Properties

Label 1008.2.t
Level 1008
Weight 2
Character orbit t
Rep. character \(\chi_{1008}(193,\cdot)\)
Character field \(\Q(\zeta_{3})\)
Dimension 92
Newform subspaces 12
Sturm bound 384
Trace bound 5

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Defining parameters

Level: \( N \) = \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1008.t (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 63 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 12 \)
Sturm bound: \(384\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1008, [\chi])\).

Total New Old
Modular forms 408 100 308
Cusp forms 360 92 268
Eisenstein series 48 8 40

Trace form

\( 92q + q^{3} - 2q^{5} + q^{7} - q^{9} + O(q^{10}) \) \( 92q + q^{3} - 2q^{5} + q^{7} - q^{9} + 2q^{11} - 2q^{13} + 7q^{15} - 2q^{17} + 2q^{19} + 2q^{23} + 74q^{25} + 4q^{27} - 6q^{29} - 7q^{31} - 13q^{33} + 9q^{35} - 2q^{37} + 7q^{39} - 2q^{41} - 4q^{43} - 17q^{45} - 21q^{47} - q^{49} - 13q^{51} - 2q^{53} + 18q^{55} - 6q^{57} + 35q^{59} + q^{61} + 33q^{63} - q^{65} - q^{67} - 31q^{69} + 32q^{71} - 2q^{73} + 54q^{75} - 21q^{77} - q^{79} - 13q^{81} + 28q^{83} + 3q^{85} - 14q^{87} + 2q^{89} - 4q^{91} - 11q^{93} + 27q^{95} - 2q^{97} + 61q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1008, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
1008.2.t.a \(2\) \(8.049\) \(\Q(\sqrt{-3}) \) None \(0\) \(-3\) \(-2\) \(1\) \(q+(-2+\zeta_{6})q^{3}-q^{5}+(-1+3\zeta_{6})q^{7}+\cdots\)
1008.2.t.b \(2\) \(8.049\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(4\) \(-4\) \(q+(1-2\zeta_{6})q^{3}+2q^{5}+(-3+2\zeta_{6})q^{7}+\cdots\)
1008.2.t.c \(2\) \(8.049\) \(\Q(\sqrt{-3}) \) None \(0\) \(3\) \(-6\) \(1\) \(q+(2-\zeta_{6})q^{3}-3q^{5}+(-1+3\zeta_{6})q^{7}+\cdots\)
1008.2.t.d \(2\) \(8.049\) \(\Q(\sqrt{-3}) \) None \(0\) \(3\) \(-2\) \(-1\) \(q+(2-\zeta_{6})q^{3}-q^{5}+(1-3\zeta_{6})q^{7}+(3+\cdots)q^{9}+\cdots\)
1008.2.t.e \(2\) \(8.049\) \(\Q(\sqrt{-3}) \) None \(0\) \(3\) \(2\) \(5\) \(q+(2-\zeta_{6})q^{3}+q^{5}+(3-\zeta_{6})q^{7}+(3+\cdots)q^{9}+\cdots\)
1008.2.t.f \(2\) \(8.049\) \(\Q(\sqrt{-3}) \) None \(0\) \(3\) \(6\) \(-5\) \(q+(2-\zeta_{6})q^{3}+3q^{5}+(-3+\zeta_{6})q^{7}+\cdots\)
1008.2.t.g \(6\) \(8.049\) 6.0.309123.1 None \(0\) \(-4\) \(10\) \(2\) \(q+(-1+\beta _{5})q^{3}+(2+\beta _{2}+\beta _{3})q^{5}+\cdots\)
1008.2.t.h \(6\) \(8.049\) 6.0.309123.1 None \(0\) \(-2\) \(-2\) \(4\) \(q+(\beta _{3}-\beta _{4}-\beta _{5})q^{3}+(-1+\beta _{1}+\beta _{2}+\cdots)q^{5}+\cdots\)
1008.2.t.i \(10\) \(8.049\) 10.0.\(\cdots\).1 None \(0\) \(-2\) \(-8\) \(1\) \(q+(-\beta _{2}+\beta _{7})q^{3}+(-\beta _{1}-\beta _{2}-\beta _{4}+\cdots)q^{5}+\cdots\)
1008.2.t.j \(14\) \(8.049\) \(\mathbb{Q}[x]/(x^{14} - \cdots)\) None \(0\) \(0\) \(4\) \(3\) \(q+(\beta _{1}+\beta _{3})q^{3}+\beta _{11}q^{5}-\beta _{7}q^{7}+(1+\cdots)q^{9}+\cdots\)
1008.2.t.k \(22\) \(8.049\) None \(0\) \(-2\) \(-6\) \(-7\)
1008.2.t.l \(22\) \(8.049\) None \(0\) \(2\) \(-2\) \(1\)

Decomposition of \(S_{2}^{\mathrm{old}}(1008, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1008, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(126, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(252, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(504, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))(\( \))
$3$ (\( 1 + 3 T + 3 T^{2} \))(\( 1 + 3 T^{2} \))(\( 1 - 3 T + 3 T^{2} \))(\( 1 - 3 T + 3 T^{2} \))(\( 1 - 3 T + 3 T^{2} \))(\( 1 - 3 T + 3 T^{2} \))(\( 1 + 4 T + 10 T^{2} + 21 T^{3} + 30 T^{4} + 36 T^{5} + 27 T^{6} \))(\( 1 + 2 T + 4 T^{2} + 3 T^{3} + 12 T^{4} + 18 T^{5} + 27 T^{6} \))(\( 1 + 2 T + 4 T^{2} + 3 T^{3} - 9 T^{5} + 27 T^{7} + 108 T^{8} + 162 T^{9} + 243 T^{10} \))(\( 1 - 5 T^{2} + 3 T^{3} + 7 T^{4} - 30 T^{5} + 117 T^{7} - 270 T^{9} + 189 T^{10} + 243 T^{11} - 1215 T^{12} + 2187 T^{14} \))
$5$ (\( ( 1 + T + 5 T^{2} )^{2} \))(\( ( 1 - 2 T + 5 T^{2} )^{2} \))(\( ( 1 + 3 T + 5 T^{2} )^{2} \))(\( ( 1 + T + 5 T^{2} )^{2} \))(\( ( 1 - T + 5 T^{2} )^{2} \))(\( ( 1 - 3 T + 5 T^{2} )^{2} \))(\( ( 1 - 5 T + 19 T^{2} - 47 T^{3} + 95 T^{4} - 125 T^{5} + 125 T^{6} )^{2} \))(\( ( 1 + T + 9 T^{2} + 13 T^{3} + 45 T^{4} + 25 T^{5} + 125 T^{6} )^{2} \))(\( ( 1 + 4 T + 20 T^{2} + 62 T^{3} + 193 T^{4} + 423 T^{5} + 965 T^{6} + 1550 T^{7} + 2500 T^{8} + 2500 T^{9} + 3125 T^{10} )^{2} \))(\( ( 1 - 2 T + 15 T^{2} - 48 T^{3} + 154 T^{4} - 429 T^{5} + 1202 T^{6} - 2471 T^{7} + 6010 T^{8} - 10725 T^{9} + 19250 T^{10} - 30000 T^{11} + 46875 T^{12} - 31250 T^{13} + 78125 T^{14} )^{2} \))
$7$ (\( 1 - T + 7 T^{2} \))(\( 1 + 4 T + 7 T^{2} \))(\( 1 - T + 7 T^{2} \))(\( 1 + T + 7 T^{2} \))(\( 1 - 5 T + 7 T^{2} \))(\( 1 + 5 T + 7 T^{2} \))(\( 1 - 2 T + 2 T^{2} + 19 T^{3} + 14 T^{4} - 98 T^{5} + 343 T^{6} \))(\( 1 - 4 T + 14 T^{2} - 55 T^{3} + 98 T^{4} - 196 T^{5} + 343 T^{6} \))(\( 1 - T - 12 T^{2} - 8 T^{3} + 62 T^{4} + 75 T^{5} + 434 T^{6} - 392 T^{7} - 4116 T^{8} - 2401 T^{9} + 16807 T^{10} \))(\( 1 - 3 T + 11 T^{2} + 5 T^{3} - 15 T^{4} + 88 T^{5} + 312 T^{6} - 753 T^{7} + 2184 T^{8} + 4312 T^{9} - 5145 T^{10} + 12005 T^{11} + 184877 T^{12} - 352947 T^{13} + 823543 T^{14} \))
$11$ (\( ( 1 + 3 T + 11 T^{2} )^{2} \))(\( ( 1 + 4 T + 11 T^{2} )^{2} \))(\( ( 1 - 3 T + 11 T^{2} )^{2} \))(\( ( 1 + 5 T + 11 T^{2} )^{2} \))(\( ( 1 - 3 T + 11 T^{2} )^{2} \))(\( ( 1 - 3 T + 11 T^{2} )^{2} \))(\( ( 1 + T + 7 T^{2} + 55 T^{3} + 77 T^{4} + 121 T^{5} + 1331 T^{6} )^{2} \))(\( ( 1 + T + 27 T^{2} + 25 T^{3} + 297 T^{4} + 121 T^{5} + 1331 T^{6} )^{2} \))(\( ( 1 - 4 T + 47 T^{2} - 161 T^{3} + 958 T^{4} - 2589 T^{5} + 10538 T^{6} - 19481 T^{7} + 62557 T^{8} - 58564 T^{9} + 161051 T^{10} )^{2} \))(\( ( 1 - 2 T + 36 T^{2} - 57 T^{3} + 460 T^{4} - 402 T^{5} + 2501 T^{6} - 455 T^{7} + 27511 T^{8} - 48642 T^{9} + 612260 T^{10} - 834537 T^{11} + 5797836 T^{12} - 3543122 T^{13} + 19487171 T^{14} )^{2} \))
$13$ (\( 1 + 3 T - 4 T^{2} + 39 T^{3} + 169 T^{4} \))(\( 1 + 3 T - 4 T^{2} + 39 T^{3} + 169 T^{4} \))(\( ( 1 - 2 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} ) \))(\( ( 1 - 7 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} ) \))(\( 1 + T - 12 T^{2} + 13 T^{3} + 169 T^{4} \))(\( 1 - T - 12 T^{2} - 13 T^{3} + 169 T^{4} \))(\( 1 + 2 T - 32 T^{2} - 26 T^{3} + 730 T^{4} + 230 T^{5} - 10729 T^{6} + 2990 T^{7} + 123370 T^{8} - 57122 T^{9} - 913952 T^{10} + 742586 T^{11} + 4826809 T^{12} \))(\( 1 - 8 T + 24 T^{2} - 42 T^{3} - 32 T^{4} + 1408 T^{5} - 7901 T^{6} + 18304 T^{7} - 5408 T^{8} - 92274 T^{9} + 685464 T^{10} - 2970344 T^{11} + 4826809 T^{12} \))(\( 1 + 8 T - 14 T^{2} - 182 T^{3} + 686 T^{4} + 4429 T^{5} - 12871 T^{6} - 43199 T^{7} + 305249 T^{8} + 358672 T^{9} - 3841969 T^{10} + 4662736 T^{11} + 51587081 T^{12} - 94908203 T^{13} - 367608631 T^{14} + 1644456697 T^{15} + 3311190974 T^{16} - 11420230094 T^{17} - 11420230094 T^{18} + 84835994984 T^{19} + 137858491849 T^{20} \))(\( 1 - 2 T - 25 T^{2} - 20 T^{3} + 172 T^{4} + 1281 T^{5} + 1882 T^{6} - 1142 T^{7} - 27931 T^{8} - 309997 T^{9} + 9092 T^{10} + 1859069 T^{11} + 4530175 T^{12} + 5185603 T^{13} - 73974588 T^{14} + 67412839 T^{15} + 765599575 T^{16} + 4084374593 T^{17} + 259676612 T^{18} - 115099716121 T^{19} - 134817602179 T^{20} - 71658806414 T^{21} + 1535205216922 T^{22} + 13584363696813 T^{23} + 23711660598028 T^{24} - 35843207880740 T^{25} - 582452128062025 T^{26} - 605750213184506 T^{27} + 3937376385699289 T^{28} \))
$17$ (\( 1 - 5 T + 8 T^{2} - 85 T^{3} + 289 T^{4} \))(\( 1 + 7 T + 32 T^{2} + 119 T^{3} + 289 T^{4} \))(\( 1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4} \))(\( 1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4} \))(\( 1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4} \))(\( 1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4} \))(\( 1 + 4 T + 9 T^{2} + 92 T^{3} + 58 T^{4} - 20 T^{5} + 5393 T^{6} - 340 T^{7} + 16762 T^{8} + 451996 T^{9} + 751689 T^{10} + 5679428 T^{11} + 24137569 T^{12} \))(\( 1 + 4 T - 23 T^{2} - 68 T^{3} + 410 T^{4} + 220 T^{5} - 8111 T^{6} + 3740 T^{7} + 118490 T^{8} - 334084 T^{9} - 1920983 T^{10} + 5679428 T^{11} + 24137569 T^{12} \))(\( 1 - 12 T + 14 T^{2} + 192 T^{3} + 1185 T^{4} - 11847 T^{5} - 6180 T^{6} + 65736 T^{7} + 1002861 T^{8} - 2436261 T^{9} - 7749777 T^{10} - 41416437 T^{11} + 289826829 T^{12} + 322960968 T^{13} - 516159780 T^{14} - 16821045879 T^{15} + 28603019265 T^{16} + 78785025216 T^{17} + 97660604174 T^{18} - 1423054517964 T^{19} + 2015993900449 T^{20} \))(\( 1 - 2 T - 65 T^{2} + 210 T^{3} + 2087 T^{4} - 9143 T^{5} - 37340 T^{6} + 240381 T^{7} + 293834 T^{8} - 4176065 T^{9} + 3382763 T^{10} + 47662233 T^{11} - 157347285 T^{12} - 273515658 T^{13} + 3170983122 T^{14} - 4649766186 T^{15} - 45473365365 T^{16} + 234164550729 T^{17} + 282531748523 T^{18} - 5929415122705 T^{19} + 7092438449546 T^{20} + 98637620554413 T^{21} - 260474782846940 T^{22} - 1084248954812071 T^{23} + 4207379270237063 T^{24} + 7197098224602930 T^{25} - 37870445419934465 T^{26} - 19809156065811874 T^{27} + 168377826559400929 T^{28} \))
$19$ (\( ( 1 - 8 T + 19 T^{2} )( 1 + T + 19 T^{2} ) \))(\( 1 - 5 T + 6 T^{2} - 95 T^{3} + 361 T^{4} \))(\( 1 - 5 T + 6 T^{2} - 95 T^{3} + 361 T^{4} \))(\( ( 1 - 8 T + 19 T^{2} )( 1 + 7 T + 19 T^{2} ) \))(\( 1 - 5 T + 6 T^{2} - 95 T^{3} + 361 T^{4} \))(\( ( 1 - T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \))(\( 1 - 3 T - 42 T^{2} + 61 T^{3} + 1311 T^{4} - 726 T^{5} - 27501 T^{6} - 13794 T^{7} + 473271 T^{8} + 418399 T^{9} - 5473482 T^{10} - 7428297 T^{11} + 47045881 T^{12} \))(\( 1 - 3 T - 12 T^{2} + 67 T^{3} - 153 T^{4} - 54 T^{5} + 6315 T^{6} - 1026 T^{7} - 55233 T^{8} + 459553 T^{9} - 1563852 T^{10} - 7428297 T^{11} + 47045881 T^{12} \))(\( 1 + T - 53 T^{2} - 190 T^{3} + 1262 T^{4} + 7007 T^{5} - 13111 T^{6} - 116110 T^{7} + 67964 T^{8} + 721616 T^{9} - 440023 T^{10} + 13710704 T^{11} + 24535004 T^{12} - 796398490 T^{13} - 1708638631 T^{14} + 17350025693 T^{15} + 59371901822 T^{16} - 169835630410 T^{17} - 900128841173 T^{18} + 322687697779 T^{19} + 6131066257801 T^{20} \))(\( 1 + 7 T - 54 T^{2} - 381 T^{3} + 1875 T^{4} + 9873 T^{5} - 65652 T^{6} - 221430 T^{7} + 1870425 T^{8} + 4319703 T^{9} - 46476858 T^{10} - 61637031 T^{11} + 1073881146 T^{12} + 457871775 T^{13} - 21789737442 T^{14} + 8699563725 T^{15} + 387671093706 T^{16} - 422768395629 T^{17} - 6056910611418 T^{18} + 10696012278597 T^{19} + 87995791969425 T^{20} - 197930019166770 T^{21} - 1115004880767732 T^{22} + 3185895640172067 T^{23} + 11495749233376875 T^{24} - 44382788640221439 T^{25} - 119519005629572694 T^{26} + 294370884235799413 T^{27} + 799006685782884121 T^{28} \))
$23$ (\( ( 1 + 5 T + 23 T^{2} )^{2} \))(\( ( 1 + 4 T + 23 T^{2} )^{2} \))(\( ( 1 - 3 T + 23 T^{2} )^{2} \))(\( ( 1 + 3 T + 23 T^{2} )^{2} \))(\( ( 1 + T + 23 T^{2} )^{2} \))(\( ( 1 - 9 T + 23 T^{2} )^{2} \))(\( ( 1 + 7 T + 73 T^{2} + 319 T^{3} + 1679 T^{4} + 3703 T^{5} + 12167 T^{6} )^{2} \))(\( ( 1 + 7 T + 81 T^{2} + 325 T^{3} + 1863 T^{4} + 3703 T^{5} + 12167 T^{6} )^{2} \))(\( ( 1 - 3 T + 52 T^{2} - 225 T^{3} + 2023 T^{4} - 5565 T^{5} + 46529 T^{6} - 119025 T^{7} + 632684 T^{8} - 839523 T^{9} + 6436343 T^{10} )^{2} \))(\( ( 1 - 11 T + 129 T^{2} - 876 T^{3} + 6604 T^{4} - 35691 T^{5} + 217568 T^{6} - 992939 T^{7} + 5004064 T^{8} - 18880539 T^{9} + 80350868 T^{10} - 245140716 T^{11} + 830288247 T^{12} - 1628394779 T^{13} + 3404825447 T^{14} )^{2} \))
$29$ (\( 1 - T - 28 T^{2} - 29 T^{3} + 841 T^{4} \))(\( 1 - T - 28 T^{2} - 29 T^{3} + 841 T^{4} \))(\( 1 - 3 T - 20 T^{2} - 87 T^{3} + 841 T^{4} \))(\( 1 - T - 28 T^{2} - 29 T^{3} + 841 T^{4} \))(\( 1 + 9 T + 52 T^{2} + 261 T^{3} + 841 T^{4} \))(\( 1 + 3 T - 20 T^{2} + 87 T^{3} + 841 T^{4} \))(\( 1 + 5 T - 30 T^{2} - 371 T^{3} - 185 T^{4} + 6020 T^{5} + 44357 T^{6} + 174580 T^{7} - 155585 T^{8} - 9048319 T^{9} - 21218430 T^{10} + 102555745 T^{11} + 594823321 T^{12} \))(\( 1 + 5 T + 4 T^{2} + 251 T^{3} + 197 T^{4} - 3418 T^{5} + 20293 T^{6} - 99122 T^{7} + 165677 T^{8} + 6121639 T^{9} + 2829124 T^{10} + 102555745 T^{11} + 594823321 T^{12} \))(\( 1 - 7 T - 76 T^{2} + 419 T^{3} + 4561 T^{4} - 15146 T^{5} - 199563 T^{6} + 341373 T^{7} + 6918636 T^{8} - 2570041 T^{9} - 219913241 T^{10} - 74531189 T^{11} + 5818572876 T^{12} + 8325746097 T^{13} - 141147118203 T^{14} - 310661862754 T^{15} + 2712989167081 T^{16} + 7227698173471 T^{17} - 38018727385036 T^{18} - 101550021831083 T^{19} + 420707233300201 T^{20} \))(\( 1 - T - 89 T^{2} + 606 T^{3} + 3413 T^{4} - 45595 T^{5} + 49603 T^{6} + 1643802 T^{7} - 7893892 T^{8} - 19552444 T^{9} + 271585946 T^{10} - 420585531 T^{11} - 3441402105 T^{12} + 10956432363 T^{13} + 10513309734 T^{14} + 317736538527 T^{15} - 2894219170305 T^{16} - 10257660515559 T^{17} + 192087579472826 T^{18} - 401043092198156 T^{19} - 4695471055055332 T^{20} + 28355381176486818 T^{21} + 24813722822104483 T^{22} - 661453320769747055 T^{23} + 1435873787253586013 T^{24} + 7393508918017732374 T^{25} - 31489515705286744649 T^{26} - 10260628712958602189 T^{27} + \)\(29\!\cdots\!81\)\( T^{28} \))
$31$ (\( 1 + 8 T + 33 T^{2} + 248 T^{3} + 961 T^{4} \))(\( 1 + 3 T - 22 T^{2} + 93 T^{3} + 961 T^{4} \))(\( ( 1 - 7 T + 31 T^{2} )( 1 + 11 T + 31 T^{2} ) \))(\( 1 - 31 T^{2} + 961 T^{4} \))(\( ( 1 - 11 T + 31 T^{2} )( 1 + 7 T + 31 T^{2} ) \))(\( 1 - 8 T + 33 T^{2} - 248 T^{3} + 961 T^{4} \))(\( 1 - 14 T + 58 T^{2} - 250 T^{3} + 2992 T^{4} - 9728 T^{5} - 11857 T^{6} - 301568 T^{7} + 2875312 T^{8} - 7447750 T^{9} + 53564218 T^{10} - 400808114 T^{11} + 887503681 T^{12} \))(\( 1 + 20 T + 186 T^{2} + 1398 T^{3} + 10342 T^{4} + 62234 T^{5} + 331987 T^{6} + 1929254 T^{7} + 9938662 T^{8} + 41647818 T^{9} + 171774906 T^{10} + 572583020 T^{11} + 887503681 T^{12} \))(\( 1 - 3 T - 125 T^{2} + 214 T^{3} + 9282 T^{4} - 8387 T^{5} - 503981 T^{6} + 245082 T^{7} + 21459514 T^{8} - 3619498 T^{9} - 734820027 T^{10} - 112204438 T^{11} + 20622592954 T^{12} + 7301237862 T^{13} - 465437037101 T^{14} - 240112689437 T^{15} + 8237809167042 T^{16} + 5887699419754 T^{17} - 106611379680125 T^{18} - 79318866482013 T^{19} + 819628286980801 T^{20} \))(\( 1 - T - 85 T^{2} + 302 T^{3} + 2323 T^{4} - 16596 T^{5} - 8720 T^{6} + 121502 T^{7} + 336059 T^{8} + 5031700 T^{9} - 52734649 T^{10} + 249456712 T^{11} - 356372186 T^{12} - 8588183707 T^{13} + 82956488229 T^{14} - 266233694917 T^{15} - 342473670746 T^{16} + 7431564907192 T^{17} - 48701555779129 T^{18} + 144053299086700 T^{19} + 298253599533179 T^{20} + 3342837639714722 T^{21} - 7437209846485520 T^{22} - 438791969378495916 T^{23} + 1903996510656400723 T^{24} + 7673360022714258962 T^{25} - 66951336622026729685 T^{26} - 24417546297445042591 T^{27} + \)\(75\!\cdots\!21\)\( T^{28} \))
$37$ (\( 1 + 3 T - 28 T^{2} + 111 T^{3} + 1369 T^{4} \))(\( ( 1 + T + 37 T^{2} )( 1 + 10 T + 37 T^{2} ) \))(\( 1 - 7 T + 12 T^{2} - 259 T^{3} + 1369 T^{4} \))(\( 1 + 3 T - 28 T^{2} + 111 T^{3} + 1369 T^{4} \))(\( 1 + 5 T - 12 T^{2} + 185 T^{3} + 1369 T^{4} \))(\( ( 1 - 11 T + 37 T^{2} )( 1 + 10 T + 37 T^{2} ) \))(\( 1 + 9 T - 21 T^{2} - 268 T^{3} + 1293 T^{4} + 4875 T^{5} - 42882 T^{6} + 180375 T^{7} + 1770117 T^{8} - 13575004 T^{9} - 39357381 T^{10} + 624095613 T^{11} + 2565726409 T^{12} \))(\( ( 1 - 11 T + 37 T^{2} )^{3}( 1 + 10 T + 37 T^{2} )^{3} \))(\( 1 - 89 T^{2} + 560 T^{3} + 4503 T^{4} - 45352 T^{5} + 27130 T^{6} + 2296536 T^{7} - 9801827 T^{8} - 33131096 T^{9} + 610977105 T^{10} - 1225850552 T^{11} - 13418701163 T^{12} + 116326438008 T^{13} + 50845987930 T^{14} - 3144887137864 T^{15} + 11553466019727 T^{16} + 53161851194480 T^{17} - 312610671398969 T^{18} + 4808584372417849 T^{20} \))(\( 1 - 10 T - 84 T^{2} + 1296 T^{3} + 1134 T^{4} - 66630 T^{5} + 77382 T^{6} + 1851174 T^{7} - 1251993 T^{8} - 21837810 T^{9} - 268575252 T^{10} - 774884982 T^{11} + 27666538119 T^{12} + 26159905206 T^{13} - 1381876923558 T^{14} + 967916492622 T^{15} + 37875490684911 T^{16} - 39250248993246 T^{17} - 503353262863572 T^{18} - 1514320157614170 T^{19} - 3212271503983137 T^{20} + 175735422719804142 T^{21} + 271802685103314822 T^{22} - 8659350722545980510 T^{23} + 5452934678321840766 T^{24} + \)\(23\!\cdots\!48\)\( T^{25} - \)\(55\!\cdots\!04\)\( T^{26} - \)\(24\!\cdots\!70\)\( T^{27} + \)\(90\!\cdots\!89\)\( T^{28} \))
$41$ (\( 1 - 5 T - 16 T^{2} - 205 T^{3} + 1681 T^{4} \))(\( 1 - 9 T + 40 T^{2} - 369 T^{3} + 1681 T^{4} \))(\( 1 - 9 T + 40 T^{2} - 369 T^{3} + 1681 T^{4} \))(\( 1 - 5 T - 16 T^{2} - 205 T^{3} + 1681 T^{4} \))(\( 1 + 7 T + 8 T^{2} + 287 T^{3} + 1681 T^{4} \))(\( 1 + 3 T - 32 T^{2} + 123 T^{3} + 1681 T^{4} \))(\( 1 + 12 T - 18 T^{2} - 78 T^{3} + 7470 T^{4} + 24546 T^{5} - 158105 T^{6} + 1006386 T^{7} + 12557070 T^{8} - 5375838 T^{9} - 50863698 T^{10} + 1390274412 T^{11} + 4750104241 T^{12} \))(\( 1 - 90 T^{2} + 18 T^{3} + 4410 T^{4} - 810 T^{5} - 194177 T^{6} - 33210 T^{7} + 7413210 T^{8} + 1240578 T^{9} - 254318490 T^{10} + 4750104241 T^{12} \))(\( 1 - 5 T - 136 T^{2} + 733 T^{3} + 10507 T^{4} - 54412 T^{5} - 554055 T^{6} + 2345451 T^{7} + 23706084 T^{8} - 41392439 T^{9} - 952045937 T^{10} - 1697089999 T^{11} + 39849927204 T^{12} + 161650828371 T^{13} - 1565627010855 T^{14} - 6303967608812 T^{15} + 49909345260187 T^{16} + 142754882754773 T^{17} - 1085949831160456 T^{18} - 1636909671969805 T^{19} + 13422659310152401 T^{20} \))(\( 1 + 33 T + 463 T^{2} + 3882 T^{3} + 26359 T^{4} + 177381 T^{5} + 987377 T^{6} + 3338436 T^{7} - 1489480 T^{8} - 146370792 T^{9} - 1532374696 T^{10} - 10553354379 T^{11} - 66691501599 T^{12} - 460808734581 T^{13} - 3110830401306 T^{14} - 18893158117821 T^{15} - 112108414187919 T^{16} - 727347737155059 T^{17} - 4330124653343656 T^{18} - 16957963898481192 T^{19} - 7075185264884680 T^{20} + 650174679078190116 T^{21} + 7884131517953805617 T^{22} + 58071334904735196141 T^{23} + \)\(35\!\cdots\!59\)\( T^{24} + \)\(21\!\cdots\!62\)\( T^{25} + \)\(10\!\cdots\!03\)\( T^{26} + \)\(30\!\cdots\!93\)\( T^{27} + \)\(37\!\cdots\!61\)\( T^{28} \))
$43$ (\( 1 + 7 T + 6 T^{2} + 301 T^{3} + 1849 T^{4} \))(\( ( 1 - 13 T + 43 T^{2} )( 1 + 8 T + 43 T^{2} ) \))(\( 1 - 11 T + 78 T^{2} - 473 T^{3} + 1849 T^{4} \))(\( 1 + T - 42 T^{2} + 43 T^{3} + 1849 T^{4} \))(\( 1 - 3 T - 34 T^{2} - 129 T^{3} + 1849 T^{4} \))(\( 1 + T - 42 T^{2} + 43 T^{3} + 1849 T^{4} \))(\( 1 + 18 T + 114 T^{2} + 682 T^{3} + 7188 T^{4} + 33492 T^{5} + 63039 T^{6} + 1440156 T^{7} + 13290612 T^{8} + 54223774 T^{9} + 389743314 T^{10} + 2646151974 T^{11} + 6321363049 T^{12} \))(\( ( 1 - 18 T + 198 T^{2} - 1519 T^{3} + 8514 T^{4} - 33282 T^{5} + 79507 T^{6} )( 1 + 12 T - 6 T^{2} - 547 T^{3} - 258 T^{4} + 22188 T^{5} + 79507 T^{6} ) \))(\( 1 - 7 T - 77 T^{2} + 66 T^{3} + 7014 T^{4} + 3843 T^{5} - 95427 T^{6} - 1632678 T^{7} - 3708600 T^{8} + 15416324 T^{9} + 670279801 T^{10} + 662901932 T^{11} - 6857201400 T^{12} - 129809329746 T^{13} - 326245923027 T^{14} + 564953446449 T^{15} + 44338040425686 T^{16} + 17940028333062 T^{17} - 899991421375277 T^{18} - 3518148283557901 T^{19} + 21611482313284249 T^{20} \))(\( 1 + 7 T - 222 T^{2} - 1221 T^{3} + 30003 T^{4} + 121065 T^{5} - 2964828 T^{6} - 8400318 T^{7} + 232132089 T^{8} + 430114695 T^{9} - 14993117802 T^{10} - 15328887375 T^{11} + 817956723570 T^{12} + 259275458991 T^{13} - 38016795252930 T^{14} + 11148844736613 T^{15} + 1512401981880930 T^{16} - 1218753848524125 T^{17} - 51258486134595402 T^{18} + 63230491623369885 T^{19} + 1467391209891779361 T^{20} - 2283362771617132026 T^{21} - 34653503452639217628 T^{22} + 60846374564133897795 T^{23} + \)\(64\!\cdots\!47\)\( T^{24} - \)\(11\!\cdots\!47\)\( T^{25} - \)\(88\!\cdots\!22\)\( T^{26} + \)\(12\!\cdots\!01\)\( T^{27} + \)\(73\!\cdots\!49\)\( T^{28} \))
$47$ (\( 1 - 8 T + 17 T^{2} - 376 T^{3} + 2209 T^{4} \))(\( 1 - 3 T - 38 T^{2} - 141 T^{3} + 2209 T^{4} \))(\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 - 8 T + 17 T^{2} - 376 T^{3} + 2209 T^{4} \))(\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 + 3 T - 108 T^{2} - 267 T^{3} + 7263 T^{4} + 9786 T^{5} - 360137 T^{6} + 459942 T^{7} + 16043967 T^{8} - 27720741 T^{9} - 527005548 T^{10} + 688035021 T^{11} + 10779215329 T^{12} \))(\( 1 - 9 T - 6 T^{2} + 531 T^{3} - 2433 T^{4} - 3438 T^{5} + 104623 T^{6} - 161586 T^{7} - 5374497 T^{8} + 55130013 T^{9} - 29278086 T^{10} - 2064105063 T^{11} + 10779215329 T^{12} \))(\( 1 + 27 T + 281 T^{2} + 1758 T^{3} + 13050 T^{4} + 78783 T^{5} - 25248 T^{6} - 1518381 T^{7} + 9454350 T^{8} + 53043051 T^{9} - 242331903 T^{10} + 2493023397 T^{11} + 20884659150 T^{12} - 157642870563 T^{13} - 123202185888 T^{14} + 18068487686481 T^{15} + 140668760043450 T^{16} + 890643445773954 T^{17} + 6690971551954841 T^{18} + 30216522773774709 T^{19} + 52599132235830049 T^{20} \))(\( 1 - 3 T - 215 T^{2} + 750 T^{3} + 23197 T^{4} - 82998 T^{5} - 1793695 T^{6} + 5388753 T^{7} + 119401781 T^{8} - 227010657 T^{9} - 7448336422 T^{10} + 6436178352 T^{11} + 428559183834 T^{12} - 95001497937 T^{13} - 21681767924409 T^{14} - 4465070403039 T^{15} + 946687237089306 T^{16} + 668223345039696 T^{17} - 36345505720041382 T^{18} - 52063760718739599 T^{19} + 1287057508065100949 T^{20} + 2730066860264352639 T^{21} - 42710185828767396895 T^{22} - 92885591006583455466 T^{23} + \)\(12\!\cdots\!53\)\( T^{24} + \)\(18\!\cdots\!50\)\( T^{25} - \)\(24\!\cdots\!15\)\( T^{26} - \)\(16\!\cdots\!81\)\( T^{27} + \)\(25\!\cdots\!69\)\( T^{28} \))
$53$ (\( 1 - T - 52 T^{2} - 53 T^{3} + 2809 T^{4} \))(\( 1 + 3 T - 44 T^{2} + 159 T^{3} + 2809 T^{4} \))(\( 1 - 3 T - 44 T^{2} - 159 T^{3} + 2809 T^{4} \))(\( 1 - 9 T + 28 T^{2} - 477 T^{3} + 2809 T^{4} \))(\( 1 + 9 T + 28 T^{2} + 477 T^{3} + 2809 T^{4} \))(\( 1 + 3 T - 44 T^{2} + 159 T^{3} + 2809 T^{4} \))(\( 1 - 9 T - 36 T^{2} + 873 T^{3} - 1179 T^{4} - 26334 T^{5} + 272077 T^{6} - 1395702 T^{7} - 3311811 T^{8} + 129969621 T^{9} - 284057316 T^{10} - 3763759437 T^{11} + 22164361129 T^{12} \))(\( 1 - 15 T - 33 T^{3} + 13635 T^{4} - 60360 T^{5} - 225155 T^{6} - 3199080 T^{7} + 38300715 T^{8} - 4912941 T^{9} - 6272932395 T^{11} + 22164361129 T^{12} \))(\( 1 + 21 T + 41 T^{2} - 924 T^{3} + 12966 T^{4} + 177027 T^{5} - 601755 T^{6} - 3783942 T^{7} + 110973258 T^{8} + 340111866 T^{9} - 4044436041 T^{10} + 18025928898 T^{11} + 311723881722 T^{12} - 563341933134 T^{13} - 4748136394155 T^{14} + 74031893539311 T^{15} + 287383106398614 T^{16} - 1085433093209388 T^{17} + 2552647306865801 T^{18} + 69295035427844793 T^{19} + 174887470365513049 T^{20} \))(\( 1 + 15 T - 44 T^{2} - 1563 T^{3} + 1621 T^{4} + 132831 T^{5} + 280796 T^{6} - 6875916 T^{7} - 32544895 T^{8} + 334617081 T^{9} + 3332374934 T^{10} - 9404405181 T^{11} - 219667861866 T^{12} + 268474558113 T^{13} + 14089335458730 T^{14} + 14229151579989 T^{15} - 617047023981594 T^{16} - 1400099630131737 T^{17} + 26294041101603254 T^{18} + 139935355155015933 T^{19} - 721336805685386455 T^{20} - 8077215121783465692 T^{21} + 17482272028748523356 T^{22} + \)\(43\!\cdots\!23\)\( T^{23} + \)\(28\!\cdots\!29\)\( T^{24} - \)\(14\!\cdots\!11\)\( T^{25} - \)\(21\!\cdots\!04\)\( T^{26} + \)\(39\!\cdots\!95\)\( T^{27} + \)\(13\!\cdots\!69\)\( T^{28} \))
$59$ (\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 + 7 T - 10 T^{2} + 413 T^{3} + 3481 T^{4} \))(\( 1 - 12 T + 85 T^{2} - 708 T^{3} + 3481 T^{4} \))(\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 + 4 T - 43 T^{2} + 236 T^{3} + 3481 T^{4} \))(\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 + 4 T - 60 T^{2} - 994 T^{3} - 1304 T^{4} + 27376 T^{5} + 435479 T^{6} + 1615184 T^{7} - 4539224 T^{8} - 204146726 T^{9} - 727041660 T^{10} + 2859697196 T^{11} + 42180533641 T^{12} \))(\( 1 - 14 T - 20 T^{2} + 154 T^{3} + 11666 T^{4} - 35126 T^{5} - 499301 T^{6} - 2072434 T^{7} + 40609346 T^{8} + 31628366 T^{9} - 242347220 T^{10} - 10008940186 T^{11} + 42180533641 T^{12} \))(\( 1 + 30 T + 299 T^{2} + 1644 T^{3} + 26547 T^{4} + 344442 T^{5} + 1635267 T^{6} + 9620487 T^{7} + 170035344 T^{8} + 1056366303 T^{9} + 3109579647 T^{10} + 62325611877 T^{11} + 591893032464 T^{12} + 1975845999573 T^{13} + 19815120570387 T^{14} + 246249955396158 T^{15} + 1119766626567627 T^{16} + 4091343041042436 T^{17} + 43902300843691979 T^{18} + 259889874559648170 T^{19} + 511116753300641401 T^{20} \))(\( 1 - 14 T - 41 T^{2} + 1932 T^{3} - 9268 T^{4} - 67958 T^{5} + 1000462 T^{6} - 4853283 T^{7} - 9914932 T^{8} + 321187219 T^{9} - 1782148489 T^{10} + 6021692682 T^{11} - 34607406657 T^{12} - 638777810472 T^{13} + 11047710270819 T^{14} - 37687890817848 T^{15} - 120468382573017 T^{16} + 1236729221336478 T^{17} - 21594936596817529 T^{18} + 229624547391334481 T^{19} - 418217122774227412 T^{20} - 12078129944196810777 T^{21} + \)\(14\!\cdots\!02\)\( T^{22} - \)\(58\!\cdots\!62\)\( T^{23} - \)\(47\!\cdots\!68\)\( T^{24} + \)\(58\!\cdots\!88\)\( T^{25} - \)\(72\!\cdots\!21\)\( T^{26} - \)\(14\!\cdots\!06\)\( T^{27} + \)\(61\!\cdots\!61\)\( T^{28} \))
$61$ (\( 1 + 10 T + 39 T^{2} + 610 T^{3} + 3721 T^{4} \))(\( 1 + 3 T - 52 T^{2} + 183 T^{3} + 3721 T^{4} \))(\( 1 + 2 T - 57 T^{2} + 122 T^{3} + 3721 T^{4} \))(\( ( 1 - 13 T + 61 T^{2} )( 1 - T + 61 T^{2} ) \))(\( 1 + 2 T - 57 T^{2} + 122 T^{3} + 3721 T^{4} \))(\( 1 + 2 T - 57 T^{2} + 122 T^{3} + 3721 T^{4} \))(\( 1 - 4 T - 32 T^{2} - 650 T^{3} + 292 T^{4} + 19532 T^{5} + 306323 T^{6} + 1191452 T^{7} + 1086532 T^{8} - 147537650 T^{9} - 443066912 T^{10} - 3378385204 T^{11} + 51520374361 T^{12} \))(\( 1 - 8 T - 114 T^{2} + 342 T^{3} + 13762 T^{4} - 13214 T^{5} - 937217 T^{6} - 806054 T^{7} + 51208402 T^{8} + 77627502 T^{9} - 1578425874 T^{10} - 6756770408 T^{11} + 51520374361 T^{12} \))(\( 1 + 14 T - 143 T^{2} - 2072 T^{3} + 23777 T^{4} + 251656 T^{5} - 2164351 T^{6} - 13562879 T^{7} + 202896254 T^{8} + 466067647 T^{9} - 12461386219 T^{10} + 28430126467 T^{11} + 754976961134 T^{12} - 3078515838299 T^{13} - 29967259814191 T^{14} + 212547726724456 T^{15} + 1224999941181497 T^{16} - 6511763156235512 T^{17} - 27414145758611183 T^{18} + 163718045299677974 T^{19} + 713342911662882601 T^{20} \))(\( 1 + 10 T - 277 T^{2} - 2192 T^{3} + 49786 T^{4} + 275394 T^{5} - 6629804 T^{6} - 25018331 T^{7} + 694393718 T^{8} + 1681932119 T^{9} - 61000901305 T^{10} - 82163567860 T^{11} + 4597121803909 T^{12} + 1969709923870 T^{13} - 300133324238415 T^{14} + 120152305356070 T^{15} + 17105890232345389 T^{16} - 18649568796430660 T^{17} - 844608780325722505 T^{18} + 1420553646240491819 T^{19} + 35775424305286664198 T^{20} - 78626180519452100951 T^{21} - \)\(12\!\cdots\!24\)\( T^{22} + \)\(32\!\cdots\!54\)\( T^{23} + \)\(35\!\cdots\!86\)\( T^{24} - \)\(95\!\cdots\!12\)\( T^{25} - \)\(73\!\cdots\!17\)\( T^{26} + \)\(16\!\cdots\!10\)\( T^{27} + \)\(98\!\cdots\!41\)\( T^{28} \))
$67$ (\( 1 + 12 T + 77 T^{2} + 804 T^{3} + 4489 T^{4} \))(\( 1 - 13 T + 102 T^{2} - 871 T^{3} + 4489 T^{4} \))(\( 1 + 4 T - 51 T^{2} + 268 T^{3} + 4489 T^{4} \))(\( 1 - 4 T - 51 T^{2} - 268 T^{3} + 4489 T^{4} \))(\( 1 - 12 T + 77 T^{2} - 804 T^{3} + 4489 T^{4} \))(\( 1 + 4 T - 51 T^{2} + 268 T^{3} + 4489 T^{4} \))(\( 1 + 5 T - 118 T^{2} - 327 T^{3} + 8263 T^{4} + 1138 T^{5} - 609341 T^{6} + 76246 T^{7} + 37092607 T^{8} - 98349501 T^{9} - 2377832278 T^{10} + 6750625535 T^{11} + 90458382169 T^{12} \))(\( 1 + T - 88 T^{2} + 243 T^{3} + 2035 T^{4} - 14290 T^{5} + 72259 T^{6} - 957430 T^{7} + 9135115 T^{8} + 73085409 T^{9} - 1773298648 T^{10} + 1350125107 T^{11} + 90458382169 T^{12} \))(\( 1 - 2 T - 128 T^{2} + 128 T^{3} + 6161 T^{4} + 2183 T^{5} + 29300 T^{6} - 394018 T^{7} - 17169907 T^{8} + 2850929 T^{9} + 1197895103 T^{10} + 191012243 T^{11} - 77075712523 T^{12} - 118506035734 T^{13} + 590427845300 T^{14} + 2947323108581 T^{15} + 557314092543209 T^{16} + 775771085481344 T^{17} - 51976662727250048 T^{18} - 54413068792589894 T^{19} + 1822837804551761449 T^{20} \))(\( 1 + 6 T - 308 T^{2} - 164 T^{3} + 62908 T^{4} - 146213 T^{5} - 7475396 T^{6} + 42326464 T^{7} + 603793243 T^{8} - 5085070936 T^{9} - 25351846493 T^{10} + 374911832519 T^{11} + 131329441787 T^{12} - 10392590626498 T^{13} + 54642703536353 T^{14} - 696303571975366 T^{15} + 589537864181843 T^{16} + 112759607483911997 T^{17} - 510868126253868653 T^{18} - 6865481941569590152 T^{19} + 54618159926353884067 T^{20} + \)\(25\!\cdots\!72\)\( T^{21} - \)\(30\!\cdots\!36\)\( T^{22} - \)\(39\!\cdots\!11\)\( T^{23} + \)\(11\!\cdots\!92\)\( T^{24} - \)\(20\!\cdots\!12\)\( T^{25} - \)\(25\!\cdots\!88\)\( T^{26} + \)\(32\!\cdots\!22\)\( T^{27} + \)\(36\!\cdots\!29\)\( T^{28} \))
$71$ (\( ( 1 + 12 T + 71 T^{2} )^{2} \))(\( ( 1 - 8 T + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 - 12 T + 71 T^{2} )^{2} \))(\( ( 1 + 8 T + 71 T^{2} )^{2} \))(\( ( 1 + 12 T + 71 T^{2} )^{2} \))(\( ( 1 + 7 T + 163 T^{2} + 895 T^{3} + 11573 T^{4} + 35287 T^{5} + 357911 T^{6} )^{2} \))(\( ( 1 + 7 T + 15 T^{2} - 599 T^{3} + 1065 T^{4} + 35287 T^{5} + 357911 T^{6} )^{2} \))(\( ( 1 - 3 T + 187 T^{2} - 285 T^{3} + 15679 T^{4} - 10143 T^{5} + 1113209 T^{6} - 1436685 T^{7} + 66929357 T^{8} - 76235043 T^{9} + 1804229351 T^{10} )^{2} \))(\( ( 1 + T + 381 T^{2} + 417 T^{3} + 66850 T^{4} + 68550 T^{5} + 7142942 T^{6} + 6246700 T^{7} + 507148882 T^{8} + 345560550 T^{9} + 23926350350 T^{10} + 10596670977 T^{11} + 687411382731 T^{12} + 128100283921 T^{13} + 9095120158391 T^{14} )^{2} \))
$73$ (\( 1 - 5 T - 48 T^{2} - 365 T^{3} + 5329 T^{4} \))(\( ( 1 - 10 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} ) \))(\( 1 + 11 T + 48 T^{2} + 803 T^{3} + 5329 T^{4} \))(\( 1 + 3 T - 64 T^{2} + 219 T^{3} + 5329 T^{4} \))(\( 1 - 13 T + 96 T^{2} - 949 T^{3} + 5329 T^{4} \))(\( 1 + 11 T + 48 T^{2} + 803 T^{3} + 5329 T^{4} \))(\( 1 + 25 T + 254 T^{2} + 2073 T^{3} + 20533 T^{4} + 115046 T^{5} + 366817 T^{6} + 8398358 T^{7} + 109420357 T^{8} + 806432241 T^{9} + 7213153214 T^{10} + 51826789825 T^{11} + 151334226289 T^{12} \))(\( 1 - 19 T + 134 T^{2} - 27 T^{3} - 5759 T^{4} + 41986 T^{5} - 314903 T^{6} + 3064978 T^{7} - 30689711 T^{8} - 10503459 T^{9} + 3805364294 T^{10} - 39388360267 T^{11} + 151334226289 T^{12} \))(\( 1 - 15 T - 134 T^{2} + 2501 T^{3} + 16563 T^{4} - 235276 T^{5} - 2002535 T^{6} + 9021201 T^{7} + 288508378 T^{8} - 238799411 T^{9} - 25271949561 T^{10} - 17432357003 T^{11} + 1537461146362 T^{12} + 3509400549417 T^{13} - 56868471540935 T^{14} - 487743992114668 T^{15} + 2506548790024707 T^{16} + 27629543696261597 T^{17} - 108065652313806854 T^{18} - 883073800624018695 T^{19} + 4297625829703557649 T^{20} \))(\( 1 - 21 T - 107 T^{2} + 4532 T^{3} + 8593 T^{4} - 574123 T^{5} - 1508435 T^{6} + 52395764 T^{7} + 342478696 T^{8} - 4227610868 T^{9} - 46283154524 T^{10} + 253115011813 T^{11} + 4673586532751 T^{12} - 7744045394141 T^{13} - 369527474872486 T^{14} - 565315313772293 T^{15} + 24905542633030079 T^{16} + 98466042550457821 T^{17} - 1314360176412792284 T^{18} - 8764139996708872724 T^{19} + 51828748479625639144 T^{20} + \)\(57\!\cdots\!08\)\( T^{21} - \)\(12\!\cdots\!35\)\( T^{22} - \)\(33\!\cdots\!99\)\( T^{23} + \)\(36\!\cdots\!57\)\( T^{24} + \)\(14\!\cdots\!64\)\( T^{25} - \)\(24\!\cdots\!47\)\( T^{26} - \)\(35\!\cdots\!93\)\( T^{27} + \)\(12\!\cdots\!09\)\( T^{28} \))
$79$ (\( 1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4} \))(\( 1 + 9 T + 2 T^{2} + 711 T^{3} + 6241 T^{4} \))(\( 1 - 8 T - 15 T^{2} - 632 T^{3} + 6241 T^{4} \))(\( 1 - 8 T - 15 T^{2} - 632 T^{3} + 6241 T^{4} \))(\( 1 - 8 T - 15 T^{2} - 632 T^{3} + 6241 T^{4} \))(\( 1 + 16 T + 177 T^{2} + 1264 T^{3} + 6241 T^{4} \))(\( 1 + 7 T - 44 T^{2} - 19 T^{3} - 1043 T^{4} - 28016 T^{5} + 109223 T^{6} - 2213264 T^{7} - 6509363 T^{8} - 9367741 T^{9} - 1713803564 T^{10} + 21539394793 T^{11} + 243087455521 T^{12} \))(\( 1 + 5 T - 138 T^{2} - 123 T^{3} + 11347 T^{4} - 21118 T^{5} - 1048937 T^{6} - 1668322 T^{7} + 70816627 T^{8} - 60643797 T^{9} - 5375111178 T^{10} + 15385281995 T^{11} + 243087455521 T^{12} \))(\( 1 - 4 T - 284 T^{2} + 1776 T^{3} + 44175 T^{4} - 312399 T^{5} - 4187754 T^{6} + 29772300 T^{7} + 295992489 T^{8} - 1067553919 T^{9} - 20151634301 T^{10} - 84336759601 T^{11} + 1847289123849 T^{12} + 14678905019700 T^{13} - 163113357508074 T^{14} - 961269341991201 T^{15} + 10738388347640175 T^{16} + 34106142359418384 T^{17} - 430858902013463324 T^{18} - 479406383930473276 T^{19} + 9468276082626847201 T^{20} \))(\( 1 - 10 T - 226 T^{2} + 4280 T^{3} + 9610 T^{4} - 654627 T^{5} + 3201292 T^{6} + 42192356 T^{7} - 519045919 T^{8} + 901423894 T^{9} + 24953123963 T^{10} - 320449224329 T^{11} + 1694205545953 T^{12} + 14615843505140 T^{13} - 287108375549019 T^{14} + 1154651636906060 T^{15} + 10573536812292673 T^{16} - 157993965113945831 T^{17} + 971926199561891003 T^{18} + 2773732161244197706 T^{19} - \)\(12\!\cdots\!99\)\( T^{20} + \)\(81\!\cdots\!04\)\( T^{21} + \)\(48\!\cdots\!12\)\( T^{22} - \)\(78\!\cdots\!13\)\( T^{23} + \)\(90\!\cdots\!10\)\( T^{24} + \)\(32\!\cdots\!20\)\( T^{25} - \)\(13\!\cdots\!66\)\( T^{26} - \)\(46\!\cdots\!90\)\( T^{27} + \)\(36\!\cdots\!81\)\( T^{28} \))
$83$ (\( 1 + 15 T + 142 T^{2} + 1245 T^{3} + 6889 T^{4} \))(\( 1 - T - 82 T^{2} - 83 T^{3} + 6889 T^{4} \))(\( 1 - 3 T - 74 T^{2} - 249 T^{3} + 6889 T^{4} \))(\( 1 + 9 T - 2 T^{2} + 747 T^{3} + 6889 T^{4} \))(\( 1 + 13 T + 86 T^{2} + 1079 T^{3} + 6889 T^{4} \))(\( 1 + 9 T - 2 T^{2} + 747 T^{3} + 6889 T^{4} \))(\( 1 + 8 T - 180 T^{2} - 518 T^{3} + 29404 T^{4} + 32420 T^{5} - 2713585 T^{6} + 2690860 T^{7} + 202564156 T^{8} - 296185666 T^{9} - 8542497780 T^{10} + 31512325144 T^{11} + 326940373369 T^{12} \))(\( 1 + 2 T - 182 T^{2} + 2 T^{3} + 18788 T^{4} - 13564 T^{5} - 1721225 T^{6} - 1125812 T^{7} + 129430532 T^{8} + 1143574 T^{9} - 8637414422 T^{10} + 7878081286 T^{11} + 326940373369 T^{12} \))(\( 1 + 9 T - 148 T^{2} + 297 T^{3} + 24654 T^{4} - 118125 T^{5} - 807174 T^{6} + 21382137 T^{7} - 37648479 T^{8} - 452536146 T^{9} + 15509586612 T^{10} - 37560500118 T^{11} - 259360371831 T^{12} + 12226027968819 T^{13} - 38307122794854 T^{14} - 465299175954375 T^{15} + 8060387965039326 T^{16} + 8059407143919219 T^{17} - 333339250356578068 T^{18} + 1682462297407863627 T^{19} + 15516041187205853449 T^{20} \))(\( 1 - 25 T + 157 T^{2} + 750 T^{3} - 6622 T^{4} - 69964 T^{5} + 1466905 T^{6} - 20424981 T^{7} + 74872112 T^{8} + 997051343 T^{9} + 7203133487 T^{10} - 269189728764 T^{11} + 1307717205141 T^{12} + 2467734607815 T^{13} - 39825488981322 T^{14} + 204821972448645 T^{15} + 9008863826216349 T^{16} - 153919187440781268 T^{17} + 341848621231895327 T^{18} + 3927425763234733549 T^{19} + 24478716252205585328 T^{20} - \)\(55\!\cdots\!87\)\( T^{21} + \)\(33\!\cdots\!05\)\( T^{22} - \)\(13\!\cdots\!92\)\( T^{23} - \)\(10\!\cdots\!78\)\( T^{24} + \)\(96\!\cdots\!50\)\( T^{25} + \)\(16\!\cdots\!77\)\( T^{26} - \)\(22\!\cdots\!75\)\( T^{27} + \)\(73\!\cdots\!29\)\( T^{28} \))
$89$ (\( 1 - 5 T - 64 T^{2} - 445 T^{3} + 7921 T^{4} \))(\( 1 + 15 T + 136 T^{2} + 1335 T^{3} + 7921 T^{4} \))(\( 1 + 15 T + 136 T^{2} + 1335 T^{3} + 7921 T^{4} \))(\( 1 - 13 T + 80 T^{2} - 1157 T^{3} + 7921 T^{4} \))(\( 1 - 9 T - 8 T^{2} - 801 T^{3} + 7921 T^{4} \))(\( 1 + 3 T - 80 T^{2} + 267 T^{3} + 7921 T^{4} \))(\( 1 + 9 T - 180 T^{2} - 729 T^{3} + 31041 T^{4} + 54846 T^{5} - 2925911 T^{6} + 4881294 T^{7} + 245875761 T^{8} - 513922401 T^{9} - 11293603380 T^{10} + 50256535041 T^{11} + 496981290961 T^{12} \))(\( 1 + 9 T - 144 T^{2} - 1197 T^{3} + 16101 T^{4} + 73314 T^{5} - 1141967 T^{6} + 6524946 T^{7} + 127536021 T^{8} - 843847893 T^{9} - 9034882704 T^{10} + 50256535041 T^{11} + 496981290961 T^{12} \))(\( 1 - 28 T + 104 T^{2} + 1736 T^{3} + 31273 T^{4} - 611939 T^{5} - 1780638 T^{6} + 18973932 T^{7} + 740914101 T^{8} - 3271180573 T^{9} - 40614588329 T^{10} - 291135070997 T^{11} + 5868780594021 T^{12} + 13376033868108 T^{13} - 111721218529758 T^{14} - 3417103755161611 T^{15} + 15542095912223353 T^{16} + 76785597378638344 T^{17} + 409405235793016424 T^{18} - 9809979303809585852 T^{19} + 31181719929966183601 T^{20} \))(\( 1 + 6 T - 395 T^{2} - 1770 T^{3} + 83413 T^{4} + 244017 T^{5} - 12791578 T^{6} - 19742595 T^{7} + 1607888462 T^{8} + 933579267 T^{9} - 173735706451 T^{10} - 11863751967 T^{11} + 16888438123377 T^{12} - 583089996186 T^{13} - 1538438146646466 T^{14} - 51895009660554 T^{15} + 133773318375269217 T^{16} - 8363577360424023 T^{17} - 10900567564453896691 T^{18} + 5213162127281843883 T^{19} + \)\(79\!\cdots\!82\)\( T^{20} - \)\(87\!\cdots\!55\)\( T^{21} - \)\(50\!\cdots\!18\)\( T^{22} + \)\(85\!\cdots\!53\)\( T^{23} + \)\(26\!\cdots\!13\)\( T^{24} - \)\(49\!\cdots\!30\)\( T^{25} - \)\(97\!\cdots\!95\)\( T^{26} + \)\(13\!\cdots\!14\)\( T^{27} + \)\(19\!\cdots\!41\)\( T^{28} \))
$97$ (\( 1 + 7 T - 48 T^{2} + 679 T^{3} + 9409 T^{4} \))(\( 1 - 17 T + 192 T^{2} - 1649 T^{3} + 9409 T^{4} \))(\( 1 - T - 96 T^{2} - 97 T^{3} + 9409 T^{4} \))(\( 1 - 9 T - 16 T^{2} - 873 T^{3} + 9409 T^{4} \))(\( 1 - 17 T + 192 T^{2} - 1649 T^{3} + 9409 T^{4} \))(\( 1 - T - 96 T^{2} - 97 T^{3} + 9409 T^{4} \))(\( 1 + 28 T + 257 T^{2} + 2820 T^{3} + 59506 T^{4} + 545924 T^{5} + 3126001 T^{6} + 52954628 T^{7} + 559891954 T^{8} + 2573737860 T^{9} + 22752025217 T^{10} + 240445527196 T^{11} + 832972004929 T^{12} \))(\( 1 - 28 T + 281 T^{2} - 2724 T^{3} + 45178 T^{4} - 388196 T^{5} + 2169217 T^{6} - 37655012 T^{7} + 425079802 T^{8} - 2486121252 T^{9} + 24876727961 T^{10} - 240445527196 T^{11} + 832972004929 T^{12} \))(\( 1 + 12 T - 197 T^{2} - 1534 T^{3} + 27813 T^{4} + 14090 T^{5} - 4545035 T^{6} - 6881349 T^{7} + 472663750 T^{8} + 908843245 T^{9} - 38512186359 T^{10} + 88157794765 T^{11} + 4447293223750 T^{12} - 6280421435877 T^{13} - 402368680669835 T^{14} + 120995624221130 T^{15} + 23167450373090277 T^{16} - 123944568389425342 T^{17} - 1543974418092261317 T^{18} + 9122772703854782604 T^{19} + 73742412689492826049 T^{20} \))(\( 1 + 18 T - 332 T^{2} - 7282 T^{3} + 70792 T^{4} + 1694642 T^{5} - 11251793 T^{6} - 262778647 T^{7} + 1667779429 T^{8} + 30411590872 T^{9} - 218639545013 T^{10} - 2436064838528 T^{11} + 25890765026051 T^{12} + 94489464293929 T^{13} - 2645836565169718 T^{14} + 9165478036511113 T^{15} + 243606208130113859 T^{16} - 2223330604373865344 T^{17} - 19356001718168025653 T^{18} + \)\(26\!\cdots\!04\)\( T^{19} + \)\(13\!\cdots\!41\)\( T^{20} - \)\(21\!\cdots\!11\)\( T^{21} - \)\(88\!\cdots\!73\)\( T^{22} + \)\(12\!\cdots\!14\)\( T^{23} + \)\(52\!\cdots\!08\)\( T^{24} - \)\(52\!\cdots\!46\)\( T^{25} - \)\(23\!\cdots\!12\)\( T^{26} + \)\(12\!\cdots\!86\)\( T^{27} + \)\(65\!\cdots\!69\)\( T^{28} \))
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