Properties

Label 432.6.i.d
Level 432
Weight 6
Character orbit 432.i
Analytic conductor 69.286
Analytic rank 0
Dimension 10
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 432 = 2^{4} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 432.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(69.2858101592\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{16} \)
Twist minimal: no (minimal twist has level 36)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 4 - 4 \beta_{1} + \beta_{7} ) q^{5} + ( -6 \beta_{1} - \beta_{2} - \beta_{7} - \beta_{8} ) q^{7} +O(q^{10})\) \( q + ( 4 - 4 \beta_{1} + \beta_{7} ) q^{5} + ( -6 \beta_{1} - \beta_{2} - \beta_{7} - \beta_{8} ) q^{7} + ( 36 \beta_{1} + \beta_{2} + \beta_{7} + 3 \beta_{8} - \beta_{9} ) q^{11} + ( -34 + 34 \beta_{1} - \beta_{3} - \beta_{5} + \beta_{6} - 8 \beta_{7} + \beta_{8} + \beta_{9} ) q^{13} + ( -229 + 2 \beta_{2} - 3 \beta_{4} + 3 \beta_{5} - 5 \beta_{6} ) q^{17} + ( 86 + 17 \beta_{2} + 7 \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{19} + ( 78 - 78 \beta_{1} + 15 \beta_{3} + 6 \beta_{5} + 5 \beta_{6} + 4 \beta_{7} - 6 \beta_{8} + 5 \beta_{9} ) q^{23} + ( -955 \beta_{1} + 33 \beta_{2} + 14 \beta_{3} - 14 \beta_{4} + 33 \beta_{7} - 12 \beta_{8} + 22 \beta_{9} ) q^{25} + ( 1198 \beta_{1} + 2 \beta_{2} - 15 \beta_{3} + 15 \beta_{4} + 2 \beta_{7} + 27 \beta_{8} + 15 \beta_{9} ) q^{29} + ( -562 + 562 \beta_{1} + 7 \beta_{3} - 12 \beta_{5} - 25 \beta_{6} - 6 \beta_{7} + 12 \beta_{8} - 25 \beta_{9} ) q^{31} + ( 3726 + 7 \beta_{2} + 30 \beta_{4} + 21 \beta_{5} + 40 \beta_{6} ) q^{35} + ( -1512 + 23 \beta_{2} - 49 \beta_{4} - 13 \beta_{5} - 23 \beta_{6} ) q^{37} + ( 3693 - 3693 \beta_{1} - 45 \beta_{3} + 51 \beta_{5} - 19 \beta_{6} - 23 \beta_{7} - 51 \beta_{8} - 19 \beta_{9} ) q^{41} + ( -256 \beta_{1} + 162 \beta_{2} - 21 \beta_{3} + 21 \beta_{4} + 162 \beta_{7} - 54 \beta_{8} - 24 \beta_{9} ) q^{43} + ( -5026 \beta_{1} - 33 \beta_{2} - 18 \beta_{3} + 18 \beta_{4} - 33 \beta_{7} + 87 \beta_{8} - 38 \beta_{9} ) q^{47} + ( -817 + 817 \beta_{1} - 35 \beta_{3} - 53 \beta_{5} - \beta_{6} + 62 \beta_{7} + 53 \beta_{8} - \beta_{9} ) q^{49} + ( -11704 - 53 \beta_{2} + 15 \beta_{4} + 27 \beta_{5} - 15 \beta_{6} ) q^{53} + ( -1542 - 264 \beta_{2} - 41 \beta_{4} - 66 \beta_{5} - 49 \beta_{6} ) q^{55} + ( -18076 + 18076 \beta_{1} - 75 \beta_{3} + 138 \beta_{5} - 50 \beta_{6} - 182 \beta_{7} - 138 \beta_{8} - 50 \beta_{9} ) q^{59} + ( 304 \beta_{1} + 120 \beta_{2} - 49 \beta_{3} + 49 \beta_{4} + 120 \beta_{7} - 87 \beta_{8} - 23 \beta_{9} ) q^{61} + ( 29692 \beta_{1} - 140 \beta_{2} - 45 \beta_{3} + 45 \beta_{4} - 140 \beta_{7} + 87 \beta_{8} - 119 \beta_{9} ) q^{65} + ( -2692 + 2692 \beta_{1} - 138 \beta_{3} - 75 \beta_{5} + 57 \beta_{6} - 195 \beta_{7} + 75 \beta_{8} + 57 \beta_{9} ) q^{67} + ( 22780 - 347 \beta_{2} - 81 \beta_{4} - 129 \beta_{5} - 259 \beta_{6} ) q^{71} + ( 1331 - 690 \beta_{2} + 133 \beta_{4} - 141 \beta_{5} - 61 \beta_{6} ) q^{73} + ( 42430 - 42430 \beta_{1} + 132 \beta_{3} + 24 \beta_{5} - 52 \beta_{6} - 403 \beta_{7} - 24 \beta_{8} - 52 \beta_{9} ) q^{77} + ( -6134 \beta_{1} - 122 \beta_{2} + 91 \beta_{3} - 91 \beta_{4} - 122 \beta_{7} + 112 \beta_{8} + 323 \beta_{9} ) q^{79} + ( -45886 \beta_{1} + 21 \beta_{2} - 240 \beta_{3} + 240 \beta_{4} + 21 \beta_{7} - 129 \beta_{8} + 130 \beta_{9} ) q^{83} + ( -9780 + 9780 \beta_{1} - 91 \beta_{3} + 165 \beta_{5} - 197 \beta_{6} - 1065 \beta_{7} - 165 \beta_{8} - 197 \beta_{9} ) q^{85} + ( -59962 - 529 \beta_{2} + 225 \beta_{4} - 375 \beta_{5} + 55 \beta_{6} ) q^{89} + ( -12562 - 84 \beta_{2} - 357 \beta_{4} + 24 \beta_{5} - 309 \beta_{6} ) q^{91} + ( -79024 + 79024 \beta_{1} - 390 \beta_{3} - 438 \beta_{5} - 386 \beta_{6} - 26 \beta_{7} + 438 \beta_{8} - 386 \beta_{9} ) q^{95} + ( 8219 \beta_{1} + 749 \beta_{2} - 281 \beta_{3} + 281 \beta_{4} + 749 \beta_{7} + 605 \beta_{8} - 43 \beta_{9} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q + 21q^{5} - 29q^{7} + O(q^{10}) \) \( 10q + 21q^{5} - 29q^{7} + 177q^{11} - 181q^{13} - 2280q^{17} + 832q^{19} + 399q^{23} - 4778q^{25} + 6033q^{29} - 2759q^{31} + 37146q^{35} - 15172q^{37} + 18435q^{41} - 1469q^{43} - 25155q^{47} - 4056q^{49} - 116844q^{53} - 14778q^{55} - 90537q^{59} + 1403q^{61} + 148407q^{65} - 13907q^{67} + 229368q^{71} + 15200q^{73} + 211983q^{77} - 29993q^{79} - 228951q^{83} - 49662q^{85} - 598332q^{89} - 124930q^{91} - 394764q^{95} + 40541q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} + 175 x^{8} + 8800 x^{6} + 124623 x^{4} + 498609 x^{2} + 442368\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -99 \nu^{9} - 23021 \nu^{7} - 1847072 \nu^{5} - 56550029 \nu^{3} - 389674035 \nu + 207097728 \)\()/ 414195456 \)
\(\beta_{2}\)\(=\)\((\)\( -47225 \nu^{8} - 9472484 \nu^{6} - 609253100 \nu^{4} - 13835630451 \nu^{2} - 61649294112 \)\()/ 427139064 \)
\(\beta_{3}\)\(=\)\((\)\(1089 \nu^{9} + 102528 \nu^{8} + 253231 \nu^{7} + 17565696 \nu^{6} + 20317792 \nu^{5} + 795822336 \nu^{4} + 622050319 \nu^{3} + 6125611392 \nu^{2} - 6896862927 \nu + 1489775232\)\()/ 414195456 \)
\(\beta_{4}\)\(=\)\((\)\( 267 \nu^{8} + 45744 \nu^{6} + 2072454 \nu^{4} + 15952113 \nu^{2} + 3879623 \)\()/539317\)
\(\beta_{5}\)\(=\)\((\)\( -497 \nu^{8} - 86492 \nu^{6} - 4241924 \nu^{4} - 51195963 \nu^{2} - 57765168 \)\()/563508\)
\(\beta_{6}\)\(=\)\((\)\( 136611 \nu^{8} + 23186844 \nu^{6} + 1089497700 \nu^{4} + 12357836001 \nu^{2} + 24330997448 \)\()/47459896\)
\(\beta_{7}\)\(=\)\((\)\(1172161 \nu^{9} + 188900 \nu^{8} + 202512139 \nu^{7} + 37889936 \nu^{6} + 9878166064 \nu^{5} + 2437012400 \nu^{4} + 126820981119 \nu^{3} + 55342521804 \nu^{2} + 425243755557 \nu + 246597176448\)\()/ 3417112512 \)
\(\beta_{8}\)\(=\)\((\)\(1720421 \nu^{9} - 1506904 \nu^{8} + 291062195 \nu^{7} - 262243744 \nu^{6} + 13499523392 \nu^{5} - 12861513568 \nu^{4} + 142053572235 \nu^{3} - 155226159816 \nu^{2} + 159206103117 \nu - 175143989376\)\()/ 3417112512 \)
\(\beta_{9}\)\(=\)\((\)\(6094287 \nu^{9} - 6557328 \nu^{8} + 1053931697 \nu^{7} - 1112968512 \nu^{6} + 51382170848 \nu^{5} - 52295889600 \nu^{4} + 643758588833 \nu^{3} - 593176128048 \nu^{2} + 1517046597423 \nu - 1167887877504\)\()/ 4556150016 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{4} - 2 \beta_{3} - 22 \beta_{1} + 11\)\()/54\)
\(\nu^{2}\)\(=\)\((\)\(5 \beta_{6} + 27 \beta_{5} + 13 \beta_{4} - 27 \beta_{2} - 3786\)\()/108\)
\(\nu^{3}\)\(=\)\((\)\(-18 \beta_{9} - 36 \beta_{8} + 126 \beta_{7} - 9 \beta_{6} + 18 \beta_{5} - 74 \beta_{4} + 148 \beta_{3} + 63 \beta_{2} + 5894 \beta_{1} - 2947\)\()/54\)
\(\nu^{4}\)\(=\)\((\)\(-182 \beta_{6} - 1188 \beta_{5} - 805 \beta_{4} + 1134 \beta_{2} + 140985\)\()/54\)
\(\nu^{5}\)\(=\)\((\)\(2338 \beta_{9} + 11106 \beta_{8} - 26370 \beta_{7} + 1169 \beta_{6} - 5553 \beta_{5} + 12955 \beta_{4} - 25910 \beta_{3} - 13185 \beta_{2} - 1652084 \beta_{1} + 826042\)\()/108\)
\(\nu^{6}\)\(=\)\((\)\(12040 \beta_{6} + 101061 \beta_{5} + 85859 \beta_{4} - 108270 \beta_{2} - 12057045\)\()/54\)
\(\nu^{7}\)\(=\)\((\)\(-60644 \beta_{9} - 671580 \beta_{8} + 1278036 \beta_{7} - 30322 \beta_{6} + 335790 \beta_{5} - 595221 \beta_{4} + 1190442 \beta_{3} + 639018 \beta_{2} + 93262802 \beta_{1} - 46631401\)\()/54\)
\(\nu^{8}\)\(=\)\((\)\(-1598887 \beta_{6} - 17799345 \beta_{5} - 17481455 \beta_{4} + 21107817 \beta_{2} + 2167343742\)\()/108\)
\(\nu^{9}\)\(=\)\((\)\(2573322 \beta_{9} + 73125792 \beta_{8} - 123164910 \beta_{7} + 1286661 \beta_{6} - 36562896 \beta_{5} + 55890928 \beta_{4} - 111781856 \beta_{3} - 61582455 \beta_{2} - 9781247602 \beta_{1} + 4890623801\)\()/54\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/432\mathbb{Z}\right)^\times\).

\(n\) \(271\) \(325\) \(353\)
\(\chi(n)\) \(1\) \(1\) \(-1 + \beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
145.1
3.71922i
9.84603i
7.64342i
2.13639i
1.11227i
3.71922i
9.84603i
7.64342i
2.13639i
1.11227i
0 0 0 −40.7270 + 70.5412i 0 −89.6312 155.246i 0 0 0
145.2 0 0 0 −13.1603 + 22.7942i 0 31.6287 + 54.7826i 0 0 0
145.3 0 0 0 −4.88422 + 8.45972i 0 68.3340 + 118.358i 0 0 0
145.4 0 0 0 14.0718 24.3731i 0 −75.7039 131.123i 0 0 0
145.5 0 0 0 55.1996 95.6086i 0 50.8724 + 88.1135i 0 0 0
289.1 0 0 0 −40.7270 70.5412i 0 −89.6312 + 155.246i 0 0 0
289.2 0 0 0 −13.1603 22.7942i 0 31.6287 54.7826i 0 0 0
289.3 0 0 0 −4.88422 8.45972i 0 68.3340 118.358i 0 0 0
289.4 0 0 0 14.0718 + 24.3731i 0 −75.7039 + 131.123i 0 0 0
289.5 0 0 0 55.1996 + 95.6086i 0 50.8724 88.1135i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 289.5
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 432.6.i.d 10
3.b odd 2 1 144.6.i.d 10
4.b odd 2 1 108.6.e.a 10
9.c even 3 1 inner 432.6.i.d 10
9.d odd 6 1 144.6.i.d 10
12.b even 2 1 36.6.e.a 10
36.f odd 6 1 108.6.e.a 10
36.f odd 6 1 324.6.a.d 5
36.h even 6 1 36.6.e.a 10
36.h even 6 1 324.6.a.e 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.6.e.a 10 12.b even 2 1
36.6.e.a 10 36.h even 6 1
108.6.e.a 10 4.b odd 2 1
108.6.e.a 10 36.f odd 6 1
144.6.i.d 10 3.b odd 2 1
144.6.i.d 10 9.d odd 6 1
324.6.a.d 5 36.f odd 6 1
324.6.a.e 5 36.h even 6 1
432.6.i.d 10 1.a even 1 1 trivial
432.6.i.d 10 9.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{5}^{10} - \cdots\) acting on \(S_{6}^{\mathrm{new}}(432, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 - 21 T - 5203 T^{2} + 519930 T^{3} + 14035794 T^{4} - 2854822770 T^{5} + 76722872007 T^{6} + 9761967315441 T^{7} - 599011867854189 T^{8} - 11924309583255600 T^{9} + 2533145723872694124 T^{10} - 37263467447673750000 T^{11} - \)\(58\!\cdots\!25\)\( T^{12} + \)\(29\!\cdots\!25\)\( T^{13} + \)\(73\!\cdots\!75\)\( T^{14} - \)\(85\!\cdots\!50\)\( T^{15} + \)\(13\!\cdots\!50\)\( T^{16} + \)\(15\!\cdots\!50\)\( T^{17} - \)\(47\!\cdots\!75\)\( T^{18} - \)\(59\!\cdots\!25\)\( T^{19} + \)\(88\!\cdots\!25\)\( T^{20} \)
$7$ \( 1 + 29 T - 39569 T^{2} - 3762444 T^{3} + 440397336 T^{4} + 77352503496 T^{5} - 2769093584103 T^{6} + 560172784984473 T^{7} + 238615372451780007 T^{8} - 16031898530170676332 T^{9} - \)\(69\!\cdots\!60\)\( T^{10} - \)\(26\!\cdots\!24\)\( T^{11} + \)\(67\!\cdots\!43\)\( T^{12} + \)\(26\!\cdots\!39\)\( T^{13} - \)\(22\!\cdots\!03\)\( T^{14} + \)\(10\!\cdots\!72\)\( T^{15} + \)\(99\!\cdots\!64\)\( T^{16} - \)\(14\!\cdots\!92\)\( T^{17} - \)\(25\!\cdots\!69\)\( T^{18} + \)\(31\!\cdots\!03\)\( T^{19} + \)\(17\!\cdots\!49\)\( T^{20} \)
$11$ \( 1 - 177 T - 396232 T^{2} - 71434269 T^{3} + 104816625882 T^{4} + 33726096455301 T^{5} - 6913987980717606 T^{6} - 8552599160812456257 T^{7} - \)\(10\!\cdots\!51\)\( T^{8} + \)\(50\!\cdots\!82\)\( T^{9} + \)\(47\!\cdots\!16\)\( T^{10} + \)\(81\!\cdots\!82\)\( T^{11} - \)\(27\!\cdots\!51\)\( T^{12} - \)\(35\!\cdots\!07\)\( T^{13} - \)\(46\!\cdots\!06\)\( T^{14} + \)\(36\!\cdots\!51\)\( T^{15} + \)\(18\!\cdots\!82\)\( T^{16} - \)\(20\!\cdots\!19\)\( T^{17} - \)\(17\!\cdots\!32\)\( T^{18} - \)\(12\!\cdots\!27\)\( T^{19} + \)\(11\!\cdots\!01\)\( T^{20} \)
$13$ \( 1 + 181 T - 1012331 T^{2} + 14482182 T^{3} + 446454243174 T^{4} - 84043375137762 T^{5} - 192479505557683773 T^{6} - 802846347498861897 T^{7} + \)\(98\!\cdots\!51\)\( T^{8} + \)\(77\!\cdots\!72\)\( T^{9} - \)\(41\!\cdots\!24\)\( T^{10} + \)\(28\!\cdots\!96\)\( T^{11} + \)\(13\!\cdots\!99\)\( T^{12} - \)\(41\!\cdots\!29\)\( T^{13} - \)\(36\!\cdots\!73\)\( T^{14} - \)\(59\!\cdots\!66\)\( T^{15} + \)\(11\!\cdots\!26\)\( T^{16} + \)\(14\!\cdots\!74\)\( T^{17} - \)\(36\!\cdots\!31\)\( T^{18} + \)\(24\!\cdots\!33\)\( T^{19} + \)\(49\!\cdots\!49\)\( T^{20} \)
$17$ \( ( 1 + 1140 T + 4980550 T^{2} + 3443850354 T^{3} + 10068870522169 T^{4} + 5069379208548852 T^{5} + 14296356292995309833 T^{6} + \)\(69\!\cdots\!46\)\( T^{7} + \)\(14\!\cdots\!50\)\( T^{8} + \)\(46\!\cdots\!40\)\( T^{9} + \)\(57\!\cdots\!57\)\( T^{10} )^{2} \)
$19$ \( ( 1 - 416 T + 5046258 T^{2} - 6215761044 T^{3} + 20272296121125 T^{4} - 15898268281316088 T^{5} + 50196212153221491375 T^{6} - \)\(38\!\cdots\!44\)\( T^{7} + \)\(76\!\cdots\!42\)\( T^{8} - \)\(15\!\cdots\!16\)\( T^{9} + \)\(93\!\cdots\!99\)\( T^{10} )^{2} \)
$23$ \( 1 - 399 T - 16077241 T^{2} - 38108825820 T^{3} + 155650662506976 T^{4} + 562944417983120520 T^{5} - 48522958516353490863 T^{6} - \)\(48\!\cdots\!51\)\( T^{7} - \)\(71\!\cdots\!41\)\( T^{8} + \)\(11\!\cdots\!52\)\( T^{9} + \)\(81\!\cdots\!80\)\( T^{10} + \)\(74\!\cdots\!36\)\( T^{11} - \)\(29\!\cdots\!09\)\( T^{12} - \)\(12\!\cdots\!57\)\( T^{13} - \)\(83\!\cdots\!63\)\( T^{14} + \)\(62\!\cdots\!60\)\( T^{15} + \)\(11\!\cdots\!24\)\( T^{16} - \)\(17\!\cdots\!40\)\( T^{17} - \)\(47\!\cdots\!41\)\( T^{18} - \)\(75\!\cdots\!57\)\( T^{19} + \)\(12\!\cdots\!49\)\( T^{20} \)
$29$ \( 1 - 6033 T + 3652157 T^{2} - 31641196734 T^{3} + 283528398607854 T^{4} - 668469168127712358 T^{5} + \)\(64\!\cdots\!39\)\( T^{6} - \)\(82\!\cdots\!55\)\( T^{7} - \)\(90\!\cdots\!17\)\( T^{8} - \)\(14\!\cdots\!60\)\( T^{9} + \)\(58\!\cdots\!16\)\( T^{10} - \)\(29\!\cdots\!40\)\( T^{11} - \)\(38\!\cdots\!17\)\( T^{12} - \)\(71\!\cdots\!95\)\( T^{13} + \)\(11\!\cdots\!39\)\( T^{14} - \)\(24\!\cdots\!42\)\( T^{15} + \)\(21\!\cdots\!54\)\( T^{16} - \)\(48\!\cdots\!66\)\( T^{17} + \)\(11\!\cdots\!57\)\( T^{18} - \)\(38\!\cdots\!17\)\( T^{19} + \)\(13\!\cdots\!01\)\( T^{20} \)
$31$ \( 1 + 2759 T - 54902477 T^{2} + 189444651072 T^{3} + 2052158291804100 T^{4} - 13274031992302596720 T^{5} - \)\(32\!\cdots\!47\)\( T^{6} + \)\(42\!\cdots\!39\)\( T^{7} - \)\(13\!\cdots\!49\)\( T^{8} - \)\(36\!\cdots\!48\)\( T^{9} + \)\(63\!\cdots\!24\)\( T^{10} - \)\(10\!\cdots\!48\)\( T^{11} - \)\(11\!\cdots\!49\)\( T^{12} + \)\(99\!\cdots\!89\)\( T^{13} - \)\(21\!\cdots\!47\)\( T^{14} - \)\(25\!\cdots\!20\)\( T^{15} + \)\(11\!\cdots\!00\)\( T^{16} + \)\(29\!\cdots\!72\)\( T^{17} - \)\(24\!\cdots\!77\)\( T^{18} + \)\(35\!\cdots\!09\)\( T^{19} + \)\(36\!\cdots\!01\)\( T^{20} \)
$37$ \( ( 1 + 7586 T + 201201093 T^{2} + 803146672896 T^{3} + 19241810738464926 T^{4} + 60351714230064941916 T^{5} + \)\(13\!\cdots\!82\)\( T^{6} + \)\(38\!\cdots\!04\)\( T^{7} + \)\(67\!\cdots\!49\)\( T^{8} + \)\(17\!\cdots\!86\)\( T^{9} + \)\(16\!\cdots\!57\)\( T^{10} )^{2} \)
$41$ \( 1 - 18435 T - 117679042 T^{2} + 4344492069675 T^{3} - 505249106564622 T^{4} - \)\(52\!\cdots\!97\)\( T^{5} + \)\(16\!\cdots\!40\)\( T^{6} + \)\(39\!\cdots\!43\)\( T^{7} - \)\(31\!\cdots\!03\)\( T^{8} - \)\(10\!\cdots\!42\)\( T^{9} + \)\(29\!\cdots\!40\)\( T^{10} - \)\(11\!\cdots\!42\)\( T^{11} - \)\(41\!\cdots\!03\)\( T^{12} + \)\(62\!\cdots\!43\)\( T^{13} + \)\(30\!\cdots\!40\)\( T^{14} - \)\(10\!\cdots\!97\)\( T^{15} - \)\(12\!\cdots\!22\)\( T^{16} + \)\(12\!\cdots\!75\)\( T^{17} - \)\(38\!\cdots\!42\)\( T^{18} - \)\(69\!\cdots\!35\)\( T^{19} + \)\(43\!\cdots\!01\)\( T^{20} \)
$43$ \( 1 + 1469 T - 271863536 T^{2} - 4016430594327 T^{3} + 12129147672135834 T^{4} + \)\(75\!\cdots\!27\)\( T^{5} + \)\(55\!\cdots\!62\)\( T^{6} + \)\(12\!\cdots\!57\)\( T^{7} - \)\(29\!\cdots\!39\)\( T^{8} - \)\(61\!\cdots\!62\)\( T^{9} - \)\(11\!\cdots\!84\)\( T^{10} - \)\(90\!\cdots\!66\)\( T^{11} - \)\(64\!\cdots\!11\)\( T^{12} + \)\(38\!\cdots\!99\)\( T^{13} + \)\(26\!\cdots\!62\)\( T^{14} + \)\(52\!\cdots\!61\)\( T^{15} + \)\(12\!\cdots\!66\)\( T^{16} - \)\(59\!\cdots\!89\)\( T^{17} - \)\(59\!\cdots\!36\)\( T^{18} + \)\(47\!\cdots\!67\)\( T^{19} + \)\(47\!\cdots\!49\)\( T^{20} \)
$47$ \( 1 + 25155 T - 401246233 T^{2} - 14349179861244 T^{3} + 97557609874842960 T^{4} + \)\(41\!\cdots\!12\)\( T^{5} - \)\(25\!\cdots\!27\)\( T^{6} - \)\(55\!\cdots\!85\)\( T^{7} + \)\(12\!\cdots\!11\)\( T^{8} + \)\(42\!\cdots\!56\)\( T^{9} - \)\(37\!\cdots\!60\)\( T^{10} + \)\(96\!\cdots\!92\)\( T^{11} + \)\(67\!\cdots\!39\)\( T^{12} - \)\(67\!\cdots\!55\)\( T^{13} - \)\(71\!\cdots\!27\)\( T^{14} + \)\(26\!\cdots\!84\)\( T^{15} + \)\(14\!\cdots\!40\)\( T^{16} - \)\(47\!\cdots\!92\)\( T^{17} - \)\(30\!\cdots\!33\)\( T^{18} + \)\(44\!\cdots\!85\)\( T^{19} + \)\(40\!\cdots\!49\)\( T^{20} \)
$53$ \( ( 1 + 58422 T + 3354568213 T^{2} + 110313236959296 T^{3} + 3390725554692289246 T^{4} + \)\(71\!\cdots\!28\)\( T^{5} + \)\(14\!\cdots\!78\)\( T^{6} + \)\(19\!\cdots\!04\)\( T^{7} + \)\(24\!\cdots\!41\)\( T^{8} + \)\(17\!\cdots\!22\)\( T^{9} + \)\(12\!\cdots\!93\)\( T^{10} )^{2} \)
$59$ \( 1 + 90537 T + 2831117840 T^{2} + 13805150996349 T^{3} - 966660594685472478 T^{4} - \)\(10\!\cdots\!09\)\( T^{5} + \)\(69\!\cdots\!78\)\( T^{6} + \)\(39\!\cdots\!93\)\( T^{7} + \)\(84\!\cdots\!01\)\( T^{8} - \)\(17\!\cdots\!74\)\( T^{9} - \)\(12\!\cdots\!20\)\( T^{10} - \)\(12\!\cdots\!26\)\( T^{11} + \)\(42\!\cdots\!01\)\( T^{12} + \)\(14\!\cdots\!07\)\( T^{13} + \)\(18\!\cdots\!78\)\( T^{14} - \)\(19\!\cdots\!91\)\( T^{15} - \)\(12\!\cdots\!78\)\( T^{16} + \)\(13\!\cdots\!51\)\( T^{17} + \)\(19\!\cdots\!40\)\( T^{18} + \)\(44\!\cdots\!63\)\( T^{19} + \)\(34\!\cdots\!01\)\( T^{20} \)
$61$ \( 1 - 1403 T - 3536905883 T^{2} - 452840008146 T^{3} + 7065863261737144698 T^{4} + \)\(54\!\cdots\!90\)\( T^{5} - \)\(10\!\cdots\!33\)\( T^{6} - \)\(70\!\cdots\!89\)\( T^{7} + \)\(11\!\cdots\!67\)\( T^{8} + \)\(32\!\cdots\!04\)\( T^{9} - \)\(10\!\cdots\!12\)\( T^{10} + \)\(27\!\cdots\!04\)\( T^{11} + \)\(80\!\cdots\!67\)\( T^{12} - \)\(42\!\cdots\!89\)\( T^{13} - \)\(51\!\cdots\!33\)\( T^{14} + \)\(23\!\cdots\!90\)\( T^{15} + \)\(25\!\cdots\!98\)\( T^{16} - \)\(13\!\cdots\!46\)\( T^{17} - \)\(91\!\cdots\!83\)\( T^{18} - \)\(30\!\cdots\!03\)\( T^{19} + \)\(18\!\cdots\!01\)\( T^{20} \)
$67$ \( 1 + 13907 T - 3876685544 T^{2} + 77425491657903 T^{3} + 10014688417385231130 T^{4} - \)\(30\!\cdots\!39\)\( T^{5} - \)\(79\!\cdots\!54\)\( T^{6} + \)\(69\!\cdots\!51\)\( T^{7} - \)\(33\!\cdots\!67\)\( T^{8} - \)\(37\!\cdots\!46\)\( T^{9} + \)\(22\!\cdots\!76\)\( T^{10} - \)\(51\!\cdots\!22\)\( T^{11} - \)\(60\!\cdots\!83\)\( T^{12} + \)\(16\!\cdots\!93\)\( T^{13} - \)\(26\!\cdots\!54\)\( T^{14} - \)\(13\!\cdots\!73\)\( T^{15} + \)\(60\!\cdots\!70\)\( T^{16} + \)\(63\!\cdots\!29\)\( T^{17} - \)\(42\!\cdots\!44\)\( T^{18} + \)\(20\!\cdots\!49\)\( T^{19} + \)\(20\!\cdots\!49\)\( T^{20} \)
$71$ \( ( 1 - 114684 T + 7758380659 T^{2} - 426246123888336 T^{3} + 19260501229393543450 T^{4} - \)\(77\!\cdots\!40\)\( T^{5} + \)\(34\!\cdots\!50\)\( T^{6} - \)\(13\!\cdots\!36\)\( T^{7} + \)\(45\!\cdots\!09\)\( T^{8} - \)\(12\!\cdots\!84\)\( T^{9} + \)\(19\!\cdots\!51\)\( T^{10} )^{2} \)
$73$ \( ( 1 - 7600 T + 3606834246 T^{2} - 31056473559714 T^{3} + 12288417972789256281 T^{4} - \)\(80\!\cdots\!84\)\( T^{5} + \)\(25\!\cdots\!33\)\( T^{6} - \)\(13\!\cdots\!86\)\( T^{7} + \)\(32\!\cdots\!22\)\( T^{8} - \)\(14\!\cdots\!00\)\( T^{9} + \)\(38\!\cdots\!93\)\( T^{10} )^{2} \)
$79$ \( 1 + 29993 T - 5352351629 T^{2} + 358913063028768 T^{3} + 26234825811851125236 T^{4} - \)\(21\!\cdots\!52\)\( T^{5} + \)\(27\!\cdots\!85\)\( T^{6} + \)\(84\!\cdots\!45\)\( T^{7} - \)\(35\!\cdots\!45\)\( T^{8} - \)\(81\!\cdots\!80\)\( T^{9} + \)\(16\!\cdots\!00\)\( T^{10} - \)\(25\!\cdots\!20\)\( T^{11} - \)\(33\!\cdots\!45\)\( T^{12} + \)\(24\!\cdots\!55\)\( T^{13} + \)\(24\!\cdots\!85\)\( T^{14} - \)\(60\!\cdots\!48\)\( T^{15} + \)\(22\!\cdots\!36\)\( T^{16} + \)\(93\!\cdots\!32\)\( T^{17} - \)\(43\!\cdots\!29\)\( T^{18} + \)\(74\!\cdots\!07\)\( T^{19} + \)\(76\!\cdots\!01\)\( T^{20} \)
$83$ \( 1 + 228951 T + 21403431983 T^{2} + 1202282302650156 T^{3} + 62567029919071222368 T^{4} + \)\(36\!\cdots\!68\)\( T^{5} + \)\(11\!\cdots\!01\)\( T^{6} - \)\(11\!\cdots\!41\)\( T^{7} - \)\(18\!\cdots\!73\)\( T^{8} - \)\(14\!\cdots\!84\)\( T^{9} - \)\(88\!\cdots\!72\)\( T^{10} - \)\(55\!\cdots\!12\)\( T^{11} - \)\(28\!\cdots\!77\)\( T^{12} - \)\(67\!\cdots\!87\)\( T^{13} + \)\(28\!\cdots\!01\)\( T^{14} + \)\(34\!\cdots\!24\)\( T^{15} + \)\(23\!\cdots\!32\)\( T^{16} + \)\(17\!\cdots\!92\)\( T^{17} + \)\(12\!\cdots\!83\)\( T^{18} + \)\(52\!\cdots\!93\)\( T^{19} + \)\(89\!\cdots\!49\)\( T^{20} \)
$89$ \( ( 1 + 299166 T + 52616244181 T^{2} + 6660261403977288 T^{3} + \)\(67\!\cdots\!10\)\( T^{4} + \)\(55\!\cdots\!64\)\( T^{5} + \)\(37\!\cdots\!90\)\( T^{6} + \)\(20\!\cdots\!88\)\( T^{7} + \)\(91\!\cdots\!69\)\( T^{8} + \)\(29\!\cdots\!66\)\( T^{9} + \)\(54\!\cdots\!49\)\( T^{10} )^{2} \)
$97$ \( 1 - 40541 T - 17893496138 T^{2} + 2263333692661293 T^{3} + 99710157551726941410 T^{4} - \)\(30\!\cdots\!95\)\( T^{5} + \)\(10\!\cdots\!20\)\( T^{6} + \)\(21\!\cdots\!29\)\( T^{7} - \)\(20\!\cdots\!15\)\( T^{8} - \)\(65\!\cdots\!06\)\( T^{9} + \)\(19\!\cdots\!00\)\( T^{10} - \)\(56\!\cdots\!42\)\( T^{11} - \)\(15\!\cdots\!35\)\( T^{12} + \)\(13\!\cdots\!97\)\( T^{13} + \)\(56\!\cdots\!20\)\( T^{14} - \)\(14\!\cdots\!15\)\( T^{15} + \)\(39\!\cdots\!90\)\( T^{16} + \)\(77\!\cdots\!49\)\( T^{17} - \)\(52\!\cdots\!38\)\( T^{18} - \)\(10\!\cdots\!37\)\( T^{19} + \)\(21\!\cdots\!49\)\( T^{20} \)
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