Properties

Label 272.10.a.g
Level $272$
Weight $10$
Character orbit 272.a
Self dual yes
Analytic conductor $140.090$
Analytic rank $1$
Dimension $7$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 272 = 2^{4} \cdot 17 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 272.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(140.089747437\)
Analytic rank: \(1\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Defining polynomial: \(x^{7} - x^{6} - 2986 x^{5} + 8252 x^{4} + 2252056 x^{3} - 10388768 x^{2} - 243559296 x - 675998208\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{19}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 17)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -13 + \beta_{1} ) q^{3} + ( 195 - \beta_{1} + \beta_{3} ) q^{5} + ( -1336 - 17 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 4 \beta_{4} - 3 \beta_{6} ) q^{7} + ( 11615 + 27 \beta_{1} + 6 \beta_{2} + 3 \beta_{3} - 21 \beta_{4} - \beta_{5} - 6 \beta_{6} ) q^{9} +O(q^{10})\) \( q + ( -13 + \beta_{1} ) q^{3} + ( 195 - \beta_{1} + \beta_{3} ) q^{5} + ( -1336 - 17 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 4 \beta_{4} - 3 \beta_{6} ) q^{7} + ( 11615 + 27 \beta_{1} + 6 \beta_{2} + 3 \beta_{3} - 21 \beta_{4} - \beta_{5} - 6 \beta_{6} ) q^{9} + ( -19391 + 70 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} - 50 \beta_{4} - 7 \beta_{5} - 7 \beta_{6} ) q^{11} + ( 23910 - 399 \beta_{1} + 26 \beta_{2} - 14 \beta_{3} + 33 \beta_{4} - 16 \beta_{5} + 27 \beta_{6} ) q^{13} + ( -22992 + 653 \beta_{1} - 71 \beta_{2} - 36 \beta_{3} - 52 \beta_{4} + 53 \beta_{5} + 3 \beta_{6} ) q^{15} + 83521 q^{17} + ( -111040 + 150 \beta_{1} - 32 \beta_{2} + 316 \beta_{3} + 858 \beta_{4} + 102 \beta_{5} + 184 \beta_{6} ) q^{19} + ( -486348 - 1665 \beta_{1} - 450 \beta_{2} - 84 \beta_{3} + 1263 \beta_{4} - 10 \beta_{5} + 431 \beta_{6} ) q^{21} + ( -194142 + 886 \beta_{1} - 445 \beta_{2} + 82 \beta_{3} + 1464 \beta_{4} - 191 \beta_{5} - 160 \beta_{6} ) q^{23} + ( 153341 - 3812 \beta_{1} + 568 \beta_{2} - 744 \beta_{3} + 166 \beta_{4} + 586 \beta_{5} - 124 \beta_{6} ) q^{25} + ( 643604 + 7333 \beta_{1} - 1071 \beta_{2} + 246 \beta_{3} - 3750 \beta_{4} - 307 \beta_{5} + 1343 \beta_{6} ) q^{27} + ( 134085 + 8657 \beta_{1} + 484 \beta_{2} - 1291 \beta_{3} - 1388 \beta_{4} + 390 \beta_{5} + 62 \beta_{6} ) q^{29} + ( -503048 - 4284 \beta_{1} - 1989 \beta_{2} + 538 \beta_{3} - 96 \beta_{4} + 333 \beta_{5} + 2528 \beta_{6} ) q^{31} + ( 1702804 + 3849 \beta_{1} + 1934 \beta_{2} + 351 \beta_{3} - 8141 \beta_{4} - 951 \beta_{5} + 2036 \beta_{6} ) q^{33} + ( 81276 - 21095 \beta_{1} + 3513 \beta_{2} - 2846 \beta_{3} - 5134 \beta_{4} + 1001 \beta_{5} - 3849 \beta_{6} ) q^{35} + ( 2615249 + 561 \beta_{1} - 3644 \beta_{2} - 4133 \beta_{3} + 1972 \beta_{4} - 2004 \beta_{5} - 1672 \beta_{6} ) q^{37} + ( -12326386 + 41 \beta_{1} - 3259 \beta_{2} - 5532 \beta_{3} + 5788 \beta_{4} - 3031 \beta_{5} + 1141 \beta_{6} ) q^{39} + ( 1472850 + 9578 \beta_{1} - 8224 \beta_{2} + 594 \beta_{3} + 9334 \beta_{4} - 2782 \beta_{5} + 6660 \beta_{6} ) q^{41} + ( -3127762 - 22405 \beta_{1} + 2487 \beta_{2} - 1942 \beta_{3} - 6236 \beta_{4} + 3029 \beta_{5} - 10599 \beta_{6} ) q^{43} + ( 15569589 - 49763 \beta_{1} + 12532 \beta_{2} - 3771 \beta_{3} - 4696 \beta_{4} + 3154 \beta_{5} - 10506 \beta_{6} ) q^{45} + ( -8040328 - 109707 \beta_{1} + 5819 \beta_{2} + 9432 \beta_{3} + 136 \beta_{4} - 1133 \beta_{5} + 14147 \beta_{6} ) q^{47} + ( 3859153 - 2481 \beta_{1} + 15854 \beta_{2} + 12353 \beta_{3} - 17547 \beta_{4} - 5785 \beta_{5} + 8828 \beta_{6} ) q^{49} + ( -1085773 + 83521 \beta_{1} ) q^{51} + ( 17435798 - 81986 \beta_{1} - 23880 \beta_{2} - 9568 \beta_{3} - 25054 \beta_{4} + 14620 \beta_{5} - 12238 \beta_{6} ) q^{53} + ( -5842192 + 20553 \beta_{1} + 6989 \beta_{2} - 44856 \beta_{3} - 11192 \beta_{4} - 2627 \beta_{5} - 9157 \beta_{6} ) q^{55} + ( 22081128 - 377142 \beta_{1} - 24252 \beta_{2} - 13590 \beta_{3} + 88674 \beta_{4} + 30070 \beta_{5} - 53012 \beta_{6} ) q^{57} + ( -4266090 + 180517 \beta_{1} + 28225 \beta_{2} - 52206 \beta_{3} - 27520 \beta_{4} - 6665 \beta_{5} - 11337 \beta_{6} ) q^{59} + ( -7109461 + 31677 \beta_{1} - 18700 \beta_{2} - 12405 \beta_{3} - 52550 \beta_{4} - 28450 \beta_{5} + 34400 \beta_{6} ) q^{61} + ( 728808 - 1065572 \beta_{1} + 45285 \beta_{2} + 9306 \beta_{3} + 131592 \beta_{4} + 20835 \beta_{5} - 27352 \beta_{6} ) q^{63} + ( -17284516 - 505064 \beta_{1} + 29992 \beta_{2} + 50360 \beta_{3} - 30276 \beta_{4} - 36896 \beta_{5} + 35384 \beta_{6} ) q^{65} + ( -42904650 - 491092 \beta_{1} - 30776 \beta_{2} + 52320 \beta_{3} - 68642 \beta_{4} + 6482 \beta_{5} - 5096 \beta_{6} ) q^{67} + ( 54412134 - 554945 \beta_{1} + 88274 \beta_{2} - 57630 \beta_{3} + 227521 \beta_{4} + 40522 \beta_{5} - 25647 \beta_{6} ) q^{69} + ( -92916154 - 607093 \beta_{1} - 33372 \beta_{2} - 30566 \beta_{3} + 143100 \beta_{4} + 2306 \beta_{5} + 48795 \beta_{6} ) q^{71} + ( 43890584 - 132432 \beta_{1} - 29256 \beta_{2} - 56004 \beta_{3} + 11318 \beta_{4} - 45658 \beta_{5} - 14764 \beta_{6} ) q^{73} + ( -131022003 - 460353 \beta_{1} - 151284 \beta_{2} + 106356 \beta_{3} + 190992 \beta_{4} - 18748 \beta_{5} + 8912 \beta_{6} ) q^{75} + ( -14168826 - 906907 \beta_{1} - 95750 \beta_{2} + 113938 \beta_{3} + 191235 \beta_{4} + 39842 \beta_{5} + 124103 \beta_{6} ) q^{77} + ( -136830538 - 277604 \beta_{1} - 60991 \beta_{2} + 112146 \beta_{3} - 100828 \beta_{4} - 43493 \beta_{5} + 62618 \beta_{6} ) q^{79} + ( -53829727 + 615337 \beta_{1} + 202194 \beta_{2} - 6711 \beta_{3} - 400353 \beta_{4} - 49189 \beta_{5} - 5362 \beta_{6} ) q^{81} + ( 216547450 - 988939 \beta_{1} + 50969 \beta_{2} + 155046 \beta_{3} - 272412 \beta_{4} + 52595 \beta_{5} + 76919 \beta_{6} ) q^{83} + ( 16286595 - 83521 \beta_{1} + 83521 \beta_{3} ) q^{85} + ( 230303520 + 193207 \beta_{1} + 12359 \beta_{2} + 161880 \beta_{3} - 294248 \beta_{4} - 97861 \beta_{5} - 34367 \beta_{6} ) q^{87} + ( -281242542 - 555299 \beta_{1} - 232618 \beta_{2} + 91143 \beta_{3} - 367269 \beta_{4} - 1787 \beta_{5} + 86736 \beta_{6} ) q^{89} + ( 150844066 + 988543 \beta_{1} + 263763 \beta_{2} - 159702 \beta_{3} - 186030 \beta_{4} + 43859 \beta_{5} - 421969 \beta_{6} ) q^{91} + ( -113182078 - 2865519 \beta_{1} + 337930 \beta_{2} + 43452 \beta_{3} - 46921 \beta_{4} + 51080 \beta_{5} - 368379 \beta_{6} ) q^{93} + ( 464087856 + 504382 \beta_{1} - 184462 \beta_{2} + 81232 \beta_{3} + 203716 \beta_{4} + 252626 \beta_{5} + 233706 \beta_{6} ) q^{95} + ( 285998180 + 1621400 \beta_{1} - 305556 \beta_{2} + 71428 \beta_{3} - 971726 \beta_{4} + 92782 \beta_{5} - 104372 \beta_{6} ) q^{97} + ( 367148563 + 3530067 \beta_{1} - 48148 \beta_{2} - 19128 \beta_{3} - 879032 \beta_{4} - 194172 \beta_{5} + 247970 \beta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q - 88q^{3} + 1362q^{5} - 9388q^{7} + 81419q^{9} + O(q^{10}) \) \( 7q - 88q^{3} + 1362q^{5} - 9388q^{7} + 81419q^{9} - 135536q^{11} + 166122q^{13} - 159048q^{15} + 584647q^{17} - 777172q^{19} - 3412104q^{21} - 1357764q^{23} + 1065785q^{25} + 4519064q^{27} + 967002q^{29} - 3546740q^{31} + 11928016q^{33} + 530736q^{35} + 18296498q^{37} - 86306872q^{39} + 10285686q^{41} - 21913204q^{43} + 108916410q^{45} - 56639800q^{47} + 27010351q^{49} - 7349848q^{51} + 121813562q^{53} - 40793128q^{55} + 153612960q^{57} - 29222388q^{59} - 49915846q^{61} + 2185356q^{63} - 122633668q^{65} - 301863420q^{67} + 379683432q^{69} - 652473940q^{71} + 306656342q^{73} - 919071912q^{75} - 102442536q^{77} - 959147884q^{79} - 374486977q^{81} + 1512945268q^{83} + 113755602q^{85} + 1612550856q^{87} - 1971327114q^{89} + 1061062864q^{91} - 798598936q^{93} + 3249631512q^{95} + 2006526254q^{97} + 2579159272q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7} - x^{6} - 2986 x^{5} + 8252 x^{4} + 2252056 x^{3} - 10388768 x^{2} - 243559296 x - 675998208\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -41053 \nu^{6} + 292545 \nu^{5} + 124232878 \nu^{4} - 1085670148 \nu^{3} - 93593409896 \nu^{2} + 949237501248 \nu + 7076106009216 \)\()/ 6260544000 \)
\(\beta_{2}\)\(=\)\((\)\( -17729 \nu^{6} - 178065 \nu^{5} + 51179204 \nu^{4} + 202357936 \nu^{3} - 36808140928 \nu^{2} + 212935482864 \nu + 2974683182688 \)\()/ 1956420000 \)
\(\beta_{3}\)\(=\)\((\)\( 332131 \nu^{6} - 7237215 \nu^{5} - 1044625906 \nu^{4} + 19522779196 \nu^{3} + 814970893592 \nu^{2} - 12834968872896 \nu - 58233632015232 \)\()/ 31302720000 \)
\(\beta_{4}\)\(=\)\((\)\( 843359 \nu^{6} - 2902635 \nu^{5} - 2465651834 \nu^{4} + 12878937644 \nu^{3} + 1752000948088 \nu^{2} - 13345412266944 \nu - 113709524082048 \)\()/ 31302720000 \)
\(\beta_{5}\)\(=\)\((\)\( 420671 \nu^{6} - 3690315 \nu^{5} - 1212277946 \nu^{4} + 12473409836 \nu^{3} + 871829642872 \nu^{2} - 9989779801536 \nu - 75110133002112 \)\()/ 10434240000 \)
\(\beta_{6}\)\(=\)\((\)\( 875471 \nu^{6} - 4842315 \nu^{5} - 2538432746 \nu^{4} + 19805426636 \nu^{3} + 1773587676472 \nu^{2} - 18497798358336 \nu - 100813095107712 \)\()/ 15651360000 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{3} + \beta_{2} - 3 \beta_{1} + 10\)\()/64\)
\(\nu^{2}\)\(=\)\((\)\(-16 \beta_{6} + 16 \beta_{5} - 9 \beta_{3} - 7 \beta_{2} - 43 \beta_{1} + 54618\)\()/64\)
\(\nu^{3}\)\(=\)\((\)\(304 \beta_{6} - 48 \beta_{5} - 832 \beta_{4} - 1479 \beta_{3} + 1143 \beta_{2} - 5093 \beta_{1} - 142586\)\()/64\)
\(\nu^{4}\)\(=\)\((\)\(-22192 \beta_{6} + 32944 \beta_{5} - 2240 \beta_{4} - 14133 \beta_{3} - 14075 \beta_{2} + 625 \beta_{1} + 80467666\)\()/64\)
\(\nu^{5}\)\(=\)\((\)\(660720 \beta_{6} - 183536 \beta_{5} - 1816384 \beta_{4} - 2606055 \beta_{3} + 1493079 \beta_{2} - 9235333 \beta_{1} - 343420858\)\()/64\)
\(\nu^{6}\)\(=\)\((\)\(-34010544 \beta_{6} + 63178160 \beta_{5} + 2280512 \beta_{4} - 24830397 \beta_{3} - 23099763 \beta_{2} + 89672921 \beta_{1} + 131574874914\)\()/64\)

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
28.6400
−34.1532
42.3973
−5.44491
16.8116
−43.1213
−4.12962
0 −243.971 0 1776.79 0 9598.61 0 39838.7 0
1.2 0 −169.801 0 195.287 0 356.628 0 9149.54 0
1.3 0 −109.740 0 −2498.37 0 −2872.61 0 −7640.20 0
1.4 0 −106.475 0 1303.94 0 −9199.27 0 −8346.12 0
1.5 0 116.887 0 −1103.40 0 5164.29 0 −6020.47 0
1.6 0 171.025 0 1536.21 0 −3027.69 0 9566.70 0
1.7 0 254.074 0 151.544 0 −9407.97 0 44870.8 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(17\) \(-1\)

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 272.10.a.g 7
4.b odd 2 1 17.10.a.b 7
12.b even 2 1 153.10.a.f 7
68.d odd 2 1 289.10.a.b 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.10.a.b 7 4.b odd 2 1
153.10.a.f 7 12.b even 2 1
272.10.a.g 7 1.a even 1 1 trivial
289.10.a.b 7 68.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{7} + 88 T_{3}^{6} - 105728 T_{3}^{5} - 9882840 T_{3}^{4} + 3088987488 T_{3}^{3} + 298088384256 T_{3}^{2} - \)\(24\!\cdots\!80\)\( T_{3} - \)\(24\!\cdots\!00\)\( \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(272))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} \)
$3$ \( -2458538542080000 - 24964560910080 T + 298088384256 T^{2} + 3088987488 T^{3} - 9882840 T^{4} - 105728 T^{5} + 88 T^{6} + T^{7} \)
$5$ \( -\)\(29\!\cdots\!00\)\( + 3598666811889720000 T - 11836046160700000 T^{2} + 3254663233200 T^{3} + 11440345400 T^{4} - 6441308 T^{5} - 1362 T^{6} + T^{7} \)
$7$ \( -\)\(13\!\cdots\!04\)\( + \)\(29\!\cdots\!24\)\( T + 21870118739669049472 T^{2} + 1394754860681056 T^{3} - 1101314051384 T^{4} - 110675528 T^{5} + 9388 T^{6} + T^{7} \)
$11$ \( \)\(12\!\cdots\!00\)\( + \)\(89\!\cdots\!80\)\( T - \)\(10\!\cdots\!12\)\( T^{2} - 20473108288822205856 T^{3} - 420218473383624 T^{4} + 2210349776 T^{5} + 135536 T^{6} + T^{7} \)
$13$ \( \)\(51\!\cdots\!32\)\( + \)\(71\!\cdots\!52\)\( T - \)\(18\!\cdots\!76\)\( T^{2} - \)\(11\!\cdots\!64\)\( T^{3} + 5366197083279160 T^{4} - 23816588620 T^{5} - 166122 T^{6} + T^{7} \)
$17$ \( ( -83521 + T )^{7} \)
$19$ \( -\)\(35\!\cdots\!00\)\( + \)\(53\!\cdots\!40\)\( T + \)\(82\!\cdots\!08\)\( T^{2} + \)\(82\!\cdots\!84\)\( T^{3} - 1573658158893097216 T^{4} - 1884318745136 T^{5} + 777172 T^{6} + T^{7} \)
$23$ \( -\)\(31\!\cdots\!72\)\( - \)\(29\!\cdots\!76\)\( T + \)\(11\!\cdots\!64\)\( T^{2} + \)\(89\!\cdots\!52\)\( T^{3} - 7952165690620827416 T^{4} - 6093544607192 T^{5} + 1357764 T^{6} + T^{7} \)
$29$ \( \)\(34\!\cdots\!60\)\( - \)\(54\!\cdots\!64\)\( T + \)\(17\!\cdots\!32\)\( T^{2} + \)\(16\!\cdots\!00\)\( T^{3} - 9055408693964232360 T^{4} - 26389618394492 T^{5} - 967002 T^{6} + T^{7} \)
$31$ \( -\)\(15\!\cdots\!00\)\( + \)\(25\!\cdots\!76\)\( T + \)\(57\!\cdots\!84\)\( T^{2} + \)\(15\!\cdots\!28\)\( T^{3} - \)\(27\!\cdots\!20\)\( T^{4} - 87320100382184 T^{5} + 3546740 T^{6} + T^{7} \)
$37$ \( \)\(26\!\cdots\!04\)\( - \)\(37\!\cdots\!60\)\( T - \)\(30\!\cdots\!32\)\( T^{2} + \)\(27\!\cdots\!60\)\( T^{3} + \)\(28\!\cdots\!76\)\( T^{4} - 244849889723324 T^{5} - 18296498 T^{6} + T^{7} \)
$41$ \( \)\(11\!\cdots\!40\)\( - \)\(21\!\cdots\!92\)\( T - \)\(85\!\cdots\!60\)\( T^{2} + \)\(15\!\cdots\!24\)\( T^{3} + \)\(15\!\cdots\!12\)\( T^{4} - 908777787122252 T^{5} - 10285686 T^{6} + T^{7} \)
$43$ \( -\)\(32\!\cdots\!64\)\( - \)\(66\!\cdots\!80\)\( T + \)\(13\!\cdots\!08\)\( T^{2} + \)\(14\!\cdots\!64\)\( T^{3} - \)\(12\!\cdots\!56\)\( T^{4} - 890347107965552 T^{5} + 21913204 T^{6} + T^{7} \)
$47$ \( \)\(18\!\cdots\!00\)\( + \)\(13\!\cdots\!80\)\( T + \)\(63\!\cdots\!64\)\( T^{2} - \)\(26\!\cdots\!88\)\( T^{3} - \)\(20\!\cdots\!76\)\( T^{4} - 2369048503609792 T^{5} + 56639800 T^{6} + T^{7} \)
$53$ \( -\)\(68\!\cdots\!00\)\( - \)\(64\!\cdots\!76\)\( T - \)\(12\!\cdots\!64\)\( T^{2} + \)\(31\!\cdots\!80\)\( T^{3} + \)\(81\!\cdots\!16\)\( T^{4} - 7137963194574220 T^{5} - 121813562 T^{6} + T^{7} \)
$59$ \( -\)\(53\!\cdots\!60\)\( - \)\(33\!\cdots\!36\)\( T + \)\(24\!\cdots\!08\)\( T^{2} + \)\(15\!\cdots\!40\)\( T^{3} - \)\(46\!\cdots\!60\)\( T^{4} - 21897186066653360 T^{5} + 29222388 T^{6} + T^{7} \)
$61$ \( -\)\(16\!\cdots\!00\)\( + \)\(26\!\cdots\!60\)\( T - \)\(13\!\cdots\!28\)\( T^{2} + \)\(21\!\cdots\!28\)\( T^{3} + \)\(57\!\cdots\!00\)\( T^{4} - 30822482868012572 T^{5} + 49915846 T^{6} + T^{7} \)
$67$ \( -\)\(18\!\cdots\!16\)\( + \)\(60\!\cdots\!88\)\( T + \)\(36\!\cdots\!72\)\( T^{2} - \)\(28\!\cdots\!64\)\( T^{3} - \)\(71\!\cdots\!68\)\( T^{4} - 16130575396626608 T^{5} + 301863420 T^{6} + T^{7} \)
$71$ \( \)\(21\!\cdots\!16\)\( + \)\(61\!\cdots\!28\)\( T + \)\(11\!\cdots\!12\)\( T^{2} - \)\(18\!\cdots\!64\)\( T^{3} - \)\(14\!\cdots\!84\)\( T^{4} + 74845948151756744 T^{5} + 652473940 T^{6} + T^{7} \)
$73$ \( \)\(86\!\cdots\!04\)\( - \)\(10\!\cdots\!20\)\( T - \)\(41\!\cdots\!32\)\( T^{2} + \)\(31\!\cdots\!04\)\( T^{3} + \)\(13\!\cdots\!60\)\( T^{4} - 43284425854107020 T^{5} - 306656342 T^{6} + T^{7} \)
$79$ \( \)\(18\!\cdots\!00\)\( + \)\(44\!\cdots\!84\)\( T - \)\(46\!\cdots\!68\)\( T^{2} - \)\(78\!\cdots\!16\)\( T^{3} + \)\(11\!\cdots\!52\)\( T^{4} + 266761687131878824 T^{5} + 959147884 T^{6} + T^{7} \)
$83$ \( \)\(61\!\cdots\!12\)\( - \)\(43\!\cdots\!48\)\( T + \)\(53\!\cdots\!68\)\( T^{2} - \)\(24\!\cdots\!48\)\( T^{3} + \)\(33\!\cdots\!88\)\( T^{4} + 442229020241797520 T^{5} - 1512945268 T^{6} + T^{7} \)
$89$ \( -\)\(45\!\cdots\!00\)\( - \)\(23\!\cdots\!00\)\( T - \)\(36\!\cdots\!80\)\( T^{2} - \)\(17\!\cdots\!56\)\( T^{3} - \)\(74\!\cdots\!60\)\( T^{4} + 1038612277666203316 T^{5} + 1971327114 T^{6} + T^{7} \)
$97$ \( \)\(15\!\cdots\!00\)\( + \)\(19\!\cdots\!00\)\( T - \)\(11\!\cdots\!20\)\( T^{2} - \)\(55\!\cdots\!76\)\( T^{3} + \)\(31\!\cdots\!16\)\( T^{4} - 711563132961471020 T^{5} - 2006526254 T^{6} + T^{7} \)
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