Properties

Label 3200.2.a.bf.1.2
Level $3200$
Weight $2$
Character 3200.1
Self dual yes
Analytic conductor $25.552$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3200 = 2^{7} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3200.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(25.5521286468\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 640)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 3200.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.23607 q^{3} +3.23607 q^{7} -1.47214 q^{9} +O(q^{10})\) \(q+1.23607 q^{3} +3.23607 q^{7} -1.47214 q^{9} +2.00000 q^{11} -4.47214 q^{13} -4.47214 q^{17} +4.47214 q^{19} +4.00000 q^{21} +4.76393 q^{23} -5.52786 q^{27} +2.00000 q^{29} +6.47214 q^{31} +2.47214 q^{33} +6.94427 q^{37} -5.52786 q^{39} +12.4721 q^{41} -7.70820 q^{43} +7.23607 q^{47} +3.47214 q^{49} -5.52786 q^{51} +0.472136 q^{53} +5.52786 q^{57} +8.47214 q^{59} +6.00000 q^{61} -4.76393 q^{63} +7.70820 q^{67} +5.88854 q^{69} -2.47214 q^{71} -4.47214 q^{73} +6.47214 q^{77} +12.9443 q^{79} -2.41641 q^{81} -3.70820 q^{83} +2.47214 q^{87} -14.9443 q^{89} -14.4721 q^{91} +8.00000 q^{93} +16.4721 q^{97} -2.94427 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 2 q^{7} + 6 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{3} + 2 q^{7} + 6 q^{9} + 4 q^{11} + 8 q^{21} + 14 q^{23} - 20 q^{27} + 4 q^{29} + 4 q^{31} - 4 q^{33} - 4 q^{37} - 20 q^{39} + 16 q^{41} - 2 q^{43} + 10 q^{47} - 2 q^{49} - 20 q^{51} - 8 q^{53} + 20 q^{57} + 8 q^{59} + 12 q^{61} - 14 q^{63} + 2 q^{67} - 24 q^{69} + 4 q^{71} + 4 q^{77} + 8 q^{79} + 22 q^{81} + 6 q^{83} - 4 q^{87} - 12 q^{89} - 20 q^{91} + 16 q^{93} + 24 q^{97} + 12 q^{99} + O(q^{100}) \)

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
\(3\) 1.23607 0.713644 0.356822 0.934172i \(-0.383860\pi\)
0.356822 + 0.934172i \(0.383860\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 3.23607 1.22312 0.611559 0.791199i \(-0.290543\pi\)
0.611559 + 0.791199i \(0.290543\pi\)
\(8\) 0 0
\(9\) −1.47214 −0.490712
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 0 0
\(13\) −4.47214 −1.24035 −0.620174 0.784465i \(-0.712938\pi\)
−0.620174 + 0.784465i \(0.712938\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −4.47214 −1.08465 −0.542326 0.840168i \(-0.682456\pi\)
−0.542326 + 0.840168i \(0.682456\pi\)
\(18\) 0 0
\(19\) 4.47214 1.02598 0.512989 0.858395i \(-0.328538\pi\)
0.512989 + 0.858395i \(0.328538\pi\)
\(20\) 0 0
\(21\) 4.00000 0.872872
\(22\) 0 0
\(23\) 4.76393 0.993348 0.496674 0.867937i \(-0.334554\pi\)
0.496674 + 0.867937i \(0.334554\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −5.52786 −1.06384
\(28\) 0 0
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) 6.47214 1.16243 0.581215 0.813750i \(-0.302578\pi\)
0.581215 + 0.813750i \(0.302578\pi\)
\(32\) 0 0
\(33\) 2.47214 0.430344
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 6.94427 1.14163 0.570816 0.821078i \(-0.306627\pi\)
0.570816 + 0.821078i \(0.306627\pi\)
\(38\) 0 0
\(39\) −5.52786 −0.885167
\(40\) 0 0
\(41\) 12.4721 1.94782 0.973910 0.226934i \(-0.0728701\pi\)
0.973910 + 0.226934i \(0.0728701\pi\)
\(42\) 0 0
\(43\) −7.70820 −1.17549 −0.587745 0.809046i \(-0.699984\pi\)
−0.587745 + 0.809046i \(0.699984\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 7.23607 1.05549 0.527744 0.849403i \(-0.323038\pi\)
0.527744 + 0.849403i \(0.323038\pi\)
\(48\) 0 0
\(49\) 3.47214 0.496019
\(50\) 0 0
\(51\) −5.52786 −0.774056
\(52\) 0 0
\(53\) 0.472136 0.0648529 0.0324264 0.999474i \(-0.489677\pi\)
0.0324264 + 0.999474i \(0.489677\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 5.52786 0.732183
\(58\) 0 0
\(59\) 8.47214 1.10298 0.551489 0.834182i \(-0.314060\pi\)
0.551489 + 0.834182i \(0.314060\pi\)
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 0 0
\(63\) −4.76393 −0.600199
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 7.70820 0.941707 0.470853 0.882211i \(-0.343946\pi\)
0.470853 + 0.882211i \(0.343946\pi\)
\(68\) 0 0
\(69\) 5.88854 0.708897
\(70\) 0 0
\(71\) −2.47214 −0.293389 −0.146694 0.989182i \(-0.546863\pi\)
−0.146694 + 0.989182i \(0.546863\pi\)
\(72\) 0 0
\(73\) −4.47214 −0.523424 −0.261712 0.965146i \(-0.584287\pi\)
−0.261712 + 0.965146i \(0.584287\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 6.47214 0.737568
\(78\) 0 0
\(79\) 12.9443 1.45634 0.728172 0.685394i \(-0.240370\pi\)
0.728172 + 0.685394i \(0.240370\pi\)
\(80\) 0 0
\(81\) −2.41641 −0.268490
\(82\) 0 0
\(83\) −3.70820 −0.407028 −0.203514 0.979072i \(-0.565236\pi\)
−0.203514 + 0.979072i \(0.565236\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 2.47214 0.265041
\(88\) 0 0
\(89\) −14.9443 −1.58409 −0.792045 0.610463i \(-0.790983\pi\)
−0.792045 + 0.610463i \(0.790983\pi\)
\(90\) 0 0
\(91\) −14.4721 −1.51709
\(92\) 0 0
\(93\) 8.00000 0.829561
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 16.4721 1.67249 0.836246 0.548354i \(-0.184746\pi\)
0.836246 + 0.548354i \(0.184746\pi\)
\(98\) 0 0
\(99\) −2.94427 −0.295910
\(100\) 0 0
\(101\) 10.0000 0.995037 0.497519 0.867453i \(-0.334245\pi\)
0.497519 + 0.867453i \(0.334245\pi\)
\(102\) 0 0
\(103\) −3.23607 −0.318859 −0.159430 0.987209i \(-0.550966\pi\)
−0.159430 + 0.987209i \(0.550966\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −17.2361 −1.66627 −0.833137 0.553067i \(-0.813457\pi\)
−0.833137 + 0.553067i \(0.813457\pi\)
\(108\) 0 0
\(109\) −14.9443 −1.43140 −0.715701 0.698407i \(-0.753893\pi\)
−0.715701 + 0.698407i \(0.753893\pi\)
\(110\) 0 0
\(111\) 8.58359 0.814719
\(112\) 0 0
\(113\) 2.94427 0.276974 0.138487 0.990364i \(-0.455776\pi\)
0.138487 + 0.990364i \(0.455776\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 6.58359 0.608653
\(118\) 0 0
\(119\) −14.4721 −1.32666
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) 15.4164 1.39005
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 20.1803 1.79072 0.895358 0.445348i \(-0.146920\pi\)
0.895358 + 0.445348i \(0.146920\pi\)
\(128\) 0 0
\(129\) −9.52786 −0.838882
\(130\) 0 0
\(131\) −14.9443 −1.30569 −0.652844 0.757493i \(-0.726424\pi\)
−0.652844 + 0.757493i \(0.726424\pi\)
\(132\) 0 0
\(133\) 14.4721 1.25489
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −2.00000 −0.170872 −0.0854358 0.996344i \(-0.527228\pi\)
−0.0854358 + 0.996344i \(0.527228\pi\)
\(138\) 0 0
\(139\) −20.4721 −1.73642 −0.868212 0.496194i \(-0.834731\pi\)
−0.868212 + 0.496194i \(0.834731\pi\)
\(140\) 0 0
\(141\) 8.94427 0.753244
\(142\) 0 0
\(143\) −8.94427 −0.747958
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 4.29180 0.353981
\(148\) 0 0
\(149\) −6.94427 −0.568897 −0.284448 0.958691i \(-0.591810\pi\)
−0.284448 + 0.958691i \(0.591810\pi\)
\(150\) 0 0
\(151\) 23.4164 1.90560 0.952800 0.303598i \(-0.0981881\pi\)
0.952800 + 0.303598i \(0.0981881\pi\)
\(152\) 0 0
\(153\) 6.58359 0.532252
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 5.05573 0.403491 0.201746 0.979438i \(-0.435339\pi\)
0.201746 + 0.979438i \(0.435339\pi\)
\(158\) 0 0
\(159\) 0.583592 0.0462819
\(160\) 0 0
\(161\) 15.4164 1.21498
\(162\) 0 0
\(163\) −11.7082 −0.917057 −0.458529 0.888680i \(-0.651623\pi\)
−0.458529 + 0.888680i \(0.651623\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −14.6525 −1.13384 −0.566921 0.823772i \(-0.691866\pi\)
−0.566921 + 0.823772i \(0.691866\pi\)
\(168\) 0 0
\(169\) 7.00000 0.538462
\(170\) 0 0
\(171\) −6.58359 −0.503460
\(172\) 0 0
\(173\) −10.9443 −0.832078 −0.416039 0.909347i \(-0.636582\pi\)
−0.416039 + 0.909347i \(0.636582\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 10.4721 0.787134
\(178\) 0 0
\(179\) 20.4721 1.53016 0.765080 0.643936i \(-0.222700\pi\)
0.765080 + 0.643936i \(0.222700\pi\)
\(180\) 0 0
\(181\) −10.9443 −0.813481 −0.406741 0.913544i \(-0.633335\pi\)
−0.406741 + 0.913544i \(0.633335\pi\)
\(182\) 0 0
\(183\) 7.41641 0.548237
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −8.94427 −0.654070
\(188\) 0 0
\(189\) −17.8885 −1.30120
\(190\) 0 0
\(191\) 11.4164 0.826062 0.413031 0.910717i \(-0.364470\pi\)
0.413031 + 0.910717i \(0.364470\pi\)
\(192\) 0 0
\(193\) 11.5279 0.829794 0.414897 0.909868i \(-0.363818\pi\)
0.414897 + 0.909868i \(0.363818\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0.472136 0.0336383 0.0168191 0.999859i \(-0.494646\pi\)
0.0168191 + 0.999859i \(0.494646\pi\)
\(198\) 0 0
\(199\) −0.944272 −0.0669377 −0.0334688 0.999440i \(-0.510655\pi\)
−0.0334688 + 0.999440i \(0.510655\pi\)
\(200\) 0 0
\(201\) 9.52786 0.672044
\(202\) 0 0
\(203\) 6.47214 0.454255
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −7.01316 −0.487448
\(208\) 0 0
\(209\) 8.94427 0.618688
\(210\) 0 0
\(211\) 1.05573 0.0726793 0.0363397 0.999339i \(-0.488430\pi\)
0.0363397 + 0.999339i \(0.488430\pi\)
\(212\) 0 0
\(213\) −3.05573 −0.209375
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 20.9443 1.42179
\(218\) 0 0
\(219\) −5.52786 −0.373538
\(220\) 0 0
\(221\) 20.0000 1.34535
\(222\) 0 0
\(223\) 8.76393 0.586876 0.293438 0.955978i \(-0.405201\pi\)
0.293438 + 0.955978i \(0.405201\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −10.1803 −0.675693 −0.337846 0.941201i \(-0.609698\pi\)
−0.337846 + 0.941201i \(0.609698\pi\)
\(228\) 0 0
\(229\) −2.94427 −0.194563 −0.0972815 0.995257i \(-0.531015\pi\)
−0.0972815 + 0.995257i \(0.531015\pi\)
\(230\) 0 0
\(231\) 8.00000 0.526361
\(232\) 0 0
\(233\) −15.5279 −1.01726 −0.508632 0.860984i \(-0.669849\pi\)
−0.508632 + 0.860984i \(0.669849\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 16.0000 1.03931
\(238\) 0 0
\(239\) −12.9443 −0.837295 −0.418648 0.908149i \(-0.637496\pi\)
−0.418648 + 0.908149i \(0.637496\pi\)
\(240\) 0 0
\(241\) 26.3607 1.69804 0.849020 0.528360i \(-0.177193\pi\)
0.849020 + 0.528360i \(0.177193\pi\)
\(242\) 0 0
\(243\) 13.5967 0.872232
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −20.0000 −1.27257
\(248\) 0 0
\(249\) −4.58359 −0.290473
\(250\) 0 0
\(251\) 2.00000 0.126239 0.0631194 0.998006i \(-0.479895\pi\)
0.0631194 + 0.998006i \(0.479895\pi\)
\(252\) 0 0
\(253\) 9.52786 0.599012
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 2.94427 0.183659 0.0918293 0.995775i \(-0.470729\pi\)
0.0918293 + 0.995775i \(0.470729\pi\)
\(258\) 0 0
\(259\) 22.4721 1.39635
\(260\) 0 0
\(261\) −2.94427 −0.182246
\(262\) 0 0
\(263\) 17.7082 1.09193 0.545967 0.837806i \(-0.316162\pi\)
0.545967 + 0.837806i \(0.316162\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −18.4721 −1.13048
\(268\) 0 0
\(269\) 23.8885 1.45651 0.728255 0.685306i \(-0.240332\pi\)
0.728255 + 0.685306i \(0.240332\pi\)
\(270\) 0 0
\(271\) −24.3607 −1.47981 −0.739903 0.672714i \(-0.765129\pi\)
−0.739903 + 0.672714i \(0.765129\pi\)
\(272\) 0 0
\(273\) −17.8885 −1.08266
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −10.9443 −0.657578 −0.328789 0.944403i \(-0.606640\pi\)
−0.328789 + 0.944403i \(0.606640\pi\)
\(278\) 0 0
\(279\) −9.52786 −0.570418
\(280\) 0 0
\(281\) 3.52786 0.210455 0.105227 0.994448i \(-0.466443\pi\)
0.105227 + 0.994448i \(0.466443\pi\)
\(282\) 0 0
\(283\) 8.29180 0.492896 0.246448 0.969156i \(-0.420737\pi\)
0.246448 + 0.969156i \(0.420737\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 40.3607 2.38242
\(288\) 0 0
\(289\) 3.00000 0.176471
\(290\) 0 0
\(291\) 20.3607 1.19356
\(292\) 0 0
\(293\) −23.8885 −1.39558 −0.697792 0.716301i \(-0.745834\pi\)
−0.697792 + 0.716301i \(0.745834\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −11.0557 −0.641518
\(298\) 0 0
\(299\) −21.3050 −1.23210
\(300\) 0 0
\(301\) −24.9443 −1.43776
\(302\) 0 0
\(303\) 12.3607 0.710102
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −0.291796 −0.0166537 −0.00832684 0.999965i \(-0.502651\pi\)
−0.00832684 + 0.999965i \(0.502651\pi\)
\(308\) 0 0
\(309\) −4.00000 −0.227552
\(310\) 0 0
\(311\) −15.4164 −0.874184 −0.437092 0.899417i \(-0.643992\pi\)
−0.437092 + 0.899417i \(0.643992\pi\)
\(312\) 0 0
\(313\) 28.8328 1.62973 0.814864 0.579653i \(-0.196812\pi\)
0.814864 + 0.579653i \(0.196812\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −20.4721 −1.14983 −0.574915 0.818213i \(-0.694965\pi\)
−0.574915 + 0.818213i \(0.694965\pi\)
\(318\) 0 0
\(319\) 4.00000 0.223957
\(320\) 0 0
\(321\) −21.3050 −1.18913
\(322\) 0 0
\(323\) −20.0000 −1.11283
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −18.4721 −1.02151
\(328\) 0 0
\(329\) 23.4164 1.29099
\(330\) 0 0
\(331\) −15.8885 −0.873313 −0.436657 0.899628i \(-0.643838\pi\)
−0.436657 + 0.899628i \(0.643838\pi\)
\(332\) 0 0
\(333\) −10.2229 −0.560212
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −5.05573 −0.275403 −0.137702 0.990474i \(-0.543971\pi\)
−0.137702 + 0.990474i \(0.543971\pi\)
\(338\) 0 0
\(339\) 3.63932 0.197661
\(340\) 0 0
\(341\) 12.9443 0.700972
\(342\) 0 0
\(343\) −11.4164 −0.616428
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −17.2361 −0.925281 −0.462640 0.886546i \(-0.653098\pi\)
−0.462640 + 0.886546i \(0.653098\pi\)
\(348\) 0 0
\(349\) 14.9443 0.799949 0.399974 0.916526i \(-0.369019\pi\)
0.399974 + 0.916526i \(0.369019\pi\)
\(350\) 0 0
\(351\) 24.7214 1.31953
\(352\) 0 0
\(353\) −5.05573 −0.269089 −0.134545 0.990908i \(-0.542957\pi\)
−0.134545 + 0.990908i \(0.542957\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −17.8885 −0.946762
\(358\) 0 0
\(359\) −15.0557 −0.794611 −0.397305 0.917686i \(-0.630055\pi\)
−0.397305 + 0.917686i \(0.630055\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −8.65248 −0.454137
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 18.2918 0.954824 0.477412 0.878680i \(-0.341575\pi\)
0.477412 + 0.878680i \(0.341575\pi\)
\(368\) 0 0
\(369\) −18.3607 −0.955819
\(370\) 0 0
\(371\) 1.52786 0.0793227
\(372\) 0 0
\(373\) 5.05573 0.261776 0.130888 0.991397i \(-0.458217\pi\)
0.130888 + 0.991397i \(0.458217\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −8.94427 −0.460653
\(378\) 0 0
\(379\) −15.5279 −0.797613 −0.398806 0.917035i \(-0.630575\pi\)
−0.398806 + 0.917035i \(0.630575\pi\)
\(380\) 0 0
\(381\) 24.9443 1.27793
\(382\) 0 0
\(383\) −10.2918 −0.525886 −0.262943 0.964811i \(-0.584693\pi\)
−0.262943 + 0.964811i \(0.584693\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 11.3475 0.576827
\(388\) 0 0
\(389\) −11.8885 −0.602773 −0.301387 0.953502i \(-0.597449\pi\)
−0.301387 + 0.953502i \(0.597449\pi\)
\(390\) 0 0
\(391\) −21.3050 −1.07744
\(392\) 0 0
\(393\) −18.4721 −0.931796
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −12.4721 −0.625959 −0.312979 0.949760i \(-0.601327\pi\)
−0.312979 + 0.949760i \(0.601327\pi\)
\(398\) 0 0
\(399\) 17.8885 0.895547
\(400\) 0 0
\(401\) −2.00000 −0.0998752 −0.0499376 0.998752i \(-0.515902\pi\)
−0.0499376 + 0.998752i \(0.515902\pi\)
\(402\) 0 0
\(403\) −28.9443 −1.44182
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 13.8885 0.688430
\(408\) 0 0
\(409\) 17.4164 0.861186 0.430593 0.902546i \(-0.358304\pi\)
0.430593 + 0.902546i \(0.358304\pi\)
\(410\) 0 0
\(411\) −2.47214 −0.121941
\(412\) 0 0
\(413\) 27.4164 1.34907
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −25.3050 −1.23919
\(418\) 0 0
\(419\) 4.47214 0.218478 0.109239 0.994016i \(-0.465159\pi\)
0.109239 + 0.994016i \(0.465159\pi\)
\(420\) 0 0
\(421\) 28.8328 1.40523 0.702613 0.711572i \(-0.252017\pi\)
0.702613 + 0.711572i \(0.252017\pi\)
\(422\) 0 0
\(423\) −10.6525 −0.517941
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 19.4164 0.939626
\(428\) 0 0
\(429\) −11.0557 −0.533776
\(430\) 0 0
\(431\) −24.3607 −1.17341 −0.586706 0.809800i \(-0.699576\pi\)
−0.586706 + 0.809800i \(0.699576\pi\)
\(432\) 0 0
\(433\) 0.472136 0.0226894 0.0113447 0.999936i \(-0.496389\pi\)
0.0113447 + 0.999936i \(0.496389\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 21.3050 1.01915
\(438\) 0 0
\(439\) 15.0557 0.718571 0.359285 0.933228i \(-0.383020\pi\)
0.359285 + 0.933228i \(0.383020\pi\)
\(440\) 0 0
\(441\) −5.11146 −0.243403
\(442\) 0 0
\(443\) −4.65248 −0.221046 −0.110523 0.993874i \(-0.535253\pi\)
−0.110523 + 0.993874i \(0.535253\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −8.58359 −0.405990
\(448\) 0 0
\(449\) −1.41641 −0.0668444 −0.0334222 0.999441i \(-0.510641\pi\)
−0.0334222 + 0.999441i \(0.510641\pi\)
\(450\) 0 0
\(451\) 24.9443 1.17458
\(452\) 0 0
\(453\) 28.9443 1.35992
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −3.88854 −0.181898 −0.0909492 0.995856i \(-0.528990\pi\)
−0.0909492 + 0.995856i \(0.528990\pi\)
\(458\) 0 0
\(459\) 24.7214 1.15389
\(460\) 0 0
\(461\) −41.7771 −1.94575 −0.972876 0.231325i \(-0.925694\pi\)
−0.972876 + 0.231325i \(0.925694\pi\)
\(462\) 0 0
\(463\) −1.12461 −0.0522651 −0.0261326 0.999658i \(-0.508319\pi\)
−0.0261326 + 0.999658i \(0.508319\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 41.5967 1.92487 0.962434 0.271516i \(-0.0875249\pi\)
0.962434 + 0.271516i \(0.0875249\pi\)
\(468\) 0 0
\(469\) 24.9443 1.15182
\(470\) 0 0
\(471\) 6.24922 0.287949
\(472\) 0 0
\(473\) −15.4164 −0.708847
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −0.695048 −0.0318241
\(478\) 0 0
\(479\) 12.9443 0.591439 0.295719 0.955275i \(-0.404441\pi\)
0.295719 + 0.955275i \(0.404441\pi\)
\(480\) 0 0
\(481\) −31.0557 −1.41602
\(482\) 0 0
\(483\) 19.0557 0.867066
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −12.7639 −0.578389 −0.289194 0.957270i \(-0.593387\pi\)
−0.289194 + 0.957270i \(0.593387\pi\)
\(488\) 0 0
\(489\) −14.4721 −0.654453
\(490\) 0 0
\(491\) 21.0557 0.950232 0.475116 0.879923i \(-0.342406\pi\)
0.475116 + 0.879923i \(0.342406\pi\)
\(492\) 0 0
\(493\) −8.94427 −0.402830
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −8.00000 −0.358849
\(498\) 0 0
\(499\) −14.5836 −0.652851 −0.326426 0.945223i \(-0.605844\pi\)
−0.326426 + 0.945223i \(0.605844\pi\)
\(500\) 0 0
\(501\) −18.1115 −0.809160
\(502\) 0 0
\(503\) −5.12461 −0.228495 −0.114248 0.993452i \(-0.536446\pi\)
−0.114248 + 0.993452i \(0.536446\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 8.65248 0.384270
\(508\) 0 0
\(509\) 16.8328 0.746101 0.373051 0.927811i \(-0.378312\pi\)
0.373051 + 0.927811i \(0.378312\pi\)
\(510\) 0 0
\(511\) −14.4721 −0.640210
\(512\) 0 0
\(513\) −24.7214 −1.09147
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 14.4721 0.636484
\(518\) 0 0
\(519\) −13.5279 −0.593807
\(520\) 0 0
\(521\) −15.8885 −0.696090 −0.348045 0.937478i \(-0.613154\pi\)
−0.348045 + 0.937478i \(0.613154\pi\)
\(522\) 0 0
\(523\) 20.0689 0.877551 0.438776 0.898597i \(-0.355412\pi\)
0.438776 + 0.898597i \(0.355412\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −28.9443 −1.26083
\(528\) 0 0
\(529\) −0.304952 −0.0132588
\(530\) 0 0
\(531\) −12.4721 −0.541245
\(532\) 0 0
\(533\) −55.7771 −2.41597
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 25.3050 1.09199
\(538\) 0 0
\(539\) 6.94427 0.299111
\(540\) 0 0
\(541\) −23.8885 −1.02705 −0.513524 0.858075i \(-0.671660\pi\)
−0.513524 + 0.858075i \(0.671660\pi\)
\(542\) 0 0
\(543\) −13.5279 −0.580536
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 20.6525 0.883036 0.441518 0.897252i \(-0.354440\pi\)
0.441518 + 0.897252i \(0.354440\pi\)
\(548\) 0 0
\(549\) −8.83282 −0.376975
\(550\) 0 0
\(551\) 8.94427 0.381039
\(552\) 0 0
\(553\) 41.8885 1.78128
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 13.0557 0.553189 0.276594 0.960987i \(-0.410794\pi\)
0.276594 + 0.960987i \(0.410794\pi\)
\(558\) 0 0
\(559\) 34.4721 1.45802
\(560\) 0 0
\(561\) −11.0557 −0.466773
\(562\) 0 0
\(563\) 4.29180 0.180878 0.0904388 0.995902i \(-0.471173\pi\)
0.0904388 + 0.995902i \(0.471173\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −7.81966 −0.328395
\(568\) 0 0
\(569\) −34.3607 −1.44047 −0.720237 0.693728i \(-0.755967\pi\)
−0.720237 + 0.693728i \(0.755967\pi\)
\(570\) 0 0
\(571\) 16.8328 0.704431 0.352216 0.935919i \(-0.385428\pi\)
0.352216 + 0.935919i \(0.385428\pi\)
\(572\) 0 0
\(573\) 14.1115 0.589515
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −19.8885 −0.827971 −0.413985 0.910283i \(-0.635864\pi\)
−0.413985 + 0.910283i \(0.635864\pi\)
\(578\) 0 0
\(579\) 14.2492 0.592178
\(580\) 0 0
\(581\) −12.0000 −0.497844
\(582\) 0 0
\(583\) 0.944272 0.0391077
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 8.65248 0.357126 0.178563 0.983928i \(-0.442855\pi\)
0.178563 + 0.983928i \(0.442855\pi\)
\(588\) 0 0
\(589\) 28.9443 1.19263
\(590\) 0 0
\(591\) 0.583592 0.0240058
\(592\) 0 0
\(593\) 14.0000 0.574911 0.287456 0.957794i \(-0.407191\pi\)
0.287456 + 0.957794i \(0.407191\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −1.16718 −0.0477697
\(598\) 0 0
\(599\) −21.8885 −0.894342 −0.447171 0.894449i \(-0.647568\pi\)
−0.447171 + 0.894449i \(0.647568\pi\)
\(600\) 0 0
\(601\) −22.3607 −0.912111 −0.456056 0.889951i \(-0.650738\pi\)
−0.456056 + 0.889951i \(0.650738\pi\)
\(602\) 0 0
\(603\) −11.3475 −0.462107
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −6.87539 −0.279063 −0.139532 0.990218i \(-0.544560\pi\)
−0.139532 + 0.990218i \(0.544560\pi\)
\(608\) 0 0
\(609\) 8.00000 0.324176
\(610\) 0 0
\(611\) −32.3607 −1.30917
\(612\) 0 0
\(613\) 19.5279 0.788723 0.394362 0.918955i \(-0.370966\pi\)
0.394362 + 0.918955i \(0.370966\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −25.4164 −1.02323 −0.511613 0.859216i \(-0.670952\pi\)
−0.511613 + 0.859216i \(0.670952\pi\)
\(618\) 0 0
\(619\) −20.4721 −0.822845 −0.411422 0.911445i \(-0.634968\pi\)
−0.411422 + 0.911445i \(0.634968\pi\)
\(620\) 0 0
\(621\) −26.3344 −1.05676
\(622\) 0 0
\(623\) −48.3607 −1.93753
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 11.0557 0.441523
\(628\) 0 0
\(629\) −31.0557 −1.23827
\(630\) 0 0
\(631\) −34.4721 −1.37231 −0.686157 0.727453i \(-0.740704\pi\)
−0.686157 + 0.727453i \(0.740704\pi\)
\(632\) 0 0
\(633\) 1.30495 0.0518672
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −15.5279 −0.615236
\(638\) 0 0
\(639\) 3.63932 0.143969
\(640\) 0 0
\(641\) 0.472136 0.0186482 0.00932412 0.999957i \(-0.497032\pi\)
0.00932412 + 0.999957i \(0.497032\pi\)
\(642\) 0 0
\(643\) 22.1803 0.874707 0.437354 0.899290i \(-0.355916\pi\)
0.437354 + 0.899290i \(0.355916\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −12.7639 −0.501802 −0.250901 0.968013i \(-0.580727\pi\)
−0.250901 + 0.968013i \(0.580727\pi\)
\(648\) 0 0
\(649\) 16.9443 0.665121
\(650\) 0 0
\(651\) 25.8885 1.01465
\(652\) 0 0
\(653\) −49.4164 −1.93381 −0.966907 0.255130i \(-0.917882\pi\)
−0.966907 + 0.255130i \(0.917882\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 6.58359 0.256850
\(658\) 0 0
\(659\) −21.4164 −0.834265 −0.417132 0.908846i \(-0.636965\pi\)
−0.417132 + 0.908846i \(0.636965\pi\)
\(660\) 0 0
\(661\) −35.8885 −1.39590 −0.697951 0.716145i \(-0.745905\pi\)
−0.697951 + 0.716145i \(0.745905\pi\)
\(662\) 0 0
\(663\) 24.7214 0.960098
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 9.52786 0.368920
\(668\) 0 0
\(669\) 10.8328 0.418821
\(670\) 0 0
\(671\) 12.0000 0.463255
\(672\) 0 0
\(673\) 23.3050 0.898340 0.449170 0.893446i \(-0.351720\pi\)
0.449170 + 0.893446i \(0.351720\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −30.3607 −1.16686 −0.583428 0.812165i \(-0.698289\pi\)
−0.583428 + 0.812165i \(0.698289\pi\)
\(678\) 0 0
\(679\) 53.3050 2.04566
\(680\) 0 0
\(681\) −12.5836 −0.482204
\(682\) 0 0
\(683\) 24.2918 0.929500 0.464750 0.885442i \(-0.346144\pi\)
0.464750 + 0.885442i \(0.346144\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −3.63932 −0.138849
\(688\) 0 0
\(689\) −2.11146 −0.0804401
\(690\) 0 0
\(691\) 30.0000 1.14125 0.570627 0.821209i \(-0.306700\pi\)
0.570627 + 0.821209i \(0.306700\pi\)
\(692\) 0 0
\(693\) −9.52786 −0.361934
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −55.7771 −2.11271
\(698\) 0 0
\(699\) −19.1935 −0.725965
\(700\) 0 0
\(701\) 9.05573 0.342030 0.171015 0.985268i \(-0.445295\pi\)
0.171015 + 0.985268i \(0.445295\pi\)
\(702\) 0 0
\(703\) 31.0557 1.17129
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 32.3607 1.21705
\(708\) 0 0
\(709\) 18.0000 0.676004 0.338002 0.941145i \(-0.390249\pi\)
0.338002 + 0.941145i \(0.390249\pi\)
\(710\) 0 0
\(711\) −19.0557 −0.714646
\(712\) 0 0
\(713\) 30.8328 1.15470
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −16.0000 −0.597531
\(718\) 0 0
\(719\) 22.8328 0.851520 0.425760 0.904836i \(-0.360007\pi\)
0.425760 + 0.904836i \(0.360007\pi\)
\(720\) 0 0
\(721\) −10.4721 −0.390003
\(722\) 0 0
\(723\) 32.5836 1.21180
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −1.70820 −0.0633538 −0.0316769 0.999498i \(-0.510085\pi\)
−0.0316769 + 0.999498i \(0.510085\pi\)
\(728\) 0 0
\(729\) 24.0557 0.890953
\(730\) 0 0
\(731\) 34.4721 1.27500
\(732\) 0 0
\(733\) 51.8885 1.91655 0.958274 0.285853i \(-0.0922768\pi\)
0.958274 + 0.285853i \(0.0922768\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 15.4164 0.567871
\(738\) 0 0
\(739\) −49.1935 −1.80961 −0.904806 0.425824i \(-0.859984\pi\)
−0.904806 + 0.425824i \(0.859984\pi\)
\(740\) 0 0
\(741\) −24.7214 −0.908162
\(742\) 0 0
\(743\) 30.6525 1.12453 0.562265 0.826957i \(-0.309930\pi\)
0.562265 + 0.826957i \(0.309930\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 5.45898 0.199734
\(748\) 0 0
\(749\) −55.7771 −2.03805
\(750\) 0 0
\(751\) 16.3607 0.597010 0.298505 0.954408i \(-0.403512\pi\)
0.298505 + 0.954408i \(0.403512\pi\)
\(752\) 0 0
\(753\) 2.47214 0.0900896
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 19.8885 0.722861 0.361431 0.932399i \(-0.382288\pi\)
0.361431 + 0.932399i \(0.382288\pi\)
\(758\) 0 0
\(759\) 11.7771 0.427481
\(760\) 0 0
\(761\) 3.88854 0.140960 0.0704798 0.997513i \(-0.477547\pi\)
0.0704798 + 0.997513i \(0.477547\pi\)
\(762\) 0 0
\(763\) −48.3607 −1.75077
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −37.8885 −1.36808
\(768\) 0 0
\(769\) −14.9443 −0.538904 −0.269452 0.963014i \(-0.586843\pi\)
−0.269452 + 0.963014i \(0.586843\pi\)
\(770\) 0 0
\(771\) 3.63932 0.131067
\(772\) 0 0
\(773\) 18.3607 0.660388 0.330194 0.943913i \(-0.392886\pi\)
0.330194 + 0.943913i \(0.392886\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 27.7771 0.996497
\(778\) 0 0
\(779\) 55.7771 1.99842
\(780\) 0 0
\(781\) −4.94427 −0.176920
\(782\) 0 0
\(783\) −11.0557 −0.395099
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −44.0689 −1.57089 −0.785443 0.618934i \(-0.787565\pi\)
−0.785443 + 0.618934i \(0.787565\pi\)
\(788\) 0 0
\(789\) 21.8885 0.779253
\(790\) 0 0
\(791\) 9.52786 0.338772
\(792\) 0 0
\(793\) −26.8328 −0.952861
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −25.4164 −0.900295 −0.450148 0.892954i \(-0.648629\pi\)
−0.450148 + 0.892954i \(0.648629\pi\)
\(798\) 0 0
\(799\) −32.3607 −1.14484
\(800\) 0 0
\(801\) 22.0000 0.777332
\(802\) 0 0
\(803\) −8.94427 −0.315637
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 29.5279 1.03943
\(808\) 0 0
\(809\) −12.1115 −0.425816 −0.212908 0.977072i \(-0.568293\pi\)
−0.212908 + 0.977072i \(0.568293\pi\)
\(810\) 0 0
\(811\) 2.00000 0.0702295 0.0351147 0.999383i \(-0.488820\pi\)
0.0351147 + 0.999383i \(0.488820\pi\)
\(812\) 0 0
\(813\) −30.1115 −1.05605
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −34.4721 −1.20603
\(818\) 0 0
\(819\) 21.3050 0.744455
\(820\) 0 0
\(821\) −21.0557 −0.734850 −0.367425 0.930053i \(-0.619761\pi\)
−0.367425 + 0.930053i \(0.619761\pi\)
\(822\) 0 0
\(823\) 1.70820 0.0595442 0.0297721 0.999557i \(-0.490522\pi\)
0.0297721 + 0.999557i \(0.490522\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −6.18034 −0.214911 −0.107456 0.994210i \(-0.534270\pi\)
−0.107456 + 0.994210i \(0.534270\pi\)
\(828\) 0 0
\(829\) 38.0000 1.31979 0.659897 0.751356i \(-0.270600\pi\)
0.659897 + 0.751356i \(0.270600\pi\)
\(830\) 0 0
\(831\) −13.5279 −0.469276
\(832\) 0 0
\(833\) −15.5279 −0.538009
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −35.7771 −1.23664
\(838\) 0 0
\(839\) 4.00000 0.138095 0.0690477 0.997613i \(-0.478004\pi\)
0.0690477 + 0.997613i \(0.478004\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 0 0
\(843\) 4.36068 0.150190
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −22.6525 −0.778348
\(848\) 0 0
\(849\) 10.2492 0.351752
\(850\) 0 0
\(851\) 33.0820 1.13404
\(852\) 0 0
\(853\) −7.52786 −0.257749 −0.128875 0.991661i \(-0.541136\pi\)
−0.128875 + 0.991661i \(0.541136\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −24.8328 −0.848273 −0.424136 0.905598i \(-0.639422\pi\)
−0.424136 + 0.905598i \(0.639422\pi\)
\(858\) 0 0
\(859\) −49.4164 −1.68607 −0.843033 0.537862i \(-0.819232\pi\)
−0.843033 + 0.537862i \(0.819232\pi\)
\(860\) 0 0
\(861\) 49.8885 1.70020
\(862\) 0 0
\(863\) −18.2918 −0.622660 −0.311330 0.950302i \(-0.600774\pi\)
−0.311330 + 0.950302i \(0.600774\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 3.70820 0.125937
\(868\) 0 0
\(869\) 25.8885 0.878209
\(870\) 0 0
\(871\) −34.4721 −1.16804
\(872\) 0 0
\(873\) −24.2492 −0.820712
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 51.8885 1.75215 0.876076 0.482173i \(-0.160152\pi\)
0.876076 + 0.482173i \(0.160152\pi\)
\(878\) 0 0
\(879\) −29.5279 −0.995950
\(880\) 0 0
\(881\) −24.4721 −0.824487 −0.412244 0.911074i \(-0.635255\pi\)
−0.412244 + 0.911074i \(0.635255\pi\)
\(882\) 0 0
\(883\) −27.7082 −0.932455 −0.466228 0.884665i \(-0.654387\pi\)
−0.466228 + 0.884665i \(0.654387\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −11.5967 −0.389381 −0.194690 0.980865i \(-0.562370\pi\)
−0.194690 + 0.980865i \(0.562370\pi\)
\(888\) 0 0
\(889\) 65.3050 2.19026
\(890\) 0 0
\(891\) −4.83282 −0.161905
\(892\) 0 0
\(893\) 32.3607 1.08291
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −26.3344 −0.879279
\(898\) 0 0
\(899\) 12.9443 0.431716
\(900\) 0 0
\(901\) −2.11146 −0.0703428
\(902\) 0 0
\(903\) −30.8328 −1.02605
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −9.23607 −0.306679 −0.153339 0.988174i \(-0.549003\pi\)
−0.153339 + 0.988174i \(0.549003\pi\)
\(908\) 0 0
\(909\) −14.7214 −0.488277
\(910\) 0 0
\(911\) −53.3050 −1.76607 −0.883036 0.469305i \(-0.844504\pi\)
−0.883036 + 0.469305i \(0.844504\pi\)
\(912\) 0 0
\(913\) −7.41641 −0.245447
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −48.3607 −1.59701
\(918\) 0 0
\(919\) −5.88854 −0.194245 −0.0971226 0.995272i \(-0.530964\pi\)
−0.0971226 + 0.995272i \(0.530964\pi\)
\(920\) 0 0
\(921\) −0.360680 −0.0118848
\(922\) 0 0
\(923\) 11.0557 0.363904
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 4.76393 0.156468
\(928\) 0 0
\(929\) 24.4721 0.802905 0.401452 0.915880i \(-0.368506\pi\)
0.401452 + 0.915880i \(0.368506\pi\)
\(930\) 0 0
\(931\) 15.5279 0.508905
\(932\) 0 0
\(933\) −19.0557 −0.623857
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 3.52786 0.115250 0.0576251 0.998338i \(-0.481647\pi\)
0.0576251 + 0.998338i \(0.481647\pi\)
\(938\) 0 0
\(939\) 35.6393 1.16305
\(940\) 0 0
\(941\) −44.8328 −1.46151 −0.730754 0.682641i \(-0.760831\pi\)
−0.730754 + 0.682641i \(0.760831\pi\)
\(942\) 0 0
\(943\) 59.4164 1.93486
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 21.8197 0.709044 0.354522 0.935048i \(-0.384644\pi\)
0.354522 + 0.935048i \(0.384644\pi\)
\(948\) 0 0
\(949\) 20.0000 0.649227
\(950\) 0 0
\(951\) −25.3050 −0.820569
\(952\) 0 0
\(953\) −45.7771 −1.48287 −0.741433 0.671027i \(-0.765853\pi\)
−0.741433 + 0.671027i \(0.765853\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 4.94427 0.159826
\(958\) 0 0
\(959\) −6.47214 −0.208996
\(960\) 0 0
\(961\) 10.8885 0.351243
\(962\) 0 0
\(963\) 25.3738 0.817660
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 1.34752 0.0433335 0.0216667 0.999765i \(-0.493103\pi\)
0.0216667 + 0.999765i \(0.493103\pi\)
\(968\) 0 0
\(969\) −24.7214 −0.794164
\(970\) 0 0
\(971\) −23.8885 −0.766620 −0.383310 0.923620i \(-0.625216\pi\)
−0.383310 + 0.923620i \(0.625216\pi\)
\(972\) 0 0
\(973\) −66.2492 −2.12385
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 39.3050 1.25748 0.628738 0.777617i \(-0.283572\pi\)
0.628738 + 0.777617i \(0.283572\pi\)
\(978\) 0 0
\(979\) −29.8885 −0.955242
\(980\) 0 0
\(981\) 22.0000 0.702406
\(982\) 0 0
\(983\) −39.0132 −1.24433 −0.622163 0.782888i \(-0.713746\pi\)
−0.622163 + 0.782888i \(0.713746\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 28.9443 0.921306
\(988\) 0 0
\(989\) −36.7214 −1.16767
\(990\) 0 0
\(991\) −17.5279 −0.556791 −0.278395 0.960467i \(-0.589803\pi\)
−0.278395 + 0.960467i \(0.589803\pi\)
\(992\) 0 0
\(993\) −19.6393 −0.623235
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −34.5836 −1.09527 −0.547637 0.836716i \(-0.684472\pi\)
−0.547637 + 0.836716i \(0.684472\pi\)
\(998\) 0 0
\(999\) −38.3870 −1.21451
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 3200.2.a.bf.1.2 2
4.3 odd 2 3200.2.a.bk.1.1 2
5.2 odd 4 3200.2.c.x.2049.2 4
5.3 odd 4 3200.2.c.x.2049.3 4
5.4 even 2 640.2.a.l.1.1 yes 2
8.3 odd 2 3200.2.a.be.1.2 2
8.5 even 2 3200.2.a.bl.1.1 2
15.14 odd 2 5760.2.a.bw.1.1 2
20.3 even 4 3200.2.c.v.2049.2 4
20.7 even 4 3200.2.c.v.2049.3 4
20.19 odd 2 640.2.a.j.1.2 yes 2
40.3 even 4 3200.2.c.w.2049.3 4
40.13 odd 4 3200.2.c.u.2049.2 4
40.19 odd 2 640.2.a.k.1.1 yes 2
40.27 even 4 3200.2.c.w.2049.2 4
40.29 even 2 640.2.a.i.1.2 2
40.37 odd 4 3200.2.c.u.2049.3 4
60.59 even 2 5760.2.a.cd.1.2 2
80.19 odd 4 1280.2.d.k.641.3 4
80.29 even 4 1280.2.d.m.641.2 4
80.59 odd 4 1280.2.d.k.641.2 4
80.69 even 4 1280.2.d.m.641.3 4
120.29 odd 2 5760.2.a.ch.1.1 2
120.59 even 2 5760.2.a.ci.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
640.2.a.i.1.2 2 40.29 even 2
640.2.a.j.1.2 yes 2 20.19 odd 2
640.2.a.k.1.1 yes 2 40.19 odd 2
640.2.a.l.1.1 yes 2 5.4 even 2
1280.2.d.k.641.2 4 80.59 odd 4
1280.2.d.k.641.3 4 80.19 odd 4
1280.2.d.m.641.2 4 80.29 even 4
1280.2.d.m.641.3 4 80.69 even 4
3200.2.a.be.1.2 2 8.3 odd 2
3200.2.a.bf.1.2 2 1.1 even 1 trivial
3200.2.a.bk.1.1 2 4.3 odd 2
3200.2.a.bl.1.1 2 8.5 even 2
3200.2.c.u.2049.2 4 40.13 odd 4
3200.2.c.u.2049.3 4 40.37 odd 4
3200.2.c.v.2049.2 4 20.3 even 4
3200.2.c.v.2049.3 4 20.7 even 4
3200.2.c.w.2049.2 4 40.27 even 4
3200.2.c.w.2049.3 4 40.3 even 4
3200.2.c.x.2049.2 4 5.2 odd 4
3200.2.c.x.2049.3 4 5.3 odd 4
5760.2.a.bw.1.1 2 15.14 odd 2
5760.2.a.cd.1.2 2 60.59 even 2
5760.2.a.ch.1.1 2 120.29 odd 2
5760.2.a.ci.1.2 2 120.59 even 2