Properties

Label 3024.2.k.h.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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: no (minimal twist has level 756)
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.h.1889.4

$q$-expansion

\(f(q)\) \(=\) \(q+1.73205 q^{5} +(1.00000 - 2.44949i) q^{7} +O(q^{10})\) \(q+1.73205 q^{5} +(1.00000 - 2.44949i) q^{7} -4.24264i q^{11} -2.44949i q^{13} +1.73205 q^{17} -2.44949i q^{19} -8.48528i q^{23} -2.00000 q^{25} +4.24264i q^{29} +7.34847i q^{31} +(1.73205 - 4.24264i) q^{35} +1.00000 q^{37} -1.73205 q^{41} -7.00000 q^{43} -12.1244 q^{47} +(-5.00000 - 4.89898i) q^{49} -7.34847i q^{55} -8.66025 q^{59} +2.44949i q^{61} -4.24264i q^{65} +10.0000 q^{67} -8.48528i q^{71} +9.79796i q^{73} +(-10.3923 - 4.24264i) q^{77} -5.00000 q^{79} +12.1244 q^{83} +3.00000 q^{85} +10.3923 q^{89} +(-6.00000 - 2.44949i) q^{91} -4.24264i q^{95} -2.44949i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{7} + O(q^{10}) \) \( 4q + 4q^{7} - 8q^{25} + 4q^{37} - 28q^{43} - 20q^{49} + 40q^{67} - 20q^{79} + 12q^{85} - 24q^{91} + O(q^{100}) \)

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
\(11\) 4.24264i 1.27920i −0.768706 0.639602i \(-0.779099\pi\)
0.768706 0.639602i \(-0.220901\pi\)
\(12\) 0 0
\(13\) 2.44949i 0.679366i −0.940540 0.339683i \(-0.889680\pi\)
0.940540 0.339683i \(-0.110320\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.73205 0.420084 0.210042 0.977692i \(-0.432640\pi\)
0.210042 + 0.977692i \(0.432640\pi\)
\(18\) 0 0
\(19\) 2.44949i 0.561951i −0.959715 0.280976i \(-0.909342\pi\)
0.959715 0.280976i \(-0.0906580\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 8.48528i 1.76930i −0.466252 0.884652i \(-0.654396\pi\)
0.466252 0.884652i \(-0.345604\pi\)
\(24\) 0 0
\(25\) −2.00000 −0.400000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 4.24264i 0.787839i 0.919145 + 0.393919i \(0.128881\pi\)
−0.919145 + 0.393919i \(0.871119\pi\)
\(30\) 0 0
\(31\) 7.34847i 1.31982i 0.751343 + 0.659912i \(0.229406\pi\)
−0.751343 + 0.659912i \(0.770594\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.73205 4.24264i 0.292770 0.717137i
\(36\) 0 0
\(37\) 1.00000 0.164399 0.0821995 0.996616i \(-0.473806\pi\)
0.0821995 + 0.996616i \(0.473806\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −1.73205 −0.270501 −0.135250 0.990811i \(-0.543184\pi\)
−0.135250 + 0.990811i \(0.543184\pi\)
\(42\) 0 0
\(43\) −7.00000 −1.06749 −0.533745 0.845645i \(-0.679216\pi\)
−0.533745 + 0.845645i \(0.679216\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −12.1244 −1.76852 −0.884260 0.466996i \(-0.845336\pi\)
−0.884260 + 0.466996i \(0.845336\pi\)
\(48\) 0 0
\(49\) −5.00000 4.89898i −0.714286 0.699854i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 7.34847i 0.990867i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −8.66025 −1.12747 −0.563735 0.825956i \(-0.690636\pi\)
−0.563735 + 0.825956i \(0.690636\pi\)
\(60\) 0 0
\(61\) 2.44949i 0.313625i 0.987628 + 0.156813i \(0.0501218\pi\)
−0.987628 + 0.156813i \(0.949878\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 4.24264i 0.526235i
\(66\) 0 0
\(67\) 10.0000 1.22169 0.610847 0.791748i \(-0.290829\pi\)
0.610847 + 0.791748i \(0.290829\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 8.48528i 1.00702i −0.863990 0.503509i \(-0.832042\pi\)
0.863990 0.503509i \(-0.167958\pi\)
\(72\) 0 0
\(73\) 9.79796i 1.14676i 0.819288 + 0.573382i \(0.194369\pi\)
−0.819288 + 0.573382i \(0.805631\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −10.3923 4.24264i −1.18431 0.483494i
\(78\) 0 0
\(79\) −5.00000 −0.562544 −0.281272 0.959628i \(-0.590756\pi\)
−0.281272 + 0.959628i \(0.590756\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 12.1244 1.33082 0.665410 0.746478i \(-0.268257\pi\)
0.665410 + 0.746478i \(0.268257\pi\)
\(84\) 0 0
\(85\) 3.00000 0.325396
\(86\) 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
\(95\) 4.24264i 0.435286i
\(96\) 0 0
\(97\) 2.44949i 0.248708i −0.992238 0.124354i \(-0.960314\pi\)
0.992238 0.124354i \(-0.0396858\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −10.3923 −1.03407 −0.517036 0.855963i \(-0.672965\pi\)
−0.517036 + 0.855963i \(0.672965\pi\)
\(102\) 0 0
\(103\) 19.5959i 1.93084i 0.260694 + 0.965422i \(0.416049\pi\)
−0.260694 + 0.965422i \(0.583951\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 8.48528i 0.820303i 0.912017 + 0.410152i \(0.134524\pi\)
−0.912017 + 0.410152i \(0.865476\pi\)
\(108\) 0 0
\(109\) 13.0000 1.24517 0.622587 0.782551i \(-0.286082\pi\)
0.622587 + 0.782551i \(0.286082\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 12.7279i 1.19734i −0.800995 0.598671i \(-0.795696\pi\)
0.800995 0.598671i \(-0.204304\pi\)
\(114\) 0 0
\(115\) 14.6969i 1.37050i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 1.73205 4.24264i 0.158777 0.388922i
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −12.1244 −1.08444
\(126\) 0 0
\(127\) 13.0000 1.15356 0.576782 0.816898i \(-0.304308\pi\)
0.576782 + 0.816898i \(0.304308\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −10.3923 −0.907980 −0.453990 0.891007i \(-0.650000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(132\) 0 0
\(133\) −6.00000 2.44949i −0.520266 0.212398i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 21.2132i 1.81237i −0.422885 0.906183i \(-0.638983\pi\)
0.422885 0.906183i \(-0.361017\pi\)
\(138\) 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
\(143\) −10.3923 −0.869048
\(144\) 0 0
\(145\) 7.34847i 0.610257i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 12.7279i 1.04271i −0.853339 0.521356i \(-0.825426\pi\)
0.853339 0.521356i \(-0.174574\pi\)
\(150\) 0 0
\(151\) 5.00000 0.406894 0.203447 0.979086i \(-0.434786\pi\)
0.203447 + 0.979086i \(0.434786\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 12.7279i 1.02233i
\(156\) 0 0
\(157\) 7.34847i 0.586472i −0.956040 0.293236i \(-0.905268\pi\)
0.956040 0.293236i \(-0.0947321\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −20.7846 8.48528i −1.63806 0.668734i
\(162\) 0 0
\(163\) 1.00000 0.0783260 0.0391630 0.999233i \(-0.487531\pi\)
0.0391630 + 0.999233i \(0.487531\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 19.0526 1.47433 0.737166 0.675712i \(-0.236164\pi\)
0.737166 + 0.675712i \(0.236164\pi\)
\(168\) 0 0
\(169\) 7.00000 0.538462
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 20.7846 1.58022 0.790112 0.612962i \(-0.210022\pi\)
0.790112 + 0.612962i \(0.210022\pi\)
\(174\) 0 0
\(175\) −2.00000 + 4.89898i −0.151186 + 0.370328i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 4.24264i 0.317110i 0.987350 + 0.158555i \(0.0506835\pi\)
−0.987350 + 0.158555i \(0.949317\pi\)
\(180\) 0 0
\(181\) 4.89898i 0.364138i 0.983286 + 0.182069i \(0.0582795\pi\)
−0.983286 + 0.182069i \(0.941721\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 1.73205 0.127343
\(186\) 0 0
\(187\) 7.34847i 0.537373i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 8.48528i 0.613973i −0.951714 0.306987i \(-0.900679\pi\)
0.951714 0.306987i \(-0.0993207\pi\)
\(192\) 0 0
\(193\) 7.00000 0.503871 0.251936 0.967744i \(-0.418933\pi\)
0.251936 + 0.967744i \(0.418933\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 21.2132i 1.51138i −0.654931 0.755689i \(-0.727302\pi\)
0.654931 0.755689i \(-0.272698\pi\)
\(198\) 0 0
\(199\) 9.79796i 0.694559i 0.937762 + 0.347279i \(0.112894\pi\)
−0.937762 + 0.347279i \(0.887106\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 10.3923 + 4.24264i 0.729397 + 0.297775i
\(204\) 0 0
\(205\) −3.00000 −0.209529
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −10.3923 −0.718851
\(210\) 0 0
\(211\) 8.00000 0.550743 0.275371 0.961338i \(-0.411199\pi\)
0.275371 + 0.961338i \(0.411199\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −12.1244 −0.826874
\(216\) 0 0
\(217\) 18.0000 + 7.34847i 1.22192 + 0.498847i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 4.24264i 0.285391i
\(222\) 0 0
\(223\) 22.0454i 1.47627i −0.674653 0.738135i \(-0.735707\pi\)
0.674653 0.738135i \(-0.264293\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) 14.6969i 0.971201i 0.874181 + 0.485601i \(0.161399\pi\)
−0.874181 + 0.485601i \(0.838601\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 4.24264i 0.277945i 0.990296 + 0.138972i \(0.0443799\pi\)
−0.990296 + 0.138972i \(0.955620\pi\)
\(234\) 0 0
\(235\) −21.0000 −1.36989
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 21.2132i 1.37217i −0.727522 0.686084i \(-0.759328\pi\)
0.727522 0.686084i \(-0.240672\pi\)
\(240\) 0 0
\(241\) 26.9444i 1.73564i −0.496878 0.867820i \(-0.665520\pi\)
0.496878 0.867820i \(-0.334480\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −8.66025 8.48528i −0.553283 0.542105i
\(246\) 0 0
\(247\) −6.00000 −0.381771
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 19.0526 1.20259 0.601293 0.799028i \(-0.294652\pi\)
0.601293 + 0.799028i \(0.294652\pi\)
\(252\) 0 0
\(253\) −36.0000 −2.26330
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 10.3923 0.648254 0.324127 0.946014i \(-0.394929\pi\)
0.324127 + 0.946014i \(0.394929\pi\)
\(258\) 0 0
\(259\) 1.00000 2.44949i 0.0621370 0.152204i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 21.2132i 1.30806i −0.756468 0.654031i \(-0.773077\pi\)
0.756468 0.654031i \(-0.226923\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −1.73205 −0.105605 −0.0528025 0.998605i \(-0.516815\pi\)
−0.0528025 + 0.998605i \(0.516815\pi\)
\(270\) 0 0
\(271\) 4.89898i 0.297592i −0.988868 0.148796i \(-0.952460\pi\)
0.988868 0.148796i \(-0.0475397\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 8.48528i 0.511682i
\(276\) 0 0
\(277\) −7.00000 −0.420589 −0.210295 0.977638i \(-0.567442\pi\)
−0.210295 + 0.977638i \(0.567442\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 8.48528i 0.506189i 0.967442 + 0.253095i \(0.0814484\pi\)
−0.967442 + 0.253095i \(0.918552\pi\)
\(282\) 0 0
\(283\) 7.34847i 0.436821i −0.975857 0.218411i \(-0.929913\pi\)
0.975857 0.218411i \(-0.0700872\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −1.73205 + 4.24264i −0.102240 + 0.250435i
\(288\) 0 0
\(289\) −14.0000 −0.823529
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 8.66025 0.505937 0.252969 0.967474i \(-0.418593\pi\)
0.252969 + 0.967474i \(0.418593\pi\)
\(294\) 0 0
\(295\) −15.0000 −0.873334
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −20.7846 −1.20201
\(300\) 0 0
\(301\) −7.00000 + 17.1464i −0.403473 + 0.988304i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 4.24264i 0.242933i
\(306\) 0 0
\(307\) 29.3939i 1.67760i 0.544442 + 0.838799i \(0.316741\pi\)
−0.544442 + 0.838799i \(0.683259\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −22.5167 −1.27680 −0.638401 0.769704i \(-0.720404\pi\)
−0.638401 + 0.769704i \(0.720404\pi\)
\(312\) 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
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) 18.0000 1.00781
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 4.24264i 0.236067i
\(324\) 0 0
\(325\) 4.89898i 0.271746i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −12.1244 + 29.6985i −0.668437 + 1.63733i
\(330\) 0 0
\(331\) −13.0000 −0.714545 −0.357272 0.934000i \(-0.616293\pi\)
−0.357272 + 0.934000i \(0.616293\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 17.3205 0.946320
\(336\) 0 0
\(337\) 25.0000 1.36184 0.680918 0.732359i \(-0.261581\pi\)
0.680918 + 0.732359i \(0.261581\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 31.1769 1.68832
\(342\) 0 0
\(343\) −17.0000 + 7.34847i −0.917914 + 0.396780i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 21.2132i 1.13878i 0.822066 + 0.569392i \(0.192821\pi\)
−0.822066 + 0.569392i \(0.807179\pi\)
\(348\) 0 0
\(349\) 19.5959i 1.04895i −0.851427 0.524473i \(-0.824262\pi\)
0.851427 0.524473i \(-0.175738\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 1.73205 0.0921878 0.0460939 0.998937i \(-0.485323\pi\)
0.0460939 + 0.998937i \(0.485323\pi\)
\(354\) 0 0
\(355\) 14.6969i 0.780033i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 33.9411i 1.79134i −0.444715 0.895672i \(-0.646695\pi\)
0.444715 0.895672i \(-0.353305\pi\)
\(360\) 0 0
\(361\) 13.0000 0.684211
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 16.9706i 0.888280i
\(366\) 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
\(371\) 0 0
\(372\) 0 0
\(373\) 11.0000 0.569558 0.284779 0.958593i \(-0.408080\pi\)
0.284779 + 0.958593i \(0.408080\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 10.3923 0.535231
\(378\) 0 0
\(379\) 35.0000 1.79783 0.898915 0.438124i \(-0.144357\pi\)
0.898915 + 0.438124i \(0.144357\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −19.0526 −0.973540 −0.486770 0.873530i \(-0.661825\pi\)
−0.486770 + 0.873530i \(0.661825\pi\)
\(384\) 0 0
\(385\) −18.0000 7.34847i −0.917365 0.374513i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 29.6985i 1.50577i 0.658150 + 0.752886i \(0.271339\pi\)
−0.658150 + 0.752886i \(0.728661\pi\)
\(390\) 0 0
\(391\) 14.6969i 0.743256i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −8.66025 −0.435745
\(396\) 0 0
\(397\) 29.3939i 1.47524i −0.675218 0.737618i \(-0.735950\pi\)
0.675218 0.737618i \(-0.264050\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 8.48528i 0.423735i 0.977298 + 0.211867i \(0.0679545\pi\)
−0.977298 + 0.211867i \(0.932046\pi\)
\(402\) 0 0
\(403\) 18.0000 0.896644
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 4.24264i 0.210300i
\(408\) 0 0
\(409\) 39.1918i 1.93791i 0.247234 + 0.968956i \(0.420478\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −8.66025 + 21.2132i −0.426143 + 1.04383i
\(414\) 0 0
\(415\) 21.0000 1.03085
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 29.4449 1.43848 0.719238 0.694764i \(-0.244491\pi\)
0.719238 + 0.694764i \(0.244491\pi\)
\(420\) 0 0
\(421\) 20.0000 0.974740 0.487370 0.873195i \(-0.337956\pi\)
0.487370 + 0.873195i \(0.337956\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −3.46410 −0.168034
\(426\) 0 0
\(427\) 6.00000 + 2.44949i 0.290360 + 0.118539i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 16.9706i 0.817443i −0.912659 0.408722i \(-0.865975\pi\)
0.912659 0.408722i \(-0.134025\pi\)
\(432\) 0 0
\(433\) 22.0454i 1.05943i 0.848174 + 0.529717i \(0.177702\pi\)
−0.848174 + 0.529717i \(0.822298\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −20.7846 −0.994263
\(438\) 0 0
\(439\) 29.3939i 1.40289i −0.712722 0.701447i \(-0.752538\pi\)
0.712722 0.701447i \(-0.247462\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 12.7279i 0.604722i 0.953194 + 0.302361i \(0.0977748\pi\)
−0.953194 + 0.302361i \(0.902225\pi\)
\(444\) 0 0
\(445\) 18.0000 0.853282
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 21.2132i 1.00111i 0.865704 + 0.500556i \(0.166871\pi\)
−0.865704 + 0.500556i \(0.833129\pi\)
\(450\) 0 0
\(451\) 7.34847i 0.346026i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −10.3923 4.24264i −0.487199 0.198898i
\(456\) 0 0
\(457\) −8.00000 −0.374224 −0.187112 0.982339i \(-0.559913\pi\)
−0.187112 + 0.982339i \(0.559913\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 29.4449 1.37138 0.685692 0.727892i \(-0.259500\pi\)
0.685692 + 0.727892i \(0.259500\pi\)
\(462\) 0 0
\(463\) −29.0000 −1.34774 −0.673872 0.738848i \(-0.735370\pi\)
−0.673872 + 0.738848i \(0.735370\pi\)
\(464\) 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
\(469\) 10.0000 24.4949i 0.461757 1.13107i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 29.6985i 1.36554i
\(474\) 0 0
\(475\) 4.89898i 0.224781i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −39.8372 −1.82021 −0.910103 0.414381i \(-0.863998\pi\)
−0.910103 + 0.414381i \(0.863998\pi\)
\(480\) 0 0
\(481\) 2.44949i 0.111687i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 4.24264i 0.192648i
\(486\) 0 0
\(487\) −22.0000 −0.996915 −0.498458 0.866914i \(-0.666100\pi\)
−0.498458 + 0.866914i \(0.666100\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 12.7279i 0.574403i 0.957870 + 0.287202i \(0.0927249\pi\)
−0.957870 + 0.287202i \(0.907275\pi\)
\(492\) 0 0
\(493\) 7.34847i 0.330958i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −20.7846 8.48528i −0.932317 0.380617i
\(498\) 0 0
\(499\) −19.0000 −0.850557 −0.425278 0.905063i \(-0.639824\pi\)
−0.425278 + 0.905063i \(0.639824\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 1.73205 0.0772283 0.0386142 0.999254i \(-0.487706\pi\)
0.0386142 + 0.999254i \(0.487706\pi\)
\(504\) 0 0
\(505\) −18.0000 −0.800989
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 19.0526 0.844490 0.422245 0.906482i \(-0.361242\pi\)
0.422245 + 0.906482i \(0.361242\pi\)
\(510\) 0 0
\(511\) 24.0000 + 9.79796i 1.06170 + 0.433436i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 33.9411i 1.49562i
\(516\) 0 0
\(517\) 51.4393i 2.26230i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −1.73205 −0.0758825 −0.0379413 0.999280i \(-0.512080\pi\)
−0.0379413 + 0.999280i \(0.512080\pi\)
\(522\) 0 0
\(523\) 31.8434i 1.39241i 0.717841 + 0.696207i \(0.245130\pi\)
−0.717841 + 0.696207i \(0.754870\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 12.7279i 0.554437i
\(528\) 0 0
\(529\) −49.0000 −2.13043
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 4.24264i 0.183769i
\(534\) 0 0
\(535\) 14.6969i 0.635404i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −20.7846 + 21.2132i −0.895257 + 0.913717i
\(540\) 0 0
\(541\) 17.0000 0.730887 0.365444 0.930834i \(-0.380917\pi\)
0.365444 + 0.930834i \(0.380917\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 22.5167 0.964508
\(546\) 0 0
\(547\) −41.0000 −1.75303 −0.876517 0.481371i \(-0.840139\pi\)
−0.876517 + 0.481371i \(0.840139\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 10.3923 0.442727
\(552\) 0 0
\(553\) −5.00000 + 12.2474i −0.212622 + 0.520814i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 8.48528i 0.359533i −0.983709 0.179766i \(-0.942466\pi\)
0.983709 0.179766i \(-0.0575342\pi\)
\(558\) 0 0
\(559\) 17.1464i 0.725217i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 20.7846 0.875967 0.437983 0.898983i \(-0.355693\pi\)
0.437983 + 0.898983i \(0.355693\pi\)
\(564\) 0 0
\(565\) 22.0454i 0.927457i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 33.9411i 1.42289i 0.702744 + 0.711443i \(0.251958\pi\)
−0.702744 + 0.711443i \(0.748042\pi\)
\(570\) 0 0
\(571\) 25.0000 1.04622 0.523109 0.852266i \(-0.324772\pi\)
0.523109 + 0.852266i \(0.324772\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 16.9706i 0.707721i
\(576\) 0 0
\(577\) 19.5959i 0.815789i −0.913029 0.407894i \(-0.866263\pi\)
0.913029 0.407894i \(-0.133737\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 12.1244 29.6985i 0.503003 1.23210i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 41.5692 1.71575 0.857873 0.513862i \(-0.171786\pi\)
0.857873 + 0.513862i \(0.171786\pi\)
\(588\) 0 0
\(589\) 18.0000 0.741677
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 8.66025 0.355634 0.177817 0.984064i \(-0.443096\pi\)
0.177817 + 0.984064i \(0.443096\pi\)
\(594\) 0 0
\(595\) 3.00000 7.34847i 0.122988 0.301258i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 4.24264i 0.173350i −0.996237 0.0866748i \(-0.972376\pi\)
0.996237 0.0866748i \(-0.0276241\pi\)
\(600\) 0 0
\(601\) 22.0454i 0.899251i −0.893217 0.449625i \(-0.851557\pi\)
0.893217 0.449625i \(-0.148443\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −12.1244 −0.492925
\(606\) 0 0
\(607\) 26.9444i 1.09364i −0.837251 0.546819i \(-0.815838\pi\)
0.837251 0.546819i \(-0.184162\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 29.6985i 1.20147i
\(612\) 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
\(617\) 25.4558i 1.02481i −0.858743 0.512407i \(-0.828754\pi\)
0.858743 0.512407i \(-0.171246\pi\)
\(618\) 0 0
\(619\) 2.44949i 0.0984533i 0.998788 + 0.0492267i \(0.0156757\pi\)
−0.998788 + 0.0492267i \(0.984324\pi\)
\(620\) 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
\(629\) 1.73205 0.0690614
\(630\) 0 0
\(631\) 37.0000 1.47295 0.736473 0.676467i \(-0.236490\pi\)
0.736473 + 0.676467i \(0.236490\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 22.5167 0.893546
\(636\) 0 0
\(637\) −12.0000 + 12.2474i −0.475457 + 0.485262i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 21.2132i 0.837871i −0.908016 0.418936i \(-0.862403\pi\)
0.908016 0.418936i \(-0.137597\pi\)
\(642\) 0 0
\(643\) 14.6969i 0.579591i 0.957089 + 0.289795i \(0.0935872\pi\)
−0.957089 + 0.289795i \(0.906413\pi\)
\(644\) 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
\(649\) 36.7423i 1.44226i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 29.6985i 1.16219i 0.813835 + 0.581096i \(0.197376\pi\)
−0.813835 + 0.581096i \(0.802624\pi\)
\(654\) 0 0
\(655\) −18.0000 −0.703318
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 16.9706i 0.661079i 0.943792 + 0.330540i \(0.107231\pi\)
−0.943792 + 0.330540i \(0.892769\pi\)
\(660\) 0 0
\(661\) 12.2474i 0.476371i 0.971220 + 0.238185i \(0.0765525\pi\)
−0.971220 + 0.238185i \(0.923447\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −10.3923 4.24264i −0.402996 0.164523i
\(666\) 0 0
\(667\) 36.0000 1.39393
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 10.3923 0.401190
\(672\) 0 0
\(673\) 16.0000 0.616755 0.308377 0.951264i \(-0.400214\pi\)
0.308377 + 0.951264i \(0.400214\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −31.1769 −1.19823 −0.599113 0.800664i \(-0.704480\pi\)
−0.599113 + 0.800664i \(0.704480\pi\)
\(678\) 0 0
\(679\) −6.00000 2.44949i −0.230259 0.0940028i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 38.1838i 1.46106i 0.682880 + 0.730531i \(0.260727\pi\)
−0.682880 + 0.730531i \(0.739273\pi\)
\(684\) 0 0
\(685\) 36.7423i 1.40385i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 22.0454i 0.838647i 0.907837 + 0.419323i \(0.137733\pi\)
−0.907837 + 0.419323i \(0.862267\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 21.2132i 0.804663i
\(696\) 0 0
\(697\) −3.00000 −0.113633
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 12.7279i 0.480727i 0.970683 + 0.240363i \(0.0772666\pi\)
−0.970683 + 0.240363i \(0.922733\pi\)
\(702\) 0 0
\(703\) 2.44949i 0.0923843i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −10.3923 + 25.4558i −0.390843 + 0.957366i
\(708\) 0 0
\(709\) 49.0000 1.84023 0.920117 0.391644i \(-0.128094\pi\)
0.920117 + 0.391644i \(0.128094\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 62.3538 2.33517
\(714\) 0 0
\(715\) −18.0000 −0.673162
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 32.9090 1.22730 0.613649 0.789579i \(-0.289701\pi\)
0.613649 + 0.789579i \(0.289701\pi\)
\(720\) 0 0
\(721\) 48.0000 + 19.5959i 1.78761 + 0.729790i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 8.48528i 0.315135i
\(726\) 0 0
\(727\) 7.34847i 0.272540i 0.990672 + 0.136270i \(0.0435114\pi\)
−0.990672 + 0.136270i \(0.956489\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −12.1244 −0.448435
\(732\) 0 0
\(733\) 29.3939i 1.08569i 0.839834 + 0.542844i \(0.182652\pi\)
−0.839834 + 0.542844i \(0.817348\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 42.4264i 1.56280i
\(738\) 0 0
\(739\) −38.0000 −1.39785 −0.698926 0.715194i \(-0.746338\pi\)
−0.698926 + 0.715194i \(0.746338\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 21.2132i 0.778237i 0.921188 + 0.389118i \(0.127220\pi\)
−0.921188 + 0.389118i \(0.872780\pi\)
\(744\) 0 0
\(745\) 22.0454i 0.807681i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 20.7846 + 8.48528i 0.759453 + 0.310045i
\(750\) 0 0
\(751\) −10.0000 −0.364905 −0.182453 0.983215i \(-0.558404\pi\)
−0.182453 + 0.983215i \(0.558404\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 8.66025 0.315179
\(756\) 0 0
\(757\) −53.0000 −1.92632 −0.963159 0.268933i \(-0.913329\pi\)
−0.963159 + 0.268933i \(0.913329\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −22.5167 −0.816228 −0.408114 0.912931i \(-0.633813\pi\)
−0.408114 + 0.912931i \(0.633813\pi\)
\(762\) 0 0
\(763\) 13.0000 31.8434i 0.470632 1.15281i
\(764\) 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
\(773\) −39.8372 −1.43284 −0.716422 0.697668i \(-0.754221\pi\)
−0.716422 + 0.697668i \(0.754221\pi\)
\(774\) 0 0
\(775\) 14.6969i 0.527930i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 4.24264i 0.152008i
\(780\) 0 0
\(781\) −36.0000 −1.28818
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 12.7279i 0.454279i
\(786\) 0 0
\(787\) 4.89898i 0.174630i −0.996181 0.0873149i \(-0.972171\pi\)
0.996181 0.0873149i \(-0.0278286\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −31.1769 12.7279i −1.10852 0.452553i
\(792\) 0 0
\(793\) 6.00000 0.213066
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 10.3923 0.368114 0.184057 0.982916i \(-0.441077\pi\)
0.184057 + 0.982916i \(0.441077\pi\)
\(798\) 0 0
\(799\) −21.0000 −0.742927
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 41.5692 1.46695
\(804\) 0 0
\(805\) −36.0000 14.6969i −1.26883 0.517999i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 25.4558i 0.894980i 0.894289 + 0.447490i \(0.147682\pi\)
−0.894289 + 0.447490i \(0.852318\pi\)
\(810\) 0 0
\(811\) 51.4393i 1.80628i −0.429349 0.903139i \(-0.641257\pi\)
0.429349 0.903139i \(-0.358743\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 1.73205 0.0606711
\(816\) 0