Properties

Label 153.10.a.f
Level $153$
Weight $10$
Character orbit 153.a
Self dual yes
Analytic conductor $78.800$
Analytic rank $0$
Dimension $7$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 153 = 3^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 153.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(78.8004829331\)
Analytic rank: \(0\)
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: \( 2^{10}\cdot 3^{5} \)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( 342 - 3 \beta_{1} + \beta_{2} ) q^{4} + ( -198 + 26 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} ) q^{5} + ( 1349 - 69 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} - 6 \beta_{4} - 7 \beta_{5} + 6 \beta_{6} ) q^{7} + ( -2468 + 464 \beta_{1} - \beta_{2} + 10 \beta_{3} - 17 \beta_{5} + 8 \beta_{6} ) q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( 342 - 3 \beta_{1} + \beta_{2} ) q^{4} + ( -198 + 26 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} ) q^{5} + ( 1349 - 69 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} - 6 \beta_{4} - 7 \beta_{5} + 6 \beta_{6} ) q^{7} + ( -2468 + 464 \beta_{1} - \beta_{2} + 10 \beta_{3} - 17 \beta_{5} + 8 \beta_{6} ) q^{8} + ( 22030 + 45 \beta_{1} + 46 \beta_{2} + 26 \beta_{3} - 40 \beta_{4} - 7 \beta_{5} + 28 \beta_{6} ) q^{10} + ( -19325 - 206 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 31 \beta_{4} - 36 \beta_{5} - 28 \beta_{6} ) q^{11} + ( 23374 + 2444 \beta_{1} - 71 \beta_{2} + 15 \beta_{3} + 133 \beta_{4} + 22 \beta_{5} + 22 \beta_{6} ) q^{13} + ( -63925 + 336 \beta_{1} + 52 \beta_{2} + 105 \beta_{3} - 56 \beta_{4} - 70 \beta_{5} + 55 \beta_{6} ) q^{14} + ( 209348 - 2152 \beta_{1} + 281 \beta_{2} + 94 \beta_{3} - 488 \beta_{4} - 203 \beta_{5} + 336 \beta_{6} ) q^{16} -83521 q^{17} + ( 110154 + 8144 \beta_{1} + 264 \beta_{2} + 100 \beta_{3} - 294 \beta_{4} - 572 \beta_{5} - 572 \beta_{6} ) q^{19} + ( 126088 + 35484 \beta_{1} + 254 \beta_{2} + 924 \beta_{3} - 24 \beta_{4} - 970 \beta_{5} + 720 \beta_{6} ) q^{20} + ( -170949 - 27903 \beta_{1} - 106 \beta_{2} - 447 \beta_{3} - 392 \beta_{4} + 173 \beta_{5} + 401 \beta_{6} ) q^{22} + ( -190141 - 26063 \beta_{1} + 141 \beta_{2} + 203 \beta_{3} + 640 \beta_{4} + 1815 \beta_{5} - 702 \beta_{6} ) q^{23} + ( 146203 + 43904 \beta_{1} + 1472 \beta_{2} + 52 \beta_{3} + 96 \beta_{4} - 296 \beta_{5} + 924 \beta_{6} ) q^{25} + ( 2056560 - 13901 \beta_{1} + 2506 \beta_{2} - 1296 \beta_{3} - 1256 \beta_{4} + 803 \beta_{5} - 198 \beta_{6} ) q^{26} + ( -473602 - 6878 \beta_{1} + 4576 \beta_{2} + 738 \beta_{3} - 1200 \beta_{4} + 2078 \beta_{5} - 94 \beta_{6} ) q^{28} + ( -132294 - 40114 \beta_{1} - 1525 \beta_{2} - 869 \beta_{3} + 4781 \beta_{4} + 1840 \beta_{5} - 904 \beta_{6} ) q^{29} + ( 506299 + 6735 \beta_{1} + 411 \beta_{2} - 3979 \beta_{3} - 1242 \beta_{4} + 2957 \beta_{5} - 5722 \beta_{6} ) q^{31} + ( -712284 + 181628 \beta_{1} + 3379 \beta_{2} + 7234 \beta_{3} + 3144 \beta_{4} - 1021 \beta_{5} - 1472 \beta_{6} ) q^{32} -83521 \beta_{1} q^{34} + ( 56522 + 157146 \beta_{1} + 5184 \beta_{2} - 4516 \beta_{3} + 2534 \beta_{4} - 2286 \beta_{5} - 5696 \beta_{6} ) q^{35} + ( 2624162 - 45102 \beta_{1} + 5441 \beta_{2} + 6105 \beta_{3} + 1491 \beta_{4} + 5648 \beta_{5} - 7352 \beta_{6} ) q^{37} + ( 7036138 + 84622 \beta_{1} + 12188 \beta_{2} - 7170 \beta_{3} - 16176 \beta_{4} - 46 \beta_{5} + 13254 \beta_{6} ) q^{38} + ( 18144320 + 192340 \beta_{1} + 38742 \beta_{2} + 2332 \beta_{3} - 19920 \beta_{4} - 14694 \beta_{5} + 13536 \beta_{6} ) q^{40} + ( -1491302 + 165676 \beta_{1} + 4594 \beta_{2} - 14290 \beta_{3} - 3618 \beta_{4} - 5456 \beta_{5} - 7756 \beta_{6} ) q^{41} + ( 3116746 + 42186 \beta_{1} - 16112 \beta_{2} + 11144 \beta_{3} - 3770 \beta_{4} - 1334 \beta_{5} + 15140 \beta_{6} ) q^{43} + ( -13831034 + 98624 \beta_{1} - 34744 \beta_{2} + 8218 \beta_{3} + 48264 \beta_{4} + 13996 \beta_{5} - 2506 \beta_{6} ) q^{44} + ( -21620799 - 269466 \beta_{1} - 37640 \beta_{2} - 3973 \beta_{3} - 11576 \beta_{4} + 25784 \beta_{5} - 5519 \beta_{6} ) q^{46} + ( -8165356 + 451462 \beta_{1} - 31664 \beta_{2} - 1104 \beta_{3} + 37380 \beta_{4} - 12878 \beta_{5} + 26028 \beta_{6} ) q^{47} + ( 3789989 + 373028 \beta_{1} - 48605 \beta_{2} - 19379 \beta_{3} + 771 \beta_{4} - 20590 \beta_{5} + 6086 \beta_{6} ) q^{49} + ( 37162216 + 1181463 \beta_{1} + 53552 \beta_{2} + 28152 \beta_{3} + 9152 \beta_{4} - 41768 \beta_{5} + 9752 \beta_{6} ) q^{50} + ( -22651292 + 2467632 \beta_{1} - 2278 \beta_{2} + 21064 \beta_{3} + 7416 \beta_{4} - 48170 \beta_{5} - 31436 \beta_{6} ) q^{52} + ( -17494218 + 446976 \beta_{1} - 74926 \beta_{2} - 21210 \beta_{3} - 10018 \beta_{4} + 27436 \beta_{5} - 4764 \beta_{6} ) q^{53} + ( 5987298 - 1239616 \beta_{1} - 41770 \beta_{2} - 28710 \beta_{3} + 37136 \beta_{4} - 592 \beta_{5} + 23568 \beta_{6} ) q^{55} + ( 27761156 + 1839768 \beta_{1} - 25988 \beta_{2} + 18836 \beta_{3} + 17968 \beta_{4} - 10996 \beta_{5} + 1700 \beta_{6} ) q^{56} + ( -33770102 - 636397 \beta_{1} - 75302 \beta_{2} - 68690 \beta_{3} + 3720 \beta_{4} + 37459 \beta_{5} - 29916 \beta_{6} ) q^{58} + ( -4419522 + 1749298 \beta_{1} + 21988 \beta_{2} + 12644 \beta_{3} - 103114 \beta_{4} - 9518 \beta_{5} - 36004 \beta_{6} ) q^{59} + ( -7219378 + 586486 \beta_{1} - 60195 \beta_{2} + 65505 \beta_{3} - 13129 \beta_{4} - 58500 \beta_{5} + 11900 \beta_{6} ) q^{61} + ( 10508925 - 151632 \beta_{1} - 107652 \beta_{2} - 92097 \beta_{3} + 133032 \beta_{4} + 84286 \beta_{5} - 81471 \beta_{6} ) q^{62} + ( 45348908 + 452052 \beta_{1} + 175985 \beta_{2} - 23146 \beta_{3} - 154248 \beta_{4} + 99569 \beta_{5} + 51328 \beta_{6} ) q^{64} + ( 17729960 - 1293780 \beta_{1} + 189528 \beta_{2} + 44968 \beta_{3} - 206940 \beta_{4} + 28764 \beta_{5} + 3024 \beta_{6} ) q^{65} + ( 42907640 + 1297892 \beta_{1} + 3484 \beta_{2} + 26640 \beta_{3} - 181172 \beta_{4} + 70028 \beta_{5} - 2772 \beta_{6} ) q^{67} + ( -28564182 + 250563 \beta_{1} - 83521 \beta_{2} ) q^{68} + ( 137689006 + 2674638 \beta_{1} + 79444 \beta_{2} - 113126 \beta_{3} + 87072 \beta_{4} - 64762 \beta_{5} + 7938 \beta_{6} ) q^{70} + ( -93452091 + 1768777 \beta_{1} + 19755 \beta_{2} + 112733 \beta_{3} + 222240 \beta_{4} + 91999 \beta_{5} + 102202 \beta_{6} ) q^{71} + ( 43791474 + 444796 \beta_{1} + 8708 \beta_{2} + 70496 \beta_{3} + 70912 \beta_{4} + 71740 \beta_{5} - 120844 \beta_{6} ) q^{73} + ( -36133758 + 2911773 \beta_{1} + 6726 \beta_{2} - 7370 \beta_{3} - 377416 \beta_{4} + 112525 \beta_{5} + 183400 \beta_{6} ) q^{74} + ( 16598076 + 13477332 \beta_{1} - 101892 \beta_{2} + 391116 \beta_{3} + 835792 \beta_{4} - 115264 \beta_{5} + 84612 \beta_{6} ) q^{76} + ( 14414776 + 1848760 \beta_{1} + 62607 \beta_{2} - 105915 \beta_{3} - 392713 \beta_{4} - 27290 \beta_{5} - 327890 \beta_{6} ) q^{77} + ( 136645065 + 2802391 \beta_{1} + 200759 \beta_{2} - 11463 \beta_{3} - 176640 \beta_{4} + 119953 \beta_{5} - 38250 \beta_{6} ) q^{79} + ( 95485368 + 25325304 \beta_{1} + 338754 \beta_{2} + 251884 \beta_{3} + 158736 \beta_{4} - 435870 \beta_{5} - 12640 \beta_{6} ) q^{80} + ( 149338292 + 1777172 \beta_{1} - 75188 \beta_{2} - 186116 \beta_{3} + 547120 \beta_{4} - 105494 \beta_{5} - 193920 \beta_{6} ) q^{82} + ( 215811674 + 1909818 \beta_{1} - 177744 \beta_{2} - 129096 \beta_{3} + 246982 \beta_{4} - 401926 \beta_{5} + 259028 \beta_{6} ) q^{83} + ( 16537158 - 2171546 \beta_{1} - 83521 \beta_{2} - 83521 \beta_{3} + 83521 \beta_{4} ) q^{85} + ( 24445798 - 5849150 \beta_{1} + 280452 \beta_{2} + 200194 \beta_{3} - 273184 \beta_{4} + 122782 \beta_{5} + 23466 \beta_{6} ) q^{86} + ( 164444756 - 19838428 \beta_{1} + 123336 \beta_{2} - 454052 \beta_{3} - 508000 \beta_{4} + 592356 \beta_{5} - 271876 \beta_{6} ) q^{88} + ( 280768818 + 5328572 \beta_{1} - 51165 \beta_{2} - 228211 \beta_{3} - 173853 \beta_{4} + 452218 \beta_{5} - 169898 \beta_{6} ) q^{89} + ( -151541070 - 1391840 \beta_{1} - 405626 \beta_{2} + 526030 \beta_{3} + 489116 \beta_{4} - 192080 \beta_{5} + 756220 \beta_{6} ) q^{91} + ( -124050502 - 33968878 \beta_{1} - 691016 \beta_{2} - 343082 \beta_{3} + 73600 \beta_{4} + 58142 \beta_{5} - 287010 \beta_{6} ) q^{92} + ( 364924540 - 18202972 \beta_{1} + 562520 \beta_{2} - 295580 \beta_{3} + 67376 \beta_{4} - 100780 \beta_{5} - 219980 \beta_{6} ) q^{94} + ( 464576476 - 3295152 \beta_{1} + 374776 \beta_{2} + 336936 \beta_{3} - 315108 \beta_{4} - 282616 \beta_{5} + 972664 \beta_{6} ) q^{95} + ( 288793462 - 13616496 \beta_{1} + 524064 \beta_{2} + 129756 \beta_{3} - 772896 \beta_{4} - 960136 \beta_{5} - 23180 \beta_{6} ) q^{97} + ( 313296656 - 21090920 \beta_{1} + 95470 \beta_{2} - 648720 \beta_{3} + 856680 \beta_{4} + 391129 \beta_{5} - 668946 \beta_{6} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q + q^{2} + 2389q^{4} - 1362q^{5} + 9388q^{7} - 16821q^{8} + O(q^{10}) \) \( 7q + q^{2} + 2389q^{4} - 1362q^{5} + 9388q^{7} - 16821q^{8} + 154226q^{10} - 135536q^{11} + 166122q^{13} - 447252q^{14} + 1463585q^{16} - 584647q^{17} + 777172q^{19} + 917162q^{20} - 1222520q^{22} - 1357764q^{23} + 1065785q^{25} + 14379966q^{26} - 3328892q^{28} - 967002q^{29} + 3546740q^{31} - 4825461q^{32} - 83521q^{34} + 530736q^{35} + 18296498q^{37} + 49363020q^{38} + 127155062q^{40} - 10285686q^{41} + 21913204q^{43} - 96696624q^{44} - 151509484q^{46} - 56639800q^{47} + 27010351q^{49} + 261150303q^{50} - 156226378q^{52} - 121813562q^{53} + 40793128q^{55} + 196175436q^{56} - 236833910q^{58} - 29222388q^{59} - 49915846q^{61} + 73506556q^{62} + 317922057q^{64} + 122633668q^{65} + 301863420q^{67} - 199531669q^{68} + 966315960q^{70} - 652473940q^{71} + 306656342q^{73} - 249173874q^{74} + 128694700q^{76} + 102442536q^{77} + 959147884q^{79} + 692173602q^{80} + 1046441254q^{82} + 1512945268q^{83} + 113755602q^{85} + 164953236q^{86} + 1132038848q^{88} + 1971327114q^{89} - 1061062864q^{91} - 901186756q^{92} + 2534831232q^{94} + 3249631512q^{95} + 2006526254q^{97} + 2170640009q^{98} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + 3 \nu - 854 \)
\(\beta_{3}\)\(=\)\((\)\( 8711 \nu^{6} + 479085 \nu^{5} - 21966986 \nu^{4} - 962897524 \nu^{3} + 9962276152 \nu^{2} + 249666517824 \nu + 1595734267008 \)\()/ 3478080000 \)
\(\beta_{4}\)\(=\)\((\)\( 41053 \nu^{6} - 292545 \nu^{5} - 124232878 \nu^{4} + 1085670148 \nu^{3} + 93593409896 \nu^{2} - 961758589248 \nu - 7071932313216 \)\()/ 2086848000 \)
\(\beta_{5}\)\(=\)\((\)\( 106349 \nu^{6} - 239985 \nu^{5} - 307414574 \nu^{4} + 1241764484 \nu^{3} + 214005649768 \nu^{2} - 1407118526784 \nu - 13060049459328 \)\()/ 5217120000 \)
\(\beta_{6}\)\(=\)\((\)\( 419317 \nu^{6} - 2816505 \nu^{5} - 1224135742 \nu^{4} + 10192644772 \nu^{3} + 873469755944 \nu^{2} - 8853358980672 \nu - 59384105783424 \)\()/ 10434240000 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 3 \beta_{1} + 854\)
\(\nu^{3}\)\(=\)\(8 \beta_{6} - 17 \beta_{5} + 10 \beta_{3} - \beta_{2} + 1488 \beta_{1} - 2468\)
\(\nu^{4}\)\(=\)\(336 \beta_{6} - 203 \beta_{5} - 488 \beta_{4} + 94 \beta_{3} + 1817 \beta_{2} - 6760 \beta_{1} + 1258948\)
\(\nu^{5}\)\(=\)\(14912 \beta_{6} - 35837 \beta_{5} + 3144 \beta_{4} + 27714 \beta_{3} + 1331 \beta_{2} + 2442620 \beta_{1} - 5766748\)
\(\nu^{6}\)\(=\)\(911488 \beta_{6} - 420111 \beta_{5} - 1403528 \beta_{4} + 217494 \beta_{3} + 3254641 \beta_{2} - 12134956 \beta_{1} + 2059247660\)

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
−43.1213
−34.1532
−5.44491
−4.12962
16.8116
28.6400
42.3973
−43.1213 0 1347.45 −1536.21 0 3027.69 −36025.5 0 66243.5
1.2 −34.1532 0 654.438 −195.287 0 −356.628 −4864.71 0 6669.66
1.3 −5.44491 0 −482.353 −1303.94 0 9199.27 5414.17 0 7099.84
1.4 −4.12962 0 −494.946 −151.544 0 9407.97 4158.31 0 625.818
1.5 16.8116 0 −229.369 1103.40 0 −5164.29 −12463.6 0 18549.9
1.6 28.6400 0 308.250 −1776.79 0 −9598.61 −5835.40 0 −50887.2
1.7 42.3973 0 1285.53 2498.37 0 2872.61 32795.8 0 105924.
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(17\) \(1\)

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 153.10.a.f 7
3.b odd 2 1 17.10.a.b 7
12.b even 2 1 272.10.a.g 7
51.c 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 3.b odd 2 1
153.10.a.f 7 1.a even 1 1 trivial
272.10.a.g 7 12.b even 2 1
289.10.a.b 7 51.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{7} - T_{2}^{6} - 2986 T_{2}^{5} + 8252 T_{2}^{4} + 2252056 T_{2}^{3} - 10388768 T_{2}^{2} - 243559296 T_{2} - 675998208 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(153))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -675998208 - 243559296 T - 10388768 T^{2} + 2252056 T^{3} + 8252 T^{4} - 2986 T^{5} - T^{6} + T^{7} \)
$3$ \( T^{7} \)
$5$ \( \)\(29\!\cdots\!00\)\( + 3598666811889720000 T + 11836046160700000 T^{2} + 3254663233200 T^{3} - 11440345400 T^{4} - 6441308 T^{5} + 1362 T^{6} + T^{7} \)
$7$ \( \)\(13\!\cdots\!04\)\( + \)\(29\!\cdots\!24\)\( T - 21870118739669049472 T^{2} + 1394754860681056 T^{3} + 1101314051384 T^{4} - 110675528 T^{5} - 9388 T^{6} + T^{7} \)
$11$ \( \)\(12\!\cdots\!00\)\( + \)\(89\!\cdots\!80\)\( T - \)\(10\!\cdots\!12\)\( T^{2} - 20473108288822205856 T^{3} - 420218473383624 T^{4} + 2210349776 T^{5} + 135536 T^{6} + T^{7} \)
$13$ \( \)\(51\!\cdots\!32\)\( + \)\(71\!\cdots\!52\)\( T - \)\(18\!\cdots\!76\)\( T^{2} - \)\(11\!\cdots\!64\)\( T^{3} + 5366197083279160 T^{4} - 23816588620 T^{5} - 166122 T^{6} + T^{7} \)
$17$ \( ( 83521 + T )^{7} \)
$19$ \( \)\(35\!\cdots\!00\)\( + \)\(53\!\cdots\!40\)\( T - \)\(82\!\cdots\!08\)\( T^{2} + \)\(82\!\cdots\!84\)\( T^{3} + 1573658158893097216 T^{4} - 1884318745136 T^{5} - 777172 T^{6} + T^{7} \)
$23$ \( -\)\(31\!\cdots\!72\)\( - \)\(29\!\cdots\!76\)\( T + \)\(11\!\cdots\!64\)\( T^{2} + \)\(89\!\cdots\!52\)\( T^{3} - 7952165690620827416 T^{4} - 6093544607192 T^{5} + 1357764 T^{6} + T^{7} \)
$29$ \( -\)\(34\!\cdots\!60\)\( - \)\(54\!\cdots\!64\)\( T - \)\(17\!\cdots\!32\)\( T^{2} + \)\(16\!\cdots\!00\)\( T^{3} + 9055408693964232360 T^{4} - 26389618394492 T^{5} + 967002 T^{6} + T^{7} \)
$31$ \( \)\(15\!\cdots\!00\)\( + \)\(25\!\cdots\!76\)\( T - \)\(57\!\cdots\!84\)\( T^{2} + \)\(15\!\cdots\!28\)\( T^{3} + \)\(27\!\cdots\!20\)\( T^{4} - 87320100382184 T^{5} - 3546740 T^{6} + T^{7} \)
$37$ \( \)\(26\!\cdots\!04\)\( - \)\(37\!\cdots\!60\)\( T - \)\(30\!\cdots\!32\)\( T^{2} + \)\(27\!\cdots\!60\)\( T^{3} + \)\(28\!\cdots\!76\)\( T^{4} - 244849889723324 T^{5} - 18296498 T^{6} + T^{7} \)
$41$ \( -\)\(11\!\cdots\!40\)\( - \)\(21\!\cdots\!92\)\( T + \)\(85\!\cdots\!60\)\( T^{2} + \)\(15\!\cdots\!24\)\( T^{3} - \)\(15\!\cdots\!12\)\( T^{4} - 908777787122252 T^{5} + 10285686 T^{6} + T^{7} \)
$43$ \( \)\(32\!\cdots\!64\)\( - \)\(66\!\cdots\!80\)\( T - \)\(13\!\cdots\!08\)\( T^{2} + \)\(14\!\cdots\!64\)\( T^{3} + \)\(12\!\cdots\!56\)\( T^{4} - 890347107965552 T^{5} - 21913204 T^{6} + T^{7} \)
$47$ \( \)\(18\!\cdots\!00\)\( + \)\(13\!\cdots\!80\)\( T + \)\(63\!\cdots\!64\)\( T^{2} - \)\(26\!\cdots\!88\)\( T^{3} - \)\(20\!\cdots\!76\)\( T^{4} - 2369048503609792 T^{5} + 56639800 T^{6} + T^{7} \)
$53$ \( \)\(68\!\cdots\!00\)\( - \)\(64\!\cdots\!76\)\( T + \)\(12\!\cdots\!64\)\( T^{2} + \)\(31\!\cdots\!80\)\( T^{3} - \)\(81\!\cdots\!16\)\( T^{4} - 7137963194574220 T^{5} + 121813562 T^{6} + T^{7} \)
$59$ \( -\)\(53\!\cdots\!60\)\( - \)\(33\!\cdots\!36\)\( T + \)\(24\!\cdots\!08\)\( T^{2} + \)\(15\!\cdots\!40\)\( T^{3} - \)\(46\!\cdots\!60\)\( T^{4} - 21897186066653360 T^{5} + 29222388 T^{6} + T^{7} \)
$61$ \( -\)\(16\!\cdots\!00\)\( + \)\(26\!\cdots\!60\)\( T - \)\(13\!\cdots\!28\)\( T^{2} + \)\(21\!\cdots\!28\)\( T^{3} + \)\(57\!\cdots\!00\)\( T^{4} - 30822482868012572 T^{5} + 49915846 T^{6} + T^{7} \)
$67$ \( \)\(18\!\cdots\!16\)\( + \)\(60\!\cdots\!88\)\( T - \)\(36\!\cdots\!72\)\( T^{2} - \)\(28\!\cdots\!64\)\( T^{3} + \)\(71\!\cdots\!68\)\( T^{4} - 16130575396626608 T^{5} - 301863420 T^{6} + T^{7} \)
$71$ \( \)\(21\!\cdots\!16\)\( + \)\(61\!\cdots\!28\)\( T + \)\(11\!\cdots\!12\)\( T^{2} - \)\(18\!\cdots\!64\)\( T^{3} - \)\(14\!\cdots\!84\)\( T^{4} + 74845948151756744 T^{5} + 652473940 T^{6} + T^{7} \)
$73$ \( \)\(86\!\cdots\!04\)\( - \)\(10\!\cdots\!20\)\( T - \)\(41\!\cdots\!32\)\( T^{2} + \)\(31\!\cdots\!04\)\( T^{3} + \)\(13\!\cdots\!60\)\( T^{4} - 43284425854107020 T^{5} - 306656342 T^{6} + T^{7} \)
$79$ \( -\)\(18\!\cdots\!00\)\( + \)\(44\!\cdots\!84\)\( T + \)\(46\!\cdots\!68\)\( T^{2} - \)\(78\!\cdots\!16\)\( T^{3} - \)\(11\!\cdots\!52\)\( T^{4} + 266761687131878824 T^{5} - 959147884 T^{6} + T^{7} \)
$83$ \( \)\(61\!\cdots\!12\)\( - \)\(43\!\cdots\!48\)\( T + \)\(53\!\cdots\!68\)\( T^{2} - \)\(24\!\cdots\!48\)\( T^{3} + \)\(33\!\cdots\!88\)\( T^{4} + 442229020241797520 T^{5} - 1512945268 T^{6} + T^{7} \)
$89$ \( \)\(45\!\cdots\!00\)\( - \)\(23\!\cdots\!00\)\( T + \)\(36\!\cdots\!80\)\( T^{2} - \)\(17\!\cdots\!56\)\( T^{3} + \)\(74\!\cdots\!60\)\( T^{4} + 1038612277666203316 T^{5} - 1971327114 T^{6} + T^{7} \)
$97$ \( \)\(15\!\cdots\!00\)\( + \)\(19\!\cdots\!00\)\( T - \)\(11\!\cdots\!20\)\( T^{2} - \)\(55\!\cdots\!76\)\( T^{3} + \)\(31\!\cdots\!16\)\( T^{4} - 711563132961471020 T^{5} - 2006526254 T^{6} + T^{7} \)
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