Properties

Label 8550.2.a.cj.1.1
Level $8550$
Weight $2$
Character 8550.1
Self dual yes
Analytic conductor $68.272$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8550.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(68.2720937282\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.257.1
Defining polynomial: \(x^{3} - x^{2} - 4 x + 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 950)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(0.713538\) of defining polynomial
Character \(\chi\) \(=\) 8550.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -4.69527 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -4.69527 q^{7} -1.00000 q^{8} -6.40880 q^{11} +1.06379 q^{13} +4.69527 q^{14} +1.00000 q^{16} -1.91794 q^{17} -1.00000 q^{19} +6.40880 q^{22} -1.79560 q^{23} -1.06379 q^{26} -4.69527 q^{28} -2.93621 q^{29} -5.55465 q^{31} -1.00000 q^{32} +1.91794 q^{34} -11.4088 q^{37} +1.00000 q^{38} +1.14585 q^{41} -3.55465 q^{43} -6.40880 q^{44} +1.79560 q^{46} -10.8359 q^{47} +15.0455 q^{49} +1.06379 q^{52} +8.69527 q^{53} +4.69527 q^{56} +2.93621 q^{58} +5.63148 q^{59} -3.39053 q^{61} +5.55465 q^{62} +1.00000 q^{64} -8.82284 q^{67} -1.91794 q^{68} +1.42708 q^{71} -12.6132 q^{73} +11.4088 q^{74} -1.00000 q^{76} +30.0910 q^{77} -1.96345 q^{79} -1.14585 q^{82} -16.2447 q^{83} +3.55465 q^{86} +6.40880 q^{88} +10.0000 q^{89} -4.99477 q^{91} -1.79560 q^{92} +10.8359 q^{94} -14.9452 q^{97} -15.0455 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{2} + 3q^{4} + 2q^{7} - 3q^{8} + O(q^{10}) \) \( 3q - 3q^{2} + 3q^{4} + 2q^{7} - 3q^{8} - 2q^{11} - 2q^{13} - 2q^{14} + 3q^{16} + 4q^{17} - 3q^{19} + 2q^{22} - 14q^{23} + 2q^{26} + 2q^{28} - 14q^{29} - 4q^{31} - 3q^{32} - 4q^{34} - 17q^{37} + 3q^{38} + 8q^{41} + 2q^{43} - 2q^{44} + 14q^{46} - 13q^{47} + 25q^{49} - 2q^{52} + 10q^{53} - 2q^{56} + 14q^{58} + 6q^{59} + 22q^{61} + 4q^{62} + 3q^{64} + 4q^{68} + 2q^{71} - 12q^{73} + 17q^{74} - 3q^{76} + 50q^{77} + 24q^{79} - 8q^{82} - 12q^{83} - 2q^{86} + 2q^{88} + 30q^{89} - 7q^{91} - 14q^{92} + 13q^{94} - 25q^{98} + O(q^{100}) \)

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
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) −4.69527 −1.77464 −0.887322 0.461151i \(-0.847437\pi\)
−0.887322 + 0.461151i \(0.847437\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) −6.40880 −1.93233 −0.966163 0.257931i \(-0.916959\pi\)
−0.966163 + 0.257931i \(0.916959\pi\)
\(12\) 0 0
\(13\) 1.06379 0.295042 0.147521 0.989059i \(-0.452871\pi\)
0.147521 + 0.989059i \(0.452871\pi\)
\(14\) 4.69527 1.25486
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −1.91794 −0.465169 −0.232584 0.972576i \(-0.574718\pi\)
−0.232584 + 0.972576i \(0.574718\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) 6.40880 1.36636
\(23\) −1.79560 −0.374408 −0.187204 0.982321i \(-0.559943\pi\)
−0.187204 + 0.982321i \(0.559943\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −1.06379 −0.208626
\(27\) 0 0
\(28\) −4.69527 −0.887322
\(29\) −2.93621 −0.545241 −0.272620 0.962122i \(-0.587890\pi\)
−0.272620 + 0.962122i \(0.587890\pi\)
\(30\) 0 0
\(31\) −5.55465 −0.997645 −0.498822 0.866704i \(-0.666234\pi\)
−0.498822 + 0.866704i \(0.666234\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 1.91794 0.328924
\(35\) 0 0
\(36\) 0 0
\(37\) −11.4088 −1.87560 −0.937798 0.347182i \(-0.887139\pi\)
−0.937798 + 0.347182i \(0.887139\pi\)
\(38\) 1.00000 0.162221
\(39\) 0 0
\(40\) 0 0
\(41\) 1.14585 0.178951 0.0894757 0.995989i \(-0.471481\pi\)
0.0894757 + 0.995989i \(0.471481\pi\)
\(42\) 0 0
\(43\) −3.55465 −0.542079 −0.271040 0.962568i \(-0.587367\pi\)
−0.271040 + 0.962568i \(0.587367\pi\)
\(44\) −6.40880 −0.966163
\(45\) 0 0
\(46\) 1.79560 0.264747
\(47\) −10.8359 −1.58058 −0.790288 0.612736i \(-0.790069\pi\)
−0.790288 + 0.612736i \(0.790069\pi\)
\(48\) 0 0
\(49\) 15.0455 2.14936
\(50\) 0 0
\(51\) 0 0
\(52\) 1.06379 0.147521
\(53\) 8.69527 1.19439 0.597193 0.802097i \(-0.296283\pi\)
0.597193 + 0.802097i \(0.296283\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 4.69527 0.627431
\(57\) 0 0
\(58\) 2.93621 0.385544
\(59\) 5.63148 0.733156 0.366578 0.930387i \(-0.380529\pi\)
0.366578 + 0.930387i \(0.380529\pi\)
\(60\) 0 0
\(61\) −3.39053 −0.434113 −0.217056 0.976159i \(-0.569646\pi\)
−0.217056 + 0.976159i \(0.569646\pi\)
\(62\) 5.55465 0.705441
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −8.82284 −1.07788 −0.538941 0.842344i \(-0.681175\pi\)
−0.538941 + 0.842344i \(0.681175\pi\)
\(68\) −1.91794 −0.232584
\(69\) 0 0
\(70\) 0 0
\(71\) 1.42708 0.169363 0.0846814 0.996408i \(-0.473013\pi\)
0.0846814 + 0.996408i \(0.473013\pi\)
\(72\) 0 0
\(73\) −12.6132 −1.47626 −0.738132 0.674656i \(-0.764292\pi\)
−0.738132 + 0.674656i \(0.764292\pi\)
\(74\) 11.4088 1.32625
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) 30.0910 3.42919
\(78\) 0 0
\(79\) −1.96345 −0.220906 −0.110453 0.993881i \(-0.535230\pi\)
−0.110453 + 0.993881i \(0.535230\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −1.14585 −0.126538
\(83\) −16.2447 −1.78309 −0.891543 0.452937i \(-0.850376\pi\)
−0.891543 + 0.452937i \(0.850376\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 3.55465 0.383308
\(87\) 0 0
\(88\) 6.40880 0.683181
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) −4.99477 −0.523594
\(92\) −1.79560 −0.187204
\(93\) 0 0
\(94\) 10.8359 1.11764
\(95\) 0 0
\(96\) 0 0
\(97\) −14.9452 −1.51745 −0.758727 0.651409i \(-0.774178\pi\)
−0.758727 + 0.651409i \(0.774178\pi\)
\(98\) −15.0455 −1.51983
\(99\) 0 0
\(100\) 0 0
\(101\) −10.5364 −1.04841 −0.524204 0.851592i \(-0.675637\pi\)
−0.524204 + 0.851592i \(0.675637\pi\)
\(102\) 0 0
\(103\) −16.9817 −1.67326 −0.836630 0.547769i \(-0.815477\pi\)
−0.836630 + 0.547769i \(0.815477\pi\)
\(104\) −1.06379 −0.104313
\(105\) 0 0
\(106\) −8.69527 −0.844559
\(107\) 1.79036 0.173081 0.0865405 0.996248i \(-0.472419\pi\)
0.0865405 + 0.996248i \(0.472419\pi\)
\(108\) 0 0
\(109\) 2.41404 0.231223 0.115611 0.993295i \(-0.463117\pi\)
0.115611 + 0.993295i \(0.463117\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −4.69527 −0.443661
\(113\) −7.14585 −0.672225 −0.336112 0.941822i \(-0.609112\pi\)
−0.336112 + 0.941822i \(0.609112\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −2.93621 −0.272620
\(117\) 0 0
\(118\) −5.63148 −0.518420
\(119\) 9.00523 0.825508
\(120\) 0 0
\(121\) 30.0728 2.73389
\(122\) 3.39053 0.306964
\(123\) 0 0
\(124\) −5.55465 −0.498822
\(125\) 0 0
\(126\) 0 0
\(127\) 9.26295 0.821954 0.410977 0.911646i \(-0.365188\pi\)
0.410977 + 0.911646i \(0.365188\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) −11.5181 −1.00634 −0.503171 0.864187i \(-0.667833\pi\)
−0.503171 + 0.864187i \(0.667833\pi\)
\(132\) 0 0
\(133\) 4.69527 0.407131
\(134\) 8.82284 0.762177
\(135\) 0 0
\(136\) 1.91794 0.164462
\(137\) −15.1041 −1.29043 −0.645214 0.764002i \(-0.723232\pi\)
−0.645214 + 0.764002i \(0.723232\pi\)
\(138\) 0 0
\(139\) −0.700500 −0.0594156 −0.0297078 0.999559i \(-0.509458\pi\)
−0.0297078 + 0.999559i \(0.509458\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −1.42708 −0.119758
\(143\) −6.81761 −0.570117
\(144\) 0 0
\(145\) 0 0
\(146\) 12.6132 1.04388
\(147\) 0 0
\(148\) −11.4088 −0.937798
\(149\) −12.9452 −1.06051 −0.530255 0.847838i \(-0.677904\pi\)
−0.530255 + 0.847838i \(0.677904\pi\)
\(150\) 0 0
\(151\) 5.70830 0.464535 0.232268 0.972652i \(-0.425385\pi\)
0.232268 + 0.972652i \(0.425385\pi\)
\(152\) 1.00000 0.0811107
\(153\) 0 0
\(154\) −30.0910 −2.42480
\(155\) 0 0
\(156\) 0 0
\(157\) 9.26295 0.739264 0.369632 0.929178i \(-0.379484\pi\)
0.369632 + 0.929178i \(0.379484\pi\)
\(158\) 1.96345 0.156204
\(159\) 0 0
\(160\) 0 0
\(161\) 8.43081 0.664441
\(162\) 0 0
\(163\) 4.28123 0.335332 0.167666 0.985844i \(-0.446377\pi\)
0.167666 + 0.985844i \(0.446377\pi\)
\(164\) 1.14585 0.0894757
\(165\) 0 0
\(166\) 16.2447 1.26083
\(167\) 15.2264 1.17825 0.589127 0.808040i \(-0.299472\pi\)
0.589127 + 0.808040i \(0.299472\pi\)
\(168\) 0 0
\(169\) −11.8684 −0.912950
\(170\) 0 0
\(171\) 0 0
\(172\) −3.55465 −0.271040
\(173\) 12.2630 0.932335 0.466168 0.884696i \(-0.345634\pi\)
0.466168 + 0.884696i \(0.345634\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −6.40880 −0.483082
\(177\) 0 0
\(178\) −10.0000 −0.749532
\(179\) 1.57292 0.117566 0.0587829 0.998271i \(-0.481278\pi\)
0.0587829 + 0.998271i \(0.481278\pi\)
\(180\) 0 0
\(181\) 11.4088 0.848010 0.424005 0.905660i \(-0.360624\pi\)
0.424005 + 0.905660i \(0.360624\pi\)
\(182\) 4.99477 0.370237
\(183\) 0 0
\(184\) 1.79560 0.132373
\(185\) 0 0
\(186\) 0 0
\(187\) 12.2917 0.898858
\(188\) −10.8359 −0.790288
\(189\) 0 0
\(190\) 0 0
\(191\) 6.35805 0.460053 0.230026 0.973184i \(-0.426119\pi\)
0.230026 + 0.973184i \(0.426119\pi\)
\(192\) 0 0
\(193\) 14.4998 1.04372 0.521860 0.853031i \(-0.325238\pi\)
0.521860 + 0.853031i \(0.325238\pi\)
\(194\) 14.9452 1.07300
\(195\) 0 0
\(196\) 15.0455 1.07468
\(197\) 5.14585 0.366627 0.183313 0.983055i \(-0.441318\pi\)
0.183313 + 0.983055i \(0.441318\pi\)
\(198\) 0 0
\(199\) 3.87766 0.274880 0.137440 0.990510i \(-0.456113\pi\)
0.137440 + 0.990510i \(0.456113\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 10.5364 0.741337
\(203\) 13.7863 0.967608
\(204\) 0 0
\(205\) 0 0
\(206\) 16.9817 1.18317
\(207\) 0 0
\(208\) 1.06379 0.0737604
\(209\) 6.40880 0.443306
\(210\) 0 0
\(211\) 17.1496 1.18063 0.590313 0.807174i \(-0.299004\pi\)
0.590313 + 0.807174i \(0.299004\pi\)
\(212\) 8.69527 0.597193
\(213\) 0 0
\(214\) −1.79036 −0.122387
\(215\) 0 0
\(216\) 0 0
\(217\) 26.0806 1.77046
\(218\) −2.41404 −0.163499
\(219\) 0 0
\(220\) 0 0
\(221\) −2.04028 −0.137244
\(222\) 0 0
\(223\) 7.84635 0.525430 0.262715 0.964873i \(-0.415382\pi\)
0.262715 + 0.964873i \(0.415382\pi\)
\(224\) 4.69527 0.313716
\(225\) 0 0
\(226\) 7.14585 0.475335
\(227\) −6.28646 −0.417247 −0.208624 0.977996i \(-0.566898\pi\)
−0.208624 + 0.977996i \(0.566898\pi\)
\(228\) 0 0
\(229\) −3.14585 −0.207884 −0.103942 0.994583i \(-0.533146\pi\)
−0.103942 + 0.994583i \(0.533146\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 2.93621 0.192772
\(233\) −0.182394 −0.0119490 −0.00597451 0.999982i \(-0.501902\pi\)
−0.00597451 + 0.999982i \(0.501902\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 5.63148 0.366578
\(237\) 0 0
\(238\) −9.00523 −0.583723
\(239\) 11.5039 0.744126 0.372063 0.928208i \(-0.378651\pi\)
0.372063 + 0.928208i \(0.378651\pi\)
\(240\) 0 0
\(241\) −0.445349 −0.0286874 −0.0143437 0.999897i \(-0.504566\pi\)
−0.0143437 + 0.999897i \(0.504566\pi\)
\(242\) −30.0728 −1.93315
\(243\) 0 0
\(244\) −3.39053 −0.217056
\(245\) 0 0
\(246\) 0 0
\(247\) −1.06379 −0.0676872
\(248\) 5.55465 0.352721
\(249\) 0 0
\(250\) 0 0
\(251\) −13.2630 −0.837150 −0.418575 0.908182i \(-0.637470\pi\)
−0.418575 + 0.908182i \(0.637470\pi\)
\(252\) 0 0
\(253\) 11.5076 0.723479
\(254\) −9.26295 −0.581209
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 23.4816 1.46474 0.732370 0.680907i \(-0.238414\pi\)
0.732370 + 0.680907i \(0.238414\pi\)
\(258\) 0 0
\(259\) 53.5674 3.32851
\(260\) 0 0
\(261\) 0 0
\(262\) 11.5181 0.711591
\(263\) −27.9269 −1.72205 −0.861023 0.508565i \(-0.830176\pi\)
−0.861023 + 0.508565i \(0.830176\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −4.69527 −0.287885
\(267\) 0 0
\(268\) −8.82284 −0.538941
\(269\) −16.4816 −1.00490 −0.502449 0.864607i \(-0.667568\pi\)
−0.502449 + 0.864607i \(0.667568\pi\)
\(270\) 0 0
\(271\) −1.46736 −0.0891356 −0.0445678 0.999006i \(-0.514191\pi\)
−0.0445678 + 0.999006i \(0.514191\pi\)
\(272\) −1.91794 −0.116292
\(273\) 0 0
\(274\) 15.1041 0.912470
\(275\) 0 0
\(276\) 0 0
\(277\) 15.8359 0.951486 0.475743 0.879584i \(-0.342179\pi\)
0.475743 + 0.879584i \(0.342179\pi\)
\(278\) 0.700500 0.0420132
\(279\) 0 0
\(280\) 0 0
\(281\) 6.25515 0.373151 0.186576 0.982441i \(-0.440261\pi\)
0.186576 + 0.982441i \(0.440261\pi\)
\(282\) 0 0
\(283\) −16.7005 −0.992742 −0.496371 0.868111i \(-0.665334\pi\)
−0.496371 + 0.868111i \(0.665334\pi\)
\(284\) 1.42708 0.0846814
\(285\) 0 0
\(286\) 6.81761 0.403134
\(287\) −5.38006 −0.317575
\(288\) 0 0
\(289\) −13.3215 −0.783618
\(290\) 0 0
\(291\) 0 0
\(292\) −12.6132 −0.738132
\(293\) −0.644516 −0.0376530 −0.0188265 0.999823i \(-0.505993\pi\)
−0.0188265 + 0.999823i \(0.505993\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 11.4088 0.663123
\(297\) 0 0
\(298\) 12.9452 0.749894
\(299\) −1.91014 −0.110466
\(300\) 0 0
\(301\) 16.6900 0.961997
\(302\) −5.70830 −0.328476
\(303\) 0 0
\(304\) −1.00000 −0.0573539
\(305\) 0 0
\(306\) 0 0
\(307\) 26.6457 1.52075 0.760375 0.649485i \(-0.225015\pi\)
0.760375 + 0.649485i \(0.225015\pi\)
\(308\) 30.0910 1.71460
\(309\) 0 0
\(310\) 0 0
\(311\) −11.5494 −0.654907 −0.327454 0.944867i \(-0.606191\pi\)
−0.327454 + 0.944867i \(0.606191\pi\)
\(312\) 0 0
\(313\) −23.0638 −1.30364 −0.651821 0.758373i \(-0.725995\pi\)
−0.651821 + 0.758373i \(0.725995\pi\)
\(314\) −9.26295 −0.522739
\(315\) 0 0
\(316\) −1.96345 −0.110453
\(317\) 13.9232 0.782003 0.391002 0.920390i \(-0.372129\pi\)
0.391002 + 0.920390i \(0.372129\pi\)
\(318\) 0 0
\(319\) 18.8176 1.05358
\(320\) 0 0
\(321\) 0 0
\(322\) −8.43081 −0.469831
\(323\) 1.91794 0.106717
\(324\) 0 0
\(325\) 0 0
\(326\) −4.28123 −0.237115
\(327\) 0 0
\(328\) −1.14585 −0.0632689
\(329\) 50.8773 2.80496
\(330\) 0 0
\(331\) −0.735546 −0.0404292 −0.0202146 0.999796i \(-0.506435\pi\)
−0.0202146 + 0.999796i \(0.506435\pi\)
\(332\) −16.2447 −0.891543
\(333\) 0 0
\(334\) −15.2264 −0.833152
\(335\) 0 0
\(336\) 0 0
\(337\) −2.28123 −0.124266 −0.0621332 0.998068i \(-0.519790\pi\)
−0.0621332 + 0.998068i \(0.519790\pi\)
\(338\) 11.8684 0.645553
\(339\) 0 0
\(340\) 0 0
\(341\) 35.5987 1.92778
\(342\) 0 0
\(343\) −37.7758 −2.03970
\(344\) 3.55465 0.191654
\(345\) 0 0
\(346\) −12.2630 −0.659261
\(347\) 2.56246 0.137560 0.0687799 0.997632i \(-0.478089\pi\)
0.0687799 + 0.997632i \(0.478089\pi\)
\(348\) 0 0
\(349\) −5.67176 −0.303602 −0.151801 0.988411i \(-0.548507\pi\)
−0.151801 + 0.988411i \(0.548507\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 6.40880 0.341590
\(353\) 23.6665 1.25964 0.629821 0.776740i \(-0.283128\pi\)
0.629821 + 0.776740i \(0.283128\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 10.0000 0.529999
\(357\) 0 0
\(358\) −1.57292 −0.0831316
\(359\) −31.9724 −1.68744 −0.843720 0.536784i \(-0.819639\pi\)
−0.843720 + 0.536784i \(0.819639\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −11.4088 −0.599633
\(363\) 0 0
\(364\) −4.99477 −0.261797
\(365\) 0 0
\(366\) 0 0
\(367\) 1.14585 0.0598128 0.0299064 0.999553i \(-0.490479\pi\)
0.0299064 + 0.999553i \(0.490479\pi\)
\(368\) −1.79560 −0.0936020
\(369\) 0 0
\(370\) 0 0
\(371\) −40.8266 −2.11961
\(372\) 0 0
\(373\) 32.8579 1.70132 0.850658 0.525719i \(-0.176204\pi\)
0.850658 + 0.525719i \(0.176204\pi\)
\(374\) −12.2917 −0.635588
\(375\) 0 0
\(376\) 10.8359 0.558818
\(377\) −3.12351 −0.160869
\(378\) 0 0
\(379\) −18.2865 −0.939312 −0.469656 0.882849i \(-0.655622\pi\)
−0.469656 + 0.882849i \(0.655622\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −6.35805 −0.325306
\(383\) −20.8542 −1.06560 −0.532799 0.846242i \(-0.678860\pi\)
−0.532799 + 0.846242i \(0.678860\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −14.4998 −0.738022
\(387\) 0 0
\(388\) −14.9452 −0.758727
\(389\) 24.9086 1.26292 0.631459 0.775409i \(-0.282456\pi\)
0.631459 + 0.775409i \(0.282456\pi\)
\(390\) 0 0
\(391\) 3.44385 0.174163
\(392\) −15.0455 −0.759913
\(393\) 0 0
\(394\) −5.14585 −0.259244
\(395\) 0 0
\(396\) 0 0
\(397\) −24.7445 −1.24189 −0.620946 0.783853i \(-0.713251\pi\)
−0.620946 + 0.783853i \(0.713251\pi\)
\(398\) −3.87766 −0.194369
\(399\) 0 0
\(400\) 0 0
\(401\) −15.6457 −0.781308 −0.390654 0.920538i \(-0.627751\pi\)
−0.390654 + 0.920538i \(0.627751\pi\)
\(402\) 0 0
\(403\) −5.90897 −0.294347
\(404\) −10.5364 −0.524204
\(405\) 0 0
\(406\) −13.7863 −0.684202
\(407\) 73.1168 3.62426
\(408\) 0 0
\(409\) 30.5804 1.51210 0.756052 0.654512i \(-0.227126\pi\)
0.756052 + 0.654512i \(0.227126\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −16.9817 −0.836630
\(413\) −26.4413 −1.30109
\(414\) 0 0
\(415\) 0 0
\(416\) −1.06379 −0.0521565
\(417\) 0 0
\(418\) −6.40880 −0.313465
\(419\) −1.96345 −0.0959210 −0.0479605 0.998849i \(-0.515272\pi\)
−0.0479605 + 0.998849i \(0.515272\pi\)
\(420\) 0 0
\(421\) −25.5949 −1.24742 −0.623710 0.781656i \(-0.714376\pi\)
−0.623710 + 0.781656i \(0.714376\pi\)
\(422\) −17.1496 −0.834829
\(423\) 0 0
\(424\) −8.69527 −0.422279
\(425\) 0 0
\(426\) 0 0
\(427\) 15.9194 0.770396
\(428\) 1.79036 0.0865405
\(429\) 0 0
\(430\) 0 0
\(431\) 15.3540 0.739575 0.369788 0.929116i \(-0.379430\pi\)
0.369788 + 0.929116i \(0.379430\pi\)
\(432\) 0 0
\(433\) 16.6640 0.800819 0.400409 0.916336i \(-0.368868\pi\)
0.400409 + 0.916336i \(0.368868\pi\)
\(434\) −26.0806 −1.25191
\(435\) 0 0
\(436\) 2.41404 0.115611
\(437\) 1.79560 0.0858951
\(438\) 0 0
\(439\) −19.5987 −0.935393 −0.467697 0.883889i \(-0.654916\pi\)
−0.467697 + 0.883889i \(0.654916\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 2.04028 0.0970462
\(443\) −13.0183 −0.618517 −0.309258 0.950978i \(-0.600081\pi\)
−0.309258 + 0.950978i \(0.600081\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −7.84635 −0.371535
\(447\) 0 0
\(448\) −4.69527 −0.221830
\(449\) −6.77359 −0.319666 −0.159833 0.987144i \(-0.551095\pi\)
−0.159833 + 0.987144i \(0.551095\pi\)
\(450\) 0 0
\(451\) −7.34352 −0.345793
\(452\) −7.14585 −0.336112
\(453\) 0 0
\(454\) 6.28646 0.295038
\(455\) 0 0
\(456\) 0 0
\(457\) −27.3189 −1.27793 −0.638963 0.769237i \(-0.720636\pi\)
−0.638963 + 0.769237i \(0.720636\pi\)
\(458\) 3.14585 0.146996
\(459\) 0 0
\(460\) 0 0
\(461\) 27.7993 1.29474 0.647372 0.762174i \(-0.275868\pi\)
0.647372 + 0.762174i \(0.275868\pi\)
\(462\) 0 0
\(463\) 38.2369 1.77702 0.888509 0.458859i \(-0.151742\pi\)
0.888509 + 0.458859i \(0.151742\pi\)
\(464\) −2.93621 −0.136310
\(465\) 0 0
\(466\) 0.182394 0.00844923
\(467\) −23.4711 −1.08611 −0.543056 0.839696i \(-0.682733\pi\)
−0.543056 + 0.839696i \(0.682733\pi\)
\(468\) 0 0
\(469\) 41.4256 1.91286
\(470\) 0 0
\(471\) 0 0
\(472\) −5.63148 −0.259210
\(473\) 22.7811 1.04747
\(474\) 0 0
\(475\) 0 0
\(476\) 9.00523 0.412754
\(477\) 0 0
\(478\) −11.5039 −0.526176
\(479\) 19.6900 0.899660 0.449830 0.893114i \(-0.351484\pi\)
0.449830 + 0.893114i \(0.351484\pi\)
\(480\) 0 0
\(481\) −12.1365 −0.553379
\(482\) 0.445349 0.0202851
\(483\) 0 0
\(484\) 30.0728 1.36694
\(485\) 0 0
\(486\) 0 0
\(487\) 16.3357 0.740242 0.370121 0.928984i \(-0.379316\pi\)
0.370121 + 0.928984i \(0.379316\pi\)
\(488\) 3.39053 0.153482
\(489\) 0 0
\(490\) 0 0
\(491\) −27.1093 −1.22343 −0.611713 0.791080i \(-0.709519\pi\)
−0.611713 + 0.791080i \(0.709519\pi\)
\(492\) 0 0
\(493\) 5.63148 0.253629
\(494\) 1.06379 0.0478621
\(495\) 0 0
\(496\) −5.55465 −0.249411
\(497\) −6.70050 −0.300558
\(498\) 0 0
\(499\) −4.69003 −0.209955 −0.104977 0.994475i \(-0.533477\pi\)
−0.104977 + 0.994475i \(0.533477\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 13.2630 0.591955
\(503\) −5.19136 −0.231471 −0.115736 0.993280i \(-0.536923\pi\)
−0.115736 + 0.993280i \(0.536923\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −11.5076 −0.511577
\(507\) 0 0
\(508\) 9.26295 0.410977
\(509\) 4.51811 0.200262 0.100131 0.994974i \(-0.468074\pi\)
0.100131 + 0.994974i \(0.468074\pi\)
\(510\) 0 0
\(511\) 59.2223 2.61984
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −23.4816 −1.03573
\(515\) 0 0
\(516\) 0 0
\(517\) 69.4450 3.05419
\(518\) −53.5674 −2.35361
\(519\) 0 0
\(520\) 0 0
\(521\) −14.4453 −0.632862 −0.316431 0.948615i \(-0.602485\pi\)
−0.316431 + 0.948615i \(0.602485\pi\)
\(522\) 0 0
\(523\) −18.2134 −0.796415 −0.398208 0.917295i \(-0.630368\pi\)
−0.398208 + 0.917295i \(0.630368\pi\)
\(524\) −11.5181 −0.503171
\(525\) 0 0
\(526\) 27.9269 1.21767
\(527\) 10.6535 0.464073
\(528\) 0 0
\(529\) −19.7758 −0.859819
\(530\) 0 0
\(531\) 0 0
\(532\) 4.69527 0.203566
\(533\) 1.21894 0.0527981
\(534\) 0 0
\(535\) 0 0
\(536\) 8.82284 0.381089
\(537\) 0 0
\(538\) 16.4816 0.710571
\(539\) −96.4237 −4.15326
\(540\) 0 0
\(541\) 37.3905 1.60754 0.803772 0.594937i \(-0.202823\pi\)
0.803772 + 0.594937i \(0.202823\pi\)
\(542\) 1.46736 0.0630284
\(543\) 0 0
\(544\) 1.91794 0.0822310
\(545\) 0 0
\(546\) 0 0
\(547\) 29.5621 1.26399 0.631993 0.774974i \(-0.282237\pi\)
0.631993 + 0.774974i \(0.282237\pi\)
\(548\) −15.1041 −0.645214
\(549\) 0 0
\(550\) 0 0
\(551\) 2.93621 0.125087
\(552\) 0 0
\(553\) 9.21894 0.392029
\(554\) −15.8359 −0.672802
\(555\) 0 0
\(556\) −0.700500 −0.0297078
\(557\) −14.3723 −0.608972 −0.304486 0.952517i \(-0.598485\pi\)
−0.304486 + 0.952517i \(0.598485\pi\)
\(558\) 0 0
\(559\) −3.78139 −0.159936
\(560\) 0 0
\(561\) 0 0
\(562\) −6.25515 −0.263858
\(563\) 6.39053 0.269329 0.134664 0.990891i \(-0.457004\pi\)
0.134664 + 0.990891i \(0.457004\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 16.7005 0.701974
\(567\) 0 0
\(568\) −1.42708 −0.0598788
\(569\) 44.9086 1.88267 0.941334 0.337476i \(-0.109573\pi\)
0.941334 + 0.337476i \(0.109573\pi\)
\(570\) 0 0
\(571\) 12.4193 0.519730 0.259865 0.965645i \(-0.416322\pi\)
0.259865 + 0.965645i \(0.416322\pi\)
\(572\) −6.81761 −0.285058
\(573\) 0 0
\(574\) 5.38006 0.224559
\(575\) 0 0
\(576\) 0 0
\(577\) −4.20440 −0.175032 −0.0875158 0.996163i \(-0.527893\pi\)
−0.0875158 + 0.996163i \(0.527893\pi\)
\(578\) 13.3215 0.554102
\(579\) 0 0
\(580\) 0 0
\(581\) 76.2731 3.16434
\(582\) 0 0
\(583\) −55.7262 −2.30795
\(584\) 12.6132 0.521938
\(585\) 0 0
\(586\) 0.644516 0.0266247
\(587\) −24.7915 −1.02326 −0.511628 0.859207i \(-0.670957\pi\)
−0.511628 + 0.859207i \(0.670957\pi\)
\(588\) 0 0
\(589\) 5.55465 0.228875
\(590\) 0 0
\(591\) 0 0
\(592\) −11.4088 −0.468899
\(593\) −4.67176 −0.191846 −0.0959231 0.995389i \(-0.530580\pi\)
−0.0959231 + 0.995389i \(0.530580\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −12.9452 −0.530255
\(597\) 0 0
\(598\) 1.91014 0.0781113
\(599\) −39.2369 −1.60318 −0.801588 0.597877i \(-0.796011\pi\)
−0.801588 + 0.597877i \(0.796011\pi\)
\(600\) 0 0
\(601\) −42.4267 −1.73062 −0.865311 0.501235i \(-0.832879\pi\)
−0.865311 + 0.501235i \(0.832879\pi\)
\(602\) −16.6900 −0.680235
\(603\) 0 0
\(604\) 5.70830 0.232268
\(605\) 0 0
\(606\) 0 0
\(607\) −2.98173 −0.121025 −0.0605123 0.998167i \(-0.519273\pi\)
−0.0605123 + 0.998167i \(0.519273\pi\)
\(608\) 1.00000 0.0405554
\(609\) 0 0
\(610\) 0 0
\(611\) −11.5271 −0.466336
\(612\) 0 0
\(613\) −28.9452 −1.16908 −0.584542 0.811363i \(-0.698726\pi\)
−0.584542 + 0.811363i \(0.698726\pi\)
\(614\) −26.6457 −1.07533
\(615\) 0 0
\(616\) −30.0910 −1.21240
\(617\) 15.0365 0.605349 0.302674 0.953094i \(-0.402121\pi\)
0.302674 + 0.953094i \(0.402121\pi\)
\(618\) 0 0
\(619\) 8.25515 0.331803 0.165901 0.986142i \(-0.446947\pi\)
0.165901 + 0.986142i \(0.446947\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 11.5494 0.463089
\(623\) −46.9527 −1.88112
\(624\) 0 0
\(625\) 0 0
\(626\) 23.0638 0.921814
\(627\) 0 0
\(628\) 9.26295 0.369632
\(629\) 21.8814 0.872468
\(630\) 0 0
\(631\) −9.09103 −0.361908 −0.180954 0.983492i \(-0.557919\pi\)
−0.180954 + 0.983492i \(0.557919\pi\)
\(632\) 1.96345 0.0781020
\(633\) 0 0
\(634\) −13.9232 −0.552960
\(635\) 0 0
\(636\) 0 0
\(637\) 16.0052 0.634150
\(638\) −18.8176 −0.744996
\(639\) 0 0
\(640\) 0 0
\(641\) −30.1171 −1.18955 −0.594777 0.803891i \(-0.702760\pi\)
−0.594777 + 0.803891i \(0.702760\pi\)
\(642\) 0 0
\(643\) −35.3174 −1.39278 −0.696392 0.717662i \(-0.745212\pi\)
−0.696392 + 0.717662i \(0.745212\pi\)
\(644\) 8.43081 0.332220
\(645\) 0 0
\(646\) −1.91794 −0.0754603
\(647\) −4.12234 −0.162066 −0.0810330 0.996711i \(-0.525822\pi\)
−0.0810330 + 0.996711i \(0.525822\pi\)
\(648\) 0 0
\(649\) −36.0910 −1.41670
\(650\) 0 0
\(651\) 0 0
\(652\) 4.28123 0.167666
\(653\) −8.62741 −0.337617 −0.168808 0.985649i \(-0.553992\pi\)
−0.168808 + 0.985649i \(0.553992\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 1.14585 0.0447379
\(657\) 0 0
\(658\) −50.8773 −1.98340
\(659\) 37.3853 1.45632 0.728162 0.685405i \(-0.240375\pi\)
0.728162 + 0.685405i \(0.240375\pi\)
\(660\) 0 0
\(661\) −12.8997 −0.501739 −0.250869 0.968021i \(-0.580716\pi\)
−0.250869 + 0.968021i \(0.580716\pi\)
\(662\) 0.735546 0.0285878
\(663\) 0 0
\(664\) 16.2447 0.630416
\(665\) 0 0
\(666\) 0 0
\(667\) 5.27226 0.204143
\(668\) 15.2264 0.589127
\(669\) 0 0
\(670\) 0 0
\(671\) 21.7292 0.838848
\(672\) 0 0
\(673\) 10.4349 0.402235 0.201118 0.979567i \(-0.435543\pi\)
0.201118 + 0.979567i \(0.435543\pi\)
\(674\) 2.28123 0.0878696
\(675\) 0 0
\(676\) −11.8684 −0.456475
\(677\) −12.4401 −0.478112 −0.239056 0.971006i \(-0.576838\pi\)
−0.239056 + 0.971006i \(0.576838\pi\)
\(678\) 0 0
\(679\) 70.1716 2.69294
\(680\) 0 0
\(681\) 0 0
\(682\) −35.5987 −1.36314
\(683\) −17.5364 −0.671011 −0.335505 0.942038i \(-0.608907\pi\)
−0.335505 + 0.942038i \(0.608907\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 37.7758 1.44229
\(687\) 0 0
\(688\) −3.55465 −0.135520
\(689\) 9.24992 0.352394
\(690\) 0 0
\(691\) −25.2734 −0.961446 −0.480723 0.876872i \(-0.659626\pi\)
−0.480723 + 0.876872i \(0.659626\pi\)
\(692\) 12.2630 0.466168
\(693\) 0 0
\(694\) −2.56246 −0.0972695
\(695\) 0 0
\(696\) 0 0
\(697\) −2.19767 −0.0832426
\(698\) 5.67176 0.214679
\(699\) 0 0
\(700\) 0 0
\(701\) 16.0545 0.606370 0.303185 0.952932i \(-0.401950\pi\)
0.303185 + 0.952932i \(0.401950\pi\)
\(702\) 0 0
\(703\) 11.4088 0.430291
\(704\) −6.40880 −0.241541
\(705\) 0 0
\(706\) −23.6665 −0.890701
\(707\) 49.4711 1.86055
\(708\) 0 0
\(709\) 13.5076 0.507290 0.253645 0.967297i \(-0.418371\pi\)
0.253645 + 0.967297i \(0.418371\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −10.0000 −0.374766
\(713\) 9.97392 0.373526
\(714\) 0 0
\(715\) 0 0
\(716\) 1.57292 0.0587829
\(717\) 0 0
\(718\) 31.9724 1.19320
\(719\) −0.803402 −0.0299619 −0.0149809 0.999888i \(-0.504769\pi\)
−0.0149809 + 0.999888i \(0.504769\pi\)
\(720\) 0 0
\(721\) 79.7337 2.96944
\(722\) −1.00000 −0.0372161
\(723\) 0 0
\(724\) 11.4088 0.424005
\(725\) 0 0
\(726\) 0 0
\(727\) 51.6860 1.91693 0.958463 0.285217i \(-0.0920655\pi\)
0.958463 + 0.285217i \(0.0920655\pi\)
\(728\) 4.99477 0.185118
\(729\) 0 0
\(730\) 0 0
\(731\) 6.81761 0.252158
\(732\) 0 0
\(733\) −22.3723 −0.826338 −0.413169 0.910654i \(-0.635578\pi\)
−0.413169 + 0.910654i \(0.635578\pi\)
\(734\) −1.14585 −0.0422940
\(735\) 0 0
\(736\) 1.79560 0.0661866
\(737\) 56.5438 2.08282
\(738\) 0 0
\(739\) 33.4271 1.22963 0.614817 0.788669i \(-0.289230\pi\)
0.614817 + 0.788669i \(0.289230\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 40.8266 1.49879
\(743\) −14.4998 −0.531947 −0.265974 0.963980i \(-0.585693\pi\)
−0.265974 + 0.963980i \(0.585693\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −32.8579 −1.20301
\(747\) 0 0
\(748\) 12.2917 0.449429
\(749\) −8.40623 −0.307157
\(750\) 0 0
\(751\) −26.6169 −0.971266 −0.485633 0.874163i \(-0.661411\pi\)
−0.485633 + 0.874163i \(0.661411\pi\)
\(752\) −10.8359 −0.395144
\(753\) 0 0
\(754\) 3.12351 0.113751
\(755\) 0 0
\(756\) 0 0
\(757\) −31.0362 −1.12803 −0.564015 0.825764i \(-0.690744\pi\)
−0.564015 + 0.825764i \(0.690744\pi\)
\(758\) 18.2865 0.664194
\(759\) 0 0
\(760\) 0 0
\(761\) 27.8083 1.00805 0.504025 0.863689i \(-0.331852\pi\)
0.504025 + 0.863689i \(0.331852\pi\)
\(762\) 0 0
\(763\) −11.3345 −0.410338
\(764\) 6.35805 0.230026
\(765\) 0 0
\(766\) 20.8542 0.753491
\(767\) 5.99070 0.216312
\(768\) 0 0
\(769\) −19.2279 −0.693376 −0.346688 0.937980i \(-0.612694\pi\)
−0.346688 + 0.937980i \(0.612694\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 14.4998 0.521860
\(773\) −6.00373 −0.215939 −0.107970 0.994154i \(-0.534435\pi\)
−0.107970 + 0.994154i \(0.534435\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 14.9452 0.536501
\(777\) 0 0
\(778\) −24.9086 −0.893018
\(779\) −1.14585 −0.0410543
\(780\) 0 0
\(781\) −9.14585 −0.327264
\(782\) −3.44385 −0.123152
\(783\) 0 0
\(784\) 15.0455 0.537340
\(785\) 0 0
\(786\) 0 0
\(787\) 22.2797 0.794187 0.397093 0.917778i \(-0.370019\pi\)
0.397093 + 0.917778i \(0.370019\pi\)
\(788\) 5.14585 0.183313
\(789\) 0 0
\(790\) 0 0
\(791\) 33.5517 1.19296
\(792\) 0 0
\(793\) −3.60680 −0.128081
\(794\) 24.7445 0.878150
\(795\) 0 0
\(796\) 3.87766 0.137440
\(797\) −26.6259 −0.943138 −0.471569 0.881829i \(-0.656312\pi\)
−0.471569 + 0.881829i \(0.656312\pi\)
\(798\) 0 0
\(799\) 20.7826 0.735234
\(800\) 0 0
\(801\) 0 0
\(802\) 15.6457 0.552468
\(803\) 80.8355 2.85262
\(804\) 0 0
\(805\) 0 0
\(806\) 5.90897 0.208135
\(807\) 0 0
\(808\) 10.5364 0.370669
\(809\) −0.651250 −0.0228967 −0.0114484 0.999934i \(-0.503644\pi\)
−0.0114484 + 0.999934i \(0.503644\pi\)
\(810\) 0 0
\(811\) −13.0780 −0.459230 −0.229615 0.973281i \(-0.573747\pi\)
−0.229615 + 0.973281i \(0.573747\pi\)
\(812\) 13.7863 0.483804
\(813\) 0 0
\(814\) −73.1168 −2.56274
\(815\) 0 0
\(816\) 0 0
\(817\) 3.55465 0.124362
\(818\) −30.5804 −1.06922
\(819\) 0 0
\(820\) 0 0
\(821\) −14.6640 −0.511776 −0.255888 0.966706i \(-0.582368\pi\)
−0.255888 + 0.966706i \(0.582368\pi\)
\(822\) 0 0
\(823\) −18.2850 −0.637374 −0.318687 0.947860i \(-0.603242\pi\)
−0.318687 + 0.947860i \(0.603242\pi\)
\(824\) 16.9817 0.591587
\(825\) 0 0
\(826\) 26.4413 0.920010
\(827\) −40.1910 −1.39758 −0.698790 0.715327i \(-0.746278\pi\)
−0.698790 + 0.715327i \(0.746278\pi\)
\(828\) 0 0
\(829\) 28.3760 0.985539 0.492769 0.870160i \(-0.335985\pi\)
0.492769 + 0.870160i \(0.335985\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 1.06379 0.0368802
\(833\) −28.8564 −0.999815
\(834\) 0 0
\(835\) 0 0
\(836\) 6.40880 0.221653
\(837\) 0 0
\(838\) 1.96345 0.0678264
\(839\) −6.93471 −0.239413 −0.119706 0.992809i \(-0.538195\pi\)
−0.119706 + 0.992809i \(0.538195\pi\)
\(840\) 0 0
\(841\) −20.3787 −0.702712
\(842\) 25.5949 0.882060
\(843\) 0 0
\(844\) 17.1496 0.590313
\(845\) 0 0
\(846\) 0 0
\(847\) −141.200 −4.85167
\(848\) 8.69527 0.298597
\(849\) 0 0
\(850\) 0 0
\(851\) 20.4856 0.702238
\(852\) 0 0
\(853\) 21.9739 0.752373 0.376186 0.926544i \(-0.377235\pi\)
0.376186 + 0.926544i \(0.377235\pi\)
\(854\) −15.9194 −0.544752
\(855\) 0 0
\(856\) −1.79036 −0.0611934
\(857\) −13.6091 −0.464879 −0.232440 0.972611i \(-0.574671\pi\)
−0.232440 + 0.972611i \(0.574671\pi\)
\(858\) 0 0
\(859\) 5.17192 0.176464 0.0882319 0.996100i \(-0.471878\pi\)
0.0882319 + 0.996100i \(0.471878\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −15.3540 −0.522959
\(863\) −15.9635 −0.543402 −0.271701 0.962382i \(-0.587586\pi\)
−0.271701 + 0.962382i \(0.587586\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −16.6640 −0.566264
\(867\) 0 0
\(868\) 26.0806 0.885232
\(869\) 12.5834 0.426862
\(870\) 0 0
\(871\) −9.38563 −0.318020
\(872\) −2.41404 −0.0817496
\(873\) 0 0
\(874\) −1.79560 −0.0607370
\(875\) 0 0
\(876\) 0 0
\(877\) −15.8866 −0.536453 −0.268227 0.963356i \(-0.586438\pi\)
−0.268227 + 0.963356i \(0.586438\pi\)
\(878\) 19.5987 0.661423
\(879\) 0 0
\(880\) 0 0
\(881\) 16.1458 0.543967 0.271984 0.962302i \(-0.412320\pi\)
0.271984 + 0.962302i \(0.412320\pi\)
\(882\) 0 0
\(883\) 38.2887 1.28852 0.644259 0.764808i \(-0.277166\pi\)
0.644259 + 0.764808i \(0.277166\pi\)
\(884\) −2.04028 −0.0686221
\(885\) 0 0
\(886\) 13.0183 0.437357
\(887\) −19.2809 −0.647389 −0.323695 0.946162i \(-0.604925\pi\)
−0.323695 + 0.946162i \(0.604925\pi\)
\(888\) 0 0
\(889\) −43.4920 −1.45868
\(890\) 0 0
\(891\) 0 0
\(892\) 7.84635 0.262715
\(893\) 10.8359 0.362609
\(894\) 0 0
\(895\) 0 0
\(896\) 4.69527 0.156858
\(897\) 0 0
\(898\) 6.77359 0.226038
\(899\) 16.3096 0.543957
\(900\) 0 0
\(901\) −16.6770 −0.555591
\(902\) 7.34352 0.244512
\(903\) 0 0
\(904\) 7.14585 0.237667
\(905\) 0 0
\(906\) 0 0
\(907\) 43.0205 1.42847 0.714236 0.699905i \(-0.246774\pi\)
0.714236 + 0.699905i \(0.246774\pi\)
\(908\) −6.28646 −0.208624
\(909\) 0 0
\(910\) 0 0
\(911\) 25.7733 0.853906 0.426953 0.904274i \(-0.359587\pi\)
0.426953 + 0.904274i \(0.359587\pi\)
\(912\) 0 0
\(913\) 104.109 3.44550
\(914\) 27.3189 0.903630
\(915\) 0 0
\(916\) −3.14585 −0.103942
\(917\) 54.0806 1.78590
\(918\) 0 0
\(919\) −29.5897 −0.976074 −0.488037 0.872823i \(-0.662287\pi\)
−0.488037 + 0.872823i \(0.662287\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −27.7993 −0.915522
\(923\) 1.51811 0.0499691
\(924\) 0 0
\(925\) 0 0
\(926\) −38.2369 −1.25654
\(927\) 0 0
\(928\) 2.93621 0.0963859
\(929\) 1.42334 0.0466983 0.0233491 0.999727i \(-0.492567\pi\)
0.0233491 + 0.999727i \(0.492567\pi\)
\(930\) 0 0
\(931\) −15.0455 −0.493097
\(932\) −0.182394 −0.00597451
\(933\) 0 0
\(934\) 23.4711 0.767998
\(935\) 0 0
\(936\) 0 0
\(937\) 28.2276 0.922155 0.461077 0.887360i \(-0.347463\pi\)
0.461077 + 0.887360i \(0.347463\pi\)
\(938\) −41.4256 −1.35259
\(939\) 0 0
\(940\) 0 0
\(941\) 15.8672 0.517256 0.258628 0.965977i \(-0.416730\pi\)
0.258628 + 0.965977i \(0.416730\pi\)
\(942\) 0 0
\(943\) −2.05748 −0.0670009
\(944\) 5.63148 0.183289
\(945\) 0 0
\(946\) −22.7811 −0.740676
\(947\) −43.6718 −1.41914 −0.709571 0.704634i \(-0.751111\pi\)
−0.709571 + 0.704634i \(0.751111\pi\)
\(948\) 0 0
\(949\) −13.4178 −0.435559
\(950\) 0 0
\(951\) 0 0
\(952\) −9.00523 −0.291861
\(953\) −16.0261 −0.519136 −0.259568 0.965725i \(-0.583580\pi\)
−0.259568 + 0.965725i \(0.583580\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 11.5039 0.372063
\(957\) 0 0
\(958\) −19.6900 −0.636156
\(959\) 70.9176 2.29005
\(960\) 0 0
\(961\) −0.145848 −0.00470478
\(962\) 12.1365 0.391298
\(963\) 0 0
\(964\) −0.445349 −0.0143437
\(965\) 0 0
\(966\) 0 0
\(967\) −11.5987 −0.372988 −0.186494 0.982456i \(-0.559712\pi\)
−0.186494 + 0.982456i \(0.559712\pi\)
\(968\) −30.0728 −0.966575
\(969\) 0 0
\(970\) 0 0
\(971\) 27.0362 0.867633 0.433817 0.901001i \(-0.357167\pi\)
0.433817 + 0.901001i \(0.357167\pi\)
\(972\) 0 0
\(973\) 3.28903 0.105442
\(974\) −16.3357 −0.523430
\(975\) 0 0
\(976\) −3.39053 −0.108528
\(977\) −14.1537 −0.452815 −0.226408 0.974033i \(-0.572698\pi\)
−0.226408 + 0.974033i \(0.572698\pi\)
\(978\) 0 0
\(979\) −64.0880 −2.04826
\(980\) 0 0
\(981\) 0 0
\(982\) 27.1093 0.865093
\(983\) 32.8542 1.04788 0.523942 0.851754i \(-0.324461\pi\)
0.523942 + 0.851754i \(0.324461\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −5.63148 −0.179343
\(987\) 0 0
\(988\) −1.06379 −0.0338436
\(989\) 6.38273 0.202959
\(990\) 0 0
\(991\) 47.9709 1.52385 0.761923 0.647667i \(-0.224255\pi\)
0.761923 + 0.647667i \(0.224255\pi\)
\(992\) 5.55465 0.176360
\(993\) 0 0
\(994\) 6.70050 0.212527
\(995\) 0 0
\(996\) 0 0
\(997\) 44.1716 1.39893 0.699464 0.714668i \(-0.253422\pi\)
0.699464 + 0.714668i \(0.253422\pi\)
\(998\) 4.69003 0.148460
\(999\) 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 8550.2.a.cj.1.1 3
3.2 odd 2 950.2.a.m.1.3 yes 3
5.4 even 2 8550.2.a.co.1.3 3
12.11 even 2 7600.2.a.bm.1.1 3
15.2 even 4 950.2.b.g.799.4 6
15.8 even 4 950.2.b.g.799.3 6
15.14 odd 2 950.2.a.k.1.1 3
60.59 even 2 7600.2.a.cb.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
950.2.a.k.1.1 3 15.14 odd 2
950.2.a.m.1.3 yes 3 3.2 odd 2
950.2.b.g.799.3 6 15.8 even 4
950.2.b.g.799.4 6 15.2 even 4
7600.2.a.bm.1.1 3 12.11 even 2
7600.2.a.cb.1.3 3 60.59 even 2
8550.2.a.cj.1.1 3 1.1 even 1 trivial
8550.2.a.co.1.3 3 5.4 even 2