Properties

Label 1764.2.t.c.521.4
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 521.4
Root \(-0.130526 - 0.991445i\) of defining polynomial
Character \(\chi\) \(=\) 1764.521
Dual form 1764.2.t.c.1097.4

$q$-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 0.737848 0.674967i \(-0.235842\pi\)
−0.215615 + 0.976478i \(0.569176\pi\)
\(12\) 0 0
\(13\) 0.317025i 0.0879270i 0.999033 + 0.0439635i \(0.0139985\pi\)
−0.999033 + 0.0439635i \(0.986001\pi\)
\(14\) 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 −0.819966 0.572412i \(-0.806008\pi\)
0.0857408 + 0.996317i \(0.472674\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −2.74666 + 1.58579i −0.572719 + 0.330659i −0.758234 0.651982i \(-0.773938\pi\)
0.185516 + 0.982641i \(0.440604\pi\)
\(24\) 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 \(0.218590\pi\)
−0.773330 + 0.634004i \(0.781410\pi\)
\(30\) 0 0
\(31\) 5.85172 + 3.37849i 1.05100 + 0.606795i 0.922929 0.384970i \(-0.125788\pi\)
0.128071 + 0.991765i \(0.459121\pi\)
\(32\) 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 0.998830 0.0483676i \(-0.0154019\pi\)
0.457527 + 0.889196i \(0.348735\pi\)
\(54\) 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 −0.684415 0.729093i \(-0.739942\pi\)
0.289205 + 0.957267i \(0.406609\pi\)
\(62\) 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 \(0.186390\pi\)
−0.833402 + 0.552667i \(0.813610\pi\)
\(72\) 0 0
\(73\) 10.2641 + 5.92596i 1.20132 + 0.693581i 0.960848 0.277075i \(-0.0893652\pi\)
0.240470 + 0.970657i \(0.422699\pi\)
\(74\) 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 \(0.205506\pi\)
−0.798730 + 0.601690i \(0.794494\pi\)
\(98\) 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 −0.829767 0.558110i \(-0.811527\pi\)
0.0684542 + 0.997654i \(0.478193\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −6.21076 + 3.58579i −0.600417 + 0.346651i −0.769206 0.639001i \(-0.779348\pi\)
0.168788 + 0.985652i \(0.446015\pi\)
\(108\) 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 \(0.165900\pi\)
−0.867227 + 0.497914i \(0.834100\pi\)
\(114\) 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 0.843887 0.536521i \(-0.180262\pi\)
−0.0426968 + 0.999088i \(0.513595\pi\)
\(138\) 0 0
\(139\) 15.6788i 1.32985i −0.746908 0.664927i \(-0.768462\pi\)
0.746908 0.664927i \(-0.231538\pi\)
\(140\) 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 −0.227730 0.973724i \(-0.573130\pi\)
0.729405 + 0.684082i \(0.239797\pi\)
\(150\) 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 0.698048 0.716051i \(-0.254052\pi\)
−0.271094 + 0.962553i \(0.587385\pi\)
\(158\) 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 0.167297 0.985907i \(-0.446496\pi\)
−0.770171 + 0.637837i \(0.779829\pi\)
\(180\) 0 0
\(181\) 1.66205i 0.123539i −0.998090 0.0617696i \(-0.980326\pi\)
0.998090 0.0617696i \(-0.0196744\pi\)
\(182\) 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 −0.932224 0.361883i \(-0.882134\pi\)
−0.152712 + 0.988271i \(0.548801\pi\)
\(192\) 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 \(-0.965106\pi\)
0.993998 0.109402i \(-0.0348937\pi\)
\(198\) 0 0
\(199\) −6.40083 3.69552i −0.453742 0.261968i 0.255667 0.966765i \(-0.417705\pi\)
−0.709409 + 0.704797i \(0.751038\pi\)
\(200\) 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 \(0.260468\pi\)
−0.683474 + 0.729975i \(0.739532\pi\)
\(224\) 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 0.0404854 0.999180i \(-0.487110\pi\)
0.885558 + 0.464529i \(0.153776\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 13.1390 7.58579i 0.860762 0.496961i −0.00350513 0.999994i \(-0.501116\pi\)
0.864268 + 0.503032i \(0.167782\pi\)
\(234\) 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 \(-0.907830\pi\)
0.958369 0.285532i \(-0.0921702\pi\)
\(240\) 0 0
\(241\) −22.1283 12.7758i −1.42541 0.822962i −0.428657 0.903467i \(-0.641013\pi\)
−0.996754 + 0.0805055i \(0.974347\pi\)
\(242\) 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 −0.127137 0.991885i \(-0.540579\pi\)
−0.922566 + 0.385838i \(0.873912\pi\)
\(264\) 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 −0.637380 0.770549i \(-0.719982\pi\)
0.348625 + 0.937262i \(0.386649\pi\)
\(272\) 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 \(-0.898750\pi\)
0.949836 0.312750i \(-0.101250\pi\)
\(282\) 0 0
\(283\) 3.52207 + 2.03347i 0.209365 + 0.120877i 0.601016 0.799237i \(-0.294763\pi\)
−0.391651 + 0.920114i \(0.628096\pi\)
\(284\) 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 \(-0.718058\pi\)
0.632712 0.774387i \(-0.281942\pi\)
\(308\) 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 0.417770 0.908553i \(-0.362812\pi\)
0.995715 + 0.0924774i \(0.0294786\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 13.6463 7.87868i 0.766451 0.442511i −0.0651561 0.997875i \(-0.520755\pi\)
0.831607 + 0.555364i \(0.187421\pi\)
\(318\) 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 −0.392586 0.919715i \(-0.628419\pi\)
−0.992790 + 0.119869i \(0.961753\pi\)
\(348\) 0 0
\(349\) 31.9372i 1.70956i 0.518993 + 0.854779i \(0.326307\pi\)
−0.518993 + 0.854779i \(0.673693\pi\)
\(350\) 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 −0.873704 0.486459i \(-0.838288\pi\)
−0.0155661 + 0.999879i \(0.504955\pi\)
\(360\) 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 \(-0.943109\pi\)
−0.645996 0.763341i \(-0.723558\pi\)
\(368\) 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 0.474058 0.880494i \(-0.342789\pi\)
−0.525501 + 0.850793i \(0.676122\pi\)
\(390\) 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 −0.163434 0.986554i \(-0.552257\pi\)
0.772664 + 0.634815i \(0.218924\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 4.77589 2.75736i 0.238496 0.137696i −0.375989 0.926624i \(-0.622697\pi\)
0.614485 + 0.788928i \(0.289364\pi\)
\(402\) 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 0.524142 0.851631i \(-0.324386\pi\)
−0.475463 + 0.879736i \(0.657719\pi\)
\(410\) 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 −0.396402 0.918077i \(-0.629741\pi\)
−0.993279 + 0.115745i \(0.963075\pi\)
\(432\) 0 0
\(433\) 0.502734i 0.0241599i 0.999927 + 0.0120799i \(0.00384526\pi\)
−0.999927 + 0.0120799i \(0.996155\pi\)
\(434\) 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 −0.758716 0.651421i \(-0.774173\pi\)
0.184789 + 0.982778i \(0.440840\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 25.3864 14.6569i 1.20615 0.696368i 0.244230 0.969717i \(-0.421465\pi\)
0.961915 + 0.273349i \(0.0881314\pi\)
\(444\) 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 \(-0.912087\pi\)
0.962102 0.272689i \(-0.0879130\pi\)
\(450\) 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 \(-0.716159\pi\)
0.628079 0.778149i \(-0.283841\pi\)
\(492\) 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 −0.113752 0.993509i \(-0.536287\pi\)
0.803528 + 0.595267i \(0.202954\pi\)
\(524\) 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 0.160636 0.987014i \(-0.448646\pi\)
−0.774461 + 0.632621i \(0.781979\pi\)
\(558\) 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 −0.491165 0.871067i \(-0.663429\pi\)
0.508783 + 0.860895i \(0.330095\pi\)
\(570\) 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 0.813510 0.581551i \(-0.197554\pi\)
−0.0968824 + 0.995296i \(0.530887\pi\)
\(578\) 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 0.655648 0.755067i \(-0.272396\pi\)
−0.326083 + 0.945341i \(0.605729\pi\)
\(600\) 0 0
\(601\) 23.9121i 0.975394i 0.873013 + 0.487697i \(0.162163\pi\)
−0.873013 + 0.487697i \(0.837837\pi\)
\(602\) 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 −0.705085 0.709123i \(-0.749091\pi\)
0.261576 + 0.965183i \(0.415758\pi\)
\(608\) 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 \(0.114285\pi\)
−0.936236 + 0.351373i \(0.885715\pi\)
\(618\) 0 0
\(619\) 1.55310 + 0.896683i 0.0624244 + 0.0360407i 0.530887 0.847442i \(-0.321859\pi\)
−0.468463 + 0.883483i \(0.655192\pi\)
\(620\) 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 0.0419557 0.999119i \(-0.486641\pi\)
−0.844285 + 0.535894i \(0.819975\pi\)
\(642\) 0 0
\(643\) 32.5168i 1.28234i −0.767400 0.641169i \(-0.778450\pi\)
0.767400 0.641169i \(-0.221550\pi\)
\(644\) 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 −0.397443 0.917627i \(-0.630102\pi\)
0.595967 + 0.803009i \(0.296769\pi\)
\(654\) 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 \(0.0184272\pi\)
−0.998325 + 0.0578583i \(0.981573\pi\)
\(660\) 0 0
\(661\) 20.2536 + 11.6934i 0.787773 + 0.454821i 0.839178 0.543857i \(-0.183037\pi\)
−0.0514050 + 0.998678i \(0.516370\pi\)
\(662\) 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 \(0.0221099\pi\)
0.558900 + 0.829235i \(0.311223\pi\)
\(684\) 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 −0.858484 0.512840i \(-0.828593\pi\)
0.0148906 + 0.999889i \(0.495260\pi\)
\(692\) 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 \(0.0942807\pi\)
−0.956455 + 0.291880i \(0.905719\pi\)
\(702\) 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 \(0.477671\pi\)
−0.0700903 + 0.997541i \(0.522329\pi\)
\(728\) 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 −0.0988808 0.995099i \(-0.531526\pi\)
0.812341 + 0.583183i \(0.198193\pi\)
\(734\) 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 \(0.295596\pi\)
−0.598922 + 0.800807i \(0.704404\pi\)
\(744\) 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 \(-0.917113\pi\)
0.966288 0.257463i \(-0.0828866\pi\)
\(770\) 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 \(0.0227470\pi\)
0.560559 + 0.828115i \(0.310586\pi\)
\(788\) 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
\(807\) 0