Properties

Label 4864.2.a.v.1.1
Level $4864$
Weight $2$
Character 4864.1
Self dual yes
Analytic conductor $38.839$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4864 = 2^{8} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4864.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(38.8392355432\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 2432)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 4864.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} -2.00000 q^{5} -4.46410 q^{7} -2.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -2.00000 q^{5} -4.46410 q^{7} -2.00000 q^{9} -4.00000 q^{11} +4.46410 q^{13} -2.00000 q^{15} -7.92820 q^{17} +1.00000 q^{19} -4.46410 q^{21} +6.46410 q^{23} -1.00000 q^{25} -5.00000 q^{27} -2.46410 q^{29} -4.92820 q^{31} -4.00000 q^{33} +8.92820 q^{35} -2.00000 q^{37} +4.46410 q^{39} -6.92820 q^{41} +4.92820 q^{43} +4.00000 q^{45} +6.92820 q^{47} +12.9282 q^{49} -7.92820 q^{51} +12.4641 q^{53} +8.00000 q^{55} +1.00000 q^{57} -9.92820 q^{59} -8.92820 q^{61} +8.92820 q^{63} -8.92820 q^{65} -5.92820 q^{67} +6.46410 q^{69} +14.0000 q^{71} -7.00000 q^{73} -1.00000 q^{75} +17.8564 q^{77} -2.00000 q^{79} +1.00000 q^{81} +2.92820 q^{83} +15.8564 q^{85} -2.46410 q^{87} -0.928203 q^{89} -19.9282 q^{91} -4.92820 q^{93} -2.00000 q^{95} +12.9282 q^{97} +8.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} - 4q^{5} - 2q^{7} - 4q^{9} + O(q^{10}) \) \( 2q + 2q^{3} - 4q^{5} - 2q^{7} - 4q^{9} - 8q^{11} + 2q^{13} - 4q^{15} - 2q^{17} + 2q^{19} - 2q^{21} + 6q^{23} - 2q^{25} - 10q^{27} + 2q^{29} + 4q^{31} - 8q^{33} + 4q^{35} - 4q^{37} + 2q^{39} - 4q^{43} + 8q^{45} + 12q^{49} - 2q^{51} + 18q^{53} + 16q^{55} + 2q^{57} - 6q^{59} - 4q^{61} + 4q^{63} - 4q^{65} + 2q^{67} + 6q^{69} + 28q^{71} - 14q^{73} - 2q^{75} + 8q^{77} - 4q^{79} + 2q^{81} - 8q^{83} + 4q^{85} + 2q^{87} + 12q^{89} - 26q^{91} + 4q^{93} - 4q^{95} + 12q^{97} + 16q^{99} + O(q^{100}) \)

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
\(3\) 1.00000 0.577350 0.288675 0.957427i \(-0.406785\pi\)
0.288675 + 0.957427i \(0.406785\pi\)
\(4\) 0 0
\(5\) −2.00000 −0.894427 −0.447214 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) 0 0
\(7\) −4.46410 −1.68727 −0.843636 0.536916i \(-0.819589\pi\)
−0.843636 + 0.536916i \(0.819589\pi\)
\(8\) 0 0
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 0 0
\(13\) 4.46410 1.23812 0.619060 0.785344i \(-0.287514\pi\)
0.619060 + 0.785344i \(0.287514\pi\)
\(14\) 0 0
\(15\) −2.00000 −0.516398
\(16\) 0 0
\(17\) −7.92820 −1.92287 −0.961436 0.275029i \(-0.911312\pi\)
−0.961436 + 0.275029i \(0.911312\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −4.46410 −0.974147
\(22\) 0 0
\(23\) 6.46410 1.34786 0.673929 0.738796i \(-0.264605\pi\)
0.673929 + 0.738796i \(0.264605\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) −5.00000 −0.962250
\(28\) 0 0
\(29\) −2.46410 −0.457572 −0.228786 0.973477i \(-0.573476\pi\)
−0.228786 + 0.973477i \(0.573476\pi\)
\(30\) 0 0
\(31\) −4.92820 −0.885131 −0.442566 0.896736i \(-0.645932\pi\)
−0.442566 + 0.896736i \(0.645932\pi\)
\(32\) 0 0
\(33\) −4.00000 −0.696311
\(34\) 0 0
\(35\) 8.92820 1.50914
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 0 0
\(39\) 4.46410 0.714828
\(40\) 0 0
\(41\) −6.92820 −1.08200 −0.541002 0.841021i \(-0.681955\pi\)
−0.541002 + 0.841021i \(0.681955\pi\)
\(42\) 0 0
\(43\) 4.92820 0.751544 0.375772 0.926712i \(-0.377378\pi\)
0.375772 + 0.926712i \(0.377378\pi\)
\(44\) 0 0
\(45\) 4.00000 0.596285
\(46\) 0 0
\(47\) 6.92820 1.01058 0.505291 0.862949i \(-0.331385\pi\)
0.505291 + 0.862949i \(0.331385\pi\)
\(48\) 0 0
\(49\) 12.9282 1.84689
\(50\) 0 0
\(51\) −7.92820 −1.11017
\(52\) 0 0
\(53\) 12.4641 1.71208 0.856038 0.516913i \(-0.172919\pi\)
0.856038 + 0.516913i \(0.172919\pi\)
\(54\) 0 0
\(55\) 8.00000 1.07872
\(56\) 0 0
\(57\) 1.00000 0.132453
\(58\) 0 0
\(59\) −9.92820 −1.29254 −0.646271 0.763108i \(-0.723672\pi\)
−0.646271 + 0.763108i \(0.723672\pi\)
\(60\) 0 0
\(61\) −8.92820 −1.14314 −0.571570 0.820554i \(-0.693665\pi\)
−0.571570 + 0.820554i \(0.693665\pi\)
\(62\) 0 0
\(63\) 8.92820 1.12485
\(64\) 0 0
\(65\) −8.92820 −1.10741
\(66\) 0 0
\(67\) −5.92820 −0.724245 −0.362123 0.932130i \(-0.617948\pi\)
−0.362123 + 0.932130i \(0.617948\pi\)
\(68\) 0 0
\(69\) 6.46410 0.778186
\(70\) 0 0
\(71\) 14.0000 1.66149 0.830747 0.556650i \(-0.187914\pi\)
0.830747 + 0.556650i \(0.187914\pi\)
\(72\) 0 0
\(73\) −7.00000 −0.819288 −0.409644 0.912245i \(-0.634347\pi\)
−0.409644 + 0.912245i \(0.634347\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) 17.8564 2.03493
\(78\) 0 0
\(79\) −2.00000 −0.225018 −0.112509 0.993651i \(-0.535889\pi\)
−0.112509 + 0.993651i \(0.535889\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 2.92820 0.321412 0.160706 0.987002i \(-0.448623\pi\)
0.160706 + 0.987002i \(0.448623\pi\)
\(84\) 0 0
\(85\) 15.8564 1.71987
\(86\) 0 0
\(87\) −2.46410 −0.264179
\(88\) 0 0
\(89\) −0.928203 −0.0983893 −0.0491947 0.998789i \(-0.515665\pi\)
−0.0491947 + 0.998789i \(0.515665\pi\)
\(90\) 0 0
\(91\) −19.9282 −2.08904
\(92\) 0 0
\(93\) −4.92820 −0.511031
\(94\) 0 0
\(95\) −2.00000 −0.205196
\(96\) 0 0
\(97\) 12.9282 1.31266 0.656330 0.754474i \(-0.272108\pi\)
0.656330 + 0.754474i \(0.272108\pi\)
\(98\) 0 0
\(99\) 8.00000 0.804030
\(100\) 0 0
\(101\) −9.85641 −0.980749 −0.490375 0.871512i \(-0.663140\pi\)
−0.490375 + 0.871512i \(0.663140\pi\)
\(102\) 0 0
\(103\) −2.92820 −0.288524 −0.144262 0.989539i \(-0.546081\pi\)
−0.144262 + 0.989539i \(0.546081\pi\)
\(104\) 0 0
\(105\) 8.92820 0.871303
\(106\) 0 0
\(107\) 13.9282 1.34649 0.673245 0.739419i \(-0.264900\pi\)
0.673245 + 0.739419i \(0.264900\pi\)
\(108\) 0 0
\(109\) −7.39230 −0.708054 −0.354027 0.935235i \(-0.615188\pi\)
−0.354027 + 0.935235i \(0.615188\pi\)
\(110\) 0 0
\(111\) −2.00000 −0.189832
\(112\) 0 0
\(113\) 12.9282 1.21618 0.608092 0.793867i \(-0.291935\pi\)
0.608092 + 0.793867i \(0.291935\pi\)
\(114\) 0 0
\(115\) −12.9282 −1.20556
\(116\) 0 0
\(117\) −8.92820 −0.825413
\(118\) 0 0
\(119\) 35.3923 3.24441
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 0 0
\(123\) −6.92820 −0.624695
\(124\) 0 0
\(125\) 12.0000 1.07331
\(126\) 0 0
\(127\) −0.928203 −0.0823647 −0.0411824 0.999152i \(-0.513112\pi\)
−0.0411824 + 0.999152i \(0.513112\pi\)
\(128\) 0 0
\(129\) 4.92820 0.433904
\(130\) 0 0
\(131\) 7.07180 0.617866 0.308933 0.951084i \(-0.400028\pi\)
0.308933 + 0.951084i \(0.400028\pi\)
\(132\) 0 0
\(133\) −4.46410 −0.387087
\(134\) 0 0
\(135\) 10.0000 0.860663
\(136\) 0 0
\(137\) 8.85641 0.756654 0.378327 0.925672i \(-0.376500\pi\)
0.378327 + 0.925672i \(0.376500\pi\)
\(138\) 0 0
\(139\) −15.8564 −1.34492 −0.672461 0.740132i \(-0.734763\pi\)
−0.672461 + 0.740132i \(0.734763\pi\)
\(140\) 0 0
\(141\) 6.92820 0.583460
\(142\) 0 0
\(143\) −17.8564 −1.49323
\(144\) 0 0
\(145\) 4.92820 0.409265
\(146\) 0 0
\(147\) 12.9282 1.06630
\(148\) 0 0
\(149\) 10.9282 0.895273 0.447637 0.894216i \(-0.352266\pi\)
0.447637 + 0.894216i \(0.352266\pi\)
\(150\) 0 0
\(151\) 16.0000 1.30206 0.651031 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) 0 0
\(153\) 15.8564 1.28191
\(154\) 0 0
\(155\) 9.85641 0.791686
\(156\) 0 0
\(157\) −10.0000 −0.798087 −0.399043 0.916932i \(-0.630658\pi\)
−0.399043 + 0.916932i \(0.630658\pi\)
\(158\) 0 0
\(159\) 12.4641 0.988468
\(160\) 0 0
\(161\) −28.8564 −2.27420
\(162\) 0 0
\(163\) 23.8564 1.86858 0.934289 0.356517i \(-0.116036\pi\)
0.934289 + 0.356517i \(0.116036\pi\)
\(164\) 0 0
\(165\) 8.00000 0.622799
\(166\) 0 0
\(167\) 5.07180 0.392467 0.196234 0.980557i \(-0.437129\pi\)
0.196234 + 0.980557i \(0.437129\pi\)
\(168\) 0 0
\(169\) 6.92820 0.532939
\(170\) 0 0
\(171\) −2.00000 −0.152944
\(172\) 0 0
\(173\) −10.0000 −0.760286 −0.380143 0.924928i \(-0.624125\pi\)
−0.380143 + 0.924928i \(0.624125\pi\)
\(174\) 0 0
\(175\) 4.46410 0.337454
\(176\) 0 0
\(177\) −9.92820 −0.746249
\(178\) 0 0
\(179\) 25.8564 1.93260 0.966299 0.257421i \(-0.0828728\pi\)
0.966299 + 0.257421i \(0.0828728\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 0 0
\(183\) −8.92820 −0.659992
\(184\) 0 0
\(185\) 4.00000 0.294086
\(186\) 0 0
\(187\) 31.7128 2.31907
\(188\) 0 0
\(189\) 22.3205 1.62358
\(190\) 0 0
\(191\) 3.53590 0.255849 0.127924 0.991784i \(-0.459169\pi\)
0.127924 + 0.991784i \(0.459169\pi\)
\(192\) 0 0
\(193\) 9.85641 0.709480 0.354740 0.934965i \(-0.384569\pi\)
0.354740 + 0.934965i \(0.384569\pi\)
\(194\) 0 0
\(195\) −8.92820 −0.639362
\(196\) 0 0
\(197\) −6.92820 −0.493614 −0.246807 0.969065i \(-0.579381\pi\)
−0.246807 + 0.969065i \(0.579381\pi\)
\(198\) 0 0
\(199\) −13.5359 −0.959534 −0.479767 0.877396i \(-0.659279\pi\)
−0.479767 + 0.877396i \(0.659279\pi\)
\(200\) 0 0
\(201\) −5.92820 −0.418143
\(202\) 0 0
\(203\) 11.0000 0.772049
\(204\) 0 0
\(205\) 13.8564 0.967773
\(206\) 0 0
\(207\) −12.9282 −0.898572
\(208\) 0 0
\(209\) −4.00000 −0.276686
\(210\) 0 0
\(211\) −24.8564 −1.71119 −0.855593 0.517649i \(-0.826807\pi\)
−0.855593 + 0.517649i \(0.826807\pi\)
\(212\) 0 0
\(213\) 14.0000 0.959264
\(214\) 0 0
\(215\) −9.85641 −0.672201
\(216\) 0 0
\(217\) 22.0000 1.49346
\(218\) 0 0
\(219\) −7.00000 −0.473016
\(220\) 0 0
\(221\) −35.3923 −2.38074
\(222\) 0 0
\(223\) 26.9282 1.80325 0.901623 0.432523i \(-0.142377\pi\)
0.901623 + 0.432523i \(0.142377\pi\)
\(224\) 0 0
\(225\) 2.00000 0.133333
\(226\) 0 0
\(227\) −1.92820 −0.127979 −0.0639897 0.997951i \(-0.520382\pi\)
−0.0639897 + 0.997951i \(0.520382\pi\)
\(228\) 0 0
\(229\) −3.07180 −0.202990 −0.101495 0.994836i \(-0.532363\pi\)
−0.101495 + 0.994836i \(0.532363\pi\)
\(230\) 0 0
\(231\) 17.8564 1.17487
\(232\) 0 0
\(233\) −15.8564 −1.03879 −0.519394 0.854535i \(-0.673842\pi\)
−0.519394 + 0.854535i \(0.673842\pi\)
\(234\) 0 0
\(235\) −13.8564 −0.903892
\(236\) 0 0
\(237\) −2.00000 −0.129914
\(238\) 0 0
\(239\) 16.3205 1.05569 0.527843 0.849342i \(-0.323001\pi\)
0.527843 + 0.849342i \(0.323001\pi\)
\(240\) 0 0
\(241\) −24.7846 −1.59652 −0.798259 0.602315i \(-0.794245\pi\)
−0.798259 + 0.602315i \(0.794245\pi\)
\(242\) 0 0
\(243\) 16.0000 1.02640
\(244\) 0 0
\(245\) −25.8564 −1.65191
\(246\) 0 0
\(247\) 4.46410 0.284044
\(248\) 0 0
\(249\) 2.92820 0.185567
\(250\) 0 0
\(251\) 12.9282 0.816021 0.408010 0.912977i \(-0.366223\pi\)
0.408010 + 0.912977i \(0.366223\pi\)
\(252\) 0 0
\(253\) −25.8564 −1.62558
\(254\) 0 0
\(255\) 15.8564 0.992967
\(256\) 0 0
\(257\) −0.928203 −0.0578997 −0.0289499 0.999581i \(-0.509216\pi\)
−0.0289499 + 0.999581i \(0.509216\pi\)
\(258\) 0 0
\(259\) 8.92820 0.554772
\(260\) 0 0
\(261\) 4.92820 0.305048
\(262\) 0 0
\(263\) 2.92820 0.180561 0.0902804 0.995916i \(-0.471224\pi\)
0.0902804 + 0.995916i \(0.471224\pi\)
\(264\) 0 0
\(265\) −24.9282 −1.53133
\(266\) 0 0
\(267\) −0.928203 −0.0568051
\(268\) 0 0
\(269\) −7.85641 −0.479014 −0.239507 0.970895i \(-0.576986\pi\)
−0.239507 + 0.970895i \(0.576986\pi\)
\(270\) 0 0
\(271\) 4.60770 0.279898 0.139949 0.990159i \(-0.455306\pi\)
0.139949 + 0.990159i \(0.455306\pi\)
\(272\) 0 0
\(273\) −19.9282 −1.20611
\(274\) 0 0
\(275\) 4.00000 0.241209
\(276\) 0 0
\(277\) −24.9282 −1.49779 −0.748895 0.662688i \(-0.769415\pi\)
−0.748895 + 0.662688i \(0.769415\pi\)
\(278\) 0 0
\(279\) 9.85641 0.590088
\(280\) 0 0
\(281\) −4.00000 −0.238620 −0.119310 0.992857i \(-0.538068\pi\)
−0.119310 + 0.992857i \(0.538068\pi\)
\(282\) 0 0
\(283\) −18.9282 −1.12516 −0.562582 0.826741i \(-0.690192\pi\)
−0.562582 + 0.826741i \(0.690192\pi\)
\(284\) 0 0
\(285\) −2.00000 −0.118470
\(286\) 0 0
\(287\) 30.9282 1.82563
\(288\) 0 0
\(289\) 45.8564 2.69744
\(290\) 0 0
\(291\) 12.9282 0.757865
\(292\) 0 0
\(293\) 8.32051 0.486089 0.243045 0.970015i \(-0.421854\pi\)
0.243045 + 0.970015i \(0.421854\pi\)
\(294\) 0 0
\(295\) 19.8564 1.15608
\(296\) 0 0
\(297\) 20.0000 1.16052
\(298\) 0 0
\(299\) 28.8564 1.66881
\(300\) 0 0
\(301\) −22.0000 −1.26806
\(302\) 0 0
\(303\) −9.85641 −0.566236
\(304\) 0 0
\(305\) 17.8564 1.02245
\(306\) 0 0
\(307\) 1.85641 0.105951 0.0529754 0.998596i \(-0.483130\pi\)
0.0529754 + 0.998596i \(0.483130\pi\)
\(308\) 0 0
\(309\) −2.92820 −0.166580
\(310\) 0 0
\(311\) −7.39230 −0.419179 −0.209590 0.977789i \(-0.567213\pi\)
−0.209590 + 0.977789i \(0.567213\pi\)
\(312\) 0 0
\(313\) −16.8564 −0.952780 −0.476390 0.879234i \(-0.658055\pi\)
−0.476390 + 0.879234i \(0.658055\pi\)
\(314\) 0 0
\(315\) −17.8564 −1.00609
\(316\) 0 0
\(317\) 14.4641 0.812385 0.406192 0.913788i \(-0.366856\pi\)
0.406192 + 0.913788i \(0.366856\pi\)
\(318\) 0 0
\(319\) 9.85641 0.551853
\(320\) 0 0
\(321\) 13.9282 0.777396
\(322\) 0 0
\(323\) −7.92820 −0.441137
\(324\) 0 0
\(325\) −4.46410 −0.247624
\(326\) 0 0
\(327\) −7.39230 −0.408795
\(328\) 0 0
\(329\) −30.9282 −1.70513
\(330\) 0 0
\(331\) −9.78461 −0.537811 −0.268905 0.963167i \(-0.586662\pi\)
−0.268905 + 0.963167i \(0.586662\pi\)
\(332\) 0 0
\(333\) 4.00000 0.219199
\(334\) 0 0
\(335\) 11.8564 0.647785
\(336\) 0 0
\(337\) −7.07180 −0.385225 −0.192613 0.981275i \(-0.561696\pi\)
−0.192613 + 0.981275i \(0.561696\pi\)
\(338\) 0 0
\(339\) 12.9282 0.702164
\(340\) 0 0
\(341\) 19.7128 1.06751
\(342\) 0 0
\(343\) −26.4641 −1.42893
\(344\) 0 0
\(345\) −12.9282 −0.696031
\(346\) 0 0
\(347\) −27.8564 −1.49541 −0.747705 0.664031i \(-0.768844\pi\)
−0.747705 + 0.664031i \(0.768844\pi\)
\(348\) 0 0
\(349\) 14.0000 0.749403 0.374701 0.927146i \(-0.377745\pi\)
0.374701 + 0.927146i \(0.377745\pi\)
\(350\) 0 0
\(351\) −22.3205 −1.19138
\(352\) 0 0
\(353\) 10.0718 0.536068 0.268034 0.963410i \(-0.413626\pi\)
0.268034 + 0.963410i \(0.413626\pi\)
\(354\) 0 0
\(355\) −28.0000 −1.48609
\(356\) 0 0
\(357\) 35.3923 1.87316
\(358\) 0 0
\(359\) 24.4641 1.29117 0.645583 0.763690i \(-0.276614\pi\)
0.645583 + 0.763690i \(0.276614\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 5.00000 0.262432
\(364\) 0 0
\(365\) 14.0000 0.732793
\(366\) 0 0
\(367\) 13.0718 0.682342 0.341171 0.940001i \(-0.389176\pi\)
0.341171 + 0.940001i \(0.389176\pi\)
\(368\) 0 0
\(369\) 13.8564 0.721336
\(370\) 0 0
\(371\) −55.6410 −2.88874
\(372\) 0 0
\(373\) −11.3923 −0.589871 −0.294936 0.955517i \(-0.595298\pi\)
−0.294936 + 0.955517i \(0.595298\pi\)
\(374\) 0 0
\(375\) 12.0000 0.619677
\(376\) 0 0
\(377\) −11.0000 −0.566529
\(378\) 0 0
\(379\) 6.85641 0.352190 0.176095 0.984373i \(-0.443653\pi\)
0.176095 + 0.984373i \(0.443653\pi\)
\(380\) 0 0
\(381\) −0.928203 −0.0475533
\(382\) 0 0
\(383\) −24.7846 −1.26643 −0.633217 0.773974i \(-0.718266\pi\)
−0.633217 + 0.773974i \(0.718266\pi\)
\(384\) 0 0
\(385\) −35.7128 −1.82009
\(386\) 0 0
\(387\) −9.85641 −0.501029
\(388\) 0 0
\(389\) −25.8564 −1.31097 −0.655486 0.755207i \(-0.727536\pi\)
−0.655486 + 0.755207i \(0.727536\pi\)
\(390\) 0 0
\(391\) −51.2487 −2.59176
\(392\) 0 0
\(393\) 7.07180 0.356725
\(394\) 0 0
\(395\) 4.00000 0.201262
\(396\) 0 0
\(397\) −8.92820 −0.448094 −0.224047 0.974578i \(-0.571927\pi\)
−0.224047 + 0.974578i \(0.571927\pi\)
\(398\) 0 0
\(399\) −4.46410 −0.223485
\(400\) 0 0
\(401\) −12.0000 −0.599251 −0.299626 0.954057i \(-0.596862\pi\)
−0.299626 + 0.954057i \(0.596862\pi\)
\(402\) 0 0
\(403\) −22.0000 −1.09590
\(404\) 0 0
\(405\) −2.00000 −0.0993808
\(406\) 0 0
\(407\) 8.00000 0.396545
\(408\) 0 0
\(409\) 6.00000 0.296681 0.148340 0.988936i \(-0.452607\pi\)
0.148340 + 0.988936i \(0.452607\pi\)
\(410\) 0 0
\(411\) 8.85641 0.436854
\(412\) 0 0
\(413\) 44.3205 2.18087
\(414\) 0 0
\(415\) −5.85641 −0.287480
\(416\) 0 0
\(417\) −15.8564 −0.776492
\(418\) 0 0
\(419\) −30.7846 −1.50393 −0.751963 0.659205i \(-0.770893\pi\)
−0.751963 + 0.659205i \(0.770893\pi\)
\(420\) 0 0
\(421\) −11.5359 −0.562225 −0.281113 0.959675i \(-0.590703\pi\)
−0.281113 + 0.959675i \(0.590703\pi\)
\(422\) 0 0
\(423\) −13.8564 −0.673722
\(424\) 0 0
\(425\) 7.92820 0.384574
\(426\) 0 0
\(427\) 39.8564 1.92879
\(428\) 0 0
\(429\) −17.8564 −0.862115
\(430\) 0 0
\(431\) 12.9282 0.622730 0.311365 0.950290i \(-0.399214\pi\)
0.311365 + 0.950290i \(0.399214\pi\)
\(432\) 0 0
\(433\) −24.7846 −1.19107 −0.595536 0.803328i \(-0.703060\pi\)
−0.595536 + 0.803328i \(0.703060\pi\)
\(434\) 0 0
\(435\) 4.92820 0.236289
\(436\) 0 0
\(437\) 6.46410 0.309220
\(438\) 0 0
\(439\) −4.00000 −0.190910 −0.0954548 0.995434i \(-0.530431\pi\)
−0.0954548 + 0.995434i \(0.530431\pi\)
\(440\) 0 0
\(441\) −25.8564 −1.23126
\(442\) 0 0
\(443\) −17.8564 −0.848383 −0.424192 0.905572i \(-0.639442\pi\)
−0.424192 + 0.905572i \(0.639442\pi\)
\(444\) 0 0
\(445\) 1.85641 0.0880021
\(446\) 0 0
\(447\) 10.9282 0.516886
\(448\) 0 0
\(449\) 20.0000 0.943858 0.471929 0.881636i \(-0.343558\pi\)
0.471929 + 0.881636i \(0.343558\pi\)
\(450\) 0 0
\(451\) 27.7128 1.30495
\(452\) 0 0
\(453\) 16.0000 0.751746
\(454\) 0 0
\(455\) 39.8564 1.86850
\(456\) 0 0
\(457\) −31.0000 −1.45012 −0.725059 0.688686i \(-0.758188\pi\)
−0.725059 + 0.688686i \(0.758188\pi\)
\(458\) 0 0
\(459\) 39.6410 1.85028
\(460\) 0 0
\(461\) 13.8564 0.645357 0.322679 0.946509i \(-0.395417\pi\)
0.322679 + 0.946509i \(0.395417\pi\)
\(462\) 0 0
\(463\) 16.7846 0.780047 0.390023 0.920805i \(-0.372467\pi\)
0.390023 + 0.920805i \(0.372467\pi\)
\(464\) 0 0
\(465\) 9.85641 0.457080
\(466\) 0 0
\(467\) 2.78461 0.128856 0.0644282 0.997922i \(-0.479478\pi\)
0.0644282 + 0.997922i \(0.479478\pi\)
\(468\) 0 0
\(469\) 26.4641 1.22200
\(470\) 0 0
\(471\) −10.0000 −0.460776
\(472\) 0 0
\(473\) −19.7128 −0.906396
\(474\) 0 0
\(475\) −1.00000 −0.0458831
\(476\) 0 0
\(477\) −24.9282 −1.14138
\(478\) 0 0
\(479\) −16.7846 −0.766908 −0.383454 0.923560i \(-0.625266\pi\)
−0.383454 + 0.923560i \(0.625266\pi\)
\(480\) 0 0
\(481\) −8.92820 −0.407091
\(482\) 0 0
\(483\) −28.8564 −1.31301
\(484\) 0 0
\(485\) −25.8564 −1.17408
\(486\) 0 0
\(487\) 34.6410 1.56973 0.784867 0.619664i \(-0.212731\pi\)
0.784867 + 0.619664i \(0.212731\pi\)
\(488\) 0 0
\(489\) 23.8564 1.07882
\(490\) 0 0
\(491\) 14.1436 0.638291 0.319146 0.947706i \(-0.396604\pi\)
0.319146 + 0.947706i \(0.396604\pi\)
\(492\) 0 0
\(493\) 19.5359 0.879853
\(494\) 0 0
\(495\) −16.0000 −0.719147
\(496\) 0 0
\(497\) −62.4974 −2.80339
\(498\) 0 0
\(499\) −22.7846 −1.01998 −0.509990 0.860181i \(-0.670351\pi\)
−0.509990 + 0.860181i \(0.670351\pi\)
\(500\) 0 0
\(501\) 5.07180 0.226591
\(502\) 0 0
\(503\) −22.3205 −0.995222 −0.497611 0.867400i \(-0.665789\pi\)
−0.497611 + 0.867400i \(0.665789\pi\)
\(504\) 0 0
\(505\) 19.7128 0.877209
\(506\) 0 0
\(507\) 6.92820 0.307692
\(508\) 0 0
\(509\) −18.0000 −0.797836 −0.398918 0.916987i \(-0.630614\pi\)
−0.398918 + 0.916987i \(0.630614\pi\)
\(510\) 0 0
\(511\) 31.2487 1.38236
\(512\) 0 0
\(513\) −5.00000 −0.220755
\(514\) 0 0
\(515\) 5.85641 0.258064
\(516\) 0 0
\(517\) −27.7128 −1.21881
\(518\) 0 0
\(519\) −10.0000 −0.438951
\(520\) 0 0
\(521\) 13.0718 0.572686 0.286343 0.958127i \(-0.407560\pi\)
0.286343 + 0.958127i \(0.407560\pi\)
\(522\) 0 0
\(523\) 31.0000 1.35554 0.677768 0.735276i \(-0.262948\pi\)
0.677768 + 0.735276i \(0.262948\pi\)
\(524\) 0 0
\(525\) 4.46410 0.194829
\(526\) 0 0
\(527\) 39.0718 1.70199
\(528\) 0 0
\(529\) 18.7846 0.816722
\(530\) 0 0
\(531\) 19.8564 0.861695
\(532\) 0 0
\(533\) −30.9282 −1.33965
\(534\) 0 0
\(535\) −27.8564 −1.20434
\(536\) 0 0
\(537\) 25.8564 1.11579
\(538\) 0 0
\(539\) −51.7128 −2.22743
\(540\) 0 0
\(541\) 12.7846 0.549653 0.274827 0.961494i \(-0.411380\pi\)
0.274827 + 0.961494i \(0.411380\pi\)
\(542\) 0 0
\(543\) −10.0000 −0.429141
\(544\) 0 0
\(545\) 14.7846 0.633303
\(546\) 0 0
\(547\) −6.14359 −0.262681 −0.131341 0.991337i \(-0.541928\pi\)
−0.131341 + 0.991337i \(0.541928\pi\)
\(548\) 0 0
\(549\) 17.8564 0.762093
\(550\) 0 0
\(551\) −2.46410 −0.104974
\(552\) 0 0
\(553\) 8.92820 0.379666
\(554\) 0 0
\(555\) 4.00000 0.169791
\(556\) 0 0
\(557\) −22.0000 −0.932170 −0.466085 0.884740i \(-0.654336\pi\)
−0.466085 + 0.884740i \(0.654336\pi\)
\(558\) 0 0
\(559\) 22.0000 0.930501
\(560\) 0 0
\(561\) 31.7128 1.33892
\(562\) 0 0
\(563\) 31.7128 1.33654 0.668268 0.743921i \(-0.267036\pi\)
0.668268 + 0.743921i \(0.267036\pi\)
\(564\) 0 0
\(565\) −25.8564 −1.08779
\(566\) 0 0
\(567\) −4.46410 −0.187475
\(568\) 0 0
\(569\) 8.78461 0.368270 0.184135 0.982901i \(-0.441052\pi\)
0.184135 + 0.982901i \(0.441052\pi\)
\(570\) 0 0
\(571\) −28.7846 −1.20460 −0.602299 0.798270i \(-0.705749\pi\)
−0.602299 + 0.798270i \(0.705749\pi\)
\(572\) 0 0
\(573\) 3.53590 0.147714
\(574\) 0 0
\(575\) −6.46410 −0.269572
\(576\) 0 0
\(577\) 11.7846 0.490600 0.245300 0.969447i \(-0.421114\pi\)
0.245300 + 0.969447i \(0.421114\pi\)
\(578\) 0 0
\(579\) 9.85641 0.409618
\(580\) 0 0
\(581\) −13.0718 −0.542310
\(582\) 0 0
\(583\) −49.8564 −2.06484
\(584\) 0 0
\(585\) 17.8564 0.738272
\(586\) 0 0
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 0 0
\(589\) −4.92820 −0.203063
\(590\) 0 0
\(591\) −6.92820 −0.284988
\(592\) 0 0
\(593\) 16.1436 0.662938 0.331469 0.943466i \(-0.392456\pi\)
0.331469 + 0.943466i \(0.392456\pi\)
\(594\) 0 0
\(595\) −70.7846 −2.90189
\(596\) 0 0
\(597\) −13.5359 −0.553987
\(598\) 0 0
\(599\) 9.85641 0.402722 0.201361 0.979517i \(-0.435464\pi\)
0.201361 + 0.979517i \(0.435464\pi\)
\(600\) 0 0
\(601\) 27.8564 1.13629 0.568143 0.822930i \(-0.307662\pi\)
0.568143 + 0.822930i \(0.307662\pi\)
\(602\) 0 0
\(603\) 11.8564 0.482830
\(604\) 0 0
\(605\) −10.0000 −0.406558
\(606\) 0 0
\(607\) 39.8564 1.61772 0.808861 0.588000i \(-0.200085\pi\)
0.808861 + 0.588000i \(0.200085\pi\)
\(608\) 0 0
\(609\) 11.0000 0.445742
\(610\) 0 0
\(611\) 30.9282 1.25122
\(612\) 0 0
\(613\) 34.6410 1.39914 0.699569 0.714565i \(-0.253375\pi\)
0.699569 + 0.714565i \(0.253375\pi\)
\(614\) 0 0
\(615\) 13.8564 0.558744
\(616\) 0 0
\(617\) −14.0000 −0.563619 −0.281809 0.959470i \(-0.590935\pi\)
−0.281809 + 0.959470i \(0.590935\pi\)
\(618\) 0 0
\(619\) 27.8564 1.11964 0.559822 0.828613i \(-0.310870\pi\)
0.559822 + 0.828613i \(0.310870\pi\)
\(620\) 0 0
\(621\) −32.3205 −1.29698
\(622\) 0 0
\(623\) 4.14359 0.166010
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 0 0
\(627\) −4.00000 −0.159745
\(628\) 0 0
\(629\) 15.8564 0.632236
\(630\) 0 0
\(631\) 38.6410 1.53827 0.769137 0.639084i \(-0.220686\pi\)
0.769137 + 0.639084i \(0.220686\pi\)
\(632\) 0 0
\(633\) −24.8564 −0.987953
\(634\) 0 0
\(635\) 1.85641 0.0736692
\(636\) 0 0
\(637\) 57.7128 2.28666
\(638\) 0 0
\(639\) −28.0000 −1.10766
\(640\) 0 0
\(641\) 8.78461 0.346971 0.173486 0.984836i \(-0.444497\pi\)
0.173486 + 0.984836i \(0.444497\pi\)
\(642\) 0 0
\(643\) −32.9282 −1.29856 −0.649281 0.760549i \(-0.724930\pi\)
−0.649281 + 0.760549i \(0.724930\pi\)
\(644\) 0 0
\(645\) −9.85641 −0.388096
\(646\) 0 0
\(647\) −13.5359 −0.532151 −0.266076 0.963952i \(-0.585727\pi\)
−0.266076 + 0.963952i \(0.585727\pi\)
\(648\) 0 0
\(649\) 39.7128 1.55886
\(650\) 0 0
\(651\) 22.0000 0.862248
\(652\) 0 0
\(653\) −6.78461 −0.265502 −0.132751 0.991149i \(-0.542381\pi\)
−0.132751 + 0.991149i \(0.542381\pi\)
\(654\) 0 0
\(655\) −14.1436 −0.552636
\(656\) 0 0
\(657\) 14.0000 0.546192
\(658\) 0 0
\(659\) −2.85641 −0.111270 −0.0556349 0.998451i \(-0.517718\pi\)
−0.0556349 + 0.998451i \(0.517718\pi\)
\(660\) 0 0
\(661\) 30.4641 1.18492 0.592458 0.805601i \(-0.298158\pi\)
0.592458 + 0.805601i \(0.298158\pi\)
\(662\) 0 0
\(663\) −35.3923 −1.37452
\(664\) 0 0
\(665\) 8.92820 0.346221
\(666\) 0 0
\(667\) −15.9282 −0.616742
\(668\) 0 0
\(669\) 26.9282 1.04110
\(670\) 0 0
\(671\) 35.7128 1.37868
\(672\) 0 0
\(673\) 9.07180 0.349692 0.174846 0.984596i \(-0.444057\pi\)
0.174846 + 0.984596i \(0.444057\pi\)
\(674\) 0 0
\(675\) 5.00000 0.192450
\(676\) 0 0
\(677\) −7.39230 −0.284109 −0.142055 0.989859i \(-0.545371\pi\)
−0.142055 + 0.989859i \(0.545371\pi\)
\(678\) 0 0
\(679\) −57.7128 −2.21481
\(680\) 0 0
\(681\) −1.92820 −0.0738889
\(682\) 0 0
\(683\) 17.8564 0.683256 0.341628 0.939835i \(-0.389022\pi\)
0.341628 + 0.939835i \(0.389022\pi\)
\(684\) 0 0
\(685\) −17.7128 −0.676772
\(686\) 0 0
\(687\) −3.07180 −0.117196
\(688\) 0 0
\(689\) 55.6410 2.11975
\(690\) 0 0
\(691\) −7.85641 −0.298872 −0.149436 0.988771i \(-0.547746\pi\)
−0.149436 + 0.988771i \(0.547746\pi\)
\(692\) 0 0
\(693\) −35.7128 −1.35662
\(694\) 0 0
\(695\) 31.7128 1.20294
\(696\) 0 0
\(697\) 54.9282 2.08055
\(698\) 0 0
\(699\) −15.8564 −0.599744
\(700\) 0 0
\(701\) 7.71281 0.291309 0.145654 0.989336i \(-0.453471\pi\)
0.145654 + 0.989336i \(0.453471\pi\)
\(702\) 0 0
\(703\) −2.00000 −0.0754314
\(704\) 0 0
\(705\) −13.8564 −0.521862
\(706\) 0 0
\(707\) 44.0000 1.65479
\(708\) 0 0
\(709\) 10.7846 0.405025 0.202512 0.979280i \(-0.435089\pi\)
0.202512 + 0.979280i \(0.435089\pi\)
\(710\) 0 0
\(711\) 4.00000 0.150012
\(712\) 0 0
\(713\) −31.8564 −1.19303
\(714\) 0 0
\(715\) 35.7128 1.33558
\(716\) 0 0
\(717\) 16.3205 0.609501
\(718\) 0 0
\(719\) 13.3923 0.499449 0.249724 0.968317i \(-0.419660\pi\)
0.249724 + 0.968317i \(0.419660\pi\)
\(720\) 0 0
\(721\) 13.0718 0.486819
\(722\) 0 0
\(723\) −24.7846 −0.921750
\(724\) 0 0
\(725\) 2.46410 0.0915144
\(726\) 0 0
\(727\) 0.320508 0.0118870 0.00594349 0.999982i \(-0.498108\pi\)
0.00594349 + 0.999982i \(0.498108\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) −39.0718 −1.44512
\(732\) 0 0
\(733\) 43.8564 1.61987 0.809937 0.586517i \(-0.199501\pi\)
0.809937 + 0.586517i \(0.199501\pi\)
\(734\) 0 0
\(735\) −25.8564 −0.953728
\(736\) 0 0
\(737\) 23.7128 0.873473
\(738\) 0 0
\(739\) −9.85641 −0.362574 −0.181287 0.983430i \(-0.558026\pi\)
−0.181287 + 0.983430i \(0.558026\pi\)
\(740\) 0 0
\(741\) 4.46410 0.163993
\(742\) 0 0
\(743\) −16.9282 −0.621036 −0.310518 0.950568i \(-0.600502\pi\)
−0.310518 + 0.950568i \(0.600502\pi\)
\(744\) 0 0
\(745\) −21.8564 −0.800757
\(746\) 0 0
\(747\) −5.85641 −0.214275
\(748\) 0 0
\(749\) −62.1769 −2.27190
\(750\) 0 0
\(751\) 33.0718 1.20681 0.603404 0.797436i \(-0.293811\pi\)
0.603404 + 0.797436i \(0.293811\pi\)
\(752\) 0 0
\(753\) 12.9282 0.471130
\(754\) 0 0
\(755\) −32.0000 −1.16460
\(756\) 0 0
\(757\) 1.21539 0.0441741 0.0220871 0.999756i \(-0.492969\pi\)
0.0220871 + 0.999756i \(0.492969\pi\)
\(758\) 0 0
\(759\) −25.8564 −0.938528
\(760\) 0 0
\(761\) −48.7128 −1.76584 −0.882919 0.469525i \(-0.844425\pi\)
−0.882919 + 0.469525i \(0.844425\pi\)
\(762\) 0 0
\(763\) 33.0000 1.19468
\(764\) 0 0
\(765\) −31.7128 −1.14658
\(766\) 0 0
\(767\) −44.3205 −1.60032
\(768\) 0 0
\(769\) −47.0000 −1.69486 −0.847432 0.530904i \(-0.821852\pi\)
−0.847432 + 0.530904i \(0.821852\pi\)
\(770\) 0 0
\(771\) −0.928203 −0.0334284
\(772\) 0 0
\(773\) 54.4641 1.95894 0.979469 0.201596i \(-0.0646128\pi\)
0.979469 + 0.201596i \(0.0646128\pi\)
\(774\) 0 0
\(775\) 4.92820 0.177026
\(776\) 0 0
\(777\) 8.92820 0.320298
\(778\) 0 0
\(779\) −6.92820 −0.248229
\(780\) 0 0
\(781\) −56.0000 −2.00384
\(782\) 0 0
\(783\) 12.3205 0.440299
\(784\) 0 0
\(785\) 20.0000 0.713831
\(786\) 0 0
\(787\) −19.7846 −0.705245 −0.352623 0.935766i \(-0.614710\pi\)
−0.352623 + 0.935766i \(0.614710\pi\)
\(788\) 0 0
\(789\) 2.92820 0.104247
\(790\) 0 0
\(791\) −57.7128 −2.05203
\(792\) 0 0
\(793\) −39.8564 −1.41534
\(794\) 0 0
\(795\) −24.9282 −0.884112
\(796\) 0 0
\(797\) −30.1769 −1.06892 −0.534461 0.845193i \(-0.679485\pi\)
−0.534461 + 0.845193i \(0.679485\pi\)
\(798\) 0 0
\(799\) −54.9282 −1.94322
\(800\) 0 0
\(801\) 1.85641 0.0655929
\(802\) 0 0
\(803\) 28.0000 0.988099
\(804\) 0 0
\(805\) 57.7128 2.03411
\(806\) 0 0
\(807\) −7.85641 −0.276559
\(808\) 0 0
\(809\) 18.0718 0.635371 0.317685 0.948196i \(-0.397094\pi\)
0.317685 + 0.948196i \(0.397094\pi\)
\(810\) 0 0
\(811\) 1.14359 0.0401570 0.0200785 0.999798i \(-0.493608\pi\)
0.0200785 + 0.999798i \(0.493608\pi\)
\(812\) 0 0
\(813\) 4.60770 0.161599
\(814\) 0 0
\(815\) −47.7128 −1.67131
\(816\) 0 0
\(817\) 4.92820 0.172416
\(818\) 0 0
\(819\) 39.8564 1.39270
\(820\) 0 0
\(821\) 44.6410 1.55798 0.778991 0.627035i \(-0.215732\pi\)
0.778991 + 0.627035i \(0.215732\pi\)
\(822\) 0 0
\(823\) −23.5359 −0.820410 −0.410205 0.911993i \(-0.634543\pi\)
−0.410205 + 0.911993i \(0.634543\pi\)
\(824\) 0 0
\(825\) 4.00000 0.139262
\(826\) 0 0
\(827\) −14.7128 −0.511615 −0.255807 0.966728i \(-0.582341\pi\)
−0.255807 + 0.966728i \(0.582341\pi\)
\(828\) 0 0
\(829\) 12.3205 0.427909 0.213954 0.976844i \(-0.431366\pi\)
0.213954 + 0.976844i \(0.431366\pi\)
\(830\) 0 0
\(831\) −24.9282 −0.864750
\(832\) 0 0
\(833\) −102.497 −3.55133
\(834\) 0 0
\(835\) −10.1436 −0.351034
\(836\) 0 0
\(837\) 24.6410 0.851718
\(838\) 0 0
\(839\) 22.0000 0.759524 0.379762 0.925084i \(-0.376006\pi\)
0.379762 + 0.925084i \(0.376006\pi\)
\(840\) 0 0
\(841\) −22.9282 −0.790628
\(842\) 0 0
\(843\) −4.00000 −0.137767
\(844\) 0 0
\(845\) −13.8564 −0.476675
\(846\) 0 0
\(847\) −22.3205 −0.766942
\(848\) 0 0
\(849\) −18.9282 −0.649614
\(850\) 0 0
\(851\) −12.9282 −0.443173
\(852\) 0 0
\(853\) 13.0718 0.447570 0.223785 0.974639i \(-0.428159\pi\)
0.223785 + 0.974639i \(0.428159\pi\)
\(854\) 0 0
\(855\) 4.00000 0.136797
\(856\) 0 0
\(857\) 49.8564 1.70306 0.851531 0.524304i \(-0.175674\pi\)
0.851531 + 0.524304i \(0.175674\pi\)
\(858\) 0 0
\(859\) −21.0718 −0.718960 −0.359480 0.933153i \(-0.617046\pi\)
−0.359480 + 0.933153i \(0.617046\pi\)
\(860\) 0 0
\(861\) 30.9282 1.05403
\(862\) 0 0
\(863\) −14.7846 −0.503274 −0.251637 0.967822i \(-0.580969\pi\)
−0.251637 + 0.967822i \(0.580969\pi\)
\(864\) 0 0
\(865\) 20.0000 0.680020
\(866\) 0 0
\(867\) 45.8564 1.55737
\(868\) 0 0
\(869\) 8.00000 0.271381
\(870\) 0 0
\(871\) −26.4641 −0.896702
\(872\) 0 0
\(873\) −25.8564 −0.875107
\(874\) 0 0
\(875\) −53.5692 −1.81097
\(876\) 0 0
\(877\) 31.2487 1.05519 0.527597 0.849495i \(-0.323093\pi\)
0.527597 + 0.849495i \(0.323093\pi\)
\(878\) 0 0
\(879\) 8.32051 0.280644
\(880\) 0 0
\(881\) −22.0000 −0.741199 −0.370599 0.928793i \(-0.620848\pi\)
−0.370599 + 0.928793i \(0.620848\pi\)
\(882\) 0 0
\(883\) 49.7128 1.67297 0.836485 0.547990i \(-0.184607\pi\)
0.836485 + 0.547990i \(0.184607\pi\)
\(884\) 0 0
\(885\) 19.8564 0.667466
\(886\) 0 0
\(887\) 12.9282 0.434087 0.217043 0.976162i \(-0.430359\pi\)
0.217043 + 0.976162i \(0.430359\pi\)
\(888\) 0 0
\(889\) 4.14359 0.138972
\(890\) 0 0
\(891\) −4.00000 −0.134005
\(892\) 0 0
\(893\) 6.92820 0.231843
\(894\) 0 0
\(895\) −51.7128 −1.72857
\(896\) 0 0
\(897\) 28.8564 0.963487
\(898\) 0 0
\(899\) 12.1436 0.405012
\(900\) 0 0
\(901\) −98.8179 −3.29210
\(902\) 0 0
\(903\) −22.0000 −0.732114
\(904\) 0 0
\(905\) 20.0000 0.664822
\(906\) 0 0
\(907\) 13.1436 0.436426 0.218213 0.975901i \(-0.429977\pi\)
0.218213 + 0.975901i \(0.429977\pi\)
\(908\) 0 0
\(909\) 19.7128 0.653833
\(910\) 0 0
\(911\) 7.85641 0.260294 0.130147 0.991495i \(-0.458455\pi\)
0.130147 + 0.991495i \(0.458455\pi\)
\(912\) 0 0
\(913\) −11.7128 −0.387638
\(914\) 0 0
\(915\) 17.8564 0.590315
\(916\) 0 0
\(917\) −31.5692 −1.04251
\(918\) 0 0
\(919\) −5.39230 −0.177876 −0.0889379 0.996037i \(-0.528347\pi\)
−0.0889379 + 0.996037i \(0.528347\pi\)
\(920\) 0 0
\(921\) 1.85641 0.0611707
\(922\) 0 0
\(923\) 62.4974 2.05713
\(924\) 0 0
\(925\) 2.00000 0.0657596
\(926\) 0 0
\(927\) 5.85641 0.192350
\(928\) 0 0
\(929\) 8.71281 0.285858 0.142929 0.989733i \(-0.454348\pi\)
0.142929 + 0.989733i \(0.454348\pi\)
\(930\) 0 0
\(931\) 12.9282 0.423705
\(932\) 0 0
\(933\) −7.39230 −0.242013
\(934\) 0 0
\(935\) −63.4256 −2.07424
\(936\) 0 0
\(937\) −29.9282 −0.977712 −0.488856 0.872365i \(-0.662586\pi\)
−0.488856 + 0.872365i \(0.662586\pi\)
\(938\) 0 0
\(939\) −16.8564 −0.550088
\(940\) 0 0
\(941\) −41.1051 −1.33999 −0.669994 0.742366i \(-0.733703\pi\)
−0.669994 + 0.742366i \(0.733703\pi\)
\(942\) 0 0
\(943\) −44.7846 −1.45839
\(944\) 0 0
\(945\) −44.6410 −1.45217
\(946\) 0 0
\(947\) 16.9282 0.550093 0.275046 0.961431i \(-0.411307\pi\)
0.275046 + 0.961431i \(0.411307\pi\)
\(948\) 0 0
\(949\) −31.2487 −1.01438
\(950\) 0 0
\(951\) 14.4641 0.469031
\(952\) 0 0
\(953\) −7.21539 −0.233729 −0.116865 0.993148i \(-0.537284\pi\)
−0.116865 + 0.993148i \(0.537284\pi\)
\(954\) 0 0
\(955\) −7.07180 −0.228838
\(956\) 0 0
\(957\) 9.85641 0.318612
\(958\) 0 0
\(959\) −39.5359 −1.27668
\(960\) 0 0
\(961\) −6.71281 −0.216542
\(962\) 0 0
\(963\) −27.8564 −0.897660
\(964\) 0 0
\(965\) −19.7128 −0.634578
\(966\) 0 0
\(967\) 24.7846 0.797019 0.398510 0.917164i \(-0.369528\pi\)
0.398510 + 0.917164i \(0.369528\pi\)
\(968\) 0 0
\(969\) −7.92820 −0.254691
\(970\) 0 0
\(971\) 25.8564 0.829772 0.414886 0.909873i \(-0.363822\pi\)
0.414886 + 0.909873i \(0.363822\pi\)
\(972\) 0 0
\(973\) 70.7846 2.26925
\(974\) 0 0
\(975\) −4.46410 −0.142966
\(976\) 0 0
\(977\) 6.00000 0.191957 0.0959785 0.995383i \(-0.469402\pi\)
0.0959785 + 0.995383i \(0.469402\pi\)
\(978\) 0 0
\(979\) 3.71281 0.118662
\(980\) 0 0
\(981\) 14.7846 0.472036
\(982\) 0 0
\(983\) 20.6410 0.658346 0.329173 0.944270i \(-0.393230\pi\)
0.329173 + 0.944270i \(0.393230\pi\)
\(984\) 0 0
\(985\) 13.8564 0.441502
\(986\) 0 0
\(987\) −30.9282 −0.984456
\(988\) 0 0
\(989\) 31.8564 1.01297
\(990\) 0 0
\(991\) 43.5692 1.38402 0.692011 0.721887i \(-0.256725\pi\)
0.692011 + 0.721887i \(0.256725\pi\)
\(992\) 0 0
\(993\) −9.78461 −0.310505
\(994\) 0 0
\(995\) 27.0718 0.858234
\(996\) 0 0
\(997\) 42.7846 1.35500 0.677501 0.735522i \(-0.263063\pi\)
0.677501 + 0.735522i \(0.263063\pi\)
\(998\) 0 0
\(999\) 10.0000 0.316386
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 4864.2.a.v.1.1 2
4.3 odd 2 4864.2.a.s.1.2 2
8.3 odd 2 4864.2.a.x.1.2 2
8.5 even 2 4864.2.a.u.1.1 2
16.3 odd 4 2432.2.c.c.1217.1 4
16.5 even 4 2432.2.c.d.1217.2 yes 4
16.11 odd 4 2432.2.c.c.1217.3 yes 4
16.13 even 4 2432.2.c.d.1217.4 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2432.2.c.c.1217.1 4 16.3 odd 4
2432.2.c.c.1217.3 yes 4 16.11 odd 4
2432.2.c.d.1217.2 yes 4 16.5 even 4
2432.2.c.d.1217.4 yes 4 16.13 even 4
4864.2.a.s.1.2 2 4.3 odd 2
4864.2.a.u.1.1 2 8.5 even 2
4864.2.a.v.1.1 2 1.1 even 1 trivial
4864.2.a.x.1.2 2 8.3 odd 2