# Properties

 Label 804.2.i Level 804 Weight 2 Character orbit i Rep. character $$\chi_{804}(37,\cdot)$$ Character field $$\Q(\zeta_{3})$$ Dimension 22 Newform subspaces 5 Sturm bound 272 Trace bound 5

# Related objects

## Defining parameters

 Level: $$N$$ = $$804 = 2^{2} \cdot 3 \cdot 67$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 804.i (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$67$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$5$$ Sturm bound: $$272$$ Trace bound: $$5$$ Distinguishing $$T_p$$: $$5$$, $$7$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(804, [\chi])$$.

Total New Old
Modular forms 284 22 262
Cusp forms 260 22 238
Eisenstein series 24 0 24

## Trace form

 $$22q - 2q^{3} + 2q^{7} + 22q^{9} + O(q^{10})$$ $$22q - 2q^{3} + 2q^{7} + 22q^{9} + 2q^{11} + q^{13} - 4q^{15} - 4q^{17} - 2q^{19} + 4q^{21} - 6q^{23} + 14q^{25} - 2q^{27} - 14q^{29} - 3q^{31} + 2q^{33} + 2q^{35} + 8q^{37} + 5q^{39} + 6q^{41} + 22q^{43} + 14q^{47} - 9q^{49} + 4q^{51} + 8q^{53} + 12q^{57} - 12q^{59} + 11q^{61} + 2q^{63} - 24q^{65} + 35q^{67} + 16q^{69} + 4q^{71} - 13q^{73} + 18q^{75} - 20q^{77} - q^{79} + 22q^{81} - 18q^{83} + 6q^{85} + 6q^{87} - 72q^{89} + 24q^{91} + 7q^{93} - 46q^{95} + 49q^{97} + 2q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(804, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
804.2.i.a $$2$$ $$6.420$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-2$$ $$-4$$ $$3$$ $$q-q^{3}-2q^{5}+3\zeta_{6}q^{7}+q^{9}-3\zeta_{6}q^{11}+\cdots$$
804.2.i.b $$2$$ $$6.420$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-2$$ $$4$$ $$-5$$ $$q-q^{3}+2q^{5}-5\zeta_{6}q^{7}+q^{9}-5\zeta_{6}q^{11}+\cdots$$
804.2.i.c $$2$$ $$6.420$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$2$$ $$4$$ $$3$$ $$q+q^{3}+2q^{5}+3\zeta_{6}q^{7}+q^{9}-\zeta_{6}q^{11}+\cdots$$
804.2.i.d $$8$$ $$6.420$$ $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ None $$0$$ $$-8$$ $$2$$ $$1$$ $$q-q^{3}+\beta _{6}q^{5}+(-\beta _{1}+\beta _{6})q^{7}+q^{9}+\cdots$$
804.2.i.e $$8$$ $$6.420$$ $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ None $$0$$ $$8$$ $$-6$$ $$0$$ $$q+q^{3}+(-1-\beta _{2}+\beta _{5}-\beta _{6})q^{5}-\beta _{3}q^{7}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(804, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(804, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(67, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(134, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(201, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(268, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(402, [\chi])$$$$^{\oplus 2}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ()()()()()
$3$ ($$( 1 + T )^{2}$$)($$( 1 + T )^{2}$$)($$( 1 - T )^{2}$$)($$( 1 + T )^{8}$$)($$( 1 - T )^{8}$$)
$5$ ($$( 1 + 2 T + 5 T^{2} )^{2}$$)($$( 1 - 2 T + 5 T^{2} )^{2}$$)($$( 1 - 2 T + 5 T^{2} )^{2}$$)($$( 1 - T + 10 T^{2} - 12 T^{3} + 62 T^{4} - 60 T^{5} + 250 T^{6} - 125 T^{7} + 625 T^{8} )^{2}$$)($$( 1 + 3 T + 10 T^{2} + 22 T^{3} + 56 T^{4} + 110 T^{5} + 250 T^{6} + 375 T^{7} + 625 T^{8} )^{2}$$)
$7$ ($$1 - 3 T + 2 T^{2} - 21 T^{3} + 49 T^{4}$$)($$( 1 + T + 7 T^{2} )( 1 + 4 T + 7 T^{2} )$$)($$1 - 3 T + 2 T^{2} - 21 T^{3} + 49 T^{4}$$)($$1 - T - 17 T^{2} + 18 T^{3} + 140 T^{4} - 118 T^{5} - 1026 T^{6} + 323 T^{7} + 7697 T^{8} + 2261 T^{9} - 50274 T^{10} - 40474 T^{11} + 336140 T^{12} + 302526 T^{13} - 2000033 T^{14} - 823543 T^{15} + 5764801 T^{16}$$)($$1 - 17 T^{2} + 14 T^{3} + 150 T^{4} - 168 T^{5} - 648 T^{6} + 693 T^{7} + 2416 T^{8} + 4851 T^{9} - 31752 T^{10} - 57624 T^{11} + 360150 T^{12} + 235298 T^{13} - 2000033 T^{14} + 5764801 T^{16}$$)
$11$ ($$1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4}$$)($$1 + 5 T + 14 T^{2} + 55 T^{3} + 121 T^{4}$$)($$1 + T - 10 T^{2} + 11 T^{3} + 121 T^{4}$$)($$( 1 - 2 T - 7 T^{2} - 22 T^{3} + 121 T^{4} )^{4}$$)($$1 - 3 T - 13 T^{2} + 44 T^{3} + 5 T^{4} + 69 T^{5} - 374 T^{6} - 2629 T^{7} + 16074 T^{8} - 28919 T^{9} - 45254 T^{10} + 91839 T^{11} + 73205 T^{12} + 7086244 T^{13} - 23030293 T^{14} - 58461513 T^{15} + 214358881 T^{16}$$)
$13$ ($$1 - T - 12 T^{2} - 13 T^{3} + 169 T^{4}$$)($$1 + 3 T - 4 T^{2} + 39 T^{3} + 169 T^{4}$$)($$( 1 - 7 T + 13 T^{2} )( 1 + 2 T + 13 T^{2} )$$)($$1 - 23 T^{2} - 6 T^{3} + 94 T^{4} + 108 T^{5} - 2222 T^{6} - 489 T^{7} + 71146 T^{8} - 6357 T^{9} - 375518 T^{10} + 237276 T^{11} + 2684734 T^{12} - 2227758 T^{13} - 111016607 T^{14} + 815730721 T^{16}$$)($$1 + 2 T - 17 T^{2} + 56 T^{3} + 242 T^{4} - 1090 T^{5} + 2856 T^{6} + 13823 T^{7} - 40782 T^{8} + 179699 T^{9} + 482664 T^{10} - 2394730 T^{11} + 6911762 T^{12} + 20792408 T^{13} - 82055753 T^{14} + 125497034 T^{15} + 815730721 T^{16}$$)
$17$ ($$1 + 5 T + 8 T^{2} + 85 T^{3} + 289 T^{4}$$)($$1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4}$$)($$1 - T - 16 T^{2} - 17 T^{3} + 289 T^{4}$$)($$1 - 4 T - 29 T^{2} + 126 T^{3} + 311 T^{4} - 670 T^{5} - 10575 T^{6} - 6188 T^{7} + 304921 T^{8} - 105196 T^{9} - 3056175 T^{10} - 3291710 T^{11} + 25975031 T^{12} + 178901982 T^{13} - 699989501 T^{14} - 1641354692 T^{15} + 6975757441 T^{16}$$)($$1 + T - 60 T^{2} - 25 T^{3} + 2179 T^{4} + 356 T^{5} - 55063 T^{6} - 2183 T^{7} + 1066504 T^{8} - 37111 T^{9} - 15913207 T^{10} + 1749028 T^{11} + 181992259 T^{12} - 35496425 T^{13} - 1448254140 T^{14} + 410338673 T^{15} + 6975757441 T^{16}$$)
$19$ ($$1 - 5 T + 6 T^{2} - 95 T^{3} + 361 T^{4}$$)($$( 1 - T + 19 T^{2} )( 1 + 8 T + 19 T^{2} )$$)($$( 1 - 8 T + 19 T^{2} )( 1 + 7 T + 19 T^{2} )$$)($$1 + 5 T - 25 T^{2} - 242 T^{3} - 48 T^{4} + 3794 T^{5} + 7346 T^{6} - 18627 T^{7} - 102551 T^{8} - 353913 T^{9} + 2651906 T^{10} + 26023046 T^{11} - 6255408 T^{12} - 599215958 T^{13} - 1176147025 T^{14} + 4469358695 T^{15} + 16983563041 T^{16}$$)($$1 - 4 T - 33 T^{2} + 246 T^{3} + 294 T^{4} - 5244 T^{5} + 7496 T^{6} + 43027 T^{7} - 209844 T^{8} + 817513 T^{9} + 2706056 T^{10} - 35968596 T^{11} + 38314374 T^{12} + 609120354 T^{13} - 1552514073 T^{14} - 3575486956 T^{15} + 16983563041 T^{16}$$)
$23$ ($$1 + 9 T + 58 T^{2} + 207 T^{3} + 529 T^{4}$$)($$1 + 3 T - 14 T^{2} + 69 T^{3} + 529 T^{4}$$)($$1 + 7 T + 26 T^{2} + 161 T^{3} + 529 T^{4}$$)($$1 - T - 24 T^{2} + 31 T^{3} + 279 T^{4} - 708 T^{5} + 18640 T^{6} - 2096 T^{7} - 503040 T^{8} - 48208 T^{9} + 9860560 T^{10} - 8614236 T^{11} + 78075639 T^{12} + 199526633 T^{13} - 3552861336 T^{14} - 3404825447 T^{15} + 78310985281 T^{16}$$)($$( 1 - 3 T - 14 T^{2} - 69 T^{3} + 529 T^{4} )^{4}$$)
$29$ ($$1 + 9 T + 52 T^{2} + 261 T^{3} + 841 T^{4}$$)($$1 - T - 28 T^{2} - 29 T^{3} + 841 T^{4}$$)($$1 - 5 T - 4 T^{2} - 145 T^{3} + 841 T^{4}$$)($$1 + 2 T - 43 T^{2} - 96 T^{3} - 115 T^{4} - 1316 T^{5} - 17729 T^{6} + 71592 T^{7} + 1922293 T^{8} + 2076168 T^{9} - 14910089 T^{10} - 32095924 T^{11} - 81337315 T^{12} - 1969070304 T^{13} - 25577402803 T^{14} + 34499752618 T^{15} + 500246412961 T^{16}$$)($$1 + 9 T - 52 T^{2} - 349 T^{3} + 4715 T^{4} + 16920 T^{5} - 180779 T^{6} - 85603 T^{7} + 7243236 T^{8} - 2482487 T^{9} - 152035139 T^{10} + 412661880 T^{11} + 3334829915 T^{12} - 7158391001 T^{13} - 30930812692 T^{14} + 155248886781 T^{15} + 500246412961 T^{16}$$)
$31$ ($$1 + 9 T + 50 T^{2} + 279 T^{3} + 961 T^{4}$$)($$1 + 5 T - 6 T^{2} + 155 T^{3} + 961 T^{4}$$)($$1 - 3 T - 22 T^{2} - 93 T^{3} + 961 T^{4}$$)($$1 - 9 T + 21 T^{2} + 334 T^{3} - 3603 T^{4} + 19093 T^{5} - 11834 T^{6} - 590409 T^{7} + 5215882 T^{8} - 18302679 T^{9} - 11372474 T^{10} + 568799563 T^{11} - 3327446163 T^{12} + 9562136434 T^{13} + 18637577301 T^{14} - 247613526999 T^{15} + 852891037441 T^{16}$$)($$1 + T - 59 T^{2} - 414 T^{3} + 1629 T^{4} + 18099 T^{5} + 60102 T^{6} - 383231 T^{7} - 3058022 T^{8} - 11880161 T^{9} + 57758022 T^{10} + 539187309 T^{11} + 1504415709 T^{12} - 11852468514 T^{13} - 52362717179 T^{14} + 27512614111 T^{15} + 852891037441 T^{16}$$)
$37$ ($$1 + 3 T - 28 T^{2} + 111 T^{3} + 1369 T^{4}$$)($$1 - 5 T - 12 T^{2} - 185 T^{3} + 1369 T^{4}$$)($$1 + 3 T - 28 T^{2} + 111 T^{3} + 1369 T^{4}$$)($$1 + 5 T - 23 T^{2} + 836 T^{3} + 4710 T^{4} - 16170 T^{5} + 319326 T^{6} + 1913289 T^{7} - 4942379 T^{8} + 70791693 T^{9} + 437157294 T^{10} - 819059010 T^{11} + 8827298310 T^{12} + 57971548052 T^{13} - 59011707407 T^{14} + 474659385665 T^{15} + 3512479453921 T^{16}$$)($$1 - 14 T + 69 T^{2} - 320 T^{3} + 1338 T^{4} + 8824 T^{5} - 80750 T^{6} + 365453 T^{7} - 2739878 T^{8} + 13521761 T^{9} - 110546750 T^{10} + 446962072 T^{11} + 2507627418 T^{12} - 22190066240 T^{13} + 177035122221 T^{14} - 1329046279862 T^{15} + 3512479453921 T^{16}$$)
$41$ ($$1 - 3 T - 32 T^{2} - 123 T^{3} + 1681 T^{4}$$)($$1 + 11 T + 80 T^{2} + 451 T^{3} + 1681 T^{4}$$)($$1 + 7 T + 8 T^{2} + 287 T^{3} + 1681 T^{4}$$)($$1 - 11 T - 5 T^{2} - 66 T^{3} + 3342 T^{4} + 5808 T^{5} - 102972 T^{6} + 250283 T^{7} - 2878959 T^{8} + 10261603 T^{9} - 173095932 T^{10} + 400293168 T^{11} + 9443693262 T^{12} - 7646509266 T^{13} - 23750521205 T^{14} - 2142297012691 T^{15} + 7984925229121 T^{16}$$)($$1 - 10 T - 49 T^{2} + 596 T^{3} + 2552 T^{4} - 13790 T^{5} - 212474 T^{6} - 65943 T^{7} + 14026710 T^{8} - 2703663 T^{9} - 357168794 T^{10} - 950420590 T^{11} + 7211342072 T^{12} + 69050295796 T^{13} - 232755107809 T^{14} - 1947542738810 T^{15} + 7984925229121 T^{16}$$)
$43$ ($$( 1 + 4 T + 43 T^{2} )^{2}$$)($$( 1 - 4 T + 43 T^{2} )^{2}$$)($$( 1 - 4 T + 43 T^{2} )^{2}$$)($$( 1 - 3 T + 57 T^{2} + 133 T^{3} + 1551 T^{4} + 5719 T^{5} + 105393 T^{6} - 238521 T^{7} + 3418801 T^{8} )^{2}$$)($$( 1 - 4 T + 165 T^{2} - 489 T^{3} + 10476 T^{4} - 21027 T^{5} + 305085 T^{6} - 318028 T^{7} + 3418801 T^{8} )^{2}$$)
$47$ ($$1 + 7 T + 2 T^{2} + 329 T^{3} + 2209 T^{4}$$)($$1 - 3 T - 38 T^{2} - 141 T^{3} + 2209 T^{4}$$)($$1 + 9 T + 34 T^{2} + 423 T^{3} + 2209 T^{4}$$)($$1 - 11 T - 29 T^{2} + 66 T^{3} + 5076 T^{4} + 6666 T^{5} - 242622 T^{6} + 458909 T^{7} - 280527 T^{8} + 21568723 T^{9} - 535951998 T^{10} + 692084118 T^{11} + 24769260756 T^{12} + 15136770462 T^{13} - 312597244541 T^{14} - 5572854325093 T^{15} + 23811286661761 T^{16}$$)($$1 - 16 T - 15 T^{2} + 490 T^{3} + 13708 T^{4} - 91334 T^{5} - 519604 T^{6} - 745255 T^{7} + 58651408 T^{8} - 35026985 T^{9} - 1147805236 T^{10} - 9482569882 T^{11} + 66890667148 T^{12} + 112379053430 T^{13} - 161688229935 T^{14} - 8105969927408 T^{15} + 23811286661761 T^{16}$$)
$53$ ($$( 1 + 6 T + 53 T^{2} )^{2}$$)($$( 1 + 6 T + 53 T^{2} )^{2}$$)($$( 1 - 2 T + 53 T^{2} )^{2}$$)($$( 1 - 20 T + 333 T^{2} - 3393 T^{3} + 29770 T^{4} - 179829 T^{5} + 935397 T^{6} - 2977540 T^{7} + 7890481 T^{8} )^{2}$$)($$( 1 + 6 T + 73 T^{2} + 691 T^{3} + 5558 T^{4} + 36623 T^{5} + 205057 T^{6} + 893262 T^{7} + 7890481 T^{8} )^{2}$$)
$59$ ($$( 1 + 12 T + 59 T^{2} )^{2}$$)($$( 1 + 8 T + 59 T^{2} )^{2}$$)($$( 1 + 59 T^{2} )^{2}$$)($$( 1 - 14 T + 201 T^{2} - 1563 T^{3} + 14548 T^{4} - 92217 T^{5} + 699681 T^{6} - 2875306 T^{7} + 12117361 T^{8} )^{2}$$)($$( 1 + 203 T^{2} - 55 T^{3} + 17052 T^{4} - 3245 T^{5} + 706643 T^{6} + 12117361 T^{8} )^{2}$$)
$61$ ($$1 + 15 T + 164 T^{2} + 915 T^{3} + 3721 T^{4}$$)($$( 1 - 14 T + 61 T^{2} )( 1 + 13 T + 61 T^{2} )$$)($$( 1 - 14 T + 61 T^{2} )( 1 + 13 T + 61 T^{2} )$$)($$1 - 20 T + 241 T^{2} - 2946 T^{3} + 27920 T^{4} - 224900 T^{5} + 1845888 T^{6} - 13520291 T^{7} + 97623770 T^{8} - 824737751 T^{9} + 6868549248 T^{10} - 51048026900 T^{11} + 386575880720 T^{12} - 2488180702746 T^{13} + 12416410221001 T^{14} - 62854856720420 T^{15} + 191707312997281 T^{16}$$)($$1 - 4 T - 107 T^{2} - 178 T^{3} + 5888 T^{4} + 40016 T^{5} - 157692 T^{6} - 1734955 T^{7} + 9802722 T^{8} - 105832255 T^{9} - 586771932 T^{10} + 9082871696 T^{11} + 81524311808 T^{12} - 150338141578 T^{13} - 5512680056627 T^{14} - 12570971344084 T^{15} + 191707312997281 T^{16}$$)
$67$ ($$1 - 16 T + 67 T^{2}$$)($$1 - 16 T + 67 T^{2}$$)($$1 - 16 T + 67 T^{2}$$)($$1 + 33 T + 545 T^{2} + 6105 T^{3} + 54459 T^{4} + 409035 T^{5} + 2446505 T^{6} + 9925179 T^{7} + 20151121 T^{8}$$)($$1 - 20 T + 301 T^{2} - 3485 T^{3} + 29812 T^{4} - 233495 T^{5} + 1351189 T^{6} - 6015260 T^{7} + 20151121 T^{8}$$)
$71$ ($$1 + 7 T - 22 T^{2} + 497 T^{3} + 5041 T^{4}$$)($$1 + T - 70 T^{2} + 71 T^{3} + 5041 T^{4}$$)($$1 + 5 T - 46 T^{2} + 355 T^{3} + 5041 T^{4}$$)($$1 - 11 T + 54 T^{2} + 1111 T^{3} - 19219 T^{4} + 170456 T^{5} - 99822 T^{6} - 11375958 T^{7} + 161395572 T^{8} - 807693018 T^{9} - 503202702 T^{10} + 61008077416 T^{11} - 488387097139 T^{12} + 2004498808961 T^{13} + 6917415331734 T^{14} - 100046321742301 T^{15} + 645753531245761 T^{16}$$)($$1 - 6 T - 160 T^{2} + 920 T^{3} + 13043 T^{4} - 50796 T^{5} - 1112732 T^{6} + 1091174 T^{7} + 92960496 T^{8} + 77473354 T^{9} - 5609282012 T^{10} - 18180447156 T^{11} + 331444555283 T^{12} + 1659891002920 T^{13} - 20496045427360 T^{14} - 54570720950346 T^{15} + 645753531245761 T^{16}$$)
$73$ ($$1 + 11 T + 48 T^{2} + 803 T^{3} + 5329 T^{4}$$)($$1 - 9 T + 8 T^{2} - 657 T^{3} + 5329 T^{4}$$)($$1 - 9 T + 8 T^{2} - 657 T^{3} + 5329 T^{4}$$)($$1 + 7 T - 130 T^{2} - 2149 T^{3} + 4271 T^{4} + 206570 T^{5} + 1092495 T^{6} - 8070615 T^{7} - 120604568 T^{8} - 589154895 T^{9} + 5821905855 T^{10} + 80359241690 T^{11} + 121288887311 T^{12} - 4455030853357 T^{13} - 19673449417570 T^{14} + 77331789633679 T^{15} + 806460091894081 T^{16}$$)($$1 + 13 T - 38 T^{2} - 1143 T^{3} - 4293 T^{4} - 40986 T^{5} - 537885 T^{6} + 4059643 T^{7} + 100253128 T^{8} + 296353939 T^{9} - 2866389165 T^{10} - 15944250762 T^{11} - 121913648613 T^{12} - 2369520830799 T^{13} - 5750700598982 T^{14} + 143616180748261 T^{15} + 806460091894081 T^{16}$$)
$79$ ($$1 + 9 T + 2 T^{2} + 711 T^{3} + 6241 T^{4}$$)($$1 + T - 78 T^{2} + 79 T^{3} + 6241 T^{4}$$)($$1 - 15 T + 146 T^{2} - 1185 T^{3} + 6241 T^{4}$$)($$1 - 12 T - 47 T^{2} - 486 T^{3} + 12854 T^{4} + 63174 T^{5} - 99654 T^{6} - 4520379 T^{7} - 25440400 T^{8} - 357109941 T^{9} - 621940614 T^{10} + 31147245786 T^{11} + 500664341174 T^{12} - 1495449409914 T^{13} - 11425110409487 T^{14} - 230446907833908 T^{15} + 1517108809906561 T^{16}$$)($$1 + 18 T - 61 T^{2} - 1412 T^{3} + 24774 T^{4} + 193916 T^{5} - 2351956 T^{6} + 166685 T^{7} + 321032056 T^{8} + 13168115 T^{9} - 14678557396 T^{10} + 95608150724 T^{11} + 964949306694 T^{12} - 4344803635388 T^{13} - 14828334786781 T^{14} + 345670361750862 T^{15} + 1517108809906561 T^{16}$$)
$83$ ($$1 - 3 T - 74 T^{2} - 249 T^{3} + 6889 T^{4}$$)($$1 - 9 T - 2 T^{2} - 747 T^{3} + 6889 T^{4}$$)($$1 + 11 T + 38 T^{2} + 913 T^{3} + 6889 T^{4}$$)($$1 + 4 T - 293 T^{2} - 642 T^{3} + 54065 T^{4} + 66808 T^{5} - 6809565 T^{6} - 2000374 T^{7} + 658163011 T^{8} - 166031042 T^{9} - 46911093285 T^{10} + 38199945896 T^{11} + 2565834124865 T^{12} - 2528864092806 T^{13} - 95793529397117 T^{14} + 108544203958508 T^{15} + 2252292232139041 T^{16}$$)($$1 + 15 T - 142 T^{2} - 1553 T^{3} + 34937 T^{4} + 183540 T^{5} - 4065077 T^{6} - 3275795 T^{7} + 440135994 T^{8} - 271890985 T^{9} - 28004315453 T^{10} + 104945785980 T^{11} + 1658051360777 T^{12} - 6117330118579 T^{13} - 46425533018398 T^{14} + 407040764844405 T^{15} + 2252292232139041 T^{16}$$)
$89$ ($$( 1 - 10 T + 89 T^{2} )^{2}$$)($$( 1 + 18 T + 89 T^{2} )^{2}$$)($$( 1 + 18 T + 89 T^{2} )^{2}$$)($$( 1 + 8 T + 295 T^{2} + 1841 T^{3} + 36468 T^{4} + 163849 T^{5} + 2336695 T^{6} + 5639752 T^{7} + 62742241 T^{8} )^{2}$$)($$( 1 + 2 T + 193 T^{2} - 325 T^{3} + 19010 T^{4} - 28925 T^{5} + 1528753 T^{6} + 1409938 T^{7} + 62742241 T^{8} )^{2}$$)
$97$ ($$1 + 3 T - 88 T^{2} + 291 T^{3} + 9409 T^{4}$$)($$1 - 17 T + 192 T^{2} - 1649 T^{3} + 9409 T^{4}$$)($$1 + 15 T + 128 T^{2} + 1455 T^{3} + 9409 T^{4}$$)($$1 - 20 T - 53 T^{2} + 1854 T^{3} + 28472 T^{4} - 297470 T^{5} - 2904618 T^{6} + 7196245 T^{7} + 391603694 T^{8} + 698035765 T^{9} - 27329550762 T^{10} - 271492837310 T^{11} + 2520605688632 T^{12} + 15920928836478 T^{13} - 44147516261237 T^{14} - 1615965689562260 T^{15} + 7837433594376961 T^{16}$$)($$1 - 30 T + 257 T^{2} - 980 T^{3} + 33000 T^{4} - 426760 T^{5} + 1244192 T^{6} - 35456635 T^{7} + 727608454 T^{8} - 3439293595 T^{9} + 11706602528 T^{10} - 389492329480 T^{11} + 2921466273000 T^{12} - 8415593451860 T^{13} + 214073805266753 T^{14} - 2423948534343390 T^{15} + 7837433594376961 T^{16}$$)