Properties

Label 1764.3.z.l.901.3
Level $1764$
Weight $3$
Character 1764.901
Analytic conductor $48.066$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1764.z (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(48.0655186332\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.339738624.1
Defining polynomial: \(x^{8} - 4 x^{6} + 14 x^{4} - 8 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 7^{2} \)
Twist minimal: no (minimal twist has level 588)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 901.3
Root \(0.662827 - 0.382683i\) of defining polynomial
Character \(\chi\) \(=\) 1764.901
Dual form 1764.3.z.l.325.3

$q$-expansion

\(f(q)\) \(=\) \(q+(0.416265 + 0.240331i) q^{5} +O(q^{10})\) \(q+(0.416265 + 0.240331i) q^{5} +(2.88158 + 4.99104i) q^{11} -1.41991i q^{13} +(-19.9767 + 11.5336i) q^{17} +(-9.24384 - 5.33693i) q^{19} +(-3.29990 + 5.71559i) q^{23} +(-12.3845 - 21.4506i) q^{25} +6.20258 q^{29} +(36.3142 - 20.9660i) q^{31} +(30.0464 - 52.0419i) q^{37} -48.8250i q^{41} -51.5603 q^{43} +(-16.6641 - 9.62104i) q^{47} +(41.0672 + 71.1306i) q^{53} +2.77013i q^{55} +(-80.1766 + 46.2900i) q^{59} +(-4.32309 - 2.49594i) q^{61} +(0.341248 - 0.591060i) q^{65} +(1.10350 + 1.91131i) q^{67} +80.5899 q^{71} +(-12.0454 + 6.95439i) q^{73} +(32.4442 - 56.1951i) q^{79} -118.005i q^{83} -11.0875 q^{85} +(-90.2959 - 52.1324i) q^{89} +(-2.56526 - 4.44316i) q^{95} -31.7875i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q - 48q^{17} - 96q^{19} - 8q^{23} - 36q^{25} - 80q^{29} + 48q^{31} - 64q^{37} - 112q^{43} + 264q^{47} - 72q^{53} - 168q^{59} - 144q^{61} + 120q^{65} + 32q^{67} - 224q^{71} + 336q^{73} + 216q^{79} - 96q^{85} - 96q^{89} - 136q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{5}{6}\right)\)

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
\(5\) 0.416265 + 0.240331i 0.0832530 + 0.0480662i 0.541049 0.840991i \(-0.318028\pi\)
−0.457796 + 0.889057i \(0.651361\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.88158 + 4.99104i 0.261962 + 0.453731i 0.966763 0.255673i \(-0.0822972\pi\)
−0.704801 + 0.709405i \(0.748964\pi\)
\(12\) 0 0
\(13\) 1.41991i 0.109224i −0.998508 0.0546120i \(-0.982608\pi\)
0.998508 0.0546120i \(-0.0173922\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −19.9767 + 11.5336i −1.17510 + 0.678445i −0.954876 0.297004i \(-0.904012\pi\)
−0.220225 + 0.975449i \(0.570679\pi\)
\(18\) 0 0
\(19\) −9.24384 5.33693i −0.486518 0.280891i 0.236611 0.971605i \(-0.423963\pi\)
−0.723129 + 0.690713i \(0.757297\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −3.29990 + 5.71559i −0.143474 + 0.248504i −0.928803 0.370575i \(-0.879161\pi\)
0.785329 + 0.619079i \(0.212494\pi\)
\(24\) 0 0
\(25\) −12.3845 21.4506i −0.495379 0.858022i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 6.20258 0.213882 0.106941 0.994265i \(-0.465894\pi\)
0.106941 + 0.994265i \(0.465894\pi\)
\(30\) 0 0
\(31\) 36.3142 20.9660i 1.17143 0.676323i 0.217410 0.976080i \(-0.430239\pi\)
0.954016 + 0.299757i \(0.0969057\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 30.0464 52.0419i 0.812066 1.40654i −0.0993502 0.995053i \(-0.531676\pi\)
0.911416 0.411486i \(-0.134990\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 48.8250i 1.19085i −0.803409 0.595427i \(-0.796983\pi\)
0.803409 0.595427i \(-0.203017\pi\)
\(42\) 0 0
\(43\) −51.5603 −1.19908 −0.599539 0.800346i \(-0.704649\pi\)
−0.599539 + 0.800346i \(0.704649\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −16.6641 9.62104i −0.354556 0.204703i 0.312134 0.950038i \(-0.398956\pi\)
−0.666690 + 0.745335i \(0.732290\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 41.0672 + 71.1306i 0.774854 + 1.34209i 0.934877 + 0.354973i \(0.115510\pi\)
−0.160023 + 0.987113i \(0.551157\pi\)
\(54\) 0 0
\(55\) 2.77013i 0.0503660i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −80.1766 + 46.2900i −1.35893 + 0.784576i −0.989479 0.144676i \(-0.953786\pi\)
−0.369447 + 0.929252i \(0.620453\pi\)
\(60\) 0 0
\(61\) −4.32309 2.49594i −0.0708703 0.0409170i 0.464146 0.885759i \(-0.346361\pi\)
−0.535016 + 0.844842i \(0.679695\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0.341248 0.591060i 0.00524998 0.00909323i
\(66\) 0 0
\(67\) 1.10350 + 1.91131i 0.0164701 + 0.0285270i 0.874143 0.485669i \(-0.161424\pi\)
−0.857673 + 0.514196i \(0.828091\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 80.5899 1.13507 0.567535 0.823349i \(-0.307897\pi\)
0.567535 + 0.823349i \(0.307897\pi\)
\(72\) 0 0
\(73\) −12.0454 + 6.95439i −0.165005 + 0.0952657i −0.580228 0.814454i \(-0.697037\pi\)
0.415223 + 0.909719i \(0.363703\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 32.4442 56.1951i 0.410687 0.711330i −0.584278 0.811553i \(-0.698622\pi\)
0.994965 + 0.100223i \(0.0319557\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 118.005i 1.42174i −0.703322 0.710872i \(-0.748301\pi\)
0.703322 0.710872i \(-0.251699\pi\)
\(84\) 0 0
\(85\) −11.0875 −0.130441
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −90.2959 52.1324i −1.01456 0.585757i −0.102037 0.994781i \(-0.532536\pi\)
−0.912524 + 0.409024i \(0.865869\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −2.56526 4.44316i −0.0270027 0.0467701i
\(96\) 0 0
\(97\) 31.7875i 0.327706i −0.986485 0.163853i \(-0.947608\pi\)
0.986485 0.163853i \(-0.0523923\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −62.7901 + 36.2519i −0.621684 + 0.358929i −0.777524 0.628853i \(-0.783525\pi\)
0.155840 + 0.987782i \(0.450191\pi\)
\(102\) 0 0
\(103\) −92.6323 53.4813i −0.899343 0.519236i −0.0223558 0.999750i \(-0.507117\pi\)
−0.876987 + 0.480514i \(0.840450\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 98.3097 170.277i 0.918782 1.59138i 0.117515 0.993071i \(-0.462507\pi\)
0.801268 0.598306i \(-0.204159\pi\)
\(108\) 0 0
\(109\) −21.3461 36.9726i −0.195836 0.339198i 0.751338 0.659917i \(-0.229409\pi\)
−0.947174 + 0.320719i \(0.896075\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −175.501 −1.55310 −0.776552 0.630053i \(-0.783033\pi\)
−0.776552 + 0.630053i \(0.783033\pi\)
\(114\) 0 0
\(115\) −2.74726 + 1.58613i −0.0238893 + 0.0137925i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 43.8930 76.0249i 0.362752 0.628305i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 23.9220i 0.191376i
\(126\) 0 0
\(127\) 31.0434 0.244436 0.122218 0.992503i \(-0.460999\pi\)
0.122218 + 0.992503i \(0.460999\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −40.2784 23.2547i −0.307469 0.177517i 0.338325 0.941029i \(-0.390140\pi\)
−0.645793 + 0.763512i \(0.723473\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 22.7327 + 39.3742i 0.165932 + 0.287403i 0.936986 0.349367i \(-0.113603\pi\)
−0.771054 + 0.636770i \(0.780270\pi\)
\(138\) 0 0
\(139\) 138.075i 0.993343i −0.867939 0.496672i \(-0.834555\pi\)
0.867939 0.496672i \(-0.165445\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 7.08684 4.09159i 0.0495583 0.0286125i
\(144\) 0 0
\(145\) 2.58192 + 1.49067i 0.0178063 + 0.0102805i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 119.614 207.178i 0.802780 1.39045i −0.115000 0.993365i \(-0.536687\pi\)
0.917780 0.397089i \(-0.129980\pi\)
\(150\) 0 0
\(151\) 94.0732 + 162.940i 0.623001 + 1.07907i 0.988924 + 0.148425i \(0.0474202\pi\)
−0.365922 + 0.930645i \(0.619246\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 20.1551 0.130033
\(156\) 0 0
\(157\) −186.402 + 107.619i −1.18728 + 0.685474i −0.957686 0.287814i \(-0.907072\pi\)
−0.229589 + 0.973288i \(0.573738\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −74.9053 + 129.740i −0.459542 + 0.795949i −0.998937 0.0461035i \(-0.985320\pi\)
0.539395 + 0.842053i \(0.318653\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 137.195i 0.821528i −0.911742 0.410764i \(-0.865262\pi\)
0.911742 0.410764i \(-0.134738\pi\)
\(168\) 0 0
\(169\) 166.984 0.988070
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −217.529 125.591i −1.25739 0.725957i −0.284828 0.958579i \(-0.591936\pi\)
−0.972567 + 0.232621i \(0.925270\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −58.3967 101.146i −0.326239 0.565062i 0.655524 0.755175i \(-0.272448\pi\)
−0.981762 + 0.190113i \(0.939115\pi\)
\(180\) 0 0
\(181\) 117.148i 0.647228i −0.946189 0.323614i \(-0.895102\pi\)
0.946189 0.323614i \(-0.104898\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 25.0146 14.4422i 0.135214 0.0780657i
\(186\) 0 0
\(187\) −115.129 66.4698i −0.615663 0.355453i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 170.909 296.023i 0.894812 1.54986i 0.0607755 0.998151i \(-0.480643\pi\)
0.834037 0.551709i \(-0.186024\pi\)
\(192\) 0 0
\(193\) 112.275 + 194.467i 0.581738 + 1.00760i 0.995273 + 0.0971119i \(0.0309605\pi\)
−0.413535 + 0.910488i \(0.635706\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −257.109 −1.30512 −0.652560 0.757737i \(-0.726305\pi\)
−0.652560 + 0.757737i \(0.726305\pi\)
\(198\) 0 0
\(199\) −212.591 + 122.740i −1.06830 + 0.616782i −0.927716 0.373287i \(-0.878231\pi\)
−0.140582 + 0.990069i \(0.544897\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 11.7342 20.3241i 0.0572398 0.0991422i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 61.5152i 0.294331i
\(210\) 0 0
\(211\) −95.8210 −0.454128 −0.227064 0.973880i \(-0.572913\pi\)
−0.227064 + 0.973880i \(0.572913\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −21.4628 12.3915i −0.0998268 0.0576350i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 16.3766 + 28.3652i 0.0741024 + 0.128349i
\(222\) 0 0
\(223\) 94.2091i 0.422462i −0.977436 0.211231i \(-0.932253\pi\)
0.977436 0.211231i \(-0.0677473\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 194.209 112.127i 0.855545 0.493949i −0.00697266 0.999976i \(-0.502219\pi\)
0.862518 + 0.506026i \(0.168886\pi\)
\(228\) 0 0
\(229\) 317.592 + 183.362i 1.38687 + 0.800708i 0.992961 0.118443i \(-0.0377902\pi\)
0.393906 + 0.919151i \(0.371124\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 34.8370 60.3395i 0.149515 0.258968i −0.781533 0.623864i \(-0.785562\pi\)
0.931048 + 0.364896i \(0.118895\pi\)
\(234\) 0 0
\(235\) −4.62447 8.00981i −0.0196786 0.0340843i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 214.544 0.897674 0.448837 0.893614i \(-0.351838\pi\)
0.448837 + 0.893614i \(0.351838\pi\)
\(240\) 0 0
\(241\) −141.504 + 81.6976i −0.587155 + 0.338994i −0.763972 0.645250i \(-0.776753\pi\)
0.176817 + 0.984244i \(0.443420\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −7.57798 + 13.1254i −0.0306801 + 0.0531394i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 330.546i 1.31692i −0.752617 0.658458i \(-0.771209\pi\)
0.752617 0.658458i \(-0.228791\pi\)
\(252\) 0 0
\(253\) −38.0357 −0.150339
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −139.022 80.2644i −0.540942 0.312313i 0.204519 0.978863i \(-0.434437\pi\)
−0.745461 + 0.666550i \(0.767770\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −53.2577 92.2451i −0.202501 0.350742i 0.746833 0.665012i \(-0.231574\pi\)
−0.949334 + 0.314270i \(0.898240\pi\)
\(264\) 0 0
\(265\) 39.4789i 0.148977i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −134.627 + 77.7268i −0.500471 + 0.288947i −0.728908 0.684611i \(-0.759972\pi\)
0.228437 + 0.973559i \(0.426639\pi\)
\(270\) 0 0
\(271\) −446.938 258.040i −1.64922 0.952176i −0.977383 0.211475i \(-0.932173\pi\)
−0.671834 0.740702i \(-0.734493\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 71.3738 123.623i 0.259541 0.449538i
\(276\) 0 0
\(277\) −100.101 173.380i −0.361375 0.625919i 0.626813 0.779170i \(-0.284359\pi\)
−0.988187 + 0.153251i \(0.951026\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −228.093 −0.811720 −0.405860 0.913935i \(-0.633028\pi\)
−0.405860 + 0.913935i \(0.633028\pi\)
\(282\) 0 0
\(283\) 225.781 130.355i 0.797814 0.460618i −0.0448922 0.998992i \(-0.514294\pi\)
0.842706 + 0.538374i \(0.180961\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 121.546 210.524i 0.420575 0.728457i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 349.885i 1.19415i 0.802186 + 0.597074i \(0.203670\pi\)
−0.802186 + 0.597074i \(0.796330\pi\)
\(294\) 0 0
\(295\) −44.4996 −0.150846
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 8.11563 + 4.68556i 0.0271426 + 0.0156708i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −1.19970 2.07794i −0.00393345 0.00681293i
\(306\) 0 0
\(307\) 146.898i 0.478495i 0.970959 + 0.239247i \(0.0769007\pi\)
−0.970959 + 0.239247i \(0.923099\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −69.9177 + 40.3670i −0.224816 + 0.129797i −0.608178 0.793801i \(-0.708099\pi\)
0.383362 + 0.923598i \(0.374766\pi\)
\(312\) 0 0
\(313\) 133.661 + 77.1695i 0.427033 + 0.246548i 0.698082 0.716018i \(-0.254037\pi\)
−0.271049 + 0.962566i \(0.587370\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −94.2726 + 163.285i −0.297390 + 0.515094i −0.975538 0.219831i \(-0.929449\pi\)
0.678148 + 0.734925i \(0.262783\pi\)
\(318\) 0 0
\(319\) 17.8732 + 30.9574i 0.0560290 + 0.0970450i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 246.216 0.762277
\(324\) 0 0
\(325\) −30.4579 + 17.5849i −0.0937166 + 0.0541073i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −73.3936 + 127.121i −0.221733 + 0.384053i −0.955334 0.295527i \(-0.904505\pi\)
0.733601 + 0.679580i \(0.237838\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 1.06082i 0.00316662i
\(336\) 0 0
\(337\) 101.231 0.300388 0.150194 0.988657i \(-0.452010\pi\)
0.150194 + 0.988657i \(0.452010\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 209.285 + 120.831i 0.613738 + 0.354342i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 175.844 + 304.571i 0.506756 + 0.877727i 0.999969 + 0.00781897i \(0.00248888\pi\)
−0.493213 + 0.869908i \(0.664178\pi\)
\(348\) 0 0
\(349\) 88.3780i 0.253232i −0.991952 0.126616i \(-0.959588\pi\)
0.991952 0.126616i \(-0.0404116\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −450.803 + 260.271i −1.27706 + 0.737312i −0.976307 0.216390i \(-0.930572\pi\)
−0.300755 + 0.953702i \(0.597239\pi\)
\(354\) 0 0
\(355\) 33.5468 + 19.3682i 0.0944980 + 0.0545584i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −257.132 + 445.366i −0.716246 + 1.24058i 0.246231 + 0.969211i \(0.420808\pi\)
−0.962477 + 0.271364i \(0.912525\pi\)
\(360\) 0 0
\(361\) −123.534 213.968i −0.342200 0.592708i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −6.68542 −0.0183162
\(366\) 0 0
\(367\) 361.726 208.843i 0.985630 0.569054i 0.0816651 0.996660i \(-0.473976\pi\)
0.903965 + 0.427606i \(0.140643\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 70.1568 121.515i 0.188088 0.325778i −0.756525 0.653965i \(-0.773104\pi\)
0.944613 + 0.328187i \(0.106438\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 8.80712i 0.0233611i
\(378\) 0 0
\(379\) 153.298 0.404480 0.202240 0.979336i \(-0.435178\pi\)
0.202240 + 0.979336i \(0.435178\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 105.410 + 60.8586i 0.275222 + 0.158900i 0.631258 0.775573i \(-0.282539\pi\)
−0.356036 + 0.934472i \(0.615872\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 200.636 + 347.511i 0.515773 + 0.893345i 0.999832 + 0.0183096i \(0.00582845\pi\)
−0.484060 + 0.875035i \(0.660838\pi\)
\(390\) 0 0
\(391\) 152.238i 0.389356i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 27.0108 15.5947i 0.0683818 0.0394802i
\(396\) 0 0
\(397\) −52.1564 30.1125i −0.131376 0.0758502i 0.432871 0.901456i \(-0.357500\pi\)
−0.564248 + 0.825605i \(0.690834\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −187.154 + 324.160i −0.466718 + 0.808380i −0.999277 0.0380132i \(-0.987897\pi\)
0.532559 + 0.846393i \(0.321230\pi\)
\(402\) 0 0
\(403\) −29.7699 51.5630i −0.0738707 0.127948i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 346.325 0.850921
\(408\) 0 0
\(409\) 532.267 307.305i 1.30139 0.751356i 0.320745 0.947166i \(-0.396067\pi\)
0.980642 + 0.195810i \(0.0627335\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 28.3602 49.1212i 0.0683377 0.118364i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 129.067i 0.308035i −0.988068 0.154017i \(-0.950779\pi\)
0.988068 0.154017i \(-0.0492212\pi\)
\(420\) 0 0
\(421\) −697.880 −1.65767 −0.828836 0.559492i \(-0.810996\pi\)
−0.828836 + 0.559492i \(0.810996\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 494.803 + 285.674i 1.16424 + 0.672175i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −138.546 239.969i −0.321453 0.556773i 0.659335 0.751849i \(-0.270838\pi\)
−0.980788 + 0.195076i \(0.937505\pi\)
\(432\) 0 0
\(433\) 822.794i 1.90022i 0.311919 + 0.950109i \(0.399028\pi\)
−0.311919 + 0.950109i \(0.600972\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 61.0075 35.2227i 0.139605 0.0806011i
\(438\) 0 0
\(439\) 316.816 + 182.914i 0.721676 + 0.416660i 0.815369 0.578941i \(-0.196534\pi\)
−0.0936932 + 0.995601i \(0.529867\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −286.696 + 496.572i −0.647169 + 1.12093i 0.336626 + 0.941638i \(0.390714\pi\)
−0.983796 + 0.179292i \(0.942619\pi\)
\(444\) 0 0
\(445\) −25.0580 43.4018i −0.0563102 0.0975321i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −227.961 −0.507708 −0.253854 0.967243i \(-0.581698\pi\)
−0.253854 + 0.967243i \(0.581698\pi\)
\(450\) 0 0
\(451\) 243.688 140.693i 0.540328 0.311958i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 171.562 297.154i 0.375409 0.650228i −0.614979 0.788544i \(-0.710836\pi\)
0.990388 + 0.138316i \(0.0441688\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 611.314i 1.32606i 0.748593 + 0.663030i \(0.230730\pi\)
−0.748593 + 0.663030i \(0.769270\pi\)
\(462\) 0 0
\(463\) −67.2682 −0.145288 −0.0726439 0.997358i \(-0.523144\pi\)
−0.0726439 + 0.997358i \(0.523144\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −81.1897 46.8749i −0.173854 0.100375i 0.410548 0.911839i \(-0.365337\pi\)
−0.584402 + 0.811464i \(0.698671\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −148.575 257.340i −0.314113 0.544059i
\(474\) 0 0
\(475\) 264.381i 0.556591i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 306.396 176.898i 0.639658 0.369306i −0.144825 0.989457i \(-0.546262\pi\)
0.784483 + 0.620151i \(0.212929\pi\)
\(480\) 0 0
\(481\) −73.8950 42.6633i −0.153628 0.0886970i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 7.63952 13.2320i 0.0157516 0.0272825i
\(486\) 0 0
\(487\) −476.338 825.042i −0.978107 1.69413i −0.669274 0.743016i \(-0.733395\pi\)
−0.308833 0.951116i \(-0.599939\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 818.649 1.66731 0.833655 0.552286i \(-0.186244\pi\)
0.833655 + 0.552286i \(0.186244\pi\)
\(492\) 0 0
\(493\) −123.907 + 71.5379i −0.251333 + 0.145107i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 12.0411 20.8557i 0.0241304 0.0417950i −0.853708 0.520752i \(-0.825652\pi\)
0.877838 + 0.478957i \(0.158985\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 420.445i 0.835874i 0.908476 + 0.417937i \(0.137247\pi\)
−0.908476 + 0.417937i \(0.862753\pi\)
\(504\) 0 0
\(505\) −34.8498 −0.0690094
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 207.732 + 119.934i 0.408119 + 0.235627i 0.689981 0.723827i \(-0.257619\pi\)
−0.281862 + 0.959455i \(0.590952\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −25.7064 44.5248i −0.0499153 0.0864559i
\(516\) 0 0
\(517\) 110.895i 0.214498i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 23.2584 13.4282i 0.0446418 0.0257739i −0.477513 0.878625i \(-0.658462\pi\)
0.522155 + 0.852851i \(0.325128\pi\)
\(522\) 0 0
\(523\) 193.762 + 111.869i 0.370482 + 0.213898i 0.673669 0.739033i \(-0.264717\pi\)
−0.303187 + 0.952931i \(0.598051\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −483.626 + 837.664i −0.917696 + 1.58950i
\(528\) 0 0
\(529\) 242.721 + 420.406i 0.458831 + 0.794718i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −69.3272 −0.130070
\(534\) 0 0
\(535\) 81.8458 47.2537i 0.152983 0.0883246i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −493.095 + 854.065i −0.911451 + 1.57868i −0.0994342 + 0.995044i \(0.531703\pi\)
−0.812016 + 0.583635i \(0.801630\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 20.5205i 0.0376523i
\(546\) 0 0
\(547\) 735.369 1.34437 0.672183 0.740385i \(-0.265357\pi\)
0.672183 + 0.740385i \(0.265357\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −57.3357 33.1028i −0.104058 0.0600776i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 209.284 + 362.491i 0.375735 + 0.650791i 0.990437 0.137968i \(-0.0440571\pi\)
−0.614702 + 0.788759i \(0.710724\pi\)
\(558\) 0 0
\(559\) 73.2111i 0.130968i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 646.694 373.369i 1.14866 0.663177i 0.200097 0.979776i \(-0.435874\pi\)
0.948560 + 0.316599i \(0.102541\pi\)
\(564\) 0 0
\(565\) −73.0549 42.1782i −0.129301 0.0746518i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −144.355 + 250.030i −0.253699 + 0.439420i −0.964541 0.263932i \(-0.914981\pi\)
0.710842 + 0.703351i \(0.248314\pi\)
\(570\) 0 0
\(571\) 458.104 + 793.459i 0.802283 + 1.38960i 0.918110 + 0.396326i \(0.129715\pi\)
−0.115827 + 0.993269i \(0.536952\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 163.470 0.284296
\(576\) 0 0
\(577\) −859.156 + 496.034i −1.48901 + 0.859678i −0.999921 0.0125584i \(-0.996002\pi\)
−0.489085 + 0.872236i \(0.662669\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −236.677 + 409.937i −0.405964 + 0.703151i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 154.965i 0.263996i 0.991250 + 0.131998i \(0.0421392\pi\)
−0.991250 + 0.131998i \(0.957861\pi\)
\(588\) 0 0
\(589\) −447.577 −0.759893
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 869.636 + 502.084i 1.46650 + 0.846685i 0.999298 0.0374640i \(-0.0119280\pi\)
0.467204 + 0.884149i \(0.345261\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −398.459 690.150i −0.665206 1.15217i −0.979229 0.202756i \(-0.935010\pi\)
0.314023 0.949415i \(-0.398323\pi\)
\(600\) 0 0
\(601\) 467.002i 0.777041i 0.921440 + 0.388521i \(0.127014\pi\)
−0.921440 + 0.388521i \(0.872986\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 36.5422 21.0977i 0.0604004 0.0348722i
\(606\) 0 0
\(607\) 788.270 + 455.108i 1.29863 + 0.749766i 0.980168 0.198169i \(-0.0634993\pi\)
0.318465 + 0.947935i \(0.396833\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −13.6610 + 23.6616i −0.0223585 + 0.0387260i
\(612\) 0 0
\(613\) −387.900 671.863i −0.632790 1.09602i −0.986979 0.160850i \(-0.948576\pi\)
0.354189 0.935174i \(-0.384757\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −974.803 −1.57991 −0.789954 0.613166i \(-0.789896\pi\)
−0.789954 + 0.613166i \(0.789896\pi\)
\(618\) 0 0
\(619\) 172.564 99.6300i 0.278779 0.160953i −0.354092 0.935211i \(-0.615210\pi\)
0.632870 + 0.774258i \(0.281877\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −303.863 + 526.306i −0.486181 + 0.842089i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 1386.17i 2.20377i
\(630\) 0 0
\(631\) 751.062 1.19027 0.595136 0.803625i \(-0.297098\pi\)
0.595136 + 0.803625i \(0.297098\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 12.9223 + 7.46068i 0.0203500 + 0.0117491i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −569.138 985.776i −0.887891 1.53787i −0.842364 0.538910i \(-0.818836\pi\)
−0.0455278 0.998963i \(-0.514497\pi\)
\(642\) 0 0
\(643\) 647.823i 1.00750i −0.863849 0.503751i \(-0.831953\pi\)
0.863849 0.503751i \(-0.168047\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 764.651 441.471i 1.18184 0.682336i 0.225401 0.974266i \(-0.427631\pi\)
0.956440 + 0.291930i \(0.0942974\pi\)
\(648\) 0 0
\(649\) −462.071 266.777i −0.711974 0.411058i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 272.558 472.085i 0.417394 0.722948i −0.578282 0.815837i \(-0.696277\pi\)
0.995676 + 0.0928890i \(0.0296102\pi\)
\(654\) 0 0
\(655\) −11.1777 19.3603i −0.0170651 0.0295577i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −698.290 −1.05962 −0.529810 0.848116i \(-0.677737\pi\)
−0.529810 + 0.848116i \(0.677737\pi\)
\(660\) 0 0
\(661\) 495.128 285.863i 0.749060 0.432470i −0.0762943 0.997085i \(-0.524309\pi\)
0.825354 + 0.564616i \(0.190976\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −20.4679 + 35.4514i −0.0306865 + 0.0531505i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 28.7690i 0.0428748i
\(672\) 0 0
\(673\) 221.015 0.328403 0.164202 0.986427i \(-0.447495\pi\)
0.164202 + 0.986427i \(0.447495\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 407.608 + 235.333i 0.602080 + 0.347611i 0.769860 0.638213i \(-0.220326\pi\)
−0.167779 + 0.985825i \(0.553660\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 107.980 + 187.027i 0.158097 + 0.273831i 0.934182 0.356796i \(-0.116131\pi\)
−0.776086 + 0.630628i \(0.782798\pi\)
\(684\) 0 0
\(685\) 21.8535i 0.0319029i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 100.999 58.3119i 0.146588 0.0846326i
\(690\) 0 0
\(691\) −69.3808 40.0570i −0.100406 0.0579697i 0.448956 0.893554i \(-0.351796\pi\)
−0.549362 + 0.835584i \(0.685129\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 33.1836 57.4757i 0.0477462 0.0826988i
\(696\) 0 0
\(697\) 563.126 + 975.363i 0.807929 + 1.39937i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 821.973 1.17257 0.586286 0.810104i \(-0.300589\pi\)
0.586286 + 0.810104i \(0.300589\pi\)
\(702\) 0 0
\(703\) −555.489 + 320.712i −0.790169 + 0.456204i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −287.728 + 498.359i −0.405822 + 0.702904i −0.994417 0.105524i \(-0.966348\pi\)
0.588595 + 0.808428i \(0.299681\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 276.743i 0.388139i
\(714\) 0 0
\(715\) 3.93334 0.00550117
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −1051.29 606.964i −1.46216 0.844178i −0.463048 0.886333i \(-0.653244\pi\)
−0.999111 + 0.0421556i \(0.986577\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −76.8158 133.049i −0.105953 0.183516i
\(726\) 0 0
\(727\) 379.498i 0.522005i −0.965338 0.261003i \(-0.915947\pi\)
0.965338 0.261003i \(-0.0840532\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 1030.01 594.674i 1.40904 0.813508i
\(732\) 0 0
\(733\) 1082.93 + 625.230i 1.47739 + 0.852973i 0.999674 0.0255391i \(-0.00813022\pi\)
0.477719 + 0.878512i \(0.341464\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −6.35963 + 11.0152i −0.00862908 + 0.0149460i
\(738\) 0 0
\(739\) −250.617 434.081i −0.339130 0.587390i 0.645139 0.764065i \(-0.276799\pi\)
−0.984269 + 0.176675i \(0.943466\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −444.584 −0.598363 −0.299181 0.954196i \(-0.596714\pi\)
−0.299181 + 0.954196i \(0.596714\pi\)
\(744\) 0 0
\(745\) 99.5824 57.4939i 0.133668 0.0771730i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −119.667 + 207.270i −0.159344 + 0.275992i −0.934632 0.355616i \(-0.884271\pi\)
0.775288 + 0.631608i \(0.217605\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 90.4347i 0.119781i
\(756\) 0 0
\(757\) 249.486 0.329572 0.164786 0.986329i \(-0.447307\pi\)
0.164786 + 0.986329i \(0.447307\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 1191.37 + 687.840i 1.56554 + 0.903864i 0.996679 + 0.0814280i \(0.0259481\pi\)
0.568858 + 0.822436i \(0.307385\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 65.7277 + 113.844i 0.0856945 + 0.148427i
\(768\) 0 0
\(769\) 528.594i 0.687379i 0.939083 + 0.343689i \(0.111677\pi\)
−0.939083 + 0.343689i \(0.888323\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −53.6879 + 30.9967i −0.0694539 + 0.0400992i −0.534325 0.845279i \(-0.679434\pi\)
0.464871 + 0.885378i \(0.346101\pi\)
\(774\) 0 0
\(775\) −899.465 519.307i −1.16060 0.670073i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −260.576 + 451.331i −0.334501 + 0.579372i
\(780\) 0 0
\(781\) 232.226 + 402.228i 0.297345 + 0.515017i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −103.457 −0.131792
\(786\) 0 0
\(787\) −141.654 + 81.7839i −0.179992 + 0.103919i −0.587289 0.809377i \(-0.699805\pi\)
0.407297 + 0.913296i \(0.366471\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −3.54401 + 6.13841i −0.00446912 + 0.00774074i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 533.394i 0.669253i 0.942351 + 0.334626i \(0.108610\pi\)
−0.942351 + 0.334626i \(0.891390\pi\)
\(798\) 0 0
\(799\) 443.860 0.555519
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −69.4194 40.0793i −0.0864500 0.0499119i
\(804\) 0