Properties

Label 1040.2.k.c.961.3
Level $1040$
Weight $2$
Character 1040.961
Analytic conductor $8.304$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1040 = 2^{4} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1040.k (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.30444181021\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.9144576.1
Defining polynomial: \(x^{6} + 12 x^{4} + 36 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 260)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 961.3
Root \(0.339877i\) of defining polynomial
Character \(\chi\) \(=\) 1040.961
Dual form 1040.2.k.c.961.4

$q$-expansion

\(f(q)\) \(=\) \(q+0.339877 q^{3} -1.00000i q^{5} +3.88448i q^{7} -2.88448 q^{9} +O(q^{10})\) \(q+0.339877 q^{3} -1.00000i q^{5} +3.88448i q^{7} -2.88448 q^{9} +1.54461i q^{11} +(3.54461 - 0.660123i) q^{13} -0.339877i q^{15} +2.86485i q^{19} +1.32025i q^{21} -5.42909 q^{23} -1.00000 q^{25} -2.00000 q^{27} -5.20473 q^{29} +6.22436i q^{31} +0.524976i q^{33} +3.88448 q^{35} +8.56424i q^{37} +(1.20473 - 0.224361i) q^{39} +9.08921i q^{41} -0.980369 q^{43} +2.88448i q^{45} -6.52498i q^{47} -8.08921 q^{49} +6.44872 q^{53} +1.54461 q^{55} +0.973697i q^{57} +4.45539i q^{59} +9.65345 q^{61} -11.2047i q^{63} +(-0.660123 - 3.54461i) q^{65} +6.97370i q^{67} -1.84522 q^{69} -12.6731i q^{71} +3.43576i q^{73} -0.339877 q^{75} -6.00000 q^{77} +13.1285 q^{79} +7.97370 q^{81} +8.56424i q^{83} -1.76897 q^{87} +17.1285i q^{89} +(2.56424 + 13.7690i) q^{91} +2.11552i q^{93} +2.86485 q^{95} -13.7690i q^{97} -4.45539i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 6q^{9} + O(q^{10}) \) \( 6q + 6q^{9} + 12q^{23} - 6q^{25} - 12q^{27} - 12q^{29} - 12q^{39} - 12q^{43} - 6q^{49} - 12q^{53} - 12q^{55} - 12q^{61} - 6q^{65} - 36q^{77} + 24q^{79} - 18q^{81} + 36q^{87} - 12q^{91} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1040\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(417\) \(561\) \(911\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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
\(3\) 0.339877 0.196228 0.0981140 0.995175i \(-0.468719\pi\)
0.0981140 + 0.995175i \(0.468719\pi\)
\(4\) 0 0
\(5\) 1.00000i 0.447214i
\(6\) 0 0
\(7\) 3.88448i 1.46820i 0.679043 + 0.734098i \(0.262395\pi\)
−0.679043 + 0.734098i \(0.737605\pi\)
\(8\) 0 0
\(9\) −2.88448 −0.961495
\(10\) 0 0
\(11\) 1.54461i 0.465716i 0.972511 + 0.232858i \(0.0748078\pi\)
−0.972511 + 0.232858i \(0.925192\pi\)
\(12\) 0 0
\(13\) 3.54461 0.660123i 0.983097 0.183085i
\(14\) 0 0
\(15\) 0.339877i 0.0877558i
\(16\) 0 0
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 2.86485i 0.657242i 0.944462 + 0.328621i \(0.106584\pi\)
−0.944462 + 0.328621i \(0.893416\pi\)
\(20\) 0 0
\(21\) 1.32025i 0.288101i
\(22\) 0 0
\(23\) −5.42909 −1.13204 −0.566022 0.824390i \(-0.691518\pi\)
−0.566022 + 0.824390i \(0.691518\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) −2.00000 −0.384900
\(28\) 0 0
\(29\) −5.20473 −0.966494 −0.483247 0.875484i \(-0.660543\pi\)
−0.483247 + 0.875484i \(0.660543\pi\)
\(30\) 0 0
\(31\) 6.22436i 1.11793i 0.829192 + 0.558964i \(0.188801\pi\)
−0.829192 + 0.558964i \(0.811199\pi\)
\(32\) 0 0
\(33\) 0.524976i 0.0913866i
\(34\) 0 0
\(35\) 3.88448 0.656598
\(36\) 0 0
\(37\) 8.56424i 1.40795i 0.710224 + 0.703976i \(0.248594\pi\)
−0.710224 + 0.703976i \(0.751406\pi\)
\(38\) 0 0
\(39\) 1.20473 0.224361i 0.192911 0.0359264i
\(40\) 0 0
\(41\) 9.08921i 1.41950i 0.704455 + 0.709748i \(0.251191\pi\)
−0.704455 + 0.709748i \(0.748809\pi\)
\(42\) 0 0
\(43\) −0.980369 −0.149505 −0.0747525 0.997202i \(-0.523817\pi\)
−0.0747525 + 0.997202i \(0.523817\pi\)
\(44\) 0 0
\(45\) 2.88448i 0.429993i
\(46\) 0 0
\(47\) 6.52498i 0.951766i −0.879509 0.475883i \(-0.842129\pi\)
0.879509 0.475883i \(-0.157871\pi\)
\(48\) 0 0
\(49\) −8.08921 −1.15560
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 6.44872 0.885800 0.442900 0.896571i \(-0.353950\pi\)
0.442900 + 0.896571i \(0.353950\pi\)
\(54\) 0 0
\(55\) 1.54461 0.208275
\(56\) 0 0
\(57\) 0.973697i 0.128969i
\(58\) 0 0
\(59\) 4.45539i 0.580043i 0.957020 + 0.290021i \(0.0936624\pi\)
−0.957020 + 0.290021i \(0.906338\pi\)
\(60\) 0 0
\(61\) 9.65345 1.23600 0.617999 0.786179i \(-0.287944\pi\)
0.617999 + 0.786179i \(0.287944\pi\)
\(62\) 0 0
\(63\) 11.2047i 1.41166i
\(64\) 0 0
\(65\) −0.660123 3.54461i −0.0818782 0.439654i
\(66\) 0 0
\(67\) 6.97370i 0.851973i 0.904730 + 0.425986i \(0.140073\pi\)
−0.904730 + 0.425986i \(0.859927\pi\)
\(68\) 0 0
\(69\) −1.84522 −0.222139
\(70\) 0 0
\(71\) 12.6731i 1.50402i −0.659153 0.752009i \(-0.729085\pi\)
0.659153 0.752009i \(-0.270915\pi\)
\(72\) 0 0
\(73\) 3.43576i 0.402126i 0.979578 + 0.201063i \(0.0644395\pi\)
−0.979578 + 0.201063i \(0.935560\pi\)
\(74\) 0 0
\(75\) −0.339877 −0.0392456
\(76\) 0 0
\(77\) −6.00000 −0.683763
\(78\) 0 0
\(79\) 13.1285 1.47707 0.738534 0.674216i \(-0.235518\pi\)
0.738534 + 0.674216i \(0.235518\pi\)
\(80\) 0 0
\(81\) 7.97370 0.885966
\(82\) 0 0
\(83\) 8.56424i 0.940047i 0.882654 + 0.470024i \(0.155755\pi\)
−0.882654 + 0.470024i \(0.844245\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −1.76897 −0.189653
\(88\) 0 0
\(89\) 17.1285i 1.81561i 0.419387 + 0.907807i \(0.362245\pi\)
−0.419387 + 0.907807i \(0.637755\pi\)
\(90\) 0 0
\(91\) 2.56424 + 13.7690i 0.268805 + 1.44338i
\(92\) 0 0
\(93\) 2.11552i 0.219369i
\(94\) 0 0
\(95\) 2.86485 0.293928
\(96\) 0 0
\(97\) 13.7690i 1.39803i −0.715109 0.699013i \(-0.753623\pi\)
0.715109 0.699013i \(-0.246377\pi\)
\(98\) 0 0
\(99\) 4.45539i 0.447784i
\(100\) 0 0
\(101\) −6.44872 −0.641672 −0.320836 0.947135i \(-0.603964\pi\)
−0.320836 + 0.947135i \(0.603964\pi\)
\(102\) 0 0
\(103\) −12.1088 −1.19312 −0.596560 0.802569i \(-0.703466\pi\)
−0.596560 + 0.802569i \(0.703466\pi\)
\(104\) 0 0
\(105\) 1.32025 0.128843
\(106\) 0 0
\(107\) −17.4291 −1.68493 −0.842467 0.538748i \(-0.818897\pi\)
−0.842467 + 0.538748i \(0.818897\pi\)
\(108\) 0 0
\(109\) 16.4095i 1.57174i −0.618391 0.785871i \(-0.712215\pi\)
0.618391 0.785871i \(-0.287785\pi\)
\(110\) 0 0
\(111\) 2.91079i 0.276280i
\(112\) 0 0
\(113\) 1.59054 0.149625 0.0748127 0.997198i \(-0.476164\pi\)
0.0748127 + 0.997198i \(0.476164\pi\)
\(114\) 0 0
\(115\) 5.42909i 0.506265i
\(116\) 0 0
\(117\) −10.2244 + 1.90411i −0.945242 + 0.176035i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 8.61419 0.783108
\(122\) 0 0
\(123\) 3.08921i 0.278545i
\(124\) 0 0
\(125\) 1.00000i 0.0894427i
\(126\) 0 0
\(127\) 0.108844 0.00965837 0.00482918 0.999988i \(-0.498463\pi\)
0.00482918 + 0.999988i \(0.498463\pi\)
\(128\) 0 0
\(129\) −0.333205 −0.0293371
\(130\) 0 0
\(131\) 10.4095 0.909479 0.454739 0.890625i \(-0.349732\pi\)
0.454739 + 0.890625i \(0.349732\pi\)
\(132\) 0 0
\(133\) −11.1285 −0.964961
\(134\) 0 0
\(135\) 2.00000i 0.172133i
\(136\) 0 0
\(137\) 9.08921i 0.776544i −0.921545 0.388272i \(-0.873072\pi\)
0.921545 0.388272i \(-0.126928\pi\)
\(138\) 0 0
\(139\) −9.04995 −0.767607 −0.383803 0.923415i \(-0.625386\pi\)
−0.383803 + 0.923415i \(0.625386\pi\)
\(140\) 0 0
\(141\) 2.21769i 0.186763i
\(142\) 0 0
\(143\) 1.01963 + 5.47502i 0.0852658 + 0.457844i
\(144\) 0 0
\(145\) 5.20473i 0.432229i
\(146\) 0 0
\(147\) −2.74934 −0.226761
\(148\) 0 0
\(149\) 11.1285i 0.911680i −0.890062 0.455840i \(-0.849339\pi\)
0.890062 0.455840i \(-0.150661\pi\)
\(150\) 0 0
\(151\) 9.13515i 0.743408i −0.928351 0.371704i \(-0.878774\pi\)
0.928351 0.371704i \(-0.121226\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 6.22436 0.499953
\(156\) 0 0
\(157\) 4.91079 0.391923 0.195962 0.980612i \(-0.437217\pi\)
0.195962 + 0.980612i \(0.437217\pi\)
\(158\) 0 0
\(159\) 2.19177 0.173819
\(160\) 0 0
\(161\) 21.0892i 1.66206i
\(162\) 0 0
\(163\) 16.9344i 1.32641i 0.748439 + 0.663204i \(0.230804\pi\)
−0.748439 + 0.663204i \(0.769196\pi\)
\(164\) 0 0
\(165\) 0.524976 0.0408693
\(166\) 0 0
\(167\) 21.6142i 1.67256i −0.548306 0.836278i \(-0.684727\pi\)
0.548306 0.836278i \(-0.315273\pi\)
\(168\) 0 0
\(169\) 12.1285 4.67975i 0.932960 0.359981i
\(170\) 0 0
\(171\) 8.26362i 0.631935i
\(172\) 0 0
\(173\) −8.03926 −0.611214 −0.305607 0.952158i \(-0.598859\pi\)
−0.305607 + 0.952158i \(0.598859\pi\)
\(174\) 0 0
\(175\) 3.88448i 0.293639i
\(176\) 0 0
\(177\) 1.51429i 0.113821i
\(178\) 0 0
\(179\) −10.8582 −0.811579 −0.405789 0.913967i \(-0.633003\pi\)
−0.405789 + 0.913967i \(0.633003\pi\)
\(180\) 0 0
\(181\) 10.5642 0.785234 0.392617 0.919702i \(-0.371570\pi\)
0.392617 + 0.919702i \(0.371570\pi\)
\(182\) 0 0
\(183\) 3.28098 0.242537
\(184\) 0 0
\(185\) 8.56424 0.629655
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 7.76897i 0.565109i
\(190\) 0 0
\(191\) 8.81892 0.638115 0.319057 0.947735i \(-0.396634\pi\)
0.319057 + 0.947735i \(0.396634\pi\)
\(192\) 0 0
\(193\) 0.270294i 0.0194562i 0.999953 + 0.00972809i \(0.00309660\pi\)
−0.999953 + 0.00972809i \(0.996903\pi\)
\(194\) 0 0
\(195\) −0.224361 1.20473i −0.0160668 0.0862725i
\(196\) 0 0
\(197\) 3.96074i 0.282191i 0.989996 + 0.141095i \(0.0450625\pi\)
−0.989996 + 0.141095i \(0.954938\pi\)
\(198\) 0 0
\(199\) −6.03926 −0.428112 −0.214056 0.976821i \(-0.568667\pi\)
−0.214056 + 0.976821i \(0.568667\pi\)
\(200\) 0 0
\(201\) 2.37020i 0.167181i
\(202\) 0 0
\(203\) 20.2177i 1.41900i
\(204\) 0 0
\(205\) 9.08921 0.634818
\(206\) 0 0
\(207\) 15.6601 1.08845
\(208\) 0 0
\(209\) −4.42507 −0.306089
\(210\) 0 0
\(211\) 16.2177 1.11647 0.558236 0.829682i \(-0.311478\pi\)
0.558236 + 0.829682i \(0.311478\pi\)
\(212\) 0 0
\(213\) 4.30729i 0.295130i
\(214\) 0 0
\(215\) 0.980369i 0.0668606i
\(216\) 0 0
\(217\) −24.1784 −1.64134
\(218\) 0 0
\(219\) 1.16774i 0.0789083i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 10.0629i 0.673862i −0.941529 0.336931i \(-0.890611\pi\)
0.941529 0.336931i \(-0.109389\pi\)
\(224\) 0 0
\(225\) 2.88448 0.192299
\(226\) 0 0
\(227\) 3.43576i 0.228040i −0.993478 0.114020i \(-0.963627\pi\)
0.993478 0.114020i \(-0.0363727\pi\)
\(228\) 0 0
\(229\) 22.8582i 1.51051i 0.655430 + 0.755256i \(0.272487\pi\)
−0.655430 + 0.755256i \(0.727513\pi\)
\(230\) 0 0
\(231\) −2.03926 −0.134174
\(232\) 0 0
\(233\) −4.40946 −0.288873 −0.144437 0.989514i \(-0.546137\pi\)
−0.144437 + 0.989514i \(0.546137\pi\)
\(234\) 0 0
\(235\) −6.52498 −0.425643
\(236\) 0 0
\(237\) 4.46207 0.289842
\(238\) 0 0
\(239\) 22.8122i 1.47560i −0.675018 0.737801i \(-0.735864\pi\)
0.675018 0.737801i \(-0.264136\pi\)
\(240\) 0 0
\(241\) 10.6798i 0.687943i 0.938980 + 0.343972i \(0.111772\pi\)
−0.938980 + 0.343972i \(0.888228\pi\)
\(242\) 0 0
\(243\) 8.71008 0.558752
\(244\) 0 0
\(245\) 8.08921i 0.516801i
\(246\) 0 0
\(247\) 1.89116 + 10.1548i 0.120331 + 0.646133i
\(248\) 0 0
\(249\) 2.91079i 0.184464i
\(250\) 0 0
\(251\) 21.2676 1.34240 0.671201 0.741276i \(-0.265779\pi\)
0.671201 + 0.741276i \(0.265779\pi\)
\(252\) 0 0
\(253\) 8.38581i 0.527211i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 18.8974 1.17879 0.589395 0.807845i \(-0.299366\pi\)
0.589395 + 0.807845i \(0.299366\pi\)
\(258\) 0 0
\(259\) −33.2676 −2.06715
\(260\) 0 0
\(261\) 15.0130 0.929279
\(262\) 0 0
\(263\) 13.0196 0.802825 0.401412 0.915897i \(-0.368519\pi\)
0.401412 + 0.915897i \(0.368519\pi\)
\(264\) 0 0
\(265\) 6.44872i 0.396142i
\(266\) 0 0
\(267\) 5.82157i 0.356274i
\(268\) 0 0
\(269\) 14.3702 0.876166 0.438083 0.898934i \(-0.355658\pi\)
0.438083 + 0.898934i \(0.355658\pi\)
\(270\) 0 0
\(271\) 13.7230i 0.833615i −0.908995 0.416807i \(-0.863149\pi\)
0.908995 0.416807i \(-0.136851\pi\)
\(272\) 0 0
\(273\) 0.871525 + 4.67975i 0.0527471 + 0.283232i
\(274\) 0 0
\(275\) 1.54461i 0.0931433i
\(276\) 0 0
\(277\) −19.2676 −1.15768 −0.578840 0.815441i \(-0.696494\pi\)
−0.578840 + 0.815441i \(0.696494\pi\)
\(278\) 0 0
\(279\) 17.9541i 1.07488i
\(280\) 0 0
\(281\) 3.96074i 0.236278i −0.992997 0.118139i \(-0.962307\pi\)
0.992997 0.118139i \(-0.0376928\pi\)
\(282\) 0 0
\(283\) 21.1981 1.26009 0.630047 0.776557i \(-0.283036\pi\)
0.630047 + 0.776557i \(0.283036\pi\)
\(284\) 0 0
\(285\) 0.973697 0.0576769
\(286\) 0 0
\(287\) −35.3069 −2.08410
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) 4.67975i 0.274332i
\(292\) 0 0
\(293\) 13.6927i 0.799937i 0.916529 + 0.399968i \(0.130979\pi\)
−0.916529 + 0.399968i \(0.869021\pi\)
\(294\) 0 0
\(295\) 4.45539 0.259403
\(296\) 0 0
\(297\) 3.08921i 0.179254i
\(298\) 0 0
\(299\) −19.2440 + 3.58387i −1.11291 + 0.207260i
\(300\) 0 0
\(301\) 3.80823i 0.219503i
\(302\) 0 0
\(303\) −2.19177 −0.125914
\(304\) 0 0
\(305\) 9.65345i 0.552755i
\(306\) 0 0
\(307\) 12.7953i 0.730265i −0.930956 0.365132i \(-0.881024\pi\)
0.930956 0.365132i \(-0.118976\pi\)
\(308\) 0 0
\(309\) −4.11552 −0.234124
\(310\) 0 0
\(311\) 22.8582 1.29617 0.648084 0.761569i \(-0.275570\pi\)
0.648084 + 0.761569i \(0.275570\pi\)
\(312\) 0 0
\(313\) −27.2284 −1.53904 −0.769520 0.638623i \(-0.779504\pi\)
−0.769520 + 0.638623i \(0.779504\pi\)
\(314\) 0 0
\(315\) −11.2047 −0.631315
\(316\) 0 0
\(317\) 27.7926i 1.56099i −0.625163 0.780494i \(-0.714967\pi\)
0.625163 0.780494i \(-0.285033\pi\)
\(318\) 0 0
\(319\) 8.03926i 0.450112i
\(320\) 0 0
\(321\) −5.92375 −0.330631
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −3.54461 + 0.660123i −0.196619 + 0.0366170i
\(326\) 0 0
\(327\) 5.57720i 0.308420i
\(328\) 0 0
\(329\) 25.3462 1.39738
\(330\) 0 0
\(331\) 19.7230i 1.08408i 0.840354 + 0.542038i \(0.182347\pi\)
−0.840354 + 0.542038i \(0.817653\pi\)
\(332\) 0 0
\(333\) 24.7034i 1.35374i
\(334\) 0 0
\(335\) 6.97370 0.381014
\(336\) 0 0
\(337\) −20.0000 −1.08947 −0.544735 0.838608i \(-0.683370\pi\)
−0.544735 + 0.838608i \(0.683370\pi\)
\(338\) 0 0
\(339\) 0.540588 0.0293607
\(340\) 0 0
\(341\) −9.61419 −0.520638
\(342\) 0 0
\(343\) 4.23103i 0.228454i
\(344\) 0 0
\(345\) 1.84522i 0.0993434i
\(346\) 0 0
\(347\) 2.61017 0.140121 0.0700607 0.997543i \(-0.477681\pi\)
0.0700607 + 0.997543i \(0.477681\pi\)
\(348\) 0 0
\(349\) 3.08921i 0.165362i 0.996576 + 0.0826809i \(0.0263482\pi\)
−0.996576 + 0.0826809i \(0.973652\pi\)
\(350\) 0 0
\(351\) −7.08921 + 1.32025i −0.378394 + 0.0704695i
\(352\) 0 0
\(353\) 20.9211i 1.11352i −0.830674 0.556759i \(-0.812045\pi\)
0.830674 0.556759i \(-0.187955\pi\)
\(354\) 0 0
\(355\) −12.6731 −0.672617
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 30.8515i 1.62828i 0.580668 + 0.814140i \(0.302791\pi\)
−0.580668 + 0.814140i \(0.697209\pi\)
\(360\) 0 0
\(361\) 10.7926 0.568032
\(362\) 0 0
\(363\) 2.92776 0.153668
\(364\) 0 0
\(365\) 3.43576 0.179836
\(366\) 0 0
\(367\) −21.0196 −1.09722 −0.548608 0.836080i \(-0.684842\pi\)
−0.548608 + 0.836080i \(0.684842\pi\)
\(368\) 0 0
\(369\) 26.2177i 1.36484i
\(370\) 0 0
\(371\) 25.0500i 1.30053i
\(372\) 0 0
\(373\) 29.3462 1.51949 0.759743 0.650223i \(-0.225325\pi\)
0.759743 + 0.650223i \(0.225325\pi\)
\(374\) 0 0
\(375\) 0.339877i 0.0175512i
\(376\) 0 0
\(377\) −18.4487 + 3.43576i −0.950157 + 0.176951i
\(378\) 0 0
\(379\) 11.3528i 0.583156i 0.956547 + 0.291578i \(0.0941803\pi\)
−0.956547 + 0.291578i \(0.905820\pi\)
\(380\) 0 0
\(381\) 0.0369937 0.00189524
\(382\) 0 0
\(383\) 20.5642i 1.05078i −0.850860 0.525392i \(-0.823919\pi\)
0.850860 0.525392i \(-0.176081\pi\)
\(384\) 0 0
\(385\) 6.00000i 0.305788i
\(386\) 0 0
\(387\) 2.82786 0.143748
\(388\) 0 0
\(389\) 26.8189 1.35977 0.679887 0.733317i \(-0.262029\pi\)
0.679887 + 0.733317i \(0.262029\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 3.53793 0.178465
\(394\) 0 0
\(395\) 13.1285i 0.660565i
\(396\) 0 0
\(397\) 36.1022i 1.81192i 0.423367 + 0.905958i \(0.360848\pi\)
−0.423367 + 0.905958i \(0.639152\pi\)
\(398\) 0 0
\(399\) −3.78231 −0.189352
\(400\) 0 0
\(401\) 4.07852i 0.203672i 0.994801 + 0.101836i \(0.0324716\pi\)
−0.994801 + 0.101836i \(0.967528\pi\)
\(402\) 0 0
\(403\) 4.10884 + 22.0629i 0.204676 + 1.09903i
\(404\) 0 0
\(405\) 7.97370i 0.396216i
\(406\) 0 0
\(407\) −13.2284 −0.655706
\(408\) 0 0
\(409\) 0.627148i 0.0310105i 0.999880 + 0.0155052i \(0.00493567\pi\)
−0.999880 + 0.0155052i \(0.995064\pi\)
\(410\) 0 0
\(411\) 3.08921i 0.152380i
\(412\) 0 0
\(413\) −17.3069 −0.851617
\(414\) 0 0
\(415\) 8.56424 0.420402
\(416\) 0 0
\(417\) −3.07587 −0.150626
\(418\) 0 0
\(419\) 11.5513 0.564317 0.282158 0.959368i \(-0.408950\pi\)
0.282158 + 0.959368i \(0.408950\pi\)
\(420\) 0 0
\(421\) 35.4853i 1.72945i 0.502246 + 0.864725i \(0.332507\pi\)
−0.502246 + 0.864725i \(0.667493\pi\)
\(422\) 0 0
\(423\) 18.8212i 0.915117i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 37.4987i 1.81469i
\(428\) 0 0
\(429\) 0.346549 + 1.86083i 0.0167315 + 0.0898419i
\(430\) 0 0
\(431\) 33.7623i 1.62627i 0.582073 + 0.813136i \(0.302242\pi\)
−0.582073 + 0.813136i \(0.697758\pi\)
\(432\) 0 0
\(433\) −2.00000 −0.0961139 −0.0480569 0.998845i \(-0.515303\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) 0 0
\(435\) 1.76897i 0.0848155i
\(436\) 0 0
\(437\) 15.5535i 0.744027i
\(438\) 0 0
\(439\) 19.0892 0.911078 0.455539 0.890216i \(-0.349446\pi\)
0.455539 + 0.890216i \(0.349446\pi\)
\(440\) 0 0
\(441\) 23.3332 1.11110
\(442\) 0 0
\(443\) 10.2873 0.488763 0.244382 0.969679i \(-0.421415\pi\)
0.244382 + 0.969679i \(0.421415\pi\)
\(444\) 0 0
\(445\) 17.1285 0.811968
\(446\) 0 0
\(447\) 3.78231i 0.178897i
\(448\) 0 0
\(449\) 2.21769i 0.104659i −0.998630 0.0523296i \(-0.983335\pi\)
0.998630 0.0523296i \(-0.0166646\pi\)
\(450\) 0 0
\(451\) −14.0393 −0.661083
\(452\) 0 0
\(453\) 3.10483i 0.145877i
\(454\) 0 0
\(455\) 13.7690 2.56424i 0.645499 0.120213i
\(456\) 0 0
\(457\) 17.3069i 0.809583i 0.914409 + 0.404791i \(0.132656\pi\)
−0.914409 + 0.404791i \(0.867344\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 16.1392i 0.751676i −0.926685 0.375838i \(-0.877355\pi\)
0.926685 0.375838i \(-0.122645\pi\)
\(462\) 0 0
\(463\) 6.52498i 0.303241i −0.988439 0.151621i \(-0.951551\pi\)
0.988439 0.151621i \(-0.0484492\pi\)
\(464\) 0 0
\(465\) 2.11552 0.0981047
\(466\) 0 0
\(467\) 35.4291 1.63946 0.819731 0.572748i \(-0.194123\pi\)
0.819731 + 0.572748i \(0.194123\pi\)
\(468\) 0 0
\(469\) −27.0892 −1.25086
\(470\) 0 0
\(471\) 1.66906 0.0769064
\(472\) 0 0
\(473\) 1.51429i 0.0696269i
\(474\) 0 0
\(475\) 2.86485i 0.131448i
\(476\) 0 0
\(477\) −18.6012 −0.851692
\(478\) 0 0
\(479\) 26.7730i 1.22329i 0.791133 + 0.611644i \(0.209492\pi\)
−0.791133 + 0.611644i \(0.790508\pi\)
\(480\) 0 0
\(481\) 5.65345 + 30.3569i 0.257775 + 1.38415i
\(482\) 0 0
\(483\) 7.16774i 0.326143i
\(484\) 0 0
\(485\) −13.7690 −0.625217
\(486\) 0 0
\(487\) 18.3725i 0.832536i −0.909242 0.416268i \(-0.863338\pi\)
0.909242 0.416268i \(-0.136662\pi\)
\(488\) 0 0
\(489\) 5.75562i 0.260278i
\(490\) 0 0
\(491\) 8.81892 0.397992 0.198996 0.980000i \(-0.436232\pi\)
0.198996 + 0.980000i \(0.436232\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) −4.45539 −0.200255
\(496\) 0 0
\(497\) 49.2284 2.20819
\(498\) 0 0
\(499\) 23.7756i 1.06434i −0.846636 0.532172i \(-0.821376\pi\)
0.846636 0.532172i \(-0.178624\pi\)
\(500\) 0 0
\(501\) 7.34616i 0.328202i
\(502\) 0 0
\(503\) −39.1455 −1.74541 −0.872705 0.488248i \(-0.837636\pi\)
−0.872705 + 0.488248i \(0.837636\pi\)
\(504\) 0 0
\(505\) 6.44872i 0.286964i
\(506\) 0 0
\(507\) 4.12219 1.59054i 0.183073 0.0706384i
\(508\) 0 0
\(509\) 44.5745i 1.97573i 0.155309 + 0.987866i \(0.450363\pi\)
−0.155309 + 0.987866i \(0.549637\pi\)
\(510\) 0 0
\(511\) −13.3462 −0.590400
\(512\) 0 0
\(513\) 5.72971i 0.252973i
\(514\) 0 0
\(515\) 12.1088i 0.533579i
\(516\) 0 0
\(517\) 10.0785 0.443253
\(518\) 0 0
\(519\) −2.73236 −0.119937
\(520\) 0 0
\(521\) −17.2047 −0.753753 −0.376876 0.926264i \(-0.623002\pi\)
−0.376876 + 0.926264i \(0.623002\pi\)
\(522\) 0 0
\(523\) −3.19806 −0.139841 −0.0699207 0.997553i \(-0.522275\pi\)
−0.0699207 + 0.997553i \(0.522275\pi\)
\(524\) 0 0
\(525\) 1.32025i 0.0576203i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 6.47502 0.281523
\(530\) 0 0
\(531\) 12.8515i 0.557708i
\(532\) 0 0
\(533\) 6.00000 + 32.2177i 0.259889 + 1.39550i
\(534\) 0 0
\(535\) 17.4291i 0.753525i
\(536\) 0 0
\(537\) −3.69044 −0.159254
\(538\) 0 0
\(539\) 12.4947i 0.538183i
\(540\) 0 0
\(541\) 1.65118i 0.0709899i 0.999370 + 0.0354950i \(0.0113008\pi\)
−0.999370 + 0.0354950i \(0.988699\pi\)
\(542\) 0 0
\(543\) 3.59054 0.154085
\(544\) 0 0
\(545\) −16.4095 −0.702904
\(546\) 0 0
\(547\) −38.3265 −1.63872 −0.819362 0.573276i \(-0.805672\pi\)
−0.819362 + 0.573276i \(0.805672\pi\)
\(548\) 0 0
\(549\) −27.8452 −1.18841
\(550\) 0 0
\(551\) 14.9108i 0.635221i
\(552\) 0 0
\(553\) 50.9973i 2.16863i
\(554\) 0 0
\(555\) 2.91079 0.123556
\(556\) 0 0
\(557\) 31.8711i 1.35042i −0.737624 0.675212i \(-0.764052\pi\)
0.737624 0.675212i \(-0.235948\pi\)
\(558\) 0 0
\(559\) −3.47502 + 0.647164i −0.146978 + 0.0273721i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −5.87781 −0.247720 −0.123860 0.992300i \(-0.539527\pi\)
−0.123860 + 0.992300i \(0.539527\pi\)
\(564\) 0 0
\(565\) 1.59054i 0.0669145i
\(566\) 0 0
\(567\) 30.9737i 1.30077i
\(568\) 0 0
\(569\) −16.0629 −0.673392 −0.336696 0.941613i \(-0.609310\pi\)
−0.336696 + 0.941613i \(0.609310\pi\)
\(570\) 0 0
\(571\) −5.04995 −0.211334 −0.105667 0.994402i \(-0.533698\pi\)
−0.105667 + 0.994402i \(0.533698\pi\)
\(572\) 0 0
\(573\) 2.99735 0.125216
\(574\) 0 0
\(575\) 5.42909 0.226409
\(576\) 0 0
\(577\) 17.3832i 0.723670i 0.932242 + 0.361835i \(0.117850\pi\)
−0.932242 + 0.361835i \(0.882150\pi\)
\(578\) 0 0
\(579\) 0.0918667i 0.00381785i
\(580\) 0 0
\(581\) −33.2676 −1.38017
\(582\) 0 0
\(583\) 9.96074i 0.412532i
\(584\) 0 0
\(585\) 1.90411 + 10.2244i 0.0787254 + 0.422725i
\(586\) 0 0
\(587\) 7.57493i 0.312651i −0.987706 0.156325i \(-0.950035\pi\)
0.987706 0.156325i \(-0.0499648\pi\)
\(588\) 0 0
\(589\) −17.8319 −0.734750
\(590\) 0 0
\(591\) 1.34616i 0.0553738i
\(592\) 0 0
\(593\) 16.9500i 0.696055i −0.937484 0.348028i \(-0.886852\pi\)
0.937484 0.348028i \(-0.113148\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −2.05261 −0.0840075
\(598\) 0 0
\(599\) 43.6771 1.78460 0.892299 0.451445i \(-0.149091\pi\)
0.892299 + 0.451445i \(0.149091\pi\)
\(600\) 0 0
\(601\) 2.07852 0.0847847 0.0423924 0.999101i \(-0.486502\pi\)
0.0423924 + 0.999101i \(0.486502\pi\)
\(602\) 0 0
\(603\) 20.1155i 0.819167i
\(604\) 0 0
\(605\) 8.61419i 0.350217i
\(606\) 0 0
\(607\) 34.3265 1.39327 0.696635 0.717425i \(-0.254680\pi\)
0.696635 + 0.717425i \(0.254680\pi\)
\(608\) 0 0
\(609\) 6.87153i 0.278448i
\(610\) 0 0
\(611\) −4.30729 23.1285i −0.174254 0.935678i
\(612\) 0 0
\(613\) 18.3569i 0.741426i 0.928747 + 0.370713i \(0.120887\pi\)
−0.928747 + 0.370713i \(0.879113\pi\)
\(614\) 0 0
\(615\) 3.08921 0.124569
\(616\) 0 0
\(617\) 37.5246i 1.51068i 0.655331 + 0.755342i \(0.272529\pi\)
−0.655331 + 0.755342i \(0.727471\pi\)
\(618\) 0 0
\(619\) 12.8256i 0.515504i −0.966211 0.257752i \(-0.917018\pi\)
0.966211 0.257752i \(-0.0829818\pi\)
\(620\) 0 0
\(621\) 10.8582 0.435724
\(622\) 0 0
\(623\) −66.5353 −2.66568
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −1.50398 −0.0600632
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 12.6472i 0.503476i −0.967795 0.251738i \(-0.918998\pi\)
0.967795 0.251738i \(-0.0810021\pi\)
\(632\) 0 0
\(633\) 5.51202 0.219083
\(634\) 0 0
\(635\) 0.108844i 0.00431935i
\(636\) 0 0
\(637\) −28.6731 + 5.33988i −1.13607 + 0.211574i
\(638\) 0 0
\(639\) 36.5553i 1.44611i
\(640\) 0 0
\(641\) 5.10256 0.201539 0.100769 0.994910i \(-0.467870\pi\)
0.100769 + 0.994910i \(0.467870\pi\)
\(642\) 0 0
\(643\) 0.346549i 0.0136666i 0.999977 + 0.00683328i \(0.00217512\pi\)
−0.999977 + 0.00683328i \(0.997825\pi\)
\(644\) 0 0
\(645\) 0.333205i 0.0131199i
\(646\) 0 0
\(647\) −44.2480 −1.73957 −0.869784 0.493432i \(-0.835742\pi\)
−0.869784 + 0.493432i \(0.835742\pi\)
\(648\) 0 0
\(649\) −6.88183 −0.270135
\(650\) 0 0
\(651\) −8.21769 −0.322077
\(652\) 0 0
\(653\) −14.0393 −0.549399 −0.274699 0.961530i \(-0.588578\pi\)
−0.274699 + 0.961530i \(0.588578\pi\)
\(654\) 0 0
\(655\) 10.4095i 0.406731i
\(656\) 0 0
\(657\) 9.91040i 0.386642i
\(658\) 0 0
\(659\) −12.4487 −0.484933 −0.242467 0.970160i \(-0.577957\pi\)
−0.242467 + 0.970160i \(0.577957\pi\)
\(660\) 0 0
\(661\) 11.8216i 0.459806i −0.973214 0.229903i \(-0.926159\pi\)
0.973214 0.229903i \(-0.0738409\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 11.1285i 0.431544i
\(666\) 0 0
\(667\) 28.2569 1.09411
\(668\) 0 0
\(669\) 3.42015i 0.132231i
\(670\) 0 0
\(671\) 14.9108i 0.575625i
\(672\) 0 0
\(673\) 31.3069 1.20679 0.603396 0.797442i \(-0.293814\pi\)
0.603396 + 0.797442i \(0.293814\pi\)
\(674\) 0 0
\(675\) 2.00000 0.0769800
\(676\) 0 0
\(677\) 28.8582 1.10911 0.554555 0.832147i \(-0.312889\pi\)
0.554555 + 0.832147i \(0.312889\pi\)
\(678\) 0 0
\(679\) 53.4853 2.05258
\(680\) 0 0
\(681\) 1.16774i 0.0447478i
\(682\) 0 0
\(683\) 20.5642i 0.786869i 0.919353 + 0.393434i \(0.128713\pi\)
−0.919353 + 0.393434i \(0.871287\pi\)
\(684\) 0 0
\(685\) −9.08921 −0.347281
\(686\) 0 0
\(687\) 7.76897i 0.296405i
\(688\) 0 0
\(689\) 22.8582 4.25695i 0.870827 0.162177i
\(690\) 0 0
\(691\) 6.67308i 0.253856i 0.991912 + 0.126928i \(0.0405117\pi\)
−0.991912 + 0.126928i \(0.959488\pi\)
\(692\) 0 0
\(693\) 17.3069 0.657435
\(694\) 0 0
\(695\) 9.04995i 0.343284i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) −1.49867 −0.0566850
\(700\) 0 0
\(701\) 9.62980 0.363713 0.181856 0.983325i \(-0.441789\pi\)
0.181856 + 0.983325i \(0.441789\pi\)
\(702\) 0 0
\(703\) −24.5353 −0.925366
\(704\) 0 0
\(705\) −2.21769 −0.0835230
\(706\) 0 0
\(707\) 25.0500i 0.942100i
\(708\) 0 0
\(709\) 33.8082i 1.26969i 0.772638 + 0.634847i \(0.218937\pi\)
−0.772638 + 0.634847i \(0.781063\pi\)
\(710\) 0 0
\(711\) −37.8689 −1.42019
\(712\) 0 0
\(713\) 33.7926i 1.26554i
\(714\) 0 0
\(715\) 5.47502 1.01963i 0.204754 0.0381320i
\(716\) 0 0
\(717\) 7.75336i 0.289554i
\(718\) 0 0
\(719\) 36.2043 1.35019 0.675097 0.737729i \(-0.264102\pi\)
0.675097 + 0.737729i \(0.264102\pi\)
\(720\) 0 0
\(721\) 47.0366i 1.75173i
\(722\) 0 0
\(723\) 3.62980i 0.134994i
\(724\) 0 0
\(725\) 5.20473 0.193299
\(726\) 0 0
\(727\) 46.2480 1.71524 0.857622 0.514281i \(-0.171941\pi\)
0.857622 + 0.514281i \(0.171941\pi\)
\(728\) 0 0
\(729\) −20.9607 −0.776324
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 20.0393i 0.740167i 0.928998 + 0.370084i \(0.120671\pi\)
−0.928998 + 0.370084i \(0.879329\pi\)
\(734\) 0 0
\(735\) 2.74934i 0.101411i
\(736\) 0 0
\(737\) −10.7716 −0.396778
\(738\) 0 0
\(739\) 52.6338i 1.93617i 0.250629 + 0.968083i \(0.419363\pi\)
−0.250629 + 0.968083i \(0.580637\pi\)
\(740\) 0 0
\(741\) 0.642760 + 3.45137i 0.0236124 + 0.126789i
\(742\) 0 0
\(743\) 47.9497i 1.75910i 0.475804 + 0.879551i \(0.342157\pi\)
−0.475804 + 0.879551i \(0.657843\pi\)
\(744\) 0 0
\(745\) −11.1285 −0.407716
\(746\) 0 0
\(747\) 24.7034i 0.903850i
\(748\) 0 0
\(749\) 67.7030i 2.47381i
\(750\) 0 0
\(751\) −23.0892 −0.842537 −0.421269 0.906936i \(-0.638415\pi\)
−0.421269 + 0.906936i \(0.638415\pi\)
\(752\) 0 0
\(753\) 7.22838 0.263417
\(754\) 0 0
\(755\) −9.13515 −0.332462
\(756\) 0 0
\(757\) 44.3176 1.61075 0.805375 0.592765i \(-0.201964\pi\)
0.805375 + 0.592765i \(0.201964\pi\)
\(758\) 0 0
\(759\) 2.85014i 0.103454i
\(760\) 0 0
\(761\) 41.4853i 1.50384i 0.659253 + 0.751921i \(0.270873\pi\)
−0.659253 + 0.751921i \(0.729127\pi\)
\(762\) 0 0
\(763\) 63.7423 2.30763
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 2.94111 + 15.7926i 0.106197 + 0.570238i
\(768\) 0 0
\(769\) 28.5879i 1.03091i −0.856918 0.515453i \(-0.827624\pi\)
0.856918 0.515453i \(-0.172376\pi\)
\(770\) 0 0
\(771\) 6.42280 0.231312
\(772\) 0 0
\(773\) 15.4358i 0.555186i 0.960699 + 0.277593i \(0.0895366\pi\)
−0.960699 + 0.277593i \(0.910463\pi\)
\(774\) 0 0
\(775\) 6.22436i 0.223586i
\(776\) 0 0
\(777\) −11.3069 −0.405633
\(778\) 0 0
\(779\) −26.0393 −0.932953
\(780\) 0 0
\(781\) 19.5749 0.700446
\(782\) 0 0
\(783\) 10.4095 0.372004
\(784\) 0 0
\(785\) 4.91079i 0.175273i
\(786\) 0 0
\(787\) 5.38316i 0.191889i 0.995387 + 0.0959444i \(0.0305871\pi\)
−0.995387 + 0.0959444i \(0.969413\pi\)
\(788\) 0 0
\(789\) 4.42507 0.157537
\(790\) 0 0
\(791\) 6.17843i 0.219680i
\(792\) 0 0
\(793\) 34.2177 6.37247i 1.21511 0.226293i
\(794\) 0 0
\(795\) 2.19177i 0.0777341i
\(796\) 0 0
\(797\) 35.7556 1.26653 0.633265 0.773935i \(-0.281714\pi\)
0.633265 + 0.773935i \(0.281714\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 49.4068i 1.74570i
\(802\) 0 0
\(803\) −5.30690 −0.187277
\(804\) 0 0
\(805\) −21.0892 −0.743297
\(806\)