# Properties

 Label 688.6.a.e Level $688$ Weight $6$ Character orbit 688.a Self dual yes Analytic conductor $110.344$ Analytic rank $1$ Dimension $8$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$688 = 2^{4} \cdot 43$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 688.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$110.344068031$$ Analytic rank: $$1$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - 4 x^{7} - 173 x^{6} + 462 x^{5} + 9118 x^{4} - 14192 x^{3} - 167688 x^{2} + 106368 x + 681984$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: no (minimal twist has level 43) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 3 + \beta_{4} ) q^{3} + ( -27 + \beta_{4} - \beta_{6} + \beta_{7} ) q^{5} + ( 17 + 4 \beta_{1} + 6 \beta_{2} + 4 \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{7} + ( 68 + 9 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} + 9 \beta_{4} - 5 \beta_{5} + \beta_{6} - 4 \beta_{7} ) q^{9} +O(q^{10})$$ $$q + ( 3 + \beta_{4} ) q^{3} + ( -27 + \beta_{4} - \beta_{6} + \beta_{7} ) q^{5} + ( 17 + 4 \beta_{1} + 6 \beta_{2} + 4 \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{7} + ( 68 + 9 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} + 9 \beta_{4} - 5 \beta_{5} + \beta_{6} - 4 \beta_{7} ) q^{9} + ( 70 - 9 \beta_{1} - 3 \beta_{2} - 10 \beta_{3} - 7 \beta_{4} - 11 \beta_{6} - \beta_{7} ) q^{11} + ( -316 - 13 \beta_{1} - 13 \beta_{2} - 15 \beta_{3} - 11 \beta_{4} + 15 \beta_{5} + 4 \beta_{6} - \beta_{7} ) q^{13} + ( 225 + 9 \beta_{1} - 10 \beta_{2} - 7 \beta_{3} - 43 \beta_{4} + 8 \beta_{5} - 4 \beta_{6} - 2 \beta_{7} ) q^{15} + ( -318 + 12 \beta_{1} + 38 \beta_{2} + 13 \beta_{3} + 37 \beta_{4} - 5 \beta_{5} + 4 \beta_{6} + 4 \beta_{7} ) q^{17} + ( 231 - 62 \beta_{1} - 56 \beta_{2} + 4 \beta_{3} - 3 \beta_{4} - 54 \beta_{5} + 13 \beta_{6} + 32 \beta_{7} ) q^{19} + ( -272 - 87 \beta_{1} - 33 \beta_{2} + 37 \beta_{3} + 58 \beta_{4} - 16 \beta_{5} + 7 \beta_{6} - 12 \beta_{7} ) q^{21} + ( 270 + 112 \beta_{1} + 40 \beta_{2} + 61 \beta_{3} - 27 \beta_{4} + 77 \beta_{5} + 35 \beta_{6} - 35 \beta_{7} ) q^{23} + ( 527 + 21 \beta_{1} - 65 \beta_{2} + 42 \beta_{3} - 72 \beta_{4} + 42 \beta_{5} + 68 \beta_{6} - 17 \beta_{7} ) q^{25} + ( 1133 - 74 \beta_{1} - 132 \beta_{2} - 65 \beta_{3} + 107 \beta_{4} - 110 \beta_{5} + 28 \beta_{6} + 3 \beta_{7} ) q^{27} + ( -470 - 54 \beta_{1} + 41 \beta_{2} + 22 \beta_{3} - 114 \beta_{4} - 76 \beta_{5} + 10 \beta_{6} + 61 \beta_{7} ) q^{29} + ( -683 - 140 \beta_{1} - 73 \beta_{2} - 83 \beta_{3} - 46 \beta_{4} + 47 \beta_{5} - 44 \beta_{6} - 101 \beta_{7} ) q^{31} + ( -1687 + 34 \beta_{1} + 42 \beta_{2} - 162 \beta_{3} - 155 \beta_{4} + 213 \beta_{5} - 16 \beta_{6} + 59 \beta_{7} ) q^{33} + ( -652 - 323 \beta_{1} - 301 \beta_{2} - 207 \beta_{3} - 50 \beta_{4} - 84 \beta_{5} - 135 \beta_{6} + 28 \beta_{7} ) q^{35} + ( -293 - 114 \beta_{1} + 45 \beta_{2} - 12 \beta_{3} - 321 \beta_{4} - 47 \beta_{5} - 110 \beta_{6} - 140 \beta_{7} ) q^{37} + ( -1261 + 242 \beta_{1} + 214 \beta_{2} - 100 \beta_{3} - 669 \beta_{4} + 63 \beta_{5} - 130 \beta_{6} - \beta_{7} ) q^{39} + ( -1325 + 173 \beta_{1} + 123 \beta_{2} - 148 \beta_{3} - 190 \beta_{4} + 80 \beta_{5} + 66 \beta_{6} - 69 \beta_{7} ) q^{41} + 1849 q^{43} + ( -5642 - 404 \beta_{1} - 353 \beta_{2} - 377 \beta_{3} - 330 \beta_{4} + 240 \beta_{5} + 60 \beta_{6} + 13 \beta_{7} ) q^{45} + ( 9581 + 193 \beta_{1} - 136 \beta_{2} - 51 \beta_{3} + 183 \beta_{4} + 123 \beta_{5} + 57 \beta_{6} - 150 \beta_{7} ) q^{47} + ( 1243 - 205 \beta_{1} - 321 \beta_{2} + 19 \beta_{3} - 16 \beta_{4} - 610 \beta_{5} - 143 \beta_{6} - 258 \beta_{7} ) q^{49} + ( 10170 - 54 \beta_{1} + 22 \beta_{2} + 182 \beta_{3} + 206 \beta_{4} - 223 \beta_{5} + 161 \beta_{6} - 241 \beta_{7} ) q^{51} + ( -7910 + 100 \beta_{1} - 372 \beta_{2} - 320 \beta_{3} - 279 \beta_{4} + 237 \beta_{5} + 103 \beta_{6} - 203 \beta_{7} ) q^{53} + ( 5843 + 516 \beta_{1} + 398 \beta_{2} + 499 \beta_{3} + 237 \beta_{4} + 444 \beta_{5} + 743 \beta_{6} + 161 \beta_{7} ) q^{55} + ( 59 + 1038 \beta_{1} + 638 \beta_{2} + 886 \beta_{3} - 67 \beta_{4} - 399 \beta_{5} + 269 \beta_{6} - 175 \beta_{7} ) q^{57} + ( 3624 - 445 \beta_{1} + 285 \beta_{2} - 815 \beta_{3} - 922 \beta_{4} - 48 \beta_{5} - 427 \beta_{6} + 364 \beta_{7} ) q^{59} + ( -10623 + 296 \beta_{1} - 542 \beta_{2} + 203 \beta_{3} - 387 \beta_{4} + 224 \beta_{5} - \beta_{6} + 323 \beta_{7} ) q^{61} + ( 8183 + 484 \beta_{1} + 738 \beta_{2} + 1309 \beta_{3} - 511 \beta_{4} - 188 \beta_{5} + 233 \beta_{6} - 707 \beta_{7} ) q^{63} + ( -119 + 796 \beta_{1} + 1158 \beta_{2} + 811 \beta_{3} + 883 \beta_{4} - 984 \beta_{5} - 323 \beta_{6} - 577 \beta_{7} ) q^{65} + ( -3142 + 77 \beta_{1} + 635 \beta_{2} - 90 \beta_{3} - 1225 \beta_{4} + 448 \beta_{5} + 69 \beta_{6} - 437 \beta_{7} ) q^{67} + ( -11760 - 1728 \beta_{1} - 969 \beta_{2} + 77 \beta_{3} + 810 \beta_{4} - 94 \beta_{5} - 372 \beta_{6} + 289 \beta_{7} ) q^{69} + ( 1108 + 816 \beta_{1} + 334 \beta_{2} + 291 \beta_{3} - 1250 \beta_{4} + 133 \beta_{5} + 903 \beta_{6} + 822 \beta_{7} ) q^{71} + ( 2089 + 280 \beta_{1} + 878 \beta_{2} + 212 \beta_{3} - 167 \beta_{4} - 409 \beta_{5} + 244 \beta_{6} + 357 \beta_{7} ) q^{73} + ( -20682 - 276 \beta_{1} + 170 \beta_{2} + 1008 \beta_{3} + 212 \beta_{4} - 492 \beta_{5} - 287 \beta_{6} + 236 \beta_{7} ) q^{75} + ( -27853 + 1714 \beta_{1} + 1798 \beta_{2} - 179 \beta_{3} - 1179 \beta_{4} + 1342 \beta_{5} + 149 \beta_{6} + 921 \beta_{7} ) q^{77} + ( -19370 - 639 \beta_{1} - 157 \beta_{2} - 625 \beta_{3} - 86 \beta_{4} - 1457 \beta_{5} - 430 \beta_{6} + 838 \beta_{7} ) q^{79} + ( 19968 + 467 \beta_{1} - 249 \beta_{2} - 653 \beta_{3} + 527 \beta_{4} - 591 \beta_{5} + 48 \beta_{6} + 605 \beta_{7} ) q^{81} + ( 9418 + 884 \beta_{1} + 408 \beta_{2} + 1917 \beta_{3} + 1755 \beta_{4} - 138 \beta_{5} + 32 \beta_{6} - 1039 \beta_{7} ) q^{83} + ( 18334 - 1274 \beta_{1} - 1523 \beta_{2} - 1672 \beta_{3} - 3606 \beta_{4} + 451 \beta_{5} + 633 \beta_{6} - 333 \beta_{7} ) q^{85} + ( -33148 - 639 \beta_{1} + 175 \beta_{2} + 791 \beta_{3} - 1044 \beta_{4} + 211 \beta_{5} + 464 \beta_{6} - 40 \beta_{7} ) q^{87} + ( -33025 - 1528 \beta_{1} - 346 \beta_{2} + 1173 \beta_{3} + 1221 \beta_{4} - 728 \beta_{5} - 343 \beta_{6} + 671 \beta_{7} ) q^{89} + ( -51135 - 2070 \beta_{1} - 2670 \beta_{2} - 2583 \beta_{3} + 327 \beta_{4} + 1296 \beta_{5} - 531 \beta_{6} + 447 \beta_{7} ) q^{91} + ( -15393 + 1312 \beta_{1} + 2872 \beta_{2} + 315 \beta_{3} - 2599 \beta_{4} + 1741 \beta_{5} + 290 \beta_{6} + 319 \beta_{7} ) q^{93} + ( -16497 + 2330 \beta_{1} + 3631 \beta_{2} + 142 \beta_{3} - 1309 \beta_{4} - 195 \beta_{5} - 1008 \beta_{6} + 1046 \beta_{7} ) q^{95} + ( 17967 - 1116 \beta_{1} - 2094 \beta_{2} + 163 \beta_{3} - 986 \beta_{4} + 900 \beta_{5} + 923 \beta_{6} - 1289 \beta_{7} ) q^{97} + ( -30494 + 1668 \beta_{1} - 732 \beta_{2} - 1801 \beta_{3} - 3541 \beta_{4} + 1398 \beta_{5} + 304 \beta_{6} + 563 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 26q^{3} - 212q^{5} + 136q^{7} + 546q^{9} + O(q^{10})$$ $$8q + 26q^{3} - 212q^{5} + 136q^{7} + 546q^{9} + 532q^{11} - 2492q^{13} + 1780q^{15} - 2534q^{17} + 1678q^{19} - 2256q^{21} + 2488q^{23} + 4378q^{25} + 8960q^{27} - 4360q^{29} - 5704q^{31} - 12852q^{33} - 5640q^{35} - 3772q^{37} - 11120q^{39} - 10698q^{41} + 14792q^{43} - 44912q^{45} + 77864q^{47} + 7188q^{49} + 80246q^{51} - 62352q^{53} + 49552q^{55} - 808q^{57} + 26224q^{59} - 82540q^{61} + 61768q^{63} - 5000q^{65} - 27784q^{67} - 93776q^{69} + 9504q^{71} + 14260q^{73} - 167420q^{75} - 218140q^{77} - 160248q^{79} + 161076q^{81} + 77176q^{83} + 141096q^{85} - 268136q^{87} - 265692q^{89} - 401148q^{91} - 123860q^{93} - 135884q^{95} + 144742q^{97} - 239516q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4 x^{7} - 173 x^{6} + 462 x^{5} + 9118 x^{4} - 14192 x^{3} - 167688 x^{2} + 106368 x + 681984$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-12949 \nu^{7} + 681148 \nu^{6} - 2601727 \nu^{5} - 80176782 \nu^{4} + 378607946 \nu^{3} + 1837549088 \nu^{2} - 8768955576 \nu - 1162574592$$$$)/ 547265856$$ $$\beta_{2}$$ $$=$$ $$($$$$36205 \nu^{7} - 1830508 \nu^{6} - 539057 \nu^{5} + 265985910 \nu^{4} - 188725898 \nu^{3} - 9341109104 \nu^{2} + 5760293592 \nu + 58885512576$$$$)/ 1094531712$$ $$\beta_{3}$$ $$=$$ $$($$$$-67489 \nu^{7} + 623356 \nu^{6} + 8964549 \nu^{5} - 78907518 \nu^{4} - 250466958 \nu^{3} + 2271218384 \nu^{2} - 114021816 \nu - 8596825856$$$$)/ 364843904$$ $$\beta_{4}$$ $$=$$ $$($$$$-204587 \nu^{7} + 1315940 \nu^{6} + 28380103 \nu^{5} - 147318570 \nu^{4} - 993160922 \nu^{3} + 4310420080 \nu^{2} + 8605716312 \nu - 37440050688$$$$)/ 1094531712$$ $$\beta_{5}$$ $$=$$ $$($$$$-309497 \nu^{7} + 2935676 \nu^{6} + 38047933 \nu^{5} - 372264846 \nu^{4} - 955734974 \nu^{3} + 11960213584 \nu^{2} + 2287797192 \nu - 76606929024$$$$)/ 1094531712$$ $$\beta_{6}$$ $$=$$ $$($$$$73259 \nu^{7} - 190788 \nu^{6} - 10643911 \nu^{5} + 12835930 \nu^{4} + 347431274 \nu^{3} - 15275088 \nu^{2} - 785010872 \nu - 3291957568$$$$)/ 182421952$$ $$\beta_{7}$$ $$=$$ $$($$$$-34066 \nu^{7} + 259912 \nu^{6} + 4750721 \nu^{5} - 30187209 \nu^{4} - 176761426 \nu^{3} + 777094526 \nu^{2} + 1922056332 \nu - 2232499728$$$$)/68408232$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{6} + \beta_{5} + \beta_{3} + 2 \beta_{2} + 4$$$$)/8$$ $$\nu^{2}$$ $$=$$ $$($$$$-2 \beta_{7} + \beta_{6} - \beta_{5} + 8 \beta_{4} + \beta_{3} + 180$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$-14 \beta_{7} + 33 \beta_{6} + 15 \beta_{5} + 32 \beta_{4} + 65 \beta_{3} + 64 \beta_{2} + 4 \beta_{1} + 380$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$-99 \beta_{7} + 84 \beta_{6} - 117 \beta_{5} + 488 \beta_{4} + 140 \beta_{3} + 63 \beta_{2} + 46 \beta_{1} + 6796$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-869 \beta_{7} + 1527 \beta_{6} - 276 \beta_{5} + 2768 \beta_{4} + 3711 \beta_{3} + 2515 \beta_{2} + 142 \beta_{1} + 27000$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$-10339 \beta_{7} + 11590 \beta_{6} - 16231 \beta_{5} + 55680 \beta_{4} + 22478 \beta_{3} + 10127 \beta_{2} + 6850 \beta_{1} + 639708$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-102849 \beta_{7} + 166838 \beta_{6} - 94865 \beta_{5} + 386624 \beta_{4} + 421838 \beta_{3} + 234341 \beta_{2} + 20926 \beta_{1} + 3616388$$$$)/2$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −2.08717 −6.09504 5.65705 2.58275 −5.06235 7.21373 10.9591 −9.16809
0 −25.6605 0 −61.4284 0 184.774 0 415.460 0
1.2 0 −11.1803 0 −63.3756 0 −223.489 0 −118.000 0
1.3 0 −7.84314 0 −107.102 0 25.5214 0 −181.485 0
1.4 0 −3.05838 0 27.7074 0 103.690 0 −233.646 0
1.5 0 9.19190 0 73.4416 0 4.24720 0 −158.509 0
1.6 0 11.2683 0 −9.80186 0 11.1041 0 −116.025 0
1.7 0 25.1057 0 −61.2927 0 166.001 0 387.294 0
1.8 0 28.1764 0 −10.1483 0 −135.849 0 550.912 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$43$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 688.6.a.e 8
4.b odd 2 1 43.6.a.a 8
12.b even 2 1 387.6.a.c 8
20.d odd 2 1 1075.6.a.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.6.a.a 8 4.b odd 2 1
387.6.a.c 8 12.b even 2 1
688.6.a.e 8 1.a even 1 1 trivial
1075.6.a.a 8 20.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{8} - \cdots$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(688))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$504223128 + 156321960 T - 14906374 T^{2} - 3627880 T^{3} + 184435 T^{4} + 22242 T^{5} - 907 T^{6} - 26 T^{7} + T^{8}$$
$5$ $$5172924974752 + 1078880170848 T + 54379881442 T^{2} - 883062964 T^{3} - 81381385 T^{4} - 1016128 T^{5} + 7783 T^{6} + 212 T^{7} + T^{8}$$
$7$ $$116222354316288 - 43459429678848 T + 4329955769184 T^{2} - 129844227552 T^{3} + 573109808 T^{4} + 9763880 T^{5} - 61574 T^{6} - 136 T^{7} + T^{8}$$
$11$ $$16753808850459134976 - 624343314702054528 T + 8339795208994704 T^{2} - 45000841976988 T^{3} + 46616679553 T^{4} + 322346400 T^{5} - 558018 T^{6} - 532 T^{7} + T^{8}$$
$13$ $$-$$$$20\!\cdots\!44$$$$+ 7478047501760359440 T + 6802553866384368 T^{2} - 370621947684948 T^{3} - 1445788688279 T^{4} - 1375580104 T^{5} + 1195110 T^{6} + 2492 T^{7} + T^{8}$$
$17$ $$37\!\cdots\!33$$$$-$$$$29\!\cdots\!02$$$$T + 196078714064136693 T^{2} + 2317785169226190 T^{3} - 1750346026308 T^{4} - 6313795694 T^{5} - 1270643 T^{6} + 2534 T^{7} + T^{8}$$
$19$ $$20\!\cdots\!68$$$$+$$$$57\!\cdots\!56$$$$T - 30881750045973655680 T^{2} - 103160205820683760 T^{3} + 57520540020045 T^{4} + 25395335782 T^{5} - 15002885 T^{6} - 1678 T^{7} + T^{8}$$
$23$ $$-$$$$18\!\cdots\!83$$$$+$$$$85\!\cdots\!36$$$$T -$$$$82\!\cdots\!05$$$$T^{2} - 532680614614099516 T^{3} + 296571412629528 T^{4} + 70979412460 T^{5} - 31683297 T^{6} - 2488 T^{7} + T^{8}$$
$29$ $$-$$$$39\!\cdots\!36$$$$-$$$$43\!\cdots\!72$$$$T +$$$$23\!\cdots\!22$$$$T^{2} + 764932230807720084 T^{3} - 102909589543017 T^{4} - 237071828860 T^{5} - 40668269 T^{6} + 4360 T^{7} + T^{8}$$
$31$ $$10\!\cdots\!13$$$$+$$$$41\!\cdots\!72$$$$T +$$$$37\!\cdots\!35$$$$T^{2} + 11705033314802358652 T^{3} + 55447831561440 T^{4} - 606806231860 T^{5} - 74495573 T^{6} + 5704 T^{7} + T^{8}$$
$37$ $$-$$$$10\!\cdots\!04$$$$-$$$$55\!\cdots\!12$$$$T -$$$$13\!\cdots\!08$$$$T^{2} +$$$$28\!\cdots\!80$$$$T^{3} + 33804397527288373 T^{4} - 2110832852496 T^{5} - 363678631 T^{6} + 3772 T^{7} + T^{8}$$
$41$ $$17\!\cdots\!57$$$$-$$$$14\!\cdots\!18$$$$T -$$$$29\!\cdots\!19$$$$T^{2} + 97139036739429657510 T^{3} + 13684977957931252 T^{4} - 1850906058934 T^{5} - 208766819 T^{6} + 10698 T^{7} + T^{8}$$
$43$ $$( -1849 + T )^{8}$$
$47$ $$-$$$$19\!\cdots\!04$$$$+$$$$10\!\cdots\!72$$$$T -$$$$20\!\cdots\!32$$$$T^{2} +$$$$10\!\cdots\!88$$$$T^{3} + 184514805755443205 T^{4} - 34288673842832 T^{5} + 2375107969 T^{6} - 77864 T^{7} + T^{8}$$
$53$ $$55\!\cdots\!84$$$$+$$$$73\!\cdots\!52$$$$T +$$$$13\!\cdots\!56$$$$T^{2} -$$$$12\!\cdots\!60$$$$T^{3} - 391239772753902151 T^{4} - 17547197551260 T^{5} + 759433570 T^{6} + 62352 T^{7} + T^{8}$$
$59$ $$-$$$$68\!\cdots\!68$$$$+$$$$10\!\cdots\!48$$$$T +$$$$57\!\cdots\!08$$$$T^{2} -$$$$58\!\cdots\!44$$$$T^{3} + 1572083239240996464 T^{4} + 85817413362304 T^{5} - 2944879380 T^{6} - 26224 T^{7} + T^{8}$$
$61$ $$23\!\cdots\!84$$$$+$$$$63\!\cdots\!84$$$$T +$$$$33\!\cdots\!20$$$$T^{2} -$$$$40\!\cdots\!76$$$$T^{3} - 4757919938906569384 T^{4} - 146433759288344 T^{5} + 310045522 T^{6} + 82540 T^{7} + T^{8}$$
$67$ $$69\!\cdots\!48$$$$-$$$$69\!\cdots\!68$$$$T -$$$$61\!\cdots\!84$$$$T^{2} +$$$$44\!\cdots\!08$$$$T^{3} + 2253346285790674161 T^{4} - 71143970372344 T^{5} - 2814867770 T^{6} + 27784 T^{7} + T^{8}$$
$71$ $$-$$$$15\!\cdots\!64$$$$-$$$$58\!\cdots\!60$$$$T -$$$$54\!\cdots\!80$$$$T^{2} +$$$$72\!\cdots\!20$$$$T^{3} + 15044739621952512640 T^{4} - 3736008685216 T^{5} - 7671997080 T^{6} - 9504 T^{7} + T^{8}$$
$73$ $$80\!\cdots\!92$$$$-$$$$13\!\cdots\!44$$$$T -$$$$28\!\cdots\!56$$$$T^{2} -$$$$95\!\cdots\!68$$$$T^{3} + 1545293860286742712 T^{4} + 23615008093864 T^{5} - 2401149454 T^{6} - 14260 T^{7} + T^{8}$$
$79$ $$19\!\cdots\!12$$$$+$$$$47\!\cdots\!36$$$$T +$$$$22\!\cdots\!36$$$$T^{2} -$$$$29\!\cdots\!88$$$$T^{3} - 38441230790897773763 T^{4} - 637841681315296 T^{5} + 3511120093 T^{6} + 160248 T^{7} + T^{8}$$
$83$ $$-$$$$19\!\cdots\!08$$$$+$$$$56\!\cdots\!20$$$$T -$$$$34\!\cdots\!60$$$$T^{2} -$$$$10\!\cdots\!80$$$$T^{3} + 37880169769568642313 T^{4} + 551728279600100 T^{5} - 11424361174 T^{6} - 77176 T^{7} + T^{8}$$
$89$ $$-$$$$13\!\cdots\!64$$$$-$$$$12\!\cdots\!52$$$$T +$$$$83\!\cdots\!12$$$$T^{2} +$$$$14\!\cdots\!28$$$$T^{3} - 72968680454010964328 T^{4} - 877164197088320 T^{5} + 15614540094 T^{6} + 265692 T^{7} + T^{8}$$
$97$ $$38\!\cdots\!17$$$$+$$$$16\!\cdots\!26$$$$T -$$$$55\!\cdots\!79$$$$T^{2} -$$$$36\!\cdots\!58$$$$T^{3} + 81215980328465108268 T^{4} + 1748586412389886 T^{5} - 15555161907 T^{6} - 144742 T^{7} + T^{8}$$