Properties

Label 147.7.d.b
Level $147$
Weight $7$
Character orbit 147.d
Analytic conductor $33.818$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 147.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(33.8179502921\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - x^{7} + 212 x^{6} + 473 x^{5} + 39800 x^{4} + 36821 x^{3} + 985651 x^{2} - 601290 x + 21068100\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6}\cdot 3\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 21)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( 1 - \beta_{3} ) q^{2} + 9 \beta_{1} q^{3} + ( 42 - 4 \beta_{3} - \beta_{4} ) q^{4} + ( -25 \beta_{1} + 3 \beta_{2} - \beta_{5} ) q^{5} + ( 9 \beta_{1} + 9 \beta_{2} ) q^{6} + ( 404 - 50 \beta_{3} - 5 \beta_{4} + 4 \beta_{7} ) q^{8} -243 q^{9} +O(q^{10})\) \( q + ( 1 - \beta_{3} ) q^{2} + 9 \beta_{1} q^{3} + ( 42 - 4 \beta_{3} - \beta_{4} ) q^{4} + ( -25 \beta_{1} + 3 \beta_{2} - \beta_{5} ) q^{5} + ( 9 \beta_{1} + 9 \beta_{2} ) q^{6} + ( 404 - 50 \beta_{3} - 5 \beta_{4} + 4 \beta_{7} ) q^{8} -243 q^{9} + ( 290 \beta_{1} - 32 \beta_{2} + 9 \beta_{5} + 5 \beta_{6} ) q^{10} + ( 55 - 103 \beta_{3} + 11 \beta_{4} - \beta_{7} ) q^{11} + ( 378 \beta_{1} + 36 \beta_{2} + 9 \beta_{5} + 9 \beta_{6} ) q^{12} + ( 264 \beta_{1} + 3 \beta_{2} + 19 \beta_{5} + 20 \beta_{6} ) q^{13} + ( 675 + 81 \beta_{3} - 9 \beta_{4} - 9 \beta_{7} ) q^{15} + ( 3020 - 646 \beta_{3} - 35 \beta_{4} + 20 \beta_{7} ) q^{16} + ( 442 \beta_{1} + 82 \beta_{2} - 6 \beta_{5} - 24 \beta_{6} ) q^{17} + ( -243 + 243 \beta_{3} ) q^{18} + ( -1550 \beta_{1} + 85 \beta_{2} + 37 \beta_{5} - 44 \beta_{6} ) q^{19} + ( -1440 \beta_{1} + 366 \beta_{2} - 27 \beta_{5} - 15 \beta_{6} ) q^{20} + ( 10798 + 308 \beta_{3} - 81 \beta_{4} - 44 \beta_{7} ) q^{22} + ( -1088 - 280 \beta_{3} - 16 \beta_{4} - 64 \beta_{7} ) q^{23} + ( 3636 \beta_{1} + 450 \beta_{2} - 27 \beta_{5} + 81 \beta_{6} ) q^{24} + ( 4030 - 1107 \beta_{3} + 103 \beta_{4} + 43 \beta_{7} ) q^{25} + ( 699 \beta_{1} + 1457 \beta_{2} - 131 \beta_{5} + 105 \beta_{6} ) q^{26} -2187 \beta_{1} q^{27} + ( -605 + 1247 \beta_{3} + 193 \beta_{4} + 145 \beta_{7} ) q^{29} + ( -7830 - 864 \beta_{3} + 171 \beta_{4} + 36 \beta_{7} ) q^{30} + ( 7331 \beta_{1} + 530 \beta_{2} - 18 \beta_{5} - 44 \beta_{6} ) q^{31} + ( 45324 - 4098 \beta_{3} - 581 \beta_{4} - 116 \beta_{7} ) q^{32} + ( 495 \beta_{1} + 927 \beta_{2} - 81 \beta_{5} - 108 \beta_{6} ) q^{33} + ( 8908 \beta_{1} - 464 \beta_{2} + 142 \beta_{5} - 74 \beta_{6} ) q^{34} + ( -10206 + 972 \beta_{3} + 243 \beta_{4} ) q^{36} + ( -16510 - 1255 \beta_{3} + 147 \beta_{4} - 113 \beta_{7} ) q^{37} + ( 7111 \beta_{1} - 2639 \beta_{2} - 93 \beta_{5} - 297 \beta_{6} ) q^{38} + ( -7128 + 81 \beta_{3} + 531 \beta_{4} - 9 \beta_{7} ) q^{39} + ( 18340 \beta_{1} + 614 \beta_{2} - 33 \beta_{5} - 5 \beta_{6} ) q^{40} + ( -7782 \beta_{1} - 1052 \beta_{2} - 652 \beta_{5} - 432 \beta_{6} ) q^{41} + ( 6182 + 1901 \beta_{3} + 399 \beta_{4} - 221 \beta_{7} ) q^{43} + ( -24840 - 7614 \beta_{3} + 7 \beta_{4} + 388 \beta_{7} ) q^{44} + ( 6075 \beta_{1} - 729 \beta_{2} + 243 \beta_{5} ) q^{45} + ( 28024 + 56 \beta_{3} + 408 \beta_{4} + 64 \beta_{7} ) q^{46} + ( 39992 \beta_{1} + 82 \beta_{2} + 1042 \beta_{5} - 96 \beta_{6} ) q^{47} + ( 27180 \beta_{1} + 5814 \beta_{2} - 45 \beta_{5} + 495 \beta_{6} ) q^{48} + ( 119905 - 1687 \beta_{3} - 1477 \beta_{4} - 412 \beta_{7} ) q^{50} + ( -11934 + 2214 \beta_{3} - 486 \beta_{4} + 162 \beta_{7} ) q^{51} + ( 137418 \beta_{1} + 7402 \beta_{2} + 922 \beta_{5} + 1174 \beta_{6} ) q^{52} + ( 137125 + 6857 \beta_{3} - 353 \beta_{4} + 511 \beta_{7} ) q^{53} + ( -2187 \beta_{1} - 2187 \beta_{2} ) q^{54} + ( 43985 \beta_{1} - 7433 \beta_{2} + 1251 \beta_{5} + 920 \beta_{6} ) q^{55} + ( 41850 + 2295 \beta_{3} - 459 \beta_{4} + 729 \beta_{7} ) q^{57} + ( -131828 + 14186 \beta_{3} - 155 \beta_{4} - 772 \beta_{7} ) q^{58} + ( 15871 \beta_{1} - 571 \beta_{2} - 2323 \beta_{5} - 756 \beta_{6} ) q^{59} + ( 38880 + 9882 \beta_{3} - 513 \beta_{4} - 108 \beta_{7} ) q^{60} + ( 38980 \beta_{1} - 15208 \beta_{2} - 1304 \beta_{5} + 720 \beta_{6} ) q^{61} + ( 62717 \beta_{1} + 6697 \beta_{2} + 682 \beta_{5} + 258 \beta_{6} ) q^{62} + ( 285124 - 49742 \beta_{3} - 1163 \beta_{4} + 1044 \beta_{7} ) q^{64} + ( 109080 + 24294 \beta_{3} - 2086 \beta_{4} - 46 \beta_{7} ) q^{65} + ( 97182 \beta_{1} - 2772 \beta_{2} + 1521 \beta_{5} + 333 \beta_{6} ) q^{66} + ( 291068 + 6357 \beta_{3} + 1103 \beta_{4} - 1045 \beta_{7} ) q^{67} + ( -68544 \beta_{1} + 1284 \beta_{2} - 858 \beta_{5} + 270 \beta_{6} ) q^{68} + ( -9792 \beta_{1} + 2520 \beta_{2} + 1296 \beta_{5} - 432 \beta_{6} ) q^{69} + ( 186172 + 22382 \beta_{3} + 2650 \beta_{4} - 758 \beta_{7} ) q^{71} + ( -98172 + 12150 \beta_{3} + 1215 \beta_{4} - 972 \beta_{7} ) q^{72} + ( 35228 \beta_{1} - 14317 \beta_{2} - 373 \beta_{5} - 1924 \beta_{6} ) q^{73} + ( 113705 + 22921 \beta_{3} + 135 \beta_{4} - 588 \beta_{7} ) q^{74} + ( 36270 \beta_{1} + 9963 \beta_{2} - 1701 \beta_{5} - 540 \beta_{6} ) q^{75} + ( -172566 \beta_{1} - 20802 \beta_{2} - 4152 \beta_{5} - 1716 \beta_{6} ) q^{76} + ( -18873 + 39339 \beta_{3} + 711 \beta_{4} - 2124 \beta_{7} ) q^{78} + ( -152983 - 7998 \beta_{3} + 2018 \beta_{4} + 2750 \beta_{7} ) q^{79} + ( 174940 \beta_{1} - 3990 \beta_{2} + 2545 \beta_{5} + 1605 \beta_{6} ) q^{80} + 59049 q^{81} + ( -120834 \beta_{1} - 40378 \beta_{2} + 3292 \beta_{5} - 2772 \beta_{6} ) q^{82} + ( -95781 \beta_{1} - 15973 \beta_{2} - 197 \beta_{5} - 1092 \beta_{6} ) q^{83} + ( -39210 - 21786 \beta_{3} + 3154 \beta_{4} - 326 \beta_{7} ) q^{85} + ( -197143 + 26113 \beta_{3} + 4731 \beta_{4} - 1596 \beta_{7} ) q^{86} + ( -5445 \beta_{1} - 11223 \beta_{2} - 4347 \beta_{5} - 432 \beta_{6} ) q^{87} + ( 85844 - 21950 \beta_{3} - 6691 \beta_{4} + 2788 \beta_{7} ) q^{88} + ( -81874 \beta_{1} - 3154 \beta_{2} + 918 \beta_{5} + 192 \beta_{6} ) q^{89} + ( -70470 \beta_{1} + 7776 \beta_{2} - 2187 \beta_{5} - 1215 \beta_{6} ) q^{90} + ( 89712 + 13776 \beta_{3} + 784 \beta_{4} + 2464 \beta_{7} ) q^{92} + ( -197937 + 14310 \beta_{3} - 954 \beta_{4} + 234 \beta_{7} ) q^{93} + ( 48026 \beta_{1} + 52686 \beta_{2} - 6074 \beta_{5} - 2674 \beta_{6} ) q^{94} + ( 143310 - 27108 \beta_{3} + 5972 \beta_{4} + 1292 \beta_{7} ) q^{95} + ( 407916 \beta_{1} + 36882 \beta_{2} + 7317 \beta_{5} + 4185 \beta_{6} ) q^{96} + ( -114285 \beta_{1} - 33555 \beta_{2} - 4027 \beta_{5} + 4 \beta_{6} ) q^{97} + ( -13365 + 25029 \beta_{3} - 2673 \beta_{4} + 243 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 10q^{2} + 346q^{4} + 3326q^{8} - 1944q^{9} + O(q^{10}) \) \( 8q + 10q^{2} + 346q^{4} + 3326q^{8} - 1944q^{9} + 628q^{11} + 5292q^{15} + 25442q^{16} - 2430q^{18} + 86106q^{22} - 7856q^{23} + 34076q^{25} - 8300q^{29} - 61398q^{30} + 372414q^{32} - 84078q^{36} - 129412q^{37} - 58212q^{39} + 45740q^{43} - 185058q^{44} + 223008q^{46} + 967216q^{50} - 99576q^{51} + 1081948q^{53} + 328212q^{57} - 1079598q^{58} + 292734q^{60} + 2378626q^{64} + 828408q^{65} + 2317804q^{67} + 1442344q^{71} - 808218q^{72} + 865880q^{74} - 222588q^{78} - 1222904q^{79} + 472392q^{81} - 275112q^{85} - 1632448q^{86} + 732882q^{88} + 678720q^{92} - 1611144q^{93} + 1183584q^{95} - 152604q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - x^{7} + 212 x^{6} + 473 x^{5} + 39800 x^{4} + 36821 x^{3} + 985651 x^{2} - 601290 x + 21068100\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(405518177 \nu^{7} + 12595814413 \nu^{6} + 65484675874 \nu^{5} + 2673339327091 \nu^{4} + 20860832470090 \nu^{3} + 529666993003657 \nu^{2} + 423382833624317 \nu + 6787296640242135\)\()/ 5770013111021565 \)
\(\beta_{2}\)\(=\)\((\)\(8497603 \nu^{7} - 13389005 \nu^{6} + 1621914529 \nu^{5} + 3085757533 \nu^{4} + 336428371378 \nu^{3} + 15479370553 \nu^{2} + 15909261436572 \nu - 5583985297290\)\()/ 7542500798721 \)
\(\beta_{3}\)\(=\)\((\)\(8497603 \nu^{7} - 13389005 \nu^{6} + 1621914529 \nu^{5} + 3085757533 \nu^{4} + 336428371378 \nu^{3} + 15479370553 \nu^{2} + 824259839130 \nu - 5583985297290\)\()/ 7542500798721 \)
\(\beta_{4}\)\(=\)\((\)\(-7295536 \nu^{7} - 50933099 \nu^{6} - 1392479248 \nu^{5} - 2649247696 \nu^{4} - 323422537422 \nu^{3} - 13289665936 \nu^{2} - 707660422560 \nu + 208034068231995\)\()/ 2514166932907 \)
\(\beta_{5}\)\(=\)\((\)\(65881610383 \nu^{7} - 648926656693 \nu^{6} + 19199653600241 \nu^{5} - 98352920361781 \nu^{4} + 3069708627551885 \nu^{3} - 15721266306367057 \nu^{2} + 162140643786549358 \nu - 277910537962726260\)\()/ 7693350814695420 \)
\(\beta_{6}\)\(=\)\((\)\(-135000035243 \nu^{7} - 904004937787 \nu^{6} - 30723933002041 \nu^{5} - 280397778215839 \nu^{4} - 6373179964075405 \nu^{3} - 43067900538846163 \nu^{2} - 253385183003352218 \nu - 478310744905115040\)\()/ 7693350814695420 \)
\(\beta_{7}\)\(=\)\((\)\(1392321691 \nu^{7} - 2568340439 \nu^{6} + 265748679913 \nu^{5} + 505597536901 \nu^{4} + 47373358739629 \nu^{3} + 2536275627841 \nu^{2} + 135053950277610 \nu - 1324669278373776\)\()/ 30170003194884 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{3} + \beta_{2}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{6} + \beta_{5} + \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 105 \beta_{1} - 105\)\()/2\)
\(\nu^{3}\)\(=\)\(-4 \beta_{7} + 2 \beta_{4} + 169 \beta_{3} - 216\)
\(\nu^{4}\)\(=\)\((\)\(-4 \beta_{7} - 217 \beta_{6} - 205 \beta_{5} + 213 \beta_{4} + 722 \beta_{3} - 722 \beta_{2} - 17781 \beta_{1} - 17781\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(848 \beta_{7} - 1614 \beta_{6} + 930 \beta_{5} - 766 \beta_{4} - 32053 \beta_{3} - 32053 \beta_{2} - 77112 \beta_{1} + 77112\)\()/2\)
\(\nu^{6}\)\(=\)\(2216 \beta_{7} - 41381 \beta_{4} - 197354 \beta_{3} + 3375249\)
\(\nu^{7}\)\(=\)\((\)\(163308 \beta_{7} + 385038 \beta_{6} - 104886 \beta_{5} - 221730 \beta_{4} - 6279409 \beta_{3} + 6279409 \beta_{2} + 20983752 \beta_{1} + 20983752\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/147\mathbb{Z}\right)^\times\).

\(n\) \(50\) \(52\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1
−6.30797 + 10.9257i
−6.30797 10.9257i
−2.75320 + 4.76869i
−2.75320 4.76869i
2.26350 3.92050i
2.26350 + 3.92050i
7.29767 12.6399i
7.29767 + 12.6399i
−11.6159 15.5885i 70.9299 190.875i 181.074i 0 −80.4975 −243.000 2217.19i
97.2 −11.6159 15.5885i 70.9299 190.875i 181.074i 0 −80.4975 −243.000 2217.19i
97.3 −4.50641 15.5885i −43.6923 62.2665i 70.2479i 0 485.305 −243.000 280.598i
97.4 −4.50641 15.5885i −43.6923 62.2665i 70.2479i 0 485.305 −243.000 280.598i
97.5 5.52700 15.5885i −33.4522 66.9661i 86.1574i 0 −538.619 −243.000 370.122i
97.6 5.52700 15.5885i −33.4522 66.9661i 86.1574i 0 −538.619 −243.000 370.122i
97.7 15.5953 15.5885i 179.215 25.8331i 243.107i 0 1796.81 −243.000 402.876i
97.8 15.5953 15.5885i 179.215 25.8331i 243.107i 0 1796.81 −243.000 402.876i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 97.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 147.7.d.b 8
3.b odd 2 1 441.7.d.c 8
7.b odd 2 1 inner 147.7.d.b 8
7.c even 3 1 21.7.f.a 8
7.c even 3 1 147.7.f.d 8
7.d odd 6 1 21.7.f.a 8
7.d odd 6 1 147.7.f.d 8
21.c even 2 1 441.7.d.c 8
21.g even 6 1 63.7.m.d 8
21.h odd 6 1 63.7.m.d 8
28.f even 6 1 336.7.bh.d 8
28.g odd 6 1 336.7.bh.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.7.f.a 8 7.c even 3 1
21.7.f.a 8 7.d odd 6 1
63.7.m.d 8 21.g even 6 1
63.7.m.d 8 21.h odd 6 1
147.7.d.b 8 1.a even 1 1 trivial
147.7.d.b 8 7.b odd 2 1 inner
147.7.f.d 8 7.c even 3 1
147.7.f.d 8 7.d odd 6 1
336.7.bh.d 8 28.f even 6 1
336.7.bh.d 8 28.g odd 6 1
441.7.d.c 8 3.b odd 2 1
441.7.d.c 8 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 5 T_{2}^{3} - 202 T_{2}^{2} + 284 T_{2} + 4512 \) acting on \(S_{7}^{\mathrm{new}}(147, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 4512 + 284 T - 202 T^{2} - 5 T^{3} + T^{4} )^{2} \)
$3$ \( ( 243 + T^{2} )^{4} \)
$5$ \( 422734160250000 + 848355795000 T^{2} + 351918225 T^{4} + 45462 T^{6} + T^{8} \)
$7$ \( T^{8} \)
$11$ \( ( -43002975612 + 2285742356 T - 3746347 T^{2} - 314 T^{3} + T^{4} )^{2} \)
$13$ \( \)\(21\!\cdots\!56\)\( + \)\(14\!\cdots\!08\)\( T^{2} + 331924469476329 T^{4} + 30553062 T^{6} + T^{8} \)
$17$ \( \)\(14\!\cdots\!04\)\( + 45391770927437501952 T^{2} + 313740876622608 T^{4} + 46058328 T^{6} + T^{8} \)
$19$ \( \)\(40\!\cdots\!56\)\( + \)\(46\!\cdots\!68\)\( T^{2} + 16571381197978233 T^{4} + 221900526 T^{6} + T^{8} \)
$23$ \( ( 3970225373184 + 88355144192 T - 145891648 T^{2} + 3928 T^{3} + T^{4} )^{2} \)
$29$ \( ( -131918946476762880 - 26271861557728 T - 1346898667 T^{2} + 4150 T^{3} + T^{4} )^{2} \)
$31$ \( \)\(16\!\cdots\!01\)\( + \)\(23\!\cdots\!04\)\( T^{2} + 354646576456751070 T^{4} + 1135098420 T^{6} + T^{8} \)
$37$ \( ( -483334842751696444 - 34229202109340 T + 453289125 T^{2} + 64706 T^{3} + T^{4} )^{2} \)
$41$ \( \)\(36\!\cdots\!44\)\( + \)\(43\!\cdots\!00\)\( T^{2} + \)\(15\!\cdots\!32\)\( T^{4} + 21593029296 T^{6} + T^{8} \)
$43$ \( ( -293597400770976796 + 85125133057276 T - 4841210811 T^{2} - 22870 T^{3} + T^{4} )^{2} \)
$47$ \( \)\(99\!\cdots\!96\)\( + \)\(50\!\cdots\!80\)\( T^{2} + \)\(84\!\cdots\!48\)\( T^{4} + 50915681112 T^{6} + T^{8} \)
$53$ \( ( -85899317769490075392 - 3583685222151280 T + 88674528125 T^{2} - 540974 T^{3} + T^{4} )^{2} \)
$59$ \( \)\(20\!\cdots\!00\)\( + \)\(67\!\cdots\!12\)\( T^{2} + \)\(61\!\cdots\!89\)\( T^{4} + 158965820454 T^{6} + T^{8} \)
$61$ \( \)\(68\!\cdots\!00\)\( + \)\(32\!\cdots\!00\)\( T^{2} + \)\(55\!\cdots\!00\)\( T^{4} + 398165036352 T^{6} + T^{8} \)
$67$ \( ( \)\(17\!\cdots\!52\)\( - 58128338394558464 T + 435008587497 T^{2} - 1158902 T^{3} + T^{4} )^{2} \)
$71$ \( ( -\)\(19\!\cdots\!12\)\( + 115286062429846880 T - 36558200716 T^{2} - 721172 T^{3} + T^{4} )^{2} \)
$73$ \( \)\(77\!\cdots\!24\)\( + \)\(42\!\cdots\!48\)\( T^{2} + \)\(76\!\cdots\!01\)\( T^{4} + 514015226550 T^{6} + T^{8} \)
$79$ \( ( -\)\(99\!\cdots\!51\)\( - 96707301792932732 T - 147338397318 T^{2} + 611452 T^{3} + T^{4} )^{2} \)
$83$ \( \)\(52\!\cdots\!24\)\( + \)\(31\!\cdots\!16\)\( T^{2} + \)\(85\!\cdots\!29\)\( T^{4} + 524747194014 T^{6} + T^{8} \)
$89$ \( \)\(87\!\cdots\!56\)\( + \)\(52\!\cdots\!48\)\( T^{2} + \)\(43\!\cdots\!28\)\( T^{4} + 116188680600 T^{6} + T^{8} \)
$97$ \( \)\(90\!\cdots\!76\)\( + \)\(26\!\cdots\!08\)\( T^{2} + \)\(12\!\cdots\!73\)\( T^{4} + 2056709392566 T^{6} + T^{8} \)
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