Properties

Label 1200.3.c.f.449.4
Level $1200$
Weight $3$
Character 1200.449
Analytic conductor $32.698$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1200.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(32.6976317232\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Defining polynomial: \(x^{4} + 3 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 15)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 449.4
Root \(0.618034i\) of defining polynomial
Character \(\chi\) \(=\) 1200.449
Dual form 1200.3.c.f.449.3

$q$-expansion

\(f(q)\) \(=\) \(q+(2.23607 + 2.00000i) q^{3} -6.00000i q^{7} +(1.00000 + 8.94427i) q^{9} +O(q^{10})\) \(q+(2.23607 + 2.00000i) q^{3} -6.00000i q^{7} +(1.00000 + 8.94427i) q^{9} -4.47214i q^{11} +16.0000i q^{13} +4.47214 q^{17} -2.00000 q^{19} +(12.0000 - 13.4164i) q^{21} +13.4164 q^{23} +(-15.6525 + 22.0000i) q^{27} +31.3050i q^{29} +18.0000 q^{31} +(8.94427 - 10.0000i) q^{33} +16.0000i q^{37} +(-32.0000 + 35.7771i) q^{39} +62.6099i q^{41} -16.0000i q^{43} +49.1935 q^{47} +13.0000 q^{49} +(10.0000 + 8.94427i) q^{51} -4.47214 q^{53} +(-4.47214 - 4.00000i) q^{57} +4.47214i q^{59} +82.0000 q^{61} +(53.6656 - 6.00000i) q^{63} +24.0000i q^{67} +(30.0000 + 26.8328i) q^{69} +125.220i q^{71} -74.0000i q^{73} -26.8328 q^{77} +138.000 q^{79} +(-79.0000 + 17.8885i) q^{81} -93.9149 q^{83} +(-62.6099 + 70.0000i) q^{87} -107.331i q^{89} +96.0000 q^{91} +(40.2492 + 36.0000i) q^{93} +166.000i q^{97} +(40.0000 - 4.47214i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{9} + O(q^{10}) \) \( 4 q + 4 q^{9} - 8 q^{19} + 48 q^{21} + 72 q^{31} - 128 q^{39} + 52 q^{49} + 40 q^{51} + 328 q^{61} + 120 q^{69} + 552 q^{79} - 316 q^{81} + 384 q^{91} + 160 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times\).

\(n\) \(401\) \(577\) \(751\) \(901\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(1\)

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
\(3\) 2.23607 + 2.00000i 0.745356 + 0.666667i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 6.00000i 0.857143i −0.903508 0.428571i \(-0.859017\pi\)
0.903508 0.428571i \(-0.140983\pi\)
\(8\) 0 0
\(9\) 1.00000 + 8.94427i 0.111111 + 0.993808i
\(10\) 0 0
\(11\) 4.47214i 0.406558i −0.979121 0.203279i \(-0.934840\pi\)
0.979121 0.203279i \(-0.0651598\pi\)
\(12\) 0 0
\(13\) 16.0000i 1.23077i 0.788227 + 0.615385i \(0.210999\pi\)
−0.788227 + 0.615385i \(0.789001\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.47214 0.263067 0.131533 0.991312i \(-0.458010\pi\)
0.131533 + 0.991312i \(0.458010\pi\)
\(18\) 0 0
\(19\) −2.00000 −0.105263 −0.0526316 0.998614i \(-0.516761\pi\)
−0.0526316 + 0.998614i \(0.516761\pi\)
\(20\) 0 0
\(21\) 12.0000 13.4164i 0.571429 0.638877i
\(22\) 0 0
\(23\) 13.4164 0.583322 0.291661 0.956522i \(-0.405792\pi\)
0.291661 + 0.956522i \(0.405792\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −15.6525 + 22.0000i −0.579721 + 0.814815i
\(28\) 0 0
\(29\) 31.3050i 1.07948i 0.841831 + 0.539741i \(0.181478\pi\)
−0.841831 + 0.539741i \(0.818522\pi\)
\(30\) 0 0
\(31\) 18.0000 0.580645 0.290323 0.956929i \(-0.406237\pi\)
0.290323 + 0.956929i \(0.406237\pi\)
\(32\) 0 0
\(33\) 8.94427 10.0000i 0.271039 0.303030i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 16.0000i 0.432432i 0.976346 + 0.216216i \(0.0693716\pi\)
−0.976346 + 0.216216i \(0.930628\pi\)
\(38\) 0 0
\(39\) −32.0000 + 35.7771i −0.820513 + 0.917361i
\(40\) 0 0
\(41\) 62.6099i 1.52707i 0.645766 + 0.763535i \(0.276538\pi\)
−0.645766 + 0.763535i \(0.723462\pi\)
\(42\) 0 0
\(43\) 16.0000i 0.372093i −0.982541 0.186047i \(-0.940432\pi\)
0.982541 0.186047i \(-0.0595675\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 49.1935 1.04667 0.523335 0.852127i \(-0.324688\pi\)
0.523335 + 0.852127i \(0.324688\pi\)
\(48\) 0 0
\(49\) 13.0000 0.265306
\(50\) 0 0
\(51\) 10.0000 + 8.94427i 0.196078 + 0.175378i
\(52\) 0 0
\(53\) −4.47214 −0.0843799 −0.0421900 0.999110i \(-0.513433\pi\)
−0.0421900 + 0.999110i \(0.513433\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −4.47214 4.00000i −0.0784585 0.0701754i
\(58\) 0 0
\(59\) 4.47214i 0.0757989i 0.999282 + 0.0378995i \(0.0120667\pi\)
−0.999282 + 0.0378995i \(0.987933\pi\)
\(60\) 0 0
\(61\) 82.0000 1.34426 0.672131 0.740432i \(-0.265379\pi\)
0.672131 + 0.740432i \(0.265379\pi\)
\(62\) 0 0
\(63\) 53.6656 6.00000i 0.851835 0.0952381i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 24.0000i 0.358209i 0.983830 + 0.179104i \(0.0573200\pi\)
−0.983830 + 0.179104i \(0.942680\pi\)
\(68\) 0 0
\(69\) 30.0000 + 26.8328i 0.434783 + 0.388881i
\(70\) 0 0
\(71\) 125.220i 1.76366i 0.471568 + 0.881830i \(0.343688\pi\)
−0.471568 + 0.881830i \(0.656312\pi\)
\(72\) 0 0
\(73\) 74.0000i 1.01370i −0.862035 0.506849i \(-0.830810\pi\)
0.862035 0.506849i \(-0.169190\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −26.8328 −0.348478
\(78\) 0 0
\(79\) 138.000 1.74684 0.873418 0.486972i \(-0.161899\pi\)
0.873418 + 0.486972i \(0.161899\pi\)
\(80\) 0 0
\(81\) −79.0000 + 17.8885i −0.975309 + 0.220846i
\(82\) 0 0
\(83\) −93.9149 −1.13150 −0.565752 0.824575i \(-0.691414\pi\)
−0.565752 + 0.824575i \(0.691414\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −62.6099 + 70.0000i −0.719654 + 0.804598i
\(88\) 0 0
\(89\) 107.331i 1.20597i −0.797753 0.602985i \(-0.793978\pi\)
0.797753 0.602985i \(-0.206022\pi\)
\(90\) 0 0
\(91\) 96.0000 1.05495
\(92\) 0 0
\(93\) 40.2492 + 36.0000i 0.432787 + 0.387097i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 166.000i 1.71134i 0.517522 + 0.855670i \(0.326855\pi\)
−0.517522 + 0.855670i \(0.673145\pi\)
\(98\) 0 0
\(99\) 40.0000 4.47214i 0.404040 0.0451731i
\(100\) 0 0
\(101\) 67.0820i 0.664179i 0.943248 + 0.332089i \(0.107754\pi\)
−0.943248 + 0.332089i \(0.892246\pi\)
\(102\) 0 0
\(103\) 26.0000i 0.252427i −0.992003 0.126214i \(-0.959718\pi\)
0.992003 0.126214i \(-0.0402825\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −201.246 −1.88080 −0.940402 0.340064i \(-0.889551\pi\)
−0.940402 + 0.340064i \(0.889551\pi\)
\(108\) 0 0
\(109\) −38.0000 −0.348624 −0.174312 0.984690i \(-0.555770\pi\)
−0.174312 + 0.984690i \(0.555770\pi\)
\(110\) 0 0
\(111\) −32.0000 + 35.7771i −0.288288 + 0.322316i
\(112\) 0 0
\(113\) −31.3050 −0.277035 −0.138517 0.990360i \(-0.544234\pi\)
−0.138517 + 0.990360i \(0.544234\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −143.108 + 16.0000i −1.22315 + 0.136752i
\(118\) 0 0
\(119\) 26.8328i 0.225486i
\(120\) 0 0
\(121\) 101.000 0.834711
\(122\) 0 0
\(123\) −125.220 + 140.000i −1.01805 + 1.13821i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 26.0000i 0.204724i −0.994747 0.102362i \(-0.967360\pi\)
0.994747 0.102362i \(-0.0326401\pi\)
\(128\) 0 0
\(129\) 32.0000 35.7771i 0.248062 0.277342i
\(130\) 0 0
\(131\) 13.4164i 0.102415i −0.998688 0.0512077i \(-0.983693\pi\)
0.998688 0.0512077i \(-0.0163070\pi\)
\(132\) 0 0
\(133\) 12.0000i 0.0902256i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −120.748 −0.881370 −0.440685 0.897662i \(-0.645264\pi\)
−0.440685 + 0.897662i \(0.645264\pi\)
\(138\) 0 0
\(139\) −82.0000 −0.589928 −0.294964 0.955508i \(-0.595308\pi\)
−0.294964 + 0.955508i \(0.595308\pi\)
\(140\) 0 0
\(141\) 110.000 + 98.3870i 0.780142 + 0.697780i
\(142\) 0 0
\(143\) 71.5542 0.500379
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 29.0689 + 26.0000i 0.197748 + 0.176871i
\(148\) 0 0
\(149\) 111.803i 0.750358i 0.926952 + 0.375179i \(0.122419\pi\)
−0.926952 + 0.375179i \(0.877581\pi\)
\(150\) 0 0
\(151\) 158.000 1.04636 0.523179 0.852223i \(-0.324746\pi\)
0.523179 + 0.852223i \(0.324746\pi\)
\(152\) 0 0
\(153\) 4.47214 + 40.0000i 0.0292296 + 0.261438i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 164.000i 1.04459i −0.852766 0.522293i \(-0.825077\pi\)
0.852766 0.522293i \(-0.174923\pi\)
\(158\) 0 0
\(159\) −10.0000 8.94427i −0.0628931 0.0562533i
\(160\) 0 0
\(161\) 80.4984i 0.499990i
\(162\) 0 0
\(163\) 236.000i 1.44785i −0.689877 0.723926i \(-0.742336\pi\)
0.689877 0.723926i \(-0.257664\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −93.9149 −0.562364 −0.281182 0.959654i \(-0.590727\pi\)
−0.281182 + 0.959654i \(0.590727\pi\)
\(168\) 0 0
\(169\) −87.0000 −0.514793
\(170\) 0 0
\(171\) −2.00000 17.8885i −0.0116959 0.104611i
\(172\) 0 0
\(173\) −13.4164 −0.0775515 −0.0387757 0.999248i \(-0.512346\pi\)
−0.0387757 + 0.999248i \(0.512346\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −8.94427 + 10.0000i −0.0505326 + 0.0564972i
\(178\) 0 0
\(179\) 192.302i 1.07431i 0.843483 + 0.537156i \(0.180501\pi\)
−0.843483 + 0.537156i \(0.819499\pi\)
\(180\) 0 0
\(181\) 2.00000 0.0110497 0.00552486 0.999985i \(-0.498241\pi\)
0.00552486 + 0.999985i \(0.498241\pi\)
\(182\) 0 0
\(183\) 183.358 + 164.000i 1.00195 + 0.896175i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 20.0000i 0.106952i
\(188\) 0 0
\(189\) 132.000 + 93.9149i 0.698413 + 0.496904i
\(190\) 0 0
\(191\) 205.718i 1.07706i 0.842607 + 0.538529i \(0.181020\pi\)
−0.842607 + 0.538529i \(0.818980\pi\)
\(192\) 0 0
\(193\) 214.000i 1.10881i −0.832248 0.554404i \(-0.812946\pi\)
0.832248 0.554404i \(-0.187054\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −93.9149 −0.476725 −0.238363 0.971176i \(-0.576611\pi\)
−0.238363 + 0.971176i \(0.576611\pi\)
\(198\) 0 0
\(199\) −242.000 −1.21608 −0.608040 0.793906i \(-0.708044\pi\)
−0.608040 + 0.793906i \(0.708044\pi\)
\(200\) 0 0
\(201\) −48.0000 + 53.6656i −0.238806 + 0.266993i
\(202\) 0 0
\(203\) 187.830 0.925270
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 13.4164 + 120.000i 0.0648136 + 0.579710i
\(208\) 0 0
\(209\) 8.94427i 0.0427956i
\(210\) 0 0
\(211\) −2.00000 −0.00947867 −0.00473934 0.999989i \(-0.501509\pi\)
−0.00473934 + 0.999989i \(0.501509\pi\)
\(212\) 0 0
\(213\) −250.440 + 280.000i −1.17577 + 1.31455i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 108.000i 0.497696i
\(218\) 0 0
\(219\) 148.000 165.469i 0.675799 0.755566i
\(220\) 0 0
\(221\) 71.5542i 0.323775i
\(222\) 0 0
\(223\) 86.0000i 0.385650i −0.981233 0.192825i \(-0.938235\pi\)
0.981233 0.192825i \(-0.0617650\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 58.1378 0.256114 0.128057 0.991767i \(-0.459126\pi\)
0.128057 + 0.991767i \(0.459126\pi\)
\(228\) 0 0
\(229\) 282.000 1.23144 0.615721 0.787965i \(-0.288865\pi\)
0.615721 + 0.787965i \(0.288865\pi\)
\(230\) 0 0
\(231\) −60.0000 53.6656i −0.259740 0.232319i
\(232\) 0 0
\(233\) 362.243 1.55469 0.777346 0.629074i \(-0.216566\pi\)
0.777346 + 0.629074i \(0.216566\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 308.577 + 276.000i 1.30201 + 1.16456i
\(238\) 0 0
\(239\) 250.440i 1.04786i −0.851760 0.523932i \(-0.824464\pi\)
0.851760 0.523932i \(-0.175536\pi\)
\(240\) 0 0
\(241\) 262.000 1.08714 0.543568 0.839365i \(-0.317073\pi\)
0.543568 + 0.839365i \(0.317073\pi\)
\(242\) 0 0
\(243\) −212.426 118.000i −0.874183 0.485597i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 32.0000i 0.129555i
\(248\) 0 0
\(249\) −210.000 187.830i −0.843373 0.754336i
\(250\) 0 0
\(251\) 469.574i 1.87081i −0.353573 0.935407i \(-0.615033\pi\)
0.353573 0.935407i \(-0.384967\pi\)
\(252\) 0 0
\(253\) 60.0000i 0.237154i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 201.246 0.783059 0.391529 0.920166i \(-0.371946\pi\)
0.391529 + 0.920166i \(0.371946\pi\)
\(258\) 0 0
\(259\) 96.0000 0.370656
\(260\) 0 0
\(261\) −280.000 + 31.3050i −1.07280 + 0.119942i
\(262\) 0 0
\(263\) −58.1378 −0.221056 −0.110528 0.993873i \(-0.535254\pi\)
−0.110528 + 0.993873i \(0.535254\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 214.663 240.000i 0.803979 0.898876i
\(268\) 0 0
\(269\) 371.187i 1.37988i 0.723867 + 0.689939i \(0.242363\pi\)
−0.723867 + 0.689939i \(0.757637\pi\)
\(270\) 0 0
\(271\) −82.0000 −0.302583 −0.151292 0.988489i \(-0.548343\pi\)
−0.151292 + 0.988489i \(0.548343\pi\)
\(272\) 0 0
\(273\) 214.663 + 192.000i 0.786310 + 0.703297i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 24.0000i 0.0866426i −0.999061 0.0433213i \(-0.986206\pi\)
0.999061 0.0433213i \(-0.0137939\pi\)
\(278\) 0 0
\(279\) 18.0000 + 160.997i 0.0645161 + 0.577050i
\(280\) 0 0
\(281\) 187.830i 0.668433i −0.942496 0.334217i \(-0.891528\pi\)
0.942496 0.334217i \(-0.108472\pi\)
\(282\) 0 0
\(283\) 144.000i 0.508834i 0.967095 + 0.254417i \(0.0818836\pi\)
−0.967095 + 0.254417i \(0.918116\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 375.659 1.30892
\(288\) 0 0
\(289\) −269.000 −0.930796
\(290\) 0 0
\(291\) −332.000 + 371.187i −1.14089 + 1.27556i
\(292\) 0 0
\(293\) −469.574 −1.60264 −0.801321 0.598234i \(-0.795869\pi\)
−0.801321 + 0.598234i \(0.795869\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 98.3870 + 70.0000i 0.331269 + 0.235690i
\(298\) 0 0
\(299\) 214.663i 0.717935i
\(300\) 0 0
\(301\) −96.0000 −0.318937
\(302\) 0 0
\(303\) −134.164 + 150.000i −0.442786 + 0.495050i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 184.000i 0.599349i 0.954042 + 0.299674i \(0.0968780\pi\)
−0.954042 + 0.299674i \(0.903122\pi\)
\(308\) 0 0
\(309\) 52.0000 58.1378i 0.168285 0.188148i
\(310\) 0 0
\(311\) 160.997i 0.517675i −0.965921 0.258837i \(-0.916661\pi\)
0.965921 0.258837i \(-0.0833394\pi\)
\(312\) 0 0
\(313\) 394.000i 1.25879i −0.777087 0.629393i \(-0.783304\pi\)
0.777087 0.629393i \(-0.216696\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 451.686 1.42488 0.712438 0.701735i \(-0.247591\pi\)
0.712438 + 0.701735i \(0.247591\pi\)
\(318\) 0 0
\(319\) 140.000 0.438871
\(320\) 0 0
\(321\) −450.000 402.492i −1.40187 1.25387i
\(322\) 0 0
\(323\) −8.94427 −0.0276912
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −84.9706 76.0000i −0.259849 0.232416i
\(328\) 0 0
\(329\) 295.161i 0.897146i
\(330\) 0 0
\(331\) 198.000 0.598187 0.299094 0.954224i \(-0.403316\pi\)
0.299094 + 0.954224i \(0.403316\pi\)
\(332\) 0 0
\(333\) −143.108 + 16.0000i −0.429755 + 0.0480480i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 394.000i 1.16914i −0.811343 0.584570i \(-0.801263\pi\)
0.811343 0.584570i \(-0.198737\pi\)
\(338\) 0 0
\(339\) −70.0000 62.6099i −0.206490 0.184690i
\(340\) 0 0
\(341\) 80.4984i 0.236066i
\(342\) 0 0
\(343\) 372.000i 1.08455i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 183.358 0.528408 0.264204 0.964467i \(-0.414891\pi\)
0.264204 + 0.964467i \(0.414891\pi\)
\(348\) 0 0
\(349\) 362.000 1.03725 0.518625 0.855002i \(-0.326444\pi\)
0.518625 + 0.855002i \(0.326444\pi\)
\(350\) 0 0
\(351\) −352.000 250.440i −1.00285 0.713503i
\(352\) 0 0
\(353\) 308.577 0.874157 0.437078 0.899423i \(-0.356013\pi\)
0.437078 + 0.899423i \(0.356013\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 53.6656 60.0000i 0.150324 0.168067i
\(358\) 0 0
\(359\) 295.161i 0.822175i 0.911596 + 0.411088i \(0.134851\pi\)
−0.911596 + 0.411088i \(0.865149\pi\)
\(360\) 0 0
\(361\) −357.000 −0.988920
\(362\) 0 0
\(363\) 225.843 + 202.000i 0.622157 + 0.556474i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 186.000i 0.506812i −0.967360 0.253406i \(-0.918449\pi\)
0.967360 0.253406i \(-0.0815509\pi\)
\(368\) 0 0
\(369\) −560.000 + 62.6099i −1.51762 + 0.169675i
\(370\) 0 0
\(371\) 26.8328i 0.0723256i
\(372\) 0 0
\(373\) 44.0000i 0.117962i −0.998259 0.0589812i \(-0.981215\pi\)
0.998259 0.0589812i \(-0.0187852\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −500.879 −1.32859
\(378\) 0 0
\(379\) −362.000 −0.955145 −0.477573 0.878592i \(-0.658483\pi\)
−0.477573 + 0.878592i \(0.658483\pi\)
\(380\) 0 0
\(381\) 52.0000 58.1378i 0.136483 0.152593i
\(382\) 0 0
\(383\) −362.243 −0.945804 −0.472902 0.881115i \(-0.656794\pi\)
−0.472902 + 0.881115i \(0.656794\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 143.108 16.0000i 0.369789 0.0413437i
\(388\) 0 0
\(389\) 442.741i 1.13815i −0.822285 0.569076i \(-0.807301\pi\)
0.822285 0.569076i \(-0.192699\pi\)
\(390\) 0 0
\(391\) 60.0000 0.153453
\(392\) 0 0
\(393\) 26.8328 30.0000i 0.0682769 0.0763359i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 124.000i 0.312343i −0.987730 0.156171i \(-0.950085\pi\)
0.987730 0.156171i \(-0.0499152\pi\)
\(398\) 0 0
\(399\) −24.0000 + 26.8328i −0.0601504 + 0.0672502i
\(400\) 0 0
\(401\) 268.328i 0.669148i −0.942370 0.334574i \(-0.891408\pi\)
0.942370 0.334574i \(-0.108592\pi\)
\(402\) 0 0
\(403\) 288.000i 0.714640i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 71.5542 0.175809
\(408\) 0 0
\(409\) −458.000 −1.11980 −0.559902 0.828559i \(-0.689161\pi\)
−0.559902 + 0.828559i \(0.689161\pi\)
\(410\) 0 0
\(411\) −270.000 241.495i −0.656934 0.587580i
\(412\) 0 0
\(413\) 26.8328 0.0649705
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −183.358 164.000i −0.439706 0.393285i
\(418\) 0 0
\(419\) 594.794i 1.41956i −0.704425 0.709778i \(-0.748795\pi\)
0.704425 0.709778i \(-0.251205\pi\)
\(420\) 0 0
\(421\) 562.000 1.33492 0.667458 0.744647i \(-0.267382\pi\)
0.667458 + 0.744647i \(0.267382\pi\)
\(422\) 0 0
\(423\) 49.1935 + 440.000i 0.116297 + 1.04019i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 492.000i 1.15222i
\(428\) 0 0
\(429\) 160.000 + 143.108i 0.372960 + 0.333586i
\(430\) 0 0
\(431\) 348.827i 0.809342i −0.914462 0.404671i \(-0.867386\pi\)
0.914462 0.404671i \(-0.132614\pi\)
\(432\) 0 0
\(433\) 226.000i 0.521940i 0.965347 + 0.260970i \(0.0840424\pi\)
−0.965347 + 0.260970i \(0.915958\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −26.8328 −0.0614023
\(438\) 0 0
\(439\) −2.00000 −0.00455581 −0.00227790 0.999997i \(-0.500725\pi\)
−0.00227790 + 0.999997i \(0.500725\pi\)
\(440\) 0 0
\(441\) 13.0000 + 116.276i 0.0294785 + 0.263663i
\(442\) 0 0
\(443\) 201.246 0.454280 0.227140 0.973862i \(-0.427062\pi\)
0.227140 + 0.973862i \(0.427062\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −223.607 + 250.000i −0.500239 + 0.559284i
\(448\) 0 0
\(449\) 313.050i 0.697215i −0.937269 0.348607i \(-0.886655\pi\)
0.937269 0.348607i \(-0.113345\pi\)
\(450\) 0 0
\(451\) 280.000 0.620843
\(452\) 0 0
\(453\) 353.299 + 316.000i 0.779909 + 0.697572i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 334.000i 0.730853i −0.930840 0.365427i \(-0.880923\pi\)
0.930840 0.365427i \(-0.119077\pi\)
\(458\) 0 0
\(459\) −70.0000 + 98.3870i −0.152505 + 0.214351i
\(460\) 0 0
\(461\) 93.9149i 0.203720i 0.994799 + 0.101860i \(0.0324794\pi\)
−0.994799 + 0.101860i \(0.967521\pi\)
\(462\) 0 0
\(463\) 366.000i 0.790497i −0.918574 0.395248i \(-0.870659\pi\)
0.918574 0.395248i \(-0.129341\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −451.686 −0.967207 −0.483604 0.875287i \(-0.660672\pi\)
−0.483604 + 0.875287i \(0.660672\pi\)
\(468\) 0 0
\(469\) 144.000 0.307036
\(470\) 0 0
\(471\) 328.000 366.715i 0.696391 0.778588i
\(472\) 0 0
\(473\) −71.5542 −0.151277
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −4.47214 40.0000i −0.00937555 0.0838574i
\(478\) 0 0
\(479\) 590.322i 1.23240i −0.787588 0.616202i \(-0.788670\pi\)
0.787588 0.616202i \(-0.211330\pi\)
\(480\) 0 0
\(481\) −256.000 −0.532225
\(482\) 0 0
\(483\) 160.997 180.000i 0.333327 0.372671i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 886.000i 1.81930i −0.415374 0.909651i \(-0.636349\pi\)
0.415374 0.909651i \(-0.363651\pi\)
\(488\) 0 0
\(489\) 472.000 527.712i 0.965235 1.07917i
\(490\) 0 0
\(491\) 406.964i 0.828848i 0.910084 + 0.414424i \(0.136017\pi\)
−0.910084 + 0.414424i \(0.863983\pi\)
\(492\) 0 0
\(493\) 140.000i 0.283976i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 751.319 1.51171
\(498\) 0 0
\(499\) −2.00000 −0.00400802 −0.00200401 0.999998i \(-0.500638\pi\)
−0.00200401 + 0.999998i \(0.500638\pi\)
\(500\) 0 0
\(501\) −210.000 187.830i −0.419162 0.374910i
\(502\) 0 0
\(503\) −219.135 −0.435655 −0.217828 0.975987i \(-0.569897\pi\)
−0.217828 + 0.975987i \(0.569897\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −194.538 174.000i −0.383704 0.343195i
\(508\) 0 0
\(509\) 800.512i 1.57272i 0.617771 + 0.786358i \(0.288036\pi\)
−0.617771 + 0.786358i \(0.711964\pi\)
\(510\) 0 0
\(511\) −444.000 −0.868885
\(512\) 0 0
\(513\) 31.3050 44.0000i 0.0610233 0.0857700i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 220.000i 0.425532i
\(518\) 0 0
\(519\) −30.0000 26.8328i −0.0578035 0.0517010i
\(520\) 0 0
\(521\) 527.712i 1.01288i −0.862274 0.506441i \(-0.830961\pi\)
0.862274 0.506441i \(-0.169039\pi\)
\(522\) 0 0
\(523\) 376.000i 0.718929i −0.933159 0.359465i \(-0.882959\pi\)
0.933159 0.359465i \(-0.117041\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 80.4984 0.152748
\(528\) 0 0
\(529\) −349.000 −0.659735
\(530\) 0 0
\(531\) −40.0000 + 4.47214i −0.0753296 + 0.00842210i
\(532\) 0 0
\(533\) −1001.76 −1.87947
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −384.604 + 430.000i −0.716208 + 0.800745i
\(538\) 0 0
\(539\) 58.1378i 0.107862i
\(540\) 0 0
\(541\) −198.000 −0.365989 −0.182994 0.983114i \(-0.558579\pi\)
−0.182994 + 0.983114i \(0.558579\pi\)
\(542\) 0 0
\(543\) 4.47214 + 4.00000i 0.00823598 + 0.00736648i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 1024.00i 1.87203i 0.351961 + 0.936015i \(0.385515\pi\)
−0.351961 + 0.936015i \(0.614485\pi\)
\(548\) 0 0
\(549\) 82.0000 + 733.430i 0.149362 + 1.33594i
\(550\) 0 0
\(551\) 62.6099i 0.113630i
\(552\) 0 0
\(553\) 828.000i 1.49729i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 67.0820 0.120435 0.0602173 0.998185i \(-0.480821\pi\)
0.0602173 + 0.998185i \(0.480821\pi\)
\(558\) 0 0
\(559\) 256.000 0.457961
\(560\) 0 0
\(561\) 40.0000 44.7214i 0.0713012 0.0797172i
\(562\) 0 0
\(563\) 254.912 0.452774 0.226387 0.974037i \(-0.427309\pi\)
0.226387 + 0.974037i \(0.427309\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 107.331 + 474.000i 0.189297 + 0.835979i
\(568\) 0 0
\(569\) 858.650i 1.50905i −0.656271 0.754526i \(-0.727867\pi\)
0.656271 0.754526i \(-0.272133\pi\)
\(570\) 0 0
\(571\) −962.000 −1.68476 −0.842382 0.538881i \(-0.818847\pi\)
−0.842382 + 0.538881i \(0.818847\pi\)
\(572\) 0 0
\(573\) −411.437 + 460.000i −0.718039 + 0.802792i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 886.000i 1.53553i 0.640732 + 0.767764i \(0.278631\pi\)
−0.640732 + 0.767764i \(0.721369\pi\)
\(578\) 0 0
\(579\) 428.000 478.519i 0.739206 0.826457i
\(580\) 0 0
\(581\) 563.489i 0.969861i
\(582\) 0 0
\(583\) 20.0000i 0.0343053i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 657.404 1.11994 0.559969 0.828513i \(-0.310813\pi\)
0.559969 + 0.828513i \(0.310813\pi\)
\(588\) 0 0
\(589\) −36.0000 −0.0611205
\(590\) 0 0
\(591\) −210.000 187.830i −0.355330 0.317817i
\(592\) 0 0
\(593\) 111.803 0.188539 0.0942693 0.995547i \(-0.469949\pi\)
0.0942693 + 0.995547i \(0.469949\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −541.128 484.000i −0.906413 0.810720i
\(598\) 0 0
\(599\) 223.607i 0.373300i 0.982426 + 0.186650i \(0.0597631\pi\)
−0.982426 + 0.186650i \(0.940237\pi\)
\(600\) 0 0
\(601\) 2.00000 0.00332779 0.00166389 0.999999i \(-0.499470\pi\)
0.00166389 + 0.999999i \(0.499470\pi\)
\(602\) 0 0
\(603\) −214.663 + 24.0000i −0.355991 + 0.0398010i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 506.000i 0.833608i −0.908996 0.416804i \(-0.863150\pi\)
0.908996 0.416804i \(-0.136850\pi\)
\(608\) 0 0
\(609\) 420.000 + 375.659i 0.689655 + 0.616846i
\(610\) 0 0
\(611\) 787.096i 1.28821i
\(612\) 0 0
\(613\) 556.000i 0.907015i 0.891253 + 0.453507i \(0.149827\pi\)
−0.891253 + 0.453507i \(0.850173\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 93.9149 0.152212 0.0761060 0.997100i \(-0.475751\pi\)
0.0761060 + 0.997100i \(0.475751\pi\)
\(618\) 0 0
\(619\) −802.000 −1.29564 −0.647819 0.761794i \(-0.724319\pi\)
−0.647819 + 0.761794i \(0.724319\pi\)
\(620\) 0 0
\(621\) −210.000 + 295.161i −0.338164 + 0.475299i
\(622\) 0 0
\(623\) −643.988 −1.03369
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −17.8885 + 20.0000i −0.0285304 + 0.0318979i
\(628\) 0 0
\(629\) 71.5542i 0.113759i
\(630\) 0 0
\(631\) 698.000 1.10618 0.553090 0.833121i \(-0.313448\pi\)
0.553090 + 0.833121i \(0.313448\pi\)
\(632\) 0 0
\(633\) −4.47214 4.00000i −0.00706499 0.00631912i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 208.000i 0.326531i
\(638\) 0 0
\(639\) −1120.00 + 125.220i −1.75274 + 0.195962i
\(640\) 0 0
\(641\) 912.316i 1.42327i 0.702550 + 0.711635i \(0.252045\pi\)
−0.702550 + 0.711635i \(0.747955\pi\)
\(642\) 0 0
\(643\) 156.000i 0.242613i −0.992615 0.121306i \(-0.961292\pi\)
0.992615 0.121306i \(-0.0387084\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −755.791 −1.16815 −0.584073 0.811701i \(-0.698542\pi\)
−0.584073 + 0.811701i \(0.698542\pi\)
\(648\) 0 0
\(649\) 20.0000 0.0308166
\(650\) 0 0
\(651\) 216.000 241.495i 0.331797 0.370961i
\(652\) 0 0
\(653\) 487.463 0.746497 0.373249 0.927731i \(-0.378244\pi\)
0.373249 + 0.927731i \(0.378244\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 661.876 74.0000i 1.00742 0.112633i
\(658\) 0 0
\(659\) 406.964i 0.617548i 0.951135 + 0.308774i \(0.0999187\pi\)
−0.951135 + 0.308774i \(0.900081\pi\)
\(660\) 0 0
\(661\) 682.000 1.03177 0.515885 0.856658i \(-0.327463\pi\)
0.515885 + 0.856658i \(0.327463\pi\)
\(662\) 0 0
\(663\) −143.108 + 160.000i −0.215850 + 0.241327i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 420.000i 0.629685i
\(668\) 0 0
\(669\) 172.000 192.302i 0.257100 0.287447i
\(670\) 0 0
\(671\) 366.715i 0.546520i
\(672\) 0 0
\(673\) 894.000i 1.32838i −0.747564 0.664190i \(-0.768777\pi\)
0.747564 0.664190i \(-0.231223\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −550.073 −0.812515 −0.406258 0.913759i \(-0.633166\pi\)
−0.406258 + 0.913759i \(0.633166\pi\)
\(678\) 0 0
\(679\) 996.000 1.46686
\(680\) 0 0
\(681\) 130.000 + 116.276i 0.190896 + 0.170742i
\(682\) 0 0
\(683\) 442.741 0.648231 0.324115 0.946018i \(-0.394933\pi\)
0.324115 + 0.946018i \(0.394933\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 630.571 + 564.000i 0.917862 + 0.820961i
\(688\) 0 0
\(689\) 71.5542i 0.103852i
\(690\) 0 0
\(691\) 758.000 1.09696 0.548480 0.836163i \(-0.315207\pi\)
0.548480 + 0.836163i \(0.315207\pi\)
\(692\) 0 0
\(693\) −26.8328 240.000i −0.0387198 0.346320i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 280.000i 0.401722i
\(698\) 0 0
\(699\) 810.000 + 724.486i 1.15880 + 1.03646i
\(700\) 0 0
\(701\) 782.624i 1.11644i −0.829693 0.558220i \(-0.811485\pi\)
0.829693 0.558220i \(-0.188515\pi\)
\(702\) 0 0
\(703\) 32.0000i 0.0455192i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 402.492 0.569296
\(708\) 0 0
\(709\) 2.00000 0.00282087 0.00141044 0.999999i \(-0.499551\pi\)
0.00141044 + 0.999999i \(0.499551\pi\)
\(710\) 0 0
\(711\) 138.000 + 1234.31i 0.194093 + 1.73602i
\(712\) 0 0
\(713\) 241.495 0.338703
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 500.879 560.000i 0.698576 0.781032i
\(718\) 0 0
\(719\) 858.650i 1.19423i 0.802156 + 0.597114i \(0.203686\pi\)
−0.802156 + 0.597114i \(0.796314\pi\)
\(720\) 0 0
\(721\) −156.000 −0.216366
\(722\) 0 0
\(723\) 585.850 + 524.000i 0.810304 + 0.724758i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 674.000i 0.927098i 0.886071 + 0.463549i \(0.153424\pi\)
−0.886071 + 0.463549i \(0.846576\pi\)
\(728\) 0 0
\(729\) −239.000 688.709i −0.327846 0.944731i
\(730\) 0 0
\(731\) 71.5542i 0.0978853i
\(732\) 0 0
\(733\) 656.000i 0.894952i 0.894296 + 0.447476i \(0.147677\pi\)
−0.894296 + 0.447476i \(0.852323\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 107.331 0.145633
\(738\) 0 0
\(739\) 598.000 0.809202 0.404601 0.914493i \(-0.367410\pi\)
0.404601 + 0.914493i \(0.367410\pi\)
\(740\) 0 0
\(741\) 64.0000 71.5542i 0.0863698 0.0965643i
\(742\) 0 0
\(743\) 782.624 1.05333 0.526665 0.850073i \(-0.323442\pi\)
0.526665 + 0.850073i \(0.323442\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −93.9149 840.000i −0.125723 1.12450i
\(748\) 0 0
\(749\) 1207.48i 1.61212i
\(750\) 0 0
\(751\) 338.000 0.450067 0.225033 0.974351i \(-0.427751\pi\)
0.225033 + 0.974351i \(0.427751\pi\)
\(752\) 0 0
\(753\) 939.149 1050.00i 1.24721 1.39442i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 656.000i 0.866579i 0.901255 + 0.433289i \(0.142647\pi\)
−0.901255 + 0.433289i \(0.857353\pi\)
\(758\) 0 0
\(759\) 120.000 134.164i 0.158103 0.176764i
\(760\) 0 0
\(761\) 295.161i 0.387859i 0.981015 + 0.193930i \(0.0621234\pi\)
−0.981015 + 0.193930i \(0.937877\pi\)
\(762\) 0 0
\(763\) 228.000i 0.298820i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −71.5542 −0.0932910
\(768\) 0 0
\(769\) 82.0000 0.106632 0.0533160 0.998578i \(-0.483021\pi\)
0.0533160 + 0.998578i \(0.483021\pi\)
\(770\) 0 0
\(771\) 450.000 + 402.492i 0.583658 + 0.522039i
\(772\) 0 0
\(773\) 1059.90 1.37115 0.685573 0.728004i \(-0.259552\pi\)
0.685573 + 0.728004i \(0.259552\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 214.663 + 192.000i 0.276271 + 0.247104i
\(778\) 0 0
\(779\) 125.220i 0.160744i
\(780\) 0 0
\(781\) 560.000 0.717029
\(782\) 0 0
\(783\) −688.709 490.000i −0.879577 0.625798i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 536.000i 0.681067i −0.940232 0.340534i \(-0.889392\pi\)
0.940232 0.340534i \(-0.110608\pi\)
\(788\) 0 0
\(789\) −130.000 116.276i −0.164766 0.147371i
\(790\) 0 0
\(791\) 187.830i 0.237459i
\(792\) 0 0
\(793\) 1312.00i 1.65448i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −406.964 −0.510620 −0.255310 0.966859i \(-0.582178\pi\)
−0.255310 + 0.966859i \(0.582178\pi\)
\(798\) 0 0
\(799\) 220.000 0.275344
\(800\) 0 0
\(801\) 960.000 107.331i 1.19850 0.133997i
\(802\) 0 0
\(803\) −330.938 −0.412127