Properties

Label 2070.2.a.z.1.3
Level $2070$
Weight $2$
Character 2070.1
Self dual yes
Analytic conductor $16.529$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2070.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(16.5290332184\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.1101.1
Defining polynomial: \(x^{3} - x^{2} - 9 x + 12\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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 \(\chi\) \(=\) 2070.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +4.50973 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +4.50973 q^{7} -1.00000 q^{8} -1.00000 q^{10} -4.33763 q^{11} -3.72833 q^{13} -4.50973 q^{14} +1.00000 q^{16} -1.11903 q^{17} +4.50973 q^{19} +1.00000 q^{20} +4.33763 q^{22} +1.00000 q^{23} +1.00000 q^{25} +3.72833 q^{26} +4.50973 q^{28} +8.23805 q^{29} +1.72833 q^{31} -1.00000 q^{32} +1.11903 q^{34} +4.50973 q^{35} -0.781399 q^{37} -4.50973 q^{38} -1.00000 q^{40} -3.90043 q^{41} +8.00000 q^{43} -4.33763 q^{44} -1.00000 q^{46} +11.4567 q^{47} +13.3376 q^{49} -1.00000 q^{50} -3.72833 q^{52} +6.00000 q^{53} -4.33763 q^{55} -4.50973 q^{56} -8.23805 q^{58} +2.23805 q^{59} +3.55623 q^{61} -1.72833 q^{62} +1.00000 q^{64} -3.72833 q^{65} +2.43720 q^{67} -1.11903 q^{68} -4.50973 q^{70} -7.11903 q^{71} -9.45665 q^{73} +0.781399 q^{74} +4.50973 q^{76} -19.5615 q^{77} -14.9133 q^{79} +1.00000 q^{80} +3.90043 q^{82} -2.78140 q^{83} -1.11903 q^{85} -8.00000 q^{86} +4.33763 q^{88} +7.69471 q^{89} -16.8137 q^{91} +1.00000 q^{92} -11.4567 q^{94} +4.50973 q^{95} -0.642920 q^{97} -13.3376 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{2} + 3q^{4} + 3q^{5} + 3q^{7} - 3q^{8} + O(q^{10}) \) \( 3q - 3q^{2} + 3q^{4} + 3q^{5} + 3q^{7} - 3q^{8} - 3q^{10} - 3q^{11} - q^{13} - 3q^{14} + 3q^{16} + 7q^{17} + 3q^{19} + 3q^{20} + 3q^{22} + 3q^{23} + 3q^{25} + q^{26} + 3q^{28} + 4q^{29} - 5q^{31} - 3q^{32} - 7q^{34} + 3q^{35} - 2q^{37} - 3q^{38} - 3q^{40} - q^{41} + 24q^{43} - 3q^{44} - 3q^{46} + 14q^{47} + 30q^{49} - 3q^{50} - q^{52} + 18q^{53} - 3q^{55} - 3q^{56} - 4q^{58} - 14q^{59} + q^{61} + 5q^{62} + 3q^{64} - q^{65} + 8q^{67} + 7q^{68} - 3q^{70} - 11q^{71} - 8q^{73} + 2q^{74} + 3q^{76} + 24q^{77} - 4q^{79} + 3q^{80} + q^{82} - 8q^{83} + 7q^{85} - 24q^{86} + 3q^{88} - 18q^{89} + q^{91} + 3q^{92} - 14q^{94} + 3q^{95} - 33q^{97} - 30q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 4.50973 1.70452 0.852258 0.523122i \(-0.175233\pi\)
0.852258 + 0.523122i \(0.175233\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) −1.00000 −0.316228
\(11\) −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 \(0.672961\pi\)
\(14\) −4.50973 −1.20527
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −1.11903 −0.271404 −0.135702 0.990750i \(-0.543329\pi\)
−0.135702 + 0.990750i \(0.543329\pi\)
\(18\) 0 0
\(19\) 4.50973 1.03460 0.517301 0.855803i \(-0.326937\pi\)
0.517301 + 0.855803i \(0.326937\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) 4.33763 0.924785
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 3.72833 0.731185
\(27\) 0 0
\(28\) 4.50973 0.852258
\(29\) 8.23805 1.52977 0.764884 0.644168i \(-0.222796\pi\)
0.764884 + 0.644168i \(0.222796\pi\)
\(30\) 0 0
\(31\) 1.72833 0.310417 0.155208 0.987882i \(-0.450395\pi\)
0.155208 + 0.987882i \(0.450395\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 1.11903 0.191911
\(35\) 4.50973 0.762283
\(36\) 0 0
\(37\) −0.781399 −0.128461 −0.0642306 0.997935i \(-0.520459\pi\)
−0.0642306 + 0.997935i \(0.520459\pi\)
\(38\) −4.50973 −0.731574
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) −3.90043 −0.609144 −0.304572 0.952489i \(-0.598513\pi\)
−0.304572 + 0.952489i \(0.598513\pi\)
\(42\) 0 0
\(43\) 8.00000 1.21999 0.609994 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) −4.33763 −0.653922
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) 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
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) −3.72833 −0.517026
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) −4.33763 −0.584886
\(56\) −4.50973 −0.602637
\(57\) 0 0
\(58\) −8.23805 −1.08171
\(59\) 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 \(0.426891\pi\)
\(62\) −1.72833 −0.219498
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −3.72833 −0.462442
\(66\) 0 0
\(67\) 2.43720 0.297752 0.148876 0.988856i \(-0.452435\pi\)
0.148876 + 0.988856i \(0.452435\pi\)
\(68\) −1.11903 −0.135702
\(69\) 0 0
\(70\) −4.50973 −0.539015
\(71\) −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 \(0.686673\pi\)
\(74\) 0.781399 0.0908357
\(75\) 0 0
\(76\) 4.50973 0.517301
\(77\) −19.5615 −2.22924
\(78\) 0 0
\(79\) −14.9133 −1.67788 −0.838939 0.544225i \(-0.816824\pi\)
−0.838939 + 0.544225i \(0.816824\pi\)
\(80\) 1.00000 0.111803
\(81\) 0 0
\(82\) 3.90043 0.430730
\(83\) −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
\(86\) −8.00000 −0.862662
\(87\) 0 0
\(88\) 4.33763 0.462393
\(89\) 7.69471 0.815637 0.407819 0.913063i \(-0.366290\pi\)
0.407819 + 0.913063i \(0.366290\pi\)
\(90\) 0 0
\(91\) −16.8137 −1.76256
\(92\) 1.00000 0.104257
\(93\) 0 0
\(94\) −11.4567 −1.18166
\(95\) 4.50973 0.462688
\(96\) 0 0
\(97\) −0.642920 −0.0652786 −0.0326393 0.999467i \(-0.510391\pi\)
−0.0326393 + 0.999467i \(0.510391\pi\)
\(98\) −13.3376 −1.34730
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 8.23805 0.819717 0.409858 0.912149i \(-0.365578\pi\)
0.409858 + 0.912149i \(0.365578\pi\)
\(102\) 0 0
\(103\) 12.3376 1.21566 0.607831 0.794066i \(-0.292040\pi\)
0.607831 + 0.794066i \(0.292040\pi\)
\(104\) 3.72833 0.365593
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) 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 \(0.522737\pi\)
\(110\) 4.33763 0.413577
\(111\) 0 0
\(112\) 4.50973 0.426129
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) 1.00000 0.0932505
\(116\) 8.23805 0.764884
\(117\) 0 0
\(118\) −2.23805 −0.206030
\(119\) −5.04650 −0.462612
\(120\) 0 0
\(121\) 7.81502 0.710456
\(122\) −3.55623 −0.321966
\(123\) 0 0
\(124\) 1.72833 0.155208
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −0.675256 −0.0599193 −0.0299597 0.999551i \(-0.509538\pi\)
−0.0299597 + 0.999551i \(0.509538\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 3.72833 0.326996
\(131\) 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
\(134\) −2.43720 −0.210542
\(135\) 0 0
\(136\) 1.11903 0.0959557
\(137\) −7.52918 −0.643261 −0.321631 0.946865i \(-0.604231\pi\)
−0.321631 + 0.946865i \(0.604231\pi\)
\(138\) 0 0
\(139\) 4.67526 0.396550 0.198275 0.980146i \(-0.436466\pi\)
0.198275 + 0.980146i \(0.436466\pi\)
\(140\) 4.50973 0.381141
\(141\) 0 0
\(142\) 7.11903 0.597415
\(143\) 16.1721 1.35238
\(144\) 0 0
\(145\) 8.23805 0.684133
\(146\) 9.45665 0.782638
\(147\) 0 0
\(148\) −0.781399 −0.0642306
\(149\) −7.52918 −0.616814 −0.308407 0.951254i \(-0.599796\pi\)
−0.308407 + 0.951254i \(0.599796\pi\)
\(150\) 0 0
\(151\) −13.3571 −1.08698 −0.543492 0.839414i \(-0.682898\pi\)
−0.543492 + 0.839414i \(0.682898\pi\)
\(152\) −4.50973 −0.365787
\(153\) 0 0
\(154\) 19.5615 1.57631
\(155\) 1.72833 0.138823
\(156\) 0 0
\(157\) 16.2381 1.29594 0.647969 0.761667i \(-0.275619\pi\)
0.647969 + 0.761667i \(0.275619\pi\)
\(158\) 14.9133 1.18644
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) 4.50973 0.355416
\(162\) 0 0
\(163\) −3.29112 −0.257781 −0.128890 0.991659i \(-0.541142\pi\)
−0.128890 + 0.991659i \(0.541142\pi\)
\(164\) −3.90043 −0.304572
\(165\) 0 0
\(166\) 2.78140 0.215878
\(167\) −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
\(170\) 1.11903 0.0858254
\(171\) 0 0
\(172\) 8.00000 0.609994
\(173\) −0.575681 −0.0437683 −0.0218841 0.999761i \(-0.506966\pi\)
−0.0218841 + 0.999761i \(0.506966\pi\)
\(174\) 0 0
\(175\) 4.50973 0.340903
\(176\) −4.33763 −0.326961
\(177\) 0 0
\(178\) −7.69471 −0.576743
\(179\) −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 \(0.640950\pi\)
\(182\) 16.8137 1.24632
\(183\) 0 0
\(184\) −1.00000 −0.0737210
\(185\) −0.781399 −0.0574496
\(186\) 0 0
\(187\) 4.85392 0.354954
\(188\) 11.4567 0.835562
\(189\) 0 0
\(190\) −4.50973 −0.327170
\(191\) 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 \(0.179229\pi\)
\(194\) 0.642920 0.0461590
\(195\) 0 0
\(196\) 13.3376 0.952688
\(197\) 18.1385 1.29231 0.646157 0.763205i \(-0.276375\pi\)
0.646157 + 0.763205i \(0.276375\pi\)
\(198\) 0 0
\(199\) −23.2575 −1.64868 −0.824340 0.566094i \(-0.808454\pi\)
−0.824340 + 0.566094i \(0.808454\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 0 0
\(202\) −8.23805 −0.579627
\(203\) 37.1514 2.60751
\(204\) 0 0
\(205\) −3.90043 −0.272418
\(206\) −12.3376 −0.859603
\(207\) 0 0
\(208\) −3.72833 −0.258513
\(209\) −19.5615 −1.35310
\(210\) 0 0
\(211\) 4.34420 0.299067 0.149533 0.988757i \(-0.452223\pi\)
0.149533 + 0.988757i \(0.452223\pi\)
\(212\) 6.00000 0.412082
\(213\) 0 0
\(214\) −15.9328 −1.08914
\(215\) 8.00000 0.545595
\(216\) 0 0
\(217\) 7.79428 0.529110
\(218\) 1.49027 0.100934
\(219\) 0 0
\(220\) −4.33763 −0.292443
\(221\) 4.17210 0.280646
\(222\) 0 0
\(223\) 12.4761 0.835462 0.417731 0.908571i \(-0.362825\pi\)
0.417731 + 0.908571i \(0.362825\pi\)
\(224\) −4.50973 −0.301319
\(225\) 0 0
\(226\) −6.00000 −0.399114
\(227\) −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 \(0.537558\pi\)
\(230\) −1.00000 −0.0659380
\(231\) 0 0
\(232\) −8.23805 −0.540855
\(233\) 27.4956 1.80129 0.900647 0.434552i \(-0.143093\pi\)
0.900647 + 0.434552i \(0.143093\pi\)
\(234\) 0 0
\(235\) 11.4567 0.747350
\(236\) 2.23805 0.145685
\(237\) 0 0
\(238\) 5.04650 0.327116
\(239\) −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 \(0.776351\pi\)
\(242\) −7.81502 −0.502368
\(243\) 0 0
\(244\) 3.55623 0.227664
\(245\) 13.3376 0.852110
\(246\) 0 0
\(247\) −16.8137 −1.06983
\(248\) −1.72833 −0.109749
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) −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
\(254\) 0.675256 0.0423693
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −5.45665 −0.340377 −0.170188 0.985412i \(-0.554438\pi\)
−0.170188 + 0.985412i \(0.554438\pi\)
\(258\) 0 0
\(259\) −3.52389 −0.218964
\(260\) −3.72833 −0.231221
\(261\) 0 0
\(262\) −13.6947 −0.846062
\(263\) −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
\(266\) −20.3376 −1.24698
\(267\) 0 0
\(268\) 2.43720 0.148876
\(269\) −14.6753 −0.894766 −0.447383 0.894342i \(-0.647644\pi\)
−0.447383 + 0.894342i \(0.647644\pi\)
\(270\) 0 0
\(271\) −8.31058 −0.504832 −0.252416 0.967619i \(-0.581225\pi\)
−0.252416 + 0.967619i \(0.581225\pi\)
\(272\) −1.11903 −0.0678510
\(273\) 0 0
\(274\) 7.52918 0.454854
\(275\) −4.33763 −0.261569
\(276\) 0 0
\(277\) 12.9133 0.775886 0.387943 0.921683i \(-0.373186\pi\)
0.387943 + 0.921683i \(0.373186\pi\)
\(278\) −4.67526 −0.280403
\(279\) 0 0
\(280\) −4.50973 −0.269508
\(281\) −2.67526 −0.159592 −0.0797962 0.996811i \(-0.525427\pi\)
−0.0797962 + 0.996811i \(0.525427\pi\)
\(282\) 0 0
\(283\) 0.742495 0.0441367 0.0220684 0.999756i \(-0.492975\pi\)
0.0220684 + 0.999756i \(0.492975\pi\)
\(284\) −7.11903 −0.422437
\(285\) 0 0
\(286\) −16.1721 −0.956276
\(287\) −17.5898 −1.03830
\(288\) 0 0
\(289\) −15.7478 −0.926340
\(290\) −8.23805 −0.483755
\(291\) 0 0
\(292\) −9.45665 −0.553409
\(293\) 6.00000 0.350524 0.175262 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 0 0
\(295\) 2.23805 0.130305
\(296\) 0.781399 0.0454179
\(297\) 0 0
\(298\) 7.52918 0.436154
\(299\) −3.72833 −0.215615
\(300\) 0 0
\(301\) 36.0778 2.07949
\(302\) 13.3571 0.768614
\(303\) 0 0
\(304\) 4.50973 0.258651
\(305\) 3.55623 0.203629
\(306\) 0 0
\(307\) 30.5084 1.74121 0.870604 0.491984i \(-0.163728\pi\)
0.870604 + 0.491984i \(0.163728\pi\)
\(308\) −19.5615 −1.11462
\(309\) 0 0
\(310\) −1.72833 −0.0981624
\(311\) −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 \(0.463282\pi\)
\(314\) −16.2381 −0.916366
\(315\) 0 0
\(316\) −14.9133 −0.838939
\(317\) 6.16553 0.346291 0.173145 0.984896i \(-0.444607\pi\)
0.173145 + 0.984896i \(0.444607\pi\)
\(318\) 0 0
\(319\) −35.7336 −2.00070
\(320\) 1.00000 0.0559017
\(321\) 0 0
\(322\) −4.50973 −0.251317
\(323\) −5.04650 −0.280795
\(324\) 0 0
\(325\) −3.72833 −0.206810
\(326\) 3.29112 0.182279
\(327\) 0 0
\(328\) 3.90043 0.215365
\(329\) 51.6664 2.84846
\(330\) 0 0
\(331\) 27.5886 1.51640 0.758202 0.652019i \(-0.226078\pi\)
0.758202 + 0.652019i \(0.226078\pi\)
\(332\) −2.78140 −0.152649
\(333\) 0 0
\(334\) 22.9133 1.25376
\(335\) 2.43720 0.133159
\(336\) 0 0
\(337\) −17.4230 −0.949093 −0.474547 0.880230i \(-0.657388\pi\)
−0.474547 + 0.880230i \(0.657388\pi\)
\(338\) −0.900425 −0.0489767
\(339\) 0 0
\(340\) −1.11903 −0.0606877
\(341\) −7.49684 −0.405977
\(342\) 0 0
\(343\) 28.5810 1.54323
\(344\) −8.00000 −0.431331
\(345\) 0 0
\(346\) 0.575681 0.0309488
\(347\) −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 \(0.277529\pi\)
\(350\) −4.50973 −0.241055
\(351\) 0 0
\(352\) 4.33763 0.231196
\(353\) 14.3442 0.763464 0.381732 0.924273i \(-0.375328\pi\)
0.381732 + 0.924273i \(0.375328\pi\)
\(354\) 0 0
\(355\) −7.11903 −0.377839
\(356\) 7.69471 0.407819
\(357\) 0 0
\(358\) 5.01945 0.265286
\(359\) −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
\(362\) 11.5292 0.605960
\(363\) 0 0
\(364\) −16.8137 −0.881279
\(365\) −9.45665 −0.494984
\(366\) 0 0
\(367\) −20.4761 −1.06884 −0.534422 0.845218i \(-0.679471\pi\)
−0.534422 + 0.845218i \(0.679471\pi\)
\(368\) 1.00000 0.0521286
\(369\) 0 0
\(370\) 0.781399 0.0406230
\(371\) 27.0584 1.40480
\(372\) 0 0
\(373\) −3.89386 −0.201616 −0.100808 0.994906i \(-0.532143\pi\)
−0.100808 + 0.994906i \(0.532143\pi\)
\(374\) −4.85392 −0.250990
\(375\) 0 0
\(376\) −11.4567 −0.590832
\(377\) −30.7142 −1.58186
\(378\) 0 0
\(379\) −30.3765 −1.56034 −0.780169 0.625569i \(-0.784867\pi\)
−0.780169 + 0.625569i \(0.784867\pi\)
\(380\) 4.50973 0.231344
\(381\) 0 0
\(382\) −18.7142 −0.957500
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) −19.5615 −0.996947
\(386\) −23.4956 −1.19589
\(387\) 0 0
\(388\) −0.642920 −0.0326393
\(389\) 18.6818 0.947206 0.473603 0.880738i \(-0.342953\pi\)
0.473603 + 0.880738i \(0.342953\pi\)
\(390\) 0 0
\(391\) −1.11903 −0.0565916
\(392\) −13.3376 −0.673652
\(393\) 0 0
\(394\) −18.1385 −0.913803
\(395\) −14.9133 −0.750370
\(396\) 0 0
\(397\) −28.5757 −1.43417 −0.717086 0.696985i \(-0.754525\pi\)
−0.717086 + 0.696985i \(0.754525\pi\)
\(398\) 23.2575 1.16579
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −12.1061 −0.604552 −0.302276 0.953220i \(-0.597746\pi\)
−0.302276 + 0.953220i \(0.597746\pi\)
\(402\) 0 0
\(403\) −6.44377 −0.320987
\(404\) 8.23805 0.409858
\(405\) 0 0
\(406\) −37.1514 −1.84379
\(407\) 3.38942 0.168007
\(408\) 0 0
\(409\) 25.2911 1.25057 0.625283 0.780398i \(-0.284984\pi\)
0.625283 + 0.780398i \(0.284984\pi\)
\(410\) 3.90043 0.192628
\(411\) 0 0
\(412\) 12.3376 0.607831
\(413\) 10.0930 0.496644
\(414\) 0 0
\(415\) −2.78140 −0.136533
\(416\) 3.72833 0.182796
\(417\) 0 0
\(418\) 19.5615 0.956785
\(419\) 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 \(0.325168\pi\)
\(422\) −4.34420 −0.211472
\(423\) 0 0
\(424\) −6.00000 −0.291386
\(425\) −1.11903 −0.0542808
\(426\) 0 0
\(427\) 16.0376 0.776115
\(428\) 15.9328 0.770139
\(429\) 0 0
\(430\) −8.00000 −0.385794
\(431\) −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 \(0.492242\pi\)
\(434\) −7.79428 −0.374138
\(435\) 0 0
\(436\) −1.49027 −0.0713712
\(437\) 4.50973 0.215729
\(438\) 0 0
\(439\) −26.7478 −1.27660 −0.638301 0.769787i \(-0.720362\pi\)
−0.638301 + 0.769787i \(0.720362\pi\)
\(440\) 4.33763 0.206788
\(441\) 0 0
\(442\) −4.17210 −0.198446
\(443\) −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
\(446\) −12.4761 −0.590761
\(447\) 0 0
\(448\) 4.50973 0.213065
\(449\) −38.7867 −1.83046 −0.915228 0.402936i \(-0.867990\pi\)
−0.915228 + 0.402936i \(0.867990\pi\)
\(450\) 0 0
\(451\) 16.9186 0.796665
\(452\) 6.00000 0.282216
\(453\) 0 0
\(454\) 15.9328 0.747762
\(455\) −16.8137 −0.788240
\(456\) 0 0
\(457\) 34.9522 1.63500 0.817498 0.575932i \(-0.195361\pi\)
0.817498 + 0.575932i \(0.195361\pi\)
\(458\) 3.56280 0.166479
\(459\) 0 0
\(460\) 1.00000 0.0466252
\(461\) −16.3700 −0.762425 −0.381213 0.924487i \(-0.624493\pi\)
−0.381213 + 0.924487i \(0.624493\pi\)
\(462\) 0 0
\(463\) 29.2186 1.35790 0.678952 0.734183i \(-0.262435\pi\)
0.678952 + 0.734183i \(0.262435\pi\)
\(464\) 8.23805 0.382442
\(465\) 0 0
\(466\) −27.4956 −1.27371
\(467\) −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
\(470\) −11.4567 −0.528456
\(471\) 0 0
\(472\) −2.23805 −0.103015
\(473\) −34.7010 −1.59555
\(474\) 0 0
\(475\) 4.50973 0.206920
\(476\) −5.04650 −0.231306
\(477\) 0 0
\(478\) 10.0389 0.459169
\(479\) −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
\(482\) 23.6947 1.07926
\(483\) 0 0
\(484\) 7.81502 0.355228
\(485\) −0.642920 −0.0291935
\(486\) 0 0
\(487\) −30.2381 −1.37022 −0.685108 0.728441i \(-0.740245\pi\)
−0.685108 + 0.728441i \(0.740245\pi\)
\(488\) −3.55623 −0.160983
\(489\) 0 0
\(490\) −13.3376 −0.602533
\(491\) −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
\(494\) 16.8137 0.756486
\(495\) 0 0
\(496\) 1.72833 0.0776042
\(497\) −32.1049 −1.44010
\(498\) 0 0
\(499\) −26.9133 −1.20481 −0.602403 0.798192i \(-0.705790\pi\)
−0.602403 + 0.798192i \(0.705790\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) 12.4425 0.555335
\(503\) −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
\(506\) 4.33763 0.192831
\(507\) 0 0
\(508\) −0.675256 −0.0299597
\(509\) 36.7142 1.62733 0.813663 0.581336i \(-0.197470\pi\)
0.813663 + 0.581336i \(0.197470\pi\)
\(510\) 0 0
\(511\) −42.6469 −1.88659
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 5.45665 0.240683
\(515\) 12.3376 0.543661
\(516\) 0 0
\(517\) −49.6947 −2.18557
\(518\) 3.52389 0.154831
\(519\) 0 0
\(520\) 3.72833 0.163498
\(521\) 4.91331 0.215256 0.107628 0.994191i \(-0.465674\pi\)
0.107628 + 0.994191i \(0.465674\pi\)
\(522\) 0 0
\(523\) −0.344196 −0.0150506 −0.00752531 0.999972i \(-0.502395\pi\)
−0.00752531 + 0.999972i \(0.502395\pi\)
\(524\) 13.6947 0.598256
\(525\) 0 0
\(526\) 0.138479 0.00603795
\(527\) −1.93404 −0.0842483
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) −6.00000 −0.260623
\(531\) 0 0
\(532\) 20.3376 0.881748
\(533\) 14.5421 0.629887
\(534\) 0 0
\(535\) 15.9328 0.688833
\(536\) −2.43720 −0.105271
\(537\) 0 0
\(538\) 14.6753 0.632695
\(539\) −57.8537 −2.49193
\(540\) 0 0
\(541\) −6.13191 −0.263631 −0.131816 0.991274i \(-0.542081\pi\)
−0.131816 + 0.991274i \(0.542081\pi\)
\(542\) 8.31058 0.356970
\(543\) 0 0
\(544\) 1.11903 0.0479779
\(545\) −1.49027 −0.0638363
\(546\) 0 0
\(547\) −9.18498 −0.392721 −0.196361 0.980532i \(-0.562912\pi\)
−0.196361 + 0.980532i \(0.562912\pi\)
\(548\) −7.52918 −0.321631
\(549\) 0 0
\(550\) 4.33763 0.184957
\(551\) 37.1514 1.58270
\(552\) 0 0
\(553\) −67.2549 −2.85997
\(554\) −12.9133 −0.548634
\(555\) 0 0
\(556\) 4.67526 0.198275
\(557\) 4.30529 0.182421 0.0912105 0.995832i \(-0.470926\pi\)
0.0912105 + 0.995832i \(0.470926\pi\)
\(558\) 0 0
\(559\) −29.8266 −1.26153
\(560\) 4.50973 0.190571
\(561\) 0 0
\(562\) 2.67526 0.112849
\(563\) −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
\(566\) −0.742495 −0.0312094
\(567\) 0 0
\(568\) 7.11903 0.298708
\(569\) 16.0389 0.672386 0.336193 0.941793i \(-0.390861\pi\)
0.336193 + 0.941793i \(0.390861\pi\)
\(570\) 0 0
\(571\) 17.9004 0.749109 0.374555 0.927205i \(-0.377796\pi\)
0.374555 + 0.927205i \(0.377796\pi\)
\(572\) 16.1721 0.676189
\(573\) 0 0
\(574\) 17.5898 0.734186
\(575\) 1.00000 0.0417029
\(576\) 0 0
\(577\) −9.12559 −0.379903 −0.189952 0.981793i \(-0.560833\pi\)
−0.189952 + 0.981793i \(0.560833\pi\)
\(578\) 15.7478 0.655021
\(579\) 0 0
\(580\) 8.23805 0.342067
\(581\) −12.5433 −0.520386
\(582\) 0 0
\(583\) −26.0258 −1.07788
\(584\) 9.45665 0.391319
\(585\) 0 0
\(586\) −6.00000 −0.247858
\(587\) 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
\(590\) −2.23805 −0.0921392
\(591\) 0 0
\(592\) −0.781399 −0.0321153
\(593\) 17.4567 0.716859 0.358429 0.933557i \(-0.383312\pi\)
0.358429 + 0.933557i \(0.383312\pi\)
\(594\) 0 0
\(595\) −5.04650 −0.206886
\(596\) −7.52918 −0.308407
\(597\) 0 0
\(598\) 3.72833 0.152463
\(599\) 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 \(0.723452\pi\)
\(602\) −36.0778 −1.47042
\(603\) 0 0
\(604\) −13.3571 −0.543492
\(605\) 7.81502 0.317726
\(606\) 0 0
\(607\) −36.0778 −1.46435 −0.732177 0.681115i \(-0.761495\pi\)
−0.732177 + 0.681115i \(0.761495\pi\)
\(608\) −4.50973 −0.182894
\(609\) 0 0
\(610\) −3.55623 −0.143988
\(611\) −42.7142 −1.72803
\(612\) 0 0
\(613\) −32.0389 −1.29404 −0.647020 0.762473i \(-0.723985\pi\)
−0.647020 + 0.762473i \(0.723985\pi\)
\(614\) −30.5084 −1.23122
\(615\) 0 0
\(616\) 19.5615 0.788156
\(617\) 13.3960 0.539302 0.269651 0.962958i \(-0.413092\pi\)
0.269651 + 0.962958i \(0.413092\pi\)
\(618\) 0 0
\(619\) 37.1309 1.49242 0.746208 0.665713i \(-0.231872\pi\)
0.746208 + 0.665713i \(0.231872\pi\)
\(620\) 1.72833 0.0694113
\(621\) 0 0
\(622\) 5.56280 0.223048
\(623\) 34.7010 1.39027
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −4.07252 −0.162771
\(627\) 0 0
\(628\) 16.2381 0.647969
\(629\) 0.874406 0.0348648
\(630\) 0 0
\(631\) 11.1125 0.442380 0.221190 0.975231i \(-0.429006\pi\)
0.221190 + 0.975231i \(0.429006\pi\)
\(632\) 14.9133 0.593220
\(633\) 0 0
\(634\) −6.16553 −0.244864
\(635\) −0.675256 −0.0267967
\(636\) 0 0
\(637\) −49.7270 −1.97026
\(638\) 35.7336 1.41471
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) −12.3831 −0.489103 −0.244552 0.969636i \(-0.578641\pi\)
−0.244552 + 0.969636i \(0.578641\pi\)
\(642\) 0 0
\(643\) −37.4956 −1.47868 −0.739340 0.673332i \(-0.764862\pi\)
−0.739340 + 0.673332i \(0.764862\pi\)
\(644\) 4.50973 0.177708
\(645\) 0 0
\(646\) 5.04650 0.198552
\(647\) −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
\(650\) 3.72833 0.146237
\(651\) 0 0
\(652\) −3.29112 −0.128890
\(653\) 4.41672 0.172840 0.0864198 0.996259i \(-0.472457\pi\)
0.0864198 + 0.996259i \(0.472457\pi\)
\(654\) 0 0
\(655\) 13.6947 0.535097
\(656\) −3.90043 −0.152286
\(657\) 0 0
\(658\) −51.6664 −2.01416
\(659\) 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 \(0.277236\pi\)
\(662\) −27.5886 −1.07226
\(663\) 0 0
\(664\) 2.78140 0.107939
\(665\) 20.3376 0.788659
\(666\) 0 0
\(667\) 8.23805 0.318979
\(668\) −22.9133 −0.886543
\(669\) 0 0
\(670\) −2.43720 −0.0941574
\(671\) −15.4256 −0.595499
\(672\) 0 0
\(673\) 19.3505 0.745907 0.372954 0.927850i \(-0.378345\pi\)
0.372954 + 0.927850i \(0.378345\pi\)
\(674\) 17.4230 0.671110
\(675\) 0 0
\(676\) 0.900425 0.0346317
\(677\) 6.00000 0.230599 0.115299 0.993331i \(-0.463217\pi\)
0.115299 + 0.993331i \(0.463217\pi\)
\(678\) 0 0
\(679\) −2.89939 −0.111268
\(680\) 1.11903 0.0429127
\(681\) 0 0
\(682\) 7.49684 0.287069
\(683\) −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
\(686\) −28.5810 −1.09123
\(687\) 0 0
\(688\) 8.00000 0.304997
\(689\) −22.3700 −0.852228
\(690\) 0 0
\(691\) −21.0195 −0.799618 −0.399809 0.916599i \(-0.630923\pi\)
−0.399809 + 0.916599i \(0.630923\pi\)
\(692\) −0.575681 −0.0218841
\(693\) 0 0
\(694\) 4.88097 0.185279
\(695\) 4.67526 0.177343
\(696\) 0 0
\(697\) 4.36468 0.165324
\(698\) −24.0389 −0.909886
\(699\) 0 0
\(700\) 4.50973 0.170452
\(701\) −19.3169 −0.729589 −0.364794 0.931088i \(-0.618861\pi\)
−0.364794 + 0.931088i \(0.618861\pi\)
\(702\) 0 0
\(703\) −3.52389 −0.132906
\(704\) −4.33763 −0.163481
\(705\) 0 0
\(706\) −14.3442 −0.539851
\(707\) 37.1514 1.39722
\(708\) 0 0
\(709\) 12.2315 0.459363 0.229682 0.973266i \(-0.426232\pi\)
0.229682 + 0.973266i \(0.426232\pi\)
\(710\) 7.11903 0.267172
\(711\) 0 0
\(712\) −7.69471 −0.288371
\(713\) 1.72833 0.0647264
\(714\) 0 0
\(715\) 16.1721 0.604802
\(716\) −5.01945 −0.187586
\(717\) 0 0
\(718\) 26.7814 0.999473
\(719\) 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
\(722\) −1.33763 −0.0497814
\(723\) 0 0
\(724\) −11.5292 −0.428479
\(725\) 8.23805 0.305954
\(726\) 0 0
\(727\) −23.4501 −0.869716 −0.434858 0.900499i \(-0.643201\pi\)
−0.434858 + 0.900499i \(0.643201\pi\)
\(728\) 16.8137 0.623158
\(729\) 0 0
\(730\) 9.45665 0.350006
\(731\) −8.95221 −0.331110
\(732\) 0 0
\(733\) −12.5150 −0.462252 −0.231126 0.972924i \(-0.574241\pi\)
−0.231126 + 0.972924i \(0.574241\pi\)
\(734\) 20.4761 0.755787
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) −10.5717 −0.389413
\(738\) 0 0
\(739\) −21.3505 −0.785391 −0.392696 0.919668i \(-0.628457\pi\)
−0.392696 + 0.919668i \(0.628457\pi\)
\(740\) −0.781399 −0.0287248
\(741\) 0 0
\(742\) −27.0584 −0.993343
\(743\) −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
\(746\) 3.89386 0.142564
\(747\) 0 0
\(748\) 4.85392 0.177477
\(749\) 71.8524 2.62543
\(750\) 0 0
\(751\) −33.6275 −1.22708 −0.613542 0.789662i \(-0.710256\pi\)
−0.613542 + 0.789662i \(0.710256\pi\)
\(752\) 11.4567 0.417781
\(753\) 0 0
\(754\) 30.7142 1.11854
\(755\) −13.3571 −0.486114
\(756\) 0 0
\(757\) −37.1230 −1.34926 −0.674630 0.738156i \(-0.735697\pi\)
−0.674630 + 0.738156i \(0.735697\pi\)
\(758\) 30.3765 1.10333
\(759\) 0 0
\(760\) −4.50973 −0.163585
\(761\) 3.87337 0.140410 0.0702048 0.997533i \(-0.477635\pi\)
0.0702048 + 0.997533i \(0.477635\pi\)
\(762\) 0 0
\(763\) −6.72073 −0.243307
\(764\) 18.7142 0.677055
\(765\) 0 0
\(766\) 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 \(0.362848\pi\)
\(770\) 19.5615 0.704948
\(771\) 0 0
\(772\) 23.4956 0.845624
\(773\) 8.78140 0.315845 0.157922 0.987452i \(-0.449520\pi\)
0.157922 + 0.987452i \(0.449520\pi\)
\(774\) 0 0
\(775\) 1.72833 0.0620834
\(776\) 0.642920 0.0230795
\(777\) 0 0
\(778\) −18.6818 −0.669776
\(779\) −17.5898 −0.630222
\(780\) 0 0
\(781\) 30.8797 1.10496
\(782\) 1.11903 0.0400163
\(783\) 0 0
\(784\) 13.3376 0.476344
\(785\) 16.2381 0.579561
\(786\) 0 0
\(787\) 49.6275 1.76903 0.884514 0.466513i \(-0.154490\pi\)
0.884514 + 0.466513i \(0.154490\pi\)
\(788\) 18.1385 0.646157
\(789\) 0 0
\(790\) 14.9133 0.530592
\(791\) 27.0584 0.962084
\(792\) 0 0
\(793\) −13.2588 −0.470833
\(794\) 28.5757 1.01411
\(795\) 0 0
\(796\) −23.2575 −0.824340
\(797\) −18.3311 −0.649319 −0.324660 0.945831i \(-0.605250\pi\)
−0.324660 + 0.945831i \(0.605250\pi\)
\(798\) 0 0
\(799\) −12.8203 −0.453550
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) 12.1061 0.427483
\(803\) 41.0195 1.44755
\(804\) 0 0
\(805\) 4.50973 0.158947
\(806\) 6.44377 0.226972
\(807\) 0 0
\(808\) −8.23805 −0.289814
\(809\) 1.93933 0.0681832 0.0340916 0.999419i \(-0.489146\pi\)
0.0340916 + 0.999419i \(0.489146\pi\)
\(810\) 0 0
\(811\) −5.41775 −0.190243 −0.0951215 0.995466i \(-0.530324\pi\)
−0.0951215 + 0.995466i \(0.530324\pi\)
\(812\) 37.1514 1.30376
\(813\) 0 0
\(814\) −3.38942 −0.118799
\(815\) −3.29112 −0.115283
\(816\) 0 0
\(817\) 36.0778 1.26220
\(818\) −25.2911 −0.884283
\(819\) 0 0
\(820\) −3.90043 −0.136209
\(821\) −37.8655 −1.32152 −0.660758 0.750599i \(-0.729765\pi\)
−0.660758 + 0.750599i \(0.729765\pi\)
\(822\) 0 0
\(823\) 43.7336 1.52446 0.762229 0.647308i \(-0.224105\pi\)
0.762229 + 0.647308i \(0.224105\pi\)
\(824\) −12.3376 −0.429802
\(825\) 0 0
\(826\) −10.0930 −0.351181
\(827\) −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 \(0.493778\pi\)
\(830\) 2.78140 0.0965438
\(831\) 0 0
\(832\) −3.72833 −0.129256
\(833\) −14.9252 −0.517126
\(834\) 0 0
\(835\) −22.9133 −0.792948
\(836\) −19.5615 −0.676549
\(837\) 0 0
\(838\) −17.3505 −0.599364
\(839\) −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
\(842\) −21.4230 −0.738287
\(843\) 0 0
\(844\) 4.34420 0.149533
\(845\) 0.900425 0.0309756
\(846\) 0 0
\(847\) 35.2436 1.21098
\(848\) 6.00000 0.206041
\(849\) 0 0
\(850\) 1.11903 0.0383823
\(851\) −0.781399 −0.0267860
\(852\) 0 0
\(853\) 47.9921 1.64322 0.821610 0.570050i \(-0.193076\pi\)
0.821610 + 0.570050i \(0.193076\pi\)
\(854\) −16.0376 −0.548796
\(855\) 0 0
\(856\) −15.9328 −0.544571
\(857\) 43.4283 1.48348 0.741742 0.670686i \(-0.234000\pi\)
0.741742 + 0.670686i \(0.234000\pi\)
\(858\) 0 0
\(859\) 32.5433 1.11036 0.555182 0.831729i \(-0.312648\pi\)
0.555182 + 0.831729i \(0.312648\pi\)
\(860\) 8.00000 0.272798
\(861\) 0 0
\(862\) 22.5822 0.769154
\(863\) −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
\(866\) −1.01417 −0.0344628
\(867\) 0 0
\(868\) 7.79428 0.264555
\(869\) 64.6884 2.19440
\(870\) 0 0
\(871\) −9.08669 −0.307891
\(872\) 1.49027 0.0504670
\(873\) 0 0
\(874\) −4.50973 −0.152544
\(875\) 4.50973 0.152457
\(876\) 0 0
\(877\) −24.0996 −0.813785 −0.406892 0.913476i \(-0.633388\pi\)
−0.406892 + 0.913476i \(0.633388\pi\)
\(878\) 26.7478 0.902694
\(879\) 0 0
\(880\) −4.33763 −0.146221
\(881\) −2.34420 −0.0789780 −0.0394890 0.999220i \(-0.512573\pi\)
−0.0394890 + 0.999220i \(0.512573\pi\)
\(882\) 0 0
\(883\) −41.0505 −1.38146 −0.690730 0.723113i \(-0.742711\pi\)
−0.690730 + 0.723113i \(0.742711\pi\)
\(884\) 4.17210 0.140323
\(885\) 0 0
\(886\) 10.2044 0.342825
\(887\) −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
\(890\) −7.69471 −0.257927
\(891\) 0 0
\(892\) 12.4761 0.417731
\(893\) 51.6664 1.72895
\(894\) 0 0
\(895\) −5.01945 −0.167782
\(896\) −4.50973 −0.150659
\(897\) 0 0
\(898\) 38.7867 1.29433
\(899\) 14.2381 0.474866
\(900\) 0 0
\(901\) −6.71416 −0.223681
\(902\) −16.9186 −0.563328
\(903\) 0 0
\(904\) −6.00000 −0.199557
\(905\) −11.5292 −0.383243
\(906\) 0 0
\(907\) −10.1061 −0.335569 −0.167784 0.985824i \(-0.553661\pi\)
−0.167784 + 0.985824i \(0.553661\pi\)
\(908\) −15.9328 −0.528747
\(909\) 0 0
\(910\) 16.8137 0.557370
\(911\) 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
\(914\) −34.9522 −1.15612
\(915\) 0 0
\(916\) −3.56280 −0.117718
\(917\) 61.7594 2.03947
\(918\) 0 0
\(919\) 23.6017 0.778548 0.389274 0.921122i \(-0.372726\pi\)
0.389274 + 0.921122i \(0.372726\pi\)
\(920\) −1.00000 −0.0329690
\(921\) 0 0
\(922\) 16.3700 0.539116
\(923\) 26.5421 0.873643
\(924\) 0 0
\(925\) −0.781399 −0.0256922
\(926\) −29.2186 −0.960183
\(927\) 0 0
\(928\) −8.23805 −0.270427
\(929\) 7.08669 0.232507 0.116253 0.993220i \(-0.462912\pi\)
0.116253 + 0.993220i \(0.462912\pi\)
\(930\) 0 0
\(931\) 60.1490 1.97131
\(932\) 27.4956 0.900647
\(933\) 0 0
\(934\) 24.2770 0.794366
\(935\) 4.85392 0.158740
\(936\) 0 0
\(937\) 27.3169 0.892404 0.446202 0.894932i \(-0.352776\pi\)
0.446202 + 0.894932i \(0.352776\pi\)
\(938\) −10.9911 −0.358873
\(939\) 0 0
\(940\) 11.4567 0.373675
\(941\) −55.8979 −1.82222 −0.911109 0.412165i \(-0.864773\pi\)
−0.911109 + 0.412165i \(0.864773\pi\)
\(942\) 0 0
\(943\) −3.90043 −0.127015
\(944\) 2.23805 0.0728424
\(945\) 0 0
\(946\) 34.7010 1.12823
\(947\) −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
\(950\) −4.50973 −0.146315
\(951\) 0 0
\(952\) 5.04650 0.163558
\(953\) 29.3828 0.951804 0.475902 0.879498i \(-0.342122\pi\)
0.475902 + 0.879498i \(0.342122\pi\)
\(954\) 0 0
\(955\) 18.7142 0.605576
\(956\) −10.0389 −0.324681
\(957\) 0 0
\(958\) 24.6080 0.795049
\(959\) −33.9545 −1.09645
\(960\) 0 0
\(961\) −28.0129 −0.903641
\(962\) −2.91331 −0.0939289
\(963\) 0 0
\(964\) −23.6947 −0.763155
\(965\) 23.4956 0.756349
\(966\) 0 0
\(967\) −49.2292 −1.58310 −0.791552 0.611102i \(-0.790726\pi\)
−0.791552 + 0.611102i \(0.790726\pi\)
\(968\) −7.81502 −0.251184
\(969\) 0 0
\(970\) 0.642920 0.0206429
\(971\) −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
\(974\) 30.2381 0.968890
\(975\) 0 0
\(976\) 3.55623 0.113832
\(977\) −18.9858 −0.607411 −0.303705 0.952766i \(-0.598224\pi\)
−0.303705 + 0.952766i \(0.598224\pi\)
\(978\) 0 0
\(979\) −33.3768 −1.06673
\(980\) 13.3376 0.426055
\(981\) 0 0
\(982\) 12.3311 0.393500
\(983\) −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
\(986\) 9.21860 0.293580
\(987\) 0 0
\(988\) −16.8137 −0.534916
\(989\) 8.00000 0.254385
\(990\) 0 0
\(991\) 1.96766 0.0625049 0.0312524 0.999512i \(-0.490050\pi\)
0.0312524 + 0.999512i \(0.490050\pi\)
\(992\) −1.72833 −0.0548744
\(993\) 0 0
\(994\) 32.1049 1.01830
\(995\) −23.2575 −0.737312
\(996\) 0 0
\(997\) 3.96110 0.125449 0.0627246 0.998031i \(-0.480021\pi\)
0.0627246 + 0.998031i \(0.480021\pi\)
\(998\) 26.9133 0.851926
\(999\) 0 0
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 2070.2.a.z.1.3 3
3.2 odd 2 230.2.a.d.1.1 3
12.11 even 2 1840.2.a.r.1.3 3
15.2 even 4 1150.2.b.j.599.6 6
15.8 even 4 1150.2.b.j.599.1 6
15.14 odd 2 1150.2.a.q.1.3 3
24.5 odd 2 7360.2.a.bz.1.3 3
24.11 even 2 7360.2.a.ce.1.1 3
60.59 even 2 9200.2.a.cf.1.1 3
69.68 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 3.2 odd 2
1150.2.a.q.1.3 3 15.14 odd 2
1150.2.b.j.599.1 6 15.8 even 4
1150.2.b.j.599.6 6 15.2 even 4
1840.2.a.r.1.3 3 12.11 even 2
2070.2.a.z.1.3 3 1.1 even 1 trivial
5290.2.a.r.1.1 3 69.68 even 2
7360.2.a.bz.1.3 3 24.5 odd 2
7360.2.a.ce.1.1 3 24.11 even 2
9200.2.a.cf.1.1 3 60.59 even 2