Properties

Label 3024.2.k.k.1889.15
Level $3024$
Weight $2$
Character 3024.1889
Analytic conductor $24.147$
Analytic rank $0$
Dimension $16$
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: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 24 x^{14} + 230 x^{12} - 1052 x^{10} + 2139 x^{8} - 1244 x^{6} + 1134 x^{4} - 104 x^{2} + 169\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{20} \)
Twist minimal: no (minimal twist has level 1512)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1889.15
Root \(-2.62616 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 3024.1889
Dual form 3024.2.k.k.1889.16

$q$-expansion

\(f(q)\) \(=\) \(q+2.86833 q^{5} +(-2.43500 - 1.03478i) q^{7} +O(q^{10})\) \(q+2.86833 q^{5} +(-2.43500 - 1.03478i) q^{7} +3.32538i q^{11} -0.821835i q^{13} +3.52027 q^{17} +5.25233i q^{19} +5.12921i q^{23} +3.22729 q^{25} +1.64271i q^{29} -2.55389i q^{31} +(-6.98438 - 2.96809i) q^{35} -8.93617 q^{37} +3.49721 q^{41} +0.161123 q^{43} -5.34071 q^{47} +(4.85846 + 5.03938i) q^{49} +3.28542i q^{53} +9.53826i q^{55} +4.08295 q^{59} +8.43509i q^{61} -2.35729i q^{65} +11.8780 q^{67} +5.68267i q^{71} +7.86238i q^{73} +(3.44103 - 8.09729i) q^{77} +15.1635 q^{79} -15.6686 q^{83} +10.0973 q^{85} +16.2082 q^{89} +(-0.850418 + 2.00117i) q^{91} +15.0654i q^{95} +12.8678i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 2q^{7} + O(q^{10}) \) \( 16q + 2q^{7} - 12q^{25} - 8q^{37} - 8q^{43} + 2q^{49} + 28q^{67} + 44q^{79} + 16q^{85} - 18q^{91} + O(q^{100}) \)

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
\(5\) 2.86833 1.28275 0.641377 0.767226i \(-0.278363\pi\)
0.641377 + 0.767226i \(0.278363\pi\)
\(6\) 0 0
\(7\) −2.43500 1.03478i −0.920344 0.391110i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.32538i 1.00264i 0.865262 + 0.501319i \(0.167152\pi\)
−0.865262 + 0.501319i \(0.832848\pi\)
\(12\) 0 0
\(13\) 0.821835i 0.227936i −0.993484 0.113968i \(-0.963644\pi\)
0.993484 0.113968i \(-0.0363561\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.52027 0.853792 0.426896 0.904301i \(-0.359607\pi\)
0.426896 + 0.904301i \(0.359607\pi\)
\(18\) 0 0
\(19\) 5.25233i 1.20497i 0.798132 + 0.602483i \(0.205822\pi\)
−0.798132 + 0.602483i \(0.794178\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 5.12921i 1.06951i 0.845006 + 0.534757i \(0.179597\pi\)
−0.845006 + 0.534757i \(0.820403\pi\)
\(24\) 0 0
\(25\) 3.22729 0.645459
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1.64271i 0.305044i 0.988300 + 0.152522i \(0.0487394\pi\)
−0.988300 + 0.152522i \(0.951261\pi\)
\(30\) 0 0
\(31\) 2.55389i 0.458691i −0.973345 0.229346i \(-0.926341\pi\)
0.973345 0.229346i \(-0.0736586\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −6.98438 2.96809i −1.18058 0.501698i
\(36\) 0 0
\(37\) −8.93617 −1.46910 −0.734549 0.678556i \(-0.762606\pi\)
−0.734549 + 0.678556i \(0.762606\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 3.49721 0.546172 0.273086 0.961990i \(-0.411956\pi\)
0.273086 + 0.961990i \(0.411956\pi\)
\(42\) 0 0
\(43\) 0.161123 0.0245710 0.0122855 0.999925i \(-0.496089\pi\)
0.0122855 + 0.999925i \(0.496089\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −5.34071 −0.779022 −0.389511 0.921022i \(-0.627356\pi\)
−0.389511 + 0.921022i \(0.627356\pi\)
\(48\) 0 0
\(49\) 4.85846 + 5.03938i 0.694066 + 0.719912i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 3.28542i 0.451287i 0.974210 + 0.225644i \(0.0724485\pi\)
−0.974210 + 0.225644i \(0.927552\pi\)
\(54\) 0 0
\(55\) 9.53826i 1.28614i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 4.08295 0.531554 0.265777 0.964034i \(-0.414371\pi\)
0.265777 + 0.964034i \(0.414371\pi\)
\(60\) 0 0
\(61\) 8.43509i 1.08000i 0.841664 + 0.540001i \(0.181576\pi\)
−0.841664 + 0.540001i \(0.818424\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 2.35729i 0.292386i
\(66\) 0 0
\(67\) 11.8780 1.45113 0.725567 0.688151i \(-0.241578\pi\)
0.725567 + 0.688151i \(0.241578\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 5.68267i 0.674408i 0.941432 + 0.337204i \(0.109481\pi\)
−0.941432 + 0.337204i \(0.890519\pi\)
\(72\) 0 0
\(73\) 7.86238i 0.920223i 0.887861 + 0.460111i \(0.152190\pi\)
−0.887861 + 0.460111i \(0.847810\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.44103 8.09729i 0.392142 0.922772i
\(78\) 0 0
\(79\) 15.1635 1.70602 0.853012 0.521892i \(-0.174774\pi\)
0.853012 + 0.521892i \(0.174774\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −15.6686 −1.71985 −0.859926 0.510419i \(-0.829490\pi\)
−0.859926 + 0.510419i \(0.829490\pi\)
\(84\) 0 0
\(85\) 10.0973 1.09521
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 16.2082 1.71807 0.859033 0.511921i \(-0.171066\pi\)
0.859033 + 0.511921i \(0.171066\pi\)
\(90\) 0 0
\(91\) −0.850418 + 2.00117i −0.0891481 + 0.209780i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 15.0654i 1.54568i
\(96\) 0 0
\(97\) 12.8678i 1.30653i 0.757129 + 0.653266i \(0.226602\pi\)
−0.757129 + 0.653266i \(0.773398\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −3.40793 −0.339102 −0.169551 0.985521i \(-0.554232\pi\)
−0.169551 + 0.985521i \(0.554232\pi\)
\(102\) 0 0
\(103\) 0.484326i 0.0477220i 0.999715 + 0.0238610i \(0.00759592\pi\)
−0.999715 + 0.0238610i \(0.992404\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 17.3827i 1.68045i −0.542238 0.840225i \(-0.682423\pi\)
0.542238 0.840225i \(-0.317577\pi\)
\(108\) 0 0
\(109\) 1.13804 0.109005 0.0545023 0.998514i \(-0.482643\pi\)
0.0545023 + 0.998514i \(0.482643\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 16.1946i 1.52346i −0.647895 0.761729i \(-0.724351\pi\)
0.647895 0.761729i \(-0.275649\pi\)
\(114\) 0 0
\(115\) 14.7122i 1.37192i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −8.57187 3.64271i −0.785782 0.333927i
\(120\) 0 0
\(121\) −0.0581269 −0.00528426
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −5.08470 −0.454790
\(126\) 0 0
\(127\) −16.0335 −1.42274 −0.711370 0.702818i \(-0.751925\pi\)
−0.711370 + 0.702818i \(0.751925\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −7.27801 −0.635883 −0.317941 0.948110i \(-0.602992\pi\)
−0.317941 + 0.948110i \(0.602992\pi\)
\(132\) 0 0
\(133\) 5.43500 12.7894i 0.471274 1.10898i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 14.2296i 1.21572i 0.794044 + 0.607860i \(0.207972\pi\)
−0.794044 + 0.607860i \(0.792028\pi\)
\(138\) 0 0
\(139\) 21.7288i 1.84301i 0.388361 + 0.921507i \(0.373042\pi\)
−0.388361 + 0.921507i \(0.626958\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 2.73291 0.228537
\(144\) 0 0
\(145\) 4.71183i 0.391296i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 19.8723i 1.62801i −0.580861 0.814003i \(-0.697284\pi\)
0.580861 0.814003i \(-0.302716\pi\)
\(150\) 0 0
\(151\) 0.870002 0.0707998 0.0353999 0.999373i \(-0.488730\pi\)
0.0353999 + 0.999373i \(0.488730\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 7.32538i 0.588388i
\(156\) 0 0
\(157\) 3.47414i 0.277266i 0.990344 + 0.138633i \(0.0442709\pi\)
−0.990344 + 0.138633i \(0.955729\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 5.30760 12.4896i 0.418298 0.984321i
\(162\) 0 0
\(163\) 2.13804 0.167464 0.0837321 0.996488i \(-0.473316\pi\)
0.0837321 + 0.996488i \(0.473316\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 19.7515 1.52842 0.764210 0.644967i \(-0.223129\pi\)
0.764210 + 0.644967i \(0.223129\pi\)
\(168\) 0 0
\(169\) 12.3246 0.948045
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 14.5084 1.10305 0.551525 0.834158i \(-0.314046\pi\)
0.551525 + 0.834158i \(0.314046\pi\)
\(174\) 0 0
\(175\) −7.85846 3.33954i −0.594044 0.252445i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 6.81188i 0.509143i −0.967054 0.254572i \(-0.918066\pi\)
0.967054 0.254572i \(-0.0819345\pi\)
\(180\) 0 0
\(181\) 23.6415i 1.75726i 0.477502 + 0.878631i \(0.341542\pi\)
−0.477502 + 0.878631i \(0.658458\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −25.6319 −1.88449
\(186\) 0 0
\(187\) 11.7062i 0.856045i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 7.06538i 0.511233i −0.966778 0.255617i \(-0.917722\pi\)
0.966778 0.255617i \(-0.0822784\pi\)
\(192\) 0 0
\(193\) −5.71692 −0.411513 −0.205757 0.978603i \(-0.565965\pi\)
−0.205757 + 0.978603i \(0.565965\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 24.7815i 1.76561i 0.469740 + 0.882805i \(0.344348\pi\)
−0.469740 + 0.882805i \(0.655652\pi\)
\(198\) 0 0
\(199\) 5.15001i 0.365075i −0.983199 0.182537i \(-0.941569\pi\)
0.983199 0.182537i \(-0.0584311\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 1.69984 4.00000i 0.119306 0.280745i
\(204\) 0 0
\(205\) 10.0311 0.700604
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −17.4660 −1.20815
\(210\) 0 0
\(211\) 12.0392 0.828810 0.414405 0.910092i \(-0.363990\pi\)
0.414405 + 0.910092i \(0.363990\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0.462153 0.0315186
\(216\) 0 0
\(217\) −2.64271 + 6.21871i −0.179399 + 0.422154i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 2.89308i 0.194610i
\(222\) 0 0
\(223\) 22.7676i 1.52463i −0.647206 0.762315i \(-0.724063\pi\)
0.647206 0.762315i \(-0.275937\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 25.3297 1.68119 0.840596 0.541663i \(-0.182205\pi\)
0.840596 + 0.541663i \(0.182205\pi\)
\(228\) 0 0
\(229\) 2.49545i 0.164904i −0.996595 0.0824520i \(-0.973725\pi\)
0.996595 0.0824520i \(-0.0262751\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 16.0000i 1.04819i 0.851658 + 0.524097i \(0.175597\pi\)
−0.851658 + 0.524097i \(0.824403\pi\)
\(234\) 0 0
\(235\) −15.3189 −0.999294
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 8.66813i 0.560695i 0.959899 + 0.280347i \(0.0904497\pi\)
−0.959899 + 0.280347i \(0.909550\pi\)
\(240\) 0 0
\(241\) 30.2907i 1.95119i −0.219571 0.975597i \(-0.570466\pi\)
0.219571 0.975597i \(-0.429534\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 13.9356 + 14.4546i 0.890316 + 0.923470i
\(246\) 0 0
\(247\) 4.31654 0.274655
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −5.78279 −0.365006 −0.182503 0.983205i \(-0.558420\pi\)
−0.182503 + 0.983205i \(0.558420\pi\)
\(252\) 0 0
\(253\) −17.0565 −1.07234
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −17.3454 −1.08197 −0.540987 0.841031i \(-0.681949\pi\)
−0.540987 + 0.841031i \(0.681949\pi\)
\(258\) 0 0
\(259\) 21.7596 + 9.24697i 1.35208 + 0.574579i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 8.80696i 0.543061i 0.962430 + 0.271530i \(0.0875297\pi\)
−0.962430 + 0.271530i \(0.912470\pi\)
\(264\) 0 0
\(265\) 9.42365i 0.578890i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −28.4310 −1.73347 −0.866733 0.498772i \(-0.833785\pi\)
−0.866733 + 0.498772i \(0.833785\pi\)
\(270\) 0 0
\(271\) 2.47554i 0.150378i −0.997169 0.0751892i \(-0.976044\pi\)
0.997169 0.0751892i \(-0.0239561\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 10.7320i 0.647162i
\(276\) 0 0
\(277\) −5.93383 −0.356529 −0.178265 0.983983i \(-0.557048\pi\)
−0.178265 + 0.983983i \(0.557048\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 9.96496i 0.594459i 0.954806 + 0.297230i \(0.0960627\pi\)
−0.954806 + 0.297230i \(0.903937\pi\)
\(282\) 0 0
\(283\) 13.6796i 0.813170i −0.913613 0.406585i \(-0.866719\pi\)
0.913613 0.406585i \(-0.133281\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −8.51570 3.61884i −0.502666 0.213613i
\(288\) 0 0
\(289\) −4.60767 −0.271039
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −1.47062 −0.0859148 −0.0429574 0.999077i \(-0.513678\pi\)
−0.0429574 + 0.999077i \(0.513678\pi\)
\(294\) 0 0
\(295\) 11.7112 0.681854
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 4.21536 0.243781
\(300\) 0 0
\(301\) −0.392335 0.166727i −0.0226138 0.00960997i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 24.1946i 1.38538i
\(306\) 0 0
\(307\) 31.6829i 1.80824i 0.427277 + 0.904121i \(0.359473\pi\)
−0.427277 + 0.904121i \(0.640527\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 19.1971 1.08857 0.544284 0.838901i \(-0.316801\pi\)
0.544284 + 0.838901i \(0.316801\pi\)
\(312\) 0 0
\(313\) 13.4406i 0.759705i 0.925047 + 0.379852i \(0.124025\pi\)
−0.925047 + 0.379852i \(0.875975\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 29.5151i 1.65773i −0.559448 0.828865i \(-0.688987\pi\)
0.559448 0.828865i \(-0.311013\pi\)
\(318\) 0 0
\(319\) −5.46263 −0.305848
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 18.4896i 1.02879i
\(324\) 0 0
\(325\) 2.65230i 0.147123i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 13.0046 + 5.52646i 0.716968 + 0.304683i
\(330\) 0 0
\(331\) −0.547756 −0.0301074 −0.0150537 0.999887i \(-0.504792\pi\)
−0.0150537 + 0.999887i \(0.504792\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 34.0701 1.86145
\(336\) 0 0
\(337\) −17.1946 −0.936649 −0.468324 0.883557i \(-0.655142\pi\)
−0.468324 + 0.883557i \(0.655142\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 8.49263 0.459902
\(342\) 0 0
\(343\) −6.61571 17.2983i −0.357215 0.934022i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 22.6443i 1.21561i 0.794087 + 0.607804i \(0.207949\pi\)
−0.794087 + 0.607804i \(0.792051\pi\)
\(348\) 0 0
\(349\) 24.2288i 1.29694i 0.761241 + 0.648469i \(0.224590\pi\)
−0.761241 + 0.648469i \(0.775410\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −16.0497 −0.854240 −0.427120 0.904195i \(-0.640472\pi\)
−0.427120 + 0.904195i \(0.640472\pi\)
\(354\) 0 0
\(355\) 16.2997i 0.865100i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 26.8119i 1.41508i −0.706675 0.707538i \(-0.749806\pi\)
0.706675 0.707538i \(-0.250194\pi\)
\(360\) 0 0
\(361\) −8.58692 −0.451943
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 22.5519i 1.18042i
\(366\) 0 0
\(367\) 22.9643i 1.19872i −0.800478 0.599362i \(-0.795421\pi\)
0.800478 0.599362i \(-0.204579\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 3.39969 8.00000i 0.176503 0.415339i
\(372\) 0 0
\(373\) 17.6043 0.911516 0.455758 0.890104i \(-0.349368\pi\)
0.455758 + 0.890104i \(0.349368\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 1.35004 0.0695304
\(378\) 0 0
\(379\) 14.7250 0.756371 0.378185 0.925730i \(-0.376548\pi\)
0.378185 + 0.925730i \(0.376548\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −36.8030 −1.88055 −0.940273 0.340421i \(-0.889430\pi\)
−0.940273 + 0.340421i \(0.889430\pi\)
\(384\) 0 0
\(385\) 9.87000 23.2257i 0.503022 1.18369i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 4.32225i 0.219147i 0.993979 + 0.109573i \(0.0349484\pi\)
−0.993979 + 0.109573i \(0.965052\pi\)
\(390\) 0 0
\(391\) 18.0562i 0.913143i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 43.4938 2.18841
\(396\) 0 0
\(397\) 3.84549i 0.192999i 0.995333 + 0.0964997i \(0.0307647\pi\)
−0.995333 + 0.0964997i \(0.969235\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 3.33655i 0.166619i 0.996524 + 0.0833097i \(0.0265491\pi\)
−0.996524 + 0.0833097i \(0.973451\pi\)
\(402\) 0 0
\(403\) −2.09887 −0.104552
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 29.7161i 1.47297i
\(408\) 0 0
\(409\) 5.99402i 0.296385i −0.988959 0.148193i \(-0.952654\pi\)
0.988959 0.148193i \(-0.0473456\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −9.94198 4.22495i −0.489213 0.207896i
\(414\) 0 0
\(415\) −44.9426 −2.20615
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 31.1326 1.52093 0.760463 0.649382i \(-0.224972\pi\)
0.760463 + 0.649382i \(0.224972\pi\)
\(420\) 0 0
\(421\) 2.93617 0.143100 0.0715501 0.997437i \(-0.477205\pi\)
0.0715501 + 0.997437i \(0.477205\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 11.3610 0.551087
\(426\) 0 0
\(427\) 8.72846 20.5395i 0.422400 0.993974i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 23.0654i 1.11102i −0.831510 0.555510i \(-0.812523\pi\)
0.831510 0.555510i \(-0.187477\pi\)
\(432\) 0 0
\(433\) 27.8831i 1.33998i −0.742371 0.669989i \(-0.766299\pi\)
0.742371 0.669989i \(-0.233701\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −26.9403 −1.28873
\(438\) 0 0
\(439\) 25.7312i 1.22808i −0.789274 0.614041i \(-0.789543\pi\)
0.789274 0.614041i \(-0.210457\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 6.73845i 0.320154i 0.987105 + 0.160077i \(0.0511742\pi\)
−0.987105 + 0.160077i \(0.948826\pi\)
\(444\) 0 0
\(445\) 46.4904 2.20386
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 38.7654i 1.82945i −0.404073 0.914727i \(-0.632406\pi\)
0.404073 0.914727i \(-0.367594\pi\)
\(450\) 0 0
\(451\) 11.6295i 0.547613i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −2.43928 + 5.74000i −0.114355 + 0.269096i
\(456\) 0 0
\(457\) 31.7423 1.48484 0.742422 0.669932i \(-0.233677\pi\)
0.742422 + 0.669932i \(0.233677\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 3.14740 0.146589 0.0732945 0.997310i \(-0.476649\pi\)
0.0732945 + 0.997310i \(0.476649\pi\)
\(462\) 0 0
\(463\) −5.46833 −0.254135 −0.127067 0.991894i \(-0.540556\pi\)
−0.127067 + 0.991894i \(0.540556\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −20.5471 −0.950809 −0.475404 0.879767i \(-0.657698\pi\)
−0.475404 + 0.879767i \(0.657698\pi\)
\(468\) 0 0
\(469\) −28.9231 12.2912i −1.33554 0.567553i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0.535795i 0.0246359i
\(474\) 0 0
\(475\) 16.9508i 0.777756i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 13.0645 0.596933 0.298466 0.954420i \(-0.403525\pi\)
0.298466 + 0.954420i \(0.403525\pi\)
\(480\) 0 0
\(481\) 7.34406i 0.334860i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 36.9092i 1.67596i
\(486\) 0 0
\(487\) −1.74571 −0.0791055 −0.0395527 0.999217i \(-0.512593\pi\)
−0.0395527 + 0.999217i \(0.512593\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 4.51350i 0.203691i −0.994800 0.101846i \(-0.967525\pi\)
0.994800 0.101846i \(-0.0324748\pi\)
\(492\) 0 0
\(493\) 5.78279i 0.260444i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 5.88031 13.8373i 0.263768 0.620688i
\(498\) 0 0
\(499\) −14.7320 −0.659493 −0.329747 0.944069i \(-0.606963\pi\)
−0.329747 + 0.944069i \(0.606963\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −8.64812 −0.385600 −0.192800 0.981238i \(-0.561757\pi\)
−0.192800 + 0.981238i \(0.561757\pi\)
\(504\) 0 0
\(505\) −9.77505 −0.434984
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −29.8666 −1.32381 −0.661906 0.749587i \(-0.730252\pi\)
−0.661906 + 0.749587i \(0.730252\pi\)
\(510\) 0 0
\(511\) 8.13584 19.1449i 0.359908 0.846921i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 1.38920i 0.0612157i
\(516\) 0 0
\(517\) 17.7599i 0.781078i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −12.5756 −0.550946 −0.275473 0.961309i \(-0.588834\pi\)
−0.275473 + 0.961309i \(0.588834\pi\)
\(522\) 0 0
\(523\) 24.9477i 1.09089i 0.838148 + 0.545443i \(0.183639\pi\)
−0.838148 + 0.545443i \(0.816361\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 8.99038i 0.391627i
\(528\) 0 0
\(529\) −3.30878 −0.143860
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 2.87413i 0.124492i
\(534\) 0 0
\(535\) 49.8593i 2.15560i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −16.7578 + 16.1562i −0.721811 + 0.695897i
\(540\) 0 0
\(541\) −12.5438 −0.539302 −0.269651 0.962958i \(-0.586908\pi\)
−0.269651 + 0.962958i \(0.586908\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 3.26427 0.139826
\(546\) 0 0
\(547\) 8.87000 0.379254 0.189627 0.981856i \(-0.439272\pi\)
0.189627 + 0.981856i \(0.439272\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −8.62805 −0.367567
\(552\) 0 0
\(553\) −36.9231 15.6908i −1.57013 0.667243i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 32.7815i 1.38900i −0.719494 0.694499i \(-0.755626\pi\)
0.719494 0.694499i \(-0.244374\pi\)
\(558\) 0 0
\(559\) 0.132416i 0.00560062i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 3.97060 0.167341 0.0836704 0.996493i \(-0.473336\pi\)
0.0836704 + 0.996493i \(0.473336\pi\)
\(564\) 0 0
\(565\) 46.4514i 1.95422i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 20.0669i 0.841250i 0.907235 + 0.420625i \(0.138189\pi\)
−0.907235 + 0.420625i \(0.861811\pi\)
\(570\) 0 0
\(571\) 22.9323 0.959685 0.479842 0.877355i \(-0.340694\pi\)
0.479842 + 0.877355i \(0.340694\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 16.5535i 0.690327i
\(576\) 0 0
\(577\) 27.5760i 1.14801i −0.818853 0.574003i \(-0.805390\pi\)
0.818853 0.574003i \(-0.194610\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 38.1530 + 16.2135i 1.58286 + 0.672651i
\(582\) 0 0
\(583\) −10.9253 −0.452478
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −14.6684 −0.605428 −0.302714 0.953081i \(-0.597893\pi\)
−0.302714 + 0.953081i \(0.597893\pi\)
\(588\) 0 0
\(589\) 13.4138 0.552708
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 27.7560 1.13980 0.569900 0.821714i \(-0.306982\pi\)
0.569900 + 0.821714i \(0.306982\pi\)
\(594\) 0 0
\(595\) −24.5869 10.4485i −1.00797 0.428346i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 31.2600i 1.27725i −0.769519 0.638624i \(-0.779504\pi\)
0.769519 0.638624i \(-0.220496\pi\)
\(600\) 0 0
\(601\) 26.8665i 1.09591i 0.836508 + 0.547955i \(0.184593\pi\)
−0.836508 + 0.547955i \(0.815407\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −0.166727 −0.00677841
\(606\) 0 0
\(607\) 22.6331i 0.918648i −0.888269 0.459324i \(-0.848092\pi\)
0.888269 0.459324i \(-0.151908\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 4.38918i 0.177567i
\(612\) 0 0
\(613\) −42.8518 −1.73077 −0.865384 0.501109i \(-0.832926\pi\)
−0.865384 + 0.501109i \(0.832926\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 10.9442i 0.440597i −0.975432 0.220299i \(-0.929297\pi\)
0.975432 0.220299i \(-0.0707032\pi\)
\(618\) 0 0
\(619\) 14.4430i 0.580515i −0.956949 0.290257i \(-0.906259\pi\)
0.956949 0.290257i \(-0.0937409\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −39.4670 16.7719i −1.58121 0.671953i
\(624\) 0 0
\(625\) −30.7210 −1.22884
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −31.4578 −1.25430
\(630\) 0 0
\(631\) −14.3326 −0.570573 −0.285286 0.958442i \(-0.592089\pi\)
−0.285286 + 0.958442i \(0.592089\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −45.9892 −1.82503
\(636\) 0 0
\(637\) 4.14154 3.99285i 0.164094 0.158203i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 31.8023i 1.25611i −0.778168 0.628057i \(-0.783851\pi\)
0.778168 0.628057i \(-0.216149\pi\)
\(642\) 0 0
\(643\) 0.345751i 0.0136351i 0.999977 + 0.00681755i \(0.00217011\pi\)
−0.999977 + 0.00681755i \(0.997830\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −8.85274 −0.348037 −0.174019 0.984742i \(-0.555675\pi\)
−0.174019 + 0.984742i \(0.555675\pi\)
\(648\) 0 0
\(649\) 13.5773i 0.532957i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 40.9603i 1.60290i −0.598062 0.801450i \(-0.704062\pi\)
0.598062 0.801450i \(-0.295938\pi\)
\(654\) 0 0
\(655\) −20.8757 −0.815681
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 5.38920i 0.209934i −0.994476 0.104967i \(-0.966526\pi\)
0.994476 0.104967i \(-0.0334736\pi\)
\(660\) 0 0
\(661\) 25.9703i 1.01013i −0.863083 0.505063i \(-0.831469\pi\)
0.863083 0.505063i \(-0.168531\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 15.5894 36.6842i 0.604529 1.42255i
\(666\) 0 0
\(667\) −8.42580 −0.326248
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −28.0499 −1.08285
\(672\) 0 0
\(673\) −28.0553 −1.08145 −0.540725 0.841199i \(-0.681850\pi\)
−0.540725 + 0.841199i \(0.681850\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 31.5977 1.21440 0.607200 0.794549i \(-0.292293\pi\)
0.607200 + 0.794549i \(0.292293\pi\)
\(678\) 0 0
\(679\) 13.3154 31.3332i 0.510998 1.20246i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 8.55679i 0.327417i 0.986509 + 0.163708i \(0.0523456\pi\)
−0.986509 + 0.163708i \(0.947654\pi\)
\(684\) 0 0
\(685\) 40.8152i 1.55947i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 2.70007 0.102865
\(690\) 0 0
\(691\) 19.5748i 0.744660i −0.928100 0.372330i \(-0.878559\pi\)
0.928100 0.372330i \(-0.121441\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 62.3254i 2.36414i
\(696\) 0 0
\(697\) 12.3111 0.466317
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 31.9810i 1.20791i −0.797019 0.603954i \(-0.793591\pi\)
0.797019 0.603954i \(-0.206409\pi\)
\(702\) 0 0
\(703\) 46.9357i 1.77021i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 8.29831 + 3.52646i 0.312090 + 0.132626i
\(708\) 0 0
\(709\) −2.07029 −0.0777515 −0.0388757 0.999244i \(-0.512378\pi\)
−0.0388757 + 0.999244i \(0.512378\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 13.0994 0.490577
\(714\) 0 0
\(715\) 7.83888 0.293157
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 33.4828 1.24870 0.624350 0.781145i \(-0.285364\pi\)
0.624350 + 0.781145i \(0.285364\pi\)
\(720\) 0 0
\(721\) 0.501171 1.17933i 0.0186646 0.0439207i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 5.30150i 0.196893i
\(726\) 0 0
\(727\) 12.7754i 0.473814i 0.971532 + 0.236907i \(0.0761336\pi\)
−0.971532 + 0.236907i \(0.923866\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0.567197 0.0209785
\(732\) 0 0
\(733\) 30.2462i 1.11717i −0.829448 0.558584i \(-0.811345\pi\)
0.829448 0.558584i \(-0.188655\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 39.4990i 1.45496i
\(738\) 0 0
\(739\) 9.10692 0.335003 0.167502 0.985872i \(-0.446430\pi\)
0.167502 + 0.985872i \(0.446430\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 36.9551i 1.35575i −0.735177 0.677876i \(-0.762901\pi\)
0.735177 0.677876i \(-0.237099\pi\)
\(744\) 0 0
\(745\) 57.0004i 2.08833i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −17.9873 + 42.3269i −0.657241 + 1.54659i
\(750\) 0 0
\(751\) −6.89879 −0.251740 −0.125870 0.992047i \(-0.540172\pi\)
−0.125870 + 0.992047i \(0.540172\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 2.49545 0.0908187
\(756\) 0 0
\(757\) −12.8861 −0.468353 −0.234176 0.972194i \(-0.575239\pi\)
−0.234176 + 0.972194i \(0.575239\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 32.8078 1.18928 0.594641 0.803991i \(-0.297294\pi\)
0.594641 + 0.803991i \(0.297294\pi\)
\(762\) 0 0
\(763\) −2.77113 1.17762i −0.100322 0.0426328i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 3.35551i 0.121160i
\(768\) 0 0
\(769\) 18.9096i 0.681899i 0.940082 + 0.340949i \(0.110749\pi\)
−0.940082 + 0.340949i \(0.889251\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 15.6195 0.561793 0.280896 0.959738i \(-0.409368\pi\)
0.280896 + 0.959738i \(0.409368\pi\)
\(774\) 0 0
\(775\) 8.24214i 0.296066i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 18.3685i 0.658118i
\(780\) 0 0
\(781\) −18.8970 −0.676188
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 9.96496i 0.355665i
\(786\) 0 0
\(787\) 14.4471i 0.514983i 0.966281 + 0.257492i \(0.0828960\pi\)
−0.966281 + 0.257492i \(0.917104\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −16.7578 + 39.4338i −0.595840 + 1.40211i
\(792\) 0 0
\(793\) 6.93225 0.246172
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 46.1246 1.63382 0.816909 0.576767i \(-0.195686\pi\)
0.816909 + 0.576767i \(0.195686\pi\)
\(798\) 0 0
\(799\) −18.8008 −0.665123
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −26.1454 −0.922651
\(804\) 0