Properties

Label 6034.2.a.k
Level 6034
Weight 2
Character orbit 6034.a
Self dual yes
Analytic conductor 48.182
Analytic rank 1
Dimension 20
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6034 = 2 \cdot 7 \cdot 431 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6034.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.1817325796\)
Analytic rank: \(1\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + \beta_{1} q^{3} + q^{4} -\beta_{15} q^{5} -\beta_{1} q^{6} + q^{7} - q^{8} + ( 1 + \beta_{2} ) q^{9} +O(q^{10})\) \( q - q^{2} + \beta_{1} q^{3} + q^{4} -\beta_{15} q^{5} -\beta_{1} q^{6} + q^{7} - q^{8} + ( 1 + \beta_{2} ) q^{9} + \beta_{15} q^{10} + ( -1 - \beta_{4} - \beta_{10} - \beta_{16} ) q^{11} + \beta_{1} q^{12} + ( -\beta_{3} + \beta_{15} ) q^{13} - q^{14} + ( -2 - \beta_{2} + \beta_{6} + \beta_{10} - \beta_{11} + \beta_{17} - \beta_{18} - \beta_{19} ) q^{15} + q^{16} + ( 3 - \beta_{1} + \beta_{4} - \beta_{6} - \beta_{8} - \beta_{9} + \beta_{10} - \beta_{12} + 2 \beta_{13} - \beta_{18} ) q^{17} + ( -1 - \beta_{2} ) q^{18} + ( \beta_{2} + \beta_{10} + \beta_{13} + \beta_{17} ) q^{19} -\beta_{15} q^{20} + \beta_{1} q^{21} + ( 1 + \beta_{4} + \beta_{10} + \beta_{16} ) q^{22} + ( -2 - \beta_{1} - \beta_{2} - \beta_{8} - \beta_{10} + \beta_{15} - \beta_{16} ) q^{23} -\beta_{1} q^{24} + ( 2 + \beta_{2} + \beta_{5} + \beta_{12} - \beta_{13} + \beta_{16} + \beta_{19} ) q^{25} + ( \beta_{3} - \beta_{15} ) q^{26} + ( \beta_{2} + \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} + \beta_{10} + 2 \beta_{11} + \beta_{13} + \beta_{14} + \beta_{15} ) q^{27} + q^{28} + ( -2 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} - \beta_{12} + \beta_{13} - \beta_{17} ) q^{29} + ( 2 + \beta_{2} - \beta_{6} - \beta_{10} + \beta_{11} - \beta_{17} + \beta_{18} + \beta_{19} ) q^{30} + ( -2 + \beta_{1} - \beta_{4} - \beta_{7} + 2 \beta_{8} + \beta_{9} - \beta_{10} + \beta_{12} - 3 \beta_{13} - \beta_{15} + \beta_{16} - \beta_{17} + 2 \beta_{18} + \beta_{19} ) q^{31} - q^{32} + ( -4 - \beta_{1} - \beta_{2} + \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} + \beta_{12} - 2 \beta_{13} - \beta_{14} - \beta_{15} + \beta_{16} + \beta_{18} ) q^{33} + ( -3 + \beta_{1} - \beta_{4} + \beta_{6} + \beta_{8} + \beta_{9} - \beta_{10} + \beta_{12} - 2 \beta_{13} + \beta_{18} ) q^{34} -\beta_{15} q^{35} + ( 1 + \beta_{2} ) q^{36} + ( -2 - \beta_{1} - 2 \beta_{2} + 2 \beta_{6} + 2 \beta_{8} + \beta_{9} - \beta_{10} - 3 \beta_{13} + \beta_{15} + 2 \beta_{16} - 2 \beta_{17} + \beta_{18} ) q^{37} + ( -\beta_{2} - \beta_{10} - \beta_{13} - \beta_{17} ) q^{38} + ( -1 - \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{10} + \beta_{11} - \beta_{13} + \beta_{15} - 2 \beta_{17} + 2 \beta_{18} ) q^{39} + \beta_{15} q^{40} + ( -5 - \beta_{1} - 2 \beta_{2} - \beta_{5} + 2 \beta_{6} + \beta_{8} + \beta_{9} - 2 \beta_{10} - 3 \beta_{13} - 2 \beta_{14} + \beta_{15} + \beta_{16} - 2 \beta_{17} + 2 \beta_{18} - \beta_{19} ) q^{41} -\beta_{1} q^{42} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{6} + \beta_{10} + \beta_{11} + \beta_{14} - \beta_{15} + \beta_{19} ) q^{43} + ( -1 - \beta_{4} - \beta_{10} - \beta_{16} ) q^{44} + ( -2 - 2 \beta_{1} - \beta_{4} + 2 \beta_{5} - \beta_{7} - 3 \beta_{10} - \beta_{11} + \beta_{12} - 2 \beta_{13} - \beta_{14} - 2 \beta_{15} + \beta_{16} + \beta_{17} + \beta_{18} + \beta_{19} ) q^{45} + ( 2 + \beta_{1} + \beta_{2} + \beta_{8} + \beta_{10} - \beta_{15} + \beta_{16} ) q^{46} + ( -3 - \beta_{1} - 2 \beta_{2} + \beta_{6} - 2 \beta_{7} + \beta_{8} - 2 \beta_{10} - 3 \beta_{13} - \beta_{14} - 2 \beta_{17} + \beta_{18} ) q^{47} + \beta_{1} q^{48} + q^{49} + ( -2 - \beta_{2} - \beta_{5} - \beta_{12} + \beta_{13} - \beta_{16} - \beta_{19} ) q^{50} + ( 2 + 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{6} + 2 \beta_{7} - \beta_{9} + 4 \beta_{10} + 2 \beta_{11} + \beta_{12} + 3 \beta_{13} + 2 \beta_{14} + \beta_{15} + \beta_{17} - 2 \beta_{18} ) q^{51} + ( -\beta_{3} + \beta_{15} ) q^{52} + ( -\beta_{9} + \beta_{11} + \beta_{13} + \beta_{14} + \beta_{15} + \beta_{16} - \beta_{17} + \beta_{19} ) q^{53} + ( -\beta_{2} - \beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} - \beta_{10} - 2 \beta_{11} - \beta_{13} - \beta_{14} - \beta_{15} ) q^{54} + ( 2 + \beta_{2} + \beta_{3} - \beta_{6} + \beta_{7} - 2 \beta_{8} - \beta_{9} + \beta_{10} + 2 \beta_{11} - \beta_{12} + 4 \beta_{13} + 2 \beta_{14} + 3 \beta_{15} - 2 \beta_{16} - \beta_{18} ) q^{55} - q^{56} + ( -3 - \beta_{1} - \beta_{7} + \beta_{9} - \beta_{10} - \beta_{11} - \beta_{13} - 2 \beta_{14} - 2 \beta_{15} + 2 \beta_{18} ) q^{57} + ( 2 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} + \beta_{12} - \beta_{13} + \beta_{17} ) q^{58} + ( -4 - 2 \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{11} + \beta_{12} - \beta_{13} - \beta_{14} - \beta_{16} + \beta_{18} - \beta_{19} ) q^{59} + ( -2 - \beta_{2} + \beta_{6} + \beta_{10} - \beta_{11} + \beta_{17} - \beta_{18} - \beta_{19} ) q^{60} + ( 3 + \beta_{2} - \beta_{6} - \beta_{8} + 3 \beta_{10} - \beta_{11} + 2 \beta_{13} + 2 \beta_{14} + \beta_{15} + \beta_{16} + \beta_{17} - 2 \beta_{18} ) q^{61} + ( 2 - \beta_{1} + \beta_{4} + \beta_{7} - 2 \beta_{8} - \beta_{9} + \beta_{10} - \beta_{12} + 3 \beta_{13} + \beta_{15} - \beta_{16} + \beta_{17} - 2 \beta_{18} - \beta_{19} ) q^{62} + ( 1 + \beta_{2} ) q^{63} + q^{64} + ( -2 - 2 \beta_{1} + \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + \beta_{7} - 2 \beta_{8} - \beta_{9} + 2 \beta_{10} + \beta_{11} - 2 \beta_{12} + 4 \beta_{13} + \beta_{15} - \beta_{16} + \beta_{17} - \beta_{19} ) q^{65} + ( 4 + \beta_{1} + \beta_{2} - \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} - \beta_{12} + 2 \beta_{13} + \beta_{14} + \beta_{15} - \beta_{16} - \beta_{18} ) q^{66} + ( -2 \beta_{1} + \beta_{4} + \beta_{5} + \beta_{9} - 2 \beta_{11} - \beta_{13} - 2 \beta_{14} + \beta_{16} + 2 \beta_{17} ) q^{67} + ( 3 - \beta_{1} + \beta_{4} - \beta_{6} - \beta_{8} - \beta_{9} + \beta_{10} - \beta_{12} + 2 \beta_{13} - \beta_{18} ) q^{68} + ( -1 - 4 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} - \beta_{13} - \beta_{14} + \beta_{16} + \beta_{19} ) q^{69} + \beta_{15} q^{70} + ( \beta_{1} + 4 \beta_{2} + \beta_{3} + 3 \beta_{5} - 2 \beta_{6} + \beta_{7} + 2 \beta_{10} + \beta_{12} + 2 \beta_{13} + 2 \beta_{14} - 2 \beta_{15} + \beta_{16} + 4 \beta_{17} - 3 \beta_{18} + \beta_{19} ) q^{71} + ( -1 - \beta_{2} ) q^{72} + ( 1 + 2 \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{7} + \beta_{8} + \beta_{9} + 3 \beta_{10} + \beta_{12} + \beta_{13} + \beta_{14} - \beta_{15} + \beta_{17} - 2 \beta_{18} ) q^{73} + ( 2 + \beta_{1} + 2 \beta_{2} - 2 \beta_{6} - 2 \beta_{8} - \beta_{9} + \beta_{10} + 3 \beta_{13} - \beta_{15} - 2 \beta_{16} + 2 \beta_{17} - \beta_{18} ) q^{74} + ( 4 + 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - 2 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} - 2 \beta_{12} + 3 \beta_{13} + 2 \beta_{14} + 3 \beta_{15} - \beta_{16} - 2 \beta_{18} + \beta_{19} ) q^{75} + ( \beta_{2} + \beta_{10} + \beta_{13} + \beta_{17} ) q^{76} + ( -1 - \beta_{4} - \beta_{10} - \beta_{16} ) q^{77} + ( 1 + \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{10} - \beta_{11} + \beta_{13} - \beta_{15} + 2 \beta_{17} - 2 \beta_{18} ) q^{78} + ( -5 - 2 \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} + 2 \beta_{9} - 2 \beta_{10} - 3 \beta_{13} - 2 \beta_{14} + 2 \beta_{16} + \beta_{18} ) q^{79} -\beta_{15} q^{80} + ( 3 - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{10} - 2 \beta_{12} + 2 \beta_{13} + \beta_{14} + 2 \beta_{15} - \beta_{16} - 2 \beta_{17} - \beta_{18} ) q^{81} + ( 5 + \beta_{1} + 2 \beta_{2} + \beta_{5} - 2 \beta_{6} - \beta_{8} - \beta_{9} + 2 \beta_{10} + 3 \beta_{13} + 2 \beta_{14} - \beta_{15} - \beta_{16} + 2 \beta_{17} - 2 \beta_{18} + \beta_{19} ) q^{82} + ( -2 + \beta_{6} + \beta_{9} + \beta_{10} - 2 \beta_{11} + \beta_{12} - \beta_{13} + \beta_{14} - \beta_{15} + \beta_{17} - 2 \beta_{18} ) q^{83} + \beta_{1} q^{84} + ( -2 - \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{10} - 2 \beta_{11} + 2 \beta_{12} - 2 \beta_{13} - 3 \beta_{14} + \beta_{16} + \beta_{17} + \beta_{18} - \beta_{19} ) q^{85} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{6} - \beta_{10} - \beta_{11} - \beta_{14} + \beta_{15} - \beta_{19} ) q^{86} + ( -1 - 2 \beta_{1} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{10} + \beta_{12} + 2 \beta_{13} + \beta_{17} - \beta_{18} ) q^{87} + ( 1 + \beta_{4} + \beta_{10} + \beta_{16} ) q^{88} + ( -1 - 2 \beta_{1} - 3 \beta_{2} + 3 \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} - 2 \beta_{8} - \beta_{9} + 2 \beta_{10} - 2 \beta_{12} + 3 \beta_{13} - \beta_{14} + 3 \beta_{15} - \beta_{18} - 2 \beta_{19} ) q^{89} + ( 2 + 2 \beta_{1} + \beta_{4} - 2 \beta_{5} + \beta_{7} + 3 \beta_{10} + \beta_{11} - \beta_{12} + 2 \beta_{13} + \beta_{14} + 2 \beta_{15} - \beta_{16} - \beta_{17} - \beta_{18} - \beta_{19} ) q^{90} + ( -\beta_{3} + \beta_{15} ) q^{91} + ( -2 - \beta_{1} - \beta_{2} - \beta_{8} - \beta_{10} + \beta_{15} - \beta_{16} ) q^{92} + ( -\beta_{3} - 3 \beta_{4} + \beta_{6} - 2 \beta_{7} - \beta_{9} - 2 \beta_{10} - \beta_{11} - \beta_{13} + \beta_{15} - 2 \beta_{16} - \beta_{17} + \beta_{18} ) q^{93} + ( 3 + \beta_{1} + 2 \beta_{2} - \beta_{6} + 2 \beta_{7} - \beta_{8} + 2 \beta_{10} + 3 \beta_{13} + \beta_{14} + 2 \beta_{17} - \beta_{18} ) q^{94} + ( -1 + 4 \beta_{1} + 2 \beta_{2} - \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} + 2 \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} + \beta_{14} - \beta_{16} + \beta_{19} ) q^{95} -\beta_{1} q^{96} + ( 4 - \beta_{1} + 3 \beta_{2} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - 3 \beta_{9} + \beta_{10} + \beta_{11} + 2 \beta_{13} + \beta_{14} + \beta_{15} - \beta_{16} + 2 \beta_{17} - \beta_{18} + \beta_{19} ) q^{97} - q^{98} + ( -1 - 2 \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{6} - 2 \beta_{7} - \beta_{9} - \beta_{11} - \beta_{12} - \beta_{14} + \beta_{15} - \beta_{17} - \beta_{19} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q - 20q^{2} + 3q^{3} + 20q^{4} - 3q^{5} - 3q^{6} + 20q^{7} - 20q^{8} + 13q^{9} + O(q^{10}) \) \( 20q - 20q^{2} + 3q^{3} + 20q^{4} - 3q^{5} - 3q^{6} + 20q^{7} - 20q^{8} + 13q^{9} + 3q^{10} - 8q^{11} + 3q^{12} - 4q^{13} - 20q^{14} - 25q^{15} + 20q^{16} + 9q^{17} - 13q^{18} - 14q^{19} - 3q^{20} + 3q^{21} + 8q^{22} - 23q^{23} - 3q^{24} + 31q^{25} + 4q^{26} - 21q^{27} + 20q^{28} - 48q^{29} + 25q^{30} - q^{31} - 20q^{32} - 29q^{33} - 9q^{34} - 3q^{35} + 13q^{36} - q^{37} + 14q^{38} - q^{39} + 3q^{40} - 27q^{41} - 3q^{42} - 3q^{43} - 8q^{44} - 12q^{45} + 23q^{46} - 26q^{47} + 3q^{48} + 20q^{49} - 31q^{50} - 17q^{51} - 4q^{52} - 43q^{53} + 21q^{54} - 16q^{55} - 20q^{56} - 25q^{57} + 48q^{58} - 19q^{59} - 25q^{60} + 9q^{61} + q^{62} + 13q^{63} + 20q^{64} - 87q^{65} + 29q^{66} + 32q^{67} + 9q^{68} - 23q^{69} + 3q^{70} - 63q^{71} - 13q^{72} + 2q^{73} + q^{74} - 8q^{75} - 14q^{76} - 8q^{77} + q^{78} - 51q^{79} - 3q^{80} + 4q^{81} + 27q^{82} - 24q^{83} + 3q^{84} + 31q^{85} + 3q^{86} - 33q^{87} + 8q^{88} - 35q^{89} + 12q^{90} - 4q^{91} - 23q^{92} + 17q^{93} + 26q^{94} - 30q^{95} - 3q^{96} + 5q^{97} - 20q^{98} - 31q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} - 3 x^{19} - 32 x^{18} + 106 x^{17} + 382 x^{16} - 1495 x^{15} - 1963 x^{14} + 10784 x^{13} + 2170 x^{12} - 42069 x^{11} + 19553 x^{10} + 84697 x^{9} - 82713 x^{8} - 68589 x^{7} + 118845 x^{6} - 10051 x^{5} - 53572 x^{4} + 25598 x^{3} - 557 x^{2} - 1040 x - 44\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 4 \)
\(\beta_{3}\)\(=\)\((\)\(1884208619073815 \nu^{19} - 6772049431260751 \nu^{18} - 54819832348153622 \nu^{17} + 244497232318481630 \nu^{16} + 531834645279097790 \nu^{15} - 3539175730986085457 \nu^{14} - 1113396279392493815 \nu^{13} + 26359431408583195074 \nu^{12} - 14331109638941604574 \nu^{11} - 107267874290971032547 \nu^{10} + 109977128581407295541 \nu^{9} + 230857985966499041093 \nu^{8} - 316718406459312876665 \nu^{7} - 220585853219456888561 \nu^{6} + 404156120962248189289 \nu^{5} + 27939152610740301313 \nu^{4} - 181665080056500571426 \nu^{3} + 41411235428956984938 \nu^{2} + 5216975976415944925 \nu + 67218999701674198\)\()/ 113435235445735860 \)
\(\beta_{4}\)\(=\)\((\)\(-1646489955919853 \nu^{19} + 4183343716317277 \nu^{18} + 56342467239727746 \nu^{17} - 148521462379189690 \nu^{16} - 757000768750022846 \nu^{15} + 2115406329855661411 \nu^{14} + 5031539410354637667 \nu^{13} - 15563451891548516804 \nu^{12} - 16648452991686642704 \nu^{11} + 63238650629312797919 \nu^{10} + 20399478263473531751 \nu^{9} - 139757278490216706639 \nu^{8} + 20786416910545552897 \nu^{7} + 150864790001142980735 \nu^{6} - 71612347865554683003 \nu^{5} - 55260926422933630697 \nu^{4} + 42615529150975749982 \nu^{3} - 3488125857404902184 \nu^{2} - 1537307441254348881 \nu - 70564680236336004\)\()/ 56717617722867930 \)
\(\beta_{5}\)\(=\)\((\)\(3127104033838755 \nu^{19} - 3540140078642762 \nu^{18} - 108807566211895234 \nu^{17} + 132019632435244250 \nu^{16} + 1513198132916894710 \nu^{15} - 1984932905299763089 \nu^{14} - 10803722285236563090 \nu^{13} + 15511667094461435358 \nu^{12} + 42065511186223993522 \nu^{11} - 67633902391246240399 \nu^{10} - 86002892896791014258 \nu^{9} + 164168401698048247281 \nu^{8} + 74528931910043807460 \nu^{7} - 207565952904505687307 \nu^{6} + 3813604006572639198 \nu^{5} + 115278628048640600541 \nu^{4} - 29683598894102148527 \nu^{3} - 20249899522952731459 \nu^{2} + 6039841121644550400 \nu + 835262515354061076\)\()/ 56717617722867930 \)
\(\beta_{6}\)\(=\)\((\)\(10065141187344371 \nu^{19} - 3991346206018785 \nu^{18} - 343220790204299614 \nu^{17} + 172548994397585930 \nu^{16} + 4657159059159682402 \nu^{15} - 2935114539028035169 \nu^{14} - 32271571631138111809 \nu^{13} + 25322430827364838482 \nu^{12} + 121093907308705820874 \nu^{11} - 118727806455225918255 \nu^{10} - 235459091984761509221 \nu^{9} + 300975641977135175641 \nu^{8} + 185972147390093472361 \nu^{7} - 380635973235985493941 \nu^{6} + 25255588012663275495 \nu^{5} + 189757253919079795777 \nu^{4} - 70981567614451844400 \nu^{3} - 18133147962913477724 \nu^{2} + 7172369417865667487 \nu + 537124289860931266\)\()/ 113435235445735860 \)
\(\beta_{7}\)\(=\)\((\)\(1119536860327121 \nu^{19} - 2438333220022678 \nu^{18} - 37840877805593040 \nu^{17} + 87597900041745720 \nu^{16} + 499904042406259194 \nu^{15} - 1263930365044317699 \nu^{14} - 3236734930360921036 \nu^{13} + 9428248289577918396 \nu^{12} + 10147403027698270934 \nu^{11} - 38863125151886120345 \nu^{10} - 9724746283820681574 \nu^{9} + 87184380574671031621 \nu^{8} - 22413709883206621946 \nu^{7} - 95678215384117260935 \nu^{6} + 57717793420680133428 \nu^{5} + 35928090013923519289 \nu^{4} - 33968674222941907429 \nu^{3} + 1876574736206580737 \nu^{2} + 2258491597010611886 \nu + 58867169842075096\)\()/ 11343523544573586 \)
\(\beta_{8}\)\(=\)\((\)\(-10503448216358573 \nu^{19} + 5034051088678948 \nu^{18} + 363424214178179798 \nu^{17} - 204385495015574670 \nu^{16} - 5033069516739563416 \nu^{15} + 3304033166645627703 \nu^{14} + 35938085817581442262 \nu^{13} - 27333349626656604218 \nu^{12} - 141491884224648904880 \nu^{11} + 123629291047411873291 \nu^{10} + 301486860612581278080 \nu^{9} - 302859648982824319957 \nu^{8} - 307154999607483199868 \nu^{7} + 368008791233657177981 \nu^{6} + 91295869333194184928 \nu^{5} - 170606770584699719105 \nu^{4} + 20986728065886996523 \nu^{3} + 10693047212385667203 \nu^{2} - 343771353327302546 \nu - 109311142072143852\)\()/ 56717617722867930 \)
\(\beta_{9}\)\(=\)\((\)\(11879423578954700 \nu^{19} - 26399200742569021 \nu^{18} - 399380682611257162 \nu^{17} + 948762142305284300 \nu^{16} + 5231711985000248940 \nu^{15} - 13692638884638157382 \nu^{14} - 33366011599506042935 \nu^{13} + 102128444313634036454 \nu^{12} + 100947764831013868166 \nu^{11} - 420635238281782202832 \nu^{10} - 78031514542204280329 \nu^{9} + 941503286636183226728 \nu^{8} - 287795403625904923975 \nu^{7} - 1026673138313829839246 \nu^{6} + 658177758251583145949 \nu^{5} + 375056080009787972978 \nu^{4} - 370408125699181241641 \nu^{3} + 26838729095394594943 \nu^{2} + 19638808582601284345 \nu + 1194417972855740628\)\()/ 56717617722867930 \)
\(\beta_{10}\)\(=\)\((\)\(25085191831479371 \nu^{19} - 19034873443334565 \nu^{18} - 867850713408271494 \nu^{17} + 728957321221247750 \nu^{16} + 11994831050850637262 \nu^{15} - 11198820939293132989 \nu^{14} - 85136639406827835829 \nu^{13} + 88702496070239696822 \nu^{12} + 330325542765500062434 \nu^{11} - 386842852178757781115 \nu^{10} - 679216189447299489521 \nu^{9} + 918068582305768390561 \nu^{8} + 622042653413978308181 \nu^{7} - 1080397092158436110881 \nu^{6} - 74213307447322281605 \nu^{5} + 475492920356722312937 \nu^{4} - 115343226461668973760 \nu^{3} - 17751734706099632224 \nu^{2} + 2277763214801098667 \nu + 35268733993435026\)\()/ 113435235445735860 \)
\(\beta_{11}\)\(=\)\((\)\(-31966208645463627 \nu^{19} + 72498357937682183 \nu^{18} + 1077136798741435754 \nu^{17} - 2602117595074175970 \nu^{16} - 14154077632308236534 \nu^{15} + 37504207045011304289 \nu^{14} + 90699297037338387083 \nu^{13} - 279370097860565801106 \nu^{12} - 277045272916197744106 \nu^{11} + 1149237242011364868311 \nu^{10} + 226468875709856889979 \nu^{9} - 2568991581728831247401 \nu^{8} + 749482882951131681213 \nu^{7} + 2795731775654655622205 \nu^{6} - 1753709700170495919137 \nu^{5} - 1014599839635315715593 \nu^{4} + 988854651337288080098 \nu^{3} - 77202290690818616026 \nu^{2} - 50629726658386178209 \nu - 2220405302942957286\)\()/ 113435235445735860 \)
\(\beta_{12}\)\(=\)\((\)\(-16224647473810125 \nu^{19} + 25866429321884853 \nu^{18} + 553852546619031056 \nu^{17} - 940299913509390050 \nu^{16} - 7460969028510329990 \nu^{15} + 13729731111826327841 \nu^{14} + 50339956735608067405 \nu^{13} - 103591966370155386702 \nu^{12} - 175101874019574190308 \nu^{11} + 431385711403507642631 \nu^{10} + 267079743961638057517 \nu^{9} - 976132120602824220979 \nu^{8} + 14549830779549933435 \nu^{7} + 1078448534742318966363 \nu^{6} - 451109215052237945047 \nu^{5} - 406200616592431367949 \nu^{4} + 299198586394309623888 \nu^{3} - 21990502912283119654 \nu^{2} - 12266222197926562235 \nu - 249743705922629934\)\()/ 56717617722867930 \)
\(\beta_{13}\)\(=\)\((\)\(-16342144147451935 \nu^{19} + 20104009767025802 \nu^{18} + 563322951257545704 \nu^{17} - 738800036979872060 \nu^{16} - 7718195745488889130 \nu^{15} + 10904462281102297799 \nu^{14} + 53742761929014228840 \nu^{13} - 83134202697530401288 \nu^{12} - 199850736345345756232 \nu^{11} + 349590177495387192079 \nu^{10} + 369300234338226285628 \nu^{9} - 798817401655150548711 \nu^{8} - 219590243417063266600 \nu^{7} + 893801568877545104677 \nu^{6} - 180515428448785359558 \nu^{5} - 347965140931814392621 \nu^{4} + 177530071668141828347 \nu^{3} - 11817748904901276961 \nu^{2} - 4270452584235103680 \nu + 257541633675664884\)\()/ 56717617722867930 \)
\(\beta_{14}\)\(=\)\((\)\(17224340615133150 \nu^{19} - 41406843424153057 \nu^{18} - 580666147649447414 \nu^{17} + 1481853849782149650 \nu^{16} + 7628601459071785650 \nu^{15} - 21292905889765348234 \nu^{14} - 48785177018579809945 \nu^{13} + 158132035611333153118 \nu^{12} + 147817116983655916622 \nu^{11} - 648698659677830432714 \nu^{10} - 113001640915608876683 \nu^{9} + 1446833818111464209566 \nu^{8} - 429755687216334791905 \nu^{7} - 1572900099036513843592 \nu^{6} + 978720279096516244953 \nu^{5} + 573124862483069279336 \nu^{4} - 547957516221886200967 \nu^{3} + 41380605186174422011 \nu^{2} + 27289472051371045745 \nu + 1280728335623362936\)\()/ 56717617722867930 \)
\(\beta_{15}\)\(=\)\((\)\(37361376400924505 \nu^{19} - 64558827124941099 \nu^{18} - 1274413605570467678 \nu^{17} + 2338285929742575490 \nu^{16} + 17132798313918081130 \nu^{15} - 34020153750713470263 \nu^{14} - 115031029357563459095 \nu^{13} + 255849454129385894786 \nu^{12} + 395080134444313106514 \nu^{11} - 1062720649383104752333 \nu^{10} - 575306463072558160851 \nu^{9} + 2401670155483626014147 \nu^{8} - 137061739189064090905 \nu^{7} - 2657866945010579613199 \nu^{6} + 1172369235067575467021 \nu^{5} + 1016446369826370907787 \nu^{4} - 756258555500456403984 \nu^{3} + 41583289250618372652 \nu^{2} + 34708132108877259925 \nu + 1805105827288545462\)\()/ 113435235445735860 \)
\(\beta_{16}\)\(=\)\((\)\(-10746543134042912 \nu^{19} + 18260962849193807 \nu^{18} + 365880217180962962 \nu^{17} - 662073116063725145 \nu^{16} - 4905429074201272744 \nu^{15} + 9640762564957410867 \nu^{14} + 32793031411241113008 \nu^{13} - 72537072214922240777 \nu^{12} - 111679607285072540890 \nu^{11} + 301201657390889352304 \nu^{10} + 158285459780581389870 \nu^{9} - 679374153241403694603 \nu^{8} + 53864470795574235898 \nu^{7} + 747063820491258359964 \nu^{6} - 351730890493329935193 \nu^{5} - 277526287381350090395 \nu^{4} + 223107187522592085377 \nu^{3} - 17550715465077968808 \nu^{2} - 10012177462068633729 \nu - 292259905165321848\)\()/ 28358808861433965 \)
\(\beta_{17}\)\(=\)\((\)\(45818802149595219 \nu^{19} - 80423199849046123 \nu^{18} - 1560542818850246142 \nu^{17} + 2909813195872212810 \nu^{16} + 20926938449972393498 \nu^{15} - 42280004014645994737 \nu^{14} - 139868223871836622291 \nu^{13} + 317423446216747218470 \nu^{12} + 475696139112830000294 \nu^{11} - 1315248817160925041811 \nu^{10} - 670170696264940021031 \nu^{9} + 2960230531782877081933 \nu^{8} - 243960778983622631921 \nu^{7} - 3247270958174814193717 \nu^{6} + 1514528377374218926757 \nu^{5} + 1201151737800137645677 \nu^{4} - 949901414726717269498 \nu^{3} + 78033602764254232158 \nu^{2} + 37794653657177039453 \nu + 1374880185302607158\)\()/ 113435235445735860 \)
\(\beta_{18}\)\(=\)\((\)\(14690039489903837 \nu^{19} - 25137550241914713 \nu^{18} - 500532558632426530 \nu^{17} + 911713206433600330 \nu^{16} + 6718854045484071566 \nu^{15} - 13284570343241530587 \nu^{14} - 45009368063324320529 \nu^{13} + 100070008563587669410 \nu^{12} + 153953951875942323930 \nu^{11} - 416411828992692958941 \nu^{10} - 221438087034559720857 \nu^{9} + 943112291717055668147 \nu^{8} - 62796098856213684591 \nu^{7} - 1047076905137305110331 \nu^{6} + 469634877785439511647 \nu^{5} + 403684530372989121723 \nu^{4} - 301623431917841174850 \nu^{3} + 14800765298119527046 \nu^{2} + 14632200301001467927 \nu + 728488020471033914\)\()/ 22687047089147172 \)
\(\beta_{19}\)\(=\)\((\)\(87010448442195819 \nu^{19} - 129109586967391609 \nu^{18} - 2968108296797542394 \nu^{17} + 4715623603162141030 \nu^{16} + 39973839933539256578 \nu^{15} - 69185942304319470029 \nu^{14} - 269984520608703321501 \nu^{13} + 524474375811286138494 \nu^{12} + 943426197150910256570 \nu^{11} - 2193831028682643821903 \nu^{10} - 1466850759779536091305 \nu^{9} + 4985914666269938045961 \nu^{8} + 33264134102817150609 \nu^{7} - 5536494816837663198793 \nu^{6} + 2263071533214921389091 \nu^{5} + 2107844200896181039245 \nu^{4} - 1531756924411167399844 \nu^{3} + 100752167824208576356 \nu^{2} + 65374448727472674843 \nu + 2396009442276911106\)\()/ 113435235445735860 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{15} + \beta_{14} + \beta_{13} + 2 \beta_{11} + \beta_{10} + \beta_{7} - \beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} + 6 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-\beta_{18} - 2 \beta_{17} - \beta_{16} + 2 \beta_{15} + \beta_{14} + 2 \beta_{13} - 2 \beta_{12} + \beta_{10} - \beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + 8 \beta_{2} - \beta_{1} + 30\)
\(\nu^{5}\)\(=\)\(\beta_{19} + 13 \beta_{15} + 14 \beta_{14} + 15 \beta_{13} + 2 \beta_{12} + 25 \beta_{11} + 14 \beta_{10} - 2 \beta_{9} - \beta_{8} + 14 \beta_{7} - 14 \beta_{6} - 2 \beta_{5} + 15 \beta_{4} + 11 \beta_{3} + 12 \beta_{2} + 47 \beta_{1} + 5\)
\(\nu^{6}\)\(=\)\(\beta_{19} - 19 \beta_{18} - 32 \beta_{17} - 16 \beta_{16} + 34 \beta_{15} + 20 \beta_{14} + 32 \beta_{13} - 31 \beta_{12} + 3 \beta_{11} + 21 \beta_{10} - 3 \beta_{9} - 11 \beta_{8} - 12 \beta_{7} - 13 \beta_{6} - 15 \beta_{5} + 12 \beta_{4} - 13 \beta_{3} + 68 \beta_{2} - 9 \beta_{1} + 271\)
\(\nu^{7}\)\(=\)\(27 \beta_{19} - 2 \beta_{18} - \beta_{17} + 3 \beta_{16} + 133 \beta_{15} + 169 \beta_{14} + 176 \beta_{13} + 41 \beta_{12} + 274 \beta_{11} + 159 \beta_{10} - 39 \beta_{9} - 17 \beta_{8} + 159 \beta_{7} - 168 \beta_{6} - 23 \beta_{5} + 170 \beta_{4} + 116 \beta_{3} + 139 \beta_{2} + 423 \beta_{1} + 96\)
\(\nu^{8}\)\(=\)\(21 \beta_{19} - 283 \beta_{18} - 374 \beta_{17} - 194 \beta_{16} + 437 \beta_{15} + 293 \beta_{14} + 414 \beta_{13} - 377 \beta_{12} + 54 \beta_{11} + 324 \beta_{10} - 66 \beta_{9} - 106 \beta_{8} - 114 \beta_{7} - 149 \beta_{6} - 162 \beta_{5} + 118 \beta_{4} - 132 \beta_{3} + 634 \beta_{2} - 64 \beta_{1} + 2653\)
\(\nu^{9}\)\(=\)\(442 \beta_{19} - 55 \beta_{18} - 4 \beta_{17} + 52 \beta_{16} + 1278 \beta_{15} + 1948 \beta_{14} + 1960 \beta_{13} + 586 \beta_{12} + 2926 \beta_{11} + 1723 \beta_{10} - 569 \beta_{9} - 234 \beta_{8} + 1717 \beta_{7} - 1927 \beta_{6} - 178 \beta_{5} + 1794 \beta_{4} + 1237 \beta_{3} + 1595 \beta_{2} + 4082 \beta_{1} + 1355\)
\(\nu^{10}\)\(=\)\(311 \beta_{19} - 3711 \beta_{18} - 3982 \beta_{17} - 2151 \beta_{16} + 5059 \beta_{15} + 3749 \beta_{14} + 4901 \beta_{13} - 4227 \beta_{12} + 673 \beta_{11} + 4316 \beta_{10} - 982 \beta_{9} - 1001 \beta_{8} - 1034 \beta_{7} - 1644 \beta_{6} - 1587 \beta_{5} + 1106 \beta_{4} - 1272 \beta_{3} + 6249 \beta_{2} - 401 \beta_{1} + 26891\)
\(\nu^{11}\)\(=\)\(5973 \beta_{19} - 950 \beta_{18} + 144 \beta_{17} + 629 \beta_{16} + 12135 \beta_{15} + 21873 \beta_{14} + 21423 \beta_{13} + 7325 \beta_{12} + 31033 \beta_{11} + 18388 \beta_{10} - 7356 \beta_{9} - 2938 \beta_{8} + 18252 \beta_{7} - 21538 \beta_{6} - 984 \beta_{5} + 18553 \beta_{4} + 13269 \beta_{3} + 17955 \beta_{2} + 40837 \beta_{1} + 16814\)
\(\nu^{12}\)\(=\)\(4036 \beta_{19} - 45126 \beta_{18} - 41127 \beta_{17} - 22989 \beta_{16} + 55796 \beta_{15} + 44573 \beta_{14} + 55287 \beta_{13} - 45816 \beta_{12} + 7243 \beta_{11} + 52839 \beta_{10} - 12501 \beta_{9} - 9447 \beta_{8} - 9478 \beta_{7} - 17737 \beta_{6} - 15056 \beta_{5} + 10241 \beta_{4} - 12272 \beta_{3} + 63373 \beta_{2} - 2277 \beta_{1} + 276951\)
\(\nu^{13}\)\(=\)\(73359 \beta_{19} - 13355 \beta_{18} + 3932 \beta_{17} + 6655 \beta_{16} + 115919 \beta_{15} + 241127 \beta_{14} + 231627 \beta_{13} + 85929 \beta_{12} + 328309 \beta_{11} + 194892 \beta_{10} - 88857 \beta_{9} - 34872 \beta_{8} + 192957 \beta_{7} - 236457 \beta_{6} - 1529 \beta_{5} + 191058 \beta_{4} + 142383 \beta_{3} + 198268 \beta_{2} + 416650 \beta_{1} + 194430\)
\(\nu^{14}\)\(=\)\(48819 \beta_{19} - 523562 \beta_{18} - 421467 \beta_{17} - 241793 \beta_{16} + 600443 \beta_{15} + 507097 \beta_{14} + 606259 \beta_{13} - 489238 \beta_{12} + 72100 \beta_{11} + 614185 \beta_{10} - 147152 \beta_{9} - 89667 \beta_{8} - 89678 \beta_{7} - 188368 \beta_{6} - 141988 \beta_{5} + 94621 \beta_{4} - 120618 \beta_{3} + 651720 \beta_{2} - 12659 \beta_{1} + 2875172\)
\(\nu^{15}\)\(=\)\(853415 \beta_{19} - 167406 \beta_{18} + 67283 \beta_{17} + 66789 \beta_{16} + 1120251 \beta_{15} + 2622414 \beta_{14} + 2484457 \beta_{13} + 973999 \beta_{12} + 3468459 \beta_{11} + 2057148 \beta_{10} - 1028364 \beta_{9} - 398820 \beta_{8} + 2035783 \beta_{7} - 2563179 \beta_{6} + 62165 \beta_{5} + 1969706 \beta_{4} + 1524649 \beta_{3} + 2155493 \beta_{2} + 4299408 \beta_{1} + 2153054\)
\(\nu^{16}\)\(=\)\(564122 \beta_{19} - 5892624 \beta_{18} - 4322419 \beta_{17} - 2526440 \beta_{16} + 6377698 \beta_{15} + 5608818 \beta_{14} + 6532467 \beta_{13} - 5190357 \beta_{12} + 680828 \beta_{11} + 6904859 \beta_{10} - 1656732 \beta_{9} - 858346 \beta_{8} - 881306 \beta_{7} - 1976489 \beta_{6} - 1347379 \beta_{5} + 873837 \beta_{4} - 1212843 \beta_{3} + 6747202 \beta_{2} - 91446 \beta_{1} + 29977633\)
\(\nu^{17}\)\(=\)\(9603713 \beta_{19} - 1952160 \beta_{18} + 967181 \beta_{17} + 662764 \beta_{16} + 10957986 \beta_{15} + 28233929 \beta_{14} + 26481842 \beta_{13} + 10818987 \beta_{12} + 36605770 \beta_{11} + 21648397 \beta_{10} - 11565912 \beta_{9} - 4447179 \beta_{8} + 21462182 \beta_{7} - 27534436 \beta_{6} + 1314514 \beta_{5} + 20362624 \beta_{4} + 16281417 \beta_{3} + 23160667 \beta_{2} + 44669704 \beta_{1} + 23175369\)
\(\nu^{18}\)\(=\)\(6307049 \beta_{19} - 64988626 \beta_{18} - 44493263 \beta_{17} - 26344094 \beta_{16} + 67260110 \beta_{15} + 60878095 \beta_{14} + 69592767 \beta_{13} - 54920440 \beta_{12} + 6138442 \beta_{11} + 75920355 \beta_{10} - 18146095 \beta_{9} - 8294730 \beta_{8} - 8980126 \beta_{7} - 20536089 \beta_{6} - 12940136 \beta_{5} + 8054508 \beta_{4} - 12460124 \beta_{3} + 70068848 \beta_{2} - 1129161 \beta_{1} + 313328762\)
\(\nu^{19}\)\(=\)\(105786370 \beta_{19} - 21660576 \beta_{18} + 12709219 \beta_{17} + 6661747 \beta_{16} + 108325526 \beta_{15} + 301687105 \beta_{14} + 280866325 \beta_{13} + 118663031 \beta_{12} + 386022666 \beta_{11} + 227255704 \beta_{10} - 127534047 \beta_{9} - 48723586 \beta_{8} + 226200288 \beta_{7} - 293880176 \beta_{6} + 19279075 \beta_{5} + 211140577 \beta_{4} + 173412295 \beta_{3} + 246738511 \beta_{2} + 466092254 \beta_{1} + 244640595\)

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
−3.26119
−2.73475
−2.57684
−2.11643
−1.69612
−1.63154
−0.937572
−0.141161
−0.0460830
0.509862
0.751545
0.814827
1.20597
1.26295
1.67912
1.69438
2.00937
2.16480
2.82470
3.22416
−1.00000 −3.26119 1.00000 −1.57113 3.26119 1.00000 −1.00000 7.63538 1.57113
1.2 −1.00000 −2.73475 1.00000 3.55748 2.73475 1.00000 −1.00000 4.47885 −3.55748
1.3 −1.00000 −2.57684 1.00000 4.04233 2.57684 1.00000 −1.00000 3.64010 −4.04233
1.4 −1.00000 −2.11643 1.00000 −3.30251 2.11643 1.00000 −1.00000 1.47927 3.30251
1.5 −1.00000 −1.69612 1.00000 −1.00716 1.69612 1.00000 −1.00000 −0.123166 1.00716
1.6 −1.00000 −1.63154 1.00000 1.13171 1.63154 1.00000 −1.00000 −0.338090 −1.13171
1.7 −1.00000 −0.937572 1.00000 2.71354 0.937572 1.00000 −1.00000 −2.12096 −2.71354
1.8 −1.00000 −0.141161 1.00000 −1.57700 0.141161 1.00000 −1.00000 −2.98007 1.57700
1.9 −1.00000 −0.0460830 1.00000 −3.28191 0.0460830 1.00000 −1.00000 −2.99788 3.28191
1.10 −1.00000 0.509862 1.00000 0.844791 −0.509862 1.00000 −1.00000 −2.74004 −0.844791
1.11 −1.00000 0.751545 1.00000 2.13649 −0.751545 1.00000 −1.00000 −2.43518 −2.13649
1.12 −1.00000 0.814827 1.00000 0.275205 −0.814827 1.00000 −1.00000 −2.33606 −0.275205
1.13 −1.00000 1.20597 1.00000 2.17113 −1.20597 1.00000 −1.00000 −1.54564 −2.17113
1.14 −1.00000 1.26295 1.00000 −3.04196 −1.26295 1.00000 −1.00000 −1.40496 3.04196
1.15 −1.00000 1.67912 1.00000 −1.69524 −1.67912 1.00000 −1.00000 −0.180554 1.69524
1.16 −1.00000 1.69438 1.00000 −4.04996 −1.69438 1.00000 −1.00000 −0.129071 4.04996
1.17 −1.00000 2.00937 1.00000 2.56029 −2.00937 1.00000 −1.00000 1.03758 −2.56029
1.18 −1.00000 2.16480 1.00000 2.31808 −2.16480 1.00000 −1.00000 1.68636 −2.31808
1.19 −1.00000 2.82470 1.00000 −1.73446 −2.82470 1.00000 −1.00000 4.97893 1.73446
1.20 −1.00000 3.22416 1.00000 −3.48974 −3.22416 1.00000 −1.00000 7.39519 3.48974
\(n\): 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 6034.2.a.k 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6034.2.a.k 20 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(7\) \(-1\)
\(431\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6034))\):

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{20} \)
$3$ \( 1 - 3 T + 28 T^{2} - 65 T^{3} + 364 T^{4} - 706 T^{5} + 3089 T^{6} - 5236 T^{7} + 19705 T^{8} - 29976 T^{9} + 101768 T^{10} - 141419 T^{11} + 446205 T^{12} - 573732 T^{13} + 1719711 T^{14} - 2068126 T^{15} + 5997992 T^{16} - 6831643 T^{17} + 19432591 T^{18} - 21297686 T^{19} + 59659216 T^{20} - 63893058 T^{21} + 174893319 T^{22} - 184454361 T^{23} + 485837352 T^{24} - 502554618 T^{25} + 1253669319 T^{26} - 1254751884 T^{27} + 2927551005 T^{28} - 2783550177 T^{29} + 6009298632 T^{30} - 5310158472 T^{31} + 10472044905 T^{32} - 8347875228 T^{33} + 14774591241 T^{34} - 10130328342 T^{35} + 15669006444 T^{36} - 8394110595 T^{37} + 10847773692 T^{38} - 3486784401 T^{39} + 3486784401 T^{40} \)
$5$ \( 1 + 3 T + 39 T^{2} + 108 T^{3} + 817 T^{4} + 2107 T^{5} + 11941 T^{6} + 28708 T^{7} + 135297 T^{8} + 304398 T^{9} + 1259877 T^{10} + 2667375 T^{11} + 10026753 T^{12} + 20096125 T^{13} + 70131392 T^{14} + 133790972 T^{15} + 439873189 T^{16} + 802071079 T^{17} + 2507143507 T^{18} + 4379636533 T^{19} + 13084661286 T^{20} + 21898182665 T^{21} + 62678587675 T^{22} + 100258884875 T^{23} + 274920743125 T^{24} + 418096787500 T^{25} + 1095803000000 T^{26} + 1570009765625 T^{27} + 3916700390625 T^{28} + 5209716796875 T^{29} + 12303486328125 T^{30} + 14863183593750 T^{31} + 33031494140625 T^{32} + 35043945312500 T^{33} + 72882080078125 T^{34} + 64300537109375 T^{35} + 124664306640625 T^{36} + 82397460937500 T^{37} + 148773193359375 T^{38} + 57220458984375 T^{39} + 95367431640625 T^{40} \)
$7$ \( ( 1 - T )^{20} \)
$11$ \( 1 + 8 T + 140 T^{2} + 939 T^{3} + 9366 T^{4} + 55388 T^{5} + 410177 T^{6} + 2196368 T^{7} + 13368877 T^{8} + 65716524 T^{9} + 346543738 T^{10} + 1575291082 T^{11} + 7424879135 T^{12} + 31343609145 T^{13} + 134651335874 T^{14} + 529316957787 T^{15} + 2098083701251 T^{16} + 7694903887899 T^{17} + 28347234822907 T^{18} + 97113608753892 T^{19} + 333763669757804 T^{20} + 1068249696292812 T^{21} + 3430015413571747 T^{22} + 10241917074793569 T^{23} + 30718043470015891 T^{24} + 85247025368554137 T^{25} + 238543055232279314 T^{26} + 610798271165778795 T^{27} + 1591588782938847935 T^{28} + 3714453969454791662 T^{29} + 8988452075323698538 T^{30} + 18749691249187876164 T^{31} + 41957262941692712317 T^{32} + 75824580226141442608 T^{33} + \)\(15\!\cdots\!57\)\( T^{34} + \)\(23\!\cdots\!88\)\( T^{35} + \)\(43\!\cdots\!26\)\( T^{36} + \)\(47\!\cdots\!69\)\( T^{37} + \)\(77\!\cdots\!40\)\( T^{38} + \)\(48\!\cdots\!28\)\( T^{39} + \)\(67\!\cdots\!01\)\( T^{40} \)
$13$ \( 1 + 4 T + 132 T^{2} + 564 T^{3} + 8951 T^{4} + 40249 T^{5} + 415446 T^{6} + 1923821 T^{7} + 14798019 T^{8} + 68861141 T^{9} + 429190736 T^{10} + 1959746410 T^{11} + 10481850070 T^{12} + 46025161340 T^{13} + 219756843420 T^{14} + 914452403460 T^{15} + 3997955235824 T^{16} + 15632818172165 T^{17} + 63473203101242 T^{18} + 232433233404674 T^{19} + 881776264255822 T^{20} + 3021632034260762 T^{21} + 10726971324109898 T^{22} + 34345301524246505 T^{23} + 114185599490369264 T^{24} + 339529776237873780 T^{25} + 1060724309631246780 T^{26} + 2888010618770732780 T^{27} + 8550367115015000470 T^{28} + 20782129576084000930 T^{29} + 59167587580522310864 T^{30} + \)\(12\!\cdots\!17\)\( T^{31} + \)\(34\!\cdots\!39\)\( T^{32} + \)\(58\!\cdots\!13\)\( T^{33} + \)\(16\!\cdots\!94\)\( T^{34} + \)\(20\!\cdots\!93\)\( T^{35} + \)\(59\!\cdots\!91\)\( T^{36} + \)\(48\!\cdots\!12\)\( T^{37} + \)\(14\!\cdots\!28\)\( T^{38} + \)\(58\!\cdots\!08\)\( T^{39} + \)\(19\!\cdots\!01\)\( T^{40} \)
$17$ \( 1 - 9 T + 184 T^{2} - 1440 T^{3} + 16993 T^{4} - 119879 T^{5} + 1059399 T^{6} - 6843270 T^{7} + 50015090 T^{8} - 298464090 T^{9} + 1898772812 T^{10} - 10533591155 T^{11} + 60110793680 T^{12} - 311447964519 T^{13} + 1624654847934 T^{14} - 7888739043337 T^{15} + 38081711212563 T^{16} - 173676417582704 T^{17} + 782021142841479 T^{18} - 3352872772726425 T^{19} + 14149762733970938 T^{20} - 56998837136349225 T^{21} + 226004110281187431 T^{22} - 853272239583824752 T^{23} + 3180622602184474323 T^{24} - 11200881351855342809 T^{25} + 39215218493191432446 T^{26} - \)\(12\!\cdots\!87\)\( T^{27} + \)\(41\!\cdots\!80\)\( T^{28} - \)\(12\!\cdots\!35\)\( T^{29} + \)\(38\!\cdots\!88\)\( T^{30} - \)\(10\!\cdots\!70\)\( T^{31} + \)\(29\!\cdots\!90\)\( T^{32} - \)\(67\!\cdots\!90\)\( T^{33} + \)\(17\!\cdots\!71\)\( T^{34} - \)\(34\!\cdots\!47\)\( T^{35} + \)\(82\!\cdots\!33\)\( T^{36} - \)\(11\!\cdots\!80\)\( T^{37} + \)\(25\!\cdots\!56\)\( T^{38} - \)\(21\!\cdots\!77\)\( T^{39} + \)\(40\!\cdots\!01\)\( T^{40} \)
$19$ \( 1 + 14 T + 302 T^{2} + 3316 T^{3} + 42874 T^{4} + 393657 T^{5} + 3888694 T^{6} + 30990371 T^{7} + 255396027 T^{8} + 1808310675 T^{9} + 12980392812 T^{10} + 82920069768 T^{11} + 531281573616 T^{12} + 3093900462249 T^{13} + 17961317941689 T^{14} + 96014355480539 T^{15} + 509878107377489 T^{16} + 2512945431314093 T^{17} + 12279947818998052 T^{18} + 55929527033992206 T^{19} + 252354225732093328 T^{20} + 1062661013645851914 T^{21} + 4433061162658296772 T^{22} + 17236292713383363887 T^{23} + 66447824831541743969 T^{24} + \)\(23\!\cdots\!61\)\( T^{25} + \)\(84\!\cdots\!09\)\( T^{26} + \)\(27\!\cdots\!11\)\( T^{27} + \)\(90\!\cdots\!56\)\( T^{28} + \)\(26\!\cdots\!72\)\( T^{29} + \)\(79\!\cdots\!12\)\( T^{30} + \)\(21\!\cdots\!25\)\( T^{31} + \)\(56\!\cdots\!47\)\( T^{32} + \)\(13\!\cdots\!89\)\( T^{33} + \)\(31\!\cdots\!74\)\( T^{34} + \)\(59\!\cdots\!43\)\( T^{35} + \)\(12\!\cdots\!94\)\( T^{36} + \)\(18\!\cdots\!24\)\( T^{37} + \)\(31\!\cdots\!82\)\( T^{38} + \)\(27\!\cdots\!06\)\( T^{39} + \)\(37\!\cdots\!01\)\( T^{40} \)
$23$ \( 1 + 23 T + 526 T^{2} + 8018 T^{3} + 113820 T^{4} + 1342316 T^{5} + 14729349 T^{6} + 144219170 T^{7} + 1323457000 T^{8} + 11187207820 T^{9} + 89277363106 T^{10} + 667345560610 T^{11} + 4737331592218 T^{12} + 31806788973124 T^{13} + 203711141953691 T^{14} + 1241300048102910 T^{15} + 7238022538305513 T^{16} + 40297569099850057 T^{17} + 215127326176378688 T^{18} + 1098665123324327784 T^{19} + 5385425031261705872 T^{20} + 25269297836459539032 T^{21} + \)\(11\!\cdots\!52\)\( T^{22} + \)\(49\!\cdots\!19\)\( T^{23} + \)\(20\!\cdots\!33\)\( T^{24} + \)\(79\!\cdots\!30\)\( T^{25} + \)\(30\!\cdots\!99\)\( T^{26} + \)\(10\!\cdots\!28\)\( T^{27} + \)\(37\!\cdots\!58\)\( T^{28} + \)\(12\!\cdots\!30\)\( T^{29} + \)\(36\!\cdots\!94\)\( T^{30} + \)\(10\!\cdots\!40\)\( T^{31} + \)\(29\!\cdots\!00\)\( T^{32} + \)\(72\!\cdots\!10\)\( T^{33} + \)\(17\!\cdots\!41\)\( T^{34} + \)\(35\!\cdots\!12\)\( T^{35} + \)\(69\!\cdots\!20\)\( T^{36} + \)\(11\!\cdots\!54\)\( T^{37} + \)\(17\!\cdots\!94\)\( T^{38} + \)\(17\!\cdots\!01\)\( T^{39} + \)\(17\!\cdots\!01\)\( T^{40} \)
$29$ \( 1 + 48 T + 1353 T^{2} + 27732 T^{3} + 456810 T^{4} + 6348864 T^{5} + 76871432 T^{6} + 827567313 T^{7} + 8041564724 T^{8} + 71289464870 T^{9} + 581447459475 T^{10} + 4391398864975 T^{11} + 30882595512991 T^{12} + 203230757115912 T^{13} + 1258099116029142 T^{14} + 7371922146999644 T^{15} + 41231745948397930 T^{16} + 222658885505596087 T^{17} + 1178617734056166657 T^{18} + 6220312799616522101 T^{19} + 33209360205374604426 T^{20} + \)\(18\!\cdots\!29\)\( T^{21} + \)\(99\!\cdots\!37\)\( T^{22} + \)\(54\!\cdots\!43\)\( T^{23} + \)\(29\!\cdots\!30\)\( T^{24} + \)\(15\!\cdots\!56\)\( T^{25} + \)\(74\!\cdots\!82\)\( T^{26} + \)\(35\!\cdots\!08\)\( T^{27} + \)\(15\!\cdots\!51\)\( T^{28} + \)\(63\!\cdots\!75\)\( T^{29} + \)\(24\!\cdots\!75\)\( T^{30} + \)\(86\!\cdots\!30\)\( T^{31} + \)\(28\!\cdots\!84\)\( T^{32} + \)\(84\!\cdots\!57\)\( T^{33} + \)\(22\!\cdots\!92\)\( T^{34} + \)\(54\!\cdots\!36\)\( T^{35} + \)\(11\!\cdots\!10\)\( T^{36} + \)\(20\!\cdots\!88\)\( T^{37} + \)\(28\!\cdots\!33\)\( T^{38} + \)\(29\!\cdots\!12\)\( T^{39} + \)\(17\!\cdots\!01\)\( T^{40} \)
$31$ \( 1 + T + 285 T^{2} + 176 T^{3} + 39107 T^{4} + 8321 T^{5} + 3508964 T^{6} - 695843 T^{7} + 237797146 T^{8} - 134201775 T^{9} + 13337625778 T^{10} - 10969165065 T^{11} + 653895388530 T^{12} - 625494006380 T^{13} + 28654125347408 T^{14} - 29075673705196 T^{15} + 1128395218835725 T^{16} - 1180166460088061 T^{17} + 40138880555402909 T^{18} - 42163320493475310 T^{19} + 1300735807038958470 T^{20} - 1307062935297734610 T^{21} + 38573464213742195549 T^{22} - 35158339012483425251 T^{23} + \)\(10\!\cdots\!25\)\( T^{24} - \)\(83\!\cdots\!96\)\( T^{25} + \)\(25\!\cdots\!48\)\( T^{26} - \)\(17\!\cdots\!80\)\( T^{27} + \)\(55\!\cdots\!30\)\( T^{28} - \)\(29\!\cdots\!15\)\( T^{29} + \)\(10\!\cdots\!78\)\( T^{30} - \)\(34\!\cdots\!25\)\( T^{31} + \)\(18\!\cdots\!06\)\( T^{32} - \)\(16\!\cdots\!13\)\( T^{33} + \)\(26\!\cdots\!44\)\( T^{34} + \)\(19\!\cdots\!71\)\( T^{35} + \)\(28\!\cdots\!67\)\( T^{36} + \)\(39\!\cdots\!36\)\( T^{37} + \)\(19\!\cdots\!85\)\( T^{38} + \)\(21\!\cdots\!71\)\( T^{39} + \)\(67\!\cdots\!01\)\( T^{40} \)
$37$ \( 1 + T + 296 T^{2} + 133 T^{3} + 45741 T^{4} + 1727 T^{5} + 4952171 T^{6} - 1294600 T^{7} + 422987199 T^{8} - 197230555 T^{9} + 30221184088 T^{10} - 18175202246 T^{11} + 1862768405654 T^{12} - 1281745771401 T^{13} + 100911495543277 T^{14} - 74376551420695 T^{15} + 4865340256877280 T^{16} - 3662278308718901 T^{17} + 210453645060256512 T^{18} - 155620885053085659 T^{19} + 8199946665463158122 T^{20} - 5757972746964169383 T^{21} + \)\(28\!\cdots\!28\)\( T^{22} - \)\(18\!\cdots\!53\)\( T^{23} + \)\(91\!\cdots\!80\)\( T^{24} - \)\(51\!\cdots\!15\)\( T^{25} + \)\(25\!\cdots\!93\)\( T^{26} - \)\(12\!\cdots\!33\)\( T^{27} + \)\(65\!\cdots\!34\)\( T^{28} - \)\(23\!\cdots\!42\)\( T^{29} + \)\(14\!\cdots\!12\)\( T^{30} - \)\(35\!\cdots\!15\)\( T^{31} + \)\(27\!\cdots\!19\)\( T^{32} - \)\(31\!\cdots\!00\)\( T^{33} + \)\(44\!\cdots\!19\)\( T^{34} + \)\(57\!\cdots\!11\)\( T^{35} + \)\(56\!\cdots\!81\)\( T^{36} + \)\(60\!\cdots\!61\)\( T^{37} + \)\(49\!\cdots\!84\)\( T^{38} + \)\(62\!\cdots\!73\)\( T^{39} + \)\(23\!\cdots\!01\)\( T^{40} \)
$41$ \( 1 + 27 T + 774 T^{2} + 14532 T^{3} + 258755 T^{4} + 3803036 T^{5} + 52421787 T^{6} + 641655818 T^{7} + 7399417360 T^{8} + 78273668227 T^{9} + 785279960194 T^{10} + 7358080805253 T^{11} + 65810302738713 T^{12} + 556225219431264 T^{13} + 4514508445601170 T^{14} + 34926329481128410 T^{15} + 260911148443213679 T^{16} + 1870196019006706434 T^{17} + 13003029741424314011 T^{18} + 87133350535381041751 T^{19} + \)\(56\!\cdots\!04\)\( T^{20} + \)\(35\!\cdots\!91\)\( T^{21} + \)\(21\!\cdots\!91\)\( T^{22} + \)\(12\!\cdots\!14\)\( T^{23} + \)\(73\!\cdots\!19\)\( T^{24} + \)\(40\!\cdots\!10\)\( T^{25} + \)\(21\!\cdots\!70\)\( T^{26} + \)\(10\!\cdots\!84\)\( T^{27} + \)\(52\!\cdots\!73\)\( T^{28} + \)\(24\!\cdots\!33\)\( T^{29} + \)\(10\!\cdots\!94\)\( T^{30} + \)\(43\!\cdots\!07\)\( T^{31} + \)\(16\!\cdots\!60\)\( T^{32} + \)\(59\!\cdots\!78\)\( T^{33} + \)\(19\!\cdots\!07\)\( T^{34} + \)\(59\!\cdots\!36\)\( T^{35} + \)\(16\!\cdots\!55\)\( T^{36} + \)\(37\!\cdots\!92\)\( T^{37} + \)\(82\!\cdots\!54\)\( T^{38} + \)\(11\!\cdots\!47\)\( T^{39} + \)\(18\!\cdots\!01\)\( T^{40} \)
$43$ \( 1 + 3 T + 558 T^{2} + 1656 T^{3} + 153841 T^{4} + 452729 T^{5} + 27954138 T^{6} + 81488561 T^{7} + 3763100568 T^{8} + 10821506526 T^{9} + 399482237502 T^{10} + 1125911273674 T^{11} + 34724963965388 T^{12} + 95140084034553 T^{13} + 2531616375644668 T^{14} + 6680060038546284 T^{15} + 157221572199050892 T^{16} + 395447334067480645 T^{17} + 8398790516685027018 T^{18} + 19910611412079661941 T^{19} + \)\(38\!\cdots\!56\)\( T^{20} + \)\(85\!\cdots\!63\)\( T^{21} + \)\(15\!\cdots\!82\)\( T^{22} + \)\(31\!\cdots\!15\)\( T^{23} + \)\(53\!\cdots\!92\)\( T^{24} + \)\(98\!\cdots\!12\)\( T^{25} + \)\(16\!\cdots\!32\)\( T^{26} + \)\(25\!\cdots\!71\)\( T^{27} + \)\(40\!\cdots\!88\)\( T^{28} + \)\(56\!\cdots\!82\)\( T^{29} + \)\(86\!\cdots\!98\)\( T^{30} + \)\(10\!\cdots\!82\)\( T^{31} + \)\(15\!\cdots\!68\)\( T^{32} + \)\(14\!\cdots\!23\)\( T^{33} + \)\(20\!\cdots\!62\)\( T^{34} + \)\(14\!\cdots\!03\)\( T^{35} + \)\(21\!\cdots\!41\)\( T^{36} + \)\(97\!\cdots\!08\)\( T^{37} + \)\(14\!\cdots\!42\)\( T^{38} + \)\(32\!\cdots\!21\)\( T^{39} + \)\(46\!\cdots\!01\)\( T^{40} \)
$47$ \( 1 + 26 T + 870 T^{2} + 16028 T^{3} + 323324 T^{4} + 4760423 T^{5} + 73299837 T^{6} + 915091024 T^{7} + 11737390110 T^{8} + 128615641451 T^{9} + 1436695805698 T^{10} + 14127103487306 T^{11} + 141084168158631 T^{12} + 1263349735301445 T^{13} + 11467991447802372 T^{14} + 94431252400095490 T^{15} + 787427240076973813 T^{16} + 5998764376175557137 T^{17} + 46250752972555591671 T^{18} + \)\(32\!\cdots\!90\)\( T^{19} + \)\(23\!\cdots\!78\)\( T^{20} + \)\(15\!\cdots\!30\)\( T^{21} + \)\(10\!\cdots\!39\)\( T^{22} + \)\(62\!\cdots\!51\)\( T^{23} + \)\(38\!\cdots\!53\)\( T^{24} + \)\(21\!\cdots\!30\)\( T^{25} + \)\(12\!\cdots\!88\)\( T^{26} + \)\(64\!\cdots\!35\)\( T^{27} + \)\(33\!\cdots\!91\)\( T^{28} + \)\(15\!\cdots\!02\)\( T^{29} + \)\(75\!\cdots\!02\)\( T^{30} + \)\(31\!\cdots\!53\)\( T^{31} + \)\(13\!\cdots\!10\)\( T^{32} + \)\(49\!\cdots\!48\)\( T^{33} + \)\(18\!\cdots\!53\)\( T^{34} + \)\(57\!\cdots\!89\)\( T^{35} + \)\(18\!\cdots\!04\)\( T^{36} + \)\(42\!\cdots\!36\)\( T^{37} + \)\(10\!\cdots\!30\)\( T^{38} + \)\(15\!\cdots\!58\)\( T^{39} + \)\(27\!\cdots\!01\)\( T^{40} \)
$53$ \( 1 + 43 T + 1530 T^{2} + 39285 T^{3} + 879273 T^{4} + 16840653 T^{5} + 291271014 T^{6} + 4547549347 T^{7} + 65507491971 T^{8} + 872935495638 T^{9} + 10878870175952 T^{10} + 127130132132097 T^{11} + 1401701761509948 T^{12} + 14612111461151600 T^{13} + 144560728611303627 T^{14} + 1359274835965237897 T^{15} + 12175844476370566970 T^{16} + \)\(10\!\cdots\!67\)\( T^{17} + \)\(84\!\cdots\!97\)\( T^{18} + \)\(66\!\cdots\!25\)\( T^{19} + \)\(49\!\cdots\!46\)\( T^{20} + \)\(35\!\cdots\!25\)\( T^{21} + \)\(23\!\cdots\!73\)\( T^{22} + \)\(15\!\cdots\!59\)\( T^{23} + \)\(96\!\cdots\!70\)\( T^{24} + \)\(56\!\cdots\!21\)\( T^{25} + \)\(32\!\cdots\!83\)\( T^{26} + \)\(17\!\cdots\!00\)\( T^{27} + \)\(87\!\cdots\!28\)\( T^{28} + \)\(41\!\cdots\!01\)\( T^{29} + \)\(19\!\cdots\!48\)\( T^{30} + \)\(80\!\cdots\!86\)\( T^{31} + \)\(32\!\cdots\!11\)\( T^{32} + \)\(11\!\cdots\!31\)\( T^{33} + \)\(40\!\cdots\!66\)\( T^{34} + \)\(12\!\cdots\!21\)\( T^{35} + \)\(34\!\cdots\!33\)\( T^{36} + \)\(80\!\cdots\!05\)\( T^{37} + \)\(16\!\cdots\!70\)\( T^{38} + \)\(24\!\cdots\!31\)\( T^{39} + \)\(30\!\cdots\!01\)\( T^{40} \)
$59$ \( 1 + 19 T + 832 T^{2} + 13548 T^{3} + 334812 T^{4} + 4745378 T^{5} + 86524705 T^{6} + 1083465250 T^{7} + 16132053456 T^{8} + 180842322444 T^{9} + 2315582566712 T^{10} + 23509046090814 T^{11} + 267133486389490 T^{12} + 2482084976802684 T^{13} + 25578850503211333 T^{14} + 219648018901596450 T^{15} + 2086623370752209129 T^{16} + 16707465905786005641 T^{17} + \)\(14\!\cdots\!66\)\( T^{18} + \)\(11\!\cdots\!60\)\( T^{19} + \)\(92\!\cdots\!96\)\( T^{20} + \)\(65\!\cdots\!40\)\( T^{21} + \)\(51\!\cdots\!46\)\( T^{22} + \)\(34\!\cdots\!39\)\( T^{23} + \)\(25\!\cdots\!69\)\( T^{24} + \)\(15\!\cdots\!50\)\( T^{25} + \)\(10\!\cdots\!53\)\( T^{26} + \)\(61\!\cdots\!96\)\( T^{27} + \)\(39\!\cdots\!90\)\( T^{28} + \)\(20\!\cdots\!46\)\( T^{29} + \)\(11\!\cdots\!12\)\( T^{30} + \)\(54\!\cdots\!96\)\( T^{31} + \)\(28\!\cdots\!36\)\( T^{32} + \)\(11\!\cdots\!50\)\( T^{33} + \)\(53\!\cdots\!05\)\( T^{34} + \)\(17\!\cdots\!22\)\( T^{35} + \)\(72\!\cdots\!92\)\( T^{36} + \)\(17\!\cdots\!12\)\( T^{37} + \)\(62\!\cdots\!72\)\( T^{38} + \)\(84\!\cdots\!41\)\( T^{39} + \)\(26\!\cdots\!01\)\( T^{40} \)
$61$ \( 1 - 9 T + 508 T^{2} - 4787 T^{3} + 144127 T^{4} - 1351547 T^{5} + 29141343 T^{6} - 266514305 T^{7} + 4615724786 T^{8} - 40772935581 T^{9} + 601588075837 T^{10} - 5103411121928 T^{11} + 66443196593385 T^{12} - 539008807097107 T^{13} + 6335124221033490 T^{14} - 48969219312859203 T^{15} + 527749044167027137 T^{16} - 3873042911804161424 T^{17} + 38704277966369843318 T^{18} - \)\(26\!\cdots\!65\)\( T^{19} + \)\(25\!\cdots\!56\)\( T^{20} - \)\(16\!\cdots\!65\)\( T^{21} + \)\(14\!\cdots\!78\)\( T^{22} - \)\(87\!\cdots\!44\)\( T^{23} + \)\(73\!\cdots\!17\)\( T^{24} - \)\(41\!\cdots\!03\)\( T^{25} + \)\(32\!\cdots\!90\)\( T^{26} - \)\(16\!\cdots\!47\)\( T^{27} + \)\(12\!\cdots\!85\)\( T^{28} - \)\(59\!\cdots\!48\)\( T^{29} + \)\(42\!\cdots\!37\)\( T^{30} - \)\(17\!\cdots\!41\)\( T^{31} + \)\(12\!\cdots\!06\)\( T^{32} - \)\(43\!\cdots\!05\)\( T^{33} + \)\(28\!\cdots\!63\)\( T^{34} - \)\(81\!\cdots\!47\)\( T^{35} + \)\(52\!\cdots\!47\)\( T^{36} - \)\(10\!\cdots\!27\)\( T^{37} + \)\(69\!\cdots\!48\)\( T^{38} - \)\(75\!\cdots\!69\)\( T^{39} + \)\(50\!\cdots\!01\)\( T^{40} \)
$67$ \( 1 - 32 T + 1112 T^{2} - 23468 T^{3} + 497446 T^{4} - 8099073 T^{5} + 130680226 T^{6} - 1751658262 T^{7} + 23250328684 T^{8} - 264532526128 T^{9} + 2990992768223 T^{10} - 29204225424926 T^{11} + 285423175237208 T^{12} - 2376444870244100 T^{13} + 20094287030906486 T^{14} - 138152863466680044 T^{15} + 1001766052993471615 T^{16} - 5204620332618989941 T^{17} + 33303985072537596289 T^{18} - \)\(11\!\cdots\!18\)\( T^{19} + \)\(11\!\cdots\!72\)\( T^{20} - \)\(74\!\cdots\!06\)\( T^{21} + \)\(14\!\cdots\!21\)\( T^{22} - \)\(15\!\cdots\!83\)\( T^{23} + \)\(20\!\cdots\!15\)\( T^{24} - \)\(18\!\cdots\!08\)\( T^{25} + \)\(18\!\cdots\!34\)\( T^{26} - \)\(14\!\cdots\!00\)\( T^{27} + \)\(11\!\cdots\!28\)\( T^{28} - \)\(79\!\cdots\!22\)\( T^{29} + \)\(54\!\cdots\!27\)\( T^{30} - \)\(32\!\cdots\!24\)\( T^{31} + \)\(19\!\cdots\!24\)\( T^{32} - \)\(96\!\cdots\!94\)\( T^{33} + \)\(48\!\cdots\!54\)\( T^{34} - \)\(19\!\cdots\!39\)\( T^{35} + \)\(82\!\cdots\!26\)\( T^{36} - \)\(25\!\cdots\!36\)\( T^{37} + \)\(82\!\cdots\!08\)\( T^{38} - \)\(15\!\cdots\!96\)\( T^{39} + \)\(33\!\cdots\!01\)\( T^{40} \)
$71$ \( 1 + 63 T + 2518 T^{2} + 74082 T^{3} + 1782795 T^{4} + 36452494 T^{5} + 653912116 T^{6} + 10460483723 T^{7} + 151353221623 T^{8} + 1996721921137 T^{9} + 24184583913520 T^{10} + 269897237750697 T^{11} + 2783009454987840 T^{12} + 26526204127340368 T^{13} + 233706270987460399 T^{14} + 1900535402625981538 T^{15} + 14271080490309951872 T^{16} + 99474146655483725344 T^{17} + \)\(65\!\cdots\!34\)\( T^{18} + \)\(44\!\cdots\!39\)\( T^{19} + \)\(33\!\cdots\!82\)\( T^{20} + \)\(31\!\cdots\!69\)\( T^{21} + \)\(33\!\cdots\!94\)\( T^{22} + \)\(35\!\cdots\!84\)\( T^{23} + \)\(36\!\cdots\!32\)\( T^{24} + \)\(34\!\cdots\!38\)\( T^{25} + \)\(29\!\cdots\!79\)\( T^{26} + \)\(24\!\cdots\!88\)\( T^{27} + \)\(17\!\cdots\!40\)\( T^{28} + \)\(12\!\cdots\!07\)\( T^{29} + \)\(78\!\cdots\!20\)\( T^{30} + \)\(46\!\cdots\!27\)\( T^{31} + \)\(24\!\cdots\!43\)\( T^{32} + \)\(12\!\cdots\!53\)\( T^{33} + \)\(54\!\cdots\!96\)\( T^{34} + \)\(21\!\cdots\!94\)\( T^{35} + \)\(74\!\cdots\!95\)\( T^{36} + \)\(21\!\cdots\!62\)\( T^{37} + \)\(52\!\cdots\!98\)\( T^{38} + \)\(94\!\cdots\!53\)\( T^{39} + \)\(10\!\cdots\!01\)\( T^{40} \)
$73$ \( 1 - 2 T + 866 T^{2} - 1669 T^{3} + 375737 T^{4} - 696928 T^{5} + 108609361 T^{6} - 192694508 T^{7} + 23462719065 T^{8} - 39498583560 T^{9} + 4028236229230 T^{10} - 6384547360326 T^{11} + 570621016377144 T^{12} - 846382220943314 T^{13} + 68338671697256618 T^{14} - 94547873439756587 T^{15} + 7032462886975158022 T^{16} - 9070716393887543956 T^{17} + \)\(62\!\cdots\!49\)\( T^{18} - \)\(75\!\cdots\!10\)\( T^{19} + \)\(49\!\cdots\!82\)\( T^{20} - \)\(55\!\cdots\!30\)\( T^{21} + \)\(33\!\cdots\!21\)\( T^{22} - \)\(35\!\cdots\!52\)\( T^{23} + \)\(19\!\cdots\!02\)\( T^{24} - \)\(19\!\cdots\!91\)\( T^{25} + \)\(10\!\cdots\!02\)\( T^{26} - \)\(93\!\cdots\!58\)\( T^{27} + \)\(46\!\cdots\!64\)\( T^{28} - \)\(37\!\cdots\!38\)\( T^{29} + \)\(17\!\cdots\!70\)\( T^{30} - \)\(12\!\cdots\!20\)\( T^{31} + \)\(53\!\cdots\!65\)\( T^{32} - \)\(32\!\cdots\!64\)\( T^{33} + \)\(13\!\cdots\!49\)\( T^{34} - \)\(62\!\cdots\!96\)\( T^{35} + \)\(24\!\cdots\!57\)\( T^{36} - \)\(79\!\cdots\!57\)\( T^{37} + \)\(30\!\cdots\!54\)\( T^{38} - \)\(50\!\cdots\!74\)\( T^{39} + \)\(18\!\cdots\!01\)\( T^{40} \)
$79$ \( 1 + 51 T + 1946 T^{2} + 53328 T^{3} + 1241359 T^{4} + 24354996 T^{5} + 425888792 T^{6} + 6618912981 T^{7} + 94069200667 T^{8} + 1220904292410 T^{9} + 14763848146647 T^{10} + 166428976040832 T^{11} + 1782482712564755 T^{12} + 18196006956762979 T^{13} + 180643917455780259 T^{14} + 1748734852772312345 T^{15} + 16772107573320232209 T^{16} + \)\(15\!\cdots\!23\)\( T^{17} + \)\(14\!\cdots\!27\)\( T^{18} + \)\(13\!\cdots\!24\)\( T^{19} + \)\(12\!\cdots\!94\)\( T^{20} + \)\(10\!\cdots\!96\)\( T^{21} + \)\(92\!\cdots\!07\)\( T^{22} + \)\(78\!\cdots\!97\)\( T^{23} + \)\(65\!\cdots\!29\)\( T^{24} + \)\(53\!\cdots\!55\)\( T^{25} + \)\(43\!\cdots\!39\)\( T^{26} + \)\(34\!\cdots\!61\)\( T^{27} + \)\(27\!\cdots\!55\)\( T^{28} + \)\(19\!\cdots\!08\)\( T^{29} + \)\(13\!\cdots\!47\)\( T^{30} + \)\(91\!\cdots\!90\)\( T^{31} + \)\(55\!\cdots\!47\)\( T^{32} + \)\(30\!\cdots\!59\)\( T^{33} + \)\(15\!\cdots\!52\)\( T^{34} + \)\(70\!\cdots\!04\)\( T^{35} + \)\(28\!\cdots\!39\)\( T^{36} + \)\(96\!\cdots\!52\)\( T^{37} + \)\(27\!\cdots\!06\)\( T^{38} + \)\(57\!\cdots\!69\)\( T^{39} + \)\(89\!\cdots\!01\)\( T^{40} \)
$83$ \( 1 + 24 T + 1179 T^{2} + 22714 T^{3} + 645080 T^{4} + 10491061 T^{5} + 222764419 T^{6} + 3158693369 T^{7} + 55336501390 T^{8} + 700168103713 T^{9} + 10659454682119 T^{10} + 122521789341917 T^{11} + 1673591654305387 T^{12} + 17719762458155814 T^{13} + 221722289906739377 T^{14} + 2184505203090054674 T^{15} + 25368211803915792777 T^{16} + \)\(23\!\cdots\!30\)\( T^{17} + \)\(25\!\cdots\!18\)\( T^{18} + \)\(22\!\cdots\!96\)\( T^{19} + \)\(22\!\cdots\!10\)\( T^{20} + \)\(18\!\cdots\!68\)\( T^{21} + \)\(17\!\cdots\!02\)\( T^{22} + \)\(13\!\cdots\!10\)\( T^{23} + \)\(12\!\cdots\!17\)\( T^{24} + \)\(86\!\cdots\!82\)\( T^{25} + \)\(72\!\cdots\!13\)\( T^{26} + \)\(48\!\cdots\!78\)\( T^{27} + \)\(37\!\cdots\!67\)\( T^{28} + \)\(22\!\cdots\!51\)\( T^{29} + \)\(16\!\cdots\!31\)\( T^{30} + \)\(90\!\cdots\!71\)\( T^{31} + \)\(59\!\cdots\!90\)\( T^{32} + \)\(28\!\cdots\!47\)\( T^{33} + \)\(16\!\cdots\!51\)\( T^{34} + \)\(64\!\cdots\!27\)\( T^{35} + \)\(32\!\cdots\!80\)\( T^{36} + \)\(95\!\cdots\!22\)\( T^{37} + \)\(41\!\cdots\!11\)\( T^{38} + \)\(69\!\cdots\!28\)\( T^{39} + \)\(24\!\cdots\!01\)\( T^{40} \)
$89$ \( 1 + 35 T + 1600 T^{2} + 39208 T^{3} + 1076260 T^{4} + 20977795 T^{5} + 439282053 T^{6} + 7254499616 T^{7} + 126754959351 T^{8} + 1839739158549 T^{9} + 28077501214504 T^{10} + 366571289455701 T^{11} + 5020279200234507 T^{12} + 59873533298255596 T^{13} + 748486005450311070 T^{14} + 8238988711689507237 T^{15} + 95052294386951211664 T^{16} + \)\(97\!\cdots\!33\)\( T^{17} + \)\(10\!\cdots\!33\)\( T^{18} + \)\(99\!\cdots\!82\)\( T^{19} + \)\(99\!\cdots\!10\)\( T^{20} + \)\(88\!\cdots\!98\)\( T^{21} + \)\(82\!\cdots\!93\)\( T^{22} + \)\(68\!\cdots\!77\)\( T^{23} + \)\(59\!\cdots\!24\)\( T^{24} + \)\(46\!\cdots\!13\)\( T^{25} + \)\(37\!\cdots\!70\)\( T^{26} + \)\(26\!\cdots\!84\)\( T^{27} + \)\(19\!\cdots\!67\)\( T^{28} + \)\(12\!\cdots\!09\)\( T^{29} + \)\(87\!\cdots\!04\)\( T^{30} + \)\(51\!\cdots\!61\)\( T^{31} + \)\(31\!\cdots\!71\)\( T^{32} + \)\(15\!\cdots\!04\)\( T^{33} + \)\(85\!\cdots\!73\)\( T^{34} + \)\(36\!\cdots\!55\)\( T^{35} + \)\(16\!\cdots\!60\)\( T^{36} + \)\(54\!\cdots\!32\)\( T^{37} + \)\(19\!\cdots\!00\)\( T^{38} + \)\(38\!\cdots\!15\)\( T^{39} + \)\(97\!\cdots\!01\)\( T^{40} \)
$97$ \( 1 - 5 T + 829 T^{2} - 2628 T^{3} + 353537 T^{4} - 489605 T^{5} + 103059371 T^{6} + 28234141 T^{7} + 23280950019 T^{8} + 39398107160 T^{9} + 4374817922566 T^{10} + 12081634912204 T^{11} + 711670430127279 T^{12} + 2457359162566182 T^{13} + 102168120522774385 T^{14} + 390654997180949146 T^{15} + 13050824163721920838 T^{16} + 51640241245899070650 T^{17} + \)\(14\!\cdots\!17\)\( T^{18} + \)\(58\!\cdots\!27\)\( T^{19} + \)\(15\!\cdots\!44\)\( T^{20} + \)\(56\!\cdots\!19\)\( T^{21} + \)\(14\!\cdots\!53\)\( T^{22} + \)\(47\!\cdots\!50\)\( T^{23} + \)\(11\!\cdots\!78\)\( T^{24} + \)\(33\!\cdots\!22\)\( T^{25} + \)\(85\!\cdots\!65\)\( T^{26} + \)\(19\!\cdots\!66\)\( T^{27} + \)\(55\!\cdots\!19\)\( T^{28} + \)\(91\!\cdots\!68\)\( T^{29} + \)\(32\!\cdots\!34\)\( T^{30} + \)\(28\!\cdots\!80\)\( T^{31} + \)\(16\!\cdots\!79\)\( T^{32} + \)\(19\!\cdots\!57\)\( T^{33} + \)\(67\!\cdots\!99\)\( T^{34} - \)\(31\!\cdots\!65\)\( T^{35} + \)\(21\!\cdots\!77\)\( T^{36} - \)\(15\!\cdots\!36\)\( T^{37} + \)\(47\!\cdots\!81\)\( T^{38} - \)\(28\!\cdots\!65\)\( T^{39} + \)\(54\!\cdots\!01\)\( T^{40} \)
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