Properties

Label 1200.3.l.g.401.2
Level $1200$
Weight $3$
Character 1200.401
Analytic conductor $32.698$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(32.6976317232\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-5}) \)
Defining polynomial: \(x^{2} + 5\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 401.2
Root \(2.23607i\) of defining polynomial
Character \(\chi\) \(=\) 1200.401
Dual form 1200.3.l.g.401.1

$q$-expansion

\(f(q)\) \(=\) \(q+(-2.00000 + 2.23607i) q^{3} -6.00000 q^{7} +(-1.00000 - 8.94427i) q^{9} +O(q^{10})\) \(q+(-2.00000 + 2.23607i) q^{3} -6.00000 q^{7} +(-1.00000 - 8.94427i) q^{9} -4.47214i q^{11} -16.0000 q^{13} -4.47214i q^{17} +2.00000 q^{19} +(12.0000 - 13.4164i) q^{21} +13.4164i q^{23} +(22.0000 + 15.6525i) q^{27} -31.3050i q^{29} +18.0000 q^{31} +(10.0000 + 8.94427i) q^{33} +16.0000 q^{37} +(32.0000 - 35.7771i) q^{39} +62.6099i q^{41} +16.0000 q^{43} -49.1935i q^{47} -13.0000 q^{49} +(10.0000 + 8.94427i) q^{51} -4.47214i q^{53} +(-4.00000 + 4.47214i) q^{57} -4.47214i q^{59} +82.0000 q^{61} +(6.00000 + 53.6656i) q^{63} +24.0000 q^{67} +(-30.0000 - 26.8328i) q^{69} +125.220i q^{71} +74.0000 q^{73} +26.8328i q^{77} -138.000 q^{79} +(-79.0000 + 17.8885i) q^{81} -93.9149i q^{83} +(70.0000 + 62.6099i) q^{87} +107.331i q^{89} +96.0000 q^{91} +(-36.0000 + 40.2492i) q^{93} +166.000 q^{97} +(-40.0000 + 4.47214i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{3} - 12q^{7} - 2q^{9} + O(q^{10}) \) \( 2q - 4q^{3} - 12q^{7} - 2q^{9} - 32q^{13} + 4q^{19} + 24q^{21} + 44q^{27} + 36q^{31} + 20q^{33} + 32q^{37} + 64q^{39} + 32q^{43} - 26q^{49} + 20q^{51} - 8q^{57} + 164q^{61} + 12q^{63} + 48q^{67} - 60q^{69} + 148q^{73} - 276q^{79} - 158q^{81} + 140q^{87} + 192q^{91} - 72q^{93} + 332q^{97} - 80q^{99} + O(q^{100}) \)

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
\(3\) −2.00000 + 2.23607i −0.666667 + 0.745356i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −6.00000 −0.857143 −0.428571 0.903508i \(-0.640983\pi\)
−0.428571 + 0.903508i \(0.640983\pi\)
\(8\) 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
\(13\) −16.0000 −1.23077 −0.615385 0.788227i \(-0.710999\pi\)
−0.615385 + 0.788227i \(0.710999\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.47214i 0.263067i −0.991312 0.131533i \(-0.958010\pi\)
0.991312 0.131533i \(-0.0419901\pi\)
\(18\) 0 0
\(19\) 2.00000 0.105263 0.0526316 0.998614i \(-0.483239\pi\)
0.0526316 + 0.998614i \(0.483239\pi\)
\(20\) 0 0
\(21\) 12.0000 13.4164i 0.571429 0.638877i
\(22\) 0 0
\(23\) 13.4164i 0.583322i 0.956522 + 0.291661i \(0.0942079\pi\)
−0.956522 + 0.291661i \(0.905792\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 22.0000 + 15.6525i 0.814815 + 0.579721i
\(28\) 0 0
\(29\) 31.3050i 1.07948i −0.841831 0.539741i \(-0.818522\pi\)
0.841831 0.539741i \(-0.181478\pi\)
\(30\) 0 0
\(31\) 18.0000 0.580645 0.290323 0.956929i \(-0.406237\pi\)
0.290323 + 0.956929i \(0.406237\pi\)
\(32\) 0 0
\(33\) 10.0000 + 8.94427i 0.303030 + 0.271039i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 16.0000 0.432432 0.216216 0.976346i \(-0.430628\pi\)
0.216216 + 0.976346i \(0.430628\pi\)
\(38\) 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
\(43\) 16.0000 0.372093 0.186047 0.982541i \(-0.440432\pi\)
0.186047 + 0.982541i \(0.440432\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 49.1935i 1.04667i −0.852127 0.523335i \(-0.824688\pi\)
0.852127 0.523335i \(-0.175312\pi\)
\(48\) 0 0
\(49\) −13.0000 −0.265306
\(50\) 0 0
\(51\) 10.0000 + 8.94427i 0.196078 + 0.175378i
\(52\) 0 0
\(53\) 4.47214i 0.0843799i −0.999110 0.0421900i \(-0.986567\pi\)
0.999110 0.0421900i \(-0.0134335\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −4.00000 + 4.47214i −0.0701754 + 0.0784585i
\(58\) 0 0
\(59\) 4.47214i 0.0757989i −0.999282 0.0378995i \(-0.987933\pi\)
0.999282 0.0378995i \(-0.0120667\pi\)
\(60\) 0 0
\(61\) 82.0000 1.34426 0.672131 0.740432i \(-0.265379\pi\)
0.672131 + 0.740432i \(0.265379\pi\)
\(62\) 0 0
\(63\) 6.00000 + 53.6656i 0.0952381 + 0.851835i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 24.0000 0.358209 0.179104 0.983830i \(-0.442680\pi\)
0.179104 + 0.983830i \(0.442680\pi\)
\(68\) 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
\(73\) 74.0000 1.01370 0.506849 0.862035i \(-0.330810\pi\)
0.506849 + 0.862035i \(0.330810\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 26.8328i 0.348478i
\(78\) 0 0
\(79\) −138.000 −1.74684 −0.873418 0.486972i \(-0.838101\pi\)
−0.873418 + 0.486972i \(0.838101\pi\)
\(80\) 0 0
\(81\) −79.0000 + 17.8885i −0.975309 + 0.220846i
\(82\) 0 0
\(83\) 93.9149i 1.13150i −0.824575 0.565752i \(-0.808586\pi\)
0.824575 0.565752i \(-0.191414\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 70.0000 + 62.6099i 0.804598 + 0.719654i
\(88\) 0 0
\(89\) 107.331i 1.20597i 0.797753 + 0.602985i \(0.206022\pi\)
−0.797753 + 0.602985i \(0.793978\pi\)
\(90\) 0 0
\(91\) 96.0000 1.05495
\(92\) 0 0
\(93\) −36.0000 + 40.2492i −0.387097 + 0.432787i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 166.000 1.71134 0.855670 0.517522i \(-0.173145\pi\)
0.855670 + 0.517522i \(0.173145\pi\)
\(98\) 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
\(103\) 26.0000 0.252427 0.126214 0.992003i \(-0.459718\pi\)
0.126214 + 0.992003i \(0.459718\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 201.246i 1.88080i 0.340064 + 0.940402i \(0.389551\pi\)
−0.340064 + 0.940402i \(0.610449\pi\)
\(108\) 0 0
\(109\) 38.0000 0.348624 0.174312 0.984690i \(-0.444230\pi\)
0.174312 + 0.984690i \(0.444230\pi\)
\(110\) 0 0
\(111\) −32.0000 + 35.7771i −0.288288 + 0.322316i
\(112\) 0 0
\(113\) 31.3050i 0.277035i −0.990360 0.138517i \(-0.955766\pi\)
0.990360 0.138517i \(-0.0442337\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 16.0000 + 143.108i 0.136752 + 1.22315i
\(118\) 0 0
\(119\) 26.8328i 0.225486i
\(120\) 0 0
\(121\) 101.000 0.834711
\(122\) 0 0
\(123\) −140.000 125.220i −1.13821 1.01805i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −26.0000 −0.204724 −0.102362 0.994747i \(-0.532640\pi\)
−0.102362 + 0.994747i \(0.532640\pi\)
\(128\) 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
\(133\) −12.0000 −0.0902256
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 120.748i 0.881370i 0.897662 + 0.440685i \(0.145264\pi\)
−0.897662 + 0.440685i \(0.854736\pi\)
\(138\) 0 0
\(139\) 82.0000 0.589928 0.294964 0.955508i \(-0.404692\pi\)
0.294964 + 0.955508i \(0.404692\pi\)
\(140\) 0 0
\(141\) 110.000 + 98.3870i 0.780142 + 0.697780i
\(142\) 0 0
\(143\) 71.5542i 0.500379i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 26.0000 29.0689i 0.176871 0.197748i
\(148\) 0 0
\(149\) 111.803i 0.750358i −0.926952 0.375179i \(-0.877581\pi\)
0.926952 0.375179i \(-0.122419\pi\)
\(150\) 0 0
\(151\) 158.000 1.04636 0.523179 0.852223i \(-0.324746\pi\)
0.523179 + 0.852223i \(0.324746\pi\)
\(152\) 0 0
\(153\) −40.0000 + 4.47214i −0.261438 + 0.0292296i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −164.000 −1.04459 −0.522293 0.852766i \(-0.674923\pi\)
−0.522293 + 0.852766i \(0.674923\pi\)
\(158\) 0 0
\(159\) 10.0000 + 8.94427i 0.0628931 + 0.0562533i
\(160\) 0 0
\(161\) 80.4984i 0.499990i
\(162\) 0 0
\(163\) 236.000 1.44785 0.723926 0.689877i \(-0.242336\pi\)
0.723926 + 0.689877i \(0.242336\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 93.9149i 0.562364i 0.959654 + 0.281182i \(0.0907265\pi\)
−0.959654 + 0.281182i \(0.909273\pi\)
\(168\) 0 0
\(169\) 87.0000 0.514793
\(170\) 0 0
\(171\) −2.00000 17.8885i −0.0116959 0.104611i
\(172\) 0 0
\(173\) 13.4164i 0.0775515i −0.999248 0.0387757i \(-0.987654\pi\)
0.999248 0.0387757i \(-0.0123458\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 10.0000 + 8.94427i 0.0564972 + 0.0505326i
\(178\) 0 0
\(179\) 192.302i 1.07431i −0.843483 0.537156i \(-0.819499\pi\)
0.843483 0.537156i \(-0.180501\pi\)
\(180\) 0 0
\(181\) 2.00000 0.0110497 0.00552486 0.999985i \(-0.498241\pi\)
0.00552486 + 0.999985i \(0.498241\pi\)
\(182\) 0 0
\(183\) −164.000 + 183.358i −0.896175 + 1.00195i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −20.0000 −0.106952
\(188\) 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
\(193\) 214.000 1.10881 0.554404 0.832248i \(-0.312946\pi\)
0.554404 + 0.832248i \(0.312946\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 93.9149i 0.476725i 0.971176 + 0.238363i \(0.0766107\pi\)
−0.971176 + 0.238363i \(0.923389\pi\)
\(198\) 0 0
\(199\) 242.000 1.21608 0.608040 0.793906i \(-0.291956\pi\)
0.608040 + 0.793906i \(0.291956\pi\)
\(200\) 0 0
\(201\) −48.0000 + 53.6656i −0.238806 + 0.266993i
\(202\) 0 0
\(203\) 187.830i 0.925270i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 120.000 13.4164i 0.579710 0.0648136i
\(208\) 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
\(213\) −280.000 250.440i −1.31455 1.17577i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −108.000 −0.497696
\(218\) 0 0
\(219\) −148.000 + 165.469i −0.675799 + 0.755566i
\(220\) 0 0
\(221\) 71.5542i 0.323775i
\(222\) 0 0
\(223\) 86.0000 0.385650 0.192825 0.981233i \(-0.438235\pi\)
0.192825 + 0.981233i \(0.438235\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 58.1378i 0.256114i −0.991767 0.128057i \(-0.959126\pi\)
0.991767 0.128057i \(-0.0408740\pi\)
\(228\) 0 0
\(229\) −282.000 −1.23144 −0.615721 0.787965i \(-0.711135\pi\)
−0.615721 + 0.787965i \(0.711135\pi\)
\(230\) 0 0
\(231\) −60.0000 53.6656i −0.259740 0.232319i
\(232\) 0 0
\(233\) 362.243i 1.55469i 0.629074 + 0.777346i \(0.283434\pi\)
−0.629074 + 0.777346i \(0.716566\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 276.000 308.577i 1.16456 1.30201i
\(238\) 0 0
\(239\) 250.440i 1.04786i 0.851760 + 0.523932i \(0.175536\pi\)
−0.851760 + 0.523932i \(0.824464\pi\)
\(240\) 0 0
\(241\) 262.000 1.08714 0.543568 0.839365i \(-0.317073\pi\)
0.543568 + 0.839365i \(0.317073\pi\)
\(242\) 0 0
\(243\) 118.000 212.426i 0.485597 0.874183i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −32.0000 −0.129555
\(248\) 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
\(253\) 60.0000 0.237154
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 201.246i 0.783059i −0.920166 0.391529i \(-0.871946\pi\)
0.920166 0.391529i \(-0.128054\pi\)
\(258\) 0 0
\(259\) −96.0000 −0.370656
\(260\) 0 0
\(261\) −280.000 + 31.3050i −1.07280 + 0.119942i
\(262\) 0 0
\(263\) 58.1378i 0.221056i −0.993873 0.110528i \(-0.964746\pi\)
0.993873 0.110528i \(-0.0352542\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −240.000 214.663i −0.898876 0.803979i
\(268\) 0 0
\(269\) 371.187i 1.37988i −0.723867 0.689939i \(-0.757637\pi\)
0.723867 0.689939i \(-0.242363\pi\)
\(270\) 0 0
\(271\) −82.0000 −0.302583 −0.151292 0.988489i \(-0.548343\pi\)
−0.151292 + 0.988489i \(0.548343\pi\)
\(272\) 0 0
\(273\) −192.000 + 214.663i −0.703297 + 0.786310i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −24.0000 −0.0866426 −0.0433213 0.999061i \(-0.513794\pi\)
−0.0433213 + 0.999061i \(0.513794\pi\)
\(278\) 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
\(283\) −144.000 −0.508834 −0.254417 0.967095i \(-0.581884\pi\)
−0.254417 + 0.967095i \(0.581884\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 375.659i 1.30892i
\(288\) 0 0
\(289\) 269.000 0.930796
\(290\) 0 0
\(291\) −332.000 + 371.187i −1.14089 + 1.27556i
\(292\) 0 0
\(293\) 469.574i 1.60264i −0.598234 0.801321i \(-0.704131\pi\)
0.598234 0.801321i \(-0.295869\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 70.0000 98.3870i 0.235690 0.331269i
\(298\) 0 0
\(299\) 214.663i 0.717935i
\(300\) 0 0
\(301\) −96.0000 −0.318937
\(302\) 0 0
\(303\) −150.000 134.164i −0.495050 0.442786i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 184.000 0.599349 0.299674 0.954042i \(-0.403122\pi\)
0.299674 + 0.954042i \(0.403122\pi\)
\(308\) 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
\(313\) 394.000 1.25879 0.629393 0.777087i \(-0.283304\pi\)
0.629393 + 0.777087i \(0.283304\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 451.686i 1.42488i −0.701735 0.712438i \(-0.747591\pi\)
0.701735 0.712438i \(-0.252409\pi\)
\(318\) 0 0
\(319\) −140.000 −0.438871
\(320\) 0 0
\(321\) −450.000 402.492i −1.40187 1.25387i
\(322\) 0 0
\(323\) 8.94427i 0.0276912i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −76.0000 + 84.9706i −0.232416 + 0.259849i
\(328\) 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
\(333\) −16.0000 143.108i −0.0480480 0.429755i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −394.000 −1.16914 −0.584570 0.811343i \(-0.698737\pi\)
−0.584570 + 0.811343i \(0.698737\pi\)
\(338\) 0 0
\(339\) 70.0000 + 62.6099i 0.206490 + 0.184690i
\(340\) 0 0
\(341\) 80.4984i 0.236066i
\(342\) 0 0
\(343\) 372.000 1.08455
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 183.358i 0.528408i −0.964467 0.264204i \(-0.914891\pi\)
0.964467 0.264204i \(-0.0851092\pi\)
\(348\) 0 0
\(349\) −362.000 −1.03725 −0.518625 0.855002i \(-0.673556\pi\)
−0.518625 + 0.855002i \(0.673556\pi\)
\(350\) 0 0
\(351\) −352.000 250.440i −1.00285 0.713503i
\(352\) 0 0
\(353\) 308.577i 0.874157i 0.899423 + 0.437078i \(0.143987\pi\)
−0.899423 + 0.437078i \(0.856013\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −60.0000 53.6656i −0.168067 0.150324i
\(358\) 0 0
\(359\) 295.161i 0.822175i −0.911596 0.411088i \(-0.865149\pi\)
0.911596 0.411088i \(-0.134851\pi\)
\(360\) 0 0
\(361\) −357.000 −0.988920
\(362\) 0 0
\(363\) −202.000 + 225.843i −0.556474 + 0.622157i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −186.000 −0.506812 −0.253406 0.967360i \(-0.581551\pi\)
−0.253406 + 0.967360i \(0.581551\pi\)
\(368\) 0 0
\(369\) 560.000 62.6099i 1.51762 0.169675i
\(370\) 0 0
\(371\) 26.8328i 0.0723256i
\(372\) 0 0
\(373\) 44.0000 0.117962 0.0589812 0.998259i \(-0.481215\pi\)
0.0589812 + 0.998259i \(0.481215\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 500.879i 1.32859i
\(378\) 0 0
\(379\) 362.000 0.955145 0.477573 0.878592i \(-0.341517\pi\)
0.477573 + 0.878592i \(0.341517\pi\)
\(380\) 0 0
\(381\) 52.0000 58.1378i 0.136483 0.152593i
\(382\) 0 0
\(383\) 362.243i 0.945804i −0.881115 0.472902i \(-0.843206\pi\)
0.881115 0.472902i \(-0.156794\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −16.0000 143.108i −0.0413437 0.369789i
\(388\) 0 0
\(389\) 442.741i 1.13815i 0.822285 + 0.569076i \(0.192699\pi\)
−0.822285 + 0.569076i \(0.807301\pi\)
\(390\) 0 0
\(391\) 60.0000 0.153453
\(392\) 0 0
\(393\) 30.0000 + 26.8328i 0.0763359 + 0.0682769i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −124.000 −0.312343 −0.156171 0.987730i \(-0.549915\pi\)
−0.156171 + 0.987730i \(0.549915\pi\)
\(398\) 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
\(403\) −288.000 −0.714640
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 71.5542i 0.175809i
\(408\) 0 0
\(409\) 458.000 1.11980 0.559902 0.828559i \(-0.310839\pi\)
0.559902 + 0.828559i \(0.310839\pi\)
\(410\) 0 0
\(411\) −270.000 241.495i −0.656934 0.587580i
\(412\) 0 0
\(413\) 26.8328i 0.0649705i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −164.000 + 183.358i −0.393285 + 0.439706i
\(418\) 0 0
\(419\) 594.794i 1.41956i 0.704425 + 0.709778i \(0.251205\pi\)
−0.704425 + 0.709778i \(0.748795\pi\)
\(420\) 0 0
\(421\) 562.000 1.33492 0.667458 0.744647i \(-0.267382\pi\)
0.667458 + 0.744647i \(0.267382\pi\)
\(422\) 0 0
\(423\) −440.000 + 49.1935i −1.04019 + 0.116297i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −492.000 −1.15222
\(428\) 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
\(433\) −226.000 −0.521940 −0.260970 0.965347i \(-0.584042\pi\)
−0.260970 + 0.965347i \(0.584042\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 26.8328i 0.0614023i
\(438\) 0 0
\(439\) 2.00000 0.00455581 0.00227790 0.999997i \(-0.499275\pi\)
0.00227790 + 0.999997i \(0.499275\pi\)
\(440\) 0 0
\(441\) 13.0000 + 116.276i 0.0294785 + 0.263663i
\(442\) 0 0
\(443\) 201.246i 0.454280i 0.973862 + 0.227140i \(0.0729375\pi\)
−0.973862 + 0.227140i \(0.927062\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 250.000 + 223.607i 0.559284 + 0.500239i
\(448\) 0 0
\(449\) 313.050i 0.697215i 0.937269 + 0.348607i \(0.113345\pi\)
−0.937269 + 0.348607i \(0.886655\pi\)
\(450\) 0 0
\(451\) 280.000 0.620843
\(452\) 0 0
\(453\) −316.000 + 353.299i −0.697572 + 0.779909i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −334.000 −0.730853 −0.365427 0.930840i \(-0.619077\pi\)
−0.365427 + 0.930840i \(0.619077\pi\)
\(458\) 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
\(463\) 366.000 0.790497 0.395248 0.918574i \(-0.370659\pi\)
0.395248 + 0.918574i \(0.370659\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 451.686i 0.967207i 0.875287 + 0.483604i \(0.160672\pi\)
−0.875287 + 0.483604i \(0.839328\pi\)
\(468\) 0 0
\(469\) −144.000 −0.307036
\(470\) 0 0
\(471\) 328.000 366.715i 0.696391 0.778588i
\(472\) 0 0
\(473\) 71.5542i 0.151277i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −40.0000 + 4.47214i −0.0838574 + 0.00937555i
\(478\) 0 0
\(479\) 590.322i 1.23240i 0.787588 + 0.616202i \(0.211330\pi\)
−0.787588 + 0.616202i \(0.788670\pi\)
\(480\) 0 0
\(481\) −256.000 −0.532225
\(482\) 0 0
\(483\) 180.000 + 160.997i 0.372671 + 0.333327i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −886.000 −1.81930 −0.909651 0.415374i \(-0.863651\pi\)
−0.909651 + 0.415374i \(0.863651\pi\)
\(488\) 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
\(493\) −140.000 −0.283976
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 751.319i 1.51171i
\(498\) 0 0
\(499\) 2.00000 0.00400802 0.00200401 0.999998i \(-0.499362\pi\)
0.00200401 + 0.999998i \(0.499362\pi\)
\(500\) 0 0
\(501\) −210.000 187.830i −0.419162 0.374910i
\(502\) 0 0
\(503\) 219.135i 0.435655i −0.975987 0.217828i \(-0.930103\pi\)
0.975987 0.217828i \(-0.0698971\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −174.000 + 194.538i −0.343195 + 0.383704i
\(508\) 0 0
\(509\) 800.512i 1.57272i −0.617771 0.786358i \(-0.711964\pi\)
0.617771 0.786358i \(-0.288036\pi\)
\(510\) 0 0
\(511\) −444.000 −0.868885
\(512\) 0 0
\(513\) 44.0000 + 31.3050i 0.0857700 + 0.0610233i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −220.000 −0.425532
\(518\) 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
\(523\) 376.000 0.718929 0.359465 0.933159i \(-0.382959\pi\)
0.359465 + 0.933159i \(0.382959\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 80.4984i 0.152748i
\(528\) 0 0
\(529\) 349.000 0.659735
\(530\) 0 0
\(531\) −40.0000 + 4.47214i −0.0753296 + 0.00842210i
\(532\) 0 0
\(533\) 1001.76i 1.87947i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 430.000 + 384.604i 0.800745 + 0.716208i
\(538\) 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
\(543\) −4.00000 + 4.47214i −0.00736648 + 0.00823598i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 1024.00 1.87203 0.936015 0.351961i \(-0.114485\pi\)
0.936015 + 0.351961i \(0.114485\pi\)
\(548\) 0 0
\(549\) −82.0000 733.430i −0.149362 1.33594i
\(550\) 0 0
\(551\) 62.6099i 0.113630i
\(552\) 0 0
\(553\) 828.000 1.49729
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 67.0820i 0.120435i −0.998185 0.0602173i \(-0.980821\pi\)
0.998185 0.0602173i \(-0.0191794\pi\)
\(558\) 0 0
\(559\) −256.000 −0.457961
\(560\) 0 0
\(561\) 40.0000 44.7214i 0.0713012 0.0797172i
\(562\) 0 0
\(563\) 254.912i 0.452774i 0.974037 + 0.226387i \(0.0726914\pi\)
−0.974037 + 0.226387i \(0.927309\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 474.000 107.331i 0.835979 0.189297i
\(568\) 0 0
\(569\) 858.650i 1.50905i 0.656271 + 0.754526i \(0.272133\pi\)
−0.656271 + 0.754526i \(0.727867\pi\)
\(570\) 0 0
\(571\) −962.000 −1.68476 −0.842382 0.538881i \(-0.818847\pi\)
−0.842382 + 0.538881i \(0.818847\pi\)
\(572\) 0 0
\(573\) −460.000 411.437i −0.802792 0.718039i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 886.000 1.53553 0.767764 0.640732i \(-0.221369\pi\)
0.767764 + 0.640732i \(0.221369\pi\)
\(578\) 0 0
\(579\) −428.000 + 478.519i −0.739206 + 0.826457i
\(580\) 0 0
\(581\) 563.489i 0.969861i
\(582\) 0 0
\(583\) −20.0000 −0.0343053
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 657.404i 1.11994i −0.828513 0.559969i \(-0.810813\pi\)
0.828513 0.559969i \(-0.189187\pi\)
\(588\) 0 0
\(589\) 36.0000 0.0611205
\(590\) 0 0
\(591\) −210.000 187.830i −0.355330 0.317817i
\(592\) 0 0
\(593\) 111.803i 0.188539i 0.995547 + 0.0942693i \(0.0300515\pi\)
−0.995547 + 0.0942693i \(0.969949\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −484.000 + 541.128i −0.810720 + 0.906413i
\(598\) 0 0
\(599\) 223.607i 0.373300i −0.982426 0.186650i \(-0.940237\pi\)
0.982426 0.186650i \(-0.0597631\pi\)
\(600\) 0 0
\(601\) 2.00000 0.00332779 0.00166389 0.999999i \(-0.499470\pi\)
0.00166389 + 0.999999i \(0.499470\pi\)
\(602\) 0 0
\(603\) −24.0000 214.663i −0.0398010 0.355991i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −506.000 −0.833608 −0.416804 0.908996i \(-0.636850\pi\)
−0.416804 + 0.908996i \(0.636850\pi\)
\(608\) 0 0
\(609\) −420.000 375.659i −0.689655 0.616846i
\(610\) 0 0
\(611\) 787.096i 1.28821i
\(612\) 0 0
\(613\) −556.000 −0.907015 −0.453507 0.891253i \(-0.649827\pi\)
−0.453507 + 0.891253i \(0.649827\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 93.9149i 0.152212i −0.997100 0.0761060i \(-0.975751\pi\)
0.997100 0.0761060i \(-0.0242488\pi\)
\(618\) 0 0
\(619\) 802.000 1.29564 0.647819 0.761794i \(-0.275681\pi\)
0.647819 + 0.761794i \(0.275681\pi\)
\(620\) 0 0
\(621\) −210.000 + 295.161i −0.338164 + 0.475299i
\(622\) 0 0
\(623\) 643.988i 1.03369i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 20.0000 + 17.8885i 0.0318979 + 0.0285304i
\(628\) 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
\(633\) 4.00000 4.47214i 0.00631912 0.00706499i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 208.000 0.326531
\(638\) 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
\(643\) 156.000 0.242613 0.121306 0.992615i \(-0.461292\pi\)
0.121306 + 0.992615i \(0.461292\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 755.791i 1.16815i 0.811701 + 0.584073i \(0.198542\pi\)
−0.811701 + 0.584073i \(0.801458\pi\)
\(648\) 0 0
\(649\) −20.0000 −0.0308166
\(650\) 0 0
\(651\) 216.000 241.495i 0.331797 0.370961i
\(652\) 0 0
\(653\) 487.463i 0.746497i 0.927731 + 0.373249i \(0.121756\pi\)
−0.927731 + 0.373249i \(0.878244\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −74.0000 661.876i −0.112633 1.00742i
\(658\) 0 0
\(659\) 406.964i 0.617548i −0.951135 0.308774i \(-0.900081\pi\)
0.951135 0.308774i \(-0.0999187\pi\)
\(660\) 0 0
\(661\) 682.000 1.03177 0.515885 0.856658i \(-0.327463\pi\)
0.515885 + 0.856658i \(0.327463\pi\)
\(662\) 0 0
\(663\) −160.000 143.108i −0.241327 0.215850i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 420.000 0.629685
\(668\) 0 0
\(669\) −172.000 + 192.302i −0.257100 + 0.287447i
\(670\) 0 0
\(671\) 366.715i 0.546520i
\(672\) 0 0
\(673\) 894.000 1.32838 0.664190 0.747564i \(-0.268777\pi\)
0.664190 + 0.747564i \(0.268777\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 550.073i 0.812515i 0.913759 + 0.406258i \(0.133166\pi\)
−0.913759 + 0.406258i \(0.866834\pi\)
\(678\) 0 0
\(679\) −996.000 −1.46686
\(680\) 0 0
\(681\) 130.000 + 116.276i 0.190896 + 0.170742i
\(682\) 0 0
\(683\) 442.741i 0.648231i 0.946018 + 0.324115i \(0.105067\pi\)
−0.946018 + 0.324115i \(0.894933\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 564.000 630.571i 0.820961 0.917862i
\(688\) 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
\(693\) 240.000 26.8328i 0.346320 0.0387198i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 280.000 0.401722
\(698\) 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
\(703\) 32.0000 0.0455192
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 402.492i 0.569296i
\(708\) 0 0
\(709\) −2.00000 −0.00282087 −0.00141044 0.999999i \(-0.500449\pi\)
−0.00141044 + 0.999999i \(0.500449\pi\)
\(710\) 0 0
\(711\) 138.000 + 1234.31i 0.194093 + 1.73602i
\(712\) 0 0
\(713\) 241.495i 0.338703i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −560.000 500.879i −0.781032 0.698576i
\(718\) 0 0
\(719\) 858.650i 1.19423i −0.802156 0.597114i \(-0.796314\pi\)
0.802156 0.597114i \(-0.203686\pi\)
\(720\) 0 0
\(721\) −156.000 −0.216366
\(722\) 0 0
\(723\) −524.000 + 585.850i −0.724758 + 0.810304i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 674.000 0.927098 0.463549 0.886071i \(-0.346576\pi\)
0.463549 + 0.886071i \(0.346576\pi\)
\(728\) 0 0
\(729\) 239.000 + 688.709i 0.327846 + 0.944731i
\(730\) 0 0
\(731\) 71.5542i 0.0978853i
\(732\) 0 0
\(733\) −656.000 −0.894952 −0.447476 0.894296i \(-0.647677\pi\)
−0.447476 + 0.894296i \(0.647677\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 107.331i 0.145633i
\(738\) 0 0
\(739\) −598.000 −0.809202 −0.404601 0.914493i \(-0.632590\pi\)
−0.404601 + 0.914493i \(0.632590\pi\)
\(740\) 0 0
\(741\) 64.0000 71.5542i 0.0863698 0.0965643i
\(742\) 0 0
\(743\) 782.624i 1.05333i 0.850073 + 0.526665i \(0.176558\pi\)
−0.850073 + 0.526665i \(0.823442\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −840.000 + 93.9149i −1.12450 + 0.125723i
\(748\) 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
\(753\) 1050.00 + 939.149i 1.39442 + 1.24721i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 656.000 0.866579 0.433289 0.901255i \(-0.357353\pi\)
0.433289 + 0.901255i \(0.357353\pi\)
\(758\) 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
\(763\) −228.000 −0.298820
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 71.5542i 0.0932910i
\(768\) 0 0
\(769\) −82.0000 −0.106632 −0.0533160 0.998578i \(-0.516979\pi\)
−0.0533160 + 0.998578i \(0.516979\pi\)
\(770\) 0 0
\(771\) 450.000 + 402.492i 0.583658 + 0.522039i
\(772\) 0 0
\(773\) 1059.90i 1.37115i 0.728004 + 0.685573i \(0.240448\pi\)
−0.728004 + 0.685573i \(0.759552\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 192.000 214.663i 0.247104 0.276271i
\(778\) 0 0
\(779\) 125.220i 0.160744i
\(780\) 0 0
\(781\) 560.000 0.717029
\(782\) 0 0
\(783\) 490.000 688.709i 0.625798 0.879577i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −536.000 −0.681067 −0.340534 0.940232i \(-0.610608\pi\)
−0.340534 + 0.940232i \(0.610608\pi\)
\(788\) 0 0
\(789\) 130.000 + 116.276i 0.164766 + 0.147371i
\(790\) 0 0
\(791\) 187.830i 0.237459i
\(792\) 0 0
\(793\) −1312.00 −1.65448
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 406.964i 0.510620i 0.966859 + 0.255310i \(0.0821776\pi\)
−0.966859 + 0.255310i \(0.917822\pi\)
\(798\) 0 0
\(799\) −220.000 −0.275344
\(800\) 0