Properties

Label 63.6.e.e
Level $63$
Weight $6$
Character orbit 63.e
Analytic conductor $10.104$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 63.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.1041806482\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - x^{7} + 98 x^{6} + 83 x^{5} + 9122 x^{4} - 91 x^{3} + 28567 x^{2} + 2058 x + 86436\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 21)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( 1 - \beta_{1} + \beta_{2} ) q^{2} + ( -\beta_{1} + 18 \beta_{2} + \beta_{3} + \beta_{6} ) q^{4} + ( -2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{5} + ( 53 - 2 \beta_{1} + 45 \beta_{2} + \beta_{4} + \beta_{5} + 6 \beta_{6} + \beta_{7} ) q^{7} + ( -37 - \beta_{4} - 2 \beta_{5} + 27 \beta_{6} + 2 \beta_{7} ) q^{8} +O(q^{10})\) \( q + ( 1 - \beta_{1} + \beta_{2} ) q^{2} + ( -\beta_{1} + 18 \beta_{2} + \beta_{3} + \beta_{6} ) q^{4} + ( -2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{5} + ( 53 - 2 \beta_{1} + 45 \beta_{2} + \beta_{4} + \beta_{5} + 6 \beta_{6} + \beta_{7} ) q^{7} + ( -37 - \beta_{4} - 2 \beta_{5} + 27 \beta_{6} + 2 \beta_{7} ) q^{8} + ( -23 \beta_{1} + 76 \beta_{2} - \beta_{3} + 23 \beta_{6} ) q^{10} + ( 8 \beta_{1} - 107 \beta_{2} - 9 \beta_{3} - 8 \beta_{6} + 3 \beta_{7} ) q^{11} + ( 97 - \beta_{4} - 9 \beta_{5} + 76 \beta_{6} + 9 \beta_{7} ) q^{13} + ( -299 - 114 \beta_{1} - 113 \beta_{2} - \beta_{3} - 9 \beta_{4} + 83 \beta_{6} + 4 \beta_{7} ) q^{14} + ( -839 + 85 \beta_{1} - 840 \beta_{2} - \beta_{3} - \beta_{4} - 6 \beta_{5} ) q^{16} + ( 108 \beta_{1} - 92 \beta_{2} + 8 \beta_{3} - 108 \beta_{6} + 4 \beta_{7} ) q^{17} + ( -73 - 220 \beta_{1} - 72 \beta_{2} + \beta_{3} + \beta_{4} - 15 \beta_{5} ) q^{19} + ( -1177 + 9 \beta_{4} - 30 \beta_{5} - 29 \beta_{6} + 30 \beta_{7} ) q^{20} + ( 449 - \beta_{4} + 12 \beta_{5} - 419 \beta_{6} - 12 \beta_{7} ) q^{22} + ( 1796 - 300 \beta_{1} + 1804 \beta_{2} + 8 \beta_{3} + 8 \beta_{4} - 20 \beta_{5} ) q^{23} + ( 80 \beta_{1} + 636 \beta_{2} - 95 \beta_{3} - 80 \beta_{6} + 45 \beta_{7} ) q^{25} + ( -3762 + 116 \beta_{1} - 3865 \beta_{2} - 103 \beta_{3} - 103 \beta_{4} - 20 \beta_{5} ) q^{26} + ( -2855 + 485 \beta_{1} + 896 \beta_{2} - \beta_{3} - 95 \beta_{4} + 40 \beta_{5} + 50 \beta_{6} - 26 \beta_{7} ) q^{28} + ( -250 + 103 \beta_{4} + 5 \beta_{5} + 254 \beta_{6} - 5 \beta_{7} ) q^{29} + ( -130 \beta_{1} - 1523 \beta_{2} + 94 \beta_{3} + 130 \beta_{6} - 24 \beta_{7} ) q^{31} + ( 131 \beta_{1} - 3948 \beta_{2} - 71 \beta_{3} - 131 \beta_{6} + 50 \beta_{7} ) q^{32} + ( 5160 + 96 \beta_{4} - 24 \beta_{5} + 312 \beta_{6} + 24 \beta_{7} ) q^{34} + ( 3240 + 24 \beta_{1} - 1706 \beta_{2} + 104 \beta_{3} - 71 \beta_{4} + 29 \beta_{5} + 208 \beta_{6} - 37 \beta_{7} ) q^{35} + ( -3603 - 1028 \beta_{1} - 3508 \beta_{2} + 95 \beta_{3} + 95 \beta_{4} - 9 \beta_{5} ) q^{37} + ( 316 \beta_{1} + 10321 \beta_{2} + 175 \beta_{3} - 316 \beta_{6} - 28 \beta_{7} ) q^{38} + ( 2025 + 633 \beta_{1} + 1932 \beta_{2} - 93 \beta_{3} - 93 \beta_{4} - 42 \beta_{5} ) q^{40} + ( -1190 + 62 \beta_{4} + 142 \beta_{5} + 328 \beta_{6} - 142 \beta_{7} ) q^{41} + ( 7049 + 93 \beta_{4} + 33 \beta_{5} + 816 \beta_{6} - 33 \beta_{7} ) q^{43} + ( 17697 - 379 \beta_{1} + 17864 \beta_{2} + 167 \beta_{3} + 167 \beta_{4} + 118 \beta_{5} ) q^{44} + ( -1752 \beta_{1} + 16008 \beta_{2} + 240 \beta_{3} + 1752 \beta_{6} - 24 \beta_{7} ) q^{46} + ( -3874 + 324 \beta_{1} - 3818 \beta_{2} + 56 \beta_{3} + 56 \beta_{4} + 28 \beta_{5} ) q^{47} + ( -5892 - 152 \beta_{1} + 5315 \beta_{2} + 94 \beta_{3} + 239 \beta_{4} - 61 \beta_{5} + 1576 \beta_{6} + 11 \beta_{7} ) q^{49} + ( 2454 - 55 \beta_{4} + 100 \beta_{5} - 2404 \beta_{6} - 100 \beta_{7} ) q^{50} + ( 6036 \beta_{1} - 13450 \beta_{2} - 144 \beta_{3} - 6036 \beta_{6} + 42 \beta_{7} ) q^{52} + ( -1338 \beta_{1} - 3095 \beta_{2} - 13 \beta_{3} + 1338 \beta_{6} - 239 \beta_{7} ) q^{53} + ( 18322 - 335 \beta_{4} + 315 \beta_{5} + 506 \beta_{6} - 315 \beta_{7} ) q^{55} + ( -9514 + 3429 \beta_{1} - 22170 \beta_{2} - 159 \beta_{3} - 4 \beta_{4} - 8 \beta_{5} + 129 \beta_{6} + 26 \beta_{7} ) q^{56} + ( -11915 - 4171 \beta_{1} - 12154 \beta_{2} - 239 \beta_{3} - 239 \beta_{4} + 216 \beta_{5} ) q^{58} + ( 888 \beta_{1} - 9011 \beta_{2} - 163 \beta_{3} - 888 \beta_{6} - 71 \beta_{7} ) q^{59} + ( 1514 - 796 \beta_{1} + 1566 \beta_{2} + 52 \beta_{3} + 52 \beta_{4} + 240 \beta_{5} ) q^{61} + ( -4447 - 58 \beta_{4} - 140 \beta_{5} + 1875 \beta_{6} + 140 \beta_{7} ) q^{62} + ( -17549 - 51 \beta_{4} - 150 \beta_{5} - 3189 \beta_{6} + 150 \beta_{7} ) q^{64} + ( 16466 + 1660 \beta_{1} + 16136 \beta_{2} - 330 \beta_{3} - 330 \beta_{4} - 246 \beta_{5} ) q^{65} + ( 2764 \beta_{1} - 2286 \beta_{2} - 193 \beta_{3} - 2764 \beta_{6} - 465 \beta_{7} ) q^{67} + ( -13184 - 5280 \beta_{1} - 13312 \beta_{2} - 128 \beta_{3} - 128 \beta_{4} + 272 \beta_{5} ) q^{68} + ( -4783 - 809 \beta_{1} - 7658 \beta_{2} - 145 \beta_{3} - 97 \beta_{4} - 276 \beta_{5} + 2406 \beta_{6} + 192 \beta_{7} ) q^{70} + ( -21414 + 24 \beta_{4} - 180 \beta_{5} - 3660 \beta_{6} + 180 \beta_{7} ) q^{71} + ( 3056 \beta_{1} + 14074 \beta_{2} - 143 \beta_{3} - 3056 \beta_{6} + 93 \beta_{7} ) q^{73} + ( -216 \beta_{1} + 46915 \beta_{2} + 1001 \beta_{3} + 216 \beta_{6} + 172 \beta_{7} ) q^{74} + ( 2630 + 432 \beta_{4} - 774 \beta_{5} + 9924 \beta_{6} + 774 \beta_{7} ) q^{76} + ( 1232 - 3730 \beta_{1} - 16115 \beta_{2} + 183 \beta_{3} + 1063 \beta_{4} - 367 \beta_{5} + 228 \beta_{6} + 196 \beta_{7} ) q^{77} + ( 11539 - 2286 \beta_{1} + 11635 \beta_{2} + 96 \beta_{3} + 96 \beta_{4} - 786 \beta_{5} ) q^{79} + ( 3607 \beta_{1} + 6920 \beta_{2} - 1047 \beta_{3} - 3607 \beta_{6} + 690 \beta_{7} ) q^{80} + ( -13420 - 4112 \beta_{1} - 13322 \beta_{2} + 98 \beta_{3} + 98 \beta_{4} + 408 \beta_{5} ) q^{82} + ( 51006 - 1005 \beta_{4} - 129 \beta_{5} + 6432 \beta_{6} + 129 \beta_{7} ) q^{83} + ( 9924 - 192 \beta_{4} - 372 \beta_{5} - 2988 \beta_{6} + 372 \beta_{7} ) q^{85} + ( -30988 - 11582 \beta_{1} - 31705 \beta_{2} - 717 \beta_{3} - 717 \beta_{4} + 252 \beta_{5} ) q^{86} + ( -13545 \beta_{1} + 25668 \beta_{2} + 765 \beta_{3} + 13545 \beta_{6} + 186 \beta_{7} ) q^{88} + ( -22548 + 9508 \beta_{1} - 20966 \beta_{2} + 1582 \beta_{3} + 1582 \beta_{4} - 622 \beta_{5} ) q^{89} + ( -25799 + 9748 \beta_{1} - 73960 \beta_{2} - 47 \beta_{3} + 431 \beta_{4} + 203 \beta_{5} + 1948 \beta_{6} + 230 \beta_{7} ) q^{91} + ( -42800 - 1424 \beta_{4} - 1072 \beta_{5} + 15792 \beta_{6} + 1072 \beta_{7} ) q^{92} + ( 990 \beta_{1} - 18798 \beta_{2} - 240 \beta_{3} - 990 \beta_{6} + 168 \beta_{7} ) q^{94} + ( -3820 \beta_{1} + 55528 \beta_{2} - 1542 \beta_{3} + 3820 \beta_{6} + 294 \beta_{7} ) q^{95} + ( -46842 - 863 \beta_{4} + 669 \beta_{5} + 464 \beta_{6} - 669 \beta_{7} ) q^{97} + ( -86434 - 2987 \beta_{1} - 76298 \beta_{2} - 1607 \beta_{3} - 1609 \beta_{4} + 268 \beta_{5} + 8219 \beta_{6} + 88 \beta_{7} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 3q^{2} - 69q^{4} + 258q^{7} - 246q^{8} + O(q^{10}) \) \( 8q + 3q^{2} - 69q^{4} + 258q^{7} - 246q^{8} - 283q^{10} + 402q^{11} + 924q^{13} - 1926q^{14} - 3273q^{16} + 276q^{17} - 510q^{19} - 9438q^{20} + 2750q^{22} + 6900q^{23} - 2814q^{25} - 15138q^{26} - 26221q^{28} - 1080q^{29} + 6410q^{31} + 15519q^{32} + 42288q^{34} + 33108q^{35} - 15250q^{37} - 41250q^{38} + 8547q^{40} - 8616q^{41} + 58396q^{43} + 70743q^{44} - 61800q^{46} - 15060q^{47} - 64252q^{49} + 14604q^{50} + 47476q^{52} + 13692q^{53} + 146248q^{55} + 15921q^{56} - 52309q^{58} + 34830q^{59} + 5364q^{61} - 32058q^{62} - 146974q^{64} + 66864q^{65} + 5994q^{67} - 58272q^{68} - 4307q^{70} - 178536q^{71} - 59638q^{73} - 185442q^{74} + 42616q^{76} + 75660q^{77} + 44062q^{79} - 33381q^{80} - 57596q^{82} + 416892q^{83} + 72648q^{85} - 136968q^{86} - 87597q^{88} - 77520q^{89} + 104722q^{91} - 316512q^{92} + 73722q^{94} - 221376q^{95} - 377260q^{97} - 382479q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - x^{7} + 98 x^{6} + 83 x^{5} + 9122 x^{4} - 91 x^{3} + 28567 x^{2} + 2058 x + 86436\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -8905 \nu^{7} + 366059 \nu^{6} - 1191820 \nu^{5} + 33153695 \nu^{4} - 47979268 \nu^{3} + 3262309295 \nu^{2} - 141627885 \nu + 307508418 \)\()/ 9888988410 \)
\(\beta_{3}\)\(=\)\((\)\( 1257 \nu^{7} - 279647 \nu^{6} + 388990 \nu^{5} - 26314019 \nu^{4} - 14452616 \nu^{3} - 2382173465 \nu^{2} - 51985031 \nu - 251390874 \)\()/ 156968070 \)
\(\beta_{4}\)\(=\)\((\)\( 22795 \nu^{7} + 56368 \nu^{6} + 2163175 \nu^{5} + 2122285 \nu^{4} + 222588334 \nu^{3} + 7196875 \nu^{2} + 20796090 \nu - 31645054854 \)\()/ 706356315 \)
\(\beta_{5}\)\(=\)\((\)\( -341623 \nu^{7} - 8468248 \nu^{6} + 27571040 \nu^{5} - 966332554 \nu^{4} + 1109931296 \nu^{3} - 75468829240 \nu^{2} + 428233011339 \nu - 235881275616 \)\()/ 9888988410 \)
\(\beta_{6}\)\(=\)\((\)\( -25511 \nu^{7} + 22795 \nu^{6} - 2420915 \nu^{5} - 2375153 \nu^{4} - 232964210 \nu^{3} - 8054375 \nu^{2} - 23273922 \nu - 54979470 \)\()/ 706356315 \)
\(\beta_{7}\)\(=\)\((\)\(15589754 \nu^{7} - 23777815 \nu^{6} + 1539409445 \nu^{5} + 516927917 \nu^{4} + 141447402185 \nu^{3} - 70438948615 \nu^{2} + 442767354393 \nu + 24613669710\)\()/ 9888988410 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{6} + \beta_{3} + 49 \beta_{2} + \beta_{1}\)
\(\nu^{3}\)\(=\)\(-2 \beta_{7} - 91 \beta_{6} + 2 \beta_{5} - 2 \beta_{4} - 44\)
\(\nu^{4}\)\(=\)\(2 \beta_{5} - 99 \beta_{4} - 99 \beta_{3} - 4505 \beta_{2} - 181 \beta_{1} - 4406\)
\(\nu^{5}\)\(=\)\(196 \beta_{7} + 8707 \beta_{6} - 286 \beta_{3} - 8624 \beta_{2} - 8707 \beta_{1}\)
\(\nu^{6}\)\(=\)\(376 \beta_{7} + 25333 \beta_{6} - 376 \beta_{5} + 9581 \beta_{4} + 421300\)
\(\nu^{7}\)\(=\)\(-18786 \beta_{5} + 36042 \beta_{4} + 36042 \beta_{3} + 1222350 \beta_{2} + 841891 \beta_{1} + 1186308\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/63\mathbb{Z}\right)^\times\).

\(n\) \(10\) \(29\)
\(\chi(n)\) \(-1 - \beta_{2}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
5.09061 8.81720i
0.895402 1.55088i
−0.874091 + 1.51397i
−4.61193 + 7.98809i
5.09061 + 8.81720i
0.895402 + 1.55088i
−0.874091 1.51397i
−4.61193 7.98809i
−4.59061 + 7.95118i 0 −26.1475 45.2888i 11.0358 19.1146i 0 126.882 26.6059i 186.333 0 101.322 + 175.495i
37.2 −0.395402 + 0.684857i 0 15.6873 + 27.1712i −52.0958 + 90.2327i 0 −7.12980 129.446i −50.1170 0 −41.1977 71.3564i
37.3 1.37409 2.37999i 0 12.2237 + 21.1722i 29.1836 50.5475i 0 −21.4366 + 127.857i 155.128 0 −80.2019 138.914i
37.4 5.11193 8.85412i 0 −36.2636 62.8104i 11.8764 20.5705i 0 30.6840 125.958i −414.344 0 −121.423 210.310i
46.1 −4.59061 7.95118i 0 −26.1475 + 45.2888i 11.0358 + 19.1146i 0 126.882 + 26.6059i 186.333 0 101.322 175.495i
46.2 −0.395402 0.684857i 0 15.6873 27.1712i −52.0958 90.2327i 0 −7.12980 + 129.446i −50.1170 0 −41.1977 + 71.3564i
46.3 1.37409 + 2.37999i 0 12.2237 21.1722i 29.1836 + 50.5475i 0 −21.4366 127.857i 155.128 0 −80.2019 + 138.914i
46.4 5.11193 + 8.85412i 0 −36.2636 + 62.8104i 11.8764 + 20.5705i 0 30.6840 + 125.958i −414.344 0 −121.423 + 210.310i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 46.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 63.6.e.e 8
3.b odd 2 1 21.6.e.c 8
7.c even 3 1 inner 63.6.e.e 8
7.c even 3 1 441.6.a.w 4
7.d odd 6 1 441.6.a.v 4
12.b even 2 1 336.6.q.j 8
21.c even 2 1 147.6.e.o 8
21.g even 6 1 147.6.a.l 4
21.g even 6 1 147.6.e.o 8
21.h odd 6 1 21.6.e.c 8
21.h odd 6 1 147.6.a.m 4
84.n even 6 1 336.6.q.j 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.e.c 8 3.b odd 2 1
21.6.e.c 8 21.h odd 6 1
63.6.e.e 8 1.a even 1 1 trivial
63.6.e.e 8 7.c even 3 1 inner
147.6.a.l 4 21.g even 6 1
147.6.a.m 4 21.h odd 6 1
147.6.e.o 8 21.c even 2 1
147.6.e.o 8 21.g even 6 1
336.6.q.j 8 12.b even 2 1
336.6.q.j 8 84.n even 6 1
441.6.a.v 4 7.d odd 6 1
441.6.a.w 4 7.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{8} - \cdots\) acting on \(S_{6}^{\mathrm{new}}(63, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 41616 + 37944 T + 53772 T^{2} - 16260 T^{3} + 9190 T^{4} - 90 T^{5} + 103 T^{6} - 3 T^{7} + T^{8} \)
$3$ \( T^{8} \)
$5$ \( 10164899803536 - 965081458800 T + 67214905692 T^{2} - 2317773900 T^{3} + 61817893 T^{4} - 605400 T^{5} + 7657 T^{6} + T^{8} \)
$7$ \( 79792266297612001 - 1224870869565294 T + 18476141086592 T^{2} - 206920120008 T^{3} + 1643194665 T^{4} - 12311544 T^{5} + 65408 T^{6} - 258 T^{7} + T^{8} \)
$11$ \( 2829568482592751376 - 169576701600014928 T + 9761813367494172 T^{2} - 25381944601332 T^{3} + 99024901837 T^{4} - 105799218 T^{5} + 399967 T^{6} - 402 T^{7} + T^{8} \)
$13$ \( ( 149501563456 + 515112852 T - 1148423 T^{2} - 462 T^{3} + T^{4} )^{2} \)
$17$ \( \)\(25\!\cdots\!24\)\( - 44837110146970681344 T + 720903316973027328 T^{2} - 1454766301642752 T^{3} + 2840324474880 T^{4} - 1349604864 T^{5} + 1670928 T^{6} - 276 T^{7} + T^{8} \)
$19$ \( \)\(54\!\cdots\!76\)\( - \)\(58\!\cdots\!80\)\( T + 44221112569626422096 T^{2} - 2834297273515140 T^{3} + 27788839301865 T^{4} - 1411756650 T^{5} + 6156971 T^{6} + 510 T^{7} + T^{8} \)
$23$ \( \)\(90\!\cdots\!76\)\( + \)\(51\!\cdots\!20\)\( T + \)\(27\!\cdots\!44\)\( T^{2} + 163456653366558720 T^{3} + 165861153884160 T^{4} - 83386679040 T^{5} + 40488144 T^{6} - 6900 T^{7} + T^{8} \)
$29$ \( ( 408027025117872 - 62527747272 T - 52650397 T^{2} + 540 T^{3} + T^{4} )^{2} \)
$31$ \( \)\(75\!\cdots\!09\)\( + \)\(50\!\cdots\!98\)\( T + \)\(49\!\cdots\!52\)\( T^{2} + 73339378003949028 T^{3} + 602809801419817 T^{4} - 3691164188 T^{5} + 58801968 T^{6} - 6410 T^{7} + T^{8} \)
$37$ \( \)\(25\!\cdots\!36\)\( + \)\(75\!\cdots\!80\)\( T + \)\(20\!\cdots\!08\)\( T^{2} + \)\(22\!\cdots\!60\)\( T^{3} + 30068709801011305 T^{4} + 2279577843610 T^{5} + 279418707 T^{6} + 15250 T^{7} + T^{8} \)
$41$ \( ( -1856858915261952 + 1101575496480 T - 192741244 T^{2} + 4308 T^{3} + T^{4} )^{2} \)
$43$ \( ( -991662745581932 - 199921376588 T + 199961493 T^{2} - 29198 T^{3} + T^{4} )^{2} \)
$47$ \( \)\(73\!\cdots\!36\)\( + \)\(12\!\cdots\!48\)\( T + \)\(15\!\cdots\!28\)\( T^{2} + 5876954381324343168 T^{3} + 3512848312989072 T^{4} + 852661119456 T^{5} + 176153856 T^{6} + 15060 T^{7} + T^{8} \)
$53$ \( \)\(72\!\cdots\!84\)\( - \)\(68\!\cdots\!40\)\( T + \)\(60\!\cdots\!12\)\( T^{2} - \)\(42\!\cdots\!28\)\( T^{3} + 366485627229994953 T^{4} - 9260410268772 T^{5} + 685378293 T^{6} - 13692 T^{7} + T^{8} \)
$59$ \( \)\(66\!\cdots\!96\)\( - \)\(11\!\cdots\!60\)\( T + \)\(69\!\cdots\!76\)\( T^{2} - 92739830872274379000 T^{3} + 54097578476747217 T^{4} - 7480212648750 T^{5} + 1024872459 T^{6} - 34830 T^{7} + T^{8} \)
$61$ \( \)\(32\!\cdots\!76\)\( + \)\(60\!\cdots\!60\)\( T + \)\(20\!\cdots\!24\)\( T^{2} - \)\(16\!\cdots\!68\)\( T^{3} + 283845325040842512 T^{4} - 3902598174816 T^{5} + 561368672 T^{6} - 5364 T^{7} + T^{8} \)
$67$ \( \)\(30\!\cdots\!56\)\( - \)\(27\!\cdots\!00\)\( T + \)\(15\!\cdots\!44\)\( T^{2} + \)\(20\!\cdots\!92\)\( T^{3} + 7519978747196869197 T^{4} + 26942404854654 T^{5} + 2881922927 T^{6} - 5994 T^{7} + T^{8} \)
$71$ \( ( 21932335650275568 - 16377596837712 T + 1521744768 T^{2} + 89268 T^{3} + T^{4} )^{2} \)
$73$ \( \)\(14\!\cdots\!00\)\( + \)\(24\!\cdots\!00\)\( T + \)\(45\!\cdots\!00\)\( T^{2} + \)\(71\!\cdots\!80\)\( T^{3} + 1466139740779982221 T^{4} + 62299772588518 T^{5} + 3193750863 T^{6} + 59638 T^{7} + T^{8} \)
$79$ \( \)\(27\!\cdots\!81\)\( - \)\(24\!\cdots\!06\)\( T + \)\(84\!\cdots\!28\)\( T^{2} + \)\(70\!\cdots\!88\)\( T^{3} + 13482185316532207897 T^{4} + 196348017119564 T^{5} + 5723264752 T^{6} - 44062 T^{7} + T^{8} \)
$83$ \( ( -41533908097096407132 + 738604511000820 T + 7607249829 T^{2} - 208446 T^{3} + T^{4} )^{2} \)
$89$ \( \)\(22\!\cdots\!24\)\( + \)\(88\!\cdots\!72\)\( T + \)\(26\!\cdots\!72\)\( T^{2} + \)\(38\!\cdots\!52\)\( T^{3} + \)\(47\!\cdots\!12\)\( T^{4} + 2429594520301248 T^{5} + 22815563908 T^{6} + 77520 T^{7} + T^{8} \)
$97$ \( ( -11638556269792123644 - 99054118022220 T + 9271508101 T^{2} + 188630 T^{3} + T^{4} )^{2} \)
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