Properties

Label 69.8.a.b
Level $69$
Weight $8$
Character orbit 69.a
Self dual yes
Analytic conductor $21.555$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 69 = 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 69.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(21.5545667584\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \(x^{6} - 2 x^{5} - 466 x^{4} + 540 x^{3} + 48973 x^{2} - 77282 x - 1061812\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{5}\cdot 3^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 - \beta_{1} ) q^{2} + 27 q^{3} + ( 29 + 3 \beta_{1} + \beta_{2} ) q^{4} + ( -66 + 14 \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{5} + ( -27 - 27 \beta_{1} ) q^{6} + ( -191 + 8 \beta_{1} - 7 \beta_{2} - \beta_{3} - 7 \beta_{4} ) q^{7} + ( -326 - 8 \beta_{1} - 9 \beta_{2} - 6 \beta_{3} - 5 \beta_{4} - 10 \beta_{5} ) q^{8} + 729 q^{9} +O(q^{10})\) \( q + ( -1 - \beta_{1} ) q^{2} + 27 q^{3} + ( 29 + 3 \beta_{1} + \beta_{2} ) q^{4} + ( -66 + 14 \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{5} + ( -27 - 27 \beta_{1} ) q^{6} + ( -191 + 8 \beta_{1} - 7 \beta_{2} - \beta_{3} - 7 \beta_{4} ) q^{7} + ( -326 - 8 \beta_{1} - 9 \beta_{2} - 6 \beta_{3} - 5 \beta_{4} - 10 \beta_{5} ) q^{8} + 729 q^{9} + ( -2167 - 27 \beta_{1} - 34 \beta_{2} - 20 \beta_{3} + 7 \beta_{4} ) q^{10} + ( -2463 - 72 \beta_{1} - 14 \beta_{2} + 26 \beta_{3} - 9 \beta_{4} - 3 \beta_{5} ) q^{11} + ( 783 + 81 \beta_{1} + 27 \beta_{2} ) q^{12} + ( -225 + 333 \beta_{1} + 21 \beta_{2} - 38 \beta_{3} - 23 \beta_{4} + 6 \beta_{5} ) q^{13} + ( -917 + 769 \beta_{1} + 58 \beta_{2} + 84 \beta_{3} + 7 \beta_{4} + 26 \beta_{5} ) q^{14} + ( -1782 + 378 \beta_{1} + 27 \beta_{2} + 27 \beta_{3} + 54 \beta_{4} + 27 \beta_{5} ) q^{15} + ( -2459 + 751 \beta_{1} + 17 \beta_{2} + 104 \beta_{3} - 28 \beta_{4} + 88 \beta_{5} ) q^{16} + ( -11359 - 306 \beta_{1} - 37 \beta_{2} - 221 \beta_{3} + 175 \beta_{4} - 76 \beta_{5} ) q^{17} + ( -729 - 729 \beta_{1} ) q^{18} + ( -7250 - 40 \beta_{1} + 87 \beta_{2} + 13 \beta_{3} + 68 \beta_{4} - 251 \beta_{5} ) q^{19} + ( 11895 + 3955 \beta_{1} + 142 \beta_{2} + 34 \beta_{3} + 89 \beta_{4} + 214 \beta_{5} ) q^{20} + ( -5157 + 216 \beta_{1} - 189 \beta_{2} - 27 \beta_{3} - 189 \beta_{4} ) q^{21} + ( 15000 + 3894 \beta_{1} + 120 \beta_{2} + 144 \beta_{3} - 184 \beta_{4} + 150 \beta_{5} ) q^{22} -12167 q^{23} + ( -8802 - 216 \beta_{1} - 243 \beta_{2} - 162 \beta_{3} - 135 \beta_{4} - 270 \beta_{5} ) q^{24} + ( 37698 + 953 \beta_{1} + 501 \beta_{2} - 210 \beta_{3} - 673 \beta_{4} - 20 \beta_{5} ) q^{25} + ( -51078 - 2950 \beta_{1} - 306 \beta_{2} + 100 \beta_{4} - 448 \beta_{5} ) q^{26} + 19683 q^{27} + ( -88505 - 7229 \beta_{1} - 624 \beta_{2} - 314 \beta_{3} + 287 \beta_{4} - 474 \beta_{5} ) q^{28} + ( -50238 - 5706 \beta_{1} - 132 \beta_{2} + 518 \beta_{3} - 220 \beta_{4} + 900 \beta_{5} ) q^{29} + ( -58509 - 729 \beta_{1} - 918 \beta_{2} - 540 \beta_{3} + 189 \beta_{4} ) q^{30} + ( -39806 - 3100 \beta_{1} + 362 \beta_{2} + 186 \beta_{3} - 70 \beta_{4} + 232 \beta_{5} ) q^{31} + ( -65270 + 184 \beta_{1} - 369 \beta_{2} + 658 \beta_{3} + 479 \beta_{4} + 798 \beta_{5} ) q^{32} + ( -66501 - 1944 \beta_{1} - 378 \beta_{2} + 702 \beta_{3} - 243 \beta_{4} - 81 \beta_{5} ) q^{33} + ( 34501 + 18059 \beta_{1} + 1046 \beta_{2} - 676 \beta_{3} + 1923 \beta_{4} + 1282 \beta_{5} ) q^{34} + ( -215129 - 21279 \beta_{1} + 3 \beta_{2} + 1512 \beta_{3} + 471 \beta_{4} - 1058 \beta_{5} ) q^{35} + ( 21141 + 2187 \beta_{1} + 729 \beta_{2} ) q^{36} + ( -103739 + 590 \beta_{1} + 1642 \beta_{2} - 1220 \beta_{3} + 1533 \beta_{4} - 1657 \beta_{5} ) q^{37} + ( 13359 - 1109 \beta_{1} + 530 \beta_{2} - 428 \beta_{3} - 2445 \beta_{4} + 568 \beta_{5} ) q^{38} + ( -6075 + 8991 \beta_{1} + 567 \beta_{2} - 1026 \beta_{3} - 621 \beta_{4} + 162 \beta_{5} ) q^{39} + ( -350171 - 28591 \beta_{1} - 1894 \beta_{2} + 746 \beta_{3} + 527 \beta_{4} - 1674 \beta_{5} ) q^{40} + ( -152190 - 22330 \beta_{1} + 498 \beta_{2} - 4688 \beta_{3} + 516 \beta_{4} - 394 \beta_{5} ) q^{41} + ( -24759 + 20763 \beta_{1} + 1566 \beta_{2} + 2268 \beta_{3} + 189 \beta_{4} + 702 \beta_{5} ) q^{42} + ( -300672 + 6572 \beta_{1} - 1101 \beta_{2} + 6105 \beta_{3} - 3134 \beta_{4} + 2763 \beta_{5} ) q^{43} + ( -282254 - 28042 \beta_{1} - 3452 \beta_{2} - 3244 \beta_{3} - 26 \beta_{4} - 2232 \beta_{5} ) q^{44} + ( -48114 + 10206 \beta_{1} + 729 \beta_{2} + 729 \beta_{3} + 1458 \beta_{4} + 729 \beta_{5} ) q^{45} + ( 12167 + 12167 \beta_{1} ) q^{46} + ( -230319 - 13753 \beta_{1} + 1117 \beta_{2} + 2432 \beta_{3} - 3383 \beta_{4} - 62 \beta_{5} ) q^{47} + ( -66393 + 20277 \beta_{1} + 459 \beta_{2} + 2808 \beta_{3} - 756 \beta_{4} + 2376 \beta_{5} ) q^{48} + ( 321014 + 36799 \beta_{1} + 259 \beta_{2} - 7742 \beta_{3} - 959 \beta_{4} + 3108 \beta_{5} ) q^{49} + ( -127983 - 101243 \beta_{1} - 1210 \beta_{2} + 1072 \beta_{3} - 4580 \beta_{4} - 9388 \beta_{5} ) q^{50} + ( -306693 - 8262 \beta_{1} - 999 \beta_{2} - 5967 \beta_{3} + 4725 \beta_{4} - 2052 \beta_{5} ) q^{51} + ( 520368 + 45516 \beta_{1} + 4038 \beta_{2} + 6996 \beta_{3} + 942 \beta_{4} + 4684 \beta_{5} ) q^{52} + ( -254730 - 13476 \beta_{1} + 1257 \beta_{2} - 657 \beta_{3} + 7178 \beta_{4} + 2769 \beta_{5} ) q^{53} + ( -19683 - 19683 \beta_{1} ) q^{54} + ( -60540 - 43700 \beta_{1} - 8592 \beta_{2} - 5964 \beta_{3} - 7044 \beta_{4} - 3884 \beta_{5} ) q^{55} + ( 1271771 + 70411 \beta_{1} + 6000 \beta_{2} - 7782 \beta_{3} + 1591 \beta_{4} + 5902 \beta_{5} ) q^{56} + ( -195750 - 1080 \beta_{1} + 2349 \beta_{2} + 351 \beta_{3} + 1836 \beta_{4} - 6777 \beta_{5} ) q^{57} + ( 974580 + 74680 \beta_{1} + 1104 \beta_{2} + 312 \beta_{3} + 4034 \beta_{4} - 2564 \beta_{5} ) q^{58} + ( -246135 + 18509 \beta_{1} + 255 \beta_{2} + 14064 \beta_{3} - 927 \beta_{4} + 9836 \beta_{5} ) q^{59} + ( 321165 + 106785 \beta_{1} + 3834 \beta_{2} + 918 \beta_{3} + 2403 \beta_{4} + 5778 \beta_{5} ) q^{60} + ( -161799 - 7428 \beta_{1} - 5328 \beta_{2} + 13464 \beta_{3} - 2947 \beta_{4} - 7987 \beta_{5} ) q^{61} + ( 552754 + 9206 \beta_{1} - 580 \beta_{2} - 2216 \beta_{3} - 1374 \beta_{4} - 4596 \beta_{5} ) q^{62} + ( -139239 + 5832 \beta_{1} - 5103 \beta_{2} - 729 \beta_{3} - 5103 \beta_{4} ) q^{63} + ( 333365 + 16759 \beta_{1} - 7547 \beta_{2} - 15568 \beta_{3} + 10400 \beta_{4} - 6576 \beta_{5} ) q^{64} + ( -83978 + 49702 \beta_{1} + 16112 \beta_{2} + 14686 \beta_{3} + 6814 \beta_{4} + 4166 \beta_{5} ) q^{65} + ( 405000 + 105138 \beta_{1} + 3240 \beta_{2} + 3888 \beta_{3} - 4968 \beta_{4} + 4050 \beta_{5} ) q^{66} + ( -81432 + 9948 \beta_{1} - 22791 \beta_{2} - 6113 \beta_{3} + 8366 \beta_{4} - 2163 \beta_{5} ) q^{67} + ( -1521791 - 103159 \beta_{1} - 29750 \beta_{2} + 7910 \beta_{3} - 1745 \beta_{4} + 4326 \beta_{5} ) q^{68} -328509 q^{69} + ( 3576958 + 263258 \beta_{1} + 20602 \beta_{2} - 728 \beta_{3} - 17766 \beta_{4} + 10052 \beta_{5} ) q^{70} + ( 44200 - 27934 \beta_{1} + 4394 \beta_{2} - 29276 \beta_{3} + 9532 \beta_{4} - 10786 \beta_{5} ) q^{71} + ( -237654 - 5832 \beta_{1} - 6561 \beta_{2} - 4374 \beta_{3} - 3645 \beta_{4} - 7290 \beta_{5} ) q^{72} + ( 2732064 + 53982 \beta_{1} + 8254 \beta_{2} + 3388 \beta_{3} - 50 \beta_{4} + 19732 \beta_{5} ) q^{73} + ( -83128 - 44946 \beta_{1} - 3096 \beta_{2} - 15736 \beta_{3} - 6918 \beta_{4} - 3034 \beta_{5} ) q^{74} + ( 1017846 + 25731 \beta_{1} + 13527 \beta_{2} - 5670 \beta_{3} - 18171 \beta_{4} - 540 \beta_{5} ) q^{75} + ( 1260045 - 97395 \beta_{1} - 7428 \beta_{2} + 8690 \beta_{3} - 15471 \beta_{4} + 9030 \beta_{5} ) q^{76} + ( 1224764 - 5498 \beta_{1} + 32658 \beta_{2} + 17240 \beta_{3} + 33528 \beta_{4} + 13546 \beta_{5} ) q^{77} + ( -1379106 - 79650 \beta_{1} - 8262 \beta_{2} + 2700 \beta_{4} - 12096 \beta_{5} ) q^{78} + ( 516947 - 327508 \beta_{1} - 2505 \beta_{2} + 13009 \beta_{3} + 9643 \beta_{4} - 15928 \beta_{5} ) q^{79} + ( 3207703 + 96563 \beta_{1} + 26330 \beta_{2} + 7198 \beta_{3} - 19575 \beta_{4} + 2898 \beta_{5} ) q^{80} + 531441 q^{81} + ( 3391646 + 148938 \beta_{1} + 33828 \beta_{2} - 5296 \beta_{3} + 29360 \beta_{4} - 9684 \beta_{5} ) q^{82} + ( -1162323 - 339538 \beta_{1} + 1992 \beta_{2} + 21206 \beta_{3} - 24065 \beta_{4} - 4669 \beta_{5} ) q^{83} + ( -2389635 - 195183 \beta_{1} - 16848 \beta_{2} - 8478 \beta_{3} + 7749 \beta_{4} - 12798 \beta_{5} ) q^{84} + ( 1178367 - 643063 \beta_{1} - 8727 \beta_{2} - 14542 \beta_{3} - 22649 \beta_{4} - 33272 \beta_{5} ) q^{85} + ( -256141 + 362979 \beta_{1} - 22694 \beta_{2} + 19884 \beta_{3} - 28033 \beta_{4} - 6636 \beta_{5} ) q^{86} + ( -1356426 - 154062 \beta_{1} - 3564 \beta_{2} + 13986 \beta_{3} - 5940 \beta_{4} + 24300 \beta_{5} ) q^{87} + ( 2433466 + 178202 \beta_{1} + 54364 \beta_{2} + 6900 \beta_{3} + 43302 \beta_{4} + 17604 \beta_{5} ) q^{88} + ( -3360189 - 73700 \beta_{1} - 22369 \beta_{2} - 17595 \beta_{3} + 34525 \beta_{4} + 16246 \beta_{5} ) q^{89} + ( -1579743 - 19683 \beta_{1} - 24786 \beta_{2} - 14580 \beta_{3} + 5103 \beta_{4} ) q^{90} + ( 1534858 - 179312 \beta_{1} - 44746 \beta_{2} - 75690 \beta_{3} - 13470 \beta_{4} - 22672 \beta_{5} ) q^{91} + ( -352843 - 36501 \beta_{1} - 12167 \beta_{2} ) q^{92} + ( -1074762 - 83700 \beta_{1} + 9774 \beta_{2} + 5022 \beta_{3} - 1890 \beta_{4} + 6264 \beta_{5} ) q^{93} + ( 2779858 + 93126 \beta_{1} + 10214 \beta_{2} + 13720 \beta_{3} - 40082 \beta_{4} - 26356 \beta_{5} ) q^{94} + ( 718967 + 385777 \beta_{1} - 52921 \beta_{2} - 46900 \beta_{3} - 23217 \beta_{4} + 7530 \beta_{5} ) q^{95} + ( -1762290 + 4968 \beta_{1} - 9963 \beta_{2} + 17766 \beta_{3} + 12933 \beta_{4} + 21546 \beta_{5} ) q^{96} + ( 2027878 + 194752 \beta_{1} + 15406 \beta_{2} + 53778 \beta_{3} + 23272 \beta_{4} + 104514 \beta_{5} ) q^{97} + ( -6365857 - 434533 \beta_{1} - 27790 \beta_{2} - 2016 \beta_{3} + 76076 \beta_{4} - 36260 \beta_{5} ) q^{98} + ( -1795527 - 52488 \beta_{1} - 10206 \beta_{2} + 18954 \beta_{3} - 6561 \beta_{4} - 2187 \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 8q^{2} + 162q^{3} + 178q^{4} - 372q^{5} - 216q^{6} - 1104q^{7} - 1956q^{8} + 4374q^{9} + O(q^{10}) \) \( 6q - 8q^{2} + 162q^{3} + 178q^{4} - 372q^{5} - 216q^{6} - 1104q^{7} - 1956q^{8} + 4374q^{9} - 13042q^{10} - 14824q^{11} + 4806q^{12} - 756q^{13} - 3926q^{14} - 10044q^{15} - 13022q^{16} - 69484q^{17} - 5832q^{18} - 43864q^{19} + 78886q^{20} - 29808q^{21} + 98204q^{22} - 73002q^{23} - 52812q^{24} + 228018q^{25} - 311956q^{26} + 118098q^{27} - 545442q^{28} - 311100q^{29} - 352134q^{30} - 245248q^{31} - 390156q^{32} - 400248q^{33} + 235834q^{34} - 1331256q^{35} + 129762q^{36} - 630044q^{37} + 80910q^{38} - 20412q^{39} - 2153982q^{40} - 969204q^{41} - 106002q^{42} - 1770208q^{43} - 1749140q^{44} - 271188q^{45} + 97336q^{46} - 1400024q^{47} - 351594q^{48} + 1985598q^{49} - 956660q^{50} - 1876068q^{51} + 3217272q^{52} - 1573516q^{53} - 157464q^{54} - 431296q^{55} + 7740702q^{56} - 1184328q^{57} + 5987188q^{58} - 1410320q^{59} + 2129922q^{60} - 942172q^{61} + 3334412q^{62} - 804816q^{63} + 1996866q^{64} - 420944q^{65} + 2651508q^{66} - 452072q^{67} - 9258254q^{68} - 1971054q^{69} + 21981136q^{70} + 122928q^{71} - 1425924q^{72} + 16490716q^{73} - 600104q^{74} + 6156486q^{75} + 7428658q^{76} + 7239696q^{77} - 8422812q^{78} + 2458408q^{79} + 19440230q^{80} + 3188646q^{81} + 20510784q^{82} - 7566456q^{83} - 14726934q^{84} + 5817744q^{85} - 669666q^{86} - 8399700q^{87} + 14775668q^{88} - 20368036q^{89} - 9507618q^{90} + 8815576q^{91} - 2165726q^{92} - 6621696q^{93} + 16952576q^{94} + 5143832q^{95} - 10534212q^{96} + 12586972q^{97} - 39164812q^{98} - 10806696q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 2 x^{5} - 466 x^{4} + 540 x^{3} + 48973 x^{2} - 77282 x - 1061812\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 156 \)
\(\beta_{3}\)\(=\)\((\)\( -45 \nu^{5} - 175 \nu^{4} + 18759 \nu^{3} + 89711 \nu^{2} - 1226918 \nu - 4103988 \)\()/10624\)
\(\beta_{4}\)\(=\)\((\)\( -2 \nu^{5} - 17 \nu^{4} + 878 \nu^{3} + 7143 \nu^{2} - 61166 \nu - 310308 \)\()/664\)
\(\beta_{5}\)\(=\)\((\)\( 43 \nu^{5} + 241 \nu^{4} - 17217 \nu^{3} - 117345 \nu^{2} + 957754 \nu + 5819212 \)\()/10624\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 156\)
\(\nu^{3}\)\(=\)\(10 \beta_{5} + 5 \beta_{4} + 6 \beta_{3} + 6 \beta_{2} + 258 \beta_{1} + 113\)
\(\nu^{4}\)\(=\)\(48 \beta_{5} - 48 \beta_{4} + 80 \beta_{3} + 371 \beta_{2} + 861 \beta_{1} + 40056\)
\(\nu^{5}\)\(=\)\(3982 \beta_{5} + 2271 \beta_{4} + 1954 \beta_{3} + 3052 \beta_{2} + 78932 \beta_{1} + 111131\)

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
18.9294
8.60678
7.53998
−4.35108
−11.6010
−17.1241
−19.9294 27.0000 269.181 466.102 −538.094 −1376.23 −2813.66 729.000 −9289.15
1.2 −9.60678 27.0000 −35.7097 26.4395 −259.383 −248.118 1572.72 729.000 −253.998
1.3 −8.53998 27.0000 −55.0688 −446.195 −230.579 1770.31 1563.40 729.000 3810.49
1.4 3.35108 27.0000 −116.770 0.849066 90.4791 405.736 −820.245 729.000 2.84529
1.5 10.6010 27.0000 −15.6190 100.131 286.227 −1216.42 −1522.50 729.000 1061.49
1.6 16.1241 27.0000 131.987 −519.327 435.351 −439.281 64.2798 729.000 −8373.68
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(23\) \(1\)

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 69.8.a.b 6
3.b odd 2 1 207.8.a.c 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
69.8.a.b 6 1.a even 1 1 trivial
207.8.a.c 6 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + 8 T_{2}^{5} - 441 T_{2}^{4} - 2364 T_{2}^{3} + 44592 T_{2}^{2} + 171760 T_{2} - 936560 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(69))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -936560 + 171760 T + 44592 T^{2} - 2364 T^{3} - 441 T^{4} + 8 T^{5} + T^{6} \)
$3$ \( ( -27 + T )^{6} \)
$5$ \( 242778320000 - 297051936000 T + 13159502400 T^{2} - 78815080 T^{3} - 279192 T^{4} + 372 T^{5} + T^{6} \)
$7$ \( 131059980861522560 + 632489980356960 T - 375707069088 T^{2} - 3968841320 T^{3} - 2854020 T^{4} + 1104 T^{5} + T^{6} \)
$11$ \( 22649012314313113600 - 500937794004170240 T - 515981318342016 T^{2} - 23477126560 T^{3} + 61017264 T^{4} + 14824 T^{5} + T^{6} \)
$13$ \( \)\(20\!\cdots\!40\)\( + 29810155193197369920 T + 6932554982479728 T^{2} - 401215491168 T^{3} - 169949444 T^{4} + 756 T^{5} + T^{6} \)
$17$ \( \)\(48\!\cdots\!00\)\( + \)\(41\!\cdots\!00\)\( T - 863489499776379040 T^{2} - 59790513849160 T^{3} + 297715144 T^{4} + 69484 T^{5} + T^{6} \)
$19$ \( \)\(23\!\cdots\!60\)\( + \)\(52\!\cdots\!40\)\( T + 696855618708199232 T^{2} - 115428389953800 T^{3} - 2470866420 T^{4} + 43864 T^{5} + T^{6} \)
$23$ \( ( 12167 + T )^{6} \)
$29$ \( \)\(30\!\cdots\!80\)\( + \)\(95\!\cdots\!20\)\( T - \)\(22\!\cdots\!56\)\( T^{2} - 5676705177070368 T^{3} - 4490567636 T^{4} + 311100 T^{5} + T^{6} \)
$31$ \( -\)\(34\!\cdots\!72\)\( - \)\(78\!\cdots\!52\)\( T - 48095784417975349248 T^{2} - 671248498891584 T^{3} + 12109867280 T^{4} + 245248 T^{5} + T^{6} \)
$37$ \( \)\(67\!\cdots\!00\)\( - \)\(43\!\cdots\!00\)\( T + \)\(79\!\cdots\!00\)\( T^{2} - 42799391786207040 T^{3} - 71923792308 T^{4} + 630044 T^{5} + T^{6} \)
$41$ \( -\)\(86\!\cdots\!64\)\( + \)\(52\!\cdots\!88\)\( T + \)\(68\!\cdots\!12\)\( T^{2} - 441437544100255136 T^{3} - 394550526852 T^{4} + 969204 T^{5} + T^{6} \)
$43$ \( -\)\(85\!\cdots\!12\)\( - \)\(12\!\cdots\!04\)\( T - \)\(57\!\cdots\!12\)\( T^{2} - 826174322461436424 T^{3} + 398128610572 T^{4} + 1770208 T^{5} + T^{6} \)
$47$ \( -\)\(28\!\cdots\!20\)\( - \)\(57\!\cdots\!00\)\( T - \)\(32\!\cdots\!44\)\( T^{2} - 602857942417180544 T^{3} + 126851022576 T^{4} + 1400024 T^{5} + T^{6} \)
$53$ \( -\)\(44\!\cdots\!80\)\( - \)\(13\!\cdots\!20\)\( T - \)\(11\!\cdots\!84\)\( T^{2} - 2756186606651180184 T^{3} - 1098196833624 T^{4} + 1573516 T^{5} + T^{6} \)
$59$ \( \)\(30\!\cdots\!68\)\( + \)\(75\!\cdots\!20\)\( T + \)\(48\!\cdots\!20\)\( T^{2} - 9426346423109773440 T^{3} - 5045958681184 T^{4} + 1410320 T^{5} + T^{6} \)
$61$ \( -\)\(23\!\cdots\!80\)\( + \)\(19\!\cdots\!00\)\( T + \)\(31\!\cdots\!36\)\( T^{2} - 10312137706013580288 T^{3} - 11163017823636 T^{4} + 942172 T^{5} + T^{6} \)
$67$ \( \)\(41\!\cdots\!92\)\( - \)\(62\!\cdots\!44\)\( T + \)\(24\!\cdots\!32\)\( T^{2} + 34825564601748577224 T^{3} - 30664395846612 T^{4} + 452072 T^{5} + T^{6} \)
$71$ \( \)\(26\!\cdots\!44\)\( - \)\(15\!\cdots\!40\)\( T + \)\(14\!\cdots\!52\)\( T^{2} + 29974990955905967360 T^{3} - 19393248315520 T^{4} - 122928 T^{5} + T^{6} \)
$73$ \( -\)\(42\!\cdots\!88\)\( + \)\(54\!\cdots\!88\)\( T - \)\(26\!\cdots\!52\)\( T^{2} - \)\(20\!\cdots\!88\)\( T^{3} + 95366424742748 T^{4} - 16490716 T^{5} + T^{6} \)
$79$ \( \)\(84\!\cdots\!32\)\( - \)\(68\!\cdots\!44\)\( T + \)\(12\!\cdots\!12\)\( T^{2} + \)\(26\!\cdots\!64\)\( T^{3} - 75605234542372 T^{4} - 2458408 T^{5} + T^{6} \)
$83$ \( \)\(69\!\cdots\!40\)\( + \)\(20\!\cdots\!60\)\( T - \)\(15\!\cdots\!52\)\( T^{2} - \)\(72\!\cdots\!68\)\( T^{3} - 65307299700304 T^{4} + 7566456 T^{5} + T^{6} \)
$89$ \( -\)\(10\!\cdots\!44\)\( - \)\(51\!\cdots\!04\)\( T - \)\(37\!\cdots\!36\)\( T^{2} - \)\(63\!\cdots\!72\)\( T^{3} + 68784980062984 T^{4} + 20368036 T^{5} + T^{6} \)
$97$ \( -\)\(11\!\cdots\!08\)\( - \)\(31\!\cdots\!56\)\( T + \)\(41\!\cdots\!04\)\( T^{2} + \)\(40\!\cdots\!88\)\( T^{3} - 374790141510324 T^{4} - 12586972 T^{5} + T^{6} \)
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