Properties

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

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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: \(x^{8} - 2 x^{7} + 193 x^{6} + 306 x^{5} + 29845 x^{4} + 16988 x^{3} + 1125468 x^{2} + 214128 x + 35378704\)
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 42)
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.

\(f(q)\) \(=\) \( q + ( 9 - 9 \beta_{1} ) q^{3} + ( 77 + 39 \beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{5} + ( -12 + 121 \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{7} -243 \beta_{1} q^{9} +O(q^{10})\) \( q + ( 9 - 9 \beta_{1} ) q^{3} + ( 77 + 39 \beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{5} + ( -12 + 121 \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{7} -243 \beta_{1} q^{9} + ( -448 - 455 \beta_{1} - 14 \beta_{2} - 20 \beta_{3} + 14 \beta_{4} - 12 \beta_{5} - 6 \beta_{6} - 7 \beta_{7} ) q^{11} + ( 692 + 1393 \beta_{1} - 9 \beta_{2} + 18 \beta_{3} - 9 \beta_{4} + 14 \beta_{5} ) q^{13} + ( 1044 + 9 \beta_{1} + 9 \beta_{2} - 9 \beta_{4} - 9 \beta_{5} - 18 \beta_{6} + 18 \beta_{7} ) q^{15} + ( 780 - 814 \beta_{1} - 94 \beta_{2} - 47 \beta_{3} + 28 \beta_{6} - 34 \beta_{7} ) q^{17} + ( -3837 - 1905 \beta_{1} - 2 \beta_{2} - 58 \beta_{3} - 27 \beta_{4} - 56 \beta_{5} - 56 \beta_{6} + 27 \beta_{7} ) q^{19} + ( 981 + 2286 \beta_{1} - 36 \beta_{2} + 9 \beta_{3} - 45 \beta_{5} - 9 \beta_{6} + 27 \beta_{7} ) q^{21} + ( 24 + 612 \beta_{1} - 267 \beta_{2} + 24 \beta_{4} - 28 \beta_{5} + 28 \beta_{6} + 24 \beta_{7} ) q^{23} + ( 8161 + 8220 \beta_{1} + 118 \beta_{2} + 548 \beta_{3} - 118 \beta_{4} - 124 \beta_{5} - 62 \beta_{6} + 59 \beta_{7} ) q^{25} + ( -2187 - 4374 \beta_{1} ) q^{27} + ( -322 - 23 \beta_{1} - 80 \beta_{2} - 57 \beta_{3} + 23 \beta_{4} - 173 \beta_{5} - 346 \beta_{6} - 46 \beta_{7} ) q^{29} + ( -12290 + 12446 \beta_{1} + 68 \beta_{2} + 34 \beta_{3} - 51 \beta_{6} + 156 \beta_{7} ) q^{31} + ( -8127 - 4158 \beta_{1} - 369 \beta_{2} - 360 \beta_{3} + 189 \beta_{4} - 162 \beta_{5} - 162 \beta_{6} - 189 \beta_{7} ) q^{33} + ( 13541 + 30033 \beta_{1} - 381 \beta_{2} + 373 \beta_{3} - 19 \beta_{4} + 60 \beta_{5} + 460 \beta_{6} - 158 \beta_{7} ) q^{35} + ( -45 + 20989 \beta_{1} + 1443 \beta_{2} - 45 \beta_{4} - 160 \beta_{5} + 160 \beta_{6} - 45 \beta_{7} ) q^{37} + ( 18765 + 18846 \beta_{1} + 162 \beta_{2} + 486 \beta_{3} - 162 \beta_{4} + 252 \beta_{5} + 126 \beta_{6} + 81 \beta_{7} ) q^{39} + ( 8906 + 17768 \beta_{1} - 226 \beta_{2} + 182 \beta_{3} + 44 \beta_{4} - 700 \beta_{5} ) q^{41} + ( -11732 - 221 \beta_{1} + 284 \beta_{2} + 505 \beta_{3} + 221 \beta_{4} - 94 \beta_{5} - 188 \beta_{6} - 442 \beta_{7} ) q^{43} + ( 9477 - 9234 \beta_{1} - 243 \beta_{6} + 243 \beta_{7} ) q^{45} + ( 53854 + 26844 \beta_{1} - 893 \beta_{2} - 1454 \beta_{3} + 166 \beta_{4} + 922 \beta_{5} + 922 \beta_{6} - 166 \beta_{7} ) q^{47} + ( -36018 - 69890 \beta_{1} - 1341 \beta_{2} + 756 \beta_{3} + 305 \beta_{4} + 462 \beta_{5} + 910 \beta_{6} - 580 \beta_{7} ) q^{49} + ( -306 - 21672 \beta_{1} - 963 \beta_{2} - 306 \beta_{4} - 252 \beta_{5} + 252 \beta_{6} - 306 \beta_{7} ) q^{51} + ( 89965 + 89078 \beta_{1} - 1774 \beta_{2} - 3460 \beta_{3} + 1774 \beta_{4} + 502 \beta_{5} + 251 \beta_{6} - 887 \beta_{7} ) q^{53} + ( -56742 - 115203 \beta_{1} + 776 \beta_{2} - 2495 \beta_{3} + 1719 \beta_{4} - 171 \beta_{5} ) q^{55} + ( -51678 + 243 \beta_{1} - 540 \beta_{2} - 783 \beta_{3} - 243 \beta_{4} - 504 \beta_{5} - 1008 \beta_{6} + 486 \beta_{7} ) q^{57} + ( -59760 + 60137 \beta_{1} - 838 \beta_{2} - 419 \beta_{3} + 586 \beta_{6} + 377 \beta_{7} ) q^{59} + ( 70256 + 35188 \beta_{1} + 2512 \beta_{2} + 4784 \beta_{3} - 120 \beta_{4} + 3320 \beta_{5} + 3320 \beta_{6} + 120 \beta_{7} ) q^{61} + ( 29403 + 32319 \beta_{1} - 486 \beta_{2} + 486 \beta_{3} + 243 \beta_{4} - 729 \beta_{5} - 486 \beta_{6} + 243 \beta_{7} ) q^{63} + ( -1598 + 79886 \beta_{1} - 1855 \beta_{2} - 1598 \beta_{4} - 436 \beta_{5} + 436 \beta_{6} - 1598 \beta_{7} ) q^{65} + ( -66137 - 67368 \beta_{1} - 2462 \beta_{2} - 427 \beta_{3} + 2462 \beta_{4} - 2216 \beta_{5} - 1108 \beta_{6} - 1231 \beta_{7} ) q^{67} + ( 5724 + 10800 \beta_{1} - 2835 \beta_{2} + 2187 \beta_{3} + 648 \beta_{4} - 756 \beta_{5} ) q^{69} + ( 57942 - 282 \beta_{1} + 6697 \beta_{2} + 6979 \beta_{3} + 282 \beta_{4} - 598 \beta_{5} - 1196 \beta_{6} - 564 \beta_{7} ) q^{71} + ( 162483 - 161598 \beta_{1} + 2836 \beta_{2} + 1418 \beta_{3} + 976 \beta_{6} + 885 \beta_{7} ) q^{73} + ( 147429 + 74511 \beta_{1} + 6525 \beta_{2} + 9864 \beta_{3} - 1593 \beta_{4} - 1674 \beta_{5} - 1674 \beta_{6} + 1593 \beta_{7} ) q^{75} + ( -304801 - 182291 \beta_{1} + 14252 \beta_{2} + 14496 \beta_{3} + 263 \beta_{4} - 765 \beta_{5} - 1413 \beta_{6} + 1601 \beta_{7} ) q^{77} + ( -2508 - 172730 \beta_{1} - 6036 \beta_{2} - 2508 \beta_{4} - 109 \beta_{5} + 109 \beta_{6} - 2508 \beta_{7} ) q^{79} + ( -59049 - 59049 \beta_{1} ) q^{81} + ( 24001 + 49693 \beta_{1} + 5776 \beta_{2} - 4085 \beta_{3} - 1691 \beta_{4} - 5336 \beta_{5} ) q^{83} + ( -485326 - 1610 \beta_{1} - 25948 \beta_{2} - 24338 \beta_{3} + 1610 \beta_{4} + 1148 \beta_{5} + 2296 \beta_{6} - 3220 \beta_{7} ) q^{85} + ( -3105 + 2484 \beta_{1} - 1026 \beta_{2} - 513 \beta_{3} - 4671 \beta_{6} - 621 \beta_{7} ) q^{87} + ( 688434 + 345966 \beta_{1} - 284 \beta_{2} - 7564 \beta_{3} - 3498 \beta_{4} - 6330 \beta_{5} - 6330 \beta_{6} + 3498 \beta_{7} ) q^{89} + ( 117009 + 115101 \beta_{1} - 6472 \beta_{2} - 20618 \beta_{3} - 1055 \beta_{4} + 890 \beta_{5} + 2292 \beta_{6} - 2189 \beta_{7} ) q^{91} + ( 1404 + 334638 \beta_{1} - 486 \beta_{2} + 1404 \beta_{4} + 459 \beta_{5} - 459 \beta_{6} + 1404 \beta_{7} ) q^{93} + ( 527556 + 525060 \beta_{1} - 4992 \beta_{2} + 15753 \beta_{3} + 4992 \beta_{4} + 1756 \beta_{5} + 878 \beta_{6} - 2496 \beta_{7} ) q^{95} + ( -6797 - 20527 \beta_{1} - 7281 \beta_{2} + 348 \beta_{3} + 6933 \beta_{4} - 2656 \beta_{5} ) q^{97} + ( -110565 - 1701 \beta_{1} - 6561 \beta_{2} - 4860 \beta_{3} + 1701 \beta_{4} - 1458 \beta_{5} - 2916 \beta_{6} - 3402 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 108 q^{3} + 462 q^{5} - 580 q^{7} + 972 q^{9} + O(q^{10}) \) \( 8 q + 108 q^{3} + 462 q^{5} - 580 q^{7} + 972 q^{9} - 1806 q^{11} + 8316 q^{15} + 9564 q^{17} - 23022 q^{19} - 1350 q^{21} - 2400 q^{23} + 32762 q^{25} - 2484 q^{29} - 148416 q^{31} - 48762 q^{33} - 11412 q^{35} - 84046 q^{37} + 75222 q^{39} - 92972 q^{43} + 112266 q^{45} + 323124 q^{47} - 8644 q^{49} + 86076 q^{51} + 358086 q^{53} - 414396 q^{57} - 719382 q^{59} + 421536 q^{61} + 104490 q^{63} - 322740 q^{65} - 267010 q^{67} + 464664 q^{71} + 1944486 q^{73} + 884574 q^{75} - 1713498 q^{77} + 685904 q^{79} - 236196 q^{81} - 3876168 q^{85} - 33534 q^{87} + 4130604 q^{89} + 484266 q^{91} - 1335744 q^{93} + 2105232 q^{95} - 877716 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 2 x^{7} + 193 x^{6} + 306 x^{5} + 29845 x^{4} + 16988 x^{3} + 1125468 x^{2} + 214128 x + 35378704\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(1636073 \nu^{7} - 177960445 \nu^{6} + 557057579 \nu^{5} - 27650057549 \nu^{4} - 22043091213 \nu^{3} - 5229627581347 \nu^{2} - 437053447260 \nu - 197665427324196\)\()/ 164920217026180 \)
\(\beta_{2}\)\(=\)\((\)\(-5696588781 \nu^{7} - 629632122285 \nu^{6} + 1970894968827 \nu^{5} - 157398820606197 \nu^{4} - 77989456039869 \nu^{3} - 18502659469095411 \nu^{2} + 1213473343058610 \nu - 699349243077611748\)\()/ 14306828827021115 \)
\(\beta_{3}\)\(=\)\((\)\(-31448254527 \nu^{7} + 1968223271760 \nu^{6} - 10845607382811 \nu^{5} + 297416713541916 \nu^{4} - 165575031962703 \nu^{3} + 36446552419931358 \nu^{2} - 10543462268940180 \nu + 354239377144818384\)\()/ 28613657654042230 \)
\(\beta_{4}\)\(=\)\((\)\(-369495651653 \nu^{7} + 2826191904405 \nu^{6} - 143960676182679 \nu^{5} + 245097685796969 \nu^{4} - 17480931945541207 \nu^{3} - 6861081933950833 \nu^{2} - 1146402000884955580 \nu - 945621140688043924\)\()/ 57227315308084460 \)
\(\beta_{5}\)\(=\)\((\)\(474632212627 \nu^{7} - 1876757118200 \nu^{6} + 111593476518171 \nu^{5} - 61817459723356 \nu^{4} + 17037777662459983 \nu^{3} - 8110688256075038 \nu^{2} + 1160586789283378480 \nu - 279233571033238804\)\()/ 57227315308084460 \)
\(\beta_{6}\)\(=\)\((\)\(156913428428 \nu^{7} - 201842164189 \nu^{6} + 21775570970562 \nu^{5} + 89098109702407 \nu^{4} + 3429047277089390 \nu^{3} + 3476685341973407 \nu^{2} - 77338538608714000 \nu + 136874662243533112\)\()/ 11445463061616892 \)
\(\beta_{7}\)\(=\)\((\)\(580833884751 \nu^{7} - 646807832245 \nu^{6} + 69581678069333 \nu^{5} + 491941235511447 \nu^{4} + 8835382339852409 \nu^{3} + 25949036508994771 \nu^{2} - 161285957516505460 \nu + 1295333628407081938\)\()/ 28613657654042230 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(4 \beta_{7} - 10 \beta_{6} + 10 \beta_{5} + 4 \beta_{4} + \beta_{2} - 82 \beta_{1} + 4\)\()/168\)
\(\nu^{2}\)\(=\)\((\)\(-8 \beta_{7} + 6 \beta_{6} + 12 \beta_{5} + 16 \beta_{4} - 143 \beta_{3} - 16 \beta_{2} - 8026 \beta_{1} - 8018\)\()/84\)
\(\nu^{3}\)\(=\)\((\)\(-248 \beta_{7} + 300 \beta_{6} + 150 \beta_{5} + 124 \beta_{4} - 287 \beta_{3} - 411 \beta_{2} - 124 \beta_{1} - 6266\)\()/24\)
\(\nu^{4}\)\(=\)\((\)\(-1154 \beta_{7} + 1002 \beta_{6} - 1002 \beta_{5} - 1154 \beta_{4} - 13319 \beta_{2} + 528644 \beta_{1} - 1154\)\()/42\)
\(\nu^{5}\)\(=\)\((\)\(148916 \beta_{7} - 146554 \beta_{6} - 293108 \beta_{5} - 297832 \beta_{4} + 455449 \beta_{3} + 297832 \beta_{2} + 11438902 \beta_{1} + 11289986\)\()/168\)
\(\nu^{6}\)\(=\)\((\)\(149120 \beta_{7} - 134492 \beta_{6} - 67246 \beta_{5} - 74560 \beta_{4} + 719931 \beta_{3} + 794491 \beta_{2} + 74560 \beta_{1} + 23310626\)\()/12\)
\(\nu^{7}\)\(=\)\((\)\(24903764 \beta_{7} - 23191906 \beta_{6} + 23191906 \beta_{5} + 24903764 \beta_{4} + 67300521 \beta_{2} - 2452564154 \beta_{1} + 24903764\)\()/168\)

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
−5.50376 + 9.53280i
−3.27778 + 5.67728i
3.07067 5.31856i
6.71087 11.6236i
−5.50376 9.53280i
−3.27778 5.67728i
3.07067 + 5.31856i
6.71087 + 11.6236i
0 13.5000 7.79423i 0 −124.321 71.7768i 0 −342.924 + 7.21089i 0 121.500 210.444i 0
145.2 0 13.5000 7.79423i 0 63.2658 + 36.5265i 0 271.352 209.802i 0 121.500 210.444i 0
145.3 0 13.5000 7.79423i 0 77.6900 + 44.8543i 0 −204.220 + 275.578i 0 121.500 210.444i 0
145.4 0 13.5000 7.79423i 0 214.365 + 123.764i 0 −14.2078 + 342.706i 0 121.500 210.444i 0
241.1 0 13.5000 + 7.79423i 0 −124.321 + 71.7768i 0 −342.924 7.21089i 0 121.500 + 210.444i 0
241.2 0 13.5000 + 7.79423i 0 63.2658 36.5265i 0 271.352 + 209.802i 0 121.500 + 210.444i 0
241.3 0 13.5000 + 7.79423i 0 77.6900 44.8543i 0 −204.220 275.578i 0 121.500 + 210.444i 0
241.4 0 13.5000 + 7.79423i 0 214.365 123.764i 0 −14.2078 342.706i 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:

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.f 8
4.b odd 2 1 42.7.g.a 8
7.d odd 6 1 inner 336.7.bh.f 8
12.b even 2 1 126.7.n.a 8
28.d even 2 1 294.7.g.d 8
28.f even 6 1 42.7.g.a 8
28.f even 6 1 294.7.c.b 8
28.g odd 6 1 294.7.c.b 8
28.g odd 6 1 294.7.g.d 8
84.j odd 6 1 126.7.n.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.7.g.a 8 4.b odd 2 1
42.7.g.a 8 28.f even 6 1
126.7.n.a 8 12.b even 2 1
126.7.n.a 8 84.j odd 6 1
294.7.c.b 8 28.f even 6 1
294.7.c.b 8 28.g odd 6 1
294.7.g.d 8 28.d even 2 1
294.7.g.d 8 28.g odd 6 1
336.7.bh.f 8 1.a even 1 1 trivial
336.7.bh.f 8 7.d odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(28\!\cdots\!00\)\( T_{5}^{2} - \)\(20\!\cdots\!00\)\( T_{5} + \)\(54\!\cdots\!00\)\( \)">\(T_{5}^{8} - \cdots\) acting on \(S_{7}^{\mathrm{new}}(336, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( ( 243 - 27 T + T^{2} )^{4} \)
$5$ \( 54227831439690000 - 2057883988770000 T + 28839143385900 T^{2} - 106548914700 T^{3} - 982673451 T^{4} + 5570334 T^{5} + 59091 T^{6} - 462 T^{7} + T^{8} \)
$7$ \( \)\(19\!\cdots\!01\)\( + 944479886788060420 T + 2387926550490922 T^{2} + 5149120253200 T^{3} + 8591553523 T^{4} + 43766800 T^{5} + 172522 T^{6} + 580 T^{7} + T^{8} \)
$11$ \( \)\(17\!\cdots\!64\)\( - \)\(98\!\cdots\!44\)\( T + 54077440614303771900 T^{2} + 30005611306535700 T^{3} + 30448136405709 T^{4} + 7104353454 T^{5} + 7429383 T^{6} + 1806 T^{7} + T^{8} \)
$13$ \( \)\(32\!\cdots\!36\)\( + \)\(32\!\cdots\!20\)\( T^{2} + 115909128245961 T^{4} + 17872710 T^{6} + T^{8} \)
$17$ \( \)\(24\!\cdots\!44\)\( - \)\(25\!\cdots\!24\)\( T + \)\(10\!\cdots\!04\)\( T^{2} - 19488328264554742656 T^{3} + 263622724215120 T^{4} + 360052350192 T^{5} - 7156596 T^{6} - 9564 T^{7} + T^{8} \)
$19$ \( \)\(32\!\cdots\!44\)\( - \)\(44\!\cdots\!52\)\( T + \)\(20\!\cdots\!24\)\( T^{2} + 719089227569418084 T^{3} - 2088732176628075 T^{4} - 66251077038 T^{5} + 173793099 T^{6} + 23022 T^{7} + T^{8} \)
$23$ \( \)\(23\!\cdots\!44\)\( - \)\(74\!\cdots\!52\)\( T + \)\(24\!\cdots\!32\)\( T^{2} + \)\(37\!\cdots\!28\)\( T^{3} + 63536314779021312 T^{4} + 2464722542592 T^{5} + 251434368 T^{6} + 2400 T^{7} + T^{8} \)
$29$ \( ( 239034049865996544 - 273243932064 T - 1288601559 T^{2} + 1242 T^{3} + T^{4} )^{2} \)
$31$ \( \)\(37\!\cdots\!49\)\( - \)\(43\!\cdots\!24\)\( T + \)\(12\!\cdots\!26\)\( T^{2} + \)\(48\!\cdots\!32\)\( T^{3} + 5644393027773678315 T^{4} + 315040279526784 T^{5} + 9465120426 T^{6} + 148416 T^{7} + T^{8} \)
$37$ \( \)\(22\!\cdots\!96\)\( + \)\(11\!\cdots\!52\)\( T + \)\(58\!\cdots\!44\)\( T^{2} + \)\(83\!\cdots\!72\)\( T^{3} + 30663682554432297133 T^{4} + 228209921786734 T^{5} + 10167761071 T^{6} + 84046 T^{7} + T^{8} \)
$41$ \( \)\(16\!\cdots\!64\)\( + \)\(14\!\cdots\!92\)\( T^{2} + 81598816321320479328 T^{4} + 15587495664 T^{6} + T^{8} \)
$43$ \( ( 1403954794589481412 - 63724778626892 T - 4423971819 T^{2} + 46486 T^{3} + T^{4} )^{2} \)
$47$ \( \)\(32\!\cdots\!76\)\( - \)\(38\!\cdots\!20\)\( T + \)\(15\!\cdots\!08\)\( T^{2} - \)\(71\!\cdots\!60\)\( T^{3} - \)\(20\!\cdots\!52\)\( T^{4} + 1076696955454608 T^{5} + 31470891900 T^{6} - 323124 T^{7} + T^{8} \)
$53$ \( \)\(14\!\cdots\!76\)\( - \)\(21\!\cdots\!88\)\( T + \)\(30\!\cdots\!28\)\( T^{2} - \)\(13\!\cdots\!80\)\( T^{3} + \)\(85\!\cdots\!25\)\( T^{4} - 26665435271316150 T^{5} + 156312293787 T^{6} - 358086 T^{7} + T^{8} \)
$59$ \( \)\(42\!\cdots\!84\)\( - \)\(19\!\cdots\!92\)\( T - \)\(65\!\cdots\!04\)\( T^{2} + \)\(15\!\cdots\!68\)\( T^{3} + \)\(36\!\cdots\!25\)\( T^{4} + 38853189456604866 T^{5} + 226512608271 T^{6} + 719382 T^{7} + T^{8} \)
$61$ \( \)\(10\!\cdots\!76\)\( + \)\(67\!\cdots\!48\)\( T + \)\(12\!\cdots\!24\)\( T^{2} - \)\(10\!\cdots\!12\)\( T^{3} + \)\(15\!\cdots\!88\)\( T^{4} + 68038017176226816 T^{5} - 102174131424 T^{6} - 421536 T^{7} + T^{8} \)
$67$ \( \)\(68\!\cdots\!16\)\( - \)\(31\!\cdots\!40\)\( T + \)\(50\!\cdots\!60\)\( T^{2} + \)\(25\!\cdots\!80\)\( T^{3} + \)\(19\!\cdots\!29\)\( T^{4} - 12847625511907070 T^{5} + 209320076635 T^{6} + 267010 T^{7} + T^{8} \)
$71$ \( ( \)\(16\!\cdots\!64\)\( + 26717778340003296 T - 127035305388 T^{2} - 232332 T^{3} + T^{4} )^{2} \)
$73$ \( \)\(43\!\cdots\!96\)\( - \)\(22\!\cdots\!24\)\( T + \)\(48\!\cdots\!08\)\( T^{2} - \)\(46\!\cdots\!64\)\( T^{3} + \)\(25\!\cdots\!13\)\( T^{4} - 841530567991907106 T^{5} + 1693119834903 T^{6} - 1944486 T^{7} + T^{8} \)
$79$ \( \)\(59\!\cdots\!49\)\( - \)\(59\!\cdots\!56\)\( T + \)\(59\!\cdots\!02\)\( T^{2} - \)\(93\!\cdots\!84\)\( T^{3} + \)\(30\!\cdots\!95\)\( T^{4} - 237712310143098752 T^{5} + 839176048882 T^{6} - 685904 T^{7} + T^{8} \)
$83$ \( \)\(21\!\cdots\!04\)\( + \)\(38\!\cdots\!16\)\( T^{2} + \)\(41\!\cdots\!37\)\( T^{4} + 1184254748142 T^{6} + T^{8} \)
$89$ \( \)\(17\!\cdots\!36\)\( - \)\(49\!\cdots\!52\)\( T - \)\(85\!\cdots\!12\)\( T^{2} + \)\(13\!\cdots\!00\)\( T^{3} + \)\(89\!\cdots\!00\)\( T^{4} - 4714272501192968400 T^{5} + 6828599895372 T^{6} - 4130604 T^{7} + T^{8} \)
$97$ \( \)\(12\!\cdots\!24\)\( + \)\(93\!\cdots\!60\)\( T^{2} + \)\(24\!\cdots\!21\)\( T^{4} + 2655305939382 T^{6} + T^{8} \)
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