Properties

Label 400.6.c.o.49.2
Level $400$
Weight $6$
Character 400.49
Analytic conductor $64.154$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 400.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(64.1535279252\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{241})\)
Defining polynomial: \( x^{4} + 121x^{2} + 3600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 200)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.2
Root \(8.26209i\) of defining polynomial
Character \(\chi\) \(=\) 400.49
Dual form 400.6.c.o.49.3

$q$-expansion

\(f(q)\) \(=\) \(q-11.5242i q^{3} -27.0483i q^{7} +110.193 q^{9} +O(q^{10})\) \(q-11.5242i q^{3} -27.0483i q^{7} +110.193 q^{9} -226.008 q^{11} -511.257i q^{13} +387.387i q^{17} -1335.93 q^{19} -311.710 q^{21} +545.369i q^{23} -4070.26i q^{27} +4637.58 q^{29} -2991.56 q^{31} +2604.55i q^{33} -1263.70i q^{37} -5891.82 q^{39} -17197.6 q^{41} -16592.0i q^{43} -13036.0i q^{47} +16075.4 q^{49} +4464.31 q^{51} +28994.7i q^{53} +15395.4i q^{57} -34429.9 q^{59} -24149.1 q^{61} -2980.55i q^{63} -29389.7i q^{67} +6284.93 q^{69} -9064.32 q^{71} +55528.7i q^{73} +6113.13i q^{77} -101587. q^{79} -20129.4 q^{81} +73240.4i q^{83} -53444.3i q^{87} +42498.5 q^{89} -13828.7 q^{91} +34475.2i q^{93} +10565.9i q^{97} -24904.6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 56 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 56 q^{9} + 400 q^{11} - 1680 q^{19} - 1992 q^{21} + 9360 q^{29} + 10016 q^{31} - 54864 q^{39} - 10668 q^{41} + 63308 q^{49} + 13200 q^{51} - 163552 q^{59} - 93864 q^{61} + 110088 q^{69} - 14896 q^{71} - 216208 q^{79} - 187324 q^{81} - 141980 q^{89} - 104992 q^{91} - 167552 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/400\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\) \(351\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) − 11.5242i − 0.739276i −0.929176 0.369638i \(-0.879482\pi\)
0.929176 0.369638i \(-0.120518\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 27.0483i − 0.208639i −0.994544 0.104320i \(-0.966734\pi\)
0.994544 0.104320i \(-0.0332665\pi\)
\(8\) 0 0
\(9\) 110.193 0.453471
\(10\) 0 0
\(11\) −226.008 −0.563173 −0.281586 0.959536i \(-0.590861\pi\)
−0.281586 + 0.959536i \(0.590861\pi\)
\(12\) 0 0
\(13\) − 511.257i − 0.839037i −0.907747 0.419518i \(-0.862199\pi\)
0.907747 0.419518i \(-0.137801\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 387.387i 0.325104i 0.986700 + 0.162552i \(0.0519725\pi\)
−0.986700 + 0.162552i \(0.948027\pi\)
\(18\) 0 0
\(19\) −1335.93 −0.848982 −0.424491 0.905432i \(-0.639547\pi\)
−0.424491 + 0.905432i \(0.639547\pi\)
\(20\) 0 0
\(21\) −311.710 −0.154242
\(22\) 0 0
\(23\) 545.369i 0.214967i 0.994207 + 0.107483i \(0.0342792\pi\)
−0.994207 + 0.107483i \(0.965721\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 4070.26i − 1.07452i
\(28\) 0 0
\(29\) 4637.58 1.02399 0.511996 0.858988i \(-0.328906\pi\)
0.511996 + 0.858988i \(0.328906\pi\)
\(30\) 0 0
\(31\) −2991.56 −0.559105 −0.279552 0.960130i \(-0.590186\pi\)
−0.279552 + 0.960130i \(0.590186\pi\)
\(32\) 0 0
\(33\) 2604.55i 0.416340i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 1263.70i − 0.151754i −0.997117 0.0758770i \(-0.975824\pi\)
0.997117 0.0758770i \(-0.0241756\pi\)
\(38\) 0 0
\(39\) −5891.82 −0.620280
\(40\) 0 0
\(41\) −17197.6 −1.59775 −0.798875 0.601497i \(-0.794571\pi\)
−0.798875 + 0.601497i \(0.794571\pi\)
\(42\) 0 0
\(43\) − 16592.0i − 1.36845i −0.729272 0.684223i \(-0.760141\pi\)
0.729272 0.684223i \(-0.239859\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 13036.0i − 0.860795i −0.902639 0.430397i \(-0.858373\pi\)
0.902639 0.430397i \(-0.141627\pi\)
\(48\) 0 0
\(49\) 16075.4 0.956470
\(50\) 0 0
\(51\) 4464.31 0.240342
\(52\) 0 0
\(53\) 28994.7i 1.41785i 0.705286 + 0.708923i \(0.250819\pi\)
−0.705286 + 0.708923i \(0.749181\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 15395.4i 0.627632i
\(58\) 0 0
\(59\) −34429.9 −1.28768 −0.643838 0.765162i \(-0.722659\pi\)
−0.643838 + 0.765162i \(0.722659\pi\)
\(60\) 0 0
\(61\) −24149.1 −0.830952 −0.415476 0.909604i \(-0.636385\pi\)
−0.415476 + 0.909604i \(0.636385\pi\)
\(62\) 0 0
\(63\) − 2980.55i − 0.0946117i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 29389.7i − 0.799849i −0.916548 0.399925i \(-0.869036\pi\)
0.916548 0.399925i \(-0.130964\pi\)
\(68\) 0 0
\(69\) 6284.93 0.158920
\(70\) 0 0
\(71\) −9064.32 −0.213397 −0.106699 0.994291i \(-0.534028\pi\)
−0.106699 + 0.994291i \(0.534028\pi\)
\(72\) 0 0
\(73\) 55528.7i 1.21958i 0.792563 + 0.609791i \(0.208746\pi\)
−0.792563 + 0.609791i \(0.791254\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 6113.13i 0.117500i
\(78\) 0 0
\(79\) −101587. −1.83135 −0.915673 0.401924i \(-0.868342\pi\)
−0.915673 + 0.401924i \(0.868342\pi\)
\(80\) 0 0
\(81\) −20129.4 −0.340893
\(82\) 0 0
\(83\) 73240.4i 1.16696i 0.812128 + 0.583479i \(0.198309\pi\)
−0.812128 + 0.583479i \(0.801691\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 53444.3i − 0.757012i
\(88\) 0 0
\(89\) 42498.5 0.568719 0.284360 0.958718i \(-0.408219\pi\)
0.284360 + 0.958718i \(0.408219\pi\)
\(90\) 0 0
\(91\) −13828.7 −0.175056
\(92\) 0 0
\(93\) 34475.2i 0.413333i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 10565.9i 0.114019i 0.998374 + 0.0570093i \(0.0181565\pi\)
−0.998374 + 0.0570093i \(0.981844\pi\)
\(98\) 0 0
\(99\) −24904.6 −0.255382
\(100\) 0 0
\(101\) 46486.0 0.453439 0.226719 0.973960i \(-0.427200\pi\)
0.226719 + 0.973960i \(0.427200\pi\)
\(102\) 0 0
\(103\) 119526.i 1.11012i 0.831811 + 0.555059i \(0.187304\pi\)
−0.831811 + 0.555059i \(0.812696\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 22922.4i 0.193553i 0.995306 + 0.0967765i \(0.0308532\pi\)
−0.995306 + 0.0967765i \(0.969147\pi\)
\(108\) 0 0
\(109\) −210121. −1.69396 −0.846980 0.531624i \(-0.821582\pi\)
−0.846980 + 0.531624i \(0.821582\pi\)
\(110\) 0 0
\(111\) −14563.1 −0.112188
\(112\) 0 0
\(113\) 203886.i 1.50208i 0.660258 + 0.751039i \(0.270447\pi\)
−0.660258 + 0.751039i \(0.729553\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 56337.2i − 0.380479i
\(118\) 0 0
\(119\) 10478.2 0.0678294
\(120\) 0 0
\(121\) −109972. −0.682837
\(122\) 0 0
\(123\) 198188.i 1.18118i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 175022.i 0.962905i 0.876472 + 0.481453i \(0.159891\pi\)
−0.876472 + 0.481453i \(0.840109\pi\)
\(128\) 0 0
\(129\) −191209. −1.01166
\(130\) 0 0
\(131\) −149218. −0.759701 −0.379850 0.925048i \(-0.624025\pi\)
−0.379850 + 0.925048i \(0.624025\pi\)
\(132\) 0 0
\(133\) 36134.6i 0.177131i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 212180.i − 0.965836i −0.875666 0.482918i \(-0.839577\pi\)
0.875666 0.482918i \(-0.160423\pi\)
\(138\) 0 0
\(139\) 77082.5 0.338391 0.169195 0.985583i \(-0.445883\pi\)
0.169195 + 0.985583i \(0.445883\pi\)
\(140\) 0 0
\(141\) −150229. −0.636365
\(142\) 0 0
\(143\) 115548.i 0.472522i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 185256.i − 0.707095i
\(148\) 0 0
\(149\) −470139. −1.73485 −0.867423 0.497571i \(-0.834225\pi\)
−0.867423 + 0.497571i \(0.834225\pi\)
\(150\) 0 0
\(151\) −311118. −1.11041 −0.555205 0.831714i \(-0.687360\pi\)
−0.555205 + 0.831714i \(0.687360\pi\)
\(152\) 0 0
\(153\) 42687.5i 0.147425i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 543649.i − 1.76023i −0.474760 0.880115i \(-0.657465\pi\)
0.474760 0.880115i \(-0.342535\pi\)
\(158\) 0 0
\(159\) 334140. 1.04818
\(160\) 0 0
\(161\) 14751.3 0.0448504
\(162\) 0 0
\(163\) 298673.i 0.880495i 0.897876 + 0.440248i \(0.145109\pi\)
−0.897876 + 0.440248i \(0.854891\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 65486.6i 0.181703i 0.995864 + 0.0908513i \(0.0289588\pi\)
−0.995864 + 0.0908513i \(0.971041\pi\)
\(168\) 0 0
\(169\) 109909. 0.296017
\(170\) 0 0
\(171\) −147210. −0.384989
\(172\) 0 0
\(173\) − 355092.i − 0.902040i −0.892514 0.451020i \(-0.851060\pi\)
0.892514 0.451020i \(-0.148940\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 396777.i 0.951947i
\(178\) 0 0
\(179\) −39265.7 −0.0915970 −0.0457985 0.998951i \(-0.514583\pi\)
−0.0457985 + 0.998951i \(0.514583\pi\)
\(180\) 0 0
\(181\) 99638.2 0.226063 0.113031 0.993591i \(-0.463944\pi\)
0.113031 + 0.993591i \(0.463944\pi\)
\(182\) 0 0
\(183\) 278298.i 0.614303i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 87552.4i − 0.183090i
\(188\) 0 0
\(189\) −110094. −0.224186
\(190\) 0 0
\(191\) −380336. −0.754369 −0.377185 0.926138i \(-0.623108\pi\)
−0.377185 + 0.926138i \(0.623108\pi\)
\(192\) 0 0
\(193\) − 126401.i − 0.244262i −0.992514 0.122131i \(-0.961027\pi\)
0.992514 0.122131i \(-0.0389728\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 326136.i − 0.598733i −0.954138 0.299366i \(-0.903225\pi\)
0.954138 0.299366i \(-0.0967753\pi\)
\(198\) 0 0
\(199\) −498559. −0.892451 −0.446225 0.894921i \(-0.647232\pi\)
−0.446225 + 0.894921i \(0.647232\pi\)
\(200\) 0 0
\(201\) −338692. −0.591309
\(202\) 0 0
\(203\) − 125439.i − 0.213645i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 60096.1i 0.0974811i
\(208\) 0 0
\(209\) 301930. 0.478123
\(210\) 0 0
\(211\) 310763. 0.480532 0.240266 0.970707i \(-0.422765\pi\)
0.240266 + 0.970707i \(0.422765\pi\)
\(212\) 0 0
\(213\) 104459.i 0.157760i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 80916.7i 0.116651i
\(218\) 0 0
\(219\) 639923. 0.901607
\(220\) 0 0
\(221\) 198054. 0.272774
\(222\) 0 0
\(223\) − 627285.i − 0.844700i −0.906433 0.422350i \(-0.861205\pi\)
0.906433 0.422350i \(-0.138795\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 450131.i − 0.579794i −0.957058 0.289897i \(-0.906379\pi\)
0.957058 0.289897i \(-0.0936211\pi\)
\(228\) 0 0
\(229\) 461485. 0.581526 0.290763 0.956795i \(-0.406091\pi\)
0.290763 + 0.956795i \(0.406091\pi\)
\(230\) 0 0
\(231\) 70448.8 0.0868648
\(232\) 0 0
\(233\) 675812.i 0.815523i 0.913089 + 0.407761i \(0.133690\pi\)
−0.913089 + 0.407761i \(0.866310\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 1.17071e6i 1.35387i
\(238\) 0 0
\(239\) −1.00730e6 −1.14068 −0.570338 0.821410i \(-0.693188\pi\)
−0.570338 + 0.821410i \(0.693188\pi\)
\(240\) 0 0
\(241\) −232091. −0.257404 −0.128702 0.991683i \(-0.541081\pi\)
−0.128702 + 0.991683i \(0.541081\pi\)
\(242\) 0 0
\(243\) − 757099.i − 0.822502i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 683002.i 0.712327i
\(248\) 0 0
\(249\) 844035. 0.862704
\(250\) 0 0
\(251\) 1.77096e6 1.77429 0.887143 0.461494i \(-0.152686\pi\)
0.887143 + 0.461494i \(0.152686\pi\)
\(252\) 0 0
\(253\) − 123258.i − 0.121063i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 528458.i 0.499088i 0.968363 + 0.249544i \(0.0802808\pi\)
−0.968363 + 0.249544i \(0.919719\pi\)
\(258\) 0 0
\(259\) −34181.0 −0.0316618
\(260\) 0 0
\(261\) 511030. 0.464350
\(262\) 0 0
\(263\) − 1.69907e6i − 1.51469i −0.653016 0.757344i \(-0.726497\pi\)
0.653016 0.757344i \(-0.273503\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 489760.i − 0.420441i
\(268\) 0 0
\(269\) 648393. 0.546333 0.273167 0.961967i \(-0.411929\pi\)
0.273167 + 0.961967i \(0.411929\pi\)
\(270\) 0 0
\(271\) 1.94947e6 1.61248 0.806239 0.591590i \(-0.201500\pi\)
0.806239 + 0.591590i \(0.201500\pi\)
\(272\) 0 0
\(273\) 159364.i 0.129415i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 313195.i 0.245254i 0.992453 + 0.122627i \(0.0391319\pi\)
−0.992453 + 0.122627i \(0.960868\pi\)
\(278\) 0 0
\(279\) −329650. −0.253538
\(280\) 0 0
\(281\) 1.72743e6 1.30507 0.652537 0.757757i \(-0.273705\pi\)
0.652537 + 0.757757i \(0.273705\pi\)
\(282\) 0 0
\(283\) − 205142.i − 0.152261i −0.997098 0.0761306i \(-0.975743\pi\)
0.997098 0.0761306i \(-0.0242566\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 465167.i 0.333353i
\(288\) 0 0
\(289\) 1.26979e6 0.894307
\(290\) 0 0
\(291\) 121763. 0.0842912
\(292\) 0 0
\(293\) − 390309.i − 0.265607i −0.991142 0.132804i \(-0.957602\pi\)
0.991142 0.132804i \(-0.0423979\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 919911.i 0.605138i
\(298\) 0 0
\(299\) 278824. 0.180365
\(300\) 0 0
\(301\) −448787. −0.285511
\(302\) 0 0
\(303\) − 535713.i − 0.335217i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 1.10708e6i − 0.670401i −0.942147 0.335201i \(-0.891196\pi\)
0.942147 0.335201i \(-0.108804\pi\)
\(308\) 0 0
\(309\) 1.37744e6 0.820684
\(310\) 0 0
\(311\) −2.19106e6 −1.28455 −0.642277 0.766472i \(-0.722010\pi\)
−0.642277 + 0.766472i \(0.722010\pi\)
\(312\) 0 0
\(313\) − 2.89050e6i − 1.66768i −0.552006 0.833840i \(-0.686138\pi\)
0.552006 0.833840i \(-0.313862\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1.46652e6i 0.819671i 0.912160 + 0.409835i \(0.134414\pi\)
−0.912160 + 0.409835i \(0.865586\pi\)
\(318\) 0 0
\(319\) −1.04813e6 −0.576684
\(320\) 0 0
\(321\) 264161. 0.143089
\(322\) 0 0
\(323\) − 517520.i − 0.276008i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 2.42147e6i 1.25230i
\(328\) 0 0
\(329\) −352602. −0.179595
\(330\) 0 0
\(331\) 2.23500e6 1.12126 0.560632 0.828065i \(-0.310558\pi\)
0.560632 + 0.828065i \(0.310558\pi\)
\(332\) 0 0
\(333\) − 139251.i − 0.0688160i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 1.48851e6i − 0.713966i −0.934111 0.356983i \(-0.883805\pi\)
0.934111 0.356983i \(-0.116195\pi\)
\(338\) 0 0
\(339\) 2.34962e6 1.11045
\(340\) 0 0
\(341\) 676115. 0.314872
\(342\) 0 0
\(343\) − 889414.i − 0.408196i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 2.34245e6i − 1.04435i −0.852837 0.522177i \(-0.825120\pi\)
0.852837 0.522177i \(-0.174880\pi\)
\(348\) 0 0
\(349\) 119901. 0.0526940 0.0263470 0.999653i \(-0.491613\pi\)
0.0263470 + 0.999653i \(0.491613\pi\)
\(350\) 0 0
\(351\) −2.08095e6 −0.901559
\(352\) 0 0
\(353\) − 2.49528e6i − 1.06582i −0.846173 0.532909i \(-0.821099\pi\)
0.846173 0.532909i \(-0.178901\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 120752.i − 0.0501447i
\(358\) 0 0
\(359\) 3.43398e6 1.40625 0.703124 0.711068i \(-0.251788\pi\)
0.703124 + 0.711068i \(0.251788\pi\)
\(360\) 0 0
\(361\) −691400. −0.279230
\(362\) 0 0
\(363\) 1.26733e6i 0.504805i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 4.67310e6i 1.81109i 0.424249 + 0.905545i \(0.360538\pi\)
−0.424249 + 0.905545i \(0.639462\pi\)
\(368\) 0 0
\(369\) −1.89507e6 −0.724533
\(370\) 0 0
\(371\) 784259. 0.295818
\(372\) 0 0
\(373\) − 2.03766e6i − 0.758333i −0.925328 0.379167i \(-0.876211\pi\)
0.925328 0.379167i \(-0.123789\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 2.37099e6i − 0.859166i
\(378\) 0 0
\(379\) 1.78978e6 0.640032 0.320016 0.947412i \(-0.396312\pi\)
0.320016 + 0.947412i \(0.396312\pi\)
\(380\) 0 0
\(381\) 2.01698e6 0.711853
\(382\) 0 0
\(383\) − 2.04521e6i − 0.712429i −0.934404 0.356214i \(-0.884067\pi\)
0.934404 0.356214i \(-0.115933\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 1.82833e6i − 0.620551i
\(388\) 0 0
\(389\) −234191. −0.0784687 −0.0392344 0.999230i \(-0.512492\pi\)
−0.0392344 + 0.999230i \(0.512492\pi\)
\(390\) 0 0
\(391\) −211269. −0.0698865
\(392\) 0 0
\(393\) 1.71961e6i 0.561629i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 2.81706e6i − 0.897058i −0.893768 0.448529i \(-0.851948\pi\)
0.893768 0.448529i \(-0.148052\pi\)
\(398\) 0 0
\(399\) 416421. 0.130949
\(400\) 0 0
\(401\) 735044. 0.228272 0.114136 0.993465i \(-0.463590\pi\)
0.114136 + 0.993465i \(0.463590\pi\)
\(402\) 0 0
\(403\) 1.52946e6i 0.469109i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 285606.i 0.0854636i
\(408\) 0 0
\(409\) −6.08894e6 −1.79984 −0.899918 0.436058i \(-0.856374\pi\)
−0.899918 + 0.436058i \(0.856374\pi\)
\(410\) 0 0
\(411\) −2.44520e6 −0.714020
\(412\) 0 0
\(413\) 931273.i 0.268659i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 888312.i − 0.250164i
\(418\) 0 0
\(419\) 5.65964e6 1.57490 0.787452 0.616376i \(-0.211400\pi\)
0.787452 + 0.616376i \(0.211400\pi\)
\(420\) 0 0
\(421\) −6.34902e6 −1.74583 −0.872914 0.487873i \(-0.837773\pi\)
−0.872914 + 0.487873i \(0.837773\pi\)
\(422\) 0 0
\(423\) − 1.43648e6i − 0.390345i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 653192.i 0.173369i
\(428\) 0 0
\(429\) 1.33160e6 0.349325
\(430\) 0 0
\(431\) −3.03516e6 −0.787024 −0.393512 0.919320i \(-0.628740\pi\)
−0.393512 + 0.919320i \(0.628740\pi\)
\(432\) 0 0
\(433\) − 3.50633e6i − 0.898739i −0.893346 0.449369i \(-0.851649\pi\)
0.893346 0.449369i \(-0.148351\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 728573.i − 0.182503i
\(438\) 0 0
\(439\) −6.08849e6 −1.50782 −0.753908 0.656980i \(-0.771834\pi\)
−0.753908 + 0.656980i \(0.771834\pi\)
\(440\) 0 0
\(441\) 1.77140e6 0.433731
\(442\) 0 0
\(443\) 6.03357e6i 1.46071i 0.683066 + 0.730357i \(0.260646\pi\)
−0.683066 + 0.730357i \(0.739354\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 5.41797e6i 1.28253i
\(448\) 0 0
\(449\) −3.97943e6 −0.931548 −0.465774 0.884904i \(-0.654224\pi\)
−0.465774 + 0.884904i \(0.654224\pi\)
\(450\) 0 0
\(451\) 3.88680e6 0.899809
\(452\) 0 0
\(453\) 3.58538e6i 0.820899i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 430533.i 0.0964308i 0.998837 + 0.0482154i \(0.0153534\pi\)
−0.998837 + 0.0482154i \(0.984647\pi\)
\(458\) 0 0
\(459\) 1.57677e6 0.349330
\(460\) 0 0
\(461\) 3.80383e6 0.833622 0.416811 0.908993i \(-0.363148\pi\)
0.416811 + 0.908993i \(0.363148\pi\)
\(462\) 0 0
\(463\) 7.62853e6i 1.65382i 0.562333 + 0.826911i \(0.309904\pi\)
−0.562333 + 0.826911i \(0.690096\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 9.05102e6i − 1.92046i −0.279209 0.960230i \(-0.590072\pi\)
0.279209 0.960230i \(-0.409928\pi\)
\(468\) 0 0
\(469\) −794943. −0.166880
\(470\) 0 0
\(471\) −6.26511e6 −1.30130
\(472\) 0 0
\(473\) 3.74992e6i 0.770672i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 3.19502e6i 0.642952i
\(478\) 0 0
\(479\) −8.51621e6 −1.69593 −0.847964 0.530054i \(-0.822172\pi\)
−0.847964 + 0.530054i \(0.822172\pi\)
\(480\) 0 0
\(481\) −646076. −0.127327
\(482\) 0 0
\(483\) − 169997.i − 0.0331569i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 2.65770e6i 0.507789i 0.967232 + 0.253895i \(0.0817117\pi\)
−0.967232 + 0.253895i \(0.918288\pi\)
\(488\) 0 0
\(489\) 3.44196e6 0.650929
\(490\) 0 0
\(491\) 5.05116e6 0.945557 0.472779 0.881181i \(-0.343251\pi\)
0.472779 + 0.881181i \(0.343251\pi\)
\(492\) 0 0
\(493\) 1.79654e6i 0.332904i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 245175.i 0.0445230i
\(498\) 0 0
\(499\) −2.15003e6 −0.386538 −0.193269 0.981146i \(-0.561909\pi\)
−0.193269 + 0.981146i \(0.561909\pi\)
\(500\) 0 0
\(501\) 754679. 0.134328
\(502\) 0 0
\(503\) − 1.79475e6i − 0.316289i −0.987416 0.158144i \(-0.949449\pi\)
0.987416 0.158144i \(-0.0505511\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 1.26661e6i − 0.218839i
\(508\) 0 0
\(509\) −697940. −0.119405 −0.0597027 0.998216i \(-0.519015\pi\)
−0.0597027 + 0.998216i \(0.519015\pi\)
\(510\) 0 0
\(511\) 1.50196e6 0.254452
\(512\) 0 0
\(513\) 5.43757e6i 0.912245i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 2.94624e6i 0.484776i
\(518\) 0 0
\(519\) −4.09214e6 −0.666857
\(520\) 0 0
\(521\) −1.14928e6 −0.185495 −0.0927475 0.995690i \(-0.529565\pi\)
−0.0927475 + 0.995690i \(0.529565\pi\)
\(522\) 0 0
\(523\) 7.22071e6i 1.15432i 0.816632 + 0.577159i \(0.195839\pi\)
−0.816632 + 0.577159i \(0.804161\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 1.15889e6i − 0.181767i
\(528\) 0 0
\(529\) 6.13892e6 0.953789
\(530\) 0 0
\(531\) −3.79395e6 −0.583923
\(532\) 0 0
\(533\) 8.79241e6i 1.34057i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 452505.i 0.0677155i
\(538\) 0 0
\(539\) −3.63316e6 −0.538657
\(540\) 0 0
\(541\) −1.24720e7 −1.83207 −0.916035 0.401099i \(-0.868628\pi\)
−0.916035 + 0.401099i \(0.868628\pi\)
\(542\) 0 0
\(543\) − 1.14825e6i − 0.167123i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 8.58966e6i − 1.22746i −0.789516 0.613730i \(-0.789668\pi\)
0.789516 0.613730i \(-0.210332\pi\)
\(548\) 0 0
\(549\) −2.66107e6 −0.376812
\(550\) 0 0
\(551\) −6.19546e6 −0.869350
\(552\) 0 0
\(553\) 2.74776e6i 0.382090i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 3.38207e6i − 0.461896i −0.972966 0.230948i \(-0.925817\pi\)
0.972966 0.230948i \(-0.0741827\pi\)
\(558\) 0 0
\(559\) −8.48278e6 −1.14818
\(560\) 0 0
\(561\) −1.00897e6 −0.135354
\(562\) 0 0
\(563\) − 1.50247e6i − 0.199772i −0.994999 0.0998859i \(-0.968152\pi\)
0.994999 0.0998859i \(-0.0318478\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 544468.i 0.0711237i
\(568\) 0 0
\(569\) −1.87838e6 −0.243222 −0.121611 0.992578i \(-0.538806\pi\)
−0.121611 + 0.992578i \(0.538806\pi\)
\(570\) 0 0
\(571\) 1.49628e7 1.92053 0.960267 0.279082i \(-0.0900301\pi\)
0.960267 + 0.279082i \(0.0900301\pi\)
\(572\) 0 0
\(573\) 4.38306e6i 0.557687i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 4.03660e6i − 0.504750i −0.967630 0.252375i \(-0.918788\pi\)
0.967630 0.252375i \(-0.0812117\pi\)
\(578\) 0 0
\(579\) −1.45666e6 −0.180577
\(580\) 0 0
\(581\) 1.98103e6 0.243473
\(582\) 0 0
\(583\) − 6.55303e6i − 0.798492i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 1.57983e7i − 1.89241i −0.323574 0.946203i \(-0.604885\pi\)
0.323574 0.946203i \(-0.395115\pi\)
\(588\) 0 0
\(589\) 3.99650e6 0.474670
\(590\) 0 0
\(591\) −3.75845e6 −0.442629
\(592\) 0 0
\(593\) 1.37947e7i 1.61092i 0.592647 + 0.805462i \(0.298083\pi\)
−0.592647 + 0.805462i \(0.701917\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 5.74549e6i 0.659768i
\(598\) 0 0
\(599\) −814106. −0.0927073 −0.0463537 0.998925i \(-0.514760\pi\)
−0.0463537 + 0.998925i \(0.514760\pi\)
\(600\) 0 0
\(601\) −1.55613e7 −1.75736 −0.878679 0.477413i \(-0.841574\pi\)
−0.878679 + 0.477413i \(0.841574\pi\)
\(602\) 0 0
\(603\) − 3.23855e6i − 0.362708i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 1.63191e7i 1.79773i 0.438226 + 0.898865i \(0.355607\pi\)
−0.438226 + 0.898865i \(0.644393\pi\)
\(608\) 0 0
\(609\) −1.44558e6 −0.157942
\(610\) 0 0
\(611\) −6.66475e6 −0.722239
\(612\) 0 0
\(613\) − 5.38233e6i − 0.578520i −0.957250 0.289260i \(-0.906591\pi\)
0.957250 0.289260i \(-0.0934093\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 5.83565e6i 0.617130i 0.951203 + 0.308565i \(0.0998487\pi\)
−0.951203 + 0.308565i \(0.900151\pi\)
\(618\) 0 0
\(619\) −3.36289e6 −0.352765 −0.176383 0.984322i \(-0.556440\pi\)
−0.176383 + 0.984322i \(0.556440\pi\)
\(620\) 0 0
\(621\) 2.21980e6 0.230985
\(622\) 0 0
\(623\) − 1.14951e6i − 0.118657i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 3.47949e6i − 0.353465i
\(628\) 0 0
\(629\) 489541. 0.0493358
\(630\) 0 0
\(631\) −8.10425e6 −0.810288 −0.405144 0.914253i \(-0.632779\pi\)
−0.405144 + 0.914253i \(0.632779\pi\)
\(632\) 0 0
\(633\) − 3.58128e6i − 0.355246i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 8.21866e6i − 0.802513i
\(638\) 0 0
\(639\) −998828. −0.0967695
\(640\) 0 0
\(641\) 7.08523e6 0.681097 0.340548 0.940227i \(-0.389387\pi\)
0.340548 + 0.940227i \(0.389387\pi\)
\(642\) 0 0
\(643\) 5.72946e6i 0.546495i 0.961944 + 0.273247i \(0.0880978\pi\)
−0.961944 + 0.273247i \(0.911902\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 2.17751e6i − 0.204503i −0.994759 0.102251i \(-0.967395\pi\)
0.994759 0.102251i \(-0.0326046\pi\)
\(648\) 0 0
\(649\) 7.78143e6 0.725183
\(650\) 0 0
\(651\) 932498. 0.0862374
\(652\) 0 0
\(653\) − 1.42048e7i − 1.30363i −0.758379 0.651813i \(-0.774009\pi\)
0.758379 0.651813i \(-0.225991\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 6.11890e6i 0.553044i
\(658\) 0 0
\(659\) 1.54033e7 1.38166 0.690829 0.723018i \(-0.257246\pi\)
0.690829 + 0.723018i \(0.257246\pi\)
\(660\) 0 0
\(661\) −544048. −0.0484322 −0.0242161 0.999707i \(-0.507709\pi\)
−0.0242161 + 0.999707i \(0.507709\pi\)
\(662\) 0 0
\(663\) − 2.28241e6i − 0.201656i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 2.52919e6i 0.220124i
\(668\) 0 0
\(669\) −7.22894e6 −0.624467
\(670\) 0 0
\(671\) 5.45787e6 0.467969
\(672\) 0 0
\(673\) − 1.37093e7i − 1.16675i −0.812203 0.583374i \(-0.801732\pi\)
0.812203 0.583374i \(-0.198268\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 1.49910e7i − 1.25707i −0.777781 0.628536i \(-0.783655\pi\)
0.777781 0.628536i \(-0.216345\pi\)
\(678\) 0 0
\(679\) 285789. 0.0237887
\(680\) 0 0
\(681\) −5.18738e6 −0.428628
\(682\) 0 0
\(683\) 3.47996e6i 0.285445i 0.989763 + 0.142723i \(0.0455857\pi\)
−0.989763 + 0.142723i \(0.954414\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 5.31824e6i − 0.429908i
\(688\) 0 0
\(689\) 1.48237e7 1.18962
\(690\) 0 0
\(691\) −5.09436e6 −0.405877 −0.202939 0.979191i \(-0.565049\pi\)
−0.202939 + 0.979191i \(0.565049\pi\)
\(692\) 0 0
\(693\) 673627.i 0.0532827i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 6.66213e6i − 0.519435i
\(698\) 0 0
\(699\) 7.78817e6 0.602896
\(700\) 0 0
\(701\) 1.75839e7 1.35151 0.675755 0.737126i \(-0.263818\pi\)
0.675755 + 0.737126i \(0.263818\pi\)
\(702\) 0 0
\(703\) 1.68821e6i 0.128836i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 1.25737e6i − 0.0946051i
\(708\) 0 0
\(709\) −1.35383e7 −1.01146 −0.505730 0.862692i \(-0.668777\pi\)
−0.505730 + 0.862692i \(0.668777\pi\)
\(710\) 0 0
\(711\) −1.11942e7 −0.830462
\(712\) 0 0
\(713\) − 1.63150e6i − 0.120189i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 1.16083e7i 0.843275i
\(718\) 0 0
\(719\) 1.27911e7 0.922756 0.461378 0.887204i \(-0.347355\pi\)
0.461378 + 0.887204i \(0.347355\pi\)
\(720\) 0 0
\(721\) 3.23298e6 0.231614
\(722\) 0 0
\(723\) 2.67466e6i 0.190293i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 2.11509e7i 1.48420i 0.670289 + 0.742100i \(0.266170\pi\)
−0.670289 + 0.742100i \(0.733830\pi\)
\(728\) 0 0
\(729\) −1.36164e7 −0.948949
\(730\) 0 0
\(731\) 6.42753e6 0.444888
\(732\) 0 0
\(733\) − 1.23015e7i − 0.845667i −0.906207 0.422833i \(-0.861036\pi\)
0.906207 0.422833i \(-0.138964\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 6.64230e6i 0.450453i
\(738\) 0 0
\(739\) 1.04425e7 0.703383 0.351691 0.936116i \(-0.385607\pi\)
0.351691 + 0.936116i \(0.385607\pi\)
\(740\) 0 0
\(741\) 7.87103e6 0.526606
\(742\) 0 0
\(743\) − 3.63296e6i − 0.241429i −0.992687 0.120714i \(-0.961481\pi\)
0.992687 0.120714i \(-0.0385185\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 8.07060e6i 0.529181i
\(748\) 0 0
\(749\) 620012. 0.0403827
\(750\) 0 0
\(751\) 1.77346e7 1.14741 0.573707 0.819060i \(-0.305505\pi\)
0.573707 + 0.819060i \(0.305505\pi\)
\(752\) 0 0
\(753\) − 2.04088e7i − 1.31169i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 7.89094e6i − 0.500483i −0.968183 0.250241i \(-0.919490\pi\)
0.968183 0.250241i \(-0.0805100\pi\)
\(758\) 0 0
\(759\) −1.42044e6 −0.0894992
\(760\) 0 0
\(761\) 1.98455e7 1.24223 0.621114 0.783721i \(-0.286681\pi\)
0.621114 + 0.783721i \(0.286681\pi\)
\(762\) 0 0
\(763\) 5.68343e6i 0.353426i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 1.76026e7i 1.08041i
\(768\) 0 0
\(769\) −3.89976e6 −0.237806 −0.118903 0.992906i \(-0.537938\pi\)
−0.118903 + 0.992906i \(0.537938\pi\)
\(770\) 0 0
\(771\) 6.09004e6 0.368964
\(772\) 0 0
\(773\) 2.33736e6i 0.140695i 0.997523 + 0.0703474i \(0.0224108\pi\)
−0.997523 + 0.0703474i \(0.977589\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 393908.i 0.0234068i
\(778\) 0 0
\(779\) 2.29748e7 1.35646
\(780\) 0 0
\(781\) 2.04860e6 0.120180
\(782\) 0 0
\(783\) − 1.88762e7i − 1.10030i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 1.52903e7i 0.879993i 0.897999 + 0.439997i \(0.145020\pi\)
−0.897999 + 0.439997i \(0.854980\pi\)
\(788\) 0 0
\(789\) −1.95804e7 −1.11977
\(790\) 0 0
\(791\) 5.51479e6 0.313392
\(792\) 0 0
\(793\) 1.23464e7i 0.697199i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 1.25972e7i − 0.702468i −0.936288 0.351234i \(-0.885762\pi\)
0.936288 0.351234i \(-0.114238\pi\)
\(798\) 0 0
\(799\) 5.04997e6 0.279848
\(800\) 0 0
\(801\) 4.68305e6 0.257898
\(802\) 0 0
\(803\) − 1.25499e7i − 0.686835i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 7.47219e6i − 0.403891i
\(808\) 0 0
\(809\) 1.13476e7 0.609584 0.304792 0.952419i \(-0.401413\pi\)
0.304792 + 0.952419i \(0.401413\pi\)
\(810\) 0 0
\(811\) 1.91000e7 1.01972 0.509860 0.860257i \(-0.329697\pi\)
0.509860 + 0.860257i \(0.329697\pi\)
\(812\) 0 0
\(813\) − 2.24661e7i − 1.19207i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 2.21657e7i 1.16179i
\(818\) 0 0
\(819\) −1.52383e6 −0.0793827
\(820\) 0 0
\(821\) 1.25668e6 0.0650678 0.0325339 0.999471i \(-0.489642\pi\)
0.0325339 + 0.999471i \(0.489642\pi\)
\(822\) 0 0
\(823\) − 6.95590e6i − 0.357976i −0.983851 0.178988i \(-0.942718\pi\)
0.983851 0.178988i \(-0.0572823\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 9.04638e6i 0.459950i 0.973197 + 0.229975i \(0.0738645\pi\)
−0.973197 + 0.229975i \(0.926136\pi\)
\(828\) 0 0
\(829\) 1.54026e7 0.778407 0.389203 0.921152i \(-0.372750\pi\)
0.389203 + 0.921152i \(0.372750\pi\)
\(830\) 0 0
\(831\) 3.60932e6 0.181310
\(832\) 0 0
\(833\) 6.22739e6i 0.310952i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 1.21764e7i 0.600767i
\(838\) 0 0
\(839\) 1.75941e7 0.862903 0.431451 0.902136i \(-0.358002\pi\)
0.431451 + 0.902136i \(0.358002\pi\)
\(840\) 0 0
\(841\) 995979. 0.0485580
\(842\) 0 0
\(843\) − 1.99072e7i − 0.964809i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 2.97455e6i 0.142466i
\(848\) 0 0
\(849\) −2.36410e6 −0.112563
\(850\) 0 0
\(851\) 689183. 0.0326220
\(852\) 0 0
\(853\) − 2.06755e7i − 0.972934i −0.873699 0.486467i \(-0.838285\pi\)
0.873699 0.486467i \(-0.161715\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 2.16205e7i − 1.00557i −0.864411 0.502786i \(-0.832308\pi\)
0.864411 0.502786i \(-0.167692\pi\)
\(858\) 0 0
\(859\) −1.70061e7 −0.786362 −0.393181 0.919461i \(-0.628625\pi\)
−0.393181 + 0.919461i \(0.628625\pi\)
\(860\) 0 0
\(861\) 5.36067e6 0.246440
\(862\) 0 0
\(863\) 2.96043e7i 1.35309i 0.736399 + 0.676547i \(0.236525\pi\)
−0.736399 + 0.676547i \(0.763475\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 1.46333e7i − 0.661140i
\(868\) 0 0
\(869\) 2.29594e7 1.03136
\(870\) 0 0
\(871\) −1.50257e7 −0.671103
\(872\) 0 0
\(873\) 1.16429e6i 0.0517041i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 2.77691e7i 1.21917i 0.792722 + 0.609583i \(0.208663\pi\)
−0.792722 + 0.609583i \(0.791337\pi\)
\(878\) 0 0
\(879\) −4.49799e6 −0.196357
\(880\) 0 0
\(881\) 1.16824e7 0.507099 0.253550 0.967322i \(-0.418402\pi\)
0.253550 + 0.967322i \(0.418402\pi\)
\(882\) 0 0
\(883\) − 2.66919e7i − 1.15207i −0.817426 0.576034i \(-0.804600\pi\)
0.817426 0.576034i \(-0.195400\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 3.95714e7i − 1.68878i −0.535733 0.844388i \(-0.679964\pi\)
0.535733 0.844388i \(-0.320036\pi\)
\(888\) 0 0
\(889\) 4.73406e6 0.200900
\(890\) 0 0
\(891\) 4.54940e6 0.191982
\(892\) 0 0
\(893\) 1.74151e7i 0.730799i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 3.21321e6i − 0.133339i
\(898\) 0 0
\(899\) −1.38736e7 −0.572518
\(900\) 0 0
\(901\) −1.12322e7 −0.460948
\(902\) 0 0
\(903\) 5.17189e6i 0.211072i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 2.79635e7i 1.12869i 0.825541 + 0.564343i \(0.190870\pi\)
−0.825541 + 0.564343i \(0.809130\pi\)
\(908\) 0 0
\(909\) 5.12245e6 0.205621
\(910\) 0 0
\(911\) 1.87296e7 0.747710 0.373855 0.927487i \(-0.378036\pi\)
0.373855 + 0.927487i \(0.378036\pi\)
\(912\) 0 0
\(913\) − 1.65529e7i − 0.657199i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 4.03610e6i 0.158503i
\(918\) 0 0
\(919\) −7.77613e6 −0.303721 −0.151861 0.988402i \(-0.548526\pi\)
−0.151861 + 0.988402i \(0.548526\pi\)
\(920\) 0 0
\(921\) −1.27582e7 −0.495612
\(922\) 0 0
\(923\) 4.63420e6i 0.179048i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 1.31710e7i 0.503406i
\(928\) 0 0
\(929\) 3.88269e7 1.47602 0.738011 0.674788i \(-0.235765\pi\)
0.738011 + 0.674788i \(0.235765\pi\)
\(930\) 0 0
\(931\) −2.14755e7 −0.812026
\(932\) 0 0
\(933\) 2.52501e7i 0.949640i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 1.75302e7i 0.652288i 0.945320 + 0.326144i \(0.105749\pi\)
−0.945320 + 0.326144i \(0.894251\pi\)
\(938\) 0 0
\(939\) −3.33107e7 −1.23288
\(940\) 0 0
\(941\) −4.29638e6 −0.158172 −0.0790859 0.996868i \(-0.525200\pi\)
−0.0790859 + 0.996868i \(0.525200\pi\)
\(942\) 0 0
\(943\) − 9.37905e6i − 0.343463i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 4.43637e7i − 1.60751i −0.594962 0.803754i \(-0.702833\pi\)
0.594962 0.803754i \(-0.297167\pi\)
\(948\) 0 0
\(949\) 2.83895e7 1.02327
\(950\) 0 0
\(951\) 1.69004e7 0.605963
\(952\) 0 0
\(953\) 5.71990e6i 0.204012i 0.994784 + 0.102006i \(0.0325261\pi\)
−0.994784 + 0.102006i \(0.967474\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 1.20788e7i 0.426329i
\(958\) 0 0
\(959\) −5.73912e6 −0.201511
\(960\) 0 0
\(961\) −1.96797e7 −0.687402
\(962\) 0 0
\(963\) 2.52589e6i 0.0877706i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 3.00548e7i − 1.03359i −0.856110 0.516794i \(-0.827125\pi\)
0.856110 0.516794i \(-0.172875\pi\)
\(968\) 0 0
\(969\) −5.96399e6 −0.204046
\(970\) 0 0
\(971\) −1.89662e7 −0.645552 −0.322776 0.946475i \(-0.604616\pi\)
−0.322776 + 0.946475i \(0.604616\pi\)
\(972\) 0 0
\(973\) − 2.08495e6i − 0.0706015i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 2.45541e7i − 0.822975i −0.911415 0.411488i \(-0.865009\pi\)
0.911415 0.411488i \(-0.134991\pi\)
\(978\) 0 0
\(979\) −9.60498e6 −0.320287
\(980\) 0 0
\(981\) −2.31540e7 −0.768162
\(982\) 0 0
\(983\) − 3.35539e7i − 1.10754i −0.832670 0.553769i \(-0.813189\pi\)
0.832670 0.553769i \(-0.186811\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 4.06345e6i 0.132771i
\(988\) 0 0
\(989\) 9.04877e6 0.294170
\(990\) 0 0
\(991\) −2.05230e7 −0.663830 −0.331915 0.943309i \(-0.607695\pi\)
−0.331915 + 0.943309i \(0.607695\pi\)
\(992\) 0 0
\(993\) − 2.57565e7i − 0.828924i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 1.25729e7i − 0.400589i −0.979736 0.200294i \(-0.935810\pi\)
0.979736 0.200294i \(-0.0641898\pi\)
\(998\) 0 0
\(999\) −5.14359e6 −0.163062
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 400.6.c.o.49.2 4
4.3 odd 2 200.6.c.f.49.3 4
5.2 odd 4 400.6.a.u.1.1 2
5.3 odd 4 400.6.a.r.1.2 2
5.4 even 2 inner 400.6.c.o.49.3 4
20.3 even 4 200.6.a.f.1.1 yes 2
20.7 even 4 200.6.a.e.1.2 2
20.19 odd 2 200.6.c.f.49.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
200.6.a.e.1.2 2 20.7 even 4
200.6.a.f.1.1 yes 2 20.3 even 4
200.6.c.f.49.2 4 20.19 odd 2
200.6.c.f.49.3 4 4.3 odd 2
400.6.a.r.1.2 2 5.3 odd 4
400.6.a.u.1.1 2 5.2 odd 4
400.6.c.o.49.2 4 1.1 even 1 trivial
400.6.c.o.49.3 4 5.4 even 2 inner