Properties

Label 320.3.e.b.159.7
Level $320$
Weight $3$
Character 320.159
Analytic conductor $8.719$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 320.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.71936845953\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial: \(x^{16} + 19 x^{14} + 301 x^{12} + 1102 x^{10} + 3238 x^{8} + 1102 x^{6} + 301 x^{4} + 19 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{34} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 159.7
Root \(-0.128617 - 0.222771i\) of defining polynomial
Character \(\chi\) \(=\) 320.159
Dual form 320.3.e.b.159.12

$q$-expansion

\(f(q)\) \(=\) \(q-1.90845i q^{3} +(4.37937 - 2.41269i) q^{5} -3.95502 q^{7} +5.35782 q^{9} +O(q^{10})\) \(q-1.90845i q^{3} +(4.37937 - 2.41269i) q^{5} -3.95502 q^{7} +5.35782 q^{9} +6.18667 q^{11} +4.94185 q^{13} +(-4.60451 - 8.35782i) q^{15} -22.4311i q^{17} +10.3923 q^{19} +7.54796i q^{21} -39.6083 q^{23} +(13.3578 - 21.1322i) q^{25} -27.4012i q^{27} +30.4354i q^{29} -24.7156i q^{31} -11.8070i q^{33} +(-17.3205 + 9.54225i) q^{35} +24.0264 q^{37} -9.43127i q^{39} +31.0735 q^{41} -52.2110i q^{43} +(23.4639 - 12.9268i) q^{45} -13.1640 q^{47} -33.3578 q^{49} -42.8087 q^{51} -17.9596 q^{53} +(27.0938 - 14.9265i) q^{55} -19.8332i q^{57} +104.421 q^{59} -57.2849i q^{61} -21.1903 q^{63} +(21.6422 - 11.9232i) q^{65} +99.3796i q^{67} +75.5905i q^{69} +16.7156i q^{71} +96.3357i q^{73} +(-40.3297 - 25.4927i) q^{75} -24.4684 q^{77} +139.578i q^{79} -4.07345 q^{81} +91.7458i q^{83} +(-54.1195 - 98.2344i) q^{85} +58.0844 q^{87} -94.7156 q^{89} -19.5451 q^{91} -47.1686 q^{93} +(45.5118 - 25.0735i) q^{95} +143.796i q^{97} +33.1471 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 96q^{9} + O(q^{10}) \) \( 16q - 96q^{9} + 32q^{25} - 48q^{41} - 352q^{49} + 528q^{65} + 480q^{81} - 1152q^{89} + O(q^{100}) \)

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
\(3\) 1.90845i 0.636150i −0.948066 0.318075i \(-0.896964\pi\)
0.948066 0.318075i \(-0.103036\pi\)
\(4\) 0 0
\(5\) 4.37937 2.41269i 0.875875 0.482539i
\(6\) 0 0
\(7\) −3.95502 −0.565003 −0.282501 0.959267i \(-0.591164\pi\)
−0.282501 + 0.959267i \(0.591164\pi\)
\(8\) 0 0
\(9\) 5.35782 0.595313
\(10\) 0 0
\(11\) 6.18667 0.562425 0.281212 0.959646i \(-0.409263\pi\)
0.281212 + 0.959646i \(0.409263\pi\)
\(12\) 0 0
\(13\) 4.94185 0.380142 0.190071 0.981770i \(-0.439128\pi\)
0.190071 + 0.981770i \(0.439128\pi\)
\(14\) 0 0
\(15\) −4.60451 8.35782i −0.306967 0.557188i
\(16\) 0 0
\(17\) 22.4311i 1.31948i −0.751494 0.659740i \(-0.770667\pi\)
0.751494 0.659740i \(-0.229333\pi\)
\(18\) 0 0
\(19\) 10.3923 0.546963 0.273482 0.961877i \(-0.411825\pi\)
0.273482 + 0.961877i \(0.411825\pi\)
\(20\) 0 0
\(21\) 7.54796i 0.359427i
\(22\) 0 0
\(23\) −39.6083 −1.72210 −0.861050 0.508520i \(-0.830193\pi\)
−0.861050 + 0.508520i \(0.830193\pi\)
\(24\) 0 0
\(25\) 13.3578 21.1322i 0.534313 0.845287i
\(26\) 0 0
\(27\) 27.4012i 1.01486i
\(28\) 0 0
\(29\) 30.4354i 1.04950i 0.851258 + 0.524748i \(0.175840\pi\)
−0.851258 + 0.524748i \(0.824160\pi\)
\(30\) 0 0
\(31\) 24.7156i 0.797278i −0.917108 0.398639i \(-0.869483\pi\)
0.917108 0.398639i \(-0.130517\pi\)
\(32\) 0 0
\(33\) 11.8070i 0.357787i
\(34\) 0 0
\(35\) −17.3205 + 9.54225i −0.494872 + 0.272636i
\(36\) 0 0
\(37\) 24.0264 0.649361 0.324680 0.945824i \(-0.394743\pi\)
0.324680 + 0.945824i \(0.394743\pi\)
\(38\) 0 0
\(39\) 9.43127i 0.241827i
\(40\) 0 0
\(41\) 31.0735 0.757889 0.378945 0.925419i \(-0.376287\pi\)
0.378945 + 0.925419i \(0.376287\pi\)
\(42\) 0 0
\(43\) 52.2110i 1.21421i −0.794622 0.607105i \(-0.792331\pi\)
0.794622 0.607105i \(-0.207669\pi\)
\(44\) 0 0
\(45\) 23.4639 12.9268i 0.521420 0.287262i
\(46\) 0 0
\(47\) −13.1640 −0.280086 −0.140043 0.990145i \(-0.544724\pi\)
−0.140043 + 0.990145i \(0.544724\pi\)
\(48\) 0 0
\(49\) −33.3578 −0.680772
\(50\) 0 0
\(51\) −42.8087 −0.839387
\(52\) 0 0
\(53\) −17.9596 −0.338860 −0.169430 0.985542i \(-0.554193\pi\)
−0.169430 + 0.985542i \(0.554193\pi\)
\(54\) 0 0
\(55\) 27.0938 14.9265i 0.492614 0.271392i
\(56\) 0 0
\(57\) 19.8332i 0.347951i
\(58\) 0 0
\(59\) 104.421 1.76985 0.884924 0.465735i \(-0.154210\pi\)
0.884924 + 0.465735i \(0.154210\pi\)
\(60\) 0 0
\(61\) 57.2849i 0.939097i −0.882907 0.469548i \(-0.844417\pi\)
0.882907 0.469548i \(-0.155583\pi\)
\(62\) 0 0
\(63\) −21.1903 −0.336354
\(64\) 0 0
\(65\) 21.6422 11.9232i 0.332957 0.183433i
\(66\) 0 0
\(67\) 99.3796i 1.48328i 0.670799 + 0.741639i \(0.265951\pi\)
−0.670799 + 0.741639i \(0.734049\pi\)
\(68\) 0 0
\(69\) 75.5905i 1.09551i
\(70\) 0 0
\(71\) 16.7156i 0.235431i 0.993047 + 0.117716i \(0.0375572\pi\)
−0.993047 + 0.117716i \(0.962443\pi\)
\(72\) 0 0
\(73\) 96.3357i 1.31967i 0.751412 + 0.659833i \(0.229373\pi\)
−0.751412 + 0.659833i \(0.770627\pi\)
\(74\) 0 0
\(75\) −40.3297 25.4927i −0.537729 0.339903i
\(76\) 0 0
\(77\) −24.4684 −0.317772
\(78\) 0 0
\(79\) 139.578i 1.76681i 0.468608 + 0.883406i \(0.344756\pi\)
−0.468608 + 0.883406i \(0.655244\pi\)
\(80\) 0 0
\(81\) −4.07345 −0.0502895
\(82\) 0 0
\(83\) 91.7458i 1.10537i 0.833390 + 0.552686i \(0.186397\pi\)
−0.833390 + 0.552686i \(0.813603\pi\)
\(84\) 0 0
\(85\) −54.1195 98.2344i −0.636700 1.15570i
\(86\) 0 0
\(87\) 58.0844 0.667637
\(88\) 0 0
\(89\) −94.7156 −1.06422 −0.532110 0.846675i \(-0.678601\pi\)
−0.532110 + 0.846675i \(0.678601\pi\)
\(90\) 0 0
\(91\) −19.5451 −0.214781
\(92\) 0 0
\(93\) −47.1686 −0.507189
\(94\) 0 0
\(95\) 45.5118 25.0735i 0.479071 0.263931i
\(96\) 0 0
\(97\) 143.796i 1.48243i 0.671267 + 0.741216i \(0.265751\pi\)
−0.671267 + 0.741216i \(0.734249\pi\)
\(98\) 0 0
\(99\) 33.1471 0.334819
\(100\) 0 0
\(101\) 66.5595i 0.659005i −0.944155 0.329502i \(-0.893119\pi\)
0.944155 0.329502i \(-0.106881\pi\)
\(102\) 0 0
\(103\) 153.063 1.48605 0.743024 0.669264i \(-0.233391\pi\)
0.743024 + 0.669264i \(0.233391\pi\)
\(104\) 0 0
\(105\) 18.2109 + 33.0553i 0.173437 + 0.314813i
\(106\) 0 0
\(107\) 75.7953i 0.708367i −0.935176 0.354184i \(-0.884759\pi\)
0.935176 0.354184i \(-0.115241\pi\)
\(108\) 0 0
\(109\) 124.088i 1.13842i 0.822192 + 0.569211i \(0.192751\pi\)
−0.822192 + 0.569211i \(0.807249\pi\)
\(110\) 0 0
\(111\) 45.8531i 0.413091i
\(112\) 0 0
\(113\) 5.19589i 0.0459814i 0.999736 + 0.0229907i \(0.00731881\pi\)
−0.999736 + 0.0229907i \(0.992681\pi\)
\(114\) 0 0
\(115\) −173.460 + 95.5627i −1.50834 + 0.830980i
\(116\) 0 0
\(117\) 26.4775 0.226303
\(118\) 0 0
\(119\) 88.7156i 0.745510i
\(120\) 0 0
\(121\) −82.7251 −0.683678
\(122\) 0 0
\(123\) 59.3021i 0.482131i
\(124\) 0 0
\(125\) 7.51340 124.774i 0.0601072 0.998192i
\(126\) 0 0
\(127\) 109.383 0.861287 0.430644 0.902522i \(-0.358287\pi\)
0.430644 + 0.902522i \(0.358287\pi\)
\(128\) 0 0
\(129\) −99.6422 −0.772420
\(130\) 0 0
\(131\) −139.062 −1.06154 −0.530771 0.847515i \(-0.678098\pi\)
−0.530771 + 0.847515i \(0.678098\pi\)
\(132\) 0 0
\(133\) −41.1018 −0.309036
\(134\) 0 0
\(135\) −66.1107 120.000i −0.489709 0.888889i
\(136\) 0 0
\(137\) 5.42829i 0.0396226i 0.999804 + 0.0198113i \(0.00630654\pi\)
−0.999804 + 0.0198113i \(0.993693\pi\)
\(138\) 0 0
\(139\) 191.765 1.37961 0.689803 0.723998i \(-0.257697\pi\)
0.689803 + 0.723998i \(0.257697\pi\)
\(140\) 0 0
\(141\) 25.1229i 0.178177i
\(142\) 0 0
\(143\) 30.5736 0.213801
\(144\) 0 0
\(145\) 73.4313 + 133.288i 0.506423 + 0.919227i
\(146\) 0 0
\(147\) 63.6617i 0.433073i
\(148\) 0 0
\(149\) 184.959i 1.24133i 0.784075 + 0.620667i \(0.213138\pi\)
−0.784075 + 0.620667i \(0.786862\pi\)
\(150\) 0 0
\(151\) 121.431i 0.804181i 0.915600 + 0.402090i \(0.131716\pi\)
−0.915600 + 0.402090i \(0.868284\pi\)
\(152\) 0 0
\(153\) 120.182i 0.785503i
\(154\) 0 0
\(155\) −59.6313 108.239i −0.384718 0.698316i
\(156\) 0 0
\(157\) −261.156 −1.66341 −0.831706 0.555216i \(-0.812636\pi\)
−0.831706 + 0.555216i \(0.812636\pi\)
\(158\) 0 0
\(159\) 34.2749i 0.215566i
\(160\) 0 0
\(161\) 156.652 0.972991
\(162\) 0 0
\(163\) 199.302i 1.22271i 0.791356 + 0.611356i \(0.209375\pi\)
−0.791356 + 0.611356i \(0.790625\pi\)
\(164\) 0 0
\(165\) −28.4866 51.7071i −0.172646 0.313376i
\(166\) 0 0
\(167\) −168.999 −1.01197 −0.505986 0.862542i \(-0.668871\pi\)
−0.505986 + 0.862542i \(0.668871\pi\)
\(168\) 0 0
\(169\) −144.578 −0.855492
\(170\) 0 0
\(171\) 55.6801 0.325614
\(172\) 0 0
\(173\) 295.307 1.70697 0.853487 0.521113i \(-0.174483\pi\)
0.853487 + 0.521113i \(0.174483\pi\)
\(174\) 0 0
\(175\) −52.8304 + 83.5782i −0.301888 + 0.477590i
\(176\) 0 0
\(177\) 199.282i 1.12589i
\(178\) 0 0
\(179\) 241.258 1.34781 0.673906 0.738817i \(-0.264615\pi\)
0.673906 + 0.738817i \(0.264615\pi\)
\(180\) 0 0
\(181\) 137.833i 0.761511i −0.924676 0.380755i \(-0.875664\pi\)
0.924676 0.380755i \(-0.124336\pi\)
\(182\) 0 0
\(183\) −109.325 −0.597406
\(184\) 0 0
\(185\) 105.220 57.9682i 0.568759 0.313342i
\(186\) 0 0
\(187\) 138.774i 0.742108i
\(188\) 0 0
\(189\) 108.372i 0.573398i
\(190\) 0 0
\(191\) 340.294i 1.78164i −0.454354 0.890821i \(-0.650130\pi\)
0.454354 0.890821i \(-0.349870\pi\)
\(192\) 0 0
\(193\) 96.5680i 0.500353i −0.968200 0.250176i \(-0.919511\pi\)
0.968200 0.250176i \(-0.0804886\pi\)
\(194\) 0 0
\(195\) −22.7548 41.3030i −0.116691 0.211810i
\(196\) 0 0
\(197\) 229.054 1.16271 0.581354 0.813651i \(-0.302523\pi\)
0.581354 + 0.813651i \(0.302523\pi\)
\(198\) 0 0
\(199\) 51.5782i 0.259187i 0.991567 + 0.129593i \(0.0413672\pi\)
−0.991567 + 0.129593i \(0.958633\pi\)
\(200\) 0 0
\(201\) 189.661 0.943587
\(202\) 0 0
\(203\) 120.373i 0.592968i
\(204\) 0 0
\(205\) 136.082 74.9707i 0.663816 0.365711i
\(206\) 0 0
\(207\) −212.214 −1.02519
\(208\) 0 0
\(209\) 64.2938 0.307626
\(210\) 0 0
\(211\) −77.7042 −0.368266 −0.184133 0.982901i \(-0.558948\pi\)
−0.184133 + 0.982901i \(0.558948\pi\)
\(212\) 0 0
\(213\) 31.9010 0.149770
\(214\) 0 0
\(215\) −125.969 228.652i −0.585904 1.06350i
\(216\) 0 0
\(217\) 97.7508i 0.450465i
\(218\) 0 0
\(219\) 183.852 0.839506
\(220\) 0 0
\(221\) 110.851i 0.501589i
\(222\) 0 0
\(223\) 98.9917 0.443909 0.221954 0.975057i \(-0.428756\pi\)
0.221954 + 0.975057i \(0.428756\pi\)
\(224\) 0 0
\(225\) 71.5687 113.222i 0.318083 0.503210i
\(226\) 0 0
\(227\) 172.584i 0.760280i −0.924929 0.380140i \(-0.875876\pi\)
0.924929 0.380140i \(-0.124124\pi\)
\(228\) 0 0
\(229\) 151.690i 0.662401i 0.943560 + 0.331201i \(0.107454\pi\)
−0.943560 + 0.331201i \(0.892546\pi\)
\(230\) 0 0
\(231\) 46.6968i 0.202151i
\(232\) 0 0
\(233\) 289.007i 1.24037i −0.784454 0.620187i \(-0.787057\pi\)
0.784454 0.620187i \(-0.212943\pi\)
\(234\) 0 0
\(235\) −57.6502 + 31.7608i −0.245320 + 0.135152i
\(236\) 0 0
\(237\) 266.378 1.12396
\(238\) 0 0
\(239\) 17.5593i 0.0734699i 0.999325 + 0.0367349i \(0.0116957\pi\)
−0.999325 + 0.0367349i \(0.988304\pi\)
\(240\) 0 0
\(241\) −101.936 −0.422971 −0.211485 0.977381i \(-0.567830\pi\)
−0.211485 + 0.977381i \(0.567830\pi\)
\(242\) 0 0
\(243\) 238.837i 0.982867i
\(244\) 0 0
\(245\) −146.086 + 80.4822i −0.596271 + 0.328499i
\(246\) 0 0
\(247\) 51.3572 0.207924
\(248\) 0 0
\(249\) 175.092 0.703182
\(250\) 0 0
\(251\) −123.966 −0.493889 −0.246944 0.969030i \(-0.579427\pi\)
−0.246944 + 0.969030i \(0.579427\pi\)
\(252\) 0 0
\(253\) −245.044 −0.968552
\(254\) 0 0
\(255\) −187.475 + 103.284i −0.735198 + 0.405037i
\(256\) 0 0
\(257\) 118.534i 0.461223i −0.973046 0.230612i \(-0.925927\pi\)
0.973046 0.230612i \(-0.0740727\pi\)
\(258\) 0 0
\(259\) −95.0247 −0.366891
\(260\) 0 0
\(261\) 163.067i 0.624779i
\(262\) 0 0
\(263\) −223.071 −0.848177 −0.424089 0.905621i \(-0.639405\pi\)
−0.424089 + 0.905621i \(0.639405\pi\)
\(264\) 0 0
\(265\) −78.6516 + 43.3309i −0.296799 + 0.163513i
\(266\) 0 0
\(267\) 180.760i 0.677004i
\(268\) 0 0
\(269\) 249.061i 0.925877i 0.886390 + 0.462938i \(0.153205\pi\)
−0.886390 + 0.462938i \(0.846795\pi\)
\(270\) 0 0
\(271\) 130.991i 0.483360i −0.970356 0.241680i \(-0.922302\pi\)
0.970356 0.241680i \(-0.0776984\pi\)
\(272\) 0 0
\(273\) 37.3008i 0.136633i
\(274\) 0 0
\(275\) 82.6405 130.738i 0.300511 0.475410i
\(276\) 0 0
\(277\) −39.7756 −0.143594 −0.0717971 0.997419i \(-0.522873\pi\)
−0.0717971 + 0.997419i \(0.522873\pi\)
\(278\) 0 0
\(279\) 132.422i 0.474630i
\(280\) 0 0
\(281\) 58.0640 0.206634 0.103317 0.994649i \(-0.467054\pi\)
0.103317 + 0.994649i \(0.467054\pi\)
\(282\) 0 0
\(283\) 241.288i 0.852607i 0.904580 + 0.426304i \(0.140184\pi\)
−0.904580 + 0.426304i \(0.859816\pi\)
\(284\) 0 0
\(285\) −47.8514 86.8570i −0.167900 0.304761i
\(286\) 0 0
\(287\) −122.896 −0.428209
\(288\) 0 0
\(289\) −214.156 −0.741025
\(290\) 0 0
\(291\) 274.427 0.943049
\(292\) 0 0
\(293\) 87.8283 0.299755 0.149878 0.988705i \(-0.452112\pi\)
0.149878 + 0.988705i \(0.452112\pi\)
\(294\) 0 0
\(295\) 457.299 251.936i 1.55016 0.854020i
\(296\) 0 0
\(297\) 169.522i 0.570782i
\(298\) 0 0
\(299\) −195.738 −0.654642
\(300\) 0 0
\(301\) 206.496i 0.686032i
\(302\) 0 0
\(303\) −127.025 −0.419226
\(304\) 0 0
\(305\) −138.211 250.872i −0.453151 0.822531i
\(306\) 0 0
\(307\) 138.914i 0.452490i 0.974070 + 0.226245i \(0.0726450\pi\)
−0.974070 + 0.226245i \(0.927355\pi\)
\(308\) 0 0
\(309\) 292.113i 0.945350i
\(310\) 0 0
\(311\) 330.863i 1.06387i 0.846786 + 0.531933i \(0.178534\pi\)
−0.846786 + 0.531933i \(0.821466\pi\)
\(312\) 0 0
\(313\) 385.575i 1.23187i 0.787797 + 0.615935i \(0.211221\pi\)
−0.787797 + 0.615935i \(0.788779\pi\)
\(314\) 0 0
\(315\) −92.8001 + 51.1256i −0.294604 + 0.162304i
\(316\) 0 0
\(317\) −414.112 −1.30635 −0.653174 0.757208i \(-0.726563\pi\)
−0.653174 + 0.757208i \(0.726563\pi\)
\(318\) 0 0
\(319\) 188.294i 0.590263i
\(320\) 0 0
\(321\) −144.652 −0.450628
\(322\) 0 0
\(323\) 233.111i 0.721707i
\(324\) 0 0
\(325\) 66.0123 104.432i 0.203115 0.321329i
\(326\) 0 0
\(327\) 236.816 0.724207
\(328\) 0 0
\(329\) 52.0640 0.158249
\(330\) 0 0
\(331\) −52.4704 −0.158521 −0.0792604 0.996854i \(-0.525256\pi\)
−0.0792604 + 0.996854i \(0.525256\pi\)
\(332\) 0 0
\(333\) 128.729 0.386573
\(334\) 0 0
\(335\) 239.773 + 435.220i 0.715739 + 1.29917i
\(336\) 0 0
\(337\) 469.871i 1.39428i 0.716937 + 0.697138i \(0.245544\pi\)
−0.716937 + 0.697138i \(0.754456\pi\)
\(338\) 0 0
\(339\) 9.91611 0.0292511
\(340\) 0 0
\(341\) 152.908i 0.448409i
\(342\) 0 0
\(343\) 325.727 0.949641
\(344\) 0 0
\(345\) 182.377 + 331.039i 0.528628 + 0.959533i
\(346\) 0 0
\(347\) 267.604i 0.771192i 0.922668 + 0.385596i \(0.126004\pi\)
−0.922668 + 0.385596i \(0.873996\pi\)
\(348\) 0 0
\(349\) 662.916i 1.89947i −0.313053 0.949736i \(-0.601352\pi\)
0.313053 0.949736i \(-0.398648\pi\)
\(350\) 0 0
\(351\) 135.412i 0.385790i
\(352\) 0 0
\(353\) 366.925i 1.03945i 0.854335 + 0.519723i \(0.173965\pi\)
−0.854335 + 0.519723i \(0.826035\pi\)
\(354\) 0 0
\(355\) 40.3297 + 73.2040i 0.113605 + 0.206208i
\(356\) 0 0
\(357\) 169.309 0.474256
\(358\) 0 0
\(359\) 234.863i 0.654213i −0.944987 0.327107i \(-0.893926\pi\)
0.944987 0.327107i \(-0.106074\pi\)
\(360\) 0 0
\(361\) −253.000 −0.700831
\(362\) 0 0
\(363\) 157.877i 0.434922i
\(364\) 0 0
\(365\) 232.428 + 421.890i 0.636790 + 1.15586i
\(366\) 0 0
\(367\) 455.060 1.23994 0.619972 0.784624i \(-0.287144\pi\)
0.619972 + 0.784624i \(0.287144\pi\)
\(368\) 0 0
\(369\) 166.486 0.451181
\(370\) 0 0
\(371\) 71.0304 0.191457
\(372\) 0 0
\(373\) −518.534 −1.39017 −0.695086 0.718926i \(-0.744634\pi\)
−0.695086 + 0.718926i \(0.744634\pi\)
\(374\) 0 0
\(375\) −238.125 14.3389i −0.635000 0.0382372i
\(376\) 0 0
\(377\) 150.407i 0.398957i
\(378\) 0 0
\(379\) −563.918 −1.48791 −0.743955 0.668230i \(-0.767052\pi\)
−0.743955 + 0.668230i \(0.767052\pi\)
\(380\) 0 0
\(381\) 208.753i 0.547908i
\(382\) 0 0
\(383\) −56.8436 −0.148417 −0.0742083 0.997243i \(-0.523643\pi\)
−0.0742083 + 0.997243i \(0.523643\pi\)
\(384\) 0 0
\(385\) −107.156 + 59.0348i −0.278328 + 0.153337i
\(386\) 0 0
\(387\) 279.737i 0.722835i
\(388\) 0 0
\(389\) 188.677i 0.485031i −0.970147 0.242516i \(-0.922027\pi\)
0.970147 0.242516i \(-0.0779726\pi\)
\(390\) 0 0
\(391\) 888.460i 2.27228i
\(392\) 0 0
\(393\) 265.393i 0.675300i
\(394\) 0 0
\(395\) 336.759 + 611.265i 0.852555 + 1.54751i
\(396\) 0 0
\(397\) 143.716 0.362005 0.181003 0.983483i \(-0.442066\pi\)
0.181003 + 0.983483i \(0.442066\pi\)
\(398\) 0 0
\(399\) 78.4407i 0.196593i
\(400\) 0 0
\(401\) 1.28437 0.00320291 0.00160145 0.999999i \(-0.499490\pi\)
0.00160145 + 0.999999i \(0.499490\pi\)
\(402\) 0 0
\(403\) 122.141i 0.303079i
\(404\) 0 0
\(405\) −17.8392 + 9.82799i −0.0440473 + 0.0242666i
\(406\) 0 0
\(407\) 148.643 0.365217
\(408\) 0 0
\(409\) −76.3767 −0.186740 −0.0933700 0.995631i \(-0.529764\pi\)
−0.0933700 + 0.995631i \(0.529764\pi\)
\(410\) 0 0
\(411\) 10.3596 0.0252059
\(412\) 0 0
\(413\) −412.987 −0.999969
\(414\) 0 0
\(415\) 221.355 + 401.789i 0.533384 + 0.968166i
\(416\) 0 0
\(417\) 365.974i 0.877636i
\(418\) 0 0
\(419\) −33.3906 −0.0796912 −0.0398456 0.999206i \(-0.512687\pi\)
−0.0398456 + 0.999206i \(0.512687\pi\)
\(420\) 0 0
\(421\) 392.318i 0.931871i 0.884819 + 0.465935i \(0.154282\pi\)
−0.884819 + 0.465935i \(0.845718\pi\)
\(422\) 0 0
\(423\) −70.5305 −0.166739
\(424\) 0 0
\(425\) −474.019 299.631i −1.11534 0.705014i
\(426\) 0 0
\(427\) 226.563i 0.530592i
\(428\) 0 0
\(429\) 58.3482i 0.136010i
\(430\) 0 0
\(431\) 840.460i 1.95002i −0.222156 0.975011i \(-0.571309\pi\)
0.222156 0.975011i \(-0.428691\pi\)
\(432\) 0 0
\(433\) 275.552i 0.636380i 0.948027 + 0.318190i \(0.103075\pi\)
−0.948027 + 0.318190i \(0.896925\pi\)
\(434\) 0 0
\(435\) 254.373 140.140i 0.584766 0.322161i
\(436\) 0 0
\(437\) −411.622 −0.941926
\(438\) 0 0
\(439\) 477.725i 1.08821i −0.839016 0.544106i \(-0.816869\pi\)
0.839016 0.544106i \(-0.183131\pi\)
\(440\) 0 0
\(441\) −178.725 −0.405272
\(442\) 0 0
\(443\) 625.709i 1.41244i 0.707994 + 0.706218i \(0.249600\pi\)
−0.707994 + 0.706218i \(0.750400\pi\)
\(444\) 0 0
\(445\) −414.795 + 228.520i −0.932124 + 0.513528i
\(446\) 0 0
\(447\) 352.984 0.789674
\(448\) 0 0
\(449\) 104.377 0.232465 0.116232 0.993222i \(-0.462918\pi\)
0.116232 + 0.993222i \(0.462918\pi\)
\(450\) 0 0
\(451\) 192.241 0.426256
\(452\) 0 0
\(453\) 231.746 0.511580
\(454\) 0 0
\(455\) −85.5953 + 47.1563i −0.188121 + 0.103640i
\(456\) 0 0
\(457\) 198.564i 0.434495i 0.976117 + 0.217248i \(0.0697079\pi\)
−0.976117 + 0.217248i \(0.930292\pi\)
\(458\) 0 0
\(459\) −614.640 −1.33908
\(460\) 0 0
\(461\) 11.1338i 0.0241515i 0.999927 + 0.0120757i \(0.00384392\pi\)
−0.999927 + 0.0120757i \(0.996156\pi\)
\(462\) 0 0
\(463\) −15.9944 −0.0345451 −0.0172725 0.999851i \(-0.505498\pi\)
−0.0172725 + 0.999851i \(0.505498\pi\)
\(464\) 0 0
\(465\) −206.569 + 113.803i −0.444234 + 0.244738i
\(466\) 0 0
\(467\) 539.567i 1.15539i −0.816253 0.577695i \(-0.803952\pi\)
0.816253 0.577695i \(-0.196048\pi\)
\(468\) 0 0
\(469\) 393.048i 0.838056i
\(470\) 0 0
\(471\) 498.403i 1.05818i
\(472\) 0 0
\(473\) 323.013i 0.682902i
\(474\) 0 0
\(475\) 138.819 219.612i 0.292249 0.462341i
\(476\) 0 0
\(477\) −96.2240 −0.201728
\(478\) 0 0
\(479\) 127.744i 0.266689i 0.991070 + 0.133344i \(0.0425716\pi\)
−0.991070 + 0.133344i \(0.957428\pi\)
\(480\) 0 0
\(481\) 118.735 0.246849
\(482\) 0 0
\(483\) 298.962i 0.618969i
\(484\) 0 0
\(485\) 346.935 + 629.736i 0.715331 + 1.29842i
\(486\) 0 0
\(487\) −624.466 −1.28227 −0.641135 0.767428i \(-0.721536\pi\)
−0.641135 + 0.767428i \(0.721536\pi\)
\(488\) 0 0
\(489\) 380.358 0.777828
\(490\) 0 0
\(491\) −424.601 −0.864769 −0.432384 0.901689i \(-0.642328\pi\)
−0.432384 + 0.901689i \(0.642328\pi\)
\(492\) 0 0
\(493\) 682.701 1.38479
\(494\) 0 0
\(495\) 145.163 79.9737i 0.293259 0.161563i
\(496\) 0 0
\(497\) 66.1107i 0.133019i
\(498\) 0 0
\(499\) −591.631 −1.18563 −0.592816 0.805338i \(-0.701984\pi\)
−0.592816 + 0.805338i \(0.701984\pi\)
\(500\) 0 0
\(501\) 322.527i 0.643766i
\(502\) 0 0
\(503\) 5.71879 0.0113694 0.00568468 0.999984i \(-0.498191\pi\)
0.00568468 + 0.999984i \(0.498191\pi\)
\(504\) 0 0
\(505\) −160.588 291.489i −0.317995 0.577205i
\(506\) 0 0
\(507\) 275.920i 0.544221i
\(508\) 0 0
\(509\) 439.708i 0.863867i −0.901905 0.431933i \(-0.857832\pi\)
0.901905 0.431933i \(-0.142168\pi\)
\(510\) 0 0
\(511\) 381.009i 0.745615i
\(512\) 0 0
\(513\) 284.761i 0.555091i
\(514\) 0 0
\(515\) 670.320 369.294i 1.30159 0.717076i
\(516\) 0 0
\(517\) −81.4416 −0.157527
\(518\) 0 0
\(519\) 563.578i 1.08589i
\(520\) 0 0
\(521\) 80.5687 0.154642 0.0773212 0.997006i \(-0.475363\pi\)
0.0773212 + 0.997006i \(0.475363\pi\)
\(522\) 0 0
\(523\) 585.089i 1.11872i −0.828926 0.559359i \(-0.811047\pi\)
0.828926 0.559359i \(-0.188953\pi\)
\(524\) 0 0
\(525\) 159.505 + 100.824i 0.303819 + 0.192046i
\(526\) 0 0
\(527\) −554.400 −1.05199
\(528\) 0 0
\(529\) 1039.82 1.96563
\(530\) 0 0
\(531\) 559.469 1.05361
\(532\) 0 0
\(533\) 153.560 0.288105
\(534\) 0 0
\(535\) −182.871 331.936i −0.341815 0.620441i
\(536\) 0 0
\(537\) 460.430i 0.857411i
\(538\) 0 0
\(539\) −206.374 −0.382883
\(540\) 0 0
\(541\) 248.685i 0.459676i 0.973229 + 0.229838i \(0.0738196\pi\)
−0.973229 + 0.229838i \(0.926180\pi\)
\(542\) 0 0
\(543\) −263.048 −0.484435
\(544\) 0 0
\(545\) 299.386 + 543.427i 0.549332 + 0.997114i
\(546\) 0 0
\(547\) 47.9916i 0.0877360i 0.999037 + 0.0438680i \(0.0139681\pi\)
−0.999037 + 0.0438680i \(0.986032\pi\)
\(548\) 0 0
\(549\) 306.922i 0.559056i
\(550\) 0 0
\(551\) 316.294i 0.574036i
\(552\) 0 0
\(553\) 552.034i 0.998254i
\(554\) 0 0
\(555\) −110.629 200.808i −0.199332 0.361816i
\(556\) 0 0
\(557\) 1091.71 1.95998 0.979990 0.199045i \(-0.0637840\pi\)
0.979990 + 0.199045i \(0.0637840\pi\)
\(558\) 0 0
\(559\) 258.019i 0.461572i
\(560\) 0 0
\(561\) −264.844 −0.472092
\(562\) 0 0
\(563\) 427.231i 0.758846i −0.925223 0.379423i \(-0.876122\pi\)
0.925223 0.379423i \(-0.123878\pi\)
\(564\) 0 0
\(565\) 12.5361 + 22.7548i 0.0221878 + 0.0402739i
\(566\) 0 0
\(567\) 16.1106 0.0284137
\(568\) 0 0
\(569\) 54.2298 0.0953072 0.0476536 0.998864i \(-0.484826\pi\)
0.0476536 + 0.998864i \(0.484826\pi\)
\(570\) 0 0
\(571\) 1050.42 1.83961 0.919806 0.392373i \(-0.128346\pi\)
0.919806 + 0.392373i \(0.128346\pi\)
\(572\) 0 0
\(573\) −649.434 −1.13339
\(574\) 0 0
\(575\) −529.080 + 837.009i −0.920140 + 1.45567i
\(576\) 0 0
\(577\) 161.031i 0.279083i −0.990216 0.139542i \(-0.955437\pi\)
0.990216 0.139542i \(-0.0445629\pi\)
\(578\) 0 0
\(579\) −184.295 −0.318299
\(580\) 0 0
\(581\) 362.856i 0.624538i
\(582\) 0 0
\(583\) −111.110 −0.190583
\(584\) 0 0
\(585\) 115.955 63.8821i 0.198213 0.109200i
\(586\) 0 0
\(587\) 186.363i 0.317484i 0.987320 + 0.158742i \(0.0507439\pi\)
−0.987320 + 0.158742i \(0.949256\pi\)
\(588\) 0 0
\(589\) 256.852i 0.436082i
\(590\) 0 0
\(591\) 437.137i 0.739657i
\(592\) 0 0
\(593\) 1011.51i 1.70576i −0.522109 0.852879i \(-0.674855\pi\)
0.522109 0.852879i \(-0.325145\pi\)
\(594\) 0 0
\(595\) 214.044 + 388.519i 0.359737 + 0.652973i
\(596\) 0 0
\(597\) 98.4344 0.164882
\(598\) 0 0
\(599\) 647.616i 1.08116i −0.841292 0.540581i \(-0.818204\pi\)
0.841292 0.540581i \(-0.181796\pi\)
\(600\) 0 0
\(601\) 40.3767 0.0671825 0.0335913 0.999436i \(-0.489306\pi\)
0.0335913 + 0.999436i \(0.489306\pi\)
\(602\) 0 0
\(603\) 532.458i 0.883014i
\(604\) 0 0
\(605\) −362.284 + 199.590i −0.598816 + 0.329901i
\(606\) 0 0
\(607\) −799.902 −1.31780 −0.658898 0.752233i \(-0.728977\pi\)
−0.658898 + 0.752233i \(0.728977\pi\)
\(608\) 0 0
\(609\) −229.725 −0.377217
\(610\) 0 0
\(611\) −65.0546 −0.106472
\(612\) 0 0
\(613\) 900.584 1.46914 0.734571 0.678532i \(-0.237384\pi\)
0.734571 + 0.678532i \(0.237384\pi\)
\(614\) 0 0
\(615\) −143.078 259.706i −0.232647 0.422287i
\(616\) 0 0
\(617\) 255.001i 0.413292i 0.978416 + 0.206646i \(0.0662549\pi\)
−0.978416 + 0.206646i \(0.933745\pi\)
\(618\) 0 0
\(619\) 168.789 0.272679 0.136340 0.990662i \(-0.456466\pi\)
0.136340 + 0.990662i \(0.456466\pi\)
\(620\) 0 0
\(621\) 1085.31i 1.74769i
\(622\) 0 0
\(623\) 374.602 0.601288
\(624\) 0 0
\(625\) −268.137 564.559i −0.429020 0.903295i
\(626\) 0 0
\(627\) 122.702i 0.195696i
\(628\) 0 0
\(629\) 538.939i 0.856818i
\(630\) 0 0
\(631\) 665.431i 1.05457i 0.849690 + 0.527283i \(0.176789\pi\)
−0.849690 + 0.527283i \(0.823211\pi\)
\(632\) 0 0
\(633\) 148.295i 0.234273i
\(634\) 0 0
\(635\) 479.031 263.909i 0.754380 0.415605i
\(636\) 0 0
\(637\) −164.849 −0.258790
\(638\) 0 0
\(639\) 89.5593i 0.140155i
\(640\) 0 0
\(641\) 552.633 0.862142 0.431071 0.902318i \(-0.358136\pi\)
0.431071 + 0.902318i \(0.358136\pi\)
\(642\) 0 0
\(643\) 901.490i 1.40201i 0.713159 + 0.701003i \(0.247264\pi\)
−0.713159 + 0.701003i \(0.752736\pi\)
\(644\) 0 0
\(645\) −436.370 + 240.406i −0.676543 + 0.372723i
\(646\) 0 0
\(647\) 67.8163 0.104817 0.0524083 0.998626i \(-0.483310\pi\)
0.0524083 + 0.998626i \(0.483310\pi\)
\(648\) 0 0
\(649\) 646.019 0.995407
\(650\) 0 0
\(651\) 186.553 0.286563
\(652\) 0 0
\(653\) 124.351 0.190431 0.0952153 0.995457i \(-0.469646\pi\)
0.0952153 + 0.995457i \(0.469646\pi\)
\(654\) 0 0
\(655\) −609.005 + 335.514i −0.929778 + 0.512235i
\(656\) 0 0
\(657\) 516.149i 0.785615i
\(658\) 0 0
\(659\) −260.582 −0.395420 −0.197710 0.980261i \(-0.563350\pi\)
−0.197710 + 0.980261i \(0.563350\pi\)
\(660\) 0 0
\(661\) 447.965i 0.677708i 0.940839 + 0.338854i \(0.110039\pi\)
−0.940839 + 0.338854i \(0.889961\pi\)
\(662\) 0 0
\(663\) −211.554 −0.319086
\(664\) 0 0
\(665\) −180.000 + 99.1660i −0.270677 + 0.149122i
\(666\) 0 0
\(667\) 1205.49i 1.80734i
\(668\) 0 0
\(669\) 188.921i 0.282393i
\(670\) 0 0
\(671\) 354.403i 0.528171i
\(672\) 0 0
\(673\) 288.775i 0.429086i −0.976715 0.214543i \(-0.931174\pi\)
0.976715 0.214543i \(-0.0688261\pi\)
\(674\) 0 0
\(675\) −579.047 366.020i −0.857847 0.542252i
\(676\) 0 0
\(677\) −248.339 −0.366823 −0.183412 0.983036i \(-0.558714\pi\)
−0.183412 + 0.983036i \(0.558714\pi\)
\(678\) 0 0
\(679\) 568.716i 0.837578i
\(680\) 0 0
\(681\) −329.367 −0.483652
\(682\) 0 0
\(683\) 451.900i 0.661640i −0.943694 0.330820i \(-0.892675\pi\)
0.943694 0.330820i \(-0.107325\pi\)
\(684\) 0 0
\(685\) 13.0968 + 23.7725i 0.0191194 + 0.0347044i
\(686\) 0 0
\(687\) 289.493 0.421387
\(688\) 0 0
\(689\) −88.7534 −0.128815
\(690\) 0 0
\(691\) −193.513 −0.280048 −0.140024 0.990148i \(-0.544718\pi\)
−0.140024 + 0.990148i \(0.544718\pi\)
\(692\) 0 0
\(693\) −131.097 −0.189174
\(694\) 0 0
\(695\) 839.811 462.670i 1.20836 0.665713i
\(696\) 0 0
\(697\) 697.013i 1.00002i
\(698\) 0 0
\(699\) −551.555 −0.789064
\(700\) 0 0
\(701\) 928.025i 1.32386i −0.749566 0.661929i \(-0.769738\pi\)
0.749566 0.661929i \(-0.230262\pi\)
\(702\) 0 0
\(703\) 249.689 0.355177
\(704\) 0 0
\(705\) 60.6139 + 110.023i 0.0859771 + 0.156060i
\(706\) 0 0
\(707\) 263.244i 0.372339i
\(708\) 0 0
\(709\) 75.9885i 0.107177i 0.998563 + 0.0535885i \(0.0170659\pi\)
−0.998563 + 0.0535885i \(0.982934\pi\)
\(710\) 0 0
\(711\) 747.834i 1.05181i
\(712\) 0 0
\(713\) 978.944i 1.37299i
\(714\) 0 0
\(715\) 133.893 73.7647i 0.187263 0.103167i
\(716\) 0 0
\(717\) 33.5111 0.0467379
\(718\) 0 0
\(719\) 395.578i 0.550178i 0.961419 + 0.275089i \(0.0887074\pi\)
−0.961419 + 0.275089i \(0.911293\pi\)
\(720\) 0 0
\(721\) −605.367 −0.839622
\(722\) 0 0
\(723\) 194.540i 0.269073i
\(724\) 0 0
\(725\) 643.166 + 406.550i 0.887125 + 0.560759i
\(726\) 0 0
\(727\) 352.113 0.484337 0.242168 0.970234i \(-0.422141\pi\)
0.242168 + 0.970234i \(0.422141\pi\)
\(728\) 0 0
\(729\) −492.469 −0.675540
\(730\) 0 0
\(731\) −1171.15 −1.60213
\(732\) 0 0
\(733\) 156.532 0.213550 0.106775 0.994283i \(-0.465947\pi\)
0.106775 + 0.994283i \(0.465947\pi\)
\(734\) 0 0
\(735\) 153.596 + 278.799i 0.208975 + 0.379318i
\(736\) 0 0
\(737\) 614.829i 0.834232i
\(738\) 0 0
\(739\) −183.376 −0.248140 −0.124070 0.992273i \(-0.539595\pi\)
−0.124070 + 0.992273i \(0.539595\pi\)
\(740\) 0 0
\(741\) 98.0126i 0.132271i
\(742\) 0 0
\(743\) 1028.34 1.38404 0.692021 0.721878i \(-0.256721\pi\)
0.692021 + 0.721878i \(0.256721\pi\)
\(744\) 0 0
\(745\) 446.249 + 810.003i 0.598991 + 1.08725i
\(746\) 0 0
\(747\) 491.557i 0.658042i
\(748\) 0 0
\(749\) 299.772i 0.400230i
\(750\) 0 0
\(751\) 740.038i 0.985403i −0.870198 0.492702i \(-0.836009\pi\)
0.870198 0.492702i \(-0.163991\pi\)
\(752\) 0 0
\(753\) 236.583i 0.314188i
\(754\) 0 0
\(755\) 292.976 + 531.793i 0.388048 + 0.704361i
\(756\) 0 0
\(757\) −934.857 −1.23495 −0.617475 0.786591i \(-0.711844\pi\)
−0.617475 + 0.786591i \(0.711844\pi\)
\(758\) 0 0
\(759\) 467.654i 0.616144i
\(760\) 0 0
\(761\) 436.735 0.573896 0.286948 0.957946i \(-0.407359\pi\)
0.286948 + 0.957946i \(0.407359\pi\)
\(762\) 0 0
\(763\) 490.770i 0.643211i
\(764\) 0 0
\(765\) −289.962 526.322i −0.379036 0.688002i
\(766\) 0 0
\(767\) 516.033 0.672793
\(768\) 0 0
\(769\) −768.275 −0.999057 −0.499529 0.866297i \(-0.666493\pi\)
−0.499529 + 0.866297i \(0.666493\pi\)
\(770\) 0 0
\(771\) −226.217 −0.293407
\(772\) 0 0
\(773\) −792.266 −1.02492 −0.512462 0.858710i \(-0.671266\pi\)
−0.512462 + 0.858710i \(0.671266\pi\)
\(774\) 0 0
\(775\) −522.295 330.147i −0.673929 0.425996i
\(776\) 0 0
\(777\) 181.350i 0.233398i
\(778\) 0 0
\(779\) 322.925 0.414538
\(780\) 0 0
\(781\) 103.414i 0.132413i
\(782\) 0 0
\(783\) 833.966 1.06509
\(784\) 0 0
\(785\) −1143.70 + 630.089i −1.45694 + 0.802661i
\(786\) 0 0
\(787\) 965.852i 1.22726i 0.789594 + 0.613629i \(0.210291\pi\)
−0.789594 + 0.613629i \(0.789709\pi\)
\(788\) 0 0
\(789\) 425.719i 0.539568i
\(790\) 0 0
\(791\) 20.5499i 0.0259796i
\(792\) 0 0
\(793\) 283.093i 0.356990i
\(794\) 0 0
\(795\) 82.6949 + 150.103i 0.104019 + 0.188808i
\(796\) 0 0
\(797\) −358.505 −0.449818 −0.224909 0.974380i \(-0.572208\pi\)
−0.224909 + 0.974380i \(0.572208\pi\)
\(798\) 0 0
\(799\) 295.284i 0.369567i
\(800\) 0 0