Properties

Label 38.8.a.e
Level $38$
Weight $8$
Character orbit 38.a
Self dual yes
Analytic conductor $11.871$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 38 = 2 \cdot 19 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 38.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(11.8706309684\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \(x^{4} - 9097 x^{2} - 110520 x + 10368000\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 3 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 8 q^{2} + ( 3 - \beta_{1} ) q^{3} + 64 q^{4} + ( -70 - 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{5} + ( 24 - 8 \beta_{1} ) q^{6} + ( 622 + \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{7} + 512 q^{8} + ( 2370 + 17 \beta_{1} + 14 \beta_{2} + 6 \beta_{3} ) q^{9} +O(q^{10})\) \( q + 8 q^{2} + ( 3 - \beta_{1} ) q^{3} + 64 q^{4} + ( -70 - 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{5} + ( 24 - 8 \beta_{1} ) q^{6} + ( 622 + \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{7} + 512 q^{8} + ( 2370 + 17 \beta_{1} + 14 \beta_{2} + 6 \beta_{3} ) q^{9} + ( -560 - 16 \beta_{1} - 8 \beta_{2} - 8 \beta_{3} ) q^{10} + ( 1324 - 10 \beta_{1} - 13 \beta_{2} + 7 \beta_{3} ) q^{11} + ( 192 - 64 \beta_{1} ) q^{12} + ( 1346 + 87 \beta_{1} - 20 \beta_{2} - 21 \beta_{3} ) q^{13} + ( 4976 + 8 \beta_{1} + 8 \beta_{2} + 16 \beta_{3} ) q^{14} + ( 6630 + 264 \beta_{1} + 42 \beta_{2} - 48 \beta_{3} ) q^{15} + 4096 q^{16} + ( 5767 - 43 \beta_{1} - 85 \beta_{2} + 49 \beta_{3} ) q^{17} + ( 18960 + 136 \beta_{1} + 112 \beta_{2} + 48 \beta_{3} ) q^{18} + 6859 q^{19} + ( -4480 - 128 \beta_{1} - 64 \beta_{2} - 64 \beta_{3} ) q^{20} + ( 774 - 833 \beta_{1} - 8 \beta_{2} + 117 \beta_{3} ) q^{21} + ( 10592 - 80 \beta_{1} - 104 \beta_{2} + 56 \beta_{3} ) q^{22} + ( 682 + 137 \beta_{1} + 230 \beta_{2} - 203 \beta_{3} ) q^{23} + ( 1536 - 512 \beta_{1} ) q^{24} + ( 28075 + 288 \beta_{1} + 279 \beta_{2} + 9 \beta_{3} ) q^{25} + ( 10768 + 696 \beta_{1} - 160 \beta_{2} - 168 \beta_{3} ) q^{26} + ( -54783 - 2383 \beta_{1} - 594 \beta_{2} + 234 \beta_{3} ) q^{27} + ( 39808 + 64 \beta_{1} + 64 \beta_{2} + 128 \beta_{3} ) q^{28} + ( -30622 + 643 \beta_{1} - 238 \beta_{2} - 295 \beta_{3} ) q^{29} + ( 53040 + 2112 \beta_{1} + 336 \beta_{2} - 384 \beta_{3} ) q^{30} + ( 56446 + 2524 \beta_{1} + 84 \beta_{2} + 194 \beta_{3} ) q^{31} + 32768 q^{32} + ( 44124 + 138 \beta_{1} + 722 \beta_{2} + 540 \beta_{3} ) q^{33} + ( 46136 - 344 \beta_{1} - 680 \beta_{2} + 392 \beta_{3} ) q^{34} + ( -196490 - 268 \beta_{1} - 899 \beta_{2} - 539 \beta_{3} ) q^{35} + ( 151680 + 1088 \beta_{1} + 896 \beta_{2} + 384 \beta_{3} ) q^{36} + ( 38424 + 308 \beta_{1} + 1980 \beta_{2} + 790 \beta_{3} ) q^{37} + 54872 q^{38} + ( -437958 + 31 \beta_{1} - 958 \beta_{2} - 1785 \beta_{3} ) q^{39} + ( -35840 - 1024 \beta_{1} - 512 \beta_{2} - 512 \beta_{3} ) q^{40} + ( -263400 - 2634 \beta_{1} - 92 \beta_{2} + 468 \beta_{3} ) q^{41} + ( 6192 - 6664 \beta_{1} - 64 \beta_{2} + 936 \beta_{3} ) q^{42} + ( -209726 + 6160 \beta_{1} - 909 \beta_{2} + 491 \beta_{3} ) q^{43} + ( 84736 - 640 \beta_{1} - 832 \beta_{2} + 448 \beta_{3} ) q^{44} + ( -1040940 - 10674 \beta_{1} - 3897 \beta_{2} - 2547 \beta_{3} ) q^{45} + ( 5456 + 1096 \beta_{1} + 1840 \beta_{2} - 1624 \beta_{3} ) q^{46} + ( -255892 + 4636 \beta_{1} + 1905 \beta_{2} + 107 \beta_{3} ) q^{47} + ( 12288 - 4096 \beta_{1} ) q^{48} + ( -154678 - 1811 \beta_{1} + 1661 \beta_{2} + 2195 \beta_{3} ) q^{49} + ( 224600 + 2304 \beta_{1} + 2232 \beta_{2} + 72 \beta_{3} ) q^{50} + ( 181905 + 3225 \beta_{1} + 4472 \beta_{2} + 3600 \beta_{3} ) q^{51} + ( 86144 + 5568 \beta_{1} - 1280 \beta_{2} - 1344 \beta_{3} ) q^{52} + ( -81770 + 6143 \beta_{1} - 420 \beta_{2} - 329 \beta_{3} ) q^{53} + ( -438264 - 19064 \beta_{1} - 4752 \beta_{2} + 1872 \beta_{3} ) q^{54} + ( -55020 + 12036 \beta_{1} - 1587 \beta_{2} - 57 \beta_{3} ) q^{55} + ( 318464 + 512 \beta_{1} + 512 \beta_{2} + 1024 \beta_{3} ) q^{56} + ( 20577 - 6859 \beta_{1} ) q^{57} + ( -244976 + 5144 \beta_{1} - 1904 \beta_{2} - 2360 \beta_{3} ) q^{58} + ( 103503 + 15159 \beta_{1} - 1988 \beta_{2} - 4680 \beta_{3} ) q^{59} + ( 424320 + 16896 \beta_{1} + 2688 \beta_{2} - 3072 \beta_{3} ) q^{60} + ( 28156 - 5612 \beta_{1} - 4129 \beta_{2} - 4165 \beta_{3} ) q^{61} + ( 451568 + 20192 \beta_{1} + 672 \beta_{2} + 1552 \beta_{3} ) q^{62} + ( 2562444 + 10306 \beta_{1} + 12087 \beta_{2} + 8019 \beta_{3} ) q^{63} + 262144 q^{64} + ( 1121590 - 34876 \beta_{1} - 2768 \beta_{2} + 3082 \beta_{3} ) q^{65} + ( 352992 + 1104 \beta_{1} + 5776 \beta_{2} + 4320 \beta_{3} ) q^{66} + ( 1449819 - 6397 \beta_{1} - 202 \beta_{2} + 2254 \beta_{3} ) q^{67} + ( 369088 - 2752 \beta_{1} - 5440 \beta_{2} + 3136 \beta_{3} ) q^{68} + ( -621750 - 22821 \beta_{1} - 13798 \beta_{2} - 14301 \beta_{3} ) q^{69} + ( -1571920 - 2144 \beta_{1} - 7192 \beta_{2} - 4312 \beta_{3} ) q^{70} + ( 2648208 + 22494 \beta_{1} - 3046 \beta_{2} - 420 \beta_{3} ) q^{71} + ( 1213440 + 8704 \beta_{1} + 7168 \beta_{2} + 3072 \beta_{3} ) q^{72} + ( 986517 - 38101 \beta_{1} + 14475 \beta_{2} - 5423 \beta_{3} ) q^{73} + ( 307392 + 2464 \beta_{1} + 15840 \beta_{2} + 6320 \beta_{3} ) q^{74} + ( -920175 - 66811 \beta_{1} - 13338 \beta_{2} - 1998 \beta_{3} ) q^{75} + 438976 q^{76} + ( 1455110 - 23774 \beta_{1} - 4857 \beta_{2} + 4505 \beta_{3} ) q^{77} + ( -3503664 + 248 \beta_{1} - 7664 \beta_{2} - 14280 \beta_{3} ) q^{78} + ( 1407140 - 27438 \beta_{1} - 820 \beta_{2} + 14970 \beta_{3} ) q^{79} + ( -286720 - 8192 \beta_{1} - 4096 \beta_{2} - 4096 \beta_{3} ) q^{80} + ( 5143881 + 126104 \beta_{1} + 27620 \beta_{2} + 17700 \beta_{3} ) q^{81} + ( -2107200 - 21072 \beta_{1} - 736 \beta_{2} + 3744 \beta_{3} ) q^{82} + ( 1160924 + 33706 \beta_{1} + 11380 \beta_{2} - 5362 \beta_{3} ) q^{83} + ( 49536 - 53312 \beta_{1} - 512 \beta_{2} + 7488 \beta_{3} ) q^{84} + ( -499530 + 81192 \beta_{1} - 8859 \beta_{2} + 3741 \beta_{3} ) q^{85} + ( -1677808 + 49280 \beta_{1} - 7272 \beta_{2} + 3928 \beta_{3} ) q^{86} + ( -3621558 + 56523 \beta_{1} - 6810 \beta_{2} - 21729 \beta_{3} ) q^{87} + ( 677888 - 5120 \beta_{1} - 6656 \beta_{2} + 3584 \beta_{3} ) q^{88} + ( -703018 + 29866 \beta_{1} + 31242 \beta_{2} + 6692 \beta_{3} ) q^{89} + ( -8327520 - 85392 \beta_{1} - 31176 \beta_{2} - 20376 \beta_{3} ) q^{90} + ( -2213227 + 105353 \beta_{1} - 14058 \beta_{2} - 19826 \beta_{3} ) q^{91} + ( 43648 + 8768 \beta_{1} + 14720 \beta_{2} - 12992 \beta_{3} ) q^{92} + ( -10988310 - 123932 \beta_{1} - 34312 \beta_{2} - 3174 \beta_{3} ) q^{93} + ( -2047136 + 37088 \beta_{1} + 15240 \beta_{2} + 856 \beta_{3} ) q^{94} + ( -480130 - 13718 \beta_{1} - 6859 \beta_{2} - 6859 \beta_{3} ) q^{95} + ( 98304 - 32768 \beta_{1} ) q^{96} + ( -3596098 + 27372 \beta_{1} - 20838 \beta_{2} + 19950 \beta_{3} ) q^{97} + ( -1237424 - 14488 \beta_{1} + 13288 \beta_{2} + 17560 \beta_{3} ) q^{98} + ( -1980408 - 129468 \beta_{1} + 12751 \beta_{2} + 15717 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 32q^{2} + 12q^{3} + 256q^{4} - 279q^{5} + 96q^{6} + 2485q^{7} + 2048q^{8} + 9482q^{9} + O(q^{10}) \) \( 4q + 32q^{2} + 12q^{3} + 256q^{4} - 279q^{5} + 96q^{6} + 2485q^{7} + 2048q^{8} + 9482q^{9} - 2232q^{10} + 5269q^{11} + 768q^{12} + 5406q^{13} + 19880q^{14} + 26658q^{15} + 16384q^{16} + 22885q^{17} + 75856q^{18} + 27436q^{19} - 17856q^{20} + 2854q^{21} + 42152q^{22} + 3364q^{23} + 6144q^{24} + 112561q^{25} + 43248q^{26} - 220194q^{27} + 159040q^{28} - 122136q^{29} + 213264q^{30} + 225480q^{31} + 131072q^{32} + 176138q^{33} + 183080q^{34} - 785781q^{35} + 606848q^{36} + 154096q^{37} + 219488q^{38} - 1749220q^{39} - 142848q^{40} - 1054628q^{41} + 22832q^{42} - 840795q^{43} + 337216q^{44} - 4162563q^{45} + 26912q^{46} - 1021877q^{47} + 49152q^{48} - 621441q^{49} + 900488q^{50} + 724892q^{51} + 345984q^{52} - 326842q^{53} - 1761552q^{54} - 221553q^{55} + 1272320q^{56} + 82308q^{57} - 977088q^{58} + 421384q^{59} + 1706112q^{60} + 116825q^{61} + 1803840q^{62} + 10245825q^{63} + 1048576q^{64} + 4477428q^{65} + 1409104q^{66} + 5794566q^{67} + 1464640q^{68} - 2472196q^{69} - 6286248q^{70} + 10590626q^{71} + 4854784q^{72} + 3971389q^{73} + 1232768q^{74} - 3690042q^{75} + 1755904q^{76} + 5806573q^{77} - 13993760q^{78} + 5597800q^{79} - 1142784q^{80} + 20567744q^{81} - 8437024q^{82} + 4665800q^{83} + 182656q^{84} - 2014461q^{85} - 6726360q^{86} - 14449584q^{87} + 2697728q^{88} - 2794214q^{89} - 33300504q^{90} - 8827314q^{91} + 215296q^{92} - 43981204q^{93} - 8175016q^{94} - 1913661q^{95} + 393216q^{96} - 14445130q^{97} - 4971528q^{98} - 7940315q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 9097 x^{2} - 110520 x + 10368000\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 30 \nu^{2} - 7627 \nu - 219060 \)\()/1140\)
\(\beta_{3}\)\(=\)\((\)\( -7 \nu^{3} + 360 \nu^{2} + 40279 \nu - 1058940 \)\()/3420\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(6 \beta_{3} + 14 \beta_{2} + 23 \beta_{1} + 4548\)
\(\nu^{3}\)\(=\)\(-180 \beta_{3} + 720 \beta_{2} + 6937 \beta_{1} + 82620\)

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
95.4845
29.4051
−48.0684
−76.8211
8.00000 −92.4845 64.0000 −426.367 −739.876 875.686 512.000 6366.38 −3410.93
1.2 8.00000 −26.4051 64.0000 139.358 −211.240 458.899 512.000 −1489.77 1114.87
1.3 8.00000 51.0684 64.0000 338.533 408.547 −143.677 512.000 420.979 2708.27
1.4 8.00000 79.8211 64.0000 −330.525 638.569 1294.09 512.000 4184.42 −2644.20
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(19\) \(-1\)

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 38.8.a.e 4
3.b odd 2 1 342.8.a.o 4
4.b odd 2 1 304.8.a.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.8.a.e 4 1.a even 1 1 trivial
304.8.a.e 4 4.b odd 2 1
342.8.a.o 4 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 12 T_{3}^{3} - 9043 T_{3}^{2} + 164994 T_{3} + 9954648 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(38))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 8 T )^{4} \)
$3$ \( 1 - 12 T - 295 T^{2} + 86262 T^{3} - 901620 T^{4} + 188654994 T^{5} - 1410975855 T^{6} - 125524238436 T^{7} + 22876792454961 T^{8} \)
$5$ \( 1 + 279 T + 138890 T^{2} + 33752025 T^{3} + 16143011250 T^{4} + 2636876953125 T^{5} + 847717285156250 T^{6} + 133037567138671875 T^{7} + 37252902984619140625 T^{8} \)
$7$ \( 1 - 2485 T + 5045419 T^{2} - 6353668364 T^{3} + 6879076274144 T^{4} - 5232519105493652 T^{5} + 3421919577990728731 T^{6} - \)\(13\!\cdots\!95\)\( T^{7} + \)\(45\!\cdots\!01\)\( T^{8} \)
$11$ \( 1 - 5269 T + 63505124 T^{2} - 263253354585 T^{3} + 1803977512557558 T^{4} - 5130063137121529035 T^{5} + \)\(24\!\cdots\!84\)\( T^{6} - \)\(38\!\cdots\!59\)\( T^{7} + \)\(14\!\cdots\!81\)\( T^{8} \)
$13$ \( 1 - 5406 T + 98520175 T^{2} + 402392756440 T^{3} + 1378699571519292 T^{4} + 25249548718152199480 T^{5} + \)\(38\!\cdots\!75\)\( T^{6} - \)\(13\!\cdots\!78\)\( T^{7} + \)\(15\!\cdots\!21\)\( T^{8} \)
$17$ \( 1 - 22885 T + 739754033 T^{2} - 19792467663090 T^{3} + 456808640899749018 T^{4} - \)\(81\!\cdots\!70\)\( T^{5} + \)\(12\!\cdots\!57\)\( T^{6} - \)\(15\!\cdots\!45\)\( T^{7} + \)\(28\!\cdots\!41\)\( T^{8} \)
$19$ \( ( 1 - 6859 T )^{4} \)
$23$ \( 1 - 3364 T + 1546353407 T^{2} - 48695367446844 T^{3} + 16006630067785073616 T^{4} - \)\(16\!\cdots\!68\)\( T^{5} + \)\(17\!\cdots\!63\)\( T^{6} - \)\(13\!\cdots\!72\)\( T^{7} + \)\(13\!\cdots\!81\)\( T^{8} \)
$29$ \( 1 + 122136 T + 55100568923 T^{2} + 6538523897630628 T^{3} + \)\(13\!\cdots\!72\)\( T^{4} + \)\(11\!\cdots\!52\)\( T^{5} + \)\(16\!\cdots\!63\)\( T^{6} + \)\(62\!\cdots\!44\)\( T^{7} + \)\(88\!\cdots\!61\)\( T^{8} \)
$31$ \( 1 - 225480 T + 69094969516 T^{2} - 15136676725778216 T^{3} + \)\(26\!\cdots\!38\)\( T^{4} - \)\(41\!\cdots\!76\)\( T^{5} + \)\(52\!\cdots\!36\)\( T^{6} - \)\(46\!\cdots\!80\)\( T^{7} + \)\(57\!\cdots\!41\)\( T^{8} \)
$37$ \( 1 - 154096 T + 43964761900 T^{2} + 22759340129468464 T^{3} + \)\(17\!\cdots\!50\)\( T^{4} + \)\(21\!\cdots\!12\)\( T^{5} + \)\(39\!\cdots\!00\)\( T^{6} - \)\(13\!\cdots\!52\)\( T^{7} + \)\(81\!\cdots\!21\)\( T^{8} \)
$41$ \( 1 + 1054628 T + 1094991857624 T^{2} + 628358421549336204 T^{3} + \)\(34\!\cdots\!66\)\( T^{4} + \)\(12\!\cdots\!24\)\( T^{5} + \)\(41\!\cdots\!64\)\( T^{6} + \)\(77\!\cdots\!48\)\( T^{7} + \)\(14\!\cdots\!21\)\( T^{8} \)
$43$ \( 1 + 840795 T + 879527106724 T^{2} + 530888279209871203 T^{3} + \)\(36\!\cdots\!94\)\( T^{4} + \)\(14\!\cdots\!21\)\( T^{5} + \)\(64\!\cdots\!76\)\( T^{6} + \)\(16\!\cdots\!85\)\( T^{7} + \)\(54\!\cdots\!01\)\( T^{8} \)
$47$ \( 1 + 1021877 T + 1972473290900 T^{2} + 1286140928961756945 T^{3} + \)\(14\!\cdots\!82\)\( T^{4} + \)\(65\!\cdots\!35\)\( T^{5} + \)\(50\!\cdots\!00\)\( T^{6} + \)\(13\!\cdots\!19\)\( T^{7} + \)\(65\!\cdots\!61\)\( T^{8} \)
$53$ \( 1 + 326842 T + 4350954270263 T^{2} + 1116130028742868416 T^{3} + \)\(74\!\cdots\!72\)\( T^{4} + \)\(13\!\cdots\!92\)\( T^{5} + \)\(60\!\cdots\!47\)\( T^{6} + \)\(52\!\cdots\!26\)\( T^{7} + \)\(19\!\cdots\!61\)\( T^{8} \)
$59$ \( 1 - 421384 T + 4416605620265 T^{2} + 5125206026271641286 T^{3} + \)\(62\!\cdots\!48\)\( T^{4} + \)\(12\!\cdots\!34\)\( T^{5} + \)\(27\!\cdots\!65\)\( T^{6} - \)\(64\!\cdots\!56\)\( T^{7} + \)\(38\!\cdots\!21\)\( T^{8} \)
$61$ \( 1 - 116825 T + 9319855015114 T^{2} - 1161006564160465283 T^{3} + \)\(41\!\cdots\!66\)\( T^{4} - \)\(36\!\cdots\!43\)\( T^{5} + \)\(92\!\cdots\!74\)\( T^{6} - \)\(36\!\cdots\!25\)\( T^{7} + \)\(97\!\cdots\!81\)\( T^{8} \)
$67$ \( 1 - 5794566 T + 35662430474173 T^{2} - \)\(11\!\cdots\!86\)\( T^{3} + \)\(36\!\cdots\!80\)\( T^{4} - \)\(69\!\cdots\!78\)\( T^{5} + \)\(13\!\cdots\!17\)\( T^{6} - \)\(12\!\cdots\!22\)\( T^{7} + \)\(13\!\cdots\!41\)\( T^{8} \)
$71$ \( 1 - 10590626 T + 72774649146920 T^{2} - \)\(33\!\cdots\!66\)\( T^{3} + \)\(11\!\cdots\!82\)\( T^{4} - \)\(30\!\cdots\!06\)\( T^{5} + \)\(60\!\cdots\!20\)\( T^{6} - \)\(79\!\cdots\!46\)\( T^{7} + \)\(68\!\cdots\!61\)\( T^{8} \)
$73$ \( 1 - 3971389 T + 11335631623573 T^{2} - 21295071158761180526 T^{3} + \)\(19\!\cdots\!02\)\( T^{4} - \)\(23\!\cdots\!22\)\( T^{5} + \)\(13\!\cdots\!57\)\( T^{6} - \)\(53\!\cdots\!97\)\( T^{7} + \)\(14\!\cdots\!81\)\( T^{8} \)
$79$ \( 1 - 5597800 T + 47755668016648 T^{2} - \)\(29\!\cdots\!80\)\( T^{3} + \)\(11\!\cdots\!02\)\( T^{4} - \)\(56\!\cdots\!20\)\( T^{5} + \)\(17\!\cdots\!88\)\( T^{6} - \)\(39\!\cdots\!00\)\( T^{7} + \)\(13\!\cdots\!61\)\( T^{8} \)
$83$ \( 1 - 4665800 T + 90191405937032 T^{2} - \)\(34\!\cdots\!20\)\( T^{3} + \)\(35\!\cdots\!58\)\( T^{4} - \)\(94\!\cdots\!40\)\( T^{5} + \)\(66\!\cdots\!28\)\( T^{6} - \)\(93\!\cdots\!00\)\( T^{7} + \)\(54\!\cdots\!41\)\( T^{8} \)
$89$ \( 1 + 2794214 T + 98763681921620 T^{2} + \)\(34\!\cdots\!66\)\( T^{3} + \)\(50\!\cdots\!58\)\( T^{4} + \)\(15\!\cdots\!14\)\( T^{5} + \)\(19\!\cdots\!20\)\( T^{6} + \)\(24\!\cdots\!46\)\( T^{7} + \)\(38\!\cdots\!81\)\( T^{8} \)
$97$ \( 1 + 14445130 T + 287388834797392 T^{2} + \)\(30\!\cdots\!06\)\( T^{3} + \)\(33\!\cdots\!74\)\( T^{4} + \)\(24\!\cdots\!78\)\( T^{5} + \)\(18\!\cdots\!48\)\( T^{6} + \)\(76\!\cdots\!10\)\( T^{7} + \)\(42\!\cdots\!61\)\( T^{8} \)
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