Properties

Label 7623.2.a.w.1.1
Level 7623
Weight 2
Character 7623.1
Self dual yes
Analytic conductor 60.870
Analytic rank 1
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 7623.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(60.8699614608\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 231)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-0.618034\)
Character \(\chi\) = 7623.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{4} -3.85410 q^{5} -1.00000 q^{7} +3.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{4} -3.85410 q^{5} -1.00000 q^{7} +3.00000 q^{8} +3.85410 q^{10} -3.23607 q^{13} +1.00000 q^{14} -1.00000 q^{16} +1.14590 q^{17} -0.381966 q^{19} +3.85410 q^{20} -8.09017 q^{23} +9.85410 q^{25} +3.23607 q^{26} +1.00000 q^{28} -2.00000 q^{29} +8.32624 q^{31} -5.00000 q^{32} -1.14590 q^{34} +3.85410 q^{35} -0.909830 q^{37} +0.381966 q^{38} -11.5623 q^{40} -4.14590 q^{41} -3.23607 q^{43} +8.09017 q^{46} -2.00000 q^{47} +1.00000 q^{49} -9.85410 q^{50} +3.23607 q^{52} +10.1803 q^{53} -3.00000 q^{56} +2.00000 q^{58} -11.2361 q^{59} +8.94427 q^{61} -8.32624 q^{62} +7.00000 q^{64} +12.4721 q^{65} +8.00000 q^{67} -1.14590 q^{68} -3.85410 q^{70} +8.94427 q^{71} +0.763932 q^{73} +0.909830 q^{74} +0.381966 q^{76} +14.0000 q^{79} +3.85410 q^{80} +4.14590 q^{82} -14.9443 q^{83} -4.41641 q^{85} +3.23607 q^{86} +10.0902 q^{89} +3.23607 q^{91} +8.09017 q^{92} +2.00000 q^{94} +1.47214 q^{95} -12.4721 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} - 2q^{4} - q^{5} - 2q^{7} + 6q^{8} + O(q^{10}) \) \( 2q - 2q^{2} - 2q^{4} - q^{5} - 2q^{7} + 6q^{8} + q^{10} - 2q^{13} + 2q^{14} - 2q^{16} + 9q^{17} - 3q^{19} + q^{20} - 5q^{23} + 13q^{25} + 2q^{26} + 2q^{28} - 4q^{29} + q^{31} - 10q^{32} - 9q^{34} + q^{35} - 13q^{37} + 3q^{38} - 3q^{40} - 15q^{41} - 2q^{43} + 5q^{46} - 4q^{47} + 2q^{49} - 13q^{50} + 2q^{52} - 2q^{53} - 6q^{56} + 4q^{58} - 18q^{59} - q^{62} + 14q^{64} + 16q^{65} + 16q^{67} - 9q^{68} - q^{70} + 6q^{73} + 13q^{74} + 3q^{76} + 28q^{79} + q^{80} + 15q^{82} - 12q^{83} + 18q^{85} + 2q^{86} + 9q^{89} + 2q^{91} + 5q^{92} + 4q^{94} - 6q^{95} - 16q^{97} - 2q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107 −0.353553 0.935414i \(-0.615027\pi\)
−0.353553 + 0.935414i \(0.615027\pi\)
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) −3.85410 −1.72361 −0.861803 0.507242i \(-0.830665\pi\)
−0.861803 + 0.507242i \(0.830665\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 3.00000 1.06066
\(9\) 0 0
\(10\) 3.85410 1.21877
\(11\) 0 0
\(12\) 0 0
\(13\) −3.23607 −0.897524 −0.448762 0.893651i \(-0.648135\pi\)
−0.448762 + 0.893651i \(0.648135\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) 1.14590 0.277921 0.138961 0.990298i \(-0.455624\pi\)
0.138961 + 0.990298i \(0.455624\pi\)
\(18\) 0 0
\(19\) −0.381966 −0.0876290 −0.0438145 0.999040i \(-0.513951\pi\)
−0.0438145 + 0.999040i \(0.513951\pi\)
\(20\) 3.85410 0.861803
\(21\) 0 0
\(22\) 0 0
\(23\) −8.09017 −1.68692 −0.843459 0.537194i \(-0.819484\pi\)
−0.843459 + 0.537194i \(0.819484\pi\)
\(24\) 0 0
\(25\) 9.85410 1.97082
\(26\) 3.23607 0.634645
\(27\) 0 0
\(28\) 1.00000 0.188982
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) 8.32624 1.49544 0.747718 0.664016i \(-0.231149\pi\)
0.747718 + 0.664016i \(0.231149\pi\)
\(32\) −5.00000 −0.883883
\(33\) 0 0
\(34\) −1.14590 −0.196520
\(35\) 3.85410 0.651462
\(36\) 0 0
\(37\) −0.909830 −0.149575 −0.0747876 0.997199i \(-0.523828\pi\)
−0.0747876 + 0.997199i \(0.523828\pi\)
\(38\) 0.381966 0.0619631
\(39\) 0 0
\(40\) −11.5623 −1.82816
\(41\) −4.14590 −0.647480 −0.323740 0.946146i \(-0.604940\pi\)
−0.323740 + 0.946146i \(0.604940\pi\)
\(42\) 0 0
\(43\) −3.23607 −0.493496 −0.246748 0.969080i \(-0.579362\pi\)
−0.246748 + 0.969080i \(0.579362\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 8.09017 1.19283
\(47\) −2.00000 −0.291730 −0.145865 0.989305i \(-0.546597\pi\)
−0.145865 + 0.989305i \(0.546597\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −9.85410 −1.39358
\(51\) 0 0
\(52\) 3.23607 0.448762
\(53\) 10.1803 1.39838 0.699189 0.714937i \(-0.253545\pi\)
0.699189 + 0.714937i \(0.253545\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −3.00000 −0.400892
\(57\) 0 0
\(58\) 2.00000 0.262613
\(59\) −11.2361 −1.46281 −0.731406 0.681943i \(-0.761135\pi\)
−0.731406 + 0.681943i \(0.761135\pi\)
\(60\) 0 0
\(61\) 8.94427 1.14520 0.572598 0.819836i \(-0.305935\pi\)
0.572598 + 0.819836i \(0.305935\pi\)
\(62\) −8.32624 −1.05743
\(63\) 0 0
\(64\) 7.00000 0.875000
\(65\) 12.4721 1.54698
\(66\) 0 0
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) −1.14590 −0.138961
\(69\) 0 0
\(70\) −3.85410 −0.460653
\(71\) 8.94427 1.06149 0.530745 0.847532i \(-0.321912\pi\)
0.530745 + 0.847532i \(0.321912\pi\)
\(72\) 0 0
\(73\) 0.763932 0.0894115 0.0447057 0.999000i \(-0.485765\pi\)
0.0447057 + 0.999000i \(0.485765\pi\)
\(74\) 0.909830 0.105766
\(75\) 0 0
\(76\) 0.381966 0.0438145
\(77\) 0 0
\(78\) 0 0
\(79\) 14.0000 1.57512 0.787562 0.616236i \(-0.211343\pi\)
0.787562 + 0.616236i \(0.211343\pi\)
\(80\) 3.85410 0.430902
\(81\) 0 0
\(82\) 4.14590 0.457838
\(83\) −14.9443 −1.64035 −0.820173 0.572115i \(-0.806123\pi\)
−0.820173 + 0.572115i \(0.806123\pi\)
\(84\) 0 0
\(85\) −4.41641 −0.479027
\(86\) 3.23607 0.348954
\(87\) 0 0
\(88\) 0 0
\(89\) 10.0902 1.06956 0.534778 0.844993i \(-0.320395\pi\)
0.534778 + 0.844993i \(0.320395\pi\)
\(90\) 0 0
\(91\) 3.23607 0.339232
\(92\) 8.09017 0.843459
\(93\) 0 0
\(94\) 2.00000 0.206284
\(95\) 1.47214 0.151038
\(96\) 0 0
\(97\) −12.4721 −1.26635 −0.633177 0.774007i \(-0.718249\pi\)
−0.633177 + 0.774007i \(0.718249\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) −9.85410 −0.985410
\(101\) 9.09017 0.904506 0.452253 0.891890i \(-0.350620\pi\)
0.452253 + 0.891890i \(0.350620\pi\)
\(102\) 0 0
\(103\) 5.32624 0.524810 0.262405 0.964958i \(-0.415484\pi\)
0.262405 + 0.964958i \(0.415484\pi\)
\(104\) −9.70820 −0.951968
\(105\) 0 0
\(106\) −10.1803 −0.988802
\(107\) 16.0902 1.55550 0.777748 0.628577i \(-0.216362\pi\)
0.777748 + 0.628577i \(0.216362\pi\)
\(108\) 0 0
\(109\) 9.56231 0.915903 0.457951 0.888977i \(-0.348583\pi\)
0.457951 + 0.888977i \(0.348583\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1.00000 0.0944911
\(113\) −5.23607 −0.492568 −0.246284 0.969198i \(-0.579210\pi\)
−0.246284 + 0.969198i \(0.579210\pi\)
\(114\) 0 0
\(115\) 31.1803 2.90758
\(116\) 2.00000 0.185695
\(117\) 0 0
\(118\) 11.2361 1.03436
\(119\) −1.14590 −0.105044
\(120\) 0 0
\(121\) 0 0
\(122\) −8.94427 −0.809776
\(123\) 0 0
\(124\) −8.32624 −0.747718
\(125\) −18.7082 −1.67331
\(126\) 0 0
\(127\) −1.23607 −0.109683 −0.0548416 0.998495i \(-0.517465\pi\)
−0.0548416 + 0.998495i \(0.517465\pi\)
\(128\) 3.00000 0.265165
\(129\) 0 0
\(130\) −12.4721 −1.09388
\(131\) 15.7082 1.37243 0.686216 0.727398i \(-0.259270\pi\)
0.686216 + 0.727398i \(0.259270\pi\)
\(132\) 0 0
\(133\) 0.381966 0.0331207
\(134\) −8.00000 −0.691095
\(135\) 0 0
\(136\) 3.43769 0.294780
\(137\) −11.2361 −0.959962 −0.479981 0.877279i \(-0.659356\pi\)
−0.479981 + 0.877279i \(0.659356\pi\)
\(138\) 0 0
\(139\) 19.7984 1.67928 0.839638 0.543146i \(-0.182767\pi\)
0.839638 + 0.543146i \(0.182767\pi\)
\(140\) −3.85410 −0.325731
\(141\) 0 0
\(142\) −8.94427 −0.750587
\(143\) 0 0
\(144\) 0 0
\(145\) 7.70820 0.640131
\(146\) −0.763932 −0.0632235
\(147\) 0 0
\(148\) 0.909830 0.0747876
\(149\) −20.0000 −1.63846 −0.819232 0.573462i \(-0.805600\pi\)
−0.819232 + 0.573462i \(0.805600\pi\)
\(150\) 0 0
\(151\) 2.00000 0.162758 0.0813788 0.996683i \(-0.474068\pi\)
0.0813788 + 0.996683i \(0.474068\pi\)
\(152\) −1.14590 −0.0929446
\(153\) 0 0
\(154\) 0 0
\(155\) −32.0902 −2.57754
\(156\) 0 0
\(157\) −3.70820 −0.295947 −0.147973 0.988991i \(-0.547275\pi\)
−0.147973 + 0.988991i \(0.547275\pi\)
\(158\) −14.0000 −1.11378
\(159\) 0 0
\(160\) 19.2705 1.52347
\(161\) 8.09017 0.637595
\(162\) 0 0
\(163\) 21.4164 1.67746 0.838731 0.544546i \(-0.183298\pi\)
0.838731 + 0.544546i \(0.183298\pi\)
\(164\) 4.14590 0.323740
\(165\) 0 0
\(166\) 14.9443 1.15990
\(167\) −14.1803 −1.09731 −0.548654 0.836050i \(-0.684859\pi\)
−0.548654 + 0.836050i \(0.684859\pi\)
\(168\) 0 0
\(169\) −2.52786 −0.194451
\(170\) 4.41641 0.338723
\(171\) 0 0
\(172\) 3.23607 0.246748
\(173\) 13.1459 0.999464 0.499732 0.866180i \(-0.333432\pi\)
0.499732 + 0.866180i \(0.333432\pi\)
\(174\) 0 0
\(175\) −9.85410 −0.744900
\(176\) 0 0
\(177\) 0 0
\(178\) −10.0902 −0.756290
\(179\) −10.6180 −0.793629 −0.396815 0.917899i \(-0.629884\pi\)
−0.396815 + 0.917899i \(0.629884\pi\)
\(180\) 0 0
\(181\) −1.05573 −0.0784717 −0.0392358 0.999230i \(-0.512492\pi\)
−0.0392358 + 0.999230i \(0.512492\pi\)
\(182\) −3.23607 −0.239873
\(183\) 0 0
\(184\) −24.2705 −1.78925
\(185\) 3.50658 0.257809
\(186\) 0 0
\(187\) 0 0
\(188\) 2.00000 0.145865
\(189\) 0 0
\(190\) −1.47214 −0.106800
\(191\) −15.2705 −1.10494 −0.552468 0.833534i \(-0.686314\pi\)
−0.552468 + 0.833534i \(0.686314\pi\)
\(192\) 0 0
\(193\) −20.6180 −1.48412 −0.742059 0.670334i \(-0.766151\pi\)
−0.742059 + 0.670334i \(0.766151\pi\)
\(194\) 12.4721 0.895447
\(195\) 0 0
\(196\) −1.00000 −0.0714286
\(197\) 4.00000 0.284988 0.142494 0.989796i \(-0.454488\pi\)
0.142494 + 0.989796i \(0.454488\pi\)
\(198\) 0 0
\(199\) −0.381966 −0.0270769 −0.0135384 0.999908i \(-0.504310\pi\)
−0.0135384 + 0.999908i \(0.504310\pi\)
\(200\) 29.5623 2.09037
\(201\) 0 0
\(202\) −9.09017 −0.639582
\(203\) 2.00000 0.140372
\(204\) 0 0
\(205\) 15.9787 1.11600
\(206\) −5.32624 −0.371097
\(207\) 0 0
\(208\) 3.23607 0.224381
\(209\) 0 0
\(210\) 0 0
\(211\) 2.00000 0.137686 0.0688428 0.997628i \(-0.478069\pi\)
0.0688428 + 0.997628i \(0.478069\pi\)
\(212\) −10.1803 −0.699189
\(213\) 0 0
\(214\) −16.0902 −1.09990
\(215\) 12.4721 0.850593
\(216\) 0 0
\(217\) −8.32624 −0.565222
\(218\) −9.56231 −0.647641
\(219\) 0 0
\(220\) 0 0
\(221\) −3.70820 −0.249441
\(222\) 0 0
\(223\) −3.79837 −0.254358 −0.127179 0.991880i \(-0.540592\pi\)
−0.127179 + 0.991880i \(0.540592\pi\)
\(224\) 5.00000 0.334077
\(225\) 0 0
\(226\) 5.23607 0.348298
\(227\) −5.41641 −0.359500 −0.179750 0.983712i \(-0.557529\pi\)
−0.179750 + 0.983712i \(0.557529\pi\)
\(228\) 0 0
\(229\) 16.7639 1.10779 0.553896 0.832586i \(-0.313141\pi\)
0.553896 + 0.832586i \(0.313141\pi\)
\(230\) −31.1803 −2.05597
\(231\) 0 0
\(232\) −6.00000 −0.393919
\(233\) 7.70820 0.504981 0.252491 0.967599i \(-0.418750\pi\)
0.252491 + 0.967599i \(0.418750\pi\)
\(234\) 0 0
\(235\) 7.70820 0.502828
\(236\) 11.2361 0.731406
\(237\) 0 0
\(238\) 1.14590 0.0742775
\(239\) 18.5623 1.20070 0.600348 0.799739i \(-0.295029\pi\)
0.600348 + 0.799739i \(0.295029\pi\)
\(240\) 0 0
\(241\) 12.9443 0.833814 0.416907 0.908949i \(-0.363114\pi\)
0.416907 + 0.908949i \(0.363114\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −8.94427 −0.572598
\(245\) −3.85410 −0.246230
\(246\) 0 0
\(247\) 1.23607 0.0786491
\(248\) 24.9787 1.58615
\(249\) 0 0
\(250\) 18.7082 1.18321
\(251\) −1.81966 −0.114856 −0.0574280 0.998350i \(-0.518290\pi\)
−0.0574280 + 0.998350i \(0.518290\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 1.23607 0.0775578
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) −0.381966 −0.0238264 −0.0119132 0.999929i \(-0.503792\pi\)
−0.0119132 + 0.999929i \(0.503792\pi\)
\(258\) 0 0
\(259\) 0.909830 0.0565341
\(260\) −12.4721 −0.773489
\(261\) 0 0
\(262\) −15.7082 −0.970456
\(263\) 22.5623 1.39125 0.695626 0.718404i \(-0.255127\pi\)
0.695626 + 0.718404i \(0.255127\pi\)
\(264\) 0 0
\(265\) −39.2361 −2.41025
\(266\) −0.381966 −0.0234198
\(267\) 0 0
\(268\) −8.00000 −0.488678
\(269\) 7.88854 0.480973 0.240487 0.970652i \(-0.422693\pi\)
0.240487 + 0.970652i \(0.422693\pi\)
\(270\) 0 0
\(271\) −12.9098 −0.784216 −0.392108 0.919919i \(-0.628254\pi\)
−0.392108 + 0.919919i \(0.628254\pi\)
\(272\) −1.14590 −0.0694803
\(273\) 0 0
\(274\) 11.2361 0.678796
\(275\) 0 0
\(276\) 0 0
\(277\) 30.5623 1.83631 0.918155 0.396220i \(-0.129678\pi\)
0.918155 + 0.396220i \(0.129678\pi\)
\(278\) −19.7984 −1.18743
\(279\) 0 0
\(280\) 11.5623 0.690980
\(281\) 18.9443 1.13012 0.565060 0.825050i \(-0.308853\pi\)
0.565060 + 0.825050i \(0.308853\pi\)
\(282\) 0 0
\(283\) −33.5623 −1.99507 −0.997536 0.0701565i \(-0.977650\pi\)
−0.997536 + 0.0701565i \(0.977650\pi\)
\(284\) −8.94427 −0.530745
\(285\) 0 0
\(286\) 0 0
\(287\) 4.14590 0.244725
\(288\) 0 0
\(289\) −15.6869 −0.922760
\(290\) −7.70820 −0.452641
\(291\) 0 0
\(292\) −0.763932 −0.0447057
\(293\) −4.79837 −0.280324 −0.140162 0.990129i \(-0.544762\pi\)
−0.140162 + 0.990129i \(0.544762\pi\)
\(294\) 0 0
\(295\) 43.3050 2.52131
\(296\) −2.72949 −0.158648
\(297\) 0 0
\(298\) 20.0000 1.15857
\(299\) 26.1803 1.51405
\(300\) 0 0
\(301\) 3.23607 0.186524
\(302\) −2.00000 −0.115087
\(303\) 0 0
\(304\) 0.381966 0.0219073
\(305\) −34.4721 −1.97387
\(306\) 0 0
\(307\) −9.27051 −0.529096 −0.264548 0.964373i \(-0.585223\pi\)
−0.264548 + 0.964373i \(0.585223\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 32.0902 1.82260
\(311\) 5.23607 0.296910 0.148455 0.988919i \(-0.452570\pi\)
0.148455 + 0.988919i \(0.452570\pi\)
\(312\) 0 0
\(313\) −32.1803 −1.81894 −0.909470 0.415769i \(-0.863512\pi\)
−0.909470 + 0.415769i \(0.863512\pi\)
\(314\) 3.70820 0.209266
\(315\) 0 0
\(316\) −14.0000 −0.787562
\(317\) −21.5967 −1.21299 −0.606497 0.795086i \(-0.707426\pi\)
−0.606497 + 0.795086i \(0.707426\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −26.9787 −1.50816
\(321\) 0 0
\(322\) −8.09017 −0.450848
\(323\) −0.437694 −0.0243540
\(324\) 0 0
\(325\) −31.8885 −1.76886
\(326\) −21.4164 −1.18615
\(327\) 0 0
\(328\) −12.4377 −0.686757
\(329\) 2.00000 0.110264
\(330\) 0 0
\(331\) −14.3607 −0.789334 −0.394667 0.918824i \(-0.629140\pi\)
−0.394667 + 0.918824i \(0.629140\pi\)
\(332\) 14.9443 0.820173
\(333\) 0 0
\(334\) 14.1803 0.775914
\(335\) −30.8328 −1.68458
\(336\) 0 0
\(337\) 33.8541 1.84415 0.922075 0.387011i \(-0.126492\pi\)
0.922075 + 0.387011i \(0.126492\pi\)
\(338\) 2.52786 0.137498
\(339\) 0 0
\(340\) 4.41641 0.239513
\(341\) 0 0
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) −9.70820 −0.523431
\(345\) 0 0
\(346\) −13.1459 −0.706728
\(347\) −14.6180 −0.784737 −0.392369 0.919808i \(-0.628344\pi\)
−0.392369 + 0.919808i \(0.628344\pi\)
\(348\) 0 0
\(349\) −27.2361 −1.45791 −0.728957 0.684560i \(-0.759994\pi\)
−0.728957 + 0.684560i \(0.759994\pi\)
\(350\) 9.85410 0.526724
\(351\) 0 0
\(352\) 0 0
\(353\) −3.52786 −0.187769 −0.0938846 0.995583i \(-0.529928\pi\)
−0.0938846 + 0.995583i \(0.529928\pi\)
\(354\) 0 0
\(355\) −34.4721 −1.82959
\(356\) −10.0902 −0.534778
\(357\) 0 0
\(358\) 10.6180 0.561181
\(359\) −22.0344 −1.16293 −0.581467 0.813570i \(-0.697521\pi\)
−0.581467 + 0.813570i \(0.697521\pi\)
\(360\) 0 0
\(361\) −18.8541 −0.992321
\(362\) 1.05573 0.0554878
\(363\) 0 0
\(364\) −3.23607 −0.169616
\(365\) −2.94427 −0.154110
\(366\) 0 0
\(367\) −17.2705 −0.901513 −0.450757 0.892647i \(-0.648846\pi\)
−0.450757 + 0.892647i \(0.648846\pi\)
\(368\) 8.09017 0.421729
\(369\) 0 0
\(370\) −3.50658 −0.182298
\(371\) −10.1803 −0.528537
\(372\) 0 0
\(373\) 7.67376 0.397332 0.198666 0.980067i \(-0.436339\pi\)
0.198666 + 0.980067i \(0.436339\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −6.00000 −0.309426
\(377\) 6.47214 0.333332
\(378\) 0 0
\(379\) −6.47214 −0.332451 −0.166226 0.986088i \(-0.553158\pi\)
−0.166226 + 0.986088i \(0.553158\pi\)
\(380\) −1.47214 −0.0755190
\(381\) 0 0
\(382\) 15.2705 0.781307
\(383\) 30.9443 1.58118 0.790589 0.612347i \(-0.209774\pi\)
0.790589 + 0.612347i \(0.209774\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 20.6180 1.04943
\(387\) 0 0
\(388\) 12.4721 0.633177
\(389\) 0.180340 0.00914360 0.00457180 0.999990i \(-0.498545\pi\)
0.00457180 + 0.999990i \(0.498545\pi\)
\(390\) 0 0
\(391\) −9.27051 −0.468830
\(392\) 3.00000 0.151523
\(393\) 0 0
\(394\) −4.00000 −0.201517
\(395\) −53.9574 −2.71489
\(396\) 0 0
\(397\) −16.6525 −0.835764 −0.417882 0.908501i \(-0.637227\pi\)
−0.417882 + 0.908501i \(0.637227\pi\)
\(398\) 0.381966 0.0191462
\(399\) 0 0
\(400\) −9.85410 −0.492705
\(401\) −9.52786 −0.475799 −0.237899 0.971290i \(-0.576459\pi\)
−0.237899 + 0.971290i \(0.576459\pi\)
\(402\) 0 0
\(403\) −26.9443 −1.34219
\(404\) −9.09017 −0.452253
\(405\) 0 0
\(406\) −2.00000 −0.0992583
\(407\) 0 0
\(408\) 0 0
\(409\) −33.2361 −1.64342 −0.821709 0.569907i \(-0.806979\pi\)
−0.821709 + 0.569907i \(0.806979\pi\)
\(410\) −15.9787 −0.789132
\(411\) 0 0
\(412\) −5.32624 −0.262405
\(413\) 11.2361 0.552891
\(414\) 0 0
\(415\) 57.5967 2.82731
\(416\) 16.1803 0.793306
\(417\) 0 0
\(418\) 0 0
\(419\) −19.2361 −0.939743 −0.469872 0.882735i \(-0.655700\pi\)
−0.469872 + 0.882735i \(0.655700\pi\)
\(420\) 0 0
\(421\) 11.3262 0.552007 0.276004 0.961157i \(-0.410990\pi\)
0.276004 + 0.961157i \(0.410990\pi\)
\(422\) −2.00000 −0.0973585
\(423\) 0 0
\(424\) 30.5410 1.48320
\(425\) 11.2918 0.547733
\(426\) 0 0
\(427\) −8.94427 −0.432844
\(428\) −16.0902 −0.777748
\(429\) 0 0
\(430\) −12.4721 −0.601460
\(431\) 9.09017 0.437858 0.218929 0.975741i \(-0.429744\pi\)
0.218929 + 0.975741i \(0.429744\pi\)
\(432\) 0 0
\(433\) −12.7639 −0.613395 −0.306698 0.951807i \(-0.599224\pi\)
−0.306698 + 0.951807i \(0.599224\pi\)
\(434\) 8.32624 0.399672
\(435\) 0 0
\(436\) −9.56231 −0.457951
\(437\) 3.09017 0.147823
\(438\) 0 0
\(439\) −2.79837 −0.133559 −0.0667795 0.997768i \(-0.521272\pi\)
−0.0667795 + 0.997768i \(0.521272\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 3.70820 0.176381
\(443\) −16.0902 −0.764467 −0.382234 0.924066i \(-0.624845\pi\)
−0.382234 + 0.924066i \(0.624845\pi\)
\(444\) 0 0
\(445\) −38.8885 −1.84349
\(446\) 3.79837 0.179858
\(447\) 0 0
\(448\) −7.00000 −0.330719
\(449\) −6.11146 −0.288417 −0.144209 0.989547i \(-0.546064\pi\)
−0.144209 + 0.989547i \(0.546064\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 5.23607 0.246284
\(453\) 0 0
\(454\) 5.41641 0.254205
\(455\) −12.4721 −0.584703
\(456\) 0 0
\(457\) −3.52786 −0.165027 −0.0825133 0.996590i \(-0.526295\pi\)
−0.0825133 + 0.996590i \(0.526295\pi\)
\(458\) −16.7639 −0.783327
\(459\) 0 0
\(460\) −31.1803 −1.45379
\(461\) 38.3607 1.78663 0.893317 0.449426i \(-0.148372\pi\)
0.893317 + 0.449426i \(0.148372\pi\)
\(462\) 0 0
\(463\) −15.4164 −0.716461 −0.358231 0.933633i \(-0.616620\pi\)
−0.358231 + 0.933633i \(0.616620\pi\)
\(464\) 2.00000 0.0928477
\(465\) 0 0
\(466\) −7.70820 −0.357076
\(467\) −14.7639 −0.683193 −0.341597 0.939847i \(-0.610968\pi\)
−0.341597 + 0.939847i \(0.610968\pi\)
\(468\) 0 0
\(469\) −8.00000 −0.369406
\(470\) −7.70820 −0.355553
\(471\) 0 0
\(472\) −33.7082 −1.55155
\(473\) 0 0
\(474\) 0 0
\(475\) −3.76393 −0.172701
\(476\) 1.14590 0.0525222
\(477\) 0 0
\(478\) −18.5623 −0.849020
\(479\) −28.9443 −1.32250 −0.661249 0.750167i \(-0.729973\pi\)
−0.661249 + 0.750167i \(0.729973\pi\)
\(480\) 0 0
\(481\) 2.94427 0.134247
\(482\) −12.9443 −0.589595
\(483\) 0 0
\(484\) 0 0
\(485\) 48.0689 2.18270
\(486\) 0 0
\(487\) −34.7639 −1.57530 −0.787652 0.616120i \(-0.788704\pi\)
−0.787652 + 0.616120i \(0.788704\pi\)
\(488\) 26.8328 1.21466
\(489\) 0 0
\(490\) 3.85410 0.174111
\(491\) 15.3262 0.691663 0.345832 0.938297i \(-0.387597\pi\)
0.345832 + 0.938297i \(0.387597\pi\)
\(492\) 0 0
\(493\) −2.29180 −0.103217
\(494\) −1.23607 −0.0556133
\(495\) 0 0
\(496\) −8.32624 −0.373859
\(497\) −8.94427 −0.401205
\(498\) 0 0
\(499\) 3.81966 0.170991 0.0854957 0.996339i \(-0.472753\pi\)
0.0854957 + 0.996339i \(0.472753\pi\)
\(500\) 18.7082 0.836656
\(501\) 0 0
\(502\) 1.81966 0.0812154
\(503\) −27.4164 −1.22244 −0.611219 0.791462i \(-0.709320\pi\)
−0.611219 + 0.791462i \(0.709320\pi\)
\(504\) 0 0
\(505\) −35.0344 −1.55901
\(506\) 0 0
\(507\) 0 0
\(508\) 1.23607 0.0548416
\(509\) 17.0902 0.757508 0.378754 0.925497i \(-0.376353\pi\)
0.378754 + 0.925497i \(0.376353\pi\)
\(510\) 0 0
\(511\) −0.763932 −0.0337944
\(512\) 11.0000 0.486136
\(513\) 0 0
\(514\) 0.381966 0.0168478
\(515\) −20.5279 −0.904566
\(516\) 0 0
\(517\) 0 0
\(518\) −0.909830 −0.0399756
\(519\) 0 0
\(520\) 37.4164 1.64082
\(521\) 9.38197 0.411031 0.205516 0.978654i \(-0.434113\pi\)
0.205516 + 0.978654i \(0.434113\pi\)
\(522\) 0 0
\(523\) −14.0902 −0.616120 −0.308060 0.951367i \(-0.599680\pi\)
−0.308060 + 0.951367i \(0.599680\pi\)
\(524\) −15.7082 −0.686216
\(525\) 0 0
\(526\) −22.5623 −0.983763
\(527\) 9.54102 0.415613
\(528\) 0 0
\(529\) 42.4508 1.84569
\(530\) 39.2361 1.70431
\(531\) 0 0
\(532\) −0.381966 −0.0165603
\(533\) 13.4164 0.581129
\(534\) 0 0
\(535\) −62.0132 −2.68106
\(536\) 24.0000 1.03664
\(537\) 0 0
\(538\) −7.88854 −0.340099
\(539\) 0 0
\(540\) 0 0
\(541\) −10.1459 −0.436206 −0.218103 0.975926i \(-0.569987\pi\)
−0.218103 + 0.975926i \(0.569987\pi\)
\(542\) 12.9098 0.554525
\(543\) 0 0
\(544\) −5.72949 −0.245650
\(545\) −36.8541 −1.57866
\(546\) 0 0
\(547\) 27.5279 1.17701 0.588503 0.808495i \(-0.299717\pi\)
0.588503 + 0.808495i \(0.299717\pi\)
\(548\) 11.2361 0.479981
\(549\) 0 0
\(550\) 0 0
\(551\) 0.763932 0.0325446
\(552\) 0 0
\(553\) −14.0000 −0.595341
\(554\) −30.5623 −1.29847
\(555\) 0 0
\(556\) −19.7984 −0.839638
\(557\) 5.41641 0.229501 0.114750 0.993394i \(-0.463393\pi\)
0.114750 + 0.993394i \(0.463393\pi\)
\(558\) 0 0
\(559\) 10.4721 0.442924
\(560\) −3.85410 −0.162866
\(561\) 0 0
\(562\) −18.9443 −0.799116
\(563\) −4.36068 −0.183781 −0.0918904 0.995769i \(-0.529291\pi\)
−0.0918904 + 0.995769i \(0.529291\pi\)
\(564\) 0 0
\(565\) 20.1803 0.848993
\(566\) 33.5623 1.41073
\(567\) 0 0
\(568\) 26.8328 1.12588
\(569\) 4.18034 0.175249 0.0876245 0.996154i \(-0.472072\pi\)
0.0876245 + 0.996154i \(0.472072\pi\)
\(570\) 0 0
\(571\) 38.5410 1.61289 0.806446 0.591308i \(-0.201388\pi\)
0.806446 + 0.591308i \(0.201388\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −4.14590 −0.173046
\(575\) −79.7214 −3.32461
\(576\) 0 0
\(577\) 16.6525 0.693252 0.346626 0.938003i \(-0.387327\pi\)
0.346626 + 0.938003i \(0.387327\pi\)
\(578\) 15.6869 0.652490
\(579\) 0 0
\(580\) −7.70820 −0.320066
\(581\) 14.9443 0.619993
\(582\) 0 0
\(583\) 0 0
\(584\) 2.29180 0.0948352
\(585\) 0 0
\(586\) 4.79837 0.198219
\(587\) 31.8885 1.31618 0.658091 0.752939i \(-0.271364\pi\)
0.658091 + 0.752939i \(0.271364\pi\)
\(588\) 0 0
\(589\) −3.18034 −0.131044
\(590\) −43.3050 −1.78284
\(591\) 0 0
\(592\) 0.909830 0.0373938
\(593\) 23.5623 0.967588 0.483794 0.875182i \(-0.339258\pi\)
0.483794 + 0.875182i \(0.339258\pi\)
\(594\) 0 0
\(595\) 4.41641 0.181055
\(596\) 20.0000 0.819232
\(597\) 0 0
\(598\) −26.1803 −1.07059
\(599\) −18.5623 −0.758435 −0.379218 0.925308i \(-0.623807\pi\)
−0.379218 + 0.925308i \(0.623807\pi\)
\(600\) 0 0
\(601\) −6.00000 −0.244745 −0.122373 0.992484i \(-0.539050\pi\)
−0.122373 + 0.992484i \(0.539050\pi\)
\(602\) −3.23607 −0.131892
\(603\) 0 0
\(604\) −2.00000 −0.0813788
\(605\) 0 0
\(606\) 0 0
\(607\) −5.32624 −0.216185 −0.108093 0.994141i \(-0.534474\pi\)
−0.108093 + 0.994141i \(0.534474\pi\)
\(608\) 1.90983 0.0774538
\(609\) 0 0
\(610\) 34.4721 1.39574
\(611\) 6.47214 0.261835
\(612\) 0 0
\(613\) −0.854102 −0.0344969 −0.0172484 0.999851i \(-0.505491\pi\)
−0.0172484 + 0.999851i \(0.505491\pi\)
\(614\) 9.27051 0.374127
\(615\) 0 0
\(616\) 0 0
\(617\) −13.3475 −0.537351 −0.268676 0.963231i \(-0.586586\pi\)
−0.268676 + 0.963231i \(0.586586\pi\)
\(618\) 0 0
\(619\) 26.5066 1.06539 0.532695 0.846308i \(-0.321179\pi\)
0.532695 + 0.846308i \(0.321179\pi\)
\(620\) 32.0902 1.28877
\(621\) 0 0
\(622\) −5.23607 −0.209947
\(623\) −10.0902 −0.404254
\(624\) 0 0
\(625\) 22.8328 0.913313
\(626\) 32.1803 1.28619
\(627\) 0 0
\(628\) 3.70820 0.147973
\(629\) −1.04257 −0.0415701
\(630\) 0 0
\(631\) −5.88854 −0.234419 −0.117210 0.993107i \(-0.537395\pi\)
−0.117210 + 0.993107i \(0.537395\pi\)
\(632\) 42.0000 1.67067
\(633\) 0 0
\(634\) 21.5967 0.857716
\(635\) 4.76393 0.189051
\(636\) 0 0
\(637\) −3.23607 −0.128218
\(638\) 0 0
\(639\) 0 0
\(640\) −11.5623 −0.457040
\(641\) 22.6525 0.894719 0.447360 0.894354i \(-0.352364\pi\)
0.447360 + 0.894354i \(0.352364\pi\)
\(642\) 0 0
\(643\) −16.9230 −0.667377 −0.333689 0.942683i \(-0.608293\pi\)
−0.333689 + 0.942683i \(0.608293\pi\)
\(644\) −8.09017 −0.318797
\(645\) 0 0
\(646\) 0.437694 0.0172208
\(647\) −14.3607 −0.564577 −0.282288 0.959330i \(-0.591093\pi\)
−0.282288 + 0.959330i \(0.591093\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 31.8885 1.25077
\(651\) 0 0
\(652\) −21.4164 −0.838731
\(653\) −9.88854 −0.386969 −0.193484 0.981103i \(-0.561979\pi\)
−0.193484 + 0.981103i \(0.561979\pi\)
\(654\) 0 0
\(655\) −60.5410 −2.36553
\(656\) 4.14590 0.161870
\(657\) 0 0
\(658\) −2.00000 −0.0779681
\(659\) −5.96556 −0.232385 −0.116193 0.993227i \(-0.537069\pi\)
−0.116193 + 0.993227i \(0.537069\pi\)
\(660\) 0 0
\(661\) 21.7082 0.844351 0.422176 0.906514i \(-0.361267\pi\)
0.422176 + 0.906514i \(0.361267\pi\)
\(662\) 14.3607 0.558144
\(663\) 0 0
\(664\) −44.8328 −1.73985
\(665\) −1.47214 −0.0570870
\(666\) 0 0
\(667\) 16.1803 0.626505
\(668\) 14.1803 0.548654
\(669\) 0 0
\(670\) 30.8328 1.19118
\(671\) 0 0
\(672\) 0 0
\(673\) −0.111456 −0.00429632 −0.00214816 0.999998i \(-0.500684\pi\)
−0.00214816 + 0.999998i \(0.500684\pi\)
\(674\) −33.8541 −1.30401
\(675\) 0 0
\(676\) 2.52786 0.0972255
\(677\) −11.8885 −0.456914 −0.228457 0.973554i \(-0.573368\pi\)
−0.228457 + 0.973554i \(0.573368\pi\)
\(678\) 0 0
\(679\) 12.4721 0.478637
\(680\) −13.2492 −0.508085
\(681\) 0 0
\(682\) 0 0
\(683\) 23.9230 0.915388 0.457694 0.889110i \(-0.348676\pi\)
0.457694 + 0.889110i \(0.348676\pi\)
\(684\) 0 0
\(685\) 43.3050 1.65460
\(686\) 1.00000 0.0381802
\(687\) 0 0
\(688\) 3.23607 0.123374
\(689\) −32.9443 −1.25508
\(690\) 0 0
\(691\) −24.9230 −0.948115 −0.474058 0.880494i \(-0.657211\pi\)
−0.474058 + 0.880494i \(0.657211\pi\)
\(692\) −13.1459 −0.499732
\(693\) 0 0
\(694\) 14.6180 0.554893
\(695\) −76.3050 −2.89441
\(696\) 0 0
\(697\) −4.75078 −0.179948
\(698\) 27.2361 1.03090
\(699\) 0 0
\(700\) 9.85410 0.372450
\(701\) −34.0689 −1.28676 −0.643382 0.765545i \(-0.722469\pi\)
−0.643382 + 0.765545i \(0.722469\pi\)
\(702\) 0 0
\(703\) 0.347524 0.0131071
\(704\) 0 0
\(705\) 0 0
\(706\) 3.52786 0.132773
\(707\) −9.09017 −0.341871
\(708\) 0 0
\(709\) 40.6869 1.52803 0.764015 0.645199i \(-0.223226\pi\)
0.764015 + 0.645199i \(0.223226\pi\)
\(710\) 34.4721 1.29372
\(711\) 0 0
\(712\) 30.2705 1.13444
\(713\) −67.3607 −2.52268
\(714\) 0 0
\(715\) 0 0
\(716\) 10.6180 0.396815
\(717\) 0 0
\(718\) 22.0344 0.822318
\(719\) −45.1246 −1.68286 −0.841432 0.540363i \(-0.818287\pi\)
−0.841432 + 0.540363i \(0.818287\pi\)
\(720\) 0 0
\(721\) −5.32624 −0.198359
\(722\) 18.8541 0.701677
\(723\) 0 0
\(724\) 1.05573 0.0392358
\(725\) −19.7082 −0.731944
\(726\) 0 0
\(727\) −16.7426 −0.620950 −0.310475 0.950581i \(-0.600488\pi\)
−0.310475 + 0.950581i \(0.600488\pi\)
\(728\) 9.70820 0.359810
\(729\) 0 0
\(730\) 2.94427 0.108972
\(731\) −3.70820 −0.137153
\(732\) 0 0
\(733\) 9.12461 0.337025 0.168513 0.985699i \(-0.446104\pi\)
0.168513 + 0.985699i \(0.446104\pi\)
\(734\) 17.2705 0.637466
\(735\) 0 0
\(736\) 40.4508 1.49104
\(737\) 0 0
\(738\) 0 0
\(739\) 27.0132 0.993695 0.496847 0.867838i \(-0.334491\pi\)
0.496847 + 0.867838i \(0.334491\pi\)
\(740\) −3.50658 −0.128904
\(741\) 0 0
\(742\) 10.1803 0.373732
\(743\) 1.67376 0.0614044 0.0307022 0.999529i \(-0.490226\pi\)
0.0307022 + 0.999529i \(0.490226\pi\)
\(744\) 0 0
\(745\) 77.0820 2.82407
\(746\) −7.67376 −0.280956
\(747\) 0 0
\(748\) 0 0
\(749\) −16.0902 −0.587922
\(750\) 0 0
\(751\) −1.52786 −0.0557526 −0.0278763 0.999611i \(-0.508874\pi\)
−0.0278763 + 0.999611i \(0.508874\pi\)
\(752\) 2.00000 0.0729325
\(753\) 0 0
\(754\) −6.47214 −0.235701
\(755\) −7.70820 −0.280530
\(756\) 0 0
\(757\) −15.3262 −0.557042 −0.278521 0.960430i \(-0.589844\pi\)
−0.278521 + 0.960430i \(0.589844\pi\)
\(758\) 6.47214 0.235079
\(759\) 0 0
\(760\) 4.41641 0.160200
\(761\) −25.7771 −0.934419 −0.467209 0.884147i \(-0.654741\pi\)
−0.467209 + 0.884147i \(0.654741\pi\)
\(762\) 0 0
\(763\) −9.56231 −0.346179
\(764\) 15.2705 0.552468
\(765\) 0 0
\(766\) −30.9443 −1.11806
\(767\) 36.3607 1.31291
\(768\) 0 0
\(769\) −41.4164 −1.49351 −0.746757 0.665097i \(-0.768390\pi\)
−0.746757 + 0.665097i \(0.768390\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 20.6180 0.742059
\(773\) 21.0557 0.757322 0.378661 0.925535i \(-0.376385\pi\)
0.378661 + 0.925535i \(0.376385\pi\)
\(774\) 0 0
\(775\) 82.0476 2.94724
\(776\) −37.4164 −1.34317
\(777\) 0 0
\(778\) −0.180340 −0.00646550
\(779\) 1.58359 0.0567381
\(780\) 0 0
\(781\) 0 0
\(782\) 9.27051 0.331513
\(783\) 0 0
\(784\) −1.00000 −0.0357143
\(785\) 14.2918 0.510096
\(786\) 0 0
\(787\) 21.2705 0.758212 0.379106 0.925353i \(-0.376232\pi\)
0.379106 + 0.925353i \(0.376232\pi\)
\(788\) −4.00000 −0.142494
\(789\) 0 0
\(790\) 53.9574 1.91972
\(791\) 5.23607 0.186173
\(792\) 0 0
\(793\) −28.9443 −1.02784
\(794\) 16.6525 0.590974
\(795\) 0 0
\(796\) 0.381966 0.0135384
\(797\) 23.2705 0.824284 0.412142 0.911120i \(-0.364781\pi\)
0.412142 + 0.911120i \(0.364781\pi\)
\(798\) 0 0
\(799\) −2.29180 −0.0810779
\(800\) −49.2705 −1.74198
\(801\) 0 0
\(802\) 9.52786 0.336441
\(803\) 0 0
\(804\) 0 0
\(805\) −31.1803 −1.09896
\(806\) 26.9443 0.949071
\(807\) 0 0
\(808\) 27.2705 0.959373
\(809\) 2.65248 0.0932561 0.0466280 0.998912i \(-0.485152\pi\)
0.0466280 + 0.998912i \(0.485152\pi\)
\(810\) 0 0
\(811\) −28.3607 −0.995878 −0.497939 0.867212i \(-0.665910\pi\)
−0.497939 + 0.867212i \(0.665910\pi\)
\(812\) −2.00000 −0.0701862
\(813\) 0 0
\(814\) 0 0
\(815\) −82.5410 −2.89129
\(816\) 0 0
\(817\) 1.23607 0.0432445
\(818\) 33.2361 1.16207
\(819\) 0 0
\(820\) −15.9787 −0.558001
\(821\) −14.3607 −0.501191 −0.250596 0.968092i \(-0.580626\pi\)
−0.250596 + 0.968092i \(0.580626\pi\)
\(822\) 0 0
\(823\) 15.5279 0.541267 0.270634 0.962682i \(-0.412767\pi\)
0.270634 + 0.962682i \(0.412767\pi\)
\(824\) 15.9787 0.556645
\(825\) 0 0
\(826\) −11.2361 −0.390953
\(827\) 11.3262 0.393852 0.196926 0.980418i \(-0.436904\pi\)
0.196926 + 0.980418i \(0.436904\pi\)
\(828\) 0 0
\(829\) 11.0557 0.383981 0.191991 0.981397i \(-0.438506\pi\)
0.191991 + 0.981397i \(0.438506\pi\)
\(830\) −57.5967 −1.99921
\(831\) 0 0
\(832\) −22.6525 −0.785333
\(833\) 1.14590 0.0397030
\(834\) 0 0
\(835\) 54.6525 1.89133
\(836\) 0 0
\(837\) 0 0
\(838\) 19.2361 0.664499
\(839\) −6.58359 −0.227291 −0.113645 0.993521i \(-0.536253\pi\)
−0.113645 + 0.993521i \(0.536253\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) −11.3262 −0.390328
\(843\) 0 0
\(844\) −2.00000 −0.0688428
\(845\) 9.74265 0.335157
\(846\) 0 0
\(847\) 0 0
\(848\) −10.1803 −0.349594
\(849\) 0 0
\(850\) −11.2918 −0.387305
\(851\) 7.36068 0.252321
\(852\) 0 0
\(853\) 2.29180 0.0784696 0.0392348 0.999230i \(-0.487508\pi\)
0.0392348 + 0.999230i \(0.487508\pi\)
\(854\) 8.94427 0.306067
\(855\) 0 0
\(856\) 48.2705 1.64985
\(857\) −1.05573 −0.0360630 −0.0180315 0.999837i \(-0.505740\pi\)
−0.0180315 + 0.999837i \(0.505740\pi\)
\(858\) 0 0
\(859\) 29.8885 1.01978 0.509892 0.860238i \(-0.329685\pi\)
0.509892 + 0.860238i \(0.329685\pi\)
\(860\) −12.4721 −0.425296
\(861\) 0 0
\(862\) −9.09017 −0.309612
\(863\) 22.6869 0.772272 0.386136 0.922442i \(-0.373810\pi\)
0.386136 + 0.922442i \(0.373810\pi\)
\(864\) 0 0
\(865\) −50.6656 −1.72268
\(866\) 12.7639 0.433736
\(867\) 0 0
\(868\) 8.32624 0.282611
\(869\) 0 0
\(870\) 0 0
\(871\) −25.8885 −0.877200
\(872\) 28.6869 0.971462
\(873\) 0 0
\(874\) −3.09017 −0.104527
\(875\) 18.7082 0.632453
\(876\) 0 0
\(877\) 1.63932 0.0553559 0.0276780 0.999617i \(-0.491189\pi\)
0.0276780 + 0.999617i \(0.491189\pi\)
\(878\) 2.79837 0.0944405
\(879\) 0 0
\(880\) 0 0
\(881\) −22.4508 −0.756388 −0.378194 0.925726i \(-0.623455\pi\)
−0.378194 + 0.925726i \(0.623455\pi\)
\(882\) 0 0
\(883\) −5.70820 −0.192096 −0.0960482 0.995377i \(-0.530620\pi\)
−0.0960482 + 0.995377i \(0.530620\pi\)
\(884\) 3.70820 0.124720
\(885\) 0 0
\(886\) 16.0902 0.540560
\(887\) 20.5410 0.689700 0.344850 0.938658i \(-0.387930\pi\)
0.344850 + 0.938658i \(0.387930\pi\)
\(888\) 0 0
\(889\) 1.23607 0.0414564
\(890\) 38.8885 1.30355
\(891\) 0 0
\(892\) 3.79837 0.127179
\(893\) 0.763932 0.0255640
\(894\) 0 0
\(895\) 40.9230 1.36790
\(896\) −3.00000 −0.100223
\(897\) 0 0
\(898\) 6.11146 0.203942
\(899\) −16.6525 −0.555391
\(900\) 0 0
\(901\) 11.6656 0.388639
\(902\) 0 0
\(903\) 0 0
\(904\) −15.7082 −0.522447
\(905\) 4.06888 0.135254
\(906\) 0 0
\(907\) −42.7639 −1.41995 −0.709977 0.704225i \(-0.751294\pi\)
−0.709977 + 0.704225i \(0.751294\pi\)
\(908\) 5.41641 0.179750
\(909\) 0 0
\(910\) 12.4721 0.413447
\(911\) 28.9443 0.958967 0.479483 0.877551i \(-0.340824\pi\)
0.479483 + 0.877551i \(0.340824\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 3.52786 0.116691
\(915\) 0 0
\(916\) −16.7639 −0.553896
\(917\) −15.7082 −0.518731
\(918\) 0 0
\(919\) 19.4164 0.640488 0.320244 0.947335i \(-0.396235\pi\)
0.320244 + 0.947335i \(0.396235\pi\)
\(920\) 93.5410 3.08396
\(921\) 0 0
\(922\) −38.3607 −1.26334
\(923\) −28.9443 −0.952712
\(924\) 0 0
\(925\) −8.96556 −0.294786
\(926\) 15.4164 0.506615
\(927\) 0 0
\(928\) 10.0000 0.328266
\(929\) 4.20163 0.137851 0.0689254 0.997622i \(-0.478043\pi\)
0.0689254 + 0.997622i \(0.478043\pi\)
\(930\) 0 0
\(931\) −0.381966 −0.0125184
\(932\) −7.70820 −0.252491
\(933\) 0 0
\(934\) 14.7639 0.483091
\(935\) 0 0
\(936\) 0 0
\(937\) 1.70820 0.0558046 0.0279023 0.999611i \(-0.491117\pi\)
0.0279023 + 0.999611i \(0.491117\pi\)
\(938\) 8.00000 0.261209
\(939\) 0 0
\(940\) −7.70820 −0.251414
\(941\) −5.14590 −0.167751 −0.0838757 0.996476i \(-0.526730\pi\)
−0.0838757 + 0.996476i \(0.526730\pi\)
\(942\) 0 0
\(943\) 33.5410 1.09225
\(944\) 11.2361 0.365703
\(945\) 0 0
\(946\) 0 0
\(947\) 43.5623 1.41558 0.707792 0.706421i \(-0.249691\pi\)
0.707792 + 0.706421i \(0.249691\pi\)
\(948\) 0 0
\(949\) −2.47214 −0.0802489
\(950\) 3.76393 0.122118
\(951\) 0 0
\(952\) −3.43769 −0.111416
\(953\) −48.8328 −1.58185 −0.790925 0.611913i \(-0.790400\pi\)
−0.790925 + 0.611913i \(0.790400\pi\)
\(954\) 0 0
\(955\) 58.8541 1.90447
\(956\) −18.5623 −0.600348
\(957\) 0 0
\(958\) 28.9443 0.935147
\(959\) 11.2361 0.362832
\(960\) 0 0
\(961\) 38.3262 1.23633
\(962\) −2.94427 −0.0949271
\(963\) 0 0
\(964\) −12.9443 −0.416907
\(965\) 79.4640 2.55804
\(966\) 0 0
\(967\) 13.3050 0.427858 0.213929 0.976849i \(-0.431374\pi\)
0.213929 + 0.976849i \(0.431374\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −48.0689 −1.54340
\(971\) 16.0689 0.515675 0.257838 0.966188i \(-0.416990\pi\)
0.257838 + 0.966188i \(0.416990\pi\)
\(972\) 0 0
\(973\) −19.7984 −0.634707
\(974\) 34.7639 1.11391
\(975\) 0 0
\(976\) −8.94427 −0.286299
\(977\) 53.4164 1.70894 0.854471 0.519499i \(-0.173881\pi\)
0.854471 + 0.519499i \(0.173881\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 3.85410 0.123115
\(981\) 0 0
\(982\) −15.3262 −0.489080
\(983\) −39.5279 −1.26074 −0.630372 0.776294i \(-0.717097\pi\)
−0.630372 + 0.776294i \(0.717097\pi\)
\(984\) 0 0
\(985\) −15.4164 −0.491208
\(986\) 2.29180 0.0729857
\(987\) 0 0
\(988\) −1.23607 −0.0393246
\(989\) 26.1803 0.832486
\(990\) 0 0
\(991\) −12.6525 −0.401919 −0.200960 0.979600i \(-0.564406\pi\)
−0.200960 + 0.979600i \(0.564406\pi\)
\(992\) −41.6312 −1.32179
\(993\) 0 0
\(994\) 8.94427 0.283695
\(995\) 1.47214 0.0466698
\(996\) 0 0
\(997\) −14.5410 −0.460519 −0.230259 0.973129i \(-0.573957\pi\)
−0.230259 + 0.973129i \(0.573957\pi\)
\(998\) −3.81966 −0.120909
\(999\) 0 0
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 7623.2.a.w.1.1 2
3.2 odd 2 2541.2.a.bd.1.2 2
11.7 odd 10 693.2.m.c.379.1 4
11.8 odd 10 693.2.m.c.64.1 4
11.10 odd 2 7623.2.a.bu.1.1 2
33.8 even 10 231.2.j.c.64.1 4
33.29 even 10 231.2.j.c.148.1 yes 4
33.32 even 2 2541.2.a.n.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
231.2.j.c.64.1 4 33.8 even 10
231.2.j.c.148.1 yes 4 33.29 even 10
693.2.m.c.64.1 4 11.8 odd 10
693.2.m.c.379.1 4 11.7 odd 10
2541.2.a.n.1.2 2 33.32 even 2
2541.2.a.bd.1.2 2 3.2 odd 2
7623.2.a.w.1.1 2 1.1 even 1 trivial
7623.2.a.bu.1.1 2 11.10 odd 2