Properties

Label 45.9.g.a
Level $45$
Weight $9$
Character orbit 45.g
Analytic conductor $18.332$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 45 = 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 45.g (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(18.3320374528\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \(x^{6} - 2 x^{5} + 2 x^{4} - 30 x^{3} + 1089 x^{2} - 3168 x + 4608\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 5)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + ( -10 \beta_{1} - \beta_{2} - 5 \beta_{3} + \beta_{4} + 5 \beta_{5} ) q^{4} + ( -48 - 39 \beta_{1} - 3 \beta_{2} + 15 \beta_{3} + 4 \beta_{4} + 20 \beta_{5} ) q^{5} + ( -434 + 434 \beta_{1} + 119 \beta_{3} - 7 \beta_{4} ) q^{7} + ( 1326 + 1326 \beta_{1} - 2 \beta_{2} + 130 \beta_{5} ) q^{8} +O(q^{10})\) \( q + \beta_{3} q^{2} + ( -10 \beta_{1} - \beta_{2} - 5 \beta_{3} + \beta_{4} + 5 \beta_{5} ) q^{4} + ( -48 - 39 \beta_{1} - 3 \beta_{2} + 15 \beta_{3} + 4 \beta_{4} + 20 \beta_{5} ) q^{5} + ( -434 + 434 \beta_{1} + 119 \beta_{3} - 7 \beta_{4} ) q^{7} + ( 1326 + 1326 \beta_{1} - 2 \beta_{2} + 130 \beta_{5} ) q^{8} + ( 5308 - 4006 \beta_{1} - 37 \beta_{2} - 190 \beta_{3} + 41 \beta_{4} - 295 \beta_{5} ) q^{10} + ( -3792 + 85 \beta_{2} - 25 \beta_{3} + 85 \beta_{4} - 25 \beta_{5} ) q^{11} + ( -20015 - 20015 \beta_{1} + 80 \beta_{2} + 554 \beta_{5} ) q^{13} + ( -31626 \beta_{1} - 77 \beta_{2} - 1260 \beta_{3} + 77 \beta_{4} + 1260 \beta_{5} ) q^{14} + ( 37132 + 374 \beta_{2} - 670 \beta_{3} + 374 \beta_{4} - 670 \beta_{5} ) q^{16} + ( 43617 - 43617 \beta_{1} + 2366 \beta_{3} + 466 \beta_{4} ) q^{17} + ( 54264 \beta_{1} - 684 \beta_{2} + 930 \beta_{3} + 684 \beta_{4} - 930 \beta_{5} ) q^{19} + ( -68634 + 38088 \beta_{1} + 451 \beta_{2} + 2745 \beta_{3} + 307 \beta_{4} - 2215 \beta_{5} ) q^{20} + ( -6310 + 6310 \beta_{1} + 2068 \beta_{3} + 970 \beta_{4} ) q^{22} + ( -4722 - 4722 \beta_{1} - 1581 \beta_{2} - 1859 \beta_{5} ) q^{23} + ( -51065 - 12420 \beta_{1} + 785 \beta_{2} + 4825 \beta_{3} + 2245 \beta_{4} - 18775 \beta_{5} ) q^{25} + ( 147684 + 1034 \beta_{2} - 20145 \beta_{3} + 1034 \beta_{4} - 20145 \beta_{5} ) q^{26} + ( 223748 + 223748 \beta_{1} - 196 \beta_{2} - 18844 \beta_{5} ) q^{28} + ( 451044 \beta_{1} + 2386 \beta_{2} + 15980 \beta_{3} - 2386 \beta_{4} - 15980 \beta_{5} ) q^{29} + ( -99628 + 3045 \beta_{2} - 34425 \beta_{3} + 3045 \beta_{4} - 34425 \beta_{5} ) q^{31} + ( -516180 + 516180 \beta_{1} + 35236 \beta_{3} + 3660 \beta_{4} ) q^{32} + ( -631220 \beta_{1} - 5162 \beta_{2} + 47165 \beta_{3} + 5162 \beta_{4} - 47165 \beta_{5} ) q^{34} + ( 854112 - 233184 \beta_{1} - 2968 \beta_{2} - 47285 \beta_{3} + 3549 \beta_{4} - 28630 \beta_{5} ) q^{35} + ( -72653 + 72653 \beta_{1} - 7536 \beta_{3} + 1506 \beta_{4} ) q^{37} + ( -250116 - 250116 \beta_{1} - 10068 \beta_{2} + 18420 \beta_{5} ) q^{38} + ( 438150 + 627450 \beta_{1} + 6400 \beta_{2} + 29250 \beta_{3} + 14550 \beta_{4} + 19000 \beta_{5} ) q^{40} + ( -422352 + 12505 \beta_{2} + 23675 \beta_{3} + 12505 \beta_{4} + 23675 \beta_{5} ) q^{41} + ( 105376 + 105376 \beta_{1} + 5323 \beta_{2} + 85519 \beta_{5} ) q^{43} + ( -1524720 \beta_{1} + 13872 \beta_{2} + 21760 \beta_{3} - 13872 \beta_{4} - 21760 \beta_{5} ) q^{44} + ( -500818 - 11345 \beta_{2} - 47600 \beta_{3} - 11345 \beta_{4} - 47600 \beta_{5} ) q^{46} + ( 2583906 - 2583906 \beta_{1} - 90779 \beta_{3} + 4263 \beta_{4} ) q^{47} + ( 1195649 \beta_{1} - 8575 \beta_{2} - 219275 \beta_{3} + 8575 \beta_{4} + 219275 \beta_{5} ) q^{49} + ( -4991010 - 1292430 \beta_{1} - 32360 \beta_{2} + 118675 \beta_{3} + 4230 \beta_{4} + 57400 \beta_{5} ) q^{50} + ( -230594 + 230594 \beta_{1} + 275554 \beta_{3} - 48362 \beta_{4} ) q^{52} + ( 2162457 + 2162457 \beta_{1} + 4186 \beta_{2} + 271376 \beta_{5} ) q^{53} + ( 710936 - 5132052 \beta_{1} + 41621 \beta_{2} + 160645 \beta_{3} - 25703 \beta_{4} - 53515 \beta_{5} ) q^{55} + ( 3082968 - 308 \beta_{2} - 11060 \beta_{3} - 308 \beta_{4} - 11060 \beta_{5} ) q^{56} + ( -4241136 - 4241136 \beta_{1} - 3328 \beta_{2} + 768320 \beta_{5} ) q^{58} + ( 1372608 \beta_{1} + 30152 \beta_{2} - 508490 \beta_{3} - 30152 \beta_{4} + 508490 \beta_{5} ) q^{59} + ( 4266032 - 95625 \beta_{2} - 466875 \beta_{3} - 95625 \beta_{4} - 466875 \beta_{5} ) q^{61} + ( -9144870 + 9144870 \beta_{1} + 445592 \beta_{3} - 32310 \beta_{4} ) q^{62} + ( 118376 \beta_{1} + 38548 \beta_{2} - 400060 \beta_{3} - 38548 \beta_{4} + 400060 \beta_{5} ) q^{64} + ( 5227299 + 2571057 \beta_{1} + 12939 \beta_{2} + 362305 \beta_{3} - 137927 \beta_{4} - 697385 \beta_{5} ) q^{65} + ( -5539832 + 5539832 \beta_{1} + 101959 \beta_{3} - 103661 \beta_{4} ) q^{67} + ( -1400586 - 1400586 \beta_{1} - 36978 \beta_{2} + 105434 \beta_{5} ) q^{68} + ( -7627452 + 12563614 \beta_{1} - 20447 \beta_{2} + 1252860 \beta_{3} - 72429 \beta_{4} - 541520 \beta_{5} ) q^{70} + ( 2912988 - 78125 \beta_{2} - 949375 \beta_{3} - 78125 \beta_{4} - 949375 \beta_{5} ) q^{71} + ( 18480997 + 18480997 \beta_{1} + 12756 \beta_{2} + 499584 \beta_{5} ) q^{73} + ( 1998552 \beta_{1} - 1500 \beta_{2} + 14725 \beta_{3} + 1500 \beta_{4} - 14725 \beta_{5} ) q^{74} + ( -9032136 + 133116 \beta_{2} - 436380 \beta_{3} + 133116 \beta_{4} - 436380 \beta_{5} ) q^{76} + ( -4353552 + 4353552 \beta_{1} + 41342 \beta_{3} + 110754 \beta_{4} ) q^{77} + ( -22879824 \beta_{1} - 122456 \beta_{2} - 1313680 \beta_{3} + 122456 \beta_{4} + 1313680 \beta_{5} ) q^{79} + ( -4670928 - 25409004 \beta_{1} + 19442 \beta_{2} + 1455290 \beta_{3} + 58494 \beta_{4} + 1112470 \beta_{5} ) q^{80} + ( 6347570 - 6347570 \beta_{1} + 166228 \beta_{3} + 197410 \beta_{4} ) q^{82} + ( 2441256 + 2441256 \beta_{1} + 212813 \beta_{2} + 273261 \beta_{5} ) q^{83} + ( -3531827 - 25557811 \beta_{1} - 286147 \beta_{2} + 1708485 \beta_{3} + 200271 \beta_{4} - 1055395 \beta_{5} ) q^{85} + ( 22769346 + 117457 \beta_{2} - 146560 \beta_{3} + 117457 \beta_{4} - 146560 \beta_{5} ) q^{86} + ( -7348032 - 7348032 \beta_{1} + 371264 \beta_{2} + 137840 \beta_{5} ) q^{88} + ( -38523288 \beta_{1} - 442572 \beta_{2} - 666960 \beta_{3} + 442572 \beta_{4} + 666960 \beta_{5} ) q^{89} + ( 29991584 + 224203 \beta_{2} - 2877665 \beta_{3} + 224203 \beta_{4} - 2877665 \beta_{5} ) q^{91} + ( -11498148 + 11498148 \beta_{1} - 297684 \beta_{3} + 173396 \beta_{4} ) q^{92} + ( 24130162 \beta_{1} + 65201 \beta_{2} + 3178480 \beta_{3} - 65201 \beta_{4} - 3178480 \beta_{5} ) q^{94} + ( -38898900 - 17396700 \beta_{1} + 213600 \beta_{2} - 1454250 \beta_{3} + 410700 \beta_{4} - 1055250 \beta_{5} ) q^{95} + ( -31017911 + 31017911 \beta_{1} + 586904 \beta_{3} + 861272 \beta_{4} ) q^{97} + ( 58292850 + 58292850 \beta_{1} + 335650 \beta_{2} - 1563051 \beta_{5} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 2q^{2} - 220q^{5} - 2352q^{7} + 8220q^{8} + O(q^{10}) \) \( 6q + 2q^{2} - 220q^{5} - 2352q^{7} + 8220q^{8} + 30870q^{10} - 23192q^{11} - 119142q^{13} + 218616q^{16} + 265502q^{17} - 412260q^{20} - 35664q^{22} - 28888q^{23} - 340350q^{25} + 801388q^{26} + 1305192q^{28} - 747648q^{31} - 3033928q^{32} + 4971680q^{35} - 454002q^{37} - 1443720q^{38} + 2683500q^{40} - 2489432q^{41} + 792648q^{43} - 3149928q^{46} + 15313352q^{47} - 29537650q^{50} - 735732q^{52} + 13509122q^{53} + 4448040q^{55} + 18454800q^{56} - 23903520q^{58} + 24111192q^{61} - 53913416q^{62} + 30943610q^{65} - 32827752q^{67} - 8118692q^{68} - 44156280q^{70} + 13992928q^{71} + 111859638q^{73} - 56470800q^{76} - 26260136q^{77} - 23045920q^{80} + 38023056q^{82} + 14768432q^{83} - 19713030q^{85} + 135560008q^{86} - 44555040q^{88} + 167542032q^{91} - 69931048q^{92} - 239661000q^{95} - 186656202q^{97} + 345959698q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 2 x^{5} + 2 x^{4} - 30 x^{3} + 1089 x^{2} - 3168 x + 4608\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 39 \nu^{5} - 22 \nu^{4} - 10 \nu^{3} + 790 \nu^{2} + 41631 \nu - 62928 \)\()/66000\)
\(\beta_{2}\)\(=\)\((\)\( -273 \nu^{5} + 154 \nu^{4} + 70 \nu^{3} - 5530 \nu^{2} + 1028583 \nu - 21504 \)\()/66000\)
\(\beta_{3}\)\(=\)\((\)\( 761 \nu^{5} - 2178 \nu^{4} - 4990 \nu^{3} - 45790 \nu^{2} + 838569 \nu - 2347872 \)\()/66000\)
\(\beta_{4}\)\(=\)\((\)\( -77 \nu^{5} + 146 \nu^{4} - 3570 \nu^{3} + 2030 \nu^{2} - 83733 \nu + 244704 \)\()/6000\)
\(\beta_{5}\)\(=\)\((\)\( -921 \nu^{5} - 1342 \nu^{4} - 610 \nu^{3} + 48190 \nu^{2} - 823209 \nu + 187392 \)\()/66000\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + 7 \beta_{1} + 7\)\()/20\)
\(\nu^{2}\)\(=\)\((\)\(10 \beta_{5} + \beta_{4} - 10 \beta_{3} - \beta_{2} + 446 \beta_{1}\)\()/20\)
\(\nu^{3}\)\(=\)\((\)\(-31 \beta_{4} - 20 \beta_{3} - 283 \beta_{1} + 283\)\()/20\)
\(\nu^{4}\)\(=\)\((\)\(-35 \beta_{5} + 5 \beta_{4} - 35 \beta_{3} + 5 \beta_{2} - 1348\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-400 \beta_{5} - 1019 \beta_{2} + 17267 \beta_{1} + 17267\)\()/20\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/45\mathbb{Z}\right)^\times\).

\(n\) \(11\) \(37\)
\(\chi(n)\) \(1\) \(-\beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
28.1
1.52966 + 1.52966i
−4.23471 4.23471i
3.70505 + 3.70505i
1.52966 1.52966i
−4.23471 + 4.23471i
3.70505 3.70505i
−15.2610 + 15.2610i 0 209.796i −558.542 280.457i 0 −2415.21 + 2415.21i −705.116 705.116i 0 12804.0 4243.86i
28.2 4.39608 4.39608i 0 217.349i 14.1685 + 624.839i 0 730.992 730.992i 2080.88 + 2080.88i 0 2809.13 + 2684.56i
28.3 11.8649 11.8649i 0 25.5528i 434.373 449.383i 0 508.219 508.219i 2734.24 + 2734.24i 0 −178.084 10485.7i
37.1 −15.2610 15.2610i 0 209.796i −558.542 + 280.457i 0 −2415.21 2415.21i −705.116 + 705.116i 0 12804.0 + 4243.86i
37.2 4.39608 + 4.39608i 0 217.349i 14.1685 624.839i 0 730.992 + 730.992i 2080.88 2080.88i 0 2809.13 2684.56i
37.3 11.8649 + 11.8649i 0 25.5528i 434.373 + 449.383i 0 508.219 + 508.219i 2734.24 2734.24i 0 −178.084 + 10485.7i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 45.9.g.a 6
3.b odd 2 1 5.9.c.a 6
5.c odd 4 1 inner 45.9.g.a 6
12.b even 2 1 80.9.p.c 6
15.d odd 2 1 25.9.c.b 6
15.e even 4 1 5.9.c.a 6
15.e even 4 1 25.9.c.b 6
60.l odd 4 1 80.9.p.c 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.9.c.a 6 3.b odd 2 1
5.9.c.a 6 15.e even 4 1
25.9.c.b 6 15.d odd 2 1
25.9.c.b 6 15.e even 4 1
45.9.g.a 6 1.a even 1 1 trivial
45.9.g.a 6 5.c odd 4 1 inner
80.9.p.c 6 12.b even 2 1
80.9.p.c 6 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} - 2 T_{2}^{5} + 2 T_{2}^{4} - 2400 T_{2}^{3} + 153664 T_{2}^{2} - 1248128 T_{2} + 5068928 \) acting on \(S_{9}^{\mathrm{new}}(45, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2 T + 2 T^{2} - 2912 T^{3} - 51136 T^{4} + 1135744 T^{5} + 2070656 T^{6} + 290750464 T^{7} - 3351248896 T^{8} - 48855252992 T^{9} + 8589934592 T^{10} - 2199023255552 T^{11} + 281474976710656 T^{12} \)
$3$ 1
$5$ \( 1 + 220 T + 194375 T^{2} + 199375000 T^{3} + 75927734375 T^{4} + 33569335937500 T^{5} + 59604644775390625 T^{6} \)
$7$ \( 1 + 2352 T + 2765952 T^{2} - 2361533048 T^{3} + 6289666320399 T^{4} + 128493496173721896 T^{5} + \)\(28\!\cdots\!96\)\( T^{6} + \)\(74\!\cdots\!96\)\( T^{7} + \)\(20\!\cdots\!99\)\( T^{8} - \)\(45\!\cdots\!48\)\( T^{9} + \)\(30\!\cdots\!52\)\( T^{10} + \)\(14\!\cdots\!52\)\( T^{11} + \)\(36\!\cdots\!01\)\( T^{12} \)
$11$ \( ( 1 + 11596 T + 493098215 T^{2} + 5106545516920 T^{3} + 105699981590497415 T^{4} + \)\(53\!\cdots\!56\)\( T^{5} + \)\(98\!\cdots\!41\)\( T^{6} )^{2} \)
$13$ \( 1 + 119142 T + 7097408082 T^{2} + 332686420223782 T^{3} + 13680551086291514559 T^{4} + \)\(46\!\cdots\!56\)\( T^{5} + \)\(13\!\cdots\!76\)\( T^{6} + \)\(38\!\cdots\!76\)\( T^{7} + \)\(91\!\cdots\!19\)\( T^{8} + \)\(18\!\cdots\!02\)\( T^{9} + \)\(31\!\cdots\!42\)\( T^{10} + \)\(43\!\cdots\!42\)\( T^{11} + \)\(29\!\cdots\!21\)\( T^{12} \)
$17$ \( 1 - 265502 T + 35245656002 T^{2} - 3505767378301982 T^{3} + \)\(37\!\cdots\!19\)\( T^{4} - \)\(40\!\cdots\!76\)\( T^{5} + \)\(37\!\cdots\!76\)\( T^{6} - \)\(28\!\cdots\!16\)\( T^{7} + \)\(18\!\cdots\!39\)\( T^{8} - \)\(11\!\cdots\!22\)\( T^{9} + \)\(83\!\cdots\!22\)\( T^{10} - \)\(43\!\cdots\!02\)\( T^{11} + \)\(11\!\cdots\!41\)\( T^{12} \)
$19$ \( 1 - 66391003446 T^{2} + \)\(21\!\cdots\!15\)\( T^{4} - \)\(42\!\cdots\!20\)\( T^{6} + \)\(60\!\cdots\!15\)\( T^{8} - \)\(55\!\cdots\!06\)\( T^{10} + \)\(23\!\cdots\!41\)\( T^{12} \)
$23$ \( 1 + 28888 T + 417258272 T^{2} + 2207314762520128 T^{3} + \)\(86\!\cdots\!39\)\( T^{4} + \)\(83\!\cdots\!64\)\( T^{5} + \)\(12\!\cdots\!16\)\( T^{6} + \)\(65\!\cdots\!84\)\( T^{7} + \)\(52\!\cdots\!79\)\( T^{8} + \)\(10\!\cdots\!48\)\( T^{9} + \)\(15\!\cdots\!12\)\( T^{10} + \)\(85\!\cdots\!88\)\( T^{11} + \)\(23\!\cdots\!81\)\( T^{12} \)
$29$ \( 1 - 1726260912966 T^{2} + \)\(13\!\cdots\!15\)\( T^{4} - \)\(77\!\cdots\!20\)\( T^{6} + \)\(34\!\cdots\!15\)\( T^{8} - \)\(10\!\cdots\!06\)\( T^{10} + \)\(15\!\cdots\!61\)\( T^{12} \)
$31$ \( ( 1 + 373824 T + 1438821265815 T^{2} + 103488660558742480 T^{3} + \)\(12\!\cdots\!15\)\( T^{4} + \)\(27\!\cdots\!44\)\( T^{5} + \)\(62\!\cdots\!21\)\( T^{6} )^{2} \)
$37$ \( 1 + 454002 T + 103058908002 T^{2} + 1591257258997412242 T^{3} + \)\(36\!\cdots\!59\)\( T^{4} + \)\(11\!\cdots\!36\)\( T^{5} + \)\(25\!\cdots\!36\)\( T^{6} + \)\(39\!\cdots\!56\)\( T^{7} + \)\(45\!\cdots\!19\)\( T^{8} + \)\(68\!\cdots\!62\)\( T^{9} + \)\(15\!\cdots\!62\)\( T^{10} + \)\(24\!\cdots\!02\)\( T^{11} + \)\(18\!\cdots\!21\)\( T^{12} \)
$41$ \( ( 1 + 1244716 T + 19779856453415 T^{2} + 17348876621060070520 T^{3} + \)\(15\!\cdots\!15\)\( T^{4} + \)\(79\!\cdots\!56\)\( T^{5} + \)\(50\!\cdots\!61\)\( T^{6} )^{2} \)
$43$ \( 1 - 792648 T + 314145425952 T^{2} - 6701894073526462448 T^{3} + \)\(27\!\cdots\!99\)\( T^{4} - \)\(18\!\cdots\!04\)\( T^{5} + \)\(86\!\cdots\!96\)\( T^{6} - \)\(22\!\cdots\!04\)\( T^{7} + \)\(37\!\cdots\!99\)\( T^{8} - \)\(10\!\cdots\!48\)\( T^{9} + \)\(58\!\cdots\!52\)\( T^{10} - \)\(17\!\cdots\!48\)\( T^{11} + \)\(25\!\cdots\!01\)\( T^{12} \)
$47$ \( 1 - 15313352 T + 117249374737952 T^{2} - \)\(85\!\cdots\!72\)\( T^{3} + \)\(63\!\cdots\!79\)\( T^{4} - \)\(35\!\cdots\!16\)\( T^{5} + \)\(17\!\cdots\!16\)\( T^{6} - \)\(85\!\cdots\!76\)\( T^{7} + \)\(36\!\cdots\!59\)\( T^{8} - \)\(11\!\cdots\!32\)\( T^{9} + \)\(37\!\cdots\!32\)\( T^{10} - \)\(11\!\cdots\!52\)\( T^{11} + \)\(18\!\cdots\!61\)\( T^{12} \)
$53$ \( 1 - 13509122 T + 91248188605442 T^{2} - \)\(98\!\cdots\!42\)\( T^{3} + \)\(11\!\cdots\!79\)\( T^{4} - \)\(81\!\cdots\!76\)\( T^{5} + \)\(50\!\cdots\!36\)\( T^{6} - \)\(51\!\cdots\!36\)\( T^{7} + \)\(46\!\cdots\!59\)\( T^{8} - \)\(23\!\cdots\!02\)\( T^{9} + \)\(13\!\cdots\!22\)\( T^{10} - \)\(12\!\cdots\!22\)\( T^{11} + \)\(58\!\cdots\!61\)\( T^{12} \)
$59$ \( 1 - 413223229068726 T^{2} + \)\(98\!\cdots\!15\)\( T^{4} - \)\(17\!\cdots\!20\)\( T^{6} + \)\(21\!\cdots\!15\)\( T^{8} - \)\(19\!\cdots\!06\)\( T^{10} + \)\(10\!\cdots\!21\)\( T^{12} \)
$61$ \( ( 1 - 12055596 T + 200152007609415 T^{2} + \)\(24\!\cdots\!80\)\( T^{3} + \)\(38\!\cdots\!15\)\( T^{4} - \)\(44\!\cdots\!56\)\( T^{5} + \)\(70\!\cdots\!41\)\( T^{6} )^{2} \)
$67$ \( 1 + 32827752 T + 538830650686752 T^{2} + \)\(16\!\cdots\!32\)\( T^{3} + \)\(54\!\cdots\!19\)\( T^{4} + \)\(88\!\cdots\!76\)\( T^{5} + \)\(12\!\cdots\!76\)\( T^{6} + \)\(36\!\cdots\!16\)\( T^{7} + \)\(90\!\cdots\!39\)\( T^{8} + \)\(10\!\cdots\!72\)\( T^{9} + \)\(14\!\cdots\!72\)\( T^{10} + \)\(36\!\cdots\!52\)\( T^{11} + \)\(44\!\cdots\!41\)\( T^{12} \)
$71$ \( ( 1 - 6996464 T + 1071469029384215 T^{2} - \)\(14\!\cdots\!80\)\( T^{3} + \)\(69\!\cdots\!15\)\( T^{4} - \)\(29\!\cdots\!44\)\( T^{5} + \)\(26\!\cdots\!81\)\( T^{6} )^{2} \)
$73$ \( 1 - 111859638 T + 6256289306745522 T^{2} - \)\(29\!\cdots\!78\)\( T^{3} + \)\(12\!\cdots\!39\)\( T^{4} - \)\(42\!\cdots\!64\)\( T^{5} + \)\(12\!\cdots\!16\)\( T^{6} - \)\(33\!\cdots\!84\)\( T^{7} + \)\(79\!\cdots\!79\)\( T^{8} - \)\(15\!\cdots\!98\)\( T^{9} + \)\(26\!\cdots\!62\)\( T^{10} - \)\(38\!\cdots\!38\)\( T^{11} + \)\(27\!\cdots\!81\)\( T^{12} \)
$79$ \( 1 - 4185433961698566 T^{2} + \)\(55\!\cdots\!15\)\( T^{4} - \)\(46\!\cdots\!20\)\( T^{6} + \)\(12\!\cdots\!15\)\( T^{8} - \)\(22\!\cdots\!06\)\( T^{10} + \)\(12\!\cdots\!61\)\( T^{12} \)
$83$ \( 1 - 14768432 T + 109053291869312 T^{2} - \)\(28\!\cdots\!12\)\( T^{3} + \)\(10\!\cdots\!19\)\( T^{4} - \)\(10\!\cdots\!16\)\( T^{5} + \)\(83\!\cdots\!56\)\( T^{6} - \)\(23\!\cdots\!56\)\( T^{7} + \)\(51\!\cdots\!39\)\( T^{8} - \)\(32\!\cdots\!52\)\( T^{9} + \)\(28\!\cdots\!32\)\( T^{10} - \)\(85\!\cdots\!32\)\( T^{11} + \)\(13\!\cdots\!41\)\( T^{12} \)
$89$ \( 1 - 8165894455313286 T^{2} + \)\(62\!\cdots\!15\)\( T^{4} - \)\(26\!\cdots\!20\)\( T^{6} + \)\(97\!\cdots\!15\)\( T^{8} - \)\(19\!\cdots\!06\)\( T^{10} + \)\(37\!\cdots\!81\)\( T^{12} \)
$97$ \( 1 + 186656202 T + 17420268872532402 T^{2} + \)\(93\!\cdots\!22\)\( T^{3} + \)\(66\!\cdots\!79\)\( T^{4} + \)\(11\!\cdots\!16\)\( T^{5} + \)\(13\!\cdots\!16\)\( T^{6} + \)\(87\!\cdots\!76\)\( T^{7} + \)\(41\!\cdots\!59\)\( T^{8} + \)\(45\!\cdots\!82\)\( T^{9} + \)\(65\!\cdots\!82\)\( T^{10} + \)\(55\!\cdots\!02\)\( T^{11} + \)\(23\!\cdots\!61\)\( T^{12} \)
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