Properties

Label 8001.2.a.l.1.5
Level 8001
Weight 2
Character 8001.1
Self dual yes
Analytic conductor 63.888
Analytic rank 1
Dimension 7
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8001.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.8883066572\)
Analytic rank: \(1\)
Dimension: \(7\)
Coefficient field: 7.7.118870813.1
Defining polynomial: \(x^{7} - 9 x^{5} - 3 x^{4} + 20 x^{3} + 7 x^{2} - 13 x - 4\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2667)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(1.20244\) of defining polynomial
Character \(\chi\) \(=\) 8001.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.24280 q^{2} -0.455452 q^{4} +3.33400 q^{5} +1.00000 q^{7} -3.05163 q^{8} +O(q^{10})\) \(q+1.24280 q^{2} -0.455452 q^{4} +3.33400 q^{5} +1.00000 q^{7} -3.05163 q^{8} +4.14349 q^{10} -2.90717 q^{11} -4.59254 q^{13} +1.24280 q^{14} -2.88166 q^{16} +5.29249 q^{17} +1.86822 q^{19} -1.51848 q^{20} -3.61302 q^{22} -5.03184 q^{23} +6.11557 q^{25} -5.70761 q^{26} -0.455452 q^{28} -3.69938 q^{29} +0.968597 q^{31} +2.52194 q^{32} +6.57750 q^{34} +3.33400 q^{35} -10.5381 q^{37} +2.32183 q^{38} -10.1741 q^{40} -11.5606 q^{41} +2.54868 q^{43} +1.32408 q^{44} -6.25356 q^{46} -7.34702 q^{47} +1.00000 q^{49} +7.60043 q^{50} +2.09168 q^{52} +11.2837 q^{53} -9.69250 q^{55} -3.05163 q^{56} -4.59758 q^{58} +13.3410 q^{59} +0.666722 q^{61} +1.20377 q^{62} +8.89758 q^{64} -15.3116 q^{65} -14.0335 q^{67} -2.41048 q^{68} +4.14349 q^{70} -9.88541 q^{71} -2.57222 q^{73} -13.0968 q^{74} -0.850887 q^{76} -2.90717 q^{77} +14.5706 q^{79} -9.60746 q^{80} -14.3675 q^{82} -0.493442 q^{83} +17.6452 q^{85} +3.16749 q^{86} +8.87160 q^{88} -12.1652 q^{89} -4.59254 q^{91} +2.29176 q^{92} -9.13087 q^{94} +6.22866 q^{95} -14.7006 q^{97} +1.24280 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q + 2q^{2} + 4q^{4} + 8q^{5} + 7q^{7} + 9q^{8} + O(q^{10}) \) \( 7q + 2q^{2} + 4q^{4} + 8q^{5} + 7q^{7} + 9q^{8} + 3q^{11} - 23q^{13} + 2q^{14} + 2q^{16} - 3q^{17} - 9q^{19} + 9q^{20} - 19q^{22} - 12q^{23} + 3q^{25} - 18q^{26} + 4q^{28} + 9q^{29} - 33q^{31} - 10q^{32} - 2q^{34} + 8q^{35} - 33q^{37} + 3q^{38} - 9q^{40} + 3q^{41} - 9q^{43} - 2q^{44} - 32q^{46} - 11q^{47} + 7q^{49} - 29q^{50} - 21q^{52} - q^{53} - 16q^{55} + 9q^{56} - 5q^{58} + 30q^{59} - 19q^{61} - 3q^{62} - 21q^{64} - 14q^{65} - 30q^{67} - 24q^{68} - 8q^{71} - 20q^{73} + 9q^{74} - 42q^{76} + 3q^{77} + 8q^{79} - 12q^{80} + 10q^{82} + 34q^{83} - 28q^{85} - 24q^{86} - q^{88} + 12q^{89} - 23q^{91} - 60q^{92} - 3q^{94} - 12q^{95} + 7q^{97} + 2q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.24280 0.878791 0.439396 0.898294i \(-0.355193\pi\)
0.439396 + 0.898294i \(0.355193\pi\)
\(3\) 0 0
\(4\) −0.455452 −0.227726
\(5\) 3.33400 1.49101 0.745506 0.666499i \(-0.232208\pi\)
0.745506 + 0.666499i \(0.232208\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) −3.05163 −1.07891
\(9\) 0 0
\(10\) 4.14349 1.31029
\(11\) −2.90717 −0.876544 −0.438272 0.898842i \(-0.644409\pi\)
−0.438272 + 0.898842i \(0.644409\pi\)
\(12\) 0 0
\(13\) −4.59254 −1.27374 −0.636871 0.770970i \(-0.719772\pi\)
−0.636871 + 0.770970i \(0.719772\pi\)
\(14\) 1.24280 0.332152
\(15\) 0 0
\(16\) −2.88166 −0.720415
\(17\) 5.29249 1.28362 0.641809 0.766865i \(-0.278184\pi\)
0.641809 + 0.766865i \(0.278184\pi\)
\(18\) 0 0
\(19\) 1.86822 0.428600 0.214300 0.976768i \(-0.431253\pi\)
0.214300 + 0.976768i \(0.431253\pi\)
\(20\) −1.51848 −0.339542
\(21\) 0 0
\(22\) −3.61302 −0.770299
\(23\) −5.03184 −1.04921 −0.524606 0.851345i \(-0.675787\pi\)
−0.524606 + 0.851345i \(0.675787\pi\)
\(24\) 0 0
\(25\) 6.11557 1.22311
\(26\) −5.70761 −1.11935
\(27\) 0 0
\(28\) −0.455452 −0.0860724
\(29\) −3.69938 −0.686957 −0.343478 0.939161i \(-0.611605\pi\)
−0.343478 + 0.939161i \(0.611605\pi\)
\(30\) 0 0
\(31\) 0.968597 0.173965 0.0869826 0.996210i \(-0.472278\pi\)
0.0869826 + 0.996210i \(0.472278\pi\)
\(32\) 2.52194 0.445821
\(33\) 0 0
\(34\) 6.57750 1.12803
\(35\) 3.33400 0.563549
\(36\) 0 0
\(37\) −10.5381 −1.73246 −0.866230 0.499645i \(-0.833464\pi\)
−0.866230 + 0.499645i \(0.833464\pi\)
\(38\) 2.32183 0.376650
\(39\) 0 0
\(40\) −10.1741 −1.60867
\(41\) −11.5606 −1.80546 −0.902731 0.430205i \(-0.858441\pi\)
−0.902731 + 0.430205i \(0.858441\pi\)
\(42\) 0 0
\(43\) 2.54868 0.388670 0.194335 0.980935i \(-0.437745\pi\)
0.194335 + 0.980935i \(0.437745\pi\)
\(44\) 1.32408 0.199612
\(45\) 0 0
\(46\) −6.25356 −0.922038
\(47\) −7.34702 −1.07167 −0.535837 0.844322i \(-0.680004\pi\)
−0.535837 + 0.844322i \(0.680004\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 7.60043 1.07486
\(51\) 0 0
\(52\) 2.09168 0.290064
\(53\) 11.2837 1.54993 0.774966 0.632003i \(-0.217767\pi\)
0.774966 + 0.632003i \(0.217767\pi\)
\(54\) 0 0
\(55\) −9.69250 −1.30694
\(56\) −3.05163 −0.407791
\(57\) 0 0
\(58\) −4.59758 −0.603691
\(59\) 13.3410 1.73685 0.868425 0.495820i \(-0.165132\pi\)
0.868425 + 0.495820i \(0.165132\pi\)
\(60\) 0 0
\(61\) 0.666722 0.0853651 0.0426825 0.999089i \(-0.486410\pi\)
0.0426825 + 0.999089i \(0.486410\pi\)
\(62\) 1.20377 0.152879
\(63\) 0 0
\(64\) 8.89758 1.11220
\(65\) −15.3116 −1.89916
\(66\) 0 0
\(67\) −14.0335 −1.71447 −0.857233 0.514929i \(-0.827818\pi\)
−0.857233 + 0.514929i \(0.827818\pi\)
\(68\) −2.41048 −0.292313
\(69\) 0 0
\(70\) 4.14349 0.495242
\(71\) −9.88541 −1.17318 −0.586591 0.809883i \(-0.699530\pi\)
−0.586591 + 0.809883i \(0.699530\pi\)
\(72\) 0 0
\(73\) −2.57222 −0.301056 −0.150528 0.988606i \(-0.548097\pi\)
−0.150528 + 0.988606i \(0.548097\pi\)
\(74\) −13.0968 −1.52247
\(75\) 0 0
\(76\) −0.850887 −0.0976034
\(77\) −2.90717 −0.331302
\(78\) 0 0
\(79\) 14.5706 1.63932 0.819661 0.572849i \(-0.194162\pi\)
0.819661 + 0.572849i \(0.194162\pi\)
\(80\) −9.60746 −1.07415
\(81\) 0 0
\(82\) −14.3675 −1.58662
\(83\) −0.493442 −0.0541623 −0.0270811 0.999633i \(-0.508621\pi\)
−0.0270811 + 0.999633i \(0.508621\pi\)
\(84\) 0 0
\(85\) 17.6452 1.91389
\(86\) 3.16749 0.341560
\(87\) 0 0
\(88\) 8.87160 0.945716
\(89\) −12.1652 −1.28951 −0.644754 0.764390i \(-0.723040\pi\)
−0.644754 + 0.764390i \(0.723040\pi\)
\(90\) 0 0
\(91\) −4.59254 −0.481429
\(92\) 2.29176 0.238933
\(93\) 0 0
\(94\) −9.13087 −0.941777
\(95\) 6.22866 0.639047
\(96\) 0 0
\(97\) −14.7006 −1.49262 −0.746312 0.665596i \(-0.768177\pi\)
−0.746312 + 0.665596i \(0.768177\pi\)
\(98\) 1.24280 0.125542
\(99\) 0 0
\(100\) −2.78535 −0.278535
\(101\) 7.69701 0.765881 0.382941 0.923773i \(-0.374911\pi\)
0.382941 + 0.923773i \(0.374911\pi\)
\(102\) 0 0
\(103\) −18.0952 −1.78297 −0.891486 0.453048i \(-0.850337\pi\)
−0.891486 + 0.453048i \(0.850337\pi\)
\(104\) 14.0148 1.37426
\(105\) 0 0
\(106\) 14.0233 1.36207
\(107\) 2.30658 0.222986 0.111493 0.993765i \(-0.464437\pi\)
0.111493 + 0.993765i \(0.464437\pi\)
\(108\) 0 0
\(109\) −0.892079 −0.0854457 −0.0427228 0.999087i \(-0.513603\pi\)
−0.0427228 + 0.999087i \(0.513603\pi\)
\(110\) −12.0458 −1.14852
\(111\) 0 0
\(112\) −2.88166 −0.272291
\(113\) −4.35699 −0.409872 −0.204936 0.978775i \(-0.565699\pi\)
−0.204936 + 0.978775i \(0.565699\pi\)
\(114\) 0 0
\(115\) −16.7762 −1.56439
\(116\) 1.68489 0.156438
\(117\) 0 0
\(118\) 16.5802 1.52633
\(119\) 5.29249 0.485162
\(120\) 0 0
\(121\) −2.54838 −0.231671
\(122\) 0.828601 0.0750181
\(123\) 0 0
\(124\) −0.441150 −0.0396164
\(125\) 3.71933 0.332667
\(126\) 0 0
\(127\) −1.00000 −0.0887357
\(128\) 6.01402 0.531569
\(129\) 0 0
\(130\) −19.0292 −1.66897
\(131\) −8.71812 −0.761706 −0.380853 0.924636i \(-0.624370\pi\)
−0.380853 + 0.924636i \(0.624370\pi\)
\(132\) 0 0
\(133\) 1.86822 0.161996
\(134\) −17.4408 −1.50666
\(135\) 0 0
\(136\) −16.1507 −1.38491
\(137\) 4.66285 0.398374 0.199187 0.979962i \(-0.436170\pi\)
0.199187 + 0.979962i \(0.436170\pi\)
\(138\) 0 0
\(139\) −19.7441 −1.67467 −0.837334 0.546691i \(-0.815887\pi\)
−0.837334 + 0.546691i \(0.815887\pi\)
\(140\) −1.51848 −0.128335
\(141\) 0 0
\(142\) −12.2856 −1.03098
\(143\) 13.3513 1.11649
\(144\) 0 0
\(145\) −12.3337 −1.02426
\(146\) −3.19675 −0.264565
\(147\) 0 0
\(148\) 4.79962 0.394526
\(149\) 5.49864 0.450467 0.225233 0.974305i \(-0.427686\pi\)
0.225233 + 0.974305i \(0.427686\pi\)
\(150\) 0 0
\(151\) −11.8169 −0.961645 −0.480822 0.876818i \(-0.659662\pi\)
−0.480822 + 0.876818i \(0.659662\pi\)
\(152\) −5.70113 −0.462423
\(153\) 0 0
\(154\) −3.61302 −0.291146
\(155\) 3.22931 0.259384
\(156\) 0 0
\(157\) 3.77762 0.301487 0.150744 0.988573i \(-0.451833\pi\)
0.150744 + 0.988573i \(0.451833\pi\)
\(158\) 18.1083 1.44062
\(159\) 0 0
\(160\) 8.40817 0.664724
\(161\) −5.03184 −0.396565
\(162\) 0 0
\(163\) −20.0111 −1.56739 −0.783694 0.621147i \(-0.786667\pi\)
−0.783694 + 0.621147i \(0.786667\pi\)
\(164\) 5.26530 0.411151
\(165\) 0 0
\(166\) −0.613249 −0.0475973
\(167\) 16.4295 1.27135 0.635675 0.771957i \(-0.280722\pi\)
0.635675 + 0.771957i \(0.280722\pi\)
\(168\) 0 0
\(169\) 8.09145 0.622419
\(170\) 21.9294 1.68191
\(171\) 0 0
\(172\) −1.16080 −0.0885103
\(173\) 16.2436 1.23498 0.617489 0.786580i \(-0.288150\pi\)
0.617489 + 0.786580i \(0.288150\pi\)
\(174\) 0 0
\(175\) 6.11557 0.462294
\(176\) 8.37746 0.631475
\(177\) 0 0
\(178\) −15.1189 −1.13321
\(179\) 4.00381 0.299259 0.149630 0.988742i \(-0.452192\pi\)
0.149630 + 0.988742i \(0.452192\pi\)
\(180\) 0 0
\(181\) −24.4061 −1.81409 −0.907044 0.421035i \(-0.861667\pi\)
−0.907044 + 0.421035i \(0.861667\pi\)
\(182\) −5.70761 −0.423076
\(183\) 0 0
\(184\) 15.3553 1.13201
\(185\) −35.1342 −2.58312
\(186\) 0 0
\(187\) −15.3862 −1.12515
\(188\) 3.34622 0.244048
\(189\) 0 0
\(190\) 7.74097 0.561589
\(191\) 22.5671 1.63290 0.816450 0.577416i \(-0.195939\pi\)
0.816450 + 0.577416i \(0.195939\pi\)
\(192\) 0 0
\(193\) 20.7333 1.49242 0.746208 0.665713i \(-0.231873\pi\)
0.746208 + 0.665713i \(0.231873\pi\)
\(194\) −18.2699 −1.31170
\(195\) 0 0
\(196\) −0.455452 −0.0325323
\(197\) 3.77768 0.269148 0.134574 0.990904i \(-0.457033\pi\)
0.134574 + 0.990904i \(0.457033\pi\)
\(198\) 0 0
\(199\) −9.73569 −0.690145 −0.345072 0.938576i \(-0.612146\pi\)
−0.345072 + 0.938576i \(0.612146\pi\)
\(200\) −18.6625 −1.31964
\(201\) 0 0
\(202\) 9.56583 0.673050
\(203\) −3.69938 −0.259645
\(204\) 0 0
\(205\) −38.5431 −2.69196
\(206\) −22.4887 −1.56686
\(207\) 0 0
\(208\) 13.2341 0.917623
\(209\) −5.43124 −0.375687
\(210\) 0 0
\(211\) 4.66669 0.321268 0.160634 0.987014i \(-0.448646\pi\)
0.160634 + 0.987014i \(0.448646\pi\)
\(212\) −5.13918 −0.352960
\(213\) 0 0
\(214\) 2.86662 0.195958
\(215\) 8.49730 0.579511
\(216\) 0 0
\(217\) 0.968597 0.0657527
\(218\) −1.10867 −0.0750889
\(219\) 0 0
\(220\) 4.41447 0.297624
\(221\) −24.3060 −1.63500
\(222\) 0 0
\(223\) 5.37863 0.360180 0.180090 0.983650i \(-0.442361\pi\)
0.180090 + 0.983650i \(0.442361\pi\)
\(224\) 2.52194 0.168504
\(225\) 0 0
\(226\) −5.41486 −0.360191
\(227\) 11.5084 0.763837 0.381918 0.924196i \(-0.375264\pi\)
0.381918 + 0.924196i \(0.375264\pi\)
\(228\) 0 0
\(229\) −1.87862 −0.124142 −0.0620712 0.998072i \(-0.519771\pi\)
−0.0620712 + 0.998072i \(0.519771\pi\)
\(230\) −20.8494 −1.37477
\(231\) 0 0
\(232\) 11.2891 0.741168
\(233\) −13.5663 −0.888755 −0.444378 0.895840i \(-0.646575\pi\)
−0.444378 + 0.895840i \(0.646575\pi\)
\(234\) 0 0
\(235\) −24.4950 −1.59788
\(236\) −6.07619 −0.395526
\(237\) 0 0
\(238\) 6.57750 0.426356
\(239\) −22.4619 −1.45294 −0.726468 0.687200i \(-0.758840\pi\)
−0.726468 + 0.687200i \(0.758840\pi\)
\(240\) 0 0
\(241\) 12.0289 0.774848 0.387424 0.921902i \(-0.373365\pi\)
0.387424 + 0.921902i \(0.373365\pi\)
\(242\) −3.16712 −0.203590
\(243\) 0 0
\(244\) −0.303660 −0.0194399
\(245\) 3.33400 0.213002
\(246\) 0 0
\(247\) −8.57990 −0.545926
\(248\) −2.95580 −0.187694
\(249\) 0 0
\(250\) 4.62237 0.292345
\(251\) −5.27452 −0.332924 −0.166462 0.986048i \(-0.553234\pi\)
−0.166462 + 0.986048i \(0.553234\pi\)
\(252\) 0 0
\(253\) 14.6284 0.919680
\(254\) −1.24280 −0.0779801
\(255\) 0 0
\(256\) −10.3210 −0.645060
\(257\) 28.6479 1.78701 0.893504 0.449055i \(-0.148239\pi\)
0.893504 + 0.449055i \(0.148239\pi\)
\(258\) 0 0
\(259\) −10.5381 −0.654808
\(260\) 6.97368 0.432489
\(261\) 0 0
\(262\) −10.8349 −0.669380
\(263\) 6.33902 0.390881 0.195440 0.980716i \(-0.437386\pi\)
0.195440 + 0.980716i \(0.437386\pi\)
\(264\) 0 0
\(265\) 37.6198 2.31097
\(266\) 2.32183 0.142360
\(267\) 0 0
\(268\) 6.39159 0.390429
\(269\) −1.26346 −0.0770348 −0.0385174 0.999258i \(-0.512264\pi\)
−0.0385174 + 0.999258i \(0.512264\pi\)
\(270\) 0 0
\(271\) −5.15641 −0.313230 −0.156615 0.987660i \(-0.550058\pi\)
−0.156615 + 0.987660i \(0.550058\pi\)
\(272\) −15.2512 −0.924737
\(273\) 0 0
\(274\) 5.79498 0.350087
\(275\) −17.7790 −1.07211
\(276\) 0 0
\(277\) 2.77471 0.166716 0.0833580 0.996520i \(-0.473435\pi\)
0.0833580 + 0.996520i \(0.473435\pi\)
\(278\) −24.5379 −1.47168
\(279\) 0 0
\(280\) −10.1741 −0.608022
\(281\) −5.01385 −0.299101 −0.149551 0.988754i \(-0.547783\pi\)
−0.149551 + 0.988754i \(0.547783\pi\)
\(282\) 0 0
\(283\) −8.84064 −0.525521 −0.262761 0.964861i \(-0.584633\pi\)
−0.262761 + 0.964861i \(0.584633\pi\)
\(284\) 4.50233 0.267164
\(285\) 0 0
\(286\) 16.5930 0.981162
\(287\) −11.5606 −0.682401
\(288\) 0 0
\(289\) 11.0105 0.647674
\(290\) −15.3283 −0.900111
\(291\) 0 0
\(292\) 1.17152 0.0685583
\(293\) 16.5617 0.967545 0.483772 0.875194i \(-0.339266\pi\)
0.483772 + 0.875194i \(0.339266\pi\)
\(294\) 0 0
\(295\) 44.4789 2.58966
\(296\) 32.1585 1.86918
\(297\) 0 0
\(298\) 6.83371 0.395866
\(299\) 23.1089 1.33643
\(300\) 0 0
\(301\) 2.54868 0.146903
\(302\) −14.6860 −0.845085
\(303\) 0 0
\(304\) −5.38358 −0.308770
\(305\) 2.22285 0.127280
\(306\) 0 0
\(307\) −23.4209 −1.33670 −0.668352 0.743845i \(-0.733000\pi\)
−0.668352 + 0.743845i \(0.733000\pi\)
\(308\) 1.32408 0.0754462
\(309\) 0 0
\(310\) 4.01338 0.227944
\(311\) 19.5726 1.10986 0.554929 0.831898i \(-0.312745\pi\)
0.554929 + 0.831898i \(0.312745\pi\)
\(312\) 0 0
\(313\) 5.10217 0.288392 0.144196 0.989549i \(-0.453940\pi\)
0.144196 + 0.989549i \(0.453940\pi\)
\(314\) 4.69483 0.264944
\(315\) 0 0
\(316\) −6.63622 −0.373316
\(317\) 19.6691 1.10472 0.552362 0.833604i \(-0.313727\pi\)
0.552362 + 0.833604i \(0.313727\pi\)
\(318\) 0 0
\(319\) 10.7547 0.602148
\(320\) 29.6646 1.65830
\(321\) 0 0
\(322\) −6.25356 −0.348497
\(323\) 9.88756 0.550158
\(324\) 0 0
\(325\) −28.0860 −1.55793
\(326\) −24.8697 −1.37741
\(327\) 0 0
\(328\) 35.2787 1.94794
\(329\) −7.34702 −0.405055
\(330\) 0 0
\(331\) 15.1305 0.831648 0.415824 0.909445i \(-0.363493\pi\)
0.415824 + 0.909445i \(0.363493\pi\)
\(332\) 0.224739 0.0123342
\(333\) 0 0
\(334\) 20.4185 1.11725
\(335\) −46.7877 −2.55629
\(336\) 0 0
\(337\) −7.21235 −0.392882 −0.196441 0.980516i \(-0.562938\pi\)
−0.196441 + 0.980516i \(0.562938\pi\)
\(338\) 10.0560 0.546977
\(339\) 0 0
\(340\) −8.03654 −0.435842
\(341\) −2.81587 −0.152488
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) −7.77763 −0.419342
\(345\) 0 0
\(346\) 20.1875 1.08529
\(347\) −28.7813 −1.54506 −0.772531 0.634977i \(-0.781010\pi\)
−0.772531 + 0.634977i \(0.781010\pi\)
\(348\) 0 0
\(349\) −28.8201 −1.54270 −0.771352 0.636409i \(-0.780419\pi\)
−0.771352 + 0.636409i \(0.780419\pi\)
\(350\) 7.60043 0.406260
\(351\) 0 0
\(352\) −7.33171 −0.390782
\(353\) −26.6489 −1.41838 −0.709188 0.705019i \(-0.750938\pi\)
−0.709188 + 0.705019i \(0.750938\pi\)
\(354\) 0 0
\(355\) −32.9580 −1.74923
\(356\) 5.54066 0.293655
\(357\) 0 0
\(358\) 4.97593 0.262986
\(359\) 14.4278 0.761469 0.380734 0.924684i \(-0.375671\pi\)
0.380734 + 0.924684i \(0.375671\pi\)
\(360\) 0 0
\(361\) −15.5097 −0.816302
\(362\) −30.3318 −1.59421
\(363\) 0 0
\(364\) 2.09168 0.109634
\(365\) −8.57580 −0.448878
\(366\) 0 0
\(367\) −8.35795 −0.436282 −0.218141 0.975917i \(-0.569999\pi\)
−0.218141 + 0.975917i \(0.569999\pi\)
\(368\) 14.5000 0.755867
\(369\) 0 0
\(370\) −43.6647 −2.27002
\(371\) 11.2837 0.585819
\(372\) 0 0
\(373\) 30.3153 1.56966 0.784832 0.619708i \(-0.212749\pi\)
0.784832 + 0.619708i \(0.212749\pi\)
\(374\) −19.1219 −0.988769
\(375\) 0 0
\(376\) 22.4204 1.15624
\(377\) 16.9895 0.875006
\(378\) 0 0
\(379\) 2.57847 0.132447 0.0662235 0.997805i \(-0.478905\pi\)
0.0662235 + 0.997805i \(0.478905\pi\)
\(380\) −2.83686 −0.145528
\(381\) 0 0
\(382\) 28.0464 1.43498
\(383\) 14.6277 0.747438 0.373719 0.927542i \(-0.378082\pi\)
0.373719 + 0.927542i \(0.378082\pi\)
\(384\) 0 0
\(385\) −9.69250 −0.493976
\(386\) 25.7673 1.31152
\(387\) 0 0
\(388\) 6.69544 0.339909
\(389\) −19.5586 −0.991663 −0.495831 0.868419i \(-0.665137\pi\)
−0.495831 + 0.868419i \(0.665137\pi\)
\(390\) 0 0
\(391\) −26.6310 −1.34679
\(392\) −3.05163 −0.154131
\(393\) 0 0
\(394\) 4.69489 0.236525
\(395\) 48.5784 2.44425
\(396\) 0 0
\(397\) −8.35455 −0.419303 −0.209651 0.977776i \(-0.567233\pi\)
−0.209651 + 0.977776i \(0.567233\pi\)
\(398\) −12.0995 −0.606493
\(399\) 0 0
\(400\) −17.6230 −0.881150
\(401\) −7.25894 −0.362494 −0.181247 0.983438i \(-0.558013\pi\)
−0.181247 + 0.983438i \(0.558013\pi\)
\(402\) 0 0
\(403\) −4.44832 −0.221587
\(404\) −3.50562 −0.174411
\(405\) 0 0
\(406\) −4.59758 −0.228174
\(407\) 30.6361 1.51858
\(408\) 0 0
\(409\) 8.46320 0.418478 0.209239 0.977864i \(-0.432901\pi\)
0.209239 + 0.977864i \(0.432901\pi\)
\(410\) −47.9013 −2.36567
\(411\) 0 0
\(412\) 8.24150 0.406029
\(413\) 13.3410 0.656468
\(414\) 0 0
\(415\) −1.64514 −0.0807566
\(416\) −11.5821 −0.567861
\(417\) 0 0
\(418\) −6.74994 −0.330150
\(419\) −26.7971 −1.30912 −0.654561 0.756009i \(-0.727146\pi\)
−0.654561 + 0.756009i \(0.727146\pi\)
\(420\) 0 0
\(421\) 14.1261 0.688463 0.344231 0.938885i \(-0.388140\pi\)
0.344231 + 0.938885i \(0.388140\pi\)
\(422\) 5.79976 0.282328
\(423\) 0 0
\(424\) −34.4336 −1.67225
\(425\) 32.3666 1.57001
\(426\) 0 0
\(427\) 0.666722 0.0322650
\(428\) −1.05054 −0.0507797
\(429\) 0 0
\(430\) 10.5604 0.509269
\(431\) 16.6128 0.800211 0.400105 0.916469i \(-0.368974\pi\)
0.400105 + 0.916469i \(0.368974\pi\)
\(432\) 0 0
\(433\) 24.5878 1.18161 0.590807 0.806813i \(-0.298810\pi\)
0.590807 + 0.806813i \(0.298810\pi\)
\(434\) 1.20377 0.0577829
\(435\) 0 0
\(436\) 0.406300 0.0194582
\(437\) −9.40061 −0.449692
\(438\) 0 0
\(439\) −38.1271 −1.81971 −0.909853 0.414930i \(-0.863806\pi\)
−0.909853 + 0.414930i \(0.863806\pi\)
\(440\) 29.5780 1.41007
\(441\) 0 0
\(442\) −30.2074 −1.43682
\(443\) −22.7120 −1.07908 −0.539541 0.841959i \(-0.681402\pi\)
−0.539541 + 0.841959i \(0.681402\pi\)
\(444\) 0 0
\(445\) −40.5588 −1.92267
\(446\) 6.68455 0.316523
\(447\) 0 0
\(448\) 8.89758 0.420371
\(449\) 29.0754 1.37215 0.686075 0.727531i \(-0.259332\pi\)
0.686075 + 0.727531i \(0.259332\pi\)
\(450\) 0 0
\(451\) 33.6086 1.58257
\(452\) 1.98440 0.0933385
\(453\) 0 0
\(454\) 14.3026 0.671253
\(455\) −15.3116 −0.717817
\(456\) 0 0
\(457\) 25.5196 1.19376 0.596879 0.802332i \(-0.296407\pi\)
0.596879 + 0.802332i \(0.296407\pi\)
\(458\) −2.33474 −0.109095
\(459\) 0 0
\(460\) 7.64075 0.356252
\(461\) 10.8134 0.503631 0.251815 0.967775i \(-0.418972\pi\)
0.251815 + 0.967775i \(0.418972\pi\)
\(462\) 0 0
\(463\) 5.98793 0.278283 0.139141 0.990273i \(-0.455566\pi\)
0.139141 + 0.990273i \(0.455566\pi\)
\(464\) 10.6603 0.494894
\(465\) 0 0
\(466\) −16.8601 −0.781030
\(467\) −41.9564 −1.94151 −0.970755 0.240074i \(-0.922828\pi\)
−0.970755 + 0.240074i \(0.922828\pi\)
\(468\) 0 0
\(469\) −14.0335 −0.648007
\(470\) −30.4423 −1.40420
\(471\) 0 0
\(472\) −40.7118 −1.87391
\(473\) −7.40943 −0.340686
\(474\) 0 0
\(475\) 11.4253 0.524227
\(476\) −2.41048 −0.110484
\(477\) 0 0
\(478\) −27.9156 −1.27683
\(479\) 9.65088 0.440960 0.220480 0.975392i \(-0.429238\pi\)
0.220480 + 0.975392i \(0.429238\pi\)
\(480\) 0 0
\(481\) 48.3969 2.20671
\(482\) 14.9495 0.680930
\(483\) 0 0
\(484\) 1.16067 0.0527575
\(485\) −49.0120 −2.22552
\(486\) 0 0
\(487\) −17.7189 −0.802920 −0.401460 0.915877i \(-0.631497\pi\)
−0.401460 + 0.915877i \(0.631497\pi\)
\(488\) −2.03459 −0.0921016
\(489\) 0 0
\(490\) 4.14349 0.187184
\(491\) 4.97599 0.224563 0.112282 0.993676i \(-0.464184\pi\)
0.112282 + 0.993676i \(0.464184\pi\)
\(492\) 0 0
\(493\) −19.5789 −0.881790
\(494\) −10.6631 −0.479755
\(495\) 0 0
\(496\) −2.79117 −0.125327
\(497\) −9.88541 −0.443421
\(498\) 0 0
\(499\) 44.0654 1.97264 0.986318 0.164851i \(-0.0527143\pi\)
0.986318 + 0.164851i \(0.0527143\pi\)
\(500\) −1.69398 −0.0757569
\(501\) 0 0
\(502\) −6.55516 −0.292571
\(503\) −5.31840 −0.237136 −0.118568 0.992946i \(-0.537830\pi\)
−0.118568 + 0.992946i \(0.537830\pi\)
\(504\) 0 0
\(505\) 25.6619 1.14194
\(506\) 18.1802 0.808207
\(507\) 0 0
\(508\) 0.455452 0.0202074
\(509\) −10.4607 −0.463661 −0.231830 0.972756i \(-0.574471\pi\)
−0.231830 + 0.972756i \(0.574471\pi\)
\(510\) 0 0
\(511\) −2.57222 −0.113788
\(512\) −24.8549 −1.09844
\(513\) 0 0
\(514\) 35.6036 1.57041
\(515\) −60.3294 −2.65843
\(516\) 0 0
\(517\) 21.3590 0.939369
\(518\) −13.0968 −0.575440
\(519\) 0 0
\(520\) 46.7252 2.04904
\(521\) −35.6052 −1.55989 −0.779947 0.625846i \(-0.784754\pi\)
−0.779947 + 0.625846i \(0.784754\pi\)
\(522\) 0 0
\(523\) 23.5608 1.03024 0.515120 0.857118i \(-0.327747\pi\)
0.515120 + 0.857118i \(0.327747\pi\)
\(524\) 3.97069 0.173460
\(525\) 0 0
\(526\) 7.87813 0.343503
\(527\) 5.12629 0.223305
\(528\) 0 0
\(529\) 2.31943 0.100845
\(530\) 46.7538 2.03086
\(531\) 0 0
\(532\) −0.850887 −0.0368906
\(533\) 53.0926 2.29969
\(534\) 0 0
\(535\) 7.69015 0.332474
\(536\) 42.8251 1.84976
\(537\) 0 0
\(538\) −1.57023 −0.0676975
\(539\) −2.90717 −0.125221
\(540\) 0 0
\(541\) 7.71667 0.331765 0.165883 0.986145i \(-0.446953\pi\)
0.165883 + 0.986145i \(0.446953\pi\)
\(542\) −6.40838 −0.275264
\(543\) 0 0
\(544\) 13.3474 0.572263
\(545\) −2.97419 −0.127400
\(546\) 0 0
\(547\) −15.9737 −0.682987 −0.341494 0.939884i \(-0.610933\pi\)
−0.341494 + 0.939884i \(0.610933\pi\)
\(548\) −2.12370 −0.0907201
\(549\) 0 0
\(550\) −22.0957 −0.942164
\(551\) −6.91126 −0.294430
\(552\) 0 0
\(553\) 14.5706 0.619605
\(554\) 3.44840 0.146509
\(555\) 0 0
\(556\) 8.99247 0.381366
\(557\) −0.464136 −0.0196661 −0.00983304 0.999952i \(-0.503130\pi\)
−0.00983304 + 0.999952i \(0.503130\pi\)
\(558\) 0 0
\(559\) −11.7049 −0.495065
\(560\) −9.60746 −0.405989
\(561\) 0 0
\(562\) −6.23121 −0.262848
\(563\) −35.9885 −1.51673 −0.758367 0.651828i \(-0.774002\pi\)
−0.758367 + 0.651828i \(0.774002\pi\)
\(564\) 0 0
\(565\) −14.5262 −0.611123
\(566\) −10.9871 −0.461824
\(567\) 0 0
\(568\) 30.1666 1.26576
\(569\) 11.1032 0.465469 0.232735 0.972540i \(-0.425233\pi\)
0.232735 + 0.972540i \(0.425233\pi\)
\(570\) 0 0
\(571\) 10.2214 0.427753 0.213877 0.976861i \(-0.431391\pi\)
0.213877 + 0.976861i \(0.431391\pi\)
\(572\) −6.08088 −0.254254
\(573\) 0 0
\(574\) −14.3675 −0.599688
\(575\) −30.7726 −1.28331
\(576\) 0 0
\(577\) 10.4490 0.434999 0.217500 0.976060i \(-0.430210\pi\)
0.217500 + 0.976060i \(0.430210\pi\)
\(578\) 13.6838 0.569170
\(579\) 0 0
\(580\) 5.61742 0.233251
\(581\) −0.493442 −0.0204714
\(582\) 0 0
\(583\) −32.8035 −1.35858
\(584\) 7.84948 0.324814
\(585\) 0 0
\(586\) 20.5829 0.850270
\(587\) −37.8825 −1.56358 −0.781789 0.623543i \(-0.785693\pi\)
−0.781789 + 0.623543i \(0.785693\pi\)
\(588\) 0 0
\(589\) 1.80956 0.0745615
\(590\) 55.2783 2.27577
\(591\) 0 0
\(592\) 30.3673 1.24809
\(593\) 43.2711 1.77693 0.888466 0.458942i \(-0.151772\pi\)
0.888466 + 0.458942i \(0.151772\pi\)
\(594\) 0 0
\(595\) 17.6452 0.723382
\(596\) −2.50437 −0.102583
\(597\) 0 0
\(598\) 28.7198 1.17444
\(599\) 28.8279 1.17787 0.588937 0.808179i \(-0.299547\pi\)
0.588937 + 0.808179i \(0.299547\pi\)
\(600\) 0 0
\(601\) 29.8137 1.21612 0.608062 0.793889i \(-0.291947\pi\)
0.608062 + 0.793889i \(0.291947\pi\)
\(602\) 3.16749 0.129097
\(603\) 0 0
\(604\) 5.38203 0.218992
\(605\) −8.49630 −0.345424
\(606\) 0 0
\(607\) −15.0035 −0.608974 −0.304487 0.952516i \(-0.598485\pi\)
−0.304487 + 0.952516i \(0.598485\pi\)
\(608\) 4.71156 0.191079
\(609\) 0 0
\(610\) 2.76256 0.111853
\(611\) 33.7415 1.36504
\(612\) 0 0
\(613\) −5.68365 −0.229561 −0.114780 0.993391i \(-0.536616\pi\)
−0.114780 + 0.993391i \(0.536616\pi\)
\(614\) −29.1075 −1.17468
\(615\) 0 0
\(616\) 8.87160 0.357447
\(617\) −13.5363 −0.544952 −0.272476 0.962163i \(-0.587842\pi\)
−0.272476 + 0.962163i \(0.587842\pi\)
\(618\) 0 0
\(619\) −27.0321 −1.08651 −0.543256 0.839567i \(-0.682809\pi\)
−0.543256 + 0.839567i \(0.682809\pi\)
\(620\) −1.47079 −0.0590685
\(621\) 0 0
\(622\) 24.3247 0.975334
\(623\) −12.1652 −0.487388
\(624\) 0 0
\(625\) −18.1776 −0.727105
\(626\) 6.34097 0.253436
\(627\) 0 0
\(628\) −1.72053 −0.0686565
\(629\) −55.7730 −2.22382
\(630\) 0 0
\(631\) 2.75817 0.109801 0.0549004 0.998492i \(-0.482516\pi\)
0.0549004 + 0.998492i \(0.482516\pi\)
\(632\) −44.4641 −1.76869
\(633\) 0 0
\(634\) 24.4447 0.970822
\(635\) −3.33400 −0.132306
\(636\) 0 0
\(637\) −4.59254 −0.181963
\(638\) 13.3659 0.529162
\(639\) 0 0
\(640\) 20.0507 0.792575
\(641\) 10.2155 0.403489 0.201744 0.979438i \(-0.435339\pi\)
0.201744 + 0.979438i \(0.435339\pi\)
\(642\) 0 0
\(643\) −25.3762 −1.00074 −0.500370 0.865812i \(-0.666803\pi\)
−0.500370 + 0.865812i \(0.666803\pi\)
\(644\) 2.29176 0.0903081
\(645\) 0 0
\(646\) 12.2882 0.483474
\(647\) 22.1567 0.871068 0.435534 0.900172i \(-0.356560\pi\)
0.435534 + 0.900172i \(0.356560\pi\)
\(648\) 0 0
\(649\) −38.7845 −1.52243
\(650\) −34.9053 −1.36910
\(651\) 0 0
\(652\) 9.11409 0.356935
\(653\) 9.65817 0.377954 0.188977 0.981982i \(-0.439483\pi\)
0.188977 + 0.981982i \(0.439483\pi\)
\(654\) 0 0
\(655\) −29.0662 −1.13571
\(656\) 33.3137 1.30068
\(657\) 0 0
\(658\) −9.13087 −0.355958
\(659\) 14.6717 0.571527 0.285764 0.958300i \(-0.407753\pi\)
0.285764 + 0.958300i \(0.407753\pi\)
\(660\) 0 0
\(661\) 33.3172 1.29589 0.647945 0.761687i \(-0.275629\pi\)
0.647945 + 0.761687i \(0.275629\pi\)
\(662\) 18.8042 0.730845
\(663\) 0 0
\(664\) 1.50580 0.0584365
\(665\) 6.22866 0.241537
\(666\) 0 0
\(667\) 18.6147 0.720763
\(668\) −7.48283 −0.289519
\(669\) 0 0
\(670\) −58.1477 −2.24644
\(671\) −1.93827 −0.0748262
\(672\) 0 0
\(673\) 1.32670 0.0511404 0.0255702 0.999673i \(-0.491860\pi\)
0.0255702 + 0.999673i \(0.491860\pi\)
\(674\) −8.96350 −0.345261
\(675\) 0 0
\(676\) −3.68527 −0.141741
\(677\) −11.9106 −0.457761 −0.228881 0.973454i \(-0.573507\pi\)
−0.228881 + 0.973454i \(0.573507\pi\)
\(678\) 0 0
\(679\) −14.7006 −0.564159
\(680\) −53.8466 −2.06492
\(681\) 0 0
\(682\) −3.49956 −0.134005
\(683\) 6.04534 0.231318 0.115659 0.993289i \(-0.463102\pi\)
0.115659 + 0.993289i \(0.463102\pi\)
\(684\) 0 0
\(685\) 15.5459 0.593980
\(686\) 1.24280 0.0474503
\(687\) 0 0
\(688\) −7.34442 −0.280003
\(689\) −51.8208 −1.97421
\(690\) 0 0
\(691\) −26.4623 −1.00667 −0.503336 0.864090i \(-0.667894\pi\)
−0.503336 + 0.864090i \(0.667894\pi\)
\(692\) −7.39818 −0.281237
\(693\) 0 0
\(694\) −35.7694 −1.35779
\(695\) −65.8267 −2.49695
\(696\) 0 0
\(697\) −61.1844 −2.31752
\(698\) −35.8176 −1.35571
\(699\) 0 0
\(700\) −2.78535 −0.105276
\(701\) −27.8873 −1.05329 −0.526645 0.850085i \(-0.676550\pi\)
−0.526645 + 0.850085i \(0.676550\pi\)
\(702\) 0 0
\(703\) −19.6876 −0.742532
\(704\) −25.8668 −0.974890
\(705\) 0 0
\(706\) −33.1192 −1.24646
\(707\) 7.69701 0.289476
\(708\) 0 0
\(709\) −21.7892 −0.818312 −0.409156 0.912465i \(-0.634177\pi\)
−0.409156 + 0.912465i \(0.634177\pi\)
\(710\) −40.9601 −1.53721
\(711\) 0 0
\(712\) 37.1237 1.39127
\(713\) −4.87383 −0.182526
\(714\) 0 0
\(715\) 44.5132 1.66470
\(716\) −1.82355 −0.0681491
\(717\) 0 0
\(718\) 17.9308 0.669172
\(719\) −0.503768 −0.0187874 −0.00939368 0.999956i \(-0.502990\pi\)
−0.00939368 + 0.999956i \(0.502990\pi\)
\(720\) 0 0
\(721\) −18.0952 −0.673900
\(722\) −19.2755 −0.717359
\(723\) 0 0
\(724\) 11.1158 0.413115
\(725\) −22.6238 −0.840227
\(726\) 0 0
\(727\) 23.4153 0.868427 0.434213 0.900810i \(-0.357026\pi\)
0.434213 + 0.900810i \(0.357026\pi\)
\(728\) 14.0148 0.519421
\(729\) 0 0
\(730\) −10.6580 −0.394470
\(731\) 13.4889 0.498903
\(732\) 0 0
\(733\) 52.3311 1.93289 0.966446 0.256871i \(-0.0826915\pi\)
0.966446 + 0.256871i \(0.0826915\pi\)
\(734\) −10.3872 −0.383400
\(735\) 0 0
\(736\) −12.6900 −0.467760
\(737\) 40.7978 1.50280
\(738\) 0 0
\(739\) −11.4312 −0.420504 −0.210252 0.977647i \(-0.567429\pi\)
−0.210252 + 0.977647i \(0.567429\pi\)
\(740\) 16.0020 0.588243
\(741\) 0 0
\(742\) 14.0233 0.514813
\(743\) −33.8238 −1.24087 −0.620437 0.784256i \(-0.713045\pi\)
−0.620437 + 0.784256i \(0.713045\pi\)
\(744\) 0 0
\(745\) 18.3325 0.671651
\(746\) 37.6758 1.37941
\(747\) 0 0
\(748\) 7.00766 0.256225
\(749\) 2.30658 0.0842807
\(750\) 0 0
\(751\) −50.0895 −1.82779 −0.913896 0.405949i \(-0.866941\pi\)
−0.913896 + 0.405949i \(0.866941\pi\)
\(752\) 21.1716 0.772049
\(753\) 0 0
\(754\) 21.1146 0.768947
\(755\) −39.3975 −1.43382
\(756\) 0 0
\(757\) 16.9391 0.615663 0.307831 0.951441i \(-0.400397\pi\)
0.307831 + 0.951441i \(0.400397\pi\)
\(758\) 3.20451 0.116393
\(759\) 0 0
\(760\) −19.0076 −0.689478
\(761\) 32.7577 1.18747 0.593734 0.804662i \(-0.297653\pi\)
0.593734 + 0.804662i \(0.297653\pi\)
\(762\) 0 0
\(763\) −0.892079 −0.0322954
\(764\) −10.2783 −0.371854
\(765\) 0 0
\(766\) 18.1792 0.656842
\(767\) −61.2691 −2.21230
\(768\) 0 0
\(769\) 47.6458 1.71815 0.859075 0.511850i \(-0.171040\pi\)
0.859075 + 0.511850i \(0.171040\pi\)
\(770\) −12.0458 −0.434101
\(771\) 0 0
\(772\) −9.44303 −0.339862
\(773\) 47.1769 1.69683 0.848417 0.529328i \(-0.177556\pi\)
0.848417 + 0.529328i \(0.177556\pi\)
\(774\) 0 0
\(775\) 5.92353 0.212779
\(776\) 44.8609 1.61041
\(777\) 0 0
\(778\) −24.3075 −0.871464
\(779\) −21.5978 −0.773821
\(780\) 0 0
\(781\) 28.7385 1.02835
\(782\) −33.0969 −1.18354
\(783\) 0 0
\(784\) −2.88166 −0.102916
\(785\) 12.5946 0.449521
\(786\) 0 0
\(787\) −15.6611 −0.558257 −0.279129 0.960254i \(-0.590046\pi\)
−0.279129 + 0.960254i \(0.590046\pi\)
\(788\) −1.72055 −0.0612921
\(789\) 0 0
\(790\) 60.3732 2.14798
\(791\) −4.35699 −0.154917
\(792\) 0 0
\(793\) −3.06195 −0.108733
\(794\) −10.3830 −0.368479
\(795\) 0 0
\(796\) 4.43414 0.157164
\(797\) 2.11085 0.0747702 0.0373851 0.999301i \(-0.488097\pi\)
0.0373851 + 0.999301i \(0.488097\pi\)
\(798\) 0 0
\(799\) −38.8841 −1.37562
\(800\) 15.4231 0.545290
\(801\) 0 0
\(802\) −9.02139 −0.318556
\(803\) 7.47788 0.263889
\(804\) 0 0
\(805\) −16.7762 −0.591282
\(806\) −5.52837 −0.194729
\(807\) 0 0
\(808\) −23.4884 −0.826321
\(809\) 10.8674 0.382079 0.191039 0.981582i \(-0.438814\pi\)
0.191039 + 0.981582i \(0.438814\pi\)
\(810\) 0 0
\(811\) −46.4996 −1.63282 −0.816411 0.577472i \(-0.804039\pi\)
−0.816411 + 0.577472i \(0.804039\pi\)
\(812\) 1.68489 0.0591280
\(813\) 0 0
\(814\) 38.0746 1.33451
\(815\) −66.7170 −2.33699
\(816\) 0 0
\(817\) 4.76150 0.166584
\(818\) 10.5181 0.367755
\(819\) 0 0
\(820\) 17.5545 0.613031
\(821\) 26.4225 0.922152 0.461076 0.887361i \(-0.347464\pi\)
0.461076 + 0.887361i \(0.347464\pi\)
\(822\) 0 0
\(823\) −26.4500 −0.921989 −0.460995 0.887403i \(-0.652507\pi\)
−0.460995 + 0.887403i \(0.652507\pi\)
\(824\) 55.2199 1.92368
\(825\) 0 0
\(826\) 16.5802 0.576898
\(827\) −31.2695 −1.08735 −0.543673 0.839297i \(-0.682967\pi\)
−0.543673 + 0.839297i \(0.682967\pi\)
\(828\) 0 0
\(829\) −52.6203 −1.82758 −0.913790 0.406187i \(-0.866858\pi\)
−0.913790 + 0.406187i \(0.866858\pi\)
\(830\) −2.04457 −0.0709682
\(831\) 0 0
\(832\) −40.8625 −1.41665
\(833\) 5.29249 0.183374
\(834\) 0 0
\(835\) 54.7758 1.89560
\(836\) 2.47367 0.0855537
\(837\) 0 0
\(838\) −33.3033 −1.15044
\(839\) 29.2755 1.01070 0.505352 0.862913i \(-0.331363\pi\)
0.505352 + 0.862913i \(0.331363\pi\)
\(840\) 0 0
\(841\) −15.3146 −0.528090
\(842\) 17.5559 0.605015
\(843\) 0 0
\(844\) −2.12546 −0.0731612
\(845\) 26.9769 0.928034
\(846\) 0 0
\(847\) −2.54838 −0.0875633
\(848\) −32.5157 −1.11659
\(849\) 0 0
\(850\) 40.2252 1.37971
\(851\) 53.0263 1.81772
\(852\) 0 0
\(853\) −22.4780 −0.769631 −0.384816 0.922994i \(-0.625735\pi\)
−0.384816 + 0.922994i \(0.625735\pi\)
\(854\) 0.828601 0.0283542
\(855\) 0 0
\(856\) −7.03884 −0.240583
\(857\) 35.5634 1.21482 0.607412 0.794387i \(-0.292208\pi\)
0.607412 + 0.794387i \(0.292208\pi\)
\(858\) 0 0
\(859\) 38.2979 1.30671 0.653354 0.757053i \(-0.273361\pi\)
0.653354 + 0.757053i \(0.273361\pi\)
\(860\) −3.87011 −0.131970
\(861\) 0 0
\(862\) 20.6464 0.703218
\(863\) −47.7198 −1.62440 −0.812201 0.583378i \(-0.801730\pi\)
−0.812201 + 0.583378i \(0.801730\pi\)
\(864\) 0 0
\(865\) 54.1562 1.84137
\(866\) 30.5577 1.03839
\(867\) 0 0
\(868\) −0.441150 −0.0149736
\(869\) −42.3592 −1.43694
\(870\) 0 0
\(871\) 64.4495 2.18379
\(872\) 2.72230 0.0921886
\(873\) 0 0
\(874\) −11.6831 −0.395185
\(875\) 3.71933 0.125736
\(876\) 0 0
\(877\) −20.9916 −0.708837 −0.354418 0.935087i \(-0.615321\pi\)
−0.354418 + 0.935087i \(0.615321\pi\)
\(878\) −47.3843 −1.59914
\(879\) 0 0
\(880\) 27.9305 0.941536
\(881\) 47.8891 1.61343 0.806713 0.590943i \(-0.201244\pi\)
0.806713 + 0.590943i \(0.201244\pi\)
\(882\) 0 0
\(883\) 10.1634 0.342026 0.171013 0.985269i \(-0.445296\pi\)
0.171013 + 0.985269i \(0.445296\pi\)
\(884\) 11.0702 0.372332
\(885\) 0 0
\(886\) −28.2265 −0.948287
\(887\) 49.2135 1.65243 0.826213 0.563357i \(-0.190490\pi\)
0.826213 + 0.563357i \(0.190490\pi\)
\(888\) 0 0
\(889\) −1.00000 −0.0335389
\(890\) −50.4064 −1.68963
\(891\) 0 0
\(892\) −2.44971 −0.0820223
\(893\) −13.7259 −0.459319
\(894\) 0 0
\(895\) 13.3487 0.446199
\(896\) 6.01402 0.200914
\(897\) 0 0
\(898\) 36.1348 1.20583
\(899\) −3.58320 −0.119507
\(900\) 0 0
\(901\) 59.7188 1.98952
\(902\) 41.7687 1.39075
\(903\) 0 0
\(904\) 13.2959 0.442216
\(905\) −81.3699 −2.70483
\(906\) 0 0
\(907\) 50.4233 1.67428 0.837139 0.546991i \(-0.184227\pi\)
0.837139 + 0.546991i \(0.184227\pi\)
\(908\) −5.24151 −0.173946
\(909\) 0 0
\(910\) −19.0292 −0.630811
\(911\) −19.4045 −0.642899 −0.321449 0.946927i \(-0.604170\pi\)
−0.321449 + 0.946927i \(0.604170\pi\)
\(912\) 0 0
\(913\) 1.43452 0.0474756
\(914\) 31.7157 1.04906
\(915\) 0 0
\(916\) 0.855620 0.0282705
\(917\) −8.71812 −0.287898
\(918\) 0 0
\(919\) −38.9767 −1.28572 −0.642861 0.765983i \(-0.722253\pi\)
−0.642861 + 0.765983i \(0.722253\pi\)
\(920\) 51.1947 1.68784
\(921\) 0 0
\(922\) 13.4389 0.442586
\(923\) 45.3992 1.49433
\(924\) 0 0
\(925\) −64.4468 −2.11900
\(926\) 7.44179 0.244552
\(927\) 0 0
\(928\) −9.32962 −0.306260
\(929\) 2.48794 0.0816266 0.0408133 0.999167i \(-0.487005\pi\)
0.0408133 + 0.999167i \(0.487005\pi\)
\(930\) 0 0
\(931\) 1.86822 0.0612286
\(932\) 6.17878 0.202393
\(933\) 0 0
\(934\) −52.1433 −1.70618
\(935\) −51.2975 −1.67761
\(936\) 0 0
\(937\) 42.9656 1.40362 0.701812 0.712362i \(-0.252375\pi\)
0.701812 + 0.712362i \(0.252375\pi\)
\(938\) −17.4408 −0.569463
\(939\) 0 0
\(940\) 11.1563 0.363878
\(941\) −49.3601 −1.60909 −0.804547 0.593889i \(-0.797592\pi\)
−0.804547 + 0.593889i \(0.797592\pi\)
\(942\) 0 0
\(943\) 58.1711 1.89431
\(944\) −38.4442 −1.25125
\(945\) 0 0
\(946\) −9.20843 −0.299392
\(947\) 7.03435 0.228586 0.114293 0.993447i \(-0.463540\pi\)
0.114293 + 0.993447i \(0.463540\pi\)
\(948\) 0 0
\(949\) 11.8130 0.383468
\(950\) 14.1993 0.460686
\(951\) 0 0
\(952\) −16.1507 −0.523448
\(953\) −12.7667 −0.413553 −0.206777 0.978388i \(-0.566297\pi\)
−0.206777 + 0.978388i \(0.566297\pi\)
\(954\) 0 0
\(955\) 75.2389 2.43467
\(956\) 10.2303 0.330872
\(957\) 0 0
\(958\) 11.9941 0.387512
\(959\) 4.66285 0.150571
\(960\) 0 0
\(961\) −30.0618 −0.969736
\(962\) 60.1476 1.93924
\(963\) 0 0
\(964\) −5.47858 −0.176453
\(965\) 69.1249 2.22521
\(966\) 0 0
\(967\) 53.8825 1.73274 0.866372 0.499400i \(-0.166446\pi\)
0.866372 + 0.499400i \(0.166446\pi\)
\(968\) 7.77671 0.249953
\(969\) 0 0
\(970\) −60.9120 −1.95577
\(971\) 55.2100 1.77177 0.885887 0.463901i \(-0.153551\pi\)
0.885887 + 0.463901i \(0.153551\pi\)
\(972\) 0 0
\(973\) −19.7441 −0.632965
\(974\) −22.0210 −0.705599
\(975\) 0 0
\(976\) −1.92127 −0.0614982
\(977\) 38.3352 1.22645 0.613226 0.789908i \(-0.289872\pi\)
0.613226 + 0.789908i \(0.289872\pi\)
\(978\) 0 0
\(979\) 35.3662 1.13031
\(980\) −1.51848 −0.0485060
\(981\) 0 0
\(982\) 6.18415 0.197344
\(983\) 19.8976 0.634635 0.317317 0.948319i \(-0.397218\pi\)
0.317317 + 0.948319i \(0.397218\pi\)
\(984\) 0 0
\(985\) 12.5948 0.401303
\(986\) −24.3326 −0.774909
\(987\) 0 0
\(988\) 3.90774 0.124322
\(989\) −12.8245 −0.407797
\(990\) 0 0
\(991\) 5.55649 0.176508 0.0882538 0.996098i \(-0.471871\pi\)
0.0882538 + 0.996098i \(0.471871\pi\)
\(992\) 2.44275 0.0775573
\(993\) 0 0
\(994\) −12.2856 −0.389675
\(995\) −32.4588 −1.02901
\(996\) 0 0
\(997\) 4.32640 0.137019 0.0685093 0.997650i \(-0.478176\pi\)
0.0685093 + 0.997650i \(0.478176\pi\)
\(998\) 54.7644 1.73354
\(999\) 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 8001.2.a.l.1.5 7
3.2 odd 2 2667.2.a.j.1.3 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2667.2.a.j.1.3 7 3.2 odd 2
8001.2.a.l.1.5 7 1.1 even 1 trivial