Properties

Label 784.4.a.y.1.1
Level $784$
Weight $4$
Character 784.1
Self dual yes
Analytic conductor $46.257$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 784.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(46.2574974445\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 98)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 784.1

$q$-expansion

\(f(q)\) \(=\) \(q-7.07107 q^{3} +19.7990 q^{5} +23.0000 q^{9} +O(q^{10})\) \(q-7.07107 q^{3} +19.7990 q^{5} +23.0000 q^{9} +14.0000 q^{11} -50.9117 q^{13} -140.000 q^{15} -1.41421 q^{17} -1.41421 q^{19} -140.000 q^{23} +267.000 q^{25} +28.2843 q^{27} -286.000 q^{29} -93.3381 q^{31} -98.9949 q^{33} -38.0000 q^{37} +360.000 q^{39} +125.865 q^{41} +34.0000 q^{43} +455.377 q^{45} +523.259 q^{47} +10.0000 q^{51} -74.0000 q^{53} +277.186 q^{55} +10.0000 q^{57} +434.164 q^{59} -14.1421 q^{61} -1008.00 q^{65} -684.000 q^{67} +989.949 q^{69} -588.000 q^{71} +270.115 q^{73} -1887.98 q^{75} -1220.00 q^{79} -821.000 q^{81} +422.850 q^{83} -28.0000 q^{85} +2022.33 q^{87} -618.011 q^{89} +660.000 q^{93} -28.0000 q^{95} -1483.51 q^{97} +322.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 46q^{9} + O(q^{10}) \) \( 2q + 46q^{9} + 28q^{11} - 280q^{15} - 280q^{23} + 534q^{25} - 572q^{29} - 76q^{37} + 720q^{39} + 68q^{43} + 20q^{51} - 148q^{53} + 20q^{57} - 2016q^{65} - 1368q^{67} - 1176q^{71} - 2440q^{79} - 1642q^{81} - 56q^{85} + 1320q^{93} - 56q^{95} + 644q^{99} + O(q^{100}) \)

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
\(3\) −7.07107 −1.36083 −0.680414 0.732828i \(-0.738200\pi\)
−0.680414 + 0.732828i \(0.738200\pi\)
\(4\) 0 0
\(5\) 19.7990 1.77088 0.885438 0.464758i \(-0.153859\pi\)
0.885438 + 0.464758i \(0.153859\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 23.0000 0.851852
\(10\) 0 0
\(11\) 14.0000 0.383742 0.191871 0.981420i \(-0.438545\pi\)
0.191871 + 0.981420i \(0.438545\pi\)
\(12\) 0 0
\(13\) −50.9117 −1.08618 −0.543091 0.839674i \(-0.682746\pi\)
−0.543091 + 0.839674i \(0.682746\pi\)
\(14\) 0 0
\(15\) −140.000 −2.40986
\(16\) 0 0
\(17\) −1.41421 −0.0201763 −0.0100882 0.999949i \(-0.503211\pi\)
−0.0100882 + 0.999949i \(0.503211\pi\)
\(18\) 0 0
\(19\) −1.41421 −0.0170759 −0.00853797 0.999964i \(-0.502718\pi\)
−0.00853797 + 0.999964i \(0.502718\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −140.000 −1.26922 −0.634609 0.772833i \(-0.718839\pi\)
−0.634609 + 0.772833i \(0.718839\pi\)
\(24\) 0 0
\(25\) 267.000 2.13600
\(26\) 0 0
\(27\) 28.2843 0.201604
\(28\) 0 0
\(29\) −286.000 −1.83134 −0.915670 0.401931i \(-0.868339\pi\)
−0.915670 + 0.401931i \(0.868339\pi\)
\(30\) 0 0
\(31\) −93.3381 −0.540775 −0.270387 0.962752i \(-0.587152\pi\)
−0.270387 + 0.962752i \(0.587152\pi\)
\(32\) 0 0
\(33\) −98.9949 −0.522206
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −38.0000 −0.168842 −0.0844211 0.996430i \(-0.526904\pi\)
−0.0844211 + 0.996430i \(0.526904\pi\)
\(38\) 0 0
\(39\) 360.000 1.47811
\(40\) 0 0
\(41\) 125.865 0.479434 0.239717 0.970843i \(-0.422945\pi\)
0.239717 + 0.970843i \(0.422945\pi\)
\(42\) 0 0
\(43\) 34.0000 0.120580 0.0602901 0.998181i \(-0.480797\pi\)
0.0602901 + 0.998181i \(0.480797\pi\)
\(44\) 0 0
\(45\) 455.377 1.50852
\(46\) 0 0
\(47\) 523.259 1.62394 0.811970 0.583699i \(-0.198395\pi\)
0.811970 + 0.583699i \(0.198395\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 10.0000 0.0274565
\(52\) 0 0
\(53\) −74.0000 −0.191786 −0.0958932 0.995392i \(-0.530571\pi\)
−0.0958932 + 0.995392i \(0.530571\pi\)
\(54\) 0 0
\(55\) 277.186 0.679559
\(56\) 0 0
\(57\) 10.0000 0.0232374
\(58\) 0 0
\(59\) 434.164 0.958022 0.479011 0.877809i \(-0.340995\pi\)
0.479011 + 0.877809i \(0.340995\pi\)
\(60\) 0 0
\(61\) −14.1421 −0.0296839 −0.0148419 0.999890i \(-0.504725\pi\)
−0.0148419 + 0.999890i \(0.504725\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −1008.00 −1.92349
\(66\) 0 0
\(67\) −684.000 −1.24722 −0.623611 0.781735i \(-0.714335\pi\)
−0.623611 + 0.781735i \(0.714335\pi\)
\(68\) 0 0
\(69\) 989.949 1.72719
\(70\) 0 0
\(71\) −588.000 −0.982856 −0.491428 0.870918i \(-0.663525\pi\)
−0.491428 + 0.870918i \(0.663525\pi\)
\(72\) 0 0
\(73\) 270.115 0.433076 0.216538 0.976274i \(-0.430523\pi\)
0.216538 + 0.976274i \(0.430523\pi\)
\(74\) 0 0
\(75\) −1887.98 −2.90673
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −1220.00 −1.73748 −0.868739 0.495271i \(-0.835069\pi\)
−0.868739 + 0.495271i \(0.835069\pi\)
\(80\) 0 0
\(81\) −821.000 −1.12620
\(82\) 0 0
\(83\) 422.850 0.559202 0.279601 0.960116i \(-0.409798\pi\)
0.279601 + 0.960116i \(0.409798\pi\)
\(84\) 0 0
\(85\) −28.0000 −0.0357297
\(86\) 0 0
\(87\) 2022.33 2.49214
\(88\) 0 0
\(89\) −618.011 −0.736057 −0.368028 0.929815i \(-0.619967\pi\)
−0.368028 + 0.929815i \(0.619967\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 660.000 0.735901
\(94\) 0 0
\(95\) −28.0000 −0.0302394
\(96\) 0 0
\(97\) −1483.51 −1.55286 −0.776431 0.630202i \(-0.782972\pi\)
−0.776431 + 0.630202i \(0.782972\pi\)
\(98\) 0 0
\(99\) 322.000 0.326891
\(100\) 0 0
\(101\) −1128.54 −1.11182 −0.555912 0.831241i \(-0.687631\pi\)
−0.555912 + 0.831241i \(0.687631\pi\)
\(102\) 0 0
\(103\) −868.327 −0.830668 −0.415334 0.909669i \(-0.636335\pi\)
−0.415334 + 0.909669i \(0.636335\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1684.00 1.52148 0.760740 0.649056i \(-0.224836\pi\)
0.760740 + 0.649056i \(0.224836\pi\)
\(108\) 0 0
\(109\) −818.000 −0.718809 −0.359405 0.933182i \(-0.617020\pi\)
−0.359405 + 0.933182i \(0.617020\pi\)
\(110\) 0 0
\(111\) 268.701 0.229765
\(112\) 0 0
\(113\) −540.000 −0.449548 −0.224774 0.974411i \(-0.572164\pi\)
−0.224774 + 0.974411i \(0.572164\pi\)
\(114\) 0 0
\(115\) −2771.86 −2.24763
\(116\) 0 0
\(117\) −1170.97 −0.925266
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −1135.00 −0.852742
\(122\) 0 0
\(123\) −890.000 −0.652428
\(124\) 0 0
\(125\) 2811.46 2.01171
\(126\) 0 0
\(127\) −1720.00 −1.20177 −0.600887 0.799334i \(-0.705186\pi\)
−0.600887 + 0.799334i \(0.705186\pi\)
\(128\) 0 0
\(129\) −240.416 −0.164089
\(130\) 0 0
\(131\) −1735.24 −1.15732 −0.578659 0.815570i \(-0.696424\pi\)
−0.578659 + 0.815570i \(0.696424\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 560.000 0.357016
\(136\) 0 0
\(137\) 828.000 0.516356 0.258178 0.966097i \(-0.416878\pi\)
0.258178 + 0.966097i \(0.416878\pi\)
\(138\) 0 0
\(139\) −425.678 −0.259752 −0.129876 0.991530i \(-0.541458\pi\)
−0.129876 + 0.991530i \(0.541458\pi\)
\(140\) 0 0
\(141\) −3700.00 −2.20990
\(142\) 0 0
\(143\) −712.764 −0.416813
\(144\) 0 0
\(145\) −5662.51 −3.24308
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 2050.00 1.12713 0.563566 0.826071i \(-0.309429\pi\)
0.563566 + 0.826071i \(0.309429\pi\)
\(150\) 0 0
\(151\) 472.000 0.254376 0.127188 0.991879i \(-0.459405\pi\)
0.127188 + 0.991879i \(0.459405\pi\)
\(152\) 0 0
\(153\) −32.5269 −0.0171872
\(154\) 0 0
\(155\) −1848.00 −0.957645
\(156\) 0 0
\(157\) 2211.83 1.12435 0.562176 0.827018i \(-0.309964\pi\)
0.562176 + 0.827018i \(0.309964\pi\)
\(158\) 0 0
\(159\) 523.259 0.260988
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −3286.00 −1.57901 −0.789507 0.613741i \(-0.789664\pi\)
−0.789507 + 0.613741i \(0.789664\pi\)
\(164\) 0 0
\(165\) −1960.00 −0.924762
\(166\) 0 0
\(167\) −1490.58 −0.690686 −0.345343 0.938476i \(-0.612237\pi\)
−0.345343 + 0.938476i \(0.612237\pi\)
\(168\) 0 0
\(169\) 395.000 0.179791
\(170\) 0 0
\(171\) −32.5269 −0.0145462
\(172\) 0 0
\(173\) 2070.41 0.909886 0.454943 0.890521i \(-0.349660\pi\)
0.454943 + 0.890521i \(0.349660\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −3070.00 −1.30370
\(178\) 0 0
\(179\) −540.000 −0.225483 −0.112742 0.993624i \(-0.535963\pi\)
−0.112742 + 0.993624i \(0.535963\pi\)
\(180\) 0 0
\(181\) −3784.44 −1.55412 −0.777058 0.629429i \(-0.783289\pi\)
−0.777058 + 0.629429i \(0.783289\pi\)
\(182\) 0 0
\(183\) 100.000 0.0403946
\(184\) 0 0
\(185\) −752.362 −0.298999
\(186\) 0 0
\(187\) −19.7990 −0.00774249
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −1028.00 −0.389442 −0.194721 0.980859i \(-0.562380\pi\)
−0.194721 + 0.980859i \(0.562380\pi\)
\(192\) 0 0
\(193\) 4592.00 1.71264 0.856320 0.516446i \(-0.172745\pi\)
0.856320 + 0.516446i \(0.172745\pi\)
\(194\) 0 0
\(195\) 7127.64 2.61754
\(196\) 0 0
\(197\) 794.000 0.287158 0.143579 0.989639i \(-0.454139\pi\)
0.143579 + 0.989639i \(0.454139\pi\)
\(198\) 0 0
\(199\) 2486.19 0.885634 0.442817 0.896612i \(-0.353979\pi\)
0.442817 + 0.896612i \(0.353979\pi\)
\(200\) 0 0
\(201\) 4836.61 1.69725
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 2492.00 0.849019
\(206\) 0 0
\(207\) −3220.00 −1.08119
\(208\) 0 0
\(209\) −19.7990 −0.00655275
\(210\) 0 0
\(211\) 2748.00 0.896588 0.448294 0.893886i \(-0.352032\pi\)
0.448294 + 0.893886i \(0.352032\pi\)
\(212\) 0 0
\(213\) 4157.79 1.33750
\(214\) 0 0
\(215\) 673.166 0.213533
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −1910.00 −0.589342
\(220\) 0 0
\(221\) 72.0000 0.0219151
\(222\) 0 0
\(223\) −3428.05 −1.02941 −0.514707 0.857366i \(-0.672099\pi\)
−0.514707 + 0.857366i \(0.672099\pi\)
\(224\) 0 0
\(225\) 6141.00 1.81956
\(226\) 0 0
\(227\) 5290.57 1.54691 0.773453 0.633854i \(-0.218528\pi\)
0.773453 + 0.633854i \(0.218528\pi\)
\(228\) 0 0
\(229\) 2749.23 0.793338 0.396669 0.917962i \(-0.370166\pi\)
0.396669 + 0.917962i \(0.370166\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 72.0000 0.0202441 0.0101221 0.999949i \(-0.496778\pi\)
0.0101221 + 0.999949i \(0.496778\pi\)
\(234\) 0 0
\(235\) 10360.0 2.87580
\(236\) 0 0
\(237\) 8626.70 2.36441
\(238\) 0 0
\(239\) −4308.00 −1.16595 −0.582974 0.812491i \(-0.698111\pi\)
−0.582974 + 0.812491i \(0.698111\pi\)
\(240\) 0 0
\(241\) 1540.08 0.411640 0.205820 0.978590i \(-0.434014\pi\)
0.205820 + 0.978590i \(0.434014\pi\)
\(242\) 0 0
\(243\) 5041.67 1.33096
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 72.0000 0.0185476
\(248\) 0 0
\(249\) −2990.00 −0.760978
\(250\) 0 0
\(251\) 931.967 0.234363 0.117182 0.993110i \(-0.462614\pi\)
0.117182 + 0.993110i \(0.462614\pi\)
\(252\) 0 0
\(253\) −1960.00 −0.487052
\(254\) 0 0
\(255\) 197.990 0.0486220
\(256\) 0 0
\(257\) −937.624 −0.227577 −0.113789 0.993505i \(-0.536299\pi\)
−0.113789 + 0.993505i \(0.536299\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −6578.00 −1.56003
\(262\) 0 0
\(263\) −7140.00 −1.67404 −0.837018 0.547176i \(-0.815703\pi\)
−0.837018 + 0.547176i \(0.815703\pi\)
\(264\) 0 0
\(265\) −1465.13 −0.339630
\(266\) 0 0
\(267\) 4370.00 1.00165
\(268\) 0 0
\(269\) −4610.34 −1.04497 −0.522485 0.852648i \(-0.674995\pi\)
−0.522485 + 0.852648i \(0.674995\pi\)
\(270\) 0 0
\(271\) −2364.57 −0.530026 −0.265013 0.964245i \(-0.585376\pi\)
−0.265013 + 0.964245i \(0.585376\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 3738.00 0.819672
\(276\) 0 0
\(277\) 4006.00 0.868943 0.434472 0.900686i \(-0.356935\pi\)
0.434472 + 0.900686i \(0.356935\pi\)
\(278\) 0 0
\(279\) −2146.78 −0.460660
\(280\) 0 0
\(281\) −5984.00 −1.27038 −0.635188 0.772358i \(-0.719077\pi\)
−0.635188 + 0.772358i \(0.719077\pi\)
\(282\) 0 0
\(283\) −4928.53 −1.03523 −0.517617 0.855613i \(-0.673181\pi\)
−0.517617 + 0.855613i \(0.673181\pi\)
\(284\) 0 0
\(285\) 197.990 0.0411506
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −4911.00 −0.999593
\(290\) 0 0
\(291\) 10490.0 2.11318
\(292\) 0 0
\(293\) −1971.41 −0.393076 −0.196538 0.980496i \(-0.562970\pi\)
−0.196538 + 0.980496i \(0.562970\pi\)
\(294\) 0 0
\(295\) 8596.00 1.69654
\(296\) 0 0
\(297\) 395.980 0.0773639
\(298\) 0 0
\(299\) 7127.64 1.37860
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 7980.00 1.51300
\(304\) 0 0
\(305\) −280.000 −0.0525664
\(306\) 0 0
\(307\) −4767.31 −0.886270 −0.443135 0.896455i \(-0.646134\pi\)
−0.443135 + 0.896455i \(0.646134\pi\)
\(308\) 0 0
\(309\) 6140.00 1.13040
\(310\) 0 0
\(311\) −6776.91 −1.23564 −0.617819 0.786320i \(-0.711984\pi\)
−0.617819 + 0.786320i \(0.711984\pi\)
\(312\) 0 0
\(313\) 6190.01 1.11783 0.558914 0.829226i \(-0.311218\pi\)
0.558914 + 0.829226i \(0.311218\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 9826.00 1.74096 0.870478 0.492207i \(-0.163810\pi\)
0.870478 + 0.492207i \(0.163810\pi\)
\(318\) 0 0
\(319\) −4004.00 −0.702762
\(320\) 0 0
\(321\) −11907.7 −2.07047
\(322\) 0 0
\(323\) 2.00000 0.000344529 0
\(324\) 0 0
\(325\) −13593.4 −2.32008
\(326\) 0 0
\(327\) 5784.13 0.978175
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 5738.00 0.952837 0.476418 0.879219i \(-0.341935\pi\)
0.476418 + 0.879219i \(0.341935\pi\)
\(332\) 0 0
\(333\) −874.000 −0.143829
\(334\) 0 0
\(335\) −13542.5 −2.20868
\(336\) 0 0
\(337\) −2254.00 −0.364342 −0.182171 0.983267i \(-0.558312\pi\)
−0.182171 + 0.983267i \(0.558312\pi\)
\(338\) 0 0
\(339\) 3818.38 0.611757
\(340\) 0 0
\(341\) −1306.73 −0.207518
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 19600.0 3.05863
\(346\) 0 0
\(347\) 1986.00 0.307245 0.153623 0.988130i \(-0.450906\pi\)
0.153623 + 0.988130i \(0.450906\pi\)
\(348\) 0 0
\(349\) −6771.25 −1.03856 −0.519279 0.854605i \(-0.673800\pi\)
−0.519279 + 0.854605i \(0.673800\pi\)
\(350\) 0 0
\(351\) −1440.00 −0.218979
\(352\) 0 0
\(353\) −6993.29 −1.05443 −0.527217 0.849731i \(-0.676764\pi\)
−0.527217 + 0.849731i \(0.676764\pi\)
\(354\) 0 0
\(355\) −11641.8 −1.74052
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −5944.00 −0.873850 −0.436925 0.899498i \(-0.643933\pi\)
−0.436925 + 0.899498i \(0.643933\pi\)
\(360\) 0 0
\(361\) −6857.00 −0.999708
\(362\) 0 0
\(363\) 8025.66 1.16044
\(364\) 0 0
\(365\) 5348.00 0.766924
\(366\) 0 0
\(367\) −842.871 −0.119884 −0.0599421 0.998202i \(-0.519092\pi\)
−0.0599421 + 0.998202i \(0.519092\pi\)
\(368\) 0 0
\(369\) 2894.90 0.408407
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −5726.00 −0.794855 −0.397428 0.917634i \(-0.630097\pi\)
−0.397428 + 0.917634i \(0.630097\pi\)
\(374\) 0 0
\(375\) −19880.0 −2.73760
\(376\) 0 0
\(377\) 14560.7 1.98917
\(378\) 0 0
\(379\) −10330.0 −1.40004 −0.700022 0.714122i \(-0.746826\pi\)
−0.700022 + 0.714122i \(0.746826\pi\)
\(380\) 0 0
\(381\) 12162.2 1.63541
\(382\) 0 0
\(383\) 1004.09 0.133960 0.0669800 0.997754i \(-0.478664\pi\)
0.0669800 + 0.997754i \(0.478664\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 782.000 0.102717
\(388\) 0 0
\(389\) 5210.00 0.679068 0.339534 0.940594i \(-0.389731\pi\)
0.339534 + 0.940594i \(0.389731\pi\)
\(390\) 0 0
\(391\) 197.990 0.0256081
\(392\) 0 0
\(393\) 12270.0 1.57491
\(394\) 0 0
\(395\) −24154.8 −3.07686
\(396\) 0 0
\(397\) 73.5391 0.00929678 0.00464839 0.999989i \(-0.498520\pi\)
0.00464839 + 0.999989i \(0.498520\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −498.000 −0.0620173 −0.0310086 0.999519i \(-0.509872\pi\)
−0.0310086 + 0.999519i \(0.509872\pi\)
\(402\) 0 0
\(403\) 4752.00 0.587380
\(404\) 0 0
\(405\) −16255.0 −1.99436
\(406\) 0 0
\(407\) −532.000 −0.0647918
\(408\) 0 0
\(409\) −3355.93 −0.405721 −0.202861 0.979208i \(-0.565024\pi\)
−0.202861 + 0.979208i \(0.565024\pi\)
\(410\) 0 0
\(411\) −5854.84 −0.702672
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 8372.00 0.990278
\(416\) 0 0
\(417\) 3010.00 0.353478
\(418\) 0 0
\(419\) 14545.2 1.69589 0.847946 0.530082i \(-0.177839\pi\)
0.847946 + 0.530082i \(0.177839\pi\)
\(420\) 0 0
\(421\) 10854.0 1.25651 0.628256 0.778007i \(-0.283769\pi\)
0.628256 + 0.778007i \(0.283769\pi\)
\(422\) 0 0
\(423\) 12035.0 1.38336
\(424\) 0 0
\(425\) −377.595 −0.0430966
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 5040.00 0.567211
\(430\) 0 0
\(431\) 5364.00 0.599477 0.299739 0.954021i \(-0.403100\pi\)
0.299739 + 0.954021i \(0.403100\pi\)
\(432\) 0 0
\(433\) 6487.00 0.719966 0.359983 0.932959i \(-0.382783\pi\)
0.359983 + 0.932959i \(0.382783\pi\)
\(434\) 0 0
\(435\) 40040.0 4.41327
\(436\) 0 0
\(437\) 197.990 0.0216731
\(438\) 0 0
\(439\) 13932.8 1.51476 0.757378 0.652977i \(-0.226480\pi\)
0.757378 + 0.652977i \(0.226480\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −5996.00 −0.643067 −0.321533 0.946898i \(-0.604198\pi\)
−0.321533 + 0.946898i \(0.604198\pi\)
\(444\) 0 0
\(445\) −12236.0 −1.30347
\(446\) 0 0
\(447\) −14495.7 −1.53383
\(448\) 0 0
\(449\) 2622.00 0.275590 0.137795 0.990461i \(-0.455999\pi\)
0.137795 + 0.990461i \(0.455999\pi\)
\(450\) 0 0
\(451\) 1762.11 0.183979
\(452\) 0 0
\(453\) −3337.54 −0.346162
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 11208.0 1.14724 0.573619 0.819122i \(-0.305539\pi\)
0.573619 + 0.819122i \(0.305539\pi\)
\(458\) 0 0
\(459\) −40.0000 −0.00406763
\(460\) 0 0
\(461\) −9786.36 −0.988712 −0.494356 0.869260i \(-0.664596\pi\)
−0.494356 + 0.869260i \(0.664596\pi\)
\(462\) 0 0
\(463\) −3952.00 −0.396685 −0.198342 0.980133i \(-0.563556\pi\)
−0.198342 + 0.980133i \(0.563556\pi\)
\(464\) 0 0
\(465\) 13067.3 1.30319
\(466\) 0 0
\(467\) 17506.5 1.73470 0.867352 0.497696i \(-0.165820\pi\)
0.867352 + 0.497696i \(0.165820\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −15640.0 −1.53005
\(472\) 0 0
\(473\) 476.000 0.0462717
\(474\) 0 0
\(475\) −377.595 −0.0364742
\(476\) 0 0
\(477\) −1702.00 −0.163374
\(478\) 0 0
\(479\) 2288.20 0.218268 0.109134 0.994027i \(-0.465192\pi\)
0.109134 + 0.994027i \(0.465192\pi\)
\(480\) 0 0
\(481\) 1934.64 0.183393
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −29372.0 −2.74993
\(486\) 0 0
\(487\) −972.000 −0.0904426 −0.0452213 0.998977i \(-0.514399\pi\)
−0.0452213 + 0.998977i \(0.514399\pi\)
\(488\) 0 0
\(489\) 23235.5 2.14877
\(490\) 0 0
\(491\) 7404.00 0.680525 0.340263 0.940330i \(-0.389484\pi\)
0.340263 + 0.940330i \(0.389484\pi\)
\(492\) 0 0
\(493\) 404.465 0.0369497
\(494\) 0 0
\(495\) 6375.27 0.578883
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 12244.0 1.09843 0.549215 0.835681i \(-0.314927\pi\)
0.549215 + 0.835681i \(0.314927\pi\)
\(500\) 0 0
\(501\) 10540.0 0.939905
\(502\) 0 0
\(503\) −2415.48 −0.214117 −0.107058 0.994253i \(-0.534143\pi\)
−0.107058 + 0.994253i \(0.534143\pi\)
\(504\) 0 0
\(505\) −22344.0 −1.96890
\(506\) 0 0
\(507\) −2793.07 −0.244664
\(508\) 0 0
\(509\) 5707.77 0.497038 0.248519 0.968627i \(-0.420056\pi\)
0.248519 + 0.968627i \(0.420056\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −40.0000 −0.00344258
\(514\) 0 0
\(515\) −17192.0 −1.47101
\(516\) 0 0
\(517\) 7325.63 0.623173
\(518\) 0 0
\(519\) −14640.0 −1.23820
\(520\) 0 0
\(521\) −1.41421 −0.000118921 0 −5.94605e−5 1.00000i \(-0.500019\pi\)
−5.94605e−5 1.00000i \(0.500019\pi\)
\(522\) 0 0
\(523\) −12257.0 −1.02478 −0.512391 0.858752i \(-0.671240\pi\)
−0.512391 + 0.858752i \(0.671240\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 132.000 0.0109108
\(528\) 0 0
\(529\) 7433.00 0.610915
\(530\) 0 0
\(531\) 9985.76 0.816093
\(532\) 0 0
\(533\) −6408.00 −0.520753
\(534\) 0 0
\(535\) 33341.5 2.69435
\(536\) 0 0
\(537\) 3818.38 0.306844
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 2050.00 0.162914 0.0814569 0.996677i \(-0.474043\pi\)
0.0814569 + 0.996677i \(0.474043\pi\)
\(542\) 0 0
\(543\) 26760.0 2.11488
\(544\) 0 0
\(545\) −16195.6 −1.27292
\(546\) 0 0
\(547\) −14554.0 −1.13763 −0.568815 0.822465i \(-0.692598\pi\)
−0.568815 + 0.822465i \(0.692598\pi\)
\(548\) 0 0
\(549\) −325.269 −0.0252862
\(550\) 0 0
\(551\) 404.465 0.0312719
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 5320.00 0.406885
\(556\) 0 0
\(557\) 6954.00 0.528995 0.264498 0.964386i \(-0.414794\pi\)
0.264498 + 0.964386i \(0.414794\pi\)
\(558\) 0 0
\(559\) −1731.00 −0.130972
\(560\) 0 0
\(561\) 140.000 0.0105362
\(562\) 0 0
\(563\) −1636.25 −0.122486 −0.0612429 0.998123i \(-0.519506\pi\)
−0.0612429 + 0.998123i \(0.519506\pi\)
\(564\) 0 0
\(565\) −10691.5 −0.796094
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −7142.00 −0.526201 −0.263100 0.964768i \(-0.584745\pi\)
−0.263100 + 0.964768i \(0.584745\pi\)
\(570\) 0 0
\(571\) 20606.0 1.51022 0.755109 0.655599i \(-0.227584\pi\)
0.755109 + 0.655599i \(0.227584\pi\)
\(572\) 0 0
\(573\) 7269.06 0.529964
\(574\) 0 0
\(575\) −37380.0 −2.71105
\(576\) 0 0
\(577\) 8803.48 0.635171 0.317585 0.948230i \(-0.397128\pi\)
0.317585 + 0.948230i \(0.397128\pi\)
\(578\) 0 0
\(579\) −32470.3 −2.33061
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −1036.00 −0.0735965
\(584\) 0 0
\(585\) −23184.0 −1.63853
\(586\) 0 0
\(587\) 6503.97 0.457321 0.228661 0.973506i \(-0.426565\pi\)
0.228661 + 0.973506i \(0.426565\pi\)
\(588\) 0 0
\(589\) 132.000 0.00923424
\(590\) 0 0
\(591\) −5614.43 −0.390773
\(592\) 0 0
\(593\) 23140.8 1.60249 0.801246 0.598335i \(-0.204171\pi\)
0.801246 + 0.598335i \(0.204171\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −17580.0 −1.20520
\(598\) 0 0
\(599\) 11296.0 0.770521 0.385260 0.922808i \(-0.374112\pi\)
0.385260 + 0.922808i \(0.374112\pi\)
\(600\) 0 0
\(601\) −8727.11 −0.592323 −0.296162 0.955138i \(-0.595707\pi\)
−0.296162 + 0.955138i \(0.595707\pi\)
\(602\) 0 0
\(603\) −15732.0 −1.06245
\(604\) 0 0
\(605\) −22471.9 −1.51010
\(606\) 0 0
\(607\) 19736.8 1.31975 0.659877 0.751374i \(-0.270608\pi\)
0.659877 + 0.751374i \(0.270608\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −26640.0 −1.76389
\(612\) 0 0
\(613\) 16962.0 1.11760 0.558800 0.829302i \(-0.311262\pi\)
0.558800 + 0.829302i \(0.311262\pi\)
\(614\) 0 0
\(615\) −17621.1 −1.15537
\(616\) 0 0
\(617\) −19034.0 −1.24194 −0.620972 0.783832i \(-0.713262\pi\)
−0.620972 + 0.783832i \(0.713262\pi\)
\(618\) 0 0
\(619\) −18677.5 −1.21278 −0.606392 0.795166i \(-0.707384\pi\)
−0.606392 + 0.795166i \(0.707384\pi\)
\(620\) 0 0
\(621\) −3959.80 −0.255880
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 22289.0 1.42650
\(626\) 0 0
\(627\) 140.000 0.00891716
\(628\) 0 0
\(629\) 53.7401 0.00340661
\(630\) 0 0
\(631\) 14716.0 0.928423 0.464211 0.885724i \(-0.346338\pi\)
0.464211 + 0.885724i \(0.346338\pi\)
\(632\) 0 0
\(633\) −19431.3 −1.22010
\(634\) 0 0
\(635\) −34054.3 −2.12819
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −13524.0 −0.837248
\(640\) 0 0
\(641\) 4730.00 0.291457 0.145728 0.989325i \(-0.453447\pi\)
0.145728 + 0.989325i \(0.453447\pi\)
\(642\) 0 0
\(643\) 19056.5 1.16877 0.584383 0.811478i \(-0.301337\pi\)
0.584383 + 0.811478i \(0.301337\pi\)
\(644\) 0 0
\(645\) −4760.00 −0.290581
\(646\) 0 0
\(647\) 9342.29 0.567672 0.283836 0.958873i \(-0.408393\pi\)
0.283836 + 0.958873i \(0.408393\pi\)
\(648\) 0 0
\(649\) 6078.29 0.367633
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 3774.00 0.226169 0.113084 0.993585i \(-0.463927\pi\)
0.113084 + 0.993585i \(0.463927\pi\)
\(654\) 0 0
\(655\) −34356.0 −2.04947
\(656\) 0 0
\(657\) 6212.64 0.368917
\(658\) 0 0
\(659\) 21150.0 1.25021 0.625104 0.780541i \(-0.285057\pi\)
0.625104 + 0.780541i \(0.285057\pi\)
\(660\) 0 0
\(661\) 10377.5 0.610647 0.305324 0.952249i \(-0.401235\pi\)
0.305324 + 0.952249i \(0.401235\pi\)
\(662\) 0 0
\(663\) −509.117 −0.0298227
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 40040.0 2.32437
\(668\) 0 0
\(669\) 24240.0 1.40086
\(670\) 0 0
\(671\) −197.990 −0.0113909
\(672\) 0 0
\(673\) −1164.00 −0.0666700 −0.0333350 0.999444i \(-0.510613\pi\)
−0.0333350 + 0.999444i \(0.510613\pi\)
\(674\) 0 0
\(675\) 7551.90 0.430626
\(676\) 0 0
\(677\) −27152.9 −1.54146 −0.770732 0.637160i \(-0.780109\pi\)
−0.770732 + 0.637160i \(0.780109\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −37410.0 −2.10507
\(682\) 0 0
\(683\) 16596.0 0.929763 0.464882 0.885373i \(-0.346097\pi\)
0.464882 + 0.885373i \(0.346097\pi\)
\(684\) 0 0
\(685\) 16393.6 0.914403
\(686\) 0 0
\(687\) −19440.0 −1.07960
\(688\) 0 0
\(689\) 3767.46 0.208315
\(690\) 0 0
\(691\) 11298.2 0.622000 0.311000 0.950410i \(-0.399336\pi\)
0.311000 + 0.950410i \(0.399336\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −8428.00 −0.459989
\(696\) 0 0
\(697\) −178.000 −0.00967321
\(698\) 0 0
\(699\) −509.117 −0.0275487
\(700\) 0 0
\(701\) −2754.00 −0.148384 −0.0741920 0.997244i \(-0.523638\pi\)
−0.0741920 + 0.997244i \(0.523638\pi\)
\(702\) 0 0
\(703\) 53.7401 0.00288314
\(704\) 0 0
\(705\) −73256.3 −3.91346
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 29434.0 1.55912 0.779561 0.626327i \(-0.215442\pi\)
0.779561 + 0.626327i \(0.215442\pi\)
\(710\) 0 0
\(711\) −28060.0 −1.48007
\(712\) 0 0
\(713\) 13067.3 0.686361
\(714\) 0 0
\(715\) −14112.0 −0.738124
\(716\) 0 0
\(717\) 30462.2 1.58665
\(718\) 0 0
\(719\) −17669.2 −0.916480 −0.458240 0.888828i \(-0.651520\pi\)
−0.458240 + 0.888828i \(0.651520\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −10890.0 −0.560171
\(724\) 0 0
\(725\) −76362.0 −3.91174
\(726\) 0 0
\(727\) 28445.5 1.45115 0.725574 0.688144i \(-0.241574\pi\)
0.725574 + 0.688144i \(0.241574\pi\)
\(728\) 0 0
\(729\) −13483.0 −0.685007
\(730\) 0 0
\(731\) −48.0833 −0.00243286
\(732\) 0 0
\(733\) −22341.7 −1.12580 −0.562900 0.826525i \(-0.690314\pi\)
−0.562900 + 0.826525i \(0.690314\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −9576.00 −0.478611
\(738\) 0 0
\(739\) −20670.0 −1.02890 −0.514451 0.857520i \(-0.672004\pi\)
−0.514451 + 0.857520i \(0.672004\pi\)
\(740\) 0 0
\(741\) −509.117 −0.0252400
\(742\) 0 0
\(743\) 25400.0 1.25415 0.627076 0.778958i \(-0.284251\pi\)
0.627076 + 0.778958i \(0.284251\pi\)
\(744\) 0 0
\(745\) 40587.9 1.99601
\(746\) 0 0
\(747\) 9725.55 0.476358
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −29180.0 −1.41783 −0.708917 0.705292i \(-0.750816\pi\)
−0.708917 + 0.705292i \(0.750816\pi\)
\(752\) 0 0
\(753\) −6590.00 −0.318928
\(754\) 0 0
\(755\) 9345.12 0.450469
\(756\) 0 0
\(757\) −26206.0 −1.25822 −0.629110 0.777316i \(-0.716581\pi\)
−0.629110 + 0.777316i \(0.716581\pi\)
\(758\) 0 0
\(759\) 13859.3 0.662794
\(760\) 0 0
\(761\) −6863.18 −0.326925 −0.163463 0.986550i \(-0.552266\pi\)
−0.163463 + 0.986550i \(0.552266\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −644.000 −0.0304364
\(766\) 0 0
\(767\) −22104.0 −1.04059
\(768\) 0 0
\(769\) 9058.04 0.424761 0.212380 0.977187i \(-0.431878\pi\)
0.212380 + 0.977187i \(0.431878\pi\)
\(770\) 0 0
\(771\) 6630.00 0.309693
\(772\) 0 0
\(773\) 132.936 0.00618548 0.00309274 0.999995i \(-0.499016\pi\)
0.00309274 + 0.999995i \(0.499016\pi\)
\(774\) 0 0
\(775\) −24921.3 −1.15509
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −178.000 −0.00818679
\(780\) 0 0
\(781\) −8232.00 −0.377163
\(782\) 0 0
\(783\) −8089.30 −0.369206
\(784\) 0 0
\(785\) 43792.0 1.99109
\(786\) 0 0
\(787\) −8729.94 −0.395411 −0.197706 0.980261i \(-0.563349\pi\)
−0.197706 + 0.980261i \(0.563349\pi\)
\(788\) 0 0
\(789\) 50487.4 2.27807
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 720.000 0.0322421
\(794\) 0 0
\(795\) 10360.0 0.462178
\(796\) 0 0
\(797\) −7517.96 −0.334128 −0.167064 0.985946i \(-0.553429\pi\)
−0.167064 + 0.985946i \(0.553429\pi\)
\(798\) 0 0
\(799\) −740.000 −0.0327651
\(800\) 0 0
\(801\) −14214.3 −0.627011
\(802\) 0 0
\(803\) 3781.61 0.166189
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 32600.0 1.42203
\(808\) 0 0
\(809\) 3776.00 0.164100 0.0820501 0.996628i \(-0.473853\pi\)
0.0820501 + 0.996628i \(0.473853\pi\)
\(810\) 0 0
\(811\) −36227.9 −1.56860 −0.784300 0.620382i \(-0.786977\pi\)
−0.784300 + 0.620382i \(0.786977\pi\)
\(812\) 0 0
\(813\) 16720.0 0.721274
\(814\) 0 0
\(815\) −65059.5 −2.79624
\(816\) 0 0
\(817\) −48.0833 −0.00205902
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −16410.0 −0.697580 −0.348790 0.937201i \(-0.613407\pi\)
−0.348790 + 0.937201i \(0.613407\pi\)
\(822\) 0 0
\(823\) −22072.0 −0.934850 −0.467425 0.884033i \(-0.654818\pi\)
−0.467425 + 0.884033i \(0.654818\pi\)
\(824\) 0 0
\(825\) −26431.7 −1.11543
\(826\) 0 0
\(827\) 11628.0 0.488930 0.244465 0.969658i \(-0.421388\pi\)
0.244465 + 0.969658i \(0.421388\pi\)
\(828\) 0 0
\(829\) 30906.2 1.29483 0.647417 0.762136i \(-0.275849\pi\)
0.647417 + 0.762136i \(0.275849\pi\)
\(830\) 0 0
\(831\) −28326.7 −1.18248
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −29512.0 −1.22312
\(836\) 0 0
\(837\) −2640.00 −0.109022
\(838\) 0 0
\(839\) 17884.1 0.735911 0.367955 0.929843i \(-0.380058\pi\)
0.367955 + 0.929843i \(0.380058\pi\)
\(840\) 0 0
\(841\) 57407.0 2.35381
\(842\) 0 0
\(843\) 42313.3 1.72876
\(844\) 0 0
\(845\) 7820.60 0.318387
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 34850.0 1.40877
\(850\) 0 0
\(851\) 5320.00 0.214298
\(852\) 0 0
\(853\) 20755.0 0.833104 0.416552 0.909112i \(-0.363238\pi\)
0.416552 + 0.909112i \(0.363238\pi\)
\(854\) 0 0
\(855\) −644.000 −0.0257595
\(856\) 0 0
\(857\) −44919.7 −1.79046 −0.895231 0.445602i \(-0.852990\pi\)
−0.895231 + 0.445602i \(0.852990\pi\)
\(858\) 0 0
\(859\) 69.2965 0.00275246 0.00137623 0.999999i \(-0.499562\pi\)
0.00137623 + 0.999999i \(0.499562\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 5452.00 0.215050 0.107525 0.994202i \(-0.465707\pi\)
0.107525 + 0.994202i \(0.465707\pi\)
\(864\) 0 0
\(865\) 40992.0 1.61129
\(866\) 0 0
\(867\) 34726.0 1.36027
\(868\) 0 0
\(869\) −17080.0 −0.666743
\(870\) 0 0
\(871\) 34823.6 1.35471
\(872\) 0 0
\(873\) −34120.7 −1.32281
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 31106.0 1.19769 0.598845 0.800865i \(-0.295626\pi\)
0.598845 + 0.800865i \(0.295626\pi\)
\(878\) 0 0
\(879\) 13940.0 0.534908
\(880\) 0 0
\(881\) −5943.94 −0.227306 −0.113653 0.993521i \(-0.536255\pi\)
−0.113653 + 0.993521i \(0.536255\pi\)
\(882\) 0 0
\(883\) −34796.0 −1.32614 −0.663068 0.748559i \(-0.730746\pi\)
−0.663068 + 0.748559i \(0.730746\pi\)
\(884\) 0 0
\(885\) −60782.9 −2.30869
\(886\) 0 0
\(887\) 9964.55 0.377200 0.188600 0.982054i \(-0.439605\pi\)
0.188600 + 0.982054i \(0.439605\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −11494.0 −0.432170
\(892\) 0 0
\(893\) −740.000 −0.0277303
\(894\) 0 0
\(895\) −10691.5 −0.399303
\(896\) 0 0
\(897\) −50400.0 −1.87604
\(898\) 0 0
\(899\) 26694.7 0.990343
\(900\) 0 0
\(901\) 104.652 0.00386954
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −74928.0 −2.75214
\(906\) 0 0
\(907\) 29756.0 1.08934 0.544670 0.838650i \(-0.316655\pi\)
0.544670 + 0.838650i \(0.316655\pi\)
\(908\) 0 0
\(909\) −25956.5 −0.947109
\(910\) 0 0
\(911\) −21440.0 −0.779735 −0.389868 0.920871i \(-0.627479\pi\)
−0.389868 + 0.920871i \(0.627479\pi\)
\(912\) 0 0
\(913\) 5919.90 0.214589
\(914\) 0 0
\(915\) 1979.90 0.0715338
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 8288.00 0.297493 0.148746 0.988875i \(-0.452476\pi\)
0.148746 + 0.988875i \(0.452476\pi\)
\(920\) 0 0
\(921\) 33710.0 1.20606
\(922\) 0 0
\(923\) 29936.1 1.06756
\(924\) 0 0
\(925\) −10146.0 −0.360647
\(926\) 0 0
\(927\) −19971.5 −0.707606
\(928\) 0 0
\(929\) 45581.5 1.60978 0.804888 0.593427i \(-0.202226\pi\)
0.804888 + 0.593427i \(0.202226\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 47920.0 1.68149
\(934\) 0 0
\(935\) −392.000 −0.0137110
\(936\) 0 0
\(937\) −11665.8 −0.406731 −0.203365 0.979103i \(-0.565188\pi\)
−0.203365 + 0.979103i \(0.565188\pi\)
\(938\) 0 0
\(939\) −43770.0 −1.52117
\(940\) 0 0
\(941\) −14.1421 −0.000489926 0 −0.000244963 1.00000i \(-0.500078\pi\)
−0.000244963 1.00000i \(0.500078\pi\)
\(942\) 0 0
\(943\) −17621.1 −0.608507
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −14034.0 −0.481567 −0.240783 0.970579i \(-0.577404\pi\)
−0.240783 + 0.970579i \(0.577404\pi\)
\(948\) 0 0
\(949\) −13752.0 −0.470399
\(950\) 0 0
\(951\) −69480.3 −2.36914
\(952\) 0 0
\(953\) −42698.0 −1.45134 −0.725668 0.688045i \(-0.758469\pi\)
−0.725668 + 0.688045i \(0.758469\pi\)
\(954\) 0 0
\(955\) −20353.4 −0.689654
\(956\) 0 0
\(957\) 28312.6 0.956337
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −21079.0 −0.707563
\(962\) 0 0
\(963\) 38732.0 1.29608
\(964\) 0 0
\(965\) 90917.0 3.03287
\(966\) 0 0
\(967\) 48492.0 1.61261 0.806307 0.591497i \(-0.201463\pi\)
0.806307 + 0.591497i \(0.201463\pi\)
\(968\) 0 0
\(969\) −14.1421 −0.000468845 0
\(970\) 0 0
\(971\) 52669.6 1.74073 0.870364 0.492409i \(-0.163884\pi\)
0.870364 + 0.492409i \(0.163884\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 96120.0 3.15723
\(976\) 0 0
\(977\) −55380.0 −1.81347 −0.906737 0.421698i \(-0.861434\pi\)
−0.906737 + 0.421698i \(0.861434\pi\)
\(978\) 0 0
\(979\) −8652.16 −0.282456
\(980\) 0 0
\(981\) −18814.0 −0.612319
\(982\) 0 0
\(983\) −50535.5 −1.63971 −0.819854 0.572573i \(-0.805945\pi\)
−0.819854 + 0.572573i \(0.805945\pi\)
\(984\) 0 0
\(985\) 15720.4 0.508521
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −4760.00 −0.153043
\(990\) 0 0
\(991\) 39712.0 1.27295 0.636475 0.771297i \(-0.280392\pi\)
0.636475 + 0.771297i \(0.280392\pi\)
\(992\) 0 0
\(993\) −40573.8 −1.29665
\(994\) 0 0
\(995\) 49224.0 1.56835
\(996\) 0 0
\(997\) −2186.37 −0.0694515 −0.0347258 0.999397i \(-0.511056\pi\)
−0.0347258 + 0.999397i \(0.511056\pi\)
\(998\) 0 0
\(999\) −1074.80 −0.0340393
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 784.4.a.y.1.1 2
4.3 odd 2 98.4.a.g.1.2 yes 2
7.6 odd 2 inner 784.4.a.y.1.2 2
12.11 even 2 882.4.a.bg.1.1 2
20.19 odd 2 2450.4.a.bx.1.1 2
28.3 even 6 98.4.c.h.79.2 4
28.11 odd 6 98.4.c.h.79.1 4
28.19 even 6 98.4.c.h.67.2 4
28.23 odd 6 98.4.c.h.67.1 4
28.27 even 2 98.4.a.g.1.1 2
84.11 even 6 882.4.g.ba.667.2 4
84.23 even 6 882.4.g.ba.361.2 4
84.47 odd 6 882.4.g.ba.361.1 4
84.59 odd 6 882.4.g.ba.667.1 4
84.83 odd 2 882.4.a.bg.1.2 2
140.139 even 2 2450.4.a.bx.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
98.4.a.g.1.1 2 28.27 even 2
98.4.a.g.1.2 yes 2 4.3 odd 2
98.4.c.h.67.1 4 28.23 odd 6
98.4.c.h.67.2 4 28.19 even 6
98.4.c.h.79.1 4 28.11 odd 6
98.4.c.h.79.2 4 28.3 even 6
784.4.a.y.1.1 2 1.1 even 1 trivial
784.4.a.y.1.2 2 7.6 odd 2 inner
882.4.a.bg.1.1 2 12.11 even 2
882.4.a.bg.1.2 2 84.83 odd 2
882.4.g.ba.361.1 4 84.47 odd 6
882.4.g.ba.361.2 4 84.23 even 6
882.4.g.ba.667.1 4 84.59 odd 6
882.4.g.ba.667.2 4 84.11 even 6
2450.4.a.bx.1.1 2 20.19 odd 2
2450.4.a.bx.1.2 2 140.139 even 2