Properties

Label 4704.2.a.by.1.2
Level $4704$
Weight $2$
Character 4704.1
Self dual yes
Analytic conductor $37.562$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4704 = 2^{5} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4704.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(37.5616291108\)
Analytic rank: \(1\)
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: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.22833\) of defining polynomial
Character \(\chi\) \(=\) 4704.1

$q$-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 \(-0.738155\pi\)
−0.680311 + 0.732924i \(0.738155\pi\)
\(6\) 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 \(-0.737112\pi\)
−0.677906 + 0.735149i \(0.737112\pi\)
\(14\) 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 \(-0.809871\pi\)
−0.826853 + 0.562418i \(0.809871\pi\)
\(24\) 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 \(-0.327364\pi\)
0.516152 + 0.856497i \(0.327364\pi\)
\(30\) 0 0
\(31\) −0.645810 −0.115991 −0.0579954 0.998317i \(-0.518471\pi\)
−0.0579954 + 0.998317i \(0.518471\pi\)
\(32\) 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 \(-0.346058\pi\)
0.464991 + 0.885316i \(0.346058\pi\)
\(38\) 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 \(-0.660137\pi\)
−0.482131 + 0.876099i \(0.660137\pi\)
\(48\) 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 \(-0.472498\pi\)
0.0862939 + 0.996270i \(0.472498\pi\)
\(54\) 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 \(-0.439038\pi\)
0.190348 + 0.981717i \(0.439038\pi\)
\(62\) 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 \(-0.798525\pi\)
−0.806284 + 0.591528i \(0.798525\pi\)
\(72\) 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 \(-0.518840\pi\)
−0.0591530 + 0.998249i \(0.518840\pi\)
\(80\) 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 \(-0.594350\pi\)
−0.292087 + 0.956392i \(0.594350\pi\)
\(102\) 0 0
\(103\) 15.0778 1.48566 0.742828 0.669482i \(-0.233484\pi\)
0.742828 + 0.669482i \(0.233484\pi\)
\(104\) 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 \(-0.712233\pi\)
−0.618436 + 0.785836i \(0.712233\pi\)
\(110\) 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 \(-0.447623\pi\)
0.163807 + 0.986492i \(0.447623\pi\)
\(128\) 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 \(-0.304688\pi\)
0.575807 + 0.817585i \(0.304688\pi\)
\(150\) 0 0
\(151\) 22.5702 1.83673 0.918367 0.395730i \(-0.129508\pi\)
0.918367 + 0.395730i \(0.129508\pi\)
\(152\) 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 \(-0.336482\pi\)
0.491409 + 0.870929i \(0.336482\pi\)
\(158\) 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 \(-0.660792\pi\)
−0.483933 + 0.875105i \(0.660792\pi\)
\(168\) 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 \(-0.758651\pi\)
−0.726061 + 0.687630i \(0.758651\pi\)
\(174\) 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 \(-0.361778\pi\)
0.420720 + 0.907191i \(0.361778\pi\)
\(182\) 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 \(-0.709423\pi\)
−0.611473 + 0.791265i \(0.709423\pi\)
\(192\) 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 \(-0.629193\pi\)
−0.394819 + 0.918759i \(0.629193\pi\)
\(198\) 0 0
\(199\) −6.51292 −0.461688 −0.230844 0.972991i \(-0.574149\pi\)
−0.230844 + 0.972991i \(0.574149\pi\)
\(200\) 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 \(-0.363602\pi\)
0.415513 + 0.909587i \(0.363602\pi\)
\(224\) 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 \(-0.746705\pi\)
−0.699750 + 0.714388i \(0.746705\pi\)
\(230\) 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 \(-0.582570\pi\)
−0.256503 + 0.966543i \(0.582570\pi\)
\(240\) 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 \(-0.243271\pi\)
0.721896 + 0.692002i \(0.243271\pi\)
\(264\) 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 \(-0.556714\pi\)
−0.177231 + 0.984169i \(0.556714\pi\)
\(270\) 0 0
\(271\) −24.3694 −1.48033 −0.740167 0.672423i \(-0.765254\pi\)
−0.740167 + 0.672423i \(0.765254\pi\)
\(272\) 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 \(-0.343017\pi\)
0.473425 + 0.880834i \(0.343017\pi\)
\(278\) 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 \(-0.691009\pi\)
−0.564701 + 0.825295i \(0.691009\pi\)
\(294\) 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 \(-0.531570\pi\)
−0.0990180 + 0.995086i \(0.531570\pi\)
\(312\) 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 \(-0.564383\pi\)
−0.200888 + 0.979614i \(0.564383\pi\)
\(318\) 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 \(-0.638737\pi\)
−0.422185 + 0.906510i \(0.638737\pi\)
\(350\) 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 \(-0.344757\pi\)
0.468603 + 0.883409i \(0.344757\pi\)
\(360\) 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 \(-0.315900\pi\)
0.546657 + 0.837357i \(0.315900\pi\)
\(368\) 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 \(-0.675525\pi\)
−0.523903 + 0.851778i \(0.675525\pi\)
\(374\) 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 \(-0.524211\pi\)
−0.0759864 + 0.997109i \(0.524211\pi\)
\(384\) 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 \(-0.517475\pi\)
−0.0548710 + 0.998493i \(0.517475\pi\)
\(390\) 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 \(-0.402427\pi\)
0.301755 + 0.953385i \(0.402427\pi\)
\(398\) 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 \(-0.480769\pi\)
0.0603792 + 0.998176i \(0.480769\pi\)
\(422\) 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 \(-0.662963\pi\)
−0.489890 + 0.871784i \(0.662963\pi\)
\(432\) 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 \(-0.597902\pi\)
−0.302742 + 0.953073i \(0.597902\pi\)
\(440\) 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 \(-0.595037\pi\)
−0.294152 + 0.955759i \(0.595037\pi\)
\(462\) 0 0
\(463\) 33.6884 1.56563 0.782817 0.622252i \(-0.213782\pi\)
0.782817 + 0.622252i \(0.213782\pi\)
\(464\) 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 \(-0.500711\pi\)
−0.00223274 + 0.999998i \(0.500711\pi\)
\(480\) 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 \(-0.161589\pi\)
0.873891 + 0.486123i \(0.161589\pi\)
\(488\) 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 \(-0.464838\pi\)
0.110240 + 0.993905i \(0.464838\pi\)
\(504\) 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 \(-0.808706\pi\)
−0.824788 + 0.565442i \(0.808706\pi\)
\(510\) 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 \(-0.582801\pi\)
−0.257204 + 0.966357i \(0.582801\pi\)
\(542\) 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 \(-0.289575\pi\)
0.613962 + 0.789336i \(0.289575\pi\)
\(558\) 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 \(-0.534194\pi\)
−0.107218 + 0.994236i \(0.534194\pi\)
\(600\) 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 \(-0.625463\pi\)
−0.384026 + 0.923322i \(0.625463\pi\)
\(608\) 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 \(-0.418816\pi\)
0.252290 + 0.967652i \(0.418816\pi\)
\(614\) 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 \(-0.331654\pi\)
0.504561 + 0.863376i \(0.331654\pi\)
\(632\) 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 \(-0.600664\pi\)
−0.311000 + 0.950410i \(0.600664\pi\)
\(648\) 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 \(-0.375687\pi\)
0.380687 + 0.924704i \(0.375687\pi\)
\(654\) 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 \(-0.562037\pi\)
−0.193663 + 0.981068i \(0.562037\pi\)
\(662\) 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 \(-0.471456\pi\)
0.0895533 + 0.995982i \(0.471456\pi\)
\(678\) 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 \(-0.395534\pi\)
0.322330 + 0.946627i \(0.395534\pi\)
\(702\) 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 \(-0.670644\pi\)
−0.510781 + 0.859711i \(0.670644\pi\)
\(710\) 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 \(-0.348745\pi\)
0.457501 + 0.889209i \(0.348745\pi\)
\(720\) 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 \(-0.772780\pi\)
−0.755858 + 0.654735i \(0.772780\pi\)
\(728\) 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 \(-0.326105\pi\)
0.519537 + 0.854448i \(0.326105\pi\)
\(734\) 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 \(-0.578735\pi\)
−0.244839 + 0.969564i \(0.578735\pi\)
\(744\) 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 \(-0.672600\pi\)
−0.516054 + 0.856556i \(0.672600\pi\)
\(752\) 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 \(-0.443512\pi\)
0.176533 + 0.984295i \(0.443512\pi\)
\(758\) 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 \(-0.0259503\pi\)
0.996679 + 0.0814351i \(0.0259503\pi\)
\(774\) 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 \(-0.344596\pi\)
0.469051 + 0.883171i \(0.344596\pi\)
\(798\) 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 \(-0.769456\pi\)
−0.748979 + 0.662593i \(0.769456\pi\)
\(822\) 0 0
\(823\) −12.0888 −0.421389 −0.210695 0.977552i \(-0.567573\pi\)
−0.210695 + 0.977552i \(0.567573\pi\)
\(824\) 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 \(-0.488610\pi\)
0.0357738 + 0.999360i \(0.488610\pi\)
\(830\) 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 \(-0.532998\pi\)
−0.103480 + 0.994632i \(0.532998\pi\)
\(840\) 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 \(-0.257614\pi\)
0.689992 + 0.723817i \(0.257614\pi\)
\(854\) 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 \(-0.812983\pi\)
−0.832312 + 0.554307i \(0.812983\pi\)
\(864\) 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 \(-0.682616\pi\)
−0.542748 + 0.839896i \(0.682616\pi\)
\(878\) 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 \(-0.529946\pi\)
−0.0939385 + 0.995578i \(0.529946\pi\)
\(888\) 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 \(-0.377305\pi\)
0.375985 + 0.926626i \(0.377305\pi\)
\(912\) 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 \(-0.568697\pi\)
−0.214145 + 0.976802i \(0.568697\pi\)
\(920\) 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 \(-0.442822\pi\)
0.178667 + 0.983910i \(0.442822\pi\)
\(942\) 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 \(-0.744956\pi\)
−0.695814 + 0.718222i \(0.744956\pi\)
\(968\) 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 \(-0.531388\pi\)
−0.0984500 + 0.995142i \(0.531388\pi\)
\(984\) 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 \(-0.474089\pi\)
0.0813118 + 0.996689i \(0.474089\pi\)
\(992\) 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 \(-0.788737\pi\)
−0.787717 + 0.616037i \(0.788737\pi\)
\(998\) 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 4704.2.a.by.1.2 yes 4
4.3 odd 2 4704.2.a.bw.1.2 4
7.6 odd 2 4704.2.a.bx.1.3 yes 4
8.3 odd 2 9408.2.a.en.1.3 4
8.5 even 2 9408.2.a.el.1.3 4
28.27 even 2 4704.2.a.bz.1.3 yes 4
56.13 odd 2 9408.2.a.em.1.2 4
56.27 even 2 9408.2.a.ek.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4704.2.a.bw.1.2 4 4.3 odd 2
4704.2.a.bx.1.3 yes 4 7.6 odd 2
4704.2.a.by.1.2 yes 4 1.1 even 1 trivial
4704.2.a.bz.1.3 yes 4 28.27 even 2
9408.2.a.ek.1.2 4 56.27 even 2
9408.2.a.el.1.3 4 8.5 even 2
9408.2.a.em.1.2 4 56.13 odd 2
9408.2.a.en.1.3 4 8.3 odd 2