# Properties

 Label 1008.2.q Level 1008 Weight 2 Character orbit q Rep. character $$\chi_{1008}(529,\cdot)$$ Character field $$\Q(\zeta_{3})$$ Dimension 92 Newform subspaces 12 Sturm bound 384 Trace bound 7

# Related objects

## Defining parameters

 Level: $$N$$ = $$1008 = 2^{4} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 1008.q (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: $$7$$ 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

 $$92q + q^{3} + q^{5} + q^{7} - q^{9} + O(q^{10})$$ $$92q + q^{3} + q^{5} + q^{7} - q^{9} - q^{11} - 2q^{13} + 7q^{15} - 2q^{17} + 2q^{19} - 3q^{21} - q^{23} - 37q^{25} + 4q^{27} - 6q^{29} + 14q^{31} + 5q^{33} + 9q^{35} - 2q^{37} + 16q^{39} - 2q^{41} - 4q^{43} + 13q^{45} + 42q^{47} - q^{49} - 19q^{51} - 2q^{53} + 18q^{55} - 6q^{57} - 70q^{59} - 2q^{61} + 2q^{65} + 2q^{67} - 31q^{69} + 32q^{71} - 2q^{73} - 33q^{75} + 21q^{77} + 2q^{79} - q^{81} + 28q^{83} + 3q^{85} + 31q^{87} + 2q^{89} - 4q^{91} - 5q^{93} - 54q^{95} - 2q^{97} + 61q^{99} + O(q^{100})$$

## 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.q.a $$2$$ $$8.049$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$-3$$ $$4$$ $$q+(1-2\zeta_{6})q^{3}+(-3+3\zeta_{6})q^{5}+(1+\cdots)q^{7}+\cdots$$
1008.2.q.b $$2$$ $$8.049$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$-1$$ $$-4$$ $$q+(1-2\zeta_{6})q^{3}+(-1+\zeta_{6})q^{5}+(-1+\cdots)q^{7}+\cdots$$
1008.2.q.c $$2$$ $$8.049$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$1$$ $$-4$$ $$q+(1-2\zeta_{6})q^{3}+(1-\zeta_{6})q^{5}+(-3+2\zeta_{6})q^{7}+\cdots$$
1008.2.q.d $$2$$ $$8.049$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$1$$ $$4$$ $$q+(-1+2\zeta_{6})q^{3}+(1-\zeta_{6})q^{5}+(3-2\zeta_{6})q^{7}+\cdots$$
1008.2.q.e $$2$$ $$8.049$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$3$$ $$4$$ $$q+(1-2\zeta_{6})q^{3}+(3-3\zeta_{6})q^{5}+(3-2\zeta_{6})q^{7}+\cdots$$
1008.2.q.f $$2$$ $$8.049$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$3$$ $$-2$$ $$5$$ $$q+(2-\zeta_{6})q^{3}+(-2+2\zeta_{6})q^{5}+(2+\zeta_{6})q^{7}+\cdots$$
1008.2.q.g $$6$$ $$8.049$$ 6.0.309123.1 None $$0$$ $$-2$$ $$1$$ $$-2$$ $$q+(\beta _{2}+\beta _{4})q^{3}+(1-\beta _{1}+\beta _{3}-\beta _{4}+\cdots)q^{5}+\cdots$$
1008.2.q.h $$6$$ $$8.049$$ 6.0.309123.1 None $$0$$ $$2$$ $$-5$$ $$-4$$ $$q+(-\beta _{3}+\beta _{4}+\beta _{5})q^{3}+(\beta _{1}+2\beta _{4}+\cdots)q^{5}+\cdots$$
1008.2.q.i $$10$$ $$8.049$$ 10.0.$$\cdots$$.1 None $$0$$ $$1$$ $$4$$ $$4$$ $$q+\beta _{8}q^{3}+(\beta _{1}+\beta _{2}+\beta _{4})q^{5}+(1-\beta _{1}+\cdots)q^{7}+\cdots$$
1008.2.q.j $$14$$ $$8.049$$ $$\mathbb{Q}[x]/(x^{14} - \cdots)$$ None $$0$$ $$-3$$ $$-2$$ $$-6$$ $$q-\beta _{3}q^{3}+\beta _{4}q^{5}+\beta _{12}q^{7}+(\beta _{1}+\beta _{2}+\cdots)q^{9}+\cdots$$
1008.2.q.k $$22$$ $$8.049$$ None $$0$$ $$-2$$ $$3$$ $$5$$
1008.2.q.l $$22$$ $$8.049$$ None $$0$$ $$2$$ $$1$$ $$-5$$

## 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

$p$ $F_p(T)$
$2$ ()()()()()()()()()()
$3$ ($$1 + 3 T^{2}$$)($$1 + 3 T^{2}$$)($$1 + 3 T^{2}$$)($$1 + 3 T^{2}$$)($$1 + 3 T^{2}$$)($$1 - 3 T + 3 T^{2}$$)($$1 + 2 T - 2 T^{2} - 9 T^{3} - 6 T^{4} + 18 T^{5} + 27 T^{6}$$)($$1 - 2 T + 4 T^{2} - 3 T^{3} + 12 T^{4} - 18 T^{5} + 27 T^{6}$$)($$1 - T - 5 T^{2} + 3 T^{3} + 9 T^{4} - 9 T^{5} + 27 T^{6} + 27 T^{7} - 135 T^{8} - 81 T^{9} + 243 T^{10}$$)($$1 + 3 T + 7 T^{2} + 15 T^{3} + 34 T^{4} + 78 T^{5} + 144 T^{6} + 252 T^{7} + 432 T^{8} + 702 T^{9} + 918 T^{10} + 1215 T^{11} + 1701 T^{12} + 2187 T^{13} + 2187 T^{14}$$)
$5$ ($$1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4}$$)($$1 + T - 4 T^{2} + 5 T^{3} + 25 T^{4}$$)($$1 - T - 4 T^{2} - 5 T^{3} + 25 T^{4}$$)($$1 - T - 4 T^{2} - 5 T^{3} + 25 T^{4}$$)($$1 - 3 T + 4 T^{2} - 15 T^{3} + 25 T^{4}$$)($$1 + 2 T - T^{2} + 10 T^{3} + 25 T^{4}$$)($$1 - T - 8 T^{2} + 17 T^{3} + 23 T^{4} - 52 T^{5} - 11 T^{6} - 260 T^{7} + 575 T^{8} + 2125 T^{9} - 5000 T^{10} - 3125 T^{11} + 15625 T^{12}$$)($$1 + 5 T + 6 T^{2} + T^{3} + 31 T^{4} + 68 T^{5} + 29 T^{6} + 340 T^{7} + 775 T^{8} + 125 T^{9} + 3750 T^{10} + 15625 T^{11} + 15625 T^{12}$$)($$1 - 4 T - 4 T^{2} + 44 T^{3} - 41 T^{4} - 119 T^{5} + 222 T^{6} - 456 T^{7} + 1623 T^{8} + 2021 T^{9} - 16541 T^{10} + 10105 T^{11} + 40575 T^{12} - 57000 T^{13} + 138750 T^{14} - 371875 T^{15} - 640625 T^{16} + 3437500 T^{17} - 1562500 T^{18} - 7812500 T^{19} + 9765625 T^{20}$$)($$1 + 2 T - 11 T^{2} - 66 T^{3} - 25 T^{4} + 533 T^{5} + 1540 T^{6} - 603 T^{7} - 11032 T^{8} - 21691 T^{9} + 19925 T^{10} + 156945 T^{11} + 233805 T^{12} - 385788 T^{13} - 1886454 T^{14} - 1928940 T^{15} + 5845125 T^{16} + 19618125 T^{17} + 12453125 T^{18} - 67784375 T^{19} - 172375000 T^{20} - 47109375 T^{21} + 601562500 T^{22} + 1041015625 T^{23} - 244140625 T^{24} - 3222656250 T^{25} - 2685546875 T^{26} + 2441406250 T^{27} + 6103515625 T^{28}$$)
$7$ ($$1 - 4 T + 7 T^{2}$$)($$1 + 4 T + 7 T^{2}$$)($$1 + 4 T + 7 T^{2}$$)($$1 - 4 T + 7 T^{2}$$)($$1 - 4 T + 7 T^{2}$$)($$1 - 5 T + 7 T^{2}$$)($$1 + 2 T - 4 T^{2} - 31 T^{3} - 28 T^{4} + 98 T^{5} + 343 T^{6}$$)($$1 + 4 T + 14 T^{2} + 55 T^{3} + 98 T^{4} + 196 T^{5} + 343 T^{6}$$)($$1 - 4 T + 12 T^{2} - 47 T^{3} + 146 T^{4} - 309 T^{5} + 1022 T^{6} - 2303 T^{7} + 4116 T^{8} - 9604 T^{9} + 16807 T^{10}$$)($$1 + 6 T + 20 T^{2} + 77 T^{3} + 309 T^{4} + 961 T^{5} + 2706 T^{6} + 7572 T^{7} + 18942 T^{8} + 47089 T^{9} + 105987 T^{10} + 184877 T^{11} + 336140 T^{12} + 705894 T^{13} + 823543 T^{14}$$)
$11$ ($$1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4}$$)($$1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4}$$)($$1 - 5 T + 14 T^{2} - 55 T^{3} + 121 T^{4}$$)($$1 - 3 T - 2 T^{2} - 33 T^{3} + 121 T^{4}$$)($$1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4}$$)($$1 - 4 T + 5 T^{2} - 44 T^{3} + 121 T^{4}$$)($$1 - T - 26 T^{2} + 23 T^{3} + 407 T^{4} - 202 T^{5} - 4853 T^{6} - 2222 T^{7} + 49247 T^{8} + 30613 T^{9} - 380666 T^{10} - 161051 T^{11} + 1771561 T^{12}$$)($$1 - T - 6 T^{2} + 103 T^{3} - 83 T^{4} - 352 T^{5} + 5027 T^{6} - 3872 T^{7} - 10043 T^{8} + 137093 T^{9} - 87846 T^{10} - 161051 T^{11} + 1771561 T^{12}$$)($$1 + 4 T - 31 T^{2} - 134 T^{3} + 607 T^{4} + 2492 T^{5} - 8385 T^{6} - 27495 T^{7} + 98940 T^{8} + 135733 T^{9} - 1043873 T^{10} + 1493063 T^{11} + 11971740 T^{12} - 36595845 T^{13} - 122764785 T^{14} + 401339092 T^{15} + 1075337527 T^{16} - 2611280914 T^{17} - 6645125311 T^{18} + 9431790764 T^{19} + 25937424601 T^{20}$$)($$1 + 2 T - 32 T^{2} - 42 T^{3} + 722 T^{4} + 614 T^{5} - 9113 T^{6} + 2733 T^{7} + 72959 T^{8} - 126076 T^{9} + 256457 T^{10} + 2291538 T^{11} - 13152543 T^{12} - 8654619 T^{13} + 213521742 T^{14} - 95200809 T^{15} - 1591457703 T^{16} + 3050037078 T^{17} + 3754786937 T^{18} - 20304665876 T^{19} + 129251318999 T^{20} + 53258438343 T^{21} - 1953452482553 T^{22} + 1447779882274 T^{23} + 18726820561922 T^{24} - 11983090165662 T^{25} - 100429708055072 T^{26} + 69045424287862 T^{27} + 379749833583241 T^{28}$$)
$13$ ($$1 - T - 12 T^{2} - 13 T^{3} + 169 T^{4}$$)($$1 + T - 12 T^{2} + 13 T^{3} + 169 T^{4}$$)($$( 1 - 7 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} )$$)($$1 + 3 T - 4 T^{2} + 39 T^{3} + 169 T^{4}$$)($$( 1 - 2 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} )$$)($$1 + 3 T - 4 T^{2} + 39 T^{3} + 169 T^{4}$$)($$1 - 8 T + 24 T^{2} - 42 T^{3} - 32 T^{4} + 1408 T^{5} - 7901 T^{6} + 18304 T^{7} - 5408 T^{8} - 92274 T^{9} + 685464 T^{10} - 2970344 T^{11} + 4826809 T^{12}$$)($$1 + 2 T - 32 T^{2} - 26 T^{3} + 730 T^{4} + 230 T^{5} - 10729 T^{6} + 2990 T^{7} + 123370 T^{8} - 57122 T^{9} - 913952 T^{10} + 742586 T^{11} + 4826809 T^{12}$$)($$1 + 8 T - 14 T^{2} - 182 T^{3} + 686 T^{4} + 4429 T^{5} - 12871 T^{6} - 43199 T^{7} + 305249 T^{8} + 358672 T^{9} - 3841969 T^{10} + 4662736 T^{11} + 51587081 T^{12} - 94908203 T^{13} - 367608631 T^{14} + 1644456697 T^{15} + 3311190974 T^{16} - 11420230094 T^{17} - 11420230094 T^{18} + 84835994984 T^{19} + 137858491849 T^{20}$$)($$1 - 2 T - 25 T^{2} - 20 T^{3} + 172 T^{4} + 1281 T^{5} + 1882 T^{6} - 1142 T^{7} - 27931 T^{8} - 309997 T^{9} + 9092 T^{10} + 1859069 T^{11} + 4530175 T^{12} + 5185603 T^{13} - 73974588 T^{14} + 67412839 T^{15} + 765599575 T^{16} + 4084374593 T^{17} + 259676612 T^{18} - 115099716121 T^{19} - 134817602179 T^{20} - 71658806414 T^{21} + 1535205216922 T^{22} + 13584363696813 T^{23} + 23711660598028 T^{24} - 35843207880740 T^{25} - 582452128062025 T^{26} - 605750213184506 T^{27} + 3937376385699289 T^{28}$$)
$17$ ($$1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4}$$)($$1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4}$$)($$1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4}$$)($$1 - 5 T + 8 T^{2} - 85 T^{3} + 289 T^{4}$$)($$1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4}$$)($$1 + 7 T + 32 T^{2} + 119 T^{3} + 289 T^{4}$$)($$1 + 4 T - 23 T^{2} - 68 T^{3} + 410 T^{4} + 220 T^{5} - 8111 T^{6} + 3740 T^{7} + 118490 T^{8} - 334084 T^{9} - 1920983 T^{10} + 5679428 T^{11} + 24137569 T^{12}$$)($$1 + 4 T + 9 T^{2} + 92 T^{3} + 58 T^{4} - 20 T^{5} + 5393 T^{6} - 340 T^{7} + 16762 T^{8} + 451996 T^{9} + 751689 T^{10} + 5679428 T^{11} + 24137569 T^{12}$$)($$1 - 12 T + 14 T^{2} + 192 T^{3} + 1185 T^{4} - 11847 T^{5} - 6180 T^{6} + 65736 T^{7} + 1002861 T^{8} - 2436261 T^{9} - 7749777 T^{10} - 41416437 T^{11} + 289826829 T^{12} + 322960968 T^{13} - 516159780 T^{14} - 16821045879 T^{15} + 28603019265 T^{16} + 78785025216 T^{17} + 97660604174 T^{18} - 1423054517964 T^{19} + 2015993900449 T^{20}$$)($$1 - 2 T - 65 T^{2} + 210 T^{3} + 2087 T^{4} - 9143 T^{5} - 37340 T^{6} + 240381 T^{7} + 293834 T^{8} - 4176065 T^{9} + 3382763 T^{10} + 47662233 T^{11} - 157347285 T^{12} - 273515658 T^{13} + 3170983122 T^{14} - 4649766186 T^{15} - 45473365365 T^{16} + 234164550729 T^{17} + 282531748523 T^{18} - 5929415122705 T^{19} + 7092438449546 T^{20} + 98637620554413 T^{21} - 260474782846940 T^{22} - 1084248954812071 T^{23} + 4207379270237063 T^{24} + 7197098224602930 T^{25} - 37870445419934465 T^{26} - 19809156065811874 T^{27} + 168377826559400929 T^{28}$$)
$19$ ($$( 1 - T + 19 T^{2} )( 1 + 8 T + 19 T^{2} )$$)($$1 - 5 T + 6 T^{2} - 95 T^{3} + 361 T^{4}$$)($$( 1 - 8 T + 19 T^{2} )( 1 + 7 T + 19 T^{2} )$$)($$( 1 - 8 T + 19 T^{2} )( 1 + T + 19 T^{2} )$$)($$1 - 5 T + 6 T^{2} - 95 T^{3} + 361 T^{4}$$)($$1 - 5 T + 6 T^{2} - 95 T^{3} + 361 T^{4}$$)($$1 - 3 T - 12 T^{2} + 67 T^{3} - 153 T^{4} - 54 T^{5} + 6315 T^{6} - 1026 T^{7} - 55233 T^{8} + 459553 T^{9} - 1563852 T^{10} - 7428297 T^{11} + 47045881 T^{12}$$)($$1 - 3 T - 42 T^{2} + 61 T^{3} + 1311 T^{4} - 726 T^{5} - 27501 T^{6} - 13794 T^{7} + 473271 T^{8} + 418399 T^{9} - 5473482 T^{10} - 7428297 T^{11} + 47045881 T^{12}$$)($$1 + T - 53 T^{2} - 190 T^{3} + 1262 T^{4} + 7007 T^{5} - 13111 T^{6} - 116110 T^{7} + 67964 T^{8} + 721616 T^{9} - 440023 T^{10} + 13710704 T^{11} + 24535004 T^{12} - 796398490 T^{13} - 1708638631 T^{14} + 17350025693 T^{15} + 59371901822 T^{16} - 169835630410 T^{17} - 900128841173 T^{18} + 322687697779 T^{19} + 6131066257801 T^{20}$$)($$1 + 7 T - 54 T^{2} - 381 T^{3} + 1875 T^{4} + 9873 T^{5} - 65652 T^{6} - 221430 T^{7} + 1870425 T^{8} + 4319703 T^{9} - 46476858 T^{10} - 61637031 T^{11} + 1073881146 T^{12} + 457871775 T^{13} - 21789737442 T^{14} + 8699563725 T^{15} + 387671093706 T^{16} - 422768395629 T^{17} - 6056910611418 T^{18} + 10696012278597 T^{19} + 87995791969425 T^{20} - 197930019166770 T^{21} - 1115004880767732 T^{22} + 3185895640172067 T^{23} + 11495749233376875 T^{24} - 44382788640221439 T^{25} - 119519005629572694 T^{26} + 294370884235799413 T^{27} + 799006685782884121 T^{28}$$)
$23$ ($$1 + 9 T + 58 T^{2} + 207 T^{3} + 529 T^{4}$$)($$1 - T - 22 T^{2} - 23 T^{3} + 529 T^{4}$$)($$1 - 3 T - 14 T^{2} - 69 T^{3} + 529 T^{4}$$)($$1 - 5 T + 2 T^{2} - 115 T^{3} + 529 T^{4}$$)($$1 + 3 T - 14 T^{2} + 69 T^{3} + 529 T^{4}$$)($$1 - 4 T - 7 T^{2} - 92 T^{3} + 529 T^{4}$$)($$1 - 7 T - 32 T^{2} + 83 T^{3} + 2423 T^{4} - 3946 T^{5} - 46865 T^{6} - 90758 T^{7} + 1281767 T^{8} + 1009861 T^{9} - 8954912 T^{10} - 45054401 T^{11} + 148035889 T^{12}$$)($$1 - 7 T - 24 T^{2} + 127 T^{3} + 1417 T^{4} - 3484 T^{5} - 22393 T^{6} - 80132 T^{7} + 749593 T^{8} + 1545209 T^{9} - 6716184 T^{10} - 45054401 T^{11} + 148035889 T^{12}$$)($$1 + 3 T - 43 T^{2} - 294 T^{3} + 6 T^{4} + 5127 T^{5} + 21792 T^{6} + 135027 T^{7} + 502362 T^{8} - 3271749 T^{9} - 33095343 T^{10} - 75250227 T^{11} + 265749498 T^{12} + 1642873509 T^{13} + 6098295072 T^{14} + 32999130561 T^{15} + 888215334 T^{16} - 1001018681418 T^{17} - 3367372367083 T^{18} + 5403457984389 T^{19} + 41426511213649 T^{20}$$)($$1 + 11 T - 8 T^{2} - 333 T^{3} + 401 T^{4} + 3407 T^{5} - 42005 T^{6} - 36987 T^{7} + 1121822 T^{8} - 103015 T^{9} - 29774440 T^{10} - 52947609 T^{11} + 527481093 T^{12} + 1421356596 T^{13} - 4817982924 T^{14} + 32691201708 T^{15} + 279037498197 T^{16} - 644213558703 T^{17} - 8332109064040 T^{18} - 663039874145 T^{19} + 166069917069758 T^{20} - 125934278808189 T^{21} - 3289452936728405 T^{22} + 6136527117604441 T^{23} + 16612030996673249 T^{24} - 317285649385337691 T^{25} - 175316995456162568 T^{26} + 5544399981301141213 T^{27} + 11592836324538749809 T^{28}$$)
$29$ ($$1 + 3 T - 20 T^{2} + 87 T^{3} + 841 T^{4}$$)($$1 + 9 T + 52 T^{2} + 261 T^{3} + 841 T^{4}$$)($$1 - T - 28 T^{2} - 29 T^{3} + 841 T^{4}$$)($$1 - T - 28 T^{2} - 29 T^{3} + 841 T^{4}$$)($$1 - 3 T - 20 T^{2} - 87 T^{3} + 841 T^{4}$$)($$1 - T - 28 T^{2} - 29 T^{3} + 841 T^{4}$$)($$1 + 5 T + 4 T^{2} + 251 T^{3} + 197 T^{4} - 3418 T^{5} + 20293 T^{6} - 99122 T^{7} + 165677 T^{8} + 6121639 T^{9} + 2829124 T^{10} + 102555745 T^{11} + 594823321 T^{12}$$)($$1 + 5 T - 30 T^{2} - 371 T^{3} - 185 T^{4} + 6020 T^{5} + 44357 T^{6} + 174580 T^{7} - 155585 T^{8} - 9048319 T^{9} - 21218430 T^{10} + 102555745 T^{11} + 594823321 T^{12}$$)($$1 - 7 T - 76 T^{2} + 419 T^{3} + 4561 T^{4} - 15146 T^{5} - 199563 T^{6} + 341373 T^{7} + 6918636 T^{8} - 2570041 T^{9} - 219913241 T^{10} - 74531189 T^{11} + 5818572876 T^{12} + 8325746097 T^{13} - 141147118203 T^{14} - 310661862754 T^{15} + 2712989167081 T^{16} + 7227698173471 T^{17} - 38018727385036 T^{18} - 101550021831083 T^{19} + 420707233300201 T^{20}$$)($$1 - T - 89 T^{2} + 606 T^{3} + 3413 T^{4} - 45595 T^{5} + 49603 T^{6} + 1643802 T^{7} - 7893892 T^{8} - 19552444 T^{9} + 271585946 T^{10} - 420585531 T^{11} - 3441402105 T^{12} + 10956432363 T^{13} + 10513309734 T^{14} + 317736538527 T^{15} - 2894219170305 T^{16} - 10257660515559 T^{17} + 192087579472826 T^{18} - 401043092198156 T^{19} - 4695471055055332 T^{20} + 28355381176486818 T^{21} + 24813722822104483 T^{22} - 661453320769747055 T^{23} + 1435873787253586013 T^{24} + 7393508918017732374 T^{25} - 31489515705286744649 T^{26} - 10260628712958602189 T^{27} +$$$$29\!\cdots\!81$$$$T^{28}$$)
$31$ ($$( 1 + 8 T + 31 T^{2} )^{2}$$)($$( 1 + 4 T + 31 T^{2} )^{2}$$)($$( 1 + 31 T^{2} )^{2}$$)($$( 1 - 8 T + 31 T^{2} )^{2}$$)($$( 1 - 4 T + 31 T^{2} )^{2}$$)($$( 1 - 3 T + 31 T^{2} )^{2}$$)($$( 1 - 20 T + 214 T^{2} - 1441 T^{3} + 6634 T^{4} - 19220 T^{5} + 29791 T^{6} )^{2}$$)($$( 1 + 14 T + 138 T^{2} + 841 T^{3} + 4278 T^{4} + 13454 T^{5} + 29791 T^{6} )^{2}$$)($$( 1 + 3 T + 134 T^{2} + 308 T^{3} + 7750 T^{4} + 13615 T^{5} + 240250 T^{6} + 295988 T^{7} + 3991994 T^{8} + 2770563 T^{9} + 28629151 T^{10} )^{2}$$)($$( 1 + T + 86 T^{2} + 194 T^{3} + 4879 T^{4} + 9670 T^{5} + 191454 T^{6} + 403758 T^{7} + 5935074 T^{8} + 9292870 T^{9} + 145350289 T^{10} + 179163074 T^{11} + 2462106986 T^{12} + 887503681 T^{13} + 27512614111 T^{14} )^{2}$$)
$37$ ($$( 1 - 11 T + 37 T^{2} )( 1 + 10 T + 37 T^{2} )$$)($$1 + 5 T - 12 T^{2} + 185 T^{3} + 1369 T^{4}$$)($$1 + 3 T - 28 T^{2} + 111 T^{3} + 1369 T^{4}$$)($$1 + 3 T - 28 T^{2} + 111 T^{3} + 1369 T^{4}$$)($$1 - 7 T + 12 T^{2} - 259 T^{3} + 1369 T^{4}$$)($$( 1 + T + 37 T^{2} )( 1 + 10 T + 37 T^{2} )$$)($$( 1 - 11 T + 37 T^{2} )^{3}( 1 + 10 T + 37 T^{2} )^{3}$$)($$1 + 9 T - 21 T^{2} - 268 T^{3} + 1293 T^{4} + 4875 T^{5} - 42882 T^{6} + 180375 T^{7} + 1770117 T^{8} - 13575004 T^{9} - 39357381 T^{10} + 624095613 T^{11} + 2565726409 T^{12}$$)($$1 - 89 T^{2} + 560 T^{3} + 4503 T^{4} - 45352 T^{5} + 27130 T^{6} + 2296536 T^{7} - 9801827 T^{8} - 33131096 T^{9} + 610977105 T^{10} - 1225850552 T^{11} - 13418701163 T^{12} + 116326438008 T^{13} + 50845987930 T^{14} - 3144887137864 T^{15} + 11553466019727 T^{16} + 53161851194480 T^{17} - 312610671398969 T^{18} + 4808584372417849 T^{20}$$)($$1 - 10 T - 84 T^{2} + 1296 T^{3} + 1134 T^{4} - 66630 T^{5} + 77382 T^{6} + 1851174 T^{7} - 1251993 T^{8} - 21837810 T^{9} - 268575252 T^{10} - 774884982 T^{11} + 27666538119 T^{12} + 26159905206 T^{13} - 1381876923558 T^{14} + 967916492622 T^{15} + 37875490684911 T^{16} - 39250248993246 T^{17} - 503353262863572 T^{18} - 1514320157614170 T^{19} - 3212271503983137 T^{20} + 175735422719804142 T^{21} + 271802685103314822 T^{22} - 8659350722545980510 T^{23} + 5452934678321840766 T^{24} +$$$$23\!\cdots\!48$$$$T^{25} -$$$$55\!\cdots\!04$$$$T^{26} -$$$$24\!\cdots\!70$$$$T^{27} +$$$$90\!\cdots\!89$$$$T^{28}$$)
$41$ ($$1 + 3 T - 32 T^{2} + 123 T^{3} + 1681 T^{4}$$)($$1 + 7 T + 8 T^{2} + 287 T^{3} + 1681 T^{4}$$)($$1 - 5 T - 16 T^{2} - 205 T^{3} + 1681 T^{4}$$)($$1 - 5 T - 16 T^{2} - 205 T^{3} + 1681 T^{4}$$)($$1 - 9 T + 40 T^{2} - 369 T^{3} + 1681 T^{4}$$)($$1 - 9 T + 40 T^{2} - 369 T^{3} + 1681 T^{4}$$)($$1 - 90 T^{2} + 18 T^{3} + 4410 T^{4} - 810 T^{5} - 194177 T^{6} - 33210 T^{7} + 7413210 T^{8} + 1240578 T^{9} - 254318490 T^{10} + 4750104241 T^{12}$$)($$1 + 12 T - 18 T^{2} - 78 T^{3} + 7470 T^{4} + 24546 T^{5} - 158105 T^{6} + 1006386 T^{7} + 12557070 T^{8} - 5375838 T^{9} - 50863698 T^{10} + 1390274412 T^{11} + 4750104241 T^{12}$$)($$1 - 5 T - 136 T^{2} + 733 T^{3} + 10507 T^{4} - 54412 T^{5} - 554055 T^{6} + 2345451 T^{7} + 23706084 T^{8} - 41392439 T^{9} - 952045937 T^{10} - 1697089999 T^{11} + 39849927204 T^{12} + 161650828371 T^{13} - 1565627010855 T^{14} - 6303967608812 T^{15} + 49909345260187 T^{16} + 142754882754773 T^{17} - 1085949831160456 T^{18} - 1636909671969805 T^{19} + 13422659310152401 T^{20}$$)($$1 + 33 T + 463 T^{2} + 3882 T^{3} + 26359 T^{4} + 177381 T^{5} + 987377 T^{6} + 3338436 T^{7} - 1489480 T^{8} - 146370792 T^{9} - 1532374696 T^{10} - 10553354379 T^{11} - 66691501599 T^{12} - 460808734581 T^{13} - 3110830401306 T^{14} - 18893158117821 T^{15} - 112108414187919 T^{16} - 727347737155059 T^{17} - 4330124653343656 T^{18} - 16957963898481192 T^{19} - 7075185264884680 T^{20} + 650174679078190116 T^{21} + 7884131517953805617 T^{22} + 58071334904735196141 T^{23} +$$$$35\!\cdots\!59$$$$T^{24} +$$$$21\!\cdots\!62$$$$T^{25} +$$$$10\!\cdots\!03$$$$T^{26} +$$$$30\!\cdots\!93$$$$T^{27} +$$$$37\!\cdots\!61$$$$T^{28}$$)
$43$ ($$1 + T - 42 T^{2} + 43 T^{3} + 1849 T^{4}$$)($$1 - 3 T - 34 T^{2} - 129 T^{3} + 1849 T^{4}$$)($$1 + T - 42 T^{2} + 43 T^{3} + 1849 T^{4}$$)($$1 + 7 T + 6 T^{2} + 301 T^{3} + 1849 T^{4}$$)($$1 - 11 T + 78 T^{2} - 473 T^{3} + 1849 T^{4}$$)($$( 1 - 13 T + 43 T^{2} )( 1 + 8 T + 43 T^{2} )$$)($$( 1 - 18 T + 198 T^{2} - 1519 T^{3} + 8514 T^{4} - 33282 T^{5} + 79507 T^{6} )( 1 + 12 T - 6 T^{2} - 547 T^{3} - 258 T^{4} + 22188 T^{5} + 79507 T^{6} )$$)($$1 + 18 T + 114 T^{2} + 682 T^{3} + 7188 T^{4} + 33492 T^{5} + 63039 T^{6} + 1440156 T^{7} + 13290612 T^{8} + 54223774 T^{9} + 389743314 T^{10} + 2646151974 T^{11} + 6321363049 T^{12}$$)($$1 - 7 T - 77 T^{2} + 66 T^{3} + 7014 T^{4} + 3843 T^{5} - 95427 T^{6} - 1632678 T^{7} - 3708600 T^{8} + 15416324 T^{9} + 670279801 T^{10} + 662901932 T^{11} - 6857201400 T^{12} - 129809329746 T^{13} - 326245923027 T^{14} + 564953446449 T^{15} + 44338040425686 T^{16} + 17940028333062 T^{17} - 899991421375277 T^{18} - 3518148283557901 T^{19} + 21611482313284249 T^{20}$$)($$1 + 7 T - 222 T^{2} - 1221 T^{3} + 30003 T^{4} + 121065 T^{5} - 2964828 T^{6} - 8400318 T^{7} + 232132089 T^{8} + 430114695 T^{9} - 14993117802 T^{10} - 15328887375 T^{11} + 817956723570 T^{12} + 259275458991 T^{13} - 38016795252930 T^{14} + 11148844736613 T^{15} + 1512401981880930 T^{16} - 1218753848524125 T^{17} - 51258486134595402 T^{18} + 63230491623369885 T^{19} + 1467391209891779361 T^{20} - 2283362771617132026 T^{21} - 34653503452639217628 T^{22} + 60846374564133897795 T^{23} +$$$$64\!\cdots\!47$$$$T^{24} -$$$$11\!\cdots\!47$$$$T^{25} -$$$$88\!\cdots\!22$$$$T^{26} +$$$$12\!\cdots\!01$$$$T^{27} +$$$$73\!\cdots\!49$$$$T^{28}$$)
$47$ ($$( 1 + 47 T^{2} )^{2}$$)($$( 1 + 8 T + 47 T^{2} )^{2}$$)($$( 1 + 47 T^{2} )^{2}$$)($$( 1 + 8 T + 47 T^{2} )^{2}$$)($$( 1 + 47 T^{2} )^{2}$$)($$( 1 + 3 T + 47 T^{2} )^{2}$$)($$( 1 + 9 T + 87 T^{2} + 657 T^{3} + 4089 T^{4} + 19881 T^{5} + 103823 T^{6} )^{2}$$)($$( 1 - 3 T + 117 T^{2} - 309 T^{3} + 5499 T^{4} - 6627 T^{5} + 103823 T^{6} )^{2}$$)($$( 1 - 27 T + 448 T^{2} - 5169 T^{3} + 48091 T^{4} - 359985 T^{5} + 2260277 T^{6} - 11418321 T^{7} + 46512704 T^{8} - 131751387 T^{9} + 229345007 T^{10} )^{2}$$)($$( 1 + 3 T + 224 T^{2} + 711 T^{3} + 24846 T^{4} + 72810 T^{5} + 1742359 T^{6} + 4337544 T^{7} + 81890873 T^{8} + 160837290 T^{9} + 2579586258 T^{10} + 3469453191 T^{11} + 51373281568 T^{12} + 32337645987 T^{13} + 506623120463 T^{14} )^{2}$$)
$53$ ($$1 + 3 T - 44 T^{2} + 159 T^{3} + 2809 T^{4}$$)($$1 + 9 T + 28 T^{2} + 477 T^{3} + 2809 T^{4}$$)($$1 - 9 T + 28 T^{2} - 477 T^{3} + 2809 T^{4}$$)($$1 - T - 52 T^{2} - 53 T^{3} + 2809 T^{4}$$)($$1 - 3 T - 44 T^{2} - 159 T^{3} + 2809 T^{4}$$)($$1 + 3 T - 44 T^{2} + 159 T^{3} + 2809 T^{4}$$)($$1 - 15 T - 33 T^{3} + 13635 T^{4} - 60360 T^{5} - 225155 T^{6} - 3199080 T^{7} + 38300715 T^{8} - 4912941 T^{9} - 6272932395 T^{11} + 22164361129 T^{12}$$)($$1 - 9 T - 36 T^{2} + 873 T^{3} - 1179 T^{4} - 26334 T^{5} + 272077 T^{6} - 1395702 T^{7} - 3311811 T^{8} + 129969621 T^{9} - 284057316 T^{10} - 3763759437 T^{11} + 22164361129 T^{12}$$)($$1 + 21 T + 41 T^{2} - 924 T^{3} + 12966 T^{4} + 177027 T^{5} - 601755 T^{6} - 3783942 T^{7} + 110973258 T^{8} + 340111866 T^{9} - 4044436041 T^{10} + 18025928898 T^{11} + 311723881722 T^{12} - 563341933134 T^{13} - 4748136394155 T^{14} + 74031893539311 T^{15} + 287383106398614 T^{16} - 1085433093209388 T^{17} + 2552647306865801 T^{18} + 69295035427844793 T^{19} + 174887470365513049 T^{20}$$)($$1 + 15 T - 44 T^{2} - 1563 T^{3} + 1621 T^{4} + 132831 T^{5} + 280796 T^{6} - 6875916 T^{7} - 32544895 T^{8} + 334617081 T^{9} + 3332374934 T^{10} - 9404405181 T^{11} - 219667861866 T^{12} + 268474558113 T^{13} + 14089335458730 T^{14} + 14229151579989 T^{15} - 617047023981594 T^{16} - 1400099630131737 T^{17} + 26294041101603254 T^{18} + 139935355155015933 T^{19} - 721336805685386455 T^{20} - 8077215121783465692 T^{21} + 17482272028748523356 T^{22} +$$$$43\!\cdots\!23$$$$T^{23} +$$$$28\!\cdots\!29$$$$T^{24} -$$$$14\!\cdots\!11$$$$T^{25} -$$$$21\!\cdots\!04$$$$T^{26} +$$$$39\!\cdots\!95$$$$T^{27} +$$$$13\!\cdots\!69$$$$T^{28}$$)
$59$ ($$( 1 + 59 T^{2} )^{2}$$)($$( 1 - 4 T + 59 T^{2} )^{2}$$)($$( 1 + 59 T^{2} )^{2}$$)($$( 1 + 59 T^{2} )^{2}$$)($$( 1 + 12 T + 59 T^{2} )^{2}$$)($$( 1 - 7 T + 59 T^{2} )^{2}$$)($$( 1 + 14 T + 216 T^{2} + 1589 T^{3} + 12744 T^{4} + 48734 T^{5} + 205379 T^{6} )^{2}$$)($$( 1 - 4 T + 76 T^{2} - 649 T^{3} + 4484 T^{4} - 13924 T^{5} + 205379 T^{6} )^{2}$$)($$( 1 - 30 T + 601 T^{2} - 8193 T^{3} + 88864 T^{4} - 752289 T^{5} + 5242976 T^{6} - 28519833 T^{7} + 123432779 T^{8} - 363520830 T^{9} + 714924299 T^{10} )^{2}$$)($$( 1 + 14 T + 237 T^{2} + 2625 T^{3} + 28687 T^{4} + 249069 T^{5} + 2197811 T^{6} + 16839260 T^{7} + 129670849 T^{8} + 867009189 T^{9} + 5891707373 T^{10} + 31808072625 T^{11} + 169437058863 T^{12} + 590527470974 T^{13} + 2488651484819 T^{14} )^{2}$$)
$61$ ($$( 1 - 2 T + 61 T^{2} )^{2}$$)($$( 1 - 2 T + 61 T^{2} )^{2}$$)($$( 1 + 14 T + 61 T^{2} )^{2}$$)($$( 1 - 10 T + 61 T^{2} )^{2}$$)($$( 1 - 2 T + 61 T^{2} )^{2}$$)($$( 1 - 3 T + 61 T^{2} )^{2}$$)($$( 1 + 8 T + 178 T^{2} + 883 T^{3} + 10858 T^{4} + 29768 T^{5} + 226981 T^{6} )^{2}$$)($$( 1 + 4 T + 48 T^{2} - 229 T^{3} + 2928 T^{4} + 14884 T^{5} + 226981 T^{6} )^{2}$$)($$( 1 - 14 T + 339 T^{2} - 3409 T^{3} + 43418 T^{4} - 311709 T^{5} + 2648498 T^{6} - 12684889 T^{7} + 76946559 T^{8} - 193841774 T^{9} + 844596301 T^{10} )^{2}$$)($$( 1 - 10 T + 377 T^{2} - 2981 T^{3} + 62533 T^{4} - 402217 T^{5} + 6040473 T^{6} - 31437684 T^{7} + 368468853 T^{8} - 1496649457 T^{9} + 14193802873 T^{10} - 41274452021 T^{11} + 318412805477 T^{12} - 515203743610 T^{13} + 3142742836021 T^{14} )^{2}$$)
$67$ ($$( 1 - 4 T + 67 T^{2} )^{2}$$)($$( 1 + 12 T + 67 T^{2} )^{2}$$)($$( 1 + 4 T + 67 T^{2} )^{2}$$)($$( 1 - 12 T + 67 T^{2} )^{2}$$)($$( 1 - 4 T + 67 T^{2} )^{2}$$)($$( 1 + 13 T + 67 T^{2} )^{2}$$)($$( 1 - T + 89 T^{2} + 77 T^{3} + 5963 T^{4} - 4489 T^{5} + 300763 T^{6} )^{2}$$)($$( 1 - 5 T + 143 T^{2} - 521 T^{3} + 9581 T^{4} - 22445 T^{5} + 300763 T^{6} )^{2}$$)($$( 1 + 2 T + 132 T^{2} + 196 T^{3} + 10871 T^{4} + 15429 T^{5} + 728357 T^{6} + 879844 T^{7} + 39700716 T^{8} + 40302242 T^{9} + 1350125107 T^{10} )^{2}$$)($$( 1 - 6 T + 344 T^{2} - 1114 T^{3} + 48744 T^{4} - 55499 T^{5} + 4192269 T^{6} - 1055346 T^{7} + 280882023 T^{8} - 249135011 T^{9} + 14660391672 T^{10} - 22448348794 T^{11} + 464443036808 T^{12} - 542750293014 T^{13} + 6060711605323 T^{14} )^{2}$$)
$71$ ($$( 1 + 12 T + 71 T^{2} )^{2}$$)($$( 1 + 8 T + 71 T^{2} )^{2}$$)($$( 1 - 12 T + 71 T^{2} )^{2}$$)($$( 1 + 12 T + 71 T^{2} )^{2}$$)($$( 1 + 71 T^{2} )^{2}$$)($$( 1 - 8 T + 71 T^{2} )^{2}$$)($$( 1 + 7 T + 15 T^{2} - 599 T^{3} + 1065 T^{4} + 35287 T^{5} + 357911 T^{6} )^{2}$$)($$( 1 + 7 T + 163 T^{2} + 895 T^{3} + 11573 T^{4} + 35287 T^{5} + 357911 T^{6} )^{2}$$)($$( 1 - 3 T + 187 T^{2} - 285 T^{3} + 15679 T^{4} - 10143 T^{5} + 1113209 T^{6} - 1436685 T^{7} + 66929357 T^{8} - 76235043 T^{9} + 1804229351 T^{10} )^{2}$$)($$( 1 + T + 381 T^{2} + 417 T^{3} + 66850 T^{4} + 68550 T^{5} + 7142942 T^{6} + 6246700 T^{7} + 507148882 T^{8} + 345560550 T^{9} + 23926350350 T^{10} + 10596670977 T^{11} + 687411382731 T^{12} + 128100283921 T^{13} + 9095120158391 T^{14} )^{2}$$)
$73$ ($$1 + 11 T + 48 T^{2} + 803 T^{3} + 5329 T^{4}$$)($$1 - 13 T + 96 T^{2} - 949 T^{3} + 5329 T^{4}$$)($$1 + 3 T - 64 T^{2} + 219 T^{3} + 5329 T^{4}$$)($$1 - 5 T - 48 T^{2} - 365 T^{3} + 5329 T^{4}$$)($$1 + 11 T + 48 T^{2} + 803 T^{3} + 5329 T^{4}$$)($$( 1 - 10 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} )$$)($$1 - 19 T + 134 T^{2} - 27 T^{3} - 5759 T^{4} + 41986 T^{5} - 314903 T^{6} + 3064978 T^{7} - 30689711 T^{8} - 10503459 T^{9} + 3805364294 T^{10} - 39388360267 T^{11} + 151334226289 T^{12}$$)($$1 + 25 T + 254 T^{2} + 2073 T^{3} + 20533 T^{4} + 115046 T^{5} + 366817 T^{6} + 8398358 T^{7} + 109420357 T^{8} + 806432241 T^{9} + 7213153214 T^{10} + 51826789825 T^{11} + 151334226289 T^{12}$$)($$1 - 15 T - 134 T^{2} + 2501 T^{3} + 16563 T^{4} - 235276 T^{5} - 2002535 T^{6} + 9021201 T^{7} + 288508378 T^{8} - 238799411 T^{9} - 25271949561 T^{10} - 17432357003 T^{11} + 1537461146362 T^{12} + 3509400549417 T^{13} - 56868471540935 T^{14} - 487743992114668 T^{15} + 2506548790024707 T^{16} + 27629543696261597 T^{17} - 108065652313806854 T^{18} - 883073800624018695 T^{19} + 4297625829703557649 T^{20}$$)($$1 - 21 T - 107 T^{2} + 4532 T^{3} + 8593 T^{4} - 574123 T^{5} - 1508435 T^{6} + 52395764 T^{7} + 342478696 T^{8} - 4227610868 T^{9} - 46283154524 T^{10} + 253115011813 T^{11} + 4673586532751 T^{12} - 7744045394141 T^{13} - 369527474872486 T^{14} - 565315313772293 T^{15} + 24905542633030079 T^{16} + 98466042550457821 T^{17} - 1314360176412792284 T^{18} - 8764139996708872724 T^{19} + 51828748479625639144 T^{20} +$$$$57\!\cdots\!08$$$$T^{21} -$$$$12\!\cdots\!35$$$$T^{22} -$$$$33\!\cdots\!99$$$$T^{23} +$$$$36\!\cdots\!57$$$$T^{24} +$$$$14\!\cdots\!64$$$$T^{25} -$$$$24\!\cdots\!47$$$$T^{26} -$$$$35\!\cdots\!93$$$$T^{27} +$$$$12\!\cdots\!09$$$$T^{28}$$)
$79$ ($$( 1 - 16 T + 79 T^{2} )^{2}$$)($$( 1 + 8 T + 79 T^{2} )^{2}$$)($$( 1 + 8 T + 79 T^{2} )^{2}$$)($$( 1 - 8 T + 79 T^{2} )^{2}$$)($$( 1 + 8 T + 79 T^{2} )^{2}$$)($$( 1 - 9 T + 79 T^{2} )^{2}$$)($$( 1 - 5 T + 163 T^{2} - 469 T^{3} + 12877 T^{4} - 31205 T^{5} + 493039 T^{6} )^{2}$$)($$( 1 - 7 T + 93 T^{2} - 335 T^{3} + 7347 T^{4} - 43687 T^{5} + 493039 T^{6} )^{2}$$)($$( 1 + 4 T + 300 T^{2} + 1488 T^{3} + 39873 T^{4} + 184983 T^{5} + 3149967 T^{6} + 9286608 T^{7} + 147911700 T^{8} + 155800324 T^{9} + 3077056399 T^{10} )^{2}$$)($$( 1 + 10 T + 326 T^{2} + 3770 T^{3} + 58966 T^{4} + 604927 T^{5} + 7130289 T^{6} + 58615338 T^{7} + 563292831 T^{8} + 3775349407 T^{9} + 29072537674 T^{10} + 146841805370 T^{11} + 1003120386074 T^{12} + 2430874555210 T^{13} + 19203908986159 T^{14} )^{2}$$)
$83$ ($$1 + 9 T - 2 T^{2} + 747 T^{3} + 6889 T^{4}$$)($$1 + 13 T + 86 T^{2} + 1079 T^{3} + 6889 T^{4}$$)($$1 + 9 T - 2 T^{2} + 747 T^{3} + 6889 T^{4}$$)($$1 + 15 T + 142 T^{2} + 1245 T^{3} + 6889 T^{4}$$)($$1 - 3 T - 74 T^{2} - 249 T^{3} + 6889 T^{4}$$)($$1 - T - 82 T^{2} - 83 T^{3} + 6889 T^{4}$$)($$1 + 2 T - 182 T^{2} + 2 T^{3} + 18788 T^{4} - 13564 T^{5} - 1721225 T^{6} - 1125812 T^{7} + 129430532 T^{8} + 1143574 T^{9} - 8637414422 T^{10} + 7878081286 T^{11} + 326940373369 T^{12}$$)($$1 + 8 T - 180 T^{2} - 518 T^{3} + 29404 T^{4} + 32420 T^{5} - 2713585 T^{6} + 2690860 T^{7} + 202564156 T^{8} - 296185666 T^{9} - 8542497780 T^{10} + 31512325144 T^{11} + 326940373369 T^{12}$$)($$1 + 9 T - 148 T^{2} + 297 T^{3} + 24654 T^{4} - 118125 T^{5} - 807174 T^{6} + 21382137 T^{7} - 37648479 T^{8} - 452536146 T^{9} + 15509586612 T^{10} - 37560500118 T^{11} - 259360371831 T^{12} + 12226027968819 T^{13} - 38307122794854 T^{14} - 465299175954375 T^{15} + 8060387965039326 T^{16} + 8059407143919219 T^{17} - 333339250356578068 T^{18} + 1682462297407863627 T^{19} + 15516041187205853449 T^{20}$$)($$1 - 25 T + 157 T^{2} + 750 T^{3} - 6622 T^{4} - 69964 T^{5} + 1466905 T^{6} - 20424981 T^{7} + 74872112 T^{8} + 997051343 T^{9} + 7203133487 T^{10} - 269189728764 T^{11} + 1307717205141 T^{12} + 2467734607815 T^{13} - 39825488981322 T^{14} + 204821972448645 T^{15} + 9008863826216349 T^{16} - 153919187440781268 T^{17} + 341848621231895327 T^{18} + 3927425763234733549 T^{19} + 24478716252205585328 T^{20} -$$$$55\!\cdots\!87$$$$T^{21} +$$$$33\!\cdots\!05$$$$T^{22} -$$$$13\!\cdots\!92$$$$T^{23} -$$$$10\!\cdots\!78$$$$T^{24} +$$$$96\!\cdots\!50$$$$T^{25} +$$$$16\!\cdots\!77$$$$T^{26} -$$$$22\!\cdots\!75$$$$T^{27} +$$$$73\!\cdots\!29$$$$T^{28}$$)
$89$ ($$1 + 3 T - 80 T^{2} + 267 T^{3} + 7921 T^{4}$$)($$1 - 9 T - 8 T^{2} - 801 T^{3} + 7921 T^{4}$$)($$1 - 13 T + 80 T^{2} - 1157 T^{3} + 7921 T^{4}$$)($$1 - 5 T - 64 T^{2} - 445 T^{3} + 7921 T^{4}$$)($$1 + 15 T + 136 T^{2} + 1335 T^{3} + 7921 T^{4}$$)($$1 + 15 T + 136 T^{2} + 1335 T^{3} + 7921 T^{4}$$)($$1 + 9 T - 144 T^{2} - 1197 T^{3} + 16101 T^{4} + 73314 T^{5} - 1141967 T^{6} + 6524946 T^{7} + 127536021 T^{8} - 843847893 T^{9} - 9034882704 T^{10} + 50256535041 T^{11} + 496981290961 T^{12}$$)($$1 + 9 T - 180 T^{2} - 729 T^{3} + 31041 T^{4} + 54846 T^{5} - 2925911 T^{6} + 4881294 T^{7} + 245875761 T^{8} - 513922401 T^{9} - 11293603380 T^{10} + 50256535041 T^{11} + 496981290961 T^{12}$$)($$1 - 28 T + 104 T^{2} + 1736 T^{3} + 31273 T^{4} - 611939 T^{5} - 1780638 T^{6} + 18973932 T^{7} + 740914101 T^{8} - 3271180573 T^{9} - 40614588329 T^{10} - 291135070997 T^{11} + 5868780594021 T^{12} + 13376033868108 T^{13} - 111721218529758 T^{14} - 3417103755161611 T^{15} + 15542095912223353 T^{16} + 76785597378638344 T^{17} + 409405235793016424 T^{18} - 9809979303809585852 T^{19} + 31181719929966183601 T^{20}$$)($$1 + 6 T - 395 T^{2} - 1770 T^{3} + 83413 T^{4} + 244017 T^{5} - 12791578 T^{6} - 19742595 T^{7} + 1607888462 T^{8} + 933579267 T^{9} - 173735706451 T^{10} - 11863751967 T^{11} + 16888438123377 T^{12} - 583089996186 T^{13} - 1538438146646466 T^{14} - 51895009660554 T^{15} + 133773318375269217 T^{16} - 8363577360424023 T^{17} - 10900567564453896691 T^{18} + 5213162127281843883 T^{19} +$$$$79\!\cdots\!82$$$$T^{20} -$$$$87\!\cdots\!55$$$$T^{21} -$$$$50\!\cdots\!18$$$$T^{22} +$$$$85\!\cdots\!53$$$$T^{23} +$$$$26\!\cdots\!13$$$$T^{24} -$$$$49\!\cdots\!30$$$$T^{25} -$$$$97\!\cdots\!95$$$$T^{26} +$$$$13\!\cdots\!14$$$$T^{27} +$$$$19\!\cdots\!41$$$$T^{28}$$)
$97$ ($$1 - T - 96 T^{2} - 97 T^{3} + 9409 T^{4}$$)($$1 - 17 T + 192 T^{2} - 1649 T^{3} + 9409 T^{4}$$)($$1 - 9 T - 16 T^{2} - 873 T^{3} + 9409 T^{4}$$)($$1 + 7 T - 48 T^{2} + 679 T^{3} + 9409 T^{4}$$)($$1 - T - 96 T^{2} - 97 T^{3} + 9409 T^{4}$$)($$1 - 17 T + 192 T^{2} - 1649 T^{3} + 9409 T^{4}$$)($$1 - 28 T + 281 T^{2} - 2724 T^{3} + 45178 T^{4} - 388196 T^{5} + 2169217 T^{6} - 37655012 T^{7} + 425079802 T^{8} - 2486121252 T^{9} + 24876727961 T^{10} - 240445527196 T^{11} + 832972004929 T^{12}$$)($$1 + 28 T + 257 T^{2} + 2820 T^{3} + 59506 T^{4} + 545924 T^{5} + 3126001 T^{6} + 52954628 T^{7} + 559891954 T^{8} + 2573737860 T^{9} + 22752025217 T^{10} + 240445527196 T^{11} + 832972004929 T^{12}$$)($$1 + 12 T - 197 T^{2} - 1534 T^{3} + 27813 T^{4} + 14090 T^{5} - 4545035 T^{6} - 6881349 T^{7} + 472663750 T^{8} + 908843245 T^{9} - 38512186359 T^{10} + 88157794765 T^{11} + 4447293223750 T^{12} - 6280421435877 T^{13} - 402368680669835 T^{14} + 120995624221130 T^{15} + 23167450373090277 T^{16} - 123944568389425342 T^{17} - 1543974418092261317 T^{18} + 9122772703854782604 T^{19} + 73742412689492826049 T^{20}$$)($$1 + 18 T - 332 T^{2} - 7282 T^{3} + 70792 T^{4} + 1694642 T^{5} - 11251793 T^{6} - 262778647 T^{7} + 1667779429 T^{8} + 30411590872 T^{9} - 218639545013 T^{10} - 2436064838528 T^{11} + 25890765026051 T^{12} + 94489464293929 T^{13} - 2645836565169718 T^{14} + 9165478036511113 T^{15} + 243606208130113859 T^{16} - 2223330604373865344 T^{17} - 19356001718168025653 T^{18} +$$$$26\!\cdots\!04$$$$T^{19} +$$$$13\!\cdots\!41$$$$T^{20} -$$$$21\!\cdots\!11$$$$T^{21} -$$$$88\!\cdots\!73$$$$T^{22} +$$$$12\!\cdots\!14$$$$T^{23} +$$$$52\!\cdots\!08$$$$T^{24} -$$$$52\!\cdots\!46$$$$T^{25} -$$$$23\!\cdots\!12$$$$T^{26} +$$$$12\!\cdots\!86$$$$T^{27} +$$$$65\!\cdots\!69$$$$T^{28}$$)