Properties

Label 6034.2.a.l
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: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial: \(x^{20} - 3 x^{19} - 36 x^{18} + 97 x^{17} + 573 x^{16} - 1292 x^{15} - 5329 x^{14} + 9121 x^{13} + 31784 x^{12} - 36075 x^{11} - 124276 x^{10} + 74594 x^{9} + 312410 x^{8} - 47208 x^{7} - 477646 x^{6} - 101137 x^{5} + 391391 x^{4} + 205294 x^{3} - 112848 x^{2} - 109144 x - 21776\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} - 3 x^{19} - 36 x^{18} + 97 x^{17} + 573 x^{16} - 1292 x^{15} - 5329 x^{14} + 9121 x^{13} + 31784 x^{12} - 36075 x^{11} - 124276 x^{10} + 74594 x^{9} + 312410 x^{8} - 47208 x^{7} - 477646 x^{6} - 101137 x^{5} + 391391 x^{4} + 205294 x^{3} - 112848 x^{2} - 109144 x - 21776\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 4 \)
\(\beta_{3}\)\(=\)\((\)\(-3960343 \nu^{19} + 614360943 \nu^{18} - 1462616462 \nu^{17} - 19663940787 \nu^{16} + 43709410711 \nu^{15} + 268014365098 \nu^{14} - 504766627077 \nu^{13} - 2047106484441 \nu^{12} + 2948268091502 \nu^{11} + 9589231744201 \nu^{10} - 9140268418762 \nu^{9} - 28036276822386 \nu^{8} + 13552222306822 \nu^{7} + 49475485804836 \nu^{6} - 3408061802102 \nu^{5} - 47354517541381 \nu^{4} - 13461222095163 \nu^{3} + 17476023229232 \nu^{2} + 11129234398640 \nu + 1809016113384\)\()/ 3040618192 \)
\(\beta_{4}\)\(=\)\((\)\(16644951 \nu^{19} - 81044325 \nu^{18} - 577306280 \nu^{17} + 2574973259 \nu^{16} + 9155450243 \nu^{15} - 32772811792 \nu^{14} - 87337002091 \nu^{13} + 208383070343 \nu^{12} + 530311897740 \nu^{11} - 641981540601 \nu^{10} - 1972951737436 \nu^{9} + 524324073778 \nu^{8} + 4051274477878 \nu^{7} + 1855579562496 \nu^{6} - 3563958517610 \nu^{5} - 4498670133799 \nu^{4} - 389081593375 \nu^{3} + 2608653060254 \nu^{2} + 1778261525636 \nu + 335027839864\)\()/ 1520309096 \)
\(\beta_{5}\)\(=\)\((\)\(34635180 \nu^{19} - 101591649 \nu^{18} - 1492020713 \nu^{17} + 4034068656 \nu^{16} + 27150657839 \nu^{15} - 65128150081 \nu^{14} - 275173394620 \nu^{13} + 554472938489 \nu^{12} + 1715817274811 \nu^{11} - 2697245206372 \nu^{10} - 6808718394193 \nu^{9} + 7567057791828 \nu^{8} + 17092287131486 \nu^{7} - 11605231024486 \nu^{6} - 26098069899884 \nu^{5} + 7824943444266 \nu^{4} + 22030650504305 \nu^{3} + 423891695997 \nu^{2} - 7872031684318 \nu - 2198305565808\)\()/ 760154548 \)
\(\beta_{6}\)\(=\)\((\)\(24749436 \nu^{19} - 91273626 \nu^{18} - 806665861 \nu^{17} + 2893666957 \nu^{16} + 11460917862 \nu^{15} - 37991837591 \nu^{14} - 94984626483 \nu^{13} + 268421167748 \nu^{12} + 510837094381 \nu^{11} - 1104063935083 \nu^{10} - 1833575792166 \nu^{9} + 2655993621793 \nu^{8} + 4292697618894 \nu^{7} - 3510755640666 \nu^{6} - 6173786067610 \nu^{5} + 2015862861242 \nu^{4} + 4863664147150 \nu^{3} + 140670079897 \nu^{2} - 1577973995415 \nu - 434932067360\)\()/ 380077274 \)
\(\beta_{7}\)\(=\)\((\)\(100169175 \nu^{19} - 425556827 \nu^{18} - 3269423778 \nu^{17} + 14088317343 \nu^{16} + 46757370765 \nu^{15} - 195724039610 \nu^{14} - 395224629647 \nu^{13} + 1488124645517 \nu^{12} + 2218857147522 \nu^{11} - 6735921243469 \nu^{10} - 8570078476466 \nu^{9} + 18400675058302 \nu^{8} + 22292286183426 \nu^{7} - 29044895255988 \nu^{6} - 36728041496050 \nu^{5} + 22694772257685 \nu^{4} + 34185117118823 \nu^{3} - 3701357233648 \nu^{2} - 13584964832756 \nu - 3456518219736\)\()/ 1520309096 \)
\(\beta_{8}\)\(=\)\((\)\(264438933 \nu^{19} - 677141369 \nu^{18} - 9871031970 \nu^{17} + 22739279353 \nu^{16} + 159530819887 \nu^{15} - 315571793034 \nu^{14} - 1464415784497 \nu^{13} + 2340509122487 \nu^{12} + 8371072116058 \nu^{11} - 9986677320235 \nu^{10} - 30600553577726 \nu^{9} + 24434559246422 \nu^{8} + 70662011375414 \nu^{7} - 31448906153828 \nu^{6} - 98399132432526 \nu^{5} + 14323866889735 \nu^{4} + 74468175567565 \nu^{3} + 7430391971228 \nu^{2} - 23206861195928 \nu - 6827044858744\)\()/ 3040618192 \)
\(\beta_{9}\)\(=\)\((\)\(84627259 \nu^{19} - 299043217 \nu^{18} - 2760722388 \nu^{17} + 9521282651 \nu^{16} + 38959352751 \nu^{15} - 126057411084 \nu^{14} - 317530122351 \nu^{13} + 905324889811 \nu^{12} + 1669140863184 \nu^{11} - 3841472821561 \nu^{10} - 5893209464936 \nu^{9} + 9778625961422 \nu^{8} + 13891466884386 \nu^{7} - 14287583245004 \nu^{6} - 20850420947606 \nu^{5} + 10123784644597 \nu^{4} + 17888086708617 \nu^{3} - 1039701681626 \nu^{2} - 6630457002476 \nu - 1733210336848\)\()/ 760154548 \)
\(\beta_{10}\)\(=\)\((\)\(112313458 \nu^{19} - 531994539 \nu^{18} - 3327989547 \nu^{17} + 17119353404 \nu^{16} + 41775750129 \nu^{15} - 230447535695 \nu^{14} - 303351747064 \nu^{13} + 1695729550403 \nu^{12} + 1489592807765 \nu^{11} - 7441733031364 \nu^{10} - 5373267279201 \nu^{9} + 19833998059986 \nu^{8} + 14086688762286 \nu^{7} - 31007101914510 \nu^{6} - 24546970398728 \nu^{5} + 25037390039884 \nu^{4} + 24561108636495 \nu^{3} - 6016009624925 \nu^{2} - 10464986875224 \nu - 2401691382064\)\()/ 760154548 \)
\(\beta_{11}\)\(=\)\((\)\(248833715 \nu^{19} - 1291516455 \nu^{18} - 7059152210 \nu^{17} + 41300189627 \nu^{16} + 83884106449 \nu^{15} - 552284871818 \nu^{14} - 577091775883 \nu^{13} + 4035613393209 \nu^{12} + 2766913680226 \nu^{11} - 17571019110121 \nu^{10} - 10240996838690 \nu^{9} + 46386347886198 \nu^{8} + 28224832586138 \nu^{7} - 71711913218188 \nu^{6} - 51199959091866 \nu^{5} + 57289050280281 \nu^{4} + 52250106029851 \nu^{3} - 13916717002464 \nu^{2} - 22319090928660 \nu - 5022246308728\)\()/ 1520309096 \)
\(\beta_{12}\)\(=\)\((\)\(315317385 \nu^{19} - 1293022755 \nu^{18} - 9626405628 \nu^{17} + 40646537601 \nu^{16} + 125166030661 \nu^{15} - 529755791300 \nu^{14} - 933086775793 \nu^{13} + 3730825904025 \nu^{12} + 4524429755200 \nu^{11} - 15440497361475 \nu^{10} - 15072335059492 \nu^{9} + 38096694582042 \nu^{8} + 34290680964482 \nu^{7} - 53852893948784 \nu^{6} - 50203746507230 \nu^{5} + 37945554762711 \nu^{4} + 41819809511415 \nu^{3} - 6866102538522 \nu^{2} - 14788998243232 \nu - 3367570379784\)\()/ 1520309096 \)
\(\beta_{13}\)\(=\)\((\)\(189297804 \nu^{19} - 776958427 \nu^{18} - 6117797523 \nu^{17} + 25168764824 \nu^{16} + 86433528077 \nu^{15} - 340072159095 \nu^{14} - 718699678124 \nu^{13} + 2495980296331 \nu^{12} + 3937430053521 \nu^{11} - 10805281715184 \nu^{10} - 14657547984787 \nu^{9} + 27913124781788 \nu^{8} + 36210148063178 \nu^{7} - 41105211794078 \nu^{6} - 55931170695232 \nu^{5} + 29241168607046 \nu^{4} + 48367939070307 \nu^{3} - 3319275783001 \nu^{2} - 17766040350958 \nu - 4582432715876\)\()/ 760154548 \)
\(\beta_{14}\)\(=\)\((\)\(835590203 \nu^{19} - 3826397139 \nu^{18} - 25262436066 \nu^{17} + 123370536655 \nu^{16} + 327212594709 \nu^{15} - 1666268316042 \nu^{14} - 2478852923031 \nu^{13} + 12324747482549 \nu^{12} + 12707364215778 \nu^{11} - 54481031610765 \nu^{10} - 47030140681646 \nu^{9} + 146507264783378 \nu^{8} + 123486106092610 \nu^{7} - 230993811060164 \nu^{6} - 212352079365906 \nu^{5} + 186645175207745 \nu^{4} + 209047703145903 \nu^{3} - 41734671242616 \nu^{2} - 87948678687112 \nu - 20980523930824\)\()/ 3040618192 \)
\(\beta_{15}\)\(=\)\((\)\(-1027938729 \nu^{19} + 4718179533 \nu^{18} + 30511731298 \nu^{17} - 149948611701 \nu^{16} - 385986008075 \nu^{15} + 1989813546586 \nu^{14} + 2844198628989 \nu^{13} - 14409203636595 \nu^{12} - 14170320998026 \nu^{11} + 62115207658047 \nu^{10} + 51071504508166 \nu^{9} - 162220478086054 \nu^{8} - 130670424420302 \nu^{7} + 247481464312132 \nu^{6} + 218562657331126 \nu^{5} - 193027604258035 \nu^{4} - 208738589070129 \nu^{3} + 41787055500476 \nu^{2} + 85005626154864 \nu + 19966247957256\)\()/ 3040618192 \)
\(\beta_{16}\)\(=\)\((\)\(519967205 \nu^{19} - 2072767045 \nu^{18} - 15916321158 \nu^{17} + 64771735209 \nu^{16} + 208012032923 \nu^{15} - 839298144558 \nu^{14} - 1562543194369 \nu^{13} + 5883644866955 \nu^{12} + 7642558381022 \nu^{11} - 24302200221347 \nu^{10} - 25658775696594 \nu^{9} + 60093801761878 \nu^{8} + 58785626890430 \nu^{7} - 85545372062572 \nu^{6} - 86809450322110 \nu^{5} + 60797505146367 \nu^{4} + 73339444737417 \nu^{3} - 10490555195360 \nu^{2} - 26573508701288 \nu - 6299303348056\)\()/ 1520309096 \)
\(\beta_{17}\)\(=\)\((\)\(-274087579 \nu^{19} + 1083920107 \nu^{18} + 8622426264 \nu^{17} - 34420744897 \nu^{16} - 117006906077 \nu^{15} + 454747815080 \nu^{14} + 921739033949 \nu^{13} - 3260250462441 \nu^{12} - 4750930316416 \nu^{11} + 13806485341891 \nu^{10} + 16739364778226 \nu^{9} - 35051513158072 \nu^{8} - 39808268971858 \nu^{7} + 51180850872076 \nu^{6} + 60341382365326 \nu^{5} - 36936455863561 \nu^{4} - 51989409745755 \nu^{3} + 5630388544830 \nu^{2} + 19202507682118 \nu + 4753242395944\)\()/ 760154548 \)
\(\beta_{18}\)\(=\)\((\)\(-290115955 \nu^{19} + 1153750064 \nu^{18} + 9150086029 \nu^{17} - 36865983801 \nu^{16} - 124471421788 \nu^{15} + 491110264125 \nu^{14} + 983119021817 \nu^{13} - 3560419069126 \nu^{12} - 5090068253835 \nu^{11} + 15306357628247 \nu^{10} + 18105460574843 \nu^{9} - 39653842084616 \nu^{8} - 43818194314372 \nu^{7} + 59478980973710 \nu^{6} + 68183513110990 \nu^{5} - 44579225210911 \nu^{4} - 60749876331836 \nu^{3} + 7613668131281 \nu^{2} + 23329830760960 \nu + 5764506969456\)\()/ 760154548 \)
\(\beta_{19}\)\(=\)\((\)\(-317683195 \nu^{19} + 1370868643 \nu^{18} + 9610163980 \nu^{17} - 43406255205 \nu^{16} - 124377259165 \nu^{15} + 572895705936 \nu^{14} + 935317640093 \nu^{13} - 4117650204609 \nu^{12} - 4686813636124 \nu^{11} + 17575111887427 \nu^{10} + 16601593774334 \nu^{9} - 45314356507148 \nu^{8} - 41023320358782 \nu^{7} + 67960867918620 \nu^{6} + 65906318041966 \nu^{5} - 51620949387853 \nu^{4} - 60574697135363 \nu^{3} + 10163029426582 \nu^{2} + 23834003996814 \nu + 5699083771564\)\()/ 760154548 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 4\)
\(\nu^{3}\)\(=\)\(-\beta_{18} + \beta_{17} - \beta_{16} + \beta_{14} + \beta_{12} - \beta_{10} + \beta_{8} - \beta_{7} - \beta_{5} + \beta_{4} + 7 \beta_{1} + 2\)
\(\nu^{4}\)\(=\)\(-\beta_{19} + \beta_{18} - \beta_{16} + \beta_{14} + \beta_{13} + \beta_{12} - \beta_{10} - \beta_{9} - \beta_{8} - \beta_{7} - \beta_{4} + 2 \beta_{3} + 10 \beta_{2} + 10 \beta_{1} + 26\)
\(\nu^{5}\)\(=\)\(2 \beta_{19} - 12 \beta_{18} + 12 \beta_{17} - 9 \beta_{16} - \beta_{15} + 10 \beta_{14} - \beta_{13} + 10 \beta_{12} - 10 \beta_{10} + \beta_{9} + 14 \beta_{8} - 11 \beta_{7} - 11 \beta_{5} + 11 \beta_{4} - 3 \beta_{3} - 2 \beta_{2} + 54 \beta_{1} + 20\)
\(\nu^{6}\)\(=\)\(-16 \beta_{19} + 14 \beta_{18} + 2 \beta_{17} - 12 \beta_{16} + 2 \beta_{15} + 13 \beta_{14} + 15 \beta_{13} + 13 \beta_{12} - 13 \beta_{10} - 13 \beta_{9} - 14 \beta_{8} - 14 \beta_{7} - 3 \beta_{6} + \beta_{5} - 14 \beta_{4} + 27 \beta_{3} + 87 \beta_{2} + 87 \beta_{1} + 201\)
\(\nu^{7}\)\(=\)\(30 \beta_{19} - 117 \beta_{18} + 121 \beta_{17} - 72 \beta_{16} - 17 \beta_{15} + 88 \beta_{14} - 13 \beta_{13} + 88 \beta_{12} - \beta_{11} - 86 \beta_{10} + 13 \beta_{9} + 150 \beta_{8} - 104 \beta_{7} - 104 \beta_{5} + 101 \beta_{4} - 43 \beta_{3} - 36 \beta_{2} + 431 \beta_{1} + 174\)
\(\nu^{8}\)\(=\)\(-184 \beta_{19} + 151 \beta_{18} + 35 \beta_{17} - 113 \beta_{16} + 36 \beta_{15} + 132 \beta_{14} + 170 \beta_{13} + 130 \beta_{12} + 5 \beta_{11} - 132 \beta_{10} - 135 \beta_{9} - 145 \beta_{8} - 155 \beta_{7} - 54 \beta_{6} + 16 \beta_{5} - 156 \beta_{4} + 283 \beta_{3} + 736 \beta_{2} + 730 \beta_{1} + 1650\)
\(\nu^{9}\)\(=\)\(325 \beta_{19} - 1069 \beta_{18} + 1158 \beta_{17} - 573 \beta_{16} - 199 \beta_{15} + 766 \beta_{14} - 121 \beta_{13} + 762 \beta_{12} - 14 \beta_{11} - 727 \beta_{10} + 126 \beta_{9} + 1469 \beta_{8} - 952 \beta_{7} - 12 \beta_{6} - 948 \beta_{5} + 885 \beta_{4} - 454 \beta_{3} - 454 \beta_{2} + 3503 \beta_{1} + 1449\)
\(\nu^{10}\)\(=\)\(-1876 \beta_{19} + 1481 \beta_{18} + 454 \beta_{17} - 993 \beta_{16} + 437 \beta_{15} + 1236 \beta_{14} + 1737 \beta_{13} + 1204 \beta_{12} + 96 \beta_{11} - 1235 \beta_{10} - 1298 \beta_{9} - 1346 \beta_{8} - 1574 \beta_{7} - 689 \beta_{6} + 179 \beta_{5} - 1587 \beta_{4} + 2727 \beta_{3} + 6193 \beta_{2} + 6037 \beta_{1} + 13856\)
\(\nu^{11}\)\(=\)\(3124 \beta_{19} - 9506 \beta_{18} + 10811 \beta_{17} - 4614 \beta_{16} - 2023 \beta_{15} + 6699 \beta_{14} - 979 \beta_{13} + 6603 \beta_{12} - 109 \beta_{11} - 6196 \beta_{10} + 1107 \beta_{9} + 13814 \beta_{8} - 8638 \beta_{7} - 305 \beta_{6} - 8520 \beta_{5} + 7623 \beta_{4} - 4296 \beta_{3} - 4998 \beta_{2} + 28805 \beta_{1} + 11824\)
\(\nu^{12}\)\(=\)\(-18080 \beta_{19} + 13930 \beta_{18} + 5196 \beta_{17} - 8507 \beta_{16} + 4540 \beta_{15} + 11184 \beta_{14} + 16876 \beta_{13} + 10827 \beta_{12} + 1275 \beta_{11} - 11159 \beta_{10} - 12034 \beta_{9} - 11872 \beta_{8} - 15282 \beta_{7} - 7709 \beta_{6} + 1746 \beta_{5} - 15402 \beta_{4} + 25346 \beta_{3} + 52157 \beta_{2} + 49597 \beta_{1} + 117536\)
\(\nu^{13}\)\(=\)\(28391 \beta_{19} - 83477 \beta_{18} + 99476 \beta_{17} - 37639 \beta_{16} - 19267 \beta_{15} + 58810 \beta_{14} - 7258 \beta_{13} + 57364 \beta_{12} - 407 \beta_{11} - 53359 \beta_{10} + 9382 \beta_{9} + 127136 \beta_{8} - 78074 \beta_{7} - 5009 \beta_{6} - 76015 \beta_{5} + 65224 \beta_{4} - 38661 \beta_{3} - 51390 \beta_{2} + 238836 \beta_{1} + 95468\)
\(\nu^{14}\)\(=\)\(-169112 \beta_{19} + 128427 \beta_{18} + 55330 \beta_{17} - 72118 \beta_{16} + 43714 \beta_{15} + 99559 \beta_{14} + 159398 \beta_{13} + 96015 \beta_{12} + 14756 \beta_{11} - 99224 \beta_{10} - 109366 \beta_{9} - 101994 \beta_{8} - 144399 \beta_{7} - 80806 \beta_{6} + 15956 \beta_{5} - 145468 \beta_{4} + 231523 \beta_{3} + 440534 \beta_{2} + 406385 \beta_{1} + 1002422\)
\(\nu^{15}\)\(=\)\(250476 \beta_{19} - 728577 \beta_{18} + 906580 \beta_{17} - 310590 \beta_{16} - 177563 \beta_{15} + 517053 \beta_{14} - 49812 \beta_{13} + 499319 \beta_{12} + 3792 \beta_{11} - 463184 \beta_{10} + 78792 \beta_{9} + 1155425 \beta_{8} - 703530 \beta_{7} - 67724 \beta_{6} - 675304 \beta_{5} + 556960 \beta_{4} - 338913 \beta_{3} - 507490 \beta_{2} + 1992955 \beta_{1} + 766341\)
\(\nu^{16}\)\(=\)\(-1554790 \beta_{19} + 1171964 \beta_{18} + 562008 \beta_{17} - 608905 \beta_{16} + 403928 \beta_{15} + 878774 \beta_{14} + 1479616 \beta_{13} + 844891 \beta_{12} + 160064 \beta_{11} - 875917 \beta_{10} - 982050 \beta_{9} - 864197 \beta_{8} - 1340293 \beta_{7} - 815737 \beta_{6} + 140953 \beta_{5} - 1350710 \beta_{4} + 2096109 \beta_{3} + 3734079 \beta_{2} + 3328578 \beta_{1} + 8579038\)
\(\nu^{17}\)\(=\)\(2173001 \beta_{19} - 6339442 \beta_{18} + 8205746 \beta_{17} - 2588120 \beta_{16} - 1608507 \beta_{15} + 4545938 \beta_{14} - 310890 \beta_{13} + 4351431 \beta_{12} + 112121 \beta_{11} - 4042355 \beta_{10} + 664382 \beta_{9} + 10417814 \beta_{8} - 6321045 \beta_{7} - 820712 \beta_{6} - 5983425 \beta_{5} + 4757646 \beta_{4} - 2926613 \beta_{3} - 4880221 \beta_{2} + 16716643 \beta_{1} + 6131100\)
\(\nu^{18}\)\(=\)\(-14145184 \beta_{19} + 10634440 \beta_{18} + 5523546 \beta_{17} - 5137667 \beta_{16} + 3644424 \beta_{15} + 7719830 \beta_{14} + 13577614 \beta_{13} + 7398958 \beta_{12} + 1674987 \beta_{11} - 7710410 \beta_{10} - 8749493 \beta_{9} - 7271553 \beta_{8} - 12287437 \beta_{7} - 8039128 \beta_{6} + 1222620 \beta_{5} - 12399171 \beta_{4} + 18886903 \beta_{3} + 31768221 \beta_{2} + 27292204 \beta_{1} + 73611932\)
\(\nu^{19}\)\(=\)\(18666972 \beta_{19} - 55075743 \beta_{18} + 73890253 \beta_{17} - 21746507 \beta_{16} - 14440173 \beta_{15} + 39941739 \beta_{14} - 1649264 \beta_{13} + 37946875 \beta_{12} + 1754403 \beta_{11} - 35404964 \beta_{10} + 5661014 \beta_{9} + 93439765 \beta_{8} - 56633328 \beta_{7} - 9278750 \beta_{6} - 52925812 \beta_{5} + 40703498 \beta_{4} - 25037221 \beta_{3} - 46057226 \beta_{2} + 140839698 \beta_{1} + 48954545\)

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
2.98166
2.88014
2.84514
2.80072
2.78441
1.59091
1.42043
1.35617
1.32183
−0.395900
−0.860159
−0.903145
−1.01642
−1.35221
−1.37338
−1.76208
−1.78235
−2.22609
−2.34324
−2.96642
1.00000 −2.98166 1.00000 −1.98359 −2.98166 −1.00000 1.00000 5.89028 −1.98359
1.2 1.00000 −2.88014 1.00000 −1.11289 −2.88014 −1.00000 1.00000 5.29519 −1.11289
1.3 1.00000 −2.84514 1.00000 −4.15918 −2.84514 −1.00000 1.00000 5.09480 −4.15918
1.4 1.00000 −2.80072 1.00000 0.235711 −2.80072 −1.00000 1.00000 4.84406 0.235711
1.5 1.00000 −2.78441 1.00000 1.22741 −2.78441 −1.00000 1.00000 4.75292 1.22741
1.6 1.00000 −1.59091 1.00000 3.74979 −1.59091 −1.00000 1.00000 −0.468990 3.74979
1.7 1.00000 −1.42043 1.00000 −4.18942 −1.42043 −1.00000 1.00000 −0.982389 −4.18942
1.8 1.00000 −1.35617 1.00000 1.14574 −1.35617 −1.00000 1.00000 −1.16079 1.14574
1.9 1.00000 −1.32183 1.00000 1.76551 −1.32183 −1.00000 1.00000 −1.25276 1.76551
1.10 1.00000 0.395900 1.00000 0.151677 0.395900 −1.00000 1.00000 −2.84326 0.151677
1.11 1.00000 0.860159 1.00000 −0.340662 0.860159 −1.00000 1.00000 −2.26013 −0.340662
1.12 1.00000 0.903145 1.00000 −3.69995 0.903145 −1.00000 1.00000 −2.18433 −3.69995
1.13 1.00000 1.01642 1.00000 1.24888 1.01642 −1.00000 1.00000 −1.96689 1.24888
1.14 1.00000 1.35221 1.00000 2.46727 1.35221 −1.00000 1.00000 −1.17152 2.46727
1.15 1.00000 1.37338 1.00000 1.18048 1.37338 −1.00000 1.00000 −1.11382 1.18048
1.16 1.00000 1.76208 1.00000 −1.26930 1.76208 −1.00000 1.00000 0.104927 −1.26930
1.17 1.00000 1.78235 1.00000 −0.422890 1.78235 −1.00000 1.00000 0.176787 −0.422890
1.18 1.00000 2.22609 1.00000 −2.43328 2.22609 −1.00000 1.00000 1.95547 −2.43328
1.19 1.00000 2.34324 1.00000 −0.555415 2.34324 −1.00000 1.00000 2.49080 −0.555415
1.20 1.00000 2.96642 1.00000 −3.00590 2.96642 −1.00000 1.00000 5.79966 −3.00590
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.20
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6034.2.a.l 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6034.2.a.l 20 1.a even 1 1 trivial

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

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{20} \)
$3$ \( 1 + 3 T + 24 T^{2} + 74 T^{3} + 339 T^{4} + 962 T^{5} + 3383 T^{6} + 8780 T^{7} + 26099 T^{8} + 62244 T^{9} + 163769 T^{10} + 362311 T^{11} + 865250 T^{12} + 1788249 T^{13} + 3943301 T^{14} + 7656907 T^{15} + 15762167 T^{16} + 28879544 T^{17} + 55914327 T^{18} + 96915322 T^{19} + 177227896 T^{20} + 290745966 T^{21} + 503228943 T^{22} + 779747688 T^{23} + 1276735527 T^{24} + 1860628401 T^{25} + 2874666429 T^{26} + 3910900563 T^{27} + 5676905250 T^{28} + 7131367413 T^{29} + 9670395681 T^{30} + 11026337868 T^{31} + 13870078659 T^{32} + 13998155940 T^{33} + 16180784127 T^{34} + 13803648534 T^{35} + 14592838419 T^{36} + 9556372062 T^{37} + 9298091736 T^{38} + 3486784401 T^{39} + 3486784401 T^{40} \)
$5$ \( 1 + 10 T + 100 T^{2} + 673 T^{3} + 4184 T^{4} + 21792 T^{5} + 104853 T^{6} + 452522 T^{7} + 1822359 T^{8} + 6782363 T^{9} + 23788863 T^{10} + 78378628 T^{11} + 245415333 T^{12} + 729456057 T^{13} + 2074808653 T^{14} + 5644597267 T^{15} + 14777577245 T^{16} + 37210029366 T^{17} + 90537843008 T^{18} + 212634700082 T^{19} + 483649864702 T^{20} + 1063173500410 T^{21} + 2263446075200 T^{22} + 4651253670750 T^{23} + 9235985778125 T^{24} + 17639366459375 T^{25} + 32418885203125 T^{26} + 56988754453125 T^{27} + 95865364453125 T^{28} + 153083257812500 T^{29} + 232313115234375 T^{30} + 331170068359375 T^{31} + 444911865234375 T^{32} + 552395019531250 T^{33} + 639971923828125 T^{34} + 665039062500000 T^{35} + 638427734375000 T^{36} + 513458251953125 T^{37} + 381469726562500 T^{38} + 190734863281250 T^{39} + 95367431640625 T^{40} \)
$7$ \( ( 1 + T )^{20} \)
$11$ \( 1 + 17 T + 262 T^{2} + 2779 T^{3} + 26652 T^{4} + 213525 T^{5} + 1573352 T^{6} + 10314287 T^{7} + 63054195 T^{8} + 353193376 T^{9} + 1864328516 T^{10} + 9171756147 T^{11} + 42880912543 T^{12} + 189038802409 T^{13} + 797906422256 T^{14} + 3204614953882 T^{15} + 12407669044248 T^{16} + 46052940406230 T^{17} + 165745632043125 T^{18} + 574888558062998 T^{19} + 1940309111902644 T^{20} + 6323774138692978 T^{21} + 20055221477218125 T^{22} + 61296463680692130 T^{23} + 181660682476834968 T^{24} + 516106442937649982 T^{25} + 1413539899318261616 T^{26} + 3683831468179394939 T^{27} + 9191904428976344383 T^{28} + 21626521229233706577 T^{29} + 48355880315244222116 T^{30} + \)\(10\!\cdots\!36\)\( T^{31} + \)\(19\!\cdots\!95\)\( T^{32} + \)\(35\!\cdots\!97\)\( T^{33} + \)\(59\!\cdots\!32\)\( T^{34} + \)\(89\!\cdots\!75\)\( T^{35} + \)\(12\!\cdots\!72\)\( T^{36} + \)\(14\!\cdots\!09\)\( T^{37} + \)\(14\!\cdots\!22\)\( T^{38} + \)\(10\!\cdots\!47\)\( T^{39} + \)\(67\!\cdots\!01\)\( T^{40} \)
$13$ \( 1 + 23 T + 392 T^{2} + 4926 T^{3} + 52532 T^{4} + 480757 T^{5} + 3934217 T^{6} + 29044664 T^{7} + 197015736 T^{8} + 1235565090 T^{9} + 7238080160 T^{10} + 39784666665 T^{11} + 206559196772 T^{12} + 1016381381784 T^{13} + 4762361875974 T^{14} + 21302343672547 T^{15} + 91285097803496 T^{16} + 375395005457828 T^{17} + 1485118564474467 T^{18} + 5656884167302232 T^{19} + 20774512439851354 T^{20} + 73539494174929016 T^{21} + 250985037396184923 T^{22} + 824742826990848116 T^{23} + 2607193678365649256 T^{24} + 7909411089210993271 T^{25} + 22987011164208186966 T^{26} + 63776424413356814328 T^{27} + \)\(16\!\cdots\!12\)\( T^{28} + \)\(42\!\cdots\!45\)\( T^{29} + \)\(99\!\cdots\!40\)\( T^{30} + \)\(22\!\cdots\!30\)\( T^{31} + \)\(45\!\cdots\!16\)\( T^{32} + \)\(87\!\cdots\!92\)\( T^{33} + \)\(15\!\cdots\!13\)\( T^{34} + \)\(24\!\cdots\!49\)\( T^{35} + \)\(34\!\cdots\!12\)\( T^{36} + \)\(42\!\cdots\!58\)\( T^{37} + \)\(44\!\cdots\!68\)\( T^{38} + \)\(33\!\cdots\!71\)\( T^{39} + \)\(19\!\cdots\!01\)\( T^{40} \)
$17$ \( 1 + 21 T + 415 T^{2} + 5504 T^{3} + 66910 T^{4} + 673732 T^{5} + 6279118 T^{6} + 52029319 T^{7} + 403459248 T^{8} + 2876014812 T^{9} + 19356717713 T^{10} + 122011298457 T^{11} + 730976601406 T^{12} + 4148161195383 T^{13} + 22481358998766 T^{14} + 116231956518111 T^{15} + 575788238408049 T^{16} + 2732903332778520 T^{17} + 12453611580210147 T^{18} + 54501867715981501 T^{19} + 229219753391983326 T^{20} + 926531751171685517 T^{21} + 3599093746680732483 T^{22} + 13426754073940868760 T^{23} + 48090409460078660529 T^{24} + \)\(16\!\cdots\!27\)\( T^{25} + \)\(54\!\cdots\!54\)\( T^{26} + \)\(17\!\cdots\!59\)\( T^{27} + \)\(50\!\cdots\!46\)\( T^{28} + \)\(14\!\cdots\!29\)\( T^{29} + \)\(39\!\cdots\!37\)\( T^{30} + \)\(98\!\cdots\!96\)\( T^{31} + \)\(23\!\cdots\!28\)\( T^{32} + \)\(51\!\cdots\!03\)\( T^{33} + \)\(10\!\cdots\!22\)\( T^{34} + \)\(19\!\cdots\!76\)\( T^{35} + \)\(32\!\cdots\!10\)\( T^{36} + \)\(45\!\cdots\!08\)\( T^{37} + \)\(58\!\cdots\!35\)\( T^{38} + \)\(50\!\cdots\!13\)\( T^{39} + \)\(40\!\cdots\!01\)\( T^{40} \)
$19$ \( 1 + 22 T + 410 T^{2} + 5271 T^{3} + 60022 T^{4} + 568447 T^{5} + 4902807 T^{6} + 37414104 T^{7} + 264231175 T^{8} + 1698101660 T^{9} + 10189807352 T^{10} + 56428534625 T^{11} + 292953307368 T^{12} + 1413134982330 T^{13} + 6397411260923 T^{14} + 27035266401000 T^{15} + 107683427946459 T^{16} + 407277069652428 T^{17} + 1503168432695800 T^{18} + 5684369830268923 T^{19} + 23355210441560646 T^{20} + 108003026775109537 T^{21} + 542643804203183800 T^{22} + 2793513420746003652 T^{23} + 14033412013410483339 T^{24} + 66941996100249699000 T^{25} + \)\(30\!\cdots\!63\)\( T^{26} + \)\(12\!\cdots\!70\)\( T^{27} + \)\(49\!\cdots\!88\)\( T^{28} + \)\(18\!\cdots\!75\)\( T^{29} + \)\(62\!\cdots\!52\)\( T^{30} + \)\(19\!\cdots\!40\)\( T^{31} + \)\(58\!\cdots\!75\)\( T^{32} + \)\(15\!\cdots\!36\)\( T^{33} + \)\(39\!\cdots\!47\)\( T^{34} + \)\(86\!\cdots\!53\)\( T^{35} + \)\(17\!\cdots\!82\)\( T^{36} + \)\(28\!\cdots\!69\)\( T^{37} + \)\(42\!\cdots\!10\)\( T^{38} + \)\(43\!\cdots\!38\)\( T^{39} + \)\(37\!\cdots\!01\)\( T^{40} \)
$23$ \( 1 - 15 T + 362 T^{2} - 4573 T^{3} + 64776 T^{4} - 694425 T^{5} + 7507590 T^{6} - 69633959 T^{7} + 629711091 T^{8} - 5150134823 T^{9} + 40635830348 T^{10} - 297498648538 T^{11} + 2095355167701 T^{12} - 13883837108274 T^{13} + 88485499271484 T^{14} - 534735606071258 T^{15} + 3109967920401641 T^{16} - 17228510967691727 T^{17} + 91905513586302764 T^{18} - 468104666251087544 T^{19} + 2296633639861040916 T^{20} - 10766407323775013512 T^{21} + 48618016687154162156 T^{22} - \)\(20\!\cdots\!09\)\( T^{23} + \)\(87\!\cdots\!81\)\( T^{24} - \)\(34\!\cdots\!94\)\( T^{25} + \)\(13\!\cdots\!76\)\( T^{26} - \)\(47\!\cdots\!78\)\( T^{27} + \)\(16\!\cdots\!81\)\( T^{28} - \)\(53\!\cdots\!94\)\( T^{29} + \)\(16\!\cdots\!52\)\( T^{30} - \)\(49\!\cdots\!21\)\( T^{31} + \)\(13\!\cdots\!11\)\( T^{32} - \)\(35\!\cdots\!97\)\( T^{33} + \)\(87\!\cdots\!10\)\( T^{34} - \)\(18\!\cdots\!75\)\( T^{35} + \)\(39\!\cdots\!36\)\( T^{36} - \)\(64\!\cdots\!19\)\( T^{37} + \)\(11\!\cdots\!78\)\( T^{38} - \)\(11\!\cdots\!05\)\( T^{39} + \)\(17\!\cdots\!01\)\( T^{40} \)
$29$ \( 1 + 3 T + 331 T^{2} + 423 T^{3} + 51662 T^{4} - 18520 T^{5} + 5230121 T^{6} - 9858076 T^{7} + 398403663 T^{8} - 1265816593 T^{9} + 24835338885 T^{10} - 101887175104 T^{11} + 1322294531857 T^{12} - 6113268305200 T^{13} + 61193236320988 T^{14} - 293936705461645 T^{15} + 2476236442769677 T^{16} - 11747320602378593 T^{17} + 87739751316611995 T^{18} - 398052468669020587 T^{19} + 2720647792635462696 T^{20} - 11543521591401597023 T^{21} + 73789130857270687795 T^{22} - \)\(28\!\cdots\!77\)\( T^{23} + \)\(17\!\cdots\!37\)\( T^{24} - \)\(60\!\cdots\!05\)\( T^{25} + \)\(36\!\cdots\!48\)\( T^{26} - \)\(10\!\cdots\!00\)\( T^{27} + \)\(66\!\cdots\!77\)\( T^{28} - \)\(14\!\cdots\!76\)\( T^{29} + \)\(10\!\cdots\!85\)\( T^{30} - \)\(15\!\cdots\!97\)\( T^{31} + \)\(14\!\cdots\!83\)\( T^{32} - \)\(10\!\cdots\!64\)\( T^{33} + \)\(15\!\cdots\!01\)\( T^{34} - \)\(15\!\cdots\!80\)\( T^{35} + \)\(12\!\cdots\!02\)\( T^{36} + \)\(30\!\cdots\!07\)\( T^{37} + \)\(69\!\cdots\!91\)\( T^{38} + \)\(18\!\cdots\!07\)\( T^{39} + \)\(17\!\cdots\!01\)\( T^{40} \)
$31$ \( 1 + 3 T + 318 T^{2} + 900 T^{3} + 50501 T^{4} + 141431 T^{5} + 5319189 T^{6} + 15364657 T^{7} + 416599638 T^{8} + 1284572237 T^{9} + 25830096352 T^{10} + 87163735283 T^{11} + 1321652239222 T^{12} + 4942928994600 T^{13} + 57675670777759 T^{14} + 238319294150951 T^{15} + 2212397453382722 T^{16} + 9869949624379216 T^{17} + 76755480374810286 T^{18} + 353178396508938568 T^{19} + 2463138350234883576 T^{20} + 10948530291777095608 T^{21} + 73762016640192684846 T^{22} + \)\(29\!\cdots\!56\)\( T^{23} + \)\(20\!\cdots\!62\)\( T^{24} + \)\(68\!\cdots\!01\)\( T^{25} + \)\(51\!\cdots\!79\)\( T^{26} + \)\(13\!\cdots\!00\)\( T^{27} + \)\(11\!\cdots\!02\)\( T^{28} + \)\(23\!\cdots\!93\)\( T^{29} + \)\(21\!\cdots\!52\)\( T^{30} + \)\(32\!\cdots\!47\)\( T^{31} + \)\(32\!\cdots\!18\)\( T^{32} + \)\(37\!\cdots\!87\)\( T^{33} + \)\(40\!\cdots\!69\)\( T^{34} + \)\(33\!\cdots\!81\)\( T^{35} + \)\(36\!\cdots\!81\)\( T^{36} + \)\(20\!\cdots\!00\)\( T^{37} + \)\(22\!\cdots\!38\)\( T^{38} + \)\(65\!\cdots\!13\)\( T^{39} + \)\(67\!\cdots\!01\)\( T^{40} \)
$37$ \( 1 + 14 T + 478 T^{2} + 5702 T^{3} + 109560 T^{4} + 1138869 T^{5} + 16076020 T^{6} + 148019276 T^{7} + 1700124271 T^{8} + 14039136258 T^{9} + 138411321670 T^{10} + 1035816187370 T^{11} + 9067771898549 T^{12} + 62130058954610 T^{13} + 495417304548222 T^{14} + 3144471431175427 T^{15} + 23344187851097695 T^{16} + 139195340121285919 T^{17} + 980454372544125171 T^{18} + 5572252652363538529 T^{19} + 37706149083875925300 T^{20} + \)\(20\!\cdots\!73\)\( T^{21} + \)\(13\!\cdots\!99\)\( T^{22} + \)\(70\!\cdots\!07\)\( T^{23} + \)\(43\!\cdots\!95\)\( T^{24} + \)\(21\!\cdots\!39\)\( T^{25} + \)\(12\!\cdots\!98\)\( T^{26} + \)\(58\!\cdots\!30\)\( T^{27} + \)\(31\!\cdots\!29\)\( T^{28} + \)\(13\!\cdots\!90\)\( T^{29} + \)\(66\!\cdots\!30\)\( T^{30} + \)\(24\!\cdots\!54\)\( T^{31} + \)\(11\!\cdots\!51\)\( T^{32} + \)\(36\!\cdots\!72\)\( T^{33} + \)\(14\!\cdots\!80\)\( T^{34} + \)\(37\!\cdots\!17\)\( T^{35} + \)\(13\!\cdots\!60\)\( T^{36} + \)\(26\!\cdots\!34\)\( T^{37} + \)\(80\!\cdots\!62\)\( T^{38} + \)\(87\!\cdots\!22\)\( T^{39} + \)\(23\!\cdots\!01\)\( T^{40} \)
$41$ \( 1 + 37 T + 1139 T^{2} + 24656 T^{3} + 472054 T^{4} + 7614375 T^{5} + 112223046 T^{6} + 1479800004 T^{7} + 18171684649 T^{8} + 205405186521 T^{9} + 2188376696738 T^{10} + 21805953366345 T^{11} + 206430025795442 T^{12} + 1845593687985130 T^{13} + 15759975623287710 T^{14} + 127879708450293802 T^{15} + 994537712033321889 T^{16} + 7377463235002134926 T^{17} + 52563825903578027799 T^{18} + \)\(35\!\cdots\!84\)\( T^{19} + \)\(23\!\cdots\!70\)\( T^{20} + \)\(14\!\cdots\!44\)\( T^{21} + \)\(88\!\cdots\!19\)\( T^{22} + \)\(50\!\cdots\!46\)\( T^{23} + \)\(28\!\cdots\!29\)\( T^{24} + \)\(14\!\cdots\!02\)\( T^{25} + \)\(74\!\cdots\!10\)\( T^{26} + \)\(35\!\cdots\!30\)\( T^{27} + \)\(16\!\cdots\!82\)\( T^{28} + \)\(71\!\cdots\!45\)\( T^{29} + \)\(29\!\cdots\!38\)\( T^{30} + \)\(11\!\cdots\!61\)\( T^{31} + \)\(41\!\cdots\!69\)\( T^{32} + \)\(13\!\cdots\!84\)\( T^{33} + \)\(42\!\cdots\!06\)\( T^{34} + \)\(11\!\cdots\!75\)\( T^{35} + \)\(30\!\cdots\!14\)\( T^{36} + \)\(64\!\cdots\!36\)\( T^{37} + \)\(12\!\cdots\!19\)\( T^{38} + \)\(16\!\cdots\!57\)\( T^{39} + \)\(18\!\cdots\!01\)\( T^{40} \)
$43$ \( 1 + 5 T + 564 T^{2} + 3348 T^{3} + 157972 T^{4} + 1034064 T^{5} + 29342290 T^{6} + 200617849 T^{7} + 4051399264 T^{8} + 27842427634 T^{9} + 440846284132 T^{10} + 2967797910895 T^{11} + 39130680758781 T^{12} + 253757295317484 T^{13} + 2897921022280148 T^{14} + 17898474415146579 T^{15} + 181853735941488851 T^{16} + 1060583722015327077 T^{17} + 9770998761396000098 T^{18} + 53391621194146534697 T^{19} + \)\(45\!\cdots\!18\)\( T^{20} + \)\(22\!\cdots\!71\)\( T^{21} + \)\(18\!\cdots\!02\)\( T^{22} + \)\(84\!\cdots\!39\)\( T^{23} + \)\(62\!\cdots\!51\)\( T^{24} + \)\(26\!\cdots\!97\)\( T^{25} + \)\(18\!\cdots\!52\)\( T^{26} + \)\(68\!\cdots\!88\)\( T^{27} + \)\(45\!\cdots\!81\)\( T^{28} + \)\(14\!\cdots\!85\)\( T^{29} + \)\(95\!\cdots\!68\)\( T^{30} + \)\(25\!\cdots\!38\)\( T^{31} + \)\(16\!\cdots\!64\)\( T^{32} + \)\(34\!\cdots\!07\)\( T^{33} + \)\(21\!\cdots\!10\)\( T^{34} + \)\(32\!\cdots\!48\)\( T^{35} + \)\(21\!\cdots\!72\)\( T^{36} + \)\(19\!\cdots\!64\)\( T^{37} + \)\(14\!\cdots\!36\)\( T^{38} + \)\(54\!\cdots\!35\)\( T^{39} + \)\(46\!\cdots\!01\)\( T^{40} \)
$47$ \( 1 + 29 T + 836 T^{2} + 16006 T^{3} + 294378 T^{4} + 4444483 T^{5} + 64396752 T^{6} + 823142364 T^{7} + 10129643860 T^{8} + 113744809482 T^{9} + 1233670062815 T^{10} + 12437434377662 T^{11} + 121440235879608 T^{12} + 1114487302876067 T^{13} + 9926053734937470 T^{14} + 83666217492160808 T^{15} + 685403554140201658 T^{16} + 5336323680174475005 T^{17} + 40417249015765775359 T^{18} + \)\(29\!\cdots\!74\)\( T^{19} + \)\(20\!\cdots\!58\)\( T^{20} + \)\(13\!\cdots\!78\)\( T^{21} + \)\(89\!\cdots\!31\)\( T^{22} + \)\(55\!\cdots\!15\)\( T^{23} + \)\(33\!\cdots\!98\)\( T^{24} + \)\(19\!\cdots\!56\)\( T^{25} + \)\(10\!\cdots\!30\)\( T^{26} + \)\(56\!\cdots\!21\)\( T^{27} + \)\(28\!\cdots\!88\)\( T^{28} + \)\(13\!\cdots\!54\)\( T^{29} + \)\(64\!\cdots\!35\)\( T^{30} + \)\(28\!\cdots\!46\)\( T^{31} + \)\(11\!\cdots\!60\)\( T^{32} + \)\(44\!\cdots\!28\)\( T^{33} + \)\(16\!\cdots\!88\)\( T^{34} + \)\(53\!\cdots\!69\)\( T^{35} + \)\(16\!\cdots\!38\)\( T^{36} + \)\(42\!\cdots\!22\)\( T^{37} + \)\(10\!\cdots\!04\)\( T^{38} + \)\(17\!\cdots\!07\)\( T^{39} + \)\(27\!\cdots\!01\)\( T^{40} \)
$53$ \( 1 + 28 T + 855 T^{2} + 16971 T^{3} + 318453 T^{4} + 4989886 T^{5} + 72859181 T^{6} + 959522258 T^{7} + 11879169618 T^{8} + 136930261681 T^{9} + 1499641108359 T^{10} + 15540991233654 T^{11} + 154241728666272 T^{12} + 1462317158506270 T^{13} + 13348321475007699 T^{14} + 117074942188318834 T^{15} + 992078761734708850 T^{16} + 8105974828203638503 T^{17} + 64120597949792064174 T^{18} + \)\(48\!\cdots\!27\)\( T^{19} + \)\(36\!\cdots\!28\)\( T^{20} + \)\(25\!\cdots\!31\)\( T^{21} + \)\(18\!\cdots\!66\)\( T^{22} + \)\(12\!\cdots\!31\)\( T^{23} + \)\(78\!\cdots\!50\)\( T^{24} + \)\(48\!\cdots\!62\)\( T^{25} + \)\(29\!\cdots\!71\)\( T^{26} + \)\(17\!\cdots\!90\)\( T^{27} + \)\(96\!\cdots\!92\)\( T^{28} + \)\(51\!\cdots\!82\)\( T^{29} + \)\(26\!\cdots\!91\)\( T^{30} + \)\(12\!\cdots\!57\)\( T^{31} + \)\(58\!\cdots\!38\)\( T^{32} + \)\(24\!\cdots\!34\)\( T^{33} + \)\(10\!\cdots\!89\)\( T^{34} + \)\(36\!\cdots\!02\)\( T^{35} + \)\(12\!\cdots\!13\)\( T^{36} + \)\(34\!\cdots\!23\)\( T^{37} + \)\(93\!\cdots\!95\)\( T^{38} + \)\(16\!\cdots\!76\)\( T^{39} + \)\(30\!\cdots\!01\)\( T^{40} \)
$59$ \( 1 + 47 T + 1788 T^{2} + 48119 T^{3} + 1126986 T^{4} + 22230123 T^{5} + 395860894 T^{6} + 6289733217 T^{7} + 92169409959 T^{8} + 1239099217883 T^{9} + 15580319162288 T^{10} + 182712600996040 T^{11} + 2024636845963401 T^{12} + 21162946437227494 T^{13} + 210740088201407780 T^{14} + 1996876129903102148 T^{15} + 18150432938574101217 T^{16} + \)\(15\!\cdots\!85\)\( T^{17} + \)\(13\!\cdots\!02\)\( T^{18} + \)\(10\!\cdots\!04\)\( T^{19} + \)\(83\!\cdots\!72\)\( T^{20} + \)\(63\!\cdots\!36\)\( T^{21} + \)\(46\!\cdots\!62\)\( T^{22} + \)\(32\!\cdots\!15\)\( T^{23} + \)\(21\!\cdots\!37\)\( T^{24} + \)\(14\!\cdots\!52\)\( T^{25} + \)\(88\!\cdots\!80\)\( T^{26} + \)\(52\!\cdots\!86\)\( T^{27} + \)\(29\!\cdots\!21\)\( T^{28} + \)\(15\!\cdots\!60\)\( T^{29} + \)\(79\!\cdots\!88\)\( T^{30} + \)\(37\!\cdots\!97\)\( T^{31} + \)\(16\!\cdots\!79\)\( T^{32} + \)\(66\!\cdots\!43\)\( T^{33} + \)\(24\!\cdots\!34\)\( T^{34} + \)\(81\!\cdots\!77\)\( T^{35} + \)\(24\!\cdots\!26\)\( T^{36} + \)\(61\!\cdots\!61\)\( T^{37} + \)\(13\!\cdots\!48\)\( T^{38} + \)\(20\!\cdots\!33\)\( T^{39} + \)\(26\!\cdots\!01\)\( T^{40} \)
$61$ \( 1 + 13 T + 543 T^{2} + 5537 T^{3} + 135356 T^{4} + 1158259 T^{5} + 22151239 T^{6} + 169961620 T^{7} + 2842190167 T^{8} + 20535982161 T^{9} + 311621710393 T^{10} + 2156274759569 T^{11} + 30008264843466 T^{12} + 198638743651611 T^{13} + 2566312885568540 T^{14} + 16252555189553676 T^{15} + 197668959333327174 T^{16} + 1201249528469311764 T^{17} + 13861604702787399114 T^{18} + 80762667542124740506 T^{19} + \)\(88\!\cdots\!22\)\( T^{20} + \)\(49\!\cdots\!66\)\( T^{21} + \)\(51\!\cdots\!94\)\( T^{22} + \)\(27\!\cdots\!84\)\( T^{23} + \)\(27\!\cdots\!34\)\( T^{24} + \)\(13\!\cdots\!76\)\( T^{25} + \)\(13\!\cdots\!40\)\( T^{26} + \)\(62\!\cdots\!31\)\( T^{27} + \)\(57\!\cdots\!46\)\( T^{28} + \)\(25\!\cdots\!29\)\( T^{29} + \)\(22\!\cdots\!93\)\( T^{30} + \)\(89\!\cdots\!21\)\( T^{31} + \)\(75\!\cdots\!07\)\( T^{32} + \)\(27\!\cdots\!20\)\( T^{33} + \)\(21\!\cdots\!99\)\( T^{34} + \)\(69\!\cdots\!59\)\( T^{35} + \)\(49\!\cdots\!16\)\( T^{36} + \)\(12\!\cdots\!77\)\( T^{37} + \)\(74\!\cdots\!83\)\( T^{38} + \)\(10\!\cdots\!33\)\( T^{39} + \)\(50\!\cdots\!01\)\( T^{40} \)
$67$ \( 1 + 24 T + 655 T^{2} + 11807 T^{3} + 206811 T^{4} + 3136202 T^{5} + 44832552 T^{6} + 596798322 T^{7} + 7501071068 T^{8} + 89816005294 T^{9} + 1026691105841 T^{10} + 11272421156478 T^{11} + 119262652414856 T^{12} + 1216266583348386 T^{13} + 12022878639293925 T^{14} + 114938154792189479 T^{15} + 1068172138954193764 T^{16} + 9630693245059776611 T^{17} + 84464162833322292609 T^{18} + \)\(72\!\cdots\!87\)\( T^{19} + \)\(59\!\cdots\!32\)\( T^{20} + \)\(48\!\cdots\!29\)\( T^{21} + \)\(37\!\cdots\!01\)\( T^{22} + \)\(28\!\cdots\!93\)\( T^{23} + \)\(21\!\cdots\!44\)\( T^{24} + \)\(15\!\cdots\!53\)\( T^{25} + \)\(10\!\cdots\!25\)\( T^{26} + \)\(73\!\cdots\!78\)\( T^{27} + \)\(48\!\cdots\!96\)\( T^{28} + \)\(30\!\cdots\!66\)\( T^{29} + \)\(18\!\cdots\!09\)\( T^{30} + \)\(10\!\cdots\!02\)\( T^{31} + \)\(61\!\cdots\!48\)\( T^{32} + \)\(32\!\cdots\!14\)\( T^{33} + \)\(16\!\cdots\!08\)\( T^{34} + \)\(77\!\cdots\!86\)\( T^{35} + \)\(34\!\cdots\!91\)\( T^{36} + \)\(13\!\cdots\!89\)\( T^{37} + \)\(48\!\cdots\!95\)\( T^{38} + \)\(11\!\cdots\!72\)\( T^{39} + \)\(33\!\cdots\!01\)\( T^{40} \)
$71$ \( 1 + 22 T + 913 T^{2} + 15614 T^{3} + 375610 T^{4} + 5418429 T^{5} + 97821779 T^{6} + 1250498050 T^{7} + 18689775413 T^{8} + 218083949274 T^{9} + 2834871504001 T^{10} + 30720985748034 T^{11} + 357651564487925 T^{12} + 3637271856489393 T^{13} + 38684748866261100 T^{14} + 371541335571880405 T^{15} + 3661669132324374044 T^{16} + 33317878341909397282 T^{17} + \)\(30\!\cdots\!56\)\( T^{18} + \)\(26\!\cdots\!21\)\( T^{19} + \)\(23\!\cdots\!52\)\( T^{20} + \)\(18\!\cdots\!91\)\( T^{21} + \)\(15\!\cdots\!96\)\( T^{22} + \)\(11\!\cdots\!02\)\( T^{23} + \)\(93\!\cdots\!64\)\( T^{24} + \)\(67\!\cdots\!55\)\( T^{25} + \)\(49\!\cdots\!00\)\( T^{26} + \)\(33\!\cdots\!63\)\( T^{27} + \)\(23\!\cdots\!25\)\( T^{28} + \)\(14\!\cdots\!54\)\( T^{29} + \)\(92\!\cdots\!01\)\( T^{30} + \)\(50\!\cdots\!54\)\( T^{31} + \)\(30\!\cdots\!33\)\( T^{32} + \)\(14\!\cdots\!50\)\( T^{33} + \)\(80\!\cdots\!99\)\( T^{34} + \)\(31\!\cdots\!79\)\( T^{35} + \)\(15\!\cdots\!10\)\( T^{36} + \)\(46\!\cdots\!74\)\( T^{37} + \)\(19\!\cdots\!93\)\( T^{38} + \)\(32\!\cdots\!82\)\( T^{39} + \)\(10\!\cdots\!01\)\( T^{40} \)
$73$ \( 1 + 37 T + 1322 T^{2} + 31457 T^{3} + 707479 T^{4} + 13098092 T^{5} + 230783295 T^{6} + 3598777065 T^{7} + 53811554660 T^{8} + 736820848086 T^{9} + 9729356612710 T^{10} + 119864595167890 T^{11} + 1429796735217682 T^{12} + 16090479386235061 T^{13} + 175808744649259140 T^{14} + 1824721474066883767 T^{15} + 18421784068159602034 T^{16} + \)\(17\!\cdots\!61\)\( T^{17} + \)\(16\!\cdots\!17\)\( T^{18} + \)\(14\!\cdots\!96\)\( T^{19} + \)\(13\!\cdots\!40\)\( T^{20} + \)\(10\!\cdots\!08\)\( T^{21} + \)\(88\!\cdots\!93\)\( T^{22} + \)\(69\!\cdots\!37\)\( T^{23} + \)\(52\!\cdots\!94\)\( T^{24} + \)\(37\!\cdots\!31\)\( T^{25} + \)\(26\!\cdots\!60\)\( T^{26} + \)\(17\!\cdots\!17\)\( T^{27} + \)\(11\!\cdots\!42\)\( T^{28} + \)\(70\!\cdots\!70\)\( T^{29} + \)\(41\!\cdots\!90\)\( T^{30} + \)\(23\!\cdots\!22\)\( T^{31} + \)\(12\!\cdots\!60\)\( T^{32} + \)\(60\!\cdots\!45\)\( T^{33} + \)\(28\!\cdots\!55\)\( T^{34} + \)\(11\!\cdots\!44\)\( T^{35} + \)\(46\!\cdots\!19\)\( T^{36} + \)\(14\!\cdots\!21\)\( T^{37} + \)\(45\!\cdots\!18\)\( T^{38} + \)\(93\!\cdots\!69\)\( T^{39} + \)\(18\!\cdots\!01\)\( T^{40} \)
$79$ \( 1 - 25 T + 1250 T^{2} - 24578 T^{3} + 714684 T^{4} - 11827289 T^{5} + 257795206 T^{6} - 3730126203 T^{7} + 67020308433 T^{8} - 867930014509 T^{9} + 13487302851081 T^{10} - 158728718389989 T^{11} + 2194288625379001 T^{12} - 23710528353777159 T^{13} + 296877642502245672 T^{14} - 2965745503247346096 T^{15} + 34035469574124251962 T^{16} - \)\(31\!\cdots\!87\)\( T^{17} + \)\(33\!\cdots\!83\)\( T^{18} - \)\(28\!\cdots\!15\)\( T^{19} + \)\(28\!\cdots\!22\)\( T^{20} - \)\(22\!\cdots\!85\)\( T^{21} + \)\(20\!\cdots\!03\)\( T^{22} - \)\(15\!\cdots\!93\)\( T^{23} + \)\(13\!\cdots\!22\)\( T^{24} - \)\(91\!\cdots\!04\)\( T^{25} + \)\(72\!\cdots\!12\)\( T^{26} - \)\(45\!\cdots\!81\)\( T^{27} + \)\(33\!\cdots\!61\)\( T^{28} - \)\(19\!\cdots\!91\)\( T^{29} + \)\(12\!\cdots\!81\)\( T^{30} - \)\(64\!\cdots\!11\)\( T^{31} + \)\(39\!\cdots\!53\)\( T^{32} - \)\(17\!\cdots\!17\)\( T^{33} + \)\(95\!\cdots\!86\)\( T^{34} - \)\(34\!\cdots\!11\)\( T^{35} + \)\(16\!\cdots\!64\)\( T^{36} - \)\(44\!\cdots\!02\)\( T^{37} + \)\(17\!\cdots\!50\)\( T^{38} - \)\(28\!\cdots\!75\)\( T^{39} + \)\(89\!\cdots\!01\)\( T^{40} \)
$83$ \( 1 + 33 T + 1134 T^{2} + 25077 T^{3} + 538999 T^{4} + 9350713 T^{5} + 157289285 T^{6} + 2310569800 T^{7} + 33131147213 T^{8} + 431195103972 T^{9} + 5516658793839 T^{10} + 65485766141129 T^{11} + 768275389211482 T^{12} + 8473190596979554 T^{13} + 92684268230633733 T^{14} + 960676708836587901 T^{15} + 9899744401622803947 T^{16} + 97159557348164041570 T^{17} + \)\(94\!\cdots\!95\)\( T^{18} + \)\(88\!\cdots\!99\)\( T^{19} + \)\(82\!\cdots\!80\)\( T^{20} + \)\(73\!\cdots\!17\)\( T^{21} + \)\(65\!\cdots\!55\)\( T^{22} + \)\(55\!\cdots\!90\)\( T^{23} + \)\(46\!\cdots\!87\)\( T^{24} + \)\(37\!\cdots\!43\)\( T^{25} + \)\(30\!\cdots\!77\)\( T^{26} + \)\(22\!\cdots\!58\)\( T^{27} + \)\(17\!\cdots\!62\)\( T^{28} + \)\(12\!\cdots\!87\)\( T^{29} + \)\(85\!\cdots\!11\)\( T^{30} + \)\(55\!\cdots\!24\)\( T^{31} + \)\(35\!\cdots\!93\)\( T^{32} + \)\(20\!\cdots\!00\)\( T^{33} + \)\(11\!\cdots\!65\)\( T^{34} + \)\(57\!\cdots\!91\)\( T^{35} + \)\(27\!\cdots\!19\)\( T^{36} + \)\(10\!\cdots\!71\)\( T^{37} + \)\(39\!\cdots\!06\)\( T^{38} + \)\(95\!\cdots\!51\)\( T^{39} + \)\(24\!\cdots\!01\)\( T^{40} \)
$89$ \( 1 + 71 T + 3343 T^{2} + 117202 T^{3} + 3412879 T^{4} + 85408826 T^{5} + 1899135453 T^{6} + 38133185174 T^{7} + 702129680973 T^{8} + 11963638122922 T^{9} + 190258595791697 T^{10} + 2839839224626253 T^{11} + 39991336529210688 T^{12} + 533225123678772761 T^{13} + 6753936578406729148 T^{14} + 81449969478068436467 T^{15} + \)\(93\!\cdots\!31\)\( T^{16} + \)\(10\!\cdots\!51\)\( T^{17} + \)\(10\!\cdots\!14\)\( T^{18} + \)\(10\!\cdots\!93\)\( T^{19} + \)\(10\!\cdots\!78\)\( T^{20} + \)\(97\!\cdots\!77\)\( T^{21} + \)\(85\!\cdots\!94\)\( T^{22} + \)\(72\!\cdots\!19\)\( T^{23} + \)\(58\!\cdots\!71\)\( T^{24} + \)\(45\!\cdots\!83\)\( T^{25} + \)\(33\!\cdots\!28\)\( T^{26} + \)\(23\!\cdots\!69\)\( T^{27} + \)\(15\!\cdots\!28\)\( T^{28} + \)\(99\!\cdots\!77\)\( T^{29} + \)\(59\!\cdots\!97\)\( T^{30} + \)\(33\!\cdots\!58\)\( T^{31} + \)\(17\!\cdots\!33\)\( T^{32} + \)\(83\!\cdots\!06\)\( T^{33} + \)\(37\!\cdots\!73\)\( T^{34} + \)\(14\!\cdots\!74\)\( T^{35} + \)\(52\!\cdots\!19\)\( T^{36} + \)\(16\!\cdots\!58\)\( T^{37} + \)\(41\!\cdots\!83\)\( T^{38} + \)\(77\!\cdots\!39\)\( T^{39} + \)\(97\!\cdots\!01\)\( T^{40} \)
$97$ \( 1 + 51 T + 2293 T^{2} + 71265 T^{3} + 2008195 T^{4} + 47361843 T^{5} + 1035451775 T^{6} + 20193675075 T^{7} + 370369666479 T^{8} + 6244076099934 T^{9} + 99953768435771 T^{10} + 1495502248233244 T^{11} + 21385023953653722 T^{12} + 288724751613226927 T^{13} + 3742241751744269966 T^{14} + 46088072796316313612 T^{15} + \)\(54\!\cdots\!93\)\( T^{16} + \)\(61\!\cdots\!58\)\( T^{17} + \)\(67\!\cdots\!19\)\( T^{18} + \)\(70\!\cdots\!07\)\( T^{19} + \)\(70\!\cdots\!68\)\( T^{20} + \)\(68\!\cdots\!79\)\( T^{21} + \)\(63\!\cdots\!71\)\( T^{22} + \)\(56\!\cdots\!34\)\( T^{23} + \)\(48\!\cdots\!33\)\( T^{24} + \)\(39\!\cdots\!84\)\( T^{25} + \)\(31\!\cdots\!14\)\( T^{26} + \)\(23\!\cdots\!51\)\( T^{27} + \)\(16\!\cdots\!42\)\( T^{28} + \)\(11\!\cdots\!48\)\( T^{29} + \)\(73\!\cdots\!79\)\( T^{30} + \)\(44\!\cdots\!02\)\( T^{31} + \)\(25\!\cdots\!39\)\( T^{32} + \)\(13\!\cdots\!75\)\( T^{33} + \)\(67\!\cdots\!75\)\( T^{34} + \)\(29\!\cdots\!99\)\( T^{35} + \)\(12\!\cdots\!95\)\( T^{36} + \)\(42\!\cdots\!05\)\( T^{37} + \)\(13\!\cdots\!77\)\( T^{38} + \)\(28\!\cdots\!83\)\( T^{39} + \)\(54\!\cdots\!01\)\( T^{40} \)
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