Properties

Label 1440.2.o.a.1439.4
Level $1440$
Weight $2$
Character 1440.1439
Analytic conductor $11.498$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1440.o (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.4984578911\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: 12.0.426337261060096.1
Defining polynomial: \(x^{12} - 4 x^{9} - 3 x^{8} + 4 x^{7} + 8 x^{6} + 8 x^{5} - 12 x^{4} - 32 x^{3} + 64\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1439.4
Root \(-0.760198 - 1.19252i\) of defining polynomial
Character \(\chi\) \(=\) 1440.1439
Dual form 1440.2.o.a.1439.3

$q$-expansion

\(f(q)\) \(=\) \(q+(-1.24561 + 1.85700i) q^{5} +0.864641 q^{7} +O(q^{10})\) \(q+(-1.24561 + 1.85700i) q^{5} +0.864641 q^{7} -3.90543 q^{11} +1.13536i q^{13} -3.71400 q^{17} +1.72928i q^{19} -9.03365i q^{23} +(-1.89692 - 4.62620i) q^{25} +1.26843i q^{29} -3.25240i q^{31} +(-1.07700 + 1.60564i) q^{35} -6.38776i q^{37} +6.39665i q^{41} +4.77551 q^{43} -4.59958i q^{47} -6.25240 q^{49} -8.98801 q^{53} +(4.86464 - 7.25240i) q^{55} -8.50501 q^{59} -9.04623 q^{61} +(-2.10836 - 1.41421i) q^{65} +11.0462 q^{67} +8.10243 q^{71} +4.47689i q^{73} -3.37680 q^{77} -14.2986i q^{79} -8.10243i q^{83} +(4.62620 - 6.89692i) q^{85} +3.56822i q^{89} +0.981678i q^{91} +(-3.21128 - 2.15401i) q^{95} -10.9817i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + O(q^{10}) \) \( 12q - 8q^{25} - 64q^{43} - 4q^{49} + 48q^{55} - 8q^{61} + 32q^{67} + 20q^{85} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1440\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(641\) \(901\) \(991\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(-1\)

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 0
\(4\) 0 0
\(5\) −1.24561 + 1.85700i −0.557053 + 0.830477i
\(6\) 0 0
\(7\) 0.864641 0.326804 0.163402 0.986560i \(-0.447753\pi\)
0.163402 + 0.986560i \(0.447753\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −3.90543 −1.17753 −0.588766 0.808304i \(-0.700386\pi\)
−0.588766 + 0.808304i \(0.700386\pi\)
\(12\) 0 0
\(13\) 1.13536i 0.314892i 0.987528 + 0.157446i \(0.0503260\pi\)
−0.987528 + 0.157446i \(0.949674\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.71400 −0.900779 −0.450389 0.892832i \(-0.648715\pi\)
−0.450389 + 0.892832i \(0.648715\pi\)
\(18\) 0 0
\(19\) 1.72928i 0.396724i 0.980129 + 0.198362i \(0.0635622\pi\)
−0.980129 + 0.198362i \(0.936438\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 9.03365i 1.88365i −0.336109 0.941823i \(-0.609111\pi\)
0.336109 0.941823i \(-0.390889\pi\)
\(24\) 0 0
\(25\) −1.89692 4.62620i −0.379383 0.925240i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1.26843i 0.235542i 0.993041 + 0.117771i \(0.0375748\pi\)
−0.993041 + 0.117771i \(0.962425\pi\)
\(30\) 0 0
\(31\) 3.25240i 0.584148i −0.956396 0.292074i \(-0.905655\pi\)
0.956396 0.292074i \(-0.0943453\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1.07700 + 1.60564i −0.182047 + 0.271403i
\(36\) 0 0
\(37\) 6.38776i 1.05014i −0.851059 0.525070i \(-0.824039\pi\)
0.851059 0.525070i \(-0.175961\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.39665i 0.998989i 0.866317 + 0.499494i \(0.166481\pi\)
−0.866317 + 0.499494i \(0.833519\pi\)
\(42\) 0 0
\(43\) 4.77551 0.728259 0.364129 0.931348i \(-0.381367\pi\)
0.364129 + 0.931348i \(0.381367\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.59958i 0.670918i −0.942055 0.335459i \(-0.891109\pi\)
0.942055 0.335459i \(-0.108891\pi\)
\(48\) 0 0
\(49\) −6.25240 −0.893199
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −8.98801 −1.23460 −0.617299 0.786729i \(-0.711773\pi\)
−0.617299 + 0.786729i \(0.711773\pi\)
\(54\) 0 0
\(55\) 4.86464 7.25240i 0.655948 0.977913i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −8.50501 −1.10726 −0.553629 0.832763i \(-0.686758\pi\)
−0.553629 + 0.832763i \(0.686758\pi\)
\(60\) 0 0
\(61\) −9.04623 −1.15825 −0.579125 0.815238i \(-0.696606\pi\)
−0.579125 + 0.815238i \(0.696606\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −2.10836 1.41421i −0.261510 0.175412i
\(66\) 0 0
\(67\) 11.0462 1.34951 0.674756 0.738041i \(-0.264249\pi\)
0.674756 + 0.738041i \(0.264249\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 8.10243 0.961581 0.480791 0.876835i \(-0.340350\pi\)
0.480791 + 0.876835i \(0.340350\pi\)
\(72\) 0 0
\(73\) 4.47689i 0.523980i 0.965071 + 0.261990i \(0.0843787\pi\)
−0.965071 + 0.261990i \(0.915621\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −3.37680 −0.384822
\(78\) 0 0
\(79\) 14.2986i 1.60872i −0.594142 0.804360i \(-0.702508\pi\)
0.594142 0.804360i \(-0.297492\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 8.10243i 0.889357i −0.895690 0.444679i \(-0.853318\pi\)
0.895690 0.444679i \(-0.146682\pi\)
\(84\) 0 0
\(85\) 4.62620 6.89692i 0.501782 0.748076i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 3.56822i 0.378231i 0.981955 + 0.189115i \(0.0605620\pi\)
−0.981955 + 0.189115i \(0.939438\pi\)
\(90\) 0 0
\(91\) 0.981678i 0.102908i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −3.21128 2.15401i −0.329470 0.220997i
\(96\) 0 0
\(97\) 10.9817i 1.11502i −0.830170 0.557510i \(-0.811757\pi\)
0.830170 0.557510i \(-0.188243\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0.885578i 0.0881183i 0.999029 + 0.0440591i \(0.0140290\pi\)
−0.999029 + 0.0440591i \(0.985971\pi\)
\(102\) 0 0
\(103\) −18.1816 −1.79149 −0.895743 0.444573i \(-0.853355\pi\)
−0.895743 + 0.444573i \(0.853355\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 2.44557i 0.236423i −0.992988 0.118211i \(-0.962284\pi\)
0.992988 0.118211i \(-0.0377160\pi\)
\(108\) 0 0
\(109\) −3.25240 −0.311523 −0.155762 0.987795i \(-0.549783\pi\)
−0.155762 + 0.987795i \(0.549783\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −17.5646 −1.65234 −0.826168 0.563424i \(-0.809484\pi\)
−0.826168 + 0.563424i \(0.809484\pi\)
\(114\) 0 0
\(115\) 16.7755 + 11.2524i 1.56432 + 1.04929i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −3.21128 −0.294378
\(120\) 0 0
\(121\) 4.25240 0.386581
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 10.9537 + 2.23986i 0.979727 + 0.200339i
\(126\) 0 0
\(127\) −2.18159 −0.193585 −0.0967923 0.995305i \(-0.530858\pi\)
−0.0967923 + 0.995305i \(0.530858\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −22.2643 −1.94524 −0.972620 0.232400i \(-0.925342\pi\)
−0.972620 + 0.232400i \(0.925342\pi\)
\(132\) 0 0
\(133\) 1.49521i 0.129651i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 6.92529 0.591667 0.295834 0.955240i \(-0.404403\pi\)
0.295834 + 0.955240i \(0.404403\pi\)
\(138\) 0 0
\(139\) 18.2986i 1.55207i 0.630690 + 0.776035i \(0.282772\pi\)
−0.630690 + 0.776035i \(0.717228\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 4.43407i 0.370795i
\(144\) 0 0
\(145\) −2.35548 1.57997i −0.195612 0.131209i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 8.98801i 0.736326i 0.929761 + 0.368163i \(0.120013\pi\)
−0.929761 + 0.368163i \(0.879987\pi\)
\(150\) 0 0
\(151\) 6.29862i 0.512575i −0.966601 0.256287i \(-0.917501\pi\)
0.966601 0.256287i \(-0.0824994\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 6.03971 + 4.05121i 0.485121 + 0.325401i
\(156\) 0 0
\(157\) 11.1633i 0.890926i 0.895300 + 0.445463i \(0.146961\pi\)
−0.895300 + 0.445463i \(0.853039\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 7.81086i 0.615582i
\(162\) 0 0
\(163\) −17.7293 −1.38866 −0.694332 0.719655i \(-0.744300\pi\)
−0.694332 + 0.719655i \(0.744300\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1.51435i 0.117184i 0.998282 + 0.0585920i \(0.0186611\pi\)
−0.998282 + 0.0585920i \(0.981339\pi\)
\(168\) 0 0
\(169\) 11.7110 0.900843
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −15.4106 −1.17164 −0.585822 0.810440i \(-0.699228\pi\)
−0.585822 + 0.810440i \(0.699228\pi\)
\(174\) 0 0
\(175\) −1.64015 4.00000i −0.123984 0.302372i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 9.56229 0.714719 0.357359 0.933967i \(-0.383677\pi\)
0.357359 + 0.933967i \(0.383677\pi\)
\(180\) 0 0
\(181\) 14.2986 1.06281 0.531404 0.847118i \(-0.321665\pi\)
0.531404 + 0.847118i \(0.321665\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 11.8621 + 7.95665i 0.872117 + 0.584984i
\(186\) 0 0
\(187\) 14.5048 1.06070
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 19.1246 1.38381 0.691903 0.721991i \(-0.256773\pi\)
0.691903 + 0.721991i \(0.256773\pi\)
\(192\) 0 0
\(193\) 22.5048i 1.61993i 0.586478 + 0.809965i \(0.300514\pi\)
−0.586478 + 0.809965i \(0.699486\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 9.91923 0.706716 0.353358 0.935488i \(-0.385040\pi\)
0.353358 + 0.935488i \(0.385040\pi\)
\(198\) 0 0
\(199\) 3.66473i 0.259786i −0.991528 0.129893i \(-0.958537\pi\)
0.991528 0.129893i \(-0.0414634\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 1.09674i 0.0769759i
\(204\) 0 0
\(205\) −11.8786 7.96772i −0.829637 0.556490i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 6.75359i 0.467156i
\(210\) 0 0
\(211\) 2.29862i 0.158244i 0.996865 + 0.0791219i \(0.0252116\pi\)
−0.996865 + 0.0791219i \(0.974788\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −5.94842 + 8.86813i −0.405679 + 0.604802i
\(216\) 0 0
\(217\) 2.81215i 0.190901i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 4.21673i 0.283648i
\(222\) 0 0
\(223\) −15.1354 −1.01354 −0.506769 0.862082i \(-0.669160\pi\)
−0.506769 + 0.862082i \(0.669160\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 26.1697i 1.73695i 0.495737 + 0.868473i \(0.334898\pi\)
−0.495737 + 0.868473i \(0.665102\pi\)
\(228\) 0 0
\(229\) 9.75719 0.644773 0.322387 0.946608i \(-0.395515\pi\)
0.322387 + 0.946608i \(0.395515\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 8.02202 0.525540 0.262770 0.964858i \(-0.415364\pi\)
0.262770 + 0.964858i \(0.415364\pi\)
\(234\) 0 0
\(235\) 8.54144 + 5.72928i 0.557182 + 0.373737i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 17.0100 1.10029 0.550144 0.835070i \(-0.314573\pi\)
0.550144 + 0.835070i \(0.314573\pi\)
\(240\) 0 0
\(241\) −21.0096 −1.35335 −0.676673 0.736284i \(-0.736579\pi\)
−0.676673 + 0.736284i \(0.736579\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 7.78804 11.6107i 0.497560 0.741781i
\(246\) 0 0
\(247\) −1.96336 −0.124925
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −8.46555 −0.534341 −0.267170 0.963649i \(-0.586089\pi\)
−0.267170 + 0.963649i \(0.586089\pi\)
\(252\) 0 0
\(253\) 35.2803i 2.21805i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 11.8164 0.737089 0.368544 0.929610i \(-0.379856\pi\)
0.368544 + 0.929610i \(0.379856\pi\)
\(258\) 0 0
\(259\) 5.52311i 0.343190i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 7.93691i 0.489411i −0.969597 0.244705i \(-0.921309\pi\)
0.969597 0.244705i \(-0.0786913\pi\)
\(264\) 0 0
\(265\) 11.1955 16.6907i 0.687737 1.02530i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 29.9750i 1.82761i −0.406154 0.913805i \(-0.633130\pi\)
0.406154 0.913805i \(-0.366870\pi\)
\(270\) 0 0
\(271\) 26.8401i 1.63042i 0.579167 + 0.815209i \(0.303378\pi\)
−0.579167 + 0.815209i \(0.696622\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 7.40828 + 18.0673i 0.446736 + 1.08950i
\(276\) 0 0
\(277\) 24.9571i 1.49953i −0.661706 0.749763i \(-0.730167\pi\)
0.661706 0.749763i \(-0.269833\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 1.12265i 0.0669716i −0.999439 0.0334858i \(-0.989339\pi\)
0.999439 0.0334858i \(-0.0106609\pi\)
\(282\) 0 0
\(283\) −25.9634 −1.54336 −0.771681 0.636010i \(-0.780584\pi\)
−0.771681 + 0.636010i \(0.780584\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 5.53080i 0.326473i
\(288\) 0 0
\(289\) −3.20617 −0.188598
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 16.6728 0.974037 0.487018 0.873392i \(-0.338085\pi\)
0.487018 + 0.873392i \(0.338085\pi\)
\(294\) 0 0
\(295\) 10.5939 15.7938i 0.616802 0.919552i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 10.2564 0.593145
\(300\) 0 0
\(301\) 4.12910 0.237997
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 11.2681 16.7989i 0.645207 0.961900i
\(306\) 0 0
\(307\) −21.0096 −1.19908 −0.599540 0.800345i \(-0.704650\pi\)
−0.599540 + 0.800345i \(0.704650\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 34.7463 1.97028 0.985141 0.171748i \(-0.0549415\pi\)
0.985141 + 0.171748i \(0.0549415\pi\)
\(312\) 0 0
\(313\) 9.49521i 0.536701i 0.963321 + 0.268350i \(0.0864785\pi\)
−0.963321 + 0.268350i \(0.913521\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 16.2505 0.912719 0.456360 0.889795i \(-0.349153\pi\)
0.456360 + 0.889795i \(0.349153\pi\)
\(318\) 0 0
\(319\) 4.95377i 0.277358i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 6.42256i 0.357361i
\(324\) 0 0
\(325\) 5.25240 2.15368i 0.291351 0.119465i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 3.97699i 0.219258i
\(330\) 0 0
\(331\) 20.7755i 1.14193i −0.820976 0.570963i \(-0.806570\pi\)
0.820976 0.570963i \(-0.193430\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −13.7593 + 20.5129i −0.751750 + 1.12074i
\(336\) 0 0
\(337\) 28.1204i 1.53181i 0.642951 + 0.765907i \(0.277710\pi\)
−0.642951 + 0.765907i \(0.722290\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 12.7020i 0.687852i
\(342\) 0 0
\(343\) −11.4586 −0.618704
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 18.8330i 1.01101i 0.862824 + 0.505504i \(0.168694\pi\)
−0.862824 + 0.505504i \(0.831306\pi\)
\(348\) 0 0
\(349\) −11.5510 −0.618312 −0.309156 0.951011i \(-0.600047\pi\)
−0.309156 + 0.951011i \(0.600047\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −2.61727 −0.139303 −0.0696515 0.997571i \(-0.522189\pi\)
−0.0696515 + 0.997571i \(0.522189\pi\)
\(354\) 0 0
\(355\) −10.0925 + 15.0462i −0.535652 + 0.798571i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −20.8044 −1.09802 −0.549008 0.835817i \(-0.684994\pi\)
−0.549008 + 0.835817i \(0.684994\pi\)
\(360\) 0 0
\(361\) 16.0096 0.842610
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −8.31359 5.57645i −0.435153 0.291885i
\(366\) 0 0
\(367\) 15.3694 0.802278 0.401139 0.916017i \(-0.368614\pi\)
0.401139 + 0.916017i \(0.368614\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −7.77140 −0.403471
\(372\) 0 0
\(373\) 29.3049i 1.51735i 0.651470 + 0.758675i \(0.274153\pi\)
−0.651470 + 0.758675i \(0.725847\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −1.44012 −0.0741702
\(378\) 0 0
\(379\) 11.7938i 0.605808i 0.953021 + 0.302904i \(0.0979562\pi\)
−0.953021 + 0.302904i \(0.902044\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 16.6790i 0.852257i −0.904663 0.426128i \(-0.859877\pi\)
0.904663 0.426128i \(-0.140123\pi\)
\(384\) 0 0
\(385\) 4.20617 6.27072i 0.214366 0.319585i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 23.2214i 1.17737i 0.808361 + 0.588687i \(0.200355\pi\)
−0.808361 + 0.588687i \(0.799645\pi\)
\(390\) 0 0
\(391\) 33.5510i 1.69675i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 26.5526 + 17.8105i 1.33601 + 0.896143i
\(396\) 0 0
\(397\) 2.02458i 0.101611i −0.998709 0.0508054i \(-0.983821\pi\)
0.998709 0.0508054i \(-0.0161788\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 28.9722i 1.44680i −0.690427 0.723402i \(-0.742577\pi\)
0.690427 0.723402i \(-0.257423\pi\)
\(402\) 0 0
\(403\) 3.69264 0.183943
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 24.9469i 1.23657i
\(408\) 0 0
\(409\) 25.8863 1.27999 0.639997 0.768377i \(-0.278935\pi\)
0.639997 + 0.768377i \(0.278935\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −7.35378 −0.361856
\(414\) 0 0
\(415\) 15.0462 + 10.0925i 0.738590 + 0.495419i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 12.7736 0.624030 0.312015 0.950077i \(-0.398996\pi\)
0.312015 + 0.950077i \(0.398996\pi\)
\(420\) 0 0
\(421\) 18.1695 0.885528 0.442764 0.896638i \(-0.353998\pi\)
0.442764 + 0.896638i \(0.353998\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 7.04516 + 17.1817i 0.341740 + 0.833436i
\(426\) 0 0
\(427\) −7.82174 −0.378520
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 2.40611 0.115898 0.0579491 0.998320i \(-0.481544\pi\)
0.0579491 + 0.998320i \(0.481544\pi\)
\(432\) 0 0
\(433\) 31.4094i 1.50944i 0.656047 + 0.754720i \(0.272227\pi\)
−0.656047 + 0.754720i \(0.727773\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 15.6217 0.747289
\(438\) 0 0
\(439\) 18.1695i 0.867184i 0.901109 + 0.433592i \(0.142754\pi\)
−0.901109 + 0.433592i \(0.857246\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 18.8330i 0.894783i −0.894338 0.447392i \(-0.852353\pi\)
0.894338 0.447392i \(-0.147647\pi\)
\(444\) 0 0
\(445\) −6.62620 4.44461i −0.314112 0.210695i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 20.9216i 0.987353i 0.869646 + 0.493677i \(0.164347\pi\)
−0.869646 + 0.493677i \(0.835653\pi\)
\(450\) 0 0
\(451\) 24.9817i 1.17634i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −1.82298 1.22279i −0.0854625 0.0573251i
\(456\) 0 0
\(457\) 35.5789i 1.66431i −0.554542 0.832156i \(-0.687106\pi\)
0.554542 0.832156i \(-0.312894\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 14.0617i 0.654920i −0.944865 0.327460i \(-0.893807\pi\)
0.944865 0.327460i \(-0.106193\pi\)
\(462\) 0 0
\(463\) −33.0987 −1.53823 −0.769114 0.639112i \(-0.779302\pi\)
−0.769114 + 0.639112i \(0.779302\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 32.0092i 1.48121i −0.671942 0.740604i \(-0.734540\pi\)
0.671942 0.740604i \(-0.265460\pi\)
\(468\) 0 0
\(469\) 9.55102 0.441025
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −18.6504 −0.857548
\(474\) 0 0
\(475\) 8.00000 3.28030i 0.367065 0.150511i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −28.6153 −1.30747 −0.653733 0.756725i \(-0.726798\pi\)
−0.653733 + 0.756725i \(0.726798\pi\)
\(480\) 0 0
\(481\) 7.25240 0.330681
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 20.3930 + 13.6789i 0.925999 + 0.621126i
\(486\) 0 0
\(487\) 2.18159 0.0988572 0.0494286 0.998778i \(-0.484260\pi\)
0.0494286 + 0.998778i \(0.484260\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −28.2127 −1.27322 −0.636611 0.771185i \(-0.719664\pi\)
−0.636611 + 0.771185i \(0.719664\pi\)
\(492\) 0 0
\(493\) 4.71096i 0.212171i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 7.00569 0.314248
\(498\) 0 0
\(499\) 0.234074i 0.0104786i −0.999986 0.00523930i \(-0.998332\pi\)
0.999986 0.00523930i \(-0.00166773\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 15.3302i 0.683538i 0.939784 + 0.341769i \(0.111026\pi\)
−0.939784 + 0.341769i \(0.888974\pi\)
\(504\) 0 0
\(505\) −1.64452 1.10308i −0.0731802 0.0490866i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 22.7473i 1.00826i −0.863629 0.504128i \(-0.831814\pi\)
0.863629 0.504128i \(-0.168186\pi\)
\(510\) 0 0
\(511\) 3.87090i 0.171238i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 22.6471 33.7633i 0.997953 1.48779i
\(516\) 0 0
\(517\) 17.9634i 0.790027i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 23.9898i 1.05101i 0.850790 + 0.525506i \(0.176124\pi\)
−0.850790 + 0.525506i \(0.823876\pi\)
\(522\) 0 0
\(523\) −28.3632 −1.24024 −0.620118 0.784509i \(-0.712915\pi\)
−0.620118 + 0.784509i \(0.712915\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 12.0794i 0.526188i
\(528\) 0 0
\(529\) −58.6068 −2.54812
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −7.26249 −0.314574
\(534\) 0 0
\(535\) 4.54144 + 3.04623i 0.196343 + 0.131700i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 24.4183 1.05177
\(540\) 0 0
\(541\) −36.8034 −1.58230 −0.791151 0.611621i \(-0.790518\pi\)
−0.791151 + 0.611621i \(0.790518\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 4.05121 6.03971i 0.173535 0.258713i
\(546\) 0 0
\(547\) 24.2341 1.03617 0.518087 0.855328i \(-0.326644\pi\)
0.518087 + 0.855328i \(0.326644\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −2.19347 −0.0934452
\(552\) 0 0
\(553\) 12.3632i 0.525736i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −43.3515 −1.83686 −0.918430 0.395584i \(-0.870542\pi\)
−0.918430 + 0.395584i \(0.870542\pi\)
\(558\) 0 0
\(559\) 5.42192i 0.229323i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 2.62815i 0.110763i 0.998465 + 0.0553817i \(0.0176376\pi\)
−0.998465 + 0.0553817i \(0.982362\pi\)
\(564\) 0 0
\(565\) 21.8786 32.6175i 0.920439 1.37223i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 23.6588i 0.991828i −0.868372 0.495914i \(-0.834833\pi\)
0.868372 0.495914i \(-0.165167\pi\)
\(570\) 0 0
\(571\) 35.6926i 1.49369i −0.664998 0.746845i \(-0.731568\pi\)
0.664998 0.746845i \(-0.268432\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −41.7915 + 17.1361i −1.74282 + 0.714624i
\(576\) 0 0
\(577\) 0.646409i 0.0269104i 0.999909 + 0.0134552i \(0.00428304\pi\)
−0.999909 + 0.0134552i \(0.995717\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 7.00569i 0.290645i
\(582\) 0 0
\(583\) 35.1020 1.45378
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 22.3753i 0.923528i −0.887003 0.461764i \(-0.847217\pi\)
0.887003 0.461764i \(-0.152783\pi\)
\(588\) 0 0
\(589\) 5.62431 0.231746
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −20.4325 −0.839061 −0.419530 0.907741i \(-0.637805\pi\)
−0.419530 + 0.907741i \(0.637805\pi\)
\(594\) 0 0
\(595\) 4.00000 5.96336i 0.163984 0.244474i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −6.13100 −0.250506 −0.125253 0.992125i \(-0.539974\pi\)
−0.125253 + 0.992125i \(0.539974\pi\)
\(600\) 0 0
\(601\) −1.79383 −0.0731720 −0.0365860 0.999331i \(-0.511648\pi\)
−0.0365860 + 0.999331i \(0.511648\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −5.29682 + 7.89671i −0.215346 + 0.321047i
\(606\) 0 0
\(607\) 16.6864 0.677279 0.338640 0.940916i \(-0.390033\pi\)
0.338640 + 0.940916i \(0.390033\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 5.22218 0.211267
\(612\) 0 0
\(613\) 23.5264i 0.950224i −0.879925 0.475112i \(-0.842408\pi\)
0.879925 0.475112i \(-0.157592\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 23.3127 0.938535 0.469267 0.883056i \(-0.344518\pi\)
0.469267 + 0.883056i \(0.344518\pi\)
\(618\) 0 0
\(619\) 32.3911i 1.30191i −0.759117 0.650954i \(-0.774369\pi\)
0.759117 0.650954i \(-0.225631\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 3.08523i 0.123607i
\(624\) 0 0
\(625\) −17.8034 + 17.5510i −0.712137 + 0.702041i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 23.7242i 0.945944i
\(630\) 0 0
\(631\) 19.2524i 0.766426i 0.923660 + 0.383213i \(0.125182\pi\)
−0.923660 + 0.383213i \(0.874818\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 2.71741 4.05121i 0.107837 0.160768i
\(636\) 0 0
\(637\) 7.09871i 0.281261i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 25.9435i 1.02471i 0.858774 + 0.512354i \(0.171226\pi\)
−0.858774 + 0.512354i \(0.828774\pi\)
\(642\) 0 0
\(643\) −11.2803 −0.444852 −0.222426 0.974950i \(-0.571398\pi\)
−0.222426 + 0.974950i \(0.571398\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 11.6053i 0.456250i −0.973632 0.228125i \(-0.926740\pi\)
0.973632 0.228125i \(-0.0732595\pi\)
\(648\) 0 0
\(649\) 33.2158 1.30383
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −35.4145 −1.38588 −0.692939 0.720996i \(-0.743684\pi\)
−0.692939 + 0.720996i \(0.743684\pi\)
\(654\) 0 0
\(655\) 27.7326 41.3449i 1.08360 1.61548i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 12.0079 0.467760 0.233880 0.972265i \(-0.424858\pi\)
0.233880 + 0.972265i \(0.424858\pi\)
\(660\) 0 0
\(661\) −3.08287 −0.119910 −0.0599549 0.998201i \(-0.519096\pi\)
−0.0599549 + 0.998201i \(0.519096\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −2.77660 1.86244i −0.107672 0.0722225i
\(666\) 0 0
\(667\) 11.4586 0.443677
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 35.3294 1.36388
\(672\) 0 0
\(673\) 25.7851i 0.993942i −0.867767 0.496971i \(-0.834445\pi\)
0.867767 0.496971i \(-0.165555\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 2.16019 0.0830227 0.0415114 0.999138i \(-0.486783\pi\)
0.0415114 + 0.999138i \(0.486783\pi\)
\(678\) 0 0
\(679\) 9.49521i 0.364393i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 28.3632i 1.08529i −0.839963 0.542644i \(-0.817423\pi\)
0.839963 0.542644i \(-0.182577\pi\)
\(684\) 0 0
\(685\) −8.62620 + 12.8603i −0.329590 + 0.491366i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 10.2046i 0.388765i
\(690\) 0 0
\(691\) 37.9142i 1.44232i −0.692766 0.721162i \(-0.743608\pi\)
0.692766 0.721162i \(-0.256392\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −33.9806 22.7929i −1.28896 0.864585i
\(696\) 0 0
\(697\) 23.7572i 0.899868i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 12.9650i 0.489681i 0.969563 + 0.244841i \(0.0787356\pi\)
−0.969563 + 0.244841i \(0.921264\pi\)
\(702\) 0 0
\(703\) 11.0462 0.416616
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0.765707i 0.0287974i
\(708\) 0 0
\(709\) −26.8401 −1.00800 −0.504000 0.863704i \(-0.668139\pi\)
−0.504000 + 0.863704i \(0.668139\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −29.3810 −1.10033
\(714\) 0 0
\(715\) 8.23407 + 5.52311i 0.307937 + 0.206553i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −40.7342 −1.51913 −0.759564 0.650432i \(-0.774588\pi\)
−0.759564 + 0.650432i \(0.774588\pi\)
\(720\) 0 0
\(721\) −15.7205 −0.585464
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 5.86801 2.40611i 0.217933 0.0893606i
\(726\) 0 0
\(727\) 24.9205 0.924248 0.462124 0.886815i \(-0.347087\pi\)
0.462124 + 0.886815i \(0.347087\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −17.7363 −0.656000
\(732\) 0 0
\(733\) 11.2278i 0.414709i 0.978266 + 0.207354i \(0.0664853\pi\)
−0.978266 + 0.207354i \(0.933515\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −43.1403 −1.58909
\(738\) 0 0
\(739\) 3.22449i 0.118615i 0.998240 + 0.0593074i \(0.0188892\pi\)
−0.998240 + 0.0593074i \(0.981111\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 14.5645i 0.534318i −0.963652 0.267159i \(-0.913915\pi\)
0.963652 0.267159i \(-0.0860849\pi\)
\(744\) 0 0
\(745\) −16.6907 11.1955i −0.611502 0.410173i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 2.11454i 0.0772637i
\(750\) 0 0
\(751\) 52.8034i 1.92682i 0.268025 + 0.963412i \(0.413629\pi\)
−0.268025 + 0.963412i \(0.586371\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 11.6966 + 7.84562i 0.425681 + 0.285531i
\(756\) 0 0
\(757\) 8.82800i 0.320859i −0.987047 0.160429i \(-0.948712\pi\)
0.987047 0.160429i \(-0.0512879\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 32.6577i 1.18384i 0.805997 + 0.591920i \(0.201630\pi\)
−0.805997 + 0.591920i \(0.798370\pi\)
\(762\) 0 0
\(763\) −2.81215 −0.101807
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 9.65625i 0.348667i
\(768\) 0 0
\(769\) −7.83048 −0.282374 −0.141187 0.989983i \(-0.545092\pi\)
−0.141187 + 0.989983i \(0.545092\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 8.14807 0.293066 0.146533 0.989206i \(-0.453189\pi\)
0.146533 + 0.989206i \(0.453189\pi\)
\(774\) 0 0
\(775\) −15.0462 + 6.16952i −0.540476 + 0.221616i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −11.0616 −0.396323
\(780\) 0 0
\(781\) −31.6435 −1.13229
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −20.7302 13.9051i −0.739893 0.496293i
\(786\) 0 0
\(787\) −35.2803 −1.25761 −0.628803 0.777564i \(-0.716455\pi\)
−0.628803 + 0.777564i \(0.716455\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −15.1871 −0.539989
\(792\) 0 0
\(793\) 10.2707i 0.364724i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 4.67999 0.165774 0.0828868 0.996559i \(-0.473586\pi\)
0.0828868 + 0.996559i \(0.473586\pi\)
\(798\) 0 0
\(799\) 17.0829i 0.604349i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 17.4842i 0.617003i
\(804\) 0 0
\(805\) 14.5048 + 9.72928i 0.511227 + 0.342912i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 19.2294i 0.676070i