Properties

Label 8016.2.a.w.1.5
Level $8016$
Weight $2$
Character 8016.1
Self dual yes
Analytic conductor $64.008$
Analytic rank $1$
Dimension $7$
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: \(1\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Defining polynomial: \(x^{7} - 2 x^{6} - 7 x^{5} + 11 x^{4} + 12 x^{3} - 14 x^{2} - 6 x + 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 4008)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(1.31154\) of defining polynomial
Character \(\chi\) \(=\) 8016.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +0.0621653 q^{5} -0.782308 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +0.0621653 q^{5} -0.782308 q^{7} +1.00000 q^{9} -2.68525 q^{11} +6.79339 q^{13} +0.0621653 q^{15} +4.20707 q^{17} -7.94188 q^{19} -0.782308 q^{21} -5.42476 q^{23} -4.99614 q^{25} +1.00000 q^{27} +1.95740 q^{29} -2.86384 q^{31} -2.68525 q^{33} -0.0486324 q^{35} -7.68582 q^{37} +6.79339 q^{39} -2.03431 q^{41} +5.17082 q^{43} +0.0621653 q^{45} +7.62796 q^{47} -6.38799 q^{49} +4.20707 q^{51} +3.26794 q^{53} -0.166929 q^{55} -7.94188 q^{57} -2.33656 q^{59} -12.7201 q^{61} -0.782308 q^{63} +0.422313 q^{65} +1.25024 q^{67} -5.42476 q^{69} -12.3346 q^{71} +10.0502 q^{73} -4.99614 q^{75} +2.10069 q^{77} -3.71570 q^{79} +1.00000 q^{81} -10.1647 q^{83} +0.261533 q^{85} +1.95740 q^{87} -2.74777 q^{89} -5.31452 q^{91} -2.86384 q^{93} -0.493709 q^{95} +1.54807 q^{97} -2.68525 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q + 7q^{3} - 3q^{5} - 8q^{7} + 7q^{9} + O(q^{10}) \) \( 7q + 7q^{3} - 3q^{5} - 8q^{7} + 7q^{9} - q^{11} - 2q^{13} - 3q^{15} + 11q^{17} - 2q^{19} - 8q^{21} - 17q^{23} + 4q^{25} + 7q^{27} - 7q^{29} - 10q^{31} - q^{33} - 10q^{35} - 21q^{37} - 2q^{39} + 8q^{41} + 12q^{43} - 3q^{45} - 25q^{47} - 7q^{49} + 11q^{51} - 7q^{53} - 15q^{55} - 2q^{57} - 3q^{59} - 14q^{61} - 8q^{63} + 4q^{65} - 4q^{67} - 17q^{69} - 27q^{71} - 12q^{73} + 4q^{75} + 16q^{77} - 8q^{79} + 7q^{81} - 15q^{83} - 3q^{85} - 7q^{87} + 14q^{89} + 3q^{91} - 10q^{93} - 37q^{95} + 3q^{97} - q^{99} + O(q^{100}) \)

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
\(5\) 0.0621653 0.0278011 0.0139006 0.999903i \(-0.495575\pi\)
0.0139006 + 0.999903i \(0.495575\pi\)
\(6\) 0 0
\(7\) −0.782308 −0.295685 −0.147842 0.989011i \(-0.547233\pi\)
−0.147842 + 0.989011i \(0.547233\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −2.68525 −0.809633 −0.404817 0.914398i \(-0.632665\pi\)
−0.404817 + 0.914398i \(0.632665\pi\)
\(12\) 0 0
\(13\) 6.79339 1.88415 0.942073 0.335407i \(-0.108874\pi\)
0.942073 + 0.335407i \(0.108874\pi\)
\(14\) 0 0
\(15\) 0.0621653 0.0160510
\(16\) 0 0
\(17\) 4.20707 1.02036 0.510182 0.860067i \(-0.329578\pi\)
0.510182 + 0.860067i \(0.329578\pi\)
\(18\) 0 0
\(19\) −7.94188 −1.82199 −0.910996 0.412416i \(-0.864685\pi\)
−0.910996 + 0.412416i \(0.864685\pi\)
\(20\) 0 0
\(21\) −0.782308 −0.170714
\(22\) 0 0
\(23\) −5.42476 −1.13114 −0.565570 0.824700i \(-0.691344\pi\)
−0.565570 + 0.824700i \(0.691344\pi\)
\(24\) 0 0
\(25\) −4.99614 −0.999227
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 1.95740 0.363480 0.181740 0.983347i \(-0.441827\pi\)
0.181740 + 0.983347i \(0.441827\pi\)
\(30\) 0 0
\(31\) −2.86384 −0.514361 −0.257180 0.966363i \(-0.582793\pi\)
−0.257180 + 0.966363i \(0.582793\pi\)
\(32\) 0 0
\(33\) −2.68525 −0.467442
\(34\) 0 0
\(35\) −0.0486324 −0.00822037
\(36\) 0 0
\(37\) −7.68582 −1.26354 −0.631771 0.775155i \(-0.717672\pi\)
−0.631771 + 0.775155i \(0.717672\pi\)
\(38\) 0 0
\(39\) 6.79339 1.08781
\(40\) 0 0
\(41\) −2.03431 −0.317706 −0.158853 0.987302i \(-0.550780\pi\)
−0.158853 + 0.987302i \(0.550780\pi\)
\(42\) 0 0
\(43\) 5.17082 0.788543 0.394271 0.918994i \(-0.370997\pi\)
0.394271 + 0.918994i \(0.370997\pi\)
\(44\) 0 0
\(45\) 0.0621653 0.00926705
\(46\) 0 0
\(47\) 7.62796 1.11265 0.556326 0.830964i \(-0.312210\pi\)
0.556326 + 0.830964i \(0.312210\pi\)
\(48\) 0 0
\(49\) −6.38799 −0.912571
\(50\) 0 0
\(51\) 4.20707 0.589107
\(52\) 0 0
\(53\) 3.26794 0.448886 0.224443 0.974487i \(-0.427944\pi\)
0.224443 + 0.974487i \(0.427944\pi\)
\(54\) 0 0
\(55\) −0.166929 −0.0225087
\(56\) 0 0
\(57\) −7.94188 −1.05193
\(58\) 0 0
\(59\) −2.33656 −0.304195 −0.152097 0.988366i \(-0.548603\pi\)
−0.152097 + 0.988366i \(0.548603\pi\)
\(60\) 0 0
\(61\) −12.7201 −1.62864 −0.814322 0.580413i \(-0.802891\pi\)
−0.814322 + 0.580413i \(0.802891\pi\)
\(62\) 0 0
\(63\) −0.782308 −0.0985615
\(64\) 0 0
\(65\) 0.422313 0.0523814
\(66\) 0 0
\(67\) 1.25024 0.152741 0.0763706 0.997080i \(-0.475667\pi\)
0.0763706 + 0.997080i \(0.475667\pi\)
\(68\) 0 0
\(69\) −5.42476 −0.653064
\(70\) 0 0
\(71\) −12.3346 −1.46384 −0.731922 0.681388i \(-0.761377\pi\)
−0.731922 + 0.681388i \(0.761377\pi\)
\(72\) 0 0
\(73\) 10.0502 1.17628 0.588141 0.808758i \(-0.299860\pi\)
0.588141 + 0.808758i \(0.299860\pi\)
\(74\) 0 0
\(75\) −4.99614 −0.576904
\(76\) 0 0
\(77\) 2.10069 0.239396
\(78\) 0 0
\(79\) −3.71570 −0.418049 −0.209025 0.977910i \(-0.567029\pi\)
−0.209025 + 0.977910i \(0.567029\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −10.1647 −1.11572 −0.557861 0.829934i \(-0.688378\pi\)
−0.557861 + 0.829934i \(0.688378\pi\)
\(84\) 0 0
\(85\) 0.261533 0.0283673
\(86\) 0 0
\(87\) 1.95740 0.209855
\(88\) 0 0
\(89\) −2.74777 −0.291263 −0.145631 0.989339i \(-0.546521\pi\)
−0.145631 + 0.989339i \(0.546521\pi\)
\(90\) 0 0
\(91\) −5.31452 −0.557113
\(92\) 0 0
\(93\) −2.86384 −0.296966
\(94\) 0 0
\(95\) −0.493709 −0.0506534
\(96\) 0 0
\(97\) 1.54807 0.157183 0.0785914 0.996907i \(-0.474958\pi\)
0.0785914 + 0.996907i \(0.474958\pi\)
\(98\) 0 0
\(99\) −2.68525 −0.269878
\(100\) 0 0
\(101\) −11.5022 −1.14451 −0.572254 0.820076i \(-0.693931\pi\)
−0.572254 + 0.820076i \(0.693931\pi\)
\(102\) 0 0
\(103\) 11.8029 1.16298 0.581489 0.813554i \(-0.302470\pi\)
0.581489 + 0.813554i \(0.302470\pi\)
\(104\) 0 0
\(105\) −0.0486324 −0.00474603
\(106\) 0 0
\(107\) −15.5983 −1.50795 −0.753974 0.656904i \(-0.771866\pi\)
−0.753974 + 0.656904i \(0.771866\pi\)
\(108\) 0 0
\(109\) 17.4215 1.66867 0.834337 0.551255i \(-0.185851\pi\)
0.834337 + 0.551255i \(0.185851\pi\)
\(110\) 0 0
\(111\) −7.68582 −0.729506
\(112\) 0 0
\(113\) −4.23856 −0.398731 −0.199365 0.979925i \(-0.563888\pi\)
−0.199365 + 0.979925i \(0.563888\pi\)
\(114\) 0 0
\(115\) −0.337232 −0.0314470
\(116\) 0 0
\(117\) 6.79339 0.628049
\(118\) 0 0
\(119\) −3.29122 −0.301706
\(120\) 0 0
\(121\) −3.78944 −0.344494
\(122\) 0 0
\(123\) −2.03431 −0.183428
\(124\) 0 0
\(125\) −0.621412 −0.0555808
\(126\) 0 0
\(127\) −6.23657 −0.553406 −0.276703 0.960955i \(-0.589242\pi\)
−0.276703 + 0.960955i \(0.589242\pi\)
\(128\) 0 0
\(129\) 5.17082 0.455265
\(130\) 0 0
\(131\) 6.03122 0.526950 0.263475 0.964666i \(-0.415131\pi\)
0.263475 + 0.964666i \(0.415131\pi\)
\(132\) 0 0
\(133\) 6.21299 0.538735
\(134\) 0 0
\(135\) 0.0621653 0.00535033
\(136\) 0 0
\(137\) −10.9123 −0.932301 −0.466151 0.884705i \(-0.654359\pi\)
−0.466151 + 0.884705i \(0.654359\pi\)
\(138\) 0 0
\(139\) 17.5444 1.48810 0.744049 0.668125i \(-0.232903\pi\)
0.744049 + 0.668125i \(0.232903\pi\)
\(140\) 0 0
\(141\) 7.62796 0.642390
\(142\) 0 0
\(143\) −18.2419 −1.52547
\(144\) 0 0
\(145\) 0.121682 0.0101052
\(146\) 0 0
\(147\) −6.38799 −0.526873
\(148\) 0 0
\(149\) 13.5444 1.10960 0.554802 0.831982i \(-0.312794\pi\)
0.554802 + 0.831982i \(0.312794\pi\)
\(150\) 0 0
\(151\) 4.98785 0.405905 0.202953 0.979189i \(-0.434946\pi\)
0.202953 + 0.979189i \(0.434946\pi\)
\(152\) 0 0
\(153\) 4.20707 0.340121
\(154\) 0 0
\(155\) −0.178031 −0.0142998
\(156\) 0 0
\(157\) −10.1442 −0.809593 −0.404797 0.914407i \(-0.632658\pi\)
−0.404797 + 0.914407i \(0.632658\pi\)
\(158\) 0 0
\(159\) 3.26794 0.259164
\(160\) 0 0
\(161\) 4.24383 0.334461
\(162\) 0 0
\(163\) −13.5338 −1.06005 −0.530025 0.847982i \(-0.677817\pi\)
−0.530025 + 0.847982i \(0.677817\pi\)
\(164\) 0 0
\(165\) −0.166929 −0.0129954
\(166\) 0 0
\(167\) 1.00000 0.0773823
\(168\) 0 0
\(169\) 33.1501 2.55001
\(170\) 0 0
\(171\) −7.94188 −0.607330
\(172\) 0 0
\(173\) 12.5805 0.956476 0.478238 0.878230i \(-0.341276\pi\)
0.478238 + 0.878230i \(0.341276\pi\)
\(174\) 0 0
\(175\) 3.90852 0.295456
\(176\) 0 0
\(177\) −2.33656 −0.175627
\(178\) 0 0
\(179\) 2.52638 0.188830 0.0944151 0.995533i \(-0.469902\pi\)
0.0944151 + 0.995533i \(0.469902\pi\)
\(180\) 0 0
\(181\) 9.84207 0.731555 0.365778 0.930702i \(-0.380803\pi\)
0.365778 + 0.930702i \(0.380803\pi\)
\(182\) 0 0
\(183\) −12.7201 −0.940298
\(184\) 0 0
\(185\) −0.477791 −0.0351279
\(186\) 0 0
\(187\) −11.2970 −0.826120
\(188\) 0 0
\(189\) −0.782308 −0.0569045
\(190\) 0 0
\(191\) −1.47483 −0.106715 −0.0533574 0.998575i \(-0.516992\pi\)
−0.0533574 + 0.998575i \(0.516992\pi\)
\(192\) 0 0
\(193\) −15.2332 −1.09651 −0.548256 0.836311i \(-0.684708\pi\)
−0.548256 + 0.836311i \(0.684708\pi\)
\(194\) 0 0
\(195\) 0.422313 0.0302424
\(196\) 0 0
\(197\) −26.2117 −1.86750 −0.933752 0.357920i \(-0.883486\pi\)
−0.933752 + 0.357920i \(0.883486\pi\)
\(198\) 0 0
\(199\) 17.1733 1.21739 0.608693 0.793406i \(-0.291694\pi\)
0.608693 + 0.793406i \(0.291694\pi\)
\(200\) 0 0
\(201\) 1.25024 0.0881852
\(202\) 0 0
\(203\) −1.53129 −0.107476
\(204\) 0 0
\(205\) −0.126463 −0.00883259
\(206\) 0 0
\(207\) −5.42476 −0.377047
\(208\) 0 0
\(209\) 21.3259 1.47514
\(210\) 0 0
\(211\) 17.7930 1.22492 0.612461 0.790501i \(-0.290180\pi\)
0.612461 + 0.790501i \(0.290180\pi\)
\(212\) 0 0
\(213\) −12.3346 −0.845151
\(214\) 0 0
\(215\) 0.321445 0.0219224
\(216\) 0 0
\(217\) 2.24040 0.152089
\(218\) 0 0
\(219\) 10.0502 0.679127
\(220\) 0 0
\(221\) 28.5802 1.92251
\(222\) 0 0
\(223\) −17.3886 −1.16443 −0.582214 0.813035i \(-0.697813\pi\)
−0.582214 + 0.813035i \(0.697813\pi\)
\(224\) 0 0
\(225\) −4.99614 −0.333076
\(226\) 0 0
\(227\) −12.9318 −0.858313 −0.429157 0.903230i \(-0.641189\pi\)
−0.429157 + 0.903230i \(0.641189\pi\)
\(228\) 0 0
\(229\) −23.6209 −1.56092 −0.780458 0.625209i \(-0.785014\pi\)
−0.780458 + 0.625209i \(0.785014\pi\)
\(230\) 0 0
\(231\) 2.10069 0.138215
\(232\) 0 0
\(233\) −21.4494 −1.40520 −0.702598 0.711587i \(-0.747977\pi\)
−0.702598 + 0.711587i \(0.747977\pi\)
\(234\) 0 0
\(235\) 0.474194 0.0309330
\(236\) 0 0
\(237\) −3.71570 −0.241361
\(238\) 0 0
\(239\) −24.3499 −1.57506 −0.787531 0.616275i \(-0.788641\pi\)
−0.787531 + 0.616275i \(0.788641\pi\)
\(240\) 0 0
\(241\) 14.1838 0.913656 0.456828 0.889555i \(-0.348985\pi\)
0.456828 + 0.889555i \(0.348985\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −0.397111 −0.0253705
\(246\) 0 0
\(247\) −53.9522 −3.43290
\(248\) 0 0
\(249\) −10.1647 −0.644162
\(250\) 0 0
\(251\) 8.79375 0.555057 0.277528 0.960717i \(-0.410485\pi\)
0.277528 + 0.960717i \(0.410485\pi\)
\(252\) 0 0
\(253\) 14.5668 0.915809
\(254\) 0 0
\(255\) 0.261533 0.0163779
\(256\) 0 0
\(257\) 0.693124 0.0432359 0.0216179 0.999766i \(-0.493118\pi\)
0.0216179 + 0.999766i \(0.493118\pi\)
\(258\) 0 0
\(259\) 6.01268 0.373610
\(260\) 0 0
\(261\) 1.95740 0.121160
\(262\) 0 0
\(263\) −14.8980 −0.918653 −0.459326 0.888268i \(-0.651909\pi\)
−0.459326 + 0.888268i \(0.651909\pi\)
\(264\) 0 0
\(265\) 0.203152 0.0124795
\(266\) 0 0
\(267\) −2.74777 −0.168161
\(268\) 0 0
\(269\) −4.68185 −0.285458 −0.142729 0.989762i \(-0.545588\pi\)
−0.142729 + 0.989762i \(0.545588\pi\)
\(270\) 0 0
\(271\) −7.76324 −0.471583 −0.235792 0.971804i \(-0.575768\pi\)
−0.235792 + 0.971804i \(0.575768\pi\)
\(272\) 0 0
\(273\) −5.31452 −0.321649
\(274\) 0 0
\(275\) 13.4159 0.809007
\(276\) 0 0
\(277\) −15.5566 −0.934703 −0.467352 0.884072i \(-0.654792\pi\)
−0.467352 + 0.884072i \(0.654792\pi\)
\(278\) 0 0
\(279\) −2.86384 −0.171454
\(280\) 0 0
\(281\) −9.94616 −0.593338 −0.296669 0.954980i \(-0.595876\pi\)
−0.296669 + 0.954980i \(0.595876\pi\)
\(282\) 0 0
\(283\) −10.1239 −0.601802 −0.300901 0.953655i \(-0.597287\pi\)
−0.300901 + 0.953655i \(0.597287\pi\)
\(284\) 0 0
\(285\) −0.493709 −0.0292448
\(286\) 0 0
\(287\) 1.59146 0.0939407
\(288\) 0 0
\(289\) 0.699411 0.0411418
\(290\) 0 0
\(291\) 1.54807 0.0907495
\(292\) 0 0
\(293\) −18.1886 −1.06259 −0.531294 0.847187i \(-0.678294\pi\)
−0.531294 + 0.847187i \(0.678294\pi\)
\(294\) 0 0
\(295\) −0.145253 −0.00845696
\(296\) 0 0
\(297\) −2.68525 −0.155814
\(298\) 0 0
\(299\) −36.8525 −2.13123
\(300\) 0 0
\(301\) −4.04517 −0.233160
\(302\) 0 0
\(303\) −11.5022 −0.660782
\(304\) 0 0
\(305\) −0.790749 −0.0452782
\(306\) 0 0
\(307\) 8.83095 0.504009 0.252004 0.967726i \(-0.418910\pi\)
0.252004 + 0.967726i \(0.418910\pi\)
\(308\) 0 0
\(309\) 11.8029 0.671446
\(310\) 0 0
\(311\) 15.6371 0.886699 0.443349 0.896349i \(-0.353790\pi\)
0.443349 + 0.896349i \(0.353790\pi\)
\(312\) 0 0
\(313\) −14.0920 −0.796526 −0.398263 0.917271i \(-0.630387\pi\)
−0.398263 + 0.917271i \(0.630387\pi\)
\(314\) 0 0
\(315\) −0.0486324 −0.00274012
\(316\) 0 0
\(317\) 16.3138 0.916273 0.458136 0.888882i \(-0.348517\pi\)
0.458136 + 0.888882i \(0.348517\pi\)
\(318\) 0 0
\(319\) −5.25611 −0.294286
\(320\) 0 0
\(321\) −15.5983 −0.870615
\(322\) 0 0
\(323\) −33.4120 −1.85909
\(324\) 0 0
\(325\) −33.9407 −1.88269
\(326\) 0 0
\(327\) 17.4215 0.963409
\(328\) 0 0
\(329\) −5.96741 −0.328994
\(330\) 0 0
\(331\) 8.18087 0.449661 0.224831 0.974398i \(-0.427817\pi\)
0.224831 + 0.974398i \(0.427817\pi\)
\(332\) 0 0
\(333\) −7.68582 −0.421181
\(334\) 0 0
\(335\) 0.0777215 0.00424638
\(336\) 0 0
\(337\) 13.5232 0.736654 0.368327 0.929696i \(-0.379931\pi\)
0.368327 + 0.929696i \(0.379931\pi\)
\(338\) 0 0
\(339\) −4.23856 −0.230207
\(340\) 0 0
\(341\) 7.69012 0.416444
\(342\) 0 0
\(343\) 10.4735 0.565518
\(344\) 0 0
\(345\) −0.337232 −0.0181559
\(346\) 0 0
\(347\) −11.4880 −0.616706 −0.308353 0.951272i \(-0.599778\pi\)
−0.308353 + 0.951272i \(0.599778\pi\)
\(348\) 0 0
\(349\) −10.7858 −0.577351 −0.288676 0.957427i \(-0.593215\pi\)
−0.288676 + 0.957427i \(0.593215\pi\)
\(350\) 0 0
\(351\) 6.79339 0.362604
\(352\) 0 0
\(353\) 27.6103 1.46955 0.734773 0.678313i \(-0.237289\pi\)
0.734773 + 0.678313i \(0.237289\pi\)
\(354\) 0 0
\(355\) −0.766782 −0.0406965
\(356\) 0 0
\(357\) −3.29122 −0.174190
\(358\) 0 0
\(359\) −22.3864 −1.18151 −0.590755 0.806851i \(-0.701170\pi\)
−0.590755 + 0.806851i \(0.701170\pi\)
\(360\) 0 0
\(361\) 44.0734 2.31965
\(362\) 0 0
\(363\) −3.78944 −0.198894
\(364\) 0 0
\(365\) 0.624771 0.0327020
\(366\) 0 0
\(367\) −2.00665 −0.104746 −0.0523731 0.998628i \(-0.516678\pi\)
−0.0523731 + 0.998628i \(0.516678\pi\)
\(368\) 0 0
\(369\) −2.03431 −0.105902
\(370\) 0 0
\(371\) −2.55653 −0.132729
\(372\) 0 0
\(373\) −13.7031 −0.709519 −0.354759 0.934958i \(-0.615437\pi\)
−0.354759 + 0.934958i \(0.615437\pi\)
\(374\) 0 0
\(375\) −0.621412 −0.0320896
\(376\) 0 0
\(377\) 13.2974 0.684850
\(378\) 0 0
\(379\) 16.3044 0.837499 0.418749 0.908102i \(-0.362469\pi\)
0.418749 + 0.908102i \(0.362469\pi\)
\(380\) 0 0
\(381\) −6.23657 −0.319509
\(382\) 0 0
\(383\) −18.4410 −0.942293 −0.471146 0.882055i \(-0.656160\pi\)
−0.471146 + 0.882055i \(0.656160\pi\)
\(384\) 0 0
\(385\) 0.130590 0.00665548
\(386\) 0 0
\(387\) 5.17082 0.262848
\(388\) 0 0
\(389\) −20.0693 −1.01756 −0.508778 0.860898i \(-0.669902\pi\)
−0.508778 + 0.860898i \(0.669902\pi\)
\(390\) 0 0
\(391\) −22.8223 −1.15417
\(392\) 0 0
\(393\) 6.03122 0.304235
\(394\) 0 0
\(395\) −0.230988 −0.0116222
\(396\) 0 0
\(397\) −5.72441 −0.287300 −0.143650 0.989629i \(-0.545884\pi\)
−0.143650 + 0.989629i \(0.545884\pi\)
\(398\) 0 0
\(399\) 6.21299 0.311039
\(400\) 0 0
\(401\) 10.6836 0.533512 0.266756 0.963764i \(-0.414048\pi\)
0.266756 + 0.963764i \(0.414048\pi\)
\(402\) 0 0
\(403\) −19.4552 −0.969131
\(404\) 0 0
\(405\) 0.0621653 0.00308902
\(406\) 0 0
\(407\) 20.6384 1.02301
\(408\) 0 0
\(409\) 7.95014 0.393109 0.196554 0.980493i \(-0.437025\pi\)
0.196554 + 0.980493i \(0.437025\pi\)
\(410\) 0 0
\(411\) −10.9123 −0.538264
\(412\) 0 0
\(413\) 1.82791 0.0899457
\(414\) 0 0
\(415\) −0.631892 −0.0310183
\(416\) 0 0
\(417\) 17.5444 0.859154
\(418\) 0 0
\(419\) −4.68925 −0.229085 −0.114542 0.993418i \(-0.536540\pi\)
−0.114542 + 0.993418i \(0.536540\pi\)
\(420\) 0 0
\(421\) −9.84889 −0.480006 −0.240003 0.970772i \(-0.577148\pi\)
−0.240003 + 0.970772i \(0.577148\pi\)
\(422\) 0 0
\(423\) 7.62796 0.370884
\(424\) 0 0
\(425\) −21.0191 −1.01957
\(426\) 0 0
\(427\) 9.95105 0.481565
\(428\) 0 0
\(429\) −18.2419 −0.880729
\(430\) 0 0
\(431\) −31.8797 −1.53559 −0.767796 0.640694i \(-0.778647\pi\)
−0.767796 + 0.640694i \(0.778647\pi\)
\(432\) 0 0
\(433\) −5.80718 −0.279075 −0.139538 0.990217i \(-0.544562\pi\)
−0.139538 + 0.990217i \(0.544562\pi\)
\(434\) 0 0
\(435\) 0.121682 0.00583422
\(436\) 0 0
\(437\) 43.0828 2.06093
\(438\) 0 0
\(439\) 25.0785 1.19693 0.598465 0.801149i \(-0.295778\pi\)
0.598465 + 0.801149i \(0.295778\pi\)
\(440\) 0 0
\(441\) −6.38799 −0.304190
\(442\) 0 0
\(443\) 31.2338 1.48396 0.741981 0.670421i \(-0.233886\pi\)
0.741981 + 0.670421i \(0.233886\pi\)
\(444\) 0 0
\(445\) −0.170816 −0.00809743
\(446\) 0 0
\(447\) 13.5444 0.640630
\(448\) 0 0
\(449\) 4.12427 0.194636 0.0973182 0.995253i \(-0.468974\pi\)
0.0973182 + 0.995253i \(0.468974\pi\)
\(450\) 0 0
\(451\) 5.46263 0.257225
\(452\) 0 0
\(453\) 4.98785 0.234350
\(454\) 0 0
\(455\) −0.330378 −0.0154884
\(456\) 0 0
\(457\) 0.0521369 0.00243886 0.00121943 0.999999i \(-0.499612\pi\)
0.00121943 + 0.999999i \(0.499612\pi\)
\(458\) 0 0
\(459\) 4.20707 0.196369
\(460\) 0 0
\(461\) −5.15218 −0.239961 −0.119981 0.992776i \(-0.538283\pi\)
−0.119981 + 0.992776i \(0.538283\pi\)
\(462\) 0 0
\(463\) 30.4668 1.41591 0.707957 0.706256i \(-0.249617\pi\)
0.707957 + 0.706256i \(0.249617\pi\)
\(464\) 0 0
\(465\) −0.178031 −0.00825601
\(466\) 0 0
\(467\) 14.8990 0.689444 0.344722 0.938705i \(-0.387973\pi\)
0.344722 + 0.938705i \(0.387973\pi\)
\(468\) 0 0
\(469\) −0.978073 −0.0451632
\(470\) 0 0
\(471\) −10.1442 −0.467419
\(472\) 0 0
\(473\) −13.8849 −0.638430
\(474\) 0 0
\(475\) 39.6787 1.82058
\(476\) 0 0
\(477\) 3.26794 0.149629
\(478\) 0 0
\(479\) −10.7395 −0.490702 −0.245351 0.969434i \(-0.578903\pi\)
−0.245351 + 0.969434i \(0.578903\pi\)
\(480\) 0 0
\(481\) −52.2128 −2.38070
\(482\) 0 0
\(483\) 4.24383 0.193101
\(484\) 0 0
\(485\) 0.0962362 0.00436986
\(486\) 0 0
\(487\) −27.7875 −1.25917 −0.629587 0.776930i \(-0.716776\pi\)
−0.629587 + 0.776930i \(0.716776\pi\)
\(488\) 0 0
\(489\) −13.5338 −0.612020
\(490\) 0 0
\(491\) −20.3242 −0.917216 −0.458608 0.888639i \(-0.651652\pi\)
−0.458608 + 0.888639i \(0.651652\pi\)
\(492\) 0 0
\(493\) 8.23492 0.370882
\(494\) 0 0
\(495\) −0.166929 −0.00750291
\(496\) 0 0
\(497\) 9.64943 0.432836
\(498\) 0 0
\(499\) −25.4727 −1.14032 −0.570158 0.821535i \(-0.693118\pi\)
−0.570158 + 0.821535i \(0.693118\pi\)
\(500\) 0 0
\(501\) 1.00000 0.0446767
\(502\) 0 0
\(503\) −16.3937 −0.730959 −0.365480 0.930819i \(-0.619095\pi\)
−0.365480 + 0.930819i \(0.619095\pi\)
\(504\) 0 0
\(505\) −0.715035 −0.0318186
\(506\) 0 0
\(507\) 33.1501 1.47225
\(508\) 0 0
\(509\) 27.3496 1.21225 0.606124 0.795370i \(-0.292723\pi\)
0.606124 + 0.795370i \(0.292723\pi\)
\(510\) 0 0
\(511\) −7.86232 −0.347809
\(512\) 0 0
\(513\) −7.94188 −0.350642
\(514\) 0 0
\(515\) 0.733733 0.0323321
\(516\) 0 0
\(517\) −20.4830 −0.900840
\(518\) 0 0
\(519\) 12.5805 0.552222
\(520\) 0 0
\(521\) 18.2555 0.799788 0.399894 0.916562i \(-0.369047\pi\)
0.399894 + 0.916562i \(0.369047\pi\)
\(522\) 0 0
\(523\) −14.8011 −0.647205 −0.323603 0.946193i \(-0.604894\pi\)
−0.323603 + 0.946193i \(0.604894\pi\)
\(524\) 0 0
\(525\) 3.90852 0.170582
\(526\) 0 0
\(527\) −12.0484 −0.524835
\(528\) 0 0
\(529\) 6.42801 0.279479
\(530\) 0 0
\(531\) −2.33656 −0.101398
\(532\) 0 0
\(533\) −13.8199 −0.598604
\(534\) 0 0
\(535\) −0.969675 −0.0419227
\(536\) 0 0
\(537\) 2.52638 0.109021
\(538\) 0 0
\(539\) 17.1534 0.738847
\(540\) 0 0
\(541\) 0.165065 0.00709671 0.00354835 0.999994i \(-0.498871\pi\)
0.00354835 + 0.999994i \(0.498871\pi\)
\(542\) 0 0
\(543\) 9.84207 0.422363
\(544\) 0 0
\(545\) 1.08301 0.0463910
\(546\) 0 0
\(547\) 14.3907 0.615300 0.307650 0.951500i \(-0.400457\pi\)
0.307650 + 0.951500i \(0.400457\pi\)
\(548\) 0 0
\(549\) −12.7201 −0.542881
\(550\) 0 0
\(551\) −15.5454 −0.662258
\(552\) 0 0
\(553\) 2.90682 0.123611
\(554\) 0 0
\(555\) −0.477791 −0.0202811
\(556\) 0 0
\(557\) 3.70371 0.156931 0.0784656 0.996917i \(-0.474998\pi\)
0.0784656 + 0.996917i \(0.474998\pi\)
\(558\) 0 0
\(559\) 35.1274 1.48573
\(560\) 0 0
\(561\) −11.2970 −0.476961
\(562\) 0 0
\(563\) 39.3783 1.65960 0.829798 0.558064i \(-0.188456\pi\)
0.829798 + 0.558064i \(0.188456\pi\)
\(564\) 0 0
\(565\) −0.263491 −0.0110852
\(566\) 0 0
\(567\) −0.782308 −0.0328538
\(568\) 0 0
\(569\) −19.8473 −0.832041 −0.416020 0.909355i \(-0.636575\pi\)
−0.416020 + 0.909355i \(0.636575\pi\)
\(570\) 0 0
\(571\) 14.1013 0.590120 0.295060 0.955479i \(-0.404660\pi\)
0.295060 + 0.955479i \(0.404660\pi\)
\(572\) 0 0
\(573\) −1.47483 −0.0616118
\(574\) 0 0
\(575\) 27.1028 1.13027
\(576\) 0 0
\(577\) 25.5547 1.06385 0.531927 0.846790i \(-0.321468\pi\)
0.531927 + 0.846790i \(0.321468\pi\)
\(578\) 0 0
\(579\) −15.2332 −0.633071
\(580\) 0 0
\(581\) 7.95193 0.329902
\(582\) 0 0
\(583\) −8.77522 −0.363433
\(584\) 0 0
\(585\) 0.422313 0.0174605
\(586\) 0 0
\(587\) −24.2117 −0.999323 −0.499661 0.866221i \(-0.666542\pi\)
−0.499661 + 0.866221i \(0.666542\pi\)
\(588\) 0 0
\(589\) 22.7443 0.937161
\(590\) 0 0
\(591\) −26.2117 −1.07820
\(592\) 0 0
\(593\) −19.2622 −0.791005 −0.395503 0.918465i \(-0.629430\pi\)
−0.395503 + 0.918465i \(0.629430\pi\)
\(594\) 0 0
\(595\) −0.204600 −0.00838777
\(596\) 0 0
\(597\) 17.1733 0.702858
\(598\) 0 0
\(599\) −3.33279 −0.136174 −0.0680870 0.997679i \(-0.521690\pi\)
−0.0680870 + 0.997679i \(0.521690\pi\)
\(600\) 0 0
\(601\) 23.2581 0.948719 0.474359 0.880331i \(-0.342680\pi\)
0.474359 + 0.880331i \(0.342680\pi\)
\(602\) 0 0
\(603\) 1.25024 0.0509137
\(604\) 0 0
\(605\) −0.235571 −0.00957733
\(606\) 0 0
\(607\) −9.74996 −0.395739 −0.197869 0.980228i \(-0.563402\pi\)
−0.197869 + 0.980228i \(0.563402\pi\)
\(608\) 0 0
\(609\) −1.53129 −0.0620510
\(610\) 0 0
\(611\) 51.8197 2.09640
\(612\) 0 0
\(613\) 5.41453 0.218691 0.109345 0.994004i \(-0.465125\pi\)
0.109345 + 0.994004i \(0.465125\pi\)
\(614\) 0 0
\(615\) −0.126463 −0.00509950
\(616\) 0 0
\(617\) 37.5966 1.51358 0.756792 0.653656i \(-0.226766\pi\)
0.756792 + 0.653656i \(0.226766\pi\)
\(618\) 0 0
\(619\) 22.5427 0.906066 0.453033 0.891494i \(-0.350342\pi\)
0.453033 + 0.891494i \(0.350342\pi\)
\(620\) 0 0
\(621\) −5.42476 −0.217688
\(622\) 0 0
\(623\) 2.14960 0.0861219
\(624\) 0 0
\(625\) 24.9420 0.997682
\(626\) 0 0
\(627\) 21.3259 0.851675
\(628\) 0 0
\(629\) −32.3348 −1.28927
\(630\) 0 0
\(631\) −34.8468 −1.38723 −0.693615 0.720346i \(-0.743983\pi\)
−0.693615 + 0.720346i \(0.743983\pi\)
\(632\) 0 0
\(633\) 17.7930 0.707209
\(634\) 0 0
\(635\) −0.387698 −0.0153853
\(636\) 0 0
\(637\) −43.3961 −1.71942
\(638\) 0 0
\(639\) −12.3346 −0.487948
\(640\) 0 0
\(641\) 23.8042 0.940211 0.470106 0.882610i \(-0.344216\pi\)
0.470106 + 0.882610i \(0.344216\pi\)
\(642\) 0 0
\(643\) 43.8213 1.72814 0.864072 0.503369i \(-0.167906\pi\)
0.864072 + 0.503369i \(0.167906\pi\)
\(644\) 0 0
\(645\) 0.321445 0.0126569
\(646\) 0 0
\(647\) −25.9087 −1.01858 −0.509288 0.860596i \(-0.670091\pi\)
−0.509288 + 0.860596i \(0.670091\pi\)
\(648\) 0 0
\(649\) 6.27426 0.246286
\(650\) 0 0
\(651\) 2.24040 0.0878084
\(652\) 0 0
\(653\) −1.32912 −0.0520125 −0.0260063 0.999662i \(-0.508279\pi\)
−0.0260063 + 0.999662i \(0.508279\pi\)
\(654\) 0 0
\(655\) 0.374932 0.0146498
\(656\) 0 0
\(657\) 10.0502 0.392094
\(658\) 0 0
\(659\) −3.44424 −0.134168 −0.0670842 0.997747i \(-0.521370\pi\)
−0.0670842 + 0.997747i \(0.521370\pi\)
\(660\) 0 0
\(661\) −43.5969 −1.69572 −0.847862 0.530217i \(-0.822111\pi\)
−0.847862 + 0.530217i \(0.822111\pi\)
\(662\) 0 0
\(663\) 28.5802 1.10996
\(664\) 0 0
\(665\) 0.386232 0.0149774
\(666\) 0 0
\(667\) −10.6184 −0.411147
\(668\) 0 0
\(669\) −17.3886 −0.672283
\(670\) 0 0
\(671\) 34.1567 1.31860
\(672\) 0 0
\(673\) −15.8422 −0.610674 −0.305337 0.952244i \(-0.598769\pi\)
−0.305337 + 0.952244i \(0.598769\pi\)
\(674\) 0 0
\(675\) −4.99614 −0.192301
\(676\) 0 0
\(677\) −33.0836 −1.27151 −0.635753 0.771893i \(-0.719310\pi\)
−0.635753 + 0.771893i \(0.719310\pi\)
\(678\) 0 0
\(679\) −1.21107 −0.0464765
\(680\) 0 0
\(681\) −12.9318 −0.495547
\(682\) 0 0
\(683\) −7.85853 −0.300698 −0.150349 0.988633i \(-0.548040\pi\)
−0.150349 + 0.988633i \(0.548040\pi\)
\(684\) 0 0
\(685\) −0.678366 −0.0259190
\(686\) 0 0
\(687\) −23.6209 −0.901195
\(688\) 0 0
\(689\) 22.2004 0.845766
\(690\) 0 0
\(691\) 30.6961 1.16773 0.583867 0.811849i \(-0.301539\pi\)
0.583867 + 0.811849i \(0.301539\pi\)
\(692\) 0 0
\(693\) 2.10069 0.0797987
\(694\) 0 0
\(695\) 1.09065 0.0413708
\(696\) 0 0
\(697\) −8.55848 −0.324176
\(698\) 0 0
\(699\) −21.4494 −0.811290
\(700\) 0 0
\(701\) −19.9235 −0.752498 −0.376249 0.926519i \(-0.622786\pi\)
−0.376249 + 0.926519i \(0.622786\pi\)
\(702\) 0 0
\(703\) 61.0399 2.30216
\(704\) 0 0
\(705\) 0.474194 0.0178592
\(706\) 0 0
\(707\) 8.99823 0.338413
\(708\) 0 0
\(709\) 16.7984 0.630879 0.315439 0.948946i \(-0.397848\pi\)
0.315439 + 0.948946i \(0.397848\pi\)
\(710\) 0 0
\(711\) −3.71570 −0.139350
\(712\) 0 0
\(713\) 15.5356 0.581814
\(714\) 0 0
\(715\) −1.13401 −0.0424097
\(716\) 0 0
\(717\) −24.3499 −0.909362
\(718\) 0 0
\(719\) −32.9584 −1.22914 −0.614571 0.788862i \(-0.710671\pi\)
−0.614571 + 0.788862i \(0.710671\pi\)
\(720\) 0 0
\(721\) −9.23354 −0.343875
\(722\) 0 0
\(723\) 14.1838 0.527500
\(724\) 0 0
\(725\) −9.77944 −0.363199
\(726\) 0 0
\(727\) 3.70734 0.137498 0.0687488 0.997634i \(-0.478099\pi\)
0.0687488 + 0.997634i \(0.478099\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 21.7540 0.804600
\(732\) 0 0
\(733\) 36.2151 1.33763 0.668817 0.743427i \(-0.266801\pi\)
0.668817 + 0.743427i \(0.266801\pi\)
\(734\) 0 0
\(735\) −0.397111 −0.0146477
\(736\) 0 0
\(737\) −3.35721 −0.123664
\(738\) 0 0
\(739\) 10.7819 0.396620 0.198310 0.980139i \(-0.436455\pi\)
0.198310 + 0.980139i \(0.436455\pi\)
\(740\) 0 0
\(741\) −53.9522 −1.98198
\(742\) 0 0
\(743\) 19.6242 0.719941 0.359970 0.932964i \(-0.382787\pi\)
0.359970 + 0.932964i \(0.382787\pi\)
\(744\) 0 0
\(745\) 0.841994 0.0308483
\(746\) 0 0
\(747\) −10.1647 −0.371907
\(748\) 0 0
\(749\) 12.2027 0.445877
\(750\) 0 0
\(751\) 4.30083 0.156939 0.0784697 0.996916i \(-0.474997\pi\)
0.0784697 + 0.996916i \(0.474997\pi\)
\(752\) 0 0
\(753\) 8.79375 0.320462
\(754\) 0 0
\(755\) 0.310071 0.0112846
\(756\) 0 0
\(757\) −7.62298 −0.277062 −0.138531 0.990358i \(-0.544238\pi\)
−0.138531 + 0.990358i \(0.544238\pi\)
\(758\) 0 0
\(759\) 14.5668 0.528742
\(760\) 0 0
\(761\) −17.0907 −0.619538 −0.309769 0.950812i \(-0.600252\pi\)
−0.309769 + 0.950812i \(0.600252\pi\)
\(762\) 0 0
\(763\) −13.6289 −0.493401
\(764\) 0 0
\(765\) 0.261533 0.00945576
\(766\) 0 0
\(767\) −15.8732 −0.573147
\(768\) 0 0
\(769\) 2.19109 0.0790126 0.0395063 0.999219i \(-0.487421\pi\)
0.0395063 + 0.999219i \(0.487421\pi\)
\(770\) 0 0
\(771\) 0.693124 0.0249623
\(772\) 0 0
\(773\) 9.07474 0.326396 0.163198 0.986593i \(-0.447819\pi\)
0.163198 + 0.986593i \(0.447819\pi\)
\(774\) 0 0
\(775\) 14.3081 0.513963
\(776\) 0 0
\(777\) 6.01268 0.215704
\(778\) 0 0
\(779\) 16.1562 0.578857
\(780\) 0 0
\(781\) 33.1214 1.18518
\(782\) 0 0
\(783\) 1.95740 0.0699518
\(784\) 0 0
\(785\) −0.630615 −0.0225076
\(786\) 0 0
\(787\) −16.6175 −0.592349 −0.296174 0.955134i \(-0.595711\pi\)
−0.296174 + 0.955134i \(0.595711\pi\)
\(788\) 0 0
\(789\) −14.8980 −0.530384
\(790\) 0 0
\(791\) 3.31586 0.117898
\(792\) 0 0
\(793\) −86.4127 −3.06860
\(794\) 0 0
\(795\) 0.203152 0.00720506
\(796\) 0 0
\(797\) −30.1950 −1.06956 −0.534781 0.844991i \(-0.679606\pi\)
−0.534781 + 0.844991i \(0.679606\pi\)
\(798\) 0 0
\(799\) 32.0913 1.13531
\(800\) 0 0
\(801\) −2.74777 −0.0970875
\(802\) 0 0
\(803\) −26.9872 −0.952358
\(804\) 0 0
\(805\) 0.263819 0.00929839
\(806\) 0 0
\(807\) −4.68185 −0.164809
\(808\) 0 0
\(809\) −41.1345 −1.44621 −0.723106 0.690737i \(-0.757286\pi\)
−0.723106 + 0.690737i \(0.757286\pi\)
\(810\) 0 0
\(811\) −20.9200 −0.734599 −0.367299 0.930103i \(-0.619718\pi\)
−0.367299 + 0.930103i \(0.619718\pi\)
\(812\) 0 0
\(813\) −7.76324 −0.272269
\(814\) 0 0
\(815\) −0.841332 −0.0294706
\(816\) 0 0
\(817\) −41.0660 −1.43672
\(818\) 0 0
\(819\) −5.31452 −0.185704
\(820\) 0 0
\(821\) −53.3825 −1.86306 −0.931531 0.363661i \(-0.881527\pi\)
−0.931531 + 0.363661i \(0.881527\pi\)
\(822\) 0 0
\(823\) −38.6317 −1.34661 −0.673307 0.739363i \(-0.735127\pi\)
−0.673307 + 0.739363i \(0.735127\pi\)
\(824\) 0 0
\(825\) 13.4159 0.467081
\(826\) 0 0
\(827\) 29.9769 1.04240 0.521199 0.853435i \(-0.325485\pi\)
0.521199 + 0.853435i \(0.325485\pi\)
\(828\) 0 0
\(829\) 35.8928 1.24661 0.623304 0.781980i \(-0.285790\pi\)
0.623304 + 0.781980i \(0.285790\pi\)
\(830\) 0 0
\(831\) −15.5566 −0.539651
\(832\) 0 0
\(833\) −26.8747 −0.931154
\(834\) 0 0
\(835\) 0.0621653 0.00215132
\(836\) 0 0
\(837\) −2.86384 −0.0989888
\(838\) 0 0
\(839\) −30.0957 −1.03902 −0.519510 0.854464i \(-0.673885\pi\)
−0.519510 + 0.854464i \(0.673885\pi\)
\(840\) 0 0
\(841\) −25.1686 −0.867882
\(842\) 0 0
\(843\) −9.94616 −0.342564
\(844\) 0 0
\(845\) 2.06078 0.0708931
\(846\) 0 0
\(847\) 2.96451 0.101862
\(848\) 0 0
\(849\) −10.1239 −0.347450
\(850\) 0 0
\(851\) 41.6937 1.42924
\(852\) 0 0
\(853\) −7.80634 −0.267284 −0.133642 0.991030i \(-0.542667\pi\)
−0.133642 + 0.991030i \(0.542667\pi\)
\(854\) 0 0
\(855\) −0.493709 −0.0168845
\(856\) 0 0
\(857\) 47.4720 1.62161 0.810806 0.585315i \(-0.199029\pi\)
0.810806 + 0.585315i \(0.199029\pi\)
\(858\) 0 0
\(859\) 14.5484 0.496385 0.248192 0.968711i \(-0.420164\pi\)
0.248192 + 0.968711i \(0.420164\pi\)
\(860\) 0 0
\(861\) 1.59146 0.0542367
\(862\) 0 0
\(863\) −43.9903 −1.49745 −0.748724 0.662882i \(-0.769333\pi\)
−0.748724 + 0.662882i \(0.769333\pi\)
\(864\) 0 0
\(865\) 0.782069 0.0265911
\(866\) 0 0
\(867\) 0.699411 0.0237532
\(868\) 0 0
\(869\) 9.97759 0.338467
\(870\) 0 0
\(871\) 8.49337 0.287787
\(872\) 0 0
\(873\) 1.54807 0.0523943
\(874\) 0 0
\(875\) 0.486136 0.0164344
\(876\) 0 0
\(877\) −18.2054 −0.614754 −0.307377 0.951588i \(-0.599451\pi\)
−0.307377 + 0.951588i \(0.599451\pi\)
\(878\) 0 0
\(879\) −18.1886 −0.613486
\(880\) 0 0
\(881\) −54.0285 −1.82027 −0.910134 0.414315i \(-0.864021\pi\)
−0.910134 + 0.414315i \(0.864021\pi\)
\(882\) 0 0
\(883\) 7.34816 0.247285 0.123643 0.992327i \(-0.460542\pi\)
0.123643 + 0.992327i \(0.460542\pi\)
\(884\) 0 0
\(885\) −0.145253 −0.00488263
\(886\) 0 0
\(887\) −9.37946 −0.314931 −0.157466 0.987524i \(-0.550332\pi\)
−0.157466 + 0.987524i \(0.550332\pi\)
\(888\) 0 0
\(889\) 4.87892 0.163634
\(890\) 0 0
\(891\) −2.68525 −0.0899592
\(892\) 0 0
\(893\) −60.5803 −2.02724
\(894\) 0 0
\(895\) 0.157053 0.00524970
\(896\) 0 0
\(897\) −36.8525 −1.23047
\(898\) 0 0
\(899\) −5.60568 −0.186960
\(900\) 0 0
\(901\) 13.7484 0.458026
\(902\) 0 0
\(903\) −4.04517 −0.134615
\(904\) 0 0
\(905\) 0.611835 0.0203381
\(906\) 0 0
\(907\) 43.2416 1.43582 0.717908 0.696138i \(-0.245100\pi\)
0.717908 + 0.696138i \(0.245100\pi\)
\(908\) 0 0
\(909\) −11.5022 −0.381503
\(910\) 0 0
\(911\) 32.3176 1.07073 0.535366 0.844620i \(-0.320174\pi\)
0.535366 + 0.844620i \(0.320174\pi\)
\(912\) 0 0
\(913\) 27.2948 0.903325
\(914\) 0 0
\(915\) −0.790749 −0.0261414
\(916\) 0 0
\(917\) −4.71827 −0.155811
\(918\) 0 0
\(919\) 27.5180 0.907735 0.453868 0.891069i \(-0.350044\pi\)
0.453868 + 0.891069i \(0.350044\pi\)
\(920\) 0 0
\(921\) 8.83095 0.290990
\(922\) 0 0
\(923\) −83.7935 −2.75810
\(924\) 0 0
\(925\) 38.3994 1.26257
\(926\) 0 0
\(927\) 11.8029 0.387660
\(928\) 0 0
\(929\) 56.3400 1.84846 0.924228 0.381841i \(-0.124710\pi\)
0.924228 + 0.381841i \(0.124710\pi\)
\(930\) 0 0
\(931\) 50.7327 1.66270
\(932\) 0 0
\(933\) 15.6371 0.511936
\(934\) 0 0
\(935\) −0.702282 −0.0229671
\(936\) 0 0
\(937\) 21.2162 0.693104 0.346552 0.938031i \(-0.387352\pi\)
0.346552 + 0.938031i \(0.387352\pi\)
\(938\) 0 0
\(939\) −14.0920 −0.459874
\(940\) 0 0
\(941\) 51.1429 1.66721 0.833606 0.552360i \(-0.186273\pi\)
0.833606 + 0.552360i \(0.186273\pi\)
\(942\) 0 0
\(943\) 11.0356 0.359370
\(944\) 0 0
\(945\) −0.0486324 −0.00158201
\(946\) 0 0
\(947\) 30.3320 0.985658 0.492829 0.870126i \(-0.335963\pi\)
0.492829 + 0.870126i \(0.335963\pi\)
\(948\) 0 0
\(949\) 68.2746 2.21629
\(950\) 0 0
\(951\) 16.3138 0.529010
\(952\) 0 0
\(953\) −30.9277 −1.00184 −0.500922 0.865492i \(-0.667006\pi\)
−0.500922 + 0.865492i \(0.667006\pi\)
\(954\) 0 0
\(955\) −0.0916830 −0.00296679
\(956\) 0 0
\(957\) −5.25611 −0.169906
\(958\) 0 0
\(959\) 8.53678 0.275667
\(960\) 0 0
\(961\) −22.7984 −0.735433
\(962\) 0 0
\(963\) −15.5983 −0.502650
\(964\) 0 0
\(965\) −0.946977 −0.0304843
\(966\) 0 0
\(967\) −5.89168 −0.189464 −0.0947318 0.995503i \(-0.530199\pi\)
−0.0947318 + 0.995503i \(0.530199\pi\)
\(968\) 0 0
\(969\) −33.4120 −1.07335
\(970\) 0 0
\(971\) −27.9307 −0.896340 −0.448170 0.893948i \(-0.647924\pi\)
−0.448170 + 0.893948i \(0.647924\pi\)
\(972\) 0 0
\(973\) −13.7251 −0.440008
\(974\) 0 0
\(975\) −33.9407 −1.08697
\(976\) 0 0
\(977\) 17.6646 0.565141 0.282570 0.959247i \(-0.408813\pi\)
0.282570 + 0.959247i \(0.408813\pi\)
\(978\) 0 0
\(979\) 7.37844 0.235816
\(980\) 0 0
\(981\) 17.4215 0.556224
\(982\) 0 0
\(983\) −26.1753 −0.834863 −0.417432 0.908708i \(-0.637070\pi\)
−0.417432 + 0.908708i \(0.637070\pi\)
\(984\) 0 0
\(985\) −1.62946 −0.0519188
\(986\) 0 0
\(987\) −5.96741 −0.189945
\(988\) 0 0
\(989\) −28.0505 −0.891953
\(990\) 0 0
\(991\) 0.799067 0.0253832 0.0126916 0.999919i \(-0.495960\pi\)
0.0126916 + 0.999919i \(0.495960\pi\)
\(992\) 0 0
\(993\) 8.18087 0.259612
\(994\) 0 0
\(995\) 1.06758 0.0338447
\(996\) 0 0
\(997\) 48.0295 1.52111 0.760555 0.649274i \(-0.224927\pi\)
0.760555 + 0.649274i \(0.224927\pi\)
\(998\) 0 0
\(999\) −7.68582 −0.243169
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.w.1.5 7
4.3 odd 2 4008.2.a.g.1.5 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4008.2.a.g.1.5 7 4.3 odd 2
8016.2.a.w.1.5 7 1.1 even 1 trivial