Properties

Label 784.4.a.y.1.2
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 1.2
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 \(-0.261800\pi\)
0.680414 + 0.732828i \(0.261800\pi\)
\(4\) 0 0
\(5\) −19.7990 −1.77088 −0.885438 0.464758i \(-0.846141\pi\)
−0.885438 + 0.464758i \(0.846141\pi\)
\(6\) 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 \(-0.317254\pi\)
0.543091 + 0.839674i \(0.317254\pi\)
\(14\) 0 0
\(15\) −140.000 −2.40986
\(16\) 0 0
\(17\) 1.41421 0.0201763 0.0100882 0.999949i \(-0.496789\pi\)
0.0100882 + 0.999949i \(0.496789\pi\)
\(18\) 0 0
\(19\) 1.41421 0.0170759 0.00853797 0.999964i \(-0.497282\pi\)
0.00853797 + 0.999964i \(0.497282\pi\)
\(20\) 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 \(-0.412848\pi\)
0.270387 + 0.962752i \(0.412848\pi\)
\(32\) 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 \(-0.577055\pi\)
−0.239717 + 0.970843i \(0.577055\pi\)
\(42\) 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 \(-0.801605\pi\)
−0.811970 + 0.583699i \(0.801605\pi\)
\(48\) 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 \(-0.659005\pi\)
−0.479011 + 0.877809i \(0.659005\pi\)
\(60\) 0 0
\(61\) 14.1421 0.0296839 0.0148419 0.999890i \(-0.495275\pi\)
0.0148419 + 0.999890i \(0.495275\pi\)
\(62\) 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 \(-0.569477\pi\)
−0.216538 + 0.976274i \(0.569477\pi\)
\(74\) 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 \(-0.590202\pi\)
−0.279601 + 0.960116i \(0.590202\pi\)
\(84\) 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 \(-0.380033\pi\)
0.368028 + 0.929815i \(0.380033\pi\)
\(90\) 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 \(-0.217028\pi\)
0.776431 + 0.630202i \(0.217028\pi\)
\(98\) 0 0
\(99\) 322.000 0.326891
\(100\) 0 0
\(101\) 1128.54 1.11182 0.555912 0.831241i \(-0.312369\pi\)
0.555912 + 0.831241i \(0.312369\pi\)
\(102\) 0 0
\(103\) 868.327 0.830668 0.415334 0.909669i \(-0.363665\pi\)
0.415334 + 0.909669i \(0.363665\pi\)
\(104\) 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 \(-0.303576\pi\)
0.578659 + 0.815570i \(0.303576\pi\)
\(132\) 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 \(-0.458542\pi\)
0.129876 + 0.991530i \(0.458542\pi\)
\(140\) 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 \(-0.690036\pi\)
−0.562176 + 0.827018i \(0.690036\pi\)
\(158\) 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 \(-0.387763\pi\)
0.345343 + 0.938476i \(0.387763\pi\)
\(168\) 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 \(-0.650340\pi\)
−0.454943 + 0.890521i \(0.650340\pi\)
\(174\) 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 \(-0.216711\pi\)
0.777058 + 0.629429i \(0.216711\pi\)
\(182\) 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 \(-0.646021\pi\)
−0.442817 + 0.896612i \(0.646021\pi\)
\(200\) 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 \(-0.327901\pi\)
0.514707 + 0.857366i \(0.327901\pi\)
\(224\) 0 0
\(225\) 6141.00 1.81956
\(226\) 0 0
\(227\) −5290.57 −1.54691 −0.773453 0.633854i \(-0.781472\pi\)
−0.773453 + 0.633854i \(0.781472\pi\)
\(228\) 0 0
\(229\) −2749.23 −0.793338 −0.396669 0.917962i \(-0.629834\pi\)
−0.396669 + 0.917962i \(0.629834\pi\)
\(230\) 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 \(-0.565986\pi\)
−0.205820 + 0.978590i \(0.565986\pi\)
\(242\) 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 \(-0.537386\pi\)
−0.117182 + 0.993110i \(0.537386\pi\)
\(252\) 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 \(-0.463701\pi\)
0.113789 + 0.993505i \(0.463701\pi\)
\(258\) 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 \(-0.325005\pi\)
0.522485 + 0.852648i \(0.325005\pi\)
\(270\) 0 0
\(271\) 2364.57 0.530026 0.265013 0.964245i \(-0.414624\pi\)
0.265013 + 0.964245i \(0.414624\pi\)
\(272\) 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 \(-0.326819\pi\)
0.517617 + 0.855613i \(0.326819\pi\)
\(284\) 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 \(-0.437030\pi\)
0.196538 + 0.980496i \(0.437030\pi\)
\(294\) 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 \(-0.353866\pi\)
0.443135 + 0.896455i \(0.353866\pi\)
\(308\) 0 0
\(309\) 6140.00 1.13040
\(310\) 0 0
\(311\) 6776.91 1.23564 0.617819 0.786320i \(-0.288016\pi\)
0.617819 + 0.786320i \(0.288016\pi\)
\(312\) 0 0
\(313\) −6190.01 −1.11783 −0.558914 0.829226i \(-0.688782\pi\)
−0.558914 + 0.829226i \(0.688782\pi\)
\(314\) 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 \(-0.326200\pi\)
0.519279 + 0.854605i \(0.326200\pi\)
\(350\) 0 0
\(351\) −1440.00 −0.218979
\(352\) 0 0
\(353\) 6993.29 1.05443 0.527217 0.849731i \(-0.323236\pi\)
0.527217 + 0.849731i \(0.323236\pi\)
\(354\) 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 \(-0.480908\pi\)
0.0599421 + 0.998202i \(0.480908\pi\)
\(368\) 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 \(-0.521336\pi\)
−0.0669800 + 0.997754i \(0.521336\pi\)
\(384\) 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 \(-0.501480\pi\)
−0.00464839 + 0.999989i \(0.501480\pi\)
\(398\) 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 \(-0.434976\pi\)
0.202861 + 0.979208i \(0.434976\pi\)
\(410\) 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 \(-0.822161\pi\)
−0.847946 + 0.530082i \(0.822161\pi\)
\(420\) 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 \(-0.617217\pi\)
−0.359983 + 0.932959i \(0.617217\pi\)
\(434\) 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 \(-0.773520\pi\)
−0.757378 + 0.652977i \(0.773520\pi\)
\(440\) 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 \(-0.335404\pi\)
0.494356 + 0.869260i \(0.335404\pi\)
\(462\) 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 \(-0.834180\pi\)
−0.867352 + 0.497696i \(0.834180\pi\)
\(468\) 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 \(-0.534808\pi\)
−0.109134 + 0.994027i \(0.534808\pi\)
\(480\) 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 \(-0.465857\pi\)
0.107058 + 0.994253i \(0.465857\pi\)
\(504\) 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 \(-0.579944\pi\)
−0.248519 + 0.968627i \(0.579944\pi\)
\(510\) 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 \(-0.499981\pi\)
5.94605e−5 1.00000i \(0.499981\pi\)
\(522\) 0 0
\(523\) 12257.0 1.02478 0.512391 0.858752i \(-0.328760\pi\)
0.512391 + 0.858752i \(0.328760\pi\)
\(524\) 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 \(-0.480494\pi\)
0.0612429 + 0.998123i \(0.480494\pi\)
\(564\) 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 \(-0.602872\pi\)
−0.317585 + 0.948230i \(0.602872\pi\)
\(578\) 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 \(-0.573435\pi\)
−0.228661 + 0.973506i \(0.573435\pi\)
\(588\) 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 \(-0.795829\pi\)
−0.801246 + 0.598335i \(0.795829\pi\)
\(594\) 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 \(-0.404293\pi\)
0.296162 + 0.955138i \(0.404293\pi\)
\(602\) 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 \(-0.729392\pi\)
−0.659877 + 0.751374i \(0.729392\pi\)
\(608\) 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 \(-0.292616\pi\)
0.606392 + 0.795166i \(0.292616\pi\)
\(620\) 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 \(-0.698663\pi\)
−0.584383 + 0.811478i \(0.698663\pi\)
\(644\) 0 0
\(645\) −4760.00 −0.290581
\(646\) 0 0
\(647\) −9342.29 −0.567672 −0.283836 0.958873i \(-0.591607\pi\)
−0.283836 + 0.958873i \(0.591607\pi\)
\(648\) 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 \(-0.598765\pi\)
−0.305324 + 0.952249i \(0.598765\pi\)
\(662\) 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 \(-0.219891\pi\)
0.770732 + 0.637160i \(0.219891\pi\)
\(678\) 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 \(-0.600664\pi\)
−0.311000 + 0.950410i \(0.600664\pi\)
\(692\) 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 \(-0.348480\pi\)
0.458240 + 0.888828i \(0.348480\pi\)
\(720\) 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 \(-0.758426\pi\)
−0.725574 + 0.688144i \(0.758426\pi\)
\(728\) 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 \(-0.309686\pi\)
0.562900 + 0.826525i \(0.309686\pi\)
\(734\) 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 \(-0.447734\pi\)
0.163463 + 0.986550i \(0.447734\pi\)
\(762\) 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 \(-0.568122\pi\)
−0.212380 + 0.977187i \(0.568122\pi\)
\(770\) 0 0
\(771\) 6630.00 0.309693
\(772\) 0 0
\(773\) −132.936 −0.00618548 −0.00309274 0.999995i \(-0.500984\pi\)
−0.00309274 + 0.999995i \(0.500984\pi\)
\(774\) 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 \(-0.436651\pi\)
0.197706 + 0.980261i \(0.436651\pi\)
\(788\) 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 \(-0.446571\pi\)
0.167064 + 0.985946i \(0.446571\pi\)
\(798\) 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 \(-0.213023\pi\)
0.784300 + 0.620382i \(0.213023\pi\)
\(812\) 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 \(-0.724151\pi\)
−0.647417 + 0.762136i \(0.724151\pi\)
\(830\) 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 \(-0.619942\pi\)
−0.367955 + 0.929843i \(0.619942\pi\)
\(840\) 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 \(-0.636762\pi\)
−0.416552 + 0.909112i \(0.636762\pi\)
\(854\) 0 0
\(855\) −644.000 −0.0257595
\(856\) 0 0
\(857\) 44919.7 1.79046 0.895231 0.445602i \(-0.147010\pi\)
0.895231 + 0.445602i \(0.147010\pi\)
\(858\) 0 0
\(859\) −69.2965 −0.00275246 −0.00137623 0.999999i \(-0.500438\pi\)
−0.00137623 + 0.999999i \(0.500438\pi\)
\(860\) 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 \(-0.463745\pi\)
0.113653 + 0.993521i \(0.463745\pi\)
\(882\) 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 \(-0.560395\pi\)
−0.188600 + 0.982054i \(0.560395\pi\)
\(888\) 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 \(-0.797774\pi\)
−0.804888 + 0.593427i \(0.797774\pi\)
\(930\) 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 \(-0.434812\pi\)
0.203365 + 0.979103i \(0.434812\pi\)
\(938\) 0 0
\(939\) −43770.0 −1.52117
\(940\) 0 0
\(941\) 14.1421 0.000489926 0 0.000244963 1.00000i \(-0.499922\pi\)
0.000244963 1.00000i \(0.499922\pi\)
\(942\) 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 \(-0.836116\pi\)
−0.870364 + 0.492409i \(0.836116\pi\)
\(972\) 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 \(-0.194055\pi\)
0.819854 + 0.572573i \(0.194055\pi\)
\(984\) 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 \(-0.488944\pi\)
0.0347258 + 0.999397i \(0.488944\pi\)
\(998\) 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.1.2 2
4.3 odd 2 98.4.a.g.1.1 2
7.6 odd 2 inner 784.4.a.y.1.1 2
12.11 even 2 882.4.a.bg.1.2 2
20.19 odd 2 2450.4.a.bx.1.2 2
28.3 even 6 98.4.c.h.79.1 4
28.11 odd 6 98.4.c.h.79.2 4
28.19 even 6 98.4.c.h.67.1 4
28.23 odd 6 98.4.c.h.67.2 4
28.27 even 2 98.4.a.g.1.2 yes 2
84.11 even 6 882.4.g.ba.667.1 4
84.23 even 6 882.4.g.ba.361.1 4
84.47 odd 6 882.4.g.ba.361.2 4
84.59 odd 6 882.4.g.ba.667.2 4
84.83 odd 2 882.4.a.bg.1.1 2
140.139 even 2 2450.4.a.bx.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
98.4.a.g.1.1 2 4.3 odd 2
98.4.a.g.1.2 yes 2 28.27 even 2
98.4.c.h.67.1 4 28.19 even 6
98.4.c.h.67.2 4 28.23 odd 6
98.4.c.h.79.1 4 28.3 even 6
98.4.c.h.79.2 4 28.11 odd 6
784.4.a.y.1.1 2 7.6 odd 2 inner
784.4.a.y.1.2 2 1.1 even 1 trivial
882.4.a.bg.1.1 2 84.83 odd 2
882.4.a.bg.1.2 2 12.11 even 2
882.4.g.ba.361.1 4 84.23 even 6
882.4.g.ba.361.2 4 84.47 odd 6
882.4.g.ba.667.1 4 84.11 even 6
882.4.g.ba.667.2 4 84.59 odd 6
2450.4.a.bx.1.1 2 140.139 even 2
2450.4.a.bx.1.2 2 20.19 odd 2