Properties

Label 69.8.a.a
Level $69$
Weight $8$
Character orbit 69.a
Self dual yes
Analytic conductor $21.555$
Analytic rank $1$
Dimension $5$
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: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Defining polynomial: \(x^{5} - 455 x^{3} - 474 x^{2} + 42284 x + 127016\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} -27 q^{3} + ( 54 + 2 \beta_{1} + \beta_{3} + \beta_{4} ) q^{4} + ( -54 + 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} ) q^{5} -27 \beta_{1} q^{6} + ( -96 + 3 \beta_{1} - \beta_{2} - 12 \beta_{3} - 4 \beta_{4} ) q^{7} + ( 284 + 9 \beta_{1} + 11 \beta_{2} - \beta_{4} ) q^{8} + 729 q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} -27 q^{3} + ( 54 + 2 \beta_{1} + \beta_{3} + \beta_{4} ) q^{4} + ( -54 + 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} ) q^{5} -27 \beta_{1} q^{6} + ( -96 + 3 \beta_{1} - \beta_{2} - 12 \beta_{3} - 4 \beta_{4} ) q^{7} + ( 284 + 9 \beta_{1} + 11 \beta_{2} - \beta_{4} ) q^{8} + 729 q^{9} + ( 296 - 72 \beta_{1} + 8 \beta_{2} - 10 \beta_{3} + 4 \beta_{4} ) q^{10} + ( -220 - 117 \beta_{1} + 4 \beta_{2} - 29 \beta_{3} - 5 \beta_{4} ) q^{11} + ( -1458 - 54 \beta_{1} - 27 \beta_{3} - 27 \beta_{4} ) q^{12} + ( -172 - 372 \beta_{1} - 13 \beta_{2} + 82 \beta_{3} - 27 \beta_{4} ) q^{13} + ( 1158 - 589 \beta_{1} - 125 \beta_{2} - 23 \beta_{3} - 6 \beta_{4} ) q^{14} + ( 1458 - 54 \beta_{1} + 27 \beta_{2} - 27 \beta_{3} + 27 \beta_{4} ) q^{15} + ( -4578 - 108 \beta_{1} + 10 \beta_{2} + 83 \beta_{3} - 59 \beta_{4} ) q^{16} + ( -1126 - 1103 \beta_{1} + 65 \beta_{2} + 76 \beta_{3} + 160 \beta_{4} ) q^{17} + 729 \beta_{1} q^{18} + ( -1208 - 598 \beta_{1} + 121 \beta_{2} + 127 \beta_{3} + 125 \beta_{4} ) q^{19} + ( -5416 - 196 \beta_{1} + 40 \beta_{2} - 92 \beta_{3} + 56 \beta_{4} ) q^{20} + ( 2592 - 81 \beta_{1} + 27 \beta_{2} + 324 \beta_{3} + 108 \beta_{4} ) q^{21} + ( -19598 - 1461 \beta_{1} - 291 \beta_{2} - 83 \beta_{3} - 130 \beta_{4} ) q^{22} + 12167 q^{23} + ( -7668 - 243 \beta_{1} - 297 \beta_{2} + 27 \beta_{4} ) q^{24} + ( -52391 - 1198 \beta_{1} + 257 \beta_{2} - 280 \beta_{3} - 19 \beta_{4} ) q^{25} + ( -71120 - 316 \beta_{1} + 780 \beta_{2} - 334 \beta_{3} - 138 \beta_{4} ) q^{26} -19683 q^{27} + ( -101182 - 161 \beta_{1} - 233 \beta_{2} - 1325 \beta_{3} - 718 \beta_{4} ) q^{28} + ( -33850 - 3746 \beta_{1} + 286 \beta_{2} + 540 \beta_{3} + 496 \beta_{4} ) q^{29} + ( -7992 + 1944 \beta_{1} - 216 \beta_{2} + 270 \beta_{3} - 108 \beta_{4} ) q^{30} + ( -39540 + 4906 \beta_{1} - 744 \beta_{2} - 1400 \beta_{3} - 430 \beta_{4} ) q^{31} + ( -56936 - 7289 \beta_{1} - 627 \beta_{2} + 474 \beta_{3} + 531 \beta_{4} ) q^{32} + ( 5940 + 3159 \beta_{1} - 108 \beta_{2} + 783 \beta_{3} + 135 \beta_{4} ) q^{33} + ( -205950 + 6707 \beta_{1} + 985 \beta_{2} - 421 \beta_{3} - 1426 \beta_{4} ) q^{34} + ( -28046 + 2746 \beta_{1} - 517 \beta_{2} + 1166 \beta_{3} - 109 \beta_{4} ) q^{35} + ( 39366 + 1458 \beta_{1} + 729 \beta_{3} + 729 \beta_{4} ) q^{36} + ( -40902 + 13985 \beta_{1} - 660 \beta_{2} + 1431 \beta_{3} + 177 \beta_{4} ) q^{37} + ( -111664 + 6430 \beta_{1} + 1516 \beta_{2} + 1334 \beta_{3} - 364 \beta_{4} ) q^{38} + ( 4644 + 10044 \beta_{1} + 351 \beta_{2} - 2214 \beta_{3} + 729 \beta_{4} ) q^{39} + ( -69128 + 3920 \beta_{1} - 1848 \beta_{2} + 1396 \beta_{3} - 972 \beta_{4} ) q^{40} + ( 107914 + 8482 \beta_{1} - 2772 \beta_{2} + 422 \beta_{3} - 434 \beta_{4} ) q^{41} + ( -31266 + 15903 \beta_{1} + 3375 \beta_{2} + 621 \beta_{3} + 162 \beta_{4} ) q^{42} + ( -180296 + 25824 \beta_{1} + 5109 \beta_{2} - 1083 \beta_{3} + 2975 \beta_{4} ) q^{43} + ( -246870 - 14067 \beta_{1} - 1763 \beta_{2} - 2633 \beta_{3} - 1792 \beta_{4} ) q^{44} + ( -39366 + 1458 \beta_{1} - 729 \beta_{2} + 729 \beta_{3} - 729 \beta_{4} ) q^{45} + 12167 \beta_{1} q^{46} + ( 20306 - 9026 \beta_{1} + 99 \beta_{2} - 7350 \beta_{3} + 3311 \beta_{4} ) q^{47} + ( 123606 + 2916 \beta_{1} - 270 \beta_{2} - 2241 \beta_{3} + 1593 \beta_{4} ) q^{48} + ( 170843 + 14966 \beta_{1} - 1043 \beta_{2} + 3080 \beta_{3} + 4893 \beta_{4} ) q^{49} + ( -189612 - 64865 \beta_{1} - 2562 \beta_{2} + 2944 \beta_{3} - 378 \beta_{4} ) q^{50} + ( 30402 + 29781 \beta_{1} - 1755 \beta_{2} - 2052 \beta_{3} - 4320 \beta_{4} ) q^{51} + ( 30968 - 46762 \beta_{1} - 1034 \beta_{2} + 3112 \beta_{3} + 7062 \beta_{4} ) q^{52} + ( -61330 - 10232 \beta_{1} + 2343 \beta_{2} + 8805 \beta_{3} - 5451 \beta_{4} ) q^{53} -19683 \beta_{1} q^{54} + ( -188820 + 19000 \beta_{1} - 3110 \beta_{2} + 4700 \beta_{3} - 570 \beta_{4} ) q^{55} + ( -107358 - 95305 \beta_{1} + 1799 \beta_{2} - 1189 \beta_{3} + 382 \beta_{4} ) q^{56} + ( 32616 + 16146 \beta_{1} - 3267 \beta_{2} - 3429 \beta_{3} - 3375 \beta_{4} ) q^{57} + ( -706228 - 3676 \beta_{1} + 6182 \beta_{2} + 498 \beta_{3} - 3716 \beta_{4} ) q^{58} + ( -634970 - 39800 \beta_{1} + 12851 \beta_{2} - 6800 \beta_{3} - 5535 \beta_{4} ) q^{59} + ( 146232 + 5292 \beta_{1} - 1080 \beta_{2} + 2484 \beta_{3} - 1512 \beta_{4} ) q^{60} + ( 23102 + 50535 \beta_{1} + 7862 \beta_{2} + 20597 \beta_{3} + 4461 \beta_{4} ) q^{61} + ( 925332 - 80282 \beta_{1} - 15174 \beta_{2} - 9566 \beta_{3} + 536 \beta_{4} ) q^{62} + ( -69984 + 2187 \beta_{1} - 729 \beta_{2} - 8748 \beta_{3} - 2916 \beta_{4} ) q^{63} + ( -818206 - 11466 \beta_{1} + 3364 \beta_{2} - 30375 \beta_{3} - 4579 \beta_{4} ) q^{64} + ( 776048 - 13664 \beta_{1} + 3432 \beta_{2} - 6662 \beta_{3} - 5208 \beta_{4} ) q^{65} + ( 529146 + 39447 \beta_{1} + 7857 \beta_{2} + 2241 \beta_{3} + 3510 \beta_{4} ) q^{66} + ( -96120 - 38948 \beta_{1} - 32053 \beta_{2} + 1711 \beta_{3} - 6131 \beta_{4} ) q^{67} + ( 1493762 - 148751 \beta_{1} - 12971 \beta_{2} + 19571 \beta_{3} - 2560 \beta_{4} ) q^{68} -328509 q^{69} + ( 421372 + 4088 \beta_{1} + 11034 \beta_{2} - 3792 \beta_{3} + 3038 \beta_{4} ) q^{70} + ( 322352 - 18550 \beta_{1} + 7316 \beta_{2} - 14726 \beta_{3} + 2758 \beta_{4} ) q^{71} + ( 207036 + 6561 \beta_{1} + 8019 \beta_{2} - 729 \beta_{4} ) q^{72} + ( -11110 - 15000 \beta_{1} - 14012 \beta_{2} + 11284 \beta_{3} + 2604 \beta_{4} ) q^{73} + ( 2436334 + 37087 \beta_{1} + 13827 \beta_{2} + 4259 \beta_{3} + 12662 \beta_{4} ) q^{74} + ( 1414557 + 32346 \beta_{1} - 6939 \beta_{2} + 7560 \beta_{3} + 513 \beta_{4} ) q^{75} + ( 1370252 - 23908 \beta_{1} - 996 \beta_{2} + 21586 \beta_{3} + 2498 \beta_{4} ) q^{76} + ( 2068324 + 69014 \beta_{1} + 15122 \beta_{2} + 8018 \beta_{3} + 12636 \beta_{4} ) q^{77} + ( 1920240 + 8532 \beta_{1} - 21060 \beta_{2} + 9018 \beta_{3} + 3726 \beta_{4} ) q^{78} + ( -782888 + 122297 \beta_{1} + 21401 \beta_{2} - 7904 \beta_{3} - 8456 \beta_{4} ) q^{79} + ( 1269376 - 39524 \beta_{1} + 6020 \beta_{2} - 10888 \beta_{3} - 4836 \beta_{4} ) q^{80} + 531441 q^{81} + ( 1374460 + 136084 \beta_{1} + 1014 \beta_{2} - 38834 \beta_{3} - 2364 \beta_{4} ) q^{82} + ( 3176088 + 72749 \beta_{1} - 178 \beta_{2} - 25355 \beta_{3} + 27507 \beta_{4} ) q^{83} + ( 2731914 + 4347 \beta_{1} + 6291 \beta_{2} + 35775 \beta_{3} + 19386 \beta_{4} ) q^{84} + ( -1702262 + 106802 \beta_{1} - 9729 \beta_{2} + 6172 \beta_{3} + 10887 \beta_{4} ) q^{85} + ( 4947060 - 36996 \beta_{1} - 2746 \beta_{2} + 103720 \beta_{3} + 34328 \beta_{4} ) q^{86} + ( 913950 + 101142 \beta_{1} - 7722 \beta_{2} - 14580 \beta_{3} - 13392 \beta_{4} ) q^{87} + ( 22934 - 233881 \beta_{1} + 7363 \beta_{2} - 33275 \beta_{3} - 2548 \beta_{4} ) q^{88} + ( 221606 + 375585 \beta_{1} - 12543 \beta_{2} - 33662 \beta_{3} - 37888 \beta_{4} ) q^{89} + ( 215784 - 52488 \beta_{1} + 5832 \beta_{2} - 7290 \beta_{3} + 2916 \beta_{4} ) q^{90} + ( -5495056 + 131090 \beta_{1} + 43414 \beta_{2} + 30604 \beta_{3} - 22260 \beta_{4} ) q^{91} + ( 657018 + 24334 \beta_{1} + 12167 \beta_{3} + 12167 \beta_{4} ) q^{92} + ( 1067580 - 132462 \beta_{1} + 20088 \beta_{2} + 37800 \beta_{3} + 11610 \beta_{4} ) q^{93} + ( -1432588 + 7068 \beta_{1} - 70090 \beta_{2} - 35188 \beta_{3} - 39786 \beta_{4} ) q^{94} + ( -1212362 + 66138 \beta_{1} - 11495 \beta_{2} + 1866 \beta_{3} + 4877 \beta_{4} ) q^{95} + ( 1537272 + 196803 \beta_{1} + 16929 \beta_{2} - 12798 \beta_{3} - 14337 \beta_{4} ) q^{96} + ( -3153854 + 490412 \beta_{1} + 4426 \beta_{2} - 782 \beta_{3} - 16134 \beta_{4} ) q^{97} + ( 2349564 + 553197 \beta_{1} + 34650 \beta_{2} - 17220 \beta_{3} - 8554 \beta_{4} ) q^{98} + ( -160380 - 85293 \beta_{1} + 2916 \beta_{2} - 21141 \beta_{3} - 3645 \beta_{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q - 135q^{3} + 270q^{4} - 266q^{5} - 496q^{7} + 1422q^{8} + 3645q^{9} + O(q^{10}) \) \( 5q - 135q^{3} + 270q^{4} - 266q^{5} - 496q^{7} + 1422q^{8} + 3645q^{9} + 1452q^{10} - 1148q^{11} - 7290q^{12} - 642q^{13} + 5756q^{14} + 7182q^{15} - 22606q^{16} - 5798q^{17} - 6036q^{19} - 27376q^{20} + 13392q^{21} - 97896q^{22} + 60835q^{23} - 38394q^{24} - 262477q^{25} - 355992q^{26} - 98415q^{27} - 507124q^{28} - 169162q^{29} - 39204q^{30} - 199640q^{31} - 284794q^{32} + 30996q^{33} - 1027740q^{34} - 137680q^{35} + 196830q^{36} - 202002q^{37} - 554924q^{38} + 17334q^{39} - 340904q^{40} + 541282q^{41} - 155412q^{42} - 909596q^{43} - 1236032q^{44} - 193914q^{45} + 80208q^{47} + 610362q^{48} + 850589q^{49} - 941416q^{50} + 156546q^{51} + 146940q^{52} - 278138q^{53} - 933560q^{55} - 539932q^{56} + 162972q^{57} - 3522712q^{58} - 3177380q^{59} + 739152q^{60} + 147782q^{61} + 4606456q^{62} - 361584q^{63} - 4142622q^{64} + 3877332q^{65} + 2643192q^{66} - 464916q^{67} + 7513072q^{68} - 1642545q^{69} + 2093200q^{70} + 1576792q^{71} + 1036638q^{72} - 38190q^{73} + 12164864q^{74} + 7086879q^{75} + 6889436q^{76} + 10332384q^{77} + 9611784q^{78} - 3913336q^{79} + 6334776q^{80} + 2657205q^{81} + 6799360q^{82} + 15774716q^{83} + 13692348q^{84} - 8520740q^{85} + 24874084q^{86} + 4567374q^{87} + 53216q^{88} + 1116482q^{89} + 1058508q^{90} - 27369552q^{91} + 3285090q^{92} + 5390280q^{93} - 7153744q^{94} - 6067832q^{95} + 7689438q^{96} - 15738566q^{97} + 11730488q^{98} - 836892q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - 455 x^{3} - 474 x^{2} + 42284 x + 127016\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{4} + 142 \nu^{3} + 467 \nu^{2} - 37904 \nu - 76396 \)\()/1552\)
\(\beta_{3}\)\(=\)\((\)\( 11 \nu^{4} - 10 \nu^{3} - 3585 \nu^{2} + 2560 \nu + 117124 \)\()/1552\)
\(\beta_{4}\)\(=\)\((\)\( -11 \nu^{4} + 10 \nu^{3} + 5137 \nu^{2} - 5664 \nu - 399588 \)\()/1552\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{4} + \beta_{3} + 2 \beta_{1} + 182\)
\(\nu^{3}\)\(=\)\(-\beta_{4} + 11 \beta_{2} + 265 \beta_{1} + 284\)
\(\nu^{4}\)\(=\)\(325 \beta_{4} + 467 \beta_{3} + 10 \beta_{2} + 660 \beta_{1} + 48926\)

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
−17.8260
−9.64907
−3.24502
12.4672
18.2528
−17.8260 −27.0000 189.767 −31.0428 481.302 −1247.35 −1101.05 729.000 553.369
1.2 −9.64907 −27.0000 −34.8955 −306.939 260.525 733.069 1571.79 729.000 2961.68
1.3 −3.24502 −27.0000 −117.470 168.083 87.6156 149.817 796.555 729.000 −545.434
1.4 12.4672 −27.0000 27.4323 −40.8788 −336.616 1126.70 −1253.80 729.000 −509.647
1.5 18.2528 −27.0000 205.166 −55.2224 −492.827 −1258.24 1408.51 729.000 −1007.97
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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.a 5
3.b odd 2 1 207.8.a.a 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
69.8.a.a 5 1.a even 1 1 trivial
207.8.a.a 5 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{5} - 455 T_{2}^{3} - 474 T_{2}^{2} + 42284 T_{2} + 127016 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(69))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 127016 + 42284 T - 474 T^{2} - 455 T^{3} + T^{5} \)
$3$ \( ( 27 + T )^{5} \)
$5$ \( -3615360000 - 260643200 T - 5761760 T^{2} - 28696 T^{3} + 266 T^{4} + T^{5} \)
$7$ \( -194204974150720 + 1423528608720 T - 510098608 T^{2} - 2361144 T^{3} + 496 T^{4} + T^{5} \)
$11$ \( 154691619430400 + 8190801003520 T - 8448479456 T^{2} - 17900584 T^{3} + 1148 T^{4} + T^{5} \)
$13$ \( 16949964907397805984 + 8797382529202512 T - 46113337584 T^{2} - 206597592 T^{3} + 642 T^{4} + T^{5} \)
$17$ \( \)\(11\!\cdots\!80\)\( + 269929356591517120 T - 11180076292112 T^{2} - 1285951984 T^{3} + 5798 T^{4} + T^{5} \)
$19$ \( -\)\(14\!\cdots\!36\)\( + 185171606250158832 T + 3874831298480 T^{2} - 976972672 T^{3} + 6036 T^{4} + T^{5} \)
$23$ \( ( -12167 + T )^{5} \)
$29$ \( \)\(24\!\cdots\!88\)\( - 20455752789771350832 T - 1428794454126960 T^{2} - 5923396536 T^{3} + 169162 T^{4} + T^{5} \)
$31$ \( -\)\(23\!\cdots\!28\)\( - \)\(10\!\cdots\!88\)\( T - 13426584591784320 T^{2} - 43994925216 T^{3} + 199640 T^{4} + T^{5} \)
$37$ \( \)\(82\!\cdots\!20\)\( + \)\(80\!\cdots\!60\)\( T - 12266792719102368 T^{2} - 105154566272 T^{3} + 202002 T^{4} + T^{5} \)
$41$ \( \)\(16\!\cdots\!52\)\( - \)\(70\!\cdots\!52\)\( T + 66617897199338800 T^{2} - 103444637496 T^{3} - 541282 T^{4} + T^{5} \)
$43$ \( \)\(66\!\cdots\!00\)\( + \)\(89\!\cdots\!12\)\( T - 636892817243404752 T^{2} - 720482936448 T^{3} + 909596 T^{4} + T^{5} \)
$47$ \( -\)\(24\!\cdots\!00\)\( + \)\(44\!\cdots\!60\)\( T + 143091146720317440 T^{2} - 1392905639296 T^{3} - 80208 T^{4} + T^{5} \)
$53$ \( \)\(10\!\cdots\!60\)\( + \)\(79\!\cdots\!40\)\( T - 499432731442902464 T^{2} - 3057292616632 T^{3} + 278138 T^{4} + T^{5} \)
$59$ \( \)\(75\!\cdots\!64\)\( - \)\(53\!\cdots\!24\)\( T - 13664970599888805632 T^{2} - 3306712956160 T^{3} + 3177380 T^{4} + T^{5} \)
$61$ \( -\)\(14\!\cdots\!84\)\( + \)\(22\!\cdots\!08\)\( T - 7818212974425020704 T^{2} - 9781093585984 T^{3} - 147782 T^{4} + T^{5} \)
$67$ \( -\)\(79\!\cdots\!40\)\( + \)\(18\!\cdots\!32\)\( T - 290932021540136880 T^{2} - 27027287614400 T^{3} + 464916 T^{4} + T^{5} \)
$71$ \( \)\(49\!\cdots\!52\)\( + \)\(15\!\cdots\!48\)\( T - 179997306641763584 T^{2} - 3496700575968 T^{3} - 1576792 T^{4} + T^{5} \)
$73$ \( \)\(11\!\cdots\!64\)\( + \)\(36\!\cdots\!64\)\( T - 1042990056947080848 T^{2} - 7488464098136 T^{3} + 38190 T^{4} + T^{5} \)
$79$ \( -\)\(36\!\cdots\!20\)\( + \)\(60\!\cdots\!88\)\( T - 7813983134821600528 T^{2} - 20007982009272 T^{3} + 3913336 T^{4} + T^{5} \)
$83$ \( \)\(18\!\cdots\!20\)\( - \)\(12\!\cdots\!40\)\( T + \)\(15\!\cdots\!08\)\( T^{2} + 58536185513432 T^{3} - 15774716 T^{4} + T^{5} \)
$89$ \( -\)\(25\!\cdots\!96\)\( + \)\(95\!\cdots\!32\)\( T + \)\(40\!\cdots\!64\)\( T^{2} - 117042154036144 T^{3} - 1116482 T^{4} + T^{5} \)
$97$ \( \)\(11\!\cdots\!20\)\( + \)\(21\!\cdots\!28\)\( T - \)\(62\!\cdots\!88\)\( T^{2} - 22269450612824 T^{3} + 15738566 T^{4} + T^{5} \)
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