Properties

Label 1440.2.f.d.289.2
Level $1440$
Weight $2$
Character 1440.289
Analytic conductor $11.498$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(11.4984578911\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 480)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 289.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1440.289
Dual form 1440.2.f.d.289.1

$q$-expansion

\(f(q)\) \(=\) \(q+(-1.00000 + 2.00000i) q^{5} +4.00000i q^{7} +O(q^{10})\) \(q+(-1.00000 + 2.00000i) q^{5} +4.00000i q^{7} +4.00000i q^{13} +8.00000 q^{19} -4.00000i q^{23} +(-3.00000 - 4.00000i) q^{25} -6.00000 q^{29} -8.00000 q^{31} +(-8.00000 - 4.00000i) q^{35} +4.00000i q^{37} -6.00000 q^{41} +4.00000i q^{43} -4.00000i q^{47} -9.00000 q^{49} +12.0000i q^{53} -6.00000 q^{61} +(-8.00000 - 4.00000i) q^{65} -12.0000i q^{67} +16.0000 q^{71} +8.00000 q^{79} +12.0000i q^{83} -10.0000 q^{89} -16.0000 q^{91} +(-8.00000 + 16.0000i) q^{95} -8.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{5} + O(q^{10}) \) \( 2q - 2q^{5} + 16q^{19} - 6q^{25} - 12q^{29} - 16q^{31} - 16q^{35} - 12q^{41} - 18q^{49} - 12q^{61} - 16q^{65} + 32q^{71} + 16q^{79} - 20q^{89} - 32q^{91} - 16q^{95} + O(q^{100}) \)

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
\(5\) −1.00000 + 2.00000i −0.447214 + 0.894427i
\(6\) 0 0
\(7\) 4.00000i 1.51186i 0.654654 + 0.755929i \(0.272814\pi\)
−0.654654 + 0.755929i \(0.727186\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 4.00000i 1.10940i 0.832050 + 0.554700i \(0.187167\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 8.00000 1.83533 0.917663 0.397360i \(-0.130073\pi\)
0.917663 + 0.397360i \(0.130073\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.00000i 0.834058i −0.908893 0.417029i \(-0.863071\pi\)
0.908893 0.417029i \(-0.136929\pi\)
\(24\) 0 0
\(25\) −3.00000 4.00000i −0.600000 0.800000i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −8.00000 4.00000i −1.35225 0.676123i
\(36\) 0 0
\(37\) 4.00000i 0.657596i 0.944400 + 0.328798i \(0.106644\pi\)
−0.944400 + 0.328798i \(0.893356\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 4.00000i 0.609994i 0.952353 + 0.304997i \(0.0986555\pi\)
−0.952353 + 0.304997i \(0.901344\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.00000i 0.583460i −0.956501 0.291730i \(-0.905769\pi\)
0.956501 0.291730i \(-0.0942309\pi\)
\(48\) 0 0
\(49\) −9.00000 −1.28571
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 12.0000i 1.64833i 0.566352 + 0.824163i \(0.308354\pi\)
−0.566352 + 0.824163i \(0.691646\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −6.00000 −0.768221 −0.384111 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −8.00000 4.00000i −0.992278 0.496139i
\(66\) 0 0
\(67\) 12.0000i 1.46603i −0.680211 0.733017i \(-0.738112\pi\)
0.680211 0.733017i \(-0.261888\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 16.0000 1.89885 0.949425 0.313993i \(-0.101667\pi\)
0.949425 + 0.313993i \(0.101667\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 12.0000i 1.31717i 0.752506 + 0.658586i \(0.228845\pi\)
−0.752506 + 0.658586i \(0.771155\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) −16.0000 −1.67726
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −8.00000 + 16.0000i −0.820783 + 1.64157i
\(96\) 0 0
\(97\) 8.00000i 0.812277i −0.913812 0.406138i \(-0.866875\pi\)
0.913812 0.406138i \(-0.133125\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −14.0000 −1.39305 −0.696526 0.717532i \(-0.745272\pi\)
−0.696526 + 0.717532i \(0.745272\pi\)
\(102\) 0 0
\(103\) 12.0000i 1.18240i −0.806527 0.591198i \(-0.798655\pi\)
0.806527 0.591198i \(-0.201345\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 12.0000i 1.16008i 0.814587 + 0.580042i \(0.196964\pi\)
−0.814587 + 0.580042i \(0.803036\pi\)
\(108\) 0 0
\(109\) 10.0000 0.957826 0.478913 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 8.00000i 0.752577i 0.926503 + 0.376288i \(0.122800\pi\)
−0.926503 + 0.376288i \(0.877200\pi\)
\(114\) 0 0
\(115\) 8.00000 + 4.00000i 0.746004 + 0.373002i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 11.0000 2.00000i 0.983870 0.178885i
\(126\) 0 0
\(127\) 4.00000i 0.354943i 0.984126 + 0.177471i \(0.0567917\pi\)
−0.984126 + 0.177471i \(0.943208\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 16.0000 1.39793 0.698963 0.715158i \(-0.253645\pi\)
0.698963 + 0.715158i \(0.253645\pi\)
\(132\) 0 0
\(133\) 32.0000i 2.77475i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) −8.00000 −0.678551 −0.339276 0.940687i \(-0.610182\pi\)
−0.339276 + 0.940687i \(0.610182\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 6.00000 12.0000i 0.498273 0.996546i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 14.0000 1.14692 0.573462 0.819232i \(-0.305600\pi\)
0.573462 + 0.819232i \(0.305600\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 8.00000 16.0000i 0.642575 1.28515i
\(156\) 0 0
\(157\) 4.00000i 0.319235i −0.987179 0.159617i \(-0.948974\pi\)
0.987179 0.159617i \(-0.0510260\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 16.0000 1.26098
\(162\) 0 0
\(163\) 4.00000i 0.313304i 0.987654 + 0.156652i \(0.0500701\pi\)
−0.987654 + 0.156652i \(0.949930\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 4.00000i 0.309529i −0.987951 0.154765i \(-0.950538\pi\)
0.987951 0.154765i \(-0.0494619\pi\)
\(168\) 0 0
\(169\) −3.00000 −0.230769
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 4.00000i 0.304114i −0.988372 0.152057i \(-0.951410\pi\)
0.988372 0.152057i \(-0.0485898\pi\)
\(174\) 0 0
\(175\) 16.0000 12.0000i 1.20949 0.907115i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −16.0000 −1.19590 −0.597948 0.801535i \(-0.704017\pi\)
−0.597948 + 0.801535i \(0.704017\pi\)
\(180\) 0 0
\(181\) 14.0000 1.04061 0.520306 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −8.00000 4.00000i −0.588172 0.294086i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −16.0000 −1.15772 −0.578860 0.815427i \(-0.696502\pi\)
−0.578860 + 0.815427i \(0.696502\pi\)
\(192\) 0 0
\(193\) 24.0000i 1.72756i 0.503871 + 0.863779i \(0.331909\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 12.0000i 0.854965i 0.904024 + 0.427482i \(0.140599\pi\)
−0.904024 + 0.427482i \(0.859401\pi\)
\(198\) 0 0
\(199\) −8.00000 −0.567105 −0.283552 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 24.0000i 1.68447i
\(204\) 0 0
\(205\) 6.00000 12.0000i 0.419058 0.838116i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 8.00000 0.550743 0.275371 0.961338i \(-0.411199\pi\)
0.275371 + 0.961338i \(0.411199\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −8.00000 4.00000i −0.545595 0.272798i
\(216\) 0 0
\(217\) 32.0000i 2.17230i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 4.00000i 0.267860i 0.990991 + 0.133930i \(0.0427597\pi\)
−0.990991 + 0.133930i \(0.957240\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 4.00000i 0.265489i −0.991150 0.132745i \(-0.957621\pi\)
0.991150 0.132745i \(-0.0423790\pi\)
\(228\) 0 0
\(229\) −2.00000 −0.132164 −0.0660819 0.997814i \(-0.521050\pi\)
−0.0660819 + 0.997814i \(0.521050\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 8.00000i 0.524097i −0.965055 0.262049i \(-0.915602\pi\)
0.965055 0.262049i \(-0.0843981\pi\)
\(234\) 0 0
\(235\) 8.00000 + 4.00000i 0.521862 + 0.260931i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −16.0000 −1.03495 −0.517477 0.855697i \(-0.673129\pi\)
−0.517477 + 0.855697i \(0.673129\pi\)
\(240\) 0 0
\(241\) 14.0000 0.901819 0.450910 0.892570i \(-0.351100\pi\)
0.450910 + 0.892570i \(0.351100\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 9.00000 18.0000i 0.574989 1.14998i
\(246\) 0 0
\(247\) 32.0000i 2.03611i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 16.0000 1.00991 0.504956 0.863145i \(-0.331509\pi\)
0.504956 + 0.863145i \(0.331509\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 24.0000i 1.49708i −0.663090 0.748539i \(-0.730755\pi\)
0.663090 0.748539i \(-0.269245\pi\)
\(258\) 0 0
\(259\) −16.0000 −0.994192
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 12.0000i 0.739952i 0.929041 + 0.369976i \(0.120634\pi\)
−0.929041 + 0.369976i \(0.879366\pi\)
\(264\) 0 0
\(265\) −24.0000 12.0000i −1.47431 0.737154i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 6.00000 0.365826 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) 0 0
\(271\) 24.0000 1.45790 0.728948 0.684569i \(-0.240010\pi\)
0.728948 + 0.684569i \(0.240010\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 4.00000i 0.240337i 0.992754 + 0.120168i \(0.0383434\pi\)
−0.992754 + 0.120168i \(0.961657\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 26.0000 1.55103 0.775515 0.631329i \(-0.217490\pi\)
0.775515 + 0.631329i \(0.217490\pi\)
\(282\) 0 0
\(283\) 20.0000i 1.18888i 0.804141 + 0.594438i \(0.202626\pi\)
−0.804141 + 0.594438i \(0.797374\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 24.0000i 1.41668i
\(288\) 0 0
\(289\) 17.0000 1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 20.0000i 1.16841i 0.811605 + 0.584206i \(0.198594\pi\)
−0.811605 + 0.584206i \(0.801406\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 16.0000 0.925304
\(300\) 0 0
\(301\) −16.0000 −0.922225
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 6.00000 12.0000i 0.343559 0.687118i
\(306\) 0 0
\(307\) 12.0000i 0.684876i −0.939540 0.342438i \(-0.888747\pi\)
0.939540 0.342438i \(-0.111253\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 8.00000i 0.452187i −0.974106 0.226093i \(-0.927405\pi\)
0.974106 0.226093i \(-0.0725954\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 12.0000i 0.673987i −0.941507 0.336994i \(-0.890590\pi\)
0.941507 0.336994i \(-0.109410\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 16.0000 12.0000i 0.887520 0.665640i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 16.0000 0.882109
\(330\) 0 0
\(331\) 8.00000 0.439720 0.219860 0.975531i \(-0.429440\pi\)
0.219860 + 0.975531i \(0.429440\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 24.0000 + 12.0000i 1.31126 + 0.655630i
\(336\) 0 0
\(337\) 16.0000i 0.871576i 0.900049 + 0.435788i \(0.143530\pi\)
−0.900049 + 0.435788i \(0.856470\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 8.00000i 0.431959i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 28.0000i 1.50312i 0.659665 + 0.751559i \(0.270698\pi\)
−0.659665 + 0.751559i \(0.729302\pi\)
\(348\) 0 0
\(349\) −10.0000 −0.535288 −0.267644 0.963518i \(-0.586245\pi\)
−0.267644 + 0.963518i \(0.586245\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 24.0000i 1.27739i 0.769460 + 0.638696i \(0.220526\pi\)
−0.769460 + 0.638696i \(0.779474\pi\)
\(354\) 0 0
\(355\) −16.0000 + 32.0000i −0.849192 + 1.69838i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) 45.0000 2.36842
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 28.0000i 1.46159i −0.682598 0.730794i \(-0.739150\pi\)
0.682598 0.730794i \(-0.260850\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −48.0000 −2.49204
\(372\) 0 0
\(373\) 20.0000i 1.03556i 0.855514 + 0.517780i \(0.173242\pi\)
−0.855514 + 0.517780i \(0.826758\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 24.0000i 1.23606i
\(378\) 0 0
\(379\) 8.00000 0.410932 0.205466 0.978664i \(-0.434129\pi\)
0.205466 + 0.978664i \(0.434129\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 12.0000i 0.613171i 0.951843 + 0.306586i \(0.0991866\pi\)
−0.951843 + 0.306586i \(0.900813\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 30.0000 1.52106 0.760530 0.649303i \(-0.224939\pi\)
0.760530 + 0.649303i \(0.224939\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −8.00000 + 16.0000i −0.402524 + 0.805047i
\(396\) 0 0
\(397\) 28.0000i 1.40528i −0.711546 0.702640i \(-0.752005\pi\)
0.711546 0.702640i \(-0.247995\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 30.0000 1.49813 0.749064 0.662497i \(-0.230503\pi\)
0.749064 + 0.662497i \(0.230503\pi\)
\(402\) 0 0
\(403\) 32.0000i 1.59403i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −10.0000 −0.494468 −0.247234 0.968956i \(-0.579522\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −24.0000 12.0000i −1.17811 0.589057i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 16.0000 0.781651 0.390826 0.920465i \(-0.372190\pi\)
0.390826 + 0.920465i \(0.372190\pi\)
\(420\) 0 0
\(421\) −14.0000 −0.682318 −0.341159 0.940006i \(-0.610819\pi\)
−0.341159 + 0.940006i \(0.610819\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 24.0000i 1.16144i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 16.0000 0.770693 0.385346 0.922772i \(-0.374082\pi\)
0.385346 + 0.922772i \(0.374082\pi\)
\(432\) 0 0
\(433\) 40.0000i 1.92228i 0.276066 + 0.961139i \(0.410969\pi\)
−0.276066 + 0.961139i \(0.589031\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 32.0000i 1.53077i
\(438\) 0 0
\(439\) −8.00000 −0.381819 −0.190910 0.981608i \(-0.561144\pi\)
−0.190910 + 0.981608i \(0.561144\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 20.0000i 0.950229i −0.879924 0.475114i \(-0.842407\pi\)
0.879924 0.475114i \(-0.157593\pi\)
\(444\) 0 0
\(445\) 10.0000 20.0000i 0.474045 0.948091i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −30.0000 −1.41579 −0.707894 0.706319i \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 16.0000 32.0000i 0.750092 1.50018i
\(456\) 0 0
\(457\) 8.00000i 0.374224i 0.982339 + 0.187112i \(0.0599128\pi\)
−0.982339 + 0.187112i \(0.940087\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −22.0000 −1.02464 −0.512321 0.858794i \(-0.671214\pi\)
−0.512321 + 0.858794i \(0.671214\pi\)
\(462\) 0 0
\(463\) 20.0000i 0.929479i 0.885448 + 0.464739i \(0.153852\pi\)
−0.885448 + 0.464739i \(0.846148\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 4.00000i 0.185098i −0.995708 0.0925490i \(-0.970499\pi\)
0.995708 0.0925490i \(-0.0295015\pi\)
\(468\) 0 0
\(469\) 48.0000 2.21643
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −24.0000 32.0000i −1.10120 1.46826i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 32.0000 1.46212 0.731059 0.682315i \(-0.239027\pi\)
0.731059 + 0.682315i \(0.239027\pi\)
\(480\) 0 0
\(481\) −16.0000 −0.729537
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 16.0000 + 8.00000i 0.726523 + 0.363261i
\(486\) 0 0
\(487\) 4.00000i 0.181257i 0.995885 + 0.0906287i \(0.0288876\pi\)
−0.995885 + 0.0906287i \(0.971112\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −32.0000 −1.44414 −0.722070 0.691820i \(-0.756809\pi\)
−0.722070 + 0.691820i \(0.756809\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 64.0000i 2.87079i
\(498\) 0 0
\(499\) 24.0000 1.07439 0.537194 0.843459i \(-0.319484\pi\)
0.537194 + 0.843459i \(0.319484\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 28.0000i 1.24846i 0.781241 + 0.624229i \(0.214587\pi\)
−0.781241 + 0.624229i \(0.785413\pi\)
\(504\) 0 0
\(505\) 14.0000 28.0000i 0.622992 1.24598i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −6.00000 −0.265945 −0.132973 0.991120i \(-0.542452\pi\)
−0.132973 + 0.991120i \(0.542452\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 24.0000 + 12.0000i 1.05757 + 0.528783i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −10.0000 −0.438108 −0.219054 0.975713i \(-0.570297\pi\)
−0.219054 + 0.975713i \(0.570297\pi\)
\(522\) 0 0
\(523\) 12.0000i 0.524723i −0.964970 0.262362i \(-0.915499\pi\)
0.964970 0.262362i \(-0.0845013\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 7.00000 0.304348
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 24.0000i 1.03956i
\(534\) 0 0
\(535\) −24.0000 12.0000i −1.03761 0.518805i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −10.0000 −0.429934 −0.214967 0.976621i \(-0.568964\pi\)
−0.214967 + 0.976621i \(0.568964\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −10.0000 + 20.0000i −0.428353 + 0.856706i
\(546\) 0 0
\(547\) 36.0000i 1.53925i 0.638497 + 0.769624i \(0.279557\pi\)
−0.638497 + 0.769624i \(0.720443\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −48.0000 −2.04487
\(552\) 0 0
\(553\) 32.0000i 1.36078i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 4.00000i 0.169485i −0.996403 0.0847427i \(-0.972993\pi\)
0.996403 0.0847427i \(-0.0270068\pi\)
\(558\) 0 0
\(559\) −16.0000 −0.676728
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 28.0000i 1.18006i 0.807382 + 0.590030i \(0.200884\pi\)
−0.807382 + 0.590030i \(0.799116\pi\)
\(564\) 0 0
\(565\) −16.0000 8.00000i −0.673125 0.336563i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 10.0000 0.419222 0.209611 0.977785i \(-0.432780\pi\)
0.209611 + 0.977785i \(0.432780\pi\)
\(570\) 0 0
\(571\) 40.0000 1.67395 0.836974 0.547243i \(-0.184323\pi\)
0.836974 + 0.547243i \(0.184323\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −16.0000 + 12.0000i −0.667246 + 0.500435i
\(576\) 0 0
\(577\) 16.0000i 0.666089i 0.942911 + 0.333044i \(0.108076\pi\)
−0.942911 + 0.333044i \(0.891924\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −48.0000 −1.99138
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 28.0000i 1.15568i 0.816149 + 0.577842i \(0.196105\pi\)
−0.816149 + 0.577842i \(0.803895\pi\)
\(588\) 0 0
\(589\) −64.0000 −2.63707
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 8.00000i 0.328521i 0.986417 + 0.164260i \(0.0525237\pi\)
−0.986417 + 0.164260i \(0.947476\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 32.0000 1.30748 0.653742 0.756717i \(-0.273198\pi\)
0.653742 + 0.756717i \(0.273198\pi\)
\(600\) 0 0
\(601\) −26.0000 −1.06056 −0.530281 0.847822i \(-0.677914\pi\)
−0.530281 + 0.847822i \(0.677914\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 11.0000 22.0000i 0.447214 0.894427i
\(606\) 0 0
\(607\) 20.0000i 0.811775i 0.913923 + 0.405887i \(0.133038\pi\)
−0.913923 + 0.405887i \(0.866962\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 16.0000 0.647291
\(612\) 0 0
\(613\) 20.0000i 0.807792i −0.914805 0.403896i \(-0.867656\pi\)
0.914805 0.403896i \(-0.132344\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 8.00000i 0.322068i −0.986949 0.161034i \(-0.948517\pi\)
0.986949 0.161034i \(-0.0514829\pi\)
\(618\) 0 0
\(619\) 24.0000 0.964641 0.482321 0.875995i \(-0.339794\pi\)
0.482321 + 0.875995i \(0.339794\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 40.0000i 1.60257i
\(624\) 0 0
\(625\) −7.00000 + 24.0000i −0.280000 + 0.960000i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 8.00000 0.318475 0.159237 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −8.00000 4.00000i −0.317470 0.158735i
\(636\) 0 0
\(637\) 36.0000i 1.42637i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 34.0000 1.34292 0.671460 0.741041i \(-0.265668\pi\)
0.671460 + 0.741041i \(0.265668\pi\)
\(642\) 0 0
\(643\) 12.0000i 0.473234i −0.971603 0.236617i \(-0.923961\pi\)
0.971603 0.236617i \(-0.0760386\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 12.0000i 0.471769i 0.971781 + 0.235884i \(0.0757987\pi\)
−0.971781 + 0.235884i \(0.924201\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 20.0000i 0.782660i 0.920250 + 0.391330i \(0.127985\pi\)
−0.920250 + 0.391330i \(0.872015\pi\)
\(654\) 0 0
\(655\) −16.0000 + 32.0000i −0.625172 + 1.25034i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) −30.0000 −1.16686 −0.583432 0.812162i \(-0.698291\pi\)
−0.583432 + 0.812162i \(0.698291\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −64.0000 32.0000i −2.48181 1.24091i
\(666\) 0 0
\(667\) 24.0000i 0.929284i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 8.00000i 0.308377i 0.988041 + 0.154189i \(0.0492764\pi\)
−0.988041 + 0.154189i \(0.950724\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 12.0000i 0.461197i 0.973049 + 0.230599i \(0.0740685\pi\)
−0.973049 + 0.230599i \(0.925932\pi\)
\(678\) 0 0
\(679\) 32.0000 1.22805
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 36.0000i 1.37750i −0.724998 0.688751i \(-0.758159\pi\)
0.724998 0.688751i \(-0.241841\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −48.0000 −1.82865
\(690\) 0 0
\(691\) 8.00000 0.304334 0.152167 0.988355i \(-0.451375\pi\)
0.152167 + 0.988355i \(0.451375\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 8.00000 16.0000i 0.303457 0.606915i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −42.0000 −1.58632 −0.793159 0.609015i \(-0.791565\pi\)
−0.793159 + 0.609015i \(0.791565\pi\)
\(702\) 0 0
\(703\) 32.0000i 1.20690i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 56.0000i 2.10610i
\(708\) 0 0
\(709\) −2.00000 −0.0751116 −0.0375558 0.999295i \(-0.511957\pi\)
−0.0375558 + 0.999295i \(0.511957\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 32.0000i 1.19841i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −16.0000 −0.596699 −0.298350 0.954457i \(-0.596436\pi\)
−0.298350 + 0.954457i \(0.596436\pi\)
\(720\) 0 0
\(721\) 48.0000 1.78761
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 18.0000 + 24.0000i 0.668503 + 0.891338i
\(726\) 0 0
\(727\) 28.0000i 1.03846i −0.854634 0.519231i \(-0.826218\pi\)
0.854634 0.519231i \(-0.173782\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 4.00000i 0.147743i −0.997268 0.0738717i \(-0.976464\pi\)
0.997268 0.0738717i \(-0.0235355\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −40.0000 −1.47142 −0.735712 0.677295i \(-0.763152\pi\)
−0.735712 + 0.677295i \(0.763152\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 12.0000i 0.440237i 0.975473 + 0.220119i \(0.0706445\pi\)
−0.975473 + 0.220119i \(0.929356\pi\)
\(744\) 0 0
\(745\) −14.0000 + 28.0000i −0.512920 + 1.02584i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −48.0000 −1.75388
\(750\) 0 0
\(751\) −8.00000 −0.291924 −0.145962 0.989290i \(-0.546628\pi\)
−0.145962 + 0.989290i \(0.546628\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 8.00000 16.0000i 0.291150 0.582300i
\(756\) 0 0
\(757\) 12.0000i 0.436147i −0.975932 0.218074i \(-0.930023\pi\)
0.975932 0.218074i \(-0.0699773\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 6.00000 0.217500 0.108750 0.994069i \(-0.465315\pi\)
0.108750 + 0.994069i \(0.465315\pi\)
\(762\) 0 0
\(763\) 40.0000i 1.44810i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −14.0000 −0.504853 −0.252426 0.967616i \(-0.581229\pi\)
−0.252426 + 0.967616i \(0.581229\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 28.0000i 1.00709i 0.863969 + 0.503545i \(0.167971\pi\)
−0.863969 + 0.503545i \(0.832029\pi\)
\(774\) 0 0
\(775\) 24.0000 + 32.0000i 0.862105 + 1.14947i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −48.0000 −1.71978
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 8.00000 + 4.00000i 0.285532 + 0.142766i
\(786\) 0 0
\(787\) 12.0000i 0.427754i −0.976861 0.213877i \(-0.931391\pi\)
0.976861 0.213877i \(-0.0686091\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −32.0000 −1.13779
\(792\) 0 0
\(793\) 24.0000i 0.852265i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 12.0000i 0.425062i −0.977154 0.212531i \(-0.931829\pi\)
0.977154 0.212531i \(-0.0681706\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) −16.0000 + 32.0000i −0.563926 + 1.12785i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\)