Properties

Label 8016.2.a.ba.1.2
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: \(x^{9} - 3 x^{8} - 16 x^{7} + 45 x^{6} + 67 x^{5} - 166 x^{4} - 83 x^{3} + 152 x^{2} + 51 x - 10\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 4008)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.99382\) of defining polynomial
Character \(\chi\) \(=\) 8016.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -2.99382 q^{5} +4.65862 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -2.99382 q^{5} +4.65862 q^{7} +1.00000 q^{9} -0.962769 q^{11} +0.976361 q^{13} +2.99382 q^{15} +2.56202 q^{17} -8.42193 q^{19} -4.65862 q^{21} -6.29084 q^{23} +3.96298 q^{25} -1.00000 q^{27} -3.04318 q^{29} +10.1421 q^{31} +0.962769 q^{33} -13.9471 q^{35} +10.7136 q^{37} -0.976361 q^{39} +0.413133 q^{41} +5.35521 q^{43} -2.99382 q^{45} +8.47126 q^{47} +14.7028 q^{49} -2.56202 q^{51} -13.1750 q^{53} +2.88236 q^{55} +8.42193 q^{57} +0.101385 q^{59} -14.3711 q^{61} +4.65862 q^{63} -2.92305 q^{65} +11.7656 q^{67} +6.29084 q^{69} -12.5261 q^{71} -11.0294 q^{73} -3.96298 q^{75} -4.48518 q^{77} +1.47183 q^{79} +1.00000 q^{81} +4.56878 q^{83} -7.67023 q^{85} +3.04318 q^{87} +8.20803 q^{89} +4.54850 q^{91} -10.1421 q^{93} +25.2138 q^{95} +11.1603 q^{97} -0.962769 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9q - 9q^{3} - 6q^{5} + 11q^{7} + 9q^{9} + O(q^{10}) \) \( 9q - 9q^{3} - 6q^{5} + 11q^{7} + 9q^{9} - q^{11} - 4q^{13} + 6q^{15} - 9q^{17} + 8q^{19} - 11q^{21} + 7q^{23} - q^{25} - 9q^{27} - 9q^{29} + 25q^{31} + q^{33} - 5q^{35} - 6q^{37} + 4q^{39} - 4q^{41} + 24q^{43} - 6q^{45} + 16q^{47} + 4q^{49} + 9q^{51} - 26q^{53} + 29q^{55} - 8q^{57} + 4q^{59} - 20q^{61} + 11q^{63} - 8q^{65} + 25q^{67} - 7q^{69} + 15q^{71} - 10q^{73} + q^{75} - 20q^{77} + 34q^{79} + 9q^{81} + 4q^{83} - 13q^{85} + 9q^{87} + 13q^{89} + 21q^{91} - 25q^{93} + 7q^{95} - 4q^{97} - q^{99} + O(q^{100}) \)

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
\(5\) −2.99382 −1.33888 −0.669439 0.742867i \(-0.733466\pi\)
−0.669439 + 0.742867i \(0.733466\pi\)
\(6\) 0 0
\(7\) 4.65862 1.76079 0.880397 0.474238i \(-0.157276\pi\)
0.880397 + 0.474238i \(0.157276\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −0.962769 −0.290286 −0.145143 0.989411i \(-0.546364\pi\)
−0.145143 + 0.989411i \(0.546364\pi\)
\(12\) 0 0
\(13\) 0.976361 0.270794 0.135397 0.990791i \(-0.456769\pi\)
0.135397 + 0.990791i \(0.456769\pi\)
\(14\) 0 0
\(15\) 2.99382 0.773002
\(16\) 0 0
\(17\) 2.56202 0.621381 0.310690 0.950511i \(-0.399440\pi\)
0.310690 + 0.950511i \(0.399440\pi\)
\(18\) 0 0
\(19\) −8.42193 −1.93212 −0.966062 0.258311i \(-0.916834\pi\)
−0.966062 + 0.258311i \(0.916834\pi\)
\(20\) 0 0
\(21\) −4.65862 −1.01659
\(22\) 0 0
\(23\) −6.29084 −1.31173 −0.655865 0.754878i \(-0.727696\pi\)
−0.655865 + 0.754878i \(0.727696\pi\)
\(24\) 0 0
\(25\) 3.96298 0.792596
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −3.04318 −0.565104 −0.282552 0.959252i \(-0.591181\pi\)
−0.282552 + 0.959252i \(0.591181\pi\)
\(30\) 0 0
\(31\) 10.1421 1.82158 0.910788 0.412875i \(-0.135475\pi\)
0.910788 + 0.412875i \(0.135475\pi\)
\(32\) 0 0
\(33\) 0.962769 0.167597
\(34\) 0 0
\(35\) −13.9471 −2.35749
\(36\) 0 0
\(37\) 10.7136 1.76130 0.880651 0.473766i \(-0.157106\pi\)
0.880651 + 0.473766i \(0.157106\pi\)
\(38\) 0 0
\(39\) −0.976361 −0.156343
\(40\) 0 0
\(41\) 0.413133 0.0645205 0.0322602 0.999480i \(-0.489729\pi\)
0.0322602 + 0.999480i \(0.489729\pi\)
\(42\) 0 0
\(43\) 5.35521 0.816662 0.408331 0.912834i \(-0.366111\pi\)
0.408331 + 0.912834i \(0.366111\pi\)
\(44\) 0 0
\(45\) −2.99382 −0.446293
\(46\) 0 0
\(47\) 8.47126 1.23566 0.617830 0.786312i \(-0.288012\pi\)
0.617830 + 0.786312i \(0.288012\pi\)
\(48\) 0 0
\(49\) 14.7028 2.10039
\(50\) 0 0
\(51\) −2.56202 −0.358754
\(52\) 0 0
\(53\) −13.1750 −1.80973 −0.904865 0.425699i \(-0.860028\pi\)
−0.904865 + 0.425699i \(0.860028\pi\)
\(54\) 0 0
\(55\) 2.88236 0.388658
\(56\) 0 0
\(57\) 8.42193 1.11551
\(58\) 0 0
\(59\) 0.101385 0.0131992 0.00659962 0.999978i \(-0.497899\pi\)
0.00659962 + 0.999978i \(0.497899\pi\)
\(60\) 0 0
\(61\) −14.3711 −1.84003 −0.920014 0.391886i \(-0.871823\pi\)
−0.920014 + 0.391886i \(0.871823\pi\)
\(62\) 0 0
\(63\) 4.65862 0.586931
\(64\) 0 0
\(65\) −2.92305 −0.362560
\(66\) 0 0
\(67\) 11.7656 1.43740 0.718698 0.695323i \(-0.244739\pi\)
0.718698 + 0.695323i \(0.244739\pi\)
\(68\) 0 0
\(69\) 6.29084 0.757328
\(70\) 0 0
\(71\) −12.5261 −1.48657 −0.743285 0.668975i \(-0.766733\pi\)
−0.743285 + 0.668975i \(0.766733\pi\)
\(72\) 0 0
\(73\) −11.0294 −1.29090 −0.645449 0.763803i \(-0.723330\pi\)
−0.645449 + 0.763803i \(0.723330\pi\)
\(74\) 0 0
\(75\) −3.96298 −0.457606
\(76\) 0 0
\(77\) −4.48518 −0.511134
\(78\) 0 0
\(79\) 1.47183 0.165594 0.0827969 0.996566i \(-0.473615\pi\)
0.0827969 + 0.996566i \(0.473615\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 4.56878 0.501489 0.250744 0.968053i \(-0.419325\pi\)
0.250744 + 0.968053i \(0.419325\pi\)
\(84\) 0 0
\(85\) −7.67023 −0.831953
\(86\) 0 0
\(87\) 3.04318 0.326263
\(88\) 0 0
\(89\) 8.20803 0.870049 0.435025 0.900419i \(-0.356740\pi\)
0.435025 + 0.900419i \(0.356740\pi\)
\(90\) 0 0
\(91\) 4.54850 0.476812
\(92\) 0 0
\(93\) −10.1421 −1.05169
\(94\) 0 0
\(95\) 25.2138 2.58688
\(96\) 0 0
\(97\) 11.1603 1.13316 0.566579 0.824007i \(-0.308266\pi\)
0.566579 + 0.824007i \(0.308266\pi\)
\(98\) 0 0
\(99\) −0.962769 −0.0967620
\(100\) 0 0
\(101\) −5.03063 −0.500566 −0.250283 0.968173i \(-0.580524\pi\)
−0.250283 + 0.968173i \(0.580524\pi\)
\(102\) 0 0
\(103\) −7.19615 −0.709057 −0.354529 0.935045i \(-0.615359\pi\)
−0.354529 + 0.935045i \(0.615359\pi\)
\(104\) 0 0
\(105\) 13.9471 1.36110
\(106\) 0 0
\(107\) 14.9736 1.44755 0.723777 0.690034i \(-0.242405\pi\)
0.723777 + 0.690034i \(0.242405\pi\)
\(108\) 0 0
\(109\) −1.32527 −0.126938 −0.0634690 0.997984i \(-0.520216\pi\)
−0.0634690 + 0.997984i \(0.520216\pi\)
\(110\) 0 0
\(111\) −10.7136 −1.01689
\(112\) 0 0
\(113\) 8.09741 0.761740 0.380870 0.924629i \(-0.375625\pi\)
0.380870 + 0.924629i \(0.375625\pi\)
\(114\) 0 0
\(115\) 18.8337 1.75625
\(116\) 0 0
\(117\) 0.976361 0.0902646
\(118\) 0 0
\(119\) 11.9355 1.09412
\(120\) 0 0
\(121\) −10.0731 −0.915734
\(122\) 0 0
\(123\) −0.413133 −0.0372509
\(124\) 0 0
\(125\) 3.10465 0.277688
\(126\) 0 0
\(127\) 15.8311 1.40479 0.702393 0.711789i \(-0.252115\pi\)
0.702393 + 0.711789i \(0.252115\pi\)
\(128\) 0 0
\(129\) −5.35521 −0.471500
\(130\) 0 0
\(131\) 5.80759 0.507412 0.253706 0.967281i \(-0.418351\pi\)
0.253706 + 0.967281i \(0.418351\pi\)
\(132\) 0 0
\(133\) −39.2346 −3.40207
\(134\) 0 0
\(135\) 2.99382 0.257667
\(136\) 0 0
\(137\) −20.6048 −1.76039 −0.880195 0.474612i \(-0.842589\pi\)
−0.880195 + 0.474612i \(0.842589\pi\)
\(138\) 0 0
\(139\) 7.58360 0.643233 0.321616 0.946870i \(-0.395774\pi\)
0.321616 + 0.946870i \(0.395774\pi\)
\(140\) 0 0
\(141\) −8.47126 −0.713409
\(142\) 0 0
\(143\) −0.940010 −0.0786076
\(144\) 0 0
\(145\) 9.11074 0.756605
\(146\) 0 0
\(147\) −14.7028 −1.21266
\(148\) 0 0
\(149\) 12.7087 1.04114 0.520569 0.853819i \(-0.325720\pi\)
0.520569 + 0.853819i \(0.325720\pi\)
\(150\) 0 0
\(151\) −14.4023 −1.17204 −0.586021 0.810296i \(-0.699306\pi\)
−0.586021 + 0.810296i \(0.699306\pi\)
\(152\) 0 0
\(153\) 2.56202 0.207127
\(154\) 0 0
\(155\) −30.3637 −2.43887
\(156\) 0 0
\(157\) −7.84981 −0.626483 −0.313241 0.949674i \(-0.601415\pi\)
−0.313241 + 0.949674i \(0.601415\pi\)
\(158\) 0 0
\(159\) 13.1750 1.04485
\(160\) 0 0
\(161\) −29.3067 −2.30969
\(162\) 0 0
\(163\) −0.151488 −0.0118654 −0.00593272 0.999982i \(-0.501888\pi\)
−0.00593272 + 0.999982i \(0.501888\pi\)
\(164\) 0 0
\(165\) −2.88236 −0.224392
\(166\) 0 0
\(167\) 1.00000 0.0773823
\(168\) 0 0
\(169\) −12.0467 −0.926671
\(170\) 0 0
\(171\) −8.42193 −0.644041
\(172\) 0 0
\(173\) 10.9592 0.833212 0.416606 0.909087i \(-0.363220\pi\)
0.416606 + 0.909087i \(0.363220\pi\)
\(174\) 0 0
\(175\) 18.4620 1.39560
\(176\) 0 0
\(177\) −0.101385 −0.00762059
\(178\) 0 0
\(179\) −14.6334 −1.09375 −0.546874 0.837215i \(-0.684182\pi\)
−0.546874 + 0.837215i \(0.684182\pi\)
\(180\) 0 0
\(181\) −4.60022 −0.341932 −0.170966 0.985277i \(-0.554689\pi\)
−0.170966 + 0.985277i \(0.554689\pi\)
\(182\) 0 0
\(183\) 14.3711 1.06234
\(184\) 0 0
\(185\) −32.0746 −2.35817
\(186\) 0 0
\(187\) −2.46663 −0.180378
\(188\) 0 0
\(189\) −4.65862 −0.338865
\(190\) 0 0
\(191\) 17.2571 1.24868 0.624339 0.781153i \(-0.285368\pi\)
0.624339 + 0.781153i \(0.285368\pi\)
\(192\) 0 0
\(193\) −11.6499 −0.838580 −0.419290 0.907852i \(-0.637721\pi\)
−0.419290 + 0.907852i \(0.637721\pi\)
\(194\) 0 0
\(195\) 2.92305 0.209324
\(196\) 0 0
\(197\) 10.8741 0.774747 0.387373 0.921923i \(-0.373382\pi\)
0.387373 + 0.921923i \(0.373382\pi\)
\(198\) 0 0
\(199\) −9.11256 −0.645972 −0.322986 0.946404i \(-0.604687\pi\)
−0.322986 + 0.946404i \(0.604687\pi\)
\(200\) 0 0
\(201\) −11.7656 −0.829881
\(202\) 0 0
\(203\) −14.1770 −0.995031
\(204\) 0 0
\(205\) −1.23685 −0.0863851
\(206\) 0 0
\(207\) −6.29084 −0.437244
\(208\) 0 0
\(209\) 8.10838 0.560868
\(210\) 0 0
\(211\) 18.1269 1.24791 0.623953 0.781462i \(-0.285526\pi\)
0.623953 + 0.781462i \(0.285526\pi\)
\(212\) 0 0
\(213\) 12.5261 0.858272
\(214\) 0 0
\(215\) −16.0326 −1.09341
\(216\) 0 0
\(217\) 47.2482 3.20742
\(218\) 0 0
\(219\) 11.0294 0.745301
\(220\) 0 0
\(221\) 2.50146 0.168266
\(222\) 0 0
\(223\) 14.2420 0.953716 0.476858 0.878980i \(-0.341776\pi\)
0.476858 + 0.878980i \(0.341776\pi\)
\(224\) 0 0
\(225\) 3.96298 0.264199
\(226\) 0 0
\(227\) −10.5377 −0.699410 −0.349705 0.936860i \(-0.613718\pi\)
−0.349705 + 0.936860i \(0.613718\pi\)
\(228\) 0 0
\(229\) 20.2304 1.33686 0.668431 0.743774i \(-0.266966\pi\)
0.668431 + 0.743774i \(0.266966\pi\)
\(230\) 0 0
\(231\) 4.48518 0.295103
\(232\) 0 0
\(233\) 23.4478 1.53612 0.768059 0.640379i \(-0.221223\pi\)
0.768059 + 0.640379i \(0.221223\pi\)
\(234\) 0 0
\(235\) −25.3615 −1.65440
\(236\) 0 0
\(237\) −1.47183 −0.0956056
\(238\) 0 0
\(239\) −8.49855 −0.549726 −0.274863 0.961483i \(-0.588632\pi\)
−0.274863 + 0.961483i \(0.588632\pi\)
\(240\) 0 0
\(241\) −18.5989 −1.19806 −0.599031 0.800726i \(-0.704447\pi\)
−0.599031 + 0.800726i \(0.704447\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −44.0175 −2.81217
\(246\) 0 0
\(247\) −8.22285 −0.523207
\(248\) 0 0
\(249\) −4.56878 −0.289535
\(250\) 0 0
\(251\) 2.98372 0.188331 0.0941655 0.995557i \(-0.469982\pi\)
0.0941655 + 0.995557i \(0.469982\pi\)
\(252\) 0 0
\(253\) 6.05663 0.380777
\(254\) 0 0
\(255\) 7.67023 0.480329
\(256\) 0 0
\(257\) 4.66532 0.291014 0.145507 0.989357i \(-0.453519\pi\)
0.145507 + 0.989357i \(0.453519\pi\)
\(258\) 0 0
\(259\) 49.9105 3.10129
\(260\) 0 0
\(261\) −3.04318 −0.188368
\(262\) 0 0
\(263\) 11.4032 0.703151 0.351576 0.936159i \(-0.385646\pi\)
0.351576 + 0.936159i \(0.385646\pi\)
\(264\) 0 0
\(265\) 39.4437 2.42301
\(266\) 0 0
\(267\) −8.20803 −0.502323
\(268\) 0 0
\(269\) −17.5202 −1.06823 −0.534114 0.845413i \(-0.679355\pi\)
−0.534114 + 0.845413i \(0.679355\pi\)
\(270\) 0 0
\(271\) 4.07499 0.247538 0.123769 0.992311i \(-0.460502\pi\)
0.123769 + 0.992311i \(0.460502\pi\)
\(272\) 0 0
\(273\) −4.54850 −0.275288
\(274\) 0 0
\(275\) −3.81544 −0.230079
\(276\) 0 0
\(277\) 22.9380 1.37821 0.689105 0.724661i \(-0.258004\pi\)
0.689105 + 0.724661i \(0.258004\pi\)
\(278\) 0 0
\(279\) 10.1421 0.607192
\(280\) 0 0
\(281\) −12.4485 −0.742615 −0.371308 0.928510i \(-0.621091\pi\)
−0.371308 + 0.928510i \(0.621091\pi\)
\(282\) 0 0
\(283\) 21.2448 1.26287 0.631436 0.775428i \(-0.282466\pi\)
0.631436 + 0.775428i \(0.282466\pi\)
\(284\) 0 0
\(285\) −25.2138 −1.49354
\(286\) 0 0
\(287\) 1.92463 0.113607
\(288\) 0 0
\(289\) −10.4361 −0.613886
\(290\) 0 0
\(291\) −11.1603 −0.654229
\(292\) 0 0
\(293\) 3.52870 0.206149 0.103074 0.994674i \(-0.467132\pi\)
0.103074 + 0.994674i \(0.467132\pi\)
\(294\) 0 0
\(295\) −0.303530 −0.0176722
\(296\) 0 0
\(297\) 0.962769 0.0558655
\(298\) 0 0
\(299\) −6.14213 −0.355209
\(300\) 0 0
\(301\) 24.9479 1.43797
\(302\) 0 0
\(303\) 5.03063 0.289002
\(304\) 0 0
\(305\) 43.0245 2.46357
\(306\) 0 0
\(307\) 15.5946 0.890031 0.445016 0.895523i \(-0.353198\pi\)
0.445016 + 0.895523i \(0.353198\pi\)
\(308\) 0 0
\(309\) 7.19615 0.409374
\(310\) 0 0
\(311\) 0.111395 0.00631664 0.00315832 0.999995i \(-0.498995\pi\)
0.00315832 + 0.999995i \(0.498995\pi\)
\(312\) 0 0
\(313\) −15.8427 −0.895484 −0.447742 0.894163i \(-0.647772\pi\)
−0.447742 + 0.894163i \(0.647772\pi\)
\(314\) 0 0
\(315\) −13.9471 −0.785830
\(316\) 0 0
\(317\) 18.3265 1.02932 0.514658 0.857395i \(-0.327919\pi\)
0.514658 + 0.857395i \(0.327919\pi\)
\(318\) 0 0
\(319\) 2.92988 0.164042
\(320\) 0 0
\(321\) −14.9736 −0.835745
\(322\) 0 0
\(323\) −21.5771 −1.20058
\(324\) 0 0
\(325\) 3.86930 0.214630
\(326\) 0 0
\(327\) 1.32527 0.0732877
\(328\) 0 0
\(329\) 39.4644 2.17574
\(330\) 0 0
\(331\) 6.26802 0.344521 0.172261 0.985051i \(-0.444893\pi\)
0.172261 + 0.985051i \(0.444893\pi\)
\(332\) 0 0
\(333\) 10.7136 0.587101
\(334\) 0 0
\(335\) −35.2241 −1.92450
\(336\) 0 0
\(337\) −9.78190 −0.532854 −0.266427 0.963855i \(-0.585843\pi\)
−0.266427 + 0.963855i \(0.585843\pi\)
\(338\) 0 0
\(339\) −8.09741 −0.439791
\(340\) 0 0
\(341\) −9.76450 −0.528778
\(342\) 0 0
\(343\) 35.8843 1.93757
\(344\) 0 0
\(345\) −18.8337 −1.01397
\(346\) 0 0
\(347\) −2.68557 −0.144169 −0.0720845 0.997399i \(-0.522965\pi\)
−0.0720845 + 0.997399i \(0.522965\pi\)
\(348\) 0 0
\(349\) −16.1287 −0.863351 −0.431675 0.902029i \(-0.642077\pi\)
−0.431675 + 0.902029i \(0.642077\pi\)
\(350\) 0 0
\(351\) −0.976361 −0.0521143
\(352\) 0 0
\(353\) 20.8777 1.11121 0.555604 0.831447i \(-0.312487\pi\)
0.555604 + 0.831447i \(0.312487\pi\)
\(354\) 0 0
\(355\) 37.5008 1.99034
\(356\) 0 0
\(357\) −11.9355 −0.631692
\(358\) 0 0
\(359\) 5.52417 0.291555 0.145777 0.989317i \(-0.453432\pi\)
0.145777 + 0.989317i \(0.453432\pi\)
\(360\) 0 0
\(361\) 51.9289 2.73310
\(362\) 0 0
\(363\) 10.0731 0.528699
\(364\) 0 0
\(365\) 33.0202 1.72836
\(366\) 0 0
\(367\) 26.8197 1.39998 0.699988 0.714155i \(-0.253189\pi\)
0.699988 + 0.714155i \(0.253189\pi\)
\(368\) 0 0
\(369\) 0.413133 0.0215068
\(370\) 0 0
\(371\) −61.3775 −3.18656
\(372\) 0 0
\(373\) −2.13709 −0.110655 −0.0553273 0.998468i \(-0.517620\pi\)
−0.0553273 + 0.998468i \(0.517620\pi\)
\(374\) 0 0
\(375\) −3.10465 −0.160324
\(376\) 0 0
\(377\) −2.97124 −0.153027
\(378\) 0 0
\(379\) 15.5192 0.797167 0.398583 0.917132i \(-0.369502\pi\)
0.398583 + 0.917132i \(0.369502\pi\)
\(380\) 0 0
\(381\) −15.8311 −0.811054
\(382\) 0 0
\(383\) 31.7717 1.62346 0.811730 0.584033i \(-0.198526\pi\)
0.811730 + 0.584033i \(0.198526\pi\)
\(384\) 0 0
\(385\) 13.4278 0.684346
\(386\) 0 0
\(387\) 5.35521 0.272221
\(388\) 0 0
\(389\) 27.7661 1.40780 0.703898 0.710301i \(-0.251441\pi\)
0.703898 + 0.710301i \(0.251441\pi\)
\(390\) 0 0
\(391\) −16.1172 −0.815084
\(392\) 0 0
\(393\) −5.80759 −0.292954
\(394\) 0 0
\(395\) −4.40640 −0.221710
\(396\) 0 0
\(397\) 24.2186 1.21550 0.607748 0.794130i \(-0.292073\pi\)
0.607748 + 0.794130i \(0.292073\pi\)
\(398\) 0 0
\(399\) 39.2346 1.96419
\(400\) 0 0
\(401\) 36.8639 1.84089 0.920447 0.390867i \(-0.127825\pi\)
0.920447 + 0.390867i \(0.127825\pi\)
\(402\) 0 0
\(403\) 9.90235 0.493271
\(404\) 0 0
\(405\) −2.99382 −0.148764
\(406\) 0 0
\(407\) −10.3147 −0.511281
\(408\) 0 0
\(409\) −7.05203 −0.348700 −0.174350 0.984684i \(-0.555782\pi\)
−0.174350 + 0.984684i \(0.555782\pi\)
\(410\) 0 0
\(411\) 20.6048 1.01636
\(412\) 0 0
\(413\) 0.472316 0.0232411
\(414\) 0 0
\(415\) −13.6781 −0.671433
\(416\) 0 0
\(417\) −7.58360 −0.371371
\(418\) 0 0
\(419\) −27.1446 −1.32610 −0.663051 0.748574i \(-0.730739\pi\)
−0.663051 + 0.748574i \(0.730739\pi\)
\(420\) 0 0
\(421\) 22.1971 1.08182 0.540910 0.841080i \(-0.318080\pi\)
0.540910 + 0.841080i \(0.318080\pi\)
\(422\) 0 0
\(423\) 8.47126 0.411887
\(424\) 0 0
\(425\) 10.1532 0.492504
\(426\) 0 0
\(427\) −66.9494 −3.23991
\(428\) 0 0
\(429\) 0.940010 0.0453841
\(430\) 0 0
\(431\) −13.0366 −0.627952 −0.313976 0.949431i \(-0.601661\pi\)
−0.313976 + 0.949431i \(0.601661\pi\)
\(432\) 0 0
\(433\) 7.48494 0.359703 0.179852 0.983694i \(-0.442438\pi\)
0.179852 + 0.983694i \(0.442438\pi\)
\(434\) 0 0
\(435\) −9.11074 −0.436826
\(436\) 0 0
\(437\) 52.9810 2.53443
\(438\) 0 0
\(439\) 31.8500 1.52012 0.760059 0.649854i \(-0.225170\pi\)
0.760059 + 0.649854i \(0.225170\pi\)
\(440\) 0 0
\(441\) 14.7028 0.700131
\(442\) 0 0
\(443\) −8.08246 −0.384009 −0.192005 0.981394i \(-0.561499\pi\)
−0.192005 + 0.981394i \(0.561499\pi\)
\(444\) 0 0
\(445\) −24.5734 −1.16489
\(446\) 0 0
\(447\) −12.7087 −0.601102
\(448\) 0 0
\(449\) −37.0405 −1.74805 −0.874024 0.485884i \(-0.838498\pi\)
−0.874024 + 0.485884i \(0.838498\pi\)
\(450\) 0 0
\(451\) −0.397751 −0.0187294
\(452\) 0 0
\(453\) 14.4023 0.676679
\(454\) 0 0
\(455\) −13.6174 −0.638394
\(456\) 0 0
\(457\) 17.4599 0.816739 0.408370 0.912817i \(-0.366097\pi\)
0.408370 + 0.912817i \(0.366097\pi\)
\(458\) 0 0
\(459\) −2.56202 −0.119585
\(460\) 0 0
\(461\) −12.6672 −0.589972 −0.294986 0.955502i \(-0.595315\pi\)
−0.294986 + 0.955502i \(0.595315\pi\)
\(462\) 0 0
\(463\) 15.9913 0.743177 0.371588 0.928398i \(-0.378813\pi\)
0.371588 + 0.928398i \(0.378813\pi\)
\(464\) 0 0
\(465\) 30.3637 1.40808
\(466\) 0 0
\(467\) 30.8847 1.42917 0.714587 0.699547i \(-0.246615\pi\)
0.714587 + 0.699547i \(0.246615\pi\)
\(468\) 0 0
\(469\) 54.8114 2.53096
\(470\) 0 0
\(471\) 7.84981 0.361700
\(472\) 0 0
\(473\) −5.15583 −0.237065
\(474\) 0 0
\(475\) −33.3760 −1.53139
\(476\) 0 0
\(477\) −13.1750 −0.603243
\(478\) 0 0
\(479\) 18.4760 0.844188 0.422094 0.906552i \(-0.361295\pi\)
0.422094 + 0.906552i \(0.361295\pi\)
\(480\) 0 0
\(481\) 10.4603 0.476950
\(482\) 0 0
\(483\) 29.3067 1.33350
\(484\) 0 0
\(485\) −33.4120 −1.51716
\(486\) 0 0
\(487\) 11.9533 0.541656 0.270828 0.962628i \(-0.412703\pi\)
0.270828 + 0.962628i \(0.412703\pi\)
\(488\) 0 0
\(489\) 0.151488 0.00685051
\(490\) 0 0
\(491\) −6.63468 −0.299419 −0.149709 0.988730i \(-0.547834\pi\)
−0.149709 + 0.988730i \(0.547834\pi\)
\(492\) 0 0
\(493\) −7.79668 −0.351145
\(494\) 0 0
\(495\) 2.88236 0.129553
\(496\) 0 0
\(497\) −58.3542 −2.61754
\(498\) 0 0
\(499\) 42.3783 1.89711 0.948557 0.316605i \(-0.102543\pi\)
0.948557 + 0.316605i \(0.102543\pi\)
\(500\) 0 0
\(501\) −1.00000 −0.0446767
\(502\) 0 0
\(503\) 24.3222 1.08447 0.542237 0.840226i \(-0.317578\pi\)
0.542237 + 0.840226i \(0.317578\pi\)
\(504\) 0 0
\(505\) 15.0608 0.670197
\(506\) 0 0
\(507\) 12.0467 0.535014
\(508\) 0 0
\(509\) −7.02710 −0.311471 −0.155735 0.987799i \(-0.549775\pi\)
−0.155735 + 0.987799i \(0.549775\pi\)
\(510\) 0 0
\(511\) −51.3820 −2.27301
\(512\) 0 0
\(513\) 8.42193 0.371837
\(514\) 0 0
\(515\) 21.5440 0.949342
\(516\) 0 0
\(517\) −8.15587 −0.358695
\(518\) 0 0
\(519\) −10.9592 −0.481055
\(520\) 0 0
\(521\) 23.5312 1.03092 0.515461 0.856913i \(-0.327621\pi\)
0.515461 + 0.856913i \(0.327621\pi\)
\(522\) 0 0
\(523\) 43.3371 1.89500 0.947500 0.319755i \(-0.103601\pi\)
0.947500 + 0.319755i \(0.103601\pi\)
\(524\) 0 0
\(525\) −18.4620 −0.805749
\(526\) 0 0
\(527\) 25.9843 1.13189
\(528\) 0 0
\(529\) 16.5747 0.720638
\(530\) 0 0
\(531\) 0.101385 0.00439975
\(532\) 0 0
\(533\) 0.403367 0.0174717
\(534\) 0 0
\(535\) −44.8283 −1.93810
\(536\) 0 0
\(537\) 14.6334 0.631476
\(538\) 0 0
\(539\) −14.1554 −0.609715
\(540\) 0 0
\(541\) −25.0364 −1.07640 −0.538201 0.842817i \(-0.680896\pi\)
−0.538201 + 0.842817i \(0.680896\pi\)
\(542\) 0 0
\(543\) 4.60022 0.197414
\(544\) 0 0
\(545\) 3.96763 0.169955
\(546\) 0 0
\(547\) −4.78599 −0.204634 −0.102317 0.994752i \(-0.532626\pi\)
−0.102317 + 0.994752i \(0.532626\pi\)
\(548\) 0 0
\(549\) −14.3711 −0.613342
\(550\) 0 0
\(551\) 25.6294 1.09185
\(552\) 0 0
\(553\) 6.85670 0.291576
\(554\) 0 0
\(555\) 32.0746 1.36149
\(556\) 0 0
\(557\) −5.00703 −0.212155 −0.106077 0.994358i \(-0.533829\pi\)
−0.106077 + 0.994358i \(0.533829\pi\)
\(558\) 0 0
\(559\) 5.22862 0.221147
\(560\) 0 0
\(561\) 2.46663 0.104141
\(562\) 0 0
\(563\) 36.9366 1.55669 0.778346 0.627836i \(-0.216059\pi\)
0.778346 + 0.627836i \(0.216059\pi\)
\(564\) 0 0
\(565\) −24.2422 −1.01988
\(566\) 0 0
\(567\) 4.65862 0.195644
\(568\) 0 0
\(569\) −28.3118 −1.18689 −0.593447 0.804873i \(-0.702233\pi\)
−0.593447 + 0.804873i \(0.702233\pi\)
\(570\) 0 0
\(571\) −6.56835 −0.274877 −0.137438 0.990510i \(-0.543887\pi\)
−0.137438 + 0.990510i \(0.543887\pi\)
\(572\) 0 0
\(573\) −17.2571 −0.720925
\(574\) 0 0
\(575\) −24.9305 −1.03967
\(576\) 0 0
\(577\) −33.2317 −1.38345 −0.691727 0.722160i \(-0.743150\pi\)
−0.691727 + 0.722160i \(0.743150\pi\)
\(578\) 0 0
\(579\) 11.6499 0.484154
\(580\) 0 0
\(581\) 21.2842 0.883018
\(582\) 0 0
\(583\) 12.6845 0.525339
\(584\) 0 0
\(585\) −2.92305 −0.120853
\(586\) 0 0
\(587\) 13.5810 0.560546 0.280273 0.959920i \(-0.409575\pi\)
0.280273 + 0.959920i \(0.409575\pi\)
\(588\) 0 0
\(589\) −85.4161 −3.51951
\(590\) 0 0
\(591\) −10.8741 −0.447300
\(592\) 0 0
\(593\) 5.20329 0.213674 0.106837 0.994277i \(-0.465928\pi\)
0.106837 + 0.994277i \(0.465928\pi\)
\(594\) 0 0
\(595\) −35.7327 −1.46490
\(596\) 0 0
\(597\) 9.11256 0.372952
\(598\) 0 0
\(599\) 10.2141 0.417337 0.208668 0.977986i \(-0.433087\pi\)
0.208668 + 0.977986i \(0.433087\pi\)
\(600\) 0 0
\(601\) 17.4052 0.709971 0.354986 0.934872i \(-0.384486\pi\)
0.354986 + 0.934872i \(0.384486\pi\)
\(602\) 0 0
\(603\) 11.7656 0.479132
\(604\) 0 0
\(605\) 30.1570 1.22606
\(606\) 0 0
\(607\) −8.73864 −0.354691 −0.177345 0.984149i \(-0.556751\pi\)
−0.177345 + 0.984149i \(0.556751\pi\)
\(608\) 0 0
\(609\) 14.1770 0.574482
\(610\) 0 0
\(611\) 8.27101 0.334609
\(612\) 0 0
\(613\) 27.9839 1.13026 0.565130 0.825002i \(-0.308826\pi\)
0.565130 + 0.825002i \(0.308826\pi\)
\(614\) 0 0
\(615\) 1.23685 0.0498745
\(616\) 0 0
\(617\) −35.8615 −1.44373 −0.721864 0.692035i \(-0.756714\pi\)
−0.721864 + 0.692035i \(0.756714\pi\)
\(618\) 0 0
\(619\) −25.4287 −1.02207 −0.511033 0.859561i \(-0.670737\pi\)
−0.511033 + 0.859561i \(0.670737\pi\)
\(620\) 0 0
\(621\) 6.29084 0.252443
\(622\) 0 0
\(623\) 38.2381 1.53198
\(624\) 0 0
\(625\) −29.1097 −1.16439
\(626\) 0 0
\(627\) −8.10838 −0.323817
\(628\) 0 0
\(629\) 27.4484 1.09444
\(630\) 0 0
\(631\) 34.3659 1.36808 0.684042 0.729442i \(-0.260220\pi\)
0.684042 + 0.729442i \(0.260220\pi\)
\(632\) 0 0
\(633\) −18.1269 −0.720479
\(634\) 0 0
\(635\) −47.3956 −1.88084
\(636\) 0 0
\(637\) 14.3552 0.568774
\(638\) 0 0
\(639\) −12.5261 −0.495523
\(640\) 0 0
\(641\) −27.6147 −1.09071 −0.545357 0.838204i \(-0.683606\pi\)
−0.545357 + 0.838204i \(0.683606\pi\)
\(642\) 0 0
\(643\) 1.24094 0.0489379 0.0244689 0.999701i \(-0.492211\pi\)
0.0244689 + 0.999701i \(0.492211\pi\)
\(644\) 0 0
\(645\) 16.0326 0.631281
\(646\) 0 0
\(647\) 25.0085 0.983186 0.491593 0.870825i \(-0.336415\pi\)
0.491593 + 0.870825i \(0.336415\pi\)
\(648\) 0 0
\(649\) −0.0976106 −0.00383155
\(650\) 0 0
\(651\) −47.2482 −1.85180
\(652\) 0 0
\(653\) 27.7223 1.08486 0.542429 0.840101i \(-0.317505\pi\)
0.542429 + 0.840101i \(0.317505\pi\)
\(654\) 0 0
\(655\) −17.3869 −0.679363
\(656\) 0 0
\(657\) −11.0294 −0.430300
\(658\) 0 0
\(659\) −10.4052 −0.405330 −0.202665 0.979248i \(-0.564960\pi\)
−0.202665 + 0.979248i \(0.564960\pi\)
\(660\) 0 0
\(661\) −14.4554 −0.562251 −0.281126 0.959671i \(-0.590708\pi\)
−0.281126 + 0.959671i \(0.590708\pi\)
\(662\) 0 0
\(663\) −2.50146 −0.0971485
\(664\) 0 0
\(665\) 117.461 4.55496
\(666\) 0 0
\(667\) 19.1441 0.741264
\(668\) 0 0
\(669\) −14.2420 −0.550628
\(670\) 0 0
\(671\) 13.8360 0.534134
\(672\) 0 0
\(673\) 9.62407 0.370981 0.185490 0.982646i \(-0.440613\pi\)
0.185490 + 0.982646i \(0.440613\pi\)
\(674\) 0 0
\(675\) −3.96298 −0.152535
\(676\) 0 0
\(677\) 28.4117 1.09195 0.545974 0.837802i \(-0.316160\pi\)
0.545974 + 0.837802i \(0.316160\pi\)
\(678\) 0 0
\(679\) 51.9917 1.99526
\(680\) 0 0
\(681\) 10.5377 0.403805
\(682\) 0 0
\(683\) 3.05896 0.117048 0.0585239 0.998286i \(-0.481361\pi\)
0.0585239 + 0.998286i \(0.481361\pi\)
\(684\) 0 0
\(685\) 61.6873 2.35695
\(686\) 0 0
\(687\) −20.2304 −0.771838
\(688\) 0 0
\(689\) −12.8636 −0.490064
\(690\) 0 0
\(691\) 20.0725 0.763594 0.381797 0.924246i \(-0.375305\pi\)
0.381797 + 0.924246i \(0.375305\pi\)
\(692\) 0 0
\(693\) −4.48518 −0.170378
\(694\) 0 0
\(695\) −22.7040 −0.861211
\(696\) 0 0
\(697\) 1.05845 0.0400918
\(698\) 0 0
\(699\) −23.4478 −0.886878
\(700\) 0 0
\(701\) −25.9984 −0.981947 −0.490974 0.871174i \(-0.663359\pi\)
−0.490974 + 0.871174i \(0.663359\pi\)
\(702\) 0 0
\(703\) −90.2291 −3.40305
\(704\) 0 0
\(705\) 25.3615 0.955168
\(706\) 0 0
\(707\) −23.4358 −0.881393
\(708\) 0 0
\(709\) 9.35327 0.351270 0.175635 0.984455i \(-0.443802\pi\)
0.175635 + 0.984455i \(0.443802\pi\)
\(710\) 0 0
\(711\) 1.47183 0.0551979
\(712\) 0 0
\(713\) −63.8024 −2.38942
\(714\) 0 0
\(715\) 2.81423 0.105246
\(716\) 0 0
\(717\) 8.49855 0.317384
\(718\) 0 0
\(719\) 36.5227 1.36207 0.681034 0.732252i \(-0.261531\pi\)
0.681034 + 0.732252i \(0.261531\pi\)
\(720\) 0 0
\(721\) −33.5241 −1.24850
\(722\) 0 0
\(723\) 18.5989 0.691701
\(724\) 0 0
\(725\) −12.0601 −0.447899
\(726\) 0 0
\(727\) 26.4767 0.981966 0.490983 0.871169i \(-0.336638\pi\)
0.490983 + 0.871169i \(0.336638\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 13.7201 0.507458
\(732\) 0 0
\(733\) −27.7117 −1.02355 −0.511777 0.859118i \(-0.671013\pi\)
−0.511777 + 0.859118i \(0.671013\pi\)
\(734\) 0 0
\(735\) 44.0175 1.62361
\(736\) 0 0
\(737\) −11.3275 −0.417255
\(738\) 0 0
\(739\) −35.5867 −1.30908 −0.654538 0.756029i \(-0.727137\pi\)
−0.654538 + 0.756029i \(0.727137\pi\)
\(740\) 0 0
\(741\) 8.22285 0.302074
\(742\) 0 0
\(743\) −41.8212 −1.53427 −0.767135 0.641485i \(-0.778319\pi\)
−0.767135 + 0.641485i \(0.778319\pi\)
\(744\) 0 0
\(745\) −38.0477 −1.39396
\(746\) 0 0
\(747\) 4.56878 0.167163
\(748\) 0 0
\(749\) 69.7564 2.54884
\(750\) 0 0
\(751\) −38.9341 −1.42073 −0.710363 0.703836i \(-0.751469\pi\)
−0.710363 + 0.703836i \(0.751469\pi\)
\(752\) 0 0
\(753\) −2.98372 −0.108733
\(754\) 0 0
\(755\) 43.1179 1.56922
\(756\) 0 0
\(757\) −13.3933 −0.486787 −0.243394 0.969928i \(-0.578261\pi\)
−0.243394 + 0.969928i \(0.578261\pi\)
\(758\) 0 0
\(759\) −6.05663 −0.219842
\(760\) 0 0
\(761\) 53.8875 1.95342 0.976710 0.214564i \(-0.0688332\pi\)
0.976710 + 0.214564i \(0.0688332\pi\)
\(762\) 0 0
\(763\) −6.17394 −0.223512
\(764\) 0 0
\(765\) −7.67023 −0.277318
\(766\) 0 0
\(767\) 0.0989887 0.00357427
\(768\) 0 0
\(769\) −17.2334 −0.621453 −0.310727 0.950499i \(-0.600572\pi\)
−0.310727 + 0.950499i \(0.600572\pi\)
\(770\) 0 0
\(771\) −4.66532 −0.168017
\(772\) 0 0
\(773\) −50.4756 −1.81548 −0.907741 0.419532i \(-0.862194\pi\)
−0.907741 + 0.419532i \(0.862194\pi\)
\(774\) 0 0
\(775\) 40.1930 1.44377
\(776\) 0 0
\(777\) −49.9105 −1.79053
\(778\) 0 0
\(779\) −3.47937 −0.124662
\(780\) 0 0
\(781\) 12.0597 0.431530
\(782\) 0 0
\(783\) 3.04318 0.108754
\(784\) 0 0
\(785\) 23.5009 0.838784
\(786\) 0 0
\(787\) −29.2874 −1.04398 −0.521991 0.852951i \(-0.674811\pi\)
−0.521991 + 0.852951i \(0.674811\pi\)
\(788\) 0 0
\(789\) −11.4032 −0.405965
\(790\) 0 0
\(791\) 37.7228 1.34127
\(792\) 0 0
\(793\) −14.0314 −0.498268
\(794\) 0 0
\(795\) −39.4437 −1.39892
\(796\) 0 0
\(797\) −1.07432 −0.0380543 −0.0190272 0.999819i \(-0.506057\pi\)
−0.0190272 + 0.999819i \(0.506057\pi\)
\(798\) 0 0
\(799\) 21.7035 0.767815
\(800\) 0 0
\(801\) 8.20803 0.290016
\(802\) 0 0
\(803\) 10.6188 0.374730
\(804\) 0 0
\(805\) 87.7389 3.09239
\(806\) 0 0
\(807\) 17.5202 0.616741
\(808\) 0 0
\(809\) 22.8689 0.804028 0.402014 0.915634i \(-0.368310\pi\)
0.402014 + 0.915634i \(0.368310\pi\)
\(810\) 0 0
\(811\) −20.4412 −0.717788 −0.358894 0.933378i \(-0.616846\pi\)
−0.358894 + 0.933378i \(0.616846\pi\)
\(812\) 0 0
\(813\) −4.07499 −0.142916
\(814\) 0 0
\(815\) 0.453527 0.0158864
\(816\) 0 0
\(817\) −45.1012 −1.57789
\(818\) 0 0
\(819\) 4.54850 0.158937
\(820\) 0 0
\(821\) 40.3421 1.40795 0.703974 0.710226i \(-0.251407\pi\)
0.703974 + 0.710226i \(0.251407\pi\)
\(822\) 0 0
\(823\) 48.6160 1.69465 0.847323 0.531078i \(-0.178213\pi\)
0.847323 + 0.531078i \(0.178213\pi\)
\(824\) 0 0
\(825\) 3.81544 0.132836
\(826\) 0 0
\(827\) 39.6734 1.37958 0.689790 0.724009i \(-0.257703\pi\)
0.689790 + 0.724009i \(0.257703\pi\)
\(828\) 0 0
\(829\) 47.4157 1.64681 0.823407 0.567451i \(-0.192071\pi\)
0.823407 + 0.567451i \(0.192071\pi\)
\(830\) 0 0
\(831\) −22.9380 −0.795710
\(832\) 0 0
\(833\) 37.6687 1.30514
\(834\) 0 0
\(835\) −2.99382 −0.103606
\(836\) 0 0
\(837\) −10.1421 −0.350562
\(838\) 0 0
\(839\) 8.74261 0.301828 0.150914 0.988547i \(-0.451778\pi\)
0.150914 + 0.988547i \(0.451778\pi\)
\(840\) 0 0
\(841\) −19.7391 −0.680658
\(842\) 0 0
\(843\) 12.4485 0.428749
\(844\) 0 0
\(845\) 36.0658 1.24070
\(846\) 0 0
\(847\) −46.9267 −1.61242
\(848\) 0 0
\(849\) −21.2448 −0.729119
\(850\) 0 0
\(851\) −67.3974 −2.31035
\(852\) 0 0
\(853\) −2.82336 −0.0966701 −0.0483350 0.998831i \(-0.515392\pi\)
−0.0483350 + 0.998831i \(0.515392\pi\)
\(854\) 0 0
\(855\) 25.2138 0.862293
\(856\) 0 0
\(857\) −39.6895 −1.35577 −0.677883 0.735170i \(-0.737103\pi\)
−0.677883 + 0.735170i \(0.737103\pi\)
\(858\) 0 0
\(859\) 0.796057 0.0271611 0.0135805 0.999908i \(-0.495677\pi\)
0.0135805 + 0.999908i \(0.495677\pi\)
\(860\) 0 0
\(861\) −1.92463 −0.0655912
\(862\) 0 0
\(863\) −7.43568 −0.253113 −0.126557 0.991959i \(-0.540393\pi\)
−0.126557 + 0.991959i \(0.540393\pi\)
\(864\) 0 0
\(865\) −32.8099 −1.11557
\(866\) 0 0
\(867\) 10.4361 0.354427
\(868\) 0 0
\(869\) −1.41703 −0.0480695
\(870\) 0 0
\(871\) 11.4875 0.389238
\(872\) 0 0
\(873\) 11.1603 0.377720
\(874\) 0 0
\(875\) 14.4634 0.488952
\(876\) 0 0
\(877\) −13.0468 −0.440558 −0.220279 0.975437i \(-0.570697\pi\)
−0.220279 + 0.975437i \(0.570697\pi\)
\(878\) 0 0
\(879\) −3.52870 −0.119020
\(880\) 0 0
\(881\) 39.0090 1.31425 0.657123 0.753783i \(-0.271773\pi\)
0.657123 + 0.753783i \(0.271773\pi\)
\(882\) 0 0
\(883\) 3.17514 0.106852 0.0534261 0.998572i \(-0.482986\pi\)
0.0534261 + 0.998572i \(0.482986\pi\)
\(884\) 0 0
\(885\) 0.303530 0.0102030
\(886\) 0 0
\(887\) 19.6773 0.660698 0.330349 0.943859i \(-0.392834\pi\)
0.330349 + 0.943859i \(0.392834\pi\)
\(888\) 0 0
\(889\) 73.7513 2.47354
\(890\) 0 0
\(891\) −0.962769 −0.0322540
\(892\) 0 0
\(893\) −71.3444 −2.38745
\(894\) 0 0
\(895\) 43.8097 1.46440
\(896\) 0 0
\(897\) 6.14213 0.205080
\(898\) 0 0
\(899\) −30.8642 −1.02938
\(900\) 0 0
\(901\) −33.7547 −1.12453
\(902\) 0 0
\(903\) −24.9479 −0.830214
\(904\) 0 0
\(905\) 13.7723 0.457805
\(906\) 0 0
\(907\) 13.3941 0.444745 0.222372 0.974962i \(-0.428620\pi\)
0.222372 + 0.974962i \(0.428620\pi\)
\(908\) 0 0
\(909\) −5.03063 −0.166855
\(910\) 0 0
\(911\) −14.3696 −0.476085 −0.238043 0.971255i \(-0.576506\pi\)
−0.238043 + 0.971255i \(0.576506\pi\)
\(912\) 0 0
\(913\) −4.39868 −0.145575
\(914\) 0 0
\(915\) −43.0245 −1.42234
\(916\) 0 0
\(917\) 27.0554 0.893447
\(918\) 0 0
\(919\) 16.6199 0.548241 0.274121 0.961695i \(-0.411613\pi\)
0.274121 + 0.961695i \(0.411613\pi\)
\(920\) 0 0
\(921\) −15.5946 −0.513860
\(922\) 0 0
\(923\) −12.2300 −0.402554
\(924\) 0 0
\(925\) 42.4577 1.39600
\(926\) 0 0
\(927\) −7.19615 −0.236352
\(928\) 0 0
\(929\) −17.7181 −0.581313 −0.290657 0.956827i \(-0.593874\pi\)
−0.290657 + 0.956827i \(0.593874\pi\)
\(930\) 0 0
\(931\) −123.826 −4.05822
\(932\) 0 0
\(933\) −0.111395 −0.00364691
\(934\) 0 0
\(935\) 7.38466 0.241504
\(936\) 0 0
\(937\) 56.4997 1.84576 0.922882 0.385083i \(-0.125827\pi\)
0.922882 + 0.385083i \(0.125827\pi\)
\(938\) 0 0
\(939\) 15.8427 0.517008
\(940\) 0 0
\(941\) 28.4198 0.926460 0.463230 0.886238i \(-0.346690\pi\)
0.463230 + 0.886238i \(0.346690\pi\)
\(942\) 0 0
\(943\) −2.59895 −0.0846335
\(944\) 0 0
\(945\) 13.9471 0.453699
\(946\) 0 0
\(947\) −27.1922 −0.883628 −0.441814 0.897107i \(-0.645665\pi\)
−0.441814 + 0.897107i \(0.645665\pi\)
\(948\) 0 0
\(949\) −10.7687 −0.349567
\(950\) 0 0
\(951\) −18.3265 −0.594276
\(952\) 0 0
\(953\) −40.0549 −1.29751 −0.648753 0.760999i \(-0.724709\pi\)
−0.648753 + 0.760999i \(0.724709\pi\)
\(954\) 0 0
\(955\) −51.6647 −1.67183
\(956\) 0 0
\(957\) −2.92988 −0.0947095
\(958\) 0 0
\(959\) −95.9902 −3.09968
\(960\) 0 0
\(961\) 71.8622 2.31814
\(962\) 0 0
\(963\) 14.9736 0.482518
\(964\) 0 0
\(965\) 34.8778 1.12276
\(966\) 0 0
\(967\) −28.2604 −0.908795 −0.454397 0.890799i \(-0.650145\pi\)
−0.454397 + 0.890799i \(0.650145\pi\)
\(968\) 0 0
\(969\) 21.5771 0.693158
\(970\) 0 0
\(971\) −57.3946 −1.84188 −0.920940 0.389704i \(-0.872577\pi\)
−0.920940 + 0.389704i \(0.872577\pi\)
\(972\) 0 0
\(973\) 35.3291 1.13260
\(974\) 0 0
\(975\) −3.86930 −0.123917
\(976\) 0 0
\(977\) 32.6415 1.04429 0.522147 0.852855i \(-0.325131\pi\)
0.522147 + 0.852855i \(0.325131\pi\)
\(978\) 0 0
\(979\) −7.90244 −0.252563
\(980\) 0 0
\(981\) −1.32527 −0.0423127
\(982\) 0 0
\(983\) 6.11976 0.195190 0.0975950 0.995226i \(-0.468885\pi\)
0.0975950 + 0.995226i \(0.468885\pi\)
\(984\) 0 0
\(985\) −32.5551 −1.03729
\(986\) 0 0
\(987\) −39.4644 −1.25617
\(988\) 0 0
\(989\) −33.6888 −1.07124
\(990\) 0 0
\(991\) 31.2483 0.992636 0.496318 0.868141i \(-0.334685\pi\)
0.496318 + 0.868141i \(0.334685\pi\)
\(992\) 0 0
\(993\) −6.26802 −0.198910
\(994\) 0 0
\(995\) 27.2814 0.864878
\(996\) 0 0
\(997\) 15.4045 0.487866 0.243933 0.969792i \(-0.421562\pi\)
0.243933 + 0.969792i \(0.421562\pi\)
\(998\) 0 0
\(999\) −10.7136 −0.338963
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.ba.1.2 9
4.3 odd 2 4008.2.a.i.1.2 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4008.2.a.i.1.2 9 4.3 odd 2
8016.2.a.ba.1.2 9 1.1 even 1 trivial