Properties

Label 63.7.m.c
Level $63$
Weight $7$
Character orbit 63.m
Analytic conductor $14.493$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 63.m (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.4934072681\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - x^{7} + 212 x^{6} - 787 x^{5} + 38792 x^{4} - 92833 x^{3} + 1563109 x^{2} + 3107772 x + 38787984\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 21)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( \beta_{1} + \beta_{2} ) q^{2} + ( -43 + 43 \beta_{2} - \beta_{5} + \beta_{6} ) q^{4} + ( 6 - 3 \beta_{2} + \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{5} + ( 77 + 5 \beta_{1} + 35 \beta_{2} - 10 \beta_{3} - \beta_{4} + 3 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{7} + ( 61 - 2 \beta_{1} - 14 \beta_{3} - 2 \beta_{4} + 3 \beta_{5} - 4 \beta_{7} ) q^{8} +O(q^{10})\) \( q + ( \beta_{1} + \beta_{2} ) q^{2} + ( -43 + 43 \beta_{2} - \beta_{5} + \beta_{6} ) q^{4} + ( 6 - 3 \beta_{2} + \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{5} + ( 77 + 5 \beta_{1} + 35 \beta_{2} - 10 \beta_{3} - \beta_{4} + 3 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{7} + ( 61 - 2 \beta_{1} - 14 \beta_{3} - 2 \beta_{4} + 3 \beta_{5} - 4 \beta_{7} ) q^{8} + ( 9 + 48 \beta_{1} + 9 \beta_{2} + 34 \beta_{3} + 3 \beta_{5} + 3 \beta_{6} + 14 \beta_{7} ) q^{10} + ( 265 + \beta_{1} - 265 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 9 \beta_{5} - 9 \beta_{6} + \beta_{7} ) q^{11} + ( 864 - 11 \beta_{1} - 1728 \beta_{2} + 14 \beta_{3} - 17 \beta_{4} + 5 \beta_{5} - 10 \beta_{6} ) q^{13} + ( -511 + 159 \beta_{1} - 546 \beta_{2} + 53 \beta_{3} - 22 \beta_{4} - 32 \beta_{5} + 19 \beta_{6} + 2 \beta_{7} ) q^{14} + ( 184 \beta_{1} + 933 \beta_{2} + 10 \beta_{4} + 5 \beta_{6} - 10 \beta_{7} ) q^{16} + ( -626 + 124 \beta_{1} - 626 \beta_{2} + 70 \beta_{3} - 24 \beta_{5} - 24 \beta_{6} + 54 \beta_{7} ) q^{17} + ( -3908 + 360 \beta_{1} + 1954 \beta_{2} - 720 \beta_{3} - 27 \beta_{4} + 2 \beta_{5} - \beta_{6} - 27 \beta_{7} ) q^{19} + ( -3945 + 114 \beta_{1} + 7890 \beta_{2} - 24 \beta_{3} - 66 \beta_{4} - 45 \beta_{5} + 90 \beta_{6} ) q^{20} + ( -169 + 26 \beta_{1} + 570 \beta_{3} + 26 \beta_{4} - 11 \beta_{5} + 52 \beta_{7} ) q^{22} + ( -1302 \beta_{1} + 4144 \beta_{2} - 4 \beta_{4} - 54 \beta_{6} + 4 \beta_{7} ) q^{23} + ( 5198 - 1399 \beta_{1} - 5198 \beta_{2} + 1398 \beta_{3} + 2 \beta_{4} + 81 \beta_{5} - 81 \beta_{6} + \beta_{7} ) q^{25} + ( 2732 - 1093 \beta_{1} - 1366 \beta_{2} + 2186 \beta_{3} - 106 \beta_{4} - 206 \beta_{5} + 103 \beta_{6} - 106 \beta_{7} ) q^{26} + ( -7952 - 1174 \beta_{1} + 15659 \beta_{2} - 312 \beta_{3} - 76 \beta_{4} - 290 \beta_{5} + 191 \beta_{6} - 86 \beta_{7} ) q^{28} + ( -3949 + 61 \beta_{1} - 466 \beta_{3} + 61 \beta_{4} - 183 \beta_{5} + 122 \beta_{7} ) q^{29} + ( 3401 + 78 \beta_{1} + 3401 \beta_{2} + 122 \beta_{3} - 282 \beta_{5} - 282 \beta_{6} - 44 \beta_{7} ) q^{31} + ( -17073 + 1978 \beta_{1} + 17073 \beta_{2} - 1920 \beta_{3} - 116 \beta_{4} + 183 \beta_{5} - 183 \beta_{6} - 58 \beta_{7} ) q^{32} + ( -10238 - 3216 \beta_{1} + 20476 \beta_{2} + 1752 \beta_{3} - 288 \beta_{4} - 466 \beta_{5} + 932 \beta_{6} ) q^{34} + ( 34419 + 1529 \beta_{1} - 9786 \beta_{2} + 2136 \beta_{3} - 9 \beta_{4} + 293 \beta_{5} + 38 \beta_{6} - 94 \beta_{7} ) q^{35} + ( 2756 \beta_{1} - 11970 \beta_{2} + 71 \beta_{4} - 475 \beta_{6} - 71 \beta_{7} ) q^{37} + ( -41396 - 1661 \beta_{1} - 41396 \beta_{2} - 1451 \beta_{3} - 525 \beta_{5} - 525 \beta_{6} - 210 \beta_{7} ) q^{38} + ( -14298 + 3056 \beta_{1} + 7149 \beta_{2} - 6112 \beta_{3} + 98 \beta_{4} - 186 \beta_{5} + 93 \beta_{6} + 98 \beta_{7} ) q^{40} + ( -6362 - 424 \beta_{1} + 12724 \beta_{2} + 340 \beta_{3} - 256 \beta_{4} - 436 \beta_{5} + 872 \beta_{6} ) q^{41} + ( -57678 + 257 \beta_{1} + 6060 \beta_{3} + 257 \beta_{4} + 1051 \beta_{5} + 514 \beta_{7} ) q^{43} + ( -1064 \beta_{1} + 47319 \beta_{2} - 122 \beta_{4} + 495 \beta_{6} + 122 \beta_{7} ) q^{44} + ( 132004 + 3440 \beta_{1} - 132004 \beta_{2} - 3516 \beta_{3} + 152 \beta_{4} + 1076 \beta_{5} - 1076 \beta_{6} + 76 \beta_{7} ) q^{46} + ( -13376 + 1702 \beta_{1} + 6688 \beta_{2} - 3404 \beta_{3} + 770 \beta_{4} + 112 \beta_{5} - 56 \beta_{6} + 770 \beta_{7} ) q^{47} + ( -20356 + 9179 \beta_{1} + 45535 \beta_{2} - 2762 \beta_{3} - 139 \beta_{4} + 1187 \beta_{5} - 586 \beta_{6} - 240 \beta_{7} ) q^{49} + ( 150284 + 170 \beta_{1} + 6623 \beta_{3} + 170 \beta_{4} + 1173 \beta_{5} + 340 \beta_{7} ) q^{50} + ( 61172 - 4182 \beta_{1} + 61172 \beta_{2} - 3804 \beta_{3} + 446 \beta_{5} + 446 \beta_{6} - 378 \beta_{7} ) q^{52} + ( 3055 + 1947 \beta_{1} - 3055 \beta_{2} - 2096 \beta_{3} + 298 \beta_{4} + 2463 \beta_{5} - 2463 \beta_{6} + 149 \beta_{7} ) q^{53} + ( 44475 - 3319 \beta_{1} - 88950 \beta_{2} + 1474 \beta_{3} + 371 \beta_{4} + 345 \beta_{5} - 690 \beta_{6} ) q^{55} + ( 44681 + 2652 \beta_{1} - 116501 \beta_{2} - 11212 \beta_{3} + 24 \beta_{4} + 509 \beta_{5} + 293 \beta_{6} - 34 \beta_{7} ) q^{56} + ( -10376 \beta_{1} - 37607 \beta_{2} - 122 \beta_{4} + 1181 \beta_{6} + 122 \beta_{7} ) q^{58} + ( -151147 + 757 \beta_{1} - 151147 \beta_{2} + 1460 \beta_{3} + 377 \beta_{5} + 377 \beta_{6} - 703 \beta_{7} ) q^{59} + ( -67056 - 2144 \beta_{1} + 33528 \beta_{2} + 4288 \beta_{3} + 964 \beta_{4} + 104 \beta_{5} - 52 \beta_{6} + 964 \beta_{7} ) q^{61} + ( -25589 - 14874 \beta_{1} + 51178 \beta_{2} + 6415 \beta_{3} + 2044 \beta_{4} - 704 \beta_{5} + 1408 \beta_{6} ) q^{62} + ( -176205 - 738 \beta_{1} + 5004 \beta_{3} - 738 \beta_{4} - 3193 \beta_{5} - 1476 \beta_{7} ) q^{64} + ( 17582 \beta_{1} + 93648 \beta_{2} + 1586 \beta_{4} - 1584 \beta_{6} - 1586 \beta_{7} ) q^{65} + ( 101116 + 18853 \beta_{1} - 101116 \beta_{2} - 19020 \beta_{3} + 334 \beta_{4} - 1851 \beta_{5} + 1851 \beta_{6} + 167 \beta_{7} ) q^{67} + ( 281604 + 22668 \beta_{1} - 140802 \beta_{2} - 45336 \beta_{3} - 1644 \beta_{4} - 228 \beta_{5} + 114 \beta_{6} - 1644 \beta_{7} ) q^{68} + ( -156282 + 33680 \beta_{1} + 400659 \beta_{2} + 9766 \beta_{3} - 58 \beta_{4} - 1044 \beta_{5} + 1683 \beta_{6} + 438 \beta_{7} ) q^{70} + ( -28372 - 1706 \beta_{1} + 7094 \beta_{3} - 1706 \beta_{4} - 1332 \beta_{5} - 3412 \beta_{7} ) q^{71} + ( 77984 - 10023 \beta_{1} + 77984 \beta_{2} - 10218 \beta_{3} - 887 \beta_{5} - 887 \beta_{6} + 195 \beta_{7} ) q^{73} + ( -304420 - 32159 \beta_{1} + 304420 \beta_{2} + 30641 \beta_{3} + 3036 \beta_{4} - 3045 \beta_{5} + 3045 \beta_{6} + 1518 \beta_{7} ) q^{74} + ( 58806 - 71982 \beta_{1} - 117612 \beta_{2} + 34440 \beta_{3} + 3102 \beta_{4} + 1200 \beta_{5} - 2400 \beta_{6} ) q^{76} + ( 107856 + 6826 \beta_{1} - 140273 \beta_{2} + 7488 \beta_{3} + 305 \beta_{4} + 2326 \beta_{5} - 1427 \beta_{6} + 293 \beta_{7} ) q^{77} + ( -51686 \beta_{1} - 127061 \beta_{2} + 484 \beta_{4} + 822 \beta_{6} - 484 \beta_{7} ) q^{79} + ( -70377 - 1754 \beta_{1} - 70377 \beta_{2} - 6204 \beta_{3} + 691 \beta_{5} + 691 \beta_{6} + 4450 \beta_{7} ) q^{80} + ( 70684 + 20938 \beta_{1} - 35342 \beta_{2} - 41876 \beta_{3} - 4664 \beta_{4} + 224 \beta_{5} - 112 \beta_{6} - 4664 \beta_{7} ) q^{82} + ( 73913 - 49367 \beta_{1} - 147826 \beta_{2} + 25496 \beta_{3} - 1625 \beta_{4} + 4843 \beta_{5} - 9686 \beta_{6} ) q^{83} + ( 225666 - 1618 \beta_{1} + 77016 \beta_{3} - 1618 \beta_{4} + 1962 \beta_{5} - 3236 \beta_{7} ) q^{85} + ( -21721 \beta_{1} + 578108 \beta_{2} - 4158 \beta_{4} + 7533 \beta_{6} + 4158 \beta_{7} ) q^{86} + ( 83477 + 28310 \beta_{1} - 83477 \beta_{2} - 28008 \beta_{3} - 604 \beta_{4} - 107 \beta_{5} + 107 \beta_{6} - 302 \beta_{7} ) q^{88} + ( 296972 + 3980 \beta_{1} - 148486 \beta_{2} - 7960 \beta_{3} - 78 \beta_{4} + 252 \beta_{5} - 126 \beta_{6} - 78 \beta_{7} ) q^{89} + ( -420476 + 47800 \beta_{1} + 96628 \beta_{2} - 22996 \beta_{3} + 5581 \beta_{4} + 3690 \beta_{5} - 5505 \beta_{6} + 3657 \beta_{7} ) q^{91} + ( -8028 + 3016 \beta_{1} + 88428 \beta_{3} + 3016 \beta_{4} - 8832 \beta_{5} + 6032 \beta_{7} ) q^{92} + ( -153920 + 390 \beta_{1} - 153920 \beta_{2} - 6106 \beta_{3} + 2750 \beta_{5} + 2750 \beta_{6} + 6496 \beta_{7} ) q^{94} + ( -265014 + 78386 \beta_{1} + 265014 \beta_{2} - 72450 \beta_{3} - 11872 \beta_{4} - 9798 \beta_{5} + 9798 \beta_{6} - 5936 \beta_{7} ) q^{95} + ( 135633 - 217 \beta_{1} - 271266 \beta_{2} + 3574 \beta_{3} - 6931 \beta_{4} + 1627 \beta_{5} - 3254 \beta_{6} ) q^{97} + ( -1054403 + 42716 \beta_{1} + 676361 \beta_{2} - 15593 \beta_{3} + 778 \beta_{4} - 11266 \beta_{5} + 2669 \beta_{6} + 2434 \beta_{7} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 5q^{2} - 173q^{4} + 42q^{5} + 748q^{7} + 454q^{8} + O(q^{10}) \) \( 8q + 5q^{2} - 173q^{4} + 42q^{5} + 748q^{7} + 454q^{8} + 261q^{10} + 1070q^{11} - 6070q^{14} + 3911q^{16} - 7212q^{17} - 24606q^{19} - 78q^{22} + 15224q^{23} + 22274q^{25} + 19044q^{26} - 3415q^{28} - 32524q^{29} + 40200q^{31} - 70203q^{32} + 242436q^{35} - 45670q^{37} - 503310q^{38} - 94941q^{40} - 445660q^{43} + 188829q^{44} + 525804q^{46} - 82884q^{47} + 24116q^{49} + 1218884q^{50} + 722856q^{52} + 13034q^{53} - 127061q^{56} - 159501q^{58} - 1810362q^{59} - 392856q^{61} - 1410446q^{64} + 389004q^{65} + 384094q^{67} + 1616346q^{68} + 406005q^{70} - 225688q^{71} + 903078q^{73} - 1185530q^{74} + 327674q^{77} - 559592q^{79} - 847713q^{80} + 347634q^{82} + 1953576q^{85} + 2302402q^{86} + 304887q^{88} + 1770036q^{89} - 2960718q^{91} + 113064q^{92} - 1837620q^{94} - 1160112q^{95} - 5732467q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - x^{7} + 212 x^{6} - 787 x^{5} + 38792 x^{4} - 92833 x^{3} + 1563109 x^{2} + 3107772 x + 38787984\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-1585013359 \nu^{7} + 18232571539 \nu^{6} - 303603349712 \nu^{5} + 4245938445433 \nu^{4} - 65749585575908 \nu^{3} + 780366646056751 \nu^{2} - 2109023702500351 \nu + 22182278913510768\)\()/ 23802631772961636 \)
\(\beta_{3}\)\(=\)\((\)\(-8019055 \nu^{7} - 15616321 \nu^{6} - 1444380025 \nu^{5} + 2053828205 \nu^{4} - 305021724904 \nu^{3} - 177516832405 \nu^{2} - 1592248206780 \nu - 29614389599556\)\()/ 11465622241311 \)
\(\beta_{4}\)\(=\)\((\)\(27322708193 \nu^{7} + 1000103862583 \nu^{6} + 22563530233273 \nu^{5} + 219959885092027 \nu^{4} + 2723907868150765 \nu^{3} + 19609334703803473 \nu^{2} + 313020438506182988 \nu + 699314583065439780\)\()/ 11901315886480818 \)
\(\beta_{5}\)\(=\)\((\)\(-39673486 \nu^{7} + 224426993 \nu^{6} - 7145928130 \nu^{5} + 10161113066 \nu^{4} - 1531993215028 \nu^{3} - 878247070906 \nu^{2} - 7877491417656 \nu - 963541759742958\)\()/ 11465622241311 \)
\(\beta_{6}\)\(=\)\((\)\(59472187739 \nu^{7} - 700951590437 \nu^{6} + 11672037067696 \nu^{5} - 218751249599021 \nu^{4} + 2527744179224764 \nu^{3} - 30001211870054633 \nu^{2} + 130709559191228591 \nu - 852798172253616144\)\()/ 11901315886480818 \)
\(\beta_{7}\)\(=\)\((\)\(226304020214 \nu^{7} + 436984015237 \nu^{6} + 31940434175641 \nu^{5} - 171439510285889 \nu^{4} + 4754608612331077 \nu^{3} - 4492583429983241 \nu^{2} - 114813811933007845 \nu + 14206534053079428\)\()/ 11901315886480818 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{6} - \beta_{5} + 2 \beta_{3} + 106 \beta_{2} - 2 \beta_{1} - 106\)
\(\nu^{3}\)\(=\)\(-4 \beta_{7} + 6 \beta_{5} - 2 \beta_{4} - 145 \beta_{3} - 2 \beta_{1} + 252\)
\(\nu^{4}\)\(=\)\(-2 \beta_{7} - 205 \beta_{6} + 2 \beta_{4} - 15886 \beta_{2} + 772 \beta_{1}\)
\(\nu^{5}\)\(=\)\(424 \beta_{7} + 1560 \beta_{6} - 1560 \beta_{5} + 848 \beta_{4} + 24589 \beta_{3} + 90180 \beta_{2} - 25013 \beta_{1} - 90180\)
\(\nu^{6}\)\(=\)\(-304 \beta_{7} + 38461 \beta_{5} - 152 \beta_{4} - 199046 \beta_{3} - 152 \beta_{1} + 2727346\)
\(\nu^{7}\)\(=\)\(75858 \beta_{7} - 355626 \beta_{6} - 75858 \beta_{4} - 22658292 \beta_{2} + 4625113 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/63\mathbb{Z}\right)^\times\).

\(n\) \(10\) \(29\)
\(\chi(n)\) \(\beta_{2}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1
−7.08935 12.2791i
−2.30325 3.98935i
4.15432 + 7.19549i
5.73828 + 9.93899i
−7.08935 + 12.2791i
−2.30325 + 3.98935i
4.15432 7.19549i
5.73828 9.93899i
−6.58935 11.4131i 0 −54.8390 + 94.9839i 68.9069 39.7834i 0 284.244 191.975i 601.976 0 −908.103 524.293i
10.2 −1.80325 3.12332i 0 25.4966 44.1614i −71.9311 + 41.5295i 0 77.0894 + 334.225i −414.723 0 259.420 + 149.776i
10.3 4.65432 + 8.06151i 0 −11.3253 + 19.6160i −151.343 + 87.3778i 0 −271.614 209.463i 384.906 0 −1408.79 813.368i
10.4 6.23828 + 10.8050i 0 −45.8323 + 79.3839i 175.367 101.248i 0 284.280 + 191.921i −345.159 0 2187.98 + 1263.23i
19.1 −6.58935 + 11.4131i 0 −54.8390 94.9839i 68.9069 + 39.7834i 0 284.244 + 191.975i 601.976 0 −908.103 + 524.293i
19.2 −1.80325 + 3.12332i 0 25.4966 + 44.1614i −71.9311 41.5295i 0 77.0894 334.225i −414.723 0 259.420 149.776i
19.3 4.65432 8.06151i 0 −11.3253 19.6160i −151.343 87.3778i 0 −271.614 + 209.463i 384.906 0 −1408.79 + 813.368i
19.4 6.23828 10.8050i 0 −45.8323 79.3839i 175.367 + 101.248i 0 284.280 191.921i −345.159 0 2187.98 1263.23i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 63.7.m.c 8
3.b odd 2 1 21.7.f.b 8
7.c even 3 1 441.7.d.d 8
7.d odd 6 1 inner 63.7.m.c 8
7.d odd 6 1 441.7.d.d 8
12.b even 2 1 336.7.bh.b 8
21.c even 2 1 147.7.f.a 8
21.g even 6 1 21.7.f.b 8
21.g even 6 1 147.7.d.a 8
21.h odd 6 1 147.7.d.a 8
21.h odd 6 1 147.7.f.a 8
84.j odd 6 1 336.7.bh.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.7.f.b 8 3.b odd 2 1
21.7.f.b 8 21.g even 6 1
63.7.m.c 8 1.a even 1 1 trivial
63.7.m.c 8 7.d odd 6 1 inner
147.7.d.a 8 21.g even 6 1
147.7.d.a 8 21.h odd 6 1
147.7.f.a 8 21.c even 2 1
147.7.f.a 8 21.h odd 6 1
336.7.bh.b 8 12.b even 2 1
336.7.bh.b 8 84.j odd 6 1
441.7.d.d 8 7.c even 3 1
441.7.d.d 8 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{8} - \cdots\) acting on \(S_{7}^{\mathrm{new}}(63, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 30470400 + 5045280 T + 1950436 T^{2} - 129428 T^{3} + 39854 T^{4} - 818 T^{5} + 227 T^{6} - 5 T^{7} + T^{8} \)
$3$ \( T^{8} \)
$5$ \( 54693259182810000 + 24205120650000 T - 9840546578700 T^{2} - 4356625500 T^{3} + 1536505749 T^{4} + 1767906 T^{5} - 41505 T^{6} - 42 T^{7} + T^{8} \)
$7$ \( \)\(19\!\cdots\!01\)\( - 1218053371237015852 T + 3705229535984494 T^{2} - 5061633633224 T^{3} + 8234236703 T^{4} - 43023176 T^{5} + 267694 T^{6} - 748 T^{7} + T^{8} \)
$11$ \( \)\(34\!\cdots\!00\)\( + 18488408967989050800 T + 113509429921169980 T^{2} + 43785002211340 T^{3} + 345504620141 T^{4} - 343064750 T^{5} + 1409111 T^{6} - 1070 T^{7} + T^{8} \)
$13$ \( \)\(34\!\cdots\!00\)\( + 53257929335845404768 T^{2} + 105787465152777 T^{4} + 22315686 T^{6} + T^{8} \)
$17$ \( \)\(22\!\cdots\!00\)\( - \)\(46\!\cdots\!60\)\( T - \)\(83\!\cdots\!72\)\( T^{2} + 23263548118111082880 T^{3} + 3514480347514032 T^{4} - 547654326480 T^{5} - 58598892 T^{6} + 7212 T^{7} + T^{8} \)
$19$ \( \)\(12\!\cdots\!96\)\( + \)\(11\!\cdots\!92\)\( T + \)\(36\!\cdots\!68\)\( T^{2} + 27399467456974423980 T^{3} - 6359170662393867 T^{4} - 670975723710 T^{5} + 174549627 T^{6} + 24606 T^{7} + T^{8} \)
$23$ \( \)\(63\!\cdots\!00\)\( + \)\(71\!\cdots\!00\)\( T + \)\(15\!\cdots\!00\)\( T^{2} - 51980122560355404800 T^{3} + 101423010754761920 T^{4} - 1256314418240 T^{5} + 520738856 T^{6} - 15224 T^{7} + T^{8} \)
$29$ \( ( -39242852020022400 - 9430137809600 T - 442580539 T^{2} + 16262 T^{3} + T^{4} )^{2} \)
$31$ \( \)\(42\!\cdots\!21\)\( + \)\(16\!\cdots\!80\)\( T - \)\(92\!\cdots\!06\)\( T^{2} - \)\(43\!\cdots\!80\)\( T^{3} + 2063880899710463727 T^{4} + 70137742150800 T^{5} - 1206039954 T^{6} - 40200 T^{7} + T^{8} \)
$37$ \( \)\(25\!\cdots\!00\)\( - \)\(12\!\cdots\!00\)\( T + \)\(21\!\cdots\!00\)\( T^{2} + \)\(27\!\cdots\!00\)\( T^{3} + 10133981768848670525 T^{4} - 93188930684650 T^{5} + 5175060655 T^{6} + 45670 T^{7} + T^{8} \)
$41$ \( \)\(53\!\cdots\!16\)\( + \)\(87\!\cdots\!08\)\( T^{2} + 50479416076831845984 T^{4} + 11994986352 T^{6} + T^{8} \)
$43$ \( ( -\)\(14\!\cdots\!84\)\( - 2845100102192540 T - 1577056827 T^{2} + 222830 T^{3} + T^{4} )^{2} \)
$47$ \( \)\(30\!\cdots\!00\)\( - \)\(12\!\cdots\!00\)\( T + \)\(18\!\cdots\!00\)\( T^{2} - \)\(87\!\cdots\!00\)\( T^{3} + \)\(17\!\cdots\!20\)\( T^{4} - 1025758292062320 T^{5} - 10085910828 T^{6} + 82884 T^{7} + T^{8} \)
$53$ \( \)\(23\!\cdots\!00\)\( + \)\(38\!\cdots\!00\)\( T + \)\(25\!\cdots\!64\)\( T^{2} - \)\(27\!\cdots\!44\)\( T^{3} + \)\(20\!\cdots\!21\)\( T^{4} - 946587441999554 T^{5} + 50746432799 T^{6} - 13034 T^{7} + T^{8} \)
$59$ \( \)\(13\!\cdots\!36\)\( + \)\(41\!\cdots\!92\)\( T + \)\(55\!\cdots\!32\)\( T^{2} + \)\(43\!\cdots\!28\)\( T^{3} + \)\(21\!\cdots\!13\)\( T^{4} + 715051889313496398 T^{5} + 1487447487327 T^{6} + 1810362 T^{7} + T^{8} \)
$61$ \( \)\(78\!\cdots\!00\)\( + \)\(11\!\cdots\!40\)\( T + \)\(61\!\cdots\!72\)\( T^{2} + \)\(15\!\cdots\!00\)\( T^{3} - \)\(51\!\cdots\!48\)\( T^{4} - 1477092202992000 T^{5} + 47685396912 T^{6} + 392856 T^{7} + T^{8} \)
$67$ \( \)\(93\!\cdots\!00\)\( + \)\(14\!\cdots\!40\)\( T + \)\(25\!\cdots\!04\)\( T^{2} + \)\(80\!\cdots\!20\)\( T^{3} + \)\(67\!\cdots\!57\)\( T^{4} - 12184102069585486 T^{5} + 192425428651 T^{6} - 384094 T^{7} + T^{8} \)
$71$ \( ( -\)\(25\!\cdots\!12\)\( - 44439430271275520 T - 187952640388 T^{2} + 112844 T^{3} + T^{4} )^{2} \)
$73$ \( \)\(12\!\cdots\!24\)\( + \)\(24\!\cdots\!32\)\( T + \)\(16\!\cdots\!20\)\( T^{2} + \)\(44\!\cdots\!04\)\( T^{3} - \)\(63\!\cdots\!51\)\( T^{4} - 18164126696429538 T^{5} + 291963532599 T^{6} - 903078 T^{7} + T^{8} \)
$79$ \( \)\(58\!\cdots\!25\)\( + \)\(18\!\cdots\!80\)\( T + \)\(57\!\cdots\!34\)\( T^{2} + \)\(12\!\cdots\!12\)\( T^{3} + \)\(36\!\cdots\!07\)\( T^{4} + 233175179655084368 T^{5} + 779488395718 T^{6} + 559592 T^{7} + T^{8} \)
$83$ \( \)\(36\!\cdots\!24\)\( + \)\(24\!\cdots\!96\)\( T^{2} + \)\(12\!\cdots\!25\)\( T^{4} + 2038066317246 T^{6} + T^{8} \)
$89$ \( \)\(13\!\cdots\!44\)\( - \)\(38\!\cdots\!40\)\( T + \)\(51\!\cdots\!48\)\( T^{2} - \)\(40\!\cdots\!80\)\( T^{3} + \)\(20\!\cdots\!44\)\( T^{4} - 673833907778767344 T^{5} + 1425031860636 T^{6} - 1770036 T^{7} + T^{8} \)
$97$ \( \)\(22\!\cdots\!00\)\( + \)\(33\!\cdots\!12\)\( T^{2} + \)\(14\!\cdots\!01\)\( T^{4} + 2348711138742 T^{6} + T^{8} \)
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