Properties

Label 900.3.g.d.701.3
Level $900$
Weight $3$
Character 900.701
Analytic conductor $24.523$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 900.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(24.5232237924\)
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: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{3} \)
Twist minimal: no (minimal twist has level 180)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 701.3
Root \(-1.58114 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 900.701
Dual form 900.3.g.d.701.4

$q$-expansion

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/900\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(451\) \(577\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(5\) 0 0
\(6\) 0 0
\(7\) 13.4868 1.92669 0.963345 0.268265i \(-0.0864502\pi\)
0.963345 + 0.268265i \(0.0864502\pi\)
\(8\) 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 \(-0.407025\pi\)
0.287955 + 0.957644i \(0.407025\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 16.9706i − 0.998268i −0.866525 0.499134i \(-0.833651\pi\)
0.866525 0.499134i \(-0.166349\pi\)
\(18\) 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 \(0.157959\pi\)
−0.879378 + 0.476124i \(0.842041\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 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
\(35\) 0 0
\(36\) 0 0
\(37\) 5.53950 0.149716 0.0748581 0.997194i \(-0.476150\pi\)
0.0748581 + 0.997194i \(0.476150\pi\)
\(38\) 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 \(-0.649711\pi\)
−0.453182 + 0.891418i \(0.649711\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 32.5642i − 0.692854i −0.938077 0.346427i \(-0.887395\pi\)
0.938077 0.346427i \(-0.112605\pi\)
\(48\) 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 \(-0.966186\pi\)
0.994363 0.106030i \(-0.0338141\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 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
\(65\) 0 0
\(66\) 0 0
\(67\) 76.0000 1.13433 0.567164 0.823605i \(-0.308040\pi\)
0.567164 + 0.823605i \(0.308040\pi\)
\(68\) 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 \(-0.725291\pi\)
−0.650144 + 0.759811i \(0.725291\pi\)
\(74\) 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 \(0.122152\pi\)
−0.927266 + 0.374403i \(0.877848\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 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
\(95\) 0 0
\(96\) 0 0
\(97\) 124.974 1.28839 0.644194 0.764862i \(-0.277193\pi\)
0.644194 + 0.764862i \(0.277193\pi\)
\(98\) 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 \(-0.190209\pi\)
0.826711 + 0.562627i \(0.190209\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 93.3381i − 0.872319i −0.899869 0.436159i \(-0.856338\pi\)
0.899869 0.436159i \(-0.143662\pi\)
\(108\) 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 \(0.162307\pi\)
−0.872793 + 0.488091i \(0.837693\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 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
\(125\) 0 0
\(126\) 0 0
\(127\) 65.3815 0.514815 0.257407 0.966303i \(-0.417132\pi\)
0.257407 + 0.966303i \(0.417132\pi\)
\(128\) 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 \(0.289124\pi\)
−0.615079 + 0.788465i \(0.710876\pi\)
\(138\) 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
\(145\) 0 0
\(146\) 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
\(155\) 0 0
\(156\) 0 0
\(157\) 31.4868 0.200553 0.100277 0.994960i \(-0.468027\pi\)
0.100277 + 0.994960i \(0.468027\pi\)
\(158\) 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 \(-0.399597\pi\)
0.310220 + 0.950665i \(0.399597\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 147.804i 0.885054i 0.896755 + 0.442527i \(0.145918\pi\)
−0.896755 + 0.442527i \(0.854082\pi\)
\(168\) 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 \(0.409401\pi\)
−0.280797 + 0.959767i \(0.590599\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 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
\(185\) 0 0
\(186\) 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 \(-0.591985\pi\)
−0.284974 + 0.958535i \(0.591985\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 211.108i − 1.07162i −0.844340 0.535808i \(-0.820007\pi\)
0.844340 0.535808i \(-0.179993\pi\)
\(198\) 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
\(205\) 0 0
\(206\) 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
\(215\) 0 0
\(216\) 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 \(-0.542469\pi\)
−0.133024 + 0.991113i \(0.542469\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 115.593i − 0.509221i −0.967044 0.254610i \(-0.918053\pi\)
0.967044 0.254610i \(-0.0819472\pi\)
\(228\) 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 \(0.146057\pi\)
−0.896561 + 0.442920i \(0.853943\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 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
\(245\) 0 0
\(246\) 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 \(0.192793\pi\)
−0.822116 + 0.569320i \(0.807207\pi\)
\(258\) 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 \(-0.827748\pi\)
0.857119 0.515118i \(-0.172252\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 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
\(275\) 0 0
\(276\) 0 0
\(277\) 342.039 1.23480 0.617399 0.786650i \(-0.288186\pi\)
0.617399 + 0.786650i \(0.288186\pi\)
\(278\) 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 \(-0.645392\pi\)
−0.441045 + 0.897485i \(0.645392\pi\)
\(284\) 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 \(0.0203810\pi\)
−0.997951 + 0.0639851i \(0.979619\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 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
\(305\) 0 0
\(306\) 0 0
\(307\) −457.842 −1.49134 −0.745671 0.666314i \(-0.767871\pi\)
−0.745671 + 0.666314i \(0.767871\pi\)
\(308\) 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 \(-0.499197\pi\)
0.00252181 + 0.999997i \(0.499197\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 529.195i 1.66938i 0.550717 + 0.834692i \(0.314354\pi\)
−0.550717 + 0.834692i \(0.685646\pi\)
\(318\) 0 0
\(319\) −836.289 −2.62160
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 186.229i 0.576561i
\(324\) 0 0
\(325\) 0 0
\(326\) 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
\(335\) 0 0
\(336\) 0 0
\(337\) −414.710 −1.23059 −0.615297 0.788295i \(-0.710964\pi\)
−0.615297 + 0.788295i \(0.710964\pi\)
\(338\) 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 \(0.0593528\pi\)
−0.982666 + 0.185384i \(0.940647\pi\)
\(348\) 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 \(-0.912993\pi\)
0.962875 0.269949i \(-0.0870067\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 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
\(365\) 0 0
\(366\) 0 0
\(367\) −300.460 −0.818693 −0.409347 0.912379i \(-0.634243\pi\)
−0.409347 + 0.912379i \(0.634243\pi\)
\(368\) 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 \(-0.892799\pi\)
−0.943823 + 0.330452i \(0.892799\pi\)
\(374\) 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 \(0.110673\pi\)
−0.940163 + 0.340726i \(0.889327\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 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
\(395\) 0 0
\(396\) 0 0
\(397\) 90.3552 0.227595 0.113797 0.993504i \(-0.463699\pi\)
0.113797 + 0.993504i \(0.463699\pi\)
\(398\) 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
\(415\) 0 0
\(416\) 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
\(425\) 0 0
\(426\) 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 \(-0.456678\pi\)
0.135682 + 0.990752i \(0.456678\pi\)
\(434\) 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 \(-0.797320\pi\)
0.804040 0.594576i \(-0.202680\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 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
\(455\) 0 0
\(456\) 0 0
\(457\) 433.079 0.947656 0.473828 0.880617i \(-0.342872\pi\)
0.473828 + 0.880617i \(0.342872\pi\)
\(458\) 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 \(-0.651092\pi\)
−0.457045 + 0.889444i \(0.651092\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 295.831i − 0.633472i −0.948514 0.316736i \(-0.897413\pi\)
0.948514 0.316736i \(-0.102587\pi\)
\(468\) 0 0
\(469\) 1025.00 2.18550
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 688.238i 1.45505i
\(474\) 0 0
\(475\) 0 0
\(476\) 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
\(485\) 0 0
\(486\) 0 0
\(487\) −247.750 −0.508727 −0.254364 0.967109i \(-0.581866\pi\)
−0.254364 + 0.967109i \(0.581866\pi\)
\(488\) 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 \(-0.896603\pi\)
0.947705 0.319149i \(-0.103397\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 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
\(515\) 0 0
\(516\) 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 \(-0.190282\pi\)
0.826582 + 0.562816i \(0.190282\pi\)
\(524\) 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
\(535\) 0 0
\(536\) 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
\(545\) 0 0
\(546\) 0 0
\(547\) 559.079 1.02208 0.511041 0.859556i \(-0.329260\pi\)
0.511041 + 0.859556i \(0.329260\pi\)
\(548\) 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 \(0.0898547\pi\)
−0.960421 + 0.278553i \(0.910145\pi\)
\(558\) 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 \(0.167872\pi\)
−0.864125 + 0.503277i \(0.832128\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 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
\(575\) 0 0
\(576\) 0 0
\(577\) −76.7630 −0.133038 −0.0665191 0.997785i \(-0.521189\pi\)
−0.0665191 + 0.997785i \(0.521189\pi\)
\(578\) 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 \(-0.801841\pi\)
0.812404 0.583095i \(-0.198159\pi\)
\(588\) 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 \(0.166402\pi\)
−0.866441 + 0.499280i \(0.833598\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 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
\(605\) 0 0
\(606\) 0 0
\(607\) 799.828 1.31767 0.658837 0.752285i \(-0.271049\pi\)
0.658837 + 0.752285i \(0.271049\pi\)
\(608\) 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 \(-0.312758\pi\)
0.554896 + 0.831920i \(0.312758\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 264.868i − 0.429283i −0.976693 0.214641i \(-0.931142\pi\)
0.976693 0.214641i \(-0.0688583\pi\)
\(618\) 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
\(625\) 0 0
\(626\) 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
\(635\) 0 0
\(636\) 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 \(-0.538557\pi\)
−0.120836 + 0.992672i \(0.538557\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 142.649i 0.220478i 0.993905 + 0.110239i \(0.0351617\pi\)
−0.993905 + 0.110239i \(0.964838\pi\)
\(648\) 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 \(0.250378\pi\)
−0.706266 + 0.707947i \(0.749622\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 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
\(665\) 0 0
\(666\) 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 \(-0.760489\pi\)
−0.730019 + 0.683427i \(0.760489\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 386.769i − 0.571298i −0.958334 0.285649i \(-0.907791\pi\)
0.958334 0.285649i \(-0.0922092\pi\)
\(678\) 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 \(0.120421\pi\)
−0.929288 + 0.369355i \(0.879579\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 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
\(695\) 0 0
\(696\) 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
\(715\) 0 0
\(716\) 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
\(725\) 0 0
\(726\) 0 0
\(727\) −1265.04 −1.74008 −0.870040 0.492981i \(-0.835907\pi\)
−0.870040 + 0.492981i \(0.835907\pi\)
\(728\) 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 \(-0.368927\pi\)
0.400238 + 0.916411i \(0.368927\pi\)
\(734\) 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 \(-0.867709\pi\)
0.914873 0.403742i \(-0.132291\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 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
\(755\) 0 0
\(756\) 0 0
\(757\) −1070.43 −1.41405 −0.707023 0.707190i \(-0.749962\pi\)
−0.707023 + 0.707190i \(0.749962\pi\)
\(758\) 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 \(-0.833882\pi\)
0.866886 0.498506i \(-0.166118\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 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
\(785\) 0 0
\(786\) 0 0
\(787\) 253.869 0.322578 0.161289 0.986907i \(-0.448435\pi\)
0.161289 + 0.986907i \(0.448435\pi\)
\(788\) 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 \(0.230460\pi\)
−0.749154 + 0.662396i \(0.769540\pi\)
\(798\) 0 0
\(799\) −552.632 −0.691655
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 1676.21i 2.08744i
\(804\) 0 0
\(805\) 0 0
\(806\) 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
\(815\) 0 0
\(816\) 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 \(-0.428424\pi\)
0.222973 + 0.974825i \(0.428424\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 122.254i 0.147829i 0.997265 + 0.0739144i \(0.0235492\pi\)
−0.997265 + 0.0739144i \(0.976451\pi\)
\(828\) 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
\(835\) 0 0
\(836\) 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
\(845\) 0 0
\(846\) 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 \(-0.647927\pi\)
−0.448178 + 0.893944i \(0.647927\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 1516.15i 1.76913i 0.466413 + 0.884567i \(0.345546\pi\)
−0.466413 + 0.884567i \(0.654454\pi\)
\(858\) 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 \(0.259275\pi\)
−0.686206 + 0.727407i \(0.740725\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 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
\(875\) 0 0
\(876\) 0 0
\(877\) −1003.59 −1.14435 −0.572173 0.820133i \(-0.693900\pi\)
−0.572173 + 0.820133i \(0.693900\pi\)
\(878\) 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 \(-0.561006\pi\)
−0.190484 + 0.981690i \(0.561006\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 793.132i 0.894174i 0.894491 + 0.447087i \(0.147539\pi\)
−0.894491 + 0.447087i \(0.852461\pi\)
\(888\) 0 0
\(889\) 881.789 0.991889
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 357.348i 0.400166i
\(894\) 0 0
\(895\) 0 0
\(896\) 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
\(905\) 0 0
\(906\) 0 0
\(907\) 1079.66 1.19036 0.595180 0.803592i \(-0.297081\pi\)
0.595180 + 0.803592i \(0.297081\pi\)
\(908\) 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
\(925\) 0 0
\(926\) 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
\(935\) 0 0
\(936\) 0 0
\(937\) 1258.05 1.34264 0.671319 0.741169i \(-0.265728\pi\)
0.671319 + 0.741169i \(0.265728\pi\)
\(938\) 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 \(0.181734\pi\)
−0.841396 + 0.540419i \(0.818266\pi\)
\(948\) 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 \(-0.989351\pi\)
0.999440 0.0334477i \(-0.0106487\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 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
\(965\) 0 0
\(966\) 0 0
\(967\) 505.986 0.523254 0.261627 0.965169i \(-0.415741\pi\)
0.261627 + 0.965169i \(0.415741\pi\)
\(968\) 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 \(0.392953\pi\)
−0.329994 + 0.943983i \(0.607047\pi\)
\(978\) 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 \(-0.902615\pi\)
0.953563 0.301193i \(-0.0973849\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 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
\(995\) 0 0
\(996\) 0 0
\(997\) −899.670 −0.902378 −0.451189 0.892429i \(-0.649000\pi\)
−0.451189 + 0.892429i \(0.649000\pi\)
\(998\) 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 900.3.g.d.701.3 4
3.2 odd 2 inner 900.3.g.d.701.4 4
4.3 odd 2 3600.3.l.n.1601.2 4
5.2 odd 4 900.3.b.b.449.7 8
5.3 odd 4 900.3.b.b.449.1 8
5.4 even 2 180.3.g.a.161.1 4
12.11 even 2 3600.3.l.n.1601.1 4
15.2 even 4 900.3.b.b.449.8 8
15.8 even 4 900.3.b.b.449.2 8
15.14 odd 2 180.3.g.a.161.3 yes 4
20.3 even 4 3600.3.c.k.449.8 8
20.7 even 4 3600.3.c.k.449.2 8
20.19 odd 2 720.3.l.c.161.2 4
40.19 odd 2 2880.3.l.f.1601.4 4
40.29 even 2 2880.3.l.b.1601.3 4
45.4 even 6 1620.3.o.f.1241.2 8
45.14 odd 6 1620.3.o.f.1241.4 8
45.29 odd 6 1620.3.o.f.701.2 8
45.34 even 6 1620.3.o.f.701.4 8
60.23 odd 4 3600.3.c.k.449.7 8
60.47 odd 4 3600.3.c.k.449.1 8
60.59 even 2 720.3.l.c.161.4 4
120.29 odd 2 2880.3.l.b.1601.1 4
120.59 even 2 2880.3.l.f.1601.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
180.3.g.a.161.1 4 5.4 even 2
180.3.g.a.161.3 yes 4 15.14 odd 2
720.3.l.c.161.2 4 20.19 odd 2
720.3.l.c.161.4 4 60.59 even 2
900.3.b.b.449.1 8 5.3 odd 4
900.3.b.b.449.2 8 15.8 even 4
900.3.b.b.449.7 8 5.2 odd 4
900.3.b.b.449.8 8 15.2 even 4
900.3.g.d.701.3 4 1.1 even 1 trivial
900.3.g.d.701.4 4 3.2 odd 2 inner
1620.3.o.f.701.2 8 45.29 odd 6
1620.3.o.f.701.4 8 45.34 even 6
1620.3.o.f.1241.2 8 45.4 even 6
1620.3.o.f.1241.4 8 45.14 odd 6
2880.3.l.b.1601.1 4 120.29 odd 2
2880.3.l.b.1601.3 4 40.29 even 2
2880.3.l.f.1601.2 4 120.59 even 2
2880.3.l.f.1601.4 4 40.19 odd 2
3600.3.c.k.449.1 8 60.47 odd 4
3600.3.c.k.449.2 8 20.7 even 4
3600.3.c.k.449.7 8 60.23 odd 4
3600.3.c.k.449.8 8 20.3 even 4
3600.3.l.n.1601.1 4 12.11 even 2
3600.3.l.n.1601.2 4 4.3 odd 2