Properties

Label 180.3.g.a.161.1
Level $180$
Weight $3$
Character 180.161
Analytic conductor $4.905$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 180.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.90464475849\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-5})\)
Defining polynomial: \( x^{4} - 4x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 161.1
Root \(1.58114 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 180.161
Dual form 180.3.g.a.161.3

$q$-expansion

\(f(q)\) \(=\) \(q-2.23607i q^{5} -13.4868 q^{7} +O(q^{10})\) \(q-2.23607i q^{5} -13.4868 q^{7} -17.6590i q^{11} -7.48683 q^{13} +16.9706i q^{17} -10.9737 q^{19} -21.9017i q^{23} -5.00000 q^{25} -47.3575i q^{29} -16.9737 q^{31} +30.1575i q^{35} -5.53950 q^{37} +66.3936i q^{41} +38.9737 q^{43} +32.5642i q^{47} +132.895 q^{49} +11.2392i q^{53} -39.4868 q^{55} -31.8757i q^{59} +46.9210 q^{61} +16.7411i q^{65} -76.0000 q^{67} -77.7445i q^{71} +94.9210 q^{73} +238.165i q^{77} -6.92100 q^{79} -62.1509i q^{83} +37.9473 q^{85} -62.2626i q^{89} +100.974 q^{91} +24.5379i q^{95} -124.974 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 16 q^{7} + 8 q^{13} + 32 q^{19} - 20 q^{25} + 8 q^{31} - 136 q^{37} + 80 q^{43} + 228 q^{49} - 120 q^{55} - 40 q^{61} - 304 q^{67} + 152 q^{73} + 200 q^{79} + 328 q^{91} - 424 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/180\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(91\) \(101\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) − 2.23607i − 0.447214i
\(6\) 0 0
\(7\) −13.4868 −1.92669 −0.963345 0.268265i \(-0.913550\pi\)
−0.963345 + 0.268265i \(0.913550\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 17.6590i − 1.60537i −0.596405 0.802684i \(-0.703405\pi\)
0.596405 0.802684i \(-0.296595\pi\)
\(12\) 0 0
\(13\) −7.48683 −0.575910 −0.287955 0.957644i \(-0.592975\pi\)
−0.287955 + 0.957644i \(0.592975\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 16.9706i 0.998268i 0.866525 + 0.499134i \(0.166349\pi\)
−0.866525 + 0.499134i \(0.833651\pi\)
\(18\) 0 0
\(19\) −10.9737 −0.577561 −0.288781 0.957395i \(-0.593250\pi\)
−0.288781 + 0.957395i \(0.593250\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 21.9017i − 0.952247i −0.879378 0.476124i \(-0.842041\pi\)
0.879378 0.476124i \(-0.157959\pi\)
\(24\) 0 0
\(25\) −5.00000 −0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 47.3575i − 1.63302i −0.577332 0.816509i \(-0.695906\pi\)
0.577332 0.816509i \(-0.304094\pi\)
\(30\) 0 0
\(31\) −16.9737 −0.547538 −0.273769 0.961796i \(-0.588270\pi\)
−0.273769 + 0.961796i \(0.588270\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 30.1575i 0.861642i
\(36\) 0 0
\(37\) −5.53950 −0.149716 −0.0748581 0.997194i \(-0.523850\pi\)
−0.0748581 + 0.997194i \(0.523850\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 66.3936i 1.61935i 0.586875 + 0.809677i \(0.300358\pi\)
−0.586875 + 0.809677i \(0.699642\pi\)
\(42\) 0 0
\(43\) 38.9737 0.906364 0.453182 0.891418i \(-0.350289\pi\)
0.453182 + 0.891418i \(0.350289\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 32.5642i 0.692854i 0.938077 + 0.346427i \(0.112605\pi\)
−0.938077 + 0.346427i \(0.887395\pi\)
\(48\) 0 0
\(49\) 132.895 2.71214
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 11.2392i 0.212061i 0.994363 + 0.106030i \(0.0338141\pi\)
−0.994363 + 0.106030i \(0.966186\pi\)
\(54\) 0 0
\(55\) −39.4868 −0.717942
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 31.8757i − 0.540266i −0.962823 0.270133i \(-0.912932\pi\)
0.962823 0.270133i \(-0.0870676\pi\)
\(60\) 0 0
\(61\) 46.9210 0.769197 0.384598 0.923084i \(-0.374340\pi\)
0.384598 + 0.923084i \(0.374340\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 16.7411i 0.257555i
\(66\) 0 0
\(67\) −76.0000 −1.13433 −0.567164 0.823605i \(-0.691960\pi\)
−0.567164 + 0.823605i \(0.691960\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 77.7445i − 1.09499i −0.836808 0.547497i \(-0.815581\pi\)
0.836808 0.547497i \(-0.184419\pi\)
\(72\) 0 0
\(73\) 94.9210 1.30029 0.650144 0.759811i \(-0.274709\pi\)
0.650144 + 0.759811i \(0.274709\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 238.165i 3.09305i
\(78\) 0 0
\(79\) −6.92100 −0.0876076 −0.0438038 0.999040i \(-0.513948\pi\)
−0.0438038 + 0.999040i \(0.513948\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 62.1509i − 0.748806i −0.927266 0.374403i \(-0.877848\pi\)
0.927266 0.374403i \(-0.122152\pi\)
\(84\) 0 0
\(85\) 37.9473 0.446439
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 62.2626i − 0.699580i −0.936828 0.349790i \(-0.886253\pi\)
0.936828 0.349790i \(-0.113747\pi\)
\(90\) 0 0
\(91\) 100.974 1.10960
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 24.5379i 0.258293i
\(96\) 0 0
\(97\) −124.974 −1.28839 −0.644194 0.764862i \(-0.722807\pi\)
−0.644194 + 0.764862i \(0.722807\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 112.262i − 1.11151i −0.831347 0.555754i \(-0.812429\pi\)
0.831347 0.555754i \(-0.187571\pi\)
\(102\) 0 0
\(103\) −170.302 −1.65342 −0.826711 0.562627i \(-0.809791\pi\)
−0.826711 + 0.562627i \(0.809791\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 93.3381i 0.872319i 0.899869 + 0.436159i \(0.143662\pi\)
−0.899869 + 0.436159i \(0.856338\pi\)
\(108\) 0 0
\(109\) −68.8683 −0.631820 −0.315910 0.948789i \(-0.602310\pi\)
−0.315910 + 0.948789i \(0.602310\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 110.309i − 0.976183i −0.872793 0.488091i \(-0.837693\pi\)
0.872793 0.488091i \(-0.162307\pi\)
\(114\) 0 0
\(115\) −48.9737 −0.425858
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) − 228.879i − 1.92335i
\(120\) 0 0
\(121\) −190.842 −1.57721
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 11.1803i 0.0894427i
\(126\) 0 0
\(127\) −65.3815 −0.514815 −0.257407 0.966303i \(-0.582868\pi\)
−0.257407 + 0.966303i \(0.582868\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 47.2458i 0.360655i 0.983607 + 0.180328i \(0.0577158\pi\)
−0.983607 + 0.180328i \(0.942284\pi\)
\(132\) 0 0
\(133\) 148.000 1.11278
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 216.039i − 1.57693i −0.615079 0.788465i \(-0.710876\pi\)
0.615079 0.788465i \(-0.289124\pi\)
\(138\) 0 0
\(139\) 211.842 1.52404 0.762022 0.647552i \(-0.224207\pi\)
0.762022 + 0.647552i \(0.224207\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 132.210i 0.924548i
\(144\) 0 0
\(145\) −105.895 −0.730308
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 92.5379i 0.621060i 0.950564 + 0.310530i \(0.100506\pi\)
−0.950564 + 0.310530i \(0.899494\pi\)
\(150\) 0 0
\(151\) −123.842 −0.820146 −0.410073 0.912053i \(-0.634497\pi\)
−0.410073 + 0.912053i \(0.634497\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 37.9543i 0.244866i
\(156\) 0 0
\(157\) −31.4868 −0.200553 −0.100277 0.994960i \(-0.531973\pi\)
−0.100277 + 0.994960i \(0.531973\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 295.384i 1.83469i
\(162\) 0 0
\(163\) −101.132 −0.620440 −0.310220 0.950665i \(-0.600403\pi\)
−0.310220 + 0.950665i \(0.600403\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 147.804i − 0.885054i −0.896755 0.442527i \(-0.854082\pi\)
0.896755 0.442527i \(-0.145918\pi\)
\(168\) 0 0
\(169\) −112.947 −0.668327
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 332.079i − 1.91953i −0.280797 0.959767i \(-0.590599\pi\)
0.280797 0.959767i \(-0.409401\pi\)
\(174\) 0 0
\(175\) 67.4342 0.385338
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 45.8688i 0.256250i 0.991758 + 0.128125i \(0.0408959\pi\)
−0.991758 + 0.128125i \(0.959104\pi\)
\(180\) 0 0
\(181\) 132.868 0.734079 0.367040 0.930205i \(-0.380371\pi\)
0.367040 + 0.930205i \(0.380371\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 12.3867i 0.0669551i
\(186\) 0 0
\(187\) 299.684 1.60259
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) − 163.974i − 0.858504i −0.903185 0.429252i \(-0.858777\pi\)
0.903185 0.429252i \(-0.141223\pi\)
\(192\) 0 0
\(193\) 110.000 0.569948 0.284974 0.958535i \(-0.408015\pi\)
0.284974 + 0.958535i \(0.408015\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 211.108i 1.07162i 0.844340 + 0.535808i \(0.179993\pi\)
−0.844340 + 0.535808i \(0.820007\pi\)
\(198\) 0 0
\(199\) −18.1053 −0.0909816 −0.0454908 0.998965i \(-0.514485\pi\)
−0.0454908 + 0.998965i \(0.514485\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 638.703i 3.14632i
\(204\) 0 0
\(205\) 148.460 0.724198
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 193.785i 0.927199i
\(210\) 0 0
\(211\) 341.579 1.61886 0.809428 0.587219i \(-0.199777\pi\)
0.809428 + 0.587219i \(0.199777\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) − 87.1478i − 0.405338i
\(216\) 0 0
\(217\) 228.921 1.05494
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 127.056i − 0.574913i
\(222\) 0 0
\(223\) 59.3288 0.266049 0.133024 0.991113i \(-0.457531\pi\)
0.133024 + 0.991113i \(0.457531\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 115.593i 0.509221i 0.967044 + 0.254610i \(0.0819472\pi\)
−0.967044 + 0.254610i \(0.918053\pi\)
\(228\) 0 0
\(229\) −253.895 −1.10871 −0.554355 0.832280i \(-0.687035\pi\)
−0.554355 + 0.832280i \(0.687035\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 206.401i − 0.885840i −0.896561 0.442920i \(-0.853943\pi\)
0.896561 0.442920i \(-0.146057\pi\)
\(234\) 0 0
\(235\) 72.8157 0.309854
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 346.296i 1.44894i 0.689308 + 0.724469i \(0.257915\pi\)
−0.689308 + 0.724469i \(0.742085\pi\)
\(240\) 0 0
\(241\) −182.053 −0.755405 −0.377703 0.925927i \(-0.623286\pi\)
−0.377703 + 0.925927i \(0.623286\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 297.162i − 1.21290i
\(246\) 0 0
\(247\) 82.1580 0.332623
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 140.807i − 0.560985i −0.959856 0.280493i \(-0.909502\pi\)
0.959856 0.280493i \(-0.0904979\pi\)
\(252\) 0 0
\(253\) −386.763 −1.52871
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 292.630i − 1.13864i −0.822116 0.569320i \(-0.807207\pi\)
0.822116 0.569320i \(-0.192793\pi\)
\(258\) 0 0
\(259\) 74.7103 0.288457
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 270.952i 1.03024i 0.857119 + 0.515118i \(0.172252\pi\)
−0.857119 + 0.515118i \(0.827748\pi\)
\(264\) 0 0
\(265\) 25.1317 0.0948365
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 275.083i 1.02261i 0.859398 + 0.511307i \(0.170838\pi\)
−0.859398 + 0.511307i \(0.829162\pi\)
\(270\) 0 0
\(271\) 248.158 0.915712 0.457856 0.889026i \(-0.348617\pi\)
0.457856 + 0.889026i \(0.348617\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 88.2952i 0.321074i
\(276\) 0 0
\(277\) −342.039 −1.23480 −0.617399 0.786650i \(-0.711814\pi\)
−0.617399 + 0.786650i \(0.711814\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 352.139i 1.25316i 0.779355 + 0.626582i \(0.215547\pi\)
−0.779355 + 0.626582i \(0.784453\pi\)
\(282\) 0 0
\(283\) 249.631 0.882089 0.441045 0.897485i \(-0.354608\pi\)
0.441045 + 0.897485i \(0.354608\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 895.439i − 3.12000i
\(288\) 0 0
\(289\) 1.00000 0.00346021
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 37.4953i − 0.127970i −0.997951 0.0639851i \(-0.979619\pi\)
0.997951 0.0639851i \(-0.0203810\pi\)
\(294\) 0 0
\(295\) −71.2762 −0.241614
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 163.974i 0.548409i
\(300\) 0 0
\(301\) −525.631 −1.74628
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) − 104.919i − 0.343995i
\(306\) 0 0
\(307\) 457.842 1.49134 0.745671 0.666314i \(-0.232129\pi\)
0.745671 + 0.666314i \(0.232129\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) − 8.48528i − 0.0272839i −0.999907 0.0136419i \(-0.995658\pi\)
0.999907 0.0136419i \(-0.00434250\pi\)
\(312\) 0 0
\(313\) −1.57866 −0.00504363 −0.00252181 0.999997i \(-0.500803\pi\)
−0.00252181 + 0.999997i \(0.500803\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 529.195i − 1.66938i −0.550717 0.834692i \(-0.685646\pi\)
0.550717 0.834692i \(-0.314354\pi\)
\(318\) 0 0
\(319\) −836.289 −2.62160
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 186.229i − 0.576561i
\(324\) 0 0
\(325\) 37.4342 0.115182
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) − 439.187i − 1.33492i
\(330\) 0 0
\(331\) −390.763 −1.18055 −0.590276 0.807201i \(-0.700981\pi\)
−0.590276 + 0.807201i \(0.700981\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 169.941i 0.507287i
\(336\) 0 0
\(337\) 414.710 1.23059 0.615297 0.788295i \(-0.289036\pi\)
0.615297 + 0.788295i \(0.289036\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 299.739i 0.878999i
\(342\) 0 0
\(343\) −1131.47 −3.29876
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 128.656i − 0.370767i −0.982666 0.185384i \(-0.940647\pi\)
0.982666 0.185384i \(-0.0593528\pi\)
\(348\) 0 0
\(349\) −37.0790 −0.106244 −0.0531218 0.998588i \(-0.516917\pi\)
−0.0531218 + 0.998588i \(0.516917\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 190.584i 0.539897i 0.962875 + 0.269949i \(0.0870067\pi\)
−0.962875 + 0.269949i \(0.912993\pi\)
\(354\) 0 0
\(355\) −173.842 −0.489696
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 254.782i 0.709699i 0.934923 + 0.354849i \(0.115468\pi\)
−0.934923 + 0.354849i \(0.884532\pi\)
\(360\) 0 0
\(361\) −240.579 −0.666423
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) − 212.250i − 0.581506i
\(366\) 0 0
\(367\) 300.460 0.818693 0.409347 0.912379i \(-0.365757\pi\)
0.409347 + 0.912379i \(0.365757\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) − 151.582i − 0.408576i
\(372\) 0 0
\(373\) 704.092 1.88765 0.943823 0.330452i \(-0.107201\pi\)
0.943823 + 0.330452i \(0.107201\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 354.558i 0.940472i
\(378\) 0 0
\(379\) 333.789 0.880711 0.440355 0.897824i \(-0.354852\pi\)
0.440355 + 0.897824i \(0.354852\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) − 260.996i − 0.681453i −0.940163 0.340726i \(-0.889327\pi\)
0.940163 0.340726i \(-0.110673\pi\)
\(384\) 0 0
\(385\) 532.552 1.38325
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 35.8949i 0.0922747i 0.998935 + 0.0461373i \(0.0146912\pi\)
−0.998935 + 0.0461373i \(0.985309\pi\)
\(390\) 0 0
\(391\) 371.684 0.950598
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 15.4758i 0.0391793i
\(396\) 0 0
\(397\) −90.3552 −0.227595 −0.113797 0.993504i \(-0.536301\pi\)
−0.113797 + 0.993504i \(0.536301\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) − 43.6917i − 0.108957i −0.998515 0.0544784i \(-0.982650\pi\)
0.998515 0.0544784i \(-0.0173496\pi\)
\(402\) 0 0
\(403\) 127.079 0.315333
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 97.8223i 0.240350i
\(408\) 0 0
\(409\) −260.053 −0.635826 −0.317913 0.948120i \(-0.602982\pi\)
−0.317913 + 0.948120i \(0.602982\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 429.902i 1.04092i
\(414\) 0 0
\(415\) −138.974 −0.334876
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 263.285i 0.628366i 0.949362 + 0.314183i \(0.101731\pi\)
−0.949362 + 0.314183i \(0.898269\pi\)
\(420\) 0 0
\(421\) 367.210 0.872233 0.436116 0.899890i \(-0.356354\pi\)
0.436116 + 0.899890i \(0.356354\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) − 84.8528i − 0.199654i
\(426\) 0 0
\(427\) −632.816 −1.48200
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) − 690.768i − 1.60271i −0.598189 0.801355i \(-0.704113\pi\)
0.598189 0.801355i \(-0.295887\pi\)
\(432\) 0 0
\(433\) −117.500 −0.271363 −0.135682 0.990752i \(-0.543322\pi\)
−0.135682 + 0.990752i \(0.543322\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 240.342i 0.549981i
\(438\) 0 0
\(439\) 50.0000 0.113895 0.0569476 0.998377i \(-0.481863\pi\)
0.0569476 + 0.998377i \(0.481863\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 526.794i 1.18915i 0.804040 + 0.594576i \(0.202680\pi\)
−0.804040 + 0.594576i \(0.797320\pi\)
\(444\) 0 0
\(445\) −139.223 −0.312862
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 321.882i 0.716887i 0.933552 + 0.358443i \(0.116692\pi\)
−0.933552 + 0.358443i \(0.883308\pi\)
\(450\) 0 0
\(451\) 1172.45 2.59966
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) − 225.784i − 0.496229i
\(456\) 0 0
\(457\) −433.079 −0.947656 −0.473828 0.880617i \(-0.657128\pi\)
−0.473828 + 0.880617i \(0.657128\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 486.768i 1.05590i 0.849277 + 0.527948i \(0.177039\pi\)
−0.849277 + 0.527948i \(0.822961\pi\)
\(462\) 0 0
\(463\) 423.223 0.914090 0.457045 0.889444i \(-0.348908\pi\)
0.457045 + 0.889444i \(0.348908\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 295.831i 0.633472i 0.948514 + 0.316736i \(0.102587\pi\)
−0.948514 + 0.316736i \(0.897413\pi\)
\(468\) 0 0
\(469\) 1025.00 2.18550
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) − 688.238i − 1.45505i
\(474\) 0 0
\(475\) 54.8683 0.115512
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 266.728i 0.556843i 0.960459 + 0.278421i \(0.0898112\pi\)
−0.960459 + 0.278421i \(0.910189\pi\)
\(480\) 0 0
\(481\) 41.4733 0.0862231
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 279.450i 0.576185i
\(486\) 0 0
\(487\) 247.750 0.508727 0.254364 0.967109i \(-0.418134\pi\)
0.254364 + 0.967109i \(0.418134\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 170.394i − 0.347035i −0.984831 0.173517i \(-0.944487\pi\)
0.984831 0.173517i \(-0.0555133\pi\)
\(492\) 0 0
\(493\) 803.684 1.63019
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 1048.53i 2.10971i
\(498\) 0 0
\(499\) 47.3950 0.0949800 0.0474900 0.998872i \(-0.484878\pi\)
0.0474900 + 0.998872i \(0.484878\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 321.064i 0.638298i 0.947705 + 0.319149i \(0.103397\pi\)
−0.947705 + 0.319149i \(0.896603\pi\)
\(504\) 0 0
\(505\) −251.026 −0.497082
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 122.125i − 0.239931i −0.992778 0.119965i \(-0.961722\pi\)
0.992778 0.119965i \(-0.0382783\pi\)
\(510\) 0 0
\(511\) −1280.18 −2.50525
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 380.808i 0.739433i
\(516\) 0 0
\(517\) 575.052 1.11229
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) − 79.4567i − 0.152508i −0.997088 0.0762540i \(-0.975704\pi\)
0.997088 0.0762540i \(-0.0242960\pi\)
\(522\) 0 0
\(523\) −864.605 −1.65316 −0.826582 0.562816i \(-0.809718\pi\)
−0.826582 + 0.562816i \(0.809718\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 288.053i − 0.546589i
\(528\) 0 0
\(529\) 49.3160 0.0932250
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 497.077i − 0.932603i
\(534\) 0 0
\(535\) 208.710 0.390113
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) − 2346.79i − 4.35398i
\(540\) 0 0
\(541\) 33.6057 0.0621177 0.0310589 0.999518i \(-0.490112\pi\)
0.0310589 + 0.999518i \(0.490112\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 153.994i 0.282558i
\(546\) 0 0
\(547\) −559.079 −1.02208 −0.511041 0.859556i \(-0.670740\pi\)
−0.511041 + 0.859556i \(0.670740\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 519.686i 0.943168i
\(552\) 0 0
\(553\) 93.3423 0.168793
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 310.308i − 0.557105i −0.960421 0.278553i \(-0.910145\pi\)
0.960421 0.278553i \(-0.0898547\pi\)
\(558\) 0 0
\(559\) −291.789 −0.521984
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 566.690i − 1.00655i −0.864125 0.503277i \(-0.832128\pi\)
0.864125 0.503277i \(-0.167872\pi\)
\(564\) 0 0
\(565\) −246.658 −0.436562
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 134.052i 0.235593i 0.993038 + 0.117796i \(0.0375830\pi\)
−0.993038 + 0.117796i \(0.962417\pi\)
\(570\) 0 0
\(571\) −661.920 −1.15923 −0.579615 0.814890i \(-0.696797\pi\)
−0.579615 + 0.814890i \(0.696797\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 109.508i 0.190449i
\(576\) 0 0
\(577\) 76.7630 0.133038 0.0665191 0.997785i \(-0.478811\pi\)
0.0665191 + 0.997785i \(0.478811\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 838.219i 1.44272i
\(582\) 0 0
\(583\) 198.474 0.340436
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 684.554i 1.16619i 0.812404 + 0.583095i \(0.198159\pi\)
−0.812404 + 0.583095i \(0.801841\pi\)
\(588\) 0 0
\(589\) 186.263 0.316237
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 592.146i − 0.998560i −0.866441 0.499280i \(-0.833598\pi\)
0.866441 0.499280i \(-0.166402\pi\)
\(594\) 0 0
\(595\) −511.789 −0.860150
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) − 1012.76i − 1.69075i −0.534169 0.845377i \(-0.679376\pi\)
0.534169 0.845377i \(-0.320624\pi\)
\(600\) 0 0
\(601\) 323.579 0.538400 0.269200 0.963084i \(-0.413241\pi\)
0.269200 + 0.963084i \(0.413241\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 426.736i 0.705348i
\(606\) 0 0
\(607\) −799.828 −1.31767 −0.658837 0.752285i \(-0.728951\pi\)
−0.658837 + 0.752285i \(0.728951\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 243.802i − 0.399022i
\(612\) 0 0
\(613\) −680.302 −1.10979 −0.554896 0.831920i \(-0.687242\pi\)
−0.554896 + 0.831920i \(0.687242\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 264.868i 0.429283i 0.976693 + 0.214641i \(0.0688583\pi\)
−0.976693 + 0.214641i \(0.931142\pi\)
\(618\) 0 0
\(619\) 535.842 0.865658 0.432829 0.901476i \(-0.357515\pi\)
0.432829 + 0.901476i \(0.357515\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 839.726i 1.34787i
\(624\) 0 0
\(625\) 25.0000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 94.0085i − 0.149457i
\(630\) 0 0
\(631\) 307.026 0.486571 0.243286 0.969955i \(-0.421775\pi\)
0.243286 + 0.969955i \(0.421775\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 146.197i 0.230232i
\(636\) 0 0
\(637\) −994.960 −1.56195
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) − 1246.33i − 1.94435i −0.234248 0.972177i \(-0.575263\pi\)
0.234248 0.972177i \(-0.424737\pi\)
\(642\) 0 0
\(643\) 155.395 0.241672 0.120836 0.992672i \(-0.461443\pi\)
0.120836 + 0.992672i \(0.461443\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 142.649i − 0.220478i −0.993905 0.110239i \(-0.964838\pi\)
0.993905 0.110239i \(-0.0351617\pi\)
\(648\) 0 0
\(649\) −562.894 −0.867325
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 924.579i − 1.41589i −0.706266 0.707947i \(-0.749622\pi\)
0.706266 0.707947i \(-0.250378\pi\)
\(654\) 0 0
\(655\) 105.645 0.161290
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 352.046i 0.534212i 0.963667 + 0.267106i \(0.0860674\pi\)
−0.963667 + 0.267106i \(0.913933\pi\)
\(660\) 0 0
\(661\) −642.921 −0.972649 −0.486325 0.873778i \(-0.661663\pi\)
−0.486325 + 0.873778i \(0.661663\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 330.938i − 0.497651i
\(666\) 0 0
\(667\) −1037.21 −1.55504
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) − 828.580i − 1.23484i
\(672\) 0 0
\(673\) 982.605 1.46004 0.730019 0.683427i \(-0.239511\pi\)
0.730019 + 0.683427i \(0.239511\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 386.769i 0.571298i 0.958334 + 0.285649i \(0.0922092\pi\)
−0.958334 + 0.285649i \(0.907791\pi\)
\(678\) 0 0
\(679\) 1685.50 2.48233
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 504.539i − 0.738710i −0.929288 0.369355i \(-0.879579\pi\)
0.929288 0.369355i \(-0.120421\pi\)
\(684\) 0 0
\(685\) −483.079 −0.705225
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 84.1462i − 0.122128i
\(690\) 0 0
\(691\) −271.079 −0.392300 −0.196150 0.980574i \(-0.562844\pi\)
−0.196150 + 0.980574i \(0.562844\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) − 473.693i − 0.681573i
\(696\) 0 0
\(697\) −1126.74 −1.61655
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) − 358.559i − 0.511496i −0.966743 0.255748i \(-0.917678\pi\)
0.966743 0.255748i \(-0.0823218\pi\)
\(702\) 0 0
\(703\) 60.7886 0.0864703
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 1514.06i 2.14153i
\(708\) 0 0
\(709\) −733.579 −1.03467 −0.517333 0.855784i \(-0.673075\pi\)
−0.517333 + 0.855784i \(0.673075\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 371.752i 0.521391i
\(714\) 0 0
\(715\) 295.631 0.413470
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 1108.37i 1.54155i 0.637110 + 0.770773i \(0.280130\pi\)
−0.637110 + 0.770773i \(0.719870\pi\)
\(720\) 0 0
\(721\) 2296.84 3.18563
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 236.788i 0.326604i
\(726\) 0 0
\(727\) 1265.04 1.74008 0.870040 0.492981i \(-0.164093\pi\)
0.870040 + 0.492981i \(0.164093\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 661.405i 0.904795i
\(732\) 0 0
\(733\) −586.749 −0.800477 −0.400238 0.916411i \(-0.631073\pi\)
−0.400238 + 0.916411i \(0.631073\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1342.09i 1.82101i
\(738\) 0 0
\(739\) −215.973 −0.292250 −0.146125 0.989266i \(-0.546680\pi\)
−0.146125 + 0.989266i \(0.546680\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 599.961i 0.807484i 0.914873 + 0.403742i \(0.132291\pi\)
−0.914873 + 0.403742i \(0.867709\pi\)
\(744\) 0 0
\(745\) 206.921 0.277746
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 1258.84i − 1.68069i
\(750\) 0 0
\(751\) 813.657 1.08343 0.541716 0.840562i \(-0.317775\pi\)
0.541716 + 0.840562i \(0.317775\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 276.919i 0.366780i
\(756\) 0 0
\(757\) 1070.43 1.41405 0.707023 0.707190i \(-0.250038\pi\)
0.707023 + 0.707190i \(0.250038\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) − 654.892i − 0.860567i −0.902694 0.430284i \(-0.858414\pi\)
0.902694 0.430284i \(-0.141586\pi\)
\(762\) 0 0
\(763\) 928.816 1.21732
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 238.648i 0.311144i
\(768\) 0 0
\(769\) 682.631 0.887686 0.443843 0.896105i \(-0.353615\pi\)
0.443843 + 0.896105i \(0.353615\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 770.690i 0.997012i 0.866886 + 0.498506i \(0.166118\pi\)
−0.866886 + 0.498506i \(0.833882\pi\)
\(774\) 0 0
\(775\) 84.8683 0.109508
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 728.581i − 0.935277i
\(780\) 0 0
\(781\) −1372.89 −1.75787
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 70.4067i 0.0896901i
\(786\) 0 0
\(787\) −253.869 −0.322578 −0.161289 0.986907i \(-0.551565\pi\)
−0.161289 + 0.986907i \(0.551565\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 1487.71i 1.88080i
\(792\) 0 0
\(793\) −351.290 −0.442988
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 1055.86i − 1.32479i −0.749154 0.662396i \(-0.769540\pi\)
0.749154 0.662396i \(-0.230460\pi\)
\(798\) 0 0
\(799\) −552.632 −0.691655
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 1676.21i − 2.08744i
\(804\) 0 0
\(805\) 660.500 0.820496
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 506.698i 0.626326i 0.949699 + 0.313163i \(0.101389\pi\)
−0.949699 + 0.313163i \(0.898611\pi\)
\(810\) 0 0
\(811\) 537.473 0.662729 0.331365 0.943503i \(-0.392491\pi\)
0.331365 + 0.943503i \(0.392491\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 226.137i 0.277469i
\(816\) 0 0
\(817\) −427.684 −0.523481
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 257.219i 0.313299i 0.987654 + 0.156650i \(0.0500694\pi\)
−0.987654 + 0.156650i \(0.949931\pi\)
\(822\) 0 0
\(823\) −367.013 −0.445945 −0.222973 0.974825i \(-0.571576\pi\)
−0.222973 + 0.974825i \(0.571576\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 122.254i − 0.147829i −0.997265 0.0739144i \(-0.976451\pi\)
0.997265 0.0739144i \(-0.0235492\pi\)
\(828\) 0 0
\(829\) 335.237 0.404387 0.202194 0.979346i \(-0.435193\pi\)
0.202194 + 0.979346i \(0.435193\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 2255.30i 2.70744i
\(834\) 0 0
\(835\) −330.500 −0.395808
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 774.021i 0.922552i 0.887257 + 0.461276i \(0.152608\pi\)
−0.887257 + 0.461276i \(0.847392\pi\)
\(840\) 0 0
\(841\) −1401.74 −1.66675
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 252.558i 0.298885i
\(846\) 0 0
\(847\) 2573.85 3.03879
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 121.324i 0.142567i
\(852\) 0 0
\(853\) 764.591 0.896356 0.448178 0.893944i \(-0.352073\pi\)
0.448178 + 0.893944i \(0.352073\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 1516.15i − 1.76913i −0.466413 0.884567i \(-0.654454\pi\)
0.466413 0.884567i \(-0.345546\pi\)
\(858\) 0 0
\(859\) 433.684 0.504871 0.252435 0.967614i \(-0.418768\pi\)
0.252435 + 0.967614i \(0.418768\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) − 1255.50i − 1.45481i −0.686206 0.727407i \(-0.740725\pi\)
0.686206 0.727407i \(-0.259275\pi\)
\(864\) 0 0
\(865\) −742.552 −0.858442
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 122.218i 0.140642i
\(870\) 0 0
\(871\) 568.999 0.653271
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) − 150.787i − 0.172328i
\(876\) 0 0
\(877\) 1003.59 1.14435 0.572173 0.820133i \(-0.306100\pi\)
0.572173 + 0.820133i \(0.306100\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 737.214i 0.836792i 0.908265 + 0.418396i \(0.137408\pi\)
−0.908265 + 0.418396i \(0.862592\pi\)
\(882\) 0 0
\(883\) 336.394 0.380968 0.190484 0.981690i \(-0.438994\pi\)
0.190484 + 0.981690i \(0.438994\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 793.132i − 0.894174i −0.894491 0.447087i \(-0.852461\pi\)
0.894491 0.447087i \(-0.147539\pi\)
\(888\) 0 0
\(889\) 881.789 0.991889
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 357.348i − 0.400166i
\(894\) 0 0
\(895\) 102.566 0.114599
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 803.831i 0.894139i
\(900\) 0 0
\(901\) −190.736 −0.211694
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) − 297.103i − 0.328290i
\(906\) 0 0
\(907\) −1079.66 −1.19036 −0.595180 0.803592i \(-0.702919\pi\)
−0.595180 + 0.803592i \(0.702919\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 1473.69i 1.61766i 0.588045 + 0.808828i \(0.299898\pi\)
−0.588045 + 0.808828i \(0.700102\pi\)
\(912\) 0 0
\(913\) −1097.53 −1.20211
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 637.196i − 0.694871i
\(918\) 0 0
\(919\) −1602.92 −1.74420 −0.872101 0.489327i \(-0.837243\pi\)
−0.872101 + 0.489327i \(0.837243\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 582.060i 0.630618i
\(924\) 0 0
\(925\) 27.6975 0.0299432
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) − 1033.98i − 1.11300i −0.830848 0.556499i \(-0.812144\pi\)
0.830848 0.556499i \(-0.187856\pi\)
\(930\) 0 0
\(931\) −1458.34 −1.56642
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) − 670.114i − 0.716699i
\(936\) 0 0
\(937\) −1258.05 −1.34264 −0.671319 0.741169i \(-0.734272\pi\)
−0.671319 + 0.741169i \(0.734272\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 762.875i − 0.810707i −0.914160 0.405353i \(-0.867148\pi\)
0.914160 0.405353i \(-0.132852\pi\)
\(942\) 0 0
\(943\) 1454.13 1.54203
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 1023.55i − 1.08084i −0.841396 0.540419i \(-0.818266\pi\)
0.841396 0.540419i \(-0.181734\pi\)
\(948\) 0 0
\(949\) −710.658 −0.748849
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 63.7513i 0.0668954i 0.999440 + 0.0334477i \(0.0106487\pi\)
−0.999440 + 0.0334477i \(0.989351\pi\)
\(954\) 0 0
\(955\) −366.658 −0.383935
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 2913.69i 3.03826i
\(960\) 0 0
\(961\) −672.895 −0.700203
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 245.967i − 0.254889i
\(966\) 0 0
\(967\) −505.986 −0.523254 −0.261627 0.965169i \(-0.584259\pi\)
−0.261627 + 0.965169i \(0.584259\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) − 1803.51i − 1.85738i −0.370862 0.928688i \(-0.620938\pi\)
0.370862 0.928688i \(-0.379062\pi\)
\(972\) 0 0
\(973\) −2857.08 −2.93636
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 1844.54i − 1.88797i −0.329994 0.943983i \(-0.607047\pi\)
0.329994 0.943983i \(-0.392953\pi\)
\(978\) 0 0
\(979\) −1099.50 −1.12308
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 592.146i 0.602386i 0.953563 + 0.301193i \(0.0973849\pi\)
−0.953563 + 0.301193i \(0.902615\pi\)
\(984\) 0 0
\(985\) 472.053 0.479241
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 853.589i − 0.863083i
\(990\) 0 0
\(991\) 961.684 0.970418 0.485209 0.874398i \(-0.338744\pi\)
0.485209 + 0.874398i \(0.338744\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 40.4848i 0.0406882i
\(996\) 0 0
\(997\) 899.670 0.902378 0.451189 0.892429i \(-0.351000\pi\)
0.451189 + 0.892429i \(0.351000\pi\)
\(998\) 0 0
\(999\) 0 0
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 \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 180.3.g.a.161.1 4
3.2 odd 2 inner 180.3.g.a.161.3 yes 4
4.3 odd 2 720.3.l.c.161.2 4
5.2 odd 4 900.3.b.b.449.1 8
5.3 odd 4 900.3.b.b.449.7 8
5.4 even 2 900.3.g.d.701.3 4
8.3 odd 2 2880.3.l.f.1601.4 4
8.5 even 2 2880.3.l.b.1601.3 4
9.2 odd 6 1620.3.o.f.701.2 8
9.4 even 3 1620.3.o.f.1241.2 8
9.5 odd 6 1620.3.o.f.1241.4 8
9.7 even 3 1620.3.o.f.701.4 8
12.11 even 2 720.3.l.c.161.4 4
15.2 even 4 900.3.b.b.449.2 8
15.8 even 4 900.3.b.b.449.8 8
15.14 odd 2 900.3.g.d.701.4 4
20.3 even 4 3600.3.c.k.449.2 8
20.7 even 4 3600.3.c.k.449.8 8
20.19 odd 2 3600.3.l.n.1601.2 4
24.5 odd 2 2880.3.l.b.1601.1 4
24.11 even 2 2880.3.l.f.1601.2 4
60.23 odd 4 3600.3.c.k.449.1 8
60.47 odd 4 3600.3.c.k.449.7 8
60.59 even 2 3600.3.l.n.1601.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
180.3.g.a.161.1 4 1.1 even 1 trivial
180.3.g.a.161.3 yes 4 3.2 odd 2 inner
720.3.l.c.161.2 4 4.3 odd 2
720.3.l.c.161.4 4 12.11 even 2
900.3.b.b.449.1 8 5.2 odd 4
900.3.b.b.449.2 8 15.2 even 4
900.3.b.b.449.7 8 5.3 odd 4
900.3.b.b.449.8 8 15.8 even 4
900.3.g.d.701.3 4 5.4 even 2
900.3.g.d.701.4 4 15.14 odd 2
1620.3.o.f.701.2 8 9.2 odd 6
1620.3.o.f.701.4 8 9.7 even 3
1620.3.o.f.1241.2 8 9.4 even 3
1620.3.o.f.1241.4 8 9.5 odd 6
2880.3.l.b.1601.1 4 24.5 odd 2
2880.3.l.b.1601.3 4 8.5 even 2
2880.3.l.f.1601.2 4 24.11 even 2
2880.3.l.f.1601.4 4 8.3 odd 2
3600.3.c.k.449.1 8 60.23 odd 4
3600.3.c.k.449.2 8 20.3 even 4
3600.3.c.k.449.7 8 60.47 odd 4
3600.3.c.k.449.8 8 20.7 even 4
3600.3.l.n.1601.1 4 60.59 even 2
3600.3.l.n.1601.2 4 20.19 odd 2