Properties

Label 320.3.e.b.159.9
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 159.9
Root \(-1.94376 - 3.36668i\) of defining polynomial
Character \(\chi\) \(=\) 320.159
Dual form 320.3.e.b.159.6

$q$-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 \(0.103036\pi\)
−0.948066 + 0.318075i \(0.896964\pi\)
\(4\) 0 0
\(5\) −4.37937 2.41269i −0.875875 0.482539i
\(6\) 0 0
\(7\) 3.95502 0.565003 0.282501 0.959267i \(-0.408836\pi\)
0.282501 + 0.959267i \(0.408836\pi\)
\(8\) 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 \(-0.560872\pi\)
−0.190071 + 0.981770i \(0.560872\pi\)
\(14\) 0 0
\(15\) 4.60451 8.35782i 0.306967 0.557188i
\(16\) 0 0
\(17\) 22.4311i 1.31948i 0.751494 + 0.659740i \(0.229333\pi\)
−0.751494 + 0.659740i \(0.770667\pi\)
\(18\) 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 \(-0.169807\pi\)
0.861050 + 0.508520i \(0.169807\pi\)
\(24\) 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 \(-0.605257\pi\)
−0.324680 + 0.945824i \(0.605257\pi\)
\(38\) 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 \(0.207669\pi\)
−0.794622 + 0.607105i \(0.792331\pi\)
\(44\) 0 0
\(45\) −23.4639 12.9268i −0.521420 0.287262i
\(46\) 0 0
\(47\) 13.1640 0.280086 0.140043 0.990145i \(-0.455276\pi\)
0.140043 + 0.990145i \(0.455276\pi\)
\(48\) 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 \(-0.445807\pi\)
0.169430 + 0.985542i \(0.445807\pi\)
\(54\) 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 \(-0.734049\pi\)
0.670799 0.741639i \(-0.265951\pi\)
\(68\) 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 \(-0.770627\pi\)
0.751412 0.659833i \(-0.229373\pi\)
\(74\) 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 \(-0.813603\pi\)
0.833390 0.552686i \(-0.186397\pi\)
\(84\) 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 \(-0.734249\pi\)
0.671267 0.741216i \(-0.265751\pi\)
\(98\) 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 \(-0.766609\pi\)
−0.743024 + 0.669264i \(0.766609\pi\)
\(104\) 0 0
\(105\) 18.2109 33.0553i 0.173437 0.314813i
\(106\) 0 0
\(107\) 75.7953i 0.708367i 0.935176 + 0.354184i \(0.115241\pi\)
−0.935176 + 0.354184i \(0.884759\pi\)
\(108\) 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 \(-0.992681\pi\)
0.999736 0.0229907i \(-0.00731881\pi\)
\(114\) 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 \(-0.641713\pi\)
−0.430644 + 0.902522i \(0.641713\pi\)
\(128\) 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 \(-0.993693\pi\)
0.999804 0.0198113i \(-0.00630654\pi\)
\(138\) 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 \(-0.187364\pi\)
0.831706 + 0.555216i \(0.187364\pi\)
\(158\) 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 \(-0.790625\pi\)
0.791356 0.611356i \(-0.209375\pi\)
\(164\) 0 0
\(165\) 28.4866 51.7071i 0.172646 0.313376i
\(166\) 0 0
\(167\) 168.999 1.01197 0.505986 0.862542i \(-0.331129\pi\)
0.505986 + 0.862542i \(0.331129\pi\)
\(168\) 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 \(-0.825517\pi\)
−0.853487 + 0.521113i \(0.825517\pi\)
\(174\) 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 \(0.0804886\pi\)
−0.968200 + 0.250176i \(0.919511\pi\)
\(194\) 0 0
\(195\) −22.7548 + 41.3030i −0.116691 + 0.211810i
\(196\) 0 0
\(197\) −229.054 −1.16271 −0.581354 0.813651i \(-0.697477\pi\)
−0.581354 + 0.813651i \(0.697477\pi\)
\(198\) 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 \(-0.571244\pi\)
−0.221954 + 0.975057i \(0.571244\pi\)
\(224\) 0 0
\(225\) 71.5687 + 113.222i 0.318083 + 0.503210i
\(226\) 0 0
\(227\) 172.584i 0.760280i 0.924929 + 0.380140i \(0.124124\pi\)
−0.924929 + 0.380140i \(0.875876\pi\)
\(228\) 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 \(0.212943\pi\)
−0.784454 + 0.620187i \(0.787057\pi\)
\(234\) 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 \(0.0740727\pi\)
−0.973046 + 0.230612i \(0.925927\pi\)
\(258\) 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 \(-0.360595\pi\)
0.424089 + 0.905621i \(0.360595\pi\)
\(264\) 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 \(-0.477127\pi\)
0.0717971 + 0.997419i \(0.477127\pi\)
\(278\) 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 \(-0.859816\pi\)
0.904580 0.426304i \(-0.140184\pi\)
\(284\) 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 \(-0.547888\pi\)
−0.149878 + 0.988705i \(0.547888\pi\)
\(294\) 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 \(-0.927355\pi\)
0.974070 0.226245i \(-0.0726450\pi\)
\(308\) 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 \(-0.788779\pi\)
0.787797 0.615935i \(-0.211221\pi\)
\(314\) 0 0
\(315\) −92.8001 51.1256i −0.294604 0.162304i
\(316\) 0 0
\(317\) 414.112 1.30635 0.653174 0.757208i \(-0.273437\pi\)
0.653174 + 0.757208i \(0.273437\pi\)
\(318\) 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 \(-0.754456\pi\)
0.716937 0.697138i \(-0.245544\pi\)
\(338\) 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 \(-0.873996\pi\)
0.922668 0.385596i \(-0.126004\pi\)
\(348\) 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 \(-0.826035\pi\)
0.854335 0.519723i \(-0.173965\pi\)
\(354\) 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 \(-0.712856\pi\)
−0.619972 + 0.784624i \(0.712856\pi\)
\(368\) 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 \(-0.255366\pi\)
0.695086 + 0.718926i \(0.255366\pi\)
\(374\) 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 \(-0.476357\pi\)
0.0742083 + 0.997243i \(0.476357\pi\)
\(384\) 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 \(-0.557934\pi\)
−0.181003 + 0.983483i \(0.557934\pi\)
\(398\) 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 \(-0.896925\pi\)
0.948027 0.318190i \(-0.103075\pi\)
\(434\) 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 \(-0.750400\pi\)
0.707994 0.706218i \(-0.249600\pi\)
\(444\) 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 \(-0.930292\pi\)
0.976117 0.217248i \(-0.0697079\pi\)
\(458\) 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 \(-0.494502\pi\)
0.0172725 + 0.999851i \(0.494502\pi\)
\(464\) 0 0
\(465\) −206.569 113.803i −0.444234 0.244738i
\(466\) 0 0
\(467\) 539.567i 1.15539i 0.816253 + 0.577695i \(0.196048\pi\)
−0.816253 + 0.577695i \(0.803952\pi\)
\(468\) 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 \(-0.278464\pi\)
0.641135 + 0.767428i \(0.278464\pi\)
\(488\) 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 \(-0.501809\pi\)
−0.00568468 + 0.999984i \(0.501809\pi\)
\(504\) 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 \(0.188953\pi\)
−0.828926 + 0.559359i \(0.811047\pi\)
\(524\) 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 \(-0.986032\pi\)
0.999037 0.0438680i \(-0.0139681\pi\)
\(548\) 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 \(-0.936216\pi\)
−0.979990 + 0.199045i \(0.936216\pi\)
\(558\) 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 \(0.123878\pi\)
−0.925223 + 0.379423i \(0.876122\pi\)
\(564\) 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 \(0.0445629\pi\)
−0.990216 + 0.139542i \(0.955437\pi\)
\(578\) 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 \(-0.949256\pi\)
0.987320 0.158742i \(-0.0507439\pi\)
\(588\) 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 \(0.325145\pi\)
−0.522109 + 0.852879i \(0.674855\pi\)
\(594\) 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 \(-0.271023\pi\)
0.658898 + 0.752233i \(0.271023\pi\)
\(608\) 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 \(-0.762616\pi\)
−0.734571 + 0.678532i \(0.762616\pi\)
\(614\) 0 0
\(615\) 143.078 259.706i 0.232647 0.422287i
\(616\) 0 0
\(617\) 255.001i 0.413292i −0.978416 0.206646i \(-0.933745\pi\)
0.978416 0.206646i \(-0.0662549\pi\)
\(618\) 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 \(-0.752736\pi\)
0.713159 0.701003i \(-0.247264\pi\)
\(644\) 0 0
\(645\) 436.370 + 240.406i 0.676543 + 0.372723i
\(646\) 0 0
\(647\) −67.8163 −0.104817 −0.0524083 0.998626i \(-0.516690\pi\)
−0.0524083 + 0.998626i \(0.516690\pi\)
\(648\) 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 \(-0.530354\pi\)
−0.0952153 + 0.995457i \(0.530354\pi\)
\(654\) 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 \(0.0688261\pi\)
−0.976715 + 0.214543i \(0.931174\pi\)
\(674\) 0 0
\(675\) −579.047 + 366.020i −0.857847 + 0.542252i
\(676\) 0 0
\(677\) 248.339 0.366823 0.183412 0.983036i \(-0.441286\pi\)
0.183412 + 0.983036i \(0.441286\pi\)
\(678\) 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 \(0.107325\pi\)
−0.943694 + 0.330820i \(0.892675\pi\)
\(684\) 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 \(-0.577859\pi\)
−0.242168 + 0.970234i \(0.577859\pi\)
\(728\) 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 \(-0.534053\pi\)
−0.106775 + 0.994283i \(0.534053\pi\)
\(734\) 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 \(-0.743279\pi\)
−0.692021 + 0.721878i \(0.743279\pi\)
\(744\) 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 \(-0.288156\pi\)
0.617475 + 0.786591i \(0.288156\pi\)
\(758\) 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 \(-0.328734\pi\)
0.512462 + 0.858710i \(0.328734\pi\)
\(774\) 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 \(-0.789709\pi\)
0.789594 0.613629i \(-0.210291\pi\)
\(788\) 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 \(-0.427792\pi\)
0.224909 + 0.974380i \(0.427792\pi\)
\(798\) 0 0
\(799\) 295.284i 0.369567i
\(800\) 0 0