Properties

Label 8016.2.a.bb.1.4
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 1.4
Root \(1.61974\) of defining polynomial
Character \(\chi\) \(=\) 8016.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -0.619742 q^{5} -1.05844 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -0.619742 q^{5} -1.05844 q^{7} +1.00000 q^{9} +2.94492 q^{11} -6.55099 q^{13} +0.619742 q^{15} +6.19312 q^{17} -7.10438 q^{19} +1.05844 q^{21} -4.16491 q^{23} -4.61592 q^{25} -1.00000 q^{27} -0.996993 q^{29} -7.31224 q^{31} -2.94492 q^{33} +0.655958 q^{35} +5.89613 q^{37} +6.55099 q^{39} +9.96986 q^{41} -4.82896 q^{43} -0.619742 q^{45} -6.61501 q^{47} -5.87971 q^{49} -6.19312 q^{51} +10.9128 q^{53} -1.82509 q^{55} +7.10438 q^{57} +9.89351 q^{59} +10.2882 q^{61} -1.05844 q^{63} +4.05992 q^{65} +10.2505 q^{67} +4.16491 q^{69} -11.7281 q^{71} -12.2938 q^{73} +4.61592 q^{75} -3.11701 q^{77} +3.71458 q^{79} +1.00000 q^{81} +5.90125 q^{83} -3.83814 q^{85} +0.996993 q^{87} +9.88073 q^{89} +6.93380 q^{91} +7.31224 q^{93} +4.40288 q^{95} -8.96090 q^{97} +2.94492 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9q - 9q^{3} + 9q^{5} - 2q^{7} + 9q^{9} + O(q^{10}) \) \( 9q - 9q^{3} + 9q^{5} - 2q^{7} + 9q^{9} - 7q^{11} + 6q^{13} - 9q^{15} + 7q^{17} - 2q^{19} + 2q^{21} - 19q^{23} + 22q^{25} - 9q^{27} + 13q^{29} - 12q^{31} + 7q^{33} - 4q^{35} + 15q^{37} - 6q^{39} + 18q^{41} + 6q^{43} + 9q^{45} - 25q^{47} + 19q^{49} - 7q^{51} + 17q^{53} + 3q^{55} + 2q^{57} - 3q^{59} + 14q^{61} - 2q^{63} + 14q^{65} + 4q^{67} + 19q^{69} - 17q^{71} - 20q^{73} - 22q^{75} + 14q^{77} + 8q^{79} + 9q^{81} + q^{83} + 5q^{85} - 13q^{87} + 36q^{89} + 41q^{91} + 12q^{93} - 5q^{95} + 31q^{97} - 7q^{99} + O(q^{100}) \)

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
\(5\) −0.619742 −0.277157 −0.138579 0.990351i \(-0.544253\pi\)
−0.138579 + 0.990351i \(0.544253\pi\)
\(6\) 0 0
\(7\) −1.05844 −0.400051 −0.200026 0.979791i \(-0.564103\pi\)
−0.200026 + 0.979791i \(0.564103\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.94492 0.887927 0.443963 0.896045i \(-0.353572\pi\)
0.443963 + 0.896045i \(0.353572\pi\)
\(12\) 0 0
\(13\) −6.55099 −1.81692 −0.908459 0.417975i \(-0.862740\pi\)
−0.908459 + 0.417975i \(0.862740\pi\)
\(14\) 0 0
\(15\) 0.619742 0.160017
\(16\) 0 0
\(17\) 6.19312 1.50205 0.751026 0.660272i \(-0.229559\pi\)
0.751026 + 0.660272i \(0.229559\pi\)
\(18\) 0 0
\(19\) −7.10438 −1.62986 −0.814928 0.579562i \(-0.803224\pi\)
−0.814928 + 0.579562i \(0.803224\pi\)
\(20\) 0 0
\(21\) 1.05844 0.230970
\(22\) 0 0
\(23\) −4.16491 −0.868444 −0.434222 0.900806i \(-0.642977\pi\)
−0.434222 + 0.900806i \(0.642977\pi\)
\(24\) 0 0
\(25\) −4.61592 −0.923184
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −0.996993 −0.185137 −0.0925685 0.995706i \(-0.529508\pi\)
−0.0925685 + 0.995706i \(0.529508\pi\)
\(30\) 0 0
\(31\) −7.31224 −1.31332 −0.656659 0.754188i \(-0.728031\pi\)
−0.656659 + 0.754188i \(0.728031\pi\)
\(32\) 0 0
\(33\) −2.94492 −0.512645
\(34\) 0 0
\(35\) 0.655958 0.110877
\(36\) 0 0
\(37\) 5.89613 0.969317 0.484659 0.874703i \(-0.338944\pi\)
0.484659 + 0.874703i \(0.338944\pi\)
\(38\) 0 0
\(39\) 6.55099 1.04900
\(40\) 0 0
\(41\) 9.96986 1.55703 0.778515 0.627626i \(-0.215973\pi\)
0.778515 + 0.627626i \(0.215973\pi\)
\(42\) 0 0
\(43\) −4.82896 −0.736409 −0.368205 0.929745i \(-0.620027\pi\)
−0.368205 + 0.929745i \(0.620027\pi\)
\(44\) 0 0
\(45\) −0.619742 −0.0923857
\(46\) 0 0
\(47\) −6.61501 −0.964898 −0.482449 0.875924i \(-0.660253\pi\)
−0.482449 + 0.875924i \(0.660253\pi\)
\(48\) 0 0
\(49\) −5.87971 −0.839959
\(50\) 0 0
\(51\) −6.19312 −0.867211
\(52\) 0 0
\(53\) 10.9128 1.49899 0.749497 0.662007i \(-0.230295\pi\)
0.749497 + 0.662007i \(0.230295\pi\)
\(54\) 0 0
\(55\) −1.82509 −0.246095
\(56\) 0 0
\(57\) 7.10438 0.940998
\(58\) 0 0
\(59\) 9.89351 1.28803 0.644013 0.765015i \(-0.277268\pi\)
0.644013 + 0.765015i \(0.277268\pi\)
\(60\) 0 0
\(61\) 10.2882 1.31727 0.658633 0.752464i \(-0.271135\pi\)
0.658633 + 0.752464i \(0.271135\pi\)
\(62\) 0 0
\(63\) −1.05844 −0.133350
\(64\) 0 0
\(65\) 4.05992 0.503572
\(66\) 0 0
\(67\) 10.2505 1.25230 0.626148 0.779705i \(-0.284631\pi\)
0.626148 + 0.779705i \(0.284631\pi\)
\(68\) 0 0
\(69\) 4.16491 0.501396
\(70\) 0 0
\(71\) −11.7281 −1.39187 −0.695936 0.718103i \(-0.745010\pi\)
−0.695936 + 0.718103i \(0.745010\pi\)
\(72\) 0 0
\(73\) −12.2938 −1.43888 −0.719442 0.694552i \(-0.755602\pi\)
−0.719442 + 0.694552i \(0.755602\pi\)
\(74\) 0 0
\(75\) 4.61592 0.533000
\(76\) 0 0
\(77\) −3.11701 −0.355216
\(78\) 0 0
\(79\) 3.71458 0.417923 0.208962 0.977924i \(-0.432992\pi\)
0.208962 + 0.977924i \(0.432992\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 5.90125 0.647747 0.323873 0.946100i \(-0.395015\pi\)
0.323873 + 0.946100i \(0.395015\pi\)
\(84\) 0 0
\(85\) −3.83814 −0.416305
\(86\) 0 0
\(87\) 0.996993 0.106889
\(88\) 0 0
\(89\) 9.88073 1.04735 0.523677 0.851917i \(-0.324560\pi\)
0.523677 + 0.851917i \(0.324560\pi\)
\(90\) 0 0
\(91\) 6.93380 0.726860
\(92\) 0 0
\(93\) 7.31224 0.758244
\(94\) 0 0
\(95\) 4.40288 0.451726
\(96\) 0 0
\(97\) −8.96090 −0.909841 −0.454921 0.890532i \(-0.650332\pi\)
−0.454921 + 0.890532i \(0.650332\pi\)
\(98\) 0 0
\(99\) 2.94492 0.295976
\(100\) 0 0
\(101\) −4.91541 −0.489101 −0.244551 0.969637i \(-0.578640\pi\)
−0.244551 + 0.969637i \(0.578640\pi\)
\(102\) 0 0
\(103\) −4.72990 −0.466051 −0.233026 0.972471i \(-0.574863\pi\)
−0.233026 + 0.972471i \(0.574863\pi\)
\(104\) 0 0
\(105\) −0.655958 −0.0640149
\(106\) 0 0
\(107\) −2.10628 −0.203622 −0.101811 0.994804i \(-0.532464\pi\)
−0.101811 + 0.994804i \(0.532464\pi\)
\(108\) 0 0
\(109\) −6.73056 −0.644671 −0.322336 0.946625i \(-0.604468\pi\)
−0.322336 + 0.946625i \(0.604468\pi\)
\(110\) 0 0
\(111\) −5.89613 −0.559636
\(112\) 0 0
\(113\) 4.21493 0.396507 0.198254 0.980151i \(-0.436473\pi\)
0.198254 + 0.980151i \(0.436473\pi\)
\(114\) 0 0
\(115\) 2.58117 0.240695
\(116\) 0 0
\(117\) −6.55099 −0.605639
\(118\) 0 0
\(119\) −6.55503 −0.600898
\(120\) 0 0
\(121\) −2.32744 −0.211586
\(122\) 0 0
\(123\) −9.96986 −0.898952
\(124\) 0 0
\(125\) 5.95939 0.533024
\(126\) 0 0
\(127\) −11.2289 −0.996401 −0.498201 0.867062i \(-0.666006\pi\)
−0.498201 + 0.867062i \(0.666006\pi\)
\(128\) 0 0
\(129\) 4.82896 0.425166
\(130\) 0 0
\(131\) 4.86472 0.425032 0.212516 0.977158i \(-0.431834\pi\)
0.212516 + 0.977158i \(0.431834\pi\)
\(132\) 0 0
\(133\) 7.51953 0.652026
\(134\) 0 0
\(135\) 0.619742 0.0533389
\(136\) 0 0
\(137\) −1.48097 −0.126528 −0.0632640 0.997997i \(-0.520151\pi\)
−0.0632640 + 0.997997i \(0.520151\pi\)
\(138\) 0 0
\(139\) 11.8564 1.00565 0.502823 0.864389i \(-0.332295\pi\)
0.502823 + 0.864389i \(0.332295\pi\)
\(140\) 0 0
\(141\) 6.61501 0.557084
\(142\) 0 0
\(143\) −19.2921 −1.61329
\(144\) 0 0
\(145\) 0.617879 0.0513120
\(146\) 0 0
\(147\) 5.87971 0.484951
\(148\) 0 0
\(149\) 1.25014 0.102415 0.0512075 0.998688i \(-0.483693\pi\)
0.0512075 + 0.998688i \(0.483693\pi\)
\(150\) 0 0
\(151\) −8.74306 −0.711500 −0.355750 0.934581i \(-0.615775\pi\)
−0.355750 + 0.934581i \(0.615775\pi\)
\(152\) 0 0
\(153\) 6.19312 0.500684
\(154\) 0 0
\(155\) 4.53170 0.363995
\(156\) 0 0
\(157\) 1.28243 0.102349 0.0511747 0.998690i \(-0.483703\pi\)
0.0511747 + 0.998690i \(0.483703\pi\)
\(158\) 0 0
\(159\) −10.9128 −0.865445
\(160\) 0 0
\(161\) 4.40829 0.347422
\(162\) 0 0
\(163\) −10.2657 −0.804074 −0.402037 0.915623i \(-0.631698\pi\)
−0.402037 + 0.915623i \(0.631698\pi\)
\(164\) 0 0
\(165\) 1.82509 0.142083
\(166\) 0 0
\(167\) −1.00000 −0.0773823
\(168\) 0 0
\(169\) 29.9154 2.30119
\(170\) 0 0
\(171\) −7.10438 −0.543285
\(172\) 0 0
\(173\) −1.26599 −0.0962515 −0.0481257 0.998841i \(-0.515325\pi\)
−0.0481257 + 0.998841i \(0.515325\pi\)
\(174\) 0 0
\(175\) 4.88566 0.369321
\(176\) 0 0
\(177\) −9.89351 −0.743642
\(178\) 0 0
\(179\) 7.48456 0.559422 0.279711 0.960084i \(-0.409761\pi\)
0.279711 + 0.960084i \(0.409761\pi\)
\(180\) 0 0
\(181\) −4.86471 −0.361591 −0.180796 0.983521i \(-0.557867\pi\)
−0.180796 + 0.983521i \(0.557867\pi\)
\(182\) 0 0
\(183\) −10.2882 −0.760524
\(184\) 0 0
\(185\) −3.65408 −0.268653
\(186\) 0 0
\(187\) 18.2383 1.33371
\(188\) 0 0
\(189\) 1.05844 0.0769899
\(190\) 0 0
\(191\) −8.37459 −0.605964 −0.302982 0.952996i \(-0.597982\pi\)
−0.302982 + 0.952996i \(0.597982\pi\)
\(192\) 0 0
\(193\) −6.53375 −0.470309 −0.235155 0.971958i \(-0.575560\pi\)
−0.235155 + 0.971958i \(0.575560\pi\)
\(194\) 0 0
\(195\) −4.05992 −0.290737
\(196\) 0 0
\(197\) 2.01768 0.143754 0.0718768 0.997414i \(-0.477101\pi\)
0.0718768 + 0.997414i \(0.477101\pi\)
\(198\) 0 0
\(199\) 18.2648 1.29476 0.647379 0.762168i \(-0.275865\pi\)
0.647379 + 0.762168i \(0.275865\pi\)
\(200\) 0 0
\(201\) −10.2505 −0.723013
\(202\) 0 0
\(203\) 1.05525 0.0740643
\(204\) 0 0
\(205\) −6.17874 −0.431542
\(206\) 0 0
\(207\) −4.16491 −0.289481
\(208\) 0 0
\(209\) −20.9218 −1.44719
\(210\) 0 0
\(211\) −8.93507 −0.615116 −0.307558 0.951529i \(-0.599512\pi\)
−0.307558 + 0.951529i \(0.599512\pi\)
\(212\) 0 0
\(213\) 11.7281 0.803598
\(214\) 0 0
\(215\) 2.99271 0.204101
\(216\) 0 0
\(217\) 7.73954 0.525394
\(218\) 0 0
\(219\) 12.2938 0.830740
\(220\) 0 0
\(221\) −40.5711 −2.72911
\(222\) 0 0
\(223\) 17.7607 1.18934 0.594671 0.803969i \(-0.297282\pi\)
0.594671 + 0.803969i \(0.297282\pi\)
\(224\) 0 0
\(225\) −4.61592 −0.307728
\(226\) 0 0
\(227\) 11.3559 0.753716 0.376858 0.926271i \(-0.377004\pi\)
0.376858 + 0.926271i \(0.377004\pi\)
\(228\) 0 0
\(229\) 14.3141 0.945900 0.472950 0.881089i \(-0.343189\pi\)
0.472950 + 0.881089i \(0.343189\pi\)
\(230\) 0 0
\(231\) 3.11701 0.205084
\(232\) 0 0
\(233\) 20.6800 1.35479 0.677395 0.735620i \(-0.263109\pi\)
0.677395 + 0.735620i \(0.263109\pi\)
\(234\) 0 0
\(235\) 4.09960 0.267428
\(236\) 0 0
\(237\) −3.71458 −0.241288
\(238\) 0 0
\(239\) −9.67387 −0.625750 −0.312875 0.949794i \(-0.601292\pi\)
−0.312875 + 0.949794i \(0.601292\pi\)
\(240\) 0 0
\(241\) 7.89334 0.508455 0.254227 0.967144i \(-0.418179\pi\)
0.254227 + 0.967144i \(0.418179\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 3.64391 0.232801
\(246\) 0 0
\(247\) 46.5407 2.96131
\(248\) 0 0
\(249\) −5.90125 −0.373977
\(250\) 0 0
\(251\) 30.8625 1.94802 0.974011 0.226500i \(-0.0727282\pi\)
0.974011 + 0.226500i \(0.0727282\pi\)
\(252\) 0 0
\(253\) −12.2653 −0.771115
\(254\) 0 0
\(255\) 3.83814 0.240354
\(256\) 0 0
\(257\) −16.5611 −1.03305 −0.516525 0.856272i \(-0.672775\pi\)
−0.516525 + 0.856272i \(0.672775\pi\)
\(258\) 0 0
\(259\) −6.24067 −0.387777
\(260\) 0 0
\(261\) −0.996993 −0.0617123
\(262\) 0 0
\(263\) 2.43522 0.150162 0.0750812 0.997177i \(-0.476078\pi\)
0.0750812 + 0.997177i \(0.476078\pi\)
\(264\) 0 0
\(265\) −6.76315 −0.415457
\(266\) 0 0
\(267\) −9.88073 −0.604691
\(268\) 0 0
\(269\) 32.6765 1.99232 0.996162 0.0875285i \(-0.0278969\pi\)
0.996162 + 0.0875285i \(0.0278969\pi\)
\(270\) 0 0
\(271\) 0.00431859 0.000262335 0 0.000131168 1.00000i \(-0.499958\pi\)
0.000131168 1.00000i \(0.499958\pi\)
\(272\) 0 0
\(273\) −6.93380 −0.419653
\(274\) 0 0
\(275\) −13.5935 −0.819720
\(276\) 0 0
\(277\) −25.8726 −1.55453 −0.777266 0.629172i \(-0.783394\pi\)
−0.777266 + 0.629172i \(0.783394\pi\)
\(278\) 0 0
\(279\) −7.31224 −0.437772
\(280\) 0 0
\(281\) 23.4091 1.39647 0.698236 0.715868i \(-0.253969\pi\)
0.698236 + 0.715868i \(0.253969\pi\)
\(282\) 0 0
\(283\) 26.8435 1.59568 0.797842 0.602867i \(-0.205975\pi\)
0.797842 + 0.602867i \(0.205975\pi\)
\(284\) 0 0
\(285\) −4.40288 −0.260804
\(286\) 0 0
\(287\) −10.5525 −0.622892
\(288\) 0 0
\(289\) 21.3548 1.25616
\(290\) 0 0
\(291\) 8.96090 0.525297
\(292\) 0 0
\(293\) 23.7852 1.38954 0.694772 0.719230i \(-0.255505\pi\)
0.694772 + 0.719230i \(0.255505\pi\)
\(294\) 0 0
\(295\) −6.13143 −0.356985
\(296\) 0 0
\(297\) −2.94492 −0.170882
\(298\) 0 0
\(299\) 27.2843 1.57789
\(300\) 0 0
\(301\) 5.11115 0.294602
\(302\) 0 0
\(303\) 4.91541 0.282383
\(304\) 0 0
\(305\) −6.37602 −0.365090
\(306\) 0 0
\(307\) 10.0558 0.573914 0.286957 0.957943i \(-0.407356\pi\)
0.286957 + 0.957943i \(0.407356\pi\)
\(308\) 0 0
\(309\) 4.72990 0.269075
\(310\) 0 0
\(311\) −18.7831 −1.06509 −0.532547 0.846401i \(-0.678765\pi\)
−0.532547 + 0.846401i \(0.678765\pi\)
\(312\) 0 0
\(313\) 14.5269 0.821110 0.410555 0.911836i \(-0.365335\pi\)
0.410555 + 0.911836i \(0.365335\pi\)
\(314\) 0 0
\(315\) 0.655958 0.0369590
\(316\) 0 0
\(317\) −8.74455 −0.491143 −0.245571 0.969379i \(-0.578976\pi\)
−0.245571 + 0.969379i \(0.578976\pi\)
\(318\) 0 0
\(319\) −2.93606 −0.164388
\(320\) 0 0
\(321\) 2.10628 0.117561
\(322\) 0 0
\(323\) −43.9983 −2.44813
\(324\) 0 0
\(325\) 30.2388 1.67735
\(326\) 0 0
\(327\) 6.73056 0.372201
\(328\) 0 0
\(329\) 7.00156 0.386009
\(330\) 0 0
\(331\) 4.49622 0.247135 0.123567 0.992336i \(-0.460567\pi\)
0.123567 + 0.992336i \(0.460567\pi\)
\(332\) 0 0
\(333\) 5.89613 0.323106
\(334\) 0 0
\(335\) −6.35265 −0.347082
\(336\) 0 0
\(337\) 11.1496 0.607359 0.303680 0.952774i \(-0.401785\pi\)
0.303680 + 0.952774i \(0.401785\pi\)
\(338\) 0 0
\(339\) −4.21493 −0.228923
\(340\) 0 0
\(341\) −21.5340 −1.16613
\(342\) 0 0
\(343\) 13.6324 0.736078
\(344\) 0 0
\(345\) −2.58117 −0.138966
\(346\) 0 0
\(347\) −31.7784 −1.70595 −0.852977 0.521949i \(-0.825205\pi\)
−0.852977 + 0.521949i \(0.825205\pi\)
\(348\) 0 0
\(349\) −11.4428 −0.612521 −0.306261 0.951948i \(-0.599078\pi\)
−0.306261 + 0.951948i \(0.599078\pi\)
\(350\) 0 0
\(351\) 6.55099 0.349666
\(352\) 0 0
\(353\) 5.68738 0.302709 0.151354 0.988480i \(-0.451637\pi\)
0.151354 + 0.988480i \(0.451637\pi\)
\(354\) 0 0
\(355\) 7.26842 0.385767
\(356\) 0 0
\(357\) 6.55503 0.346929
\(358\) 0 0
\(359\) 36.8239 1.94349 0.971745 0.236031i \(-0.0758468\pi\)
0.971745 + 0.236031i \(0.0758468\pi\)
\(360\) 0 0
\(361\) 31.4722 1.65643
\(362\) 0 0
\(363\) 2.32744 0.122159
\(364\) 0 0
\(365\) 7.61901 0.398797
\(366\) 0 0
\(367\) 22.4128 1.16994 0.584968 0.811056i \(-0.301107\pi\)
0.584968 + 0.811056i \(0.301107\pi\)
\(368\) 0 0
\(369\) 9.96986 0.519010
\(370\) 0 0
\(371\) −11.5506 −0.599675
\(372\) 0 0
\(373\) −20.0874 −1.04009 −0.520044 0.854140i \(-0.674084\pi\)
−0.520044 + 0.854140i \(0.674084\pi\)
\(374\) 0 0
\(375\) −5.95939 −0.307742
\(376\) 0 0
\(377\) 6.53129 0.336378
\(378\) 0 0
\(379\) 13.4368 0.690203 0.345102 0.938565i \(-0.387844\pi\)
0.345102 + 0.938565i \(0.387844\pi\)
\(380\) 0 0
\(381\) 11.2289 0.575273
\(382\) 0 0
\(383\) 21.1688 1.08168 0.540838 0.841127i \(-0.318107\pi\)
0.540838 + 0.841127i \(0.318107\pi\)
\(384\) 0 0
\(385\) 1.93174 0.0984508
\(386\) 0 0
\(387\) −4.82896 −0.245470
\(388\) 0 0
\(389\) 13.8032 0.699849 0.349924 0.936778i \(-0.386207\pi\)
0.349924 + 0.936778i \(0.386207\pi\)
\(390\) 0 0
\(391\) −25.7938 −1.30445
\(392\) 0 0
\(393\) −4.86472 −0.245393
\(394\) 0 0
\(395\) −2.30208 −0.115830
\(396\) 0 0
\(397\) 23.8793 1.19847 0.599234 0.800574i \(-0.295472\pi\)
0.599234 + 0.800574i \(0.295472\pi\)
\(398\) 0 0
\(399\) −7.51953 −0.376447
\(400\) 0 0
\(401\) −31.6924 −1.58265 −0.791323 0.611399i \(-0.790607\pi\)
−0.791323 + 0.611399i \(0.790607\pi\)
\(402\) 0 0
\(403\) 47.9024 2.38619
\(404\) 0 0
\(405\) −0.619742 −0.0307952
\(406\) 0 0
\(407\) 17.3636 0.860683
\(408\) 0 0
\(409\) −10.5695 −0.522626 −0.261313 0.965254i \(-0.584156\pi\)
−0.261313 + 0.965254i \(0.584156\pi\)
\(410\) 0 0
\(411\) 1.48097 0.0730510
\(412\) 0 0
\(413\) −10.4717 −0.515276
\(414\) 0 0
\(415\) −3.65726 −0.179528
\(416\) 0 0
\(417\) −11.8564 −0.580610
\(418\) 0 0
\(419\) 6.09660 0.297839 0.148919 0.988849i \(-0.452421\pi\)
0.148919 + 0.988849i \(0.452421\pi\)
\(420\) 0 0
\(421\) −8.05451 −0.392553 −0.196276 0.980549i \(-0.562885\pi\)
−0.196276 + 0.980549i \(0.562885\pi\)
\(422\) 0 0
\(423\) −6.61501 −0.321633
\(424\) 0 0
\(425\) −28.5870 −1.38667
\(426\) 0 0
\(427\) −10.8894 −0.526974
\(428\) 0 0
\(429\) 19.2921 0.931433
\(430\) 0 0
\(431\) −7.89755 −0.380412 −0.190206 0.981744i \(-0.560916\pi\)
−0.190206 + 0.981744i \(0.560916\pi\)
\(432\) 0 0
\(433\) −20.1848 −0.970021 −0.485010 0.874508i \(-0.661184\pi\)
−0.485010 + 0.874508i \(0.661184\pi\)
\(434\) 0 0
\(435\) −0.617879 −0.0296250
\(436\) 0 0
\(437\) 29.5891 1.41544
\(438\) 0 0
\(439\) 30.9441 1.47688 0.738442 0.674317i \(-0.235562\pi\)
0.738442 + 0.674317i \(0.235562\pi\)
\(440\) 0 0
\(441\) −5.87971 −0.279986
\(442\) 0 0
\(443\) 33.5088 1.59205 0.796025 0.605264i \(-0.206933\pi\)
0.796025 + 0.605264i \(0.206933\pi\)
\(444\) 0 0
\(445\) −6.12350 −0.290282
\(446\) 0 0
\(447\) −1.25014 −0.0591294
\(448\) 0 0
\(449\) −21.8767 −1.03243 −0.516213 0.856460i \(-0.672659\pi\)
−0.516213 + 0.856460i \(0.672659\pi\)
\(450\) 0 0
\(451\) 29.3604 1.38253
\(452\) 0 0
\(453\) 8.74306 0.410785
\(454\) 0 0
\(455\) −4.29717 −0.201454
\(456\) 0 0
\(457\) 21.2520 0.994128 0.497064 0.867714i \(-0.334411\pi\)
0.497064 + 0.867714i \(0.334411\pi\)
\(458\) 0 0
\(459\) −6.19312 −0.289070
\(460\) 0 0
\(461\) −2.74130 −0.127675 −0.0638375 0.997960i \(-0.520334\pi\)
−0.0638375 + 0.997960i \(0.520334\pi\)
\(462\) 0 0
\(463\) −0.939856 −0.0436788 −0.0218394 0.999761i \(-0.506952\pi\)
−0.0218394 + 0.999761i \(0.506952\pi\)
\(464\) 0 0
\(465\) −4.53170 −0.210153
\(466\) 0 0
\(467\) −4.00247 −0.185212 −0.0926061 0.995703i \(-0.529520\pi\)
−0.0926061 + 0.995703i \(0.529520\pi\)
\(468\) 0 0
\(469\) −10.8495 −0.500982
\(470\) 0 0
\(471\) −1.28243 −0.0590915
\(472\) 0 0
\(473\) −14.2209 −0.653878
\(474\) 0 0
\(475\) 32.7932 1.50466
\(476\) 0 0
\(477\) 10.9128 0.499665
\(478\) 0 0
\(479\) 9.13204 0.417254 0.208627 0.977995i \(-0.433101\pi\)
0.208627 + 0.977995i \(0.433101\pi\)
\(480\) 0 0
\(481\) −38.6255 −1.76117
\(482\) 0 0
\(483\) −4.40829 −0.200584
\(484\) 0 0
\(485\) 5.55345 0.252169
\(486\) 0 0
\(487\) 35.6996 1.61770 0.808852 0.588012i \(-0.200089\pi\)
0.808852 + 0.588012i \(0.200089\pi\)
\(488\) 0 0
\(489\) 10.2657 0.464232
\(490\) 0 0
\(491\) −38.1762 −1.72287 −0.861434 0.507870i \(-0.830433\pi\)
−0.861434 + 0.507870i \(0.830433\pi\)
\(492\) 0 0
\(493\) −6.17450 −0.278085
\(494\) 0 0
\(495\) −1.82509 −0.0820318
\(496\) 0 0
\(497\) 12.4135 0.556821
\(498\) 0 0
\(499\) 1.03459 0.0463144 0.0231572 0.999732i \(-0.492628\pi\)
0.0231572 + 0.999732i \(0.492628\pi\)
\(500\) 0 0
\(501\) 1.00000 0.0446767
\(502\) 0 0
\(503\) −28.2992 −1.26180 −0.630900 0.775864i \(-0.717314\pi\)
−0.630900 + 0.775864i \(0.717314\pi\)
\(504\) 0 0
\(505\) 3.04629 0.135558
\(506\) 0 0
\(507\) −29.9154 −1.32859
\(508\) 0 0
\(509\) 9.65536 0.427966 0.213983 0.976837i \(-0.431356\pi\)
0.213983 + 0.976837i \(0.431356\pi\)
\(510\) 0 0
\(511\) 13.0122 0.575627
\(512\) 0 0
\(513\) 7.10438 0.313666
\(514\) 0 0
\(515\) 2.93132 0.129169
\(516\) 0 0
\(517\) −19.4807 −0.856759
\(518\) 0 0
\(519\) 1.26599 0.0555708
\(520\) 0 0
\(521\) −0.841870 −0.0368830 −0.0184415 0.999830i \(-0.505870\pi\)
−0.0184415 + 0.999830i \(0.505870\pi\)
\(522\) 0 0
\(523\) 12.6622 0.553679 0.276840 0.960916i \(-0.410713\pi\)
0.276840 + 0.960916i \(0.410713\pi\)
\(524\) 0 0
\(525\) −4.88566 −0.213228
\(526\) 0 0
\(527\) −45.2856 −1.97267
\(528\) 0 0
\(529\) −5.65352 −0.245805
\(530\) 0 0
\(531\) 9.89351 0.429342
\(532\) 0 0
\(533\) −65.3124 −2.82900
\(534\) 0 0
\(535\) 1.30535 0.0564352
\(536\) 0 0
\(537\) −7.48456 −0.322982
\(538\) 0 0
\(539\) −17.3153 −0.745822
\(540\) 0 0
\(541\) 10.6265 0.456869 0.228435 0.973559i \(-0.426639\pi\)
0.228435 + 0.973559i \(0.426639\pi\)
\(542\) 0 0
\(543\) 4.86471 0.208765
\(544\) 0 0
\(545\) 4.17122 0.178675
\(546\) 0 0
\(547\) −46.1517 −1.97330 −0.986652 0.162841i \(-0.947934\pi\)
−0.986652 + 0.162841i \(0.947934\pi\)
\(548\) 0 0
\(549\) 10.2882 0.439089
\(550\) 0 0
\(551\) 7.08301 0.301747
\(552\) 0 0
\(553\) −3.93165 −0.167191
\(554\) 0 0
\(555\) 3.65408 0.155107
\(556\) 0 0
\(557\) −19.0103 −0.805494 −0.402747 0.915311i \(-0.631944\pi\)
−0.402747 + 0.915311i \(0.631944\pi\)
\(558\) 0 0
\(559\) 31.6345 1.33800
\(560\) 0 0
\(561\) −18.2383 −0.770020
\(562\) 0 0
\(563\) 24.1476 1.01770 0.508850 0.860855i \(-0.330071\pi\)
0.508850 + 0.860855i \(0.330071\pi\)
\(564\) 0 0
\(565\) −2.61217 −0.109895
\(566\) 0 0
\(567\) −1.05844 −0.0444501
\(568\) 0 0
\(569\) −22.7175 −0.952369 −0.476185 0.879345i \(-0.657981\pi\)
−0.476185 + 0.879345i \(0.657981\pi\)
\(570\) 0 0
\(571\) 12.5374 0.524673 0.262337 0.964976i \(-0.415507\pi\)
0.262337 + 0.964976i \(0.415507\pi\)
\(572\) 0 0
\(573\) 8.37459 0.349854
\(574\) 0 0
\(575\) 19.2249 0.801733
\(576\) 0 0
\(577\) 41.0884 1.71053 0.855267 0.518188i \(-0.173393\pi\)
0.855267 + 0.518188i \(0.173393\pi\)
\(578\) 0 0
\(579\) 6.53375 0.271533
\(580\) 0 0
\(581\) −6.24610 −0.259132
\(582\) 0 0
\(583\) 32.1375 1.33100
\(584\) 0 0
\(585\) 4.05992 0.167857
\(586\) 0 0
\(587\) 1.04147 0.0429861 0.0214931 0.999769i \(-0.493158\pi\)
0.0214931 + 0.999769i \(0.493158\pi\)
\(588\) 0 0
\(589\) 51.9489 2.14052
\(590\) 0 0
\(591\) −2.01768 −0.0829962
\(592\) 0 0
\(593\) 34.8470 1.43100 0.715498 0.698615i \(-0.246200\pi\)
0.715498 + 0.698615i \(0.246200\pi\)
\(594\) 0 0
\(595\) 4.06243 0.166543
\(596\) 0 0
\(597\) −18.2648 −0.747529
\(598\) 0 0
\(599\) 39.0595 1.59593 0.797965 0.602704i \(-0.205910\pi\)
0.797965 + 0.602704i \(0.205910\pi\)
\(600\) 0 0
\(601\) 9.49751 0.387411 0.193706 0.981060i \(-0.437949\pi\)
0.193706 + 0.981060i \(0.437949\pi\)
\(602\) 0 0
\(603\) 10.2505 0.417432
\(604\) 0 0
\(605\) 1.44241 0.0586425
\(606\) 0 0
\(607\) −18.2746 −0.741742 −0.370871 0.928684i \(-0.620941\pi\)
−0.370871 + 0.928684i \(0.620941\pi\)
\(608\) 0 0
\(609\) −1.05525 −0.0427610
\(610\) 0 0
\(611\) 43.3348 1.75314
\(612\) 0 0
\(613\) −33.5367 −1.35454 −0.677268 0.735736i \(-0.736836\pi\)
−0.677268 + 0.735736i \(0.736836\pi\)
\(614\) 0 0
\(615\) 6.17874 0.249151
\(616\) 0 0
\(617\) 1.66780 0.0671429 0.0335715 0.999436i \(-0.489312\pi\)
0.0335715 + 0.999436i \(0.489312\pi\)
\(618\) 0 0
\(619\) 22.5977 0.908278 0.454139 0.890931i \(-0.349947\pi\)
0.454139 + 0.890931i \(0.349947\pi\)
\(620\) 0 0
\(621\) 4.16491 0.167132
\(622\) 0 0
\(623\) −10.4581 −0.418996
\(624\) 0 0
\(625\) 19.3863 0.775452
\(626\) 0 0
\(627\) 20.9218 0.835538
\(628\) 0 0
\(629\) 36.5154 1.45597
\(630\) 0 0
\(631\) −9.50108 −0.378232 −0.189116 0.981955i \(-0.560562\pi\)
−0.189116 + 0.981955i \(0.560562\pi\)
\(632\) 0 0
\(633\) 8.93507 0.355137
\(634\) 0 0
\(635\) 6.95901 0.276160
\(636\) 0 0
\(637\) 38.5179 1.52614
\(638\) 0 0
\(639\) −11.7281 −0.463958
\(640\) 0 0
\(641\) 21.9083 0.865325 0.432663 0.901556i \(-0.357574\pi\)
0.432663 + 0.901556i \(0.357574\pi\)
\(642\) 0 0
\(643\) −41.7641 −1.64701 −0.823507 0.567306i \(-0.807986\pi\)
−0.823507 + 0.567306i \(0.807986\pi\)
\(644\) 0 0
\(645\) −2.99271 −0.117838
\(646\) 0 0
\(647\) 4.32773 0.170141 0.0850703 0.996375i \(-0.472889\pi\)
0.0850703 + 0.996375i \(0.472889\pi\)
\(648\) 0 0
\(649\) 29.1356 1.14367
\(650\) 0 0
\(651\) −7.73954 −0.303336
\(652\) 0 0
\(653\) 39.5986 1.54961 0.774807 0.632198i \(-0.217847\pi\)
0.774807 + 0.632198i \(0.217847\pi\)
\(654\) 0 0
\(655\) −3.01487 −0.117801
\(656\) 0 0
\(657\) −12.2938 −0.479628
\(658\) 0 0
\(659\) 33.0788 1.28857 0.644284 0.764786i \(-0.277155\pi\)
0.644284 + 0.764786i \(0.277155\pi\)
\(660\) 0 0
\(661\) −14.2897 −0.555803 −0.277902 0.960610i \(-0.589639\pi\)
−0.277902 + 0.960610i \(0.589639\pi\)
\(662\) 0 0
\(663\) 40.5711 1.57565
\(664\) 0 0
\(665\) −4.66017 −0.180714
\(666\) 0 0
\(667\) 4.15239 0.160781
\(668\) 0 0
\(669\) −17.7607 −0.686667
\(670\) 0 0
\(671\) 30.2979 1.16964
\(672\) 0 0
\(673\) −43.1679 −1.66400 −0.832000 0.554776i \(-0.812804\pi\)
−0.832000 + 0.554776i \(0.812804\pi\)
\(674\) 0 0
\(675\) 4.61592 0.177667
\(676\) 0 0
\(677\) 28.8027 1.10698 0.553489 0.832857i \(-0.313296\pi\)
0.553489 + 0.832857i \(0.313296\pi\)
\(678\) 0 0
\(679\) 9.48454 0.363983
\(680\) 0 0
\(681\) −11.3559 −0.435158
\(682\) 0 0
\(683\) 28.6388 1.09583 0.547916 0.836533i \(-0.315421\pi\)
0.547916 + 0.836533i \(0.315421\pi\)
\(684\) 0 0
\(685\) 0.917821 0.0350682
\(686\) 0 0
\(687\) −14.3141 −0.546116
\(688\) 0 0
\(689\) −71.4899 −2.72355
\(690\) 0 0
\(691\) −38.6489 −1.47027 −0.735137 0.677919i \(-0.762882\pi\)
−0.735137 + 0.677919i \(0.762882\pi\)
\(692\) 0 0
\(693\) −3.11701 −0.118405
\(694\) 0 0
\(695\) −7.34791 −0.278722
\(696\) 0 0
\(697\) 61.7446 2.33874
\(698\) 0 0
\(699\) −20.6800 −0.782188
\(700\) 0 0
\(701\) −3.43773 −0.129841 −0.0649206 0.997890i \(-0.520679\pi\)
−0.0649206 + 0.997890i \(0.520679\pi\)
\(702\) 0 0
\(703\) −41.8883 −1.57985
\(704\) 0 0
\(705\) −4.09960 −0.154400
\(706\) 0 0
\(707\) 5.20265 0.195666
\(708\) 0 0
\(709\) 20.2732 0.761375 0.380687 0.924704i \(-0.375687\pi\)
0.380687 + 0.924704i \(0.375687\pi\)
\(710\) 0 0
\(711\) 3.71458 0.139308
\(712\) 0 0
\(713\) 30.4548 1.14054
\(714\) 0 0
\(715\) 11.9562 0.447135
\(716\) 0 0
\(717\) 9.67387 0.361277
\(718\) 0 0
\(719\) 30.6813 1.14422 0.572110 0.820177i \(-0.306125\pi\)
0.572110 + 0.820177i \(0.306125\pi\)
\(720\) 0 0
\(721\) 5.00630 0.186444
\(722\) 0 0
\(723\) −7.89334 −0.293557
\(724\) 0 0
\(725\) 4.60204 0.170915
\(726\) 0 0
\(727\) −11.9584 −0.443512 −0.221756 0.975102i \(-0.571179\pi\)
−0.221756 + 0.975102i \(0.571179\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −29.9063 −1.10613
\(732\) 0 0
\(733\) 9.76983 0.360857 0.180429 0.983588i \(-0.442252\pi\)
0.180429 + 0.983588i \(0.442252\pi\)
\(734\) 0 0
\(735\) −3.64391 −0.134407
\(736\) 0 0
\(737\) 30.1868 1.11195
\(738\) 0 0
\(739\) 36.8792 1.35662 0.678312 0.734774i \(-0.262712\pi\)
0.678312 + 0.734774i \(0.262712\pi\)
\(740\) 0 0
\(741\) −46.5407 −1.70972
\(742\) 0 0
\(743\) 31.8422 1.16818 0.584089 0.811690i \(-0.301452\pi\)
0.584089 + 0.811690i \(0.301452\pi\)
\(744\) 0 0
\(745\) −0.774761 −0.0283851
\(746\) 0 0
\(747\) 5.90125 0.215916
\(748\) 0 0
\(749\) 2.22936 0.0814591
\(750\) 0 0
\(751\) −7.33017 −0.267482 −0.133741 0.991016i \(-0.542699\pi\)
−0.133741 + 0.991016i \(0.542699\pi\)
\(752\) 0 0
\(753\) −30.8625 −1.12469
\(754\) 0 0
\(755\) 5.41844 0.197197
\(756\) 0 0
\(757\) −45.1792 −1.64206 −0.821032 0.570882i \(-0.806602\pi\)
−0.821032 + 0.570882i \(0.806602\pi\)
\(758\) 0 0
\(759\) 12.2653 0.445203
\(760\) 0 0
\(761\) −28.8565 −1.04605 −0.523024 0.852318i \(-0.675196\pi\)
−0.523024 + 0.852318i \(0.675196\pi\)
\(762\) 0 0
\(763\) 7.12387 0.257902
\(764\) 0 0
\(765\) −3.83814 −0.138768
\(766\) 0 0
\(767\) −64.8123 −2.34024
\(768\) 0 0
\(769\) 11.3707 0.410039 0.205019 0.978758i \(-0.434274\pi\)
0.205019 + 0.978758i \(0.434274\pi\)
\(770\) 0 0
\(771\) 16.5611 0.596432
\(772\) 0 0
\(773\) −20.9885 −0.754904 −0.377452 0.926029i \(-0.623200\pi\)
−0.377452 + 0.926029i \(0.623200\pi\)
\(774\) 0 0
\(775\) 33.7527 1.21243
\(776\) 0 0
\(777\) 6.24067 0.223883
\(778\) 0 0
\(779\) −70.8297 −2.53774
\(780\) 0 0
\(781\) −34.5384 −1.23588
\(782\) 0 0
\(783\) 0.996993 0.0356296
\(784\) 0 0
\(785\) −0.794779 −0.0283669
\(786\) 0 0
\(787\) −41.3812 −1.47508 −0.737540 0.675304i \(-0.764012\pi\)
−0.737540 + 0.675304i \(0.764012\pi\)
\(788\) 0 0
\(789\) −2.43522 −0.0866963
\(790\) 0 0
\(791\) −4.46123 −0.158623
\(792\) 0 0
\(793\) −67.3977 −2.39336
\(794\) 0 0
\(795\) 6.76315 0.239864
\(796\) 0 0
\(797\) −27.3144 −0.967525 −0.483762 0.875199i \(-0.660730\pi\)
−0.483762 + 0.875199i \(0.660730\pi\)
\(798\) 0 0
\(799\) −40.9676 −1.44933
\(800\) 0 0
\(801\) 9.88073 0.349118
\(802\) 0 0
\(803\) −36.2044 −1.27762
\(804\) 0 0
\(805\) −2.73200 −0.0962905
\(806\) 0 0
\(807\) −32.6765 −1.15027
\(808\) 0 0
\(809\) 33.5282 1.17879 0.589395 0.807845i \(-0.299366\pi\)
0.589395 + 0.807845i \(0.299366\pi\)
\(810\) 0 0
\(811\) −33.7840 −1.18632 −0.593159 0.805086i \(-0.702119\pi\)
−0.593159 + 0.805086i \(0.702119\pi\)
\(812\) 0 0
\(813\) −0.00431859 −0.000151459 0
\(814\) 0 0
\(815\) 6.36211 0.222855
\(816\) 0 0
\(817\) 34.3068 1.20024
\(818\) 0 0
\(819\) 6.93380 0.242287
\(820\) 0 0
\(821\) 26.8743 0.937919 0.468959 0.883220i \(-0.344629\pi\)
0.468959 + 0.883220i \(0.344629\pi\)
\(822\) 0 0
\(823\) −10.8499 −0.378205 −0.189103 0.981957i \(-0.560558\pi\)
−0.189103 + 0.981957i \(0.560558\pi\)
\(824\) 0 0
\(825\) 13.5935 0.473266
\(826\) 0 0
\(827\) 45.4359 1.57996 0.789981 0.613131i \(-0.210090\pi\)
0.789981 + 0.613131i \(0.210090\pi\)
\(828\) 0 0
\(829\) 37.4911 1.30212 0.651060 0.759026i \(-0.274325\pi\)
0.651060 + 0.759026i \(0.274325\pi\)
\(830\) 0 0
\(831\) 25.8726 0.897509
\(832\) 0 0
\(833\) −36.4138 −1.26166
\(834\) 0 0
\(835\) 0.619742 0.0214471
\(836\) 0 0
\(837\) 7.31224 0.252748
\(838\) 0 0
\(839\) −23.1675 −0.799831 −0.399916 0.916552i \(-0.630961\pi\)
−0.399916 + 0.916552i \(0.630961\pi\)
\(840\) 0 0
\(841\) −28.0060 −0.965724
\(842\) 0 0
\(843\) −23.4091 −0.806253
\(844\) 0 0
\(845\) −18.5399 −0.637791
\(846\) 0 0
\(847\) 2.46345 0.0846451
\(848\) 0 0
\(849\) −26.8435 −0.921268
\(850\) 0 0
\(851\) −24.5568 −0.841798
\(852\) 0 0
\(853\) 24.1133 0.825624 0.412812 0.910816i \(-0.364547\pi\)
0.412812 + 0.910816i \(0.364547\pi\)
\(854\) 0 0
\(855\) 4.40288 0.150575
\(856\) 0 0
\(857\) −10.6072 −0.362335 −0.181168 0.983452i \(-0.557988\pi\)
−0.181168 + 0.983452i \(0.557988\pi\)
\(858\) 0 0
\(859\) −37.8886 −1.29274 −0.646372 0.763022i \(-0.723715\pi\)
−0.646372 + 0.763022i \(0.723715\pi\)
\(860\) 0 0
\(861\) 10.5525 0.359627
\(862\) 0 0
\(863\) −12.3480 −0.420332 −0.210166 0.977666i \(-0.567400\pi\)
−0.210166 + 0.977666i \(0.567400\pi\)
\(864\) 0 0
\(865\) 0.784588 0.0266768
\(866\) 0 0
\(867\) −21.3548 −0.725246
\(868\) 0 0
\(869\) 10.9392 0.371085
\(870\) 0 0
\(871\) −67.1508 −2.27532
\(872\) 0 0
\(873\) −8.96090 −0.303280
\(874\) 0 0
\(875\) −6.30764 −0.213237
\(876\) 0 0
\(877\) −49.0918 −1.65771 −0.828856 0.559462i \(-0.811008\pi\)
−0.828856 + 0.559462i \(0.811008\pi\)
\(878\) 0 0
\(879\) −23.7852 −0.802254
\(880\) 0 0
\(881\) 27.4045 0.923280 0.461640 0.887067i \(-0.347261\pi\)
0.461640 + 0.887067i \(0.347261\pi\)
\(882\) 0 0
\(883\) 34.8248 1.17195 0.585974 0.810330i \(-0.300712\pi\)
0.585974 + 0.810330i \(0.300712\pi\)
\(884\) 0 0
\(885\) 6.13143 0.206106
\(886\) 0 0
\(887\) −0.157909 −0.00530207 −0.00265104 0.999996i \(-0.500844\pi\)
−0.00265104 + 0.999996i \(0.500844\pi\)
\(888\) 0 0
\(889\) 11.8850 0.398612
\(890\) 0 0
\(891\) 2.94492 0.0986586
\(892\) 0 0
\(893\) 46.9955 1.57265
\(894\) 0 0
\(895\) −4.63849 −0.155048
\(896\) 0 0
\(897\) −27.2843 −0.910996
\(898\) 0 0
\(899\) 7.29025 0.243143
\(900\) 0 0
\(901\) 67.5846 2.25157
\(902\) 0 0
\(903\) −5.11115 −0.170088
\(904\) 0 0
\(905\) 3.01487 0.100218
\(906\) 0 0
\(907\) 3.38987 0.112559 0.0562794 0.998415i \(-0.482076\pi\)
0.0562794 + 0.998415i \(0.482076\pi\)
\(908\) 0 0
\(909\) −4.91541 −0.163034
\(910\) 0 0
\(911\) −37.5128 −1.24285 −0.621427 0.783472i \(-0.713447\pi\)
−0.621427 + 0.783472i \(0.713447\pi\)
\(912\) 0 0
\(913\) 17.3787 0.575152
\(914\) 0 0
\(915\) 6.37602 0.210785
\(916\) 0 0
\(917\) −5.14899 −0.170035
\(918\) 0 0
\(919\) −3.48351 −0.114910 −0.0574552 0.998348i \(-0.518299\pi\)
−0.0574552 + 0.998348i \(0.518299\pi\)
\(920\) 0 0
\(921\) −10.0558 −0.331349
\(922\) 0 0
\(923\) 76.8308 2.52892
\(924\) 0 0
\(925\) −27.2160 −0.894858
\(926\) 0 0
\(927\) −4.72990 −0.155350
\(928\) 0 0
\(929\) 16.6068 0.544851 0.272425 0.962177i \(-0.412174\pi\)
0.272425 + 0.962177i \(0.412174\pi\)
\(930\) 0 0
\(931\) 41.7717 1.36901
\(932\) 0 0
\(933\) 18.7831 0.614932
\(934\) 0 0
\(935\) −11.3030 −0.369648
\(936\) 0 0
\(937\) −41.7410 −1.36362 −0.681810 0.731529i \(-0.738807\pi\)
−0.681810 + 0.731529i \(0.738807\pi\)
\(938\) 0 0
\(939\) −14.5269 −0.474068
\(940\) 0 0
\(941\) 60.1088 1.95949 0.979746 0.200246i \(-0.0641741\pi\)
0.979746 + 0.200246i \(0.0641741\pi\)
\(942\) 0 0
\(943\) −41.5236 −1.35219
\(944\) 0 0
\(945\) −0.655958 −0.0213383
\(946\) 0 0
\(947\) 14.3268 0.465558 0.232779 0.972530i \(-0.425218\pi\)
0.232779 + 0.972530i \(0.425218\pi\)
\(948\) 0 0
\(949\) 80.5367 2.61433
\(950\) 0 0
\(951\) 8.74455 0.283561
\(952\) 0 0
\(953\) −53.4836 −1.73250 −0.866252 0.499608i \(-0.833478\pi\)
−0.866252 + 0.499608i \(0.833478\pi\)
\(954\) 0 0
\(955\) 5.19009 0.167947
\(956\) 0 0
\(957\) 2.93606 0.0949095
\(958\) 0 0
\(959\) 1.56752 0.0506177
\(960\) 0 0
\(961\) 22.4688 0.724802
\(962\) 0 0
\(963\) −2.10628 −0.0678739
\(964\) 0 0
\(965\) 4.04924 0.130350
\(966\) 0 0
\(967\) −31.0170 −0.997440 −0.498720 0.866763i \(-0.666196\pi\)
−0.498720 + 0.866763i \(0.666196\pi\)
\(968\) 0 0
\(969\) 43.9983 1.41343
\(970\) 0 0
\(971\) −19.6921 −0.631949 −0.315974 0.948768i \(-0.602331\pi\)
−0.315974 + 0.948768i \(0.602331\pi\)
\(972\) 0 0
\(973\) −12.5492 −0.402310
\(974\) 0 0
\(975\) −30.2388 −0.968418
\(976\) 0 0
\(977\) −53.4720 −1.71072 −0.855360 0.518033i \(-0.826664\pi\)
−0.855360 + 0.518033i \(0.826664\pi\)
\(978\) 0 0
\(979\) 29.0980 0.929975
\(980\) 0 0
\(981\) −6.73056 −0.214890
\(982\) 0 0
\(983\) −10.0426 −0.320309 −0.160154 0.987092i \(-0.551199\pi\)
−0.160154 + 0.987092i \(0.551199\pi\)
\(984\) 0 0
\(985\) −1.25044 −0.0398424
\(986\) 0 0
\(987\) −7.00156 −0.222862
\(988\) 0 0
\(989\) 20.1122 0.639530
\(990\) 0 0
\(991\) 23.1529 0.735477 0.367739 0.929929i \(-0.380132\pi\)
0.367739 + 0.929929i \(0.380132\pi\)
\(992\) 0 0
\(993\) −4.49622 −0.142683
\(994\) 0 0
\(995\) −11.3195 −0.358852
\(996\) 0 0
\(997\) −14.0835 −0.446029 −0.223014 0.974815i \(-0.571590\pi\)
−0.223014 + 0.974815i \(0.571590\pi\)
\(998\) 0 0
\(999\) −5.89613 −0.186545
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.1.4 9
4.3 odd 2 2004.2.a.d.1.4 9
12.11 even 2 6012.2.a.h.1.6 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2004.2.a.d.1.4 9 4.3 odd 2
6012.2.a.h.1.6 9 12.11 even 2
8016.2.a.bb.1.4 9 1.1 even 1 trivial