Properties

Label 567.4.a.i
Level $567$
Weight $4$
Character orbit 567.a
Self dual yes
Analytic conductor $33.454$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 567 = 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 567.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(33.4540829733\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 3 x^{7} - 49 x^{6} + 138 x^{5} + 708 x^{4} - 1941 x^{3} - 2506 x^{2} + 8592 x - 4616\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 3^{5} \)
Twist minimal: no (minimal twist has level 63)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( 5 + \beta_{2} ) q^{4} + ( 4 + \beta_{4} ) q^{5} -7 q^{7} + ( -1 + 4 \beta_{1} + \beta_{2} + \beta_{3} ) q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( 5 + \beta_{2} ) q^{4} + ( 4 + \beta_{4} ) q^{5} -7 q^{7} + ( -1 + 4 \beta_{1} + \beta_{2} + \beta_{3} ) q^{8} + ( -3 + 6 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{10} + ( 2 + \beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} ) q^{11} + ( 7 + 7 \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{7} ) q^{13} -7 \beta_{1} q^{14} + ( 9 + 8 \beta_{1} + 3 \beta_{2} + 2 \beta_{4} + 2 \beta_{5} + \beta_{7} ) q^{16} + ( 21 - 4 \beta_{1} + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{7} ) q^{17} + ( 17 + 11 \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} + 3 \beta_{7} ) q^{19} + ( 42 - 7 \beta_{1} + 8 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{20} + ( 18 + 4 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{22} + ( 25 + 4 \beta_{1} + 5 \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + 4 \beta_{7} ) q^{23} + ( 29 + 11 \beta_{1} - \beta_{2} - 3 \beta_{3} + 9 \beta_{4} - \beta_{5} + \beta_{6} - 3 \beta_{7} ) q^{25} + ( 94 - 2 \beta_{1} + 14 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{26} + ( -35 - 7 \beta_{2} ) q^{28} + ( 80 - 4 \beta_{1} - 5 \beta_{2} - 5 \beta_{4} + \beta_{5} + \beta_{6} - 4 \beta_{7} ) q^{29} + ( 3 + 10 \beta_{1} + 5 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} + \beta_{5} - 4 \beta_{6} + 3 \beta_{7} ) q^{31} + ( 94 + 5 \beta_{1} + 3 \beta_{3} - 4 \beta_{4} + 4 \beta_{5} + 2 \beta_{6} + 4 \beta_{7} ) q^{32} + ( -49 + 35 \beta_{1} - 3 \beta_{2} + 5 \beta_{3} - 14 \beta_{4} + 6 \beta_{5} + 4 \beta_{6} ) q^{34} + ( -28 - 7 \beta_{4} ) q^{35} + ( 31 - 2 \beta_{1} - 13 \beta_{2} - 5 \beta_{3} - \beta_{4} + 3 \beta_{5} + 2 \beta_{6} - 3 \beta_{7} ) q^{37} + ( 122 + 21 \beta_{1} + 6 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} + 3 \beta_{5} - 6 \beta_{6} ) q^{38} + ( -63 + 48 \beta_{1} - 7 \beta_{2} - 6 \beta_{4} - 9 \beta_{5} - 3 \beta_{7} ) q^{40} + ( 62 - 3 \beta_{1} - 15 \beta_{2} + 5 \beta_{4} - 3 \beta_{5} - 6 \beta_{7} ) q^{41} + ( -7 - 2 \beta_{1} + 7 \beta_{2} - 7 \beta_{3} - 9 \beta_{4} - 7 \beta_{5} + 4 \beta_{6} + \beta_{7} ) q^{43} + ( 46 + 28 \beta_{1} - 14 \beta_{2} + 5 \beta_{3} + 14 \beta_{4} - 6 \beta_{5} + 4 \beta_{6} - \beta_{7} ) q^{44} + ( 25 + 60 \beta_{1} + 11 \beta_{2} + 9 \beta_{3} + 2 \beta_{4} + 6 \beta_{5} - 4 \beta_{6} + 7 \beta_{7} ) q^{46} + ( 80 - 33 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 8 \beta_{4} - \beta_{6} + 4 \beta_{7} ) q^{47} + 49 q^{49} + ( 135 + 37 \beta_{1} - 11 \beta_{2} + 6 \beta_{3} + 16 \beta_{4} - 18 \beta_{5} + 2 \beta_{6} - 8 \beta_{7} ) q^{50} + ( -105 + 131 \beta_{1} + 3 \beta_{2} + 13 \beta_{3} - 10 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + 3 \beta_{7} ) q^{52} + ( 89 - 23 \beta_{1} - 25 \beta_{2} - 9 \beta_{3} + 9 \beta_{4} - \beta_{5} - 7 \beta_{6} - 5 \beta_{7} ) q^{53} + ( -131 + 54 \beta_{1} + 24 \beta_{2} + 5 \beta_{3} - 14 \beta_{5} - 7 \beta_{6} + \beta_{7} ) q^{55} + ( 7 - 28 \beta_{1} - 7 \beta_{2} - 7 \beta_{3} ) q^{56} + ( -23 + 47 \beta_{1} + 7 \beta_{2} - 9 \beta_{3} - 14 \beta_{4} + 8 \beta_{5} + 8 \beta_{6} - \beta_{7} ) q^{58} + ( -16 - 10 \beta_{1} - 17 \beta_{2} - 10 \beta_{3} - 13 \beta_{4} + 5 \beta_{5} + 5 \beta_{6} + 4 \beta_{7} ) q^{59} + ( 78 - 5 \beta_{1} + 4 \beta_{2} - 6 \beta_{3} + 7 \beta_{4} - 2 \beta_{5} - 9 \beta_{6} - 4 \beta_{7} ) q^{61} + ( 128 + 32 \beta_{1} + 16 \beta_{3} - 16 \beta_{4} + 4 \beta_{6} + 11 \beta_{7} ) q^{62} + ( -25 + 22 \beta_{1} - 9 \beta_{2} + \beta_{3} - 34 \beta_{4} + 6 \beta_{5} - 4 \beta_{6} - 3 \beta_{7} ) q^{64} + ( -61 + 90 \beta_{1} - 35 \beta_{2} + 13 \beta_{3} + 27 \beta_{4} - 3 \beta_{5} + 6 \beta_{6} + 3 \beta_{7} ) q^{65} + ( -93 + 4 \beta_{1} - 28 \beta_{2} + 19 \beta_{3} + 34 \beta_{4} + \beta_{6} + \beta_{7} ) q^{67} + ( 304 - 50 \beta_{1} + 58 \beta_{2} - 4 \beta_{3} - 26 \beta_{4} + 26 \beta_{5} - 4 \beta_{6} + 13 \beta_{7} ) q^{68} + ( 21 - 42 \beta_{1} + 7 \beta_{2} - 7 \beta_{3} - 14 \beta_{4} + 7 \beta_{5} ) q^{70} + ( 122 - 19 \beta_{1} - 3 \beta_{2} + \beta_{3} - \beta_{4} + 9 \beta_{5} - 17 \beta_{6} + 11 \beta_{7} ) q^{71} + ( 110 + 13 \beta_{1} - 6 \beta_{3} - 8 \beta_{4} + 10 \beta_{5} + 21 \beta_{6} - 16 \beta_{7} ) q^{73} + ( -60 \beta_{1} - 36 \beta_{2} - 4 \beta_{3} - 24 \beta_{4} + 8 \beta_{6} - 5 \beta_{7} ) q^{74} + ( 110 + 113 \beta_{1} + 50 \beta_{2} + 18 \beta_{3} - 18 \beta_{4} + 5 \beta_{5} + 10 \beta_{6} + 3 \beta_{7} ) q^{76} + ( -14 - 7 \beta_{1} - 7 \beta_{2} + 7 \beta_{4} + 7 \beta_{5} ) q^{77} + ( 38 - 60 \beta_{1} - 60 \beta_{2} + 7 \beta_{3} + 36 \beta_{4} + 2 \beta_{5} - 11 \beta_{6} + 5 \beta_{7} ) q^{79} + ( 349 - 86 \beta_{1} - 17 \beta_{2} - 16 \beta_{3} + 8 \beta_{4} + 3 \beta_{5} + 4 \beta_{6} - 10 \beta_{7} ) q^{80} + ( -12 - 24 \beta_{1} - 8 \beta_{2} - 19 \beta_{3} + 22 \beta_{4} - 14 \beta_{5} + 6 \beta_{6} - 6 \beta_{7} ) q^{82} + ( 361 + 20 \beta_{1} - 2 \beta_{2} - \beta_{3} + 30 \beta_{4} + 4 \beta_{5} + 15 \beta_{6} + 15 \beta_{7} ) q^{83} + ( -17 - 193 \beta_{1} - 16 \beta_{2} - 3 \beta_{3} + 20 \beta_{4} + 26 \beta_{5} + 13 \beta_{7} ) q^{85} + ( 26 - 26 \beta_{1} - 46 \beta_{2} - 24 \beta_{3} - 4 \beta_{4} - 26 \beta_{5} - 24 \beta_{6} - 33 \beta_{7} ) q^{86} + ( 203 - 73 \beta_{1} + 3 \beta_{2} - 7 \beta_{3} + 46 \beta_{4} - 6 \beta_{5} - 2 \beta_{6} - 3 \beta_{7} ) q^{88} + ( 538 - 9 \beta_{1} + 31 \beta_{2} - 2 \beta_{3} - 35 \beta_{4} - 25 \beta_{5} - 2 \beta_{6} - 28 \beta_{7} ) q^{89} + ( -49 - 49 \beta_{1} + 14 \beta_{2} - 7 \beta_{3} + 7 \beta_{7} ) q^{91} + ( 506 + 101 \beta_{1} + 76 \beta_{2} + 14 \beta_{3} - 10 \beta_{4} + 26 \beta_{5} + 14 \beta_{6} + \beta_{7} ) q^{92} + ( -415 + 65 \beta_{1} - 45 \beta_{2} - 2 \beta_{3} - 20 \beta_{4} + 4 \beta_{5} - 6 \beta_{6} + \beta_{7} ) q^{94} + ( 114 + 64 \beta_{1} - 45 \beta_{2} + \beta_{3} + 13 \beta_{4} - 5 \beta_{5} + 6 \beta_{6} - 15 \beta_{7} ) q^{95} + ( 44 - 164 \beta_{1} + 45 \beta_{2} + 18 \beta_{3} + \beta_{4} - 5 \beta_{5} - 21 \beta_{6} + 14 \beta_{7} ) q^{97} + 49 \beta_{1} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 3q^{2} + 43q^{4} + 30q^{5} - 56q^{7} + 6q^{8} + O(q^{10}) \) \( 8q + 3q^{2} + 43q^{4} + 30q^{5} - 56q^{7} + 6q^{8} - 14q^{10} + 24q^{11} + 68q^{13} - 21q^{14} + 103q^{16} + 168q^{17} + 176q^{19} + 330q^{20} + 151q^{22} + 228q^{23} + 244q^{25} + 795q^{26} - 301q^{28} + 618q^{29} + 72q^{31} + 786q^{32} - 261q^{34} - 210q^{35} + 210q^{37} + 1032q^{38} - 375q^{40} + 420q^{41} - 2q^{43} + 387q^{44} + 402q^{46} + 570q^{47} + 392q^{49} + 1110q^{50} - 431q^{52} + 528q^{53} - 838q^{55} - 42q^{56} + 37q^{58} - 150q^{59} + 578q^{61} + 1170q^{62} - 112q^{64} - 366q^{65} - 898q^{67} + 2526q^{68} + 98q^{70} + 882q^{71} + 972q^{73} - 222q^{74} + 1423q^{76} - 168q^{77} - 158q^{79} + 2475q^{80} - 211q^{82} + 2958q^{83} - 774q^{85} - 114q^{86} + 1317q^{88} + 4380q^{89} - 476q^{91} + 4629q^{92} - 3234q^{94} + 930q^{95} - 60q^{97} + 147q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 3 x^{7} - 49 x^{6} + 138 x^{5} + 708 x^{4} - 1941 x^{3} - 2506 x^{2} + 8592 x - 4616\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 13 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 20 \nu + 14 \)
\(\beta_{4}\)\(=\)\((\)\( \nu^{7} - 14 \nu^{6} - 81 \nu^{5} + 657 \nu^{4} + 1851 \nu^{3} - 8352 \nu^{2} - 11632 \nu + 24200 \)\()/744\)
\(\beta_{5}\)\(=\)\((\)\( 13 \nu^{7} + 4 \nu^{6} - 681 \nu^{5} + 171 \nu^{4} + 10857 \nu^{3} - 9066 \nu^{2} - 52264 \nu + 61640 \)\()/744\)
\(\beta_{6}\)\(=\)\((\)\( 8 \nu^{7} - 19 \nu^{6} - 369 \nu^{5} + 699 \nu^{4} + 4950 \nu^{3} - 6738 \nu^{2} - 17633 \nu + 19504 \)\()/186\)
\(\beta_{7}\)\(=\)\((\)\( -7 \nu^{7} + 5 \nu^{6} + 381 \nu^{5} - 228 \nu^{4} - 6354 \nu^{3} + 3687 \nu^{2} + 30460 \nu - 25436 \)\()/186\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 13\)
\(\nu^{3}\)\(=\)\(\beta_{3} + \beta_{2} + 20 \beta_{1} - 1\)
\(\nu^{4}\)\(=\)\(\beta_{7} + 2 \beta_{5} + 2 \beta_{4} + 27 \beta_{2} + 8 \beta_{1} + 257\)
\(\nu^{5}\)\(=\)\(4 \beta_{7} + 2 \beta_{6} + 4 \beta_{5} - 4 \beta_{4} + 35 \beta_{3} + 32 \beta_{2} + 453 \beta_{1} + 62\)
\(\nu^{6}\)\(=\)\(37 \beta_{7} - 4 \beta_{6} + 86 \beta_{5} + 46 \beta_{4} + \beta_{3} + 687 \beta_{2} + 342 \beta_{1} + 5775\)
\(\nu^{7}\)\(=\)\(185 \beta_{7} + 106 \beta_{6} + 214 \beta_{5} - 250 \beta_{4} + 998 \beta_{3} + 972 \beta_{2} + 10837 \beta_{1} + 3250\)

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
−4.95181
−3.93417
−2.93948
0.807372
1.59498
2.61577
4.56358
5.24375
−4.95181 0 16.5205 16.1400 0 −7.00000 −42.1917 0 −79.9221
1.2 −3.93417 0 7.47771 −2.43142 0 −7.00000 2.05480 0 9.56561
1.3 −2.93948 0 0.640533 −2.56885 0 −7.00000 21.6330 0 7.55109
1.4 0.807372 0 −7.34815 18.2289 0 −7.00000 −12.3917 0 14.7175
1.5 1.59498 0 −5.45602 −2.55632 0 −7.00000 −21.4622 0 −4.07730
1.6 2.61577 0 −1.15774 −13.5431 0 −7.00000 −23.9546 0 −35.4255
1.7 4.56358 0 12.8263 20.7910 0 −7.00000 22.0250 0 94.8813
1.8 5.24375 0 19.4970 −4.06017 0 −7.00000 60.2873 0 −21.2906
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(7\) \(1\)

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 567.4.a.i 8
3.b odd 2 1 567.4.a.g 8
9.c even 3 2 63.4.f.b 16
9.d odd 6 2 189.4.f.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.4.f.b 16 9.c even 3 2
189.4.f.b 16 9.d odd 6 2
567.4.a.g 8 3.b odd 2 1
567.4.a.i 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(567))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -4616 + 8592 T - 2506 T^{2} - 1941 T^{3} + 708 T^{4} + 138 T^{5} - 49 T^{6} - 3 T^{7} + T^{8} \)
$3$ \( T^{8} \)
$5$ \( -5370473 - 7233858 T - 3397711 T^{2} - 565608 T^{3} + 14001 T^{4} + 8610 T^{5} - 172 T^{6} - 30 T^{7} + T^{8} \)
$7$ \( ( 7 + T )^{8} \)
$11$ \( 4399098628 + 8149275276 T - 820937803 T^{2} - 64097388 T^{3} + 3617979 T^{4} + 84834 T^{5} - 3760 T^{6} - 24 T^{7} + T^{8} \)
$13$ \( -8057410599908 + 466823643556 T + 11935027315 T^{2} - 1046864588 T^{3} + 3974167 T^{4} + 584044 T^{5} - 6887 T^{6} - 68 T^{7} + T^{8} \)
$17$ \( 2073513126204 + 1554538753260 T + 250373208411 T^{2} - 8466840468 T^{3} - 32700159 T^{4} + 2480508 T^{5} - 8910 T^{6} - 168 T^{7} + T^{8} \)
$19$ \( -604815137888177 + 18413307722920 T + 473899377403 T^{2} - 15172506716 T^{3} - 63882233 T^{4} + 3616846 T^{5} - 12980 T^{6} - 176 T^{7} + T^{8} \)
$23$ \( -145237005548415 + 15365757330018 T + 426462975270 T^{2} - 22928842098 T^{3} - 89682147 T^{4} + 6216822 T^{5} - 19827 T^{6} - 228 T^{7} + T^{8} \)
$29$ \( 498541969683460 - 111593220428004 T + 4126180974341 T^{2} + 42151101756 T^{3} - 1333718022 T^{4} + 218022 T^{5} + 108797 T^{6} - 618 T^{7} + T^{8} \)
$31$ \( -2141134930990692 - 865622373907284 T - 29397434125473 T^{2} - 17995301658 T^{3} + 3419732898 T^{4} + 4044006 T^{5} - 110799 T^{6} - 72 T^{7} + T^{8} \)
$37$ \( 71324346320230788 - 1595546991527796 T - 52771182253767 T^{2} + 124534542204 T^{3} + 6138784098 T^{4} + 12199410 T^{5} - 131643 T^{6} - 210 T^{7} + T^{8} \)
$41$ \( 462426168557412736 + 10682988414253200 T - 6676629991213 T^{2} - 1199664512340 T^{3} - 128239347 T^{4} + 49443456 T^{5} - 87130 T^{6} - 420 T^{7} + T^{8} \)
$43$ \( 20950193983088735284 - 18965485935823156 T - 3087591835842791 T^{2} + 1857073655618 T^{3} + 67028138194 T^{4} - 10001572 T^{5} - 461129 T^{6} + 2 T^{7} + T^{8} \)
$47$ \( -266172075699679296 + 3690234827458704 T + 72863146602747 T^{2} - 849034787274 T^{3} - 3679740576 T^{4} + 43446078 T^{5} - 1665 T^{6} - 570 T^{7} + T^{8} \)
$53$ \( 86158340642449528308 - 2650393862685934092 T + 24282335794603149 T^{2} - 81691443067554 T^{3} + 17633836809 T^{4} + 437892894 T^{5} - 587934 T^{6} - 528 T^{7} + T^{8} \)
$59$ \( 3497507042491117152 + 800025206240455272 T - 7516085639118957 T^{2} - 3568497587424 T^{3} + 134276268996 T^{4} - 42907140 T^{5} - 674415 T^{6} + 150 T^{7} + T^{8} \)
$61$ \( 797499529610837935 + 34334165070805642 T + 265758786651520 T^{2} - 3900655026314 T^{3} - 24959688881 T^{4} + 239974072 T^{5} - 336029 T^{6} - 578 T^{7} + T^{8} \)
$67$ \( 16406775796815316768 - 4042724170048486616 T - 4376198717597015 T^{2} + 110804701702978 T^{3} + 159691772083 T^{4} - 649638344 T^{5} - 808250 T^{6} + 898 T^{7} + T^{8} \)
$71$ \( \)\(36\!\cdots\!25\)\( + 41762863933845347934 T - 84625618285280943 T^{2} - 370893752864550 T^{3} + 541546496631 T^{4} + 1037538936 T^{5} - 1297368 T^{6} - 882 T^{7} + T^{8} \)
$73$ \( \)\(45\!\cdots\!76\)\( + 20102375258878654764 T + 7761474434502279 T^{2} - 334588931457660 T^{3} + 98139802230 T^{4} + 1566589032 T^{5} - 1393785 T^{6} - 972 T^{7} + T^{8} \)
$79$ \( \)\(13\!\cdots\!09\)\( - 36002589298497772636 T - 396022336235969075 T^{2} + 137375639612150 T^{3} + 1655367536629 T^{4} - 251111914 T^{5} - 2287370 T^{6} + 158 T^{7} + T^{8} \)
$83$ \( -\)\(13\!\cdots\!60\)\( + 12507738718368381408 T - 213297172017370389 T^{2} + 1160957901101148 T^{3} - 2454567763617 T^{4} + 1328931702 T^{5} + 2129274 T^{6} - 2958 T^{7} + T^{8} \)
$89$ \( \)\(44\!\cdots\!84\)\( - \)\(56\!\cdots\!32\)\( T + 1426979281925511527 T^{2} + 1461374513042748 T^{3} - 5942862637383 T^{4} + 1699594764 T^{5} + 5183942 T^{6} - 4380 T^{7} + T^{8} \)
$97$ \( -\)\(22\!\cdots\!00\)\( + \)\(19\!\cdots\!28\)\( T - 75028456400690025 T^{2} - 2329113227232420 T^{3} + 4347204763950 T^{4} + 1124173728 T^{5} - 4188249 T^{6} + 60 T^{7} + T^{8} \)
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