Properties

Label 731.2.a.b.1.1
Level 731
Weight 2
Character 731.1
Self dual yes
Analytic conductor 5.837
Analytic rank 1
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 731 = 17 \cdot 43 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 731.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.83706438776\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.56155\)
Character \(\chi\) = 731.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.56155 q^{3} -1.00000 q^{4} +1.56155 q^{5} +1.56155 q^{6} +3.00000 q^{8} -0.561553 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.56155 q^{3} -1.00000 q^{4} +1.56155 q^{5} +1.56155 q^{6} +3.00000 q^{8} -0.561553 q^{9} -1.56155 q^{10} -2.00000 q^{11} +1.56155 q^{12} -0.438447 q^{13} -2.43845 q^{15} -1.00000 q^{16} -1.00000 q^{17} +0.561553 q^{18} +7.12311 q^{19} -1.56155 q^{20} +2.00000 q^{22} +2.00000 q^{23} -4.68466 q^{24} -2.56155 q^{25} +0.438447 q^{26} +5.56155 q^{27} +3.12311 q^{29} +2.43845 q^{30} -6.00000 q^{31} -5.00000 q^{32} +3.12311 q^{33} +1.00000 q^{34} +0.561553 q^{36} -5.56155 q^{37} -7.12311 q^{38} +0.684658 q^{39} +4.68466 q^{40} -5.12311 q^{41} -1.00000 q^{43} +2.00000 q^{44} -0.876894 q^{45} -2.00000 q^{46} -9.56155 q^{47} +1.56155 q^{48} -7.00000 q^{49} +2.56155 q^{50} +1.56155 q^{51} +0.438447 q^{52} -10.6847 q^{53} -5.56155 q^{54} -3.12311 q^{55} -11.1231 q^{57} -3.12311 q^{58} -11.8078 q^{59} +2.43845 q^{60} +0.684658 q^{61} +6.00000 q^{62} +7.00000 q^{64} -0.684658 q^{65} -3.12311 q^{66} +5.56155 q^{67} +1.00000 q^{68} -3.12311 q^{69} +5.56155 q^{71} -1.68466 q^{72} -11.8078 q^{73} +5.56155 q^{74} +4.00000 q^{75} -7.12311 q^{76} -0.684658 q^{78} -9.12311 q^{79} -1.56155 q^{80} -7.00000 q^{81} +5.12311 q^{82} +15.8078 q^{83} -1.56155 q^{85} +1.00000 q^{86} -4.87689 q^{87} -6.00000 q^{88} -0.246211 q^{89} +0.876894 q^{90} -2.00000 q^{92} +9.36932 q^{93} +9.56155 q^{94} +11.1231 q^{95} +7.80776 q^{96} +16.2462 q^{97} +7.00000 q^{98} +1.12311 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + q^{3} - 2q^{4} - q^{5} - q^{6} + 6q^{8} + 3q^{9} + O(q^{10}) \) \( 2q - 2q^{2} + q^{3} - 2q^{4} - q^{5} - q^{6} + 6q^{8} + 3q^{9} + q^{10} - 4q^{11} - q^{12} - 5q^{13} - 9q^{15} - 2q^{16} - 2q^{17} - 3q^{18} + 6q^{19} + q^{20} + 4q^{22} + 4q^{23} + 3q^{24} - q^{25} + 5q^{26} + 7q^{27} - 2q^{29} + 9q^{30} - 12q^{31} - 10q^{32} - 2q^{33} + 2q^{34} - 3q^{36} - 7q^{37} - 6q^{38} - 11q^{39} - 3q^{40} - 2q^{41} - 2q^{43} + 4q^{44} - 10q^{45} - 4q^{46} - 15q^{47} - q^{48} - 14q^{49} + q^{50} - q^{51} + 5q^{52} - 9q^{53} - 7q^{54} + 2q^{55} - 14q^{57} + 2q^{58} - 3q^{59} + 9q^{60} - 11q^{61} + 12q^{62} + 14q^{64} + 11q^{65} + 2q^{66} + 7q^{67} + 2q^{68} + 2q^{69} + 7q^{71} + 9q^{72} - 3q^{73} + 7q^{74} + 8q^{75} - 6q^{76} + 11q^{78} - 10q^{79} + q^{80} - 14q^{81} + 2q^{82} + 11q^{83} + q^{85} + 2q^{86} - 18q^{87} - 12q^{88} + 16q^{89} + 10q^{90} - 4q^{92} - 6q^{93} + 15q^{94} + 14q^{95} - 5q^{96} + 16q^{97} + 14q^{98} - 6q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107 −0.353553 0.935414i \(-0.615027\pi\)
−0.353553 + 0.935414i \(0.615027\pi\)
\(3\) −1.56155 −0.901563 −0.450781 0.892634i \(-0.648855\pi\)
−0.450781 + 0.892634i \(0.648855\pi\)
\(4\) −1.00000 −0.500000
\(5\) 1.56155 0.698348 0.349174 0.937058i \(-0.386462\pi\)
0.349174 + 0.937058i \(0.386462\pi\)
\(6\) 1.56155 0.637501
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 3.00000 1.06066
\(9\) −0.561553 −0.187184
\(10\) −1.56155 −0.493806
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 1.56155 0.450781
\(13\) −0.438447 −0.121603 −0.0608017 0.998150i \(-0.519366\pi\)
−0.0608017 + 0.998150i \(0.519366\pi\)
\(14\) 0 0
\(15\) −2.43845 −0.629604
\(16\) −1.00000 −0.250000
\(17\) −1.00000 −0.242536
\(18\) 0.561553 0.132359
\(19\) 7.12311 1.63415 0.817076 0.576530i \(-0.195593\pi\)
0.817076 + 0.576530i \(0.195593\pi\)
\(20\) −1.56155 −0.349174
\(21\) 0 0
\(22\) 2.00000 0.426401
\(23\) 2.00000 0.417029 0.208514 0.978019i \(-0.433137\pi\)
0.208514 + 0.978019i \(0.433137\pi\)
\(24\) −4.68466 −0.956252
\(25\) −2.56155 −0.512311
\(26\) 0.438447 0.0859866
\(27\) 5.56155 1.07032
\(28\) 0 0
\(29\) 3.12311 0.579946 0.289973 0.957035i \(-0.406354\pi\)
0.289973 + 0.957035i \(0.406354\pi\)
\(30\) 2.43845 0.445198
\(31\) −6.00000 −1.07763 −0.538816 0.842424i \(-0.681128\pi\)
−0.538816 + 0.842424i \(0.681128\pi\)
\(32\) −5.00000 −0.883883
\(33\) 3.12311 0.543663
\(34\) 1.00000 0.171499
\(35\) 0 0
\(36\) 0.561553 0.0935921
\(37\) −5.56155 −0.914314 −0.457157 0.889386i \(-0.651132\pi\)
−0.457157 + 0.889386i \(0.651132\pi\)
\(38\) −7.12311 −1.15552
\(39\) 0.684658 0.109633
\(40\) 4.68466 0.740710
\(41\) −5.12311 −0.800095 −0.400047 0.916494i \(-0.631006\pi\)
−0.400047 + 0.916494i \(0.631006\pi\)
\(42\) 0 0
\(43\) −1.00000 −0.152499
\(44\) 2.00000 0.301511
\(45\) −0.876894 −0.130720
\(46\) −2.00000 −0.294884
\(47\) −9.56155 −1.39470 −0.697348 0.716733i \(-0.745637\pi\)
−0.697348 + 0.716733i \(0.745637\pi\)
\(48\) 1.56155 0.225391
\(49\) −7.00000 −1.00000
\(50\) 2.56155 0.362258
\(51\) 1.56155 0.218661
\(52\) 0.438447 0.0608017
\(53\) −10.6847 −1.46765 −0.733825 0.679338i \(-0.762267\pi\)
−0.733825 + 0.679338i \(0.762267\pi\)
\(54\) −5.56155 −0.756831
\(55\) −3.12311 −0.421119
\(56\) 0 0
\(57\) −11.1231 −1.47329
\(58\) −3.12311 −0.410084
\(59\) −11.8078 −1.53724 −0.768620 0.639706i \(-0.779056\pi\)
−0.768620 + 0.639706i \(0.779056\pi\)
\(60\) 2.43845 0.314802
\(61\) 0.684658 0.0876615 0.0438308 0.999039i \(-0.486044\pi\)
0.0438308 + 0.999039i \(0.486044\pi\)
\(62\) 6.00000 0.762001
\(63\) 0 0
\(64\) 7.00000 0.875000
\(65\) −0.684658 −0.0849214
\(66\) −3.12311 −0.384428
\(67\) 5.56155 0.679452 0.339726 0.940524i \(-0.389666\pi\)
0.339726 + 0.940524i \(0.389666\pi\)
\(68\) 1.00000 0.121268
\(69\) −3.12311 −0.375978
\(70\) 0 0
\(71\) 5.56155 0.660035 0.330017 0.943975i \(-0.392945\pi\)
0.330017 + 0.943975i \(0.392945\pi\)
\(72\) −1.68466 −0.198539
\(73\) −11.8078 −1.38199 −0.690997 0.722858i \(-0.742828\pi\)
−0.690997 + 0.722858i \(0.742828\pi\)
\(74\) 5.56155 0.646517
\(75\) 4.00000 0.461880
\(76\) −7.12311 −0.817076
\(77\) 0 0
\(78\) −0.684658 −0.0775223
\(79\) −9.12311 −1.02643 −0.513215 0.858260i \(-0.671546\pi\)
−0.513215 + 0.858260i \(0.671546\pi\)
\(80\) −1.56155 −0.174587
\(81\) −7.00000 −0.777778
\(82\) 5.12311 0.565752
\(83\) 15.8078 1.73513 0.867564 0.497326i \(-0.165685\pi\)
0.867564 + 0.497326i \(0.165685\pi\)
\(84\) 0 0
\(85\) −1.56155 −0.169374
\(86\) 1.00000 0.107833
\(87\) −4.87689 −0.522858
\(88\) −6.00000 −0.639602
\(89\) −0.246211 −0.0260983 −0.0130492 0.999915i \(-0.504154\pi\)
−0.0130492 + 0.999915i \(0.504154\pi\)
\(90\) 0.876894 0.0924328
\(91\) 0 0
\(92\) −2.00000 −0.208514
\(93\) 9.36932 0.971553
\(94\) 9.56155 0.986199
\(95\) 11.1231 1.14121
\(96\) 7.80776 0.796877
\(97\) 16.2462 1.64955 0.824776 0.565459i \(-0.191301\pi\)
0.824776 + 0.565459i \(0.191301\pi\)
\(98\) 7.00000 0.707107
\(99\) 1.12311 0.112876
\(100\) 2.56155 0.256155
\(101\) −2.00000 −0.199007 −0.0995037 0.995037i \(-0.531726\pi\)
−0.0995037 + 0.995037i \(0.531726\pi\)
\(102\) −1.56155 −0.154617
\(103\) −17.5616 −1.73039 −0.865196 0.501435i \(-0.832806\pi\)
−0.865196 + 0.501435i \(0.832806\pi\)
\(104\) −1.31534 −0.128980
\(105\) 0 0
\(106\) 10.6847 1.03779
\(107\) 2.00000 0.193347 0.0966736 0.995316i \(-0.469180\pi\)
0.0966736 + 0.995316i \(0.469180\pi\)
\(108\) −5.56155 −0.535161
\(109\) −4.24621 −0.406713 −0.203357 0.979105i \(-0.565185\pi\)
−0.203357 + 0.979105i \(0.565185\pi\)
\(110\) 3.12311 0.297776
\(111\) 8.68466 0.824311
\(112\) 0 0
\(113\) −11.8078 −1.11078 −0.555391 0.831590i \(-0.687431\pi\)
−0.555391 + 0.831590i \(0.687431\pi\)
\(114\) 11.1231 1.04177
\(115\) 3.12311 0.291231
\(116\) −3.12311 −0.289973
\(117\) 0.246211 0.0227622
\(118\) 11.8078 1.08699
\(119\) 0 0
\(120\) −7.31534 −0.667796
\(121\) −7.00000 −0.636364
\(122\) −0.684658 −0.0619861
\(123\) 8.00000 0.721336
\(124\) 6.00000 0.538816
\(125\) −11.8078 −1.05612
\(126\) 0 0
\(127\) 22.0540 1.95697 0.978487 0.206309i \(-0.0661452\pi\)
0.978487 + 0.206309i \(0.0661452\pi\)
\(128\) 3.00000 0.265165
\(129\) 1.56155 0.137487
\(130\) 0.684658 0.0600485
\(131\) 11.8078 1.03165 0.515825 0.856694i \(-0.327486\pi\)
0.515825 + 0.856694i \(0.327486\pi\)
\(132\) −3.12311 −0.271831
\(133\) 0 0
\(134\) −5.56155 −0.480445
\(135\) 8.68466 0.747456
\(136\) −3.00000 −0.257248
\(137\) 12.2462 1.04626 0.523132 0.852252i \(-0.324763\pi\)
0.523132 + 0.852252i \(0.324763\pi\)
\(138\) 3.12311 0.265856
\(139\) −19.3693 −1.64288 −0.821442 0.570292i \(-0.806830\pi\)
−0.821442 + 0.570292i \(0.806830\pi\)
\(140\) 0 0
\(141\) 14.9309 1.25741
\(142\) −5.56155 −0.466715
\(143\) 0.876894 0.0733296
\(144\) 0.561553 0.0467961
\(145\) 4.87689 0.405004
\(146\) 11.8078 0.977218
\(147\) 10.9309 0.901563
\(148\) 5.56155 0.457157
\(149\) −8.24621 −0.675556 −0.337778 0.941226i \(-0.609675\pi\)
−0.337778 + 0.941226i \(0.609675\pi\)
\(150\) −4.00000 −0.326599
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 21.3693 1.73328
\(153\) 0.561553 0.0453989
\(154\) 0 0
\(155\) −9.36932 −0.752562
\(156\) −0.684658 −0.0548165
\(157\) −22.4924 −1.79509 −0.897545 0.440922i \(-0.854651\pi\)
−0.897545 + 0.440922i \(0.854651\pi\)
\(158\) 9.12311 0.725795
\(159\) 16.6847 1.32318
\(160\) −7.80776 −0.617258
\(161\) 0 0
\(162\) 7.00000 0.549972
\(163\) −21.5616 −1.68883 −0.844416 0.535689i \(-0.820052\pi\)
−0.844416 + 0.535689i \(0.820052\pi\)
\(164\) 5.12311 0.400047
\(165\) 4.87689 0.379666
\(166\) −15.8078 −1.22692
\(167\) 22.0000 1.70241 0.851206 0.524832i \(-0.175872\pi\)
0.851206 + 0.524832i \(0.175872\pi\)
\(168\) 0 0
\(169\) −12.8078 −0.985213
\(170\) 1.56155 0.119766
\(171\) −4.00000 −0.305888
\(172\) 1.00000 0.0762493
\(173\) −5.12311 −0.389503 −0.194751 0.980853i \(-0.562390\pi\)
−0.194751 + 0.980853i \(0.562390\pi\)
\(174\) 4.87689 0.369716
\(175\) 0 0
\(176\) 2.00000 0.150756
\(177\) 18.4384 1.38592
\(178\) 0.246211 0.0184543
\(179\) 8.87689 0.663490 0.331745 0.943369i \(-0.392363\pi\)
0.331745 + 0.943369i \(0.392363\pi\)
\(180\) 0.876894 0.0653598
\(181\) −19.3693 −1.43971 −0.719855 0.694124i \(-0.755792\pi\)
−0.719855 + 0.694124i \(0.755792\pi\)
\(182\) 0 0
\(183\) −1.06913 −0.0790324
\(184\) 6.00000 0.442326
\(185\) −8.68466 −0.638509
\(186\) −9.36932 −0.686992
\(187\) 2.00000 0.146254
\(188\) 9.56155 0.697348
\(189\) 0 0
\(190\) −11.1231 −0.806955
\(191\) 11.1231 0.804840 0.402420 0.915455i \(-0.368169\pi\)
0.402420 + 0.915455i \(0.368169\pi\)
\(192\) −10.9309 −0.788868
\(193\) 2.00000 0.143963 0.0719816 0.997406i \(-0.477068\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) −16.2462 −1.16641
\(195\) 1.06913 0.0765620
\(196\) 7.00000 0.500000
\(197\) 17.6155 1.25505 0.627527 0.778595i \(-0.284067\pi\)
0.627527 + 0.778595i \(0.284067\pi\)
\(198\) −1.12311 −0.0798156
\(199\) −2.93087 −0.207764 −0.103882 0.994590i \(-0.533126\pi\)
−0.103882 + 0.994590i \(0.533126\pi\)
\(200\) −7.68466 −0.543387
\(201\) −8.68466 −0.612569
\(202\) 2.00000 0.140720
\(203\) 0 0
\(204\) −1.56155 −0.109331
\(205\) −8.00000 −0.558744
\(206\) 17.5616 1.22357
\(207\) −1.12311 −0.0780612
\(208\) 0.438447 0.0304008
\(209\) −14.2462 −0.985431
\(210\) 0 0
\(211\) −4.19224 −0.288605 −0.144303 0.989534i \(-0.546094\pi\)
−0.144303 + 0.989534i \(0.546094\pi\)
\(212\) 10.6847 0.733825
\(213\) −8.68466 −0.595063
\(214\) −2.00000 −0.136717
\(215\) −1.56155 −0.106497
\(216\) 16.6847 1.13525
\(217\) 0 0
\(218\) 4.24621 0.287590
\(219\) 18.4384 1.24595
\(220\) 3.12311 0.210560
\(221\) 0.438447 0.0294931
\(222\) −8.68466 −0.582876
\(223\) −16.0000 −1.07144 −0.535720 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) 0 0
\(225\) 1.43845 0.0958965
\(226\) 11.8078 0.785441
\(227\) 20.4924 1.36013 0.680065 0.733152i \(-0.261952\pi\)
0.680065 + 0.733152i \(0.261952\pi\)
\(228\) 11.1231 0.736646
\(229\) 5.80776 0.383788 0.191894 0.981416i \(-0.438537\pi\)
0.191894 + 0.981416i \(0.438537\pi\)
\(230\) −3.12311 −0.205931
\(231\) 0 0
\(232\) 9.36932 0.615126
\(233\) 19.1231 1.25280 0.626398 0.779503i \(-0.284528\pi\)
0.626398 + 0.779503i \(0.284528\pi\)
\(234\) −0.246211 −0.0160953
\(235\) −14.9309 −0.973983
\(236\) 11.8078 0.768620
\(237\) 14.2462 0.925391
\(238\) 0 0
\(239\) −8.00000 −0.517477 −0.258738 0.965947i \(-0.583307\pi\)
−0.258738 + 0.965947i \(0.583307\pi\)
\(240\) 2.43845 0.157401
\(241\) −5.36932 −0.345868 −0.172934 0.984933i \(-0.555325\pi\)
−0.172934 + 0.984933i \(0.555325\pi\)
\(242\) 7.00000 0.449977
\(243\) −5.75379 −0.369106
\(244\) −0.684658 −0.0438308
\(245\) −10.9309 −0.698348
\(246\) −8.00000 −0.510061
\(247\) −3.12311 −0.198718
\(248\) −18.0000 −1.14300
\(249\) −24.6847 −1.56433
\(250\) 11.8078 0.746789
\(251\) −6.43845 −0.406391 −0.203196 0.979138i \(-0.565133\pi\)
−0.203196 + 0.979138i \(0.565133\pi\)
\(252\) 0 0
\(253\) −4.00000 −0.251478
\(254\) −22.0540 −1.38379
\(255\) 2.43845 0.152701
\(256\) −17.0000 −1.06250
\(257\) 26.0000 1.62184 0.810918 0.585160i \(-0.198968\pi\)
0.810918 + 0.585160i \(0.198968\pi\)
\(258\) −1.56155 −0.0972180
\(259\) 0 0
\(260\) 0.684658 0.0424607
\(261\) −1.75379 −0.108557
\(262\) −11.8078 −0.729486
\(263\) 12.4924 0.770316 0.385158 0.922851i \(-0.374147\pi\)
0.385158 + 0.922851i \(0.374147\pi\)
\(264\) 9.36932 0.576642
\(265\) −16.6847 −1.02493
\(266\) 0 0
\(267\) 0.384472 0.0235293
\(268\) −5.56155 −0.339726
\(269\) 8.24621 0.502780 0.251390 0.967886i \(-0.419112\pi\)
0.251390 + 0.967886i \(0.419112\pi\)
\(270\) −8.68466 −0.528531
\(271\) −18.9309 −1.14997 −0.574984 0.818164i \(-0.694992\pi\)
−0.574984 + 0.818164i \(0.694992\pi\)
\(272\) 1.00000 0.0606339
\(273\) 0 0
\(274\) −12.2462 −0.739821
\(275\) 5.12311 0.308935
\(276\) 3.12311 0.187989
\(277\) 30.9309 1.85846 0.929228 0.369507i \(-0.120473\pi\)
0.929228 + 0.369507i \(0.120473\pi\)
\(278\) 19.3693 1.16169
\(279\) 3.36932 0.201716
\(280\) 0 0
\(281\) −20.9309 −1.24863 −0.624316 0.781172i \(-0.714622\pi\)
−0.624316 + 0.781172i \(0.714622\pi\)
\(282\) −14.9309 −0.889120
\(283\) 12.2462 0.727962 0.363981 0.931406i \(-0.381417\pi\)
0.363981 + 0.931406i \(0.381417\pi\)
\(284\) −5.56155 −0.330017
\(285\) −17.3693 −1.02887
\(286\) −0.876894 −0.0518519
\(287\) 0 0
\(288\) 2.80776 0.165449
\(289\) 1.00000 0.0588235
\(290\) −4.87689 −0.286381
\(291\) −25.3693 −1.48718
\(292\) 11.8078 0.690997
\(293\) −0.246211 −0.0143838 −0.00719191 0.999974i \(-0.502289\pi\)
−0.00719191 + 0.999974i \(0.502289\pi\)
\(294\) −10.9309 −0.637501
\(295\) −18.4384 −1.07353
\(296\) −16.6847 −0.969776
\(297\) −11.1231 −0.645428
\(298\) 8.24621 0.477690
\(299\) −0.876894 −0.0507121
\(300\) −4.00000 −0.230940
\(301\) 0 0
\(302\) −8.00000 −0.460348
\(303\) 3.12311 0.179418
\(304\) −7.12311 −0.408538
\(305\) 1.06913 0.0612182
\(306\) −0.561553 −0.0321018
\(307\) −14.2462 −0.813074 −0.406537 0.913634i \(-0.633264\pi\)
−0.406537 + 0.913634i \(0.633264\pi\)
\(308\) 0 0
\(309\) 27.4233 1.56006
\(310\) 9.36932 0.532141
\(311\) 15.3693 0.871514 0.435757 0.900064i \(-0.356481\pi\)
0.435757 + 0.900064i \(0.356481\pi\)
\(312\) 2.05398 0.116283
\(313\) 15.8078 0.893508 0.446754 0.894657i \(-0.352580\pi\)
0.446754 + 0.894657i \(0.352580\pi\)
\(314\) 22.4924 1.26932
\(315\) 0 0
\(316\) 9.12311 0.513215
\(317\) −19.3693 −1.08789 −0.543945 0.839121i \(-0.683070\pi\)
−0.543945 + 0.839121i \(0.683070\pi\)
\(318\) −16.6847 −0.935629
\(319\) −6.24621 −0.349721
\(320\) 10.9309 0.611054
\(321\) −3.12311 −0.174315
\(322\) 0 0
\(323\) −7.12311 −0.396340
\(324\) 7.00000 0.388889
\(325\) 1.12311 0.0622987
\(326\) 21.5616 1.19418
\(327\) 6.63068 0.366678
\(328\) −15.3693 −0.848629
\(329\) 0 0
\(330\) −4.87689 −0.268464
\(331\) 12.0000 0.659580 0.329790 0.944054i \(-0.393022\pi\)
0.329790 + 0.944054i \(0.393022\pi\)
\(332\) −15.8078 −0.867564
\(333\) 3.12311 0.171145
\(334\) −22.0000 −1.20379
\(335\) 8.68466 0.474494
\(336\) 0 0
\(337\) −3.75379 −0.204482 −0.102241 0.994760i \(-0.532601\pi\)
−0.102241 + 0.994760i \(0.532601\pi\)
\(338\) 12.8078 0.696651
\(339\) 18.4384 1.00144
\(340\) 1.56155 0.0846871
\(341\) 12.0000 0.649836
\(342\) 4.00000 0.216295
\(343\) 0 0
\(344\) −3.00000 −0.161749
\(345\) −4.87689 −0.262563
\(346\) 5.12311 0.275420
\(347\) −18.4384 −0.989828 −0.494914 0.868942i \(-0.664800\pi\)
−0.494914 + 0.868942i \(0.664800\pi\)
\(348\) 4.87689 0.261429
\(349\) −10.8769 −0.582227 −0.291113 0.956689i \(-0.594026\pi\)
−0.291113 + 0.956689i \(0.594026\pi\)
\(350\) 0 0
\(351\) −2.43845 −0.130155
\(352\) 10.0000 0.533002
\(353\) −19.1771 −1.02069 −0.510347 0.859969i \(-0.670483\pi\)
−0.510347 + 0.859969i \(0.670483\pi\)
\(354\) −18.4384 −0.979992
\(355\) 8.68466 0.460934
\(356\) 0.246211 0.0130492
\(357\) 0 0
\(358\) −8.87689 −0.469158
\(359\) −3.80776 −0.200966 −0.100483 0.994939i \(-0.532039\pi\)
−0.100483 + 0.994939i \(0.532039\pi\)
\(360\) −2.63068 −0.138649
\(361\) 31.7386 1.67045
\(362\) 19.3693 1.01803
\(363\) 10.9309 0.573722
\(364\) 0 0
\(365\) −18.4384 −0.965112
\(366\) 1.06913 0.0558843
\(367\) 18.4924 0.965297 0.482648 0.875814i \(-0.339675\pi\)
0.482648 + 0.875814i \(0.339675\pi\)
\(368\) −2.00000 −0.104257
\(369\) 2.87689 0.149765
\(370\) 8.68466 0.451494
\(371\) 0 0
\(372\) −9.36932 −0.485776
\(373\) −12.6307 −0.653992 −0.326996 0.945026i \(-0.606036\pi\)
−0.326996 + 0.945026i \(0.606036\pi\)
\(374\) −2.00000 −0.103418
\(375\) 18.4384 0.952157
\(376\) −28.6847 −1.47930
\(377\) −1.36932 −0.0705234
\(378\) 0 0
\(379\) −9.12311 −0.468622 −0.234311 0.972162i \(-0.575283\pi\)
−0.234311 + 0.972162i \(0.575283\pi\)
\(380\) −11.1231 −0.570603
\(381\) −34.4384 −1.76434
\(382\) −11.1231 −0.569108
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) −4.68466 −0.239063
\(385\) 0 0
\(386\) −2.00000 −0.101797
\(387\) 0.561553 0.0285453
\(388\) −16.2462 −0.824776
\(389\) 2.49242 0.126371 0.0631854 0.998002i \(-0.479874\pi\)
0.0631854 + 0.998002i \(0.479874\pi\)
\(390\) −1.06913 −0.0541375
\(391\) −2.00000 −0.101144
\(392\) −21.0000 −1.06066
\(393\) −18.4384 −0.930097
\(394\) −17.6155 −0.887457
\(395\) −14.2462 −0.716805
\(396\) −1.12311 −0.0564382
\(397\) 6.87689 0.345141 0.172571 0.984997i \(-0.444793\pi\)
0.172571 + 0.984997i \(0.444793\pi\)
\(398\) 2.93087 0.146911
\(399\) 0 0
\(400\) 2.56155 0.128078
\(401\) −3.75379 −0.187455 −0.0937276 0.995598i \(-0.529878\pi\)
−0.0937276 + 0.995598i \(0.529878\pi\)
\(402\) 8.68466 0.433151
\(403\) 2.63068 0.131044
\(404\) 2.00000 0.0995037
\(405\) −10.9309 −0.543159
\(406\) 0 0
\(407\) 11.1231 0.551352
\(408\) 4.68466 0.231925
\(409\) −35.8617 −1.77325 −0.886624 0.462490i \(-0.846956\pi\)
−0.886624 + 0.462490i \(0.846956\pi\)
\(410\) 8.00000 0.395092
\(411\) −19.1231 −0.943273
\(412\) 17.5616 0.865196
\(413\) 0 0
\(414\) 1.12311 0.0551976
\(415\) 24.6847 1.21172
\(416\) 2.19224 0.107483
\(417\) 30.2462 1.48116
\(418\) 14.2462 0.696805
\(419\) −20.0000 −0.977064 −0.488532 0.872546i \(-0.662467\pi\)
−0.488532 + 0.872546i \(0.662467\pi\)
\(420\) 0 0
\(421\) 12.7386 0.620843 0.310422 0.950599i \(-0.399530\pi\)
0.310422 + 0.950599i \(0.399530\pi\)
\(422\) 4.19224 0.204075
\(423\) 5.36932 0.261065
\(424\) −32.0540 −1.55668
\(425\) 2.56155 0.124254
\(426\) 8.68466 0.420773
\(427\) 0 0
\(428\) −2.00000 −0.0966736
\(429\) −1.36932 −0.0661112
\(430\) 1.56155 0.0753048
\(431\) −13.1231 −0.632118 −0.316059 0.948740i \(-0.602360\pi\)
−0.316059 + 0.948740i \(0.602360\pi\)
\(432\) −5.56155 −0.267580
\(433\) 11.3693 0.546375 0.273187 0.961961i \(-0.411922\pi\)
0.273187 + 0.961961i \(0.411922\pi\)
\(434\) 0 0
\(435\) −7.61553 −0.365137
\(436\) 4.24621 0.203357
\(437\) 14.2462 0.681489
\(438\) −18.4384 −0.881023
\(439\) 15.3693 0.733537 0.366769 0.930312i \(-0.380464\pi\)
0.366769 + 0.930312i \(0.380464\pi\)
\(440\) −9.36932 −0.446665
\(441\) 3.93087 0.187184
\(442\) −0.438447 −0.0208548
\(443\) −31.4233 −1.49297 −0.746483 0.665405i \(-0.768259\pi\)
−0.746483 + 0.665405i \(0.768259\pi\)
\(444\) −8.68466 −0.412156
\(445\) −0.384472 −0.0182257
\(446\) 16.0000 0.757622
\(447\) 12.8769 0.609056
\(448\) 0 0
\(449\) −30.4384 −1.43648 −0.718240 0.695796i \(-0.755052\pi\)
−0.718240 + 0.695796i \(0.755052\pi\)
\(450\) −1.43845 −0.0678091
\(451\) 10.2462 0.482475
\(452\) 11.8078 0.555391
\(453\) −12.4924 −0.586945
\(454\) −20.4924 −0.961757
\(455\) 0 0
\(456\) −33.3693 −1.56266
\(457\) −4.63068 −0.216614 −0.108307 0.994117i \(-0.534543\pi\)
−0.108307 + 0.994117i \(0.534543\pi\)
\(458\) −5.80776 −0.271379
\(459\) −5.56155 −0.259591
\(460\) −3.12311 −0.145616
\(461\) −17.8078 −0.829390 −0.414695 0.909960i \(-0.636112\pi\)
−0.414695 + 0.909960i \(0.636112\pi\)
\(462\) 0 0
\(463\) −8.00000 −0.371792 −0.185896 0.982569i \(-0.559519\pi\)
−0.185896 + 0.982569i \(0.559519\pi\)
\(464\) −3.12311 −0.144987
\(465\) 14.6307 0.678482
\(466\) −19.1231 −0.885861
\(467\) −32.4924 −1.50357 −0.751785 0.659408i \(-0.770807\pi\)
−0.751785 + 0.659408i \(0.770807\pi\)
\(468\) −0.246211 −0.0113811
\(469\) 0 0
\(470\) 14.9309 0.688710
\(471\) 35.1231 1.61839
\(472\) −35.4233 −1.63049
\(473\) 2.00000 0.0919601
\(474\) −14.2462 −0.654350
\(475\) −18.2462 −0.837194
\(476\) 0 0
\(477\) 6.00000 0.274721
\(478\) 8.00000 0.365911
\(479\) 31.3693 1.43330 0.716650 0.697433i \(-0.245674\pi\)
0.716650 + 0.697433i \(0.245674\pi\)
\(480\) 12.1922 0.556497
\(481\) 2.43845 0.111184
\(482\) 5.36932 0.244566
\(483\) 0 0
\(484\) 7.00000 0.318182
\(485\) 25.3693 1.15196
\(486\) 5.75379 0.260997
\(487\) −26.4924 −1.20049 −0.600243 0.799818i \(-0.704930\pi\)
−0.600243 + 0.799818i \(0.704930\pi\)
\(488\) 2.05398 0.0929791
\(489\) 33.6695 1.52259
\(490\) 10.9309 0.493806
\(491\) −5.36932 −0.242314 −0.121157 0.992633i \(-0.538660\pi\)
−0.121157 + 0.992633i \(0.538660\pi\)
\(492\) −8.00000 −0.360668
\(493\) −3.12311 −0.140658
\(494\) 3.12311 0.140515
\(495\) 1.75379 0.0788269
\(496\) 6.00000 0.269408
\(497\) 0 0
\(498\) 24.6847 1.10615
\(499\) 4.68466 0.209714 0.104857 0.994487i \(-0.466561\pi\)
0.104857 + 0.994487i \(0.466561\pi\)
\(500\) 11.8078 0.528059
\(501\) −34.3542 −1.53483
\(502\) 6.43845 0.287362
\(503\) 39.4233 1.75780 0.878899 0.477008i \(-0.158279\pi\)
0.878899 + 0.477008i \(0.158279\pi\)
\(504\) 0 0
\(505\) −3.12311 −0.138976
\(506\) 4.00000 0.177822
\(507\) 20.0000 0.888231
\(508\) −22.0540 −0.978487
\(509\) 4.05398 0.179689 0.0898446 0.995956i \(-0.471363\pi\)
0.0898446 + 0.995956i \(0.471363\pi\)
\(510\) −2.43845 −0.107976
\(511\) 0 0
\(512\) 11.0000 0.486136
\(513\) 39.6155 1.74907
\(514\) −26.0000 −1.14681
\(515\) −27.4233 −1.20841
\(516\) −1.56155 −0.0687435
\(517\) 19.1231 0.841033
\(518\) 0 0
\(519\) 8.00000 0.351161
\(520\) −2.05398 −0.0900728
\(521\) −2.63068 −0.115252 −0.0576262 0.998338i \(-0.518353\pi\)
−0.0576262 + 0.998338i \(0.518353\pi\)
\(522\) 1.75379 0.0767612
\(523\) 39.1231 1.71073 0.855367 0.518023i \(-0.173332\pi\)
0.855367 + 0.518023i \(0.173332\pi\)
\(524\) −11.8078 −0.515825
\(525\) 0 0
\(526\) −12.4924 −0.544696
\(527\) 6.00000 0.261364
\(528\) −3.12311 −0.135916
\(529\) −19.0000 −0.826087
\(530\) 16.6847 0.724735
\(531\) 6.63068 0.287747
\(532\) 0 0
\(533\) 2.24621 0.0972942
\(534\) −0.384472 −0.0166377
\(535\) 3.12311 0.135024
\(536\) 16.6847 0.720667
\(537\) −13.8617 −0.598178
\(538\) −8.24621 −0.355519
\(539\) 14.0000 0.603023
\(540\) −8.68466 −0.373728
\(541\) 21.6155 0.929324 0.464662 0.885488i \(-0.346176\pi\)
0.464662 + 0.885488i \(0.346176\pi\)
\(542\) 18.9309 0.813150
\(543\) 30.2462 1.29799
\(544\) 5.00000 0.214373
\(545\) −6.63068 −0.284027
\(546\) 0 0
\(547\) 34.4924 1.47479 0.737395 0.675462i \(-0.236056\pi\)
0.737395 + 0.675462i \(0.236056\pi\)
\(548\) −12.2462 −0.523132
\(549\) −0.384472 −0.0164089
\(550\) −5.12311 −0.218450
\(551\) 22.2462 0.947720
\(552\) −9.36932 −0.398785
\(553\) 0 0
\(554\) −30.9309 −1.31413
\(555\) 13.5616 0.575656
\(556\) 19.3693 0.821442
\(557\) 19.1771 0.812559 0.406279 0.913749i \(-0.366826\pi\)
0.406279 + 0.913749i \(0.366826\pi\)
\(558\) −3.36932 −0.142635
\(559\) 0.438447 0.0185443
\(560\) 0 0
\(561\) −3.12311 −0.131858
\(562\) 20.9309 0.882915
\(563\) −1.75379 −0.0739134 −0.0369567 0.999317i \(-0.511766\pi\)
−0.0369567 + 0.999317i \(0.511766\pi\)
\(564\) −14.9309 −0.628703
\(565\) −18.4384 −0.775711
\(566\) −12.2462 −0.514747
\(567\) 0 0
\(568\) 16.6847 0.700073
\(569\) 3.56155 0.149308 0.0746540 0.997209i \(-0.476215\pi\)
0.0746540 + 0.997209i \(0.476215\pi\)
\(570\) 17.3693 0.727521
\(571\) −30.2462 −1.26576 −0.632882 0.774248i \(-0.718128\pi\)
−0.632882 + 0.774248i \(0.718128\pi\)
\(572\) −0.876894 −0.0366648
\(573\) −17.3693 −0.725614
\(574\) 0 0
\(575\) −5.12311 −0.213648
\(576\) −3.93087 −0.163786
\(577\) −4.73863 −0.197272 −0.0986360 0.995124i \(-0.531448\pi\)
−0.0986360 + 0.995124i \(0.531448\pi\)
\(578\) −1.00000 −0.0415945
\(579\) −3.12311 −0.129792
\(580\) −4.87689 −0.202502
\(581\) 0 0
\(582\) 25.3693 1.05159
\(583\) 21.3693 0.885027
\(584\) −35.4233 −1.46583
\(585\) 0.384472 0.0158960
\(586\) 0.246211 0.0101709
\(587\) −2.63068 −0.108580 −0.0542900 0.998525i \(-0.517290\pi\)
−0.0542900 + 0.998525i \(0.517290\pi\)
\(588\) −10.9309 −0.450781
\(589\) −42.7386 −1.76101
\(590\) 18.4384 0.759099
\(591\) −27.5076 −1.13151
\(592\) 5.56155 0.228578
\(593\) 12.2462 0.502892 0.251446 0.967871i \(-0.419094\pi\)
0.251446 + 0.967871i \(0.419094\pi\)
\(594\) 11.1231 0.456387
\(595\) 0 0
\(596\) 8.24621 0.337778
\(597\) 4.57671 0.187312
\(598\) 0.876894 0.0358589
\(599\) 24.6847 1.00859 0.504294 0.863532i \(-0.331753\pi\)
0.504294 + 0.863532i \(0.331753\pi\)
\(600\) 12.0000 0.489898
\(601\) 2.43845 0.0994663 0.0497332 0.998763i \(-0.484163\pi\)
0.0497332 + 0.998763i \(0.484163\pi\)
\(602\) 0 0
\(603\) −3.12311 −0.127183
\(604\) −8.00000 −0.325515
\(605\) −10.9309 −0.444403
\(606\) −3.12311 −0.126867
\(607\) 19.3153 0.783986 0.391993 0.919968i \(-0.371786\pi\)
0.391993 + 0.919968i \(0.371786\pi\)
\(608\) −35.6155 −1.44440
\(609\) 0 0
\(610\) −1.06913 −0.0432878
\(611\) 4.19224 0.169600
\(612\) −0.561553 −0.0226994
\(613\) −1.50758 −0.0608905 −0.0304452 0.999536i \(-0.509693\pi\)
−0.0304452 + 0.999536i \(0.509693\pi\)
\(614\) 14.2462 0.574930
\(615\) 12.4924 0.503743
\(616\) 0 0
\(617\) 40.2462 1.62025 0.810126 0.586256i \(-0.199399\pi\)
0.810126 + 0.586256i \(0.199399\pi\)
\(618\) −27.4233 −1.10313
\(619\) 48.7386 1.95897 0.979486 0.201514i \(-0.0645863\pi\)
0.979486 + 0.201514i \(0.0645863\pi\)
\(620\) 9.36932 0.376281
\(621\) 11.1231 0.446355
\(622\) −15.3693 −0.616253
\(623\) 0 0
\(624\) −0.684658 −0.0274083
\(625\) −5.63068 −0.225227
\(626\) −15.8078 −0.631805
\(627\) 22.2462 0.888428
\(628\) 22.4924 0.897545
\(629\) 5.56155 0.221754
\(630\) 0 0
\(631\) −0.384472 −0.0153056 −0.00765279 0.999971i \(-0.502436\pi\)
−0.00765279 + 0.999971i \(0.502436\pi\)
\(632\) −27.3693 −1.08869
\(633\) 6.54640 0.260196
\(634\) 19.3693 0.769254
\(635\) 34.4384 1.36665
\(636\) −16.6847 −0.661590
\(637\) 3.06913 0.121603
\(638\) 6.24621 0.247290
\(639\) −3.12311 −0.123548
\(640\) 4.68466 0.185177
\(641\) −15.4233 −0.609183 −0.304592 0.952483i \(-0.598520\pi\)
−0.304592 + 0.952483i \(0.598520\pi\)
\(642\) 3.12311 0.123259
\(643\) 44.2462 1.74490 0.872450 0.488703i \(-0.162530\pi\)
0.872450 + 0.488703i \(0.162530\pi\)
\(644\) 0 0
\(645\) 2.43845 0.0960138
\(646\) 7.12311 0.280255
\(647\) −36.1080 −1.41955 −0.709775 0.704428i \(-0.751203\pi\)
−0.709775 + 0.704428i \(0.751203\pi\)
\(648\) −21.0000 −0.824958
\(649\) 23.6155 0.926991
\(650\) −1.12311 −0.0440518
\(651\) 0 0
\(652\) 21.5616 0.844416
\(653\) −11.1231 −0.435281 −0.217640 0.976029i \(-0.569836\pi\)
−0.217640 + 0.976029i \(0.569836\pi\)
\(654\) −6.63068 −0.259280
\(655\) 18.4384 0.720450
\(656\) 5.12311 0.200024
\(657\) 6.63068 0.258688
\(658\) 0 0
\(659\) −46.2462 −1.80150 −0.900748 0.434341i \(-0.856981\pi\)
−0.900748 + 0.434341i \(0.856981\pi\)
\(660\) −4.87689 −0.189833
\(661\) −40.4384 −1.57287 −0.786437 0.617671i \(-0.788076\pi\)
−0.786437 + 0.617671i \(0.788076\pi\)
\(662\) −12.0000 −0.466393
\(663\) −0.684658 −0.0265899
\(664\) 47.4233 1.84038
\(665\) 0 0
\(666\) −3.12311 −0.121018
\(667\) 6.24621 0.241854
\(668\) −22.0000 −0.851206
\(669\) 24.9848 0.965970
\(670\) −8.68466 −0.335518
\(671\) −1.36932 −0.0528619
\(672\) 0 0
\(673\) 28.1922 1.08673 0.543365 0.839496i \(-0.317150\pi\)
0.543365 + 0.839496i \(0.317150\pi\)
\(674\) 3.75379 0.144591
\(675\) −14.2462 −0.548337
\(676\) 12.8078 0.492606
\(677\) −0.384472 −0.0147765 −0.00738823 0.999973i \(-0.502352\pi\)
−0.00738823 + 0.999973i \(0.502352\pi\)
\(678\) −18.4384 −0.708124
\(679\) 0 0
\(680\) −4.68466 −0.179648
\(681\) −32.0000 −1.22624
\(682\) −12.0000 −0.459504
\(683\) 43.3693 1.65948 0.829740 0.558150i \(-0.188488\pi\)
0.829740 + 0.558150i \(0.188488\pi\)
\(684\) 4.00000 0.152944
\(685\) 19.1231 0.730656
\(686\) 0 0
\(687\) −9.06913 −0.346009
\(688\) 1.00000 0.0381246
\(689\) 4.68466 0.178471
\(690\) 4.87689 0.185660
\(691\) −0.492423 −0.0187326 −0.00936632 0.999956i \(-0.502981\pi\)
−0.00936632 + 0.999956i \(0.502981\pi\)
\(692\) 5.12311 0.194751
\(693\) 0 0
\(694\) 18.4384 0.699914
\(695\) −30.2462 −1.14730
\(696\) −14.6307 −0.554575
\(697\) 5.12311 0.194051
\(698\) 10.8769 0.411697
\(699\) −29.8617 −1.12947
\(700\) 0 0
\(701\) 28.0540 1.05958 0.529792 0.848128i \(-0.322270\pi\)
0.529792 + 0.848128i \(0.322270\pi\)
\(702\) 2.43845 0.0920333
\(703\) −39.6155 −1.49413
\(704\) −14.0000 −0.527645
\(705\) 23.3153 0.878107
\(706\) 19.1771 0.721739
\(707\) 0 0
\(708\) −18.4384 −0.692959
\(709\) 5.50758 0.206841 0.103421 0.994638i \(-0.467021\pi\)
0.103421 + 0.994638i \(0.467021\pi\)
\(710\) −8.68466 −0.325929
\(711\) 5.12311 0.192131
\(712\) −0.738634 −0.0276815
\(713\) −12.0000 −0.449404
\(714\) 0 0
\(715\) 1.36932 0.0512095
\(716\) −8.87689 −0.331745
\(717\) 12.4924 0.466538
\(718\) 3.80776 0.142104
\(719\) −6.00000 −0.223762 −0.111881 0.993722i \(-0.535688\pi\)
−0.111881 + 0.993722i \(0.535688\pi\)
\(720\) 0.876894 0.0326799
\(721\) 0 0
\(722\) −31.7386 −1.18119
\(723\) 8.38447 0.311822
\(724\) 19.3693 0.719855
\(725\) −8.00000 −0.297113
\(726\) −10.9309 −0.405683
\(727\) −50.7386 −1.88179 −0.940896 0.338696i \(-0.890014\pi\)
−0.940896 + 0.338696i \(0.890014\pi\)
\(728\) 0 0
\(729\) 29.9848 1.11055
\(730\) 18.4384 0.682438
\(731\) 1.00000 0.0369863
\(732\) 1.06913 0.0395162
\(733\) 8.24621 0.304581 0.152290 0.988336i \(-0.451335\pi\)
0.152290 + 0.988336i \(0.451335\pi\)
\(734\) −18.4924 −0.682568
\(735\) 17.0691 0.629604
\(736\) −10.0000 −0.368605
\(737\) −11.1231 −0.409725
\(738\) −2.87689 −0.105900
\(739\) −40.1080 −1.47539 −0.737697 0.675131i \(-0.764087\pi\)
−0.737697 + 0.675131i \(0.764087\pi\)
\(740\) 8.68466 0.319254
\(741\) 4.87689 0.179157
\(742\) 0 0
\(743\) 34.5464 1.26738 0.633692 0.773585i \(-0.281539\pi\)
0.633692 + 0.773585i \(0.281539\pi\)
\(744\) 28.1080 1.03049
\(745\) −12.8769 −0.471773
\(746\) 12.6307 0.462442
\(747\) −8.87689 −0.324789
\(748\) −2.00000 −0.0731272
\(749\) 0 0
\(750\) −18.4384 −0.673277
\(751\) −30.0540 −1.09669 −0.548343 0.836254i \(-0.684741\pi\)
−0.548343 + 0.836254i \(0.684741\pi\)
\(752\) 9.56155 0.348674
\(753\) 10.0540 0.366387
\(754\) 1.36932 0.0498676
\(755\) 12.4924 0.454646
\(756\) 0 0
\(757\) 26.4924 0.962883 0.481442 0.876478i \(-0.340113\pi\)
0.481442 + 0.876478i \(0.340113\pi\)
\(758\) 9.12311 0.331366
\(759\) 6.24621 0.226723
\(760\) 33.3693 1.21043
\(761\) 54.6004 1.97926 0.989631 0.143633i \(-0.0458786\pi\)
0.989631 + 0.143633i \(0.0458786\pi\)
\(762\) 34.4384 1.24757
\(763\) 0 0
\(764\) −11.1231 −0.402420
\(765\) 0.876894 0.0317042
\(766\) 0 0
\(767\) 5.17708 0.186934
\(768\) 26.5464 0.957911
\(769\) 24.5464 0.885166 0.442583 0.896728i \(-0.354062\pi\)
0.442583 + 0.896728i \(0.354062\pi\)
\(770\) 0 0
\(771\) −40.6004 −1.46219
\(772\) −2.00000 −0.0719816
\(773\) −22.4924 −0.808996 −0.404498 0.914539i \(-0.632554\pi\)
−0.404498 + 0.914539i \(0.632554\pi\)
\(774\) −0.561553 −0.0201846
\(775\) 15.3693 0.552082
\(776\) 48.7386 1.74961
\(777\) 0 0
\(778\) −2.49242 −0.0893577
\(779\) −36.4924 −1.30748
\(780\) −1.06913 −0.0382810
\(781\) −11.1231 −0.398016
\(782\) 2.00000 0.0715199
\(783\) 17.3693 0.620729
\(784\) 7.00000 0.250000
\(785\) −35.1231 −1.25360
\(786\) 18.4384 0.657678
\(787\) −34.9848 −1.24708 −0.623538 0.781793i \(-0.714305\pi\)
−0.623538 + 0.781793i \(0.714305\pi\)
\(788\) −17.6155 −0.627527
\(789\) −19.5076 −0.694488
\(790\) 14.2462 0.506857
\(791\) 0 0
\(792\) 3.36932 0.119723
\(793\) −0.300187 −0.0106599
\(794\) −6.87689 −0.244052
\(795\) 26.0540 0.924039
\(796\) 2.93087 0.103882
\(797\) 12.9309 0.458035 0.229017 0.973422i \(-0.426449\pi\)
0.229017 + 0.973422i \(0.426449\pi\)
\(798\) 0 0
\(799\) 9.56155 0.338263
\(800\) 12.8078 0.452823
\(801\) 0.138261 0.00488520
\(802\) 3.75379 0.132551
\(803\) 23.6155 0.833374
\(804\) 8.68466 0.306284
\(805\) 0 0
\(806\) −2.63068 −0.0926619
\(807\) −12.8769 −0.453288
\(808\) −6.00000 −0.211079
\(809\) 39.3693 1.38415 0.692076 0.721825i \(-0.256696\pi\)
0.692076 + 0.721825i \(0.256696\pi\)
\(810\) 10.9309 0.384072
\(811\) −7.50758 −0.263627 −0.131813 0.991275i \(-0.542080\pi\)
−0.131813 + 0.991275i \(0.542080\pi\)
\(812\) 0 0
\(813\) 29.5616 1.03677
\(814\) −11.1231 −0.389865
\(815\) −33.6695 −1.17939
\(816\) −1.56155 −0.0546653
\(817\) −7.12311 −0.249206
\(818\) 35.8617 1.25388
\(819\) 0 0
\(820\) 8.00000 0.279372
\(821\) −16.2462 −0.566997 −0.283498 0.958973i \(-0.591495\pi\)
−0.283498 + 0.958973i \(0.591495\pi\)
\(822\) 19.1231 0.666995
\(823\) 9.50758 0.331413 0.165707 0.986175i \(-0.447010\pi\)
0.165707 + 0.986175i \(0.447010\pi\)
\(824\) −52.6847 −1.83536
\(825\) −8.00000 −0.278524
\(826\) 0 0
\(827\) −28.7386 −0.999340 −0.499670 0.866216i \(-0.666545\pi\)
−0.499670 + 0.866216i \(0.666545\pi\)
\(828\) 1.12311 0.0390306
\(829\) 41.1231 1.42826 0.714132 0.700011i \(-0.246822\pi\)
0.714132 + 0.700011i \(0.246822\pi\)
\(830\) −24.6847 −0.856817
\(831\) −48.3002 −1.67551
\(832\) −3.06913 −0.106403
\(833\) 7.00000 0.242536
\(834\) −30.2462 −1.04734
\(835\) 34.3542 1.18887
\(836\) 14.2462 0.492716
\(837\) −33.3693 −1.15341
\(838\) 20.0000 0.690889
\(839\) 19.8078 0.683840 0.341920 0.939729i \(-0.388923\pi\)
0.341920 + 0.939729i \(0.388923\pi\)
\(840\) 0 0
\(841\) −19.2462 −0.663662
\(842\) −12.7386 −0.439002
\(843\) 32.6847 1.12572
\(844\) 4.19224 0.144303
\(845\) −20.0000 −0.688021
\(846\) −5.36932 −0.184601
\(847\) 0 0
\(848\) 10.6847 0.366913
\(849\) −19.1231 −0.656303
\(850\) −2.56155 −0.0878605
\(851\) −11.1231 −0.381295
\(852\) 8.68466 0.297531
\(853\) −54.9848 −1.88265 −0.941323 0.337508i \(-0.890416\pi\)
−0.941323 + 0.337508i \(0.890416\pi\)
\(854\) 0 0
\(855\) −6.24621 −0.213616
\(856\) 6.00000 0.205076
\(857\) 37.1231 1.26810 0.634051 0.773292i \(-0.281391\pi\)
0.634051 + 0.773292i \(0.281391\pi\)
\(858\) 1.36932 0.0467477
\(859\) 2.63068 0.0897577 0.0448789 0.998992i \(-0.485710\pi\)
0.0448789 + 0.998992i \(0.485710\pi\)
\(860\) 1.56155 0.0532485
\(861\) 0 0
\(862\) 13.1231 0.446975
\(863\) −12.4924 −0.425247 −0.212624 0.977134i \(-0.568201\pi\)
−0.212624 + 0.977134i \(0.568201\pi\)
\(864\) −27.8078 −0.946039
\(865\) −8.00000 −0.272008
\(866\) −11.3693 −0.386345
\(867\) −1.56155 −0.0530331
\(868\) 0 0
\(869\) 18.2462 0.618960
\(870\) 7.61553 0.258191
\(871\) −2.43845 −0.0826236
\(872\) −12.7386 −0.431385
\(873\) −9.12311 −0.308770
\(874\) −14.2462 −0.481885
\(875\) 0 0
\(876\) −18.4384 −0.622977
\(877\) −21.1231 −0.713277 −0.356638 0.934243i \(-0.616077\pi\)
−0.356638 + 0.934243i \(0.616077\pi\)
\(878\) −15.3693 −0.518689
\(879\) 0.384472 0.0129679
\(880\) 3.12311 0.105280
\(881\) −26.0000 −0.875962 −0.437981 0.898984i \(-0.644306\pi\)
−0.437981 + 0.898984i \(0.644306\pi\)
\(882\) −3.93087 −0.132359
\(883\) −51.9157 −1.74710 −0.873551 0.486732i \(-0.838189\pi\)
−0.873551 + 0.486732i \(0.838189\pi\)
\(884\) −0.438447 −0.0147466
\(885\) 28.7926 0.967853
\(886\) 31.4233 1.05569
\(887\) 38.2462 1.28418 0.642091 0.766628i \(-0.278067\pi\)
0.642091 + 0.766628i \(0.278067\pi\)
\(888\) 26.0540 0.874314
\(889\) 0 0
\(890\) 0.384472 0.0128875
\(891\) 14.0000 0.469018
\(892\) 16.0000 0.535720
\(893\) −68.1080 −2.27915
\(894\) −12.8769 −0.430668
\(895\) 13.8617 0.463347
\(896\) 0 0
\(897\) 1.36932 0.0457202
\(898\) 30.4384 1.01574
\(899\) −18.7386 −0.624968
\(900\) −1.43845 −0.0479482
\(901\) 10.6847 0.355958
\(902\) −10.2462 −0.341162
\(903\) 0 0
\(904\) −35.4233 −1.17816
\(905\) −30.2462 −1.00542
\(906\) 12.4924 0.415033
\(907\) 22.0000 0.730498 0.365249 0.930910i \(-0.380984\pi\)
0.365249 + 0.930910i \(0.380984\pi\)
\(908\) −20.4924 −0.680065
\(909\) 1.12311 0.0372511
\(910\) 0 0
\(911\) 31.8078 1.05384 0.526919 0.849915i \(-0.323347\pi\)
0.526919 + 0.849915i \(0.323347\pi\)
\(912\) 11.1231 0.368323
\(913\) −31.6155 −1.04632
\(914\) 4.63068 0.153169
\(915\) −1.66950 −0.0551921
\(916\) −5.80776 −0.191894
\(917\) 0 0
\(918\) 5.56155 0.183559
\(919\) −11.8078 −0.389502 −0.194751 0.980853i \(-0.562390\pi\)
−0.194751 + 0.980853i \(0.562390\pi\)
\(920\) 9.36932 0.308897
\(921\) 22.2462 0.733038
\(922\) 17.8078 0.586467
\(923\) −2.43845 −0.0802625
\(924\) 0 0
\(925\) 14.2462 0.468413
\(926\) 8.00000 0.262896
\(927\) 9.86174 0.323902
\(928\) −15.6155 −0.512605
\(929\) −25.8617 −0.848496 −0.424248 0.905546i \(-0.639462\pi\)
−0.424248 + 0.905546i \(0.639462\pi\)
\(930\) −14.6307 −0.479759
\(931\) −49.8617 −1.63415
\(932\) −19.1231 −0.626398
\(933\) −24.0000 −0.785725
\(934\) 32.4924 1.06318
\(935\) 3.12311 0.102136
\(936\) 0.738634 0.0241430
\(937\) 34.1080 1.11426 0.557129 0.830426i \(-0.311903\pi\)
0.557129 + 0.830426i \(0.311903\pi\)
\(938\) 0 0
\(939\) −24.6847 −0.805553
\(940\) 14.9309 0.486991
\(941\) 28.6307 0.933334 0.466667 0.884433i \(-0.345455\pi\)
0.466667 + 0.884433i \(0.345455\pi\)
\(942\) −35.1231 −1.14437
\(943\) −10.2462 −0.333663
\(944\) 11.8078 0.384310
\(945\) 0 0
\(946\) −2.00000 −0.0650256
\(947\) 0.246211 0.00800079 0.00400040 0.999992i \(-0.498727\pi\)
0.00400040 + 0.999992i \(0.498727\pi\)
\(948\) −14.2462 −0.462695
\(949\) 5.17708 0.168055
\(950\) 18.2462 0.591985
\(951\) 30.2462 0.980801
\(952\) 0 0
\(953\) −29.1231 −0.943390 −0.471695 0.881762i \(-0.656358\pi\)
−0.471695 + 0.881762i \(0.656358\pi\)
\(954\) −6.00000 −0.194257
\(955\) 17.3693 0.562058
\(956\) 8.00000 0.258738
\(957\) 9.75379 0.315295
\(958\) −31.3693 −1.01350
\(959\) 0 0
\(960\) −17.0691 −0.550904
\(961\) 5.00000 0.161290
\(962\) −2.43845 −0.0786187
\(963\) −1.12311 −0.0361916
\(964\) 5.36932 0.172934
\(965\) 3.12311 0.100536
\(966\) 0 0
\(967\) −24.9848 −0.803458 −0.401729 0.915758i \(-0.631591\pi\)
−0.401729 + 0.915758i \(0.631591\pi\)
\(968\) −21.0000 −0.674966
\(969\) 11.1231 0.357326
\(970\) −25.3693 −0.814560
\(971\) 17.1771 0.551239 0.275619 0.961267i \(-0.411117\pi\)
0.275619 + 0.961267i \(0.411117\pi\)
\(972\) 5.75379 0.184553
\(973\) 0 0
\(974\) 26.4924 0.848872
\(975\) −1.75379 −0.0561662
\(976\) −0.684658 −0.0219154
\(977\) 23.7538 0.759951 0.379976 0.924997i \(-0.375932\pi\)
0.379976 + 0.924997i \(0.375932\pi\)
\(978\) −33.6695 −1.07663
\(979\) 0.492423 0.0157379
\(980\) 10.9309 0.349174
\(981\) 2.38447 0.0761303
\(982\) 5.36932 0.171342
\(983\) −42.5464 −1.35702 −0.678510 0.734591i \(-0.737374\pi\)
−0.678510 + 0.734591i \(0.737374\pi\)
\(984\) 24.0000 0.765092
\(985\) 27.5076 0.876464
\(986\) 3.12311 0.0994599
\(987\) 0 0
\(988\) 3.12311 0.0993592
\(989\) −2.00000 −0.0635963
\(990\) −1.75379 −0.0557391
\(991\) −55.9157 −1.77622 −0.888111 0.459630i \(-0.847982\pi\)
−0.888111 + 0.459630i \(0.847982\pi\)
\(992\) 30.0000 0.952501
\(993\) −18.7386 −0.594653
\(994\) 0 0
\(995\) −4.57671 −0.145091
\(996\) 24.6847 0.782163
\(997\) −0.876894 −0.0277715 −0.0138858 0.999904i \(-0.504420\pi\)
−0.0138858 + 0.999904i \(0.504420\pi\)
\(998\) −4.68466 −0.148290
\(999\) −30.9309 −0.978609
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 731.2.a.b.1.1 2
3.2 odd 2 6579.2.a.f.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
731.2.a.b.1.1 2 1.1 even 1 trivial
6579.2.a.f.1.1 2 3.2 odd 2