Properties

Label 4008.2.a.j
Level 4008
Weight 2
Character orbit 4008.a
Self dual yes
Analytic conductor 32.004
Analytic rank 1
Dimension 10
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4008 = 2^{3} \cdot 3 \cdot 167 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4008.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.0040411301\)
Analytic rank: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q - q^{3} + ( -1 - \beta_{1} ) q^{5} + ( \beta_{2} + \beta_{4} + \beta_{7} ) q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} + ( -1 - \beta_{1} ) q^{5} + ( \beta_{2} + \beta_{4} + \beta_{7} ) q^{7} + q^{9} + ( -\beta_{4} - \beta_{7} ) q^{11} + ( -1 + \beta_{1} + \beta_{8} ) q^{13} + ( 1 + \beta_{1} ) q^{15} + ( \beta_{3} - \beta_{8} ) q^{17} + ( 1 - \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{8} - \beta_{9} ) q^{19} + ( -\beta_{2} - \beta_{4} - \beta_{7} ) q^{21} + ( \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} ) q^{23} + ( \beta_{1} - 2 \beta_{3} + \beta_{4} + \beta_{7} + \beta_{9} ) q^{25} - q^{27} + ( -1 - \beta_{4} - \beta_{6} - \beta_{8} - \beta_{9} ) q^{29} + ( 2 - \beta_{2} - \beta_{5} + \beta_{7} + \beta_{9} ) q^{31} + ( \beta_{4} + \beta_{7} ) q^{33} + ( -1 - 2 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} ) q^{35} + ( -1 + 2 \beta_{1} - \beta_{3} - \beta_{7} ) q^{37} + ( 1 - \beta_{1} - \beta_{8} ) q^{39} + ( 2 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{6} - 2 \beta_{7} - \beta_{9} ) q^{41} + ( 1 - \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} - 2 \beta_{7} - \beta_{9} ) q^{43} + ( -1 - \beta_{1} ) q^{45} + ( 1 - \beta_{2} - \beta_{4} ) q^{47} + ( 3 \beta_{1} + \beta_{2} + \beta_{4} - 2 \beta_{5} + \beta_{7} + \beta_{8} ) q^{49} + ( -\beta_{3} + \beta_{8} ) q^{51} + ( -2 + \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{8} + \beta_{9} ) q^{53} + ( 3 + 2 \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{8} - \beta_{9} ) q^{55} + ( -1 + \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{8} + \beta_{9} ) q^{57} + ( -2 - 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} + 2 \beta_{9} ) q^{59} + ( 1 + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} ) q^{61} + ( \beta_{2} + \beta_{4} + \beta_{7} ) q^{63} + ( -1 + 2 \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} ) q^{65} + ( -2 + 2 \beta_{2} + \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + 3 \beta_{7} + \beta_{8} + 2 \beta_{9} ) q^{67} + ( -\beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{69} + ( 3 + \beta_{1} - 2 \beta_{4} + \beta_{6} - 2 \beta_{7} - \beta_{9} ) q^{71} + ( -2 + \beta_{1} + 2 \beta_{2} + 2 \beta_{4} - \beta_{5} - 2 \beta_{6} + 2 \beta_{7} + \beta_{8} + \beta_{9} ) q^{73} + ( -\beta_{1} + 2 \beta_{3} - \beta_{4} - \beta_{7} - \beta_{9} ) q^{75} + ( -4 - \beta_{1} - 2 \beta_{2} + 2 \beta_{5} + \beta_{6} - \beta_{7} - \beta_{9} ) q^{77} + ( 3 - \beta_{2} - \beta_{3} + \beta_{6} - \beta_{7} - \beta_{8} - 2 \beta_{9} ) q^{79} + q^{81} + ( 1 + 2 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} + \beta_{5} + 2 \beta_{6} - 3 \beta_{7} - \beta_{9} ) q^{83} + ( -2 + 2 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{7} + 2 \beta_{8} ) q^{85} + ( 1 + \beta_{4} + \beta_{6} + \beta_{8} + \beta_{9} ) q^{87} + ( -5 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} + 3 \beta_{7} + \beta_{9} ) q^{89} + ( -1 + 2 \beta_{1} + \beta_{2} + 2 \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{7} + 2 \beta_{8} ) q^{91} + ( -2 + \beta_{2} + \beta_{5} - \beta_{7} - \beta_{9} ) q^{93} + ( \beta_{1} + \beta_{2} + \beta_{4} + 2 \beta_{6} - 2 \beta_{7} + \beta_{9} ) q^{95} + ( -2 + \beta_{1} - \beta_{2} + 2 \beta_{3} + 3 \beta_{5} - 2 \beta_{7} + \beta_{8} - \beta_{9} ) q^{97} + ( -\beta_{4} - \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q - 10q^{3} - 10q^{5} + q^{7} + 10q^{9} + O(q^{10}) \) \( 10q - 10q^{3} - 10q^{5} + q^{7} + 10q^{9} - q^{11} - 6q^{13} + 10q^{15} - 9q^{17} + 2q^{19} - q^{21} + 7q^{23} + 12q^{25} - 10q^{27} - 13q^{29} + 23q^{31} + q^{33} + q^{35} - 6q^{37} + 6q^{39} - 12q^{41} - 10q^{45} + 10q^{47} + 7q^{49} + 9q^{51} - 26q^{53} + 11q^{55} - 2q^{57} - 10q^{59} - 10q^{61} + q^{63} - 22q^{65} - 5q^{67} - 7q^{69} + 25q^{71} - 8q^{73} - 12q^{75} - 46q^{77} + 26q^{79} + 10q^{81} - 14q^{83} + 9q^{85} + 13q^{87} - 31q^{89} - 3q^{91} - 23q^{93} - 5q^{95} - 32q^{97} - q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} - 26 x^{8} - 3 x^{7} + 220 x^{6} + 42 x^{5} - 675 x^{4} - 67 x^{3} + 628 x^{2} - 48 x - 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 31274 \nu^{9} + 188403 \nu^{8} - 799085 \nu^{7} - 4445172 \nu^{6} + 5651462 \nu^{5} + 31352276 \nu^{4} - 4209233 \nu^{3} - 62769314 \nu^{2} - 26831938 \nu + 17959625 \)\()/6386707\)
\(\beta_{3}\)\(=\)\((\)\( 45076 \nu^{9} - 26608 \nu^{8} - 1096194 \nu^{7} + 535650 \nu^{6} + 9006576 \nu^{5} - 3054363 \nu^{4} - 31921253 \nu^{3} + 4461694 \nu^{2} + 45128857 \nu + 3445799 \)\()/6386707\)
\(\beta_{4}\)\(=\)\((\)\(-140074 \nu^{9} - 197290 \nu^{8} + 4018521 \nu^{7} + 4959639 \nu^{6} - 37660725 \nu^{5} - 36535738 \nu^{4} + 125932814 \nu^{3} + 66704449 \nu^{2} - 128029606 \nu - 3074005\)\()/6386707\)
\(\beta_{5}\)\(=\)\((\)\(-252127 \nu^{9} - 49534 \nu^{8} + 6474592 \nu^{7} + 2278935 \nu^{6} - 53041455 \nu^{5} - 25138542 \nu^{4} + 150476284 \nu^{3} + 61255546 \nu^{2} - 122806757 \nu - 20051371\)\()/6386707\)
\(\beta_{6}\)\(=\)\((\)\( -400908 \nu^{9} + 156741 \nu^{8} + 10079448 \nu^{7} - 2045962 \nu^{6} - 81378952 \nu^{5} + 2450805 \nu^{4} + 233226318 \nu^{3} - 5489909 \nu^{2} - 195374082 \nu + 15210925 \)\()/6386707\)
\(\beta_{7}\)\(=\)\((\)\( -446372 \nu^{9} + 13553 \nu^{8} + 11479221 \nu^{7} + 1022842 \nu^{6} - 95483893 \nu^{5} - 17429855 \nu^{4} + 282874842 \nu^{3} + 36533942 \nu^{2} - 233347950 \nu + 3313559 \)\()/6386707\)
\(\beta_{8}\)\(=\)\((\)\( -581351 \nu^{9} + 4817 \nu^{8} + 14786968 \nu^{7} + 1678767 \nu^{6} - 120466276 \nu^{5} - 24247147 \nu^{4} + 344299563 \nu^{3} + 37019495 \nu^{2} - 291434843 \nu + 24062029 \)\()/6386707\)
\(\beta_{9}\)\(=\)\((\)\(676598 \nu^{9} + 130521 \nu^{8} - 17690130 \nu^{7} - 4911181 \nu^{6} + 151157770 \nu^{5} + 47856867 \nu^{4} - 472650162 \nu^{3} - 87928296 \nu^{2} + 458021977 \nu - 18894784\)\()/6386707\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{9} + \beta_{7} + \beta_{4} - 2 \beta_{3} - \beta_{1} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{3} + \beta_{2} + 9 \beta_{1}\)
\(\nu^{4}\)\(=\)\(12 \beta_{9} + \beta_{8} + 9 \beta_{7} + 3 \beta_{6} - 3 \beta_{5} + 14 \beta_{4} - 24 \beta_{3} - \beta_{2} - 6 \beta_{1} + 33\)
\(\nu^{5}\)\(=\)\(4 \beta_{9} + 18 \beta_{8} - 11 \beta_{7} - 18 \beta_{6} + 16 \beta_{5} + 2 \beta_{4} - 13 \beta_{3} + 17 \beta_{2} + 93 \beta_{1} + 3\)
\(\nu^{6}\)\(=\)\(132 \beta_{9} + 6 \beta_{8} + 81 \beta_{7} + 59 \beta_{6} - 40 \beta_{5} + 167 \beta_{4} - 268 \beta_{3} - 20 \beta_{2} - 40 \beta_{1} + 341\)
\(\nu^{7}\)\(=\)\(81 \beta_{9} + 246 \beta_{8} - 103 \beta_{7} - 253 \beta_{6} + 212 \beta_{5} + 51 \beta_{4} - 139 \beta_{3} + 245 \beta_{2} + 1024 \beta_{1} + 45\)
\(\nu^{8}\)\(=\)\(1477 \beta_{9} - 11 \beta_{8} + 784 \beta_{7} + 876 \beta_{6} - 447 \beta_{5} + 1957 \beta_{4} - 2992 \beta_{3} - 265 \beta_{2} - 319 \beta_{1} + 3816\)
\(\nu^{9}\)\(=\)\(1188 \beta_{9} + 3084 \beta_{8} - 1004 \beta_{7} - 3245 \beta_{6} + 2675 \beta_{5} + 861 \beta_{4} - 1359 \beta_{3} + 3283 \beta_{2} + 11672 \beta_{1} + 459\)

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
3.44741
3.20371
1.76045
1.15141
0.108417
−0.0299246
−1.39234
−2.25873
−2.53098
−3.45943
0 −1.00000 0 −4.44741 0 2.42708 0 1.00000 0
1.2 0 −1.00000 0 −4.20371 0 −3.93441 0 1.00000 0
1.3 0 −1.00000 0 −2.76045 0 3.28542 0 1.00000 0
1.4 0 −1.00000 0 −2.15141 0 −2.22742 0 1.00000 0
1.5 0 −1.00000 0 −1.10842 0 −3.58606 0 1.00000 0
1.6 0 −1.00000 0 −0.970075 0 4.67245 0 1.00000 0
1.7 0 −1.00000 0 0.392343 0 0.0476950 0 1.00000 0
1.8 0 −1.00000 0 1.25873 0 −0.924269 0 1.00000 0
1.9 0 −1.00000 0 1.53098 0 1.95354 0 1.00000 0
1.10 0 −1.00000 0 2.45943 0 −0.714036 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

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 4008.2.a.j 10
4.b odd 2 1 8016.2.a.bd 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4008.2.a.j 10 1.a even 1 1 trivial
8016.2.a.bd 10 4.b odd 2 1

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4008))\):

\(T_{5}^{10} + \cdots\)
\(T_{7}^{10} - \cdots\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( ( 1 + T )^{10} \)
$5$ \( 1 + 10 T + 69 T^{2} + 365 T^{3} + 1608 T^{4} + 6162 T^{5} + 20900 T^{6} + 63916 T^{7} + 177614 T^{8} + 451353 T^{9} + 1053362 T^{10} + 2256765 T^{11} + 4440350 T^{12} + 7989500 T^{13} + 13062500 T^{14} + 19256250 T^{15} + 25125000 T^{16} + 28515625 T^{17} + 26953125 T^{18} + 19531250 T^{19} + 9765625 T^{20} \)
$7$ \( 1 - T + 32 T^{2} - 38 T^{3} + 544 T^{4} - 725 T^{5} + 6513 T^{6} - 9463 T^{7} + 61102 T^{8} - 89363 T^{9} + 470216 T^{10} - 625541 T^{11} + 2993998 T^{12} - 3245809 T^{13} + 15637713 T^{14} - 12185075 T^{15} + 64001056 T^{16} - 31294634 T^{17} + 184473632 T^{18} - 40353607 T^{19} + 282475249 T^{20} \)
$11$ \( 1 + T + 57 T^{2} + 23 T^{3} + 1562 T^{4} - 101 T^{5} + 28412 T^{6} - 11647 T^{7} + 399505 T^{8} - 230292 T^{9} + 4729134 T^{10} - 2533212 T^{11} + 48340105 T^{12} - 15502157 T^{13} + 415980092 T^{14} - 16266151 T^{15} + 2767178282 T^{16} + 448204933 T^{17} + 12218456217 T^{18} + 2357947691 T^{19} + 25937424601 T^{20} \)
$13$ \( 1 + 6 T + 83 T^{2} + 478 T^{3} + 3637 T^{4} + 18563 T^{5} + 104668 T^{6} + 465567 T^{7} + 2135350 T^{8} + 8277926 T^{9} + 32123658 T^{10} + 107613038 T^{11} + 360874150 T^{12} + 1022850699 T^{13} + 2989422748 T^{14} + 6892311959 T^{15} + 17555104333 T^{16} + 29993791126 T^{17} + 67705649843 T^{18} + 63626996238 T^{19} + 137858491849 T^{20} \)
$17$ \( 1 + 9 T + 118 T^{2} + 837 T^{3} + 6250 T^{4} + 35453 T^{5} + 196924 T^{6} + 935647 T^{7} + 4342997 T^{8} + 18507198 T^{9} + 78377748 T^{10} + 314622366 T^{11} + 1255126133 T^{12} + 4596833711 T^{13} + 16447289404 T^{14} + 50338190221 T^{15} + 150859806250 T^{16} + 343453469301 T^{17} + 823139378038 T^{18} + 1067290888473 T^{19} + 2015993900449 T^{20} \)
$19$ \( 1 - 2 T + 85 T^{2} - 2 T^{3} + 3473 T^{4} + 5923 T^{5} + 99516 T^{6} + 314949 T^{7} + 2275338 T^{8} + 9282398 T^{9} + 45237014 T^{10} + 176365562 T^{11} + 821397018 T^{12} + 2160235191 T^{13} + 12969024636 T^{14} + 14665934377 T^{15} + 163390344713 T^{16} - 1787743478 T^{17} + 1443602858485 T^{18} - 645375395558 T^{19} + 6131066257801 T^{20} \)
$23$ \( 1 - 7 T + 95 T^{2} - 569 T^{3} + 5344 T^{4} - 29007 T^{5} + 221988 T^{6} - 1080533 T^{7} + 7067347 T^{8} - 31317240 T^{9} + 181558482 T^{10} - 720296520 T^{11} + 3738626563 T^{12} - 13146845011 T^{13} + 62121343908 T^{14} - 186699001401 T^{15} + 791103790816 T^{16} - 1937345679343 T^{17} + 7439543601695 T^{18} - 12608068630241 T^{19} + 41426511213649 T^{20} \)
$29$ \( 1 + 13 T + 159 T^{2} + 939 T^{3} + 6582 T^{4} + 25017 T^{5} + 196396 T^{6} + 616513 T^{7} + 4522549 T^{8} + 29470 T^{9} + 53516466 T^{10} + 854630 T^{11} + 3803463709 T^{12} + 15036135557 T^{13} + 138907159276 T^{14} + 513127414533 T^{15} + 3915127098822 T^{16} + 16197633854151 T^{17} + 79539179660799 T^{18} + 188592897686297 T^{19} + 420707233300201 T^{20} \)
$31$ \( 1 - 23 T + 381 T^{2} - 4487 T^{3} + 44144 T^{4} - 360340 T^{5} + 2585914 T^{6} - 16406937 T^{7} + 96372624 T^{8} - 537949941 T^{9} + 2990336088 T^{10} - 16676448171 T^{11} + 92614091664 T^{12} - 488779060167 T^{13} + 2388145883194 T^{14} - 10316228271340 T^{15} + 39177962494064 T^{16} - 123449099516057 T^{17} + 324951485265021 T^{18} - 608111309695433 T^{19} + 819628286980801 T^{20} \)
$37$ \( 1 + 6 T + 193 T^{2} + 441 T^{3} + 14860 T^{4} - 11088 T^{5} + 798266 T^{6} - 1849094 T^{7} + 42290634 T^{8} - 74596125 T^{9} + 1868536464 T^{10} - 2760056625 T^{11} + 57895877946 T^{12} - 93662158382 T^{13} + 1496079004826 T^{14} - 768885795216 T^{15} + 38126694437740 T^{16} + 41864957815653 T^{17} + 677908534606753 T^{18} + 779770438770462 T^{19} + 4808584372417849 T^{20} \)
$41$ \( 1 + 12 T + 294 T^{2} + 3115 T^{3} + 43285 T^{4} + 398010 T^{5} + 4063184 T^{6} + 32482454 T^{7} + 266719470 T^{8} + 1847808865 T^{9} + 12746910316 T^{10} + 75760163465 T^{11} + 448355429070 T^{12} + 2238723212134 T^{13} + 11481586883024 T^{14} + 46111926560010 T^{15} + 205608262071685 T^{16} + 606659563139315 T^{17} + 2347568017361574 T^{18} + 3928583212727532 T^{19} + 13422659310152401 T^{20} \)
$43$ \( 1 + 252 T^{2} - 279 T^{3} + 32851 T^{4} - 55784 T^{5} + 2880724 T^{6} - 5482364 T^{7} + 186179700 T^{8} - 341733543 T^{9} + 9146061736 T^{10} - 14694542349 T^{11} + 344246265300 T^{12} - 435886314548 T^{13} + 9848622091924 T^{14} - 8200718984312 T^{15} + 207663097522699 T^{16} - 75837392498853 T^{17} + 2945426469955452 T^{18} + 21611482313284249 T^{20} \)
$47$ \( 1 - 10 T + 441 T^{2} - 3860 T^{3} + 88852 T^{4} - 677217 T^{5} + 10763404 T^{6} - 70906584 T^{7} + 867634490 T^{8} - 4884101757 T^{9} + 48609497732 T^{10} - 229552782579 T^{11} + 1916604588410 T^{12} - 7361734270632 T^{13} + 52521977994124 T^{14} - 155316337605519 T^{15} + 957754840412308 T^{16} - 1955565244987180 T^{17} + 10500777417836601 T^{18} - 11191304731027670 T^{19} + 52599132235830049 T^{20} \)
$53$ \( 1 + 26 T + 577 T^{2} + 8762 T^{3} + 121850 T^{4} + 1386171 T^{5} + 14819742 T^{6} + 137932446 T^{7} + 1227120420 T^{8} + 9761201687 T^{9} + 74837899402 T^{10} + 517343689411 T^{11} + 3446981259780 T^{12} + 20534968763142 T^{13} + 116934892675902 T^{14} + 579690464727303 T^{15} + 2700727403568650 T^{16} + 10292819007251794 T^{17} + 35923841367355297 T^{18} + 85793853386855458 T^{19} + 174887470365513049 T^{20} \)
$59$ \( 1 + 10 T + 413 T^{2} + 3896 T^{3} + 85628 T^{4} + 727669 T^{5} + 11417612 T^{6} + 85854922 T^{7} + 1070499278 T^{8} + 7042654803 T^{9} + 73409697452 T^{10} + 415516633377 T^{11} + 3726407986718 T^{12} + 17632798025438 T^{13} + 138351326361932 T^{14} + 520228249729031 T^{15} + 3611834734611548 T^{16} + 9695786184854824 T^{17} + 60640970730584573 T^{18} + 86629958186549390 T^{19} + 511116753300641401 T^{20} \)
$61$ \( 1 + 10 T + 531 T^{2} + 4616 T^{3} + 130505 T^{4} + 988345 T^{5} + 19590364 T^{6} + 128900293 T^{7} + 1991769246 T^{8} + 11288878956 T^{9} + 143433528194 T^{10} + 688621616316 T^{11} + 7411373364366 T^{12} + 29257917405433 T^{13} + 271245065076124 T^{14} + 834752531111845 T^{15} + 6723666455982305 T^{16} + 14506900931072936 T^{17} + 101796583201556211 T^{18} + 116941460928341410 T^{19} + 713342911662882601 T^{20} \)
$67$ \( 1 + 5 T + 356 T^{2} + 1893 T^{3} + 65824 T^{4} + 371066 T^{5} + 8314295 T^{6} + 48002653 T^{7} + 791130118 T^{8} + 4397053995 T^{9} + 59302045306 T^{10} + 294602617665 T^{11} + 3551383099702 T^{12} + 14437421924239 T^{13} + 167542364574695 T^{14} + 500985522954062 T^{15} + 5954332547892256 T^{16} + 11472927068876439 T^{17} + 144560093210164196 T^{18} + 136032671981474735 T^{19} + 1822837804551761449 T^{20} \)
$71$ \( 1 - 25 T + 758 T^{2} - 12926 T^{3} + 235700 T^{4} - 3124326 T^{5} + 42759652 T^{6} - 464718772 T^{7} + 5145626719 T^{8} - 46901534195 T^{9} + 433380203268 T^{10} - 3330008927845 T^{11} + 25939104290479 T^{12} - 166327960405292 T^{13} + 1086594636295012 T^{14} - 5637000671292426 T^{15} + 30193236920179700 T^{16} - 117563523167362066 T^{17} + 489481176684286838 T^{18} - 1146212517961225775 T^{19} + 3255243551009881201 T^{20} \)
$73$ \( 1 + 8 T + 557 T^{2} + 4242 T^{3} + 145301 T^{4} + 1047687 T^{5} + 23623980 T^{6} + 158940879 T^{7} + 2687146702 T^{8} + 16400535416 T^{9} + 226096563398 T^{10} + 1197239085368 T^{11} + 14319804774958 T^{12} + 61830703925943 T^{13} + 670879477419180 T^{14} + 2171930158055391 T^{15} + 21989014414017989 T^{16} + 46863064518009474 T^{17} + 449198271185003117 T^{18} + 470972693666143304 T^{19} + 4297625829703557649 T^{20} \)
$79$ \( 1 - 26 T + 782 T^{2} - 14711 T^{3} + 270063 T^{4} - 3959612 T^{5} + 55142006 T^{6} - 659168568 T^{7} + 7454929856 T^{8} - 74328091409 T^{9} + 701127963032 T^{10} - 5871919221311 T^{11} + 46526217231296 T^{12} - 324995811598152 T^{13} + 2147785600202486 T^{14} - 12183949442157188 T^{15} + 65648927500367823 T^{16} - 282508705095385049 T^{17} + 1186379089346930702 T^{18} - 3116141495548076294 T^{19} + 9468276082626847201 T^{20} \)
$83$ \( 1 + 14 T + 577 T^{2} + 5800 T^{3} + 138728 T^{4} + 1019691 T^{5} + 19211234 T^{6} + 101335232 T^{7} + 1848101498 T^{8} + 7313651899 T^{9} + 153849005048 T^{10} + 607033107617 T^{11} + 12731571219722 T^{12} + 57942168299584 T^{13} + 911732909978114 T^{14} + 4016604292301313 T^{15} + 45355784116734632 T^{16} + 157389095739836600 T^{17} + 1299572617944226657 T^{18} + 2617163573745565642 T^{19} + 15516041187205853449 T^{20} \)
$89$ \( 1 + 31 T + 1084 T^{2} + 21819 T^{3} + 452262 T^{4} + 6916306 T^{5} + 106735781 T^{6} + 1322186569 T^{7} + 16410110830 T^{8} + 169159033935 T^{9} + 1742700155052 T^{10} + 15055154020215 T^{11} + 129984487884430 T^{12} + 932100543361361 T^{13} + 6696842094825221 T^{14} + 38621063871475394 T^{15} + 224765752612603782 T^{16} + 965083496085547251 T^{17} + 4267262265381055804 T^{18} + 10861048514932041479 T^{19} + 31181719929966183601 T^{20} \)
$97$ \( 1 + 32 T + 740 T^{2} + 12795 T^{3} + 186194 T^{4} + 2199458 T^{5} + 22630839 T^{6} + 194197764 T^{7} + 1475234704 T^{8} + 10082483539 T^{9} + 85871535452 T^{10} + 978000903283 T^{11} + 13880483329936 T^{12} + 177239055863172 T^{13} + 2003491905096759 T^{14} + 18887494226980706 T^{15} + 155094389485750226 T^{16} + 1033814049897455835 T^{17} + 5799700859838951140 T^{18} + 24327393876946086944 T^{19} + 73742412689492826049 T^{20} \)
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