Properties

Label 3381.2.a.bi.1.2
Level $3381$
Weight $2$
Character 3381.1
Self dual yes
Analytic conductor $26.997$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3381 = 3 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3381.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.9974209234\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \(x^{10} - 3 x^{9} - 13 x^{8} + 41 x^{7} + 47 x^{6} - 165 x^{5} - 45 x^{4} + 207 x^{3} + 12 x^{2} - 76 x - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 483)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-2.06506\) of defining polynomial
Character \(\chi\) \(=\) 3381.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.06506 q^{2} -1.00000 q^{3} +2.26446 q^{4} -0.608853 q^{5} +2.06506 q^{6} -0.546135 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-2.06506 q^{2} -1.00000 q^{3} +2.26446 q^{4} -0.608853 q^{5} +2.06506 q^{6} -0.546135 q^{8} +1.00000 q^{9} +1.25732 q^{10} -1.10243 q^{11} -2.26446 q^{12} +1.84712 q^{13} +0.608853 q^{15} -3.40113 q^{16} +5.88576 q^{17} -2.06506 q^{18} +1.08814 q^{19} -1.37873 q^{20} +2.27659 q^{22} +1.00000 q^{23} +0.546135 q^{24} -4.62930 q^{25} -3.81441 q^{26} -1.00000 q^{27} -0.804169 q^{29} -1.25732 q^{30} +8.77360 q^{31} +8.11580 q^{32} +1.10243 q^{33} -12.1544 q^{34} +2.26446 q^{36} +9.11528 q^{37} -2.24707 q^{38} -1.84712 q^{39} +0.332516 q^{40} -9.79559 q^{41} +5.23469 q^{43} -2.49642 q^{44} -0.608853 q^{45} -2.06506 q^{46} -5.90878 q^{47} +3.40113 q^{48} +9.55977 q^{50} -5.88576 q^{51} +4.18274 q^{52} +4.51013 q^{53} +2.06506 q^{54} +0.671220 q^{55} -1.08814 q^{57} +1.66066 q^{58} -7.40064 q^{59} +1.37873 q^{60} +5.02768 q^{61} -18.1180 q^{62} -9.95734 q^{64} -1.12462 q^{65} -2.27659 q^{66} +2.06305 q^{67} +13.3281 q^{68} -1.00000 q^{69} +4.30092 q^{71} -0.546135 q^{72} -1.15613 q^{73} -18.8236 q^{74} +4.62930 q^{75} +2.46406 q^{76} +3.81441 q^{78} +15.1716 q^{79} +2.07079 q^{80} +1.00000 q^{81} +20.2285 q^{82} -10.7196 q^{83} -3.58356 q^{85} -10.8099 q^{86} +0.804169 q^{87} +0.602078 q^{88} -7.92167 q^{89} +1.25732 q^{90} +2.26446 q^{92} -8.77360 q^{93} +12.2020 q^{94} -0.662518 q^{95} -8.11580 q^{96} -17.0648 q^{97} -1.10243 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q + 3q^{2} - 10q^{3} + 15q^{4} - 5q^{5} - 3q^{6} + 9q^{8} + 10q^{9} + O(q^{10}) \) \( 10q + 3q^{2} - 10q^{3} + 15q^{4} - 5q^{5} - 3q^{6} + 9q^{8} + 10q^{9} + 11q^{10} + 8q^{11} - 15q^{12} + 5q^{15} + 37q^{16} - 11q^{17} + 3q^{18} + q^{19} - 15q^{20} + 6q^{22} + 10q^{23} - 9q^{24} + 21q^{25} + q^{26} - 10q^{27} + 22q^{29} - 11q^{30} - 3q^{31} + 11q^{32} - 8q^{33} - 3q^{34} + 15q^{36} - 3q^{37} + 16q^{38} + 39q^{40} - 26q^{41} + 27q^{43} + 16q^{44} - 5q^{45} + 3q^{46} + 11q^{47} - 37q^{48} + 2q^{50} + 11q^{51} + 29q^{52} + 5q^{53} - 3q^{54} - 18q^{55} - q^{57} + 16q^{58} - 10q^{59} + 15q^{60} + 22q^{61} - 32q^{62} + 69q^{64} - 11q^{65} - 6q^{66} - 2q^{67} - 21q^{68} - 10q^{69} + 27q^{71} + 9q^{72} - 8q^{73} + 14q^{74} - 21q^{75} - 22q^{76} - q^{78} + 21q^{79} - 53q^{80} + 10q^{81} + 36q^{82} - 12q^{83} + 23q^{85} + 18q^{86} - 22q^{87} - 10q^{88} + 6q^{89} + 11q^{90} + 15q^{92} + 3q^{93} + 35q^{94} + 44q^{95} - 11q^{96} + 6q^{97} + 8q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.06506 −1.46022 −0.730108 0.683331i \(-0.760530\pi\)
−0.730108 + 0.683331i \(0.760530\pi\)
\(3\) −1.00000 −0.577350
\(4\) 2.26446 1.13223
\(5\) −0.608853 −0.272287 −0.136144 0.990689i \(-0.543471\pi\)
−0.136144 + 0.990689i \(0.543471\pi\)
\(6\) 2.06506 0.843056
\(7\) 0 0
\(8\) −0.546135 −0.193088
\(9\) 1.00000 0.333333
\(10\) 1.25732 0.397599
\(11\) −1.10243 −0.332396 −0.166198 0.986092i \(-0.553149\pi\)
−0.166198 + 0.986092i \(0.553149\pi\)
\(12\) −2.26446 −0.653695
\(13\) 1.84712 0.512299 0.256149 0.966637i \(-0.417546\pi\)
0.256149 + 0.966637i \(0.417546\pi\)
\(14\) 0 0
\(15\) 0.608853 0.157205
\(16\) −3.40113 −0.850282
\(17\) 5.88576 1.42751 0.713753 0.700397i \(-0.246994\pi\)
0.713753 + 0.700397i \(0.246994\pi\)
\(18\) −2.06506 −0.486739
\(19\) 1.08814 0.249637 0.124818 0.992180i \(-0.460165\pi\)
0.124818 + 0.992180i \(0.460165\pi\)
\(20\) −1.37873 −0.308293
\(21\) 0 0
\(22\) 2.27659 0.485370
\(23\) 1.00000 0.208514
\(24\) 0.546135 0.111479
\(25\) −4.62930 −0.925860
\(26\) −3.81441 −0.748067
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −0.804169 −0.149330 −0.0746652 0.997209i \(-0.523789\pi\)
−0.0746652 + 0.997209i \(0.523789\pi\)
\(30\) −1.25732 −0.229554
\(31\) 8.77360 1.57579 0.787893 0.615813i \(-0.211172\pi\)
0.787893 + 0.615813i \(0.211172\pi\)
\(32\) 8.11580 1.43468
\(33\) 1.10243 0.191909
\(34\) −12.1544 −2.08447
\(35\) 0 0
\(36\) 2.26446 0.377411
\(37\) 9.11528 1.49854 0.749271 0.662263i \(-0.230404\pi\)
0.749271 + 0.662263i \(0.230404\pi\)
\(38\) −2.24707 −0.364524
\(39\) −1.84712 −0.295776
\(40\) 0.332516 0.0525754
\(41\) −9.79559 −1.52981 −0.764907 0.644140i \(-0.777215\pi\)
−0.764907 + 0.644140i \(0.777215\pi\)
\(42\) 0 0
\(43\) 5.23469 0.798283 0.399142 0.916889i \(-0.369308\pi\)
0.399142 + 0.916889i \(0.369308\pi\)
\(44\) −2.49642 −0.376350
\(45\) −0.608853 −0.0907625
\(46\) −2.06506 −0.304476
\(47\) −5.90878 −0.861885 −0.430942 0.902379i \(-0.641819\pi\)
−0.430942 + 0.902379i \(0.641819\pi\)
\(48\) 3.40113 0.490911
\(49\) 0 0
\(50\) 9.55977 1.35196
\(51\) −5.88576 −0.824171
\(52\) 4.18274 0.580041
\(53\) 4.51013 0.619515 0.309757 0.950816i \(-0.399752\pi\)
0.309757 + 0.950816i \(0.399752\pi\)
\(54\) 2.06506 0.281019
\(55\) 0.671220 0.0905072
\(56\) 0 0
\(57\) −1.08814 −0.144128
\(58\) 1.66066 0.218055
\(59\) −7.40064 −0.963481 −0.481741 0.876314i \(-0.659995\pi\)
−0.481741 + 0.876314i \(0.659995\pi\)
\(60\) 1.37873 0.177993
\(61\) 5.02768 0.643729 0.321864 0.946786i \(-0.395690\pi\)
0.321864 + 0.946786i \(0.395690\pi\)
\(62\) −18.1180 −2.30099
\(63\) 0 0
\(64\) −9.95734 −1.24467
\(65\) −1.12462 −0.139492
\(66\) −2.27659 −0.280229
\(67\) 2.06305 0.252042 0.126021 0.992028i \(-0.459779\pi\)
0.126021 + 0.992028i \(0.459779\pi\)
\(68\) 13.3281 1.61627
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 4.30092 0.510425 0.255213 0.966885i \(-0.417855\pi\)
0.255213 + 0.966885i \(0.417855\pi\)
\(72\) −0.546135 −0.0643627
\(73\) −1.15613 −0.135315 −0.0676577 0.997709i \(-0.521553\pi\)
−0.0676577 + 0.997709i \(0.521553\pi\)
\(74\) −18.8236 −2.18820
\(75\) 4.62930 0.534545
\(76\) 2.46406 0.282647
\(77\) 0 0
\(78\) 3.81441 0.431897
\(79\) 15.1716 1.70694 0.853469 0.521144i \(-0.174495\pi\)
0.853469 + 0.521144i \(0.174495\pi\)
\(80\) 2.07079 0.231521
\(81\) 1.00000 0.111111
\(82\) 20.2285 2.23386
\(83\) −10.7196 −1.17663 −0.588315 0.808632i \(-0.700209\pi\)
−0.588315 + 0.808632i \(0.700209\pi\)
\(84\) 0 0
\(85\) −3.58356 −0.388692
\(86\) −10.8099 −1.16567
\(87\) 0.804169 0.0862160
\(88\) 0.602078 0.0641817
\(89\) −7.92167 −0.839695 −0.419847 0.907595i \(-0.637916\pi\)
−0.419847 + 0.907595i \(0.637916\pi\)
\(90\) 1.25732 0.132533
\(91\) 0 0
\(92\) 2.26446 0.236087
\(93\) −8.77360 −0.909780
\(94\) 12.2020 1.25854
\(95\) −0.662518 −0.0679729
\(96\) −8.11580 −0.828315
\(97\) −17.0648 −1.73267 −0.866336 0.499461i \(-0.833531\pi\)
−0.866336 + 0.499461i \(0.833531\pi\)
\(98\) 0 0
\(99\) −1.10243 −0.110799
\(100\) −10.4829 −1.04829
\(101\) −8.56557 −0.852306 −0.426153 0.904651i \(-0.640131\pi\)
−0.426153 + 0.904651i \(0.640131\pi\)
\(102\) 12.1544 1.20347
\(103\) 10.4373 1.02842 0.514208 0.857666i \(-0.328086\pi\)
0.514208 + 0.857666i \(0.328086\pi\)
\(104\) −1.00878 −0.0989187
\(105\) 0 0
\(106\) −9.31369 −0.904625
\(107\) −8.07268 −0.780415 −0.390208 0.920727i \(-0.627597\pi\)
−0.390208 + 0.920727i \(0.627597\pi\)
\(108\) −2.26446 −0.217898
\(109\) 3.81163 0.365088 0.182544 0.983198i \(-0.441567\pi\)
0.182544 + 0.983198i \(0.441567\pi\)
\(110\) −1.38611 −0.132160
\(111\) −9.11528 −0.865184
\(112\) 0 0
\(113\) 8.89428 0.836703 0.418352 0.908285i \(-0.362608\pi\)
0.418352 + 0.908285i \(0.362608\pi\)
\(114\) 2.24707 0.210458
\(115\) −0.608853 −0.0567758
\(116\) −1.82101 −0.169077
\(117\) 1.84712 0.170766
\(118\) 15.2827 1.40689
\(119\) 0 0
\(120\) −0.332516 −0.0303544
\(121\) −9.78464 −0.889513
\(122\) −10.3825 −0.939984
\(123\) 9.79559 0.883239
\(124\) 19.8675 1.78416
\(125\) 5.86283 0.524387
\(126\) 0 0
\(127\) 7.91043 0.701937 0.350969 0.936387i \(-0.385852\pi\)
0.350969 + 0.936387i \(0.385852\pi\)
\(128\) 4.33089 0.382800
\(129\) −5.23469 −0.460889
\(130\) 2.32241 0.203689
\(131\) 0.475706 0.0415627 0.0207813 0.999784i \(-0.493385\pi\)
0.0207813 + 0.999784i \(0.493385\pi\)
\(132\) 2.49642 0.217286
\(133\) 0 0
\(134\) −4.26033 −0.368036
\(135\) 0.608853 0.0524017
\(136\) −3.21442 −0.275634
\(137\) −8.27302 −0.706812 −0.353406 0.935470i \(-0.614977\pi\)
−0.353406 + 0.935470i \(0.614977\pi\)
\(138\) 2.06506 0.175789
\(139\) −2.48191 −0.210513 −0.105257 0.994445i \(-0.533566\pi\)
−0.105257 + 0.994445i \(0.533566\pi\)
\(140\) 0 0
\(141\) 5.90878 0.497609
\(142\) −8.88165 −0.745331
\(143\) −2.03633 −0.170286
\(144\) −3.40113 −0.283427
\(145\) 0.489621 0.0406608
\(146\) 2.38749 0.197590
\(147\) 0 0
\(148\) 20.6412 1.69670
\(149\) 20.5957 1.68726 0.843631 0.536924i \(-0.180414\pi\)
0.843631 + 0.536924i \(0.180414\pi\)
\(150\) −9.55977 −0.780552
\(151\) 12.7948 1.04123 0.520613 0.853793i \(-0.325703\pi\)
0.520613 + 0.853793i \(0.325703\pi\)
\(152\) −0.594272 −0.0482018
\(153\) 5.88576 0.475835
\(154\) 0 0
\(155\) −5.34183 −0.429066
\(156\) −4.18274 −0.334887
\(157\) −4.57005 −0.364730 −0.182365 0.983231i \(-0.558375\pi\)
−0.182365 + 0.983231i \(0.558375\pi\)
\(158\) −31.3302 −2.49250
\(159\) −4.51013 −0.357677
\(160\) −4.94133 −0.390646
\(161\) 0 0
\(162\) −2.06506 −0.162246
\(163\) −21.7544 −1.70393 −0.851967 0.523596i \(-0.824590\pi\)
−0.851967 + 0.523596i \(0.824590\pi\)
\(164\) −22.1818 −1.73211
\(165\) −0.671220 −0.0522544
\(166\) 22.1366 1.71814
\(167\) −3.92341 −0.303602 −0.151801 0.988411i \(-0.548507\pi\)
−0.151801 + 0.988411i \(0.548507\pi\)
\(168\) 0 0
\(169\) −9.58815 −0.737550
\(170\) 7.40026 0.567574
\(171\) 1.08814 0.0832122
\(172\) 11.8538 0.903842
\(173\) −24.0171 −1.82599 −0.912993 0.407976i \(-0.866235\pi\)
−0.912993 + 0.407976i \(0.866235\pi\)
\(174\) −1.66066 −0.125894
\(175\) 0 0
\(176\) 3.74952 0.282630
\(177\) 7.40064 0.556266
\(178\) 16.3587 1.22614
\(179\) 20.1690 1.50750 0.753752 0.657159i \(-0.228242\pi\)
0.753752 + 0.657159i \(0.228242\pi\)
\(180\) −1.37873 −0.102764
\(181\) 16.7087 1.24195 0.620973 0.783832i \(-0.286738\pi\)
0.620973 + 0.783832i \(0.286738\pi\)
\(182\) 0 0
\(183\) −5.02768 −0.371657
\(184\) −0.546135 −0.0402616
\(185\) −5.54986 −0.408034
\(186\) 18.1180 1.32848
\(187\) −6.48865 −0.474497
\(188\) −13.3802 −0.975854
\(189\) 0 0
\(190\) 1.36814 0.0992551
\(191\) 3.50958 0.253944 0.126972 0.991906i \(-0.459474\pi\)
0.126972 + 0.991906i \(0.459474\pi\)
\(192\) 9.95734 0.718609
\(193\) −13.6422 −0.981987 −0.490993 0.871163i \(-0.663366\pi\)
−0.490993 + 0.871163i \(0.663366\pi\)
\(194\) 35.2399 2.53008
\(195\) 1.12462 0.0805360
\(196\) 0 0
\(197\) −13.4651 −0.959349 −0.479675 0.877447i \(-0.659245\pi\)
−0.479675 + 0.877447i \(0.659245\pi\)
\(198\) 2.27659 0.161790
\(199\) 14.2974 1.01352 0.506760 0.862087i \(-0.330843\pi\)
0.506760 + 0.862087i \(0.330843\pi\)
\(200\) 2.52822 0.178772
\(201\) −2.06305 −0.145517
\(202\) 17.6884 1.24455
\(203\) 0 0
\(204\) −13.3281 −0.933153
\(205\) 5.96408 0.416549
\(206\) −21.5536 −1.50171
\(207\) 1.00000 0.0695048
\(208\) −6.28229 −0.435598
\(209\) −1.19960 −0.0829782
\(210\) 0 0
\(211\) 2.83108 0.194900 0.0974499 0.995240i \(-0.468931\pi\)
0.0974499 + 0.995240i \(0.468931\pi\)
\(212\) 10.2130 0.701435
\(213\) −4.30092 −0.294694
\(214\) 16.6706 1.13958
\(215\) −3.18716 −0.217362
\(216\) 0.546135 0.0371598
\(217\) 0 0
\(218\) −7.87123 −0.533107
\(219\) 1.15613 0.0781243
\(220\) 1.51995 0.102475
\(221\) 10.8717 0.731310
\(222\) 18.8236 1.26336
\(223\) 4.78780 0.320615 0.160307 0.987067i \(-0.448751\pi\)
0.160307 + 0.987067i \(0.448751\pi\)
\(224\) 0 0
\(225\) −4.62930 −0.308620
\(226\) −18.3672 −1.22177
\(227\) 19.2951 1.28066 0.640331 0.768099i \(-0.278797\pi\)
0.640331 + 0.768099i \(0.278797\pi\)
\(228\) −2.46406 −0.163186
\(229\) 8.12954 0.537215 0.268607 0.963250i \(-0.413437\pi\)
0.268607 + 0.963250i \(0.413437\pi\)
\(230\) 1.25732 0.0829050
\(231\) 0 0
\(232\) 0.439185 0.0288339
\(233\) 7.94940 0.520783 0.260391 0.965503i \(-0.416148\pi\)
0.260391 + 0.965503i \(0.416148\pi\)
\(234\) −3.81441 −0.249356
\(235\) 3.59758 0.234680
\(236\) −16.7585 −1.09088
\(237\) −15.1716 −0.985501
\(238\) 0 0
\(239\) 15.7789 1.02065 0.510326 0.859981i \(-0.329525\pi\)
0.510326 + 0.859981i \(0.329525\pi\)
\(240\) −2.07079 −0.133669
\(241\) −11.2656 −0.725680 −0.362840 0.931851i \(-0.618193\pi\)
−0.362840 + 0.931851i \(0.618193\pi\)
\(242\) 20.2059 1.29888
\(243\) −1.00000 −0.0641500
\(244\) 11.3850 0.728851
\(245\) 0 0
\(246\) −20.2285 −1.28972
\(247\) 2.00993 0.127889
\(248\) −4.79157 −0.304265
\(249\) 10.7196 0.679328
\(250\) −12.1071 −0.765719
\(251\) −2.27255 −0.143442 −0.0717209 0.997425i \(-0.522849\pi\)
−0.0717209 + 0.997425i \(0.522849\pi\)
\(252\) 0 0
\(253\) −1.10243 −0.0693094
\(254\) −16.3355 −1.02498
\(255\) 3.58356 0.224411
\(256\) 10.9711 0.685697
\(257\) −0.871285 −0.0543492 −0.0271746 0.999631i \(-0.508651\pi\)
−0.0271746 + 0.999631i \(0.508651\pi\)
\(258\) 10.8099 0.672998
\(259\) 0 0
\(260\) −2.54667 −0.157938
\(261\) −0.804169 −0.0497768
\(262\) −0.982361 −0.0606905
\(263\) −5.33476 −0.328956 −0.164478 0.986381i \(-0.552594\pi\)
−0.164478 + 0.986381i \(0.552594\pi\)
\(264\) −0.602078 −0.0370553
\(265\) −2.74601 −0.168686
\(266\) 0 0
\(267\) 7.92167 0.484798
\(268\) 4.67171 0.285370
\(269\) 22.7388 1.38641 0.693203 0.720742i \(-0.256199\pi\)
0.693203 + 0.720742i \(0.256199\pi\)
\(270\) −1.25732 −0.0765179
\(271\) −6.42151 −0.390079 −0.195039 0.980795i \(-0.562483\pi\)
−0.195039 + 0.980795i \(0.562483\pi\)
\(272\) −20.0182 −1.21378
\(273\) 0 0
\(274\) 17.0843 1.03210
\(275\) 5.10349 0.307752
\(276\) −2.26446 −0.136305
\(277\) 14.0353 0.843298 0.421649 0.906759i \(-0.361452\pi\)
0.421649 + 0.906759i \(0.361452\pi\)
\(278\) 5.12530 0.307395
\(279\) 8.77360 0.525262
\(280\) 0 0
\(281\) 24.5278 1.46321 0.731603 0.681731i \(-0.238773\pi\)
0.731603 + 0.681731i \(0.238773\pi\)
\(282\) −12.2020 −0.726617
\(283\) −4.39941 −0.261518 −0.130759 0.991414i \(-0.541741\pi\)
−0.130759 + 0.991414i \(0.541741\pi\)
\(284\) 9.73928 0.577920
\(285\) 0.662518 0.0392442
\(286\) 4.20513 0.248655
\(287\) 0 0
\(288\) 8.11580 0.478228
\(289\) 17.6422 1.03777
\(290\) −1.01110 −0.0593736
\(291\) 17.0648 1.00036
\(292\) −2.61803 −0.153208
\(293\) 13.3413 0.779407 0.389704 0.920940i \(-0.372577\pi\)
0.389704 + 0.920940i \(0.372577\pi\)
\(294\) 0 0
\(295\) 4.50590 0.262344
\(296\) −4.97818 −0.289351
\(297\) 1.10243 0.0639696
\(298\) −42.5312 −2.46377
\(299\) 1.84712 0.106822
\(300\) 10.4829 0.605230
\(301\) 0 0
\(302\) −26.4220 −1.52042
\(303\) 8.56557 0.492079
\(304\) −3.70091 −0.212262
\(305\) −3.06112 −0.175279
\(306\) −12.1544 −0.694823
\(307\) 24.5499 1.40114 0.700569 0.713585i \(-0.252930\pi\)
0.700569 + 0.713585i \(0.252930\pi\)
\(308\) 0 0
\(309\) −10.4373 −0.593756
\(310\) 11.0312 0.626530
\(311\) 26.0773 1.47871 0.739355 0.673315i \(-0.235130\pi\)
0.739355 + 0.673315i \(0.235130\pi\)
\(312\) 1.00878 0.0571108
\(313\) 24.1973 1.36771 0.683857 0.729616i \(-0.260301\pi\)
0.683857 + 0.729616i \(0.260301\pi\)
\(314\) 9.43741 0.532584
\(315\) 0 0
\(316\) 34.3555 1.93265
\(317\) 3.60976 0.202744 0.101372 0.994849i \(-0.467677\pi\)
0.101372 + 0.994849i \(0.467677\pi\)
\(318\) 9.31369 0.522286
\(319\) 0.886542 0.0496368
\(320\) 6.06256 0.338907
\(321\) 8.07268 0.450573
\(322\) 0 0
\(323\) 6.40453 0.356358
\(324\) 2.26446 0.125804
\(325\) −8.55087 −0.474317
\(326\) 44.9240 2.48811
\(327\) −3.81163 −0.210784
\(328\) 5.34972 0.295389
\(329\) 0 0
\(330\) 1.38611 0.0763027
\(331\) 8.23405 0.452584 0.226292 0.974059i \(-0.427340\pi\)
0.226292 + 0.974059i \(0.427340\pi\)
\(332\) −24.2742 −1.33222
\(333\) 9.11528 0.499514
\(334\) 8.10206 0.443325
\(335\) −1.25610 −0.0686279
\(336\) 0 0
\(337\) 26.1865 1.42647 0.713235 0.700925i \(-0.247229\pi\)
0.713235 + 0.700925i \(0.247229\pi\)
\(338\) 19.8001 1.07698
\(339\) −8.89428 −0.483071
\(340\) −8.11485 −0.440090
\(341\) −9.67231 −0.523785
\(342\) −2.24707 −0.121508
\(343\) 0 0
\(344\) −2.85885 −0.154139
\(345\) 0.608853 0.0327795
\(346\) 49.5967 2.66633
\(347\) −5.92735 −0.318197 −0.159098 0.987263i \(-0.550859\pi\)
−0.159098 + 0.987263i \(0.550859\pi\)
\(348\) 1.82101 0.0976165
\(349\) 29.6319 1.58616 0.793080 0.609118i \(-0.208476\pi\)
0.793080 + 0.609118i \(0.208476\pi\)
\(350\) 0 0
\(351\) −1.84712 −0.0985919
\(352\) −8.94712 −0.476883
\(353\) −14.2388 −0.757853 −0.378926 0.925427i \(-0.623707\pi\)
−0.378926 + 0.925427i \(0.623707\pi\)
\(354\) −15.2827 −0.812269
\(355\) −2.61863 −0.138982
\(356\) −17.9383 −0.950730
\(357\) 0 0
\(358\) −41.6502 −2.20128
\(359\) 2.78230 0.146844 0.0734220 0.997301i \(-0.476608\pi\)
0.0734220 + 0.997301i \(0.476608\pi\)
\(360\) 0.332516 0.0175251
\(361\) −17.8159 −0.937682
\(362\) −34.5044 −1.81351
\(363\) 9.78464 0.513561
\(364\) 0 0
\(365\) 0.703916 0.0368447
\(366\) 10.3825 0.542700
\(367\) −6.66447 −0.347883 −0.173941 0.984756i \(-0.555650\pi\)
−0.173941 + 0.984756i \(0.555650\pi\)
\(368\) −3.40113 −0.177296
\(369\) −9.79559 −0.509938
\(370\) 11.4608 0.595818
\(371\) 0 0
\(372\) −19.8675 −1.03008
\(373\) −19.0824 −0.988048 −0.494024 0.869448i \(-0.664475\pi\)
−0.494024 + 0.869448i \(0.664475\pi\)
\(374\) 13.3994 0.692869
\(375\) −5.86283 −0.302755
\(376\) 3.22700 0.166420
\(377\) −1.48540 −0.0765018
\(378\) 0 0
\(379\) 13.2631 0.681282 0.340641 0.940193i \(-0.389356\pi\)
0.340641 + 0.940193i \(0.389356\pi\)
\(380\) −1.50025 −0.0769611
\(381\) −7.91043 −0.405264
\(382\) −7.24748 −0.370813
\(383\) −3.11602 −0.159221 −0.0796105 0.996826i \(-0.525368\pi\)
−0.0796105 + 0.996826i \(0.525368\pi\)
\(384\) −4.33089 −0.221010
\(385\) 0 0
\(386\) 28.1719 1.43391
\(387\) 5.23469 0.266094
\(388\) −38.6427 −1.96179
\(389\) −9.98409 −0.506214 −0.253107 0.967438i \(-0.581452\pi\)
−0.253107 + 0.967438i \(0.581452\pi\)
\(390\) −2.32241 −0.117600
\(391\) 5.88576 0.297656
\(392\) 0 0
\(393\) −0.475706 −0.0239962
\(394\) 27.8062 1.40086
\(395\) −9.23727 −0.464777
\(396\) −2.49642 −0.125450
\(397\) 5.64444 0.283286 0.141643 0.989918i \(-0.454761\pi\)
0.141643 + 0.989918i \(0.454761\pi\)
\(398\) −29.5251 −1.47996
\(399\) 0 0
\(400\) 15.7448 0.787242
\(401\) 38.5622 1.92571 0.962853 0.270028i \(-0.0870328\pi\)
0.962853 + 0.270028i \(0.0870328\pi\)
\(402\) 4.26033 0.212486
\(403\) 16.2059 0.807273
\(404\) −19.3964 −0.965008
\(405\) −0.608853 −0.0302542
\(406\) 0 0
\(407\) −10.0490 −0.498110
\(408\) 3.21442 0.159138
\(409\) −36.8071 −1.81999 −0.909997 0.414614i \(-0.863917\pi\)
−0.909997 + 0.414614i \(0.863917\pi\)
\(410\) −12.3162 −0.608252
\(411\) 8.27302 0.408078
\(412\) 23.6349 1.16441
\(413\) 0 0
\(414\) −2.06506 −0.101492
\(415\) 6.52667 0.320382
\(416\) 14.9908 0.734987
\(417\) 2.48191 0.121540
\(418\) 2.47725 0.121166
\(419\) −6.45390 −0.315293 −0.157647 0.987496i \(-0.550391\pi\)
−0.157647 + 0.987496i \(0.550391\pi\)
\(420\) 0 0
\(421\) 29.9287 1.45863 0.729317 0.684176i \(-0.239838\pi\)
0.729317 + 0.684176i \(0.239838\pi\)
\(422\) −5.84635 −0.284596
\(423\) −5.90878 −0.287295
\(424\) −2.46314 −0.119621
\(425\) −27.2469 −1.32167
\(426\) 8.88165 0.430317
\(427\) 0 0
\(428\) −18.2803 −0.883612
\(429\) 2.03633 0.0983147
\(430\) 6.58167 0.317396
\(431\) 7.74839 0.373227 0.186613 0.982433i \(-0.440249\pi\)
0.186613 + 0.982433i \(0.440249\pi\)
\(432\) 3.40113 0.163637
\(433\) 5.17696 0.248789 0.124394 0.992233i \(-0.460301\pi\)
0.124394 + 0.992233i \(0.460301\pi\)
\(434\) 0 0
\(435\) −0.489621 −0.0234755
\(436\) 8.63130 0.413364
\(437\) 1.08814 0.0520528
\(438\) −2.38749 −0.114078
\(439\) −30.2921 −1.44576 −0.722881 0.690973i \(-0.757182\pi\)
−0.722881 + 0.690973i \(0.757182\pi\)
\(440\) −0.366577 −0.0174759
\(441\) 0 0
\(442\) −22.4507 −1.06787
\(443\) 38.2021 1.81504 0.907519 0.420012i \(-0.137974\pi\)
0.907519 + 0.420012i \(0.137974\pi\)
\(444\) −20.6412 −0.979589
\(445\) 4.82313 0.228638
\(446\) −9.88708 −0.468167
\(447\) −20.5957 −0.974141
\(448\) 0 0
\(449\) 2.95568 0.139487 0.0697435 0.997565i \(-0.477782\pi\)
0.0697435 + 0.997565i \(0.477782\pi\)
\(450\) 9.55977 0.450652
\(451\) 10.7990 0.508504
\(452\) 20.1408 0.947343
\(453\) −12.7948 −0.601152
\(454\) −39.8455 −1.87004
\(455\) 0 0
\(456\) 0.594272 0.0278293
\(457\) 17.6954 0.827758 0.413879 0.910332i \(-0.364174\pi\)
0.413879 + 0.910332i \(0.364174\pi\)
\(458\) −16.7880 −0.784450
\(459\) −5.88576 −0.274724
\(460\) −1.37873 −0.0642834
\(461\) −11.7447 −0.547005 −0.273503 0.961871i \(-0.588182\pi\)
−0.273503 + 0.961871i \(0.588182\pi\)
\(462\) 0 0
\(463\) 27.0872 1.25885 0.629424 0.777062i \(-0.283291\pi\)
0.629424 + 0.777062i \(0.283291\pi\)
\(464\) 2.73508 0.126973
\(465\) 5.34183 0.247722
\(466\) −16.4160 −0.760455
\(467\) 23.1520 1.07135 0.535673 0.844426i \(-0.320058\pi\)
0.535673 + 0.844426i \(0.320058\pi\)
\(468\) 4.18274 0.193347
\(469\) 0 0
\(470\) −7.42921 −0.342684
\(471\) 4.57005 0.210577
\(472\) 4.04175 0.186037
\(473\) −5.77090 −0.265346
\(474\) 31.3302 1.43904
\(475\) −5.03733 −0.231128
\(476\) 0 0
\(477\) 4.51013 0.206505
\(478\) −32.5843 −1.49037
\(479\) 22.3675 1.02200 0.510999 0.859581i \(-0.329275\pi\)
0.510999 + 0.859581i \(0.329275\pi\)
\(480\) 4.94133 0.225540
\(481\) 16.8370 0.767701
\(482\) 23.2641 1.05965
\(483\) 0 0
\(484\) −22.1570 −1.00714
\(485\) 10.3900 0.471785
\(486\) 2.06506 0.0936729
\(487\) 41.8522 1.89650 0.948252 0.317518i \(-0.102849\pi\)
0.948252 + 0.317518i \(0.102849\pi\)
\(488\) −2.74580 −0.124296
\(489\) 21.7544 0.983766
\(490\) 0 0
\(491\) 16.2658 0.734063 0.367032 0.930209i \(-0.380374\pi\)
0.367032 + 0.930209i \(0.380374\pi\)
\(492\) 22.1818 1.00003
\(493\) −4.73315 −0.213170
\(494\) −4.15061 −0.186745
\(495\) 0.671220 0.0301691
\(496\) −29.8401 −1.33986
\(497\) 0 0
\(498\) −22.1366 −0.991966
\(499\) −19.8250 −0.887489 −0.443744 0.896153i \(-0.646350\pi\)
−0.443744 + 0.896153i \(0.646350\pi\)
\(500\) 13.2762 0.593728
\(501\) 3.92341 0.175285
\(502\) 4.69294 0.209456
\(503\) 9.18201 0.409406 0.204703 0.978824i \(-0.434377\pi\)
0.204703 + 0.978824i \(0.434377\pi\)
\(504\) 0 0
\(505\) 5.21517 0.232072
\(506\) 2.27659 0.101207
\(507\) 9.58815 0.425825
\(508\) 17.9129 0.794756
\(509\) −32.8014 −1.45390 −0.726948 0.686692i \(-0.759062\pi\)
−0.726948 + 0.686692i \(0.759062\pi\)
\(510\) −7.40026 −0.327689
\(511\) 0 0
\(512\) −31.3178 −1.38407
\(513\) −1.08814 −0.0480426
\(514\) 1.79925 0.0793617
\(515\) −6.35477 −0.280025
\(516\) −11.8538 −0.521834
\(517\) 6.51404 0.286487
\(518\) 0 0
\(519\) 24.0171 1.05423
\(520\) 0.614197 0.0269343
\(521\) 6.86356 0.300698 0.150349 0.988633i \(-0.451960\pi\)
0.150349 + 0.988633i \(0.451960\pi\)
\(522\) 1.66066 0.0726849
\(523\) 9.76622 0.427047 0.213524 0.976938i \(-0.431506\pi\)
0.213524 + 0.976938i \(0.431506\pi\)
\(524\) 1.07722 0.0470586
\(525\) 0 0
\(526\) 11.0166 0.480347
\(527\) 51.6393 2.24944
\(528\) −3.74952 −0.163177
\(529\) 1.00000 0.0434783
\(530\) 5.67067 0.246318
\(531\) −7.40064 −0.321160
\(532\) 0 0
\(533\) −18.0936 −0.783722
\(534\) −16.3587 −0.707910
\(535\) 4.91508 0.212497
\(536\) −1.12671 −0.0486663
\(537\) −20.1690 −0.870357
\(538\) −46.9568 −2.02445
\(539\) 0 0
\(540\) 1.37873 0.0593309
\(541\) −4.64207 −0.199578 −0.0997891 0.995009i \(-0.531817\pi\)
−0.0997891 + 0.995009i \(0.531817\pi\)
\(542\) 13.2608 0.569600
\(543\) −16.7087 −0.717038
\(544\) 47.7676 2.04802
\(545\) −2.32072 −0.0994088
\(546\) 0 0
\(547\) 21.6976 0.927722 0.463861 0.885908i \(-0.346464\pi\)
0.463861 + 0.885908i \(0.346464\pi\)
\(548\) −18.7340 −0.800275
\(549\) 5.02768 0.214576
\(550\) −10.5390 −0.449385
\(551\) −0.875049 −0.0372783
\(552\) 0.546135 0.0232451
\(553\) 0 0
\(554\) −28.9836 −1.23140
\(555\) 5.54986 0.235579
\(556\) −5.62021 −0.238350
\(557\) −32.1596 −1.36265 −0.681323 0.731983i \(-0.738595\pi\)
−0.681323 + 0.731983i \(0.738595\pi\)
\(558\) −18.1180 −0.766996
\(559\) 9.66910 0.408960
\(560\) 0 0
\(561\) 6.48865 0.273951
\(562\) −50.6513 −2.13660
\(563\) −12.3159 −0.519055 −0.259528 0.965736i \(-0.583567\pi\)
−0.259528 + 0.965736i \(0.583567\pi\)
\(564\) 13.3802 0.563409
\(565\) −5.41531 −0.227824
\(566\) 9.08503 0.381872
\(567\) 0 0
\(568\) −2.34888 −0.0985570
\(569\) −26.2350 −1.09983 −0.549914 0.835221i \(-0.685340\pi\)
−0.549914 + 0.835221i \(0.685340\pi\)
\(570\) −1.36814 −0.0573050
\(571\) −5.59829 −0.234281 −0.117141 0.993115i \(-0.537373\pi\)
−0.117141 + 0.993115i \(0.537373\pi\)
\(572\) −4.61119 −0.192803
\(573\) −3.50958 −0.146615
\(574\) 0 0
\(575\) −4.62930 −0.193055
\(576\) −9.95734 −0.414889
\(577\) −10.1348 −0.421915 −0.210958 0.977495i \(-0.567658\pi\)
−0.210958 + 0.977495i \(0.567658\pi\)
\(578\) −36.4321 −1.51538
\(579\) 13.6422 0.566950
\(580\) 1.10873 0.0460375
\(581\) 0 0
\(582\) −35.2399 −1.46074
\(583\) −4.97212 −0.205924
\(584\) 0.631406 0.0261278
\(585\) −1.12462 −0.0464975
\(586\) −27.5506 −1.13810
\(587\) 27.2323 1.12400 0.561999 0.827138i \(-0.310032\pi\)
0.561999 + 0.827138i \(0.310032\pi\)
\(588\) 0 0
\(589\) 9.54691 0.393374
\(590\) −9.30495 −0.383079
\(591\) 13.4651 0.553880
\(592\) −31.0022 −1.27418
\(593\) 38.3547 1.57504 0.787519 0.616290i \(-0.211365\pi\)
0.787519 + 0.616290i \(0.211365\pi\)
\(594\) −2.27659 −0.0934095
\(595\) 0 0
\(596\) 46.6381 1.91037
\(597\) −14.2974 −0.585155
\(598\) −3.81441 −0.155983
\(599\) −6.88378 −0.281264 −0.140632 0.990062i \(-0.544913\pi\)
−0.140632 + 0.990062i \(0.544913\pi\)
\(600\) −2.52822 −0.103214
\(601\) 1.80012 0.0734283 0.0367141 0.999326i \(-0.488311\pi\)
0.0367141 + 0.999326i \(0.488311\pi\)
\(602\) 0 0
\(603\) 2.06305 0.0840141
\(604\) 28.9734 1.17891
\(605\) 5.95741 0.242203
\(606\) −17.6884 −0.718542
\(607\) 2.93517 0.119135 0.0595674 0.998224i \(-0.481028\pi\)
0.0595674 + 0.998224i \(0.481028\pi\)
\(608\) 8.83113 0.358150
\(609\) 0 0
\(610\) 6.32139 0.255946
\(611\) −10.9142 −0.441542
\(612\) 13.3281 0.538756
\(613\) 31.7390 1.28192 0.640962 0.767572i \(-0.278535\pi\)
0.640962 + 0.767572i \(0.278535\pi\)
\(614\) −50.6970 −2.04596
\(615\) −5.96408 −0.240495
\(616\) 0 0
\(617\) −41.7807 −1.68203 −0.841013 0.541015i \(-0.818040\pi\)
−0.841013 + 0.541015i \(0.818040\pi\)
\(618\) 21.5536 0.867012
\(619\) 27.2170 1.09394 0.546972 0.837151i \(-0.315780\pi\)
0.546972 + 0.837151i \(0.315780\pi\)
\(620\) −12.0964 −0.485803
\(621\) −1.00000 −0.0401286
\(622\) −53.8512 −2.15924
\(623\) 0 0
\(624\) 6.28229 0.251493
\(625\) 19.5769 0.783076
\(626\) −49.9689 −1.99716
\(627\) 1.19960 0.0479075
\(628\) −10.3487 −0.412959
\(629\) 53.6503 2.13918
\(630\) 0 0
\(631\) 25.8728 1.02998 0.514990 0.857196i \(-0.327795\pi\)
0.514990 + 0.857196i \(0.327795\pi\)
\(632\) −8.28574 −0.329589
\(633\) −2.83108 −0.112525
\(634\) −7.45437 −0.296051
\(635\) −4.81629 −0.191129
\(636\) −10.2130 −0.404973
\(637\) 0 0
\(638\) −1.83076 −0.0724805
\(639\) 4.30092 0.170142
\(640\) −2.63687 −0.104232
\(641\) −13.3591 −0.527651 −0.263826 0.964570i \(-0.584984\pi\)
−0.263826 + 0.964570i \(0.584984\pi\)
\(642\) −16.6706 −0.657934
\(643\) 43.4719 1.71436 0.857182 0.515014i \(-0.172213\pi\)
0.857182 + 0.515014i \(0.172213\pi\)
\(644\) 0 0
\(645\) 3.18716 0.125494
\(646\) −13.2257 −0.520360
\(647\) −46.1515 −1.81440 −0.907201 0.420698i \(-0.861785\pi\)
−0.907201 + 0.420698i \(0.861785\pi\)
\(648\) −0.546135 −0.0214542
\(649\) 8.15871 0.320257
\(650\) 17.6580 0.692605
\(651\) 0 0
\(652\) −49.2620 −1.92925
\(653\) −33.5965 −1.31473 −0.657367 0.753570i \(-0.728330\pi\)
−0.657367 + 0.753570i \(0.728330\pi\)
\(654\) 7.87123 0.307790
\(655\) −0.289635 −0.0113170
\(656\) 33.3161 1.30077
\(657\) −1.15613 −0.0451051
\(658\) 0 0
\(659\) 10.2122 0.397809 0.198905 0.980019i \(-0.436262\pi\)
0.198905 + 0.980019i \(0.436262\pi\)
\(660\) −1.51995 −0.0591641
\(661\) 18.0200 0.700895 0.350448 0.936582i \(-0.386029\pi\)
0.350448 + 0.936582i \(0.386029\pi\)
\(662\) −17.0038 −0.660871
\(663\) −10.8717 −0.422222
\(664\) 5.85436 0.227193
\(665\) 0 0
\(666\) −18.8236 −0.729399
\(667\) −0.804169 −0.0311375
\(668\) −8.88442 −0.343748
\(669\) −4.78780 −0.185107
\(670\) 2.59391 0.100212
\(671\) −5.54268 −0.213973
\(672\) 0 0
\(673\) −40.5777 −1.56415 −0.782077 0.623181i \(-0.785840\pi\)
−0.782077 + 0.623181i \(0.785840\pi\)
\(674\) −54.0767 −2.08296
\(675\) 4.62930 0.178182
\(676\) −21.7120 −0.835078
\(677\) 25.9423 0.997044 0.498522 0.866877i \(-0.333876\pi\)
0.498522 + 0.866877i \(0.333876\pi\)
\(678\) 18.3672 0.705388
\(679\) 0 0
\(680\) 1.95711 0.0750518
\(681\) −19.2951 −0.739390
\(682\) 19.9739 0.764839
\(683\) −25.8167 −0.987848 −0.493924 0.869505i \(-0.664438\pi\)
−0.493924 + 0.869505i \(0.664438\pi\)
\(684\) 2.46406 0.0942156
\(685\) 5.03705 0.192456
\(686\) 0 0
\(687\) −8.12954 −0.310161
\(688\) −17.8039 −0.678766
\(689\) 8.33076 0.317377
\(690\) −1.25732 −0.0478652
\(691\) −15.6020 −0.593528 −0.296764 0.954951i \(-0.595907\pi\)
−0.296764 + 0.954951i \(0.595907\pi\)
\(692\) −54.3859 −2.06744
\(693\) 0 0
\(694\) 12.2403 0.464636
\(695\) 1.51112 0.0573201
\(696\) −0.439185 −0.0166473
\(697\) −57.6545 −2.18382
\(698\) −61.1916 −2.31614
\(699\) −7.94940 −0.300674
\(700\) 0 0
\(701\) 38.6350 1.45922 0.729612 0.683861i \(-0.239701\pi\)
0.729612 + 0.683861i \(0.239701\pi\)
\(702\) 3.81441 0.143966
\(703\) 9.91871 0.374091
\(704\) 10.9773 0.413722
\(705\) −3.59758 −0.135493
\(706\) 29.4039 1.10663
\(707\) 0 0
\(708\) 16.7585 0.629823
\(709\) −43.3432 −1.62779 −0.813894 0.581013i \(-0.802656\pi\)
−0.813894 + 0.581013i \(0.802656\pi\)
\(710\) 5.40762 0.202944
\(711\) 15.1716 0.568979
\(712\) 4.32630 0.162135
\(713\) 8.77360 0.328574
\(714\) 0 0
\(715\) 1.23982 0.0463667
\(716\) 45.6720 1.70684
\(717\) −15.7789 −0.589274
\(718\) −5.74560 −0.214424
\(719\) −29.4485 −1.09824 −0.549122 0.835742i \(-0.685038\pi\)
−0.549122 + 0.835742i \(0.685038\pi\)
\(720\) 2.07079 0.0771737
\(721\) 0 0
\(722\) 36.7910 1.36922
\(723\) 11.2656 0.418971
\(724\) 37.8362 1.40617
\(725\) 3.72274 0.138259
\(726\) −20.2059 −0.749910
\(727\) −50.1254 −1.85905 −0.929524 0.368763i \(-0.879782\pi\)
−0.929524 + 0.368763i \(0.879782\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −1.45363 −0.0538012
\(731\) 30.8101 1.13955
\(732\) −11.3850 −0.420802
\(733\) −20.9760 −0.774766 −0.387383 0.921919i \(-0.626621\pi\)
−0.387383 + 0.921919i \(0.626621\pi\)
\(734\) 13.7625 0.507984
\(735\) 0 0
\(736\) 8.11580 0.299152
\(737\) −2.27438 −0.0837778
\(738\) 20.2285 0.744620
\(739\) 11.5852 0.426168 0.213084 0.977034i \(-0.431649\pi\)
0.213084 + 0.977034i \(0.431649\pi\)
\(740\) −12.5675 −0.461990
\(741\) −2.00993 −0.0738365
\(742\) 0 0
\(743\) −44.7847 −1.64299 −0.821495 0.570216i \(-0.806860\pi\)
−0.821495 + 0.570216i \(0.806860\pi\)
\(744\) 4.79157 0.175668
\(745\) −12.5397 −0.459420
\(746\) 39.4062 1.44276
\(747\) −10.7196 −0.392210
\(748\) −14.6933 −0.537241
\(749\) 0 0
\(750\) 12.1071 0.442088
\(751\) −22.2887 −0.813325 −0.406662 0.913579i \(-0.633307\pi\)
−0.406662 + 0.913579i \(0.633307\pi\)
\(752\) 20.0965 0.732845
\(753\) 2.27255 0.0828162
\(754\) 3.06743 0.111709
\(755\) −7.79015 −0.283513
\(756\) 0 0
\(757\) −34.3724 −1.24929 −0.624643 0.780910i \(-0.714756\pi\)
−0.624643 + 0.780910i \(0.714756\pi\)
\(758\) −27.3892 −0.994820
\(759\) 1.10243 0.0400158
\(760\) 0.361824 0.0131248
\(761\) 43.7206 1.58487 0.792435 0.609956i \(-0.208813\pi\)
0.792435 + 0.609956i \(0.208813\pi\)
\(762\) 16.3355 0.591773
\(763\) 0 0
\(764\) 7.94731 0.287524
\(765\) −3.58356 −0.129564
\(766\) 6.43476 0.232497
\(767\) −13.6699 −0.493590
\(768\) −10.9711 −0.395887
\(769\) −2.96450 −0.106903 −0.0534513 0.998570i \(-0.517022\pi\)
−0.0534513 + 0.998570i \(0.517022\pi\)
\(770\) 0 0
\(771\) 0.871285 0.0313785
\(772\) −30.8923 −1.11184
\(773\) 22.2038 0.798615 0.399307 0.916817i \(-0.369251\pi\)
0.399307 + 0.916817i \(0.369251\pi\)
\(774\) −10.8099 −0.388556
\(775\) −40.6156 −1.45896
\(776\) 9.31971 0.334558
\(777\) 0 0
\(778\) 20.6177 0.739181
\(779\) −10.6590 −0.381898
\(780\) 2.54667 0.0911855
\(781\) −4.74147 −0.169663
\(782\) −12.1544 −0.434642
\(783\) 0.804169 0.0287387
\(784\) 0 0
\(785\) 2.78249 0.0993112
\(786\) 0.982361 0.0350397
\(787\) −34.6515 −1.23519 −0.617596 0.786495i \(-0.711893\pi\)
−0.617596 + 0.786495i \(0.711893\pi\)
\(788\) −30.4913 −1.08621
\(789\) 5.33476 0.189923
\(790\) 19.0755 0.678676
\(791\) 0 0
\(792\) 0.602078 0.0213939
\(793\) 9.28673 0.329781
\(794\) −11.6561 −0.413660
\(795\) 2.74601 0.0973909
\(796\) 32.3761 1.14754
\(797\) −22.4200 −0.794158 −0.397079 0.917784i \(-0.629976\pi\)
−0.397079 + 0.917784i \(0.629976\pi\)
\(798\) 0 0
\(799\) −34.7777 −1.23035
\(800\) −37.5705 −1.32832
\(801\) −7.92167 −0.279898
\(802\) −79.6332 −2.81195
\(803\) 1.27456 0.0449783
\(804\) −4.67171 −0.164759
\(805\) 0 0
\(806\) −33.4661 −1.17879
\(807\) −22.7388 −0.800442
\(808\) 4.67796 0.164570
\(809\) 26.0244 0.914970 0.457485 0.889217i \(-0.348750\pi\)
0.457485 + 0.889217i \(0.348750\pi\)
\(810\) 1.25732 0.0441776
\(811\) −45.2644 −1.58945 −0.794724 0.606970i \(-0.792385\pi\)
−0.794724 + 0.606970i \(0.792385\pi\)
\(812\) 0 0
\(813\) 6.42151 0.225212
\(814\) 20.7517 0.727348
\(815\) 13.2452 0.463959
\(816\) 20.0182 0.700778
\(817\) 5.69608 0.199281
\(818\) 76.0088 2.65759
\(819\) 0 0
\(820\) 13.5054 0.471630
\(821\) 7.28729 0.254328 0.127164 0.991882i \(-0.459413\pi\)
0.127164 + 0.991882i \(0.459413\pi\)
\(822\) −17.0843 −0.595882
\(823\) 3.29869 0.114985 0.0574925 0.998346i \(-0.481689\pi\)
0.0574925 + 0.998346i \(0.481689\pi\)
\(824\) −5.70017 −0.198575
\(825\) −5.10349 −0.177681
\(826\) 0 0
\(827\) −48.8022 −1.69702 −0.848510 0.529179i \(-0.822500\pi\)
−0.848510 + 0.529179i \(0.822500\pi\)
\(828\) 2.26446 0.0786956
\(829\) 24.7978 0.861263 0.430631 0.902528i \(-0.358291\pi\)
0.430631 + 0.902528i \(0.358291\pi\)
\(830\) −13.4780 −0.467827
\(831\) −14.0353 −0.486878
\(832\) −18.3924 −0.637642
\(833\) 0 0
\(834\) −5.12530 −0.177475
\(835\) 2.38878 0.0826671
\(836\) −2.71646 −0.0939506
\(837\) −8.77360 −0.303260
\(838\) 13.3277 0.460397
\(839\) −6.27230 −0.216544 −0.108272 0.994121i \(-0.534532\pi\)
−0.108272 + 0.994121i \(0.534532\pi\)
\(840\) 0 0
\(841\) −28.3533 −0.977700
\(842\) −61.8044 −2.12992
\(843\) −24.5278 −0.844782
\(844\) 6.41089 0.220672
\(845\) 5.83777 0.200826
\(846\) 12.2020 0.419513
\(847\) 0 0
\(848\) −15.3395 −0.526762
\(849\) 4.39941 0.150987
\(850\) 56.2665 1.92993
\(851\) 9.11528 0.312468
\(852\) −9.73928 −0.333662
\(853\) −1.75188 −0.0599834 −0.0299917 0.999550i \(-0.509548\pi\)
−0.0299917 + 0.999550i \(0.509548\pi\)
\(854\) 0 0
\(855\) −0.662518 −0.0226576
\(856\) 4.40878 0.150689
\(857\) −15.0955 −0.515653 −0.257826 0.966191i \(-0.583006\pi\)
−0.257826 + 0.966191i \(0.583006\pi\)
\(858\) −4.20513 −0.143561
\(859\) 9.58801 0.327139 0.163569 0.986532i \(-0.447699\pi\)
0.163569 + 0.986532i \(0.447699\pi\)
\(860\) −7.21721 −0.246105
\(861\) 0 0
\(862\) −16.0009 −0.544992
\(863\) 34.3335 1.16872 0.584362 0.811493i \(-0.301345\pi\)
0.584362 + 0.811493i \(0.301345\pi\)
\(864\) −8.11580 −0.276105
\(865\) 14.6229 0.497193
\(866\) −10.6907 −0.363285
\(867\) −17.6422 −0.599159
\(868\) 0 0
\(869\) −16.7257 −0.567379
\(870\) 1.01110 0.0342793
\(871\) 3.81071 0.129121
\(872\) −2.08166 −0.0704941
\(873\) −17.0648 −0.577557
\(874\) −2.24707 −0.0760084
\(875\) 0 0
\(876\) 2.61803 0.0884549
\(877\) 17.6058 0.594505 0.297252 0.954799i \(-0.403930\pi\)
0.297252 + 0.954799i \(0.403930\pi\)
\(878\) 62.5549 2.11112
\(879\) −13.3413 −0.449991
\(880\) −2.28290 −0.0769567
\(881\) 39.5494 1.33245 0.666227 0.745749i \(-0.267908\pi\)
0.666227 + 0.745749i \(0.267908\pi\)
\(882\) 0 0
\(883\) 0.340528 0.0114597 0.00572983 0.999984i \(-0.498176\pi\)
0.00572983 + 0.999984i \(0.498176\pi\)
\(884\) 24.6186 0.828013
\(885\) −4.50590 −0.151464
\(886\) −78.8896 −2.65035
\(887\) 30.0962 1.01053 0.505266 0.862964i \(-0.331394\pi\)
0.505266 + 0.862964i \(0.331394\pi\)
\(888\) 4.97818 0.167057
\(889\) 0 0
\(890\) −9.96004 −0.333861
\(891\) −1.10243 −0.0369329
\(892\) 10.8418 0.363010
\(893\) −6.42959 −0.215158
\(894\) 42.5312 1.42246
\(895\) −12.2800 −0.410474
\(896\) 0 0
\(897\) −1.84712 −0.0616735
\(898\) −6.10364 −0.203681
\(899\) −7.05546 −0.235313
\(900\) −10.4829 −0.349429
\(901\) 26.5456 0.884361
\(902\) −22.3005 −0.742526
\(903\) 0 0
\(904\) −4.85748 −0.161557
\(905\) −10.1731 −0.338166
\(906\) 26.4220 0.877812
\(907\) 31.8853 1.05873 0.529366 0.848393i \(-0.322430\pi\)
0.529366 + 0.848393i \(0.322430\pi\)
\(908\) 43.6931 1.45001
\(909\) −8.56557 −0.284102
\(910\) 0 0
\(911\) 36.7297 1.21691 0.608454 0.793589i \(-0.291790\pi\)
0.608454 + 0.793589i \(0.291790\pi\)
\(912\) 3.70091 0.122549
\(913\) 11.8177 0.391107
\(914\) −36.5421 −1.20871
\(915\) 3.06112 0.101198
\(916\) 18.4090 0.608252
\(917\) 0 0
\(918\) 12.1544 0.401156
\(919\) −29.3315 −0.967556 −0.483778 0.875191i \(-0.660736\pi\)
−0.483778 + 0.875191i \(0.660736\pi\)
\(920\) 0.332516 0.0109627
\(921\) −24.5499 −0.808947
\(922\) 24.2535 0.798746
\(923\) 7.94431 0.261490
\(924\) 0 0
\(925\) −42.1973 −1.38744
\(926\) −55.9366 −1.83819
\(927\) 10.4373 0.342805
\(928\) −6.52647 −0.214242
\(929\) −1.30704 −0.0428826 −0.0214413 0.999770i \(-0.506826\pi\)
−0.0214413 + 0.999770i \(0.506826\pi\)
\(930\) −11.0312 −0.361727
\(931\) 0 0
\(932\) 18.0011 0.589647
\(933\) −26.0773 −0.853734
\(934\) −47.8102 −1.56440
\(935\) 3.95064 0.129200
\(936\) −1.00878 −0.0329729
\(937\) −4.77786 −0.156086 −0.0780430 0.996950i \(-0.524867\pi\)
−0.0780430 + 0.996950i \(0.524867\pi\)
\(938\) 0 0
\(939\) −24.1973 −0.789650
\(940\) 8.14660 0.265713
\(941\) −33.4004 −1.08882 −0.544411 0.838819i \(-0.683247\pi\)
−0.544411 + 0.838819i \(0.683247\pi\)
\(942\) −9.43741 −0.307488
\(943\) −9.79559 −0.318988
\(944\) 25.1705 0.819231
\(945\) 0 0
\(946\) 11.9172 0.387463
\(947\) 35.8911 1.16630 0.583152 0.812363i \(-0.301819\pi\)
0.583152 + 0.812363i \(0.301819\pi\)
\(948\) −34.3555 −1.11582
\(949\) −2.13552 −0.0693219
\(950\) 10.4024 0.337498
\(951\) −3.60976 −0.117055
\(952\) 0 0
\(953\) −42.4876 −1.37631 −0.688154 0.725564i \(-0.741579\pi\)
−0.688154 + 0.725564i \(0.741579\pi\)
\(954\) −9.31369 −0.301542
\(955\) −2.13682 −0.0691457
\(956\) 35.7308 1.15562
\(957\) −0.886542 −0.0286578
\(958\) −46.1903 −1.49234
\(959\) 0 0
\(960\) −6.06256 −0.195668
\(961\) 45.9761 1.48310
\(962\) −34.7694 −1.12101
\(963\) −8.07268 −0.260138
\(964\) −25.5105 −0.821638
\(965\) 8.30609 0.267383
\(966\) 0 0
\(967\) −50.9129 −1.63725 −0.818624 0.574330i \(-0.805263\pi\)
−0.818624 + 0.574330i \(0.805263\pi\)
\(968\) 5.34374 0.171754
\(969\) −6.40453 −0.205743
\(970\) −21.4559 −0.688908
\(971\) −16.7372 −0.537123 −0.268561 0.963263i \(-0.586548\pi\)
−0.268561 + 0.963263i \(0.586548\pi\)
\(972\) −2.26446 −0.0726327
\(973\) 0 0
\(974\) −86.4272 −2.76931
\(975\) 8.55087 0.273847
\(976\) −17.0998 −0.547351
\(977\) 47.3729 1.51559 0.757797 0.652491i \(-0.226276\pi\)
0.757797 + 0.652491i \(0.226276\pi\)
\(978\) −44.9240 −1.43651
\(979\) 8.73310 0.279111
\(980\) 0 0
\(981\) 3.81163 0.121696
\(982\) −33.5897 −1.07189
\(983\) 35.2609 1.12465 0.562323 0.826918i \(-0.309908\pi\)
0.562323 + 0.826918i \(0.309908\pi\)
\(984\) −5.34972 −0.170543
\(985\) 8.19827 0.261219
\(986\) 9.77422 0.311275
\(987\) 0 0
\(988\) 4.55141 0.144800
\(989\) 5.23469 0.166454
\(990\) −1.38611 −0.0440534
\(991\) 16.9164 0.537369 0.268684 0.963228i \(-0.413411\pi\)
0.268684 + 0.963228i \(0.413411\pi\)
\(992\) 71.2048 2.26075
\(993\) −8.23405 −0.261300
\(994\) 0 0
\(995\) −8.70504 −0.275968
\(996\) 24.2742 0.769157
\(997\) −52.3647 −1.65841 −0.829203 0.558947i \(-0.811205\pi\)
−0.829203 + 0.558947i \(0.811205\pi\)
\(998\) 40.9398 1.29593
\(999\) −9.11528 −0.288395
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 3381.2.a.bi.1.2 10
7.2 even 3 483.2.i.h.277.9 20
7.4 even 3 483.2.i.h.415.9 yes 20
7.6 odd 2 3381.2.a.bj.1.2 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
483.2.i.h.277.9 20 7.2 even 3
483.2.i.h.415.9 yes 20 7.4 even 3
3381.2.a.bi.1.2 10 1.1 even 1 trivial
3381.2.a.bj.1.2 10 7.6 odd 2