Properties

Label 8470.2.a.df.1.4
Level $8470$
Weight $2$
Character 8470.1
Self dual yes
Analytic conductor $67.633$
Analytic rank $0$
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: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.10784448.1
Defining polynomial: \(x^{6} - 11 x^{4} - 4 x^{3} + 31 x^{2} + 22 x - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-0.930827\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +0.529317 q^{3} +1.00000 q^{4} +1.00000 q^{5} +0.529317 q^{6} +1.00000 q^{7} +1.00000 q^{8} -2.71982 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +0.529317 q^{3} +1.00000 q^{4} +1.00000 q^{5} +0.529317 q^{6} +1.00000 q^{7} +1.00000 q^{8} -2.71982 q^{9} +1.00000 q^{10} +0.529317 q^{12} +4.81243 q^{13} +1.00000 q^{14} +0.529317 q^{15} +1.00000 q^{16} +6.02760 q^{17} -2.71982 q^{18} +1.96321 q^{19} +1.00000 q^{20} +0.529317 q^{21} +2.56329 q^{23} +0.529317 q^{24} +1.00000 q^{25} +4.81243 q^{26} -3.02760 q^{27} +1.00000 q^{28} +8.14477 q^{29} +0.529317 q^{30} -2.24914 q^{31} +1.00000 q^{32} +6.02760 q^{34} +1.00000 q^{35} -2.71982 q^{36} -6.14477 q^{37} +1.96321 q^{38} +2.54730 q^{39} +1.00000 q^{40} -5.35973 q^{41} +0.529317 q^{42} -5.58127 q^{43} -2.71982 q^{45} +2.56329 q^{46} +6.23315 q^{47} +0.529317 q^{48} +1.00000 q^{49} +1.00000 q^{50} +3.19051 q^{51} +4.81243 q^{52} -0.648854 q^{53} -3.02760 q^{54} +1.00000 q^{56} +1.03916 q^{57} +8.14477 q^{58} -6.84003 q^{59} +0.529317 q^{60} +1.67669 q^{61} -2.24914 q^{62} -2.71982 q^{63} +1.00000 q^{64} +4.81243 q^{65} +7.80030 q^{67} +6.02760 q^{68} +1.35679 q^{69} +1.00000 q^{70} +8.56040 q^{71} -2.71982 q^{72} -5.50769 q^{73} -6.14477 q^{74} +0.529317 q^{75} +1.96321 q^{76} +2.54730 q^{78} +3.71916 q^{79} +1.00000 q^{80} +6.55691 q^{81} -5.35973 q^{82} -11.5079 q^{83} +0.529317 q^{84} +6.02760 q^{85} -5.58127 q^{86} +4.31116 q^{87} -14.2096 q^{89} -2.71982 q^{90} +4.81243 q^{91} +2.56329 q^{92} -1.19051 q^{93} +6.23315 q^{94} +1.96321 q^{95} +0.529317 q^{96} -3.26750 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 6q^{2} + 4q^{3} + 6q^{4} + 6q^{5} + 4q^{6} + 6q^{7} + 6q^{8} + 2q^{9} + O(q^{10}) \) \( 6q + 6q^{2} + 4q^{3} + 6q^{4} + 6q^{5} + 4q^{6} + 6q^{7} + 6q^{8} + 2q^{9} + 6q^{10} + 4q^{12} + 6q^{14} + 4q^{15} + 6q^{16} + 2q^{17} + 2q^{18} + 6q^{20} + 4q^{21} + 4q^{23} + 4q^{24} + 6q^{25} + 16q^{27} + 6q^{28} + 8q^{29} + 4q^{30} + 4q^{31} + 6q^{32} + 2q^{34} + 6q^{35} + 2q^{36} + 4q^{37} + 6q^{40} + 12q^{41} + 4q^{42} - 6q^{43} + 2q^{45} + 4q^{46} + 16q^{47} + 4q^{48} + 6q^{49} + 6q^{50} + 12q^{53} + 16q^{54} + 6q^{56} + 8q^{57} + 8q^{58} + 22q^{59} + 4q^{60} - 4q^{61} + 4q^{62} + 2q^{63} + 6q^{64} + 20q^{67} + 2q^{68} + 12q^{69} + 6q^{70} + 14q^{71} + 2q^{72} + 18q^{73} + 4q^{74} + 4q^{75} + 32q^{79} + 6q^{80} + 6q^{81} + 12q^{82} - 16q^{83} + 4q^{84} + 2q^{85} - 6q^{86} - 4q^{87} + 4q^{89} + 2q^{90} + 4q^{92} + 12q^{93} + 16q^{94} + 4q^{96} + 4q^{97} + 6q^{98} + O(q^{100}) \)

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
\(3\) 0.529317 0.305601 0.152801 0.988257i \(-0.451171\pi\)
0.152801 + 0.988257i \(0.451171\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 0.529317 0.216093
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) −2.71982 −0.906608
\(10\) 1.00000 0.316228
\(11\) 0 0
\(12\) 0.529317 0.152801
\(13\) 4.81243 1.33473 0.667364 0.744732i \(-0.267423\pi\)
0.667364 + 0.744732i \(0.267423\pi\)
\(14\) 1.00000 0.267261
\(15\) 0.529317 0.136669
\(16\) 1.00000 0.250000
\(17\) 6.02760 1.46191 0.730954 0.682427i \(-0.239076\pi\)
0.730954 + 0.682427i \(0.239076\pi\)
\(18\) −2.71982 −0.641069
\(19\) 1.96321 0.450391 0.225196 0.974314i \(-0.427698\pi\)
0.225196 + 0.974314i \(0.427698\pi\)
\(20\) 1.00000 0.223607
\(21\) 0.529317 0.115506
\(22\) 0 0
\(23\) 2.56329 0.534483 0.267241 0.963630i \(-0.413888\pi\)
0.267241 + 0.963630i \(0.413888\pi\)
\(24\) 0.529317 0.108046
\(25\) 1.00000 0.200000
\(26\) 4.81243 0.943795
\(27\) −3.02760 −0.582661
\(28\) 1.00000 0.188982
\(29\) 8.14477 1.51245 0.756223 0.654314i \(-0.227043\pi\)
0.756223 + 0.654314i \(0.227043\pi\)
\(30\) 0.529317 0.0966395
\(31\) −2.24914 −0.403958 −0.201979 0.979390i \(-0.564737\pi\)
−0.201979 + 0.979390i \(0.564737\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 6.02760 1.03372
\(35\) 1.00000 0.169031
\(36\) −2.71982 −0.453304
\(37\) −6.14477 −1.01019 −0.505097 0.863063i \(-0.668543\pi\)
−0.505097 + 0.863063i \(0.668543\pi\)
\(38\) 1.96321 0.318475
\(39\) 2.54730 0.407894
\(40\) 1.00000 0.158114
\(41\) −5.35973 −0.837049 −0.418524 0.908206i \(-0.637453\pi\)
−0.418524 + 0.908206i \(0.637453\pi\)
\(42\) 0.529317 0.0816753
\(43\) −5.58127 −0.851136 −0.425568 0.904927i \(-0.639926\pi\)
−0.425568 + 0.904927i \(0.639926\pi\)
\(44\) 0 0
\(45\) −2.71982 −0.405447
\(46\) 2.56329 0.377936
\(47\) 6.23315 0.909198 0.454599 0.890696i \(-0.349782\pi\)
0.454599 + 0.890696i \(0.349782\pi\)
\(48\) 0.529317 0.0764003
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 3.19051 0.446760
\(52\) 4.81243 0.667364
\(53\) −0.648854 −0.0891270 −0.0445635 0.999007i \(-0.514190\pi\)
−0.0445635 + 0.999007i \(0.514190\pi\)
\(54\) −3.02760 −0.412004
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 1.03916 0.137640
\(58\) 8.14477 1.06946
\(59\) −6.84003 −0.890496 −0.445248 0.895407i \(-0.646884\pi\)
−0.445248 + 0.895407i \(0.646884\pi\)
\(60\) 0.529317 0.0683345
\(61\) 1.67669 0.214679 0.107339 0.994222i \(-0.465767\pi\)
0.107339 + 0.994222i \(0.465767\pi\)
\(62\) −2.24914 −0.285641
\(63\) −2.71982 −0.342666
\(64\) 1.00000 0.125000
\(65\) 4.81243 0.596908
\(66\) 0 0
\(67\) 7.80030 0.952958 0.476479 0.879186i \(-0.341913\pi\)
0.476479 + 0.879186i \(0.341913\pi\)
\(68\) 6.02760 0.730954
\(69\) 1.35679 0.163338
\(70\) 1.00000 0.119523
\(71\) 8.56040 1.01593 0.507966 0.861377i \(-0.330397\pi\)
0.507966 + 0.861377i \(0.330397\pi\)
\(72\) −2.71982 −0.320534
\(73\) −5.50769 −0.644626 −0.322313 0.946633i \(-0.604460\pi\)
−0.322313 + 0.946633i \(0.604460\pi\)
\(74\) −6.14477 −0.714315
\(75\) 0.529317 0.0611202
\(76\) 1.96321 0.225196
\(77\) 0 0
\(78\) 2.54730 0.288425
\(79\) 3.71916 0.418438 0.209219 0.977869i \(-0.432908\pi\)
0.209219 + 0.977869i \(0.432908\pi\)
\(80\) 1.00000 0.111803
\(81\) 6.55691 0.728546
\(82\) −5.35973 −0.591883
\(83\) −11.5079 −1.26316 −0.631578 0.775312i \(-0.717592\pi\)
−0.631578 + 0.775312i \(0.717592\pi\)
\(84\) 0.529317 0.0577532
\(85\) 6.02760 0.653785
\(86\) −5.58127 −0.601844
\(87\) 4.31116 0.462205
\(88\) 0 0
\(89\) −14.2096 −1.50622 −0.753108 0.657897i \(-0.771446\pi\)
−0.753108 + 0.657897i \(0.771446\pi\)
\(90\) −2.71982 −0.286695
\(91\) 4.81243 0.504480
\(92\) 2.56329 0.267241
\(93\) −1.19051 −0.123450
\(94\) 6.23315 0.642900
\(95\) 1.96321 0.201421
\(96\) 0.529317 0.0540231
\(97\) −3.26750 −0.331764 −0.165882 0.986146i \(-0.553047\pi\)
−0.165882 + 0.986146i \(0.553047\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 0.584368 0.0581468 0.0290734 0.999577i \(-0.490744\pi\)
0.0290734 + 0.999577i \(0.490744\pi\)
\(102\) 3.19051 0.315907
\(103\) 12.8427 1.26543 0.632717 0.774383i \(-0.281940\pi\)
0.632717 + 0.774383i \(0.281940\pi\)
\(104\) 4.81243 0.471897
\(105\) 0.529317 0.0516560
\(106\) −0.648854 −0.0630223
\(107\) −4.82110 −0.466073 −0.233037 0.972468i \(-0.574866\pi\)
−0.233037 + 0.972468i \(0.574866\pi\)
\(108\) −3.02760 −0.291331
\(109\) −9.52895 −0.912708 −0.456354 0.889798i \(-0.650845\pi\)
−0.456354 + 0.889798i \(0.650845\pi\)
\(110\) 0 0
\(111\) −3.25253 −0.308716
\(112\) 1.00000 0.0944911
\(113\) −2.71812 −0.255699 −0.127850 0.991794i \(-0.540807\pi\)
−0.127850 + 0.991794i \(0.540807\pi\)
\(114\) 1.03916 0.0973262
\(115\) 2.56329 0.239028
\(116\) 8.14477 0.756223
\(117\) −13.0890 −1.21007
\(118\) −6.84003 −0.629675
\(119\) 6.02760 0.552549
\(120\) 0.529317 0.0483198
\(121\) 0 0
\(122\) 1.67669 0.151801
\(123\) −2.83699 −0.255803
\(124\) −2.24914 −0.201979
\(125\) 1.00000 0.0894427
\(126\) −2.71982 −0.242301
\(127\) −3.60394 −0.319798 −0.159899 0.987133i \(-0.551117\pi\)
−0.159899 + 0.987133i \(0.551117\pi\)
\(128\) 1.00000 0.0883883
\(129\) −2.95426 −0.260108
\(130\) 4.81243 0.422078
\(131\) −5.17800 −0.452404 −0.226202 0.974080i \(-0.572631\pi\)
−0.226202 + 0.974080i \(0.572631\pi\)
\(132\) 0 0
\(133\) 1.96321 0.170232
\(134\) 7.80030 0.673843
\(135\) −3.02760 −0.260574
\(136\) 6.02760 0.516862
\(137\) 7.79465 0.665942 0.332971 0.942937i \(-0.391949\pi\)
0.332971 + 0.942937i \(0.391949\pi\)
\(138\) 1.35679 0.115498
\(139\) 9.73564 0.825767 0.412883 0.910784i \(-0.364522\pi\)
0.412883 + 0.910784i \(0.364522\pi\)
\(140\) 1.00000 0.0845154
\(141\) 3.29931 0.277852
\(142\) 8.56040 0.718373
\(143\) 0 0
\(144\) −2.71982 −0.226652
\(145\) 8.14477 0.676386
\(146\) −5.50769 −0.455820
\(147\) 0.529317 0.0436573
\(148\) −6.14477 −0.505097
\(149\) 5.26382 0.431229 0.215614 0.976479i \(-0.430825\pi\)
0.215614 + 0.976479i \(0.430825\pi\)
\(150\) 0.529317 0.0432185
\(151\) −3.80904 −0.309976 −0.154988 0.987916i \(-0.549534\pi\)
−0.154988 + 0.987916i \(0.549534\pi\)
\(152\) 1.96321 0.159237
\(153\) −16.3940 −1.32538
\(154\) 0 0
\(155\) −2.24914 −0.180655
\(156\) 2.54730 0.203947
\(157\) 15.8237 1.26287 0.631436 0.775428i \(-0.282466\pi\)
0.631436 + 0.775428i \(0.282466\pi\)
\(158\) 3.71916 0.295880
\(159\) −0.343449 −0.0272373
\(160\) 1.00000 0.0790569
\(161\) 2.56329 0.202015
\(162\) 6.55691 0.515160
\(163\) 10.0410 0.786470 0.393235 0.919438i \(-0.371356\pi\)
0.393235 + 0.919438i \(0.371356\pi\)
\(164\) −5.35973 −0.418524
\(165\) 0 0
\(166\) −11.5079 −0.893186
\(167\) 5.62687 0.435420 0.217710 0.976013i \(-0.430141\pi\)
0.217710 + 0.976013i \(0.430141\pi\)
\(168\) 0.529317 0.0408377
\(169\) 10.1595 0.781498
\(170\) 6.02760 0.462296
\(171\) −5.33958 −0.408328
\(172\) −5.58127 −0.425568
\(173\) 11.3235 0.860909 0.430455 0.902612i \(-0.358353\pi\)
0.430455 + 0.902612i \(0.358353\pi\)
\(174\) 4.31116 0.326828
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) −3.62054 −0.272136
\(178\) −14.2096 −1.06506
\(179\) 23.7596 1.77587 0.887936 0.459966i \(-0.152139\pi\)
0.887936 + 0.459966i \(0.152139\pi\)
\(180\) −2.71982 −0.202724
\(181\) −1.06147 −0.0788986 −0.0394493 0.999222i \(-0.512560\pi\)
−0.0394493 + 0.999222i \(0.512560\pi\)
\(182\) 4.81243 0.356721
\(183\) 0.887502 0.0656060
\(184\) 2.56329 0.188968
\(185\) −6.14477 −0.451772
\(186\) −1.19051 −0.0872922
\(187\) 0 0
\(188\) 6.23315 0.454599
\(189\) −3.02760 −0.220225
\(190\) 1.96321 0.142426
\(191\) 16.9651 1.22755 0.613776 0.789480i \(-0.289650\pi\)
0.613776 + 0.789480i \(0.289650\pi\)
\(192\) 0.529317 0.0382001
\(193\) 18.4425 1.32752 0.663761 0.747945i \(-0.268959\pi\)
0.663761 + 0.747945i \(0.268959\pi\)
\(194\) −3.26750 −0.234593
\(195\) 2.54730 0.182416
\(196\) 1.00000 0.0714286
\(197\) −20.3968 −1.45321 −0.726607 0.687053i \(-0.758904\pi\)
−0.726607 + 0.687053i \(0.758904\pi\)
\(198\) 0 0
\(199\) 6.47192 0.458782 0.229391 0.973334i \(-0.426327\pi\)
0.229391 + 0.973334i \(0.426327\pi\)
\(200\) 1.00000 0.0707107
\(201\) 4.12883 0.291225
\(202\) 0.584368 0.0411160
\(203\) 8.14477 0.571651
\(204\) 3.19051 0.223380
\(205\) −5.35973 −0.374340
\(206\) 12.8427 0.894797
\(207\) −6.97169 −0.484566
\(208\) 4.81243 0.333682
\(209\) 0 0
\(210\) 0.529317 0.0365263
\(211\) 17.7130 1.21941 0.609706 0.792628i \(-0.291288\pi\)
0.609706 + 0.792628i \(0.291288\pi\)
\(212\) −0.648854 −0.0445635
\(213\) 4.53116 0.310470
\(214\) −4.82110 −0.329564
\(215\) −5.58127 −0.380639
\(216\) −3.02760 −0.206002
\(217\) −2.24914 −0.152682
\(218\) −9.52895 −0.645382
\(219\) −2.91531 −0.196998
\(220\) 0 0
\(221\) 29.0074 1.95125
\(222\) −3.25253 −0.218295
\(223\) 21.6290 1.44839 0.724193 0.689598i \(-0.242213\pi\)
0.724193 + 0.689598i \(0.242213\pi\)
\(224\) 1.00000 0.0668153
\(225\) −2.71982 −0.181322
\(226\) −2.71812 −0.180807
\(227\) −15.3595 −1.01944 −0.509721 0.860339i \(-0.670252\pi\)
−0.509721 + 0.860339i \(0.670252\pi\)
\(228\) 1.03916 0.0688200
\(229\) 21.2484 1.40414 0.702068 0.712110i \(-0.252260\pi\)
0.702068 + 0.712110i \(0.252260\pi\)
\(230\) 2.56329 0.169018
\(231\) 0 0
\(232\) 8.14477 0.534730
\(233\) −18.8912 −1.23761 −0.618803 0.785546i \(-0.712382\pi\)
−0.618803 + 0.785546i \(0.712382\pi\)
\(234\) −13.0890 −0.855652
\(235\) 6.23315 0.406606
\(236\) −6.84003 −0.445248
\(237\) 1.96861 0.127875
\(238\) 6.02760 0.390711
\(239\) 25.4078 1.64349 0.821747 0.569853i \(-0.193000\pi\)
0.821747 + 0.569853i \(0.193000\pi\)
\(240\) 0.529317 0.0341672
\(241\) 10.1147 0.651544 0.325772 0.945448i \(-0.394376\pi\)
0.325772 + 0.945448i \(0.394376\pi\)
\(242\) 0 0
\(243\) 12.5535 0.805306
\(244\) 1.67669 0.107339
\(245\) 1.00000 0.0638877
\(246\) −2.83699 −0.180880
\(247\) 9.44781 0.601150
\(248\) −2.24914 −0.142821
\(249\) −6.09132 −0.386022
\(250\) 1.00000 0.0632456
\(251\) −15.5061 −0.978736 −0.489368 0.872078i \(-0.662772\pi\)
−0.489368 + 0.872078i \(0.662772\pi\)
\(252\) −2.71982 −0.171333
\(253\) 0 0
\(254\) −3.60394 −0.226131
\(255\) 3.19051 0.199797
\(256\) 1.00000 0.0625000
\(257\) −18.6645 −1.16426 −0.582130 0.813095i \(-0.697781\pi\)
−0.582130 + 0.813095i \(0.697781\pi\)
\(258\) −2.95426 −0.183924
\(259\) −6.14477 −0.381817
\(260\) 4.81243 0.298454
\(261\) −22.1523 −1.37119
\(262\) −5.17800 −0.319898
\(263\) −2.35968 −0.145504 −0.0727521 0.997350i \(-0.523178\pi\)
−0.0727521 + 0.997350i \(0.523178\pi\)
\(264\) 0 0
\(265\) −0.648854 −0.0398588
\(266\) 1.96321 0.120372
\(267\) −7.52138 −0.460301
\(268\) 7.80030 0.476479
\(269\) −14.5570 −0.887556 −0.443778 0.896137i \(-0.646362\pi\)
−0.443778 + 0.896137i \(0.646362\pi\)
\(270\) −3.02760 −0.184254
\(271\) −10.4172 −0.632798 −0.316399 0.948626i \(-0.602474\pi\)
−0.316399 + 0.948626i \(0.602474\pi\)
\(272\) 6.02760 0.365477
\(273\) 2.54730 0.154170
\(274\) 7.79465 0.470892
\(275\) 0 0
\(276\) 1.35679 0.0816692
\(277\) 14.9701 0.899466 0.449733 0.893163i \(-0.351519\pi\)
0.449733 + 0.893163i \(0.351519\pi\)
\(278\) 9.73564 0.583905
\(279\) 6.11727 0.366231
\(280\) 1.00000 0.0597614
\(281\) 8.00750 0.477688 0.238844 0.971058i \(-0.423232\pi\)
0.238844 + 0.971058i \(0.423232\pi\)
\(282\) 3.29931 0.196471
\(283\) 22.3793 1.33031 0.665154 0.746706i \(-0.268366\pi\)
0.665154 + 0.746706i \(0.268366\pi\)
\(284\) 8.56040 0.507966
\(285\) 1.03916 0.0615545
\(286\) 0 0
\(287\) −5.35973 −0.316375
\(288\) −2.71982 −0.160267
\(289\) 19.3319 1.13717
\(290\) 8.14477 0.478277
\(291\) −1.72954 −0.101388
\(292\) −5.50769 −0.322313
\(293\) 2.78343 0.162610 0.0813049 0.996689i \(-0.474091\pi\)
0.0813049 + 0.996689i \(0.474091\pi\)
\(294\) 0.529317 0.0308704
\(295\) −6.84003 −0.398242
\(296\) −6.14477 −0.357157
\(297\) 0 0
\(298\) 5.26382 0.304925
\(299\) 12.3356 0.713389
\(300\) 0.529317 0.0305601
\(301\) −5.58127 −0.321699
\(302\) −3.80904 −0.219186
\(303\) 0.309316 0.0177697
\(304\) 1.96321 0.112598
\(305\) 1.67669 0.0960072
\(306\) −16.3940 −0.937183
\(307\) −24.7810 −1.41433 −0.707164 0.707049i \(-0.750026\pi\)
−0.707164 + 0.707049i \(0.750026\pi\)
\(308\) 0 0
\(309\) 6.79788 0.386718
\(310\) −2.24914 −0.127743
\(311\) 18.8355 1.06807 0.534033 0.845464i \(-0.320676\pi\)
0.534033 + 0.845464i \(0.320676\pi\)
\(312\) 2.54730 0.144212
\(313\) −12.1788 −0.688388 −0.344194 0.938899i \(-0.611848\pi\)
−0.344194 + 0.938899i \(0.611848\pi\)
\(314\) 15.8237 0.892985
\(315\) −2.71982 −0.153245
\(316\) 3.71916 0.209219
\(317\) −29.5674 −1.66067 −0.830334 0.557266i \(-0.811850\pi\)
−0.830334 + 0.557266i \(0.811850\pi\)
\(318\) −0.343449 −0.0192597
\(319\) 0 0
\(320\) 1.00000 0.0559017
\(321\) −2.55189 −0.142433
\(322\) 2.56329 0.142846
\(323\) 11.8334 0.658430
\(324\) 6.55691 0.364273
\(325\) 4.81243 0.266946
\(326\) 10.0410 0.556118
\(327\) −5.04383 −0.278924
\(328\) −5.35973 −0.295941
\(329\) 6.23315 0.343645
\(330\) 0 0
\(331\) −25.7588 −1.41583 −0.707916 0.706297i \(-0.750364\pi\)
−0.707916 + 0.706297i \(0.750364\pi\)
\(332\) −11.5079 −0.631578
\(333\) 16.7127 0.915849
\(334\) 5.62687 0.307888
\(335\) 7.80030 0.426176
\(336\) 0.529317 0.0288766
\(337\) −19.1851 −1.04508 −0.522539 0.852615i \(-0.675015\pi\)
−0.522539 + 0.852615i \(0.675015\pi\)
\(338\) 10.1595 0.552602
\(339\) −1.43875 −0.0781420
\(340\) 6.02760 0.326892
\(341\) 0 0
\(342\) −5.33958 −0.288732
\(343\) 1.00000 0.0539949
\(344\) −5.58127 −0.300922
\(345\) 1.35679 0.0730472
\(346\) 11.3235 0.608755
\(347\) −27.6006 −1.48168 −0.740840 0.671681i \(-0.765572\pi\)
−0.740840 + 0.671681i \(0.765572\pi\)
\(348\) 4.31116 0.231102
\(349\) −2.01175 −0.107687 −0.0538433 0.998549i \(-0.517147\pi\)
−0.0538433 + 0.998549i \(0.517147\pi\)
\(350\) 1.00000 0.0534522
\(351\) −14.5701 −0.777694
\(352\) 0 0
\(353\) −26.8808 −1.43072 −0.715360 0.698756i \(-0.753737\pi\)
−0.715360 + 0.698756i \(0.753737\pi\)
\(354\) −3.62054 −0.192430
\(355\) 8.56040 0.454339
\(356\) −14.2096 −0.753108
\(357\) 3.19051 0.168860
\(358\) 23.7596 1.25573
\(359\) −11.7756 −0.621491 −0.310746 0.950493i \(-0.600579\pi\)
−0.310746 + 0.950493i \(0.600579\pi\)
\(360\) −2.71982 −0.143347
\(361\) −15.1458 −0.797148
\(362\) −1.06147 −0.0557897
\(363\) 0 0
\(364\) 4.81243 0.252240
\(365\) −5.50769 −0.288286
\(366\) 0.887502 0.0463905
\(367\) 1.62917 0.0850419 0.0425210 0.999096i \(-0.486461\pi\)
0.0425210 + 0.999096i \(0.486461\pi\)
\(368\) 2.56329 0.133621
\(369\) 14.5775 0.758875
\(370\) −6.14477 −0.319451
\(371\) −0.648854 −0.0336868
\(372\) −1.19051 −0.0617249
\(373\) −28.4460 −1.47288 −0.736438 0.676505i \(-0.763494\pi\)
−0.736438 + 0.676505i \(0.763494\pi\)
\(374\) 0 0
\(375\) 0.529317 0.0273338
\(376\) 6.23315 0.321450
\(377\) 39.1961 2.01870
\(378\) −3.02760 −0.155723
\(379\) −10.4292 −0.535711 −0.267855 0.963459i \(-0.586315\pi\)
−0.267855 + 0.963459i \(0.586315\pi\)
\(380\) 1.96321 0.100711
\(381\) −1.90763 −0.0977306
\(382\) 16.9651 0.868011
\(383\) 6.50389 0.332333 0.166167 0.986098i \(-0.446861\pi\)
0.166167 + 0.986098i \(0.446861\pi\)
\(384\) 0.529317 0.0270116
\(385\) 0 0
\(386\) 18.4425 0.938700
\(387\) 15.1801 0.771646
\(388\) −3.26750 −0.165882
\(389\) 33.4810 1.69755 0.848776 0.528753i \(-0.177340\pi\)
0.848776 + 0.528753i \(0.177340\pi\)
\(390\) 2.54730 0.128987
\(391\) 15.4505 0.781364
\(392\) 1.00000 0.0505076
\(393\) −2.74080 −0.138255
\(394\) −20.3968 −1.02758
\(395\) 3.71916 0.187131
\(396\) 0 0
\(397\) −24.7013 −1.23972 −0.619860 0.784712i \(-0.712811\pi\)
−0.619860 + 0.784712i \(0.712811\pi\)
\(398\) 6.47192 0.324408
\(399\) 1.03916 0.0520230
\(400\) 1.00000 0.0500000
\(401\) 17.7421 0.885999 0.443000 0.896522i \(-0.353914\pi\)
0.443000 + 0.896522i \(0.353914\pi\)
\(402\) 4.12883 0.205927
\(403\) −10.8238 −0.539173
\(404\) 0.584368 0.0290734
\(405\) 6.55691 0.325816
\(406\) 8.14477 0.404218
\(407\) 0 0
\(408\) 3.19051 0.157954
\(409\) 32.8021 1.62196 0.810981 0.585073i \(-0.198934\pi\)
0.810981 + 0.585073i \(0.198934\pi\)
\(410\) −5.35973 −0.264698
\(411\) 4.12584 0.203513
\(412\) 12.8427 0.632717
\(413\) −6.84003 −0.336576
\(414\) −6.97169 −0.342640
\(415\) −11.5079 −0.564900
\(416\) 4.81243 0.235949
\(417\) 5.15324 0.252355
\(418\) 0 0
\(419\) −27.0785 −1.32287 −0.661437 0.750001i \(-0.730053\pi\)
−0.661437 + 0.750001i \(0.730053\pi\)
\(420\) 0.529317 0.0258280
\(421\) −9.64201 −0.469923 −0.234961 0.972005i \(-0.575496\pi\)
−0.234961 + 0.972005i \(0.575496\pi\)
\(422\) 17.7130 0.862254
\(423\) −16.9531 −0.824287
\(424\) −0.648854 −0.0315111
\(425\) 6.02760 0.292381
\(426\) 4.53116 0.219535
\(427\) 1.67669 0.0811409
\(428\) −4.82110 −0.233037
\(429\) 0 0
\(430\) −5.58127 −0.269153
\(431\) 9.17233 0.441815 0.220908 0.975295i \(-0.429098\pi\)
0.220908 + 0.975295i \(0.429098\pi\)
\(432\) −3.02760 −0.145665
\(433\) −29.1569 −1.40119 −0.700595 0.713559i \(-0.747082\pi\)
−0.700595 + 0.713559i \(0.747082\pi\)
\(434\) −2.24914 −0.107962
\(435\) 4.31116 0.206704
\(436\) −9.52895 −0.456354
\(437\) 5.03227 0.240726
\(438\) −2.91531 −0.139299
\(439\) 3.57840 0.170788 0.0853938 0.996347i \(-0.472785\pi\)
0.0853938 + 0.996347i \(0.472785\pi\)
\(440\) 0 0
\(441\) −2.71982 −0.129515
\(442\) 29.0074 1.37974
\(443\) 4.36344 0.207313 0.103657 0.994613i \(-0.466946\pi\)
0.103657 + 0.994613i \(0.466946\pi\)
\(444\) −3.25253 −0.154358
\(445\) −14.2096 −0.673600
\(446\) 21.6290 1.02416
\(447\) 2.78623 0.131784
\(448\) 1.00000 0.0472456
\(449\) 0.918697 0.0433560 0.0216780 0.999765i \(-0.493099\pi\)
0.0216780 + 0.999765i \(0.493099\pi\)
\(450\) −2.71982 −0.128214
\(451\) 0 0
\(452\) −2.71812 −0.127850
\(453\) −2.01619 −0.0947288
\(454\) −15.3595 −0.720855
\(455\) 4.81243 0.225610
\(456\) 1.03916 0.0486631
\(457\) −1.28030 −0.0598901 −0.0299450 0.999552i \(-0.509533\pi\)
−0.0299450 + 0.999552i \(0.509533\pi\)
\(458\) 21.2484 0.992874
\(459\) −18.2491 −0.851797
\(460\) 2.56329 0.119514
\(461\) −3.42591 −0.159561 −0.0797803 0.996812i \(-0.525422\pi\)
−0.0797803 + 0.996812i \(0.525422\pi\)
\(462\) 0 0
\(463\) −35.9279 −1.66971 −0.834855 0.550469i \(-0.814449\pi\)
−0.834855 + 0.550469i \(0.814449\pi\)
\(464\) 8.14477 0.378111
\(465\) −1.19051 −0.0552085
\(466\) −18.8912 −0.875120
\(467\) −0.633419 −0.0293111 −0.0146556 0.999893i \(-0.504665\pi\)
−0.0146556 + 0.999893i \(0.504665\pi\)
\(468\) −13.0890 −0.605037
\(469\) 7.80030 0.360184
\(470\) 6.23315 0.287514
\(471\) 8.37576 0.385935
\(472\) −6.84003 −0.314838
\(473\) 0 0
\(474\) 1.96861 0.0904214
\(475\) 1.96321 0.0900782
\(476\) 6.02760 0.276274
\(477\) 1.76477 0.0808032
\(478\) 25.4078 1.16213
\(479\) 32.8050 1.49890 0.749450 0.662061i \(-0.230318\pi\)
0.749450 + 0.662061i \(0.230318\pi\)
\(480\) 0.529317 0.0241599
\(481\) −29.5713 −1.34833
\(482\) 10.1147 0.460711
\(483\) 1.35679 0.0617361
\(484\) 0 0
\(485\) −3.26750 −0.148370
\(486\) 12.5535 0.569437
\(487\) −38.3866 −1.73946 −0.869732 0.493525i \(-0.835708\pi\)
−0.869732 + 0.493525i \(0.835708\pi\)
\(488\) 1.67669 0.0759004
\(489\) 5.31486 0.240346
\(490\) 1.00000 0.0451754
\(491\) −37.5492 −1.69457 −0.847285 0.531139i \(-0.821764\pi\)
−0.847285 + 0.531139i \(0.821764\pi\)
\(492\) −2.83699 −0.127902
\(493\) 49.0934 2.21105
\(494\) 9.44781 0.425077
\(495\) 0 0
\(496\) −2.24914 −0.100989
\(497\) 8.56040 0.383986
\(498\) −6.09132 −0.272959
\(499\) 26.5551 1.18877 0.594386 0.804180i \(-0.297395\pi\)
0.594386 + 0.804180i \(0.297395\pi\)
\(500\) 1.00000 0.0447214
\(501\) 2.97839 0.133065
\(502\) −15.5061 −0.692071
\(503\) 19.0642 0.850030 0.425015 0.905186i \(-0.360269\pi\)
0.425015 + 0.905186i \(0.360269\pi\)
\(504\) −2.71982 −0.121151
\(505\) 0.584368 0.0260040
\(506\) 0 0
\(507\) 5.37758 0.238827
\(508\) −3.60394 −0.159899
\(509\) 15.8369 0.701960 0.350980 0.936383i \(-0.385849\pi\)
0.350980 + 0.936383i \(0.385849\pi\)
\(510\) 3.19051 0.141278
\(511\) −5.50769 −0.243646
\(512\) 1.00000 0.0441942
\(513\) −5.94381 −0.262426
\(514\) −18.6645 −0.823257
\(515\) 12.8427 0.565919
\(516\) −2.95426 −0.130054
\(517\) 0 0
\(518\) −6.14477 −0.269986
\(519\) 5.99371 0.263095
\(520\) 4.81243 0.211039
\(521\) −38.6470 −1.69316 −0.846579 0.532263i \(-0.821342\pi\)
−0.846579 + 0.532263i \(0.821342\pi\)
\(522\) −22.1523 −0.969581
\(523\) −24.0284 −1.05069 −0.525344 0.850890i \(-0.676063\pi\)
−0.525344 + 0.850890i \(0.676063\pi\)
\(524\) −5.17800 −0.226202
\(525\) 0.529317 0.0231013
\(526\) −2.35968 −0.102887
\(527\) −13.5569 −0.590548
\(528\) 0 0
\(529\) −16.4296 −0.714328
\(530\) −0.648854 −0.0281844
\(531\) 18.6037 0.807330
\(532\) 1.96321 0.0851159
\(533\) −25.7933 −1.11723
\(534\) −7.52138 −0.325482
\(535\) −4.82110 −0.208434
\(536\) 7.80030 0.336922
\(537\) 12.5763 0.542709
\(538\) −14.5570 −0.627597
\(539\) 0 0
\(540\) −3.02760 −0.130287
\(541\) −18.9007 −0.812604 −0.406302 0.913739i \(-0.633182\pi\)
−0.406302 + 0.913739i \(0.633182\pi\)
\(542\) −10.4172 −0.447456
\(543\) −0.561855 −0.0241115
\(544\) 6.02760 0.258431
\(545\) −9.52895 −0.408175
\(546\) 2.54730 0.109014
\(547\) 0.760577 0.0325199 0.0162600 0.999868i \(-0.494824\pi\)
0.0162600 + 0.999868i \(0.494824\pi\)
\(548\) 7.79465 0.332971
\(549\) −4.56031 −0.194629
\(550\) 0 0
\(551\) 15.9899 0.681192
\(552\) 1.35679 0.0577489
\(553\) 3.71916 0.158155
\(554\) 14.9701 0.636018
\(555\) −3.25253 −0.138062
\(556\) 9.73564 0.412883
\(557\) 6.85003 0.290245 0.145123 0.989414i \(-0.453642\pi\)
0.145123 + 0.989414i \(0.453642\pi\)
\(558\) 6.11727 0.258965
\(559\) −26.8595 −1.13603
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) 8.00750 0.337776
\(563\) −37.3212 −1.57290 −0.786451 0.617652i \(-0.788084\pi\)
−0.786451 + 0.617652i \(0.788084\pi\)
\(564\) 3.29931 0.138926
\(565\) −2.71812 −0.114352
\(566\) 22.3793 0.940670
\(567\) 6.55691 0.275365
\(568\) 8.56040 0.359186
\(569\) −46.3025 −1.94110 −0.970551 0.240897i \(-0.922559\pi\)
−0.970551 + 0.240897i \(0.922559\pi\)
\(570\) 1.03916 0.0435256
\(571\) −23.7033 −0.991951 −0.495975 0.868337i \(-0.665189\pi\)
−0.495975 + 0.868337i \(0.665189\pi\)
\(572\) 0 0
\(573\) 8.97991 0.375141
\(574\) −5.35973 −0.223711
\(575\) 2.56329 0.106897
\(576\) −2.71982 −0.113326
\(577\) 30.3334 1.26280 0.631399 0.775458i \(-0.282481\pi\)
0.631399 + 0.775458i \(0.282481\pi\)
\(578\) 19.3319 0.804102
\(579\) 9.76193 0.405692
\(580\) 8.14477 0.338193
\(581\) −11.5079 −0.477428
\(582\) −1.72954 −0.0716918
\(583\) 0 0
\(584\) −5.50769 −0.227910
\(585\) −13.0890 −0.541162
\(586\) 2.78343 0.114983
\(587\) 28.8863 1.19227 0.596133 0.802886i \(-0.296703\pi\)
0.596133 + 0.802886i \(0.296703\pi\)
\(588\) 0.529317 0.0218286
\(589\) −4.41553 −0.181939
\(590\) −6.84003 −0.281599
\(591\) −10.7964 −0.444104
\(592\) −6.14477 −0.252548
\(593\) 34.0215 1.39710 0.698548 0.715563i \(-0.253830\pi\)
0.698548 + 0.715563i \(0.253830\pi\)
\(594\) 0 0
\(595\) 6.02760 0.247107
\(596\) 5.26382 0.215614
\(597\) 3.42569 0.140204
\(598\) 12.3356 0.504442
\(599\) −12.8416 −0.524694 −0.262347 0.964974i \(-0.584496\pi\)
−0.262347 + 0.964974i \(0.584496\pi\)
\(600\) 0.529317 0.0216093
\(601\) −22.9731 −0.937094 −0.468547 0.883439i \(-0.655222\pi\)
−0.468547 + 0.883439i \(0.655222\pi\)
\(602\) −5.58127 −0.227476
\(603\) −21.2154 −0.863960
\(604\) −3.80904 −0.154988
\(605\) 0 0
\(606\) 0.309316 0.0125651
\(607\) −1.84682 −0.0749600 −0.0374800 0.999297i \(-0.511933\pi\)
−0.0374800 + 0.999297i \(0.511933\pi\)
\(608\) 1.96321 0.0796187
\(609\) 4.31116 0.174697
\(610\) 1.67669 0.0678873
\(611\) 29.9966 1.21353
\(612\) −16.3940 −0.662688
\(613\) 22.4753 0.907769 0.453884 0.891061i \(-0.350038\pi\)
0.453884 + 0.891061i \(0.350038\pi\)
\(614\) −24.7810 −1.00008
\(615\) −2.83699 −0.114399
\(616\) 0 0
\(617\) −12.3558 −0.497425 −0.248713 0.968577i \(-0.580007\pi\)
−0.248713 + 0.968577i \(0.580007\pi\)
\(618\) 6.79788 0.273451
\(619\) −44.0581 −1.77085 −0.885423 0.464786i \(-0.846131\pi\)
−0.885423 + 0.464786i \(0.846131\pi\)
\(620\) −2.24914 −0.0903277
\(621\) −7.76061 −0.311422
\(622\) 18.8355 0.755236
\(623\) −14.2096 −0.569296
\(624\) 2.54730 0.101974
\(625\) 1.00000 0.0400000
\(626\) −12.1788 −0.486763
\(627\) 0 0
\(628\) 15.8237 0.631436
\(629\) −37.0382 −1.47681
\(630\) −2.71982 −0.108360
\(631\) −13.8538 −0.551510 −0.275755 0.961228i \(-0.588928\pi\)
−0.275755 + 0.961228i \(0.588928\pi\)
\(632\) 3.71916 0.147940
\(633\) 9.37577 0.372653
\(634\) −29.5674 −1.17427
\(635\) −3.60394 −0.143018
\(636\) −0.343449 −0.0136186
\(637\) 4.81243 0.190675
\(638\) 0 0
\(639\) −23.2828 −0.921052
\(640\) 1.00000 0.0395285
\(641\) 22.3155 0.881409 0.440704 0.897652i \(-0.354729\pi\)
0.440704 + 0.897652i \(0.354729\pi\)
\(642\) −2.55189 −0.100715
\(643\) 43.6797 1.72256 0.861280 0.508130i \(-0.169663\pi\)
0.861280 + 0.508130i \(0.169663\pi\)
\(644\) 2.56329 0.101008
\(645\) −2.95426 −0.116324
\(646\) 11.8334 0.465580
\(647\) 35.7267 1.40456 0.702280 0.711901i \(-0.252166\pi\)
0.702280 + 0.711901i \(0.252166\pi\)
\(648\) 6.55691 0.257580
\(649\) 0 0
\(650\) 4.81243 0.188759
\(651\) −1.19051 −0.0466597
\(652\) 10.0410 0.393235
\(653\) 16.0177 0.626822 0.313411 0.949618i \(-0.398528\pi\)
0.313411 + 0.949618i \(0.398528\pi\)
\(654\) −5.04383 −0.197229
\(655\) −5.17800 −0.202321
\(656\) −5.35973 −0.209262
\(657\) 14.9799 0.584423
\(658\) 6.23315 0.242994
\(659\) 32.8329 1.27899 0.639494 0.768796i \(-0.279144\pi\)
0.639494 + 0.768796i \(0.279144\pi\)
\(660\) 0 0
\(661\) 4.83580 0.188091 0.0940454 0.995568i \(-0.470020\pi\)
0.0940454 + 0.995568i \(0.470020\pi\)
\(662\) −25.7588 −1.00114
\(663\) 15.3541 0.596303
\(664\) −11.5079 −0.446593
\(665\) 1.96321 0.0761300
\(666\) 16.7127 0.647603
\(667\) 20.8774 0.808376
\(668\) 5.62687 0.217710
\(669\) 11.4486 0.442628
\(670\) 7.80030 0.301352
\(671\) 0 0
\(672\) 0.529317 0.0204188
\(673\) −47.4627 −1.82955 −0.914777 0.403959i \(-0.867634\pi\)
−0.914777 + 0.403959i \(0.867634\pi\)
\(674\) −19.1851 −0.738982
\(675\) −3.02760 −0.116532
\(676\) 10.1595 0.390749
\(677\) 7.70258 0.296034 0.148017 0.988985i \(-0.452711\pi\)
0.148017 + 0.988985i \(0.452711\pi\)
\(678\) −1.43875 −0.0552547
\(679\) −3.26750 −0.125395
\(680\) 6.02760 0.231148
\(681\) −8.13001 −0.311543
\(682\) 0 0
\(683\) −10.0811 −0.385742 −0.192871 0.981224i \(-0.561780\pi\)
−0.192871 + 0.981224i \(0.561780\pi\)
\(684\) −5.33958 −0.204164
\(685\) 7.79465 0.297818
\(686\) 1.00000 0.0381802
\(687\) 11.2471 0.429105
\(688\) −5.58127 −0.212784
\(689\) −3.12256 −0.118960
\(690\) 1.35679 0.0516522
\(691\) 1.47391 0.0560703 0.0280351 0.999607i \(-0.491075\pi\)
0.0280351 + 0.999607i \(0.491075\pi\)
\(692\) 11.3235 0.430455
\(693\) 0 0
\(694\) −27.6006 −1.04771
\(695\) 9.73564 0.369294
\(696\) 4.31116 0.163414
\(697\) −32.3063 −1.22369
\(698\) −2.01175 −0.0761459
\(699\) −9.99945 −0.378214
\(700\) 1.00000 0.0377964
\(701\) 22.3369 0.843653 0.421826 0.906677i \(-0.361389\pi\)
0.421826 + 0.906677i \(0.361389\pi\)
\(702\) −14.5701 −0.549913
\(703\) −12.0635 −0.454982
\(704\) 0 0
\(705\) 3.29931 0.124259
\(706\) −26.8808 −1.01167
\(707\) 0.584368 0.0219774
\(708\) −3.62054 −0.136068
\(709\) −1.82779 −0.0686440 −0.0343220 0.999411i \(-0.510927\pi\)
−0.0343220 + 0.999411i \(0.510927\pi\)
\(710\) 8.56040 0.321266
\(711\) −10.1155 −0.379359
\(712\) −14.2096 −0.532528
\(713\) −5.76520 −0.215908
\(714\) 3.19051 0.119402
\(715\) 0 0
\(716\) 23.7596 0.887936
\(717\) 13.4488 0.502253
\(718\) −11.7756 −0.439461
\(719\) 6.70006 0.249870 0.124935 0.992165i \(-0.460128\pi\)
0.124935 + 0.992165i \(0.460128\pi\)
\(720\) −2.71982 −0.101362
\(721\) 12.8427 0.478289
\(722\) −15.1458 −0.563669
\(723\) 5.35387 0.199112
\(724\) −1.06147 −0.0394493
\(725\) 8.14477 0.302489
\(726\) 0 0
\(727\) −14.5561 −0.539855 −0.269928 0.962881i \(-0.587000\pi\)
−0.269928 + 0.962881i \(0.587000\pi\)
\(728\) 4.81243 0.178360
\(729\) −13.0260 −0.482444
\(730\) −5.50769 −0.203849
\(731\) −33.6417 −1.24428
\(732\) 0.887502 0.0328030
\(733\) −39.6761 −1.46547 −0.732736 0.680513i \(-0.761757\pi\)
−0.732736 + 0.680513i \(0.761757\pi\)
\(734\) 1.62917 0.0601337
\(735\) 0.529317 0.0195241
\(736\) 2.56329 0.0944841
\(737\) 0 0
\(738\) 14.5775 0.536606
\(739\) 38.6096 1.42028 0.710139 0.704061i \(-0.248632\pi\)
0.710139 + 0.704061i \(0.248632\pi\)
\(740\) −6.14477 −0.225886
\(741\) 5.00088 0.183712
\(742\) −0.648854 −0.0238202
\(743\) −18.3540 −0.673342 −0.336671 0.941622i \(-0.609301\pi\)
−0.336671 + 0.941622i \(0.609301\pi\)
\(744\) −1.19051 −0.0436461
\(745\) 5.26382 0.192851
\(746\) −28.4460 −1.04148
\(747\) 31.2995 1.14519
\(748\) 0 0
\(749\) −4.82110 −0.176159
\(750\) 0.529317 0.0193279
\(751\) −37.3350 −1.36238 −0.681188 0.732109i \(-0.738536\pi\)
−0.681188 + 0.732109i \(0.738536\pi\)
\(752\) 6.23315 0.227300
\(753\) −8.20763 −0.299103
\(754\) 39.1961 1.42744
\(755\) −3.80904 −0.138625
\(756\) −3.02760 −0.110113
\(757\) 11.2902 0.410350 0.205175 0.978725i \(-0.434224\pi\)
0.205175 + 0.978725i \(0.434224\pi\)
\(758\) −10.4292 −0.378805
\(759\) 0 0
\(760\) 1.96321 0.0712131
\(761\) −52.6886 −1.90996 −0.954980 0.296669i \(-0.904124\pi\)
−0.954980 + 0.296669i \(0.904124\pi\)
\(762\) −1.90763 −0.0691060
\(763\) −9.52895 −0.344971
\(764\) 16.9651 0.613776
\(765\) −16.3940 −0.592726
\(766\) 6.50389 0.234995
\(767\) −32.9171 −1.18857
\(768\) 0.529317 0.0191001
\(769\) 2.47046 0.0890870 0.0445435 0.999007i \(-0.485817\pi\)
0.0445435 + 0.999007i \(0.485817\pi\)
\(770\) 0 0
\(771\) −9.87944 −0.355799
\(772\) 18.4425 0.663761
\(773\) 34.3463 1.23535 0.617676 0.786433i \(-0.288074\pi\)
0.617676 + 0.786433i \(0.288074\pi\)
\(774\) 15.1801 0.545636
\(775\) −2.24914 −0.0807915
\(776\) −3.26750 −0.117296
\(777\) −3.25253 −0.116684
\(778\) 33.4810 1.20035
\(779\) −10.5223 −0.376999
\(780\) 2.54730 0.0912079
\(781\) 0 0
\(782\) 15.4505 0.552508
\(783\) −24.6591 −0.881243
\(784\) 1.00000 0.0357143
\(785\) 15.8237 0.564773
\(786\) −2.74080 −0.0977612
\(787\) 41.2984 1.47213 0.736064 0.676912i \(-0.236682\pi\)
0.736064 + 0.676912i \(0.236682\pi\)
\(788\) −20.3968 −0.726607
\(789\) −1.24902 −0.0444662
\(790\) 3.71916 0.132322
\(791\) −2.71812 −0.0966452
\(792\) 0 0
\(793\) 8.06897 0.286538
\(794\) −24.7013 −0.876615
\(795\) −0.343449 −0.0121809
\(796\) 6.47192 0.229391
\(797\) 12.0500 0.426832 0.213416 0.976961i \(-0.431541\pi\)
0.213416 + 0.976961i \(0.431541\pi\)
\(798\) 1.03916 0.0367858
\(799\) 37.5709 1.32916
\(800\) 1.00000 0.0353553
\(801\) 38.6476 1.36555
\(802\) 17.7421 0.626496
\(803\) 0 0
\(804\) 4.12883 0.145613
\(805\) 2.56329 0.0903440
\(806\) −10.8238 −0.381253
\(807\) −7.70526 −0.271238
\(808\) 0.584368 0.0205580
\(809\) −2.86537 −0.100741 −0.0503705 0.998731i \(-0.516040\pi\)
−0.0503705 + 0.998731i \(0.516040\pi\)
\(810\) 6.55691 0.230386
\(811\) −47.4986 −1.66790 −0.833952 0.551838i \(-0.813927\pi\)
−0.833952 + 0.551838i \(0.813927\pi\)
\(812\) 8.14477 0.285825
\(813\) −5.51398 −0.193384
\(814\) 0 0
\(815\) 10.0410 0.351720
\(816\) 3.19051 0.111690
\(817\) −10.9572 −0.383344
\(818\) 32.8021 1.14690
\(819\) −13.0890 −0.457365
\(820\) −5.35973 −0.187170
\(821\) 20.5458 0.717052 0.358526 0.933520i \(-0.383279\pi\)
0.358526 + 0.933520i \(0.383279\pi\)
\(822\) 4.12584 0.143905
\(823\) 24.5540 0.855897 0.427949 0.903803i \(-0.359236\pi\)
0.427949 + 0.903803i \(0.359236\pi\)
\(824\) 12.8427 0.447398
\(825\) 0 0
\(826\) −6.84003 −0.237995
\(827\) 54.8582 1.90761 0.953803 0.300434i \(-0.0971315\pi\)
0.953803 + 0.300434i \(0.0971315\pi\)
\(828\) −6.97169 −0.242283
\(829\) 4.58811 0.159351 0.0796757 0.996821i \(-0.474612\pi\)
0.0796757 + 0.996821i \(0.474612\pi\)
\(830\) −11.5079 −0.399445
\(831\) 7.92392 0.274878
\(832\) 4.81243 0.166841
\(833\) 6.02760 0.208844
\(834\) 5.15324 0.178442
\(835\) 5.62687 0.194726
\(836\) 0 0
\(837\) 6.80949 0.235370
\(838\) −27.0785 −0.935413
\(839\) 4.66116 0.160921 0.0804605 0.996758i \(-0.474361\pi\)
0.0804605 + 0.996758i \(0.474361\pi\)
\(840\) 0.529317 0.0182632
\(841\) 37.3372 1.28749
\(842\) −9.64201 −0.332285
\(843\) 4.23851 0.145982
\(844\) 17.7130 0.609706
\(845\) 10.1595 0.349496
\(846\) −16.9531 −0.582859
\(847\) 0 0
\(848\) −0.648854 −0.0222817
\(849\) 11.8457 0.406544
\(850\) 6.02760 0.206745
\(851\) −15.7508 −0.539931
\(852\) 4.53116 0.155235
\(853\) −9.02516 −0.309016 −0.154508 0.987992i \(-0.549379\pi\)
−0.154508 + 0.987992i \(0.549379\pi\)
\(854\) 1.67669 0.0573753
\(855\) −5.33958 −0.182610
\(856\) −4.82110 −0.164782
\(857\) −19.8659 −0.678605 −0.339303 0.940677i \(-0.610191\pi\)
−0.339303 + 0.940677i \(0.610191\pi\)
\(858\) 0 0
\(859\) −4.98915 −0.170228 −0.0851138 0.996371i \(-0.527125\pi\)
−0.0851138 + 0.996371i \(0.527125\pi\)
\(860\) −5.58127 −0.190320
\(861\) −2.83699 −0.0966844
\(862\) 9.17233 0.312411
\(863\) 35.2641 1.20040 0.600202 0.799849i \(-0.295087\pi\)
0.600202 + 0.799849i \(0.295087\pi\)
\(864\) −3.02760 −0.103001
\(865\) 11.3235 0.385010
\(866\) −29.1569 −0.990791
\(867\) 10.2327 0.347521
\(868\) −2.24914 −0.0763408
\(869\) 0 0
\(870\) 4.31116 0.146162
\(871\) 37.5384 1.27194
\(872\) −9.52895 −0.322691
\(873\) 8.88702 0.300780
\(874\) 5.03227 0.170219
\(875\) 1.00000 0.0338062
\(876\) −2.91531 −0.0984992
\(877\) 18.2415 0.615971 0.307985 0.951391i \(-0.400345\pi\)
0.307985 + 0.951391i \(0.400345\pi\)
\(878\) 3.57840 0.120765
\(879\) 1.47332 0.0496938
\(880\) 0 0
\(881\) 26.1005 0.879349 0.439675 0.898157i \(-0.355094\pi\)
0.439675 + 0.898157i \(0.355094\pi\)
\(882\) −2.71982 −0.0915812
\(883\) 11.8714 0.399506 0.199753 0.979846i \(-0.435986\pi\)
0.199753 + 0.979846i \(0.435986\pi\)
\(884\) 29.0074 0.975624
\(885\) −3.62054 −0.121703
\(886\) 4.36344 0.146593
\(887\) 41.1812 1.38273 0.691364 0.722506i \(-0.257010\pi\)
0.691364 + 0.722506i \(0.257010\pi\)
\(888\) −3.25253 −0.109148
\(889\) −3.60394 −0.120872
\(890\) −14.2096 −0.476307
\(891\) 0 0
\(892\) 21.6290 0.724193
\(893\) 12.2370 0.409495
\(894\) 2.78623 0.0931853
\(895\) 23.7596 0.794194
\(896\) 1.00000 0.0334077
\(897\) 6.52946 0.218012
\(898\) 0.918697 0.0306573
\(899\) −18.3187 −0.610964
\(900\) −2.71982 −0.0906608
\(901\) −3.91103 −0.130295
\(902\) 0 0
\(903\) −2.95426 −0.0983116
\(904\) −2.71812 −0.0904033
\(905\) −1.06147 −0.0352845
\(906\) −2.01619 −0.0669834
\(907\) −35.7007 −1.18542 −0.592711 0.805415i \(-0.701942\pi\)
−0.592711 + 0.805415i \(0.701942\pi\)
\(908\) −15.3595 −0.509721
\(909\) −1.58938 −0.0527164
\(910\) 4.81243 0.159530
\(911\) −30.1957 −1.00043 −0.500214 0.865902i \(-0.666745\pi\)
−0.500214 + 0.865902i \(0.666745\pi\)
\(912\) 1.03916 0.0344100
\(913\) 0 0
\(914\) −1.28030 −0.0423487
\(915\) 0.887502 0.0293399
\(916\) 21.2484 0.702068
\(917\) −5.17800 −0.170993
\(918\) −18.2491 −0.602311
\(919\) −46.5051 −1.53406 −0.767030 0.641611i \(-0.778266\pi\)
−0.767030 + 0.641611i \(0.778266\pi\)
\(920\) 2.56329 0.0845091
\(921\) −13.1170 −0.432220
\(922\) −3.42591 −0.112826
\(923\) 41.1963 1.35599
\(924\) 0 0
\(925\) −6.14477 −0.202039
\(926\) −35.9279 −1.18066
\(927\) −34.9300 −1.14725
\(928\) 8.14477 0.267365
\(929\) 52.8390 1.73359 0.866796 0.498664i \(-0.166176\pi\)
0.866796 + 0.498664i \(0.166176\pi\)
\(930\) −1.19051 −0.0390383
\(931\) 1.96321 0.0643416
\(932\) −18.8912 −0.618803
\(933\) 9.96996 0.326402
\(934\) −0.633419 −0.0207261
\(935\) 0 0
\(936\) −13.0890 −0.427826
\(937\) −10.7968 −0.352717 −0.176358 0.984326i \(-0.556432\pi\)
−0.176358 + 0.984326i \(0.556432\pi\)
\(938\) 7.80030 0.254689
\(939\) −6.44645 −0.210372
\(940\) 6.23315 0.203303
\(941\) −15.5858 −0.508082 −0.254041 0.967194i \(-0.581760\pi\)
−0.254041 + 0.967194i \(0.581760\pi\)
\(942\) 8.37576 0.272897
\(943\) −13.7385 −0.447388
\(944\) −6.84003 −0.222624
\(945\) −3.02760 −0.0984878
\(946\) 0 0
\(947\) 4.99210 0.162221 0.0811107 0.996705i \(-0.474153\pi\)
0.0811107 + 0.996705i \(0.474153\pi\)
\(948\) 1.96861 0.0639376
\(949\) −26.5054 −0.860400
\(950\) 1.96321 0.0636949
\(951\) −15.6505 −0.507502
\(952\) 6.02760 0.195356
\(953\) 19.6682 0.637117 0.318558 0.947903i \(-0.396801\pi\)
0.318558 + 0.947903i \(0.396801\pi\)
\(954\) 1.76477 0.0571365
\(955\) 16.9651 0.548978
\(956\) 25.4078 0.821747
\(957\) 0 0
\(958\) 32.8050 1.05988
\(959\) 7.79465 0.251703
\(960\) 0.529317 0.0170836
\(961\) −25.9414 −0.836818
\(962\) −29.5713 −0.953415
\(963\) 13.1125 0.422546
\(964\) 10.1147 0.325772
\(965\) 18.4425 0.593686
\(966\) 1.35679 0.0436540
\(967\) −44.8169 −1.44122 −0.720608 0.693343i \(-0.756137\pi\)
−0.720608 + 0.693343i \(0.756137\pi\)
\(968\) 0 0
\(969\) 6.26363 0.201217
\(970\) −3.26750 −0.104913
\(971\) 38.2665 1.22803 0.614015 0.789294i \(-0.289553\pi\)
0.614015 + 0.789294i \(0.289553\pi\)
\(972\) 12.5535 0.402653
\(973\) 9.73564 0.312110
\(974\) −38.3866 −1.22999
\(975\) 2.54730 0.0815788
\(976\) 1.67669 0.0536697
\(977\) 45.4470 1.45398 0.726989 0.686649i \(-0.240919\pi\)
0.726989 + 0.686649i \(0.240919\pi\)
\(978\) 5.31486 0.169950
\(979\) 0 0
\(980\) 1.00000 0.0319438
\(981\) 25.9171 0.827468
\(982\) −37.5492 −1.19824
\(983\) 3.97367 0.126740 0.0633702 0.997990i \(-0.479815\pi\)
0.0633702 + 0.997990i \(0.479815\pi\)
\(984\) −2.83699 −0.0904400
\(985\) −20.3968 −0.649897
\(986\) 49.0934 1.56345
\(987\) 3.29931 0.105018
\(988\) 9.44781 0.300575
\(989\) −14.3064 −0.454917
\(990\) 0 0
\(991\) −28.4012 −0.902193 −0.451097 0.892475i \(-0.648967\pi\)
−0.451097 + 0.892475i \(0.648967\pi\)
\(992\) −2.24914 −0.0714103
\(993\) −13.6346 −0.432680
\(994\) 8.56040 0.271519
\(995\) 6.47192 0.205174
\(996\) −6.09132 −0.193011
\(997\) −34.4767 −1.09189 −0.545944 0.837821i \(-0.683829\pi\)
−0.545944 + 0.837821i \(0.683829\pi\)
\(998\) 26.5551 0.840588
\(999\) 18.6039 0.588601
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.df.1.4 yes 6
11.10 odd 2 8470.2.a.cz.1.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8470.2.a.cz.1.3 6 11.10 odd 2
8470.2.a.df.1.4 yes 6 1.1 even 1 trivial