Properties

Label 1620.3.o.f.701.2
Level $1620$
Weight $3$
Character 1620.701
Analytic conductor $44.142$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1620.o (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(44.1418028264\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.3317760000.8
Defining polynomial: \( x^{8} + 4x^{6} + 7x^{4} + 36x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 180)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 701.2
Root \(1.40294 - 1.01575i\) of defining polynomial
Character \(\chi\) \(=\) 1620.701
Dual form 1620.3.o.f.1241.2

$q$-expansion

\(f(q)\) \(=\) \(q+(-1.93649 - 1.11803i) q^{5} +(6.74342 + 11.6799i) q^{7} +O(q^{10})\) \(q+(-1.93649 - 1.11803i) q^{5} +(6.74342 + 11.6799i) q^{7} +(15.2932 - 8.82952i) q^{11} +(3.74342 - 6.48379i) q^{13} -16.9706i q^{17} -10.9737 q^{19} +(-18.9674 - 10.9508i) q^{23} +(2.50000 + 4.33013i) q^{25} +(41.0128 - 23.6788i) q^{29} +(8.48683 - 14.6996i) q^{31} -30.1575i q^{35} -5.53950 q^{37} +(57.4985 + 33.1968i) q^{41} +(-19.4868 - 33.7522i) q^{43} +(-28.2014 + 16.2821i) q^{47} +(-66.4473 + 115.090i) q^{49} -11.2392i q^{53} -39.4868 q^{55} +(-27.6051 - 15.9378i) q^{59} +(-23.4605 - 40.6348i) q^{61} +(-14.4982 + 8.37053i) q^{65} +(38.0000 - 65.8179i) q^{67} +77.7445i q^{71} +94.9210 q^{73} +(206.257 + 119.082i) q^{77} +(3.46050 + 5.99376i) q^{79} +(53.8243 - 31.0755i) q^{83} +(-18.9737 + 32.8634i) q^{85} +62.2626i q^{89} +100.974 q^{91} +(21.2504 + 12.2689i) q^{95} +(62.4868 + 108.230i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 16 q^{7} - 8 q^{13} + 64 q^{19} + 20 q^{25} - 8 q^{31} - 272 q^{37} - 80 q^{43} - 228 q^{49} - 240 q^{55} + 40 q^{61} + 304 q^{67} + 304 q^{73} - 200 q^{79} + 656 q^{91} + 424 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times\).

\(n\) \(811\) \(1297\) \(1541\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{5}{6}\right)\)

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 0
\(4\) 0 0
\(5\) −1.93649 1.11803i −0.387298 0.223607i
\(6\) 0 0
\(7\) 6.74342 + 11.6799i 0.963345 + 1.66856i 0.713997 + 0.700149i \(0.246883\pi\)
0.249348 + 0.968414i \(0.419784\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 15.2932 8.82952i 1.39029 0.802684i 0.396942 0.917843i \(-0.370071\pi\)
0.993347 + 0.115159i \(0.0367379\pi\)
\(12\) 0 0
\(13\) 3.74342 6.48379i 0.287955 0.498753i −0.685366 0.728198i \(-0.740358\pi\)
0.973322 + 0.229446i \(0.0736913\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 16.9706i 0.998268i −0.866525 0.499134i \(-0.833651\pi\)
0.866525 0.499134i \(-0.166349\pi\)
\(18\) 0 0
\(19\) −10.9737 −0.577561 −0.288781 0.957395i \(-0.593250\pi\)
−0.288781 + 0.957395i \(0.593250\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −18.9674 10.9508i −0.824670 0.476124i 0.0273540 0.999626i \(-0.491292\pi\)
−0.852024 + 0.523502i \(0.824625\pi\)
\(24\) 0 0
\(25\) 2.50000 + 4.33013i 0.100000 + 0.173205i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 41.0128 23.6788i 1.41424 0.816509i 0.418451 0.908239i \(-0.362573\pi\)
0.995784 + 0.0917300i \(0.0292397\pi\)
\(30\) 0 0
\(31\) 8.48683 14.6996i 0.273769 0.474181i −0.696055 0.717989i \(-0.745063\pi\)
0.969824 + 0.243807i \(0.0783964\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 30.1575i 0.861642i
\(36\) 0 0
\(37\) −5.53950 −0.149716 −0.0748581 0.997194i \(-0.523850\pi\)
−0.0748581 + 0.997194i \(0.523850\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 57.4985 + 33.1968i 1.40240 + 0.809677i 0.994639 0.103410i \(-0.0329754\pi\)
0.407764 + 0.913087i \(0.366309\pi\)
\(42\) 0 0
\(43\) −19.4868 33.7522i −0.453182 0.784935i 0.545399 0.838176i \(-0.316378\pi\)
−0.998582 + 0.0532417i \(0.983045\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −28.2014 + 16.2821i −0.600029 + 0.346427i −0.769053 0.639185i \(-0.779272\pi\)
0.169024 + 0.985612i \(0.445939\pi\)
\(48\) 0 0
\(49\) −66.4473 + 115.090i −1.35607 + 2.34878i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 11.2392i 0.212061i −0.994363 0.106030i \(-0.966186\pi\)
0.994363 0.106030i \(-0.0338141\pi\)
\(54\) 0 0
\(55\) −39.4868 −0.717942
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −27.6051 15.9378i −0.467884 0.270133i 0.247470 0.968896i \(-0.420401\pi\)
−0.715353 + 0.698763i \(0.753734\pi\)
\(60\) 0 0
\(61\) −23.4605 40.6348i −0.384598 0.666144i 0.607115 0.794614i \(-0.292327\pi\)
−0.991713 + 0.128470i \(0.958993\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −14.4982 + 8.37053i −0.223049 + 0.128777i
\(66\) 0 0
\(67\) 38.0000 65.8179i 0.567164 0.982357i −0.429681 0.902981i \(-0.641374\pi\)
0.996845 0.0793762i \(-0.0252928\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 77.7445i 1.09499i 0.836808 + 0.547497i \(0.184419\pi\)
−0.836808 + 0.547497i \(0.815581\pi\)
\(72\) 0 0
\(73\) 94.9210 1.30029 0.650144 0.759811i \(-0.274709\pi\)
0.650144 + 0.759811i \(0.274709\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 206.257 + 119.082i 2.67866 + 1.54652i
\(78\) 0 0
\(79\) 3.46050 + 5.99376i 0.0438038 + 0.0758704i 0.887096 0.461585i \(-0.152719\pi\)
−0.843292 + 0.537455i \(0.819386\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 53.8243 31.0755i 0.648485 0.374403i −0.139390 0.990237i \(-0.544514\pi\)
0.787876 + 0.615834i \(0.211181\pi\)
\(84\) 0 0
\(85\) −18.9737 + 32.8634i −0.223220 + 0.386628i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 62.2626i 0.699580i 0.936828 + 0.349790i \(0.113747\pi\)
−0.936828 + 0.349790i \(0.886253\pi\)
\(90\) 0 0
\(91\) 100.974 1.10960
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 21.2504 + 12.2689i 0.223689 + 0.129147i
\(96\) 0 0
\(97\) 62.4868 + 108.230i 0.644194 + 1.11578i 0.984487 + 0.175457i \(0.0561405\pi\)
−0.340293 + 0.940320i \(0.610526\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 97.2221 56.1312i 0.962595 0.555754i 0.0656242 0.997844i \(-0.479096\pi\)
0.896971 + 0.442090i \(0.145763\pi\)
\(102\) 0 0
\(103\) 85.1512 147.486i 0.826711 1.43191i −0.0738934 0.997266i \(-0.523542\pi\)
0.900604 0.434640i \(-0.143124\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 93.3381i 0.872319i −0.899869 0.436159i \(-0.856338\pi\)
0.899869 0.436159i \(-0.143662\pi\)
\(108\) 0 0
\(109\) −68.8683 −0.631820 −0.315910 0.948789i \(-0.602310\pi\)
−0.315910 + 0.948789i \(0.602310\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −95.5301 55.1543i −0.845399 0.488091i 0.0136967 0.999906i \(-0.495640\pi\)
−0.859096 + 0.511815i \(0.828973\pi\)
\(114\) 0 0
\(115\) 24.4868 + 42.4124i 0.212929 + 0.368804i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 198.215 114.440i 1.66567 0.961677i
\(120\) 0 0
\(121\) 95.4210 165.274i 0.788603 1.36590i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 11.1803i 0.0894427i
\(126\) 0 0
\(127\) −65.3815 −0.514815 −0.257407 0.966303i \(-0.582868\pi\)
−0.257407 + 0.966303i \(0.582868\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 40.9161 + 23.6229i 0.312336 + 0.180328i 0.647972 0.761665i \(-0.275618\pi\)
−0.335635 + 0.941992i \(0.608951\pi\)
\(132\) 0 0
\(133\) −74.0000 128.172i −0.556391 0.963697i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 187.096 108.020i 1.36566 0.788465i 0.375291 0.926907i \(-0.377543\pi\)
0.990371 + 0.138442i \(0.0442093\pi\)
\(138\) 0 0
\(139\) −105.921 + 183.461i −0.762022 + 1.31986i 0.179785 + 0.983706i \(0.442460\pi\)
−0.941807 + 0.336154i \(0.890874\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 132.210i 0.924548i
\(144\) 0 0
\(145\) −105.895 −0.730308
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 80.1402 + 46.2689i 0.537853 + 0.310530i 0.744209 0.667947i \(-0.232827\pi\)
−0.206355 + 0.978477i \(0.566160\pi\)
\(150\) 0 0
\(151\) 61.9210 + 107.250i 0.410073 + 0.710267i 0.994897 0.100893i \(-0.0321699\pi\)
−0.584824 + 0.811160i \(0.698837\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −32.8694 + 18.9771i −0.212060 + 0.122433i
\(156\) 0 0
\(157\) 15.7434 27.2684i 0.100277 0.173684i −0.811522 0.584322i \(-0.801361\pi\)
0.911799 + 0.410638i \(0.134694\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 295.384i 1.83469i
\(162\) 0 0
\(163\) −101.132 −0.620440 −0.310220 0.950665i \(-0.600403\pi\)
−0.310220 + 0.950665i \(0.600403\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −128.002 73.9020i −0.766479 0.442527i 0.0651382 0.997876i \(-0.479251\pi\)
−0.831617 + 0.555349i \(0.812584\pi\)
\(168\) 0 0
\(169\) 56.4737 + 97.8153i 0.334164 + 0.578789i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 287.589 166.040i 1.66237 0.959767i 0.690785 0.723061i \(-0.257265\pi\)
0.971581 0.236707i \(-0.0760681\pi\)
\(174\) 0 0
\(175\) −33.7171 + 58.3997i −0.192669 + 0.333713i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 45.8688i 0.256250i −0.991758 0.128125i \(-0.959104\pi\)
0.991758 0.128125i \(-0.0408959\pi\)
\(180\) 0 0
\(181\) 132.868 0.734079 0.367040 0.930205i \(-0.380371\pi\)
0.367040 + 0.930205i \(0.380371\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 10.7272 + 6.19335i 0.0579849 + 0.0334776i
\(186\) 0 0
\(187\) −149.842 259.534i −0.801294 1.38788i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 142.006 81.9871i 0.743486 0.429252i −0.0798492 0.996807i \(-0.525444\pi\)
0.823336 + 0.567555i \(0.192111\pi\)
\(192\) 0 0
\(193\) −55.0000 + 95.2628i −0.284974 + 0.493590i −0.972603 0.232473i \(-0.925318\pi\)
0.687629 + 0.726062i \(0.258652\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 211.108i 1.07162i −0.844340 0.535808i \(-0.820007\pi\)
0.844340 0.535808i \(-0.179993\pi\)
\(198\) 0 0
\(199\) −18.1053 −0.0909816 −0.0454908 0.998965i \(-0.514485\pi\)
−0.0454908 + 0.998965i \(0.514485\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 553.133 + 319.352i 2.72479 + 1.57316i
\(204\) 0 0
\(205\) −74.2302 128.571i −0.362099 0.627173i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −167.822 + 96.8923i −0.802978 + 0.463599i
\(210\) 0 0
\(211\) −170.789 + 295.816i −0.809428 + 1.40197i 0.103833 + 0.994595i \(0.466889\pi\)
−0.913261 + 0.407376i \(0.866444\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 87.1478i 0.405338i
\(216\) 0 0
\(217\) 228.921 1.05494
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −110.034 63.5279i −0.497889 0.287456i
\(222\) 0 0
\(223\) −29.6644 51.3803i −0.133024 0.230405i 0.791817 0.610759i \(-0.209135\pi\)
−0.924841 + 0.380354i \(0.875802\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −100.107 + 57.7965i −0.440998 + 0.254610i −0.704021 0.710179i \(-0.748614\pi\)
0.263023 + 0.964790i \(0.415281\pi\)
\(228\) 0 0
\(229\) 126.947 219.879i 0.554355 0.960171i −0.443598 0.896226i \(-0.646298\pi\)
0.997953 0.0639455i \(-0.0203684\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 206.401i 0.885840i 0.896561 + 0.442920i \(0.146057\pi\)
−0.896561 + 0.442920i \(0.853943\pi\)
\(234\) 0 0
\(235\) 72.8157 0.309854
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 299.901 + 173.148i 1.25482 + 0.724469i 0.972062 0.234723i \(-0.0754184\pi\)
0.282755 + 0.959192i \(0.408752\pi\)
\(240\) 0 0
\(241\) 91.0263 + 157.662i 0.377703 + 0.654200i 0.990728 0.135863i \(-0.0433808\pi\)
−0.613025 + 0.790064i \(0.710047\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 257.349 148.581i 1.05041 0.606452i
\(246\) 0 0
\(247\) −41.0790 + 71.1509i −0.166312 + 0.288060i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 140.807i 0.560985i 0.959856 + 0.280493i \(0.0904979\pi\)
−0.959856 + 0.280493i \(0.909502\pi\)
\(252\) 0 0
\(253\) −386.763 −1.52871
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −253.425 146.315i −0.986091 0.569320i −0.0819876 0.996633i \(-0.526127\pi\)
−0.904104 + 0.427313i \(0.859460\pi\)
\(258\) 0 0
\(259\) −37.3552 64.7010i −0.144228 0.249811i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −234.652 + 135.476i −0.892211 + 0.515118i −0.874665 0.484728i \(-0.838919\pi\)
−0.0175460 + 0.999846i \(0.505585\pi\)
\(264\) 0 0
\(265\) −12.5658 + 21.7647i −0.0474182 + 0.0821308i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 275.083i 1.02261i −0.859398 0.511307i \(-0.829162\pi\)
0.859398 0.511307i \(-0.170838\pi\)
\(270\) 0 0
\(271\) 248.158 0.915712 0.457856 0.889026i \(-0.348617\pi\)
0.457856 + 0.889026i \(0.348617\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 76.4659 + 44.1476i 0.278058 + 0.160537i
\(276\) 0 0
\(277\) 171.020 + 296.215i 0.617399 + 1.06937i 0.989958 + 0.141358i \(0.0451470\pi\)
−0.372559 + 0.928008i \(0.621520\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −304.961 + 176.070i −1.08527 + 0.626582i −0.932314 0.361650i \(-0.882213\pi\)
−0.152958 + 0.988233i \(0.548880\pi\)
\(282\) 0 0
\(283\) −124.816 + 216.187i −0.441045 + 0.763912i −0.997767 0.0667866i \(-0.978725\pi\)
0.556723 + 0.830699i \(0.312059\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 895.439i 3.12000i
\(288\) 0 0
\(289\) 1.00000 0.00346021
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −32.4719 18.7476i −0.110825 0.0639851i 0.443563 0.896243i \(-0.353714\pi\)
−0.554388 + 0.832258i \(0.687048\pi\)
\(294\) 0 0
\(295\) 35.6381 + 61.7270i 0.120807 + 0.209244i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −142.006 + 81.9871i −0.474936 + 0.274204i
\(300\) 0 0
\(301\) 262.816 455.210i 0.873142 1.51233i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 104.919i 0.343995i
\(306\) 0 0
\(307\) 457.842 1.49134 0.745671 0.666314i \(-0.232129\pi\)
0.745671 + 0.666314i \(0.232129\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −7.34847 4.24264i −0.0236285 0.0136419i 0.488139 0.872766i \(-0.337676\pi\)
−0.511768 + 0.859124i \(0.671009\pi\)
\(312\) 0 0
\(313\) 0.789328 + 1.36716i 0.00252181 + 0.00436791i 0.867284 0.497814i \(-0.165864\pi\)
−0.864762 + 0.502182i \(0.832531\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 458.296 264.597i 1.44573 0.834692i 0.447506 0.894281i \(-0.352312\pi\)
0.998223 + 0.0595890i \(0.0189790\pi\)
\(318\) 0 0
\(319\) 418.144 724.248i 1.31080 2.27037i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 186.229i 0.576561i
\(324\) 0 0
\(325\) 37.4342 0.115182
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −380.347 219.594i −1.15607 0.667458i
\(330\) 0 0
\(331\) 195.381 + 338.411i 0.590276 + 1.02239i 0.994195 + 0.107594i \(0.0343146\pi\)
−0.403919 + 0.914795i \(0.632352\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −147.173 + 84.9706i −0.439323 + 0.253644i
\(336\) 0 0
\(337\) −207.355 + 359.150i −0.615297 + 1.06573i 0.375035 + 0.927011i \(0.377631\pi\)
−0.990332 + 0.138715i \(0.955703\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 299.739i 0.878999i
\(342\) 0 0
\(343\) −1131.47 −3.29876
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −111.420 64.3281i −0.321094 0.185384i 0.330786 0.943706i \(-0.392686\pi\)
−0.651880 + 0.758322i \(0.726019\pi\)
\(348\) 0 0
\(349\) 18.5395 + 32.1114i 0.0531218 + 0.0920096i 0.891364 0.453289i \(-0.149750\pi\)
−0.838242 + 0.545299i \(0.816416\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −165.050 + 95.2918i −0.467565 + 0.269949i −0.715220 0.698900i \(-0.753673\pi\)
0.247655 + 0.968848i \(0.420340\pi\)
\(354\) 0 0
\(355\) 86.9210 150.552i 0.244848 0.424089i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 254.782i 0.709699i −0.934923 0.354849i \(-0.884532\pi\)
0.934923 0.354849i \(-0.115468\pi\)
\(360\) 0 0
\(361\) −240.579 −0.666423
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −183.814 106.125i −0.503599 0.290753i
\(366\) 0 0
\(367\) −150.230 260.206i −0.409347 0.709009i 0.585470 0.810694i \(-0.300910\pi\)
−0.994817 + 0.101685i \(0.967577\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 131.273 75.7908i 0.353837 0.204288i
\(372\) 0 0
\(373\) −352.046 + 609.761i −0.943823 + 1.63475i −0.185732 + 0.982601i \(0.559466\pi\)
−0.758091 + 0.652149i \(0.773868\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 354.558i 0.940472i
\(378\) 0 0
\(379\) 333.789 0.880711 0.440355 0.897824i \(-0.354852\pi\)
0.440355 + 0.897824i \(0.354852\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −226.030 130.498i −0.590155 0.340726i 0.175004 0.984568i \(-0.444006\pi\)
−0.765159 + 0.643841i \(0.777340\pi\)
\(384\) 0 0
\(385\) −266.276 461.204i −0.691626 1.19793i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −31.0859 + 17.9474i −0.0799122 + 0.0461373i −0.539424 0.842035i \(-0.681358\pi\)
0.459511 + 0.888172i \(0.348025\pi\)
\(390\) 0 0
\(391\) −185.842 + 321.888i −0.475299 + 0.823242i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 15.4758i 0.0391793i
\(396\) 0 0
\(397\) −90.3552 −0.227595 −0.113797 0.993504i \(-0.536301\pi\)
−0.113797 + 0.993504i \(0.536301\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −37.8381 21.8458i −0.0943593 0.0544784i 0.452078 0.891978i \(-0.350683\pi\)
−0.546437 + 0.837500i \(0.684016\pi\)
\(402\) 0 0
\(403\) −63.5395 110.054i −0.157666 0.273086i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −84.7166 + 48.9112i −0.208149 + 0.120175i
\(408\) 0 0
\(409\) 130.026 225.212i 0.317913 0.550641i −0.662140 0.749381i \(-0.730351\pi\)
0.980052 + 0.198739i \(0.0636847\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 429.902i 1.04092i
\(414\) 0 0
\(415\) −138.974 −0.334876
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 228.012 + 131.643i 0.544181 + 0.314183i 0.746772 0.665081i \(-0.231603\pi\)
−0.202591 + 0.979263i \(0.564936\pi\)
\(420\) 0 0
\(421\) −183.605 318.013i −0.436116 0.755376i 0.561270 0.827633i \(-0.310313\pi\)
−0.997386 + 0.0722573i \(0.976980\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 73.4847 42.4264i 0.172905 0.0998268i
\(426\) 0 0
\(427\) 316.408 548.034i 0.741002 1.28345i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 690.768i 1.60271i 0.598189 + 0.801355i \(0.295887\pi\)
−0.598189 + 0.801355i \(0.704113\pi\)
\(432\) 0 0
\(433\) −117.500 −0.271363 −0.135682 0.990752i \(-0.543322\pi\)
−0.135682 + 0.990752i \(0.543322\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 208.142 + 120.171i 0.476298 + 0.274991i
\(438\) 0 0
\(439\) −25.0000 43.3013i −0.0569476 0.0986362i 0.836146 0.548507i \(-0.184803\pi\)
−0.893094 + 0.449871i \(0.851470\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −456.217 + 263.397i −1.02984 + 0.594576i −0.916937 0.399031i \(-0.869347\pi\)
−0.112898 + 0.993607i \(0.536013\pi\)
\(444\) 0 0
\(445\) 69.6117 120.571i 0.156431 0.270946i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 321.882i 0.716887i −0.933552 0.358443i \(-0.883308\pi\)
0.933552 0.358443i \(-0.116692\pi\)
\(450\) 0 0
\(451\) 1172.45 2.59966
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −195.535 112.892i −0.429747 0.248114i
\(456\) 0 0
\(457\) 216.540 + 375.057i 0.473828 + 0.820695i 0.999551 0.0299614i \(-0.00953842\pi\)
−0.525723 + 0.850656i \(0.676205\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −421.554 + 243.384i −0.914433 + 0.527948i −0.881855 0.471521i \(-0.843705\pi\)
−0.0325783 + 0.999469i \(0.510372\pi\)
\(462\) 0 0
\(463\) −211.612 + 366.522i −0.457045 + 0.791625i −0.998803 0.0489094i \(-0.984425\pi\)
0.541758 + 0.840534i \(0.317759\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 295.831i 0.633472i −0.948514 0.316736i \(-0.897413\pi\)
0.948514 0.316736i \(-0.102587\pi\)
\(468\) 0 0
\(469\) 1025.00 2.18550
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −596.031 344.119i −1.26011 0.727524i
\(474\) 0 0
\(475\) −27.4342 47.5174i −0.0577561 0.100037i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −230.993 + 133.364i −0.482240 + 0.278421i −0.721350 0.692571i \(-0.756478\pi\)
0.239109 + 0.970993i \(0.423145\pi\)
\(480\) 0 0
\(481\) −20.7367 + 35.9169i −0.0431116 + 0.0746714i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 279.450i 0.576185i
\(486\) 0 0
\(487\) 247.750 0.508727 0.254364 0.967109i \(-0.418134\pi\)
0.254364 + 0.967109i \(0.418134\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −147.566 85.1971i −0.300541 0.173517i 0.342145 0.939647i \(-0.388847\pi\)
−0.642686 + 0.766130i \(0.722180\pi\)
\(492\) 0 0
\(493\) −401.842 696.011i −0.815095 1.41179i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −908.051 + 524.264i −1.82706 + 1.05486i
\(498\) 0 0
\(499\) −23.6975 + 41.0453i −0.0474900 + 0.0822551i −0.888793 0.458308i \(-0.848456\pi\)
0.841303 + 0.540563i \(0.181789\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 321.064i 0.638298i −0.947705 0.319149i \(-0.896603\pi\)
0.947705 0.319149i \(-0.103397\pi\)
\(504\) 0 0
\(505\) −251.026 −0.497082
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −105.763 61.0623i −0.207786 0.119965i 0.392496 0.919754i \(-0.371612\pi\)
−0.600282 + 0.799788i \(0.704945\pi\)
\(510\) 0 0
\(511\) 640.092 + 1108.67i 1.25263 + 2.16961i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −329.789 + 190.404i −0.640368 + 0.369716i
\(516\) 0 0
\(517\) −287.526 + 498.010i −0.556143 + 0.963268i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 79.4567i 0.152508i 0.997088 + 0.0762540i \(0.0242960\pi\)
−0.997088 + 0.0762540i \(0.975704\pi\)
\(522\) 0 0
\(523\) −864.605 −1.65316 −0.826582 0.562816i \(-0.809718\pi\)
−0.826582 + 0.562816i \(0.809718\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −249.461 144.026i −0.473360 0.273295i
\(528\) 0 0
\(529\) −24.6580 42.7089i −0.0466125 0.0807352i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 430.482 248.539i 0.807658 0.466302i
\(534\) 0 0
\(535\) −104.355 + 180.748i −0.195056 + 0.337848i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 2346.79i 4.35398i
\(540\) 0 0
\(541\) 33.6057 0.0621177 0.0310589 0.999518i \(-0.490112\pi\)
0.0310589 + 0.999518i \(0.490112\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 133.363 + 76.9971i 0.244703 + 0.141279i
\(546\) 0 0
\(547\) 279.540 + 484.177i 0.511041 + 0.885149i 0.999918 + 0.0127965i \(0.00407336\pi\)
−0.488877 + 0.872353i \(0.662593\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −450.061 + 259.843i −0.816808 + 0.471584i
\(552\) 0 0
\(553\) −46.6712 + 80.8368i −0.0843963 + 0.146179i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 310.308i 0.557105i 0.960421 + 0.278553i \(0.0898547\pi\)
−0.960421 + 0.278553i \(0.910145\pi\)
\(558\) 0 0
\(559\) −291.789 −0.521984
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −490.768 283.345i −0.871701 0.503277i −0.00378817 0.999993i \(-0.501206\pi\)
−0.867913 + 0.496716i \(0.834539\pi\)
\(564\) 0 0
\(565\) 123.329 + 213.612i 0.218281 + 0.378074i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −116.093 + 67.0262i −0.204029 + 0.117796i −0.598534 0.801098i \(-0.704250\pi\)
0.394504 + 0.918894i \(0.370916\pi\)
\(570\) 0 0
\(571\) 330.960 573.240i 0.579615 1.00392i −0.415908 0.909407i \(-0.636536\pi\)
0.995523 0.0945161i \(-0.0301304\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 109.508i 0.190449i
\(576\) 0 0
\(577\) 76.7630 0.133038 0.0665191 0.997785i \(-0.478811\pi\)
0.0665191 + 0.997785i \(0.478811\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 725.919 + 419.109i 1.24943 + 0.721359i
\(582\) 0 0
\(583\) −99.2370 171.884i −0.170218 0.294826i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −592.841 + 342.277i −1.00995 + 0.583095i −0.911177 0.412015i \(-0.864825\pi\)
−0.0987734 + 0.995110i \(0.531492\pi\)
\(588\) 0 0
\(589\) −93.1317 + 161.309i −0.158118 + 0.273869i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 592.146i 0.998560i 0.866441 + 0.499280i \(0.166402\pi\)
−0.866441 + 0.499280i \(0.833598\pi\)
\(594\) 0 0
\(595\) −511.789 −0.860150
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −877.078 506.381i −1.46424 0.845377i −0.465034 0.885293i \(-0.653958\pi\)
−0.999203 + 0.0399156i \(0.987291\pi\)
\(600\) 0 0
\(601\) −161.789 280.227i −0.269200 0.466268i 0.699455 0.714676i \(-0.253426\pi\)
−0.968656 + 0.248408i \(0.920093\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −369.564 + 213.368i −0.610849 + 0.352674i
\(606\) 0 0
\(607\) 399.914 692.672i 0.658837 1.14114i −0.322080 0.946713i \(-0.604382\pi\)
0.980917 0.194427i \(-0.0622848\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 243.802i 0.399022i
\(612\) 0 0
\(613\) −680.302 −1.10979 −0.554896 0.831920i \(-0.687242\pi\)
−0.554896 + 0.831920i \(0.687242\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 229.382 + 132.434i 0.371770 + 0.214641i 0.674231 0.738520i \(-0.264475\pi\)
−0.302461 + 0.953162i \(0.597808\pi\)
\(618\) 0 0
\(619\) −267.921 464.053i −0.432829 0.749681i 0.564287 0.825579i \(-0.309151\pi\)
−0.997116 + 0.0758974i \(0.975818\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −727.224 + 419.863i −1.16729 + 0.673937i
\(624\) 0 0
\(625\) −12.5000 + 21.6506i −0.0200000 + 0.0346410i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 94.0085i 0.149457i
\(630\) 0 0
\(631\) 307.026 0.486571 0.243286 0.969955i \(-0.421775\pi\)
0.243286 + 0.969955i \(0.421775\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 126.611 + 73.0987i 0.199387 + 0.115116i
\(636\) 0 0
\(637\) 497.480 + 861.661i 0.780973 + 1.35269i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 1079.35 623.165i 1.68386 0.972177i 0.724806 0.688953i \(-0.241929\pi\)
0.959054 0.283224i \(-0.0914040\pi\)
\(642\) 0 0
\(643\) −77.6975 + 134.576i −0.120836 + 0.209294i −0.920098 0.391689i \(-0.871891\pi\)
0.799262 + 0.600983i \(0.205224\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 142.649i 0.220478i 0.993905 + 0.110239i \(0.0351617\pi\)
−0.993905 + 0.110239i \(0.964838\pi\)
\(648\) 0 0
\(649\) −562.894 −0.867325
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −800.709 462.289i −1.22620 0.707947i −0.259967 0.965617i \(-0.583712\pi\)
−0.966233 + 0.257671i \(0.917045\pi\)
\(654\) 0 0
\(655\) −52.8224 91.4911i −0.0806449 0.139681i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −304.880 + 176.023i −0.462641 + 0.267106i −0.713154 0.701007i \(-0.752734\pi\)
0.250513 + 0.968113i \(0.419401\pi\)
\(660\) 0 0
\(661\) 321.460 556.786i 0.486325 0.842339i −0.513552 0.858059i \(-0.671671\pi\)
0.999876 + 0.0157198i \(0.00500396\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 330.938i 0.497651i
\(666\) 0 0
\(667\) −1037.21 −1.55504
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −717.571 414.290i −1.06941 0.617422i
\(672\) 0 0
\(673\) −491.302 850.961i −0.730019 1.26443i −0.956875 0.290501i \(-0.906178\pi\)
0.226856 0.973928i \(-0.427155\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −334.952 + 193.384i −0.494759 + 0.285649i −0.726546 0.687117i \(-0.758876\pi\)
0.231788 + 0.972766i \(0.425542\pi\)
\(678\) 0 0
\(679\) −842.749 + 1459.68i −1.24116 + 2.14976i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 504.539i 0.738710i 0.929288 + 0.369355i \(0.120421\pi\)
−0.929288 + 0.369355i \(0.879579\pi\)
\(684\) 0 0
\(685\) −483.079 −0.705225
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −72.8727 42.0731i −0.105766 0.0610640i
\(690\) 0 0
\(691\) 135.540 + 234.761i 0.196150 + 0.339741i 0.947277 0.320416i \(-0.103823\pi\)
−0.751127 + 0.660158i \(0.770489\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 410.230 236.847i 0.590259 0.340786i
\(696\) 0 0
\(697\) 563.368 975.782i 0.808275 1.39997i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 358.559i 0.511496i 0.966743 + 0.255748i \(0.0823218\pi\)
−0.966743 + 0.255748i \(0.917678\pi\)
\(702\) 0 0
\(703\) 60.7886 0.0864703
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 1311.22 + 757.032i 1.85462 + 1.07077i
\(708\) 0 0
\(709\) 366.789 + 635.298i 0.517333 + 0.896048i 0.999797 + 0.0201318i \(0.00640859\pi\)
−0.482464 + 0.875916i \(0.660258\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −321.947 + 185.876i −0.451538 + 0.260696i
\(714\) 0 0
\(715\) −147.816 + 256.024i −0.206735 + 0.358076i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 1108.37i 1.54155i −0.637110 0.770773i \(-0.719870\pi\)
0.637110 0.770773i \(-0.280130\pi\)
\(720\) 0 0
\(721\) 2296.84 3.18563
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 205.064 + 118.394i 0.282847 + 0.163302i
\(726\) 0 0
\(727\) −632.519 1095.56i −0.870040 1.50695i −0.861954 0.506987i \(-0.830759\pi\)
−0.00808623 0.999967i \(-0.502574\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −572.794 + 330.703i −0.783575 + 0.452397i
\(732\) 0 0
\(733\) 293.375 508.140i 0.400238 0.693233i −0.593516 0.804822i \(-0.702261\pi\)
0.993754 + 0.111589i \(0.0355940\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1342.09i 1.82101i
\(738\) 0 0
\(739\) −215.973 −0.292250 −0.146125 0.989266i \(-0.546680\pi\)
−0.146125 + 0.989266i \(0.546680\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 519.581 + 299.980i 0.699302 + 0.403742i 0.807087 0.590432i \(-0.201043\pi\)
−0.107786 + 0.994174i \(0.534376\pi\)
\(744\) 0 0
\(745\) −103.460 179.199i −0.138873 0.240535i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 1090.18 629.418i 1.45552 0.840344i
\(750\) 0 0
\(751\) −406.828 + 704.648i −0.541716 + 0.938279i 0.457090 + 0.889420i \(0.348892\pi\)
−0.998806 + 0.0488587i \(0.984442\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 276.919i 0.366780i
\(756\) 0 0
\(757\) 1070.43 1.41405 0.707023 0.707190i \(-0.250038\pi\)
0.707023 + 0.707190i \(0.250038\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −567.153 327.446i −0.745273 0.430284i 0.0787104 0.996898i \(-0.474920\pi\)
−0.823983 + 0.566614i \(0.808253\pi\)
\(762\) 0 0
\(763\) −464.408 804.378i −0.608660 1.05423i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −206.675 + 119.324i −0.269459 + 0.155572i
\(768\) 0 0
\(769\) −341.315 + 591.175i −0.443843 + 0.768759i −0.997971 0.0636729i \(-0.979719\pi\)
0.554128 + 0.832432i \(0.313052\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 770.690i 0.997012i −0.866886 0.498506i \(-0.833882\pi\)
0.866886 0.498506i \(-0.166118\pi\)
\(774\) 0 0
\(775\) 84.8683 0.109508
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −630.969 364.290i −0.809974 0.467638i
\(780\) 0 0
\(781\) 686.447 + 1188.96i 0.878933 + 1.52236i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −60.9740 + 35.2033i −0.0776739 + 0.0448450i
\(786\) 0 0
\(787\) 126.935 219.857i 0.161289 0.279361i −0.774042 0.633134i \(-0.781768\pi\)
0.935331 + 0.353773i \(0.115102\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 1487.71i 1.88080i
\(792\) 0 0
\(793\) −351.290 −0.442988
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −914.401 527.929i −1.14730 0.662396i −0.199075 0.979984i \(-0.563794\pi\)
−0.948229 + 0.317589i \(0.897127\pi\)
\(798\) 0 0
\(799\) 276.316 + 478.593i 0.345827 + 0.598990i
\(800\) 0 0