Properties

Label 6020.2.a.k
Level $6020$
Weight $2$
Character orbit 6020.a
Self dual yes
Analytic conductor $48.070$
Analytic rank $0$
Dimension $13$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6020 = 2^{2} \cdot 5 \cdot 7 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6020.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.0699420168\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Defining polynomial: \(x^{13} - 27 x^{11} - 2 x^{10} + 268 x^{9} + 37 x^{8} - 1201 x^{7} - 189 x^{6} + 2384 x^{5} + 231 x^{4} - 1729 x^{3} + 20 x^{2} + 105 x - 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{12}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + q^{5} - q^{7} + ( 1 + \beta_{2} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + q^{5} - q^{7} + ( 1 + \beta_{2} ) q^{9} + \beta_{8} q^{11} + ( 1 + \beta_{10} ) q^{13} + \beta_{1} q^{15} + ( 1 + \beta_{11} ) q^{17} + \beta_{9} q^{19} -\beta_{1} q^{21} + ( \beta_{1} + \beta_{3} - \beta_{5} - \beta_{6} ) q^{23} + q^{25} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{6} - \beta_{10} - \beta_{11} ) q^{27} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{9} + \beta_{12} ) q^{29} + ( \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} - \beta_{12} ) q^{31} + ( 1 - \beta_{1} + \beta_{3} + \beta_{5} + \beta_{6} + \beta_{8} - \beta_{11} - \beta_{12} ) q^{33} - q^{35} + ( 1 - \beta_{3} + \beta_{5} + \beta_{11} + \beta_{12} ) q^{37} + ( -1 - 2 \beta_{2} + 2 \beta_{4} + \beta_{11} ) q^{39} + ( 1 - \beta_{4} + \beta_{5} + \beta_{7} ) q^{41} + q^{43} + ( 1 + \beta_{2} ) q^{45} + ( -\beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{11} ) q^{47} + q^{49} + ( 1 + 3 \beta_{1} + \beta_{6} - \beta_{9} - \beta_{11} ) q^{51} + ( 2 + \beta_{1} - \beta_{4} - \beta_{5} + \beta_{8} - \beta_{9} - \beta_{10} + \beta_{12} ) q^{53} + \beta_{8} q^{55} + ( 1 - \beta_{1} + \beta_{2} + \beta_{5} + \beta_{7} - \beta_{11} ) q^{57} + ( -1 + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{7} - \beta_{12} ) q^{59} + ( 1 + 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{6} - \beta_{9} + \beta_{10} ) q^{61} + ( -1 - \beta_{2} ) q^{63} + ( 1 + \beta_{10} ) q^{65} + ( 1 - \beta_{1} + \beta_{3} + 2 \beta_{4} - \beta_{6} + \beta_{9} + \beta_{10} ) q^{67} + ( 2 - \beta_{1} - \beta_{2} + 2 \beta_{4} + \beta_{6} - \beta_{7} - \beta_{12} ) q^{69} + ( 1 + 2 \beta_{1} + \beta_{2} - \beta_{5} - \beta_{9} - \beta_{11} + \beta_{12} ) q^{71} + ( 3 + \beta_{2} + \beta_{4} - \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} - \beta_{12} ) q^{73} + \beta_{1} q^{75} -\beta_{8} q^{77} + ( -1 + 2 \beta_{1} - \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - \beta_{9} - 2 \beta_{10} + \beta_{11} + \beta_{12} ) q^{79} + ( \beta_{1} + \beta_{2} - 2 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{10} + \beta_{12} ) q^{81} + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{9} - \beta_{12} ) q^{83} + ( 1 + \beta_{11} ) q^{85} + ( 1 + 3 \beta_{1} + 2 \beta_{6} - \beta_{7} - \beta_{9} - \beta_{10} + \beta_{12} ) q^{87} + ( 1 + \beta_{1} - 3 \beta_{3} - 2 \beta_{4} + \beta_{6} - \beta_{9} - \beta_{10} ) q^{89} + ( -1 - \beta_{10} ) q^{91} + ( 2 - \beta_{1} - \beta_{2} - \beta_{3} + 3 \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{8} + \beta_{11} - \beta_{12} ) q^{93} + \beta_{9} q^{95} + ( 4 - \beta_{1} + \beta_{6} - \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} ) q^{97} + ( -1 + \beta_{2} - \beta_{3} - 3 \beta_{4} + \beta_{5} + \beta_{7} - 2 \beta_{10} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13q + 13q^{5} - 13q^{7} + 15q^{9} + O(q^{10}) \) \( 13q + 13q^{5} - 13q^{7} + 15q^{9} + 11q^{13} + 16q^{17} + 3q^{23} + 13q^{25} + 6q^{27} + 10q^{29} - q^{31} + 14q^{33} - 13q^{35} + 16q^{37} - 14q^{39} + 23q^{41} + 13q^{43} + 15q^{45} + 2q^{47} + 13q^{49} + 4q^{51} + 20q^{53} + 22q^{57} + 2q^{59} + 5q^{61} - 15q^{63} + 11q^{65} + 19q^{67} + 16q^{69} + 4q^{71} + 34q^{73} - 15q^{79} + 17q^{81} + 27q^{83} + 16q^{85} - 5q^{87} + 3q^{89} - 11q^{91} + 35q^{93} + 45q^{97} + q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{13} - 27 x^{11} - 2 x^{10} + 268 x^{9} + 37 x^{8} - 1201 x^{7} - 189 x^{6} + 2384 x^{5} + 231 x^{4} - 1729 x^{3} + 20 x^{2} + 105 x - 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 4 \)
\(\beta_{3}\)\(=\)\((\)\(-1537654 \nu^{12} - 33962 \nu^{11} + 42386523 \nu^{10} + 2329850 \nu^{9} - 430739109 \nu^{8} - 29216373 \nu^{7} + 1977473357 \nu^{6} + 64922637 \nu^{5} - 3993847798 \nu^{4} + 274187663 \nu^{3} + 2815717179 \nu^{2} - 483490486 \nu - 61265760\)\()/42614911\)
\(\beta_{4}\)\(=\)\((\)\(-3379072 \nu^{12} + 657965 \nu^{11} + 90038475 \nu^{10} - 6078641 \nu^{9} - 883869052 \nu^{8} - 53662673 \nu^{7} + 3942799622 \nu^{6} + 586775704 \nu^{5} - 7864942625 \nu^{4} - 1213473653 \nu^{3} + 5777558205 \nu^{2} + 572663033 \nu - 407264550\)\()/85229822\)
\(\beta_{5}\)\(=\)\((\)\(3634961 \nu^{12} - 1032220 \nu^{11} - 98400044 \nu^{10} + 19594321 \nu^{9} + 981840509 \nu^{8} - 115849143 \nu^{7} - 4449167492 \nu^{6} + 327739797 \nu^{5} + 9057266266 \nu^{4} - 950176204 \nu^{3} - 6983810052 \nu^{2} + 1396689219 \nu + 532793576\)\()/85229822\)
\(\beta_{6}\)\(=\)\((\)\(4068317 \nu^{12} + 2428772 \nu^{11} - 108151396 \nu^{10} - 71287003 \nu^{9} + 1045858413 \nu^{8} + 738896929 \nu^{7} - 4477485048 \nu^{6} - 3114270009 \nu^{5} + 8198411856 \nu^{4} + 4641122190 \nu^{3} - 5196373544 \nu^{2} - 1743391215 \nu - 56489344\)\()/85229822\)
\(\beta_{7}\)\(=\)\((\)\(4383099 \nu^{12} + 2682991 \nu^{11} - 114991907 \nu^{10} - 79451502 \nu^{9} + 1081452331 \nu^{8} + 830625394 \nu^{7} - 4359230160 \nu^{6} - 3559234085 \nu^{5} + 6955695027 \nu^{4} + 5618110765 \nu^{3} - 3259290077 \nu^{2} - 2683469426 \nu - 56900248\)\()/85229822\)
\(\beta_{8}\)\(=\)\((\)\(-2205646 \nu^{12} + 200934 \nu^{11} + 58451359 \nu^{10} + 796160 \nu^{9} - 567105876 \nu^{8} - 68621002 \nu^{7} + 2478773524 \nu^{6} + 489147473 \nu^{5} - 4827397410 \nu^{4} - 998832979 \nu^{3} + 3550774102 \nu^{2} + 617582318 \nu - 238977264\)\()/42614911\)
\(\beta_{9}\)\(=\)\((\)\(-2264279 \nu^{12} + 1297400 \nu^{11} + 60374111 \nu^{10} - 27283586 \nu^{9} - 591766125 \nu^{8} + 184521935 \nu^{7} + 2625107725 \nu^{6} - 452829515 \nu^{5} - 5160860541 \nu^{4} + 270284028 \nu^{3} + 3575600842 \nu^{2} + 187215299 \nu - 115635328\)\()/42614911\)
\(\beta_{10}\)\(=\)\((\)\(5099437 \nu^{12} - 1334884 \nu^{11} - 133195254 \nu^{10} + 20110413 \nu^{9} + 1264513359 \nu^{8} - 52366611 \nu^{7} - 5328390224 \nu^{6} - 125035509 \nu^{5} + 9624753738 \nu^{4} - 125618964 \nu^{3} - 5897264220 \nu^{2} + 863773441 \nu - 20700834\)\()/85229822\)
\(\beta_{11}\)\(=\)\((\)\(5423260 \nu^{12} + 3173615 \nu^{11} - 149767663 \nu^{10} - 89978475 \nu^{9} + 1526692324 \nu^{8} + 903358959 \nu^{7} - 7046841160 \nu^{6} - 3705855478 \nu^{5} + 14426296339 \nu^{4} + 5346609659 \nu^{3} - 10622872065 \nu^{2} - 1616237963 \nu + 238318094\)\()/85229822\)
\(\beta_{12}\)\(=\)\((\)\(-5608450 \nu^{12} + 759047 \nu^{11} + 152122251 \nu^{10} - 3476691 \nu^{9} - 1520659172 \nu^{8} - 135571279 \nu^{7} + 6888121624 \nu^{6} + 1165740178 \nu^{5} - 13834451127 \nu^{4} - 2579378641 \nu^{3} + 9852280555 \nu^{2} + 1467258677 \nu - 259624920\)\()/85229822\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 4\)
\(\nu^{3}\)\(=\)\(-\beta_{11} - \beta_{10} + \beta_{6} - \beta_{4} - \beta_{3} + \beta_{2} + 7 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(\beta_{12} - \beta_{10} + \beta_{7} - \beta_{6} + \beta_{5} - 2 \beta_{4} + 10 \beta_{2} + \beta_{1} + 27\)
\(\nu^{5}\)\(=\)\(-\beta_{12} - 13 \beta_{11} - 12 \beta_{10} + 2 \beta_{8} - \beta_{7} + 15 \beta_{6} + 2 \beta_{5} - 11 \beta_{4} - 11 \beta_{3} + 11 \beta_{2} + 55 \beta_{1} + 14\)
\(\nu^{6}\)\(=\)\(16 \beta_{12} + \beta_{11} - 14 \beta_{10} - 3 \beta_{9} + 5 \beta_{8} + 15 \beta_{7} - 14 \beta_{6} + 13 \beta_{5} - 32 \beta_{4} + 96 \beta_{2} + 14 \beta_{1} + 213\)
\(\nu^{7}\)\(=\)\(-16 \beta_{12} - 140 \beta_{11} - 126 \beta_{10} - 5 \beta_{9} + 35 \beta_{8} - 17 \beta_{7} + 177 \beta_{6} + 34 \beta_{5} - 105 \beta_{4} - 104 \beta_{3} + 102 \beta_{2} + 466 \beta_{1} + 145\)
\(\nu^{8}\)\(=\)\(200 \beta_{12} + 25 \beta_{11} - 154 \beta_{10} - 56 \beta_{9} + 87 \beta_{8} + 172 \beta_{7} - 161 \beta_{6} + 135 \beta_{5} - 391 \beta_{4} + 3 \beta_{3} + 921 \beta_{2} + 148 \beta_{1} + 1822\)
\(\nu^{9}\)\(=\)\(-178 \beta_{12} - 1427 \beta_{11} - 1278 \beta_{10} - 114 \beta_{9} + 444 \beta_{8} - 217 \beta_{7} + 1923 \beta_{6} + 422 \beta_{5} - 978 \beta_{4} - 972 \beta_{3} + 904 \beta_{2} + 4153 \beta_{1} + 1355\)
\(\nu^{10}\)\(=\)\(2291 \beta_{12} + 416 \beta_{11} - 1549 \beta_{10} - 758 \beta_{9} + 1089 \beta_{8} + 1809 \beta_{7} - 1752 \beta_{6} + 1308 \beta_{5} - 4316 \beta_{4} + 90 \beta_{3} + 8866 \beta_{2} + 1400 \beta_{1} + 16329\)
\(\nu^{11}\)\(=\)\(-1730 \beta_{12} - 14222 \beta_{11} - 12771 \beta_{10} - 1743 \beta_{9} + 4994 \beta_{8} - 2510 \beta_{7} + 20128 \beta_{6} + 4647 \beta_{5} - 9084 \beta_{4} - 9175 \beta_{3} + 7846 \beta_{2} + 38282 \beta_{1} + 12073\)
\(\nu^{12}\)\(=\)\(25137 \beta_{12} + 5835 \beta_{11} - 14912 \beta_{10} - 9105 \beta_{9} + 12060 \beta_{8} + 18385 \beta_{7} - 18684 \beta_{6} + 12335 \beta_{5} - 45331 \beta_{4} + 1676 \beta_{3} + 85618 \beta_{2} + 12389 \beta_{1} + 150610\)

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
−3.18036
−2.54322
−2.12558
−1.90637
−1.30831
−0.265767
0.0387611
0.247295
1.13823
1.47357
2.41335
2.91397
3.10444
0 −3.18036 0 1.00000 0 −1.00000 0 7.11471 0
1.2 0 −2.54322 0 1.00000 0 −1.00000 0 3.46796 0
1.3 0 −2.12558 0 1.00000 0 −1.00000 0 1.51810 0
1.4 0 −1.90637 0 1.00000 0 −1.00000 0 0.634257 0
1.5 0 −1.30831 0 1.00000 0 −1.00000 0 −1.28831 0
1.6 0 −0.265767 0 1.00000 0 −1.00000 0 −2.92937 0
1.7 0 0.0387611 0 1.00000 0 −1.00000 0 −2.99850 0
1.8 0 0.247295 0 1.00000 0 −1.00000 0 −2.93885 0
1.9 0 1.13823 0 1.00000 0 −1.00000 0 −1.70443 0
1.10 0 1.47357 0 1.00000 0 −1.00000 0 −0.828594 0
1.11 0 2.41335 0 1.00000 0 −1.00000 0 2.82427 0
1.12 0 2.91397 0 1.00000 0 −1.00000 0 5.49125 0
1.13 0 3.10444 0 1.00000 0 −1.00000 0 6.63752 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.13
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)
\(7\) \(1\)
\(43\) \(-1\)

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 6020.2.a.k 13
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6020.2.a.k 13 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6020))\):

\(T_{3}^{13} - \cdots\)
\(T_{11}^{13} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 12 T^{2} - 2 T^{3} + 79 T^{4} - 23 T^{5} + 392 T^{6} - 111 T^{7} + 1625 T^{8} - 327 T^{9} + 5897 T^{10} - 799 T^{11} + 19149 T^{12} - 2152 T^{13} + 57447 T^{14} - 7191 T^{15} + 159219 T^{16} - 26487 T^{17} + 394875 T^{18} - 80919 T^{19} + 857304 T^{20} - 150903 T^{21} + 1554957 T^{22} - 118098 T^{23} + 2125764 T^{24} + 1594323 T^{26} \)
$5$ \( ( 1 - T )^{13} \)
$7$ \( ( 1 + T )^{13} \)
$11$ \( 1 + 53 T^{2} + 50 T^{3} + 1501 T^{4} + 2028 T^{5} + 30139 T^{6} + 49789 T^{7} + 460566 T^{8} + 887374 T^{9} + 5984420 T^{10} + 12445289 T^{11} + 69492496 T^{12} + 149524172 T^{13} + 764417456 T^{14} + 1505879969 T^{15} + 7965263020 T^{16} + 12992042734 T^{17} + 74174614866 T^{18} + 88204250629 T^{19} + 587323846769 T^{20} + 434719810668 T^{21} + 3539279484191 T^{22} + 1296871230050 T^{23} + 15121518542383 T^{24} + 34522712143931 T^{26} \)
$13$ \( 1 - 11 T + 124 T^{2} - 925 T^{3} + 6357 T^{4} - 36721 T^{5} + 195108 T^{6} - 942729 T^{7} + 4270443 T^{8} - 18191133 T^{9} + 74302953 T^{10} - 289333586 T^{11} + 1099364059 T^{12} - 3999583790 T^{13} + 14291732767 T^{14} - 48897376034 T^{15} + 163243587741 T^{16} - 519556949613 T^{17} + 1585585592799 T^{18} - 4550372821761 T^{19} + 12242737654836 T^{20} - 29954447805841 T^{21} + 67412802514161 T^{22} - 127519104960325 T^{23} + 222227888860588 T^{24} - 256278936347291 T^{25} + 302875106592253 T^{26} \)
$17$ \( 1 - 16 T + 232 T^{2} - 2164 T^{3} + 18346 T^{4} - 123150 T^{5} + 772393 T^{6} - 4181271 T^{7} + 21988401 T^{8} - 105957487 T^{9} + 510803344 T^{10} - 2309289270 T^{11} + 10360350472 T^{12} - 43159109488 T^{13} + 176125958024 T^{14} - 667384599030 T^{15} + 2509576829072 T^{16} - 8849675271727 T^{17} + 31220385078657 T^{18} - 100925717270199 T^{19} + 316942718654489 T^{20} - 859064528859150 T^{21} + 2175613182213962 T^{22} - 4362610800571636 T^{23} + 7951079943370856 T^{24} - 9321955795676176 T^{25} + 9904578032905937 T^{26} \)
$19$ \( 1 + 117 T^{2} + 5 T^{3} + 6835 T^{4} - 997 T^{5} + 266583 T^{6} - 134458 T^{7} + 7847403 T^{8} - 7348506 T^{9} + 188460095 T^{10} - 243708059 T^{11} + 3937295160 T^{12} - 5517755554 T^{13} + 74808608040 T^{14} - 87978609299 T^{15} + 1292647791605 T^{16} - 957664650426 T^{17} + 19430946720897 T^{18} - 6325695067498 T^{19} + 238291009797837 T^{20} - 16932612351877 T^{21} + 2205570414319465 T^{22} + 30655331289005 T^{23} + 13629360291091623 T^{24} + 42052983462257059 T^{26} \)
$23$ \( 1 - 3 T + 151 T^{2} - 443 T^{3} + 11161 T^{4} - 32060 T^{5} + 548870 T^{6} - 1558229 T^{7} + 20629834 T^{8} - 58509997 T^{9} + 639250817 T^{10} - 1798029221 T^{11} + 16949966784 T^{12} - 45597290078 T^{13} + 389849236032 T^{14} - 951157457909 T^{15} + 7777764690439 T^{16} - 16373496070477 T^{17} + 132780687657062 T^{18} - 230673815280581 T^{19} + 1868806543094890 T^{20} - 2510650188108860 T^{21} + 20102664854588543 T^{22} - 18351944467646507 T^{23} + 143874273445002977 T^{24} - 65743873296060963 T^{25} + 504036361936467383 T^{26} \)
$29$ \( 1 - 10 T + 238 T^{2} - 2079 T^{3} + 28142 T^{4} - 215709 T^{5} + 2184543 T^{6} - 14852326 T^{7} + 124454688 T^{8} - 758718278 T^{9} + 5514778387 T^{10} - 30368027851 T^{11} + 196241084625 T^{12} - 977916685190 T^{13} + 5690991454125 T^{14} - 25539511422691 T^{15} + 134499930080543 T^{16} - 536627022382118 T^{17} + 2552708649316512 T^{18} - 8834509875894646 T^{19} + 37683096541691787 T^{20} - 107907653493404349 T^{21} + 408260102052905398 T^{22} - 874650338031117879 T^{23} + 2903721324237987302 T^{24} - 3538147832054690410 T^{25} + 10260628712958602189 T^{26} \)
$31$ \( 1 + T + 88 T^{2} + 108 T^{3} + 5936 T^{4} + 5938 T^{5} + 335461 T^{6} + 258870 T^{7} + 14930920 T^{8} + 9479568 T^{9} + 590741078 T^{10} + 294357379 T^{11} + 20604001480 T^{12} + 8929202624 T^{13} + 638724045880 T^{14} + 282877441219 T^{15} + 17598767454698 T^{16} + 8754580118928 T^{17} + 427459563248920 T^{18} + 229748077900470 T^{19} + 9229409042290171 T^{20} + 5064466980324658 T^{21} + 156945597145743056 T^{22} + 88519854993926508 T^{23} + 2235945966883625128 T^{24} + 787662783788549761 T^{25} + 24417546297445042591 T^{26} \)
$37$ \( 1 - 16 T + 335 T^{2} - 3583 T^{3} + 47783 T^{4} - 421511 T^{5} + 4504539 T^{6} - 34659239 T^{7} + 316406640 T^{8} - 2174885757 T^{9} + 17559909034 T^{10} - 109093513955 T^{11} + 790552591108 T^{12} - 4453783675410 T^{13} + 29250445870996 T^{14} - 149349020604395 T^{15} + 889462072299202 T^{16} - 4076086065224877 T^{17} + 21940888438674480 T^{18} - 88926124818142751 T^{19} + 427624342888806687 T^{20} - 1480548727101694631 T^{21} + 6209961812628164291 T^{22} - 17229157806373152967 T^{23} + 59602403296119238355 T^{24} - \)\(10\!\cdots\!96\)\( T^{25} + \)\(24\!\cdots\!97\)\( T^{26} \)
$41$ \( 1 - 23 T + 576 T^{2} - 9070 T^{3} + 139315 T^{4} - 1704579 T^{5} + 19979305 T^{6} - 201725431 T^{7} + 1947185284 T^{8} - 16761244471 T^{9} + 138092320179 T^{10} - 1031759307419 T^{11} + 7387434553068 T^{12} - 48308950864830 T^{13} + 302884816675788 T^{14} - 1734387395771339 T^{15} + 9517460799056859 T^{16} - 47363270937617431 T^{17} + 225593489647346084 T^{18} - 958216825310652871 T^{19} + 3891055037922032705 T^{20} - 13610935862129845059 T^{21} + 45609214190094676715 T^{22} - \)\(12\!\cdots\!70\)\( T^{23} + \)\(31\!\cdots\!16\)\( T^{24} - \)\(51\!\cdots\!63\)\( T^{25} + \)\(92\!\cdots\!21\)\( T^{26} \)
$43$ \( ( 1 - T )^{13} \)
$47$ \( 1 - 2 T + 284 T^{2} - 299 T^{3} + 41950 T^{4} - 5019 T^{5} + 4220701 T^{6} + 3181572 T^{7} + 326022532 T^{8} + 474687616 T^{9} + 20670190075 T^{10} + 38654405367 T^{11} + 1118458410385 T^{12} + 2149968933466 T^{13} + 52567545288095 T^{14} + 85387581455703 T^{15} + 2146041144156725 T^{16} + 2316324140730496 T^{17} + 74771639883697724 T^{18} + 34294849672717188 T^{19} + 2138304711161304563 T^{20} - 119508847755378459 T^{21} + 46947523346661075650 T^{22} - 15727140538513184651 T^{23} + \)\(70\!\cdots\!52\)\( T^{24} - \)\(23\!\cdots\!82\)\( T^{25} + \)\(54\!\cdots\!27\)\( T^{26} \)
$53$ \( 1 - 20 T + 570 T^{2} - 8239 T^{3} + 140200 T^{4} - 1616147 T^{5} + 20795049 T^{6} - 201364088 T^{7} + 2145810135 T^{8} - 18082107742 T^{9} + 168072325522 T^{10} - 1267913455673 T^{11} + 10633847752991 T^{12} - 73271137966998 T^{13} + 563593930908523 T^{14} - 3561568896985457 T^{15} + 25022103606738794 T^{16} - 142676527578203902 T^{17} + 897368127290721555 T^{18} - 4463106364843735352 T^{19} + 24428175713756267013 T^{20} - \)\(10\!\cdots\!67\)\( T^{21} + \)\(46\!\cdots\!00\)\( T^{22} - \)\(14\!\cdots\!11\)\( T^{23} + \)\(52\!\cdots\!90\)\( T^{24} - \)\(98\!\cdots\!20\)\( T^{25} + \)\(26\!\cdots\!73\)\( T^{26} \)
$59$ \( 1 - 2 T + 336 T^{2} + 278 T^{3} + 57047 T^{4} + 172755 T^{5} + 6988059 T^{6} + 30304986 T^{7} + 699474600 T^{8} + 3308404742 T^{9} + 58857012758 T^{10} + 270061385137 T^{11} + 4148204927199 T^{12} + 17669714125088 T^{13} + 244744090704741 T^{14} + 940083681661897 T^{15} + 12087994423225282 T^{16} + 40089134592925862 T^{17} + 500071388073305400 T^{18} + 1278280481463034026 T^{19} + 17390843406352776321 T^{20} + 25365692248334474355 T^{21} + \)\(49\!\cdots\!33\)\( T^{22} + \)\(14\!\cdots\!78\)\( T^{23} + \)\(10\!\cdots\!24\)\( T^{24} - \)\(35\!\cdots\!62\)\( T^{25} + \)\(10\!\cdots\!79\)\( T^{26} \)
$61$ \( 1 - 5 T + 385 T^{2} - 1392 T^{3} + 78249 T^{4} - 228619 T^{5} + 11220912 T^{6} - 27733767 T^{7} + 1238659618 T^{8} - 2654743135 T^{9} + 110130687753 T^{10} - 209812101594 T^{11} + 8071813843508 T^{12} - 13922354346548 T^{13} + 492380644453988 T^{14} - 780710830031274 T^{15} + 24997573636863693 T^{16} - 36757151343051535 T^{17} + 1046167331560873018 T^{18} - 1428854058280747887 T^{19} + 35264440801622071152 T^{20} - 43827934190125384939 T^{21} + \)\(91\!\cdots\!09\)\( T^{22} - \)\(99\!\cdots\!92\)\( T^{23} + \)\(16\!\cdots\!85\)\( T^{24} - \)\(13\!\cdots\!05\)\( T^{25} + \)\(16\!\cdots\!81\)\( T^{26} \)
$67$ \( 1 - 19 T + 665 T^{2} - 10361 T^{3} + 214846 T^{4} - 2819094 T^{5} + 44046119 T^{6} - 499516783 T^{7} + 6414440638 T^{8} - 63876028473 T^{9} + 702928421016 T^{10} - 6203811416389 T^{11} + 59811288016869 T^{12} - 469196028137730 T^{13} + 4007356297130223 T^{14} - 27848909448170221 T^{15} + 211414860690035208 T^{16} - 1287173578758868233 T^{17} + 8660297352724898266 T^{18} - 45185480056443442327 T^{19} + \)\(26\!\cdots\!37\)\( T^{20} - \)\(11\!\cdots\!54\)\( T^{21} + \)\(58\!\cdots\!62\)\( T^{22} - \)\(18\!\cdots\!89\)\( T^{23} + \)\(81\!\cdots\!95\)\( T^{24} - \)\(15\!\cdots\!59\)\( T^{25} + \)\(54\!\cdots\!87\)\( T^{26} \)
$71$ \( 1 - 4 T + 548 T^{2} - 2789 T^{3} + 147676 T^{4} - 863499 T^{5} + 26477214 T^{6} - 164426873 T^{7} + 3562074814 T^{8} - 22118088775 T^{9} + 379875465159 T^{10} - 2254879842485 T^{11} + 32902726454248 T^{12} - 180052027863694 T^{13} + 2336093578251608 T^{14} - 11366849285966885 T^{15} + 135961607610522849 T^{16} - 562057816279980775 T^{17} + 6426799929876665714 T^{18} - 21063129115542209033 T^{19} + \)\(24\!\cdots\!74\)\( T^{20} - \)\(55\!\cdots\!39\)\( T^{21} + \)\(67\!\cdots\!56\)\( T^{22} - \)\(90\!\cdots\!89\)\( T^{23} + \)\(12\!\cdots\!08\)\( T^{24} - \)\(65\!\cdots\!64\)\( T^{25} + \)\(11\!\cdots\!11\)\( T^{26} \)
$73$ \( 1 - 34 T + 1121 T^{2} - 25315 T^{3} + 516240 T^{4} - 8885498 T^{5} + 138834580 T^{6} - 1945250236 T^{7} + 25048831315 T^{8} - 296064712316 T^{9} + 3247541880683 T^{10} - 33025369477201 T^{11} + 313501550116924 T^{12} - 2769720931551088 T^{13} + 22885613158535452 T^{14} - 175992193944004129 T^{15} + 1263348999797658611 T^{16} - 8407717051945436156 T^{17} + 51928020636975334795 T^{18} - \)\(29\!\cdots\!04\)\( T^{19} + \)\(15\!\cdots\!60\)\( T^{20} - \)\(71\!\cdots\!38\)\( T^{21} + \)\(30\!\cdots\!20\)\( T^{22} - \)\(10\!\cdots\!35\)\( T^{23} + \)\(35\!\cdots\!17\)\( T^{24} - \)\(77\!\cdots\!14\)\( T^{25} + \)\(16\!\cdots\!33\)\( T^{26} \)
$79$ \( 1 + 15 T + 443 T^{2} + 5563 T^{3} + 99432 T^{4} + 962780 T^{5} + 12890577 T^{6} + 90762914 T^{7} + 956871031 T^{8} + 2892970095 T^{9} + 23658872691 T^{10} - 363925121613 T^{11} - 2458340352577 T^{12} - 52611160040276 T^{13} - 194208887853583 T^{14} - 2271256683986733 T^{15} + 11664746932697949 T^{16} + 112681419530827695 T^{17} + 2944346128956277369 T^{18} + 22063325819931348194 T^{19} + \)\(24\!\cdots\!43\)\( T^{20} + \)\(14\!\cdots\!80\)\( T^{21} + \)\(11\!\cdots\!08\)\( T^{22} + \)\(52\!\cdots\!63\)\( T^{23} + \)\(33\!\cdots\!97\)\( T^{24} + \)\(88\!\cdots\!15\)\( T^{25} + \)\(46\!\cdots\!39\)\( T^{26} \)
$83$ \( 1 - 27 T + 934 T^{2} - 17473 T^{3} + 368115 T^{4} - 5500034 T^{5} + 89024957 T^{6} - 1128150732 T^{7} + 15245376228 T^{8} - 169102543191 T^{9} + 1986785395163 T^{10} - 19626628915027 T^{11} + 204813701011982 T^{12} - 1816049205194936 T^{13} + 16999537183994506 T^{14} - 135207846595621003 T^{15} + 1136018060744066281 T^{16} - 8025322776674842311 T^{17} + 60052156579918034604 T^{18} - \)\(36\!\cdots\!08\)\( T^{19} + \)\(24\!\cdots\!39\)\( T^{20} - \)\(12\!\cdots\!94\)\( T^{21} + \)\(68\!\cdots\!45\)\( T^{22} - \)\(27\!\cdots\!77\)\( T^{23} + \)\(12\!\cdots\!78\)\( T^{24} - \)\(28\!\cdots\!47\)\( T^{25} + \)\(88\!\cdots\!63\)\( T^{26} \)
$89$ \( 1 - 3 T + 525 T^{2} - 2921 T^{3} + 144772 T^{4} - 1059644 T^{5} + 28951997 T^{6} - 228043921 T^{7} + 4615052546 T^{8} - 35443806123 T^{9} + 597964680032 T^{10} - 4340614081579 T^{11} + 63785482146699 T^{12} - 429326946412826 T^{13} + 5676907911056211 T^{14} - 34382004140187259 T^{15} + 421546562517479008 T^{16} - 2223823825726541643 T^{17} + 25770727777122807154 T^{18} - \)\(11\!\cdots\!81\)\( T^{19} + \)\(12\!\cdots\!13\)\( T^{20} - \)\(41\!\cdots\!64\)\( T^{21} + \)\(50\!\cdots\!48\)\( T^{22} - \)\(91\!\cdots\!21\)\( T^{23} + \)\(14\!\cdots\!25\)\( T^{24} - \)\(74\!\cdots\!63\)\( T^{25} + \)\(21\!\cdots\!69\)\( T^{26} \)
$97$ \( 1 - 45 T + 1601 T^{2} - 38478 T^{3} + 793352 T^{4} - 13225379 T^{5} + 198615715 T^{6} - 2604983062 T^{7} + 32405292753 T^{8} - 374378589486 T^{9} + 4279452478888 T^{10} - 46557894756050 T^{11} + 497998439882051 T^{12} - 4986376893479044 T^{13} + 48305848668558947 T^{14} - 438063231759674450 T^{15} + 3905740732264147624 T^{16} - 33143467348989739566 T^{17} + \)\(27\!\cdots\!21\)\( T^{18} - \)\(21\!\cdots\!98\)\( T^{19} + \)\(16\!\cdots\!95\)\( T^{20} - \)\(10\!\cdots\!19\)\( T^{21} + \)\(60\!\cdots\!84\)\( T^{22} - \)\(28\!\cdots\!22\)\( T^{23} + \)\(11\!\cdots\!53\)\( T^{24} - \)\(31\!\cdots\!45\)\( T^{25} + \)\(67\!\cdots\!77\)\( T^{26} \)
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