Properties

Label 245.6.a.i
Level $245$
Weight $6$
Character orbit 245.a
Self dual yes
Analytic conductor $39.294$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 245.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(39.2940358542\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \(x^{6} - x^{5} - 109 x^{4} + 41 x^{3} + 2208 x^{2} - 3204 x + 560\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 35)
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.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{1} ) q^{2} + ( 3 + \beta_{1} - \beta_{3} ) q^{3} + ( 5 - \beta_{1} + \beta_{2} ) q^{4} -25 q^{5} + ( 16 + 5 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{6} + ( -23 + 5 \beta_{1} + 4 \beta_{3} + 2 \beta_{5} ) q^{8} + ( 65 + 2 \beta_{1} - 4 \beta_{2} - 3 \beta_{3} - 2 \beta_{4} - 3 \beta_{5} ) q^{9} +O(q^{10})\) \( q + ( -1 + \beta_{1} ) q^{2} + ( 3 + \beta_{1} - \beta_{3} ) q^{3} + ( 5 - \beta_{1} + \beta_{2} ) q^{4} -25 q^{5} + ( 16 + 5 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{6} + ( -23 + 5 \beta_{1} + 4 \beta_{3} + 2 \beta_{5} ) q^{8} + ( 65 + 2 \beta_{1} - 4 \beta_{2} - 3 \beta_{3} - 2 \beta_{4} - 3 \beta_{5} ) q^{9} + ( 25 - 25 \beta_{1} ) q^{10} + ( -151 - 32 \beta_{1} - 4 \beta_{2} - 9 \beta_{3} + 4 \beta_{4} + \beta_{5} ) q^{11} + ( 64 - 27 \beta_{1} + 10 \beta_{2} + 13 \beta_{3} + 5 \beta_{4} - 9 \beta_{5} ) q^{12} + ( -29 + 4 \beta_{1} + 7 \beta_{3} + 10 \beta_{4} - 3 \beta_{5} ) q^{13} + ( -75 - 25 \beta_{1} + 25 \beta_{3} ) q^{15} + ( 69 + 19 \beta_{1} - 11 \beta_{2} - 8 \beta_{3} + 8 \beta_{4} - 8 \beta_{5} ) q^{16} + ( 256 + 2 \beta_{1} + 20 \beta_{2} + 34 \beta_{3} - 4 \beta_{5} ) q^{17} + ( 47 - 56 \beta_{1} - 20 \beta_{2} - 31 \beta_{3} - 5 \beta_{4} - 7 \beta_{5} ) q^{18} + ( 67 - 230 \beta_{1} - 8 \beta_{2} + 43 \beta_{3} - 8 \beta_{4} + 3 \beta_{5} ) q^{19} + ( -125 + 25 \beta_{1} - 25 \beta_{2} ) q^{20} + ( -1063 - 308 \beta_{1} - 50 \beta_{2} + 39 \beta_{3} - 15 \beta_{4} + 15 \beta_{5} ) q^{22} + ( -646 + 49 \beta_{1} - 60 \beta_{2} - 34 \beta_{3} - 20 \beta_{4} + 53 \beta_{5} ) q^{23} + ( -1208 + 47 \beta_{1} + 5 \beta_{2} + 73 \beta_{3} + 17 \beta_{4} + 13 \beta_{5} ) q^{24} + 625 q^{25} + ( 495 - 210 \beta_{1} + 6 \beta_{2} + 119 \beta_{3} - 19 \beta_{4} + 39 \beta_{5} ) q^{26} + ( 375 + 103 \beta_{1} - 60 \beta_{2} - 13 \beta_{3} - 60 \beta_{4} - 6 \beta_{5} ) q^{27} + ( 240 - 264 \beta_{1} + 112 \beta_{2} + 147 \beta_{3} + 38 \beta_{4} - 103 \beta_{5} ) q^{29} + ( -400 - 125 \beta_{1} + 25 \beta_{2} - 25 \beta_{3} + 25 \beta_{4} - 25 \beta_{5} ) q^{30} + ( -1206 + 434 \beta_{1} - 12 \beta_{2} + 148 \beta_{3} - 68 \beta_{4} + 18 \beta_{5} ) q^{31} + ( 1649 - 611 \beta_{1} - 40 \beta_{2} - 52 \beta_{3} - 40 \beta_{4} - 30 \beta_{5} ) q^{32} + ( 866 - 704 \beta_{1} + 192 \beta_{3} + 40 \beta_{4} + 40 \beta_{5} ) q^{33} + ( 198 + 792 \beta_{1} + 74 \beta_{2} + 54 \beta_{3} + 26 \beta_{4} + 14 \beta_{5} ) q^{34} + ( -4313 - 588 \beta_{1} - 38 \beta_{2} + \beta_{3} + 29 \beta_{4} + 81 \beta_{5} ) q^{36} + ( -2445 + 14 \beta_{1} + 220 \beta_{2} + 9 \beta_{3} + 10 \beta_{4} + 25 \beta_{5} ) q^{37} + ( -7679 - 136 \beta_{1} - 140 \beta_{2} - 177 \beta_{3} + 65 \beta_{4} - 97 \beta_{5} ) q^{38} + ( -2460 - 994 \beta_{1} + 100 \beta_{2} + 174 \beta_{3} + 36 \beta_{4} + 244 \beta_{5} ) q^{39} + ( 575 - 125 \beta_{1} - 100 \beta_{3} - 50 \beta_{5} ) q^{40} + ( 607 + 44 \beta_{1} + 168 \beta_{2} + 198 \beta_{3} + 112 \beta_{4} - 62 \beta_{5} ) q^{41} + ( -3909 - 871 \beta_{1} - 40 \beta_{2} - 193 \beta_{3} - 120 \beta_{4} + 112 \beta_{5} ) q^{43} + ( -4355 - 1372 \beta_{1} - 92 \beta_{2} - 161 \beta_{3} - 29 \beta_{4} - 261 \beta_{5} ) q^{44} + ( -1625 - 50 \beta_{1} + 100 \beta_{2} + 75 \beta_{3} + 50 \beta_{4} + 75 \beta_{5} ) q^{45} + ( 1279 - 1741 \beta_{1} + 133 \beta_{2} - 552 \beta_{3} + 112 \beta_{4} - 272 \beta_{5} ) q^{46} + ( 7483 - 1402 \beta_{1} - 120 \beta_{2} - 59 \beta_{3} - 80 \beta_{4} - 135 \beta_{5} ) q^{47} + ( 1988 - 451 \beta_{1} - 70 \beta_{2} - 291 \beta_{3} - 95 \beta_{4} + 267 \beta_{5} ) q^{48} + ( -625 + 625 \beta_{1} ) q^{50} + ( -7126 - 38 \beta_{1} + 284 \beta_{2} + 4 \beta_{3} + 76 \beta_{4} - 6 \beta_{5} ) q^{51} + ( -6273 + 938 \beta_{1} + 190 \beta_{2} - 625 \beta_{3} - 85 \beta_{4} - 165 \beta_{5} ) q^{52} + ( -135 - 1068 \beta_{1} + 240 \beta_{2} + 567 \beta_{3} - 110 \beta_{4} - 155 \beta_{5} ) q^{53} + ( 3238 - 733 \beta_{1} - 7 \beta_{2} - 935 \beta_{3} + 95 \beta_{4} - 335 \beta_{5} ) q^{54} + ( 3775 + 800 \beta_{1} + 100 \beta_{2} + 225 \beta_{3} - 100 \beta_{4} - 25 \beta_{5} ) q^{55} + ( -16396 + 30 \beta_{1} + 220 \beta_{2} - 312 \beta_{3} + 220 \beta_{4} - 166 \beta_{5} ) q^{57} + ( -6118 + 2071 \beta_{1} - 270 \beta_{2} + 963 \beta_{3} - 135 \beta_{4} + 435 \beta_{5} ) q^{58} + ( -10892 + 968 \beta_{1} - 476 \beta_{2} - 342 \beta_{3} - 68 \beta_{4} + 598 \beta_{5} ) q^{59} + ( -1600 + 675 \beta_{1} - 250 \beta_{2} - 325 \beta_{3} - 125 \beta_{4} + 225 \beta_{5} ) q^{60} + ( 11694 + 2444 \beta_{1} - 112 \beta_{2} - 727 \beta_{3} + 22 \beta_{4} + 443 \beta_{5} ) q^{61} + ( 18184 - 772 \beta_{1} + 790 \beta_{2} - 1048 \beta_{3} + 320 \beta_{4} - 480 \beta_{5} ) q^{62} + ( -26107 + 155 \beta_{1} - 523 \beta_{2} - 272 \beta_{3} - 288 \beta_{4} + 128 \beta_{5} ) q^{64} + ( 725 - 100 \beta_{1} - 175 \beta_{3} - 250 \beta_{4} + 75 \beta_{5} ) q^{65} + ( -23226 + 282 \beta_{1} - 160 \beta_{2} + 208 \beta_{3} + 192 \beta_{4} - 112 \beta_{5} ) q^{66} + ( -23351 - 239 \beta_{1} - 460 \beta_{2} + 1901 \beta_{3} + 260 \beta_{4} - 234 \beta_{5} ) q^{67} + ( 20074 + 2156 \beta_{1} + 390 \beta_{2} - 562 \beta_{3} + 30 \beta_{4} + 298 \beta_{5} ) q^{68} + ( 2732 + 3328 \beta_{1} - 904 \beta_{2} + 451 \beta_{3} + 34 \beta_{4} - 219 \beta_{5} ) q^{69} + ( -26588 + 3068 \beta_{1} + 396 \beta_{2} - 1874 \beta_{3} - 116 \beta_{4} - 114 \beta_{5} ) q^{71} + ( -19105 - 3416 \beta_{1} + 340 \beta_{2} + 1025 \beta_{3} + 265 \beta_{4} + 101 \beta_{5} ) q^{72} + ( -882 - 34 \beta_{1} + 500 \beta_{2} - 276 \beta_{3} - 20 \beta_{4} - 1010 \beta_{5} ) q^{73} + ( -363 + 4662 \beta_{1} + 352 \beta_{2} + 941 \beta_{3} + 39 \beta_{4} + 421 \beta_{5} ) q^{74} + ( 1875 + 625 \beta_{1} - 625 \beta_{3} ) q^{75} + ( 2537 - 6228 \beta_{1} - 762 \beta_{2} - 785 \beta_{3} - 245 \beta_{4} + 255 \beta_{5} ) q^{76} + ( -36386 + 2084 \beta_{1} + 430 \beta_{2} + 170 \beta_{3} + 590 \beta_{4} - 318 \beta_{5} ) q^{78} + ( -37458 + 2576 \beta_{1} - 1116 \beta_{2} - 820 \beta_{3} + 380 \beta_{4} - 60 \beta_{5} ) q^{79} + ( -1725 - 475 \beta_{1} + 275 \beta_{2} + 200 \beta_{3} - 200 \beta_{4} + 200 \beta_{5} ) q^{80} + ( -8851 + 8060 \beta_{1} - 184 \beta_{2} - 2196 \beta_{3} - 284 \beta_{4} + 84 \beta_{5} ) q^{81} + ( 4861 + 3461 \beta_{1} + 360 \beta_{2} + 1942 \beta_{3} - 150 \beta_{4} + 710 \beta_{5} ) q^{82} + ( 31123 - 1645 \beta_{1} - 1420 \beta_{2} + 1275 \beta_{3} - 540 \beta_{4} + 222 \beta_{5} ) q^{83} + ( -6400 - 50 \beta_{1} - 500 \beta_{2} - 850 \beta_{3} + 100 \beta_{5} ) q^{85} + ( -34200 - 2115 \beta_{1} - 849 \beta_{2} - 1631 \beta_{3} + 271 \beta_{4} - 591 \beta_{5} ) q^{86} + ( -34477 - 7721 \beta_{1} + 2120 \beta_{2} + 37 \beta_{3} + 160 \beta_{4} + 600 \beta_{5} ) q^{87} + ( -7395 + 936 \beta_{1} - 1230 \beta_{2} - 1281 \beta_{3} - 145 \beta_{4} - 97 \beta_{5} ) q^{88} + ( 6542 + 6634 \beta_{1} + 140 \beta_{2} + 3531 \beta_{3} - 186 \beta_{4} - 429 \beta_{5} ) q^{89} + ( -1175 + 1400 \beta_{1} + 500 \beta_{2} + 775 \beta_{3} + 125 \beta_{4} + 175 \beta_{5} ) q^{90} + ( -47249 + 1055 \beta_{1} - 1880 \beta_{2} + 4060 \beta_{3} - 680 \beta_{4} + 114 \beta_{5} ) q^{92} + ( -34166 + 9790 \beta_{1} - 660 \beta_{2} - 56 \beta_{3} - 580 \beta_{4} - 310 \beta_{5} ) q^{93} + ( -55563 + 3666 \beta_{1} - 2180 \beta_{2} - 1111 \beta_{3} - 169 \beta_{4} - 231 \beta_{5} ) q^{94} + ( -1675 + 5750 \beta_{1} + 200 \beta_{2} - 1075 \beta_{3} + 200 \beta_{4} - 75 \beta_{5} ) q^{95} + ( 9528 + 2559 \beta_{1} - 195 \beta_{2} - 3999 \beta_{3} - 111 \beta_{4} - 1179 \beta_{5} ) q^{96} + ( -12738 - 2152 \beta_{1} + 1480 \beta_{2} + 464 \beta_{3} + 120 \beta_{4} + 1744 \beta_{5} ) q^{97} + ( -32765 + 1922 \beta_{1} + 2532 \beta_{2} + 3201 \beta_{3} + 524 \beta_{4} + 461 \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 5 q^{2} + 20 q^{3} + 31 q^{4} - 150 q^{5} + 96 q^{6} - 135 q^{8} + 378 q^{9} + O(q^{10}) \) \( 6 q - 5 q^{2} + 20 q^{3} + 31 q^{4} - 150 q^{5} + 96 q^{6} - 135 q^{8} + 378 q^{9} + 125 q^{10} - 924 q^{11} + 370 q^{12} - 150 q^{13} - 500 q^{15} + 435 q^{16} + 1540 q^{17} + 195 q^{18} + 92 q^{19} - 775 q^{20} - 6855 q^{22} - 3920 q^{23} - 7200 q^{24} + 3750 q^{25} + 2635 q^{26} + 2060 q^{27} + 1264 q^{29} - 2400 q^{30} - 7160 q^{31} + 9105 q^{32} + 4460 q^{33} + 2166 q^{34} - 26375 q^{36} - 14170 q^{37} - 46215 q^{38} - 15376 q^{39} + 3375 q^{40} + 4098 q^{41} - 24460 q^{43} - 27873 q^{44} - 9450 q^{45} + 6815 q^{46} + 42940 q^{47} + 11610 q^{48} - 3125 q^{50} - 42008 q^{51} - 36115 q^{52} - 2450 q^{53} + 19566 q^{54} + 23100 q^{55} - 97100 q^{57} - 36110 q^{58} - 64600 q^{59} - 9250 q^{60} + 73620 q^{61} + 111440 q^{62} - 157997 q^{64} + 3750 q^{65} - 139138 q^{66} - 142620 q^{67} + 124330 q^{68} + 17344 q^{69} - 154256 q^{71} - 117495 q^{72} - 5120 q^{73} + 2785 q^{74} + 12500 q^{75} + 7775 q^{76} - 214090 q^{78} - 222504 q^{79} - 10875 q^{80} - 43986 q^{81} + 31665 q^{82} + 179580 q^{83} - 38500 q^{85} - 207160 q^{86} - 209300 q^{87} - 45145 q^{88} + 41648 q^{89} - 4875 q^{90} - 292185 q^{92} - 198520 q^{93} - 333699 q^{94} - 2300 q^{95} + 61824 q^{96} - 73980 q^{97} - 190772 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - x^{5} - 109 x^{4} + 41 x^{3} + 2208 x^{2} - 3204 x + 560\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 36 \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{5} - 103 \nu^{3} - 86 \nu^{2} + 1744 \nu - 704 \)\()/48\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{5} + 6 \nu^{4} + 103 \nu^{3} - 460 \nu^{2} - 2380 \nu + 5552 \)\()/48\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{5} + 115 \nu^{3} + 50 \nu^{2} - 2536 \nu + 1736 \)\()/24\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 36\)
\(\nu^{3}\)\(=\)\(2 \beta_{5} + 4 \beta_{3} + 3 \beta_{2} + 69 \beta_{1} + 22\)
\(\nu^{4}\)\(=\)\(8 \beta_{4} + 8 \beta_{3} + 91 \beta_{2} + 197 \beta_{1} + 2468\)
\(\nu^{5}\)\(=\)\(206 \beta_{5} + 460 \beta_{3} + 395 \beta_{2} + 5449 \beta_{1} + 6066\)

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
−8.17253
−6.22733
0.203319
1.35048
4.12050
9.72556
−9.17253 14.3258 52.1353 −25.0000 −131.404 0 −184.691 −37.7723 229.313
1.2 −7.22733 −15.9209 20.2343 −25.0000 115.066 0 85.0347 10.4759 180.683
1.3 −0.796681 10.5748 −31.3653 −25.0000 −8.42477 0 50.4819 −131.173 19.9170
1.4 0.350479 −21.5910 −31.8772 −25.0000 −7.56720 0 −22.3876 223.173 −8.76197
1.5 3.12050 27.8717 −22.2625 −25.0000 86.9736 0 −169.326 533.832 −78.0125
1.6 8.72556 4.73965 44.1354 −25.0000 41.3561 0 105.888 −220.536 −218.139
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(7\) \(1\)

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 245.6.a.i 6
7.b odd 2 1 245.6.a.h 6
7.c even 3 2 35.6.e.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.6.e.a 12 7.c even 3 2
245.6.a.h 6 7.b odd 2 1
245.6.a.i 6 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(245))\):

\( T_{2}^{6} + 5 T_{2}^{5} - 99 T_{2}^{4} - 385 T_{2}^{3} + 1682 T_{2}^{2} + 900 T_{2} - 504 \)
\( T_{3}^{6} - 20 T_{3}^{5} - 718 T_{3}^{4} + 13100 T_{3}^{3} + 87921 T_{3}^{2} - 2078280 T_{3} + 6879276 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -504 + 900 T + 1682 T^{2} - 385 T^{3} - 99 T^{4} + 5 T^{5} + T^{6} \)
$3$ \( 6879276 - 2078280 T + 87921 T^{2} + 13100 T^{3} - 718 T^{4} - 20 T^{5} + T^{6} \)
$5$ \( ( 25 + T )^{6} \)
$7$ \( T^{6} \)
$11$ \( 2328635198430000 + 11999598762720 T - 46979552744 T^{2} - 296171976 T^{3} - 140589 T^{4} + 924 T^{5} + T^{6} \)
$13$ \( 3794636714817536 - 94970094602560 T + 522164709184 T^{2} + 19388360 T^{3} - 1431471 T^{4} + 150 T^{5} + T^{6} \)
$17$ \( 188618121955528704 - 666515335668480 T - 65330890624 T^{2} + 2061212640 T^{3} - 1067612 T^{4} - 1540 T^{5} + T^{6} \)
$19$ \( -1519818793649959696 + 1256689884274752 T + 6461324409960 T^{2} - 2546951792 T^{3} - 7344133 T^{4} - 92 T^{5} + T^{6} \)
$23$ \( -\)\(17\!\cdots\!01\)\( + 123922496362540320 T + 82710904155947 T^{2} - 44620272240 T^{3} - 14581871 T^{4} + 3920 T^{5} + T^{6} \)
$29$ \( -\)\(10\!\cdots\!00\)\( + 876618789962911500 T + 2234807351887825 T^{2} + 31871142140 T^{3} - 88728846 T^{4} - 1264 T^{5} + T^{6} \)
$31$ \( \)\(60\!\cdots\!44\)\( - 4350430229710967808 T + 2457044782150144 T^{2} - 211048112704 T^{3} - 80054416 T^{4} + 7160 T^{5} + T^{6} \)
$37$ \( \)\(68\!\cdots\!84\)\( + 44252140175226923520 T + 1022308764651296 T^{2} - 1720837960800 T^{3} - 97250327 T^{4} + 14170 T^{5} + T^{6} \)
$41$ \( \)\(10\!\cdots\!25\)\( - 77638260349353131010 T + 8391490364069951 T^{2} + 1580290318468 T^{3} - 233696641 T^{4} - 4098 T^{5} + T^{6} \)
$43$ \( \)\(19\!\cdots\!56\)\( + \)\(10\!\cdots\!80\)\( T - 12489811241143791 T^{2} - 5053525366340 T^{3} - 114323566 T^{4} + 24460 T^{5} + T^{6} \)
$47$ \( \)\(31\!\cdots\!00\)\( + \)\(31\!\cdots\!60\)\( T - 157744383016419376 T^{2} + 10172976238440 T^{3} + 286429267 T^{4} - 42940 T^{5} + T^{6} \)
$53$ \( \)\(26\!\cdots\!16\)\( - \)\(23\!\cdots\!60\)\( T + 10803666284913016 T^{2} + 7434915535840 T^{3} - 777874183 T^{4} + 2450 T^{5} + T^{6} \)
$59$ \( \)\(36\!\cdots\!00\)\( + \)\(24\!\cdots\!40\)\( T - 799743651108132656 T^{2} - 71039249553920 T^{3} - 286494424 T^{4} + 64600 T^{5} + T^{6} \)
$61$ \( \)\(21\!\cdots\!04\)\( - \)\(37\!\cdots\!48\)\( T - 598009689069958391 T^{2} + 48460451292756 T^{3} + 527816974 T^{4} - 73620 T^{5} + T^{6} \)
$67$ \( -\)\(18\!\cdots\!00\)\( - \)\(47\!\cdots\!20\)\( T - 27290003520934171671 T^{2} - 477188461901340 T^{3} + 2533120822 T^{4} + 142620 T^{5} + T^{6} \)
$71$ \( \)\(99\!\cdots\!24\)\( + \)\(26\!\cdots\!16\)\( T - 6554225661784715056 T^{2} - 152072627420928 T^{3} + 5078085496 T^{4} + 154256 T^{5} + T^{6} \)
$73$ \( -\)\(51\!\cdots\!04\)\( - \)\(12\!\cdots\!60\)\( T + 7018304398510967296 T^{2} + 57971304970560 T^{3} - 5532951728 T^{4} + 5120 T^{5} + T^{6} \)
$79$ \( \)\(71\!\cdots\!36\)\( - \)\(36\!\cdots\!84\)\( T - 32929806511168284752 T^{2} - 277156452678016 T^{3} + 12328150740 T^{4} + 222504 T^{5} + T^{6} \)
$83$ \( \)\(35\!\cdots\!64\)\( - \)\(11\!\cdots\!80\)\( T - 29886635061159702727 T^{2} + 893393938898060 T^{3} + 2780781062 T^{4} - 179580 T^{5} + T^{6} \)
$89$ \( -\)\(46\!\cdots\!56\)\( - \)\(86\!\cdots\!12\)\( T + 78803056260791799145 T^{2} + 450849137206852 T^{3} - 18519083398 T^{4} - 41648 T^{5} + T^{6} \)
$97$ \( \)\(99\!\cdots\!84\)\( - \)\(92\!\cdots\!40\)\( T + \)\(23\!\cdots\!24\)\( T^{2} - 419241276168160 T^{3} - 27433506084 T^{4} + 73980 T^{5} + T^{6} \)
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