Properties

Label 387.2.h
Level 387
Weight 2
Character orbit h
Rep. character \(\chi_{387}(208,\cdot)\)
Character field \(\Q(\zeta_{3})\)
Dimension 34
Newform subspaces 7
Sturm bound 88
Trace bound 2

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Defining parameters

Level: \( N \) \(=\) \( 387 = 3^{2} \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 387.h (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 43 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 7 \)
Sturm bound: \(88\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(387, [\chi])\).

Total New Old
Modular forms 96 38 58
Cusp forms 80 34 46
Eisenstein series 16 4 12

Trace form

\( 34q + 8q^{2} + 28q^{4} + q^{5} + 3q^{7} + 18q^{8} + O(q^{10}) \) \( 34q + 8q^{2} + 28q^{4} + q^{5} + 3q^{7} + 18q^{8} + q^{10} - 6q^{11} - 2q^{13} - 6q^{14} + 16q^{16} + 4q^{17} + q^{19} - 9q^{20} - 14q^{22} - 2q^{23} - 4q^{25} + 9q^{26} + 2q^{28} - 7q^{29} + q^{31} + 24q^{32} + 2q^{34} - 2q^{35} + 10q^{37} + 5q^{38} - 9q^{40} + 4q^{41} + 26q^{43} + 44q^{44} - 6q^{46} - 10q^{47} + 14q^{49} - 25q^{50} - 12q^{52} - 8q^{53} - 24q^{55} - 7q^{56} - 34q^{58} - 34q^{59} + 10q^{61} - 39q^{62} - 26q^{64} + 38q^{65} - 6q^{67} - 37q^{68} - 70q^{70} - 12q^{71} + 14q^{73} - 36q^{74} - 33q^{76} - 5q^{77} - 22q^{79} - 17q^{80} - 22q^{82} - 6q^{83} + 26q^{85} - 73q^{86} - 126q^{88} + 30q^{89} - 18q^{91} + 11q^{92} - 138q^{94} - q^{95} + 32q^{97} + 18q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(387, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
387.2.h.a \(2\) \(3.090\) \(\Q(\sqrt{-3}) \) None \(-2\) \(0\) \(-1\) \(-3\) \(q-q^{2}-q^{4}+(-1+\zeta_{6})q^{5}-3\zeta_{6}q^{7}+\cdots\)
387.2.h.b \(2\) \(3.090\) \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(1\) \(q-2q^{4}+\zeta_{6}q^{7}+7\zeta_{6}q^{13}+4q^{16}+\cdots\)
387.2.h.c \(2\) \(3.090\) \(\Q(\sqrt{-3}) \) None \(4\) \(0\) \(2\) \(-3\) \(q+2q^{2}+2q^{4}+(2-2\zeta_{6})q^{5}-3\zeta_{6}q^{7}+\cdots\)
387.2.h.d \(4\) \(3.090\) \(\Q(\sqrt{-3}, \sqrt{5})\) None \(6\) \(0\) \(-2\) \(4\) \(q+(1-\beta _{2})q^{2}-3\beta _{2}q^{4}-2\beta _{1}q^{5}+(2\beta _{1}+\cdots)q^{7}+\cdots\)
387.2.h.e \(6\) \(3.090\) 6.0.64827.1 None \(-2\) \(0\) \(-1\) \(5\) \(q-\beta _{2}q^{2}+(-1+\beta _{2}-\beta _{3})q^{4}+(\beta _{1}+\cdots)q^{5}+\cdots\)
387.2.h.f \(6\) \(3.090\) 6.0.1783323.2 None \(2\) \(0\) \(3\) \(-3\) \(q-\beta _{2}q^{2}+(1+\beta _{3})q^{4}+(\beta _{3}+\beta _{4}-\beta _{5})q^{5}+\cdots\)
387.2.h.g \(12\) \(3.090\) \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None \(0\) \(0\) \(0\) \(2\) \(q+(-\beta _{1}-\beta _{6})q^{2}+(2+\beta _{2}+\beta _{4}+\beta _{7}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(387, [\chi])\) into lower level spaces

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 + T + 2 T^{2} )^{2} \))(\( ( 1 + 2 T^{2} )^{2} \))(\( ( 1 - 2 T + 2 T^{2} )^{2} \))(\( ( 1 - 3 T + 5 T^{2} - 6 T^{3} + 4 T^{4} )^{2} \))(\( ( 1 + T + 4 T^{2} + 3 T^{3} + 8 T^{4} + 4 T^{5} + 8 T^{6} )^{2} \))(\( ( 1 - T + 2 T^{2} - T^{3} + 4 T^{4} - 4 T^{5} + 8 T^{6} )^{2} \))(\( ( 1 + T^{2} + 4 T^{4} + 5 T^{6} + 16 T^{8} + 16 T^{10} + 64 T^{12} )^{2} \))
$3$ 1
$5$ (\( 1 + T - 4 T^{2} + 5 T^{3} + 25 T^{4} \))(\( 1 - 5 T^{2} + 25 T^{4} \))(\( 1 - 2 T - T^{2} - 10 T^{3} + 25 T^{4} \))(\( 1 + 2 T - 2 T^{2} - 8 T^{3} - 9 T^{4} - 40 T^{5} - 50 T^{6} + 250 T^{7} + 625 T^{8} \))(\( 1 + T + 2 T^{2} + 37 T^{3} + 25 T^{4} + 54 T^{5} + 581 T^{6} + 270 T^{7} + 625 T^{8} + 4625 T^{9} + 1250 T^{10} + 3125 T^{11} + 15625 T^{12} \))(\( 1 - 3 T - 4 T^{2} + 7 T^{3} + 35 T^{4} + 16 T^{5} - 291 T^{6} + 80 T^{7} + 875 T^{8} + 875 T^{9} - 2500 T^{10} - 9375 T^{11} + 15625 T^{12} \))(\( 1 - 17 T^{2} + 140 T^{4} - 791 T^{6} + 3669 T^{8} - 13754 T^{10} + 54241 T^{12} - 343850 T^{14} + 2293125 T^{16} - 12359375 T^{18} + 54687500 T^{20} - 166015625 T^{22} + 244140625 T^{24} \))
$7$ (\( 1 + 3 T + 2 T^{2} + 21 T^{3} + 49 T^{4} \))(\( ( 1 - 5 T + 7 T^{2} )( 1 + 4 T + 7 T^{2} ) \))(\( 1 + 3 T + 2 T^{2} + 21 T^{3} + 49 T^{4} \))(\( 1 - 4 T + 3 T^{2} + 4 T^{3} + 8 T^{4} + 28 T^{5} + 147 T^{6} - 1372 T^{7} + 2401 T^{8} \))(\( 1 - 5 T - 2 T^{2} + 7 T^{3} + 185 T^{4} - 272 T^{5} - 601 T^{6} - 1904 T^{7} + 9065 T^{8} + 2401 T^{9} - 4802 T^{10} - 84035 T^{11} + 117649 T^{12} \))(\( 1 + 3 T - 10 T^{2} - 25 T^{3} + 105 T^{4} + 128 T^{5} - 601 T^{6} + 896 T^{7} + 5145 T^{8} - 8575 T^{9} - 24010 T^{10} + 50421 T^{11} + 117649 T^{12} \))(\( ( 1 - T - 10 T^{2} + 35 T^{3} + 21 T^{4} - 148 T^{5} + 319 T^{6} - 1036 T^{7} + 1029 T^{8} + 12005 T^{9} - 24010 T^{10} - 16807 T^{11} + 117649 T^{12} )^{2} \))
$11$ (\( ( 1 + 11 T^{2} )^{2} \))(\( ( 1 + 11 T^{2} )^{2} \))(\( ( 1 - 6 T + 11 T^{2} )^{2} \))(\( ( 1 - 5 T + 27 T^{2} - 55 T^{3} + 121 T^{4} )^{2} \))(\( ( 1 + 10 T + 50 T^{2} + 179 T^{3} + 550 T^{4} + 1210 T^{5} + 1331 T^{6} )^{2} \))(\( ( 1 + 4 T + 34 T^{2} + 83 T^{3} + 374 T^{4} + 484 T^{5} + 1331 T^{6} )^{2} \))(\( ( 1 + 24 T^{2} + 406 T^{4} + 5615 T^{6} + 49126 T^{8} + 351384 T^{10} + 1771561 T^{12} )^{2} \))
$13$ (\( ( 1 - 7 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} ) \))(\( ( 1 - 5 T + 13 T^{2} )( 1 - 2 T + 13 T^{2} ) \))(\( 1 + T - 12 T^{2} + 13 T^{3} + 169 T^{4} \))(\( 1 + 5 T - 6 T^{2} + 25 T^{3} + 467 T^{4} + 325 T^{5} - 1014 T^{6} + 10985 T^{7} + 28561 T^{8} \))(\( 1 + 3 T - 12 T^{2} - 39 T^{3} + 15 T^{4} - 60 T^{5} - 259 T^{6} - 780 T^{7} + 2535 T^{8} - 85683 T^{9} - 342732 T^{10} + 1113879 T^{11} + 4826809 T^{12} \))(\( 1 - T - 12 T^{2} + 69 T^{3} - 41 T^{4} - 364 T^{5} + 3709 T^{6} - 4732 T^{7} - 6929 T^{8} + 151593 T^{9} - 342732 T^{10} - 371293 T^{11} + 4826809 T^{12} \))(\( ( 1 + 3 T - 28 T^{2} - 43 T^{3} + 657 T^{4} + 470 T^{5} - 8995 T^{6} + 6110 T^{7} + 111033 T^{8} - 94471 T^{9} - 799708 T^{10} + 1113879 T^{11} + 4826809 T^{12} )^{2} \))
$17$ (\( 1 - 3 T - 8 T^{2} - 51 T^{3} + 289 T^{4} \))(\( 1 - 17 T^{2} + 289 T^{4} \))(\( 1 - 6 T + 19 T^{2} - 102 T^{3} + 289 T^{4} \))(\( 1 - T - 2 T^{2} + 31 T^{3} - 297 T^{4} + 527 T^{5} - 578 T^{6} - 4913 T^{7} + 83521 T^{8} \))(\( 1 - 4 T - 24 T^{2} + 110 T^{3} + 380 T^{4} - 1116 T^{5} - 3773 T^{6} - 18972 T^{7} + 109820 T^{8} + 540430 T^{9} - 2004504 T^{10} - 5679428 T^{11} + 24137569 T^{12} \))(\( 1 + 10 T + 20 T^{2} + 78 T^{3} + 1430 T^{4} + 4570 T^{5} + 2447 T^{6} + 77690 T^{7} + 413270 T^{8} + 383214 T^{9} + 1670420 T^{10} + 14198570 T^{11} + 24137569 T^{12} \))(\( 1 - 60 T^{2} + 1682 T^{4} - 34706 T^{6} + 713202 T^{8} - 15573686 T^{10} + 299443271 T^{12} - 4500795254 T^{14} + 59567344242 T^{16} - 837718469714 T^{18} + 11733224015762 T^{20} - 120959634026940 T^{22} + 582622237229761 T^{24} \))
$19$ (\( ( 1 - 7 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \))(\( ( 1 + T + 19 T^{2} )( 1 + 7 T + 19 T^{2} ) \))(\( 1 + 4 T - 3 T^{2} + 76 T^{3} + 361 T^{4} \))(\( 1 + 2 T - 30 T^{2} - 8 T^{3} + 719 T^{4} - 152 T^{5} - 10830 T^{6} + 13718 T^{7} + 130321 T^{8} \))(\( 1 + 5 T - 10 T^{2} - 11 T^{3} + 95 T^{4} - 1590 T^{5} - 9933 T^{6} - 30210 T^{7} + 34295 T^{8} - 75449 T^{9} - 1303210 T^{10} + 12380495 T^{11} + 47045881 T^{12} \))(\( 1 - 3 T - 46 T^{2} + 49 T^{3} + 1659 T^{4} - 698 T^{5} - 35557 T^{6} - 13262 T^{7} + 598899 T^{8} + 336091 T^{9} - 5994766 T^{10} - 7428297 T^{11} + 47045881 T^{12} \))(\( ( 1 - 9 T + 38 T^{2} - 97 T^{3} - 273 T^{4} + 5222 T^{5} - 29629 T^{6} + 99218 T^{7} - 98553 T^{8} - 665323 T^{9} + 4952198 T^{10} - 22284891 T^{11} + 47045881 T^{12} )^{2} \))
$23$ (\( 1 + 7 T + 26 T^{2} + 161 T^{3} + 529 T^{4} \))(\( 1 - 23 T^{2} + 529 T^{4} \))(\( 1 + 4 T - 7 T^{2} + 92 T^{3} + 529 T^{4} \))(\( 1 - 11 T + 46 T^{2} - 319 T^{3} + 2313 T^{4} - 7337 T^{5} + 24334 T^{6} - 133837 T^{7} + 279841 T^{8} \))(\( 1 - 2 T - 36 T^{2} - 38 T^{3} + 638 T^{4} + 1782 T^{5} - 14141 T^{6} + 40986 T^{7} + 337502 T^{8} - 462346 T^{9} - 10074276 T^{10} - 12872686 T^{11} + 148035889 T^{12} \))(\( 1 + 4 T - 50 T^{2} - 86 T^{3} + 2138 T^{4} + 1522 T^{5} - 53693 T^{6} + 35006 T^{7} + 1131002 T^{8} - 1046362 T^{9} - 13992050 T^{10} + 25745372 T^{11} + 148035889 T^{12} \))(\( 1 - 110 T^{2} + 6548 T^{4} - 284066 T^{6} + 9921726 T^{8} - 291432254 T^{10} + 7279436191 T^{12} - 154167662366 T^{14} + 2776505725566 T^{16} - 42051962844674 T^{18} + 512780331619988 T^{20} - 4556916233501390 T^{22} + 21914624432020321 T^{24} \))
$29$ (\( 1 - 3 T - 20 T^{2} - 87 T^{3} + 841 T^{4} \))(\( 1 - 29 T^{2} + 841 T^{4} \))(\( 1 + 6 T + 7 T^{2} + 174 T^{3} + 841 T^{4} \))(\( ( 1 - 3 T - 20 T^{2} - 87 T^{3} + 841 T^{4} )^{2} \))(\( 1 + 4 T - 60 T^{2} - 158 T^{3} + 2648 T^{4} + 3288 T^{5} - 78821 T^{6} + 95352 T^{7} + 2226968 T^{8} - 3853462 T^{9} - 42436860 T^{10} + 82044596 T^{11} + 594823321 T^{12} \))(\( 1 + 6 T + 30 T^{2} + 366 T^{3} + 852 T^{4} + 1968 T^{5} + 44683 T^{6} + 57072 T^{7} + 716532 T^{8} + 8926374 T^{9} + 21218430 T^{10} + 123066894 T^{11} + 594823321 T^{12} \))(\( 1 + 6 T^{2} - 528 T^{4} - 62770 T^{6} - 354690 T^{8} + 17207226 T^{10} + 1990753719 T^{12} + 14471277066 T^{14} - 250865497890 T^{16} - 37337059859170 T^{18} - 264130106043408 T^{20} + 2524243399801206 T^{22} + 353814783205469041 T^{24} \))
$31$ (\( 1 + 5 T - 6 T^{2} + 155 T^{3} + 961 T^{4} \))(\( ( 1 - 11 T + 31 T^{2} )( 1 + 4 T + 31 T^{2} ) \))(\( 1 + 5 T - 6 T^{2} + 155 T^{3} + 961 T^{4} \))(\( ( 1 - 31 T^{2} + 961 T^{4} )^{2} \))(\( 1 + T - 76 T^{2} + 11 T^{3} + 3509 T^{4} - 1272 T^{5} - 124089 T^{6} - 39432 T^{7} + 3372149 T^{8} + 327701 T^{9} - 70187596 T^{10} + 28629151 T^{11} + 887503681 T^{12} \))(\( 1 - 3 T - 48 T^{2} + 211 T^{3} + 909 T^{4} - 3168 T^{5} - 13305 T^{6} - 98208 T^{7} + 873549 T^{8} + 6285901 T^{9} - 44329008 T^{10} - 85887453 T^{11} + 887503681 T^{12} \))(\( ( 1 - T - 68 T^{2} - 35 T^{3} + 2605 T^{4} + 2144 T^{5} - 88681 T^{6} + 66464 T^{7} + 2503405 T^{8} - 1042685 T^{9} - 62799428 T^{10} - 28629151 T^{11} + 887503681 T^{12} )^{2} \))
$37$ (\( 1 - 9 T + 44 T^{2} - 333 T^{3} + 1369 T^{4} \))(\( ( 1 + T + 37 T^{2} )( 1 + 10 T + 37 T^{2} ) \))(\( 1 + 3 T - 28 T^{2} + 111 T^{3} + 1369 T^{4} \))(\( 1 - 3 T - 56 T^{2} + 27 T^{3} + 2523 T^{4} + 999 T^{5} - 76664 T^{6} - 151959 T^{7} + 1874161 T^{8} \))(\( 1 + 9 T - 8 T^{2} - 193 T^{3} + 155 T^{4} - 2720 T^{5} - 55651 T^{6} - 100640 T^{7} + 212195 T^{8} - 9776029 T^{9} - 14993288 T^{10} + 624095613 T^{11} + 2565726409 T^{12} \))(\( 1 + 9 T - 16 T^{2} - 65 T^{3} + 1599 T^{4} - 6788 T^{5} - 137755 T^{6} - 251156 T^{7} + 2189031 T^{8} - 3292445 T^{9} - 29986576 T^{10} + 624095613 T^{11} + 2565726409 T^{12} \))(\( ( 1 - 15 T + 78 T^{2} - 39 T^{3} - 75 T^{4} - 16566 T^{5} + 166637 T^{6} - 612942 T^{7} - 102675 T^{8} - 1975467 T^{9} + 146184558 T^{10} - 1040159355 T^{11} + 2565726409 T^{12} )^{2} \))
$41$ (\( ( 1 - 10 T + 41 T^{2} )^{2} \))(\( ( 1 + 41 T^{2} )^{2} \))(\( ( 1 + 2 T + 41 T^{2} )^{2} \))(\( ( 1 + 10 T + 87 T^{2} + 410 T^{3} + 1681 T^{4} )^{2} \))(\( ( 1 + 6 T + 2 T^{2} - 67 T^{3} + 82 T^{4} + 10086 T^{5} + 68921 T^{6} )^{2} \))(\( ( 1 - 10 T + 100 T^{2} - 575 T^{3} + 4100 T^{4} - 16810 T^{5} + 68921 T^{6} )^{2} \))(\( ( 1 + 134 T^{2} + 9422 T^{4} + 448063 T^{6} + 15838382 T^{8} + 378651974 T^{10} + 4750104241 T^{12} )^{2} \))
$43$ (\( 1 + 8 T + 43 T^{2} \))(\( 1 - 8 T + 43 T^{2} \))(\( 1 + 8 T + 43 T^{2} \))(\( ( 1 + 13 T + 43 T^{2} )^{2} \))(\( 1 - 18 T + 216 T^{2} - 1645 T^{3} + 9288 T^{4} - 33282 T^{5} + 79507 T^{6} \))(\( 1 - 2 T - 4 T^{2} - 221 T^{3} - 172 T^{4} - 3698 T^{5} + 79507 T^{6} \))(\( ( 1 - 20 T + 182 T^{2} - 1205 T^{3} + 7826 T^{4} - 36980 T^{5} + 79507 T^{6} )^{2} \))
$47$ (\( ( 1 - 8 T + 47 T^{2} )^{2} \))(\( ( 1 + 47 T^{2} )^{2} \))(\( ( 1 - 8 T + 47 T^{2} )^{2} \))(\( ( 1 + 9 T + 103 T^{2} + 423 T^{3} + 2209 T^{4} )^{2} \))(\( ( 1 + 4 T + 74 T^{2} + 347 T^{3} + 3478 T^{4} + 8836 T^{5} + 103823 T^{6} )^{2} \))(\( ( 1 + 8 T + 60 T^{2} + 691 T^{3} + 2820 T^{4} + 17672 T^{5} + 103823 T^{6} )^{2} \))(\( ( 1 + 134 T^{2} + 6814 T^{4} + 254611 T^{6} + 15052126 T^{8} + 653877254 T^{10} + 10779215329 T^{12} )^{2} \))
$53$ (\( 1 + 5 T - 28 T^{2} + 265 T^{3} + 2809 T^{4} \))(\( 1 - 53 T^{2} + 2809 T^{4} \))(\( 1 - 10 T + 47 T^{2} - 530 T^{3} + 2809 T^{4} \))(\( 1 + 5 T - 86 T^{2} + 25 T^{3} + 8187 T^{4} + 1325 T^{5} - 241574 T^{6} + 744385 T^{7} + 7890481 T^{8} \))(\( 1 + 5 T - 112 T^{2} - 349 T^{3} + 8923 T^{4} + 12264 T^{5} - 499939 T^{6} + 649992 T^{7} + 25064707 T^{8} - 51958073 T^{9} - 883733872 T^{10} + 2090977465 T^{11} + 22164361129 T^{12} \))(\( 1 + 3 T - 46 T^{2} - 621 T^{3} - 1069 T^{4} + 12552 T^{5} + 266597 T^{6} + 665256 T^{7} - 3002821 T^{8} - 92452617 T^{9} - 362962126 T^{10} + 1254586479 T^{11} + 22164361129 T^{12} \))(\( 1 - 263 T^{2} + 37922 T^{4} - 3954107 T^{6} + 327905385 T^{8} - 22523014574 T^{10} + 1301027375017 T^{12} - 63267147938366 T^{14} + 2587331210140185 T^{16} - 87640255490706803 T^{18} + 2361011979779631842 T^{20} - 45995404706129931887 T^{22} + \)\(49\!\cdots\!41\)\( T^{24} \))
$59$ (\( ( 1 + 12 T + 59 T^{2} )^{2} \))(\( ( 1 + 59 T^{2} )^{2} \))(\( ( 1 + 59 T^{2} )^{2} \))(\( ( 1 - T + 87 T^{2} - 59 T^{3} + 3481 T^{4} )^{2} \))(\( ( 1 - 7 T + 93 T^{2} - 525 T^{3} + 5487 T^{4} - 24367 T^{5} + 205379 T^{6} )^{2} \))(\( ( 1 + 13 T + 155 T^{2} + 1351 T^{3} + 9145 T^{4} + 45253 T^{5} + 205379 T^{6} )^{2} \))(\( ( 1 + 189 T^{2} + 19729 T^{4} + 1361231 T^{6} + 68676649 T^{8} + 2290181229 T^{10} + 42180533641 T^{12} )^{2} \))
$61$ (\( ( 1 - 14 T + 61 T^{2} )( 1 + T + 61 T^{2} ) \))(\( ( 1 + T + 61 T^{2} )( 1 + 13 T + 61 T^{2} ) \))(\( 1 + 2 T - 57 T^{2} + 122 T^{3} + 3721 T^{4} \))(\( 1 - T - 110 T^{2} + 11 T^{3} + 8539 T^{4} + 671 T^{5} - 409310 T^{6} - 226981 T^{7} + 13845841 T^{8} \))(\( 1 - T - 124 T^{2} + 145 T^{3} + 7865 T^{4} - 5346 T^{5} - 484659 T^{6} - 326106 T^{7} + 29265665 T^{8} + 32912245 T^{9} - 1716884284 T^{10} - 844596301 T^{11} + 51520374361 T^{12} \))(\( 1 + 11 T - 92 T^{2} - 339 T^{3} + 17625 T^{4} + 40718 T^{5} - 965507 T^{6} + 2483798 T^{7} + 65582625 T^{8} - 76946559 T^{9} - 1273817372 T^{10} + 9290559311 T^{11} + 51520374361 T^{12} \))(\( ( 1 - 11 T - 22 T^{2} + 69 T^{3} + 2695 T^{4} + 33572 T^{5} - 569627 T^{6} + 2047892 T^{7} + 10028095 T^{8} + 15661689 T^{9} - 304608502 T^{10} - 9290559311 T^{11} + 51520374361 T^{12} )^{2} \))
$67$ (\( 1 - 3 T - 58 T^{2} - 201 T^{3} + 4489 T^{4} \))(\( ( 1 - 11 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \))(\( 1 + 9 T + 14 T^{2} + 603 T^{3} + 4489 T^{4} \))(\( 1 - T - 122 T^{2} + 11 T^{3} + 10573 T^{4} + 737 T^{5} - 547658 T^{6} - 300763 T^{7} + 20151121 T^{8} \))(\( 1 + 13 T - 72 T^{2} - 325 T^{3} + 19457 T^{4} + 55224 T^{5} - 1059101 T^{6} + 3700008 T^{7} + 87342473 T^{8} - 97747975 T^{9} - 1450880712 T^{10} + 17551626391 T^{11} + 90458382169 T^{12} \))(\( 1 - 15 T + 132 T^{2} - 669 T^{3} - 3027 T^{4} + 85836 T^{5} - 856213 T^{6} + 5751012 T^{7} - 13588203 T^{8} - 201210447 T^{9} + 2659947972 T^{10} - 20251876605 T^{11} + 90458382169 T^{12} \))(\( ( 1 - T - 66 T^{2} - 165 T^{3} + 49 T^{4} + 7772 T^{5} + 298675 T^{6} + 520724 T^{7} + 219961 T^{8} - 49625895 T^{9} - 1329973986 T^{10} - 1350125107 T^{11} + 90458382169 T^{12} )^{2} \))
$71$ (\( 1 + T - 70 T^{2} + 71 T^{3} + 5041 T^{4} \))(\( 1 - 71 T^{2} + 5041 T^{4} \))(\( 1 - 2 T - 67 T^{2} - 142 T^{3} + 5041 T^{4} \))(\( 1 - 3 T + 16 T^{2} + 447 T^{3} - 5631 T^{4} + 31737 T^{5} + 80656 T^{6} - 1073733 T^{7} + 25411681 T^{8} \))(\( 1 + 8 T - 161 T^{2} - 456 T^{3} + 25626 T^{4} + 38528 T^{5} - 1928793 T^{6} + 2735488 T^{7} + 129180666 T^{8} - 163207416 T^{9} - 4091280641 T^{10} + 14433834808 T^{11} + 128100283921 T^{12} \))(\( 1 + 8 T - 129 T^{2} - 744 T^{3} + 14394 T^{4} + 41872 T^{5} - 942745 T^{6} + 2972912 T^{7} + 72560154 T^{8} - 266285784 T^{9} - 3278106849 T^{10} + 14433834808 T^{11} + 128100283921 T^{12} \))(\( 1 - 250 T^{2} + 28437 T^{4} - 2570382 T^{6} + 245405386 T^{8} - 21755663842 T^{10} + 1655809017469 T^{12} - 109670301427522 T^{14} + 6236163384713866 T^{16} - 329266663985427822 T^{18} + 18363293168035705557 T^{20} - \)\(81\!\cdots\!50\)\( T^{22} + \)\(16\!\cdots\!41\)\( T^{24} \))
$73$ (\( 1 + 11 T + 48 T^{2} + 803 T^{3} + 5329 T^{4} \))(\( ( 1 - 17 T + 73 T^{2} )( 1 + 10 T + 73 T^{2} ) \))(\( ( 1 - 17 T + 73 T^{2} )( 1 + 10 T + 73 T^{2} ) \))(\( 1 - 3 T - 128 T^{2} + 27 T^{3} + 12783 T^{4} + 1971 T^{5} - 682112 T^{6} - 1167051 T^{7} + 28398241 T^{8} \))(\( 1 - 7 T - 30 T^{2} + 1897 T^{3} - 8101 T^{4} - 53340 T^{5} + 1561945 T^{6} - 3893820 T^{7} - 43170229 T^{8} + 737965249 T^{9} - 851947230 T^{10} - 14511501151 T^{11} + 151334226289 T^{12} \))(\( 1 + T - 130 T^{2} - 35 T^{3} + 7515 T^{4} - 2924 T^{5} - 473159 T^{6} - 213452 T^{7} + 40047435 T^{8} - 13615595 T^{9} - 3691771330 T^{10} + 2073071593 T^{11} + 151334226289 T^{12} \))(\( ( 1 - T - 194 T^{2} + 7 T^{3} + 23689 T^{4} + 3446 T^{5} - 1992919 T^{6} + 251558 T^{7} + 126238681 T^{8} + 2723119 T^{9} - 5509258754 T^{10} - 2073071593 T^{11} + 151334226289 T^{12} )^{2} \))
$79$ (\( 1 - 5 T - 54 T^{2} - 395 T^{3} + 6241 T^{4} \))(\( ( 1 - 17 T + 79 T^{2} )( 1 + 13 T + 79 T^{2} ) \))(\( ( 1 - 13 T + 79 T^{2} )( 1 + 17 T + 79 T^{2} ) \))(\( 1 - 5 T - 138 T^{2} - 25 T^{3} + 18353 T^{4} - 1975 T^{5} - 861258 T^{6} - 2465195 T^{7} + 38950081 T^{8} \))(\( 1 + 12 T + 34 T^{2} + 550 T^{3} - 1210 T^{4} - 91318 T^{5} - 720301 T^{6} - 7214122 T^{7} - 7551610 T^{8} + 271171450 T^{9} + 1324302754 T^{10} + 36924676788 T^{11} + 243087455521 T^{12} \))(\( 1 + 4 T + 38 T^{2} + 1998 T^{3} + 6394 T^{4} + 61814 T^{5} + 1935835 T^{6} + 4883306 T^{7} + 39904954 T^{8} + 985091922 T^{9} + 1480103078 T^{10} + 12308225596 T^{11} + 243087455521 T^{12} \))(\( ( 1 + 8 T - 20 T^{2} - 378 T^{3} - 3780 T^{4} - 12148 T^{5} + 304855 T^{6} - 959692 T^{7} - 23590980 T^{8} - 186368742 T^{9} - 779001620 T^{10} + 24616451192 T^{11} + 243087455521 T^{12} )^{2} \))
$83$ (\( 1 - 9 T - 2 T^{2} - 747 T^{3} + 6889 T^{4} \))(\( 1 - 83 T^{2} + 6889 T^{4} \))(\( 1 + 12 T + 61 T^{2} + 996 T^{3} + 6889 T^{4} \))(\( 1 - 3 T + 52 T^{2} + 627 T^{3} - 5787 T^{4} + 52041 T^{5} + 358228 T^{6} - 1715361 T^{7} + 47458321 T^{8} \))(\( 1 + T - 232 T^{2} - 41 T^{3} + 34813 T^{4} + 132 T^{5} - 3350533 T^{6} + 10956 T^{7} + 239826757 T^{8} - 23443267 T^{9} - 11010330472 T^{10} + 3939040643 T^{11} + 326940373369 T^{12} \))(\( 1 + 5 T - 130 T^{2} - 939 T^{3} + 6875 T^{4} + 38630 T^{5} - 288277 T^{6} + 3206290 T^{7} + 47361875 T^{8} - 536907993 T^{9} - 6169581730 T^{10} + 19695203215 T^{11} + 326940373369 T^{12} \))(\( 1 - 149 T^{2} + 5744 T^{4} + 618033 T^{6} - 71263811 T^{8} + 1451115734 T^{10} + 90909479425 T^{12} + 9996736291526 T^{14} - 3382060818121331 T^{16} + 202059939774363177 T^{18} + 12937166581406651504 T^{20} - \)\(23\!\cdots\!01\)\( T^{22} + \)\(10\!\cdots\!61\)\( T^{24} \))
$89$ (\( 1 + T - 88 T^{2} + 89 T^{3} + 7921 T^{4} \))(\( 1 - 89 T^{2} + 7921 T^{4} \))(\( 1 - 8 T - 25 T^{2} - 712 T^{3} + 7921 T^{4} \))(\( 1 - 3 T - 160 T^{2} + 27 T^{3} + 19839 T^{4} + 2403 T^{5} - 1267360 T^{6} - 2114907 T^{7} + 62742241 T^{8} \))(\( 1 - 10 T - 191 T^{2} + 666 T^{3} + 40902 T^{4} - 81538 T^{5} - 3721827 T^{6} - 7256882 T^{7} + 323984742 T^{8} + 469509354 T^{9} - 11983768031 T^{10} - 55840594490 T^{11} + 496981290961 T^{12} \))(\( 1 - 10 T - 95 T^{2} + 602 T^{3} + 7910 T^{4} + 19070 T^{5} - 1138211 T^{6} + 1697230 T^{7} + 62655110 T^{8} + 424391338 T^{9} - 5960512895 T^{10} - 55840594490 T^{11} + 496981290961 T^{12} \))(\( 1 + 58 T^{2} - 15291 T^{4} - 226770 T^{6} + 162288730 T^{8} - 1294376702 T^{10} - 1545468619427 T^{12} - 10252757856542 T^{14} + 10182358609243930 T^{16} - 112700447351225970 T^{18} - 60194379427990520571 T^{20} + \)\(18\!\cdots\!58\)\( T^{22} + \)\(24\!\cdots\!21\)\( T^{24} \))
$97$ (\( ( 1 + 2 T + 97 T^{2} )^{2} \))(\( ( 1 - 14 T + 97 T^{2} )^{2} \))(\( ( 1 - 10 T + 97 T^{2} )^{2} \))(\( ( 1 + 14 T + 238 T^{2} + 1358 T^{3} + 9409 T^{4} )^{2} \))(\( ( 1 - 16 T + 136 T^{2} - 1045 T^{3} + 13192 T^{4} - 150544 T^{5} + 912673 T^{6} )^{2} \))(\( ( 1 + 160 T^{2} - 469 T^{3} + 15520 T^{4} + 912673 T^{6} )^{2} \))(\( ( 1 + 4 T + 276 T^{2} + 783 T^{3} + 26772 T^{4} + 37636 T^{5} + 912673 T^{6} )^{4} \))
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