Properties

Label 2672.2.a.c.1.1
Level 2672
Weight 2
Character 2672.1
Self dual yes
Analytic conductor 21.336
Analytic rank 1
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 2672 = 2^{4} \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2672.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(21.3360274201\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Defining polynomial: \(x^{2} - x - 3\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 668)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.30278\) of defining polynomial
Character \(\chi\) \(=\) 2672.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.30278 q^{3} -3.00000 q^{5} -3.30278 q^{7} +2.30278 q^{9} +O(q^{10})\) \(q-2.30278 q^{3} -3.00000 q^{5} -3.30278 q^{7} +2.30278 q^{9} +6.30278 q^{13} +6.90833 q^{15} -1.30278 q^{17} -2.00000 q^{19} +7.60555 q^{21} -1.30278 q^{23} +4.00000 q^{25} +1.60555 q^{27} +0.394449 q^{29} +0.605551 q^{31} +9.90833 q^{35} +7.60555 q^{37} -14.5139 q^{39} +8.21110 q^{41} -2.39445 q^{43} -6.90833 q^{45} +5.60555 q^{47} +3.90833 q^{49} +3.00000 q^{51} +2.60555 q^{53} +4.60555 q^{57} +13.8167 q^{59} -14.8167 q^{61} -7.60555 q^{63} -18.9083 q^{65} +6.21110 q^{67} +3.00000 q^{69} +6.90833 q^{71} -13.9083 q^{73} -9.21110 q^{75} -13.6056 q^{79} -10.6056 q^{81} -5.60555 q^{83} +3.90833 q^{85} -0.908327 q^{87} +7.81665 q^{89} -20.8167 q^{91} -1.39445 q^{93} +6.00000 q^{95} -13.5139 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{3} - 6q^{5} - 3q^{7} + q^{9} + O(q^{10}) \) \( 2q - q^{3} - 6q^{5} - 3q^{7} + q^{9} + 9q^{13} + 3q^{15} + q^{17} - 4q^{19} + 8q^{21} + q^{23} + 8q^{25} - 4q^{27} + 8q^{29} - 6q^{31} + 9q^{35} + 8q^{37} - 11q^{39} + 2q^{41} - 12q^{43} - 3q^{45} + 4q^{47} - 3q^{49} + 6q^{51} - 2q^{53} + 2q^{57} + 6q^{59} - 8q^{61} - 8q^{63} - 27q^{65} - 2q^{67} + 6q^{69} + 3q^{71} - 17q^{73} - 4q^{75} - 20q^{79} - 14q^{81} - 4q^{83} - 3q^{85} + 9q^{87} - 6q^{89} - 20q^{91} - 10q^{93} + 12q^{95} - 9q^{97} + O(q^{100}) \)

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
\(3\) −2.30278 −1.32951 −0.664754 0.747062i \(-0.731464\pi\)
−0.664754 + 0.747062i \(0.731464\pi\)
\(4\) 0 0
\(5\) −3.00000 −1.34164 −0.670820 0.741620i \(-0.734058\pi\)
−0.670820 + 0.741620i \(0.734058\pi\)
\(6\) 0 0
\(7\) −3.30278 −1.24833 −0.624166 0.781292i \(-0.714561\pi\)
−0.624166 + 0.781292i \(0.714561\pi\)
\(8\) 0 0
\(9\) 2.30278 0.767592
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 6.30278 1.74808 0.874038 0.485858i \(-0.161493\pi\)
0.874038 + 0.485858i \(0.161493\pi\)
\(14\) 0 0
\(15\) 6.90833 1.78372
\(16\) 0 0
\(17\) −1.30278 −0.315970 −0.157985 0.987442i \(-0.550500\pi\)
−0.157985 + 0.987442i \(0.550500\pi\)
\(18\) 0 0
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) 7.60555 1.65967
\(22\) 0 0
\(23\) −1.30278 −0.271647 −0.135824 0.990733i \(-0.543368\pi\)
−0.135824 + 0.990733i \(0.543368\pi\)
\(24\) 0 0
\(25\) 4.00000 0.800000
\(26\) 0 0
\(27\) 1.60555 0.308988
\(28\) 0 0
\(29\) 0.394449 0.0732473 0.0366236 0.999329i \(-0.488340\pi\)
0.0366236 + 0.999329i \(0.488340\pi\)
\(30\) 0 0
\(31\) 0.605551 0.108760 0.0543801 0.998520i \(-0.482682\pi\)
0.0543801 + 0.998520i \(0.482682\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 9.90833 1.67481
\(36\) 0 0
\(37\) 7.60555 1.25034 0.625172 0.780487i \(-0.285029\pi\)
0.625172 + 0.780487i \(0.285029\pi\)
\(38\) 0 0
\(39\) −14.5139 −2.32408
\(40\) 0 0
\(41\) 8.21110 1.28236 0.641179 0.767391i \(-0.278445\pi\)
0.641179 + 0.767391i \(0.278445\pi\)
\(42\) 0 0
\(43\) −2.39445 −0.365150 −0.182575 0.983192i \(-0.558443\pi\)
−0.182575 + 0.983192i \(0.558443\pi\)
\(44\) 0 0
\(45\) −6.90833 −1.02983
\(46\) 0 0
\(47\) 5.60555 0.817654 0.408827 0.912612i \(-0.365938\pi\)
0.408827 + 0.912612i \(0.365938\pi\)
\(48\) 0 0
\(49\) 3.90833 0.558332
\(50\) 0 0
\(51\) 3.00000 0.420084
\(52\) 0 0
\(53\) 2.60555 0.357900 0.178950 0.983858i \(-0.442730\pi\)
0.178950 + 0.983858i \(0.442730\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 4.60555 0.610020
\(58\) 0 0
\(59\) 13.8167 1.79878 0.899388 0.437152i \(-0.144013\pi\)
0.899388 + 0.437152i \(0.144013\pi\)
\(60\) 0 0
\(61\) −14.8167 −1.89708 −0.948539 0.316660i \(-0.897439\pi\)
−0.948539 + 0.316660i \(0.897439\pi\)
\(62\) 0 0
\(63\) −7.60555 −0.958209
\(64\) 0 0
\(65\) −18.9083 −2.34529
\(66\) 0 0
\(67\) 6.21110 0.758807 0.379403 0.925231i \(-0.376129\pi\)
0.379403 + 0.925231i \(0.376129\pi\)
\(68\) 0 0
\(69\) 3.00000 0.361158
\(70\) 0 0
\(71\) 6.90833 0.819868 0.409934 0.912115i \(-0.365552\pi\)
0.409934 + 0.912115i \(0.365552\pi\)
\(72\) 0 0
\(73\) −13.9083 −1.62785 −0.813923 0.580972i \(-0.802672\pi\)
−0.813923 + 0.580972i \(0.802672\pi\)
\(74\) 0 0
\(75\) −9.21110 −1.06361
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −13.6056 −1.53074 −0.765372 0.643588i \(-0.777445\pi\)
−0.765372 + 0.643588i \(0.777445\pi\)
\(80\) 0 0
\(81\) −10.6056 −1.17839
\(82\) 0 0
\(83\) −5.60555 −0.615289 −0.307645 0.951501i \(-0.599541\pi\)
−0.307645 + 0.951501i \(0.599541\pi\)
\(84\) 0 0
\(85\) 3.90833 0.423918
\(86\) 0 0
\(87\) −0.908327 −0.0973829
\(88\) 0 0
\(89\) 7.81665 0.828564 0.414282 0.910149i \(-0.364033\pi\)
0.414282 + 0.910149i \(0.364033\pi\)
\(90\) 0 0
\(91\) −20.8167 −2.18218
\(92\) 0 0
\(93\) −1.39445 −0.144598
\(94\) 0 0
\(95\) 6.00000 0.615587
\(96\) 0 0
\(97\) −13.5139 −1.37213 −0.686063 0.727542i \(-0.740663\pi\)
−0.686063 + 0.727542i \(0.740663\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 10.8167 1.07630 0.538149 0.842850i \(-0.319124\pi\)
0.538149 + 0.842850i \(0.319124\pi\)
\(102\) 0 0
\(103\) −0.302776 −0.0298334 −0.0149167 0.999889i \(-0.504748\pi\)
−0.0149167 + 0.999889i \(0.504748\pi\)
\(104\) 0 0
\(105\) −22.8167 −2.22668
\(106\) 0 0
\(107\) −0.394449 −0.0381328 −0.0190664 0.999818i \(-0.506069\pi\)
−0.0190664 + 0.999818i \(0.506069\pi\)
\(108\) 0 0
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) −17.5139 −1.66234
\(112\) 0 0
\(113\) −19.4222 −1.82709 −0.913544 0.406741i \(-0.866665\pi\)
−0.913544 + 0.406741i \(0.866665\pi\)
\(114\) 0 0
\(115\) 3.90833 0.364453
\(116\) 0 0
\(117\) 14.5139 1.34181
\(118\) 0 0
\(119\) 4.30278 0.394435
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) −18.9083 −1.70491
\(124\) 0 0
\(125\) 3.00000 0.268328
\(126\) 0 0
\(127\) 0.211103 0.0187323 0.00936616 0.999956i \(-0.497019\pi\)
0.00936616 + 0.999956i \(0.497019\pi\)
\(128\) 0 0
\(129\) 5.51388 0.485470
\(130\) 0 0
\(131\) −16.0278 −1.40035 −0.700176 0.713970i \(-0.746895\pi\)
−0.700176 + 0.713970i \(0.746895\pi\)
\(132\) 0 0
\(133\) 6.60555 0.572774
\(134\) 0 0
\(135\) −4.81665 −0.414552
\(136\) 0 0
\(137\) −17.2111 −1.47044 −0.735222 0.677827i \(-0.762922\pi\)
−0.735222 + 0.677827i \(0.762922\pi\)
\(138\) 0 0
\(139\) 4.90833 0.416319 0.208159 0.978095i \(-0.433253\pi\)
0.208159 + 0.978095i \(0.433253\pi\)
\(140\) 0 0
\(141\) −12.9083 −1.08708
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −1.18335 −0.0982716
\(146\) 0 0
\(147\) −9.00000 −0.742307
\(148\) 0 0
\(149\) −0.513878 −0.0420985 −0.0210493 0.999778i \(-0.506701\pi\)
−0.0210493 + 0.999778i \(0.506701\pi\)
\(150\) 0 0
\(151\) −1.48612 −0.120939 −0.0604694 0.998170i \(-0.519260\pi\)
−0.0604694 + 0.998170i \(0.519260\pi\)
\(152\) 0 0
\(153\) −3.00000 −0.242536
\(154\) 0 0
\(155\) −1.81665 −0.145917
\(156\) 0 0
\(157\) −20.8167 −1.66135 −0.830675 0.556758i \(-0.812045\pi\)
−0.830675 + 0.556758i \(0.812045\pi\)
\(158\) 0 0
\(159\) −6.00000 −0.475831
\(160\) 0 0
\(161\) 4.30278 0.339106
\(162\) 0 0
\(163\) 12.6056 0.987343 0.493671 0.869648i \(-0.335655\pi\)
0.493671 + 0.869648i \(0.335655\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1.00000 0.0773823
\(168\) 0 0
\(169\) 26.7250 2.05577
\(170\) 0 0
\(171\) −4.60555 −0.352195
\(172\) 0 0
\(173\) 20.2111 1.53662 0.768311 0.640077i \(-0.221098\pi\)
0.768311 + 0.640077i \(0.221098\pi\)
\(174\) 0 0
\(175\) −13.2111 −0.998665
\(176\) 0 0
\(177\) −31.8167 −2.39149
\(178\) 0 0
\(179\) 9.90833 0.740583 0.370292 0.928916i \(-0.379258\pi\)
0.370292 + 0.928916i \(0.379258\pi\)
\(180\) 0 0
\(181\) 7.21110 0.535997 0.267999 0.963419i \(-0.413638\pi\)
0.267999 + 0.963419i \(0.413638\pi\)
\(182\) 0 0
\(183\) 34.1194 2.52218
\(184\) 0 0
\(185\) −22.8167 −1.67751
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −5.30278 −0.385720
\(190\) 0 0
\(191\) 7.69722 0.556952 0.278476 0.960443i \(-0.410171\pi\)
0.278476 + 0.960443i \(0.410171\pi\)
\(192\) 0 0
\(193\) −4.90833 −0.353309 −0.176655 0.984273i \(-0.556528\pi\)
−0.176655 + 0.984273i \(0.556528\pi\)
\(194\) 0 0
\(195\) 43.5416 3.11808
\(196\) 0 0
\(197\) 7.30278 0.520301 0.260151 0.965568i \(-0.416228\pi\)
0.260151 + 0.965568i \(0.416228\pi\)
\(198\) 0 0
\(199\) 14.3028 1.01390 0.506948 0.861976i \(-0.330773\pi\)
0.506948 + 0.861976i \(0.330773\pi\)
\(200\) 0 0
\(201\) −14.3028 −1.00884
\(202\) 0 0
\(203\) −1.30278 −0.0914369
\(204\) 0 0
\(205\) −24.6333 −1.72046
\(206\) 0 0
\(207\) −3.00000 −0.208514
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 4.51388 0.310748 0.155374 0.987856i \(-0.450342\pi\)
0.155374 + 0.987856i \(0.450342\pi\)
\(212\) 0 0
\(213\) −15.9083 −1.09002
\(214\) 0 0
\(215\) 7.18335 0.489900
\(216\) 0 0
\(217\) −2.00000 −0.135769
\(218\) 0 0
\(219\) 32.0278 2.16424
\(220\) 0 0
\(221\) −8.21110 −0.552339
\(222\) 0 0
\(223\) −17.5139 −1.17282 −0.586408 0.810016i \(-0.699458\pi\)
−0.586408 + 0.810016i \(0.699458\pi\)
\(224\) 0 0
\(225\) 9.21110 0.614074
\(226\) 0 0
\(227\) −22.8167 −1.51439 −0.757197 0.653186i \(-0.773432\pi\)
−0.757197 + 0.653186i \(0.773432\pi\)
\(228\) 0 0
\(229\) 12.3028 0.812990 0.406495 0.913653i \(-0.366751\pi\)
0.406495 + 0.913653i \(0.366751\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −3.90833 −0.256043 −0.128022 0.991771i \(-0.540863\pi\)
−0.128022 + 0.991771i \(0.540863\pi\)
\(234\) 0 0
\(235\) −16.8167 −1.09700
\(236\) 0 0
\(237\) 31.3305 2.03514
\(238\) 0 0
\(239\) −21.9083 −1.41713 −0.708566 0.705645i \(-0.750658\pi\)
−0.708566 + 0.705645i \(0.750658\pi\)
\(240\) 0 0
\(241\) 6.30278 0.405997 0.202999 0.979179i \(-0.434931\pi\)
0.202999 + 0.979179i \(0.434931\pi\)
\(242\) 0 0
\(243\) 19.6056 1.25770
\(244\) 0 0
\(245\) −11.7250 −0.749082
\(246\) 0 0
\(247\) −12.6056 −0.802072
\(248\) 0 0
\(249\) 12.9083 0.818032
\(250\) 0 0
\(251\) 17.6056 1.11125 0.555626 0.831432i \(-0.312479\pi\)
0.555626 + 0.831432i \(0.312479\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) −9.00000 −0.563602
\(256\) 0 0
\(257\) −6.11943 −0.381720 −0.190860 0.981617i \(-0.561128\pi\)
−0.190860 + 0.981617i \(0.561128\pi\)
\(258\) 0 0
\(259\) −25.1194 −1.56085
\(260\) 0 0
\(261\) 0.908327 0.0562240
\(262\) 0 0
\(263\) 2.21110 0.136342 0.0681712 0.997674i \(-0.478284\pi\)
0.0681712 + 0.997674i \(0.478284\pi\)
\(264\) 0 0
\(265\) −7.81665 −0.480173
\(266\) 0 0
\(267\) −18.0000 −1.10158
\(268\) 0 0
\(269\) 1.69722 0.103482 0.0517408 0.998661i \(-0.483523\pi\)
0.0517408 + 0.998661i \(0.483523\pi\)
\(270\) 0 0
\(271\) 14.4222 0.876087 0.438043 0.898954i \(-0.355672\pi\)
0.438043 + 0.898954i \(0.355672\pi\)
\(272\) 0 0
\(273\) 47.9361 2.90122
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 7.48612 0.449797 0.224899 0.974382i \(-0.427795\pi\)
0.224899 + 0.974382i \(0.427795\pi\)
\(278\) 0 0
\(279\) 1.39445 0.0834835
\(280\) 0 0
\(281\) 13.4222 0.800702 0.400351 0.916362i \(-0.368888\pi\)
0.400351 + 0.916362i \(0.368888\pi\)
\(282\) 0 0
\(283\) −18.4222 −1.09509 −0.547543 0.836777i \(-0.684437\pi\)
−0.547543 + 0.836777i \(0.684437\pi\)
\(284\) 0 0
\(285\) −13.8167 −0.818428
\(286\) 0 0
\(287\) −27.1194 −1.60081
\(288\) 0 0
\(289\) −15.3028 −0.900163
\(290\) 0 0
\(291\) 31.1194 1.82425
\(292\) 0 0
\(293\) 18.0000 1.05157 0.525786 0.850617i \(-0.323771\pi\)
0.525786 + 0.850617i \(0.323771\pi\)
\(294\) 0 0
\(295\) −41.4500 −2.41331
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −8.21110 −0.474860
\(300\) 0 0
\(301\) 7.90833 0.455828
\(302\) 0 0
\(303\) −24.9083 −1.43095
\(304\) 0 0
\(305\) 44.4500 2.54520
\(306\) 0 0
\(307\) 20.9361 1.19489 0.597443 0.801912i \(-0.296184\pi\)
0.597443 + 0.801912i \(0.296184\pi\)
\(308\) 0 0
\(309\) 0.697224 0.0396637
\(310\) 0 0
\(311\) 8.21110 0.465609 0.232804 0.972524i \(-0.425210\pi\)
0.232804 + 0.972524i \(0.425210\pi\)
\(312\) 0 0
\(313\) −7.51388 −0.424710 −0.212355 0.977193i \(-0.568113\pi\)
−0.212355 + 0.977193i \(0.568113\pi\)
\(314\) 0 0
\(315\) 22.8167 1.28557
\(316\) 0 0
\(317\) −25.5416 −1.43456 −0.717281 0.696784i \(-0.754613\pi\)
−0.717281 + 0.696784i \(0.754613\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0.908327 0.0506979
\(322\) 0 0
\(323\) 2.60555 0.144977
\(324\) 0 0
\(325\) 25.2111 1.39846
\(326\) 0 0
\(327\) −4.60555 −0.254688
\(328\) 0 0
\(329\) −18.5139 −1.02070
\(330\) 0 0
\(331\) −30.8167 −1.69384 −0.846918 0.531723i \(-0.821545\pi\)
−0.846918 + 0.531723i \(0.821545\pi\)
\(332\) 0 0
\(333\) 17.5139 0.959755
\(334\) 0 0
\(335\) −18.6333 −1.01805
\(336\) 0 0
\(337\) 20.3944 1.11096 0.555478 0.831531i \(-0.312535\pi\)
0.555478 + 0.831531i \(0.312535\pi\)
\(338\) 0 0
\(339\) 44.7250 2.42913
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 10.2111 0.551348
\(344\) 0 0
\(345\) −9.00000 −0.484544
\(346\) 0 0
\(347\) −11.0917 −0.595432 −0.297716 0.954654i \(-0.596225\pi\)
−0.297716 + 0.954654i \(0.596225\pi\)
\(348\) 0 0
\(349\) 28.4500 1.52289 0.761446 0.648229i \(-0.224490\pi\)
0.761446 + 0.648229i \(0.224490\pi\)
\(350\) 0 0
\(351\) 10.1194 0.540135
\(352\) 0 0
\(353\) −25.8167 −1.37408 −0.687041 0.726619i \(-0.741091\pi\)
−0.687041 + 0.726619i \(0.741091\pi\)
\(354\) 0 0
\(355\) −20.7250 −1.09997
\(356\) 0 0
\(357\) −9.90833 −0.524404
\(358\) 0 0
\(359\) −22.8167 −1.20422 −0.602108 0.798414i \(-0.705673\pi\)
−0.602108 + 0.798414i \(0.705673\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) 25.3305 1.32951
\(364\) 0 0
\(365\) 41.7250 2.18399
\(366\) 0 0
\(367\) 2.81665 0.147028 0.0735141 0.997294i \(-0.476579\pi\)
0.0735141 + 0.997294i \(0.476579\pi\)
\(368\) 0 0
\(369\) 18.9083 0.984328
\(370\) 0 0
\(371\) −8.60555 −0.446778
\(372\) 0 0
\(373\) −34.5139 −1.78706 −0.893530 0.449003i \(-0.851779\pi\)
−0.893530 + 0.449003i \(0.851779\pi\)
\(374\) 0 0
\(375\) −6.90833 −0.356744
\(376\) 0 0
\(377\) 2.48612 0.128042
\(378\) 0 0
\(379\) −36.0278 −1.85062 −0.925311 0.379210i \(-0.876196\pi\)
−0.925311 + 0.379210i \(0.876196\pi\)
\(380\) 0 0
\(381\) −0.486122 −0.0249048
\(382\) 0 0
\(383\) −29.8444 −1.52498 −0.762489 0.647001i \(-0.776023\pi\)
−0.762489 + 0.647001i \(0.776023\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −5.51388 −0.280286
\(388\) 0 0
\(389\) 16.8167 0.852638 0.426319 0.904573i \(-0.359810\pi\)
0.426319 + 0.904573i \(0.359810\pi\)
\(390\) 0 0
\(391\) 1.69722 0.0858323
\(392\) 0 0
\(393\) 36.9083 1.86178
\(394\) 0 0
\(395\) 40.8167 2.05371
\(396\) 0 0
\(397\) −22.1194 −1.11014 −0.555071 0.831803i \(-0.687309\pi\)
−0.555071 + 0.831803i \(0.687309\pi\)
\(398\) 0 0
\(399\) −15.2111 −0.761508
\(400\) 0 0
\(401\) −25.3028 −1.26356 −0.631780 0.775148i \(-0.717675\pi\)
−0.631780 + 0.775148i \(0.717675\pi\)
\(402\) 0 0
\(403\) 3.81665 0.190121
\(404\) 0 0
\(405\) 31.8167 1.58098
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 29.9083 1.47887 0.739436 0.673227i \(-0.235092\pi\)
0.739436 + 0.673227i \(0.235092\pi\)
\(410\) 0 0
\(411\) 39.6333 1.95497
\(412\) 0 0
\(413\) −45.6333 −2.24547
\(414\) 0 0
\(415\) 16.8167 0.825497
\(416\) 0 0
\(417\) −11.3028 −0.553499
\(418\) 0 0
\(419\) −2.21110 −0.108019 −0.0540097 0.998540i \(-0.517200\pi\)
−0.0540097 + 0.998540i \(0.517200\pi\)
\(420\) 0 0
\(421\) 5.39445 0.262909 0.131455 0.991322i \(-0.458035\pi\)
0.131455 + 0.991322i \(0.458035\pi\)
\(422\) 0 0
\(423\) 12.9083 0.627624
\(424\) 0 0
\(425\) −5.21110 −0.252776
\(426\) 0 0
\(427\) 48.9361 2.36818
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 19.0278 0.916535 0.458267 0.888814i \(-0.348470\pi\)
0.458267 + 0.888814i \(0.348470\pi\)
\(432\) 0 0
\(433\) 10.3305 0.496454 0.248227 0.968702i \(-0.420152\pi\)
0.248227 + 0.968702i \(0.420152\pi\)
\(434\) 0 0
\(435\) 2.72498 0.130653
\(436\) 0 0
\(437\) 2.60555 0.124640
\(438\) 0 0
\(439\) 7.78890 0.371744 0.185872 0.982574i \(-0.440489\pi\)
0.185872 + 0.982574i \(0.440489\pi\)
\(440\) 0 0
\(441\) 9.00000 0.428571
\(442\) 0 0
\(443\) −32.7250 −1.55481 −0.777405 0.629000i \(-0.783465\pi\)
−0.777405 + 0.629000i \(0.783465\pi\)
\(444\) 0 0
\(445\) −23.4500 −1.11163
\(446\) 0 0
\(447\) 1.18335 0.0559704
\(448\) 0 0
\(449\) 40.0278 1.88903 0.944513 0.328473i \(-0.106534\pi\)
0.944513 + 0.328473i \(0.106534\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 3.42221 0.160789
\(454\) 0 0
\(455\) 62.4500 2.92770
\(456\) 0 0
\(457\) 32.1194 1.50248 0.751242 0.660027i \(-0.229455\pi\)
0.751242 + 0.660027i \(0.229455\pi\)
\(458\) 0 0
\(459\) −2.09167 −0.0976309
\(460\) 0 0
\(461\) −25.3028 −1.17847 −0.589234 0.807963i \(-0.700570\pi\)
−0.589234 + 0.807963i \(0.700570\pi\)
\(462\) 0 0
\(463\) −16.2111 −0.753394 −0.376697 0.926337i \(-0.622940\pi\)
−0.376697 + 0.926337i \(0.622940\pi\)
\(464\) 0 0
\(465\) 4.18335 0.193998
\(466\) 0 0
\(467\) −12.1194 −0.560820 −0.280410 0.959880i \(-0.590470\pi\)
−0.280410 + 0.959880i \(0.590470\pi\)
\(468\) 0 0
\(469\) −20.5139 −0.947243
\(470\) 0 0
\(471\) 47.9361 2.20878
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −8.00000 −0.367065
\(476\) 0 0
\(477\) 6.00000 0.274721
\(478\) 0 0
\(479\) 39.6333 1.81089 0.905446 0.424461i \(-0.139537\pi\)
0.905446 + 0.424461i \(0.139537\pi\)
\(480\) 0 0
\(481\) 47.9361 2.18570
\(482\) 0 0
\(483\) −9.90833 −0.450844
\(484\) 0 0
\(485\) 40.5416 1.84090
\(486\) 0 0
\(487\) −9.18335 −0.416137 −0.208069 0.978114i \(-0.566718\pi\)
−0.208069 + 0.978114i \(0.566718\pi\)
\(488\) 0 0
\(489\) −29.0278 −1.31268
\(490\) 0 0
\(491\) 42.6333 1.92401 0.962007 0.273024i \(-0.0880240\pi\)
0.962007 + 0.273024i \(0.0880240\pi\)
\(492\) 0 0
\(493\) −0.513878 −0.0231439
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −22.8167 −1.02347
\(498\) 0 0
\(499\) −40.3305 −1.80544 −0.902721 0.430226i \(-0.858434\pi\)
−0.902721 + 0.430226i \(0.858434\pi\)
\(500\) 0 0
\(501\) −2.30278 −0.102880
\(502\) 0 0
\(503\) 18.5139 0.825493 0.412747 0.910846i \(-0.364570\pi\)
0.412747 + 0.910846i \(0.364570\pi\)
\(504\) 0 0
\(505\) −32.4500 −1.44400
\(506\) 0 0
\(507\) −61.5416 −2.73316
\(508\) 0 0
\(509\) 14.0917 0.624602 0.312301 0.949983i \(-0.398900\pi\)
0.312301 + 0.949983i \(0.398900\pi\)
\(510\) 0 0
\(511\) 45.9361 2.03209
\(512\) 0 0
\(513\) −3.21110 −0.141774
\(514\) 0 0
\(515\) 0.908327 0.0400257
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −46.5416 −2.04295
\(520\) 0 0
\(521\) −22.1833 −0.971870 −0.485935 0.873995i \(-0.661521\pi\)
−0.485935 + 0.873995i \(0.661521\pi\)
\(522\) 0 0
\(523\) −12.8167 −0.560433 −0.280217 0.959937i \(-0.590406\pi\)
−0.280217 + 0.959937i \(0.590406\pi\)
\(524\) 0 0
\(525\) 30.4222 1.32773
\(526\) 0 0
\(527\) −0.788897 −0.0343649
\(528\) 0 0
\(529\) −21.3028 −0.926208
\(530\) 0 0
\(531\) 31.8167 1.38073
\(532\) 0 0
\(533\) 51.7527 2.24166
\(534\) 0 0
\(535\) 1.18335 0.0511605
\(536\) 0 0
\(537\) −22.8167 −0.984611
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −28.3944 −1.22077 −0.610386 0.792104i \(-0.708986\pi\)
−0.610386 + 0.792104i \(0.708986\pi\)
\(542\) 0 0
\(543\) −16.6056 −0.712612
\(544\) 0 0
\(545\) −6.00000 −0.257012
\(546\) 0 0
\(547\) −41.3583 −1.76835 −0.884176 0.467153i \(-0.845280\pi\)
−0.884176 + 0.467153i \(0.845280\pi\)
\(548\) 0 0
\(549\) −34.1194 −1.45618
\(550\) 0 0
\(551\) −0.788897 −0.0336082
\(552\) 0 0
\(553\) 44.9361 1.91088
\(554\) 0 0
\(555\) 52.5416 2.23027
\(556\) 0 0
\(557\) −10.8167 −0.458316 −0.229158 0.973389i \(-0.573597\pi\)
−0.229158 + 0.973389i \(0.573597\pi\)
\(558\) 0 0
\(559\) −15.0917 −0.638310
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −5.48612 −0.231212 −0.115606 0.993295i \(-0.536881\pi\)
−0.115606 + 0.993295i \(0.536881\pi\)
\(564\) 0 0
\(565\) 58.2666 2.45129
\(566\) 0 0
\(567\) 35.0278 1.47103
\(568\) 0 0
\(569\) 21.7889 0.913438 0.456719 0.889611i \(-0.349024\pi\)
0.456719 + 0.889611i \(0.349024\pi\)
\(570\) 0 0
\(571\) −39.5416 −1.65477 −0.827383 0.561638i \(-0.810171\pi\)
−0.827383 + 0.561638i \(0.810171\pi\)
\(572\) 0 0
\(573\) −17.7250 −0.740472
\(574\) 0 0
\(575\) −5.21110 −0.217318
\(576\) 0 0
\(577\) −6.48612 −0.270021 −0.135010 0.990844i \(-0.543107\pi\)
−0.135010 + 0.990844i \(0.543107\pi\)
\(578\) 0 0
\(579\) 11.3028 0.469727
\(580\) 0 0
\(581\) 18.5139 0.768085
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −43.5416 −1.80023
\(586\) 0 0
\(587\) 32.0917 1.32457 0.662283 0.749254i \(-0.269588\pi\)
0.662283 + 0.749254i \(0.269588\pi\)
\(588\) 0 0
\(589\) −1.21110 −0.0499026
\(590\) 0 0
\(591\) −16.8167 −0.691745
\(592\) 0 0
\(593\) −35.6056 −1.46214 −0.731072 0.682300i \(-0.760980\pi\)
−0.731072 + 0.682300i \(0.760980\pi\)
\(594\) 0 0
\(595\) −12.9083 −0.529190
\(596\) 0 0
\(597\) −32.9361 −1.34798
\(598\) 0 0
\(599\) 36.9083 1.50803 0.754017 0.656855i \(-0.228114\pi\)
0.754017 + 0.656855i \(0.228114\pi\)
\(600\) 0 0
\(601\) −29.5416 −1.20503 −0.602514 0.798108i \(-0.705834\pi\)
−0.602514 + 0.798108i \(0.705834\pi\)
\(602\) 0 0
\(603\) 14.3028 0.582454
\(604\) 0 0
\(605\) 33.0000 1.34164
\(606\) 0 0
\(607\) 18.7250 0.760024 0.380012 0.924982i \(-0.375920\pi\)
0.380012 + 0.924982i \(0.375920\pi\)
\(608\) 0 0
\(609\) 3.00000 0.121566
\(610\) 0 0
\(611\) 35.3305 1.42932
\(612\) 0 0
\(613\) 1.33053 0.0537397 0.0268698 0.999639i \(-0.491446\pi\)
0.0268698 + 0.999639i \(0.491446\pi\)
\(614\) 0 0
\(615\) 56.7250 2.28737
\(616\) 0 0
\(617\) 26.4500 1.06484 0.532418 0.846482i \(-0.321284\pi\)
0.532418 + 0.846482i \(0.321284\pi\)
\(618\) 0 0
\(619\) 11.0278 0.443243 0.221621 0.975133i \(-0.428865\pi\)
0.221621 + 0.975133i \(0.428865\pi\)
\(620\) 0 0
\(621\) −2.09167 −0.0839359
\(622\) 0 0
\(623\) −25.8167 −1.03432
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −9.90833 −0.395071
\(630\) 0 0
\(631\) −12.9361 −0.514977 −0.257489 0.966281i \(-0.582895\pi\)
−0.257489 + 0.966281i \(0.582895\pi\)
\(632\) 0 0
\(633\) −10.3944 −0.413142
\(634\) 0 0
\(635\) −0.633308 −0.0251320
\(636\) 0 0
\(637\) 24.6333 0.976007
\(638\) 0 0
\(639\) 15.9083 0.629324
\(640\) 0 0
\(641\) −7.69722 −0.304022 −0.152011 0.988379i \(-0.548575\pi\)
−0.152011 + 0.988379i \(0.548575\pi\)
\(642\) 0 0
\(643\) −5.51388 −0.217446 −0.108723 0.994072i \(-0.534676\pi\)
−0.108723 + 0.994072i \(0.534676\pi\)
\(644\) 0 0
\(645\) −16.5416 −0.651326
\(646\) 0 0
\(647\) −10.9361 −0.429942 −0.214971 0.976620i \(-0.568966\pi\)
−0.214971 + 0.976620i \(0.568966\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 4.60555 0.180506
\(652\) 0 0
\(653\) 12.1194 0.474270 0.237135 0.971477i \(-0.423792\pi\)
0.237135 + 0.971477i \(0.423792\pi\)
\(654\) 0 0
\(655\) 48.0833 1.87877
\(656\) 0 0
\(657\) −32.0278 −1.24952
\(658\) 0 0
\(659\) −1.81665 −0.0707668 −0.0353834 0.999374i \(-0.511265\pi\)
−0.0353834 + 0.999374i \(0.511265\pi\)
\(660\) 0 0
\(661\) −12.8806 −0.500996 −0.250498 0.968117i \(-0.580594\pi\)
−0.250498 + 0.968117i \(0.580594\pi\)
\(662\) 0 0
\(663\) 18.9083 0.734339
\(664\) 0 0
\(665\) −19.8167 −0.768457
\(666\) 0 0
\(667\) −0.513878 −0.0198974
\(668\) 0 0
\(669\) 40.3305 1.55927
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −1.63331 −0.0629594 −0.0314797 0.999504i \(-0.510022\pi\)
−0.0314797 + 0.999504i \(0.510022\pi\)
\(674\) 0 0
\(675\) 6.42221 0.247191
\(676\) 0 0
\(677\) 7.81665 0.300418 0.150209 0.988654i \(-0.452005\pi\)
0.150209 + 0.988654i \(0.452005\pi\)
\(678\) 0 0
\(679\) 44.6333 1.71287
\(680\) 0 0
\(681\) 52.5416 2.01340
\(682\) 0 0
\(683\) −4.54163 −0.173781 −0.0868904 0.996218i \(-0.527693\pi\)
−0.0868904 + 0.996218i \(0.527693\pi\)
\(684\) 0 0
\(685\) 51.6333 1.97281
\(686\) 0 0
\(687\) −28.3305 −1.08088
\(688\) 0 0
\(689\) 16.4222 0.625636
\(690\) 0 0
\(691\) −22.4500 −0.854037 −0.427018 0.904243i \(-0.640436\pi\)
−0.427018 + 0.904243i \(0.640436\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −14.7250 −0.558550
\(696\) 0 0
\(697\) −10.6972 −0.405186
\(698\) 0 0
\(699\) 9.00000 0.340411
\(700\) 0 0
\(701\) −28.0278 −1.05859 −0.529297 0.848437i \(-0.677544\pi\)
−0.529297 + 0.848437i \(0.677544\pi\)
\(702\) 0 0
\(703\) −15.2111 −0.573698
\(704\) 0 0
\(705\) 38.7250 1.45847
\(706\) 0 0
\(707\) −35.7250 −1.34358
\(708\) 0 0
\(709\) −11.1472 −0.418641 −0.209321 0.977847i \(-0.567125\pi\)
−0.209321 + 0.977847i \(0.567125\pi\)
\(710\) 0 0
\(711\) −31.3305 −1.17499
\(712\) 0 0
\(713\) −0.788897 −0.0295444
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 50.4500 1.88409
\(718\) 0 0
\(719\) 48.1194 1.79455 0.897276 0.441470i \(-0.145543\pi\)
0.897276 + 0.441470i \(0.145543\pi\)
\(720\) 0 0
\(721\) 1.00000 0.0372419
\(722\) 0 0
\(723\) −14.5139 −0.539777
\(724\) 0 0
\(725\) 1.57779 0.0585978
\(726\) 0 0
\(727\) 3.72498 0.138152 0.0690759 0.997611i \(-0.477995\pi\)
0.0690759 + 0.997611i \(0.477995\pi\)
\(728\) 0 0
\(729\) −13.3305 −0.493723
\(730\) 0 0
\(731\) 3.11943 0.115376
\(732\) 0 0
\(733\) 14.0000 0.517102 0.258551 0.965998i \(-0.416755\pi\)
0.258551 + 0.965998i \(0.416755\pi\)
\(734\) 0 0
\(735\) 27.0000 0.995910
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −42.9361 −1.57943 −0.789715 0.613474i \(-0.789771\pi\)
−0.789715 + 0.613474i \(0.789771\pi\)
\(740\) 0 0
\(741\) 29.0278 1.06636
\(742\) 0 0
\(743\) −26.8806 −0.986152 −0.493076 0.869986i \(-0.664128\pi\)
−0.493076 + 0.869986i \(0.664128\pi\)
\(744\) 0 0
\(745\) 1.54163 0.0564811
\(746\) 0 0
\(747\) −12.9083 −0.472291
\(748\) 0 0
\(749\) 1.30278 0.0476024
\(750\) 0 0
\(751\) 16.3944 0.598242 0.299121 0.954215i \(-0.403307\pi\)
0.299121 + 0.954215i \(0.403307\pi\)
\(752\) 0 0
\(753\) −40.5416 −1.47742
\(754\) 0 0
\(755\) 4.45837 0.162257
\(756\) 0 0
\(757\) −9.72498 −0.353460 −0.176730 0.984259i \(-0.556552\pi\)
−0.176730 + 0.984259i \(0.556552\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −7.30278 −0.264725 −0.132363 0.991201i \(-0.542256\pi\)
−0.132363 + 0.991201i \(0.542256\pi\)
\(762\) 0 0
\(763\) −6.60555 −0.239137
\(764\) 0 0
\(765\) 9.00000 0.325396
\(766\) 0 0
\(767\) 87.0833 3.14439
\(768\) 0 0
\(769\) 6.81665 0.245815 0.122907 0.992418i \(-0.460778\pi\)
0.122907 + 0.992418i \(0.460778\pi\)
\(770\) 0 0
\(771\) 14.0917 0.507499
\(772\) 0 0
\(773\) 3.39445 0.122090 0.0610449 0.998135i \(-0.480557\pi\)
0.0610449 + 0.998135i \(0.480557\pi\)
\(774\) 0 0
\(775\) 2.42221 0.0870082
\(776\) 0 0
\(777\) 57.8444 2.07516
\(778\) 0 0
\(779\) −16.4222 −0.588387
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0.633308 0.0226326
\(784\) 0 0
\(785\) 62.4500 2.22893
\(786\) 0 0
\(787\) 21.4500 0.764609 0.382304 0.924036i \(-0.375131\pi\)
0.382304 + 0.924036i \(0.375131\pi\)
\(788\) 0 0
\(789\) −5.09167 −0.181268
\(790\) 0 0
\(791\) 64.1472 2.28081
\(792\) 0 0
\(793\) −93.3860 −3.31624
\(794\) 0 0
\(795\) 18.0000 0.638394
\(796\) 0 0
\(797\) −6.63331 −0.234964 −0.117482 0.993075i \(-0.537482\pi\)
−0.117482 + 0.993075i \(0.537482\pi\)
\(798\) 0 0
\(799\) −7.30278 −0.258354
\(800\) 0 0
\(801\) 18.0000 0.635999
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) −12.9083 −0.454959
\(806\) 0 0
\(807\) −3.90833 −0.137580
\(808\) 0 0
\(809\) −8.72498 −0.306754 −0.153377 0.988168i \(-0.549015\pi\)
−0.153377 + 0.988168i \(0.549015\pi\)
\(810\) 0 0
\(811\) −19.8444 −0.696831 −0.348416 0.937340i \(-0.613280\pi\)
−0.348416 + 0.937340i \(0.613280\pi\)
\(812\) 0 0
\(813\) −33.2111 −1.16476
\(814\) 0 0
\(815\) −37.8167 −1.32466
\(816\) 0 0
\(817\) 4.78890 0.167542
\(818\) 0 0
\(819\) −47.9361 −1.67502
\(820\) 0 0
\(821\) −17.7250 −0.618606 −0.309303 0.950964i \(-0.600096\pi\)
−0.309303 + 0.950964i \(0.600096\pi\)
\(822\) 0 0
\(823\) 17.6972 0.616886 0.308443 0.951243i \(-0.400192\pi\)
0.308443 + 0.951243i \(0.400192\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −55.5416 −1.93137 −0.965686 0.259713i \(-0.916372\pi\)
−0.965686 + 0.259713i \(0.916372\pi\)
\(828\) 0 0
\(829\) −55.8722 −1.94052 −0.970260 0.242064i \(-0.922176\pi\)
−0.970260 + 0.242064i \(0.922176\pi\)
\(830\) 0 0
\(831\) −17.2389 −0.598009
\(832\) 0 0
\(833\) −5.09167 −0.176416
\(834\) 0 0
\(835\) −3.00000 −0.103819
\(836\) 0 0
\(837\) 0.972244 0.0336057
\(838\) 0 0
\(839\) −47.6056 −1.64353 −0.821763 0.569829i \(-0.807009\pi\)
−0.821763 + 0.569829i \(0.807009\pi\)
\(840\) 0 0
\(841\) −28.8444 −0.994635
\(842\) 0 0
\(843\) −30.9083 −1.06454
\(844\) 0 0
\(845\) −80.1749 −2.75810
\(846\) 0 0
\(847\) 36.3305 1.24833
\(848\) 0 0
\(849\) 42.4222 1.45593
\(850\) 0 0
\(851\) −9.90833 −0.339653
\(852\) 0 0
\(853\) 8.27502 0.283331 0.141666 0.989915i \(-0.454754\pi\)
0.141666 + 0.989915i \(0.454754\pi\)
\(854\) 0 0
\(855\) 13.8167 0.472520
\(856\) 0 0
\(857\) −28.5416 −0.974964 −0.487482 0.873133i \(-0.662084\pi\)
−0.487482 + 0.873133i \(0.662084\pi\)
\(858\) 0 0
\(859\) 48.6056 1.65840 0.829200 0.558952i \(-0.188796\pi\)
0.829200 + 0.558952i \(0.188796\pi\)
\(860\) 0 0
\(861\) 62.4500 2.12829
\(862\) 0 0
\(863\) −6.51388 −0.221735 −0.110867 0.993835i \(-0.535363\pi\)
−0.110867 + 0.993835i \(0.535363\pi\)
\(864\) 0 0
\(865\) −60.6333 −2.06159
\(866\) 0 0
\(867\) 35.2389 1.19677
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 39.1472 1.32645
\(872\) 0 0
\(873\) −31.1194 −1.05323
\(874\) 0 0
\(875\) −9.90833 −0.334963
\(876\) 0 0
\(877\) 1.48612 0.0501828 0.0250914 0.999685i \(-0.492012\pi\)
0.0250914 + 0.999685i \(0.492012\pi\)
\(878\) 0 0
\(879\) −41.4500 −1.39807
\(880\) 0 0
\(881\) −29.2111 −0.984147 −0.492074 0.870554i \(-0.663761\pi\)
−0.492074 + 0.870554i \(0.663761\pi\)
\(882\) 0 0
\(883\) −18.9361 −0.637250 −0.318625 0.947881i \(-0.603221\pi\)
−0.318625 + 0.947881i \(0.603221\pi\)
\(884\) 0 0
\(885\) 95.4500 3.20852
\(886\) 0 0
\(887\) 10.8167 0.363188 0.181594 0.983374i \(-0.441874\pi\)
0.181594 + 0.983374i \(0.441874\pi\)
\(888\) 0 0
\(889\) −0.697224 −0.0233842
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −11.2111 −0.375165
\(894\) 0 0
\(895\) −29.7250 −0.993597
\(896\) 0 0
\(897\) 18.9083 0.631331
\(898\) 0 0
\(899\) 0.238859 0.00796639
\(900\) 0 0
\(901\) −3.39445 −0.113085
\(902\) 0 0
\(903\) −18.2111 −0.606028
\(904\) 0 0
\(905\) −21.6333 −0.719115
\(906\) 0 0
\(907\) −38.6333 −1.28280 −0.641399 0.767208i \(-0.721646\pi\)
−0.641399 + 0.767208i \(0.721646\pi\)
\(908\) 0 0
\(909\) 24.9083 0.826157
\(910\) 0 0
\(911\) −28.8167 −0.954738 −0.477369 0.878703i \(-0.658410\pi\)
−0.477369 + 0.878703i \(0.658410\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) −102.358 −3.38386
\(916\) 0 0
\(917\) 52.9361 1.74810
\(918\) 0 0
\(919\) 10.9083 0.359833 0.179916 0.983682i \(-0.442417\pi\)
0.179916 + 0.983682i \(0.442417\pi\)
\(920\) 0 0
\(921\) −48.2111 −1.58861
\(922\) 0 0
\(923\) 43.5416 1.43319
\(924\) 0 0
\(925\) 30.4222 1.00028
\(926\) 0 0
\(927\) −0.697224 −0.0228999
\(928\) 0 0
\(929\) 33.5139 1.09955 0.549777 0.835311i \(-0.314713\pi\)
0.549777 + 0.835311i \(0.314713\pi\)
\(930\) 0 0
\(931\) −7.81665 −0.256180
\(932\) 0 0
\(933\) −18.9083 −0.619031
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 13.4500 0.439391 0.219696 0.975568i \(-0.429494\pi\)
0.219696 + 0.975568i \(0.429494\pi\)
\(938\) 0 0
\(939\) 17.3028 0.564655
\(940\) 0 0
\(941\) 8.76114 0.285605 0.142803 0.989751i \(-0.454389\pi\)
0.142803 + 0.989751i \(0.454389\pi\)
\(942\) 0 0
\(943\) −10.6972 −0.348350
\(944\) 0 0
\(945\) 15.9083 0.517498
\(946\) 0 0
\(947\) −24.9083 −0.809412 −0.404706 0.914447i \(-0.632626\pi\)
−0.404706 + 0.914447i \(0.632626\pi\)
\(948\) 0 0
\(949\) −87.6611 −2.84560
\(950\) 0 0
\(951\) 58.8167 1.90726
\(952\) 0 0
\(953\) −33.3583 −1.08058 −0.540290 0.841479i \(-0.681686\pi\)
−0.540290 + 0.841479i \(0.681686\pi\)
\(954\) 0 0
\(955\) −23.0917 −0.747229
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 56.8444 1.83560
\(960\) 0 0
\(961\) −30.6333 −0.988171
\(962\) 0 0
\(963\) −0.908327 −0.0292704
\(964\) 0 0
\(965\) 14.7250 0.474014
\(966\) 0 0
\(967\) 43.5139 1.39931 0.699656 0.714480i \(-0.253337\pi\)
0.699656 + 0.714480i \(0.253337\pi\)
\(968\) 0 0
\(969\) −6.00000 −0.192748
\(970\) 0 0
\(971\) 51.4777 1.65200 0.825999 0.563671i \(-0.190611\pi\)
0.825999 + 0.563671i \(0.190611\pi\)
\(972\) 0 0
\(973\) −16.2111 −0.519704
\(974\) 0 0
\(975\) −58.0555 −1.85926
\(976\) 0 0
\(977\) 47.5694 1.52188 0.760940 0.648822i \(-0.224738\pi\)
0.760940 + 0.648822i \(0.224738\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 4.60555 0.147044
\(982\) 0 0
\(983\) −55.2666 −1.76273 −0.881366 0.472435i \(-0.843375\pi\)
−0.881366 + 0.472435i \(0.843375\pi\)
\(984\) 0 0
\(985\) −21.9083 −0.698057
\(986\) 0 0
\(987\) 42.6333 1.35703
\(988\) 0 0
\(989\) 3.11943 0.0991921
\(990\) 0 0
\(991\) −23.6333 −0.750737 −0.375368 0.926876i \(-0.622484\pi\)
−0.375368 + 0.926876i \(0.622484\pi\)
\(992\) 0 0
\(993\) 70.9638 2.25197
\(994\) 0 0
\(995\) −42.9083 −1.36029
\(996\) 0 0
\(997\) −39.2111 −1.24183 −0.620914 0.783879i \(-0.713238\pi\)
−0.620914 + 0.783879i \(0.713238\pi\)
\(998\) 0 0
\(999\) 12.2111 0.386342
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 2672.2.a.c.1.1 2
4.3 odd 2 668.2.a.a.1.2 2
12.11 even 2 6012.2.a.a.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
668.2.a.a.1.2 2 4.3 odd 2
2672.2.a.c.1.1 2 1.1 even 1 trivial
6012.2.a.a.1.2 2 12.11 even 2