Properties

Label 336.7.bh.d
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: \(x^{8} - x^{7} + 212 x^{6} + 473 x^{5} + 39800 x^{4} + 36821 x^{3} + 985651 x^{2} - 601290 x + 21068100\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{12}\cdot 3\cdot 7 \)
Twist minimal: no (minimal twist has level 21)
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.

\(f(q)\) \(=\) \( q + ( 18 - 9 \beta_{1} ) q^{3} + ( -24 - 25 \beta_{1} + \beta_{2} + \beta_{6} ) q^{5} + ( 76 + 10 \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} ) q^{7} + ( 243 - 243 \beta_{1} ) q^{9} +O(q^{10})\) \( q + ( 18 - 9 \beta_{1} ) q^{3} + ( -24 - 25 \beta_{1} + \beta_{2} + \beta_{6} ) q^{5} + ( 76 + 10 \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} ) q^{7} + ( 243 - 243 \beta_{1} ) q^{9} + ( 2 + 79 \beta_{1} + 9 \beta_{2} + 6 \beta_{3} - \beta_{4} + 3 \beta_{5} + 2 \beta_{6} + 3 \beta_{7} ) q^{11} + ( 289 - 544 \beta_{1} - 6 \beta_{2} + 25 \beta_{3} + 4 \beta_{4} + 5 \beta_{6} - 5 \beta_{7} ) q^{13} + ( -657 - 9 \beta_{1} + 9 \beta_{2} - 9 \beta_{4} + 18 \beta_{6} ) q^{15} + ( -892 + 440 \beta_{1} - 11 \beta_{2} + 24 \beta_{3} - 6 \beta_{4} + 16 \beta_{5} + 12 \beta_{6} - 30 \beta_{7} ) q^{17} + ( 1571 + 1468 \beta_{1} + 21 \beta_{2} - 11 \beta_{3} + 11 \beta_{4} + 8 \beta_{5} + 59 \beta_{6} - 44 \beta_{7} ) q^{19} + ( 1449 - 585 \beta_{1} - 36 \beta_{2} + 9 \beta_{3} - 36 \beta_{5} - 54 \beta_{6} + 9 \beta_{7} ) q^{21} + ( -986 + 1026 \beta_{1} + 64 \beta_{2} - 36 \beta_{3} + 116 \beta_{4} - 95 \beta_{5} - 64 \beta_{6} + 24 \beta_{7} ) q^{23} + ( -58 - 4238 \beta_{1} - 161 \beta_{2} - 30 \beta_{3} - 43 \beta_{4} - 15 \beta_{5} - 58 \beta_{6} - 15 \beta_{7} ) q^{25} + ( 2187 - 4374 \beta_{1} ) q^{27} + ( -989 - 133 \beta_{1} - 186 \beta_{2} - 12 \beta_{3} - 157 \beta_{4} + 343 \beta_{5} + 302 \beta_{6} + 36 \beta_{7} ) q^{29} + ( 14863 - 7424 \beta_{1} + 65 \beta_{2} - 44 \beta_{3} + 4 \beta_{4} - 119 \beta_{5} - 15 \beta_{6} + 55 \beta_{7} ) q^{31} + ( 666 + 747 \beta_{1} + 171 \beta_{2} + 27 \beta_{3} - 27 \beta_{4} - 45 \beta_{5} + 27 \beta_{6} + 108 \beta_{7} ) q^{33} + ( 19340 - 31998 \beta_{1} + 207 \beta_{2} + 45 \beta_{3} + 152 \beta_{4} + 60 \beta_{5} - 35 \beta_{6} - 75 \beta_{7} ) q^{35} + ( 16315 - 16298 \beta_{1} - 113 \beta_{2} + 195 \beta_{3} - 161 \beta_{4} + 219 \beta_{5} + 113 \beta_{6} - 130 \beta_{7} ) q^{37} + ( 126 - 7272 \beta_{1} - 153 \beta_{2} + 270 \beta_{3} - 9 \beta_{4} + 135 \beta_{5} + 126 \beta_{6} + 135 \beta_{7} ) q^{39} + ( -8750 + 16524 \beta_{1} + 102 \beta_{2} - 540 \beta_{3} - 328 \beta_{4} + 214 \beta_{5} - 108 \beta_{6} + 108 \beta_{7} ) q^{41} + ( -5297 - 376 \beta_{1} + 365 \beta_{2} + 155 \beta_{3} - 66 \beta_{4} - 299 \beta_{5} + 287 \beta_{6} - 465 \beta_{7} ) q^{43} + ( -11907 + 5832 \beta_{1} - 243 \beta_{4} + 243 \beta_{6} ) q^{45} + ( -39680 - 40866 \beta_{1} + 312 \beta_{2} - 24 \beta_{3} + 24 \beta_{4} + 365 \beta_{5} + 1090 \beta_{6} - 96 \beta_{7} ) q^{47} + ( -65748 + 54327 \beta_{1} - 522 \beta_{2} + 591 \beta_{3} - 176 \beta_{4} - 424 \beta_{5} + 285 \beta_{6} + 111 \beta_{7} ) q^{49} + ( -12042 + 12204 \beta_{1} - 162 \beta_{2} + 486 \beta_{3} - 162 \beta_{4} + 297 \beta_{5} + 162 \beta_{6} - 324 \beta_{7} ) q^{51} + ( -295 - 134988 \beta_{1} + 194 \beta_{2} + 432 \beta_{3} - 511 \beta_{4} + 216 \beta_{5} - 295 \beta_{6} + 216 \beta_{7} ) q^{53} + ( -43525 + 85109 \beta_{1} - 906 \beta_{2} - 1150 \beta_{3} - 561 \beta_{4} - 805 \beta_{5} - 230 \beta_{6} + 230 \beta_{7} ) q^{55} + ( 41985 - 1026 \beta_{1} - 81 \beta_{2} + 297 \beta_{3} - 432 \beta_{4} + 513 \beta_{5} + 1161 \beta_{6} - 891 \beta_{7} ) q^{57} + ( 29350 - 15553 \beta_{1} - 848 \beta_{2} - 756 \beta_{3} - 1945 \beta_{4} + 1885 \beta_{5} + 1756 \beta_{6} + 945 \beta_{7} ) q^{59} + ( 36044 + 33660 \beta_{1} - 2936 \beta_{2} - 180 \beta_{3} + 180 \beta_{4} + 2120 \beta_{5} + 1664 \beta_{6} - 720 \beta_{7} ) q^{61} + ( 20655 - 18225 \beta_{1} - 486 \beta_{2} + 486 \beta_{4} - 486 \beta_{5} - 972 \beta_{6} + 243 \beta_{7} ) q^{63} + ( -105104 + 104038 \beta_{1} - 46 \beta_{2} - 1530 \beta_{3} - 602 \beta_{4} - 2475 \beta_{5} + 46 \beta_{6} + 1020 \beta_{7} ) q^{65} + ( -508 + 290248 \beta_{1} - 2246 \beta_{2} + 1074 \beta_{3} - 1045 \beta_{4} + 537 \beta_{5} - 508 \beta_{6} + 537 \beta_{7} ) q^{67} + ( -8406 + 17784 \beta_{1} + 873 \beta_{2} - 540 \beta_{3} + 1620 \beta_{4} - 963 \beta_{5} - 108 \beta_{6} + 108 \beta_{7} ) q^{69} + ( -178210 - 1610 \beta_{1} + 3555 \beta_{2} + 852 \beta_{3} + 94 \beta_{4} - 3649 \beta_{5} + 664 \beta_{6} - 2556 \beta_{7} ) q^{71} + ( -60706 + 29818 \beta_{1} + 1689 \beta_{2} + 1924 \beta_{3} - 589 \beta_{4} - 3859 \beta_{5} + 1070 \beta_{6} - 2405 \beta_{7} ) q^{73} + ( -38781 - 37890 \beta_{1} - 2511 \beta_{2} - 135 \beta_{3} + 135 \beta_{4} + 405 \beta_{5} - 1431 \beta_{6} - 540 \beta_{7} ) q^{75} + ( 126646 + 18339 \beta_{1} - 175 \beta_{2} + 3744 \beta_{3} + 144 \beta_{4} + 1883 \beta_{5} - 163 \beta_{6} - 2652 \beta_{7} ) q^{77} + ( -153689 + 151305 \beta_{1} - 2750 \beta_{2} + 549 \beta_{3} - 5317 \beta_{4} + 1911 \beta_{5} + 2750 \beta_{6} - 366 \beta_{7} ) q^{79} -59049 \beta_{1} q^{81} + ( 100468 - 199920 \beta_{1} - 2829 \beta_{2} + 1365 \beta_{3} - 622 \beta_{4} - 1661 \beta_{5} + 273 \beta_{6} - 273 \beta_{7} ) q^{83} + ( -36292 + 1196 \beta_{1} + 1894 \beta_{2} - 870 \beta_{3} - 544 \beta_{4} - 1350 \beta_{5} + 218 \beta_{6} + 2610 \beta_{7} ) q^{85} + ( -19215 + 7596 \beta_{1} - 2979 \beta_{2} - 432 \beta_{3} - 4131 \beta_{4} + 6066 \beta_{5} + 4023 \beta_{6} + 540 \beta_{7} ) q^{87} + ( -82748 - 82118 \beta_{1} - 874 \beta_{2} - 48 \beta_{3} + 48 \beta_{4} - 22 \beta_{5} - 822 \beta_{6} - 192 \beta_{7} ) q^{89} + ( 60863 - 193672 \beta_{1} - 12573 \beta_{2} + 5491 \beta_{3} - 3903 \beta_{4} + 10702 \beta_{5} + 2153 \beta_{6} - 3002 \beta_{7} ) q^{91} + ( 200619 - 200979 \beta_{1} + 234 \beta_{2} - 891 \beta_{3} + 171 \beta_{4} - 1755 \beta_{5} - 234 \beta_{6} + 594 \beta_{7} ) q^{93} + ( 2462 + 147302 \beta_{1} + 3349 \beta_{2} + 2340 \beta_{3} + 1292 \beta_{4} + 1170 \beta_{5} + 2462 \beta_{6} + 1170 \beta_{7} ) q^{95} + ( -125692 + 247360 \beta_{1} + 1674 \beta_{2} + 5 \beta_{3} - 4030 \beta_{4} + 5706 \beta_{5} + \beta_{6} - \beta_{7} ) q^{97} + ( 17496 + 972 \beta_{1} + 2430 \beta_{2} - 729 \beta_{3} - 486 \beta_{4} - 1944 \beta_{5} + 243 \beta_{6} + 2187 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 108 q^{3} - 294 q^{5} + 656 q^{7} + 972 q^{9} + O(q^{10}) \) \( 8 q + 108 q^{3} - 294 q^{5} + 656 q^{7} + 972 q^{9} + 314 q^{11} - 5292 q^{15} - 5532 q^{17} + 18234 q^{19} + 9342 q^{21} - 3928 q^{23} - 17038 q^{25} - 8300 q^{29} + 89508 q^{31} + 8478 q^{33} + 25860 q^{35} + 64706 q^{37} - 29106 q^{39} - 45740 q^{43} - 71442 q^{45} - 483276 q^{47} - 310684 q^{49} - 49788 q^{51} - 540974 q^{53} + 328212 q^{57} + 181770 q^{59} + 418224 q^{61} + 92826 q^{63} - 414204 q^{65} + 1158902 q^{67} - 1442344 q^{71} - 378666 q^{73} - 460026 q^{75} + 1065994 q^{77} - 611452 q^{79} - 236196 q^{81} - 275112 q^{85} - 112050 q^{87} - 989196 q^{89} - 304446 q^{91} + 805572 q^{93} + 591792 q^{95} + 152604 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - x^{7} + 212 x^{6} + 473 x^{5} + 39800 x^{4} + 36821 x^{3} + 985651 x^{2} - 601290 x + 21068100\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(405518177 \nu^{7} + 12595814413 \nu^{6} + 65484675874 \nu^{5} + 2673339327091 \nu^{4} + 20860832470090 \nu^{3} + 529666993003657 \nu^{2} + 423382833624317 \nu + 12557309751263700\)\()/ 11540026222043130 \)
\(\beta_{2}\)\(=\)\((\)\(-405518177 \nu^{7} - 12595814413 \nu^{6} - 65484675874 \nu^{5} - 2673339327091 \nu^{4} - 20860832470090 \nu^{3} - 529666993003657 \nu^{2} + 45736722054548203 \nu - 12557309751263700\)\()/ 5770013111021565 \)
\(\beta_{3}\)\(=\)\((\)\(364310749 \nu^{7} + 24961807811 \nu^{6} + 79053499373 \nu^{5} + 5590904486327 \nu^{4} + 32457316245365 \nu^{3} + 931390008804239 \nu^{2} + 2868690525093844 \nu + 14323863372251580\)\()/ 157007159483580 \)
\(\beta_{4}\)\(=\)\((\)\(-23765159851 \nu^{7} + 369812106601 \nu^{6} - 8515763214551 \nu^{5} + 65746538778493 \nu^{4} - 1215798353693087 \nu^{3} + 9673046705357005 \nu^{2} - 104833713051471712 \nu + 161517049672316796\)\()/ 4616010488817252 \)
\(\beta_{5}\)\(=\)\((\)\(51599812183 \nu^{7} - 94536525013 \nu^{6} + 9860632241606 \nu^{5} + 16211496774869 \nu^{4} + 2038080800363270 \nu^{3} - 434933245219297 \nu^{2} + 50781192270023803 \nu - 35191273548635370\)\()/ 5770013111021565 \)
\(\beta_{6}\)\(=\)\((\)\(323868198161 \nu^{7} - 69488833736 \nu^{6} + 51830673245317 \nu^{5} + 311126494638088 \nu^{4} + 9026268210443605 \nu^{3} + 26467592670891616 \nu^{2} - 316074427247336044 \nu + 943466352741067860\)\()/ 23080052444086260 \)
\(\beta_{7}\)\(=\)\((\)\(-357421335253 \nu^{7} + 1466129677768 \nu^{6} - 71066142961631 \nu^{5} + 198818924395276 \nu^{4} - 12879973820900975 \nu^{3} + 52875146540226352 \nu^{2} - 12177112382292298 \nu + 3294721428011371440\)\()/ 23080052444086260 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + 2 \beta_{1}\)\()/8\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{7} + 2 \beta_{5} - \beta_{4} - 3 \beta_{3} + 424 \beta_{1} - 426\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(36 \beta_{7} - 52 \beta_{6} + 157 \beta_{5} + 20 \beta_{4} - 12 \beta_{3} - 177 \beta_{2} + 44 \beta_{1} - 2094\)\()/8\)
\(\nu^{4}\)\(=\)\((\)\(217 \beta_{7} + 201 \beta_{6} + 217 \beta_{5} - 16 \beta_{4} + 434 \beta_{3} - 592 \beta_{2} - 72043 \beta_{1} + 201\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(-6456 \beta_{7} + 6784 \beta_{6} - 27649 \beta_{5} - 10340 \beta_{4} + 9684 \beta_{3} - 6784 \beta_{2} - 685762 \beta_{1} + 685434\)\()/8\)
\(\nu^{6}\)\(=\)\((\)\(-130791 \beta_{7} - 25869 \beta_{6} - 175899 \beta_{5} + 34733 \beta_{4} + 43597 \beta_{3} + 141166 \beta_{2} - 52461 \beta_{1} + 13826925\)\()/4\)
\(\nu^{7}\)\(=\)\((\)\(-770076 \beta_{7} + 536388 \beta_{6} - 770076 \beta_{5} + 1306464 \beta_{4} - 1540152 \beta_{3} + 8192641 \beta_{2} + 179565830 \beta_{1} + 536388\)\()/8\)

Character values

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

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

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
−6.30797 10.9257i
2.26350 + 3.92050i
7.29767 + 12.6399i
−2.75320 4.76869i
−6.30797 + 10.9257i
2.26350 3.92050i
7.29767 12.6399i
−2.75320 + 4.76869i
0 13.5000 7.79423i 0 −165.302 95.4373i 0 103.254 + 327.089i 0 121.500 210.444i 0
145.2 0 13.5000 7.79423i 0 −57.9943 33.4830i 0 240.457 + 244.600i 0 121.500 210.444i 0
145.3 0 13.5000 7.79423i 0 22.3721 + 12.9165i 0 203.121 276.389i 0 121.500 210.444i 0
145.4 0 13.5000 7.79423i 0 53.9244 + 31.1333i 0 −218.833 264.124i 0 121.500 210.444i 0
241.1 0 13.5000 + 7.79423i 0 −165.302 + 95.4373i 0 103.254 327.089i 0 121.500 + 210.444i 0
241.2 0 13.5000 + 7.79423i 0 −57.9943 + 33.4830i 0 240.457 244.600i 0 121.500 + 210.444i 0
241.3 0 13.5000 + 7.79423i 0 22.3721 12.9165i 0 203.121 + 276.389i 0 121.500 + 210.444i 0
241.4 0 13.5000 + 7.79423i 0 53.9244 31.1333i 0 −218.833 + 264.124i 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.d 8
4.b odd 2 1 21.7.f.a 8
7.d odd 6 1 inner 336.7.bh.d 8
12.b even 2 1 63.7.m.d 8
28.d even 2 1 147.7.f.d 8
28.f even 6 1 21.7.f.a 8
28.f even 6 1 147.7.d.b 8
28.g odd 6 1 147.7.d.b 8
28.g odd 6 1 147.7.f.d 8
84.j odd 6 1 63.7.m.d 8
84.j odd 6 1 441.7.d.c 8
84.n even 6 1 441.7.d.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.7.f.a 8 4.b odd 2 1
21.7.f.a 8 28.f even 6 1
63.7.m.d 8 12.b even 2 1
63.7.m.d 8 84.j odd 6 1
147.7.d.b 8 28.f even 6 1
147.7.d.b 8 28.g odd 6 1
147.7.f.d 8 28.d even 2 1
147.7.f.d 8 28.g odd 6 1
336.7.bh.d 8 1.a even 1 1 trivial
336.7.bh.d 8 7.d odd 6 1 inner
441.7.d.c 8 84.j odd 6 1
441.7.d.c 8 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(25\!\cdots\!00\)\( T_{5} + \)\(42\!\cdots\!00\)\( \)">\(T_{5}^{8} + \cdots\) acting on \(S_{7}^{\mathrm{new}}(336, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( ( 243 - 27 T + T^{2} )^{4} \)
$5$ \( 422734160250000 - 25332592050000 T + 334857307500 T^{2} + 10257232500 T^{3} - 72000675 T^{4} - 2447550 T^{5} + 20487 T^{6} + 294 T^{7} + T^{8} \)
$7$ \( \)\(19\!\cdots\!01\)\( - 1068239320229254544 T + 5128335320842510 T^{2} - 12493153898344 T^{3} + 41840034887 T^{4} - 106190056 T^{5} + 370510 T^{6} - 656 T^{7} + T^{8} \)
$11$ \( \)\(18\!\cdots\!44\)\( - 98293722790383421872 T + 5063514049337341372 T^{2} - 8590189886857868 T^{3} + 14795841919805 T^{4} - 3395131754 T^{5} + 3844943 T^{6} - 314 T^{7} + T^{8} \)
$13$ \( \)\(21\!\cdots\!56\)\( + \)\(14\!\cdots\!08\)\( T^{2} + 331924469476329 T^{4} + 30553062 T^{6} + T^{8} \)
$17$ \( \)\(14\!\cdots\!04\)\( + \)\(32\!\cdots\!48\)\( T - 18954316226540153088 T^{2} - 49044433474028160 T^{3} + 325284835782096 T^{4} - 99181347120 T^{5} - 7727652 T^{6} + 5532 T^{7} + T^{8} \)
$19$ \( \)\(40\!\cdots\!56\)\( - \)\(29\!\cdots\!56\)\( T + \)\(80\!\cdots\!44\)\( T^{2} - 80019415606831775892 T^{3} - 3659130219753963 T^{4} + 1012664156058 T^{5} + 55289115 T^{6} - 18234 T^{7} + T^{8} \)
$23$ \( \)\(15\!\cdots\!56\)\( + \)\(35\!\cdots\!28\)\( T + \)\(83\!\cdots\!96\)\( T^{2} - 12921467685980241920 T^{3} + 20933343724396544 T^{4} - 749772681728 T^{5} + 161320832 T^{6} + 3928 T^{7} + T^{8} \)
$29$ \( ( -131918946476762880 - 26271861557728 T - 1346898667 T^{2} + 4150 T^{3} + T^{4} )^{2} \)
$31$ \( \)\(16\!\cdots\!01\)\( + \)\(45\!\cdots\!96\)\( T - \)\(58\!\cdots\!94\)\( T^{2} - \)\(26\!\cdots\!36\)\( T^{3} + 706793352941013399 T^{4} - 68718078342072 T^{5} + 3438291822 T^{6} - 89508 T^{7} + T^{8} \)
$37$ \( \)\(23\!\cdots\!36\)\( - \)\(16\!\cdots\!60\)\( T + \)\(13\!\cdots\!00\)\( T^{2} - \)\(47\!\cdots\!28\)\( T^{3} + 2903640625281916109 T^{4} - 97788930340930 T^{5} + 3733577311 T^{6} - 64706 T^{7} + T^{8} \)
$41$ \( \)\(36\!\cdots\!44\)\( + \)\(43\!\cdots\!00\)\( T^{2} + \)\(15\!\cdots\!32\)\( T^{4} + 21593029296 T^{6} + T^{8} \)
$43$ \( ( -293597400770976796 - 85125133057276 T - 4841210811 T^{2} + 22870 T^{3} + T^{4} )^{2} \)
$47$ \( \)\(99\!\cdots\!96\)\( + \)\(26\!\cdots\!80\)\( T + \)\(10\!\cdots\!60\)\( T^{2} - \)\(36\!\cdots\!00\)\( T^{3} - \)\(15\!\cdots\!24\)\( T^{4} + 6508813429466640 T^{5} + 91320005532 T^{6} + 483276 T^{7} + T^{8} \)
$53$ \( \)\(73\!\cdots\!64\)\( - \)\(30\!\cdots\!60\)\( T + \)\(20\!\cdots\!00\)\( T^{2} + \)\(41\!\cdots\!16\)\( T^{3} + \)\(60\!\cdots\!97\)\( T^{4} + 40803243733591190 T^{5} + 203978340551 T^{6} + 540974 T^{7} + T^{8} \)
$59$ \( \)\(20\!\cdots\!00\)\( - \)\(67\!\cdots\!60\)\( T - \)\(32\!\cdots\!28\)\( T^{2} + \)\(11\!\cdots\!68\)\( T^{3} + \)\(51\!\cdots\!49\)\( T^{4} + 13446651706756290 T^{5} - 62962743777 T^{6} - 181770 T^{7} + T^{8} \)
$61$ \( \)\(68\!\cdots\!00\)\( + \)\(30\!\cdots\!00\)\( T - \)\(93\!\cdots\!00\)\( T^{2} - \)\(63\!\cdots\!00\)\( T^{3} + \)\(15\!\cdots\!00\)\( T^{4} + 71069068838315520 T^{5} - 111626861088 T^{6} - 418224 T^{7} + T^{8} \)
$67$ \( \)\(29\!\cdots\!04\)\( - \)\(10\!\cdots\!28\)\( T + \)\(26\!\cdots\!52\)\( T^{2} - \)\(21\!\cdots\!00\)\( T^{3} + \)\(12\!\cdots\!29\)\( T^{4} - 387875645278331366 T^{5} + 908045258107 T^{6} - 1158902 T^{7} + T^{8} \)
$71$ \( ( -\)\(19\!\cdots\!12\)\( - 115286062429846880 T - 36558200716 T^{2} + 721172 T^{3} + T^{4} )^{2} \)
$73$ \( \)\(77\!\cdots\!24\)\( + \)\(13\!\cdots\!92\)\( T + \)\(10\!\cdots\!44\)\( T^{2} + \)\(36\!\cdots\!56\)\( T^{3} + \)\(43\!\cdots\!01\)\( T^{4} - 88270688635072434 T^{5} - 185313643497 T^{6} + 378666 T^{7} + T^{8} \)
$79$ \( \)\(99\!\cdots\!01\)\( + \)\(96\!\cdots\!32\)\( T + \)\(92\!\cdots\!06\)\( T^{2} + \)\(15\!\cdots\!80\)\( T^{3} + \)\(81\!\cdots\!39\)\( T^{4} + 103324245868979728 T^{5} + 521211945622 T^{6} + 611452 T^{7} + T^{8} \)
$83$ \( \)\(52\!\cdots\!24\)\( + \)\(31\!\cdots\!16\)\( T^{2} + \)\(85\!\cdots\!29\)\( T^{4} + 524747194014 T^{6} + T^{8} \)
$89$ \( \)\(87\!\cdots\!56\)\( - \)\(92\!\cdots\!00\)\( T + \)\(22\!\cdots\!76\)\( T^{2} + \)\(10\!\cdots\!00\)\( T^{3} + \)\(14\!\cdots\!80\)\( T^{4} + 103856130641901456 T^{5} + 431160022908 T^{6} + 989196 T^{7} + T^{8} \)
$97$ \( \)\(90\!\cdots\!76\)\( + \)\(26\!\cdots\!08\)\( T^{2} + \)\(12\!\cdots\!73\)\( T^{4} + 2056709392566 T^{6} + T^{8} \)
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