Properties

Label 531.4.a.e
Level $531$
Weight $4$
Character orbit 531.a
Self dual yes
Analytic conductor $31.330$
Analytic rank $1$
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: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 2 x^{7} - 45 x^{6} + 47 x^{5} + 654 x^{4} - 157 x^{3} - 2898 x^{2} + 96 x + 2432\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
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.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{1} ) q^{2} + ( 5 - \beta_{1} + \beta_{2} ) q^{4} + ( -5 - \beta_{6} ) q^{5} + ( 7 + \beta_{7} ) q^{7} + ( -7 + 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{8} +O(q^{10})\) \( q + ( -1 + \beta_{1} ) q^{2} + ( 5 - \beta_{1} + \beta_{2} ) q^{4} + ( -5 - \beta_{6} ) q^{5} + ( 7 + \beta_{7} ) q^{7} + ( -7 + 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{8} + ( 2 - 5 \beta_{1} - 2 \beta_{2} - \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{10} + ( -10 - \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{11} + ( 4 - \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{6} ) q^{13} + ( -11 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + 3 \beta_{4} + 2 \beta_{5} + 4 \beta_{6} ) q^{14} + ( -3 - 7 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{16} + ( -16 - 4 \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{17} + ( 3 - 2 \beta_{1} - 7 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} + 2 \beta_{6} - 3 \beta_{7} ) q^{19} + ( -16 + 3 \beta_{1} - 3 \beta_{2} - 6 \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{20} + ( -11 - 7 \beta_{1} - 7 \beta_{2} - 5 \beta_{4} - 2 \beta_{6} + 2 \beta_{7} ) q^{22} + ( -25 - 14 \beta_{1} + \beta_{2} + 2 \beta_{3} + 5 \beta_{4} - \beta_{5} + 2 \beta_{6} - 3 \beta_{7} ) q^{23} + ( 8 + 2 \beta_{1} - 4 \beta_{2} + \beta_{3} - \beta_{5} + 8 \beta_{6} ) q^{25} + ( -7 - 5 \beta_{1} - 7 \beta_{2} - 3 \beta_{3} + \beta_{4} - 9 \beta_{5} + \beta_{6} - 4 \beta_{7} ) q^{26} + ( 13 - 3 \beta_{1} + 5 \beta_{2} + 6 \beta_{3} + \beta_{4} - 4 \beta_{6} + 6 \beta_{7} ) q^{28} + ( -48 - 22 \beta_{1} + 3 \beta_{2} + 4 \beta_{3} - \beta_{4} + 7 \beta_{5} - \beta_{6} + 3 \beta_{7} ) q^{29} + ( 10 + 12 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} - 9 \beta_{4} + \beta_{5} + 5 \beta_{6} - \beta_{7} ) q^{31} + ( -23 - 17 \beta_{1} - 13 \beta_{2} - 4 \beta_{3} - 2 \beta_{4} - 8 \beta_{5} + 4 \beta_{7} ) q^{32} + ( -9 - 5 \beta_{1} + 2 \beta_{3} - \beta_{4} - 2 \beta_{5} - 6 \beta_{6} + 4 \beta_{7} ) q^{34} + ( -56 - 36 \beta_{1} - \beta_{2} + 5 \beta_{3} - \beta_{4} + 2 \beta_{5} - 12 \beta_{6} + \beta_{7} ) q^{35} + ( 12 + 2 \beta_{1} + 2 \beta_{2} + 16 \beta_{4} - 6 \beta_{5} + \beta_{6} - 13 \beta_{7} ) q^{37} + ( -18 - 11 \beta_{1} + 5 \beta_{2} + \beta_{3} - 10 \beta_{4} + 10 \beta_{5} - 14 \beta_{6} + 10 \beta_{7} ) q^{38} + ( -28 + 4 \beta_{1} - \beta_{2} - 11 \beta_{3} - 5 \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{40} + ( -47 - 42 \beta_{1} - 8 \beta_{2} + 4 \beta_{4} - 10 \beta_{5} + 2 \beta_{6} - 3 \beta_{7} ) q^{41} + ( -16 + 8 \beta_{1} + 11 \beta_{2} - 8 \beta_{3} + 5 \beta_{4} + 5 \beta_{5} - 4 \beta_{7} ) q^{43} + ( -61 - 51 \beta_{1} - 14 \beta_{2} + 2 \beta_{3} + 9 \beta_{4} - 6 \beta_{6} - 14 \beta_{7} ) q^{44} + ( -68 - 15 \beta_{1} + 15 \beta_{2} + 7 \beta_{3} + 14 \beta_{5} - 14 \beta_{6} + 14 \beta_{7} ) q^{46} + ( -26 - 40 \beta_{1} + 23 \beta_{2} + 11 \beta_{3} + 9 \beta_{4} + 10 \beta_{5} + 2 \beta_{6} + 7 \beta_{7} ) q^{47} + ( 33 + 30 \beta_{1} + 16 \beta_{2} + \beta_{3} - 18 \beta_{4} - 3 \beta_{5} - 16 \beta_{6} + 17 \beta_{7} ) q^{49} + ( 39 - 14 \beta_{1} + 28 \beta_{2} - 5 \beta_{3} + 2 \beta_{4} + 14 \beta_{5} - 8 \beta_{6} + 16 \beta_{7} ) q^{50} + ( -105 - 5 \beta_{1} + 18 \beta_{2} - 7 \beta_{3} - 9 \beta_{4} + 9 \beta_{5} - 9 \beta_{6} + 4 \beta_{7} ) q^{52} + ( -23 - 20 \beta_{1} - 12 \beta_{3} - 12 \beta_{4} - 4 \beta_{5} + 29 \beta_{6} - 12 \beta_{7} ) q^{53} + ( -53 - 6 \beta_{1} + 32 \beta_{2} + 27 \beta_{3} + 8 \beta_{4} + 17 \beta_{5} - 4 \beta_{7} ) q^{55} + ( 69 - 27 \beta_{1} + 24 \beta_{2} + 2 \beta_{3} + 7 \beta_{4} + 6 \beta_{5} - 4 \beta_{6} - 6 \beta_{7} ) q^{56} + ( -194 - 66 \beta_{1} - 25 \beta_{2} - 5 \beta_{3} + 16 \beta_{4} - 17 \beta_{5} + 13 \beta_{6} - 4 \beta_{7} ) q^{58} -59 q^{59} + ( -150 - 44 \beta_{1} + 3 \beta_{2} - 18 \beta_{3} + 23 \beta_{4} - \beta_{5} - 11 \beta_{6} - 5 \beta_{7} ) q^{61} + ( 92 + 2 \beta_{1} + \beta_{2} - 13 \beta_{3} - 8 \beta_{4} - 15 \beta_{5} - 9 \beta_{6} - 8 \beta_{7} ) q^{62} + ( -255 - 30 \beta_{1} - 7 \beta_{2} + \beta_{3} - 14 \beta_{4} + 12 \beta_{5} + 16 \beta_{6} - 4 \beta_{7} ) q^{64} + ( 87 + 8 \beta_{2} - 21 \beta_{3} + 26 \beta_{4} - 31 \beta_{5} + 4 \beta_{6} - 10 \beta_{7} ) q^{65} + ( -133 - 14 \beta_{1} - 2 \beta_{2} + 5 \beta_{3} - 22 \beta_{4} - 9 \beta_{5} + 10 \beta_{6} - 26 \beta_{7} ) q^{67} + ( 49 - 8 \beta_{2} - 15 \beta_{3} + 15 \beta_{4} + 4 \beta_{5} + 6 \beta_{6} - 6 \beta_{7} ) q^{68} + ( -384 - 82 \beta_{1} - 39 \beta_{2} - 8 \beta_{3} + 12 \beta_{4} - 10 \beta_{5} + 16 \beta_{6} - 26 \beta_{7} ) q^{70} + ( 16 + 48 \beta_{1} + 23 \beta_{2} + 25 \beta_{3} - 47 \beta_{4} + 8 \beta_{5} - 9 \beta_{6} + 10 \beta_{7} ) q^{71} + ( -136 - 30 \beta_{1} - 17 \beta_{2} + 5 \beta_{3} - 23 \beta_{4} - 26 \beta_{5} - 16 \beta_{6} + 9 \beta_{7} ) q^{73} + ( 193 + 77 \beta_{1} + 50 \beta_{2} + 31 \beta_{3} - 23 \beta_{4} + 31 \beta_{5} - 53 \beta_{6} + 34 \beta_{7} ) q^{74} + ( -272 - 30 \beta_{1} - 27 \beta_{2} - 2 \beta_{4} - 28 \beta_{5} + 38 \beta_{6} - 24 \beta_{7} ) q^{76} + ( -119 + 50 \beta_{1} - 54 \beta_{2} - 32 \beta_{3} - 20 \beta_{4} - 14 \beta_{5} + 40 \beta_{6} - 11 \beta_{7} ) q^{77} + ( -74 + 14 \beta_{1} + 65 \beta_{2} + 6 \beta_{3} - 9 \beta_{4} + 5 \beta_{5} + 12 \beta_{7} ) q^{79} + ( 104 - 29 \beta_{1} - 12 \beta_{2} + 8 \beta_{3} + 32 \beta_{4} - 5 \beta_{5} + 15 \beta_{6} - 14 \beta_{7} ) q^{80} + ( -433 - 66 \beta_{1} + 8 \beta_{2} - \beta_{3} - 5 \beta_{4} + 44 \beta_{5} - 14 \beta_{6} + 12 \beta_{7} ) q^{82} + ( -125 + 66 \beta_{1} - 24 \beta_{2} - 16 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} - 8 \beta_{6} + 7 \beta_{7} ) q^{83} + ( -129 + 48 \beta_{1} + 3 \beta_{2} - 24 \beta_{3} - 5 \beta_{4} - 11 \beta_{5} + 26 \beta_{6} + 5 \beta_{7} ) q^{85} + ( 131 + 73 \beta_{1} - 53 \beta_{2} + 28 \beta_{3} - 23 \beta_{4} - 34 \beta_{5} - 16 \beta_{6} + 10 \beta_{7} ) q^{86} + ( -365 - 20 \beta_{1} + 4 \beta_{2} + 7 \beta_{3} + 11 \beta_{4} - 12 \beta_{5} - 34 \beta_{6} - 10 \beta_{7} ) q^{88} + ( -150 + 44 \beta_{1} - 25 \beta_{2} + 41 \beta_{3} + 23 \beta_{4} + 24 \beta_{6} - 33 \beta_{7} ) q^{89} + ( -273 - 4 \beta_{1} - 24 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} - 8 \beta_{6} + 3 \beta_{7} ) q^{91} + ( 90 + 14 \beta_{1} - 37 \beta_{2} - 22 \beta_{3} + 16 \beta_{4} - 20 \beta_{5} + 54 \beta_{6} - 4 \beta_{7} ) q^{92} + ( -260 + 2 \beta_{1} + 31 \beta_{2} + 14 \beta_{3} + 52 \beta_{4} + 16 \beta_{5} + 26 \beta_{6} + 22 \beta_{7} ) q^{94} + ( -231 + 116 \beta_{1} + 21 \beta_{2} + 20 \beta_{3} + 15 \beta_{4} + 49 \beta_{5} - 36 \beta_{6} + 13 \beta_{7} ) q^{95} + ( -57 + 72 \beta_{1} + 88 \beta_{2} - 21 \beta_{3} + 7 \beta_{5} + 14 \beta_{6} + 34 \beta_{7} ) q^{97} + ( 104 + 46 \beta_{1} - 4 \beta_{2} - 20 \beta_{3} + 35 \beta_{4} - 4 \beta_{5} + 84 \beta_{6} - 68 \beta_{7} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 6q^{2} + 34q^{4} - 42q^{5} + 53q^{7} - 51q^{8} + O(q^{10}) \) \( 8q - 6q^{2} + 34q^{4} - 42q^{5} + 53q^{7} - 51q^{8} + 21q^{10} - 67q^{11} + 33q^{13} - 79q^{14} - 30q^{16} - 139q^{17} + 64q^{19} - 117q^{20} - 84q^{22} - 226q^{23} + 96q^{25} - 24q^{26} + 34q^{28} - 456q^{29} + 124q^{31} - 174q^{32} - 114q^{34} - 556q^{35} + 127q^{37} - 237q^{38} - 188q^{40} - 425q^{41} - 115q^{43} - 510q^{44} - 711q^{46} - 420q^{47} + 171q^{49} + 137q^{50} - 922q^{52} - 98q^{53} - 616q^{55} + 412q^{56} - 1548q^{58} - 472q^{59} - 1254q^{61} + 766q^{62} - 2019q^{64} + 734q^{65} - 1010q^{67} + 503q^{68} - 2956q^{70} + 17q^{71} - 1180q^{73} + 1228q^{74} - 2008q^{76} - 441q^{77} - 873q^{79} + 865q^{80} - 3645q^{82} - 759q^{83} - 850q^{85} + 1226q^{86} - 3047q^{88} - 988q^{89} - 2111q^{91} + 1062q^{92} - 2240q^{94} - 1822q^{95} - 668q^{97} + 1368q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 2 x^{7} - 45 x^{6} + 47 x^{5} + 654 x^{4} - 157 x^{3} - 2898 x^{2} + 96 x + 2432\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 12 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 2 \nu^{2} - 16 \nu + 10 \)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{7} + 5 \nu^{6} + 30 \nu^{5} - 135 \nu^{4} - 247 \nu^{3} + 842 \nu^{2} + 294 \nu - 748 \)\()/4\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{7} - 5 \nu^{6} - 30 \nu^{5} + 137 \nu^{4} + 243 \nu^{3} - 886 \nu^{2} - 256 \nu + 876 \)\()/4\)
\(\beta_{6}\)\(=\)\((\)\( -5 \nu^{7} + 26 \nu^{6} + 145 \nu^{5} - 707 \nu^{4} - 1110 \nu^{3} + 4513 \nu^{2} + 898 \nu - 4128 \)\()/16\)
\(\beta_{7}\)\(=\)\((\)\( 3 \nu^{7} - 15 \nu^{6} - 88 \nu^{5} + 403 \nu^{4} + 689 \nu^{3} - 2520 \nu^{2} - 648 \nu + 2248 \)\()/8\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 12\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 2 \beta_{2} + 18 \beta_{1} + 14\)
\(\nu^{4}\)\(=\)\(2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + 26 \beta_{2} + 39 \beta_{1} + 228\)
\(\nu^{5}\)\(=\)\(4 \beta_{7} + 2 \beta_{5} + 8 \beta_{4} + 28 \beta_{3} + 75 \beta_{2} + 387 \beta_{1} + 554\)
\(\nu^{6}\)\(=\)\(20 \beta_{7} + 16 \beta_{6} + 74 \beta_{5} + 84 \beta_{4} + 79 \beta_{3} + 654 \beta_{2} + 1202 \beta_{1} + 5068\)
\(\nu^{7}\)\(=\)\(220 \beta_{7} + 80 \beta_{6} + 160 \beta_{5} + 386 \beta_{4} + 718 \beta_{3} + 2358 \beta_{2} + 9045 \beta_{1} + 17078\)

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
−4.15242
−3.08481
−3.06139
−1.04902
1.03574
2.26905
4.77847
5.26439
−5.15242 0 18.5474 −1.69104 0 −11.0943 −54.3449 0 8.71294
1.2 −4.08481 0 8.68564 7.45529 0 34.0237 −2.80073 0 −30.4534
1.3 −4.06139 0 8.49485 −16.2722 0 6.77038 −2.00979 0 66.0875
1.4 −2.04902 0 −3.80150 −16.1855 0 −1.13960 24.1816 0 33.1645
1.5 0.0357401 0 −7.99872 7.80970 0 4.90513 −0.571796 0 0.279120
1.6 1.26905 0 −6.38952 −14.8820 0 22.4903 −18.2610 0 −18.8859
1.7 3.77847 0 6.27681 5.74283 0 −24.4226 −6.51103 0 21.6991
1.8 4.26439 0 10.1850 −13.9771 0 21.4669 9.31764 0 −59.6039
\(n\): 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.e 8
3.b odd 2 1 177.4.a.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.4.a.d 8 3.b odd 2 1
531.4.a.e 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

$p$ $F_p(T)$
$2$ \( 128 - 3596 T + 336 T^{2} + 2015 T^{3} + 214 T^{4} - 209 T^{5} - 31 T^{6} + 6 T^{7} + T^{8} \)
$3$ \( T^{8} \)
$5$ \( -30976256 - 12917800 T + 3991700 T^{2} + 367590 T^{3} - 72995 T^{4} - 6168 T^{5} + 334 T^{6} + 42 T^{7} + T^{8} \)
$7$ \( -168443392 - 90382784 T + 47188312 T^{2} - 3320285 T^{3} - 403973 T^{4} + 37385 T^{5} - 53 T^{6} - 53 T^{7} + T^{8} \)
$11$ \( -229585463488 - 9449588432 T + 3022828836 T^{2} + 119167539 T^{3} - 6345239 T^{4} - 369558 T^{5} - 3550 T^{6} + 67 T^{7} + T^{8} \)
$13$ \( 1833054248 + 14789588684 T + 1293985870 T^{2} - 252026325 T^{3} + 4859041 T^{4} + 274826 T^{5} - 7352 T^{6} - 33 T^{7} + T^{8} \)
$17$ \( 75388834904 + 52431883284 T + 7784736238 T^{2} + 164499319 T^{3} - 12512297 T^{4} - 361925 T^{5} + 2411 T^{6} + 139 T^{7} + T^{8} \)
$19$ \( -40302801741824 - 5192817609408 T - 154443288784 T^{2} + 2808233428 T^{3} + 139661467 T^{4} + 319846 T^{5} - 21833 T^{6} - 64 T^{7} + T^{8} \)
$23$ \( -7187882575616 + 648142884608 T + 203986831672 T^{2} + 4164936686 T^{3} - 248339695 T^{4} - 5441286 T^{5} - 14471 T^{6} + 226 T^{7} + T^{8} \)
$29$ \( -19854632566319120 + 187515874935424 T + 17664968321008 T^{2} + 106916884912 T^{3} - 1773352167 T^{4} - 16368136 T^{5} + 17205 T^{6} + 456 T^{7} + T^{8} \)
$31$ \( -17327669038916864 - 907557921670416 T - 13211527133376 T^{2} + 29866593002 T^{3} + 1737947581 T^{4} + 3736652 T^{5} - 71125 T^{6} - 124 T^{7} + T^{8} \)
$37$ \( -401645346932032424 + 75371891736410972 T - 452970379143374 T^{2} - 3545535731997 T^{3} + 23110606355 T^{4} + 37672165 T^{5} - 285407 T^{6} - 127 T^{7} + T^{8} \)
$41$ \( 1423396374229159160 + 46939808868435804 T + 447341600539054 T^{2} - 39435535087 T^{3} - 23637153001 T^{4} - 128729927 T^{5} - 167793 T^{6} + 425 T^{7} + T^{8} \)
$43$ \( -99433287705157472 + 7671686227387200 T - 176637968565798 T^{2} + 790160019323 T^{3} + 10003070517 T^{4} - 40068734 T^{5} - 248960 T^{6} + 115 T^{7} + T^{8} \)
$47$ \( 1665526666113050624 + 26784216636081152 T - 789651439272992 T^{2} + 1906338567880 T^{3} + 30299943485 T^{4} - 92714016 T^{5} - 320551 T^{6} + 420 T^{7} + T^{8} \)
$53$ \( -\)\(29\!\cdots\!24\)\( + 3918441368015949672 T - 12576257813716916 T^{2} - 28371371768774 T^{3} + 171942191505 T^{4} + 25742844 T^{5} - 714158 T^{6} + 98 T^{7} + T^{8} \)
$59$ \( ( 59 + T )^{8} \)
$61$ \( \)\(17\!\cdots\!44\)\( + 4210289853656183552 T + 29077707639110096 T^{2} + 50002829460912 T^{3} - 166679626533 T^{4} - 602461440 T^{5} - 107335 T^{6} + 1254 T^{7} + T^{8} \)
$67$ \( \)\(50\!\cdots\!16\)\( - 10750574728687103584 T + 10824161043281816 T^{2} + 201263807544250 T^{3} - 586320783 T^{4} - 1089160844 T^{5} - 862426 T^{6} + 1010 T^{7} + T^{8} \)
$71$ \( -\)\(14\!\cdots\!12\)\( - 41066458800632805384 T - 191811050101998892 T^{2} + 79334849159143 T^{3} + 1094725739177 T^{4} + 6039618 T^{5} - 1903918 T^{6} - 17 T^{7} + T^{8} \)
$73$ \( -\)\(24\!\cdots\!28\)\( + 16651647386262236184 T + 178437112187809844 T^{2} + 436690506525874 T^{3} - 268549934245 T^{4} - 1880284694 T^{5} - 1165931 T^{6} + 1180 T^{7} + T^{8} \)
$79$ \( \)\(58\!\cdots\!76\)\( - 4315764007461827520 T - 12762134903028948 T^{2} + 95166585626737 T^{3} + 131145611305 T^{4} - 611378026 T^{5} - 770822 T^{6} + 873 T^{7} + T^{8} \)
$83$ \( -31977395506814839744 + 804160107703445712 T + 13983042273881372 T^{2} + 53374289286583 T^{3} - 35750882483 T^{4} - 496474997 T^{5} - 500127 T^{6} + 759 T^{7} + T^{8} \)
$89$ \( -\)\(98\!\cdots\!28\)\( + \)\(27\!\cdots\!52\)\( T - 1744244355968310004 T^{2} + 1194359312678276 T^{3} + 3942370515055 T^{4} - 2508938974 T^{5} - 3436617 T^{6} + 988 T^{7} + T^{8} \)
$97$ \( \)\(24\!\cdots\!72\)\( - \)\(11\!\cdots\!12\)\( T - 501222560134990132 T^{2} + 913194018545046 T^{3} + 2195419241175 T^{4} - 1864665330 T^{5} - 3082044 T^{6} + 668 T^{7} + T^{8} \)
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