Properties

Label 256.6.b.b.129.2
Level $256$
Weight $6$
Character 256.129
Analytic conductor $41.058$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 256 = 2^{8} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 256.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(41.0582578721\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 32)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 129.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 256.129
Dual form 256.6.b.b.129.1

$q$-expansion

\(f(q)\) \(=\) \(q+8.00000i q^{3} -14.0000i q^{5} -208.000 q^{7} +179.000 q^{9} +O(q^{10})\) \(q+8.00000i q^{3} -14.0000i q^{5} -208.000 q^{7} +179.000 q^{9} -536.000i q^{11} +694.000i q^{13} +112.000 q^{15} -1278.00 q^{17} -1112.00i q^{19} -1664.00i q^{21} +3216.00 q^{23} +2929.00 q^{25} +3376.00i q^{27} +2918.00i q^{29} +2624.00 q^{31} +4288.00 q^{33} +2912.00i q^{35} +9458.00i q^{37} -5552.00 q^{39} -170.000 q^{41} -19928.0i q^{43} -2506.00i q^{45} -32.0000 q^{47} +26457.0 q^{49} -10224.0i q^{51} +22178.0i q^{53} -7504.00 q^{55} +8896.00 q^{57} +41480.0i q^{59} +15462.0i q^{61} -37232.0 q^{63} +9716.00 q^{65} +20744.0i q^{67} +25728.0i q^{69} +28592.0 q^{71} +53670.0 q^{73} +23432.0i q^{75} +111488. i q^{77} +69152.0 q^{79} +16489.0 q^{81} +37800.0i q^{83} +17892.0i q^{85} -23344.0 q^{87} +126806. q^{89} -144352. i q^{91} +20992.0i q^{93} -15568.0 q^{95} +62290.0 q^{97} -95944.0i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 416q^{7} + 358q^{9} + O(q^{10}) \) \( 2q - 416q^{7} + 358q^{9} + 224q^{15} - 2556q^{17} + 6432q^{23} + 5858q^{25} + 5248q^{31} + 8576q^{33} - 11104q^{39} - 340q^{41} - 64q^{47} + 52914q^{49} - 15008q^{55} + 17792q^{57} - 74464q^{63} + 19432q^{65} + 57184q^{71} + 107340q^{73} + 138304q^{79} + 32978q^{81} - 46688q^{87} + 253612q^{89} - 31136q^{95} + 124580q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/256\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(255\)
\(\chi(n)\) \(-1\) \(1\)

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
\(3\) 8.00000i 0.513200i 0.966518 + 0.256600i \(0.0826023\pi\)
−0.966518 + 0.256600i \(0.917398\pi\)
\(4\) 0 0
\(5\) − 14.0000i − 0.250440i −0.992129 0.125220i \(-0.960036\pi\)
0.992129 0.125220i \(-0.0399636\pi\)
\(6\) 0 0
\(7\) −208.000 −1.60442 −0.802210 0.597042i \(-0.796343\pi\)
−0.802210 + 0.597042i \(0.796343\pi\)
\(8\) 0 0
\(9\) 179.000 0.736626
\(10\) 0 0
\(11\) − 536.000i − 1.33562i −0.744332 0.667810i \(-0.767232\pi\)
0.744332 0.667810i \(-0.232768\pi\)
\(12\) 0 0
\(13\) 694.000i 1.13894i 0.822012 + 0.569470i \(0.192852\pi\)
−0.822012 + 0.569470i \(0.807148\pi\)
\(14\) 0 0
\(15\) 112.000 0.128526
\(16\) 0 0
\(17\) −1278.00 −1.07253 −0.536264 0.844050i \(-0.680165\pi\)
−0.536264 + 0.844050i \(0.680165\pi\)
\(18\) 0 0
\(19\) − 1112.00i − 0.706677i −0.935496 0.353338i \(-0.885046\pi\)
0.935496 0.353338i \(-0.114954\pi\)
\(20\) 0 0
\(21\) − 1664.00i − 0.823389i
\(22\) 0 0
\(23\) 3216.00 1.26764 0.633821 0.773480i \(-0.281486\pi\)
0.633821 + 0.773480i \(0.281486\pi\)
\(24\) 0 0
\(25\) 2929.00 0.937280
\(26\) 0 0
\(27\) 3376.00i 0.891237i
\(28\) 0 0
\(29\) 2918.00i 0.644303i 0.946688 + 0.322152i \(0.104406\pi\)
−0.946688 + 0.322152i \(0.895594\pi\)
\(30\) 0 0
\(31\) 2624.00 0.490410 0.245205 0.969471i \(-0.421145\pi\)
0.245205 + 0.969471i \(0.421145\pi\)
\(32\) 0 0
\(33\) 4288.00 0.685441
\(34\) 0 0
\(35\) 2912.00i 0.401810i
\(36\) 0 0
\(37\) 9458.00i 1.13578i 0.823104 + 0.567891i \(0.192241\pi\)
−0.823104 + 0.567891i \(0.807759\pi\)
\(38\) 0 0
\(39\) −5552.00 −0.584505
\(40\) 0 0
\(41\) −170.000 −0.0157939 −0.00789695 0.999969i \(-0.502514\pi\)
−0.00789695 + 0.999969i \(0.502514\pi\)
\(42\) 0 0
\(43\) − 19928.0i − 1.64359i −0.569786 0.821793i \(-0.692974\pi\)
0.569786 0.821793i \(-0.307026\pi\)
\(44\) 0 0
\(45\) − 2506.00i − 0.184480i
\(46\) 0 0
\(47\) −32.0000 −0.00211303 −0.00105651 0.999999i \(-0.500336\pi\)
−0.00105651 + 0.999999i \(0.500336\pi\)
\(48\) 0 0
\(49\) 26457.0 1.57417
\(50\) 0 0
\(51\) − 10224.0i − 0.550422i
\(52\) 0 0
\(53\) 22178.0i 1.08451i 0.840215 + 0.542254i \(0.182429\pi\)
−0.840215 + 0.542254i \(0.817571\pi\)
\(54\) 0 0
\(55\) −7504.00 −0.334492
\(56\) 0 0
\(57\) 8896.00 0.362667
\(58\) 0 0
\(59\) 41480.0i 1.55135i 0.631135 + 0.775673i \(0.282589\pi\)
−0.631135 + 0.775673i \(0.717411\pi\)
\(60\) 0 0
\(61\) 15462.0i 0.532036i 0.963968 + 0.266018i \(0.0857081\pi\)
−0.963968 + 0.266018i \(0.914292\pi\)
\(62\) 0 0
\(63\) −37232.0 −1.18186
\(64\) 0 0
\(65\) 9716.00 0.285236
\(66\) 0 0
\(67\) 20744.0i 0.564554i 0.959333 + 0.282277i \(0.0910897\pi\)
−0.959333 + 0.282277i \(0.908910\pi\)
\(68\) 0 0
\(69\) 25728.0i 0.650554i
\(70\) 0 0
\(71\) 28592.0 0.673130 0.336565 0.941660i \(-0.390735\pi\)
0.336565 + 0.941660i \(0.390735\pi\)
\(72\) 0 0
\(73\) 53670.0 1.17876 0.589379 0.807857i \(-0.299373\pi\)
0.589379 + 0.807857i \(0.299373\pi\)
\(74\) 0 0
\(75\) 23432.0i 0.481012i
\(76\) 0 0
\(77\) 111488.i 2.14290i
\(78\) 0 0
\(79\) 69152.0 1.24663 0.623314 0.781971i \(-0.285786\pi\)
0.623314 + 0.781971i \(0.285786\pi\)
\(80\) 0 0
\(81\) 16489.0 0.279243
\(82\) 0 0
\(83\) 37800.0i 0.602277i 0.953580 + 0.301139i \(0.0973667\pi\)
−0.953580 + 0.301139i \(0.902633\pi\)
\(84\) 0 0
\(85\) 17892.0i 0.268603i
\(86\) 0 0
\(87\) −23344.0 −0.330657
\(88\) 0 0
\(89\) 126806. 1.69693 0.848467 0.529249i \(-0.177526\pi\)
0.848467 + 0.529249i \(0.177526\pi\)
\(90\) 0 0
\(91\) − 144352.i − 1.82734i
\(92\) 0 0
\(93\) 20992.0i 0.251679i
\(94\) 0 0
\(95\) −15568.0 −0.176980
\(96\) 0 0
\(97\) 62290.0 0.672185 0.336093 0.941829i \(-0.390894\pi\)
0.336093 + 0.941829i \(0.390894\pi\)
\(98\) 0 0
\(99\) − 95944.0i − 0.983852i
\(100\) 0 0
\(101\) − 6414.00i − 0.0625641i −0.999511 0.0312821i \(-0.990041\pi\)
0.999511 0.0312821i \(-0.00995902\pi\)
\(102\) 0 0
\(103\) −108432. −1.00708 −0.503541 0.863972i \(-0.667970\pi\)
−0.503541 + 0.863972i \(0.667970\pi\)
\(104\) 0 0
\(105\) −23296.0 −0.206209
\(106\) 0 0
\(107\) 103976.i 0.877958i 0.898497 + 0.438979i \(0.144660\pi\)
−0.898497 + 0.438979i \(0.855340\pi\)
\(108\) 0 0
\(109\) 2486.00i 0.0200417i 0.999950 + 0.0100209i \(0.00318979\pi\)
−0.999950 + 0.0100209i \(0.996810\pi\)
\(110\) 0 0
\(111\) −75664.0 −0.582884
\(112\) 0 0
\(113\) 15794.0 0.116358 0.0581790 0.998306i \(-0.481471\pi\)
0.0581790 + 0.998306i \(0.481471\pi\)
\(114\) 0 0
\(115\) − 45024.0i − 0.317468i
\(116\) 0 0
\(117\) 124226.i 0.838973i
\(118\) 0 0
\(119\) 265824. 1.72079
\(120\) 0 0
\(121\) −126245. −0.783882
\(122\) 0 0
\(123\) − 1360.00i − 0.00810543i
\(124\) 0 0
\(125\) − 84756.0i − 0.485172i
\(126\) 0 0
\(127\) −1024.00 −0.00563366 −0.00281683 0.999996i \(-0.500897\pi\)
−0.00281683 + 0.999996i \(0.500897\pi\)
\(128\) 0 0
\(129\) 159424. 0.843489
\(130\) 0 0
\(131\) 22664.0i 0.115387i 0.998334 + 0.0576937i \(0.0183747\pi\)
−0.998334 + 0.0576937i \(0.981625\pi\)
\(132\) 0 0
\(133\) 231296.i 1.13381i
\(134\) 0 0
\(135\) 47264.0 0.223201
\(136\) 0 0
\(137\) 53238.0 0.242337 0.121169 0.992632i \(-0.461336\pi\)
0.121169 + 0.992632i \(0.461336\pi\)
\(138\) 0 0
\(139\) 19816.0i 0.0869919i 0.999054 + 0.0434960i \(0.0138496\pi\)
−0.999054 + 0.0434960i \(0.986150\pi\)
\(140\) 0 0
\(141\) − 256.000i − 0.00108441i
\(142\) 0 0
\(143\) 371984. 1.52119
\(144\) 0 0
\(145\) 40852.0 0.161359
\(146\) 0 0
\(147\) 211656.i 0.807862i
\(148\) 0 0
\(149\) − 452190.i − 1.66861i −0.551302 0.834306i \(-0.685869\pi\)
0.551302 0.834306i \(-0.314131\pi\)
\(150\) 0 0
\(151\) −263280. −0.939670 −0.469835 0.882754i \(-0.655687\pi\)
−0.469835 + 0.882754i \(0.655687\pi\)
\(152\) 0 0
\(153\) −228762. −0.790051
\(154\) 0 0
\(155\) − 36736.0i − 0.122818i
\(156\) 0 0
\(157\) − 353530.i − 1.14466i −0.820023 0.572331i \(-0.806039\pi\)
0.820023 0.572331i \(-0.193961\pi\)
\(158\) 0 0
\(159\) −177424. −0.556570
\(160\) 0 0
\(161\) −668928. −2.03383
\(162\) 0 0
\(163\) 100936.i 0.297562i 0.988870 + 0.148781i \(0.0475349\pi\)
−0.988870 + 0.148781i \(0.952465\pi\)
\(164\) 0 0
\(165\) − 60032.0i − 0.171662i
\(166\) 0 0
\(167\) −284944. −0.790621 −0.395310 0.918548i \(-0.629363\pi\)
−0.395310 + 0.918548i \(0.629363\pi\)
\(168\) 0 0
\(169\) −110343. −0.297186
\(170\) 0 0
\(171\) − 199048.i − 0.520556i
\(172\) 0 0
\(173\) 484374.i 1.23045i 0.788350 + 0.615227i \(0.210936\pi\)
−0.788350 + 0.615227i \(0.789064\pi\)
\(174\) 0 0
\(175\) −609232. −1.50379
\(176\) 0 0
\(177\) −331840. −0.796151
\(178\) 0 0
\(179\) − 406680.i − 0.948681i −0.880341 0.474341i \(-0.842687\pi\)
0.880341 0.474341i \(-0.157313\pi\)
\(180\) 0 0
\(181\) − 570302.i − 1.29392i −0.762523 0.646962i \(-0.776039\pi\)
0.762523 0.646962i \(-0.223961\pi\)
\(182\) 0 0
\(183\) −123696. −0.273041
\(184\) 0 0
\(185\) 132412. 0.284445
\(186\) 0 0
\(187\) 685008.i 1.43249i
\(188\) 0 0
\(189\) − 702208.i − 1.42992i
\(190\) 0 0
\(191\) −138624. −0.274951 −0.137475 0.990505i \(-0.543899\pi\)
−0.137475 + 0.990505i \(0.543899\pi\)
\(192\) 0 0
\(193\) 34482.0 0.0666345 0.0333173 0.999445i \(-0.489393\pi\)
0.0333173 + 0.999445i \(0.489393\pi\)
\(194\) 0 0
\(195\) 77728.0i 0.146383i
\(196\) 0 0
\(197\) − 643598.i − 1.18154i −0.806839 0.590771i \(-0.798824\pi\)
0.806839 0.590771i \(-0.201176\pi\)
\(198\) 0 0
\(199\) 1.10738e6 1.98227 0.991134 0.132865i \(-0.0424178\pi\)
0.991134 + 0.132865i \(0.0424178\pi\)
\(200\) 0 0
\(201\) −165952. −0.289729
\(202\) 0 0
\(203\) − 606944.i − 1.03373i
\(204\) 0 0
\(205\) 2380.00i 0.00395542i
\(206\) 0 0
\(207\) 575664. 0.933777
\(208\) 0 0
\(209\) −596032. −0.943852
\(210\) 0 0
\(211\) − 229976.i − 0.355612i −0.984066 0.177806i \(-0.943100\pi\)
0.984066 0.177806i \(-0.0568999\pi\)
\(212\) 0 0
\(213\) 228736.i 0.345450i
\(214\) 0 0
\(215\) −278992. −0.411619
\(216\) 0 0
\(217\) −545792. −0.786824
\(218\) 0 0
\(219\) 429360.i 0.604939i
\(220\) 0 0
\(221\) − 886932.i − 1.22155i
\(222\) 0 0
\(223\) 1.08947e6 1.46708 0.733540 0.679646i \(-0.237867\pi\)
0.733540 + 0.679646i \(0.237867\pi\)
\(224\) 0 0
\(225\) 524291. 0.690424
\(226\) 0 0
\(227\) 687048.i 0.884958i 0.896779 + 0.442479i \(0.145901\pi\)
−0.896779 + 0.442479i \(0.854099\pi\)
\(228\) 0 0
\(229\) 699730.i 0.881743i 0.897570 + 0.440871i \(0.145330\pi\)
−0.897570 + 0.440871i \(0.854670\pi\)
\(230\) 0 0
\(231\) −891904. −1.09974
\(232\) 0 0
\(233\) −937722. −1.13158 −0.565789 0.824550i \(-0.691428\pi\)
−0.565789 + 0.824550i \(0.691428\pi\)
\(234\) 0 0
\(235\) 448.000i 0 0.000529186i
\(236\) 0 0
\(237\) 553216.i 0.639770i
\(238\) 0 0
\(239\) −643488. −0.728695 −0.364347 0.931263i \(-0.618708\pi\)
−0.364347 + 0.931263i \(0.618708\pi\)
\(240\) 0 0
\(241\) 157282. 0.174436 0.0872181 0.996189i \(-0.472202\pi\)
0.0872181 + 0.996189i \(0.472202\pi\)
\(242\) 0 0
\(243\) 952280.i 1.03454i
\(244\) 0 0
\(245\) − 370398.i − 0.394233i
\(246\) 0 0
\(247\) 771728. 0.804863
\(248\) 0 0
\(249\) −302400. −0.309089
\(250\) 0 0
\(251\) − 1.58604e6i − 1.58902i −0.607250 0.794511i \(-0.707727\pi\)
0.607250 0.794511i \(-0.292273\pi\)
\(252\) 0 0
\(253\) − 1.72378e6i − 1.69309i
\(254\) 0 0
\(255\) −143136. −0.137847
\(256\) 0 0
\(257\) −654334. −0.617969 −0.308984 0.951067i \(-0.599989\pi\)
−0.308984 + 0.951067i \(0.599989\pi\)
\(258\) 0 0
\(259\) − 1.96726e6i − 1.82227i
\(260\) 0 0
\(261\) 522322.i 0.474610i
\(262\) 0 0
\(263\) −330192. −0.294359 −0.147179 0.989110i \(-0.547019\pi\)
−0.147179 + 0.989110i \(0.547019\pi\)
\(264\) 0 0
\(265\) 310492. 0.271604
\(266\) 0 0
\(267\) 1.01445e6i 0.870867i
\(268\) 0 0
\(269\) 1.56956e6i 1.32250i 0.750164 + 0.661252i \(0.229974\pi\)
−0.750164 + 0.661252i \(0.770026\pi\)
\(270\) 0 0
\(271\) −957792. −0.792224 −0.396112 0.918202i \(-0.629641\pi\)
−0.396112 + 0.918202i \(0.629641\pi\)
\(272\) 0 0
\(273\) 1.15482e6 0.937791
\(274\) 0 0
\(275\) − 1.56994e6i − 1.25185i
\(276\) 0 0
\(277\) − 565438.i − 0.442778i −0.975186 0.221389i \(-0.928941\pi\)
0.975186 0.221389i \(-0.0710590\pi\)
\(278\) 0 0
\(279\) 469696. 0.361249
\(280\) 0 0
\(281\) 1.34127e6 1.01333 0.506664 0.862143i \(-0.330878\pi\)
0.506664 + 0.862143i \(0.330878\pi\)
\(282\) 0 0
\(283\) − 734264.i − 0.544987i −0.962158 0.272494i \(-0.912152\pi\)
0.962158 0.272494i \(-0.0878483\pi\)
\(284\) 0 0
\(285\) − 124544.i − 0.0908261i
\(286\) 0 0
\(287\) 35360.0 0.0253401
\(288\) 0 0
\(289\) 213427. 0.150316
\(290\) 0 0
\(291\) 498320.i 0.344966i
\(292\) 0 0
\(293\) 1.13320e6i 0.771149i 0.922677 + 0.385574i \(0.125997\pi\)
−0.922677 + 0.385574i \(0.874003\pi\)
\(294\) 0 0
\(295\) 580720. 0.388519
\(296\) 0 0
\(297\) 1.80954e6 1.19035
\(298\) 0 0
\(299\) 2.23190e6i 1.44377i
\(300\) 0 0
\(301\) 4.14502e6i 2.63700i
\(302\) 0 0
\(303\) 51312.0 0.0321079
\(304\) 0 0
\(305\) 216468. 0.133243
\(306\) 0 0
\(307\) 2.91377e6i 1.76445i 0.470829 + 0.882224i \(0.343955\pi\)
−0.470829 + 0.882224i \(0.656045\pi\)
\(308\) 0 0
\(309\) − 867456.i − 0.516834i
\(310\) 0 0
\(311\) −1.43813e6 −0.843134 −0.421567 0.906797i \(-0.638520\pi\)
−0.421567 + 0.906797i \(0.638520\pi\)
\(312\) 0 0
\(313\) 1.37601e6 0.793888 0.396944 0.917843i \(-0.370071\pi\)
0.396944 + 0.917843i \(0.370071\pi\)
\(314\) 0 0
\(315\) 521248.i 0.295984i
\(316\) 0 0
\(317\) − 1.23494e6i − 0.690235i −0.938559 0.345118i \(-0.887839\pi\)
0.938559 0.345118i \(-0.112161\pi\)
\(318\) 0 0
\(319\) 1.56405e6 0.860545
\(320\) 0 0
\(321\) −831808. −0.450568
\(322\) 0 0
\(323\) 1.42114e6i 0.757930i
\(324\) 0 0
\(325\) 2.03273e6i 1.06751i
\(326\) 0 0
\(327\) −19888.0 −0.0102854
\(328\) 0 0
\(329\) 6656.00 0.00339019
\(330\) 0 0
\(331\) − 1.48930e6i − 0.747160i −0.927598 0.373580i \(-0.878130\pi\)
0.927598 0.373580i \(-0.121870\pi\)
\(332\) 0 0
\(333\) 1.69298e6i 0.836646i
\(334\) 0 0
\(335\) 290416. 0.141387
\(336\) 0 0
\(337\) 838226. 0.402056 0.201028 0.979586i \(-0.435572\pi\)
0.201028 + 0.979586i \(0.435572\pi\)
\(338\) 0 0
\(339\) 126352.i 0.0597149i
\(340\) 0 0
\(341\) − 1.40646e6i − 0.655002i
\(342\) 0 0
\(343\) −2.00720e6 −0.921203
\(344\) 0 0
\(345\) 360192. 0.162924
\(346\) 0 0
\(347\) − 350008.i − 0.156047i −0.996952 0.0780233i \(-0.975139\pi\)
0.996952 0.0780233i \(-0.0248609\pi\)
\(348\) 0 0
\(349\) − 383642.i − 0.168602i −0.996440 0.0843010i \(-0.973134\pi\)
0.996440 0.0843010i \(-0.0268657\pi\)
\(350\) 0 0
\(351\) −2.34294e6 −1.01507
\(352\) 0 0
\(353\) 4.09309e6 1.74829 0.874147 0.485661i \(-0.161421\pi\)
0.874147 + 0.485661i \(0.161421\pi\)
\(354\) 0 0
\(355\) − 400288.i − 0.168578i
\(356\) 0 0
\(357\) 2.12659e6i 0.883108i
\(358\) 0 0
\(359\) 3.14430e6 1.28762 0.643811 0.765185i \(-0.277352\pi\)
0.643811 + 0.765185i \(0.277352\pi\)
\(360\) 0 0
\(361\) 1.23955e6 0.500608
\(362\) 0 0
\(363\) − 1.00996e6i − 0.402288i
\(364\) 0 0
\(365\) − 751380.i − 0.295208i
\(366\) 0 0
\(367\) 1.47619e6 0.572108 0.286054 0.958214i \(-0.407656\pi\)
0.286054 + 0.958214i \(0.407656\pi\)
\(368\) 0 0
\(369\) −30430.0 −0.0116342
\(370\) 0 0
\(371\) − 4.61302e6i − 1.74001i
\(372\) 0 0
\(373\) 3.73981e6i 1.39180i 0.718138 + 0.695901i \(0.244995\pi\)
−0.718138 + 0.695901i \(0.755005\pi\)
\(374\) 0 0
\(375\) 678048. 0.248990
\(376\) 0 0
\(377\) −2.02509e6 −0.733823
\(378\) 0 0
\(379\) 1.89966e6i 0.679324i 0.940548 + 0.339662i \(0.110313\pi\)
−0.940548 + 0.339662i \(0.889687\pi\)
\(380\) 0 0
\(381\) − 8192.00i − 0.00289120i
\(382\) 0 0
\(383\) −1.74310e6 −0.607192 −0.303596 0.952801i \(-0.598187\pi\)
−0.303596 + 0.952801i \(0.598187\pi\)
\(384\) 0 0
\(385\) 1.56083e6 0.536666
\(386\) 0 0
\(387\) − 3.56711e6i − 1.21071i
\(388\) 0 0
\(389\) 2.69147e6i 0.901812i 0.892571 + 0.450906i \(0.148899\pi\)
−0.892571 + 0.450906i \(0.851101\pi\)
\(390\) 0 0
\(391\) −4.11005e6 −1.35958
\(392\) 0 0
\(393\) −181312. −0.0592168
\(394\) 0 0
\(395\) − 968128.i − 0.312205i
\(396\) 0 0
\(397\) 5.37353e6i 1.71113i 0.517695 + 0.855565i \(0.326790\pi\)
−0.517695 + 0.855565i \(0.673210\pi\)
\(398\) 0 0
\(399\) −1.85037e6 −0.581870
\(400\) 0 0
\(401\) 156418. 0.0485765 0.0242882 0.999705i \(-0.492268\pi\)
0.0242882 + 0.999705i \(0.492268\pi\)
\(402\) 0 0
\(403\) 1.82106e6i 0.558548i
\(404\) 0 0
\(405\) − 230846.i − 0.0699334i
\(406\) 0 0
\(407\) 5.06949e6 1.51697
\(408\) 0 0
\(409\) 306086. 0.0904764 0.0452382 0.998976i \(-0.485595\pi\)
0.0452382 + 0.998976i \(0.485595\pi\)
\(410\) 0 0
\(411\) 425904.i 0.124368i
\(412\) 0 0
\(413\) − 8.62784e6i − 2.48901i
\(414\) 0 0
\(415\) 529200. 0.150834
\(416\) 0 0
\(417\) −158528. −0.0446443
\(418\) 0 0
\(419\) 6.70868e6i 1.86682i 0.358814 + 0.933409i \(0.383181\pi\)
−0.358814 + 0.933409i \(0.616819\pi\)
\(420\) 0 0
\(421\) − 4.02347e6i − 1.10636i −0.833063 0.553179i \(-0.813415\pi\)
0.833063 0.553179i \(-0.186585\pi\)
\(422\) 0 0
\(423\) −5728.00 −0.00155651
\(424\) 0 0
\(425\) −3.74326e6 −1.00526
\(426\) 0 0
\(427\) − 3.21610e6i − 0.853610i
\(428\) 0 0
\(429\) 2.97587e6i 0.780676i
\(430\) 0 0
\(431\) 7.04304e6 1.82628 0.913139 0.407648i \(-0.133651\pi\)
0.913139 + 0.407648i \(0.133651\pi\)
\(432\) 0 0
\(433\) −1.25142e6 −0.320763 −0.160381 0.987055i \(-0.551272\pi\)
−0.160381 + 0.987055i \(0.551272\pi\)
\(434\) 0 0
\(435\) 326816.i 0.0828095i
\(436\) 0 0
\(437\) − 3.57619e6i − 0.895813i
\(438\) 0 0
\(439\) −1.25406e6 −0.310569 −0.155285 0.987870i \(-0.549629\pi\)
−0.155285 + 0.987870i \(0.549629\pi\)
\(440\) 0 0
\(441\) 4.73580e6 1.15957
\(442\) 0 0
\(443\) 5.18081e6i 1.25426i 0.778914 + 0.627131i \(0.215771\pi\)
−0.778914 + 0.627131i \(0.784229\pi\)
\(444\) 0 0
\(445\) − 1.77528e6i − 0.424979i
\(446\) 0 0
\(447\) 3.61752e6 0.856332
\(448\) 0 0
\(449\) −5.73064e6 −1.34149 −0.670745 0.741688i \(-0.734025\pi\)
−0.670745 + 0.741688i \(0.734025\pi\)
\(450\) 0 0
\(451\) 91120.0i 0.0210947i
\(452\) 0 0
\(453\) − 2.10624e6i − 0.482239i
\(454\) 0 0
\(455\) −2.02093e6 −0.457638
\(456\) 0 0
\(457\) 5.24153e6 1.17400 0.586999 0.809588i \(-0.300309\pi\)
0.586999 + 0.809588i \(0.300309\pi\)
\(458\) 0 0
\(459\) − 4.31453e6i − 0.955876i
\(460\) 0 0
\(461\) − 173994.i − 0.0381313i −0.999818 0.0190657i \(-0.993931\pi\)
0.999818 0.0190657i \(-0.00606916\pi\)
\(462\) 0 0
\(463\) −4.01277e6 −0.869945 −0.434972 0.900444i \(-0.643242\pi\)
−0.434972 + 0.900444i \(0.643242\pi\)
\(464\) 0 0
\(465\) 293888. 0.0630303
\(466\) 0 0
\(467\) − 774616.i − 0.164359i −0.996618 0.0821796i \(-0.973812\pi\)
0.996618 0.0821796i \(-0.0261881\pi\)
\(468\) 0 0
\(469\) − 4.31475e6i − 0.905782i
\(470\) 0 0
\(471\) 2.82824e6 0.587441
\(472\) 0 0
\(473\) −1.06814e7 −2.19521
\(474\) 0 0
\(475\) − 3.25705e6i − 0.662354i
\(476\) 0 0
\(477\) 3.96986e6i 0.798876i
\(478\) 0 0
\(479\) −2.33530e6 −0.465054 −0.232527 0.972590i \(-0.574699\pi\)
−0.232527 + 0.972590i \(0.574699\pi\)
\(480\) 0 0
\(481\) −6.56385e6 −1.29359
\(482\) 0 0
\(483\) − 5.35142e6i − 1.04376i
\(484\) 0 0
\(485\) − 872060.i − 0.168342i
\(486\) 0 0
\(487\) 1.03947e6 0.198605 0.0993025 0.995057i \(-0.468339\pi\)
0.0993025 + 0.995057i \(0.468339\pi\)
\(488\) 0 0
\(489\) −807488. −0.152709
\(490\) 0 0
\(491\) 7.85092e6i 1.46966i 0.678251 + 0.734830i \(0.262738\pi\)
−0.678251 + 0.734830i \(0.737262\pi\)
\(492\) 0 0
\(493\) − 3.72920e6i − 0.691033i
\(494\) 0 0
\(495\) −1.34322e6 −0.246396
\(496\) 0 0
\(497\) −5.94714e6 −1.07998
\(498\) 0 0
\(499\) − 2.71644e6i − 0.488370i −0.969729 0.244185i \(-0.921480\pi\)
0.969729 0.244185i \(-0.0785204\pi\)
\(500\) 0 0
\(501\) − 2.27955e6i − 0.405747i
\(502\) 0 0
\(503\) −4.62034e6 −0.814242 −0.407121 0.913374i \(-0.633467\pi\)
−0.407121 + 0.913374i \(0.633467\pi\)
\(504\) 0 0
\(505\) −89796.0 −0.0156685
\(506\) 0 0
\(507\) − 882744.i − 0.152516i
\(508\) 0 0
\(509\) − 4.46198e6i − 0.763366i −0.924293 0.381683i \(-0.875345\pi\)
0.924293 0.381683i \(-0.124655\pi\)
\(510\) 0 0
\(511\) −1.11634e7 −1.89122
\(512\) 0 0
\(513\) 3.75411e6 0.629816
\(514\) 0 0
\(515\) 1.51805e6i 0.252213i
\(516\) 0 0
\(517\) 17152.0i 0.00282220i
\(518\) 0 0
\(519\) −3.87499e6 −0.631470
\(520\) 0 0
\(521\) −3.74375e6 −0.604245 −0.302122 0.953269i \(-0.597695\pi\)
−0.302122 + 0.953269i \(0.597695\pi\)
\(522\) 0 0
\(523\) 9.28433e6i 1.48421i 0.670282 + 0.742107i \(0.266173\pi\)
−0.670282 + 0.742107i \(0.733827\pi\)
\(524\) 0 0
\(525\) − 4.87386e6i − 0.771746i
\(526\) 0 0
\(527\) −3.35347e6 −0.525979
\(528\) 0 0
\(529\) 3.90631e6 0.606915
\(530\) 0 0
\(531\) 7.42492e6i 1.14276i
\(532\) 0 0
\(533\) − 117980.i − 0.0179883i
\(534\) 0 0
\(535\) 1.45566e6 0.219875
\(536\) 0 0
\(537\) 3.25344e6 0.486863
\(538\) 0 0
\(539\) − 1.41810e7i − 2.10249i
\(540\) 0 0
\(541\) 862150.i 0.126645i 0.997993 + 0.0633227i \(0.0201697\pi\)
−0.997993 + 0.0633227i \(0.979830\pi\)
\(542\) 0 0
\(543\) 4.56242e6 0.664042
\(544\) 0 0
\(545\) 34804.0 0.00501924
\(546\) 0 0
\(547\) − 1.75442e6i − 0.250707i −0.992112 0.125353i \(-0.959994\pi\)
0.992112 0.125353i \(-0.0400065\pi\)
\(548\) 0 0
\(549\) 2.76770e6i 0.391911i
\(550\) 0 0
\(551\) 3.24482e6 0.455314
\(552\) 0 0
\(553\) −1.43836e7 −2.00012
\(554\) 0 0
\(555\) 1.05930e6i 0.145977i
\(556\) 0 0
\(557\) − 1.00292e7i − 1.36971i −0.728678 0.684856i \(-0.759865\pi\)
0.728678 0.684856i \(-0.240135\pi\)
\(558\) 0 0
\(559\) 1.38300e7 1.87195
\(560\) 0 0
\(561\) −5.48006e6 −0.735154
\(562\) 0 0
\(563\) 5.27460e6i 0.701324i 0.936502 + 0.350662i \(0.114043\pi\)
−0.936502 + 0.350662i \(0.885957\pi\)
\(564\) 0 0
\(565\) − 221116.i − 0.0291406i
\(566\) 0 0
\(567\) −3.42971e6 −0.448023
\(568\) 0 0
\(569\) −8.36940e6 −1.08371 −0.541856 0.840471i \(-0.682278\pi\)
−0.541856 + 0.840471i \(0.682278\pi\)
\(570\) 0 0
\(571\) 4.02702e6i 0.516884i 0.966027 + 0.258442i \(0.0832091\pi\)
−0.966027 + 0.258442i \(0.916791\pi\)
\(572\) 0 0
\(573\) − 1.10899e6i − 0.141105i
\(574\) 0 0
\(575\) 9.41966e6 1.18814
\(576\) 0 0
\(577\) 2.37568e6 0.297063 0.148532 0.988908i \(-0.452545\pi\)
0.148532 + 0.988908i \(0.452545\pi\)
\(578\) 0 0
\(579\) 275856.i 0.0341968i
\(580\) 0 0
\(581\) − 7.86240e6i − 0.966306i
\(582\) 0 0
\(583\) 1.18874e7 1.44849
\(584\) 0 0
\(585\) 1.73916e6 0.210112
\(586\) 0 0
\(587\) − 3.44028e6i − 0.412096i −0.978542 0.206048i \(-0.933940\pi\)
0.978542 0.206048i \(-0.0660603\pi\)
\(588\) 0 0
\(589\) − 2.91789e6i − 0.346562i
\(590\) 0 0
\(591\) 5.14878e6 0.606368
\(592\) 0 0
\(593\) −3.22942e6 −0.377127 −0.188564 0.982061i \(-0.560383\pi\)
−0.188564 + 0.982061i \(0.560383\pi\)
\(594\) 0 0
\(595\) − 3.72154e6i − 0.430953i
\(596\) 0 0
\(597\) 8.85901e6i 1.01730i
\(598\) 0 0
\(599\) −9.29714e6 −1.05872 −0.529361 0.848397i \(-0.677568\pi\)
−0.529361 + 0.848397i \(0.677568\pi\)
\(600\) 0 0
\(601\) 1.12782e7 1.27366 0.636828 0.771006i \(-0.280246\pi\)
0.636828 + 0.771006i \(0.280246\pi\)
\(602\) 0 0
\(603\) 3.71318e6i 0.415865i
\(604\) 0 0
\(605\) 1.76743e6i 0.196315i
\(606\) 0 0
\(607\) −7.00115e6 −0.771255 −0.385627 0.922655i \(-0.626015\pi\)
−0.385627 + 0.922655i \(0.626015\pi\)
\(608\) 0 0
\(609\) 4.85555e6 0.530512
\(610\) 0 0
\(611\) − 22208.0i − 0.00240661i
\(612\) 0 0
\(613\) 6.19432e6i 0.665798i 0.942962 + 0.332899i \(0.108027\pi\)
−0.942962 + 0.332899i \(0.891973\pi\)
\(614\) 0 0
\(615\) −19040.0 −0.00202992
\(616\) 0 0
\(617\) 2.62407e6 0.277500 0.138750 0.990327i \(-0.455692\pi\)
0.138750 + 0.990327i \(0.455692\pi\)
\(618\) 0 0
\(619\) − 2.83721e6i − 0.297622i −0.988866 0.148811i \(-0.952455\pi\)
0.988866 0.148811i \(-0.0475446\pi\)
\(620\) 0 0
\(621\) 1.08572e7i 1.12977i
\(622\) 0 0
\(623\) −2.63756e7 −2.72259
\(624\) 0 0
\(625\) 7.96654e6 0.815774
\(626\) 0 0
\(627\) − 4.76826e6i − 0.484385i
\(628\) 0 0
\(629\) − 1.20873e7i − 1.21816i
\(630\) 0 0
\(631\) 1.29656e7 1.29634 0.648170 0.761496i \(-0.275535\pi\)
0.648170 + 0.761496i \(0.275535\pi\)
\(632\) 0 0
\(633\) 1.83981e6 0.182500
\(634\) 0 0
\(635\) 14336.0i 0.00141089i
\(636\) 0 0
\(637\) 1.83612e7i 1.79288i
\(638\) 0 0
\(639\) 5.11797e6 0.495844
\(640\) 0 0
\(641\) −1.16798e7 −1.12276 −0.561382 0.827557i \(-0.689730\pi\)
−0.561382 + 0.827557i \(0.689730\pi\)
\(642\) 0 0
\(643\) − 7.02732e6i − 0.670289i −0.942167 0.335145i \(-0.891215\pi\)
0.942167 0.335145i \(-0.108785\pi\)
\(644\) 0 0
\(645\) − 2.23194e6i − 0.211243i
\(646\) 0 0
\(647\) −9.72821e6 −0.913634 −0.456817 0.889561i \(-0.651011\pi\)
−0.456817 + 0.889561i \(0.651011\pi\)
\(648\) 0 0
\(649\) 2.22333e7 2.07201
\(650\) 0 0
\(651\) − 4.36634e6i − 0.403798i
\(652\) 0 0
\(653\) − 9.81425e6i − 0.900688i −0.892855 0.450344i \(-0.851301\pi\)
0.892855 0.450344i \(-0.148699\pi\)
\(654\) 0 0
\(655\) 317296. 0.0288976
\(656\) 0 0
\(657\) 9.60693e6 0.868303
\(658\) 0 0
\(659\) − 1.46652e7i − 1.31545i −0.753259 0.657724i \(-0.771519\pi\)
0.753259 0.657724i \(-0.228481\pi\)
\(660\) 0 0
\(661\) 1.41837e7i 1.26265i 0.775517 + 0.631327i \(0.217489\pi\)
−0.775517 + 0.631327i \(0.782511\pi\)
\(662\) 0 0
\(663\) 7.09546e6 0.626897
\(664\) 0 0
\(665\) 3.23814e6 0.283950
\(666\) 0 0
\(667\) 9.38429e6i 0.816745i
\(668\) 0 0
\(669\) 8.71578e6i 0.752906i
\(670\) 0 0
\(671\) 8.28763e6 0.710598
\(672\) 0 0
\(673\) 5.49941e6 0.468035 0.234018 0.972232i \(-0.424813\pi\)
0.234018 + 0.972232i \(0.424813\pi\)
\(674\) 0 0
\(675\) 9.88830e6i 0.835338i
\(676\) 0 0
\(677\) − 1.77375e7i − 1.48737i −0.668528 0.743687i \(-0.733075\pi\)
0.668528 0.743687i \(-0.266925\pi\)
\(678\) 0 0
\(679\) −1.29563e7 −1.07847
\(680\) 0 0
\(681\) −5.49638e6 −0.454160
\(682\) 0 0
\(683\) − 9.39670e6i − 0.770768i −0.922756 0.385384i \(-0.874069\pi\)
0.922756 0.385384i \(-0.125931\pi\)
\(684\) 0 0
\(685\) − 745332.i − 0.0606909i
\(686\) 0 0
\(687\) −5.59784e6 −0.452510
\(688\) 0 0
\(689\) −1.53915e7 −1.23519
\(690\) 0 0
\(691\) 1.34767e7i 1.07371i 0.843673 + 0.536857i \(0.180389\pi\)
−0.843673 + 0.536857i \(0.819611\pi\)
\(692\) 0 0
\(693\) 1.99564e7i 1.57851i
\(694\) 0 0
\(695\) 277424. 0.0217862
\(696\) 0 0
\(697\) 217260. 0.0169394
\(698\) 0 0
\(699\) − 7.50178e6i − 0.580726i
\(700\) 0 0
\(701\) 2.15594e7i 1.65707i 0.559935 + 0.828536i \(0.310826\pi\)
−0.559935 + 0.828536i \(0.689174\pi\)
\(702\) 0 0
\(703\) 1.05173e7 0.802631
\(704\) 0 0
\(705\) −3584.00 −0.000271578 0
\(706\) 0 0
\(707\) 1.33411e6i 0.100379i
\(708\) 0 0
\(709\) 6.38165e6i 0.476779i 0.971170 + 0.238390i \(0.0766195\pi\)
−0.971170 + 0.238390i \(0.923380\pi\)
\(710\) 0 0
\(711\) 1.23782e7 0.918298
\(712\) 0 0
\(713\) 8.43878e6 0.621664
\(714\) 0 0
\(715\) − 5.20778e6i − 0.380967i
\(716\) 0 0
\(717\) − 5.14790e6i − 0.373966i
\(718\) 0 0
\(719\) 1.63566e7 1.17997 0.589986 0.807413i \(-0.299133\pi\)
0.589986 + 0.807413i \(0.299133\pi\)
\(720\) 0 0
\(721\) 2.25539e7 1.61578
\(722\) 0 0
\(723\) 1.25826e6i 0.0895207i
\(724\) 0 0
\(725\) 8.54682e6i 0.603893i
\(726\) 0 0
\(727\) −2.13130e7 −1.49558 −0.747788 0.663937i \(-0.768884\pi\)
−0.747788 + 0.663937i \(0.768884\pi\)
\(728\) 0 0
\(729\) −3.61141e6 −0.251686
\(730\) 0 0
\(731\) 2.54680e7i 1.76279i
\(732\) 0 0
\(733\) 1.21571e7i 0.835737i 0.908507 + 0.417869i \(0.137223\pi\)
−0.908507 + 0.417869i \(0.862777\pi\)
\(734\) 0 0
\(735\) 2.96318e6 0.202321
\(736\) 0 0
\(737\) 1.11188e7 0.754030
\(738\) 0 0
\(739\) 1.92337e7i 1.29555i 0.761834 + 0.647773i \(0.224299\pi\)
−0.761834 + 0.647773i \(0.775701\pi\)
\(740\) 0 0
\(741\) 6.17382e6i 0.413056i
\(742\) 0 0
\(743\) 1.66565e6 0.110691 0.0553454 0.998467i \(-0.482374\pi\)
0.0553454 + 0.998467i \(0.482374\pi\)
\(744\) 0 0
\(745\) −6.33066e6 −0.417886
\(746\) 0 0
\(747\) 6.76620e6i 0.443653i
\(748\) 0 0
\(749\) − 2.16270e7i − 1.40861i
\(750\) 0 0
\(751\) −9.81290e6 −0.634888 −0.317444 0.948277i \(-0.602825\pi\)
−0.317444 + 0.948277i \(0.602825\pi\)
\(752\) 0 0
\(753\) 1.26883e7 0.815486
\(754\) 0 0
\(755\) 3.68592e6i 0.235331i
\(756\) 0 0
\(757\) 1.92753e7i 1.22254i 0.791423 + 0.611269i \(0.209341\pi\)
−0.791423 + 0.611269i \(0.790659\pi\)
\(758\) 0 0
\(759\) 1.37902e7 0.868893
\(760\) 0 0
\(761\) 1.17863e7 0.737762 0.368881 0.929477i \(-0.379741\pi\)
0.368881 + 0.929477i \(0.379741\pi\)
\(762\) 0 0
\(763\) − 517088.i − 0.0321553i
\(764\) 0 0
\(765\) 3.20267e6i 0.197860i
\(766\) 0 0
\(767\) −2.87871e7 −1.76689
\(768\) 0 0
\(769\) 1.22941e7 0.749690 0.374845 0.927087i \(-0.377696\pi\)
0.374845 + 0.927087i \(0.377696\pi\)
\(770\) 0 0
\(771\) − 5.23467e6i − 0.317142i
\(772\) 0 0
\(773\) − 2.57086e6i − 0.154750i −0.997002 0.0773749i \(-0.975346\pi\)
0.997002 0.0773749i \(-0.0246538\pi\)
\(774\) 0 0
\(775\) 7.68570e6 0.459652
\(776\) 0 0
\(777\) 1.57381e7 0.935190
\(778\) 0 0
\(779\) 189040.i 0.0111612i
\(780\) 0 0
\(781\) − 1.53253e7i − 0.899046i
\(782\) 0 0
\(783\) −9.85117e6 −0.574227
\(784\) 0 0
\(785\) −4.94942e6 −0.286669
\(786\) 0 0
\(787\) − 5.54594e6i − 0.319182i −0.987183 0.159591i \(-0.948982\pi\)
0.987183 0.159591i \(-0.0510176\pi\)
\(788\) 0 0
\(789\) − 2.64154e6i − 0.151065i
\(790\) 0 0
\(791\) −3.28515e6 −0.186687
\(792\) 0 0
\(793\) −1.07306e7 −0.605958
\(794\) 0 0
\(795\) 2.48394e6i 0.139387i
\(796\) 0 0
\(797\) 2.09610e7i 1.16887i 0.811440 + 0.584436i \(0.198684\pi\)
−0.811440 + 0.584436i \(0.801316\pi\)
\(798\) 0 0
\(799\) 40896.0 0.00226628
\(800\) 0 0
\(801\) 2.26983e7 1.25000
\(802\) 0 0
\(803\) − 2.87671e7i − 1.57437i
\(804\) 0 0
\(805\) 9.36499e6i 0.509352i
\(806\) 0 0
\(807\) −1.25565e7 −0.678709
\(808\) 0 0
\(809\) −2.70297e7 −1.45201 −0.726005 0.687690i \(-0.758625\pi\)
−0.726005 + 0.687690i \(0.758625\pi\)
\(810\) 0 0
\(811\) − 2.13052e6i − 0.113745i −0.998381 0.0568727i \(-0.981887\pi\)
0.998381 0.0568727i \(-0.0181129\pi\)
\(812\) 0 0
\(813\) − 7.66234e6i − 0.406570i
\(814\) 0 0
\(815\) 1.41310e6 0.0745212
\(816\) 0 0
\(817\) −2.21599e7 −1.16148
\(818\) 0 0
\(819\) − 2.58390e7i − 1.34607i
\(820\) 0 0
\(821\) − 1.58060e7i − 0.818396i −0.912446 0.409198i \(-0.865809\pi\)
0.912446 0.409198i \(-0.134191\pi\)
\(822\) 0 0
\(823\) −2.28848e7 −1.17773 −0.588867 0.808230i \(-0.700426\pi\)
−0.588867 + 0.808230i \(0.700426\pi\)
\(824\) 0 0
\(825\) 1.25596e7 0.642450
\(826\) 0 0
\(827\) − 2.55336e7i − 1.29822i −0.760695 0.649109i \(-0.775142\pi\)
0.760695 0.649109i \(-0.224858\pi\)
\(828\) 0 0
\(829\) 8.31786e6i 0.420364i 0.977662 + 0.210182i \(0.0674056\pi\)
−0.977662 + 0.210182i \(0.932594\pi\)
\(830\) 0 0
\(831\) 4.52350e6 0.227234
\(832\) 0 0
\(833\) −3.38120e7 −1.68834
\(834\) 0 0
\(835\) 3.98922e6i 0.198003i
\(836\) 0 0
\(837\) 8.85862e6i 0.437072i
\(838\) 0 0
\(839\) 3.66261e7 1.79633 0.898164 0.439660i \(-0.144901\pi\)
0.898164 + 0.439660i \(0.144901\pi\)
\(840\) 0 0
\(841\) 1.19964e7 0.584873
\(842\) 0 0
\(843\) 1.07302e7i 0.520041i
\(844\) 0 0
\(845\) 1.54480e6i 0.0744271i
\(846\) 0 0
\(847\) 2.62590e7 1.25768
\(848\) 0 0
\(849\) 5.87411e6 0.279687
\(850\) 0 0
\(851\) 3.04169e7i 1.43976i
\(852\) 0 0
\(853\) − 1.74802e7i − 0.822571i −0.911507 0.411286i \(-0.865080\pi\)
0.911507 0.411286i \(-0.134920\pi\)
\(854\) 0 0
\(855\) −2.78667e6 −0.130368
\(856\) 0 0
\(857\) −9.31062e6 −0.433038 −0.216519 0.976278i \(-0.569470\pi\)
−0.216519 + 0.976278i \(0.569470\pi\)
\(858\) 0 0
\(859\) − 3.49525e7i − 1.61620i −0.589045 0.808101i \(-0.700496\pi\)
0.589045 0.808101i \(-0.299504\pi\)
\(860\) 0 0
\(861\) 282880.i 0.0130045i
\(862\) 0 0
\(863\) 2.02349e7 0.924858 0.462429 0.886656i \(-0.346978\pi\)
0.462429 + 0.886656i \(0.346978\pi\)
\(864\) 0 0
\(865\) 6.78124e6 0.308155
\(866\) 0 0
\(867\) 1.70742e6i 0.0771421i
\(868\) 0 0
\(869\) − 3.70655e7i − 1.66502i
\(870\) 0 0
\(871\) −1.43963e7 −0.642994
\(872\) 0 0
\(873\) 1.11499e7 0.495149
\(874\) 0 0
\(875\) 1.76292e7i 0.778419i
\(876\) 0 0
\(877\) − 1.98343e7i − 0.870797i −0.900238 0.435398i \(-0.856608\pi\)
0.900238 0.435398i \(-0.143392\pi\)
\(878\) 0 0
\(879\) −9.06562e6 −0.395754
\(880\) 0 0
\(881\) −3.81023e7 −1.65391 −0.826954 0.562270i \(-0.809928\pi\)
−0.826954 + 0.562270i \(0.809928\pi\)
\(882\) 0 0
\(883\) 2.41560e7i 1.04261i 0.853369 + 0.521307i \(0.174555\pi\)
−0.853369 + 0.521307i \(0.825445\pi\)
\(884\) 0 0
\(885\) 4.64576e6i 0.199388i
\(886\) 0 0
\(887\) 2.02368e7 0.863638 0.431819 0.901960i \(-0.357872\pi\)
0.431819 + 0.901960i \(0.357872\pi\)
\(888\) 0 0
\(889\) 212992. 0.00903876
\(890\) 0 0
\(891\) − 8.83810e6i − 0.372962i
\(892\) 0 0
\(893\) 35584.0i 0.00149323i
\(894\) 0 0
\(895\) −5.69352e6 −0.237587
\(896\) 0 0
\(897\) −1.78552e7 −0.740942
\(898\) 0 0
\(899\) 7.65683e6i 0.315973i
\(900\) 0 0
\(901\) − 2.83435e7i − 1.16316i
\(902\) 0 0
\(903\) −3.31602e7 −1.35331
\(904\) 0 0
\(905\) −7.98423e6 −0.324050
\(906\) 0 0
\(907\) 1.97976e7i 0.799088i 0.916714 + 0.399544i \(0.130831\pi\)
−0.916714 + 0.399544i \(0.869169\pi\)
\(908\) 0 0
\(909\) − 1.14811e6i − 0.0460863i
\(910\) 0 0
\(911\) 2.13242e7 0.851288 0.425644 0.904891i \(-0.360048\pi\)
0.425644 + 0.904891i \(0.360048\pi\)
\(912\) 0 0
\(913\) 2.02608e7 0.804414
\(914\) 0 0
\(915\) 1.73174e6i 0.0683803i
\(916\) 0 0
\(917\) − 4.71411e6i − 0.185130i
\(918\) 0 0
\(919\) 3.49941e7 1.36680 0.683401 0.730043i \(-0.260500\pi\)
0.683401 + 0.730043i \(0.260500\pi\)
\(920\) 0 0
\(921\) −2.33101e7 −0.905515
\(922\) 0 0
\(923\) 1.98428e7i 0.766655i
\(924\) 0 0
\(925\) 2.77025e7i 1.06455i
\(926\) 0 0
\(927\) −1.94093e7 −0.741842
\(928\) 0 0
\(929\) −2.88107e7 −1.09525 −0.547627 0.836722i \(-0.684469\pi\)
−0.547627 + 0.836722i \(0.684469\pi\)
\(930\) 0 0
\(931\) − 2.94202e7i − 1.11243i
\(932\) 0 0
\(933\) − 1.15050e7i − 0.432697i
\(934\) 0 0
\(935\) 9.59011e6 0.358752
\(936\) 0 0
\(937\) −1.28854e7 −0.479457 −0.239729 0.970840i \(-0.577059\pi\)
−0.239729 + 0.970840i \(0.577059\pi\)
\(938\) 0 0
\(939\) 1.10080e7i 0.407424i
\(940\) 0 0
\(941\) − 5.27615e7i − 1.94242i −0.238227 0.971210i \(-0.576566\pi\)
0.238227 0.971210i \(-0.423434\pi\)
\(942\) 0 0
\(943\) −546720. −0.0200210
\(944\) 0 0
\(945\) −9.83091e6 −0.358108
\(946\) 0 0
\(947\) − 5.53961e6i − 0.200726i −0.994951 0.100363i \(-0.968000\pi\)
0.994951 0.100363i \(-0.0320004\pi\)
\(948\) 0 0
\(949\) 3.72470e7i 1.34253i
\(950\) 0 0
\(951\) 9.87950e6 0.354229
\(952\) 0 0
\(953\) 228102. 0.00813574 0.00406787 0.999992i \(-0.498705\pi\)
0.00406787 + 0.999992i \(0.498705\pi\)
\(954\) 0 0
\(955\) 1.94074e6i 0.0688586i
\(956\) 0 0
\(957\) 1.25124e7i 0.441632i
\(958\) 0 0
\(959\) −1.10735e7 −0.388811
\(960\) 0 0
\(961\) −2.17438e7 −0.759498
\(962\) 0 0
\(963\) 1.86117e7i 0.646726i
\(964\) 0 0
\(965\) − 482748.i − 0.0166879i
\(966\) 0 0
\(967\) 7.36709e6 0.253355 0.126678 0.991944i \(-0.459569\pi\)
0.126678 + 0.991944i \(0.459569\pi\)
\(968\) 0 0
\(969\) −1.13691e7 −0.388970
\(970\) 0 0
\(971\) − 1.91161e7i − 0.650654i −0.945602 0.325327i \(-0.894526\pi\)
0.945602 0.325327i \(-0.105474\pi\)
\(972\) 0 0
\(973\) − 4.12173e6i − 0.139572i
\(974\) 0 0
\(975\) −1.62618e7 −0.547844
\(976\) 0 0
\(977\) 5.57040e7 1.86702 0.933512 0.358546i \(-0.116727\pi\)
0.933512 + 0.358546i \(0.116727\pi\)
\(978\) 0 0
\(979\) − 6.79680e7i − 2.26646i
\(980\) 0 0
\(981\) 444994.i 0.0147632i
\(982\) 0 0
\(983\) −1.55469e7 −0.513167 −0.256584 0.966522i \(-0.582597\pi\)
−0.256584 + 0.966522i \(0.582597\pi\)
\(984\) 0 0
\(985\) −9.01037e6 −0.295905
\(986\) 0 0
\(987\) 53248.0i 0.00173984i
\(988\) 0 0
\(989\) − 6.40884e7i − 2.08348i
\(990\) 0 0
\(991\) −2.36890e7 −0.766237 −0.383118 0.923699i \(-0.625150\pi\)
−0.383118 + 0.923699i \(0.625150\pi\)
\(992\) 0 0
\(993\) 1.19144e7 0.383442
\(994\) 0 0
\(995\) − 1.55033e7i − 0.496438i
\(996\) 0 0
\(997\) − 3.71720e6i − 0.118434i −0.998245 0.0592172i \(-0.981140\pi\)
0.998245 0.0592172i \(-0.0188605\pi\)
\(998\) 0 0
\(999\) −3.19302e7 −1.01225
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 256.6.b.b.129.2 2
4.3 odd 2 256.6.b.h.129.1 2
8.3 odd 2 256.6.b.h.129.2 2
8.5 even 2 inner 256.6.b.b.129.1 2
16.3 odd 4 64.6.a.e.1.1 1
16.5 even 4 32.6.a.c.1.1 yes 1
16.11 odd 4 32.6.a.a.1.1 1
16.13 even 4 64.6.a.c.1.1 1
48.5 odd 4 288.6.a.e.1.1 1
48.11 even 4 288.6.a.d.1.1 1
48.29 odd 4 576.6.a.v.1.1 1
48.35 even 4 576.6.a.u.1.1 1
80.27 even 4 800.6.c.a.449.2 2
80.37 odd 4 800.6.c.b.449.1 2
80.43 even 4 800.6.c.a.449.1 2
80.53 odd 4 800.6.c.b.449.2 2
80.59 odd 4 800.6.a.e.1.1 1
80.69 even 4 800.6.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
32.6.a.a.1.1 1 16.11 odd 4
32.6.a.c.1.1 yes 1 16.5 even 4
64.6.a.c.1.1 1 16.13 even 4
64.6.a.e.1.1 1 16.3 odd 4
256.6.b.b.129.1 2 8.5 even 2 inner
256.6.b.b.129.2 2 1.1 even 1 trivial
256.6.b.h.129.1 2 4.3 odd 2
256.6.b.h.129.2 2 8.3 odd 2
288.6.a.d.1.1 1 48.11 even 4
288.6.a.e.1.1 1 48.5 odd 4
576.6.a.u.1.1 1 48.35 even 4
576.6.a.v.1.1 1 48.29 odd 4
800.6.a.a.1.1 1 80.69 even 4
800.6.a.e.1.1 1 80.59 odd 4
800.6.c.a.449.1 2 80.43 even 4
800.6.c.a.449.2 2 80.27 even 4
800.6.c.b.449.1 2 80.37 odd 4
800.6.c.b.449.2 2 80.53 odd 4