Properties

Label 320.2.d.b.161.3
Level $320$
Weight $2$
Character 320.161
Analytic conductor $2.555$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 320.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.55521286468\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 161.3
Root \(0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 320.161
Dual form 320.2.d.b.161.2

$q$-expansion

\(f(q)\) \(=\) \(q+0.732051i q^{3} -1.00000i q^{5} +1.26795 q^{7} +2.46410 q^{9} +O(q^{10})\) \(q+0.732051i q^{3} -1.00000i q^{5} +1.26795 q^{7} +2.46410 q^{9} +3.46410i q^{11} -3.46410i q^{13} +0.732051 q^{15} +3.46410 q^{17} +2.00000i q^{19} +0.928203i q^{21} +8.19615 q^{23} -1.00000 q^{25} +4.00000i q^{27} -9.46410 q^{31} -2.53590 q^{33} -1.26795i q^{35} +6.00000i q^{37} +2.53590 q^{39} +2.53590 q^{41} -10.1962i q^{43} -2.46410i q^{45} -8.19615 q^{47} -5.39230 q^{49} +2.53590i q^{51} -10.3923i q^{53} +3.46410 q^{55} -1.46410 q^{57} -6.00000i q^{59} +12.9282i q^{61} +3.12436 q^{63} -3.46410 q^{65} -10.1962i q^{67} +6.00000i q^{69} -4.39230 q^{71} -14.3923 q^{73} -0.732051i q^{75} +4.39230i q^{77} -12.0000 q^{79} +4.46410 q^{81} -4.73205i q^{83} -3.46410i q^{85} +0.928203 q^{89} -4.39230i q^{91} -6.92820i q^{93} +2.00000 q^{95} -6.39230 q^{97} +8.53590i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 12 q^{7} - 4 q^{9} + O(q^{10}) \) \( 4 q + 12 q^{7} - 4 q^{9} - 4 q^{15} + 12 q^{23} - 4 q^{25} - 24 q^{31} - 24 q^{33} + 24 q^{39} + 24 q^{41} - 12 q^{47} + 20 q^{49} + 8 q^{57} - 36 q^{63} + 24 q^{71} - 16 q^{73} - 48 q^{79} + 4 q^{81} - 24 q^{89} + 8 q^{95} + 16 q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/320\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(257\) \(261\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) 0.732051i 0.422650i 0.977416 + 0.211325i \(0.0677778\pi\)
−0.977416 + 0.211325i \(0.932222\pi\)
\(4\) 0 0
\(5\) − 1.00000i − 0.447214i
\(6\) 0 0
\(7\) 1.26795 0.479240 0.239620 0.970867i \(-0.422977\pi\)
0.239620 + 0.970867i \(0.422977\pi\)
\(8\) 0 0
\(9\) 2.46410 0.821367
\(10\) 0 0
\(11\) 3.46410i 1.04447i 0.852803 + 0.522233i \(0.174901\pi\)
−0.852803 + 0.522233i \(0.825099\pi\)
\(12\) 0 0
\(13\) − 3.46410i − 0.960769i −0.877058 0.480384i \(-0.840497\pi\)
0.877058 0.480384i \(-0.159503\pi\)
\(14\) 0 0
\(15\) 0.732051 0.189015
\(16\) 0 0
\(17\) 3.46410 0.840168 0.420084 0.907485i \(-0.362001\pi\)
0.420084 + 0.907485i \(0.362001\pi\)
\(18\) 0 0
\(19\) 2.00000i 0.458831i 0.973329 + 0.229416i \(0.0736815\pi\)
−0.973329 + 0.229416i \(0.926318\pi\)
\(20\) 0 0
\(21\) 0.928203i 0.202551i
\(22\) 0 0
\(23\) 8.19615 1.70902 0.854508 0.519438i \(-0.173859\pi\)
0.854508 + 0.519438i \(0.173859\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 4.00000i 0.769800i
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −9.46410 −1.69980 −0.849901 0.526942i \(-0.823339\pi\)
−0.849901 + 0.526942i \(0.823339\pi\)
\(32\) 0 0
\(33\) −2.53590 −0.441443
\(34\) 0 0
\(35\) − 1.26795i − 0.214323i
\(36\) 0 0
\(37\) 6.00000i 0.986394i 0.869918 + 0.493197i \(0.164172\pi\)
−0.869918 + 0.493197i \(0.835828\pi\)
\(38\) 0 0
\(39\) 2.53590 0.406069
\(40\) 0 0
\(41\) 2.53590 0.396041 0.198020 0.980198i \(-0.436549\pi\)
0.198020 + 0.980198i \(0.436549\pi\)
\(42\) 0 0
\(43\) − 10.1962i − 1.55490i −0.628946 0.777449i \(-0.716513\pi\)
0.628946 0.777449i \(-0.283487\pi\)
\(44\) 0 0
\(45\) − 2.46410i − 0.367327i
\(46\) 0 0
\(47\) −8.19615 −1.19553 −0.597766 0.801671i \(-0.703945\pi\)
−0.597766 + 0.801671i \(0.703945\pi\)
\(48\) 0 0
\(49\) −5.39230 −0.770329
\(50\) 0 0
\(51\) 2.53590i 0.355097i
\(52\) 0 0
\(53\) − 10.3923i − 1.42749i −0.700404 0.713746i \(-0.746997\pi\)
0.700404 0.713746i \(-0.253003\pi\)
\(54\) 0 0
\(55\) 3.46410 0.467099
\(56\) 0 0
\(57\) −1.46410 −0.193925
\(58\) 0 0
\(59\) − 6.00000i − 0.781133i −0.920575 0.390567i \(-0.872279\pi\)
0.920575 0.390567i \(-0.127721\pi\)
\(60\) 0 0
\(61\) 12.9282i 1.65529i 0.561254 + 0.827643i \(0.310319\pi\)
−0.561254 + 0.827643i \(0.689681\pi\)
\(62\) 0 0
\(63\) 3.12436 0.393632
\(64\) 0 0
\(65\) −3.46410 −0.429669
\(66\) 0 0
\(67\) − 10.1962i − 1.24566i −0.782358 0.622829i \(-0.785983\pi\)
0.782358 0.622829i \(-0.214017\pi\)
\(68\) 0 0
\(69\) 6.00000i 0.722315i
\(70\) 0 0
\(71\) −4.39230 −0.521271 −0.260635 0.965437i \(-0.583932\pi\)
−0.260635 + 0.965437i \(0.583932\pi\)
\(72\) 0 0
\(73\) −14.3923 −1.68449 −0.842246 0.539093i \(-0.818767\pi\)
−0.842246 + 0.539093i \(0.818767\pi\)
\(74\) 0 0
\(75\) − 0.732051i − 0.0845299i
\(76\) 0 0
\(77\) 4.39230i 0.500550i
\(78\) 0 0
\(79\) −12.0000 −1.35011 −0.675053 0.737769i \(-0.735879\pi\)
−0.675053 + 0.737769i \(0.735879\pi\)
\(80\) 0 0
\(81\) 4.46410 0.496011
\(82\) 0 0
\(83\) − 4.73205i − 0.519410i −0.965688 0.259705i \(-0.916375\pi\)
0.965688 0.259705i \(-0.0836253\pi\)
\(84\) 0 0
\(85\) − 3.46410i − 0.375735i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0.928203 0.0983893 0.0491947 0.998789i \(-0.484335\pi\)
0.0491947 + 0.998789i \(0.484335\pi\)
\(90\) 0 0
\(91\) − 4.39230i − 0.460439i
\(92\) 0 0
\(93\) − 6.92820i − 0.718421i
\(94\) 0 0
\(95\) 2.00000 0.205196
\(96\) 0 0
\(97\) −6.39230 −0.649040 −0.324520 0.945879i \(-0.605203\pi\)
−0.324520 + 0.945879i \(0.605203\pi\)
\(98\) 0 0
\(99\) 8.53590i 0.857890i
\(100\) 0 0
\(101\) 12.0000i 1.19404i 0.802225 + 0.597022i \(0.203650\pi\)
−0.802225 + 0.597022i \(0.796350\pi\)
\(102\) 0 0
\(103\) 8.19615 0.807591 0.403795 0.914849i \(-0.367691\pi\)
0.403795 + 0.914849i \(0.367691\pi\)
\(104\) 0 0
\(105\) 0.928203 0.0905834
\(106\) 0 0
\(107\) 16.7321i 1.61755i 0.588119 + 0.808774i \(0.299869\pi\)
−0.588119 + 0.808774i \(0.700131\pi\)
\(108\) 0 0
\(109\) 0.928203i 0.0889057i 0.999011 + 0.0444529i \(0.0141545\pi\)
−0.999011 + 0.0444529i \(0.985846\pi\)
\(110\) 0 0
\(111\) −4.39230 −0.416899
\(112\) 0 0
\(113\) 0.928203 0.0873180 0.0436590 0.999046i \(-0.486098\pi\)
0.0436590 + 0.999046i \(0.486098\pi\)
\(114\) 0 0
\(115\) − 8.19615i − 0.764295i
\(116\) 0 0
\(117\) − 8.53590i − 0.789144i
\(118\) 0 0
\(119\) 4.39230 0.402642
\(120\) 0 0
\(121\) −1.00000 −0.0909091
\(122\) 0 0
\(123\) 1.85641i 0.167387i
\(124\) 0 0
\(125\) 1.00000i 0.0894427i
\(126\) 0 0
\(127\) −3.80385 −0.337537 −0.168768 0.985656i \(-0.553979\pi\)
−0.168768 + 0.985656i \(0.553979\pi\)
\(128\) 0 0
\(129\) 7.46410 0.657178
\(130\) 0 0
\(131\) 10.3923i 0.907980i 0.891007 + 0.453990i \(0.150000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(132\) 0 0
\(133\) 2.53590i 0.219890i
\(134\) 0 0
\(135\) 4.00000 0.344265
\(136\) 0 0
\(137\) 12.9282 1.10453 0.552265 0.833668i \(-0.313763\pi\)
0.552265 + 0.833668i \(0.313763\pi\)
\(138\) 0 0
\(139\) − 10.0000i − 0.848189i −0.905618 0.424094i \(-0.860592\pi\)
0.905618 0.424094i \(-0.139408\pi\)
\(140\) 0 0
\(141\) − 6.00000i − 0.505291i
\(142\) 0 0
\(143\) 12.0000 1.00349
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 3.94744i − 0.325579i
\(148\) 0 0
\(149\) − 18.0000i − 1.47462i −0.675556 0.737309i \(-0.736096\pi\)
0.675556 0.737309i \(-0.263904\pi\)
\(150\) 0 0
\(151\) 2.53590 0.206368 0.103184 0.994662i \(-0.467097\pi\)
0.103184 + 0.994662i \(0.467097\pi\)
\(152\) 0 0
\(153\) 8.53590 0.690086
\(154\) 0 0
\(155\) 9.46410i 0.760175i
\(156\) 0 0
\(157\) − 0.928203i − 0.0740787i −0.999314 0.0370393i \(-0.988207\pi\)
0.999314 0.0370393i \(-0.0117927\pi\)
\(158\) 0 0
\(159\) 7.60770 0.603329
\(160\) 0 0
\(161\) 10.3923 0.819028
\(162\) 0 0
\(163\) 5.80385i 0.454592i 0.973826 + 0.227296i \(0.0729886\pi\)
−0.973826 + 0.227296i \(0.927011\pi\)
\(164\) 0 0
\(165\) 2.53590i 0.197419i
\(166\) 0 0
\(167\) −8.19615 −0.634237 −0.317119 0.948386i \(-0.602715\pi\)
−0.317119 + 0.948386i \(0.602715\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 4.92820i 0.376869i
\(172\) 0 0
\(173\) − 6.00000i − 0.456172i −0.973641 0.228086i \(-0.926753\pi\)
0.973641 0.228086i \(-0.0732467\pi\)
\(174\) 0 0
\(175\) −1.26795 −0.0958479
\(176\) 0 0
\(177\) 4.39230 0.330146
\(178\) 0 0
\(179\) 19.8564i 1.48414i 0.670324 + 0.742069i \(0.266155\pi\)
−0.670324 + 0.742069i \(0.733845\pi\)
\(180\) 0 0
\(181\) − 6.92820i − 0.514969i −0.966282 0.257485i \(-0.917106\pi\)
0.966282 0.257485i \(-0.0828937\pi\)
\(182\) 0 0
\(183\) −9.46410 −0.699607
\(184\) 0 0
\(185\) 6.00000 0.441129
\(186\) 0 0
\(187\) 12.0000i 0.877527i
\(188\) 0 0
\(189\) 5.07180i 0.368919i
\(190\) 0 0
\(191\) 16.3923 1.18611 0.593053 0.805164i \(-0.297923\pi\)
0.593053 + 0.805164i \(0.297923\pi\)
\(192\) 0 0
\(193\) 6.39230 0.460128 0.230064 0.973175i \(-0.426106\pi\)
0.230064 + 0.973175i \(0.426106\pi\)
\(194\) 0 0
\(195\) − 2.53590i − 0.181599i
\(196\) 0 0
\(197\) 10.3923i 0.740421i 0.928948 + 0.370211i \(0.120714\pi\)
−0.928948 + 0.370211i \(0.879286\pi\)
\(198\) 0 0
\(199\) −6.92820 −0.491127 −0.245564 0.969380i \(-0.578973\pi\)
−0.245564 + 0.969380i \(0.578973\pi\)
\(200\) 0 0
\(201\) 7.46410 0.526477
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) − 2.53590i − 0.177115i
\(206\) 0 0
\(207\) 20.1962 1.40373
\(208\) 0 0
\(209\) −6.92820 −0.479234
\(210\) 0 0
\(211\) − 6.39230i − 0.440064i −0.975493 0.220032i \(-0.929384\pi\)
0.975493 0.220032i \(-0.0706162\pi\)
\(212\) 0 0
\(213\) − 3.21539i − 0.220315i
\(214\) 0 0
\(215\) −10.1962 −0.695372
\(216\) 0 0
\(217\) −12.0000 −0.814613
\(218\) 0 0
\(219\) − 10.5359i − 0.711950i
\(220\) 0 0
\(221\) − 12.0000i − 0.807207i
\(222\) 0 0
\(223\) −15.1244 −1.01280 −0.506401 0.862298i \(-0.669024\pi\)
−0.506401 + 0.862298i \(0.669024\pi\)
\(224\) 0 0
\(225\) −2.46410 −0.164273
\(226\) 0 0
\(227\) − 18.5885i − 1.23376i −0.787058 0.616880i \(-0.788397\pi\)
0.787058 0.616880i \(-0.211603\pi\)
\(228\) 0 0
\(229\) 18.9282i 1.25081i 0.780300 + 0.625405i \(0.215066\pi\)
−0.780300 + 0.625405i \(0.784934\pi\)
\(230\) 0 0
\(231\) −3.21539 −0.211557
\(232\) 0 0
\(233\) −1.60770 −0.105324 −0.0526618 0.998612i \(-0.516771\pi\)
−0.0526618 + 0.998612i \(0.516771\pi\)
\(234\) 0 0
\(235\) 8.19615i 0.534658i
\(236\) 0 0
\(237\) − 8.78461i − 0.570622i
\(238\) 0 0
\(239\) 20.7846 1.34444 0.672222 0.740349i \(-0.265340\pi\)
0.672222 + 0.740349i \(0.265340\pi\)
\(240\) 0 0
\(241\) −20.3923 −1.31358 −0.656792 0.754072i \(-0.728087\pi\)
−0.656792 + 0.754072i \(0.728087\pi\)
\(242\) 0 0
\(243\) 15.2679i 0.979439i
\(244\) 0 0
\(245\) 5.39230i 0.344502i
\(246\) 0 0
\(247\) 6.92820 0.440831
\(248\) 0 0
\(249\) 3.46410 0.219529
\(250\) 0 0
\(251\) − 15.4641i − 0.976085i −0.872820 0.488043i \(-0.837711\pi\)
0.872820 0.488043i \(-0.162289\pi\)
\(252\) 0 0
\(253\) 28.3923i 1.78501i
\(254\) 0 0
\(255\) 2.53590 0.158804
\(256\) 0 0
\(257\) 6.00000 0.374270 0.187135 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) 0 0
\(259\) 7.60770i 0.472719i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −20.1962 −1.24535 −0.622674 0.782481i \(-0.713954\pi\)
−0.622674 + 0.782481i \(0.713954\pi\)
\(264\) 0 0
\(265\) −10.3923 −0.638394
\(266\) 0 0
\(267\) 0.679492i 0.0415842i
\(268\) 0 0
\(269\) − 14.7846i − 0.901434i −0.892667 0.450717i \(-0.851168\pi\)
0.892667 0.450717i \(-0.148832\pi\)
\(270\) 0 0
\(271\) 4.39230 0.266814 0.133407 0.991061i \(-0.457408\pi\)
0.133407 + 0.991061i \(0.457408\pi\)
\(272\) 0 0
\(273\) 3.21539 0.194604
\(274\) 0 0
\(275\) − 3.46410i − 0.208893i
\(276\) 0 0
\(277\) − 19.8564i − 1.19306i −0.802592 0.596528i \(-0.796546\pi\)
0.802592 0.596528i \(-0.203454\pi\)
\(278\) 0 0
\(279\) −23.3205 −1.39616
\(280\) 0 0
\(281\) −28.3923 −1.69374 −0.846871 0.531798i \(-0.821517\pi\)
−0.846871 + 0.531798i \(0.821517\pi\)
\(282\) 0 0
\(283\) − 10.5885i − 0.629418i −0.949188 0.314709i \(-0.898093\pi\)
0.949188 0.314709i \(-0.101907\pi\)
\(284\) 0 0
\(285\) 1.46410i 0.0867259i
\(286\) 0 0
\(287\) 3.21539 0.189798
\(288\) 0 0
\(289\) −5.00000 −0.294118
\(290\) 0 0
\(291\) − 4.67949i − 0.274317i
\(292\) 0 0
\(293\) 30.0000i 1.75262i 0.481749 + 0.876309i \(0.340002\pi\)
−0.481749 + 0.876309i \(0.659998\pi\)
\(294\) 0 0
\(295\) −6.00000 −0.349334
\(296\) 0 0
\(297\) −13.8564 −0.804030
\(298\) 0 0
\(299\) − 28.3923i − 1.64197i
\(300\) 0 0
\(301\) − 12.9282i − 0.745169i
\(302\) 0 0
\(303\) −8.78461 −0.504663
\(304\) 0 0
\(305\) 12.9282 0.740267
\(306\) 0 0
\(307\) − 34.1962i − 1.95168i −0.218492 0.975839i \(-0.570114\pi\)
0.218492 0.975839i \(-0.429886\pi\)
\(308\) 0 0
\(309\) 6.00000i 0.341328i
\(310\) 0 0
\(311\) −7.60770 −0.431393 −0.215696 0.976460i \(-0.569202\pi\)
−0.215696 + 0.976460i \(0.569202\pi\)
\(312\) 0 0
\(313\) 22.0000 1.24351 0.621757 0.783210i \(-0.286419\pi\)
0.621757 + 0.783210i \(0.286419\pi\)
\(314\) 0 0
\(315\) − 3.12436i − 0.176037i
\(316\) 0 0
\(317\) 22.3923i 1.25768i 0.777536 + 0.628839i \(0.216469\pi\)
−0.777536 + 0.628839i \(0.783531\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −12.2487 −0.683656
\(322\) 0 0
\(323\) 6.92820i 0.385496i
\(324\) 0 0
\(325\) 3.46410i 0.192154i
\(326\) 0 0
\(327\) −0.679492 −0.0375760
\(328\) 0 0
\(329\) −10.3923 −0.572946
\(330\) 0 0
\(331\) − 5.60770i − 0.308227i −0.988053 0.154113i \(-0.950748\pi\)
0.988053 0.154113i \(-0.0492521\pi\)
\(332\) 0 0
\(333\) 14.7846i 0.810192i
\(334\) 0 0
\(335\) −10.1962 −0.557075
\(336\) 0 0
\(337\) −14.0000 −0.762629 −0.381314 0.924445i \(-0.624528\pi\)
−0.381314 + 0.924445i \(0.624528\pi\)
\(338\) 0 0
\(339\) 0.679492i 0.0369049i
\(340\) 0 0
\(341\) − 32.7846i − 1.77539i
\(342\) 0 0
\(343\) −15.7128 −0.848412
\(344\) 0 0
\(345\) 6.00000 0.323029
\(346\) 0 0
\(347\) 28.0526i 1.50594i 0.658055 + 0.752970i \(0.271380\pi\)
−0.658055 + 0.752970i \(0.728620\pi\)
\(348\) 0 0
\(349\) 8.78461i 0.470229i 0.971968 + 0.235115i \(0.0755466\pi\)
−0.971968 + 0.235115i \(0.924453\pi\)
\(350\) 0 0
\(351\) 13.8564 0.739600
\(352\) 0 0
\(353\) −14.7846 −0.786905 −0.393453 0.919345i \(-0.628719\pi\)
−0.393453 + 0.919345i \(0.628719\pi\)
\(354\) 0 0
\(355\) 4.39230i 0.233119i
\(356\) 0 0
\(357\) 3.21539i 0.170177i
\(358\) 0 0
\(359\) 8.78461 0.463634 0.231817 0.972759i \(-0.425533\pi\)
0.231817 + 0.972759i \(0.425533\pi\)
\(360\) 0 0
\(361\) 15.0000 0.789474
\(362\) 0 0
\(363\) − 0.732051i − 0.0384227i
\(364\) 0 0
\(365\) 14.3923i 0.753328i
\(366\) 0 0
\(367\) 10.0526 0.524739 0.262370 0.964967i \(-0.415496\pi\)
0.262370 + 0.964967i \(0.415496\pi\)
\(368\) 0 0
\(369\) 6.24871 0.325295
\(370\) 0 0
\(371\) − 13.1769i − 0.684111i
\(372\) 0 0
\(373\) 7.85641i 0.406789i 0.979097 + 0.203395i \(0.0651974\pi\)
−0.979097 + 0.203395i \(0.934803\pi\)
\(374\) 0 0
\(375\) −0.732051 −0.0378029
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) − 2.00000i − 0.102733i −0.998680 0.0513665i \(-0.983642\pi\)
0.998680 0.0513665i \(-0.0163577\pi\)
\(380\) 0 0
\(381\) − 2.78461i − 0.142660i
\(382\) 0 0
\(383\) 3.80385 0.194368 0.0971838 0.995266i \(-0.469017\pi\)
0.0971838 + 0.995266i \(0.469017\pi\)
\(384\) 0 0
\(385\) 4.39230 0.223853
\(386\) 0 0
\(387\) − 25.1244i − 1.27714i
\(388\) 0 0
\(389\) − 26.7846i − 1.35803i −0.734123 0.679017i \(-0.762406\pi\)
0.734123 0.679017i \(-0.237594\pi\)
\(390\) 0 0
\(391\) 28.3923 1.43586
\(392\) 0 0
\(393\) −7.60770 −0.383757
\(394\) 0 0
\(395\) 12.0000i 0.603786i
\(396\) 0 0
\(397\) − 27.4641i − 1.37838i −0.724579 0.689192i \(-0.757966\pi\)
0.724579 0.689192i \(-0.242034\pi\)
\(398\) 0 0
\(399\) −1.85641 −0.0929366
\(400\) 0 0
\(401\) −4.14359 −0.206921 −0.103461 0.994634i \(-0.532992\pi\)
−0.103461 + 0.994634i \(0.532992\pi\)
\(402\) 0 0
\(403\) 32.7846i 1.63312i
\(404\) 0 0
\(405\) − 4.46410i − 0.221823i
\(406\) 0 0
\(407\) −20.7846 −1.03025
\(408\) 0 0
\(409\) 3.60770 0.178389 0.0891945 0.996014i \(-0.471571\pi\)
0.0891945 + 0.996014i \(0.471571\pi\)
\(410\) 0 0
\(411\) 9.46410i 0.466830i
\(412\) 0 0
\(413\) − 7.60770i − 0.374350i
\(414\) 0 0
\(415\) −4.73205 −0.232287
\(416\) 0 0
\(417\) 7.32051 0.358487
\(418\) 0 0
\(419\) 0.928203i 0.0453457i 0.999743 + 0.0226728i \(0.00721761\pi\)
−0.999743 + 0.0226728i \(0.992782\pi\)
\(420\) 0 0
\(421\) 6.00000i 0.292422i 0.989253 + 0.146211i \(0.0467079\pi\)
−0.989253 + 0.146211i \(0.953292\pi\)
\(422\) 0 0
\(423\) −20.1962 −0.981971
\(424\) 0 0
\(425\) −3.46410 −0.168034
\(426\) 0 0
\(427\) 16.3923i 0.793279i
\(428\) 0 0
\(429\) 8.78461i 0.424125i
\(430\) 0 0
\(431\) 28.3923 1.36761 0.683805 0.729665i \(-0.260324\pi\)
0.683805 + 0.729665i \(0.260324\pi\)
\(432\) 0 0
\(433\) 26.3923 1.26833 0.634167 0.773196i \(-0.281343\pi\)
0.634167 + 0.773196i \(0.281343\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 16.3923i 0.784150i
\(438\) 0 0
\(439\) 18.9282 0.903394 0.451697 0.892171i \(-0.350819\pi\)
0.451697 + 0.892171i \(0.350819\pi\)
\(440\) 0 0
\(441\) −13.2872 −0.632723
\(442\) 0 0
\(443\) 0.339746i 0.0161418i 0.999967 + 0.00807091i \(0.00256908\pi\)
−0.999967 + 0.00807091i \(0.997431\pi\)
\(444\) 0 0
\(445\) − 0.928203i − 0.0440011i
\(446\) 0 0
\(447\) 13.1769 0.623247
\(448\) 0 0
\(449\) −2.53590 −0.119676 −0.0598382 0.998208i \(-0.519058\pi\)
−0.0598382 + 0.998208i \(0.519058\pi\)
\(450\) 0 0
\(451\) 8.78461i 0.413651i
\(452\) 0 0
\(453\) 1.85641i 0.0872216i
\(454\) 0 0
\(455\) −4.39230 −0.205914
\(456\) 0 0
\(457\) 22.7846 1.06582 0.532910 0.846172i \(-0.321099\pi\)
0.532910 + 0.846172i \(0.321099\pi\)
\(458\) 0 0
\(459\) 13.8564i 0.646762i
\(460\) 0 0
\(461\) 12.0000i 0.558896i 0.960161 + 0.279448i \(0.0901514\pi\)
−0.960161 + 0.279448i \(0.909849\pi\)
\(462\) 0 0
\(463\) −15.8038 −0.734467 −0.367234 0.930129i \(-0.619695\pi\)
−0.367234 + 0.930129i \(0.619695\pi\)
\(464\) 0 0
\(465\) −6.92820 −0.321288
\(466\) 0 0
\(467\) 22.9808i 1.06342i 0.846926 + 0.531711i \(0.178451\pi\)
−0.846926 + 0.531711i \(0.821549\pi\)
\(468\) 0 0
\(469\) − 12.9282i − 0.596969i
\(470\) 0 0
\(471\) 0.679492 0.0313093
\(472\) 0 0
\(473\) 35.3205 1.62404
\(474\) 0 0
\(475\) − 2.00000i − 0.0917663i
\(476\) 0 0
\(477\) − 25.6077i − 1.17250i
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 20.7846 0.947697
\(482\) 0 0
\(483\) 7.60770i 0.346162i
\(484\) 0 0
\(485\) 6.39230i 0.290260i
\(486\) 0 0
\(487\) 39.1244 1.77289 0.886447 0.462830i \(-0.153166\pi\)
0.886447 + 0.462830i \(0.153166\pi\)
\(488\) 0 0
\(489\) −4.24871 −0.192133
\(490\) 0 0
\(491\) − 22.3923i − 1.01055i −0.862958 0.505275i \(-0.831391\pi\)
0.862958 0.505275i \(-0.168609\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 8.53590 0.383660
\(496\) 0 0
\(497\) −5.56922 −0.249814
\(498\) 0 0
\(499\) 39.5692i 1.77136i 0.464295 + 0.885681i \(0.346308\pi\)
−0.464295 + 0.885681i \(0.653692\pi\)
\(500\) 0 0
\(501\) − 6.00000i − 0.268060i
\(502\) 0 0
\(503\) 8.19615 0.365448 0.182724 0.983164i \(-0.441508\pi\)
0.182724 + 0.983164i \(0.441508\pi\)
\(504\) 0 0
\(505\) 12.0000 0.533993
\(506\) 0 0
\(507\) 0.732051i 0.0325115i
\(508\) 0 0
\(509\) − 8.78461i − 0.389371i −0.980866 0.194685i \(-0.937631\pi\)
0.980866 0.194685i \(-0.0623686\pi\)
\(510\) 0 0
\(511\) −18.2487 −0.807275
\(512\) 0 0
\(513\) −8.00000 −0.353209
\(514\) 0 0
\(515\) − 8.19615i − 0.361166i
\(516\) 0 0
\(517\) − 28.3923i − 1.24869i
\(518\) 0 0
\(519\) 4.39230 0.192801
\(520\) 0 0
\(521\) 31.8564 1.39565 0.697827 0.716266i \(-0.254150\pi\)
0.697827 + 0.716266i \(0.254150\pi\)
\(522\) 0 0
\(523\) 14.9808i 0.655063i 0.944840 + 0.327531i \(0.106217\pi\)
−0.944840 + 0.327531i \(0.893783\pi\)
\(524\) 0 0
\(525\) − 0.928203i − 0.0405101i
\(526\) 0 0
\(527\) −32.7846 −1.42812
\(528\) 0 0
\(529\) 44.1769 1.92074
\(530\) 0 0
\(531\) − 14.7846i − 0.641597i
\(532\) 0 0
\(533\) − 8.78461i − 0.380504i
\(534\) 0 0
\(535\) 16.7321 0.723390
\(536\) 0 0
\(537\) −14.5359 −0.627270
\(538\) 0 0
\(539\) − 18.6795i − 0.804583i
\(540\) 0 0
\(541\) − 15.7128i − 0.675547i −0.941227 0.337773i \(-0.890326\pi\)
0.941227 0.337773i \(-0.109674\pi\)
\(542\) 0 0
\(543\) 5.07180 0.217652
\(544\) 0 0
\(545\) 0.928203 0.0397599
\(546\) 0 0
\(547\) − 1.80385i − 0.0771270i −0.999256 0.0385635i \(-0.987722\pi\)
0.999256 0.0385635i \(-0.0122782\pi\)
\(548\) 0 0
\(549\) 31.8564i 1.35960i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −15.2154 −0.647024
\(554\) 0 0
\(555\) 4.39230i 0.186443i
\(556\) 0 0
\(557\) 14.7846i 0.626444i 0.949680 + 0.313222i \(0.101408\pi\)
−0.949680 + 0.313222i \(0.898592\pi\)
\(558\) 0 0
\(559\) −35.3205 −1.49390
\(560\) 0 0
\(561\) −8.78461 −0.370887
\(562\) 0 0
\(563\) − 26.1962i − 1.10404i −0.833832 0.552018i \(-0.813858\pi\)
0.833832 0.552018i \(-0.186142\pi\)
\(564\) 0 0
\(565\) − 0.928203i − 0.0390498i
\(566\) 0 0
\(567\) 5.66025 0.237708
\(568\) 0 0
\(569\) −42.2487 −1.77116 −0.885579 0.464489i \(-0.846238\pi\)
−0.885579 + 0.464489i \(0.846238\pi\)
\(570\) 0 0
\(571\) 30.3923i 1.27188i 0.771739 + 0.635939i \(0.219387\pi\)
−0.771739 + 0.635939i \(0.780613\pi\)
\(572\) 0 0
\(573\) 12.0000i 0.501307i
\(574\) 0 0
\(575\) −8.19615 −0.341803
\(576\) 0 0
\(577\) −2.00000 −0.0832611 −0.0416305 0.999133i \(-0.513255\pi\)
−0.0416305 + 0.999133i \(0.513255\pi\)
\(578\) 0 0
\(579\) 4.67949i 0.194473i
\(580\) 0 0
\(581\) − 6.00000i − 0.248922i
\(582\) 0 0
\(583\) 36.0000 1.49097
\(584\) 0 0
\(585\) −8.53590 −0.352916
\(586\) 0 0
\(587\) − 13.5167i − 0.557892i −0.960307 0.278946i \(-0.910015\pi\)
0.960307 0.278946i \(-0.0899851\pi\)
\(588\) 0 0
\(589\) − 18.9282i − 0.779923i
\(590\) 0 0
\(591\) −7.60770 −0.312939
\(592\) 0 0
\(593\) −0.928203 −0.0381167 −0.0190584 0.999818i \(-0.506067\pi\)
−0.0190584 + 0.999818i \(0.506067\pi\)
\(594\) 0 0
\(595\) − 4.39230i − 0.180067i
\(596\) 0 0
\(597\) − 5.07180i − 0.207575i
\(598\) 0 0
\(599\) 12.0000 0.490307 0.245153 0.969484i \(-0.421162\pi\)
0.245153 + 0.969484i \(0.421162\pi\)
\(600\) 0 0
\(601\) −20.3923 −0.831819 −0.415910 0.909406i \(-0.636537\pi\)
−0.415910 + 0.909406i \(0.636537\pi\)
\(602\) 0 0
\(603\) − 25.1244i − 1.02314i
\(604\) 0 0
\(605\) 1.00000i 0.0406558i
\(606\) 0 0
\(607\) −8.19615 −0.332672 −0.166336 0.986069i \(-0.553194\pi\)
−0.166336 + 0.986069i \(0.553194\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 28.3923i 1.14863i
\(612\) 0 0
\(613\) − 13.6077i − 0.549610i −0.961500 0.274805i \(-0.911387\pi\)
0.961500 0.274805i \(-0.0886132\pi\)
\(614\) 0 0
\(615\) 1.85641 0.0748575
\(616\) 0 0
\(617\) 13.6077 0.547825 0.273913 0.961755i \(-0.411682\pi\)
0.273913 + 0.961755i \(0.411682\pi\)
\(618\) 0 0
\(619\) − 6.78461i − 0.272696i −0.990661 0.136348i \(-0.956463\pi\)
0.990661 0.136348i \(-0.0435366\pi\)
\(620\) 0 0
\(621\) 32.7846i 1.31560i
\(622\) 0 0
\(623\) 1.17691 0.0471521
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) − 5.07180i − 0.202548i
\(628\) 0 0
\(629\) 20.7846i 0.828737i
\(630\) 0 0
\(631\) 21.4641 0.854472 0.427236 0.904140i \(-0.359487\pi\)
0.427236 + 0.904140i \(0.359487\pi\)
\(632\) 0 0
\(633\) 4.67949 0.185993
\(634\) 0 0
\(635\) 3.80385i 0.150951i
\(636\) 0 0
\(637\) 18.6795i 0.740108i
\(638\) 0 0
\(639\) −10.8231 −0.428155
\(640\) 0 0
\(641\) −4.39230 −0.173486 −0.0867428 0.996231i \(-0.527646\pi\)
−0.0867428 + 0.996231i \(0.527646\pi\)
\(642\) 0 0
\(643\) − 10.5885i − 0.417568i −0.977962 0.208784i \(-0.933049\pi\)
0.977962 0.208784i \(-0.0669506\pi\)
\(644\) 0 0
\(645\) − 7.46410i − 0.293899i
\(646\) 0 0
\(647\) −36.5885 −1.43844 −0.719220 0.694782i \(-0.755501\pi\)
−0.719220 + 0.694782i \(0.755501\pi\)
\(648\) 0 0
\(649\) 20.7846 0.815867
\(650\) 0 0
\(651\) − 8.78461i − 0.344296i
\(652\) 0 0
\(653\) − 19.1769i − 0.750451i −0.926934 0.375225i \(-0.877565\pi\)
0.926934 0.375225i \(-0.122435\pi\)
\(654\) 0 0
\(655\) 10.3923 0.406061
\(656\) 0 0
\(657\) −35.4641 −1.38359
\(658\) 0 0
\(659\) − 40.6410i − 1.58315i −0.611073 0.791575i \(-0.709262\pi\)
0.611073 0.791575i \(-0.290738\pi\)
\(660\) 0 0
\(661\) 35.5692i 1.38348i 0.722146 + 0.691741i \(0.243156\pi\)
−0.722146 + 0.691741i \(0.756844\pi\)
\(662\) 0 0
\(663\) 8.78461 0.341166
\(664\) 0 0
\(665\) 2.53590 0.0983379
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) − 11.0718i − 0.428060i
\(670\) 0 0
\(671\) −44.7846 −1.72889
\(672\) 0 0
\(673\) −39.1769 −1.51016 −0.755080 0.655633i \(-0.772402\pi\)
−0.755080 + 0.655633i \(0.772402\pi\)
\(674\) 0 0
\(675\) − 4.00000i − 0.153960i
\(676\) 0 0
\(677\) 43.1769i 1.65942i 0.558192 + 0.829712i \(0.311495\pi\)
−0.558192 + 0.829712i \(0.688505\pi\)
\(678\) 0 0
\(679\) −8.10512 −0.311046
\(680\) 0 0
\(681\) 13.6077 0.521448
\(682\) 0 0
\(683\) 11.6603i 0.446167i 0.974799 + 0.223084i \(0.0716123\pi\)
−0.974799 + 0.223084i \(0.928388\pi\)
\(684\) 0 0
\(685\) − 12.9282i − 0.493961i
\(686\) 0 0
\(687\) −13.8564 −0.528655
\(688\) 0 0
\(689\) −36.0000 −1.37149
\(690\) 0 0
\(691\) − 5.60770i − 0.213327i −0.994295 0.106663i \(-0.965983\pi\)
0.994295 0.106663i \(-0.0340167\pi\)
\(692\) 0 0
\(693\) 10.8231i 0.411135i
\(694\) 0 0
\(695\) −10.0000 −0.379322
\(696\) 0 0
\(697\) 8.78461 0.332741
\(698\) 0 0
\(699\) − 1.17691i − 0.0445150i
\(700\) 0 0
\(701\) 14.7846i 0.558407i 0.960232 + 0.279204i \(0.0900704\pi\)
−0.960232 + 0.279204i \(0.909930\pi\)
\(702\) 0 0
\(703\) −12.0000 −0.452589
\(704\) 0 0
\(705\) −6.00000 −0.225973
\(706\) 0 0
\(707\) 15.2154i 0.572234i
\(708\) 0 0
\(709\) − 10.1436i − 0.380951i −0.981692 0.190475i \(-0.938997\pi\)
0.981692 0.190475i \(-0.0610029\pi\)
\(710\) 0 0
\(711\) −29.5692 −1.10893
\(712\) 0 0
\(713\) −77.5692 −2.90499
\(714\) 0 0
\(715\) − 12.0000i − 0.448775i
\(716\) 0 0
\(717\) 15.2154i 0.568229i
\(718\) 0 0
\(719\) 44.7846 1.67018 0.835092 0.550110i \(-0.185414\pi\)
0.835092 + 0.550110i \(0.185414\pi\)
\(720\) 0 0
\(721\) 10.3923 0.387030
\(722\) 0 0
\(723\) − 14.9282i − 0.555186i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 34.0526 1.26294 0.631470 0.775401i \(-0.282452\pi\)
0.631470 + 0.775401i \(0.282452\pi\)
\(728\) 0 0
\(729\) 2.21539 0.0820515
\(730\) 0 0
\(731\) − 35.3205i − 1.30638i
\(732\) 0 0
\(733\) − 38.7846i − 1.43254i −0.697822 0.716271i \(-0.745847\pi\)
0.697822 0.716271i \(-0.254153\pi\)
\(734\) 0 0
\(735\) −3.94744 −0.145604
\(736\) 0 0
\(737\) 35.3205 1.30105
\(738\) 0 0
\(739\) − 2.00000i − 0.0735712i −0.999323 0.0367856i \(-0.988288\pi\)
0.999323 0.0367856i \(-0.0117119\pi\)
\(740\) 0 0
\(741\) 5.07180i 0.186317i
\(742\) 0 0
\(743\) 36.5885 1.34230 0.671150 0.741321i \(-0.265801\pi\)
0.671150 + 0.741321i \(0.265801\pi\)
\(744\) 0 0
\(745\) −18.0000 −0.659469
\(746\) 0 0
\(747\) − 11.6603i − 0.426626i
\(748\) 0 0
\(749\) 21.2154i 0.775193i
\(750\) 0 0
\(751\) −40.3923 −1.47394 −0.736968 0.675928i \(-0.763743\pi\)
−0.736968 + 0.675928i \(0.763743\pi\)
\(752\) 0 0
\(753\) 11.3205 0.412542
\(754\) 0 0
\(755\) − 2.53590i − 0.0922908i
\(756\) 0 0
\(757\) 23.0718i 0.838559i 0.907857 + 0.419279i \(0.137717\pi\)
−0.907857 + 0.419279i \(0.862283\pi\)
\(758\) 0 0
\(759\) −20.7846 −0.754434
\(760\) 0 0
\(761\) −21.7128 −0.787089 −0.393544 0.919306i \(-0.628751\pi\)
−0.393544 + 0.919306i \(0.628751\pi\)
\(762\) 0 0
\(763\) 1.17691i 0.0426072i
\(764\) 0 0
\(765\) − 8.53590i − 0.308616i
\(766\) 0 0
\(767\) −20.7846 −0.750489
\(768\) 0 0
\(769\) −6.78461 −0.244659 −0.122330 0.992490i \(-0.539037\pi\)
−0.122330 + 0.992490i \(0.539037\pi\)
\(770\) 0 0
\(771\) 4.39230i 0.158185i
\(772\) 0 0
\(773\) − 46.3923i − 1.66862i −0.551299 0.834308i \(-0.685868\pi\)
0.551299 0.834308i \(-0.314132\pi\)
\(774\) 0 0
\(775\) 9.46410 0.339961
\(776\) 0 0
\(777\) −5.56922 −0.199795
\(778\) 0 0
\(779\) 5.07180i 0.181716i
\(780\) 0 0
\(781\) − 15.2154i − 0.544449i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −0.928203 −0.0331290
\(786\) 0 0
\(787\) 5.80385i 0.206885i 0.994635 + 0.103442i \(0.0329857\pi\)
−0.994635 + 0.103442i \(0.967014\pi\)
\(788\) 0 0
\(789\) − 14.7846i − 0.526346i
\(790\) 0 0
\(791\) 1.17691 0.0418463
\(792\) 0 0
\(793\) 44.7846 1.59035
\(794\) 0 0
\(795\) − 7.60770i − 0.269817i
\(796\) 0 0
\(797\) − 1.60770i − 0.0569475i −0.999595 0.0284737i \(-0.990935\pi\)
0.999595 0.0284737i \(-0.00906470\pi\)
\(798\) 0 0
\(799\) −28.3923 −1.00445
\(800\) 0 0
\(801\) 2.28719 0.0808138
\(802\) 0 0
\(803\) − 49.8564i − 1.75939i
\(804\) 0 0
\(805\) − 10.3923i − 0.366281i
\(806\) 0 0
\(807\) 10.8231 0.380991
\(808\) 0 0
\(809\) 35.5692 1.25055 0.625274 0.780406i \(-0.284987\pi\)
0.625274 + 0.780406i \(0.284987\pi\)
\(810\) 0 0
\(811\) 38.3923i 1.34814i 0.738669 + 0.674068i \(0.235455\pi\)
−0.738669 + 0.674068i \(0.764545\pi\)
\(812\) 0 0
\(813\) 3.21539i 0.112769i
\(814\) 0 0
\(815\) 5.80385 0.203300
\(816\) 0 0
\(817\) 20.3923 0.713436
\(818\) 0 0
\(819\) − 10.8231i − 0.378189i
\(820\) 0 0
\(821\) 50.7846i 1.77240i 0.463308 + 0.886198i \(0.346663\pi\)
−0.463308 + 0.886198i \(0.653337\pi\)
\(822\) 0 0
\(823\) 30.8372 1.07492 0.537458 0.843290i \(-0.319385\pi\)
0.537458 + 0.843290i \(0.319385\pi\)
\(824\) 0 0
\(825\) 2.53590 0.0882886
\(826\) 0 0
\(827\) − 4.73205i − 0.164550i −0.996610 0.0822748i \(-0.973781\pi\)
0.996610 0.0822748i \(-0.0262185\pi\)
\(828\) 0 0
\(829\) 50.7846i 1.76382i 0.471416 + 0.881911i \(0.343743\pi\)
−0.471416 + 0.881911i \(0.656257\pi\)
\(830\) 0 0
\(831\) 14.5359 0.504245
\(832\) 0 0
\(833\) −18.6795 −0.647206
\(834\) 0 0
\(835\) 8.19615i 0.283640i
\(836\) 0 0
\(837\) − 37.8564i − 1.30851i
\(838\) 0 0
\(839\) 8.78461 0.303278 0.151639 0.988436i \(-0.451545\pi\)
0.151639 + 0.988436i \(0.451545\pi\)
\(840\) 0 0
\(841\) 29.0000 1.00000
\(842\) 0 0
\(843\) − 20.7846i − 0.715860i
\(844\) 0 0
\(845\) − 1.00000i − 0.0344010i
\(846\) 0 0
\(847\) −1.26795 −0.0435672
\(848\) 0 0
\(849\) 7.75129 0.266024
\(850\) 0 0
\(851\) 49.1769i 1.68576i
\(852\) 0 0
\(853\) − 3.46410i − 0.118609i −0.998240 0.0593043i \(-0.981112\pi\)
0.998240 0.0593043i \(-0.0188882\pi\)
\(854\) 0 0
\(855\) 4.92820 0.168541
\(856\) 0 0
\(857\) −16.1436 −0.551455 −0.275727 0.961236i \(-0.588919\pi\)
−0.275727 + 0.961236i \(0.588919\pi\)
\(858\) 0 0
\(859\) 51.5692i 1.75952i 0.475419 + 0.879760i \(0.342296\pi\)
−0.475419 + 0.879760i \(0.657704\pi\)
\(860\) 0 0
\(861\) 2.35383i 0.0802183i
\(862\) 0 0
\(863\) 40.9808 1.39500 0.697501 0.716584i \(-0.254295\pi\)
0.697501 + 0.716584i \(0.254295\pi\)
\(864\) 0 0
\(865\) −6.00000 −0.204006
\(866\) 0 0
\(867\) − 3.66025i − 0.124309i
\(868\) 0 0
\(869\) − 41.5692i − 1.41014i
\(870\) 0 0
\(871\) −35.3205 −1.19679
\(872\) 0 0
\(873\) −15.7513 −0.533100
\(874\) 0 0
\(875\) 1.26795i 0.0428645i
\(876\) 0 0
\(877\) 7.85641i 0.265292i 0.991163 + 0.132646i \(0.0423473\pi\)
−0.991163 + 0.132646i \(0.957653\pi\)
\(878\) 0 0
\(879\) −21.9615 −0.740744
\(880\) 0 0
\(881\) −7.60770 −0.256310 −0.128155 0.991754i \(-0.540905\pi\)
−0.128155 + 0.991754i \(0.540905\pi\)
\(882\) 0 0
\(883\) 5.80385i 0.195315i 0.995220 + 0.0976575i \(0.0311350\pi\)
−0.995220 + 0.0976575i \(0.968865\pi\)
\(884\) 0 0
\(885\) − 4.39230i − 0.147646i
\(886\) 0 0
\(887\) −21.3731 −0.717637 −0.358819 0.933407i \(-0.616820\pi\)
−0.358819 + 0.933407i \(0.616820\pi\)
\(888\) 0 0
\(889\) −4.82309 −0.161761
\(890\) 0 0
\(891\) 15.4641i 0.518067i
\(892\) 0 0
\(893\) − 16.3923i − 0.548548i
\(894\) 0 0
\(895\) 19.8564 0.663726
\(896\) 0 0
\(897\) 20.7846 0.693978
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) − 36.0000i − 1.19933i
\(902\) 0 0
\(903\) 9.46410 0.314946
\(904\) 0 0
\(905\) −6.92820 −0.230301
\(906\) 0 0
\(907\) − 1.41154i − 0.0468695i −0.999725 0.0234348i \(-0.992540\pi\)
0.999725 0.0234348i \(-0.00746020\pi\)
\(908\) 0 0
\(909\) 29.5692i 0.980749i
\(910\) 0 0
\(911\) −37.1769 −1.23173 −0.615863 0.787853i \(-0.711193\pi\)
−0.615863 + 0.787853i \(0.711193\pi\)
\(912\) 0 0
\(913\) 16.3923 0.542506
\(914\) 0 0
\(915\) 9.46410i 0.312874i
\(916\) 0 0
\(917\) 13.1769i 0.435140i
\(918\) 0 0
\(919\) 39.7128 1.31000 0.655002 0.755627i \(-0.272668\pi\)
0.655002 + 0.755627i \(0.272668\pi\)
\(920\) 0 0
\(921\) 25.0333 0.824876
\(922\) 0 0
\(923\) 15.2154i 0.500821i
\(924\) 0 0
\(925\) − 6.00000i − 0.197279i
\(926\) 0 0
\(927\) 20.1962 0.663329
\(928\) 0 0
\(929\) 7.60770 0.249600 0.124800 0.992182i \(-0.460171\pi\)
0.124800 + 0.992182i \(0.460171\pi\)
\(930\) 0 0
\(931\) − 10.7846i − 0.353451i
\(932\) 0 0
\(933\) − 5.56922i − 0.182328i
\(934\) 0 0
\(935\) 12.0000 0.392442
\(936\) 0 0
\(937\) −9.60770 −0.313870 −0.156935 0.987609i \(-0.550161\pi\)
−0.156935 + 0.987609i \(0.550161\pi\)
\(938\) 0 0
\(939\) 16.1051i 0.525571i
\(940\) 0 0
\(941\) − 20.7846i − 0.677559i −0.940866 0.338779i \(-0.889986\pi\)
0.940866 0.338779i \(-0.110014\pi\)
\(942\) 0 0
\(943\) 20.7846 0.676840
\(944\) 0 0
\(945\) 5.07180 0.164986
\(946\) 0 0
\(947\) 45.8038i 1.48843i 0.667943 + 0.744213i \(0.267175\pi\)
−0.667943 + 0.744213i \(0.732825\pi\)
\(948\) 0 0
\(949\) 49.8564i 1.61841i
\(950\) 0 0
\(951\) −16.3923 −0.531557
\(952\) 0 0
\(953\) 24.9282 0.807504 0.403752 0.914869i \(-0.367706\pi\)
0.403752 + 0.914869i \(0.367706\pi\)
\(954\) 0 0
\(955\) − 16.3923i − 0.530443i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 16.3923 0.529335
\(960\) 0 0
\(961\) 58.5692 1.88933
\(962\) 0 0
\(963\) 41.2295i 1.32860i
\(964\) 0 0
\(965\) − 6.39230i − 0.205776i
\(966\) 0 0
\(967\) −8.19615 −0.263570 −0.131785 0.991278i \(-0.542071\pi\)
−0.131785 + 0.991278i \(0.542071\pi\)
\(968\) 0 0
\(969\) −5.07180 −0.162930
\(970\) 0 0
\(971\) 33.0333i 1.06009i 0.847970 + 0.530045i \(0.177825\pi\)
−0.847970 + 0.530045i \(0.822175\pi\)
\(972\) 0 0
\(973\) − 12.6795i − 0.406486i
\(974\) 0 0
\(975\) −2.53590 −0.0812137
\(976\) 0 0
\(977\) 5.32051 0.170218 0.0851091 0.996372i \(-0.472876\pi\)
0.0851091 + 0.996372i \(0.472876\pi\)
\(978\) 0 0
\(979\) 3.21539i 0.102764i
\(980\) 0 0
\(981\) 2.28719i 0.0730243i
\(982\) 0 0
\(983\) −11.4115 −0.363972 −0.181986 0.983301i \(-0.558253\pi\)
−0.181986 + 0.983301i \(0.558253\pi\)
\(984\) 0 0
\(985\) 10.3923 0.331126
\(986\) 0 0
\(987\) − 7.60770i − 0.242156i
\(988\) 0 0
\(989\) − 83.5692i − 2.65735i
\(990\) 0 0
\(991\) 44.1051 1.40105 0.700523 0.713630i \(-0.252950\pi\)
0.700523 + 0.713630i \(0.252950\pi\)
\(992\) 0 0
\(993\) 4.10512 0.130272
\(994\) 0 0
\(995\) 6.92820i 0.219639i
\(996\) 0 0
\(997\) − 25.6077i − 0.811004i −0.914094 0.405502i \(-0.867097\pi\)
0.914094 0.405502i \(-0.132903\pi\)
\(998\) 0 0
\(999\) −24.0000 −0.759326
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 320.2.d.b.161.3 yes 4
3.2 odd 2 2880.2.k.l.1441.3 4
4.3 odd 2 320.2.d.a.161.2 4
5.2 odd 4 1600.2.f.d.1249.4 4
5.3 odd 4 1600.2.f.h.1249.1 4
5.4 even 2 1600.2.d.b.801.2 4
8.3 odd 2 320.2.d.a.161.3 yes 4
8.5 even 2 inner 320.2.d.b.161.2 yes 4
12.11 even 2 2880.2.k.e.1441.4 4
16.3 odd 4 1280.2.a.b.1.2 2
16.5 even 4 1280.2.a.c.1.2 2
16.11 odd 4 1280.2.a.p.1.1 2
16.13 even 4 1280.2.a.m.1.1 2
20.3 even 4 1600.2.f.e.1249.4 4
20.7 even 4 1600.2.f.i.1249.1 4
20.19 odd 2 1600.2.d.h.801.3 4
24.5 odd 2 2880.2.k.l.1441.1 4
24.11 even 2 2880.2.k.e.1441.2 4
40.3 even 4 1600.2.f.i.1249.2 4
40.13 odd 4 1600.2.f.d.1249.3 4
40.19 odd 2 1600.2.d.h.801.2 4
40.27 even 4 1600.2.f.e.1249.3 4
40.29 even 2 1600.2.d.b.801.3 4
40.37 odd 4 1600.2.f.h.1249.2 4
80.19 odd 4 6400.2.a.cd.1.1 2
80.29 even 4 6400.2.a.bf.1.2 2
80.59 odd 4 6400.2.a.y.1.2 2
80.69 even 4 6400.2.a.ck.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
320.2.d.a.161.2 4 4.3 odd 2
320.2.d.a.161.3 yes 4 8.3 odd 2
320.2.d.b.161.2 yes 4 8.5 even 2 inner
320.2.d.b.161.3 yes 4 1.1 even 1 trivial
1280.2.a.b.1.2 2 16.3 odd 4
1280.2.a.c.1.2 2 16.5 even 4
1280.2.a.m.1.1 2 16.13 even 4
1280.2.a.p.1.1 2 16.11 odd 4
1600.2.d.b.801.2 4 5.4 even 2
1600.2.d.b.801.3 4 40.29 even 2
1600.2.d.h.801.2 4 40.19 odd 2
1600.2.d.h.801.3 4 20.19 odd 2
1600.2.f.d.1249.3 4 40.13 odd 4
1600.2.f.d.1249.4 4 5.2 odd 4
1600.2.f.e.1249.3 4 40.27 even 4
1600.2.f.e.1249.4 4 20.3 even 4
1600.2.f.h.1249.1 4 5.3 odd 4
1600.2.f.h.1249.2 4 40.37 odd 4
1600.2.f.i.1249.1 4 20.7 even 4
1600.2.f.i.1249.2 4 40.3 even 4
2880.2.k.e.1441.2 4 24.11 even 2
2880.2.k.e.1441.4 4 12.11 even 2
2880.2.k.l.1441.1 4 24.5 odd 2
2880.2.k.l.1441.3 4 3.2 odd 2
6400.2.a.y.1.2 2 80.59 odd 4
6400.2.a.bf.1.2 2 80.29 even 4
6400.2.a.cd.1.1 2 80.19 odd 4
6400.2.a.ck.1.1 2 80.69 even 4