Properties

Label 2394.2.a.s.1.2
Level $2394$
Weight $2$
Character 2394.1
Self dual yes
Analytic conductor $19.116$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2394.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(19.1161862439\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 2394.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +3.23607 q^{5} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +3.23607 q^{5} -1.00000 q^{7} -1.00000 q^{8} -3.23607 q^{10} +0.763932 q^{11} +1.23607 q^{13} +1.00000 q^{14} +1.00000 q^{16} -7.70820 q^{17} +1.00000 q^{19} +3.23607 q^{20} -0.763932 q^{22} +5.23607 q^{23} +5.47214 q^{25} -1.23607 q^{26} -1.00000 q^{28} +8.47214 q^{29} -2.00000 q^{31} -1.00000 q^{32} +7.70820 q^{34} -3.23607 q^{35} -10.4721 q^{37} -1.00000 q^{38} -3.23607 q^{40} +6.00000 q^{41} +8.94427 q^{43} +0.763932 q^{44} -5.23607 q^{46} +8.94427 q^{47} +1.00000 q^{49} -5.47214 q^{50} +1.23607 q^{52} +12.4721 q^{53} +2.47214 q^{55} +1.00000 q^{56} -8.47214 q^{58} +8.00000 q^{59} -0.472136 q^{61} +2.00000 q^{62} +1.00000 q^{64} +4.00000 q^{65} +0.291796 q^{67} -7.70820 q^{68} +3.23607 q^{70} +10.4721 q^{71} -12.4721 q^{73} +10.4721 q^{74} +1.00000 q^{76} -0.763932 q^{77} +0.763932 q^{79} +3.23607 q^{80} -6.00000 q^{82} -10.0000 q^{83} -24.9443 q^{85} -8.94427 q^{86} -0.763932 q^{88} +6.00000 q^{89} -1.23607 q^{91} +5.23607 q^{92} -8.94427 q^{94} +3.23607 q^{95} +4.76393 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 2 q^{7} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 2 q^{7} - 2 q^{8} - 2 q^{10} + 6 q^{11} - 2 q^{13} + 2 q^{14} + 2 q^{16} - 2 q^{17} + 2 q^{19} + 2 q^{20} - 6 q^{22} + 6 q^{23} + 2 q^{25} + 2 q^{26} - 2 q^{28} + 8 q^{29} - 4 q^{31} - 2 q^{32} + 2 q^{34} - 2 q^{35} - 12 q^{37} - 2 q^{38} - 2 q^{40} + 12 q^{41} + 6 q^{44} - 6 q^{46} + 2 q^{49} - 2 q^{50} - 2 q^{52} + 16 q^{53} - 4 q^{55} + 2 q^{56} - 8 q^{58} + 16 q^{59} + 8 q^{61} + 4 q^{62} + 2 q^{64} + 8 q^{65} + 14 q^{67} - 2 q^{68} + 2 q^{70} + 12 q^{71} - 16 q^{73} + 12 q^{74} + 2 q^{76} - 6 q^{77} + 6 q^{79} + 2 q^{80} - 12 q^{82} - 20 q^{83} - 32 q^{85} - 6 q^{88} + 12 q^{89} + 2 q^{91} + 6 q^{92} + 2 q^{95} + 14 q^{97} - 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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
\(5\) 3.23607 1.44721 0.723607 0.690212i \(-0.242483\pi\)
0.723607 + 0.690212i \(0.242483\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) −3.23607 −1.02333
\(11\) 0.763932 0.230334 0.115167 0.993346i \(-0.463260\pi\)
0.115167 + 0.993346i \(0.463260\pi\)
\(12\) 0 0
\(13\) 1.23607 0.342824 0.171412 0.985199i \(-0.445167\pi\)
0.171412 + 0.985199i \(0.445167\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −7.70820 −1.86951 −0.934757 0.355288i \(-0.884383\pi\)
−0.934757 + 0.355288i \(0.884383\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 3.23607 0.723607
\(21\) 0 0
\(22\) −0.763932 −0.162871
\(23\) 5.23607 1.09180 0.545898 0.837852i \(-0.316189\pi\)
0.545898 + 0.837852i \(0.316189\pi\)
\(24\) 0 0
\(25\) 5.47214 1.09443
\(26\) −1.23607 −0.242413
\(27\) 0 0
\(28\) −1.00000 −0.188982
\(29\) 8.47214 1.57324 0.786618 0.617440i \(-0.211830\pi\)
0.786618 + 0.617440i \(0.211830\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 7.70820 1.32195
\(35\) −3.23607 −0.546995
\(36\) 0 0
\(37\) −10.4721 −1.72161 −0.860804 0.508936i \(-0.830039\pi\)
−0.860804 + 0.508936i \(0.830039\pi\)
\(38\) −1.00000 −0.162221
\(39\) 0 0
\(40\) −3.23607 −0.511667
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) 8.94427 1.36399 0.681994 0.731357i \(-0.261113\pi\)
0.681994 + 0.731357i \(0.261113\pi\)
\(44\) 0.763932 0.115167
\(45\) 0 0
\(46\) −5.23607 −0.772016
\(47\) 8.94427 1.30466 0.652328 0.757937i \(-0.273792\pi\)
0.652328 + 0.757937i \(0.273792\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −5.47214 −0.773877
\(51\) 0 0
\(52\) 1.23607 0.171412
\(53\) 12.4721 1.71318 0.856590 0.515998i \(-0.172579\pi\)
0.856590 + 0.515998i \(0.172579\pi\)
\(54\) 0 0
\(55\) 2.47214 0.333343
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) −8.47214 −1.11245
\(59\) 8.00000 1.04151 0.520756 0.853706i \(-0.325650\pi\)
0.520756 + 0.853706i \(0.325650\pi\)
\(60\) 0 0
\(61\) −0.472136 −0.0604508 −0.0302254 0.999543i \(-0.509623\pi\)
−0.0302254 + 0.999543i \(0.509623\pi\)
\(62\) 2.00000 0.254000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 4.00000 0.496139
\(66\) 0 0
\(67\) 0.291796 0.0356486 0.0178243 0.999841i \(-0.494326\pi\)
0.0178243 + 0.999841i \(0.494326\pi\)
\(68\) −7.70820 −0.934757
\(69\) 0 0
\(70\) 3.23607 0.386784
\(71\) 10.4721 1.24281 0.621407 0.783488i \(-0.286561\pi\)
0.621407 + 0.783488i \(0.286561\pi\)
\(72\) 0 0
\(73\) −12.4721 −1.45975 −0.729877 0.683579i \(-0.760422\pi\)
−0.729877 + 0.683579i \(0.760422\pi\)
\(74\) 10.4721 1.21736
\(75\) 0 0
\(76\) 1.00000 0.114708
\(77\) −0.763932 −0.0870581
\(78\) 0 0
\(79\) 0.763932 0.0859491 0.0429745 0.999076i \(-0.486317\pi\)
0.0429745 + 0.999076i \(0.486317\pi\)
\(80\) 3.23607 0.361803
\(81\) 0 0
\(82\) −6.00000 −0.662589
\(83\) −10.0000 −1.09764 −0.548821 0.835940i \(-0.684923\pi\)
−0.548821 + 0.835940i \(0.684923\pi\)
\(84\) 0 0
\(85\) −24.9443 −2.70559
\(86\) −8.94427 −0.964486
\(87\) 0 0
\(88\) −0.763932 −0.0814354
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) −1.23607 −0.129575
\(92\) 5.23607 0.545898
\(93\) 0 0
\(94\) −8.94427 −0.922531
\(95\) 3.23607 0.332014
\(96\) 0 0
\(97\) 4.76393 0.483704 0.241852 0.970313i \(-0.422245\pi\)
0.241852 + 0.970313i \(0.422245\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 5.47214 0.547214
\(101\) −13.7082 −1.36402 −0.682009 0.731344i \(-0.738893\pi\)
−0.682009 + 0.731344i \(0.738893\pi\)
\(102\) 0 0
\(103\) −6.94427 −0.684239 −0.342120 0.939656i \(-0.611145\pi\)
−0.342120 + 0.939656i \(0.611145\pi\)
\(104\) −1.23607 −0.121206
\(105\) 0 0
\(106\) −12.4721 −1.21140
\(107\) 5.52786 0.534399 0.267199 0.963641i \(-0.413902\pi\)
0.267199 + 0.963641i \(0.413902\pi\)
\(108\) 0 0
\(109\) 18.4721 1.76931 0.884655 0.466246i \(-0.154394\pi\)
0.884655 + 0.466246i \(0.154394\pi\)
\(110\) −2.47214 −0.235709
\(111\) 0 0
\(112\) −1.00000 −0.0944911
\(113\) 10.0000 0.940721 0.470360 0.882474i \(-0.344124\pi\)
0.470360 + 0.882474i \(0.344124\pi\)
\(114\) 0 0
\(115\) 16.9443 1.58006
\(116\) 8.47214 0.786618
\(117\) 0 0
\(118\) −8.00000 −0.736460
\(119\) 7.70820 0.706610
\(120\) 0 0
\(121\) −10.4164 −0.946946
\(122\) 0.472136 0.0427452
\(123\) 0 0
\(124\) −2.00000 −0.179605
\(125\) 1.52786 0.136656
\(126\) 0 0
\(127\) −9.70820 −0.861464 −0.430732 0.902480i \(-0.641745\pi\)
−0.430732 + 0.902480i \(0.641745\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) −4.00000 −0.350823
\(131\) −16.4721 −1.43918 −0.719589 0.694401i \(-0.755670\pi\)
−0.719589 + 0.694401i \(0.755670\pi\)
\(132\) 0 0
\(133\) −1.00000 −0.0867110
\(134\) −0.291796 −0.0252073
\(135\) 0 0
\(136\) 7.70820 0.660973
\(137\) −15.4164 −1.31711 −0.658556 0.752531i \(-0.728833\pi\)
−0.658556 + 0.752531i \(0.728833\pi\)
\(138\) 0 0
\(139\) 21.8885 1.85656 0.928281 0.371879i \(-0.121287\pi\)
0.928281 + 0.371879i \(0.121287\pi\)
\(140\) −3.23607 −0.273498
\(141\) 0 0
\(142\) −10.4721 −0.878802
\(143\) 0.944272 0.0789640
\(144\) 0 0
\(145\) 27.4164 2.27681
\(146\) 12.4721 1.03220
\(147\) 0 0
\(148\) −10.4721 −0.860804
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) 0.763932 0.0621679 0.0310840 0.999517i \(-0.490104\pi\)
0.0310840 + 0.999517i \(0.490104\pi\)
\(152\) −1.00000 −0.0811107
\(153\) 0 0
\(154\) 0.763932 0.0615594
\(155\) −6.47214 −0.519854
\(156\) 0 0
\(157\) 13.4164 1.07075 0.535373 0.844616i \(-0.320171\pi\)
0.535373 + 0.844616i \(0.320171\pi\)
\(158\) −0.763932 −0.0607752
\(159\) 0 0
\(160\) −3.23607 −0.255834
\(161\) −5.23607 −0.412660
\(162\) 0 0
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) 6.00000 0.468521
\(165\) 0 0
\(166\) 10.0000 0.776151
\(167\) 16.9443 1.31119 0.655594 0.755114i \(-0.272418\pi\)
0.655594 + 0.755114i \(0.272418\pi\)
\(168\) 0 0
\(169\) −11.4721 −0.882472
\(170\) 24.9443 1.91314
\(171\) 0 0
\(172\) 8.94427 0.681994
\(173\) −2.00000 −0.152057 −0.0760286 0.997106i \(-0.524224\pi\)
−0.0760286 + 0.997106i \(0.524224\pi\)
\(174\) 0 0
\(175\) −5.47214 −0.413655
\(176\) 0.763932 0.0575835
\(177\) 0 0
\(178\) −6.00000 −0.449719
\(179\) −13.8885 −1.03808 −0.519039 0.854750i \(-0.673710\pi\)
−0.519039 + 0.854750i \(0.673710\pi\)
\(180\) 0 0
\(181\) −5.23607 −0.389194 −0.194597 0.980883i \(-0.562340\pi\)
−0.194597 + 0.980883i \(0.562340\pi\)
\(182\) 1.23607 0.0916235
\(183\) 0 0
\(184\) −5.23607 −0.386008
\(185\) −33.8885 −2.49154
\(186\) 0 0
\(187\) −5.88854 −0.430613
\(188\) 8.94427 0.652328
\(189\) 0 0
\(190\) −3.23607 −0.234769
\(191\) 17.2361 1.24716 0.623579 0.781760i \(-0.285678\pi\)
0.623579 + 0.781760i \(0.285678\pi\)
\(192\) 0 0
\(193\) −18.0000 −1.29567 −0.647834 0.761781i \(-0.724325\pi\)
−0.647834 + 0.761781i \(0.724325\pi\)
\(194\) −4.76393 −0.342030
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −10.9443 −0.779747 −0.389874 0.920868i \(-0.627481\pi\)
−0.389874 + 0.920868i \(0.627481\pi\)
\(198\) 0 0
\(199\) 15.4164 1.09284 0.546420 0.837511i \(-0.315990\pi\)
0.546420 + 0.837511i \(0.315990\pi\)
\(200\) −5.47214 −0.386938
\(201\) 0 0
\(202\) 13.7082 0.964506
\(203\) −8.47214 −0.594627
\(204\) 0 0
\(205\) 19.4164 1.35610
\(206\) 6.94427 0.483830
\(207\) 0 0
\(208\) 1.23607 0.0857059
\(209\) 0.763932 0.0528423
\(210\) 0 0
\(211\) −21.5967 −1.48678 −0.743391 0.668857i \(-0.766784\pi\)
−0.743391 + 0.668857i \(0.766784\pi\)
\(212\) 12.4721 0.856590
\(213\) 0 0
\(214\) −5.52786 −0.377877
\(215\) 28.9443 1.97398
\(216\) 0 0
\(217\) 2.00000 0.135769
\(218\) −18.4721 −1.25109
\(219\) 0 0
\(220\) 2.47214 0.166671
\(221\) −9.52786 −0.640913
\(222\) 0 0
\(223\) 28.4721 1.90664 0.953318 0.301969i \(-0.0976440\pi\)
0.953318 + 0.301969i \(0.0976440\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) −10.0000 −0.665190
\(227\) 11.4164 0.757734 0.378867 0.925451i \(-0.376314\pi\)
0.378867 + 0.925451i \(0.376314\pi\)
\(228\) 0 0
\(229\) −1.05573 −0.0697645 −0.0348822 0.999391i \(-0.511106\pi\)
−0.0348822 + 0.999391i \(0.511106\pi\)
\(230\) −16.9443 −1.11727
\(231\) 0 0
\(232\) −8.47214 −0.556223
\(233\) 5.52786 0.362142 0.181071 0.983470i \(-0.442044\pi\)
0.181071 + 0.983470i \(0.442044\pi\)
\(234\) 0 0
\(235\) 28.9443 1.88812
\(236\) 8.00000 0.520756
\(237\) 0 0
\(238\) −7.70820 −0.499649
\(239\) 26.1803 1.69347 0.846733 0.532019i \(-0.178566\pi\)
0.846733 + 0.532019i \(0.178566\pi\)
\(240\) 0 0
\(241\) −0.763932 −0.0492092 −0.0246046 0.999697i \(-0.507833\pi\)
−0.0246046 + 0.999697i \(0.507833\pi\)
\(242\) 10.4164 0.669592
\(243\) 0 0
\(244\) −0.472136 −0.0302254
\(245\) 3.23607 0.206745
\(246\) 0 0
\(247\) 1.23607 0.0786491
\(248\) 2.00000 0.127000
\(249\) 0 0
\(250\) −1.52786 −0.0966306
\(251\) −1.41641 −0.0894029 −0.0447014 0.999000i \(-0.514234\pi\)
−0.0447014 + 0.999000i \(0.514234\pi\)
\(252\) 0 0
\(253\) 4.00000 0.251478
\(254\) 9.70820 0.609147
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −3.52786 −0.220062 −0.110031 0.993928i \(-0.535095\pi\)
−0.110031 + 0.993928i \(0.535095\pi\)
\(258\) 0 0
\(259\) 10.4721 0.650707
\(260\) 4.00000 0.248069
\(261\) 0 0
\(262\) 16.4721 1.01765
\(263\) 7.70820 0.475308 0.237654 0.971350i \(-0.423622\pi\)
0.237654 + 0.971350i \(0.423622\pi\)
\(264\) 0 0
\(265\) 40.3607 2.47934
\(266\) 1.00000 0.0613139
\(267\) 0 0
\(268\) 0.291796 0.0178243
\(269\) −6.94427 −0.423400 −0.211700 0.977335i \(-0.567900\pi\)
−0.211700 + 0.977335i \(0.567900\pi\)
\(270\) 0 0
\(271\) 12.0000 0.728948 0.364474 0.931214i \(-0.381249\pi\)
0.364474 + 0.931214i \(0.381249\pi\)
\(272\) −7.70820 −0.467379
\(273\) 0 0
\(274\) 15.4164 0.931339
\(275\) 4.18034 0.252084
\(276\) 0 0
\(277\) −14.9443 −0.897914 −0.448957 0.893553i \(-0.648204\pi\)
−0.448957 + 0.893553i \(0.648204\pi\)
\(278\) −21.8885 −1.31279
\(279\) 0 0
\(280\) 3.23607 0.193392
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) −21.8885 −1.30114 −0.650569 0.759447i \(-0.725470\pi\)
−0.650569 + 0.759447i \(0.725470\pi\)
\(284\) 10.4721 0.621407
\(285\) 0 0
\(286\) −0.944272 −0.0558360
\(287\) −6.00000 −0.354169
\(288\) 0 0
\(289\) 42.4164 2.49508
\(290\) −27.4164 −1.60995
\(291\) 0 0
\(292\) −12.4721 −0.729877
\(293\) 15.8885 0.928219 0.464109 0.885778i \(-0.346374\pi\)
0.464109 + 0.885778i \(0.346374\pi\)
\(294\) 0 0
\(295\) 25.8885 1.50729
\(296\) 10.4721 0.608681
\(297\) 0 0
\(298\) −6.00000 −0.347571
\(299\) 6.47214 0.374293
\(300\) 0 0
\(301\) −8.94427 −0.515539
\(302\) −0.763932 −0.0439593
\(303\) 0 0
\(304\) 1.00000 0.0573539
\(305\) −1.52786 −0.0874852
\(306\) 0 0
\(307\) 4.58359 0.261599 0.130800 0.991409i \(-0.458246\pi\)
0.130800 + 0.991409i \(0.458246\pi\)
\(308\) −0.763932 −0.0435291
\(309\) 0 0
\(310\) 6.47214 0.367593
\(311\) 16.3607 0.927729 0.463865 0.885906i \(-0.346462\pi\)
0.463865 + 0.885906i \(0.346462\pi\)
\(312\) 0 0
\(313\) 6.94427 0.392513 0.196257 0.980553i \(-0.437121\pi\)
0.196257 + 0.980553i \(0.437121\pi\)
\(314\) −13.4164 −0.757132
\(315\) 0 0
\(316\) 0.763932 0.0429745
\(317\) −27.8885 −1.56638 −0.783188 0.621785i \(-0.786408\pi\)
−0.783188 + 0.621785i \(0.786408\pi\)
\(318\) 0 0
\(319\) 6.47214 0.362370
\(320\) 3.23607 0.180902
\(321\) 0 0
\(322\) 5.23607 0.291795
\(323\) −7.70820 −0.428896
\(324\) 0 0
\(325\) 6.76393 0.375195
\(326\) −4.00000 −0.221540
\(327\) 0 0
\(328\) −6.00000 −0.331295
\(329\) −8.94427 −0.493114
\(330\) 0 0
\(331\) 29.5967 1.62678 0.813392 0.581716i \(-0.197618\pi\)
0.813392 + 0.581716i \(0.197618\pi\)
\(332\) −10.0000 −0.548821
\(333\) 0 0
\(334\) −16.9443 −0.927149
\(335\) 0.944272 0.0515911
\(336\) 0 0
\(337\) −20.4721 −1.11519 −0.557594 0.830114i \(-0.688275\pi\)
−0.557594 + 0.830114i \(0.688275\pi\)
\(338\) 11.4721 0.624002
\(339\) 0 0
\(340\) −24.9443 −1.35279
\(341\) −1.52786 −0.0827385
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) −8.94427 −0.482243
\(345\) 0 0
\(346\) 2.00000 0.107521
\(347\) 6.29180 0.337761 0.168881 0.985637i \(-0.445985\pi\)
0.168881 + 0.985637i \(0.445985\pi\)
\(348\) 0 0
\(349\) −24.8328 −1.32927 −0.664635 0.747168i \(-0.731413\pi\)
−0.664635 + 0.747168i \(0.731413\pi\)
\(350\) 5.47214 0.292498
\(351\) 0 0
\(352\) −0.763932 −0.0407177
\(353\) −11.7082 −0.623165 −0.311582 0.950219i \(-0.600859\pi\)
−0.311582 + 0.950219i \(0.600859\pi\)
\(354\) 0 0
\(355\) 33.8885 1.79862
\(356\) 6.00000 0.317999
\(357\) 0 0
\(358\) 13.8885 0.734032
\(359\) −11.1246 −0.587135 −0.293567 0.955938i \(-0.594842\pi\)
−0.293567 + 0.955938i \(0.594842\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 5.23607 0.275202
\(363\) 0 0
\(364\) −1.23607 −0.0647876
\(365\) −40.3607 −2.11257
\(366\) 0 0
\(367\) −4.00000 −0.208798 −0.104399 0.994535i \(-0.533292\pi\)
−0.104399 + 0.994535i \(0.533292\pi\)
\(368\) 5.23607 0.272949
\(369\) 0 0
\(370\) 33.8885 1.76178
\(371\) −12.4721 −0.647521
\(372\) 0 0
\(373\) −4.94427 −0.256005 −0.128002 0.991774i \(-0.540857\pi\)
−0.128002 + 0.991774i \(0.540857\pi\)
\(374\) 5.88854 0.304489
\(375\) 0 0
\(376\) −8.94427 −0.461266
\(377\) 10.4721 0.539342
\(378\) 0 0
\(379\) −27.1246 −1.39330 −0.696649 0.717412i \(-0.745326\pi\)
−0.696649 + 0.717412i \(0.745326\pi\)
\(380\) 3.23607 0.166007
\(381\) 0 0
\(382\) −17.2361 −0.881874
\(383\) −8.00000 −0.408781 −0.204390 0.978889i \(-0.565521\pi\)
−0.204390 + 0.978889i \(0.565521\pi\)
\(384\) 0 0
\(385\) −2.47214 −0.125992
\(386\) 18.0000 0.916176
\(387\) 0 0
\(388\) 4.76393 0.241852
\(389\) −2.58359 −0.130993 −0.0654967 0.997853i \(-0.520863\pi\)
−0.0654967 + 0.997853i \(0.520863\pi\)
\(390\) 0 0
\(391\) −40.3607 −2.04113
\(392\) −1.00000 −0.0505076
\(393\) 0 0
\(394\) 10.9443 0.551364
\(395\) 2.47214 0.124387
\(396\) 0 0
\(397\) −25.4164 −1.27561 −0.637806 0.770197i \(-0.720158\pi\)
−0.637806 + 0.770197i \(0.720158\pi\)
\(398\) −15.4164 −0.772755
\(399\) 0 0
\(400\) 5.47214 0.273607
\(401\) 30.0000 1.49813 0.749064 0.662497i \(-0.230503\pi\)
0.749064 + 0.662497i \(0.230503\pi\)
\(402\) 0 0
\(403\) −2.47214 −0.123146
\(404\) −13.7082 −0.682009
\(405\) 0 0
\(406\) 8.47214 0.420465
\(407\) −8.00000 −0.396545
\(408\) 0 0
\(409\) −37.1246 −1.83569 −0.917847 0.396934i \(-0.870074\pi\)
−0.917847 + 0.396934i \(0.870074\pi\)
\(410\) −19.4164 −0.958908
\(411\) 0 0
\(412\) −6.94427 −0.342120
\(413\) −8.00000 −0.393654
\(414\) 0 0
\(415\) −32.3607 −1.58852
\(416\) −1.23607 −0.0606032
\(417\) 0 0
\(418\) −0.763932 −0.0373651
\(419\) −30.3607 −1.48322 −0.741608 0.670833i \(-0.765937\pi\)
−0.741608 + 0.670833i \(0.765937\pi\)
\(420\) 0 0
\(421\) 18.8328 0.917855 0.458928 0.888474i \(-0.348234\pi\)
0.458928 + 0.888474i \(0.348234\pi\)
\(422\) 21.5967 1.05131
\(423\) 0 0
\(424\) −12.4721 −0.605700
\(425\) −42.1803 −2.04605
\(426\) 0 0
\(427\) 0.472136 0.0228483
\(428\) 5.52786 0.267199
\(429\) 0 0
\(430\) −28.9443 −1.39582
\(431\) −30.8328 −1.48516 −0.742582 0.669755i \(-0.766399\pi\)
−0.742582 + 0.669755i \(0.766399\pi\)
\(432\) 0 0
\(433\) −23.2361 −1.11665 −0.558327 0.829621i \(-0.688557\pi\)
−0.558327 + 0.829621i \(0.688557\pi\)
\(434\) −2.00000 −0.0960031
\(435\) 0 0
\(436\) 18.4721 0.884655
\(437\) 5.23607 0.250475
\(438\) 0 0
\(439\) 10.0000 0.477274 0.238637 0.971109i \(-0.423299\pi\)
0.238637 + 0.971109i \(0.423299\pi\)
\(440\) −2.47214 −0.117854
\(441\) 0 0
\(442\) 9.52786 0.453194
\(443\) 5.70820 0.271205 0.135602 0.990763i \(-0.456703\pi\)
0.135602 + 0.990763i \(0.456703\pi\)
\(444\) 0 0
\(445\) 19.4164 0.920426
\(446\) −28.4721 −1.34819
\(447\) 0 0
\(448\) −1.00000 −0.0472456
\(449\) 20.8328 0.983161 0.491581 0.870832i \(-0.336419\pi\)
0.491581 + 0.870832i \(0.336419\pi\)
\(450\) 0 0
\(451\) 4.58359 0.215833
\(452\) 10.0000 0.470360
\(453\) 0 0
\(454\) −11.4164 −0.535799
\(455\) −4.00000 −0.187523
\(456\) 0 0
\(457\) −27.3050 −1.27727 −0.638636 0.769509i \(-0.720501\pi\)
−0.638636 + 0.769509i \(0.720501\pi\)
\(458\) 1.05573 0.0493309
\(459\) 0 0
\(460\) 16.9443 0.790031
\(461\) 24.1803 1.12619 0.563095 0.826392i \(-0.309610\pi\)
0.563095 + 0.826392i \(0.309610\pi\)
\(462\) 0 0
\(463\) 21.3050 0.990125 0.495063 0.868857i \(-0.335145\pi\)
0.495063 + 0.868857i \(0.335145\pi\)
\(464\) 8.47214 0.393309
\(465\) 0 0
\(466\) −5.52786 −0.256073
\(467\) 15.8885 0.735234 0.367617 0.929977i \(-0.380174\pi\)
0.367617 + 0.929977i \(0.380174\pi\)
\(468\) 0 0
\(469\) −0.291796 −0.0134739
\(470\) −28.9443 −1.33510
\(471\) 0 0
\(472\) −8.00000 −0.368230
\(473\) 6.83282 0.314173
\(474\) 0 0
\(475\) 5.47214 0.251079
\(476\) 7.70820 0.353305
\(477\) 0 0
\(478\) −26.1803 −1.19746
\(479\) −25.5279 −1.16640 −0.583199 0.812329i \(-0.698199\pi\)
−0.583199 + 0.812329i \(0.698199\pi\)
\(480\) 0 0
\(481\) −12.9443 −0.590208
\(482\) 0.763932 0.0347962
\(483\) 0 0
\(484\) −10.4164 −0.473473
\(485\) 15.4164 0.700023
\(486\) 0 0
\(487\) 30.0689 1.36255 0.681276 0.732027i \(-0.261426\pi\)
0.681276 + 0.732027i \(0.261426\pi\)
\(488\) 0.472136 0.0213726
\(489\) 0 0
\(490\) −3.23607 −0.146191
\(491\) −13.7082 −0.618643 −0.309321 0.950958i \(-0.600102\pi\)
−0.309321 + 0.950958i \(0.600102\pi\)
\(492\) 0 0
\(493\) −65.3050 −2.94119
\(494\) −1.23607 −0.0556133
\(495\) 0 0
\(496\) −2.00000 −0.0898027
\(497\) −10.4721 −0.469739
\(498\) 0 0
\(499\) −37.3050 −1.67000 −0.834999 0.550251i \(-0.814532\pi\)
−0.834999 + 0.550251i \(0.814532\pi\)
\(500\) 1.52786 0.0683282
\(501\) 0 0
\(502\) 1.41641 0.0632174
\(503\) −26.4721 −1.18033 −0.590167 0.807281i \(-0.700938\pi\)
−0.590167 + 0.807281i \(0.700938\pi\)
\(504\) 0 0
\(505\) −44.3607 −1.97402
\(506\) −4.00000 −0.177822
\(507\) 0 0
\(508\) −9.70820 −0.430732
\(509\) −13.4164 −0.594672 −0.297336 0.954773i \(-0.596098\pi\)
−0.297336 + 0.954773i \(0.596098\pi\)
\(510\) 0 0
\(511\) 12.4721 0.551735
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 3.52786 0.155607
\(515\) −22.4721 −0.990241
\(516\) 0 0
\(517\) 6.83282 0.300507
\(518\) −10.4721 −0.460119
\(519\) 0 0
\(520\) −4.00000 −0.175412
\(521\) 26.3607 1.15488 0.577441 0.816432i \(-0.304051\pi\)
0.577441 + 0.816432i \(0.304051\pi\)
\(522\) 0 0
\(523\) 5.52786 0.241717 0.120858 0.992670i \(-0.461435\pi\)
0.120858 + 0.992670i \(0.461435\pi\)
\(524\) −16.4721 −0.719589
\(525\) 0 0
\(526\) −7.70820 −0.336094
\(527\) 15.4164 0.671549
\(528\) 0 0
\(529\) 4.41641 0.192018
\(530\) −40.3607 −1.75316
\(531\) 0 0
\(532\) −1.00000 −0.0433555
\(533\) 7.41641 0.321240
\(534\) 0 0
\(535\) 17.8885 0.773389
\(536\) −0.291796 −0.0126037
\(537\) 0 0
\(538\) 6.94427 0.299389
\(539\) 0.763932 0.0329049
\(540\) 0 0
\(541\) 25.7771 1.10824 0.554122 0.832436i \(-0.313054\pi\)
0.554122 + 0.832436i \(0.313054\pi\)
\(542\) −12.0000 −0.515444
\(543\) 0 0
\(544\) 7.70820 0.330487
\(545\) 59.7771 2.56057
\(546\) 0 0
\(547\) 11.1246 0.475654 0.237827 0.971308i \(-0.423565\pi\)
0.237827 + 0.971308i \(0.423565\pi\)
\(548\) −15.4164 −0.658556
\(549\) 0 0
\(550\) −4.18034 −0.178250
\(551\) 8.47214 0.360925
\(552\) 0 0
\(553\) −0.763932 −0.0324857
\(554\) 14.9443 0.634921
\(555\) 0 0
\(556\) 21.8885 0.928281
\(557\) −0.111456 −0.00472255 −0.00236127 0.999997i \(-0.500752\pi\)
−0.00236127 + 0.999997i \(0.500752\pi\)
\(558\) 0 0
\(559\) 11.0557 0.467607
\(560\) −3.23607 −0.136749
\(561\) 0 0
\(562\) −6.00000 −0.253095
\(563\) −20.0000 −0.842900 −0.421450 0.906852i \(-0.638479\pi\)
−0.421450 + 0.906852i \(0.638479\pi\)
\(564\) 0 0
\(565\) 32.3607 1.36142
\(566\) 21.8885 0.920044
\(567\) 0 0
\(568\) −10.4721 −0.439401
\(569\) −2.94427 −0.123430 −0.0617151 0.998094i \(-0.519657\pi\)
−0.0617151 + 0.998094i \(0.519657\pi\)
\(570\) 0 0
\(571\) 6.47214 0.270850 0.135425 0.990788i \(-0.456760\pi\)
0.135425 + 0.990788i \(0.456760\pi\)
\(572\) 0.944272 0.0394820
\(573\) 0 0
\(574\) 6.00000 0.250435
\(575\) 28.6525 1.19489
\(576\) 0 0
\(577\) 21.4164 0.891577 0.445788 0.895138i \(-0.352923\pi\)
0.445788 + 0.895138i \(0.352923\pi\)
\(578\) −42.4164 −1.76429
\(579\) 0 0
\(580\) 27.4164 1.13840
\(581\) 10.0000 0.414870
\(582\) 0 0
\(583\) 9.52786 0.394604
\(584\) 12.4721 0.516101
\(585\) 0 0
\(586\) −15.8885 −0.656350
\(587\) −44.8328 −1.85045 −0.925224 0.379421i \(-0.876123\pi\)
−0.925224 + 0.379421i \(0.876123\pi\)
\(588\) 0 0
\(589\) −2.00000 −0.0824086
\(590\) −25.8885 −1.06581
\(591\) 0 0
\(592\) −10.4721 −0.430402
\(593\) −3.34752 −0.137466 −0.0687332 0.997635i \(-0.521896\pi\)
−0.0687332 + 0.997635i \(0.521896\pi\)
\(594\) 0 0
\(595\) 24.9443 1.02262
\(596\) 6.00000 0.245770
\(597\) 0 0
\(598\) −6.47214 −0.264665
\(599\) −39.4164 −1.61051 −0.805255 0.592928i \(-0.797972\pi\)
−0.805255 + 0.592928i \(0.797972\pi\)
\(600\) 0 0
\(601\) 1.70820 0.0696791 0.0348395 0.999393i \(-0.488908\pi\)
0.0348395 + 0.999393i \(0.488908\pi\)
\(602\) 8.94427 0.364541
\(603\) 0 0
\(604\) 0.763932 0.0310840
\(605\) −33.7082 −1.37043
\(606\) 0 0
\(607\) 25.4164 1.03162 0.515810 0.856703i \(-0.327491\pi\)
0.515810 + 0.856703i \(0.327491\pi\)
\(608\) −1.00000 −0.0405554
\(609\) 0 0
\(610\) 1.52786 0.0618614
\(611\) 11.0557 0.447267
\(612\) 0 0
\(613\) 8.11146 0.327619 0.163809 0.986492i \(-0.447622\pi\)
0.163809 + 0.986492i \(0.447622\pi\)
\(614\) −4.58359 −0.184979
\(615\) 0 0
\(616\) 0.763932 0.0307797
\(617\) −3.41641 −0.137539 −0.0687697 0.997633i \(-0.521907\pi\)
−0.0687697 + 0.997633i \(0.521907\pi\)
\(618\) 0 0
\(619\) −41.8885 −1.68364 −0.841821 0.539756i \(-0.818516\pi\)
−0.841821 + 0.539756i \(0.818516\pi\)
\(620\) −6.47214 −0.259927
\(621\) 0 0
\(622\) −16.3607 −0.656003
\(623\) −6.00000 −0.240385
\(624\) 0 0
\(625\) −22.4164 −0.896656
\(626\) −6.94427 −0.277549
\(627\) 0 0
\(628\) 13.4164 0.535373
\(629\) 80.7214 3.21857
\(630\) 0 0
\(631\) −6.47214 −0.257652 −0.128826 0.991667i \(-0.541121\pi\)
−0.128826 + 0.991667i \(0.541121\pi\)
\(632\) −0.763932 −0.0303876
\(633\) 0 0
\(634\) 27.8885 1.10760
\(635\) −31.4164 −1.24672
\(636\) 0 0
\(637\) 1.23607 0.0489748
\(638\) −6.47214 −0.256234
\(639\) 0 0
\(640\) −3.23607 −0.127917
\(641\) −47.8885 −1.89148 −0.945742 0.324919i \(-0.894663\pi\)
−0.945742 + 0.324919i \(0.894663\pi\)
\(642\) 0 0
\(643\) −27.0557 −1.06697 −0.533487 0.845808i \(-0.679119\pi\)
−0.533487 + 0.845808i \(0.679119\pi\)
\(644\) −5.23607 −0.206330
\(645\) 0 0
\(646\) 7.70820 0.303275
\(647\) −46.2492 −1.81824 −0.909122 0.416529i \(-0.863246\pi\)
−0.909122 + 0.416529i \(0.863246\pi\)
\(648\) 0 0
\(649\) 6.11146 0.239896
\(650\) −6.76393 −0.265303
\(651\) 0 0
\(652\) 4.00000 0.156652
\(653\) −9.41641 −0.368493 −0.184246 0.982880i \(-0.558984\pi\)
−0.184246 + 0.982880i \(0.558984\pi\)
\(654\) 0 0
\(655\) −53.3050 −2.08280
\(656\) 6.00000 0.234261
\(657\) 0 0
\(658\) 8.94427 0.348684
\(659\) 5.52786 0.215335 0.107668 0.994187i \(-0.465662\pi\)
0.107668 + 0.994187i \(0.465662\pi\)
\(660\) 0 0
\(661\) −45.2361 −1.75948 −0.879740 0.475456i \(-0.842283\pi\)
−0.879740 + 0.475456i \(0.842283\pi\)
\(662\) −29.5967 −1.15031
\(663\) 0 0
\(664\) 10.0000 0.388075
\(665\) −3.23607 −0.125489
\(666\) 0 0
\(667\) 44.3607 1.71765
\(668\) 16.9443 0.655594
\(669\) 0 0
\(670\) −0.944272 −0.0364804
\(671\) −0.360680 −0.0139239
\(672\) 0 0
\(673\) 36.2492 1.39730 0.698652 0.715461i \(-0.253783\pi\)
0.698652 + 0.715461i \(0.253783\pi\)
\(674\) 20.4721 0.788557
\(675\) 0 0
\(676\) −11.4721 −0.441236
\(677\) −28.4721 −1.09427 −0.547137 0.837043i \(-0.684282\pi\)
−0.547137 + 0.837043i \(0.684282\pi\)
\(678\) 0 0
\(679\) −4.76393 −0.182823
\(680\) 24.9443 0.956569
\(681\) 0 0
\(682\) 1.52786 0.0585049
\(683\) 10.1115 0.386904 0.193452 0.981110i \(-0.438032\pi\)
0.193452 + 0.981110i \(0.438032\pi\)
\(684\) 0 0
\(685\) −49.8885 −1.90614
\(686\) 1.00000 0.0381802
\(687\) 0 0
\(688\) 8.94427 0.340997
\(689\) 15.4164 0.587318
\(690\) 0 0
\(691\) −26.8328 −1.02077 −0.510384 0.859946i \(-0.670497\pi\)
−0.510384 + 0.859946i \(0.670497\pi\)
\(692\) −2.00000 −0.0760286
\(693\) 0 0
\(694\) −6.29180 −0.238833
\(695\) 70.8328 2.68684
\(696\) 0 0
\(697\) −46.2492 −1.75181
\(698\) 24.8328 0.939936
\(699\) 0 0
\(700\) −5.47214 −0.206827
\(701\) −27.5279 −1.03971 −0.519857 0.854254i \(-0.674015\pi\)
−0.519857 + 0.854254i \(0.674015\pi\)
\(702\) 0 0
\(703\) −10.4721 −0.394964
\(704\) 0.763932 0.0287918
\(705\) 0 0
\(706\) 11.7082 0.440644
\(707\) 13.7082 0.515550
\(708\) 0 0
\(709\) 42.0000 1.57734 0.788672 0.614815i \(-0.210769\pi\)
0.788672 + 0.614815i \(0.210769\pi\)
\(710\) −33.8885 −1.27181
\(711\) 0 0
\(712\) −6.00000 −0.224860
\(713\) −10.4721 −0.392185
\(714\) 0 0
\(715\) 3.05573 0.114278
\(716\) −13.8885 −0.519039
\(717\) 0 0
\(718\) 11.1246 0.415167
\(719\) −6.47214 −0.241370 −0.120685 0.992691i \(-0.538509\pi\)
−0.120685 + 0.992691i \(0.538509\pi\)
\(720\) 0 0
\(721\) 6.94427 0.258618
\(722\) −1.00000 −0.0372161
\(723\) 0 0
\(724\) −5.23607 −0.194597
\(725\) 46.3607 1.72179
\(726\) 0 0
\(727\) −27.7771 −1.03020 −0.515098 0.857132i \(-0.672244\pi\)
−0.515098 + 0.857132i \(0.672244\pi\)
\(728\) 1.23607 0.0458117
\(729\) 0 0
\(730\) 40.3607 1.49382
\(731\) −68.9443 −2.55000
\(732\) 0 0
\(733\) 12.8328 0.473991 0.236995 0.971511i \(-0.423837\pi\)
0.236995 + 0.971511i \(0.423837\pi\)
\(734\) 4.00000 0.147643
\(735\) 0 0
\(736\) −5.23607 −0.193004
\(737\) 0.222912 0.00821108
\(738\) 0 0
\(739\) −3.41641 −0.125675 −0.0628373 0.998024i \(-0.520015\pi\)
−0.0628373 + 0.998024i \(0.520015\pi\)
\(740\) −33.8885 −1.24577
\(741\) 0 0
\(742\) 12.4721 0.457867
\(743\) −23.4164 −0.859065 −0.429532 0.903051i \(-0.641322\pi\)
−0.429532 + 0.903051i \(0.641322\pi\)
\(744\) 0 0
\(745\) 19.4164 0.711362
\(746\) 4.94427 0.181023
\(747\) 0 0
\(748\) −5.88854 −0.215306
\(749\) −5.52786 −0.201984
\(750\) 0 0
\(751\) 23.0132 0.839762 0.419881 0.907579i \(-0.362072\pi\)
0.419881 + 0.907579i \(0.362072\pi\)
\(752\) 8.94427 0.326164
\(753\) 0 0
\(754\) −10.4721 −0.381373
\(755\) 2.47214 0.0899702
\(756\) 0 0
\(757\) 35.8885 1.30439 0.652196 0.758051i \(-0.273848\pi\)
0.652196 + 0.758051i \(0.273848\pi\)
\(758\) 27.1246 0.985210
\(759\) 0 0
\(760\) −3.23607 −0.117385
\(761\) 28.6525 1.03865 0.519326 0.854576i \(-0.326183\pi\)
0.519326 + 0.854576i \(0.326183\pi\)
\(762\) 0 0
\(763\) −18.4721 −0.668736
\(764\) 17.2361 0.623579
\(765\) 0 0
\(766\) 8.00000 0.289052
\(767\) 9.88854 0.357055
\(768\) 0 0
\(769\) −10.9443 −0.394661 −0.197330 0.980337i \(-0.563227\pi\)
−0.197330 + 0.980337i \(0.563227\pi\)
\(770\) 2.47214 0.0890896
\(771\) 0 0
\(772\) −18.0000 −0.647834
\(773\) −45.4164 −1.63351 −0.816757 0.576981i \(-0.804231\pi\)
−0.816757 + 0.576981i \(0.804231\pi\)
\(774\) 0 0
\(775\) −10.9443 −0.393130
\(776\) −4.76393 −0.171015
\(777\) 0 0
\(778\) 2.58359 0.0926263
\(779\) 6.00000 0.214972
\(780\) 0 0
\(781\) 8.00000 0.286263
\(782\) 40.3607 1.44329
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) 43.4164 1.54960
\(786\) 0 0
\(787\) 19.4164 0.692120 0.346060 0.938212i \(-0.387519\pi\)
0.346060 + 0.938212i \(0.387519\pi\)
\(788\) −10.9443 −0.389874
\(789\) 0 0
\(790\) −2.47214 −0.0879547
\(791\) −10.0000 −0.355559
\(792\) 0 0
\(793\) −0.583592 −0.0207240
\(794\) 25.4164 0.901995
\(795\) 0 0
\(796\) 15.4164 0.546420
\(797\) 18.5836 0.658265 0.329132 0.944284i \(-0.393244\pi\)
0.329132 + 0.944284i \(0.393244\pi\)
\(798\) 0 0
\(799\) −68.9443 −2.43907
\(800\) −5.47214 −0.193469
\(801\) 0 0
\(802\) −30.0000 −1.05934
\(803\) −9.52786 −0.336231
\(804\) 0 0
\(805\) −16.9443 −0.597207
\(806\) 2.47214 0.0870773
\(807\) 0 0
\(808\) 13.7082 0.482253
\(809\) −8.94427 −0.314464 −0.157232 0.987562i \(-0.550257\pi\)
−0.157232 + 0.987562i \(0.550257\pi\)
\(810\) 0 0
\(811\) 3.41641 0.119966 0.0599832 0.998199i \(-0.480895\pi\)
0.0599832 + 0.998199i \(0.480895\pi\)
\(812\) −8.47214 −0.297314
\(813\) 0 0
\(814\) 8.00000 0.280400
\(815\) 12.9443 0.453418
\(816\) 0 0
\(817\) 8.94427 0.312920
\(818\) 37.1246 1.29803
\(819\) 0 0
\(820\) 19.4164 0.678050
\(821\) −10.9443 −0.381958 −0.190979 0.981594i \(-0.561166\pi\)
−0.190979 + 0.981594i \(0.561166\pi\)
\(822\) 0 0
\(823\) 38.4721 1.34105 0.670527 0.741885i \(-0.266068\pi\)
0.670527 + 0.741885i \(0.266068\pi\)
\(824\) 6.94427 0.241915
\(825\) 0 0
\(826\) 8.00000 0.278356
\(827\) 10.8328 0.376694 0.188347 0.982103i \(-0.439687\pi\)
0.188347 + 0.982103i \(0.439687\pi\)
\(828\) 0 0
\(829\) −18.7639 −0.651698 −0.325849 0.945422i \(-0.605650\pi\)
−0.325849 + 0.945422i \(0.605650\pi\)
\(830\) 32.3607 1.12326
\(831\) 0 0
\(832\) 1.23607 0.0428529
\(833\) −7.70820 −0.267073
\(834\) 0 0
\(835\) 54.8328 1.89757
\(836\) 0.763932 0.0264211
\(837\) 0 0
\(838\) 30.3607 1.04879
\(839\) 18.1115 0.625277 0.312638 0.949872i \(-0.398787\pi\)
0.312638 + 0.949872i \(0.398787\pi\)
\(840\) 0 0
\(841\) 42.7771 1.47507
\(842\) −18.8328 −0.649022
\(843\) 0 0
\(844\) −21.5967 −0.743391
\(845\) −37.1246 −1.27713
\(846\) 0 0
\(847\) 10.4164 0.357912
\(848\) 12.4721 0.428295
\(849\) 0 0
\(850\) 42.1803 1.44677
\(851\) −54.8328 −1.87964
\(852\) 0 0
\(853\) −10.9443 −0.374725 −0.187362 0.982291i \(-0.559994\pi\)
−0.187362 + 0.982291i \(0.559994\pi\)
\(854\) −0.472136 −0.0161562
\(855\) 0 0
\(856\) −5.52786 −0.188939
\(857\) −3.88854 −0.132830 −0.0664151 0.997792i \(-0.521156\pi\)
−0.0664151 + 0.997792i \(0.521156\pi\)
\(858\) 0 0
\(859\) 57.8885 1.97513 0.987566 0.157206i \(-0.0502487\pi\)
0.987566 + 0.157206i \(0.0502487\pi\)
\(860\) 28.9443 0.986991
\(861\) 0 0
\(862\) 30.8328 1.05017
\(863\) 3.63932 0.123884 0.0619420 0.998080i \(-0.480271\pi\)
0.0619420 + 0.998080i \(0.480271\pi\)
\(864\) 0 0
\(865\) −6.47214 −0.220059
\(866\) 23.2361 0.789594
\(867\) 0 0
\(868\) 2.00000 0.0678844
\(869\) 0.583592 0.0197970
\(870\) 0 0
\(871\) 0.360680 0.0122212
\(872\) −18.4721 −0.625545
\(873\) 0 0
\(874\) −5.23607 −0.177113
\(875\) −1.52786 −0.0516512
\(876\) 0 0
\(877\) 7.41641 0.250434 0.125217 0.992129i \(-0.460037\pi\)
0.125217 + 0.992129i \(0.460037\pi\)
\(878\) −10.0000 −0.337484
\(879\) 0 0
\(880\) 2.47214 0.0833357
\(881\) −5.59675 −0.188559 −0.0942796 0.995546i \(-0.530055\pi\)
−0.0942796 + 0.995546i \(0.530055\pi\)
\(882\) 0 0
\(883\) −14.4721 −0.487026 −0.243513 0.969898i \(-0.578300\pi\)
−0.243513 + 0.969898i \(0.578300\pi\)
\(884\) −9.52786 −0.320457
\(885\) 0 0
\(886\) −5.70820 −0.191771
\(887\) −32.0000 −1.07445 −0.537227 0.843437i \(-0.680528\pi\)
−0.537227 + 0.843437i \(0.680528\pi\)
\(888\) 0 0
\(889\) 9.70820 0.325603
\(890\) −19.4164 −0.650839
\(891\) 0 0
\(892\) 28.4721 0.953318
\(893\) 8.94427 0.299309
\(894\) 0 0
\(895\) −44.9443 −1.50232
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) −20.8328 −0.695200
\(899\) −16.9443 −0.565123
\(900\) 0 0
\(901\) −96.1378 −3.20281
\(902\) −4.58359 −0.152617
\(903\) 0 0
\(904\) −10.0000 −0.332595
\(905\) −16.9443 −0.563247
\(906\) 0 0
\(907\) 18.7639 0.623046 0.311523 0.950239i \(-0.399161\pi\)
0.311523 + 0.950239i \(0.399161\pi\)
\(908\) 11.4164 0.378867
\(909\) 0 0
\(910\) 4.00000 0.132599
\(911\) 38.2492 1.26725 0.633627 0.773639i \(-0.281566\pi\)
0.633627 + 0.773639i \(0.281566\pi\)
\(912\) 0 0
\(913\) −7.63932 −0.252825
\(914\) 27.3050 0.903168
\(915\) 0 0
\(916\) −1.05573 −0.0348822
\(917\) 16.4721 0.543958
\(918\) 0 0
\(919\) 32.3607 1.06748 0.533740 0.845649i \(-0.320786\pi\)
0.533740 + 0.845649i \(0.320786\pi\)
\(920\) −16.9443 −0.558636
\(921\) 0 0
\(922\) −24.1803 −0.796337
\(923\) 12.9443 0.426066
\(924\) 0 0
\(925\) −57.3050 −1.88418
\(926\) −21.3050 −0.700124
\(927\) 0 0
\(928\) −8.47214 −0.278111
\(929\) 16.0689 0.527203 0.263601 0.964632i \(-0.415090\pi\)
0.263601 + 0.964632i \(0.415090\pi\)
\(930\) 0 0
\(931\) 1.00000 0.0327737
\(932\) 5.52786 0.181071
\(933\) 0 0
\(934\) −15.8885 −0.519889
\(935\) −19.0557 −0.623189
\(936\) 0 0
\(937\) −29.0557 −0.949209 −0.474605 0.880199i \(-0.657409\pi\)
−0.474605 + 0.880199i \(0.657409\pi\)
\(938\) 0.291796 0.00952748
\(939\) 0 0
\(940\) 28.9443 0.944058
\(941\) 44.4721 1.44975 0.724875 0.688880i \(-0.241897\pi\)
0.724875 + 0.688880i \(0.241897\pi\)
\(942\) 0 0
\(943\) 31.4164 1.02306
\(944\) 8.00000 0.260378
\(945\) 0 0
\(946\) −6.83282 −0.222154
\(947\) 59.0132 1.91767 0.958835 0.283964i \(-0.0916496\pi\)
0.958835 + 0.283964i \(0.0916496\pi\)
\(948\) 0 0
\(949\) −15.4164 −0.500438
\(950\) −5.47214 −0.177540
\(951\) 0 0
\(952\) −7.70820 −0.249824
\(953\) 25.0557 0.811635 0.405817 0.913954i \(-0.366987\pi\)
0.405817 + 0.913954i \(0.366987\pi\)
\(954\) 0 0
\(955\) 55.7771 1.80490
\(956\) 26.1803 0.846733
\(957\) 0 0
\(958\) 25.5279 0.824768
\(959\) 15.4164 0.497822
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 12.9443 0.417340
\(963\) 0 0
\(964\) −0.763932 −0.0246046
\(965\) −58.2492 −1.87511
\(966\) 0 0
\(967\) −22.8328 −0.734254 −0.367127 0.930171i \(-0.619659\pi\)
−0.367127 + 0.930171i \(0.619659\pi\)
\(968\) 10.4164 0.334796
\(969\) 0 0
\(970\) −15.4164 −0.494991
\(971\) 25.5279 0.819228 0.409614 0.912259i \(-0.365663\pi\)
0.409614 + 0.912259i \(0.365663\pi\)
\(972\) 0 0
\(973\) −21.8885 −0.701714
\(974\) −30.0689 −0.963469
\(975\) 0 0
\(976\) −0.472136 −0.0151127
\(977\) 7.88854 0.252377 0.126188 0.992006i \(-0.459726\pi\)
0.126188 + 0.992006i \(0.459726\pi\)
\(978\) 0 0
\(979\) 4.58359 0.146492
\(980\) 3.23607 0.103372
\(981\) 0 0
\(982\) 13.7082 0.437446
\(983\) −4.00000 −0.127580 −0.0637901 0.997963i \(-0.520319\pi\)
−0.0637901 + 0.997963i \(0.520319\pi\)
\(984\) 0 0
\(985\) −35.4164 −1.12846
\(986\) 65.3050 2.07973
\(987\) 0 0
\(988\) 1.23607 0.0393246
\(989\) 46.8328 1.48920
\(990\) 0 0
\(991\) 24.7639 0.786652 0.393326 0.919399i \(-0.371324\pi\)
0.393326 + 0.919399i \(0.371324\pi\)
\(992\) 2.00000 0.0635001
\(993\) 0 0
\(994\) 10.4721 0.332156
\(995\) 49.8885 1.58157
\(996\) 0 0
\(997\) −61.1935 −1.93802 −0.969009 0.247027i \(-0.920547\pi\)
−0.969009 + 0.247027i \(0.920547\pi\)
\(998\) 37.3050 1.18087
\(999\) 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 2394.2.a.s.1.2 2
3.2 odd 2 2394.2.a.v.1.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2394.2.a.s.1.2 2 1.1 even 1 trivial
2394.2.a.v.1.1 yes 2 3.2 odd 2