Properties

Label 531.4.a.f
Level $531$
Weight $4$
Character orbit 531.a
Self dual yes
Analytic conductor $31.330$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 531 = 3^{2} \cdot 59 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 531.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(31.3300142130\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 2 x^{7} - 49 x^{6} + 89 x^{5} + 648 x^{4} - 1023 x^{3} - 1476 x^{2} + 1940 x - 384\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 177)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + ( 5 + \beta_{2} ) q^{4} + ( 2 - \beta_{1} - \beta_{4} - \beta_{7} ) q^{5} + ( 7 - \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{5} - \beta_{7} ) q^{7} + ( -7 \beta_{1} + 3 \beta_{3} - \beta_{6} ) q^{8} +O(q^{10})\) \( q -\beta_{1} q^{2} + ( 5 + \beta_{2} ) q^{4} + ( 2 - \beta_{1} - \beta_{4} - \beta_{7} ) q^{5} + ( 7 - \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{5} - \beta_{7} ) q^{7} + ( -7 \beta_{1} + 3 \beta_{3} - \beta_{6} ) q^{8} + ( 3 + 3 \beta_{1} + 2 \beta_{2} - 3 \beta_{4} - \beta_{5} + \beta_{6} ) q^{10} + ( 7 - 6 \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} + 3 \beta_{6} - 2 \beta_{7} ) q^{11} + ( 9 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} - 3 \beta_{6} ) q^{13} + ( 8 - 12 \beta_{1} + 3 \beta_{2} - 2 \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{14} + ( 48 - \beta_{1} + 6 \beta_{2} - 2 \beta_{3} - 8 \beta_{4} + 2 \beta_{5} - 2 \beta_{7} ) q^{16} + ( -7 - 6 \beta_{1} - \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{17} + ( 36 + 7 \beta_{1} - 4 \beta_{2} - \beta_{3} + 6 \beta_{4} + 5 \beta_{6} - \beta_{7} ) q^{19} + ( -60 - 10 \beta_{1} + \beta_{2} + 8 \beta_{3} + 3 \beta_{4} - 11 \beta_{5} - \beta_{6} ) q^{20} + ( 74 - 2 \beta_{1} + \beta_{2} + \beta_{3} + 6 \beta_{4} - 8 \beta_{5} - \beta_{7} ) q^{22} + ( -28 + 7 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} + 8 \beta_{4} - 6 \beta_{5} - 3 \beta_{6} + \beta_{7} ) q^{23} + ( 41 - 11 \beta_{1} + \beta_{2} + 3 \beta_{3} - 6 \beta_{4} + 17 \beta_{5} - 7 \beta_{7} ) q^{25} + ( -30 + 2 \beta_{1} + 2 \beta_{2} - 8 \beta_{3} - 7 \beta_{4} + 15 \beta_{5} + 2 \beta_{6} + 7 \beta_{7} ) q^{26} + ( 90 - 26 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} + 6 \beta_{4} - 14 \beta_{5} - 6 \beta_{6} + 5 \beta_{7} ) q^{28} + ( 18 + 10 \beta_{1} + 2 \beta_{2} - 15 \beta_{3} - 5 \beta_{4} + 2 \beta_{5} + \beta_{6} ) q^{29} + ( 68 - 16 \beta_{1} - 4 \beta_{2} - 11 \beta_{3} + \beta_{4} - 2 \beta_{5} - 3 \beta_{6} + 14 \beta_{7} ) q^{31} + ( -35 - 26 \beta_{1} + \beta_{2} + 2 \beta_{3} - 20 \beta_{5} + 2 \beta_{6} - 8 \beta_{7} ) q^{32} + ( 59 + 11 \beta_{1} - \beta_{3} + 4 \beta_{4} + 14 \beta_{5} + 13 \beta_{7} ) q^{34} + ( 40 - 8 \beta_{1} + 17 \beta_{2} + 14 \beta_{3} - 26 \beta_{4} + 19 \beta_{5} + 7 \beta_{6} - 30 \beta_{7} ) q^{35} + ( 57 - 24 \beta_{1} - \beta_{2} - 8 \beta_{3} + 9 \beta_{4} - 25 \beta_{5} + 4 \beta_{6} - 8 \beta_{7} ) q^{37} + ( -44 - 20 \beta_{2} - 23 \beta_{3} + 16 \beta_{4} + 10 \beta_{5} + 5 \beta_{6} + 10 \beta_{7} ) q^{38} + ( 131 - 3 \beta_{1} + 30 \beta_{2} - 7 \beta_{3} - 35 \beta_{4} + 19 \beta_{5} + 2 \beta_{6} - 12 \beta_{7} ) q^{40} + ( -13 + 15 \beta_{1} - 23 \beta_{2} - 4 \beta_{3} + 26 \beta_{4} - 5 \beta_{5} - 4 \beta_{6} + 29 \beta_{7} ) q^{41} + ( 63 - 56 \beta_{1} - 15 \beta_{2} + 3 \beta_{3} + 14 \beta_{4} - 25 \beta_{5} - 3 \beta_{6} + 18 \beta_{7} ) q^{43} + ( -3 - 61 \beta_{1} + 6 \beta_{2} + 3 \beta_{3} - 14 \beta_{4} + 12 \beta_{5} - 16 \beta_{6} + 21 \beta_{7} ) q^{44} + ( -68 + 14 \beta_{1} - 6 \beta_{2} - 7 \beta_{3} - 14 \beta_{4} + 28 \beta_{5} + 7 \beta_{6} + 14 \beta_{7} ) q^{46} + ( -126 + 8 \beta_{1} - 5 \beta_{2} + 26 \beta_{3} - 21 \beta_{5} + 11 \beta_{6} + 8 \beta_{7} ) q^{47} + ( 58 - 36 \beta_{1} - 2 \beta_{2} + 7 \beta_{3} - 20 \beta_{4} + 14 \beta_{5} - 2 \beta_{6} - 22 \beta_{7} ) q^{49} + ( 63 + 16 \beta_{1} - 11 \beta_{2} - \beta_{3} + 42 \beta_{4} - 4 \beta_{5} - 11 \beta_{6} + 16 \beta_{7} ) q^{50} + ( -95 + 19 \beta_{1} - 27 \beta_{2} + 10 \beta_{3} + 63 \beta_{4} - 39 \beta_{5} - 7 \beta_{7} ) q^{52} + ( 102 + 21 \beta_{1} + 6 \beta_{2} + 2 \beta_{3} - \beta_{4} - 18 \beta_{5} + 16 \beta_{6} - 15 \beta_{7} ) q^{53} + ( 86 - 17 \beta_{1} + 11 \beta_{2} + 17 \beta_{3} - 26 \beta_{4} + 9 \beta_{5} + 28 \beta_{6} - 33 \beta_{7} ) q^{55} + ( 303 - 85 \beta_{1} + 24 \beta_{2} + 17 \beta_{3} - 30 \beta_{4} + 4 \beta_{5} + 14 \beta_{6} - 11 \beta_{7} ) q^{56} + ( -161 - 23 \beta_{1} - 40 \beta_{2} + 25 \beta_{3} + 35 \beta_{4} - 17 \beta_{5} - 4 \beta_{6} + 6 \beta_{7} ) q^{58} + 59 q^{59} + ( -72 - 32 \beta_{1} - 32 \beta_{2} + 19 \beta_{3} + 11 \beta_{4} + 22 \beta_{5} - 33 \beta_{6} + 16 \beta_{7} ) q^{61} + ( 269 - 79 \beta_{1} + 15 \beta_{3} + 37 \beta_{4} - 19 \beta_{5} - 8 \beta_{6} - 14 \beta_{7} ) q^{62} + ( -60 + 19 \beta_{1} + 20 \beta_{2} + 7 \beta_{3} - 32 \beta_{4} - 4 \beta_{5} + 27 \beta_{6} + 8 \beta_{7} ) q^{64} + ( 118 + 65 \beta_{1} + 9 \beta_{2} - 7 \beta_{3} - 38 \beta_{4} + 25 \beta_{5} - 10 \beta_{6} - 19 \beta_{7} ) q^{65} + ( 246 + 3 \beta_{1} + 7 \beta_{2} - 21 \beta_{3} - 18 \beta_{4} + 23 \beta_{5} - 16 \beta_{6} + 11 \beta_{7} ) q^{67} + ( -8 - 30 \beta_{1} - 45 \beta_{2} + 18 \beta_{3} + 72 \beta_{4} - 30 \beta_{5} - 11 \beta_{6} + 13 \beta_{7} ) q^{68} + ( -143 - 23 \beta_{1} + 17 \beta_{2} + 26 \beta_{3} - 24 \beta_{4} - 32 \beta_{5} - 6 \beta_{6} + 6 \beta_{7} ) q^{70} + ( 211 - 6 \beta_{1} - 2 \beta_{2} - 8 \beta_{3} + 21 \beta_{4} + 22 \beta_{5} - 33 \beta_{6} + 10 \beta_{7} ) q^{71} + ( 272 - 52 \beta_{1} + 23 \beta_{2} - 6 \beta_{3} - 62 \beta_{4} + 25 \beta_{5} - 13 \beta_{6} - 22 \beta_{7} ) q^{73} + ( 341 - 87 \beta_{1} + 45 \beta_{2} - 16 \beta_{3} - 83 \beta_{4} + 35 \beta_{5} + 34 \beta_{6} + 13 \beta_{7} ) q^{74} + ( -167 + 151 \beta_{1} - 55 \beta_{2} - 40 \beta_{3} + 84 \beta_{4} + 18 \beta_{5} - 40 \beta_{6} + 48 \beta_{7} ) q^{76} + ( 125 - 99 \beta_{1} + 13 \beta_{2} + 20 \beta_{3} - 56 \beta_{4} + 73 \beta_{5} - 48 \beta_{6} - 7 \beta_{7} ) q^{77} + ( 413 + 76 \beta_{1} - 21 \beta_{2} - 11 \beta_{3} + 30 \beta_{4} + 13 \beta_{5} + 33 \beta_{6} - 8 \beta_{7} ) q^{79} + ( 293 - 205 \beta_{1} - 24 \beta_{2} + 54 \beta_{3} + 11 \beta_{4} + 3 \beta_{5} - 29 \beta_{6} - 22 \beta_{7} ) q^{80} + ( 60 + 90 \beta_{1} - 35 \beta_{2} - 58 \beta_{3} + 64 \beta_{4} + 28 \beta_{5} - \beta_{6} - 3 \beta_{7} ) q^{82} + ( -279 + 117 \beta_{1} - 35 \beta_{2} - 60 \beta_{3} - 40 \beta_{4} - \beta_{5} + 24 \beta_{6} + 19 \beta_{7} ) q^{83} + ( 28 + 137 \beta_{1} + 12 \beta_{2} + 37 \beta_{3} - 40 \beta_{4} + 6 \beta_{5} + 7 \beta_{6} - 11 \beta_{7} ) q^{85} + ( 902 - 48 \beta_{1} + 101 \beta_{2} - 41 \beta_{3} - 62 \beta_{4} + 12 \beta_{5} + 22 \beta_{6} - 33 \beta_{7} ) q^{86} + ( 166 - 68 \beta_{1} + 65 \beta_{2} + 58 \beta_{3} - 10 \beta_{4} + 12 \beta_{5} - 39 \beta_{6} - 37 \beta_{7} ) q^{88} + ( -244 + 98 \beta_{1} - 67 \beta_{2} - 18 \beta_{3} + 88 \beta_{4} - 85 \beta_{5} + 31 \beta_{6} - 4 \beta_{7} ) q^{89} + ( 315 + 79 \beta_{1} + 3 \beta_{2} + 10 \beta_{3} + 28 \beta_{4} + \beta_{5} + 46 \beta_{6} - 51 \beta_{7} ) q^{91} + ( 77 + 121 \beta_{1} - 61 \beta_{2} + 34 \beta_{3} + 90 \beta_{4} - 36 \beta_{5} - 12 \beta_{6} - 36 \beta_{7} ) q^{92} + ( 57 + 121 \beta_{1} + 75 \beta_{2} - 44 \beta_{3} - 98 \beta_{4} - 38 \beta_{5} + 18 \beta_{6} - 74 \beta_{7} ) q^{94} + ( -242 + 99 \beta_{1} + 6 \beta_{2} - 15 \beta_{3} - 76 \beta_{4} + 2 \beta_{5} + 63 \beta_{6} - 99 \beta_{7} ) q^{95} + ( 172 + 27 \beta_{1} + 15 \beta_{2} + 67 \beta_{3} + 42 \beta_{4} - 101 \beta_{5} - 30 \beta_{6} + 45 \beta_{7} ) q^{97} + ( 237 + 94 \beta_{1} + 44 \beta_{2} - 13 \beta_{3} - 26 \beta_{4} - 12 \beta_{5} + 10 \beta_{6} + 13 \beta_{7} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 2q^{2} + 38q^{4} + 12q^{5} + 53q^{7} - 3q^{8} + O(q^{10}) \) \( 8q - 2q^{2} + 38q^{4} + 12q^{5} + 53q^{7} - 3q^{8} + 29q^{10} + 27q^{11} + 89q^{13} + 37q^{14} + 362q^{16} - 79q^{17} + 288q^{19} - 457q^{20} + 596q^{22} - 202q^{23} + 264q^{25} - 270q^{26} + 702q^{28} + 114q^{29} + 538q^{31} - 316q^{32} + 498q^{34} + 196q^{35} + 395q^{37} - 397q^{38} + 918q^{40} + 39q^{41} + 527q^{43} - 64q^{44} - 539q^{46} - 860q^{47} + 347q^{49} + 591q^{50} - 644q^{52} + 812q^{53} + 536q^{55} + 2218q^{56} - 1154q^{58} + 472q^{59} - 460q^{61} + 2014q^{62} - 451q^{64} + 986q^{65} + 1934q^{67} + 69q^{68} - 1028q^{70} + 1687q^{71} + 1980q^{73} + 2400q^{74} - 940q^{76} + 821q^{77} + 3319q^{79} + 2119q^{80} + 429q^{82} - 2057q^{83} + 566q^{85} + 6690q^{86} + 1189q^{88} - 1668q^{89} + 2427q^{91} + 980q^{92} + 332q^{94} - 2146q^{95} + 1956q^{97} + 2026q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 2 x^{7} - 49 x^{6} + 89 x^{5} + 648 x^{4} - 1023 x^{3} - 1476 x^{2} + 1940 x - 384\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 13 \)
\(\beta_{3}\)\(=\)\((\)\( -13 \nu^{7} + 34 \nu^{6} + 647 \nu^{5} - 1483 \nu^{4} - 9408 \nu^{3} + 16089 \nu^{2} + 36118 \nu - 20832 \)\()/2144\)
\(\beta_{4}\)\(=\)\((\)\( -18 \nu^{7} + 11 \nu^{6} + 901 \nu^{5} - 734 \nu^{4} - 12207 \nu^{3} + 12861 \nu^{2} + 28786 \nu - 33792 \)\()/2144\)
\(\beta_{5}\)\(=\)\((\)\( -7 \nu^{7} + 8 \nu^{6} + 369 \nu^{5} - 345 \nu^{4} - 5406 \nu^{3} + 3695 \nu^{2} + 15758 \nu - 6352 \)\()/536\)
\(\beta_{6}\)\(=\)\((\)\( -39 \nu^{7} + 102 \nu^{6} + 1941 \nu^{5} - 4449 \nu^{4} - 26080 \nu^{3} + 48267 \nu^{2} + 59042 \nu - 62496 \)\()/2144\)
\(\beta_{7}\)\(=\)\((\)\( 57 \nu^{7} - 46 \nu^{6} - 2775 \nu^{5} + 1967 \nu^{4} + 36612 \nu^{3} - 20593 \nu^{2} - 89302 \nu + 29824 \)\()/2144\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 13\)
\(\nu^{3}\)\(=\)\(\beta_{6} - 3 \beta_{3} + 23 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-2 \beta_{7} + 2 \beta_{5} - 8 \beta_{4} - 2 \beta_{3} + 30 \beta_{2} - \beta_{1} + 296\)
\(\nu^{5}\)\(=\)\(8 \beta_{7} + 30 \beta_{6} + 20 \beta_{5} - 98 \beta_{3} - \beta_{2} + 570 \beta_{1} + 35\)
\(\nu^{6}\)\(=\)\(-72 \beta_{7} + 27 \beta_{6} + 76 \beta_{5} - 352 \beta_{4} - 73 \beta_{3} + 836 \beta_{2} - 21 \beta_{1} + 7300\)
\(\nu^{7}\)\(=\)\(438 \beta_{7} + 840 \beta_{6} + 966 \beta_{5} - 8 \beta_{4} - 2834 \beta_{3} - 48 \beta_{2} + 14561 \beta_{1} + 1554\)

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
5.26363
4.61734
2.17127
0.780043
0.254436
−1.67303
−4.21744
−5.19624
−5.26363 0 19.7058 −11.8799 0 −3.36662 −61.6149 0 62.5311
1.2 −4.61734 0 13.3198 3.21787 0 15.9864 −24.5634 0 −14.8580
1.3 −2.17127 0 −3.28558 −9.58086 0 14.1591 24.5041 0 20.8027
1.4 −0.780043 0 −7.39153 21.9196 0 32.1153 12.0061 0 −17.0983
1.5 −0.254436 0 −7.93526 10.8225 0 −23.2950 4.05451 0 −2.75362
1.6 1.67303 0 −5.20096 6.76323 0 −19.0526 −22.0856 0 11.3151
1.7 4.21744 0 9.78684 −17.5635 0 14.8996 7.53588 0 −74.0731
1.8 5.19624 0 19.0009 8.30102 0 21.5539 57.1634 0 43.1341
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(59\) \(-1\)

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 531.4.a.f 8
3.b odd 2 1 177.4.a.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.4.a.c 8 3.b odd 2 1
531.4.a.f 8 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{8} + \cdots\) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(531))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -384 - 1940 T - 1476 T^{2} + 1023 T^{3} + 648 T^{4} - 89 T^{5} - 49 T^{6} + 2 T^{7} + T^{8} \)
$3$ \( T^{8} \)
$5$ \( -85672464 + 40405196 T - 2469078 T^{2} - 861524 T^{3} + 80277 T^{4} + 5716 T^{5} - 560 T^{6} - 12 T^{7} + T^{8} \)
$7$ \( -3488279296 - 399824208 T + 167815056 T^{2} - 8549781 T^{3} - 516133 T^{4} + 45841 T^{5} - 141 T^{6} - 53 T^{7} + T^{8} \)
$11$ \( -223404635568 + 74917280804 T - 1963574268 T^{2} - 255710059 T^{3} + 9043745 T^{4} + 148898 T^{5} - 5942 T^{6} - 27 T^{7} + T^{8} \)
$13$ \( -5466828930044 + 441869764290 T + 5449126378 T^{2} - 909465641 T^{3} + 5431885 T^{4} + 519680 T^{5} - 5244 T^{6} - 89 T^{7} + T^{8} \)
$17$ \( 34686030744 - 821633910388 T - 18768055686 T^{2} + 1716992763 T^{3} + 27757319 T^{4} - 771505 T^{5} - 10801 T^{6} + 79 T^{7} + T^{8} \)
$19$ \( 612675702830144 - 91813882891136 T + 2714482097692 T^{2} - 12538959788 T^{3} - 466181345 T^{4} + 5561678 T^{5} + 3875 T^{6} - 288 T^{7} + T^{8} \)
$23$ \( -96081715816128 + 9355764769776 T + 383069668344 T^{2} - 12488762998 T^{3} - 608438203 T^{4} - 7421234 T^{5} - 18459 T^{6} + 202 T^{7} + T^{8} \)
$29$ \( -238656589193400 + 31895285211880 T - 729683982206 T^{2} - 29947346050 T^{3} + 450175903 T^{4} + 4361778 T^{5} - 52705 T^{6} - 114 T^{7} + T^{8} \)
$31$ \( 373989746307029344 - 14671727528641988 T + 177146746026986 T^{2} - 394299187352 T^{3} - 6572943625 T^{4} + 40216226 T^{5} + 9513 T^{6} - 538 T^{7} + T^{8} \)
$37$ \( -17609718565325908 - 6802879059905102 T - 545645679592918 T^{2} - 2581093421219 T^{3} + 17078307635 T^{4} + 66279559 T^{5} - 212175 T^{6} - 395 T^{7} + T^{8} \)
$41$ \( 3467857828629798168 + 54596984145333748 T - 1094937506543542 T^{2} - 1329506773555 T^{3} + 36119965003 T^{4} + 12235701 T^{5} - 343149 T^{6} - 39 T^{7} + T^{8} \)
$43$ \( -95383866842213663104 + 769097138033222688 T + 1891831228827066 T^{2} - 26244588674487 T^{3} + 21254212085 T^{4} + 219020278 T^{5} - 345408 T^{6} - 527 T^{7} + T^{8} \)
$47$ \( -61510955775947758848 - 14995869117559264 T + 4466702502723240 T^{2} + 10608638398880 T^{3} - 83816564919 T^{4} - 369668772 T^{5} - 213287 T^{6} + 860 T^{7} + T^{8} \)
$53$ \( 2415145722211600416 - 80808123212097596 T + 886827150577794 T^{2} - 2270325034168 T^{3} - 19636881903 T^{4} + 107112256 T^{5} + 29228 T^{6} - 812 T^{7} + T^{8} \)
$59$ \( ( -59 + T )^{8} \)
$61$ \( \)\(12\!\cdots\!88\)\( - 4148264243603022992 T - 27517213686627650 T^{2} + 72787451003566 T^{3} + 240158206629 T^{4} - 377447970 T^{5} - 909985 T^{6} + 460 T^{7} + T^{8} \)
$67$ \( -7430708718423391936 - 110334409032439696 T + 783050030355816 T^{2} + 9528145887250 T^{3} - 49162745095 T^{4} - 164546676 T^{5} + 1170254 T^{6} - 1934 T^{7} + T^{8} \)
$71$ \( -90426436968280392288 - 571114471027464574 T + 15561798577343672 T^{2} - 29934011829203 T^{3} - 245163194671 T^{4} + 617779352 T^{5} + 331910 T^{6} - 1687 T^{7} + T^{8} \)
$73$ \( \)\(70\!\cdots\!32\)\( + 9801783114216604168 T + 21707139357947164 T^{2} - 118754585627498 T^{3} - 273813192733 T^{4} + 691676814 T^{5} + 653565 T^{6} - 1980 T^{7} + T^{8} \)
$79$ \( \)\(47\!\cdots\!16\)\( - \)\(21\!\cdots\!52\)\( T - 30273778868140660 T^{2} + 1580910686454401 T^{3} - 2783070666087 T^{4} + 396292006 T^{5} + 3319402 T^{6} - 3319 T^{7} + T^{8} \)
$83$ \( \)\(11\!\cdots\!52\)\( - \)\(16\!\cdots\!52\)\( T - 316851294553911156 T^{2} + 1753324230030757 T^{3} + 626066347589 T^{4} - 3409775871 T^{5} - 1145959 T^{6} + 2057 T^{7} + T^{8} \)
$89$ \( -\)\(59\!\cdots\!44\)\( + 53671090805958449312 T + 2286036609406369528 T^{2} + 5387443498831848 T^{3} + 1080354579879 T^{4} - 6020684798 T^{5} - 3091237 T^{6} + 1668 T^{7} + T^{8} \)
$97$ \( \)\(11\!\cdots\!68\)\( + \)\(47\!\cdots\!72\)\( T - 1418207248608454620 T^{2} - 5565204479475238 T^{3} + 3092278816143 T^{4} + 7008888906 T^{5} - 3509564 T^{6} - 1956 T^{7} + T^{8} \)
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