Properties

Label 3024.2.k.i.1889.2
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: \(\Q(\sqrt{-3}, \sqrt{-5})\)
Defining polynomial: \(x^{4} - 5 x^{2} + 25\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 189)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1889.2
Root \(-1.93649 + 1.11803i\) of defining polynomial
Character \(\chi\) \(=\) 3024.1889
Dual form 3024.2.k.i.1889.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.87298 q^{5} +(2.00000 + 1.73205i) q^{7} +O(q^{10})\) \(q-3.87298 q^{5} +(2.00000 + 1.73205i) q^{7} -2.23607i q^{11} +3.46410i q^{13} -5.19615i q^{19} -2.23607i q^{23} +10.0000 q^{25} -4.47214i q^{29} +1.73205i q^{31} +(-7.74597 - 6.70820i) q^{35} -1.00000 q^{37} -3.87298 q^{41} -2.00000 q^{43} +7.74597 q^{47} +(1.00000 + 6.92820i) q^{49} +8.94427i q^{53} +8.66025i q^{55} -7.74597 q^{59} +6.92820i q^{61} -13.4164i q^{65} +10.0000 q^{67} +11.1803i q^{71} +10.3923i q^{73} +(3.87298 - 4.47214i) q^{77} -2.00000 q^{79} +7.74597 q^{83} +11.6190 q^{89} +(-6.00000 + 6.92820i) q^{91} +20.1246i q^{95} -13.8564i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 8q^{7} + O(q^{10}) \) \( 4q + 8q^{7} + 40q^{25} - 4q^{37} - 8q^{43} + 4q^{49} + 40q^{67} - 8q^{79} - 24q^{91} + O(q^{100}) \)

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
\(5\) −3.87298 −1.73205 −0.866025 0.500000i \(-0.833333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(6\) 0 0
\(7\) 2.00000 + 1.73205i 0.755929 + 0.654654i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.23607i 0.674200i −0.941469 0.337100i \(-0.890554\pi\)
0.941469 0.337100i \(-0.109446\pi\)
\(12\) 0 0
\(13\) 3.46410i 0.960769i 0.877058 + 0.480384i \(0.159503\pi\)
−0.877058 + 0.480384i \(0.840497\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 5.19615i 1.19208i −0.802955 0.596040i \(-0.796740\pi\)
0.802955 0.596040i \(-0.203260\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.23607i 0.466252i −0.972446 0.233126i \(-0.925104\pi\)
0.972446 0.233126i \(-0.0748955\pi\)
\(24\) 0 0
\(25\) 10.0000 2.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 4.47214i 0.830455i −0.909718 0.415227i \(-0.863702\pi\)
0.909718 0.415227i \(-0.136298\pi\)
\(30\) 0 0
\(31\) 1.73205i 0.311086i 0.987829 + 0.155543i \(0.0497126\pi\)
−0.987829 + 0.155543i \(0.950287\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −7.74597 6.70820i −1.30931 1.13389i
\(36\) 0 0
\(37\) −1.00000 −0.164399 −0.0821995 0.996616i \(-0.526194\pi\)
−0.0821995 + 0.996616i \(0.526194\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −3.87298 −0.604858 −0.302429 0.953172i \(-0.597798\pi\)
−0.302429 + 0.953172i \(0.597798\pi\)
\(42\) 0 0
\(43\) −2.00000 −0.304997 −0.152499 0.988304i \(-0.548732\pi\)
−0.152499 + 0.988304i \(0.548732\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 7.74597 1.12987 0.564933 0.825137i \(-0.308902\pi\)
0.564933 + 0.825137i \(0.308902\pi\)
\(48\) 0 0
\(49\) 1.00000 + 6.92820i 0.142857 + 0.989743i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 8.94427i 1.22859i 0.789076 + 0.614295i \(0.210560\pi\)
−0.789076 + 0.614295i \(0.789440\pi\)
\(54\) 0 0
\(55\) 8.66025i 1.16775i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −7.74597 −1.00844 −0.504219 0.863576i \(-0.668220\pi\)
−0.504219 + 0.863576i \(0.668220\pi\)
\(60\) 0 0
\(61\) 6.92820i 0.887066i 0.896258 + 0.443533i \(0.146275\pi\)
−0.896258 + 0.443533i \(0.853725\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 13.4164i 1.66410i
\(66\) 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
\(71\) 11.1803i 1.32686i 0.748237 + 0.663431i \(0.230900\pi\)
−0.748237 + 0.663431i \(0.769100\pi\)
\(72\) 0 0
\(73\) 10.3923i 1.21633i 0.793812 + 0.608164i \(0.208094\pi\)
−0.793812 + 0.608164i \(0.791906\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.87298 4.47214i 0.441367 0.509647i
\(78\) 0 0
\(79\) −2.00000 −0.225018 −0.112509 0.993651i \(-0.535889\pi\)
−0.112509 + 0.993651i \(0.535889\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 7.74597 0.850230 0.425115 0.905139i \(-0.360234\pi\)
0.425115 + 0.905139i \(0.360234\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 11.6190 1.23161 0.615803 0.787900i \(-0.288832\pi\)
0.615803 + 0.787900i \(0.288832\pi\)
\(90\) 0 0
\(91\) −6.00000 + 6.92820i −0.628971 + 0.726273i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 20.1246i 2.06474i
\(96\) 0 0
\(97\) 13.8564i 1.40690i −0.710742 0.703452i \(-0.751641\pi\)
0.710742 0.703452i \(-0.248359\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 15.4919 1.54150 0.770752 0.637135i \(-0.219880\pi\)
0.770752 + 0.637135i \(0.219880\pi\)
\(102\) 0 0
\(103\) 1.73205i 0.170664i 0.996353 + 0.0853320i \(0.0271951\pi\)
−0.996353 + 0.0853320i \(0.972805\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 8.94427i 0.864675i −0.901712 0.432338i \(-0.857689\pi\)
0.901712 0.432338i \(-0.142311\pi\)
\(108\) 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
\(113\) 8.94427i 0.841406i 0.907198 + 0.420703i \(0.138217\pi\)
−0.907198 + 0.420703i \(0.861783\pi\)
\(114\) 0 0
\(115\) 8.66025i 0.807573i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 6.00000 0.545455
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −19.3649 −1.73205
\(126\) 0 0
\(127\) 10.0000 0.887357 0.443678 0.896186i \(-0.353673\pi\)
0.443678 + 0.896186i \(0.353673\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −7.74597 −0.676768 −0.338384 0.941008i \(-0.609880\pi\)
−0.338384 + 0.941008i \(0.609880\pi\)
\(132\) 0 0
\(133\) 9.00000 10.3923i 0.780399 0.901127i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 22.3607i 1.91040i 0.295958 + 0.955201i \(0.404361\pi\)
−0.295958 + 0.955201i \(0.595639\pi\)
\(138\) 0 0
\(139\) 3.46410i 0.293821i −0.989150 0.146911i \(-0.953067\pi\)
0.989150 0.146911i \(-0.0469330\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 7.74597 0.647750
\(144\) 0 0
\(145\) 17.3205i 1.43839i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 8.94427i 0.732743i 0.930469 + 0.366372i \(0.119400\pi\)
−0.930469 + 0.366372i \(0.880600\pi\)
\(150\) 0 0
\(151\) 16.0000 1.30206 0.651031 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 6.70820i 0.538816i
\(156\) 0 0
\(157\) 3.46410i 0.276465i 0.990400 + 0.138233i \(0.0441422\pi\)
−0.990400 + 0.138233i \(0.955858\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 3.87298 4.47214i 0.305234 0.352454i
\(162\) 0 0
\(163\) 10.0000 0.783260 0.391630 0.920123i \(-0.371911\pi\)
0.391630 + 0.920123i \(0.371911\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 15.4919 1.19880 0.599401 0.800449i \(-0.295406\pi\)
0.599401 + 0.800449i \(0.295406\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 3.87298 0.294457 0.147229 0.989102i \(-0.452965\pi\)
0.147229 + 0.989102i \(0.452965\pi\)
\(174\) 0 0
\(175\) 20.0000 + 17.3205i 1.51186 + 1.30931i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 17.8885i 1.33705i 0.743689 + 0.668526i \(0.233075\pi\)
−0.743689 + 0.668526i \(0.766925\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 3.87298 0.284747
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 2.23607i 0.161796i −0.996722 0.0808981i \(-0.974221\pi\)
0.996722 0.0808981i \(-0.0257788\pi\)
\(192\) 0 0
\(193\) 2.00000 0.143963 0.0719816 0.997406i \(-0.477068\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 8.94427i 0.637253i 0.947880 + 0.318626i \(0.103222\pi\)
−0.947880 + 0.318626i \(0.896778\pi\)
\(198\) 0 0
\(199\) 25.9808i 1.84173i 0.389885 + 0.920864i \(0.372515\pi\)
−0.389885 + 0.920864i \(0.627485\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 7.74597 8.94427i 0.543660 0.627765i
\(204\) 0 0
\(205\) 15.0000 1.04765
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −11.6190 −0.803700
\(210\) 0 0
\(211\) −8.00000 −0.550743 −0.275371 0.961338i \(-0.588801\pi\)
−0.275371 + 0.961338i \(0.588801\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 7.74597 0.528271
\(216\) 0 0
\(217\) −3.00000 + 3.46410i −0.203653 + 0.235159i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 1.73205i 0.115987i −0.998317 0.0579934i \(-0.981530\pi\)
0.998317 0.0579934i \(-0.0184702\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −15.4919 −1.02824 −0.514118 0.857720i \(-0.671881\pi\)
−0.514118 + 0.857720i \(0.671881\pi\)
\(228\) 0 0
\(229\) 13.8564i 0.915657i 0.889041 + 0.457829i \(0.151373\pi\)
−0.889041 + 0.457829i \(0.848627\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 17.8885i 1.17192i −0.810341 0.585959i \(-0.800718\pi\)
0.810341 0.585959i \(-0.199282\pi\)
\(234\) 0 0
\(235\) −30.0000 −1.95698
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 8.94427i 0.578557i −0.957245 0.289278i \(-0.906585\pi\)
0.957245 0.289278i \(-0.0934153\pi\)
\(240\) 0 0
\(241\) 17.3205i 1.11571i 0.829938 + 0.557856i \(0.188376\pi\)
−0.829938 + 0.557856i \(0.811624\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −3.87298 26.8328i −0.247436 1.71429i
\(246\) 0 0
\(247\) 18.0000 1.14531
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) −5.00000 −0.314347
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −27.1109 −1.69113 −0.845565 0.533872i \(-0.820736\pi\)
−0.845565 + 0.533872i \(0.820736\pi\)
\(258\) 0 0
\(259\) −2.00000 1.73205i −0.124274 0.107624i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 2.23607i 0.137882i −0.997621 0.0689409i \(-0.978038\pi\)
0.997621 0.0689409i \(-0.0219620\pi\)
\(264\) 0 0
\(265\) 34.6410i 2.12798i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −11.6190 −0.708420 −0.354210 0.935166i \(-0.615250\pi\)
−0.354210 + 0.935166i \(0.615250\pi\)
\(270\) 0 0
\(271\) 10.3923i 0.631288i 0.948878 + 0.315644i \(0.102220\pi\)
−0.948878 + 0.315644i \(0.897780\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 22.3607i 1.34840i
\(276\) 0 0
\(277\) 11.0000 0.660926 0.330463 0.943819i \(-0.392795\pi\)
0.330463 + 0.943819i \(0.392795\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 22.3607i 1.33393i 0.745091 + 0.666963i \(0.232406\pi\)
−0.745091 + 0.666963i \(0.767594\pi\)
\(282\) 0 0
\(283\) 24.2487i 1.44144i −0.693228 0.720718i \(-0.743812\pi\)
0.693228 0.720718i \(-0.256188\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −7.74597 6.70820i −0.457230 0.395973i
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 30.9839 1.81010 0.905048 0.425309i \(-0.139834\pi\)
0.905048 + 0.425309i \(0.139834\pi\)
\(294\) 0 0
\(295\) 30.0000 1.74667
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 7.74597 0.447961
\(300\) 0 0
\(301\) −4.00000 3.46410i −0.230556 0.199667i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 26.8328i 1.53644i
\(306\) 0 0
\(307\) 5.19615i 0.296560i −0.988945 0.148280i \(-0.952626\pi\)
0.988945 0.148280i \(-0.0473737\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −7.74597 −0.439233 −0.219617 0.975586i \(-0.570481\pi\)
−0.219617 + 0.975586i \(0.570481\pi\)
\(312\) 0 0
\(313\) 3.46410i 0.195803i −0.995196 0.0979013i \(-0.968787\pi\)
0.995196 0.0979013i \(-0.0312129\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 4.47214i 0.251180i −0.992082 0.125590i \(-0.959918\pi\)
0.992082 0.125590i \(-0.0400824\pi\)
\(318\) 0 0
\(319\) −10.0000 −0.559893
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 34.6410i 1.92154i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 15.4919 + 13.4164i 0.854098 + 0.739671i
\(330\) 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
\(335\) −38.7298 −2.11604
\(336\) 0 0
\(337\) 35.0000 1.90657 0.953286 0.302070i \(-0.0976776\pi\)
0.953286 + 0.302070i \(0.0976776\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 3.87298 0.209734
\(342\) 0 0
\(343\) −10.0000 + 15.5885i −0.539949 + 0.841698i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 11.1803i 0.600192i 0.953909 + 0.300096i \(0.0970187\pi\)
−0.953909 + 0.300096i \(0.902981\pi\)
\(348\) 0 0
\(349\) 13.8564i 0.741716i −0.928689 0.370858i \(-0.879064\pi\)
0.928689 0.370858i \(-0.120936\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −19.3649 −1.03069 −0.515345 0.856983i \(-0.672336\pi\)
−0.515345 + 0.856983i \(0.672336\pi\)
\(354\) 0 0
\(355\) 43.3013i 2.29819i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 8.94427i 0.472061i −0.971746 0.236030i \(-0.924154\pi\)
0.971746 0.236030i \(-0.0758465\pi\)
\(360\) 0 0
\(361\) −8.00000 −0.421053
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 40.2492i 2.10674i
\(366\) 0 0
\(367\) 1.73205i 0.0904123i −0.998978 0.0452062i \(-0.985606\pi\)
0.998978 0.0452062i \(-0.0143945\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −15.4919 + 17.8885i −0.804301 + 0.928727i
\(372\) 0 0
\(373\) 17.0000 0.880227 0.440113 0.897942i \(-0.354938\pi\)
0.440113 + 0.897942i \(0.354938\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 15.4919 0.797875
\(378\) 0 0
\(379\) −26.0000 −1.33553 −0.667765 0.744372i \(-0.732749\pi\)
−0.667765 + 0.744372i \(0.732749\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −30.9839 −1.58320 −0.791601 0.611039i \(-0.790752\pi\)
−0.791601 + 0.611039i \(0.790752\pi\)
\(384\) 0 0
\(385\) −15.0000 + 17.3205i −0.764471 + 0.882735i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 35.7771i 1.81397i 0.421163 + 0.906985i \(0.361622\pi\)
−0.421163 + 0.906985i \(0.638378\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 7.74597 0.389742
\(396\) 0 0
\(397\) 20.7846i 1.04315i −0.853206 0.521575i \(-0.825345\pi\)
0.853206 0.521575i \(-0.174655\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 31.3050i 1.56329i −0.623721 0.781647i \(-0.714380\pi\)
0.623721 0.781647i \(-0.285620\pi\)
\(402\) 0 0
\(403\) −6.00000 −0.298881
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 2.23607i 0.110838i
\(408\) 0 0
\(409\) 17.3205i 0.856444i −0.903674 0.428222i \(-0.859140\pi\)
0.903674 0.428222i \(-0.140860\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −15.4919 13.4164i −0.762308 0.660178i
\(414\) 0 0
\(415\) −30.0000 −1.47264
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 15.4919 0.756830 0.378415 0.925636i \(-0.376469\pi\)
0.378415 + 0.925636i \(0.376469\pi\)
\(420\) 0 0
\(421\) 11.0000 0.536107 0.268054 0.963404i \(-0.413620\pi\)
0.268054 + 0.963404i \(0.413620\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −12.0000 + 13.8564i −0.580721 + 0.670559i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 2.23607i 0.107708i −0.998549 0.0538538i \(-0.982850\pi\)
0.998549 0.0538538i \(-0.0171505\pi\)
\(432\) 0 0
\(433\) 10.3923i 0.499422i 0.968320 + 0.249711i \(0.0803357\pi\)
−0.968320 + 0.249711i \(0.919664\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −11.6190 −0.555810
\(438\) 0 0
\(439\) 38.1051i 1.81866i −0.416078 0.909329i \(-0.636596\pi\)
0.416078 0.909329i \(-0.363404\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 24.5967i 1.16863i 0.811528 + 0.584313i \(0.198636\pi\)
−0.811528 + 0.584313i \(0.801364\pi\)
\(444\) 0 0
\(445\) −45.0000 −2.13320
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 8.94427i 0.422106i 0.977475 + 0.211053i \(0.0676893\pi\)
−0.977475 + 0.211053i \(0.932311\pi\)
\(450\) 0 0
\(451\) 8.66025i 0.407795i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 23.2379 26.8328i 1.08941 1.25794i
\(456\) 0 0
\(457\) −25.0000 −1.16945 −0.584725 0.811231i \(-0.698798\pi\)
−0.584725 + 0.811231i \(0.698798\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 3.87298 0.180383 0.0901914 0.995924i \(-0.471252\pi\)
0.0901914 + 0.995924i \(0.471252\pi\)
\(462\) 0 0
\(463\) −8.00000 −0.371792 −0.185896 0.982569i \(-0.559519\pi\)
−0.185896 + 0.982569i \(0.559519\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 23.2379 1.07532 0.537661 0.843161i \(-0.319308\pi\)
0.537661 + 0.843161i \(0.319308\pi\)
\(468\) 0 0
\(469\) 20.0000 + 17.3205i 0.923514 + 0.799787i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 4.47214i 0.205629i
\(474\) 0 0
\(475\) 51.9615i 2.38416i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 30.9839 1.41569 0.707845 0.706368i \(-0.249668\pi\)
0.707845 + 0.706368i \(0.249668\pi\)
\(480\) 0 0
\(481\) 3.46410i 0.157949i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 53.6656i 2.43683i
\(486\) 0 0
\(487\) 10.0000 0.453143 0.226572 0.973995i \(-0.427248\pi\)
0.226572 + 0.973995i \(0.427248\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 38.0132i 1.71551i 0.514059 + 0.857755i \(0.328141\pi\)
−0.514059 + 0.857755i \(0.671859\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −19.3649 + 22.3607i −0.868635 + 1.00301i
\(498\) 0 0
\(499\) −38.0000 −1.70111 −0.850557 0.525883i \(-0.823735\pi\)
−0.850557 + 0.525883i \(0.823735\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −23.2379 −1.03613 −0.518063 0.855342i \(-0.673347\pi\)
−0.518063 + 0.855342i \(0.673347\pi\)
\(504\) 0 0
\(505\) −60.0000 −2.66996
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −15.4919 −0.686668 −0.343334 0.939213i \(-0.611556\pi\)
−0.343334 + 0.939213i \(0.611556\pi\)
\(510\) 0 0
\(511\) −18.0000 + 20.7846i −0.796273 + 0.919457i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 6.70820i 0.295599i
\(516\) 0 0
\(517\) 17.3205i 0.761755i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 34.8569 1.52711 0.763553 0.645745i \(-0.223453\pi\)
0.763553 + 0.645745i \(0.223453\pi\)
\(522\) 0 0
\(523\) 15.5885i 0.681636i −0.940129 0.340818i \(-0.889296\pi\)
0.940129 0.340818i \(-0.110704\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 18.0000 0.782609
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 13.4164i 0.581129i
\(534\) 0 0
\(535\) 34.6410i 1.49766i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 15.4919 2.23607i 0.667285 0.0963143i
\(540\) 0 0
\(541\) −1.00000 −0.0429934 −0.0214967 0.999769i \(-0.506843\pi\)
−0.0214967 + 0.999769i \(0.506843\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 27.1109 1.16130
\(546\) 0 0
\(547\) 16.0000 0.684111 0.342055 0.939680i \(-0.388877\pi\)
0.342055 + 0.939680i \(0.388877\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −23.2379 −0.989968
\(552\) 0 0
\(553\) −4.00000 3.46410i −0.170097 0.147309i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 8.94427i 0.378981i 0.981883 + 0.189490i \(0.0606836\pi\)
−0.981883 + 0.189490i \(0.939316\pi\)
\(558\) 0 0
\(559\) 6.92820i 0.293032i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 38.7298 1.63227 0.816134 0.577863i \(-0.196113\pi\)
0.816134 + 0.577863i \(0.196113\pi\)
\(564\) 0 0
\(565\) 34.6410i 1.45736i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 17.8885i 0.749927i −0.927040 0.374963i \(-0.877655\pi\)
0.927040 0.374963i \(-0.122345\pi\)
\(570\) 0 0
\(571\) 16.0000 0.669579 0.334790 0.942293i \(-0.391335\pi\)
0.334790 + 0.942293i \(0.391335\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 22.3607i 0.932505i
\(576\) 0 0
\(577\) 20.7846i 0.865275i −0.901568 0.432637i \(-0.857583\pi\)
0.901568 0.432637i \(-0.142417\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 15.4919 + 13.4164i 0.642714 + 0.556606i
\(582\) 0 0
\(583\) 20.0000 0.828315
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 30.9839 1.27884 0.639421 0.768857i \(-0.279174\pi\)
0.639421 + 0.768857i \(0.279174\pi\)
\(588\) 0 0
\(589\) 9.00000 0.370839
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 34.8569 1.43140 0.715700 0.698408i \(-0.246108\pi\)
0.715700 + 0.698408i \(0.246108\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 29.0689i 1.18772i −0.804568 0.593861i \(-0.797603\pi\)
0.804568 0.593861i \(-0.202397\pi\)
\(600\) 0 0
\(601\) 24.2487i 0.989126i −0.869142 0.494563i \(-0.835328\pi\)
0.869142 0.494563i \(-0.164672\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −23.2379 −0.944755
\(606\) 0 0
\(607\) 38.1051i 1.54664i 0.634017 + 0.773320i \(0.281405\pi\)
−0.634017 + 0.773320i \(0.718595\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 26.8328i 1.08554i
\(612\) 0 0
\(613\) 17.0000 0.686624 0.343312 0.939222i \(-0.388451\pi\)
0.343312 + 0.939222i \(0.388451\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 22.3607i 0.900207i 0.892976 + 0.450104i \(0.148613\pi\)
−0.892976 + 0.450104i \(0.851387\pi\)
\(618\) 0 0
\(619\) 22.5167i 0.905021i −0.891759 0.452510i \(-0.850529\pi\)
0.891759 0.452510i \(-0.149471\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 23.2379 + 20.1246i 0.931007 + 0.806276i
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −2.00000 −0.0796187 −0.0398094 0.999207i \(-0.512675\pi\)
−0.0398094 + 0.999207i \(0.512675\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −38.7298 −1.53695
\(636\) 0 0
\(637\) −24.0000 + 3.46410i −0.950915 + 0.137253i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 17.8885i 0.706555i −0.935519 0.353278i \(-0.885067\pi\)
0.935519 0.353278i \(-0.114933\pi\)
\(642\) 0 0
\(643\) 39.8372i 1.57102i −0.618846 0.785512i \(-0.712400\pi\)
0.618846 0.785512i \(-0.287600\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −23.2379 −0.913576 −0.456788 0.889576i \(-0.651000\pi\)
−0.456788 + 0.889576i \(0.651000\pi\)
\(648\) 0 0
\(649\) 17.3205i 0.679889i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 49.1935i 1.92509i 0.271122 + 0.962545i \(0.412605\pi\)
−0.271122 + 0.962545i \(0.587395\pi\)
\(654\) 0 0
\(655\) 30.0000 1.17220
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 2.23607i 0.0871048i −0.999051 0.0435524i \(-0.986132\pi\)
0.999051 0.0435524i \(-0.0138676\pi\)
\(660\) 0 0
\(661\) 45.0333i 1.75159i 0.482680 + 0.875797i \(0.339663\pi\)
−0.482680 + 0.875797i \(0.660337\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −34.8569 + 40.2492i −1.35169 + 1.56080i
\(666\) 0 0
\(667\) −10.0000 −0.387202
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 15.4919 0.598059
\(672\) 0 0
\(673\) 2.00000 0.0770943 0.0385472 0.999257i \(-0.487727\pi\)
0.0385472 + 0.999257i \(0.487727\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 27.1109 1.04196 0.520978 0.853570i \(-0.325567\pi\)
0.520978 + 0.853570i \(0.325567\pi\)
\(678\) 0 0
\(679\) 24.0000 27.7128i 0.921035 1.06352i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 29.0689i 1.11229i −0.831085 0.556145i \(-0.812280\pi\)
0.831085 0.556145i \(-0.187720\pi\)
\(684\) 0 0
\(685\) 86.6025i 3.30891i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −30.9839 −1.18039
\(690\) 0 0
\(691\) 17.3205i 0.658903i −0.944172 0.329452i \(-0.893136\pi\)
0.944172 0.329452i \(-0.106864\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 13.4164i 0.508913i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 31.3050i 1.18237i −0.806535 0.591186i \(-0.798660\pi\)
0.806535 0.591186i \(-0.201340\pi\)
\(702\) 0 0
\(703\) 5.19615i 0.195977i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 30.9839 + 26.8328i 1.16527 + 1.00915i
\(708\) 0 0
\(709\) −43.0000 −1.61490 −0.807449 0.589937i \(-0.799153\pi\)
−0.807449 + 0.589937i \(0.799153\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 3.87298 0.145044
\(714\) 0 0
\(715\) −30.0000 −1.12194
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 23.2379 0.866627 0.433314 0.901243i \(-0.357344\pi\)
0.433314 + 0.901243i \(0.357344\pi\)
\(720\) 0 0
\(721\) −3.00000 + 3.46410i −0.111726 + 0.129010i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 44.7214i 1.66091i
\(726\) 0 0
\(727\) 3.46410i 0.128476i 0.997935 + 0.0642382i \(0.0204617\pi\)
−0.997935 + 0.0642382i \(0.979538\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 38.1051i 1.40744i −0.710475 0.703722i \(-0.751520\pi\)
0.710475 0.703722i \(-0.248480\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 22.3607i 0.823666i
\(738\) 0 0
\(739\) −44.0000 −1.61857 −0.809283 0.587419i \(-0.800144\pi\)
−0.809283 + 0.587419i \(0.800144\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 11.1803i 0.410167i 0.978744 + 0.205083i \(0.0657466\pi\)
−0.978744 + 0.205083i \(0.934253\pi\)
\(744\) 0 0
\(745\) 34.6410i 1.26915i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 15.4919 17.8885i 0.566063 0.653633i
\(750\) 0 0
\(751\) 46.0000 1.67856 0.839282 0.543696i \(-0.182976\pi\)
0.839282 + 0.543696i \(0.182976\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −61.9677 −2.25524
\(756\) 0 0
\(757\) −34.0000 −1.23575 −0.617876 0.786276i \(-0.712006\pi\)
−0.617876 + 0.786276i \(0.712006\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −15.4919 −0.561582 −0.280791 0.959769i \(-0.590597\pi\)
−0.280791 + 0.959769i \(0.590597\pi\)
\(762\) 0 0
\(763\) −14.0000 12.1244i −0.506834 0.438931i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 26.8328i 0.968877i
\(768\) 0 0
\(769\) 17.3205i 0.624593i −0.949985 0.312297i \(-0.898902\pi\)
0.949985 0.312297i \(-0.101098\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 11.6190 0.417905 0.208952 0.977926i \(-0.432995\pi\)
0.208952 + 0.977926i \(0.432995\pi\)
\(774\) 0 0
\(775\) 17.3205i 0.622171i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 20.1246i 0.721039i
\(780\) 0 0
\(781\) 25.0000 0.894570
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 13.4164i 0.478852i
\(786\) 0 0
\(787\) 38.1051i 1.35830i 0.733999 + 0.679150i \(0.237652\pi\)
−0.733999 + 0.679150i \(0.762348\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −15.4919 + 17.8885i −0.550830 + 0.636043i
\(792\) 0 0
\(793\) −24.0000 −0.852265
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −3.87298 −0.137188 −0.0685941 0.997645i \(-0.521851\pi\)
−0.0685941 + 0.997645i \(0.521851\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 23.2379 0.820048
\(804\) 0 0
\(805\) −15.0000 + 17.3205i −0.528681 + 0.610468i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 31.3050i 1.10062i −0.834959 0.550312i \(-0.814509\pi\)
0.834959 0.550312i \(-0.185491\pi\)
\(810\) 0 0
\(811\) 36.3731i 1.27723i −0.769526 0.638616i \(-0.779507\pi\)
0.769526 0.638616i \(-0.220493\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −38.7298 −1.35665
\(8