Properties

Label 1764.3.z.m.325.2
Level $1764$
Weight $3$
Character 1764.325
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 325.2
Root \(-0.662827 - 0.382683i\) of defining polynomial
Character \(\chi\) \(=\) 1764.325
Dual form 1764.3.z.m.901.2

$q$-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 \(\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times\).

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

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 \(-0.681972\pi\)
0.457796 + 0.889057i \(0.348639\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.88158 4.99104i 0.261962 0.453731i −0.704801 0.709405i \(-0.748964\pi\)
0.966763 + 0.255673i \(0.0822972\pi\)
\(12\) 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 \(-0.0959876\pi\)
0.220225 + 0.975449i \(0.429321\pi\)
\(18\) 0 0
\(19\) 9.24384 5.33693i 0.486518 0.280891i −0.236611 0.971605i \(-0.576037\pi\)
0.723129 + 0.690713i \(0.242703\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −3.29990 5.71559i −0.143474 0.248504i 0.785329 0.619079i \(-0.212494\pi\)
−0.928803 + 0.370575i \(0.879161\pi\)
\(24\) 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 \(-0.569761\pi\)
−0.954016 + 0.299757i \(0.903094\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 30.0464 + 52.0419i 0.812066 + 1.40654i 0.911416 + 0.411486i \(0.134990\pi\)
−0.0993502 + 0.995053i \(0.531676\pi\)
\(38\) 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 \(-0.601044\pi\)
0.666690 + 0.745335i \(0.267710\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 41.0672 71.1306i 0.774854 1.34209i −0.160023 0.987113i \(-0.551157\pi\)
0.934877 0.354973i \(-0.115510\pi\)
\(54\) 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 \(-0.0462138\pi\)
0.369447 + 0.929252i \(0.379547\pi\)
\(60\) 0 0
\(61\) 4.32309 2.49594i 0.0708703 0.0409170i −0.464146 0.885759i \(-0.653639\pi\)
0.535016 + 0.844842i \(0.320305\pi\)
\(62\) 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 −0.857673 0.514196i \(-0.828091\pi\)
0.874143 + 0.485669i \(0.161424\pi\)
\(68\) 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 \(-0.302963\pi\)
−0.415223 + 0.909719i \(0.636297\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 32.4442 + 56.1951i 0.410687 + 0.711330i 0.994965 0.100223i \(-0.0319557\pi\)
−0.584278 + 0.811553i \(0.698622\pi\)
\(80\) 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 \(-0.467464\pi\)
0.912524 + 0.409024i \(0.134131\pi\)
\(90\) 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 \(-0.216475\pi\)
−0.155840 + 0.987782i \(0.549809\pi\)
\(102\) 0 0
\(103\) 92.6323 53.4813i 0.899343 0.519236i 0.0223558 0.999750i \(-0.492883\pi\)
0.876987 + 0.480514i \(0.159550\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 98.3097 + 170.277i 0.918782 + 1.59138i 0.801268 + 0.598306i \(0.204159\pi\)
0.117515 + 0.993071i \(0.462507\pi\)
\(108\) 0 0
\(109\) −21.3461 + 36.9726i −0.195836 + 0.339198i −0.947174 0.320719i \(-0.896075\pi\)
0.751338 + 0.659917i \(0.229409\pi\)
\(110\) 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 \(-0.609860\pi\)
0.645793 + 0.763512i \(0.276527\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 22.7327 39.3742i 0.165932 0.287403i −0.771054 0.636770i \(-0.780270\pi\)
0.936986 + 0.349367i \(0.113603\pi\)
\(138\) 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 0.917780 + 0.397089i \(0.129980\pi\)
−0.115000 + 0.993365i \(0.536687\pi\)
\(150\) 0 0
\(151\) 94.0732 162.940i 0.623001 1.07907i −0.365922 0.930645i \(-0.619246\pi\)
0.988924 0.148425i \(-0.0474202\pi\)
\(152\) 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 \(-0.0929284\pi\)
0.229589 + 0.973288i \(0.426262\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −74.9053 129.740i −0.459542 0.795949i 0.539395 0.842053i \(-0.318653\pi\)
−0.998937 + 0.0461035i \(0.985320\pi\)
\(164\) 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 \(-0.408064\pi\)
0.972567 + 0.232621i \(0.0747303\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −58.3967 + 101.146i −0.326239 + 0.565062i −0.981762 0.190113i \(-0.939115\pi\)
0.655524 + 0.755175i \(0.272448\pi\)
\(180\) 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 0.834037 + 0.551709i \(0.186024\pi\)
0.0607755 + 0.998151i \(0.480643\pi\)
\(192\) 0 0
\(193\) 112.275 194.467i 0.581738 1.00760i −0.413535 0.910488i \(-0.635706\pi\)
0.995273 0.0971119i \(-0.0309605\pi\)
\(194\) 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 \(-0.121769\pi\)
0.140582 + 0.990069i \(0.455103\pi\)
\(200\) 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 \(-0.497781\pi\)
−0.862518 + 0.506026i \(0.831114\pi\)
\(228\) 0 0
\(229\) −317.592 + 183.362i −1.38687 + 0.800708i −0.992961 0.118443i \(-0.962210\pi\)
−0.393906 + 0.919151i \(0.628876\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 34.8370 + 60.3395i 0.149515 + 0.258968i 0.931048 0.364896i \(-0.118895\pi\)
−0.781533 + 0.623864i \(0.785562\pi\)
\(234\) 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 \(-0.223247\pi\)
−0.176817 + 0.984244i \(0.556580\pi\)
\(242\) 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 \(-0.565563\pi\)
0.745461 + 0.666550i \(0.232230\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −53.2577 + 92.2451i −0.202501 + 0.350742i −0.949334 0.314270i \(-0.898240\pi\)
0.746833 + 0.665012i \(0.231574\pi\)
\(264\) 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 \(-0.240028\pi\)
−0.228437 + 0.973559i \(0.573361\pi\)
\(270\) 0 0
\(271\) 446.938 258.040i 1.64922 0.952176i 0.671834 0.740702i \(-0.265507\pi\)
0.977383 0.211475i \(-0.0678266\pi\)
\(272\) 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 −0.988187 0.153251i \(-0.951026\pi\)
0.626813 + 0.779170i \(0.284359\pi\)
\(278\) 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 \(-0.485706\pi\)
−0.842706 + 0.538374i \(0.819039\pi\)
\(284\) 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 \(-0.291901\pi\)
−0.383362 + 0.923598i \(0.625234\pi\)
\(312\) 0 0
\(313\) −133.661 + 77.1695i −0.427033 + 0.246548i −0.698082 0.716018i \(-0.745963\pi\)
0.271049 + 0.962566i \(0.412630\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −94.2726 163.285i −0.297390 0.515094i 0.678148 0.734925i \(-0.262783\pi\)
−0.975538 + 0.219831i \(0.929449\pi\)
\(318\) 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 0.733601 0.679580i \(-0.237838\pi\)
−0.955334 + 0.295527i \(0.904505\pi\)
\(332\) 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 −0.493213 0.869908i \(-0.664178\pi\)
0.999969 0.00781897i \(-0.00248888\pi\)
\(348\) 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 \(-0.0694281\pi\)
0.300755 + 0.953702i \(0.402761\pi\)
\(354\) 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 −0.962477 0.271364i \(-0.912525\pi\)
0.246231 0.969211i \(-0.420808\pi\)
\(360\) 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 \(-0.526024\pi\)
−0.903965 + 0.427606i \(0.859357\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 70.1568 + 121.515i 0.188088 + 0.325778i 0.944613 0.328187i \(-0.106438\pi\)
−0.756525 + 0.653965i \(0.773104\pi\)
\(374\) 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 \(-0.717461\pi\)
0.356036 + 0.934472i \(0.384128\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 200.636 347.511i 0.515773 0.893345i −0.484060 0.875035i \(-0.660838\pi\)
0.999832 0.0183096i \(-0.00582845\pi\)
\(390\) 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 \(-0.642500\pi\)
0.564248 + 0.825605i \(0.309166\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −187.154 324.160i −0.466718 0.808380i 0.532559 0.846393i \(-0.321230\pi\)
−0.999277 + 0.0380132i \(0.987897\pi\)
\(402\) 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 \(-0.603933\pi\)
−0.980642 + 0.195810i \(0.937267\pi\)
\(410\) 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 −0.980788 0.195076i \(-0.937505\pi\)
0.659335 + 0.751849i \(0.270838\pi\)
\(432\) 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 \(-0.803466\pi\)
0.0936932 + 0.995601i \(0.470133\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −286.696 496.572i −0.647169 1.12093i −0.983796 0.179292i \(-0.942619\pi\)
0.336626 0.941638i \(-0.390714\pi\)
\(444\) 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 0.990388 0.138316i \(-0.0441688\pi\)
−0.614979 + 0.788544i \(0.710836\pi\)
\(458\) 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 \(-0.634663\pi\)
0.584402 + 0.811464i \(0.301329\pi\)
\(468\) 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 \(-0.453738\pi\)
−0.784483 + 0.620151i \(0.787071\pi\)
\(480\) 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 −0.308833 + 0.951116i \(0.599939\pi\)
−0.669274 + 0.743016i \(0.733395\pi\)
\(488\) 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 0.877838 0.478957i \(-0.158985\pi\)
−0.853708 + 0.520752i \(0.825652\pi\)
\(500\) 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 \(-0.742381\pi\)
0.281862 + 0.959455i \(0.409048\pi\)
\(510\) 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 \(-0.341538\pi\)
−0.522155 + 0.852851i \(0.674872\pi\)
\(522\) 0 0
\(523\) −193.762 + 111.869i −0.370482 + 0.213898i −0.673669 0.739033i \(-0.735283\pi\)
0.303187 + 0.952931i \(0.401949\pi\)
\(524\) 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 −0.812016 0.583635i \(-0.801630\pi\)
−0.0994342 0.995044i \(-0.531703\pi\)
\(542\) 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 −0.614702 0.788759i \(-0.710724\pi\)
0.990437 + 0.137968i \(0.0440571\pi\)
\(558\) 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 \(-0.564126\pi\)
−0.948560 + 0.316599i \(0.897459\pi\)
\(564\) 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 0.710842 0.703351i \(-0.248314\pi\)
−0.964541 + 0.263932i \(0.914981\pi\)
\(570\) 0 0
\(571\) 458.104 793.459i 0.802283 1.38960i −0.115827 0.993269i \(-0.536952\pi\)
0.918110 0.396326i \(-0.129715\pi\)
\(572\) 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 \(-0.00399756\pi\)
0.489085 + 0.872236i \(0.337331\pi\)
\(578\) 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 \(-0.988072\pi\)
−0.467204 + 0.884149i \(0.654739\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −398.459 + 690.150i −0.665206 + 1.15217i 0.314023 + 0.949415i \(0.398323\pi\)
−0.979229 + 0.202756i \(0.935010\pi\)
\(600\) 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 \(-0.936501\pi\)
−0.318465 + 0.947935i \(0.603167\pi\)
\(608\) 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 0.354189 + 0.935174i \(0.384757\pi\)
−0.986979 + 0.160850i \(0.948576\pi\)
\(614\) 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 \(-0.384790\pi\)
−0.632870 + 0.774258i \(0.718123\pi\)
\(620\) 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 −0.0455278 + 0.998963i \(0.514497\pi\)
−0.842364 + 0.538910i \(0.818836\pi\)
\(642\) 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 \(-0.572369\pi\)
−0.956440 + 0.291930i \(0.905703\pi\)
\(648\) 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 0.995676 0.0928890i \(-0.0296102\pi\)
−0.578282 + 0.815837i \(0.696277\pi\)
\(654\) 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 \(-0.475691\pi\)
−0.825354 + 0.564616i \(0.809024\pi\)
\(662\) 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 \(-0.779674\pi\)
0.167779 + 0.985825i \(0.446340\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 107.980 187.027i 0.158097 0.273831i −0.776086 0.630628i \(-0.782798\pi\)
0.934182 + 0.356796i \(0.116131\pi\)
\(684\) 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 \(-0.648204\pi\)
0.549362 + 0.835584i \(0.314871\pi\)
\(692\) 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 0.588595 0.808428i \(-0.299681\pi\)
−0.994417 + 0.105524i \(0.966348\pi\)
\(710\) 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 \(-0.346756\pi\)
0.999111 + 0.0421556i \(0.0134225\pi\)
\(720\) 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 \(-0.991870\pi\)
−0.477719 + 0.878512i \(0.658536\pi\)
\(734\) 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 −0.984269 0.176675i \(-0.943466\pi\)
0.645139 + 0.764065i \(0.276799\pi\)
\(740\) 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 0.775288 0.631608i \(-0.217605\pi\)
−0.934632 + 0.355616i \(0.884271\pi\)
\(752\) 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 −0.568858 + 0.822436i \(0.692615\pi\)
−0.996679 + 0.0814280i \(0.974052\pi\)
\(762\) 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 \(-0.320566\pi\)
−0.464871 + 0.885378i \(0.653899\pi\)
\(774\) 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 \(-0.300195\pi\)
−0.407297 + 0.913296i \(0.633529\pi\)
\(788\) 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