Properties

Label 1764.2.t.c.521.3
Level $1764$
Weight $2$
Character 1764.521
Analytic conductor $14.086$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(14.0856109166\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{48})\)
Defining polynomial: \(x^{16} - x^{8} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 521.3
Root \(-0.793353 - 0.608761i\) of defining polynomial
Character \(\chi\) \(=\) 1764.521
Dual form 1764.2.t.c.1097.3

$q$-expansion

\(f(q)\) \(=\) \(q+(-0.382683 - 0.662827i) q^{5} +O(q^{10})\) \(q+(-0.382683 - 0.662827i) q^{5} +(-1.73205 - 1.00000i) q^{11} -0.317025i q^{13} +(-2.77164 + 4.80062i) q^{17} +(3.20041 - 1.84776i) q^{19} +(2.74666 - 1.58579i) q^{23} +(2.20711 - 3.82282i) q^{25} -6.82843i q^{29} +(-5.85172 - 3.37849i) q^{31} +(0.121320 + 0.210133i) q^{37} -2.74444 q^{41} +6.82843 q^{43} +(-5.99162 - 10.3778i) q^{47} +(-10.6024 - 6.12132i) q^{53} +1.53073i q^{55} +(-6.62567 + 11.4760i) q^{59} +(3.08669 - 1.78210i) q^{61} +(-0.210133 + 0.121320i) q^{65} +(2.24264 - 3.88437i) q^{67} -9.31371i q^{71} +(-10.2641 - 5.92596i) q^{73} +(5.65685 + 9.79796i) q^{79} +4.32957 q^{83} +4.24264 q^{85} +(-0.831025 - 1.43938i) q^{89} +(-2.44949 - 1.41421i) q^{95} -11.8519i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + O(q^{10}) \) \( 16 q + 24 q^{25} - 32 q^{37} + 64 q^{43} - 32 q^{67} + O(q^{100}) \)

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
\(5\) −0.382683 0.662827i −0.171141 0.296425i 0.767678 0.640836i \(-0.221412\pi\)
−0.938819 + 0.344411i \(0.888079\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.73205 1.00000i −0.522233 0.301511i 0.215615 0.976478i \(-0.430824\pi\)
−0.737848 + 0.674967i \(0.764158\pi\)
\(12\) 0 0
\(13\) 0.317025i 0.0879270i −0.999033 0.0439635i \(-0.986001\pi\)
0.999033 0.0439635i \(-0.0139985\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −2.77164 + 4.80062i −0.672221 + 1.16432i 0.305052 + 0.952336i \(0.401326\pi\)
−0.977273 + 0.211985i \(0.932007\pi\)
\(18\) 0 0
\(19\) 3.20041 1.84776i 0.734225 0.423905i −0.0857408 0.996317i \(-0.527326\pi\)
0.819966 + 0.572412i \(0.193992\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.74666 1.58579i 0.572719 0.330659i −0.185516 0.982641i \(-0.559396\pi\)
0.758234 + 0.651982i \(0.226062\pi\)
\(24\) 0 0
\(25\) 2.20711 3.82282i 0.441421 0.764564i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 6.82843i 1.26801i −0.773330 0.634004i \(-0.781410\pi\)
0.773330 0.634004i \(-0.218590\pi\)
\(30\) 0 0
\(31\) −5.85172 3.37849i −1.05100 0.606795i −0.128071 0.991765i \(-0.540879\pi\)
−0.922929 + 0.384970i \(0.874212\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0.121320 + 0.210133i 0.0199449 + 0.0345457i 0.875826 0.482628i \(-0.160318\pi\)
−0.855881 + 0.517173i \(0.826984\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −2.74444 −0.428610 −0.214305 0.976767i \(-0.568749\pi\)
−0.214305 + 0.976767i \(0.568749\pi\)
\(42\) 0 0
\(43\) 6.82843 1.04133 0.520663 0.853762i \(-0.325685\pi\)
0.520663 + 0.853762i \(0.325685\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −5.99162 10.3778i −0.873967 1.51376i −0.857859 0.513886i \(-0.828206\pi\)
−0.0161088 0.999870i \(-0.505128\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −10.6024 6.12132i −1.45636 0.840828i −0.457527 0.889196i \(-0.651265\pi\)
−0.998830 + 0.0483676i \(0.984598\pi\)
\(54\) 0 0
\(55\) 1.53073i 0.206404i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −6.62567 + 11.4760i −0.862589 + 1.49405i 0.00683301 + 0.999977i \(0.497825\pi\)
−0.869422 + 0.494071i \(0.835508\pi\)
\(60\) 0 0
\(61\) 3.08669 1.78210i 0.395210 0.228175i −0.289205 0.957267i \(-0.593391\pi\)
0.684415 + 0.729093i \(0.260058\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −0.210133 + 0.121320i −0.0260638 + 0.0150479i
\(66\) 0 0
\(67\) 2.24264 3.88437i 0.273982 0.474551i −0.695896 0.718143i \(-0.744992\pi\)
0.969878 + 0.243592i \(0.0783257\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 9.31371i 1.10533i −0.833402 0.552667i \(-0.813610\pi\)
0.833402 0.552667i \(-0.186390\pi\)
\(72\) 0 0
\(73\) −10.2641 5.92596i −1.20132 0.693581i −0.240470 0.970657i \(-0.577301\pi\)
−0.960848 + 0.277075i \(0.910635\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 5.65685 + 9.79796i 0.636446 + 1.10236i 0.986207 + 0.165518i \(0.0529295\pi\)
−0.349761 + 0.936839i \(0.613737\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 4.32957 0.475232 0.237616 0.971359i \(-0.423634\pi\)
0.237616 + 0.971359i \(0.423634\pi\)
\(84\) 0 0
\(85\) 4.24264 0.460179
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −0.831025 1.43938i −0.0880885 0.152574i 0.818615 0.574343i \(-0.194742\pi\)
−0.906703 + 0.421769i \(0.861409\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −2.44949 1.41421i −0.251312 0.145095i
\(96\) 0 0
\(97\) 11.8519i 1.20338i −0.798730 0.601690i \(-0.794494\pi\)
0.798730 0.601690i \(-0.205506\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −5.92596 + 10.2641i −0.589655 + 1.02131i 0.404622 + 0.914484i \(0.367403\pi\)
−0.994277 + 0.106829i \(0.965930\pi\)
\(102\) 0 0
\(103\) 7.72648 4.46088i 0.761313 0.439544i −0.0684542 0.997654i \(-0.521807\pi\)
0.829767 + 0.558110i \(0.188473\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 6.21076 3.58579i 0.600417 0.346651i −0.168788 0.985652i \(-0.553985\pi\)
0.769206 + 0.639001i \(0.220652\pi\)
\(108\) 0 0
\(109\) 5.53553 9.58783i 0.530208 0.918347i −0.469171 0.883107i \(-0.655447\pi\)
0.999379 0.0352398i \(-0.0112195\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 10.5858i 0.995827i −0.867227 0.497914i \(-0.834100\pi\)
0.867227 0.497914i \(-0.165900\pi\)
\(114\) 0 0
\(115\) −2.10220 1.21371i −0.196032 0.113179i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −3.50000 6.06218i −0.318182 0.551107i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −7.20533 −0.644464
\(126\) 0 0
\(127\) 1.17157 0.103960 0.0519801 0.998648i \(-0.483447\pi\)
0.0519801 + 0.998648i \(0.483447\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −8.47343 14.6764i −0.740327 1.28228i −0.952346 0.305018i \(-0.901337\pi\)
0.212020 0.977265i \(-0.431996\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −9.37769 5.41421i −0.801190 0.462567i 0.0426968 0.999088i \(-0.486405\pi\)
−0.843887 + 0.536521i \(0.819738\pi\)
\(138\) 0 0
\(139\) 15.6788i 1.32985i 0.746908 + 0.664927i \(0.231538\pi\)
−0.746908 + 0.664927i \(0.768462\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −0.317025 + 0.549104i −0.0265110 + 0.0459184i
\(144\) 0 0
\(145\) −4.52607 + 2.61313i −0.375869 + 0.217008i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −6.12372 + 3.53553i −0.501675 + 0.289642i −0.729405 0.684082i \(-0.760203\pi\)
0.227730 + 0.973724i \(0.426870\pi\)
\(150\) 0 0
\(151\) −5.07107 + 8.78335i −0.412678 + 0.714779i −0.995182 0.0980492i \(-0.968740\pi\)
0.582504 + 0.812828i \(0.302073\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 5.17157i 0.415391i
\(156\) 0 0
\(157\) −5.34972 3.08866i −0.426954 0.246502i 0.271094 0.962553i \(-0.412615\pi\)
−0.698048 + 0.716051i \(0.745948\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −1.41421 2.44949i −0.110770 0.191859i 0.805311 0.592852i \(-0.201998\pi\)
−0.916081 + 0.400994i \(0.868665\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 2.42742 0.187839 0.0939196 0.995580i \(-0.470060\pi\)
0.0939196 + 0.995580i \(0.470060\pi\)
\(168\) 0 0
\(169\) 12.8995 0.992269
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 9.39731 + 16.2766i 0.714464 + 1.23749i 0.963166 + 0.268908i \(0.0866628\pi\)
−0.248702 + 0.968580i \(0.580004\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 8.06591 + 4.65685i 0.602874 + 0.348070i 0.770171 0.637837i \(-0.220171\pi\)
−0.167297 + 0.985907i \(0.553504\pi\)
\(180\) 0 0
\(181\) 1.66205i 0.123539i 0.998090 + 0.0617696i \(0.0196744\pi\)
−0.998090 + 0.0617696i \(0.980326\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0.0928546 0.160829i 0.00682680 0.0118244i
\(186\) 0 0
\(187\) 9.60124 5.54328i 0.702112 0.405365i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 14.9941 8.65685i 1.08494 0.626388i 0.152712 0.988271i \(-0.451199\pi\)
0.932224 + 0.361883i \(0.117866\pi\)
\(192\) 0 0
\(193\) 4.82843 8.36308i 0.347558 0.601988i −0.638257 0.769823i \(-0.720344\pi\)
0.985815 + 0.167835i \(0.0536777\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 3.07107i 0.218805i 0.993998 + 0.109402i \(0.0348937\pi\)
−0.993998 + 0.109402i \(0.965106\pi\)
\(198\) 0 0
\(199\) 6.40083 + 3.69552i 0.453742 + 0.261968i 0.709409 0.704797i \(-0.248962\pi\)
−0.255667 + 0.966765i \(0.582295\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 1.05025 + 1.81909i 0.0733528 + 0.127051i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −7.39104 −0.511249
\(210\) 0 0
\(211\) −18.6274 −1.28236 −0.641182 0.767389i \(-0.721556\pi\)
−0.641182 + 0.767389i \(0.721556\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −2.61313 4.52607i −0.178214 0.308675i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 1.52192 + 0.878680i 0.102375 + 0.0591064i
\(222\) 0 0
\(223\) 21.8017i 1.45995i −0.683474 0.729975i \(-0.739532\pi\)
0.683474 0.729975i \(-0.260468\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 3.37849 5.85172i 0.224238 0.388392i −0.731852 0.681463i \(-0.761344\pi\)
0.956091 + 0.293071i \(0.0946772\pi\)
\(228\) 0 0
\(229\) −14.0136 + 8.09075i −0.926044 + 0.534651i −0.885558 0.464529i \(-0.846224\pi\)
−0.0404854 + 0.999180i \(0.512890\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −13.1390 + 7.58579i −0.860762 + 0.496961i −0.864268 0.503032i \(-0.832218\pi\)
0.00350513 + 0.999994i \(0.498884\pi\)
\(234\) 0 0
\(235\) −4.58579 + 7.94282i −0.299144 + 0.518132i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 8.82843i 0.571063i 0.958369 + 0.285532i \(0.0921702\pi\)
−0.958369 + 0.285532i \(0.907830\pi\)
\(240\) 0 0
\(241\) 22.1283 + 12.7758i 1.42541 + 0.822962i 0.996754 0.0805055i \(-0.0256535\pi\)
0.428657 + 0.903467i \(0.358987\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −0.585786 1.01461i −0.0372727 0.0645582i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 25.4972 1.60937 0.804685 0.593702i \(-0.202334\pi\)
0.804685 + 0.593702i \(0.202334\pi\)
\(252\) 0 0
\(253\) −6.34315 −0.398790
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 10.2940 + 17.8297i 0.642122 + 1.11219i 0.984958 + 0.172792i \(0.0552789\pi\)
−0.342837 + 0.939395i \(0.611388\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 17.0233 + 9.82843i 1.04970 + 0.606047i 0.922566 0.385838i \(-0.126088\pi\)
0.127137 + 0.991885i \(0.459421\pi\)
\(264\) 0 0
\(265\) 9.37011i 0.575601i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 8.94897 15.5001i 0.545628 0.945056i −0.452939 0.891542i \(-0.649624\pi\)
0.998567 0.0535141i \(-0.0170422\pi\)
\(270\) 0 0
\(271\) 4.75351 2.74444i 0.288755 0.166713i −0.348625 0.937262i \(-0.613351\pi\)
0.637380 + 0.770549i \(0.280018\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −7.64564 + 4.41421i −0.461050 + 0.266187i
\(276\) 0 0
\(277\) −6.48528 + 11.2328i −0.389663 + 0.674916i −0.992404 0.123021i \(-0.960742\pi\)
0.602741 + 0.797937i \(0.294075\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 10.4853i 0.625499i 0.949836 + 0.312750i \(0.101250\pi\)
−0.949836 + 0.312750i \(0.898750\pi\)
\(282\) 0 0
\(283\) −3.52207 2.03347i −0.209365 0.120877i 0.391651 0.920114i \(-0.371904\pi\)
−0.601016 + 0.799237i \(0.705237\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −6.86396 11.8887i −0.403762 0.699337i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −16.4441 −0.960676 −0.480338 0.877083i \(-0.659486\pi\)
−0.480338 + 0.877083i \(0.659486\pi\)
\(294\) 0 0
\(295\) 10.1421 0.590498
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −0.502734 0.870762i −0.0290739 0.0503574i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −2.36245 1.36396i −0.135273 0.0781002i
\(306\) 0 0
\(307\) 27.1367i 1.54877i 0.632712 + 0.774387i \(0.281942\pi\)
−0.632712 + 0.774387i \(0.718058\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 15.3617 26.6073i 0.871084 1.50876i 0.0102070 0.999948i \(-0.496751\pi\)
0.860877 0.508814i \(-0.169916\pi\)
\(312\) 0 0
\(313\) −25.0071 + 14.4379i −1.41348 + 0.816076i −0.995715 0.0924774i \(-0.970521\pi\)
−0.417770 + 0.908553i \(0.637188\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −13.6463 + 7.87868i −0.766451 + 0.442511i −0.831607 0.555364i \(-0.812579\pi\)
0.0651561 + 0.997875i \(0.479245\pi\)
\(318\) 0 0
\(319\) −6.82843 + 11.8272i −0.382319 + 0.662195i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 20.4853i 1.13983i
\(324\) 0 0
\(325\) −1.21193 0.699709i −0.0672258 0.0388129i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 13.6569 + 23.6544i 0.750649 + 1.30016i 0.947509 + 0.319730i \(0.103592\pi\)
−0.196860 + 0.980432i \(0.563074\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −3.43289 −0.187559
\(336\) 0 0
\(337\) −27.0711 −1.47466 −0.737328 0.675535i \(-0.763913\pi\)
−0.737328 + 0.675535i \(0.763913\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 6.75699 + 11.7034i 0.365911 + 0.633777i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 25.8067 + 14.8995i 1.38538 + 0.799847i 0.992790 0.119869i \(-0.0382474\pi\)
0.392586 + 0.919715i \(0.371581\pi\)
\(348\) 0 0
\(349\) 31.9372i 1.70956i −0.518993 0.854779i \(-0.673693\pi\)
0.518993 0.854779i \(-0.326307\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 10.5726 18.3122i 0.562720 0.974660i −0.434537 0.900654i \(-0.643088\pi\)
0.997258 0.0740064i \(-0.0235785\pi\)
\(354\) 0 0
\(355\) −6.17338 + 3.56420i −0.327649 + 0.189168i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 16.8493 9.72792i 0.889270 0.513420i 0.0155661 0.999879i \(-0.495045\pi\)
0.873704 + 0.486459i \(0.161712\pi\)
\(360\) 0 0
\(361\) −2.67157 + 4.62730i −0.140609 + 0.243542i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 9.07107i 0.474801i
\(366\) 0 0
\(367\) 31.2276 + 18.0292i 1.63007 + 0.941119i 0.984071 + 0.177778i \(0.0568909\pi\)
0.645996 + 0.763341i \(0.276442\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −15.3137 26.5241i −0.792914 1.37337i −0.924155 0.382017i \(-0.875230\pi\)
0.131242 0.991350i \(-0.458104\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −2.16478 −0.111492
\(378\) 0 0
\(379\) −36.2843 −1.86380 −0.931899 0.362718i \(-0.881849\pi\)
−0.931899 + 0.362718i \(0.881849\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 7.39104 + 12.8017i 0.377664 + 0.654134i 0.990722 0.135904i \(-0.0433940\pi\)
−0.613058 + 0.790038i \(0.710061\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 1.01461 + 0.585786i 0.0514429 + 0.0297006i 0.525501 0.850793i \(-0.323878\pi\)
−0.474058 + 0.880494i \(0.657211\pi\)
\(390\) 0 0
\(391\) 17.5809i 0.889105i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 4.32957 7.49903i 0.217844 0.377317i
\(396\) 0 0
\(397\) −12.1388 + 7.00835i −0.609230 + 0.351739i −0.772664 0.634815i \(-0.781076\pi\)
0.163434 + 0.986554i \(0.447743\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −4.77589 + 2.75736i −0.238496 + 0.137696i −0.614485 0.788928i \(-0.710636\pi\)
0.375989 + 0.926624i \(0.377303\pi\)
\(402\) 0 0
\(403\) −1.07107 + 1.85514i −0.0533537 + 0.0924113i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0.485281i 0.0240545i
\(408\) 0 0
\(409\) −0.984485 0.568393i −0.0486796 0.0281052i 0.475463 0.879736i \(-0.342281\pi\)
−0.524142 + 0.851631i \(0.675614\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −1.65685 2.86976i −0.0813318 0.140871i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 27.6620 1.35138 0.675688 0.737187i \(-0.263846\pi\)
0.675688 + 0.737187i \(0.263846\pi\)
\(420\) 0 0
\(421\) 15.3137 0.746344 0.373172 0.927762i \(-0.378270\pi\)
0.373172 + 0.927762i \(0.378270\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 12.2346 + 21.1910i 0.593465 + 1.02791i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 28.8505 + 16.6569i 1.38968 + 0.802332i 0.993279 0.115745i \(-0.0369255\pi\)
0.396402 + 0.918077i \(0.370259\pi\)
\(432\) 0 0
\(433\) 0.502734i 0.0241599i −0.999927 0.0120799i \(-0.996155\pi\)
0.999927 0.0120799i \(-0.00384526\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 5.86030 10.1503i 0.280336 0.485557i
\(438\) 0 0
\(439\) 12.0251 6.94269i 0.573927 0.331357i −0.184789 0.982778i \(-0.559160\pi\)
0.758716 + 0.651421i \(0.225827\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −25.3864 + 14.6569i −1.20615 + 0.696368i −0.961915 0.273349i \(-0.911869\pi\)
−0.244230 + 0.969717i \(0.578535\pi\)
\(444\) 0 0
\(445\) −0.636039 + 1.10165i −0.0301511 + 0.0522233i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 11.5563i 0.545378i 0.962102 + 0.272689i \(0.0879130\pi\)
−0.962102 + 0.272689i \(0.912087\pi\)
\(450\) 0 0
\(451\) 4.75351 + 2.74444i 0.223834 + 0.129231i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 11.0000 + 19.0526i 0.514558 + 0.891241i 0.999857 + 0.0168929i \(0.00537742\pi\)
−0.485299 + 0.874348i \(0.661289\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 5.17186 0.240877 0.120439 0.992721i \(-0.461570\pi\)
0.120439 + 0.992721i \(0.461570\pi\)
\(462\) 0 0
\(463\) −2.82843 −0.131448 −0.0657241 0.997838i \(-0.520936\pi\)
−0.0657241 + 0.997838i \(0.520936\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −3.11586 5.39683i −0.144185 0.249735i 0.784884 0.619643i \(-0.212723\pi\)
−0.929069 + 0.369908i \(0.879389\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −11.8272 6.82843i −0.543814 0.313971i
\(474\) 0 0
\(475\) 16.3128i 0.748483i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 11.6662 20.2065i 0.533043 0.923257i −0.466213 0.884673i \(-0.654382\pi\)
0.999255 0.0385845i \(-0.0122849\pi\)
\(480\) 0 0
\(481\) 0.0666175 0.0384616i 0.00303750 0.00175370i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −7.85578 + 4.53553i −0.356712 + 0.205948i
\(486\) 0 0
\(487\) −9.89949 + 17.1464i −0.448589 + 0.776979i −0.998294 0.0583797i \(-0.981407\pi\)
0.549706 + 0.835359i \(0.314740\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 34.4853i 1.55630i 0.628079 + 0.778149i \(0.283841\pi\)
−0.628079 + 0.778149i \(0.716159\pi\)
\(492\) 0 0
\(493\) 32.7807 + 18.9259i 1.47637 + 0.852381i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −13.0711 22.6398i −0.585141 1.01349i −0.994858 0.101282i \(-0.967706\pi\)
0.409716 0.912213i \(-0.365628\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −29.5641 −1.31820 −0.659100 0.752055i \(-0.729063\pi\)
−0.659100 + 0.752055i \(0.729063\pi\)
\(504\) 0 0
\(505\) 9.07107 0.403657
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 6.46716 + 11.2014i 0.286652 + 0.496495i 0.973008 0.230770i \(-0.0741244\pi\)
−0.686357 + 0.727265i \(0.740791\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −5.91359 3.41421i −0.260584 0.150448i
\(516\) 0 0
\(517\) 23.9665i 1.05404i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −1.01673 + 1.76104i −0.0445439 + 0.0771523i −0.887438 0.460928i \(-0.847517\pi\)
0.842894 + 0.538080i \(0.180850\pi\)
\(522\) 0 0
\(523\) −15.7746 + 9.10748i −0.689776 + 0.398242i −0.803528 0.595267i \(-0.797046\pi\)
0.113752 + 0.993509i \(0.463713\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 32.4377 18.7279i 1.41301 0.815801i
\(528\) 0 0
\(529\) −6.47056 + 11.2073i −0.281329 + 0.487276i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0.870058i 0.0376864i
\(534\) 0 0
\(535\) −4.75351 2.74444i −0.205512 0.118653i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −4.17157 7.22538i −0.179350 0.310643i 0.762308 0.647214i \(-0.224066\pi\)
−0.941658 + 0.336571i \(0.890733\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −8.47343 −0.362962
\(546\) 0 0
\(547\) 16.0000 0.684111 0.342055 0.939680i \(-0.388877\pi\)
0.342055 + 0.939680i \(0.388877\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −12.6173 21.8538i −0.537515 0.931003i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 14.4868 + 8.36396i 0.613826 + 0.354392i 0.774461 0.632621i \(-0.218021\pi\)
−0.160636 + 0.987014i \(0.551354\pi\)
\(558\) 0 0
\(559\) 2.16478i 0.0915606i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −11.0322 + 19.1083i −0.464950 + 0.805317i −0.999199 0.0400098i \(-0.987261\pi\)
0.534249 + 0.845327i \(0.320594\pi\)
\(564\) 0 0
\(565\) −7.01655 + 4.05101i −0.295188 + 0.170427i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −0.420266 + 0.242641i −0.0176185 + 0.0101720i −0.508783 0.860895i \(-0.669905\pi\)
0.491165 + 0.871067i \(0.336571\pi\)
\(570\) 0 0
\(571\) 1.31371 2.27541i 0.0549770 0.0952229i −0.837227 0.546855i \(-0.815825\pi\)
0.892204 + 0.451632i \(0.149158\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 14.0000i 0.583840i
\(576\) 0 0
\(577\) −17.2140 9.93850i −0.716628 0.413745i 0.0968824 0.995296i \(-0.469113\pi\)
−0.813510 + 0.581551i \(0.802446\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 12.2426 + 21.2049i 0.507038 + 0.878216i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 28.0334 1.15706 0.578531 0.815660i \(-0.303626\pi\)
0.578531 + 0.815660i \(0.303626\pi\)
\(588\) 0 0
\(589\) −24.9706 −1.02889
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 15.3889 + 26.6544i 0.631947 + 1.09457i 0.987153 + 0.159777i \(0.0510775\pi\)
−0.355206 + 0.934788i \(0.615589\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −8.06591 4.65685i −0.329564 0.190274i 0.326083 0.945341i \(-0.394271\pi\)
−0.655648 + 0.755067i \(0.727604\pi\)
\(600\) 0 0
\(601\) 23.9121i 0.975394i −0.873013 0.487697i \(-0.837837\pi\)
0.873013 0.487697i \(-0.162163\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −2.67878 + 4.63979i −0.108908 + 0.188634i
\(606\) 0 0
\(607\) 10.9269 6.30864i 0.443509 0.256060i −0.261576 0.965183i \(-0.584242\pi\)
0.705085 + 0.709123i \(0.250909\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −3.29002 + 1.89949i −0.133100 + 0.0768453i
\(612\) 0 0
\(613\) 22.6066 39.1558i 0.913072 1.58149i 0.103372 0.994643i \(-0.467037\pi\)
0.809700 0.586844i \(-0.199630\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 17.4558i 0.702746i −0.936236 0.351373i \(-0.885715\pi\)
0.936236 0.351373i \(-0.114285\pi\)
\(618\) 0 0
\(619\) −1.55310 0.896683i −0.0624244 0.0360407i 0.468463 0.883483i \(-0.344808\pi\)
−0.530887 + 0.847442i \(0.678141\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −8.27817 14.3382i −0.331127 0.573529i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −1.34502 −0.0536296
\(630\) 0 0
\(631\) −12.4853 −0.497031 −0.248516 0.968628i \(-0.579943\pi\)
−0.248516 + 0.968628i \(0.579943\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −0.448342 0.776550i −0.0177919 0.0308165i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 20.3134 + 11.7279i 0.802329 + 0.463225i 0.844285 0.535894i \(-0.180025\pi\)
−0.0419557 + 0.999119i \(0.513359\pi\)
\(642\) 0 0
\(643\) 32.5168i 1.28234i 0.767400 + 0.641169i \(0.221550\pi\)
−0.767400 + 0.641169i \(0.778450\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 3.19278 5.53006i 0.125521 0.217409i −0.796415 0.604750i \(-0.793273\pi\)
0.921937 + 0.387341i \(0.126606\pi\)
\(648\) 0 0
\(649\) 22.9520 13.2513i 0.900944 0.520161i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −5.07306 + 2.92893i −0.198524 + 0.114618i −0.595967 0.803009i \(-0.703231\pi\)
0.397443 + 0.917627i \(0.369898\pi\)
\(654\) 0 0
\(655\) −6.48528 + 11.2328i −0.253401 + 0.438903i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 2.97056i 0.115717i −0.998325 0.0578583i \(-0.981573\pi\)
0.998325 0.0578583i \(-0.0184272\pi\)
\(660\) 0 0
\(661\) −20.2536 11.6934i −0.787773 0.454821i 0.0514050 0.998678i \(-0.483630\pi\)
−0.839178 + 0.543857i \(0.816963\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −10.8284 18.7554i −0.419278 0.726211i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −7.12840 −0.275189
\(672\) 0 0
\(673\) 20.0416 0.772548 0.386274 0.922384i \(-0.373762\pi\)
0.386274 + 0.922384i \(0.373762\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −13.9895 24.2305i −0.537661 0.931255i −0.999029 0.0440470i \(-0.985975\pi\)
0.461369 0.887208i \(-0.347358\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −40.6777 23.4853i −1.55649 0.898639i −0.997589 0.0694045i \(-0.977890\pi\)
−0.558900 0.829235i \(-0.688777\pi\)
\(684\) 0 0
\(685\) 8.28772i 0.316657i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −1.94061 + 3.36124i −0.0739315 + 0.128053i
\(690\) 0 0
\(691\) 22.1754 12.8030i 0.843594 0.487049i −0.0148906 0.999889i \(-0.504740\pi\)
0.858484 + 0.512840i \(0.171407\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 10.3923 6.00000i 0.394203 0.227593i
\(696\) 0 0
\(697\) 7.60660 13.1750i 0.288121 0.499039i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 15.4558i 0.583759i −0.956455 0.291880i \(-0.905719\pi\)
0.956455 0.291880i \(-0.0942807\pi\)
\(702\) 0 0
\(703\) 0.776550 + 0.448342i 0.0292881 + 0.0169095i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 15.4350 + 26.7343i 0.579675 + 1.00403i 0.995516 + 0.0945890i \(0.0301537\pi\)
−0.415842 + 0.909437i \(0.636513\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −21.4303 −0.802570
\(714\) 0 0
\(715\) 0.485281 0.0181485
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −9.81845 17.0061i −0.366167 0.634219i 0.622796 0.782384i \(-0.285997\pi\)
−0.988963 + 0.148165i \(0.952663\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −26.1039 15.0711i −0.969473 0.559725i
\(726\) 0 0
\(727\) 53.7933i 1.99508i −0.0700903 0.997541i \(-0.522329\pi\)
0.0700903 0.997541i \(-0.477671\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −18.9259 + 32.7807i −0.700001 + 1.21244i
\(732\) 0 0
\(733\) −19.3162 + 11.1522i −0.713460 + 0.411916i −0.812341 0.583183i \(-0.801807\pi\)
0.0988808 + 0.995099i \(0.468474\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −7.76874 + 4.48528i −0.286165 + 0.165217i
\(738\) 0 0
\(739\) −1.27208 + 2.20330i −0.0467941 + 0.0810498i −0.888474 0.458927i \(-0.848234\pi\)
0.841680 + 0.539977i \(0.181567\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 43.6569i 1.60161i −0.598922 0.800807i \(-0.704404\pi\)
0.598922 0.800807i \(-0.295596\pi\)
\(744\) 0 0
\(745\) 4.68690 + 2.70598i 0.171715 + 0.0991395i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 18.1421 + 31.4231i 0.662016 + 1.14665i 0.980085 + 0.198578i \(0.0636322\pi\)
−0.318069 + 0.948067i \(0.603034\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 7.76245 0.282505
\(756\) 0 0
\(757\) 18.1005 0.657874 0.328937 0.944352i \(-0.393310\pi\)
0.328937 + 0.944352i \(0.393310\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −26.8310 46.4726i −0.972622 1.68463i −0.687570 0.726118i \(-0.741322\pi\)
−0.285052 0.958512i \(-0.592011\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 3.63818 + 2.10051i 0.131367 + 0.0758448i
\(768\) 0 0
\(769\) 14.2793i 0.514926i 0.966288 + 0.257463i \(0.0828866\pi\)
−0.966288 + 0.257463i \(0.917113\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −16.7883 + 29.0783i −0.603835 + 1.04587i 0.388400 + 0.921491i \(0.373028\pi\)
−0.992234 + 0.124381i \(0.960305\pi\)
\(774\) 0 0
\(775\) −25.8307 + 14.9134i −0.927868 + 0.535705i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −8.78335 + 5.07107i −0.314696 + 0.181690i
\(780\) 0 0
\(781\) −9.31371 + 16.1318i −0.333271 + 0.577242i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 4.72792i 0.168747i
\(786\) 0 0
\(787\) −43.7076 25.2346i −1.55801 0.899515i −0.997448 0.0714009i \(-0.977253\pi\)
−0.560559 0.828115i \(-0.689414\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −0.564971 0.978559i −0.0200627 0.0347496i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 26.3714 0.934122 0.467061 0.884225i \(-0.345313\pi\)
0.467061 + 0.884225i \(0.345313\pi\)
\(798\) 0 0
\(799\) 66.4264 2.35000
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 11.8519 + 20.5281i 0.418245 + 0.724422i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0