Properties

Label 1050.3.p.i
Level 1050
Weight 3
Character orbit 1050.p
Analytic conductor 28.610
Analytic rank 0
Dimension 16
CM no
Inner twists 2

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1050.p (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(28.6104277578\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{4}\cdot 7 \)
Twist minimal: no (minimal twist has level 210)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\beta_{1} - \beta_{5} ) q^{2} + ( 2 + \beta_{3} ) q^{3} + 2 \beta_{3} q^{4} + ( -\beta_{1} - 2 \beta_{5} ) q^{6} + ( 1 + \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{5} ) q^{7} + 2 \beta_{1} q^{8} + ( 3 + 3 \beta_{3} ) q^{9} +O(q^{10})\) \( q + ( -\beta_{1} - \beta_{5} ) q^{2} + ( 2 + \beta_{3} ) q^{3} + 2 \beta_{3} q^{4} + ( -\beta_{1} - 2 \beta_{5} ) q^{6} + ( 1 + \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{5} ) q^{7} + 2 \beta_{1} q^{8} + ( 3 + 3 \beta_{3} ) q^{9} + ( \beta_{3} + \beta_{5} + \beta_{10} + \beta_{15} ) q^{11} + ( -2 + 2 \beta_{3} ) q^{12} + ( -2 - 4 \beta_{3} + \beta_{6} - \beta_{9} - \beta_{10} - \beta_{15} ) q^{13} + ( \beta_{1} - \beta_{3} - \beta_{5} + \beta_{10} ) q^{14} + ( -4 - 4 \beta_{3} ) q^{16} + ( -1 + \beta_{1} + \beta_{5} - \beta_{8} + \beta_{9} - \beta_{10} + \beta_{13} ) q^{17} -3 \beta_{5} q^{18} + ( -4 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{6} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{19} + ( \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{5} - \beta_{9} ) q^{21} + ( 3 - \beta_{2} - \beta_{3} - \beta_{5} - \beta_{6} - \beta_{8} - \beta_{9} - \beta_{10} ) q^{22} + ( 1 + \beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} - 2 \beta_{10} ) q^{23} + ( 4 \beta_{1} + 2 \beta_{5} ) q^{24} + ( 1 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{7} + \beta_{9} + \beta_{10} + \beta_{15} ) q^{26} + ( 3 + 6 \beta_{3} ) q^{27} + ( -4 - 2 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} - 2 \beta_{5} - 2 \beta_{9} ) q^{28} + ( 3 - 7 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} - \beta_{4} - 4 \beta_{5} + \beta_{6} - \beta_{7} - 3 \beta_{9} + 2 \beta_{10} - \beta_{11} + \beta_{12} - \beta_{15} ) q^{29} + ( 10 + \beta_{1} - \beta_{2} + 5 \beta_{3} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} - \beta_{13} ) q^{31} + 4 \beta_{5} q^{32} + ( -1 - 2 \beta_{1} + \beta_{3} + \beta_{6} - \beta_{7} + 2 \beta_{10} + \beta_{15} ) q^{33} + ( -1 + \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{8} - 2 \beta_{10} - 2 \beta_{11} - 2 \beta_{12} - \beta_{15} ) q^{34} -6 q^{36} + ( -5 - 6 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} + \beta_{4} - 5 \beta_{5} + \beta_{6} + 3 \beta_{9} + \beta_{10} - 2 \beta_{13} - \beta_{14} - 2 \beta_{15} ) q^{37} + ( 6 + 5 \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} - 2 \beta_{9} - \beta_{10} + \beta_{13} + \beta_{15} ) q^{38} + ( -1 - \beta_{2} - 7 \beta_{3} + \beta_{6} + \beta_{7} - 2 \beta_{9} - 2 \beta_{10} - 2 \beta_{15} ) q^{39} + ( -3 - 5 \beta_{1} - 7 \beta_{3} + 3 \beta_{4} - 12 \beta_{5} + \beta_{6} + \beta_{7} + 2 \beta_{8} - 3 \beta_{9} + 2 \beta_{10} - \beta_{15} ) q^{41} + ( 1 + 2 \beta_{1} - \beta_{3} - \beta_{5} - \beta_{7} + 2 \beta_{10} ) q^{42} + ( 2 + 5 \beta_{1} - \beta_{3} - \beta_{4} - 2 \beta_{5} + 3 \beta_{6} - \beta_{7} + 2 \beta_{8} - \beta_{9} + 4 \beta_{10} - \beta_{15} ) q^{43} + ( -2 - 4 \beta_{1} - 2 \beta_{3} - 4 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{15} ) q^{44} + ( 2 + \beta_{1} + 3 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} + 3 \beta_{9} - \beta_{10} ) q^{46} + ( -2 \beta_{2} - 4 \beta_{3} + 2 \beta_{7} - 2 \beta_{9} - 2 \beta_{10} - \beta_{11} + 2 \beta_{14} - 2 \beta_{15} ) q^{47} + ( -4 - 8 \beta_{3} ) q^{48} + ( -5 + 11 \beta_{1} - 2 \beta_{2} - 6 \beta_{3} - 8 \beta_{5} - \beta_{8} - 3 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} + 3 \beta_{12} - \beta_{15} ) q^{49} + ( -1 + 2 \beta_{1} + \beta_{2} - \beta_{4} + 3 \beta_{5} - 2 \beta_{8} + 2 \beta_{9} - 2 \beta_{10} + 2 \beta_{13} + \beta_{14} ) q^{51} + ( 6 - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{6} + 2 \beta_{7} ) q^{52} + ( -2 - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 14 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{9} - 2 \beta_{10} ) q^{53} + ( 3 \beta_{1} - 3 \beta_{5} ) q^{54} + ( 2 + 2 \beta_{3} + 2 \beta_{5} - 2 \beta_{7} ) q^{56} + ( -11 - 6 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 3 \beta_{5} + 2 \beta_{6} - \beta_{7} - \beta_{9} + 2 \beta_{10} + \beta_{11} - \beta_{12} - \beta_{15} ) q^{57} + ( 9 - 8 \beta_{1} - 5 \beta_{2} + 7 \beta_{3} - \beta_{4} - 10 \beta_{5} - \beta_{6} - 2 \beta_{7} + \beta_{8} - 4 \beta_{9} + \beta_{10} - 2 \beta_{13} - \beta_{14} + \beta_{15} ) q^{58} + ( 8 + 10 \beta_{1} - 4 \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} + 2 \beta_{6} - 6 \beta_{7} - 2 \beta_{8} - 4 \beta_{9} + \beta_{10} - 3 \beta_{12} - 2 \beta_{13} - \beta_{15} ) q^{59} + ( 2 - 4 \beta_{1} - 5 \beta_{2} - 9 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} - \beta_{6} + 3 \beta_{7} - 2 \beta_{9} - 4 \beta_{10} + \beta_{11} - 2 \beta_{14} - 3 \beta_{15} ) q^{61} + ( -2 - 5 \beta_{1} - 4 \beta_{3} - 10 \beta_{5} + 2 \beta_{6} - 2 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} + 2 \beta_{12} - 2 \beta_{15} ) q^{62} + ( -3 - 3 \beta_{9} ) q^{63} + 8 q^{64} + ( 6 - 2 \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{6} - 2 \beta_{8} - \beta_{9} - 2 \beta_{10} + \beta_{15} ) q^{66} + ( 3 + 2 \beta_{1} + 5 \beta_{2} - 17 \beta_{3} - \beta_{4} + 29 \beta_{5} - \beta_{6} - \beta_{7} - 2 \beta_{8} + \beta_{9} + 2 \beta_{10} + \beta_{15} ) q^{67} + ( 2 + 2 \beta_{2} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{14} ) q^{68} + ( -\beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{8} - 2 \beta_{9} - 4 \beta_{10} - \beta_{15} ) q^{69} + ( -9 + 4 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} + 5 \beta_{5} + 3 \beta_{6} + \beta_{7} + 2 \beta_{8} + 6 \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} ) q^{71} + ( 6 \beta_{1} + 6 \beta_{5} ) q^{72} + ( 26 + 23 \beta_{1} - \beta_{2} + 12 \beta_{3} + 8 \beta_{5} + 5 \beta_{6} - 5 \beta_{7} + \beta_{8} - \beta_{9} + 4 \beta_{10} + \beta_{13} - 3 \beta_{15} ) q^{73} + ( -2 - 2 \beta_{2} + 12 \beta_{3} - \beta_{4} + 4 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} - 3 \beta_{9} + 4 \beta_{11} + 2 \beta_{12} + \beta_{15} ) q^{74} + ( -6 - 4 \beta_{1} - 12 \beta_{3} - 6 \beta_{5} - 2 \beta_{7} + 2 \beta_{9} - 2 \beta_{11} - 2 \beta_{12} - 2 \beta_{15} ) q^{76} + ( 4 + 29 \beta_{1} - \beta_{2} + 22 \beta_{3} - \beta_{4} + 6 \beta_{5} + 2 \beta_{6} + \beta_{8} - \beta_{9} - \beta_{10} - 2 \beta_{11} - 3 \beta_{12} - \beta_{13} - 3 \beta_{14} ) q^{77} + ( 1 - 4 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{6} - 2 \beta_{7} + \beta_{8} + 2 \beta_{10} + \beta_{15} ) q^{78} + ( 34 - 2 \beta_{1} - \beta_{2} + 34 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + \beta_{6} - 2 \beta_{7} - 2 \beta_{8} - \beta_{9} - 3 \beta_{10} - \beta_{11} - 2 \beta_{12} + 4 \beta_{13} + 2 \beta_{14} ) q^{79} + 9 \beta_{3} q^{81} + ( -11 - 4 \beta_{1} + \beta_{2} + 13 \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{6} - 4 \beta_{7} - \beta_{8} + 5 \beta_{10} + 3 \beta_{15} ) q^{82} + ( -12 + 2 \beta_{1} - \beta_{2} - 25 \beta_{3} + 4 \beta_{4} + \beta_{5} + 4 \beta_{6} + 2 \beta_{7} + 3 \beta_{8} - 8 \beta_{9} - 3 \beta_{15} ) q^{83} + ( -6 - 4 \beta_{1} - 4 \beta_{2} - 8 \beta_{3} - 4 \beta_{5} - 2 \beta_{9} ) q^{84} + ( -11 - 5 \beta_{1} - 5 \beta_{2} - 13 \beta_{3} - \beta_{4} - 7 \beta_{5} - 5 \beta_{6} + 2 \beta_{7} + \beta_{8} - 4 \beta_{9} + \beta_{10} + 5 \beta_{15} ) q^{86} + ( 6 - 15 \beta_{1} - 5 \beta_{2} - \beta_{3} - \beta_{4} - 13 \beta_{5} + \beta_{6} - 3 \beta_{7} + \beta_{8} - 4 \beta_{9} + 4 \beta_{10} + 3 \beta_{12} - 2 \beta_{15} ) q^{87} + ( 8 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{9} + 2 \beta_{15} ) q^{88} + ( -15 + 12 \beta_{1} - 3 \beta_{2} + 6 \beta_{3} - 2 \beta_{4} - 17 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} - \beta_{8} - 7 \beta_{9} - 4 \beta_{14} - 3 \beta_{15} ) q^{89} + ( -5 + 17 \beta_{1} + 24 \beta_{3} + \beta_{4} + 41 \beta_{5} - 2 \beta_{6} + 3 \beta_{7} - \beta_{8} + 4 \beta_{9} - \beta_{10} + 2 \beta_{11} + 3 \beta_{12} + \beta_{13} + 3 \beta_{14} ) q^{91} + ( -4 - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{5} + 4 \beta_{7} + 2 \beta_{8} - 2 \beta_{15} ) q^{92} + ( 16 + 2 \beta_{1} + 17 \beta_{3} - \beta_{4} + 3 \beta_{5} - 2 \beta_{6} + \beta_{7} - 2 \beta_{8} + 3 \beta_{9} - \beta_{10} - 2 \beta_{13} - \beta_{14} + \beta_{15} ) q^{93} + ( -2 \beta_{1} + 2 \beta_{2} - 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} + 4 \beta_{12} - \beta_{13} ) q^{94} + ( -4 \beta_{1} + 4 \beta_{5} ) q^{96} + ( 13 - 27 \beta_{1} + \beta_{2} + 26 \beta_{3} - \beta_{4} - 55 \beta_{5} - 3 \beta_{6} + \beta_{7} - \beta_{8} + 3 \beta_{9} + 2 \beta_{10} + 3 \beta_{13} + 3 \beta_{14} + 4 \beta_{15} ) q^{97} + ( -36 + 2 \beta_{2} - 23 \beta_{3} - \beta_{4} + 3 \beta_{5} + 2 \beta_{6} - 4 \beta_{7} + \beta_{9} + \beta_{10} - \beta_{13} - 3 \beta_{14} - \beta_{15} ) q^{98} + ( -3 - 6 \beta_{1} - 3 \beta_{5} + 3 \beta_{6} - 3 \beta_{7} + 3 \beta_{10} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 24q^{3} - 16q^{4} - 4q^{7} + 24q^{9} + O(q^{10}) \) \( 16q + 24q^{3} - 16q^{4} - 4q^{7} + 24q^{9} - 4q^{11} - 48q^{12} + 8q^{14} - 32q^{16} - 12q^{17} - 72q^{19} - 24q^{21} + 48q^{22} + 12q^{23} - 32q^{28} + 72q^{29} + 120q^{31} - 12q^{33} - 96q^{36} - 44q^{37} + 72q^{38} + 36q^{39} + 24q^{42} + 56q^{43} - 8q^{44} + 8q^{46} + 24q^{47} - 40q^{49} - 12q^{51} + 72q^{52} - 32q^{53} + 16q^{56} - 144q^{57} + 88q^{58} + 132q^{59} + 96q^{61} - 60q^{63} + 128q^{64} + 72q^{66} + 164q^{67} + 24q^{68} - 136q^{71} + 348q^{73} - 112q^{74} - 96q^{77} + 280q^{79} - 72q^{81} - 264q^{82} - 24q^{84} - 88q^{86} + 108q^{87} - 48q^{88} - 300q^{89} - 272q^{91} - 48q^{92} + 120q^{93} - 384q^{98} - 24q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} + 92 x^{14} - 112 x^{13} + 5846 x^{12} - 7728 x^{11} + 197216 x^{10} - 298200 x^{9} + 4836403 x^{8} - 6808704 x^{7} + 64376800 x^{6} - 91953512 x^{5} + 595763862 x^{4} - 630430976 x^{3} + 1087013404 x^{2} + 294123256 x + 101626561\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-57754008957283349 \nu^{15} - 676562680999836294 \nu^{14} - 8239103585403158159 \nu^{13} - 54078590776251682475 \nu^{12} - 455975368108340251668 \nu^{11} - 2793391292439344623383 \nu^{10} - 16545566601055515429559 \nu^{9} - 76261143494901216401580 \nu^{8} - 306572240997688184788012 \nu^{7} - 1467974756910134628134465 \nu^{6} - 3497231496721962844392765 \nu^{5} - 10591214719874231223793348 \nu^{4} + 4685579583089306764764583 \nu^{3} - 35874681437029057265441973 \nu^{2} - 9955887688266404429946106 \nu + 1202728311133779179554909109\)\()/ \)\(85\!\cdots\!24\)\( \)
\(\beta_{2}\)\(=\)\((\)\(\)\(46\!\cdots\!78\)\( \nu^{15} - \)\(46\!\cdots\!77\)\( \nu^{14} + \)\(51\!\cdots\!20\)\( \nu^{13} - \)\(49\!\cdots\!73\)\( \nu^{12} + \)\(39\!\cdots\!22\)\( \nu^{11} - \)\(33\!\cdots\!70\)\( \nu^{10} + \)\(17\!\cdots\!62\)\( \nu^{9} - \)\(12\!\cdots\!95\)\( \nu^{8} + \)\(51\!\cdots\!50\)\( \nu^{7} - \)\(30\!\cdots\!94\)\( \nu^{6} + \)\(10\!\cdots\!26\)\( \nu^{5} - \)\(43\!\cdots\!77\)\( \nu^{4} + \)\(12\!\cdots\!64\)\( \nu^{3} - \)\(39\!\cdots\!59\)\( \nu^{2} + \)\(90\!\cdots\!14\)\( \nu - \)\(36\!\cdots\!12\)\(\)\()/ \)\(95\!\cdots\!44\)\( \)
\(\beta_{3}\)\(=\)\((\)\(\)\(17\!\cdots\!52\)\( \nu^{15} - \)\(54\!\cdots\!68\)\( \nu^{14} + \)\(15\!\cdots\!20\)\( \nu^{13} - \)\(24\!\cdots\!26\)\( \nu^{12} + \)\(10\!\cdots\!84\)\( \nu^{11} - \)\(16\!\cdots\!08\)\( \nu^{10} + \)\(34\!\cdots\!20\)\( \nu^{9} - \)\(61\!\cdots\!59\)\( \nu^{8} + \)\(84\!\cdots\!08\)\( \nu^{7} - \)\(14\!\cdots\!20\)\( \nu^{6} + \)\(11\!\cdots\!24\)\( \nu^{5} - \)\(18\!\cdots\!30\)\( \nu^{4} + \)\(10\!\cdots\!24\)\( \nu^{3} - \)\(13\!\cdots\!92\)\( \nu^{2} + \)\(18\!\cdots\!72\)\( \nu - \)\(13\!\cdots\!85\)\(\)\()/ \)\(62\!\cdots\!01\)\( \)
\(\beta_{4}\)\(=\)\((\)\(\)\(10\!\cdots\!99\)\( \nu^{15} - \)\(49\!\cdots\!94\)\( \nu^{14} + \)\(95\!\cdots\!91\)\( \nu^{13} - \)\(53\!\cdots\!61\)\( \nu^{12} + \)\(66\!\cdots\!34\)\( \nu^{11} - \)\(34\!\cdots\!41\)\( \nu^{10} + \)\(24\!\cdots\!81\)\( \nu^{9} - \)\(11\!\cdots\!60\)\( \nu^{8} + \)\(66\!\cdots\!02\)\( \nu^{7} - \)\(25\!\cdots\!31\)\( \nu^{6} + \)\(10\!\cdots\!23\)\( \nu^{5} - \)\(30\!\cdots\!24\)\( \nu^{4} + \)\(11\!\cdots\!21\)\( \nu^{3} - \)\(24\!\cdots\!19\)\( \nu^{2} + \)\(45\!\cdots\!80\)\( \nu - \)\(10\!\cdots\!25\)\(\)\()/ \)\(31\!\cdots\!48\)\( \)
\(\beta_{5}\)\(=\)\((\)\(\)\(13\!\cdots\!02\)\( \nu^{15} - \)\(21\!\cdots\!67\)\( \nu^{14} + \)\(12\!\cdots\!27\)\( \nu^{13} - \)\(17\!\cdots\!98\)\( \nu^{12} + \)\(77\!\cdots\!07\)\( \nu^{11} - \)\(11\!\cdots\!47\)\( \nu^{10} + \)\(26\!\cdots\!76\)\( \nu^{9} - \)\(45\!\cdots\!39\)\( \nu^{8} + \)\(63\!\cdots\!65\)\( \nu^{7} - \)\(10\!\cdots\!73\)\( \nu^{6} + \)\(83\!\cdots\!68\)\( \nu^{5} - \)\(14\!\cdots\!09\)\( \nu^{4} + \)\(76\!\cdots\!85\)\( \nu^{3} - \)\(10\!\cdots\!02\)\( \nu^{2} + \)\(13\!\cdots\!99\)\( \nu - \)\(98\!\cdots\!05\)\(\)\()/ \)\(31\!\cdots\!48\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-\)\(61\!\cdots\!57\)\( \nu^{15} + \)\(28\!\cdots\!27\)\( \nu^{14} - \)\(60\!\cdots\!84\)\( \nu^{13} + \)\(32\!\cdots\!72\)\( \nu^{12} - \)\(42\!\cdots\!83\)\( \nu^{11} + \)\(21\!\cdots\!27\)\( \nu^{10} - \)\(16\!\cdots\!85\)\( \nu^{9} + \)\(72\!\cdots\!23\)\( \nu^{8} - \)\(45\!\cdots\!61\)\( \nu^{7} + \)\(17\!\cdots\!73\)\( \nu^{6} - \)\(75\!\cdots\!27\)\( \nu^{5} + \)\(22\!\cdots\!53\)\( \nu^{4} - \)\(79\!\cdots\!72\)\( \nu^{3} + \)\(19\!\cdots\!34\)\( \nu^{2} - \)\(38\!\cdots\!05\)\( \nu + \)\(12\!\cdots\!23\)\(\)\()/ \)\(13\!\cdots\!92\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-\)\(57\!\cdots\!33\)\( \nu^{15} + \)\(12\!\cdots\!58\)\( \nu^{14} - \)\(55\!\cdots\!71\)\( \nu^{13} + \)\(16\!\cdots\!15\)\( \nu^{12} - \)\(37\!\cdots\!48\)\( \nu^{11} + \)\(11\!\cdots\!81\)\( \nu^{10} - \)\(13\!\cdots\!35\)\( \nu^{9} + \)\(38\!\cdots\!58\)\( \nu^{8} - \)\(34\!\cdots\!12\)\( \nu^{7} + \)\(93\!\cdots\!91\)\( \nu^{6} - \)\(52\!\cdots\!53\)\( \nu^{5} + \)\(12\!\cdots\!70\)\( \nu^{4} - \)\(54\!\cdots\!81\)\( \nu^{3} + \)\(10\!\cdots\!23\)\( \nu^{2} - \)\(19\!\cdots\!82\)\( \nu + \)\(33\!\cdots\!81\)\(\)\()/ \)\(95\!\cdots\!44\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-\)\(59\!\cdots\!96\)\( \nu^{15} - \)\(25\!\cdots\!29\)\( \nu^{14} - \)\(53\!\cdots\!87\)\( \nu^{13} + \)\(53\!\cdots\!64\)\( \nu^{12} - \)\(33\!\cdots\!61\)\( \nu^{11} + \)\(38\!\cdots\!89\)\( \nu^{10} - \)\(10\!\cdots\!46\)\( \nu^{9} + \)\(16\!\cdots\!41\)\( \nu^{8} - \)\(25\!\cdots\!19\)\( \nu^{7} + \)\(35\!\cdots\!19\)\( \nu^{6} - \)\(32\!\cdots\!42\)\( \nu^{5} + \)\(48\!\cdots\!79\)\( \nu^{4} - \)\(29\!\cdots\!65\)\( \nu^{3} + \)\(20\!\cdots\!84\)\( \nu^{2} - \)\(34\!\cdots\!85\)\( \nu - \)\(15\!\cdots\!63\)\(\)\()/ \)\(95\!\cdots\!44\)\( \)
\(\beta_{9}\)\(=\)\((\)\(\)\(71\!\cdots\!98\)\( \nu^{15} + \)\(87\!\cdots\!62\)\( \nu^{14} + \)\(65\!\cdots\!21\)\( \nu^{13} - \)\(94\!\cdots\!62\)\( \nu^{12} + \)\(41\!\cdots\!29\)\( \nu^{11} - \)\(46\!\cdots\!36\)\( \nu^{10} + \)\(13\!\cdots\!56\)\( \nu^{9} - \)\(44\!\cdots\!94\)\( \nu^{8} + \)\(32\!\cdots\!99\)\( \nu^{7} - \)\(66\!\cdots\!68\)\( \nu^{6} + \)\(42\!\cdots\!40\)\( \nu^{5} - \)\(11\!\cdots\!38\)\( \nu^{4} + \)\(37\!\cdots\!11\)\( \nu^{3} + \)\(71\!\cdots\!82\)\( \nu^{2} + \)\(36\!\cdots\!21\)\( \nu + \)\(61\!\cdots\!20\)\(\)\()/ \)\(95\!\cdots\!44\)\( \)
\(\beta_{10}\)\(=\)\((\)\(\)\(17\!\cdots\!43\)\( \nu^{15} + \)\(18\!\cdots\!28\)\( \nu^{14} + \)\(16\!\cdots\!27\)\( \nu^{13} - \)\(39\!\cdots\!59\)\( \nu^{12} + \)\(10\!\cdots\!06\)\( \nu^{11} - \)\(39\!\cdots\!37\)\( \nu^{10} + \)\(33\!\cdots\!17\)\( \nu^{9} - \)\(22\!\cdots\!16\)\( \nu^{8} + \)\(81\!\cdots\!10\)\( \nu^{7} - \)\(49\!\cdots\!87\)\( \nu^{6} + \)\(10\!\cdots\!63\)\( \nu^{5} - \)\(78\!\cdots\!72\)\( \nu^{4} + \)\(92\!\cdots\!37\)\( \nu^{3} - \)\(38\!\cdots\!57\)\( \nu^{2} + \)\(12\!\cdots\!20\)\( \nu + \)\(82\!\cdots\!89\)\(\)\()/ \)\(95\!\cdots\!44\)\( \)
\(\beta_{11}\)\(=\)\((\)\(-\)\(15\!\cdots\!96\)\( \nu^{15} + \)\(38\!\cdots\!85\)\( \nu^{14} - \)\(13\!\cdots\!72\)\( \nu^{13} + \)\(20\!\cdots\!38\)\( \nu^{12} - \)\(88\!\cdots\!08\)\( \nu^{11} + \)\(14\!\cdots\!26\)\( \nu^{10} - \)\(29\!\cdots\!68\)\( \nu^{9} + \)\(53\!\cdots\!48\)\( \nu^{8} - \)\(73\!\cdots\!36\)\( \nu^{7} + \)\(12\!\cdots\!58\)\( \nu^{6} - \)\(99\!\cdots\!36\)\( \nu^{5} + \)\(16\!\cdots\!18\)\( \nu^{4} - \)\(93\!\cdots\!40\)\( \nu^{3} + \)\(12\!\cdots\!43\)\( \nu^{2} - \)\(16\!\cdots\!76\)\( \nu + \)\(68\!\cdots\!16\)\(\)\()/ \)\(72\!\cdots\!17\)\( \)
\(\beta_{12}\)\(=\)\((\)\(-\)\(15\!\cdots\!20\)\( \nu^{15} + \)\(67\!\cdots\!15\)\( \nu^{14} - \)\(14\!\cdots\!96\)\( \nu^{13} + \)\(23\!\cdots\!76\)\( \nu^{12} - \)\(92\!\cdots\!60\)\( \nu^{11} + \)\(15\!\cdots\!89\)\( \nu^{10} - \)\(31\!\cdots\!16\)\( \nu^{9} + \)\(59\!\cdots\!84\)\( \nu^{8} - \)\(77\!\cdots\!16\)\( \nu^{7} + \)\(13\!\cdots\!95\)\( \nu^{6} - \)\(10\!\cdots\!04\)\( \nu^{5} + \)\(17\!\cdots\!02\)\( \nu^{4} - \)\(94\!\cdots\!28\)\( \nu^{3} + \)\(12\!\cdots\!24\)\( \nu^{2} - \)\(16\!\cdots\!36\)\( \nu - \)\(10\!\cdots\!40\)\(\)\()/ \)\(72\!\cdots\!17\)\( \)
\(\beta_{13}\)\(=\)\((\)\(-\)\(37\!\cdots\!08\)\( \nu^{15} + \)\(23\!\cdots\!47\)\( \nu^{14} - \)\(34\!\cdots\!47\)\( \nu^{13} + \)\(63\!\cdots\!32\)\( \nu^{12} - \)\(22\!\cdots\!79\)\( \nu^{11} + \)\(42\!\cdots\!09\)\( \nu^{10} - \)\(74\!\cdots\!82\)\( \nu^{9} + \)\(15\!\cdots\!65\)\( \nu^{8} - \)\(18\!\cdots\!33\)\( \nu^{7} + \)\(35\!\cdots\!35\)\( \nu^{6} - \)\(24\!\cdots\!50\)\( \nu^{5} + \)\(45\!\cdots\!11\)\( \nu^{4} - \)\(22\!\cdots\!31\)\( \nu^{3} + \)\(30\!\cdots\!16\)\( \nu^{2} - \)\(38\!\cdots\!01\)\( \nu - \)\(23\!\cdots\!35\)\(\)\()/ \)\(12\!\cdots\!98\)\( \)
\(\beta_{14}\)\(=\)\((\)\(-\)\(38\!\cdots\!37\)\( \nu^{15} + \)\(24\!\cdots\!57\)\( \nu^{14} - \)\(35\!\cdots\!20\)\( \nu^{13} + \)\(47\!\cdots\!39\)\( \nu^{12} - \)\(22\!\cdots\!83\)\( \nu^{11} + \)\(32\!\cdots\!10\)\( \nu^{10} - \)\(75\!\cdots\!95\)\( \nu^{9} + \)\(12\!\cdots\!37\)\( \nu^{8} - \)\(18\!\cdots\!93\)\( \nu^{7} + \)\(28\!\cdots\!18\)\( \nu^{6} - \)\(24\!\cdots\!61\)\( \nu^{5} + \)\(41\!\cdots\!79\)\( \nu^{4} - \)\(22\!\cdots\!80\)\( \nu^{3} + \)\(29\!\cdots\!03\)\( \nu^{2} - \)\(39\!\cdots\!71\)\( \nu + \)\(16\!\cdots\!40\)\(\)\()/ \)\(12\!\cdots\!98\)\( \)
\(\beta_{15}\)\(=\)\((\)\(-\)\(37\!\cdots\!43\)\( \nu^{15} - \)\(47\!\cdots\!73\)\( \nu^{14} - \)\(34\!\cdots\!07\)\( \nu^{13} + \)\(37\!\cdots\!09\)\( \nu^{12} - \)\(21\!\cdots\!98\)\( \nu^{11} + \)\(26\!\cdots\!68\)\( \nu^{10} - \)\(74\!\cdots\!93\)\( \nu^{9} + \)\(10\!\cdots\!55\)\( \nu^{8} - \)\(18\!\cdots\!74\)\( \nu^{7} + \)\(23\!\cdots\!00\)\( \nu^{6} - \)\(24\!\cdots\!23\)\( \nu^{5} + \)\(32\!\cdots\!69\)\( \nu^{4} - \)\(23\!\cdots\!21\)\( \nu^{3} + \)\(22\!\cdots\!79\)\( \nu^{2} - \)\(51\!\cdots\!76\)\( \nu - \)\(64\!\cdots\!36\)\(\)\()/ \)\(95\!\cdots\!44\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-3 \beta_{15} - 7 \beta_{9} - \beta_{8} + 2 \beta_{7} + 2 \beta_{6} - 10 \beta_{5} + 2 \beta_{4} - 5 \beta_{3} + \beta_{1} - 1\)\()/14\)
\(\nu^{2}\)\(=\)\((\)\(-4 \beta_{15} - 14 \beta_{12} - 7 \beta_{11} - 7 \beta_{9} + \beta_{8} - 9 \beta_{7} + 5 \beta_{6} + 87 \beta_{5} - 2 \beta_{4} - 331 \beta_{3} - 7 \beta_{2} + 90 \beta_{1} - 328\)\()/14\)
\(\nu^{3}\)\(=\)\((\)\(8 \beta_{15} + 2 \beta_{14} - 2 \beta_{13} + \beta_{12} - \beta_{11} - 4 \beta_{10} + 31 \beta_{9} + 17 \beta_{8} + 13 \beta_{7} - \beta_{6} + 49 \beta_{5} + 3 \beta_{4} + 36 \beta_{3} + 33 \beta_{2} - 45 \beta_{1} + 59\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(142 \beta_{15} - 112 \beta_{14} - 56 \beta_{13} + 175 \beta_{12} + 350 \beta_{11} + 266 \beta_{10} - 56 \beta_{9} - 74 \beta_{8} - 34 \beta_{7} - 34 \beta_{6} - 2476 \beta_{5} + 50 \beta_{4} + 5783 \beta_{3} + 182 \beta_{2} + 74 \beta_{1} + 108\)\()/7\)
\(\nu^{5}\)\(=\)\((\)\(-1117 \beta_{15} + 1078 \beta_{14} + 2156 \beta_{13} - 1876 \beta_{12} - 938 \beta_{11} - 3276 \beta_{10} + 406 \beta_{9} - 4836 \beta_{8} - 7107 \beta_{7} - 443 \beta_{6} + 18469 \beta_{5} - 5413 \beta_{4} - 26959 \beta_{3} - 8127 \beta_{2} + 19046 \beta_{1} - 26382\)\()/14\)
\(\nu^{6}\)\(=\)\((\)\(-373 \beta_{15} + 1232 \beta_{14} - 1232 \beta_{13} + 2513 \beta_{12} - 2513 \beta_{11} - 2499 \beta_{10} + 3871 \beta_{9} + 2622 \beta_{8} + 5494 \beta_{7} - 1688 \beta_{6} + 8181 \beta_{5} + 811 \beta_{4} + 2687 \beta_{3} + 1876 \beta_{2} - 29397 \beta_{1} + 67276\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(28571 \beta_{15} - 133280 \beta_{14} - 66640 \beta_{13} + 72667 \beta_{12} + 145334 \beta_{11} + 288316 \beta_{10} - 352709 \beta_{9} - 19517 \beta_{8} + 46482 \beta_{7} + 46482 \beta_{6} - 1730928 \beta_{5} + 240228 \beta_{4} + 1298817 \beta_{3} - 7448 \beta_{2} + 19517 \beta_{1} - 26965\)\()/14\)
\(\nu^{8}\)\(=\)\((\)\(-315240 \beta_{15} + 262248 \beta_{14} + 524496 \beta_{13} - 900284 \beta_{12} - 450142 \beta_{11} - 520968 \beta_{10} - 254436 \beta_{9} - 478908 \beta_{8} - 1006524 \beta_{7} + 357300 \beta_{6} + 4889316 \beta_{5} - 585096 \beta_{4} - 10909879 \beta_{3} - 755412 \beta_{2} + 4995504 \beta_{1} - 10803691\)\()/7\)
\(\nu^{9}\)\(=\)\((\)\(211658 \beta_{15} + 547220 \beta_{14} - 547220 \beta_{13} + 668732 \beta_{12} - 668732 \beta_{11} - 738084 \beta_{10} + 2283949 \beta_{9} + 1846509 \beta_{8} + 2372935 \beta_{7} - 625193 \beta_{6} + 4755651 \beta_{5} + 112891 \beta_{4} + 2382716 \beta_{3} + 2269825 \beta_{2} - 8482089 \beta_{1} + 14101101\)\()/2\)
\(\nu^{10}\)\(=\)\((\)\(35876621 \beta_{15} - 59317664 \beta_{14} - 29658832 \beta_{13} + 46745419 \beta_{12} + 93490838 \beta_{11} + 105337351 \beta_{10} - 64929816 \beta_{9} - 8326683 \beta_{8} - 2265457 \beta_{7} - 2265457 \beta_{6} - 689743204 \beta_{5} + 61134047 \beta_{4} + 973723816 \beta_{3} + 18918823 \beta_{2} + 8326683 \beta_{1} + 10592140\)\()/14\)
\(\nu^{11}\)\(=\)\((\)\(-243360247 \beta_{15} + 213441354 \beta_{14} + 426882708 \beta_{13} - 555203334 \beta_{12} - 277601667 \beta_{11} - 572978994 \beta_{10} - 22383382 \beta_{9} - 623548862 \beta_{8} - 939862359 \beta_{7} + 192790379 \beta_{6} + 3158906229 \beta_{5} - 660025487 \beta_{4} - 5659756251 \beta_{3} - 783548605 \beta_{2} + 3195382854 \beta_{1} - 5623279626\)\()/14\)
\(\nu^{12}\)\(=\)\(-2844970 \beta_{15} + 116453456 \beta_{14} - 116453456 \beta_{13} + 175274631 \beta_{12} - 175274631 \beta_{11} - 153734462 \beta_{10} + 330765190 \beta_{9} + 283303276 \beta_{8} + 439882708 \beta_{7} - 128581020 \beta_{6} + 742649486 \beta_{5} + 25153442 \beta_{4} + 302766778 \beta_{3} + 277613336 \beta_{2} - 1924183986 \beta_{1} + 3692186209\)
\(\nu^{13}\)\(=\)\((\)\(11420160217 \beta_{15} - 23434733140 \beta_{14} - 11717366570 \beta_{13} + 15786097446 \beta_{12} + 31572194892 \beta_{11} + 45734611286 \beta_{10} - 38031705061 \beta_{9} - 1822501717 \beta_{8} + 1858626996 \beta_{7} + 1858626996 \beta_{6} - 259758006294 \beta_{5} + 32491949352 \beta_{4} + 277548890873 \beta_{3} + 1786376438 \beta_{2} + 1822501717 \beta_{1} - 36125279\)\()/14\)
\(\nu^{14}\)\(=\)\((\)\(-96789315536 \beta_{15} + 88554736480 \beta_{14} + 177109472960 \beta_{13} - 259719950854 \beta_{12} - 129859975427 \beta_{11} - 204691661160 \beta_{10} - 31472309085 \beta_{9} - 205388683541 \beta_{8} - 334347330543 \beta_{7} + 96092293155 \beta_{6} + 1382587776433 \beta_{5} - 221473349274 \beta_{4} - 2699608177345 \beta_{3} - 254339703121 \beta_{2} + 1398672442166 \beta_{1} - 2683523511612\)\()/14\)
\(\nu^{15}\)\(=\)\((\)\(10562294418 \beta_{15} + 91189454988 \beta_{14} - 91189454988 \beta_{13} + 125367881465 \beta_{12} - 125367881465 \beta_{11} - 116239363090 \beta_{10} + 295682469183 \beta_{9} + 257998257311 \beta_{8} + 363675325983 \beta_{7} - 102678404363 \beta_{6} + 656359130857 \beta_{5} + 13560958727 \beta_{4} + 292683804874 \beta_{3} + 279122846147 \beta_{2} - 1367996804997 \beta_{1} + 2491129308357\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(701\)
\(\chi(n)\) \(1\) \(1 + \beta_{3}\) \(1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
451.1
2.81422 4.87437i
−0.141814 + 0.245629i
−2.10711 + 3.64962i
0.848921 1.47037i
2.96377 5.13339i
−2.63284 + 4.56021i
−3.67087 + 6.35814i
1.92573 3.33546i
2.81422 + 4.87437i
−0.141814 0.245629i
−2.10711 3.64962i
0.848921 + 1.47037i
2.96377 + 5.13339i
−2.63284 4.56021i
−3.67087 6.35814i
1.92573 + 3.33546i
−0.707107 1.22474i 1.50000 + 0.866025i −1.00000 + 1.73205i 0 2.44949i −6.99242 + 0.325616i 2.82843 1.50000 + 2.59808i 0
451.2 −0.707107 1.22474i 1.50000 + 0.866025i −1.00000 + 1.73205i 0 2.44949i −4.24494 + 5.56601i 2.82843 1.50000 + 2.59808i 0
451.3 −0.707107 1.22474i 1.50000 + 0.866025i −1.00000 + 1.73205i 0 2.44949i 5.26304 + 4.61524i 2.82843 1.50000 + 2.59808i 0
451.4 −0.707107 1.22474i 1.50000 + 0.866025i −1.00000 + 1.73205i 0 2.44949i 6.38854 2.86123i 2.82843 1.50000 + 2.59808i 0
451.5 0.707107 + 1.22474i 1.50000 + 0.866025i −1.00000 + 1.73205i 0 2.44949i −5.73733 4.01037i −2.82843 1.50000 + 2.59808i 0
451.6 0.707107 + 1.22474i 1.50000 + 0.866025i −1.00000 + 1.73205i 0 2.44949i −1.94434 + 6.72455i −2.82843 1.50000 + 2.59808i 0
451.7 0.707107 + 1.22474i 1.50000 + 0.866025i −1.00000 + 1.73205i 0 2.44949i 2.59373 + 6.50174i −2.82843 1.50000 + 2.59808i 0
451.8 0.707107 + 1.22474i 1.50000 + 0.866025i −1.00000 + 1.73205i 0 2.44949i 2.67372 6.46925i −2.82843 1.50000 + 2.59808i 0
901.1 −0.707107 + 1.22474i 1.50000 0.866025i −1.00000 1.73205i 0 2.44949i −6.99242 0.325616i 2.82843 1.50000 2.59808i 0
901.2 −0.707107 + 1.22474i 1.50000 0.866025i −1.00000 1.73205i 0 2.44949i −4.24494 5.56601i 2.82843 1.50000 2.59808i 0
901.3 −0.707107 + 1.22474i 1.50000 0.866025i −1.00000 1.73205i 0 2.44949i 5.26304 4.61524i 2.82843 1.50000 2.59808i 0
901.4 −0.707107 + 1.22474i 1.50000 0.866025i −1.00000 1.73205i 0 2.44949i 6.38854 + 2.86123i 2.82843 1.50000 2.59808i 0
901.5 0.707107 1.22474i 1.50000 0.866025i −1.00000 1.73205i 0 2.44949i −5.73733 + 4.01037i −2.82843 1.50000 2.59808i 0
901.6 0.707107 1.22474i 1.50000 0.866025i −1.00000 1.73205i 0 2.44949i −1.94434 6.72455i −2.82843 1.50000 2.59808i 0
901.7 0.707107 1.22474i 1.50000 0.866025i −1.00000 1.73205i 0 2.44949i 2.59373 6.50174i −2.82843 1.50000 2.59808i 0
901.8 0.707107 1.22474i 1.50000 0.866025i −1.00000 1.73205i 0 2.44949i 2.67372 + 6.46925i −2.82843 1.50000 2.59808i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 901.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1050.3.p.i 16
5.b even 2 1 210.3.o.b 16
5.c odd 4 2 1050.3.q.e 32
7.d odd 6 1 inner 1050.3.p.i 16
15.d odd 2 1 630.3.v.c 16
35.i odd 6 1 210.3.o.b 16
35.i odd 6 1 1470.3.f.d 16
35.j even 6 1 1470.3.f.d 16
35.k even 12 2 1050.3.q.e 32
105.p even 6 1 630.3.v.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.3.o.b 16 5.b even 2 1
210.3.o.b 16 35.i odd 6 1
630.3.v.c 16 15.d odd 2 1
630.3.v.c 16 105.p even 6 1
1050.3.p.i 16 1.a even 1 1 trivial
1050.3.p.i 16 7.d odd 6 1 inner
1050.3.q.e 32 5.c odd 4 2
1050.3.q.e 32 35.k even 12 2
1470.3.f.d 16 35.i odd 6 1
1470.3.f.d 16 35.j even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1050, [\chi])\):

\(T_{11}^{16} + \cdots\)
\(T_{17}^{16} + \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + 2 T^{2} + 4 T^{4} )^{4} \)
$3$ \( ( 1 - 3 T + 3 T^{2} )^{8} \)
$5$ 1
$7$ \( 1 + 4 T + 28 T^{2} + 352 T^{3} + 2898 T^{4} + 16836 T^{5} + 70272 T^{6} + 1158948 T^{7} + 7794283 T^{8} + 56788452 T^{9} + 168723072 T^{10} + 1980738564 T^{11} + 16706393298 T^{12} + 99431287648 T^{13} + 387556041628 T^{14} + 2712892291396 T^{15} + 33232930569601 T^{16} \)
$11$ \( 1 + 4 T - 364 T^{2} - 2552 T^{3} + 54108 T^{4} + 655724 T^{5} - 3422504 T^{6} - 97911028 T^{7} - 146055574 T^{8} + 11147160032 T^{9} + 52178728220 T^{10} - 1271915177612 T^{11} - 12776124769680 T^{12} + 140823145942564 T^{13} + 3084355376442156 T^{14} - 7698059096283072 T^{15} - 472018303966569005 T^{16} - 931465150650251712 T^{17} + 45158047066489605996 T^{18} + \)\(24\!\cdots\!04\)\( T^{19} - \)\(27\!\cdots\!80\)\( T^{20} - \)\(32\!\cdots\!12\)\( T^{21} + \)\(16\!\cdots\!20\)\( T^{22} + \)\(42\!\cdots\!12\)\( T^{23} - \)\(67\!\cdots\!14\)\( T^{24} - \)\(54\!\cdots\!68\)\( T^{25} - \)\(23\!\cdots\!04\)\( T^{26} + \)\(53\!\cdots\!04\)\( T^{27} + \)\(53\!\cdots\!28\)\( T^{28} - \)\(30\!\cdots\!72\)\( T^{29} - \)\(52\!\cdots\!84\)\( T^{30} + \)\(69\!\cdots\!04\)\( T^{31} + \)\(21\!\cdots\!21\)\( T^{32} \)
$13$ \( 1 - 1336 T^{2} + 822548 T^{4} - 307870064 T^{6} + 77337463770 T^{8} - 13349067637096 T^{10} + 1484717408181296 T^{12} - 75519988778385192 T^{14} - 843289248891793565 T^{16} - \)\(21\!\cdots\!12\)\( T^{18} + \)\(12\!\cdots\!16\)\( T^{20} - \)\(31\!\cdots\!76\)\( T^{22} + \)\(51\!\cdots\!70\)\( T^{24} - \)\(58\!\cdots\!64\)\( T^{26} + \)\(44\!\cdots\!28\)\( T^{28} - \)\(20\!\cdots\!56\)\( T^{30} + \)\(44\!\cdots\!81\)\( T^{32} \)
$17$ \( 1 + 12 T + 420 T^{2} + 4464 T^{3} + 92044 T^{4} + 785292 T^{5} + 23989560 T^{6} - 227912772 T^{7} - 3236850998 T^{8} - 177940653384 T^{9} - 2004860299572 T^{10} - 43692613137348 T^{11} - 466147839356624 T^{12} - 8985915182672820 T^{13} - 29797959923559588 T^{14} + 769960022203743096 T^{15} + 34516652247342574675 T^{16} + \)\(22\!\cdots\!44\)\( T^{17} - \)\(24\!\cdots\!48\)\( T^{18} - \)\(21\!\cdots\!80\)\( T^{19} - \)\(32\!\cdots\!84\)\( T^{20} - \)\(88\!\cdots\!52\)\( T^{21} - \)\(11\!\cdots\!92\)\( T^{22} - \)\(29\!\cdots\!36\)\( T^{23} - \)\(15\!\cdots\!38\)\( T^{24} - \)\(32\!\cdots\!48\)\( T^{25} + \)\(97\!\cdots\!60\)\( T^{26} + \)\(92\!\cdots\!88\)\( T^{27} + \)\(31\!\cdots\!24\)\( T^{28} + \)\(43\!\cdots\!16\)\( T^{29} + \)\(11\!\cdots\!20\)\( T^{30} + \)\(98\!\cdots\!88\)\( T^{31} + \)\(23\!\cdots\!61\)\( T^{32} \)
$19$ \( 1 + 72 T + 3812 T^{2} + 150048 T^{3} + 4883342 T^{4} + 136887912 T^{5} + 3384543112 T^{6} + 75691203624 T^{7} + 1567163213049 T^{8} + 30524546336184 T^{9} + 578634766673960 T^{10} + 10979211701479464 T^{11} + 213751725988675694 T^{12} + 4327693265152902096 T^{13} + 88720456587214840044 T^{14} + \)\(17\!\cdots\!80\)\( T^{15} + \)\(34\!\cdots\!48\)\( T^{16} + \)\(64\!\cdots\!80\)\( T^{17} + \)\(11\!\cdots\!24\)\( T^{18} + \)\(20\!\cdots\!76\)\( T^{19} + \)\(36\!\cdots\!54\)\( T^{20} + \)\(67\!\cdots\!64\)\( T^{21} + \)\(12\!\cdots\!60\)\( T^{22} + \)\(24\!\cdots\!64\)\( T^{23} + \)\(45\!\cdots\!69\)\( T^{24} + \)\(78\!\cdots\!84\)\( T^{25} + \)\(12\!\cdots\!12\)\( T^{26} + \)\(18\!\cdots\!32\)\( T^{27} + \)\(23\!\cdots\!82\)\( T^{28} + \)\(26\!\cdots\!88\)\( T^{29} + \)\(24\!\cdots\!92\)\( T^{30} + \)\(16\!\cdots\!72\)\( T^{31} + \)\(83\!\cdots\!61\)\( T^{32} \)
$23$ \( 1 - 12 T - 1804 T^{2} + 41256 T^{3} + 1078172 T^{4} - 48109188 T^{5} - 226519400 T^{6} + 29579538300 T^{7} + 68149425194 T^{8} - 16885870838304 T^{9} + 1241401844668 T^{10} + 10296857200481412 T^{11} - 100210062266615184 T^{12} - 3750005340988049388 T^{13} + 80231845600155444876 T^{14} + \)\(51\!\cdots\!40\)\( T^{15} - \)\(38\!\cdots\!85\)\( T^{16} + \)\(27\!\cdots\!60\)\( T^{17} + \)\(22\!\cdots\!16\)\( T^{18} - \)\(55\!\cdots\!32\)\( T^{19} - \)\(78\!\cdots\!04\)\( T^{20} + \)\(42\!\cdots\!88\)\( T^{21} + \)\(27\!\cdots\!28\)\( T^{22} - \)\(19\!\cdots\!36\)\( T^{23} + \)\(41\!\cdots\!34\)\( T^{24} + \)\(95\!\cdots\!00\)\( T^{25} - \)\(38\!\cdots\!00\)\( T^{26} - \)\(43\!\cdots\!52\)\( T^{27} + \)\(51\!\cdots\!52\)\( T^{28} + \)\(10\!\cdots\!84\)\( T^{29} - \)\(24\!\cdots\!24\)\( T^{30} - \)\(85\!\cdots\!88\)\( T^{31} + \)\(37\!\cdots\!21\)\( T^{32} \)
$29$ \( ( 1 - 36 T + 1444 T^{2} + 19116 T^{3} - 1274860 T^{4} + 68800452 T^{5} + 84102156 T^{6} - 30035473452 T^{7} + 2426711187974 T^{8} - 25259833173132 T^{9} + 59483856997836 T^{10} + 40924113344941092 T^{11} - 637744142027460460 T^{12} + 8042239471766642316 T^{13} + \)\(51\!\cdots\!04\)\( T^{14} - \)\(10\!\cdots\!16\)\( T^{15} + \)\(25\!\cdots\!21\)\( T^{16} )^{2} \)
$31$ \( 1 - 120 T + 11588 T^{2} - 814560 T^{3} + 49123694 T^{4} - 2488742424 T^{5} + 113293093576 T^{6} - 4558808688792 T^{7} + 170270490089625 T^{8} - 5869377562140936 T^{9} + 196357797692622056 T^{10} - 6377368910762118360 T^{11} + \)\(21\!\cdots\!82\)\( T^{12} - \)\(69\!\cdots\!28\)\( T^{13} + \)\(23\!\cdots\!88\)\( T^{14} - \)\(74\!\cdots\!40\)\( T^{15} + \)\(23\!\cdots\!16\)\( T^{16} - \)\(71\!\cdots\!40\)\( T^{17} + \)\(21\!\cdots\!48\)\( T^{18} - \)\(61\!\cdots\!68\)\( T^{19} + \)\(17\!\cdots\!62\)\( T^{20} - \)\(52\!\cdots\!60\)\( T^{21} + \)\(15\!\cdots\!16\)\( T^{22} - \)\(44\!\cdots\!56\)\( T^{23} + \)\(12\!\cdots\!25\)\( T^{24} - \)\(31\!\cdots\!72\)\( T^{25} + \)\(76\!\cdots\!76\)\( T^{26} - \)\(16\!\cdots\!64\)\( T^{27} + \)\(30\!\cdots\!74\)\( T^{28} - \)\(48\!\cdots\!60\)\( T^{29} + \)\(66\!\cdots\!08\)\( T^{30} - \)\(66\!\cdots\!20\)\( T^{31} + \)\(52\!\cdots\!61\)\( T^{32} \)
$37$ \( 1 + 44 T - 1956 T^{2} + 11728 T^{3} + 3652806 T^{4} - 242885524 T^{5} - 1534355576 T^{6} + 317275895964 T^{7} - 9605253544351 T^{8} - 75954637404076 T^{9} + 6072970847630504 T^{10} - 248299820626414612 T^{11} + 4168440621293518230 T^{12} - \)\(29\!\cdots\!60\)\( T^{13} + \)\(10\!\cdots\!28\)\( T^{14} + \)\(69\!\cdots\!04\)\( T^{15} - \)\(27\!\cdots\!16\)\( T^{16} + \)\(95\!\cdots\!76\)\( T^{17} + \)\(20\!\cdots\!08\)\( T^{18} - \)\(75\!\cdots\!40\)\( T^{19} + \)\(14\!\cdots\!30\)\( T^{20} - \)\(11\!\cdots\!88\)\( T^{21} + \)\(39\!\cdots\!24\)\( T^{22} - \)\(68\!\cdots\!64\)\( T^{23} - \)\(11\!\cdots\!91\)\( T^{24} + \)\(53\!\cdots\!56\)\( T^{25} - \)\(35\!\cdots\!76\)\( T^{26} - \)\(76\!\cdots\!56\)\( T^{27} + \)\(15\!\cdots\!66\)\( T^{28} + \)\(69\!\cdots\!52\)\( T^{29} - \)\(15\!\cdots\!76\)\( T^{30} + \)\(48\!\cdots\!56\)\( T^{31} + \)\(15\!\cdots\!81\)\( T^{32} \)
$41$ \( 1 - 11416 T^{2} + 66218264 T^{4} - 265318066664 T^{6} + 835818107018940 T^{8} - 2208514761298495288 T^{10} + \)\(50\!\cdots\!56\)\( T^{12} - \)\(10\!\cdots\!32\)\( T^{14} + \)\(18\!\cdots\!14\)\( T^{16} - \)\(28\!\cdots\!52\)\( T^{18} + \)\(40\!\cdots\!76\)\( T^{20} - \)\(49\!\cdots\!28\)\( T^{22} + \)\(53\!\cdots\!40\)\( T^{24} - \)\(47\!\cdots\!64\)\( T^{26} + \)\(33\!\cdots\!04\)\( T^{28} - \)\(16\!\cdots\!36\)\( T^{30} + \)\(40\!\cdots\!81\)\( T^{32} \)
$43$ \( ( 1 - 28 T + 5732 T^{2} - 194288 T^{3} + 21755610 T^{4} - 646615948 T^{5} + 59916131504 T^{6} - 1611056221164 T^{7} + 122468758030675 T^{8} - 2978842952932236 T^{9} + 204841330302006704 T^{10} - 4087494160581305452 T^{11} + \)\(25\!\cdots\!10\)\( T^{12} - \)\(41\!\cdots\!12\)\( T^{13} + \)\(22\!\cdots\!32\)\( T^{14} - \)\(20\!\cdots\!72\)\( T^{15} + \)\(13\!\cdots\!01\)\( T^{16} )^{2} \)
$47$ \( 1 - 24 T + 10040 T^{2} - 236352 T^{3} + 48191588 T^{4} - 529473000 T^{5} + 142018937488 T^{6} + 1533668366952 T^{7} + 289429105887690 T^{8} + 11114511768111792 T^{9} + 616929826339364168 T^{10} + 28449623988679606104 T^{11} + \)\(19\!\cdots\!48\)\( T^{12} + \)\(39\!\cdots\!52\)\( T^{13} + \)\(60\!\cdots\!84\)\( T^{14} + \)\(28\!\cdots\!36\)\( T^{15} + \)\(14\!\cdots\!27\)\( T^{16} + \)\(62\!\cdots\!24\)\( T^{17} + \)\(29\!\cdots\!04\)\( T^{18} + \)\(42\!\cdots\!08\)\( T^{19} + \)\(46\!\cdots\!28\)\( T^{20} + \)\(14\!\cdots\!96\)\( T^{21} + \)\(71\!\cdots\!88\)\( T^{22} + \)\(28\!\cdots\!48\)\( T^{23} + \)\(16\!\cdots\!90\)\( T^{24} + \)\(19\!\cdots\!28\)\( T^{25} + \)\(39\!\cdots\!88\)\( T^{26} - \)\(32\!\cdots\!00\)\( T^{27} + \)\(65\!\cdots\!28\)\( T^{28} - \)\(70\!\cdots\!08\)\( T^{29} + \)\(66\!\cdots\!40\)\( T^{30} - \)\(34\!\cdots\!76\)\( T^{31} + \)\(32\!\cdots\!41\)\( T^{32} \)
$53$ \( 1 + 32 T - 12072 T^{2} - 345280 T^{3} + 71529764 T^{4} + 1574568864 T^{5} - 290206976304 T^{6} - 2770742930720 T^{7} + 1006589576083018 T^{8} - 3889510628997504 T^{9} - 3360408915215254232 T^{10} + 25194864185334307040 T^{11} + \)\(10\!\cdots\!32\)\( T^{12} - \)\(34\!\cdots\!88\)\( T^{13} - \)\(28\!\cdots\!00\)\( T^{14} + \)\(63\!\cdots\!04\)\( T^{15} + \)\(75\!\cdots\!55\)\( T^{16} + \)\(17\!\cdots\!36\)\( T^{17} - \)\(22\!\cdots\!00\)\( T^{18} - \)\(75\!\cdots\!52\)\( T^{19} + \)\(64\!\cdots\!52\)\( T^{20} + \)\(44\!\cdots\!60\)\( T^{21} - \)\(16\!\cdots\!12\)\( T^{22} - \)\(53\!\cdots\!76\)\( T^{23} + \)\(39\!\cdots\!78\)\( T^{24} - \)\(30\!\cdots\!80\)\( T^{25} - \)\(88\!\cdots\!04\)\( T^{26} + \)\(13\!\cdots\!76\)\( T^{27} + \)\(17\!\cdots\!84\)\( T^{28} - \)\(23\!\cdots\!20\)\( T^{29} - \)\(22\!\cdots\!92\)\( T^{30} + \)\(17\!\cdots\!68\)\( T^{31} + \)\(15\!\cdots\!41\)\( T^{32} \)
$59$ \( 1 - 132 T + 9660 T^{2} - 508464 T^{3} + 17070268 T^{4} - 1368116580 T^{5} + 85370423112 T^{6} - 5735724829716 T^{7} + 274189008275050 T^{8} - 2776853544451464 T^{9} + 245103035882958612 T^{10} + 4667418467579246988 T^{11} + \)\(23\!\cdots\!68\)\( T^{12} - \)\(15\!\cdots\!44\)\( T^{13} + \)\(45\!\cdots\!16\)\( T^{14} - \)\(18\!\cdots\!68\)\( T^{15} - \)\(23\!\cdots\!33\)\( T^{16} - \)\(62\!\cdots\!08\)\( T^{17} + \)\(55\!\cdots\!76\)\( T^{18} - \)\(65\!\cdots\!04\)\( T^{19} + \)\(34\!\cdots\!28\)\( T^{20} + \)\(23\!\cdots\!88\)\( T^{21} + \)\(43\!\cdots\!72\)\( T^{22} - \)\(17\!\cdots\!04\)\( T^{23} + \)\(59\!\cdots\!50\)\( T^{24} - \)\(43\!\cdots\!36\)\( T^{25} + \)\(22\!\cdots\!12\)\( T^{26} - \)\(12\!\cdots\!80\)\( T^{27} + \)\(54\!\cdots\!48\)\( T^{28} - \)\(56\!\cdots\!24\)\( T^{29} + \)\(37\!\cdots\!60\)\( T^{30} - \)\(17\!\cdots\!32\)\( T^{31} + \)\(46\!\cdots\!81\)\( T^{32} \)
$61$ \( 1 - 96 T + 17624 T^{2} - 1396992 T^{3} + 159354980 T^{4} - 12864972960 T^{5} + 1094820209872 T^{6} - 90213896412000 T^{7} + 6336184490820426 T^{8} - 516522392725304640 T^{9} + 33635567216910203432 T^{10} - \)\(25\!\cdots\!96\)\( T^{11} + \)\(16\!\cdots\!12\)\( T^{12} - \)\(10\!\cdots\!72\)\( T^{13} + \)\(72\!\cdots\!12\)\( T^{14} - \)\(43\!\cdots\!24\)\( T^{15} + \)\(28\!\cdots\!43\)\( T^{16} - \)\(16\!\cdots\!04\)\( T^{17} + \)\(10\!\cdots\!92\)\( T^{18} - \)\(56\!\cdots\!92\)\( T^{19} + \)\(31\!\cdots\!72\)\( T^{20} - \)\(18\!\cdots\!96\)\( T^{21} + \)\(89\!\cdots\!72\)\( T^{22} - \)\(51\!\cdots\!40\)\( T^{23} + \)\(23\!\cdots\!86\)\( T^{24} - \)\(12\!\cdots\!00\)\( T^{25} + \)\(55\!\cdots\!72\)\( T^{26} - \)\(24\!\cdots\!60\)\( T^{27} + \)\(11\!\cdots\!80\)\( T^{28} - \)\(36\!\cdots\!12\)\( T^{29} + \)\(17\!\cdots\!44\)\( T^{30} - \)\(34\!\cdots\!96\)\( T^{31} + \)\(13\!\cdots\!21\)\( T^{32} \)
$67$ \( 1 - 164 T - 6340 T^{2} + 2597968 T^{3} - 31915482 T^{4} - 22697073892 T^{5} + 967634972296 T^{6} + 123392891205548 T^{7} - 9455087268413407 T^{8} - 333882071018564764 T^{9} + 51639925158822741896 T^{10} - \)\(26\!\cdots\!32\)\( T^{11} - \)\(16\!\cdots\!86\)\( T^{12} + \)\(60\!\cdots\!36\)\( T^{13} + \)\(18\!\cdots\!60\)\( T^{14} - \)\(15\!\cdots\!28\)\( T^{15} + \)\(54\!\cdots\!88\)\( T^{16} - \)\(70\!\cdots\!92\)\( T^{17} + \)\(36\!\cdots\!60\)\( T^{18} + \)\(55\!\cdots\!84\)\( T^{19} - \)\(67\!\cdots\!26\)\( T^{20} - \)\(47\!\cdots\!68\)\( T^{21} + \)\(42\!\cdots\!56\)\( T^{22} - \)\(12\!\cdots\!56\)\( T^{23} - \)\(15\!\cdots\!67\)\( T^{24} + \)\(91\!\cdots\!32\)\( T^{25} + \)\(32\!\cdots\!96\)\( T^{26} - \)\(33\!\cdots\!88\)\( T^{27} - \)\(21\!\cdots\!22\)\( T^{28} + \)\(78\!\cdots\!92\)\( T^{29} - \)\(85\!\cdots\!40\)\( T^{30} - \)\(99\!\cdots\!36\)\( T^{31} + \)\(27\!\cdots\!61\)\( T^{32} \)
$71$ \( ( 1 + 68 T + 26092 T^{2} + 2195028 T^{3} + 351820772 T^{4} + 28773766364 T^{5} + 3188424927972 T^{6} + 217589962697708 T^{7} + 19655929349959526 T^{8} + 1096871001959146028 T^{9} + 81023237162072440932 T^{10} + \)\(36\!\cdots\!44\)\( T^{11} + \)\(22\!\cdots\!92\)\( T^{12} + \)\(71\!\cdots\!28\)\( T^{13} + \)\(42\!\cdots\!72\)\( T^{14} + \)\(56\!\cdots\!08\)\( T^{15} + \)\(41\!\cdots\!21\)\( T^{16} )^{2} \)
$73$ \( 1 - 348 T + 82708 T^{2} - 14734320 T^{3} + 2214323678 T^{4} - 290251205388 T^{5} + 34312832193656 T^{6} - 3733330194438684 T^{7} + 379285162573239545 T^{8} - 36467064302261366820 T^{9} + \)\(33\!\cdots\!56\)\( T^{10} - \)\(29\!\cdots\!16\)\( T^{11} + \)\(24\!\cdots\!70\)\( T^{12} - \)\(20\!\cdots\!16\)\( T^{13} + \)\(16\!\cdots\!72\)\( T^{14} - \)\(12\!\cdots\!12\)\( T^{15} + \)\(91\!\cdots\!12\)\( T^{16} - \)\(65\!\cdots\!48\)\( T^{17} + \)\(45\!\cdots\!52\)\( T^{18} - \)\(30\!\cdots\!24\)\( T^{19} + \)\(20\!\cdots\!70\)\( T^{20} - \)\(12\!\cdots\!84\)\( T^{21} + \)\(76\!\cdots\!76\)\( T^{22} - \)\(44\!\cdots\!80\)\( T^{23} + \)\(24\!\cdots\!45\)\( T^{24} - \)\(12\!\cdots\!96\)\( T^{25} + \)\(63\!\cdots\!56\)\( T^{26} - \)\(28\!\cdots\!52\)\( T^{27} + \)\(11\!\cdots\!98\)\( T^{28} - \)\(41\!\cdots\!80\)\( T^{29} + \)\(12\!\cdots\!48\)\( T^{30} - \)\(27\!\cdots\!52\)\( T^{31} + \)\(42\!\cdots\!21\)\( T^{32} \)
$79$ \( 1 - 280 T + 14532 T^{2} + 1622944 T^{3} + 35326638 T^{4} - 30655524856 T^{5} + 55399514824 T^{6} + 161497193433480 T^{7} + 10355496367899353 T^{8} - 1123575678148193512 T^{9} - 61265347082641448728 T^{10} + \)\(14\!\cdots\!40\)\( T^{11} + \)\(44\!\cdots\!62\)\( T^{12} - \)\(73\!\cdots\!60\)\( T^{13} + \)\(53\!\cdots\!92\)\( T^{14} - \)\(10\!\cdots\!56\)\( T^{15} + \)\(35\!\cdots\!48\)\( T^{16} - \)\(66\!\cdots\!96\)\( T^{17} + \)\(20\!\cdots\!52\)\( T^{18} - \)\(17\!\cdots\!60\)\( T^{19} + \)\(67\!\cdots\!82\)\( T^{20} + \)\(13\!\cdots\!40\)\( T^{21} - \)\(36\!\cdots\!48\)\( T^{22} - \)\(41\!\cdots\!72\)\( T^{23} + \)\(23\!\cdots\!13\)\( T^{24} + \)\(23\!\cdots\!80\)\( T^{25} + \)\(49\!\cdots\!24\)\( T^{26} - \)\(17\!\cdots\!96\)\( T^{27} + \)\(12\!\cdots\!78\)\( T^{28} + \)\(35\!\cdots\!24\)\( T^{29} + \)\(19\!\cdots\!52\)\( T^{30} - \)\(23\!\cdots\!80\)\( T^{31} + \)\(52\!\cdots\!41\)\( T^{32} \)
$83$ \( 1 - 66184 T^{2} + 2113671608 T^{4} - 43986302935160 T^{6} + 678322851950602236 T^{8} - \)\(83\!\cdots\!72\)\( T^{10} + \)\(84\!\cdots\!60\)\( T^{12} - \)\(72\!\cdots\!04\)\( T^{14} + \)\(53\!\cdots\!06\)\( T^{16} - \)\(34\!\cdots\!84\)\( T^{18} + \)\(19\!\cdots\!60\)\( T^{20} - \)\(88\!\cdots\!92\)\( T^{22} + \)\(34\!\cdots\!16\)\( T^{24} - \)\(10\!\cdots\!60\)\( T^{26} + \)\(24\!\cdots\!68\)\( T^{28} - \)\(35\!\cdots\!44\)\( T^{30} + \)\(25\!\cdots\!61\)\( T^{32} \)
$89$ \( 1 + 300 T + 54668 T^{2} + 7400400 T^{3} + 757620572 T^{4} + 53632718700 T^{5} + 870152589352 T^{6} - 480156239482500 T^{7} - 98736006237054870 T^{8} - 12632355432189921000 T^{9} - \)\(12\!\cdots\!92\)\( T^{10} - \)\(89\!\cdots\!00\)\( T^{11} - \)\(37\!\cdots\!44\)\( T^{12} + \)\(17\!\cdots\!40\)\( T^{13} + \)\(60\!\cdots\!12\)\( T^{14} + \)\(83\!\cdots\!20\)\( T^{15} + \)\(83\!\cdots\!23\)\( T^{16} + \)\(66\!\cdots\!20\)\( T^{17} + \)\(37\!\cdots\!92\)\( T^{18} + \)\(84\!\cdots\!40\)\( T^{19} - \)\(14\!\cdots\!64\)\( T^{20} - \)\(27\!\cdots\!00\)\( T^{21} - \)\(30\!\cdots\!32\)\( T^{22} - \)\(24\!\cdots\!00\)\( T^{23} - \)\(15\!\cdots\!70\)\( T^{24} - \)\(58\!\cdots\!00\)\( T^{25} + \)\(84\!\cdots\!52\)\( T^{26} + \)\(41\!\cdots\!00\)\( T^{27} + \)\(46\!\cdots\!52\)\( T^{28} + \)\(35\!\cdots\!00\)\( T^{29} + \)\(20\!\cdots\!08\)\( T^{30} + \)\(90\!\cdots\!00\)\( T^{31} + \)\(24\!\cdots\!21\)\( T^{32} \)
$97$ \( 1 - 70448 T^{2} + 2604307832 T^{4} - 66719407643536 T^{6} + 1323480529449253148 T^{8} - \)\(21\!\cdots\!72\)\( T^{10} + \)\(29\!\cdots\!68\)\( T^{12} - \)\(34\!\cdots\!84\)\( T^{14} + \)\(34\!\cdots\!38\)\( T^{16} - \)\(30\!\cdots\!04\)\( T^{18} + \)\(23\!\cdots\!48\)\( T^{20} - \)\(14\!\cdots\!52\)\( T^{22} + \)\(81\!\cdots\!08\)\( T^{24} - \)\(36\!\cdots\!36\)\( T^{26} + \)\(12\!\cdots\!92\)\( T^{28} - \)\(30\!\cdots\!28\)\( T^{30} + \)\(37\!\cdots\!41\)\( T^{32} \)
show more
show less