Properties

Label 320.6.d.c.161.12
Level 320
Weight 6
Character 320.161
Analytic conductor 51.323
Analytic rank 0
Dimension 12
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 320.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(51.3228223402\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 343 x^{10} - 696 x^{9} + 44406 x^{8} + 179640 x^{7} - 2401691 x^{6} - 15554592 x^{5} + 26901210 x^{4} + 434775816 x^{3} + 1271335685 x^{2} + 1475231592 x + 653157349\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{30} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 161.12
Root \(10.0457 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 320.161
Dual form 320.6.d.c.161.1

$q$-expansion

\(f(q)\) \(=\) \(q+30.0196i q^{3} -25.0000i q^{5} +45.4594 q^{7} -658.179 q^{9} +O(q^{10})\) \(q+30.0196i q^{3} -25.0000i q^{5} +45.4594 q^{7} -658.179 q^{9} +142.717i q^{11} +897.074i q^{13} +750.491 q^{15} +11.4886 q^{17} -1818.86i q^{19} +1364.67i q^{21} -3875.05 q^{23} -625.000 q^{25} -12463.5i q^{27} -3463.55i q^{29} -1550.37 q^{31} -4284.30 q^{33} -1136.48i q^{35} +9610.99i q^{37} -26929.8 q^{39} -14807.5 q^{41} -6324.64i q^{43} +16454.5i q^{45} +29003.7 q^{47} -14740.4 q^{49} +344.883i q^{51} -32583.1i q^{53} +3567.91 q^{55} +54601.4 q^{57} -652.257i q^{59} +4715.33i q^{61} -29920.4 q^{63} +22426.9 q^{65} -46314.1i q^{67} -116327. i q^{69} -52668.5 q^{71} +69352.8 q^{73} -18762.3i q^{75} +6487.80i q^{77} +33288.8 q^{79} +214213. q^{81} +55352.9i q^{83} -287.214i q^{85} +103975. q^{87} +33235.3 q^{89} +40780.4i q^{91} -46541.5i q^{93} -45471.4 q^{95} -82468.3 q^{97} -93933.0i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 268q^{7} - 428q^{9} + O(q^{10}) \) \( 12q - 268q^{7} - 428q^{9} + 900q^{15} + 2400q^{17} - 8108q^{23} - 7500q^{25} - 7976q^{31} - 20776q^{33} - 40984q^{39} - 56408q^{41} - 21172q^{47} - 5540q^{49} - 18400q^{55} + 39992q^{57} - 179516q^{63} + 44000q^{65} - 367704q^{71} + 58736q^{73} - 26192q^{79} + 411692q^{81} - 183200q^{87} + 87672q^{89} - 121000q^{95} - 172336q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/320\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(257\) \(261\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) 30.0196i 1.92576i 0.269928 + 0.962880i \(0.413000\pi\)
−0.269928 + 0.962880i \(0.587000\pi\)
\(4\) 0 0
\(5\) − 25.0000i − 0.447214i
\(6\) 0 0
\(7\) 45.4594 0.350654 0.175327 0.984510i \(-0.443902\pi\)
0.175327 + 0.984510i \(0.443902\pi\)
\(8\) 0 0
\(9\) −658.179 −2.70855
\(10\) 0 0
\(11\) 142.717i 0.355625i 0.984064 + 0.177813i \(0.0569021\pi\)
−0.984064 + 0.177813i \(0.943098\pi\)
\(12\) 0 0
\(13\) 897.074i 1.47221i 0.676867 + 0.736105i \(0.263337\pi\)
−0.676867 + 0.736105i \(0.736663\pi\)
\(14\) 0 0
\(15\) 750.491 0.861226
\(16\) 0 0
\(17\) 11.4886 0.00964148 0.00482074 0.999988i \(-0.498466\pi\)
0.00482074 + 0.999988i \(0.498466\pi\)
\(18\) 0 0
\(19\) − 1818.86i − 1.15588i −0.816078 0.577942i \(-0.803856\pi\)
0.816078 0.577942i \(-0.196144\pi\)
\(20\) 0 0
\(21\) 1364.67i 0.675275i
\(22\) 0 0
\(23\) −3875.05 −1.52742 −0.763708 0.645562i \(-0.776623\pi\)
−0.763708 + 0.645562i \(0.776623\pi\)
\(24\) 0 0
\(25\) −625.000 −0.200000
\(26\) 0 0
\(27\) − 12463.5i − 3.29027i
\(28\) 0 0
\(29\) − 3463.55i − 0.764763i −0.924004 0.382382i \(-0.875104\pi\)
0.924004 0.382382i \(-0.124896\pi\)
\(30\) 0 0
\(31\) −1550.37 −0.289755 −0.144877 0.989450i \(-0.546279\pi\)
−0.144877 + 0.989450i \(0.546279\pi\)
\(32\) 0 0
\(33\) −4284.30 −0.684849
\(34\) 0 0
\(35\) − 1136.48i − 0.156817i
\(36\) 0 0
\(37\) 9610.99i 1.15415i 0.816690 + 0.577077i \(0.195807\pi\)
−0.816690 + 0.577077i \(0.804193\pi\)
\(38\) 0 0
\(39\) −26929.8 −2.83513
\(40\) 0 0
\(41\) −14807.5 −1.37569 −0.687847 0.725856i \(-0.741444\pi\)
−0.687847 + 0.725856i \(0.741444\pi\)
\(42\) 0 0
\(43\) − 6324.64i − 0.521632i −0.965388 0.260816i \(-0.916008\pi\)
0.965388 0.260816i \(-0.0839916\pi\)
\(44\) 0 0
\(45\) 16454.5i 1.21130i
\(46\) 0 0
\(47\) 29003.7 1.91517 0.957587 0.288145i \(-0.0930384\pi\)
0.957587 + 0.288145i \(0.0930384\pi\)
\(48\) 0 0
\(49\) −14740.4 −0.877042
\(50\) 0 0
\(51\) 344.883i 0.0185672i
\(52\) 0 0
\(53\) − 32583.1i − 1.59332i −0.604429 0.796659i \(-0.706599\pi\)
0.604429 0.796659i \(-0.293401\pi\)
\(54\) 0 0
\(55\) 3567.91 0.159040
\(56\) 0 0
\(57\) 54601.4 2.22596
\(58\) 0 0
\(59\) − 652.257i − 0.0243943i −0.999926 0.0121972i \(-0.996117\pi\)
0.999926 0.0121972i \(-0.00388257\pi\)
\(60\) 0 0
\(61\) 4715.33i 0.162251i 0.996704 + 0.0811255i \(0.0258515\pi\)
−0.996704 + 0.0811255i \(0.974149\pi\)
\(62\) 0 0
\(63\) −29920.4 −0.949765
\(64\) 0 0
\(65\) 22426.9 0.658393
\(66\) 0 0
\(67\) − 46314.1i − 1.26045i −0.776411 0.630226i \(-0.782962\pi\)
0.776411 0.630226i \(-0.217038\pi\)
\(68\) 0 0
\(69\) − 116327.i − 2.94144i
\(70\) 0 0
\(71\) −52668.5 −1.23995 −0.619976 0.784621i \(-0.712858\pi\)
−0.619976 + 0.784621i \(0.712858\pi\)
\(72\) 0 0
\(73\) 69352.8 1.52320 0.761600 0.648048i \(-0.224414\pi\)
0.761600 + 0.648048i \(0.224414\pi\)
\(74\) 0 0
\(75\) − 18762.3i − 0.385152i
\(76\) 0 0
\(77\) 6487.80i 0.124701i
\(78\) 0 0
\(79\) 33288.8 0.600109 0.300054 0.953922i \(-0.402995\pi\)
0.300054 + 0.953922i \(0.402995\pi\)
\(80\) 0 0
\(81\) 214213. 3.62771
\(82\) 0 0
\(83\) 55352.9i 0.881953i 0.897519 + 0.440976i \(0.145368\pi\)
−0.897519 + 0.440976i \(0.854632\pi\)
\(84\) 0 0
\(85\) − 287.214i − 0.00431180i
\(86\) 0 0
\(87\) 103975. 1.47275
\(88\) 0 0
\(89\) 33235.3 0.444758 0.222379 0.974960i \(-0.428618\pi\)
0.222379 + 0.974960i \(0.428618\pi\)
\(90\) 0 0
\(91\) 40780.4i 0.516236i
\(92\) 0 0
\(93\) − 46541.5i − 0.557998i
\(94\) 0 0
\(95\) −45471.4 −0.516927
\(96\) 0 0
\(97\) −82468.3 −0.889934 −0.444967 0.895547i \(-0.646785\pi\)
−0.444967 + 0.895547i \(0.646785\pi\)
\(98\) 0 0
\(99\) − 93933.0i − 0.963231i
\(100\) 0 0
\(101\) 36413.2i 0.355185i 0.984104 + 0.177593i \(0.0568310\pi\)
−0.984104 + 0.177593i \(0.943169\pi\)
\(102\) 0 0
\(103\) 83074.5 0.771569 0.385785 0.922589i \(-0.373931\pi\)
0.385785 + 0.922589i \(0.373931\pi\)
\(104\) 0 0
\(105\) 34116.8 0.301992
\(106\) 0 0
\(107\) − 30476.7i − 0.257340i −0.991687 0.128670i \(-0.958929\pi\)
0.991687 0.128670i \(-0.0410709\pi\)
\(108\) 0 0
\(109\) 200282.i 1.61464i 0.590117 + 0.807318i \(0.299082\pi\)
−0.590117 + 0.807318i \(0.700918\pi\)
\(110\) 0 0
\(111\) −288519. −2.22263
\(112\) 0 0
\(113\) −24491.9 −0.180437 −0.0902186 0.995922i \(-0.528757\pi\)
−0.0902186 + 0.995922i \(0.528757\pi\)
\(114\) 0 0
\(115\) 96876.1i 0.683081i
\(116\) 0 0
\(117\) − 590435.i − 3.98756i
\(118\) 0 0
\(119\) 522.263 0.00338082
\(120\) 0 0
\(121\) 140683. 0.873531
\(122\) 0 0
\(123\) − 444515.i − 2.64926i
\(124\) 0 0
\(125\) 15625.0i 0.0894427i
\(126\) 0 0
\(127\) −133675. −0.735431 −0.367716 0.929938i \(-0.619860\pi\)
−0.367716 + 0.929938i \(0.619860\pi\)
\(128\) 0 0
\(129\) 189863. 1.00454
\(130\) 0 0
\(131\) − 227954.i − 1.16057i −0.814415 0.580283i \(-0.802942\pi\)
0.814415 0.580283i \(-0.197058\pi\)
\(132\) 0 0
\(133\) − 82684.1i − 0.405315i
\(134\) 0 0
\(135\) −311588. −1.47145
\(136\) 0 0
\(137\) −354970. −1.61581 −0.807905 0.589313i \(-0.799398\pi\)
−0.807905 + 0.589313i \(0.799398\pi\)
\(138\) 0 0
\(139\) − 414105.i − 1.81792i −0.416887 0.908958i \(-0.636879\pi\)
0.416887 0.908958i \(-0.363121\pi\)
\(140\) 0 0
\(141\) 870679.i 3.68817i
\(142\) 0 0
\(143\) −128027. −0.523555
\(144\) 0 0
\(145\) −86588.9 −0.342013
\(146\) 0 0
\(147\) − 442503.i − 1.68897i
\(148\) 0 0
\(149\) − 233661.i − 0.862224i −0.902298 0.431112i \(-0.858121\pi\)
0.902298 0.431112i \(-0.141879\pi\)
\(150\) 0 0
\(151\) −408202. −1.45691 −0.728455 0.685094i \(-0.759761\pi\)
−0.728455 + 0.685094i \(0.759761\pi\)
\(152\) 0 0
\(153\) −7561.53 −0.0261145
\(154\) 0 0
\(155\) 38759.2i 0.129582i
\(156\) 0 0
\(157\) − 342377.i − 1.10855i −0.832334 0.554275i \(-0.812996\pi\)
0.832334 0.554275i \(-0.187004\pi\)
\(158\) 0 0
\(159\) 978132. 3.06835
\(160\) 0 0
\(161\) −176157. −0.535594
\(162\) 0 0
\(163\) 374906.i 1.10523i 0.833436 + 0.552616i \(0.186370\pi\)
−0.833436 + 0.552616i \(0.813630\pi\)
\(164\) 0 0
\(165\) 107107.i 0.306274i
\(166\) 0 0
\(167\) 75465.0 0.209389 0.104695 0.994504i \(-0.466613\pi\)
0.104695 + 0.994504i \(0.466613\pi\)
\(168\) 0 0
\(169\) −433449. −1.16740
\(170\) 0 0
\(171\) 1.19713e6i 3.13078i
\(172\) 0 0
\(173\) 372218.i 0.945545i 0.881185 + 0.472772i \(0.156747\pi\)
−0.881185 + 0.472772i \(0.843253\pi\)
\(174\) 0 0
\(175\) −28412.1 −0.0701307
\(176\) 0 0
\(177\) 19580.5 0.0469776
\(178\) 0 0
\(179\) − 235406.i − 0.549141i −0.961567 0.274571i \(-0.911464\pi\)
0.961567 0.274571i \(-0.0885357\pi\)
\(180\) 0 0
\(181\) 625543.i 1.41926i 0.704577 + 0.709628i \(0.251137\pi\)
−0.704577 + 0.709628i \(0.748863\pi\)
\(182\) 0 0
\(183\) −141553. −0.312457
\(184\) 0 0
\(185\) 240275. 0.516154
\(186\) 0 0
\(187\) 1639.61i 0.00342875i
\(188\) 0 0
\(189\) − 566584.i − 1.15374i
\(190\) 0 0
\(191\) −51939.4 −0.103018 −0.0515090 0.998673i \(-0.516403\pi\)
−0.0515090 + 0.998673i \(0.516403\pi\)
\(192\) 0 0
\(193\) −355320. −0.686635 −0.343318 0.939219i \(-0.611551\pi\)
−0.343318 + 0.939219i \(0.611551\pi\)
\(194\) 0 0
\(195\) 673246.i 1.26791i
\(196\) 0 0
\(197\) − 125802.i − 0.230953i −0.993310 0.115476i \(-0.963161\pi\)
0.993310 0.115476i \(-0.0368394\pi\)
\(198\) 0 0
\(199\) −593616. −1.06261 −0.531304 0.847181i \(-0.678298\pi\)
−0.531304 + 0.847181i \(0.678298\pi\)
\(200\) 0 0
\(201\) 1.39033e6 2.42733
\(202\) 0 0
\(203\) − 157451.i − 0.268167i
\(204\) 0 0
\(205\) 370187.i 0.615229i
\(206\) 0 0
\(207\) 2.55047e6 4.13709
\(208\) 0 0
\(209\) 259581. 0.411062
\(210\) 0 0
\(211\) − 316256.i − 0.489027i −0.969646 0.244513i \(-0.921372\pi\)
0.969646 0.244513i \(-0.0786282\pi\)
\(212\) 0 0
\(213\) − 1.58109e6i − 2.38785i
\(214\) 0 0
\(215\) −158116. −0.233281
\(216\) 0 0
\(217\) −70478.7 −0.101603
\(218\) 0 0
\(219\) 2.08195e6i 2.93332i
\(220\) 0 0
\(221\) 10306.1i 0.0141943i
\(222\) 0 0
\(223\) 154021. 0.207405 0.103702 0.994608i \(-0.466931\pi\)
0.103702 + 0.994608i \(0.466931\pi\)
\(224\) 0 0
\(225\) 411362. 0.541711
\(226\) 0 0
\(227\) − 570137.i − 0.734370i −0.930148 0.367185i \(-0.880322\pi\)
0.930148 0.367185i \(-0.119678\pi\)
\(228\) 0 0
\(229\) − 80803.8i − 0.101822i −0.998703 0.0509111i \(-0.983787\pi\)
0.998703 0.0509111i \(-0.0162125\pi\)
\(230\) 0 0
\(231\) −194762. −0.240145
\(232\) 0 0
\(233\) −932668. −1.12548 −0.562739 0.826635i \(-0.690252\pi\)
−0.562739 + 0.826635i \(0.690252\pi\)
\(234\) 0 0
\(235\) − 725092.i − 0.856492i
\(236\) 0 0
\(237\) 999317.i 1.15567i
\(238\) 0 0
\(239\) −483104. −0.547074 −0.273537 0.961861i \(-0.588194\pi\)
−0.273537 + 0.961861i \(0.588194\pi\)
\(240\) 0 0
\(241\) −194028. −0.215190 −0.107595 0.994195i \(-0.534315\pi\)
−0.107595 + 0.994195i \(0.534315\pi\)
\(242\) 0 0
\(243\) 3.40196e6i 3.69584i
\(244\) 0 0
\(245\) 368511.i 0.392225i
\(246\) 0 0
\(247\) 1.63165e6 1.70171
\(248\) 0 0
\(249\) −1.66167e6 −1.69843
\(250\) 0 0
\(251\) 863008.i 0.864631i 0.901723 + 0.432315i \(0.142303\pi\)
−0.901723 + 0.432315i \(0.857697\pi\)
\(252\) 0 0
\(253\) − 553033.i − 0.543188i
\(254\) 0 0
\(255\) 8622.07 0.00830350
\(256\) 0 0
\(257\) −1.00468e6 −0.948841 −0.474420 0.880298i \(-0.657342\pi\)
−0.474420 + 0.880298i \(0.657342\pi\)
\(258\) 0 0
\(259\) 436910.i 0.404708i
\(260\) 0 0
\(261\) 2.27964e6i 2.07140i
\(262\) 0 0
\(263\) −1.50898e6 −1.34522 −0.672611 0.739996i \(-0.734827\pi\)
−0.672611 + 0.739996i \(0.734827\pi\)
\(264\) 0 0
\(265\) −814577. −0.712553
\(266\) 0 0
\(267\) 997711.i 0.856498i
\(268\) 0 0
\(269\) − 695284.i − 0.585844i −0.956136 0.292922i \(-0.905372\pi\)
0.956136 0.292922i \(-0.0946276\pi\)
\(270\) 0 0
\(271\) 361686. 0.299163 0.149582 0.988749i \(-0.452207\pi\)
0.149582 + 0.988749i \(0.452207\pi\)
\(272\) 0 0
\(273\) −1.22421e6 −0.994147
\(274\) 0 0
\(275\) − 89197.8i − 0.0711251i
\(276\) 0 0
\(277\) 2.07506e6i 1.62491i 0.583020 + 0.812457i \(0.301871\pi\)
−0.583020 + 0.812457i \(0.698129\pi\)
\(278\) 0 0
\(279\) 1.02042e6 0.784816
\(280\) 0 0
\(281\) −1.34117e6 −1.01325 −0.506626 0.862166i \(-0.669108\pi\)
−0.506626 + 0.862166i \(0.669108\pi\)
\(282\) 0 0
\(283\) − 2.41343e6i − 1.79130i −0.444762 0.895649i \(-0.646712\pi\)
0.444762 0.895649i \(-0.353288\pi\)
\(284\) 0 0
\(285\) − 1.36504e6i − 0.995478i
\(286\) 0 0
\(287\) −673139. −0.482392
\(288\) 0 0
\(289\) −1.41973e6 −0.999907
\(290\) 0 0
\(291\) − 2.47567e6i − 1.71380i
\(292\) 0 0
\(293\) 2.04581e6i 1.39219i 0.717952 + 0.696093i \(0.245080\pi\)
−0.717952 + 0.696093i \(0.754920\pi\)
\(294\) 0 0
\(295\) −16306.4 −0.0109095
\(296\) 0 0
\(297\) 1.77875e6 1.17010
\(298\) 0 0
\(299\) − 3.47620e6i − 2.24868i
\(300\) 0 0
\(301\) − 287514.i − 0.182912i
\(302\) 0 0
\(303\) −1.09311e6 −0.684002
\(304\) 0 0
\(305\) 117883. 0.0725609
\(306\) 0 0
\(307\) − 454883.i − 0.275457i −0.990470 0.137728i \(-0.956020\pi\)
0.990470 0.137728i \(-0.0439801\pi\)
\(308\) 0 0
\(309\) 2.49387e6i 1.48586i
\(310\) 0 0
\(311\) 1.36777e6 0.801884 0.400942 0.916103i \(-0.368683\pi\)
0.400942 + 0.916103i \(0.368683\pi\)
\(312\) 0 0
\(313\) 391986. 0.226157 0.113078 0.993586i \(-0.463929\pi\)
0.113078 + 0.993586i \(0.463929\pi\)
\(314\) 0 0
\(315\) 748010.i 0.424748i
\(316\) 0 0
\(317\) − 51502.5i − 0.0287859i −0.999896 0.0143930i \(-0.995418\pi\)
0.999896 0.0143930i \(-0.00458158\pi\)
\(318\) 0 0
\(319\) 494307. 0.271969
\(320\) 0 0
\(321\) 914898. 0.495576
\(322\) 0 0
\(323\) − 20896.1i − 0.0111444i
\(324\) 0 0
\(325\) − 560671.i − 0.294442i
\(326\) 0 0
\(327\) −6.01238e6 −3.10940
\(328\) 0 0
\(329\) 1.31849e6 0.671563
\(330\) 0 0
\(331\) − 235665.i − 0.118229i −0.998251 0.0591147i \(-0.981172\pi\)
0.998251 0.0591147i \(-0.0188278\pi\)
\(332\) 0 0
\(333\) − 6.32575e6i − 3.12609i
\(334\) 0 0
\(335\) −1.15785e6 −0.563692
\(336\) 0 0
\(337\) −1.60833e6 −0.771435 −0.385717 0.922617i \(-0.626046\pi\)
−0.385717 + 0.922617i \(0.626046\pi\)
\(338\) 0 0
\(339\) − 735237.i − 0.347479i
\(340\) 0 0
\(341\) − 221263.i − 0.103044i
\(342\) 0 0
\(343\) −1.43413e6 −0.658192
\(344\) 0 0
\(345\) −2.90819e6 −1.31545
\(346\) 0 0
\(347\) − 266162.i − 0.118665i −0.998238 0.0593324i \(-0.981103\pi\)
0.998238 0.0593324i \(-0.0188972\pi\)
\(348\) 0 0
\(349\) 4.43079e6i 1.94723i 0.228192 + 0.973616i \(0.426719\pi\)
−0.228192 + 0.973616i \(0.573281\pi\)
\(350\) 0 0
\(351\) 1.11807e7 4.84397
\(352\) 0 0
\(353\) −2.80868e6 −1.19968 −0.599840 0.800120i \(-0.704769\pi\)
−0.599840 + 0.800120i \(0.704769\pi\)
\(354\) 0 0
\(355\) 1.31671e6i 0.554523i
\(356\) 0 0
\(357\) 15678.1i 0.00651065i
\(358\) 0 0
\(359\) −2.32643e6 −0.952697 −0.476348 0.879257i \(-0.658040\pi\)
−0.476348 + 0.879257i \(0.658040\pi\)
\(360\) 0 0
\(361\) −832139. −0.336069
\(362\) 0 0
\(363\) 4.22325e6i 1.68221i
\(364\) 0 0
\(365\) − 1.73382e6i − 0.681196i
\(366\) 0 0
\(367\) −2.99596e6 −1.16110 −0.580552 0.814223i \(-0.697163\pi\)
−0.580552 + 0.814223i \(0.697163\pi\)
\(368\) 0 0
\(369\) 9.74598e6 3.72614
\(370\) 0 0
\(371\) − 1.48121e6i − 0.558703i
\(372\) 0 0
\(373\) 1.15331e6i 0.429214i 0.976700 + 0.214607i \(0.0688471\pi\)
−0.976700 + 0.214607i \(0.931153\pi\)
\(374\) 0 0
\(375\) −469057. −0.172245
\(376\) 0 0
\(377\) 3.10707e6 1.12589
\(378\) 0 0
\(379\) 2.98305e6i 1.06675i 0.845879 + 0.533375i \(0.179077\pi\)
−0.845879 + 0.533375i \(0.820923\pi\)
\(380\) 0 0
\(381\) − 4.01289e6i − 1.41626i
\(382\) 0 0
\(383\) 3.42628e6 1.19351 0.596755 0.802423i \(-0.296456\pi\)
0.596755 + 0.802423i \(0.296456\pi\)
\(384\) 0 0
\(385\) 162195. 0.0557681
\(386\) 0 0
\(387\) 4.16274e6i 1.41287i
\(388\) 0 0
\(389\) 578513.i 0.193838i 0.995292 + 0.0969191i \(0.0308988\pi\)
−0.995292 + 0.0969191i \(0.969101\pi\)
\(390\) 0 0
\(391\) −44518.7 −0.0147265
\(392\) 0 0
\(393\) 6.84311e6 2.23497
\(394\) 0 0
\(395\) − 832219.i − 0.268377i
\(396\) 0 0
\(397\) − 2.80594e6i − 0.893514i −0.894655 0.446757i \(-0.852579\pi\)
0.894655 0.446757i \(-0.147421\pi\)
\(398\) 0 0
\(399\) 2.48215e6 0.780540
\(400\) 0 0
\(401\) −333020. −0.103421 −0.0517106 0.998662i \(-0.516467\pi\)
−0.0517106 + 0.998662i \(0.516467\pi\)
\(402\) 0 0
\(403\) − 1.39079e6i − 0.426580i
\(404\) 0 0
\(405\) − 5.35532e6i − 1.62236i
\(406\) 0 0
\(407\) −1.37165e6 −0.410447
\(408\) 0 0
\(409\) −2.51090e6 −0.742201 −0.371101 0.928593i \(-0.621020\pi\)
−0.371101 + 0.928593i \(0.621020\pi\)
\(410\) 0 0
\(411\) − 1.06561e7i − 3.11166i
\(412\) 0 0
\(413\) − 29651.2i − 0.00855396i
\(414\) 0 0
\(415\) 1.38382e6 0.394421
\(416\) 0 0
\(417\) 1.24313e7 3.50087
\(418\) 0 0
\(419\) − 1.44535e6i − 0.402196i −0.979571 0.201098i \(-0.935549\pi\)
0.979571 0.201098i \(-0.0644510\pi\)
\(420\) 0 0
\(421\) − 1.66158e6i − 0.456896i −0.973556 0.228448i \(-0.926635\pi\)
0.973556 0.228448i \(-0.0733651\pi\)
\(422\) 0 0
\(423\) −1.90896e7 −5.18735
\(424\) 0 0
\(425\) −7180.35 −0.00192830
\(426\) 0 0
\(427\) 214356.i 0.0568939i
\(428\) 0 0
\(429\) − 3.84333e6i − 1.00824i
\(430\) 0 0
\(431\) 2.83017e6 0.733870 0.366935 0.930247i \(-0.380407\pi\)
0.366935 + 0.930247i \(0.380407\pi\)
\(432\) 0 0
\(433\) −3.79105e6 −0.971717 −0.485859 0.874037i \(-0.661493\pi\)
−0.485859 + 0.874037i \(0.661493\pi\)
\(434\) 0 0
\(435\) − 2.59937e6i − 0.658634i
\(436\) 0 0
\(437\) 7.04815e6i 1.76552i
\(438\) 0 0
\(439\) −651598. −0.161368 −0.0806842 0.996740i \(-0.525711\pi\)
−0.0806842 + 0.996740i \(0.525711\pi\)
\(440\) 0 0
\(441\) 9.70185e6 2.37552
\(442\) 0 0
\(443\) 5.06806e6i 1.22697i 0.789708 + 0.613483i \(0.210232\pi\)
−0.789708 + 0.613483i \(0.789768\pi\)
\(444\) 0 0
\(445\) − 830882.i − 0.198902i
\(446\) 0 0
\(447\) 7.01441e6 1.66044
\(448\) 0 0
\(449\) 4.01985e6 0.941010 0.470505 0.882397i \(-0.344072\pi\)
0.470505 + 0.882397i \(0.344072\pi\)
\(450\) 0 0
\(451\) − 2.11327e6i − 0.489232i
\(452\) 0 0
\(453\) − 1.22541e7i − 2.80566i
\(454\) 0 0
\(455\) 1.01951e6 0.230868
\(456\) 0 0
\(457\) −4.13558e6 −0.926288 −0.463144 0.886283i \(-0.653279\pi\)
−0.463144 + 0.886283i \(0.653279\pi\)
\(458\) 0 0
\(459\) − 143188.i − 0.0317230i
\(460\) 0 0
\(461\) − 710960.i − 0.155809i −0.996961 0.0779046i \(-0.975177\pi\)
0.996961 0.0779046i \(-0.0248230\pi\)
\(462\) 0 0
\(463\) −1.72786e6 −0.374589 −0.187295 0.982304i \(-0.559972\pi\)
−0.187295 + 0.982304i \(0.559972\pi\)
\(464\) 0 0
\(465\) −1.16354e6 −0.249544
\(466\) 0 0
\(467\) − 2.42693e6i − 0.514950i −0.966285 0.257475i \(-0.917110\pi\)
0.966285 0.257475i \(-0.0828905\pi\)
\(468\) 0 0
\(469\) − 2.10541e6i − 0.441982i
\(470\) 0 0
\(471\) 1.02780e7 2.13480
\(472\) 0 0
\(473\) 902630. 0.185506
\(474\) 0 0
\(475\) 1.13679e6i 0.231177i
\(476\) 0 0
\(477\) 2.14455e7i 4.31559i
\(478\) 0 0
\(479\) −7.40523e6 −1.47469 −0.737343 0.675519i \(-0.763920\pi\)
−0.737343 + 0.675519i \(0.763920\pi\)
\(480\) 0 0
\(481\) −8.62177e6 −1.69916
\(482\) 0 0
\(483\) − 5.28817e6i − 1.03143i
\(484\) 0 0
\(485\) 2.06171e6i 0.397991i
\(486\) 0 0
\(487\) −920253. −0.175827 −0.0879133 0.996128i \(-0.528020\pi\)
−0.0879133 + 0.996128i \(0.528020\pi\)
\(488\) 0 0
\(489\) −1.12545e7 −2.12841
\(490\) 0 0
\(491\) − 8.74081e6i − 1.63624i −0.575045 0.818122i \(-0.695016\pi\)
0.575045 0.818122i \(-0.304984\pi\)
\(492\) 0 0
\(493\) − 39791.3i − 0.00737345i
\(494\) 0 0
\(495\) −2.34833e6 −0.430770
\(496\) 0 0
\(497\) −2.39428e6 −0.434794
\(498\) 0 0
\(499\) 5.85693e6i 1.05298i 0.850182 + 0.526488i \(0.176492\pi\)
−0.850182 + 0.526488i \(0.823508\pi\)
\(500\) 0 0
\(501\) 2.26543e6i 0.403234i
\(502\) 0 0
\(503\) 3.54886e6 0.625416 0.312708 0.949849i \(-0.398764\pi\)
0.312708 + 0.949849i \(0.398764\pi\)
\(504\) 0 0
\(505\) 910329. 0.158844
\(506\) 0 0
\(507\) − 1.30120e7i − 2.24814i
\(508\) 0 0
\(509\) 2.90600e6i 0.497165i 0.968611 + 0.248582i \(0.0799647\pi\)
−0.968611 + 0.248582i \(0.920035\pi\)
\(510\) 0 0
\(511\) 3.15273e6 0.534115
\(512\) 0 0
\(513\) −2.26694e7 −3.80317
\(514\) 0 0
\(515\) − 2.07686e6i − 0.345056i
\(516\) 0 0
\(517\) 4.13930e6i 0.681084i
\(518\) 0 0
\(519\) −1.11739e7 −1.82089
\(520\) 0 0
\(521\) 5.04123e6 0.813658 0.406829 0.913504i \(-0.366634\pi\)
0.406829 + 0.913504i \(0.366634\pi\)
\(522\) 0 0
\(523\) 7.90605e6i 1.26388i 0.775018 + 0.631939i \(0.217741\pi\)
−0.775018 + 0.631939i \(0.782259\pi\)
\(524\) 0 0
\(525\) − 852921.i − 0.135055i
\(526\) 0 0
\(527\) −17811.5 −0.00279366
\(528\) 0 0
\(529\) 8.57964e6 1.33300
\(530\) 0 0
\(531\) 429302.i 0.0660734i
\(532\) 0 0
\(533\) − 1.32834e7i − 2.02531i
\(534\) 0 0
\(535\) −761916. −0.115086
\(536\) 0 0
\(537\) 7.06679e6 1.05751
\(538\) 0 0
\(539\) − 2.10371e6i − 0.311898i
\(540\) 0 0
\(541\) 9.78603e6i 1.43752i 0.695259 + 0.718759i \(0.255289\pi\)
−0.695259 + 0.718759i \(0.744711\pi\)
\(542\) 0 0
\(543\) −1.87786e7 −2.73315
\(544\) 0 0
\(545\) 5.00704e6 0.722087
\(546\) 0 0
\(547\) 2.64510e6i 0.377984i 0.981979 + 0.188992i \(0.0605220\pi\)
−0.981979 + 0.188992i \(0.939478\pi\)
\(548\) 0 0
\(549\) − 3.10353e6i − 0.439466i
\(550\) 0 0
\(551\) −6.29971e6 −0.883978
\(552\) 0 0
\(553\) 1.51329e6 0.210430
\(554\) 0 0
\(555\) 7.21296e6i 0.993988i
\(556\) 0 0
\(557\) − 4.84707e6i − 0.661975i −0.943635 0.330987i \(-0.892618\pi\)
0.943635 0.330987i \(-0.107382\pi\)
\(558\) 0 0
\(559\) 5.67367e6 0.767953
\(560\) 0 0
\(561\) −49220.5 −0.00660296
\(562\) 0 0
\(563\) − 7.18722e6i − 0.955631i −0.878460 0.477815i \(-0.841429\pi\)
0.878460 0.477815i \(-0.158571\pi\)
\(564\) 0 0
\(565\) 612297.i 0.0806940i
\(566\) 0 0
\(567\) 9.73798e6 1.27207
\(568\) 0 0
\(569\) −1.30594e7 −1.69100 −0.845498 0.533979i \(-0.820696\pi\)
−0.845498 + 0.533979i \(0.820696\pi\)
\(570\) 0 0
\(571\) 2.93976e6i 0.377330i 0.982041 + 0.188665i \(0.0604161\pi\)
−0.982041 + 0.188665i \(0.939584\pi\)
\(572\) 0 0
\(573\) − 1.55920e6i − 0.198388i
\(574\) 0 0
\(575\) 2.42190e6 0.305483
\(576\) 0 0
\(577\) 4.78611e6 0.598471 0.299236 0.954179i \(-0.403268\pi\)
0.299236 + 0.954179i \(0.403268\pi\)
\(578\) 0 0
\(579\) − 1.06666e7i − 1.32230i
\(580\) 0 0
\(581\) 2.51631e6i 0.309260i
\(582\) 0 0
\(583\) 4.65014e6 0.566624
\(584\) 0 0
\(585\) −1.47609e7 −1.78329
\(586\) 0 0
\(587\) − 2.07501e6i − 0.248557i −0.992247 0.124278i \(-0.960338\pi\)
0.992247 0.124278i \(-0.0396616\pi\)
\(588\) 0 0
\(589\) 2.81989e6i 0.334923i
\(590\) 0 0
\(591\) 3.77654e6 0.444759
\(592\) 0 0
\(593\) 3.08207e6 0.359920 0.179960 0.983674i \(-0.442403\pi\)
0.179960 + 0.983674i \(0.442403\pi\)
\(594\) 0 0
\(595\) − 13056.6i − 0.00151195i
\(596\) 0 0
\(597\) − 1.78201e7i − 2.04633i
\(598\) 0 0
\(599\) 1.34954e7 1.53680 0.768400 0.639970i \(-0.221053\pi\)
0.768400 + 0.639970i \(0.221053\pi\)
\(600\) 0 0
\(601\) −1.69923e6 −0.191897 −0.0959483 0.995386i \(-0.530588\pi\)
−0.0959483 + 0.995386i \(0.530588\pi\)
\(602\) 0 0
\(603\) 3.04830e7i 3.41401i
\(604\) 0 0
\(605\) − 3.51707e6i − 0.390655i
\(606\) 0 0
\(607\) 1.47300e7 1.62267 0.811335 0.584581i \(-0.198741\pi\)
0.811335 + 0.584581i \(0.198741\pi\)
\(608\) 0 0
\(609\) 4.72662e6 0.516426
\(610\) 0 0
\(611\) 2.60184e7i 2.81954i
\(612\) 0 0
\(613\) 5.34059e6i 0.574035i 0.957925 + 0.287017i \(0.0926638\pi\)
−0.957925 + 0.287017i \(0.907336\pi\)
\(614\) 0 0
\(615\) −1.11129e7 −1.18478
\(616\) 0 0
\(617\) −16763.1 −0.00177272 −0.000886361 1.00000i \(-0.500282\pi\)
−0.000886361 1.00000i \(0.500282\pi\)
\(618\) 0 0
\(619\) 1.20604e7i 1.26513i 0.774507 + 0.632566i \(0.217998\pi\)
−0.774507 + 0.632566i \(0.782002\pi\)
\(620\) 0 0
\(621\) 4.82967e7i 5.02561i
\(622\) 0 0
\(623\) 1.51085e6 0.155956
\(624\) 0 0
\(625\) 390625. 0.0400000
\(626\) 0 0
\(627\) 7.79253e6i 0.791607i
\(628\) 0 0
\(629\) 110417.i 0.0111278i
\(630\) 0 0
\(631\) 9.37030e6 0.936871 0.468436 0.883498i \(-0.344818\pi\)
0.468436 + 0.883498i \(0.344818\pi\)
\(632\) 0 0
\(633\) 9.49389e6 0.941749
\(634\) 0 0
\(635\) 3.34188e6i 0.328895i
\(636\) 0 0
\(637\) − 1.32233e7i − 1.29119i
\(638\) 0 0
\(639\) 3.46653e7 3.35848
\(640\) 0 0
\(641\) 1.24983e7 1.20145 0.600723 0.799457i \(-0.294879\pi\)
0.600723 + 0.799457i \(0.294879\pi\)
\(642\) 0 0
\(643\) 8.90727e6i 0.849605i 0.905286 + 0.424803i \(0.139656\pi\)
−0.905286 + 0.424803i \(0.860344\pi\)
\(644\) 0 0
\(645\) − 4.74658e6i − 0.449243i
\(646\) 0 0
\(647\) −4.19736e6 −0.394199 −0.197100 0.980383i \(-0.563152\pi\)
−0.197100 + 0.980383i \(0.563152\pi\)
\(648\) 0 0
\(649\) 93087.9 0.00867524
\(650\) 0 0
\(651\) − 2.11575e6i − 0.195664i
\(652\) 0 0
\(653\) − 7.16059e6i − 0.657152i −0.944477 0.328576i \(-0.893431\pi\)
0.944477 0.328576i \(-0.106569\pi\)
\(654\) 0 0
\(655\) −5.69886e6 −0.519021
\(656\) 0 0
\(657\) −4.56465e7 −4.12567
\(658\) 0 0
\(659\) 6.13902e6i 0.550662i 0.961349 + 0.275331i \(0.0887875\pi\)
−0.961349 + 0.275331i \(0.911213\pi\)
\(660\) 0 0
\(661\) − 3.75445e6i − 0.334228i −0.985938 0.167114i \(-0.946555\pi\)
0.985938 0.167114i \(-0.0534448\pi\)
\(662\) 0 0
\(663\) −309385. −0.0273348
\(664\) 0 0
\(665\) −2.06710e6 −0.181262
\(666\) 0 0
\(667\) 1.34214e7i 1.16811i
\(668\) 0 0
\(669\) 4.62367e6i 0.399412i
\(670\) 0 0
\(671\) −672956. −0.0577006
\(672\) 0 0
\(673\) −1.77234e7 −1.50838 −0.754188 0.656659i \(-0.771969\pi\)
−0.754188 + 0.656659i \(0.771969\pi\)
\(674\) 0 0
\(675\) 7.78970e6i 0.658054i
\(676\) 0 0
\(677\) − 1.16984e7i − 0.980969i −0.871450 0.490485i \(-0.836820\pi\)
0.871450 0.490485i \(-0.163180\pi\)
\(678\) 0 0
\(679\) −3.74896e6 −0.312059
\(680\) 0 0
\(681\) 1.71153e7 1.41422
\(682\) 0 0
\(683\) − 1.76343e7i − 1.44646i −0.690608 0.723230i \(-0.742657\pi\)
0.690608 0.723230i \(-0.257343\pi\)
\(684\) 0 0
\(685\) 8.87425e6i 0.722612i
\(686\) 0 0
\(687\) 2.42570e6 0.196085
\(688\) 0 0
\(689\) 2.92294e7 2.34570
\(690\) 0 0
\(691\) 6.23688e6i 0.496903i 0.968644 + 0.248452i \(0.0799217\pi\)
−0.968644 + 0.248452i \(0.920078\pi\)
\(692\) 0 0
\(693\) − 4.27014e6i − 0.337760i
\(694\) 0 0
\(695\) −1.03526e7 −0.812997
\(696\) 0 0
\(697\) −170117. −0.0132637
\(698\) 0 0
\(699\) − 2.79983e7i − 2.16740i
\(700\) 0 0
\(701\) − 1.52056e7i − 1.16872i −0.811496 0.584358i \(-0.801346\pi\)
0.811496 0.584358i \(-0.198654\pi\)
\(702\) 0 0
\(703\) 1.74810e7 1.33407
\(704\) 0 0
\(705\) 2.17670e7 1.64940
\(706\) 0 0
\(707\) 1.65532e6i 0.124547i
\(708\) 0 0
\(709\) 2.26475e7i 1.69202i 0.533170 + 0.846008i \(0.321000\pi\)
−0.533170 + 0.846008i \(0.679000\pi\)
\(710\) 0 0
\(711\) −2.19100e7 −1.62543
\(712\) 0 0
\(713\) 6.00774e6 0.442576
\(714\) 0 0
\(715\) 3.20068e6i 0.234141i
\(716\) 0 0
\(717\) − 1.45026e7i − 1.05353i
\(718\) 0 0
\(719\) −1.27400e7 −0.919063 −0.459532 0.888161i \(-0.651983\pi\)
−0.459532 + 0.888161i \(0.651983\pi\)
\(720\) 0 0
\(721\) 3.77652e6 0.270553
\(722\) 0 0
\(723\) − 5.82465e6i − 0.414404i
\(724\) 0 0
\(725\) 2.16472e6i 0.152953i
\(726\) 0 0
\(727\) 1.70816e7 1.19865 0.599325 0.800506i \(-0.295436\pi\)
0.599325 + 0.800506i \(0.295436\pi\)
\(728\) 0 0
\(729\) −5.00719e7 −3.48959
\(730\) 0 0
\(731\) − 72661.0i − 0.00502931i
\(732\) 0 0
\(733\) − 9.13585e6i − 0.628042i −0.949416 0.314021i \(-0.898324\pi\)
0.949416 0.314021i \(-0.101676\pi\)
\(734\) 0 0
\(735\) −1.10626e7 −0.755332
\(736\) 0 0
\(737\) 6.60980e6 0.448249
\(738\) 0 0
\(739\) − 5.83288e6i − 0.392891i −0.980515 0.196446i \(-0.937060\pi\)
0.980515 0.196446i \(-0.0629399\pi\)
\(740\) 0 0
\(741\) 4.89815e7i 3.27708i
\(742\) 0 0
\(743\) −2.70851e6 −0.179994 −0.0899970 0.995942i \(-0.528686\pi\)
−0.0899970 + 0.995942i \(0.528686\pi\)
\(744\) 0 0
\(745\) −5.84152e6 −0.385598
\(746\) 0 0
\(747\) − 3.64321e7i − 2.38882i
\(748\) 0 0
\(749\) − 1.38545e6i − 0.0902373i
\(750\) 0 0
\(751\) −1.54490e7 −0.999539 −0.499769 0.866159i \(-0.666582\pi\)
−0.499769 + 0.866159i \(0.666582\pi\)
\(752\) 0 0
\(753\) −2.59072e7 −1.66507
\(754\) 0 0
\(755\) 1.02050e7i 0.651550i
\(756\) 0 0
\(757\) 2.16555e7i 1.37350i 0.726893 + 0.686750i \(0.240963\pi\)
−0.726893 + 0.686750i \(0.759037\pi\)
\(758\) 0 0
\(759\) 1.66019e7 1.04605
\(760\) 0 0
\(761\) 1.80099e7 1.12732 0.563662 0.826006i \(-0.309392\pi\)
0.563662 + 0.826006i \(0.309392\pi\)
\(762\) 0 0
\(763\) 9.10467e6i 0.566178i
\(764\) 0 0
\(765\) 189038.i 0.0116787i
\(766\) 0 0
\(767\) 585123. 0.0359136
\(768\) 0 0
\(769\) −2.00138e7 −1.22043 −0.610215 0.792236i \(-0.708917\pi\)
−0.610215 + 0.792236i \(0.708917\pi\)
\(770\) 0 0
\(771\) − 3.01600e7i − 1.82724i
\(772\) 0 0
\(773\) 5.74507e6i 0.345817i 0.984938 + 0.172909i \(0.0553166\pi\)
−0.984938 + 0.172909i \(0.944683\pi\)
\(774\) 0 0
\(775\) 968979. 0.0579509
\(776\) 0 0
\(777\) −1.31159e7 −0.779372
\(778\) 0 0
\(779\) 2.69327e7i 1.59014i
\(780\) 0 0
\(781\) − 7.51666e6i − 0.440958i
\(782\) 0 0
\(783\) −4.31681e7 −2.51628
\(784\) 0 0
\(785\) −8.55941e6 −0.495758
\(786\) 0 0
\(787\) 5.04447e6i 0.290321i 0.989408 + 0.145160i \(0.0463698\pi\)
−0.989408 + 0.145160i \(0.953630\pi\)
\(788\) 0 0
\(789\) − 4.52990e7i − 2.59057i
\(790\) 0 0
\(791\) −1.11339e6 −0.0632709
\(792\) 0 0
\(793\) −4.23000e6 −0.238868
\(794\) 0 0
\(795\) − 2.44533e7i − 1.37221i
\(796\) 0 0
\(797\) 963795.i 0.0537451i 0.999639 + 0.0268726i \(0.00855483\pi\)
−0.999639 + 0.0268726i \(0.991445\pi\)
\(798\) 0 0
\(799\) 333211. 0.0184651
\(800\) 0 0
\(801\) −2.18748e7 −1.20465
\(802\) 0 0
\(803\) 9.89779e6i 0.541688i
\(804\) 0 0
\(805\) 4.40393e6i 0.239525i
\(806\) 0 0
\(807\) 2.08722e7 1.12820
\(808\) 0 0
\(809\) −6.63331e6 −0.356335 −0.178168 0.984000i \(-0.557017\pi\)
−0.178168 + 0.984000i \(0.557017\pi\)
\(810\) 0 0
\(811\) 3.04422e7i 1.62526i 0.582779 + 0.812631i \(0.301965\pi\)
−0.582779 + 0.812631i \(0.698035\pi\)
\(812\) 0 0
\(813\) 1.08577e7i 0.576117i
\(814\) 0 0
\(815\) 9.37264e6 0.494274
\(816\) 0 0
\(817\) −1.15036e7 −0.602947
\(818\) 0 0
\(819\) − 2.68408e7i − 1.39825i
\(820\) 0 0
\(821\) 9.68811e6i 0.501627i 0.968035 + 0.250814i \(0.0806981\pi\)
−0.968035 + 0.250814i \(0.919302\pi\)
\(822\) 0 0
\(823\) 8.72636e6 0.449090 0.224545 0.974464i \(-0.427910\pi\)
0.224545 + 0.974464i \(0.427910\pi\)
\(824\) 0 0
\(825\) 2.67769e6 0.136970
\(826\) 0 0
\(827\) − 2.23162e7i − 1.13464i −0.823498 0.567319i \(-0.807981\pi\)
0.823498 0.567319i \(-0.192019\pi\)
\(828\) 0 0
\(829\) − 3.36746e7i − 1.70183i −0.525302 0.850916i \(-0.676048\pi\)
0.525302 0.850916i \(-0.323952\pi\)
\(830\) 0 0
\(831\) −6.22925e7 −3.12920
\(832\) 0 0
\(833\) −169347. −0.00845598
\(834\) 0 0
\(835\) − 1.88662e6i − 0.0936417i
\(836\) 0 0
\(837\) 1.93230e7i 0.953370i
\(838\) 0 0
\(839\) −8.64415e6 −0.423952 −0.211976 0.977275i \(-0.567990\pi\)
−0.211976 + 0.977275i \(0.567990\pi\)
\(840\) 0 0
\(841\) 8.51494e6 0.415137
\(842\) 0 0
\(843\) − 4.02614e7i − 1.95128i
\(844\) 0 0
\(845\) 1.08362e7i 0.522079i
\(846\) 0 0
\(847\) 6.39536e6 0.306307
\(848\) 0 0
\(849\) 7.24502e7 3.44961
\(850\) 0 0
\(851\) − 3.72430e7i − 1.76287i
\(852\) 0 0
\(853\) − 2.91285e6i − 0.137071i −0.997649 0.0685355i \(-0.978167\pi\)
0.997649 0.0685355i \(-0.0218326\pi\)
\(854\) 0 0
\(855\) 2.99283e7 1.40013
\(856\) 0 0
\(857\) 3.92591e7 1.82595 0.912973 0.408019i \(-0.133780\pi\)
0.912973 + 0.408019i \(0.133780\pi\)
\(858\) 0 0
\(859\) 1.53343e7i 0.709059i 0.935045 + 0.354529i \(0.115359\pi\)
−0.935045 + 0.354529i \(0.884641\pi\)
\(860\) 0 0
\(861\) − 2.02074e7i − 0.928972i
\(862\) 0 0
\(863\) −1.19629e7 −0.546774 −0.273387 0.961904i \(-0.588144\pi\)
−0.273387 + 0.961904i \(0.588144\pi\)
\(864\) 0 0
\(865\) 9.30545e6 0.422861
\(866\) 0 0
\(867\) − 4.26196e7i − 1.92558i
\(868\) 0 0
\(869\) 4.75086e6i 0.213414i
\(870\) 0 0
\(871\) 4.15472e7 1.85565
\(872\) 0 0
\(873\) 5.42789e7 2.41044
\(874\) 0 0
\(875\) 710303.i 0.0313634i
\(876\) 0 0
\(877\) − 1.64985e7i − 0.724346i −0.932111 0.362173i \(-0.882035\pi\)
0.932111 0.362173i \(-0.117965\pi\)
\(878\) 0 0
\(879\) −6.14146e7 −2.68102
\(880\) 0 0
\(881\) −1.28694e7 −0.558623 −0.279312 0.960201i \(-0.590106\pi\)
−0.279312 + 0.960201i \(0.590106\pi\)
\(882\) 0 0
\(883\) 2.00179e7i 0.864005i 0.901872 + 0.432002i \(0.142193\pi\)
−0.901872 + 0.432002i \(0.857807\pi\)
\(884\) 0 0
\(885\) − 489513.i − 0.0210090i
\(886\) 0 0
\(887\) 3.41401e7 1.45699 0.728493 0.685053i \(-0.240221\pi\)
0.728493 + 0.685053i \(0.240221\pi\)
\(888\) 0 0
\(889\) −6.07680e6 −0.257882
\(890\) 0 0
\(891\) 3.05717e7i 1.29011i
\(892\) 0 0
\(893\) − 5.27535e7i − 2.21372i
\(894\) 0 0
\(895\) −5.88514e6 −0.245583
\(896\) 0 0
\(897\) 1.04354e8 4.33041
\(898\) 0 0
\(899\) 5.36978e6i 0.221594i
\(900\) 0 0
\(901\) − 374333.i − 0.0153619i
\(902\) 0 0
\(903\) 8.63107e6 0.352245
\(904\) 0 0
\(905\) 1.56386e7 0.634710
\(906\) 0 0
\(907\) 2.89420e7i 1.16818i 0.811688 + 0.584091i \(0.198549\pi\)
−0.811688 + 0.584091i \(0.801451\pi\)
\(908\) 0 0
\(909\) − 2.39664e7i − 0.962039i
\(910\) 0 0
\(911\) 3.26460e7 1.30327 0.651634 0.758534i \(-0.274084\pi\)
0.651634 + 0.758534i \(0.274084\pi\)
\(912\) 0 0
\(913\) −7.89978e6 −0.313645
\(914\) 0 0
\(915\) 3.53881e6i 0.139735i
\(916\) 0 0
\(917\) − 1.03627e7i − 0.406957i
\(918\) 0 0
\(919\) −2.68175e7 −1.04744 −0.523720 0.851890i \(-0.675456\pi\)
−0.523720 + 0.851890i \(0.675456\pi\)
\(920\) 0 0
\(921\) 1.36554e7 0.530464
\(922\) 0 0
\(923\) − 4.72475e7i − 1.82547i
\(924\) 0 0
\(925\) − 6.00687e6i − 0.230831i
\(926\) 0 0
\(927\) −5.46779e7 −2.08984
\(928\) 0 0
\(929\) 2.02792e6 0.0770923 0.0385462 0.999257i \(-0.487727\pi\)
0.0385462 + 0.999257i \(0.487727\pi\)
\(930\) 0 0
\(931\) 2.68108e7i 1.01376i
\(932\) 0 0
\(933\) 4.10599e7i 1.54424i
\(934\) 0 0
\(935\) 40990.2 0.00153339
\(936\) 0 0
\(937\) −3.18276e7 −1.18428 −0.592141 0.805834i \(-0.701717\pi\)
−0.592141 + 0.805834i \(0.701717\pi\)
\(938\) 0 0
\(939\) 1.17673e7i 0.435523i
\(940\) 0 0
\(941\) − 4.36130e7i − 1.60562i −0.596237 0.802809i \(-0.703338\pi\)
0.596237 0.802809i \(-0.296662\pi\)
\(942\) 0 0
\(943\) 5.73797e7 2.10126
\(944\) 0 0
\(945\) −1.41646e7 −0.515970
\(946\) 0 0
\(947\) 1.10000e7i 0.398582i 0.979940 + 0.199291i \(0.0638638\pi\)
−0.979940 + 0.199291i \(0.936136\pi\)
\(948\) 0 0
\(949\) 6.22146e7i 2.24247i
\(950\) 0 0
\(951\) 1.54609e6 0.0554348
\(952\) 0 0
\(953\) 2.76840e7 0.987406 0.493703 0.869630i \(-0.335643\pi\)
0.493703 + 0.869630i \(0.335643\pi\)
\(954\) 0 0
\(955\) 1.29848e6i 0.0460710i
\(956\) 0 0
\(957\) 1.48389e7i 0.523748i
\(958\) 0 0
\(959\) −1.61367e7 −0.566589
\(960\) 0 0
\(961\) −2.62255e7 −0.916042
\(962\) 0 0
\(963\) 2.00591e7i 0.697020i
\(964\) 0 0
\(965\) 8.88300e6i 0.307073i
\(966\) 0 0
\(967\) −3.67852e7 −1.26505 −0.632523 0.774541i \(-0.717981\pi\)
−0.632523 + 0.774541i \(0.717981\pi\)
\(968\) 0 0
\(969\) 627292. 0.0214615
\(970\) 0 0
\(971\) − 3.30736e6i − 0.112573i −0.998415 0.0562865i \(-0.982074\pi\)
0.998415 0.0562865i \(-0.0179260\pi\)
\(972\) 0 0
\(973\) − 1.88250e7i − 0.637459i
\(974\) 0 0
\(975\) 1.68312e7 0.567025
\(976\) 0 0
\(977\) −4.78379e7 −1.60338 −0.801689 0.597741i \(-0.796065\pi\)
−0.801689 + 0.597741i \(0.796065\pi\)
\(978\) 0 0
\(979\) 4.74322e6i 0.158167i
\(980\) 0 0
\(981\) − 1.31821e8i − 4.37333i
\(982\) 0 0
\(983\) −1.03384e7 −0.341248 −0.170624 0.985336i \(-0.554578\pi\)
−0.170624 + 0.985336i \(0.554578\pi\)
\(984\) 0 0
\(985\) −3.14506e6 −0.103285
\(986\) 0 0
\(987\) 3.95805e7i 1.29327i
\(988\) 0 0
\(989\) 2.45083e7i 0.796749i
\(990\) 0 0
\(991\) 3.54794e7 1.14760 0.573802 0.818994i \(-0.305468\pi\)
0.573802 + 0.818994i \(0.305468\pi\)
\(992\) 0 0
\(993\) 7.07459e6 0.227682
\(994\) 0 0
\(995\) 1.48404e7i 0.475213i
\(996\) 0 0
\(997\) 3.07614e7i 0.980094i 0.871696 + 0.490047i \(0.163020\pi\)
−0.871696 + 0.490047i \(0.836980\pi\)
\(998\) 0 0
\(999\) 1.19787e8 3.79748
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 \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 320.6.d.c.161.12 yes 12
4.3 odd 2 320.6.d.d.161.1 yes 12
8.3 odd 2 320.6.d.d.161.12 yes 12
8.5 even 2 inner 320.6.d.c.161.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
320.6.d.c.161.1 12 8.5 even 2 inner
320.6.d.c.161.12 yes 12 1.1 even 1 trivial
320.6.d.d.161.1 yes 12 4.3 odd 2
320.6.d.d.161.12 yes 12 8.3 odd 2