Properties

Label 1764.2.t.c.521.2
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 521.2
Root \(0.991445 - 0.130526i\) of defining polynomial
Character \(\chi\) \(=\) 1764.521
Dual form 1764.2.t.c.1097.2

$q$-expansion

\(f(q)\) \(=\) \(q+(-0.923880 - 1.60021i) q^{5} +O(q^{10})\) \(q+(-0.923880 - 1.60021i) q^{5} +(1.73205 + 1.00000i) q^{11} -4.46088i q^{13} +(1.14805 - 1.98848i) q^{17} +(1.32565 - 0.765367i) q^{19} +(-7.64564 + 4.41421i) q^{23} +(0.792893 - 1.37333i) q^{25} +1.17157i q^{29} +(5.07517 + 2.93015i) q^{31} +(-4.12132 - 7.13834i) q^{37} -11.8519 q^{41} +1.17157 q^{43} +(-4.01254 - 6.94993i) q^{47} +(3.25397 + 1.87868i) q^{53} -3.69552i q^{55} +(4.90923 - 8.50303i) q^{59} +(10.6523 - 6.15013i) q^{61} +(-7.13834 + 4.12132i) q^{65} +(-6.24264 + 10.8126i) q^{67} -13.3137i q^{71} +(-2.37676 - 1.37222i) q^{73} +(-5.65685 - 9.79796i) q^{79} -10.4525 q^{83} -4.24264 q^{85} +(-7.23252 - 12.5271i) q^{89} +(-2.44949 - 1.41421i) q^{95} -2.74444i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + O(q^{10}) \) \( 16 q + 24 q^{25} - 32 q^{37} + 64 q^{43} - 32 q^{67} + O(q^{100}) \)

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
\(5\) −0.923880 1.60021i −0.413171 0.715634i 0.582063 0.813144i \(-0.302246\pi\)
−0.995235 + 0.0975096i \(0.968912\pi\)
\(6\) 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
\(13\) 4.46088i 1.23723i −0.785695 0.618613i \(-0.787695\pi\)
0.785695 0.618613i \(-0.212305\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.14805 1.98848i 0.278443 0.482278i −0.692555 0.721365i \(-0.743515\pi\)
0.970998 + 0.239088i \(0.0768483\pi\)
\(18\) 0 0
\(19\) 1.32565 0.765367i 0.304126 0.175587i −0.340169 0.940364i \(-0.610484\pi\)
0.644295 + 0.764777i \(0.277151\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −7.64564 + 4.41421i −1.59423 + 0.920427i −0.601656 + 0.798755i \(0.705492\pi\)
−0.992570 + 0.121672i \(0.961174\pi\)
\(24\) 0 0
\(25\) 0.792893 1.37333i 0.158579 0.274666i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1.17157i 0.217556i 0.994066 + 0.108778i \(0.0346937\pi\)
−0.994066 + 0.108778i \(0.965306\pi\)
\(30\) 0 0
\(31\) 5.07517 + 2.93015i 0.911528 + 0.526271i 0.880922 0.473261i \(-0.156923\pi\)
0.0306053 + 0.999532i \(0.490257\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −4.12132 7.13834i −0.677541 1.17354i −0.975719 0.219025i \(-0.929712\pi\)
0.298178 0.954510i \(-0.403621\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −11.8519 −1.85096 −0.925480 0.378798i \(-0.876338\pi\)
−0.925480 + 0.378798i \(0.876338\pi\)
\(42\) 0 0
\(43\) 1.17157 0.178663 0.0893316 0.996002i \(-0.471527\pi\)
0.0893316 + 0.996002i \(0.471527\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −4.01254 6.94993i −0.585290 1.01375i −0.994839 0.101464i \(-0.967647\pi\)
0.409550 0.912288i \(-0.365686\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 3.25397 + 1.87868i 0.446967 + 0.258056i 0.706548 0.707665i \(-0.250251\pi\)
−0.259581 + 0.965721i \(0.583585\pi\)
\(54\) 0 0
\(55\) 3.69552i 0.498304i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 4.90923 8.50303i 0.639127 1.10700i −0.346498 0.938051i \(-0.612629\pi\)
0.985625 0.168949i \(-0.0540375\pi\)
\(60\) 0 0
\(61\) 10.6523 6.15013i 1.36389 0.787444i 0.373753 0.927528i \(-0.378071\pi\)
0.990140 + 0.140085i \(0.0447375\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −7.13834 + 4.12132i −0.885402 + 0.511187i
\(66\) 0 0
\(67\) −6.24264 + 10.8126i −0.762660 + 1.32097i 0.178815 + 0.983883i \(0.442774\pi\)
−0.941475 + 0.337083i \(0.890560\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 13.3137i 1.58005i −0.613077 0.790023i \(-0.710068\pi\)
0.613077 0.790023i \(-0.289932\pi\)
\(72\) 0 0
\(73\) −2.37676 1.37222i −0.278178 0.160606i 0.354420 0.935086i \(-0.384678\pi\)
−0.632598 + 0.774480i \(0.718012\pi\)
\(74\) 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 \(-0.947071\pi\)
0.349761 0.936839i \(-0.386263\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −10.4525 −1.14731 −0.573656 0.819097i \(-0.694475\pi\)
−0.573656 + 0.819097i \(0.694475\pi\)
\(84\) 0 0
\(85\) −4.24264 −0.460179
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −7.23252 12.5271i −0.766646 1.32787i −0.939372 0.342900i \(-0.888591\pi\)
0.172726 0.984970i \(-0.444742\pi\)
\(90\) 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
\(97\) 2.74444i 0.278656i −0.990246 0.139328i \(-0.955506\pi\)
0.990246 0.139328i \(-0.0444942\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 1.37222 2.37676i 0.136541 0.236496i −0.789644 0.613565i \(-0.789735\pi\)
0.926185 + 0.377069i \(0.123068\pi\)
\(102\) 0 0
\(103\) −0.549104 + 0.317025i −0.0541048 + 0.0312374i −0.526808 0.849984i \(-0.676611\pi\)
0.472704 + 0.881221i \(0.343278\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −11.1097 + 6.41421i −1.07402 + 0.620085i −0.929277 0.369384i \(-0.879569\pi\)
−0.144743 + 0.989469i \(0.546235\pi\)
\(108\) 0 0
\(109\) −1.53553 + 2.65962i −0.147077 + 0.254746i −0.930146 0.367190i \(-0.880320\pi\)
0.783069 + 0.621935i \(0.213653\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 13.4142i 1.26190i 0.775822 + 0.630952i \(0.217335\pi\)
−0.775822 + 0.630952i \(0.782665\pi\)
\(114\) 0 0
\(115\) 14.1273 + 8.15640i 1.31738 + 0.760589i
\(116\) 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
\(125\) −12.1689 −1.08842
\(126\) 0 0
\(127\) 6.82843 0.605925 0.302962 0.953002i \(-0.402024\pi\)
0.302962 + 0.953002i \(0.402024\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 5.67459 + 9.82868i 0.495792 + 0.858736i 0.999988 0.00485273i \(-0.00154468\pi\)
−0.504197 + 0.863589i \(0.668211\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 4.47871 + 2.58579i 0.382642 + 0.220919i 0.678967 0.734169i \(-0.262428\pi\)
−0.296325 + 0.955087i \(0.595761\pi\)
\(138\) 0 0
\(139\) 6.49435i 0.550844i −0.961323 0.275422i \(-0.911182\pi\)
0.961323 0.275422i \(-0.0888176\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 4.46088 7.72648i 0.373038 0.646121i
\(144\) 0 0
\(145\) 1.87476 1.08239i 0.155690 0.0898878i
\(146\) 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
\(151\) 9.07107 15.7116i 0.738193 1.27859i −0.215115 0.976589i \(-0.569013\pi\)
0.953308 0.301999i \(-0.0976540\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 10.8284i 0.869760i
\(156\) 0 0
\(157\) −9.71496 5.60894i −0.775338 0.447642i 0.0594373 0.998232i \(-0.481069\pi\)
−0.834776 + 0.550590i \(0.814403\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 1.41421 + 2.44949i 0.110770 + 0.191859i 0.916081 0.400994i \(-0.131335\pi\)
−0.805311 + 0.592852i \(0.798002\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 16.3128 1.26232 0.631161 0.775652i \(-0.282579\pi\)
0.631161 + 0.775652i \(0.282579\pi\)
\(168\) 0 0
\(169\) −6.89949 −0.530730
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −6.05728 10.4915i −0.460526 0.797655i 0.538461 0.842650i \(-0.319006\pi\)
−0.998987 + 0.0449956i \(0.985673\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 11.5300 + 6.65685i 0.861793 + 0.497557i 0.864612 0.502439i \(-0.167564\pi\)
−0.00281905 + 0.999996i \(0.500897\pi\)
\(180\) 0 0
\(181\) 14.4650i 1.07518i −0.843207 0.537589i \(-0.819335\pi\)
0.843207 0.537589i \(-0.180665\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −7.61521 + 13.1899i −0.559881 + 0.969743i
\(186\) 0 0
\(187\) 3.97696 2.29610i 0.290824 0.167908i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 4.60181 2.65685i 0.332975 0.192243i −0.324186 0.945993i \(-0.605090\pi\)
0.657161 + 0.753750i \(0.271757\pi\)
\(192\) 0 0
\(193\) −0.828427 + 1.43488i −0.0596315 + 0.103285i −0.894300 0.447468i \(-0.852326\pi\)
0.834669 + 0.550753i \(0.185659\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 11.0711i 0.788781i 0.918943 + 0.394390i \(0.129044\pi\)
−0.918943 + 0.394390i \(0.870956\pi\)
\(198\) 0 0
\(199\) 2.65131 + 1.53073i 0.187946 + 0.108511i 0.591021 0.806656i \(-0.298725\pi\)
−0.403074 + 0.915167i \(0.632058\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 10.9497 + 18.9655i 0.764764 + 1.32461i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 3.06147 0.211766
\(210\) 0 0
\(211\) 26.6274 1.83311 0.916553 0.399912i \(-0.130959\pi\)
0.916553 + 0.399912i \(0.130959\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −1.08239 1.87476i −0.0738185 0.127857i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −8.87039 5.12132i −0.596687 0.344497i
\(222\) 0 0
\(223\) 21.2764i 1.42477i 0.701786 + 0.712387i \(0.252386\pi\)
−0.701786 + 0.712387i \(0.747614\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 2.93015 5.07517i 0.194481 0.336851i −0.752249 0.658879i \(-0.771031\pi\)
0.946730 + 0.322028i \(0.104364\pi\)
\(228\) 0 0
\(229\) −11.4289 + 6.59847i −0.755242 + 0.436039i −0.827585 0.561340i \(-0.810286\pi\)
0.0723426 + 0.997380i \(0.476952\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 18.0379 10.4142i 1.18171 0.682258i 0.225297 0.974290i \(-0.427665\pi\)
0.956408 + 0.292032i \(0.0943315\pi\)
\(234\) 0 0
\(235\) −7.41421 + 12.8418i −0.483650 + 0.837706i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 3.17157i 0.205152i −0.994725 0.102576i \(-0.967292\pi\)
0.994725 0.102576i \(-0.0327085\pi\)
\(240\) 0 0
\(241\) 5.41634 + 3.12713i 0.348897 + 0.201436i 0.664199 0.747555i \(-0.268773\pi\)
−0.315302 + 0.948991i \(0.602106\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −3.41421 5.91359i −0.217241 0.376273i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 19.7457 1.24634 0.623169 0.782088i \(-0.285845\pi\)
0.623169 + 0.782088i \(0.285845\pi\)
\(252\) 0 0
\(253\) −17.6569 −1.11008
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 6.56001 + 11.3623i 0.409202 + 0.708759i 0.994801 0.101842i \(-0.0324737\pi\)
−0.585598 + 0.810601i \(0.699140\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −7.22538 4.17157i −0.445536 0.257230i 0.260407 0.965499i \(-0.416143\pi\)
−0.705943 + 0.708269i \(0.749477\pi\)
\(264\) 0 0
\(265\) 6.94269i 0.426486i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −12.3659 + 21.4184i −0.753964 + 1.30590i 0.191924 + 0.981410i \(0.438527\pi\)
−0.945888 + 0.324493i \(0.894806\pi\)
\(270\) 0 0
\(271\) −20.5281 + 11.8519i −1.24700 + 0.719953i −0.970509 0.241065i \(-0.922503\pi\)
−0.276486 + 0.961018i \(0.589170\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 2.74666 1.58579i 0.165630 0.0956265i
\(276\) 0 0
\(277\) 10.4853 18.1610i 0.630000 1.09119i −0.357552 0.933893i \(-0.616388\pi\)
0.987551 0.157298i \(-0.0502783\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 6.48528i 0.386879i 0.981112 + 0.193440i \(0.0619644\pi\)
−0.981112 + 0.193440i \(0.938036\pi\)
\(282\) 0 0
\(283\) −27.7055 15.9958i −1.64692 0.950850i −0.978286 0.207260i \(-0.933545\pi\)
−0.668635 0.743590i \(-0.733121\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 5.86396 + 10.1567i 0.344939 + 0.597452i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −8.34211 −0.487351 −0.243676 0.969857i \(-0.578353\pi\)
−0.243676 + 0.969857i \(0.578353\pi\)
\(294\) 0 0
\(295\) −18.1421 −1.05628
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 19.6913 + 34.1063i 1.13878 + 1.97242i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −19.6830 11.3640i −1.12704 0.650699i
\(306\) 0 0
\(307\) 28.5587i 1.62993i 0.579510 + 0.814965i \(0.303244\pi\)
−0.579510 + 0.814965i \(0.696756\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 10.9552 18.9750i 0.621215 1.07598i −0.368045 0.929808i \(-0.619973\pi\)
0.989260 0.146167i \(-0.0466938\pi\)
\(312\) 0 0
\(313\) 19.6379 11.3379i 1.11000 0.640857i 0.171169 0.985242i \(-0.445246\pi\)
0.938829 + 0.344385i \(0.111912\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 20.9947 12.1213i 1.17918 0.680801i 0.223357 0.974737i \(-0.428298\pi\)
0.955825 + 0.293936i \(0.0949651\pi\)
\(318\) 0 0
\(319\) −1.17157 + 2.02922i −0.0655955 + 0.113615i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 3.51472i 0.195564i
\(324\) 0 0
\(325\) −6.12627 3.53701i −0.339824 0.196198i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 2.34315 + 4.05845i 0.128791 + 0.223072i 0.923208 0.384300i \(-0.125557\pi\)
−0.794417 + 0.607372i \(0.792224\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 23.0698 1.26044
\(336\) 0 0
\(337\) −12.9289 −0.704284 −0.352142 0.935947i \(-0.614547\pi\)
−0.352142 + 0.935947i \(0.614547\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 5.86030 + 10.1503i 0.317353 + 0.549672i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 8.48617 + 4.89949i 0.455562 + 0.263019i 0.710176 0.704024i \(-0.248615\pi\)
−0.254615 + 0.967043i \(0.581949\pi\)
\(348\) 0 0
\(349\) 30.0669i 1.60944i 0.593652 + 0.804722i \(0.297685\pi\)
−0.593652 + 0.804722i \(0.702315\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −16.2856 + 28.2075i −0.866796 + 1.50133i −0.00154235 + 0.999999i \(0.500491\pi\)
−0.865253 + 0.501335i \(0.832842\pi\)
\(354\) 0 0
\(355\) −21.3047 + 12.3003i −1.13074 + 0.652830i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 27.2416 15.7279i 1.43775 0.830088i 0.440061 0.897968i \(-0.354957\pi\)
0.997694 + 0.0678799i \(0.0216235\pi\)
\(360\) 0 0
\(361\) −8.32843 + 14.4253i −0.438338 + 0.759224i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 5.07107i 0.265432i
\(366\) 0 0
\(367\) 24.1834 + 13.9623i 1.26237 + 0.728827i 0.973532 0.228553i \(-0.0733993\pi\)
0.288833 + 0.957379i \(0.406733\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 7.31371 + 12.6677i 0.378689 + 0.655909i 0.990872 0.134807i \(-0.0430415\pi\)
−0.612182 + 0.790717i \(0.709708\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 5.22625 0.269166
\(378\) 0 0
\(379\) 20.2843 1.04193 0.520967 0.853577i \(-0.325572\pi\)
0.520967 + 0.853577i \(0.325572\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −3.06147 5.30262i −0.156434 0.270951i 0.777146 0.629320i \(-0.216666\pi\)
−0.933580 + 0.358369i \(0.883333\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −5.91359 3.41421i −0.299831 0.173107i 0.342536 0.939505i \(-0.388714\pi\)
−0.642367 + 0.766397i \(0.722047\pi\)
\(390\) 0 0
\(391\) 20.2710i 1.02515i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −10.4525 + 18.1043i −0.525923 + 0.910925i
\(396\) 0 0
\(397\) −6.90282 + 3.98535i −0.346443 + 0.200019i −0.663117 0.748515i \(-0.730767\pi\)
0.316675 + 0.948534i \(0.397434\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 19.4728 11.2426i 0.972426 0.561431i 0.0724514 0.997372i \(-0.476918\pi\)
0.899975 + 0.435941i \(0.143584\pi\)
\(402\) 0 0
\(403\) 13.0711 22.6398i 0.651116 1.12777i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 16.4853i 0.817145i
\(408\) 0 0
\(409\) −24.7796 14.3065i −1.22527 0.707413i −0.259237 0.965814i \(-0.583471\pi\)
−0.966038 + 0.258401i \(0.916804\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 9.65685 + 16.7262i 0.474036 + 0.821055i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 14.5194 0.709321 0.354661 0.934995i \(-0.384596\pi\)
0.354661 + 0.934995i \(0.384596\pi\)
\(420\) 0 0
\(421\) −7.31371 −0.356448 −0.178224 0.983990i \(-0.557035\pi\)
−0.178224 + 0.983990i \(0.557035\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −1.82056 3.15331i −0.0883103 0.152958i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −9.25460 5.34315i −0.445778 0.257370i 0.260267 0.965537i \(-0.416189\pi\)
−0.706046 + 0.708166i \(0.749523\pi\)
\(432\) 0 0
\(433\) 19.6913i 0.946303i −0.880981 0.473152i \(-0.843116\pi\)
0.880981 0.473152i \(-0.156884\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −6.75699 + 11.7034i −0.323230 + 0.559852i
\(438\) 0 0
\(439\) 16.2295 9.37011i 0.774592 0.447211i −0.0599181 0.998203i \(-0.519084\pi\)
0.834510 + 0.550992i \(0.185751\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 5.79050 3.34315i 0.275115 0.158838i −0.356095 0.934450i \(-0.615892\pi\)
0.631210 + 0.775612i \(0.282559\pi\)
\(444\) 0 0
\(445\) −13.3640 + 23.1471i −0.633513 + 1.09728i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 19.5563i 0.922921i 0.887161 + 0.461461i \(0.152674\pi\)
−0.887161 + 0.461461i \(0.847326\pi\)
\(450\) 0 0
\(451\) −20.5281 11.8519i −0.966632 0.558085i
\(452\) 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
\(461\) 28.1647 1.31176 0.655881 0.754864i \(-0.272297\pi\)
0.655881 + 0.754864i \(0.272297\pi\)
\(462\) 0 0
\(463\) 2.82843 0.131448 0.0657241 0.997838i \(-0.479064\pi\)
0.0657241 + 0.997838i \(0.479064\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 18.6089 + 32.2316i 0.861118 + 1.49150i 0.870851 + 0.491547i \(0.163568\pi\)
−0.00973373 + 0.999953i \(0.503098\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 2.02922 + 1.17157i 0.0933038 + 0.0538690i
\(474\) 0 0
\(475\) 2.42742i 0.111378i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 12.4860 21.6263i 0.570499 0.988133i −0.426016 0.904716i \(-0.640083\pi\)
0.996515 0.0834170i \(-0.0265834\pi\)
\(480\) 0 0
\(481\) −31.8433 + 18.3847i −1.45193 + 0.838272i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −4.39167 + 2.53553i −0.199416 + 0.115133i
\(486\) 0 0
\(487\) 9.89949 17.1464i 0.448589 0.776979i −0.549706 0.835359i \(-0.685260\pi\)
0.998294 + 0.0583797i \(0.0185934\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 17.5147i 0.790428i −0.918589 0.395214i \(-0.870670\pi\)
0.918589 0.395214i \(-0.129330\pi\)
\(492\) 0 0
\(493\) 2.32965 + 1.34502i 0.104922 + 0.0605769i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 1.07107 + 1.85514i 0.0479476 + 0.0830476i 0.889003 0.457901i \(-0.151399\pi\)
−0.841056 + 0.540949i \(0.818065\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 12.2459 0.546016 0.273008 0.962012i \(-0.411981\pi\)
0.273008 + 0.962012i \(0.411981\pi\)
\(504\) 0 0
\(505\) −5.07107 −0.225660
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −2.67878 4.63979i −0.118735 0.205655i 0.800532 0.599291i \(-0.204551\pi\)
−0.919267 + 0.393635i \(0.871217\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 1.01461 + 0.585786i 0.0447091 + 0.0258128i
\(516\) 0 0
\(517\) 16.0502i 0.705886i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 7.99789 13.8528i 0.350394 0.606900i −0.635925 0.771751i \(-0.719381\pi\)
0.986318 + 0.164851i \(0.0527144\pi\)
\(522\) 0 0
\(523\) −25.2816 + 14.5964i −1.10549 + 0.638254i −0.937657 0.347561i \(-0.887010\pi\)
−0.167832 + 0.985816i \(0.553677\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 11.6531 6.72792i 0.507617 0.293073i
\(528\) 0 0
\(529\) 27.4706 47.5804i 1.19437 2.06871i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 52.8701i 2.29006i
\(534\) 0 0
\(535\) 20.5281 + 11.8519i 0.887508 + 0.512403i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −9.82843 17.0233i −0.422557 0.731890i 0.573632 0.819113i \(-0.305534\pi\)
−0.996189 + 0.0872230i \(0.972201\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 5.67459 0.243073
\(546\) 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
\(551\) 0.896683 + 1.55310i 0.0382000 + 0.0661643i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 7.55860 + 4.36396i 0.320268 + 0.184907i 0.651512 0.758638i \(-0.274135\pi\)
−0.331244 + 0.943545i \(0.607468\pi\)
\(558\) 0 0
\(559\) 5.22625i 0.221047i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −21.4077 + 37.0793i −0.902229 + 1.56271i −0.0776342 + 0.996982i \(0.524737\pi\)
−0.824595 + 0.565724i \(0.808597\pi\)
\(564\) 0 0
\(565\) 21.4655 12.3931i 0.903061 0.521382i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −14.2767 + 8.24264i −0.598509 + 0.345549i −0.768455 0.639904i \(-0.778974\pi\)
0.169946 + 0.985453i \(0.445641\pi\)
\(570\) 0 0
\(571\) −21.3137 + 36.9164i −0.891951 + 1.54490i −0.0544175 + 0.998518i \(0.517330\pi\)
−0.837533 + 0.546386i \(0.816003\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 14.0000i 0.583840i
\(576\) 0 0
\(577\) −12.7545 7.36384i −0.530979 0.306561i 0.210436 0.977608i \(-0.432512\pi\)
−0.741415 + 0.671047i \(0.765845\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 3.75736 + 6.50794i 0.155614 + 0.269531i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −15.9414 −0.657971 −0.328986 0.944335i \(-0.606707\pi\)
−0.328986 + 0.944335i \(0.606707\pi\)
\(588\) 0 0
\(589\) 8.97056 0.369626
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −2.04473 3.54158i −0.0839671 0.145435i 0.820983 0.570952i \(-0.193426\pi\)
−0.904951 + 0.425517i \(0.860092\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −11.5300 6.65685i −0.471103 0.271992i 0.245598 0.969372i \(-0.421016\pi\)
−0.716702 + 0.697380i \(0.754349\pi\)
\(600\) 0 0
\(601\) 42.0501i 1.71526i 0.514267 + 0.857630i \(0.328064\pi\)
−0.514267 + 0.857630i \(0.671936\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −6.46716 + 11.2014i −0.262927 + 0.455403i
\(606\) 0 0
\(607\) 0.776550 0.448342i 0.0315192 0.0181976i −0.484158 0.874981i \(-0.660874\pi\)
0.515677 + 0.856783i \(0.327541\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −31.0028 + 17.8995i −1.25424 + 0.724136i
\(612\) 0 0
\(613\) 1.39340 2.41344i 0.0562788 0.0974778i −0.836513 0.547946i \(-0.815410\pi\)
0.892792 + 0.450469i \(0.148743\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 33.4558i 1.34688i −0.739241 0.673441i \(-0.764816\pi\)
0.739241 0.673441i \(-0.235184\pi\)
\(618\) 0 0
\(619\) 21.8538 + 12.6173i 0.878378 + 0.507132i 0.870123 0.492834i \(-0.164039\pi\)
0.00825456 + 0.999966i \(0.497372\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 7.27817 + 12.6062i 0.291127 + 0.504247i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −18.9259 −0.754626
\(630\) 0 0
\(631\) 4.48528 0.178556 0.0892781 0.996007i \(-0.471544\pi\)
0.0892781 + 0.996007i \(0.471544\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −6.30864 10.9269i −0.250351 0.433620i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 23.7775 + 13.7279i 0.939153 + 0.542220i 0.889695 0.456556i \(-0.150917\pi\)
0.0494584 + 0.998776i \(0.484250\pi\)
\(642\) 0 0
\(643\) 47.1451i 1.85922i −0.368546 0.929610i \(-0.620144\pi\)
0.368546 0.929610i \(-0.379856\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 18.1606 31.4550i 0.713965 1.23662i −0.249392 0.968403i \(-0.580231\pi\)
0.963357 0.268222i \(-0.0864360\pi\)
\(648\) 0 0
\(649\) 17.0061 9.81845i 0.667546 0.385408i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 29.5680 17.0711i 1.15708 0.668043i 0.206480 0.978451i \(-0.433799\pi\)
0.950603 + 0.310408i \(0.100466\pi\)
\(654\) 0 0
\(655\) 10.4853 18.1610i 0.409694 0.709611i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 30.9706i 1.20644i −0.797574 0.603221i \(-0.793884\pi\)
0.797574 0.603221i \(-0.206116\pi\)
\(660\) 0 0
\(661\) −0.890273 0.514000i −0.0346276 0.0199923i 0.482586 0.875848i \(-0.339697\pi\)
−0.517214 + 0.855856i \(0.673031\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −5.17157 8.95743i −0.200244 0.346833i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 24.6005 0.949693
\(672\) 0 0
\(673\) −28.0416 −1.08093 −0.540463 0.841368i \(-0.681751\pi\)
−0.540463 + 0.841368i \(0.681751\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −5.02928 8.71096i −0.193291 0.334790i 0.753048 0.657966i \(-0.228583\pi\)
−0.946339 + 0.323176i \(0.895249\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 11.2838 + 6.51472i 0.431764 + 0.249279i 0.700098 0.714047i \(-0.253140\pi\)
−0.268334 + 0.963326i \(0.586473\pi\)
\(684\) 0 0
\(685\) 9.55582i 0.365109i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 8.38057 14.5156i 0.319274 0.553000i
\(690\) 0 0
\(691\) 27.9330 16.1271i 1.06262 0.613504i 0.136464 0.990645i \(-0.456426\pi\)
0.926156 + 0.377141i \(0.123093\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −10.3923 + 6.00000i −0.394203 + 0.227593i
\(696\) 0 0
\(697\) −13.6066 + 23.5673i −0.515387 + 0.892676i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 35.4558i 1.33915i −0.742745 0.669574i \(-0.766477\pi\)
0.742745 0.669574i \(-0.233523\pi\)
\(702\) 0 0
\(703\) −10.9269 6.30864i −0.412116 0.237935i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −11.4350 19.8061i −0.429452 0.743832i 0.567373 0.823461i \(-0.307960\pi\)
−0.996825 + 0.0796290i \(0.974626\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −51.7373 −1.93758
\(714\) 0 0
\(715\) −16.4853 −0.616515
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −13.2513 22.9520i −0.494192 0.855965i 0.505786 0.862659i \(-0.331203\pi\)
−0.999978 + 0.00669409i \(0.997869\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 1.60896 + 0.928932i 0.0597552 + 0.0344997i
\(726\) 0 0
\(727\) 25.3434i 0.939933i 0.882684 + 0.469967i \(0.155734\pi\)
−0.882684 + 0.469967i \(0.844266\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 1.34502 2.32965i 0.0497475 0.0861653i
\(732\) 0 0
\(733\) 1.37276 0.792563i 0.0507040 0.0292740i −0.474434 0.880291i \(-0.657347\pi\)
0.525138 + 0.851017i \(0.324014\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −21.6251 + 12.4853i −0.796572 + 0.459901i
\(738\) 0 0
\(739\) −26.7279 + 46.2941i −0.983203 + 1.70296i −0.333536 + 0.942737i \(0.608242\pi\)
−0.649667 + 0.760219i \(0.725091\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 32.3431i 1.18655i 0.804998 + 0.593277i \(0.202166\pi\)
−0.804998 + 0.593277i \(0.797834\pi\)
\(744\) 0 0
\(745\) 11.3152 + 6.53281i 0.414556 + 0.239344i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −10.1421 17.5667i −0.370092 0.641018i 0.619488 0.785006i \(-0.287340\pi\)
−0.989579 + 0.143989i \(0.954007\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −33.5223 −1.22000
\(756\) 0 0
\(757\) 37.8995 1.37748 0.688740 0.725008i \(-0.258164\pi\)
0.688740 + 0.725008i \(0.258164\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −7.28692 12.6213i −0.264151 0.457522i 0.703190 0.711002i \(-0.251758\pi\)
−0.967341 + 0.253480i \(0.918425\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −37.9310 21.8995i −1.36961 0.790745i
\(768\) 0 0
\(769\) 13.5684i 0.489288i −0.969613 0.244644i \(-0.921329\pi\)
0.969613 0.244644i \(-0.0786710\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 9.11874 15.7941i 0.327978 0.568075i −0.654132 0.756380i \(-0.726966\pi\)
0.982111 + 0.188305i \(0.0602993\pi\)
\(774\) 0 0
\(775\) 8.04814 4.64659i 0.289098 0.166911i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −15.7116 + 9.07107i −0.562925 + 0.325005i
\(780\) 0 0
\(781\) 13.3137 23.0600i 0.476402 0.825152i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 20.7279i 0.739811i
\(786\) 0 0
\(787\) −3.10620 1.79337i −0.110724 0.0639266i 0.443615 0.896217i \(-0.353696\pi\)
−0.554339 + 0.832291i \(0.687029\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −27.4350 47.5189i −0.974246 1.68744i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −30.4064 −1.07705 −0.538526 0.842609i \(-0.681018\pi\)
−0.538526 + 0.842609i \(0.681018\pi\)
\(798\) 0 0
\(799\) −18.4264 −0.651879
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −2.74444 4.75351i −0.0968493 0.167748i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\)