Properties

Label 8004.2.a.g.1.8
Level $8004$
Weight $2$
Character 8004.1
Self dual yes
Analytic conductor $63.912$
Analytic rank $1$
Dimension $12$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8004.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.9122617778\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 3 x^{11} - 31 x^{10} + 80 x^{9} + 347 x^{8} - 697 x^{7} - 1714 x^{6} + 2146 x^{5} + 3304 x^{4} - 1156 x^{3} - 616 x^{2} + 136 x + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(-0.413592\) of defining polynomial
Character \(\chi\) \(=\) 8004.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +0.413592 q^{5} +2.75070 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +0.413592 q^{5} +2.75070 q^{7} +1.00000 q^{9} -5.40788 q^{11} +2.14893 q^{13} -0.413592 q^{15} +5.24934 q^{17} -3.94252 q^{19} -2.75070 q^{21} +1.00000 q^{23} -4.82894 q^{25} -1.00000 q^{27} -1.00000 q^{29} +4.35163 q^{31} +5.40788 q^{33} +1.13767 q^{35} -2.72398 q^{37} -2.14893 q^{39} -6.03343 q^{41} -3.64006 q^{43} +0.413592 q^{45} +2.47763 q^{47} +0.566323 q^{49} -5.24934 q^{51} -7.66902 q^{53} -2.23666 q^{55} +3.94252 q^{57} +7.67427 q^{59} -6.93585 q^{61} +2.75070 q^{63} +0.888782 q^{65} +10.2926 q^{67} -1.00000 q^{69} -0.981264 q^{71} -6.83799 q^{73} +4.82894 q^{75} -14.8754 q^{77} +3.02058 q^{79} +1.00000 q^{81} -0.833761 q^{83} +2.17109 q^{85} +1.00000 q^{87} -5.00216 q^{89} +5.91106 q^{91} -4.35163 q^{93} -1.63060 q^{95} -11.6816 q^{97} -5.40788 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 12q^{3} - 3q^{5} + 4q^{7} + 12q^{9} + O(q^{10}) \) \( 12q - 12q^{3} - 3q^{5} + 4q^{7} + 12q^{9} - 5q^{11} - 6q^{13} + 3q^{15} - 7q^{17} - 3q^{19} - 4q^{21} + 12q^{23} + 11q^{25} - 12q^{27} - 12q^{29} + 2q^{31} + 5q^{33} - 9q^{35} - 20q^{37} + 6q^{39} - 3q^{41} + 5q^{43} - 3q^{45} - 2q^{49} + 7q^{51} - 3q^{53} + 19q^{55} + 3q^{57} - 20q^{59} - 17q^{61} + 4q^{63} - 4q^{65} - 9q^{67} - 12q^{69} + 7q^{71} - 9q^{73} - 11q^{75} - 34q^{77} + 14q^{79} + 12q^{81} + 5q^{83} - 12q^{85} + 12q^{87} - 22q^{89} - 3q^{91} - 2q^{93} - 27q^{95} + 17q^{97} - 5q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 0.413592 0.184964 0.0924820 0.995714i \(-0.470520\pi\)
0.0924820 + 0.995714i \(0.470520\pi\)
\(6\) 0 0
\(7\) 2.75070 1.03967 0.519833 0.854268i \(-0.325994\pi\)
0.519833 + 0.854268i \(0.325994\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −5.40788 −1.63054 −0.815268 0.579083i \(-0.803411\pi\)
−0.815268 + 0.579083i \(0.803411\pi\)
\(12\) 0 0
\(13\) 2.14893 0.596007 0.298004 0.954565i \(-0.403679\pi\)
0.298004 + 0.954565i \(0.403679\pi\)
\(14\) 0 0
\(15\) −0.413592 −0.106789
\(16\) 0 0
\(17\) 5.24934 1.27315 0.636576 0.771214i \(-0.280350\pi\)
0.636576 + 0.771214i \(0.280350\pi\)
\(18\) 0 0
\(19\) −3.94252 −0.904477 −0.452238 0.891897i \(-0.649374\pi\)
−0.452238 + 0.891897i \(0.649374\pi\)
\(20\) 0 0
\(21\) −2.75070 −0.600251
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) −4.82894 −0.965788
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) 4.35163 0.781576 0.390788 0.920481i \(-0.372202\pi\)
0.390788 + 0.920481i \(0.372202\pi\)
\(32\) 0 0
\(33\) 5.40788 0.941391
\(34\) 0 0
\(35\) 1.13767 0.192301
\(36\) 0 0
\(37\) −2.72398 −0.447820 −0.223910 0.974610i \(-0.571882\pi\)
−0.223910 + 0.974610i \(0.571882\pi\)
\(38\) 0 0
\(39\) −2.14893 −0.344105
\(40\) 0 0
\(41\) −6.03343 −0.942264 −0.471132 0.882063i \(-0.656154\pi\)
−0.471132 + 0.882063i \(0.656154\pi\)
\(42\) 0 0
\(43\) −3.64006 −0.555105 −0.277552 0.960711i \(-0.589523\pi\)
−0.277552 + 0.960711i \(0.589523\pi\)
\(44\) 0 0
\(45\) 0.413592 0.0616547
\(46\) 0 0
\(47\) 2.47763 0.361399 0.180700 0.983538i \(-0.442164\pi\)
0.180700 + 0.983538i \(0.442164\pi\)
\(48\) 0 0
\(49\) 0.566323 0.0809034
\(50\) 0 0
\(51\) −5.24934 −0.735055
\(52\) 0 0
\(53\) −7.66902 −1.05342 −0.526711 0.850045i \(-0.676575\pi\)
−0.526711 + 0.850045i \(0.676575\pi\)
\(54\) 0 0
\(55\) −2.23666 −0.301591
\(56\) 0 0
\(57\) 3.94252 0.522200
\(58\) 0 0
\(59\) 7.67427 0.999105 0.499552 0.866284i \(-0.333498\pi\)
0.499552 + 0.866284i \(0.333498\pi\)
\(60\) 0 0
\(61\) −6.93585 −0.888044 −0.444022 0.896016i \(-0.646449\pi\)
−0.444022 + 0.896016i \(0.646449\pi\)
\(62\) 0 0
\(63\) 2.75070 0.346555
\(64\) 0 0
\(65\) 0.888782 0.110240
\(66\) 0 0
\(67\) 10.2926 1.25744 0.628718 0.777634i \(-0.283580\pi\)
0.628718 + 0.777634i \(0.283580\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −0.981264 −0.116455 −0.0582273 0.998303i \(-0.518545\pi\)
−0.0582273 + 0.998303i \(0.518545\pi\)
\(72\) 0 0
\(73\) −6.83799 −0.800327 −0.400163 0.916444i \(-0.631047\pi\)
−0.400163 + 0.916444i \(0.631047\pi\)
\(74\) 0 0
\(75\) 4.82894 0.557598
\(76\) 0 0
\(77\) −14.8754 −1.69521
\(78\) 0 0
\(79\) 3.02058 0.339842 0.169921 0.985458i \(-0.445649\pi\)
0.169921 + 0.985458i \(0.445649\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −0.833761 −0.0915172 −0.0457586 0.998953i \(-0.514571\pi\)
−0.0457586 + 0.998953i \(0.514571\pi\)
\(84\) 0 0
\(85\) 2.17109 0.235487
\(86\) 0 0
\(87\) 1.00000 0.107211
\(88\) 0 0
\(89\) −5.00216 −0.530228 −0.265114 0.964217i \(-0.585410\pi\)
−0.265114 + 0.964217i \(0.585410\pi\)
\(90\) 0 0
\(91\) 5.91106 0.619648
\(92\) 0 0
\(93\) −4.35163 −0.451243
\(94\) 0 0
\(95\) −1.63060 −0.167296
\(96\) 0 0
\(97\) −11.6816 −1.18608 −0.593042 0.805171i \(-0.702073\pi\)
−0.593042 + 0.805171i \(0.702073\pi\)
\(98\) 0 0
\(99\) −5.40788 −0.543512
\(100\) 0 0
\(101\) −10.7048 −1.06517 −0.532586 0.846376i \(-0.678780\pi\)
−0.532586 + 0.846376i \(0.678780\pi\)
\(102\) 0 0
\(103\) 5.40129 0.532204 0.266102 0.963945i \(-0.414264\pi\)
0.266102 + 0.963945i \(0.414264\pi\)
\(104\) 0 0
\(105\) −1.13767 −0.111025
\(106\) 0 0
\(107\) 6.56999 0.635144 0.317572 0.948234i \(-0.397132\pi\)
0.317572 + 0.948234i \(0.397132\pi\)
\(108\) 0 0
\(109\) 0.0782059 0.00749076 0.00374538 0.999993i \(-0.498808\pi\)
0.00374538 + 0.999993i \(0.498808\pi\)
\(110\) 0 0
\(111\) 2.72398 0.258549
\(112\) 0 0
\(113\) −13.2337 −1.24492 −0.622460 0.782651i \(-0.713867\pi\)
−0.622460 + 0.782651i \(0.713867\pi\)
\(114\) 0 0
\(115\) 0.413592 0.0385677
\(116\) 0 0
\(117\) 2.14893 0.198669
\(118\) 0 0
\(119\) 14.4393 1.32365
\(120\) 0 0
\(121\) 18.2451 1.65865
\(122\) 0 0
\(123\) 6.03343 0.544016
\(124\) 0 0
\(125\) −4.06517 −0.363600
\(126\) 0 0
\(127\) 4.49520 0.398884 0.199442 0.979910i \(-0.436087\pi\)
0.199442 + 0.979910i \(0.436087\pi\)
\(128\) 0 0
\(129\) 3.64006 0.320490
\(130\) 0 0
\(131\) −16.6600 −1.45559 −0.727797 0.685793i \(-0.759456\pi\)
−0.727797 + 0.685793i \(0.759456\pi\)
\(132\) 0 0
\(133\) −10.8447 −0.940353
\(134\) 0 0
\(135\) −0.413592 −0.0355963
\(136\) 0 0
\(137\) −2.27426 −0.194303 −0.0971514 0.995270i \(-0.530973\pi\)
−0.0971514 + 0.995270i \(0.530973\pi\)
\(138\) 0 0
\(139\) −1.41693 −0.120183 −0.0600913 0.998193i \(-0.519139\pi\)
−0.0600913 + 0.998193i \(0.519139\pi\)
\(140\) 0 0
\(141\) −2.47763 −0.208654
\(142\) 0 0
\(143\) −11.6212 −0.971812
\(144\) 0 0
\(145\) −0.413592 −0.0343470
\(146\) 0 0
\(147\) −0.566323 −0.0467096
\(148\) 0 0
\(149\) −7.50663 −0.614967 −0.307484 0.951553i \(-0.599487\pi\)
−0.307484 + 0.951553i \(0.599487\pi\)
\(150\) 0 0
\(151\) 19.0639 1.55140 0.775698 0.631104i \(-0.217398\pi\)
0.775698 + 0.631104i \(0.217398\pi\)
\(152\) 0 0
\(153\) 5.24934 0.424384
\(154\) 0 0
\(155\) 1.79980 0.144564
\(156\) 0 0
\(157\) 17.7593 1.41735 0.708673 0.705537i \(-0.249294\pi\)
0.708673 + 0.705537i \(0.249294\pi\)
\(158\) 0 0
\(159\) 7.66902 0.608193
\(160\) 0 0
\(161\) 2.75070 0.216785
\(162\) 0 0
\(163\) −12.7907 −1.00184 −0.500921 0.865493i \(-0.667005\pi\)
−0.500921 + 0.865493i \(0.667005\pi\)
\(164\) 0 0
\(165\) 2.23666 0.174123
\(166\) 0 0
\(167\) −6.05789 −0.468774 −0.234387 0.972143i \(-0.575308\pi\)
−0.234387 + 0.972143i \(0.575308\pi\)
\(168\) 0 0
\(169\) −8.38208 −0.644775
\(170\) 0 0
\(171\) −3.94252 −0.301492
\(172\) 0 0
\(173\) −10.6465 −0.809442 −0.404721 0.914440i \(-0.632631\pi\)
−0.404721 + 0.914440i \(0.632631\pi\)
\(174\) 0 0
\(175\) −13.2829 −1.00410
\(176\) 0 0
\(177\) −7.67427 −0.576834
\(178\) 0 0
\(179\) 2.02536 0.151382 0.0756911 0.997131i \(-0.475884\pi\)
0.0756911 + 0.997131i \(0.475884\pi\)
\(180\) 0 0
\(181\) 8.61476 0.640330 0.320165 0.947362i \(-0.396262\pi\)
0.320165 + 0.947362i \(0.396262\pi\)
\(182\) 0 0
\(183\) 6.93585 0.512713
\(184\) 0 0
\(185\) −1.12662 −0.0828306
\(186\) 0 0
\(187\) −28.3878 −2.07592
\(188\) 0 0
\(189\) −2.75070 −0.200084
\(190\) 0 0
\(191\) −0.577000 −0.0417502 −0.0208751 0.999782i \(-0.506645\pi\)
−0.0208751 + 0.999782i \(0.506645\pi\)
\(192\) 0 0
\(193\) 2.92378 0.210458 0.105229 0.994448i \(-0.466442\pi\)
0.105229 + 0.994448i \(0.466442\pi\)
\(194\) 0 0
\(195\) −0.888782 −0.0636470
\(196\) 0 0
\(197\) −20.4451 −1.45665 −0.728327 0.685230i \(-0.759702\pi\)
−0.728327 + 0.685230i \(0.759702\pi\)
\(198\) 0 0
\(199\) −1.04208 −0.0738710 −0.0369355 0.999318i \(-0.511760\pi\)
−0.0369355 + 0.999318i \(0.511760\pi\)
\(200\) 0 0
\(201\) −10.2926 −0.725981
\(202\) 0 0
\(203\) −2.75070 −0.193061
\(204\) 0 0
\(205\) −2.49538 −0.174285
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) 21.3207 1.47478
\(210\) 0 0
\(211\) 16.4609 1.13322 0.566608 0.823988i \(-0.308255\pi\)
0.566608 + 0.823988i \(0.308255\pi\)
\(212\) 0 0
\(213\) 0.981264 0.0672351
\(214\) 0 0
\(215\) −1.50550 −0.102674
\(216\) 0 0
\(217\) 11.9700 0.812578
\(218\) 0 0
\(219\) 6.83799 0.462069
\(220\) 0 0
\(221\) 11.2805 0.758808
\(222\) 0 0
\(223\) 18.7123 1.25307 0.626535 0.779393i \(-0.284472\pi\)
0.626535 + 0.779393i \(0.284472\pi\)
\(224\) 0 0
\(225\) −4.82894 −0.321929
\(226\) 0 0
\(227\) 2.90845 0.193040 0.0965202 0.995331i \(-0.469229\pi\)
0.0965202 + 0.995331i \(0.469229\pi\)
\(228\) 0 0
\(229\) 7.60770 0.502731 0.251365 0.967892i \(-0.419120\pi\)
0.251365 + 0.967892i \(0.419120\pi\)
\(230\) 0 0
\(231\) 14.8754 0.978731
\(232\) 0 0
\(233\) −28.7075 −1.88069 −0.940344 0.340225i \(-0.889497\pi\)
−0.940344 + 0.340225i \(0.889497\pi\)
\(234\) 0 0
\(235\) 1.02473 0.0668459
\(236\) 0 0
\(237\) −3.02058 −0.196208
\(238\) 0 0
\(239\) 6.02058 0.389439 0.194719 0.980859i \(-0.437620\pi\)
0.194719 + 0.980859i \(0.437620\pi\)
\(240\) 0 0
\(241\) 18.4403 1.18785 0.593923 0.804522i \(-0.297578\pi\)
0.593923 + 0.804522i \(0.297578\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 0.234227 0.0149642
\(246\) 0 0
\(247\) −8.47222 −0.539075
\(248\) 0 0
\(249\) 0.833761 0.0528375
\(250\) 0 0
\(251\) −9.57148 −0.604146 −0.302073 0.953285i \(-0.597679\pi\)
−0.302073 + 0.953285i \(0.597679\pi\)
\(252\) 0 0
\(253\) −5.40788 −0.339990
\(254\) 0 0
\(255\) −2.17109 −0.135959
\(256\) 0 0
\(257\) 15.1906 0.947563 0.473781 0.880642i \(-0.342889\pi\)
0.473781 + 0.880642i \(0.342889\pi\)
\(258\) 0 0
\(259\) −7.49285 −0.465583
\(260\) 0 0
\(261\) −1.00000 −0.0618984
\(262\) 0 0
\(263\) −27.9622 −1.72422 −0.862110 0.506721i \(-0.830857\pi\)
−0.862110 + 0.506721i \(0.830857\pi\)
\(264\) 0 0
\(265\) −3.17185 −0.194845
\(266\) 0 0
\(267\) 5.00216 0.306127
\(268\) 0 0
\(269\) −22.3628 −1.36348 −0.681742 0.731592i \(-0.738777\pi\)
−0.681742 + 0.731592i \(0.738777\pi\)
\(270\) 0 0
\(271\) −2.45290 −0.149003 −0.0745017 0.997221i \(-0.523737\pi\)
−0.0745017 + 0.997221i \(0.523737\pi\)
\(272\) 0 0
\(273\) −5.91106 −0.357754
\(274\) 0 0
\(275\) 26.1143 1.57475
\(276\) 0 0
\(277\) 1.25987 0.0756982 0.0378491 0.999283i \(-0.487949\pi\)
0.0378491 + 0.999283i \(0.487949\pi\)
\(278\) 0 0
\(279\) 4.35163 0.260525
\(280\) 0 0
\(281\) 16.9691 1.01229 0.506145 0.862449i \(-0.331070\pi\)
0.506145 + 0.862449i \(0.331070\pi\)
\(282\) 0 0
\(283\) −13.9411 −0.828715 −0.414357 0.910114i \(-0.635994\pi\)
−0.414357 + 0.910114i \(0.635994\pi\)
\(284\) 0 0
\(285\) 1.63060 0.0965882
\(286\) 0 0
\(287\) −16.5961 −0.979639
\(288\) 0 0
\(289\) 10.5556 0.620918
\(290\) 0 0
\(291\) 11.6816 0.684786
\(292\) 0 0
\(293\) −22.0522 −1.28831 −0.644153 0.764897i \(-0.722790\pi\)
−0.644153 + 0.764897i \(0.722790\pi\)
\(294\) 0 0
\(295\) 3.17402 0.184798
\(296\) 0 0
\(297\) 5.40788 0.313797
\(298\) 0 0
\(299\) 2.14893 0.124276
\(300\) 0 0
\(301\) −10.0127 −0.577123
\(302\) 0 0
\(303\) 10.7048 0.614977
\(304\) 0 0
\(305\) −2.86861 −0.164256
\(306\) 0 0
\(307\) 32.1760 1.83638 0.918190 0.396139i \(-0.129650\pi\)
0.918190 + 0.396139i \(0.129650\pi\)
\(308\) 0 0
\(309\) −5.40129 −0.307268
\(310\) 0 0
\(311\) 3.57851 0.202919 0.101459 0.994840i \(-0.467649\pi\)
0.101459 + 0.994840i \(0.467649\pi\)
\(312\) 0 0
\(313\) −0.885822 −0.0500696 −0.0250348 0.999687i \(-0.507970\pi\)
−0.0250348 + 0.999687i \(0.507970\pi\)
\(314\) 0 0
\(315\) 1.13767 0.0641002
\(316\) 0 0
\(317\) 2.62770 0.147586 0.0737932 0.997274i \(-0.476490\pi\)
0.0737932 + 0.997274i \(0.476490\pi\)
\(318\) 0 0
\(319\) 5.40788 0.302783
\(320\) 0 0
\(321\) −6.56999 −0.366701
\(322\) 0 0
\(323\) −20.6957 −1.15154
\(324\) 0 0
\(325\) −10.3771 −0.575617
\(326\) 0 0
\(327\) −0.0782059 −0.00432479
\(328\) 0 0
\(329\) 6.81520 0.375734
\(330\) 0 0
\(331\) −9.42233 −0.517898 −0.258949 0.965891i \(-0.583376\pi\)
−0.258949 + 0.965891i \(0.583376\pi\)
\(332\) 0 0
\(333\) −2.72398 −0.149273
\(334\) 0 0
\(335\) 4.25692 0.232580
\(336\) 0 0
\(337\) 8.82736 0.480857 0.240428 0.970667i \(-0.422712\pi\)
0.240428 + 0.970667i \(0.422712\pi\)
\(338\) 0 0
\(339\) 13.2337 0.718755
\(340\) 0 0
\(341\) −23.5331 −1.27439
\(342\) 0 0
\(343\) −17.6971 −0.955553
\(344\) 0 0
\(345\) −0.413592 −0.0222670
\(346\) 0 0
\(347\) −9.49207 −0.509561 −0.254781 0.966999i \(-0.582003\pi\)
−0.254781 + 0.966999i \(0.582003\pi\)
\(348\) 0 0
\(349\) −9.91108 −0.530528 −0.265264 0.964176i \(-0.585459\pi\)
−0.265264 + 0.964176i \(0.585459\pi\)
\(350\) 0 0
\(351\) −2.14893 −0.114702
\(352\) 0 0
\(353\) −9.79466 −0.521317 −0.260658 0.965431i \(-0.583940\pi\)
−0.260658 + 0.965431i \(0.583940\pi\)
\(354\) 0 0
\(355\) −0.405843 −0.0215399
\(356\) 0 0
\(357\) −14.4393 −0.764211
\(358\) 0 0
\(359\) 18.3697 0.969513 0.484757 0.874649i \(-0.338908\pi\)
0.484757 + 0.874649i \(0.338908\pi\)
\(360\) 0 0
\(361\) −3.45651 −0.181922
\(362\) 0 0
\(363\) −18.2451 −0.957622
\(364\) 0 0
\(365\) −2.82814 −0.148032
\(366\) 0 0
\(367\) −30.7192 −1.60353 −0.801766 0.597639i \(-0.796106\pi\)
−0.801766 + 0.597639i \(0.796106\pi\)
\(368\) 0 0
\(369\) −6.03343 −0.314088
\(370\) 0 0
\(371\) −21.0951 −1.09521
\(372\) 0 0
\(373\) 9.55347 0.494660 0.247330 0.968931i \(-0.420447\pi\)
0.247330 + 0.968931i \(0.420447\pi\)
\(374\) 0 0
\(375\) 4.06517 0.209925
\(376\) 0 0
\(377\) −2.14893 −0.110676
\(378\) 0 0
\(379\) −18.9145 −0.971571 −0.485786 0.874078i \(-0.661466\pi\)
−0.485786 + 0.874078i \(0.661466\pi\)
\(380\) 0 0
\(381\) −4.49520 −0.230296
\(382\) 0 0
\(383\) −5.97880 −0.305502 −0.152751 0.988265i \(-0.548813\pi\)
−0.152751 + 0.988265i \(0.548813\pi\)
\(384\) 0 0
\(385\) −6.15236 −0.313553
\(386\) 0 0
\(387\) −3.64006 −0.185035
\(388\) 0 0
\(389\) −30.1804 −1.53021 −0.765103 0.643908i \(-0.777312\pi\)
−0.765103 + 0.643908i \(0.777312\pi\)
\(390\) 0 0
\(391\) 5.24934 0.265471
\(392\) 0 0
\(393\) 16.6600 0.840388
\(394\) 0 0
\(395\) 1.24929 0.0628585
\(396\) 0 0
\(397\) −24.2430 −1.21672 −0.608361 0.793660i \(-0.708173\pi\)
−0.608361 + 0.793660i \(0.708173\pi\)
\(398\) 0 0
\(399\) 10.8447 0.542913
\(400\) 0 0
\(401\) 12.0298 0.600739 0.300370 0.953823i \(-0.402890\pi\)
0.300370 + 0.953823i \(0.402890\pi\)
\(402\) 0 0
\(403\) 9.35138 0.465825
\(404\) 0 0
\(405\) 0.413592 0.0205516
\(406\) 0 0
\(407\) 14.7310 0.730187
\(408\) 0 0
\(409\) 11.8879 0.587821 0.293910 0.955833i \(-0.405043\pi\)
0.293910 + 0.955833i \(0.405043\pi\)
\(410\) 0 0
\(411\) 2.27426 0.112181
\(412\) 0 0
\(413\) 21.1096 1.03873
\(414\) 0 0
\(415\) −0.344837 −0.0169274
\(416\) 0 0
\(417\) 1.41693 0.0693874
\(418\) 0 0
\(419\) 15.8820 0.775888 0.387944 0.921683i \(-0.373185\pi\)
0.387944 + 0.921683i \(0.373185\pi\)
\(420\) 0 0
\(421\) −15.7258 −0.766430 −0.383215 0.923659i \(-0.625183\pi\)
−0.383215 + 0.923659i \(0.625183\pi\)
\(422\) 0 0
\(423\) 2.47763 0.120466
\(424\) 0 0
\(425\) −25.3488 −1.22960
\(426\) 0 0
\(427\) −19.0784 −0.923269
\(428\) 0 0
\(429\) 11.6212 0.561076
\(430\) 0 0
\(431\) −2.48721 −0.119805 −0.0599024 0.998204i \(-0.519079\pi\)
−0.0599024 + 0.998204i \(0.519079\pi\)
\(432\) 0 0
\(433\) 22.8419 1.09771 0.548857 0.835917i \(-0.315063\pi\)
0.548857 + 0.835917i \(0.315063\pi\)
\(434\) 0 0
\(435\) 0.413592 0.0198302
\(436\) 0 0
\(437\) −3.94252 −0.188596
\(438\) 0 0
\(439\) 3.32717 0.158797 0.0793987 0.996843i \(-0.474700\pi\)
0.0793987 + 0.996843i \(0.474700\pi\)
\(440\) 0 0
\(441\) 0.566323 0.0269678
\(442\) 0 0
\(443\) −4.05630 −0.192721 −0.0963604 0.995347i \(-0.530720\pi\)
−0.0963604 + 0.995347i \(0.530720\pi\)
\(444\) 0 0
\(445\) −2.06885 −0.0980731
\(446\) 0 0
\(447\) 7.50663 0.355051
\(448\) 0 0
\(449\) 36.8958 1.74122 0.870610 0.491974i \(-0.163724\pi\)
0.870610 + 0.491974i \(0.163724\pi\)
\(450\) 0 0
\(451\) 32.6281 1.53640
\(452\) 0 0
\(453\) −19.0639 −0.895699
\(454\) 0 0
\(455\) 2.44477 0.114613
\(456\) 0 0
\(457\) 22.2602 1.04129 0.520643 0.853774i \(-0.325692\pi\)
0.520643 + 0.853774i \(0.325692\pi\)
\(458\) 0 0
\(459\) −5.24934 −0.245018
\(460\) 0 0
\(461\) −10.4495 −0.486684 −0.243342 0.969941i \(-0.578244\pi\)
−0.243342 + 0.969941i \(0.578244\pi\)
\(462\) 0 0
\(463\) −40.0311 −1.86041 −0.930203 0.367047i \(-0.880369\pi\)
−0.930203 + 0.367047i \(0.880369\pi\)
\(464\) 0 0
\(465\) −1.79980 −0.0834638
\(466\) 0 0
\(467\) −15.3735 −0.711399 −0.355700 0.934600i \(-0.615757\pi\)
−0.355700 + 0.934600i \(0.615757\pi\)
\(468\) 0 0
\(469\) 28.3117 1.30731
\(470\) 0 0
\(471\) −17.7593 −0.818305
\(472\) 0 0
\(473\) 19.6850 0.905119
\(474\) 0 0
\(475\) 19.0382 0.873533
\(476\) 0 0
\(477\) −7.66902 −0.351140
\(478\) 0 0
\(479\) −11.5928 −0.529688 −0.264844 0.964291i \(-0.585320\pi\)
−0.264844 + 0.964291i \(0.585320\pi\)
\(480\) 0 0
\(481\) −5.85366 −0.266904
\(482\) 0 0
\(483\) −2.75070 −0.125161
\(484\) 0 0
\(485\) −4.83141 −0.219383
\(486\) 0 0
\(487\) −29.4897 −1.33631 −0.668154 0.744023i \(-0.732915\pi\)
−0.668154 + 0.744023i \(0.732915\pi\)
\(488\) 0 0
\(489\) 12.7907 0.578414
\(490\) 0 0
\(491\) 11.7766 0.531472 0.265736 0.964046i \(-0.414385\pi\)
0.265736 + 0.964046i \(0.414385\pi\)
\(492\) 0 0
\(493\) −5.24934 −0.236419
\(494\) 0 0
\(495\) −2.23666 −0.100530
\(496\) 0 0
\(497\) −2.69916 −0.121074
\(498\) 0 0
\(499\) −29.4491 −1.31832 −0.659160 0.752002i \(-0.729088\pi\)
−0.659160 + 0.752002i \(0.729088\pi\)
\(500\) 0 0
\(501\) 6.05789 0.270647
\(502\) 0 0
\(503\) 4.57702 0.204079 0.102040 0.994780i \(-0.467463\pi\)
0.102040 + 0.994780i \(0.467463\pi\)
\(504\) 0 0
\(505\) −4.42744 −0.197018
\(506\) 0 0
\(507\) 8.38208 0.372261
\(508\) 0 0
\(509\) −27.0861 −1.20057 −0.600285 0.799786i \(-0.704946\pi\)
−0.600285 + 0.799786i \(0.704946\pi\)
\(510\) 0 0
\(511\) −18.8092 −0.832072
\(512\) 0 0
\(513\) 3.94252 0.174067
\(514\) 0 0
\(515\) 2.23393 0.0984387
\(516\) 0 0
\(517\) −13.3987 −0.589275
\(518\) 0 0
\(519\) 10.6465 0.467331
\(520\) 0 0
\(521\) −39.6318 −1.73630 −0.868150 0.496303i \(-0.834691\pi\)
−0.868150 + 0.496303i \(0.834691\pi\)
\(522\) 0 0
\(523\) 9.48188 0.414614 0.207307 0.978276i \(-0.433530\pi\)
0.207307 + 0.978276i \(0.433530\pi\)
\(524\) 0 0
\(525\) 13.2829 0.579715
\(526\) 0 0
\(527\) 22.8432 0.995066
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 7.67427 0.333035
\(532\) 0 0
\(533\) −12.9655 −0.561596
\(534\) 0 0
\(535\) 2.71729 0.117479
\(536\) 0 0
\(537\) −2.02536 −0.0874005
\(538\) 0 0
\(539\) −3.06261 −0.131916
\(540\) 0 0
\(541\) −11.0625 −0.475613 −0.237806 0.971313i \(-0.576428\pi\)
−0.237806 + 0.971313i \(0.576428\pi\)
\(542\) 0 0
\(543\) −8.61476 −0.369695
\(544\) 0 0
\(545\) 0.0323453 0.00138552
\(546\) 0 0
\(547\) −30.6278 −1.30955 −0.654775 0.755824i \(-0.727237\pi\)
−0.654775 + 0.755824i \(0.727237\pi\)
\(548\) 0 0
\(549\) −6.93585 −0.296015
\(550\) 0 0
\(551\) 3.94252 0.167957
\(552\) 0 0
\(553\) 8.30869 0.353322
\(554\) 0 0
\(555\) 1.12662 0.0478223
\(556\) 0 0
\(557\) 3.17601 0.134572 0.0672860 0.997734i \(-0.478566\pi\)
0.0672860 + 0.997734i \(0.478566\pi\)
\(558\) 0 0
\(559\) −7.82226 −0.330846
\(560\) 0 0
\(561\) 28.3878 1.19853
\(562\) 0 0
\(563\) −12.8333 −0.540857 −0.270429 0.962740i \(-0.587165\pi\)
−0.270429 + 0.962740i \(0.587165\pi\)
\(564\) 0 0
\(565\) −5.47335 −0.230266
\(566\) 0 0
\(567\) 2.75070 0.115518
\(568\) 0 0
\(569\) 26.4461 1.10868 0.554338 0.832292i \(-0.312971\pi\)
0.554338 + 0.832292i \(0.312971\pi\)
\(570\) 0 0
\(571\) 6.77280 0.283433 0.141716 0.989907i \(-0.454738\pi\)
0.141716 + 0.989907i \(0.454738\pi\)
\(572\) 0 0
\(573\) 0.577000 0.0241045
\(574\) 0 0
\(575\) −4.82894 −0.201381
\(576\) 0 0
\(577\) 1.17680 0.0489907 0.0244954 0.999700i \(-0.492202\pi\)
0.0244954 + 0.999700i \(0.492202\pi\)
\(578\) 0 0
\(579\) −2.92378 −0.121508
\(580\) 0 0
\(581\) −2.29342 −0.0951472
\(582\) 0 0
\(583\) 41.4732 1.71764
\(584\) 0 0
\(585\) 0.888782 0.0367466
\(586\) 0 0
\(587\) −11.6449 −0.480635 −0.240318 0.970694i \(-0.577252\pi\)
−0.240318 + 0.970694i \(0.577252\pi\)
\(588\) 0 0
\(589\) −17.1564 −0.706918
\(590\) 0 0
\(591\) 20.4451 0.840999
\(592\) 0 0
\(593\) −4.15677 −0.170698 −0.0853490 0.996351i \(-0.527201\pi\)
−0.0853490 + 0.996351i \(0.527201\pi\)
\(594\) 0 0
\(595\) 5.97200 0.244828
\(596\) 0 0
\(597\) 1.04208 0.0426495
\(598\) 0 0
\(599\) 6.35672 0.259728 0.129864 0.991532i \(-0.458546\pi\)
0.129864 + 0.991532i \(0.458546\pi\)
\(600\) 0 0
\(601\) −13.7488 −0.560826 −0.280413 0.959880i \(-0.590471\pi\)
−0.280413 + 0.959880i \(0.590471\pi\)
\(602\) 0 0
\(603\) 10.2926 0.419145
\(604\) 0 0
\(605\) 7.54605 0.306791
\(606\) 0 0
\(607\) −6.97089 −0.282940 −0.141470 0.989943i \(-0.545183\pi\)
−0.141470 + 0.989943i \(0.545183\pi\)
\(608\) 0 0
\(609\) 2.75070 0.111464
\(610\) 0 0
\(611\) 5.32426 0.215397
\(612\) 0 0
\(613\) −14.5629 −0.588191 −0.294096 0.955776i \(-0.595018\pi\)
−0.294096 + 0.955776i \(0.595018\pi\)
\(614\) 0 0
\(615\) 2.49538 0.100623
\(616\) 0 0
\(617\) 10.4608 0.421138 0.210569 0.977579i \(-0.432468\pi\)
0.210569 + 0.977579i \(0.432468\pi\)
\(618\) 0 0
\(619\) −35.8262 −1.43997 −0.719987 0.693987i \(-0.755852\pi\)
−0.719987 + 0.693987i \(0.755852\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) −13.7594 −0.551259
\(624\) 0 0
\(625\) 22.4634 0.898535
\(626\) 0 0
\(627\) −21.3207 −0.851466
\(628\) 0 0
\(629\) −14.2991 −0.570144
\(630\) 0 0
\(631\) −22.4116 −0.892193 −0.446097 0.894985i \(-0.647186\pi\)
−0.446097 + 0.894985i \(0.647186\pi\)
\(632\) 0 0
\(633\) −16.4609 −0.654262
\(634\) 0 0
\(635\) 1.85918 0.0737793
\(636\) 0 0
\(637\) 1.21699 0.0482190
\(638\) 0 0
\(639\) −0.981264 −0.0388182
\(640\) 0 0
\(641\) 42.2654 1.66938 0.834692 0.550717i \(-0.185646\pi\)
0.834692 + 0.550717i \(0.185646\pi\)
\(642\) 0 0
\(643\) −25.9281 −1.02250 −0.511251 0.859431i \(-0.670818\pi\)
−0.511251 + 0.859431i \(0.670818\pi\)
\(644\) 0 0
\(645\) 1.50550 0.0592791
\(646\) 0 0
\(647\) −30.8953 −1.21462 −0.607310 0.794465i \(-0.707751\pi\)
−0.607310 + 0.794465i \(0.707751\pi\)
\(648\) 0 0
\(649\) −41.5015 −1.62908
\(650\) 0 0
\(651\) −11.9700 −0.469142
\(652\) 0 0
\(653\) 18.1136 0.708839 0.354419 0.935087i \(-0.384679\pi\)
0.354419 + 0.935087i \(0.384679\pi\)
\(654\) 0 0
\(655\) −6.89046 −0.269233
\(656\) 0 0
\(657\) −6.83799 −0.266776
\(658\) 0 0
\(659\) 22.9131 0.892567 0.446283 0.894892i \(-0.352747\pi\)
0.446283 + 0.894892i \(0.352747\pi\)
\(660\) 0 0
\(661\) −24.2173 −0.941942 −0.470971 0.882149i \(-0.656096\pi\)
−0.470971 + 0.882149i \(0.656096\pi\)
\(662\) 0 0
\(663\) −11.2805 −0.438098
\(664\) 0 0
\(665\) −4.48527 −0.173931
\(666\) 0 0
\(667\) −1.00000 −0.0387202
\(668\) 0 0
\(669\) −18.7123 −0.723460
\(670\) 0 0
\(671\) 37.5082 1.44799
\(672\) 0 0
\(673\) −47.7524 −1.84072 −0.920359 0.391073i \(-0.872104\pi\)
−0.920359 + 0.391073i \(0.872104\pi\)
\(674\) 0 0
\(675\) 4.82894 0.185866
\(676\) 0 0
\(677\) −2.08884 −0.0802808 −0.0401404 0.999194i \(-0.512781\pi\)
−0.0401404 + 0.999194i \(0.512781\pi\)
\(678\) 0 0
\(679\) −32.1325 −1.23313
\(680\) 0 0
\(681\) −2.90845 −0.111452
\(682\) 0 0
\(683\) −16.0856 −0.615498 −0.307749 0.951468i \(-0.599576\pi\)
−0.307749 + 0.951468i \(0.599576\pi\)
\(684\) 0 0
\(685\) −0.940614 −0.0359390
\(686\) 0 0
\(687\) −7.60770 −0.290252
\(688\) 0 0
\(689\) −16.4802 −0.627847
\(690\) 0 0
\(691\) −14.3136 −0.544517 −0.272258 0.962224i \(-0.587771\pi\)
−0.272258 + 0.962224i \(0.587771\pi\)
\(692\) 0 0
\(693\) −14.8754 −0.565071
\(694\) 0 0
\(695\) −0.586031 −0.0222294
\(696\) 0 0
\(697\) −31.6716 −1.19965
\(698\) 0 0
\(699\) 28.7075 1.08582
\(700\) 0 0
\(701\) −34.8008 −1.31441 −0.657205 0.753712i \(-0.728261\pi\)
−0.657205 + 0.753712i \(0.728261\pi\)
\(702\) 0 0
\(703\) 10.7394 0.405043
\(704\) 0 0
\(705\) −1.02473 −0.0385935
\(706\) 0 0
\(707\) −29.4457 −1.10742
\(708\) 0 0
\(709\) −10.2207 −0.383846 −0.191923 0.981410i \(-0.561472\pi\)
−0.191923 + 0.981410i \(0.561472\pi\)
\(710\) 0 0
\(711\) 3.02058 0.113281
\(712\) 0 0
\(713\) 4.35163 0.162970
\(714\) 0 0
\(715\) −4.80643 −0.179750
\(716\) 0 0
\(717\) −6.02058 −0.224843
\(718\) 0 0
\(719\) −12.2122 −0.455439 −0.227719 0.973727i \(-0.573127\pi\)
−0.227719 + 0.973727i \(0.573127\pi\)
\(720\) 0 0
\(721\) 14.8573 0.553314
\(722\) 0 0
\(723\) −18.4403 −0.685804
\(724\) 0 0
\(725\) 4.82894 0.179342
\(726\) 0 0
\(727\) 27.0485 1.00317 0.501586 0.865108i \(-0.332750\pi\)
0.501586 + 0.865108i \(0.332750\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −19.1079 −0.706733
\(732\) 0 0
\(733\) 11.0713 0.408928 0.204464 0.978874i \(-0.434455\pi\)
0.204464 + 0.978874i \(0.434455\pi\)
\(734\) 0 0
\(735\) −0.234227 −0.00863959
\(736\) 0 0
\(737\) −55.6609 −2.05029
\(738\) 0 0
\(739\) −26.0664 −0.958869 −0.479434 0.877578i \(-0.659158\pi\)
−0.479434 + 0.877578i \(0.659158\pi\)
\(740\) 0 0
\(741\) 8.47222 0.311235
\(742\) 0 0
\(743\) −40.1217 −1.47192 −0.735962 0.677023i \(-0.763270\pi\)
−0.735962 + 0.677023i \(0.763270\pi\)
\(744\) 0 0
\(745\) −3.10468 −0.113747
\(746\) 0 0
\(747\) −0.833761 −0.0305057
\(748\) 0 0
\(749\) 18.0720 0.660337
\(750\) 0 0
\(751\) 21.5597 0.786726 0.393363 0.919383i \(-0.371312\pi\)
0.393363 + 0.919383i \(0.371312\pi\)
\(752\) 0 0
\(753\) 9.57148 0.348804
\(754\) 0 0
\(755\) 7.88467 0.286952
\(756\) 0 0
\(757\) 35.2200 1.28009 0.640047 0.768336i \(-0.278915\pi\)
0.640047 + 0.768336i \(0.278915\pi\)
\(758\) 0 0
\(759\) 5.40788 0.196294
\(760\) 0 0
\(761\) −35.3427 −1.28117 −0.640585 0.767887i \(-0.721308\pi\)
−0.640585 + 0.767887i \(0.721308\pi\)
\(762\) 0 0
\(763\) 0.215120 0.00778788
\(764\) 0 0
\(765\) 2.17109 0.0784958
\(766\) 0 0
\(767\) 16.4915 0.595474
\(768\) 0 0
\(769\) 17.3197 0.624564 0.312282 0.949989i \(-0.398907\pi\)
0.312282 + 0.949989i \(0.398907\pi\)
\(770\) 0 0
\(771\) −15.1906 −0.547076
\(772\) 0 0
\(773\) −43.0548 −1.54857 −0.774287 0.632834i \(-0.781892\pi\)
−0.774287 + 0.632834i \(0.781892\pi\)
\(774\) 0 0
\(775\) −21.0138 −0.754837
\(776\) 0 0
\(777\) 7.49285 0.268804
\(778\) 0 0
\(779\) 23.7869 0.852256
\(780\) 0 0
\(781\) 5.30656 0.189883
\(782\) 0 0
\(783\) 1.00000 0.0357371
\(784\) 0 0
\(785\) 7.34511 0.262158
\(786\) 0 0
\(787\) −15.6413 −0.557552 −0.278776 0.960356i \(-0.589929\pi\)
−0.278776 + 0.960356i \(0.589929\pi\)
\(788\) 0 0
\(789\) 27.9622 0.995479
\(790\) 0 0
\(791\) −36.4018 −1.29430
\(792\) 0 0
\(793\) −14.9047 −0.529281
\(794\) 0 0
\(795\) 3.17185 0.112494
\(796\) 0 0
\(797\) 0.738555 0.0261610 0.0130805 0.999914i \(-0.495836\pi\)
0.0130805 + 0.999914i \(0.495836\pi\)
\(798\) 0 0
\(799\) 13.0059 0.460116
\(800\) 0 0
\(801\) −5.00216 −0.176743
\(802\) 0 0
\(803\) 36.9790 1.30496
\(804\) 0 0
\(805\) 1.13767 0.0400974
\(806\) 0 0
\(807\) 22.3628 0.787208
\(808\) 0 0
\(809\) 0.366427 0.0128829 0.00644145 0.999979i \(-0.497950\pi\)
0.00644145 + 0.999979i \(0.497950\pi\)
\(810\) 0 0
\(811\) 4.82115 0.169294 0.0846468 0.996411i \(-0.473024\pi\)
0.0846468 + 0.996411i \(0.473024\pi\)
\(812\) 0 0
\(813\) 2.45290 0.0860271
\(814\) 0 0
\(815\) −5.29012 −0.185305
\(816\) 0 0
\(817\) 14.3510 0.502079
\(818\) 0 0
\(819\) 5.91106 0.206549
\(820\) 0 0
\(821\) 47.3002 1.65079 0.825394 0.564558i \(-0.190953\pi\)
0.825394 + 0.564558i \(0.190953\pi\)
\(822\) 0 0
\(823\) −14.4791 −0.504709 −0.252354 0.967635i \(-0.581205\pi\)
−0.252354 + 0.967635i \(0.581205\pi\)
\(824\) 0 0
\(825\) −26.1143 −0.909184
\(826\) 0 0
\(827\) 34.3250 1.19360 0.596798 0.802392i \(-0.296439\pi\)
0.596798 + 0.802392i \(0.296439\pi\)
\(828\) 0 0
\(829\) 8.30962 0.288605 0.144302 0.989534i \(-0.453906\pi\)
0.144302 + 0.989534i \(0.453906\pi\)
\(830\) 0 0
\(831\) −1.25987 −0.0437044
\(832\) 0 0
\(833\) 2.97283 0.103002
\(834\) 0 0
\(835\) −2.50550 −0.0867063
\(836\) 0 0
\(837\) −4.35163 −0.150414
\(838\) 0 0
\(839\) −48.9839 −1.69111 −0.845556 0.533887i \(-0.820731\pi\)
−0.845556 + 0.533887i \(0.820731\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 0 0
\(843\) −16.9691 −0.584445
\(844\) 0 0
\(845\) −3.46676 −0.119260
\(846\) 0 0
\(847\) 50.1868 1.72444
\(848\) 0 0
\(849\) 13.9411 0.478459
\(850\) 0 0
\(851\) −2.72398 −0.0933770
\(852\) 0 0
\(853\) −21.7630 −0.745152 −0.372576 0.928002i \(-0.621525\pi\)
−0.372576 + 0.928002i \(0.621525\pi\)
\(854\) 0 0
\(855\) −1.63060 −0.0557652
\(856\) 0 0
\(857\) −9.99374 −0.341380 −0.170690 0.985325i \(-0.554600\pi\)
−0.170690 + 0.985325i \(0.554600\pi\)
\(858\) 0 0
\(859\) 51.9696 1.77318 0.886590 0.462557i \(-0.153068\pi\)
0.886590 + 0.462557i \(0.153068\pi\)
\(860\) 0 0
\(861\) 16.5961 0.565595
\(862\) 0 0
\(863\) 26.2719 0.894305 0.447152 0.894458i \(-0.352438\pi\)
0.447152 + 0.894458i \(0.352438\pi\)
\(864\) 0 0
\(865\) −4.40333 −0.149718
\(866\) 0 0
\(867\) −10.5556 −0.358487
\(868\) 0 0
\(869\) −16.3349 −0.554124
\(870\) 0 0
\(871\) 22.1180 0.749441
\(872\) 0 0
\(873\) −11.6816 −0.395362
\(874\) 0 0
\(875\) −11.1820 −0.378022
\(876\) 0 0
\(877\) 20.5178 0.692837 0.346418 0.938080i \(-0.387398\pi\)
0.346418 + 0.938080i \(0.387398\pi\)
\(878\) 0 0
\(879\) 22.0522 0.743804
\(880\) 0 0
\(881\) −30.1070 −1.01433 −0.507165 0.861849i \(-0.669306\pi\)
−0.507165 + 0.861849i \(0.669306\pi\)
\(882\) 0 0
\(883\) 39.3130 1.32299 0.661493 0.749951i \(-0.269923\pi\)
0.661493 + 0.749951i \(0.269923\pi\)
\(884\) 0 0
\(885\) −3.17402 −0.106693
\(886\) 0 0
\(887\) −56.9041 −1.91065 −0.955326 0.295555i \(-0.904495\pi\)
−0.955326 + 0.295555i \(0.904495\pi\)
\(888\) 0 0
\(889\) 12.3649 0.414706
\(890\) 0 0
\(891\) −5.40788 −0.181171
\(892\) 0 0
\(893\) −9.76811 −0.326877
\(894\) 0 0
\(895\) 0.837671 0.0280003
\(896\) 0 0
\(897\) −2.14893 −0.0717508
\(898\) 0 0
\(899\) −4.35163 −0.145135
\(900\) 0 0
\(901\) −40.2573 −1.34117
\(902\) 0 0
\(903\) 10.0127 0.333202
\(904\) 0 0
\(905\) 3.56300 0.118438
\(906\) 0 0
\(907\) 24.9777 0.829371 0.414685 0.909965i \(-0.363892\pi\)
0.414685 + 0.909965i \(0.363892\pi\)
\(908\) 0 0
\(909\) −10.7048 −0.355057
\(910\) 0 0
\(911\) 30.1327 0.998340 0.499170 0.866504i \(-0.333638\pi\)
0.499170 + 0.866504i \(0.333638\pi\)
\(912\) 0 0
\(913\) 4.50888 0.149222
\(914\) 0 0
\(915\) 2.86861 0.0948334
\(916\) 0 0
\(917\) −45.8267 −1.51333
\(918\) 0 0
\(919\) 29.8994 0.986291 0.493146 0.869947i \(-0.335847\pi\)
0.493146 + 0.869947i \(0.335847\pi\)
\(920\) 0 0
\(921\) −32.1760 −1.06024
\(922\) 0 0
\(923\) −2.10867 −0.0694078
\(924\) 0 0
\(925\) 13.1540 0.432500
\(926\) 0 0
\(927\) 5.40129 0.177401
\(928\) 0 0
\(929\) 33.0203 1.08336 0.541681 0.840584i \(-0.317788\pi\)
0.541681 + 0.840584i \(0.317788\pi\)
\(930\) 0 0
\(931\) −2.23274 −0.0731752
\(932\) 0 0
\(933\) −3.57851 −0.117155
\(934\) 0 0
\(935\) −11.7410 −0.383971
\(936\) 0 0
\(937\) −27.4957 −0.898245 −0.449122 0.893470i \(-0.648263\pi\)
−0.449122 + 0.893470i \(0.648263\pi\)
\(938\) 0 0
\(939\) 0.885822 0.0289077
\(940\) 0 0
\(941\) −6.16565 −0.200994 −0.100497 0.994937i \(-0.532043\pi\)
−0.100497 + 0.994937i \(0.532043\pi\)
\(942\) 0 0
\(943\) −6.03343 −0.196476
\(944\) 0 0
\(945\) −1.13767 −0.0370083
\(946\) 0 0
\(947\) 30.2342 0.982479 0.491239 0.871025i \(-0.336544\pi\)
0.491239 + 0.871025i \(0.336544\pi\)
\(948\) 0 0
\(949\) −14.6944 −0.477000
\(950\) 0 0
\(951\) −2.62770 −0.0852090
\(952\) 0 0
\(953\) −11.3177 −0.366616 −0.183308 0.983056i \(-0.558681\pi\)
−0.183308 + 0.983056i \(0.558681\pi\)
\(954\) 0 0
\(955\) −0.238643 −0.00772229
\(956\) 0 0
\(957\) −5.40788 −0.174812
\(958\) 0 0
\(959\) −6.25578 −0.202010
\(960\) 0 0
\(961\) −12.0633 −0.389138
\(962\) 0 0
\(963\) 6.56999 0.211715
\(964\) 0 0
\(965\) 1.20925 0.0389272
\(966\) 0 0
\(967\) 34.5548 1.11121 0.555603 0.831448i \(-0.312487\pi\)
0.555603 + 0.831448i \(0.312487\pi\)
\(968\) 0 0
\(969\) 20.6957 0.664840
\(970\) 0 0
\(971\) 17.7659 0.570135 0.285068 0.958507i \(-0.407984\pi\)
0.285068 + 0.958507i \(0.407984\pi\)
\(972\) 0 0
\(973\) −3.89755 −0.124950
\(974\) 0 0
\(975\) 10.3771 0.332333
\(976\) 0 0
\(977\) 39.9876 1.27932 0.639659 0.768659i \(-0.279076\pi\)
0.639659 + 0.768659i \(0.279076\pi\)
\(978\) 0 0
\(979\) 27.0511 0.864556
\(980\) 0 0
\(981\) 0.0782059 0.00249692
\(982\) 0 0
\(983\) 40.6221 1.29564 0.647822 0.761792i \(-0.275680\pi\)
0.647822 + 0.761792i \(0.275680\pi\)
\(984\) 0 0
\(985\) −8.45593 −0.269428
\(986\) 0 0
\(987\) −6.81520 −0.216930
\(988\) 0 0
\(989\) −3.64006 −0.115747
\(990\) 0 0
\(991\) 61.4622 1.95241 0.976205 0.216850i \(-0.0695782\pi\)
0.976205 + 0.216850i \(0.0695782\pi\)
\(992\) 0 0
\(993\) 9.42233 0.299009
\(994\) 0 0
\(995\) −0.430996 −0.0136635
\(996\) 0 0
\(997\) −16.5556 −0.524322 −0.262161 0.965024i \(-0.584435\pi\)
−0.262161 + 0.965024i \(0.584435\pi\)
\(998\) 0 0
\(999\) 2.72398 0.0861830
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 8004.2.a.g.1.8 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8004.2.a.g.1.8 12 1.1 even 1 trivial