Properties

Label 6020.2.a.j
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: \(x^{13} - x^{12} - 28 x^{11} + 26 x^{10} + 286 x^{9} - 235 x^{8} - 1298 x^{7} + 895 x^{6} + 2571 x^{5} - 1548 x^{4} - 1833 x^{3} + 1119 x^{2} + 342 x - 216\)
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_{3} q^{11} -\beta_{11} q^{13} + \beta_{1} q^{15} + ( -1 - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{10} ) q^{17} -\beta_{12} q^{19} + \beta_{1} q^{21} + ( -1 + \beta_{3} + \beta_{4} - \beta_{7} ) q^{23} + q^{25} + ( -1 - \beta_{1} + \beta_{3} - \beta_{7} + \beta_{8} + \beta_{10} - \beta_{11} ) q^{27} + ( 1 + \beta_{2} + \beta_{4} - \beta_{7} ) q^{29} + ( -1 - \beta_{1} - \beta_{6} + \beta_{7} - \beta_{8} ) q^{31} + ( 1 - 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{8} - \beta_{9} ) q^{33} + q^{35} + ( 1 - \beta_{2} + \beta_{7} - \beta_{8} - \beta_{10} + \beta_{12} ) q^{37} + ( 1 + \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{6} + \beta_{9} + \beta_{10} ) q^{39} + ( -\beta_{3} + \beta_{5} + \beta_{8} + \beta_{11} - \beta_{12} ) q^{41} - q^{43} + ( -1 - \beta_{2} ) q^{45} + ( -\beta_{1} - \beta_{2} + \beta_{5} + \beta_{6} ) q^{47} + q^{49} + ( 1 - 2 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} - 2 \beta_{8} + \beta_{9} ) q^{51} + ( \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} - \beta_{10} + \beta_{11} + \beta_{12} ) q^{53} -\beta_{3} q^{55} + ( 1 - \beta_{1} - \beta_{4} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{11} + \beta_{12} ) q^{57} + ( 1 - 3 \beta_{1} - \beta_{3} - \beta_{4} - \beta_{6} + 2 \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} ) q^{59} + ( \beta_{3} + \beta_{4} + \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} ) q^{61} + ( -1 - \beta_{2} ) q^{63} + \beta_{11} q^{65} + ( -\beta_{1} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{8} + \beta_{10} + \beta_{11} - \beta_{12} ) q^{67} + ( 2 - \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{5} + \beta_{8} - 2 \beta_{9} + \beta_{11} ) q^{69} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{10} + \beta_{12} ) q^{71} + ( 1 - \beta_{1} + \beta_{6} - \beta_{8} + \beta_{9} - \beta_{10} ) q^{73} -\beta_{1} q^{75} -\beta_{3} q^{77} + ( 2 - 2 \beta_{1} - \beta_{3} + \beta_{7} + \beta_{8} - \beta_{9} + \beta_{12} ) q^{79} + ( 2 + \beta_{2} + \beta_{3} + \beta_{6} - \beta_{7} + \beta_{10} ) q^{81} + ( -\beta_{1} - \beta_{7} - \beta_{8} - \beta_{12} ) q^{83} + ( 1 + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{10} ) q^{85} + ( -4 \beta_{1} + \beta_{4} + \beta_{5} - \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} ) q^{87} + ( 2 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} - \beta_{12} ) q^{89} + \beta_{11} q^{91} + ( 1 + \beta_{1} - \beta_{3} - \beta_{5} - \beta_{6} + \beta_{8} + \beta_{9} - \beta_{12} ) q^{93} + \beta_{12} q^{95} + ( 3 - 2 \beta_{1} - \beta_{2} + \beta_{7} - 2 \beta_{8} - \beta_{11} ) q^{97} + ( 4 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + \beta_{5} + 2 \beta_{6} - 3 \beta_{7} + 2 \beta_{9} + 2 \beta_{10} - \beta_{11} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13q - q^{3} - 13q^{5} - 13q^{7} + 18q^{9} + O(q^{10}) \) \( 13q - q^{3} - 13q^{5} - 13q^{7} + 18q^{9} + 6q^{11} + q^{13} + q^{15} - 6q^{17} + 2q^{19} + q^{21} - 6q^{23} + 13q^{25} - q^{27} + 19q^{29} - 24q^{31} + 17q^{33} + 13q^{35} + 15q^{39} + 4q^{41} - 13q^{43} - 18q^{45} - 3q^{47} + 13q^{49} + 7q^{51} - 5q^{53} - 6q^{55} + 6q^{57} - 6q^{59} + 3q^{61} - 18q^{63} - q^{65} - 2q^{67} + 20q^{69} + 18q^{71} + 14q^{73} - q^{75} - 6q^{77} + 12q^{79} + 37q^{81} + 2q^{83} + 6q^{85} - 2q^{87} + 17q^{89} - q^{91} + 15q^{93} - 2q^{95} + 17q^{97} + 80q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{13} - x^{12} - 28 x^{11} + 26 x^{10} + 286 x^{9} - 235 x^{8} - 1298 x^{7} + 895 x^{6} + 2571 x^{5} - 1548 x^{4} - 1833 x^{3} + 1119 x^{2} + 342 x - 216\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 4 \)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{12} - \nu^{11} + 28 \nu^{10} + 30 \nu^{9} - 282 \nu^{8} - 333 \nu^{7} + 1200 \nu^{6} + 1603 \nu^{5} - 1845 \nu^{4} - 2868 \nu^{3} + 297 \nu^{2} + 1011 \nu + 108 \)\()/18\)
\(\beta_{4}\)\(=\)\((\)\( 3 \nu^{12} - \nu^{11} - 87 \nu^{10} + 22 \nu^{9} + 929 \nu^{8} - 140 \nu^{7} - 4450 \nu^{6} + 193 \nu^{5} + 9321 \nu^{4} + 87 \nu^{3} - 7113 \nu^{2} + 102 \nu + 1512 \)\()/18\)
\(\beta_{5}\)\(=\)\((\)\( -25 \nu^{12} + 10 \nu^{11} + 715 \nu^{10} - 221 \nu^{9} - 7486 \nu^{8} + 1414 \nu^{7} + 34850 \nu^{6} - 1915 \nu^{5} - 69996 \nu^{4} - 1467 \nu^{3} + 49848 \nu^{2} - 318 \nu - 10242 \)\()/54\)
\(\beta_{6}\)\(=\)\((\)\( 41 \nu^{12} - 8 \nu^{11} - 1169 \nu^{10} + 151 \nu^{9} + 12194 \nu^{8} - 446 \nu^{7} - 56314 \nu^{6} - 3751 \nu^{5} + 110382 \nu^{4} + 12177 \nu^{3} - 72930 \nu^{2} - 2532 \nu + 13626 \)\()/54\)
\(\beta_{7}\)\(=\)\((\)\( -53 \nu^{12} + 17 \nu^{11} + 1487 \nu^{10} - 352 \nu^{9} - 15197 \nu^{8} + 1748 \nu^{7} + 68440 \nu^{6} + 2059 \nu^{5} - 130623 \nu^{4} - 14913 \nu^{3} + 83847 \nu^{2} + 4044 \nu - 15462 \)\()/54\)
\(\beta_{8}\)\(=\)\((\)\( -20 \nu^{12} + 6 \nu^{11} + 563 \nu^{10} - 128 \nu^{9} - 5785 \nu^{8} + 691 \nu^{7} + 26288 \nu^{6} + 246 \nu^{5} - 50976 \nu^{4} - 4119 \nu^{3} + 33882 \nu^{2} + 819 \nu - 6426 \)\()/18\)
\(\beta_{9}\)\(=\)\((\)\( -12 \nu^{12} + 4 \nu^{11} + 339 \nu^{10} - 88 \nu^{9} - 3500 \nu^{8} + 533 \nu^{7} + 16018 \nu^{6} - 412 \nu^{5} - 31434 \nu^{4} - 1437 \nu^{3} + 21423 \nu^{2} + 60 \nu - 4158 \)\()/9\)
\(\beta_{10}\)\(=\)\((\)\( -91 \nu^{12} + 28 \nu^{11} + 2572 \nu^{10} - 593 \nu^{9} - 26545 \nu^{8} + 3193 \nu^{7} + 121154 \nu^{6} + 1001 \nu^{5} - 235416 \nu^{4} - 18486 \nu^{3} + 155346 \nu^{2} + 3543 \nu - 28818 \)\()/54\)
\(\beta_{11}\)\(=\)\((\)\( -101 \nu^{12} + 26 \nu^{11} + 2858 \nu^{10} - 535 \nu^{9} - 29549 \nu^{8} + 2519 \nu^{7} + 135178 \nu^{6} + 4489 \nu^{5} - 263256 \nu^{4} - 24480 \nu^{3} + 174036 \nu^{2} + 4611 \nu - 32364 \)\()/54\)
\(\beta_{12}\)\(=\)\((\)\( -86 \nu^{12} + 29 \nu^{11} + 2429 \nu^{10} - 622 \nu^{9} - 25043 \nu^{8} + 3530 \nu^{7} + 114169 \nu^{6} - 770 \nu^{5} - 221874 \nu^{4} - 15138 \nu^{3} + 147243 \nu^{2} + 2226 \nu - 27612 \)\()/27\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{11} - \beta_{10} - \beta_{8} + \beta_{7} - \beta_{3} + 7 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(\beta_{10} - \beta_{7} + \beta_{6} + \beta_{3} + 10 \beta_{2} + 29\)
\(\nu^{5}\)\(=\)\(-\beta_{12} + 11 \beta_{11} - 10 \beta_{10} + \beta_{9} - 12 \beta_{8} + 12 \beta_{7} + \beta_{5} + \beta_{4} - 11 \beta_{3} - 2 \beta_{2} + 58 \beta_{1} + 11\)
\(\nu^{6}\)\(=\)\(-\beta_{11} + 14 \beta_{10} + \beta_{9} + \beta_{8} - 15 \beta_{7} + 14 \beta_{6} + \beta_{5} + \beta_{4} + 16 \beta_{3} + 92 \beta_{2} - 4 \beta_{1} + 241\)
\(\nu^{7}\)\(=\)\(-13 \beta_{12} + 103 \beta_{11} - 92 \beta_{10} + 19 \beta_{9} - 122 \beta_{8} + 118 \beta_{7} - \beta_{6} + 12 \beta_{5} + 15 \beta_{4} - 101 \beta_{3} - 36 \beta_{2} + 512 \beta_{1} + 81\)
\(\nu^{8}\)\(=\)\(-22 \beta_{11} + 158 \beta_{10} + 19 \beta_{9} + 22 \beta_{8} - 178 \beta_{7} + 154 \beta_{6} + 22 \beta_{5} + 26 \beta_{4} + 196 \beta_{3} + 837 \beta_{2} - 91 \beta_{1} + 2100\)
\(\nu^{9}\)\(=\)\(-132 \beta_{12} + 927 \beta_{11} - 840 \beta_{10} + 257 \beta_{9} - 1188 \beta_{8} + 1109 \beta_{7} - 24 \beta_{6} + 107 \beta_{5} + 164 \beta_{4} - 889 \beta_{3} - 483 \beta_{2} + 4637 \beta_{1} + 448\)
\(\nu^{10}\)\(=\)\(-\beta_{12} - 320 \beta_{11} + 1667 \beta_{10} + 240 \beta_{9} + 337 \beta_{8} - 1947 \beta_{7} + 1577 \beta_{6} + 334 \beta_{5} + 413 \beta_{4} + 2170 \beta_{3} + 7619 \beta_{2} - 1403 \beta_{1} + 18694\)
\(\nu^{11}\)\(=\)\(-1244 \beta_{12} + 8249 \beta_{11} - 7699 \beta_{10} + 3038 \beta_{9} - 11410 \beta_{8} + 10336 \beta_{7} - 403 \beta_{6} + 823 \beta_{5} + 1584 \beta_{4} - 7786 \beta_{3} - 5761 \beta_{2} + 42466 \beta_{1} + 1133\)
\(\nu^{12}\)\(=\)\(-18 \beta_{12} - 3929 \beta_{11} + 17048 \beta_{10} + 2510 \beta_{9} + 4460 \beta_{8} - 20467 \beta_{7} + 15699 \beta_{6} + 4342 \beta_{5} + 5376 \beta_{4} + 22809 \beta_{3} + 69598 \beta_{2} - 18365 \beta_{1} + 168322\)

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
2.98613
2.95506
2.78565
1.72520
0.700922
0.669559
0.612150
−0.538302
−0.999095
−1.84160
−1.87398
−3.05719
−3.12450
0 −2.98613 0 −1.00000 0 −1.00000 0 5.91695 0
1.2 0 −2.95506 0 −1.00000 0 −1.00000 0 5.73239 0
1.3 0 −2.78565 0 −1.00000 0 −1.00000 0 4.75984 0
1.4 0 −1.72520 0 −1.00000 0 −1.00000 0 −0.0236759 0
1.5 0 −0.700922 0 −1.00000 0 −1.00000 0 −2.50871 0
1.6 0 −0.669559 0 −1.00000 0 −1.00000 0 −2.55169 0
1.7 0 −0.612150 0 −1.00000 0 −1.00000 0 −2.62527 0
1.8 0 0.538302 0 −1.00000 0 −1.00000 0 −2.71023 0
1.9 0 0.999095 0 −1.00000 0 −1.00000 0 −2.00181 0
1.10 0 1.84160 0 −1.00000 0 −1.00000 0 0.391491 0
1.11 0 1.87398 0 −1.00000 0 −1.00000 0 0.511819 0
1.12 0 3.05719 0 −1.00000 0 −1.00000 0 6.34642 0
1.13 0 3.12450 0 −1.00000 0 −1.00000 0 6.76247 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.j 13
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6020.2.a.j 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
$3$ \( 1 + T + 11 T^{2} + 10 T^{3} + 64 T^{4} + 49 T^{5} + 286 T^{6} + 155 T^{7} + 1152 T^{8} + 513 T^{9} + 4359 T^{10} + 2148 T^{11} + 15057 T^{12} + 7704 T^{13} + 45171 T^{14} + 19332 T^{15} + 117693 T^{16} + 41553 T^{17} + 279936 T^{18} + 112995 T^{19} + 625482 T^{20} + 321489 T^{21} + 1259712 T^{22} + 590490 T^{23} + 1948617 T^{24} + 531441 T^{25} + 1594323 T^{26} \)
$5$ \( ( 1 + T )^{13} \)
$7$ \( ( 1 + T )^{13} \)
$11$ \( 1 - 6 T + 59 T^{2} - 248 T^{3} + 1581 T^{4} - 5720 T^{5} + 30871 T^{6} - 107293 T^{7} + 530010 T^{8} - 1762108 T^{9} + 7801286 T^{10} - 23734787 T^{11} + 96516880 T^{12} - 274829644 T^{13} + 1061685680 T^{14} - 2871909227 T^{15} + 10383511666 T^{16} - 25799023228 T^{17} + 85358640510 T^{18} - 190076094373 T^{19} + 601588455941 T^{20} - 1226132799320 T^{21} + 3727915299471 T^{22} - 6432481301048 T^{23} + 16833388566049 T^{24} - 18830570260326 T^{25} + 34522712143931 T^{26} \)
$13$ \( 1 - T + 42 T^{2} - 119 T^{3} + 1209 T^{4} - 4461 T^{5} + 28600 T^{6} - 113623 T^{7} + 581813 T^{8} - 2188827 T^{9} + 10323689 T^{10} - 35052724 T^{11} + 157110065 T^{12} - 486080046 T^{13} + 2042430845 T^{14} - 5923910356 T^{15} + 22681144733 T^{16} - 62515087947 T^{17} + 216023094209 T^{18} - 548436519007 T^{19} + 1794607586200 T^{20} - 3638974746381 T^{21} + 12820839741957 T^{22} - 16405160530031 T^{23} + 75270736549554 T^{24} - 23298085122481 T^{25} + 302875106592253 T^{26} \)
$17$ \( 1 + 6 T + 46 T^{2} + 118 T^{3} + 86 T^{4} - 2790 T^{5} - 13459 T^{6} - 16501 T^{7} + 265111 T^{8} + 2044419 T^{9} + 8347780 T^{10} + 5550836 T^{11} - 75048454 T^{12} - 641333272 T^{13} - 1275823718 T^{14} + 1604191604 T^{15} + 41012643140 T^{16} + 170751919299 T^{17} + 376419709127 T^{18} - 398294026069 T^{19} - 5522748199907 T^{20} - 19462363260390 T^{21} + 10198557378742 T^{22} + 237887280252982 T^{23} + 1576507230151118 T^{24} + 3495733423378566 T^{25} + 9904578032905937 T^{26} \)
$19$ \( 1 - 2 T + 121 T^{2} - 275 T^{3} + 7377 T^{4} - 18137 T^{5} + 309743 T^{6} - 784786 T^{7} + 10136193 T^{8} - 25265048 T^{9} + 272317833 T^{10} - 643724907 T^{11} + 6130202978 T^{12} - 13431972858 T^{13} + 116473856582 T^{14} - 232384691427 T^{15} + 1867828016547 T^{16} - 3292566320408 T^{17} + 25098217351107 T^{18} - 36920948766466 T^{19} + 276870514053077 T^{20} - 308030882874617 T^{21} + 2380467146515683 T^{22} - 1686043220895275 T^{23} + 14095321326684499 T^{24} - 4426629838132322 T^{25} + 42052983462257059 T^{26} \)
$23$ \( 1 + 6 T + 160 T^{2} + 863 T^{3} + 12731 T^{4} + 62527 T^{5} + 670148 T^{6} + 3066700 T^{7} + 26586161 T^{8} + 115095928 T^{9} + 854227773 T^{10} + 3502704669 T^{11} + 23066429419 T^{12} + 88301270678 T^{13} + 530527876637 T^{14} + 1852930769901 T^{15} + 10393389314091 T^{16} + 32208559587448 T^{17} + 171117651249223 T^{18} + 453981660796300 T^{19} + 2281736963656156 T^{20} + 4896550976665087 T^{21} + 22930474533085453 T^{22} + 35751079177379087 T^{23} + 152449561266228320 T^{24} + 131487746592121926 T^{25} + 504036361936467383 T^{26} \)
$29$ \( 1 - 19 T + 345 T^{2} - 4261 T^{3} + 47972 T^{4} - 457573 T^{5} + 4015220 T^{6} - 32000724 T^{7} + 237965729 T^{8} - 1649791767 T^{9} + 10790731529 T^{10} - 66417311655 T^{11} + 388361417932 T^{12} - 2143115560290 T^{13} + 11262481120028 T^{14} - 55856959101855 T^{15} + 263175151260781 T^{16} - 1166866370755527 T^{17} + 4880950524412621 T^{18} - 19034776924084404 T^{19} + 69262048353422980 T^{20} - 228899251917803653 T^{21} + 695936806754387668 T^{22} - 1792633521092156461 T^{23} + 4209175869168511005 T^{24} - 6722480880903911779 T^{25} + 10260628712958602189 T^{26} \)
$31$ \( 1 + 24 T + 431 T^{2} + 5537 T^{3} + 62031 T^{4} + 586029 T^{5} + 5058378 T^{6} + 38936718 T^{7} + 281366797 T^{8} + 1871006940 T^{9} + 11941699909 T^{10} + 71759580679 T^{11} + 421436130472 T^{12} + 2364158112576 T^{13} + 13064520044632 T^{14} + 68960957032519 T^{15} + 355755181989019 T^{16} + 1727914200235740 T^{17} + 8055292517699347 T^{18} + 34556480551058958 T^{19} + 139169201941571958 T^{20} + 499818881780511789 T^{21} + 1640076202248582801 T^{22} + 4538281825012695137 T^{23} + 10951053542350482161 T^{24} + 18903906810925194264 T^{25} + 24417546297445042591 T^{26} \)
$37$ \( 1 + 159 T^{2} - 65 T^{3} + 13687 T^{4} - 11035 T^{5} + 896027 T^{6} - 1047043 T^{7} + 48051734 T^{8} - 67756109 T^{9} + 2199084610 T^{10} - 3460205835 T^{11} + 89695119236 T^{12} - 144308592102 T^{13} + 3318719411732 T^{14} - 4737021788115 T^{15} + 111390232750330 T^{16} - 126985856999549 T^{17} + 3332097376271438 T^{18} - 2686425876458587 T^{19} + 85061525071850591 T^{20} - 38760210774018235 T^{21} + 1778786332575218899 T^{22} - 312557984207160185 T^{23} + 28288901862934205667 T^{24} + \)\(24\!\cdots\!97\)\( T^{26} \)
$41$ \( 1 - 4 T + 203 T^{2} - 615 T^{3} + 21892 T^{4} - 41473 T^{5} + 1617927 T^{6} - 1135628 T^{7} + 92776367 T^{8} + 43648960 T^{9} + 4488717943 T^{10} + 6188763195 T^{11} + 196227909099 T^{12} + 334886235642 T^{13} + 8045344273059 T^{14} + 10403310930795 T^{15} + 309366929349503 T^{16} + 123341528858560 T^{17} + 10748717423201767 T^{18} - 5394351378998348 T^{19} + 315098198077464687 T^{20} - 331158804027335233 T^{21} + 7167045307752594212 T^{22} - 8254935475743726615 T^{23} + \)\(11\!\cdots\!23\)\( T^{24} - 90253961201464744324 T^{25} + \)\(92\!\cdots\!21\)\( T^{26} \)
$43$ \( ( 1 + T )^{13} \)
$47$ \( 1 + 3 T + 257 T^{2} + 245 T^{3} + 31470 T^{4} - 25765 T^{5} + 2689464 T^{6} - 5592916 T^{7} + 191691947 T^{8} - 513903521 T^{9} + 11905217699 T^{10} - 34151915681 T^{11} + 640807014018 T^{12} - 1802610042490 T^{13} + 30117929658846 T^{14} - 75441581739329 T^{15} + 1236035417163277 T^{16} - 2507685247256801 T^{17} + 43963590926558629 T^{18} - 60287245881009364 T^{19} + 1362544644052901832 T^{20} - 613497800840272165 T^{21} + 35219035988544077490 T^{22} + 12886787397778362005 T^{23} + \)\(63\!\cdots\!71\)\( T^{24} + \)\(34\!\cdots\!23\)\( T^{25} + \)\(54\!\cdots\!27\)\( T^{26} \)
$53$ \( 1 + 5 T + 206 T^{2} + 1024 T^{3} + 28503 T^{4} + 117838 T^{5} + 2772923 T^{6} + 10154426 T^{7} + 219680255 T^{8} + 684592497 T^{9} + 14674399440 T^{10} + 40669131910 T^{11} + 864447325828 T^{12} + 2198187112888 T^{13} + 45815708268884 T^{14} + 114239591535190 T^{15} + 2184680565428880 T^{16} + 5401764090321057 T^{17} + 91869292542090715 T^{18} + 225066364921706954 T^{19} + 3257383538010233551 T^{20} + 7336557398693957518 T^{21} + 94053161657136196899 T^{22} + \)\(17\!\cdots\!76\)\( T^{23} + \)\(19\!\cdots\!82\)\( T^{24} + \)\(24\!\cdots\!05\)\( T^{25} + \)\(26\!\cdots\!73\)\( T^{26} \)
$59$ \( 1 + 6 T + 370 T^{2} + 2850 T^{3} + 74993 T^{4} + 611581 T^{5} + 10790567 T^{6} + 85030264 T^{7} + 1187320672 T^{8} + 8776364752 T^{9} + 103929218590 T^{10} + 711731131051 T^{11} + 7425858186431 T^{12} + 46556314342528 T^{13} + 438125632999429 T^{14} + 2477536067188531 T^{15} + 21344878984795610 T^{16} + 106346379967659472 T^{17} + 848844399117808928 T^{18} + 3586621911155111224 T^{19} + 26853960586588902373 T^{20} + 89798705860488241501 T^{21} + \)\(64\!\cdots\!27\)\( T^{22} + \)\(14\!\cdots\!50\)\( T^{23} + \)\(11\!\cdots\!30\)\( T^{24} + \)\(10\!\cdots\!86\)\( T^{25} + \)\(10\!\cdots\!79\)\( T^{26} \)
$61$ \( 1 - 3 T + 359 T^{2} - 1420 T^{3} + 65695 T^{4} - 276825 T^{5} + 8404468 T^{6} - 34019353 T^{7} + 849825192 T^{8} - 3234866265 T^{9} + 71604144077 T^{10} - 259411482242 T^{11} + 5118625799312 T^{12} - 17444783659600 T^{13} + 312236173758032 T^{14} - 965270125422482 T^{15} + 16252780226741537 T^{16} - 44789443961453865 T^{17} + 717759213659814792 T^{18} - 1752689802079008433 T^{19} + 26413081597567741828 T^{20} - 53069376920472312825 T^{21} + \)\(76\!\cdots\!95\)\( T^{22} - \)\(10\!\cdots\!20\)\( T^{23} + \)\(15\!\cdots\!99\)\( T^{24} - \)\(79\!\cdots\!63\)\( T^{25} + \)\(16\!\cdots\!81\)\( T^{26} \)
$67$ \( 1 + 2 T + 430 T^{2} + 1019 T^{3} + 100178 T^{4} + 235386 T^{5} + 16162269 T^{6} + 36145792 T^{7} + 1989343179 T^{8} + 4148737066 T^{9} + 195949441626 T^{10} + 377692593386 T^{11} + 15834281128678 T^{12} + 27942033674878 T^{13} + 1060896835621426 T^{14} + 1695462051709754 T^{15} + 58934341911760638 T^{16} + 83601702614150986 T^{17} + 2685862172407095153 T^{18} + 3269689866537182848 T^{19} + 97954851296652157887 T^{20} + 95582646349347498426 T^{21} + \)\(27\!\cdots\!66\)\( T^{22} + \)\(18\!\cdots\!31\)\( T^{23} + \)\(52\!\cdots\!90\)\( T^{24} + \)\(16\!\cdots\!22\)\( T^{25} + \)\(54\!\cdots\!87\)\( T^{26} \)
$71$ \( 1 - 18 T + 690 T^{2} - 11141 T^{3} + 232704 T^{4} - 3320847 T^{5} + 50625628 T^{6} - 633484607 T^{7} + 7892310806 T^{8} - 86501912601 T^{9} + 927301022969 T^{10} - 8934033500155 T^{11} + 84233671633266 T^{12} - 716796274886102 T^{13} + 5980590685961886 T^{14} - 45036462874281355 T^{15} + 331891236431857759 T^{16} - 2198159008906492281 T^{17} + 14239538803399666906 T^{18} - 81149558016283104047 T^{19} + \)\(46\!\cdots\!48\)\( T^{20} - \)\(21\!\cdots\!67\)\( T^{21} + \)\(10\!\cdots\!24\)\( T^{22} - \)\(36\!\cdots\!41\)\( T^{23} + \)\(15\!\cdots\!90\)\( T^{24} - \)\(29\!\cdots\!38\)\( T^{25} + \)\(11\!\cdots\!11\)\( T^{26} \)
$73$ \( 1 - 14 T + 525 T^{2} - 5043 T^{3} + 114146 T^{4} - 835086 T^{5} + 15908696 T^{6} - 102811410 T^{7} + 1819584051 T^{8} - 11362668704 T^{9} + 178933541731 T^{10} - 1054953366283 T^{11} + 14857808936322 T^{12} - 82146024126824 T^{13} + 1084620052351506 T^{14} - 5621846488922107 T^{15} + 69608189603568427 T^{16} - 322679804259349664 T^{17} + 3772128007203963243 T^{18} - 15558885186031157490 T^{19} + \)\(17\!\cdots\!12\)\( T^{20} - \)\(67\!\cdots\!66\)\( T^{21} + \)\(67\!\cdots\!98\)\( T^{22} - \)\(21\!\cdots\!07\)\( T^{23} + \)\(16\!\cdots\!25\)\( T^{24} - \)\(32\!\cdots\!94\)\( T^{25} + \)\(16\!\cdots\!33\)\( T^{26} \)
$79$ \( 1 - 12 T + 595 T^{2} - 6449 T^{3} + 169907 T^{4} - 1710709 T^{5} + 32178095 T^{6} - 304033412 T^{7} + 4623962789 T^{8} - 40700298130 T^{9} + 533918230395 T^{10} - 4328557359099 T^{11} + 50720396514356 T^{12} - 376280308395514 T^{13} + 4006911324634124 T^{14} - 27014526478136859 T^{15} + 263242510395720405 T^{16} - 1585279908887648530 T^{17} + 14228194288630336811 T^{18} - 73906708516447867652 T^{19} + \)\(61\!\cdots\!05\)\( T^{20} - \)\(25\!\cdots\!49\)\( T^{21} + \)\(20\!\cdots\!33\)\( T^{22} - \)\(61\!\cdots\!49\)\( T^{23} + \)\(44\!\cdots\!05\)\( T^{24} - \)\(70\!\cdots\!92\)\( T^{25} + \)\(46\!\cdots\!39\)\( T^{26} \)
$83$ \( 1 - 2 T + 727 T^{2} - 1826 T^{3} + 257099 T^{4} - 779996 T^{5} + 58919473 T^{6} - 204052465 T^{7} + 9817483748 T^{8} - 36459014816 T^{9} + 1262655137512 T^{10} - 4705699702101 T^{11} + 129500344746208 T^{12} - 451205917808004 T^{13} + 10748528613935264 T^{14} - 32417565247773789 T^{15} + 721969793112573944 T^{16} - 1730283628481483936 T^{17} + 38671467495363969964 T^{18} - 66712989093964804585 T^{19} + \)\(15\!\cdots\!71\)\( T^{20} - \)\(17\!\cdots\!36\)\( T^{21} + \)\(48\!\cdots\!97\)\( T^{22} - \)\(28\!\cdots\!74\)\( T^{23} + \)\(93\!\cdots\!09\)\( T^{24} - \)\(21\!\cdots\!22\)\( T^{25} + \)\(88\!\cdots\!63\)\( T^{26} \)
$89$ \( 1 - 17 T + 707 T^{2} - 9671 T^{3} + 235154 T^{4} - 2691872 T^{5} + 50151493 T^{6} - 498236069 T^{7} + 7876262822 T^{8} - 69633244675 T^{9} + 983243725344 T^{10} - 7882580573581 T^{11} + 102253061772539 T^{12} - 755772741431854 T^{13} + 9100522497755971 T^{14} - 62437920723335101 T^{15} + 693156345812034336 T^{16} - 4368945819010816675 T^{17} + 43981519833996505078 T^{18} - \)\(24\!\cdots\!09\)\( T^{19} + \)\(22\!\cdots\!97\)\( T^{20} - \)\(10\!\cdots\!32\)\( T^{21} + \)\(82\!\cdots\!86\)\( T^{22} - \)\(30\!\cdots\!71\)\( T^{23} + \)\(19\!\cdots\!23\)\( T^{24} - \)\(41\!\cdots\!57\)\( T^{25} + \)\(21\!\cdots\!69\)\( T^{26} \)
$97$ \( 1 - 17 T + 867 T^{2} - 11690 T^{3} + 328710 T^{4} - 3534085 T^{5} + 71374515 T^{6} - 595341282 T^{7} + 9718631709 T^{8} - 57584520574 T^{9} + 872219089956 T^{10} - 2912705250056 T^{11} + 61763545310987 T^{12} - 114855653661152 T^{13} + 5991063895165739 T^{14} - 27405643697776904 T^{15} + 796050813487412388 T^{16} - 5097916203145927294 T^{17} + 83457197317652409213 T^{18} - \)\(49\!\cdots\!78\)\( T^{19} + \)\(57\!\cdots\!95\)\( T^{20} - \)\(27\!\cdots\!85\)\( T^{21} + \)\(24\!\cdots\!70\)\( T^{22} - \)\(86\!\cdots\!10\)\( T^{23} + \)\(62\!\cdots\!51\)\( T^{24} - \)\(11\!\cdots\!97\)\( T^{25} + \)\(67\!\cdots\!77\)\( T^{26} \)
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