Properties

Label 384.7.b.d.319.1
Level $384$
Weight $7$
Character 384.319
Analytic conductor $88.341$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 384.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(88.3407681100\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.1731891456.1
Defining polynomial: \(x^{8} - 9 x^{6} + 65 x^{4} - 144 x^{2} + 256\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{32}\cdot 3^{12} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 319.1
Root \(2.21837 + 1.28078i\) of defining polynomial
Character \(\chi\) \(=\) 384.319
Dual form 384.7.b.d.319.4

$q$-expansion

\(f(q)\) \(=\) \(q-15.5885 q^{3} -20.0000i q^{5} -529.850i q^{7} +243.000 q^{9} +O(q^{10})\) \(q-15.5885 q^{3} -20.0000i q^{5} -529.850i q^{7} +243.000 q^{9} -435.847 q^{11} +341.182i q^{13} +311.769i q^{15} +7682.73 q^{17} +4300.52 q^{19} +8259.54i q^{21} +3175.32i q^{23} +15225.0 q^{25} -3788.00 q^{27} -19409.3i q^{29} -15297.3i q^{31} +6794.18 q^{33} -10597.0 q^{35} +61969.9i q^{37} -5318.50i q^{39} -33740.2 q^{41} +99312.6 q^{43} -4860.00i q^{45} +17726.4i q^{47} -163092. q^{49} -119762. q^{51} +224406. i q^{53} +8716.94i q^{55} -67038.5 q^{57} -199557. q^{59} -45671.7i q^{61} -128754. i q^{63} +6823.63 q^{65} +496811. q^{67} -49498.3i q^{69} -452032. i q^{71} +394349. q^{73} -237334. q^{75} +230933. i q^{77} -571742. i q^{79} +59049.0 q^{81} -324465. q^{83} -153655. i q^{85} +302561. i q^{87} +758607. q^{89} +180775. q^{91} +238462. i q^{93} -86010.5i q^{95} +25015.4 q^{97} -105911. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 1944 q^{9} + O(q^{10}) \) \( 8 q + 1944 q^{9} + 4464 q^{17} + 121800 q^{25} - 116640 q^{33} - 98928 q^{41} - 278776 q^{49} - 23328 q^{57} - 230400 q^{65} + 76912 q^{73} + 472392 q^{81} + 6638832 q^{89} - 4929680 q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(133\) \(257\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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
\(3\) −15.5885 −0.577350
\(4\) 0 0
\(5\) − 20.0000i − 0.160000i −0.996795 0.0800000i \(-0.974508\pi\)
0.996795 0.0800000i \(-0.0254920\pi\)
\(6\) 0 0
\(7\) − 529.850i − 1.54475i −0.635165 0.772376i \(-0.719068\pi\)
0.635165 0.772376i \(-0.280932\pi\)
\(8\) 0 0
\(9\) 243.000 0.333333
\(10\) 0 0
\(11\) −435.847 −0.327458 −0.163729 0.986505i \(-0.552352\pi\)
−0.163729 + 0.986505i \(0.552352\pi\)
\(12\) 0 0
\(13\) 341.182i 0.155294i 0.996981 + 0.0776472i \(0.0247408\pi\)
−0.996981 + 0.0776472i \(0.975259\pi\)
\(14\) 0 0
\(15\) 311.769i 0.0923760i
\(16\) 0 0
\(17\) 7682.73 1.56375 0.781877 0.623432i \(-0.214262\pi\)
0.781877 + 0.623432i \(0.214262\pi\)
\(18\) 0 0
\(19\) 4300.52 0.626990 0.313495 0.949590i \(-0.398500\pi\)
0.313495 + 0.949590i \(0.398500\pi\)
\(20\) 0 0
\(21\) 8259.54i 0.891863i
\(22\) 0 0
\(23\) 3175.32i 0.260978i 0.991450 + 0.130489i \(0.0416547\pi\)
−0.991450 + 0.130489i \(0.958345\pi\)
\(24\) 0 0
\(25\) 15225.0 0.974400
\(26\) 0 0
\(27\) −3788.00 −0.192450
\(28\) 0 0
\(29\) − 19409.3i − 0.795821i −0.917424 0.397910i \(-0.869736\pi\)
0.917424 0.397910i \(-0.130264\pi\)
\(30\) 0 0
\(31\) − 15297.3i − 0.513488i −0.966479 0.256744i \(-0.917350\pi\)
0.966479 0.256744i \(-0.0826497\pi\)
\(32\) 0 0
\(33\) 6794.18 0.189058
\(34\) 0 0
\(35\) −10597.0 −0.247160
\(36\) 0 0
\(37\) 61969.9i 1.22342i 0.791082 + 0.611710i \(0.209518\pi\)
−0.791082 + 0.611710i \(0.790482\pi\)
\(38\) 0 0
\(39\) − 5318.50i − 0.0896592i
\(40\) 0 0
\(41\) −33740.2 −0.489549 −0.244774 0.969580i \(-0.578714\pi\)
−0.244774 + 0.969580i \(0.578714\pi\)
\(42\) 0 0
\(43\) 99312.6 1.24911 0.624553 0.780983i \(-0.285281\pi\)
0.624553 + 0.780983i \(0.285281\pi\)
\(44\) 0 0
\(45\) − 4860.00i − 0.0533333i
\(46\) 0 0
\(47\) 17726.4i 0.170737i 0.996349 + 0.0853685i \(0.0272068\pi\)
−0.996349 + 0.0853685i \(0.972793\pi\)
\(48\) 0 0
\(49\) −163092. −1.38626
\(50\) 0 0
\(51\) −119762. −0.902834
\(52\) 0 0
\(53\) 224406.i 1.50733i 0.657260 + 0.753664i \(0.271715\pi\)
−0.657260 + 0.753664i \(0.728285\pi\)
\(54\) 0 0
\(55\) 8716.94i 0.0523933i
\(56\) 0 0
\(57\) −67038.5 −0.361993
\(58\) 0 0
\(59\) −199557. −0.971651 −0.485826 0.874056i \(-0.661481\pi\)
−0.485826 + 0.874056i \(0.661481\pi\)
\(60\) 0 0
\(61\) − 45671.7i − 0.201214i −0.994926 0.100607i \(-0.967922\pi\)
0.994926 0.100607i \(-0.0320784\pi\)
\(62\) 0 0
\(63\) − 128754.i − 0.514917i
\(64\) 0 0
\(65\) 6823.63 0.0248471
\(66\) 0 0
\(67\) 496811. 1.65183 0.825917 0.563791i \(-0.190658\pi\)
0.825917 + 0.563791i \(0.190658\pi\)
\(68\) 0 0
\(69\) − 49498.3i − 0.150676i
\(70\) 0 0
\(71\) − 452032.i − 1.26297i −0.775387 0.631487i \(-0.782445\pi\)
0.775387 0.631487i \(-0.217555\pi\)
\(72\) 0 0
\(73\) 394349. 1.01371 0.506853 0.862032i \(-0.330809\pi\)
0.506853 + 0.862032i \(0.330809\pi\)
\(74\) 0 0
\(75\) −237334. −0.562570
\(76\) 0 0
\(77\) 230933.i 0.505842i
\(78\) 0 0
\(79\) − 571742.i − 1.15963i −0.814749 0.579814i \(-0.803125\pi\)
0.814749 0.579814i \(-0.196875\pi\)
\(80\) 0 0
\(81\) 59049.0 0.111111
\(82\) 0 0
\(83\) −324465. −0.567458 −0.283729 0.958904i \(-0.591572\pi\)
−0.283729 + 0.958904i \(0.591572\pi\)
\(84\) 0 0
\(85\) − 153655.i − 0.250201i
\(86\) 0 0
\(87\) 302561.i 0.459467i
\(88\) 0 0
\(89\) 758607. 1.07609 0.538043 0.842918i \(-0.319164\pi\)
0.538043 + 0.842918i \(0.319164\pi\)
\(90\) 0 0
\(91\) 180775. 0.239891
\(92\) 0 0
\(93\) 238462.i 0.296462i
\(94\) 0 0
\(95\) − 86010.5i − 0.100318i
\(96\) 0 0
\(97\) 25015.4 0.0274089 0.0137045 0.999906i \(-0.495638\pi\)
0.0137045 + 0.999906i \(0.495638\pi\)
\(98\) 0 0
\(99\) −105911. −0.109153
\(100\) 0 0
\(101\) − 1.60937e6i − 1.56204i −0.624508 0.781018i \(-0.714700\pi\)
0.624508 0.781018i \(-0.285300\pi\)
\(102\) 0 0
\(103\) − 647116.i − 0.592203i −0.955156 0.296101i \(-0.904313\pi\)
0.955156 0.296101i \(-0.0956867\pi\)
\(104\) 0 0
\(105\) 165191. 0.142698
\(106\) 0 0
\(107\) 233041. 0.190231 0.0951156 0.995466i \(-0.469678\pi\)
0.0951156 + 0.995466i \(0.469678\pi\)
\(108\) 0 0
\(109\) 525849.i 0.406052i 0.979173 + 0.203026i \(0.0650776\pi\)
−0.979173 + 0.203026i \(0.934922\pi\)
\(110\) 0 0
\(111\) − 966015.i − 0.706342i
\(112\) 0 0
\(113\) −1.21523e6 −0.842216 −0.421108 0.907011i \(-0.638359\pi\)
−0.421108 + 0.907011i \(0.638359\pi\)
\(114\) 0 0
\(115\) 63506.4 0.0417565
\(116\) 0 0
\(117\) 82907.1i 0.0517648i
\(118\) 0 0
\(119\) − 4.07069e6i − 2.41561i
\(120\) 0 0
\(121\) −1.58160e6 −0.892771
\(122\) 0 0
\(123\) 525957. 0.282641
\(124\) 0 0
\(125\) − 617000.i − 0.315904i
\(126\) 0 0
\(127\) − 3.23094e6i − 1.57731i −0.614836 0.788655i \(-0.710778\pi\)
0.614836 0.788655i \(-0.289222\pi\)
\(128\) 0 0
\(129\) −1.54813e6 −0.721171
\(130\) 0 0
\(131\) 1.80119e6 0.801209 0.400604 0.916251i \(-0.368800\pi\)
0.400604 + 0.916251i \(0.368800\pi\)
\(132\) 0 0
\(133\) − 2.27863e6i − 0.968544i
\(134\) 0 0
\(135\) 75759.9i 0.0307920i
\(136\) 0 0
\(137\) 50362.3 0.0195859 0.00979295 0.999952i \(-0.496883\pi\)
0.00979295 + 0.999952i \(0.496883\pi\)
\(138\) 0 0
\(139\) −4.23758e6 −1.57788 −0.788940 0.614471i \(-0.789370\pi\)
−0.788940 + 0.614471i \(0.789370\pi\)
\(140\) 0 0
\(141\) − 276328.i − 0.0985751i
\(142\) 0 0
\(143\) − 148703.i − 0.0508524i
\(144\) 0 0
\(145\) −388185. −0.127331
\(146\) 0 0
\(147\) 2.54235e6 0.800357
\(148\) 0 0
\(149\) 4.40006e6i 1.33015i 0.746778 + 0.665073i \(0.231600\pi\)
−0.746778 + 0.665073i \(0.768400\pi\)
\(150\) 0 0
\(151\) − 2.17763e6i − 0.632488i −0.948678 0.316244i \(-0.897578\pi\)
0.948678 0.316244i \(-0.102422\pi\)
\(152\) 0 0
\(153\) 1.86690e6 0.521252
\(154\) 0 0
\(155\) −305946. −0.0821580
\(156\) 0 0
\(157\) − 5.73506e6i − 1.48197i −0.671522 0.740985i \(-0.734359\pi\)
0.671522 0.740985i \(-0.265641\pi\)
\(158\) 0 0
\(159\) − 3.49815e6i − 0.870256i
\(160\) 0 0
\(161\) 1.68244e6 0.403147
\(162\) 0 0
\(163\) −2.99695e6 −0.692016 −0.346008 0.938232i \(-0.612463\pi\)
−0.346008 + 0.938232i \(0.612463\pi\)
\(164\) 0 0
\(165\) − 135884.i − 0.0302493i
\(166\) 0 0
\(167\) − 8.03225e6i − 1.72460i −0.506400 0.862298i \(-0.669024\pi\)
0.506400 0.862298i \(-0.330976\pi\)
\(168\) 0 0
\(169\) 4.71040e6 0.975884
\(170\) 0 0
\(171\) 1.04503e6 0.208997
\(172\) 0 0
\(173\) 2.42976e6i 0.469272i 0.972083 + 0.234636i \(0.0753898\pi\)
−0.972083 + 0.234636i \(0.924610\pi\)
\(174\) 0 0
\(175\) − 8.06697e6i − 1.50521i
\(176\) 0 0
\(177\) 3.11078e6 0.560983
\(178\) 0 0
\(179\) 5.34862e6 0.932573 0.466286 0.884634i \(-0.345592\pi\)
0.466286 + 0.884634i \(0.345592\pi\)
\(180\) 0 0
\(181\) 9.11072e6i 1.53645i 0.640183 + 0.768223i \(0.278859\pi\)
−0.640183 + 0.768223i \(0.721141\pi\)
\(182\) 0 0
\(183\) 711952.i 0.116171i
\(184\) 0 0
\(185\) 1.23940e6 0.195747
\(186\) 0 0
\(187\) −3.34849e6 −0.512064
\(188\) 0 0
\(189\) 2.00707e6i 0.297288i
\(190\) 0 0
\(191\) − 1.19063e6i − 0.170875i −0.996344 0.0854373i \(-0.972771\pi\)
0.996344 0.0854373i \(-0.0272287\pi\)
\(192\) 0 0
\(193\) 2.07789e6 0.289034 0.144517 0.989502i \(-0.453837\pi\)
0.144517 + 0.989502i \(0.453837\pi\)
\(194\) 0 0
\(195\) −106370. −0.0143455
\(196\) 0 0
\(197\) − 2.19700e6i − 0.287364i −0.989624 0.143682i \(-0.954106\pi\)
0.989624 0.143682i \(-0.0458942\pi\)
\(198\) 0 0
\(199\) − 6.24588e6i − 0.792564i −0.918129 0.396282i \(-0.870300\pi\)
0.918129 0.396282i \(-0.129700\pi\)
\(200\) 0 0
\(201\) −7.74451e6 −0.953687
\(202\) 0 0
\(203\) −1.02840e7 −1.22935
\(204\) 0 0
\(205\) 674804.i 0.0783278i
\(206\) 0 0
\(207\) 771603.i 0.0869927i
\(208\) 0 0
\(209\) −1.87437e6 −0.205313
\(210\) 0 0
\(211\) 1.11658e7 1.18862 0.594311 0.804235i \(-0.297425\pi\)
0.594311 + 0.804235i \(0.297425\pi\)
\(212\) 0 0
\(213\) 7.04648e6i 0.729178i
\(214\) 0 0
\(215\) − 1.98625e6i − 0.199857i
\(216\) 0 0
\(217\) −8.10528e6 −0.793211
\(218\) 0 0
\(219\) −6.14730e6 −0.585264
\(220\) 0 0
\(221\) 2.62121e6i 0.242842i
\(222\) 0 0
\(223\) − 658854.i − 0.0594120i −0.999559 0.0297060i \(-0.990543\pi\)
0.999559 0.0297060i \(-0.00945711\pi\)
\(224\) 0 0
\(225\) 3.69967e6 0.324800
\(226\) 0 0
\(227\) 1.04595e7 0.894198 0.447099 0.894484i \(-0.352457\pi\)
0.447099 + 0.894484i \(0.352457\pi\)
\(228\) 0 0
\(229\) − 1.94802e7i − 1.62214i −0.584951 0.811069i \(-0.698886\pi\)
0.584951 0.811069i \(-0.301114\pi\)
\(230\) 0 0
\(231\) − 3.59990e6i − 0.292048i
\(232\) 0 0
\(233\) −1.62947e7 −1.28819 −0.644095 0.764946i \(-0.722766\pi\)
−0.644095 + 0.764946i \(0.722766\pi\)
\(234\) 0 0
\(235\) 354529. 0.0273179
\(236\) 0 0
\(237\) 8.91257e6i 0.669511i
\(238\) 0 0
\(239\) − 2.00798e7i − 1.47084i −0.677609 0.735422i \(-0.736984\pi\)
0.677609 0.735422i \(-0.263016\pi\)
\(240\) 0 0
\(241\) −5.97568e6 −0.426910 −0.213455 0.976953i \(-0.568472\pi\)
−0.213455 + 0.976953i \(0.568472\pi\)
\(242\) 0 0
\(243\) −920483. −0.0641500
\(244\) 0 0
\(245\) 3.26184e6i 0.221802i
\(246\) 0 0
\(247\) 1.46726e6i 0.0973680i
\(248\) 0 0
\(249\) 5.05791e6 0.327622
\(250\) 0 0
\(251\) −1.92078e7 −1.21466 −0.607331 0.794449i \(-0.707760\pi\)
−0.607331 + 0.794449i \(0.707760\pi\)
\(252\) 0 0
\(253\) − 1.38395e6i − 0.0854594i
\(254\) 0 0
\(255\) 2.39524e6i 0.144453i
\(256\) 0 0
\(257\) 5.96168e6 0.351212 0.175606 0.984461i \(-0.443811\pi\)
0.175606 + 0.984461i \(0.443811\pi\)
\(258\) 0 0
\(259\) 3.28348e7 1.88988
\(260\) 0 0
\(261\) − 4.71645e6i − 0.265274i
\(262\) 0 0
\(263\) 1.92686e7i 1.05921i 0.848243 + 0.529607i \(0.177661\pi\)
−0.848243 + 0.529607i \(0.822339\pi\)
\(264\) 0 0
\(265\) 4.48813e6 0.241172
\(266\) 0 0
\(267\) −1.18255e7 −0.621278
\(268\) 0 0
\(269\) 1.74390e7i 0.895909i 0.894056 + 0.447955i \(0.147847\pi\)
−0.894056 + 0.447955i \(0.852153\pi\)
\(270\) 0 0
\(271\) 2.41844e7i 1.21514i 0.794265 + 0.607571i \(0.207856\pi\)
−0.794265 + 0.607571i \(0.792144\pi\)
\(272\) 0 0
\(273\) −2.81800e6 −0.138501
\(274\) 0 0
\(275\) −6.63577e6 −0.319075
\(276\) 0 0
\(277\) − 3.19698e7i − 1.50418i −0.659059 0.752091i \(-0.729045\pi\)
0.659059 0.752091i \(-0.270955\pi\)
\(278\) 0 0
\(279\) − 3.71725e6i − 0.171163i
\(280\) 0 0
\(281\) 2.34865e7 1.05852 0.529259 0.848460i \(-0.322470\pi\)
0.529259 + 0.848460i \(0.322470\pi\)
\(282\) 0 0
\(283\) −7.74978e6 −0.341924 −0.170962 0.985278i \(-0.554688\pi\)
−0.170962 + 0.985278i \(0.554688\pi\)
\(284\) 0 0
\(285\) 1.34077e6i 0.0579189i
\(286\) 0 0
\(287\) 1.78772e7i 0.756231i
\(288\) 0 0
\(289\) 3.48867e7 1.44533
\(290\) 0 0
\(291\) −389951. −0.0158246
\(292\) 0 0
\(293\) 1.14848e7i 0.456584i 0.973593 + 0.228292i \(0.0733140\pi\)
−0.973593 + 0.228292i \(0.926686\pi\)
\(294\) 0 0
\(295\) 3.99114e6i 0.155464i
\(296\) 0 0
\(297\) 1.65099e6 0.0630194
\(298\) 0 0
\(299\) −1.08336e6 −0.0405284
\(300\) 0 0
\(301\) − 5.26208e7i − 1.92956i
\(302\) 0 0
\(303\) 2.50876e7i 0.901842i
\(304\) 0 0
\(305\) −913434. −0.0321942
\(306\) 0 0
\(307\) 2.30562e7 0.796841 0.398421 0.917203i \(-0.369558\pi\)
0.398421 + 0.917203i \(0.369558\pi\)
\(308\) 0 0
\(309\) 1.00875e7i 0.341909i
\(310\) 0 0
\(311\) 4.24296e7i 1.41055i 0.708935 + 0.705274i \(0.249176\pi\)
−0.708935 + 0.705274i \(0.750824\pi\)
\(312\) 0 0
\(313\) −2.01928e7 −0.658513 −0.329257 0.944240i \(-0.606798\pi\)
−0.329257 + 0.944240i \(0.606798\pi\)
\(314\) 0 0
\(315\) −2.57507e6 −0.0823868
\(316\) 0 0
\(317\) − 5.83594e6i − 0.183203i −0.995796 0.0916016i \(-0.970801\pi\)
0.995796 0.0916016i \(-0.0291986\pi\)
\(318\) 0 0
\(319\) 8.45947e6i 0.260598i
\(320\) 0 0
\(321\) −3.63276e6 −0.109830
\(322\) 0 0
\(323\) 3.30398e7 0.980458
\(324\) 0 0
\(325\) 5.19449e6i 0.151319i
\(326\) 0 0
\(327\) − 8.19718e6i − 0.234434i
\(328\) 0 0
\(329\) 9.39235e6 0.263746
\(330\) 0 0
\(331\) −6.94601e7 −1.91536 −0.957682 0.287828i \(-0.907067\pi\)
−0.957682 + 0.287828i \(0.907067\pi\)
\(332\) 0 0
\(333\) 1.50587e7i 0.407807i
\(334\) 0 0
\(335\) − 9.93621e6i − 0.264294i
\(336\) 0 0
\(337\) −2.65194e7 −0.692906 −0.346453 0.938067i \(-0.612614\pi\)
−0.346453 + 0.938067i \(0.612614\pi\)
\(338\) 0 0
\(339\) 1.89436e7 0.486254
\(340\) 0 0
\(341\) 6.66729e6i 0.168146i
\(342\) 0 0
\(343\) 2.40780e7i 0.596676i
\(344\) 0 0
\(345\) −989967. −0.0241081
\(346\) 0 0
\(347\) −5.53620e7 −1.32502 −0.662511 0.749052i \(-0.730509\pi\)
−0.662511 + 0.749052i \(0.730509\pi\)
\(348\) 0 0
\(349\) − 4.23047e7i − 0.995206i −0.867405 0.497603i \(-0.834214\pi\)
0.867405 0.497603i \(-0.165786\pi\)
\(350\) 0 0
\(351\) − 1.29239e6i − 0.0298864i
\(352\) 0 0
\(353\) −4.67810e7 −1.06352 −0.531760 0.846895i \(-0.678469\pi\)
−0.531760 + 0.846895i \(0.678469\pi\)
\(354\) 0 0
\(355\) −9.04064e6 −0.202076
\(356\) 0 0
\(357\) 6.34558e7i 1.39466i
\(358\) 0 0
\(359\) 6.79965e7i 1.46961i 0.678276 + 0.734807i \(0.262727\pi\)
−0.678276 + 0.734807i \(0.737273\pi\)
\(360\) 0 0
\(361\) −2.85514e7 −0.606884
\(362\) 0 0
\(363\) 2.46547e7 0.515442
\(364\) 0 0
\(365\) − 7.88698e6i − 0.162193i
\(366\) 0 0
\(367\) 8.52346e7i 1.72432i 0.506637 + 0.862160i \(0.330889\pi\)
−0.506637 + 0.862160i \(0.669111\pi\)
\(368\) 0 0
\(369\) −8.19886e6 −0.163183
\(370\) 0 0
\(371\) 1.18902e8 2.32845
\(372\) 0 0
\(373\) − 5.70643e7i − 1.09961i −0.835293 0.549804i \(-0.814702\pi\)
0.835293 0.549804i \(-0.185298\pi\)
\(374\) 0 0
\(375\) 9.61808e6i 0.182387i
\(376\) 0 0
\(377\) 6.62209e6 0.123586
\(378\) 0 0
\(379\) −5.18779e7 −0.952939 −0.476469 0.879191i \(-0.658084\pi\)
−0.476469 + 0.879191i \(0.658084\pi\)
\(380\) 0 0
\(381\) 5.03653e7i 0.910661i
\(382\) 0 0
\(383\) 7.83098e7i 1.39386i 0.717138 + 0.696931i \(0.245452\pi\)
−0.717138 + 0.696931i \(0.754548\pi\)
\(384\) 0 0
\(385\) 4.61867e6 0.0809347
\(386\) 0 0
\(387\) 2.41330e7 0.416369
\(388\) 0 0
\(389\) 9.23410e7i 1.56872i 0.620305 + 0.784361i \(0.287009\pi\)
−0.620305 + 0.784361i \(0.712991\pi\)
\(390\) 0 0
\(391\) 2.43951e7i 0.408106i
\(392\) 0 0
\(393\) −2.80778e7 −0.462578
\(394\) 0 0
\(395\) −1.14348e7 −0.185540
\(396\) 0 0
\(397\) − 3.75593e7i − 0.600270i −0.953897 0.300135i \(-0.902968\pi\)
0.953897 0.300135i \(-0.0970317\pi\)
\(398\) 0 0
\(399\) 3.55204e7i 0.559189i
\(400\) 0 0
\(401\) −3.85071e7 −0.597183 −0.298592 0.954381i \(-0.596517\pi\)
−0.298592 + 0.954381i \(0.596517\pi\)
\(402\) 0 0
\(403\) 5.21916e6 0.0797417
\(404\) 0 0
\(405\) − 1.18098e6i − 0.0177778i
\(406\) 0 0
\(407\) − 2.70094e7i − 0.400619i
\(408\) 0 0
\(409\) 1.07783e8 1.57536 0.787681 0.616083i \(-0.211281\pi\)
0.787681 + 0.616083i \(0.211281\pi\)
\(410\) 0 0
\(411\) −785070. −0.0113079
\(412\) 0 0
\(413\) 1.05735e8i 1.50096i
\(414\) 0 0
\(415\) 6.48931e6i 0.0907933i
\(416\) 0 0
\(417\) 6.60574e7 0.910989
\(418\) 0 0
\(419\) −7.92976e7 −1.07800 −0.538999 0.842306i \(-0.681197\pi\)
−0.538999 + 0.842306i \(0.681197\pi\)
\(420\) 0 0
\(421\) − 6.67669e7i − 0.894777i −0.894340 0.447389i \(-0.852354\pi\)
0.894340 0.447389i \(-0.147646\pi\)
\(422\) 0 0
\(423\) 4.30752e6i 0.0569123i
\(424\) 0 0
\(425\) 1.16970e8 1.52372
\(426\) 0 0
\(427\) −2.41992e7 −0.310826
\(428\) 0 0
\(429\) 2.31805e6i 0.0293596i
\(430\) 0 0
\(431\) − 5.07275e7i − 0.633595i −0.948493 0.316798i \(-0.897392\pi\)
0.948493 0.316798i \(-0.102608\pi\)
\(432\) 0 0
\(433\) −3.19683e7 −0.393782 −0.196891 0.980425i \(-0.563084\pi\)
−0.196891 + 0.980425i \(0.563084\pi\)
\(434\) 0 0
\(435\) 6.05121e6 0.0735148
\(436\) 0 0
\(437\) 1.36555e7i 0.163631i
\(438\) 0 0
\(439\) 7.14885e7i 0.844972i 0.906369 + 0.422486i \(0.138842\pi\)
−0.906369 + 0.422486i \(0.861158\pi\)
\(440\) 0 0
\(441\) −3.96314e7 −0.462087
\(442\) 0 0
\(443\) 7.60820e7 0.875127 0.437563 0.899188i \(-0.355842\pi\)
0.437563 + 0.899188i \(0.355842\pi\)
\(444\) 0 0
\(445\) − 1.51721e7i − 0.172174i
\(446\) 0 0
\(447\) − 6.85901e7i − 0.767960i
\(448\) 0 0
\(449\) 9.00650e7 0.994986 0.497493 0.867468i \(-0.334254\pi\)
0.497493 + 0.867468i \(0.334254\pi\)
\(450\) 0 0
\(451\) 1.47056e7 0.160307
\(452\) 0 0
\(453\) 3.39458e7i 0.365167i
\(454\) 0 0
\(455\) − 3.61550e6i − 0.0383826i
\(456\) 0 0
\(457\) −3.59392e7 −0.376548 −0.188274 0.982117i \(-0.560289\pi\)
−0.188274 + 0.982117i \(0.560289\pi\)
\(458\) 0 0
\(459\) −2.91021e7 −0.300945
\(460\) 0 0
\(461\) 1.09802e8i 1.12075i 0.828240 + 0.560373i \(0.189342\pi\)
−0.828240 + 0.560373i \(0.810658\pi\)
\(462\) 0 0
\(463\) − 5.53763e7i − 0.557932i −0.960301 0.278966i \(-0.910008\pi\)
0.960301 0.278966i \(-0.0899917\pi\)
\(464\) 0 0
\(465\) 4.76923e6 0.0474340
\(466\) 0 0
\(467\) 5.38248e7 0.528484 0.264242 0.964456i \(-0.414878\pi\)
0.264242 + 0.964456i \(0.414878\pi\)
\(468\) 0 0
\(469\) − 2.63235e8i − 2.55168i
\(470\) 0 0
\(471\) 8.94008e7i 0.855615i
\(472\) 0 0
\(473\) −4.32851e7 −0.409030
\(474\) 0 0
\(475\) 6.54755e7 0.610939
\(476\) 0 0
\(477\) 5.45307e7i 0.502442i
\(478\) 0 0
\(479\) 1.71807e8i 1.56328i 0.623733 + 0.781638i \(0.285615\pi\)
−0.623733 + 0.781638i \(0.714385\pi\)
\(480\) 0 0
\(481\) −2.11430e7 −0.189990
\(482\) 0 0
\(483\) −2.62267e7 −0.232757
\(484\) 0 0
\(485\) − 500308.i − 0.00438543i
\(486\) 0 0
\(487\) − 5.72463e7i − 0.495634i −0.968807 0.247817i \(-0.920287\pi\)
0.968807 0.247817i \(-0.0797131\pi\)
\(488\) 0 0
\(489\) 4.67178e7 0.399536
\(490\) 0 0
\(491\) −2.13055e8 −1.79989 −0.899946 0.436001i \(-0.856395\pi\)
−0.899946 + 0.436001i \(0.856395\pi\)
\(492\) 0 0
\(493\) − 1.49116e8i − 1.24447i
\(494\) 0 0
\(495\) 2.11822e6i 0.0174644i
\(496\) 0 0
\(497\) −2.39509e8 −1.95098
\(498\) 0 0
\(499\) 9.04068e7 0.727611 0.363806 0.931475i \(-0.381477\pi\)
0.363806 + 0.931475i \(0.381477\pi\)
\(500\) 0 0
\(501\) 1.25210e8i 0.995697i
\(502\) 0 0
\(503\) − 2.29271e8i − 1.80155i −0.434290 0.900773i \(-0.643001\pi\)
0.434290 0.900773i \(-0.356999\pi\)
\(504\) 0 0
\(505\) −3.21874e7 −0.249926
\(506\) 0 0
\(507\) −7.34279e7 −0.563427
\(508\) 0 0
\(509\) 1.29225e7i 0.0979930i 0.998799 + 0.0489965i \(0.0156023\pi\)
−0.998799 + 0.0489965i \(0.984398\pi\)
\(510\) 0 0
\(511\) − 2.08946e8i − 1.56593i
\(512\) 0 0
\(513\) −1.62904e7 −0.120664
\(514\) 0 0
\(515\) −1.29423e7 −0.0947525
\(516\) 0 0
\(517\) − 7.72601e6i − 0.0559092i
\(518\) 0 0
\(519\) − 3.78762e7i − 0.270934i
\(520\) 0 0
\(521\) −7.99955e7 −0.565656 −0.282828 0.959171i \(-0.591273\pi\)
−0.282828 + 0.959171i \(0.591273\pi\)
\(522\) 0 0
\(523\) 1.28458e8 0.897957 0.448979 0.893543i \(-0.351788\pi\)
0.448979 + 0.893543i \(0.351788\pi\)
\(524\) 0 0
\(525\) 1.25752e8i 0.869031i
\(526\) 0 0
\(527\) − 1.17525e8i − 0.802969i
\(528\) 0 0
\(529\) 1.37953e8 0.931890
\(530\) 0 0
\(531\) −4.84923e7 −0.323884
\(532\) 0 0
\(533\) − 1.15115e7i − 0.0760241i
\(534\) 0 0
\(535\) − 4.66083e6i − 0.0304370i
\(536\) 0 0
\(537\) −8.33768e7 −0.538421
\(538\) 0 0
\(539\) 7.10832e7 0.453942
\(540\) 0 0
\(541\) − 2.90048e8i − 1.83180i −0.401410 0.915899i \(-0.631480\pi\)
0.401410 0.915899i \(-0.368520\pi\)
\(542\) 0 0
\(543\) − 1.42022e8i − 0.887067i
\(544\) 0 0
\(545\) 1.05170e7 0.0649684
\(546\) 0 0
\(547\) 2.91198e7 0.177921 0.0889604 0.996035i \(-0.471646\pi\)
0.0889604 + 0.996035i \(0.471646\pi\)
\(548\) 0 0
\(549\) − 1.10982e7i − 0.0670713i
\(550\) 0 0
\(551\) − 8.34700e7i − 0.498972i
\(552\) 0 0
\(553\) −3.02937e8 −1.79134
\(554\) 0 0
\(555\) −1.93203e7 −0.113015
\(556\) 0 0
\(557\) − 1.95014e8i − 1.12849i −0.825606 0.564247i \(-0.809166\pi\)
0.825606 0.564247i \(-0.190834\pi\)
\(558\) 0 0
\(559\) 3.38836e7i 0.193979i
\(560\) 0 0
\(561\) 5.21978e7 0.295640
\(562\) 0 0
\(563\) −3.06065e8 −1.71510 −0.857548 0.514404i \(-0.828013\pi\)
−0.857548 + 0.514404i \(0.828013\pi\)
\(564\) 0 0
\(565\) 2.43046e7i 0.134755i
\(566\) 0 0
\(567\) − 3.12871e7i − 0.171639i
\(568\) 0 0
\(569\) 3.27388e8 1.77716 0.888578 0.458725i \(-0.151694\pi\)
0.888578 + 0.458725i \(0.151694\pi\)
\(570\) 0 0
\(571\) 1.37293e8 0.737461 0.368731 0.929536i \(-0.379792\pi\)
0.368731 + 0.929536i \(0.379792\pi\)
\(572\) 0 0
\(573\) 1.85601e7i 0.0986544i
\(574\) 0 0
\(575\) 4.83443e7i 0.254297i
\(576\) 0 0
\(577\) 2.84639e8 1.48172 0.740861 0.671658i \(-0.234417\pi\)
0.740861 + 0.671658i \(0.234417\pi\)
\(578\) 0 0
\(579\) −3.23910e7 −0.166874
\(580\) 0 0
\(581\) 1.71918e8i 0.876583i
\(582\) 0 0
\(583\) − 9.78068e7i − 0.493587i
\(584\) 0 0
\(585\) 1.65814e6 0.00828236
\(586\) 0 0
\(587\) 3.11468e8 1.53992 0.769962 0.638090i \(-0.220275\pi\)
0.769962 + 0.638090i \(0.220275\pi\)
\(588\) 0 0
\(589\) − 6.57865e7i − 0.321952i
\(590\) 0 0
\(591\) 3.42479e7i 0.165909i
\(592\) 0 0
\(593\) −7.54519e7 −0.361831 −0.180916 0.983499i \(-0.557906\pi\)
−0.180916 + 0.983499i \(0.557906\pi\)
\(594\) 0 0
\(595\) −8.14139e7 −0.386498
\(596\) 0 0
\(597\) 9.73636e7i 0.457587i
\(598\) 0 0
\(599\) − 2.08861e8i − 0.971800i −0.874014 0.485900i \(-0.838492\pi\)
0.874014 0.485900i \(-0.161508\pi\)
\(600\) 0 0
\(601\) −2.88006e7 −0.132672 −0.0663359 0.997797i \(-0.521131\pi\)
−0.0663359 + 0.997797i \(0.521131\pi\)
\(602\) 0 0
\(603\) 1.20725e8 0.550611
\(604\) 0 0
\(605\) 3.16320e7i 0.142843i
\(606\) 0 0
\(607\) − 2.89154e8i − 1.29290i −0.762958 0.646448i \(-0.776254\pi\)
0.762958 0.646448i \(-0.223746\pi\)
\(608\) 0 0
\(609\) 1.60312e8 0.709763
\(610\) 0 0
\(611\) −6.04793e6 −0.0265145
\(612\) 0 0
\(613\) − 3.53775e8i − 1.53584i −0.640548 0.767919i \(-0.721293\pi\)
0.640548 0.767919i \(-0.278707\pi\)
\(614\) 0 0
\(615\) − 1.05191e7i − 0.0452226i
\(616\) 0 0
\(617\) 1.88929e8 0.804348 0.402174 0.915563i \(-0.368255\pi\)
0.402174 + 0.915563i \(0.368255\pi\)
\(618\) 0 0
\(619\) −8.45244e7 −0.356377 −0.178189 0.983996i \(-0.557024\pi\)
−0.178189 + 0.983996i \(0.557024\pi\)
\(620\) 0 0
\(621\) − 1.20281e7i − 0.0502253i
\(622\) 0 0
\(623\) − 4.01948e8i − 1.66229i
\(624\) 0 0
\(625\) 2.25551e8 0.923855
\(626\) 0 0
\(627\) 2.92185e7 0.118538
\(628\) 0 0
\(629\) 4.76098e8i 1.91313i
\(630\) 0 0
\(631\) − 3.49408e8i − 1.39074i −0.718653 0.695369i \(-0.755241\pi\)
0.718653 0.695369i \(-0.244759\pi\)
\(632\) 0 0
\(633\) −1.74058e8 −0.686251
\(634\) 0 0
\(635\) −6.46187e7 −0.252370
\(636\) 0 0
\(637\) − 5.56440e7i − 0.215278i
\(638\) 0 0
\(639\) − 1.09844e8i − 0.420991i
\(640\) 0 0
\(641\) −2.70070e8 −1.02542 −0.512710 0.858562i \(-0.671359\pi\)
−0.512710 + 0.858562i \(0.671359\pi\)
\(642\) 0 0
\(643\) −4.94695e8 −1.86082 −0.930411 0.366518i \(-0.880550\pi\)
−0.930411 + 0.366518i \(0.880550\pi\)
\(644\) 0 0
\(645\) 3.09626e7i 0.115387i
\(646\) 0 0
\(647\) − 1.19819e7i − 0.0442396i −0.999755 0.0221198i \(-0.992958\pi\)
0.999755 0.0221198i \(-0.00704153\pi\)
\(648\) 0 0
\(649\) 8.69762e7 0.318175
\(650\) 0 0
\(651\) 1.26349e8 0.457961
\(652\) 0 0
\(653\) 5.21595e8i 1.87324i 0.350345 + 0.936621i \(0.386064\pi\)
−0.350345 + 0.936621i \(0.613936\pi\)
\(654\) 0 0
\(655\) − 3.60238e7i − 0.128193i
\(656\) 0 0
\(657\) 9.58269e7 0.337902
\(658\) 0 0
\(659\) 4.10833e8 1.43552 0.717759 0.696292i \(-0.245168\pi\)
0.717759 + 0.696292i \(0.245168\pi\)
\(660\) 0 0
\(661\) 8.48526e7i 0.293806i 0.989151 + 0.146903i \(0.0469305\pi\)
−0.989151 + 0.146903i \(0.953069\pi\)
\(662\) 0 0
\(663\) − 4.08605e7i − 0.140205i
\(664\) 0 0
\(665\) −4.55727e7 −0.154967
\(666\) 0 0
\(667\) 6.16306e7 0.207692
\(668\) 0 0
\(669\) 1.02705e7i 0.0343016i
\(670\) 0 0
\(671\) 1.99059e7i 0.0658891i
\(672\) 0 0
\(673\) −3.39619e8 −1.11416 −0.557079 0.830460i \(-0.688078\pi\)
−0.557079 + 0.830460i \(0.688078\pi\)
\(674\) 0 0
\(675\) −5.76722e7 −0.187523
\(676\) 0 0
\(677\) − 4.04369e8i − 1.30320i −0.758561 0.651602i \(-0.774097\pi\)
0.758561 0.651602i \(-0.225903\pi\)
\(678\) 0 0
\(679\) − 1.32544e7i − 0.0423400i
\(680\) 0 0
\(681\) −1.63048e8 −0.516266
\(682\) 0 0
\(683\) 2.05590e8 0.645268 0.322634 0.946524i \(-0.395432\pi\)
0.322634 + 0.946524i \(0.395432\pi\)
\(684\) 0 0
\(685\) − 1.00725e6i − 0.00313374i
\(686\) 0 0
\(687\) 3.03667e8i 0.936542i
\(688\) 0 0
\(689\) −7.65633e7 −0.234079
\(690\) 0 0
\(691\) −1.25719e8 −0.381037 −0.190518 0.981684i \(-0.561017\pi\)
−0.190518 + 0.981684i \(0.561017\pi\)
\(692\) 0 0
\(693\) 5.61168e7i 0.168614i
\(694\) 0 0
\(695\) 8.47517e7i 0.252461i
\(696\) 0 0
\(697\) −2.59217e8 −0.765534
\(698\) 0 0
\(699\) 2.54010e8 0.743737
\(700\) 0 0
\(701\) 1.36521e8i 0.396320i 0.980170 + 0.198160i \(0.0634966\pi\)
−0.980170 + 0.198160i \(0.936503\pi\)
\(702\) 0 0
\(703\) 2.66503e8i 0.767072i
\(704\) 0 0
\(705\) −5.52655e6 −0.0157720
\(706\) 0 0
\(707\) −8.52724e8 −2.41296
\(708\) 0 0
\(709\) − 1.19437e8i − 0.335119i −0.985862 0.167559i \(-0.946411\pi\)
0.985862 0.167559i \(-0.0535886\pi\)
\(710\) 0 0
\(711\) − 1.38933e8i − 0.386543i
\(712\) 0 0
\(713\) 4.85739e7 0.134009
\(714\) 0 0
\(715\) −2.97406e6 −0.00813638
\(716\) 0 0
\(717\) 3.13014e8i 0.849192i
\(718\) 0 0
\(719\) − 2.90874e8i − 0.782561i −0.920272 0.391280i \(-0.872032\pi\)
0.920272 0.391280i \(-0.127968\pi\)
\(720\) 0 0
\(721\) −3.42875e8 −0.914807
\(722\) 0 0
\(723\) 9.31516e7 0.246477
\(724\) 0 0
\(725\) − 2.95506e8i − 0.775448i
\(726\) 0 0
\(727\) 8.94298e7i 0.232744i 0.993206 + 0.116372i \(0.0371265\pi\)
−0.993206 + 0.116372i \(0.962873\pi\)
\(728\) 0 0
\(729\) 1.43489e7 0.0370370
\(730\) 0 0
\(731\) 7.62992e8 1.95329
\(732\) 0 0
\(733\) 1.28445e8i 0.326141i 0.986614 + 0.163071i \(0.0521398\pi\)
−0.986614 + 0.163071i \(0.947860\pi\)
\(734\) 0 0
\(735\) − 5.08471e7i − 0.128057i
\(736\) 0 0
\(737\) −2.16533e8 −0.540907
\(738\) 0 0
\(739\) −7.15451e8 −1.77275 −0.886373 0.462972i \(-0.846783\pi\)
−0.886373 + 0.462972i \(0.846783\pi\)
\(740\) 0 0
\(741\) − 2.28723e7i − 0.0562154i
\(742\) 0 0
\(743\) 3.45246e8i 0.841708i 0.907128 + 0.420854i \(0.138270\pi\)
−0.907128 + 0.420854i \(0.861730\pi\)
\(744\) 0 0
\(745\) 8.80011e7 0.212823
\(746\) 0 0
\(747\) −7.88451e7 −0.189153
\(748\) 0 0
\(749\) − 1.23477e8i − 0.293860i
\(750\) 0 0
\(751\) 1.75422e8i 0.414157i 0.978324 + 0.207079i \(0.0663956\pi\)
−0.978324 + 0.207079i \(0.933604\pi\)
\(752\) 0 0
\(753\) 2.99419e8 0.701286
\(754\) 0 0
\(755\) −4.35525e7 −0.101198
\(756\) 0 0
\(757\) 4.03668e8i 0.930544i 0.885168 + 0.465272i \(0.154044\pi\)
−0.885168 + 0.465272i \(0.845956\pi\)
\(758\) 0 0
\(759\) 2.15737e7i 0.0493400i
\(760\) 0 0
\(761\) 5.80482e8 1.31715 0.658574 0.752516i \(-0.271160\pi\)
0.658574 + 0.752516i \(0.271160\pi\)
\(762\) 0 0
\(763\) 2.78621e8 0.627250
\(764\) 0 0
\(765\) − 3.73381e7i − 0.0834002i
\(766\) 0 0
\(767\) − 6.80851e7i − 0.150892i
\(768\) 0 0
\(769\) 5.90678e8 1.29889 0.649444 0.760409i \(-0.275002\pi\)
0.649444 + 0.760409i \(0.275002\pi\)
\(770\) 0 0
\(771\) −9.29335e7 −0.202772
\(772\) 0 0
\(773\) − 3.10093e8i − 0.671357i −0.941977 0.335679i \(-0.891034\pi\)
0.941977 0.335679i \(-0.108966\pi\)
\(774\) 0 0
\(775\) − 2.32902e8i − 0.500342i
\(776\) 0 0
\(777\) −5.11843e8 −1.09112
\(778\) 0 0
\(779\) −1.45100e8 −0.306942
\(780\) 0 0
\(781\) 1.97017e8i 0.413571i
\(782\) 0 0
\(783\) 7.35222e7i 0.153156i
\(784\) 0 0
\(785\) −1.14701e8 −0.237115
\(786\) 0 0
\(787\) 5.73439e8 1.17642 0.588210 0.808708i \(-0.299833\pi\)
0.588210 + 0.808708i \(0.299833\pi\)
\(788\) 0 0
\(789\) − 3.00368e8i − 0.611538i
\(790\) 0 0
\(791\) 6.43890e8i 1.30101i
\(792\) 0 0
\(793\) 1.55824e7 0.0312474
\(794\) 0 0
\(795\) −6.99630e7 −0.139241
\(796\) 0 0
\(797\) − 6.24736e8i − 1.23402i −0.786956 0.617009i \(-0.788344\pi\)
0.786956 0.617009i \(-0.211656\pi\)
\(798\) 0 0
\(799\) 1.36187e8i 0.266991i
\(800\) 0 0
\(801\) 1.84341e8 0.358695
\(802\) 0 0
\(803\) −1.71876e8 −0.331947
\(804\) 0 0
\(805\) − 3.36489e7i − 0.0645034i
\(806\) 0 0
\(807\) − 2.71847e8i − 0.517253i
\(808\) 0 0
\(809\) 9.88696e8 1.86731 0.933657 0.358169i \(-0.116599\pi\)
0.933657 + 0.358169i \(0.116599\pi\)
\(810\) 0 0
\(811\) 4.02063e8 0.753757 0.376879 0.926263i \(-0.376997\pi\)
0.376879 + 0.926263i \(0.376997\pi\)
\(812\) 0 0
\(813\) − 3.76997e8i − 0.701563i
\(814\) 0 0
\(815\) 5.99389e7i 0.110723i
\(816\) 0 0
\(817\) 4.27096e8 0.783177
\(818\) 0 0
\(819\) 4.39284e7 0.0799638
\(820\) 0 0
\(821\) 6.26365e8i 1.13187i 0.824448 + 0.565937i \(0.191486\pi\)
−0.824448 + 0.565937i \(0.808514\pi\)
\(822\) 0 0
\(823\) 8.23953e8i 1.47810i 0.673652 + 0.739049i \(0.264725\pi\)
−0.673652 + 0.739049i \(0.735275\pi\)
\(824\) 0 0
\(825\) 1.03441e8 0.184218
\(826\) 0 0
\(827\) −3.02093e7 −0.0534102 −0.0267051 0.999643i \(-0.508502\pi\)
−0.0267051 + 0.999643i \(0.508502\pi\)
\(828\) 0 0
\(829\) − 2.17842e7i − 0.0382366i −0.999817 0.0191183i \(-0.993914\pi\)
0.999817 0.0191183i \(-0.00608591\pi\)
\(830\) 0 0
\(831\) 4.98360e8i 0.868440i
\(832\) 0 0
\(833\) −1.25299e9 −2.16777
\(834\) 0 0
\(835\) −1.60645e8 −0.275936
\(836\) 0 0
\(837\) 5.79461e7i 0.0988208i
\(838\) 0 0
\(839\) 5.51909e8i 0.934505i 0.884124 + 0.467252i \(0.154756\pi\)
−0.884124 + 0.467252i \(0.845244\pi\)
\(840\) 0 0
\(841\) 2.18104e8 0.366670
\(842\) 0 0
\(843\) −3.66118e8 −0.611136
\(844\) 0 0
\(845\) − 9.42081e7i − 0.156141i
\(846\) 0 0
\(847\) 8.38010e8i 1.37911i
\(848\) 0 0
\(849\) 1.20807e8 0.197410
\(850\) 0 0
\(851\) −1.96774e8 −0.319286
\(852\) 0 0
\(853\) − 4.15243e8i − 0.669045i −0.942388 0.334523i \(-0.891425\pi\)
0.942388 0.334523i \(-0.108575\pi\)
\(854\) 0 0
\(855\) − 2.09005e7i − 0.0334395i
\(856\) 0 0
\(857\) 4.44449e8 0.706121 0.353061 0.935600i \(-0.385141\pi\)
0.353061 + 0.935600i \(0.385141\pi\)
\(858\) 0 0
\(859\) 4.67604e8 0.737733 0.368866 0.929482i \(-0.379746\pi\)
0.368866 + 0.929482i \(0.379746\pi\)
\(860\) 0 0
\(861\) − 2.78679e8i − 0.436610i
\(862\) 0 0
\(863\) 5.95476e8i 0.926472i 0.886235 + 0.463236i \(0.153312\pi\)
−0.886235 + 0.463236i \(0.846688\pi\)
\(864\) 0 0
\(865\) 4.85952e7 0.0750835
\(866\) 0 0
\(867\) −5.43830e8 −0.834461
\(868\) 0 0
\(869\) 2.49192e8i 0.379730i
\(870\) 0 0
\(871\) 1.69503e8i 0.256521i
\(872\) 0 0
\(873\) 6.07874e6 0.00913631
\(874\) 0 0
\(875\) −3.26917e8 −0.487993
\(876\) 0 0
\(877\) − 7.21425e8i − 1.06953i −0.845001 0.534765i \(-0.820400\pi\)
0.845001 0.534765i \(-0.179600\pi\)
\(878\) 0 0
\(879\) − 1.79030e8i − 0.263609i
\(880\) 0 0
\(881\) −6.77300e8 −0.990497 −0.495248 0.868751i \(-0.664923\pi\)
−0.495248 + 0.868751i \(0.664923\pi\)
\(882\) 0 0
\(883\) 8.41700e8 1.22257 0.611287 0.791409i \(-0.290652\pi\)
0.611287 + 0.791409i \(0.290652\pi\)
\(884\) 0 0
\(885\) − 6.22157e7i − 0.0897573i
\(886\) 0 0
\(887\) 2.23932e8i 0.320882i 0.987045 + 0.160441i \(0.0512917\pi\)
−0.987045 + 0.160441i \(0.948708\pi\)
\(888\) 0 0
\(889\) −1.71191e9 −2.43655
\(890\) 0 0
\(891\) −2.57363e7 −0.0363842
\(892\) 0 0
\(893\) 7.62329e7i 0.107050i
\(894\) 0 0
\(895\) − 1.06972e8i − 0.149212i
\(896\) 0 0
\(897\) 1.68879e7 0.0233991
\(898\) 0 0
\(899\) −2.96910e8 −0.408644
\(900\) 0 0
\(901\) 1.72405e9i 2.35709i
\(902\) 0 0
\(903\) 8.20277e8i 1.11403i
\(904\) 0 0
\(905\) 1.82214e8 0.245831
\(906\) 0 0
\(907\) 6.79727e8 0.910988 0.455494 0.890239i \(-0.349463\pi\)
0.455494 + 0.890239i \(0.349463\pi\)
\(908\) 0 0
\(909\) − 3.91076e8i − 0.520679i
\(910\) 0 0
\(911\) 9.24099e8i 1.22226i 0.791530 + 0.611130i \(0.209285\pi\)
−0.791530 + 0.611130i \(0.790715\pi\)
\(912\) 0 0
\(913\) 1.41417e8 0.185819
\(914\) 0 0
\(915\) 1.42390e7 0.0185873
\(916\) 0 0
\(917\) − 9.54361e8i − 1.23767i
\(918\) 0 0
\(919\) − 3.27998e8i − 0.422595i −0.977422 0.211297i \(-0.932231\pi\)
0.977422 0.211297i \(-0.0677689\pi\)
\(920\) 0 0
\(921\) −3.59410e8 −0.460056
\(922\) 0 0
\(923\) 1.54225e8 0.196133
\(924\) 0 0
\(925\) 9.43492e8i 1.19210i
\(926\) 0 0
\(927\) − 1.57249e8i − 0.197401i
\(928\) 0 0
\(929\) −8.75671e8 −1.09218 −0.546090 0.837727i \(-0.683884\pi\)
−0.546090 + 0.837727i \(0.683884\pi\)
\(930\) 0 0
\(931\) −7.01381e8 −0.869171
\(932\) 0 0
\(933\) − 6.61412e8i − 0.814381i
\(934\) 0 0
\(935\) 6.69698e7i 0.0819303i
\(936\) 0 0
\(937\) −1.46253e8 −0.177782 −0.0888908 0.996041i \(-0.528332\pi\)
−0.0888908 + 0.996041i \(0.528332\pi\)
\(938\) 0 0
\(939\) 3.14775e8 0.380193
\(940\) 0 0
\(941\) − 9.23237e8i − 1.10801i −0.832513 0.554006i \(-0.813099\pi\)
0.832513 0.554006i \(-0.186901\pi\)
\(942\) 0 0
\(943\) − 1.07136e8i − 0.127761i
\(944\) 0 0
\(945\) 4.01414e7 0.0475660
\(946\) 0 0
\(947\) 4.29334e8 0.505529 0.252764 0.967528i \(-0.418660\pi\)
0.252764 + 0.967528i \(0.418660\pi\)
\(948\) 0 0
\(949\) 1.34545e8i 0.157423i
\(950\) 0 0
\(951\) 9.09733e7i 0.105772i
\(952\) 0 0
\(953\) 9.16653e8 1.05907 0.529537 0.848287i \(-0.322366\pi\)
0.529537 + 0.848287i \(0.322366\pi\)
\(954\) 0 0
\(955\) −2.38126e7 −0.0273399
\(956\) 0 0
\(957\) − 1.31870e8i − 0.150456i
\(958\) 0 0
\(959\) − 2.66845e7i − 0.0302554i
\(960\) 0 0
\(961\) 6.53496e8 0.736330
\(962\) 0 0
\(963\) 5.66291e7 0.0634104
\(964\) 0 0
\(965\) − 4.15577e7i − 0.0462455i
\(966\) 0 0
\(967\) 1.74117e9i 1.92558i 0.270247 + 0.962791i \(0.412895\pi\)
−0.270247 + 0.962791i \(0.587105\pi\)
\(968\) 0 0
\(969\) −5.15039e8 −0.566068
\(970\) 0 0
\(971\) −5.75119e8 −0.628203 −0.314102 0.949389i \(-0.601703\pi\)
−0.314102 + 0.949389i \(0.601703\pi\)
\(972\) 0 0
\(973\) 2.24528e9i 2.43743i
\(974\) 0 0
\(975\) − 8.09741e7i − 0.0873639i
\(976\) 0 0
\(977\) 1.49329e8 0.160125 0.0800625 0.996790i \(-0.474488\pi\)
0.0800625 + 0.996790i \(0.474488\pi\)
\(978\) 0 0
\(979\) −3.30636e8 −0.352373
\(980\) 0 0
\(981\) 1.27781e8i 0.135351i
\(982\) 0 0
\(983\) − 6.57661e8i − 0.692376i −0.938165 0.346188i \(-0.887476\pi\)
0.938165 0.346188i \(-0.112524\pi\)
\(984\) 0 0
\(985\) −4.39400e7 −0.0459782
\(986\) 0 0
\(987\) −1.46412e8 −0.152274
\(988\) 0 0
\(989\) 3.15349e8i 0.325989i
\(990\) 0 0
\(991\) 1.37005e9i 1.40771i 0.710341 + 0.703857i \(0.248541\pi\)
−0.710341 + 0.703857i \(0.751459\pi\)
\(992\) 0 0
\(993\) 1.08278e9 1.10584
\(994\) 0 0
\(995\) −1.24918e8 −0.126810
\(996\) 0 0
\(997\) 1.05005e9i 1.05956i 0.848135 + 0.529780i \(0.177726\pi\)
−0.848135 + 0.529780i \(0.822274\pi\)
\(998\) 0 0
\(999\) − 2.34742e8i − 0.235447i
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 384.7.b.d.319.1 8
4.3 odd 2 inner 384.7.b.d.319.6 yes 8
8.3 odd 2 inner 384.7.b.d.319.4 yes 8
8.5 even 2 inner 384.7.b.d.319.7 yes 8
16.3 odd 4 768.7.g.c.511.3 4
16.5 even 4 768.7.g.e.511.4 4
16.11 odd 4 768.7.g.e.511.1 4
16.13 even 4 768.7.g.c.511.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.7.b.d.319.1 8 1.1 even 1 trivial
384.7.b.d.319.4 yes 8 8.3 odd 2 inner
384.7.b.d.319.6 yes 8 4.3 odd 2 inner
384.7.b.d.319.7 yes 8 8.5 even 2 inner
768.7.g.c.511.2 4 16.13 even 4
768.7.g.c.511.3 4 16.3 odd 4
768.7.g.e.511.1 4 16.11 odd 4
768.7.g.e.511.4 4 16.5 even 4