Properties

Label 1840.2.a.r.1.3
Level $1840$
Weight $2$
Character 1840.1
Self dual yes
Analytic conductor $14.692$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1840.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.6924739719\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.1101.1
Defining polynomial: \(x^{3} - x^{2} - 9 x + 12\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 230)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-3.11903\) of defining polynomial
Character \(\chi\) \(=\) 1840.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.11903 q^{3} -1.00000 q^{5} -4.50973 q^{7} +6.72833 q^{9} +O(q^{10})\) \(q+3.11903 q^{3} -1.00000 q^{5} -4.50973 q^{7} +6.72833 q^{9} -4.33763 q^{11} -3.72833 q^{13} -3.11903 q^{15} +1.11903 q^{17} -4.50973 q^{19} -14.0660 q^{21} +1.00000 q^{23} +1.00000 q^{25} +11.6288 q^{27} -8.23805 q^{29} -1.72833 q^{31} -13.5292 q^{33} +4.50973 q^{35} -0.781399 q^{37} -11.6288 q^{39} +3.90043 q^{41} -8.00000 q^{43} -6.72833 q^{45} +11.4567 q^{47} +13.3376 q^{49} +3.49027 q^{51} -6.00000 q^{53} +4.33763 q^{55} -14.0660 q^{57} +2.23805 q^{59} +3.55623 q^{61} -30.3429 q^{63} +3.72833 q^{65} -2.43720 q^{67} +3.11903 q^{69} -7.11903 q^{71} -9.45665 q^{73} +3.11903 q^{75} +19.5615 q^{77} +14.9133 q^{79} +16.0854 q^{81} -2.78140 q^{83} -1.11903 q^{85} -25.6947 q^{87} -7.69471 q^{89} +16.8137 q^{91} -5.39070 q^{93} +4.50973 q^{95} -0.642920 q^{97} -29.1850 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - q^{3} - 3q^{5} - 3q^{7} + 10q^{9} + O(q^{10}) \) \( 3q - q^{3} - 3q^{5} - 3q^{7} + 10q^{9} - 3q^{11} - q^{13} + q^{15} - 7q^{17} - 3q^{19} - 22q^{21} + 3q^{23} + 3q^{25} + 14q^{27} - 4q^{29} + 5q^{31} - 9q^{33} + 3q^{35} - 2q^{37} - 14q^{39} + q^{41} - 24q^{43} - 10q^{45} + 14q^{47} + 30q^{49} + 21q^{51} - 18q^{53} + 3q^{55} - 22q^{57} - 14q^{59} + q^{61} - 8q^{63} + q^{65} - 8q^{67} - q^{69} - 11q^{71} - 8q^{73} - q^{75} - 24q^{77} + 4q^{79} + 7q^{81} - 8q^{83} + 7q^{85} - 36q^{87} + 18q^{89} - q^{91} - 16q^{93} + 3q^{95} - 33q^{97} - 57q^{99} + O(q^{100}) \)

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
\(3\) 3.11903 1.80077 0.900385 0.435093i \(-0.143285\pi\)
0.900385 + 0.435093i \(0.143285\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −4.50973 −1.70452 −0.852258 0.523122i \(-0.824767\pi\)
−0.852258 + 0.523122i \(0.824767\pi\)
\(8\) 0 0
\(9\) 6.72833 2.24278
\(10\) 0 0
\(11\) −4.33763 −1.30784 −0.653922 0.756562i \(-0.726878\pi\)
−0.653922 + 0.756562i \(0.726878\pi\)
\(12\) 0 0
\(13\) −3.72833 −1.03405 −0.517026 0.855970i \(-0.672961\pi\)
−0.517026 + 0.855970i \(0.672961\pi\)
\(14\) 0 0
\(15\) −3.11903 −0.805329
\(16\) 0 0
\(17\) 1.11903 0.271404 0.135702 0.990750i \(-0.456671\pi\)
0.135702 + 0.990750i \(0.456671\pi\)
\(18\) 0 0
\(19\) −4.50973 −1.03460 −0.517301 0.855803i \(-0.673063\pi\)
−0.517301 + 0.855803i \(0.673063\pi\)
\(20\) 0 0
\(21\) −14.0660 −3.06944
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 11.6288 2.23795
\(28\) 0 0
\(29\) −8.23805 −1.52977 −0.764884 0.644168i \(-0.777204\pi\)
−0.764884 + 0.644168i \(0.777204\pi\)
\(30\) 0 0
\(31\) −1.72833 −0.310417 −0.155208 0.987882i \(-0.549605\pi\)
−0.155208 + 0.987882i \(0.549605\pi\)
\(32\) 0 0
\(33\) −13.5292 −2.35513
\(34\) 0 0
\(35\) 4.50973 0.762283
\(36\) 0 0
\(37\) −0.781399 −0.128461 −0.0642306 0.997935i \(-0.520459\pi\)
−0.0642306 + 0.997935i \(0.520459\pi\)
\(38\) 0 0
\(39\) −11.6288 −1.86209
\(40\) 0 0
\(41\) 3.90043 0.609144 0.304572 0.952489i \(-0.401487\pi\)
0.304572 + 0.952489i \(0.401487\pi\)
\(42\) 0 0
\(43\) −8.00000 −1.21999 −0.609994 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 0 0
\(45\) −6.72833 −1.00300
\(46\) 0 0
\(47\) 11.4567 1.67112 0.835562 0.549396i \(-0.185142\pi\)
0.835562 + 0.549396i \(0.185142\pi\)
\(48\) 0 0
\(49\) 13.3376 1.90538
\(50\) 0 0
\(51\) 3.49027 0.488736
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 4.33763 0.584886
\(56\) 0 0
\(57\) −14.0660 −1.86308
\(58\) 0 0
\(59\) 2.23805 0.291370 0.145685 0.989331i \(-0.453461\pi\)
0.145685 + 0.989331i \(0.453461\pi\)
\(60\) 0 0
\(61\) 3.55623 0.455329 0.227664 0.973740i \(-0.426891\pi\)
0.227664 + 0.973740i \(0.426891\pi\)
\(62\) 0 0
\(63\) −30.3429 −3.82285
\(64\) 0 0
\(65\) 3.72833 0.462442
\(66\) 0 0
\(67\) −2.43720 −0.297752 −0.148876 0.988856i \(-0.547565\pi\)
−0.148876 + 0.988856i \(0.547565\pi\)
\(68\) 0 0
\(69\) 3.11903 0.375487
\(70\) 0 0
\(71\) −7.11903 −0.844873 −0.422437 0.906393i \(-0.638825\pi\)
−0.422437 + 0.906393i \(0.638825\pi\)
\(72\) 0 0
\(73\) −9.45665 −1.10682 −0.553409 0.832910i \(-0.686673\pi\)
−0.553409 + 0.832910i \(0.686673\pi\)
\(74\) 0 0
\(75\) 3.11903 0.360154
\(76\) 0 0
\(77\) 19.5615 2.22924
\(78\) 0 0
\(79\) 14.9133 1.67788 0.838939 0.544225i \(-0.183176\pi\)
0.838939 + 0.544225i \(0.183176\pi\)
\(80\) 0 0
\(81\) 16.0854 1.78727
\(82\) 0 0
\(83\) −2.78140 −0.305298 −0.152649 0.988280i \(-0.548780\pi\)
−0.152649 + 0.988280i \(0.548780\pi\)
\(84\) 0 0
\(85\) −1.11903 −0.121375
\(86\) 0 0
\(87\) −25.6947 −2.75476
\(88\) 0 0
\(89\) −7.69471 −0.815637 −0.407819 0.913063i \(-0.633710\pi\)
−0.407819 + 0.913063i \(0.633710\pi\)
\(90\) 0 0
\(91\) 16.8137 1.76256
\(92\) 0 0
\(93\) −5.39070 −0.558989
\(94\) 0 0
\(95\) 4.50973 0.462688
\(96\) 0 0
\(97\) −0.642920 −0.0652786 −0.0326393 0.999467i \(-0.510391\pi\)
−0.0326393 + 0.999467i \(0.510391\pi\)
\(98\) 0 0
\(99\) −29.1850 −2.93320
\(100\) 0 0
\(101\) −8.23805 −0.819717 −0.409858 0.912149i \(-0.634422\pi\)
−0.409858 + 0.912149i \(0.634422\pi\)
\(102\) 0 0
\(103\) −12.3376 −1.21566 −0.607831 0.794066i \(-0.707960\pi\)
−0.607831 + 0.794066i \(0.707960\pi\)
\(104\) 0 0
\(105\) 14.0660 1.37270
\(106\) 0 0
\(107\) 15.9328 1.54028 0.770139 0.637876i \(-0.220187\pi\)
0.770139 + 0.637876i \(0.220187\pi\)
\(108\) 0 0
\(109\) −1.49027 −0.142742 −0.0713712 0.997450i \(-0.522737\pi\)
−0.0713712 + 0.997450i \(0.522737\pi\)
\(110\) 0 0
\(111\) −2.43720 −0.231329
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) −1.00000 −0.0932505
\(116\) 0 0
\(117\) −25.0854 −2.31915
\(118\) 0 0
\(119\) −5.04650 −0.462612
\(120\) 0 0
\(121\) 7.81502 0.710456
\(122\) 0 0
\(123\) 12.1655 1.09693
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 0.675256 0.0599193 0.0299597 0.999551i \(-0.490462\pi\)
0.0299597 + 0.999551i \(0.490462\pi\)
\(128\) 0 0
\(129\) −24.9522 −2.19692
\(130\) 0 0
\(131\) 13.6947 1.19651 0.598256 0.801305i \(-0.295861\pi\)
0.598256 + 0.801305i \(0.295861\pi\)
\(132\) 0 0
\(133\) 20.3376 1.76350
\(134\) 0 0
\(135\) −11.6288 −1.00084
\(136\) 0 0
\(137\) 7.52918 0.643261 0.321631 0.946865i \(-0.395769\pi\)
0.321631 + 0.946865i \(0.395769\pi\)
\(138\) 0 0
\(139\) −4.67526 −0.396550 −0.198275 0.980146i \(-0.563534\pi\)
−0.198275 + 0.980146i \(0.563534\pi\)
\(140\) 0 0
\(141\) 35.7336 3.00931
\(142\) 0 0
\(143\) 16.1721 1.35238
\(144\) 0 0
\(145\) 8.23805 0.684133
\(146\) 0 0
\(147\) 41.6004 3.43114
\(148\) 0 0
\(149\) 7.52918 0.616814 0.308407 0.951254i \(-0.400204\pi\)
0.308407 + 0.951254i \(0.400204\pi\)
\(150\) 0 0
\(151\) 13.3571 1.08698 0.543492 0.839414i \(-0.317102\pi\)
0.543492 + 0.839414i \(0.317102\pi\)
\(152\) 0 0
\(153\) 7.52918 0.608698
\(154\) 0 0
\(155\) 1.72833 0.138823
\(156\) 0 0
\(157\) 16.2381 1.29594 0.647969 0.761667i \(-0.275619\pi\)
0.647969 + 0.761667i \(0.275619\pi\)
\(158\) 0 0
\(159\) −18.7142 −1.48413
\(160\) 0 0
\(161\) −4.50973 −0.355416
\(162\) 0 0
\(163\) 3.29112 0.257781 0.128890 0.991659i \(-0.458858\pi\)
0.128890 + 0.991659i \(0.458858\pi\)
\(164\) 0 0
\(165\) 13.5292 1.05325
\(166\) 0 0
\(167\) −22.9133 −1.77309 −0.886543 0.462647i \(-0.846900\pi\)
−0.886543 + 0.462647i \(0.846900\pi\)
\(168\) 0 0
\(169\) 0.900425 0.0692635
\(170\) 0 0
\(171\) −30.3429 −2.32038
\(172\) 0 0
\(173\) 0.575681 0.0437683 0.0218841 0.999761i \(-0.493034\pi\)
0.0218841 + 0.999761i \(0.493034\pi\)
\(174\) 0 0
\(175\) −4.50973 −0.340903
\(176\) 0 0
\(177\) 6.98055 0.524690
\(178\) 0 0
\(179\) −5.01945 −0.375171 −0.187586 0.982248i \(-0.560066\pi\)
−0.187586 + 0.982248i \(0.560066\pi\)
\(180\) 0 0
\(181\) −11.5292 −0.856957 −0.428479 0.903552i \(-0.640950\pi\)
−0.428479 + 0.903552i \(0.640950\pi\)
\(182\) 0 0
\(183\) 11.0920 0.819942
\(184\) 0 0
\(185\) 0.781399 0.0574496
\(186\) 0 0
\(187\) −4.85392 −0.354954
\(188\) 0 0
\(189\) −52.4425 −3.81463
\(190\) 0 0
\(191\) 18.7142 1.35411 0.677055 0.735933i \(-0.263256\pi\)
0.677055 + 0.735933i \(0.263256\pi\)
\(192\) 0 0
\(193\) 23.4956 1.69125 0.845624 0.533780i \(-0.179229\pi\)
0.845624 + 0.533780i \(0.179229\pi\)
\(194\) 0 0
\(195\) 11.6288 0.832752
\(196\) 0 0
\(197\) −18.1385 −1.29231 −0.646157 0.763205i \(-0.723625\pi\)
−0.646157 + 0.763205i \(0.723625\pi\)
\(198\) 0 0
\(199\) 23.2575 1.64868 0.824340 0.566094i \(-0.191546\pi\)
0.824340 + 0.566094i \(0.191546\pi\)
\(200\) 0 0
\(201\) −7.60170 −0.536183
\(202\) 0 0
\(203\) 37.1514 2.60751
\(204\) 0 0
\(205\) −3.90043 −0.272418
\(206\) 0 0
\(207\) 6.72833 0.467651
\(208\) 0 0
\(209\) 19.5615 1.35310
\(210\) 0 0
\(211\) −4.34420 −0.299067 −0.149533 0.988757i \(-0.547777\pi\)
−0.149533 + 0.988757i \(0.547777\pi\)
\(212\) 0 0
\(213\) −22.2044 −1.52142
\(214\) 0 0
\(215\) 8.00000 0.545595
\(216\) 0 0
\(217\) 7.79428 0.529110
\(218\) 0 0
\(219\) −29.4956 −1.99313
\(220\) 0 0
\(221\) −4.17210 −0.280646
\(222\) 0 0
\(223\) −12.4761 −0.835462 −0.417731 0.908571i \(-0.637175\pi\)
−0.417731 + 0.908571i \(0.637175\pi\)
\(224\) 0 0
\(225\) 6.72833 0.448555
\(226\) 0 0
\(227\) −15.9328 −1.05749 −0.528747 0.848779i \(-0.677338\pi\)
−0.528747 + 0.848779i \(0.677338\pi\)
\(228\) 0 0
\(229\) −3.56280 −0.235436 −0.117718 0.993047i \(-0.537558\pi\)
−0.117718 + 0.993047i \(0.537558\pi\)
\(230\) 0 0
\(231\) 61.0129 4.01435
\(232\) 0 0
\(233\) −27.4956 −1.80129 −0.900647 0.434552i \(-0.856907\pi\)
−0.900647 + 0.434552i \(0.856907\pi\)
\(234\) 0 0
\(235\) −11.4567 −0.747350
\(236\) 0 0
\(237\) 46.5150 3.02147
\(238\) 0 0
\(239\) −10.0389 −0.649363 −0.324681 0.945823i \(-0.605257\pi\)
−0.324681 + 0.945823i \(0.605257\pi\)
\(240\) 0 0
\(241\) −23.6947 −1.52631 −0.763155 0.646215i \(-0.776351\pi\)
−0.763155 + 0.646215i \(0.776351\pi\)
\(242\) 0 0
\(243\) 15.2846 0.980505
\(244\) 0 0
\(245\) −13.3376 −0.852110
\(246\) 0 0
\(247\) 16.8137 1.06983
\(248\) 0 0
\(249\) −8.67526 −0.549772
\(250\) 0 0
\(251\) −12.4425 −0.785363 −0.392681 0.919675i \(-0.628452\pi\)
−0.392681 + 0.919675i \(0.628452\pi\)
\(252\) 0 0
\(253\) −4.33763 −0.272704
\(254\) 0 0
\(255\) −3.49027 −0.218569
\(256\) 0 0
\(257\) 5.45665 0.340377 0.170188 0.985412i \(-0.445562\pi\)
0.170188 + 0.985412i \(0.445562\pi\)
\(258\) 0 0
\(259\) 3.52389 0.218964
\(260\) 0 0
\(261\) −55.4283 −3.43093
\(262\) 0 0
\(263\) −0.138479 −0.00853895 −0.00426948 0.999991i \(-0.501359\pi\)
−0.00426948 + 0.999991i \(0.501359\pi\)
\(264\) 0 0
\(265\) 6.00000 0.368577
\(266\) 0 0
\(267\) −24.0000 −1.46878
\(268\) 0 0
\(269\) 14.6753 0.894766 0.447383 0.894342i \(-0.352356\pi\)
0.447383 + 0.894342i \(0.352356\pi\)
\(270\) 0 0
\(271\) 8.31058 0.504832 0.252416 0.967619i \(-0.418775\pi\)
0.252416 + 0.967619i \(0.418775\pi\)
\(272\) 0 0
\(273\) 52.4425 3.17396
\(274\) 0 0
\(275\) −4.33763 −0.261569
\(276\) 0 0
\(277\) 12.9133 0.775886 0.387943 0.921683i \(-0.373186\pi\)
0.387943 + 0.921683i \(0.373186\pi\)
\(278\) 0 0
\(279\) −11.6288 −0.696195
\(280\) 0 0
\(281\) 2.67526 0.159592 0.0797962 0.996811i \(-0.474573\pi\)
0.0797962 + 0.996811i \(0.474573\pi\)
\(282\) 0 0
\(283\) −0.742495 −0.0441367 −0.0220684 0.999756i \(-0.507025\pi\)
−0.0220684 + 0.999756i \(0.507025\pi\)
\(284\) 0 0
\(285\) 14.0660 0.833195
\(286\) 0 0
\(287\) −17.5898 −1.03830
\(288\) 0 0
\(289\) −15.7478 −0.926340
\(290\) 0 0
\(291\) −2.00528 −0.117552
\(292\) 0 0
\(293\) −6.00000 −0.350524 −0.175262 0.984522i \(-0.556077\pi\)
−0.175262 + 0.984522i \(0.556077\pi\)
\(294\) 0 0
\(295\) −2.23805 −0.130305
\(296\) 0 0
\(297\) −50.4412 −2.92690
\(298\) 0 0
\(299\) −3.72833 −0.215615
\(300\) 0 0
\(301\) 36.0778 2.07949
\(302\) 0 0
\(303\) −25.6947 −1.47612
\(304\) 0 0
\(305\) −3.55623 −0.203629
\(306\) 0 0
\(307\) −30.5084 −1.74121 −0.870604 0.491984i \(-0.836272\pi\)
−0.870604 + 0.491984i \(0.836272\pi\)
\(308\) 0 0
\(309\) −38.4814 −2.18913
\(310\) 0 0
\(311\) −5.56280 −0.315437 −0.157719 0.987484i \(-0.550414\pi\)
−0.157719 + 0.987484i \(0.550414\pi\)
\(312\) 0 0
\(313\) 4.07252 0.230193 0.115096 0.993354i \(-0.463282\pi\)
0.115096 + 0.993354i \(0.463282\pi\)
\(314\) 0 0
\(315\) 30.3429 1.70963
\(316\) 0 0
\(317\) −6.16553 −0.346291 −0.173145 0.984896i \(-0.555393\pi\)
−0.173145 + 0.984896i \(0.555393\pi\)
\(318\) 0 0
\(319\) 35.7336 2.00070
\(320\) 0 0
\(321\) 49.6947 2.77369
\(322\) 0 0
\(323\) −5.04650 −0.280795
\(324\) 0 0
\(325\) −3.72833 −0.206810
\(326\) 0 0
\(327\) −4.64820 −0.257046
\(328\) 0 0
\(329\) −51.6664 −2.84846
\(330\) 0 0
\(331\) −27.5886 −1.51640 −0.758202 0.652019i \(-0.773922\pi\)
−0.758202 + 0.652019i \(0.773922\pi\)
\(332\) 0 0
\(333\) −5.25751 −0.288110
\(334\) 0 0
\(335\) 2.43720 0.133159
\(336\) 0 0
\(337\) −17.4230 −0.949093 −0.474547 0.880230i \(-0.657388\pi\)
−0.474547 + 0.880230i \(0.657388\pi\)
\(338\) 0 0
\(339\) −18.7142 −1.01641
\(340\) 0 0
\(341\) 7.49684 0.405977
\(342\) 0 0
\(343\) −28.5810 −1.54323
\(344\) 0 0
\(345\) −3.11903 −0.167923
\(346\) 0 0
\(347\) −4.88097 −0.262024 −0.131012 0.991381i \(-0.541823\pi\)
−0.131012 + 0.991381i \(0.541823\pi\)
\(348\) 0 0
\(349\) 24.0389 1.28677 0.643387 0.765542i \(-0.277529\pi\)
0.643387 + 0.765542i \(0.277529\pi\)
\(350\) 0 0
\(351\) −43.3558 −2.31416
\(352\) 0 0
\(353\) −14.3442 −0.763464 −0.381732 0.924273i \(-0.624672\pi\)
−0.381732 + 0.924273i \(0.624672\pi\)
\(354\) 0 0
\(355\) 7.11903 0.377839
\(356\) 0 0
\(357\) −15.7402 −0.833059
\(358\) 0 0
\(359\) −26.7814 −1.41347 −0.706734 0.707479i \(-0.749832\pi\)
−0.706734 + 0.707479i \(0.749832\pi\)
\(360\) 0 0
\(361\) 1.33763 0.0704015
\(362\) 0 0
\(363\) 24.3752 1.27937
\(364\) 0 0
\(365\) 9.45665 0.494984
\(366\) 0 0
\(367\) 20.4761 1.06884 0.534422 0.845218i \(-0.320529\pi\)
0.534422 + 0.845218i \(0.320529\pi\)
\(368\) 0 0
\(369\) 26.2433 1.36617
\(370\) 0 0
\(371\) 27.0584 1.40480
\(372\) 0 0
\(373\) −3.89386 −0.201616 −0.100808 0.994906i \(-0.532143\pi\)
−0.100808 + 0.994906i \(0.532143\pi\)
\(374\) 0 0
\(375\) −3.11903 −0.161066
\(376\) 0 0
\(377\) 30.7142 1.58186
\(378\) 0 0
\(379\) 30.3765 1.56034 0.780169 0.625569i \(-0.215133\pi\)
0.780169 + 0.625569i \(0.215133\pi\)
\(380\) 0 0
\(381\) 2.10614 0.107901
\(382\) 0 0
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) −19.5615 −0.996947
\(386\) 0 0
\(387\) −53.8266 −2.73616
\(388\) 0 0
\(389\) −18.6818 −0.947206 −0.473603 0.880738i \(-0.657047\pi\)
−0.473603 + 0.880738i \(0.657047\pi\)
\(390\) 0 0
\(391\) 1.11903 0.0565916
\(392\) 0 0
\(393\) 42.7142 2.15464
\(394\) 0 0
\(395\) −14.9133 −0.750370
\(396\) 0 0
\(397\) −28.5757 −1.43417 −0.717086 0.696985i \(-0.754525\pi\)
−0.717086 + 0.696985i \(0.754525\pi\)
\(398\) 0 0
\(399\) 63.4336 3.17565
\(400\) 0 0
\(401\) 12.1061 0.604552 0.302276 0.953220i \(-0.402254\pi\)
0.302276 + 0.953220i \(0.402254\pi\)
\(402\) 0 0
\(403\) 6.44377 0.320987
\(404\) 0 0
\(405\) −16.0854 −0.799290
\(406\) 0 0
\(407\) 3.38942 0.168007
\(408\) 0 0
\(409\) 25.2911 1.25057 0.625283 0.780398i \(-0.284984\pi\)
0.625283 + 0.780398i \(0.284984\pi\)
\(410\) 0 0
\(411\) 23.4837 1.15837
\(412\) 0 0
\(413\) −10.0930 −0.496644
\(414\) 0 0
\(415\) 2.78140 0.136533
\(416\) 0 0
\(417\) −14.5822 −0.714096
\(418\) 0 0
\(419\) 17.3505 0.847628 0.423814 0.905749i \(-0.360691\pi\)
0.423814 + 0.905749i \(0.360691\pi\)
\(420\) 0 0
\(421\) 21.4230 1.04409 0.522047 0.852916i \(-0.325168\pi\)
0.522047 + 0.852916i \(0.325168\pi\)
\(422\) 0 0
\(423\) 77.0841 3.74796
\(424\) 0 0
\(425\) 1.11903 0.0542808
\(426\) 0 0
\(427\) −16.0376 −0.776115
\(428\) 0 0
\(429\) 50.4412 2.43532
\(430\) 0 0
\(431\) −22.5822 −1.08775 −0.543874 0.839167i \(-0.683043\pi\)
−0.543874 + 0.839167i \(0.683043\pi\)
\(432\) 0 0
\(433\) 1.01417 0.0487378 0.0243689 0.999703i \(-0.492242\pi\)
0.0243689 + 0.999703i \(0.492242\pi\)
\(434\) 0 0
\(435\) 25.6947 1.23197
\(436\) 0 0
\(437\) −4.50973 −0.215729
\(438\) 0 0
\(439\) 26.7478 1.27660 0.638301 0.769787i \(-0.279638\pi\)
0.638301 + 0.769787i \(0.279638\pi\)
\(440\) 0 0
\(441\) 89.7399 4.27333
\(442\) 0 0
\(443\) −10.2044 −0.484827 −0.242414 0.970173i \(-0.577939\pi\)
−0.242414 + 0.970173i \(0.577939\pi\)
\(444\) 0 0
\(445\) 7.69471 0.364764
\(446\) 0 0
\(447\) 23.4837 1.11074
\(448\) 0 0
\(449\) 38.7867 1.83046 0.915228 0.402936i \(-0.132010\pi\)
0.915228 + 0.402936i \(0.132010\pi\)
\(450\) 0 0
\(451\) −16.9186 −0.796665
\(452\) 0 0
\(453\) 41.6611 1.95741
\(454\) 0 0
\(455\) −16.8137 −0.788240
\(456\) 0 0
\(457\) 34.9522 1.63500 0.817498 0.575932i \(-0.195361\pi\)
0.817498 + 0.575932i \(0.195361\pi\)
\(458\) 0 0
\(459\) 13.0129 0.607389
\(460\) 0 0
\(461\) 16.3700 0.762425 0.381213 0.924487i \(-0.375507\pi\)
0.381213 + 0.924487i \(0.375507\pi\)
\(462\) 0 0
\(463\) −29.2186 −1.35790 −0.678952 0.734183i \(-0.737565\pi\)
−0.678952 + 0.734183i \(0.737565\pi\)
\(464\) 0 0
\(465\) 5.39070 0.249988
\(466\) 0 0
\(467\) −24.2770 −1.12340 −0.561702 0.827340i \(-0.689853\pi\)
−0.561702 + 0.827340i \(0.689853\pi\)
\(468\) 0 0
\(469\) 10.9911 0.507523
\(470\) 0 0
\(471\) 50.6469 2.33369
\(472\) 0 0
\(473\) 34.7010 1.59555
\(474\) 0 0
\(475\) −4.50973 −0.206920
\(476\) 0 0
\(477\) −40.3700 −1.84841
\(478\) 0 0
\(479\) −24.6080 −1.12437 −0.562185 0.827012i \(-0.690039\pi\)
−0.562185 + 0.827012i \(0.690039\pi\)
\(480\) 0 0
\(481\) 2.91331 0.132835
\(482\) 0 0
\(483\) −14.0660 −0.640023
\(484\) 0 0
\(485\) 0.642920 0.0291935
\(486\) 0 0
\(487\) 30.2381 1.37022 0.685108 0.728441i \(-0.259755\pi\)
0.685108 + 0.728441i \(0.259755\pi\)
\(488\) 0 0
\(489\) 10.2651 0.464204
\(490\) 0 0
\(491\) −12.3311 −0.556493 −0.278246 0.960510i \(-0.589753\pi\)
−0.278246 + 0.960510i \(0.589753\pi\)
\(492\) 0 0
\(493\) −9.21860 −0.415185
\(494\) 0 0
\(495\) 29.1850 1.31177
\(496\) 0 0
\(497\) 32.1049 1.44010
\(498\) 0 0
\(499\) 26.9133 1.20481 0.602403 0.798192i \(-0.294210\pi\)
0.602403 + 0.798192i \(0.294210\pi\)
\(500\) 0 0
\(501\) −71.4672 −3.19292
\(502\) 0 0
\(503\) −20.5097 −0.914483 −0.457242 0.889342i \(-0.651163\pi\)
−0.457242 + 0.889342i \(0.651163\pi\)
\(504\) 0 0
\(505\) 8.23805 0.366589
\(506\) 0 0
\(507\) 2.80845 0.124728
\(508\) 0 0
\(509\) −36.7142 −1.62733 −0.813663 0.581336i \(-0.802530\pi\)
−0.813663 + 0.581336i \(0.802530\pi\)
\(510\) 0 0
\(511\) 42.6469 1.88659
\(512\) 0 0
\(513\) −52.4425 −2.31539
\(514\) 0 0
\(515\) 12.3376 0.543661
\(516\) 0 0
\(517\) −49.6947 −2.18557
\(518\) 0 0
\(519\) 1.79557 0.0788166
\(520\) 0 0
\(521\) −4.91331 −0.215256 −0.107628 0.994191i \(-0.534326\pi\)
−0.107628 + 0.994191i \(0.534326\pi\)
\(522\) 0 0
\(523\) 0.344196 0.0150506 0.00752531 0.999972i \(-0.497605\pi\)
0.00752531 + 0.999972i \(0.497605\pi\)
\(524\) 0 0
\(525\) −14.0660 −0.613889
\(526\) 0 0
\(527\) −1.93404 −0.0842483
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 15.0584 0.653477
\(532\) 0 0
\(533\) −14.5421 −0.629887
\(534\) 0 0
\(535\) −15.9328 −0.688833
\(536\) 0 0
\(537\) −15.6558 −0.675598
\(538\) 0 0
\(539\) −57.8537 −2.49193
\(540\) 0 0
\(541\) −6.13191 −0.263631 −0.131816 0.991274i \(-0.542081\pi\)
−0.131816 + 0.991274i \(0.542081\pi\)
\(542\) 0 0
\(543\) −35.9598 −1.54318
\(544\) 0 0
\(545\) 1.49027 0.0638363
\(546\) 0 0
\(547\) 9.18498 0.392721 0.196361 0.980532i \(-0.437088\pi\)
0.196361 + 0.980532i \(0.437088\pi\)
\(548\) 0 0
\(549\) 23.9275 1.02120
\(550\) 0 0
\(551\) 37.1514 1.58270
\(552\) 0 0
\(553\) −67.2549 −2.85997
\(554\) 0 0
\(555\) 2.43720 0.103454
\(556\) 0 0
\(557\) −4.30529 −0.182421 −0.0912105 0.995832i \(-0.529074\pi\)
−0.0912105 + 0.995832i \(0.529074\pi\)
\(558\) 0 0
\(559\) 29.8266 1.26153
\(560\) 0 0
\(561\) −15.1395 −0.639191
\(562\) 0 0
\(563\) −11.1256 −0.468888 −0.234444 0.972130i \(-0.575327\pi\)
−0.234444 + 0.972130i \(0.575327\pi\)
\(564\) 0 0
\(565\) 6.00000 0.252422
\(566\) 0 0
\(567\) −72.5408 −3.04643
\(568\) 0 0
\(569\) −16.0389 −0.672386 −0.336193 0.941793i \(-0.609139\pi\)
−0.336193 + 0.941793i \(0.609139\pi\)
\(570\) 0 0
\(571\) −17.9004 −0.749109 −0.374555 0.927205i \(-0.622204\pi\)
−0.374555 + 0.927205i \(0.622204\pi\)
\(572\) 0 0
\(573\) 58.3700 2.43844
\(574\) 0 0
\(575\) 1.00000 0.0417029
\(576\) 0 0
\(577\) −9.12559 −0.379903 −0.189952 0.981793i \(-0.560833\pi\)
−0.189952 + 0.981793i \(0.560833\pi\)
\(578\) 0 0
\(579\) 73.2833 3.04555
\(580\) 0 0
\(581\) 12.5433 0.520386
\(582\) 0 0
\(583\) 26.0258 1.07788
\(584\) 0 0
\(585\) 25.0854 1.03715
\(586\) 0 0
\(587\) 33.6340 1.38823 0.694113 0.719866i \(-0.255797\pi\)
0.694113 + 0.719866i \(0.255797\pi\)
\(588\) 0 0
\(589\) 7.79428 0.321158
\(590\) 0 0
\(591\) −56.5744 −2.32716
\(592\) 0 0
\(593\) −17.4567 −0.716859 −0.358429 0.933557i \(-0.616688\pi\)
−0.358429 + 0.933557i \(0.616688\pi\)
\(594\) 0 0
\(595\) 5.04650 0.206886
\(596\) 0 0
\(597\) 72.5408 2.96890
\(598\) 0 0
\(599\) 11.5951 0.473764 0.236882 0.971538i \(-0.423874\pi\)
0.236882 + 0.971538i \(0.423874\pi\)
\(600\) 0 0
\(601\) −31.6611 −1.29148 −0.645741 0.763556i \(-0.723452\pi\)
−0.645741 + 0.763556i \(0.723452\pi\)
\(602\) 0 0
\(603\) −16.3983 −0.667790
\(604\) 0 0
\(605\) −7.81502 −0.317726
\(606\) 0 0
\(607\) 36.0778 1.46435 0.732177 0.681115i \(-0.238505\pi\)
0.732177 + 0.681115i \(0.238505\pi\)
\(608\) 0 0
\(609\) 115.876 4.69554
\(610\) 0 0
\(611\) −42.7142 −1.72803
\(612\) 0 0
\(613\) −32.0389 −1.29404 −0.647020 0.762473i \(-0.723985\pi\)
−0.647020 + 0.762473i \(0.723985\pi\)
\(614\) 0 0
\(615\) −12.1655 −0.490562
\(616\) 0 0
\(617\) −13.3960 −0.539302 −0.269651 0.962958i \(-0.586908\pi\)
−0.269651 + 0.962958i \(0.586908\pi\)
\(618\) 0 0
\(619\) −37.1309 −1.49242 −0.746208 0.665713i \(-0.768128\pi\)
−0.746208 + 0.665713i \(0.768128\pi\)
\(620\) 0 0
\(621\) 11.6288 0.466646
\(622\) 0 0
\(623\) 34.7010 1.39027
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 61.0129 2.43662
\(628\) 0 0
\(629\) −0.874406 −0.0348648
\(630\) 0 0
\(631\) −11.1125 −0.442380 −0.221190 0.975231i \(-0.570994\pi\)
−0.221190 + 0.975231i \(0.570994\pi\)
\(632\) 0 0
\(633\) −13.5497 −0.538551
\(634\) 0 0
\(635\) −0.675256 −0.0267967
\(636\) 0 0
\(637\) −49.7270 −1.97026
\(638\) 0 0
\(639\) −47.8991 −1.89486
\(640\) 0 0
\(641\) 12.3831 0.489103 0.244552 0.969636i \(-0.421359\pi\)
0.244552 + 0.969636i \(0.421359\pi\)
\(642\) 0 0
\(643\) 37.4956 1.47868 0.739340 0.673332i \(-0.235138\pi\)
0.739340 + 0.673332i \(0.235138\pi\)
\(644\) 0 0
\(645\) 24.9522 0.982492
\(646\) 0 0
\(647\) −14.5691 −0.572771 −0.286385 0.958114i \(-0.592454\pi\)
−0.286385 + 0.958114i \(0.592454\pi\)
\(648\) 0 0
\(649\) −9.70784 −0.381066
\(650\) 0 0
\(651\) 24.3106 0.952807
\(652\) 0 0
\(653\) −4.41672 −0.172840 −0.0864198 0.996259i \(-0.527543\pi\)
−0.0864198 + 0.996259i \(0.527543\pi\)
\(654\) 0 0
\(655\) −13.6947 −0.535097
\(656\) 0 0
\(657\) −63.6275 −2.48234
\(658\) 0 0
\(659\) 31.8655 1.24130 0.620652 0.784086i \(-0.286868\pi\)
0.620652 + 0.784086i \(0.286868\pi\)
\(660\) 0 0
\(661\) 33.1190 1.28818 0.644090 0.764949i \(-0.277236\pi\)
0.644090 + 0.764949i \(0.277236\pi\)
\(662\) 0 0
\(663\) −13.0129 −0.505379
\(664\) 0 0
\(665\) −20.3376 −0.788659
\(666\) 0 0
\(667\) −8.23805 −0.318979
\(668\) 0 0
\(669\) −38.9133 −1.50448
\(670\) 0 0
\(671\) −15.4256 −0.595499
\(672\) 0 0
\(673\) 19.3505 0.745907 0.372954 0.927850i \(-0.378345\pi\)
0.372954 + 0.927850i \(0.378345\pi\)
\(674\) 0 0
\(675\) 11.6288 0.447591
\(676\) 0 0
\(677\) −6.00000 −0.230599 −0.115299 0.993331i \(-0.536783\pi\)
−0.115299 + 0.993331i \(0.536783\pi\)
\(678\) 0 0
\(679\) 2.89939 0.111268
\(680\) 0 0
\(681\) −49.6947 −1.90431
\(682\) 0 0
\(683\) −38.3495 −1.46740 −0.733701 0.679472i \(-0.762209\pi\)
−0.733701 + 0.679472i \(0.762209\pi\)
\(684\) 0 0
\(685\) −7.52918 −0.287675
\(686\) 0 0
\(687\) −11.1125 −0.423967
\(688\) 0 0
\(689\) 22.3700 0.852228
\(690\) 0 0
\(691\) 21.0195 0.799618 0.399809 0.916599i \(-0.369077\pi\)
0.399809 + 0.916599i \(0.369077\pi\)
\(692\) 0 0
\(693\) 131.616 4.99969
\(694\) 0 0
\(695\) 4.67526 0.177343
\(696\) 0 0
\(697\) 4.36468 0.165324
\(698\) 0 0
\(699\) −85.7594 −3.24372
\(700\) 0 0
\(701\) 19.3169 0.729589 0.364794 0.931088i \(-0.381139\pi\)
0.364794 + 0.931088i \(0.381139\pi\)
\(702\) 0 0
\(703\) 3.52389 0.132906
\(704\) 0 0
\(705\) −35.7336 −1.34581
\(706\) 0 0
\(707\) 37.1514 1.39722
\(708\) 0 0
\(709\) 12.2315 0.459363 0.229682 0.973266i \(-0.426232\pi\)
0.229682 + 0.973266i \(0.426232\pi\)
\(710\) 0 0
\(711\) 100.342 3.76311
\(712\) 0 0
\(713\) −1.72833 −0.0647264
\(714\) 0 0
\(715\) −16.1721 −0.604802
\(716\) 0 0
\(717\) −31.3116 −1.16935
\(718\) 0 0
\(719\) 40.6416 1.51568 0.757839 0.652442i \(-0.226255\pi\)
0.757839 + 0.652442i \(0.226255\pi\)
\(720\) 0 0
\(721\) 55.6393 2.07212
\(722\) 0 0
\(723\) −73.9044 −2.74854
\(724\) 0 0
\(725\) −8.23805 −0.305954
\(726\) 0 0
\(727\) 23.4501 0.869716 0.434858 0.900499i \(-0.356799\pi\)
0.434858 + 0.900499i \(0.356799\pi\)
\(728\) 0 0
\(729\) −0.583281 −0.0216030
\(730\) 0 0
\(731\) −8.95221 −0.331110
\(732\) 0 0
\(733\) −12.5150 −0.462252 −0.231126 0.972924i \(-0.574241\pi\)
−0.231126 + 0.972924i \(0.574241\pi\)
\(734\) 0 0
\(735\) −41.6004 −1.53445
\(736\) 0 0
\(737\) 10.5717 0.389413
\(738\) 0 0
\(739\) 21.3505 0.785391 0.392696 0.919668i \(-0.371543\pi\)
0.392696 + 0.919668i \(0.371543\pi\)
\(740\) 0 0
\(741\) 52.4425 1.92652
\(742\) 0 0
\(743\) −24.9858 −0.916641 −0.458321 0.888787i \(-0.651549\pi\)
−0.458321 + 0.888787i \(0.651549\pi\)
\(744\) 0 0
\(745\) −7.52918 −0.275848
\(746\) 0 0
\(747\) −18.7142 −0.684715
\(748\) 0 0
\(749\) −71.8524 −2.62543
\(750\) 0 0
\(751\) 33.6275 1.22708 0.613542 0.789662i \(-0.289744\pi\)
0.613542 + 0.789662i \(0.289744\pi\)
\(752\) 0 0
\(753\) −38.8085 −1.41426
\(754\) 0 0
\(755\) −13.3571 −0.486114
\(756\) 0 0
\(757\) −37.1230 −1.34926 −0.674630 0.738156i \(-0.735697\pi\)
−0.674630 + 0.738156i \(0.735697\pi\)
\(758\) 0 0
\(759\) −13.5292 −0.491078
\(760\) 0 0
\(761\) −3.87337 −0.140410 −0.0702048 0.997533i \(-0.522365\pi\)
−0.0702048 + 0.997533i \(0.522365\pi\)
\(762\) 0 0
\(763\) 6.72073 0.243307
\(764\) 0 0
\(765\) −7.52918 −0.272218
\(766\) 0 0
\(767\) −8.34420 −0.301291
\(768\) 0 0
\(769\) 23.1645 0.835333 0.417667 0.908600i \(-0.362848\pi\)
0.417667 + 0.908600i \(0.362848\pi\)
\(770\) 0 0
\(771\) 17.0195 0.612941
\(772\) 0 0
\(773\) −8.78140 −0.315845 −0.157922 0.987452i \(-0.550480\pi\)
−0.157922 + 0.987452i \(0.550480\pi\)
\(774\) 0 0
\(775\) −1.72833 −0.0620834
\(776\) 0 0
\(777\) 10.9911 0.394304
\(778\) 0 0
\(779\) −17.5898 −0.630222
\(780\) 0 0
\(781\) 30.8797 1.10496
\(782\) 0 0
\(783\) −95.7983 −3.42355
\(784\) 0 0
\(785\) −16.2381 −0.579561
\(786\) 0 0
\(787\) −49.6275 −1.76903 −0.884514 0.466513i \(-0.845510\pi\)
−0.884514 + 0.466513i \(0.845510\pi\)
\(788\) 0 0
\(789\) −0.431918 −0.0153767
\(790\) 0 0
\(791\) 27.0584 0.962084
\(792\) 0 0
\(793\) −13.2588 −0.470833
\(794\) 0 0
\(795\) 18.7142 0.663723
\(796\) 0 0
\(797\) 18.3311 0.649319 0.324660 0.945831i \(-0.394750\pi\)
0.324660 + 0.945831i \(0.394750\pi\)
\(798\) 0 0
\(799\) 12.8203 0.453550
\(800\) 0 0
\(801\) −51.7725 −1.82929
\(802\) 0 0
\(803\) 41.0195 1.44755
\(804\) 0 0
\(805\) 4.50973 0.158947
\(806\) 0 0
\(807\) 45.7725 1.61127
\(808\) 0 0
\(809\) −1.93933 −0.0681832 −0.0340916 0.999419i \(-0.510854\pi\)
−0.0340916 + 0.999419i \(0.510854\pi\)
\(810\) 0 0
\(811\) 5.41775 0.190243 0.0951215 0.995466i \(-0.469676\pi\)
0.0951215 + 0.995466i \(0.469676\pi\)
\(812\) 0 0
\(813\) 25.9209 0.909086
\(814\) 0 0
\(815\) −3.29112 −0.115283
\(816\) 0 0
\(817\) 36.0778 1.26220
\(818\) 0 0
\(819\) 113.128 3.95302
\(820\) 0 0
\(821\) 37.8655 1.32152 0.660758 0.750599i \(-0.270235\pi\)
0.660758 + 0.750599i \(0.270235\pi\)
\(822\) 0 0
\(823\) −43.7336 −1.52446 −0.762229 0.647308i \(-0.775895\pi\)
−0.762229 + 0.647308i \(0.775895\pi\)
\(824\) 0 0
\(825\) −13.5292 −0.471026
\(826\) 0 0
\(827\) −28.1991 −0.980581 −0.490290 0.871559i \(-0.663109\pi\)
−0.490290 + 0.871559i \(0.663109\pi\)
\(828\) 0 0
\(829\) 1.12559 0.0390935 0.0195468 0.999809i \(-0.493778\pi\)
0.0195468 + 0.999809i \(0.493778\pi\)
\(830\) 0 0
\(831\) 40.2770 1.39719
\(832\) 0 0
\(833\) 14.9252 0.517126
\(834\) 0 0
\(835\) 22.9133 0.792948
\(836\) 0 0
\(837\) −20.0983 −0.694699
\(838\) 0 0
\(839\) −39.9328 −1.37863 −0.689316 0.724461i \(-0.742089\pi\)
−0.689316 + 0.724461i \(0.742089\pi\)
\(840\) 0 0
\(841\) 38.8655 1.34019
\(842\) 0 0
\(843\) 8.34420 0.287389
\(844\) 0 0
\(845\) −0.900425 −0.0309756
\(846\) 0 0
\(847\) −35.2436 −1.21098
\(848\) 0 0
\(849\) −2.31586 −0.0794801
\(850\) 0 0
\(851\) −0.781399 −0.0267860
\(852\) 0 0
\(853\) 47.9921 1.64322 0.821610 0.570050i \(-0.193076\pi\)
0.821610 + 0.570050i \(0.193076\pi\)
\(854\) 0 0
\(855\) 30.3429 1.03771
\(856\) 0 0
\(857\) −43.4283 −1.48348 −0.741742 0.670686i \(-0.766000\pi\)
−0.741742 + 0.670686i \(0.766000\pi\)
\(858\) 0 0
\(859\) −32.5433 −1.11036 −0.555182 0.831729i \(-0.687352\pi\)
−0.555182 + 0.831729i \(0.687352\pi\)
\(860\) 0 0
\(861\) −54.8632 −1.86973
\(862\) 0 0
\(863\) −46.1036 −1.56938 −0.784692 0.619886i \(-0.787179\pi\)
−0.784692 + 0.619886i \(0.787179\pi\)
\(864\) 0 0
\(865\) −0.575681 −0.0195738
\(866\) 0 0
\(867\) −49.1177 −1.66813
\(868\) 0 0
\(869\) −64.6884 −2.19440
\(870\) 0 0
\(871\) 9.08669 0.307891
\(872\) 0 0
\(873\) −4.32578 −0.146405
\(874\) 0 0
\(875\) 4.50973 0.152457
\(876\) 0 0
\(877\) −24.0996 −0.813785 −0.406892 0.913476i \(-0.633388\pi\)
−0.406892 + 0.913476i \(0.633388\pi\)
\(878\) 0 0
\(879\) −18.7142 −0.631213
\(880\) 0 0
\(881\) 2.34420 0.0789780 0.0394890 0.999220i \(-0.487427\pi\)
0.0394890 + 0.999220i \(0.487427\pi\)
\(882\) 0 0
\(883\) 41.0505 1.38146 0.690730 0.723113i \(-0.257289\pi\)
0.690730 + 0.723113i \(0.257289\pi\)
\(884\) 0 0
\(885\) −6.98055 −0.234649
\(886\) 0 0
\(887\) −54.7788 −1.83929 −0.919647 0.392747i \(-0.871525\pi\)
−0.919647 + 0.392747i \(0.871525\pi\)
\(888\) 0 0
\(889\) −3.04522 −0.102133
\(890\) 0 0
\(891\) −69.7725 −2.33747
\(892\) 0 0
\(893\) −51.6664 −1.72895
\(894\) 0 0
\(895\) 5.01945 0.167782
\(896\) 0 0
\(897\) −11.6288 −0.388273
\(898\) 0 0
\(899\) 14.2381 0.474866
\(900\) 0 0
\(901\) −6.71416 −0.223681
\(902\) 0 0
\(903\) 112.528 3.74469
\(904\) 0 0
\(905\) 11.5292 0.383243
\(906\) 0 0
\(907\) 10.1061 0.335569 0.167784 0.985824i \(-0.446339\pi\)
0.167784 + 0.985824i \(0.446339\pi\)
\(908\) 0 0
\(909\) −55.4283 −1.83844
\(910\) 0 0
\(911\) 25.4178 0.842128 0.421064 0.907031i \(-0.361657\pi\)
0.421064 + 0.907031i \(0.361657\pi\)
\(912\) 0 0
\(913\) 12.0647 0.399282
\(914\) 0 0
\(915\) −11.0920 −0.366689
\(916\) 0 0
\(917\) −61.7594 −2.03947
\(918\) 0 0
\(919\) −23.6017 −0.778548 −0.389274 0.921122i \(-0.627274\pi\)
−0.389274 + 0.921122i \(0.627274\pi\)
\(920\) 0 0
\(921\) −95.1566 −3.13552
\(922\) 0 0
\(923\) 26.5421 0.873643
\(924\) 0 0
\(925\) −0.781399 −0.0256922
\(926\) 0 0
\(927\) −83.0116 −2.72646
\(928\) 0 0
\(929\) −7.08669 −0.232507 −0.116253 0.993220i \(-0.537088\pi\)
−0.116253 + 0.993220i \(0.537088\pi\)
\(930\) 0 0
\(931\) −60.1490 −1.97131
\(932\) 0 0
\(933\) −17.3505 −0.568030
\(934\) 0 0
\(935\) 4.85392 0.158740
\(936\) 0 0
\(937\) 27.3169 0.892404 0.446202 0.894932i \(-0.352776\pi\)
0.446202 + 0.894932i \(0.352776\pi\)
\(938\) 0 0
\(939\) 12.7023 0.414524
\(940\) 0 0
\(941\) 55.8979 1.82222 0.911109 0.412165i \(-0.135227\pi\)
0.911109 + 0.412165i \(0.135227\pi\)
\(942\) 0 0
\(943\) 3.90043 0.127015
\(944\) 0 0
\(945\) 52.4425 1.70595
\(946\) 0 0
\(947\) −37.5939 −1.22164 −0.610818 0.791771i \(-0.709159\pi\)
−0.610818 + 0.791771i \(0.709159\pi\)
\(948\) 0 0
\(949\) 35.2575 1.14451
\(950\) 0 0
\(951\) −19.2305 −0.623590
\(952\) 0 0
\(953\) −29.3828 −0.951804 −0.475902 0.879498i \(-0.657878\pi\)
−0.475902 + 0.879498i \(0.657878\pi\)
\(954\) 0 0
\(955\) −18.7142 −0.605576
\(956\) 0 0
\(957\) 111.454 3.60280
\(958\) 0 0
\(959\) −33.9545 −1.09645
\(960\) 0 0
\(961\) −28.0129 −0.903641
\(962\) 0 0
\(963\) 107.201 3.45450
\(964\) 0 0
\(965\) −23.4956 −0.756349
\(966\) 0 0
\(967\) 49.2292 1.58310 0.791552 0.611102i \(-0.209274\pi\)
0.791552 + 0.611102i \(0.209274\pi\)
\(968\) 0 0
\(969\) −15.7402 −0.505647
\(970\) 0 0
\(971\) −9.62347 −0.308832 −0.154416 0.988006i \(-0.549350\pi\)
−0.154416 + 0.988006i \(0.549350\pi\)
\(972\) 0 0
\(973\) 21.0841 0.675926
\(974\) 0 0
\(975\) −11.6288 −0.372418
\(976\) 0 0
\(977\) 18.9858 0.607411 0.303705 0.952766i \(-0.401776\pi\)
0.303705 + 0.952766i \(0.401776\pi\)
\(978\) 0 0
\(979\) 33.3768 1.06673
\(980\) 0 0
\(981\) −10.0271 −0.320139
\(982\) 0 0
\(983\) −4.33763 −0.138349 −0.0691744 0.997605i \(-0.522037\pi\)
−0.0691744 + 0.997605i \(0.522037\pi\)
\(984\) 0 0
\(985\) 18.1385 0.577940
\(986\) 0 0
\(987\) −161.149 −5.12942
\(988\) 0 0
\(989\) −8.00000 −0.254385
\(990\) 0 0
\(991\) −1.96766 −0.0625049 −0.0312524 0.999512i \(-0.509950\pi\)
−0.0312524 + 0.999512i \(0.509950\pi\)
\(992\) 0 0
\(993\) −86.0495 −2.73070
\(994\) 0 0
\(995\) −23.2575 −0.737312
\(996\) 0 0
\(997\) 3.96110 0.125449 0.0627246 0.998031i \(-0.480021\pi\)
0.0627246 + 0.998031i \(0.480021\pi\)
\(998\) 0 0
\(999\) −9.08669 −0.287490
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 1840.2.a.r.1.3 3
4.3 odd 2 230.2.a.d.1.1 3
5.4 even 2 9200.2.a.cf.1.1 3
8.3 odd 2 7360.2.a.bz.1.3 3
8.5 even 2 7360.2.a.ce.1.1 3
12.11 even 2 2070.2.a.z.1.3 3
20.3 even 4 1150.2.b.j.599.1 6
20.7 even 4 1150.2.b.j.599.6 6
20.19 odd 2 1150.2.a.q.1.3 3
92.91 even 2 5290.2.a.r.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.a.d.1.1 3 4.3 odd 2
1150.2.a.q.1.3 3 20.19 odd 2
1150.2.b.j.599.1 6 20.3 even 4
1150.2.b.j.599.6 6 20.7 even 4
1840.2.a.r.1.3 3 1.1 even 1 trivial
2070.2.a.z.1.3 3 12.11 even 2
5290.2.a.r.1.1 3 92.91 even 2
7360.2.a.bz.1.3 3 8.3 odd 2
7360.2.a.ce.1.1 3 8.5 even 2
9200.2.a.cf.1.1 3 5.4 even 2