Properties

Label 8016.2.a.bb.1.6
Level $8016$
Weight $2$
Character 8016.1
Self dual yes
Analytic conductor $64.008$
Analytic rank $0$
Dimension $9$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8016 = 2^{4} \cdot 3 \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8016.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.0080822603\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Defining polynomial: \(x^{9} - 29 x^{7} - 7 x^{6} + 266 x^{5} + 69 x^{4} - 901 x^{3} - 199 x^{2} + 875 x + 391\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2004)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(-0.907808\) of defining polynomial
Character \(\chi\) \(=\) 8016.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +1.90781 q^{5} -2.81337 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +1.90781 q^{5} -2.81337 q^{7} +1.00000 q^{9} -0.318882 q^{11} +5.12583 q^{13} -1.90781 q^{15} +3.73184 q^{17} +0.725453 q^{19} +2.81337 q^{21} -0.612463 q^{23} -1.36027 q^{25} -1.00000 q^{27} -3.87979 q^{29} +6.65817 q^{31} +0.318882 q^{33} -5.36737 q^{35} -9.04643 q^{37} -5.12583 q^{39} +9.55805 q^{41} +10.9328 q^{43} +1.90781 q^{45} -8.76460 q^{47} +0.915046 q^{49} -3.73184 q^{51} -2.46673 q^{53} -0.608366 q^{55} -0.725453 q^{57} +11.0871 q^{59} +12.5460 q^{61} -2.81337 q^{63} +9.77910 q^{65} -5.21723 q^{67} +0.612463 q^{69} +11.3217 q^{71} +2.88490 q^{73} +1.36027 q^{75} +0.897133 q^{77} -8.98655 q^{79} +1.00000 q^{81} -4.04802 q^{83} +7.11963 q^{85} +3.87979 q^{87} -9.15294 q^{89} -14.4208 q^{91} -6.65817 q^{93} +1.38402 q^{95} -9.62101 q^{97} -0.318882 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9q - 9q^{3} + 9q^{5} - 2q^{7} + 9q^{9} + O(q^{10}) \) \( 9q - 9q^{3} + 9q^{5} - 2q^{7} + 9q^{9} - 7q^{11} + 6q^{13} - 9q^{15} + 7q^{17} - 2q^{19} + 2q^{21} - 19q^{23} + 22q^{25} - 9q^{27} + 13q^{29} - 12q^{31} + 7q^{33} - 4q^{35} + 15q^{37} - 6q^{39} + 18q^{41} + 6q^{43} + 9q^{45} - 25q^{47} + 19q^{49} - 7q^{51} + 17q^{53} + 3q^{55} + 2q^{57} - 3q^{59} + 14q^{61} - 2q^{63} + 14q^{65} + 4q^{67} + 19q^{69} - 17q^{71} - 20q^{73} - 22q^{75} + 14q^{77} + 8q^{79} + 9q^{81} + q^{83} + 5q^{85} - 13q^{87} + 36q^{89} + 41q^{91} + 12q^{93} - 5q^{95} + 31q^{97} - 7q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 1.90781 0.853198 0.426599 0.904441i \(-0.359712\pi\)
0.426599 + 0.904441i \(0.359712\pi\)
\(6\) 0 0
\(7\) −2.81337 −1.06335 −0.531677 0.846947i \(-0.678438\pi\)
−0.531677 + 0.846947i \(0.678438\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −0.318882 −0.0961466 −0.0480733 0.998844i \(-0.515308\pi\)
−0.0480733 + 0.998844i \(0.515308\pi\)
\(12\) 0 0
\(13\) 5.12583 1.42165 0.710824 0.703370i \(-0.248322\pi\)
0.710824 + 0.703370i \(0.248322\pi\)
\(14\) 0 0
\(15\) −1.90781 −0.492594
\(16\) 0 0
\(17\) 3.73184 0.905104 0.452552 0.891738i \(-0.350514\pi\)
0.452552 + 0.891738i \(0.350514\pi\)
\(18\) 0 0
\(19\) 0.725453 0.166430 0.0832151 0.996532i \(-0.473481\pi\)
0.0832151 + 0.996532i \(0.473481\pi\)
\(20\) 0 0
\(21\) 2.81337 0.613927
\(22\) 0 0
\(23\) −0.612463 −0.127707 −0.0638537 0.997959i \(-0.520339\pi\)
−0.0638537 + 0.997959i \(0.520339\pi\)
\(24\) 0 0
\(25\) −1.36027 −0.272053
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −3.87979 −0.720459 −0.360229 0.932864i \(-0.617302\pi\)
−0.360229 + 0.932864i \(0.617302\pi\)
\(30\) 0 0
\(31\) 6.65817 1.19584 0.597922 0.801554i \(-0.295993\pi\)
0.597922 + 0.801554i \(0.295993\pi\)
\(32\) 0 0
\(33\) 0.318882 0.0555103
\(34\) 0 0
\(35\) −5.36737 −0.907251
\(36\) 0 0
\(37\) −9.04643 −1.48722 −0.743612 0.668612i \(-0.766889\pi\)
−0.743612 + 0.668612i \(0.766889\pi\)
\(38\) 0 0
\(39\) −5.12583 −0.820789
\(40\) 0 0
\(41\) 9.55805 1.49272 0.746358 0.665545i \(-0.231801\pi\)
0.746358 + 0.665545i \(0.231801\pi\)
\(42\) 0 0
\(43\) 10.9328 1.66724 0.833620 0.552338i \(-0.186264\pi\)
0.833620 + 0.552338i \(0.186264\pi\)
\(44\) 0 0
\(45\) 1.90781 0.284399
\(46\) 0 0
\(47\) −8.76460 −1.27845 −0.639224 0.769020i \(-0.720744\pi\)
−0.639224 + 0.769020i \(0.720744\pi\)
\(48\) 0 0
\(49\) 0.915046 0.130721
\(50\) 0 0
\(51\) −3.73184 −0.522562
\(52\) 0 0
\(53\) −2.46673 −0.338831 −0.169416 0.985545i \(-0.554188\pi\)
−0.169416 + 0.985545i \(0.554188\pi\)
\(54\) 0 0
\(55\) −0.608366 −0.0820321
\(56\) 0 0
\(57\) −0.725453 −0.0960885
\(58\) 0 0
\(59\) 11.0871 1.44341 0.721706 0.692200i \(-0.243358\pi\)
0.721706 + 0.692200i \(0.243358\pi\)
\(60\) 0 0
\(61\) 12.5460 1.60636 0.803178 0.595739i \(-0.203141\pi\)
0.803178 + 0.595739i \(0.203141\pi\)
\(62\) 0 0
\(63\) −2.81337 −0.354451
\(64\) 0 0
\(65\) 9.77910 1.21295
\(66\) 0 0
\(67\) −5.21723 −0.637387 −0.318693 0.947858i \(-0.603244\pi\)
−0.318693 + 0.947858i \(0.603244\pi\)
\(68\) 0 0
\(69\) 0.612463 0.0737319
\(70\) 0 0
\(71\) 11.3217 1.34363 0.671816 0.740718i \(-0.265514\pi\)
0.671816 + 0.740718i \(0.265514\pi\)
\(72\) 0 0
\(73\) 2.88490 0.337652 0.168826 0.985646i \(-0.446002\pi\)
0.168826 + 0.985646i \(0.446002\pi\)
\(74\) 0 0
\(75\) 1.36027 0.157070
\(76\) 0 0
\(77\) 0.897133 0.102238
\(78\) 0 0
\(79\) −8.98655 −1.01107 −0.505533 0.862807i \(-0.668704\pi\)
−0.505533 + 0.862807i \(0.668704\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −4.04802 −0.444327 −0.222164 0.975009i \(-0.571312\pi\)
−0.222164 + 0.975009i \(0.571312\pi\)
\(84\) 0 0
\(85\) 7.11963 0.772233
\(86\) 0 0
\(87\) 3.87979 0.415957
\(88\) 0 0
\(89\) −9.15294 −0.970210 −0.485105 0.874456i \(-0.661219\pi\)
−0.485105 + 0.874456i \(0.661219\pi\)
\(90\) 0 0
\(91\) −14.4208 −1.51172
\(92\) 0 0
\(93\) −6.65817 −0.690421
\(94\) 0 0
\(95\) 1.38402 0.141998
\(96\) 0 0
\(97\) −9.62101 −0.976866 −0.488433 0.872601i \(-0.662431\pi\)
−0.488433 + 0.872601i \(0.662431\pi\)
\(98\) 0 0
\(99\) −0.318882 −0.0320489
\(100\) 0 0
\(101\) −13.2834 −1.32175 −0.660875 0.750496i \(-0.729815\pi\)
−0.660875 + 0.750496i \(0.729815\pi\)
\(102\) 0 0
\(103\) 18.4226 1.81523 0.907616 0.419802i \(-0.137901\pi\)
0.907616 + 0.419802i \(0.137901\pi\)
\(104\) 0 0
\(105\) 5.36737 0.523802
\(106\) 0 0
\(107\) −11.8365 −1.14428 −0.572139 0.820157i \(-0.693886\pi\)
−0.572139 + 0.820157i \(0.693886\pi\)
\(108\) 0 0
\(109\) −2.94111 −0.281708 −0.140854 0.990030i \(-0.544985\pi\)
−0.140854 + 0.990030i \(0.544985\pi\)
\(110\) 0 0
\(111\) 9.04643 0.858649
\(112\) 0 0
\(113\) 16.9782 1.59718 0.798588 0.601878i \(-0.205581\pi\)
0.798588 + 0.601878i \(0.205581\pi\)
\(114\) 0 0
\(115\) −1.16846 −0.108960
\(116\) 0 0
\(117\) 5.12583 0.473883
\(118\) 0 0
\(119\) −10.4990 −0.962445
\(120\) 0 0
\(121\) −10.8983 −0.990756
\(122\) 0 0
\(123\) −9.55805 −0.861820
\(124\) 0 0
\(125\) −12.1342 −1.08531
\(126\) 0 0
\(127\) −16.7091 −1.48269 −0.741345 0.671124i \(-0.765812\pi\)
−0.741345 + 0.671124i \(0.765812\pi\)
\(128\) 0 0
\(129\) −10.9328 −0.962582
\(130\) 0 0
\(131\) 0.146536 0.0128029 0.00640144 0.999980i \(-0.497962\pi\)
0.00640144 + 0.999980i \(0.497962\pi\)
\(132\) 0 0
\(133\) −2.04097 −0.176974
\(134\) 0 0
\(135\) −1.90781 −0.164198
\(136\) 0 0
\(137\) 11.7309 1.00224 0.501120 0.865378i \(-0.332922\pi\)
0.501120 + 0.865378i \(0.332922\pi\)
\(138\) 0 0
\(139\) 9.09999 0.771851 0.385926 0.922530i \(-0.373882\pi\)
0.385926 + 0.922530i \(0.373882\pi\)
\(140\) 0 0
\(141\) 8.76460 0.738113
\(142\) 0 0
\(143\) −1.63454 −0.136687
\(144\) 0 0
\(145\) −7.40189 −0.614694
\(146\) 0 0
\(147\) −0.915046 −0.0754717
\(148\) 0 0
\(149\) 23.3492 1.91284 0.956422 0.291988i \(-0.0943168\pi\)
0.956422 + 0.291988i \(0.0943168\pi\)
\(150\) 0 0
\(151\) 0.859951 0.0699818 0.0349909 0.999388i \(-0.488860\pi\)
0.0349909 + 0.999388i \(0.488860\pi\)
\(152\) 0 0
\(153\) 3.73184 0.301701
\(154\) 0 0
\(155\) 12.7025 1.02029
\(156\) 0 0
\(157\) −6.36080 −0.507647 −0.253824 0.967251i \(-0.581688\pi\)
−0.253824 + 0.967251i \(0.581688\pi\)
\(158\) 0 0
\(159\) 2.46673 0.195624
\(160\) 0 0
\(161\) 1.72309 0.135798
\(162\) 0 0
\(163\) 0.722678 0.0566045 0.0283023 0.999599i \(-0.490990\pi\)
0.0283023 + 0.999599i \(0.490990\pi\)
\(164\) 0 0
\(165\) 0.608366 0.0473612
\(166\) 0 0
\(167\) −1.00000 −0.0773823
\(168\) 0 0
\(169\) 13.2741 1.02109
\(170\) 0 0
\(171\) 0.725453 0.0554767
\(172\) 0 0
\(173\) 16.1210 1.22566 0.612829 0.790216i \(-0.290032\pi\)
0.612829 + 0.790216i \(0.290032\pi\)
\(174\) 0 0
\(175\) 3.82693 0.289289
\(176\) 0 0
\(177\) −11.0871 −0.833354
\(178\) 0 0
\(179\) −11.5480 −0.863137 −0.431569 0.902080i \(-0.642040\pi\)
−0.431569 + 0.902080i \(0.642040\pi\)
\(180\) 0 0
\(181\) −3.91786 −0.291212 −0.145606 0.989343i \(-0.546513\pi\)
−0.145606 + 0.989343i \(0.546513\pi\)
\(182\) 0 0
\(183\) −12.5460 −0.927430
\(184\) 0 0
\(185\) −17.2588 −1.26890
\(186\) 0 0
\(187\) −1.19002 −0.0870226
\(188\) 0 0
\(189\) 2.81337 0.204642
\(190\) 0 0
\(191\) 19.5571 1.41510 0.707549 0.706664i \(-0.249801\pi\)
0.707549 + 0.706664i \(0.249801\pi\)
\(192\) 0 0
\(193\) 22.0005 1.58363 0.791814 0.610762i \(-0.209137\pi\)
0.791814 + 0.610762i \(0.209137\pi\)
\(194\) 0 0
\(195\) −9.77910 −0.700296
\(196\) 0 0
\(197\) 10.3862 0.739989 0.369995 0.929034i \(-0.379360\pi\)
0.369995 + 0.929034i \(0.379360\pi\)
\(198\) 0 0
\(199\) 17.3380 1.22906 0.614528 0.788895i \(-0.289347\pi\)
0.614528 + 0.788895i \(0.289347\pi\)
\(200\) 0 0
\(201\) 5.21723 0.367995
\(202\) 0 0
\(203\) 10.9153 0.766102
\(204\) 0 0
\(205\) 18.2349 1.27358
\(206\) 0 0
\(207\) −0.612463 −0.0425691
\(208\) 0 0
\(209\) −0.231334 −0.0160017
\(210\) 0 0
\(211\) 20.2222 1.39216 0.696078 0.717966i \(-0.254927\pi\)
0.696078 + 0.717966i \(0.254927\pi\)
\(212\) 0 0
\(213\) −11.3217 −0.775747
\(214\) 0 0
\(215\) 20.8577 1.42249
\(216\) 0 0
\(217\) −18.7319 −1.27160
\(218\) 0 0
\(219\) −2.88490 −0.194944
\(220\) 0 0
\(221\) 19.1288 1.28674
\(222\) 0 0
\(223\) −9.60888 −0.643458 −0.321729 0.946832i \(-0.604264\pi\)
−0.321729 + 0.946832i \(0.604264\pi\)
\(224\) 0 0
\(225\) −1.36027 −0.0906845
\(226\) 0 0
\(227\) −23.3573 −1.55028 −0.775140 0.631790i \(-0.782321\pi\)
−0.775140 + 0.631790i \(0.782321\pi\)
\(228\) 0 0
\(229\) −14.5969 −0.964591 −0.482296 0.876008i \(-0.660197\pi\)
−0.482296 + 0.876008i \(0.660197\pi\)
\(230\) 0 0
\(231\) −0.897133 −0.0590270
\(232\) 0 0
\(233\) 18.0225 1.18069 0.590345 0.807151i \(-0.298992\pi\)
0.590345 + 0.807151i \(0.298992\pi\)
\(234\) 0 0
\(235\) −16.7212 −1.09077
\(236\) 0 0
\(237\) 8.98655 0.583739
\(238\) 0 0
\(239\) 9.44595 0.611007 0.305504 0.952191i \(-0.401175\pi\)
0.305504 + 0.952191i \(0.401175\pi\)
\(240\) 0 0
\(241\) 5.88509 0.379092 0.189546 0.981872i \(-0.439298\pi\)
0.189546 + 0.981872i \(0.439298\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 1.74573 0.111531
\(246\) 0 0
\(247\) 3.71854 0.236605
\(248\) 0 0
\(249\) 4.04802 0.256533
\(250\) 0 0
\(251\) −3.14251 −0.198353 −0.0991767 0.995070i \(-0.531621\pi\)
−0.0991767 + 0.995070i \(0.531621\pi\)
\(252\) 0 0
\(253\) 0.195304 0.0122786
\(254\) 0 0
\(255\) −7.11963 −0.445849
\(256\) 0 0
\(257\) 1.52800 0.0953142 0.0476571 0.998864i \(-0.484825\pi\)
0.0476571 + 0.998864i \(0.484825\pi\)
\(258\) 0 0
\(259\) 25.4509 1.58144
\(260\) 0 0
\(261\) −3.87979 −0.240153
\(262\) 0 0
\(263\) 24.3283 1.50015 0.750075 0.661353i \(-0.230017\pi\)
0.750075 + 0.661353i \(0.230017\pi\)
\(264\) 0 0
\(265\) −4.70604 −0.289090
\(266\) 0 0
\(267\) 9.15294 0.560151
\(268\) 0 0
\(269\) 0.163618 0.00997594 0.00498797 0.999988i \(-0.498412\pi\)
0.00498797 + 0.999988i \(0.498412\pi\)
\(270\) 0 0
\(271\) 8.88391 0.539659 0.269829 0.962908i \(-0.413033\pi\)
0.269829 + 0.962908i \(0.413033\pi\)
\(272\) 0 0
\(273\) 14.4208 0.872789
\(274\) 0 0
\(275\) 0.433765 0.0261570
\(276\) 0 0
\(277\) −5.45934 −0.328020 −0.164010 0.986459i \(-0.552443\pi\)
−0.164010 + 0.986459i \(0.552443\pi\)
\(278\) 0 0
\(279\) 6.65817 0.398614
\(280\) 0 0
\(281\) −8.18240 −0.488121 −0.244061 0.969760i \(-0.578480\pi\)
−0.244061 + 0.969760i \(0.578480\pi\)
\(282\) 0 0
\(283\) −9.17416 −0.545347 −0.272674 0.962107i \(-0.587908\pi\)
−0.272674 + 0.962107i \(0.587908\pi\)
\(284\) 0 0
\(285\) −1.38402 −0.0819825
\(286\) 0 0
\(287\) −26.8903 −1.58728
\(288\) 0 0
\(289\) −3.07338 −0.180787
\(290\) 0 0
\(291\) 9.62101 0.563994
\(292\) 0 0
\(293\) −30.6446 −1.79027 −0.895137 0.445790i \(-0.852923\pi\)
−0.895137 + 0.445790i \(0.852923\pi\)
\(294\) 0 0
\(295\) 21.1520 1.23152
\(296\) 0 0
\(297\) 0.318882 0.0185034
\(298\) 0 0
\(299\) −3.13938 −0.181555
\(300\) 0 0
\(301\) −30.7581 −1.77287
\(302\) 0 0
\(303\) 13.2834 0.763113
\(304\) 0 0
\(305\) 23.9354 1.37054
\(306\) 0 0
\(307\) 25.0225 1.42811 0.714055 0.700090i \(-0.246857\pi\)
0.714055 + 0.700090i \(0.246857\pi\)
\(308\) 0 0
\(309\) −18.4226 −1.04802
\(310\) 0 0
\(311\) −26.4473 −1.49969 −0.749845 0.661613i \(-0.769872\pi\)
−0.749845 + 0.661613i \(0.769872\pi\)
\(312\) 0 0
\(313\) 22.3927 1.26571 0.632854 0.774271i \(-0.281883\pi\)
0.632854 + 0.774271i \(0.281883\pi\)
\(314\) 0 0
\(315\) −5.36737 −0.302417
\(316\) 0 0
\(317\) 24.1684 1.35743 0.678717 0.734400i \(-0.262536\pi\)
0.678717 + 0.734400i \(0.262536\pi\)
\(318\) 0 0
\(319\) 1.23720 0.0692697
\(320\) 0 0
\(321\) 11.8365 0.660649
\(322\) 0 0
\(323\) 2.70727 0.150637
\(324\) 0 0
\(325\) −6.97249 −0.386764
\(326\) 0 0
\(327\) 2.94111 0.162644
\(328\) 0 0
\(329\) 24.6581 1.35944
\(330\) 0 0
\(331\) 28.8009 1.58304 0.791521 0.611142i \(-0.209290\pi\)
0.791521 + 0.611142i \(0.209290\pi\)
\(332\) 0 0
\(333\) −9.04643 −0.495741
\(334\) 0 0
\(335\) −9.95348 −0.543817
\(336\) 0 0
\(337\) −2.40083 −0.130782 −0.0653909 0.997860i \(-0.520829\pi\)
−0.0653909 + 0.997860i \(0.520829\pi\)
\(338\) 0 0
\(339\) −16.9782 −0.922130
\(340\) 0 0
\(341\) −2.12317 −0.114976
\(342\) 0 0
\(343\) 17.1192 0.924351
\(344\) 0 0
\(345\) 1.16846 0.0629079
\(346\) 0 0
\(347\) 19.3977 1.04132 0.520662 0.853763i \(-0.325685\pi\)
0.520662 + 0.853763i \(0.325685\pi\)
\(348\) 0 0
\(349\) −36.7633 −1.96789 −0.983947 0.178462i \(-0.942888\pi\)
−0.983947 + 0.178462i \(0.942888\pi\)
\(350\) 0 0
\(351\) −5.12583 −0.273596
\(352\) 0 0
\(353\) −25.4854 −1.35645 −0.678226 0.734853i \(-0.737251\pi\)
−0.678226 + 0.734853i \(0.737251\pi\)
\(354\) 0 0
\(355\) 21.5995 1.14638
\(356\) 0 0
\(357\) 10.4990 0.555668
\(358\) 0 0
\(359\) 7.51148 0.396441 0.198220 0.980157i \(-0.436484\pi\)
0.198220 + 0.980157i \(0.436484\pi\)
\(360\) 0 0
\(361\) −18.4737 −0.972301
\(362\) 0 0
\(363\) 10.8983 0.572013
\(364\) 0 0
\(365\) 5.50384 0.288084
\(366\) 0 0
\(367\) −15.8402 −0.826853 −0.413426 0.910537i \(-0.635668\pi\)
−0.413426 + 0.910537i \(0.635668\pi\)
\(368\) 0 0
\(369\) 9.55805 0.497572
\(370\) 0 0
\(371\) 6.93981 0.360297
\(372\) 0 0
\(373\) 34.3907 1.78068 0.890340 0.455296i \(-0.150466\pi\)
0.890340 + 0.455296i \(0.150466\pi\)
\(374\) 0 0
\(375\) 12.1342 0.626606
\(376\) 0 0
\(377\) −19.8871 −1.02424
\(378\) 0 0
\(379\) 13.6557 0.701448 0.350724 0.936479i \(-0.385936\pi\)
0.350724 + 0.936479i \(0.385936\pi\)
\(380\) 0 0
\(381\) 16.7091 0.856032
\(382\) 0 0
\(383\) −12.8238 −0.655263 −0.327632 0.944806i \(-0.606250\pi\)
−0.327632 + 0.944806i \(0.606250\pi\)
\(384\) 0 0
\(385\) 1.71156 0.0872291
\(386\) 0 0
\(387\) 10.9328 0.555747
\(388\) 0 0
\(389\) −1.34694 −0.0682925 −0.0341462 0.999417i \(-0.510871\pi\)
−0.0341462 + 0.999417i \(0.510871\pi\)
\(390\) 0 0
\(391\) −2.28561 −0.115588
\(392\) 0 0
\(393\) −0.146536 −0.00739175
\(394\) 0 0
\(395\) −17.1446 −0.862639
\(396\) 0 0
\(397\) 9.21493 0.462484 0.231242 0.972896i \(-0.425721\pi\)
0.231242 + 0.972896i \(0.425721\pi\)
\(398\) 0 0
\(399\) 2.04097 0.102176
\(400\) 0 0
\(401\) 5.28032 0.263686 0.131843 0.991271i \(-0.457910\pi\)
0.131843 + 0.991271i \(0.457910\pi\)
\(402\) 0 0
\(403\) 34.1287 1.70007
\(404\) 0 0
\(405\) 1.90781 0.0947998
\(406\) 0 0
\(407\) 2.88474 0.142991
\(408\) 0 0
\(409\) 36.9707 1.82808 0.914042 0.405619i \(-0.132944\pi\)
0.914042 + 0.405619i \(0.132944\pi\)
\(410\) 0 0
\(411\) −11.7309 −0.578643
\(412\) 0 0
\(413\) −31.1920 −1.53486
\(414\) 0 0
\(415\) −7.72284 −0.379099
\(416\) 0 0
\(417\) −9.09999 −0.445629
\(418\) 0 0
\(419\) 12.2542 0.598655 0.299328 0.954150i \(-0.403238\pi\)
0.299328 + 0.954150i \(0.403238\pi\)
\(420\) 0 0
\(421\) 33.4130 1.62845 0.814225 0.580550i \(-0.197162\pi\)
0.814225 + 0.580550i \(0.197162\pi\)
\(422\) 0 0
\(423\) −8.76460 −0.426150
\(424\) 0 0
\(425\) −5.07630 −0.246237
\(426\) 0 0
\(427\) −35.2966 −1.70812
\(428\) 0 0
\(429\) 1.63454 0.0789161
\(430\) 0 0
\(431\) 15.5835 0.750632 0.375316 0.926897i \(-0.377534\pi\)
0.375316 + 0.926897i \(0.377534\pi\)
\(432\) 0 0
\(433\) −14.5888 −0.701095 −0.350547 0.936545i \(-0.614004\pi\)
−0.350547 + 0.936545i \(0.614004\pi\)
\(434\) 0 0
\(435\) 7.40189 0.354894
\(436\) 0 0
\(437\) −0.444313 −0.0212544
\(438\) 0 0
\(439\) −0.127708 −0.00609515 −0.00304758 0.999995i \(-0.500970\pi\)
−0.00304758 + 0.999995i \(0.500970\pi\)
\(440\) 0 0
\(441\) 0.915046 0.0435736
\(442\) 0 0
\(443\) −1.30938 −0.0622103 −0.0311052 0.999516i \(-0.509903\pi\)
−0.0311052 + 0.999516i \(0.509903\pi\)
\(444\) 0 0
\(445\) −17.4621 −0.827781
\(446\) 0 0
\(447\) −23.3492 −1.10438
\(448\) 0 0
\(449\) −0.999933 −0.0471898 −0.0235949 0.999722i \(-0.507511\pi\)
−0.0235949 + 0.999722i \(0.507511\pi\)
\(450\) 0 0
\(451\) −3.04789 −0.143520
\(452\) 0 0
\(453\) −0.859951 −0.0404040
\(454\) 0 0
\(455\) −27.5122 −1.28979
\(456\) 0 0
\(457\) −22.5930 −1.05686 −0.528429 0.848977i \(-0.677219\pi\)
−0.528429 + 0.848977i \(0.677219\pi\)
\(458\) 0 0
\(459\) −3.73184 −0.174187
\(460\) 0 0
\(461\) 16.4897 0.768002 0.384001 0.923333i \(-0.374546\pi\)
0.384001 + 0.923333i \(0.374546\pi\)
\(462\) 0 0
\(463\) −11.7668 −0.546850 −0.273425 0.961893i \(-0.588157\pi\)
−0.273425 + 0.961893i \(0.588157\pi\)
\(464\) 0 0
\(465\) −12.7025 −0.589065
\(466\) 0 0
\(467\) 24.6075 1.13870 0.569349 0.822096i \(-0.307195\pi\)
0.569349 + 0.822096i \(0.307195\pi\)
\(468\) 0 0
\(469\) 14.6780 0.677767
\(470\) 0 0
\(471\) 6.36080 0.293090
\(472\) 0 0
\(473\) −3.48629 −0.160300
\(474\) 0 0
\(475\) −0.986809 −0.0452779
\(476\) 0 0
\(477\) −2.46673 −0.112944
\(478\) 0 0
\(479\) 1.30934 0.0598252 0.0299126 0.999553i \(-0.490477\pi\)
0.0299126 + 0.999553i \(0.490477\pi\)
\(480\) 0 0
\(481\) −46.3704 −2.11431
\(482\) 0 0
\(483\) −1.72309 −0.0784031
\(484\) 0 0
\(485\) −18.3550 −0.833460
\(486\) 0 0
\(487\) −2.87718 −0.130377 −0.0651886 0.997873i \(-0.520765\pi\)
−0.0651886 + 0.997873i \(0.520765\pi\)
\(488\) 0 0
\(489\) −0.722678 −0.0326806
\(490\) 0 0
\(491\) 2.52068 0.113757 0.0568783 0.998381i \(-0.481885\pi\)
0.0568783 + 0.998381i \(0.481885\pi\)
\(492\) 0 0
\(493\) −14.4787 −0.652090
\(494\) 0 0
\(495\) −0.608366 −0.0273440
\(496\) 0 0
\(497\) −31.8520 −1.42876
\(498\) 0 0
\(499\) 10.9276 0.489185 0.244592 0.969626i \(-0.421346\pi\)
0.244592 + 0.969626i \(0.421346\pi\)
\(500\) 0 0
\(501\) 1.00000 0.0446767
\(502\) 0 0
\(503\) −4.15303 −0.185175 −0.0925873 0.995705i \(-0.529514\pi\)
−0.0925873 + 0.995705i \(0.529514\pi\)
\(504\) 0 0
\(505\) −25.3422 −1.12771
\(506\) 0 0
\(507\) −13.2741 −0.589524
\(508\) 0 0
\(509\) 8.73062 0.386978 0.193489 0.981102i \(-0.438020\pi\)
0.193489 + 0.981102i \(0.438020\pi\)
\(510\) 0 0
\(511\) −8.11629 −0.359044
\(512\) 0 0
\(513\) −0.725453 −0.0320295
\(514\) 0 0
\(515\) 35.1468 1.54875
\(516\) 0 0
\(517\) 2.79488 0.122918
\(518\) 0 0
\(519\) −16.1210 −0.707634
\(520\) 0 0
\(521\) 23.9413 1.04889 0.524444 0.851445i \(-0.324273\pi\)
0.524444 + 0.851445i \(0.324273\pi\)
\(522\) 0 0
\(523\) 37.1193 1.62311 0.811557 0.584273i \(-0.198620\pi\)
0.811557 + 0.584273i \(0.198620\pi\)
\(524\) 0 0
\(525\) −3.82693 −0.167021
\(526\) 0 0
\(527\) 24.8472 1.08236
\(528\) 0 0
\(529\) −22.6249 −0.983691
\(530\) 0 0
\(531\) 11.0871 0.481137
\(532\) 0 0
\(533\) 48.9929 2.12212
\(534\) 0 0
\(535\) −22.5818 −0.976295
\(536\) 0 0
\(537\) 11.5480 0.498333
\(538\) 0 0
\(539\) −0.291792 −0.0125684
\(540\) 0 0
\(541\) 33.9215 1.45840 0.729199 0.684302i \(-0.239893\pi\)
0.729199 + 0.684302i \(0.239893\pi\)
\(542\) 0 0
\(543\) 3.91786 0.168131
\(544\) 0 0
\(545\) −5.61108 −0.240352
\(546\) 0 0
\(547\) 16.6850 0.713397 0.356699 0.934219i \(-0.383902\pi\)
0.356699 + 0.934219i \(0.383902\pi\)
\(548\) 0 0
\(549\) 12.5460 0.535452
\(550\) 0 0
\(551\) −2.81460 −0.119906
\(552\) 0 0
\(553\) 25.2825 1.07512
\(554\) 0 0
\(555\) 17.2588 0.732597
\(556\) 0 0
\(557\) 30.4280 1.28928 0.644638 0.764488i \(-0.277008\pi\)
0.644638 + 0.764488i \(0.277008\pi\)
\(558\) 0 0
\(559\) 56.0398 2.37023
\(560\) 0 0
\(561\) 1.19002 0.0502425
\(562\) 0 0
\(563\) 32.0419 1.35041 0.675203 0.737632i \(-0.264056\pi\)
0.675203 + 0.737632i \(0.264056\pi\)
\(564\) 0 0
\(565\) 32.3912 1.36271
\(566\) 0 0
\(567\) −2.81337 −0.118150
\(568\) 0 0
\(569\) −27.2824 −1.14374 −0.571869 0.820345i \(-0.693782\pi\)
−0.571869 + 0.820345i \(0.693782\pi\)
\(570\) 0 0
\(571\) 12.3064 0.515007 0.257503 0.966277i \(-0.417100\pi\)
0.257503 + 0.966277i \(0.417100\pi\)
\(572\) 0 0
\(573\) −19.5571 −0.817008
\(574\) 0 0
\(575\) 0.833113 0.0347432
\(576\) 0 0
\(577\) 4.05674 0.168884 0.0844422 0.996428i \(-0.473089\pi\)
0.0844422 + 0.996428i \(0.473089\pi\)
\(578\) 0 0
\(579\) −22.0005 −0.914308
\(580\) 0 0
\(581\) 11.3886 0.472477
\(582\) 0 0
\(583\) 0.786596 0.0325775
\(584\) 0 0
\(585\) 9.77910 0.404316
\(586\) 0 0
\(587\) 21.5748 0.890489 0.445244 0.895409i \(-0.353117\pi\)
0.445244 + 0.895409i \(0.353117\pi\)
\(588\) 0 0
\(589\) 4.83019 0.199025
\(590\) 0 0
\(591\) −10.3862 −0.427233
\(592\) 0 0
\(593\) −5.81937 −0.238973 −0.119486 0.992836i \(-0.538125\pi\)
−0.119486 + 0.992836i \(0.538125\pi\)
\(594\) 0 0
\(595\) −20.0302 −0.821156
\(596\) 0 0
\(597\) −17.3380 −0.709596
\(598\) 0 0
\(599\) 14.0478 0.573978 0.286989 0.957934i \(-0.407346\pi\)
0.286989 + 0.957934i \(0.407346\pi\)
\(600\) 0 0
\(601\) 26.3024 1.07290 0.536448 0.843933i \(-0.319766\pi\)
0.536448 + 0.843933i \(0.319766\pi\)
\(602\) 0 0
\(603\) −5.21723 −0.212462
\(604\) 0 0
\(605\) −20.7919 −0.845311
\(606\) 0 0
\(607\) −16.2583 −0.659906 −0.329953 0.943997i \(-0.607033\pi\)
−0.329953 + 0.943997i \(0.607033\pi\)
\(608\) 0 0
\(609\) −10.9153 −0.442309
\(610\) 0 0
\(611\) −44.9258 −1.81750
\(612\) 0 0
\(613\) −10.3724 −0.418936 −0.209468 0.977816i \(-0.567173\pi\)
−0.209468 + 0.977816i \(0.567173\pi\)
\(614\) 0 0
\(615\) −18.2349 −0.735303
\(616\) 0 0
\(617\) −33.0424 −1.33024 −0.665118 0.746738i \(-0.731619\pi\)
−0.665118 + 0.746738i \(0.731619\pi\)
\(618\) 0 0
\(619\) 5.92967 0.238334 0.119167 0.992874i \(-0.461978\pi\)
0.119167 + 0.992874i \(0.461978\pi\)
\(620\) 0 0
\(621\) 0.612463 0.0245773
\(622\) 0 0
\(623\) 25.7506 1.03168
\(624\) 0 0
\(625\) −16.3483 −0.653934
\(626\) 0 0
\(627\) 0.231334 0.00923859
\(628\) 0 0
\(629\) −33.7598 −1.34609
\(630\) 0 0
\(631\) 35.5703 1.41603 0.708016 0.706196i \(-0.249590\pi\)
0.708016 + 0.706196i \(0.249590\pi\)
\(632\) 0 0
\(633\) −20.2222 −0.803761
\(634\) 0 0
\(635\) −31.8777 −1.26503
\(636\) 0 0
\(637\) 4.69037 0.185839
\(638\) 0 0
\(639\) 11.3217 0.447878
\(640\) 0 0
\(641\) −1.75263 −0.0692248 −0.0346124 0.999401i \(-0.511020\pi\)
−0.0346124 + 0.999401i \(0.511020\pi\)
\(642\) 0 0
\(643\) 44.6647 1.76140 0.880701 0.473672i \(-0.157072\pi\)
0.880701 + 0.473672i \(0.157072\pi\)
\(644\) 0 0
\(645\) −20.8577 −0.821273
\(646\) 0 0
\(647\) −16.7568 −0.658776 −0.329388 0.944195i \(-0.606842\pi\)
−0.329388 + 0.944195i \(0.606842\pi\)
\(648\) 0 0
\(649\) −3.53547 −0.138779
\(650\) 0 0
\(651\) 18.7319 0.734161
\(652\) 0 0
\(653\) −29.8135 −1.16669 −0.583346 0.812224i \(-0.698257\pi\)
−0.583346 + 0.812224i \(0.698257\pi\)
\(654\) 0 0
\(655\) 0.279562 0.0109234
\(656\) 0 0
\(657\) 2.88490 0.112551
\(658\) 0 0
\(659\) −40.4834 −1.57701 −0.788504 0.615030i \(-0.789144\pi\)
−0.788504 + 0.615030i \(0.789144\pi\)
\(660\) 0 0
\(661\) 16.4694 0.640586 0.320293 0.947319i \(-0.396219\pi\)
0.320293 + 0.947319i \(0.396219\pi\)
\(662\) 0 0
\(663\) −19.1288 −0.742899
\(664\) 0 0
\(665\) −3.89377 −0.150994
\(666\) 0 0
\(667\) 2.37623 0.0920079
\(668\) 0 0
\(669\) 9.60888 0.371501
\(670\) 0 0
\(671\) −4.00071 −0.154446
\(672\) 0 0
\(673\) 7.04173 0.271439 0.135719 0.990747i \(-0.456665\pi\)
0.135719 + 0.990747i \(0.456665\pi\)
\(674\) 0 0
\(675\) 1.36027 0.0523567
\(676\) 0 0
\(677\) 13.0327 0.500888 0.250444 0.968131i \(-0.419423\pi\)
0.250444 + 0.968131i \(0.419423\pi\)
\(678\) 0 0
\(679\) 27.0675 1.03875
\(680\) 0 0
\(681\) 23.3573 0.895054
\(682\) 0 0
\(683\) −44.8287 −1.71532 −0.857661 0.514216i \(-0.828083\pi\)
−0.857661 + 0.514216i \(0.828083\pi\)
\(684\) 0 0
\(685\) 22.3803 0.855109
\(686\) 0 0
\(687\) 14.5969 0.556907
\(688\) 0 0
\(689\) −12.6440 −0.481699
\(690\) 0 0
\(691\) 10.2680 0.390611 0.195306 0.980742i \(-0.437430\pi\)
0.195306 + 0.980742i \(0.437430\pi\)
\(692\) 0 0
\(693\) 0.897133 0.0340793
\(694\) 0 0
\(695\) 17.3610 0.658542
\(696\) 0 0
\(697\) 35.6691 1.35106
\(698\) 0 0
\(699\) −18.0225 −0.681672
\(700\) 0 0
\(701\) 29.6121 1.11843 0.559217 0.829021i \(-0.311102\pi\)
0.559217 + 0.829021i \(0.311102\pi\)
\(702\) 0 0
\(703\) −6.56275 −0.247519
\(704\) 0 0
\(705\) 16.7212 0.629756
\(706\) 0 0
\(707\) 37.3712 1.40549
\(708\) 0 0
\(709\) 29.7383 1.11685 0.558423 0.829556i \(-0.311406\pi\)
0.558423 + 0.829556i \(0.311406\pi\)
\(710\) 0 0
\(711\) −8.98655 −0.337022
\(712\) 0 0
\(713\) −4.07789 −0.152718
\(714\) 0 0
\(715\) −3.11838 −0.116621
\(716\) 0 0
\(717\) −9.44595 −0.352765
\(718\) 0 0
\(719\) −1.84535 −0.0688200 −0.0344100 0.999408i \(-0.510955\pi\)
−0.0344100 + 0.999408i \(0.510955\pi\)
\(720\) 0 0
\(721\) −51.8295 −1.93023
\(722\) 0 0
\(723\) −5.88509 −0.218869
\(724\) 0 0
\(725\) 5.27755 0.196003
\(726\) 0 0
\(727\) −34.6845 −1.28638 −0.643189 0.765708i \(-0.722389\pi\)
−0.643189 + 0.765708i \(0.722389\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 40.7996 1.50903
\(732\) 0 0
\(733\) −27.7444 −1.02476 −0.512381 0.858758i \(-0.671236\pi\)
−0.512381 + 0.858758i \(0.671236\pi\)
\(734\) 0 0
\(735\) −1.74573 −0.0643923
\(736\) 0 0
\(737\) 1.66368 0.0612826
\(738\) 0 0
\(739\) −3.98690 −0.146660 −0.0733302 0.997308i \(-0.523363\pi\)
−0.0733302 + 0.997308i \(0.523363\pi\)
\(740\) 0 0
\(741\) −3.71854 −0.136604
\(742\) 0 0
\(743\) −29.3305 −1.07603 −0.538016 0.842935i \(-0.680826\pi\)
−0.538016 + 0.842935i \(0.680826\pi\)
\(744\) 0 0
\(745\) 44.5459 1.63203
\(746\) 0 0
\(747\) −4.04802 −0.148109
\(748\) 0 0
\(749\) 33.3004 1.21677
\(750\) 0 0
\(751\) −17.0963 −0.623851 −0.311926 0.950107i \(-0.600974\pi\)
−0.311926 + 0.950107i \(0.600974\pi\)
\(752\) 0 0
\(753\) 3.14251 0.114519
\(754\) 0 0
\(755\) 1.64062 0.0597083
\(756\) 0 0
\(757\) −16.8059 −0.610819 −0.305410 0.952221i \(-0.598793\pi\)
−0.305410 + 0.952221i \(0.598793\pi\)
\(758\) 0 0
\(759\) −0.195304 −0.00708907
\(760\) 0 0
\(761\) −33.1343 −1.20112 −0.600559 0.799580i \(-0.705055\pi\)
−0.600559 + 0.799580i \(0.705055\pi\)
\(762\) 0 0
\(763\) 8.27444 0.299555
\(764\) 0 0
\(765\) 7.11963 0.257411
\(766\) 0 0
\(767\) 56.8303 2.05202
\(768\) 0 0
\(769\) 46.8068 1.68790 0.843949 0.536424i \(-0.180225\pi\)
0.843949 + 0.536424i \(0.180225\pi\)
\(770\) 0 0
\(771\) −1.52800 −0.0550297
\(772\) 0 0
\(773\) −36.5866 −1.31593 −0.657963 0.753050i \(-0.728582\pi\)
−0.657963 + 0.753050i \(0.728582\pi\)
\(774\) 0 0
\(775\) −9.05690 −0.325333
\(776\) 0 0
\(777\) −25.4509 −0.913047
\(778\) 0 0
\(779\) 6.93391 0.248433
\(780\) 0 0
\(781\) −3.61027 −0.129186
\(782\) 0 0
\(783\) 3.87979 0.138652
\(784\) 0 0
\(785\) −12.1352 −0.433124
\(786\) 0 0
\(787\) 47.3781 1.68885 0.844423 0.535677i \(-0.179943\pi\)
0.844423 + 0.535677i \(0.179943\pi\)
\(788\) 0 0
\(789\) −24.3283 −0.866112
\(790\) 0 0
\(791\) −47.7660 −1.69836
\(792\) 0 0
\(793\) 64.3088 2.28367
\(794\) 0 0
\(795\) 4.70604 0.166906
\(796\) 0 0
\(797\) 46.4164 1.64415 0.822077 0.569376i \(-0.192815\pi\)
0.822077 + 0.569376i \(0.192815\pi\)
\(798\) 0 0
\(799\) −32.7081 −1.15713
\(800\) 0 0
\(801\) −9.15294 −0.323403
\(802\) 0 0
\(803\) −0.919944 −0.0324641
\(804\) 0 0
\(805\) 3.28732 0.115863
\(806\) 0 0
\(807\) −0.163618 −0.00575961
\(808\) 0 0
\(809\) −36.9134 −1.29781 −0.648903 0.760871i \(-0.724772\pi\)
−0.648903 + 0.760871i \(0.724772\pi\)
\(810\) 0 0
\(811\) −7.35245 −0.258179 −0.129090 0.991633i \(-0.541206\pi\)
−0.129090 + 0.991633i \(0.541206\pi\)
\(812\) 0 0
\(813\) −8.88391 −0.311572
\(814\) 0 0
\(815\) 1.37873 0.0482948
\(816\) 0 0
\(817\) 7.93125 0.277479
\(818\) 0 0
\(819\) −14.4208 −0.503905
\(820\) 0 0
\(821\) 30.0419 1.04847 0.524235 0.851574i \(-0.324351\pi\)
0.524235 + 0.851574i \(0.324351\pi\)
\(822\) 0 0
\(823\) 27.5122 0.959016 0.479508 0.877538i \(-0.340815\pi\)
0.479508 + 0.877538i \(0.340815\pi\)
\(824\) 0 0
\(825\) −0.433765 −0.0151018
\(826\) 0 0
\(827\) −49.1971 −1.71075 −0.855375 0.518009i \(-0.826674\pi\)
−0.855375 + 0.518009i \(0.826674\pi\)
\(828\) 0 0
\(829\) 41.9008 1.45527 0.727637 0.685963i \(-0.240619\pi\)
0.727637 + 0.685963i \(0.240619\pi\)
\(830\) 0 0
\(831\) 5.45934 0.189382
\(832\) 0 0
\(833\) 3.41480 0.118316
\(834\) 0 0
\(835\) −1.90781 −0.0660224
\(836\) 0 0
\(837\) −6.65817 −0.230140
\(838\) 0 0
\(839\) −34.5508 −1.19283 −0.596413 0.802678i \(-0.703408\pi\)
−0.596413 + 0.802678i \(0.703408\pi\)
\(840\) 0 0
\(841\) −13.9472 −0.480939
\(842\) 0 0
\(843\) 8.18240 0.281817
\(844\) 0 0
\(845\) 25.3245 0.871188
\(846\) 0 0
\(847\) 30.6610 1.05352
\(848\) 0 0
\(849\) 9.17416 0.314856
\(850\) 0 0
\(851\) 5.54060 0.189929
\(852\) 0 0
\(853\) −27.6838 −0.947874 −0.473937 0.880559i \(-0.657168\pi\)
−0.473937 + 0.880559i \(0.657168\pi\)
\(854\) 0 0
\(855\) 1.38402 0.0473326
\(856\) 0 0
\(857\) −15.1732 −0.518307 −0.259154 0.965836i \(-0.583444\pi\)
−0.259154 + 0.965836i \(0.583444\pi\)
\(858\) 0 0
\(859\) −0.561134 −0.0191456 −0.00957281 0.999954i \(-0.503047\pi\)
−0.00957281 + 0.999954i \(0.503047\pi\)
\(860\) 0 0
\(861\) 26.8903 0.916419
\(862\) 0 0
\(863\) −21.7750 −0.741230 −0.370615 0.928786i \(-0.620853\pi\)
−0.370615 + 0.928786i \(0.620853\pi\)
\(864\) 0 0
\(865\) 30.7558 1.04573
\(866\) 0 0
\(867\) 3.07338 0.104378
\(868\) 0 0
\(869\) 2.86565 0.0972105
\(870\) 0 0
\(871\) −26.7426 −0.906140
\(872\) 0 0
\(873\) −9.62101 −0.325622
\(874\) 0 0
\(875\) 34.1379 1.15407
\(876\) 0 0
\(877\) 20.0002 0.675358 0.337679 0.941261i \(-0.390358\pi\)
0.337679 + 0.941261i \(0.390358\pi\)
\(878\) 0 0
\(879\) 30.6446 1.03362
\(880\) 0 0
\(881\) 5.92933 0.199764 0.0998821 0.994999i \(-0.468153\pi\)
0.0998821 + 0.994999i \(0.468153\pi\)
\(882\) 0 0
\(883\) −6.56436 −0.220908 −0.110454 0.993881i \(-0.535231\pi\)
−0.110454 + 0.993881i \(0.535231\pi\)
\(884\) 0 0
\(885\) −21.1520 −0.711016
\(886\) 0 0
\(887\) 1.74021 0.0584306 0.0292153 0.999573i \(-0.490699\pi\)
0.0292153 + 0.999573i \(0.490699\pi\)
\(888\) 0 0
\(889\) 47.0088 1.57662
\(890\) 0 0
\(891\) −0.318882 −0.0106830
\(892\) 0 0
\(893\) −6.35830 −0.212773
\(894\) 0 0
\(895\) −22.0314 −0.736427
\(896\) 0 0
\(897\) 3.13938 0.104821
\(898\) 0 0
\(899\) −25.8323 −0.861556
\(900\) 0 0
\(901\) −9.20543 −0.306677
\(902\) 0 0
\(903\) 30.7581 1.02357
\(904\) 0 0
\(905\) −7.47452 −0.248461
\(906\) 0 0
\(907\) −28.3718 −0.942071 −0.471036 0.882114i \(-0.656120\pi\)
−0.471036 + 0.882114i \(0.656120\pi\)
\(908\) 0 0
\(909\) −13.2834 −0.440583
\(910\) 0 0
\(911\) 0.700571 0.0232110 0.0116055 0.999933i \(-0.496306\pi\)
0.0116055 + 0.999933i \(0.496306\pi\)
\(912\) 0 0
\(913\) 1.29084 0.0427206
\(914\) 0 0
\(915\) −23.9354 −0.791281
\(916\) 0 0
\(917\) −0.412259 −0.0136140
\(918\) 0 0
\(919\) −17.8304 −0.588170 −0.294085 0.955779i \(-0.595015\pi\)
−0.294085 + 0.955779i \(0.595015\pi\)
\(920\) 0 0
\(921\) −25.0225 −0.824519
\(922\) 0 0
\(923\) 58.0328 1.91017
\(924\) 0 0
\(925\) 12.3056 0.404604
\(926\) 0 0
\(927\) 18.4226 0.605077
\(928\) 0 0
\(929\) −42.4672 −1.39330 −0.696652 0.717410i \(-0.745328\pi\)
−0.696652 + 0.717410i \(0.745328\pi\)
\(930\) 0 0
\(931\) 0.663822 0.0217559
\(932\) 0 0
\(933\) 26.4473 0.865847
\(934\) 0 0
\(935\) −2.27032 −0.0742475
\(936\) 0 0
\(937\) 44.9342 1.46794 0.733968 0.679184i \(-0.237666\pi\)
0.733968 + 0.679184i \(0.237666\pi\)
\(938\) 0 0
\(939\) −22.3927 −0.730757
\(940\) 0 0
\(941\) −39.9153 −1.30120 −0.650600 0.759420i \(-0.725483\pi\)
−0.650600 + 0.759420i \(0.725483\pi\)
\(942\) 0 0
\(943\) −5.85395 −0.190631
\(944\) 0 0
\(945\) 5.36737 0.174601
\(946\) 0 0
\(947\) −1.83044 −0.0594812 −0.0297406 0.999558i \(-0.509468\pi\)
−0.0297406 + 0.999558i \(0.509468\pi\)
\(948\) 0 0
\(949\) 14.7875 0.480023
\(950\) 0 0
\(951\) −24.1684 −0.783714
\(952\) 0 0
\(953\) 21.0500 0.681877 0.340938 0.940086i \(-0.389255\pi\)
0.340938 + 0.940086i \(0.389255\pi\)
\(954\) 0 0
\(955\) 37.3111 1.20736
\(956\) 0 0
\(957\) −1.23720 −0.0399929
\(958\) 0 0
\(959\) −33.0034 −1.06573
\(960\) 0 0
\(961\) 13.3313 0.430042
\(962\) 0 0
\(963\) −11.8365 −0.381426
\(964\) 0 0
\(965\) 41.9727 1.35115
\(966\) 0 0
\(967\) 49.7457 1.59972 0.799858 0.600190i \(-0.204908\pi\)
0.799858 + 0.600190i \(0.204908\pi\)
\(968\) 0 0
\(969\) −2.70727 −0.0869701
\(970\) 0 0
\(971\) 34.2216 1.09822 0.549112 0.835749i \(-0.314966\pi\)
0.549112 + 0.835749i \(0.314966\pi\)
\(972\) 0 0
\(973\) −25.6016 −0.820751
\(974\) 0 0
\(975\) 6.97249 0.223299
\(976\) 0 0
\(977\) 11.2596 0.360227 0.180113 0.983646i \(-0.442354\pi\)
0.180113 + 0.983646i \(0.442354\pi\)
\(978\) 0 0
\(979\) 2.91871 0.0932824
\(980\) 0 0
\(981\) −2.94111 −0.0939026
\(982\) 0 0
\(983\) −6.69693 −0.213599 −0.106799 0.994281i \(-0.534060\pi\)
−0.106799 + 0.994281i \(0.534060\pi\)
\(984\) 0 0
\(985\) 19.8150 0.631357
\(986\) 0 0
\(987\) −24.6581 −0.784875
\(988\) 0 0
\(989\) −6.69596 −0.212919
\(990\) 0 0
\(991\) −53.4452 −1.69774 −0.848872 0.528599i \(-0.822718\pi\)
−0.848872 + 0.528599i \(0.822718\pi\)
\(992\) 0 0
\(993\) −28.8009 −0.913969
\(994\) 0 0
\(995\) 33.0775 1.04863
\(996\) 0 0
\(997\) 52.2893 1.65602 0.828009 0.560714i \(-0.189473\pi\)
0.828009 + 0.560714i \(0.189473\pi\)
\(998\) 0 0
\(999\) 9.04643 0.286216
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 8016.2.a.bb.1.6 9
4.3 odd 2 2004.2.a.d.1.6 9
12.11 even 2 6012.2.a.h.1.4 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2004.2.a.d.1.6 9 4.3 odd 2
6012.2.a.h.1.4 9 12.11 even 2
8016.2.a.bb.1.6 9 1.1 even 1 trivial