# Properties

 Label 336.7.bh.e Level $336$ Weight $7$ Character orbit 336.bh Analytic conductor $77.298$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$336 = 2^{4} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 336.bh (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$77.2981720963$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - x^{7} + 82 x^{6} - 165 x^{5} + 5606 x^{4} - 7807 x^{3} + 102447 x^{2} + 132594 x + 1162084$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{10}\cdot 3^{4}\cdot 7^{2}$$ Twist minimal: no (minimal twist has level 84) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 9 - 9 \beta_{1} ) q^{3} + ( -7 - 3 \beta_{1} + \beta_{4} ) q^{5} + ( -54 - 132 \beta_{1} + \beta_{2} - \beta_{4} - \beta_{6} - \beta_{7} ) q^{7} -243 \beta_{1} q^{9} +O(q^{10})$$ $$q + ( 9 - 9 \beta_{1} ) q^{3} + ( -7 - 3 \beta_{1} + \beta_{4} ) q^{5} + ( -54 - 132 \beta_{1} + \beta_{2} - \beta_{4} - \beta_{6} - \beta_{7} ) q^{7} -243 \beta_{1} q^{9} + ( -35 - 32 \beta_{1} - 5 \beta_{2} + 2 \beta_{3} + \beta_{4} + 2 \beta_{5} + 2 \beta_{7} ) q^{11} + ( -276 - 546 \beta_{1} - 6 \beta_{2} + 7 \beta_{3} - 4 \beta_{4} + 7 \beta_{5} + 3 \beta_{6} ) q^{13} + ( -99 + 9 \beta_{3} + 9 \beta_{4} ) q^{15} + ( -204 + 218 \beta_{1} + 20 \beta_{2} + 6 \beta_{3} - 20 \beta_{4} - 18 \beta_{5} - 40 \beta_{6} + 18 \beta_{7} ) q^{17} + ( -3330 - 1681 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} - 29 \beta_{4} + 3 \beta_{6} + 14 \beta_{7} ) q^{19} + ( -1665 - 1881 \beta_{1} + 18 \beta_{2} - 9 \beta_{3} - 9 \beta_{4} + 9 \beta_{5} - 9 \beta_{6} - 18 \beta_{7} ) q^{21} + ( 14 - 3914 \beta_{1} + 46 \beta_{2} + 18 \beta_{3} - 32 \beta_{4} - 92 \beta_{5} - 46 \beta_{6} + 46 \beta_{7} ) q^{23} + ( 10600 + 10681 \beta_{1} - 115 \beta_{2} + 34 \beta_{3} + 47 \beta_{4} + 50 \beta_{5} + 50 \beta_{7} ) q^{25} + ( -2187 - 4374 \beta_{1} ) q^{27} + ( 16187 - 7 \beta_{3} - 57 \beta_{4} + 40 \beta_{5} - 50 \beta_{6} - 80 \beta_{7} ) q^{29} + ( 3526 - 3490 \beta_{1} + 21 \beta_{2} - 15 \beta_{3} - 21 \beta_{4} - 153 \beta_{5} - 42 \beta_{6} + 153 \beta_{7} ) q^{31} + ( -612 - 288 \beta_{1} - 45 \beta_{2} + 45 \beta_{3} - 9 \beta_{4} - 45 \beta_{6} + 54 \beta_{7} ) q^{33} + ( -15253 - 12224 \beta_{1} + 70 \beta_{2} + 187 \beta_{3} - 110 \beta_{4} - 40 \beta_{5} + 135 \beta_{6} - 70 \beta_{7} ) q^{35} + ( 116 - 2435 \beta_{1} + 285 \beta_{2} + 53 \beta_{3} - 169 \beta_{4} - 130 \beta_{5} - 285 \beta_{6} + 65 \beta_{7} ) q^{37} + ( -7362 - 7371 \beta_{1} - 81 \beta_{2} + 90 \beta_{3} - 99 \beta_{4} + 63 \beta_{5} + 63 \beta_{7} ) q^{39} + ( -6746 - 13436 \beta_{1} - 56 \beta_{2} + 388 \beta_{3} - 360 \beta_{4} + 300 \beta_{5} + 28 \beta_{6} ) q^{41} + ( 8858 + 225 \beta_{3} + 388 \beta_{4} + 26 \beta_{5} + 163 \beta_{6} - 52 \beta_{7} ) q^{43} + ( -972 + 729 \beta_{1} + 243 \beta_{3} ) q^{45} + ( -14336 - 7458 \beta_{1} + 162 \beta_{2} - 162 \beta_{3} - 418 \beta_{4} + 162 \beta_{6} - 36 \beta_{7} ) q^{47} + ( 33646 - 11635 \beta_{1} + 16 \beta_{2} - 773 \beta_{3} + 610 \beta_{4} + 136 \beta_{5} + 365 \beta_{6} - 310 \beta_{7} ) q^{49} + ( 306 + 5886 \beta_{1} + 540 \beta_{2} - 72 \beta_{3} - 234 \beta_{4} - 324 \beta_{5} - 540 \beta_{6} + 162 \beta_{7} ) q^{51} + ( 98144 + 98361 \beta_{1} + 108 \beta_{2} - 325 \beta_{3} + 542 \beta_{4} + 114 \beta_{5} + 114 \beta_{7} ) q^{53} + ( 9159 + 18078 \beta_{1} + 240 \beta_{2} - 837 \beta_{3} + 717 \beta_{4} + 180 \beta_{5} - 120 \beta_{6} ) q^{55} + ( -44838 - 315 \beta_{3} - 234 \beta_{4} - 126 \beta_{5} + 81 \beta_{6} + 252 \beta_{7} ) q^{57} + ( 46215 - 46024 \beta_{1} - 199 \beta_{2} - 390 \beta_{3} + 199 \beta_{4} - 120 \beta_{5} + 398 \beta_{6} + 120 \beta_{7} ) q^{59} + ( -168056 - 83612 \beta_{1} + 128 \beta_{2} - 128 \beta_{3} + 960 \beta_{4} + 128 \beta_{6} + 166 \beta_{7} ) q^{61} + ( -31833 - 18711 \beta_{1} + 243 \beta_{2} - 243 \beta_{3} + 243 \beta_{5} - 243 \beta_{7} ) q^{63} + ( 546 - 146884 \beta_{1} + 680 \beta_{2} - 412 \beta_{3} - 134 \beta_{4} - 220 \beta_{5} - 680 \beta_{6} + 110 \beta_{7} ) q^{65} + ( -57402 - 56627 \beta_{1} - 349 \beta_{2} - 426 \beta_{3} + 1201 \beta_{4} + 436 \beta_{5} + 436 \beta_{7} ) q^{67} + ( -34812 - 70452 \beta_{1} + 828 \beta_{2} + 36 \beta_{3} - 450 \beta_{4} - 1242 \beta_{5} - 414 \beta_{6} ) q^{69} + ( -26324 + 526 \beta_{3} + 560 \beta_{4} - 68 \beta_{5} + 34 \beta_{6} + 136 \beta_{7} ) q^{71} + ( -103952 + 104263 \beta_{1} - 651 \beta_{2} - 962 \beta_{3} + 651 \beta_{4} - 1184 \beta_{5} + 1302 \beta_{6} + 1184 \beta_{7} ) q^{73} + ( 191106 + 96129 \beta_{1} - 1035 \beta_{2} + 1035 \beta_{3} + 117 \beta_{4} - 1035 \beta_{6} + 1350 \beta_{7} ) q^{75} + ( 80997 - 90831 \beta_{1} + 378 \beta_{2} + 918 \beta_{3} - 1639 \beta_{4} + 62 \beta_{5} - 610 \beta_{6} + 896 \beta_{7} ) q^{77} + ( 1853 + 175630 \beta_{1} + 1203 \beta_{2} - 2503 \beta_{3} + 650 \beta_{4} + 1106 \beta_{5} - 1203 \beta_{6} - 553 \beta_{7} ) q^{79} + ( -59049 - 59049 \beta_{1} ) q^{81} + ( 117327 + 235032 \beta_{1} - 378 \beta_{2} + 1285 \beta_{3} - 1096 \beta_{4} - 720 \beta_{5} + 189 \beta_{6} ) q^{83} + ( -163290 + 930 \beta_{3} + 4000 \beta_{4} - 40 \beta_{5} + 3070 \beta_{6} + 80 \beta_{7} ) q^{85} + ( 146196 - 145107 \beta_{1} + 450 \beta_{2} - 639 \beta_{3} - 450 \beta_{4} + 1080 \beta_{5} - 900 \beta_{6} - 1080 \beta_{7} ) q^{87} + ( -235242 - 118698 \beta_{1} + 540 \beta_{2} - 540 \beta_{3} - 1614 \beta_{4} + 540 \beta_{6} + 1032 \beta_{7} ) q^{89} + ( 20830 - 148507 \beta_{1} - 1849 \beta_{2} + 937 \beta_{3} - 1391 \beta_{4} + 2542 \beta_{5} + 373 \beta_{6} + 36 \beta_{7} ) q^{91} + ( 513 - 94230 \beta_{1} + 567 \beta_{2} - 459 \beta_{3} - 54 \beta_{4} - 2754 \beta_{5} - 567 \beta_{6} + 1377 \beta_{7} ) q^{93} + ( -761610 - 764606 \beta_{1} + 1780 \beta_{2} + 1216 \beta_{3} - 4212 \beta_{4} - 1060 \beta_{5} - 1060 \beta_{7} ) q^{95} + ( -274155 - 548920 \beta_{1} + 610 \beta_{2} - 2295 \beta_{3} + 1990 \beta_{4} - 1490 \beta_{5} - 305 \beta_{6} ) q^{97} + ( -8019 + 729 \beta_{3} - 486 \beta_{4} - 486 \beta_{5} - 1215 \beta_{6} + 972 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 108 q^{3} - 42 q^{5} + 92 q^{7} + 972 q^{9} + O(q^{10})$$ $$8 q + 108 q^{3} - 42 q^{5} + 92 q^{7} + 972 q^{9} - 126 q^{11} - 756 q^{15} - 2532 q^{17} - 19998 q^{19} - 5886 q^{21} + 15648 q^{23} + 42698 q^{25} + 129468 q^{29} + 42096 q^{31} - 3402 q^{33} - 73524 q^{35} + 9866 q^{37} - 29106 q^{39} + 71764 q^{43} - 10206 q^{45} - 86988 q^{47} + 314588 q^{49} - 22788 q^{51} + 391710 q^{53} - 359964 q^{57} + 553434 q^{59} - 1009104 q^{61} - 181278 q^{63} + 589452 q^{65} - 229762 q^{67} - 208488 q^{71} - 1249290 q^{73} + 1152846 q^{75} + 1009566 q^{77} - 693808 q^{79} - 236196 q^{81} - 1302600 q^{85} + 1747818 q^{87} - 1414692 q^{89} + 766410 q^{91} + 378864 q^{93} - 3047568 q^{95} - 61236 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - x^{7} + 82 x^{6} - 165 x^{5} + 5606 x^{4} - 7807 x^{3} + 102447 x^{2} + 132594 x + 1162084$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-9659521 \nu^{7} + 77530555 \nu^{6} - 720692020 \nu^{5} + 6152911039 \nu^{4} - 55995523440 \nu^{3} + 452112814595 \nu^{2} - 792644803413 \nu - 195965705782$$$$)/ 6272450648002$$ $$\beta_{2}$$ $$=$$ $$($$$$3595888 \nu^{7} - 332009022 \nu^{6} + 8071513719 \nu^{5} - 71932276648 \nu^{4} + 363472370987 \nu^{3} - 4996578595810 \nu^{2} + 18306687879799 \nu - 126708615949482$$$$)/ 570222786182$$ $$\beta_{3}$$ $$=$$ $$($$$$-1501215361 \nu^{7} + 19919106314 \nu^{6} - 75483296654 \nu^{5} + 1097694255988 \nu^{4} - 11665316721070 \nu^{3} + 48825649433922 \nu^{2} - 118587635028125 \nu - 647680862469410$$$$)/ 6272450648002$$ $$\beta_{4}$$ $$=$$ $$($$$$1550256892 \nu^{7} - 2575341491 \nu^{6} + 78777555545 \nu^{5} - 1099026853639 \nu^{4} + 3220520571049 \nu^{3} - 48683343625431 \nu^{2} + 119371499117903 \nu - 1303131051441296$$$$)/ 6272450648002$$ $$\beta_{5}$$ $$=$$ $$($$$$1165235138 \nu^{7} - 24628162010 \nu^{6} - 47869345266 \nu^{5} - 2232961192814 \nu^{4} + 1362037855790 \nu^{3} - 94356613107226 \nu^{2} - 223999376170016 \nu - 974144033632308$$$$)/ 3136225324001$$ $$\beta_{6}$$ $$=$$ $$($$$$-4035826132 \nu^{7} + 13794693635 \nu^{6} - 245740299185 \nu^{5} + 1166566827679 \nu^{4} - 16462691463769 \nu^{3} + 41470866097791 \nu^{2} - 159100040013023 \nu - 971154652898584$$$$)/ 6272450648002$$ $$\beta_{7}$$ $$=$$ $$($$$$4725027148 \nu^{7} + 2418333230 \nu^{6} + 254322741186 \nu^{5} - 1229041627066 \nu^{4} + 14678747458450 \nu^{3} - 35158107913394 \nu^{2} - 45788743036606 \nu - 766559230402812$$$$)/ 3136225324001$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{7} - \beta_{5} + 6 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + 46 \beta_{1} + 42$$$$)/168$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{7} - 16 \beta_{6} + 2 \beta_{5} - 4 \beta_{4} - 8 \beta_{3} + 16 \beta_{2} + 6856 \beta_{1} + 12$$$$)/168$$ $$\nu^{3}$$ $$=$$ $$($$$$62 \beta_{7} + 202 \beta_{6} - 31 \beta_{5} - 34 \beta_{4} - 236 \beta_{3} + 5368$$$$)/168$$ $$\nu^{4}$$ $$=$$ $$($$$$-173 \beta_{7} - 173 \beta_{5} - 180 \beta_{4} + 760 \beta_{3} - 1340 \beta_{2} - 363476 \beta_{1} - 364056$$$$)/168$$ $$\nu^{5}$$ $$=$$ $$($$$$-1383 \beta_{7} - 14834 \beta_{6} + 2766 \beta_{5} - 14716 \beta_{4} + 14598 \beta_{3} + 14834 \beta_{2} + 942338 \beta_{1} + 118$$$$)/168$$ $$\nu^{6}$$ $$=$$ $$($$$$4390 \beta_{7} + 14472 \beta_{6} - 2195 \beta_{5} + 6332 \beta_{4} - 8140 \beta_{3} + 3106880$$$$)/24$$ $$\nu^{7}$$ $$=$$ $$($$$$-84519 \beta_{7} - 84519 \beta_{5} + 856582 \beta_{4} + 95366 \beta_{3} - 1047314 \beta_{2} - 93805778 \beta_{1} - 94757726$$$$)/168$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/336\mathbb{Z}\right)^\times$$.

 $$n$$ $$85$$ $$113$$ $$127$$ $$241$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-\beta_{1}$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
145.1
 −1.59227 − 2.75789i −4.24382 − 7.35052i 2.91150 + 5.04287i 3.42459 + 5.93157i −1.59227 + 2.75789i −4.24382 + 7.35052i 2.91150 − 5.04287i 3.42459 − 5.93157i
0 13.5000 7.79423i 0 −181.505 104.792i 0 223.761 259.961i 0 121.500 210.444i 0
145.2 0 13.5000 7.79423i 0 −68.1069 39.3216i 0 −335.720 + 70.2945i 0 121.500 210.444i 0
145.3 0 13.5000 7.79423i 0 27.1744 + 15.6892i 0 342.316 + 21.6478i 0 121.500 210.444i 0
145.4 0 13.5000 7.79423i 0 201.438 + 116.300i 0 −184.358 289.242i 0 121.500 210.444i 0
241.1 0 13.5000 + 7.79423i 0 −181.505 + 104.792i 0 223.761 + 259.961i 0 121.500 + 210.444i 0
241.2 0 13.5000 + 7.79423i 0 −68.1069 + 39.3216i 0 −335.720 70.2945i 0 121.500 + 210.444i 0
241.3 0 13.5000 + 7.79423i 0 27.1744 15.6892i 0 342.316 21.6478i 0 121.500 + 210.444i 0
241.4 0 13.5000 + 7.79423i 0 201.438 116.300i 0 −184.358 + 289.242i 0 121.500 + 210.444i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 241.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.7.bh.e 8
4.b odd 2 1 84.7.m.a 8
7.d odd 6 1 inner 336.7.bh.e 8
12.b even 2 1 252.7.z.d 8
28.d even 2 1 588.7.m.c 8
28.f even 6 1 84.7.m.a 8
28.f even 6 1 588.7.d.b 8
28.g odd 6 1 588.7.d.b 8
28.g odd 6 1 588.7.m.c 8
84.j odd 6 1 252.7.z.d 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.7.m.a 8 4.b odd 2 1
84.7.m.a 8 28.f even 6 1
252.7.z.d 8 12.b even 2 1
252.7.z.d 8 84.j odd 6 1
336.7.bh.e 8 1.a even 1 1 trivial
336.7.bh.e 8 7.d odd 6 1 inner
588.7.d.b 8 28.f even 6 1
588.7.d.b 8 28.g odd 6 1
588.7.m.c 8 28.d even 2 1
588.7.m.c 8 28.g odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$12\!\cdots\!00$$$$T_{5}^{2} -$$$$46\!\cdots\!00$$$$T_{5} +$$$$14\!\cdots\!00$$">$$T_{5}^{8} + \cdots$$ acting on $$S_{7}^{\mathrm{new}}(336, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$( 243 - 27 T + T^{2} )^{4}$$
$5$ $$14471729102250000 - 468261911250000 T - 1241694292500 T^{2} + 203597212500 T^{3} + 2561019525 T^{4} - 2196810 T^{5} - 51717 T^{6} + 42 T^{7} + T^{8}$$
$7$ $$19\!\cdots\!01$$$$- 149814051007761308 T - 2118575101559462 T^{2} + 697955986672 T^{3} + 12154906435 T^{4} + 5932528 T^{5} - 153062 T^{6} - 92 T^{7} + T^{8}$$
$11$ $$66\!\cdots\!36$$$$- 85930795043021000880 T + 1491749111632846140 T^{2} - 15078013454412 T^{3} + 2483177863149 T^{4} - 18003330 T^{5} + 1828791 T^{6} + 126 T^{7} + T^{8}$$
$13$ $$65\!\cdots\!96$$$$+ 8821606540309296768 T^{2} + 33404807665593 T^{4} + 12743190 T^{6} + T^{8}$$
$17$ $$10\!\cdots\!00$$$$-$$$$23\!\cdots\!60$$$$T -$$$$62\!\cdots\!08$$$$T^{2} + 18777019234121914752 T^{3} + 5160374456094288 T^{4} - 202104007056 T^{5} - 77682900 T^{6} + 2532 T^{7} + T^{8}$$
$19$ $$51\!\cdots\!44$$$$-$$$$45\!\cdots\!32$$$$T +$$$$11\!\cdots\!00$$$$T^{2} + 16833109646899298196 T^{3} - 4195989325596267 T^{4} - 535277686878 T^{5} + 106540107 T^{6} + 19998 T^{7} + T^{8}$$
$23$ $$12\!\cdots\!64$$$$-$$$$31\!\cdots\!12$$$$T +$$$$59\!\cdots\!44$$$$T^{2} -$$$$50\!\cdots\!84$$$$T^{3} + 371379144905404416 T^{4} - 10424936687616 T^{5} + 691155648 T^{6} - 15648 T^{7} + T^{8}$$
$29$ $$( 88256422643616000 + 24841946613600 T + 312177825 T^{2} - 64734 T^{3} + T^{4} )^{2}$$
$31$ $$49\!\cdots\!69$$$$-$$$$92\!\cdots\!08$$$$T +$$$$71\!\cdots\!38$$$$T^{2} -$$$$25\!\cdots\!52$$$$T^{3} + 2660518864582319835 T^{4} + 82147095114912 T^{5} - 1360731750 T^{6} - 42096 T^{7} + T^{8}$$
$37$ $$18\!\cdots\!64$$$$+$$$$29\!\cdots\!60$$$$T +$$$$27\!\cdots\!04$$$$T^{2} -$$$$27\!\cdots\!04$$$$T^{3} + 24041729406302502781 T^{4} - 85472030715482 T^{5} + 5354106919 T^{6} - 9866 T^{7} + T^{8}$$
$41$ $$16\!\cdots\!56$$$$+$$$$12\!\cdots\!52$$$$T^{2} +$$$$31\!\cdots\!68$$$$T^{4} + 32134086384 T^{6} + T^{8}$$
$43$ $$( -8603696213123190140 + 859771773405076 T - 16775839371 T^{2} - 35882 T^{3} + T^{4} )^{2}$$
$47$ $$15\!\cdots\!24$$$$-$$$$31\!\cdots\!64$$$$T -$$$$18\!\cdots\!56$$$$T^{2} +$$$$43\!\cdots\!64$$$$T^{3} +$$$$26\!\cdots\!48$$$$T^{4} - 1469385818051184 T^{5} - 14369518020 T^{6} + 86988 T^{7} + T^{8}$$
$53$ $$94\!\cdots\!56$$$$+$$$$23\!\cdots\!48$$$$T +$$$$11\!\cdots\!76$$$$T^{2} -$$$$27\!\cdots\!56$$$$T^{3} +$$$$14\!\cdots\!05$$$$T^{4} - 14733973975104414 T^{5} + 115436387763 T^{6} - 391710 T^{7} + T^{8}$$
$59$ $$90\!\cdots\!56$$$$-$$$$41\!\cdots\!76$$$$T +$$$$82\!\cdots\!44$$$$T^{2} -$$$$84\!\cdots\!52$$$$T^{3} +$$$$44\!\cdots\!57$$$$T^{4} - 10641602864760462 T^{5} + 121324711695 T^{6} - 553434 T^{7} + T^{8}$$
$61$ $$28\!\cdots\!00$$$$+$$$$28\!\cdots\!60$$$$T -$$$$29\!\cdots\!92$$$$T^{2} -$$$$39\!\cdots\!68$$$$T^{3} +$$$$39\!\cdots\!52$$$$T^{4} + 72981658926042624 T^{5} + 411753522528 T^{6} + 1009104 T^{7} + T^{8}$$
$67$ $$12\!\cdots\!04$$$$+$$$$26\!\cdots\!64$$$$T +$$$$45\!\cdots\!68$$$$T^{2} +$$$$24\!\cdots\!44$$$$T^{3} +$$$$13\!\cdots\!45$$$$T^{4} + 27603503610872578 T^{5} + 136883783947 T^{6} + 229762 T^{7} + T^{8}$$
$71$ $$( -95078277029578691520 - 4338729981277344 T - 44981910252 T^{2} + 104244 T^{3} + T^{4} )^{2}$$
$73$ $$76\!\cdots\!56$$$$+$$$$92\!\cdots\!24$$$$T +$$$$43\!\cdots\!96$$$$T^{2} +$$$$68\!\cdots\!56$$$$T^{3} -$$$$69\!\cdots\!55$$$$T^{4} - 255752941296663090 T^{5} + 315523201479 T^{6} + 1249290 T^{7} + T^{8}$$
$79$ $$19\!\cdots\!25$$$$-$$$$10\!\cdots\!20$$$$T +$$$$70\!\cdots\!46$$$$T^{2} +$$$$36\!\cdots\!56$$$$T^{3} +$$$$29\!\cdots\!83$$$$T^{4} + 160718657824073536 T^{5} + 907915485778 T^{6} + 693808 T^{7} + T^{8}$$
$83$ $$47\!\cdots\!16$$$$+$$$$14\!\cdots\!40$$$$T^{2} +$$$$67\!\cdots\!25$$$$T^{4} + 503985708366 T^{6} + T^{8}$$
$89$ $$87\!\cdots\!64$$$$+$$$$33\!\cdots\!04$$$$T +$$$$45\!\cdots\!20$$$$T^{2} +$$$$70\!\cdots\!92$$$$T^{3} -$$$$12\!\cdots\!68$$$$T^{4} - 278669060549098128 T^{5} + 470135676204 T^{6} + 1414692 T^{7} + T^{8}$$
$97$ $$50\!\cdots\!00$$$$+$$$$41\!\cdots\!00$$$$T^{2} +$$$$80\!\cdots\!25$$$$T^{4} + 1777407002550 T^{6} + T^{8}$$