Properties

Label 3024.2.k.g.1889.3
Level 3024
Weight 2
Character 3024.1889
Analytic conductor 24.147
Analytic rank 0
Dimension 4
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 3024 = 2^{4} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3024.k (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(24.1467615712\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{3})\)
Defining polynomial: \(x^{4} + 4 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 189)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1889.3
Root \(1.93185i\) of defining polynomial
Character \(\chi\) \(=\) 3024.1889
Dual form 3024.2.k.g.1889.4

$q$-expansion

\(f(q)\) \(=\) \(q+1.73205 q^{5} +(-1.00000 - 2.44949i) q^{7} +O(q^{10})\) \(q+1.73205 q^{5} +(-1.00000 - 2.44949i) q^{7} +1.41421i q^{11} +2.44949i q^{13} -5.19615 q^{17} +7.34847i q^{19} -2.82843i q^{23} -2.00000 q^{25} +7.07107i q^{29} -2.44949i q^{31} +(-1.73205 - 4.24264i) q^{35} +5.00000 q^{37} -8.66025 q^{41} -5.00000 q^{43} -8.66025 q^{47} +(-5.00000 + 4.89898i) q^{49} +11.3137i q^{53} +2.44949i q^{55} +8.66025 q^{59} -2.44949i q^{61} +4.24264i q^{65} -2.00000 q^{67} +14.1421i q^{71} +(3.46410 - 1.41421i) q^{77} +13.0000 q^{79} +1.73205 q^{83} -9.00000 q^{85} +10.3923 q^{89} +(6.00000 - 2.44949i) q^{91} +12.7279i q^{95} -17.1464i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{7} + O(q^{10}) \) \( 4q - 4q^{7} - 8q^{25} + 20q^{37} - 20q^{43} - 20q^{49} - 8q^{67} + 52q^{79} - 36q^{85} + 24q^{91} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3024\mathbb{Z}\right)^\times\).

\(n\) \(757\) \(785\) \(1135\) \(2593\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 1.73205 0.774597 0.387298 0.921954i \(-0.373408\pi\)
0.387298 + 0.921954i \(0.373408\pi\)
\(6\) 0 0
\(7\) −1.00000 2.44949i −0.377964 0.925820i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.41421i 0.426401i 0.977008 + 0.213201i \(0.0683888\pi\)
−0.977008 + 0.213201i \(0.931611\pi\)
\(12\) 0 0
\(13\) 2.44949i 0.679366i 0.940540 + 0.339683i \(0.110320\pi\)
−0.940540 + 0.339683i \(0.889680\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −5.19615 −1.26025 −0.630126 0.776493i \(-0.716997\pi\)
−0.630126 + 0.776493i \(0.716997\pi\)
\(18\) 0 0
\(19\) 7.34847i 1.68585i 0.538028 + 0.842927i \(0.319170\pi\)
−0.538028 + 0.842927i \(0.680830\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.82843i 0.589768i −0.955533 0.294884i \(-0.904719\pi\)
0.955533 0.294884i \(-0.0952810\pi\)
\(24\) 0 0
\(25\) −2.00000 −0.400000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 7.07107i 1.31306i 0.754298 + 0.656532i \(0.227977\pi\)
−0.754298 + 0.656532i \(0.772023\pi\)
\(30\) 0 0
\(31\) 2.44949i 0.439941i −0.975506 0.219971i \(-0.929404\pi\)
0.975506 0.219971i \(-0.0705962\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1.73205 4.24264i −0.292770 0.717137i
\(36\) 0 0
\(37\) 5.00000 0.821995 0.410997 0.911636i \(-0.365181\pi\)
0.410997 + 0.911636i \(0.365181\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −8.66025 −1.35250 −0.676252 0.736670i \(-0.736397\pi\)
−0.676252 + 0.736670i \(0.736397\pi\)
\(42\) 0 0
\(43\) −5.00000 −0.762493 −0.381246 0.924473i \(-0.624505\pi\)
−0.381246 + 0.924473i \(0.624505\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −8.66025 −1.26323 −0.631614 0.775283i \(-0.717607\pi\)
−0.631614 + 0.775283i \(0.717607\pi\)
\(48\) 0 0
\(49\) −5.00000 + 4.89898i −0.714286 + 0.699854i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 11.3137i 1.55406i 0.629465 + 0.777029i \(0.283274\pi\)
−0.629465 + 0.777029i \(0.716726\pi\)
\(54\) 0 0
\(55\) 2.44949i 0.330289i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 8.66025 1.12747 0.563735 0.825956i \(-0.309364\pi\)
0.563735 + 0.825956i \(0.309364\pi\)
\(60\) 0 0
\(61\) 2.44949i 0.313625i −0.987628 0.156813i \(-0.949878\pi\)
0.987628 0.156813i \(-0.0501218\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 4.24264i 0.526235i
\(66\) 0 0
\(67\) −2.00000 −0.244339 −0.122169 0.992509i \(-0.538985\pi\)
−0.122169 + 0.992509i \(0.538985\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 14.1421i 1.67836i 0.543852 + 0.839181i \(0.316965\pi\)
−0.543852 + 0.839181i \(0.683035\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.46410 1.41421i 0.394771 0.161165i
\(78\) 0 0
\(79\) 13.0000 1.46261 0.731307 0.682048i \(-0.238911\pi\)
0.731307 + 0.682048i \(0.238911\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 1.73205 0.190117 0.0950586 0.995472i \(-0.469696\pi\)
0.0950586 + 0.995472i \(0.469696\pi\)
\(84\) 0 0
\(85\) −9.00000 −0.976187
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 10.3923 1.10158 0.550791 0.834643i \(-0.314326\pi\)
0.550791 + 0.834643i \(0.314326\pi\)
\(90\) 0 0
\(91\) 6.00000 2.44949i 0.628971 0.256776i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 12.7279i 1.30586i
\(96\) 0 0
\(97\) 17.1464i 1.74096i −0.492207 0.870478i \(-0.663810\pi\)
0.492207 0.870478i \(-0.336190\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 3.46410 0.344691 0.172345 0.985037i \(-0.444865\pi\)
0.172345 + 0.985037i \(0.444865\pi\)
\(102\) 0 0
\(103\) 9.79796i 0.965422i −0.875780 0.482711i \(-0.839652\pi\)
0.875780 0.482711i \(-0.160348\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 14.1421i 1.36717i 0.729870 + 0.683586i \(0.239581\pi\)
−0.729870 + 0.683586i \(0.760419\pi\)
\(108\) 0 0
\(109\) −7.00000 −0.670478 −0.335239 0.942133i \(-0.608817\pi\)
−0.335239 + 0.942133i \(0.608817\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 7.07107i 0.665190i 0.943070 + 0.332595i \(0.107924\pi\)
−0.943070 + 0.332595i \(0.892076\pi\)
\(114\) 0 0
\(115\) 4.89898i 0.456832i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 5.19615 + 12.7279i 0.476331 + 1.16677i
\(120\) 0 0
\(121\) 9.00000 0.818182
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −12.1244 −1.08444
\(126\) 0 0
\(127\) −5.00000 −0.443678 −0.221839 0.975083i \(-0.571206\pi\)
−0.221839 + 0.975083i \(0.571206\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −17.3205 −1.51330 −0.756650 0.653820i \(-0.773165\pi\)
−0.756650 + 0.653820i \(0.773165\pi\)
\(132\) 0 0
\(133\) 18.0000 7.34847i 1.56080 0.637193i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 1.41421i 0.120824i −0.998174 0.0604122i \(-0.980758\pi\)
0.998174 0.0604122i \(-0.0192415\pi\)
\(138\) 0 0
\(139\) 12.2474i 1.03882i 0.854527 + 0.519408i \(0.173847\pi\)
−0.854527 + 0.519408i \(0.826153\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −3.46410 −0.289683
\(144\) 0 0
\(145\) 12.2474i 1.01710i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 7.07107i 0.579284i 0.957135 + 0.289642i \(0.0935363\pi\)
−0.957135 + 0.289642i \(0.906464\pi\)
\(150\) 0 0
\(151\) −5.00000 −0.406894 −0.203447 0.979086i \(-0.565214\pi\)
−0.203447 + 0.979086i \(0.565214\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 4.24264i 0.340777i
\(156\) 0 0
\(157\) 17.1464i 1.36843i 0.729279 + 0.684217i \(0.239856\pi\)
−0.729279 + 0.684217i \(0.760144\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −6.92820 + 2.82843i −0.546019 + 0.222911i
\(162\) 0 0
\(163\) −5.00000 −0.391630 −0.195815 0.980641i \(-0.562735\pi\)
−0.195815 + 0.980641i \(0.562735\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −12.1244 −0.938211 −0.469105 0.883142i \(-0.655424\pi\)
−0.469105 + 0.883142i \(0.655424\pi\)
\(168\) 0 0
\(169\) 7.00000 0.538462
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −6.92820 −0.526742 −0.263371 0.964695i \(-0.584834\pi\)
−0.263371 + 0.964695i \(0.584834\pi\)
\(174\) 0 0
\(175\) 2.00000 + 4.89898i 0.151186 + 0.370328i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 7.07107i 0.528516i −0.964452 0.264258i \(-0.914873\pi\)
0.964452 0.264258i \(-0.0851271\pi\)
\(180\) 0 0
\(181\) 14.6969i 1.09241i −0.837650 0.546207i \(-0.816071\pi\)
0.837650 0.546207i \(-0.183929\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 8.66025 0.636715
\(186\) 0 0
\(187\) 7.34847i 0.537373i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 14.1421i 1.02329i 0.859197 + 0.511645i \(0.170964\pi\)
−0.859197 + 0.511645i \(0.829036\pi\)
\(192\) 0 0
\(193\) −1.00000 −0.0719816 −0.0359908 0.999352i \(-0.511459\pi\)
−0.0359908 + 0.999352i \(0.511459\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 1.41421i 0.100759i −0.998730 0.0503793i \(-0.983957\pi\)
0.998730 0.0503793i \(-0.0160430\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 17.3205 7.07107i 1.21566 0.496292i
\(204\) 0 0
\(205\) −15.0000 −1.04765
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −10.3923 −0.718851
\(210\) 0 0
\(211\) 16.0000 1.10149 0.550743 0.834675i \(-0.314345\pi\)
0.550743 + 0.834675i \(0.314345\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −8.66025 −0.590624
\(216\) 0 0
\(217\) −6.00000 + 2.44949i −0.407307 + 0.166282i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 12.7279i 0.856173i
\(222\) 0 0
\(223\) 12.2474i 0.820150i −0.912052 0.410075i \(-0.865503\pi\)
0.912052 0.410075i \(-0.134497\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −13.8564 −0.919682 −0.459841 0.888001i \(-0.652094\pi\)
−0.459841 + 0.888001i \(0.652094\pi\)
\(228\) 0 0
\(229\) 4.89898i 0.323734i −0.986813 0.161867i \(-0.948248\pi\)
0.986813 0.161867i \(-0.0517515\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 7.07107i 0.463241i 0.972806 + 0.231621i \(0.0744028\pi\)
−0.972806 + 0.231621i \(0.925597\pi\)
\(234\) 0 0
\(235\) −15.0000 −0.978492
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 18.3848i 1.18921i 0.804017 + 0.594606i \(0.202692\pi\)
−0.804017 + 0.594606i \(0.797308\pi\)
\(240\) 0 0
\(241\) 2.44949i 0.157786i −0.996883 0.0788928i \(-0.974862\pi\)
0.996883 0.0788928i \(-0.0251385\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −8.66025 + 8.48528i −0.553283 + 0.542105i
\(246\) 0 0
\(247\) −18.0000 −1.14531
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −5.19615 −0.327978 −0.163989 0.986462i \(-0.552436\pi\)
−0.163989 + 0.986462i \(0.552436\pi\)
\(252\) 0 0
\(253\) 4.00000 0.251478
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −3.46410 −0.216085 −0.108042 0.994146i \(-0.534458\pi\)
−0.108042 + 0.994146i \(0.534458\pi\)
\(258\) 0 0
\(259\) −5.00000 12.2474i −0.310685 0.761019i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 1.41421i 0.0872041i 0.999049 + 0.0436021i \(0.0138834\pi\)
−0.999049 + 0.0436021i \(0.986117\pi\)
\(264\) 0 0
\(265\) 19.5959i 1.20377i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 25.9808 1.58408 0.792038 0.610472i \(-0.209020\pi\)
0.792038 + 0.610472i \(0.209020\pi\)
\(270\) 0 0
\(271\) 14.6969i 0.892775i 0.894840 + 0.446388i \(0.147290\pi\)
−0.894840 + 0.446388i \(0.852710\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 2.82843i 0.170561i
\(276\) 0 0
\(277\) 5.00000 0.300421 0.150210 0.988654i \(-0.452005\pi\)
0.150210 + 0.988654i \(0.452005\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 14.1421i 0.843649i −0.906677 0.421825i \(-0.861390\pi\)
0.906677 0.421825i \(-0.138610\pi\)
\(282\) 0 0
\(283\) 12.2474i 0.728035i 0.931392 + 0.364018i \(0.118595\pi\)
−0.931392 + 0.364018i \(0.881405\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 8.66025 + 21.2132i 0.511199 + 1.25218i
\(288\) 0 0
\(289\) 10.0000 0.588235
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −19.0526 −1.11306 −0.556531 0.830827i \(-0.687868\pi\)
−0.556531 + 0.830827i \(0.687868\pi\)
\(294\) 0 0
\(295\) 15.0000 0.873334
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 6.92820 0.400668
\(300\) 0 0
\(301\) 5.00000 + 12.2474i 0.288195 + 0.705931i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 4.24264i 0.242933i
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −12.1244 −0.687509 −0.343755 0.939060i \(-0.611699\pi\)
−0.343755 + 0.939060i \(0.611699\pi\)
\(312\) 0 0
\(313\) 24.4949i 1.38453i −0.721642 0.692267i \(-0.756612\pi\)
0.721642 0.692267i \(-0.243388\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 5.65685i 0.317721i −0.987301 0.158860i \(-0.949218\pi\)
0.987301 0.158860i \(-0.0507819\pi\)
\(318\) 0 0
\(319\) −10.0000 −0.559893
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 38.1838i 2.12460i
\(324\) 0 0
\(325\) 4.89898i 0.271746i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 8.66025 + 21.2132i 0.477455 + 1.16952i
\(330\) 0 0
\(331\) 25.0000 1.37412 0.687062 0.726599i \(-0.258900\pi\)
0.687062 + 0.726599i \(0.258900\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −3.46410 −0.189264
\(336\) 0 0
\(337\) −7.00000 −0.381314 −0.190657 0.981657i \(-0.561062\pi\)
−0.190657 + 0.981657i \(0.561062\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 3.46410 0.187592
\(342\) 0 0
\(343\) 17.0000 + 7.34847i 0.917914 + 0.396780i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 7.07107i 0.379595i −0.981823 0.189797i \(-0.939217\pi\)
0.981823 0.189797i \(-0.0607831\pi\)
\(348\) 0 0
\(349\) 19.5959i 1.04895i 0.851427 + 0.524473i \(0.175738\pi\)
−0.851427 + 0.524473i \(0.824262\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 8.66025 0.460939 0.230469 0.973080i \(-0.425974\pi\)
0.230469 + 0.973080i \(0.425974\pi\)
\(354\) 0 0
\(355\) 24.4949i 1.30005i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 11.3137i 0.597115i −0.954392 0.298557i \(-0.903495\pi\)
0.954392 0.298557i \(-0.0965054\pi\)
\(360\) 0 0
\(361\) −35.0000 −1.84211
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 4.89898i 0.255725i −0.991792 0.127862i \(-0.959188\pi\)
0.991792 0.127862i \(-0.0408116\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 27.7128 11.3137i 1.43878 0.587378i
\(372\) 0 0
\(373\) −25.0000 −1.29445 −0.647225 0.762299i \(-0.724071\pi\)
−0.647225 + 0.762299i \(0.724071\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −17.3205 −0.892052
\(378\) 0 0
\(379\) −23.0000 −1.18143 −0.590715 0.806880i \(-0.701154\pi\)
−0.590715 + 0.806880i \(0.701154\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −1.73205 −0.0885037 −0.0442518 0.999020i \(-0.514090\pi\)
−0.0442518 + 0.999020i \(0.514090\pi\)
\(384\) 0 0
\(385\) 6.00000 2.44949i 0.305788 0.124838i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 32.5269i 1.64918i 0.565731 + 0.824590i \(0.308594\pi\)
−0.565731 + 0.824590i \(0.691406\pi\)
\(390\) 0 0
\(391\) 14.6969i 0.743256i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 22.5167 1.13294
\(396\) 0 0
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 19.7990i 0.988714i 0.869259 + 0.494357i \(0.164597\pi\)
−0.869259 + 0.494357i \(0.835403\pi\)
\(402\) 0 0
\(403\) 6.00000 0.298881
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 7.07107i 0.350500i
\(408\) 0 0
\(409\) 19.5959i 0.968956i −0.874804 0.484478i \(-0.839010\pi\)
0.874804 0.484478i \(-0.160990\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −8.66025 21.2132i −0.426143 1.04383i
\(414\) 0 0
\(415\) 3.00000 0.147264
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −1.73205 −0.0846162 −0.0423081 0.999105i \(-0.513471\pi\)
−0.0423081 + 0.999105i \(0.513471\pi\)
\(420\) 0 0
\(421\) −4.00000 −0.194948 −0.0974740 0.995238i \(-0.531076\pi\)
−0.0974740 + 0.995238i \(0.531076\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 10.3923 0.504101
\(426\) 0 0
\(427\) −6.00000 + 2.44949i −0.290360 + 0.118539i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 5.65685i 0.272481i 0.990676 + 0.136241i \(0.0435020\pi\)
−0.990676 + 0.136241i \(0.956498\pi\)
\(432\) 0 0
\(433\) 36.7423i 1.76572i 0.469632 + 0.882862i \(0.344387\pi\)
−0.469632 + 0.882862i \(0.655613\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 20.7846 0.994263
\(438\) 0 0
\(439\) 19.5959i 0.935262i −0.883924 0.467631i \(-0.845108\pi\)
0.883924 0.467631i \(-0.154892\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 32.5269i 1.54540i −0.634771 0.772700i \(-0.718906\pi\)
0.634771 0.772700i \(-0.281094\pi\)
\(444\) 0 0
\(445\) 18.0000 0.853282
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 24.0416i 1.13459i 0.823513 + 0.567297i \(0.192011\pi\)
−0.823513 + 0.567297i \(0.807989\pi\)
\(450\) 0 0
\(451\) 12.2474i 0.576710i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 10.3923 4.24264i 0.487199 0.198898i
\(456\) 0 0
\(457\) 32.0000 1.49690 0.748448 0.663193i \(-0.230799\pi\)
0.748448 + 0.663193i \(0.230799\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −12.1244 −0.564688 −0.282344 0.959313i \(-0.591112\pi\)
−0.282344 + 0.959313i \(0.591112\pi\)
\(462\) 0 0
\(463\) −11.0000 −0.511213 −0.255607 0.966781i \(-0.582275\pi\)
−0.255607 + 0.966781i \(0.582275\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 20.7846 0.961797 0.480899 0.876776i \(-0.340311\pi\)
0.480899 + 0.876776i \(0.340311\pi\)
\(468\) 0 0
\(469\) 2.00000 + 4.89898i 0.0923514 + 0.226214i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 7.07107i 0.325128i
\(474\) 0 0
\(475\) 14.6969i 0.674342i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 32.9090 1.50365 0.751825 0.659363i \(-0.229174\pi\)
0.751825 + 0.659363i \(0.229174\pi\)
\(480\) 0 0
\(481\) 12.2474i 0.558436i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 29.6985i 1.34854i
\(486\) 0 0
\(487\) 22.0000 0.996915 0.498458 0.866914i \(-0.333900\pi\)
0.498458 + 0.866914i \(0.333900\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 15.5563i 0.702048i −0.936366 0.351024i \(-0.885834\pi\)
0.936366 0.351024i \(-0.114166\pi\)
\(492\) 0 0
\(493\) 36.7423i 1.65479i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 34.6410 14.1421i 1.55386 0.634361i
\(498\) 0 0
\(499\) −17.0000 −0.761025 −0.380512 0.924776i \(-0.624252\pi\)
−0.380512 + 0.924776i \(0.624252\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −36.3731 −1.62179 −0.810897 0.585188i \(-0.801021\pi\)
−0.810897 + 0.585188i \(0.801021\pi\)
\(504\) 0 0
\(505\) 6.00000 0.266996
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −8.66025 −0.383859 −0.191930 0.981409i \(-0.561474\pi\)
−0.191930 + 0.981409i \(0.561474\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 16.9706i 0.747812i
\(516\) 0 0
\(517\) 12.2474i 0.538642i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 5.19615 0.227648 0.113824 0.993501i \(-0.463690\pi\)
0.113824 + 0.993501i \(0.463690\pi\)
\(522\) 0 0
\(523\) 36.7423i 1.60663i −0.595554 0.803315i \(-0.703067\pi\)
0.595554 0.803315i \(-0.296933\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 12.7279i 0.554437i
\(528\) 0 0
\(529\) 15.0000 0.652174
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 21.2132i 0.918846i
\(534\) 0 0
\(535\) 24.4949i 1.05901i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −6.92820 7.07107i −0.298419 0.304572i
\(540\) 0 0
\(541\) 5.00000 0.214967 0.107483 0.994207i \(-0.465721\pi\)
0.107483 + 0.994207i \(0.465721\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −12.1244 −0.519350
\(546\) 0 0
\(547\) 13.0000 0.555840 0.277920 0.960604i \(-0.410355\pi\)
0.277920 + 0.960604i \(0.410355\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −51.9615 −2.21364
\(552\) 0 0
\(553\) −13.0000 31.8434i −0.552816 1.35412i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 14.1421i 0.599222i −0.954062 0.299611i \(-0.903143\pi\)
0.954062 0.299611i \(-0.0968568\pi\)
\(558\) 0 0
\(559\) 12.2474i 0.518012i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 34.6410 1.45994 0.729972 0.683477i \(-0.239533\pi\)
0.729972 + 0.683477i \(0.239533\pi\)
\(564\) 0 0
\(565\) 12.2474i 0.515254i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 39.5980i 1.66003i −0.557738 0.830017i \(-0.688331\pi\)
0.557738 0.830017i \(-0.311669\pi\)
\(570\) 0 0
\(571\) 19.0000 0.795125 0.397563 0.917575i \(-0.369856\pi\)
0.397563 + 0.917575i \(0.369856\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 5.65685i 0.235907i
\(576\) 0 0
\(577\) 29.3939i 1.22368i 0.790980 + 0.611842i \(0.209571\pi\)
−0.790980 + 0.611842i \(0.790429\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −1.73205 4.24264i −0.0718576 0.176014i
\(582\) 0 0
\(583\) −16.0000 −0.662652
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −13.8564 −0.571915 −0.285958 0.958242i \(-0.592312\pi\)
−0.285958 + 0.958242i \(0.592312\pi\)
\(588\) 0 0
\(589\) 18.0000 0.741677
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 15.5885 0.640141 0.320071 0.947394i \(-0.396293\pi\)
0.320071 + 0.947394i \(0.396293\pi\)
\(594\) 0 0
\(595\) 9.00000 + 22.0454i 0.368964 + 0.903774i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 18.3848i 0.751182i 0.926786 + 0.375591i \(0.122560\pi\)
−0.926786 + 0.375591i \(0.877440\pi\)
\(600\) 0 0
\(601\) 46.5403i 1.89842i −0.314645 0.949209i \(-0.601886\pi\)
0.314645 0.949209i \(-0.398114\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 15.5885 0.633761
\(606\) 0 0
\(607\) 12.2474i 0.497109i 0.968618 + 0.248554i \(0.0799554\pi\)
−0.968618 + 0.248554i \(0.920045\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 21.2132i 0.858194i
\(612\) 0 0
\(613\) −4.00000 −0.161558 −0.0807792 0.996732i \(-0.525741\pi\)
−0.0807792 + 0.996732i \(0.525741\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 48.0833i 1.93576i −0.251414 0.967880i \(-0.580896\pi\)
0.251414 0.967880i \(-0.419104\pi\)
\(618\) 0 0
\(619\) 26.9444i 1.08299i −0.840705 0.541493i \(-0.817859\pi\)
0.840705 0.541493i \(-0.182141\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −10.3923 25.4558i −0.416359 1.01987i
\(624\) 0 0
\(625\) −11.0000 −0.440000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −25.9808 −1.03592
\(630\) 0 0
\(631\) −29.0000 −1.15447 −0.577236 0.816577i \(-0.695869\pi\)
−0.577236 + 0.816577i \(0.695869\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −8.66025 −0.343672
\(636\) 0 0
\(637\) −12.0000 12.2474i −0.475457 0.485262i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 1.41421i 0.0558581i −0.999610 0.0279290i \(-0.991109\pi\)
0.999610 0.0279290i \(-0.00889125\pi\)
\(642\) 0 0
\(643\) 24.4949i 0.965984i −0.875625 0.482992i \(-0.839550\pi\)
0.875625 0.482992i \(-0.160450\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 20.7846 0.817127 0.408564 0.912730i \(-0.366030\pi\)
0.408564 + 0.912730i \(0.366030\pi\)
\(648\) 0 0
\(649\) 12.2474i 0.480754i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 18.3848i 0.719452i −0.933058 0.359726i \(-0.882870\pi\)
0.933058 0.359726i \(-0.117130\pi\)
\(654\) 0 0
\(655\) −30.0000 −1.17220
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 11.3137i 0.440720i −0.975419 0.220360i \(-0.929277\pi\)
0.975419 0.220360i \(-0.0707231\pi\)
\(660\) 0 0
\(661\) 17.1464i 0.666919i 0.942764 + 0.333459i \(0.108216\pi\)
−0.942764 + 0.333459i \(0.891784\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 31.1769 12.7279i 1.20899 0.493568i
\(666\) 0 0
\(667\) 20.0000 0.774403
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 3.46410 0.133730
\(672\) 0 0
\(673\) −40.0000 −1.54189 −0.770943 0.636904i \(-0.780215\pi\)
−0.770943 + 0.636904i \(0.780215\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −17.3205 −0.665681 −0.332841 0.942983i \(-0.608007\pi\)
−0.332841 + 0.942983i \(0.608007\pi\)
\(678\) 0 0
\(679\) −42.0000 + 17.1464i −1.61181 + 0.658020i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 24.0416i 0.919927i −0.887938 0.459964i \(-0.847862\pi\)
0.887938 0.459964i \(-0.152138\pi\)
\(684\) 0 0
\(685\) 2.44949i 0.0935902i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −27.7128 −1.05577
\(690\) 0 0
\(691\) 26.9444i 1.02501i −0.858683 0.512506i \(-0.828717\pi\)
0.858683 0.512506i \(-0.171283\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 21.2132i 0.804663i
\(696\) 0 0
\(697\) 45.0000 1.70450
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 1.41421i 0.0534141i −0.999643 0.0267071i \(-0.991498\pi\)
0.999643 0.0267071i \(-0.00850213\pi\)
\(702\) 0 0
\(703\) 36.7423i 1.38576i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −3.46410 8.48528i −0.130281 0.319122i
\(708\) 0 0
\(709\) 5.00000 0.187779 0.0938895 0.995583i \(-0.470070\pi\)
0.0938895 + 0.995583i \(0.470070\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −6.92820 −0.259463
\(714\) 0 0
\(715\) −6.00000 −0.224387
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 15.5885 0.581351 0.290676 0.956822i \(-0.406120\pi\)
0.290676 + 0.956822i \(0.406120\pi\)
\(720\) 0 0
\(721\) −24.0000 + 9.79796i −0.893807 + 0.364895i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 14.1421i 0.525226i
\(726\) 0 0
\(727\) 12.2474i 0.454233i −0.973868 0.227116i \(-0.927070\pi\)
0.973868 0.227116i \(-0.0729298\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 25.9808 0.960933
\(732\) 0 0
\(733\) 39.1918i 1.44758i 0.690018 + 0.723792i \(0.257602\pi\)
−0.690018 + 0.723792i \(0.742398\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 2.82843i 0.104186i
\(738\) 0 0
\(739\) 22.0000 0.809283 0.404642 0.914475i \(-0.367396\pi\)
0.404642 + 0.914475i \(0.367396\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 7.07107i 0.259412i −0.991552 0.129706i \(-0.958597\pi\)
0.991552 0.129706i \(-0.0414034\pi\)
\(744\) 0 0
\(745\) 12.2474i 0.448712i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 34.6410 14.1421i 1.26576 0.516742i
\(750\) 0 0
\(751\) 34.0000 1.24068 0.620339 0.784334i \(-0.286995\pi\)
0.620339 + 0.784334i \(0.286995\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −8.66025 −0.315179
\(756\) 0 0
\(757\) 47.0000 1.70824 0.854122 0.520073i \(-0.174095\pi\)
0.854122 + 0.520073i \(0.174095\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 12.1244 0.439508 0.219754 0.975555i \(-0.429475\pi\)
0.219754 + 0.975555i \(0.429475\pi\)
\(762\) 0 0
\(763\) 7.00000 + 17.1464i 0.253417 + 0.620742i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 21.2132i 0.765964i
\(768\) 0 0
\(769\) 4.89898i 0.176662i −0.996091 0.0883309i \(-0.971847\pi\)
0.996091 0.0883309i \(-0.0281533\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 15.5885 0.560678 0.280339 0.959901i \(-0.409553\pi\)
0.280339 + 0.959901i \(0.409553\pi\)
\(774\) 0 0
\(775\) 4.89898i 0.175977i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 63.6396i 2.28013i
\(780\) 0 0
\(781\) −20.0000 −0.715656
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 29.6985i 1.05998i
\(786\) 0 0
\(787\) 4.89898i 0.174630i 0.996181 + 0.0873149i \(0.0278286\pi\)
−0.996181 + 0.0873149i \(0.972171\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 17.3205 7.07107i 0.615846 0.251418i
\(792\) 0 0
\(793\) 6.00000 0.213066
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 38.1051 1.34975 0.674876 0.737931i \(-0.264197\pi\)
0.674876 + 0.737931i \(0.264197\pi\)
\(798\) 0 0
\(799\) 45.0000 1.59199
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) −12.0000 + 4.89898i −0.422944 + 0.172666i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 14.1421i 0.497211i −0.968605 0.248606i \(-0.920028\pi\)
0.968605 0.248606i \(-0.0799723\pi\)
\(810\) 0 0
\(811\) 36.7423i 1.29020i 0.764099 + 0.645099i \(0.223184\pi\)
−0.764099 + 0.645099i \(0.776816\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −8.66025 −0.303355
\(816\)