Properties

Label 211.2.a
Level 211
Weight 2
Character orbit a
Rep. character \(\chi_{211}(1,\cdot)\)
Character field \(\Q\)
Dimension 17
Newform subspaces 4
Sturm bound 35
Trace bound 2

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Defining parameters

Level: \( N \) \(=\) \( 211 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 211.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(35\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(211))\).

Total New Old
Modular forms 18 18 0
Cusp forms 17 17 0
Eisenstein series 1 1 0

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(211\)Dim.
\(+\)\(6\)
\(-\)\(11\)

Trace form

\( 17q - 2q^{2} - 2q^{3} + 12q^{4} + 4q^{5} - 6q^{6} - 2q^{7} - 12q^{8} + 13q^{9} + O(q^{10}) \) \( 17q - 2q^{2} - 2q^{3} + 12q^{4} + 4q^{5} - 6q^{6} - 2q^{7} - 12q^{8} + 13q^{9} - 8q^{10} - 4q^{11} - 10q^{12} + 2q^{13} + 6q^{14} + 8q^{15} - 2q^{16} + 4q^{17} - 2q^{18} - 12q^{19} + 22q^{20} + 2q^{22} + 2q^{23} + 10q^{24} + 33q^{25} + 8q^{26} - 8q^{27} - 14q^{28} + 2q^{30} - 8q^{31} - 6q^{32} - 14q^{33} - 12q^{34} - 18q^{35} + 10q^{36} + 4q^{37} + 14q^{38} - 34q^{39} - 36q^{40} - 6q^{41} - 30q^{42} - 12q^{43} - 20q^{44} + 14q^{45} - 10q^{46} + 12q^{47} - 34q^{48} + 3q^{49} + 8q^{50} - 16q^{51} + 15q^{53} - 30q^{54} + 25q^{55} + 36q^{56} + 4q^{57} - 4q^{58} + 7q^{59} + 22q^{60} - 6q^{61} - 16q^{62} + 14q^{63} - 14q^{64} + 3q^{65} + 34q^{66} - 20q^{67} + 16q^{68} + 10q^{69} + 14q^{70} + 20q^{71} - 34q^{72} + 11q^{73} + 34q^{74} - 18q^{75} - 48q^{76} - 22q^{77} + 36q^{78} + 56q^{80} + 9q^{81} - 42q^{82} + 21q^{83} + 4q^{84} + 32q^{85} - 38q^{86} + 32q^{87} - 48q^{88} + 22q^{89} - 90q^{90} - 34q^{91} + 42q^{92} - 46q^{93} - 18q^{94} - 17q^{95} + 36q^{96} - 10q^{97} - 12q^{98} - 16q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(211))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 211
211.2.a.a \(2\) \(1.685\) \(\Q(\sqrt{5}) \) None \(1\) \(3\) \(2\) \(1\) \(-\) \(q+\beta q^{2}+(1+\beta )q^{3}+(-1+\beta )q^{4}+(2+\cdots)q^{5}+\cdots\)
211.2.a.b \(3\) \(1.685\) \(\Q(\zeta_{14})^+\) None \(-2\) \(-1\) \(-8\) \(2\) \(+\) \(q+(-1-\beta _{2})q^{2}-\beta _{1}q^{3}+(\beta _{1}+\beta _{2})q^{4}+\cdots\)
211.2.a.c \(3\) \(1.685\) 3.3.229.1 None \(0\) \(-3\) \(-5\) \(-3\) \(+\) \(q-\beta _{1}q^{2}+(-1+\beta _{1})q^{3}+(1+\beta _{2})q^{4}+\cdots\)
211.2.a.d \(9\) \(1.685\) \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(-1\) \(-1\) \(15\) \(-2\) \(-\) \(q-\beta _{1}q^{2}+\beta _{7}q^{3}+(1+\beta _{2})q^{4}+(2+\beta _{3}+\cdots)q^{5}+\cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 - T + 3 T^{2} - 2 T^{3} + 4 T^{4} \))(\( 1 + 2 T + 5 T^{2} + 7 T^{3} + 10 T^{4} + 8 T^{5} + 8 T^{6} \))(\( 1 + 2 T^{2} + T^{3} + 4 T^{4} + 8 T^{6} \))(\( 1 + T + 4 T^{2} + 5 T^{3} + 14 T^{4} + 16 T^{5} + 33 T^{6} + 38 T^{7} + 70 T^{8} + 80 T^{9} + 140 T^{10} + 152 T^{11} + 264 T^{12} + 256 T^{13} + 448 T^{14} + 320 T^{15} + 512 T^{16} + 256 T^{17} + 512 T^{18} \))
$3$ (\( 1 - 3 T + 7 T^{2} - 9 T^{3} + 9 T^{4} \))(\( 1 + T + 7 T^{2} + 5 T^{3} + 21 T^{4} + 9 T^{5} + 27 T^{6} \))(\( 1 + 3 T + 8 T^{2} + 14 T^{3} + 24 T^{4} + 27 T^{5} + 27 T^{6} \))(\( 1 + T + 7 T^{2} + 7 T^{3} + 32 T^{4} + 26 T^{5} + 116 T^{6} + 105 T^{7} + 422 T^{8} + 346 T^{9} + 1266 T^{10} + 945 T^{11} + 3132 T^{12} + 2106 T^{13} + 7776 T^{14} + 5103 T^{15} + 15309 T^{16} + 6561 T^{17} + 19683 T^{18} \))
$5$ (\( 1 - 2 T + 6 T^{2} - 10 T^{3} + 25 T^{4} \))(\( 1 + 8 T + 34 T^{2} + 93 T^{3} + 170 T^{4} + 200 T^{5} + 125 T^{6} \))(\( 1 + 5 T + 17 T^{2} + 46 T^{3} + 85 T^{4} + 125 T^{5} + 125 T^{6} \))(\( 1 - 15 T + 128 T^{2} - 789 T^{3} + 3868 T^{4} - 15793 T^{5} + 55240 T^{6} - 168325 T^{7} + 451526 T^{8} - 1072103 T^{9} + 2257630 T^{10} - 4208125 T^{11} + 6905000 T^{12} - 9870625 T^{13} + 12087500 T^{14} - 12328125 T^{15} + 10000000 T^{16} - 5859375 T^{17} + 1953125 T^{18} \))
$7$ (\( 1 - T + 13 T^{2} - 7 T^{3} + 49 T^{4} \))(\( 1 - 2 T + 6 T^{2} + T^{3} + 42 T^{4} - 98 T^{5} + 343 T^{6} \))(\( 1 + 3 T + 20 T^{2} + 40 T^{3} + 140 T^{4} + 147 T^{5} + 343 T^{6} \))(\( 1 + 2 T + 28 T^{2} + 55 T^{3} + 371 T^{4} + 550 T^{5} + 3083 T^{6} + 2473 T^{7} + 19851 T^{8} + 8656 T^{9} + 138957 T^{10} + 121177 T^{11} + 1057469 T^{12} + 1320550 T^{13} + 6235397 T^{14} + 6470695 T^{15} + 23059204 T^{16} + 11529602 T^{17} + 40353607 T^{18} \))
$11$ (\( ( 1 + 3 T + 11 T^{2} )^{2} \))(\( 1 + 2 T + 4 T^{2} - 27 T^{3} + 44 T^{4} + 242 T^{5} + 1331 T^{6} \))(\( ( 1 + 3 T + 11 T^{2} )^{3} \))(\( 1 - 13 T + 130 T^{2} - 909 T^{3} + 5510 T^{4} - 27863 T^{5} + 128212 T^{6} - 522487 T^{7} + 1980592 T^{8} - 6791293 T^{9} + 21786512 T^{10} - 63220927 T^{11} + 170650172 T^{12} - 407942183 T^{13} + 887391010 T^{14} - 1610348949 T^{15} + 2533332230 T^{16} - 2786665453 T^{17} + 2357947691 T^{18} \))
$13$ (\( 1 - 8 T + 37 T^{2} - 104 T^{3} + 169 T^{4} \))(\( 1 + 3 T + 35 T^{2} + 79 T^{3} + 455 T^{4} + 507 T^{5} + 2197 T^{6} \))(\( 1 - T + 18 T^{2} + 11 T^{3} + 234 T^{4} - 169 T^{5} + 2197 T^{6} \))(\( 1 + 4 T + 80 T^{2} + 364 T^{3} + 3197 T^{4} + 14686 T^{5} + 82667 T^{6} + 352805 T^{7} + 1496091 T^{8} + 5579059 T^{9} + 19449183 T^{10} + 59624045 T^{11} + 181619399 T^{12} + 419446846 T^{13} + 1187023721 T^{14} + 1756958476 T^{15} + 5019881360 T^{16} + 3262922884 T^{17} + 10604499373 T^{18} \))
$17$ (\( 1 - 11 T + 63 T^{2} - 187 T^{3} + 289 T^{4} \))(\( 1 - 6 T + 56 T^{2} - 205 T^{3} + 952 T^{4} - 1734 T^{5} + 4913 T^{6} \))(\( 1 + 17 T + 142 T^{2} + 726 T^{3} + 2414 T^{4} + 4913 T^{5} + 4913 T^{6} \))(\( 1 - 4 T + 84 T^{2} - 199 T^{3} + 2931 T^{4} - 3078 T^{5} + 64145 T^{6} - 4729 T^{7} + 1169223 T^{8} + 330324 T^{9} + 19876791 T^{10} - 1366681 T^{11} + 315144385 T^{12} - 257077638 T^{13} + 4161600867 T^{14} - 4803376231 T^{15} + 34468448532 T^{16} - 27903029764 T^{17} + 118587876497 T^{18} \))
$19$ (\( 1 + 5 T + 33 T^{2} + 95 T^{3} + 361 T^{4} \))(\( 1 + 7 T + 57 T^{2} + 259 T^{3} + 1083 T^{4} + 2527 T^{5} + 6859 T^{6} \))(\( 1 - 2 T + 53 T^{2} - 69 T^{3} + 1007 T^{4} - 722 T^{5} + 6859 T^{6} \))(\( 1 + 2 T + 94 T^{2} + 92 T^{3} + 4359 T^{4} + 624 T^{5} + 131795 T^{6} - 67689 T^{7} + 3021289 T^{8} - 2189359 T^{9} + 57404491 T^{10} - 24435729 T^{11} + 903981905 T^{12} + 81320304 T^{13} + 10793315541 T^{14} + 4328221052 T^{15} + 84023943466 T^{16} + 33967126082 T^{17} + 322687697779 T^{18} \))
$23$ (\( 1 - 8 T + 57 T^{2} - 184 T^{3} + 529 T^{4} \))(\( 1 + 19 T + 187 T^{2} + 1113 T^{3} + 4301 T^{4} + 10051 T^{5} + 12167 T^{6} \))(\( 1 - 16 T + 142 T^{2} - 810 T^{3} + 3266 T^{4} - 8464 T^{5} + 12167 T^{6} \))(\( 1 + 3 T + 135 T^{2} + 335 T^{3} + 8414 T^{4} + 19206 T^{5} + 339074 T^{6} + 758305 T^{7} + 10119426 T^{8} + 21034414 T^{9} + 232746798 T^{10} + 401143345 T^{11} + 4125513358 T^{12} + 5374626246 T^{13} + 54155390002 T^{14} + 49592022815 T^{15} + 459651435345 T^{16} + 234932955843 T^{17} + 1801152661463 T^{18} \))
$29$ (\( 1 + 53 T^{2} + 841 T^{4} \))(\( 1 + 6 T + 8 T^{2} - 29 T^{3} + 232 T^{4} + 5046 T^{5} + 24389 T^{6} \))(\( 1 + 20 T + 208 T^{2} + 1386 T^{3} + 6032 T^{4} + 16820 T^{5} + 24389 T^{6} \))(\( 1 - 26 T + 482 T^{2} - 6339 T^{3} + 69613 T^{4} - 636246 T^{5} + 5118367 T^{6} - 36054501 T^{7} + 228623789 T^{8} - 1293317504 T^{9} + 6630089881 T^{10} - 30321835341 T^{11} + 124831852763 T^{12} - 450004707126 T^{13} + 1427842615337 T^{14} - 3770585031819 T^{15} + 8314440380938 T^{16} - 13006406736986 T^{17} + 14507145975869 T^{18} \))
$31$ (\( 1 + 11 T + 61 T^{2} + 341 T^{3} + 961 T^{4} \))(\( 1 + 5 T + 71 T^{2} + 297 T^{3} + 2201 T^{4} + 4805 T^{5} + 29791 T^{6} \))(\( 1 - 3 T + 48 T^{2} - 240 T^{3} + 1488 T^{4} - 2883 T^{5} + 29791 T^{6} \))(\( 1 - 5 T + 161 T^{2} - 593 T^{3} + 10904 T^{4} - 22838 T^{5} + 415280 T^{6} - 63223 T^{7} + 11471976 T^{8} + 13881958 T^{9} + 355631256 T^{10} - 60757303 T^{11} + 12371606480 T^{12} - 21091372598 T^{13} + 312172262504 T^{14} - 526289682833 T^{15} + 4429530871871 T^{16} - 4264455187205 T^{17} + 26439622160671 T^{18} \))
$37$ (\( 1 + 4 T - 2 T^{2} + 148 T^{3} + 1369 T^{4} \))(\( 1 + 2 T + 68 T^{2} + 231 T^{3} + 2516 T^{4} + 2738 T^{5} + 50653 T^{6} \))(\( 1 - 5 T + 107 T^{2} - 366 T^{3} + 3959 T^{4} - 6845 T^{5} + 50653 T^{6} \))(\( 1 - 5 T + 244 T^{2} - 1145 T^{3} + 28980 T^{4} - 123303 T^{5} + 2170930 T^{6} - 8164375 T^{7} + 112360078 T^{8} - 363884069 T^{9} + 4157322886 T^{10} - 11177029375 T^{11} + 109964117290 T^{12} - 231089673783 T^{13} + 2009587873860 T^{14} - 2937756738305 T^{15} + 23163378020452 T^{16} - 17562397269605 T^{17} + 129961739795077 T^{18} \))
$41$ (\( ( 1 + 3 T + 41 T^{2} )^{2} \))(\( 1 + 18 T + 203 T^{2} + 1468 T^{3} + 8323 T^{4} + 30258 T^{5} + 68921 T^{6} \))(\( 1 + 2 T + 34 T^{2} + 222 T^{3} + 1394 T^{4} + 3362 T^{5} + 68921 T^{6} \))(\( 1 - 20 T + 313 T^{2} - 3380 T^{3} + 33036 T^{4} - 272120 T^{5} + 2182348 T^{6} - 15965324 T^{7} + 114565302 T^{8} - 747130600 T^{9} + 4697177382 T^{10} - 26837709644 T^{11} + 150409606508 T^{12} - 768946083320 T^{13} + 3827425456236 T^{14} - 16055352334580 T^{15} + 60958087724753 T^{16} - 159698504582420 T^{17} + 327381934393961 T^{18} \))
$43$ (\( ( 1 - 9 T + 43 T^{2} )^{2} \))(\( 1 - 4 T + 118 T^{2} - 343 T^{3} + 5074 T^{4} - 7396 T^{5} + 79507 T^{6} \))(\( 1 - 3 T + 68 T^{2} - 259 T^{3} + 2924 T^{4} - 5547 T^{5} + 79507 T^{6} \))(\( 1 + 37 T + 894 T^{2} + 15365 T^{3} + 214164 T^{4} + 2472903 T^{5} + 24730478 T^{6} + 215717073 T^{7} + 1676662722 T^{8} + 11597517407 T^{9} + 72096497046 T^{10} + 398860867977 T^{11} + 1966246114346 T^{12} + 8454363249303 T^{13} + 31483916186652 T^{14} + 97127743247885 T^{15} + 243005838329658 T^{16} + 432463410271237 T^{17} + 502592611936843 T^{18} \))
$47$ (\( 1 - T + 93 T^{2} - 47 T^{3} + 2209 T^{4} \))(\( 1 - 11 T + 165 T^{2} - 1047 T^{3} + 7755 T^{4} - 24299 T^{5} + 103823 T^{6} \))(\( 1 + 4 T + 131 T^{2} + 335 T^{3} + 6157 T^{4} + 8836 T^{5} + 103823 T^{6} \))(\( 1 - 4 T + 104 T^{2} - 242 T^{3} + 6037 T^{4} + 3040 T^{5} + 380223 T^{6} - 227413 T^{7} + 25088141 T^{8} - 41059065 T^{9} + 1179142627 T^{10} - 502355317 T^{11} + 39475892529 T^{12} + 14834230240 T^{13} + 1384555807259 T^{14} - 2608570109618 T^{15} + 52688804528152 T^{16} - 95245146647044 T^{17} + 1119130473102767 T^{18} \))
$53$ (\( 1 - 13 T + 147 T^{2} - 689 T^{3} + 2809 T^{4} \))(\( 1 + 10 T + 134 T^{2} + 935 T^{3} + 7102 T^{4} + 28090 T^{5} + 148877 T^{6} \))(\( 1 + T + 154 T^{2} + 108 T^{3} + 8162 T^{4} + 2809 T^{5} + 148877 T^{6} \))(\( 1 - 13 T + 423 T^{2} - 4289 T^{3} + 79484 T^{4} - 662622 T^{5} + 8988435 T^{6} - 62982601 T^{7} + 682399049 T^{8} - 4025561062 T^{9} + 36167149597 T^{10} - 176918126209 T^{11} + 1338171237495 T^{12} - 5228406301182 T^{13} + 33239850565612 T^{14} - 95062944882281 T^{15} + 496902812151051 T^{16} - 809375975347693 T^{17} + 3299763591802133 T^{18} \))
$59$ (\( 1 + 73 T^{2} + 3481 T^{4} \))(\( 1 - 5 T + 99 T^{2} - 759 T^{3} + 5841 T^{4} - 17405 T^{5} + 205379 T^{6} \))(\( 1 + 12 T + 164 T^{2} + 1268 T^{3} + 9676 T^{4} + 41772 T^{5} + 205379 T^{6} \))(\( 1 - 14 T + 273 T^{2} - 2401 T^{3} + 29668 T^{4} - 174639 T^{5} + 1664059 T^{6} - 4749683 T^{7} + 56131267 T^{8} - 40471710 T^{9} + 3311744753 T^{10} - 16533646523 T^{11} + 341762773361 T^{12} - 2116163807679 T^{13} + 21210374102732 T^{14} - 101275461272041 T^{15} + 679401855355587 T^{16} - 2055626126460494 T^{17} + 8662995818654939 T^{18} \))
$61$ (\( ( 1 + 3 T + 61 T^{2} )^{2} \))(\( 1 + 23 T + 322 T^{2} + 2987 T^{3} + 19642 T^{4} + 85583 T^{5} + 226981 T^{6} \))(\( 1 + 126 T^{2} + 52 T^{3} + 7686 T^{4} + 226981 T^{6} \))(\( 1 - 23 T + 646 T^{2} - 10081 T^{3} + 167065 T^{4} - 1991338 T^{5} + 24260591 T^{6} - 231700919 T^{7} + 2232381533 T^{8} - 17388473502 T^{9} + 136175273513 T^{10} - 862159119599 T^{11} + 5506693205771 T^{12} - 27571749325258 T^{13} + 141102481026565 T^{14} - 519376893933241 T^{15} + 2030211872069566 T^{16} - 4409268198937463 T^{17} + 11694146092834141 T^{18} \))
$67$ (\( ( 1 + 12 T + 67 T^{2} )^{2} \))(\( 1 - 7 T + 201 T^{2} - 889 T^{3} + 13467 T^{4} - 31423 T^{5} + 300763 T^{6} \))(\( ( 1 + 67 T^{2} )^{3} \))(\( 1 + 3 T + 239 T^{2} + 17 T^{3} + 30098 T^{4} - 95826 T^{5} + 2411418 T^{6} - 17595721 T^{7} + 155819658 T^{8} - 1562789922 T^{9} + 10439917086 T^{10} - 78987191569 T^{11} + 725265311934 T^{12} - 1931001320946 T^{13} + 40636065470486 T^{14} + 1537792496873 T^{15} + 1448510073672197 T^{16} + 1218203032669923 T^{17} + 27206534396294947 T^{18} \))
$71$ (\( 1 + 6 T + 26 T^{2} + 426 T^{3} + 5041 T^{4} \))(\( 1 - 18 T + 272 T^{2} - 2429 T^{3} + 19312 T^{4} - 90738 T^{5} + 357911 T^{6} \))(\( 1 + 11 T + 95 T^{2} + 790 T^{3} + 6745 T^{4} + 55451 T^{5} + 357911 T^{6} \))(\( 1 - 19 T + 534 T^{2} - 6723 T^{3} + 110154 T^{4} - 1035141 T^{5} + 12922004 T^{6} - 98281789 T^{7} + 1079354496 T^{8} - 7356534671 T^{9} + 76634169216 T^{10} - 495438498349 T^{11} + 4624927373644 T^{12} - 26304672882021 T^{13} + 198743079930054 T^{14} - 861218208800883 T^{15} + 4856794164580794 T^{16} - 12269317093669459 T^{17} + 45848500718449031 T^{18} \))
$73$ (\( 1 + 7 T + 127 T^{2} + 511 T^{3} + 5329 T^{4} \))(\( 1 + 2 T + 43 T^{2} + 956 T^{3} + 3139 T^{4} + 10658 T^{5} + 389017 T^{6} \))(\( 1 - 3 T + 218 T^{2} - 436 T^{3} + 15914 T^{4} - 15987 T^{5} + 389017 T^{6} \))(\( 1 - 17 T + 456 T^{2} - 6651 T^{3} + 106577 T^{4} - 1299606 T^{5} + 15907139 T^{6} - 161804901 T^{7} + 1637432987 T^{8} - 14017245842 T^{9} + 119532608051 T^{10} - 862258317429 T^{11} + 6188147492363 T^{12} - 36906524393046 T^{13} + 220941751167161 T^{14} - 1006523939048139 T^{15} + 5037613724708232 T^{16} - 13709821562199377 T^{17} + 58871586708267913 T^{18} \))
$79$ (\( 1 + 10 T + 138 T^{2} + 790 T^{3} + 6241 T^{4} \))(\( 1 - 8 T + 137 T^{2} - 696 T^{3} + 10823 T^{4} - 49928 T^{5} + 493039 T^{6} \))(\( 1 + 5 T + 11 T^{2} - 822 T^{3} + 869 T^{4} + 31205 T^{5} + 493039 T^{6} \))(\( 1 - 7 T + 501 T^{2} - 3633 T^{3} + 122580 T^{4} - 870050 T^{5} + 19038015 T^{6} - 126186795 T^{7} + 2069616807 T^{8} - 12107857118 T^{9} + 163499727753 T^{10} - 787531787595 T^{11} + 9386483877585 T^{12} - 33888517974050 T^{13} + 377185573389420 T^{14} - 883136725907793 T^{15} + 9621158402065659 T^{16} - 10619761669345927 T^{17} + 119851595982618319 T^{18} \))
$83$ (\( 1 - 8 T + 162 T^{2} - 664 T^{3} + 6889 T^{4} \))(\( 1 + 21 T + 347 T^{2} + 3535 T^{3} + 28801 T^{4} + 144669 T^{5} + 571787 T^{6} \))(\( 1 - 28 T + 461 T^{2} - 5096 T^{3} + 38263 T^{4} - 192892 T^{5} + 571787 T^{6} \))(\( 1 - 6 T + 425 T^{2} - 2371 T^{3} + 98340 T^{4} - 503493 T^{5} + 15146331 T^{6} - 69364165 T^{7} + 1688312923 T^{8} - 6801028730 T^{9} + 140129972609 T^{10} - 477849732685 T^{11} + 8660475163497 T^{12} - 23894932415253 T^{13} + 387365256832620 T^{14} - 775175625257899 T^{15} + 11532821670591475 T^{16} - 13513753392834246 T^{17} + 186940255267540403 T^{18} \))
$89$ (\( 1 - 15 T + 223 T^{2} - 1335 T^{3} + 7921 T^{4} \))(\( 1 + 31 T + 571 T^{2} + 6471 T^{3} + 50819 T^{4} + 245551 T^{5} + 704969 T^{6} \))(\( 1 - 5 T + 78 T^{2} - 1626 T^{3} + 6942 T^{4} - 39605 T^{5} + 704969 T^{6} \))(\( 1 - 33 T + 855 T^{2} - 15199 T^{3} + 240940 T^{4} - 3145538 T^{5} + 38621540 T^{6} - 417369329 T^{7} + 4382266528 T^{8} - 41703298394 T^{9} + 390021720992 T^{10} - 3305982455009 T^{11} + 27226988432260 T^{12} - 197358103270658 T^{13} + 1345423283642060 T^{14} - 7553618641316239 T^{15} + 37817791335677295 T^{16} - 129907430588168673 T^{17} + 350356403707485209 T^{18} \))
$97$ (\( 1 - T + 183 T^{2} - 97 T^{3} + 9409 T^{4} \))(\( 1 + 7 T + 193 T^{2} + 721 T^{3} + 18721 T^{4} + 65863 T^{5} + 912673 T^{6} \))(\( 1 - 7 T + 222 T^{2} - 1246 T^{3} + 21534 T^{4} - 65863 T^{5} + 912673 T^{6} \))(\( 1 + 11 T + 705 T^{2} + 6575 T^{3} + 231506 T^{4} + 1858210 T^{5} + 46821726 T^{6} + 323465585 T^{7} + 6438386990 T^{8} + 37837079046 T^{9} + 624523538030 T^{10} + 3043487689265 T^{11} + 42732925133598 T^{12} + 164505995247010 T^{13} + 1988020793537042 T^{14} + 5476790932408175 T^{15} + 56962790557069665 T^{16} + 86211769538146571 T^{17} + 760231058654565217 T^{18} \))
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