Properties

Label 8470.2.a.dg.1.5
Level $8470$
Weight $2$
Character 8470.1
Self dual yes
Analytic conductor $67.633$
Analytic rank $1$
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: \(1\)
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 1.5
Root \(0.365778\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +0.365778 q^{3} +1.00000 q^{4} -1.00000 q^{5} -0.365778 q^{6} +1.00000 q^{7} -1.00000 q^{8} -2.86621 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +0.365778 q^{3} +1.00000 q^{4} -1.00000 q^{5} -0.365778 q^{6} +1.00000 q^{7} -1.00000 q^{8} -2.86621 q^{9} +1.00000 q^{10} +0.365778 q^{12} +2.20049 q^{13} -1.00000 q^{14} -0.365778 q^{15} +1.00000 q^{16} -6.26330 q^{17} +2.86621 q^{18} +6.00967 q^{19} -1.00000 q^{20} +0.365778 q^{21} +0.452126 q^{23} -0.365778 q^{24} +1.00000 q^{25} -2.20049 q^{26} -2.14573 q^{27} +1.00000 q^{28} +1.90770 q^{29} +0.365778 q^{30} -6.05116 q^{31} -1.00000 q^{32} +6.26330 q^{34} -1.00000 q^{35} -2.86621 q^{36} +3.54127 q^{37} -6.00967 q^{38} +0.804890 q^{39} +1.00000 q^{40} +0.0206760 q^{41} -0.365778 q^{42} -3.71582 q^{43} +2.86621 q^{45} -0.452126 q^{46} +8.63329 q^{47} +0.365778 q^{48} +1.00000 q^{49} -1.00000 q^{50} -2.29098 q^{51} +2.20049 q^{52} +8.03517 q^{53} +2.14573 q^{54} -1.00000 q^{56} +2.19820 q^{57} -1.90770 q^{58} -12.7279 q^{59} -0.365778 q^{60} -6.02013 q^{61} +6.05116 q^{62} -2.86621 q^{63} +1.00000 q^{64} -2.20049 q^{65} +8.19469 q^{67} -6.26330 q^{68} +0.165378 q^{69} +1.00000 q^{70} +15.7605 q^{71} +2.86621 q^{72} -12.7842 q^{73} -3.54127 q^{74} +0.365778 q^{75} +6.00967 q^{76} -0.804890 q^{78} -10.7445 q^{79} -1.00000 q^{80} +7.81376 q^{81} -0.0206760 q^{82} -10.9679 q^{83} +0.365778 q^{84} +6.26330 q^{85} +3.71582 q^{86} +0.697795 q^{87} -9.56469 q^{89} -2.86621 q^{90} +2.20049 q^{91} +0.452126 q^{92} -2.21338 q^{93} -8.63329 q^{94} -6.00967 q^{95} -0.365778 q^{96} -4.22569 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{2} + 8q^{4} - 8q^{5} + 8q^{7} - 8q^{8} + 8q^{9} + O(q^{10}) \) \( 8q - 8q^{2} + 8q^{4} - 8q^{5} + 8q^{7} - 8q^{8} + 8q^{9} + 8q^{10} + q^{13} - 8q^{14} + 8q^{16} - 6q^{17} - 8q^{18} - 5q^{19} - 8q^{20} + 10q^{23} + 8q^{25} - q^{26} + 8q^{28} - 3q^{29} - 8q^{31} - 8q^{32} + 6q^{34} - 8q^{35} + 8q^{36} - 6q^{37} + 5q^{38} - 35q^{39} + 8q^{40} - 11q^{41} + 5q^{43} - 8q^{45} - 10q^{46} - 15q^{47} + 8q^{49} - 8q^{50} + 6q^{51} + q^{52} - 16q^{53} - 8q^{56} - 38q^{57} + 3q^{58} - 9q^{59} - 32q^{61} + 8q^{62} + 8q^{63} + 8q^{64} - q^{65} + 33q^{67} - 6q^{68} - 22q^{69} + 8q^{70} + 11q^{71} - 8q^{72} + 34q^{73} + 6q^{74} - 5q^{76} + 35q^{78} - 31q^{79} - 8q^{80} + 20q^{81} + 11q^{82} - 50q^{83} + 6q^{85} - 5q^{86} + 12q^{87} + q^{89} + 8q^{90} + q^{91} + 10q^{92} + 26q^{93} + 15q^{94} + 5q^{95} - 4q^{97} - 8q^{98} + O(q^{100}) \)

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
\(3\) 0.365778 0.211182 0.105591 0.994410i \(-0.466327\pi\)
0.105591 + 0.994410i \(0.466327\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) −0.365778 −0.149328
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) −2.86621 −0.955402
\(10\) 1.00000 0.316228
\(11\) 0 0
\(12\) 0.365778 0.105591
\(13\) 2.20049 0.610306 0.305153 0.952303i \(-0.401292\pi\)
0.305153 + 0.952303i \(0.401292\pi\)
\(14\) −1.00000 −0.267261
\(15\) −0.365778 −0.0944434
\(16\) 1.00000 0.250000
\(17\) −6.26330 −1.51907 −0.759537 0.650464i \(-0.774575\pi\)
−0.759537 + 0.650464i \(0.774575\pi\)
\(18\) 2.86621 0.675571
\(19\) 6.00967 1.37871 0.689356 0.724422i \(-0.257893\pi\)
0.689356 + 0.724422i \(0.257893\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0.365778 0.0798193
\(22\) 0 0
\(23\) 0.452126 0.0942748 0.0471374 0.998888i \(-0.484990\pi\)
0.0471374 + 0.998888i \(0.484990\pi\)
\(24\) −0.365778 −0.0746641
\(25\) 1.00000 0.200000
\(26\) −2.20049 −0.431551
\(27\) −2.14573 −0.412946
\(28\) 1.00000 0.188982
\(29\) 1.90770 0.354251 0.177126 0.984188i \(-0.443320\pi\)
0.177126 + 0.984188i \(0.443320\pi\)
\(30\) 0.365778 0.0667816
\(31\) −6.05116 −1.08682 −0.543411 0.839467i \(-0.682867\pi\)
−0.543411 + 0.839467i \(0.682867\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 6.26330 1.07415
\(35\) −1.00000 −0.169031
\(36\) −2.86621 −0.477701
\(37\) 3.54127 0.582181 0.291090 0.956696i \(-0.405982\pi\)
0.291090 + 0.956696i \(0.405982\pi\)
\(38\) −6.00967 −0.974897
\(39\) 0.804890 0.128886
\(40\) 1.00000 0.158114
\(41\) 0.0206760 0.00322905 0.00161452 0.999999i \(-0.499486\pi\)
0.00161452 + 0.999999i \(0.499486\pi\)
\(42\) −0.365778 −0.0564407
\(43\) −3.71582 −0.566657 −0.283329 0.959023i \(-0.591439\pi\)
−0.283329 + 0.959023i \(0.591439\pi\)
\(44\) 0 0
\(45\) 2.86621 0.427269
\(46\) −0.452126 −0.0666624
\(47\) 8.63329 1.25929 0.629647 0.776881i \(-0.283199\pi\)
0.629647 + 0.776881i \(0.283199\pi\)
\(48\) 0.365778 0.0527955
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) −2.29098 −0.320801
\(52\) 2.20049 0.305153
\(53\) 8.03517 1.10372 0.551858 0.833938i \(-0.313919\pi\)
0.551858 + 0.833938i \(0.313919\pi\)
\(54\) 2.14573 0.291997
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) 2.19820 0.291159
\(58\) −1.90770 −0.250493
\(59\) −12.7279 −1.65703 −0.828516 0.559965i \(-0.810814\pi\)
−0.828516 + 0.559965i \(0.810814\pi\)
\(60\) −0.365778 −0.0472217
\(61\) −6.02013 −0.770798 −0.385399 0.922750i \(-0.625936\pi\)
−0.385399 + 0.922750i \(0.625936\pi\)
\(62\) 6.05116 0.768499
\(63\) −2.86621 −0.361108
\(64\) 1.00000 0.125000
\(65\) −2.20049 −0.272937
\(66\) 0 0
\(67\) 8.19469 1.00114 0.500570 0.865696i \(-0.333124\pi\)
0.500570 + 0.865696i \(0.333124\pi\)
\(68\) −6.26330 −0.759537
\(69\) 0.165378 0.0199091
\(70\) 1.00000 0.119523
\(71\) 15.7605 1.87043 0.935216 0.354077i \(-0.115205\pi\)
0.935216 + 0.354077i \(0.115205\pi\)
\(72\) 2.86621 0.337786
\(73\) −12.7842 −1.49628 −0.748139 0.663542i \(-0.769053\pi\)
−0.748139 + 0.663542i \(0.769053\pi\)
\(74\) −3.54127 −0.411664
\(75\) 0.365778 0.0422364
\(76\) 6.00967 0.689356
\(77\) 0 0
\(78\) −0.804890 −0.0911358
\(79\) −10.7445 −1.20885 −0.604425 0.796662i \(-0.706597\pi\)
−0.604425 + 0.796662i \(0.706597\pi\)
\(80\) −1.00000 −0.111803
\(81\) 7.81376 0.868196
\(82\) −0.0206760 −0.00228328
\(83\) −10.9679 −1.20388 −0.601942 0.798540i \(-0.705606\pi\)
−0.601942 + 0.798540i \(0.705606\pi\)
\(84\) 0.365778 0.0399096
\(85\) 6.26330 0.679351
\(86\) 3.71582 0.400687
\(87\) 0.697795 0.0748115
\(88\) 0 0
\(89\) −9.56469 −1.01385 −0.506927 0.861989i \(-0.669219\pi\)
−0.506927 + 0.861989i \(0.669219\pi\)
\(90\) −2.86621 −0.302125
\(91\) 2.20049 0.230674
\(92\) 0.452126 0.0471374
\(93\) −2.21338 −0.229517
\(94\) −8.63329 −0.890455
\(95\) −6.00967 −0.616579
\(96\) −0.365778 −0.0373320
\(97\) −4.22569 −0.429053 −0.214527 0.976718i \(-0.568821\pi\)
−0.214527 + 0.976718i \(0.568821\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 10.0515 1.00016 0.500080 0.865979i \(-0.333304\pi\)
0.500080 + 0.865979i \(0.333304\pi\)
\(102\) 2.29098 0.226841
\(103\) 14.6726 1.44573 0.722867 0.690987i \(-0.242824\pi\)
0.722867 + 0.690987i \(0.242824\pi\)
\(104\) −2.20049 −0.215776
\(105\) −0.365778 −0.0356963
\(106\) −8.03517 −0.780445
\(107\) 1.85065 0.178909 0.0894545 0.995991i \(-0.471488\pi\)
0.0894545 + 0.995991i \(0.471488\pi\)
\(108\) −2.14573 −0.206473
\(109\) 16.3800 1.56892 0.784459 0.620181i \(-0.212941\pi\)
0.784459 + 0.620181i \(0.212941\pi\)
\(110\) 0 0
\(111\) 1.29532 0.122946
\(112\) 1.00000 0.0944911
\(113\) −10.7686 −1.01303 −0.506514 0.862232i \(-0.669066\pi\)
−0.506514 + 0.862232i \(0.669066\pi\)
\(114\) −2.19820 −0.205881
\(115\) −0.452126 −0.0421610
\(116\) 1.90770 0.177126
\(117\) −6.30706 −0.583087
\(118\) 12.7279 1.17170
\(119\) −6.26330 −0.574156
\(120\) 0.365778 0.0333908
\(121\) 0 0
\(122\) 6.02013 0.545037
\(123\) 0.00756282 0.000681917 0
\(124\) −6.05116 −0.543411
\(125\) −1.00000 −0.0894427
\(126\) 2.86621 0.255342
\(127\) −4.44315 −0.394266 −0.197133 0.980377i \(-0.563163\pi\)
−0.197133 + 0.980377i \(0.563163\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −1.35916 −0.119668
\(130\) 2.20049 0.192996
\(131\) 2.54949 0.222750 0.111375 0.993778i \(-0.464474\pi\)
0.111375 + 0.993778i \(0.464474\pi\)
\(132\) 0 0
\(133\) 6.00967 0.521104
\(134\) −8.19469 −0.707913
\(135\) 2.14573 0.184675
\(136\) 6.26330 0.537074
\(137\) −15.1630 −1.29546 −0.647730 0.761870i \(-0.724281\pi\)
−0.647730 + 0.761870i \(0.724281\pi\)
\(138\) −0.165378 −0.0140779
\(139\) −4.56534 −0.387227 −0.193614 0.981078i \(-0.562021\pi\)
−0.193614 + 0.981078i \(0.562021\pi\)
\(140\) −1.00000 −0.0845154
\(141\) 3.15786 0.265940
\(142\) −15.7605 −1.32260
\(143\) 0 0
\(144\) −2.86621 −0.238851
\(145\) −1.90770 −0.158426
\(146\) 12.7842 1.05803
\(147\) 0.365778 0.0301688
\(148\) 3.54127 0.291090
\(149\) −13.9113 −1.13966 −0.569830 0.821762i \(-0.692991\pi\)
−0.569830 + 0.821762i \(0.692991\pi\)
\(150\) −0.365778 −0.0298656
\(151\) −3.78025 −0.307632 −0.153816 0.988099i \(-0.549156\pi\)
−0.153816 + 0.988099i \(0.549156\pi\)
\(152\) −6.00967 −0.487449
\(153\) 17.9519 1.45133
\(154\) 0 0
\(155\) 6.05116 0.486041
\(156\) 0.804890 0.0644428
\(157\) 12.6782 1.01183 0.505916 0.862583i \(-0.331154\pi\)
0.505916 + 0.862583i \(0.331154\pi\)
\(158\) 10.7445 0.854787
\(159\) 2.93909 0.233085
\(160\) 1.00000 0.0790569
\(161\) 0.452126 0.0356325
\(162\) −7.81376 −0.613907
\(163\) 18.8534 1.47671 0.738356 0.674412i \(-0.235603\pi\)
0.738356 + 0.674412i \(0.235603\pi\)
\(164\) 0.0206760 0.00161452
\(165\) 0 0
\(166\) 10.9679 0.851274
\(167\) −15.5834 −1.20588 −0.602939 0.797787i \(-0.706004\pi\)
−0.602939 + 0.797787i \(0.706004\pi\)
\(168\) −0.365778 −0.0282204
\(169\) −8.15785 −0.627527
\(170\) −6.26330 −0.480374
\(171\) −17.2250 −1.31723
\(172\) −3.71582 −0.283329
\(173\) −6.79145 −0.516344 −0.258172 0.966099i \(-0.583120\pi\)
−0.258172 + 0.966099i \(0.583120\pi\)
\(174\) −0.697795 −0.0528997
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) −4.65559 −0.349935
\(178\) 9.56469 0.716904
\(179\) −23.7293 −1.77361 −0.886807 0.462139i \(-0.847082\pi\)
−0.886807 + 0.462139i \(0.847082\pi\)
\(180\) 2.86621 0.213634
\(181\) 14.5332 1.08024 0.540122 0.841587i \(-0.318378\pi\)
0.540122 + 0.841587i \(0.318378\pi\)
\(182\) −2.20049 −0.163111
\(183\) −2.20203 −0.162779
\(184\) −0.452126 −0.0333312
\(185\) −3.54127 −0.260359
\(186\) 2.21338 0.162293
\(187\) 0 0
\(188\) 8.63329 0.629647
\(189\) −2.14573 −0.156079
\(190\) 6.00967 0.435987
\(191\) −6.74284 −0.487895 −0.243947 0.969789i \(-0.578442\pi\)
−0.243947 + 0.969789i \(0.578442\pi\)
\(192\) 0.365778 0.0263977
\(193\) 10.4918 0.755220 0.377610 0.925965i \(-0.376746\pi\)
0.377610 + 0.925965i \(0.376746\pi\)
\(194\) 4.22569 0.303387
\(195\) −0.804890 −0.0576394
\(196\) 1.00000 0.0714286
\(197\) 8.40324 0.598706 0.299353 0.954142i \(-0.403229\pi\)
0.299353 + 0.954142i \(0.403229\pi\)
\(198\) 0 0
\(199\) 5.41088 0.383567 0.191783 0.981437i \(-0.438573\pi\)
0.191783 + 0.981437i \(0.438573\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 2.99744 0.211423
\(202\) −10.0515 −0.707220
\(203\) 1.90770 0.133894
\(204\) −2.29098 −0.160401
\(205\) −0.0206760 −0.00144407
\(206\) −14.6726 −1.02229
\(207\) −1.29589 −0.0900704
\(208\) 2.20049 0.152576
\(209\) 0 0
\(210\) 0.365778 0.0252411
\(211\) −8.53391 −0.587499 −0.293749 0.955882i \(-0.594903\pi\)
−0.293749 + 0.955882i \(0.594903\pi\)
\(212\) 8.03517 0.551858
\(213\) 5.76486 0.395001
\(214\) −1.85065 −0.126508
\(215\) 3.71582 0.253417
\(216\) 2.14573 0.145998
\(217\) −6.05116 −0.410780
\(218\) −16.3800 −1.10939
\(219\) −4.67618 −0.315987
\(220\) 0 0
\(221\) −13.7823 −0.927100
\(222\) −1.29532 −0.0869360
\(223\) −19.6449 −1.31552 −0.657760 0.753228i \(-0.728496\pi\)
−0.657760 + 0.753228i \(0.728496\pi\)
\(224\) −1.00000 −0.0668153
\(225\) −2.86621 −0.191080
\(226\) 10.7686 0.716318
\(227\) −2.37742 −0.157795 −0.0788975 0.996883i \(-0.525140\pi\)
−0.0788975 + 0.996883i \(0.525140\pi\)
\(228\) 2.19820 0.145580
\(229\) 9.73039 0.643002 0.321501 0.946909i \(-0.395813\pi\)
0.321501 + 0.946909i \(0.395813\pi\)
\(230\) 0.452126 0.0298123
\(231\) 0 0
\(232\) −1.90770 −0.125247
\(233\) −18.4540 −1.20896 −0.604482 0.796619i \(-0.706620\pi\)
−0.604482 + 0.796619i \(0.706620\pi\)
\(234\) 6.30706 0.412305
\(235\) −8.63329 −0.563173
\(236\) −12.7279 −0.828516
\(237\) −3.93010 −0.255287
\(238\) 6.26330 0.405990
\(239\) −17.6068 −1.13889 −0.569445 0.822030i \(-0.692842\pi\)
−0.569445 + 0.822030i \(0.692842\pi\)
\(240\) −0.365778 −0.0236109
\(241\) −17.4835 −1.12621 −0.563105 0.826385i \(-0.690393\pi\)
−0.563105 + 0.826385i \(0.690393\pi\)
\(242\) 0 0
\(243\) 9.29528 0.596293
\(244\) −6.02013 −0.385399
\(245\) −1.00000 −0.0638877
\(246\) −0.00756282 −0.000482188 0
\(247\) 13.2242 0.841436
\(248\) 6.05116 0.384249
\(249\) −4.01182 −0.254239
\(250\) 1.00000 0.0632456
\(251\) −22.9585 −1.44913 −0.724563 0.689209i \(-0.757958\pi\)
−0.724563 + 0.689209i \(0.757958\pi\)
\(252\) −2.86621 −0.180554
\(253\) 0 0
\(254\) 4.44315 0.278788
\(255\) 2.29098 0.143467
\(256\) 1.00000 0.0625000
\(257\) 14.2609 0.889569 0.444784 0.895638i \(-0.353280\pi\)
0.444784 + 0.895638i \(0.353280\pi\)
\(258\) 1.35916 0.0846179
\(259\) 3.54127 0.220044
\(260\) −2.20049 −0.136469
\(261\) −5.46787 −0.338452
\(262\) −2.54949 −0.157508
\(263\) −9.29262 −0.573007 −0.286504 0.958079i \(-0.592493\pi\)
−0.286504 + 0.958079i \(0.592493\pi\)
\(264\) 0 0
\(265\) −8.03517 −0.493597
\(266\) −6.00967 −0.368476
\(267\) −3.49855 −0.214108
\(268\) 8.19469 0.500570
\(269\) 1.17894 0.0718811 0.0359405 0.999354i \(-0.488557\pi\)
0.0359405 + 0.999354i \(0.488557\pi\)
\(270\) −2.14573 −0.130585
\(271\) 8.12467 0.493539 0.246769 0.969074i \(-0.420631\pi\)
0.246769 + 0.969074i \(0.420631\pi\)
\(272\) −6.26330 −0.379769
\(273\) 0.804890 0.0487142
\(274\) 15.1630 0.916028
\(275\) 0 0
\(276\) 0.165378 0.00995457
\(277\) −8.23042 −0.494518 −0.247259 0.968949i \(-0.579530\pi\)
−0.247259 + 0.968949i \(0.579530\pi\)
\(278\) 4.56534 0.273811
\(279\) 17.3439 1.03835
\(280\) 1.00000 0.0597614
\(281\) −21.0726 −1.25709 −0.628543 0.777775i \(-0.716348\pi\)
−0.628543 + 0.777775i \(0.716348\pi\)
\(282\) −3.15786 −0.188048
\(283\) −0.468381 −0.0278423 −0.0139212 0.999903i \(-0.504431\pi\)
−0.0139212 + 0.999903i \(0.504431\pi\)
\(284\) 15.7605 0.935216
\(285\) −2.19820 −0.130210
\(286\) 0 0
\(287\) 0.0206760 0.00122047
\(288\) 2.86621 0.168893
\(289\) 22.2290 1.30759
\(290\) 1.90770 0.112024
\(291\) −1.54566 −0.0906083
\(292\) −12.7842 −0.748139
\(293\) 6.71911 0.392535 0.196267 0.980550i \(-0.437118\pi\)
0.196267 + 0.980550i \(0.437118\pi\)
\(294\) −0.365778 −0.0213326
\(295\) 12.7279 0.741047
\(296\) −3.54127 −0.205832
\(297\) 0 0
\(298\) 13.9113 0.805862
\(299\) 0.994899 0.0575365
\(300\) 0.365778 0.0211182
\(301\) −3.71582 −0.214176
\(302\) 3.78025 0.217529
\(303\) 3.67661 0.211216
\(304\) 6.00967 0.344678
\(305\) 6.02013 0.344712
\(306\) −17.9519 −1.02624
\(307\) −22.7190 −1.29664 −0.648321 0.761367i \(-0.724529\pi\)
−0.648321 + 0.761367i \(0.724529\pi\)
\(308\) 0 0
\(309\) 5.36691 0.305313
\(310\) −6.05116 −0.343683
\(311\) −21.2105 −1.20274 −0.601368 0.798972i \(-0.705377\pi\)
−0.601368 + 0.798972i \(0.705377\pi\)
\(312\) −0.804890 −0.0455679
\(313\) −3.70075 −0.209179 −0.104589 0.994515i \(-0.533353\pi\)
−0.104589 + 0.994515i \(0.533353\pi\)
\(314\) −12.6782 −0.715474
\(315\) 2.86621 0.161492
\(316\) −10.7445 −0.604425
\(317\) −7.71067 −0.433074 −0.216537 0.976274i \(-0.569476\pi\)
−0.216537 + 0.976274i \(0.569476\pi\)
\(318\) −2.93909 −0.164816
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) 0.676926 0.0377824
\(322\) −0.452126 −0.0251960
\(323\) −37.6404 −2.09437
\(324\) 7.81376 0.434098
\(325\) 2.20049 0.122061
\(326\) −18.8534 −1.04419
\(327\) 5.99143 0.331327
\(328\) −0.0206760 −0.00114164
\(329\) 8.63329 0.475968
\(330\) 0 0
\(331\) 32.2493 1.77258 0.886291 0.463128i \(-0.153273\pi\)
0.886291 + 0.463128i \(0.153273\pi\)
\(332\) −10.9679 −0.601942
\(333\) −10.1500 −0.556217
\(334\) 15.5834 0.852684
\(335\) −8.19469 −0.447724
\(336\) 0.365778 0.0199548
\(337\) −33.2152 −1.80935 −0.904673 0.426106i \(-0.859885\pi\)
−0.904673 + 0.426106i \(0.859885\pi\)
\(338\) 8.15785 0.443728
\(339\) −3.93892 −0.213933
\(340\) 6.26330 0.339675
\(341\) 0 0
\(342\) 17.2250 0.931419
\(343\) 1.00000 0.0539949
\(344\) 3.71582 0.200344
\(345\) −0.165378 −0.00890364
\(346\) 6.79145 0.365111
\(347\) 3.37780 0.181330 0.0906649 0.995881i \(-0.471101\pi\)
0.0906649 + 0.995881i \(0.471101\pi\)
\(348\) 0.697795 0.0374057
\(349\) 31.4117 1.68143 0.840716 0.541477i \(-0.182135\pi\)
0.840716 + 0.541477i \(0.182135\pi\)
\(350\) −1.00000 −0.0534522
\(351\) −4.72165 −0.252023
\(352\) 0 0
\(353\) 34.9041 1.85776 0.928879 0.370383i \(-0.120774\pi\)
0.928879 + 0.370383i \(0.120774\pi\)
\(354\) 4.65559 0.247442
\(355\) −15.7605 −0.836483
\(356\) −9.56469 −0.506927
\(357\) −2.29098 −0.121251
\(358\) 23.7293 1.25414
\(359\) −32.4765 −1.71404 −0.857022 0.515280i \(-0.827688\pi\)
−0.857022 + 0.515280i \(0.827688\pi\)
\(360\) −2.86621 −0.151062
\(361\) 17.1161 0.900849
\(362\) −14.5332 −0.763848
\(363\) 0 0
\(364\) 2.20049 0.115337
\(365\) 12.7842 0.669156
\(366\) 2.20203 0.115102
\(367\) −28.7289 −1.49963 −0.749817 0.661645i \(-0.769859\pi\)
−0.749817 + 0.661645i \(0.769859\pi\)
\(368\) 0.452126 0.0235687
\(369\) −0.0592617 −0.00308504
\(370\) 3.54127 0.184102
\(371\) 8.03517 0.417165
\(372\) −2.21338 −0.114758
\(373\) 36.4802 1.88887 0.944437 0.328691i \(-0.106608\pi\)
0.944437 + 0.328691i \(0.106608\pi\)
\(374\) 0 0
\(375\) −0.365778 −0.0188887
\(376\) −8.63329 −0.445228
\(377\) 4.19788 0.216202
\(378\) 2.14573 0.110364
\(379\) 0.299074 0.0153624 0.00768120 0.999970i \(-0.497555\pi\)
0.00768120 + 0.999970i \(0.497555\pi\)
\(380\) −6.00967 −0.308290
\(381\) −1.62521 −0.0832619
\(382\) 6.74284 0.344994
\(383\) −15.7020 −0.802337 −0.401168 0.916004i \(-0.631396\pi\)
−0.401168 + 0.916004i \(0.631396\pi\)
\(384\) −0.365778 −0.0186660
\(385\) 0 0
\(386\) −10.4918 −0.534021
\(387\) 10.6503 0.541386
\(388\) −4.22569 −0.214527
\(389\) −9.66386 −0.489977 −0.244989 0.969526i \(-0.578784\pi\)
−0.244989 + 0.969526i \(0.578784\pi\)
\(390\) 0.804890 0.0407572
\(391\) −2.83180 −0.143211
\(392\) −1.00000 −0.0505076
\(393\) 0.932548 0.0470408
\(394\) −8.40324 −0.423349
\(395\) 10.7445 0.540615
\(396\) 0 0
\(397\) 17.7947 0.893090 0.446545 0.894761i \(-0.352654\pi\)
0.446545 + 0.894761i \(0.352654\pi\)
\(398\) −5.41088 −0.271223
\(399\) 2.19820 0.110048
\(400\) 1.00000 0.0500000
\(401\) 37.5083 1.87308 0.936538 0.350566i \(-0.114011\pi\)
0.936538 + 0.350566i \(0.114011\pi\)
\(402\) −2.99744 −0.149498
\(403\) −13.3155 −0.663293
\(404\) 10.0515 0.500080
\(405\) −7.81376 −0.388269
\(406\) −1.90770 −0.0946776
\(407\) 0 0
\(408\) 2.29098 0.113420
\(409\) −21.9950 −1.08758 −0.543791 0.839221i \(-0.683012\pi\)
−0.543791 + 0.839221i \(0.683012\pi\)
\(410\) 0.0206760 0.00102112
\(411\) −5.54628 −0.273578
\(412\) 14.6726 0.722867
\(413\) −12.7279 −0.626299
\(414\) 1.29589 0.0636894
\(415\) 10.9679 0.538393
\(416\) −2.20049 −0.107888
\(417\) −1.66990 −0.0817754
\(418\) 0 0
\(419\) −14.6014 −0.713327 −0.356664 0.934233i \(-0.616086\pi\)
−0.356664 + 0.934233i \(0.616086\pi\)
\(420\) −0.365778 −0.0178481
\(421\) −36.1140 −1.76009 −0.880044 0.474893i \(-0.842487\pi\)
−0.880044 + 0.474893i \(0.842487\pi\)
\(422\) 8.53391 0.415424
\(423\) −24.7448 −1.20313
\(424\) −8.03517 −0.390223
\(425\) −6.26330 −0.303815
\(426\) −5.76486 −0.279308
\(427\) −6.02013 −0.291334
\(428\) 1.85065 0.0894545
\(429\) 0 0
\(430\) −3.71582 −0.179193
\(431\) −1.92226 −0.0925922 −0.0462961 0.998928i \(-0.514742\pi\)
−0.0462961 + 0.998928i \(0.514742\pi\)
\(432\) −2.14573 −0.103236
\(433\) −8.11232 −0.389853 −0.194927 0.980818i \(-0.562447\pi\)
−0.194927 + 0.980818i \(0.562447\pi\)
\(434\) 6.05116 0.290465
\(435\) −0.697795 −0.0334567
\(436\) 16.3800 0.784459
\(437\) 2.71713 0.129978
\(438\) 4.67618 0.223437
\(439\) −3.97670 −0.189797 −0.0948987 0.995487i \(-0.530253\pi\)
−0.0948987 + 0.995487i \(0.530253\pi\)
\(440\) 0 0
\(441\) −2.86621 −0.136486
\(442\) 13.7823 0.655559
\(443\) −7.61310 −0.361709 −0.180855 0.983510i \(-0.557886\pi\)
−0.180855 + 0.983510i \(0.557886\pi\)
\(444\) 1.29532 0.0614730
\(445\) 9.56469 0.453410
\(446\) 19.6449 0.930213
\(447\) −5.08846 −0.240676
\(448\) 1.00000 0.0472456
\(449\) 3.27630 0.154618 0.0773090 0.997007i \(-0.475367\pi\)
0.0773090 + 0.997007i \(0.475367\pi\)
\(450\) 2.86621 0.135114
\(451\) 0 0
\(452\) −10.7686 −0.506514
\(453\) −1.38273 −0.0649664
\(454\) 2.37742 0.111578
\(455\) −2.20049 −0.103161
\(456\) −2.19820 −0.102940
\(457\) −8.41968 −0.393856 −0.196928 0.980418i \(-0.563097\pi\)
−0.196928 + 0.980418i \(0.563097\pi\)
\(458\) −9.73039 −0.454671
\(459\) 13.4393 0.627295
\(460\) −0.452126 −0.0210805
\(461\) 22.7631 1.06018 0.530091 0.847941i \(-0.322158\pi\)
0.530091 + 0.847941i \(0.322158\pi\)
\(462\) 0 0
\(463\) 24.6401 1.14512 0.572561 0.819862i \(-0.305950\pi\)
0.572561 + 0.819862i \(0.305950\pi\)
\(464\) 1.90770 0.0885628
\(465\) 2.21338 0.102643
\(466\) 18.4540 0.854867
\(467\) 4.38924 0.203110 0.101555 0.994830i \(-0.467618\pi\)
0.101555 + 0.994830i \(0.467618\pi\)
\(468\) −6.30706 −0.291544
\(469\) 8.19469 0.378396
\(470\) 8.63329 0.398224
\(471\) 4.63741 0.213681
\(472\) 12.7279 0.585849
\(473\) 0 0
\(474\) 3.93010 0.180516
\(475\) 6.00967 0.275743
\(476\) −6.26330 −0.287078
\(477\) −23.0305 −1.05449
\(478\) 17.6068 0.805316
\(479\) 10.1427 0.463433 0.231716 0.972783i \(-0.425566\pi\)
0.231716 + 0.972783i \(0.425566\pi\)
\(480\) 0.365778 0.0166954
\(481\) 7.79252 0.355308
\(482\) 17.4835 0.796351
\(483\) 0.165378 0.00752495
\(484\) 0 0
\(485\) 4.22569 0.191879
\(486\) −9.29528 −0.421643
\(487\) −22.6195 −1.02499 −0.512494 0.858691i \(-0.671278\pi\)
−0.512494 + 0.858691i \(0.671278\pi\)
\(488\) 6.02013 0.272518
\(489\) 6.89615 0.311855
\(490\) 1.00000 0.0451754
\(491\) −2.55144 −0.115145 −0.0575724 0.998341i \(-0.518336\pi\)
−0.0575724 + 0.998341i \(0.518336\pi\)
\(492\) 0.00756282 0.000340958 0
\(493\) −11.9485 −0.538134
\(494\) −13.2242 −0.594985
\(495\) 0 0
\(496\) −6.05116 −0.271705
\(497\) 15.7605 0.706957
\(498\) 4.01182 0.179774
\(499\) 4.35625 0.195012 0.0975062 0.995235i \(-0.468913\pi\)
0.0975062 + 0.995235i \(0.468913\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −5.70005 −0.254660
\(502\) 22.9585 1.02469
\(503\) −23.2545 −1.03687 −0.518433 0.855118i \(-0.673485\pi\)
−0.518433 + 0.855118i \(0.673485\pi\)
\(504\) 2.86621 0.127671
\(505\) −10.0515 −0.447285
\(506\) 0 0
\(507\) −2.98396 −0.132522
\(508\) −4.44315 −0.197133
\(509\) −2.94487 −0.130529 −0.0652646 0.997868i \(-0.520789\pi\)
−0.0652646 + 0.997868i \(0.520789\pi\)
\(510\) −2.29098 −0.101446
\(511\) −12.7842 −0.565540
\(512\) −1.00000 −0.0441942
\(513\) −12.8951 −0.569333
\(514\) −14.2609 −0.629020
\(515\) −14.6726 −0.646552
\(516\) −1.35916 −0.0598339
\(517\) 0 0
\(518\) −3.54127 −0.155594
\(519\) −2.48416 −0.109043
\(520\) 2.20049 0.0964978
\(521\) −35.1983 −1.54207 −0.771034 0.636794i \(-0.780260\pi\)
−0.771034 + 0.636794i \(0.780260\pi\)
\(522\) 5.46787 0.239322
\(523\) −4.34730 −0.190094 −0.0950470 0.995473i \(-0.530300\pi\)
−0.0950470 + 0.995473i \(0.530300\pi\)
\(524\) 2.54949 0.111375
\(525\) 0.365778 0.0159639
\(526\) 9.29262 0.405177
\(527\) 37.9003 1.65096
\(528\) 0 0
\(529\) −22.7956 −0.991112
\(530\) 8.03517 0.349026
\(531\) 36.4808 1.58313
\(532\) 6.00967 0.260552
\(533\) 0.0454973 0.00197071
\(534\) 3.49855 0.151397
\(535\) −1.85065 −0.0800105
\(536\) −8.19469 −0.353957
\(537\) −8.67967 −0.374555
\(538\) −1.17894 −0.0508276
\(539\) 0 0
\(540\) 2.14573 0.0923374
\(541\) −19.7931 −0.850973 −0.425486 0.904965i \(-0.639897\pi\)
−0.425486 + 0.904965i \(0.639897\pi\)
\(542\) −8.12467 −0.348985
\(543\) 5.31592 0.228128
\(544\) 6.26330 0.268537
\(545\) −16.3800 −0.701641
\(546\) −0.804890 −0.0344461
\(547\) −12.4719 −0.533260 −0.266630 0.963799i \(-0.585910\pi\)
−0.266630 + 0.963799i \(0.585910\pi\)
\(548\) −15.1630 −0.647730
\(549\) 17.2549 0.736423
\(550\) 0 0
\(551\) 11.4647 0.488411
\(552\) −0.165378 −0.00703894
\(553\) −10.7445 −0.456903
\(554\) 8.23042 0.349677
\(555\) −1.29532 −0.0549831
\(556\) −4.56534 −0.193614
\(557\) −30.3225 −1.28480 −0.642402 0.766368i \(-0.722062\pi\)
−0.642402 + 0.766368i \(0.722062\pi\)
\(558\) −17.3439 −0.734225
\(559\) −8.17662 −0.345834
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) 21.0726 0.888894
\(563\) 19.1404 0.806671 0.403336 0.915052i \(-0.367851\pi\)
0.403336 + 0.915052i \(0.367851\pi\)
\(564\) 3.15786 0.132970
\(565\) 10.7686 0.453040
\(566\) 0.468381 0.0196875
\(567\) 7.81376 0.328147
\(568\) −15.7605 −0.661298
\(569\) −9.22473 −0.386721 −0.193360 0.981128i \(-0.561939\pi\)
−0.193360 + 0.981128i \(0.561939\pi\)
\(570\) 2.19820 0.0920726
\(571\) −26.6182 −1.11394 −0.556968 0.830534i \(-0.688035\pi\)
−0.556968 + 0.830534i \(0.688035\pi\)
\(572\) 0 0
\(573\) −2.46638 −0.103035
\(574\) −0.0206760 −0.000863000 0
\(575\) 0.452126 0.0188550
\(576\) −2.86621 −0.119425
\(577\) −8.14831 −0.339219 −0.169609 0.985511i \(-0.554251\pi\)
−0.169609 + 0.985511i \(0.554251\pi\)
\(578\) −22.2290 −0.924604
\(579\) 3.83768 0.159489
\(580\) −1.90770 −0.0792130
\(581\) −10.9679 −0.455025
\(582\) 1.54566 0.0640698
\(583\) 0 0
\(584\) 12.7842 0.529014
\(585\) 6.30706 0.260765
\(586\) −6.71911 −0.277564
\(587\) −18.1817 −0.750440 −0.375220 0.926936i \(-0.622433\pi\)
−0.375220 + 0.926936i \(0.622433\pi\)
\(588\) 0.365778 0.0150844
\(589\) −36.3655 −1.49841
\(590\) −12.7279 −0.524000
\(591\) 3.07372 0.126436
\(592\) 3.54127 0.145545
\(593\) 14.6040 0.599715 0.299857 0.953984i \(-0.403061\pi\)
0.299857 + 0.953984i \(0.403061\pi\)
\(594\) 0 0
\(595\) 6.26330 0.256770
\(596\) −13.9113 −0.569830
\(597\) 1.97918 0.0810024
\(598\) −0.994899 −0.0406844
\(599\) −15.9571 −0.651990 −0.325995 0.945371i \(-0.605699\pi\)
−0.325995 + 0.945371i \(0.605699\pi\)
\(600\) −0.365778 −0.0149328
\(601\) −22.0047 −0.897591 −0.448796 0.893634i \(-0.648147\pi\)
−0.448796 + 0.893634i \(0.648147\pi\)
\(602\) 3.71582 0.151446
\(603\) −23.4877 −0.956492
\(604\) −3.78025 −0.153816
\(605\) 0 0
\(606\) −3.67661 −0.149352
\(607\) 17.4148 0.706846 0.353423 0.935464i \(-0.385018\pi\)
0.353423 + 0.935464i \(0.385018\pi\)
\(608\) −6.00967 −0.243724
\(609\) 0.697795 0.0282761
\(610\) −6.02013 −0.243748
\(611\) 18.9975 0.768555
\(612\) 17.9519 0.725664
\(613\) 18.4109 0.743611 0.371805 0.928311i \(-0.378739\pi\)
0.371805 + 0.928311i \(0.378739\pi\)
\(614\) 22.7190 0.916864
\(615\) −0.00756282 −0.000304963 0
\(616\) 0 0
\(617\) −5.54431 −0.223206 −0.111603 0.993753i \(-0.535598\pi\)
−0.111603 + 0.993753i \(0.535598\pi\)
\(618\) −5.36691 −0.215889
\(619\) −27.3331 −1.09861 −0.549305 0.835622i \(-0.685108\pi\)
−0.549305 + 0.835622i \(0.685108\pi\)
\(620\) 6.05116 0.243021
\(621\) −0.970140 −0.0389304
\(622\) 21.2105 0.850463
\(623\) −9.56469 −0.383201
\(624\) 0.804890 0.0322214
\(625\) 1.00000 0.0400000
\(626\) 3.70075 0.147912
\(627\) 0 0
\(628\) 12.6782 0.505916
\(629\) −22.1800 −0.884376
\(630\) −2.86621 −0.114192
\(631\) 33.1472 1.31957 0.659785 0.751454i \(-0.270647\pi\)
0.659785 + 0.751454i \(0.270647\pi\)
\(632\) 10.7445 0.427393
\(633\) −3.12152 −0.124069
\(634\) 7.71067 0.306230
\(635\) 4.44315 0.176321
\(636\) 2.93909 0.116542
\(637\) 2.20049 0.0871865
\(638\) 0 0
\(639\) −45.1730 −1.78702
\(640\) 1.00000 0.0395285
\(641\) 1.55259 0.0613236 0.0306618 0.999530i \(-0.490239\pi\)
0.0306618 + 0.999530i \(0.490239\pi\)
\(642\) −0.676926 −0.0267162
\(643\) 7.60419 0.299880 0.149940 0.988695i \(-0.452092\pi\)
0.149940 + 0.988695i \(0.452092\pi\)
\(644\) 0.452126 0.0178163
\(645\) 1.35916 0.0535171
\(646\) 37.6404 1.48094
\(647\) 8.45292 0.332319 0.166159 0.986099i \(-0.446863\pi\)
0.166159 + 0.986099i \(0.446863\pi\)
\(648\) −7.81376 −0.306953
\(649\) 0 0
\(650\) −2.20049 −0.0863103
\(651\) −2.21338 −0.0867493
\(652\) 18.8534 0.738356
\(653\) 32.5646 1.27435 0.637175 0.770719i \(-0.280103\pi\)
0.637175 + 0.770719i \(0.280103\pi\)
\(654\) −5.99143 −0.234284
\(655\) −2.54949 −0.0996169
\(656\) 0.0206760 0.000807262 0
\(657\) 36.6422 1.42955
\(658\) −8.63329 −0.336561
\(659\) −37.9922 −1.47997 −0.739983 0.672626i \(-0.765166\pi\)
−0.739983 + 0.672626i \(0.765166\pi\)
\(660\) 0 0
\(661\) −15.0403 −0.584999 −0.292499 0.956266i \(-0.594487\pi\)
−0.292499 + 0.956266i \(0.594487\pi\)
\(662\) −32.2493 −1.25341
\(663\) −5.04127 −0.195787
\(664\) 10.9679 0.425637
\(665\) −6.00967 −0.233045
\(666\) 10.1500 0.393305
\(667\) 0.862522 0.0333970
\(668\) −15.5834 −0.602939
\(669\) −7.18567 −0.277814
\(670\) 8.19469 0.316588
\(671\) 0 0
\(672\) −0.365778 −0.0141102
\(673\) 38.3805 1.47946 0.739729 0.672905i \(-0.234954\pi\)
0.739729 + 0.672905i \(0.234954\pi\)
\(674\) 33.2152 1.27940
\(675\) −2.14573 −0.0825891
\(676\) −8.15785 −0.313763
\(677\) 30.8004 1.18375 0.591877 0.806028i \(-0.298387\pi\)
0.591877 + 0.806028i \(0.298387\pi\)
\(678\) 3.93892 0.151274
\(679\) −4.22569 −0.162167
\(680\) −6.26330 −0.240187
\(681\) −0.869608 −0.0333234
\(682\) 0 0
\(683\) 46.0974 1.76387 0.881935 0.471371i \(-0.156241\pi\)
0.881935 + 0.471371i \(0.156241\pi\)
\(684\) −17.2250 −0.658613
\(685\) 15.1630 0.579347
\(686\) −1.00000 −0.0381802
\(687\) 3.55916 0.135790
\(688\) −3.71582 −0.141664
\(689\) 17.6813 0.673604
\(690\) 0.165378 0.00629582
\(691\) −26.6683 −1.01451 −0.507255 0.861796i \(-0.669340\pi\)
−0.507255 + 0.861796i \(0.669340\pi\)
\(692\) −6.79145 −0.258172
\(693\) 0 0
\(694\) −3.37780 −0.128220
\(695\) 4.56534 0.173173
\(696\) −0.697795 −0.0264498
\(697\) −0.129500 −0.00490517
\(698\) −31.4117 −1.18895
\(699\) −6.75008 −0.255311
\(700\) 1.00000 0.0377964
\(701\) −15.2735 −0.576872 −0.288436 0.957499i \(-0.593135\pi\)
−0.288436 + 0.957499i \(0.593135\pi\)
\(702\) 4.72165 0.178207
\(703\) 21.2818 0.802660
\(704\) 0 0
\(705\) −3.15786 −0.118932
\(706\) −34.9041 −1.31363
\(707\) 10.0515 0.378025
\(708\) −4.65559 −0.174968
\(709\) −23.2892 −0.874645 −0.437323 0.899305i \(-0.644073\pi\)
−0.437323 + 0.899305i \(0.644073\pi\)
\(710\) 15.7605 0.591483
\(711\) 30.7960 1.15494
\(712\) 9.56469 0.358452
\(713\) −2.73589 −0.102460
\(714\) 2.29098 0.0857377
\(715\) 0 0
\(716\) −23.7293 −0.886807
\(717\) −6.44018 −0.240513
\(718\) 32.4765 1.21201
\(719\) 6.64924 0.247975 0.123987 0.992284i \(-0.460432\pi\)
0.123987 + 0.992284i \(0.460432\pi\)
\(720\) 2.86621 0.106817
\(721\) 14.6726 0.546436
\(722\) −17.1161 −0.636996
\(723\) −6.39507 −0.237835
\(724\) 14.5332 0.540122
\(725\) 1.90770 0.0708502
\(726\) 0 0
\(727\) −36.8003 −1.36485 −0.682424 0.730956i \(-0.739074\pi\)
−0.682424 + 0.730956i \(0.739074\pi\)
\(728\) −2.20049 −0.0815555
\(729\) −20.0413 −0.742269
\(730\) −12.7842 −0.473165
\(731\) 23.2733 0.860795
\(732\) −2.20203 −0.0813893
\(733\) 6.60763 0.244058 0.122029 0.992527i \(-0.461060\pi\)
0.122029 + 0.992527i \(0.461060\pi\)
\(734\) 28.7289 1.06040
\(735\) −0.365778 −0.0134919
\(736\) −0.452126 −0.0166656
\(737\) 0 0
\(738\) 0.0592617 0.00218145
\(739\) 14.1761 0.521476 0.260738 0.965410i \(-0.416034\pi\)
0.260738 + 0.965410i \(0.416034\pi\)
\(740\) −3.54127 −0.130180
\(741\) 4.83712 0.177696
\(742\) −8.03517 −0.294980
\(743\) 21.8533 0.801719 0.400859 0.916140i \(-0.368712\pi\)
0.400859 + 0.916140i \(0.368712\pi\)
\(744\) 2.21338 0.0811465
\(745\) 13.9113 0.509672
\(746\) −36.4802 −1.33564
\(747\) 31.4363 1.15019
\(748\) 0 0
\(749\) 1.85065 0.0676213
\(750\) 0.365778 0.0133563
\(751\) −18.3253 −0.668700 −0.334350 0.942449i \(-0.608517\pi\)
−0.334350 + 0.942449i \(0.608517\pi\)
\(752\) 8.63329 0.314824
\(753\) −8.39770 −0.306029
\(754\) −4.19788 −0.152878
\(755\) 3.78025 0.137577
\(756\) −2.14573 −0.0780394
\(757\) −11.6388 −0.423020 −0.211510 0.977376i \(-0.567838\pi\)
−0.211510 + 0.977376i \(0.567838\pi\)
\(758\) −0.299074 −0.0108629
\(759\) 0 0
\(760\) 6.00967 0.217994
\(761\) −25.4547 −0.922731 −0.461366 0.887210i \(-0.652640\pi\)
−0.461366 + 0.887210i \(0.652640\pi\)
\(762\) 1.62521 0.0588750
\(763\) 16.3800 0.592995
\(764\) −6.74284 −0.243947
\(765\) −17.9519 −0.649053
\(766\) 15.7020 0.567338
\(767\) −28.0076 −1.01130
\(768\) 0.365778 0.0131989
\(769\) −41.1644 −1.48442 −0.742212 0.670165i \(-0.766223\pi\)
−0.742212 + 0.670165i \(0.766223\pi\)
\(770\) 0 0
\(771\) 5.21631 0.187861
\(772\) 10.4918 0.377610
\(773\) 3.93221 0.141432 0.0707159 0.997496i \(-0.477472\pi\)
0.0707159 + 0.997496i \(0.477472\pi\)
\(774\) −10.6503 −0.382817
\(775\) −6.05116 −0.217364
\(776\) 4.22569 0.151693
\(777\) 1.29532 0.0464692
\(778\) 9.66386 0.346466
\(779\) 0.124256 0.00445193
\(780\) −0.804890 −0.0288197
\(781\) 0 0
\(782\) 2.83180 0.101265
\(783\) −4.09341 −0.146286
\(784\) 1.00000 0.0357143
\(785\) −12.6782 −0.452505
\(786\) −0.932548 −0.0332629
\(787\) 34.2138 1.21959 0.609796 0.792558i \(-0.291251\pi\)
0.609796 + 0.792558i \(0.291251\pi\)
\(788\) 8.40324 0.299353
\(789\) −3.39903 −0.121009
\(790\) −10.7445 −0.382272
\(791\) −10.7686 −0.382888
\(792\) 0 0
\(793\) −13.2472 −0.470423
\(794\) −17.7947 −0.631510
\(795\) −2.93909 −0.104239
\(796\) 5.41088 0.191783
\(797\) 0.946359 0.0335217 0.0167609 0.999860i \(-0.494665\pi\)
0.0167609 + 0.999860i \(0.494665\pi\)
\(798\) −2.19820 −0.0778156
\(799\) −54.0729 −1.91296
\(800\) −1.00000 −0.0353553
\(801\) 27.4144 0.968639
\(802\) −37.5083 −1.32446
\(803\) 0 0
\(804\) 2.99744 0.105711
\(805\) −0.452126 −0.0159354
\(806\) 13.3155 0.469019
\(807\) 0.431229 0.0151800
\(808\) −10.0515 −0.353610
\(809\) 12.0510 0.423692 0.211846 0.977303i \(-0.432053\pi\)
0.211846 + 0.977303i \(0.432053\pi\)
\(810\) 7.81376 0.274548
\(811\) −31.6751 −1.11226 −0.556132 0.831094i \(-0.687715\pi\)
−0.556132 + 0.831094i \(0.687715\pi\)
\(812\) 1.90770 0.0669472
\(813\) 2.97182 0.104226
\(814\) 0 0
\(815\) −18.8534 −0.660405
\(816\) −2.29098 −0.0802003
\(817\) −22.3309 −0.781258
\(818\) 21.9950 0.769037
\(819\) −6.30706 −0.220386
\(820\) −0.0206760 −0.000722037 0
\(821\) −24.4294 −0.852591 −0.426296 0.904584i \(-0.640182\pi\)
−0.426296 + 0.904584i \(0.640182\pi\)
\(822\) 5.54628 0.193449
\(823\) −28.4472 −0.991607 −0.495803 0.868435i \(-0.665126\pi\)
−0.495803 + 0.868435i \(0.665126\pi\)
\(824\) −14.6726 −0.511144
\(825\) 0 0
\(826\) 12.7279 0.442860
\(827\) −43.6508 −1.51789 −0.758943 0.651157i \(-0.774284\pi\)
−0.758943 + 0.651157i \(0.774284\pi\)
\(828\) −1.29589 −0.0450352
\(829\) 0.554570 0.0192610 0.00963050 0.999954i \(-0.496934\pi\)
0.00963050 + 0.999954i \(0.496934\pi\)
\(830\) −10.9679 −0.380701
\(831\) −3.01051 −0.104433
\(832\) 2.20049 0.0762882
\(833\) −6.26330 −0.217011
\(834\) 1.66990 0.0578239
\(835\) 15.5834 0.539285
\(836\) 0 0
\(837\) 12.9842 0.448798
\(838\) 14.6014 0.504399
\(839\) −2.69760 −0.0931315 −0.0465658 0.998915i \(-0.514828\pi\)
−0.0465658 + 0.998915i \(0.514828\pi\)
\(840\) 0.365778 0.0126205
\(841\) −25.3607 −0.874506
\(842\) 36.1140 1.24457
\(843\) −7.70789 −0.265474
\(844\) −8.53391 −0.293749
\(845\) 8.15785 0.280639
\(846\) 24.7448 0.850743
\(847\) 0 0
\(848\) 8.03517 0.275929
\(849\) −0.171323 −0.00587980
\(850\) 6.26330 0.214830
\(851\) 1.60110 0.0548850
\(852\) 5.76486 0.197501
\(853\) −9.81741 −0.336142 −0.168071 0.985775i \(-0.553754\pi\)
−0.168071 + 0.985775i \(0.553754\pi\)
\(854\) 6.02013 0.206005
\(855\) 17.2250 0.589081
\(856\) −1.85065 −0.0632539
\(857\) −38.1531 −1.30329 −0.651643 0.758526i \(-0.725920\pi\)
−0.651643 + 0.758526i \(0.725920\pi\)
\(858\) 0 0
\(859\) 7.51538 0.256421 0.128211 0.991747i \(-0.459077\pi\)
0.128211 + 0.991747i \(0.459077\pi\)
\(860\) 3.71582 0.126708
\(861\) 0.00756282 0.000257740 0
\(862\) 1.92226 0.0654726
\(863\) 57.5800 1.96005 0.980024 0.198880i \(-0.0637305\pi\)
0.980024 + 0.198880i \(0.0637305\pi\)
\(864\) 2.14573 0.0729992
\(865\) 6.79145 0.230916
\(866\) 8.11232 0.275668
\(867\) 8.13087 0.276139
\(868\) −6.05116 −0.205390
\(869\) 0 0
\(870\) 0.697795 0.0236575
\(871\) 18.0323 0.611002
\(872\) −16.3800 −0.554696
\(873\) 12.1117 0.409919
\(874\) −2.71713 −0.0919083
\(875\) −1.00000 −0.0338062
\(876\) −4.67618 −0.157994
\(877\) 29.0321 0.980343 0.490172 0.871626i \(-0.336934\pi\)
0.490172 + 0.871626i \(0.336934\pi\)
\(878\) 3.97670 0.134207
\(879\) 2.45770 0.0828963
\(880\) 0 0
\(881\) −48.6656 −1.63958 −0.819792 0.572661i \(-0.805911\pi\)
−0.819792 + 0.572661i \(0.805911\pi\)
\(882\) 2.86621 0.0965102
\(883\) 47.0038 1.58180 0.790902 0.611943i \(-0.209612\pi\)
0.790902 + 0.611943i \(0.209612\pi\)
\(884\) −13.7823 −0.463550
\(885\) 4.65559 0.156496
\(886\) 7.61310 0.255767
\(887\) −30.2777 −1.01662 −0.508312 0.861173i \(-0.669730\pi\)
−0.508312 + 0.861173i \(0.669730\pi\)
\(888\) −1.29532 −0.0434680
\(889\) −4.44315 −0.149019
\(890\) −9.56469 −0.320609
\(891\) 0 0
\(892\) −19.6449 −0.657760
\(893\) 51.8832 1.73621
\(894\) 5.08846 0.170183
\(895\) 23.7293 0.793185
\(896\) −1.00000 −0.0334077
\(897\) 0.363912 0.0121507
\(898\) −3.27630 −0.109331
\(899\) −11.5438 −0.385008
\(900\) −2.86621 −0.0955402
\(901\) −50.3267 −1.67663
\(902\) 0 0
\(903\) −1.35916 −0.0452302
\(904\) 10.7686 0.358159
\(905\) −14.5332 −0.483100
\(906\) 1.38273 0.0459382
\(907\) 48.9263 1.62457 0.812285 0.583261i \(-0.198223\pi\)
0.812285 + 0.583261i \(0.198223\pi\)
\(908\) −2.37742 −0.0788975
\(909\) −28.8096 −0.955556
\(910\) 2.20049 0.0729455
\(911\) 42.4585 1.40671 0.703357 0.710837i \(-0.251684\pi\)
0.703357 + 0.710837i \(0.251684\pi\)
\(912\) 2.19820 0.0727898
\(913\) 0 0
\(914\) 8.41968 0.278498
\(915\) 2.20203 0.0727968
\(916\) 9.73039 0.321501
\(917\) 2.54949 0.0841917
\(918\) −13.4393 −0.443565
\(919\) −40.5258 −1.33682 −0.668412 0.743791i \(-0.733026\pi\)
−0.668412 + 0.743791i \(0.733026\pi\)
\(920\) 0.452126 0.0149062
\(921\) −8.31011 −0.273827
\(922\) −22.7631 −0.749662
\(923\) 34.6809 1.14154
\(924\) 0 0
\(925\) 3.54127 0.116436
\(926\) −24.6401 −0.809724
\(927\) −42.0547 −1.38126
\(928\) −1.90770 −0.0626234
\(929\) 40.3461 1.32371 0.661856 0.749631i \(-0.269769\pi\)
0.661856 + 0.749631i \(0.269769\pi\)
\(930\) −2.21338 −0.0725796
\(931\) 6.00967 0.196959
\(932\) −18.4540 −0.604482
\(933\) −7.75833 −0.253996
\(934\) −4.38924 −0.143620
\(935\) 0 0
\(936\) 6.30706 0.206153
\(937\) −33.7379 −1.10217 −0.551084 0.834450i \(-0.685786\pi\)
−0.551084 + 0.834450i \(0.685786\pi\)
\(938\) −8.19469 −0.267566
\(939\) −1.35365 −0.0441747
\(940\) −8.63329 −0.281587
\(941\) −12.3247 −0.401773 −0.200887 0.979614i \(-0.564382\pi\)
−0.200887 + 0.979614i \(0.564382\pi\)
\(942\) −4.63741 −0.151095
\(943\) 0.00934817 0.000304418 0
\(944\) −12.7279 −0.414258
\(945\) 2.14573 0.0698005
\(946\) 0 0
\(947\) −0.817385 −0.0265615 −0.0132807 0.999912i \(-0.504228\pi\)
−0.0132807 + 0.999912i \(0.504228\pi\)
\(948\) −3.93010 −0.127644
\(949\) −28.1315 −0.913188
\(950\) −6.00967 −0.194979
\(951\) −2.82039 −0.0914575
\(952\) 6.26330 0.202995
\(953\) −6.96194 −0.225519 −0.112760 0.993622i \(-0.535969\pi\)
−0.112760 + 0.993622i \(0.535969\pi\)
\(954\) 23.0305 0.745639
\(955\) 6.74284 0.218193
\(956\) −17.6068 −0.569445
\(957\) 0 0
\(958\) −10.1427 −0.327696
\(959\) −15.1630 −0.489638
\(960\) −0.365778 −0.0118054
\(961\) 5.61659 0.181180
\(962\) −7.79252 −0.251241
\(963\) −5.30434 −0.170930
\(964\) −17.4835 −0.563105
\(965\) −10.4918 −0.337744
\(966\) −0.165378 −0.00532094
\(967\) 5.35673 0.172261 0.0861304 0.996284i \(-0.472550\pi\)
0.0861304 + 0.996284i \(0.472550\pi\)
\(968\) 0 0
\(969\) −13.7680 −0.442293
\(970\) −4.22569 −0.135679
\(971\) −17.4088 −0.558676 −0.279338 0.960193i \(-0.590115\pi\)
−0.279338 + 0.960193i \(0.590115\pi\)
\(972\) 9.29528 0.298146
\(973\) −4.56534 −0.146358
\(974\) 22.6195 0.724776
\(975\) 0.804890 0.0257771
\(976\) −6.02013 −0.192700
\(977\) 5.42269 0.173487 0.0867437 0.996231i \(-0.472354\pi\)
0.0867437 + 0.996231i \(0.472354\pi\)
\(978\) −6.89615 −0.220515
\(979\) 0 0
\(980\) −1.00000 −0.0319438
\(981\) −46.9484 −1.49895
\(982\) 2.55144 0.0814197
\(983\) 12.9202 0.412092 0.206046 0.978542i \(-0.433940\pi\)
0.206046 + 0.978542i \(0.433940\pi\)
\(984\) −0.00756282 −0.000241094 0
\(985\) −8.40324 −0.267750
\(986\) 11.9485 0.380518
\(987\) 3.15786 0.100516
\(988\) 13.2242 0.420718
\(989\) −1.68002 −0.0534215
\(990\) 0 0
\(991\) −26.6040 −0.845104 −0.422552 0.906339i \(-0.638866\pi\)
−0.422552 + 0.906339i \(0.638866\pi\)
\(992\) 6.05116 0.192125
\(993\) 11.7961 0.374337
\(994\) −15.7605 −0.499894
\(995\) −5.41088 −0.171536
\(996\) −4.01182 −0.127119
\(997\) 57.0633 1.80721 0.903606 0.428365i \(-0.140910\pi\)
0.903606 + 0.428365i \(0.140910\pi\)
\(998\) −4.35625 −0.137895
\(999\) −7.59860 −0.240409
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.dg.1.5 8
11.7 odd 10 770.2.n.k.71.3 16
11.8 odd 10 770.2.n.k.141.3 yes 16
11.10 odd 2 8470.2.a.dh.1.5 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.n.k.71.3 16 11.7 odd 10
770.2.n.k.141.3 yes 16 11.8 odd 10
8470.2.a.dg.1.5 8 1.1 even 1 trivial
8470.2.a.dh.1.5 8 11.10 odd 2