Properties

Label 4008.2.a.k
Level $4008$
Weight $2$
Character orbit 4008.a
Self dual yes
Analytic conductor $32.004$
Analytic rank $0$
Dimension $11$
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: \(0\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
Defining polynomial: \(x^{11} - x^{10} - 33 x^{9} + 22 x^{8} + 417 x^{7} - 151 x^{6} - 2470 x^{5} + 272 x^{4} + 6584 x^{3} + 292 x^{2} - 5687 x + 242\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{10}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} + ( 1 - \beta_{1} ) q^{5} + \beta_{3} q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} + ( 1 - \beta_{1} ) q^{5} + \beta_{3} q^{7} + q^{9} -\beta_{10} q^{11} + ( 1 - \beta_{4} ) q^{13} + ( -1 + \beta_{1} ) q^{15} + ( 1 - \beta_{4} + \beta_{5} + \beta_{7} ) q^{17} + ( -1 + \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{19} -\beta_{3} q^{21} + ( -1 + \beta_{6} + \beta_{8} - \beta_{9} ) q^{23} + ( 2 - 2 \beta_{1} + \beta_{2} ) q^{25} - q^{27} + ( 2 - \beta_{7} ) q^{29} + ( -\beta_{2} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} ) q^{31} + \beta_{10} q^{33} + ( 2 + 3 \beta_{3} - 2 \beta_{6} - \beta_{8} + \beta_{9} + \beta_{10} ) q^{35} + ( 1 - 2 \beta_{2} - \beta_{6} + \beta_{9} - \beta_{10} ) q^{37} + ( -1 + \beta_{4} ) q^{39} + ( 2 + \beta_{3} - \beta_{5} + \beta_{9} ) q^{41} + ( 1 - \beta_{1} + 2 \beta_{2} + \beta_{4} + \beta_{6} - \beta_{8} ) q^{43} + ( 1 - \beta_{1} ) q^{45} + ( -1 + \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - \beta_{8} ) q^{47} + ( 2 - 2 \beta_{1} + \beta_{2} + 2 \beta_{4} - \beta_{8} - \beta_{9} ) q^{49} + ( -1 + \beta_{4} - \beta_{5} - \beta_{7} ) q^{51} + ( 4 + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} - \beta_{9} + \beta_{10} ) q^{53} + ( -\beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - 3 \beta_{6} - \beta_{7} + \beta_{8} - \beta_{10} ) q^{55} + ( 1 - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{57} + ( \beta_{3} - \beta_{4} - \beta_{6} + \beta_{10} ) q^{59} + ( 2 \beta_{5} + 2 \beta_{6} + \beta_{7} + \beta_{10} ) q^{61} + \beta_{3} q^{63} + ( 1 - 2 \beta_{1} - \beta_{2} - \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{9} ) q^{65} + ( -1 - \beta_{3} + \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} ) q^{67} + ( 1 - \beta_{6} - \beta_{8} + \beta_{9} ) q^{69} + ( -2 - \beta_{2} - \beta_{4} + 2 \beta_{5} + \beta_{7} + 2 \beta_{8} + \beta_{9} + \beta_{10} ) q^{71} + ( 2 - \beta_{3} + \beta_{4} + \beta_{6} - \beta_{7} + \beta_{8} ) q^{73} + ( -2 + 2 \beta_{1} - \beta_{2} ) q^{75} + ( 4 + 2 \beta_{1} - \beta_{2} + 3 \beta_{4} - 2 \beta_{5} - 2 \beta_{7} - 2 \beta_{8} + \beta_{9} - \beta_{10} ) q^{77} + ( -1 - 2 \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} ) q^{79} + q^{81} + ( 1 + \beta_{2} - \beta_{3} + 2 \beta_{5} - \beta_{7} + \beta_{8} ) q^{83} + ( -2 \beta_{1} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} + \beta_{10} ) q^{85} + ( -2 + \beta_{7} ) q^{87} + ( 1 + 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} + 3 \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{9} ) q^{89} + ( -2 \beta_{5} + \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} ) q^{91} + ( \beta_{2} + \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} ) q^{93} + ( -1 + \beta_{2} - \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + \beta_{7} - 2 \beta_{9} + \beta_{10} ) q^{95} + ( 1 - 2 \beta_{1} - 2 \beta_{2} - \beta_{3} - 3 \beta_{6} - \beta_{10} ) q^{97} -\beta_{10} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11q - 11q^{3} + 10q^{5} - q^{7} + 11q^{9} + O(q^{10}) \) \( 11q - 11q^{3} + 10q^{5} - q^{7} + 11q^{9} - q^{11} + 10q^{13} - 10q^{15} + 17q^{17} + 2q^{19} + q^{21} - 3q^{23} + 21q^{25} - 11q^{27} + 17q^{29} - 15q^{31} + q^{33} + 11q^{35} + 4q^{37} - 10q^{39} + 16q^{41} + 10q^{43} + 10q^{45} - 16q^{47} + 22q^{49} - 17q^{51} + 42q^{53} - 5q^{55} - 2q^{57} - 2q^{59} + 12q^{61} - q^{63} + 10q^{65} - q^{67} + 3q^{69} - 9q^{71} + 24q^{73} - 21q^{75} + 22q^{77} - 30q^{79} + 11q^{81} + 16q^{83} + 25q^{85} - 17q^{87} + 37q^{89} + q^{91} + 15q^{93} + 5q^{95} + 4q^{97} - q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{11} - x^{10} - 33 x^{9} + 22 x^{8} + 417 x^{7} - 151 x^{6} - 2470 x^{5} + 272 x^{4} + 6584 x^{3} + 292 x^{2} - 5687 x + 242\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 6 \)
\(\beta_{3}\)\(=\)\((\)\(2623 \nu^{10} - 70944 \nu^{9} - 193391 \nu^{8} + 2378035 \nu^{7} + 2748114 \nu^{6} - 23458607 \nu^{5} - 15128561 \nu^{4} + 74577279 \nu^{3} + 34805559 \nu^{2} - 38238885 \nu - 7710406\)\()/10323192\)
\(\beta_{4}\)\(=\)\((\)\( -1379 \nu^{10} + 55664 \nu^{9} - 65533 \nu^{8} - 1226719 \nu^{7} + 1417390 \nu^{6} + 9091923 \nu^{5} - 7692731 \nu^{4} - 25269067 \nu^{3} + 11595133 \nu^{2} + 19499297 \nu - 2478234 \)\()/1876944\)
\(\beta_{5}\)\(=\)\((\)\(64985 \nu^{10} - 195192 \nu^{9} - 1989097 \nu^{8} + 5477813 \nu^{7} + 21384366 \nu^{6} - 52505737 \nu^{5} - 92700271 \nu^{4} + 195304833 \nu^{3} + 124569801 \nu^{2} - 220910235 \nu + 34175614\)\()/10323192\)
\(\beta_{6}\)\(=\)\((\)\(101765 \nu^{10} - 492056 \nu^{9} - 2608397 \nu^{8} + 12499993 \nu^{7} + 25094846 \nu^{6} - 110447061 \nu^{5} - 109201699 \nu^{4} + 387400045 \nu^{3} + 176633261 \nu^{2} - 427193567 \nu + 39160638\)\()/10323192\)
\(\beta_{7}\)\(=\)\((\)\(116141 \nu^{10} - 242652 \nu^{9} - 2909401 \nu^{8} + 5468837 \nu^{7} + 25357890 \nu^{6} - 43356601 \nu^{5} - 87108655 \nu^{4} + 142378425 \nu^{3} + 86939145 \nu^{2} - 163661007 \nu + 20257270\)\()/5161596\)
\(\beta_{8}\)\(=\)\((\)\(-522481 \nu^{10} + 1421984 \nu^{9} + 13457169 \nu^{8} - 32908293 \nu^{7} - 124595270 \nu^{6} + 263695489 \nu^{5} + 481600775 \nu^{4} - 850380841 \nu^{3} - 618583841 \nu^{2} + 928868747 \nu - 65756350\)\()/20646384\)
\(\beta_{9}\)\(=\)\((\)\(-596041 \nu^{10} + 2015712 \nu^{9} + 14695769 \nu^{8} - 46952653 \nu^{7} - 132016230 \nu^{6} + 379578137 \nu^{5} + 514603631 \nu^{4} - 1213924881 \nu^{3} - 743357145 \nu^{2} + 1176264339 \nu - 13787246\)\()/20646384\)
\(\beta_{10}\)\(=\)\((\)\(303385 \nu^{10} - 1032256 \nu^{9} - 7769641 \nu^{8} + 25611773 \nu^{7} + 71466478 \nu^{6} - 223268481 \nu^{5} - 278511527 \nu^{4} + 789881825 \nu^{3} + 385046857 \nu^{2} - 908813755 \nu + 47745918\)\()/10323192\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 6\)
\(\nu^{3}\)\(=\)\(\beta_{9} - \beta_{8} + \beta_{6} - \beta_{5} + \beta_{2} + 8 \beta_{1} + 3\)
\(\nu^{4}\)\(=\)\(-2 \beta_{10} + \beta_{9} + 2 \beta_{7} + 3 \beta_{6} + 2 \beta_{5} - \beta_{4} - \beta_{3} + 11 \beta_{2} + 4 \beta_{1} + 48\)
\(\nu^{5}\)\(=\)\(-4 \beta_{10} + 14 \beta_{9} - 14 \beta_{8} + 3 \beta_{7} + 19 \beta_{6} - 14 \beta_{5} + 3 \beta_{3} + 17 \beta_{2} + 73 \beta_{1} + 50\)
\(\nu^{6}\)\(=\)\(-39 \beta_{10} + 17 \beta_{9} - 2 \beta_{8} + 35 \beta_{7} + 68 \beta_{6} + 20 \beta_{5} - 9 \beta_{4} - 16 \beta_{3} + 122 \beta_{2} + 74 \beta_{1} + 432\)
\(\nu^{7}\)\(=\)\(-89 \beta_{10} + 167 \beta_{9} - 151 \beta_{8} + 77 \beta_{7} + 294 \beta_{6} - 163 \beta_{5} + 4 \beta_{4} + 54 \beta_{3} + 241 \beta_{2} + 722 \beta_{1} + 656\)
\(\nu^{8}\)\(=\)\(-576 \beta_{10} + 244 \beta_{9} - 31 \beta_{8} + 516 \beta_{7} + 1101 \beta_{6} + 116 \beta_{5} - 55 \beta_{4} - 186 \beta_{3} + 1378 \beta_{2} + 1072 \beta_{1} + 4199\)
\(\nu^{9}\)\(=\)\(-1462 \beta_{10} + 1922 \beta_{9} - 1489 \beta_{8} + 1378 \beta_{7} + 4200 \beta_{6} - 1860 \beta_{5} + 147 \beta_{4} + 704 \beta_{3} + 3209 \beta_{2} + 7555 \beta_{1} + 8094\)
\(\nu^{10}\)\(=\)\(-7771 \beta_{10} + 3303 \beta_{9} - 341 \beta_{8} + 7202 \beta_{7} + 15782 \beta_{6} - 172 \beta_{5} - 44 \beta_{4} - 1868 \beta_{3} + 15861 \beta_{2} + 14338 \beta_{1} + 43210\)

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
3.54432
3.11942
2.80306
2.38119
1.09328
0.0427374
−1.60189
−2.17907
−2.34947
−2.83671
−3.01685
0 −1.00000 0 −2.54432 0 0.525270 0 1.00000 0
1.2 0 −1.00000 0 −2.11942 0 −0.802640 0 1.00000 0
1.3 0 −1.00000 0 −1.80306 0 −4.10861 0 1.00000 0
1.4 0 −1.00000 0 −1.38119 0 −0.260099 0 1.00000 0
1.5 0 −1.00000 0 −0.0932775 0 3.86231 0 1.00000 0
1.6 0 −1.00000 0 0.957263 0 −0.898491 0 1.00000 0
1.7 0 −1.00000 0 2.60189 0 −3.58131 0 1.00000 0
1.8 0 −1.00000 0 3.17907 0 0.651548 0 1.00000 0
1.9 0 −1.00000 0 3.34947 0 3.54581 0 1.00000 0
1.10 0 −1.00000 0 3.83671 0 4.48179 0 1.00000 0
1.11 0 −1.00000 0 4.01685 0 −4.41557 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.11
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 4008.2.a.k 11
4.b odd 2 1 8016.2.a.be 11
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4008.2.a.k 11 1.a even 1 1 trivial
8016.2.a.be 11 4.b odd 2 1

Hecke kernels

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

\(T_{5}^{11} - \cdots\)
\(T_{7}^{11} + \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 + T )^{11} \)
$5$ \( 1 - 10 T + 67 T^{2} - 345 T^{3} + 1530 T^{4} - 5914 T^{5} + 20615 T^{6} - 65174 T^{7} + 189802 T^{8} - 509486 T^{9} + 1271685 T^{10} - 2945398 T^{11} + 6358425 T^{12} - 12737150 T^{13} + 23725250 T^{14} - 40733750 T^{15} + 64421875 T^{16} - 92406250 T^{17} + 119531250 T^{18} - 134765625 T^{19} + 130859375 T^{20} - 97656250 T^{21} + 48828125 T^{22} \)
$7$ \( 1 + T + 28 T^{2} + 24 T^{3} + 408 T^{4} + 321 T^{5} + 4798 T^{6} + 3540 T^{7} + 47411 T^{8} + 31342 T^{9} + 378850 T^{10} + 230184 T^{11} + 2651950 T^{12} + 1535758 T^{13} + 16261973 T^{14} + 8499540 T^{15} + 80639986 T^{16} + 37765329 T^{17} + 336005544 T^{18} + 138355224 T^{19} + 1129900996 T^{20} + 282475249 T^{21} + 1977326743 T^{22} \)
$11$ \( 1 + T + 48 T^{2} + 26 T^{3} + 1293 T^{4} + 412 T^{5} + 25538 T^{6} + 4366 T^{7} + 402125 T^{8} + 33083 T^{9} + 5243181 T^{10} + 242368 T^{11} + 57674991 T^{12} + 4003043 T^{13} + 535228375 T^{14} + 63922606 T^{15} + 4112920438 T^{16} + 729883132 T^{17} + 25196912103 T^{18} + 5573330906 T^{19} + 113181489168 T^{20} + 25937424601 T^{21} + 285311670611 T^{22} \)
$13$ \( 1 - 10 T + 116 T^{2} - 816 T^{3} + 5728 T^{4} - 31413 T^{5} + 169547 T^{6} - 774510 T^{7} + 3520316 T^{8} - 14058633 T^{9} + 56398720 T^{10} - 202572420 T^{11} + 733183360 T^{12} - 2375908977 T^{13} + 7734134252 T^{14} - 22120780110 T^{15} + 62951614271 T^{16} - 151624551117 T^{17} + 359423505376 T^{18} - 665636268336 T^{19} + 1230121927268 T^{20} - 1378584918490 T^{21} + 1792160394037 T^{22} \)
$17$ \( 1 - 17 T + 241 T^{2} - 2426 T^{3} + 21464 T^{4} - 159918 T^{5} + 1080486 T^{6} - 6482792 T^{7} + 35861503 T^{8} - 180071453 T^{9} + 840324305 T^{10} - 3589741012 T^{11} + 14285513185 T^{12} - 52040649917 T^{13} + 176187564239 T^{14} - 541449270632 T^{15} + 1534135610502 T^{16} - 3860031759342 T^{17} + 8807509277272 T^{18} - 16923187551866 T^{19} + 28579678235777 T^{20} - 34271896307633 T^{21} + 34271896307633 T^{22} \)
$19$ \( 1 - 2 T + 104 T^{2} - 96 T^{3} + 5444 T^{4} + 761 T^{5} + 186103 T^{6} + 224246 T^{7} + 4770170 T^{8} + 9884721 T^{9} + 101541836 T^{10} + 240128532 T^{11} + 1929294884 T^{12} + 3568384281 T^{13} + 32718596030 T^{14} + 29223962966 T^{15} + 460809452197 T^{16} + 35801915441 T^{17} + 4866237747116 T^{18} - 1630422051936 T^{19} + 33559520569016 T^{20} - 12262132515602 T^{21} + 116490258898219 T^{22} \)
$23$ \( 1 + 3 T + 162 T^{2} + 512 T^{3} + 12407 T^{4} + 43808 T^{5} + 605456 T^{6} + 2403850 T^{7} + 21488859 T^{8} + 91406541 T^{9} + 600918697 T^{10} + 2479545100 T^{11} + 13821130031 T^{12} + 48354060189 T^{13} + 261454947453 T^{14} + 672695787850 T^{15} + 3896922487408 T^{16} + 6485156225312 T^{17} + 42243669320929 T^{18} + 40095224463872 T^{19} + 291786731157006 T^{20} + 124279533640947 T^{21} + 952809757913927 T^{22} \)
$29$ \( 1 - 17 T + 374 T^{2} - 4462 T^{3} + 57451 T^{4} - 534092 T^{5} + 5082040 T^{6} - 38682606 T^{7} + 295198523 T^{8} - 1885119491 T^{9} + 11965413375 T^{10} - 64710360584 T^{11} + 346996987875 T^{12} - 1585385491931 T^{13} + 7199596777447 T^{14} - 27359472254286 T^{15} + 104238479663960 T^{16} - 317690377159532 T^{17} + 991022643828359 T^{18} - 2232099494631982 T^{19} + 5425672594975006 T^{20} - 7152022966103417 T^{21} + 12200509765705829 T^{22} \)
$31$ \( 1 + 15 T + 277 T^{2} + 3289 T^{3} + 37064 T^{4} + 354644 T^{5} + 3091453 T^{6} + 24533802 T^{7} + 178218002 T^{8} + 1198477245 T^{9} + 7456864249 T^{10} + 43067489466 T^{11} + 231162791719 T^{12} + 1151736632445 T^{13} + 5309292497582 T^{14} + 22657481356842 T^{15} + 88505674746403 T^{16} + 314747855444564 T^{17} + 1019727529410104 T^{18} + 2805158622143449 T^{19} + 7323775338505867 T^{20} + 12294424304712015 T^{21} + 25408476896404831 T^{22} \)
$37$ \( 1 - 4 T + 183 T^{2} - 607 T^{3} + 15052 T^{4} - 47184 T^{5} + 792027 T^{6} - 3114852 T^{7} + 34303738 T^{8} - 186828452 T^{9} + 1400308163 T^{10} - 8356445354 T^{11} + 51811402031 T^{12} - 255768150788 T^{13} + 1737587240914 T^{14} - 5837734139172 T^{15} + 54922286230839 T^{16} - 121061234882256 T^{17} + 1428914614605916 T^{18} - 2132075028530047 T^{19} + 23782998382499091 T^{20} - 19234337489671396 T^{21} + 177917621779460413 T^{22} \)
$41$ \( 1 - 16 T + 399 T^{2} - 4853 T^{3} + 70135 T^{4} - 690939 T^{5} + 7427693 T^{6} - 61625762 T^{7} + 540949428 T^{8} - 3875396693 T^{9} + 29071797776 T^{10} - 182118029858 T^{11} + 1191943708816 T^{12} - 6514541840933 T^{13} + 37282775527188 T^{14} - 174139674854882 T^{15} + 860544293174293 T^{16} - 3282032274172299 T^{17} + 13659090998643935 T^{18} - 38750842136924213 T^{19} + 130625391823190439 T^{20} - 214762548962438416 T^{21} + 550329031716248441 T^{22} \)
$43$ \( 1 - 10 T + 279 T^{2} - 2075 T^{3} + 36023 T^{4} - 221003 T^{5} + 3082429 T^{6} - 16403326 T^{7} + 200110078 T^{8} - 952121859 T^{9} + 10478092890 T^{10} - 45165864910 T^{11} + 450557994270 T^{12} - 1760473317291 T^{13} + 15910151971546 T^{14} - 56079707332126 T^{15} + 453143087948047 T^{16} - 1397040197918147 T^{17} + 9791721827907461 T^{18} - 24253015576022075 T^{19} + 140223338730379197 T^{20} - 216114823132842490 T^{21} + 929293739471222707 T^{22} \)
$47$ \( 1 + 16 T + 307 T^{2} + 3330 T^{3} + 38070 T^{4} + 298909 T^{5} + 2385155 T^{6} + 12604810 T^{7} + 66749748 T^{8} + 88508431 T^{9} - 248658699 T^{10} - 10612004112 T^{11} - 11686958853 T^{12} + 195515124079 T^{13} + 6930159086604 T^{14} + 61507451865610 T^{15} + 547023390171085 T^{16} + 3222004474776061 T^{17} + 19287142196026410 T^{18} + 79291584583664130 T^{19} + 343573055242549469 T^{20} + 841586115773280784 T^{21} + 2472159215084012303 T^{22} \)
$53$ \( 1 - 42 T + 1173 T^{2} - 23620 T^{3} + 389854 T^{4} - 5399067 T^{5} + 65490503 T^{6} - 703633882 T^{7} + 6846296048 T^{8} - 60676194697 T^{9} + 496287074001 T^{10} - 3748573238368 T^{11} + 26303214922053 T^{12} - 170439430903873 T^{13} + 1019256016738096 T^{14} - 5552009776877242 T^{15} + 27387833188902979 T^{16} - 119666870747666643 T^{17} + 457965836710013798 T^{18} - 1470573887516346820 T^{19} + 3870622693183902009 T^{20} - 7345273755351548058 T^{21} + 9269035929372191597 T^{22} \)
$59$ \( 1 + 2 T + 505 T^{2} + 666 T^{3} + 120674 T^{4} + 94581 T^{5} + 18110815 T^{6} + 7300032 T^{7} + 1907206020 T^{8} + 344138969 T^{9} + 148634314931 T^{10} + 15352893564 T^{11} + 8769424580929 T^{12} + 1197947751089 T^{13} + 391700065181580 T^{14} + 88457123055552 T^{15} + 12947861718193685 T^{16} + 3989477052299421 T^{17} + 300315529279048006 T^{18} + 97789071444477786 T^{19} + 4374812888420744195 T^{20} + 1022233506601282802 T^{21} + 30155888444737842659 T^{22} \)
$61$ \( 1 - 12 T + 340 T^{2} - 3384 T^{3} + 57488 T^{4} - 463101 T^{5} + 6208719 T^{6} - 41463550 T^{7} + 488568268 T^{8} - 2821350367 T^{9} + 32150979308 T^{10} - 172424859412 T^{11} + 1961209737788 T^{12} - 10498244715607 T^{13} + 110895714038908 T^{14} - 574097720595550 T^{15} + 5243861101348419 T^{16} - 23859136886953461 T^{17} + 180670000157175248 T^{18} - 648737547182798904 T^{19} + 3976009671563607940 T^{20} - 8560114939954591212 T^{21} + 43513917611435838661 T^{22} \)
$67$ \( 1 + T + 436 T^{2} + 865 T^{3} + 97490 T^{4} + 247038 T^{5} + 14590774 T^{6} + 40768042 T^{7} + 1610450641 T^{8} + 4515496635 T^{9} + 137204831826 T^{10} + 355748041550 T^{11} + 9192723732342 T^{12} + 20270064394515 T^{13} + 484363966139083 T^{14} + 821521747275082 T^{15} + 19699370307962818 T^{16} + 22346657814265422 T^{17} + 590858774402939270 T^{18} + 351248541086494465 T^{19} + 11862048996784596892 T^{20} + 1822837804551761449 T^{21} + \)\(12\!\cdots\!83\)\( T^{22} \)
$71$ \( 1 + 9 T + 407 T^{2} + 2019 T^{3} + 72682 T^{4} + 216594 T^{5} + 9495944 T^{6} + 23315926 T^{7} + 1016193411 T^{8} + 2055559861 T^{9} + 85180979199 T^{10} + 141342461934 T^{11} + 6047849523129 T^{12} + 10362077259301 T^{13} + 363706799924421 T^{14} + 592496873731606 T^{15} + 17132860880252344 T^{16} + 27745752895585074 T^{17} + 661051523352174662 T^{18} + 1303776379585191459 T^{19} + 18660339792408755617 T^{20} + 29297191959088930809 T^{21} + \)\(23\!\cdots\!71\)\( T^{22} \)
$73$ \( 1 - 24 T + 832 T^{2} - 14672 T^{3} + 296314 T^{4} - 4148141 T^{5} + 62082611 T^{6} - 719899178 T^{7} + 8699232388 T^{8} - 85544428603 T^{9} + 867998838438 T^{10} - 7310385685164 T^{11} + 63363915205974 T^{12} - 455866260025387 T^{13} + 3384149285882596 T^{14} - 20443870352545898 T^{15} + 128701697283369323 T^{16} - 627755708772678749 T^{17} + 3273498844787708458 T^{18} - 11832382468269956432 T^{19} + 48981160141278903616 T^{20} - \)\(10\!\cdots\!76\)\( T^{21} + \)\(31\!\cdots\!77\)\( T^{22} \)
$79$ \( 1 + 30 T + 645 T^{2} + 9619 T^{3} + 123851 T^{4} + 1379707 T^{5} + 14867779 T^{6} + 154181526 T^{7} + 1571210460 T^{8} + 15355190883 T^{9} + 144073470752 T^{10} + 1303460329558 T^{11} + 11381804189408 T^{12} + 95831746300803 T^{13} + 774668033987940 T^{14} + 6005382926403606 T^{15} + 45748994510867821 T^{16} + 335389463994512347 T^{17} + 2378423331844778309 T^{18} + 14593069642491210259 T^{19} + 77304279408788815755 T^{20} + \)\(28\!\cdots\!30\)\( T^{21} + \)\(74\!\cdots\!79\)\( T^{22} \)
$83$ \( 1 - 16 T + 559 T^{2} - 8890 T^{3} + 166834 T^{4} - 2332409 T^{5} + 33323265 T^{6} - 397711016 T^{7} + 4765849894 T^{8} - 49629010023 T^{9} + 511273556061 T^{10} - 4708369901132 T^{11} + 42435705153063 T^{12} - 341894250048447 T^{13} + 2725051013340578 T^{14} - 18874697062564136 T^{15} + 131261695192459395 T^{16} - 762558669309215921 T^{17} + 4527215930803430918 T^{18} - 20022877943716074490 T^{19} + \)\(10\!\cdots\!77\)\( T^{20} - \)\(24\!\cdots\!84\)\( T^{21} + \)\(12\!\cdots\!67\)\( T^{22} \)
$89$ \( 1 - 37 T + 1176 T^{2} - 25175 T^{3} + 484854 T^{4} - 7662396 T^{5} + 112691726 T^{6} - 1458814172 T^{7} + 17904001283 T^{8} - 198977030567 T^{9} + 2106943642192 T^{10} - 20364870673674 T^{11} + 187517984155088 T^{12} - 1576097059121207 T^{13} + 12621765880475227 T^{14} - 91529270353839452 T^{15} + 629277297394418974 T^{16} - 3808067455934402556 T^{17} + 21445739649436817766 T^{18} - 99103623183549889175 T^{19} + \)\(41\!\cdots\!84\)\( T^{20} - \)\(11\!\cdots\!37\)\( T^{21} + \)\(27\!\cdots\!89\)\( T^{22} \)
$97$ \( 1 - 4 T + 512 T^{2} - 1207 T^{3} + 132392 T^{4} - 161794 T^{5} + 24030516 T^{6} - 13362994 T^{7} + 3429379285 T^{8} - 700457026 T^{9} + 400079268110 T^{10} - 34763160478 T^{11} + 38807689006670 T^{12} - 6590600157634 T^{13} + 3129901880178805 T^{14} - 1183016250827314 T^{15} + 206358217443282612 T^{16} - 134769872565482626 T^{17} + 10697046478626336296 T^{18} - 9459782348412991927 T^{19} + \)\(38\!\cdots\!04\)\( T^{20} - \)\(29\!\cdots\!96\)\( T^{21} + \)\(71\!\cdots\!53\)\( T^{22} \)
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