Properties

Label 3822.2.a.bu.1.1
Level $3822$
Weight $2$
Character 3822.1
Self dual yes
Analytic conductor $30.519$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3822.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(30.5188236525\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 546)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 3822.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +0.585786 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +0.585786 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +0.585786 q^{10} -0.414214 q^{11} +1.00000 q^{12} -1.00000 q^{13} +0.585786 q^{15} +1.00000 q^{16} +2.41421 q^{17} +1.00000 q^{18} +2.17157 q^{19} +0.585786 q^{20} -0.414214 q^{22} +1.41421 q^{23} +1.00000 q^{24} -4.65685 q^{25} -1.00000 q^{26} +1.00000 q^{27} -1.82843 q^{29} +0.585786 q^{30} +8.48528 q^{31} +1.00000 q^{32} -0.414214 q^{33} +2.41421 q^{34} +1.00000 q^{36} +1.41421 q^{37} +2.17157 q^{38} -1.00000 q^{39} +0.585786 q^{40} +9.89949 q^{41} +6.48528 q^{43} -0.414214 q^{44} +0.585786 q^{45} +1.41421 q^{46} -1.00000 q^{47} +1.00000 q^{48} -4.65685 q^{50} +2.41421 q^{51} -1.00000 q^{52} +9.48528 q^{53} +1.00000 q^{54} -0.242641 q^{55} +2.17157 q^{57} -1.82843 q^{58} -2.07107 q^{59} +0.585786 q^{60} +4.41421 q^{61} +8.48528 q^{62} +1.00000 q^{64} -0.585786 q^{65} -0.414214 q^{66} +1.82843 q^{67} +2.41421 q^{68} +1.41421 q^{69} -5.00000 q^{71} +1.00000 q^{72} -1.41421 q^{73} +1.41421 q^{74} -4.65685 q^{75} +2.17157 q^{76} -1.00000 q^{78} -11.6569 q^{79} +0.585786 q^{80} +1.00000 q^{81} +9.89949 q^{82} -7.65685 q^{83} +1.41421 q^{85} +6.48528 q^{86} -1.82843 q^{87} -0.414214 q^{88} -2.58579 q^{89} +0.585786 q^{90} +1.41421 q^{92} +8.48528 q^{93} -1.00000 q^{94} +1.27208 q^{95} +1.00000 q^{96} +0.928932 q^{97} -0.414214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 4 q^{5} + 2 q^{6} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 4 q^{5} + 2 q^{6} + 2 q^{8} + 2 q^{9} + 4 q^{10} + 2 q^{11} + 2 q^{12} - 2 q^{13} + 4 q^{15} + 2 q^{16} + 2 q^{17} + 2 q^{18} + 10 q^{19} + 4 q^{20} + 2 q^{22} + 2 q^{24} + 2 q^{25} - 2 q^{26} + 2 q^{27} + 2 q^{29} + 4 q^{30} + 2 q^{32} + 2 q^{33} + 2 q^{34} + 2 q^{36} + 10 q^{38} - 2 q^{39} + 4 q^{40} - 4 q^{43} + 2 q^{44} + 4 q^{45} - 2 q^{47} + 2 q^{48} + 2 q^{50} + 2 q^{51} - 2 q^{52} + 2 q^{53} + 2 q^{54} + 8 q^{55} + 10 q^{57} + 2 q^{58} + 10 q^{59} + 4 q^{60} + 6 q^{61} + 2 q^{64} - 4 q^{65} + 2 q^{66} - 2 q^{67} + 2 q^{68} - 10 q^{71} + 2 q^{72} + 2 q^{75} + 10 q^{76} - 2 q^{78} - 12 q^{79} + 4 q^{80} + 2 q^{81} - 4 q^{83} - 4 q^{86} + 2 q^{87} + 2 q^{88} - 8 q^{89} + 4 q^{90} - 2 q^{94} + 28 q^{95} + 2 q^{96} + 16 q^{97} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 0.585786 0.261972 0.130986 0.991384i \(-0.458186\pi\)
0.130986 + 0.991384i \(0.458186\pi\)
\(6\) 1.00000 0.408248
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0.585786 0.185242
\(11\) −0.414214 −0.124890 −0.0624450 0.998048i \(-0.519890\pi\)
−0.0624450 + 0.998048i \(0.519890\pi\)
\(12\) 1.00000 0.288675
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) 0.585786 0.151249
\(16\) 1.00000 0.250000
\(17\) 2.41421 0.585533 0.292766 0.956184i \(-0.405424\pi\)
0.292766 + 0.956184i \(0.405424\pi\)
\(18\) 1.00000 0.235702
\(19\) 2.17157 0.498193 0.249096 0.968479i \(-0.419866\pi\)
0.249096 + 0.968479i \(0.419866\pi\)
\(20\) 0.585786 0.130986
\(21\) 0 0
\(22\) −0.414214 −0.0883106
\(23\) 1.41421 0.294884 0.147442 0.989071i \(-0.452896\pi\)
0.147442 + 0.989071i \(0.452896\pi\)
\(24\) 1.00000 0.204124
\(25\) −4.65685 −0.931371
\(26\) −1.00000 −0.196116
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −1.82843 −0.339530 −0.169765 0.985485i \(-0.554301\pi\)
−0.169765 + 0.985485i \(0.554301\pi\)
\(30\) 0.585786 0.106949
\(31\) 8.48528 1.52400 0.762001 0.647576i \(-0.224217\pi\)
0.762001 + 0.647576i \(0.224217\pi\)
\(32\) 1.00000 0.176777
\(33\) −0.414214 −0.0721053
\(34\) 2.41421 0.414034
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 1.41421 0.232495 0.116248 0.993220i \(-0.462913\pi\)
0.116248 + 0.993220i \(0.462913\pi\)
\(38\) 2.17157 0.352276
\(39\) −1.00000 −0.160128
\(40\) 0.585786 0.0926210
\(41\) 9.89949 1.54604 0.773021 0.634381i \(-0.218745\pi\)
0.773021 + 0.634381i \(0.218745\pi\)
\(42\) 0 0
\(43\) 6.48528 0.988996 0.494498 0.869179i \(-0.335352\pi\)
0.494498 + 0.869179i \(0.335352\pi\)
\(44\) −0.414214 −0.0624450
\(45\) 0.585786 0.0873239
\(46\) 1.41421 0.208514
\(47\) −1.00000 −0.145865 −0.0729325 0.997337i \(-0.523236\pi\)
−0.0729325 + 0.997337i \(0.523236\pi\)
\(48\) 1.00000 0.144338
\(49\) 0 0
\(50\) −4.65685 −0.658579
\(51\) 2.41421 0.338058
\(52\) −1.00000 −0.138675
\(53\) 9.48528 1.30290 0.651452 0.758690i \(-0.274160\pi\)
0.651452 + 0.758690i \(0.274160\pi\)
\(54\) 1.00000 0.136083
\(55\) −0.242641 −0.0327177
\(56\) 0 0
\(57\) 2.17157 0.287632
\(58\) −1.82843 −0.240084
\(59\) −2.07107 −0.269630 −0.134815 0.990871i \(-0.543044\pi\)
−0.134815 + 0.990871i \(0.543044\pi\)
\(60\) 0.585786 0.0756247
\(61\) 4.41421 0.565182 0.282591 0.959240i \(-0.408806\pi\)
0.282591 + 0.959240i \(0.408806\pi\)
\(62\) 8.48528 1.07763
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −0.585786 −0.0726579
\(66\) −0.414214 −0.0509862
\(67\) 1.82843 0.223378 0.111689 0.993743i \(-0.464374\pi\)
0.111689 + 0.993743i \(0.464374\pi\)
\(68\) 2.41421 0.292766
\(69\) 1.41421 0.170251
\(70\) 0 0
\(71\) −5.00000 −0.593391 −0.296695 0.954972i \(-0.595885\pi\)
−0.296695 + 0.954972i \(0.595885\pi\)
\(72\) 1.00000 0.117851
\(73\) −1.41421 −0.165521 −0.0827606 0.996569i \(-0.526374\pi\)
−0.0827606 + 0.996569i \(0.526374\pi\)
\(74\) 1.41421 0.164399
\(75\) −4.65685 −0.537727
\(76\) 2.17157 0.249096
\(77\) 0 0
\(78\) −1.00000 −0.113228
\(79\) −11.6569 −1.31150 −0.655749 0.754979i \(-0.727647\pi\)
−0.655749 + 0.754979i \(0.727647\pi\)
\(80\) 0.585786 0.0654929
\(81\) 1.00000 0.111111
\(82\) 9.89949 1.09322
\(83\) −7.65685 −0.840449 −0.420224 0.907420i \(-0.638049\pi\)
−0.420224 + 0.907420i \(0.638049\pi\)
\(84\) 0 0
\(85\) 1.41421 0.153393
\(86\) 6.48528 0.699326
\(87\) −1.82843 −0.196028
\(88\) −0.414214 −0.0441553
\(89\) −2.58579 −0.274093 −0.137046 0.990565i \(-0.543761\pi\)
−0.137046 + 0.990565i \(0.543761\pi\)
\(90\) 0.585786 0.0617473
\(91\) 0 0
\(92\) 1.41421 0.147442
\(93\) 8.48528 0.879883
\(94\) −1.00000 −0.103142
\(95\) 1.27208 0.130512
\(96\) 1.00000 0.102062
\(97\) 0.928932 0.0943188 0.0471594 0.998887i \(-0.484983\pi\)
0.0471594 + 0.998887i \(0.484983\pi\)
\(98\) 0 0
\(99\) −0.414214 −0.0416300
\(100\) −4.65685 −0.465685
\(101\) 14.8284 1.47548 0.737742 0.675083i \(-0.235892\pi\)
0.737742 + 0.675083i \(0.235892\pi\)
\(102\) 2.41421 0.239043
\(103\) 16.8284 1.65815 0.829077 0.559134i \(-0.188866\pi\)
0.829077 + 0.559134i \(0.188866\pi\)
\(104\) −1.00000 −0.0980581
\(105\) 0 0
\(106\) 9.48528 0.921292
\(107\) −19.3137 −1.86713 −0.933563 0.358412i \(-0.883318\pi\)
−0.933563 + 0.358412i \(0.883318\pi\)
\(108\) 1.00000 0.0962250
\(109\) −9.65685 −0.924959 −0.462479 0.886630i \(-0.653040\pi\)
−0.462479 + 0.886630i \(0.653040\pi\)
\(110\) −0.242641 −0.0231349
\(111\) 1.41421 0.134231
\(112\) 0 0
\(113\) 4.89949 0.460906 0.230453 0.973083i \(-0.425979\pi\)
0.230453 + 0.973083i \(0.425979\pi\)
\(114\) 2.17157 0.203386
\(115\) 0.828427 0.0772512
\(116\) −1.82843 −0.169765
\(117\) −1.00000 −0.0924500
\(118\) −2.07107 −0.190657
\(119\) 0 0
\(120\) 0.585786 0.0534747
\(121\) −10.8284 −0.984402
\(122\) 4.41421 0.399644
\(123\) 9.89949 0.892607
\(124\) 8.48528 0.762001
\(125\) −5.65685 −0.505964
\(126\) 0 0
\(127\) −7.89949 −0.700967 −0.350483 0.936569i \(-0.613983\pi\)
−0.350483 + 0.936569i \(0.613983\pi\)
\(128\) 1.00000 0.0883883
\(129\) 6.48528 0.570997
\(130\) −0.585786 −0.0513769
\(131\) 9.55635 0.834942 0.417471 0.908690i \(-0.362917\pi\)
0.417471 + 0.908690i \(0.362917\pi\)
\(132\) −0.414214 −0.0360527
\(133\) 0 0
\(134\) 1.82843 0.157952
\(135\) 0.585786 0.0504165
\(136\) 2.41421 0.207017
\(137\) 11.0711 0.945865 0.472933 0.881099i \(-0.343195\pi\)
0.472933 + 0.881099i \(0.343195\pi\)
\(138\) 1.41421 0.120386
\(139\) 13.0711 1.10867 0.554337 0.832292i \(-0.312972\pi\)
0.554337 + 0.832292i \(0.312972\pi\)
\(140\) 0 0
\(141\) −1.00000 −0.0842152
\(142\) −5.00000 −0.419591
\(143\) 0.414214 0.0346383
\(144\) 1.00000 0.0833333
\(145\) −1.07107 −0.0889473
\(146\) −1.41421 −0.117041
\(147\) 0 0
\(148\) 1.41421 0.116248
\(149\) −4.92893 −0.403794 −0.201897 0.979407i \(-0.564711\pi\)
−0.201897 + 0.979407i \(0.564711\pi\)
\(150\) −4.65685 −0.380231
\(151\) 4.07107 0.331299 0.165649 0.986185i \(-0.447028\pi\)
0.165649 + 0.986185i \(0.447028\pi\)
\(152\) 2.17157 0.176138
\(153\) 2.41421 0.195178
\(154\) 0 0
\(155\) 4.97056 0.399245
\(156\) −1.00000 −0.0800641
\(157\) −9.72792 −0.776373 −0.388186 0.921581i \(-0.626898\pi\)
−0.388186 + 0.921581i \(0.626898\pi\)
\(158\) −11.6569 −0.927370
\(159\) 9.48528 0.752232
\(160\) 0.585786 0.0463105
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) −18.6569 −1.46132 −0.730659 0.682743i \(-0.760787\pi\)
−0.730659 + 0.682743i \(0.760787\pi\)
\(164\) 9.89949 0.773021
\(165\) −0.242641 −0.0188896
\(166\) −7.65685 −0.594287
\(167\) 0.656854 0.0508289 0.0254145 0.999677i \(-0.491909\pi\)
0.0254145 + 0.999677i \(0.491909\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 1.41421 0.108465
\(171\) 2.17157 0.166064
\(172\) 6.48528 0.494498
\(173\) −5.48528 −0.417038 −0.208519 0.978018i \(-0.566864\pi\)
−0.208519 + 0.978018i \(0.566864\pi\)
\(174\) −1.82843 −0.138613
\(175\) 0 0
\(176\) −0.414214 −0.0312225
\(177\) −2.07107 −0.155671
\(178\) −2.58579 −0.193813
\(179\) 19.6569 1.46922 0.734611 0.678488i \(-0.237365\pi\)
0.734611 + 0.678488i \(0.237365\pi\)
\(180\) 0.585786 0.0436619
\(181\) −2.89949 −0.215518 −0.107759 0.994177i \(-0.534367\pi\)
−0.107759 + 0.994177i \(0.534367\pi\)
\(182\) 0 0
\(183\) 4.41421 0.326308
\(184\) 1.41421 0.104257
\(185\) 0.828427 0.0609072
\(186\) 8.48528 0.622171
\(187\) −1.00000 −0.0731272
\(188\) −1.00000 −0.0729325
\(189\) 0 0
\(190\) 1.27208 0.0922862
\(191\) −10.3431 −0.748404 −0.374202 0.927347i \(-0.622083\pi\)
−0.374202 + 0.927347i \(0.622083\pi\)
\(192\) 1.00000 0.0721688
\(193\) −2.92893 −0.210829 −0.105415 0.994428i \(-0.533617\pi\)
−0.105415 + 0.994428i \(0.533617\pi\)
\(194\) 0.928932 0.0666934
\(195\) −0.585786 −0.0419490
\(196\) 0 0
\(197\) −15.5563 −1.10834 −0.554172 0.832402i \(-0.686965\pi\)
−0.554172 + 0.832402i \(0.686965\pi\)
\(198\) −0.414214 −0.0294369
\(199\) −2.72792 −0.193377 −0.0966886 0.995315i \(-0.530825\pi\)
−0.0966886 + 0.995315i \(0.530825\pi\)
\(200\) −4.65685 −0.329289
\(201\) 1.82843 0.128967
\(202\) 14.8284 1.04332
\(203\) 0 0
\(204\) 2.41421 0.169029
\(205\) 5.79899 0.405019
\(206\) 16.8284 1.17249
\(207\) 1.41421 0.0982946
\(208\) −1.00000 −0.0693375
\(209\) −0.899495 −0.0622194
\(210\) 0 0
\(211\) −12.9289 −0.890064 −0.445032 0.895515i \(-0.646808\pi\)
−0.445032 + 0.895515i \(0.646808\pi\)
\(212\) 9.48528 0.651452
\(213\) −5.00000 −0.342594
\(214\) −19.3137 −1.32026
\(215\) 3.79899 0.259089
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) −9.65685 −0.654045
\(219\) −1.41421 −0.0955637
\(220\) −0.242641 −0.0163588
\(221\) −2.41421 −0.162398
\(222\) 1.41421 0.0949158
\(223\) 7.92893 0.530961 0.265480 0.964116i \(-0.414469\pi\)
0.265480 + 0.964116i \(0.414469\pi\)
\(224\) 0 0
\(225\) −4.65685 −0.310457
\(226\) 4.89949 0.325910
\(227\) 21.1716 1.40521 0.702603 0.711582i \(-0.252021\pi\)
0.702603 + 0.711582i \(0.252021\pi\)
\(228\) 2.17157 0.143816
\(229\) −26.4853 −1.75020 −0.875098 0.483945i \(-0.839203\pi\)
−0.875098 + 0.483945i \(0.839203\pi\)
\(230\) 0.828427 0.0546249
\(231\) 0 0
\(232\) −1.82843 −0.120042
\(233\) −17.7279 −1.16139 −0.580697 0.814119i \(-0.697220\pi\)
−0.580697 + 0.814119i \(0.697220\pi\)
\(234\) −1.00000 −0.0653720
\(235\) −0.585786 −0.0382125
\(236\) −2.07107 −0.134815
\(237\) −11.6569 −0.757194
\(238\) 0 0
\(239\) 4.51472 0.292033 0.146016 0.989282i \(-0.453355\pi\)
0.146016 + 0.989282i \(0.453355\pi\)
\(240\) 0.585786 0.0378124
\(241\) 2.14214 0.137987 0.0689935 0.997617i \(-0.478021\pi\)
0.0689935 + 0.997617i \(0.478021\pi\)
\(242\) −10.8284 −0.696078
\(243\) 1.00000 0.0641500
\(244\) 4.41421 0.282591
\(245\) 0 0
\(246\) 9.89949 0.631169
\(247\) −2.17157 −0.138174
\(248\) 8.48528 0.538816
\(249\) −7.65685 −0.485233
\(250\) −5.65685 −0.357771
\(251\) −24.8284 −1.56716 −0.783578 0.621293i \(-0.786608\pi\)
−0.783578 + 0.621293i \(0.786608\pi\)
\(252\) 0 0
\(253\) −0.585786 −0.0368281
\(254\) −7.89949 −0.495658
\(255\) 1.41421 0.0885615
\(256\) 1.00000 0.0625000
\(257\) −26.4853 −1.65211 −0.826053 0.563592i \(-0.809419\pi\)
−0.826053 + 0.563592i \(0.809419\pi\)
\(258\) 6.48528 0.403756
\(259\) 0 0
\(260\) −0.585786 −0.0363289
\(261\) −1.82843 −0.113177
\(262\) 9.55635 0.590393
\(263\) 20.5858 1.26937 0.634687 0.772769i \(-0.281129\pi\)
0.634687 + 0.772769i \(0.281129\pi\)
\(264\) −0.414214 −0.0254931
\(265\) 5.55635 0.341324
\(266\) 0 0
\(267\) −2.58579 −0.158248
\(268\) 1.82843 0.111689
\(269\) 9.14214 0.557406 0.278703 0.960377i \(-0.410096\pi\)
0.278703 + 0.960377i \(0.410096\pi\)
\(270\) 0.585786 0.0356498
\(271\) −26.2132 −1.59234 −0.796169 0.605074i \(-0.793144\pi\)
−0.796169 + 0.605074i \(0.793144\pi\)
\(272\) 2.41421 0.146383
\(273\) 0 0
\(274\) 11.0711 0.668828
\(275\) 1.92893 0.116319
\(276\) 1.41421 0.0851257
\(277\) 18.0711 1.08579 0.542893 0.839802i \(-0.317329\pi\)
0.542893 + 0.839802i \(0.317329\pi\)
\(278\) 13.0711 0.783951
\(279\) 8.48528 0.508001
\(280\) 0 0
\(281\) 0.686292 0.0409407 0.0204704 0.999790i \(-0.493484\pi\)
0.0204704 + 0.999790i \(0.493484\pi\)
\(282\) −1.00000 −0.0595491
\(283\) 15.7574 0.936678 0.468339 0.883549i \(-0.344853\pi\)
0.468339 + 0.883549i \(0.344853\pi\)
\(284\) −5.00000 −0.296695
\(285\) 1.27208 0.0753514
\(286\) 0.414214 0.0244930
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) −11.1716 −0.657151
\(290\) −1.07107 −0.0628953
\(291\) 0.928932 0.0544550
\(292\) −1.41421 −0.0827606
\(293\) 9.89949 0.578335 0.289167 0.957279i \(-0.406622\pi\)
0.289167 + 0.957279i \(0.406622\pi\)
\(294\) 0 0
\(295\) −1.21320 −0.0706354
\(296\) 1.41421 0.0821995
\(297\) −0.414214 −0.0240351
\(298\) −4.92893 −0.285525
\(299\) −1.41421 −0.0817861
\(300\) −4.65685 −0.268864
\(301\) 0 0
\(302\) 4.07107 0.234264
\(303\) 14.8284 0.851871
\(304\) 2.17157 0.124548
\(305\) 2.58579 0.148062
\(306\) 2.41421 0.138011
\(307\) 28.1127 1.60448 0.802238 0.597004i \(-0.203642\pi\)
0.802238 + 0.597004i \(0.203642\pi\)
\(308\) 0 0
\(309\) 16.8284 0.957336
\(310\) 4.97056 0.282309
\(311\) 8.10051 0.459338 0.229669 0.973269i \(-0.426236\pi\)
0.229669 + 0.973269i \(0.426236\pi\)
\(312\) −1.00000 −0.0566139
\(313\) −3.51472 −0.198664 −0.0993318 0.995054i \(-0.531671\pi\)
−0.0993318 + 0.995054i \(0.531671\pi\)
\(314\) −9.72792 −0.548978
\(315\) 0 0
\(316\) −11.6569 −0.655749
\(317\) 29.7990 1.67368 0.836839 0.547449i \(-0.184401\pi\)
0.836839 + 0.547449i \(0.184401\pi\)
\(318\) 9.48528 0.531908
\(319\) 0.757359 0.0424040
\(320\) 0.585786 0.0327465
\(321\) −19.3137 −1.07799
\(322\) 0 0
\(323\) 5.24264 0.291708
\(324\) 1.00000 0.0555556
\(325\) 4.65685 0.258316
\(326\) −18.6569 −1.03331
\(327\) −9.65685 −0.534025
\(328\) 9.89949 0.546608
\(329\) 0 0
\(330\) −0.242641 −0.0133569
\(331\) −22.4853 −1.23590 −0.617951 0.786216i \(-0.712037\pi\)
−0.617951 + 0.786216i \(0.712037\pi\)
\(332\) −7.65685 −0.420224
\(333\) 1.41421 0.0774984
\(334\) 0.656854 0.0359415
\(335\) 1.07107 0.0585187
\(336\) 0 0
\(337\) −20.3137 −1.10656 −0.553279 0.832996i \(-0.686624\pi\)
−0.553279 + 0.832996i \(0.686624\pi\)
\(338\) 1.00000 0.0543928
\(339\) 4.89949 0.266104
\(340\) 1.41421 0.0766965
\(341\) −3.51472 −0.190333
\(342\) 2.17157 0.117425
\(343\) 0 0
\(344\) 6.48528 0.349663
\(345\) 0.828427 0.0446010
\(346\) −5.48528 −0.294891
\(347\) −8.72792 −0.468539 −0.234270 0.972172i \(-0.575270\pi\)
−0.234270 + 0.972172i \(0.575270\pi\)
\(348\) −1.82843 −0.0980140
\(349\) 6.72792 0.360137 0.180069 0.983654i \(-0.442368\pi\)
0.180069 + 0.983654i \(0.442368\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) −0.414214 −0.0220777
\(353\) −12.6274 −0.672090 −0.336045 0.941846i \(-0.609089\pi\)
−0.336045 + 0.941846i \(0.609089\pi\)
\(354\) −2.07107 −0.110076
\(355\) −2.92893 −0.155452
\(356\) −2.58579 −0.137046
\(357\) 0 0
\(358\) 19.6569 1.03890
\(359\) −0.828427 −0.0437227 −0.0218614 0.999761i \(-0.506959\pi\)
−0.0218614 + 0.999761i \(0.506959\pi\)
\(360\) 0.585786 0.0308737
\(361\) −14.2843 −0.751804
\(362\) −2.89949 −0.152394
\(363\) −10.8284 −0.568345
\(364\) 0 0
\(365\) −0.828427 −0.0433619
\(366\) 4.41421 0.230735
\(367\) −12.6274 −0.659146 −0.329573 0.944130i \(-0.606905\pi\)
−0.329573 + 0.944130i \(0.606905\pi\)
\(368\) 1.41421 0.0737210
\(369\) 9.89949 0.515347
\(370\) 0.828427 0.0430679
\(371\) 0 0
\(372\) 8.48528 0.439941
\(373\) −5.58579 −0.289221 −0.144611 0.989489i \(-0.546193\pi\)
−0.144611 + 0.989489i \(0.546193\pi\)
\(374\) −1.00000 −0.0517088
\(375\) −5.65685 −0.292119
\(376\) −1.00000 −0.0515711
\(377\) 1.82843 0.0941688
\(378\) 0 0
\(379\) 18.6274 0.956826 0.478413 0.878135i \(-0.341212\pi\)
0.478413 + 0.878135i \(0.341212\pi\)
\(380\) 1.27208 0.0652562
\(381\) −7.89949 −0.404703
\(382\) −10.3431 −0.529201
\(383\) −11.1716 −0.570841 −0.285420 0.958402i \(-0.592133\pi\)
−0.285420 + 0.958402i \(0.592133\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −2.92893 −0.149079
\(387\) 6.48528 0.329665
\(388\) 0.928932 0.0471594
\(389\) −4.51472 −0.228905 −0.114453 0.993429i \(-0.536511\pi\)
−0.114453 + 0.993429i \(0.536511\pi\)
\(390\) −0.585786 −0.0296624
\(391\) 3.41421 0.172664
\(392\) 0 0
\(393\) 9.55635 0.482054
\(394\) −15.5563 −0.783718
\(395\) −6.82843 −0.343575
\(396\) −0.414214 −0.0208150
\(397\) 9.55635 0.479619 0.239810 0.970820i \(-0.422915\pi\)
0.239810 + 0.970820i \(0.422915\pi\)
\(398\) −2.72792 −0.136738
\(399\) 0 0
\(400\) −4.65685 −0.232843
\(401\) 19.8995 0.993733 0.496867 0.867827i \(-0.334484\pi\)
0.496867 + 0.867827i \(0.334484\pi\)
\(402\) 1.82843 0.0911937
\(403\) −8.48528 −0.422682
\(404\) 14.8284 0.737742
\(405\) 0.585786 0.0291080
\(406\) 0 0
\(407\) −0.585786 −0.0290364
\(408\) 2.41421 0.119521
\(409\) −32.5858 −1.61126 −0.805632 0.592417i \(-0.798174\pi\)
−0.805632 + 0.592417i \(0.798174\pi\)
\(410\) 5.79899 0.286392
\(411\) 11.0711 0.546096
\(412\) 16.8284 0.829077
\(413\) 0 0
\(414\) 1.41421 0.0695048
\(415\) −4.48528 −0.220174
\(416\) −1.00000 −0.0490290
\(417\) 13.0711 0.640093
\(418\) −0.899495 −0.0439957
\(419\) −27.4558 −1.34131 −0.670653 0.741771i \(-0.733986\pi\)
−0.670653 + 0.741771i \(0.733986\pi\)
\(420\) 0 0
\(421\) 1.07107 0.0522007 0.0261003 0.999659i \(-0.491691\pi\)
0.0261003 + 0.999659i \(0.491691\pi\)
\(422\) −12.9289 −0.629371
\(423\) −1.00000 −0.0486217
\(424\) 9.48528 0.460646
\(425\) −11.2426 −0.545348
\(426\) −5.00000 −0.242251
\(427\) 0 0
\(428\) −19.3137 −0.933563
\(429\) 0.414214 0.0199984
\(430\) 3.79899 0.183204
\(431\) 30.4853 1.46842 0.734212 0.678920i \(-0.237552\pi\)
0.734212 + 0.678920i \(0.237552\pi\)
\(432\) 1.00000 0.0481125
\(433\) −29.9706 −1.44029 −0.720147 0.693822i \(-0.755926\pi\)
−0.720147 + 0.693822i \(0.755926\pi\)
\(434\) 0 0
\(435\) −1.07107 −0.0513538
\(436\) −9.65685 −0.462479
\(437\) 3.07107 0.146909
\(438\) −1.41421 −0.0675737
\(439\) −16.5858 −0.791596 −0.395798 0.918338i \(-0.629532\pi\)
−0.395798 + 0.918338i \(0.629532\pi\)
\(440\) −0.242641 −0.0115674
\(441\) 0 0
\(442\) −2.41421 −0.114832
\(443\) −33.6985 −1.60106 −0.800532 0.599290i \(-0.795449\pi\)
−0.800532 + 0.599290i \(0.795449\pi\)
\(444\) 1.41421 0.0671156
\(445\) −1.51472 −0.0718045
\(446\) 7.92893 0.375446
\(447\) −4.92893 −0.233130
\(448\) 0 0
\(449\) 24.9706 1.17843 0.589217 0.807975i \(-0.299436\pi\)
0.589217 + 0.807975i \(0.299436\pi\)
\(450\) −4.65685 −0.219526
\(451\) −4.10051 −0.193085
\(452\) 4.89949 0.230453
\(453\) 4.07107 0.191275
\(454\) 21.1716 0.993631
\(455\) 0 0
\(456\) 2.17157 0.101693
\(457\) −3.07107 −0.143658 −0.0718292 0.997417i \(-0.522884\pi\)
−0.0718292 + 0.997417i \(0.522884\pi\)
\(458\) −26.4853 −1.23758
\(459\) 2.41421 0.112686
\(460\) 0.828427 0.0386256
\(461\) 4.48528 0.208900 0.104450 0.994530i \(-0.466692\pi\)
0.104450 + 0.994530i \(0.466692\pi\)
\(462\) 0 0
\(463\) −20.9706 −0.974585 −0.487292 0.873239i \(-0.662015\pi\)
−0.487292 + 0.873239i \(0.662015\pi\)
\(464\) −1.82843 −0.0848826
\(465\) 4.97056 0.230504
\(466\) −17.7279 −0.821230
\(467\) 11.4142 0.528187 0.264093 0.964497i \(-0.414927\pi\)
0.264093 + 0.964497i \(0.414927\pi\)
\(468\) −1.00000 −0.0462250
\(469\) 0 0
\(470\) −0.585786 −0.0270203
\(471\) −9.72792 −0.448239
\(472\) −2.07107 −0.0953286
\(473\) −2.68629 −0.123516
\(474\) −11.6569 −0.535417
\(475\) −10.1127 −0.464002
\(476\) 0 0
\(477\) 9.48528 0.434301
\(478\) 4.51472 0.206498
\(479\) 32.1127 1.46727 0.733633 0.679546i \(-0.237823\pi\)
0.733633 + 0.679546i \(0.237823\pi\)
\(480\) 0.585786 0.0267374
\(481\) −1.41421 −0.0644826
\(482\) 2.14214 0.0975716
\(483\) 0 0
\(484\) −10.8284 −0.492201
\(485\) 0.544156 0.0247088
\(486\) 1.00000 0.0453609
\(487\) 16.2132 0.734690 0.367345 0.930085i \(-0.380267\pi\)
0.367345 + 0.930085i \(0.380267\pi\)
\(488\) 4.41421 0.199822
\(489\) −18.6569 −0.843692
\(490\) 0 0
\(491\) 23.6569 1.06762 0.533809 0.845605i \(-0.320760\pi\)
0.533809 + 0.845605i \(0.320760\pi\)
\(492\) 9.89949 0.446304
\(493\) −4.41421 −0.198806
\(494\) −2.17157 −0.0977037
\(495\) −0.242641 −0.0109059
\(496\) 8.48528 0.381000
\(497\) 0 0
\(498\) −7.65685 −0.343112
\(499\) 3.65685 0.163703 0.0818516 0.996645i \(-0.473917\pi\)
0.0818516 + 0.996645i \(0.473917\pi\)
\(500\) −5.65685 −0.252982
\(501\) 0.656854 0.0293461
\(502\) −24.8284 −1.10815
\(503\) −26.1421 −1.16562 −0.582810 0.812608i \(-0.698047\pi\)
−0.582810 + 0.812608i \(0.698047\pi\)
\(504\) 0 0
\(505\) 8.68629 0.386535
\(506\) −0.585786 −0.0260414
\(507\) 1.00000 0.0444116
\(508\) −7.89949 −0.350483
\(509\) −10.6274 −0.471052 −0.235526 0.971868i \(-0.575681\pi\)
−0.235526 + 0.971868i \(0.575681\pi\)
\(510\) 1.41421 0.0626224
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 2.17157 0.0958773
\(514\) −26.4853 −1.16822
\(515\) 9.85786 0.434389
\(516\) 6.48528 0.285499
\(517\) 0.414214 0.0182171
\(518\) 0 0
\(519\) −5.48528 −0.240777
\(520\) −0.585786 −0.0256884
\(521\) −40.6274 −1.77992 −0.889960 0.456039i \(-0.849268\pi\)
−0.889960 + 0.456039i \(0.849268\pi\)
\(522\) −1.82843 −0.0800281
\(523\) 16.8284 0.735856 0.367928 0.929854i \(-0.380067\pi\)
0.367928 + 0.929854i \(0.380067\pi\)
\(524\) 9.55635 0.417471
\(525\) 0 0
\(526\) 20.5858 0.897583
\(527\) 20.4853 0.892353
\(528\) −0.414214 −0.0180263
\(529\) −21.0000 −0.913043
\(530\) 5.55635 0.241352
\(531\) −2.07107 −0.0898767
\(532\) 0 0
\(533\) −9.89949 −0.428795
\(534\) −2.58579 −0.111898
\(535\) −11.3137 −0.489134
\(536\) 1.82843 0.0789760
\(537\) 19.6569 0.848256
\(538\) 9.14214 0.394145
\(539\) 0 0
\(540\) 0.585786 0.0252082
\(541\) 23.0711 0.991903 0.495951 0.868350i \(-0.334819\pi\)
0.495951 + 0.868350i \(0.334819\pi\)
\(542\) −26.2132 −1.12595
\(543\) −2.89949 −0.124429
\(544\) 2.41421 0.103509
\(545\) −5.65685 −0.242313
\(546\) 0 0
\(547\) −30.8701 −1.31991 −0.659954 0.751306i \(-0.729424\pi\)
−0.659954 + 0.751306i \(0.729424\pi\)
\(548\) 11.0711 0.472933
\(549\) 4.41421 0.188394
\(550\) 1.92893 0.0822499
\(551\) −3.97056 −0.169152
\(552\) 1.41421 0.0601929
\(553\) 0 0
\(554\) 18.0711 0.767766
\(555\) 0.828427 0.0351648
\(556\) 13.0711 0.554337
\(557\) −32.6274 −1.38247 −0.691234 0.722631i \(-0.742933\pi\)
−0.691234 + 0.722631i \(0.742933\pi\)
\(558\) 8.48528 0.359211
\(559\) −6.48528 −0.274298
\(560\) 0 0
\(561\) −1.00000 −0.0422200
\(562\) 0.686292 0.0289495
\(563\) 32.7696 1.38107 0.690536 0.723298i \(-0.257375\pi\)
0.690536 + 0.723298i \(0.257375\pi\)
\(564\) −1.00000 −0.0421076
\(565\) 2.87006 0.120744
\(566\) 15.7574 0.662331
\(567\) 0 0
\(568\) −5.00000 −0.209795
\(569\) 23.7279 0.994726 0.497363 0.867542i \(-0.334302\pi\)
0.497363 + 0.867542i \(0.334302\pi\)
\(570\) 1.27208 0.0532815
\(571\) −31.3137 −1.31044 −0.655219 0.755439i \(-0.727424\pi\)
−0.655219 + 0.755439i \(0.727424\pi\)
\(572\) 0.414214 0.0173191
\(573\) −10.3431 −0.432091
\(574\) 0 0
\(575\) −6.58579 −0.274646
\(576\) 1.00000 0.0416667
\(577\) −16.3431 −0.680374 −0.340187 0.940358i \(-0.610490\pi\)
−0.340187 + 0.940358i \(0.610490\pi\)
\(578\) −11.1716 −0.464676
\(579\) −2.92893 −0.121722
\(580\) −1.07107 −0.0444737
\(581\) 0 0
\(582\) 0.928932 0.0385055
\(583\) −3.92893 −0.162720
\(584\) −1.41421 −0.0585206
\(585\) −0.585786 −0.0242193
\(586\) 9.89949 0.408944
\(587\) 1.58579 0.0654524 0.0327262 0.999464i \(-0.489581\pi\)
0.0327262 + 0.999464i \(0.489581\pi\)
\(588\) 0 0
\(589\) 18.4264 0.759247
\(590\) −1.21320 −0.0499468
\(591\) −15.5563 −0.639903
\(592\) 1.41421 0.0581238
\(593\) 34.3848 1.41201 0.706007 0.708205i \(-0.250495\pi\)
0.706007 + 0.708205i \(0.250495\pi\)
\(594\) −0.414214 −0.0169954
\(595\) 0 0
\(596\) −4.92893 −0.201897
\(597\) −2.72792 −0.111646
\(598\) −1.41421 −0.0578315
\(599\) 1.55635 0.0635907 0.0317954 0.999494i \(-0.489878\pi\)
0.0317954 + 0.999494i \(0.489878\pi\)
\(600\) −4.65685 −0.190115
\(601\) −1.20101 −0.0489902 −0.0244951 0.999700i \(-0.507798\pi\)
−0.0244951 + 0.999700i \(0.507798\pi\)
\(602\) 0 0
\(603\) 1.82843 0.0744593
\(604\) 4.07107 0.165649
\(605\) −6.34315 −0.257886
\(606\) 14.8284 0.602364
\(607\) −34.5269 −1.40140 −0.700702 0.713454i \(-0.747130\pi\)
−0.700702 + 0.713454i \(0.747130\pi\)
\(608\) 2.17157 0.0880689
\(609\) 0 0
\(610\) 2.58579 0.104695
\(611\) 1.00000 0.0404557
\(612\) 2.41421 0.0975888
\(613\) −7.79899 −0.314998 −0.157499 0.987519i \(-0.550343\pi\)
−0.157499 + 0.987519i \(0.550343\pi\)
\(614\) 28.1127 1.13454
\(615\) 5.79899 0.233838
\(616\) 0 0
\(617\) −24.1421 −0.971926 −0.485963 0.873979i \(-0.661531\pi\)
−0.485963 + 0.873979i \(0.661531\pi\)
\(618\) 16.8284 0.676939
\(619\) −20.8284 −0.837165 −0.418583 0.908179i \(-0.637473\pi\)
−0.418583 + 0.908179i \(0.637473\pi\)
\(620\) 4.97056 0.199623
\(621\) 1.41421 0.0567504
\(622\) 8.10051 0.324801
\(623\) 0 0
\(624\) −1.00000 −0.0400320
\(625\) 19.9706 0.798823
\(626\) −3.51472 −0.140476
\(627\) −0.899495 −0.0359224
\(628\) −9.72792 −0.388186
\(629\) 3.41421 0.136134
\(630\) 0 0
\(631\) 32.6274 1.29888 0.649438 0.760414i \(-0.275004\pi\)
0.649438 + 0.760414i \(0.275004\pi\)
\(632\) −11.6569 −0.463685
\(633\) −12.9289 −0.513879
\(634\) 29.7990 1.18347
\(635\) −4.62742 −0.183633
\(636\) 9.48528 0.376116
\(637\) 0 0
\(638\) 0.757359 0.0299841
\(639\) −5.00000 −0.197797
\(640\) 0.585786 0.0231552
\(641\) 31.7990 1.25598 0.627992 0.778220i \(-0.283877\pi\)
0.627992 + 0.778220i \(0.283877\pi\)
\(642\) −19.3137 −0.762251
\(643\) 2.51472 0.0991708 0.0495854 0.998770i \(-0.484210\pi\)
0.0495854 + 0.998770i \(0.484210\pi\)
\(644\) 0 0
\(645\) 3.79899 0.149585
\(646\) 5.24264 0.206269
\(647\) 9.02944 0.354984 0.177492 0.984122i \(-0.443202\pi\)
0.177492 + 0.984122i \(0.443202\pi\)
\(648\) 1.00000 0.0392837
\(649\) 0.857864 0.0336741
\(650\) 4.65685 0.182657
\(651\) 0 0
\(652\) −18.6569 −0.730659
\(653\) 22.1421 0.866489 0.433244 0.901276i \(-0.357369\pi\)
0.433244 + 0.901276i \(0.357369\pi\)
\(654\) −9.65685 −0.377613
\(655\) 5.59798 0.218731
\(656\) 9.89949 0.386510
\(657\) −1.41421 −0.0551737
\(658\) 0 0
\(659\) 10.5858 0.412364 0.206182 0.978514i \(-0.433896\pi\)
0.206182 + 0.978514i \(0.433896\pi\)
\(660\) −0.242641 −0.00944478
\(661\) −40.2426 −1.56526 −0.782629 0.622489i \(-0.786122\pi\)
−0.782629 + 0.622489i \(0.786122\pi\)
\(662\) −22.4853 −0.873915
\(663\) −2.41421 −0.0937603
\(664\) −7.65685 −0.297144
\(665\) 0 0
\(666\) 1.41421 0.0547997
\(667\) −2.58579 −0.100122
\(668\) 0.656854 0.0254145
\(669\) 7.92893 0.306550
\(670\) 1.07107 0.0413790
\(671\) −1.82843 −0.0705856
\(672\) 0 0
\(673\) 42.1421 1.62446 0.812230 0.583337i \(-0.198253\pi\)
0.812230 + 0.583337i \(0.198253\pi\)
\(674\) −20.3137 −0.782455
\(675\) −4.65685 −0.179242
\(676\) 1.00000 0.0384615
\(677\) −15.0000 −0.576497 −0.288248 0.957556i \(-0.593073\pi\)
−0.288248 + 0.957556i \(0.593073\pi\)
\(678\) 4.89949 0.188164
\(679\) 0 0
\(680\) 1.41421 0.0542326
\(681\) 21.1716 0.811296
\(682\) −3.51472 −0.134586
\(683\) −14.1421 −0.541134 −0.270567 0.962701i \(-0.587211\pi\)
−0.270567 + 0.962701i \(0.587211\pi\)
\(684\) 2.17157 0.0830322
\(685\) 6.48528 0.247790
\(686\) 0 0
\(687\) −26.4853 −1.01048
\(688\) 6.48528 0.247249
\(689\) −9.48528 −0.361360
\(690\) 0.828427 0.0315377
\(691\) 20.6569 0.785824 0.392912 0.919576i \(-0.371468\pi\)
0.392912 + 0.919576i \(0.371468\pi\)
\(692\) −5.48528 −0.208519
\(693\) 0 0
\(694\) −8.72792 −0.331307
\(695\) 7.65685 0.290441
\(696\) −1.82843 −0.0693064
\(697\) 23.8995 0.905258
\(698\) 6.72792 0.254656
\(699\) −17.7279 −0.670532
\(700\) 0 0
\(701\) −29.1716 −1.10180 −0.550898 0.834573i \(-0.685715\pi\)
−0.550898 + 0.834573i \(0.685715\pi\)
\(702\) −1.00000 −0.0377426
\(703\) 3.07107 0.115828
\(704\) −0.414214 −0.0156113
\(705\) −0.585786 −0.0220620
\(706\) −12.6274 −0.475239
\(707\) 0 0
\(708\) −2.07107 −0.0778355
\(709\) −36.8701 −1.38468 −0.692342 0.721569i \(-0.743421\pi\)
−0.692342 + 0.721569i \(0.743421\pi\)
\(710\) −2.92893 −0.109921
\(711\) −11.6569 −0.437166
\(712\) −2.58579 −0.0969064
\(713\) 12.0000 0.449404
\(714\) 0 0
\(715\) 0.242641 0.00907425
\(716\) 19.6569 0.734611
\(717\) 4.51472 0.168605
\(718\) −0.828427 −0.0309166
\(719\) −9.31371 −0.347343 −0.173671 0.984804i \(-0.555563\pi\)
−0.173671 + 0.984804i \(0.555563\pi\)
\(720\) 0.585786 0.0218310
\(721\) 0 0
\(722\) −14.2843 −0.531606
\(723\) 2.14214 0.0796669
\(724\) −2.89949 −0.107759
\(725\) 8.51472 0.316229
\(726\) −10.8284 −0.401881
\(727\) 50.4264 1.87021 0.935106 0.354368i \(-0.115304\pi\)
0.935106 + 0.354368i \(0.115304\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −0.828427 −0.0306615
\(731\) 15.6569 0.579090
\(732\) 4.41421 0.163154
\(733\) 28.5858 1.05584 0.527920 0.849294i \(-0.322972\pi\)
0.527920 + 0.849294i \(0.322972\pi\)
\(734\) −12.6274 −0.466086
\(735\) 0 0
\(736\) 1.41421 0.0521286
\(737\) −0.757359 −0.0278977
\(738\) 9.89949 0.364405
\(739\) −30.0000 −1.10357 −0.551784 0.833987i \(-0.686053\pi\)
−0.551784 + 0.833987i \(0.686053\pi\)
\(740\) 0.828427 0.0304536
\(741\) −2.17157 −0.0797747
\(742\) 0 0
\(743\) 12.1716 0.446532 0.223266 0.974758i \(-0.428328\pi\)
0.223266 + 0.974758i \(0.428328\pi\)
\(744\) 8.48528 0.311086
\(745\) −2.88730 −0.105783
\(746\) −5.58579 −0.204510
\(747\) −7.65685 −0.280150
\(748\) −1.00000 −0.0365636
\(749\) 0 0
\(750\) −5.65685 −0.206559
\(751\) −19.3137 −0.704767 −0.352384 0.935856i \(-0.614629\pi\)
−0.352384 + 0.935856i \(0.614629\pi\)
\(752\) −1.00000 −0.0364662
\(753\) −24.8284 −0.904798
\(754\) 1.82843 0.0665874
\(755\) 2.38478 0.0867909
\(756\) 0 0
\(757\) −30.7574 −1.11790 −0.558948 0.829203i \(-0.688795\pi\)
−0.558948 + 0.829203i \(0.688795\pi\)
\(758\) 18.6274 0.676578
\(759\) −0.585786 −0.0212627
\(760\) 1.27208 0.0461431
\(761\) −43.1127 −1.56283 −0.781417 0.624009i \(-0.785503\pi\)
−0.781417 + 0.624009i \(0.785503\pi\)
\(762\) −7.89949 −0.286169
\(763\) 0 0
\(764\) −10.3431 −0.374202
\(765\) 1.41421 0.0511310
\(766\) −11.1716 −0.403645
\(767\) 2.07107 0.0747819
\(768\) 1.00000 0.0360844
\(769\) −9.89949 −0.356985 −0.178492 0.983941i \(-0.557122\pi\)
−0.178492 + 0.983941i \(0.557122\pi\)
\(770\) 0 0
\(771\) −26.4853 −0.953844
\(772\) −2.92893 −0.105415
\(773\) −18.9706 −0.682324 −0.341162 0.940005i \(-0.610820\pi\)
−0.341162 + 0.940005i \(0.610820\pi\)
\(774\) 6.48528 0.233109
\(775\) −39.5147 −1.41941
\(776\) 0.928932 0.0333467
\(777\) 0 0
\(778\) −4.51472 −0.161861
\(779\) 21.4975 0.770227
\(780\) −0.585786 −0.0209745
\(781\) 2.07107 0.0741086
\(782\) 3.41421 0.122092
\(783\) −1.82843 −0.0653427
\(784\) 0 0
\(785\) −5.69848 −0.203388
\(786\) 9.55635 0.340864
\(787\) 26.3137 0.937982 0.468991 0.883203i \(-0.344618\pi\)
0.468991 + 0.883203i \(0.344618\pi\)
\(788\) −15.5563 −0.554172
\(789\) 20.5858 0.732873
\(790\) −6.82843 −0.242945
\(791\) 0 0
\(792\) −0.414214 −0.0147184
\(793\) −4.41421 −0.156753
\(794\) 9.55635 0.339142
\(795\) 5.55635 0.197063
\(796\) −2.72792 −0.0966886
\(797\) −34.3431 −1.21650 −0.608248 0.793747i \(-0.708128\pi\)
−0.608248 + 0.793747i \(0.708128\pi\)
\(798\) 0 0
\(799\) −2.41421 −0.0854087
\(800\) −4.65685 −0.164645
\(801\) −2.58579 −0.0913643
\(802\) 19.8995 0.702676
\(803\) 0.585786 0.0206720
\(804\) 1.82843 0.0644837
\(805\) 0 0
\(806\) −8.48528 −0.298881
\(807\) 9.14214 0.321818
\(808\) 14.8284 0.521662
\(809\) −17.5858 −0.618283 −0.309142 0.951016i \(-0.600042\pi\)
−0.309142 + 0.951016i \(0.600042\pi\)
\(810\) 0.585786 0.0205824
\(811\) −55.4558 −1.94732 −0.973659 0.228009i \(-0.926778\pi\)
−0.973659 + 0.228009i \(0.926778\pi\)
\(812\) 0 0
\(813\) −26.2132 −0.919337
\(814\) −0.585786 −0.0205318
\(815\) −10.9289 −0.382824
\(816\) 2.41421 0.0845144
\(817\) 14.0833 0.492711
\(818\) −32.5858 −1.13934
\(819\) 0 0
\(820\) 5.79899 0.202510
\(821\) 47.8995 1.67170 0.835852 0.548955i \(-0.184974\pi\)
0.835852 + 0.548955i \(0.184974\pi\)
\(822\) 11.0711 0.386148
\(823\) −43.8406 −1.52819 −0.764094 0.645105i \(-0.776814\pi\)
−0.764094 + 0.645105i \(0.776814\pi\)
\(824\) 16.8284 0.586246
\(825\) 1.92893 0.0671568
\(826\) 0 0
\(827\) 46.3553 1.61193 0.805967 0.591961i \(-0.201646\pi\)
0.805967 + 0.591961i \(0.201646\pi\)
\(828\) 1.41421 0.0491473
\(829\) 26.6985 0.927277 0.463638 0.886025i \(-0.346544\pi\)
0.463638 + 0.886025i \(0.346544\pi\)
\(830\) −4.48528 −0.155686
\(831\) 18.0711 0.626878
\(832\) −1.00000 −0.0346688
\(833\) 0 0
\(834\) 13.0711 0.452614
\(835\) 0.384776 0.0133157
\(836\) −0.899495 −0.0311097
\(837\) 8.48528 0.293294
\(838\) −27.4558 −0.948446
\(839\) 31.8284 1.09884 0.549420 0.835547i \(-0.314849\pi\)
0.549420 + 0.835547i \(0.314849\pi\)
\(840\) 0 0
\(841\) −25.6569 −0.884719
\(842\) 1.07107 0.0369114
\(843\) 0.686292 0.0236371
\(844\) −12.9289 −0.445032
\(845\) 0.585786 0.0201517
\(846\) −1.00000 −0.0343807
\(847\) 0 0
\(848\) 9.48528 0.325726
\(849\) 15.7574 0.540791
\(850\) −11.2426 −0.385619
\(851\) 2.00000 0.0685591
\(852\) −5.00000 −0.171297
\(853\) −6.72792 −0.230360 −0.115180 0.993345i \(-0.536744\pi\)
−0.115180 + 0.993345i \(0.536744\pi\)
\(854\) 0 0
\(855\) 1.27208 0.0435041
\(856\) −19.3137 −0.660129
\(857\) −20.5563 −0.702192 −0.351096 0.936340i \(-0.614191\pi\)
−0.351096 + 0.936340i \(0.614191\pi\)
\(858\) 0.414214 0.0141410
\(859\) 11.7990 0.402576 0.201288 0.979532i \(-0.435487\pi\)
0.201288 + 0.979532i \(0.435487\pi\)
\(860\) 3.79899 0.129544
\(861\) 0 0
\(862\) 30.4853 1.03833
\(863\) 39.3137 1.33825 0.669127 0.743148i \(-0.266668\pi\)
0.669127 + 0.743148i \(0.266668\pi\)
\(864\) 1.00000 0.0340207
\(865\) −3.21320 −0.109252
\(866\) −29.9706 −1.01844
\(867\) −11.1716 −0.379407
\(868\) 0 0
\(869\) 4.82843 0.163793
\(870\) −1.07107 −0.0363126
\(871\) −1.82843 −0.0619539
\(872\) −9.65685 −0.327022
\(873\) 0.928932 0.0314396
\(874\) 3.07107 0.103880
\(875\) 0 0
\(876\) −1.41421 −0.0477818
\(877\) −40.0416 −1.35211 −0.676055 0.736851i \(-0.736312\pi\)
−0.676055 + 0.736851i \(0.736312\pi\)
\(878\) −16.5858 −0.559743
\(879\) 9.89949 0.333902
\(880\) −0.242641 −0.00817942
\(881\) −18.4853 −0.622785 −0.311392 0.950281i \(-0.600795\pi\)
−0.311392 + 0.950281i \(0.600795\pi\)
\(882\) 0 0
\(883\) −9.75736 −0.328361 −0.164181 0.986430i \(-0.552498\pi\)
−0.164181 + 0.986430i \(0.552498\pi\)
\(884\) −2.41421 −0.0811988
\(885\) −1.21320 −0.0407814
\(886\) −33.6985 −1.13212
\(887\) −48.9117 −1.64229 −0.821147 0.570717i \(-0.806665\pi\)
−0.821147 + 0.570717i \(0.806665\pi\)
\(888\) 1.41421 0.0474579
\(889\) 0 0
\(890\) −1.51472 −0.0507735
\(891\) −0.414214 −0.0138767
\(892\) 7.92893 0.265480
\(893\) −2.17157 −0.0726689
\(894\) −4.92893 −0.164848
\(895\) 11.5147 0.384895
\(896\) 0 0
\(897\) −1.41421 −0.0472192
\(898\) 24.9706 0.833278
\(899\) −15.5147 −0.517445
\(900\) −4.65685 −0.155228
\(901\) 22.8995 0.762893
\(902\) −4.10051 −0.136532
\(903\) 0 0
\(904\) 4.89949 0.162955
\(905\) −1.69848 −0.0564595
\(906\) 4.07107 0.135252
\(907\) 54.3848 1.80582 0.902908 0.429833i \(-0.141428\pi\)
0.902908 + 0.429833i \(0.141428\pi\)
\(908\) 21.1716 0.702603
\(909\) 14.8284 0.491828
\(910\) 0 0
\(911\) −4.34315 −0.143895 −0.0719474 0.997408i \(-0.522921\pi\)
−0.0719474 + 0.997408i \(0.522921\pi\)
\(912\) 2.17157 0.0719080
\(913\) 3.17157 0.104964
\(914\) −3.07107 −0.101582
\(915\) 2.58579 0.0854835
\(916\) −26.4853 −0.875098
\(917\) 0 0
\(918\) 2.41421 0.0796809
\(919\) 1.79899 0.0593432 0.0296716 0.999560i \(-0.490554\pi\)
0.0296716 + 0.999560i \(0.490554\pi\)
\(920\) 0.828427 0.0273124
\(921\) 28.1127 0.926345
\(922\) 4.48528 0.147715
\(923\) 5.00000 0.164577
\(924\) 0 0
\(925\) −6.58579 −0.216539
\(926\) −20.9706 −0.689135
\(927\) 16.8284 0.552718
\(928\) −1.82843 −0.0600211
\(929\) −34.6863 −1.13802 −0.569010 0.822330i \(-0.692673\pi\)
−0.569010 + 0.822330i \(0.692673\pi\)
\(930\) 4.97056 0.162991
\(931\) 0 0
\(932\) −17.7279 −0.580697
\(933\) 8.10051 0.265199
\(934\) 11.4142 0.373484
\(935\) −0.585786 −0.0191573
\(936\) −1.00000 −0.0326860
\(937\) −42.6569 −1.39354 −0.696769 0.717295i \(-0.745380\pi\)
−0.696769 + 0.717295i \(0.745380\pi\)
\(938\) 0 0
\(939\) −3.51472 −0.114699
\(940\) −0.585786 −0.0191062
\(941\) 12.9706 0.422828 0.211414 0.977397i \(-0.432193\pi\)
0.211414 + 0.977397i \(0.432193\pi\)
\(942\) −9.72792 −0.316953
\(943\) 14.0000 0.455903
\(944\) −2.07107 −0.0674075
\(945\) 0 0
\(946\) −2.68629 −0.0873389
\(947\) 8.41421 0.273425 0.136713 0.990611i \(-0.456346\pi\)
0.136713 + 0.990611i \(0.456346\pi\)
\(948\) −11.6569 −0.378597
\(949\) 1.41421 0.0459073
\(950\) −10.1127 −0.328099
\(951\) 29.7990 0.966298
\(952\) 0 0
\(953\) −41.5858 −1.34710 −0.673548 0.739144i \(-0.735230\pi\)
−0.673548 + 0.739144i \(0.735230\pi\)
\(954\) 9.48528 0.307097
\(955\) −6.05887 −0.196061
\(956\) 4.51472 0.146016
\(957\) 0.757359 0.0244819
\(958\) 32.1127 1.03751
\(959\) 0 0
\(960\) 0.585786 0.0189062
\(961\) 41.0000 1.32258
\(962\) −1.41421 −0.0455961
\(963\) −19.3137 −0.622376
\(964\) 2.14214 0.0689935
\(965\) −1.71573 −0.0552313
\(966\) 0 0
\(967\) −13.5858 −0.436889 −0.218445 0.975849i \(-0.570098\pi\)
−0.218445 + 0.975849i \(0.570098\pi\)
\(968\) −10.8284 −0.348039
\(969\) 5.24264 0.168418
\(970\) 0.544156 0.0174718
\(971\) 32.5269 1.04384 0.521919 0.852995i \(-0.325216\pi\)
0.521919 + 0.852995i \(0.325216\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) 16.2132 0.519505
\(975\) 4.65685 0.149139
\(976\) 4.41421 0.141296
\(977\) 43.1716 1.38118 0.690590 0.723246i \(-0.257351\pi\)
0.690590 + 0.723246i \(0.257351\pi\)
\(978\) −18.6569 −0.596580
\(979\) 1.07107 0.0342315
\(980\) 0 0
\(981\) −9.65685 −0.308320
\(982\) 23.6569 0.754921
\(983\) 29.4853 0.940434 0.470217 0.882551i \(-0.344176\pi\)
0.470217 + 0.882551i \(0.344176\pi\)
\(984\) 9.89949 0.315584
\(985\) −9.11270 −0.290355
\(986\) −4.41421 −0.140577
\(987\) 0 0
\(988\) −2.17157 −0.0690869
\(989\) 9.17157 0.291639
\(990\) −0.242641 −0.00771163
\(991\) −0.686292 −0.0218008 −0.0109004 0.999941i \(-0.503470\pi\)
−0.0109004 + 0.999941i \(0.503470\pi\)
\(992\) 8.48528 0.269408
\(993\) −22.4853 −0.713549
\(994\) 0 0
\(995\) −1.59798 −0.0506594
\(996\) −7.65685 −0.242617
\(997\) −52.4975 −1.66261 −0.831306 0.555815i \(-0.812406\pi\)
−0.831306 + 0.555815i \(0.812406\pi\)
\(998\) 3.65685 0.115756
\(999\) 1.41421 0.0447437
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 3822.2.a.bu.1.1 2
7.3 odd 6 546.2.i.i.79.1 4
7.5 odd 6 546.2.i.i.235.1 yes 4
7.6 odd 2 3822.2.a.bn.1.2 2
21.5 even 6 1638.2.j.m.235.2 4
21.17 even 6 1638.2.j.m.1171.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.i.i.79.1 4 7.3 odd 6
546.2.i.i.235.1 yes 4 7.5 odd 6
1638.2.j.m.235.2 4 21.5 even 6
1638.2.j.m.1171.2 4 21.17 even 6
3822.2.a.bn.1.2 2 7.6 odd 2
3822.2.a.bu.1.1 2 1.1 even 1 trivial