Properties

Label 4020.2.q.k
Level 4020
Weight 2
Character orbit 4020.q
Analytic conductor 32.100
Analytic rank 0
Dimension 14
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4020.q (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(32.0998616126\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} + q^{5} -\beta_{6} q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} + q^{5} -\beta_{6} q^{7} + q^{9} + ( 1 - \beta_{5} - \beta_{10} - \beta_{12} ) q^{11} + ( -\beta_{1} - \beta_{3} + \beta_{5} ) q^{13} - q^{15} + ( -\beta_{2} - 2 \beta_{5} - \beta_{11} + \beta_{13} ) q^{17} + ( -\beta_{2} + 2 \beta_{5} - \beta_{11} ) q^{19} + \beta_{6} q^{21} + ( -\beta_{2} + 2 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{10} - \beta_{11} - \beta_{13} ) q^{23} + q^{25} - q^{27} + ( -1 - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{10} - \beta_{11} - \beta_{12} + \beta_{13} ) q^{29} + ( 1 + \beta_{1} - \beta_{5} - \beta_{6} - \beta_{10} - \beta_{11} + \beta_{12} ) q^{31} + ( -1 + \beta_{5} + \beta_{10} + \beta_{12} ) q^{33} -\beta_{6} q^{35} + ( -2 \beta_{2} + \beta_{9} - 2 \beta_{11} - \beta_{12} ) q^{37} + ( \beta_{1} + \beta_{3} - \beta_{5} ) q^{39} + ( -2 - \beta_{4} + 2 \beta_{5} + \beta_{12} + \beta_{13} ) q^{41} + ( 1 - \beta_{2} - 2 \beta_{8} + 2 \beta_{9} ) q^{43} + q^{45} + ( 2 - 2 \beta_{1} + \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} ) q^{47} + ( -\beta_{2} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} + \beta_{13} ) q^{49} + ( \beta_{2} + 2 \beta_{5} + \beta_{11} - \beta_{13} ) q^{51} + ( \beta_{2} + \beta_{3} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{53} + ( 1 - \beta_{5} - \beta_{10} - \beta_{12} ) q^{55} + ( \beta_{2} - 2 \beta_{5} + \beta_{11} ) q^{57} + ( -1 - \beta_{3} + 2 \beta_{8} + \beta_{9} ) q^{59} + ( \beta_{1} + \beta_{3} - \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{8} + \beta_{9} - 2 \beta_{10} - \beta_{12} + \beta_{13} ) q^{61} -\beta_{6} q^{63} + ( -\beta_{1} - \beta_{3} + \beta_{5} ) q^{65} + ( 3 + \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{9} - \beta_{10} + 2 \beta_{11} + \beta_{12} ) q^{67} + ( \beta_{2} - 2 \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{10} + \beta_{11} + \beta_{13} ) q^{69} + ( 4 - \beta_{1} + 2 \beta_{4} - 4 \beta_{5} + 3 \beta_{6} + \beta_{10} + 2 \beta_{11} + \beta_{12} - 2 \beta_{13} ) q^{71} + ( -\beta_{2} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} - \beta_{12} ) q^{73} - q^{75} + ( \beta_{1} + \beta_{3} + 2 \beta_{5} - 3 \beta_{6} - 3 \beta_{8} + 2 \beta_{9} - 2 \beta_{12} + \beta_{13} ) q^{77} + ( 1 + \beta_{1} - 2 \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{11} + \beta_{12} + 2 \beta_{13} ) q^{79} + q^{81} + ( -2 \beta_{1} - 2 \beta_{3} - \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} + 2 \beta_{12} - \beta_{13} ) q^{83} + ( -\beta_{2} - 2 \beta_{5} - \beta_{11} + \beta_{13} ) q^{85} + ( 1 + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} ) q^{87} + ( -1 + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{7} + \beta_{9} ) q^{89} + ( -4 - \beta_{3} + 2 \beta_{4} + \beta_{8} ) q^{91} + ( -1 - \beta_{1} + \beta_{5} + \beta_{6} + \beta_{10} + \beta_{11} - \beta_{12} ) q^{93} + ( -\beta_{2} + 2 \beta_{5} - \beta_{11} ) q^{95} + ( -2 \beta_{1} + \beta_{2} - 2 \beta_{3} + 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{10} + \beta_{11} - \beta_{13} ) q^{97} + ( 1 - \beta_{5} - \beta_{10} - \beta_{12} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14q - 14q^{3} + 14q^{5} + 3q^{7} + 14q^{9} + O(q^{10}) \) \( 14q - 14q^{3} + 14q^{5} + 3q^{7} + 14q^{9} + 6q^{11} + 9q^{13} - 14q^{15} - 12q^{17} + 14q^{19} - 3q^{21} + 6q^{23} + 14q^{25} - 14q^{27} - q^{29} + 7q^{31} - 6q^{33} + 3q^{35} - 2q^{37} - 9q^{39} - 18q^{41} - 6q^{43} + 14q^{45} + 7q^{47} + 12q^{51} - 12q^{53} + 6q^{55} - 14q^{57} - 2q^{59} + 3q^{63} + 9q^{65} + 25q^{67} - 6q^{69} + 22q^{71} - 15q^{73} - 14q^{75} + q^{77} + 9q^{79} + 14q^{81} - q^{83} - 12q^{85} + q^{87} - 12q^{89} - 38q^{91} - 7q^{93} + 14q^{95} + 16q^{97} + 6q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{14} - x^{13} + 11 x^{12} - 8 x^{11} + 88 x^{10} - 57 x^{9} + 270 x^{8} + 17 x^{7} + 458 x^{6} - 101 x^{5} + 189 x^{4} - 30 x^{3} + 54 x^{2} - 12 x + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu \)
\(\beta_{2}\)\(=\)\((\)\(-2680061185 \nu^{13} + 4118856992 \nu^{12} - 29747186705 \nu^{11} + 35610122429 \nu^{10} - 234940121607 \nu^{9} + 266952713308 \nu^{8} - 708369775075 \nu^{7} + 250820920885 \nu^{6} - 934856019808 \nu^{5} + 984764699106 \nu^{4} - 308324347672 \nu^{3} + 320958040428 \nu^{2} - 76957145272 \nu + 970249939441\)\()/ 325648644013 \)
\(\beta_{3}\)\(=\)\((\)\(3347996299 \nu^{13} - 8708118669 \nu^{12} + 45065673273 \nu^{11} - 86278343802 \nu^{10} + 365843919170 \nu^{9} - 660716032257 \nu^{8} + 1437864427346 \nu^{7} - 1359823613067 \nu^{6} + 2035024146712 \nu^{5} - 2207859665815 \nu^{4} + 2602300698723 \nu^{3} - 717088584314 \nu^{2} + 171410592976 \nu - 194090246132\)\()/ 325648644013 \)
\(\beta_{4}\)\(=\)\((\)\(-10892153560 \nu^{13} + 9758640678 \nu^{12} - 105683331260 \nu^{11} + 64289364146 \nu^{10} - 815634907035 \nu^{9} + 455144021782 \nu^{8} - 1814707096630 \nu^{7} - 908790740124 \nu^{6} - 2130418252432 \nu^{5} + 2120608715454 \nu^{4} + 2831612352657 \nu^{3} + 695977413972 \nu^{2} - 167893036528 \nu + 861786952374\)\()/ 325648644013 \)
\(\beta_{5}\)\(=\)\((\)\(-48522561533 \nu^{13} + 45174565234 \nu^{12} - 525040058194 \nu^{11} + 343114818991 \nu^{10} - 4183707071102 \nu^{9} + 2399942088211 \nu^{8} - 12440375581653 \nu^{7} - 2262747973407 \nu^{6} - 20863509569047 \nu^{5} + 2865754568121 \nu^{4} - 6962904463922 \nu^{3} - 1146623852733 \nu^{2} - 1903129738468 \nu + 410860145420\)\()/ 651297288026 \)
\(\beta_{6}\)\(=\)\((\)\(-809674271 \nu^{13} - 252390382 \nu^{12} - 8122293536 \nu^{11} - 4713678201 \nu^{10} - 65999965440 \nu^{9} - 42770485013 \nu^{8} - 184014082317 \nu^{7} - 266207177033 \nu^{6} - 473551890151 \nu^{5} - 353375346337 \nu^{4} - 160280487010 \nu^{3} - 54504988027 \nu^{2} - 43501677200 \nu - 7944658082\)\()/ 8458406338 \)
\(\beta_{7}\)\(=\)\((\)\(37194535803 \nu^{13} - 47919288537 \nu^{12} + 395084796401 \nu^{11} - 400339609502 \nu^{10} + 3098569859104 \nu^{9} - 2969882527405 \nu^{8} + 8589017027582 \nu^{7} - 1664041174406 \nu^{6} + 11026504067640 \nu^{5} - 11470995916307 \nu^{4} - 2133066776665 \nu^{3} - 3744294132514 \nu^{2} + 898967472816 \nu - 841108812233\)\()/ 325648644013 \)
\(\beta_{8}\)\(=\)\((\)\(-45711237420 \nu^{13} + 74034574851 \nu^{12} - 518899664240 \nu^{11} + 668330017670 \nu^{10} - 4117338117621 \nu^{9} + 5030477527030 \nu^{8} - 12888553943330 \nu^{7} + 6408314158958 \nu^{6} - 17209864321680 \nu^{5} + 18222273890390 \nu^{4} - 5135681407065 \nu^{3} + 5935414947940 \nu^{2} - 1422383351760 \nu + 974875365383\)\()/ 325648644013 \)
\(\beta_{9}\)\(=\)\((\)\(-58435287502 \nu^{13} + 104277787850 \nu^{12} - 683843870764 \nu^{11} + 962775171505 \nu^{10} - 5449809996738 \nu^{9} + 7279578337109 \nu^{8} - 17910517000443 \nu^{7} + 10738605919044 \nu^{6} - 24249792781624 \nu^{5} + 25830617586941 \nu^{4} - 12925259206893 \nu^{3} + 8407638741310 \nu^{2} - 2013572275960 \nu + 2201747399207\)\()/ 325648644013 \)
\(\beta_{10}\)\(=\)\((\)\(126618741863 \nu^{13} - 49216475610 \nu^{12} + 1335886940970 \nu^{11} - 183514685193 \nu^{10} + 10733169032550 \nu^{9} - 585966489615 \nu^{8} + 31403882537963 \nu^{7} + 21731699467035 \nu^{6} + 63503082991395 \nu^{5} + 22467398449373 \nu^{4} + 21347976231450 \nu^{3} + 5457745711665 \nu^{2} + 2242764960296 \nu + 903683328390\)\()/ 651297288026 \)
\(\beta_{11}\)\(=\)\((\)\(-20029651747 \nu^{13} + 18183711674 \nu^{12} - 216517971596 \nu^{11} + 136874887445 \nu^{10} - 1725891567156 \nu^{9} + 952274405431 \nu^{8} - 5129198170687 \nu^{7} - 1041412251713 \nu^{6} - 8674402381075 \nu^{5} + 946819186593 \nu^{4} - 2896009242346 \nu^{3} - 490070050147 \nu^{2} - 793639274980 \nu - 101131348946\)\()/ 93042469718 \)
\(\beta_{12}\)\(=\)\((\)\(156907511177 \nu^{13} - 63210384410 \nu^{12} + 1644341611148 \nu^{11} - 250245646301 \nu^{10} + 13207909689396 \nu^{9} - 942157914055 \nu^{8} + 38225859873625 \nu^{7} + 26110314425969 \nu^{6} + 77748959678947 \nu^{5} + 23844336362505 \nu^{4} + 26131870142026 \nu^{3} + 6616282893811 \nu^{2} + 10832044605312 \nu + 1100661461810\)\()/ 651297288026 \)
\(\beta_{13}\)\(=\)\((\)\(-87738052524 \nu^{13} + 71009258532 \nu^{12} - 933363507764 \nu^{11} + 502153407798 \nu^{10} - 7424278972227 \nu^{9} + 3401605949923 \nu^{8} - 21440487873381 \nu^{7} - 6897421130211 \nu^{6} - 36599366732449 \nu^{5} + 1482620818260 \nu^{4} - 8695369936605 \nu^{3} - 1496669907681 \nu^{2} - 2171714647326 \nu + 438504860088\)\()/ 325648644013 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/2\)
\(\nu^{2}\)\(=\)\(\beta_{11} - 3 \beta_{5} + \beta_{2}\)
\(\nu^{3}\)\(=\)\(\beta_{9} - \beta_{8} + 3 \beta_{3} - \beta_{2} + 1\)
\(\nu^{4}\)\(=\)\(\beta_{13} - \beta_{12} - 8 \beta_{11} - \beta_{6} + 18 \beta_{5} - \beta_{4} + \beta_{1} - 18\)
\(\nu^{5}\)\(=\)\(8 \beta_{12} + 10 \beta_{11} + \beta_{10} - 8 \beta_{9} + 9 \beta_{8} + \beta_{7} + 9 \beta_{6} - 12 \beta_{5} - 20 \beta_{3} + 10 \beta_{2} - 20 \beta_{1}\)
\(\nu^{6}\)\(=\)\(10 \beta_{9} - 10 \beta_{8} + \beta_{7} + 10 \beta_{4} + 13 \beta_{3} - 58 \beta_{2} + 119\)
\(\nu^{7}\)\(=\)\(-\beta_{13} - 58 \beta_{12} - 84 \beta_{11} - 11 \beta_{10} - 68 \beta_{6} + 117 \beta_{5} + \beta_{4} + 136 \beta_{1} - 117\)
\(\nu^{8}\)\(=\)\(-80 \beta_{13} + 84 \beta_{12} + 415 \beta_{11} - 12 \beta_{10} - 84 \beta_{9} + 83 \beta_{8} - 12 \beta_{7} + 83 \beta_{6} - 812 \beta_{5} - 127 \beta_{3} + 415 \beta_{2} - 127 \beta_{1}\)
\(\nu^{9}\)\(=\)\(415 \beta_{9} - 495 \beta_{8} - 92 \beta_{7} - 9 \beta_{4} + 939 \beta_{3} - 673 \beta_{2} + 1027\)
\(\nu^{10}\)\(=\)\(596 \beta_{13} - 673 \beta_{12} - 2975 \beta_{11} + 101 \beta_{10} - 664 \beta_{6} + 5649 \beta_{5} - 596 \beta_{4} + 1114 \beta_{1} - 5649\)
\(\nu^{11}\)\(=\)\(33 \beta_{13} + 2975 \beta_{12} + 5280 \beta_{11} + 697 \beta_{10} - 2975 \beta_{9} + 3571 \beta_{8} + 697 \beta_{7} + 3571 \beta_{6} - 8509 \beta_{5} - 6568 \beta_{3} + 5280 \beta_{2} - 6568 \beta_{1}\)
\(\nu^{12}\)\(=\)\(5280 \beta_{9} - 5247 \beta_{8} + 730 \beta_{7} + 4301 \beta_{4} + 9250 \beta_{3} - 21424 \beta_{2} + 39857\)
\(\nu^{13}\)\(=\)\(216 \beta_{13} - 21424 \beta_{12} - 40903 \beta_{11} - 5031 \beta_{10} - 25725 \beta_{6} + 68139 \beta_{5} - 216 \beta_{4} + 46436 \beta_{1} - 68139\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4020\mathbb{Z}\right)^\times\).

\(n\) \(1141\) \(2011\) \(2681\) \(3217\)
\(\chi(n)\) \(-1 + \beta_{5}\) \(1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
841.1
0.295122 0.511167i
−0.288886 + 0.500366i
1.10425 1.91261i
0.150493 0.260662i
1.24942 2.16407i
−1.36264 + 2.36015i
−0.647766 + 1.12196i
0.295122 + 0.511167i
−0.288886 0.500366i
1.10425 + 1.91261i
0.150493 + 0.260662i
1.24942 + 2.16407i
−1.36264 2.36015i
−0.647766 1.12196i
0 −1.00000 0 1.00000 0 −1.59463 + 2.76199i 0 1.00000 0
841.2 0 −1.00000 0 1.00000 0 −1.13810 + 1.97125i 0 1.00000 0
841.3 0 −1.00000 0 1.00000 0 −0.952306 + 1.64944i 0 1.00000 0
841.4 0 −1.00000 0 1.00000 0 0.827889 1.43395i 0 1.00000 0
841.5 0 −1.00000 0 1.00000 0 1.07806 1.86725i 0 1.00000 0
841.6 0 −1.00000 0 1.00000 0 1.26463 2.19041i 0 1.00000 0
841.7 0 −1.00000 0 1.00000 0 2.01447 3.48916i 0 1.00000 0
3781.1 0 −1.00000 0 1.00000 0 −1.59463 2.76199i 0 1.00000 0
3781.2 0 −1.00000 0 1.00000 0 −1.13810 1.97125i 0 1.00000 0
3781.3 0 −1.00000 0 1.00000 0 −0.952306 1.64944i 0 1.00000 0
3781.4 0 −1.00000 0 1.00000 0 0.827889 + 1.43395i 0 1.00000 0
3781.5 0 −1.00000 0 1.00000 0 1.07806 + 1.86725i 0 1.00000 0
3781.6 0 −1.00000 0 1.00000 0 1.26463 + 2.19041i 0 1.00000 0
3781.7 0 −1.00000 0 1.00000 0 2.01447 + 3.48916i 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3781.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
67.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4020.2.q.k 14
67.c even 3 1 inner 4020.2.q.k 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4020.2.q.k 14 1.a even 1 1 trivial
4020.2.q.k 14 67.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4020, [\chi])\):

\(T_{7}^{14} - \cdots\)
\(T_{11}^{14} - \cdots\)
\(T_{17}^{14} + \cdots\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 + T )^{14} \)
$5$ \( ( 1 - T )^{14} \)
$7$ \( 1 - 3 T - 20 T^{2} + 57 T^{3} + 177 T^{4} - 389 T^{5} - 1803 T^{6} + 3303 T^{7} + 18070 T^{8} - 36377 T^{9} - 110178 T^{10} + 123265 T^{11} + 781213 T^{12} + 114570 T^{13} - 6762664 T^{14} + 801990 T^{15} + 38279437 T^{16} + 42279895 T^{17} - 264537378 T^{18} - 611388239 T^{19} + 2125917430 T^{20} + 2720162529 T^{21} - 10393936203 T^{22} - 15697553123 T^{23} + 49998119073 T^{24} + 112707624351 T^{25} - 276825744020 T^{26} - 290667031221 T^{27} + 678223072849 T^{28} \)
$11$ \( 1 - 6 T - 11 T^{2} + 16 T^{3} + 550 T^{4} + 451 T^{5} - 6158 T^{6} - 23818 T^{7} + 26095 T^{8} + 323347 T^{9} + 407400 T^{10} - 2104801 T^{11} - 9984167 T^{12} + 14819849 T^{13} + 70409980 T^{14} + 163018339 T^{15} - 1208084207 T^{16} - 2801490131 T^{17} + 5964743400 T^{18} + 52075357697 T^{19} + 46228884295 T^{20} - 464145438878 T^{21} - 1320021989198 T^{22} + 1063434408641 T^{23} + 14265583530550 T^{24} + 4564986729776 T^{25} - 34522712143931 T^{26} - 207136272863586 T^{27} + 379749833583241 T^{28} \)
$13$ \( 1 - 9 T - 3 T^{2} + 234 T^{3} - 216 T^{4} - 3440 T^{5} + 5289 T^{6} + 22909 T^{7} - 40116 T^{8} + 156307 T^{9} - 627413 T^{10} - 5854224 T^{11} + 29555027 T^{12} + 38130675 T^{13} - 498920098 T^{14} + 495698775 T^{15} + 4994799563 T^{16} - 12861730128 T^{17} - 17919542693 T^{18} + 58035694951 T^{19} - 193632269844 T^{20} + 1437505775953 T^{21} + 4314399783369 T^{22} - 36479477843120 T^{23} - 29777434239384 T^{24} + 419365532204658 T^{25} - 69894255367443 T^{26} - 2725875959330277 T^{27} + 3937376385699289 T^{28} \)
$17$ \( 1 + 12 T + 13 T^{2} - 120 T^{3} + 1763 T^{4} + 8439 T^{5} - 29854 T^{6} + 48731 T^{7} + 1353310 T^{8} - 587405 T^{9} - 5786839 T^{10} + 84344359 T^{11} + 106817487 T^{12} - 21465324 T^{13} + 3964551862 T^{14} - 364910508 T^{15} + 30870253743 T^{16} + 414383835767 T^{17} - 483322580119 T^{18} - 834031101085 T^{19} + 32665613503390 T^{20} + 19996213873963 T^{21} - 208254262643614 T^{22} + 1000763089758183 T^{23} + 3554197246491587 T^{24} - 4112627556915960 T^{25} + 7574089083986893 T^{26} + 118854936394871244 T^{27} + 168377826559400929 T^{28} \)
$19$ \( 1 - 14 T + 8 T^{2} + 582 T^{3} + 222 T^{4} - 25358 T^{5} - 8233 T^{6} + 614419 T^{7} + 1938247 T^{8} - 16399564 T^{9} - 65078281 T^{10} + 228533498 T^{11} + 2092709725 T^{12} - 2476014457 T^{13} - 40027234346 T^{14} - 47044274683 T^{15} + 755468210725 T^{16} + 1567511262782 T^{17} - 8481066658201 T^{18} - 40606944020836 T^{19} + 91186537710607 T^{20} + 549211780004641 T^{21} - 139825674516553 T^{22} - 8182714640279882 T^{23} + 1361096709231822 T^{24} + 67797330678763458 T^{25} + 17706519352529288 T^{26} - 588741768471598826 T^{27} + 799006685782884121 T^{28} \)
$23$ \( 1 - 6 T - 55 T^{2} + 462 T^{3} + 753 T^{4} - 13307 T^{5} + 10046 T^{6} + 152185 T^{7} - 95636 T^{8} - 824917 T^{9} - 12768145 T^{10} + 51517171 T^{11} + 264832895 T^{12} - 956116592 T^{13} - 2075704102 T^{14} - 21990681616 T^{15} + 140096601455 T^{16} + 626809419557 T^{17} - 3573050464945 T^{18} - 5309448758531 T^{19} - 14157560280404 T^{20} + 518163360651695 T^{21} + 786712158132926 T^{22} - 23967938466088141 T^{23} + 31194162943877697 T^{24} + 440198108156234274 T^{25} - 1205304343761117655 T^{26} - 3024218171618804298 T^{27} + 11592836324538749809 T^{28} \)
$29$ \( 1 + T - 94 T^{2} + 111 T^{3} + 4141 T^{4} - 11695 T^{5} - 81857 T^{6} + 347769 T^{7} + 213802 T^{8} + 2744015 T^{9} + 736264 T^{10} - 378858495 T^{11} + 1902063729 T^{12} + 6351771646 T^{13} - 90593298388 T^{14} + 184201377734 T^{15} + 1599635596089 T^{16} - 9239979834555 T^{17} + 520745538184 T^{18} + 56282900523235 T^{19} + 127174415676442 T^{20} + 5998972234104621 T^{21} - 40948670625748577 T^{22} - 169661072187787955 T^{23} + 1742148653096132341 T^{24} + 1354256583993347019 T^{25} - 33258589621314089854 T^{26} + 10260628712958602189 T^{27} + \)\(29\!\cdots\!81\)\( T^{28} \)
$31$ \( 1 - 7 T - 39 T^{2} + 660 T^{3} - 2351 T^{4} - 13913 T^{5} + 146821 T^{6} - 144326 T^{7} - 3076282 T^{8} + 12618238 T^{9} + 70496210 T^{10} - 862571403 T^{11} + 1968366689 T^{12} + 19721322013 T^{13} - 189458995778 T^{14} + 611360982403 T^{15} + 1891600388129 T^{16} - 25696864666773 T^{17} + 65104730355410 T^{18} + 361249441055938 T^{19} - 2730211598794042 T^{20} - 3970785544184186 T^{21} + 125222315008125061 T^{22} - 367854463121415623 T^{23} - 1926946102691863151 T^{24} + 16769594751627188460 T^{25} - 30718848567753440679 T^{26} - \)\(17\!\cdots\!37\)\( T^{27} + \)\(75\!\cdots\!21\)\( T^{28} \)
$37$ \( 1 + 2 T - 144 T^{2} - 58 T^{3} + 11562 T^{4} - 8564 T^{5} - 567933 T^{6} + 1219349 T^{7} + 16972491 T^{8} - 72220508 T^{9} - 82031967 T^{10} + 2604100054 T^{11} - 19609237189 T^{12} - 38834898747 T^{13} + 1110997548598 T^{14} - 1436891253639 T^{15} - 26845045711741 T^{16} + 131905480035262 T^{17} - 153741113304687 T^{18} - 5008055801270156 T^{19} + 43546768385214819 T^{20} + 115755089450246417 T^{21} - 1994852993703715293 T^{22} - 1112992339605039428 T^{23} + 55596852513895170138 T^{24} - 10319222063208703954 T^{25} - \)\(94\!\cdots\!64\)\( T^{26} + \)\(48\!\cdots\!94\)\( T^{27} + \)\(90\!\cdots\!89\)\( T^{28} \)
$41$ \( 1 + 18 T - 29 T^{2} - 2200 T^{3} - 1130 T^{4} + 183989 T^{5} + 379300 T^{6} - 8058104 T^{7} - 4062155 T^{8} + 263971859 T^{9} - 1279880946 T^{10} - 2948011895 T^{11} + 160106068045 T^{12} + 57760123657 T^{13} - 7992127618244 T^{14} + 2368165069937 T^{15} + 269138300383645 T^{16} - 203179927815295 T^{17} - 3616637661849906 T^{18} + 30582776754647659 T^{19} - 19295659693099355 T^{20} - 1569350193377581624 T^{21} + 3028682139405595300 T^{22} + 60234674727210490429 T^{23} - 15167605020472213130 T^{24} - \)\(12\!\cdots\!00\)\( T^{25} - \)\(65\!\cdots\!49\)\( T^{26} + \)\(16\!\cdots\!78\)\( T^{27} + \)\(37\!\cdots\!61\)\( T^{28} \)
$43$ \( ( 1 + 3 T + 165 T^{2} + 625 T^{3} + 14336 T^{4} + 57190 T^{5} + 833372 T^{6} + 3148724 T^{7} + 35834996 T^{8} + 105744310 T^{9} + 1139812352 T^{10} + 2136750625 T^{11} + 24256393095 T^{12} + 18964089147 T^{13} + 271818611107 T^{14} )^{2} \)
$47$ \( 1 - 7 T - 153 T^{2} + 1362 T^{3} + 9935 T^{4} - 114473 T^{5} - 354685 T^{6} + 5522720 T^{7} + 11215060 T^{8} - 156793834 T^{9} - 900317666 T^{10} + 2422525493 T^{11} + 78249322737 T^{12} - 17047291839 T^{13} - 4482369027754 T^{14} - 801222716433 T^{15} + 172852753926033 T^{16} + 251513864259739 T^{17} - 4393263008744546 T^{18} - 35959882956286838 T^{19} + 120889546667654740 T^{20} + 2797937639843419360 T^{21} - 8445506209626700285 T^{22} - \)\(12\!\cdots\!91\)\( T^{23} + \)\(52\!\cdots\!15\)\( T^{24} + \)\(33\!\cdots\!86\)\( T^{25} - \)\(17\!\cdots\!73\)\( T^{26} - \)\(38\!\cdots\!89\)\( T^{27} + \)\(25\!\cdots\!69\)\( T^{28} \)
$53$ \( ( 1 + 6 T + 247 T^{2} + 1145 T^{3} + 30041 T^{4} + 116883 T^{5} + 2344387 T^{6} + 7652092 T^{7} + 124252511 T^{8} + 328324347 T^{9} + 4472413957 T^{10} + 9034600745 T^{11} + 103294286771 T^{12} + 132986166774 T^{13} + 1174711139837 T^{14} )^{2} \)
$59$ \( ( 1 + T + 289 T^{2} + 403 T^{3} + 39692 T^{4} + 63484 T^{5} + 3413740 T^{6} + 5088680 T^{7} + 201410660 T^{8} + 220987804 T^{9} + 8151903268 T^{10} + 4883296483 T^{11} + 206613122411 T^{12} + 42180533641 T^{13} + 2488651484819 T^{14} )^{2} \)
$61$ \( 1 - 250 T^{2} - 1082 T^{3} + 35548 T^{4} + 260866 T^{5} - 2717433 T^{6} - 36918565 T^{7} + 78765775 T^{8} + 3261082584 T^{9} + 10157211285 T^{10} - 189570224500 T^{11} - 1612043826243 T^{12} + 4675152702057 T^{13} + 127013861698650 T^{14} + 285184314825477 T^{15} - 5998415077450203 T^{16} - 43028839127234500 T^{17} + 140635132455515685 T^{18} + 2754298287701921784 T^{19} + 4058042214834294775 T^{20} - \)\(11\!\cdots\!65\)\( T^{21} - \)\(52\!\cdots\!73\)\( T^{22} + \)\(30\!\cdots\!06\)\( T^{23} + \)\(25\!\cdots\!48\)\( T^{24} - \)\(47\!\cdots\!02\)\( T^{25} - \)\(66\!\cdots\!50\)\( T^{26} + \)\(98\!\cdots\!41\)\( T^{28} \)
$67$ \( 1 - 25 T + 298 T^{2} - 2382 T^{3} + 12789 T^{4} - 5772 T^{5} - 884312 T^{6} + 10995854 T^{7} - 59248904 T^{8} - 25910508 T^{9} + 3846458007 T^{10} - 47999970222 T^{11} + 402337281886 T^{12} - 2261459554225 T^{13} + 6060711605323 T^{14} \)
$71$ \( 1 - 22 T - 17 T^{2} + 820 T^{3} + 49824 T^{4} - 364769 T^{5} - 2625066 T^{6} - 31439174 T^{7} + 604960277 T^{8} + 1291494445 T^{9} + 13047003346 T^{10} - 519002271697 T^{11} - 392108245783 T^{12} - 2884405046697 T^{13} + 359176316454548 T^{14} - 204792758315487 T^{15} - 1976617666992103 T^{16} - 185756622065344967 T^{17} + 331546287034484626 T^{18} + 2330152184322455195 T^{19} + 77495583244626806117 T^{20} - \)\(28\!\cdots\!34\)\( T^{21} - \)\(16\!\cdots\!26\)\( T^{22} - \)\(16\!\cdots\!39\)\( T^{23} + \)\(16\!\cdots\!24\)\( T^{24} + \)\(18\!\cdots\!20\)\( T^{25} - \)\(27\!\cdots\!97\)\( T^{26} - \)\(25\!\cdots\!42\)\( T^{27} + \)\(82\!\cdots\!81\)\( T^{28} \)
$73$ \( 1 + 15 T - 305 T^{2} - 3966 T^{3} + 75355 T^{4} + 692695 T^{5} - 13270379 T^{6} - 81309244 T^{7} + 1857676842 T^{8} + 7051146754 T^{9} - 209843950998 T^{10} - 423003183473 T^{11} + 19720732477631 T^{12} + 12001986433031 T^{13} - 1563096084841678 T^{14} + 876145009611263 T^{15} + 105091783373295599 T^{16} - 164555429425116041 T^{17} - 5959199092833394518 T^{18} + 14617532033791559122 T^{19} + \)\(28\!\cdots\!38\)\( T^{20} - \)\(89\!\cdots\!68\)\( T^{21} - \)\(10\!\cdots\!99\)\( T^{22} + \)\(40\!\cdots\!35\)\( T^{23} + \)\(32\!\cdots\!95\)\( T^{24} - \)\(12\!\cdots\!82\)\( T^{25} - \)\(69\!\cdots\!05\)\( T^{26} + \)\(25\!\cdots\!95\)\( T^{27} + \)\(12\!\cdots\!09\)\( T^{28} \)
$79$ \( 1 - 9 T - 290 T^{2} + 971 T^{3} + 58957 T^{4} + 39935 T^{5} - 7304652 T^{6} - 31245424 T^{7} + 627140605 T^{8} + 4416045423 T^{9} - 33081571720 T^{10} - 353555917085 T^{11} + 956394409296 T^{12} + 11107915331417 T^{13} - 6705152277818 T^{14} + 877525311181943 T^{15} + 5968857508416336 T^{16} - 174316855803671315 T^{17} - 1288529898101309320 T^{18} + 13588420827116811777 T^{19} + \)\(15\!\cdots\!05\)\( T^{20} - \)\(60\!\cdots\!16\)\( T^{21} - \)\(11\!\cdots\!72\)\( T^{22} + \)\(47\!\cdots\!65\)\( T^{23} + \)\(55\!\cdots\!57\)\( T^{24} + \)\(72\!\cdots\!09\)\( T^{25} - \)\(17\!\cdots\!90\)\( T^{26} - \)\(42\!\cdots\!51\)\( T^{27} + \)\(36\!\cdots\!81\)\( T^{28} \)
$83$ \( 1 + T - 299 T^{2} + 1040 T^{3} + 46121 T^{4} - 338171 T^{5} - 3970157 T^{6} + 54632160 T^{7} + 144189246 T^{8} - 5524282328 T^{9} + 13760244348 T^{10} + 359721042959 T^{11} - 3019786479455 T^{12} - 11121758694339 T^{13} + 306888352958254 T^{14} - 923105971630137 T^{15} - 20803309056965495 T^{16} + 205683815990397733 T^{17} + 653038093305819708 T^{18} - 21760372613398656904 T^{19} + 47141285923034589774 T^{20} + \)\(14\!\cdots\!20\)\( T^{21} - \)\(89\!\cdots\!37\)\( T^{22} - \)\(63\!\cdots\!13\)\( T^{23} + \)\(71\!\cdots\!29\)\( T^{24} + \)\(13\!\cdots\!80\)\( T^{25} - \)\(31\!\cdots\!39\)\( T^{26} + \)\(88\!\cdots\!63\)\( T^{27} + \)\(73\!\cdots\!29\)\( T^{28} \)
$89$ \( ( 1 + 6 T + 509 T^{2} + 2393 T^{3} + 116955 T^{4} + 439711 T^{5} + 16028883 T^{6} + 48830780 T^{7} + 1426570587 T^{8} + 3482950831 T^{9} + 82449649395 T^{10} + 150142182713 T^{11} + 2842286259541 T^{12} + 2981887745766 T^{13} + 44231334895529 T^{14} )^{2} \)
$97$ \( 1 - 16 T - 305 T^{2} + 4468 T^{3} + 66659 T^{4} - 627781 T^{5} - 13361122 T^{6} + 73927187 T^{7} + 2109835734 T^{8} - 7502842347 T^{9} - 269438721123 T^{10} + 493586727995 T^{11} + 30832175504259 T^{12} - 14324354895126 T^{13} - 3192872878082918 T^{14} - 1389462424827222 T^{15} + 290099939319572931 T^{16} + 450483279799380635 T^{17} - 23853216254578702563 T^{18} - 64429460128317463179 T^{19} + \)\(17\!\cdots\!86\)\( T^{20} + \)\(59\!\cdots\!31\)\( T^{21} - \)\(10\!\cdots\!42\)\( T^{22} - \)\(47\!\cdots\!77\)\( T^{23} + \)\(49\!\cdots\!91\)\( T^{24} + \)\(31\!\cdots\!04\)\( T^{25} - \)\(21\!\cdots\!05\)\( T^{26} - \)\(10\!\cdots\!32\)\( T^{27} + \)\(65\!\cdots\!69\)\( T^{28} \)
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