Properties

Label 9408.2.a.en.1.3
Level $9408$
Weight $2$
Character 9408.1
Self dual yes
Analytic conductor $75.123$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 9408 = 2^{6} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9408.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(75.1232582216\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.2624.1
Defining polynomial: \(x^{4} - 2 x^{3} - 3 x^{2} + 2 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 4704)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.814115\) of defining polynomial
Character \(\chi\) \(=\) 9408.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +3.04244 q^{5} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +3.04244 q^{5} +1.00000 q^{9} +3.93089 q^{11} +4.88845 q^{13} +3.04244 q^{15} +5.34511 q^{17} -2.30266 q^{19} +7.93089 q^{23} +4.25646 q^{25} +1.00000 q^{27} -5.55912 q^{29} +0.645810 q^{31} +3.93089 q^{33} -5.65685 q^{37} +4.88845 q^{39} -10.0886 q^{41} +8.91331 q^{43} +3.04244 q^{45} +6.61065 q^{47} +5.34511 q^{51} -1.25646 q^{53} +11.9595 q^{55} -2.30266 q^{57} +3.04621 q^{59} -2.97334 q^{61} +14.8728 q^{65} -13.5186 q^{67} +7.93089 q^{69} +13.5877 q^{71} -4.67067 q^{73} +4.25646 q^{75} +1.05153 q^{79} +1.00000 q^{81} +8.60533 q^{83} +16.2622 q^{85} -5.55912 q^{87} -4.85983 q^{89} +0.645810 q^{93} -7.00572 q^{95} -18.3275 q^{97} +3.93089 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{3} + 4q^{5} + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{3} + 4q^{5} + 4q^{9} - 4q^{11} + 8q^{13} + 4q^{15} - 4q^{17} + 8q^{19} + 12q^{23} + 12q^{25} + 4q^{27} + 8q^{31} - 4q^{33} + 8q^{39} - 20q^{41} + 8q^{43} + 4q^{45} + 16q^{47} - 4q^{51} + 8q^{55} + 8q^{57} + 16q^{61} - 8q^{65} + 8q^{67} + 12q^{69} + 12q^{71} - 8q^{73} + 12q^{75} + 16q^{79} + 4q^{81} + 8q^{85} - 28q^{89} + 8q^{93} + 24q^{95} - 40q^{97} - 4q^{99} + O(q^{100}) \)

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
\(5\) 3.04244 1.36062 0.680311 0.732924i \(-0.261845\pi\)
0.680311 + 0.732924i \(0.261845\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 3.93089 1.18521 0.592604 0.805494i \(-0.298100\pi\)
0.592604 + 0.805494i \(0.298100\pi\)
\(12\) 0 0
\(13\) 4.88845 1.35581 0.677906 0.735149i \(-0.262888\pi\)
0.677906 + 0.735149i \(0.262888\pi\)
\(14\) 0 0
\(15\) 3.04244 0.785555
\(16\) 0 0
\(17\) 5.34511 1.29638 0.648189 0.761479i \(-0.275526\pi\)
0.648189 + 0.761479i \(0.275526\pi\)
\(18\) 0 0
\(19\) −2.30266 −0.528267 −0.264134 0.964486i \(-0.585086\pi\)
−0.264134 + 0.964486i \(0.585086\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 7.93089 1.65371 0.826853 0.562418i \(-0.190129\pi\)
0.826853 + 0.562418i \(0.190129\pi\)
\(24\) 0 0
\(25\) 4.25646 0.851292
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −5.55912 −1.03230 −0.516152 0.856497i \(-0.672636\pi\)
−0.516152 + 0.856497i \(0.672636\pi\)
\(30\) 0 0
\(31\) 0.645810 0.115991 0.0579954 0.998317i \(-0.481529\pi\)
0.0579954 + 0.998317i \(0.481529\pi\)
\(32\) 0 0
\(33\) 3.93089 0.684281
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −5.65685 −0.929981 −0.464991 0.885316i \(-0.653942\pi\)
−0.464991 + 0.885316i \(0.653942\pi\)
\(38\) 0 0
\(39\) 4.88845 0.782779
\(40\) 0 0
\(41\) −10.0886 −1.57558 −0.787791 0.615943i \(-0.788775\pi\)
−0.787791 + 0.615943i \(0.788775\pi\)
\(42\) 0 0
\(43\) 8.91331 1.35927 0.679634 0.733552i \(-0.262139\pi\)
0.679634 + 0.733552i \(0.262139\pi\)
\(44\) 0 0
\(45\) 3.04244 0.453541
\(46\) 0 0
\(47\) 6.61065 0.964262 0.482131 0.876099i \(-0.339863\pi\)
0.482131 + 0.876099i \(0.339863\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 5.34511 0.748465
\(52\) 0 0
\(53\) −1.25646 −0.172588 −0.0862939 0.996270i \(-0.527502\pi\)
−0.0862939 + 0.996270i \(0.527502\pi\)
\(54\) 0 0
\(55\) 11.9595 1.61262
\(56\) 0 0
\(57\) −2.30266 −0.304995
\(58\) 0 0
\(59\) 3.04621 0.396582 0.198291 0.980143i \(-0.436461\pi\)
0.198291 + 0.980143i \(0.436461\pi\)
\(60\) 0 0
\(61\) −2.97334 −0.380697 −0.190348 0.981717i \(-0.560962\pi\)
−0.190348 + 0.981717i \(0.560962\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 14.8728 1.84475
\(66\) 0 0
\(67\) −13.5186 −1.65156 −0.825782 0.563989i \(-0.809266\pi\)
−0.825782 + 0.563989i \(0.809266\pi\)
\(68\) 0 0
\(69\) 7.93089 0.954767
\(70\) 0 0
\(71\) 13.5877 1.61257 0.806284 0.591528i \(-0.201475\pi\)
0.806284 + 0.591528i \(0.201475\pi\)
\(72\) 0 0
\(73\) −4.67067 −0.546661 −0.273330 0.961920i \(-0.588125\pi\)
−0.273330 + 0.961920i \(0.588125\pi\)
\(74\) 0 0
\(75\) 4.25646 0.491494
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 1.05153 0.118306 0.0591530 0.998249i \(-0.481160\pi\)
0.0591530 + 0.998249i \(0.481160\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 8.60533 0.944557 0.472279 0.881449i \(-0.343432\pi\)
0.472279 + 0.881449i \(0.343432\pi\)
\(84\) 0 0
\(85\) 16.2622 1.76388
\(86\) 0 0
\(87\) −5.55912 −0.596001
\(88\) 0 0
\(89\) −4.85983 −0.515140 −0.257570 0.966260i \(-0.582922\pi\)
−0.257570 + 0.966260i \(0.582922\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0.645810 0.0669673
\(94\) 0 0
\(95\) −7.00572 −0.718772
\(96\) 0 0
\(97\) −18.3275 −1.86088 −0.930439 0.366446i \(-0.880574\pi\)
−0.930439 + 0.366446i \(0.880574\pi\)
\(98\) 0 0
\(99\) 3.93089 0.395070
\(100\) 0 0
\(101\) 5.87087 0.584173 0.292087 0.956392i \(-0.405650\pi\)
0.292087 + 0.956392i \(0.405650\pi\)
\(102\) 0 0
\(103\) −15.0778 −1.48566 −0.742828 0.669482i \(-0.766516\pi\)
−0.742828 + 0.669482i \(0.766516\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0.238878 0.0230932 0.0115466 0.999933i \(-0.496325\pi\)
0.0115466 + 0.999933i \(0.496325\pi\)
\(108\) 0 0
\(109\) 12.9133 1.23687 0.618436 0.785836i \(-0.287767\pi\)
0.618436 + 0.785836i \(0.287767\pi\)
\(110\) 0 0
\(111\) −5.65685 −0.536925
\(112\) 0 0
\(113\) −13.3137 −1.25245 −0.626224 0.779643i \(-0.715401\pi\)
−0.626224 + 0.779643i \(0.715401\pi\)
\(114\) 0 0
\(115\) 24.1293 2.25007
\(116\) 0 0
\(117\) 4.88845 0.451937
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 4.45192 0.404720
\(122\) 0 0
\(123\) −10.0886 −0.909663
\(124\) 0 0
\(125\) −2.26218 −0.202336
\(126\) 0 0
\(127\) −3.69202 −0.327613 −0.163807 0.986492i \(-0.552377\pi\)
−0.163807 + 0.986492i \(0.552377\pi\)
\(128\) 0 0
\(129\) 8.91331 0.784773
\(130\) 0 0
\(131\) −18.4320 −1.61041 −0.805204 0.592998i \(-0.797944\pi\)
−0.805204 + 0.592998i \(0.797944\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 3.04244 0.261852
\(136\) 0 0
\(137\) −4.26750 −0.364597 −0.182299 0.983243i \(-0.558354\pi\)
−0.182299 + 0.983243i \(0.558354\pi\)
\(138\) 0 0
\(139\) −7.31371 −0.620341 −0.310170 0.950681i \(-0.600386\pi\)
−0.310170 + 0.950681i \(0.600386\pi\)
\(140\) 0 0
\(141\) 6.61065 0.556717
\(142\) 0 0
\(143\) 19.2160 1.60692
\(144\) 0 0
\(145\) −16.9133 −1.40457
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −14.0572 −1.15161 −0.575807 0.817585i \(-0.695312\pi\)
−0.575807 + 0.817585i \(0.695312\pi\)
\(150\) 0 0
\(151\) −22.5702 −1.83673 −0.918367 0.395730i \(-0.870492\pi\)
−0.918367 + 0.395730i \(0.870492\pi\)
\(152\) 0 0
\(153\) 5.34511 0.432126
\(154\) 0 0
\(155\) 1.96484 0.157820
\(156\) 0 0
\(157\) −12.3147 −0.982818 −0.491409 0.870929i \(-0.663518\pi\)
−0.491409 + 0.870929i \(0.663518\pi\)
\(158\) 0 0
\(159\) −1.25646 −0.0996436
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 3.45192 0.270375 0.135188 0.990820i \(-0.456836\pi\)
0.135188 + 0.990820i \(0.456836\pi\)
\(164\) 0 0
\(165\) 11.9595 0.931047
\(166\) 0 0
\(167\) 12.5076 0.967867 0.483933 0.875105i \(-0.339208\pi\)
0.483933 + 0.875105i \(0.339208\pi\)
\(168\) 0 0
\(169\) 10.8969 0.838227
\(170\) 0 0
\(171\) −2.30266 −0.176089
\(172\) 0 0
\(173\) 19.0997 1.45212 0.726061 0.687630i \(-0.241349\pi\)
0.726061 + 0.687630i \(0.241349\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 3.04621 0.228967
\(178\) 0 0
\(179\) −11.6908 −0.873811 −0.436906 0.899507i \(-0.643926\pi\)
−0.436906 + 0.899507i \(0.643926\pi\)
\(180\) 0 0
\(181\) −11.3204 −0.841439 −0.420720 0.907191i \(-0.638222\pi\)
−0.420720 + 0.907191i \(0.638222\pi\)
\(182\) 0 0
\(183\) −2.97334 −0.219795
\(184\) 0 0
\(185\) −17.2107 −1.26535
\(186\) 0 0
\(187\) 21.0110 1.53648
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 16.9015 1.22295 0.611473 0.791265i \(-0.290577\pi\)
0.611473 + 0.791265i \(0.290577\pi\)
\(192\) 0 0
\(193\) −23.6884 −1.70513 −0.852565 0.522622i \(-0.824954\pi\)
−0.852565 + 0.522622i \(0.824954\pi\)
\(194\) 0 0
\(195\) 14.8728 1.06507
\(196\) 0 0
\(197\) 11.0831 0.789637 0.394819 0.918759i \(-0.370807\pi\)
0.394819 + 0.918759i \(0.370807\pi\)
\(198\) 0 0
\(199\) 6.51292 0.461688 0.230844 0.972991i \(-0.425851\pi\)
0.230844 + 0.972991i \(0.425851\pi\)
\(200\) 0 0
\(201\) −13.5186 −0.953531
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −30.6941 −2.14377
\(206\) 0 0
\(207\) 7.93089 0.551235
\(208\) 0 0
\(209\) −9.05153 −0.626107
\(210\) 0 0
\(211\) 3.83023 0.263684 0.131842 0.991271i \(-0.457911\pi\)
0.131842 + 0.991271i \(0.457911\pi\)
\(212\) 0 0
\(213\) 13.5877 0.931017
\(214\) 0 0
\(215\) 27.1182 1.84945
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −4.67067 −0.315615
\(220\) 0 0
\(221\) 26.1293 1.75765
\(222\) 0 0
\(223\) −12.4099 −0.831026 −0.415513 0.909587i \(-0.636398\pi\)
−0.415513 + 0.909587i \(0.636398\pi\)
\(224\) 0 0
\(225\) 4.25646 0.283764
\(226\) 0 0
\(227\) −17.4782 −1.16007 −0.580033 0.814593i \(-0.696960\pi\)
−0.580033 + 0.814593i \(0.696960\pi\)
\(228\) 0 0
\(229\) 21.1783 1.39950 0.699750 0.714388i \(-0.253295\pi\)
0.699750 + 0.714388i \(0.253295\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 5.75459 0.376995 0.188498 0.982074i \(-0.439638\pi\)
0.188498 + 0.982074i \(0.439638\pi\)
\(234\) 0 0
\(235\) 20.1125 1.31200
\(236\) 0 0
\(237\) 1.05153 0.0683040
\(238\) 0 0
\(239\) 7.93089 0.513007 0.256503 0.966543i \(-0.417430\pi\)
0.256503 + 0.966543i \(0.417430\pi\)
\(240\) 0 0
\(241\) −20.9605 −1.35018 −0.675092 0.737734i \(-0.735896\pi\)
−0.675092 + 0.737734i \(0.735896\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −11.2565 −0.716231
\(248\) 0 0
\(249\) 8.60533 0.545340
\(250\) 0 0
\(251\) −7.45607 −0.470623 −0.235311 0.971920i \(-0.575611\pi\)
−0.235311 + 0.971920i \(0.575611\pi\)
\(252\) 0 0
\(253\) 31.1755 1.95999
\(254\) 0 0
\(255\) 16.2622 1.01838
\(256\) 0 0
\(257\) 12.3433 0.769954 0.384977 0.922926i \(-0.374209\pi\)
0.384977 + 0.922926i \(0.374209\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −5.55912 −0.344101
\(262\) 0 0
\(263\) −23.4144 −1.44379 −0.721896 0.692002i \(-0.756729\pi\)
−0.721896 + 0.692002i \(0.756729\pi\)
\(264\) 0 0
\(265\) −3.82270 −0.234827
\(266\) 0 0
\(267\) −4.85983 −0.297416
\(268\) 0 0
\(269\) 5.81362 0.354463 0.177231 0.984169i \(-0.443286\pi\)
0.177231 + 0.984169i \(0.443286\pi\)
\(270\) 0 0
\(271\) 24.3694 1.48033 0.740167 0.672423i \(-0.234746\pi\)
0.740167 + 0.672423i \(0.234746\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 16.7317 1.00896
\(276\) 0 0
\(277\) −15.7587 −0.946851 −0.473425 0.880834i \(-0.656983\pi\)
−0.473425 + 0.880834i \(0.656983\pi\)
\(278\) 0 0
\(279\) 0.645810 0.0386636
\(280\) 0 0
\(281\) −30.7698 −1.83557 −0.917786 0.397076i \(-0.870025\pi\)
−0.917786 + 0.397076i \(0.870025\pi\)
\(282\) 0 0
\(283\) 22.9080 1.36174 0.680869 0.732405i \(-0.261602\pi\)
0.680869 + 0.732405i \(0.261602\pi\)
\(284\) 0 0
\(285\) −7.00572 −0.414983
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 11.5702 0.680598
\(290\) 0 0
\(291\) −18.3275 −1.07438
\(292\) 0 0
\(293\) 19.3323 1.12940 0.564701 0.825295i \(-0.308991\pi\)
0.564701 + 0.825295i \(0.308991\pi\)
\(294\) 0 0
\(295\) 9.26791 0.539598
\(296\) 0 0
\(297\) 3.93089 0.228094
\(298\) 0 0
\(299\) 38.7698 2.24211
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 5.87087 0.337273
\(304\) 0 0
\(305\) −9.04621 −0.517984
\(306\) 0 0
\(307\) −26.8377 −1.53171 −0.765853 0.643015i \(-0.777683\pi\)
−0.765853 + 0.643015i \(0.777683\pi\)
\(308\) 0 0
\(309\) −15.0778 −0.857744
\(310\) 0 0
\(311\) 3.49240 0.198036 0.0990180 0.995086i \(-0.468430\pi\)
0.0990180 + 0.995086i \(0.468430\pi\)
\(312\) 0 0
\(313\) 2.75556 0.155753 0.0778767 0.996963i \(-0.475186\pi\)
0.0778767 + 0.996963i \(0.475186\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 7.15341 0.401775 0.200888 0.979614i \(-0.435617\pi\)
0.200888 + 0.979614i \(0.435617\pi\)
\(318\) 0 0
\(319\) −21.8523 −1.22349
\(320\) 0 0
\(321\) 0.238878 0.0133329
\(322\) 0 0
\(323\) −12.3080 −0.684835
\(324\) 0 0
\(325\) 20.8075 1.15419
\(326\) 0 0
\(327\) 12.9133 0.714108
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 31.7236 1.74369 0.871843 0.489786i \(-0.162925\pi\)
0.871843 + 0.489786i \(0.162925\pi\)
\(332\) 0 0
\(333\) −5.65685 −0.309994
\(334\) 0 0
\(335\) −41.1297 −2.24716
\(336\) 0 0
\(337\) 12.6053 0.686656 0.343328 0.939216i \(-0.388446\pi\)
0.343328 + 0.939216i \(0.388446\pi\)
\(338\) 0 0
\(339\) −13.3137 −0.723101
\(340\) 0 0
\(341\) 2.53861 0.137473
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 24.1293 1.29908
\(346\) 0 0
\(347\) −31.1064 −1.66988 −0.834939 0.550342i \(-0.814497\pi\)
−0.834939 + 0.550342i \(0.814497\pi\)
\(348\) 0 0
\(349\) 15.7741 0.844370 0.422185 0.906510i \(-0.361263\pi\)
0.422185 + 0.906510i \(0.361263\pi\)
\(350\) 0 0
\(351\) 4.88845 0.260926
\(352\) 0 0
\(353\) 18.3785 0.978187 0.489094 0.872231i \(-0.337328\pi\)
0.489094 + 0.872231i \(0.337328\pi\)
\(354\) 0 0
\(355\) 41.3399 2.19410
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −17.7575 −0.937206 −0.468603 0.883409i \(-0.655243\pi\)
−0.468603 + 0.883409i \(0.655243\pi\)
\(360\) 0 0
\(361\) −13.6977 −0.720934
\(362\) 0 0
\(363\) 4.45192 0.233665
\(364\) 0 0
\(365\) −14.2103 −0.743799
\(366\) 0 0
\(367\) −20.9449 −1.09331 −0.546657 0.837357i \(-0.684100\pi\)
−0.546657 + 0.837357i \(0.684100\pi\)
\(368\) 0 0
\(369\) −10.0886 −0.525194
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 20.2365 1.04781 0.523903 0.851778i \(-0.324475\pi\)
0.523903 + 0.851778i \(0.324475\pi\)
\(374\) 0 0
\(375\) −2.26218 −0.116819
\(376\) 0 0
\(377\) −27.1755 −1.39961
\(378\) 0 0
\(379\) 36.5244 1.87613 0.938065 0.346459i \(-0.112616\pi\)
0.938065 + 0.346459i \(0.112616\pi\)
\(380\) 0 0
\(381\) −3.69202 −0.189148
\(382\) 0 0
\(383\) 2.97417 0.151973 0.0759864 0.997109i \(-0.475789\pi\)
0.0759864 + 0.997109i \(0.475789\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 8.91331 0.453089
\(388\) 0 0
\(389\) 2.16445 0.109742 0.0548710 0.998493i \(-0.482525\pi\)
0.0548710 + 0.998493i \(0.482525\pi\)
\(390\) 0 0
\(391\) 42.3915 2.14383
\(392\) 0 0
\(393\) −18.4320 −0.929769
\(394\) 0 0
\(395\) 3.19921 0.160970
\(396\) 0 0
\(397\) −12.0249 −0.603511 −0.301755 0.953385i \(-0.597573\pi\)
−0.301755 + 0.953385i \(0.597573\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −21.5591 −1.07661 −0.538306 0.842750i \(-0.680935\pi\)
−0.538306 + 0.842750i \(0.680935\pi\)
\(402\) 0 0
\(403\) 3.15701 0.157262
\(404\) 0 0
\(405\) 3.04244 0.151180
\(406\) 0 0
\(407\) −22.2365 −1.10222
\(408\) 0 0
\(409\) 24.5143 1.21215 0.606077 0.795406i \(-0.292742\pi\)
0.606077 + 0.795406i \(0.292742\pi\)
\(410\) 0 0
\(411\) −4.26750 −0.210500
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 26.1812 1.28519
\(416\) 0 0
\(417\) −7.31371 −0.358154
\(418\) 0 0
\(419\) 35.3048 1.72475 0.862376 0.506269i \(-0.168976\pi\)
0.862376 + 0.506269i \(0.168976\pi\)
\(420\) 0 0
\(421\) −2.47776 −0.120758 −0.0603792 0.998176i \(-0.519231\pi\)
−0.0603792 + 0.998176i \(0.519231\pi\)
\(422\) 0 0
\(423\) 6.61065 0.321421
\(424\) 0 0
\(425\) 22.7512 1.10360
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 19.2160 0.927756
\(430\) 0 0
\(431\) 20.3408 0.979780 0.489890 0.871784i \(-0.337037\pi\)
0.489890 + 0.871784i \(0.337037\pi\)
\(432\) 0 0
\(433\) −15.7964 −0.759129 −0.379564 0.925165i \(-0.623926\pi\)
−0.379564 + 0.925165i \(0.623926\pi\)
\(434\) 0 0
\(435\) −16.9133 −0.810931
\(436\) 0 0
\(437\) −18.2622 −0.873599
\(438\) 0 0
\(439\) 12.6863 0.605484 0.302742 0.953073i \(-0.402098\pi\)
0.302742 + 0.953073i \(0.402098\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −25.9273 −1.23184 −0.615921 0.787808i \(-0.711216\pi\)
−0.615921 + 0.787808i \(0.711216\pi\)
\(444\) 0 0
\(445\) −14.7857 −0.700911
\(446\) 0 0
\(447\) −14.0572 −0.664885
\(448\) 0 0
\(449\) −18.0000 −0.849473 −0.424736 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) 0 0
\(451\) −39.6574 −1.86739
\(452\) 0 0
\(453\) −22.5702 −1.06044
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 29.7236 1.39041 0.695205 0.718811i \(-0.255314\pi\)
0.695205 + 0.718811i \(0.255314\pi\)
\(458\) 0 0
\(459\) 5.34511 0.249488
\(460\) 0 0
\(461\) 12.6314 0.588303 0.294152 0.955759i \(-0.404963\pi\)
0.294152 + 0.955759i \(0.404963\pi\)
\(462\) 0 0
\(463\) −33.6884 −1.56563 −0.782817 0.622252i \(-0.786218\pi\)
−0.782817 + 0.622252i \(0.786218\pi\)
\(464\) 0 0
\(465\) 1.96484 0.0911172
\(466\) 0 0
\(467\) 34.0941 1.57769 0.788844 0.614593i \(-0.210680\pi\)
0.788844 + 0.614593i \(0.210680\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −12.3147 −0.567431
\(472\) 0 0
\(473\) 35.0373 1.61102
\(474\) 0 0
\(475\) −9.80119 −0.449710
\(476\) 0 0
\(477\) −1.25646 −0.0575293
\(478\) 0 0
\(479\) 0.0977317 0.00446547 0.00223274 0.999998i \(-0.499289\pi\)
0.00223274 + 0.999998i \(0.499289\pi\)
\(480\) 0 0
\(481\) −27.6533 −1.26088
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −55.7605 −2.53195
\(486\) 0 0
\(487\) −38.5702 −1.74778 −0.873891 0.486123i \(-0.838411\pi\)
−0.873891 + 0.486123i \(0.838411\pi\)
\(488\) 0 0
\(489\) 3.45192 0.156101
\(490\) 0 0
\(491\) 19.0748 0.860835 0.430418 0.902630i \(-0.358366\pi\)
0.430418 + 0.902630i \(0.358366\pi\)
\(492\) 0 0
\(493\) −29.7141 −1.33826
\(494\) 0 0
\(495\) 11.9595 0.537540
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 8.05725 0.360692 0.180346 0.983603i \(-0.442278\pi\)
0.180346 + 0.983603i \(0.442278\pi\)
\(500\) 0 0
\(501\) 12.5076 0.558798
\(502\) 0 0
\(503\) −4.94487 −0.220481 −0.110240 0.993905i \(-0.535162\pi\)
−0.110240 + 0.993905i \(0.535162\pi\)
\(504\) 0 0
\(505\) 17.8618 0.794839
\(506\) 0 0
\(507\) 10.8969 0.483950
\(508\) 0 0
\(509\) 37.2161 1.64958 0.824788 0.565442i \(-0.191294\pi\)
0.824788 + 0.565442i \(0.191294\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −2.30266 −0.101665
\(514\) 0 0
\(515\) −45.8732 −2.02142
\(516\) 0 0
\(517\) 25.9858 1.14285
\(518\) 0 0
\(519\) 19.0997 0.838383
\(520\) 0 0
\(521\) −13.1050 −0.574141 −0.287071 0.957909i \(-0.592681\pi\)
−0.287071 + 0.957909i \(0.592681\pi\)
\(522\) 0 0
\(523\) 19.5795 0.856151 0.428076 0.903743i \(-0.359192\pi\)
0.428076 + 0.903743i \(0.359192\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 3.45192 0.150368
\(528\) 0 0
\(529\) 39.8991 1.73474
\(530\) 0 0
\(531\) 3.04621 0.132194
\(532\) 0 0
\(533\) −49.3179 −2.13619
\(534\) 0 0
\(535\) 0.726773 0.0314211
\(536\) 0 0
\(537\) −11.6908 −0.504495
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 11.9648 0.514409 0.257204 0.966357i \(-0.417199\pi\)
0.257204 + 0.966357i \(0.417199\pi\)
\(542\) 0 0
\(543\) −11.3204 −0.485805
\(544\) 0 0
\(545\) 39.2880 1.68291
\(546\) 0 0
\(547\) −39.8827 −1.70526 −0.852631 0.522514i \(-0.824994\pi\)
−0.852631 + 0.522514i \(0.824994\pi\)
\(548\) 0 0
\(549\) −2.97334 −0.126899
\(550\) 0 0
\(551\) 12.8008 0.545332
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −17.2107 −0.730552
\(556\) 0 0
\(557\) −28.9800 −1.22792 −0.613962 0.789336i \(-0.710425\pi\)
−0.613962 + 0.789336i \(0.710425\pi\)
\(558\) 0 0
\(559\) 43.5723 1.84291
\(560\) 0 0
\(561\) 21.0110 0.887087
\(562\) 0 0
\(563\) 27.3857 1.15417 0.577086 0.816683i \(-0.304190\pi\)
0.577086 + 0.816683i \(0.304190\pi\)
\(564\) 0 0
\(565\) −40.5062 −1.70411
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −6.37056 −0.267068 −0.133534 0.991044i \(-0.542632\pi\)
−0.133534 + 0.991044i \(0.542632\pi\)
\(570\) 0 0
\(571\) 14.9485 0.625574 0.312787 0.949823i \(-0.398737\pi\)
0.312787 + 0.949823i \(0.398737\pi\)
\(572\) 0 0
\(573\) 16.9015 0.706068
\(574\) 0 0
\(575\) 33.7575 1.40779
\(576\) 0 0
\(577\) 6.21500 0.258734 0.129367 0.991597i \(-0.458705\pi\)
0.129367 + 0.991597i \(0.458705\pi\)
\(578\) 0 0
\(579\) −23.6884 −0.984457
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −4.93900 −0.204553
\(584\) 0 0
\(585\) 14.8728 0.614916
\(586\) 0 0
\(587\) 27.0462 1.11632 0.558158 0.829735i \(-0.311508\pi\)
0.558158 + 0.829735i \(0.311508\pi\)
\(588\) 0 0
\(589\) −1.48708 −0.0612742
\(590\) 0 0
\(591\) 11.0831 0.455897
\(592\) 0 0
\(593\) 15.8837 0.652266 0.326133 0.945324i \(-0.394254\pi\)
0.326133 + 0.945324i \(0.394254\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 6.51292 0.266556
\(598\) 0 0
\(599\) 5.24820 0.214436 0.107218 0.994236i \(-0.465806\pi\)
0.107218 + 0.994236i \(0.465806\pi\)
\(600\) 0 0
\(601\) 33.3830 1.36172 0.680860 0.732414i \(-0.261606\pi\)
0.680860 + 0.732414i \(0.261606\pi\)
\(602\) 0 0
\(603\) −13.5186 −0.550522
\(604\) 0 0
\(605\) 13.5447 0.550671
\(606\) 0 0
\(607\) 18.9228 0.768052 0.384026 0.923322i \(-0.374537\pi\)
0.384026 + 0.923322i \(0.374537\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 32.3158 1.30736
\(612\) 0 0
\(613\) −12.4928 −0.504580 −0.252290 0.967652i \(-0.581184\pi\)
−0.252290 + 0.967652i \(0.581184\pi\)
\(614\) 0 0
\(615\) −30.6941 −1.23771
\(616\) 0 0
\(617\) 34.4230 1.38582 0.692910 0.721025i \(-0.256329\pi\)
0.692910 + 0.721025i \(0.256329\pi\)
\(618\) 0 0
\(619\) 22.6381 0.909900 0.454950 0.890517i \(-0.349657\pi\)
0.454950 + 0.890517i \(0.349657\pi\)
\(620\) 0 0
\(621\) 7.93089 0.318256
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −28.1649 −1.12659
\(626\) 0 0
\(627\) −9.05153 −0.361483
\(628\) 0 0
\(629\) −30.2365 −1.20561
\(630\) 0 0
\(631\) −25.3489 −1.00912 −0.504561 0.863376i \(-0.668346\pi\)
−0.504561 + 0.863376i \(0.668346\pi\)
\(632\) 0 0
\(633\) 3.83023 0.152238
\(634\) 0 0
\(635\) −11.2327 −0.445758
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 13.5877 0.537523
\(640\) 0 0
\(641\) −13.2196 −0.522142 −0.261071 0.965320i \(-0.584076\pi\)
−0.261071 + 0.965320i \(0.584076\pi\)
\(642\) 0 0
\(643\) −23.7194 −0.935403 −0.467701 0.883887i \(-0.654918\pi\)
−0.467701 + 0.883887i \(0.654918\pi\)
\(644\) 0 0
\(645\) 27.1182 1.06778
\(646\) 0 0
\(647\) 15.8213 0.622000 0.311000 0.950410i \(-0.399336\pi\)
0.311000 + 0.950410i \(0.399336\pi\)
\(648\) 0 0
\(649\) 11.9743 0.470033
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −19.4561 −0.761375 −0.380687 0.924704i \(-0.624313\pi\)
−0.380687 + 0.924704i \(0.624313\pi\)
\(654\) 0 0
\(655\) −56.0782 −2.19116
\(656\) 0 0
\(657\) −4.67067 −0.182220
\(658\) 0 0
\(659\) −10.5820 −0.412217 −0.206109 0.978529i \(-0.566080\pi\)
−0.206109 + 0.978529i \(0.566080\pi\)
\(660\) 0 0
\(661\) 9.95814 0.387327 0.193663 0.981068i \(-0.437963\pi\)
0.193663 + 0.981068i \(0.437963\pi\)
\(662\) 0 0
\(663\) 26.1293 1.01478
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −44.0888 −1.70713
\(668\) 0 0
\(669\) −12.4099 −0.479793
\(670\) 0 0
\(671\) −11.6879 −0.451205
\(672\) 0 0
\(673\) −21.0724 −0.812283 −0.406141 0.913810i \(-0.633126\pi\)
−0.406141 + 0.913810i \(0.633126\pi\)
\(674\) 0 0
\(675\) 4.25646 0.163831
\(676\) 0 0
\(677\) −4.66021 −0.179107 −0.0895533 0.995982i \(-0.528544\pi\)
−0.0895533 + 0.995982i \(0.528544\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −17.4782 −0.669765
\(682\) 0 0
\(683\) −9.24820 −0.353873 −0.176936 0.984222i \(-0.556619\pi\)
−0.176936 + 0.984222i \(0.556619\pi\)
\(684\) 0 0
\(685\) −12.9836 −0.496079
\(686\) 0 0
\(687\) 21.1783 0.808001
\(688\) 0 0
\(689\) −6.14214 −0.233997
\(690\) 0 0
\(691\) −10.0924 −0.383933 −0.191967 0.981401i \(-0.561487\pi\)
−0.191967 + 0.981401i \(0.561487\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −22.2515 −0.844049
\(696\) 0 0
\(697\) −53.9249 −2.04255
\(698\) 0 0
\(699\) 5.75459 0.217658
\(700\) 0 0
\(701\) −17.0683 −0.644661 −0.322330 0.946627i \(-0.604466\pi\)
−0.322330 + 0.946627i \(0.604466\pi\)
\(702\) 0 0
\(703\) 13.0258 0.491279
\(704\) 0 0
\(705\) 20.1125 0.757481
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 27.2012 1.02156 0.510781 0.859711i \(-0.329356\pi\)
0.510781 + 0.859711i \(0.329356\pi\)
\(710\) 0 0
\(711\) 1.05153 0.0394353
\(712\) 0 0
\(713\) 5.12185 0.191815
\(714\) 0 0
\(715\) 58.4635 2.18641
\(716\) 0 0
\(717\) 7.93089 0.296185
\(718\) 0 0
\(719\) −24.5350 −0.915001 −0.457501 0.889209i \(-0.651255\pi\)
−0.457501 + 0.889209i \(0.651255\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −20.9605 −0.779529
\(724\) 0 0
\(725\) −23.6622 −0.878791
\(726\) 0 0
\(727\) 40.7603 1.51172 0.755858 0.654735i \(-0.227220\pi\)
0.755858 + 0.654735i \(0.227220\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 47.6426 1.76213
\(732\) 0 0
\(733\) −28.1318 −1.03907 −0.519537 0.854448i \(-0.673895\pi\)
−0.519537 + 0.854448i \(0.673895\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −53.1403 −1.95745
\(738\) 0 0
\(739\) −12.6626 −0.465800 −0.232900 0.972501i \(-0.574822\pi\)
−0.232900 + 0.972501i \(0.574822\pi\)
\(740\) 0 0
\(741\) −11.2565 −0.413516
\(742\) 0 0
\(743\) 13.3477 0.489678 0.244839 0.969564i \(-0.421265\pi\)
0.244839 + 0.969564i \(0.421265\pi\)
\(744\) 0 0
\(745\) −42.7684 −1.56691
\(746\) 0 0
\(747\) 8.60533 0.314852
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 28.2843 1.03211 0.516054 0.856556i \(-0.327400\pi\)
0.516054 + 0.856556i \(0.327400\pi\)
\(752\) 0 0
\(753\) −7.45607 −0.271714
\(754\) 0 0
\(755\) −68.6684 −2.49910
\(756\) 0 0
\(757\) −9.71410 −0.353065 −0.176533 0.984295i \(-0.556488\pi\)
−0.176533 + 0.984295i \(0.556488\pi\)
\(758\) 0 0
\(759\) 31.1755 1.13160
\(760\) 0 0
\(761\) 16.5569 0.600188 0.300094 0.953910i \(-0.402982\pi\)
0.300094 + 0.953910i \(0.402982\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 16.2622 0.587960
\(766\) 0 0
\(767\) 14.8912 0.537691
\(768\) 0 0
\(769\) 12.3884 0.446736 0.223368 0.974734i \(-0.428295\pi\)
0.223368 + 0.974734i \(0.428295\pi\)
\(770\) 0 0
\(771\) 12.3433 0.444533
\(772\) 0 0
\(773\) −55.4211 −1.99336 −0.996679 0.0814351i \(-0.974050\pi\)
−0.996679 + 0.0814351i \(0.974050\pi\)
\(774\) 0 0
\(775\) 2.74886 0.0987421
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 23.2308 0.832329
\(780\) 0 0
\(781\) 53.4120 1.91123
\(782\) 0 0
\(783\) −5.55912 −0.198667
\(784\) 0 0
\(785\) −37.4667 −1.33724
\(786\) 0 0
\(787\) 13.4061 0.477877 0.238938 0.971035i \(-0.423201\pi\)
0.238938 + 0.971035i \(0.423201\pi\)
\(788\) 0 0
\(789\) −23.4144 −0.833574
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −14.5350 −0.516153
\(794\) 0 0
\(795\) −3.82270 −0.135577
\(796\) 0 0
\(797\) −26.4837 −0.938102 −0.469051 0.883171i \(-0.655404\pi\)
−0.469051 + 0.883171i \(0.655404\pi\)
\(798\) 0 0
\(799\) 35.3346 1.25005
\(800\) 0 0
\(801\) −4.85983 −0.171713
\(802\) 0 0
\(803\) −18.3599 −0.647907
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 5.81362 0.204649
\(808\) 0 0
\(809\) −36.5760 −1.28594 −0.642972 0.765889i \(-0.722299\pi\)
−0.642972 + 0.765889i \(0.722299\pi\)
\(810\) 0 0
\(811\) 4.67565 0.164184 0.0820921 0.996625i \(-0.473840\pi\)
0.0820921 + 0.996625i \(0.473840\pi\)
\(812\) 0 0
\(813\) 24.3694 0.854672
\(814\) 0 0
\(815\) 10.5023 0.367879
\(816\) 0 0
\(817\) −20.5244 −0.718057
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 42.9212 1.49796 0.748979 0.662593i \(-0.230544\pi\)
0.748979 + 0.662593i \(0.230544\pi\)
\(822\) 0 0
\(823\) 12.0888 0.421389 0.210695 0.977552i \(-0.432427\pi\)
0.210695 + 0.977552i \(0.432427\pi\)
\(824\) 0 0
\(825\) 16.7317 0.582522
\(826\) 0 0
\(827\) −3.23676 −0.112553 −0.0562765 0.998415i \(-0.517923\pi\)
−0.0562765 + 0.998415i \(0.517923\pi\)
\(828\) 0 0
\(829\) −2.06002 −0.0715476 −0.0357738 0.999360i \(-0.511390\pi\)
−0.0357738 + 0.999360i \(0.511390\pi\)
\(830\) 0 0
\(831\) −15.7587 −0.546664
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 38.0536 1.31690
\(836\) 0 0
\(837\) 0.645810 0.0223224
\(838\) 0 0
\(839\) 5.99468 0.206959 0.103480 0.994632i \(-0.467002\pi\)
0.103480 + 0.994632i \(0.467002\pi\)
\(840\) 0 0
\(841\) 1.90384 0.0656498
\(842\) 0 0
\(843\) −30.7698 −1.05977
\(844\) 0 0
\(845\) 33.1533 1.14051
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 22.9080 0.786200
\(850\) 0 0
\(851\) −44.8639 −1.53791
\(852\) 0 0
\(853\) −40.3040 −1.37998 −0.689992 0.723817i \(-0.742386\pi\)
−0.689992 + 0.723817i \(0.742386\pi\)
\(854\) 0 0
\(855\) −7.00572 −0.239591
\(856\) 0 0
\(857\) −24.1735 −0.825752 −0.412876 0.910787i \(-0.635476\pi\)
−0.412876 + 0.910787i \(0.635476\pi\)
\(858\) 0 0
\(859\) −22.6316 −0.772179 −0.386090 0.922461i \(-0.626174\pi\)
−0.386090 + 0.922461i \(0.626174\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 48.9015 1.66462 0.832312 0.554307i \(-0.187017\pi\)
0.832312 + 0.554307i \(0.187017\pi\)
\(864\) 0 0
\(865\) 58.1097 1.97579
\(866\) 0 0
\(867\) 11.5702 0.392943
\(868\) 0 0
\(869\) 4.13344 0.140217
\(870\) 0 0
\(871\) −66.0852 −2.23921
\(872\) 0 0
\(873\) −18.3275 −0.620293
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 32.1461 1.08550 0.542748 0.839896i \(-0.317384\pi\)
0.542748 + 0.839896i \(0.317384\pi\)
\(878\) 0 0
\(879\) 19.3323 0.652061
\(880\) 0 0
\(881\) −1.94291 −0.0654583 −0.0327291 0.999464i \(-0.510420\pi\)
−0.0327291 + 0.999464i \(0.510420\pi\)
\(882\) 0 0
\(883\) 48.8770 1.64484 0.822421 0.568880i \(-0.192623\pi\)
0.822421 + 0.568880i \(0.192623\pi\)
\(884\) 0 0
\(885\) 9.26791 0.311537
\(886\) 0 0
\(887\) 5.59546 0.187877 0.0939385 0.995578i \(-0.470054\pi\)
0.0939385 + 0.995578i \(0.470054\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 3.93089 0.131690
\(892\) 0 0
\(893\) −15.2221 −0.509388
\(894\) 0 0
\(895\) −35.5686 −1.18893
\(896\) 0 0
\(897\) 38.7698 1.29449
\(898\) 0 0
\(899\) −3.59014 −0.119738
\(900\) 0 0
\(901\) −6.71591 −0.223739
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −34.4417 −1.14488
\(906\) 0 0
\(907\) 51.0945 1.69657 0.848283 0.529543i \(-0.177637\pi\)
0.848283 + 0.529543i \(0.177637\pi\)
\(908\) 0 0
\(909\) 5.87087 0.194724
\(910\) 0 0
\(911\) −22.6965 −0.751969 −0.375985 0.926626i \(-0.622695\pi\)
−0.375985 + 0.926626i \(0.622695\pi\)
\(912\) 0 0
\(913\) 33.8266 1.11950
\(914\) 0 0
\(915\) −9.04621 −0.299058
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 12.9836 0.428291 0.214145 0.976802i \(-0.431303\pi\)
0.214145 + 0.976802i \(0.431303\pi\)
\(920\) 0 0
\(921\) −26.8377 −0.884331
\(922\) 0 0
\(923\) 66.4230 2.18634
\(924\) 0 0
\(925\) −24.0782 −0.791685
\(926\) 0 0
\(927\) −15.0778 −0.495219
\(928\) 0 0
\(929\) 1.84856 0.0606491 0.0303246 0.999540i \(-0.490346\pi\)
0.0303246 + 0.999540i \(0.490346\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 3.49240 0.114336
\(934\) 0 0
\(935\) 63.9249 2.09057
\(936\) 0 0
\(937\) 53.0177 1.73201 0.866007 0.500032i \(-0.166678\pi\)
0.866007 + 0.500032i \(0.166678\pi\)
\(938\) 0 0
\(939\) 2.75556 0.0899242
\(940\) 0 0
\(941\) −10.9615 −0.357334 −0.178667 0.983910i \(-0.557178\pi\)
−0.178667 + 0.983910i \(0.557178\pi\)
\(942\) 0 0
\(943\) −80.0120 −2.60555
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −8.61718 −0.280021 −0.140010 0.990150i \(-0.544714\pi\)
−0.140010 + 0.990150i \(0.544714\pi\)
\(948\) 0 0
\(949\) −22.8323 −0.741169
\(950\) 0 0
\(951\) 7.15341 0.231965
\(952\) 0 0
\(953\) 6.40986 0.207636 0.103818 0.994596i \(-0.466894\pi\)
0.103818 + 0.994596i \(0.466894\pi\)
\(954\) 0 0
\(955\) 51.4217 1.66397
\(956\) 0 0
\(957\) −21.8523 −0.706385
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −30.5829 −0.986546
\(962\) 0 0
\(963\) 0.238878 0.00769774
\(964\) 0 0
\(965\) −72.0706 −2.32004
\(966\) 0 0
\(967\) 43.2749 1.39163 0.695814 0.718222i \(-0.255044\pi\)
0.695814 + 0.718222i \(0.255044\pi\)
\(968\) 0 0
\(969\) −12.3080 −0.395389
\(970\) 0 0
\(971\) 21.8266 0.700450 0.350225 0.936666i \(-0.386105\pi\)
0.350225 + 0.936666i \(0.386105\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 20.8075 0.666373
\(976\) 0 0
\(977\) −58.5040 −1.87171 −0.935854 0.352387i \(-0.885370\pi\)
−0.935854 + 0.352387i \(0.885370\pi\)
\(978\) 0 0
\(979\) −19.1035 −0.610549
\(980\) 0 0
\(981\) 12.9133 0.412290
\(982\) 0 0
\(983\) 6.17337 0.196900 0.0984500 0.995142i \(-0.468612\pi\)
0.0984500 + 0.995142i \(0.468612\pi\)
\(984\) 0 0
\(985\) 33.7197 1.07440
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 70.6905 2.24783
\(990\) 0 0
\(991\) −5.11942 −0.162624 −0.0813118 0.996689i \(-0.525911\pi\)
−0.0813118 + 0.996689i \(0.525911\pi\)
\(992\) 0 0
\(993\) 31.7236 1.00672
\(994\) 0 0
\(995\) 19.8152 0.628183
\(996\) 0 0
\(997\) 49.7448 1.57543 0.787717 0.616037i \(-0.211263\pi\)
0.787717 + 0.616037i \(0.211263\pi\)
\(998\) 0 0
\(999\) −5.65685 −0.178975
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 9408.2.a.en.1.3 4
4.3 odd 2 9408.2.a.el.1.3 4
7.6 odd 2 9408.2.a.ek.1.2 4
8.3 odd 2 4704.2.a.by.1.2 yes 4
8.5 even 2 4704.2.a.bw.1.2 4
28.27 even 2 9408.2.a.em.1.2 4
56.13 odd 2 4704.2.a.bz.1.3 yes 4
56.27 even 2 4704.2.a.bx.1.3 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4704.2.a.bw.1.2 4 8.5 even 2
4704.2.a.bx.1.3 yes 4 56.27 even 2
4704.2.a.by.1.2 yes 4 8.3 odd 2
4704.2.a.bz.1.3 yes 4 56.13 odd 2
9408.2.a.ek.1.2 4 7.6 odd 2
9408.2.a.el.1.3 4 4.3 odd 2
9408.2.a.em.1.2 4 28.27 even 2
9408.2.a.en.1.3 4 1.1 even 1 trivial