Properties

Label 8470.2.a.db.1.2
Level $8470$
Weight $2$
Character 8470.1
Self dual yes
Analytic conductor $67.633$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8470.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(67.6332905120\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.4642000.1
Defining polynomial: \(x^{6} - x^{5} - 8 x^{4} + 5 x^{3} + 14 x^{2} - 9 x - 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 770)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.32077\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.32077 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.32077 q^{6} -1.00000 q^{7} +1.00000 q^{8} -1.25558 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.32077 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.32077 q^{6} -1.00000 q^{7} +1.00000 q^{8} -1.25558 q^{9} +1.00000 q^{10} -1.32077 q^{12} +5.34554 q^{13} -1.00000 q^{14} -1.32077 q^{15} +1.00000 q^{16} -6.44843 q^{17} -1.25558 q^{18} -6.42752 q^{19} +1.00000 q^{20} +1.32077 q^{21} +5.51016 q^{23} -1.32077 q^{24} +1.00000 q^{25} +5.34554 q^{26} +5.62062 q^{27} -1.00000 q^{28} -8.19519 q^{29} -1.32077 q^{30} +1.77459 q^{31} +1.00000 q^{32} -6.44843 q^{34} -1.00000 q^{35} -1.25558 q^{36} +10.9821 q^{37} -6.42752 q^{38} -7.06020 q^{39} +1.00000 q^{40} +2.08350 q^{41} +1.32077 q^{42} -2.71912 q^{43} -1.25558 q^{45} +5.51016 q^{46} +4.35747 q^{47} -1.32077 q^{48} +1.00000 q^{49} +1.00000 q^{50} +8.51686 q^{51} +5.34554 q^{52} -8.59127 q^{53} +5.62062 q^{54} -1.00000 q^{56} +8.48925 q^{57} -8.19519 q^{58} -5.44657 q^{59} -1.32077 q^{60} +13.3987 q^{61} +1.77459 q^{62} +1.25558 q^{63} +1.00000 q^{64} +5.34554 q^{65} -10.8106 q^{67} -6.44843 q^{68} -7.27763 q^{69} -1.00000 q^{70} -11.8156 q^{71} -1.25558 q^{72} -5.54852 q^{73} +10.9821 q^{74} -1.32077 q^{75} -6.42752 q^{76} -7.06020 q^{78} -7.35859 q^{79} +1.00000 q^{80} -3.65680 q^{81} +2.08350 q^{82} -7.19204 q^{83} +1.32077 q^{84} -6.44843 q^{85} -2.71912 q^{86} +10.8239 q^{87} -5.39995 q^{89} -1.25558 q^{90} -5.34554 q^{91} +5.51016 q^{92} -2.34381 q^{93} +4.35747 q^{94} -6.42752 q^{95} -1.32077 q^{96} -0.548361 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{2} - q^{3} + 6 q^{4} + 6 q^{5} - q^{6} - 6 q^{7} + 6 q^{8} - q^{9} + O(q^{10}) \) \( 6 q + 6 q^{2} - q^{3} + 6 q^{4} + 6 q^{5} - q^{6} - 6 q^{7} + 6 q^{8} - q^{9} + 6 q^{10} - q^{12} - 6 q^{13} - 6 q^{14} - q^{15} + 6 q^{16} - 21 q^{17} - q^{18} + 3 q^{19} + 6 q^{20} + q^{21} - 10 q^{23} - q^{24} + 6 q^{25} - 6 q^{26} - 4 q^{27} - 6 q^{28} - 10 q^{29} - q^{30} - 4 q^{31} + 6 q^{32} - 21 q^{34} - 6 q^{35} - q^{36} - 2 q^{37} + 3 q^{38} - 26 q^{39} + 6 q^{40} - 7 q^{41} + q^{42} - 19 q^{43} - q^{45} - 10 q^{46} + 10 q^{47} - q^{48} + 6 q^{49} + 6 q^{50} + 4 q^{51} - 6 q^{52} - 16 q^{53} - 4 q^{54} - 6 q^{56} - 16 q^{57} - 10 q^{58} - 3 q^{59} - q^{60} + 8 q^{61} - 4 q^{62} + q^{63} + 6 q^{64} - 6 q^{65} - 27 q^{67} - 21 q^{68} + 4 q^{69} - 6 q^{70} + 4 q^{71} - q^{72} - 13 q^{73} - 2 q^{74} - q^{75} + 3 q^{76} - 26 q^{78} - 14 q^{79} + 6 q^{80} - 14 q^{81} - 7 q^{82} - 51 q^{83} + q^{84} - 21 q^{85} - 19 q^{86} - 8 q^{87} + q^{89} - q^{90} + 6 q^{91} - 10 q^{92} + 4 q^{93} + 10 q^{94} + 3 q^{95} - q^{96} + 7 q^{97} + 6 q^{98} + O(q^{100}) \)

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
\(3\) −1.32077 −0.762545 −0.381272 0.924463i \(-0.624514\pi\)
−0.381272 + 0.924463i \(0.624514\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) −1.32077 −0.539201
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) −1.25558 −0.418526
\(10\) 1.00000 0.316228
\(11\) 0 0
\(12\) −1.32077 −0.381272
\(13\) 5.34554 1.48259 0.741293 0.671182i \(-0.234213\pi\)
0.741293 + 0.671182i \(0.234213\pi\)
\(14\) −1.00000 −0.267261
\(15\) −1.32077 −0.341020
\(16\) 1.00000 0.250000
\(17\) −6.44843 −1.56397 −0.781987 0.623295i \(-0.785794\pi\)
−0.781987 + 0.623295i \(0.785794\pi\)
\(18\) −1.25558 −0.295942
\(19\) −6.42752 −1.47457 −0.737287 0.675580i \(-0.763893\pi\)
−0.737287 + 0.675580i \(0.763893\pi\)
\(20\) 1.00000 0.223607
\(21\) 1.32077 0.288215
\(22\) 0 0
\(23\) 5.51016 1.14895 0.574474 0.818523i \(-0.305207\pi\)
0.574474 + 0.818523i \(0.305207\pi\)
\(24\) −1.32077 −0.269600
\(25\) 1.00000 0.200000
\(26\) 5.34554 1.04835
\(27\) 5.62062 1.08169
\(28\) −1.00000 −0.188982
\(29\) −8.19519 −1.52181 −0.760904 0.648864i \(-0.775244\pi\)
−0.760904 + 0.648864i \(0.775244\pi\)
\(30\) −1.32077 −0.241138
\(31\) 1.77459 0.318725 0.159363 0.987220i \(-0.449056\pi\)
0.159363 + 0.987220i \(0.449056\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −6.44843 −1.10590
\(35\) −1.00000 −0.169031
\(36\) −1.25558 −0.209263
\(37\) 10.9821 1.80544 0.902722 0.430225i \(-0.141566\pi\)
0.902722 + 0.430225i \(0.141566\pi\)
\(38\) −6.42752 −1.04268
\(39\) −7.06020 −1.13054
\(40\) 1.00000 0.158114
\(41\) 2.08350 0.325389 0.162694 0.986677i \(-0.447982\pi\)
0.162694 + 0.986677i \(0.447982\pi\)
\(42\) 1.32077 0.203799
\(43\) −2.71912 −0.414661 −0.207331 0.978271i \(-0.566478\pi\)
−0.207331 + 0.978271i \(0.566478\pi\)
\(44\) 0 0
\(45\) −1.25558 −0.187170
\(46\) 5.51016 0.812428
\(47\) 4.35747 0.635603 0.317801 0.948157i \(-0.397056\pi\)
0.317801 + 0.948157i \(0.397056\pi\)
\(48\) −1.32077 −0.190636
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 8.51686 1.19260
\(52\) 5.34554 0.741293
\(53\) −8.59127 −1.18010 −0.590051 0.807366i \(-0.700892\pi\)
−0.590051 + 0.807366i \(0.700892\pi\)
\(54\) 5.62062 0.764870
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) 8.48925 1.12443
\(58\) −8.19519 −1.07608
\(59\) −5.44657 −0.709083 −0.354541 0.935040i \(-0.615363\pi\)
−0.354541 + 0.935040i \(0.615363\pi\)
\(60\) −1.32077 −0.170510
\(61\) 13.3987 1.71553 0.857766 0.514040i \(-0.171852\pi\)
0.857766 + 0.514040i \(0.171852\pi\)
\(62\) 1.77459 0.225373
\(63\) 1.25558 0.158188
\(64\) 1.00000 0.125000
\(65\) 5.34554 0.663032
\(66\) 0 0
\(67\) −10.8106 −1.32073 −0.660363 0.750947i \(-0.729597\pi\)
−0.660363 + 0.750947i \(0.729597\pi\)
\(68\) −6.44843 −0.781987
\(69\) −7.27763 −0.876124
\(70\) −1.00000 −0.119523
\(71\) −11.8156 −1.40225 −0.701127 0.713036i \(-0.747319\pi\)
−0.701127 + 0.713036i \(0.747319\pi\)
\(72\) −1.25558 −0.147971
\(73\) −5.54852 −0.649405 −0.324702 0.945816i \(-0.605264\pi\)
−0.324702 + 0.945816i \(0.605264\pi\)
\(74\) 10.9821 1.27664
\(75\) −1.32077 −0.152509
\(76\) −6.42752 −0.737287
\(77\) 0 0
\(78\) −7.06020 −0.799411
\(79\) −7.35859 −0.827906 −0.413953 0.910298i \(-0.635852\pi\)
−0.413953 + 0.910298i \(0.635852\pi\)
\(80\) 1.00000 0.111803
\(81\) −3.65680 −0.406311
\(82\) 2.08350 0.230085
\(83\) −7.19204 −0.789429 −0.394714 0.918804i \(-0.629156\pi\)
−0.394714 + 0.918804i \(0.629156\pi\)
\(84\) 1.32077 0.144107
\(85\) −6.44843 −0.699430
\(86\) −2.71912 −0.293210
\(87\) 10.8239 1.16045
\(88\) 0 0
\(89\) −5.39995 −0.572393 −0.286197 0.958171i \(-0.592391\pi\)
−0.286197 + 0.958171i \(0.592391\pi\)
\(90\) −1.25558 −0.132349
\(91\) −5.34554 −0.560364
\(92\) 5.51016 0.574474
\(93\) −2.34381 −0.243042
\(94\) 4.35747 0.449439
\(95\) −6.42752 −0.659449
\(96\) −1.32077 −0.134800
\(97\) −0.548361 −0.0556776 −0.0278388 0.999612i \(-0.508863\pi\)
−0.0278388 + 0.999612i \(0.508863\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 2.44229 0.243017 0.121509 0.992590i \(-0.461227\pi\)
0.121509 + 0.992590i \(0.461227\pi\)
\(102\) 8.51686 0.843295
\(103\) 0.630289 0.0621042 0.0310521 0.999518i \(-0.490114\pi\)
0.0310521 + 0.999518i \(0.490114\pi\)
\(104\) 5.34554 0.524173
\(105\) 1.32077 0.128894
\(106\) −8.59127 −0.834458
\(107\) 11.4352 1.10549 0.552743 0.833352i \(-0.313581\pi\)
0.552743 + 0.833352i \(0.313581\pi\)
\(108\) 5.62062 0.540845
\(109\) −8.31582 −0.796512 −0.398256 0.917274i \(-0.630384\pi\)
−0.398256 + 0.917274i \(0.630384\pi\)
\(110\) 0 0
\(111\) −14.5048 −1.37673
\(112\) −1.00000 −0.0944911
\(113\) −20.1474 −1.89531 −0.947656 0.319294i \(-0.896554\pi\)
−0.947656 + 0.319294i \(0.896554\pi\)
\(114\) 8.48925 0.795091
\(115\) 5.51016 0.513825
\(116\) −8.19519 −0.760904
\(117\) −6.71173 −0.620500
\(118\) −5.44657 −0.501397
\(119\) 6.44843 0.591126
\(120\) −1.32077 −0.120569
\(121\) 0 0
\(122\) 13.3987 1.21306
\(123\) −2.75182 −0.248123
\(124\) 1.77459 0.159363
\(125\) 1.00000 0.0894427
\(126\) 1.25558 0.111856
\(127\) −11.0010 −0.976180 −0.488090 0.872793i \(-0.662306\pi\)
−0.488090 + 0.872793i \(0.662306\pi\)
\(128\) 1.00000 0.0883883
\(129\) 3.59132 0.316198
\(130\) 5.34554 0.468835
\(131\) −18.2978 −1.59868 −0.799342 0.600876i \(-0.794818\pi\)
−0.799342 + 0.600876i \(0.794818\pi\)
\(132\) 0 0
\(133\) 6.42752 0.557336
\(134\) −10.8106 −0.933894
\(135\) 5.62062 0.483746
\(136\) −6.44843 −0.552948
\(137\) 5.46086 0.466552 0.233276 0.972411i \(-0.425055\pi\)
0.233276 + 0.972411i \(0.425055\pi\)
\(138\) −7.27763 −0.619513
\(139\) −5.25491 −0.445716 −0.222858 0.974851i \(-0.571539\pi\)
−0.222858 + 0.974851i \(0.571539\pi\)
\(140\) −1.00000 −0.0845154
\(141\) −5.75520 −0.484675
\(142\) −11.8156 −0.991543
\(143\) 0 0
\(144\) −1.25558 −0.104631
\(145\) −8.19519 −0.680574
\(146\) −5.54852 −0.459199
\(147\) −1.32077 −0.108935
\(148\) 10.9821 0.902722
\(149\) 9.21250 0.754718 0.377359 0.926067i \(-0.376832\pi\)
0.377359 + 0.926067i \(0.376832\pi\)
\(150\) −1.32077 −0.107840
\(151\) 14.2773 1.16187 0.580936 0.813949i \(-0.302686\pi\)
0.580936 + 0.813949i \(0.302686\pi\)
\(152\) −6.42752 −0.521340
\(153\) 8.09650 0.654563
\(154\) 0 0
\(155\) 1.77459 0.142538
\(156\) −7.06020 −0.565269
\(157\) 14.9439 1.19265 0.596325 0.802743i \(-0.296627\pi\)
0.596325 + 0.802743i \(0.296627\pi\)
\(158\) −7.35859 −0.585418
\(159\) 11.3471 0.899880
\(160\) 1.00000 0.0790569
\(161\) −5.51016 −0.434261
\(162\) −3.65680 −0.287305
\(163\) 16.7682 1.31339 0.656695 0.754157i \(-0.271954\pi\)
0.656695 + 0.754157i \(0.271954\pi\)
\(164\) 2.08350 0.162694
\(165\) 0 0
\(166\) −7.19204 −0.558210
\(167\) 6.20304 0.480006 0.240003 0.970772i \(-0.422852\pi\)
0.240003 + 0.970772i \(0.422852\pi\)
\(168\) 1.32077 0.101899
\(169\) 15.5748 1.19806
\(170\) −6.44843 −0.494572
\(171\) 8.07024 0.617147
\(172\) −2.71912 −0.207331
\(173\) −7.35006 −0.558815 −0.279407 0.960173i \(-0.590138\pi\)
−0.279407 + 0.960173i \(0.590138\pi\)
\(174\) 10.8239 0.820560
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) 7.19364 0.540707
\(178\) −5.39995 −0.404743
\(179\) −4.40431 −0.329194 −0.164597 0.986361i \(-0.552632\pi\)
−0.164597 + 0.986361i \(0.552632\pi\)
\(180\) −1.25558 −0.0935852
\(181\) −11.0910 −0.824387 −0.412193 0.911096i \(-0.635237\pi\)
−0.412193 + 0.911096i \(0.635237\pi\)
\(182\) −5.34554 −0.396238
\(183\) −17.6966 −1.30817
\(184\) 5.51016 0.406214
\(185\) 10.9821 0.807419
\(186\) −2.34381 −0.171857
\(187\) 0 0
\(188\) 4.35747 0.317801
\(189\) −5.62062 −0.408840
\(190\) −6.42752 −0.466301
\(191\) −14.3562 −1.03878 −0.519388 0.854538i \(-0.673840\pi\)
−0.519388 + 0.854538i \(0.673840\pi\)
\(192\) −1.32077 −0.0953181
\(193\) 8.48187 0.610538 0.305269 0.952266i \(-0.401254\pi\)
0.305269 + 0.952266i \(0.401254\pi\)
\(194\) −0.548361 −0.0393700
\(195\) −7.06020 −0.505592
\(196\) 1.00000 0.0714286
\(197\) −4.97357 −0.354352 −0.177176 0.984179i \(-0.556696\pi\)
−0.177176 + 0.984179i \(0.556696\pi\)
\(198\) 0 0
\(199\) −22.6336 −1.60445 −0.802227 0.597018i \(-0.796352\pi\)
−0.802227 + 0.597018i \(0.796352\pi\)
\(200\) 1.00000 0.0707107
\(201\) 14.2783 1.00711
\(202\) 2.44229 0.171839
\(203\) 8.19519 0.575190
\(204\) 8.51686 0.596300
\(205\) 2.08350 0.145518
\(206\) 0.630289 0.0439143
\(207\) −6.91843 −0.480864
\(208\) 5.34554 0.370646
\(209\) 0 0
\(210\) 1.32077 0.0911415
\(211\) −3.78425 −0.260519 −0.130259 0.991480i \(-0.541581\pi\)
−0.130259 + 0.991480i \(0.541581\pi\)
\(212\) −8.59127 −0.590051
\(213\) 15.6056 1.06928
\(214\) 11.4352 0.781697
\(215\) −2.71912 −0.185442
\(216\) 5.62062 0.382435
\(217\) −1.77459 −0.120467
\(218\) −8.31582 −0.563219
\(219\) 7.32830 0.495200
\(220\) 0 0
\(221\) −34.4703 −2.31872
\(222\) −14.5048 −0.973496
\(223\) 22.5140 1.50765 0.753824 0.657077i \(-0.228207\pi\)
0.753824 + 0.657077i \(0.228207\pi\)
\(224\) −1.00000 −0.0668153
\(225\) −1.25558 −0.0837051
\(226\) −20.1474 −1.34019
\(227\) −10.8997 −0.723439 −0.361719 0.932287i \(-0.617810\pi\)
−0.361719 + 0.932287i \(0.617810\pi\)
\(228\) 8.48925 0.562214
\(229\) −3.17445 −0.209773 −0.104887 0.994484i \(-0.533448\pi\)
−0.104887 + 0.994484i \(0.533448\pi\)
\(230\) 5.51016 0.363329
\(231\) 0 0
\(232\) −8.19519 −0.538041
\(233\) −25.6323 −1.67923 −0.839614 0.543184i \(-0.817219\pi\)
−0.839614 + 0.543184i \(0.817219\pi\)
\(234\) −6.71173 −0.438760
\(235\) 4.35747 0.284250
\(236\) −5.44657 −0.354541
\(237\) 9.71897 0.631315
\(238\) 6.44843 0.417989
\(239\) −17.5664 −1.13628 −0.568139 0.822932i \(-0.692337\pi\)
−0.568139 + 0.822932i \(0.692337\pi\)
\(240\) −1.32077 −0.0852551
\(241\) 5.63187 0.362781 0.181390 0.983411i \(-0.441940\pi\)
0.181390 + 0.983411i \(0.441940\pi\)
\(242\) 0 0
\(243\) −12.0321 −0.771859
\(244\) 13.3987 0.857766
\(245\) 1.00000 0.0638877
\(246\) −2.75182 −0.175450
\(247\) −34.3585 −2.18618
\(248\) 1.77459 0.112686
\(249\) 9.49900 0.601975
\(250\) 1.00000 0.0632456
\(251\) 14.3083 0.903130 0.451565 0.892238i \(-0.350866\pi\)
0.451565 + 0.892238i \(0.350866\pi\)
\(252\) 1.25558 0.0790939
\(253\) 0 0
\(254\) −11.0010 −0.690263
\(255\) 8.51686 0.533347
\(256\) 1.00000 0.0625000
\(257\) −7.69294 −0.479872 −0.239936 0.970789i \(-0.577126\pi\)
−0.239936 + 0.970789i \(0.577126\pi\)
\(258\) 3.59132 0.223586
\(259\) −10.9821 −0.682393
\(260\) 5.34554 0.331516
\(261\) 10.2897 0.636916
\(262\) −18.2978 −1.13044
\(263\) −6.81613 −0.420300 −0.210150 0.977669i \(-0.567395\pi\)
−0.210150 + 0.977669i \(0.567395\pi\)
\(264\) 0 0
\(265\) −8.59127 −0.527758
\(266\) 6.42752 0.394096
\(267\) 7.13207 0.436475
\(268\) −10.8106 −0.660363
\(269\) 3.79822 0.231581 0.115791 0.993274i \(-0.463060\pi\)
0.115791 + 0.993274i \(0.463060\pi\)
\(270\) 5.62062 0.342060
\(271\) 2.35406 0.142999 0.0714996 0.997441i \(-0.477222\pi\)
0.0714996 + 0.997441i \(0.477222\pi\)
\(272\) −6.44843 −0.390993
\(273\) 7.06020 0.427303
\(274\) 5.46086 0.329902
\(275\) 0 0
\(276\) −7.27763 −0.438062
\(277\) −17.4409 −1.04792 −0.523962 0.851742i \(-0.675547\pi\)
−0.523962 + 0.851742i \(0.675547\pi\)
\(278\) −5.25491 −0.315169
\(279\) −2.22813 −0.133395
\(280\) −1.00000 −0.0597614
\(281\) −22.6310 −1.35005 −0.675027 0.737793i \(-0.735868\pi\)
−0.675027 + 0.737793i \(0.735868\pi\)
\(282\) −5.75520 −0.342717
\(283\) −27.0596 −1.60852 −0.804262 0.594274i \(-0.797439\pi\)
−0.804262 + 0.594274i \(0.797439\pi\)
\(284\) −11.8156 −0.701127
\(285\) 8.48925 0.502860
\(286\) 0 0
\(287\) −2.08350 −0.122985
\(288\) −1.25558 −0.0739856
\(289\) 24.5822 1.44601
\(290\) −8.19519 −0.481238
\(291\) 0.724257 0.0424567
\(292\) −5.54852 −0.324702
\(293\) −20.1958 −1.17985 −0.589926 0.807458i \(-0.700843\pi\)
−0.589926 + 0.807458i \(0.700843\pi\)
\(294\) −1.32077 −0.0770286
\(295\) −5.44657 −0.317111
\(296\) 10.9821 0.638321
\(297\) 0 0
\(298\) 9.21250 0.533666
\(299\) 29.4547 1.70341
\(300\) −1.32077 −0.0762545
\(301\) 2.71912 0.156727
\(302\) 14.2773 0.821568
\(303\) −3.22570 −0.185311
\(304\) −6.42752 −0.368643
\(305\) 13.3987 0.767209
\(306\) 8.09650 0.462846
\(307\) 24.4128 1.39331 0.696656 0.717405i \(-0.254670\pi\)
0.696656 + 0.717405i \(0.254670\pi\)
\(308\) 0 0
\(309\) −0.832464 −0.0473572
\(310\) 1.77459 0.100790
\(311\) 25.2667 1.43274 0.716371 0.697720i \(-0.245802\pi\)
0.716371 + 0.697720i \(0.245802\pi\)
\(312\) −7.06020 −0.399705
\(313\) −18.1747 −1.02729 −0.513647 0.858002i \(-0.671706\pi\)
−0.513647 + 0.858002i \(0.671706\pi\)
\(314\) 14.9439 0.843331
\(315\) 1.25558 0.0707437
\(316\) −7.35859 −0.413953
\(317\) 10.6739 0.599506 0.299753 0.954017i \(-0.403096\pi\)
0.299753 + 0.954017i \(0.403096\pi\)
\(318\) 11.3471 0.636312
\(319\) 0 0
\(320\) 1.00000 0.0559017
\(321\) −15.1033 −0.842983
\(322\) −5.51016 −0.307069
\(323\) 41.4474 2.30619
\(324\) −3.65680 −0.203155
\(325\) 5.34554 0.296517
\(326\) 16.7682 0.928707
\(327\) 10.9833 0.607376
\(328\) 2.08350 0.115042
\(329\) −4.35747 −0.240235
\(330\) 0 0
\(331\) −9.75660 −0.536271 −0.268136 0.963381i \(-0.586408\pi\)
−0.268136 + 0.963381i \(0.586408\pi\)
\(332\) −7.19204 −0.394714
\(333\) −13.7888 −0.755624
\(334\) 6.20304 0.339415
\(335\) −10.8106 −0.590646
\(336\) 1.32077 0.0720537
\(337\) 10.1462 0.552699 0.276350 0.961057i \(-0.410875\pi\)
0.276350 + 0.961057i \(0.410875\pi\)
\(338\) 15.5748 0.847155
\(339\) 26.6101 1.44526
\(340\) −6.44843 −0.349715
\(341\) 0 0
\(342\) 8.07024 0.436389
\(343\) −1.00000 −0.0539949
\(344\) −2.71912 −0.146605
\(345\) −7.27763 −0.391814
\(346\) −7.35006 −0.395142
\(347\) −21.1756 −1.13677 −0.568383 0.822764i \(-0.692431\pi\)
−0.568383 + 0.822764i \(0.692431\pi\)
\(348\) 10.8239 0.580224
\(349\) −4.58943 −0.245667 −0.122833 0.992427i \(-0.539198\pi\)
−0.122833 + 0.992427i \(0.539198\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 30.0452 1.60370
\(352\) 0 0
\(353\) 32.2094 1.71433 0.857166 0.515040i \(-0.172223\pi\)
0.857166 + 0.515040i \(0.172223\pi\)
\(354\) 7.19364 0.382338
\(355\) −11.8156 −0.627107
\(356\) −5.39995 −0.286197
\(357\) −8.51686 −0.450760
\(358\) −4.40431 −0.232775
\(359\) −6.54031 −0.345184 −0.172592 0.984993i \(-0.555214\pi\)
−0.172592 + 0.984993i \(0.555214\pi\)
\(360\) −1.25558 −0.0661747
\(361\) 22.3130 1.17437
\(362\) −11.0910 −0.582930
\(363\) 0 0
\(364\) −5.34554 −0.280182
\(365\) −5.54852 −0.290423
\(366\) −17.6966 −0.925016
\(367\) −2.98556 −0.155845 −0.0779225 0.996959i \(-0.524829\pi\)
−0.0779225 + 0.996959i \(0.524829\pi\)
\(368\) 5.51016 0.287237
\(369\) −2.61600 −0.136183
\(370\) 10.9821 0.570931
\(371\) 8.59127 0.446037
\(372\) −2.34381 −0.121521
\(373\) −16.8471 −0.872311 −0.436155 0.899871i \(-0.643660\pi\)
−0.436155 + 0.899871i \(0.643660\pi\)
\(374\) 0 0
\(375\) −1.32077 −0.0682041
\(376\) 4.35747 0.224719
\(377\) −43.8077 −2.25621
\(378\) −5.62062 −0.289094
\(379\) −27.4781 −1.41145 −0.705727 0.708484i \(-0.749380\pi\)
−0.705727 + 0.708484i \(0.749380\pi\)
\(380\) −6.42752 −0.329725
\(381\) 14.5297 0.744381
\(382\) −14.3562 −0.734526
\(383\) −31.3930 −1.60411 −0.802055 0.597251i \(-0.796260\pi\)
−0.802055 + 0.597251i \(0.796260\pi\)
\(384\) −1.32077 −0.0674001
\(385\) 0 0
\(386\) 8.48187 0.431716
\(387\) 3.41406 0.173546
\(388\) −0.548361 −0.0278388
\(389\) −0.115199 −0.00584080 −0.00292040 0.999996i \(-0.500930\pi\)
−0.00292040 + 0.999996i \(0.500930\pi\)
\(390\) −7.06020 −0.357507
\(391\) −35.5318 −1.79692
\(392\) 1.00000 0.0505076
\(393\) 24.1671 1.21907
\(394\) −4.97357 −0.250565
\(395\) −7.35859 −0.370251
\(396\) 0 0
\(397\) −18.2744 −0.917165 −0.458583 0.888652i \(-0.651643\pi\)
−0.458583 + 0.888652i \(0.651643\pi\)
\(398\) −22.6336 −1.13452
\(399\) −8.48925 −0.424994
\(400\) 1.00000 0.0500000
\(401\) 30.0646 1.50136 0.750678 0.660668i \(-0.229727\pi\)
0.750678 + 0.660668i \(0.229727\pi\)
\(402\) 14.2783 0.712136
\(403\) 9.48612 0.472537
\(404\) 2.44229 0.121509
\(405\) −3.65680 −0.181708
\(406\) 8.19519 0.406720
\(407\) 0 0
\(408\) 8.51686 0.421648
\(409\) 29.2637 1.44700 0.723499 0.690326i \(-0.242533\pi\)
0.723499 + 0.690326i \(0.242533\pi\)
\(410\) 2.08350 0.102897
\(411\) −7.21252 −0.355767
\(412\) 0.630289 0.0310521
\(413\) 5.44657 0.268008
\(414\) −6.91843 −0.340022
\(415\) −7.19204 −0.353043
\(416\) 5.34554 0.262086
\(417\) 6.94051 0.339878
\(418\) 0 0
\(419\) −17.3655 −0.848362 −0.424181 0.905577i \(-0.639438\pi\)
−0.424181 + 0.905577i \(0.639438\pi\)
\(420\) 1.32077 0.0644468
\(421\) 8.29303 0.404177 0.202089 0.979367i \(-0.435227\pi\)
0.202089 + 0.979367i \(0.435227\pi\)
\(422\) −3.78425 −0.184214
\(423\) −5.47114 −0.266016
\(424\) −8.59127 −0.417229
\(425\) −6.44843 −0.312795
\(426\) 15.6056 0.756096
\(427\) −13.3987 −0.648410
\(428\) 11.4352 0.552743
\(429\) 0 0
\(430\) −2.71912 −0.131127
\(431\) 13.3393 0.642530 0.321265 0.946989i \(-0.395892\pi\)
0.321265 + 0.946989i \(0.395892\pi\)
\(432\) 5.62062 0.270422
\(433\) −14.8300 −0.712685 −0.356342 0.934355i \(-0.615976\pi\)
−0.356342 + 0.934355i \(0.615976\pi\)
\(434\) −1.77459 −0.0851829
\(435\) 10.8239 0.518968
\(436\) −8.31582 −0.398256
\(437\) −35.4166 −1.69421
\(438\) 7.32830 0.350160
\(439\) −6.08156 −0.290257 −0.145128 0.989413i \(-0.546360\pi\)
−0.145128 + 0.989413i \(0.546360\pi\)
\(440\) 0 0
\(441\) −1.25558 −0.0597894
\(442\) −34.4703 −1.63959
\(443\) −19.8590 −0.943530 −0.471765 0.881724i \(-0.656383\pi\)
−0.471765 + 0.881724i \(0.656383\pi\)
\(444\) −14.5048 −0.688366
\(445\) −5.39995 −0.255982
\(446\) 22.5140 1.06607
\(447\) −12.1676 −0.575506
\(448\) −1.00000 −0.0472456
\(449\) 0.806539 0.0380629 0.0190315 0.999819i \(-0.493942\pi\)
0.0190315 + 0.999819i \(0.493942\pi\)
\(450\) −1.25558 −0.0591885
\(451\) 0 0
\(452\) −20.1474 −0.947656
\(453\) −18.8570 −0.885980
\(454\) −10.8997 −0.511548
\(455\) −5.34554 −0.250603
\(456\) 8.48925 0.397545
\(457\) −29.9349 −1.40030 −0.700149 0.713997i \(-0.746883\pi\)
−0.700149 + 0.713997i \(0.746883\pi\)
\(458\) −3.17445 −0.148332
\(459\) −36.2442 −1.69173
\(460\) 5.51016 0.256912
\(461\) 33.4831 1.55946 0.779732 0.626114i \(-0.215355\pi\)
0.779732 + 0.626114i \(0.215355\pi\)
\(462\) 0 0
\(463\) −29.7511 −1.38265 −0.691324 0.722545i \(-0.742972\pi\)
−0.691324 + 0.722545i \(0.742972\pi\)
\(464\) −8.19519 −0.380452
\(465\) −2.34381 −0.108692
\(466\) −25.6323 −1.18739
\(467\) −6.65053 −0.307750 −0.153875 0.988090i \(-0.549175\pi\)
−0.153875 + 0.988090i \(0.549175\pi\)
\(468\) −6.71173 −0.310250
\(469\) 10.8106 0.499187
\(470\) 4.35747 0.200995
\(471\) −19.7373 −0.909449
\(472\) −5.44657 −0.250699
\(473\) 0 0
\(474\) 9.71897 0.446407
\(475\) −6.42752 −0.294915
\(476\) 6.44843 0.295563
\(477\) 10.7870 0.493903
\(478\) −17.5664 −0.803470
\(479\) −17.9277 −0.819137 −0.409569 0.912279i \(-0.634321\pi\)
−0.409569 + 0.912279i \(0.634321\pi\)
\(480\) −1.32077 −0.0602845
\(481\) 58.7051 2.67672
\(482\) 5.63187 0.256525
\(483\) 7.27763 0.331144
\(484\) 0 0
\(485\) −0.548361 −0.0248998
\(486\) −12.0321 −0.545787
\(487\) 37.2239 1.68677 0.843387 0.537307i \(-0.180558\pi\)
0.843387 + 0.537307i \(0.180558\pi\)
\(488\) 13.3987 0.606532
\(489\) −22.1469 −1.00152
\(490\) 1.00000 0.0451754
\(491\) −21.7165 −0.980051 −0.490026 0.871708i \(-0.663013\pi\)
−0.490026 + 0.871708i \(0.663013\pi\)
\(492\) −2.75182 −0.124062
\(493\) 52.8461 2.38007
\(494\) −34.3585 −1.54586
\(495\) 0 0
\(496\) 1.77459 0.0796813
\(497\) 11.8156 0.530002
\(498\) 9.49900 0.425660
\(499\) −15.7823 −0.706515 −0.353257 0.935526i \(-0.614926\pi\)
−0.353257 + 0.935526i \(0.614926\pi\)
\(500\) 1.00000 0.0447214
\(501\) −8.19277 −0.366026
\(502\) 14.3083 0.638610
\(503\) −8.96024 −0.399517 −0.199759 0.979845i \(-0.564016\pi\)
−0.199759 + 0.979845i \(0.564016\pi\)
\(504\) 1.25558 0.0559278
\(505\) 2.44229 0.108681
\(506\) 0 0
\(507\) −20.5706 −0.913573
\(508\) −11.0010 −0.488090
\(509\) 43.5620 1.93085 0.965426 0.260678i \(-0.0839461\pi\)
0.965426 + 0.260678i \(0.0839461\pi\)
\(510\) 8.51686 0.377133
\(511\) 5.54852 0.245452
\(512\) 1.00000 0.0441942
\(513\) −36.1266 −1.59503
\(514\) −7.69294 −0.339321
\(515\) 0.630289 0.0277738
\(516\) 3.59132 0.158099
\(517\) 0 0
\(518\) −10.9821 −0.482525
\(519\) 9.70771 0.426121
\(520\) 5.34554 0.234417
\(521\) 0.872633 0.0382307 0.0191154 0.999817i \(-0.493915\pi\)
0.0191154 + 0.999817i \(0.493915\pi\)
\(522\) 10.2897 0.450368
\(523\) −22.4315 −0.980860 −0.490430 0.871481i \(-0.663160\pi\)
−0.490430 + 0.871481i \(0.663160\pi\)
\(524\) −18.2978 −0.799342
\(525\) 1.32077 0.0576430
\(526\) −6.81613 −0.297197
\(527\) −11.4433 −0.498478
\(528\) 0 0
\(529\) 7.36183 0.320080
\(530\) −8.59127 −0.373181
\(531\) 6.83858 0.296769
\(532\) 6.42752 0.278668
\(533\) 11.1374 0.482416
\(534\) 7.13207 0.308635
\(535\) 11.4352 0.494389
\(536\) −10.8106 −0.466947
\(537\) 5.81707 0.251025
\(538\) 3.79822 0.163753
\(539\) 0 0
\(540\) 5.62062 0.241873
\(541\) −10.0432 −0.431793 −0.215896 0.976416i \(-0.569267\pi\)
−0.215896 + 0.976416i \(0.569267\pi\)
\(542\) 2.35406 0.101116
\(543\) 14.6486 0.628632
\(544\) −6.44843 −0.276474
\(545\) −8.31582 −0.356211
\(546\) 7.06020 0.302149
\(547\) −15.3645 −0.656937 −0.328469 0.944515i \(-0.606533\pi\)
−0.328469 + 0.944515i \(0.606533\pi\)
\(548\) 5.46086 0.233276
\(549\) −16.8231 −0.717994
\(550\) 0 0
\(551\) 52.6747 2.24402
\(552\) −7.27763 −0.309756
\(553\) 7.35859 0.312919
\(554\) −17.4409 −0.740994
\(555\) −14.5048 −0.615693
\(556\) −5.25491 −0.222858
\(557\) −45.8170 −1.94133 −0.970665 0.240437i \(-0.922709\pi\)
−0.970665 + 0.240437i \(0.922709\pi\)
\(558\) −2.22813 −0.0943243
\(559\) −14.5351 −0.614770
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) −22.6310 −0.954632
\(563\) −17.1178 −0.721429 −0.360714 0.932676i \(-0.617467\pi\)
−0.360714 + 0.932676i \(0.617467\pi\)
\(564\) −5.75520 −0.242338
\(565\) −20.1474 −0.847609
\(566\) −27.0596 −1.13740
\(567\) 3.65680 0.153571
\(568\) −11.8156 −0.495772
\(569\) 18.0666 0.757389 0.378695 0.925522i \(-0.376373\pi\)
0.378695 + 0.925522i \(0.376373\pi\)
\(570\) 8.48925 0.355575
\(571\) 13.8533 0.579743 0.289872 0.957066i \(-0.406387\pi\)
0.289872 + 0.957066i \(0.406387\pi\)
\(572\) 0 0
\(573\) 18.9612 0.792114
\(574\) −2.08350 −0.0869638
\(575\) 5.51016 0.229789
\(576\) −1.25558 −0.0523157
\(577\) 39.1469 1.62971 0.814854 0.579667i \(-0.196817\pi\)
0.814854 + 0.579667i \(0.196817\pi\)
\(578\) 24.5822 1.02249
\(579\) −11.2026 −0.465563
\(580\) −8.19519 −0.340287
\(581\) 7.19204 0.298376
\(582\) 0.724257 0.0300214
\(583\) 0 0
\(584\) −5.54852 −0.229599
\(585\) −6.71173 −0.277496
\(586\) −20.1958 −0.834281
\(587\) −25.3734 −1.04727 −0.523637 0.851942i \(-0.675425\pi\)
−0.523637 + 0.851942i \(0.675425\pi\)
\(588\) −1.32077 −0.0544675
\(589\) −11.4062 −0.469984
\(590\) −5.44657 −0.224232
\(591\) 6.56892 0.270209
\(592\) 10.9821 0.451361
\(593\) 24.5601 1.00856 0.504280 0.863540i \(-0.331758\pi\)
0.504280 + 0.863540i \(0.331758\pi\)
\(594\) 0 0
\(595\) 6.44843 0.264360
\(596\) 9.21250 0.377359
\(597\) 29.8937 1.22347
\(598\) 29.4547 1.20449
\(599\) −35.4719 −1.44934 −0.724672 0.689094i \(-0.758009\pi\)
−0.724672 + 0.689094i \(0.758009\pi\)
\(600\) −1.32077 −0.0539201
\(601\) −14.7392 −0.601223 −0.300611 0.953747i \(-0.597191\pi\)
−0.300611 + 0.953747i \(0.597191\pi\)
\(602\) 2.71912 0.110823
\(603\) 13.5735 0.552757
\(604\) 14.2773 0.580936
\(605\) 0 0
\(606\) −3.22570 −0.131035
\(607\) 6.09924 0.247560 0.123780 0.992310i \(-0.460498\pi\)
0.123780 + 0.992310i \(0.460498\pi\)
\(608\) −6.42752 −0.260670
\(609\) −10.8239 −0.438608
\(610\) 13.3987 0.542499
\(611\) 23.2930 0.942335
\(612\) 8.09650 0.327281
\(613\) 18.0889 0.730604 0.365302 0.930889i \(-0.380966\pi\)
0.365302 + 0.930889i \(0.380966\pi\)
\(614\) 24.4128 0.985221
\(615\) −2.75182 −0.110964
\(616\) 0 0
\(617\) 15.9814 0.643386 0.321693 0.946844i \(-0.395748\pi\)
0.321693 + 0.946844i \(0.395748\pi\)
\(618\) −0.832464 −0.0334866
\(619\) 5.20453 0.209188 0.104594 0.994515i \(-0.466646\pi\)
0.104594 + 0.994515i \(0.466646\pi\)
\(620\) 1.77459 0.0712691
\(621\) 30.9705 1.24280
\(622\) 25.2667 1.01310
\(623\) 5.39995 0.216344
\(624\) −7.06020 −0.282634
\(625\) 1.00000 0.0400000
\(626\) −18.1747 −0.726407
\(627\) 0 0
\(628\) 14.9439 0.596325
\(629\) −70.8172 −2.82367
\(630\) 1.25558 0.0500234
\(631\) −34.2712 −1.36432 −0.682158 0.731204i \(-0.738959\pi\)
−0.682158 + 0.731204i \(0.738959\pi\)
\(632\) −7.35859 −0.292709
\(633\) 4.99811 0.198657
\(634\) 10.6739 0.423915
\(635\) −11.0010 −0.436561
\(636\) 11.3471 0.449940
\(637\) 5.34554 0.211798
\(638\) 0 0
\(639\) 14.8354 0.586879
\(640\) 1.00000 0.0395285
\(641\) 38.3894 1.51629 0.758145 0.652086i \(-0.226106\pi\)
0.758145 + 0.652086i \(0.226106\pi\)
\(642\) −15.1033 −0.596079
\(643\) 4.86445 0.191835 0.0959177 0.995389i \(-0.469421\pi\)
0.0959177 + 0.995389i \(0.469421\pi\)
\(644\) −5.51016 −0.217131
\(645\) 3.59132 0.141408
\(646\) 41.4474 1.63073
\(647\) 13.6415 0.536303 0.268151 0.963377i \(-0.413587\pi\)
0.268151 + 0.963377i \(0.413587\pi\)
\(648\) −3.65680 −0.143653
\(649\) 0 0
\(650\) 5.34554 0.209669
\(651\) 2.34381 0.0918613
\(652\) 16.7682 0.656695
\(653\) 21.3235 0.834453 0.417226 0.908803i \(-0.363002\pi\)
0.417226 + 0.908803i \(0.363002\pi\)
\(654\) 10.9833 0.429479
\(655\) −18.2978 −0.714953
\(656\) 2.08350 0.0813472
\(657\) 6.96659 0.271793
\(658\) −4.35747 −0.169872
\(659\) −12.6150 −0.491409 −0.245704 0.969345i \(-0.579019\pi\)
−0.245704 + 0.969345i \(0.579019\pi\)
\(660\) 0 0
\(661\) 32.2901 1.25594 0.627969 0.778238i \(-0.283886\pi\)
0.627969 + 0.778238i \(0.283886\pi\)
\(662\) −9.75660 −0.379201
\(663\) 45.5272 1.76813
\(664\) −7.19204 −0.279105
\(665\) 6.42752 0.249248
\(666\) −13.7888 −0.534307
\(667\) −45.1568 −1.74848
\(668\) 6.20304 0.240003
\(669\) −29.7357 −1.14965
\(670\) −10.8106 −0.417650
\(671\) 0 0
\(672\) 1.32077 0.0509497
\(673\) 7.77486 0.299699 0.149849 0.988709i \(-0.452121\pi\)
0.149849 + 0.988709i \(0.452121\pi\)
\(674\) 10.1462 0.390817
\(675\) 5.62062 0.216338
\(676\) 15.5748 0.599029
\(677\) 11.1286 0.427708 0.213854 0.976866i \(-0.431398\pi\)
0.213854 + 0.976866i \(0.431398\pi\)
\(678\) 26.6101 1.02195
\(679\) 0.548361 0.0210442
\(680\) −6.44843 −0.247286
\(681\) 14.3960 0.551654
\(682\) 0 0
\(683\) −6.91240 −0.264496 −0.132248 0.991217i \(-0.542219\pi\)
−0.132248 + 0.991217i \(0.542219\pi\)
\(684\) 8.07024 0.308573
\(685\) 5.46086 0.208649
\(686\) −1.00000 −0.0381802
\(687\) 4.19270 0.159962
\(688\) −2.71912 −0.103665
\(689\) −45.9250 −1.74960
\(690\) −7.27763 −0.277055
\(691\) 31.3199 1.19146 0.595732 0.803183i \(-0.296862\pi\)
0.595732 + 0.803183i \(0.296862\pi\)
\(692\) −7.35006 −0.279407
\(693\) 0 0
\(694\) −21.1756 −0.803815
\(695\) −5.25491 −0.199330
\(696\) 10.8239 0.410280
\(697\) −13.4353 −0.508899
\(698\) −4.58943 −0.173713
\(699\) 33.8543 1.28049
\(700\) −1.00000 −0.0377964
\(701\) −12.1542 −0.459058 −0.229529 0.973302i \(-0.573719\pi\)
−0.229529 + 0.973302i \(0.573719\pi\)
\(702\) 30.0452 1.13398
\(703\) −70.5875 −2.66226
\(704\) 0 0
\(705\) −5.75520 −0.216753
\(706\) 32.2094 1.21222
\(707\) −2.44229 −0.0918518
\(708\) 7.19364 0.270354
\(709\) −32.3970 −1.21669 −0.608347 0.793671i \(-0.708167\pi\)
−0.608347 + 0.793671i \(0.708167\pi\)
\(710\) −11.8156 −0.443432
\(711\) 9.23927 0.346500
\(712\) −5.39995 −0.202372
\(713\) 9.77825 0.366199
\(714\) −8.51686 −0.318736
\(715\) 0 0
\(716\) −4.40431 −0.164597
\(717\) 23.2012 0.866463
\(718\) −6.54031 −0.244082
\(719\) 12.3255 0.459663 0.229832 0.973230i \(-0.426182\pi\)
0.229832 + 0.973230i \(0.426182\pi\)
\(720\) −1.25558 −0.0467926
\(721\) −0.630289 −0.0234732
\(722\) 22.3130 0.830403
\(723\) −7.43839 −0.276637
\(724\) −11.0910 −0.412193
\(725\) −8.19519 −0.304362
\(726\) 0 0
\(727\) 2.42914 0.0900918 0.0450459 0.998985i \(-0.485657\pi\)
0.0450459 + 0.998985i \(0.485657\pi\)
\(728\) −5.34554 −0.198119
\(729\) 26.8620 0.994888
\(730\) −5.54852 −0.205360
\(731\) 17.5340 0.648519
\(732\) −17.6966 −0.654085
\(733\) 46.1729 1.70543 0.852717 0.522373i \(-0.174953\pi\)
0.852717 + 0.522373i \(0.174953\pi\)
\(734\) −2.98556 −0.110199
\(735\) −1.32077 −0.0487172
\(736\) 5.51016 0.203107
\(737\) 0 0
\(738\) −2.61600 −0.0962963
\(739\) 48.8700 1.79771 0.898856 0.438244i \(-0.144399\pi\)
0.898856 + 0.438244i \(0.144399\pi\)
\(740\) 10.9821 0.403709
\(741\) 45.3796 1.66706
\(742\) 8.59127 0.315395
\(743\) 19.7114 0.723141 0.361571 0.932345i \(-0.382241\pi\)
0.361571 + 0.932345i \(0.382241\pi\)
\(744\) −2.34381 −0.0859284
\(745\) 9.21250 0.337520
\(746\) −16.8471 −0.616817
\(747\) 9.03015 0.330396
\(748\) 0 0
\(749\) −11.4352 −0.417835
\(750\) −1.32077 −0.0482276
\(751\) −52.1667 −1.90359 −0.951795 0.306735i \(-0.900763\pi\)
−0.951795 + 0.306735i \(0.900763\pi\)
\(752\) 4.35747 0.158901
\(753\) −18.8979 −0.688677
\(754\) −43.8077 −1.59538
\(755\) 14.2773 0.519605
\(756\) −5.62062 −0.204420
\(757\) −32.7016 −1.18856 −0.594280 0.804258i \(-0.702563\pi\)
−0.594280 + 0.804258i \(0.702563\pi\)
\(758\) −27.4781 −0.998049
\(759\) 0 0
\(760\) −6.42752 −0.233151
\(761\) 24.3007 0.880899 0.440450 0.897777i \(-0.354819\pi\)
0.440450 + 0.897777i \(0.354819\pi\)
\(762\) 14.5297 0.526357
\(763\) 8.31582 0.301053
\(764\) −14.3562 −0.519388
\(765\) 8.09650 0.292729
\(766\) −31.3930 −1.13428
\(767\) −29.1148 −1.05128
\(768\) −1.32077 −0.0476590
\(769\) 34.3945 1.24030 0.620148 0.784485i \(-0.287073\pi\)
0.620148 + 0.784485i \(0.287073\pi\)
\(770\) 0 0
\(771\) 10.1606 0.365924
\(772\) 8.48187 0.305269
\(773\) 11.7248 0.421712 0.210856 0.977517i \(-0.432375\pi\)
0.210856 + 0.977517i \(0.432375\pi\)
\(774\) 3.41406 0.122716
\(775\) 1.77459 0.0637451
\(776\) −0.548361 −0.0196850
\(777\) 14.5048 0.520355
\(778\) −0.115199 −0.00413007
\(779\) −13.3918 −0.479810
\(780\) −7.06020 −0.252796
\(781\) 0 0
\(782\) −35.5318 −1.27062
\(783\) −46.0621 −1.64612
\(784\) 1.00000 0.0357143
\(785\) 14.9439 0.533369
\(786\) 24.1671 0.862011
\(787\) 11.1529 0.397558 0.198779 0.980044i \(-0.436302\pi\)
0.198779 + 0.980044i \(0.436302\pi\)
\(788\) −4.97357 −0.177176
\(789\) 9.00251 0.320498
\(790\) −7.35859 −0.261807
\(791\) 20.1474 0.716360
\(792\) 0 0
\(793\) 71.6234 2.54342
\(794\) −18.2744 −0.648534
\(795\) 11.3471 0.402439
\(796\) −22.6336 −0.802227
\(797\) −14.0990 −0.499411 −0.249706 0.968322i \(-0.580334\pi\)
−0.249706 + 0.968322i \(0.580334\pi\)
\(798\) −8.48925 −0.300516
\(799\) −28.0988 −0.994065
\(800\) 1.00000 0.0353553
\(801\) 6.78005 0.239561
\(802\) 30.0646 1.06162
\(803\) 0 0
\(804\) 14.2783 0.503556
\(805\) −5.51016 −0.194208
\(806\) 9.48612 0.334134
\(807\) −5.01655 −0.176591
\(808\) 2.44229 0.0859195
\(809\) 44.7742 1.57418 0.787088 0.616840i \(-0.211588\pi\)
0.787088 + 0.616840i \(0.211588\pi\)
\(810\) −3.65680 −0.128487
\(811\) 33.4692 1.17526 0.587631 0.809129i \(-0.300061\pi\)
0.587631 + 0.809129i \(0.300061\pi\)
\(812\) 8.19519 0.287595
\(813\) −3.10917 −0.109043
\(814\) 0 0
\(815\) 16.7682 0.587366
\(816\) 8.51686 0.298150
\(817\) 17.4772 0.611448
\(818\) 29.2637 1.02318
\(819\) 6.71173 0.234527
\(820\) 2.08350 0.0727591
\(821\) −44.5883 −1.55614 −0.778072 0.628175i \(-0.783802\pi\)
−0.778072 + 0.628175i \(0.783802\pi\)
\(822\) −7.21252 −0.251565
\(823\) −41.0470 −1.43081 −0.715405 0.698710i \(-0.753758\pi\)
−0.715405 + 0.698710i \(0.753758\pi\)
\(824\) 0.630289 0.0219571
\(825\) 0 0
\(826\) 5.44657 0.189510
\(827\) 34.2851 1.19221 0.596105 0.802907i \(-0.296714\pi\)
0.596105 + 0.802907i \(0.296714\pi\)
\(828\) −6.91843 −0.240432
\(829\) 15.5238 0.539162 0.269581 0.962978i \(-0.413115\pi\)
0.269581 + 0.962978i \(0.413115\pi\)
\(830\) −7.19204 −0.249639
\(831\) 23.0354 0.799089
\(832\) 5.34554 0.185323
\(833\) −6.44843 −0.223425
\(834\) 6.94051 0.240330
\(835\) 6.20304 0.214665
\(836\) 0 0
\(837\) 9.97428 0.344762
\(838\) −17.3655 −0.599883
\(839\) 12.0295 0.415304 0.207652 0.978203i \(-0.433418\pi\)
0.207652 + 0.978203i \(0.433418\pi\)
\(840\) 1.32077 0.0455708
\(841\) 38.1611 1.31590
\(842\) 8.29303 0.285797
\(843\) 29.8903 1.02948
\(844\) −3.78425 −0.130259
\(845\) 15.5748 0.535788
\(846\) −5.47114 −0.188102
\(847\) 0 0
\(848\) −8.59127 −0.295025
\(849\) 35.7394 1.22657
\(850\) −6.44843 −0.221179
\(851\) 60.5130 2.07436
\(852\) 15.6056 0.534641
\(853\) −35.5225 −1.21627 −0.608134 0.793834i \(-0.708082\pi\)
−0.608134 + 0.793834i \(0.708082\pi\)
\(854\) −13.3987 −0.458495
\(855\) 8.07024 0.275996
\(856\) 11.4352 0.390849
\(857\) −48.4891 −1.65636 −0.828178 0.560465i \(-0.810622\pi\)
−0.828178 + 0.560465i \(0.810622\pi\)
\(858\) 0 0
\(859\) −1.93679 −0.0660823 −0.0330411 0.999454i \(-0.510519\pi\)
−0.0330411 + 0.999454i \(0.510519\pi\)
\(860\) −2.71912 −0.0927211
\(861\) 2.75182 0.0937818
\(862\) 13.3393 0.454338
\(863\) 21.2807 0.724405 0.362202 0.932099i \(-0.382025\pi\)
0.362202 + 0.932099i \(0.382025\pi\)
\(864\) 5.62062 0.191217
\(865\) −7.35006 −0.249910
\(866\) −14.8300 −0.503944
\(867\) −32.4674 −1.10265
\(868\) −1.77459 −0.0602334
\(869\) 0 0
\(870\) 10.8239 0.366966
\(871\) −57.7885 −1.95809
\(872\) −8.31582 −0.281609
\(873\) 0.688510 0.0233025
\(874\) −35.4166 −1.19799
\(875\) −1.00000 −0.0338062
\(876\) 7.32830 0.247600
\(877\) 28.4478 0.960614 0.480307 0.877100i \(-0.340525\pi\)
0.480307 + 0.877100i \(0.340525\pi\)
\(878\) −6.08156 −0.205243
\(879\) 26.6739 0.899689
\(880\) 0 0
\(881\) −19.2969 −0.650127 −0.325064 0.945692i \(-0.605386\pi\)
−0.325064 + 0.945692i \(0.605386\pi\)
\(882\) −1.25558 −0.0422775
\(883\) 0.696897 0.0234524 0.0117262 0.999931i \(-0.496267\pi\)
0.0117262 + 0.999931i \(0.496267\pi\)
\(884\) −34.4703 −1.15936
\(885\) 7.19364 0.241812
\(886\) −19.8590 −0.667176
\(887\) 10.1179 0.339727 0.169864 0.985468i \(-0.445667\pi\)
0.169864 + 0.985468i \(0.445667\pi\)
\(888\) −14.5048 −0.486748
\(889\) 11.0010 0.368961
\(890\) −5.39995 −0.181007
\(891\) 0 0
\(892\) 22.5140 0.753824
\(893\) −28.0077 −0.937243
\(894\) −12.1676 −0.406944
\(895\) −4.40431 −0.147220
\(896\) −1.00000 −0.0334077
\(897\) −38.9028 −1.29893
\(898\) 0.806539 0.0269145
\(899\) −14.5431 −0.485039
\(900\) −1.25558 −0.0418526
\(901\) 55.4002 1.84565
\(902\) 0 0
\(903\) −3.59132 −0.119511
\(904\) −20.1474 −0.670094
\(905\) −11.0910 −0.368677
\(906\) −18.8570 −0.626482
\(907\) −26.1814 −0.869341 −0.434670 0.900590i \(-0.643135\pi\)
−0.434670 + 0.900590i \(0.643135\pi\)
\(908\) −10.8997 −0.361719
\(909\) −3.06648 −0.101709
\(910\) −5.34554 −0.177203
\(911\) −34.0567 −1.12835 −0.564175 0.825655i \(-0.690806\pi\)
−0.564175 + 0.825655i \(0.690806\pi\)
\(912\) 8.48925 0.281107
\(913\) 0 0
\(914\) −29.9349 −0.990160
\(915\) −17.6966 −0.585031
\(916\) −3.17445 −0.104887
\(917\) 18.2978 0.604246
\(918\) −36.2442 −1.19624
\(919\) 37.6890 1.24325 0.621623 0.783316i \(-0.286473\pi\)
0.621623 + 0.783316i \(0.286473\pi\)
\(920\) 5.51016 0.181664
\(921\) −32.2436 −1.06246
\(922\) 33.4831 1.10271
\(923\) −63.1607 −2.07896
\(924\) 0 0
\(925\) 10.9821 0.361089
\(926\) −29.7511 −0.977680
\(927\) −0.791376 −0.0259922
\(928\) −8.19519 −0.269020
\(929\) 24.5323 0.804879 0.402440 0.915446i \(-0.368162\pi\)
0.402440 + 0.915446i \(0.368162\pi\)
\(930\) −2.34381 −0.0768567
\(931\) −6.42752 −0.210653
\(932\) −25.6323 −0.839614
\(933\) −33.3714 −1.09253
\(934\) −6.65053 −0.217612
\(935\) 0 0
\(936\) −6.71173 −0.219380
\(937\) 30.8744 1.00862 0.504311 0.863522i \(-0.331746\pi\)
0.504311 + 0.863522i \(0.331746\pi\)
\(938\) 10.8106 0.352979
\(939\) 24.0045 0.783358
\(940\) 4.35747 0.142125
\(941\) −46.4060 −1.51279 −0.756396 0.654114i \(-0.773042\pi\)
−0.756396 + 0.654114i \(0.773042\pi\)
\(942\) −19.7373 −0.643078
\(943\) 11.4804 0.373854
\(944\) −5.44657 −0.177271
\(945\) −5.62062 −0.182839
\(946\) 0 0
\(947\) 8.46285 0.275006 0.137503 0.990501i \(-0.456092\pi\)
0.137503 + 0.990501i \(0.456092\pi\)
\(948\) 9.71897 0.315658
\(949\) −29.6598 −0.962798
\(950\) −6.42752 −0.208536
\(951\) −14.0977 −0.457150
\(952\) 6.44843 0.208995
\(953\) 21.1245 0.684289 0.342145 0.939647i \(-0.388847\pi\)
0.342145 + 0.939647i \(0.388847\pi\)
\(954\) 10.7870 0.349242
\(955\) −14.3562 −0.464555
\(956\) −17.5664 −0.568139
\(957\) 0 0
\(958\) −17.9277 −0.579217
\(959\) −5.46086 −0.176340
\(960\) −1.32077 −0.0426275
\(961\) −27.8508 −0.898414
\(962\) 58.7051 1.89273
\(963\) −14.3578 −0.462674
\(964\) 5.63187 0.181390
\(965\) 8.48187 0.273041
\(966\) 7.27763 0.234154
\(967\) −5.05793 −0.162652 −0.0813260 0.996688i \(-0.525915\pi\)
−0.0813260 + 0.996688i \(0.525915\pi\)
\(968\) 0 0
\(969\) −54.7423 −1.75858
\(970\) −0.548361 −0.0176068
\(971\) 14.8010 0.474988 0.237494 0.971389i \(-0.423674\pi\)
0.237494 + 0.971389i \(0.423674\pi\)
\(972\) −12.0321 −0.385930
\(973\) 5.25491 0.168465
\(974\) 37.2239 1.19273
\(975\) −7.06020 −0.226107
\(976\) 13.3987 0.428883
\(977\) −7.38146 −0.236154 −0.118077 0.993004i \(-0.537673\pi\)
−0.118077 + 0.993004i \(0.537673\pi\)
\(978\) −22.1469 −0.708180
\(979\) 0 0
\(980\) 1.00000 0.0319438
\(981\) 10.4412 0.333360
\(982\) −21.7165 −0.693001
\(983\) 1.45695 0.0464694 0.0232347 0.999730i \(-0.492603\pi\)
0.0232347 + 0.999730i \(0.492603\pi\)
\(984\) −2.75182 −0.0877249
\(985\) −4.97357 −0.158471
\(986\) 52.8461 1.68296
\(987\) 5.75520 0.183190
\(988\) −34.3585 −1.09309
\(989\) −14.9828 −0.476424
\(990\) 0 0
\(991\) 34.1561 1.08501 0.542503 0.840054i \(-0.317477\pi\)
0.542503 + 0.840054i \(0.317477\pi\)
\(992\) 1.77459 0.0563432
\(993\) 12.8862 0.408931
\(994\) 11.8156 0.374768
\(995\) −22.6336 −0.717534
\(996\) 9.49900 0.300987
\(997\) −56.9485 −1.80358 −0.901789 0.432177i \(-0.857746\pi\)
−0.901789 + 0.432177i \(0.857746\pi\)
\(998\) −15.7823 −0.499582
\(999\) 61.7261 1.95293
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 8470.2.a.db.1.2 6
11.3 even 5 770.2.n.g.141.3 yes 12
11.4 even 5 770.2.n.g.71.3 12
11.10 odd 2 8470.2.a.cv.1.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.n.g.71.3 12 11.4 even 5
770.2.n.g.141.3 yes 12 11.3 even 5
8470.2.a.cv.1.2 6 11.10 odd 2
8470.2.a.db.1.2 6 1.1 even 1 trivial