Properties

Label 3600.2.w.d.593.2
Level $3600$
Weight $2$
Character 3600.593
Analytic conductor $28.746$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3600.w (of order \(4\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(28.7461447277\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 360)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 593.2
Root \(0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 3600.593
Dual form 3600.2.w.d.1457.1

$q$-expansion

\(f(q)\) \(=\) \(q+O(q^{10})\) \(q+5.65685i q^{11} +(-3.00000 - 3.00000i) q^{13} -4.00000i q^{19} +(2.82843 - 2.82843i) q^{23} +1.41421 q^{29} +8.00000 q^{31} +(-7.00000 + 7.00000i) q^{37} +1.41421i q^{41} +(4.00000 + 4.00000i) q^{43} +(2.82843 + 2.82843i) q^{47} +7.00000i q^{49} +(8.48528 - 8.48528i) q^{53} +11.3137 q^{59} -12.0000 q^{61} +(-8.00000 + 8.00000i) q^{67} +5.65685i q^{71} +(3.00000 + 3.00000i) q^{73} +8.00000i q^{79} +(-11.3137 + 11.3137i) q^{83} -7.07107 q^{89} +(-5.00000 + 5.00000i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q - 12q^{13} + 32q^{31} - 28q^{37} + 16q^{43} - 48q^{61} - 32q^{67} + 12q^{73} - 20q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3600\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(2801\) \(3151\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(1\) \(-1\) \(1\)

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
\(7\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 5.65685i 1.70561i 0.522233 + 0.852803i \(0.325099\pi\)
−0.522233 + 0.852803i \(0.674901\pi\)
\(12\) 0 0
\(13\) −3.00000 3.00000i −0.832050 0.832050i 0.155747 0.987797i \(-0.450222\pi\)
−0.987797 + 0.155747i \(0.950222\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(18\) 0 0
\(19\) 4.00000i 0.917663i −0.888523 0.458831i \(-0.848268\pi\)
0.888523 0.458831i \(-0.151732\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.82843 2.82843i 0.589768 0.589768i −0.347801 0.937568i \(-0.613071\pi\)
0.937568 + 0.347801i \(0.113071\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1.41421 0.262613 0.131306 0.991342i \(-0.458083\pi\)
0.131306 + 0.991342i \(0.458083\pi\)
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −7.00000 + 7.00000i −1.15079 + 1.15079i −0.164399 + 0.986394i \(0.552568\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 1.41421i 0.220863i 0.993884 + 0.110432i \(0.0352233\pi\)
−0.993884 + 0.110432i \(0.964777\pi\)
\(42\) 0 0
\(43\) 4.00000 + 4.00000i 0.609994 + 0.609994i 0.942944 0.332950i \(-0.108044\pi\)
−0.332950 + 0.942944i \(0.608044\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.82843 + 2.82843i 0.412568 + 0.412568i 0.882632 0.470064i \(-0.155769\pi\)
−0.470064 + 0.882632i \(0.655769\pi\)
\(48\) 0 0
\(49\) 7.00000i 1.00000i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 8.48528 8.48528i 1.16554 1.16554i 0.182300 0.983243i \(-0.441646\pi\)
0.983243 0.182300i \(-0.0583542\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 11.3137 1.47292 0.736460 0.676481i \(-0.236496\pi\)
0.736460 + 0.676481i \(0.236496\pi\)
\(60\) 0 0
\(61\) −12.0000 −1.53644 −0.768221 0.640184i \(-0.778858\pi\)
−0.768221 + 0.640184i \(0.778858\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −8.00000 + 8.00000i −0.977356 + 0.977356i −0.999749 0.0223937i \(-0.992871\pi\)
0.0223937 + 0.999749i \(0.492871\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 5.65685i 0.671345i 0.941979 + 0.335673i \(0.108964\pi\)
−0.941979 + 0.335673i \(0.891036\pi\)
\(72\) 0 0
\(73\) 3.00000 + 3.00000i 0.351123 + 0.351123i 0.860527 0.509404i \(-0.170134\pi\)
−0.509404 + 0.860527i \(0.670134\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 8.00000i 0.900070i 0.893011 + 0.450035i \(0.148589\pi\)
−0.893011 + 0.450035i \(0.851411\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −11.3137 + 11.3137i −1.24184 + 1.24184i −0.282604 + 0.959237i \(0.591198\pi\)
−0.959237 + 0.282604i \(0.908802\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −7.07107 −0.749532 −0.374766 0.927119i \(-0.622277\pi\)
−0.374766 + 0.927119i \(0.622277\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −5.00000 + 5.00000i −0.507673 + 0.507673i −0.913812 0.406138i \(-0.866875\pi\)
0.406138 + 0.913812i \(0.366875\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 1.41421i 0.140720i −0.997522 0.0703598i \(-0.977585\pi\)
0.997522 0.0703598i \(-0.0224147\pi\)
\(102\) 0 0
\(103\) 4.00000 + 4.00000i 0.394132 + 0.394132i 0.876157 0.482025i \(-0.160099\pi\)
−0.482025 + 0.876157i \(0.660099\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 11.3137 + 11.3137i 1.09374 + 1.09374i 0.995126 + 0.0986115i \(0.0314401\pi\)
0.0986115 + 0.995126i \(0.468560\pi\)
\(108\) 0 0
\(109\) 4.00000i 0.383131i 0.981480 + 0.191565i \(0.0613564\pi\)
−0.981480 + 0.191565i \(0.938644\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 4.24264 4.24264i 0.399114 0.399114i −0.478806 0.877920i \(-0.658930\pi\)
0.877920 + 0.478806i \(0.158930\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −21.0000 −1.90909
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 12.0000 12.0000i 1.06483 1.06483i 0.0670802 0.997748i \(-0.478632\pi\)
0.997748 0.0670802i \(-0.0213683\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 5.65685i 0.494242i 0.968985 + 0.247121i \(0.0794845\pi\)
−0.968985 + 0.247121i \(0.920516\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 12.7279 + 12.7279i 1.08742 + 1.08742i 0.995793 + 0.0916263i \(0.0292065\pi\)
0.0916263 + 0.995793i \(0.470793\pi\)
\(138\) 0 0
\(139\) 20.0000i 1.69638i −0.529694 0.848189i \(-0.677693\pi\)
0.529694 0.848189i \(-0.322307\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 16.9706 16.9706i 1.41915 1.41915i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 4.24264 0.347571 0.173785 0.984784i \(-0.444400\pi\)
0.173785 + 0.984784i \(0.444400\pi\)
\(150\) 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
\(155\) 0 0
\(156\) 0 0
\(157\) 7.00000 7.00000i 0.558661 0.558661i −0.370265 0.928926i \(-0.620733\pi\)
0.928926 + 0.370265i \(0.120733\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 12.0000 + 12.0000i 0.939913 + 0.939913i 0.998294 0.0583818i \(-0.0185941\pi\)
−0.0583818 + 0.998294i \(0.518594\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 8.48528 + 8.48528i 0.656611 + 0.656611i 0.954577 0.297966i \(-0.0963081\pi\)
−0.297966 + 0.954577i \(0.596308\pi\)
\(168\) 0 0
\(169\) 5.00000i 0.384615i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −7.07107 + 7.07107i −0.537603 + 0.537603i −0.922824 0.385221i \(-0.874125\pi\)
0.385221 + 0.922824i \(0.374125\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −5.65685 −0.422813 −0.211407 0.977398i \(-0.567804\pi\)
−0.211407 + 0.977398i \(0.567804\pi\)
\(180\) 0 0
\(181\) 4.00000 0.297318 0.148659 0.988889i \(-0.452504\pi\)
0.148659 + 0.988889i \(0.452504\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 11.3137i 0.818631i 0.912393 + 0.409316i \(0.134232\pi\)
−0.912393 + 0.409316i \(0.865768\pi\)
\(192\) 0 0
\(193\) −5.00000 5.00000i −0.359908 0.359908i 0.503871 0.863779i \(-0.331909\pi\)
−0.863779 + 0.503871i \(0.831909\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 12.7279 + 12.7279i 0.906827 + 0.906827i 0.996015 0.0891879i \(-0.0284272\pi\)
−0.0891879 + 0.996015i \(0.528427\pi\)
\(198\) 0 0
\(199\) 24.0000i 1.70131i 0.525720 + 0.850657i \(0.323796\pi\)
−0.525720 + 0.850657i \(0.676204\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 22.6274 1.56517
\(210\) 0 0
\(211\) −20.0000 −1.37686 −0.688428 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(228\) 0 0
\(229\) 6.00000i 0.396491i 0.980152 + 0.198246i \(0.0635244\pi\)
−0.980152 + 0.198246i \(0.936476\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 5.65685 5.65685i 0.370593 0.370593i −0.497100 0.867693i \(-0.665602\pi\)
0.867693 + 0.497100i \(0.165602\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 16.9706 1.09773 0.548867 0.835910i \(-0.315059\pi\)
0.548867 + 0.835910i \(0.315059\pi\)
\(240\) 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
\(245\) 0 0
\(246\) 0 0
\(247\) −12.0000 + 12.0000i −0.763542 + 0.763542i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) 16.0000 + 16.0000i 1.00591 + 1.00591i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −7.07107 7.07107i −0.441081 0.441081i 0.451294 0.892375i \(-0.350963\pi\)
−0.892375 + 0.451294i \(0.850963\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 2.82843 2.82843i 0.174408 0.174408i −0.614505 0.788913i \(-0.710644\pi\)
0.788913 + 0.614505i \(0.210644\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −15.5563 −0.948487 −0.474244 0.880394i \(-0.657278\pi\)
−0.474244 + 0.880394i \(0.657278\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −11.0000 + 11.0000i −0.660926 + 0.660926i −0.955598 0.294672i \(-0.904789\pi\)
0.294672 + 0.955598i \(0.404789\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 4.24264i 0.253095i 0.991961 + 0.126547i \(0.0403896\pi\)
−0.991961 + 0.126547i \(0.959610\pi\)
\(282\) 0 0
\(283\) −16.0000 16.0000i −0.951101 0.951101i 0.0477577 0.998859i \(-0.484792\pi\)
−0.998859 + 0.0477577i \(0.984792\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 17.0000i 1.00000i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −1.41421 + 1.41421i −0.0826192 + 0.0826192i −0.747209 0.664589i \(-0.768606\pi\)
0.664589 + 0.747209i \(0.268606\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −16.9706 −0.981433
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −20.0000 + 20.0000i −1.14146 + 1.14146i −0.153277 + 0.988183i \(0.548983\pi\)
−0.988183 + 0.153277i \(0.951017\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 28.2843i 1.60385i −0.597422 0.801927i \(-0.703808\pi\)
0.597422 0.801927i \(-0.296192\pi\)
\(312\) 0 0
\(313\) 9.00000 + 9.00000i 0.508710 + 0.508710i 0.914130 0.405420i \(-0.132875\pi\)
−0.405420 + 0.914130i \(0.632875\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 2.82843 + 2.82843i 0.158860 + 0.158860i 0.782062 0.623201i \(-0.214168\pi\)
−0.623201 + 0.782062i \(0.714168\pi\)
\(318\) 0 0
\(319\) 8.00000i 0.447914i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 4.00000 0.219860 0.109930 0.993939i \(-0.464937\pi\)
0.109930 + 0.993939i \(0.464937\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −15.0000 + 15.0000i −0.817102 + 0.817102i −0.985687 0.168585i \(-0.946080\pi\)
0.168585 + 0.985687i \(0.446080\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 45.2548i 2.45069i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(348\) 0 0
\(349\) 4.00000i 0.214115i −0.994253 0.107058i \(-0.965857\pi\)
0.994253 0.107058i \(-0.0341429\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 5.65685 5.65685i 0.301084 0.301084i −0.540354 0.841438i \(-0.681710\pi\)
0.841438 + 0.540354i \(0.181710\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 5.65685 0.298557 0.149279 0.988795i \(-0.452305\pi\)
0.149279 + 0.988795i \(0.452305\pi\)
\(360\) 0 0
\(361\) 3.00000 0.157895
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −4.00000 + 4.00000i −0.208798 + 0.208798i −0.803757 0.594958i \(-0.797169\pi\)
0.594958 + 0.803757i \(0.297169\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 3.00000 + 3.00000i 0.155334 + 0.155334i 0.780496 0.625161i \(-0.214967\pi\)
−0.625161 + 0.780496i \(0.714967\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −4.24264 4.24264i −0.218507 0.218507i
\(378\) 0 0
\(379\) 12.0000i 0.616399i 0.951322 + 0.308199i \(0.0997264\pi\)
−0.951322 + 0.308199i \(0.900274\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 8.48528 8.48528i 0.433578 0.433578i −0.456266 0.889843i \(-0.650813\pi\)
0.889843 + 0.456266i \(0.150813\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 29.6985 1.50577 0.752886 0.658150i \(-0.228661\pi\)
0.752886 + 0.658150i \(0.228661\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 19.0000 19.0000i 0.953583 0.953583i −0.0453868 0.998969i \(-0.514452\pi\)
0.998969 + 0.0453868i \(0.0144520\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 26.8701i 1.34183i 0.741536 + 0.670913i \(0.234098\pi\)
−0.741536 + 0.670913i \(0.765902\pi\)
\(402\) 0 0
\(403\) −24.0000 24.0000i −1.19553 1.19553i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −39.5980 39.5980i −1.96280 1.96280i
\(408\) 0 0
\(409\) 24.0000i 1.18672i −0.804936 0.593362i \(-0.797800\pi\)
0.804936 0.593362i \(-0.202200\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −11.3137 −0.552711 −0.276355 0.961056i \(-0.589127\pi\)
−0.276355 + 0.961056i \(0.589127\pi\)
\(420\) 0 0
\(421\) 38.0000 1.85201 0.926003 0.377515i \(-0.123221\pi\)
0.926003 + 0.377515i \(0.123221\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 5.65685i 0.272481i −0.990676 0.136241i \(-0.956498\pi\)
0.990676 0.136241i \(-0.0435020\pi\)
\(432\) 0 0
\(433\) −11.0000 11.0000i −0.528626 0.528626i 0.391536 0.920163i \(-0.371944\pi\)
−0.920163 + 0.391536i \(0.871944\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −11.3137 11.3137i −0.541208 0.541208i
\(438\) 0 0
\(439\) 8.00000i 0.381819i −0.981608 0.190910i \(-0.938856\pi\)
0.981608 0.190910i \(-0.0611437\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −11.3137 + 11.3137i −0.537531 + 0.537531i −0.922803 0.385272i \(-0.874107\pi\)
0.385272 + 0.922803i \(0.374107\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −18.3848 −0.867631 −0.433816 0.901002i \(-0.642833\pi\)
−0.433816 + 0.901002i \(0.642833\pi\)
\(450\) 0 0
\(451\) −8.00000 −0.376705
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 23.0000 23.0000i 1.07589 1.07589i 0.0790217 0.996873i \(-0.474820\pi\)
0.996873 0.0790217i \(-0.0251796\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 32.5269i 1.51493i −0.652876 0.757465i \(-0.726438\pi\)
0.652876 0.757465i \(-0.273562\pi\)
\(462\) 0 0
\(463\) 4.00000 + 4.00000i 0.185896 + 0.185896i 0.793919 0.608023i \(-0.208037\pi\)
−0.608023 + 0.793919i \(0.708037\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 16.9706 + 16.9706i 0.785304 + 0.785304i 0.980720 0.195416i \(-0.0626058\pi\)
−0.195416 + 0.980720i \(0.562606\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −22.6274 + 22.6274i −1.04041 + 1.04041i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 5.65685 0.258468 0.129234 0.991614i \(-0.458748\pi\)
0.129234 + 0.991614i \(0.458748\pi\)
\(480\) 0 0
\(481\) 42.0000 1.91504
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 12.0000 12.0000i 0.543772 0.543772i −0.380861 0.924632i \(-0.624372\pi\)
0.924632 + 0.380861i \(0.124372\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 22.6274i 1.02116i 0.859830 + 0.510581i \(0.170569\pi\)
−0.859830 + 0.510581i \(0.829431\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 4.00000i 0.179065i 0.995984 + 0.0895323i \(0.0285372\pi\)
−0.995984 + 0.0895323i \(0.971463\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 25.4558 25.4558i 1.13502 1.13502i 0.145690 0.989330i \(-0.453460\pi\)
0.989330 0.145690i \(-0.0465401\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −12.7279 −0.564155 −0.282078 0.959392i \(-0.591024\pi\)
−0.282078 + 0.959392i \(0.591024\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −16.0000 + 16.0000i −0.703679 + 0.703679i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 21.2132i 0.929367i −0.885477 0.464684i \(-0.846168\pi\)
0.885477 0.464684i \(-0.153832\pi\)
\(522\) 0 0
\(523\) −24.0000 24.0000i −1.04945 1.04945i −0.998712 0.0507346i \(-0.983844\pi\)
−0.0507346 0.998712i \(-0.516156\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 7.00000i 0.304348i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 4.24264 4.24264i 0.183769 0.183769i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −39.5980 −1.70561
\(540\) 0 0
\(541\) −4.00000 −0.171973 −0.0859867 0.996296i \(-0.527404\pi\)
−0.0859867 + 0.996296i \(0.527404\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 12.0000 12.0000i 0.513083 0.513083i −0.402387 0.915470i \(-0.631819\pi\)
0.915470 + 0.402387i \(0.131819\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 5.65685i 0.240990i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 8.48528 + 8.48528i 0.359533 + 0.359533i 0.863641 0.504108i \(-0.168179\pi\)
−0.504108 + 0.863641i \(0.668179\pi\)
\(558\) 0 0
\(559\) 24.0000i 1.01509i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −5.65685 + 5.65685i −0.238408 + 0.238408i −0.816191 0.577783i \(-0.803918\pi\)
0.577783 + 0.816191i \(0.303918\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 35.3553 1.48217 0.741086 0.671410i \(-0.234311\pi\)
0.741086 + 0.671410i \(0.234311\pi\)
\(570\) 0 0
\(571\) −20.0000 −0.836974 −0.418487 0.908223i \(-0.637439\pi\)
−0.418487 + 0.908223i \(0.637439\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 9.00000 9.00000i 0.374675 0.374675i −0.494502 0.869177i \(-0.664649\pi\)
0.869177 + 0.494502i \(0.164649\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 48.0000 + 48.0000i 1.98796 + 1.98796i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 5.65685 + 5.65685i 0.233483 + 0.233483i 0.814145 0.580662i \(-0.197206\pi\)
−0.580662 + 0.814145i \(0.697206\pi\)
\(588\) 0 0
\(589\) 32.0000i 1.31854i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −18.3848 + 18.3848i −0.754972 + 0.754972i −0.975403 0.220430i \(-0.929254\pi\)
0.220430 + 0.975403i \(0.429254\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 28.2843 1.15566 0.577832 0.816156i \(-0.303899\pi\)
0.577832 + 0.816156i \(0.303899\pi\)
\(600\) 0 0
\(601\) −32.0000 −1.30531 −0.652654 0.757656i \(-0.726344\pi\)
−0.652654 + 0.757656i \(0.726344\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 8.00000 8.00000i 0.324710 0.324710i −0.525861 0.850571i \(-0.676257\pi\)
0.850571 + 0.525861i \(0.176257\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 16.9706i 0.686555i
\(612\) 0 0
\(613\) −15.0000 15.0000i −0.605844 0.605844i 0.336013 0.941857i \(-0.390921\pi\)
−0.941857 + 0.336013i \(0.890921\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 16.9706 + 16.9706i 0.683209 + 0.683209i 0.960722 0.277513i \(-0.0895101\pi\)
−0.277513 + 0.960722i \(0.589510\pi\)
\(618\) 0 0
\(619\) 12.0000i 0.482321i 0.970485 + 0.241160i \(0.0775280\pi\)
−0.970485 + 0.241160i \(0.922472\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 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
\(635\) 0 0
\(636\) 0 0
\(637\) 21.0000 21.0000i 0.832050 0.832050i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 26.8701i 1.06130i −0.847590 0.530652i \(-0.821947\pi\)
0.847590 0.530652i \(-0.178053\pi\)
\(642\) 0 0
\(643\) −28.0000 28.0000i −1.10421 1.10421i −0.993897 0.110316i \(-0.964814\pi\)
−0.110316 0.993897i \(-0.535186\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −19.7990 19.7990i −0.778379 0.778379i 0.201176 0.979555i \(-0.435524\pi\)
−0.979555 + 0.201176i \(0.935524\pi\)
\(648\) 0 0
\(649\) 64.0000i 2.51222i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −25.4558 + 25.4558i −0.996164 + 0.996164i −0.999993 0.00382851i \(-0.998781\pi\)
0.00382851 + 0.999993i \(0.498781\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −33.9411 −1.32216 −0.661079 0.750316i \(-0.729901\pi\)
−0.661079 + 0.750316i \(0.729901\pi\)
\(660\) 0 0
\(661\) −20.0000 −0.777910 −0.388955 0.921257i \(-0.627164\pi\)
−0.388955 + 0.921257i \(0.627164\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 4.00000 4.00000i 0.154881 0.154881i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 67.8823i 2.62057i
\(672\) 0 0
\(673\) 19.0000 + 19.0000i 0.732396 + 0.732396i 0.971094 0.238698i \(-0.0767205\pi\)
−0.238698 + 0.971094i \(0.576721\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −19.7990 19.7990i −0.760937 0.760937i 0.215555 0.976492i \(-0.430844\pi\)
−0.976492 + 0.215555i \(0.930844\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −33.9411 + 33.9411i −1.29872 + 1.29872i −0.369484 + 0.929237i \(0.620466\pi\)
−0.929237 + 0.369484i \(0.879534\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −50.9117 −1.93958
\(690\) 0 0
\(691\) 12.0000 0.456502 0.228251 0.973602i \(-0.426699\pi\)
0.228251 + 0.973602i \(0.426699\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 32.5269i 1.22852i −0.789102 0.614262i \(-0.789454\pi\)
0.789102 0.614262i \(-0.210546\pi\)
\(702\) 0 0
\(703\) 28.0000 + 28.0000i 1.05604 + 1.05604i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 6.00000i 0.225335i −0.993633 0.112667i \(-0.964061\pi\)
0.993633 0.112667i \(-0.0359394\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 22.6274 22.6274i 0.847403 0.847403i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 22.6274 0.843860 0.421930 0.906628i \(-0.361353\pi\)
0.421930 + 0.906628i \(0.361353\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 12.0000 12.0000i 0.445055 0.445055i −0.448651 0.893707i \(-0.648096\pi\)
0.893707 + 0.448651i \(0.148096\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −23.0000 23.0000i −0.849524 0.849524i 0.140549 0.990074i \(-0.455113\pi\)
−0.990074 + 0.140549i \(0.955113\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −45.2548 45.2548i −1.66698 1.66698i
\(738\) 0 0
\(739\) 4.00000i 0.147142i 0.997290 + 0.0735712i \(0.0234396\pi\)
−0.997290 + 0.0735712i \(0.976560\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −25.4558 + 25.4558i −0.933884 + 0.933884i −0.997946 0.0640616i \(-0.979595\pi\)
0.0640616 + 0.997946i \(0.479595\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 3.00000 3.00000i 0.109037 0.109037i −0.650484 0.759520i \(-0.725434\pi\)
0.759520 + 0.650484i \(0.225434\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 1.41421i 0.0512652i 0.999671 + 0.0256326i \(0.00816000\pi\)
−0.999671 + 0.0256326i \(0.991840\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −33.9411 33.9411i −1.22554 1.22554i
\(768\) 0 0
\(769\) 40.0000i 1.44244i −0.692708 0.721218i \(-0.743582\pi\)
0.692708 0.721218i \(-0.256418\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 14.1421 14.1421i 0.508657 0.508657i −0.405457 0.914114i \(-0.632888\pi\)
0.914114 + 0.405457i \(0.132888\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 5.65685 0.202678
\(780\) 0 0
\(781\) −32.0000 −1.14505
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 24.0000 24.0000i 0.855508 0.855508i −0.135297 0.990805i \(-0.543199\pi\)
0.990805 + 0.135297i \(0.0431990\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 36.0000 + 36.0000i 1.27840 + 1.27840i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −29.6985 29.6985i −1.05197 1.05197i −0.998573 0.0534012i \(-0.982994\pi\)
−0.0534012 0.998573i \(-0.517006\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −16.9706 + 16.9706i −0.598878 + 0.598878i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −32.5269 −1.14359 −0.571793 0.820398i