Properties

Label 121.4.c.c
Level $121$
Weight $4$
Character orbit 121.c
Analytic conductor $7.139$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 121 = 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 121.c (of order \(5\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.13923111069\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.324000000.3
Defining polynomial: \(x^{8} + 3 x^{6} + 9 x^{4} + 27 x^{2} + 81\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 11)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( \beta_{2} + \beta_{7} ) q^{2} + ( -4 \beta_{1} - \beta_{6} ) q^{3} + ( -2 \beta_{1} - 2 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} - 2 \beta_{7} ) q^{4} + ( -1 - \beta_{2} + 8 \beta_{3} - \beta_{4} - \beta_{6} ) q^{5} + ( 13 + 13 \beta_{2} - 5 \beta_{3} + 13 \beta_{4} + 13 \beta_{6} ) q^{6} + ( 4 \beta_{1} + 4 \beta_{3} + 10 \beta_{4} + 4 \beta_{5} + 4 \beta_{7} ) q^{7} + ( -10 \beta_{1} - 6 \beta_{6} ) q^{8} + ( 22 \beta_{2} + 8 \beta_{7} ) q^{9} +O(q^{10})\) \( q + ( \beta_{2} + \beta_{7} ) q^{2} + ( -4 \beta_{1} - \beta_{6} ) q^{3} + ( -2 \beta_{1} - 2 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} - 2 \beta_{7} ) q^{4} + ( -1 - \beta_{2} + 8 \beta_{3} - \beta_{4} - \beta_{6} ) q^{5} + ( 13 + 13 \beta_{2} - 5 \beta_{3} + 13 \beta_{4} + 13 \beta_{6} ) q^{6} + ( 4 \beta_{1} + 4 \beta_{3} + 10 \beta_{4} + 4 \beta_{5} + 4 \beta_{7} ) q^{7} + ( -10 \beta_{1} - 6 \beta_{6} ) q^{8} + ( 22 \beta_{2} + 8 \beta_{7} ) q^{9} + ( 25 + 9 \beta_{5} ) q^{10} + ( -20 + 14 \beta_{5} ) q^{12} + ( 40 \beta_{2} - 20 \beta_{7} ) q^{13} + ( 6 \beta_{1} - 2 \beta_{6} ) q^{14} + ( 12 \beta_{1} + 12 \beta_{3} - 97 \beta_{4} + 12 \beta_{5} + 12 \beta_{7} ) q^{15} + ( 4 + 4 \beta_{2} - 32 \beta_{3} + 4 \beta_{4} + 4 \beta_{6} ) q^{16} + ( 62 + 62 \beta_{2} + 12 \beta_{3} + 62 \beta_{4} + 62 \beta_{6} ) q^{17} + ( -30 \beta_{1} - 30 \beta_{3} + 46 \beta_{4} - 30 \beta_{5} - 30 \beta_{7} ) q^{18} + ( 60 \beta_{1} + 36 \beta_{6} ) q^{19} + ( 44 \beta_{2} - 30 \beta_{7} ) q^{20} + ( 38 - 36 \beta_{5} ) q^{21} + ( -49 - 36 \beta_{5} ) q^{23} + ( 126 \beta_{2} + 34 \beta_{7} ) q^{24} + ( 16 \beta_{1} + 68 \beta_{6} ) q^{25} + ( -20 \beta_{1} - 20 \beta_{3} - 20 \beta_{4} - 20 \beta_{5} - 20 \beta_{7} ) q^{26} + ( 91 + 91 \beta_{2} + 12 \beta_{3} + 91 \beta_{4} + 91 \beta_{6} ) q^{27} + ( 64 + 64 \beta_{2} + 36 \beta_{3} + 64 \beta_{4} + 64 \beta_{6} ) q^{28} + ( 56 \beta_{1} + 56 \beta_{3} + 72 \beta_{4} + 56 \beta_{5} + 56 \beta_{7} ) q^{29} + ( -109 \beta_{1} - 133 \beta_{6} ) q^{30} + ( -17 \beta_{2} + 28 \beta_{7} ) q^{31} + ( -52 + 44 \beta_{5} ) q^{32} + ( -26 - 50 \beta_{5} ) q^{34} + ( -86 \beta_{2} + 76 \beta_{7} ) q^{35} + ( 12 \beta_{1} - 40 \beta_{6} ) q^{36} + ( 8 \beta_{1} + 8 \beta_{3} + 27 \beta_{4} + 8 \beta_{5} + 8 \beta_{7} ) q^{37} + ( -216 - 216 \beta_{2} + 96 \beta_{3} - 216 \beta_{4} - 216 \beta_{6} ) q^{38} + ( -200 - 200 \beta_{2} - 140 \beta_{3} - 200 \beta_{4} - 200 \beta_{6} ) q^{39} + ( 58 \beta_{1} + 58 \beta_{3} - 246 \beta_{4} + 58 \beta_{5} + 58 \beta_{7} ) q^{40} + ( -4 \beta_{1} + 268 \beta_{6} ) q^{41} + ( -70 \beta_{2} + 2 \beta_{7} ) q^{42} + ( -30 - 16 \beta_{5} ) q^{43} + ( 214 + 184 \beta_{5} ) q^{45} + ( -157 \beta_{2} - 85 \beta_{7} ) q^{46} + ( -120 \beta_{1} - 136 \beta_{6} ) q^{47} + ( -48 \beta_{1} - 48 \beta_{3} + 388 \beta_{4} - 48 \beta_{5} - 48 \beta_{7} ) q^{48} + ( 195 + 195 \beta_{2} - 80 \beta_{3} + 195 \beta_{4} + 195 \beta_{6} ) q^{49} + ( -116 - 116 \beta_{2} + 84 \beta_{3} - 116 \beta_{4} - 116 \beta_{6} ) q^{50} + ( -236 \beta_{1} - 236 \beta_{3} - 82 \beta_{4} - 236 \beta_{5} - 236 \beta_{7} ) q^{51} + ( 160 \beta_{1} - 280 \beta_{6} ) q^{52} + ( -246 \beta_{2} - 56 \beta_{7} ) q^{53} + ( -55 - 79 \beta_{5} ) q^{54} + ( 60 - 76 \beta_{5} ) q^{56} + ( -756 \beta_{2} - 204 \beta_{7} ) q^{57} + ( 16 \beta_{1} - 96 \beta_{6} ) q^{58} + ( 132 \beta_{1} + 132 \beta_{3} + 317 \beta_{4} + 132 \beta_{5} + 132 \beta_{7} ) q^{59} + ( -316 - 316 \beta_{2} - 146 \beta_{3} - 316 \beta_{4} - 316 \beta_{6} ) q^{60} + ( -420 - 420 \beta_{2} + 184 \beta_{3} - 420 \beta_{4} - 420 \beta_{6} ) q^{61} + ( -11 \beta_{1} - 11 \beta_{3} + 67 \beta_{4} - 11 \beta_{5} - 11 \beta_{7} ) q^{62} + ( -8 \beta_{1} + 124 \beta_{6} ) q^{63} + ( 112 \beta_{2} + 248 \beta_{7} ) q^{64} + ( -440 + 300 \beta_{5} ) q^{65} + ( 377 - 20 \beta_{5} ) q^{67} + ( 320 \beta_{2} - 172 \beta_{7} ) q^{68} + ( 232 \beta_{1} + 481 \beta_{6} ) q^{69} + ( 10 \beta_{1} + 10 \beta_{3} + 142 \beta_{4} + 10 \beta_{5} + 10 \beta_{7} ) q^{70} + ( 339 + 339 \beta_{2} + 76 \beta_{3} + 339 \beta_{4} + 339 \beta_{6} ) q^{71} + ( 372 + 372 \beta_{2} - 268 \beta_{3} + 372 \beta_{4} + 372 \beta_{6} ) q^{72} + ( 468 \beta_{1} + 468 \beta_{3} - 200 \beta_{4} + 468 \beta_{5} + 468 \beta_{7} ) q^{73} + ( 19 \beta_{1} + 3 \beta_{6} ) q^{74} + ( -260 \beta_{2} - 288 \beta_{7} ) q^{75} + ( 216 - 168 \beta_{5} ) q^{76} + ( -220 + 60 \beta_{5} ) q^{78} + ( 158 \beta_{2} + 656 \beta_{7} ) q^{79} + ( -64 \beta_{1} - 772 \beta_{6} ) q^{80} + ( -136 \beta_{1} - 136 \beta_{3} - 647 \beta_{4} - 136 \beta_{5} - 136 \beta_{7} ) q^{81} + ( -256 - 256 \beta_{2} + 264 \beta_{3} - 256 \beta_{4} - 256 \beta_{6} ) q^{82} + ( -234 - 234 \beta_{2} + 120 \beta_{3} - 234 \beta_{4} - 234 \beta_{6} ) q^{83} + ( -220 \beta_{1} - 220 \beta_{3} - 368 \beta_{4} - 220 \beta_{5} - 220 \beta_{7} ) q^{84} + ( -484 \beta_{1} + 226 \beta_{6} ) q^{85} + ( -78 \beta_{2} - 46 \beta_{7} ) q^{86} + ( 600 - 232 \beta_{5} ) q^{87} + ( -921 - 328 \beta_{5} ) q^{89} + ( 766 \beta_{2} + 398 \beta_{7} ) q^{90} + ( -360 \beta_{1} + 640 \beta_{6} ) q^{91} + ( -46 \beta_{1} - 46 \beta_{3} - 20 \beta_{4} - 46 \beta_{5} - 46 \beta_{7} ) q^{92} + ( 319 + 319 \beta_{2} + 40 \beta_{3} + 319 \beta_{4} + 319 \beta_{6} ) q^{93} + ( 496 + 496 \beta_{2} - 256 \beta_{3} + 496 \beta_{4} + 496 \beta_{6} ) q^{94} + ( -348 \beta_{1} - 348 \beta_{3} + 1476 \beta_{4} - 348 \beta_{5} - 348 \beta_{7} ) q^{95} + ( 164 \beta_{1} - 476 \beta_{6} ) q^{96} + ( 1097 \beta_{2} + 144 \beta_{7} ) q^{97} + ( -435 - 275 \beta_{5} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{2} + 2 q^{3} + 8 q^{4} - 2 q^{5} + 26 q^{6} - 20 q^{7} + 12 q^{8} - 44 q^{9} + O(q^{10}) \) \( 8 q - 2 q^{2} + 2 q^{3} + 8 q^{4} - 2 q^{5} + 26 q^{6} - 20 q^{7} + 12 q^{8} - 44 q^{9} + 200 q^{10} - 160 q^{12} - 80 q^{13} + 4 q^{14} + 194 q^{15} + 8 q^{16} + 124 q^{17} - 92 q^{18} - 72 q^{19} - 88 q^{20} + 304 q^{21} - 392 q^{23} - 252 q^{24} - 136 q^{25} + 40 q^{26} + 182 q^{27} + 128 q^{28} - 144 q^{29} + 266 q^{30} + 34 q^{31} - 416 q^{32} - 208 q^{34} + 172 q^{35} + 80 q^{36} - 54 q^{37} - 432 q^{38} - 400 q^{39} + 492 q^{40} - 536 q^{41} + 140 q^{42} - 240 q^{43} + 1712 q^{45} + 314 q^{46} + 272 q^{47} - 776 q^{48} + 390 q^{49} - 232 q^{50} + 164 q^{51} + 560 q^{52} + 492 q^{53} - 440 q^{54} + 480 q^{56} + 1512 q^{57} + 192 q^{58} - 634 q^{59} - 632 q^{60} - 840 q^{61} - 134 q^{62} - 248 q^{63} - 224 q^{64} - 3520 q^{65} + 3016 q^{67} - 640 q^{68} - 962 q^{69} - 284 q^{70} + 678 q^{71} + 744 q^{72} + 400 q^{73} - 6 q^{74} + 520 q^{75} + 1728 q^{76} - 1760 q^{78} - 316 q^{79} + 1544 q^{80} + 1294 q^{81} - 512 q^{82} - 468 q^{83} + 736 q^{84} - 452 q^{85} + 156 q^{86} + 4800 q^{87} - 7368 q^{89} - 1532 q^{90} - 1280 q^{91} + 40 q^{92} + 638 q^{93} + 992 q^{94} - 2952 q^{95} + 952 q^{96} - 2194 q^{97} - 3480 q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 3 x^{6} + 9 x^{4} + 27 x^{2} + 81\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/3\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/3\)
\(\beta_{4}\)\(=\)\( \nu^{4} \)\(/9\)
\(\beta_{5}\)\(=\)\( \nu^{5} \)\(/9\)
\(\beta_{6}\)\(=\)\( \nu^{6} \)\(/27\)
\(\beta_{7}\)\(=\)\( \nu^{7} \)\(/27\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(3 \beta_{2}\)
\(\nu^{3}\)\(=\)\(3 \beta_{3}\)
\(\nu^{4}\)\(=\)\(9 \beta_{4}\)
\(\nu^{5}\)\(=\)\(9 \beta_{5}\)
\(\nu^{6}\)\(=\)\(27 \beta_{6}\)
\(\nu^{7}\)\(=\)\(27 \beta_{7}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/121\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(\beta_{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
0.535233 + 1.64728i
−0.535233 1.64728i
1.40126 1.01807i
−1.40126 + 1.01807i
1.40126 + 1.01807i
−1.40126 1.01807i
0.535233 1.64728i
−0.535233 + 1.64728i
−2.21028 + 1.60586i −2.44995 7.54017i −0.165602 + 0.509670i −12.0191 8.73238i 17.5235 + 12.7316i 0.949237 2.92145i −7.20643 22.1791i −29.0084 + 21.0759i 40.5885
3.2 0.592242 0.430289i 1.83192 + 5.63806i −2.30653 + 7.09878i 10.4011 + 7.55681i 3.51093 + 2.55084i 5.23110 16.0997i 3.49823 + 10.7664i −6.58831 + 4.78668i 9.41154
9.1 −0.226216 + 0.696222i −4.79602 + 3.48451i 6.03859 + 4.38729i −3.97285 12.2272i −1.34106 4.12734i −13.6952 9.95015i −9.15848 + 6.65403i 2.51651 7.74502i 9.41154
9.2 0.844250 2.59833i 6.41405 4.66008i 0.433551 + 0.314993i 4.59088 + 14.1293i −6.69339 20.6001i −2.48514 1.80556i 18.8667 13.7075i 11.0802 34.1015i 40.5885
27.1 −0.226216 0.696222i −4.79602 3.48451i 6.03859 4.38729i −3.97285 + 12.2272i −1.34106 + 4.12734i −13.6952 + 9.95015i −9.15848 6.65403i 2.51651 + 7.74502i 9.41154
27.2 0.844250 + 2.59833i 6.41405 + 4.66008i 0.433551 0.314993i 4.59088 14.1293i −6.69339 + 20.6001i −2.48514 + 1.80556i 18.8667 + 13.7075i 11.0802 + 34.1015i 40.5885
81.1 −2.21028 1.60586i −2.44995 + 7.54017i −0.165602 0.509670i −12.0191 + 8.73238i 17.5235 12.7316i 0.949237 + 2.92145i −7.20643 + 22.1791i −29.0084 21.0759i 40.5885
81.2 0.592242 + 0.430289i 1.83192 5.63806i −2.30653 7.09878i 10.4011 7.55681i 3.51093 2.55084i 5.23110 + 16.0997i 3.49823 10.7664i −6.58831 4.78668i 9.41154
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 81.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 121.4.c.c 8
11.b odd 2 1 121.4.c.f 8
11.c even 5 1 11.4.a.a 2
11.c even 5 3 inner 121.4.c.c 8
11.d odd 10 1 121.4.a.c 2
11.d odd 10 3 121.4.c.f 8
33.f even 10 1 1089.4.a.v 2
33.h odd 10 1 99.4.a.c 2
44.g even 10 1 1936.4.a.w 2
44.h odd 10 1 176.4.a.i 2
55.j even 10 1 275.4.a.b 2
55.k odd 20 2 275.4.b.c 4
77.j odd 10 1 539.4.a.e 2
88.l odd 10 1 704.4.a.n 2
88.o even 10 1 704.4.a.p 2
132.o even 10 1 1584.4.a.bc 2
143.n even 10 1 1859.4.a.a 2
165.o odd 10 1 2475.4.a.q 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.4.a.a 2 11.c even 5 1
99.4.a.c 2 33.h odd 10 1
121.4.a.c 2 11.d odd 10 1
121.4.c.c 8 1.a even 1 1 trivial
121.4.c.c 8 11.c even 5 3 inner
121.4.c.f 8 11.b odd 2 1
121.4.c.f 8 11.d odd 10 3
176.4.a.i 2 44.h odd 10 1
275.4.a.b 2 55.j even 10 1
275.4.b.c 4 55.k odd 20 2
539.4.a.e 2 77.j odd 10 1
704.4.a.n 2 88.l odd 10 1
704.4.a.p 2 88.o even 10 1
1089.4.a.v 2 33.f even 10 1
1584.4.a.bc 2 132.o even 10 1
1859.4.a.a 2 143.n even 10 1
1936.4.a.w 2 44.g even 10 1
2475.4.a.q 2 165.o odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{8} + \cdots\) acting on \(S_{4}^{\mathrm{new}}(121, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 16 - 16 T + 24 T^{2} - 32 T^{3} + 44 T^{4} + 16 T^{5} + 6 T^{6} + 2 T^{7} + T^{8} \)
$3$ \( 4879681 + 207646 T + 112659 T^{2} + 9212 T^{3} + 2789 T^{4} - 196 T^{5} + 51 T^{6} - 2 T^{7} + T^{8} \)
$5$ \( 1330863361 - 13935742 T + 7113795 T^{2} - 147452 T^{3} + 38789 T^{4} + 772 T^{5} + 195 T^{6} + 2 T^{7} + T^{8} \)
$7$ \( 7311616 + 2812160 T + 940992 T^{2} + 307840 T^{3} + 100304 T^{4} + 5920 T^{5} + 348 T^{6} + 20 T^{7} + T^{8} \)
$11$ \( T^{8} \)
$13$ \( 25600000000 + 5120000000 T + 960000000 T^{2} + 179200000 T^{3} + 33440000 T^{4} + 448000 T^{5} + 6000 T^{6} + 80 T^{7} + T^{8} \)
$17$ \( 135530203361536 - 4925482185472 T + 139281825216 T^{2} - 3618248576 T^{3} + 90674384 T^{4} - 1060448 T^{5} + 11964 T^{6} - 124 T^{7} + T^{8} \)
$19$ \( 8158789166432256 - 61809008836608 T + 1326708523008 T^{2} - 16554295296 T^{3} + 265006080 T^{4} + 1741824 T^{5} + 14688 T^{6} + 72 T^{7} + T^{8} \)
$23$ \( ( -1487 + 98 T + T^{2} )^{4} \)
$29$ \( 318343244414976 - 10852610605056 T + 445340712960 T^{2} - 17751343104 T^{3} + 710590464 T^{4} + 4202496 T^{5} + 24960 T^{6} + 144 T^{7} + T^{8} \)
$31$ \( 18113272128961 + 298522177598 T + 13699964211 T^{2} + 370490044 T^{3} + 12746789 T^{4} - 179588 T^{5} + 3219 T^{6} - 34 T^{7} + T^{8} \)
$37$ \( 83156680161 + 8362124262 T + 686029851 T^{2} + 53414316 T^{3} + 4093749 T^{4} + 99468 T^{5} + 2379 T^{6} + 54 T^{7} + T^{8} \)
$41$ \( 26540983231867518976 + 198199495824244736 T + 1110314680811520 T^{2} + 5530109968384 T^{3} + 25827912704 T^{4} + 77046784 T^{5} + 215520 T^{6} + 536 T^{7} + T^{8} \)
$43$ \( ( 132 + 60 T + T^{2} )^{4} \)
$47$ \( 372450974242963456 + 4100820312260608 T + 60228064247808 T^{2} + 829131063296 T^{3} + 11567022080 T^{4} - 33562624 T^{5} + 98688 T^{6} - 272 T^{7} + T^{8} \)
$53$ \( 6822688517501296896 - 65679790847042304 T + 498782354606784 T^{2} - 3516496979328 T^{3} + 24092787024 T^{4} - 68805216 T^{5} + 190956 T^{6} - 492 T^{7} + T^{8} \)
$59$ \( 5405062778469469921 + 71070572651754442 T + 822400404063771 T^{2} + 9339678609716 T^{3} + 105750167669 T^{4} + 193700948 T^{5} + 353739 T^{6} + 634 T^{7} + T^{8} \)
$61$ \( 31358076138306994176 + 351998930353029120 T + 3532192449196032 T^{2} + 34945514311680 T^{3} + 345066810624 T^{4} + 466986240 T^{5} + 630768 T^{6} + 840 T^{7} + T^{8} \)
$67$ \( ( 140929 - 754 T + T^{2} )^{4} \)
$71$ \( 90714074381111535201 - 630210593284289046 T + 3448697220760059 T^{2} - 17501317946892 T^{3} + 86247951669 T^{4} - 179329644 T^{5} + 362091 T^{6} - 678 T^{7} + T^{8} \)
$73$ \( \)\(14\!\cdots\!56\)\( + 93986940567982899200 T + 295891807929397248 T^{2} + 344114890547200 T^{3} + 702572413184 T^{4} - 557657600 T^{5} + 777072 T^{6} - 400 T^{7} + T^{8} \)
$79$ \( \)\(25\!\cdots\!96\)\( - \)\(64\!\cdots\!44\)\( T + 2189356595231582400 T^{2} - 1052961583413376 T^{3} + 1992104899664 T^{4} + 831694304 T^{5} + 1365900 T^{6} + 316 T^{7} + T^{8} \)
$83$ \( 17833235004170496 + 722218240044288 T + 27705512403648 T^{2} + 1059532845696 T^{3} + 40511929680 T^{4} + 91686816 T^{5} + 207468 T^{6} + 468 T^{7} + T^{8} \)
$89$ \( ( 525489 + 1842 T + T^{2} )^{4} \)
$97$ \( \)\(16\!\cdots\!01\)\( + \)\(32\!\cdots\!94\)\( T + 4782757978435716435 T^{2} + 6337691017842596 T^{3} + 7993452612389 T^{4} + 5553527396 T^{5} + 3672435 T^{6} + 2194 T^{7} + T^{8} \)
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