Properties

Label 6012.2.a.k
Level $6012$
Weight $2$
Character orbit 6012.a
Self dual yes
Analytic conductor $48.006$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6012 = 2^{2} \cdot 3^{2} \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6012.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.0060616952\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \(x^{10} - 4 x^{9} - 26 x^{8} + 82 x^{7} + 211 x^{6} - 340 x^{5} - 593 x^{4} + 192 x^{3} + 423 x^{2} + 126 x + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + \beta_{3} ) q^{5} + \beta_{1} q^{7} +O(q^{10})\) \( q + ( 1 + \beta_{3} ) q^{5} + \beta_{1} q^{7} + ( \beta_{2} - \beta_{3} - \beta_{6} - \beta_{7} ) q^{11} -\beta_{9} q^{13} + ( -\beta_{3} + \beta_{5} + \beta_{7} - \beta_{9} ) q^{17} + ( -\beta_{1} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{19} + ( 3 - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{23} + ( 2 + 2 \beta_{2} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{25} + ( 1 - \beta_{2} - \beta_{6} + \beta_{7} + \beta_{8} ) q^{29} + ( -\beta_{1} + \beta_{2} + \beta_{6} - \beta_{8} - \beta_{9} ) q^{31} + ( -1 + 3 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{35} + ( -\beta_{1} - \beta_{4} - \beta_{7} + \beta_{9} ) q^{37} + ( -3 + \beta_{1} - \beta_{2} + 2 \beta_{4} + 2 \beta_{5} + \beta_{8} + \beta_{9} ) q^{41} + ( 2 - \beta_{1} - \beta_{2} + \beta_{5} - \beta_{6} + 2 \beta_{8} ) q^{43} + ( 5 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{7} + \beta_{9} ) q^{47} + ( 1 + \beta_{1} + \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{8} ) q^{49} + ( 3 - \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{8} - \beta_{9} ) q^{53} + ( -1 + 2 \beta_{1} - \beta_{2} + 4 \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} + 2 \beta_{8} + 2 \beta_{9} ) q^{55} + ( 1 - 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{9} ) q^{59} + ( -2 + 2 \beta_{1} + \beta_{2} + 2 \beta_{4} - \beta_{5} - 3 \beta_{7} - \beta_{8} + \beta_{9} ) q^{61} + ( 1 + 3 \beta_{1} + \beta_{3} + \beta_{4} - \beta_{7} + 2 \beta_{8} + 2 \beta_{9} ) q^{65} + ( 2 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{7} - \beta_{9} ) q^{67} + ( 4 - 3 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{8} - 2 \beta_{9} ) q^{71} + ( 2 + \beta_{1} + 2 \beta_{3} + \beta_{4} - 2 \beta_{7} - \beta_{8} + 2 \beta_{9} ) q^{73} + ( 2 + \beta_{1} + \beta_{2} - \beta_{4} + \beta_{6} - 2 \beta_{8} + \beta_{9} ) q^{77} + ( -\beta_{1} - \beta_{2} + \beta_{5} - \beta_{6} ) q^{79} + ( 4 + \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{83} + ( -5 + 2 \beta_{1} + \beta_{2} + 2 \beta_{4} + 2 \beta_{5} + \beta_{6} + 2 \beta_{7} + 3 \beta_{8} + \beta_{9} ) q^{85} + ( 1 - 3 \beta_{4} - 2 \beta_{5} + 3 \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{89} + ( 5 + 2 \beta_{3} - \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{91} + ( 2 - 4 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{4} + 2 \beta_{5} - 3 \beta_{6} + 2 \beta_{7} + 3 \beta_{8} ) q^{95} + ( -1 - 3 \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q + 6q^{5} + 4q^{7} + O(q^{10}) \) \( 10q + 6q^{5} + 4q^{7} + 8q^{11} - 2q^{13} + 6q^{17} + 20q^{23} + 24q^{25} + 8q^{29} - 4q^{31} - 4q^{37} - 14q^{41} + 20q^{43} + 48q^{47} - 2q^{49} + 22q^{53} - 6q^{55} + 2q^{59} - 8q^{61} + 28q^{65} - 6q^{67} + 20q^{71} + 20q^{73} + 24q^{77} - 4q^{79} + 46q^{83} - 18q^{85} - 8q^{89} + 28q^{91} + 36q^{95} - 34q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} - 4 x^{9} - 26 x^{8} + 82 x^{7} + 211 x^{6} - 340 x^{5} - 593 x^{4} + 192 x^{3} + 423 x^{2} + 126 x + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -1438 \nu^{9} + 6681 \nu^{8} + 32441 \nu^{7} - 136919 \nu^{6} - 197274 \nu^{5} + 584470 \nu^{4} + 319314 \nu^{3} - 469017 \nu^{2} + 48114 \nu + 66348 \)\()/44982\)
\(\beta_{3}\)\(=\)\((\)\( 1499 \nu^{9} - 6987 \nu^{8} - 34945 \nu^{7} + 148276 \nu^{6} + 234612 \nu^{5} - 710756 \nu^{4} - 561669 \nu^{3} + 837054 \nu^{2} + 435294 \nu - 151056 \)\()/44982\)
\(\beta_{4}\)\(=\)\((\)\( -2840 \nu^{9} + 9945 \nu^{8} + 81184 \nu^{7} - 204046 \nu^{6} - 752934 \nu^{5} + 832712 \nu^{4} + 2390865 \nu^{3} - 463230 \nu^{2} - 1880118 \nu - 239175 \)\()/44982\)
\(\beta_{5}\)\(=\)\((\)\( -1167 \nu^{9} + 5185 \nu^{8} + 27571 \nu^{7} - 105787 \nu^{6} - 188846 \nu^{5} + 438577 \nu^{4} + 436038 \nu^{3} - 257619 \nu^{2} - 276453 \nu - 92196 \)\()/14994\)
\(\beta_{6}\)\(=\)\((\)\( 426 \nu^{9} - 1700 \nu^{8} - 11178 \nu^{7} + 35355 \nu^{6} + 91532 \nu^{5} - 154395 \nu^{4} - 254713 \nu^{3} + 119881 \nu^{2} + 168813 \nu + 29004 \)\()/4998\)
\(\beta_{7}\)\(=\)\((\)\( 1849 \nu^{9} - 8415 \nu^{8} - 43331 \nu^{7} + 174848 \nu^{6} + 291942 \nu^{5} - 775366 \nu^{4} - 667089 \nu^{3} + 651792 \nu^{2} + 460620 \nu + 60750 \)\()/14994\)
\(\beta_{8}\)\(=\)\((\)\( 2419 \nu^{9} - 9622 \nu^{8} - 63438 \nu^{7} + 198654 \nu^{6} + 521801 \nu^{5} - 846911 \nu^{4} - 1490655 \nu^{3} + 595053 \nu^{2} + 1098198 \nu + 204129 \)\()/14994\)
\(\beta_{9}\)\(=\)\((\)\( 14219 \nu^{9} - 61455 \nu^{8} - 349141 \nu^{7} + 1274200 \nu^{6} + 2574117 \nu^{5} - 5573576 \nu^{4} - 6555282 \nu^{3} + 4419333 \nu^{2} + 4422843 \nu + 679113 \)\()/44982\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-2 \beta_{8} + \beta_{6} - \beta_{5} - 2 \beta_{4} + \beta_{3} + \beta_{1} + 8\)
\(\nu^{3}\)\(=\)\(3 \beta_{9} - 3 \beta_{8} - 6 \beta_{7} + \beta_{6} - 2 \beta_{5} + 2 \beta_{3} + \beta_{2} + 18 \beta_{1} + 7\)
\(\nu^{4}\)\(=\)\(5 \beta_{9} - 43 \beta_{8} - 11 \beta_{7} + 25 \beta_{6} - 26 \beta_{5} - 34 \beta_{4} + 21 \beta_{3} + 9 \beta_{2} + 38 \beta_{1} + 126\)
\(\nu^{5}\)\(=\)\(65 \beta_{9} - 93 \beta_{8} - 146 \beta_{7} + 46 \beta_{6} - 79 \beta_{5} - 18 \beta_{4} + 77 \beta_{3} + 41 \beta_{2} + 354 \beta_{1} + 226\)
\(\nu^{6}\)\(=\)\(138 \beta_{9} - 866 \beta_{8} - 381 \beta_{7} + 571 \beta_{6} - 625 \beta_{5} - 584 \beta_{4} + 520 \beta_{3} + 264 \beta_{2} + 1030 \beta_{1} + 2345\)
\(\nu^{7}\)\(=\)\(1227 \beta_{9} - 2433 \beta_{8} - 3207 \beta_{7} + 1585 \beta_{6} - 2366 \beta_{5} - 684 \beta_{4} + 2249 \beta_{3} + 1165 \beta_{2} + 7317 \beta_{1} + 6043\)
\(\nu^{8}\)\(=\)\(3143 \beta_{9} - 17755 \beta_{8} - 10508 \beta_{7} + 13075 \beta_{6} - 15044 \beta_{5} - 10450 \beta_{4} + 13002 \beta_{3} + 6621 \beta_{2} + 25514 \beta_{1} + 46794\)
\(\nu^{9}\)\(=\)\(22925 \beta_{9} - 59655 \beta_{8} - 70667 \beta_{7} + 45883 \beta_{6} - 63610 \beta_{5} - 19074 \beta_{4} + 59723 \beta_{3} + 30131 \beta_{2} + 156123 \beta_{1} + 149656\)

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
−0.291281
−0.108196
4.25727
−0.617293
2.09707
−1.19255
−3.81111
−2.20804
4.79792
1.07621
0 0 0 −3.41200 0 −0.291281 0 0 0
1.2 0 0 0 −3.17375 0 −0.108196 0 0 0
1.3 0 0 0 −1.39178 0 4.25727 0 0 0
1.4 0 0 0 −0.860385 0 −0.617293 0 0 0
1.5 0 0 0 0.399927 0 2.09707 0 0 0
1.6 0 0 0 0.553287 0 −1.19255 0 0 0
1.7 0 0 0 2.67502 0 −3.81111 0 0 0
1.8 0 0 0 3.55061 0 −2.20804 0 0 0
1.9 0 0 0 3.67303 0 4.79792 0 0 0
1.10 0 0 0 3.98604 0 1.07621 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(167\) \(1\)

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 6012.2.a.k yes 10
3.b odd 2 1 6012.2.a.j 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6012.2.a.j 10 3.b odd 2 1
6012.2.a.k yes 10 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{5}^{10} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6012))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 - 6 T + 31 T^{2} - 118 T^{3} + 420 T^{4} - 1262 T^{5} + 3754 T^{6} - 10022 T^{7} + 26368 T^{8} - 63276 T^{9} + 148879 T^{10} - 316380 T^{11} + 659200 T^{12} - 1252750 T^{13} + 2346250 T^{14} - 3943750 T^{15} + 6562500 T^{16} - 9218750 T^{17} + 12109375 T^{18} - 11718750 T^{19} + 9765625 T^{20} \)
$7$ \( 1 - 4 T + 44 T^{2} - 170 T^{3} + 960 T^{4} - 3378 T^{5} + 13757 T^{6} - 42578 T^{7} + 143706 T^{8} - 388136 T^{9} + 1144593 T^{10} - 2716952 T^{11} + 7041594 T^{12} - 14604254 T^{13} + 33030557 T^{14} - 56774046 T^{15} + 112943040 T^{16} - 140002310 T^{17} + 253651244 T^{18} - 161414428 T^{19} + 282475249 T^{20} \)
$11$ \( 1 - 8 T + 73 T^{2} - 368 T^{3} + 2114 T^{4} - 9084 T^{5} + 43602 T^{6} - 170180 T^{7} + 687924 T^{8} - 2358932 T^{9} + 8343787 T^{10} - 25948252 T^{11} + 83238804 T^{12} - 226509580 T^{13} + 638376882 T^{14} - 1462987284 T^{15} + 3745079954 T^{16} - 7171278928 T^{17} + 15648198313 T^{18} - 18863581528 T^{19} + 25937424601 T^{20} \)
$13$ \( 1 + 2 T + 72 T^{2} + 186 T^{3} + 2796 T^{4} + 7554 T^{5} + 74088 T^{6} + 192782 T^{7} + 1444053 T^{8} + 3441600 T^{9} + 21428349 T^{10} + 44740800 T^{11} + 244044957 T^{12} + 423542054 T^{13} + 2116027368 T^{14} + 2804747322 T^{15} + 13495757964 T^{16} + 11671224162 T^{17} + 58732611912 T^{18} + 21208998746 T^{19} + 137858491849 T^{20} \)
$17$ \( 1 - 6 T + 73 T^{2} - 278 T^{3} + 2415 T^{4} - 5388 T^{5} + 43190 T^{6} - 6396 T^{7} + 466109 T^{8} + 1601566 T^{9} + 4603487 T^{10} + 27226622 T^{11} + 134705501 T^{12} - 31423548 T^{13} + 3607271990 T^{14} - 7650189516 T^{15} + 58292229135 T^{16} - 114074151094 T^{17} + 509230293193 T^{18} - 711527258982 T^{19} + 2015993900449 T^{20} \)
$19$ \( 1 + 72 T^{2} + 2510 T^{4} + 642 T^{5} + 57562 T^{6} + 39308 T^{7} + 1006725 T^{8} + 1125458 T^{9} + 17251013 T^{10} + 21383702 T^{11} + 363427725 T^{12} + 269613572 T^{13} + 7501537402 T^{14} + 1589655558 T^{15} + 118085161310 T^{16} + 1222816538952 T^{18} + 6131066257801 T^{20} \)
$23$ \( 1 - 20 T + 327 T^{2} - 3770 T^{3} + 37980 T^{4} - 319026 T^{5} + 2419166 T^{6} - 16113194 T^{7} + 98396320 T^{8} - 538676186 T^{9} + 2720812683 T^{10} - 12389552278 T^{11} + 52051653280 T^{12} - 196049231398 T^{13} + 676981832606 T^{14} - 2053360761918 T^{15} + 5622403064220 T^{16} - 12836191935190 T^{17} + 25607692186887 T^{18} - 36023053229260 T^{19} + 41426511213649 T^{20} \)
$29$ \( 1 - 8 T + 191 T^{2} - 1228 T^{3} + 16872 T^{4} - 92576 T^{5} + 948244 T^{6} - 4588668 T^{7} + 38877162 T^{8} - 168989592 T^{9} + 1253952287 T^{10} - 4900698168 T^{11} + 32695693242 T^{12} - 111913023852 T^{13} + 670674964564 T^{14} - 1898840129824 T^{15} + 10035859071912 T^{16} - 21182848107452 T^{17} + 95547064875551 T^{18} - 116057167806952 T^{19} + 420707233300201 T^{20} \)
$31$ \( 1 + 4 T + 142 T^{2} + 458 T^{3} + 11005 T^{4} + 30384 T^{5} + 601849 T^{6} + 1434880 T^{7} + 25562280 T^{8} + 54307126 T^{9} + 878289431 T^{10} + 1683520906 T^{11} + 24565351080 T^{12} + 42746510080 T^{13} + 555820190329 T^{14} + 869868123984 T^{15} + 9766978009405 T^{16} + 12600777262838 T^{17} + 121110527316622 T^{18} + 105758488642684 T^{19} + 819628286980801 T^{20} \)
$37$ \( 1 + 4 T + 203 T^{2} + 854 T^{3} + 22808 T^{4} + 91856 T^{5} + 1727746 T^{6} + 6489918 T^{7} + 96059522 T^{8} + 326243728 T^{9} + 4058751253 T^{10} + 12071017936 T^{11} + 131505485618 T^{12} + 328733816454 T^{13} + 3238074171106 T^{14} + 6369658514192 T^{15} + 58519087936472 T^{16} + 81071823071582 T^{17} + 713033329145963 T^{18} + 519846959180308 T^{19} + 4808584372417849 T^{20} \)
$41$ \( 1 + 14 T + 290 T^{2} + 2856 T^{3} + 35413 T^{4} + 279742 T^{5} + 2727407 T^{6} + 18824774 T^{7} + 159158784 T^{8} + 986229368 T^{9} + 7356881549 T^{10} + 40435404088 T^{11} + 267545915904 T^{12} + 1297422248854 T^{13} + 7707000331727 T^{14} + 32409845380142 T^{15} + 168215441486533 T^{16} + 556218206204136 T^{17} + 2315628316445090 T^{18} + 4583347081515454 T^{19} + 13422659310152401 T^{20} \)
$43$ \( 1 - 20 T + 424 T^{2} - 5856 T^{3} + 77619 T^{4} - 832522 T^{5} + 8433959 T^{6} - 74018054 T^{7} + 612099082 T^{8} - 4500726094 T^{9} + 31190623967 T^{10} - 193531222042 T^{11} + 1131771202618 T^{12} - 5884953419378 T^{13} + 28834027463159 T^{14} - 122387762983246 T^{15} + 490657878500331 T^{16} - 1591769786642592 T^{17} + 4955796917702824 T^{18} - 10051852238736860 T^{19} + 21611482313284249 T^{20} \)
$47$ \( 1 - 48 T + 1383 T^{2} - 28760 T^{3} + 477360 T^{4} - 6596704 T^{5} + 78236101 T^{6} - 809345700 T^{7} + 7394658901 T^{8} - 60054409196 T^{9} + 435533508747 T^{10} - 2822557232212 T^{11} + 16334801512309 T^{12} - 84028698611100 T^{13} + 381767215563781 T^{14} - 1512921125056928 T^{15} + 5145566229451440 T^{16} - 14570480944515880 T^{17} + 32931009453215463 T^{18} - 53718262708932816 T^{19} + 52599132235830049 T^{20} \)
$53$ \( 1 - 22 T + 551 T^{2} - 8384 T^{3} + 126720 T^{4} - 1499280 T^{5} + 17111347 T^{6} - 165696154 T^{7} + 1535860765 T^{8} - 12456885392 T^{9} + 96538460331 T^{10} - 660214925776 T^{11} + 4314232888885 T^{12} - 24668346319058 T^{13} + 135016758387907 T^{14} - 626992138745040 T^{15} + 2808667842266880 T^{16} - 9848778196393408 T^{17} + 34305089416659911 T^{18} - 72594799019646926 T^{19} + 174887470365513049 T^{20} \)
$59$ \( 1 - 2 T + 315 T^{2} - 360 T^{3} + 44416 T^{4} - 12072 T^{5} + 3832523 T^{6} + 2142498 T^{7} + 245635973 T^{8} + 288960572 T^{9} + 14343462495 T^{10} + 17048673748 T^{11} + 855058822013 T^{12} + 440024096742 T^{13} + 46440064731803 T^{14} - 8630566137528 T^{15} + 1873490582198656 T^{16} - 895914534534840 T^{17} + 46251587845361115 T^{18} - 17325991637309878 T^{19} + 511116753300641401 T^{20} \)
$61$ \( 1 + 8 T + 256 T^{2} + 2016 T^{3} + 36630 T^{4} + 291138 T^{5} + 3934246 T^{6} + 29798532 T^{7} + 333755353 T^{8} + 2290787898 T^{9} + 22604219961 T^{10} + 139738061778 T^{11} + 1241903668513 T^{12} + 6763700591892 T^{13} + 54472944570886 T^{14} + 245894077880538 T^{15} + 1887191312843430 T^{16} + 6335769557418336 T^{17} + 49077072127303936 T^{18} + 93553168742673128 T^{19} + 713342911662882601 T^{20} \)
$67$ \( 1 + 6 T + 356 T^{2} + 1810 T^{3} + 68636 T^{4} + 318454 T^{5} + 9047396 T^{6} + 37814198 T^{7} + 885832585 T^{8} + 3336093832 T^{9} + 67206009417 T^{10} + 223518286744 T^{11} + 3976502474065 T^{12} + 11373111633074 T^{13} + 182315171530916 T^{14} + 429952740824578 T^{15} + 6208701518551484 T^{16} + 10969888005634630 T^{17} + 144560093210164196 T^{18} + 163239206377769682 T^{19} + 1822837804551761449 T^{20} \)
$71$ \( 1 - 20 T + 413 T^{2} - 5416 T^{3} + 78441 T^{4} - 862990 T^{5} + 10116693 T^{6} - 96430862 T^{7} + 987629625 T^{8} - 8457342282 T^{9} + 77624229705 T^{10} - 600471302022 T^{11} + 4978640939625 T^{12} - 34513666249282 T^{13} + 257082175290933 T^{14} - 1557031887619490 T^{15} + 10048314371047161 T^{16} - 49259170777845656 T^{17} + 266696208404499293 T^{18} - 916970014368980620 T^{19} + 3255243551009881201 T^{20} \)
$73$ \( 1 - 20 T + 652 T^{2} - 9490 T^{3} + 180120 T^{4} - 2079332 T^{5} + 29480734 T^{6} - 283788426 T^{7} + 3311081589 T^{8} - 27471524666 T^{9} + 276476108261 T^{10} - 2005421300618 T^{11} + 17644753787781 T^{12} - 110398522117242 T^{13} + 837200988988894 T^{14} - 4310604101615876 T^{15} + 27258320839174680 T^{16} - 104839811946230530 T^{17} + 525811979914940812 T^{18} - 1177431734165358260 T^{19} + 4297625829703557649 T^{20} \)
$79$ \( 1 + 4 T + 684 T^{2} + 2424 T^{3} + 216539 T^{4} + 676826 T^{5} + 41907971 T^{6} + 114663634 T^{7} + 5502155218 T^{8} + 13018598954 T^{9} + 512639976643 T^{10} + 1028469317366 T^{11} + 34338950715538 T^{12} + 56533643443726 T^{13} + 1632318864995651 T^{14} + 2082631774309574 T^{15} + 52637914531061819 T^{16} + 46550275382449416 T^{17} + 1037702425976087724 T^{18} + 479406383930473276 T^{19} + 9468276082626847201 T^{20} \)
$83$ \( 1 - 46 T + 1627 T^{2} - 39946 T^{3} + 835088 T^{4} - 14345466 T^{5} + 218449939 T^{6} - 2885544604 T^{7} + 34445382845 T^{8} - 364612104940 T^{9} + 3516567220763 T^{10} - 30262804710020 T^{11} + 237294242419205 T^{12} - 1649916892487348 T^{13} + 10367267327492419 T^{14} - 56507373616774638 T^{15} + 273023982515971472 T^{16} - 1083976692831640142 T^{17} + 3664479461690219707 T^{18} - 8599251742306858538 T^{19} + 15516041187205853449 T^{20} \)
$89$ \( 1 + 8 T + 428 T^{2} + 3052 T^{3} + 101630 T^{4} + 660200 T^{5} + 17030276 T^{6} + 100257152 T^{7} + 2164606323 T^{8} + 11514510644 T^{9} + 216271050931 T^{10} + 1024791447316 T^{11} + 17145846684483 T^{12} + 70678184188288 T^{13} + 1068517681088516 T^{14} + 3686596048229800 T^{15} + 50508208600366430 T^{16} + 134994034101154508 T^{17} + 1684860008840490668 T^{18} + 2802851229659881672 T^{19} + 31181719929966183601 T^{20} \)
$97$ \( 1 + 34 T + 919 T^{2} + 17748 T^{3} + 313381 T^{4} + 4683554 T^{5} + 65950679 T^{6} + 827454592 T^{7} + 9878651509 T^{8} + 106806253562 T^{9} + 1101981063735 T^{10} + 10360206595514 T^{11} + 92948232048181 T^{12} + 755195464844416 T^{13} + 5838566193331799 T^{14} + 40219271810033378 T^{15} + 261037599876654949 T^{16} + 1434007952917549524 T^{17} + 7202601473232427159 T^{18} + 25847855994255217378 T^{19} + 73742412689492826049 T^{20} \)
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