Properties

Label 177.12.a.d.1.10
Level $177$
Weight $12$
Character 177.1
Self dual yes
Analytic conductor $135.997$
Analytic rank $0$
Dimension $28$
CM no
Inner twists $1$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 177.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(135.996742959\)
Analytic rank: \(0\)
Dimension: \(28\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.10
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-33.4729 q^{2} +243.000 q^{3} -927.563 q^{4} -11827.5 q^{5} -8133.92 q^{6} +55304.7 q^{7} +99600.8 q^{8} +59049.0 q^{9} +O(q^{10})\) \(q-33.4729 q^{2} +243.000 q^{3} -927.563 q^{4} -11827.5 q^{5} -8133.92 q^{6} +55304.7 q^{7} +99600.8 q^{8} +59049.0 q^{9} +395902. q^{10} +310193. q^{11} -225398. q^{12} +635071. q^{13} -1.85121e6 q^{14} -2.87409e6 q^{15} -1.43428e6 q^{16} +4.94920e6 q^{17} -1.97654e6 q^{18} +7.38988e6 q^{19} +1.09708e7 q^{20} +1.34390e7 q^{21} -1.03831e7 q^{22} +5.49557e7 q^{23} +2.42030e7 q^{24} +9.10623e7 q^{25} -2.12577e7 q^{26} +1.43489e7 q^{27} -5.12986e7 q^{28} -1.08586e8 q^{29} +9.62042e7 q^{30} -1.74859e8 q^{31} -1.55973e8 q^{32} +7.53769e7 q^{33} -1.65664e8 q^{34} -6.54118e8 q^{35} -5.47716e7 q^{36} +2.79905e8 q^{37} -2.47361e8 q^{38} +1.54322e8 q^{39} -1.17803e9 q^{40} +2.62906e8 q^{41} -4.49844e8 q^{42} -6.62017e8 q^{43} -2.87723e8 q^{44} -6.98404e8 q^{45} -1.83953e9 q^{46} +2.01581e9 q^{47} -3.48531e8 q^{48} +1.08128e9 q^{49} -3.04812e9 q^{50} +1.20266e9 q^{51} -5.89068e8 q^{52} +1.32226e9 q^{53} -4.80300e8 q^{54} -3.66881e9 q^{55} +5.50839e9 q^{56} +1.79574e9 q^{57} +3.63469e9 q^{58} +7.14924e8 q^{59} +2.66590e9 q^{60} -1.28279e10 q^{61} +5.85305e9 q^{62} +3.26569e9 q^{63} +8.15828e9 q^{64} -7.51131e9 q^{65} -2.52309e9 q^{66} -3.98567e9 q^{67} -4.59070e9 q^{68} +1.33542e10 q^{69} +2.18952e10 q^{70} +4.51133e9 q^{71} +5.88133e9 q^{72} -1.35618e10 q^{73} -9.36925e9 q^{74} +2.21281e10 q^{75} -6.85458e9 q^{76} +1.71551e10 q^{77} -5.16561e9 q^{78} +3.46284e10 q^{79} +1.69640e10 q^{80} +3.48678e9 q^{81} -8.80025e9 q^{82} +3.01060e10 q^{83} -1.24655e10 q^{84} -5.85369e10 q^{85} +2.21597e10 q^{86} -2.63864e10 q^{87} +3.08955e10 q^{88} +2.22704e10 q^{89} +2.33776e10 q^{90} +3.51224e10 q^{91} -5.09749e10 q^{92} -4.24908e10 q^{93} -6.74752e10 q^{94} -8.74041e10 q^{95} -3.79014e10 q^{96} -8.94449e10 q^{97} -3.61936e10 q^{98} +1.83166e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 28q + 96q^{2} + 6804q^{3} + 29214q^{4} + 26562q^{5} + 23328q^{6} + 142333q^{7} + 332331q^{8} + 1653372q^{9} + O(q^{10}) \) \( 28q + 96q^{2} + 6804q^{3} + 29214q^{4} + 26562q^{5} + 23328q^{6} + 142333q^{7} + 332331q^{8} + 1653372q^{9} + 616281q^{10} + 1082362q^{11} + 7099002q^{12} + 503712q^{13} + 1321669q^{14} + 6454566q^{15} + 34870338q^{16} + 13513579q^{17} + 5668704q^{18} + 35971687q^{19} + 96105997q^{20} + 34586919q^{21} - 47598882q^{22} + 61380539q^{23} + 80756433q^{24} + 294744746q^{25} + 62820734q^{26} + 401769396q^{27} + 148068294q^{28} + 322339307q^{29} + 149756283q^{30} + 151247077q^{31} + 466383494q^{32} + 263013966q^{33} + 684479860q^{34} + 960297361q^{35} + 1725057486q^{36} + 863508437q^{37} + 992640509q^{38} + 122402016q^{39} + 3067680252q^{40} + 3081170377q^{41} + 321165567q^{42} + 2554238300q^{43} + 4350123570q^{44} + 1568459538q^{45} - 1987059155q^{46} + 6203398333q^{47} + 8473492134q^{48} + 10327857997q^{49} + 17577682253q^{50} + 3283799697q^{51} + 32137181618q^{52} + 14571770754q^{53} + 1377495072q^{54} + 18251419334q^{55} + 33498842836q^{56} + 8741119941q^{57} + 11860778276q^{58} + 20017880372q^{59} + 23353757271q^{60} + 2761613771q^{61} + 13785829526q^{62} + 8404621317q^{63} + 86547545293q^{64} + 32034985256q^{65} - 11566528326q^{66} + 39381333296q^{67} + 38995496621q^{68} + 14915470977q^{69} + 8551800364q^{70} + 26130020296q^{71} + 19623813219q^{72} + 41382402799q^{73} + 23815315058q^{74} + 71622973278q^{75} + 10611720128q^{76} - 8426124313q^{77} + 15265438362q^{78} + 59825111206q^{79} + 4009687655q^{80} + 97629963228q^{81} - 39592715115q^{82} + 35433122727q^{83} + 35980595442q^{84} - 8950496085q^{85} - 182032360688q^{86} + 78328451601q^{87} - 220003602335q^{88} + 102303043039q^{89} + 36390776769q^{90} - 111146323655q^{91} - 163000203526q^{92} + 36753039711q^{93} - 81314346008q^{94} + 208102168887q^{95} + 113331189042q^{96} - 171891031490q^{97} + 72304707792q^{98} + 63912393738q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −33.4729 −0.739654 −0.369827 0.929101i \(-0.620583\pi\)
−0.369827 + 0.929101i \(0.620583\pi\)
\(3\) 243.000 0.577350
\(4\) −927.563 −0.452911
\(5\) −11827.5 −1.69262 −0.846309 0.532692i \(-0.821180\pi\)
−0.846309 + 0.532692i \(0.821180\pi\)
\(6\) −8133.92 −0.427040
\(7\) 55304.7 1.24372 0.621860 0.783128i \(-0.286377\pi\)
0.621860 + 0.783128i \(0.286377\pi\)
\(8\) 99600.8 1.07465
\(9\) 59049.0 0.333333
\(10\) 395902. 1.25195
\(11\) 310193. 0.580727 0.290364 0.956916i \(-0.406224\pi\)
0.290364 + 0.956916i \(0.406224\pi\)
\(12\) −225398. −0.261489
\(13\) 635071. 0.474388 0.237194 0.971462i \(-0.423772\pi\)
0.237194 + 0.971462i \(0.423772\pi\)
\(14\) −1.85121e6 −0.919923
\(15\) −2.87409e6 −0.977233
\(16\) −1.43428e6 −0.341960
\(17\) 4.94920e6 0.845408 0.422704 0.906268i \(-0.361081\pi\)
0.422704 + 0.906268i \(0.361081\pi\)
\(18\) −1.97654e6 −0.246551
\(19\) 7.38988e6 0.684688 0.342344 0.939575i \(-0.388779\pi\)
0.342344 + 0.939575i \(0.388779\pi\)
\(20\) 1.09708e7 0.766606
\(21\) 1.34390e7 0.718062
\(22\) −1.03831e7 −0.429537
\(23\) 5.49557e7 1.78037 0.890184 0.455602i \(-0.150576\pi\)
0.890184 + 0.455602i \(0.150576\pi\)
\(24\) 2.42030e7 0.620451
\(25\) 9.10623e7 1.86496
\(26\) −2.12577e7 −0.350883
\(27\) 1.43489e7 0.192450
\(28\) −5.12986e7 −0.563295
\(29\) −1.08586e8 −0.983071 −0.491536 0.870857i \(-0.663564\pi\)
−0.491536 + 0.870857i \(0.663564\pi\)
\(30\) 9.62042e7 0.722815
\(31\) −1.74859e8 −1.09698 −0.548491 0.836157i \(-0.684797\pi\)
−0.548491 + 0.836157i \(0.684797\pi\)
\(32\) −1.55973e8 −0.821720
\(33\) 7.53769e7 0.335283
\(34\) −1.65664e8 −0.625310
\(35\) −6.54118e8 −2.10514
\(36\) −5.47716e7 −0.150970
\(37\) 2.79905e8 0.663593 0.331796 0.943351i \(-0.392345\pi\)
0.331796 + 0.943351i \(0.392345\pi\)
\(38\) −2.47361e8 −0.506433
\(39\) 1.54322e8 0.273888
\(40\) −1.17803e9 −1.81898
\(41\) 2.62906e8 0.354397 0.177198 0.984175i \(-0.443297\pi\)
0.177198 + 0.984175i \(0.443297\pi\)
\(42\) −4.49844e8 −0.531118
\(43\) −6.62017e8 −0.686741 −0.343370 0.939200i \(-0.611569\pi\)
−0.343370 + 0.939200i \(0.611569\pi\)
\(44\) −2.87723e8 −0.263018
\(45\) −6.98404e8 −0.564206
\(46\) −1.83953e9 −1.31686
\(47\) 2.01581e9 1.28207 0.641035 0.767511i \(-0.278505\pi\)
0.641035 + 0.767511i \(0.278505\pi\)
\(48\) −3.48531e8 −0.197431
\(49\) 1.08128e9 0.546840
\(50\) −3.04812e9 −1.37942
\(51\) 1.20266e9 0.488097
\(52\) −5.89068e8 −0.214856
\(53\) 1.32226e9 0.434310 0.217155 0.976137i \(-0.430322\pi\)
0.217155 + 0.976137i \(0.430322\pi\)
\(54\) −4.80300e8 −0.142347
\(55\) −3.66881e9 −0.982949
\(56\) 5.50839e9 1.33657
\(57\) 1.79574e9 0.395305
\(58\) 3.63469e9 0.727133
\(59\) 7.14924e8 0.130189
\(60\) 2.66590e9 0.442600
\(61\) −1.28279e10 −1.94465 −0.972323 0.233642i \(-0.924936\pi\)
−0.972323 + 0.233642i \(0.924936\pi\)
\(62\) 5.85305e9 0.811387
\(63\) 3.26569e9 0.414573
\(64\) 8.15828e9 0.949749
\(65\) −7.51131e9 −0.802957
\(66\) −2.52309e9 −0.247993
\(67\) −3.98567e9 −0.360654 −0.180327 0.983607i \(-0.557716\pi\)
−0.180327 + 0.983607i \(0.557716\pi\)
\(68\) −4.59070e9 −0.382895
\(69\) 1.33542e10 1.02790
\(70\) 2.18952e10 1.55708
\(71\) 4.51133e9 0.296745 0.148373 0.988932i \(-0.452596\pi\)
0.148373 + 0.988932i \(0.452596\pi\)
\(72\) 5.88133e9 0.358217
\(73\) −1.35618e10 −0.765668 −0.382834 0.923817i \(-0.625052\pi\)
−0.382834 + 0.923817i \(0.625052\pi\)
\(74\) −9.36925e9 −0.490829
\(75\) 2.21281e10 1.07673
\(76\) −6.85458e9 −0.310103
\(77\) 1.71551e10 0.722262
\(78\) −5.16561e9 −0.202582
\(79\) 3.46284e10 1.26614 0.633072 0.774093i \(-0.281794\pi\)
0.633072 + 0.774093i \(0.281794\pi\)
\(80\) 1.69640e10 0.578807
\(81\) 3.48678e9 0.111111
\(82\) −8.80025e9 −0.262131
\(83\) 3.01060e10 0.838927 0.419464 0.907772i \(-0.362218\pi\)
0.419464 + 0.907772i \(0.362218\pi\)
\(84\) −1.24655e10 −0.325219
\(85\) −5.85369e10 −1.43095
\(86\) 2.21597e10 0.507951
\(87\) −2.63864e10 −0.567576
\(88\) 3.08955e10 0.624080
\(89\) 2.22704e10 0.422750 0.211375 0.977405i \(-0.432206\pi\)
0.211375 + 0.977405i \(0.432206\pi\)
\(90\) 2.33776e10 0.417317
\(91\) 3.51224e10 0.590006
\(92\) −5.09749e10 −0.806349
\(93\) −4.24908e10 −0.633343
\(94\) −6.74752e10 −0.948289
\(95\) −8.74041e10 −1.15892
\(96\) −3.79014e10 −0.474420
\(97\) −8.94449e10 −1.05758 −0.528788 0.848754i \(-0.677353\pi\)
−0.528788 + 0.848754i \(0.677353\pi\)
\(98\) −3.61936e10 −0.404472
\(99\) 1.83166e10 0.193576
\(100\) −8.44660e10 −0.844660
\(101\) −4.57588e10 −0.433219 −0.216609 0.976258i \(-0.569500\pi\)
−0.216609 + 0.976258i \(0.569500\pi\)
\(102\) −4.02565e10 −0.361023
\(103\) 2.53716e10 0.215647 0.107823 0.994170i \(-0.465612\pi\)
0.107823 + 0.994170i \(0.465612\pi\)
\(104\) 6.32535e10 0.509802
\(105\) −1.58951e11 −1.21540
\(106\) −4.42599e10 −0.321239
\(107\) 5.58310e10 0.384826 0.192413 0.981314i \(-0.438369\pi\)
0.192413 + 0.981314i \(0.438369\pi\)
\(108\) −1.33095e10 −0.0871628
\(109\) 1.74576e11 1.08677 0.543386 0.839483i \(-0.317142\pi\)
0.543386 + 0.839483i \(0.317142\pi\)
\(110\) 1.22806e11 0.727043
\(111\) 6.80170e10 0.383125
\(112\) −7.93226e10 −0.425302
\(113\) 2.62472e10 0.134014 0.0670071 0.997752i \(-0.478655\pi\)
0.0670071 + 0.997752i \(0.478655\pi\)
\(114\) −6.01088e10 −0.292389
\(115\) −6.49990e11 −3.01348
\(116\) 1.00720e11 0.445244
\(117\) 3.75003e10 0.158129
\(118\) −2.39306e10 −0.0962948
\(119\) 2.73714e11 1.05145
\(120\) −2.86262e11 −1.05019
\(121\) −1.89092e11 −0.662756
\(122\) 4.29386e11 1.43837
\(123\) 6.38863e10 0.204611
\(124\) 1.62193e11 0.496836
\(125\) −4.99526e11 −1.46404
\(126\) −1.09312e11 −0.306641
\(127\) 8.60866e10 0.231215 0.115607 0.993295i \(-0.463119\pi\)
0.115607 + 0.993295i \(0.463119\pi\)
\(128\) 4.63507e10 0.119234
\(129\) −1.60870e11 −0.396490
\(130\) 2.51426e11 0.593911
\(131\) 3.13858e11 0.710789 0.355395 0.934716i \(-0.384346\pi\)
0.355395 + 0.934716i \(0.384346\pi\)
\(132\) −6.99168e10 −0.151853
\(133\) 4.08695e11 0.851560
\(134\) 1.33412e11 0.266759
\(135\) −1.69712e11 −0.325744
\(136\) 4.92945e11 0.908520
\(137\) 6.14363e10 0.108758 0.0543791 0.998520i \(-0.482682\pi\)
0.0543791 + 0.998520i \(0.482682\pi\)
\(138\) −4.47005e11 −0.760287
\(139\) 3.91170e11 0.639417 0.319709 0.947516i \(-0.396415\pi\)
0.319709 + 0.947516i \(0.396415\pi\)
\(140\) 6.06735e11 0.953443
\(141\) 4.89843e11 0.740204
\(142\) −1.51008e11 −0.219489
\(143\) 1.96994e11 0.275490
\(144\) −8.46930e10 −0.113987
\(145\) 1.28430e12 1.66396
\(146\) 4.53952e11 0.566330
\(147\) 2.62751e11 0.315718
\(148\) −2.59630e11 −0.300549
\(149\) −1.13502e12 −1.26613 −0.633064 0.774099i \(-0.718203\pi\)
−0.633064 + 0.774099i \(0.718203\pi\)
\(150\) −7.40694e11 −0.796410
\(151\) −5.95629e11 −0.617451 −0.308726 0.951151i \(-0.599903\pi\)
−0.308726 + 0.951151i \(0.599903\pi\)
\(152\) 7.36039e11 0.735802
\(153\) 2.92246e11 0.281803
\(154\) −5.74232e11 −0.534224
\(155\) 2.06815e12 1.85677
\(156\) −1.43143e11 −0.124047
\(157\) 6.19298e11 0.518145 0.259072 0.965858i \(-0.416583\pi\)
0.259072 + 0.965858i \(0.416583\pi\)
\(158\) −1.15911e12 −0.936508
\(159\) 3.21309e11 0.250749
\(160\) 1.84477e12 1.39086
\(161\) 3.03931e12 2.21428
\(162\) −1.16713e11 −0.0821838
\(163\) −3.02889e11 −0.206183 −0.103091 0.994672i \(-0.532873\pi\)
−0.103091 + 0.994672i \(0.532873\pi\)
\(164\) −2.43862e11 −0.160510
\(165\) −8.91522e11 −0.567506
\(166\) −1.00774e12 −0.620516
\(167\) −2.88370e12 −1.71795 −0.858974 0.512020i \(-0.828897\pi\)
−0.858974 + 0.512020i \(0.828897\pi\)
\(168\) 1.33854e12 0.771667
\(169\) −1.38885e12 −0.774956
\(170\) 1.95940e12 1.05841
\(171\) 4.36365e11 0.228229
\(172\) 6.14062e11 0.311033
\(173\) 3.10578e12 1.52376 0.761881 0.647717i \(-0.224276\pi\)
0.761881 + 0.647717i \(0.224276\pi\)
\(174\) 8.83230e11 0.419810
\(175\) 5.03617e12 2.31948
\(176\) −4.44904e11 −0.198585
\(177\) 1.73727e11 0.0751646
\(178\) −7.45456e11 −0.312689
\(179\) −5.83860e11 −0.237475 −0.118737 0.992926i \(-0.537885\pi\)
−0.118737 + 0.992926i \(0.537885\pi\)
\(180\) 6.47813e11 0.255535
\(181\) −1.55134e12 −0.593573 −0.296786 0.954944i \(-0.595915\pi\)
−0.296786 + 0.954944i \(0.595915\pi\)
\(182\) −1.17565e12 −0.436400
\(183\) −3.11717e12 −1.12274
\(184\) 5.47363e12 1.91328
\(185\) −3.31059e12 −1.12321
\(186\) 1.42229e12 0.468455
\(187\) 1.53521e12 0.490951
\(188\) −1.86979e12 −0.580665
\(189\) 7.93562e11 0.239354
\(190\) 2.92567e12 0.857197
\(191\) −1.36960e12 −0.389861 −0.194931 0.980817i \(-0.562448\pi\)
−0.194931 + 0.980817i \(0.562448\pi\)
\(192\) 1.98246e12 0.548338
\(193\) 3.06839e12 0.824794 0.412397 0.911004i \(-0.364692\pi\)
0.412397 + 0.911004i \(0.364692\pi\)
\(194\) 2.99398e12 0.782240
\(195\) −1.82525e12 −0.463588
\(196\) −1.00296e12 −0.247670
\(197\) 7.65200e12 1.83743 0.918715 0.394921i \(-0.129228\pi\)
0.918715 + 0.394921i \(0.129228\pi\)
\(198\) −6.13110e11 −0.143179
\(199\) 7.77276e12 1.76556 0.882782 0.469783i \(-0.155668\pi\)
0.882782 + 0.469783i \(0.155668\pi\)
\(200\) 9.06988e12 2.00418
\(201\) −9.68519e11 −0.208224
\(202\) 1.53168e12 0.320432
\(203\) −6.00532e12 −1.22267
\(204\) −1.11554e12 −0.221065
\(205\) −3.10953e12 −0.599859
\(206\) −8.49261e11 −0.159504
\(207\) 3.24508e12 0.593456
\(208\) −9.10871e11 −0.162222
\(209\) 2.29229e12 0.397617
\(210\) 5.32054e12 0.898979
\(211\) 7.92113e12 1.30387 0.651934 0.758275i \(-0.273958\pi\)
0.651934 + 0.758275i \(0.273958\pi\)
\(212\) −1.22648e12 −0.196704
\(213\) 1.09625e12 0.171326
\(214\) −1.86883e12 −0.284638
\(215\) 7.83003e12 1.16239
\(216\) 1.42916e12 0.206817
\(217\) −9.67053e12 −1.36434
\(218\) −5.84357e12 −0.803836
\(219\) −3.29551e12 −0.442059
\(220\) 3.40306e12 0.445189
\(221\) 3.14309e12 0.401051
\(222\) −2.27673e12 −0.283380
\(223\) 1.15651e13 1.40434 0.702171 0.712008i \(-0.252214\pi\)
0.702171 + 0.712008i \(0.252214\pi\)
\(224\) −8.62603e12 −1.02199
\(225\) 5.37714e12 0.621652
\(226\) −8.78570e11 −0.0991242
\(227\) 3.55430e12 0.391392 0.195696 0.980665i \(-0.437303\pi\)
0.195696 + 0.980665i \(0.437303\pi\)
\(228\) −1.66566e12 −0.179038
\(229\) −1.38410e13 −1.45236 −0.726178 0.687507i \(-0.758705\pi\)
−0.726178 + 0.687507i \(0.758705\pi\)
\(230\) 2.17571e13 2.22894
\(231\) 4.16869e12 0.416998
\(232\) −1.08153e13 −1.05646
\(233\) −1.32391e13 −1.26299 −0.631495 0.775380i \(-0.717558\pi\)
−0.631495 + 0.775380i \(0.717558\pi\)
\(234\) −1.25524e12 −0.116961
\(235\) −2.38421e13 −2.17006
\(236\) −6.63137e11 −0.0589640
\(237\) 8.41469e12 0.731008
\(238\) −9.16202e12 −0.777710
\(239\) −4.46934e12 −0.370727 −0.185364 0.982670i \(-0.559346\pi\)
−0.185364 + 0.982670i \(0.559346\pi\)
\(240\) 4.12226e12 0.334175
\(241\) 5.74664e12 0.455324 0.227662 0.973740i \(-0.426892\pi\)
0.227662 + 0.973740i \(0.426892\pi\)
\(242\) 6.32947e12 0.490210
\(243\) 8.47289e11 0.0641500
\(244\) 1.18987e13 0.880752
\(245\) −1.27889e13 −0.925590
\(246\) −2.13846e12 −0.151342
\(247\) 4.69310e12 0.324808
\(248\) −1.74161e13 −1.17887
\(249\) 7.31577e12 0.484355
\(250\) 1.67206e13 1.08288
\(251\) 1.44705e13 0.916809 0.458404 0.888744i \(-0.348421\pi\)
0.458404 + 0.888744i \(0.348421\pi\)
\(252\) −3.02913e12 −0.187765
\(253\) 1.70469e13 1.03391
\(254\) −2.88157e12 −0.171019
\(255\) −1.42245e13 −0.826161
\(256\) −1.82597e13 −1.03794
\(257\) 2.67034e13 1.48571 0.742856 0.669451i \(-0.233471\pi\)
0.742856 + 0.669451i \(0.233471\pi\)
\(258\) 5.38480e12 0.293265
\(259\) 1.54801e13 0.825324
\(260\) 6.96721e12 0.363669
\(261\) −6.41190e12 −0.327690
\(262\) −1.05057e13 −0.525738
\(263\) 1.00444e13 0.492230 0.246115 0.969241i \(-0.420846\pi\)
0.246115 + 0.969241i \(0.420846\pi\)
\(264\) 7.50760e12 0.360313
\(265\) −1.56391e13 −0.735121
\(266\) −1.36802e13 −0.629860
\(267\) 5.41171e12 0.244075
\(268\) 3.69696e12 0.163344
\(269\) −2.88682e13 −1.24963 −0.624816 0.780772i \(-0.714826\pi\)
−0.624816 + 0.780772i \(0.714826\pi\)
\(270\) 5.68076e12 0.240938
\(271\) 2.93904e13 1.22145 0.610724 0.791844i \(-0.290879\pi\)
0.610724 + 0.791844i \(0.290879\pi\)
\(272\) −7.09856e12 −0.289096
\(273\) 8.53474e12 0.340640
\(274\) −2.05645e12 −0.0804435
\(275\) 2.82469e13 1.08303
\(276\) −1.23869e13 −0.465546
\(277\) −3.79666e13 −1.39882 −0.699412 0.714719i \(-0.746555\pi\)
−0.699412 + 0.714719i \(0.746555\pi\)
\(278\) −1.30936e13 −0.472948
\(279\) −1.03253e13 −0.365661
\(280\) −6.51506e13 −2.26230
\(281\) 4.86401e13 1.65619 0.828094 0.560589i \(-0.189425\pi\)
0.828094 + 0.560589i \(0.189425\pi\)
\(282\) −1.63965e13 −0.547495
\(283\) −2.79496e13 −0.915271 −0.457636 0.889140i \(-0.651304\pi\)
−0.457636 + 0.889140i \(0.651304\pi\)
\(284\) −4.18454e12 −0.134399
\(285\) −2.12392e13 −0.669100
\(286\) −6.59398e12 −0.203767
\(287\) 1.45400e13 0.440771
\(288\) −9.21004e12 −0.273907
\(289\) −9.77727e12 −0.285285
\(290\) −4.29894e13 −1.23076
\(291\) −2.17351e13 −0.610591
\(292\) 1.25794e13 0.346780
\(293\) 2.56382e13 0.693610 0.346805 0.937937i \(-0.387267\pi\)
0.346805 + 0.937937i \(0.387267\pi\)
\(294\) −8.79505e12 −0.233522
\(295\) −8.45579e12 −0.220360
\(296\) 2.78788e13 0.713131
\(297\) 4.45093e12 0.111761
\(298\) 3.79924e13 0.936498
\(299\) 3.49007e13 0.844585
\(300\) −2.05252e13 −0.487665
\(301\) −3.66126e13 −0.854113
\(302\) 1.99375e13 0.456701
\(303\) −1.11194e13 −0.250119
\(304\) −1.05992e13 −0.234136
\(305\) 1.51722e14 3.29154
\(306\) −9.78232e12 −0.208437
\(307\) −3.77775e13 −0.790628 −0.395314 0.918546i \(-0.629364\pi\)
−0.395314 + 0.918546i \(0.629364\pi\)
\(308\) −1.59124e13 −0.327121
\(309\) 6.16529e12 0.124504
\(310\) −6.92271e13 −1.37337
\(311\) 2.70770e13 0.527738 0.263869 0.964559i \(-0.415001\pi\)
0.263869 + 0.964559i \(0.415001\pi\)
\(312\) 1.53706e13 0.294334
\(313\) 3.66522e13 0.689615 0.344807 0.938673i \(-0.387944\pi\)
0.344807 + 0.938673i \(0.387944\pi\)
\(314\) −2.07297e13 −0.383248
\(315\) −3.86250e13 −0.701714
\(316\) −3.21200e13 −0.573451
\(317\) 2.63514e13 0.462358 0.231179 0.972911i \(-0.425742\pi\)
0.231179 + 0.972911i \(0.425742\pi\)
\(318\) −1.07552e13 −0.185468
\(319\) −3.36826e13 −0.570896
\(320\) −9.64923e13 −1.60756
\(321\) 1.35669e13 0.222179
\(322\) −1.01735e14 −1.63780
\(323\) 3.65741e13 0.578841
\(324\) −3.23421e12 −0.0503235
\(325\) 5.78310e13 0.884712
\(326\) 1.01386e13 0.152504
\(327\) 4.24219e13 0.627448
\(328\) 2.61857e13 0.380854
\(329\) 1.11484e14 1.59454
\(330\) 2.98419e13 0.419758
\(331\) 6.09860e12 0.0843677 0.0421839 0.999110i \(-0.486568\pi\)
0.0421839 + 0.999110i \(0.486568\pi\)
\(332\) −2.79252e13 −0.379960
\(333\) 1.65281e13 0.221198
\(334\) 9.65260e13 1.27069
\(335\) 4.71407e13 0.610449
\(336\) −1.92754e13 −0.245548
\(337\) 2.31559e13 0.290200 0.145100 0.989417i \(-0.453650\pi\)
0.145100 + 0.989417i \(0.453650\pi\)
\(338\) 4.64887e13 0.573200
\(339\) 6.37806e12 0.0773732
\(340\) 5.42966e13 0.648095
\(341\) −5.42401e13 −0.637047
\(342\) −1.46064e13 −0.168811
\(343\) −4.95555e13 −0.563605
\(344\) −6.59374e13 −0.738007
\(345\) −1.57948e14 −1.73983
\(346\) −1.03960e14 −1.12706
\(347\) −1.36157e14 −1.45287 −0.726436 0.687234i \(-0.758825\pi\)
−0.726436 + 0.687234i \(0.758825\pi\)
\(348\) 2.44750e13 0.257062
\(349\) 1.72629e14 1.78474 0.892369 0.451307i \(-0.149042\pi\)
0.892369 + 0.451307i \(0.149042\pi\)
\(350\) −1.68575e14 −1.71562
\(351\) 9.11257e12 0.0912960
\(352\) −4.83817e13 −0.477195
\(353\) 1.46540e14 1.42297 0.711485 0.702701i \(-0.248023\pi\)
0.711485 + 0.702701i \(0.248023\pi\)
\(354\) −5.81514e12 −0.0555958
\(355\) −5.33579e13 −0.502276
\(356\) −2.06572e13 −0.191468
\(357\) 6.65125e13 0.607055
\(358\) 1.95435e13 0.175649
\(359\) −9.00449e13 −0.796966 −0.398483 0.917176i \(-0.630463\pi\)
−0.398483 + 0.917176i \(0.630463\pi\)
\(360\) −6.95616e13 −0.606325
\(361\) −6.18799e13 −0.531202
\(362\) 5.19278e13 0.439039
\(363\) −4.59494e13 −0.382642
\(364\) −3.25782e13 −0.267220
\(365\) 1.60402e14 1.29598
\(366\) 1.04341e14 0.830441
\(367\) 4.84767e13 0.380076 0.190038 0.981777i \(-0.439139\pi\)
0.190038 + 0.981777i \(0.439139\pi\)
\(368\) −7.88220e13 −0.608814
\(369\) 1.55244e13 0.118132
\(370\) 1.10815e14 0.830786
\(371\) 7.31272e13 0.540160
\(372\) 3.94129e13 0.286848
\(373\) −1.78441e14 −1.27966 −0.639832 0.768515i \(-0.720996\pi\)
−0.639832 + 0.768515i \(0.720996\pi\)
\(374\) −5.13879e13 −0.363134
\(375\) −1.21385e14 −0.845263
\(376\) 2.00777e14 1.37778
\(377\) −6.89598e13 −0.466357
\(378\) −2.65628e13 −0.177039
\(379\) −2.86072e14 −1.87914 −0.939570 0.342357i \(-0.888775\pi\)
−0.939570 + 0.342357i \(0.888775\pi\)
\(380\) 8.10727e13 0.524886
\(381\) 2.09191e13 0.133492
\(382\) 4.58445e13 0.288362
\(383\) −2.08002e14 −1.28966 −0.644830 0.764326i \(-0.723072\pi\)
−0.644830 + 0.764326i \(0.723072\pi\)
\(384\) 1.12632e13 0.0688401
\(385\) −2.02903e14 −1.22251
\(386\) −1.02708e14 −0.610063
\(387\) −3.90915e13 −0.228914
\(388\) 8.29658e13 0.478988
\(389\) 1.11622e14 0.635372 0.317686 0.948196i \(-0.397094\pi\)
0.317686 + 0.948196i \(0.397094\pi\)
\(390\) 6.10965e13 0.342895
\(391\) 2.71987e14 1.50514
\(392\) 1.07696e14 0.587662
\(393\) 7.62675e13 0.410374
\(394\) −2.56135e14 −1.35906
\(395\) −4.09568e14 −2.14310
\(396\) −1.69898e13 −0.0876726
\(397\) −2.59151e14 −1.31888 −0.659440 0.751757i \(-0.729206\pi\)
−0.659440 + 0.751757i \(0.729206\pi\)
\(398\) −2.60177e14 −1.30591
\(399\) 9.93129e13 0.491649
\(400\) −1.30609e14 −0.637740
\(401\) 9.88216e13 0.475946 0.237973 0.971272i \(-0.423517\pi\)
0.237973 + 0.971272i \(0.423517\pi\)
\(402\) 3.24192e13 0.154013
\(403\) −1.11048e14 −0.520395
\(404\) 4.24442e13 0.196210
\(405\) −4.12400e13 −0.188069
\(406\) 2.01016e14 0.904350
\(407\) 8.68246e13 0.385366
\(408\) 1.19786e14 0.524534
\(409\) −2.00981e14 −0.868312 −0.434156 0.900838i \(-0.642953\pi\)
−0.434156 + 0.900838i \(0.642953\pi\)
\(410\) 1.04085e14 0.443688
\(411\) 1.49290e13 0.0627916
\(412\) −2.35337e13 −0.0976688
\(413\) 3.95387e13 0.161919
\(414\) −1.08622e14 −0.438952
\(415\) −3.56080e14 −1.41998
\(416\) −9.90537e13 −0.389814
\(417\) 9.50544e13 0.369168
\(418\) −7.67297e13 −0.294099
\(419\) 2.58507e14 0.977903 0.488952 0.872311i \(-0.337379\pi\)
0.488952 + 0.872311i \(0.337379\pi\)
\(420\) 1.47437e14 0.550471
\(421\) 6.18380e13 0.227879 0.113939 0.993488i \(-0.463653\pi\)
0.113939 + 0.993488i \(0.463653\pi\)
\(422\) −2.65144e14 −0.964412
\(423\) 1.19032e14 0.427357
\(424\) 1.31698e14 0.466732
\(425\) 4.50686e14 1.57665
\(426\) −3.66948e13 −0.126722
\(427\) −7.09441e14 −2.41859
\(428\) −5.17867e13 −0.174292
\(429\) 4.78696e13 0.159054
\(430\) −2.62094e14 −0.859766
\(431\) 4.14707e14 1.34313 0.671563 0.740947i \(-0.265623\pi\)
0.671563 + 0.740947i \(0.265623\pi\)
\(432\) −2.05804e13 −0.0658102
\(433\) −1.63951e14 −0.517645 −0.258822 0.965925i \(-0.583334\pi\)
−0.258822 + 0.965925i \(0.583334\pi\)
\(434\) 3.23701e14 1.00914
\(435\) 3.12086e14 0.960690
\(436\) −1.61930e14 −0.492212
\(437\) 4.06116e14 1.21900
\(438\) 1.10310e14 0.326971
\(439\) 2.29011e14 0.670349 0.335174 0.942156i \(-0.391205\pi\)
0.335174 + 0.942156i \(0.391205\pi\)
\(440\) −3.65417e14 −1.05633
\(441\) 6.38485e13 0.182280
\(442\) −1.05209e14 −0.296639
\(443\) −3.12630e14 −0.870582 −0.435291 0.900290i \(-0.643354\pi\)
−0.435291 + 0.900290i \(0.643354\pi\)
\(444\) −6.30900e13 −0.173522
\(445\) −2.63404e14 −0.715554
\(446\) −3.87118e14 −1.03873
\(447\) −2.75809e14 −0.731000
\(448\) 4.51191e14 1.18122
\(449\) −3.46003e14 −0.894797 −0.447399 0.894335i \(-0.647649\pi\)
−0.447399 + 0.894335i \(0.647649\pi\)
\(450\) −1.79989e14 −0.459807
\(451\) 8.15517e13 0.205808
\(452\) −2.43459e13 −0.0606966
\(453\) −1.44738e14 −0.356486
\(454\) −1.18973e14 −0.289495
\(455\) −4.15411e14 −0.998654
\(456\) 1.78857e14 0.424815
\(457\) 6.34100e14 1.48805 0.744027 0.668150i \(-0.232913\pi\)
0.744027 + 0.668150i \(0.232913\pi\)
\(458\) 4.63299e14 1.07424
\(459\) 7.10157e13 0.162699
\(460\) 6.02907e14 1.36484
\(461\) −9.14184e13 −0.204493 −0.102247 0.994759i \(-0.532603\pi\)
−0.102247 + 0.994759i \(0.532603\pi\)
\(462\) −1.39538e14 −0.308434
\(463\) −4.66339e14 −1.01861 −0.509303 0.860587i \(-0.670097\pi\)
−0.509303 + 0.860587i \(0.670097\pi\)
\(464\) 1.55743e14 0.336171
\(465\) 5.02561e14 1.07201
\(466\) 4.43150e14 0.934175
\(467\) −5.25176e14 −1.09411 −0.547056 0.837096i \(-0.684252\pi\)
−0.547056 + 0.837096i \(0.684252\pi\)
\(468\) −3.47839e13 −0.0716186
\(469\) −2.20426e14 −0.448552
\(470\) 7.98064e14 1.60509
\(471\) 1.50489e14 0.299151
\(472\) 7.12070e13 0.139908
\(473\) −2.05353e14 −0.398809
\(474\) −2.81664e14 −0.540693
\(475\) 6.72940e14 1.27691
\(476\) −2.53887e14 −0.476214
\(477\) 7.80782e13 0.144770
\(478\) 1.49602e14 0.274210
\(479\) −3.00365e14 −0.544257 −0.272128 0.962261i \(-0.587728\pi\)
−0.272128 + 0.962261i \(0.587728\pi\)
\(480\) 4.48280e14 0.803012
\(481\) 1.77760e14 0.314800
\(482\) −1.92357e14 −0.336782
\(483\) 7.38552e14 1.27841
\(484\) 1.75395e14 0.300170
\(485\) 1.05791e15 1.79007
\(486\) −2.83612e13 −0.0474488
\(487\) 4.61971e14 0.764197 0.382098 0.924122i \(-0.375202\pi\)
0.382098 + 0.924122i \(0.375202\pi\)
\(488\) −1.27767e15 −2.08982
\(489\) −7.36021e13 −0.119040
\(490\) 4.28081e14 0.684617
\(491\) 4.64833e14 0.735104 0.367552 0.930003i \(-0.380196\pi\)
0.367552 + 0.930003i \(0.380196\pi\)
\(492\) −5.92585e13 −0.0926708
\(493\) −5.37415e14 −0.831096
\(494\) −1.57092e14 −0.240245
\(495\) −2.16640e14 −0.327650
\(496\) 2.50798e14 0.375124
\(497\) 2.49498e14 0.369068
\(498\) −2.44880e14 −0.358255
\(499\) −8.26145e14 −1.19537 −0.597686 0.801730i \(-0.703913\pi\)
−0.597686 + 0.801730i \(0.703913\pi\)
\(500\) 4.63341e14 0.663080
\(501\) −7.00740e14 −0.991857
\(502\) −4.84371e14 −0.678121
\(503\) 4.95542e14 0.686210 0.343105 0.939297i \(-0.388521\pi\)
0.343105 + 0.939297i \(0.388521\pi\)
\(504\) 3.25265e14 0.445522
\(505\) 5.41214e14 0.733274
\(506\) −5.70609e14 −0.764734
\(507\) −3.37490e14 −0.447421
\(508\) −7.98508e13 −0.104720
\(509\) 1.10728e15 1.43651 0.718254 0.695781i \(-0.244941\pi\)
0.718254 + 0.695781i \(0.244941\pi\)
\(510\) 4.76134e14 0.611074
\(511\) −7.50029e14 −0.952277
\(512\) 5.16278e14 0.648483
\(513\) 1.06037e14 0.131768
\(514\) −8.93842e14 −1.09891
\(515\) −3.00083e14 −0.365007
\(516\) 1.49217e14 0.179575
\(517\) 6.25291e14 0.744533
\(518\) −5.18163e14 −0.610454
\(519\) 7.54705e14 0.879744
\(520\) −7.48133e14 −0.862900
\(521\) 1.05195e15 1.20057 0.600287 0.799784i \(-0.295053\pi\)
0.600287 + 0.799784i \(0.295053\pi\)
\(522\) 2.14625e14 0.242378
\(523\) −7.70218e14 −0.860705 −0.430353 0.902661i \(-0.641611\pi\)
−0.430353 + 0.902661i \(0.641611\pi\)
\(524\) −2.91123e14 −0.321925
\(525\) 1.22379e15 1.33915
\(526\) −3.36216e14 −0.364080
\(527\) −8.65414e14 −0.927397
\(528\) −1.08112e14 −0.114653
\(529\) 2.06732e15 2.16971
\(530\) 5.23486e14 0.543735
\(531\) 4.22156e13 0.0433963
\(532\) −3.79090e14 −0.385681
\(533\) 1.66964e14 0.168122
\(534\) −1.81146e14 −0.180531
\(535\) −6.60342e14 −0.651363
\(536\) −3.96976e14 −0.387577
\(537\) −1.41878e14 −0.137106
\(538\) 9.66304e14 0.924296
\(539\) 3.35406e14 0.317565
\(540\) 1.57419e14 0.147533
\(541\) −1.63413e14 −0.151601 −0.0758006 0.997123i \(-0.524151\pi\)
−0.0758006 + 0.997123i \(0.524151\pi\)
\(542\) −9.83784e14 −0.903449
\(543\) −3.76975e14 −0.342699
\(544\) −7.71941e14 −0.694689
\(545\) −2.06480e15 −1.83949
\(546\) −2.85683e14 −0.251956
\(547\) 8.97921e14 0.783985 0.391993 0.919968i \(-0.371786\pi\)
0.391993 + 0.919968i \(0.371786\pi\)
\(548\) −5.69861e13 −0.0492579
\(549\) −7.57473e14 −0.648215
\(550\) −9.45506e14 −0.801068
\(551\) −8.02438e14 −0.673097
\(552\) 1.33009e15 1.10463
\(553\) 1.91511e15 1.57473
\(554\) 1.27085e15 1.03465
\(555\) −8.04473e14 −0.648485
\(556\) −3.62835e14 −0.289599
\(557\) 1.31969e15 1.04296 0.521480 0.853264i \(-0.325380\pi\)
0.521480 + 0.853264i \(0.325380\pi\)
\(558\) 3.45617e14 0.270462
\(559\) −4.20428e14 −0.325781
\(560\) 9.38190e14 0.719874
\(561\) 3.73056e14 0.283451
\(562\) −1.62813e15 −1.22501
\(563\) −2.48165e15 −1.84903 −0.924517 0.381140i \(-0.875532\pi\)
−0.924517 + 0.381140i \(0.875532\pi\)
\(564\) −4.54360e14 −0.335247
\(565\) −3.10439e14 −0.226835
\(566\) 9.35554e14 0.676984
\(567\) 1.92835e14 0.138191
\(568\) 4.49332e14 0.318898
\(569\) 2.23206e15 1.56887 0.784436 0.620210i \(-0.212953\pi\)
0.784436 + 0.620210i \(0.212953\pi\)
\(570\) 7.10938e14 0.494903
\(571\) 1.32952e15 0.916636 0.458318 0.888788i \(-0.348452\pi\)
0.458318 + 0.888788i \(0.348452\pi\)
\(572\) −1.82725e14 −0.124773
\(573\) −3.32813e14 −0.225086
\(574\) −4.86695e14 −0.326018
\(575\) 5.00439e15 3.32031
\(576\) 4.81738e14 0.316583
\(577\) −7.60604e14 −0.495098 −0.247549 0.968875i \(-0.579625\pi\)
−0.247549 + 0.968875i \(0.579625\pi\)
\(578\) 3.27274e14 0.211012
\(579\) 7.45619e14 0.476195
\(580\) −1.19127e15 −0.753628
\(581\) 1.66501e15 1.04339
\(582\) 7.27538e14 0.451626
\(583\) 4.10156e14 0.252216
\(584\) −1.35076e15 −0.822827
\(585\) −4.43536e14 −0.267652
\(586\) −8.58185e14 −0.513032
\(587\) 2.80832e15 1.66317 0.831586 0.555396i \(-0.187433\pi\)
0.831586 + 0.555396i \(0.187433\pi\)
\(588\) −2.43718e14 −0.142992
\(589\) −1.29219e15 −0.751090
\(590\) 2.83040e14 0.162990
\(591\) 1.85944e15 1.06084
\(592\) −4.01463e14 −0.226922
\(593\) −2.39814e15 −1.34299 −0.671497 0.741007i \(-0.734349\pi\)
−0.671497 + 0.741007i \(0.734349\pi\)
\(594\) −1.48986e14 −0.0826645
\(595\) −3.23736e15 −1.77970
\(596\) 1.05280e15 0.573444
\(597\) 1.88878e15 1.01935
\(598\) −1.16823e15 −0.624701
\(599\) 2.55061e15 1.35144 0.675718 0.737160i \(-0.263834\pi\)
0.675718 + 0.737160i \(0.263834\pi\)
\(600\) 2.20398e15 1.15711
\(601\) −2.73179e15 −1.42114 −0.710571 0.703626i \(-0.751563\pi\)
−0.710571 + 0.703626i \(0.751563\pi\)
\(602\) 1.22553e15 0.631748
\(603\) −2.35350e14 −0.120218
\(604\) 5.52483e14 0.279651
\(605\) 2.23649e15 1.12179
\(606\) 3.72199e14 0.185002
\(607\) 2.31047e15 1.13806 0.569028 0.822318i \(-0.307320\pi\)
0.569028 + 0.822318i \(0.307320\pi\)
\(608\) −1.15262e15 −0.562622
\(609\) −1.45929e15 −0.705906
\(610\) −5.07858e15 −2.43460
\(611\) 1.28018e15 0.608199
\(612\) −2.71076e14 −0.127632
\(613\) 3.16315e15 1.47600 0.738002 0.674799i \(-0.235770\pi\)
0.738002 + 0.674799i \(0.235770\pi\)
\(614\) 1.26452e15 0.584792
\(615\) −7.55617e14 −0.346329
\(616\) 1.70866e15 0.776180
\(617\) −5.87068e14 −0.264314 −0.132157 0.991229i \(-0.542190\pi\)
−0.132157 + 0.991229i \(0.542190\pi\)
\(618\) −2.06370e14 −0.0920897
\(619\) 2.62487e15 1.16094 0.580469 0.814283i \(-0.302869\pi\)
0.580469 + 0.814283i \(0.302869\pi\)
\(620\) −1.91834e15 −0.840953
\(621\) 7.88554e14 0.342632
\(622\) −9.06346e14 −0.390344
\(623\) 1.23166e15 0.525782
\(624\) −2.21342e14 −0.0936587
\(625\) 1.46175e15 0.613104
\(626\) −1.22686e15 −0.510077
\(627\) 5.57026e14 0.229564
\(628\) −5.74437e14 −0.234674
\(629\) 1.38531e15 0.561007
\(630\) 1.29289e15 0.519026
\(631\) 4.20767e14 0.167448 0.0837240 0.996489i \(-0.473319\pi\)
0.0837240 + 0.996489i \(0.473319\pi\)
\(632\) 3.44901e15 1.36066
\(633\) 1.92484e15 0.752789
\(634\) −8.82060e14 −0.341985
\(635\) −1.01819e15 −0.391358
\(636\) −2.98035e14 −0.113567
\(637\) 6.86689e14 0.259414
\(638\) 1.12746e15 0.422266
\(639\) 2.66390e14 0.0989151
\(640\) −5.48215e14 −0.201818
\(641\) 2.55210e15 0.931492 0.465746 0.884919i \(-0.345786\pi\)
0.465746 + 0.884919i \(0.345786\pi\)
\(642\) −4.54125e14 −0.164336
\(643\) −1.91924e15 −0.688604 −0.344302 0.938859i \(-0.611884\pi\)
−0.344302 + 0.938859i \(0.611884\pi\)
\(644\) −2.81915e15 −1.00287
\(645\) 1.90270e15 0.671106
\(646\) −1.22424e15 −0.428142
\(647\) −1.70129e13 −0.00589935 −0.00294967 0.999996i \(-0.500939\pi\)
−0.00294967 + 0.999996i \(0.500939\pi\)
\(648\) 3.47287e14 0.119406
\(649\) 2.21764e14 0.0756042
\(650\) −1.93577e15 −0.654381
\(651\) −2.34994e15 −0.787701
\(652\) 2.80949e14 0.0933824
\(653\) 3.83593e15 1.26430 0.632148 0.774848i \(-0.282174\pi\)
0.632148 + 0.774848i \(0.282174\pi\)
\(654\) −1.41999e15 −0.464095
\(655\) −3.71216e15 −1.20309
\(656\) −3.77082e14 −0.121190
\(657\) −8.00809e14 −0.255223
\(658\) −3.73169e15 −1.17941
\(659\) −2.75101e15 −0.862230 −0.431115 0.902297i \(-0.641880\pi\)
−0.431115 + 0.902297i \(0.641880\pi\)
\(660\) 8.26943e14 0.257030
\(661\) 1.58002e15 0.487029 0.243515 0.969897i \(-0.421700\pi\)
0.243515 + 0.969897i \(0.421700\pi\)
\(662\) −2.04138e14 −0.0624030
\(663\) 7.63772e14 0.231547
\(664\) 2.99859e15 0.901555
\(665\) −4.83385e15 −1.44137
\(666\) −5.53245e14 −0.163610
\(667\) −5.96742e15 −1.75023
\(668\) 2.67481e15 0.778078
\(669\) 2.81032e15 0.810797
\(670\) −1.57794e15 −0.451521
\(671\) −3.97911e15 −1.12931
\(672\) −2.09612e15 −0.590046
\(673\) −8.65273e14 −0.241585 −0.120793 0.992678i \(-0.538544\pi\)
−0.120793 + 0.992678i \(0.538544\pi\)
\(674\) −7.75096e14 −0.214648
\(675\) 1.30664e15 0.358911
\(676\) 1.28824e15 0.350987
\(677\) −5.04092e15 −1.36230 −0.681148 0.732145i \(-0.738519\pi\)
−0.681148 + 0.732145i \(0.738519\pi\)
\(678\) −2.13492e14 −0.0572294
\(679\) −4.94672e15 −1.31533
\(680\) −5.83032e15 −1.53778
\(681\) 8.63695e14 0.225970
\(682\) 1.81558e15 0.471194
\(683\) −6.71310e15 −1.72826 −0.864131 0.503267i \(-0.832131\pi\)
−0.864131 + 0.503267i \(0.832131\pi\)
\(684\) −4.04756e14 −0.103368
\(685\) −7.26640e14 −0.184086
\(686\) 1.65877e15 0.416873
\(687\) −3.36337e15 −0.838518
\(688\) 9.49520e14 0.234838
\(689\) 8.39729e14 0.206031
\(690\) 5.28697e15 1.28688
\(691\) −1.18226e15 −0.285484 −0.142742 0.989760i \(-0.545592\pi\)
−0.142742 + 0.989760i \(0.545592\pi\)
\(692\) −2.88081e15 −0.690129
\(693\) 1.01299e15 0.240754
\(694\) 4.55757e15 1.07462
\(695\) −4.62658e15 −1.08229
\(696\) −2.62811e15 −0.609947
\(697\) 1.30118e15 0.299610
\(698\) −5.77841e15 −1.32009
\(699\) −3.21709e15 −0.729187
\(700\) −4.67136e15 −1.05052
\(701\) −4.74706e14 −0.105919 −0.0529597 0.998597i \(-0.516865\pi\)
−0.0529597 + 0.998597i \(0.516865\pi\)
\(702\) −3.05024e14 −0.0675275
\(703\) 2.06847e15 0.454354
\(704\) 2.53064e15 0.551545
\(705\) −5.79363e15 −1.25288
\(706\) −4.90513e15 −1.05251
\(707\) −2.53068e15 −0.538803
\(708\) −1.61142e14 −0.0340429
\(709\) 3.87764e15 0.812855 0.406428 0.913683i \(-0.366774\pi\)
0.406428 + 0.913683i \(0.366774\pi\)
\(710\) 1.78605e15 0.371511
\(711\) 2.04477e15 0.422048
\(712\) 2.21815e15 0.454309
\(713\) −9.60951e15 −1.95303
\(714\) −2.22637e15 −0.449011
\(715\) −2.32996e15 −0.466299
\(716\) 5.41567e14 0.107555
\(717\) −1.08605e15 −0.214040
\(718\) 3.01407e15 0.589479
\(719\) 9.10646e15 1.76742 0.883712 0.468032i \(-0.155037\pi\)
0.883712 + 0.468032i \(0.155037\pi\)
\(720\) 1.00171e15 0.192936
\(721\) 1.40317e15 0.268204
\(722\) 2.07130e15 0.392906
\(723\) 1.39643e15 0.262881
\(724\) 1.43896e15 0.268836
\(725\) −9.88809e15 −1.83338
\(726\) 1.53806e15 0.283023
\(727\) −5.94180e15 −1.08512 −0.542562 0.840016i \(-0.682546\pi\)
−0.542562 + 0.840016i \(0.682546\pi\)
\(728\) 3.49822e15 0.634051
\(729\) 2.05891e14 0.0370370
\(730\) −5.36913e15 −0.958580
\(731\) −3.27646e15 −0.580576
\(732\) 2.89137e15 0.508503
\(733\) 3.64382e15 0.636040 0.318020 0.948084i \(-0.396982\pi\)
0.318020 + 0.948084i \(0.396982\pi\)
\(734\) −1.62266e15 −0.281125
\(735\) −3.10770e15 −0.534390
\(736\) −8.57159e15 −1.46296
\(737\) −1.23633e15 −0.209441
\(738\) −5.19646e14 −0.0873771
\(739\) −1.33144e15 −0.222217 −0.111108 0.993808i \(-0.535440\pi\)
−0.111108 + 0.993808i \(0.535440\pi\)
\(740\) 3.07078e15 0.508714
\(741\) 1.14042e15 0.187528
\(742\) −2.44778e15 −0.399532
\(743\) −6.17816e14 −0.100097 −0.0500485 0.998747i \(-0.515938\pi\)
−0.0500485 + 0.998747i \(0.515938\pi\)
\(744\) −4.23212e15 −0.680623
\(745\) 1.34244e16 2.14307
\(746\) 5.97294e15 0.946508
\(747\) 1.77773e15 0.279642
\(748\) −1.42400e15 −0.222357
\(749\) 3.08771e15 0.478616
\(750\) 4.06310e15 0.625203
\(751\) −9.61547e15 −1.46876 −0.734380 0.678739i \(-0.762527\pi\)
−0.734380 + 0.678739i \(0.762527\pi\)
\(752\) −2.89125e15 −0.438417
\(753\) 3.51634e15 0.529320
\(754\) 2.30829e15 0.344943
\(755\) 7.04482e15 1.04511
\(756\) −7.36078e14 −0.108406
\(757\) 8.96873e15 1.31131 0.655653 0.755063i \(-0.272394\pi\)
0.655653 + 0.755063i \(0.272394\pi\)
\(758\) 9.57565e15 1.38991
\(759\) 4.14239e15 0.596927
\(760\) −8.70552e15 −1.24543
\(761\) 1.26989e16 1.80364 0.901822 0.432107i \(-0.142230\pi\)
0.901822 + 0.432107i \(0.142230\pi\)
\(762\) −7.00222e14 −0.0987378
\(763\) 9.65486e15 1.35164
\(764\) 1.27039e15 0.176573
\(765\) −3.45654e15 −0.476984
\(766\) 6.96245e15 0.953903
\(767\) 4.54027e14 0.0617600
\(768\) −4.43710e15 −0.599256
\(769\) 9.65709e15 1.29495 0.647473 0.762089i \(-0.275826\pi\)
0.647473 + 0.762089i \(0.275826\pi\)
\(770\) 6.79175e15 0.904237
\(771\) 6.48893e15 0.857777
\(772\) −2.84613e15 −0.373559
\(773\) −6.07111e15 −0.791190 −0.395595 0.918425i \(-0.629462\pi\)
−0.395595 + 0.918425i \(0.629462\pi\)
\(774\) 1.30851e15 0.169317
\(775\) −1.59231e16 −2.04582
\(776\) −8.90879e15 −1.13653
\(777\) 3.76166e15 0.476501
\(778\) −3.73633e15 −0.469956
\(779\) 1.94285e15 0.242651
\(780\) 1.69303e15 0.209964
\(781\) 1.39938e15 0.172328
\(782\) −9.10420e15 −1.11328
\(783\) −1.55809e15 −0.189192
\(784\) −1.55086e15 −0.186997
\(785\) −7.32476e15 −0.877021
\(786\) −2.55290e15 −0.303535
\(787\) −4.75214e15 −0.561084 −0.280542 0.959842i \(-0.590514\pi\)
−0.280542 + 0.959842i \(0.590514\pi\)
\(788\) −7.09771e15 −0.832193
\(789\) 2.44079e15 0.284189
\(790\) 1.37094e16 1.58515
\(791\) 1.45159e15 0.166676
\(792\) 1.82435e15 0.208027
\(793\) −8.14660e15 −0.922516
\(794\) 8.67455e15 0.975515
\(795\) −3.80030e15 −0.424422
\(796\) −7.20972e15 −0.799644
\(797\) −5.01714e15 −0.552631 −0.276316 0.961067i \(-0.589114\pi\)
−0.276316 + 0.961067i \(0.589114\pi\)
\(798\) −3.32430e15 −0.363650
\(799\) 9.97667e15 1.08387
\(800\) −1.42032e16 −1.53247
\(801\) 1.31505e15 0.140917
\(802\) −3.30785e15 −0.352036
\(803\) −4.20676e15 −0.444644
\(804\) 8.98362e14 0.0943068
\(805\) −3.59475e16 −3.74793
\(806\) 3.71710e15 0.384912
\(807\) −7.01498e15 −0.721476
\(808\) −4.55762e15 −0.465560
\(809\) 1.27332e16 1.29188 0.645940 0.763388i \(-0.276466\pi\)
0.645940 + 0.763388i \(0.276466\pi\)
\(810\) 1.38043e15 0.139106
\(811\) −1.14176e16 −1.14277 −0.571387 0.820681i \(-0.693595\pi\)
−0.571387 + 0.820681i \(0.693595\pi\)
\(812\) 5.57031e15 0.553759
\(813\) 7.14187e15 0.705203
\(814\) −2.90628e15 −0.285038
\(815\) 3.58243e15 0.348988
\(816\) −1.72495e15 −0.166909
\(817\) −4.89223e15 −0.470203
\(818\) 6.72741e15 0.642251
\(819\) 2.07394e15 0.196669
\(820\) 2.88429e15 0.271683
\(821\) 1.13662e16 1.06348 0.531739 0.846908i \(-0.321539\pi\)
0.531739 + 0.846908i \(0.321539\pi\)
\(822\) −4.99718e14 −0.0464441
\(823\) −1.05257e16 −0.971745 −0.485873 0.874030i \(-0.661498\pi\)
−0.485873 + 0.874030i \(0.661498\pi\)
\(824\) 2.52703e15 0.231745
\(825\) 6.86399e15 0.625288
\(826\) −1.32347e15 −0.119764
\(827\) −1.82817e16 −1.64337 −0.821686 0.569940i \(-0.806966\pi\)
−0.821686 + 0.569940i \(0.806966\pi\)
\(828\) −3.01001e15 −0.268783
\(829\) 8.77537e15 0.778423 0.389211 0.921148i \(-0.372748\pi\)
0.389211 + 0.921148i \(0.372748\pi\)
\(830\) 1.19190e16 1.05030
\(831\) −9.22588e15 −0.807611
\(832\) 5.18108e15 0.450549
\(833\) 5.35148e15 0.462303
\(834\) −3.18175e15 −0.273057
\(835\) 3.41071e16 2.90783
\(836\) −2.12624e15 −0.180085
\(837\) −2.50904e15 −0.211114
\(838\) −8.65300e15 −0.723311
\(839\) 1.66805e16 1.38522 0.692610 0.721312i \(-0.256461\pi\)
0.692610 + 0.721312i \(0.256461\pi\)
\(840\) −1.58316e16 −1.30614
\(841\) −4.09582e14 −0.0335709
\(842\) −2.06990e15 −0.168552
\(843\) 1.18195e16 0.956200
\(844\) −7.34735e15 −0.590537
\(845\) 1.64266e16 1.31170
\(846\) −3.98434e15 −0.316096
\(847\) −1.04577e16 −0.824283
\(848\) −1.89650e15 −0.148517
\(849\) −6.79175e15 −0.528432
\(850\) −1.50858e16 −1.16617
\(851\) 1.53824e16 1.18144
\(852\) −1.01684e15 −0.0775955
\(853\) 1.59893e16 1.21230 0.606150 0.795351i \(-0.292713\pi\)
0.606150 + 0.795351i \(0.292713\pi\)
\(854\) 2.37471e16 1.78892
\(855\) −5.16112e15 −0.386305
\(856\) 5.56081e15 0.413554
\(857\) 1.32251e16 0.977245 0.488622 0.872495i \(-0.337500\pi\)
0.488622 + 0.872495i \(0.337500\pi\)
\(858\) −1.60234e15 −0.117645
\(859\) 1.43479e16 1.04671 0.523354 0.852115i \(-0.324680\pi\)
0.523354 + 0.852115i \(0.324680\pi\)
\(860\) −7.26284e15 −0.526460
\(861\) 3.53321e15 0.254479
\(862\) −1.38815e16 −0.993449
\(863\) 1.05265e16 0.748555 0.374277 0.927317i \(-0.377891\pi\)
0.374277 + 0.927317i \(0.377891\pi\)
\(864\) −2.23804e15 −0.158140
\(865\) −3.67337e16 −2.57915
\(866\) 5.48793e15 0.382878
\(867\) −2.37588e15 −0.164710
\(868\) 8.97003e15 0.617924
\(869\) 1.07415e16 0.735284
\(870\) −1.04464e16 −0.710579
\(871\) −2.53118e15 −0.171090
\(872\) 1.73879e16 1.16790
\(873\) −5.28163e15 −0.352525
\(874\) −1.35939e16 −0.901636
\(875\) −2.76261e16 −1.82085
\(876\) 3.05679e15 0.200213
\(877\) −2.81599e16 −1.83288 −0.916438 0.400177i \(-0.868949\pi\)
−0.916438 + 0.400177i \(0.868949\pi\)
\(878\) −7.66566e15 −0.495826
\(879\) 6.23008e15 0.400456
\(880\) 5.26212e15 0.336129
\(881\) 2.10815e16 1.33824 0.669119 0.743155i \(-0.266672\pi\)
0.669119 + 0.743155i \(0.266672\pi\)
\(882\) −2.13720e15 −0.134824
\(883\) 1.51951e16 0.952621 0.476311 0.879277i \(-0.341974\pi\)
0.476311 + 0.879277i \(0.341974\pi\)
\(884\) −2.91542e15 −0.181641
\(885\) −2.05476e15 −0.127225
\(886\) 1.04646e16 0.643930
\(887\) −2.11674e15 −0.129445 −0.0647227 0.997903i \(-0.520616\pi\)
−0.0647227 + 0.997903i \(0.520616\pi\)
\(888\) 6.77455e15 0.411727
\(889\) 4.76099e15 0.287566
\(890\) 8.81691e15 0.529263
\(891\) 1.08158e15 0.0645252
\(892\) −1.07274e16 −0.636042
\(893\) 1.48966e16 0.877819
\(894\) 9.23214e15 0.540687
\(895\) 6.90563e15 0.401954
\(896\) 2.56341e15 0.148294
\(897\) 8.48088e15 0.487621
\(898\) 1.15817e16 0.661841
\(899\) 1.89873e16 1.07841
\(900\) −4.98763e15 −0.281553
\(901\) 6.54414e15 0.367169
\(902\) −2.72978e15 −0.152227
\(903\) −8.89687e15 −0.493122
\(904\) 2.61424e15 0.144019
\(905\) 1.83485e16 1.00469
\(906\) 4.84480e15 0.263676
\(907\) 2.67376e16 1.44638 0.723191 0.690648i \(-0.242675\pi\)
0.723191 + 0.690648i \(0.242675\pi\)
\(908\) −3.29683e15 −0.177266
\(909\) −2.70201e15 −0.144406
\(910\) 1.39050e16 0.738659
\(911\) 1.85949e16 0.981843 0.490922 0.871204i \(-0.336660\pi\)
0.490922 + 0.871204i \(0.336660\pi\)
\(912\) −2.57560e15 −0.135178
\(913\) 9.33868e15 0.487188
\(914\) −2.12252e16 −1.10065
\(915\) 3.68684e16 1.90037
\(916\) 1.28384e16 0.657788
\(917\) 1.73578e16 0.884023
\(918\) −2.37710e15 −0.120341
\(919\) −5.99557e15 −0.301714 −0.150857 0.988556i \(-0.548203\pi\)
−0.150857 + 0.988556i \(0.548203\pi\)
\(920\) −6.47395e16 −3.23845
\(921\) −9.17994e15 −0.456469
\(922\) 3.06004e15 0.151254
\(923\) 2.86501e15 0.140772
\(924\) −3.86672e15 −0.188863
\(925\) 2.54888e16 1.23757
\(926\) 1.56097e16 0.753416
\(927\) 1.49817e15 0.0718822
\(928\) 1.69365e16 0.807810
\(929\) −2.22513e16 −1.05504 −0.527521 0.849542i \(-0.676878\pi\)
−0.527521 + 0.849542i \(0.676878\pi\)
\(930\) −1.68222e16 −0.792915
\(931\) 7.99054e15 0.374415
\(932\) 1.22801e16 0.572022
\(933\) 6.57971e15 0.304690
\(934\) 1.75792e16 0.809265
\(935\) −1.81577e16 −0.830993
\(936\) 3.73506e15 0.169934
\(937\) 2.70151e16 1.22191 0.610954 0.791666i \(-0.290786\pi\)
0.610954 + 0.791666i \(0.290786\pi\)
\(938\) 7.37832e15 0.331774
\(939\) 8.90649e15 0.398149
\(940\) 2.21150e16 0.982843
\(941\) 2.48372e16 1.09739 0.548694 0.836024i \(-0.315125\pi\)
0.548694 + 0.836024i \(0.315125\pi\)
\(942\) −5.03732e15 −0.221268
\(943\) 1.44482e16 0.630957
\(944\) −1.02540e15 −0.0445194
\(945\) −9.38587e15 −0.405135
\(946\) 6.87377e15 0.294981
\(947\) −7.51480e15 −0.320621 −0.160311 0.987067i \(-0.551250\pi\)
−0.160311 + 0.987067i \(0.551250\pi\)
\(948\) −7.80515e15 −0.331082
\(949\) −8.61268e15 −0.363224
\(950\) −2.25253e16 −0.944474
\(951\) 6.40340e15 0.266942
\(952\) 2.72622e16 1.12994
\(953\) −1.72059e16 −0.709033 −0.354516 0.935050i \(-0.615354\pi\)
−0.354516 + 0.935050i \(0.615354\pi\)
\(954\) −2.61351e15 −0.107080
\(955\) 1.61990e16 0.659886
\(956\) 4.14559e15 0.167907
\(957\) −8.18488e15 −0.329607
\(958\) 1.00541e16 0.402562
\(959\) 3.39772e15 0.135265
\(960\) −2.34476e16 −0.928126
\(961\) 5.16728e15 0.203369
\(962\) −5.95014e15 −0.232843
\(963\) 3.29676e15 0.128275
\(964\) −5.33037e15 −0.206221
\(965\) −3.62915e16 −1.39606
\(966\) −2.47215e16 −0.945585
\(967\) 2.43801e16 0.927236 0.463618 0.886035i \(-0.346551\pi\)
0.463618 + 0.886035i \(0.346551\pi\)
\(968\) −1.88337e16 −0.712232
\(969\) 8.88750e15 0.334194
\(970\) −3.54114e16 −1.32403
\(971\) 1.42680e16 0.530465 0.265232 0.964185i \(-0.414551\pi\)
0.265232 + 0.964185i \(0.414551\pi\)
\(972\) −7.85913e14 −0.0290543
\(973\) 2.16335e16 0.795256
\(974\) −1.54635e16 −0.565241
\(975\) 1.40529e16 0.510789
\(976\) 1.83988e16 0.664990
\(977\) −4.63196e16 −1.66473 −0.832366 0.554227i \(-0.813014\pi\)
−0.832366 + 0.554227i \(0.813014\pi\)
\(978\) 2.46368e15 0.0880481
\(979\) 6.90813e15 0.245502
\(980\) 1.18625e16 0.419211
\(981\) 1.03085e16 0.362257
\(982\) −1.55593e16 −0.543723
\(983\) 2.48936e15 0.0865055 0.0432528 0.999064i \(-0.486228\pi\)
0.0432528 + 0.999064i \(0.486228\pi\)
\(984\) 6.36312e15 0.219886
\(985\) −9.05043e16 −3.11007
\(986\) 1.79888e16 0.614724
\(987\) 2.70906e16 0.920606
\(988\) −4.35314e15 −0.147109
\(989\) −3.63816e16 −1.22265
\(990\) 7.25157e15 0.242348
\(991\) 1.45625e16 0.483984 0.241992 0.970278i \(-0.422199\pi\)
0.241992 + 0.970278i \(0.422199\pi\)
\(992\) 2.72733e16 0.901412
\(993\) 1.48196e15 0.0487097
\(994\) −8.35142e15 −0.272983
\(995\) −9.19325e16 −2.98842
\(996\) −6.78583e15 −0.219370
\(997\) 3.41264e16 1.09715 0.548576 0.836101i \(-0.315170\pi\)
0.548576 + 0.836101i \(0.315170\pi\)
\(998\) 2.76535e16 0.884163
\(999\) 4.01634e15 0.127708
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 177.12.a.d.1.10 28
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.12.a.d.1.10 28 1.1 even 1 trivial