Properties

Label 8470.2.a.dh.1.8
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 1.8
Root \(3.18761\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +3.18761 q^{3} +1.00000 q^{4} -1.00000 q^{5} +3.18761 q^{6} -1.00000 q^{7} +1.00000 q^{8} +7.16083 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +3.18761 q^{3} +1.00000 q^{4} -1.00000 q^{5} +3.18761 q^{6} -1.00000 q^{7} +1.00000 q^{8} +7.16083 q^{9} -1.00000 q^{10} +3.18761 q^{12} +5.69102 q^{13} -1.00000 q^{14} -3.18761 q^{15} +1.00000 q^{16} -0.749022 q^{17} +7.16083 q^{18} +6.78967 q^{19} -1.00000 q^{20} -3.18761 q^{21} +3.94010 q^{23} +3.18761 q^{24} +1.00000 q^{25} +5.69102 q^{26} +13.2631 q^{27} -1.00000 q^{28} -5.73837 q^{29} -3.18761 q^{30} -7.10952 q^{31} +1.00000 q^{32} -0.749022 q^{34} +1.00000 q^{35} +7.16083 q^{36} -11.6050 q^{37} +6.78967 q^{38} +18.1407 q^{39} -1.00000 q^{40} +11.4458 q^{41} -3.18761 q^{42} +2.33345 q^{43} -7.16083 q^{45} +3.94010 q^{46} -5.00025 q^{47} +3.18761 q^{48} +1.00000 q^{49} +1.00000 q^{50} -2.38759 q^{51} +5.69102 q^{52} -10.7671 q^{53} +13.2631 q^{54} -1.00000 q^{56} +21.6428 q^{57} -5.73837 q^{58} +5.22904 q^{59} -3.18761 q^{60} +7.68398 q^{61} -7.10952 q^{62} -7.16083 q^{63} +1.00000 q^{64} -5.69102 q^{65} -7.39587 q^{67} -0.749022 q^{68} +12.5595 q^{69} +1.00000 q^{70} -0.575191 q^{71} +7.16083 q^{72} -12.5704 q^{73} -11.6050 q^{74} +3.18761 q^{75} +6.78967 q^{76} +18.1407 q^{78} +6.83481 q^{79} -1.00000 q^{80} +20.7950 q^{81} +11.4458 q^{82} +10.5252 q^{83} -3.18761 q^{84} +0.749022 q^{85} +2.33345 q^{86} -18.2916 q^{87} +4.80467 q^{89} -7.16083 q^{90} -5.69102 q^{91} +3.94010 q^{92} -22.6624 q^{93} -5.00025 q^{94} -6.78967 q^{95} +3.18761 q^{96} +16.2717 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 8q^{2} + 8q^{4} - 8q^{5} - 8q^{7} + 8q^{8} + 8q^{9} + O(q^{10}) \) \( 8q + 8q^{2} + 8q^{4} - 8q^{5} - 8q^{7} + 8q^{8} + 8q^{9} - 8q^{10} - q^{13} - 8q^{14} + 8q^{16} + 6q^{17} + 8q^{18} + 5q^{19} - 8q^{20} + 10q^{23} + 8q^{25} - q^{26} - 8q^{28} + 3q^{29} - 8q^{31} + 8q^{32} + 6q^{34} + 8q^{35} + 8q^{36} - 6q^{37} + 5q^{38} + 35q^{39} - 8q^{40} + 11q^{41} - 5q^{43} - 8q^{45} + 10q^{46} - 15q^{47} + 8q^{49} + 8q^{50} - 6q^{51} - q^{52} - 16q^{53} - 8q^{56} + 38q^{57} + 3q^{58} - 9q^{59} + 32q^{61} - 8q^{62} - 8q^{63} + 8q^{64} + q^{65} + 33q^{67} + 6q^{68} - 22q^{69} + 8q^{70} + 11q^{71} + 8q^{72} - 34q^{73} - 6q^{74} + 5q^{76} + 35q^{78} + 31q^{79} - 8q^{80} + 20q^{81} + 11q^{82} + 50q^{83} - 6q^{85} - 5q^{86} - 12q^{87} + q^{89} - 8q^{90} + q^{91} + 10q^{92} + 26q^{93} - 15q^{94} - 5q^{95} - 4q^{97} + 8q^{98} + O(q^{100}) \)

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
\(3\) 3.18761 1.84036 0.920182 0.391490i \(-0.128040\pi\)
0.920182 + 0.391490i \(0.128040\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 3.18761 1.30133
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) 7.16083 2.38694
\(10\) −1.00000 −0.316228
\(11\) 0 0
\(12\) 3.18761 0.920182
\(13\) 5.69102 1.57841 0.789203 0.614133i \(-0.210494\pi\)
0.789203 + 0.614133i \(0.210494\pi\)
\(14\) −1.00000 −0.267261
\(15\) −3.18761 −0.823036
\(16\) 1.00000 0.250000
\(17\) −0.749022 −0.181664 −0.0908322 0.995866i \(-0.528953\pi\)
−0.0908322 + 0.995866i \(0.528953\pi\)
\(18\) 7.16083 1.68782
\(19\) 6.78967 1.55766 0.778828 0.627237i \(-0.215814\pi\)
0.778828 + 0.627237i \(0.215814\pi\)
\(20\) −1.00000 −0.223607
\(21\) −3.18761 −0.695592
\(22\) 0 0
\(23\) 3.94010 0.821567 0.410783 0.911733i \(-0.365255\pi\)
0.410783 + 0.911733i \(0.365255\pi\)
\(24\) 3.18761 0.650667
\(25\) 1.00000 0.200000
\(26\) 5.69102 1.11610
\(27\) 13.2631 2.55248
\(28\) −1.00000 −0.188982
\(29\) −5.73837 −1.06559 −0.532794 0.846245i \(-0.678858\pi\)
−0.532794 + 0.846245i \(0.678858\pi\)
\(30\) −3.18761 −0.581974
\(31\) −7.10952 −1.27691 −0.638454 0.769660i \(-0.720426\pi\)
−0.638454 + 0.769660i \(0.720426\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −0.749022 −0.128456
\(35\) 1.00000 0.169031
\(36\) 7.16083 1.19347
\(37\) −11.6050 −1.90785 −0.953923 0.300052i \(-0.902996\pi\)
−0.953923 + 0.300052i \(0.902996\pi\)
\(38\) 6.78967 1.10143
\(39\) 18.1407 2.90484
\(40\) −1.00000 −0.158114
\(41\) 11.4458 1.78754 0.893768 0.448530i \(-0.148052\pi\)
0.893768 + 0.448530i \(0.148052\pi\)
\(42\) −3.18761 −0.491858
\(43\) 2.33345 0.355847 0.177924 0.984044i \(-0.443062\pi\)
0.177924 + 0.984044i \(0.443062\pi\)
\(44\) 0 0
\(45\) −7.16083 −1.06747
\(46\) 3.94010 0.580936
\(47\) −5.00025 −0.729361 −0.364680 0.931133i \(-0.618822\pi\)
−0.364680 + 0.931133i \(0.618822\pi\)
\(48\) 3.18761 0.460091
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) −2.38759 −0.334329
\(52\) 5.69102 0.789203
\(53\) −10.7671 −1.47897 −0.739487 0.673171i \(-0.764932\pi\)
−0.739487 + 0.673171i \(0.764932\pi\)
\(54\) 13.2631 1.80488
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) 21.6428 2.86666
\(58\) −5.73837 −0.753484
\(59\) 5.22904 0.680763 0.340381 0.940287i \(-0.389444\pi\)
0.340381 + 0.940287i \(0.389444\pi\)
\(60\) −3.18761 −0.411518
\(61\) 7.68398 0.983833 0.491917 0.870642i \(-0.336296\pi\)
0.491917 + 0.870642i \(0.336296\pi\)
\(62\) −7.10952 −0.902911
\(63\) −7.16083 −0.902179
\(64\) 1.00000 0.125000
\(65\) −5.69102 −0.705884
\(66\) 0 0
\(67\) −7.39587 −0.903549 −0.451775 0.892132i \(-0.649209\pi\)
−0.451775 + 0.892132i \(0.649209\pi\)
\(68\) −0.749022 −0.0908322
\(69\) 12.5595 1.51198
\(70\) 1.00000 0.119523
\(71\) −0.575191 −0.0682626 −0.0341313 0.999417i \(-0.510866\pi\)
−0.0341313 + 0.999417i \(0.510866\pi\)
\(72\) 7.16083 0.843912
\(73\) −12.5704 −1.47125 −0.735627 0.677387i \(-0.763112\pi\)
−0.735627 + 0.677387i \(0.763112\pi\)
\(74\) −11.6050 −1.34905
\(75\) 3.18761 0.368073
\(76\) 6.78967 0.778828
\(77\) 0 0
\(78\) 18.1407 2.05403
\(79\) 6.83481 0.768976 0.384488 0.923130i \(-0.374378\pi\)
0.384488 + 0.923130i \(0.374378\pi\)
\(80\) −1.00000 −0.111803
\(81\) 20.7950 2.31055
\(82\) 11.4458 1.26398
\(83\) 10.5252 1.15530 0.577648 0.816286i \(-0.303971\pi\)
0.577648 + 0.816286i \(0.303971\pi\)
\(84\) −3.18761 −0.347796
\(85\) 0.749022 0.0812428
\(86\) 2.33345 0.251622
\(87\) −18.2916 −1.96107
\(88\) 0 0
\(89\) 4.80467 0.509294 0.254647 0.967034i \(-0.418041\pi\)
0.254647 + 0.967034i \(0.418041\pi\)
\(90\) −7.16083 −0.754817
\(91\) −5.69102 −0.596581
\(92\) 3.94010 0.410783
\(93\) −22.6624 −2.34998
\(94\) −5.00025 −0.515736
\(95\) −6.78967 −0.696605
\(96\) 3.18761 0.325334
\(97\) 16.2717 1.65214 0.826071 0.563566i \(-0.190571\pi\)
0.826071 + 0.563566i \(0.190571\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −6.38488 −0.635319 −0.317659 0.948205i \(-0.602897\pi\)
−0.317659 + 0.948205i \(0.602897\pi\)
\(102\) −2.38759 −0.236406
\(103\) 2.32870 0.229454 0.114727 0.993397i \(-0.463401\pi\)
0.114727 + 0.993397i \(0.463401\pi\)
\(104\) 5.69102 0.558051
\(105\) 3.18761 0.311078
\(106\) −10.7671 −1.04579
\(107\) 1.88876 0.182593 0.0912967 0.995824i \(-0.470899\pi\)
0.0912967 + 0.995824i \(0.470899\pi\)
\(108\) 13.2631 1.27624
\(109\) 7.76554 0.743804 0.371902 0.928272i \(-0.378706\pi\)
0.371902 + 0.928272i \(0.378706\pi\)
\(110\) 0 0
\(111\) −36.9921 −3.51113
\(112\) −1.00000 −0.0944911
\(113\) 7.75283 0.729325 0.364663 0.931140i \(-0.381184\pi\)
0.364663 + 0.931140i \(0.381184\pi\)
\(114\) 21.6428 2.02703
\(115\) −3.94010 −0.367416
\(116\) −5.73837 −0.532794
\(117\) 40.7524 3.76756
\(118\) 5.22904 0.481372
\(119\) 0.749022 0.0686627
\(120\) −3.18761 −0.290987
\(121\) 0 0
\(122\) 7.68398 0.695675
\(123\) 36.4847 3.28972
\(124\) −7.10952 −0.638454
\(125\) −1.00000 −0.0894427
\(126\) −7.16083 −0.637937
\(127\) −15.5292 −1.37799 −0.688996 0.724765i \(-0.741948\pi\)
−0.688996 + 0.724765i \(0.741948\pi\)
\(128\) 1.00000 0.0883883
\(129\) 7.43810 0.654889
\(130\) −5.69102 −0.499136
\(131\) 0.957959 0.0836973 0.0418486 0.999124i \(-0.486675\pi\)
0.0418486 + 0.999124i \(0.486675\pi\)
\(132\) 0 0
\(133\) −6.78967 −0.588739
\(134\) −7.39587 −0.638906
\(135\) −13.2631 −1.14150
\(136\) −0.749022 −0.0642281
\(137\) −4.66176 −0.398281 −0.199140 0.979971i \(-0.563815\pi\)
−0.199140 + 0.979971i \(0.563815\pi\)
\(138\) 12.5595 1.06913
\(139\) 4.46428 0.378655 0.189328 0.981914i \(-0.439369\pi\)
0.189328 + 0.981914i \(0.439369\pi\)
\(140\) 1.00000 0.0845154
\(141\) −15.9388 −1.34229
\(142\) −0.575191 −0.0482690
\(143\) 0 0
\(144\) 7.16083 0.596736
\(145\) 5.73837 0.476545
\(146\) −12.5704 −1.04033
\(147\) 3.18761 0.262909
\(148\) −11.6050 −0.953923
\(149\) 15.6247 1.28003 0.640013 0.768364i \(-0.278929\pi\)
0.640013 + 0.768364i \(0.278929\pi\)
\(150\) 3.18761 0.260267
\(151\) −17.7850 −1.44732 −0.723660 0.690157i \(-0.757542\pi\)
−0.723660 + 0.690157i \(0.757542\pi\)
\(152\) 6.78967 0.550715
\(153\) −5.36361 −0.433623
\(154\) 0 0
\(155\) 7.10952 0.571051
\(156\) 18.1407 1.45242
\(157\) −7.37542 −0.588623 −0.294311 0.955710i \(-0.595090\pi\)
−0.294311 + 0.955710i \(0.595090\pi\)
\(158\) 6.83481 0.543748
\(159\) −34.3212 −2.72185
\(160\) −1.00000 −0.0790569
\(161\) −3.94010 −0.310523
\(162\) 20.7950 1.63381
\(163\) −10.4614 −0.819402 −0.409701 0.912220i \(-0.634367\pi\)
−0.409701 + 0.912220i \(0.634367\pi\)
\(164\) 11.4458 0.893768
\(165\) 0 0
\(166\) 10.5252 0.816918
\(167\) −13.8928 −1.07506 −0.537528 0.843246i \(-0.680642\pi\)
−0.537528 + 0.843246i \(0.680642\pi\)
\(168\) −3.18761 −0.245929
\(169\) 19.3877 1.49136
\(170\) 0.749022 0.0574473
\(171\) 48.6196 3.71804
\(172\) 2.33345 0.177924
\(173\) 9.79571 0.744754 0.372377 0.928082i \(-0.378543\pi\)
0.372377 + 0.928082i \(0.378543\pi\)
\(174\) −18.2916 −1.38669
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) 16.6681 1.25285
\(178\) 4.80467 0.360126
\(179\) 0.676043 0.0505298 0.0252649 0.999681i \(-0.491957\pi\)
0.0252649 + 0.999681i \(0.491957\pi\)
\(180\) −7.16083 −0.533737
\(181\) −5.50028 −0.408833 −0.204416 0.978884i \(-0.565530\pi\)
−0.204416 + 0.978884i \(0.565530\pi\)
\(182\) −5.69102 −0.421847
\(183\) 24.4935 1.81061
\(184\) 3.94010 0.290468
\(185\) 11.6050 0.853215
\(186\) −22.6624 −1.66168
\(187\) 0 0
\(188\) −5.00025 −0.364680
\(189\) −13.2631 −0.964747
\(190\) −6.78967 −0.492574
\(191\) −12.5963 −0.911434 −0.455717 0.890125i \(-0.650617\pi\)
−0.455717 + 0.890125i \(0.650617\pi\)
\(192\) 3.18761 0.230046
\(193\) −2.47246 −0.177971 −0.0889857 0.996033i \(-0.528363\pi\)
−0.0889857 + 0.996033i \(0.528363\pi\)
\(194\) 16.2717 1.16824
\(195\) −18.1407 −1.29908
\(196\) 1.00000 0.0714286
\(197\) 8.91129 0.634903 0.317451 0.948275i \(-0.397173\pi\)
0.317451 + 0.948275i \(0.397173\pi\)
\(198\) 0 0
\(199\) 26.4542 1.87529 0.937644 0.347598i \(-0.113002\pi\)
0.937644 + 0.347598i \(0.113002\pi\)
\(200\) 1.00000 0.0707107
\(201\) −23.5751 −1.66286
\(202\) −6.38488 −0.449238
\(203\) 5.73837 0.402754
\(204\) −2.38759 −0.167164
\(205\) −11.4458 −0.799410
\(206\) 2.32870 0.162248
\(207\) 28.2144 1.96103
\(208\) 5.69102 0.394601
\(209\) 0 0
\(210\) 3.18761 0.219966
\(211\) −8.29206 −0.570849 −0.285424 0.958401i \(-0.592135\pi\)
−0.285424 + 0.958401i \(0.592135\pi\)
\(212\) −10.7671 −0.739487
\(213\) −1.83348 −0.125628
\(214\) 1.88876 0.129113
\(215\) −2.33345 −0.159140
\(216\) 13.2631 0.902438
\(217\) 7.10952 0.482626
\(218\) 7.76554 0.525949
\(219\) −40.0695 −2.70764
\(220\) 0 0
\(221\) −4.26270 −0.286740
\(222\) −36.9921 −2.48275
\(223\) 0.390611 0.0261573 0.0130786 0.999914i \(-0.495837\pi\)
0.0130786 + 0.999914i \(0.495837\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 7.16083 0.477388
\(226\) 7.75283 0.515711
\(227\) −4.91748 −0.326385 −0.163192 0.986594i \(-0.552179\pi\)
−0.163192 + 0.986594i \(0.552179\pi\)
\(228\) 21.6428 1.43333
\(229\) −9.18360 −0.606869 −0.303435 0.952852i \(-0.598133\pi\)
−0.303435 + 0.952852i \(0.598133\pi\)
\(230\) −3.94010 −0.259802
\(231\) 0 0
\(232\) −5.73837 −0.376742
\(233\) −20.1599 −1.32072 −0.660359 0.750950i \(-0.729596\pi\)
−0.660359 + 0.750950i \(0.729596\pi\)
\(234\) 40.7524 2.66407
\(235\) 5.00025 0.326180
\(236\) 5.22904 0.340381
\(237\) 21.7867 1.41520
\(238\) 0.749022 0.0485519
\(239\) 9.77852 0.632520 0.316260 0.948673i \(-0.397573\pi\)
0.316260 + 0.948673i \(0.397573\pi\)
\(240\) −3.18761 −0.205759
\(241\) −12.3008 −0.792366 −0.396183 0.918172i \(-0.629665\pi\)
−0.396183 + 0.918172i \(0.629665\pi\)
\(242\) 0 0
\(243\) 26.4969 1.69978
\(244\) 7.68398 0.491917
\(245\) −1.00000 −0.0638877
\(246\) 36.4847 2.32618
\(247\) 38.6401 2.45861
\(248\) −7.10952 −0.451455
\(249\) 33.5503 2.12617
\(250\) −1.00000 −0.0632456
\(251\) −20.8441 −1.31567 −0.657833 0.753164i \(-0.728527\pi\)
−0.657833 + 0.753164i \(0.728527\pi\)
\(252\) −7.16083 −0.451090
\(253\) 0 0
\(254\) −15.5292 −0.974388
\(255\) 2.38759 0.149516
\(256\) 1.00000 0.0625000
\(257\) 7.52911 0.469653 0.234827 0.972037i \(-0.424548\pi\)
0.234827 + 0.972037i \(0.424548\pi\)
\(258\) 7.43810 0.463076
\(259\) 11.6050 0.721098
\(260\) −5.69102 −0.352942
\(261\) −41.0914 −2.54350
\(262\) 0.957959 0.0591829
\(263\) 3.27610 0.202013 0.101007 0.994886i \(-0.467794\pi\)
0.101007 + 0.994886i \(0.467794\pi\)
\(264\) 0 0
\(265\) 10.7671 0.661417
\(266\) −6.78967 −0.416301
\(267\) 15.3154 0.937287
\(268\) −7.39587 −0.451775
\(269\) 1.12272 0.0684535 0.0342267 0.999414i \(-0.489103\pi\)
0.0342267 + 0.999414i \(0.489103\pi\)
\(270\) −13.2631 −0.807165
\(271\) 9.82214 0.596652 0.298326 0.954464i \(-0.403572\pi\)
0.298326 + 0.954464i \(0.403572\pi\)
\(272\) −0.749022 −0.0454161
\(273\) −18.1407 −1.09793
\(274\) −4.66176 −0.281627
\(275\) 0 0
\(276\) 12.5595 0.755991
\(277\) −0.891218 −0.0535481 −0.0267741 0.999642i \(-0.508523\pi\)
−0.0267741 + 0.999642i \(0.508523\pi\)
\(278\) 4.46428 0.267750
\(279\) −50.9101 −3.04791
\(280\) 1.00000 0.0597614
\(281\) 17.3571 1.03544 0.517720 0.855550i \(-0.326781\pi\)
0.517720 + 0.855550i \(0.326781\pi\)
\(282\) −15.9388 −0.949142
\(283\) −2.98931 −0.177696 −0.0888481 0.996045i \(-0.528319\pi\)
−0.0888481 + 0.996045i \(0.528319\pi\)
\(284\) −0.575191 −0.0341313
\(285\) −21.6428 −1.28201
\(286\) 0 0
\(287\) −11.4458 −0.675625
\(288\) 7.16083 0.421956
\(289\) −16.4390 −0.966998
\(290\) 5.73837 0.336968
\(291\) 51.8678 3.04054
\(292\) −12.5704 −0.735627
\(293\) 17.5887 1.02754 0.513771 0.857927i \(-0.328248\pi\)
0.513771 + 0.857927i \(0.328248\pi\)
\(294\) 3.18761 0.185905
\(295\) −5.22904 −0.304446
\(296\) −11.6050 −0.674525
\(297\) 0 0
\(298\) 15.6247 0.905115
\(299\) 22.4232 1.29677
\(300\) 3.18761 0.184036
\(301\) −2.33345 −0.134498
\(302\) −17.7850 −1.02341
\(303\) −20.3525 −1.16922
\(304\) 6.78967 0.389414
\(305\) −7.68398 −0.439984
\(306\) −5.36361 −0.306617
\(307\) 1.28791 0.0735047 0.0367524 0.999324i \(-0.488299\pi\)
0.0367524 + 0.999324i \(0.488299\pi\)
\(308\) 0 0
\(309\) 7.42298 0.422279
\(310\) 7.10952 0.403794
\(311\) −0.0496330 −0.00281443 −0.00140721 0.999999i \(-0.500448\pi\)
−0.00140721 + 0.999999i \(0.500448\pi\)
\(312\) 18.1407 1.02702
\(313\) −27.6778 −1.56444 −0.782220 0.623003i \(-0.785913\pi\)
−0.782220 + 0.623003i \(0.785913\pi\)
\(314\) −7.37542 −0.416219
\(315\) 7.16083 0.403467
\(316\) 6.83481 0.384488
\(317\) 9.23290 0.518571 0.259286 0.965801i \(-0.416513\pi\)
0.259286 + 0.965801i \(0.416513\pi\)
\(318\) −34.3212 −1.92464
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) 6.02062 0.336038
\(322\) −3.94010 −0.219573
\(323\) −5.08561 −0.282971
\(324\) 20.7950 1.15528
\(325\) 5.69102 0.315681
\(326\) −10.4614 −0.579405
\(327\) 24.7535 1.36887
\(328\) 11.4458 0.631989
\(329\) 5.00025 0.275673
\(330\) 0 0
\(331\) 16.5715 0.910851 0.455425 0.890274i \(-0.349487\pi\)
0.455425 + 0.890274i \(0.349487\pi\)
\(332\) 10.5252 0.577648
\(333\) −83.1012 −4.55392
\(334\) −13.8928 −0.760180
\(335\) 7.39587 0.404080
\(336\) −3.18761 −0.173898
\(337\) −6.46713 −0.352287 −0.176144 0.984364i \(-0.556362\pi\)
−0.176144 + 0.984364i \(0.556362\pi\)
\(338\) 19.3877 1.05455
\(339\) 24.7130 1.34222
\(340\) 0.749022 0.0406214
\(341\) 0 0
\(342\) 48.6196 2.62905
\(343\) −1.00000 −0.0539949
\(344\) 2.33345 0.125811
\(345\) −12.5595 −0.676179
\(346\) 9.79571 0.526621
\(347\) −29.5567 −1.58669 −0.793344 0.608774i \(-0.791662\pi\)
−0.793344 + 0.608774i \(0.791662\pi\)
\(348\) −18.2916 −0.980535
\(349\) −25.4079 −1.36005 −0.680027 0.733187i \(-0.738032\pi\)
−0.680027 + 0.733187i \(0.738032\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 75.4804 4.02885
\(352\) 0 0
\(353\) 13.7939 0.734174 0.367087 0.930187i \(-0.380355\pi\)
0.367087 + 0.930187i \(0.380355\pi\)
\(354\) 16.6681 0.885900
\(355\) 0.575191 0.0305280
\(356\) 4.80467 0.254647
\(357\) 2.38759 0.126364
\(358\) 0.676043 0.0357300
\(359\) 22.7783 1.20219 0.601096 0.799177i \(-0.294731\pi\)
0.601096 + 0.799177i \(0.294731\pi\)
\(360\) −7.16083 −0.377409
\(361\) 27.0996 1.42629
\(362\) −5.50028 −0.289089
\(363\) 0 0
\(364\) −5.69102 −0.298291
\(365\) 12.5704 0.657965
\(366\) 24.4935 1.28030
\(367\) −9.71895 −0.507325 −0.253662 0.967293i \(-0.581635\pi\)
−0.253662 + 0.967293i \(0.581635\pi\)
\(368\) 3.94010 0.205392
\(369\) 81.9615 4.26674
\(370\) 11.6050 0.603314
\(371\) 10.7671 0.558999
\(372\) −22.6624 −1.17499
\(373\) −6.44329 −0.333621 −0.166811 0.985989i \(-0.553347\pi\)
−0.166811 + 0.985989i \(0.553347\pi\)
\(374\) 0 0
\(375\) −3.18761 −0.164607
\(376\) −5.00025 −0.257868
\(377\) −32.6572 −1.68193
\(378\) −13.2631 −0.682179
\(379\) −28.8120 −1.47997 −0.739987 0.672621i \(-0.765169\pi\)
−0.739987 + 0.672621i \(0.765169\pi\)
\(380\) −6.78967 −0.348303
\(381\) −49.5009 −2.53601
\(382\) −12.5963 −0.644481
\(383\) 19.2162 0.981903 0.490952 0.871187i \(-0.336649\pi\)
0.490952 + 0.871187i \(0.336649\pi\)
\(384\) 3.18761 0.162667
\(385\) 0 0
\(386\) −2.47246 −0.125845
\(387\) 16.7094 0.849387
\(388\) 16.2717 0.826071
\(389\) −1.16875 −0.0592582 −0.0296291 0.999561i \(-0.509433\pi\)
−0.0296291 + 0.999561i \(0.509433\pi\)
\(390\) −18.1407 −0.918592
\(391\) −2.95122 −0.149249
\(392\) 1.00000 0.0505076
\(393\) 3.05360 0.154034
\(394\) 8.91129 0.448944
\(395\) −6.83481 −0.343896
\(396\) 0 0
\(397\) −34.1835 −1.71562 −0.857811 0.513966i \(-0.828176\pi\)
−0.857811 + 0.513966i \(0.828176\pi\)
\(398\) 26.4542 1.32603
\(399\) −21.6428 −1.08349
\(400\) 1.00000 0.0500000
\(401\) 24.0445 1.20072 0.600362 0.799728i \(-0.295023\pi\)
0.600362 + 0.799728i \(0.295023\pi\)
\(402\) −23.5751 −1.17582
\(403\) −40.4605 −2.01548
\(404\) −6.38488 −0.317659
\(405\) −20.7950 −1.03331
\(406\) 5.73837 0.284790
\(407\) 0 0
\(408\) −2.38759 −0.118203
\(409\) −5.46092 −0.270025 −0.135013 0.990844i \(-0.543107\pi\)
−0.135013 + 0.990844i \(0.543107\pi\)
\(410\) −11.4458 −0.565268
\(411\) −14.8598 −0.732982
\(412\) 2.32870 0.114727
\(413\) −5.22904 −0.257304
\(414\) 28.2144 1.38666
\(415\) −10.5252 −0.516664
\(416\) 5.69102 0.279025
\(417\) 14.2304 0.696864
\(418\) 0 0
\(419\) −37.5281 −1.83337 −0.916684 0.399613i \(-0.869145\pi\)
−0.916684 + 0.399613i \(0.869145\pi\)
\(420\) 3.18761 0.155539
\(421\) 9.35357 0.455865 0.227933 0.973677i \(-0.426803\pi\)
0.227933 + 0.973677i \(0.426803\pi\)
\(422\) −8.29206 −0.403651
\(423\) −35.8059 −1.74094
\(424\) −10.7671 −0.522896
\(425\) −0.749022 −0.0363329
\(426\) −1.83348 −0.0888325
\(427\) −7.68398 −0.371854
\(428\) 1.88876 0.0912967
\(429\) 0 0
\(430\) −2.33345 −0.112529
\(431\) −5.10019 −0.245668 −0.122834 0.992427i \(-0.539198\pi\)
−0.122834 + 0.992427i \(0.539198\pi\)
\(432\) 13.2631 0.638120
\(433\) −15.0790 −0.724649 −0.362325 0.932052i \(-0.618017\pi\)
−0.362325 + 0.932052i \(0.618017\pi\)
\(434\) 7.10952 0.341268
\(435\) 18.2916 0.877017
\(436\) 7.76554 0.371902
\(437\) 26.7520 1.27972
\(438\) −40.0695 −1.91459
\(439\) −8.35774 −0.398893 −0.199447 0.979909i \(-0.563914\pi\)
−0.199447 + 0.979909i \(0.563914\pi\)
\(440\) 0 0
\(441\) 7.16083 0.340992
\(442\) −4.26270 −0.202756
\(443\) 6.68524 0.317626 0.158813 0.987309i \(-0.449233\pi\)
0.158813 + 0.987309i \(0.449233\pi\)
\(444\) −36.9921 −1.75557
\(445\) −4.80467 −0.227763
\(446\) 0.390611 0.0184960
\(447\) 49.8054 2.35572
\(448\) −1.00000 −0.0472456
\(449\) −7.97205 −0.376224 −0.188112 0.982148i \(-0.560237\pi\)
−0.188112 + 0.982148i \(0.560237\pi\)
\(450\) 7.16083 0.337565
\(451\) 0 0
\(452\) 7.75283 0.364663
\(453\) −56.6915 −2.66360
\(454\) −4.91748 −0.230789
\(455\) 5.69102 0.266799
\(456\) 21.6428 1.01352
\(457\) −2.34209 −0.109558 −0.0547791 0.998498i \(-0.517445\pi\)
−0.0547791 + 0.998498i \(0.517445\pi\)
\(458\) −9.18360 −0.429121
\(459\) −9.93433 −0.463695
\(460\) −3.94010 −0.183708
\(461\) −0.312762 −0.0145668 −0.00728340 0.999973i \(-0.502318\pi\)
−0.00728340 + 0.999973i \(0.502318\pi\)
\(462\) 0 0
\(463\) 15.8769 0.737863 0.368931 0.929457i \(-0.379724\pi\)
0.368931 + 0.929457i \(0.379724\pi\)
\(464\) −5.73837 −0.266397
\(465\) 22.6624 1.05094
\(466\) −20.1599 −0.933888
\(467\) −0.774386 −0.0358343 −0.0179172 0.999839i \(-0.505704\pi\)
−0.0179172 + 0.999839i \(0.505704\pi\)
\(468\) 40.7524 1.88378
\(469\) 7.39587 0.341510
\(470\) 5.00025 0.230644
\(471\) −23.5099 −1.08328
\(472\) 5.22904 0.240686
\(473\) 0 0
\(474\) 21.7867 1.00069
\(475\) 6.78967 0.311531
\(476\) 0.749022 0.0343313
\(477\) −77.1013 −3.53022
\(478\) 9.77852 0.447259
\(479\) 30.3787 1.38804 0.694019 0.719956i \(-0.255838\pi\)
0.694019 + 0.719956i \(0.255838\pi\)
\(480\) −3.18761 −0.145494
\(481\) −66.0441 −3.01135
\(482\) −12.3008 −0.560287
\(483\) −12.5595 −0.571476
\(484\) 0 0
\(485\) −16.2717 −0.738861
\(486\) 26.4969 1.20192
\(487\) 9.43710 0.427636 0.213818 0.976874i \(-0.431410\pi\)
0.213818 + 0.976874i \(0.431410\pi\)
\(488\) 7.68398 0.347838
\(489\) −33.3469 −1.50800
\(490\) −1.00000 −0.0451754
\(491\) 28.7129 1.29579 0.647897 0.761728i \(-0.275649\pi\)
0.647897 + 0.761728i \(0.275649\pi\)
\(492\) 36.4847 1.64486
\(493\) 4.29816 0.193579
\(494\) 38.6401 1.73850
\(495\) 0 0
\(496\) −7.10952 −0.319227
\(497\) 0.575191 0.0258008
\(498\) 33.5503 1.50343
\(499\) 27.6602 1.23824 0.619120 0.785297i \(-0.287490\pi\)
0.619120 + 0.785297i \(0.287490\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −44.2848 −1.97850
\(502\) −20.8441 −0.930316
\(503\) −1.65413 −0.0737539 −0.0368770 0.999320i \(-0.511741\pi\)
−0.0368770 + 0.999320i \(0.511741\pi\)
\(504\) −7.16083 −0.318969
\(505\) 6.38488 0.284123
\(506\) 0 0
\(507\) 61.8004 2.74465
\(508\) −15.5292 −0.688996
\(509\) 11.0335 0.489050 0.244525 0.969643i \(-0.421368\pi\)
0.244525 + 0.969643i \(0.421368\pi\)
\(510\) 2.38759 0.105724
\(511\) 12.5704 0.556082
\(512\) 1.00000 0.0441942
\(513\) 90.0519 3.97589
\(514\) 7.52911 0.332095
\(515\) −2.32870 −0.102615
\(516\) 7.43810 0.327444
\(517\) 0 0
\(518\) 11.6050 0.509893
\(519\) 31.2249 1.37062
\(520\) −5.69102 −0.249568
\(521\) 19.2485 0.843294 0.421647 0.906760i \(-0.361452\pi\)
0.421647 + 0.906760i \(0.361452\pi\)
\(522\) −41.0914 −1.79852
\(523\) 4.09540 0.179079 0.0895396 0.995983i \(-0.471460\pi\)
0.0895396 + 0.995983i \(0.471460\pi\)
\(524\) 0.957959 0.0418486
\(525\) −3.18761 −0.139118
\(526\) 3.27610 0.142845
\(527\) 5.32519 0.231969
\(528\) 0 0
\(529\) −7.47564 −0.325028
\(530\) 10.7671 0.467692
\(531\) 37.4442 1.62494
\(532\) −6.78967 −0.294369
\(533\) 65.1383 2.82146
\(534\) 15.3154 0.662762
\(535\) −1.88876 −0.0816582
\(536\) −7.39587 −0.319453
\(537\) 2.15496 0.0929933
\(538\) 1.12272 0.0484039
\(539\) 0 0
\(540\) −13.2631 −0.570752
\(541\) −12.0631 −0.518632 −0.259316 0.965793i \(-0.583497\pi\)
−0.259316 + 0.965793i \(0.583497\pi\)
\(542\) 9.82214 0.421897
\(543\) −17.5327 −0.752402
\(544\) −0.749022 −0.0321140
\(545\) −7.76554 −0.332639
\(546\) −18.1407 −0.776352
\(547\) −5.69776 −0.243619 −0.121809 0.992554i \(-0.538870\pi\)
−0.121809 + 0.992554i \(0.538870\pi\)
\(548\) −4.66176 −0.199140
\(549\) 55.0237 2.34835
\(550\) 0 0
\(551\) −38.9616 −1.65982
\(552\) 12.5595 0.534567
\(553\) −6.83481 −0.290645
\(554\) −0.891218 −0.0378642
\(555\) 36.9921 1.57023
\(556\) 4.46428 0.189328
\(557\) 8.87789 0.376168 0.188084 0.982153i \(-0.439772\pi\)
0.188084 + 0.982153i \(0.439772\pi\)
\(558\) −50.9101 −2.15520
\(559\) 13.2797 0.561671
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) 17.3571 0.732167
\(563\) 20.1280 0.848294 0.424147 0.905593i \(-0.360574\pi\)
0.424147 + 0.905593i \(0.360574\pi\)
\(564\) −15.9388 −0.671145
\(565\) −7.75283 −0.326164
\(566\) −2.98931 −0.125650
\(567\) −20.7950 −0.873306
\(568\) −0.575191 −0.0241345
\(569\) 24.8357 1.04117 0.520583 0.853811i \(-0.325715\pi\)
0.520583 + 0.853811i \(0.325715\pi\)
\(570\) −21.6428 −0.906516
\(571\) −28.6955 −1.20087 −0.600435 0.799674i \(-0.705006\pi\)
−0.600435 + 0.799674i \(0.705006\pi\)
\(572\) 0 0
\(573\) −40.1520 −1.67737
\(574\) −11.4458 −0.477739
\(575\) 3.94010 0.164313
\(576\) 7.16083 0.298368
\(577\) −15.3853 −0.640499 −0.320249 0.947333i \(-0.603767\pi\)
−0.320249 + 0.947333i \(0.603767\pi\)
\(578\) −16.4390 −0.683771
\(579\) −7.88122 −0.327532
\(580\) 5.73837 0.238273
\(581\) −10.5252 −0.436661
\(582\) 51.8678 2.14999
\(583\) 0 0
\(584\) −12.5704 −0.520167
\(585\) −40.7524 −1.68491
\(586\) 17.5887 0.726582
\(587\) −36.1775 −1.49321 −0.746603 0.665270i \(-0.768317\pi\)
−0.746603 + 0.665270i \(0.768317\pi\)
\(588\) 3.18761 0.131455
\(589\) −48.2713 −1.98898
\(590\) −5.22904 −0.215276
\(591\) 28.4057 1.16845
\(592\) −11.6050 −0.476961
\(593\) 22.6239 0.929052 0.464526 0.885560i \(-0.346225\pi\)
0.464526 + 0.885560i \(0.346225\pi\)
\(594\) 0 0
\(595\) −0.749022 −0.0307069
\(596\) 15.6247 0.640013
\(597\) 84.3255 3.45121
\(598\) 22.4232 0.916952
\(599\) 2.93462 0.119906 0.0599528 0.998201i \(-0.480905\pi\)
0.0599528 + 0.998201i \(0.480905\pi\)
\(600\) 3.18761 0.130133
\(601\) −33.3078 −1.35865 −0.679327 0.733836i \(-0.737728\pi\)
−0.679327 + 0.733836i \(0.737728\pi\)
\(602\) −2.33345 −0.0951041
\(603\) −52.9606 −2.15672
\(604\) −17.7850 −0.723660
\(605\) 0 0
\(606\) −20.3525 −0.826762
\(607\) 27.9310 1.13369 0.566843 0.823826i \(-0.308165\pi\)
0.566843 + 0.823826i \(0.308165\pi\)
\(608\) 6.78967 0.275357
\(609\) 18.2916 0.741215
\(610\) −7.68398 −0.311115
\(611\) −28.4565 −1.15123
\(612\) −5.36361 −0.216811
\(613\) 13.2976 0.537087 0.268543 0.963268i \(-0.413458\pi\)
0.268543 + 0.963268i \(0.413458\pi\)
\(614\) 1.28791 0.0519757
\(615\) −36.4847 −1.47121
\(616\) 0 0
\(617\) −17.0581 −0.686735 −0.343368 0.939201i \(-0.611568\pi\)
−0.343368 + 0.939201i \(0.611568\pi\)
\(618\) 7.42298 0.298596
\(619\) −29.0446 −1.16740 −0.583700 0.811969i \(-0.698396\pi\)
−0.583700 + 0.811969i \(0.698396\pi\)
\(620\) 7.10952 0.285525
\(621\) 52.2578 2.09703
\(622\) −0.0496330 −0.00199010
\(623\) −4.80467 −0.192495
\(624\) 18.1407 0.726210
\(625\) 1.00000 0.0400000
\(626\) −27.6778 −1.10623
\(627\) 0 0
\(628\) −7.37542 −0.294311
\(629\) 8.69238 0.346588
\(630\) 7.16083 0.285294
\(631\) 32.0364 1.27535 0.637675 0.770305i \(-0.279896\pi\)
0.637675 + 0.770305i \(0.279896\pi\)
\(632\) 6.83481 0.271874
\(633\) −26.4318 −1.05057
\(634\) 9.23290 0.366685
\(635\) 15.5292 0.616257
\(636\) −34.3212 −1.36093
\(637\) 5.69102 0.225486
\(638\) 0 0
\(639\) −4.11884 −0.162939
\(640\) −1.00000 −0.0395285
\(641\) 7.75431 0.306277 0.153138 0.988205i \(-0.451062\pi\)
0.153138 + 0.988205i \(0.451062\pi\)
\(642\) 6.02062 0.237615
\(643\) 24.2104 0.954763 0.477382 0.878696i \(-0.341586\pi\)
0.477382 + 0.878696i \(0.341586\pi\)
\(644\) −3.94010 −0.155262
\(645\) −7.43810 −0.292875
\(646\) −5.08561 −0.200091
\(647\) −0.861530 −0.0338702 −0.0169351 0.999857i \(-0.505391\pi\)
−0.0169351 + 0.999857i \(0.505391\pi\)
\(648\) 20.7950 0.816903
\(649\) 0 0
\(650\) 5.69102 0.223220
\(651\) 22.6624 0.888208
\(652\) −10.4614 −0.409701
\(653\) −21.7225 −0.850068 −0.425034 0.905177i \(-0.639738\pi\)
−0.425034 + 0.905177i \(0.639738\pi\)
\(654\) 24.7535 0.967938
\(655\) −0.957959 −0.0374306
\(656\) 11.4458 0.446884
\(657\) −90.0144 −3.51180
\(658\) 5.00025 0.194930
\(659\) −35.0148 −1.36398 −0.681991 0.731360i \(-0.738886\pi\)
−0.681991 + 0.731360i \(0.738886\pi\)
\(660\) 0 0
\(661\) 21.6301 0.841315 0.420657 0.907220i \(-0.361799\pi\)
0.420657 + 0.907220i \(0.361799\pi\)
\(662\) 16.5715 0.644069
\(663\) −13.5878 −0.527706
\(664\) 10.5252 0.408459
\(665\) 6.78967 0.263292
\(666\) −83.1012 −3.22011
\(667\) −22.6097 −0.875452
\(668\) −13.8928 −0.537528
\(669\) 1.24511 0.0481389
\(670\) 7.39587 0.285727
\(671\) 0 0
\(672\) −3.18761 −0.122965
\(673\) −34.4741 −1.32888 −0.664440 0.747342i \(-0.731330\pi\)
−0.664440 + 0.747342i \(0.731330\pi\)
\(674\) −6.46713 −0.249105
\(675\) 13.2631 0.510496
\(676\) 19.3877 0.745682
\(677\) −13.5315 −0.520058 −0.260029 0.965601i \(-0.583732\pi\)
−0.260029 + 0.965601i \(0.583732\pi\)
\(678\) 24.7130 0.949096
\(679\) −16.2717 −0.624451
\(680\) 0.749022 0.0287237
\(681\) −15.6750 −0.600667
\(682\) 0 0
\(683\) 19.6188 0.750694 0.375347 0.926884i \(-0.377524\pi\)
0.375347 + 0.926884i \(0.377524\pi\)
\(684\) 48.6196 1.85902
\(685\) 4.66176 0.178117
\(686\) −1.00000 −0.0381802
\(687\) −29.2737 −1.11686
\(688\) 2.33345 0.0889618
\(689\) −61.2757 −2.33442
\(690\) −12.5595 −0.478131
\(691\) −27.4372 −1.04376 −0.521880 0.853019i \(-0.674769\pi\)
−0.521880 + 0.853019i \(0.674769\pi\)
\(692\) 9.79571 0.372377
\(693\) 0 0
\(694\) −29.5567 −1.12196
\(695\) −4.46428 −0.169340
\(696\) −18.2916 −0.693343
\(697\) −8.57316 −0.324732
\(698\) −25.4079 −0.961703
\(699\) −64.2617 −2.43060
\(700\) −1.00000 −0.0377964
\(701\) −32.2446 −1.21786 −0.608930 0.793224i \(-0.708401\pi\)
−0.608930 + 0.793224i \(0.708401\pi\)
\(702\) 75.4804 2.84883
\(703\) −78.7939 −2.97177
\(704\) 0 0
\(705\) 15.9388 0.600290
\(706\) 13.7939 0.519139
\(707\) 6.38488 0.240128
\(708\) 16.6681 0.626426
\(709\) 20.8485 0.782982 0.391491 0.920182i \(-0.371959\pi\)
0.391491 + 0.920182i \(0.371959\pi\)
\(710\) 0.575191 0.0215865
\(711\) 48.9429 1.83550
\(712\) 4.80467 0.180063
\(713\) −28.0122 −1.04907
\(714\) 2.38759 0.0893531
\(715\) 0 0
\(716\) 0.676043 0.0252649
\(717\) 31.1701 1.16407
\(718\) 22.7783 0.850078
\(719\) −40.3188 −1.50364 −0.751818 0.659370i \(-0.770823\pi\)
−0.751818 + 0.659370i \(0.770823\pi\)
\(720\) −7.16083 −0.266868
\(721\) −2.32870 −0.0867254
\(722\) 27.0996 1.00854
\(723\) −39.2102 −1.45824
\(724\) −5.50028 −0.204416
\(725\) −5.73837 −0.213118
\(726\) 0 0
\(727\) 39.1899 1.45347 0.726736 0.686917i \(-0.241036\pi\)
0.726736 + 0.686917i \(0.241036\pi\)
\(728\) −5.69102 −0.210923
\(729\) 22.0768 0.817660
\(730\) 12.5704 0.465251
\(731\) −1.74780 −0.0646448
\(732\) 24.4935 0.905306
\(733\) −24.4077 −0.901518 −0.450759 0.892646i \(-0.648847\pi\)
−0.450759 + 0.892646i \(0.648847\pi\)
\(734\) −9.71895 −0.358733
\(735\) −3.18761 −0.117577
\(736\) 3.94010 0.145234
\(737\) 0 0
\(738\) 81.9615 3.01704
\(739\) −41.5299 −1.52770 −0.763851 0.645392i \(-0.776694\pi\)
−0.763851 + 0.645392i \(0.776694\pi\)
\(740\) 11.6050 0.426607
\(741\) 123.170 4.52475
\(742\) 10.7671 0.395272
\(743\) −22.6025 −0.829207 −0.414603 0.910002i \(-0.636080\pi\)
−0.414603 + 0.910002i \(0.636080\pi\)
\(744\) −22.6624 −0.830842
\(745\) −15.6247 −0.572445
\(746\) −6.44329 −0.235906
\(747\) 75.3695 2.75763
\(748\) 0 0
\(749\) −1.88876 −0.0690138
\(750\) −3.18761 −0.116395
\(751\) −22.7802 −0.831262 −0.415631 0.909533i \(-0.636439\pi\)
−0.415631 + 0.909533i \(0.636439\pi\)
\(752\) −5.00025 −0.182340
\(753\) −66.4426 −2.42130
\(754\) −32.6572 −1.18930
\(755\) 17.7850 0.647261
\(756\) −13.2631 −0.482373
\(757\) 6.77323 0.246177 0.123089 0.992396i \(-0.460720\pi\)
0.123089 + 0.992396i \(0.460720\pi\)
\(758\) −28.8120 −1.04650
\(759\) 0 0
\(760\) −6.78967 −0.246287
\(761\) 7.24515 0.262637 0.131318 0.991340i \(-0.458079\pi\)
0.131318 + 0.991340i \(0.458079\pi\)
\(762\) −49.5009 −1.79323
\(763\) −7.76554 −0.281132
\(764\) −12.5963 −0.455717
\(765\) 5.36361 0.193922
\(766\) 19.2162 0.694310
\(767\) 29.7586 1.07452
\(768\) 3.18761 0.115023
\(769\) −1.45679 −0.0525333 −0.0262667 0.999655i \(-0.508362\pi\)
−0.0262667 + 0.999655i \(0.508362\pi\)
\(770\) 0 0
\(771\) 23.9998 0.864333
\(772\) −2.47246 −0.0889857
\(773\) −25.4505 −0.915390 −0.457695 0.889109i \(-0.651325\pi\)
−0.457695 + 0.889109i \(0.651325\pi\)
\(774\) 16.7094 0.600607
\(775\) −7.10952 −0.255382
\(776\) 16.2717 0.584121
\(777\) 36.9921 1.32708
\(778\) −1.16875 −0.0419018
\(779\) 77.7133 2.78437
\(780\) −18.1407 −0.649542
\(781\) 0 0
\(782\) −2.95122 −0.105535
\(783\) −76.1084 −2.71989
\(784\) 1.00000 0.0357143
\(785\) 7.37542 0.263240
\(786\) 3.05360 0.108918
\(787\) 18.9942 0.677069 0.338535 0.940954i \(-0.390069\pi\)
0.338535 + 0.940954i \(0.390069\pi\)
\(788\) 8.91129 0.317451
\(789\) 10.4429 0.371778
\(790\) −6.83481 −0.243171
\(791\) −7.75283 −0.275659
\(792\) 0 0
\(793\) 43.7297 1.55289
\(794\) −34.1835 −1.21313
\(795\) 34.3212 1.21725
\(796\) 26.4542 0.937644
\(797\) −44.9953 −1.59381 −0.796907 0.604102i \(-0.793532\pi\)
−0.796907 + 0.604102i \(0.793532\pi\)
\(798\) −21.6428 −0.766146
\(799\) 3.74529 0.132499
\(800\) 1.00000 0.0353553
\(801\) 34.4054 1.21566
\(802\) 24.0445 0.849041
\(803\) 0 0
\(804\) −23.5751 −0.831430
\(805\) 3.94010 0.138870
\(806\) −40.4605 −1.42516
\(807\) 3.57879 0.125979
\(808\) −6.38488 −0.224619
\(809\) −31.2609 −1.09907 −0.549537 0.835469i \(-0.685196\pi\)
−0.549537 + 0.835469i \(0.685196\pi\)
\(810\) −20.7950 −0.730661
\(811\) 27.5975 0.969078 0.484539 0.874770i \(-0.338987\pi\)
0.484539 + 0.874770i \(0.338987\pi\)
\(812\) 5.73837 0.201377
\(813\) 31.3091 1.09806
\(814\) 0 0
\(815\) 10.4614 0.366448
\(816\) −2.38759 −0.0835822
\(817\) 15.8433 0.554288
\(818\) −5.46092 −0.190937
\(819\) −40.7524 −1.42400
\(820\) −11.4458 −0.399705
\(821\) −21.3493 −0.745096 −0.372548 0.928013i \(-0.621516\pi\)
−0.372548 + 0.928013i \(0.621516\pi\)
\(822\) −14.8598 −0.518296
\(823\) −52.8518 −1.84230 −0.921149 0.389210i \(-0.872748\pi\)
−0.921149 + 0.389210i \(0.872748\pi\)
\(824\) 2.32870 0.0811242
\(825\) 0 0
\(826\) −5.22904 −0.181941
\(827\) 17.8294 0.619990 0.309995 0.950738i \(-0.399673\pi\)
0.309995 + 0.950738i \(0.399673\pi\)
\(828\) 28.2144 0.980517
\(829\) −46.2177 −1.60521 −0.802603 0.596513i \(-0.796552\pi\)
−0.802603 + 0.596513i \(0.796552\pi\)
\(830\) −10.5252 −0.365337
\(831\) −2.84085 −0.0985481
\(832\) 5.69102 0.197301
\(833\) −0.749022 −0.0259521
\(834\) 14.2304 0.492757
\(835\) 13.8928 0.480780
\(836\) 0 0
\(837\) −94.2942 −3.25928
\(838\) −37.5281 −1.29639
\(839\) −19.4145 −0.670263 −0.335131 0.942171i \(-0.608781\pi\)
−0.335131 + 0.942171i \(0.608781\pi\)
\(840\) 3.18761 0.109983
\(841\) 3.92884 0.135477
\(842\) 9.35357 0.322345
\(843\) 55.3277 1.90559
\(844\) −8.29206 −0.285424
\(845\) −19.3877 −0.666958
\(846\) −35.8059 −1.23103
\(847\) 0 0
\(848\) −10.7671 −0.369743
\(849\) −9.52875 −0.327026
\(850\) −0.749022 −0.0256912
\(851\) −45.7247 −1.56742
\(852\) −1.83348 −0.0628141
\(853\) −23.3656 −0.800021 −0.400011 0.916510i \(-0.630994\pi\)
−0.400011 + 0.916510i \(0.630994\pi\)
\(854\) −7.68398 −0.262940
\(855\) −48.6196 −1.66276
\(856\) 1.88876 0.0645565
\(857\) −32.8025 −1.12051 −0.560256 0.828320i \(-0.689297\pi\)
−0.560256 + 0.828320i \(0.689297\pi\)
\(858\) 0 0
\(859\) −5.80773 −0.198157 −0.0990786 0.995080i \(-0.531590\pi\)
−0.0990786 + 0.995080i \(0.531590\pi\)
\(860\) −2.33345 −0.0795698
\(861\) −36.4847 −1.24340
\(862\) −5.10019 −0.173713
\(863\) −44.4137 −1.51186 −0.755931 0.654652i \(-0.772815\pi\)
−0.755931 + 0.654652i \(0.772815\pi\)
\(864\) 13.2631 0.451219
\(865\) −9.79571 −0.333064
\(866\) −15.0790 −0.512405
\(867\) −52.4009 −1.77963
\(868\) 7.10952 0.241313
\(869\) 0 0
\(870\) 18.2916 0.620145
\(871\) −42.0901 −1.42617
\(872\) 7.76554 0.262974
\(873\) 116.519 3.94357
\(874\) 26.7520 0.904898
\(875\) 1.00000 0.0338062
\(876\) −40.0695 −1.35382
\(877\) −17.3464 −0.585746 −0.292873 0.956151i \(-0.594611\pi\)
−0.292873 + 0.956151i \(0.594611\pi\)
\(878\) −8.35774 −0.282060
\(879\) 56.0658 1.89105
\(880\) 0 0
\(881\) 2.06132 0.0694478 0.0347239 0.999397i \(-0.488945\pi\)
0.0347239 + 0.999397i \(0.488945\pi\)
\(882\) 7.16083 0.241118
\(883\) −46.5639 −1.56700 −0.783501 0.621391i \(-0.786568\pi\)
−0.783501 + 0.621391i \(0.786568\pi\)
\(884\) −4.26270 −0.143370
\(885\) −16.6681 −0.560292
\(886\) 6.68524 0.224595
\(887\) 27.9025 0.936873 0.468437 0.883497i \(-0.344817\pi\)
0.468437 + 0.883497i \(0.344817\pi\)
\(888\) −36.9921 −1.24137
\(889\) 15.5292 0.520832
\(890\) −4.80467 −0.161053
\(891\) 0 0
\(892\) 0.390611 0.0130786
\(893\) −33.9500 −1.13609
\(894\) 49.8054 1.66574
\(895\) −0.676043 −0.0225976
\(896\) −1.00000 −0.0334077
\(897\) 71.4762 2.38652
\(898\) −7.97205 −0.266031
\(899\) 40.7971 1.36066
\(900\) 7.16083 0.238694
\(901\) 8.06478 0.268677
\(902\) 0 0
\(903\) −7.43810 −0.247525
\(904\) 7.75283 0.257855
\(905\) 5.50028 0.182836
\(906\) −56.6915 −1.88345
\(907\) −7.30550 −0.242575 −0.121288 0.992617i \(-0.538702\pi\)
−0.121288 + 0.992617i \(0.538702\pi\)
\(908\) −4.91748 −0.163192
\(909\) −45.7210 −1.51647
\(910\) 5.69102 0.188656
\(911\) −28.9807 −0.960174 −0.480087 0.877221i \(-0.659395\pi\)
−0.480087 + 0.877221i \(0.659395\pi\)
\(912\) 21.6428 0.716664
\(913\) 0 0
\(914\) −2.34209 −0.0774693
\(915\) −24.4935 −0.809730
\(916\) −9.18360 −0.303435
\(917\) −0.957959 −0.0316346
\(918\) −9.93433 −0.327882
\(919\) 5.06468 0.167068 0.0835342 0.996505i \(-0.473379\pi\)
0.0835342 + 0.996505i \(0.473379\pi\)
\(920\) −3.94010 −0.129901
\(921\) 4.10534 0.135276
\(922\) −0.312762 −0.0103003
\(923\) −3.27342 −0.107746
\(924\) 0 0
\(925\) −11.6050 −0.381569
\(926\) 15.8769 0.521748
\(927\) 16.6754 0.547693
\(928\) −5.73837 −0.188371
\(929\) −23.6262 −0.775151 −0.387576 0.921838i \(-0.626687\pi\)
−0.387576 + 0.921838i \(0.626687\pi\)
\(930\) 22.6624 0.743128
\(931\) 6.78967 0.222522
\(932\) −20.1599 −0.660359
\(933\) −0.158210 −0.00517957
\(934\) −0.774386 −0.0253387
\(935\) 0 0
\(936\) 40.7524 1.33203
\(937\) −39.5066 −1.29063 −0.645313 0.763919i \(-0.723273\pi\)
−0.645313 + 0.763919i \(0.723273\pi\)
\(938\) 7.39587 0.241484
\(939\) −88.2258 −2.87914
\(940\) 5.00025 0.163090
\(941\) 14.2738 0.465313 0.232656 0.972559i \(-0.425258\pi\)
0.232656 + 0.972559i \(0.425258\pi\)
\(942\) −23.5099 −0.765995
\(943\) 45.0976 1.46858
\(944\) 5.22904 0.170191
\(945\) 13.2631 0.431448
\(946\) 0 0
\(947\) 15.8228 0.514171 0.257085 0.966389i \(-0.417238\pi\)
0.257085 + 0.966389i \(0.417238\pi\)
\(948\) 21.7867 0.707598
\(949\) −71.5384 −2.32223
\(950\) 6.78967 0.220286
\(951\) 29.4308 0.954360
\(952\) 0.749022 0.0242759
\(953\) 31.9949 1.03642 0.518208 0.855255i \(-0.326599\pi\)
0.518208 + 0.855255i \(0.326599\pi\)
\(954\) −77.1013 −2.49625
\(955\) 12.5963 0.407606
\(956\) 9.77852 0.316260
\(957\) 0 0
\(958\) 30.3787 0.981491
\(959\) 4.66176 0.150536
\(960\) −3.18761 −0.102880
\(961\) 19.5453 0.630495
\(962\) −66.0441 −2.12935
\(963\) 13.5251 0.435840
\(964\) −12.3008 −0.396183
\(965\) 2.47246 0.0795912
\(966\) −12.5595 −0.404094
\(967\) −24.1383 −0.776237 −0.388118 0.921610i \(-0.626875\pi\)
−0.388118 + 0.921610i \(0.626875\pi\)
\(968\) 0 0
\(969\) −16.2109 −0.520770
\(970\) −16.2717 −0.522453
\(971\) −24.3861 −0.782587 −0.391293 0.920266i \(-0.627972\pi\)
−0.391293 + 0.920266i \(0.627972\pi\)
\(972\) 26.4969 0.849889
\(973\) −4.46428 −0.143118
\(974\) 9.43710 0.302384
\(975\) 18.1407 0.580968
\(976\) 7.68398 0.245958
\(977\) 54.4580 1.74227 0.871133 0.491046i \(-0.163385\pi\)
0.871133 + 0.491046i \(0.163385\pi\)
\(978\) −33.3469 −1.06632
\(979\) 0 0
\(980\) −1.00000 −0.0319438
\(981\) 55.6077 1.77542
\(982\) 28.7129 0.916264
\(983\) −52.4013 −1.67134 −0.835670 0.549231i \(-0.814921\pi\)
−0.835670 + 0.549231i \(0.814921\pi\)
\(984\) 36.4847 1.16309
\(985\) −8.91129 −0.283937
\(986\) 4.29816 0.136881
\(987\) 15.9388 0.507338
\(988\) 38.6401 1.22931
\(989\) 9.19400 0.292352
\(990\) 0 0
\(991\) 13.7527 0.436870 0.218435 0.975851i \(-0.429905\pi\)
0.218435 + 0.975851i \(0.429905\pi\)
\(992\) −7.10952 −0.225728
\(993\) 52.8233 1.67630
\(994\) 0.575191 0.0182440
\(995\) −26.4542 −0.838654
\(996\) 33.5503 1.06308
\(997\) 38.7130 1.22605 0.613026 0.790062i \(-0.289952\pi\)
0.613026 + 0.790062i \(0.289952\pi\)
\(998\) 27.6602 0.875568
\(999\) −153.918 −4.86974
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.1.8 8
11.3 even 5 770.2.n.k.141.1 yes 16
11.4 even 5 770.2.n.k.71.1 16
11.10 odd 2 8470.2.a.dg.1.8 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.n.k.71.1 16 11.4 even 5
770.2.n.k.141.1 yes 16 11.3 even 5
8470.2.a.dg.1.8 8 11.10 odd 2
8470.2.a.dh.1.8 8 1.1 even 1 trivial