Properties

Label 3380.2.a.m.1.3
Level $3380$
Weight $2$
Character 3380.1
Self dual yes
Analytic conductor $26.989$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3380 = 2^{2} \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3380.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.9894358832\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.756.1
Defining polynomial: \(x^{3} - 6 x - 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 260)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.60168\) of defining polynomial
Character \(\chi\) \(=\) 3380.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.60168 q^{3} -1.00000 q^{5} -2.76873 q^{7} +3.76873 q^{9} +O(q^{10})\) \(q+2.60168 q^{3} -1.00000 q^{5} -2.76873 q^{7} +3.76873 q^{9} -2.16706 q^{11} -2.60168 q^{15} -5.03630 q^{19} -7.20336 q^{21} +4.93579 q^{23} +1.00000 q^{25} +2.00000 q^{27} -4.43462 q^{29} +3.37041 q^{31} -5.63798 q^{33} +2.76873 q^{35} +3.97209 q^{37} +1.66589 q^{41} -9.80504 q^{43} -3.76873 q^{45} -11.6380 q^{47} +0.665890 q^{49} -12.7408 q^{53} +2.16706 q^{55} -13.1028 q^{57} +8.16706 q^{59} -10.3062 q^{61} -10.4346 q^{63} +7.10284 q^{67} +12.8413 q^{69} -16.1112 q^{71} -15.9721 q^{73} +2.60168 q^{75} +6.00000 q^{77} +11.9442 q^{79} -6.10284 q^{81} +3.97209 q^{83} -11.5375 q^{87} +7.94419 q^{89} +8.76873 q^{93} +5.03630 q^{95} -0.462531 q^{97} -8.16706 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{5} + 3 q^{9} + O(q^{10}) \) \( 3 q - 3 q^{5} + 3 q^{9} - 6 q^{11} - 6 q^{21} + 6 q^{23} + 3 q^{25} + 6 q^{27} - 6 q^{29} - 6 q^{31} + 6 q^{33} - 12 q^{37} + 6 q^{41} - 6 q^{43} - 3 q^{45} - 12 q^{47} + 3 q^{49} - 6 q^{53} + 6 q^{55} - 30 q^{57} + 24 q^{59} - 6 q^{61} - 24 q^{63} + 12 q^{67} - 24 q^{73} + 18 q^{77} - 12 q^{79} - 9 q^{81} - 12 q^{83} - 18 q^{87} - 24 q^{89} + 18 q^{93} - 18 q^{97} - 24 q^{99} + O(q^{100}) \)

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
\(3\) 2.60168 1.50208 0.751040 0.660257i \(-0.229552\pi\)
0.751040 + 0.660257i \(0.229552\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −2.76873 −1.04648 −0.523242 0.852184i \(-0.675277\pi\)
−0.523242 + 0.852184i \(0.675277\pi\)
\(8\) 0 0
\(9\) 3.76873 1.25624
\(10\) 0 0
\(11\) −2.16706 −0.653392 −0.326696 0.945130i \(-0.605935\pi\)
−0.326696 + 0.945130i \(0.605935\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 0 0
\(15\) −2.60168 −0.671751
\(16\) 0 0
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) −5.03630 −1.15541 −0.577704 0.816247i \(-0.696051\pi\)
−0.577704 + 0.816247i \(0.696051\pi\)
\(20\) 0 0
\(21\) −7.20336 −1.57190
\(22\) 0 0
\(23\) 4.93579 1.02918 0.514592 0.857435i \(-0.327944\pi\)
0.514592 + 0.857435i \(0.327944\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 2.00000 0.384900
\(28\) 0 0
\(29\) −4.43462 −0.823489 −0.411744 0.911299i \(-0.635080\pi\)
−0.411744 + 0.911299i \(0.635080\pi\)
\(30\) 0 0
\(31\) 3.37041 0.605344 0.302672 0.953095i \(-0.402121\pi\)
0.302672 + 0.953095i \(0.402121\pi\)
\(32\) 0 0
\(33\) −5.63798 −0.981447
\(34\) 0 0
\(35\) 2.76873 0.468002
\(36\) 0 0
\(37\) 3.97209 0.653008 0.326504 0.945196i \(-0.394129\pi\)
0.326504 + 0.945196i \(0.394129\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 1.66589 0.260168 0.130084 0.991503i \(-0.458475\pi\)
0.130084 + 0.991503i \(0.458475\pi\)
\(42\) 0 0
\(43\) −9.80504 −1.49525 −0.747627 0.664119i \(-0.768807\pi\)
−0.747627 + 0.664119i \(0.768807\pi\)
\(44\) 0 0
\(45\) −3.76873 −0.561810
\(46\) 0 0
\(47\) −11.6380 −1.69757 −0.848787 0.528735i \(-0.822667\pi\)
−0.848787 + 0.528735i \(0.822667\pi\)
\(48\) 0 0
\(49\) 0.665890 0.0951271
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −12.7408 −1.75009 −0.875044 0.484044i \(-0.839167\pi\)
−0.875044 + 0.484044i \(0.839167\pi\)
\(54\) 0 0
\(55\) 2.16706 0.292206
\(56\) 0 0
\(57\) −13.1028 −1.73551
\(58\) 0 0
\(59\) 8.16706 1.06326 0.531630 0.846977i \(-0.321580\pi\)
0.531630 + 0.846977i \(0.321580\pi\)
\(60\) 0 0
\(61\) −10.3062 −1.31957 −0.659787 0.751453i \(-0.729354\pi\)
−0.659787 + 0.751453i \(0.729354\pi\)
\(62\) 0 0
\(63\) −10.4346 −1.31464
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 7.10284 0.867751 0.433875 0.900973i \(-0.357146\pi\)
0.433875 + 0.900973i \(0.357146\pi\)
\(68\) 0 0
\(69\) 12.8413 1.54592
\(70\) 0 0
\(71\) −16.1112 −1.91205 −0.956026 0.293281i \(-0.905253\pi\)
−0.956026 + 0.293281i \(0.905253\pi\)
\(72\) 0 0
\(73\) −15.9721 −1.86939 −0.934696 0.355448i \(-0.884328\pi\)
−0.934696 + 0.355448i \(0.884328\pi\)
\(74\) 0 0
\(75\) 2.60168 0.300416
\(76\) 0 0
\(77\) 6.00000 0.683763
\(78\) 0 0
\(79\) 11.9442 1.34383 0.671913 0.740630i \(-0.265473\pi\)
0.671913 + 0.740630i \(0.265473\pi\)
\(80\) 0 0
\(81\) −6.10284 −0.678094
\(82\) 0 0
\(83\) 3.97209 0.435994 0.217997 0.975949i \(-0.430048\pi\)
0.217997 + 0.975949i \(0.430048\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −11.5375 −1.23695
\(88\) 0 0
\(89\) 7.94419 0.842082 0.421041 0.907042i \(-0.361665\pi\)
0.421041 + 0.907042i \(0.361665\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 8.76873 0.909275
\(94\) 0 0
\(95\) 5.03630 0.516714
\(96\) 0 0
\(97\) −0.462531 −0.0469629 −0.0234815 0.999724i \(-0.507475\pi\)
−0.0234815 + 0.999724i \(0.507475\pi\)
\(98\) 0 0
\(99\) −8.16706 −0.820820
\(100\) 0 0
\(101\) −12.7408 −1.26776 −0.633880 0.773432i \(-0.718539\pi\)
−0.633880 + 0.773432i \(0.718539\pi\)
\(102\) 0 0
\(103\) 4.13915 0.407842 0.203921 0.978987i \(-0.434631\pi\)
0.203921 + 0.978987i \(0.434631\pi\)
\(104\) 0 0
\(105\) 7.20336 0.702976
\(106\) 0 0
\(107\) 7.06421 0.682923 0.341462 0.939896i \(-0.389078\pi\)
0.341462 + 0.939896i \(0.389078\pi\)
\(108\) 0 0
\(109\) −14.8692 −1.42422 −0.712108 0.702070i \(-0.752259\pi\)
−0.712108 + 0.702070i \(0.752259\pi\)
\(110\) 0 0
\(111\) 10.3341 0.980870
\(112\) 0 0
\(113\) 3.13075 0.294516 0.147258 0.989098i \(-0.452955\pi\)
0.147258 + 0.989098i \(0.452955\pi\)
\(114\) 0 0
\(115\) −4.93579 −0.460265
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −6.30387 −0.573079
\(122\) 0 0
\(123\) 4.33411 0.390794
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −16.1391 −1.43212 −0.716059 0.698040i \(-0.754056\pi\)
−0.716059 + 0.698040i \(0.754056\pi\)
\(128\) 0 0
\(129\) −25.5096 −2.24599
\(130\) 0 0
\(131\) −8.86925 −0.774910 −0.387455 0.921889i \(-0.626646\pi\)
−0.387455 + 0.921889i \(0.626646\pi\)
\(132\) 0 0
\(133\) 13.9442 1.20911
\(134\) 0 0
\(135\) −2.00000 −0.172133
\(136\) 0 0
\(137\) 1.66589 0.142327 0.0711633 0.997465i \(-0.477329\pi\)
0.0711633 + 0.997465i \(0.477329\pi\)
\(138\) 0 0
\(139\) 19.2760 1.63497 0.817483 0.575953i \(-0.195369\pi\)
0.817483 + 0.575953i \(0.195369\pi\)
\(140\) 0 0
\(141\) −30.2783 −2.54989
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 4.43462 0.368275
\(146\) 0 0
\(147\) 1.73243 0.142889
\(148\) 0 0
\(149\) 13.9442 1.14235 0.571176 0.820828i \(-0.306487\pi\)
0.571176 + 0.820828i \(0.306487\pi\)
\(150\) 0 0
\(151\) −6.96370 −0.566698 −0.283349 0.959017i \(-0.591445\pi\)
−0.283349 + 0.959017i \(0.591445\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −3.37041 −0.270718
\(156\) 0 0
\(157\) 12.3341 0.984369 0.492185 0.870491i \(-0.336199\pi\)
0.492185 + 0.870491i \(0.336199\pi\)
\(158\) 0 0
\(159\) −33.1475 −2.62877
\(160\) 0 0
\(161\) −13.6659 −1.07702
\(162\) 0 0
\(163\) 20.5072 1.60625 0.803125 0.595810i \(-0.203169\pi\)
0.803125 + 0.595810i \(0.203169\pi\)
\(164\) 0 0
\(165\) 5.63798 0.438916
\(166\) 0 0
\(167\) −19.3039 −1.49378 −0.746889 0.664948i \(-0.768453\pi\)
−0.746889 + 0.664948i \(0.768453\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) −18.9805 −1.45147
\(172\) 0 0
\(173\) −9.61007 −0.730640 −0.365320 0.930882i \(-0.619041\pi\)
−0.365320 + 0.930882i \(0.619041\pi\)
\(174\) 0 0
\(175\) −2.76873 −0.209297
\(176\) 0 0
\(177\) 21.2481 1.59710
\(178\) 0 0
\(179\) 9.87158 0.737836 0.368918 0.929462i \(-0.379728\pi\)
0.368918 + 0.929462i \(0.379728\pi\)
\(180\) 0 0
\(181\) 1.97209 0.146584 0.0732922 0.997311i \(-0.476649\pi\)
0.0732922 + 0.997311i \(0.476649\pi\)
\(182\) 0 0
\(183\) −26.8134 −1.98211
\(184\) 0 0
\(185\) −3.97209 −0.292034
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −5.53747 −0.402792
\(190\) 0 0
\(191\) −5.73850 −0.415223 −0.207611 0.978211i \(-0.566569\pi\)
−0.207611 + 0.978211i \(0.566569\pi\)
\(192\) 0 0
\(193\) 4.07261 0.293153 0.146576 0.989199i \(-0.453175\pi\)
0.146576 + 0.989199i \(0.453175\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 21.6101 1.53965 0.769827 0.638253i \(-0.220342\pi\)
0.769827 + 0.638253i \(0.220342\pi\)
\(198\) 0 0
\(199\) 11.6101 0.823016 0.411508 0.911406i \(-0.365002\pi\)
0.411508 + 0.911406i \(0.365002\pi\)
\(200\) 0 0
\(201\) 18.4793 1.30343
\(202\) 0 0
\(203\) 12.2783 0.861767
\(204\) 0 0
\(205\) −1.66589 −0.116351
\(206\) 0 0
\(207\) 18.6017 1.29291
\(208\) 0 0
\(209\) 10.9139 0.754933
\(210\) 0 0
\(211\) 16.2783 1.12064 0.560322 0.828275i \(-0.310677\pi\)
0.560322 + 0.828275i \(0.310677\pi\)
\(212\) 0 0
\(213\) −41.9163 −2.87206
\(214\) 0 0
\(215\) 9.80504 0.668698
\(216\) 0 0
\(217\) −9.33178 −0.633482
\(218\) 0 0
\(219\) −41.5543 −2.80798
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −11.4370 −0.765875 −0.382938 0.923774i \(-0.625088\pi\)
−0.382938 + 0.923774i \(0.625088\pi\)
\(224\) 0 0
\(225\) 3.76873 0.251249
\(226\) 0 0
\(227\) 15.9721 1.06011 0.530053 0.847965i \(-0.322172\pi\)
0.530053 + 0.847965i \(0.322172\pi\)
\(228\) 0 0
\(229\) −2.12842 −0.140650 −0.0703250 0.997524i \(-0.522404\pi\)
−0.0703250 + 0.997524i \(0.522404\pi\)
\(230\) 0 0
\(231\) 15.6101 1.02707
\(232\) 0 0
\(233\) 2.86925 0.187971 0.0939853 0.995574i \(-0.470039\pi\)
0.0939853 + 0.995574i \(0.470039\pi\)
\(234\) 0 0
\(235\) 11.6380 0.759178
\(236\) 0 0
\(237\) 31.0749 2.01853
\(238\) 0 0
\(239\) −3.16939 −0.205011 −0.102505 0.994732i \(-0.532686\pi\)
−0.102505 + 0.994732i \(0.532686\pi\)
\(240\) 0 0
\(241\) −4.79664 −0.308979 −0.154489 0.987994i \(-0.549373\pi\)
−0.154489 + 0.987994i \(0.549373\pi\)
\(242\) 0 0
\(243\) −21.8776 −1.40345
\(244\) 0 0
\(245\) −0.665890 −0.0425421
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 10.3341 0.654898
\(250\) 0 0
\(251\) −1.00233 −0.0632666 −0.0316333 0.999500i \(-0.510071\pi\)
−0.0316333 + 0.999500i \(0.510071\pi\)
\(252\) 0 0
\(253\) −10.6961 −0.672460
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 19.4817 1.21523 0.607616 0.794231i \(-0.292126\pi\)
0.607616 + 0.794231i \(0.292126\pi\)
\(258\) 0 0
\(259\) −10.9977 −0.683362
\(260\) 0 0
\(261\) −16.7129 −1.03450
\(262\) 0 0
\(263\) −4.19496 −0.258672 −0.129336 0.991601i \(-0.541285\pi\)
−0.129336 + 0.991601i \(0.541285\pi\)
\(264\) 0 0
\(265\) 12.7408 0.782663
\(266\) 0 0
\(267\) 20.6682 1.26487
\(268\) 0 0
\(269\) 30.4793 1.85836 0.929179 0.369631i \(-0.120516\pi\)
0.929179 + 0.369631i \(0.120516\pi\)
\(270\) 0 0
\(271\) 4.83528 0.293722 0.146861 0.989157i \(-0.453083\pi\)
0.146861 + 0.989157i \(0.453083\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −2.16706 −0.130678
\(276\) 0 0
\(277\) −3.00233 −0.180393 −0.0901963 0.995924i \(-0.528749\pi\)
−0.0901963 + 0.995924i \(0.528749\pi\)
\(278\) 0 0
\(279\) 12.7022 0.760460
\(280\) 0 0
\(281\) 21.6101 1.28915 0.644574 0.764542i \(-0.277035\pi\)
0.644574 + 0.764542i \(0.277035\pi\)
\(282\) 0 0
\(283\) −2.47326 −0.147020 −0.0735100 0.997294i \(-0.523420\pi\)
−0.0735100 + 0.997294i \(0.523420\pi\)
\(284\) 0 0
\(285\) 13.1028 0.776146
\(286\) 0 0
\(287\) −4.61241 −0.272262
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) −1.20336 −0.0705421
\(292\) 0 0
\(293\) 23.9163 1.39720 0.698602 0.715511i \(-0.253806\pi\)
0.698602 + 0.715511i \(0.253806\pi\)
\(294\) 0 0
\(295\) −8.16706 −0.475504
\(296\) 0 0
\(297\) −4.33411 −0.251491
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 27.1475 1.56476
\(302\) 0 0
\(303\) −33.1475 −1.90428
\(304\) 0 0
\(305\) 10.3062 0.590131
\(306\) 0 0
\(307\) −13.5654 −0.774217 −0.387108 0.922034i \(-0.626526\pi\)
−0.387108 + 0.922034i \(0.626526\pi\)
\(308\) 0 0
\(309\) 10.7687 0.612612
\(310\) 0 0
\(311\) 2.12842 0.120692 0.0603458 0.998178i \(-0.480780\pi\)
0.0603458 + 0.998178i \(0.480780\pi\)
\(312\) 0 0
\(313\) −22.6077 −1.27787 −0.638933 0.769263i \(-0.720624\pi\)
−0.638933 + 0.769263i \(0.720624\pi\)
\(314\) 0 0
\(315\) 10.4346 0.587924
\(316\) 0 0
\(317\) −10.6357 −0.597358 −0.298679 0.954354i \(-0.596546\pi\)
−0.298679 + 0.954354i \(0.596546\pi\)
\(318\) 0 0
\(319\) 9.61007 0.538061
\(320\) 0 0
\(321\) 18.3788 1.02581
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −38.6850 −2.13929
\(328\) 0 0
\(329\) 32.2225 1.77648
\(330\) 0 0
\(331\) −1.16472 −0.0640190 −0.0320095 0.999488i \(-0.510191\pi\)
−0.0320095 + 0.999488i \(0.510191\pi\)
\(332\) 0 0
\(333\) 14.9698 0.820338
\(334\) 0 0
\(335\) −7.10284 −0.388070
\(336\) 0 0
\(337\) 20.0000 1.08947 0.544735 0.838608i \(-0.316630\pi\)
0.544735 + 0.838608i \(0.316630\pi\)
\(338\) 0 0
\(339\) 8.14521 0.442387
\(340\) 0 0
\(341\) −7.30387 −0.395527
\(342\) 0 0
\(343\) 17.5375 0.946934
\(344\) 0 0
\(345\) −12.8413 −0.691355
\(346\) 0 0
\(347\) 4.67429 0.250929 0.125464 0.992098i \(-0.459958\pi\)
0.125464 + 0.992098i \(0.459958\pi\)
\(348\) 0 0
\(349\) 4.33411 0.232000 0.116000 0.993249i \(-0.462993\pi\)
0.116000 + 0.993249i \(0.462993\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 21.3085 1.13414 0.567069 0.823670i \(-0.308077\pi\)
0.567069 + 0.823670i \(0.308077\pi\)
\(354\) 0 0
\(355\) 16.1112 0.855096
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −12.7795 −0.674474 −0.337237 0.941420i \(-0.609492\pi\)
−0.337237 + 0.941420i \(0.609492\pi\)
\(360\) 0 0
\(361\) 6.36435 0.334966
\(362\) 0 0
\(363\) −16.4007 −0.860811
\(364\) 0 0
\(365\) 15.9721 0.836018
\(366\) 0 0
\(367\) 12.1950 0.636572 0.318286 0.947995i \(-0.396893\pi\)
0.318286 + 0.947995i \(0.396893\pi\)
\(368\) 0 0
\(369\) 6.27830 0.326835
\(370\) 0 0
\(371\) 35.2760 1.83144
\(372\) 0 0
\(373\) −28.2225 −1.46130 −0.730652 0.682750i \(-0.760784\pi\)
−0.730652 + 0.682750i \(0.760784\pi\)
\(374\) 0 0
\(375\) −2.60168 −0.134350
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 23.3146 1.19759 0.598795 0.800902i \(-0.295646\pi\)
0.598795 + 0.800902i \(0.295646\pi\)
\(380\) 0 0
\(381\) −41.9889 −2.15116
\(382\) 0 0
\(383\) 8.02791 0.410207 0.205103 0.978740i \(-0.434247\pi\)
0.205103 + 0.978740i \(0.434247\pi\)
\(384\) 0 0
\(385\) −6.00000 −0.305788
\(386\) 0 0
\(387\) −36.9526 −1.87841
\(388\) 0 0
\(389\) −23.7385 −1.20359 −0.601795 0.798651i \(-0.705547\pi\)
−0.601795 + 0.798651i \(0.705547\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) −23.0749 −1.16398
\(394\) 0 0
\(395\) −11.9442 −0.600977
\(396\) 0 0
\(397\) 3.04703 0.152926 0.0764630 0.997072i \(-0.475637\pi\)
0.0764630 + 0.997072i \(0.475637\pi\)
\(398\) 0 0
\(399\) 36.2783 1.81619
\(400\) 0 0
\(401\) 31.2201 1.55906 0.779530 0.626365i \(-0.215458\pi\)
0.779530 + 0.626365i \(0.215458\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 6.10284 0.303253
\(406\) 0 0
\(407\) −8.60774 −0.426670
\(408\) 0 0
\(409\) −33.4090 −1.65197 −0.825986 0.563691i \(-0.809381\pi\)
−0.825986 + 0.563691i \(0.809381\pi\)
\(410\) 0 0
\(411\) 4.33411 0.213786
\(412\) 0 0
\(413\) −22.6124 −1.11268
\(414\) 0 0
\(415\) −3.97209 −0.194982
\(416\) 0 0
\(417\) 50.1499 2.45585
\(418\) 0 0
\(419\) −30.7408 −1.50179 −0.750894 0.660423i \(-0.770377\pi\)
−0.750894 + 0.660423i \(0.770377\pi\)
\(420\) 0 0
\(421\) −19.2806 −0.939680 −0.469840 0.882752i \(-0.655688\pi\)
−0.469840 + 0.882752i \(0.655688\pi\)
\(422\) 0 0
\(423\) −43.8605 −2.13257
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 28.5351 1.38091
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 2.44535 0.117788 0.0588942 0.998264i \(-0.481243\pi\)
0.0588942 + 0.998264i \(0.481243\pi\)
\(432\) 0 0
\(433\) 2.00000 0.0961139 0.0480569 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) 0 0
\(435\) 11.5375 0.553179
\(436\) 0 0
\(437\) −24.8581 −1.18913
\(438\) 0 0
\(439\) 11.6659 0.556783 0.278391 0.960468i \(-0.410199\pi\)
0.278391 + 0.960468i \(0.410199\pi\)
\(440\) 0 0
\(441\) 2.50956 0.119503
\(442\) 0 0
\(443\) 20.8074 0.988588 0.494294 0.869295i \(-0.335427\pi\)
0.494294 + 0.869295i \(0.335427\pi\)
\(444\) 0 0
\(445\) −7.94419 −0.376590
\(446\) 0 0
\(447\) 36.2783 1.71590
\(448\) 0 0
\(449\) −30.2783 −1.42892 −0.714461 0.699676i \(-0.753328\pi\)
−0.714461 + 0.699676i \(0.753328\pi\)
\(450\) 0 0
\(451\) −3.61007 −0.169992
\(452\) 0 0
\(453\) −18.1173 −0.851225
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −22.6124 −1.05776 −0.528882 0.848695i \(-0.677389\pi\)
−0.528882 + 0.848695i \(0.677389\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −18.9419 −0.882210 −0.441105 0.897455i \(-0.645413\pi\)
−0.441105 + 0.897455i \(0.645413\pi\)
\(462\) 0 0
\(463\) −11.6380 −0.540863 −0.270431 0.962739i \(-0.587166\pi\)
−0.270431 + 0.962739i \(0.587166\pi\)
\(464\) 0 0
\(465\) −8.76873 −0.406640
\(466\) 0 0
\(467\) 25.0642 1.15983 0.579917 0.814676i \(-0.303085\pi\)
0.579917 + 0.814676i \(0.303085\pi\)
\(468\) 0 0
\(469\) −19.6659 −0.908086
\(470\) 0 0
\(471\) 32.0894 1.47860
\(472\) 0 0
\(473\) 21.2481 0.976987
\(474\) 0 0
\(475\) −5.03630 −0.231081
\(476\) 0 0
\(477\) −48.0168 −2.19854
\(478\) 0 0
\(479\) 18.4407 0.842577 0.421288 0.906927i \(-0.361578\pi\)
0.421288 + 0.906927i \(0.361578\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) −35.5543 −1.61777
\(484\) 0 0
\(485\) 0.462531 0.0210025
\(486\) 0 0
\(487\) −25.1196 −1.13828 −0.569140 0.822241i \(-0.692724\pi\)
−0.569140 + 0.822241i \(0.692724\pi\)
\(488\) 0 0
\(489\) 53.3532 2.41272
\(490\) 0 0
\(491\) 5.73850 0.258975 0.129487 0.991581i \(-0.458667\pi\)
0.129487 + 0.991581i \(0.458667\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 8.16706 0.367082
\(496\) 0 0
\(497\) 44.6077 2.00093
\(498\) 0 0
\(499\) 33.3704 1.49386 0.746932 0.664900i \(-0.231526\pi\)
0.746932 + 0.664900i \(0.231526\pi\)
\(500\) 0 0
\(501\) −50.2225 −2.24377
\(502\) 0 0
\(503\) −12.6789 −0.565326 −0.282663 0.959219i \(-0.591218\pi\)
−0.282663 + 0.959219i \(0.591218\pi\)
\(504\) 0 0
\(505\) 12.7408 0.566959
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −17.6147 −0.780759 −0.390380 0.920654i \(-0.627656\pi\)
−0.390380 + 0.920654i \(0.627656\pi\)
\(510\) 0 0
\(511\) 44.2225 1.95629
\(512\) 0 0
\(513\) −10.0726 −0.444716
\(514\) 0 0
\(515\) −4.13915 −0.182393
\(516\) 0 0
\(517\) 25.2201 1.10918
\(518\) 0 0
\(519\) −25.0023 −1.09748
\(520\) 0 0
\(521\) −16.4346 −0.720014 −0.360007 0.932950i \(-0.617226\pi\)
−0.360007 + 0.932950i \(0.617226\pi\)
\(522\) 0 0
\(523\) −20.4733 −0.895233 −0.447617 0.894226i \(-0.647727\pi\)
−0.447617 + 0.894226i \(0.647727\pi\)
\(524\) 0 0
\(525\) −7.20336 −0.314380
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 1.36202 0.0592182
\(530\) 0 0
\(531\) 30.7795 1.33571
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −7.06421 −0.305412
\(536\) 0 0
\(537\) 25.6827 1.10829
\(538\) 0 0
\(539\) −1.44302 −0.0621553
\(540\) 0 0
\(541\) −41.2928 −1.77531 −0.887657 0.460505i \(-0.847668\pi\)
−0.887657 + 0.460505i \(0.847668\pi\)
\(542\) 0 0
\(543\) 5.13075 0.220182
\(544\) 0 0
\(545\) 14.8692 0.636929
\(546\) 0 0
\(547\) −10.4174 −0.445418 −0.222709 0.974885i \(-0.571490\pi\)
−0.222709 + 0.974885i \(0.571490\pi\)
\(548\) 0 0
\(549\) −38.8413 −1.65771
\(550\) 0 0
\(551\) 22.3341 0.951465
\(552\) 0 0
\(553\) −33.0703 −1.40629
\(554\) 0 0
\(555\) −10.3341 −0.438659
\(556\) 0 0
\(557\) −20.5845 −0.872193 −0.436097 0.899900i \(-0.643639\pi\)
−0.436097 + 0.899900i \(0.643639\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 23.6766 0.997850 0.498925 0.866645i \(-0.333728\pi\)
0.498925 + 0.866645i \(0.333728\pi\)
\(564\) 0 0
\(565\) −3.13075 −0.131712
\(566\) 0 0
\(567\) 16.8972 0.709614
\(568\) 0 0
\(569\) −5.43695 −0.227929 −0.113965 0.993485i \(-0.536355\pi\)
−0.113965 + 0.993485i \(0.536355\pi\)
\(570\) 0 0
\(571\) −15.2760 −0.639279 −0.319640 0.947539i \(-0.603562\pi\)
−0.319640 + 0.947539i \(0.603562\pi\)
\(572\) 0 0
\(573\) −14.9297 −0.623698
\(574\) 0 0
\(575\) 4.93579 0.205837
\(576\) 0 0
\(577\) 1.76640 0.0735363 0.0367682 0.999324i \(-0.488294\pi\)
0.0367682 + 0.999324i \(0.488294\pi\)
\(578\) 0 0
\(579\) 10.5956 0.440339
\(580\) 0 0
\(581\) −10.9977 −0.456260
\(582\) 0 0
\(583\) 27.6101 1.14349
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 22.9139 0.945760 0.472880 0.881127i \(-0.343214\pi\)
0.472880 + 0.881127i \(0.343214\pi\)
\(588\) 0 0
\(589\) −16.9744 −0.699419
\(590\) 0 0
\(591\) 56.2225 2.31268
\(592\) 0 0
\(593\) 6.72404 0.276123 0.138062 0.990424i \(-0.455913\pi\)
0.138062 + 0.990424i \(0.455913\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 30.2057 1.23624
\(598\) 0 0
\(599\) −19.8669 −0.811740 −0.405870 0.913931i \(-0.633031\pi\)
−0.405870 + 0.913931i \(0.633031\pi\)
\(600\) 0 0
\(601\) −33.2201 −1.35508 −0.677539 0.735487i \(-0.736954\pi\)
−0.677539 + 0.735487i \(0.736954\pi\)
\(602\) 0 0
\(603\) 26.7687 1.09011
\(604\) 0 0
\(605\) 6.30387 0.256289
\(606\) 0 0
\(607\) 14.4174 0.585186 0.292593 0.956237i \(-0.405482\pi\)
0.292593 + 0.956237i \(0.405482\pi\)
\(608\) 0 0
\(609\) 31.9442 1.29444
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −11.3364 −0.457875 −0.228937 0.973441i \(-0.573525\pi\)
−0.228937 + 0.973441i \(0.573525\pi\)
\(614\) 0 0
\(615\) −4.33411 −0.174768
\(616\) 0 0
\(617\) −34.8907 −1.40465 −0.702323 0.711858i \(-0.747854\pi\)
−0.702323 + 0.711858i \(0.747854\pi\)
\(618\) 0 0
\(619\) −32.6464 −1.31217 −0.656084 0.754688i \(-0.727788\pi\)
−0.656084 + 0.754688i \(0.727788\pi\)
\(620\) 0 0
\(621\) 9.87158 0.396133
\(622\) 0 0
\(623\) −21.9953 −0.881225
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 28.3946 1.13397
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −47.3146 −1.88356 −0.941782 0.336224i \(-0.890850\pi\)
−0.941782 + 0.336224i \(0.890850\pi\)
\(632\) 0 0
\(633\) 42.3509 1.68330
\(634\) 0 0
\(635\) 16.1391 0.640463
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −60.7190 −2.40201
\(640\) 0 0
\(641\) −43.4817 −1.71742 −0.858711 0.512460i \(-0.828734\pi\)
−0.858711 + 0.512460i \(0.828734\pi\)
\(642\) 0 0
\(643\) −20.3062 −0.800798 −0.400399 0.916341i \(-0.631129\pi\)
−0.400399 + 0.916341i \(0.631129\pi\)
\(644\) 0 0
\(645\) 25.5096 1.00444
\(646\) 0 0
\(647\) −30.8027 −1.21098 −0.605490 0.795853i \(-0.707023\pi\)
−0.605490 + 0.795853i \(0.707023\pi\)
\(648\) 0 0
\(649\) −17.6985 −0.694725
\(650\) 0 0
\(651\) −24.2783 −0.951541
\(652\) 0 0
\(653\) 3.61007 0.141273 0.0706366 0.997502i \(-0.477497\pi\)
0.0706366 + 0.997502i \(0.477497\pi\)
\(654\) 0 0
\(655\) 8.86925 0.346550
\(656\) 0 0
\(657\) −60.1946 −2.34841
\(658\) 0 0
\(659\) −6.74083 −0.262585 −0.131293 0.991344i \(-0.541913\pi\)
−0.131293 + 0.991344i \(0.541913\pi\)
\(660\) 0 0
\(661\) 26.6682 1.03727 0.518637 0.854995i \(-0.326440\pi\)
0.518637 + 0.854995i \(0.326440\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −13.9442 −0.540732
\(666\) 0 0
\(667\) −21.8884 −0.847521
\(668\) 0 0
\(669\) −29.7553 −1.15041
\(670\) 0 0
\(671\) 22.3341 0.862199
\(672\) 0 0
\(673\) 8.61241 0.331984 0.165992 0.986127i \(-0.446917\pi\)
0.165992 + 0.986127i \(0.446917\pi\)
\(674\) 0 0
\(675\) 2.00000 0.0769800
\(676\) 0 0
\(677\) 8.12842 0.312401 0.156200 0.987725i \(-0.450075\pi\)
0.156200 + 0.987725i \(0.450075\pi\)
\(678\) 0 0
\(679\) 1.28063 0.0491459
\(680\) 0 0
\(681\) 41.5543 1.59236
\(682\) 0 0
\(683\) 8.02791 0.307179 0.153590 0.988135i \(-0.450917\pi\)
0.153590 + 0.988135i \(0.450917\pi\)
\(684\) 0 0
\(685\) −1.66589 −0.0636504
\(686\) 0 0
\(687\) −5.53747 −0.211268
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 22.1112 0.841151 0.420576 0.907257i \(-0.361828\pi\)
0.420576 + 0.907257i \(0.361828\pi\)
\(692\) 0 0
\(693\) 22.6124 0.858974
\(694\) 0 0
\(695\) −19.2760 −0.731179
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 7.46486 0.282347
\(700\) 0 0
\(701\) 6.47932 0.244721 0.122360 0.992486i \(-0.460954\pi\)
0.122360 + 0.992486i \(0.460954\pi\)
\(702\) 0 0
\(703\) −20.0047 −0.754490
\(704\) 0 0
\(705\) 30.2783 1.14035
\(706\) 0 0
\(707\) 35.2760 1.32669
\(708\) 0 0
\(709\) −2.85246 −0.107126 −0.0535631 0.998564i \(-0.517058\pi\)
−0.0535631 + 0.998564i \(0.517058\pi\)
\(710\) 0 0
\(711\) 45.0145 1.68817
\(712\) 0 0
\(713\) 16.6357 0.623010
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −8.24573 −0.307942
\(718\) 0 0
\(719\) −42.0941 −1.56984 −0.784922 0.619595i \(-0.787297\pi\)
−0.784922 + 0.619595i \(0.787297\pi\)
\(720\) 0 0
\(721\) −11.4602 −0.426800
\(722\) 0 0
\(723\) −12.4793 −0.464111
\(724\) 0 0
\(725\) −4.43462 −0.164698
\(726\) 0 0
\(727\) 32.8027 1.21659 0.608293 0.793713i \(-0.291855\pi\)
0.608293 + 0.793713i \(0.291855\pi\)
\(728\) 0 0
\(729\) −38.6101 −1.43000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 2.38993 0.0882739 0.0441370 0.999025i \(-0.485946\pi\)
0.0441370 + 0.999025i \(0.485946\pi\)
\(734\) 0 0
\(735\) −1.73243 −0.0639017
\(736\) 0 0
\(737\) −15.3923 −0.566981
\(738\) 0 0
\(739\) 41.4988 1.52656 0.763280 0.646068i \(-0.223588\pi\)
0.763280 + 0.646068i \(0.223588\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 39.8046 1.46029 0.730145 0.683292i \(-0.239452\pi\)
0.730145 + 0.683292i \(0.239452\pi\)
\(744\) 0 0
\(745\) −13.9442 −0.510875
\(746\) 0 0
\(747\) 14.9698 0.547715
\(748\) 0 0
\(749\) −19.5589 −0.714667
\(750\) 0 0
\(751\) −15.6659 −0.571656 −0.285828 0.958281i \(-0.592269\pi\)
−0.285828 + 0.958281i \(0.592269\pi\)
\(752\) 0 0
\(753\) −2.60774 −0.0950315
\(754\) 0 0
\(755\) 6.96370 0.253435
\(756\) 0 0
\(757\) 32.2736 1.17301 0.586503 0.809947i \(-0.300504\pi\)
0.586503 + 0.809947i \(0.300504\pi\)
\(758\) 0 0
\(759\) −27.8279 −1.01009
\(760\) 0 0
\(761\) 13.2806 0.481422 0.240711 0.970597i \(-0.422619\pi\)
0.240711 + 0.970597i \(0.422619\pi\)
\(762\) 0 0
\(763\) 41.1690 1.49042
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −12.2010 −0.439980 −0.219990 0.975502i \(-0.570603\pi\)
−0.219990 + 0.975502i \(0.570603\pi\)
\(770\) 0 0
\(771\) 50.6850 1.82538
\(772\) 0 0
\(773\) 27.9721 1.00609 0.503043 0.864261i \(-0.332214\pi\)
0.503043 + 0.864261i \(0.332214\pi\)
\(774\) 0 0
\(775\) 3.37041 0.121069
\(776\) 0 0
\(777\) −28.6124 −1.02646
\(778\) 0 0
\(779\) −8.38993 −0.300600
\(780\) 0 0
\(781\) 34.9139 1.24932
\(782\) 0 0
\(783\) −8.86925 −0.316961
\(784\) 0 0
\(785\) −12.3341 −0.440223
\(786\) 0 0
\(787\) −10.2336 −0.364788 −0.182394 0.983225i \(-0.558385\pi\)
−0.182394 + 0.983225i \(0.558385\pi\)
\(788\) 0 0
\(789\) −10.9139 −0.388547
\(790\) 0 0
\(791\) −8.66822 −0.308206
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 33.1475 1.17562
\(796\) 0 0
\(797\) 23.3532 0.827214 0.413607 0.910456i \(-0.364269\pi\)
0.413607 + 0.910456i \(0.364269\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 29.9395 1.05786
\(802\) 0 0
\(803\) 34.6124 1.22145
\(804\) 0 0
\(805\) 13.6659 0.481659
\(806\) 0 0
\(807\) 79.2974 2.79140
\(808\) 0 0
\(809\) 37.1364 1.30565 0.652824 0.757510i \(-0.273584\pi\)
0.652824 + 0.757510i \(0.273584\pi\)
\(810\) 0 0
\(811\) 20.6296 0.724403 0.362201 0.932100i \(-0.382025\pi\)
0.362201 + 0.932100i \(0.382025\pi\)
\(812\) 0 0
\(813\) 12.5798 0.441194
\(814\) 0 0
\(815\) −20.5072 −0.718337
\(816\) 0 0
\(817\) 49.3811 1.72763
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −20.6077 −0.719215 −0.359608 0.933104i \(-0.617089\pi\)
−0.359608 + 0.933104i \(0.617089\pi\)
\(822\) 0 0
\(823\) −39.8050 −1.38752 −0.693758 0.720208i \(-0.744046\pi\)
−0.693758 + 0.720208i \(0.744046\pi\)
\(824\) 0 0
\(825\) −5.63798 −0.196289
\(826\) 0 0
\(827\) 55.8605 1.94246 0.971229 0.238146i \(-0.0765398\pi\)
0.971229 + 0.238146i \(0.0765398\pi\)
\(828\) 0 0
\(829\) 33.5822 1.16636 0.583178 0.812344i \(-0.301809\pi\)
0.583178 + 0.812344i \(0.301809\pi\)
\(830\) 0 0
\(831\) −7.81110 −0.270964
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 19.3039 0.668038
\(836\) 0 0
\(837\) 6.74083 0.232997
\(838\) 0 0
\(839\) 26.4454 0.912995 0.456497 0.889725i \(-0.349104\pi\)
0.456497 + 0.889725i \(0.349104\pi\)
\(840\) 0 0
\(841\) −9.33411 −0.321866
\(842\) 0 0
\(843\) 56.2225 1.93641
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 17.4537 0.599718
\(848\) 0 0
\(849\) −6.43462 −0.220836
\(850\) 0 0
\(851\) 19.6054 0.672065
\(852\) 0 0
\(853\) −35.7153 −1.22287 −0.611433 0.791296i \(-0.709407\pi\)
−0.611433 + 0.791296i \(0.709407\pi\)
\(854\) 0 0
\(855\) 18.9805 0.649119
\(856\) 0 0
\(857\) −22.0894 −0.754559 −0.377280 0.926099i \(-0.623140\pi\)
−0.377280 + 0.926099i \(0.623140\pi\)
\(858\) 0 0
\(859\) −39.2201 −1.33817 −0.669087 0.743184i \(-0.733315\pi\)
−0.669087 + 0.743184i \(0.733315\pi\)
\(860\) 0 0
\(861\) −12.0000 −0.408959
\(862\) 0 0
\(863\) 28.9744 0.986301 0.493150 0.869944i \(-0.335845\pi\)
0.493150 + 0.869944i \(0.335845\pi\)
\(864\) 0 0
\(865\) 9.61007 0.326752
\(866\) 0 0
\(867\) −44.2285 −1.50208
\(868\) 0 0
\(869\) −25.8837 −0.878045
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −1.74316 −0.0589970
\(874\) 0 0
\(875\) 2.76873 0.0936003
\(876\) 0 0
\(877\) −30.9251 −1.04427 −0.522133 0.852864i \(-0.674863\pi\)
−0.522133 + 0.852864i \(0.674863\pi\)
\(878\) 0 0
\(879\) 62.2225 2.09871
\(880\) 0 0
\(881\) −41.2695 −1.39041 −0.695203 0.718814i \(-0.744685\pi\)
−0.695203 + 0.718814i \(0.744685\pi\)
\(882\) 0 0
\(883\) −15.1973 −0.511430 −0.255715 0.966752i \(-0.582311\pi\)
−0.255715 + 0.966752i \(0.582311\pi\)
\(884\) 0 0
\(885\) −21.2481 −0.714246
\(886\) 0 0
\(887\) −12.4174 −0.416937 −0.208468 0.978029i \(-0.566848\pi\)
−0.208468 + 0.978029i \(0.566848\pi\)
\(888\) 0 0
\(889\) 44.6850 1.49869
\(890\) 0 0
\(891\) 13.2252 0.443061
\(892\) 0 0
\(893\) 58.6124 1.96139
\(894\) 0 0
\(895\) −9.87158 −0.329970
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −14.9465 −0.498494
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 70.6292 2.35039
\(904\) 0 0
\(905\) −1.97209 −0.0655546
\(906\) 0 0
\(907\) 11.4709 0.380886 0.190443 0.981698i \(-0.439008\pi\)
0.190443 + 0.981698i \(0.439008\pi\)
\(908\) 0 0
\(909\) −48.0168 −1.59262
\(910\) 0 0
\(911\) 54.5734 1.80810 0.904048 0.427430i \(-0.140581\pi\)
0.904048 + 0.427430i \(0.140581\pi\)
\(912\) 0 0
\(913\) −8.60774 −0.284875
\(914\) 0 0
\(915\) 26.8134 0.886425
\(916\) 0 0
\(917\) 24.5566 0.810930
\(918\) 0 0
\(919\) 9.05815 0.298801 0.149400 0.988777i \(-0.452266\pi\)
0.149400 + 0.988777i \(0.452266\pi\)
\(920\) 0 0
\(921\) −35.2928 −1.16294
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 3.97209 0.130602
\(926\) 0 0
\(927\) 15.5993 0.512350
\(928\) 0 0
\(929\) 50.2225 1.64775 0.823873 0.566774i \(-0.191809\pi\)
0.823873 + 0.566774i \(0.191809\pi\)
\(930\) 0 0
\(931\) −3.35362 −0.109911
\(932\) 0 0
\(933\) 5.53747 0.181289
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 5.38759 0.176005 0.0880025 0.996120i \(-0.471952\pi\)
0.0880025 + 0.996120i \(0.471952\pi\)
\(938\) 0 0
\(939\) −58.8181 −1.91946
\(940\) 0 0
\(941\) −58.8302 −1.91781 −0.958905 0.283726i \(-0.908429\pi\)
−0.958905 + 0.283726i \(0.908429\pi\)
\(942\) 0 0
\(943\) 8.22248 0.267761
\(944\) 0 0
\(945\) 5.53747 0.180134
\(946\) 0 0
\(947\) 12.6403 0.410755 0.205377 0.978683i \(-0.434158\pi\)
0.205377 + 0.978683i \(0.434158\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −27.6706 −0.897279
\(952\) 0 0
\(953\) 32.6077 1.05627 0.528134 0.849161i \(-0.322892\pi\)
0.528134 + 0.849161i \(0.322892\pi\)
\(954\) 0 0
\(955\) 5.73850 0.185693
\(956\) 0 0
\(957\) 25.0023 0.808211
\(958\) 0 0
\(959\) −4.61241 −0.148942
\(960\) 0 0
\(961\) −19.6403 −0.633558
\(962\) 0 0
\(963\) 26.6231 0.857918
\(964\) 0 0
\(965\) −4.07261 −0.131102
\(966\) 0 0
\(967\) −20.5845 −0.661953 −0.330976 0.943639i \(-0.607378\pi\)
−0.330976 + 0.943639i \(0.607378\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −60.8349 −1.95228 −0.976142 0.217132i \(-0.930330\pi\)
−0.976142 + 0.217132i \(0.930330\pi\)
\(972\) 0 0
\(973\) −53.3700 −1.71096
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −7.30387 −0.233672 −0.116836 0.993151i \(-0.537275\pi\)
−0.116836 + 0.993151i \(0.537275\pi\)
\(978\) 0 0
\(979\) −17.2155 −0.550209
\(980\) 0 0
\(981\) −56.0382 −1.78916
\(982\) 0 0
\(983\) 40.2504 1.28379 0.641894 0.766793i \(-0.278149\pi\)
0.641894 + 0.766793i \(0.278149\pi\)
\(984\) 0 0
\(985\) −21.6101 −0.688554
\(986\) 0 0
\(987\) 83.8326 2.66842
\(988\) 0 0
\(989\) −48.3956 −1.53889
\(990\) 0 0
\(991\) −50.2271 −1.59552 −0.797759 0.602977i \(-0.793981\pi\)
−0.797759 + 0.602977i \(0.793981\pi\)
\(992\) 0 0
\(993\) −3.03024 −0.0961617
\(994\) 0 0
\(995\) −11.6101 −0.368064
\(996\) 0 0
\(997\) 0.273633 0.00866606 0.00433303 0.999991i \(-0.498621\pi\)
0.00433303 + 0.999991i \(0.498621\pi\)
\(998\) 0 0
\(999\) 7.94419 0.251343
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 3380.2.a.m.1.3 3
13.5 odd 4 260.2.f.a.181.6 yes 6
13.8 odd 4 260.2.f.a.181.5 6
13.12 even 2 3380.2.a.n.1.3 3
39.5 even 4 2340.2.c.d.181.1 6
39.8 even 4 2340.2.c.d.181.6 6
52.31 even 4 1040.2.k.c.961.2 6
52.47 even 4 1040.2.k.c.961.1 6
65.8 even 4 1300.2.d.d.649.6 6
65.18 even 4 1300.2.d.c.649.6 6
65.34 odd 4 1300.2.f.e.701.1 6
65.44 odd 4 1300.2.f.e.701.2 6
65.47 even 4 1300.2.d.c.649.1 6
65.57 even 4 1300.2.d.d.649.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
260.2.f.a.181.5 6 13.8 odd 4
260.2.f.a.181.6 yes 6 13.5 odd 4
1040.2.k.c.961.1 6 52.47 even 4
1040.2.k.c.961.2 6 52.31 even 4
1300.2.d.c.649.1 6 65.47 even 4
1300.2.d.c.649.6 6 65.18 even 4
1300.2.d.d.649.1 6 65.57 even 4
1300.2.d.d.649.6 6 65.8 even 4
1300.2.f.e.701.1 6 65.34 odd 4
1300.2.f.e.701.2 6 65.44 odd 4
2340.2.c.d.181.1 6 39.5 even 4
2340.2.c.d.181.6 6 39.8 even 4
3380.2.a.m.1.3 3 1.1 even 1 trivial
3380.2.a.n.1.3 3 13.12 even 2