Properties

Label 33.8.a.e
Level 33
Weight 8
Character orbit 33.a
Self dual yes
Analytic conductor 10.309
Analytic rank 0
Dimension 4
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 33 = 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 33.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(10.3087058410\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \(x^{4} - x^{3} - 510 x^{2} - 1544 x + 28880\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 4 - \beta_{1} ) q^{2} + 27 q^{3} + ( 142 - 2 \beta_{1} + \beta_{2} ) q^{4} + ( 74 + 10 \beta_{1} + \beta_{3} ) q^{5} + ( 108 - 27 \beta_{1} ) q^{6} + ( 230 - 32 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{7} + ( 646 - 126 \beta_{1} + \beta_{2} - 6 \beta_{3} ) q^{8} + 729 q^{9} +O(q^{10})\) \( q + ( 4 - \beta_{1} ) q^{2} + 27 q^{3} + ( 142 - 2 \beta_{1} + \beta_{2} ) q^{4} + ( 74 + 10 \beta_{1} + \beta_{3} ) q^{5} + ( 108 - 27 \beta_{1} ) q^{6} + ( 230 - 32 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{7} + ( 646 - 126 \beta_{1} + \beta_{2} - 6 \beta_{3} ) q^{8} + 729 q^{9} + ( -2224 - 70 \beta_{1} - 20 \beta_{2} + 14 \beta_{3} ) q^{10} + 1331 q^{11} + ( 3834 - 54 \beta_{1} + 27 \beta_{2} ) q^{12} + ( -514 + 206 \beta_{1} - 28 \beta_{2} + 15 \beta_{3} ) q^{13} + ( 8844 + 18 \beta_{1} + 54 \beta_{2} - 16 \beta_{3} ) q^{14} + ( 1998 + 270 \beta_{1} + 27 \beta_{3} ) q^{15} + ( 16374 - 398 \beta_{1} + 57 \beta_{2} - 90 \beta_{3} ) q^{16} + ( 8000 + 842 \beta_{1} - 14 \beta_{2} - 21 \beta_{3} ) q^{17} + ( 2916 - 729 \beta_{1} ) q^{18} + ( -3430 + 810 \beta_{1} - 126 \beta_{2} + 43 \beta_{3} ) q^{19} + ( -1948 + 3740 \beta_{1} - 50 \beta_{2} + 188 \beta_{3} ) q^{20} + ( 6210 - 864 \beta_{1} - 54 \beta_{2} - 54 \beta_{3} ) q^{21} + ( 5324 - 1331 \beta_{1} ) q^{22} + ( 27864 + 3414 \beta_{1} + 12 \beta_{2} - 149 \beta_{3} ) q^{23} + ( 17442 - 3402 \beta_{1} + 27 \beta_{2} - 162 \beta_{3} ) q^{24} + ( 18599 - 1260 \beta_{1} + 280 \beta_{2} + 106 \beta_{3} ) q^{25} + ( -56376 + 3710 \beta_{1} - 328 \beta_{2} + 378 \beta_{3} ) q^{26} + 19683 q^{27} + ( 5472 - 11432 \beta_{1} + 344 \beta_{2} - 292 \beta_{3} ) q^{28} + ( -25452 - 3254 \beta_{1} - 110 \beta_{2} - 349 \beta_{3} ) q^{29} + ( -60048 - 1890 \beta_{1} - 540 \beta_{2} + 378 \beta_{3} ) q^{30} + ( -7016 + 3832 \beta_{1} + 744 \beta_{2} + 532 \beta_{3} ) q^{31} + ( 86774 - 8222 \beta_{1} + 1113 \beta_{2} - 834 \beta_{3} ) q^{32} + 35937 q^{33} + ( -183436 - 8564 \beta_{1} - 618 \beta_{2} - 210 \beta_{3} ) q^{34} + ( -180884 - 100 \beta_{1} - 780 \beta_{2} + 234 \beta_{3} ) q^{35} + ( 103518 - 1458 \beta_{1} + 729 \beta_{2} ) q^{36} + ( -126982 + 8708 \beta_{1} - 364 \beta_{2} - 610 \beta_{3} ) q^{37} + ( -228932 + 17458 \beta_{1} - 1114 \beta_{2} + 1358 \beta_{3} ) q^{38} + ( -13878 + 5562 \beta_{1} - 756 \beta_{2} + 405 \beta_{3} ) q^{39} + ( -673420 + 13740 \beta_{1} - 3010 \beta_{2} + 1140 \beta_{3} ) q^{40} + ( 180316 + 13602 \beta_{1} + 310 \beta_{2} - 1477 \beta_{3} ) q^{41} + ( 238788 + 486 \beta_{1} + 1458 \beta_{2} - 432 \beta_{3} ) q^{42} + ( -56034 + 25034 \beta_{1} + 2814 \beta_{2} + 119 \beta_{3} ) q^{43} + ( 189002 - 2662 \beta_{1} + 1331 \beta_{2} ) q^{44} + ( 53946 + 7290 \beta_{1} + 729 \beta_{3} ) q^{45} + ( -757696 - 39660 \beta_{1} - 1936 \beta_{2} - 2158 \beta_{3} ) q^{46} + ( 496912 + 11674 \beta_{1} + 3428 \beta_{2} + 1677 \beta_{3} ) q^{47} + ( 442098 - 10746 \beta_{1} + 1539 \beta_{2} - 2430 \beta_{3} ) q^{48} + ( -193675 + 4216 \beta_{1} + 540 \beta_{2} - 1336 \beta_{3} ) q^{49} + ( 419516 - 46015 \beta_{1} - 80 \beta_{2} - 196 \beta_{3} ) q^{50} + ( 216000 + 22734 \beta_{1} - 378 \beta_{2} - 567 \beta_{3} ) q^{51} + ( -1121388 + 69708 \beta_{1} - 3578 \beta_{2} + 5340 \beta_{3} ) q^{52} + ( 250110 - 74378 \beta_{1} - 3908 \beta_{2} + 1883 \beta_{3} ) q^{53} + ( 78732 - 19683 \beta_{1} ) q^{54} + ( 98494 + 13310 \beta_{1} + 1331 \beta_{3} ) q^{55} + ( 1815952 - 31824 \beta_{1} + 7096 \beta_{2} - 4104 \beta_{3} ) q^{56} + ( -92610 + 21870 \beta_{1} - 3402 \beta_{2} + 1161 \beta_{3} ) q^{57} + ( 708708 + 36344 \beta_{1} + 6854 \beta_{2} - 4226 \beta_{3} ) q^{58} + ( 348820 - 54068 \beta_{1} - 11944 \beta_{2} + 582 \beta_{3} ) q^{59} + ( -52596 + 100980 \beta_{1} - 1350 \beta_{2} + 5076 \beta_{3} ) q^{60} + ( 1016210 - 66010 \beta_{1} + 56 \beta_{2} - 1057 \beta_{3} ) q^{61} + ( -929744 - 74184 \beta_{1} - 9896 \beta_{2} + 2984 \beta_{3} ) q^{62} + ( 167670 - 23328 \beta_{1} - 1458 \beta_{2} - 1458 \beta_{3} ) q^{63} + ( 414198 - 168510 \beta_{1} + 8153 \beta_{2} - 6834 \beta_{3} ) q^{64} + ( 1679452 - 100100 \beta_{1} + 5600 \beta_{2} - 1322 \beta_{3} ) q^{65} + ( 143748 - 35937 \beta_{1} ) q^{66} + ( 439412 - 36688 \beta_{1} - 1048 \beta_{2} + 3760 \beta_{3} ) q^{67} + ( 362636 + 159436 \beta_{1} + 13074 \beta_{2} + 3456 \beta_{3} ) q^{68} + ( 752328 + 92178 \beta_{1} + 324 \beta_{2} - 4023 \beta_{3} ) q^{69} + ( -757416 + 277180 \beta_{1} - 1460 \beta_{2} + 7956 \beta_{3} ) q^{70} + ( 1378400 + 62262 \beta_{1} + 4004 \beta_{2} - 7581 \beta_{3} ) q^{71} + ( 470934 - 91854 \beta_{1} + 729 \beta_{2} - 4374 \beta_{3} ) q^{72} + ( 1425918 - 94460 \beta_{1} + 8684 \beta_{2} - 6122 \beta_{3} ) q^{73} + ( -2761808 + 137150 \beta_{1} - 2244 \beta_{2} - 6356 \beta_{3} ) q^{74} + ( 502173 - 34020 \beta_{1} + 7560 \beta_{2} + 2862 \beta_{3} ) q^{75} + ( -4975208 + 252152 \beta_{1} - 13796 \beta_{2} + 20192 \beta_{3} ) q^{76} + ( 306130 - 42592 \beta_{1} - 2662 \beta_{2} - 2662 \beta_{3} ) q^{77} + ( -1522152 + 100170 \beta_{1} - 8856 \beta_{2} + 10206 \beta_{3} ) q^{78} + ( -1925610 - 18508 \beta_{1} - 10722 \beta_{2} + 16 \beta_{3} ) q^{79} + ( -6158316 + 543740 \beta_{1} - 15730 \beta_{2} + 9956 \beta_{3} ) q^{80} + 531441 q^{81} + ( -2737764 - 278928 \beta_{1} + 858 \beta_{2} - 22538 \beta_{3} ) q^{82} + ( 909448 - 186716 \beta_{1} + 19220 \beta_{2} + 16670 \beta_{3} ) q^{83} + ( 147744 - 308664 \beta_{1} + 9288 \beta_{2} - 7884 \beta_{3} ) q^{84} + ( 1430656 + 202160 \beta_{1} + 15300 \beta_{2} - 3556 \beta_{3} ) q^{85} + ( -6349644 - 317602 \beta_{1} - 29038 \beta_{2} - 15218 \beta_{3} ) q^{86} + ( -687204 - 87858 \beta_{1} - 2970 \beta_{2} - 9423 \beta_{3} ) q^{87} + ( 859826 - 167706 \beta_{1} + 1331 \beta_{2} - 7986 \beta_{3} ) q^{88} + ( 1121490 + 107288 \beta_{1} - 18280 \beta_{2} + 6580 \beta_{3} ) q^{89} + ( -1621296 - 51030 \beta_{1} - 14580 \beta_{2} + 10206 \beta_{3} ) q^{90} + ( -2325516 + 243332 \beta_{1} - 26036 \beta_{2} + 6982 \beta_{3} ) q^{91} + ( 3274352 + 572808 \beta_{1} + 61640 \beta_{2} + 476 \beta_{3} ) q^{92} + ( -189432 + 103464 \beta_{1} + 20088 \beta_{2} + 14364 \beta_{3} ) q^{93} + ( -662912 - 877660 \beta_{1} - 31872 \beta_{2} + 2910 \beta_{3} ) q^{94} + ( 5561860 - 398540 \beta_{1} + 20900 \beta_{2} - 9130 \beta_{3} ) q^{95} + ( 2342898 - 221994 \beta_{1} + 30051 \beta_{2} - 22518 \beta_{3} ) q^{96} + ( 414422 - 269112 \beta_{1} - 13116 \beta_{2} + 27384 \beta_{3} ) q^{97} + ( -1828004 + 90539 \beta_{1} + 8604 \beta_{2} - 21944 \beta_{3} ) q^{98} + 970299 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 15q^{2} + 108q^{3} + 565q^{4} + 306q^{5} + 405q^{6} + 890q^{7} + 2457q^{8} + 2916q^{9} + O(q^{10}) \) \( 4q + 15q^{2} + 108q^{3} + 565q^{4} + 306q^{5} + 405q^{6} + 890q^{7} + 2457q^{8} + 2916q^{9} - 8946q^{10} + 5324q^{11} + 15255q^{12} - 1822q^{13} + 35340q^{14} + 8262q^{15} + 65041q^{16} + 32856q^{17} + 10935q^{18} - 12784q^{19} - 4002q^{20} + 24030q^{21} + 19965q^{22} + 114858q^{23} + 66339q^{24} + 72856q^{25} - 221466q^{26} + 78732q^{27} + 10112q^{28} - 104952q^{29} - 241542q^{30} - 24976q^{31} + 337761q^{32} + 143748q^{33} - 741690q^{34} - 722856q^{35} + 411885q^{36} - 498856q^{37} - 897156q^{38} - 49194q^{39} - 2676930q^{40} + 734556q^{41} + 954180q^{42} - 201916q^{43} + 752015q^{44} + 223074q^{45} - 3068508q^{46} + 1995894q^{47} + 1756107q^{48} - 771024q^{49} + 1632129q^{50} + 887112q^{51} - 4412266q^{52} + 929970q^{53} + 295245q^{54} + 407286q^{55} + 7224888q^{56} - 345168q^{57} + 2864322q^{58} + 1353156q^{59} - 108054q^{60} + 3998774q^{61} - 3783264q^{62} + 648810q^{63} + 1480129q^{64} + 6612108q^{65} + 539055q^{66} + 1722008q^{67} + 1596906q^{68} + 3101166q^{69} - 2751024q^{70} + 5571858q^{71} + 1791153q^{72} + 5600528q^{73} - 10907838q^{74} + 1967112q^{75} - 19634884q^{76} + 1184590q^{77} - 5979582q^{78} - 7710226q^{79} - 24073794q^{80} + 2125764q^{81} - 11230842q^{82} + 3431856q^{83} + 273024q^{84} + 5909484q^{85} - 25687140q^{86} - 2833704q^{87} + 3270267q^{88} + 4611528q^{89} - 6521634q^{90} - 9032696q^{91} + 13608576q^{92} - 674352q^{93} - 3497436q^{94} + 21828000q^{95} + 9119547q^{96} + 1401692q^{97} - 7230081q^{98} + 3881196q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 510 x^{2} - 1544 x + 28880\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 6 \nu - 254 \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} - 11 \nu^{2} - 340 \nu + 1352 \)\()/6\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 6 \beta_{1} + 254\)
\(\nu^{3}\)\(=\)\(6 \beta_{3} + 11 \beta_{2} + 406 \beta_{1} + 1442\)

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
23.3791
6.33327
−11.0316
−17.6807
−19.3791 27.0000 247.551 336.012 −523.236 −879.192 −2316.79 729.000 −6511.62
1.2 −2.33327 27.0000 −122.556 −27.4165 −62.9982 860.612 584.614 729.000 63.9699
1.3 15.0316 27.0000 97.9505 367.278 405.855 −91.9512 −451.694 729.000 5520.80
1.4 21.6807 27.0000 342.055 −369.874 585.380 1000.53 4640.87 729.000 −8019.14
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(11\) \(-1\)

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 33.8.a.e 4
3.b odd 2 1 99.8.a.f 4
4.b odd 2 1 528.8.a.r 4
11.b odd 2 1 363.8.a.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.8.a.e 4 1.a even 1 1 trivial
99.8.a.f 4 3.b odd 2 1
363.8.a.f 4 11.b odd 2 1
528.8.a.r 4 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 15 T_{2}^{3} - 426 T_{2}^{2} + 5416 T_{2} + 14736 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(33))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 15 T + 86 T^{2} - 344 T^{3} + 3984 T^{4} - 44032 T^{5} + 1409024 T^{6} - 31457280 T^{7} + 268435456 T^{8} \)
$3$ \( ( 1 - 27 T )^{4} \)
$5$ \( 1 - 306 T + 166640 T^{2} - 29820950 T^{3} + 15081924750 T^{4} - 2329761718750 T^{5} + 1017089843750000 T^{6} - 145912170410156250 T^{7} + 37252902984619140625 T^{8} \)
$7$ \( 1 - 890 T + 2428648 T^{2} - 1513098682 T^{3} + 2713357113806 T^{4} - 1246101827870326 T^{5} + 1647165109428578152 T^{6} - \)\(49\!\cdots\!30\)\( T^{7} + \)\(45\!\cdots\!01\)\( T^{8} \)
$11$ \( ( 1 - 1331 T )^{4} \)
$13$ \( 1 + 1822 T + 143434048 T^{2} + 743315141738 T^{3} + 9963947917232846 T^{4} + 46641922807704302546 T^{5} + \)\(56\!\cdots\!72\)\( T^{6} + \)\(45\!\cdots\!86\)\( T^{7} + \)\(15\!\cdots\!21\)\( T^{8} \)
$17$ \( 1 - 32856 T + 1617694640 T^{2} - 35580276275048 T^{3} + 1006917500477444958 T^{4} - \)\(14\!\cdots\!04\)\( T^{5} + \)\(27\!\cdots\!60\)\( T^{6} - \)\(22\!\cdots\!52\)\( T^{7} + \)\(28\!\cdots\!41\)\( T^{8} \)
$19$ \( 1 + 12784 T + 1968859072 T^{2} + 47795746564128 T^{3} + 1965145444697364174 T^{4} + \)\(42\!\cdots\!92\)\( T^{5} + \)\(15\!\cdots\!12\)\( T^{6} + \)\(91\!\cdots\!96\)\( T^{7} + \)\(63\!\cdots\!41\)\( T^{8} \)
$23$ \( 1 - 114858 T + 9605711756 T^{2} - 598055876188978 T^{3} + 37322607621124682694 T^{4} - \)\(20\!\cdots\!66\)\( T^{5} + \)\(11\!\cdots\!04\)\( T^{6} - \)\(45\!\cdots\!34\)\( T^{7} + \)\(13\!\cdots\!81\)\( T^{8} \)
$29$ \( 1 + 104952 T + 52207854272 T^{2} + 4176821606180040 T^{3} + \)\(12\!\cdots\!34\)\( T^{4} + \)\(72\!\cdots\!60\)\( T^{5} + \)\(15\!\cdots\!32\)\( T^{6} + \)\(53\!\cdots\!08\)\( T^{7} + \)\(88\!\cdots\!61\)\( T^{8} \)
$31$ \( 1 + 24976 T + 43688791996 T^{2} + 2739115850747216 T^{3} + \)\(14\!\cdots\!22\)\( T^{4} + \)\(75\!\cdots\!76\)\( T^{5} + \)\(33\!\cdots\!16\)\( T^{6} + \)\(52\!\cdots\!56\)\( T^{7} + \)\(57\!\cdots\!41\)\( T^{8} \)
$37$ \( 1 + 498856 T + 381977216092 T^{2} + 132519305800020216 T^{3} + \)\(54\!\cdots\!66\)\( T^{4} + \)\(12\!\cdots\!28\)\( T^{5} + \)\(34\!\cdots\!88\)\( T^{6} + \)\(42\!\cdots\!72\)\( T^{7} + \)\(81\!\cdots\!21\)\( T^{8} \)
$41$ \( 1 - 734556 T + 582543752696 T^{2} - 174072649937861684 T^{3} + \)\(10\!\cdots\!66\)\( T^{4} - \)\(33\!\cdots\!04\)\( T^{5} + \)\(22\!\cdots\!56\)\( T^{6} - \)\(54\!\cdots\!96\)\( T^{7} + \)\(14\!\cdots\!21\)\( T^{8} \)
$43$ \( 1 + 201916 T + 337036845088 T^{2} - 169098883219921380 T^{3} + \)\(16\!\cdots\!34\)\( T^{4} - \)\(45\!\cdots\!60\)\( T^{5} + \)\(24\!\cdots\!12\)\( T^{6} + \)\(40\!\cdots\!88\)\( T^{7} + \)\(54\!\cdots\!01\)\( T^{8} \)
$47$ \( 1 - 1995894 T + 2541886433372 T^{2} - 2356429844123408062 T^{3} + \)\(18\!\cdots\!58\)\( T^{4} - \)\(11\!\cdots\!06\)\( T^{5} + \)\(65\!\cdots\!68\)\( T^{6} - \)\(25\!\cdots\!18\)\( T^{7} + \)\(65\!\cdots\!61\)\( T^{8} \)
$53$ \( 1 - 929970 T + 696841347632 T^{2} + 410483380995330858 T^{3} - \)\(19\!\cdots\!26\)\( T^{4} + \)\(48\!\cdots\!46\)\( T^{5} + \)\(96\!\cdots\!08\)\( T^{6} - \)\(15\!\cdots\!10\)\( T^{7} + \)\(19\!\cdots\!61\)\( T^{8} \)
$59$ \( 1 - 1353156 T + 641417092700 T^{2} - 638087859099231076 T^{3} + \)\(35\!\cdots\!18\)\( T^{4} - \)\(15\!\cdots\!44\)\( T^{5} + \)\(39\!\cdots\!00\)\( T^{6} - \)\(20\!\cdots\!04\)\( T^{7} + \)\(38\!\cdots\!21\)\( T^{8} \)
$61$ \( 1 - 3998774 T + 16202477625640 T^{2} - 36808547102293149042 T^{3} + \)\(80\!\cdots\!50\)\( T^{4} - \)\(11\!\cdots\!82\)\( T^{5} + \)\(16\!\cdots\!40\)\( T^{6} - \)\(12\!\cdots\!14\)\( T^{7} + \)\(97\!\cdots\!81\)\( T^{8} \)
$67$ \( 1 - 1722008 T + 22644499591660 T^{2} - 31607526991944054296 T^{3} + \)\(20\!\cdots\!06\)\( T^{4} - \)\(19\!\cdots\!08\)\( T^{5} + \)\(83\!\cdots\!40\)\( T^{6} - \)\(38\!\cdots\!36\)\( T^{7} + \)\(13\!\cdots\!41\)\( T^{8} \)
$71$ \( 1 - 5571858 T + 36809238259148 T^{2} - \)\(11\!\cdots\!98\)\( T^{3} + \)\(45\!\cdots\!94\)\( T^{4} - \)\(10\!\cdots\!18\)\( T^{5} + \)\(30\!\cdots\!88\)\( T^{6} - \)\(41\!\cdots\!18\)\( T^{7} + \)\(68\!\cdots\!61\)\( T^{8} \)
$73$ \( 1 - 5600528 T + 40596655684732 T^{2} - \)\(18\!\cdots\!84\)\( T^{3} + \)\(65\!\cdots\!02\)\( T^{4} - \)\(20\!\cdots\!48\)\( T^{5} + \)\(49\!\cdots\!88\)\( T^{6} - \)\(75\!\cdots\!44\)\( T^{7} + \)\(14\!\cdots\!81\)\( T^{8} \)
$79$ \( 1 + 7710226 T + 92181525864424 T^{2} + \)\(44\!\cdots\!70\)\( T^{3} + \)\(27\!\cdots\!30\)\( T^{4} + \)\(85\!\cdots\!30\)\( T^{5} + \)\(33\!\cdots\!44\)\( T^{6} + \)\(54\!\cdots\!54\)\( T^{7} + \)\(13\!\cdots\!61\)\( T^{8} \)
$83$ \( 1 - 3431856 T + 43757629855388 T^{2} - 89436476395110295728 T^{3} + \)\(91\!\cdots\!66\)\( T^{4} - \)\(24\!\cdots\!56\)\( T^{5} + \)\(32\!\cdots\!52\)\( T^{6} - \)\(68\!\cdots\!48\)\( T^{7} + \)\(54\!\cdots\!41\)\( T^{8} \)
$89$ \( 1 - 4611528 T + 150275359100156 T^{2} - \)\(46\!\cdots\!64\)\( T^{3} + \)\(92\!\cdots\!06\)\( T^{4} - \)\(20\!\cdots\!56\)\( T^{5} + \)\(29\!\cdots\!96\)\( T^{6} - \)\(39\!\cdots\!92\)\( T^{7} + \)\(38\!\cdots\!81\)\( T^{8} \)
$97$ \( 1 - 1401692 T + 169506769846084 T^{2} - \)\(11\!\cdots\!24\)\( T^{3} + \)\(13\!\cdots\!10\)\( T^{4} - \)\(94\!\cdots\!12\)\( T^{5} + \)\(11\!\cdots\!96\)\( T^{6} - \)\(73\!\cdots\!24\)\( T^{7} + \)\(42\!\cdots\!61\)\( T^{8} \)
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