Properties

Label 8470.2.a.dh.1.6
Level $8470$
Weight $2$
Character 8470.1
Self dual yes
Analytic conductor $67.633$
Analytic rank $0$
Dimension $8$
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: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 16 x^{6} + 69 x^{4} - 10 x^{3} - 70 x^{2} + 10 x + 5\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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 1.6
Root \(1.22201\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.22201 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.22201 q^{6} -1.00000 q^{7} +1.00000 q^{8} -1.50670 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.22201 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.22201 q^{6} -1.00000 q^{7} +1.00000 q^{8} -1.50670 q^{9} -1.00000 q^{10} +1.22201 q^{12} +3.47816 q^{13} -1.00000 q^{14} -1.22201 q^{15} +1.00000 q^{16} -0.138127 q^{17} -1.50670 q^{18} +0.515316 q^{19} -1.00000 q^{20} -1.22201 q^{21} +1.51048 q^{23} +1.22201 q^{24} +1.00000 q^{25} +3.47816 q^{26} -5.50722 q^{27} -1.00000 q^{28} +9.24062 q^{29} -1.22201 q^{30} +10.2626 q^{31} +1.00000 q^{32} -0.138127 q^{34} +1.00000 q^{35} -1.50670 q^{36} +1.66859 q^{37} +0.515316 q^{38} +4.25033 q^{39} -1.00000 q^{40} -12.1868 q^{41} -1.22201 q^{42} -10.6660 q^{43} +1.50670 q^{45} +1.51048 q^{46} -1.80692 q^{47} +1.22201 q^{48} +1.00000 q^{49} +1.00000 q^{50} -0.168791 q^{51} +3.47816 q^{52} +1.39105 q^{53} -5.50722 q^{54} -1.00000 q^{56} +0.629719 q^{57} +9.24062 q^{58} +9.30512 q^{59} -1.22201 q^{60} +13.5218 q^{61} +10.2626 q^{62} +1.50670 q^{63} +1.00000 q^{64} -3.47816 q^{65} -1.84979 q^{67} -0.138127 q^{68} +1.84582 q^{69} +1.00000 q^{70} -6.39213 q^{71} -1.50670 q^{72} -11.0649 q^{73} +1.66859 q^{74} +1.22201 q^{75} +0.515316 q^{76} +4.25033 q^{78} +9.98676 q^{79} -1.00000 q^{80} -2.20974 q^{81} -12.1868 q^{82} +2.20436 q^{83} -1.22201 q^{84} +0.138127 q^{85} -10.6660 q^{86} +11.2921 q^{87} +1.74164 q^{89} +1.50670 q^{90} -3.47816 q^{91} +1.51048 q^{92} +12.5410 q^{93} -1.80692 q^{94} -0.515316 q^{95} +1.22201 q^{96} -4.41074 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 8q^{2} + 8q^{4} - 8q^{5} - 8q^{7} + 8q^{8} + 8q^{9} + O(q^{10}) \) \( 8q + 8q^{2} + 8q^{4} - 8q^{5} - 8q^{7} + 8q^{8} + 8q^{9} - 8q^{10} - q^{13} - 8q^{14} + 8q^{16} + 6q^{17} + 8q^{18} + 5q^{19} - 8q^{20} + 10q^{23} + 8q^{25} - q^{26} - 8q^{28} + 3q^{29} - 8q^{31} + 8q^{32} + 6q^{34} + 8q^{35} + 8q^{36} - 6q^{37} + 5q^{38} + 35q^{39} - 8q^{40} + 11q^{41} - 5q^{43} - 8q^{45} + 10q^{46} - 15q^{47} + 8q^{49} + 8q^{50} - 6q^{51} - q^{52} - 16q^{53} - 8q^{56} + 38q^{57} + 3q^{58} - 9q^{59} + 32q^{61} - 8q^{62} - 8q^{63} + 8q^{64} + q^{65} + 33q^{67} + 6q^{68} - 22q^{69} + 8q^{70} + 11q^{71} + 8q^{72} - 34q^{73} - 6q^{74} + 5q^{76} + 35q^{78} + 31q^{79} - 8q^{80} + 20q^{81} + 11q^{82} + 50q^{83} - 6q^{85} - 5q^{86} - 12q^{87} + q^{89} - 8q^{90} + q^{91} + 10q^{92} + 26q^{93} - 15q^{94} - 5q^{95} - 4q^{97} + 8q^{98} + O(q^{100}) \)

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
\(3\) 1.22201 0.705525 0.352763 0.935713i \(-0.385242\pi\)
0.352763 + 0.935713i \(0.385242\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 1.22201 0.498882
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) −1.50670 −0.502234
\(10\) −1.00000 −0.316228
\(11\) 0 0
\(12\) 1.22201 0.352763
\(13\) 3.47816 0.964669 0.482334 0.875987i \(-0.339789\pi\)
0.482334 + 0.875987i \(0.339789\pi\)
\(14\) −1.00000 −0.267261
\(15\) −1.22201 −0.315520
\(16\) 1.00000 0.250000
\(17\) −0.138127 −0.0335006 −0.0167503 0.999860i \(-0.505332\pi\)
−0.0167503 + 0.999860i \(0.505332\pi\)
\(18\) −1.50670 −0.355133
\(19\) 0.515316 0.118222 0.0591108 0.998251i \(-0.481173\pi\)
0.0591108 + 0.998251i \(0.481173\pi\)
\(20\) −1.00000 −0.223607
\(21\) −1.22201 −0.266663
\(22\) 0 0
\(23\) 1.51048 0.314957 0.157479 0.987522i \(-0.449663\pi\)
0.157479 + 0.987522i \(0.449663\pi\)
\(24\) 1.22201 0.249441
\(25\) 1.00000 0.200000
\(26\) 3.47816 0.682124
\(27\) −5.50722 −1.05986
\(28\) −1.00000 −0.188982
\(29\) 9.24062 1.71594 0.857970 0.513699i \(-0.171725\pi\)
0.857970 + 0.513699i \(0.171725\pi\)
\(30\) −1.22201 −0.223107
\(31\) 10.2626 1.84322 0.921612 0.388112i \(-0.126873\pi\)
0.921612 + 0.388112i \(0.126873\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −0.138127 −0.0236885
\(35\) 1.00000 0.169031
\(36\) −1.50670 −0.251117
\(37\) 1.66859 0.274315 0.137157 0.990549i \(-0.456203\pi\)
0.137157 + 0.990549i \(0.456203\pi\)
\(38\) 0.515316 0.0835952
\(39\) 4.25033 0.680598
\(40\) −1.00000 −0.158114
\(41\) −12.1868 −1.90326 −0.951628 0.307251i \(-0.900591\pi\)
−0.951628 + 0.307251i \(0.900591\pi\)
\(42\) −1.22201 −0.188560
\(43\) −10.6660 −1.62655 −0.813277 0.581877i \(-0.802318\pi\)
−0.813277 + 0.581877i \(0.802318\pi\)
\(44\) 0 0
\(45\) 1.50670 0.224606
\(46\) 1.51048 0.222708
\(47\) −1.80692 −0.263567 −0.131783 0.991279i \(-0.542070\pi\)
−0.131783 + 0.991279i \(0.542070\pi\)
\(48\) 1.22201 0.176381
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) −0.168791 −0.0236355
\(52\) 3.47816 0.482334
\(53\) 1.39105 0.191075 0.0955376 0.995426i \(-0.469543\pi\)
0.0955376 + 0.995426i \(0.469543\pi\)
\(54\) −5.50722 −0.749437
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) 0.629719 0.0834083
\(58\) 9.24062 1.21335
\(59\) 9.30512 1.21142 0.605712 0.795684i \(-0.292889\pi\)
0.605712 + 0.795684i \(0.292889\pi\)
\(60\) −1.22201 −0.157760
\(61\) 13.5218 1.73129 0.865646 0.500656i \(-0.166908\pi\)
0.865646 + 0.500656i \(0.166908\pi\)
\(62\) 10.2626 1.30336
\(63\) 1.50670 0.189827
\(64\) 1.00000 0.125000
\(65\) −3.47816 −0.431413
\(66\) 0 0
\(67\) −1.84979 −0.225988 −0.112994 0.993596i \(-0.536044\pi\)
−0.112994 + 0.993596i \(0.536044\pi\)
\(68\) −0.138127 −0.0167503
\(69\) 1.84582 0.222210
\(70\) 1.00000 0.119523
\(71\) −6.39213 −0.758606 −0.379303 0.925272i \(-0.623836\pi\)
−0.379303 + 0.925272i \(0.623836\pi\)
\(72\) −1.50670 −0.177567
\(73\) −11.0649 −1.29505 −0.647525 0.762044i \(-0.724196\pi\)
−0.647525 + 0.762044i \(0.724196\pi\)
\(74\) 1.66859 0.193970
\(75\) 1.22201 0.141105
\(76\) 0.515316 0.0591108
\(77\) 0 0
\(78\) 4.25033 0.481256
\(79\) 9.98676 1.12360 0.561799 0.827274i \(-0.310109\pi\)
0.561799 + 0.827274i \(0.310109\pi\)
\(80\) −1.00000 −0.111803
\(81\) −2.20974 −0.245527
\(82\) −12.1868 −1.34581
\(83\) 2.20436 0.241960 0.120980 0.992655i \(-0.461396\pi\)
0.120980 + 0.992655i \(0.461396\pi\)
\(84\) −1.22201 −0.133332
\(85\) 0.138127 0.0149819
\(86\) −10.6660 −1.15015
\(87\) 11.2921 1.21064
\(88\) 0 0
\(89\) 1.74164 0.184614 0.0923069 0.995731i \(-0.470576\pi\)
0.0923069 + 0.995731i \(0.470576\pi\)
\(90\) 1.50670 0.158820
\(91\) −3.47816 −0.364610
\(92\) 1.51048 0.157479
\(93\) 12.5410 1.30044
\(94\) −1.80692 −0.186370
\(95\) −0.515316 −0.0528703
\(96\) 1.22201 0.124720
\(97\) −4.41074 −0.447843 −0.223921 0.974607i \(-0.571886\pi\)
−0.223921 + 0.974607i \(0.571886\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 12.5813 1.25188 0.625942 0.779870i \(-0.284715\pi\)
0.625942 + 0.779870i \(0.284715\pi\)
\(102\) −0.168791 −0.0167128
\(103\) −2.98471 −0.294092 −0.147046 0.989130i \(-0.546976\pi\)
−0.147046 + 0.989130i \(0.546976\pi\)
\(104\) 3.47816 0.341062
\(105\) 1.22201 0.119256
\(106\) 1.39105 0.135111
\(107\) 13.2777 1.28360 0.641800 0.766872i \(-0.278188\pi\)
0.641800 + 0.766872i \(0.278188\pi\)
\(108\) −5.50722 −0.529932
\(109\) 0.502741 0.0481539 0.0240769 0.999710i \(-0.492335\pi\)
0.0240769 + 0.999710i \(0.492335\pi\)
\(110\) 0 0
\(111\) 2.03903 0.193536
\(112\) −1.00000 −0.0944911
\(113\) −8.91196 −0.838367 −0.419183 0.907902i \(-0.637684\pi\)
−0.419183 + 0.907902i \(0.637684\pi\)
\(114\) 0.629719 0.0589786
\(115\) −1.51048 −0.140853
\(116\) 9.24062 0.857970
\(117\) −5.24056 −0.484489
\(118\) 9.30512 0.856605
\(119\) 0.138127 0.0126620
\(120\) −1.22201 −0.111553
\(121\) 0 0
\(122\) 13.5218 1.22421
\(123\) −14.8923 −1.34280
\(124\) 10.2626 0.921612
\(125\) −1.00000 −0.0894427
\(126\) 1.50670 0.134228
\(127\) 12.6728 1.12452 0.562262 0.826959i \(-0.309931\pi\)
0.562262 + 0.826959i \(0.309931\pi\)
\(128\) 1.00000 0.0883883
\(129\) −13.0339 −1.14757
\(130\) −3.47816 −0.305055
\(131\) 15.1497 1.32364 0.661818 0.749664i \(-0.269785\pi\)
0.661818 + 0.749664i \(0.269785\pi\)
\(132\) 0 0
\(133\) −0.515316 −0.0446835
\(134\) −1.84979 −0.159798
\(135\) 5.50722 0.473986
\(136\) −0.138127 −0.0118443
\(137\) 21.8856 1.86981 0.934904 0.354900i \(-0.115485\pi\)
0.934904 + 0.354900i \(0.115485\pi\)
\(138\) 1.84582 0.157126
\(139\) 2.47370 0.209816 0.104908 0.994482i \(-0.466545\pi\)
0.104908 + 0.994482i \(0.466545\pi\)
\(140\) 1.00000 0.0845154
\(141\) −2.20807 −0.185953
\(142\) −6.39213 −0.536416
\(143\) 0 0
\(144\) −1.50670 −0.125559
\(145\) −9.24062 −0.767392
\(146\) −11.0649 −0.915738
\(147\) 1.22201 0.100789
\(148\) 1.66859 0.137157
\(149\) 17.1038 1.40120 0.700598 0.713556i \(-0.252917\pi\)
0.700598 + 0.713556i \(0.252917\pi\)
\(150\) 1.22201 0.0997763
\(151\) 3.21272 0.261447 0.130724 0.991419i \(-0.458270\pi\)
0.130724 + 0.991419i \(0.458270\pi\)
\(152\) 0.515316 0.0417976
\(153\) 0.208116 0.0168252
\(154\) 0 0
\(155\) −10.2626 −0.824315
\(156\) 4.25033 0.340299
\(157\) 2.74824 0.219333 0.109667 0.993968i \(-0.465022\pi\)
0.109667 + 0.993968i \(0.465022\pi\)
\(158\) 9.98676 0.794504
\(159\) 1.69987 0.134808
\(160\) −1.00000 −0.0790569
\(161\) −1.51048 −0.119043
\(162\) −2.20974 −0.173614
\(163\) −12.5201 −0.980649 −0.490325 0.871540i \(-0.663122\pi\)
−0.490325 + 0.871540i \(0.663122\pi\)
\(164\) −12.1868 −0.951628
\(165\) 0 0
\(166\) 2.20436 0.171091
\(167\) 17.0335 1.31809 0.659047 0.752102i \(-0.270960\pi\)
0.659047 + 0.752102i \(0.270960\pi\)
\(168\) −1.22201 −0.0942798
\(169\) −0.902388 −0.0694145
\(170\) 0.138127 0.0105938
\(171\) −0.776427 −0.0593749
\(172\) −10.6660 −0.813277
\(173\) 9.38603 0.713607 0.356803 0.934179i \(-0.383867\pi\)
0.356803 + 0.934179i \(0.383867\pi\)
\(174\) 11.2921 0.856051
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) 11.3709 0.854690
\(178\) 1.74164 0.130542
\(179\) −11.1233 −0.831397 −0.415698 0.909503i \(-0.636463\pi\)
−0.415698 + 0.909503i \(0.636463\pi\)
\(180\) 1.50670 0.112303
\(181\) 9.57646 0.711813 0.355906 0.934522i \(-0.384172\pi\)
0.355906 + 0.934522i \(0.384172\pi\)
\(182\) −3.47816 −0.257819
\(183\) 16.5237 1.22147
\(184\) 1.51048 0.111354
\(185\) −1.66859 −0.122677
\(186\) 12.5410 0.919551
\(187\) 0 0
\(188\) −1.80692 −0.131783
\(189\) 5.50722 0.400591
\(190\) −0.515316 −0.0373849
\(191\) 9.79900 0.709031 0.354515 0.935050i \(-0.384646\pi\)
0.354515 + 0.935050i \(0.384646\pi\)
\(192\) 1.22201 0.0881907
\(193\) 14.4512 1.04022 0.520111 0.854099i \(-0.325891\pi\)
0.520111 + 0.854099i \(0.325891\pi\)
\(194\) −4.41074 −0.316673
\(195\) −4.25033 −0.304373
\(196\) 1.00000 0.0714286
\(197\) 22.6950 1.61695 0.808476 0.588530i \(-0.200293\pi\)
0.808476 + 0.588530i \(0.200293\pi\)
\(198\) 0 0
\(199\) 9.43893 0.669108 0.334554 0.942377i \(-0.391414\pi\)
0.334554 + 0.942377i \(0.391414\pi\)
\(200\) 1.00000 0.0707107
\(201\) −2.26045 −0.159440
\(202\) 12.5813 0.885215
\(203\) −9.24062 −0.648565
\(204\) −0.168791 −0.0118178
\(205\) 12.1868 0.851162
\(206\) −2.98471 −0.207954
\(207\) −2.27585 −0.158182
\(208\) 3.47816 0.241167
\(209\) 0 0
\(210\) 1.22201 0.0843264
\(211\) 11.1500 0.767595 0.383797 0.923417i \(-0.374616\pi\)
0.383797 + 0.923417i \(0.374616\pi\)
\(212\) 1.39105 0.0955376
\(213\) −7.81122 −0.535216
\(214\) 13.2777 0.907642
\(215\) 10.6660 0.727417
\(216\) −5.50722 −0.374719
\(217\) −10.2626 −0.696673
\(218\) 0.502741 0.0340499
\(219\) −13.5214 −0.913690
\(220\) 0 0
\(221\) −0.480427 −0.0323170
\(222\) 2.03903 0.136851
\(223\) −3.97800 −0.266387 −0.133193 0.991090i \(-0.542523\pi\)
−0.133193 + 0.991090i \(0.542523\pi\)
\(224\) −1.00000 −0.0668153
\(225\) −1.50670 −0.100447
\(226\) −8.91196 −0.592815
\(227\) 21.7347 1.44258 0.721292 0.692631i \(-0.243548\pi\)
0.721292 + 0.692631i \(0.243548\pi\)
\(228\) 0.629719 0.0417041
\(229\) 1.64618 0.108783 0.0543913 0.998520i \(-0.482678\pi\)
0.0543913 + 0.998520i \(0.482678\pi\)
\(230\) −1.51048 −0.0995982
\(231\) 0 0
\(232\) 9.24062 0.606677
\(233\) −1.11650 −0.0731441 −0.0365720 0.999331i \(-0.511644\pi\)
−0.0365720 + 0.999331i \(0.511644\pi\)
\(234\) −5.24056 −0.342586
\(235\) 1.80692 0.117871
\(236\) 9.30512 0.605712
\(237\) 12.2039 0.792727
\(238\) 0.138127 0.00895342
\(239\) 17.3118 1.11981 0.559903 0.828558i \(-0.310839\pi\)
0.559903 + 0.828558i \(0.310839\pi\)
\(240\) −1.22201 −0.0788801
\(241\) −23.7524 −1.53003 −0.765015 0.644013i \(-0.777268\pi\)
−0.765015 + 0.644013i \(0.777268\pi\)
\(242\) 0 0
\(243\) 13.8213 0.886639
\(244\) 13.5218 0.865646
\(245\) −1.00000 −0.0638877
\(246\) −14.8923 −0.949500
\(247\) 1.79235 0.114045
\(248\) 10.2626 0.651678
\(249\) 2.69374 0.170709
\(250\) −1.00000 −0.0632456
\(251\) 12.1255 0.765357 0.382679 0.923881i \(-0.375002\pi\)
0.382679 + 0.923881i \(0.375002\pi\)
\(252\) 1.50670 0.0949133
\(253\) 0 0
\(254\) 12.6728 0.795159
\(255\) 0.168791 0.0105701
\(256\) 1.00000 0.0625000
\(257\) −28.6094 −1.78461 −0.892304 0.451435i \(-0.850912\pi\)
−0.892304 + 0.451435i \(0.850912\pi\)
\(258\) −13.0339 −0.811458
\(259\) −1.66859 −0.103681
\(260\) −3.47816 −0.215706
\(261\) −13.9229 −0.861804
\(262\) 15.1497 0.935953
\(263\) −30.4738 −1.87910 −0.939549 0.342415i \(-0.888755\pi\)
−0.939549 + 0.342415i \(0.888755\pi\)
\(264\) 0 0
\(265\) −1.39105 −0.0854514
\(266\) −0.515316 −0.0315960
\(267\) 2.12830 0.130250
\(268\) −1.84979 −0.112994
\(269\) −31.0485 −1.89306 −0.946531 0.322613i \(-0.895439\pi\)
−0.946531 + 0.322613i \(0.895439\pi\)
\(270\) 5.50722 0.335158
\(271\) 8.50957 0.516920 0.258460 0.966022i \(-0.416785\pi\)
0.258460 + 0.966022i \(0.416785\pi\)
\(272\) −0.138127 −0.00837515
\(273\) −4.25033 −0.257242
\(274\) 21.8856 1.32215
\(275\) 0 0
\(276\) 1.84582 0.111105
\(277\) −14.6394 −0.879598 −0.439799 0.898096i \(-0.644950\pi\)
−0.439799 + 0.898096i \(0.644950\pi\)
\(278\) 2.47370 0.148363
\(279\) −15.4627 −0.925730
\(280\) 1.00000 0.0597614
\(281\) −8.91851 −0.532034 −0.266017 0.963968i \(-0.585708\pi\)
−0.266017 + 0.963968i \(0.585708\pi\)
\(282\) −2.20807 −0.131489
\(283\) 19.7163 1.17201 0.586005 0.810307i \(-0.300700\pi\)
0.586005 + 0.810307i \(0.300700\pi\)
\(284\) −6.39213 −0.379303
\(285\) −0.629719 −0.0373013
\(286\) 0 0
\(287\) 12.1868 0.719363
\(288\) −1.50670 −0.0887833
\(289\) −16.9809 −0.998878
\(290\) −9.24062 −0.542628
\(291\) −5.38995 −0.315964
\(292\) −11.0649 −0.647525
\(293\) 6.92872 0.404780 0.202390 0.979305i \(-0.435129\pi\)
0.202390 + 0.979305i \(0.435129\pi\)
\(294\) 1.22201 0.0712688
\(295\) −9.30512 −0.541765
\(296\) 1.66859 0.0969850
\(297\) 0 0
\(298\) 17.1038 0.990795
\(299\) 5.25370 0.303829
\(300\) 1.22201 0.0705525
\(301\) 10.6660 0.614779
\(302\) 3.21272 0.184871
\(303\) 15.3744 0.883235
\(304\) 0.515316 0.0295554
\(305\) −13.5218 −0.774257
\(306\) 0.208116 0.0118972
\(307\) −20.6340 −1.17764 −0.588821 0.808263i \(-0.700408\pi\)
−0.588821 + 0.808263i \(0.700408\pi\)
\(308\) 0 0
\(309\) −3.64733 −0.207489
\(310\) −10.2626 −0.582879
\(311\) −20.2359 −1.14748 −0.573738 0.819039i \(-0.694507\pi\)
−0.573738 + 0.819039i \(0.694507\pi\)
\(312\) 4.25033 0.240628
\(313\) −4.40784 −0.249146 −0.124573 0.992210i \(-0.539756\pi\)
−0.124573 + 0.992210i \(0.539756\pi\)
\(314\) 2.74824 0.155092
\(315\) −1.50670 −0.0848931
\(316\) 9.98676 0.561799
\(317\) 25.9982 1.46020 0.730101 0.683339i \(-0.239473\pi\)
0.730101 + 0.683339i \(0.239473\pi\)
\(318\) 1.69987 0.0953239
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) 16.2254 0.905612
\(322\) −1.51048 −0.0841759
\(323\) −0.0711788 −0.00396049
\(324\) −2.20974 −0.122763
\(325\) 3.47816 0.192934
\(326\) −12.5201 −0.693424
\(327\) 0.614353 0.0339738
\(328\) −12.1868 −0.672903
\(329\) 1.80692 0.0996188
\(330\) 0 0
\(331\) −19.8219 −1.08951 −0.544755 0.838595i \(-0.683377\pi\)
−0.544755 + 0.838595i \(0.683377\pi\)
\(332\) 2.20436 0.120980
\(333\) −2.51407 −0.137770
\(334\) 17.0335 0.932033
\(335\) 1.84979 0.101065
\(336\) −1.22201 −0.0666659
\(337\) 30.3840 1.65512 0.827560 0.561377i \(-0.189728\pi\)
0.827560 + 0.561377i \(0.189728\pi\)
\(338\) −0.902388 −0.0490834
\(339\) −10.8905 −0.591489
\(340\) 0.138127 0.00749096
\(341\) 0 0
\(342\) −0.776427 −0.0419844
\(343\) −1.00000 −0.0539949
\(344\) −10.6660 −0.575074
\(345\) −1.84582 −0.0993755
\(346\) 9.38603 0.504596
\(347\) −16.6544 −0.894057 −0.447028 0.894520i \(-0.647518\pi\)
−0.447028 + 0.894520i \(0.647518\pi\)
\(348\) 11.2921 0.605320
\(349\) −1.70386 −0.0912057 −0.0456028 0.998960i \(-0.514521\pi\)
−0.0456028 + 0.998960i \(0.514521\pi\)
\(350\) −1.00000 −0.0534522
\(351\) −19.1550 −1.02242
\(352\) 0 0
\(353\) 10.7417 0.571724 0.285862 0.958271i \(-0.407720\pi\)
0.285862 + 0.958271i \(0.407720\pi\)
\(354\) 11.3709 0.604357
\(355\) 6.39213 0.339259
\(356\) 1.74164 0.0923069
\(357\) 0.168791 0.00893339
\(358\) −11.1233 −0.587886
\(359\) −16.2804 −0.859246 −0.429623 0.903008i \(-0.641354\pi\)
−0.429623 + 0.903008i \(0.641354\pi\)
\(360\) 1.50670 0.0794102
\(361\) −18.7344 −0.986024
\(362\) 9.57646 0.503328
\(363\) 0 0
\(364\) −3.47816 −0.182305
\(365\) 11.0649 0.579164
\(366\) 16.5237 0.863710
\(367\) −21.0947 −1.10114 −0.550568 0.834790i \(-0.685589\pi\)
−0.550568 + 0.834790i \(0.685589\pi\)
\(368\) 1.51048 0.0787393
\(369\) 18.3619 0.955881
\(370\) −1.66859 −0.0867460
\(371\) −1.39105 −0.0722196
\(372\) 12.5410 0.650221
\(373\) −13.9335 −0.721449 −0.360725 0.932672i \(-0.617471\pi\)
−0.360725 + 0.932672i \(0.617471\pi\)
\(374\) 0 0
\(375\) −1.22201 −0.0631041
\(376\) −1.80692 −0.0931849
\(377\) 32.1404 1.65531
\(378\) 5.50722 0.283261
\(379\) 22.1451 1.13752 0.568759 0.822505i \(-0.307424\pi\)
0.568759 + 0.822505i \(0.307424\pi\)
\(380\) −0.515316 −0.0264351
\(381\) 15.4862 0.793381
\(382\) 9.79900 0.501360
\(383\) −22.5171 −1.15057 −0.575284 0.817954i \(-0.695108\pi\)
−0.575284 + 0.817954i \(0.695108\pi\)
\(384\) 1.22201 0.0623602
\(385\) 0 0
\(386\) 14.4512 0.735548
\(387\) 16.0705 0.816911
\(388\) −4.41074 −0.223921
\(389\) −20.8343 −1.05634 −0.528171 0.849138i \(-0.677122\pi\)
−0.528171 + 0.849138i \(0.677122\pi\)
\(390\) −4.25033 −0.215224
\(391\) −0.208638 −0.0105513
\(392\) 1.00000 0.0505076
\(393\) 18.5130 0.933859
\(394\) 22.6950 1.14336
\(395\) −9.98676 −0.502488
\(396\) 0 0
\(397\) 19.0865 0.957924 0.478962 0.877836i \(-0.341013\pi\)
0.478962 + 0.877836i \(0.341013\pi\)
\(398\) 9.43893 0.473131
\(399\) −0.629719 −0.0315254
\(400\) 1.00000 0.0500000
\(401\) 30.1895 1.50759 0.753795 0.657110i \(-0.228221\pi\)
0.753795 + 0.657110i \(0.228221\pi\)
\(402\) −2.26045 −0.112741
\(403\) 35.6951 1.77810
\(404\) 12.5813 0.625942
\(405\) 2.20974 0.109803
\(406\) −9.24062 −0.458604
\(407\) 0 0
\(408\) −0.168791 −0.00835642
\(409\) −29.4582 −1.45662 −0.728308 0.685250i \(-0.759693\pi\)
−0.728308 + 0.685250i \(0.759693\pi\)
\(410\) 12.1868 0.601863
\(411\) 26.7443 1.31920
\(412\) −2.98471 −0.147046
\(413\) −9.30512 −0.457875
\(414\) −2.27585 −0.111852
\(415\) −2.20436 −0.108208
\(416\) 3.47816 0.170531
\(417\) 3.02287 0.148031
\(418\) 0 0
\(419\) −20.4867 −1.00084 −0.500419 0.865783i \(-0.666821\pi\)
−0.500419 + 0.865783i \(0.666821\pi\)
\(420\) 1.22201 0.0596278
\(421\) −2.61302 −0.127351 −0.0636755 0.997971i \(-0.520282\pi\)
−0.0636755 + 0.997971i \(0.520282\pi\)
\(422\) 11.1500 0.542772
\(423\) 2.72249 0.132372
\(424\) 1.39105 0.0675553
\(425\) −0.138127 −0.00670012
\(426\) −7.81122 −0.378455
\(427\) −13.5218 −0.654367
\(428\) 13.2777 0.641800
\(429\) 0 0
\(430\) 10.6660 0.514361
\(431\) −9.87795 −0.475804 −0.237902 0.971289i \(-0.576460\pi\)
−0.237902 + 0.971289i \(0.576460\pi\)
\(432\) −5.50722 −0.264966
\(433\) 19.5259 0.938356 0.469178 0.883104i \(-0.344550\pi\)
0.469178 + 0.883104i \(0.344550\pi\)
\(434\) −10.2626 −0.492623
\(435\) −11.2921 −0.541414
\(436\) 0.502741 0.0240769
\(437\) 0.778375 0.0372347
\(438\) −13.5214 −0.646077
\(439\) 3.38579 0.161595 0.0807974 0.996731i \(-0.474253\pi\)
0.0807974 + 0.996731i \(0.474253\pi\)
\(440\) 0 0
\(441\) −1.50670 −0.0717477
\(442\) −0.480427 −0.0228516
\(443\) −19.3485 −0.919276 −0.459638 0.888106i \(-0.652021\pi\)
−0.459638 + 0.888106i \(0.652021\pi\)
\(444\) 2.03903 0.0967680
\(445\) −1.74164 −0.0825618
\(446\) −3.97800 −0.188364
\(447\) 20.9009 0.988579
\(448\) −1.00000 −0.0472456
\(449\) 33.1331 1.56365 0.781824 0.623500i \(-0.214290\pi\)
0.781824 + 0.623500i \(0.214290\pi\)
\(450\) −1.50670 −0.0710266
\(451\) 0 0
\(452\) −8.91196 −0.419183
\(453\) 3.92596 0.184458
\(454\) 21.7347 1.02006
\(455\) 3.47816 0.163059
\(456\) 0.629719 0.0294893
\(457\) −12.6845 −0.593358 −0.296679 0.954977i \(-0.595879\pi\)
−0.296679 + 0.954977i \(0.595879\pi\)
\(458\) 1.64618 0.0769210
\(459\) 0.760693 0.0355061
\(460\) −1.51048 −0.0704266
\(461\) 19.4617 0.906421 0.453211 0.891403i \(-0.350279\pi\)
0.453211 + 0.891403i \(0.350279\pi\)
\(462\) 0 0
\(463\) 29.6401 1.37749 0.688746 0.725003i \(-0.258162\pi\)
0.688746 + 0.725003i \(0.258162\pi\)
\(464\) 9.24062 0.428985
\(465\) −12.5410 −0.581575
\(466\) −1.11650 −0.0517207
\(467\) 26.8009 1.24020 0.620099 0.784523i \(-0.287092\pi\)
0.620099 + 0.784523i \(0.287092\pi\)
\(468\) −5.24056 −0.242245
\(469\) 1.84979 0.0854154
\(470\) 1.80692 0.0833471
\(471\) 3.35836 0.154745
\(472\) 9.30512 0.428303
\(473\) 0 0
\(474\) 12.2039 0.560542
\(475\) 0.515316 0.0236443
\(476\) 0.138127 0.00633102
\(477\) −2.09590 −0.0959645
\(478\) 17.3118 0.791822
\(479\) 13.3413 0.609581 0.304791 0.952419i \(-0.401413\pi\)
0.304791 + 0.952419i \(0.401413\pi\)
\(480\) −1.22201 −0.0557767
\(481\) 5.80363 0.264623
\(482\) −23.7524 −1.08189
\(483\) −1.84582 −0.0839876
\(484\) 0 0
\(485\) 4.41074 0.200281
\(486\) 13.8213 0.626948
\(487\) −13.3718 −0.605932 −0.302966 0.953001i \(-0.597977\pi\)
−0.302966 + 0.953001i \(0.597977\pi\)
\(488\) 13.5218 0.612104
\(489\) −15.2996 −0.691873
\(490\) −1.00000 −0.0451754
\(491\) 25.9713 1.17207 0.586033 0.810287i \(-0.300689\pi\)
0.586033 + 0.810287i \(0.300689\pi\)
\(492\) −14.8923 −0.671398
\(493\) −1.27638 −0.0574851
\(494\) 1.79235 0.0806417
\(495\) 0 0
\(496\) 10.2626 0.460806
\(497\) 6.39213 0.286726
\(498\) 2.69374 0.120709
\(499\) −22.5866 −1.01111 −0.505557 0.862793i \(-0.668713\pi\)
−0.505557 + 0.862793i \(0.668713\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 20.8151 0.929948
\(502\) 12.1255 0.541189
\(503\) −28.7933 −1.28383 −0.641916 0.766775i \(-0.721860\pi\)
−0.641916 + 0.766775i \(0.721860\pi\)
\(504\) 1.50670 0.0671139
\(505\) −12.5813 −0.559859
\(506\) 0 0
\(507\) −1.10272 −0.0489737
\(508\) 12.6728 0.562262
\(509\) −23.9556 −1.06182 −0.530908 0.847430i \(-0.678149\pi\)
−0.530908 + 0.847430i \(0.678149\pi\)
\(510\) 0.168791 0.00747421
\(511\) 11.0649 0.489483
\(512\) 1.00000 0.0441942
\(513\) −2.83795 −0.125299
\(514\) −28.6094 −1.26191
\(515\) 2.98471 0.131522
\(516\) −13.0339 −0.573787
\(517\) 0 0
\(518\) −1.66859 −0.0733137
\(519\) 11.4698 0.503468
\(520\) −3.47816 −0.152528
\(521\) −15.7907 −0.691804 −0.345902 0.938271i \(-0.612427\pi\)
−0.345902 + 0.938271i \(0.612427\pi\)
\(522\) −13.9229 −0.609387
\(523\) −22.1852 −0.970091 −0.485045 0.874489i \(-0.661197\pi\)
−0.485045 + 0.874489i \(0.661197\pi\)
\(524\) 15.1497 0.661818
\(525\) −1.22201 −0.0533327
\(526\) −30.4738 −1.32872
\(527\) −1.41754 −0.0617492
\(528\) 0 0
\(529\) −20.7184 −0.900802
\(530\) −1.39105 −0.0604233
\(531\) −14.0200 −0.608418
\(532\) −0.515316 −0.0223418
\(533\) −42.3876 −1.83601
\(534\) 2.12830 0.0921005
\(535\) −13.2777 −0.574044
\(536\) −1.84979 −0.0798988
\(537\) −13.5928 −0.586571
\(538\) −31.0485 −1.33860
\(539\) 0 0
\(540\) 5.50722 0.236993
\(541\) −18.9510 −0.814768 −0.407384 0.913257i \(-0.633559\pi\)
−0.407384 + 0.913257i \(0.633559\pi\)
\(542\) 8.50957 0.365517
\(543\) 11.7025 0.502202
\(544\) −0.138127 −0.00592213
\(545\) −0.502741 −0.0215351
\(546\) −4.25033 −0.181897
\(547\) −31.6057 −1.35136 −0.675681 0.737194i \(-0.736150\pi\)
−0.675681 + 0.737194i \(0.736150\pi\)
\(548\) 21.8856 0.934904
\(549\) −20.3734 −0.869514
\(550\) 0 0
\(551\) 4.76184 0.202861
\(552\) 1.84582 0.0785632
\(553\) −9.98676 −0.424680
\(554\) −14.6394 −0.621970
\(555\) −2.03903 −0.0865520
\(556\) 2.47370 0.104908
\(557\) −15.7384 −0.666857 −0.333429 0.942775i \(-0.608206\pi\)
−0.333429 + 0.942775i \(0.608206\pi\)
\(558\) −15.4627 −0.654590
\(559\) −37.0982 −1.56909
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) −8.91851 −0.376205
\(563\) 1.64956 0.0695206 0.0347603 0.999396i \(-0.488933\pi\)
0.0347603 + 0.999396i \(0.488933\pi\)
\(564\) −2.20807 −0.0929765
\(565\) 8.91196 0.374929
\(566\) 19.7163 0.828736
\(567\) 2.20974 0.0928004
\(568\) −6.39213 −0.268208
\(569\) −35.6665 −1.49522 −0.747608 0.664141i \(-0.768798\pi\)
−0.747608 + 0.664141i \(0.768798\pi\)
\(570\) −0.629719 −0.0263760
\(571\) 29.5199 1.23537 0.617684 0.786427i \(-0.288071\pi\)
0.617684 + 0.786427i \(0.288071\pi\)
\(572\) 0 0
\(573\) 11.9744 0.500239
\(574\) 12.1868 0.508667
\(575\) 1.51048 0.0629915
\(576\) −1.50670 −0.0627793
\(577\) −31.1204 −1.29556 −0.647780 0.761827i \(-0.724302\pi\)
−0.647780 + 0.761827i \(0.724302\pi\)
\(578\) −16.9809 −0.706313
\(579\) 17.6595 0.733903
\(580\) −9.24062 −0.383696
\(581\) −2.20436 −0.0914521
\(582\) −5.38995 −0.223420
\(583\) 0 0
\(584\) −11.0649 −0.457869
\(585\) 5.24056 0.216670
\(586\) 6.92872 0.286223
\(587\) −36.5570 −1.50887 −0.754435 0.656374i \(-0.772089\pi\)
−0.754435 + 0.656374i \(0.772089\pi\)
\(588\) 1.22201 0.0503947
\(589\) 5.28850 0.217909
\(590\) −9.30512 −0.383086
\(591\) 27.7334 1.14080
\(592\) 1.66859 0.0685787
\(593\) −4.49251 −0.184485 −0.0922427 0.995737i \(-0.529404\pi\)
−0.0922427 + 0.995737i \(0.529404\pi\)
\(594\) 0 0
\(595\) −0.138127 −0.00566264
\(596\) 17.1038 0.700598
\(597\) 11.5344 0.472073
\(598\) 5.25370 0.214840
\(599\) −5.04343 −0.206069 −0.103035 0.994678i \(-0.532855\pi\)
−0.103035 + 0.994678i \(0.532855\pi\)
\(600\) 1.22201 0.0498882
\(601\) −27.5263 −1.12282 −0.561410 0.827538i \(-0.689741\pi\)
−0.561410 + 0.827538i \(0.689741\pi\)
\(602\) 10.6660 0.434715
\(603\) 2.78708 0.113499
\(604\) 3.21272 0.130724
\(605\) 0 0
\(606\) 15.3744 0.624542
\(607\) −3.44128 −0.139677 −0.0698387 0.997558i \(-0.522248\pi\)
−0.0698387 + 0.997558i \(0.522248\pi\)
\(608\) 0.515316 0.0208988
\(609\) −11.2921 −0.457579
\(610\) −13.5218 −0.547483
\(611\) −6.28477 −0.254254
\(612\) 0.208116 0.00841258
\(613\) −11.1154 −0.448949 −0.224474 0.974480i \(-0.572066\pi\)
−0.224474 + 0.974480i \(0.572066\pi\)
\(614\) −20.6340 −0.832719
\(615\) 14.8923 0.600517
\(616\) 0 0
\(617\) −10.7522 −0.432869 −0.216434 0.976297i \(-0.569443\pi\)
−0.216434 + 0.976297i \(0.569443\pi\)
\(618\) −3.64733 −0.146717
\(619\) 10.1391 0.407523 0.203761 0.979021i \(-0.434683\pi\)
0.203761 + 0.979021i \(0.434683\pi\)
\(620\) −10.2626 −0.412158
\(621\) −8.31855 −0.333812
\(622\) −20.2359 −0.811388
\(623\) −1.74164 −0.0697775
\(624\) 4.25033 0.170150
\(625\) 1.00000 0.0400000
\(626\) −4.40784 −0.176173
\(627\) 0 0
\(628\) 2.74824 0.109667
\(629\) −0.230477 −0.00918972
\(630\) −1.50670 −0.0600285
\(631\) −2.60248 −0.103603 −0.0518015 0.998657i \(-0.516496\pi\)
−0.0518015 + 0.998657i \(0.516496\pi\)
\(632\) 9.98676 0.397252
\(633\) 13.6253 0.541558
\(634\) 25.9982 1.03252
\(635\) −12.6728 −0.502903
\(636\) 1.69987 0.0674042
\(637\) 3.47816 0.137810
\(638\) 0 0
\(639\) 9.63104 0.380998
\(640\) −1.00000 −0.0395285
\(641\) −12.1917 −0.481543 −0.240771 0.970582i \(-0.577400\pi\)
−0.240771 + 0.970582i \(0.577400\pi\)
\(642\) 16.2254 0.640365
\(643\) −12.1108 −0.477603 −0.238802 0.971068i \(-0.576755\pi\)
−0.238802 + 0.971068i \(0.576755\pi\)
\(644\) −1.51048 −0.0595213
\(645\) 13.0339 0.513211
\(646\) −0.0711788 −0.00280049
\(647\) −4.91915 −0.193392 −0.0966959 0.995314i \(-0.530827\pi\)
−0.0966959 + 0.995314i \(0.530827\pi\)
\(648\) −2.20974 −0.0868068
\(649\) 0 0
\(650\) 3.47816 0.136425
\(651\) −12.5410 −0.491521
\(652\) −12.5201 −0.490325
\(653\) −13.2506 −0.518537 −0.259268 0.965805i \(-0.583481\pi\)
−0.259268 + 0.965805i \(0.583481\pi\)
\(654\) 0.614353 0.0240231
\(655\) −15.1497 −0.591948
\(656\) −12.1868 −0.475814
\(657\) 16.6715 0.650418
\(658\) 1.80692 0.0704412
\(659\) 30.5268 1.18916 0.594579 0.804038i \(-0.297319\pi\)
0.594579 + 0.804038i \(0.297319\pi\)
\(660\) 0 0
\(661\) −2.30028 −0.0894707 −0.0447353 0.998999i \(-0.514244\pi\)
−0.0447353 + 0.998999i \(0.514244\pi\)
\(662\) −19.8219 −0.770400
\(663\) −0.587084 −0.0228005
\(664\) 2.20436 0.0855456
\(665\) 0.515316 0.0199831
\(666\) −2.51407 −0.0974183
\(667\) 13.9578 0.540448
\(668\) 17.0335 0.659047
\(669\) −4.86114 −0.187943
\(670\) 1.84979 0.0714636
\(671\) 0 0
\(672\) −1.22201 −0.0471399
\(673\) −24.9976 −0.963588 −0.481794 0.876284i \(-0.660015\pi\)
−0.481794 + 0.876284i \(0.660015\pi\)
\(674\) 30.3840 1.17035
\(675\) −5.50722 −0.211973
\(676\) −0.902388 −0.0347072
\(677\) 16.6035 0.638123 0.319062 0.947734i \(-0.396632\pi\)
0.319062 + 0.947734i \(0.396632\pi\)
\(678\) −10.8905 −0.418246
\(679\) 4.41074 0.169269
\(680\) 0.138127 0.00529691
\(681\) 26.5600 1.01778
\(682\) 0 0
\(683\) 13.6055 0.520598 0.260299 0.965528i \(-0.416179\pi\)
0.260299 + 0.965528i \(0.416179\pi\)
\(684\) −0.776427 −0.0296874
\(685\) −21.8856 −0.836204
\(686\) −1.00000 −0.0381802
\(687\) 2.01164 0.0767489
\(688\) −10.6660 −0.406638
\(689\) 4.83829 0.184324
\(690\) −1.84582 −0.0702691
\(691\) −28.3703 −1.07926 −0.539629 0.841903i \(-0.681436\pi\)
−0.539629 + 0.841903i \(0.681436\pi\)
\(692\) 9.38603 0.356803
\(693\) 0 0
\(694\) −16.6544 −0.632194
\(695\) −2.47370 −0.0938328
\(696\) 11.2921 0.428026
\(697\) 1.68332 0.0637603
\(698\) −1.70386 −0.0644922
\(699\) −1.36436 −0.0516050
\(700\) −1.00000 −0.0377964
\(701\) 21.8649 0.825826 0.412913 0.910770i \(-0.364511\pi\)
0.412913 + 0.910770i \(0.364511\pi\)
\(702\) −19.1550 −0.722958
\(703\) 0.859852 0.0324299
\(704\) 0 0
\(705\) 2.20807 0.0831607
\(706\) 10.7417 0.404270
\(707\) −12.5813 −0.473167
\(708\) 11.3709 0.427345
\(709\) 37.6438 1.41374 0.706872 0.707342i \(-0.250106\pi\)
0.706872 + 0.707342i \(0.250106\pi\)
\(710\) 6.39213 0.239892
\(711\) −15.0471 −0.564309
\(712\) 1.74164 0.0652709
\(713\) 15.5015 0.580537
\(714\) 0.168791 0.00631686
\(715\) 0 0
\(716\) −11.1233 −0.415698
\(717\) 21.1551 0.790051
\(718\) −16.2804 −0.607579
\(719\) 8.91960 0.332645 0.166322 0.986071i \(-0.446811\pi\)
0.166322 + 0.986071i \(0.446811\pi\)
\(720\) 1.50670 0.0561515
\(721\) 2.98471 0.111156
\(722\) −18.7344 −0.697224
\(723\) −29.0256 −1.07947
\(724\) 9.57646 0.355906
\(725\) 9.24062 0.343188
\(726\) 0 0
\(727\) −21.9240 −0.813116 −0.406558 0.913625i \(-0.633271\pi\)
−0.406558 + 0.913625i \(0.633271\pi\)
\(728\) −3.47816 −0.128909
\(729\) 23.5190 0.871073
\(730\) 11.0649 0.409531
\(731\) 1.47326 0.0544905
\(732\) 16.5237 0.610735
\(733\) −29.9606 −1.10662 −0.553310 0.832975i \(-0.686636\pi\)
−0.553310 + 0.832975i \(0.686636\pi\)
\(734\) −21.0947 −0.778621
\(735\) −1.22201 −0.0450744
\(736\) 1.51048 0.0556771
\(737\) 0 0
\(738\) 18.3619 0.675910
\(739\) 25.1030 0.923428 0.461714 0.887029i \(-0.347235\pi\)
0.461714 + 0.887029i \(0.347235\pi\)
\(740\) −1.66859 −0.0613387
\(741\) 2.19026 0.0804613
\(742\) −1.39105 −0.0510670
\(743\) −36.8689 −1.35259 −0.676294 0.736632i \(-0.736415\pi\)
−0.676294 + 0.736632i \(0.736415\pi\)
\(744\) 12.5410 0.459776
\(745\) −17.1038 −0.626634
\(746\) −13.9335 −0.510142
\(747\) −3.32131 −0.121520
\(748\) 0 0
\(749\) −13.2777 −0.485155
\(750\) −1.22201 −0.0446213
\(751\) 30.4227 1.11014 0.555070 0.831803i \(-0.312691\pi\)
0.555070 + 0.831803i \(0.312691\pi\)
\(752\) −1.80692 −0.0658917
\(753\) 14.8175 0.539979
\(754\) 32.1404 1.17048
\(755\) −3.21272 −0.116923
\(756\) 5.50722 0.200295
\(757\) −12.0400 −0.437601 −0.218801 0.975770i \(-0.570214\pi\)
−0.218801 + 0.975770i \(0.570214\pi\)
\(758\) 22.1451 0.804346
\(759\) 0 0
\(760\) −0.515316 −0.0186925
\(761\) −11.4432 −0.414817 −0.207408 0.978254i \(-0.566503\pi\)
−0.207408 + 0.978254i \(0.566503\pi\)
\(762\) 15.4862 0.561005
\(763\) −0.502741 −0.0182005
\(764\) 9.79900 0.354515
\(765\) −0.208116 −0.00752444
\(766\) −22.5171 −0.813574
\(767\) 32.3647 1.16862
\(768\) 1.22201 0.0440953
\(769\) 3.36501 0.121346 0.0606728 0.998158i \(-0.480675\pi\)
0.0606728 + 0.998158i \(0.480675\pi\)
\(770\) 0 0
\(771\) −34.9609 −1.25909
\(772\) 14.4512 0.520111
\(773\) −3.19918 −0.115067 −0.0575333 0.998344i \(-0.518324\pi\)
−0.0575333 + 0.998344i \(0.518324\pi\)
\(774\) 16.0705 0.577643
\(775\) 10.2626 0.368645
\(776\) −4.41074 −0.158336
\(777\) −2.03903 −0.0731498
\(778\) −20.8343 −0.746946
\(779\) −6.28004 −0.225006
\(780\) −4.25033 −0.152186
\(781\) 0 0
\(782\) −0.208638 −0.00746087
\(783\) −50.8901 −1.81866
\(784\) 1.00000 0.0357143
\(785\) −2.74824 −0.0980889
\(786\) 18.5130 0.660338
\(787\) −5.27916 −0.188182 −0.0940908 0.995564i \(-0.529994\pi\)
−0.0940908 + 0.995564i \(0.529994\pi\)
\(788\) 22.6950 0.808476
\(789\) −37.2392 −1.32575
\(790\) −9.98676 −0.355313
\(791\) 8.91196 0.316873
\(792\) 0 0
\(793\) 47.0311 1.67012
\(794\) 19.0865 0.677355
\(795\) −1.69987 −0.0602881
\(796\) 9.43893 0.334554
\(797\) 41.1536 1.45773 0.728867 0.684655i \(-0.240047\pi\)
0.728867 + 0.684655i \(0.240047\pi\)
\(798\) −0.629719 −0.0222918
\(799\) 0.249584 0.00882964
\(800\) 1.00000 0.0353553
\(801\) −2.62414 −0.0927194
\(802\) 30.1895 1.06603
\(803\) 0 0
\(804\) −2.26045 −0.0797201
\(805\) 1.51048 0.0532375
\(806\) 35.6951 1.25731
\(807\) −37.9415 −1.33560
\(808\) 12.5813 0.442608
\(809\) −11.2803 −0.396596 −0.198298 0.980142i \(-0.563541\pi\)
−0.198298 + 0.980142i \(0.563541\pi\)
\(810\) 2.20974 0.0776424
\(811\) −37.0216 −1.30000 −0.650002 0.759933i \(-0.725232\pi\)
−0.650002 + 0.759933i \(0.725232\pi\)
\(812\) −9.24062 −0.324282
\(813\) 10.3987 0.364700
\(814\) 0 0
\(815\) 12.5201 0.438560
\(816\) −0.168791 −0.00590888
\(817\) −5.49637 −0.192294
\(818\) −29.4582 −1.02998
\(819\) 5.24056 0.183120
\(820\) 12.1868 0.425581
\(821\) 49.7341 1.73573 0.867866 0.496799i \(-0.165491\pi\)
0.867866 + 0.496799i \(0.165491\pi\)
\(822\) 26.7443 0.932813
\(823\) 34.5639 1.20482 0.602410 0.798187i \(-0.294207\pi\)
0.602410 + 0.798187i \(0.294207\pi\)
\(824\) −2.98471 −0.103977
\(825\) 0 0
\(826\) −9.30512 −0.323766
\(827\) −25.6050 −0.890374 −0.445187 0.895438i \(-0.646863\pi\)
−0.445187 + 0.895438i \(0.646863\pi\)
\(828\) −2.27585 −0.0790911
\(829\) −12.0869 −0.419797 −0.209898 0.977723i \(-0.567313\pi\)
−0.209898 + 0.977723i \(0.567313\pi\)
\(830\) −2.20436 −0.0765143
\(831\) −17.8895 −0.620579
\(832\) 3.47816 0.120584
\(833\) −0.138127 −0.00478580
\(834\) 3.02287 0.104674
\(835\) −17.0335 −0.589469
\(836\) 0 0
\(837\) −56.5186 −1.95357
\(838\) −20.4867 −0.707700
\(839\) −9.12559 −0.315050 −0.157525 0.987515i \(-0.550352\pi\)
−0.157525 + 0.987515i \(0.550352\pi\)
\(840\) 1.22201 0.0421632
\(841\) 56.3891 1.94445
\(842\) −2.61302 −0.0900507
\(843\) −10.8985 −0.375363
\(844\) 11.1500 0.383797
\(845\) 0.902388 0.0310431
\(846\) 2.72249 0.0936013
\(847\) 0 0
\(848\) 1.39105 0.0477688
\(849\) 24.0934 0.826883
\(850\) −0.138127 −0.00473770
\(851\) 2.52038 0.0863975
\(852\) −7.81122 −0.267608
\(853\) 4.18612 0.143330 0.0716650 0.997429i \(-0.477169\pi\)
0.0716650 + 0.997429i \(0.477169\pi\)
\(854\) −13.5218 −0.462707
\(855\) 0.776427 0.0265533
\(856\) 13.2777 0.453821
\(857\) −23.8579 −0.814971 −0.407485 0.913212i \(-0.633594\pi\)
−0.407485 + 0.913212i \(0.633594\pi\)
\(858\) 0 0
\(859\) −11.8765 −0.405221 −0.202610 0.979259i \(-0.564943\pi\)
−0.202610 + 0.979259i \(0.564943\pi\)
\(860\) 10.6660 0.363708
\(861\) 14.8923 0.507529
\(862\) −9.87795 −0.336444
\(863\) 16.8809 0.574633 0.287316 0.957836i \(-0.407237\pi\)
0.287316 + 0.957836i \(0.407237\pi\)
\(864\) −5.50722 −0.187359
\(865\) −9.38603 −0.319135
\(866\) 19.5259 0.663518
\(867\) −20.7508 −0.704733
\(868\) −10.2626 −0.348337
\(869\) 0 0
\(870\) −11.2921 −0.382838
\(871\) −6.43387 −0.218003
\(872\) 0.502741 0.0170250
\(873\) 6.64567 0.224922
\(874\) 0.778375 0.0263289
\(875\) 1.00000 0.0338062
\(876\) −13.5214 −0.456845
\(877\) −8.83200 −0.298236 −0.149118 0.988819i \(-0.547643\pi\)
−0.149118 + 0.988819i \(0.547643\pi\)
\(878\) 3.38579 0.114265
\(879\) 8.46693 0.285582
\(880\) 0 0
\(881\) −9.23255 −0.311052 −0.155526 0.987832i \(-0.549707\pi\)
−0.155526 + 0.987832i \(0.549707\pi\)
\(882\) −1.50670 −0.0507333
\(883\) −36.8709 −1.24080 −0.620402 0.784284i \(-0.713031\pi\)
−0.620402 + 0.784284i \(0.713031\pi\)
\(884\) −0.480427 −0.0161585
\(885\) −11.3709 −0.382229
\(886\) −19.3485 −0.650026
\(887\) −30.1184 −1.01128 −0.505638 0.862746i \(-0.668743\pi\)
−0.505638 + 0.862746i \(0.668743\pi\)
\(888\) 2.03903 0.0684253
\(889\) −12.6728 −0.425030
\(890\) −1.74164 −0.0583800
\(891\) 0 0
\(892\) −3.97800 −0.133193
\(893\) −0.931135 −0.0311593
\(894\) 20.9009 0.699031
\(895\) 11.1233 0.371812
\(896\) −1.00000 −0.0334077
\(897\) 6.42005 0.214359
\(898\) 33.1331 1.10567
\(899\) 94.8332 3.16286
\(900\) −1.50670 −0.0502234
\(901\) −0.192141 −0.00640114
\(902\) 0 0
\(903\) 13.0339 0.433742
\(904\) −8.91196 −0.296407
\(905\) −9.57646 −0.318332
\(906\) 3.92596 0.130431
\(907\) −49.5891 −1.64658 −0.823290 0.567621i \(-0.807864\pi\)
−0.823290 + 0.567621i \(0.807864\pi\)
\(908\) 21.7347 0.721292
\(909\) −18.9562 −0.628738
\(910\) 3.47816 0.115300
\(911\) −51.5703 −1.70860 −0.854301 0.519779i \(-0.826014\pi\)
−0.854301 + 0.519779i \(0.826014\pi\)
\(912\) 0.629719 0.0208521
\(913\) 0 0
\(914\) −12.6845 −0.419567
\(915\) −16.5237 −0.546258
\(916\) 1.64618 0.0543913
\(917\) −15.1497 −0.500288
\(918\) 0.760693 0.0251066
\(919\) 29.1350 0.961075 0.480537 0.876974i \(-0.340442\pi\)
0.480537 + 0.876974i \(0.340442\pi\)
\(920\) −1.51048 −0.0497991
\(921\) −25.2148 −0.830856
\(922\) 19.4617 0.640937
\(923\) −22.2329 −0.731804
\(924\) 0 0
\(925\) 1.66859 0.0548630
\(926\) 29.6401 0.974034
\(927\) 4.49706 0.147703
\(928\) 9.24062 0.303338
\(929\) −45.0244 −1.47720 −0.738602 0.674142i \(-0.764513\pi\)
−0.738602 + 0.674142i \(0.764513\pi\)
\(930\) −12.5410 −0.411236
\(931\) 0.515316 0.0168888
\(932\) −1.11650 −0.0365720
\(933\) −24.7284 −0.809573
\(934\) 26.8009 0.876953
\(935\) 0 0
\(936\) −5.24056 −0.171293
\(937\) −6.89838 −0.225360 −0.112680 0.993631i \(-0.535944\pi\)
−0.112680 + 0.993631i \(0.535944\pi\)
\(938\) 1.84979 0.0603978
\(939\) −5.38641 −0.175779
\(940\) 1.80692 0.0589353
\(941\) −6.88982 −0.224602 −0.112301 0.993674i \(-0.535822\pi\)
−0.112301 + 0.993674i \(0.535822\pi\)
\(942\) 3.35836 0.109421
\(943\) −18.4079 −0.599445
\(944\) 9.30512 0.302856
\(945\) −5.50722 −0.179150
\(946\) 0 0
\(947\) 34.2294 1.11231 0.556154 0.831079i \(-0.312277\pi\)
0.556154 + 0.831079i \(0.312277\pi\)
\(948\) 12.2039 0.396363
\(949\) −38.4856 −1.24929
\(950\) 0.515316 0.0167190
\(951\) 31.7699 1.03021
\(952\) 0.138127 0.00447671
\(953\) 57.9062 1.87577 0.937883 0.346951i \(-0.112783\pi\)
0.937883 + 0.346951i \(0.112783\pi\)
\(954\) −2.09590 −0.0678571
\(955\) −9.79900 −0.317088
\(956\) 17.3118 0.559903
\(957\) 0 0
\(958\) 13.3413 0.431039
\(959\) −21.8856 −0.706721
\(960\) −1.22201 −0.0394401
\(961\) 74.3218 2.39748
\(962\) 5.80363 0.187117
\(963\) −20.0055 −0.644668
\(964\) −23.7524 −0.765015
\(965\) −14.4512 −0.465202
\(966\) −1.84582 −0.0593882
\(967\) 39.7063 1.27687 0.638435 0.769676i \(-0.279582\pi\)
0.638435 + 0.769676i \(0.279582\pi\)
\(968\) 0 0
\(969\) −0.0869809 −0.00279423
\(970\) 4.41074 0.141620
\(971\) −7.63191 −0.244920 −0.122460 0.992473i \(-0.539078\pi\)
−0.122460 + 0.992473i \(0.539078\pi\)
\(972\) 13.8213 0.443319
\(973\) −2.47370 −0.0793032
\(974\) −13.3718 −0.428459
\(975\) 4.25033 0.136120
\(976\) 13.5218 0.432823
\(977\) 14.9297 0.477643 0.238821 0.971063i \(-0.423239\pi\)
0.238821 + 0.971063i \(0.423239\pi\)
\(978\) −15.2996 −0.489228
\(979\) 0 0
\(980\) −1.00000 −0.0319438
\(981\) −0.757481 −0.0241845
\(982\) 25.9713 0.828776
\(983\) 1.28851 0.0410971 0.0205485 0.999789i \(-0.493459\pi\)
0.0205485 + 0.999789i \(0.493459\pi\)
\(984\) −14.8923 −0.474750
\(985\) −22.6950 −0.723122
\(986\) −1.27638 −0.0406481
\(987\) 2.20807 0.0702836
\(988\) 1.79235 0.0570223
\(989\) −16.1108 −0.512295
\(990\) 0 0
\(991\) 30.8872 0.981163 0.490582 0.871395i \(-0.336784\pi\)
0.490582 + 0.871395i \(0.336784\pi\)
\(992\) 10.2626 0.325839
\(993\) −24.2225 −0.768677
\(994\) 6.39213 0.202746
\(995\) −9.43893 −0.299234
\(996\) 2.69374 0.0853543
\(997\) −44.9211 −1.42266 −0.711332 0.702856i \(-0.751908\pi\)
−0.711332 + 0.702856i \(0.751908\pi\)
\(998\) −22.5866 −0.714966
\(999\) −9.18930 −0.290736
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.dh.1.6 8
11.3 even 5 770.2.n.k.141.2 yes 16
11.4 even 5 770.2.n.k.71.2 16
11.10 odd 2 8470.2.a.dg.1.6 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.n.k.71.2 16 11.4 even 5
770.2.n.k.141.2 yes 16 11.3 even 5
8470.2.a.dg.1.6 8 11.10 odd 2
8470.2.a.dh.1.6 8 1.1 even 1 trivial