Properties

Label 200.6.c.f.49.2
Level $200$
Weight $6$
Character 200.49
Analytic conductor $32.077$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 200 = 2^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 200.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(32.0767639626\)
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: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.2
Root \(8.26209i\) of defining polynomial
Character \(\chi\) \(=\) 200.49
Dual form 200.6.c.f.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 \(\left(\mathbb{Z}/200\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(151\) \(177\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(-0.409139\pi\)
0.281586 + 0.959536i \(0.409139\pi\)
\(12\) 0 0
\(13\) 511.257i 0.839037i 0.907747 + 0.419518i \(0.137801\pi\)
−0.907747 + 0.419518i \(0.862199\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 387.387i − 0.325104i −0.986700 0.162552i \(-0.948027\pi\)
0.986700 0.162552i \(-0.0519725\pi\)
\(18\) 0 0
\(19\) 1335.93 0.848982 0.424491 0.905432i \(-0.360453\pi\)
0.424491 + 0.905432i \(0.360453\pi\)
\(20\) 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 \(-0.409814\pi\)
0.279552 + 0.960130i \(0.409814\pi\)
\(32\) 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 \(0.0241756\pi\)
−0.997117 + 0.0758770i \(0.975824\pi\)
\(38\) 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 \(-0.749181\pi\)
0.705286 0.708923i \(-0.250819\pi\)
\(54\) 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 \(-0.277341\pi\)
0.643838 + 0.765162i \(0.277341\pi\)
\(60\) 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 \(-0.465972\pi\)
0.106699 + 0.994291i \(0.465972\pi\)
\(72\) 0 0
\(73\) − 55528.7i − 1.21958i −0.792563 0.609791i \(-0.791254\pi\)
0.792563 0.609791i \(-0.208746\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 6113.13i − 0.117500i
\(78\) 0 0
\(79\) 101587. 1.83135 0.915673 0.401924i \(-0.131658\pi\)
0.915673 + 0.401924i \(0.131658\pi\)
\(80\) 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 \(-0.981844\pi\)
0.998374 0.0570093i \(-0.0181565\pi\)
\(98\) 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 \(-0.729553\pi\)
0.660258 0.751039i \(-0.270447\pi\)
\(114\) 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 \(-0.375975\pi\)
0.379850 + 0.925048i \(0.375975\pi\)
\(132\) 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 \(0.160423\pi\)
−0.875666 + 0.482918i \(0.839577\pi\)
\(138\) 0 0
\(139\) −77082.5 −0.338391 −0.169195 0.985583i \(-0.554117\pi\)
−0.169195 + 0.985583i \(0.554117\pi\)
\(140\) 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 \(-0.312640\pi\)
0.555205 + 0.831714i \(0.312640\pi\)
\(152\) 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 \(0.342535\pi\)
−0.474760 + 0.880115i \(0.657465\pi\)
\(158\) 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 \(0.148940\pi\)
−0.892514 + 0.451020i \(0.851060\pi\)
\(174\) 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 \(-0.485417\pi\)
0.0457985 + 0.998951i \(0.485417\pi\)
\(180\) 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 \(-0.376892\pi\)
0.377185 + 0.926138i \(0.376892\pi\)
\(192\) 0 0
\(193\) 126401.i 0.244262i 0.992514 + 0.122131i \(0.0389728\pi\)
−0.992514 + 0.122131i \(0.961027\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 326136.i 0.598733i 0.954138 + 0.299366i \(0.0967753\pi\)
−0.954138 + 0.299366i \(0.903225\pi\)
\(198\) 0 0
\(199\) 498559. 0.892451 0.446225 0.894921i \(-0.352768\pi\)
0.446225 + 0.894921i \(0.352768\pi\)
\(200\) 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 \(-0.577235\pi\)
−0.240266 + 0.970707i \(0.577235\pi\)
\(212\) 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 \(-0.866310\pi\)
0.913089 0.407761i \(-0.133690\pi\)
\(234\) 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 \(-0.306812\pi\)
0.570338 + 0.821410i \(0.306812\pi\)
\(240\) 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 \(-0.847314\pi\)
−0.887143 + 0.461494i \(0.847314\pi\)
\(252\) 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 \(-0.919719\pi\)
0.968363 0.249544i \(-0.0802808\pi\)
\(258\) 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 \(-0.798500\pi\)
−0.806239 + 0.591590i \(0.798500\pi\)
\(272\) 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 \(-0.960868\pi\)
0.992453 0.122627i \(-0.0391319\pi\)
\(278\) 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 \(0.0423979\pi\)
−0.991142 + 0.132804i \(0.957602\pi\)
\(294\) 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 \(-0.277990\pi\)
0.642277 + 0.766472i \(0.277990\pi\)
\(312\) 0 0
\(313\) 2.89050e6i 1.66768i 0.552006 + 0.833840i \(0.313862\pi\)
−0.552006 + 0.833840i \(0.686138\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 1.46652e6i − 0.819671i −0.912160 0.409835i \(-0.865586\pi\)
0.912160 0.409835i \(-0.134414\pi\)
\(318\) 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 \(-0.689442\pi\)
−0.560632 + 0.828065i \(0.689442\pi\)
\(332\) 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 \(0.116195\pi\)
−0.934111 + 0.356983i \(0.883805\pi\)
\(338\) 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 \(0.178901\pi\)
−0.846173 + 0.532909i \(0.821099\pi\)
\(354\) 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 \(-0.748212\pi\)
−0.703124 + 0.711068i \(0.748212\pi\)
\(360\) 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 \(0.123789\pi\)
−0.925328 + 0.379167i \(0.876211\pi\)
\(374\) 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 \(-0.603688\pi\)
−0.320016 + 0.947412i \(0.603688\pi\)
\(380\) 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 \(0.148052\pi\)
−0.893768 + 0.448529i \(0.851948\pi\)
\(398\) 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 \(-0.788600\pi\)
−0.787452 + 0.616376i \(0.788600\pi\)
\(420\) 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 \(-0.371260\pi\)
0.393512 + 0.919320i \(0.371260\pi\)
\(432\) 0 0
\(433\) 3.50633e6i 0.898739i 0.893346 + 0.449369i \(0.148351\pi\)
−0.893346 + 0.449369i \(0.851649\pi\)
\(434\) 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 \(-0.228166\pi\)
0.753908 + 0.656980i \(0.228166\pi\)
\(440\) 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 \(-0.984647\pi\)
0.998837 0.0482154i \(-0.0153534\pi\)
\(458\) 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 \(-0.177828\pi\)
0.847964 + 0.530054i \(0.177828\pi\)
\(480\) 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 \(-0.656749\pi\)
−0.472779 + 0.881181i \(0.656749\pi\)
\(492\) 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 \(-0.438091\pi\)
0.193269 + 0.981146i \(0.438091\pi\)
\(500\) 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 \(0.0741827\pi\)
−0.972966 + 0.230948i \(0.925817\pi\)
\(558\) 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 \(-0.909970\pi\)
−0.960267 + 0.279082i \(0.909970\pi\)
\(572\) 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 \(0.0812117\pi\)
−0.967630 + 0.252375i \(0.918788\pi\)
\(578\) 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 \(-0.701917\pi\)
0.592647 0.805462i \(-0.298083\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 5.74549e6i − 0.659768i
\(598\) 0 0
\(599\) 814106. 0.0927073 0.0463537 0.998925i \(-0.485240\pi\)
0.0463537 + 0.998925i \(0.485240\pi\)
\(600\) 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 \(0.0934093\pi\)
−0.957250 + 0.289260i \(0.906591\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 5.83565e6i − 0.617130i −0.951203 0.308565i \(-0.900151\pi\)
0.951203 0.308565i \(-0.0998487\pi\)
\(618\) 0 0
\(619\) 3.36289e6 0.352765 0.176383 0.984322i \(-0.443560\pi\)
0.176383 + 0.984322i \(0.443560\pi\)
\(620\) 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 \(-0.367221\pi\)
0.405144 + 0.914253i \(0.367221\pi\)
\(632\) 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 \(0.225991\pi\)
−0.758379 + 0.651813i \(0.774009\pi\)
\(654\) 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 \(-0.742754\pi\)
−0.690829 + 0.723018i \(0.742754\pi\)
\(660\) 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 \(0.198268\pi\)
−0.812203 + 0.583374i \(0.801732\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 1.49910e7i 1.25707i 0.777781 + 0.628536i \(0.216345\pi\)
−0.777781 + 0.628536i \(0.783655\pi\)
\(678\) 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 \(-0.434951\pi\)
0.202939 + 0.979191i \(0.434951\pi\)
\(692\) 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 \(-0.652645\pi\)
−0.461378 + 0.887204i \(0.652645\pi\)
\(720\) 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 \(0.138964\pi\)
−0.906207 + 0.422833i \(0.861036\pi\)
\(734\) 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 \(-0.614393\pi\)
−0.351691 + 0.936116i \(0.614393\pi\)
\(740\) 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 \(-0.694495\pi\)
−0.573707 + 0.819060i \(0.694495\pi\)
\(752\) 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 \(0.0805100\pi\)
−0.968183 + 0.250241i \(0.919490\pi\)
\(758\) 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 \(-0.977589\pi\)
0.997523 0.0703474i \(-0.0224108\pi\)
\(774\) 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 \(0.114238\pi\)
−0.936288 + 0.351234i \(0.885762\pi\)
\(798\) 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 \(-0.670303\pi\)
−0.509860 + 0.860257i \(0.670303\pi\)
\(812\) 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 \(-0.641998\pi\)
−0.431451 + 0.902136i \(0.641998\pi\)
\(840\) 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 \(0.161715\pi\)
−0.873699 + 0.486467i \(0.838285\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 2.16205e7i 1.00557i 0.864411 + 0.502786i \(0.167692\pi\)
−0.864411 + 0.502786i \(0.832308\pi\)
\(858\) 0 0
\(859\) 1.70061e7 0.786362 0.393181 0.919461i \(-0.371375\pi\)
0.393181 + 0.919461i \(0.371375\pi\)
\(860\) 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 \(-0.791337\pi\)
0.792722 0.609583i \(-0.208663\pi\)
\(878\) 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 \(-0.621964\pi\)
−0.373855 + 0.927487i \(0.621964\pi\)
\(912\) 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 \(-0.451474\pi\)
0.151861 + 0.988402i \(0.451474\pi\)
\(920\) 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 \(-0.894251\pi\)
0.945320 0.326144i \(-0.105749\pi\)
\(938\) 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 \(-0.967474\pi\)
0.994784 0.102006i \(-0.0325261\pi\)
\(954\) 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 \(-0.395384\pi\)
0.322776 + 0.946475i \(0.395384\pi\)
\(972\) 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 \(0.134991\pi\)
−0.911415 + 0.411488i \(0.865009\pi\)
\(978\) 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 \(-0.392305\pi\)
0.331915 + 0.943309i \(0.392305\pi\)
\(992\) 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 \(0.0641898\pi\)
−0.979736 + 0.200294i \(0.935810\pi\)
\(998\) 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 200.6.c.f.49.2 4
4.3 odd 2 400.6.c.o.49.3 4
5.2 odd 4 200.6.a.f.1.1 yes 2
5.3 odd 4 200.6.a.e.1.2 2
5.4 even 2 inner 200.6.c.f.49.3 4
20.3 even 4 400.6.a.u.1.1 2
20.7 even 4 400.6.a.r.1.2 2
20.19 odd 2 400.6.c.o.49.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
200.6.a.e.1.2 2 5.3 odd 4
200.6.a.f.1.1 yes 2 5.2 odd 4
200.6.c.f.49.2 4 1.1 even 1 trivial
200.6.c.f.49.3 4 5.4 even 2 inner
400.6.a.r.1.2 2 20.7 even 4
400.6.a.u.1.1 2 20.3 even 4
400.6.c.o.49.2 4 20.19 odd 2
400.6.c.o.49.3 4 4.3 odd 2