Properties

Label 8036.2.a.s
Level 8036
Weight 2
Character orbit 8036.a
Self dual yes
Analytic conductor 64.168
Analytic rank 1
Dimension 20
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 8036 = 2^{2} \cdot 7^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8036.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.1677830643\)
Analytic rank: \(1\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial: \(x^{20} - 4 x^{19} - 30 x^{18} + 128 x^{17} + 348 x^{16} - 1644 x^{15} - 1934 x^{14} + 10948 x^{13} + 4748 x^{12} - 40524 x^{11} - 220 x^{10} + 82500 x^{9} - 21992 x^{8} - 84720 x^{7} + 37544 x^{6} + 34656 x^{5} - 18823 x^{4} - 2276 x^{3} + 1130 x^{2} + 204 x + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{3} + \beta_{11} q^{5} + ( 1 + \beta_{2} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{3} + \beta_{11} q^{5} + ( 1 + \beta_{2} ) q^{9} + ( -1 + \beta_{1} - \beta_{7} ) q^{11} + ( -1 - \beta_{14} ) q^{13} + ( \beta_{1} + \beta_{6} - \beta_{11} - \beta_{15} ) q^{15} -\beta_{18} q^{17} + ( -1 + \beta_{5} + \beta_{11} + \beta_{15} - \beta_{16} ) q^{19} + ( -\beta_{2} + \beta_{4} - \beta_{6} + \beta_{9} - \beta_{10} - 2 \beta_{11} - \beta_{12} - \beta_{13} + \beta_{14} + \beta_{16} ) q^{23} + ( \beta_{1} - \beta_{2} - \beta_{6} - \beta_{10} - 2 \beta_{11} - \beta_{12} + \beta_{18} - \beta_{19} ) q^{25} + ( -1 - \beta_{1} - \beta_{3} + \beta_{10} + 2 \beta_{11} + \beta_{12} + \beta_{15} - \beta_{16} + \beta_{17} ) q^{27} + ( -\beta_{1} + \beta_{2} - \beta_{4} + \beta_{6} + \beta_{7} + \beta_{10} + \beta_{11} + 2 \beta_{12} + \beta_{13} + \beta_{15} ) q^{29} + ( \beta_{3} + \beta_{4} - \beta_{8} - \beta_{11} - \beta_{12} + \beta_{18} ) q^{31} + ( -2 - \beta_{2} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{11} + \beta_{14} - \beta_{15} + \beta_{16} - \beta_{17} + \beta_{19} ) q^{33} + ( -\beta_{2} - \beta_{5} + \beta_{6} - \beta_{7} - \beta_{9} - \beta_{11} - \beta_{13} + \beta_{14} - \beta_{15} ) q^{37} + ( 2 + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{6} + 2 \beta_{7} + \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} + \beta_{16} - \beta_{17} + \beta_{19} ) q^{39} + q^{41} + ( \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} - \beta_{11} + \beta_{13} + \beta_{14} - 2 \beta_{15} + \beta_{16} + \beta_{19} ) q^{43} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} + 2 \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} + \beta_{16} + \beta_{19} ) q^{45} + ( -\beta_{4} + \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} + \beta_{12} + \beta_{13} - \beta_{19} ) q^{47} + ( 1 + \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{10} + 2 \beta_{11} + \beta_{12} - \beta_{14} + 2 \beta_{15} - \beta_{16} - \beta_{18} - \beta_{19} ) q^{51} + ( -2 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{11} - 2 \beta_{12} - \beta_{13} - \beta_{16} + \beta_{17} + \beta_{18} ) q^{53} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{11} - \beta_{12} + \beta_{16} ) q^{55} + ( 1 + \beta_{1} + \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{8} + \beta_{9} - 2 \beta_{11} - 2 \beta_{12} + \beta_{15} + \beta_{18} ) q^{57} + ( -\beta_{1} - 2 \beta_{2} - \beta_{4} - \beta_{6} + \beta_{7} - 2 \beta_{11} - \beta_{12} - \beta_{13} - \beta_{15} + \beta_{16} - \beta_{17} + \beta_{18} - \beta_{19} ) q^{59} + ( -3 + \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} - \beta_{13} + \beta_{15} - \beta_{17} - \beta_{19} ) q^{61} + ( -1 - 2 \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{6} - \beta_{8} + \beta_{9} + \beta_{10} + \beta_{12} + \beta_{13} + \beta_{15} - \beta_{16} + \beta_{18} + \beta_{19} ) q^{65} + ( 2 \beta_{1} + \beta_{2} - \beta_{6} - \beta_{7} + \beta_{11} + \beta_{12} - \beta_{18} ) q^{67} + ( -3 + 2 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} + \beta_{5} + \beta_{6} + \beta_{8} - \beta_{9} - 2 \beta_{10} - 2 \beta_{11} - \beta_{12} + 2 \beta_{14} - \beta_{15} + \beta_{17} ) q^{69} + ( \beta_{1} + 2 \beta_{2} - \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{9} + \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} - \beta_{14} - \beta_{16} - \beta_{18} + \beta_{19} ) q^{71} + ( -3 + \beta_{2} + \beta_{4} + \beta_{7} + \beta_{8} + 2 \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} + \beta_{14} + 2 \beta_{17} ) q^{73} + ( -4 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} - \beta_{11} - \beta_{13} - \beta_{17} ) q^{75} + ( -3 + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} - \beta_{8} + 2 \beta_{9} - \beta_{10} - 2 \beta_{11} - \beta_{13} - \beta_{14} ) q^{79} + ( 3 \beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{7} - 2 \beta_{10} - \beta_{11} - \beta_{13} - \beta_{16} + \beta_{17} - \beta_{18} - 2 \beta_{19} ) q^{81} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{7} - \beta_{10} + \beta_{12} + \beta_{13} + \beta_{14} - \beta_{15} + \beta_{16} + \beta_{17} + \beta_{19} ) q^{83} + ( -2 + \beta_{2} + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{9} + \beta_{12} + \beta_{14} + \beta_{15} - \beta_{16} - \beta_{18} + \beta_{19} ) q^{85} + ( -2 + \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - \beta_{6} - 3 \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} - \beta_{13} - 2 \beta_{14} + \beta_{15} - \beta_{16} - \beta_{19} ) q^{87} + ( -4 + \beta_{3} - \beta_{5} - 2 \beta_{11} - \beta_{12} - \beta_{14} - \beta_{15} - \beta_{17} + \beta_{19} ) q^{89} + ( \beta_{1} - \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{8} - \beta_{9} + 2 \beta_{11} + \beta_{12} + \beta_{15} - \beta_{16} - \beta_{18} - \beta_{19} ) q^{93} + ( 1 + 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - 4 \beta_{6} + \beta_{7} - \beta_{11} - \beta_{12} - \beta_{13} - \beta_{15} - \beta_{18} ) q^{95} + ( -1 + \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{8} + \beta_{9} + \beta_{10} + \beta_{12} - \beta_{17} - \beta_{18} ) q^{97} + ( -2 + 4 \beta_{1} - 2 \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{9} - 3 \beta_{10} - 4 \beta_{11} - 3 \beta_{12} - \beta_{13} - \beta_{14} - \beta_{15} + \beta_{16} + \beta_{18} - \beta_{19} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q - 4q^{3} - 8q^{5} + 16q^{9} + O(q^{10}) \) \( 20q - 4q^{3} - 8q^{5} + 16q^{9} - 8q^{11} - 12q^{13} + 8q^{15} - 8q^{17} - 24q^{19} + 8q^{23} + 20q^{25} - 16q^{27} - 12q^{29} - 44q^{33} + 12q^{37} + 12q^{39} + 20q^{41} + 4q^{43} - 40q^{45} - 4q^{47} + 4q^{51} - 12q^{53} + 16q^{55} + 28q^{57} - 16q^{59} - 68q^{61} - 8q^{65} + 4q^{67} - 32q^{69} + 8q^{71} - 48q^{73} - 60q^{75} - 20q^{79} + 32q^{81} + 8q^{83} - 28q^{85} - 60q^{89} - 16q^{93} + 20q^{95} - 40q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} - 4 x^{19} - 30 x^{18} + 128 x^{17} + 348 x^{16} - 1644 x^{15} - 1934 x^{14} + 10948 x^{13} + 4748 x^{12} - 40524 x^{11} - 220 x^{10} + 82500 x^{9} - 21992 x^{8} - 84720 x^{7} + 37544 x^{6} + 34656 x^{5} - 18823 x^{4} - 2276 x^{3} + 1130 x^{2} + 204 x + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 4 \)
\(\beta_{3}\)\(=\)\((\)\(693771077180 \nu^{19} - 2778645197913 \nu^{18} - 20725264188092 \nu^{17} + 88680236252730 \nu^{16} + 238604415918489 \nu^{15} - 1134655223791407 \nu^{14} - 1304931363722533 \nu^{13} + 7513711673122948 \nu^{12} + 3042707531442513 \nu^{11} - 27567693836690814 \nu^{10} + 811667273331286 \nu^{9} + 55269502351641049 \nu^{8} - 17304021386138605 \nu^{7} - 54976710977041287 \nu^{6} + 28198262762586312 \nu^{5} + 20402313321661661 \nu^{4} - 13852123920661924 \nu^{3} - 196305191328821 \nu^{2} + 738083784597603 \nu + 59952362841513\)\()/ 8907432980628 \)
\(\beta_{4}\)\(=\)\((\)\(1567892632533415234 \nu^{19} - 6277851071446228846 \nu^{18} - 47074158679250201267 \nu^{17} + 200987274480806224623 \nu^{16} + 547064973518762056275 \nu^{15} - 2583040292296088360850 \nu^{14} - 3055149489179038791221 \nu^{13} + 17214742740867450156004 \nu^{12} + 7637148660948016713900 \nu^{11} - 63775711863953965507911 \nu^{10} - 1268832263838809426215 \nu^{9} + 129922976197309212858705 \nu^{8} - 31996738533222421425408 \nu^{7} - 133301820620661470919197 \nu^{6} + 55417302249307152484678 \nu^{5} + 53967376288662842383274 \nu^{4} - 27515570123960339467871 \nu^{3} - 2950763457356350118969 \nu^{2} + 1555738005610136792040 \nu + 190983464209108996263\)\()/ 7045699320779922348 \)
\(\beta_{5}\)\(=\)\((\)\(2338757660528812226 \nu^{19} - 9262887357902091940 \nu^{18} - 70424247764763687101 \nu^{17} + 296231174702212346977 \nu^{16} + 822244089607423348755 \nu^{15} - 3801108458647015368204 \nu^{14} - 4630220852871028587469 \nu^{13} + 25275099625931947407732 \nu^{12} + 11815139014102216431238 \nu^{11} - 93326413146857308210869 \nu^{10} - 3140475534235154283977 \nu^{9} + 189167220552118841239433 \nu^{8} - 46054021734106426917342 \nu^{7} - 192459021916121267807269 \nu^{6} + 82085159614420114983234 \nu^{5} + 76448600437103562826964 \nu^{4} - 41370391953693733063937 \nu^{3} - 3540129341608367956689 \nu^{2} + 2351603977580673921478 \nu + 237506550933062053047\)\()/ 7045699320779922348 \)
\(\beta_{6}\)\(=\)\((\)\(6661373649057 \nu^{19} - 27339265673408 \nu^{18} - 197062564273797 \nu^{17} + 873381091267388 \nu^{16} + 2229477793619106 \nu^{15} - 11189902694968197 \nu^{14} - 11748441413484831 \nu^{13} + 74233650073598569 \nu^{12} + 24114490412599688 \nu^{11} - 272988213285828381 \nu^{10} + 26102191633898274 \nu^{9} + 548751658773871214 \nu^{8} - 201766431641702593 \nu^{7} - 547047554161970435 \nu^{6} + 305071323257237295 \nu^{5} + 202658302419133080 \nu^{4} - 145789349517861572 \nu^{3} - 1309162504591808 \nu^{2} + 7723657414763231 \nu + 620836439810025\)\()/ 8907432980628 \)
\(\beta_{7}\)\(=\)\((\)\(1049275988912470441 \nu^{19} - 4334271434248616952 \nu^{18} - 30930718070891675108 \nu^{17} + 138415002843547076609 \nu^{16} + 347664092105664073677 \nu^{15} - 1772518274441041104831 \nu^{14} - 1805440454952041490734 \nu^{13} + 11749994362571690048087 \nu^{12} + 3498116413657850939414 \nu^{11} - 43154858634515689790568 \nu^{10} + 5221070179901194950473 \nu^{9} + 86537534252271920654531 \nu^{8} - 34029275141695079410549 \nu^{7} - 85783168225253530008394 \nu^{6} + 50327088938842591365967 \nu^{5} + 31172753632958704540280 \nu^{4} - 23837476286048645159857 \nu^{3} + 137683984888334896715 \nu^{2} + 1234511067892056717935 \nu + 97226989580590920354\)\()/ 1006528474397131764 \)
\(\beta_{8}\)\(=\)\((\)\(7576293982927979121 \nu^{19} - 31073635796039525650 \nu^{18} - 224074168773683551559 \nu^{17} + 992270980545485712530 \nu^{16} + 2533932755487489006282 \nu^{15} - 12705398753421574017087 \nu^{14} - 13339100027251658888871 \nu^{13} + 84209031937717862545217 \nu^{12} + 27270563865313329311810 \nu^{11} - 309205145409647896750815 \nu^{10} + 30268230823871797593612 \nu^{9} + 619876660982428449866914 \nu^{8} - 230588328077451908360595 \nu^{7} - 614376030427052055814071 \nu^{6} + 347741351999178390256559 \nu^{5} + 223334005665434926549898 \nu^{4} - 165493117300867307298002 \nu^{3} + 1023954424120529904920 \nu^{2} + 8395092563731168662665 \nu + 606826360165697573025\)\()/ 7045699320779922348 \)
\(\beta_{9}\)\(=\)\((\)\(3972783117852940809 \nu^{19} - 16342846128512544173 \nu^{18} - 117320671722447788928 \nu^{17} + 521906431334303239643 \nu^{16} + 1322869278130680137994 \nu^{15} - 6683420464723393238238 \nu^{14} - 6916239982613718566301 \nu^{13} + 44305336615419018612667 \nu^{12} + 13746983099618899161683 \nu^{11} - 162744505113663406493844 \nu^{10} + 18087044949039785162649 \nu^{9} + 326510789581180900501616 \nu^{8} - 125963171001350030109214 \nu^{7} - 324232804881156119466287 \nu^{6} + 188504634894224254795827 \nu^{5} + 118753928893976335890255 \nu^{4} - 90208099965535405712597 \nu^{3} - 113306502518376301934 \nu^{2} + 4967650318979614979306 \nu + 392584122713171677173\)\()/ 3522849660389961174 \)
\(\beta_{10}\)\(=\)\((\)\(-2047786292360142275 \nu^{19} + 8453306354806390649 \nu^{18} + 60415489542289066625 \nu^{17} - 270070665887929370564 \nu^{16} - 680096131542676025598 \nu^{15} + 3460621390478264793615 \nu^{14} + 3543492889243349642386 \nu^{13} - 22962070603119437409617 \nu^{12} - 6956162240324603946659 \nu^{11} + 84464400288516335583243 \nu^{10} - 9732855884360263281280 \nu^{9} - 169854154542396568640853 \nu^{8} + 65568896606482854054425 \nu^{7} + 169432892648625526742675 \nu^{6} - 97578804334811055019104 \nu^{5} - 62888363036257849808159 \nu^{4} + 46575172685757514399803 \nu^{3} + 517517724798107266852 \nu^{2} - 2586400883220466564331 \nu - 221061471807410959458\)\()/ 1761424830194980587 \)
\(\beta_{11}\)\(=\)\((\)\(-8356272662031765421 \nu^{19} + 34202040308351693999 \nu^{18} + 247380582079000993072 \nu^{17} - 1092181398511623458951 \nu^{16} - 2802339477585508903152 \nu^{15} + 13984839560922314660460 \nu^{14} + 14807867321195626880525 \nu^{13} - 92690601820563143085051 \nu^{12} - 30703096120479886653617 \nu^{11} + 340365600693759113900646 \nu^{10} - 31116542935035778457819 \nu^{9} - 682452603951447075395134 \nu^{8} + 249783826314754461858204 \nu^{7} + 676734395670792749969903 \nu^{6} - 378819093668096210188203 \nu^{5} - 246474731929496125697395 \nu^{4} + 180338782805028994078333 \nu^{3} - 880481015365818753984 \nu^{2} - 8943772883317561833176 \nu - 667661155757941536921\)\()/ 7045699320779922348 \)
\(\beta_{12}\)\(=\)\((\)\(10817735530643672823 \nu^{19} - 44004533606033770380 \nu^{18} - 321427966399360503593 \nu^{17} + 1406026625803290483046 \nu^{16} + 3665451754025270244468 \nu^{15} - 18018706532048828189043 \nu^{14} - 19652386941621891075063 \nu^{13} + 119580968135206563719079 \nu^{12} + 42952487320825958297536 \nu^{11} - 440051048998962173817801 \nu^{10} + 28483713793780474325016 \nu^{9} + 885892877952153906591132 \nu^{8} - 299668100923119368276203 \nu^{7} - 886587875671824490647875 \nu^{6} + 467029295366688741017921 \nu^{5} + 333187098654750974129600 \nu^{4} - 225340677579528815198014 \nu^{3} - 5004868760721322981926 \nu^{2} + 11973135027759598136281 \nu + 1039311719576074586769\)\()/ 7045699320779922348 \)
\(\beta_{13}\)\(=\)\((\)\(-5765110608764984219 \nu^{19} + 23774153729099490769 \nu^{18} + 170121063113279701288 \nu^{17} - 759339338545804496015 \nu^{16} - 1915869049907838609096 \nu^{15} + 9725925495596380955028 \nu^{14} + 9993361151771601407425 \nu^{13} - 64491433648002404390229 \nu^{12} - 19717931987831196765635 \nu^{11} + 236964510849042737550060 \nu^{10} - 26798116659087406948195 \nu^{9} - 475532953075794779675528 \nu^{8} + 182985431498476594699216 \nu^{7} + 472103252678673209814171 \nu^{6} - 272203242371124194650313 \nu^{5} - 172340908041091230121837 \nu^{4} + 129044554135267262643513 \nu^{3} - 395578244725691148738 \nu^{2} - 6622263148757199924236 \nu - 508723097353615002549\)\()/ 3522849660389961174 \)
\(\beta_{14}\)\(=\)\((\)\(-6253881503724915265 \nu^{19} + 25710747398443785873 \nu^{18} + 184817918043869208801 \nu^{17} - 821215965080328317242 \nu^{16} - 2087033475527660694129 \nu^{15} + 10518852033194202600960 \nu^{14} + 10952270309572306681580 \nu^{13} - 69753480654678076097837 \nu^{12} - 22128310064864091219739 \nu^{11} + 256332262566733625164623 \nu^{10} - 26358642828281747248718 \nu^{9} - 514564778294223298658171 \nu^{8} + 193026483803831537370618 \nu^{7} + 511321403595229157488586 \nu^{6} - 289687900253439035789069 \nu^{5} - 187322137052218898966457 \nu^{4} + 137687461555365045233094 \nu^{3} - 20762070898140250163 \nu^{2} - 7045871191970967584026 \nu - 562332436760355025320\)\()/ 3522849660389961174 \)
\(\beta_{15}\)\(=\)\((\)\(14402308926297643223 \nu^{19} - 59134751184578330885 \nu^{18} - 425833794628053440962 \nu^{17} + 1888661390075851423815 \nu^{16} + 4812963651580171485534 \nu^{15} - 24189115890691711217376 \nu^{14} - 25306907459928782093989 \nu^{13} + 160381237404775884342221 \nu^{12} + 51513448344016378444467 \nu^{11} - 589251743429612557666656 \nu^{10} + 58703032263898179944719 \nu^{9} + 1182522905208002117547780 \nu^{8} - 440588282101994854095084 \nu^{7} - 1174535280429699515543167 \nu^{6} + 663248018168538332130329 \nu^{5} + 429824287705913693338525 \nu^{4} - 315556203763661953628365 \nu^{3} + 343760481474573044810 \nu^{2} + 16097180552095863216408 \nu + 1233943646077998910485\)\()/ 7045699320779922348 \)
\(\beta_{16}\)\(=\)\((\)\(14645155040509963820 \nu^{19} - 59642851508940305086 \nu^{18} - 434846673787457040185 \nu^{17} + 1905411720410527343023 \nu^{16} + 4952712998950562047623 \nu^{15} - 24412967211259506148740 \nu^{14} - 26485294281914839965835 \nu^{13} + 161958144182482346030238 \nu^{12} + 57379737864677684063536 \nu^{11} - 595622318194356679148397 \nu^{10} + 41227112868720385687825 \nu^{9} + 1197582146857521448758737 \nu^{8} - 410474329960614160479102 \nu^{7} - 1194877112155448350622119 \nu^{6} + 635842430113801342281372 \nu^{5} + 444117927600826721299226 \nu^{4} - 304997202980737096627727 \nu^{3} - 3554413158860711081553 \nu^{2} + 15402396422972692780726 \nu + 1234585521558462095301\)\()/ 7045699320779922348 \)
\(\beta_{17}\)\(=\)\((\)\(7438783877591216548 \nu^{19} - 30459378048999778099 \nu^{18} - 220200766266302213723 \nu^{17} + 972757216930066495875 \nu^{16} + 2494047510119311879458 \nu^{15} - 12457405771199322026019 \nu^{14} - 13173947522025534613790 \nu^{13} + 82584351503069563769488 \nu^{12} + 27275698837548390607893 \nu^{11} - 303357063011471710727439 \nu^{10} + 27923448863376964351637 \nu^{9} + 608602934352236377481532 \nu^{8} - 222874255585591719674739 \nu^{7} - 604220200791226407993500 \nu^{6} + 337911546657425909550568 \nu^{5} + 220873751730400091677289 \nu^{4} - 161021264521041724703129 \nu^{3} + 321155306165435922256 \nu^{2} + 8099183834371161701361 \nu + 617682916703533320726\)\()/ 3522849660389961174 \)
\(\beta_{18}\)\(=\)\((\)\(-8138886928753770041 \nu^{19} + 33499956946277000400 \nu^{18} + 240323297279872800415 \nu^{17} - 1069812371476842628582 \nu^{16} - 2709600815139216588042 \nu^{15} + 13699510172653880436333 \nu^{14} + 14168966215045114054639 \nu^{13} - 90809978718344201679229 \nu^{12} - 28224573672325185426058 \nu^{11} + 333503362373389497835863 \nu^{10} - 36486799032282948761686 \nu^{9} - 668728059025171509164860 \nu^{8} + 255826579517142651304643 \nu^{7} + 662877997391653078852985 \nu^{6} - 381979586086710828489095 \nu^{5} - 240832709198133636810616 \nu^{4} + 181302476507307694588730 \nu^{3} - 1221891996360336135274 \nu^{2} - 9235939572202007595577 \nu - 699634305926123616771\)\()/ 3522849660389961174 \)
\(\beta_{19}\)\(=\)\((\)\(9044475989189897959 \nu^{19} - 37172639749616416747 \nu^{18} - 267337054794349577899 \nu^{17} + 1187394854241713040880 \nu^{16} + 3019802809987906445103 \nu^{15} - 15210701217109761007314 \nu^{14} - 15856792379304385164044 \nu^{13} + 100882261019315843361703 \nu^{12} + 32101655974673253786169 \nu^{11} - 370826360968046011930875 \nu^{10} + 37844023940778257308124 \nu^{9} + 744805250324323650729783 \nu^{8} - 278897838495269416179886 \nu^{7} - 741081009576028826055736 \nu^{6} + 419276455516957466764863 \nu^{5} + 272777471804394727207141 \nu^{4} - 199891469248486819547262 \nu^{3} - 748060606226458740485 \nu^{2} + 10524879908602517748106 \nu + 829762774988511791610\)\()/ 3522849660389961174 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 4\)
\(\nu^{3}\)\(=\)\(-\beta_{17} + \beta_{16} - \beta_{15} - \beta_{12} - 2 \beta_{11} - \beta_{10} + \beta_{3} + 7 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(-2 \beta_{19} - \beta_{18} + \beta_{17} - \beta_{16} - \beta_{13} - \beta_{11} - 2 \beta_{10} - 2 \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + 9 \beta_{2} + 3 \beta_{1} + 27\)
\(\nu^{5}\)\(=\)\(\beta_{19} - 10 \beta_{17} + 10 \beta_{16} - 14 \beta_{15} - \beta_{14} + \beta_{13} - 12 \beta_{12} - 23 \beta_{11} - 12 \beta_{10} - 2 \beta_{9} + 2 \beta_{8} + 3 \beta_{6} + 2 \beta_{5} - \beta_{4} + 11 \beta_{3} + 61 \beta_{1} + 14\)
\(\nu^{6}\)\(=\)\(-31 \beta_{19} - 16 \beta_{18} + 13 \beta_{17} - 18 \beta_{16} + 2 \beta_{15} - 4 \beta_{14} - 15 \beta_{13} + \beta_{12} - 18 \beta_{11} - 29 \beta_{10} - 2 \beta_{8} - 37 \beta_{7} - 8 \beta_{6} + 16 \beta_{5} + 13 \beta_{4} - 16 \beta_{3} + 85 \beta_{2} + 48 \beta_{1} + 211\)
\(\nu^{7}\)\(=\)\(9 \beta_{19} + \beta_{18} - 87 \beta_{17} + 91 \beta_{16} - 165 \beta_{15} - 14 \beta_{14} + 14 \beta_{13} - 130 \beta_{12} - 244 \beta_{11} - 134 \beta_{10} - 35 \beta_{9} + 37 \beta_{8} - 3 \beta_{7} + 67 \beta_{6} + 38 \beta_{5} - 17 \beta_{4} + 95 \beta_{3} - 8 \beta_{2} + 581 \beta_{1} + 151\)
\(\nu^{8}\)\(=\)\(-378 \beta_{19} - 197 \beta_{18} + 141 \beta_{17} - 233 \beta_{16} + 29 \beta_{15} - 78 \beta_{14} - 166 \beta_{13} + 15 \beta_{12} - 238 \beta_{11} - 343 \beta_{10} - 7 \beta_{9} - 28 \beta_{8} - 475 \beta_{7} - 25 \beta_{6} + 206 \beta_{5} + 131 \beta_{4} - 221 \beta_{3} + 824 \beta_{2} + 596 \beta_{1} + 1806\)
\(\nu^{9}\)\(=\)\(38 \beta_{19} + 7 \beta_{18} - 748 \beta_{17} + 827 \beta_{16} - 1839 \beta_{15} - 152 \beta_{14} + 178 \beta_{13} - 1361 \beta_{12} - 2555 \beta_{11} - 1473 \beta_{10} - 474 \beta_{9} + 518 \beta_{8} - 63 \beta_{7} + 1034 \beta_{6} + 519 \beta_{5} - 235 \beta_{4} + 726 \beta_{3} - 167 \beta_{2} + 5766 \beta_{1} + 1521\)
\(\nu^{10}\)\(=\)\(-4243 \beta_{19} - 2238 \beta_{18} + 1492 \beta_{17} - 2696 \beta_{16} + 233 \beta_{15} - 1046 \beta_{14} - 1629 \beta_{13} + 169 \beta_{12} - 2819 \beta_{11} - 3837 \beta_{10} - 236 \beta_{9} - 217 \beta_{8} - 5363 \beta_{7} + 473 \beta_{6} + 2452 \beta_{5} + 1173 \beta_{4} - 2886 \beta_{3} + 8071 \beta_{2} + 6869 \beta_{1} + 16429\)
\(\nu^{11}\)\(=\)\(-277 \beta_{19} - 133 \beta_{18} - 6518 \beta_{17} + 7570 \beta_{16} - 19975 \beta_{15} - 1542 \beta_{14} + 2282 \beta_{13} - 13950 \beta_{12} - 26591 \beta_{11} - 16019 \beta_{10} - 5916 \beta_{9} + 6526 \beta_{8} - 896 \beta_{7} + 13810 \beta_{6} + 6318 \beta_{5} - 3104 \beta_{4} + 4784 \beta_{3} - 2376 \beta_{2} + 58474 \beta_{1} + 15151\)
\(\nu^{12}\)\(=\)\(-45923 \beta_{19} - 24655 \beta_{18} + 15782 \beta_{17} - 29677 \beta_{16} + 569 \beta_{15} - 12100 \beta_{14} - 14928 \beta_{13} + 1815 \beta_{12} - 31788 \beta_{11} - 42101 \beta_{10} - 4876 \beta_{9} - 417 \beta_{8} - 57145 \beta_{7} + 12998 \beta_{6} + 28063 \beta_{5} + 9426 \beta_{4} - 36181 \beta_{3} + 79438 \beta_{2} + 77074 \beta_{1} + 155705\)
\(\nu^{13}\)\(=\)\(-9828 \beta_{19} - 5006 \beta_{18} - 57756 \beta_{17} + 69586 \beta_{16} - 214052 \beta_{15} - 15246 \beta_{14} + 29053 \beta_{13} - 140895 \beta_{12} - 275443 \beta_{11} - 172867 \beta_{10} - 71231 \beta_{9} + 78089 \beta_{8} - 10859 \beta_{7} + 171522 \beta_{6} + 73218 \beta_{5} - 40130 \beta_{4} + 22103 \beta_{3} - 28888 \beta_{2} + 600588 \beta_{1} + 152541\)
\(\nu^{14}\)\(=\)\(-488333 \beta_{19} - 268379 \beta_{18} + 166948 \beta_{17} - 318472 \beta_{16} - 20717 \beta_{15} - 130109 \beta_{14} - 129557 \beta_{13} + 20182 \beta_{12} - 349438 \beta_{11} - 458410 \beta_{10} - 80985 \beta_{9} + 21669 \beta_{8} - 591235 \beta_{7} + 219551 \beta_{6} + 314111 \beta_{5} + 64153 \beta_{4} - 439351 \beta_{3} + 784451 \beta_{2} + 855988 \beta_{1} + 1517375\)
\(\nu^{15}\)\(=\)\(-171630 \beta_{19} - 101254 \beta_{18} - 518951 \beta_{17} + 639870 \beta_{16} - 2276953 \beta_{15} - 148336 \beta_{14} + 362583 \beta_{13} - 1408178 \beta_{12} - 2843005 \beta_{11} - 1856361 \beta_{10} - 840656 \beta_{9} + 908234 \beta_{8} - 121133 \beta_{7} + 2043303 \beta_{6} + 828805 \beta_{5} - 508524 \beta_{4} - 43294 \beta_{3} - 323319 \beta_{2} + 6219868 \beta_{1} + 1564378\)
\(\nu^{16}\)\(=\)\(-5147120 \beta_{19} - 2909362 \beta_{18} + 1762319 \beta_{17} - 3370083 \beta_{16} - 536726 \beta_{15} - 1343624 \beta_{14} - 1059434 \beta_{13} + 236052 \beta_{12} - 3785785 \beta_{11} - 4974420 \beta_{10} - 1191284 \beta_{9} + 537912 \beta_{8} - 6019735 \beta_{7} + 3143784 \beta_{6} + 3469453 \beta_{5} + 281084 \beta_{4} - 5205679 \beta_{3} + 7769680 \beta_{2} + 9469298 \beta_{1} + 15077038\)
\(\nu^{17}\)\(=\)\(-2477970 \beta_{19} - 1649754 \beta_{18} - 4708169 \beta_{17} + 5865640 \beta_{16} - 24125681 \beta_{15} - 1422288 \beta_{14} + 4424234 \beta_{13} - 13966392 \beta_{12} - 29273289 \beta_{11} - 19880479 \beta_{10} - 9789062 \beta_{9} + 10384354 \beta_{8} - 1288423 \beta_{7} + 23714425 \beta_{6} + 9266739 \beta_{5} - 6315612 \beta_{4} - 3152036 \beta_{3} - 3439771 \beta_{2} + 64793238 \beta_{1} + 16359136\)
\(\nu^{18}\)\(=\)\(-54018690 \beta_{19} - 31510453 \beta_{18} + 18543655 \beta_{17} - 35377845 \beta_{16} - 9178237 \beta_{15} - 13545029 \beta_{14} - 7953251 \beta_{13} + 2855543 \beta_{12} - 40659906 \beta_{11} - 53893287 \beta_{10} - 16244890 \beta_{9} + 9093699 \beta_{8} - 60737222 \beta_{7} + 41391847 \beta_{6} + 38011629 \beta_{5} - 1322899 \beta_{4} - 60534715 \beta_{3} + 77183689 \beta_{2} + 104595921 \beta_{1} + 151922809\)
\(\nu^{19}\)\(=\)\(-32748466 \beta_{19} - 23979501 \beta_{18} - 42949033 \beta_{17} + 53432566 \beta_{16} - 255133669 \beta_{15} - 13432468 \beta_{14} + 52891047 \beta_{13} - 137716944 \beta_{12} - 301000564 \beta_{11} - 212638933 \beta_{10} - 112832331 \beta_{9} + 117414503 \beta_{8} - 13323065 \beta_{7} + 270495307 \beta_{6} + 102870573 \beta_{5} - 76968461 \beta_{4} - 59643231 \beta_{3} - 35332232 \beta_{2} + 677975640 \beta_{1} + 174055949\)

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
3.29362
3.09326
2.83122
2.36764
2.15119
1.53485
1.15563
1.06664
0.538932
0.476040
−0.0954148
−0.103746
−0.146508
−0.915292
−1.37724
−1.77641
−2.15199
−2.32542
−2.45978
−3.15720
0 −3.29362 0 −3.10165 0 0 0 7.84790 0
1.2 0 −3.09326 0 −2.53432 0 0 0 6.56828 0
1.3 0 −2.83122 0 2.22134 0 0 0 5.01578 0
1.4 0 −2.36764 0 −2.42438 0 0 0 2.60574 0
1.5 0 −2.15119 0 3.75581 0 0 0 1.62763 0
1.6 0 −1.53485 0 −1.74990 0 0 0 −0.644239 0
1.7 0 −1.15563 0 −3.94572 0 0 0 −1.66452 0
1.8 0 −1.06664 0 0.715473 0 0 0 −1.86229 0
1.9 0 −0.538932 0 −0.810745 0 0 0 −2.70955 0
1.10 0 −0.476040 0 2.17736 0 0 0 −2.77339 0
1.11 0 0.0954148 0 0.581124 0 0 0 −2.99090 0
1.12 0 0.103746 0 3.71451 0 0 0 −2.98924 0
1.13 0 0.146508 0 −3.56882 0 0 0 −2.97854 0
1.14 0 0.915292 0 1.31004 0 0 0 −2.16224 0
1.15 0 1.37724 0 −3.83294 0 0 0 −1.10321 0
1.16 0 1.77641 0 0.716761 0 0 0 0.155650 0
1.17 0 2.15199 0 0.0591445 0 0 0 1.63107 0
1.18 0 2.32542 0 −1.16439 0 0 0 2.40759 0
1.19 0 2.45978 0 2.10149 0 0 0 3.05053 0
1.20 0 3.15720 0 −2.22019 0 0 0 6.96793 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.20
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 8036.2.a.s 20
7.b odd 2 1 8036.2.a.t yes 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8036.2.a.s 20 1.a even 1 1 trivial
8036.2.a.t yes 20 7.b odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(1\)
\(41\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8036))\):

\(T_{3}^{20} + \cdots\)
\(T_{5}^{20} + \cdots\)
\(T_{11}^{20} + \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 4 T + 30 T^{2} + 100 T^{3} + 438 T^{4} + 1272 T^{5} + 4240 T^{6} + 11012 T^{7} + 30845 T^{8} + 72876 T^{9} + 180194 T^{10} + 392412 T^{11} + 881848 T^{12} + 1790364 T^{13} + 3730214 T^{14} + 7134852 T^{15} + 13991387 T^{16} + 25453628 T^{17} + 47522000 T^{18} + 82831548 T^{19} + 148322703 T^{20} + 248494644 T^{21} + 427698000 T^{22} + 687247956 T^{23} + 1133302347 T^{24} + 1733769036 T^{25} + 2719326006 T^{26} + 3915526068 T^{27} + 5785804728 T^{28} + 7723845396 T^{29} + 10640275506 T^{30} + 12909764772 T^{31} + 16392297645 T^{32} + 17556684876 T^{33} + 20279788560 T^{34} + 18251809704 T^{35} + 18854463798 T^{36} + 12914016300 T^{37} + 11622614670 T^{38} + 4649045868 T^{39} + 3486784401 T^{40} \)
$5$ \( 1 + 8 T + 72 T^{2} + 400 T^{3} + 2237 T^{4} + 9876 T^{5} + 42972 T^{6} + 160960 T^{7} + 592638 T^{8} + 1959204 T^{9} + 6383550 T^{10} + 19135824 T^{11} + 56758118 T^{12} + 157264968 T^{13} + 432551686 T^{14} + 1121760004 T^{15} + 2892510705 T^{16} + 7067419456 T^{17} + 17173284840 T^{18} + 39625592820 T^{19} + 90903287002 T^{20} + 198127964100 T^{21} + 429332121000 T^{22} + 883427432000 T^{23} + 1807819190625 T^{24} + 3505500012500 T^{25} + 6758620093750 T^{26} + 12286325625000 T^{27} + 22171139843750 T^{28} + 37374656250000 T^{29} + 62339355468750 T^{30} + 95664257812500 T^{31} + 144687011718750 T^{32} + 196484375000000 T^{33} + 262280273437500 T^{34} + 301391601562500 T^{35} + 341339111328125 T^{36} + 305175781250000 T^{37} + 274658203125000 T^{38} + 152587890625000 T^{39} + 95367431640625 T^{40} \)
$7$ 1
$11$ \( 1 + 8 T + 142 T^{2} + 992 T^{3} + 9964 T^{4} + 61248 T^{5} + 458342 T^{6} + 2512024 T^{7} + 15530598 T^{8} + 76936024 T^{9} + 413551562 T^{10} + 1873362464 T^{11} + 9012640270 T^{12} + 37664981344 T^{13} + 165092145546 T^{14} + 640460360552 T^{15} + 2586564514345 T^{16} + 9350628067040 T^{17} + 35040381858552 T^{18} + 118256665353312 T^{19} + 412895546420908 T^{20} + 1300823318886432 T^{21} + 4239886204884792 T^{22} + 12445685957230240 T^{23} + 37869891054525145 T^{24} + 103146781527260152 T^{25} + 292470806455617306 T^{26} + 733983932162337824 T^{27} + 1931939483132737870 T^{28} + 4417290696394870624 T^{29} + 10726462458000776762 T^{30} + 21950745537607990664 T^{31} + 48741669470646409158 T^{32} + 86721881450646126344 T^{33} + \)\(17\!\cdots\!22\)\( T^{34} + \)\(25\!\cdots\!48\)\( T^{35} + \)\(45\!\cdots\!04\)\( T^{36} + \)\(50\!\cdots\!32\)\( T^{37} + \)\(78\!\cdots\!02\)\( T^{38} + \)\(48\!\cdots\!28\)\( T^{39} + \)\(67\!\cdots\!01\)\( T^{40} \)
$13$ \( 1 + 12 T + 194 T^{2} + 1700 T^{3} + 16215 T^{4} + 114764 T^{5} + 829070 T^{6} + 5018428 T^{7} + 30262070 T^{8} + 162395528 T^{9} + 860372336 T^{10} + 4190858120 T^{11} + 20109646694 T^{12} + 90368949632 T^{13} + 400312442964 T^{14} + 1679055021608 T^{15} + 6949980791045 T^{16} + 27425632181028 T^{17} + 106863781088238 T^{18} + 398549262766884 T^{19} + 1467826932439611 T^{20} + 5181140415969492 T^{21} + 18059979003912222 T^{22} + 60254113901718516 T^{23} + 198498401373036245 T^{24} + 623421376137899144 T^{25} + 1932231702510621876 T^{26} + 5670517572255695744 T^{27} + 16404056596751886374 T^{28} + 44441952305871958760 T^{29} + \)\(11\!\cdots\!64\)\( T^{30} + \)\(29\!\cdots\!36\)\( T^{31} + \)\(70\!\cdots\!70\)\( T^{32} + \)\(15\!\cdots\!84\)\( T^{33} + \)\(32\!\cdots\!30\)\( T^{34} + \)\(58\!\cdots\!48\)\( T^{35} + \)\(10\!\cdots\!15\)\( T^{36} + \)\(14\!\cdots\!00\)\( T^{37} + \)\(21\!\cdots\!26\)\( T^{38} + \)\(17\!\cdots\!24\)\( T^{39} + \)\(19\!\cdots\!01\)\( T^{40} \)
$17$ \( 1 + 8 T + 204 T^{2} + 1364 T^{3} + 19837 T^{4} + 115236 T^{5} + 1252068 T^{6} + 6497620 T^{7} + 58571224 T^{8} + 277021276 T^{9} + 2184743874 T^{10} + 9546857708 T^{11} + 67882331020 T^{12} + 276510702884 T^{13} + 1805247496970 T^{14} + 6893991759912 T^{15} + 41807869607761 T^{16} + 150199940099848 T^{17} + 852315750839294 T^{18} + 2884906960753048 T^{19} + 15384600661717447 T^{20} + 49043418332801816 T^{21} + 246319251992555966 T^{22} + 737932305710553224 T^{23} + 3491835077509806481 T^{24} + 9788482458253372584 T^{25} + 43574286020190665930 T^{26} + \)\(11\!\cdots\!32\)\( T^{27} + \)\(47\!\cdots\!20\)\( T^{28} + \)\(11\!\cdots\!76\)\( T^{29} + \)\(44\!\cdots\!26\)\( T^{30} + \)\(94\!\cdots\!08\)\( T^{31} + \)\(34\!\cdots\!64\)\( T^{32} + \)\(64\!\cdots\!40\)\( T^{33} + \)\(21\!\cdots\!72\)\( T^{34} + \)\(32\!\cdots\!48\)\( T^{35} + \)\(96\!\cdots\!97\)\( T^{36} + \)\(11\!\cdots\!28\)\( T^{37} + \)\(28\!\cdots\!36\)\( T^{38} + \)\(19\!\cdots\!24\)\( T^{39} + \)\(40\!\cdots\!01\)\( T^{40} \)
$19$ \( 1 + 24 T + 442 T^{2} + 5948 T^{3} + 69172 T^{4} + 689232 T^{5} + 6223974 T^{6} + 50847780 T^{7} + 386362967 T^{8} + 2729413288 T^{9} + 18223922508 T^{10} + 114944678504 T^{11} + 692265483630 T^{12} + 3976725038900 T^{13} + 21951204485116 T^{14} + 116244899280332 T^{15} + 593773138113969 T^{16} + 2919711686019564 T^{17} + 13879126186324128 T^{18} + 63632736387802896 T^{19} + 282340935861651499 T^{20} + 1209021991368255024 T^{21} + 5010364553263010208 T^{22} + 20026302454408189476 T^{23} + 77381109132150554049 T^{24} + \)\(28\!\cdots\!68\)\( T^{25} + \)\(10\!\cdots\!96\)\( T^{26} + \)\(35\!\cdots\!00\)\( T^{27} + \)\(11\!\cdots\!30\)\( T^{28} + \)\(37\!\cdots\!16\)\( T^{29} + \)\(11\!\cdots\!08\)\( T^{30} + \)\(31\!\cdots\!72\)\( T^{31} + \)\(85\!\cdots\!87\)\( T^{32} + \)\(21\!\cdots\!20\)\( T^{33} + \)\(49\!\cdots\!54\)\( T^{34} + \)\(10\!\cdots\!68\)\( T^{35} + \)\(19\!\cdots\!32\)\( T^{36} + \)\(32\!\cdots\!72\)\( T^{37} + \)\(46\!\cdots\!22\)\( T^{38} + \)\(47\!\cdots\!96\)\( T^{39} + \)\(37\!\cdots\!01\)\( T^{40} \)
$23$ \( 1 - 8 T + 224 T^{2} - 1604 T^{3} + 25586 T^{4} - 164324 T^{5} + 1949088 T^{6} - 11272892 T^{7} + 110293819 T^{8} - 575766952 T^{9} + 4917180158 T^{10} - 23186904576 T^{11} + 179684057376 T^{12} - 766054796244 T^{13} + 5564463719942 T^{14} - 21547403690928 T^{15} + 151225681694129 T^{16} - 539520406114572 T^{17} + 3750834473275234 T^{18} - 12680390219549800 T^{19} + 88013214377958071 T^{20} - 291648975049645400 T^{21} + 1984191436362598786 T^{22} - 6564344781195997524 T^{23} + 42319145990966753489 T^{24} - \)\(13\!\cdots\!04\)\( T^{25} + \)\(82\!\cdots\!38\)\( T^{26} - \)\(26\!\cdots\!68\)\( T^{27} + \)\(14\!\cdots\!56\)\( T^{28} - \)\(41\!\cdots\!88\)\( T^{29} + \)\(20\!\cdots\!42\)\( T^{30} - \)\(54\!\cdots\!04\)\( T^{31} + \)\(24\!\cdots\!99\)\( T^{32} - \)\(56\!\cdots\!36\)\( T^{33} + \)\(22\!\cdots\!92\)\( T^{34} - \)\(43\!\cdots\!68\)\( T^{35} + \)\(15\!\cdots\!46\)\( T^{36} - \)\(22\!\cdots\!12\)\( T^{37} + \)\(72\!\cdots\!56\)\( T^{38} - \)\(59\!\cdots\!96\)\( T^{39} + \)\(17\!\cdots\!01\)\( T^{40} \)
$29$ \( 1 + 12 T + 372 T^{2} + 3900 T^{3} + 68255 T^{4} + 641676 T^{5} + 8277632 T^{6} + 70822400 T^{7} + 746135170 T^{8} + 5863304764 T^{9} + 53174262038 T^{10} + 386094445984 T^{11} + 3108561209242 T^{12} + 20939598180832 T^{13} + 152594143829102 T^{14} + 956108937529028 T^{15} + 6385477457957389 T^{16} + 37270237580395268 T^{17} + 229952446886288440 T^{18} + 1250667711151805704 T^{19} + 7163785976565568494 T^{20} + 36269363623402365416 T^{21} + \)\(19\!\cdots\!40\)\( T^{22} + \)\(90\!\cdots\!52\)\( T^{23} + \)\(45\!\cdots\!09\)\( T^{24} + \)\(19\!\cdots\!72\)\( T^{25} + \)\(90\!\cdots\!42\)\( T^{26} + \)\(36\!\cdots\!88\)\( T^{27} + \)\(15\!\cdots\!62\)\( T^{28} + \)\(56\!\cdots\!96\)\( T^{29} + \)\(22\!\cdots\!38\)\( T^{30} + \)\(71\!\cdots\!56\)\( T^{31} + \)\(26\!\cdots\!70\)\( T^{32} + \)\(72\!\cdots\!00\)\( T^{33} + \)\(24\!\cdots\!92\)\( T^{34} + \)\(55\!\cdots\!24\)\( T^{35} + \)\(17\!\cdots\!55\)\( T^{36} + \)\(28\!\cdots\!00\)\( T^{37} + \)\(78\!\cdots\!92\)\( T^{38} + \)\(73\!\cdots\!28\)\( T^{39} + \)\(17\!\cdots\!01\)\( T^{40} \)
$31$ \( 1 + 366 T^{2} - 200 T^{3} + 64893 T^{4} - 66076 T^{5} + 7487952 T^{6} - 10495888 T^{7} + 637833274 T^{8} - 1075441548 T^{9} + 43108352406 T^{10} - 80651055032 T^{11} + 2420742392118 T^{12} - 4765258606976 T^{13} + 116381484643210 T^{14} - 232551995261652 T^{15} + 4881847246546661 T^{16} - 9645402933660120 T^{17} + 180712343974789634 T^{18} - 344947354297833756 T^{19} + 5939483468474401498 T^{20} - 10693367983232846436 T^{21} + \)\(17\!\cdots\!74\)\( T^{22} - \)\(28\!\cdots\!20\)\( T^{23} + \)\(45\!\cdots\!81\)\( T^{24} - \)\(66\!\cdots\!52\)\( T^{25} + \)\(10\!\cdots\!10\)\( T^{26} - \)\(13\!\cdots\!36\)\( T^{27} + \)\(20\!\cdots\!38\)\( T^{28} - \)\(21\!\cdots\!72\)\( T^{29} + \)\(35\!\cdots\!06\)\( T^{30} - \)\(27\!\cdots\!88\)\( T^{31} + \)\(50\!\cdots\!14\)\( T^{32} - \)\(25\!\cdots\!08\)\( T^{33} + \)\(56\!\cdots\!92\)\( T^{34} - \)\(15\!\cdots\!76\)\( T^{35} + \)\(47\!\cdots\!33\)\( T^{36} - \)\(45\!\cdots\!00\)\( T^{37} + \)\(25\!\cdots\!06\)\( T^{38} + \)\(67\!\cdots\!01\)\( T^{40} \)
$37$ \( 1 - 12 T + 414 T^{2} - 4100 T^{3} + 82723 T^{4} - 714876 T^{5} + 10950648 T^{6} - 85102596 T^{7} + 1090533699 T^{8} - 7755996252 T^{9} + 87298936098 T^{10} - 574754531056 T^{11} + 5846021890633 T^{12} - 35909756683808 T^{13} + 335961485822006 T^{14} - 1935713494053164 T^{15} + 16848638971526184 T^{16} - 91370333355176520 T^{17} + 745252319804448866 T^{18} - 3810432221941255680 T^{19} + 29246522638093731856 T^{20} - \)\(14\!\cdots\!60\)\( T^{21} + \)\(10\!\cdots\!54\)\( T^{22} - \)\(46\!\cdots\!60\)\( T^{23} + \)\(31\!\cdots\!24\)\( T^{24} - \)\(13\!\cdots\!48\)\( T^{25} + \)\(86\!\cdots\!54\)\( T^{26} - \)\(34\!\cdots\!64\)\( T^{27} + \)\(20\!\cdots\!93\)\( T^{28} - \)\(74\!\cdots\!12\)\( T^{29} + \)\(41\!\cdots\!02\)\( T^{30} - \)\(13\!\cdots\!76\)\( T^{31} + \)\(71\!\cdots\!19\)\( T^{32} - \)\(20\!\cdots\!12\)\( T^{33} + \)\(98\!\cdots\!72\)\( T^{34} - \)\(23\!\cdots\!68\)\( T^{35} + \)\(10\!\cdots\!43\)\( T^{36} - \)\(18\!\cdots\!00\)\( T^{37} + \)\(69\!\cdots\!06\)\( T^{38} - \)\(74\!\cdots\!76\)\( T^{39} + \)\(23\!\cdots\!01\)\( T^{40} \)
$41$ \( ( 1 - T )^{20} \)
$43$ \( 1 - 4 T + 362 T^{2} - 1816 T^{3} + 68417 T^{4} - 376728 T^{5} + 8859796 T^{6} - 50098400 T^{7} + 873099406 T^{8} - 4884181768 T^{9} + 69429029716 T^{10} - 376406649008 T^{11} + 4640398136768 T^{12} - 24165516805592 T^{13} + 269513852332750 T^{14} - 1347357682359948 T^{15} + 14004381065462057 T^{16} - 67481618639288864 T^{17} + 667084060401896952 T^{18} - 3111440283755638144 T^{19} + 29620566888741233631 T^{20} - \)\(13\!\cdots\!92\)\( T^{21} + \)\(12\!\cdots\!48\)\( T^{22} - \)\(53\!\cdots\!48\)\( T^{23} + \)\(47\!\cdots\!57\)\( T^{24} - \)\(19\!\cdots\!64\)\( T^{25} + \)\(17\!\cdots\!50\)\( T^{26} - \)\(65\!\cdots\!44\)\( T^{27} + \)\(54\!\cdots\!68\)\( T^{28} - \)\(18\!\cdots\!44\)\( T^{29} + \)\(15\!\cdots\!84\)\( T^{30} - \)\(45\!\cdots\!76\)\( T^{31} + \)\(34\!\cdots\!06\)\( T^{32} - \)\(86\!\cdots\!00\)\( T^{33} + \)\(65\!\cdots\!04\)\( T^{34} - \)\(11\!\cdots\!96\)\( T^{35} + \)\(93\!\cdots\!17\)\( T^{36} - \)\(10\!\cdots\!88\)\( T^{37} + \)\(91\!\cdots\!38\)\( T^{38} - \)\(43\!\cdots\!28\)\( T^{39} + \)\(46\!\cdots\!01\)\( T^{40} \)
$47$ \( 1 + 4 T + 374 T^{2} + 1096 T^{3} + 70333 T^{4} + 138060 T^{5} + 8812902 T^{6} + 8238792 T^{7} + 825273227 T^{8} - 162954236 T^{9} + 62195044950 T^{10} - 87253393748 T^{11} + 4018555017717 T^{12} - 10192167743020 T^{13} + 235818086464330 T^{14} - 791041310737316 T^{15} + 13050043304896332 T^{16} - 48390279852871812 T^{17} + 684184059349996740 T^{18} - 2536743929071383924 T^{19} + 33454277957904126460 T^{20} - \)\(11\!\cdots\!28\)\( T^{21} + \)\(15\!\cdots\!60\)\( T^{22} - \)\(50\!\cdots\!76\)\( T^{23} + \)\(63\!\cdots\!92\)\( T^{24} - \)\(18\!\cdots\!12\)\( T^{25} + \)\(25\!\cdots\!70\)\( T^{26} - \)\(51\!\cdots\!60\)\( T^{27} + \)\(95\!\cdots\!37\)\( T^{28} - \)\(97\!\cdots\!16\)\( T^{29} + \)\(32\!\cdots\!50\)\( T^{30} - \)\(40\!\cdots\!08\)\( T^{31} + \)\(95\!\cdots\!07\)\( T^{32} + \)\(44\!\cdots\!84\)\( T^{33} + \)\(22\!\cdots\!38\)\( T^{34} + \)\(16\!\cdots\!80\)\( T^{35} + \)\(39\!\cdots\!93\)\( T^{36} + \)\(29\!\cdots\!52\)\( T^{37} + \)\(46\!\cdots\!86\)\( T^{38} + \)\(23\!\cdots\!32\)\( T^{39} + \)\(27\!\cdots\!01\)\( T^{40} \)
$53$ \( 1 + 12 T + 512 T^{2} + 5612 T^{3} + 135003 T^{4} + 1361876 T^{5} + 24146848 T^{6} + 224957376 T^{7} + 3265904782 T^{8} + 28191103556 T^{9} + 354427483702 T^{10} + 2845087467008 T^{11} + 32079538394358 T^{12} + 240579990164368 T^{13} + 2492534087391782 T^{14} + 17565005054366700 T^{15} + 170202190083621233 T^{16} + 1134513791108890324 T^{17} + 10410150847987163156 T^{18} + 66034944862846181312 T^{19} + \)\(57\!\cdots\!62\)\( T^{20} + \)\(34\!\cdots\!36\)\( T^{21} + \)\(29\!\cdots\!04\)\( T^{22} + \)\(16\!\cdots\!48\)\( T^{23} + \)\(13\!\cdots\!73\)\( T^{24} + \)\(73\!\cdots\!00\)\( T^{25} + \)\(55\!\cdots\!78\)\( T^{26} + \)\(28\!\cdots\!16\)\( T^{27} + \)\(19\!\cdots\!38\)\( T^{28} + \)\(93\!\cdots\!64\)\( T^{29} + \)\(61\!\cdots\!98\)\( T^{30} + \)\(26\!\cdots\!32\)\( T^{31} + \)\(16\!\cdots\!62\)\( T^{32} + \)\(58\!\cdots\!48\)\( T^{33} + \)\(33\!\cdots\!12\)\( T^{34} + \)\(99\!\cdots\!32\)\( T^{35} + \)\(52\!\cdots\!63\)\( T^{36} + \)\(11\!\cdots\!56\)\( T^{37} + \)\(55\!\cdots\!68\)\( T^{38} + \)\(69\!\cdots\!04\)\( T^{39} + \)\(30\!\cdots\!01\)\( T^{40} \)
$59$ \( 1 + 16 T + 600 T^{2} + 7152 T^{3} + 160149 T^{4} + 1517188 T^{5} + 26845940 T^{6} + 211678632 T^{7} + 3369110190 T^{8} + 23123338092 T^{9} + 354171762190 T^{10} + 2196705117576 T^{11} + 32918184304670 T^{12} + 187976789926624 T^{13} + 2737276590449382 T^{14} + 14486638251217924 T^{15} + 204335692284544161 T^{16} + 1010492249607485920 T^{17} + 13812880948781473744 T^{18} + 64587694050718683228 T^{19} + \)\(85\!\cdots\!74\)\( T^{20} + \)\(38\!\cdots\!52\)\( T^{21} + \)\(48\!\cdots\!64\)\( T^{22} + \)\(20\!\cdots\!80\)\( T^{23} + \)\(24\!\cdots\!21\)\( T^{24} + \)\(10\!\cdots\!76\)\( T^{25} + \)\(11\!\cdots\!62\)\( T^{26} + \)\(46\!\cdots\!56\)\( T^{27} + \)\(48\!\cdots\!70\)\( T^{28} + \)\(19\!\cdots\!64\)\( T^{29} + \)\(18\!\cdots\!90\)\( T^{30} + \)\(69\!\cdots\!28\)\( T^{31} + \)\(59\!\cdots\!90\)\( T^{32} + \)\(22\!\cdots\!28\)\( T^{33} + \)\(16\!\cdots\!40\)\( T^{34} + \)\(55\!\cdots\!12\)\( T^{35} + \)\(34\!\cdots\!09\)\( T^{36} + \)\(90\!\cdots\!88\)\( T^{37} + \)\(45\!\cdots\!00\)\( T^{38} + \)\(70\!\cdots\!24\)\( T^{39} + \)\(26\!\cdots\!01\)\( T^{40} \)
$61$ \( 1 + 68 T + 2746 T^{2} + 80188 T^{3} + 1878988 T^{4} + 37113284 T^{5} + 640367622 T^{6} + 9880068788 T^{7} + 138898695230 T^{8} + 1804249453812 T^{9} + 21905924713998 T^{10} + 250758694746204 T^{11} + 2725099373231954 T^{12} + 28254058109831964 T^{13} + 280527819775663110 T^{14} + 2674075904274351012 T^{15} + 24521277038835470769 T^{16} + \)\(21\!\cdots\!80\)\( T^{17} + \)\(18\!\cdots\!40\)\( T^{18} + \)\(15\!\cdots\!44\)\( T^{19} + \)\(12\!\cdots\!64\)\( T^{20} + \)\(92\!\cdots\!84\)\( T^{21} + \)\(68\!\cdots\!40\)\( T^{22} + \)\(49\!\cdots\!80\)\( T^{23} + \)\(33\!\cdots\!29\)\( T^{24} + \)\(22\!\cdots\!12\)\( T^{25} + \)\(14\!\cdots\!10\)\( T^{26} + \)\(88\!\cdots\!44\)\( T^{27} + \)\(52\!\cdots\!74\)\( T^{28} + \)\(29\!\cdots\!64\)\( T^{29} + \)\(15\!\cdots\!98\)\( T^{30} + \)\(78\!\cdots\!32\)\( T^{31} + \)\(36\!\cdots\!30\)\( T^{32} + \)\(15\!\cdots\!28\)\( T^{33} + \)\(63\!\cdots\!02\)\( T^{34} + \)\(22\!\cdots\!84\)\( T^{35} + \)\(69\!\cdots\!68\)\( T^{36} + \)\(17\!\cdots\!48\)\( T^{37} + \)\(37\!\cdots\!26\)\( T^{38} + \)\(56\!\cdots\!88\)\( T^{39} + \)\(50\!\cdots\!01\)\( T^{40} \)
$67$ \( 1 - 4 T + 964 T^{2} - 4156 T^{3} + 456375 T^{4} - 2061116 T^{5} + 141287868 T^{6} - 652790288 T^{7} + 32101542726 T^{8} - 148762145132 T^{9} + 5690411521150 T^{10} - 26012669512256 T^{11} + 816354810670358 T^{12} - 3628940346767312 T^{13} + 97022283322151438 T^{14} - 414055046813706044 T^{15} + 9698391419487720521 T^{16} - 39254578625509580396 T^{17} + \)\(82\!\cdots\!24\)\( T^{18} - \)\(31\!\cdots\!48\)\( T^{19} + \)\(59\!\cdots\!90\)\( T^{20} - \)\(20\!\cdots\!16\)\( T^{21} + \)\(36\!\cdots\!36\)\( T^{22} - \)\(11\!\cdots\!48\)\( T^{23} + \)\(19\!\cdots\!41\)\( T^{24} - \)\(55\!\cdots\!08\)\( T^{25} + \)\(87\!\cdots\!22\)\( T^{26} - \)\(21\!\cdots\!76\)\( T^{27} + \)\(33\!\cdots\!78\)\( T^{28} - \)\(70\!\cdots\!32\)\( T^{29} + \)\(10\!\cdots\!50\)\( T^{30} - \)\(18\!\cdots\!56\)\( T^{31} + \)\(26\!\cdots\!86\)\( T^{32} - \)\(35\!\cdots\!56\)\( T^{33} + \)\(51\!\cdots\!72\)\( T^{34} - \)\(50\!\cdots\!88\)\( T^{35} + \)\(75\!\cdots\!75\)\( T^{36} - \)\(45\!\cdots\!12\)\( T^{37} + \)\(71\!\cdots\!76\)\( T^{38} - \)\(19\!\cdots\!12\)\( T^{39} + \)\(33\!\cdots\!01\)\( T^{40} \)
$71$ \( 1 - 8 T + 874 T^{2} - 6608 T^{3} + 368744 T^{4} - 2610424 T^{5} + 100404914 T^{6} - 660904456 T^{7} + 19947410094 T^{8} - 121657957840 T^{9} + 3106217030942 T^{10} - 17569063338760 T^{11} + 398194221131546 T^{12} - 2099453527999864 T^{13} + 43527546864216302 T^{14} - 215467526265102368 T^{15} + 4154554567737912833 T^{16} - 19417901038856108816 T^{17} + \)\(35\!\cdots\!52\)\( T^{18} - \)\(15\!\cdots\!16\)\( T^{19} + \)\(26\!\cdots\!20\)\( T^{20} - \)\(11\!\cdots\!36\)\( T^{21} + \)\(17\!\cdots\!32\)\( T^{22} - \)\(69\!\cdots\!76\)\( T^{23} + \)\(10\!\cdots\!73\)\( T^{24} - \)\(38\!\cdots\!68\)\( T^{25} + \)\(55\!\cdots\!42\)\( T^{26} - \)\(19\!\cdots\!24\)\( T^{27} + \)\(25\!\cdots\!06\)\( T^{28} - \)\(80\!\cdots\!60\)\( T^{29} + \)\(10\!\cdots\!42\)\( T^{30} - \)\(28\!\cdots\!40\)\( T^{31} + \)\(32\!\cdots\!54\)\( T^{32} - \)\(77\!\cdots\!16\)\( T^{33} + \)\(83\!\cdots\!34\)\( T^{34} - \)\(15\!\cdots\!24\)\( T^{35} + \)\(15\!\cdots\!24\)\( T^{36} - \)\(19\!\cdots\!28\)\( T^{37} + \)\(18\!\cdots\!14\)\( T^{38} - \)\(11\!\cdots\!48\)\( T^{39} + \)\(10\!\cdots\!01\)\( T^{40} \)
$73$ \( 1 + 48 T + 1494 T^{2} + 33256 T^{3} + 598148 T^{4} + 8939320 T^{5} + 116257546 T^{6} + 1342575896 T^{7} + 14264801870 T^{8} + 142480859576 T^{9} + 1376060227538 T^{10} + 12947573383648 T^{11} + 119527307724954 T^{12} + 1075306586055696 T^{13} + 9483832704310266 T^{14} + 82276567800105240 T^{15} + 713707218156263457 T^{16} + 6206964800767534632 T^{17} + 54355099696009499380 T^{18} + \)\(47\!\cdots\!28\)\( T^{19} + \)\(40\!\cdots\!04\)\( T^{20} + \)\(34\!\cdots\!44\)\( T^{21} + \)\(28\!\cdots\!20\)\( T^{22} + \)\(24\!\cdots\!44\)\( T^{23} + \)\(20\!\cdots\!37\)\( T^{24} + \)\(17\!\cdots\!20\)\( T^{25} + \)\(14\!\cdots\!74\)\( T^{26} + \)\(11\!\cdots\!12\)\( T^{27} + \)\(96\!\cdots\!74\)\( T^{28} + \)\(76\!\cdots\!24\)\( T^{29} + \)\(59\!\cdots\!62\)\( T^{30} + \)\(44\!\cdots\!52\)\( T^{31} + \)\(32\!\cdots\!70\)\( T^{32} + \)\(22\!\cdots\!68\)\( T^{33} + \)\(14\!\cdots\!14\)\( T^{34} + \)\(79\!\cdots\!40\)\( T^{35} + \)\(38\!\cdots\!28\)\( T^{36} + \)\(15\!\cdots\!68\)\( T^{37} + \)\(51\!\cdots\!86\)\( T^{38} + \)\(12\!\cdots\!76\)\( T^{39} + \)\(18\!\cdots\!01\)\( T^{40} \)
$79$ \( 1 + 20 T + 936 T^{2} + 18068 T^{3} + 457182 T^{4} + 8086332 T^{5} + 151587824 T^{6} + 2407195644 T^{7} + 37630638349 T^{8} + 536777135360 T^{9} + 7375070117720 T^{10} + 95320730703424 T^{11} + 1181426005717928 T^{12} + 13965857953973408 T^{13} + 158429454783904840 T^{14} + 1725932412403134048 T^{15} + 18076051380908681650 T^{16} + \)\(18\!\cdots\!80\)\( T^{17} + \)\(17\!\cdots\!16\)\( T^{18} + \)\(16\!\cdots\!72\)\( T^{19} + \)\(15\!\cdots\!32\)\( T^{20} + \)\(13\!\cdots\!88\)\( T^{21} + \)\(11\!\cdots\!56\)\( T^{22} + \)\(89\!\cdots\!20\)\( T^{23} + \)\(70\!\cdots\!50\)\( T^{24} + \)\(53\!\cdots\!52\)\( T^{25} + \)\(38\!\cdots\!40\)\( T^{26} + \)\(26\!\cdots\!72\)\( T^{27} + \)\(17\!\cdots\!08\)\( T^{28} + \)\(11\!\cdots\!56\)\( T^{29} + \)\(69\!\cdots\!20\)\( T^{30} + \)\(40\!\cdots\!40\)\( T^{31} + \)\(22\!\cdots\!09\)\( T^{32} + \)\(11\!\cdots\!16\)\( T^{33} + \)\(55\!\cdots\!44\)\( T^{34} + \)\(23\!\cdots\!68\)\( T^{35} + \)\(10\!\cdots\!22\)\( T^{36} + \)\(32\!\cdots\!12\)\( T^{37} + \)\(13\!\cdots\!96\)\( T^{38} + \)\(22\!\cdots\!80\)\( T^{39} + \)\(89\!\cdots\!01\)\( T^{40} \)
$83$ \( 1 - 8 T + 844 T^{2} - 7192 T^{3} + 372340 T^{4} - 3255184 T^{5} + 113045612 T^{6} - 989158864 T^{7} + 26286771966 T^{8} - 226434233416 T^{9} + 4946391546212 T^{10} - 41465008968600 T^{11} + 778056222842042 T^{12} - 6291772550614808 T^{13} + 104429188772228900 T^{14} - 808563541486623240 T^{15} + 12119228017148642625 T^{16} - 89221305839101878056 T^{17} + \)\(12\!\cdots\!12\)\( T^{18} - \)\(85\!\cdots\!24\)\( T^{19} + \)\(10\!\cdots\!32\)\( T^{20} - \)\(70\!\cdots\!92\)\( T^{21} + \)\(84\!\cdots\!68\)\( T^{22} - \)\(51\!\cdots\!72\)\( T^{23} + \)\(57\!\cdots\!25\)\( T^{24} - \)\(31\!\cdots\!20\)\( T^{25} + \)\(34\!\cdots\!00\)\( T^{26} - \)\(17\!\cdots\!16\)\( T^{27} + \)\(17\!\cdots\!22\)\( T^{28} - \)\(77\!\cdots\!00\)\( T^{29} + \)\(76\!\cdots\!88\)\( T^{30} - \)\(29\!\cdots\!72\)\( T^{31} + \)\(28\!\cdots\!26\)\( T^{32} - \)\(87\!\cdots\!32\)\( T^{33} + \)\(83\!\cdots\!48\)\( T^{34} - \)\(19\!\cdots\!88\)\( T^{35} + \)\(18\!\cdots\!40\)\( T^{36} - \)\(30\!\cdots\!16\)\( T^{37} + \)\(29\!\cdots\!96\)\( T^{38} - \)\(23\!\cdots\!76\)\( T^{39} + \)\(24\!\cdots\!01\)\( T^{40} \)
$89$ \( 1 + 60 T + 3064 T^{2} + 109736 T^{3} + 3469110 T^{4} + 92247504 T^{5} + 2226138806 T^{6} + 47923058388 T^{7} + 952895395711 T^{8} + 17367055631856 T^{9} + 295721937917720 T^{10} + 4684930005199444 T^{11} + 69874023790081134 T^{12} + 978367239015143448 T^{13} + 12963332685038076396 T^{14} + \)\(16\!\cdots\!80\)\( T^{15} + \)\(19\!\cdots\!71\)\( T^{16} + \)\(21\!\cdots\!04\)\( T^{17} + \)\(23\!\cdots\!62\)\( T^{18} + \)\(23\!\cdots\!64\)\( T^{19} + \)\(22\!\cdots\!11\)\( T^{20} + \)\(20\!\cdots\!96\)\( T^{21} + \)\(18\!\cdots\!02\)\( T^{22} + \)\(15\!\cdots\!76\)\( T^{23} + \)\(12\!\cdots\!11\)\( T^{24} + \)\(90\!\cdots\!20\)\( T^{25} + \)\(64\!\cdots\!56\)\( T^{26} + \)\(43\!\cdots\!92\)\( T^{27} + \)\(27\!\cdots\!54\)\( T^{28} + \)\(16\!\cdots\!96\)\( T^{29} + \)\(92\!\cdots\!20\)\( T^{30} + \)\(48\!\cdots\!84\)\( T^{31} + \)\(23\!\cdots\!31\)\( T^{32} + \)\(10\!\cdots\!72\)\( T^{33} + \)\(43\!\cdots\!46\)\( T^{34} + \)\(16\!\cdots\!96\)\( T^{35} + \)\(53\!\cdots\!10\)\( T^{36} + \)\(15\!\cdots\!44\)\( T^{37} + \)\(37\!\cdots\!84\)\( T^{38} + \)\(65\!\cdots\!40\)\( T^{39} + \)\(97\!\cdots\!01\)\( T^{40} \)
$97$ \( 1 + 40 T + 1682 T^{2} + 45380 T^{3} + 1181790 T^{4} + 25037632 T^{5} + 506502396 T^{6} + 9044350868 T^{7} + 154470490273 T^{8} + 2419186654120 T^{9} + 36384002638630 T^{10} + 512269362167364 T^{11} + 6953901987187436 T^{12} + 89457001024798496 T^{13} + 1113239218578782266 T^{14} + 13227501904031877232 T^{15} + \)\(15\!\cdots\!63\)\( T^{16} + \)\(16\!\cdots\!56\)\( T^{17} + \)\(18\!\cdots\!84\)\( T^{18} + \)\(18\!\cdots\!40\)\( T^{19} + \)\(18\!\cdots\!39\)\( T^{20} + \)\(18\!\cdots\!80\)\( T^{21} + \)\(17\!\cdots\!56\)\( T^{22} + \)\(15\!\cdots\!88\)\( T^{23} + \)\(13\!\cdots\!03\)\( T^{24} + \)\(11\!\cdots\!24\)\( T^{25} + \)\(92\!\cdots\!14\)\( T^{26} + \)\(72\!\cdots\!48\)\( T^{27} + \)\(54\!\cdots\!96\)\( T^{28} + \)\(38\!\cdots\!88\)\( T^{29} + \)\(26\!\cdots\!70\)\( T^{30} + \)\(17\!\cdots\!60\)\( T^{31} + \)\(10\!\cdots\!93\)\( T^{32} + \)\(60\!\cdots\!36\)\( T^{33} + \)\(33\!\cdots\!24\)\( T^{34} + \)\(15\!\cdots\!76\)\( T^{35} + \)\(72\!\cdots\!90\)\( T^{36} + \)\(27\!\cdots\!60\)\( T^{37} + \)\(97\!\cdots\!98\)\( T^{38} + \)\(22\!\cdots\!20\)\( T^{39} + \)\(54\!\cdots\!01\)\( T^{40} \)
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