} - 5016 T^{4} - 1609375 T^{6} + 244140625 T^{8} \))(\( 1 + 12 T - 85 T^{2} - 252 T^{3} + 18816 T^{4} - 31500 T^{5} - 1328125 T^{6} + 23437500 T^{7} + 244140625 T^{8} \))
$7$ (\( 1 - 25 T + 282 T^{2} - 8575 T^{3} + 117649 T^{4} \))(\( ( 1 - 17 T + 343 T^{2} )( 1 + 37 T + 343 T^{2} ) \))(\( 1 - 25 T + 282 T^{2} - 8575 T^{3} + 117649 T^{4} \))(\( 1 + 10 T - 98 T^{2} - 4880 T^{3} - 112745 T^{4} - 1673840 T^{5} - 11529602 T^{6} + 403536070 T^{7} + 13841287201 T^{8} \))(\( ( 1 + 11 T - 222 T^{2} + 3773 T^{3} + 117649 T^{4} )^{2} \))(\( ( 1 - 22 T + 141 T^{2} - 7546 T^{3} + 117649 T^{4} )^{2} \))(\( 1 + 10 T - 98 T^{2} - 4880 T^{3} - 112745 T^{4} - 1673840 T^{5} - 11529602 T^{6} + 403536070 T^{7} + 13841287201 T^{8} \))
$11$ (\( 1 - 15 T - 1106 T^{2} - 19965 T^{3} + 1771561 T^{4} \))(\( 1 - 1331 T^{2} + 1771561 T^{4} \))(\( 1 + 15 T - 1106 T^{2} + 19965 T^{3} + 1771561 T^{4} \))(\( 1 - 42 T - 1282 T^{2} - 16128 T^{3} + 5158671 T^{4} - 21466368 T^{5} - 2271141202 T^{6} - 99033803022 T^{7} + 3138428376721 T^{8} \))(\( 1 - 2374 T^{2} + 3864315 T^{4} - 4205685814 T^{6} + 3138428376721 T^{8} \))(\( 1 + 806 T^{2} - 1121925 T^{4} + 1427878166 T^{6} + 3138428376721 T^{8} \))(\( 1 + 42 T - 1282 T^{2} + 16128 T^{3} + 5158671 T^{4} + 21466368 T^{5} - 2271141202 T^{6} + 99033803022 T^{7} + 3138428376721 T^{8} \))
$13$ (\( 1 + 20 T - 1797 T^{2} + 43940 T^{3} + 4826809 T^{4} \))(\( ( 1 - 89 T + 2197 T^{2} )( 1 + 19 T + 2197 T^{2} ) \))(\( 1 + 20 T - 1797 T^{2} + 43940 T^{3} + 4826809 T^{4} \))(\( 1 + 64 T - 809 T^{2} + 32704 T^{3} + 10233304 T^{4} + 71850688 T^{5} - 3904888481 T^{6} + 678687959872 T^{7} + 23298085122481 T^{8} \))(\( ( 1 + 29 T - 1356 T^{2} + 63713 T^{3} + 4826809 T^{4} )^{2} \))(\( ( 1 - 49 T + 204 T^{2} - 107653 T^{3} + 4826809 T^{4} )^{2} \))(\( 1 + 64 T - 809 T^{2} + 32704 T^{3} + 10233304 T^{4} + 71850688 T^{5} - 3904888481 T^{6} + 678687959872 T^{7} + 23298085122481 T^{8} \))
$17$ (\( ( 1 - 72 T + 4913 T^{2} )^{2} \))(\( ( 1 + 4913 T^{2} )^{2} \))(\( ( 1 + 72 T + 4913 T^{2} )^{2} \))(\( ( 1 - 18 T + 1699 T^{2} - 88434 T^{3} + 24137569 T^{4} )^{2} \))(\( ( 1 + 7234 T^{2} + 24137569 T^{4} )^{2} \))(\( ( 1 + 8503 T^{2} + 24137569 T^{4} )^{2} \))(\( ( 1 + 18 T + 1699 T^{2} + 88434 T^{3} + 24137569 T^{4} )^{2} \))
$19$ (\( ( 1 - 2 T + 6859 T^{2} )^{2} \))(\( ( 1 - 56 T + 6859 T^{2} )^{2} \))(\( ( 1 - 2 T + 6859 T^{2} )^{2} \))(\( ( 1 - 130 T + 13326 T^{2} - 891670 T^{3} + 47045881 T^{4} )^{2} \))(\( ( 1 - 29 T + 6859 T^{2} )^{4} \))(\( ( 1 + 70 T + 6859 T^{2} )^{4} \))(\( ( 1 - 130 T + 13326 T^{2} - 891670 T^{3} + 47045881 T^{4} )^{2} \))
$23$ (\( 1 + 114 T + 829 T^{2} + 1387038 T^{3} + 148035889 T^{4} \))(\( 1 - 12167 T^{2} + 148035889 T^{4} \))(\( 1 - 114 T + 829 T^{2} - 1387038 T^{3} + 148035889 T^{4} \))(\( 1 + 114 T - 7690 T^{2} - 415872 T^{3} + 121752375 T^{4} - 5059914624 T^{5} - 1138395986410 T^{6} + 205331403406782 T^{7} + 21914624432020321 T^{8} \))(\( 1 - 17134 T^{2} + 145538067 T^{4} - 2536446922126 T^{6} + 21914624432020321 T^{8} \))(\( 1 - 23746 T^{2} + 415836627 T^{4} - 3515260220194 T^{6} + 21914624432020321 T^{8} \))(\( 1 - 114 T - 7690 T^{2} + 415872 T^{3} + 121752375 T^{4} + 5059914624 T^{5} - 1138395986410 T^{6} - 205331403406782 T^{7} + 21914624432020321 T^{8} \))
$29$ (\( 1 + 30 T - 23489 T^{2} + 731670 T^{3} + 594823321 T^{4} \))(\( 1 - 24389 T^{2} + 594823321 T^{4} \))(\( 1 - 30 T - 23489 T^{2} - 731670 T^{3} + 594823321 T^{4} \))(\( 1 - 240 T + 4055 T^{2} - 1144080 T^{3} + 867437304 T^{4} - 27902967120 T^{5} + 2412008566655 T^{6} - 3481715034208560 T^{7} + 353814783205469041 T^{8} \))(\( 1 + 24950 T^{2} + 27679179 T^{4} + 14840841858950 T^{6} + 353814783205469041 T^{8} \))(\( 1 + 28985 T^{2} + 245306904 T^{4} + 17240953959185 T^{6} + 353814783205469041 T^{8} \))(\( 1 + 240 T + 4055 T^{2} + 1144080 T^{3} + 867437304 T^{4} + 27902967120 T^{5} + 2412008566655 T^{6} + 3481715034208560 T^{7} + 353814783205469041 T^{8} \))
$31$ (\( 1 + 101 T - 19590 T^{2} + 3008891 T^{3} + 887503681 T^{4} \))(\( ( 1 + 19 T + 29791 T^{2} )( 1 + 289 T + 29791 T^{2} ) \))(\( 1 + 101 T - 19590 T^{2} + 3008891 T^{3} + 887503681 T^{4} \))(\( 1 - 8 T - 26702 T^{2} + 262528 T^{3} - 172991549 T^{4} + 7820971648 T^{5} - 23698123290062 T^{6} - 211516977285368 T^{7} + 787662783788549761 T^{8} \))(\( ( 1 - 268 T + 42033 T^{2} - 7983988 T^{3} + 887503681 T^{4} )^{2} \))(\( ( 1 - 112 T - 17247 T^{2} - 3336592 T^{3} + 887503681 T^{4} )^{2} \))(\( 1 - 8 T - 26702 T^{2} + 262528 T^{3} - 172991549 T^{4} + 7820971648 T^{5} - 23698123290062 T^{6} - 211516977285368 T^{7} + 787662783788549761 T^{8} \))
$37$ (\( ( 1 + 430 T + 50653 T^{2} )^{2} \))(\( ( 1 - 110 T + 50653 T^{2} )^{2} \))(\( ( 1 + 430 T + 50653 T^{2} )^{2} \))(\( ( 1 - 184 T + 105153 T^{2} - 9320152 T^{3} + 2565726409 T^{4} )^{2} \))(\( ( 1 - 83 T + 50653 T^{2} )^{4} \))(\( ( 1 - 281 T + 50653 T^{2} )^{4} \))(\( ( 1 - 184 T + 105153 T^{2} - 9320152 T^{3} + 2565726409 T^{4} )^{2} \))
$41$ (\( 1 - 30 T - 68021 T^{2} - 2067630 T^{3} + 4750104241 T^{4} \))(\( 1 - 68921 T^{2} + 4750104241 T^{4} \))(\( 1 + 30 T - 68021 T^{2} + 2067630 T^{3} + 4750104241 T^{4} \))(\( 1 + 672 T + 212018 T^{2} + 68358528 T^{3} + 21518143251 T^{4} + 4711338108288 T^{5} + 1007107600968338 T^{6} + 220000659912741792 T^{7} + 22563490300366186081 T^{8} \))(\( 1 - 64114 T^{2} - 639499245 T^{4} - 304548183307474 T^{6} + 22563490300366186081 T^{8} \))(\( 1 - 135490 T^{2} + 13607435859 T^{4} - 643591623613090 T^{6} + 22563490300366186081 T^{8} \))(\( 1 - 672 T + 212018 T^{2} - 68358528 T^{3} + 21518143251 T^{4} - 4711338108288 T^{5} + 1007107600968338 T^{6} - 220000659912741792 T^{7} + 22563490300366186081 T^{8} \))
$43$ (\( 1 + 110 T - 67407 T^{2} + 8745770 T^{3} + 6321363049 T^{4} \))(\( ( 1 - 449 T + 79507 T^{2} )( 1 - 71 T + 79507 T^{2} ) \))(\( 1 + 110 T - 67407 T^{2} + 8745770 T^{3} + 6321363049 T^{4} \))(\( 1 + 370 T - 55826 T^{2} + 12473440 T^{3} + 19937439727 T^{4} + 991725794080 T^{5} - 352896413573474 T^{6} + 185959266416631910 T^{7} + 39959630797262576401 T^{8} \))(\( ( 1 - 232 T - 25683 T^{2} - 18445624 T^{3} + 6321363049 T^{4} )^{2} \))(\( ( 1 + 50 T - 77007 T^{2} + 3975350 T^{3} + 6321363049 T^{4} )^{2} \))(\( 1 + 370 T - 55826 T^{2} + 12473440 T^{3} + 19937439727 T^{4} + 991725794080 T^{5} - 352896413573474 T^{6} + 185959266416631910 T^{7} + 39959630797262576401 T^{8} \))
$47$ (\( 1 - 330 T + 5077 T^{2} - 34261590 T^{3} + 10779215329 T^{4} \))(\( 1 - 103823 T^{2} + 10779215329 T^{4} \))(\( 1 + 330 T + 5077 T^{2} + 34261590 T^{3} + 10779215329 T^{4} \))(\( 1 - 276 T - 150286 T^{2} - 5193216 T^{3} + 32597001267 T^{4} - 539175264768 T^{5} - 1619965154934094 T^{6} - 308880010576363692 T^{7} + \)\(11\!\cdots\!41\)\( T^{8} \))(\( 1 - 55294 T^{2} - 7721788893 T^{4} - 596025932401726 T^{6} + \)\(11\!\cdots\!41\)\( T^{8} \))(\( 1 - 148846 T^{2} + 11375916387 T^{4} - 1604443084860334 T^{6} + \)\(11\!\cdots\!41\)\( T^{8} \))(\( 1 + 276 T - 150286 T^{2} + 5193216 T^{3} + 32597001267 T^{4} + 539175264768 T^{5} - 1619965154934094 T^{6} + 308880010576363692 T^{7} + \)\(11\!\cdots\!41\)\( T^{8} \))
$53$ (\( ( 1 - 621 T + 148877 T^{2} )^{2} \))(\( ( 1 + 148877 T^{2} )^{2} \))(\( ( 1 + 621 T + 148877 T^{2} )^{2} \))(\( ( 1 - 162 T + 148877 T^{2} )^{4} \))(\( ( 1 + 204442 T^{2} + 22164361129 T^{4} )^{2} \))(\( ( 1 + 157786 T^{2} + 22164361129 T^{4} )^{2} \))(\( ( 1 + 162 T + 148877 T^{2} )^{4} \))
$59$ (\( 1 - 660 T + 230221 T^{2} - 135550140 T^{3} + 42180533641 T^{4} \))(\( 1 - 205379 T^{2} + 42180533641 T^{4} \))(\( 1 + 660 T + 230221 T^{2} + 135550140 T^{3} + 42180533641 T^{4} \))(\( 1 + 204 T - 258934 T^{2} - 22482432 T^{3} + 39602716395 T^{4} - 4617419401728 T^{5} - 10921974297798694 T^{6} + 1767251147005607556 T^{7} + \)\(17\!\cdots\!81\)\( T^{8} \))(\( 1 - 327526 T^{2} + 65092747035 T^{4} - 13815221461302166 T^{6} + \)\(17\!\cdots\!81\)\( T^{8} \))(\( 1 - 401350 T^{2} + 118901288859 T^{4} - 16929157176815350 T^{6} + \)\(17\!\cdots\!81\)\( T^{8} \))(\( 1 - 204 T - 258934 T^{2} + 22482432 T^{3} + 39602716395 T^{4} + 4617419401728 T^{5} - 10921974297798694 T^{6} - 1767251147005607556 T^{7} + \)\(17\!\cdots\!81\)\( T^{8} \))
$61$ (\( 1 - 376 T - 85605 T^{2} - 85344856 T^{3} + 51520374361 T^{4} \))(\( ( 1 - 719 T + 226981 T^{2} )( 1 + 901 T + 226981 T^{2} ) \))(\( 1 - 376 T - 85605 T^{2} - 85344856 T^{3} + 51520374361 T^{4} \))(\( 1 + 208 T - 150137 T^{2} - 54196688 T^{3} - 23936533544 T^{4} - 12301618438928 T^{5} - 7735114445437457 T^{6} + 2432382387309501328 T^{7} + \)\(26\!\cdots\!21\)\( T^{8} \))(\( ( 1 + 767 T + 361308 T^{2} + 174094427 T^{3} + 51520374361 T^{4} )^{2} \))(\( ( 1 - 679 T + 234060 T^{2} - 154120099 T^{3} + 51520374361 T^{4} )^{2} \))(\( 1 + 208 T - 150137 T^{2} - 54196688 T^{3} - 23936533544 T^{4} - 12301618438928 T^{5} - 7735114445437457 T^{6} + 2432382387309501328 T^{7} + \)\(26\!\cdots\!21\)\( T^{8} \))
$67$ (\( 1 - 250 T - 238263 T^{2} - 75190750 T^{3} + 90458382169 T^{4} \))(\( ( 1 - 1007 T + 300763 T^{2} )( 1 + 127 T + 300763 T^{2} ) \))(\( 1 - 250 T - 238263 T^{2} - 75190750 T^{3} + 90458382169 T^{4} \))(\( 1 + 802 T - 32426 T^{2} + 59431408 T^{3} + 172565550079 T^{4} + 17874768564304 T^{5} - 2933203500211994 T^{6} + 21819640585828547494 T^{7} + \)\(81\!\cdots\!61\)\( T^{8} \))(\( ( 1 - 511 T - 39642 T^{2} - 153689893 T^{3} + 90458382169 T^{4} )^{2} \))(\( ( 1 - 274 T - 225687 T^{2} - 82409062 T^{3} + 90458382169 T^{4} )^{2} \))(\( 1 + 802 T - 32426 T^{2} + 59431408 T^{3} + 172565550079 T^{4} + 17874768564304 T^{5} - 2933203500211994 T^{6} + 21819640585828547494 T^{7} + \)\(81\!\cdots\!61\)\( T^{8} \))
$71$ (\( ( 1 + 360 T + 357911 T^{2} )^{2} \))(\( ( 1 + 357911 T^{2} )^{2} \))(\( ( 1 - 360 T + 357911 T^{2} )^{2} \))(\( ( 1 - 126 T + 534598 T^{2} - 45096786 T^{3} + 128100283921 T^{4} )^{2} \))(\( ( 1 + 207790 T^{2} + 128100283921 T^{4} )^{2} \))(\( ( 1 + 516130 T^{2} + 128100283921 T^{4} )^{2} \))(\( ( 1 + 126 T + 534598 T^{2} + 45096786 T^{3} + 128100283921 T^{4} )^{2} \))
$73$ (\( ( 1 - 785 T + 389017 T^{2} )^{2} \))(\( ( 1 - 1190 T + 389017 T^{2} )^{2} \))(\( ( 1 - 785 T + 389017 T^{2} )^{2} \))(\( ( 1 + 374 T + 646791 T^{2} + 145492358 T^{3} + 151334226289 T^{4} )^{2} \))(\( ( 1 - 137 T + 389017 T^{2} )^{4} \))(\( ( 1 + 511 T + 389017 T^{2} )^{4} \))(\( ( 1 + 374 T + 646791 T^{2} + 145492358 T^{3} + 151334226289 T^{4} )^{2} \))
$79$ (\( 1 + 488 T - 254895 T^{2} + 240603032 T^{3} + 243087455521 T^{4} \))(\( ( 1 - 503 T + 493039 T^{2} )( 1 + 1387 T + 493039 T^{2} ) \))(\( 1 + 488 T - 254895 T^{2} + 240603032 T^{3} + 243087455521 T^{4} \))(\( 1 + 1378 T + 438598 T^{2} + 653458624 T^{3} + 953123877799 T^{4} + 322180586518336 T^{5} + 106617671816599558 T^{6} + \)\(16\!\cdots\!82\)\( T^{7} + \)\(59\!\cdots\!41\)\( T^{8} \))(\( ( 1 - 475 T - 267414 T^{2} - 234193525 T^{3} + 243087455521 T^{4} )^{2} \))(\( ( 1 - 526 T - 216363 T^{2} - 259338514 T^{3} + 243087455521 T^{4} )^{2} \))(\( 1 + 1378 T + 438598 T^{2} + 653458624 T^{3} + 953123877799 T^{4} + 322180586518336 T^{5} + 106617671816599558 T^{6} + \)\(16\!\cdots\!82\)\( T^{7} + \)\(59\!\cdots\!41\)\( T^{8} \))
$83$ (\( 1 + 489 T - 332666 T^{2} + 279603843 T^{3} + 326940373369 T^{4} \))(\( 1 - 571787 T^{2} + 326940373369 T^{4} \))(\( 1 - 489 T - 332666 T^{2} - 279603843 T^{3} + 326940373369 T^{4} \))(\( 1 + 1164 T + 15098 T^{2} + 228404736 T^{3} + 693406388283 T^{4} + 130598858783232 T^{5} + 4936145757125162 T^{6} + \)\(21\!\cdots\!92\)\( T^{7} + \)\(10\!\cdots\!61\)\( T^{8} \))(\( 1 - 810646 T^{2} + 330206563947 T^{4} - 265032905910086374 T^{6} + \)\(10\!\cdots\!61\)\( T^{8} \))(\( 1 - 993046 T^{2} + 659199984747 T^{4} - 324666830012591974 T^{6} + \)\(10\!\cdots\!61\)\( T^{8} \))(\( 1 - 1164 T + 15098 T^{2} - 228404736 T^{3} + 693406388283 T^{4} - 130598858783232 T^{5} + 4936145757125162 T^{6} - \)\(21\!\cdots\!92\)\( T^{7} + \)\(10\!\cdots\!61\)\( T^{8} \))
$89$ (\( ( 1 + 450 T + 704969 T^{2} )^{2} \))(\( ( 1 + 704969 T^{2} )^{2} \))(\( ( 1 - 450 T + 704969 T^{2} )^{2} \))(\( ( 1 - 954 T + 816667 T^{2} - 672540426 T^{3} + 496981290961 T^{4} )^{2} \))(\( ( 1 + 1345138 T^{2} + 496981290961 T^{4} )^{2} \))(\( ( 1 + 1408615 T^{2} + 496981290961 T^{4} )^{2} \))(\( ( 1 + 954 T + 816667 T^{2} + 672540426 T^{3} + 496981290961 T^{4} )^{2} \))
$97$ (\( 1 - 1105 T + 308352 T^{2} - 1008503665 T^{3} + 832972004929 T^{4} \))(\( ( 1 - 1853 T + 912673 T^{2} )( 1 + 523 T + 912673 T^{2} ) \))(\( 1 - 1105 T + 308352 T^{2} - 1008503665 T^{3} + 832972004929 T^{4} \))(\( 1 - 872 T - 1203758 T^{2} - 121030112 T^{3} + 2330899844803 T^{4} - 110460915409376 T^{5} - 1002696714709323182 T^{6} - \)\(66\!\cdots\!24\)\( T^{7} + \)\(69\!\cdots\!41\)\( T^{8} \))(\( ( 1 + 821 T - 238632 T^{2} + 749304533 T^{3} + 832972004929 T^{4} )^{2} \))(\( ( 1 + 1778 T + 2248611 T^{2} + 1622732594 T^{3} + 832972004929 T^{4} )^{2} \))(\( 1 - 872 T - 1203758 T^{2} - 121030112 T^{3} + 2330899844803 T^{4} - 110460915409376 T^{5} - 1002696714709323182 T^{6} - \)\(66\!\cdots\!24\)\( T^{7} + \)\(69\!\cdots\!41\)\( T^{8} \))
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