Properties

Label 1216.2.h.d.1215.6
Level $1216$
Weight $2$
Character 1216.1215
Analytic conductor $9.710$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1216.h (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.70980888579\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.14453810176.1
Defining polynomial: \(x^{8} + 3 x^{6} + 6 x^{4} + 12 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 76)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1215.6
Root \(-1.06789 + 0.927153i\) of defining polynomial
Character \(\chi\) \(=\) 1216.1215
Dual form 1216.2.h.d.1215.5

$q$-expansion

\(f(q)\) \(=\) \(q+1.19935 q^{3} -1.56155 q^{5} +0.868210i q^{7} -1.56155 q^{9} +O(q^{10})\) \(q+1.19935 q^{3} -1.56155 q^{5} +0.868210i q^{7} -1.56155 q^{9} -3.09218i q^{11} -4.74990i q^{13} -1.87285 q^{15} -1.00000 q^{17} +(-3.07221 + 3.09218i) q^{19} +1.04129i q^{21} -3.96039i q^{23} -2.56155 q^{25} -5.47091 q^{27} +8.45851i q^{29} -4.27156 q^{31} -3.70861i q^{33} -1.35576i q^{35} -3.70861i q^{37} -5.69681i q^{39} -3.70861i q^{41} -11.0129i q^{43} +2.43845 q^{45} -9.27653i q^{47} +6.24621 q^{49} -1.19935 q^{51} -1.04129i q^{53} +4.82860i q^{55} +(-3.68466 + 3.70861i) q^{57} -11.6153 q^{59} +0.684658 q^{61} -1.35576i q^{63} +7.41722i q^{65} +9.74247 q^{67} -4.74990i q^{69} -10.9418 q^{71} +8.12311 q^{73} -3.07221 q^{75} +2.68466 q^{77} -8.01726 q^{79} -1.87689 q^{81} -9.65719i q^{83} +1.56155 q^{85} +10.1447i q^{87} -5.79119i q^{89} +4.12391 q^{91} -5.12311 q^{93} +(4.79741 - 4.82860i) q^{95} +16.9170i q^{97} +4.82860i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 4q^{5} + 4q^{9} + O(q^{10}) \) \( 8q + 4q^{5} + 4q^{9} - 8q^{17} - 4q^{25} + 36q^{45} - 16q^{49} + 20q^{57} - 44q^{61} + 32q^{73} - 28q^{77} - 48q^{81} - 4q^{85} - 8q^{93} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(705\) \(837\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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
\(3\) 1.19935 0.692447 0.346223 0.938152i \(-0.387464\pi\)
0.346223 + 0.938152i \(0.387464\pi\)
\(4\) 0 0
\(5\) −1.56155 −0.698348 −0.349174 0.937058i \(-0.613538\pi\)
−0.349174 + 0.937058i \(0.613538\pi\)
\(6\) 0 0
\(7\) 0.868210i 0.328153i 0.986448 + 0.164076i \(0.0524643\pi\)
−0.986448 + 0.164076i \(0.947536\pi\)
\(8\) 0 0
\(9\) −1.56155 −0.520518
\(10\) 0 0
\(11\) 3.09218i 0.932326i −0.884699 0.466163i \(-0.845636\pi\)
0.884699 0.466163i \(-0.154364\pi\)
\(12\) 0 0
\(13\) 4.74990i 1.31739i −0.752412 0.658693i \(-0.771110\pi\)
0.752412 0.658693i \(-0.228890\pi\)
\(14\) 0 0
\(15\) −1.87285 −0.483569
\(16\) 0 0
\(17\) −1.00000 −0.242536 −0.121268 0.992620i \(-0.538696\pi\)
−0.121268 + 0.992620i \(0.538696\pi\)
\(18\) 0 0
\(19\) −3.07221 + 3.09218i −0.704812 + 0.709394i
\(20\) 0 0
\(21\) 1.04129i 0.227228i
\(22\) 0 0
\(23\) 3.96039i 0.825798i −0.910777 0.412899i \(-0.864516\pi\)
0.910777 0.412899i \(-0.135484\pi\)
\(24\) 0 0
\(25\) −2.56155 −0.512311
\(26\) 0 0
\(27\) −5.47091 −1.05288
\(28\) 0 0
\(29\) 8.45851i 1.57071i 0.619048 + 0.785353i \(0.287519\pi\)
−0.619048 + 0.785353i \(0.712481\pi\)
\(30\) 0 0
\(31\) −4.27156 −0.767195 −0.383597 0.923500i \(-0.625315\pi\)
−0.383597 + 0.923500i \(0.625315\pi\)
\(32\) 0 0
\(33\) 3.70861i 0.645586i
\(34\) 0 0
\(35\) 1.35576i 0.229165i
\(36\) 0 0
\(37\) 3.70861i 0.609692i −0.952402 0.304846i \(-0.901395\pi\)
0.952402 0.304846i \(-0.0986050\pi\)
\(38\) 0 0
\(39\) 5.69681i 0.912219i
\(40\) 0 0
\(41\) 3.70861i 0.579188i −0.957150 0.289594i \(-0.906480\pi\)
0.957150 0.289594i \(-0.0935202\pi\)
\(42\) 0 0
\(43\) 11.0129i 1.67946i −0.543005 0.839729i \(-0.682714\pi\)
0.543005 0.839729i \(-0.317286\pi\)
\(44\) 0 0
\(45\) 2.43845 0.363502
\(46\) 0 0
\(47\) 9.27653i 1.35312i −0.736387 0.676560i \(-0.763470\pi\)
0.736387 0.676560i \(-0.236530\pi\)
\(48\) 0 0
\(49\) 6.24621 0.892316
\(50\) 0 0
\(51\) −1.19935 −0.167943
\(52\) 0 0
\(53\) 1.04129i 0.143032i −0.997439 0.0715161i \(-0.977216\pi\)
0.997439 0.0715161i \(-0.0227837\pi\)
\(54\) 0 0
\(55\) 4.82860i 0.651088i
\(56\) 0 0
\(57\) −3.68466 + 3.70861i −0.488045 + 0.491217i
\(58\) 0 0
\(59\) −11.6153 −1.51219 −0.756093 0.654464i \(-0.772894\pi\)
−0.756093 + 0.654464i \(0.772894\pi\)
\(60\) 0 0
\(61\) 0.684658 0.0876615 0.0438308 0.999039i \(-0.486044\pi\)
0.0438308 + 0.999039i \(0.486044\pi\)
\(62\) 0 0
\(63\) 1.35576i 0.170809i
\(64\) 0 0
\(65\) 7.41722i 0.919993i
\(66\) 0 0
\(67\) 9.74247 1.19023 0.595116 0.803640i \(-0.297106\pi\)
0.595116 + 0.803640i \(0.297106\pi\)
\(68\) 0 0
\(69\) 4.74990i 0.571821i
\(70\) 0 0
\(71\) −10.9418 −1.29856 −0.649278 0.760551i \(-0.724929\pi\)
−0.649278 + 0.760551i \(0.724929\pi\)
\(72\) 0 0
\(73\) 8.12311 0.950738 0.475369 0.879787i \(-0.342315\pi\)
0.475369 + 0.879787i \(0.342315\pi\)
\(74\) 0 0
\(75\) −3.07221 −0.354748
\(76\) 0 0
\(77\) 2.68466 0.305945
\(78\) 0 0
\(79\) −8.01726 −0.902013 −0.451006 0.892521i \(-0.648935\pi\)
−0.451006 + 0.892521i \(0.648935\pi\)
\(80\) 0 0
\(81\) −1.87689 −0.208544
\(82\) 0 0
\(83\) 9.65719i 1.06001i −0.847993 0.530007i \(-0.822189\pi\)
0.847993 0.530007i \(-0.177811\pi\)
\(84\) 0 0
\(85\) 1.56155 0.169374
\(86\) 0 0
\(87\) 10.1447i 1.08763i
\(88\) 0 0
\(89\) 5.79119i 0.613865i −0.951731 0.306932i \(-0.900697\pi\)
0.951731 0.306932i \(-0.0993026\pi\)
\(90\) 0 0
\(91\) 4.12391 0.432303
\(92\) 0 0
\(93\) −5.12311 −0.531241
\(94\) 0 0
\(95\) 4.79741 4.82860i 0.492204 0.495404i
\(96\) 0 0
\(97\) 16.9170i 1.71766i 0.512258 + 0.858832i \(0.328809\pi\)
−0.512258 + 0.858832i \(0.671191\pi\)
\(98\) 0 0
\(99\) 4.82860i 0.485292i
\(100\) 0 0
\(101\) −8.24621 −0.820529 −0.410264 0.911967i \(-0.634564\pi\)
−0.410264 + 0.911967i \(0.634564\pi\)
\(102\) 0 0
\(103\) 3.74571 0.369075 0.184538 0.982825i \(-0.440921\pi\)
0.184538 + 0.982825i \(0.440921\pi\)
\(104\) 0 0
\(105\) 1.62603i 0.158684i
\(106\) 0 0
\(107\) −7.86962 −0.760785 −0.380392 0.924825i \(-0.624211\pi\)
−0.380392 + 0.924825i \(0.624211\pi\)
\(108\) 0 0
\(109\) 15.8757i 1.52062i 0.649561 + 0.760310i \(0.274953\pi\)
−0.649561 + 0.760310i \(0.725047\pi\)
\(110\) 0 0
\(111\) 4.44793i 0.422179i
\(112\) 0 0
\(113\) 3.70861i 0.348877i 0.984668 + 0.174438i \(0.0558110\pi\)
−0.984668 + 0.174438i \(0.944189\pi\)
\(114\) 0 0
\(115\) 6.18435i 0.576694i
\(116\) 0 0
\(117\) 7.41722i 0.685722i
\(118\) 0 0
\(119\) 0.868210i 0.0795887i
\(120\) 0 0
\(121\) 1.43845 0.130768
\(122\) 0 0
\(123\) 4.44793i 0.401057i
\(124\) 0 0
\(125\) 11.8078 1.05612
\(126\) 0 0
\(127\) 1.87285 0.166189 0.0830944 0.996542i \(-0.473520\pi\)
0.0830944 + 0.996542i \(0.473520\pi\)
\(128\) 0 0
\(129\) 13.2084i 1.16294i
\(130\) 0 0
\(131\) 4.82860i 0.421876i 0.977499 + 0.210938i \(0.0676519\pi\)
−0.977499 + 0.210938i \(0.932348\pi\)
\(132\) 0 0
\(133\) −2.68466 2.66732i −0.232789 0.231286i
\(134\) 0 0
\(135\) 8.54312 0.735274
\(136\) 0 0
\(137\) −3.87689 −0.331225 −0.165613 0.986191i \(-0.552960\pi\)
−0.165613 + 0.986191i \(0.552960\pi\)
\(138\) 0 0
\(139\) 0.380664i 0.0322875i 0.999870 + 0.0161438i \(0.00513894\pi\)
−0.999870 + 0.0161438i \(0.994861\pi\)
\(140\) 0 0
\(141\) 11.1258i 0.936964i
\(142\) 0 0
\(143\) −14.6875 −1.22823
\(144\) 0 0
\(145\) 13.2084i 1.09690i
\(146\) 0 0
\(147\) 7.49141 0.617881
\(148\) 0 0
\(149\) 17.8078 1.45887 0.729434 0.684051i \(-0.239783\pi\)
0.729434 + 0.684051i \(0.239783\pi\)
\(150\) 0 0
\(151\) 24.2824 1.97607 0.988035 0.154231i \(-0.0492900\pi\)
0.988035 + 0.154231i \(0.0492900\pi\)
\(152\) 0 0
\(153\) 1.56155 0.126244
\(154\) 0 0
\(155\) 6.67026 0.535769
\(156\) 0 0
\(157\) −6.00000 −0.478852 −0.239426 0.970915i \(-0.576959\pi\)
−0.239426 + 0.970915i \(0.576959\pi\)
\(158\) 0 0
\(159\) 1.24887i 0.0990422i
\(160\) 0 0
\(161\) 3.43845 0.270988
\(162\) 0 0
\(163\) 10.6323i 0.832785i 0.909185 + 0.416392i \(0.136706\pi\)
−0.909185 + 0.416392i \(0.863294\pi\)
\(164\) 0 0
\(165\) 5.79119i 0.450844i
\(166\) 0 0
\(167\) 0.525853 0.0406917 0.0203459 0.999793i \(-0.493523\pi\)
0.0203459 + 0.999793i \(0.493523\pi\)
\(168\) 0 0
\(169\) −9.56155 −0.735504
\(170\) 0 0
\(171\) 4.79741 4.82860i 0.366867 0.369252i
\(172\) 0 0
\(173\) 16.9170i 1.28618i −0.765792 0.643089i \(-0.777653\pi\)
0.765792 0.643089i \(-0.222347\pi\)
\(174\) 0 0
\(175\) 2.22397i 0.168116i
\(176\) 0 0
\(177\) −13.9309 −1.04711
\(178\) 0 0
\(179\) −9.06897 −0.677847 −0.338923 0.940814i \(-0.610063\pi\)
−0.338923 + 0.940814i \(0.610063\pi\)
\(180\) 0 0
\(181\) 2.08258i 0.154797i −0.997000 0.0773985i \(-0.975339\pi\)
0.997000 0.0773985i \(-0.0246614\pi\)
\(182\) 0 0
\(183\) 0.821147 0.0607009
\(184\) 0 0
\(185\) 5.79119i 0.425777i
\(186\) 0 0
\(187\) 3.09218i 0.226122i
\(188\) 0 0
\(189\) 4.74990i 0.345504i
\(190\) 0 0
\(191\) 8.78898i 0.635948i 0.948099 + 0.317974i \(0.103003\pi\)
−0.948099 + 0.317974i \(0.896997\pi\)
\(192\) 0 0
\(193\) 3.70861i 0.266952i −0.991052 0.133476i \(-0.957386\pi\)
0.991052 0.133476i \(-0.0426138\pi\)
\(194\) 0 0
\(195\) 8.89586i 0.637046i
\(196\) 0 0
\(197\) −10.4924 −0.747554 −0.373777 0.927519i \(-0.621937\pi\)
−0.373777 + 0.927519i \(0.621937\pi\)
\(198\) 0 0
\(199\) 13.2369i 0.938340i 0.883108 + 0.469170i \(0.155447\pi\)
−0.883108 + 0.469170i \(0.844553\pi\)
\(200\) 0 0
\(201\) 11.6847 0.824172
\(202\) 0 0
\(203\) −7.34376 −0.515431
\(204\) 0 0
\(205\) 5.79119i 0.404474i
\(206\) 0 0
\(207\) 6.18435i 0.429842i
\(208\) 0 0
\(209\) 9.56155 + 9.49980i 0.661386 + 0.657115i
\(210\) 0 0
\(211\) 2.02050 0.139097 0.0695485 0.997579i \(-0.477844\pi\)
0.0695485 + 0.997579i \(0.477844\pi\)
\(212\) 0 0
\(213\) −13.1231 −0.899180
\(214\) 0 0
\(215\) 17.1973i 1.17285i
\(216\) 0 0
\(217\) 3.70861i 0.251757i
\(218\) 0 0
\(219\) 9.74247 0.658335
\(220\) 0 0
\(221\) 4.74990i 0.319513i
\(222\) 0 0
\(223\) −16.2651 −1.08919 −0.544595 0.838699i \(-0.683317\pi\)
−0.544595 + 0.838699i \(0.683317\pi\)
\(224\) 0 0
\(225\) 4.00000 0.266667
\(226\) 0 0
\(227\) 17.4644 1.15916 0.579578 0.814917i \(-0.303217\pi\)
0.579578 + 0.814917i \(0.303217\pi\)
\(228\) 0 0
\(229\) 2.43845 0.161137 0.0805686 0.996749i \(-0.474326\pi\)
0.0805686 + 0.996749i \(0.474326\pi\)
\(230\) 0 0
\(231\) 3.21985 0.211851
\(232\) 0 0
\(233\) −9.80776 −0.642528 −0.321264 0.946990i \(-0.604108\pi\)
−0.321264 + 0.946990i \(0.604108\pi\)
\(234\) 0 0
\(235\) 14.4858i 0.944949i
\(236\) 0 0
\(237\) −9.61553 −0.624596
\(238\) 0 0
\(239\) 19.4213i 1.25626i −0.778110 0.628129i \(-0.783821\pi\)
0.778110 0.628129i \(-0.216179\pi\)
\(240\) 0 0
\(241\) 15.2910i 0.984979i 0.870318 + 0.492490i \(0.163913\pi\)
−0.870318 + 0.492490i \(0.836087\pi\)
\(242\) 0 0
\(243\) 14.1617 0.908472
\(244\) 0 0
\(245\) −9.75379 −0.623147
\(246\) 0 0
\(247\) 14.6875 + 14.5927i 0.934545 + 0.928509i
\(248\) 0 0
\(249\) 11.5824i 0.734004i
\(250\) 0 0
\(251\) 22.4066i 1.41429i −0.707069 0.707145i \(-0.749983\pi\)
0.707069 0.707145i \(-0.250017\pi\)
\(252\) 0 0
\(253\) −12.2462 −0.769913
\(254\) 0 0
\(255\) 1.87285 0.117283
\(256\) 0 0
\(257\) 18.9996i 1.18516i −0.805511 0.592581i \(-0.798109\pi\)
0.805511 0.592581i \(-0.201891\pi\)
\(258\) 0 0
\(259\) 3.21985 0.200072
\(260\) 0 0
\(261\) 13.2084i 0.817580i
\(262\) 0 0
\(263\) 6.56502i 0.404816i 0.979301 + 0.202408i \(0.0648768\pi\)
−0.979301 + 0.202408i \(0.935123\pi\)
\(264\) 0 0
\(265\) 1.62603i 0.0998862i
\(266\) 0 0
\(267\) 6.94568i 0.425069i
\(268\) 0 0
\(269\) 3.70861i 0.226118i 0.993588 + 0.113059i \(0.0360649\pi\)
−0.993588 + 0.113059i \(0.963935\pi\)
\(270\) 0 0
\(271\) 15.3540i 0.932689i −0.884603 0.466345i \(-0.845571\pi\)
0.884603 0.466345i \(-0.154429\pi\)
\(272\) 0 0
\(273\) 4.94602 0.299347
\(274\) 0 0
\(275\) 7.92077i 0.477641i
\(276\) 0 0
\(277\) 24.0540 1.44526 0.722632 0.691233i \(-0.242932\pi\)
0.722632 + 0.691233i \(0.242932\pi\)
\(278\) 0 0
\(279\) 6.67026 0.399338
\(280\) 0 0
\(281\) 11.5824i 0.690947i 0.938429 + 0.345473i \(0.112282\pi\)
−0.938429 + 0.345473i \(0.887718\pi\)
\(282\) 0 0
\(283\) 4.82860i 0.287030i 0.989648 + 0.143515i \(0.0458406\pi\)
−0.989648 + 0.143515i \(0.954159\pi\)
\(284\) 0 0
\(285\) 5.75379 5.79119i 0.340825 0.343041i
\(286\) 0 0
\(287\) 3.21985 0.190062
\(288\) 0 0
\(289\) −16.0000 −0.941176
\(290\) 0 0
\(291\) 20.2895i 1.18939i
\(292\) 0 0
\(293\) 4.74990i 0.277492i 0.990328 + 0.138746i \(0.0443072\pi\)
−0.990328 + 0.138746i \(0.955693\pi\)
\(294\) 0 0
\(295\) 18.1379 1.05603
\(296\) 0 0
\(297\) 16.9170i 0.981625i
\(298\) 0 0
\(299\) −18.8114 −1.08789
\(300\) 0 0
\(301\) 9.56155 0.551119
\(302\) 0 0
\(303\) −9.89012 −0.568172
\(304\) 0 0
\(305\) −1.06913 −0.0612182
\(306\) 0 0
\(307\) 8.01726 0.457569 0.228785 0.973477i \(-0.426525\pi\)
0.228785 + 0.973477i \(0.426525\pi\)
\(308\) 0 0
\(309\) 4.49242 0.255565
\(310\) 0 0
\(311\) 0.868210i 0.0492317i 0.999697 + 0.0246158i \(0.00783626\pi\)
−0.999697 + 0.0246158i \(0.992164\pi\)
\(312\) 0 0
\(313\) −4.56155 −0.257834 −0.128917 0.991655i \(-0.541150\pi\)
−0.128917 + 0.991655i \(0.541150\pi\)
\(314\) 0 0
\(315\) 2.11708i 0.119284i
\(316\) 0 0
\(317\) 4.74990i 0.266781i −0.991064 0.133390i \(-0.957414\pi\)
0.991064 0.133390i \(-0.0425864\pi\)
\(318\) 0 0
\(319\) 26.1552 1.46441
\(320\) 0 0
\(321\) −9.43845 −0.526803
\(322\) 0 0
\(323\) 3.07221 3.09218i 0.170942 0.172053i
\(324\) 0 0
\(325\) 12.1671i 0.674910i
\(326\) 0 0
\(327\) 19.0406i 1.05295i
\(328\) 0 0
\(329\) 8.05398 0.444030
\(330\) 0 0
\(331\) −2.25106 −0.123729 −0.0618647 0.998085i \(-0.519705\pi\)
−0.0618647 + 0.998085i \(0.519705\pi\)
\(332\) 0 0
\(333\) 5.79119i 0.317355i
\(334\) 0 0
\(335\) −15.2134 −0.831196
\(336\) 0 0
\(337\) 26.4168i 1.43902i −0.694484 0.719508i \(-0.744367\pi\)
0.694484 0.719508i \(-0.255633\pi\)
\(338\) 0 0
\(339\) 4.44793i 0.241579i
\(340\) 0 0
\(341\) 13.2084i 0.715276i
\(342\) 0 0
\(343\) 11.5005i 0.620968i
\(344\) 0 0
\(345\) 7.41722i 0.399330i
\(346\) 0 0
\(347\) 24.1430i 1.29606i 0.761613 + 0.648032i \(0.224408\pi\)
−0.761613 + 0.648032i \(0.775592\pi\)
\(348\) 0 0
\(349\) 28.0540 1.50169 0.750847 0.660476i \(-0.229645\pi\)
0.750847 + 0.660476i \(0.229645\pi\)
\(350\) 0 0
\(351\) 25.9863i 1.38705i
\(352\) 0 0
\(353\) 6.31534 0.336132 0.168066 0.985776i \(-0.446248\pi\)
0.168066 + 0.985776i \(0.446248\pi\)
\(354\) 0 0
\(355\) 17.0862 0.906843
\(356\) 0 0
\(357\) 1.04129i 0.0551109i
\(358\) 0 0
\(359\) 27.3420i 1.44306i −0.692384 0.721529i \(-0.743440\pi\)
0.692384 0.721529i \(-0.256560\pi\)
\(360\) 0 0
\(361\) −0.123106 18.9996i −0.00647924 0.999979i
\(362\) 0 0
\(363\) 1.72521 0.0905498
\(364\) 0 0
\(365\) −12.6847 −0.663945
\(366\) 0 0
\(367\) 14.1051i 0.736282i −0.929770 0.368141i \(-0.879994\pi\)
0.929770 0.368141i \(-0.120006\pi\)
\(368\) 0 0
\(369\) 5.79119i 0.301477i
\(370\) 0 0
\(371\) 0.904059 0.0469364
\(372\) 0 0
\(373\) 12.1671i 0.629990i 0.949093 + 0.314995i \(0.102003\pi\)
−0.949093 + 0.314995i \(0.897997\pi\)
\(374\) 0 0
\(375\) 14.1617 0.731306
\(376\) 0 0
\(377\) 40.1771 2.06922
\(378\) 0 0
\(379\) −6.81791 −0.350213 −0.175106 0.984550i \(-0.556027\pi\)
−0.175106 + 0.984550i \(0.556027\pi\)
\(380\) 0 0
\(381\) 2.24621 0.115077
\(382\) 0 0
\(383\) 16.2651 0.831107 0.415554 0.909569i \(-0.363588\pi\)
0.415554 + 0.909569i \(0.363588\pi\)
\(384\) 0 0
\(385\) −4.19224 −0.213656
\(386\) 0 0
\(387\) 17.1973i 0.874188i
\(388\) 0 0
\(389\) 5.80776 0.294465 0.147233 0.989102i \(-0.452963\pi\)
0.147233 + 0.989102i \(0.452963\pi\)
\(390\) 0 0
\(391\) 3.96039i 0.200285i
\(392\) 0 0
\(393\) 5.79119i 0.292127i
\(394\) 0 0
\(395\) 12.5194 0.629918
\(396\) 0 0
\(397\) −24.9309 −1.25124 −0.625622 0.780126i \(-0.715155\pi\)
−0.625622 + 0.780126i \(0.715155\pi\)
\(398\) 0 0
\(399\) −3.21985 3.19906i −0.161194 0.160153i
\(400\) 0 0
\(401\) 20.6256i 1.02999i −0.857192 0.514997i \(-0.827793\pi\)
0.857192 0.514997i \(-0.172207\pi\)
\(402\) 0 0
\(403\) 20.2895i 1.01069i
\(404\) 0 0
\(405\) 2.93087 0.145636
\(406\) 0 0
\(407\) −11.4677 −0.568432
\(408\) 0 0
\(409\) 18.5431i 0.916895i 0.888722 + 0.458447i \(0.151594\pi\)
−0.888722 + 0.458447i \(0.848406\pi\)
\(410\) 0 0
\(411\) −4.64976 −0.229356
\(412\) 0 0
\(413\) 10.0845i 0.496228i
\(414\) 0 0
\(415\) 15.0802i 0.740259i
\(416\) 0 0
\(417\) 0.456551i 0.0223574i
\(418\) 0 0
\(419\) 27.4489i 1.34097i −0.741924 0.670484i \(-0.766087\pi\)
0.741924 0.670484i \(-0.233913\pi\)
\(420\) 0 0
\(421\) 23.2930i 1.13523i −0.823294 0.567614i \(-0.807866\pi\)
0.823294 0.567614i \(-0.192134\pi\)
\(422\) 0 0
\(423\) 14.4858i 0.704323i
\(424\) 0 0
\(425\) 2.56155 0.124254
\(426\) 0 0
\(427\) 0.594427i 0.0287664i
\(428\) 0 0
\(429\) −17.6155 −0.850486
\(430\) 0 0
\(431\) −17.0862 −0.823015 −0.411507 0.911406i \(-0.634998\pi\)
−0.411507 + 0.911406i \(0.634998\pi\)
\(432\) 0 0
\(433\) 28.0429i 1.34765i 0.738889 + 0.673827i \(0.235351\pi\)
−0.738889 + 0.673827i \(0.764649\pi\)
\(434\) 0 0
\(435\) 15.8415i 0.759544i
\(436\) 0 0
\(437\) 12.2462 + 12.1671i 0.585816 + 0.582032i
\(438\) 0 0
\(439\) −24.8082 −1.18403 −0.592015 0.805927i \(-0.701668\pi\)
−0.592015 + 0.805927i \(0.701668\pi\)
\(440\) 0 0
\(441\) −9.75379 −0.464466
\(442\) 0 0
\(443\) 34.7753i 1.65222i −0.563507 0.826111i \(-0.690548\pi\)
0.563507 0.826111i \(-0.309452\pi\)
\(444\) 0 0
\(445\) 9.04325i 0.428691i
\(446\) 0 0
\(447\) 21.3578 1.01019
\(448\) 0 0
\(449\) 35.9166i 1.69501i 0.530787 + 0.847505i \(0.321896\pi\)
−0.530787 + 0.847505i \(0.678104\pi\)
\(450\) 0 0
\(451\) −11.4677 −0.539992
\(452\) 0 0
\(453\) 29.1231 1.36832
\(454\) 0 0
\(455\) −6.43971 −0.301898
\(456\) 0 0
\(457\) 12.1231 0.567095 0.283547 0.958958i \(-0.408489\pi\)
0.283547 + 0.958958i \(0.408489\pi\)
\(458\) 0 0
\(459\) 5.47091 0.255360
\(460\) 0 0
\(461\) −19.3153 −0.899605 −0.449803 0.893128i \(-0.648506\pi\)
−0.449803 + 0.893128i \(0.648506\pi\)
\(462\) 0 0
\(463\) 21.6452i 1.00594i −0.864304 0.502970i \(-0.832241\pi\)
0.864304 0.502970i \(-0.167759\pi\)
\(464\) 0 0
\(465\) 8.00000 0.370991
\(466\) 0 0
\(467\) 3.09218i 0.143089i −0.997437 0.0715444i \(-0.977207\pi\)
0.997437 0.0715444i \(-0.0227928\pi\)
\(468\) 0 0
\(469\) 8.45851i 0.390578i
\(470\) 0 0
\(471\) −7.19612 −0.331580
\(472\) 0 0
\(473\) −34.0540 −1.56580
\(474\) 0 0
\(475\) 7.86962 7.92077i 0.361083 0.363430i
\(476\) 0 0
\(477\) 1.62603i 0.0744508i
\(478\) 0 0
\(479\) 8.68210i 0.396695i 0.980132 + 0.198348i \(0.0635575\pi\)
−0.980132 + 0.198348i \(0.936442\pi\)
\(480\) 0 0
\(481\) −17.6155 −0.803199
\(482\) 0 0
\(483\) 4.12391 0.187644
\(484\) 0 0
\(485\) 26.4168i 1.19953i
\(486\) 0 0
\(487\) −9.59482 −0.434783 −0.217391 0.976085i \(-0.569755\pi\)
−0.217391 + 0.976085i \(0.569755\pi\)
\(488\) 0 0
\(489\) 12.7519i 0.576659i
\(490\) 0 0
\(491\) 2.71151i 0.122369i −0.998126 0.0611844i \(-0.980512\pi\)
0.998126 0.0611844i \(-0.0194878\pi\)
\(492\) 0 0
\(493\) 8.45851i 0.380952i
\(494\) 0 0
\(495\) 7.54011i 0.338903i
\(496\) 0 0
\(497\) 9.49980i 0.426124i
\(498\) 0 0
\(499\) 11.0129i 0.493007i −0.969142 0.246504i \(-0.920718\pi\)
0.969142 0.246504i \(-0.0792817\pi\)
\(500\) 0 0
\(501\) 0.630683 0.0281768
\(502\) 0 0
\(503\) 35.6435i 1.58926i −0.607091 0.794632i \(-0.707664\pi\)
0.607091 0.794632i \(-0.292336\pi\)
\(504\) 0 0
\(505\) 12.8769 0.573014
\(506\) 0 0
\(507\) −11.4677 −0.509297
\(508\) 0 0
\(509\) 9.49980i 0.421071i 0.977586 + 0.210536i \(0.0675208\pi\)
−0.977586 + 0.210536i \(0.932479\pi\)
\(510\) 0 0
\(511\) 7.05256i 0.311987i
\(512\) 0 0
\(513\) 16.8078 16.9170i 0.742081 0.746905i
\(514\) 0 0
\(515\) −5.84912 −0.257743
\(516\) 0 0
\(517\) −28.6847 −1.26155
\(518\) 0 0
\(519\) 20.2895i 0.890609i
\(520\) 0 0
\(521\) 15.2910i 0.669910i −0.942234 0.334955i \(-0.891279\pi\)
0.942234 0.334955i \(-0.108721\pi\)
\(522\) 0 0
\(523\) −33.4990 −1.46481 −0.732404 0.680871i \(-0.761602\pi\)
−0.732404 + 0.680871i \(0.761602\pi\)
\(524\) 0 0
\(525\) 2.66732i 0.116411i
\(526\) 0 0
\(527\) 4.27156 0.186072
\(528\) 0 0
\(529\) 7.31534 0.318058
\(530\) 0 0
\(531\) 18.1379 0.787120
\(532\) 0 0
\(533\) −17.6155 −0.763013
\(534\) 0 0
\(535\) 12.2888 0.531292
\(536\) 0 0
\(537\) −10.8769 −0.469373
\(538\) 0 0
\(539\) 19.3144i 0.831929i
\(540\) 0 0
\(541\) 12.1922 0.524185 0.262093 0.965043i \(-0.415587\pi\)
0.262093 + 0.965043i \(0.415587\pi\)
\(542\) 0 0
\(543\) 2.49775i 0.107189i
\(544\) 0 0
\(545\) 24.7908i 1.06192i
\(546\) 0 0
\(547\) −19.4849 −0.833116 −0.416558 0.909109i \(-0.636764\pi\)
−0.416558 + 0.909109i \(0.636764\pi\)
\(548\) 0 0
\(549\) −1.06913 −0.0456294
\(550\) 0 0
\(551\) −26.1552 25.9863i −1.11425 1.10705i
\(552\) 0 0
\(553\) 6.96067i 0.295998i
\(554\) 0 0
\(555\) 6.94568i 0.294828i
\(556\) 0 0
\(557\) 18.9309 0.802127 0.401063 0.916050i \(-0.368641\pi\)
0.401063 + 0.916050i \(0.368641\pi\)
\(558\) 0 0
\(559\) −52.3104 −2.21249
\(560\) 0 0
\(561\) 3.70861i 0.156578i
\(562\) 0 0
\(563\) −33.6466 −1.41804 −0.709018 0.705191i \(-0.750861\pi\)
−0.709018 + 0.705191i \(0.750861\pi\)
\(564\) 0 0
\(565\) 5.79119i 0.243637i
\(566\) 0 0
\(567\) 1.62954i 0.0684342i
\(568\) 0 0
\(569\) 12.7519i 0.534586i −0.963615 0.267293i \(-0.913871\pi\)
0.963615 0.267293i \(-0.0861291\pi\)
\(570\) 0 0
\(571\) 17.5780i 0.735615i 0.929902 + 0.367807i \(0.119891\pi\)
−0.929902 + 0.367807i \(0.880109\pi\)
\(572\) 0 0
\(573\) 10.5411i 0.440360i
\(574\) 0 0
\(575\) 10.1447i 0.423065i
\(576\) 0 0
\(577\) 27.0000 1.12402 0.562012 0.827129i \(-0.310027\pi\)
0.562012 + 0.827129i \(0.310027\pi\)
\(578\) 0 0
\(579\) 4.44793i 0.184850i
\(580\) 0 0
\(581\) 8.38447 0.347847
\(582\) 0 0
\(583\) −3.21985 −0.133353
\(584\) 0 0
\(585\) 11.5824i 0.478873i
\(586\) 0 0
\(587\) 4.82860i 0.199297i 0.995023 + 0.0996487i \(0.0317719\pi\)
−0.995023 + 0.0996487i \(0.968228\pi\)
\(588\) 0 0
\(589\) 13.1231 13.2084i 0.540728 0.544243i
\(590\) 0 0
\(591\) −12.5841 −0.517641
\(592\) 0 0
\(593\) −36.7386 −1.50867 −0.754337 0.656487i \(-0.772042\pi\)
−0.754337 + 0.656487i \(0.772042\pi\)
\(594\) 0 0
\(595\) 1.35576i 0.0555806i
\(596\) 0 0
\(597\) 15.8757i 0.649750i
\(598\) 0 0
\(599\) 7.72197 0.315511 0.157756 0.987478i \(-0.449574\pi\)
0.157756 + 0.987478i \(0.449574\pi\)
\(600\) 0 0
\(601\) 39.1687i 1.59772i −0.601514 0.798862i \(-0.705436\pi\)
0.601514 0.798862i \(-0.294564\pi\)
\(602\) 0 0
\(603\) −15.2134 −0.619537
\(604\) 0 0
\(605\) −2.24621 −0.0913215
\(606\) 0 0
\(607\) 13.6358 0.553461 0.276730 0.960948i \(-0.410749\pi\)
0.276730 + 0.960948i \(0.410749\pi\)
\(608\) 0 0
\(609\) −8.80776 −0.356909
\(610\) 0 0
\(611\) −44.0626 −1.78258
\(612\) 0 0
\(613\) −24.3002 −0.981475 −0.490738 0.871307i \(-0.663273\pi\)
−0.490738 + 0.871307i \(0.663273\pi\)
\(614\) 0 0
\(615\) 6.94568i 0.280077i
\(616\) 0 0
\(617\) −28.5464 −1.14923 −0.574617 0.818422i \(-0.694849\pi\)
−0.574617 + 0.818422i \(0.694849\pi\)
\(618\) 0 0
\(619\) 34.3946i 1.38244i −0.722646 0.691218i \(-0.757074\pi\)
0.722646 0.691218i \(-0.242926\pi\)
\(620\) 0 0
\(621\) 21.6669i 0.869464i
\(622\) 0 0
\(623\) 5.02797 0.201441
\(624\) 0 0
\(625\) −5.63068 −0.225227
\(626\) 0 0
\(627\) 11.4677 + 11.3936i 0.457975 + 0.455017i
\(628\) 0 0
\(629\) 3.70861i 0.147872i
\(630\) 0 0
\(631\) 12.7494i 0.507544i −0.967264 0.253772i \(-0.918329\pi\)
0.967264 0.253772i \(-0.0816713\pi\)
\(632\) 0 0
\(633\) 2.42329 0.0963172
\(634\) 0 0
\(635\) −2.92456 −0.116058
\(636\) 0 0
\(637\) 29.6689i 1.17552i
\(638\) 0 0
\(639\) 17.0862 0.675921
\(640\) 0 0
\(641\) 48.6685i 1.92229i 0.276045 + 0.961145i \(0.410976\pi\)
−0.276045 + 0.961145i \(0.589024\pi\)
\(642\) 0 0
\(643\) 6.56502i 0.258899i −0.991586 0.129449i \(-0.958679\pi\)
0.991586 0.129449i \(-0.0413210\pi\)
\(644\) 0 0
\(645\) 20.6256i 0.812133i
\(646\) 0 0
\(647\) 36.2379i 1.42466i −0.701845 0.712329i \(-0.747640\pi\)
0.701845 0.712329i \(-0.252360\pi\)
\(648\) 0 0
\(649\) 35.9166i 1.40985i
\(650\) 0 0
\(651\) 4.44793i 0.174328i
\(652\) 0 0
\(653\) 45.8078 1.79260 0.896298 0.443452i \(-0.146246\pi\)
0.896298 + 0.443452i \(0.146246\pi\)
\(654\) 0 0
\(655\) 7.54011i 0.294616i
\(656\) 0 0
\(657\) −12.6847 −0.494876
\(658\) 0 0
\(659\) −3.36750 −0.131179 −0.0655896 0.997847i \(-0.520893\pi\)
−0.0655896 + 0.997847i \(0.520893\pi\)
\(660\) 0 0
\(661\) 17.5018i 0.680740i −0.940292 0.340370i \(-0.889448\pi\)
0.940292 0.340370i \(-0.110552\pi\)
\(662\) 0 0
\(663\) 5.69681i 0.221246i
\(664\) 0 0
\(665\) 4.19224 + 4.16516i 0.162568 + 0.161518i
\(666\) 0 0
\(667\) 33.4990 1.29709
\(668\) 0 0
\(669\) −19.5076 −0.754207
\(670\) 0 0
\(671\) 2.11708i 0.0817291i
\(672\) 0 0
\(673\) 31.7515i 1.22393i −0.790885 0.611964i \(-0.790380\pi\)
0.790885 0.611964i \(-0.209620\pi\)
\(674\) 0 0
\(675\) 14.0140 0.539400
\(676\) 0 0
\(677\) 4.29335i 0.165007i −0.996591 0.0825034i \(-0.973708\pi\)
0.996591 0.0825034i \(-0.0262915\pi\)
\(678\) 0 0
\(679\) −14.6875 −0.563656
\(680\) 0 0
\(681\) 20.9460 0.802653
\(682\) 0 0
\(683\) 13.6358 0.521760 0.260880 0.965371i \(-0.415987\pi\)
0.260880 + 0.965371i \(0.415987\pi\)
\(684\) 0 0
\(685\) 6.05398 0.231311
\(686\) 0 0
\(687\) 2.92456 0.111579
\(688\) 0 0
\(689\) −4.94602 −0.188429
\(690\) 0 0
\(691\) 23.3817i 0.889480i −0.895660 0.444740i \(-0.853296\pi\)
0.895660 0.444740i \(-0.146704\pi\)
\(692\) 0 0
\(693\) −4.19224 −0.159250
\(694\) 0 0
\(695\) 0.594427i 0.0225479i
\(696\) 0 0
\(697\) 3.70861i 0.140474i
\(698\) 0 0
\(699\) −11.7630 −0.444916
\(700\) 0 0
\(701\) −29.3693 −1.10926 −0.554632 0.832096i \(-0.687141\pi\)
−0.554632 + 0.832096i \(0.687141\pi\)
\(702\) 0 0
\(703\) 11.4677 + 11.3936i 0.432512 + 0.429718i
\(704\) 0 0
\(705\) 17.3736i 0.654327i
\(706\) 0 0
\(707\) 7.15944i 0.269259i
\(708\) 0 0
\(709\) −37.3693 −1.40343 −0.701717 0.712456i \(-0.747583\pi\)
−0.701717 + 0.712456i \(0.747583\pi\)
\(710\) 0 0
\(711\) 12.5194 0.469513
\(712\) 0 0
\(713\) 16.9170i 0.633547i
\(714\) 0 0
\(715\) 22.9354 0.857733
\(716\) 0 0
\(717\) 23.2930i 0.869891i
\(718\) 0 0
\(719\) 15.9484i 0.594776i −0.954757 0.297388i \(-0.903885\pi\)
0.954757 0.297388i \(-0.0961155\pi\)
\(720\) 0 0
\(721\) 3.25206i 0.121113i
\(722\) 0 0
\(723\) 18.3393i 0.682046i
\(724\) 0 0
\(725\) 21.6669i 0.804689i
\(726\) 0 0
\(727\) 23.6554i 0.877332i 0.898650 + 0.438666i \(0.144549\pi\)
−0.898650 + 0.438666i \(0.855451\pi\)
\(728\) 0 0
\(729\) 22.6155 0.837612
\(730\) 0 0
\(731\) 11.0129i 0.407329i
\(732\) 0 0
\(733\) −3.12311 −0.115355 −0.0576773 0.998335i \(-0.518369\pi\)
−0.0576773 + 0.998335i \(0.518369\pi\)
\(734\) 0 0
\(735\) −11.6982 −0.431496
\(736\) 0 0
\(737\) 30.1254i 1.10968i
\(738\) 0 0
\(739\) 30.3273i 1.11561i −0.829972 0.557804i \(-0.811644\pi\)
0.829972 0.557804i \(-0.188356\pi\)
\(740\) 0 0
\(741\) 17.6155 + 17.5018i 0.647123 + 0.642943i
\(742\) 0 0
\(743\) −38.4440 −1.41037 −0.705187 0.709021i \(-0.749137\pi\)
−0.705187 + 0.709021i \(0.749137\pi\)
\(744\) 0 0
\(745\) −27.8078 −1.01880
\(746\) 0 0
\(747\) 15.0802i 0.551756i
\(748\) 0 0
\(749\) 6.83248i 0.249653i
\(750\) 0 0
\(751\) −8.24782 −0.300967 −0.150484 0.988612i \(-0.548083\pi\)
−0.150484 + 0.988612i \(0.548083\pi\)
\(752\) 0 0
\(753\) 26.8734i 0.979320i
\(754\) 0 0
\(755\) −37.9182 −1.37998
\(756\) 0 0
\(757\) 24.1922 0.879282 0.439641 0.898174i \(-0.355106\pi\)
0.439641 + 0.898174i \(0.355106\pi\)
\(758\) 0 0
\(759\) −14.6875 −0.533123
\(760\) 0 0
\(761\) −10.6155 −0.384813 −0.192406 0.981315i \(-0.561629\pi\)
−0.192406 + 0.981315i \(0.561629\pi\)
\(762\) 0 0
\(763\) −13.7835 −0.498995
\(764\) 0 0
\(765\) −2.43845 −0.0881622
\(766\) 0 0
\(767\) 55.1716i 1.99213i
\(768\) 0 0
\(769\) 54.2311 1.95562 0.977811 0.209489i \(-0.0671801\pi\)
0.977811 + 0.209489i \(0.0671801\pi\)
\(770\) 0 0
\(771\) 22.7872i 0.820662i
\(772\) 0 0
\(773\) 25.8321i 0.929115i −0.885543 0.464558i \(-0.846213\pi\)
0.885543 0.464558i \(-0.153787\pi\)
\(774\) 0 0
\(775\) 10.9418 0.393042
\(776\) 0 0
\(777\) 3.86174 0.138539
\(778\) 0 0
\(779\) 11.4677 + 11.3936i 0.410872 + 0.408219i
\(780\) 0 0
\(781\) 33.8340i 1.21068i
\(782\) 0 0
\(783\) 46.2758i 1.65376i
\(784\) 0 0
\(785\) 9.36932 0.334405
\(786\) 0 0
\(787\) 40.9904 1.46115 0.730575 0.682833i \(-0.239252\pi\)
0.730575 + 0.682833i \(0.239252\pi\)
\(788\) 0 0
\(789\) 7.87377i 0.280314i
\(790\) 0 0
\(791\) −3.21985 −0.114485
\(792\) 0 0
\(793\) 3.25206i 0.115484i
\(794\) 0 0
\(795\) 1.95018i 0.0691659i
\(796\) 0 0
\(797\) 23.2930i 0.825079i 0.910940 + 0.412539i \(0.135358\pi\)
−0.910940 + 0.412539i \(0.864642\pi\)
\(798\) 0 0
\(799\) 9.27653i 0.328180i
\(800\) 0 0
\(801\) 9.04325i 0.319528i
\(802\) 0 0
\(803\) 25.1181i 0.886398i
\(804\) 0 0
\(805\) −5.36932 −0.189244
\(806\)