Properties

Label 8.4.a.a.1.1
Level $8$
Weight $4$
Character 8.1
Self dual yes
Analytic conductor $0.472$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8 = 2^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 8.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.472015280046\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 8.1

$q$-expansion

\(f(q)\) \(=\) \(q-4.00000 q^{3} -2.00000 q^{5} +24.0000 q^{7} -11.0000 q^{9} +O(q^{10})\) \(q-4.00000 q^{3} -2.00000 q^{5} +24.0000 q^{7} -11.0000 q^{9} -44.0000 q^{11} +22.0000 q^{13} +8.00000 q^{15} +50.0000 q^{17} +44.0000 q^{19} -96.0000 q^{21} -56.0000 q^{23} -121.000 q^{25} +152.000 q^{27} +198.000 q^{29} -160.000 q^{31} +176.000 q^{33} -48.0000 q^{35} -162.000 q^{37} -88.0000 q^{39} -198.000 q^{41} +52.0000 q^{43} +22.0000 q^{45} +528.000 q^{47} +233.000 q^{49} -200.000 q^{51} -242.000 q^{53} +88.0000 q^{55} -176.000 q^{57} -668.000 q^{59} +550.000 q^{61} -264.000 q^{63} -44.0000 q^{65} +188.000 q^{67} +224.000 q^{69} +728.000 q^{71} +154.000 q^{73} +484.000 q^{75} -1056.00 q^{77} -656.000 q^{79} -311.000 q^{81} +236.000 q^{83} -100.000 q^{85} -792.000 q^{87} +714.000 q^{89} +528.000 q^{91} +640.000 q^{93} -88.0000 q^{95} -478.000 q^{97} +484.000 q^{99} +O(q^{100})\)

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
\(3\) −4.00000 −0.769800 −0.384900 0.922958i \(-0.625764\pi\)
−0.384900 + 0.922958i \(0.625764\pi\)
\(4\) 0 0
\(5\) −2.00000 −0.178885 −0.0894427 0.995992i \(-0.528509\pi\)
−0.0894427 + 0.995992i \(0.528509\pi\)
\(6\) 0 0
\(7\) 24.0000 1.29588 0.647939 0.761692i \(-0.275631\pi\)
0.647939 + 0.761692i \(0.275631\pi\)
\(8\) 0 0
\(9\) −11.0000 −0.407407
\(10\) 0 0
\(11\) −44.0000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 0 0
\(13\) 22.0000 0.469362 0.234681 0.972072i \(-0.424595\pi\)
0.234681 + 0.972072i \(0.424595\pi\)
\(14\) 0 0
\(15\) 8.00000 0.137706
\(16\) 0 0
\(17\) 50.0000 0.713340 0.356670 0.934230i \(-0.383912\pi\)
0.356670 + 0.934230i \(0.383912\pi\)
\(18\) 0 0
\(19\) 44.0000 0.531279 0.265639 0.964072i \(-0.414417\pi\)
0.265639 + 0.964072i \(0.414417\pi\)
\(20\) 0 0
\(21\) −96.0000 −0.997567
\(22\) 0 0
\(23\) −56.0000 −0.507687 −0.253844 0.967245i \(-0.581695\pi\)
−0.253844 + 0.967245i \(0.581695\pi\)
\(24\) 0 0
\(25\) −121.000 −0.968000
\(26\) 0 0
\(27\) 152.000 1.08342
\(28\) 0 0
\(29\) 198.000 1.26785 0.633925 0.773394i \(-0.281443\pi\)
0.633925 + 0.773394i \(0.281443\pi\)
\(30\) 0 0
\(31\) −160.000 −0.926995 −0.463498 0.886098i \(-0.653406\pi\)
−0.463498 + 0.886098i \(0.653406\pi\)
\(32\) 0 0
\(33\) 176.000 0.928414
\(34\) 0 0
\(35\) −48.0000 −0.231814
\(36\) 0 0
\(37\) −162.000 −0.719801 −0.359900 0.932991i \(-0.617189\pi\)
−0.359900 + 0.932991i \(0.617189\pi\)
\(38\) 0 0
\(39\) −88.0000 −0.361315
\(40\) 0 0
\(41\) −198.000 −0.754205 −0.377102 0.926172i \(-0.623080\pi\)
−0.377102 + 0.926172i \(0.623080\pi\)
\(42\) 0 0
\(43\) 52.0000 0.184417 0.0922084 0.995740i \(-0.470607\pi\)
0.0922084 + 0.995740i \(0.470607\pi\)
\(44\) 0 0
\(45\) 22.0000 0.0728793
\(46\) 0 0
\(47\) 528.000 1.63865 0.819327 0.573327i \(-0.194347\pi\)
0.819327 + 0.573327i \(0.194347\pi\)
\(48\) 0 0
\(49\) 233.000 0.679300
\(50\) 0 0
\(51\) −200.000 −0.549129
\(52\) 0 0
\(53\) −242.000 −0.627194 −0.313597 0.949556i \(-0.601534\pi\)
−0.313597 + 0.949556i \(0.601534\pi\)
\(54\) 0 0
\(55\) 88.0000 0.215744
\(56\) 0 0
\(57\) −176.000 −0.408978
\(58\) 0 0
\(59\) −668.000 −1.47400 −0.737002 0.675891i \(-0.763759\pi\)
−0.737002 + 0.675891i \(0.763759\pi\)
\(60\) 0 0
\(61\) 550.000 1.15443 0.577215 0.816592i \(-0.304139\pi\)
0.577215 + 0.816592i \(0.304139\pi\)
\(62\) 0 0
\(63\) −264.000 −0.527950
\(64\) 0 0
\(65\) −44.0000 −0.0839620
\(66\) 0 0
\(67\) 188.000 0.342804 0.171402 0.985201i \(-0.445170\pi\)
0.171402 + 0.985201i \(0.445170\pi\)
\(68\) 0 0
\(69\) 224.000 0.390818
\(70\) 0 0
\(71\) 728.000 1.21687 0.608435 0.793604i \(-0.291798\pi\)
0.608435 + 0.793604i \(0.291798\pi\)
\(72\) 0 0
\(73\) 154.000 0.246909 0.123454 0.992350i \(-0.460603\pi\)
0.123454 + 0.992350i \(0.460603\pi\)
\(74\) 0 0
\(75\) 484.000 0.745167
\(76\) 0 0
\(77\) −1056.00 −1.56289
\(78\) 0 0
\(79\) −656.000 −0.934250 −0.467125 0.884191i \(-0.654710\pi\)
−0.467125 + 0.884191i \(0.654710\pi\)
\(80\) 0 0
\(81\) −311.000 −0.426612
\(82\) 0 0
\(83\) 236.000 0.312101 0.156050 0.987749i \(-0.450124\pi\)
0.156050 + 0.987749i \(0.450124\pi\)
\(84\) 0 0
\(85\) −100.000 −0.127606
\(86\) 0 0
\(87\) −792.000 −0.975992
\(88\) 0 0
\(89\) 714.000 0.850380 0.425190 0.905104i \(-0.360207\pi\)
0.425190 + 0.905104i \(0.360207\pi\)
\(90\) 0 0
\(91\) 528.000 0.608236
\(92\) 0 0
\(93\) 640.000 0.713601
\(94\) 0 0
\(95\) −88.0000 −0.0950380
\(96\) 0 0
\(97\) −478.000 −0.500346 −0.250173 0.968201i \(-0.580487\pi\)
−0.250173 + 0.968201i \(0.580487\pi\)
\(98\) 0 0
\(99\) 484.000 0.491352
\(100\) 0 0
\(101\) 1566.00 1.54280 0.771400 0.636350i \(-0.219557\pi\)
0.771400 + 0.636350i \(0.219557\pi\)
\(102\) 0 0
\(103\) −968.000 −0.926018 −0.463009 0.886354i \(-0.653230\pi\)
−0.463009 + 0.886354i \(0.653230\pi\)
\(104\) 0 0
\(105\) 192.000 0.178450
\(106\) 0 0
\(107\) −780.000 −0.704724 −0.352362 0.935864i \(-0.614621\pi\)
−0.352362 + 0.935864i \(0.614621\pi\)
\(108\) 0 0
\(109\) −1994.00 −1.75221 −0.876103 0.482123i \(-0.839866\pi\)
−0.876103 + 0.482123i \(0.839866\pi\)
\(110\) 0 0
\(111\) 648.000 0.554103
\(112\) 0 0
\(113\) −942.000 −0.784212 −0.392106 0.919920i \(-0.628253\pi\)
−0.392106 + 0.919920i \(0.628253\pi\)
\(114\) 0 0
\(115\) 112.000 0.0908179
\(116\) 0 0
\(117\) −242.000 −0.191221
\(118\) 0 0
\(119\) 1200.00 0.924402
\(120\) 0 0
\(121\) 605.000 0.454545
\(122\) 0 0
\(123\) 792.000 0.580587
\(124\) 0 0
\(125\) 492.000 0.352047
\(126\) 0 0
\(127\) 1408.00 0.983778 0.491889 0.870658i \(-0.336307\pi\)
0.491889 + 0.870658i \(0.336307\pi\)
\(128\) 0 0
\(129\) −208.000 −0.141964
\(130\) 0 0
\(131\) −2692.00 −1.79543 −0.897714 0.440578i \(-0.854773\pi\)
−0.897714 + 0.440578i \(0.854773\pi\)
\(132\) 0 0
\(133\) 1056.00 0.688472
\(134\) 0 0
\(135\) −304.000 −0.193809
\(136\) 0 0
\(137\) 1626.00 1.01400 0.507002 0.861945i \(-0.330754\pi\)
0.507002 + 0.861945i \(0.330754\pi\)
\(138\) 0 0
\(139\) −684.000 −0.417382 −0.208691 0.977982i \(-0.566920\pi\)
−0.208691 + 0.977982i \(0.566920\pi\)
\(140\) 0 0
\(141\) −2112.00 −1.26144
\(142\) 0 0
\(143\) −968.000 −0.566072
\(144\) 0 0
\(145\) −396.000 −0.226800
\(146\) 0 0
\(147\) −932.000 −0.522926
\(148\) 0 0
\(149\) 302.000 0.166046 0.0830228 0.996548i \(-0.473543\pi\)
0.0830228 + 0.996548i \(0.473543\pi\)
\(150\) 0 0
\(151\) 1352.00 0.728637 0.364319 0.931274i \(-0.381302\pi\)
0.364319 + 0.931274i \(0.381302\pi\)
\(152\) 0 0
\(153\) −550.000 −0.290620
\(154\) 0 0
\(155\) 320.000 0.165826
\(156\) 0 0
\(157\) 3142.00 1.59719 0.798595 0.601868i \(-0.205577\pi\)
0.798595 + 0.601868i \(0.205577\pi\)
\(158\) 0 0
\(159\) 968.000 0.482814
\(160\) 0 0
\(161\) −1344.00 −0.657901
\(162\) 0 0
\(163\) 3036.00 1.45888 0.729441 0.684043i \(-0.239780\pi\)
0.729441 + 0.684043i \(0.239780\pi\)
\(164\) 0 0
\(165\) −352.000 −0.166080
\(166\) 0 0
\(167\) −264.000 −0.122329 −0.0611645 0.998128i \(-0.519481\pi\)
−0.0611645 + 0.998128i \(0.519481\pi\)
\(168\) 0 0
\(169\) −1713.00 −0.779700
\(170\) 0 0
\(171\) −484.000 −0.216447
\(172\) 0 0
\(173\) −2826.00 −1.24195 −0.620973 0.783832i \(-0.713263\pi\)
−0.620973 + 0.783832i \(0.713263\pi\)
\(174\) 0 0
\(175\) −2904.00 −1.25441
\(176\) 0 0
\(177\) 2672.00 1.13469
\(178\) 0 0
\(179\) 3084.00 1.28776 0.643880 0.765127i \(-0.277324\pi\)
0.643880 + 0.765127i \(0.277324\pi\)
\(180\) 0 0
\(181\) −2418.00 −0.992975 −0.496488 0.868044i \(-0.665377\pi\)
−0.496488 + 0.868044i \(0.665377\pi\)
\(182\) 0 0
\(183\) −2200.00 −0.888681
\(184\) 0 0
\(185\) 324.000 0.128762
\(186\) 0 0
\(187\) −2200.00 −0.860320
\(188\) 0 0
\(189\) 3648.00 1.40398
\(190\) 0 0
\(191\) −960.000 −0.363681 −0.181841 0.983328i \(-0.558206\pi\)
−0.181841 + 0.983328i \(0.558206\pi\)
\(192\) 0 0
\(193\) 2882.00 1.07488 0.537438 0.843304i \(-0.319392\pi\)
0.537438 + 0.843304i \(0.319392\pi\)
\(194\) 0 0
\(195\) 176.000 0.0646340
\(196\) 0 0
\(197\) 1086.00 0.392763 0.196381 0.980528i \(-0.437081\pi\)
0.196381 + 0.980528i \(0.437081\pi\)
\(198\) 0 0
\(199\) 88.0000 0.0313475 0.0156738 0.999877i \(-0.495011\pi\)
0.0156738 + 0.999877i \(0.495011\pi\)
\(200\) 0 0
\(201\) −752.000 −0.263890
\(202\) 0 0
\(203\) 4752.00 1.64298
\(204\) 0 0
\(205\) 396.000 0.134916
\(206\) 0 0
\(207\) 616.000 0.206836
\(208\) 0 0
\(209\) −1936.00 −0.640746
\(210\) 0 0
\(211\) −3476.00 −1.13411 −0.567056 0.823679i \(-0.691918\pi\)
−0.567056 + 0.823679i \(0.691918\pi\)
\(212\) 0 0
\(213\) −2912.00 −0.936746
\(214\) 0 0
\(215\) −104.000 −0.0329895
\(216\) 0 0
\(217\) −3840.00 −1.20127
\(218\) 0 0
\(219\) −616.000 −0.190070
\(220\) 0 0
\(221\) 1100.00 0.334815
\(222\) 0 0
\(223\) 928.000 0.278670 0.139335 0.990245i \(-0.455503\pi\)
0.139335 + 0.990245i \(0.455503\pi\)
\(224\) 0 0
\(225\) 1331.00 0.394370
\(226\) 0 0
\(227\) 156.000 0.0456127 0.0228064 0.999740i \(-0.492740\pi\)
0.0228064 + 0.999740i \(0.492740\pi\)
\(228\) 0 0
\(229\) −1634.00 −0.471519 −0.235759 0.971811i \(-0.575758\pi\)
−0.235759 + 0.971811i \(0.575758\pi\)
\(230\) 0 0
\(231\) 4224.00 1.20311
\(232\) 0 0
\(233\) −902.000 −0.253614 −0.126807 0.991927i \(-0.540473\pi\)
−0.126807 + 0.991927i \(0.540473\pi\)
\(234\) 0 0
\(235\) −1056.00 −0.293131
\(236\) 0 0
\(237\) 2624.00 0.719186
\(238\) 0 0
\(239\) 1616.00 0.437365 0.218683 0.975796i \(-0.429824\pi\)
0.218683 + 0.975796i \(0.429824\pi\)
\(240\) 0 0
\(241\) 4818.00 1.28778 0.643889 0.765119i \(-0.277320\pi\)
0.643889 + 0.765119i \(0.277320\pi\)
\(242\) 0 0
\(243\) −2860.00 −0.755017
\(244\) 0 0
\(245\) −466.000 −0.121517
\(246\) 0 0
\(247\) 968.000 0.249362
\(248\) 0 0
\(249\) −944.000 −0.240255
\(250\) 0 0
\(251\) −2140.00 −0.538150 −0.269075 0.963119i \(-0.586718\pi\)
−0.269075 + 0.963119i \(0.586718\pi\)
\(252\) 0 0
\(253\) 2464.00 0.612294
\(254\) 0 0
\(255\) 400.000 0.0982313
\(256\) 0 0
\(257\) 770.000 0.186892 0.0934461 0.995624i \(-0.470212\pi\)
0.0934461 + 0.995624i \(0.470212\pi\)
\(258\) 0 0
\(259\) −3888.00 −0.932774
\(260\) 0 0
\(261\) −2178.00 −0.516532
\(262\) 0 0
\(263\) −7400.00 −1.73499 −0.867497 0.497442i \(-0.834273\pi\)
−0.867497 + 0.497442i \(0.834273\pi\)
\(264\) 0 0
\(265\) 484.000 0.112196
\(266\) 0 0
\(267\) −2856.00 −0.654623
\(268\) 0 0
\(269\) −2794.00 −0.633283 −0.316642 0.948545i \(-0.602555\pi\)
−0.316642 + 0.948545i \(0.602555\pi\)
\(270\) 0 0
\(271\) 8624.00 1.93310 0.966551 0.256474i \(-0.0825608\pi\)
0.966551 + 0.256474i \(0.0825608\pi\)
\(272\) 0 0
\(273\) −2112.00 −0.468220
\(274\) 0 0
\(275\) 5324.00 1.16745
\(276\) 0 0
\(277\) −1874.00 −0.406490 −0.203245 0.979128i \(-0.565149\pi\)
−0.203245 + 0.979128i \(0.565149\pi\)
\(278\) 0 0
\(279\) 1760.00 0.377665
\(280\) 0 0
\(281\) 3338.00 0.708642 0.354321 0.935124i \(-0.384712\pi\)
0.354321 + 0.935124i \(0.384712\pi\)
\(282\) 0 0
\(283\) 7172.00 1.50647 0.753235 0.657751i \(-0.228492\pi\)
0.753235 + 0.657751i \(0.228492\pi\)
\(284\) 0 0
\(285\) 352.000 0.0731603
\(286\) 0 0
\(287\) −4752.00 −0.977358
\(288\) 0 0
\(289\) −2413.00 −0.491146
\(290\) 0 0
\(291\) 1912.00 0.385166
\(292\) 0 0
\(293\) 5214.00 1.03961 0.519804 0.854286i \(-0.326005\pi\)
0.519804 + 0.854286i \(0.326005\pi\)
\(294\) 0 0
\(295\) 1336.00 0.263678
\(296\) 0 0
\(297\) −6688.00 −1.30666
\(298\) 0 0
\(299\) −1232.00 −0.238289
\(300\) 0 0
\(301\) 1248.00 0.238982
\(302\) 0 0
\(303\) −6264.00 −1.18765
\(304\) 0 0
\(305\) −1100.00 −0.206511
\(306\) 0 0
\(307\) 396.000 0.0736186 0.0368093 0.999322i \(-0.488281\pi\)
0.0368093 + 0.999322i \(0.488281\pi\)
\(308\) 0 0
\(309\) 3872.00 0.712849
\(310\) 0 0
\(311\) −4056.00 −0.739533 −0.369766 0.929125i \(-0.620562\pi\)
−0.369766 + 0.929125i \(0.620562\pi\)
\(312\) 0 0
\(313\) 2154.00 0.388982 0.194491 0.980904i \(-0.437695\pi\)
0.194491 + 0.980904i \(0.437695\pi\)
\(314\) 0 0
\(315\) 528.000 0.0944426
\(316\) 0 0
\(317\) −7386.00 −1.30864 −0.654320 0.756217i \(-0.727045\pi\)
−0.654320 + 0.756217i \(0.727045\pi\)
\(318\) 0 0
\(319\) −8712.00 −1.52909
\(320\) 0 0
\(321\) 3120.00 0.542497
\(322\) 0 0
\(323\) 2200.00 0.378982
\(324\) 0 0
\(325\) −2662.00 −0.454342
\(326\) 0 0
\(327\) 7976.00 1.34885
\(328\) 0 0
\(329\) 12672.0 2.12350
\(330\) 0 0
\(331\) −1132.00 −0.187977 −0.0939884 0.995573i \(-0.529962\pi\)
−0.0939884 + 0.995573i \(0.529962\pi\)
\(332\) 0 0
\(333\) 1782.00 0.293252
\(334\) 0 0
\(335\) −376.000 −0.0613226
\(336\) 0 0
\(337\) −3342.00 −0.540209 −0.270104 0.962831i \(-0.587058\pi\)
−0.270104 + 0.962831i \(0.587058\pi\)
\(338\) 0 0
\(339\) 3768.00 0.603686
\(340\) 0 0
\(341\) 7040.00 1.11800
\(342\) 0 0
\(343\) −2640.00 −0.415588
\(344\) 0 0
\(345\) −448.000 −0.0699116
\(346\) 0 0
\(347\) 2244.00 0.347159 0.173580 0.984820i \(-0.444467\pi\)
0.173580 + 0.984820i \(0.444467\pi\)
\(348\) 0 0
\(349\) −6522.00 −1.00033 −0.500164 0.865931i \(-0.666727\pi\)
−0.500164 + 0.865931i \(0.666727\pi\)
\(350\) 0 0
\(351\) 3344.00 0.508517
\(352\) 0 0
\(353\) −11230.0 −1.69324 −0.846618 0.532200i \(-0.821365\pi\)
−0.846618 + 0.532200i \(0.821365\pi\)
\(354\) 0 0
\(355\) −1456.00 −0.217680
\(356\) 0 0
\(357\) −4800.00 −0.711605
\(358\) 0 0
\(359\) 1848.00 0.271682 0.135841 0.990731i \(-0.456626\pi\)
0.135841 + 0.990731i \(0.456626\pi\)
\(360\) 0 0
\(361\) −4923.00 −0.717743
\(362\) 0 0
\(363\) −2420.00 −0.349909
\(364\) 0 0
\(365\) −308.000 −0.0441684
\(366\) 0 0
\(367\) 7120.00 1.01270 0.506350 0.862328i \(-0.330994\pi\)
0.506350 + 0.862328i \(0.330994\pi\)
\(368\) 0 0
\(369\) 2178.00 0.307269
\(370\) 0 0
\(371\) −5808.00 −0.812766
\(372\) 0 0
\(373\) 6350.00 0.881476 0.440738 0.897636i \(-0.354717\pi\)
0.440738 + 0.897636i \(0.354717\pi\)
\(374\) 0 0
\(375\) −1968.00 −0.271006
\(376\) 0 0
\(377\) 4356.00 0.595081
\(378\) 0 0
\(379\) −7900.00 −1.07070 −0.535351 0.844630i \(-0.679821\pi\)
−0.535351 + 0.844630i \(0.679821\pi\)
\(380\) 0 0
\(381\) −5632.00 −0.757313
\(382\) 0 0
\(383\) 10368.0 1.38324 0.691619 0.722263i \(-0.256898\pi\)
0.691619 + 0.722263i \(0.256898\pi\)
\(384\) 0 0
\(385\) 2112.00 0.279578
\(386\) 0 0
\(387\) −572.000 −0.0751328
\(388\) 0 0
\(389\) 8830.00 1.15090 0.575448 0.817838i \(-0.304828\pi\)
0.575448 + 0.817838i \(0.304828\pi\)
\(390\) 0 0
\(391\) −2800.00 −0.362154
\(392\) 0 0
\(393\) 10768.0 1.38212
\(394\) 0 0
\(395\) 1312.00 0.167124
\(396\) 0 0
\(397\) 9878.00 1.24877 0.624386 0.781116i \(-0.285349\pi\)
0.624386 + 0.781116i \(0.285349\pi\)
\(398\) 0 0
\(399\) −4224.00 −0.529986
\(400\) 0 0
\(401\) −13134.0 −1.63561 −0.817806 0.575494i \(-0.804810\pi\)
−0.817806 + 0.575494i \(0.804810\pi\)
\(402\) 0 0
\(403\) −3520.00 −0.435096
\(404\) 0 0
\(405\) 622.000 0.0763146
\(406\) 0 0
\(407\) 7128.00 0.868113
\(408\) 0 0
\(409\) 906.000 0.109533 0.0547663 0.998499i \(-0.482559\pi\)
0.0547663 + 0.998499i \(0.482559\pi\)
\(410\) 0 0
\(411\) −6504.00 −0.780581
\(412\) 0 0
\(413\) −16032.0 −1.91013
\(414\) 0 0
\(415\) −472.000 −0.0558303
\(416\) 0 0
\(417\) 2736.00 0.321301
\(418\) 0 0
\(419\) −5412.00 −0.631011 −0.315505 0.948924i \(-0.602174\pi\)
−0.315505 + 0.948924i \(0.602174\pi\)
\(420\) 0 0
\(421\) −4642.00 −0.537381 −0.268690 0.963227i \(-0.586591\pi\)
−0.268690 + 0.963227i \(0.586591\pi\)
\(422\) 0 0
\(423\) −5808.00 −0.667600
\(424\) 0 0
\(425\) −6050.00 −0.690513
\(426\) 0 0
\(427\) 13200.0 1.49600
\(428\) 0 0
\(429\) 3872.00 0.435762
\(430\) 0 0
\(431\) 656.000 0.0733142 0.0366571 0.999328i \(-0.488329\pi\)
0.0366571 + 0.999328i \(0.488329\pi\)
\(432\) 0 0
\(433\) 9490.00 1.05326 0.526629 0.850096i \(-0.323456\pi\)
0.526629 + 0.850096i \(0.323456\pi\)
\(434\) 0 0
\(435\) 1584.00 0.174591
\(436\) 0 0
\(437\) −2464.00 −0.269723
\(438\) 0 0
\(439\) 5544.00 0.602735 0.301368 0.953508i \(-0.402557\pi\)
0.301368 + 0.953508i \(0.402557\pi\)
\(440\) 0 0
\(441\) −2563.00 −0.276752
\(442\) 0 0
\(443\) 7652.00 0.820672 0.410336 0.911935i \(-0.365412\pi\)
0.410336 + 0.911935i \(0.365412\pi\)
\(444\) 0 0
\(445\) −1428.00 −0.152121
\(446\) 0 0
\(447\) −1208.00 −0.127822
\(448\) 0 0
\(449\) −446.000 −0.0468776 −0.0234388 0.999725i \(-0.507461\pi\)
−0.0234388 + 0.999725i \(0.507461\pi\)
\(450\) 0 0
\(451\) 8712.00 0.909605
\(452\) 0 0
\(453\) −5408.00 −0.560905
\(454\) 0 0
\(455\) −1056.00 −0.108804
\(456\) 0 0
\(457\) 1562.00 0.159885 0.0799423 0.996799i \(-0.474526\pi\)
0.0799423 + 0.996799i \(0.474526\pi\)
\(458\) 0 0
\(459\) 7600.00 0.772849
\(460\) 0 0
\(461\) 10582.0 1.06910 0.534548 0.845138i \(-0.320482\pi\)
0.534548 + 0.845138i \(0.320482\pi\)
\(462\) 0 0
\(463\) −10768.0 −1.08085 −0.540423 0.841394i \(-0.681736\pi\)
−0.540423 + 0.841394i \(0.681736\pi\)
\(464\) 0 0
\(465\) −1280.00 −0.127653
\(466\) 0 0
\(467\) −9876.00 −0.978601 −0.489301 0.872115i \(-0.662748\pi\)
−0.489301 + 0.872115i \(0.662748\pi\)
\(468\) 0 0
\(469\) 4512.00 0.444232
\(470\) 0 0
\(471\) −12568.0 −1.22952
\(472\) 0 0
\(473\) −2288.00 −0.222415
\(474\) 0 0
\(475\) −5324.00 −0.514278
\(476\) 0 0
\(477\) 2662.00 0.255523
\(478\) 0 0
\(479\) −352.000 −0.0335768 −0.0167884 0.999859i \(-0.505344\pi\)
−0.0167884 + 0.999859i \(0.505344\pi\)
\(480\) 0 0
\(481\) −3564.00 −0.337847
\(482\) 0 0
\(483\) 5376.00 0.506452
\(484\) 0 0
\(485\) 956.000 0.0895046
\(486\) 0 0
\(487\) −15176.0 −1.41209 −0.706047 0.708165i \(-0.749523\pi\)
−0.706047 + 0.708165i \(0.749523\pi\)
\(488\) 0 0
\(489\) −12144.0 −1.12305
\(490\) 0 0
\(491\) −8844.00 −0.812880 −0.406440 0.913677i \(-0.633230\pi\)
−0.406440 + 0.913677i \(0.633230\pi\)
\(492\) 0 0
\(493\) 9900.00 0.904409
\(494\) 0 0
\(495\) −968.000 −0.0878957
\(496\) 0 0
\(497\) 17472.0 1.57691
\(498\) 0 0
\(499\) 19404.0 1.74077 0.870383 0.492375i \(-0.163871\pi\)
0.870383 + 0.492375i \(0.163871\pi\)
\(500\) 0 0
\(501\) 1056.00 0.0941689
\(502\) 0 0
\(503\) 16488.0 1.46156 0.730779 0.682614i \(-0.239157\pi\)
0.730779 + 0.682614i \(0.239157\pi\)
\(504\) 0 0
\(505\) −3132.00 −0.275984
\(506\) 0 0
\(507\) 6852.00 0.600213
\(508\) 0 0
\(509\) −12954.0 −1.12805 −0.564024 0.825759i \(-0.690747\pi\)
−0.564024 + 0.825759i \(0.690747\pi\)
\(510\) 0 0
\(511\) 3696.00 0.319964
\(512\) 0 0
\(513\) 6688.00 0.575599
\(514\) 0 0
\(515\) 1936.00 0.165651
\(516\) 0 0
\(517\) −23232.0 −1.97629
\(518\) 0 0
\(519\) 11304.0 0.956051
\(520\) 0 0
\(521\) 10970.0 0.922465 0.461233 0.887279i \(-0.347407\pi\)
0.461233 + 0.887279i \(0.347407\pi\)
\(522\) 0 0
\(523\) −16940.0 −1.41632 −0.708159 0.706053i \(-0.750474\pi\)
−0.708159 + 0.706053i \(0.750474\pi\)
\(524\) 0 0
\(525\) 11616.0 0.965645
\(526\) 0 0
\(527\) −8000.00 −0.661263
\(528\) 0 0
\(529\) −9031.00 −0.742254
\(530\) 0 0
\(531\) 7348.00 0.600520
\(532\) 0 0
\(533\) −4356.00 −0.353995
\(534\) 0 0
\(535\) 1560.00 0.126065
\(536\) 0 0
\(537\) −12336.0 −0.991318
\(538\) 0 0
\(539\) −10252.0 −0.819267
\(540\) 0 0
\(541\) 198.000 0.0157351 0.00786755 0.999969i \(-0.497496\pi\)
0.00786755 + 0.999969i \(0.497496\pi\)
\(542\) 0 0
\(543\) 9672.00 0.764393
\(544\) 0 0
\(545\) 3988.00 0.313444
\(546\) 0 0
\(547\) −15268.0 −1.19344 −0.596721 0.802449i \(-0.703530\pi\)
−0.596721 + 0.802449i \(0.703530\pi\)
\(548\) 0 0
\(549\) −6050.00 −0.470324
\(550\) 0 0
\(551\) 8712.00 0.673582
\(552\) 0 0
\(553\) −15744.0 −1.21067
\(554\) 0 0
\(555\) −1296.00 −0.0991210
\(556\) 0 0
\(557\) 20854.0 1.58638 0.793189 0.608976i \(-0.208419\pi\)
0.793189 + 0.608976i \(0.208419\pi\)
\(558\) 0 0
\(559\) 1144.00 0.0865582
\(560\) 0 0
\(561\) 8800.00 0.662275
\(562\) 0 0
\(563\) −19316.0 −1.44595 −0.722977 0.690872i \(-0.757227\pi\)
−0.722977 + 0.690872i \(0.757227\pi\)
\(564\) 0 0
\(565\) 1884.00 0.140284
\(566\) 0 0
\(567\) −7464.00 −0.552837
\(568\) 0 0
\(569\) 7018.00 0.517065 0.258532 0.966003i \(-0.416761\pi\)
0.258532 + 0.966003i \(0.416761\pi\)
\(570\) 0 0
\(571\) 24420.0 1.78975 0.894873 0.446320i \(-0.147266\pi\)
0.894873 + 0.446320i \(0.147266\pi\)
\(572\) 0 0
\(573\) 3840.00 0.279962
\(574\) 0 0
\(575\) 6776.00 0.491441
\(576\) 0 0
\(577\) 23234.0 1.67633 0.838166 0.545415i \(-0.183628\pi\)
0.838166 + 0.545415i \(0.183628\pi\)
\(578\) 0 0
\(579\) −11528.0 −0.827439
\(580\) 0 0
\(581\) 5664.00 0.404445
\(582\) 0 0
\(583\) 10648.0 0.756424
\(584\) 0 0
\(585\) 484.000 0.0342067
\(586\) 0 0
\(587\) −10604.0 −0.745611 −0.372806 0.927909i \(-0.621604\pi\)
−0.372806 + 0.927909i \(0.621604\pi\)
\(588\) 0 0
\(589\) −7040.00 −0.492493
\(590\) 0 0
\(591\) −4344.00 −0.302349
\(592\) 0 0
\(593\) −13838.0 −0.958277 −0.479139 0.877739i \(-0.659051\pi\)
−0.479139 + 0.877739i \(0.659051\pi\)
\(594\) 0 0
\(595\) −2400.00 −0.165362
\(596\) 0 0
\(597\) −352.000 −0.0241313
\(598\) 0 0
\(599\) −3960.00 −0.270119 −0.135059 0.990837i \(-0.543123\pi\)
−0.135059 + 0.990837i \(0.543123\pi\)
\(600\) 0 0
\(601\) −5942.00 −0.403293 −0.201647 0.979458i \(-0.564629\pi\)
−0.201647 + 0.979458i \(0.564629\pi\)
\(602\) 0 0
\(603\) −2068.00 −0.139661
\(604\) 0 0
\(605\) −1210.00 −0.0813116
\(606\) 0 0
\(607\) −3040.00 −0.203278 −0.101639 0.994821i \(-0.532409\pi\)
−0.101639 + 0.994821i \(0.532409\pi\)
\(608\) 0 0
\(609\) −19008.0 −1.26477
\(610\) 0 0
\(611\) 11616.0 0.769121
\(612\) 0 0
\(613\) −2530.00 −0.166698 −0.0833489 0.996520i \(-0.526562\pi\)
−0.0833489 + 0.996520i \(0.526562\pi\)
\(614\) 0 0
\(615\) −1584.00 −0.103859
\(616\) 0 0
\(617\) −19206.0 −1.25317 −0.626584 0.779354i \(-0.715547\pi\)
−0.626584 + 0.779354i \(0.715547\pi\)
\(618\) 0 0
\(619\) 10996.0 0.714001 0.357000 0.934104i \(-0.383799\pi\)
0.357000 + 0.934104i \(0.383799\pi\)
\(620\) 0 0
\(621\) −8512.00 −0.550040
\(622\) 0 0
\(623\) 17136.0 1.10199
\(624\) 0 0
\(625\) 14141.0 0.905024
\(626\) 0 0
\(627\) 7744.00 0.493247
\(628\) 0 0
\(629\) −8100.00 −0.513463
\(630\) 0 0
\(631\) −6680.00 −0.421437 −0.210718 0.977547i \(-0.567580\pi\)
−0.210718 + 0.977547i \(0.567580\pi\)
\(632\) 0 0
\(633\) 13904.0 0.873040
\(634\) 0 0
\(635\) −2816.00 −0.175984
\(636\) 0 0
\(637\) 5126.00 0.318838
\(638\) 0 0
\(639\) −8008.00 −0.495761
\(640\) 0 0
\(641\) 6274.00 0.386596 0.193298 0.981140i \(-0.438082\pi\)
0.193298 + 0.981140i \(0.438082\pi\)
\(642\) 0 0
\(643\) 9084.00 0.557135 0.278568 0.960417i \(-0.410140\pi\)
0.278568 + 0.960417i \(0.410140\pi\)
\(644\) 0 0
\(645\) 416.000 0.0253953
\(646\) 0 0
\(647\) −23656.0 −1.43742 −0.718712 0.695308i \(-0.755268\pi\)
−0.718712 + 0.695308i \(0.755268\pi\)
\(648\) 0 0
\(649\) 29392.0 1.77771
\(650\) 0 0
\(651\) 15360.0 0.924740
\(652\) 0 0
\(653\) −6762.00 −0.405234 −0.202617 0.979258i \(-0.564945\pi\)
−0.202617 + 0.979258i \(0.564945\pi\)
\(654\) 0 0
\(655\) 5384.00 0.321176
\(656\) 0 0
\(657\) −1694.00 −0.100592
\(658\) 0 0
\(659\) 15276.0 0.902987 0.451494 0.892274i \(-0.350891\pi\)
0.451494 + 0.892274i \(0.350891\pi\)
\(660\) 0 0
\(661\) 11054.0 0.650455 0.325228 0.945636i \(-0.394559\pi\)
0.325228 + 0.945636i \(0.394559\pi\)
\(662\) 0 0
\(663\) −4400.00 −0.257740
\(664\) 0 0
\(665\) −2112.00 −0.123158
\(666\) 0 0
\(667\) −11088.0 −0.643672
\(668\) 0 0
\(669\) −3712.00 −0.214520
\(670\) 0 0
\(671\) −24200.0 −1.39230
\(672\) 0 0
\(673\) −21278.0 −1.21873 −0.609366 0.792889i \(-0.708576\pi\)
−0.609366 + 0.792889i \(0.708576\pi\)
\(674\) 0 0
\(675\) −18392.0 −1.04875
\(676\) 0 0
\(677\) 8926.00 0.506727 0.253363 0.967371i \(-0.418463\pi\)
0.253363 + 0.967371i \(0.418463\pi\)
\(678\) 0 0
\(679\) −11472.0 −0.648387
\(680\) 0 0
\(681\) −624.000 −0.0351127
\(682\) 0 0
\(683\) 8116.00 0.454685 0.227343 0.973815i \(-0.426996\pi\)
0.227343 + 0.973815i \(0.426996\pi\)
\(684\) 0 0
\(685\) −3252.00 −0.181391
\(686\) 0 0
\(687\) 6536.00 0.362975
\(688\) 0 0
\(689\) −5324.00 −0.294381
\(690\) 0 0
\(691\) −11764.0 −0.647646 −0.323823 0.946118i \(-0.604968\pi\)
−0.323823 + 0.946118i \(0.604968\pi\)
\(692\) 0 0
\(693\) 11616.0 0.636732
\(694\) 0 0
\(695\) 1368.00 0.0746636
\(696\) 0 0
\(697\) −9900.00 −0.538005
\(698\) 0 0
\(699\) 3608.00 0.195232
\(700\) 0 0
\(701\) −4698.00 −0.253126 −0.126563 0.991959i \(-0.540395\pi\)
−0.126563 + 0.991959i \(0.540395\pi\)
\(702\) 0 0
\(703\) −7128.00 −0.382415
\(704\) 0 0
\(705\) 4224.00 0.225653
\(706\) 0 0
\(707\) 37584.0 1.99928
\(708\) 0 0
\(709\) 24638.0 1.30508 0.652538 0.757756i \(-0.273704\pi\)
0.652538 + 0.757756i \(0.273704\pi\)
\(710\) 0 0
\(711\) 7216.00 0.380620
\(712\) 0 0
\(713\) 8960.00 0.470624
\(714\) 0 0
\(715\) 1936.00 0.101262
\(716\) 0 0
\(717\) −6464.00 −0.336684
\(718\) 0 0
\(719\) 16624.0 0.862268 0.431134 0.902288i \(-0.358114\pi\)
0.431134 + 0.902288i \(0.358114\pi\)
\(720\) 0 0
\(721\) −23232.0 −1.20001
\(722\) 0 0
\(723\) −19272.0 −0.991332
\(724\) 0 0
\(725\) −23958.0 −1.22728
\(726\) 0 0
\(727\) 30216.0 1.54147 0.770735 0.637155i \(-0.219889\pi\)
0.770735 + 0.637155i \(0.219889\pi\)
\(728\) 0 0
\(729\) 19837.0 1.00782
\(730\) 0 0
\(731\) 2600.00 0.131552
\(732\) 0 0
\(733\) −3322.00 −0.167395 −0.0836977 0.996491i \(-0.526673\pi\)
−0.0836977 + 0.996491i \(0.526673\pi\)
\(734\) 0 0
\(735\) 1864.00 0.0935438
\(736\) 0 0
\(737\) −8272.00 −0.413437
\(738\) 0 0
\(739\) −14692.0 −0.731331 −0.365666 0.930746i \(-0.619159\pi\)
−0.365666 + 0.930746i \(0.619159\pi\)
\(740\) 0 0
\(741\) −3872.00 −0.191959
\(742\) 0 0
\(743\) 28600.0 1.41216 0.706078 0.708134i \(-0.250463\pi\)
0.706078 + 0.708134i \(0.250463\pi\)
\(744\) 0 0
\(745\) −604.000 −0.0297032
\(746\) 0 0
\(747\) −2596.00 −0.127152
\(748\) 0 0
\(749\) −18720.0 −0.913236
\(750\) 0 0
\(751\) −29616.0 −1.43902 −0.719509 0.694483i \(-0.755633\pi\)
−0.719509 + 0.694483i \(0.755633\pi\)
\(752\) 0 0
\(753\) 8560.00 0.414268
\(754\) 0 0
\(755\) −2704.00 −0.130343
\(756\) 0 0
\(757\) 2894.00 0.138949 0.0694744 0.997584i \(-0.477868\pi\)
0.0694744 + 0.997584i \(0.477868\pi\)
\(758\) 0 0
\(759\) −9856.00 −0.471344
\(760\) 0 0
\(761\) 14762.0 0.703183 0.351591 0.936154i \(-0.385641\pi\)
0.351591 + 0.936154i \(0.385641\pi\)
\(762\) 0 0
\(763\) −47856.0 −2.27065
\(764\) 0 0
\(765\) 1100.00 0.0519877
\(766\) 0 0
\(767\) −14696.0 −0.691841
\(768\) 0 0
\(769\) −7678.00 −0.360047 −0.180023 0.983662i \(-0.557617\pi\)
−0.180023 + 0.983662i \(0.557617\pi\)
\(770\) 0 0
\(771\) −3080.00 −0.143870
\(772\) 0 0
\(773\) 27390.0 1.27445 0.637225 0.770678i \(-0.280082\pi\)
0.637225 + 0.770678i \(0.280082\pi\)
\(774\) 0 0
\(775\) 19360.0 0.897331
\(776\) 0 0
\(777\) 15552.0 0.718050
\(778\) 0 0
\(779\) −8712.00 −0.400693
\(780\) 0 0
\(781\) −32032.0 −1.46760
\(782\) 0 0
\(783\) 30096.0 1.37362
\(784\) 0 0
\(785\) −6284.00 −0.285714
\(786\) 0 0
\(787\) 19756.0 0.894823 0.447411 0.894328i \(-0.352346\pi\)
0.447411 + 0.894328i \(0.352346\pi\)
\(788\) 0 0
\(789\) 29600.0 1.33560
\(790\) 0 0
\(791\) −22608.0 −1.01624
\(792\) 0 0
\(793\) 12100.0 0.541846
\(794\) 0 0
\(795\) −1936.00 −0.0863684
\(796\) 0 0
\(797\) 38854.0 1.72682 0.863412 0.504499i \(-0.168323\pi\)
0.863412 + 0.504499i \(0.168323\pi\)
\(798\) 0 0
\(799\) 26400.0 1.16892
\(800\) 0 0
\(801\) −7854.00 −0.346451
\(802\) 0 0
\(803\) −6776.00 −0.297783
\(804\) 0 0
\(805\) 2688.00 0.117689
\(806\) 0 0
\(807\) 11176.0 0.487502
\(808\) 0 0
\(809\) −14278.0 −0.620504 −0.310252 0.950654i \(-0.600413\pi\)
−0.310252 + 0.950654i \(0.600413\pi\)
\(810\) 0 0
\(811\) −716.000 −0.0310014 −0.0155007 0.999880i \(-0.504934\pi\)
−0.0155007 + 0.999880i \(0.504934\pi\)
\(812\) 0 0
\(813\) −34496.0 −1.48810
\(814\) 0 0
\(815\) −6072.00 −0.260973
\(816\) 0 0
\(817\) 2288.00 0.0979767
\(818\) 0 0
\(819\) −5808.00 −0.247800
\(820\) 0 0
\(821\) −23538.0 −1.00059 −0.500293 0.865856i \(-0.666775\pi\)
−0.500293 + 0.865856i \(0.666775\pi\)
\(822\) 0 0
\(823\) −6616.00 −0.280218 −0.140109 0.990136i \(-0.544745\pi\)
−0.140109 + 0.990136i \(0.544745\pi\)
\(824\) 0 0
\(825\) −21296.0 −0.898705
\(826\) 0 0
\(827\) 27236.0 1.14521 0.572605 0.819831i \(-0.305933\pi\)
0.572605 + 0.819831i \(0.305933\pi\)
\(828\) 0 0
\(829\) 12070.0 0.505680 0.252840 0.967508i \(-0.418635\pi\)
0.252840 + 0.967508i \(0.418635\pi\)
\(830\) 0 0
\(831\) 7496.00 0.312916
\(832\) 0 0
\(833\) 11650.0 0.484572
\(834\) 0 0
\(835\) 528.000 0.0218829
\(836\) 0 0
\(837\) −24320.0 −1.00433
\(838\) 0 0
\(839\) −42024.0 −1.72924 −0.864618 0.502429i \(-0.832440\pi\)
−0.864618 + 0.502429i \(0.832440\pi\)
\(840\) 0 0
\(841\) 14815.0 0.607446
\(842\) 0 0
\(843\) −13352.0 −0.545513
\(844\) 0 0
\(845\) 3426.00 0.139477
\(846\) 0 0
\(847\) 14520.0 0.589036
\(848\) 0 0
\(849\) −28688.0 −1.15968
\(850\) 0 0
\(851\) 9072.00 0.365434
\(852\) 0 0
\(853\) 2414.00 0.0968978 0.0484489 0.998826i \(-0.484572\pi\)
0.0484489 + 0.998826i \(0.484572\pi\)
\(854\) 0 0
\(855\) 968.000 0.0387192
\(856\) 0 0
\(857\) −37686.0 −1.50213 −0.751067 0.660226i \(-0.770461\pi\)
−0.751067 + 0.660226i \(0.770461\pi\)
\(858\) 0 0
\(859\) 40644.0 1.61438 0.807192 0.590289i \(-0.200986\pi\)
0.807192 + 0.590289i \(0.200986\pi\)
\(860\) 0 0
\(861\) 19008.0 0.752370
\(862\) 0 0
\(863\) −18656.0 −0.735872 −0.367936 0.929851i \(-0.619935\pi\)
−0.367936 + 0.929851i \(0.619935\pi\)
\(864\) 0 0
\(865\) 5652.00 0.222166
\(866\) 0 0
\(867\) 9652.00 0.378084
\(868\) 0 0
\(869\) 28864.0 1.12675
\(870\) 0 0
\(871\) 4136.00 0.160899
\(872\) 0 0
\(873\) 5258.00 0.203845
\(874\) 0 0
\(875\) 11808.0 0.456209
\(876\) 0 0
\(877\) −13002.0 −0.500623 −0.250311 0.968165i \(-0.580533\pi\)
−0.250311 + 0.968165i \(0.580533\pi\)
\(878\) 0 0
\(879\) −20856.0 −0.800291
\(880\) 0 0
\(881\) 49490.0 1.89258 0.946289 0.323323i \(-0.104800\pi\)
0.946289 + 0.323323i \(0.104800\pi\)
\(882\) 0 0
\(883\) 1100.00 0.0419229 0.0209615 0.999780i \(-0.493327\pi\)
0.0209615 + 0.999780i \(0.493327\pi\)
\(884\) 0 0
\(885\) −5344.00 −0.202979
\(886\) 0 0
\(887\) −14104.0 −0.533896 −0.266948 0.963711i \(-0.586015\pi\)
−0.266948 + 0.963711i \(0.586015\pi\)
\(888\) 0 0
\(889\) 33792.0 1.27486
\(890\) 0 0
\(891\) 13684.0 0.514513
\(892\) 0 0
\(893\) 23232.0 0.870581
\(894\) 0 0
\(895\) −6168.00 −0.230361
\(896\) 0 0
\(897\) 4928.00 0.183435
\(898\) 0 0
\(899\) −31680.0 −1.17529
\(900\) 0 0
\(901\) −12100.0 −0.447402
\(902\) 0 0
\(903\) −4992.00 −0.183968
\(904\) 0 0
\(905\) 4836.00 0.177629
\(906\) 0 0
\(907\) −12716.0 −0.465521 −0.232761 0.972534i \(-0.574776\pi\)
−0.232761 + 0.972534i \(0.574776\pi\)
\(908\) 0 0
\(909\) −17226.0 −0.628548
\(910\) 0 0
\(911\) −39632.0 −1.44135 −0.720673 0.693275i \(-0.756167\pi\)
−0.720673 + 0.693275i \(0.756167\pi\)
\(912\) 0 0
\(913\) −10384.0 −0.376408
\(914\) 0 0
\(915\) 4400.00 0.158972
\(916\) 0 0
\(917\) −64608.0 −2.32666
\(918\) 0 0
\(919\) 5704.00 0.204742 0.102371 0.994746i \(-0.467357\pi\)
0.102371 + 0.994746i \(0.467357\pi\)
\(920\) 0 0
\(921\) −1584.00 −0.0566716
\(922\) 0 0
\(923\) 16016.0 0.571152
\(924\) 0 0
\(925\) 19602.0 0.696767
\(926\) 0 0
\(927\) 10648.0 0.377267
\(928\) 0 0
\(929\) 8162.00 0.288252 0.144126 0.989559i \(-0.453963\pi\)
0.144126 + 0.989559i \(0.453963\pi\)
\(930\) 0 0
\(931\) 10252.0 0.360898
\(932\) 0 0
\(933\) 16224.0 0.569293
\(934\) 0 0
\(935\) 4400.00 0.153899
\(936\) 0 0
\(937\) −55110.0 −1.92141 −0.960707 0.277564i \(-0.910473\pi\)
−0.960707 + 0.277564i \(0.910473\pi\)
\(938\) 0 0
\(939\) −8616.00 −0.299438
\(940\) 0 0
\(941\) 16374.0 0.567245 0.283622 0.958936i \(-0.408464\pi\)
0.283622 + 0.958936i \(0.408464\pi\)
\(942\) 0 0
\(943\) 11088.0 0.382900
\(944\) 0 0
\(945\) −7296.00 −0.251152
\(946\) 0 0
\(947\) 8460.00 0.290299 0.145149 0.989410i \(-0.453634\pi\)
0.145149 + 0.989410i \(0.453634\pi\)
\(948\) 0 0
\(949\) 3388.00 0.115889
\(950\) 0 0
\(951\) 29544.0 1.00739
\(952\) 0 0
\(953\) −20502.0 −0.696878 −0.348439 0.937331i \(-0.613288\pi\)
−0.348439 + 0.937331i \(0.613288\pi\)
\(954\) 0 0
\(955\) 1920.00 0.0650573
\(956\) 0 0
\(957\) 34848.0 1.17709
\(958\) 0 0
\(959\) 39024.0 1.31403
\(960\) 0 0
\(961\) −4191.00 −0.140680
\(962\) 0 0
\(963\) 8580.00 0.287110
\(964\) 0 0
\(965\) −5764.00 −0.192280
\(966\) 0 0
\(967\) −36520.0 −1.21448 −0.607241 0.794518i \(-0.707724\pi\)
−0.607241 + 0.794518i \(0.707724\pi\)
\(968\) 0 0
\(969\) −8800.00 −0.291741
\(970\) 0 0
\(971\) 20244.0 0.669064 0.334532 0.942384i \(-0.391422\pi\)
0.334532 + 0.942384i \(0.391422\pi\)
\(972\) 0 0
\(973\) −16416.0 −0.540876
\(974\) 0 0
\(975\) 10648.0 0.349753
\(976\) 0 0
\(977\) 50034.0 1.63841 0.819206 0.573499i \(-0.194414\pi\)
0.819206 + 0.573499i \(0.194414\pi\)
\(978\) 0 0
\(979\) −31416.0 −1.02560
\(980\) 0 0
\(981\) 21934.0 0.713862
\(982\) 0 0
\(983\) 37128.0 1.20468 0.602339 0.798240i \(-0.294235\pi\)
0.602339 + 0.798240i \(0.294235\pi\)
\(984\) 0 0
\(985\) −2172.00 −0.0702596
\(986\) 0 0
\(987\) −50688.0 −1.63467
\(988\) 0 0
\(989\) −2912.00 −0.0936261
\(990\) 0 0
\(991\) 27808.0 0.891373 0.445686 0.895189i \(-0.352960\pi\)
0.445686 + 0.895189i \(0.352960\pi\)
\(992\) 0 0
\(993\) 4528.00 0.144705
\(994\) 0 0
\(995\) −176.000 −0.00560761
\(996\) 0 0
\(997\) −28514.0 −0.905765 −0.452882 0.891570i \(-0.649604\pi\)
−0.452882 + 0.891570i \(0.649604\pi\)
\(998\) 0 0
\(999\) −24624.0 −0.779849
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 8.4.a.a.1.1 1
3.2 odd 2 72.4.a.c.1.1 1
4.3 odd 2 16.4.a.a.1.1 1
5.2 odd 4 200.4.c.e.49.2 2
5.3 odd 4 200.4.c.e.49.1 2
5.4 even 2 200.4.a.g.1.1 1
7.2 even 3 392.4.i.g.361.1 2
7.3 odd 6 392.4.i.b.177.1 2
7.4 even 3 392.4.i.g.177.1 2
7.5 odd 6 392.4.i.b.361.1 2
7.6 odd 2 392.4.a.e.1.1 1
8.3 odd 2 64.4.a.b.1.1 1
8.5 even 2 64.4.a.d.1.1 1
9.2 odd 6 648.4.i.e.433.1 2
9.4 even 3 648.4.i.h.217.1 2
9.5 odd 6 648.4.i.e.217.1 2
9.7 even 3 648.4.i.h.433.1 2
11.10 odd 2 968.4.a.a.1.1 1
12.11 even 2 144.4.a.e.1.1 1
13.12 even 2 1352.4.a.a.1.1 1
15.2 even 4 1800.4.f.u.649.2 2
15.8 even 4 1800.4.f.u.649.1 2
15.14 odd 2 1800.4.a.d.1.1 1
16.3 odd 4 256.4.b.g.129.2 2
16.5 even 4 256.4.b.a.129.2 2
16.11 odd 4 256.4.b.g.129.1 2
16.13 even 4 256.4.b.a.129.1 2
17.16 even 2 2312.4.a.a.1.1 1
20.3 even 4 400.4.c.i.49.2 2
20.7 even 4 400.4.c.i.49.1 2
20.19 odd 2 400.4.a.g.1.1 1
24.5 odd 2 576.4.a.k.1.1 1
24.11 even 2 576.4.a.j.1.1 1
28.27 even 2 784.4.a.e.1.1 1
40.19 odd 2 1600.4.a.bm.1.1 1
40.29 even 2 1600.4.a.o.1.1 1
44.43 even 2 1936.4.a.l.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8.4.a.a.1.1 1 1.1 even 1 trivial
16.4.a.a.1.1 1 4.3 odd 2
64.4.a.b.1.1 1 8.3 odd 2
64.4.a.d.1.1 1 8.5 even 2
72.4.a.c.1.1 1 3.2 odd 2
144.4.a.e.1.1 1 12.11 even 2
200.4.a.g.1.1 1 5.4 even 2
200.4.c.e.49.1 2 5.3 odd 4
200.4.c.e.49.2 2 5.2 odd 4
256.4.b.a.129.1 2 16.13 even 4
256.4.b.a.129.2 2 16.5 even 4
256.4.b.g.129.1 2 16.11 odd 4
256.4.b.g.129.2 2 16.3 odd 4
392.4.a.e.1.1 1 7.6 odd 2
392.4.i.b.177.1 2 7.3 odd 6
392.4.i.b.361.1 2 7.5 odd 6
392.4.i.g.177.1 2 7.4 even 3
392.4.i.g.361.1 2 7.2 even 3
400.4.a.g.1.1 1 20.19 odd 2
400.4.c.i.49.1 2 20.7 even 4
400.4.c.i.49.2 2 20.3 even 4
576.4.a.j.1.1 1 24.11 even 2
576.4.a.k.1.1 1 24.5 odd 2
648.4.i.e.217.1 2 9.5 odd 6
648.4.i.e.433.1 2 9.2 odd 6
648.4.i.h.217.1 2 9.4 even 3
648.4.i.h.433.1 2 9.7 even 3
784.4.a.e.1.1 1 28.27 even 2
968.4.a.a.1.1 1 11.10 odd 2
1352.4.a.a.1.1 1 13.12 even 2
1600.4.a.o.1.1 1 40.29 even 2
1600.4.a.bm.1.1 1 40.19 odd 2
1800.4.a.d.1.1 1 15.14 odd 2
1800.4.f.u.649.1 2 15.8 even 4
1800.4.f.u.649.2 2 15.2 even 4
1936.4.a.l.1.1 1 44.43 even 2
2312.4.a.a.1.1 1 17.16 even 2