Properties

Label 38.10.a.e
Level $38$
Weight $10$
Character orbit 38.a
Self dual yes
Analytic conductor $19.571$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 38 = 2 \cdot 19 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 38.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(19.5713617742\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \(x^{4} - 2 x^{3} - 8016 x^{2} - 155839 x + 2105804\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 3 \)
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.

\(f(q)\) \(=\) \( q + 16 q^{2} + ( 57 + \beta_{2} ) q^{3} + 256 q^{4} + ( 218 - \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{5} + ( 912 + 16 \beta_{2} ) q^{6} + ( 672 + 20 \beta_{1} + 9 \beta_{2} + \beta_{3} ) q^{7} + 4096 q^{8} + ( 12678 - 95 \beta_{1} + 53 \beta_{2} - 7 \beta_{3} ) q^{9} +O(q^{10})\) \( q + 16 q^{2} + ( 57 + \beta_{2} ) q^{3} + 256 q^{4} + ( 218 - \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{5} + ( 912 + 16 \beta_{2} ) q^{6} + ( 672 + 20 \beta_{1} + 9 \beta_{2} + \beta_{3} ) q^{7} + 4096 q^{8} + ( 12678 - 95 \beta_{1} + 53 \beta_{2} - 7 \beta_{3} ) q^{9} + ( 3488 - 16 \beta_{1} + 48 \beta_{2} + 16 \beta_{3} ) q^{10} + ( 29748 + 75 \beta_{1} - 121 \beta_{2} - 23 \beta_{3} ) q^{11} + ( 14592 + 256 \beta_{2} ) q^{12} + ( -1568 + 275 \beta_{1} + 238 \beta_{2} - 48 \beta_{3} ) q^{13} + ( 10752 + 320 \beta_{1} + 144 \beta_{2} + 16 \beta_{3} ) q^{14} + ( 88972 - 489 \beta_{1} - 8 \beta_{2} + 49 \beta_{3} ) q^{15} + 65536 q^{16} + ( 168631 + 20 \beta_{1} - 2050 \beta_{2} + 164 \beta_{3} ) q^{17} + ( 202848 - 1520 \beta_{1} + 848 \beta_{2} - 112 \beta_{3} ) q^{18} -130321 q^{19} + ( 55808 - 256 \beta_{1} + 768 \beta_{2} + 256 \beta_{3} ) q^{20} + ( 394438 + 1230 \beta_{1} - 4492 \beta_{2} - 119 \beta_{3} ) q^{21} + ( 475968 + 1200 \beta_{1} - 1936 \beta_{2} - 368 \beta_{3} ) q^{22} + ( 724276 + 885 \beta_{1} - 7382 \beta_{2} - 286 \beta_{3} ) q^{23} + ( 233472 + 4096 \beta_{2} ) q^{24} + ( 67043 + 3659 \beta_{1} - 297 \beta_{2} + 1171 \beta_{3} ) q^{25} + ( -25088 + 4400 \beta_{1} + 3808 \beta_{2} - 768 \beta_{3} ) q^{26} + ( 709589 - 14725 \beta_{1} + 18149 \beta_{2} - 249 \beta_{3} ) q^{27} + ( 172032 + 5120 \beta_{1} + 2304 \beta_{2} + 256 \beta_{3} ) q^{28} + ( 2073202 + 550 \beta_{1} + 852 \beta_{2} - 2317 \beta_{3} ) q^{29} + ( 1423552 - 7824 \beta_{1} - 128 \beta_{2} + 784 \beta_{3} ) q^{30} + ( 870276 - 9775 \beta_{1} + 17818 \beta_{2} + 1575 \beta_{3} ) q^{31} + 1048576 q^{32} + ( -1318954 + 21855 \beta_{1} + 22986 \beta_{2} - 1075 \beta_{3} ) q^{33} + ( 2698096 + 320 \beta_{1} - 32800 \beta_{2} + 2624 \beta_{3} ) q^{34} + ( 1919394 - 6443 \beta_{1} - 18741 \beta_{2} + 5483 \beta_{3} ) q^{35} + ( 3245568 - 24320 \beta_{1} + 13568 \beta_{2} - 1792 \beta_{3} ) q^{36} + ( -261758 + 59215 \beta_{1} - 20754 \beta_{2} - 2407 \beta_{3} ) q^{37} -2085136 q^{38} + ( 8491292 + 11925 \beta_{1} - 45366 \beta_{2} - 6388 \beta_{3} ) q^{39} + ( 892928 - 4096 \beta_{1} + 12288 \beta_{2} + 4096 \beta_{3} ) q^{40} + ( 2148182 - 68935 \beta_{1} + 39302 \beta_{2} + 6677 \beta_{3} ) q^{41} + ( 6311008 + 19680 \beta_{1} - 71872 \beta_{2} - 1904 \beta_{3} ) q^{42} + ( 3472138 - 40335 \beta_{1} - 6087 \beta_{2} - 12765 \beta_{3} ) q^{43} + ( 7615488 + 19200 \beta_{1} - 30976 \beta_{2} - 5888 \beta_{3} ) q^{44} + ( -2179336 - 37513 \beta_{1} + 122429 \beta_{2} - 13557 \beta_{3} ) q^{45} + ( 11588416 + 14160 \beta_{1} - 118112 \beta_{2} - 4576 \beta_{3} ) q^{46} + ( -9150076 + 15365 \beta_{1} - 132675 \beta_{2} + 26833 \beta_{3} ) q^{47} + ( 3735552 + 65536 \beta_{2} ) q^{48} + ( -7756834 + 35580 \beta_{1} - 67764 \beta_{2} - 2112 \beta_{3} ) q^{49} + ( 1072688 + 58544 \beta_{1} - 4752 \beta_{2} + 18736 \beta_{3} ) q^{50} + ( -50917929 + 181350 \beta_{1} + 98679 \beta_{2} + 24726 \beta_{3} ) q^{51} + ( -401408 + 70400 \beta_{1} + 60928 \beta_{2} - 12288 \beta_{3} ) q^{52} + ( -28392182 - 211250 \beta_{1} + 200314 \beta_{2} - 5549 \beta_{3} ) q^{53} + ( 11353424 - 235600 \beta_{1} + 290384 \beta_{2} - 3984 \beta_{3} ) q^{54} + ( -44498004 - 113927 \beta_{1} + 74041 \beta_{2} + 17917 \beta_{3} ) q^{55} + ( 2752512 + 81920 \beta_{1} + 36864 \beta_{2} + 4096 \beta_{3} ) q^{56} + ( -7428297 - 130321 \beta_{2} ) q^{57} + ( 33171232 + 8800 \beta_{1} + 13632 \beta_{2} - 37072 \beta_{3} ) q^{58} + ( -99211973 + 255595 \beta_{1} - 37063 \beta_{2} - 23037 \beta_{3} ) q^{59} + ( 22776832 - 125184 \beta_{1} - 2048 \beta_{2} + 12544 \beta_{3} ) q^{60} + ( -74655920 + 201455 \beta_{1} - 215807 \beta_{2} - 40083 \beta_{3} ) q^{61} + ( 13924416 - 156400 \beta_{1} + 285088 \beta_{2} + 25200 \beta_{3} ) q^{62} + ( -114680548 + 178455 \beta_{1} + 751 \beta_{2} - 3235 \beta_{3} ) q^{63} + 16777216 q^{64} + ( -72911878 - 425824 \beta_{1} - 229918 \beta_{2} - 8146 \beta_{3} ) q^{65} + ( -21103264 + 349680 \beta_{1} + 367776 \beta_{2} - 17200 \beta_{3} ) q^{66} + ( -28773199 - 305650 \beta_{1} - 770537 \beta_{2} - 58278 \beta_{3} ) q^{67} + ( 43169536 + 5120 \beta_{1} - 524800 \beta_{2} + 41984 \beta_{3} ) q^{68} + ( -167542508 + 824925 \beta_{1} + 674842 \beta_{2} + 28060 \beta_{3} ) q^{69} + ( 30710304 - 103088 \beta_{1} - 299856 \beta_{2} + 87728 \beta_{3} ) q^{70} + ( 850912 - 669390 \beta_{1} - 627986 \beta_{2} + 94124 \beta_{3} ) q^{71} + ( 51929088 - 389120 \beta_{1} + 217088 \beta_{2} - 28672 \beta_{3} ) q^{72} + ( 33945965 - 709870 \beta_{1} - 207296 \beta_{2} - 55158 \beta_{3} ) q^{73} + ( -4188128 + 947440 \beta_{1} - 332064 \beta_{2} - 38512 \beta_{3} ) q^{74} + ( 6670777 + 315801 \beta_{1} - 1312583 \beta_{2} + 55069 \beta_{3} ) q^{75} -33362176 q^{76} + ( 30611478 + 918985 \beta_{1} + 947013 \beta_{2} - 99511 \beta_{3} ) q^{77} + ( 135860672 + 190800 \beta_{1} - 725856 \beta_{2} - 102208 \beta_{3} ) q^{78} + ( 17015018 - 62905 \beta_{1} + 402324 \beta_{2} + 39373 \beta_{3} ) q^{79} + ( 14286848 - 65536 \beta_{1} + 196608 \beta_{2} + 65536 \beta_{3} ) q^{80} + ( 247135793 - 1435640 \beta_{1} + 3150996 \beta_{2} + 83152 \beta_{3} ) q^{81} + ( 34370912 - 1102960 \beta_{1} + 628832 \beta_{2} + 106832 \beta_{3} ) q^{82} + ( 343756056 - 199770 \beta_{1} - 740496 \beta_{2} - 59700 \beta_{3} ) q^{83} + ( 100976128 + 314880 \beta_{1} - 1149952 \beta_{2} - 30464 \beta_{3} ) q^{84} + ( 159920954 + 1503517 \beta_{1} + 640819 \beta_{2} + 379483 \beta_{3} ) q^{85} + ( 55554208 - 645360 \beta_{1} - 97392 \beta_{2} - 204240 \beta_{3} ) q^{86} + ( 159148436 + 199125 \beta_{1} + 2979110 \beta_{2} - 157552 \beta_{3} ) q^{87} + ( 121847808 + 307200 \beta_{1} - 495616 \beta_{2} - 94208 \beta_{3} ) q^{88} + ( 389375140 + 1442325 \beta_{1} + 144650 \beta_{2} + 16075 \beta_{3} ) q^{89} + ( -34869376 - 600208 \beta_{1} + 1958864 \beta_{2} - 216912 \beta_{3} ) q^{90} + ( 355102557 + 871825 \beta_{1} - 1181751 \beta_{2} - 397111 \beta_{3} ) q^{91} + ( 185414656 + 226560 \beta_{1} - 1889792 \beta_{2} - 73216 \beta_{3} ) q^{92} + ( 510361898 - 2907810 \beta_{1} + 2380754 \beta_{2} + 34724 \beta_{3} ) q^{93} + ( -146401216 + 245840 \beta_{1} - 2122800 \beta_{2} + 429328 \beta_{3} ) q^{94} + ( -28409978 + 130321 \beta_{1} - 390963 \beta_{2} - 130321 \beta_{3} ) q^{95} + ( 59768832 + 1048576 \beta_{2} ) q^{96} + ( 244332850 + 247540 \beta_{1} + 756606 \beta_{2} - 183388 \beta_{3} ) q^{97} + ( -124109344 + 569280 \beta_{1} - 1084224 \beta_{2} - 33792 \beta_{3} ) q^{98} + ( 123874300 - 1175575 \beta_{1} - 3661725 \beta_{2} + 91877 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 64q^{2} + 226q^{3} + 1024q^{4} + 866q^{5} + 3616q^{6} + 2670q^{7} + 16384q^{8} + 50606q^{9} + O(q^{10}) \) \( 4q + 64q^{2} + 226q^{3} + 1024q^{4} + 866q^{5} + 3616q^{6} + 2670q^{7} + 16384q^{8} + 50606q^{9} + 13856q^{10} + 119234q^{11} + 57856q^{12} - 6748q^{13} + 42720q^{14} + 355904q^{15} + 262144q^{16} + 678624q^{17} + 809696q^{18} - 521284q^{19} + 221696q^{20} + 1586736q^{21} + 1907744q^{22} + 2911868q^{23} + 925696q^{24} + 268766q^{25} - 107968q^{26} + 2802058q^{27} + 683520q^{28} + 8291104q^{29} + 5694464q^{30} + 3445468q^{31} + 4194304q^{32} - 5321788q^{33} + 10857984q^{34} + 7715058q^{35} + 12955136q^{36} - 1005524q^{37} - 8340544q^{38} + 34055900q^{39} + 3547136q^{40} + 8514124q^{41} + 25387776q^{42} + 13900726q^{43} + 30523904q^{44} - 8962202q^{45} + 46589888q^{46} - 36334954q^{47} + 14811136q^{48} - 30891808q^{49} + 4300256q^{50} - 203869074q^{51} - 1727488q^{52} - 113969356q^{53} + 44832928q^{54} - 178140098q^{55} + 10936320q^{56} - 29452546q^{57} + 132657664q^{58} - 396773766q^{59} + 91111424q^{60} - 298192066q^{61} + 55127488q^{62} - 458723694q^{63} + 67108864q^{64} - 291187676q^{65} - 85148608q^{66} - 113551722q^{67} + 173727744q^{68} - 671519716q^{69} + 123440928q^{70} + 4659620q^{71} + 207282176q^{72} + 136198452q^{73} - 16088384q^{74} + 29308274q^{75} - 133448704q^{76} + 120551886q^{77} + 544894400q^{78} + 67255424q^{79} + 56754176q^{80} + 982241180q^{81} + 136225984q^{82} + 1376505216q^{83} + 406204416q^{84} + 638402178q^{85} + 222411616q^{86} + 630635524q^{87} + 488382464q^{88} + 1557211260q^{89} - 143395232q^{90} + 1422773730q^{91} + 745438208q^{92} + 2036686084q^{93} - 581359264q^{94} - 112857986q^{95} + 236978176q^{96} + 975818188q^{97} - 494268928q^{98} + 502820650q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{3} - 8016 x^{2} - 155839 x + 2105804\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 2 \nu^{3} - 47 \nu^{2} - 12893 \nu - 62941 \)\()/693\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{2} + 57 \nu + 3970 \)\()/21\)
\(\beta_{3}\)\(=\)\((\)\( -2 \nu^{3} + 179 \nu^{2} + 22001 \nu - 470801 \)\()/693\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + 4 \beta_{2} + \beta_{1} + 14\)\()/24\)
\(\nu^{2}\)\(=\)\((\)\(19 \beta_{3} - 92 \beta_{2} + 19 \beta_{1} + 32026\)\()/8\)
\(\nu^{3}\)\(=\)\((\)\(3893 \beta_{3} + 9650 \beta_{2} + 8051 \beta_{1} + 1551688\)\()/12\)

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
−73.1023
97.8976
−31.9926
9.19740
16.0000 −206.847 256.000 −845.315 −3309.55 −6607.98 4096.00 23102.6 −13525.0
1.2 16.0000 55.3919 256.000 2263.93 886.270 5766.09 4096.00 −16614.7 36222.9
1.3 16.0000 110.471 256.000 −1298.23 1767.54 6626.40 4096.00 −7479.16 −20771.6
1.4 16.0000 266.984 256.000 745.613 4271.74 −3114.51 4096.00 51597.3 11929.8
\(n\): 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 38.10.a.e 4
3.b odd 2 1 342.10.a.i 4
4.b odd 2 1 304.10.a.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.10.a.e 4 1.a even 1 1 trivial
304.10.a.d 4 4.b odd 2 1
342.10.a.i 4 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 226 T_{3}^{3} - 39131 T_{3}^{2} + 8791740 T_{3} - 337930956 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(38))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 16 T )^{4} \)
$3$ \( 1 - 226 T + 39601 T^{2} - 4553334 T^{3} + 446161032 T^{4} - 89623273122 T^{5} + 15342238784889 T^{6} - 1723385031607062 T^{7} + 150094635296999121 T^{8} \)
$5$ \( 1 - 866 T + 4146845 T^{2} - 4758590450 T^{3} + 10421665474000 T^{4} - 9294121972656250 T^{5} + 15818958282470703125 T^{6} - \)\(64\!\cdots\!50\)\( T^{7} + \)\(14\!\cdots\!25\)\( T^{8} \)
$7$ \( 1 - 2670 T + 99717568 T^{2} - 206796374568 T^{3} + 5577453453641097 T^{4} - 8344979628341866776 T^{5} + \)\(16\!\cdots\!32\)\( T^{6} - \)\(17\!\cdots\!10\)\( T^{7} + \)\(26\!\cdots\!01\)\( T^{8} \)
$11$ \( 1 - 119234 T + 11711266961 T^{2} - 777508728576194 T^{3} + 44417401991215600432 T^{4} - \)\(18\!\cdots\!54\)\( T^{5} + \)\(65\!\cdots\!41\)\( T^{6} - \)\(15\!\cdots\!14\)\( T^{7} + \)\(30\!\cdots\!61\)\( T^{8} \)
$13$ \( 1 + 6748 T + 20740879831 T^{2} - 1088981125496420 T^{3} + \)\(20\!\cdots\!44\)\( T^{4} - \)\(11\!\cdots\!60\)\( T^{5} + \)\(23\!\cdots\!99\)\( T^{6} + \)\(80\!\cdots\!16\)\( T^{7} + \)\(12\!\cdots\!41\)\( T^{8} \)
$17$ \( 1 - 678624 T + 301210589426 T^{2} - 139032189424965312 T^{3} + \)\(60\!\cdots\!35\)\( T^{4} - \)\(16\!\cdots\!64\)\( T^{5} + \)\(42\!\cdots\!34\)\( T^{6} - \)\(11\!\cdots\!52\)\( T^{7} + \)\(19\!\cdots\!81\)\( T^{8} \)
$19$ \( ( 1 + 130321 T )^{4} \)
$23$ \( 1 - 2911868 T + 7020626597375 T^{2} - 13006771889300755148 T^{3} + \)\(18\!\cdots\!24\)\( T^{4} - \)\(23\!\cdots\!24\)\( T^{5} + \)\(22\!\cdots\!75\)\( T^{6} - \)\(17\!\cdots\!96\)\( T^{7} + \)\(10\!\cdots\!61\)\( T^{8} \)
$29$ \( 1 - 8291104 T + 65007930953507 T^{2} - \)\(28\!\cdots\!48\)\( T^{3} + \)\(13\!\cdots\!44\)\( T^{4} - \)\(41\!\cdots\!12\)\( T^{5} + \)\(13\!\cdots\!27\)\( T^{6} - \)\(25\!\cdots\!36\)\( T^{7} + \)\(44\!\cdots\!21\)\( T^{8} \)
$31$ \( 1 - 3445468 T + 76079895372940 T^{2} - \)\(24\!\cdots\!04\)\( T^{3} + \)\(27\!\cdots\!42\)\( T^{4} - \)\(63\!\cdots\!84\)\( T^{5} + \)\(53\!\cdots\!40\)\( T^{6} - \)\(63\!\cdots\!48\)\( T^{7} + \)\(48\!\cdots\!81\)\( T^{8} \)
$37$ \( 1 + 1005524 T + 61996920196900 T^{2} - \)\(10\!\cdots\!84\)\( T^{3} - \)\(87\!\cdots\!06\)\( T^{4} - \)\(14\!\cdots\!68\)\( T^{5} + \)\(10\!\cdots\!00\)\( T^{6} + \)\(22\!\cdots\!92\)\( T^{7} + \)\(28\!\cdots\!41\)\( T^{8} \)
$41$ \( 1 - 8514124 T + 576067992126680 T^{2} - \)\(62\!\cdots\!80\)\( T^{3} + \)\(14\!\cdots\!26\)\( T^{4} - \)\(20\!\cdots\!80\)\( T^{5} + \)\(61\!\cdots\!80\)\( T^{6} - \)\(29\!\cdots\!44\)\( T^{7} + \)\(11\!\cdots\!41\)\( T^{8} \)
$43$ \( 1 - 13900726 T + 1263032063045605 T^{2} - \)\(49\!\cdots\!66\)\( T^{3} + \)\(69\!\cdots\!60\)\( T^{4} - \)\(24\!\cdots\!38\)\( T^{5} + \)\(31\!\cdots\!45\)\( T^{6} - \)\(17\!\cdots\!82\)\( T^{7} + \)\(63\!\cdots\!01\)\( T^{8} \)
$47$ \( 1 + 36334954 T + 1333040393862797 T^{2} + \)\(22\!\cdots\!54\)\( T^{3} + \)\(16\!\cdots\!20\)\( T^{4} + \)\(25\!\cdots\!18\)\( T^{5} + \)\(16\!\cdots\!33\)\( T^{6} + \)\(50\!\cdots\!02\)\( T^{7} + \)\(15\!\cdots\!21\)\( T^{8} \)
$53$ \( 1 + 113969356 T + 10662975352067663 T^{2} + \)\(54\!\cdots\!12\)\( T^{3} + \)\(35\!\cdots\!72\)\( T^{4} + \)\(17\!\cdots\!96\)\( T^{5} + \)\(11\!\cdots\!07\)\( T^{6} + \)\(40\!\cdots\!72\)\( T^{7} + \)\(11\!\cdots\!21\)\( T^{8} \)
$59$ \( 1 + 396773766 T + 84015153907615097 T^{2} + \)\(11\!\cdots\!22\)\( T^{3} + \)\(12\!\cdots\!24\)\( T^{4} + \)\(10\!\cdots\!58\)\( T^{5} + \)\(63\!\cdots\!37\)\( T^{6} + \)\(25\!\cdots\!54\)\( T^{7} + \)\(56\!\cdots\!41\)\( T^{8} \)
$61$ \( 1 + 298192066 T + 68250132645236185 T^{2} + \)\(10\!\cdots\!50\)\( T^{3} + \)\(12\!\cdots\!16\)\( T^{4} + \)\(11\!\cdots\!50\)\( T^{5} + \)\(93\!\cdots\!85\)\( T^{6} + \)\(47\!\cdots\!86\)\( T^{7} + \)\(18\!\cdots\!61\)\( T^{8} \)
$67$ \( 1 + 113551722 T + 50137220305978117 T^{2} - \)\(13\!\cdots\!94\)\( T^{3} + \)\(78\!\cdots\!32\)\( T^{4} - \)\(35\!\cdots\!18\)\( T^{5} + \)\(37\!\cdots\!53\)\( T^{6} + \)\(22\!\cdots\!06\)\( T^{7} + \)\(54\!\cdots\!81\)\( T^{8} \)
$71$ \( 1 - 4659620 T + 66622804019153564 T^{2} + \)\(12\!\cdots\!88\)\( T^{3} + \)\(17\!\cdots\!34\)\( T^{4} + \)\(56\!\cdots\!28\)\( T^{5} + \)\(14\!\cdots\!04\)\( T^{6} - \)\(44\!\cdots\!20\)\( T^{7} + \)\(44\!\cdots\!21\)\( T^{8} \)
$73$ \( 1 - 136198452 T + 159202022806579642 T^{2} - \)\(18\!\cdots\!28\)\( T^{3} + \)\(13\!\cdots\!11\)\( T^{4} - \)\(11\!\cdots\!64\)\( T^{5} + \)\(55\!\cdots\!98\)\( T^{6} - \)\(27\!\cdots\!44\)\( T^{7} + \)\(12\!\cdots\!61\)\( T^{8} \)
$79$ \( 1 - 67255424 T + 466792363237010728 T^{2} - \)\(23\!\cdots\!08\)\( T^{3} + \)\(83\!\cdots\!42\)\( T^{4} - \)\(28\!\cdots\!52\)\( T^{5} + \)\(67\!\cdots\!08\)\( T^{6} - \)\(11\!\cdots\!16\)\( T^{7} + \)\(20\!\cdots\!21\)\( T^{8} \)
$83$ \( 1 - 1376505216 T + 1405885246005088616 T^{2} - \)\(90\!\cdots\!24\)\( T^{3} + \)\(46\!\cdots\!02\)\( T^{4} - \)\(16\!\cdots\!72\)\( T^{5} + \)\(49\!\cdots\!44\)\( T^{6} - \)\(89\!\cdots\!32\)\( T^{7} + \)\(12\!\cdots\!81\)\( T^{8} \)
$89$ \( 1 - 1557211260 T + 2036231523650812436 T^{2} - \)\(16\!\cdots\!20\)\( T^{3} + \)\(11\!\cdots\!86\)\( T^{4} - \)\(58\!\cdots\!80\)\( T^{5} + \)\(24\!\cdots\!16\)\( T^{6} - \)\(66\!\cdots\!40\)\( T^{7} + \)\(15\!\cdots\!61\)\( T^{8} \)
$97$ \( 1 - 975818188 T + 3235606984725522088 T^{2} - \)\(22\!\cdots\!80\)\( T^{3} + \)\(37\!\cdots\!30\)\( T^{4} - \)\(16\!\cdots\!60\)\( T^{5} + \)\(18\!\cdots\!32\)\( T^{6} - \)\(42\!\cdots\!44\)\( T^{7} + \)\(33\!\cdots\!21\)\( T^{8} \)
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