Properties

Label 8016.2.a.be.1.8
Level 8016
Weight 2
Character 8016.1
Self dual yes
Analytic conductor 64.008
Analytic rank 0
Dimension 11
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: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
Defining polynomial: \(x^{11} - x^{10} - 33 x^{9} + 22 x^{8} + 417 x^{7} - 151 x^{6} - 2470 x^{5} + 272 x^{4} + 6584 x^{3} + 292 x^{2} - 5687 x + 242\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 4008)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(-2.17907\) of defining polynomial
Character \(\chi\) \(=\) 8016.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +3.17907 q^{5} -0.651548 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +3.17907 q^{5} -0.651548 q^{7} +1.00000 q^{9} -2.35582 q^{11} +5.77932 q^{13} +3.17907 q^{15} -0.736701 q^{17} +3.04977 q^{19} -0.651548 q^{21} +1.45399 q^{23} +5.10651 q^{25} +1.00000 q^{27} +3.59579 q^{29} -5.65723 q^{31} -2.35582 q^{33} -2.07132 q^{35} +7.15702 q^{37} +5.77932 q^{39} +8.85120 q^{41} +4.96597 q^{43} +3.17907 q^{45} -3.18019 q^{47} -6.57548 q^{49} -0.736701 q^{51} +1.50138 q^{53} -7.48933 q^{55} +3.04977 q^{57} -3.09356 q^{59} -13.8291 q^{61} -0.651548 q^{63} +18.3729 q^{65} +6.06273 q^{67} +1.45399 q^{69} +6.79383 q^{71} -1.00965 q^{73} +5.10651 q^{75} +1.53493 q^{77} -13.3793 q^{79} +1.00000 q^{81} +8.30393 q^{83} -2.34203 q^{85} +3.59579 q^{87} -11.5653 q^{89} -3.76551 q^{91} -5.65723 q^{93} +9.69545 q^{95} +9.62123 q^{97} -2.35582 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11q + 11q^{3} + 10q^{5} + q^{7} + 11q^{9} + O(q^{10}) \) \( 11q + 11q^{3} + 10q^{5} + q^{7} + 11q^{9} + q^{11} + 10q^{13} + 10q^{15} + 17q^{17} - 2q^{19} + q^{21} + 3q^{23} + 21q^{25} + 11q^{27} + 17q^{29} + 15q^{31} + q^{33} - 11q^{35} + 4q^{37} + 10q^{39} + 16q^{41} - 10q^{43} + 10q^{45} + 16q^{47} + 22q^{49} + 17q^{51} + 42q^{53} + 5q^{55} - 2q^{57} + 2q^{59} + 12q^{61} + q^{63} + 10q^{65} + q^{67} + 3q^{69} + 9q^{71} + 24q^{73} + 21q^{75} + 22q^{77} + 30q^{79} + 11q^{81} - 16q^{83} + 25q^{85} + 17q^{87} + 37q^{89} - q^{91} + 15q^{93} - 5q^{95} + 4q^{97} + q^{99} + O(q^{100}) \)

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
\(5\) 3.17907 1.42172 0.710862 0.703331i \(-0.248305\pi\)
0.710862 + 0.703331i \(0.248305\pi\)
\(6\) 0 0
\(7\) −0.651548 −0.246262 −0.123131 0.992390i \(-0.539294\pi\)
−0.123131 + 0.992390i \(0.539294\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −2.35582 −0.710307 −0.355153 0.934808i \(-0.615571\pi\)
−0.355153 + 0.934808i \(0.615571\pi\)
\(12\) 0 0
\(13\) 5.77932 1.60289 0.801447 0.598065i \(-0.204064\pi\)
0.801447 + 0.598065i \(0.204064\pi\)
\(14\) 0 0
\(15\) 3.17907 0.820833
\(16\) 0 0
\(17\) −0.736701 −0.178676 −0.0893381 0.996001i \(-0.528475\pi\)
−0.0893381 + 0.996001i \(0.528475\pi\)
\(18\) 0 0
\(19\) 3.04977 0.699666 0.349833 0.936812i \(-0.386238\pi\)
0.349833 + 0.936812i \(0.386238\pi\)
\(20\) 0 0
\(21\) −0.651548 −0.142180
\(22\) 0 0
\(23\) 1.45399 0.303177 0.151588 0.988444i \(-0.451561\pi\)
0.151588 + 0.988444i \(0.451561\pi\)
\(24\) 0 0
\(25\) 5.10651 1.02130
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 3.59579 0.667721 0.333860 0.942623i \(-0.391649\pi\)
0.333860 + 0.942623i \(0.391649\pi\)
\(30\) 0 0
\(31\) −5.65723 −1.01607 −0.508034 0.861337i \(-0.669628\pi\)
−0.508034 + 0.861337i \(0.669628\pi\)
\(32\) 0 0
\(33\) −2.35582 −0.410096
\(34\) 0 0
\(35\) −2.07132 −0.350117
\(36\) 0 0
\(37\) 7.15702 1.17661 0.588304 0.808640i \(-0.299796\pi\)
0.588304 + 0.808640i \(0.299796\pi\)
\(38\) 0 0
\(39\) 5.77932 0.925432
\(40\) 0 0
\(41\) 8.85120 1.38232 0.691162 0.722700i \(-0.257099\pi\)
0.691162 + 0.722700i \(0.257099\pi\)
\(42\) 0 0
\(43\) 4.96597 0.757304 0.378652 0.925539i \(-0.376388\pi\)
0.378652 + 0.925539i \(0.376388\pi\)
\(44\) 0 0
\(45\) 3.17907 0.473908
\(46\) 0 0
\(47\) −3.18019 −0.463878 −0.231939 0.972730i \(-0.574507\pi\)
−0.231939 + 0.972730i \(0.574507\pi\)
\(48\) 0 0
\(49\) −6.57548 −0.939355
\(50\) 0 0
\(51\) −0.736701 −0.103159
\(52\) 0 0
\(53\) 1.50138 0.206230 0.103115 0.994669i \(-0.467119\pi\)
0.103115 + 0.994669i \(0.467119\pi\)
\(54\) 0 0
\(55\) −7.48933 −1.00986
\(56\) 0 0
\(57\) 3.04977 0.403952
\(58\) 0 0
\(59\) −3.09356 −0.402747 −0.201373 0.979515i \(-0.564540\pi\)
−0.201373 + 0.979515i \(0.564540\pi\)
\(60\) 0 0
\(61\) −13.8291 −1.77063 −0.885317 0.464988i \(-0.846059\pi\)
−0.885317 + 0.464988i \(0.846059\pi\)
\(62\) 0 0
\(63\) −0.651548 −0.0820874
\(64\) 0 0
\(65\) 18.3729 2.27888
\(66\) 0 0
\(67\) 6.06273 0.740680 0.370340 0.928896i \(-0.379241\pi\)
0.370340 + 0.928896i \(0.379241\pi\)
\(68\) 0 0
\(69\) 1.45399 0.175039
\(70\) 0 0
\(71\) 6.79383 0.806279 0.403139 0.915139i \(-0.367919\pi\)
0.403139 + 0.915139i \(0.367919\pi\)
\(72\) 0 0
\(73\) −1.00965 −0.118171 −0.0590855 0.998253i \(-0.518818\pi\)
−0.0590855 + 0.998253i \(0.518818\pi\)
\(74\) 0 0
\(75\) 5.10651 0.589649
\(76\) 0 0
\(77\) 1.53493 0.174922
\(78\) 0 0
\(79\) −13.3793 −1.50529 −0.752646 0.658425i \(-0.771223\pi\)
−0.752646 + 0.658425i \(0.771223\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 8.30393 0.911475 0.455737 0.890114i \(-0.349376\pi\)
0.455737 + 0.890114i \(0.349376\pi\)
\(84\) 0 0
\(85\) −2.34203 −0.254028
\(86\) 0 0
\(87\) 3.59579 0.385509
\(88\) 0 0
\(89\) −11.5653 −1.22592 −0.612962 0.790113i \(-0.710022\pi\)
−0.612962 + 0.790113i \(0.710022\pi\)
\(90\) 0 0
\(91\) −3.76551 −0.394732
\(92\) 0 0
\(93\) −5.65723 −0.586628
\(94\) 0 0
\(95\) 9.69545 0.994732
\(96\) 0 0
\(97\) 9.62123 0.976888 0.488444 0.872595i \(-0.337565\pi\)
0.488444 + 0.872595i \(0.337565\pi\)
\(98\) 0 0
\(99\) −2.35582 −0.236769
\(100\) 0 0
\(101\) −0.374583 −0.0372724 −0.0186362 0.999826i \(-0.505932\pi\)
−0.0186362 + 0.999826i \(0.505932\pi\)
\(102\) 0 0
\(103\) −10.0604 −0.991282 −0.495641 0.868527i \(-0.665067\pi\)
−0.495641 + 0.868527i \(0.665067\pi\)
\(104\) 0 0
\(105\) −2.07132 −0.202140
\(106\) 0 0
\(107\) 14.8116 1.43189 0.715944 0.698158i \(-0.245997\pi\)
0.715944 + 0.698158i \(0.245997\pi\)
\(108\) 0 0
\(109\) 15.4949 1.48414 0.742070 0.670323i \(-0.233844\pi\)
0.742070 + 0.670323i \(0.233844\pi\)
\(110\) 0 0
\(111\) 7.15702 0.679315
\(112\) 0 0
\(113\) 1.98184 0.186436 0.0932178 0.995646i \(-0.470285\pi\)
0.0932178 + 0.995646i \(0.470285\pi\)
\(114\) 0 0
\(115\) 4.62233 0.431034
\(116\) 0 0
\(117\) 5.77932 0.534298
\(118\) 0 0
\(119\) 0.479996 0.0440012
\(120\) 0 0
\(121\) −5.45011 −0.495464
\(122\) 0 0
\(123\) 8.85120 0.798086
\(124\) 0 0
\(125\) 0.338597 0.0302851
\(126\) 0 0
\(127\) −4.57114 −0.405623 −0.202812 0.979218i \(-0.565008\pi\)
−0.202812 + 0.979218i \(0.565008\pi\)
\(128\) 0 0
\(129\) 4.96597 0.437230
\(130\) 0 0
\(131\) 8.46187 0.739317 0.369658 0.929168i \(-0.379475\pi\)
0.369658 + 0.929168i \(0.379475\pi\)
\(132\) 0 0
\(133\) −1.98707 −0.172301
\(134\) 0 0
\(135\) 3.17907 0.273611
\(136\) 0 0
\(137\) −9.15273 −0.781970 −0.390985 0.920397i \(-0.627866\pi\)
−0.390985 + 0.920397i \(0.627866\pi\)
\(138\) 0 0
\(139\) 4.94667 0.419571 0.209786 0.977747i \(-0.432723\pi\)
0.209786 + 0.977747i \(0.432723\pi\)
\(140\) 0 0
\(141\) −3.18019 −0.267820
\(142\) 0 0
\(143\) −13.6150 −1.13855
\(144\) 0 0
\(145\) 11.4313 0.949315
\(146\) 0 0
\(147\) −6.57548 −0.542337
\(148\) 0 0
\(149\) −9.47425 −0.776161 −0.388080 0.921626i \(-0.626862\pi\)
−0.388080 + 0.921626i \(0.626862\pi\)
\(150\) 0 0
\(151\) 5.22991 0.425604 0.212802 0.977095i \(-0.431741\pi\)
0.212802 + 0.977095i \(0.431741\pi\)
\(152\) 0 0
\(153\) −0.736701 −0.0595587
\(154\) 0 0
\(155\) −17.9848 −1.44457
\(156\) 0 0
\(157\) 16.0547 1.28131 0.640654 0.767830i \(-0.278663\pi\)
0.640654 + 0.767830i \(0.278663\pi\)
\(158\) 0 0
\(159\) 1.50138 0.119067
\(160\) 0 0
\(161\) −0.947342 −0.0746610
\(162\) 0 0
\(163\) 22.3004 1.74670 0.873351 0.487091i \(-0.161942\pi\)
0.873351 + 0.487091i \(0.161942\pi\)
\(164\) 0 0
\(165\) −7.48933 −0.583043
\(166\) 0 0
\(167\) −1.00000 −0.0773823
\(168\) 0 0
\(169\) 20.4005 1.56927
\(170\) 0 0
\(171\) 3.04977 0.233222
\(172\) 0 0
\(173\) 17.5505 1.33434 0.667171 0.744904i \(-0.267505\pi\)
0.667171 + 0.744904i \(0.267505\pi\)
\(174\) 0 0
\(175\) −3.32714 −0.251508
\(176\) 0 0
\(177\) −3.09356 −0.232526
\(178\) 0 0
\(179\) −21.4005 −1.59955 −0.799773 0.600303i \(-0.795047\pi\)
−0.799773 + 0.600303i \(0.795047\pi\)
\(180\) 0 0
\(181\) −10.4205 −0.774550 −0.387275 0.921964i \(-0.626584\pi\)
−0.387275 + 0.921964i \(0.626584\pi\)
\(182\) 0 0
\(183\) −13.8291 −1.02228
\(184\) 0 0
\(185\) 22.7527 1.67281
\(186\) 0 0
\(187\) 1.73554 0.126915
\(188\) 0 0
\(189\) −0.651548 −0.0473932
\(190\) 0 0
\(191\) −10.3798 −0.751059 −0.375529 0.926810i \(-0.622539\pi\)
−0.375529 + 0.926810i \(0.622539\pi\)
\(192\) 0 0
\(193\) −19.9382 −1.43518 −0.717591 0.696465i \(-0.754755\pi\)
−0.717591 + 0.696465i \(0.754755\pi\)
\(194\) 0 0
\(195\) 18.3729 1.31571
\(196\) 0 0
\(197\) 21.5273 1.53375 0.766877 0.641794i \(-0.221810\pi\)
0.766877 + 0.641794i \(0.221810\pi\)
\(198\) 0 0
\(199\) −1.57836 −0.111887 −0.0559435 0.998434i \(-0.517817\pi\)
−0.0559435 + 0.998434i \(0.517817\pi\)
\(200\) 0 0
\(201\) 6.06273 0.427632
\(202\) 0 0
\(203\) −2.34283 −0.164434
\(204\) 0 0
\(205\) 28.1386 1.96529
\(206\) 0 0
\(207\) 1.45399 0.101059
\(208\) 0 0
\(209\) −7.18472 −0.496977
\(210\) 0 0
\(211\) −16.1516 −1.11192 −0.555960 0.831209i \(-0.687649\pi\)
−0.555960 + 0.831209i \(0.687649\pi\)
\(212\) 0 0
\(213\) 6.79383 0.465505
\(214\) 0 0
\(215\) 15.7872 1.07668
\(216\) 0 0
\(217\) 3.68596 0.250219
\(218\) 0 0
\(219\) −1.00965 −0.0682260
\(220\) 0 0
\(221\) −4.25763 −0.286399
\(222\) 0 0
\(223\) 15.9061 1.06515 0.532577 0.846382i \(-0.321224\pi\)
0.532577 + 0.846382i \(0.321224\pi\)
\(224\) 0 0
\(225\) 5.10651 0.340434
\(226\) 0 0
\(227\) −25.5938 −1.69872 −0.849359 0.527816i \(-0.823011\pi\)
−0.849359 + 0.527816i \(0.823011\pi\)
\(228\) 0 0
\(229\) −3.79809 −0.250985 −0.125492 0.992095i \(-0.540051\pi\)
−0.125492 + 0.992095i \(0.540051\pi\)
\(230\) 0 0
\(231\) 1.53493 0.100991
\(232\) 0 0
\(233\) 11.9836 0.785070 0.392535 0.919737i \(-0.371598\pi\)
0.392535 + 0.919737i \(0.371598\pi\)
\(234\) 0 0
\(235\) −10.1101 −0.659507
\(236\) 0 0
\(237\) −13.3793 −0.869081
\(238\) 0 0
\(239\) −4.97611 −0.321878 −0.160939 0.986964i \(-0.551452\pi\)
−0.160939 + 0.986964i \(0.551452\pi\)
\(240\) 0 0
\(241\) 5.08605 0.327621 0.163811 0.986492i \(-0.447621\pi\)
0.163811 + 0.986492i \(0.447621\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −20.9039 −1.33550
\(246\) 0 0
\(247\) 17.6256 1.12149
\(248\) 0 0
\(249\) 8.30393 0.526240
\(250\) 0 0
\(251\) 22.9778 1.45034 0.725172 0.688568i \(-0.241760\pi\)
0.725172 + 0.688568i \(0.241760\pi\)
\(252\) 0 0
\(253\) −3.42533 −0.215349
\(254\) 0 0
\(255\) −2.34203 −0.146663
\(256\) 0 0
\(257\) 5.54328 0.345781 0.172890 0.984941i \(-0.444689\pi\)
0.172890 + 0.984941i \(0.444689\pi\)
\(258\) 0 0
\(259\) −4.66315 −0.289754
\(260\) 0 0
\(261\) 3.59579 0.222574
\(262\) 0 0
\(263\) −10.0070 −0.617060 −0.308530 0.951215i \(-0.599837\pi\)
−0.308530 + 0.951215i \(0.599837\pi\)
\(264\) 0 0
\(265\) 4.77298 0.293202
\(266\) 0 0
\(267\) −11.5653 −0.707787
\(268\) 0 0
\(269\) 28.4920 1.73719 0.868594 0.495525i \(-0.165024\pi\)
0.868594 + 0.495525i \(0.165024\pi\)
\(270\) 0 0
\(271\) 3.31835 0.201575 0.100788 0.994908i \(-0.467864\pi\)
0.100788 + 0.994908i \(0.467864\pi\)
\(272\) 0 0
\(273\) −3.76551 −0.227899
\(274\) 0 0
\(275\) −12.0300 −0.725437
\(276\) 0 0
\(277\) −3.48347 −0.209301 −0.104651 0.994509i \(-0.533372\pi\)
−0.104651 + 0.994509i \(0.533372\pi\)
\(278\) 0 0
\(279\) −5.65723 −0.338690
\(280\) 0 0
\(281\) 3.76222 0.224435 0.112218 0.993684i \(-0.464205\pi\)
0.112218 + 0.993684i \(0.464205\pi\)
\(282\) 0 0
\(283\) 16.4873 0.980066 0.490033 0.871704i \(-0.336985\pi\)
0.490033 + 0.871704i \(0.336985\pi\)
\(284\) 0 0
\(285\) 9.69545 0.574309
\(286\) 0 0
\(287\) −5.76698 −0.340414
\(288\) 0 0
\(289\) −16.4573 −0.968075
\(290\) 0 0
\(291\) 9.62123 0.564006
\(292\) 0 0
\(293\) 0.965119 0.0563828 0.0281914 0.999603i \(-0.491025\pi\)
0.0281914 + 0.999603i \(0.491025\pi\)
\(294\) 0 0
\(295\) −9.83465 −0.572595
\(296\) 0 0
\(297\) −2.35582 −0.136699
\(298\) 0 0
\(299\) 8.40305 0.485961
\(300\) 0 0
\(301\) −3.23557 −0.186495
\(302\) 0 0
\(303\) −0.374583 −0.0215192
\(304\) 0 0
\(305\) −43.9637 −2.51735
\(306\) 0 0
\(307\) 10.8989 0.622030 0.311015 0.950405i \(-0.399331\pi\)
0.311015 + 0.950405i \(0.399331\pi\)
\(308\) 0 0
\(309\) −10.0604 −0.572317
\(310\) 0 0
\(311\) 23.2877 1.32052 0.660262 0.751035i \(-0.270445\pi\)
0.660262 + 0.751035i \(0.270445\pi\)
\(312\) 0 0
\(313\) 26.2329 1.48277 0.741386 0.671078i \(-0.234169\pi\)
0.741386 + 0.671078i \(0.234169\pi\)
\(314\) 0 0
\(315\) −2.07132 −0.116706
\(316\) 0 0
\(317\) 27.5938 1.54982 0.774911 0.632070i \(-0.217795\pi\)
0.774911 + 0.632070i \(0.217795\pi\)
\(318\) 0 0
\(319\) −8.47103 −0.474287
\(320\) 0 0
\(321\) 14.8116 0.826701
\(322\) 0 0
\(323\) −2.24677 −0.125014
\(324\) 0 0
\(325\) 29.5121 1.63704
\(326\) 0 0
\(327\) 15.4949 0.856868
\(328\) 0 0
\(329\) 2.07205 0.114236
\(330\) 0 0
\(331\) −11.6057 −0.637905 −0.318952 0.947771i \(-0.603331\pi\)
−0.318952 + 0.947771i \(0.603331\pi\)
\(332\) 0 0
\(333\) 7.15702 0.392202
\(334\) 0 0
\(335\) 19.2739 1.05304
\(336\) 0 0
\(337\) −14.4885 −0.789241 −0.394620 0.918844i \(-0.629124\pi\)
−0.394620 + 0.918844i \(0.629124\pi\)
\(338\) 0 0
\(339\) 1.98184 0.107639
\(340\) 0 0
\(341\) 13.3274 0.721721
\(342\) 0 0
\(343\) 8.84508 0.477590
\(344\) 0 0
\(345\) 4.62233 0.248858
\(346\) 0 0
\(347\) −9.21580 −0.494730 −0.247365 0.968922i \(-0.579565\pi\)
−0.247365 + 0.968922i \(0.579565\pi\)
\(348\) 0 0
\(349\) −3.40211 −0.182111 −0.0910555 0.995846i \(-0.529024\pi\)
−0.0910555 + 0.995846i \(0.529024\pi\)
\(350\) 0 0
\(351\) 5.77932 0.308477
\(352\) 0 0
\(353\) −16.0856 −0.856151 −0.428075 0.903743i \(-0.640808\pi\)
−0.428075 + 0.903743i \(0.640808\pi\)
\(354\) 0 0
\(355\) 21.5981 1.14631
\(356\) 0 0
\(357\) 0.479996 0.0254041
\(358\) 0 0
\(359\) −26.0982 −1.37741 −0.688704 0.725042i \(-0.741820\pi\)
−0.688704 + 0.725042i \(0.741820\pi\)
\(360\) 0 0
\(361\) −9.69889 −0.510468
\(362\) 0 0
\(363\) −5.45011 −0.286056
\(364\) 0 0
\(365\) −3.20976 −0.168007
\(366\) 0 0
\(367\) 11.5886 0.604918 0.302459 0.953162i \(-0.402192\pi\)
0.302459 + 0.953162i \(0.402192\pi\)
\(368\) 0 0
\(369\) 8.85120 0.460775
\(370\) 0 0
\(371\) −0.978219 −0.0507866
\(372\) 0 0
\(373\) −7.42795 −0.384605 −0.192302 0.981336i \(-0.561595\pi\)
−0.192302 + 0.981336i \(0.561595\pi\)
\(374\) 0 0
\(375\) 0.338597 0.0174851
\(376\) 0 0
\(377\) 20.7812 1.07029
\(378\) 0 0
\(379\) 13.6468 0.700991 0.350495 0.936564i \(-0.386013\pi\)
0.350495 + 0.936564i \(0.386013\pi\)
\(380\) 0 0
\(381\) −4.57114 −0.234187
\(382\) 0 0
\(383\) −32.5127 −1.66132 −0.830661 0.556779i \(-0.812037\pi\)
−0.830661 + 0.556779i \(0.812037\pi\)
\(384\) 0 0
\(385\) 4.87966 0.248690
\(386\) 0 0
\(387\) 4.96597 0.252435
\(388\) 0 0
\(389\) 7.12806 0.361407 0.180703 0.983538i \(-0.442163\pi\)
0.180703 + 0.983538i \(0.442163\pi\)
\(390\) 0 0
\(391\) −1.07115 −0.0541705
\(392\) 0 0
\(393\) 8.46187 0.426845
\(394\) 0 0
\(395\) −42.5339 −2.14011
\(396\) 0 0
\(397\) −15.1636 −0.761041 −0.380520 0.924773i \(-0.624255\pi\)
−0.380520 + 0.924773i \(0.624255\pi\)
\(398\) 0 0
\(399\) −1.98707 −0.0994781
\(400\) 0 0
\(401\) 4.19754 0.209615 0.104808 0.994493i \(-0.466577\pi\)
0.104808 + 0.994493i \(0.466577\pi\)
\(402\) 0 0
\(403\) −32.6950 −1.62865
\(404\) 0 0
\(405\) 3.17907 0.157969
\(406\) 0 0
\(407\) −16.8607 −0.835752
\(408\) 0 0
\(409\) −10.9137 −0.539648 −0.269824 0.962910i \(-0.586966\pi\)
−0.269824 + 0.962910i \(0.586966\pi\)
\(410\) 0 0
\(411\) −9.15273 −0.451471
\(412\) 0 0
\(413\) 2.01560 0.0991813
\(414\) 0 0
\(415\) 26.3988 1.29587
\(416\) 0 0
\(417\) 4.94667 0.242240
\(418\) 0 0
\(419\) −15.7174 −0.767843 −0.383921 0.923366i \(-0.625427\pi\)
−0.383921 + 0.923366i \(0.625427\pi\)
\(420\) 0 0
\(421\) −6.26688 −0.305429 −0.152714 0.988270i \(-0.548801\pi\)
−0.152714 + 0.988270i \(0.548801\pi\)
\(422\) 0 0
\(423\) −3.18019 −0.154626
\(424\) 0 0
\(425\) −3.76197 −0.182482
\(426\) 0 0
\(427\) 9.01033 0.436040
\(428\) 0 0
\(429\) −13.6150 −0.657341
\(430\) 0 0
\(431\) 3.93255 0.189424 0.0947122 0.995505i \(-0.469807\pi\)
0.0947122 + 0.995505i \(0.469807\pi\)
\(432\) 0 0
\(433\) 1.01897 0.0489686 0.0244843 0.999700i \(-0.492206\pi\)
0.0244843 + 0.999700i \(0.492206\pi\)
\(434\) 0 0
\(435\) 11.4313 0.548088
\(436\) 0 0
\(437\) 4.43433 0.212123
\(438\) 0 0
\(439\) −32.3685 −1.54487 −0.772433 0.635097i \(-0.780960\pi\)
−0.772433 + 0.635097i \(0.780960\pi\)
\(440\) 0 0
\(441\) −6.57548 −0.313118
\(442\) 0 0
\(443\) 18.1882 0.864149 0.432075 0.901838i \(-0.357782\pi\)
0.432075 + 0.901838i \(0.357782\pi\)
\(444\) 0 0
\(445\) −36.7671 −1.74293
\(446\) 0 0
\(447\) −9.47425 −0.448117
\(448\) 0 0
\(449\) −17.8478 −0.842288 −0.421144 0.906994i \(-0.638371\pi\)
−0.421144 + 0.906994i \(0.638371\pi\)
\(450\) 0 0
\(451\) −20.8518 −0.981875
\(452\) 0 0
\(453\) 5.22991 0.245723
\(454\) 0 0
\(455\) −11.9708 −0.561201
\(456\) 0 0
\(457\) −32.7658 −1.53272 −0.766359 0.642413i \(-0.777934\pi\)
−0.766359 + 0.642413i \(0.777934\pi\)
\(458\) 0 0
\(459\) −0.736701 −0.0343863
\(460\) 0 0
\(461\) 30.0329 1.39877 0.699386 0.714744i \(-0.253457\pi\)
0.699386 + 0.714744i \(0.253457\pi\)
\(462\) 0 0
\(463\) −40.2896 −1.87242 −0.936208 0.351445i \(-0.885690\pi\)
−0.936208 + 0.351445i \(0.885690\pi\)
\(464\) 0 0
\(465\) −17.9848 −0.834023
\(466\) 0 0
\(467\) −3.79685 −0.175697 −0.0878485 0.996134i \(-0.527999\pi\)
−0.0878485 + 0.996134i \(0.527999\pi\)
\(468\) 0 0
\(469\) −3.95016 −0.182401
\(470\) 0 0
\(471\) 16.0547 0.739763
\(472\) 0 0
\(473\) −11.6989 −0.537918
\(474\) 0 0
\(475\) 15.5737 0.714570
\(476\) 0 0
\(477\) 1.50138 0.0687433
\(478\) 0 0
\(479\) −25.1667 −1.14989 −0.574947 0.818190i \(-0.694977\pi\)
−0.574947 + 0.818190i \(0.694977\pi\)
\(480\) 0 0
\(481\) 41.3627 1.88598
\(482\) 0 0
\(483\) −0.947342 −0.0431056
\(484\) 0 0
\(485\) 30.5866 1.38887
\(486\) 0 0
\(487\) −8.54863 −0.387375 −0.193688 0.981063i \(-0.562045\pi\)
−0.193688 + 0.981063i \(0.562045\pi\)
\(488\) 0 0
\(489\) 22.3004 1.00846
\(490\) 0 0
\(491\) 25.8474 1.16648 0.583238 0.812301i \(-0.301785\pi\)
0.583238 + 0.812301i \(0.301785\pi\)
\(492\) 0 0
\(493\) −2.64902 −0.119306
\(494\) 0 0
\(495\) −7.48933 −0.336620
\(496\) 0 0
\(497\) −4.42651 −0.198556
\(498\) 0 0
\(499\) −21.1286 −0.945847 −0.472923 0.881104i \(-0.656801\pi\)
−0.472923 + 0.881104i \(0.656801\pi\)
\(500\) 0 0
\(501\) −1.00000 −0.0446767
\(502\) 0 0
\(503\) −7.82587 −0.348938 −0.174469 0.984663i \(-0.555821\pi\)
−0.174469 + 0.984663i \(0.555821\pi\)
\(504\) 0 0
\(505\) −1.19083 −0.0529911
\(506\) 0 0
\(507\) 20.4005 0.906020
\(508\) 0 0
\(509\) −22.7245 −1.00725 −0.503623 0.863923i \(-0.668000\pi\)
−0.503623 + 0.863923i \(0.668000\pi\)
\(510\) 0 0
\(511\) 0.657838 0.0291010
\(512\) 0 0
\(513\) 3.04977 0.134651
\(514\) 0 0
\(515\) −31.9828 −1.40933
\(516\) 0 0
\(517\) 7.49195 0.329496
\(518\) 0 0
\(519\) 17.5505 0.770383
\(520\) 0 0
\(521\) −11.1453 −0.488286 −0.244143 0.969739i \(-0.578507\pi\)
−0.244143 + 0.969739i \(0.578507\pi\)
\(522\) 0 0
\(523\) 26.3399 1.15176 0.575881 0.817534i \(-0.304659\pi\)
0.575881 + 0.817534i \(0.304659\pi\)
\(524\) 0 0
\(525\) −3.32714 −0.145208
\(526\) 0 0
\(527\) 4.16769 0.181547
\(528\) 0 0
\(529\) −20.8859 −0.908084
\(530\) 0 0
\(531\) −3.09356 −0.134249
\(532\) 0 0
\(533\) 51.1539 2.21572
\(534\) 0 0
\(535\) 47.0870 2.03575
\(536\) 0 0
\(537\) −21.4005 −0.923498
\(538\) 0 0
\(539\) 15.4907 0.667230
\(540\) 0 0
\(541\) 26.6396 1.14532 0.572662 0.819791i \(-0.305911\pi\)
0.572662 + 0.819791i \(0.305911\pi\)
\(542\) 0 0
\(543\) −10.4205 −0.447187
\(544\) 0 0
\(545\) 49.2593 2.11004
\(546\) 0 0
\(547\) −2.96240 −0.126663 −0.0633315 0.997993i \(-0.520173\pi\)
−0.0633315 + 0.997993i \(0.520173\pi\)
\(548\) 0 0
\(549\) −13.8291 −0.590211
\(550\) 0 0
\(551\) 10.9663 0.467182
\(552\) 0 0
\(553\) 8.71728 0.370696
\(554\) 0 0
\(555\) 22.7527 0.965798
\(556\) 0 0
\(557\) 29.2995 1.24146 0.620730 0.784024i \(-0.286836\pi\)
0.620730 + 0.784024i \(0.286836\pi\)
\(558\) 0 0
\(559\) 28.6999 1.21388
\(560\) 0 0
\(561\) 1.73554 0.0732744
\(562\) 0 0
\(563\) 9.57725 0.403633 0.201816 0.979423i \(-0.435316\pi\)
0.201816 + 0.979423i \(0.435316\pi\)
\(564\) 0 0
\(565\) 6.30041 0.265060
\(566\) 0 0
\(567\) −0.651548 −0.0273625
\(568\) 0 0
\(569\) −10.4227 −0.436944 −0.218472 0.975843i \(-0.570107\pi\)
−0.218472 + 0.975843i \(0.570107\pi\)
\(570\) 0 0
\(571\) 14.2919 0.598097 0.299048 0.954238i \(-0.403331\pi\)
0.299048 + 0.954238i \(0.403331\pi\)
\(572\) 0 0
\(573\) −10.3798 −0.433624
\(574\) 0 0
\(575\) 7.42479 0.309635
\(576\) 0 0
\(577\) −2.12593 −0.0885038 −0.0442519 0.999020i \(-0.514090\pi\)
−0.0442519 + 0.999020i \(0.514090\pi\)
\(578\) 0 0
\(579\) −19.9382 −0.828603
\(580\) 0 0
\(581\) −5.41041 −0.224462
\(582\) 0 0
\(583\) −3.53697 −0.146486
\(584\) 0 0
\(585\) 18.3729 0.759625
\(586\) 0 0
\(587\) −30.4771 −1.25792 −0.628962 0.777436i \(-0.716520\pi\)
−0.628962 + 0.777436i \(0.716520\pi\)
\(588\) 0 0
\(589\) −17.2533 −0.710909
\(590\) 0 0
\(591\) 21.5273 0.885513
\(592\) 0 0
\(593\) −40.2084 −1.65116 −0.825580 0.564285i \(-0.809152\pi\)
−0.825580 + 0.564285i \(0.809152\pi\)
\(594\) 0 0
\(595\) 1.52594 0.0625576
\(596\) 0 0
\(597\) −1.57836 −0.0645980
\(598\) 0 0
\(599\) −0.845760 −0.0345568 −0.0172784 0.999851i \(-0.505500\pi\)
−0.0172784 + 0.999851i \(0.505500\pi\)
\(600\) 0 0
\(601\) −30.0722 −1.22667 −0.613335 0.789823i \(-0.710172\pi\)
−0.613335 + 0.789823i \(0.710172\pi\)
\(602\) 0 0
\(603\) 6.06273 0.246893
\(604\) 0 0
\(605\) −17.3263 −0.704414
\(606\) 0 0
\(607\) 31.9582 1.29714 0.648572 0.761153i \(-0.275366\pi\)
0.648572 + 0.761153i \(0.275366\pi\)
\(608\) 0 0
\(609\) −2.34283 −0.0949362
\(610\) 0 0
\(611\) −18.3793 −0.743548
\(612\) 0 0
\(613\) 15.1244 0.610868 0.305434 0.952213i \(-0.401198\pi\)
0.305434 + 0.952213i \(0.401198\pi\)
\(614\) 0 0
\(615\) 28.1386 1.13466
\(616\) 0 0
\(617\) −15.5830 −0.627350 −0.313675 0.949530i \(-0.601560\pi\)
−0.313675 + 0.949530i \(0.601560\pi\)
\(618\) 0 0
\(619\) −8.89579 −0.357552 −0.178776 0.983890i \(-0.557214\pi\)
−0.178776 + 0.983890i \(0.557214\pi\)
\(620\) 0 0
\(621\) 1.45399 0.0583464
\(622\) 0 0
\(623\) 7.53538 0.301898
\(624\) 0 0
\(625\) −24.4561 −0.978245
\(626\) 0 0
\(627\) −7.18472 −0.286930
\(628\) 0 0
\(629\) −5.27259 −0.210232
\(630\) 0 0
\(631\) 42.4222 1.68880 0.844400 0.535713i \(-0.179957\pi\)
0.844400 + 0.535713i \(0.179957\pi\)
\(632\) 0 0
\(633\) −16.1516 −0.641967
\(634\) 0 0
\(635\) −14.5320 −0.576684
\(636\) 0 0
\(637\) −38.0018 −1.50569
\(638\) 0 0
\(639\) 6.79383 0.268760
\(640\) 0 0
\(641\) 34.9454 1.38026 0.690130 0.723685i \(-0.257553\pi\)
0.690130 + 0.723685i \(0.257553\pi\)
\(642\) 0 0
\(643\) −18.2870 −0.721170 −0.360585 0.932726i \(-0.617423\pi\)
−0.360585 + 0.932726i \(0.617423\pi\)
\(644\) 0 0
\(645\) 15.7872 0.621620
\(646\) 0 0
\(647\) 28.9514 1.13820 0.569099 0.822269i \(-0.307292\pi\)
0.569099 + 0.822269i \(0.307292\pi\)
\(648\) 0 0
\(649\) 7.28787 0.286074
\(650\) 0 0
\(651\) 3.68596 0.144464
\(652\) 0 0
\(653\) −7.68083 −0.300574 −0.150287 0.988642i \(-0.548020\pi\)
−0.150287 + 0.988642i \(0.548020\pi\)
\(654\) 0 0
\(655\) 26.9009 1.05111
\(656\) 0 0
\(657\) −1.00965 −0.0393903
\(658\) 0 0
\(659\) 16.8279 0.655523 0.327761 0.944761i \(-0.393706\pi\)
0.327761 + 0.944761i \(0.393706\pi\)
\(660\) 0 0
\(661\) −34.2589 −1.33252 −0.666258 0.745721i \(-0.732105\pi\)
−0.666258 + 0.745721i \(0.732105\pi\)
\(662\) 0 0
\(663\) −4.25763 −0.165353
\(664\) 0 0
\(665\) −6.31706 −0.244965
\(666\) 0 0
\(667\) 5.22822 0.202438
\(668\) 0 0
\(669\) 15.9061 0.614967
\(670\) 0 0
\(671\) 32.5789 1.25769
\(672\) 0 0
\(673\) −3.98223 −0.153504 −0.0767519 0.997050i \(-0.524455\pi\)
−0.0767519 + 0.997050i \(0.524455\pi\)
\(674\) 0 0
\(675\) 5.10651 0.196550
\(676\) 0 0
\(677\) 23.2722 0.894424 0.447212 0.894428i \(-0.352417\pi\)
0.447212 + 0.894428i \(0.352417\pi\)
\(678\) 0 0
\(679\) −6.26870 −0.240570
\(680\) 0 0
\(681\) −25.5938 −0.980755
\(682\) 0 0
\(683\) −14.0489 −0.537564 −0.268782 0.963201i \(-0.586621\pi\)
−0.268782 + 0.963201i \(0.586621\pi\)
\(684\) 0 0
\(685\) −29.0972 −1.11175
\(686\) 0 0
\(687\) −3.79809 −0.144906
\(688\) 0 0
\(689\) 8.67693 0.330565
\(690\) 0 0
\(691\) −5.46717 −0.207981 −0.103990 0.994578i \(-0.533161\pi\)
−0.103990 + 0.994578i \(0.533161\pi\)
\(692\) 0 0
\(693\) 1.53493 0.0583072
\(694\) 0 0
\(695\) 15.7258 0.596515
\(696\) 0 0
\(697\) −6.52069 −0.246989
\(698\) 0 0
\(699\) 11.9836 0.453261
\(700\) 0 0
\(701\) 1.62817 0.0614952 0.0307476 0.999527i \(-0.490211\pi\)
0.0307476 + 0.999527i \(0.490211\pi\)
\(702\) 0 0
\(703\) 21.8273 0.823232
\(704\) 0 0
\(705\) −10.1101 −0.380767
\(706\) 0 0
\(707\) 0.244059 0.00917878
\(708\) 0 0
\(709\) −49.5471 −1.86078 −0.930390 0.366572i \(-0.880531\pi\)
−0.930390 + 0.366572i \(0.880531\pi\)
\(710\) 0 0
\(711\) −13.3793 −0.501764
\(712\) 0 0
\(713\) −8.22554 −0.308049
\(714\) 0 0
\(715\) −43.2832 −1.61870
\(716\) 0 0
\(717\) −4.97611 −0.185836
\(718\) 0 0
\(719\) 44.8137 1.67127 0.835635 0.549286i \(-0.185100\pi\)
0.835635 + 0.549286i \(0.185100\pi\)
\(720\) 0 0
\(721\) 6.55485 0.244115
\(722\) 0 0
\(723\) 5.08605 0.189152
\(724\) 0 0
\(725\) 18.3619 0.681944
\(726\) 0 0
\(727\) −4.63373 −0.171856 −0.0859278 0.996301i \(-0.527385\pi\)
−0.0859278 + 0.996301i \(0.527385\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −3.65844 −0.135312
\(732\) 0 0
\(733\) −18.5214 −0.684103 −0.342052 0.939681i \(-0.611122\pi\)
−0.342052 + 0.939681i \(0.611122\pi\)
\(734\) 0 0
\(735\) −20.9039 −0.771054
\(736\) 0 0
\(737\) −14.2827 −0.526110
\(738\) 0 0
\(739\) −12.8273 −0.471860 −0.235930 0.971770i \(-0.575814\pi\)
−0.235930 + 0.971770i \(0.575814\pi\)
\(740\) 0 0
\(741\) 17.6256 0.647493
\(742\) 0 0
\(743\) 14.8237 0.543827 0.271914 0.962322i \(-0.412344\pi\)
0.271914 + 0.962322i \(0.412344\pi\)
\(744\) 0 0
\(745\) −30.1193 −1.10349
\(746\) 0 0
\(747\) 8.30393 0.303825
\(748\) 0 0
\(749\) −9.65045 −0.352620
\(750\) 0 0
\(751\) −6.09316 −0.222343 −0.111171 0.993801i \(-0.535460\pi\)
−0.111171 + 0.993801i \(0.535460\pi\)
\(752\) 0 0
\(753\) 22.9778 0.837356
\(754\) 0 0
\(755\) 16.6263 0.605092
\(756\) 0 0
\(757\) 17.7244 0.644203 0.322101 0.946705i \(-0.395611\pi\)
0.322101 + 0.946705i \(0.395611\pi\)
\(758\) 0 0
\(759\) −3.42533 −0.124332
\(760\) 0 0
\(761\) −28.5542 −1.03509 −0.517544 0.855657i \(-0.673153\pi\)
−0.517544 + 0.855657i \(0.673153\pi\)
\(762\) 0 0
\(763\) −10.0957 −0.365487
\(764\) 0 0
\(765\) −2.34203 −0.0846762
\(766\) 0 0
\(767\) −17.8787 −0.645561
\(768\) 0 0
\(769\) −34.8197 −1.25563 −0.627815 0.778362i \(-0.716051\pi\)
−0.627815 + 0.778362i \(0.716051\pi\)
\(770\) 0 0
\(771\) 5.54328 0.199637
\(772\) 0 0
\(773\) −1.10623 −0.0397883 −0.0198941 0.999802i \(-0.506333\pi\)
−0.0198941 + 0.999802i \(0.506333\pi\)
\(774\) 0 0
\(775\) −28.8887 −1.03771
\(776\) 0 0
\(777\) −4.66315 −0.167289
\(778\) 0 0
\(779\) 26.9941 0.967165
\(780\) 0 0
\(781\) −16.0050 −0.572705
\(782\) 0 0
\(783\) 3.59579 0.128503
\(784\) 0 0
\(785\) 51.0392 1.82167
\(786\) 0 0
\(787\) 21.4837 0.765812 0.382906 0.923787i \(-0.374923\pi\)
0.382906 + 0.923787i \(0.374923\pi\)
\(788\) 0 0
\(789\) −10.0070 −0.356260
\(790\) 0 0
\(791\) −1.29126 −0.0459120
\(792\) 0 0
\(793\) −79.9228 −2.83814
\(794\) 0 0
\(795\) 4.77298 0.169280
\(796\) 0 0
\(797\) −19.3920 −0.686900 −0.343450 0.939171i \(-0.611596\pi\)
−0.343450 + 0.939171i \(0.611596\pi\)
\(798\) 0 0
\(799\) 2.34285 0.0828840
\(800\) 0 0
\(801\) −11.5653 −0.408641
\(802\) 0 0
\(803\) 2.37856 0.0839376
\(804\) 0 0
\(805\) −3.01167 −0.106147
\(806\) 0 0
\(807\) 28.4920 1.00297
\(808\) 0 0
\(809\) −1.98259 −0.0697041 −0.0348520 0.999392i \(-0.511096\pi\)
−0.0348520 + 0.999392i \(0.511096\pi\)
\(810\) 0 0
\(811\) 5.12364 0.179915 0.0899577 0.995946i \(-0.471327\pi\)
0.0899577 + 0.995946i \(0.471327\pi\)
\(812\) 0 0
\(813\) 3.31835 0.116379
\(814\) 0 0
\(815\) 70.8946 2.48333
\(816\) 0 0
\(817\) 15.1451 0.529860
\(818\) 0 0
\(819\) −3.76551 −0.131577
\(820\) 0 0
\(821\) 15.0449 0.525072 0.262536 0.964922i \(-0.415441\pi\)
0.262536 + 0.964922i \(0.415441\pi\)
\(822\) 0 0
\(823\) −8.86337 −0.308958 −0.154479 0.987996i \(-0.549370\pi\)
−0.154479 + 0.987996i \(0.549370\pi\)
\(824\) 0 0
\(825\) −12.0300 −0.418832
\(826\) 0 0
\(827\) 15.1782 0.527797 0.263898 0.964551i \(-0.414992\pi\)
0.263898 + 0.964551i \(0.414992\pi\)
\(828\) 0 0
\(829\) −19.1528 −0.665205 −0.332602 0.943067i \(-0.607927\pi\)
−0.332602 + 0.943067i \(0.607927\pi\)
\(830\) 0 0
\(831\) −3.48347 −0.120840
\(832\) 0 0
\(833\) 4.84417 0.167840
\(834\) 0 0
\(835\) −3.17907 −0.110016
\(836\) 0 0
\(837\) −5.65723 −0.195543
\(838\) 0 0
\(839\) 40.8741 1.41113 0.705565 0.708645i \(-0.250693\pi\)
0.705565 + 0.708645i \(0.250693\pi\)
\(840\) 0 0
\(841\) −16.0703 −0.554149
\(842\) 0 0
\(843\) 3.76222 0.129578
\(844\) 0 0
\(845\) 64.8548 2.23107
\(846\) 0 0
\(847\) 3.55101 0.122014
\(848\) 0 0
\(849\) 16.4873 0.565841
\(850\) 0 0
\(851\) 10.4062 0.356720
\(852\) 0 0
\(853\) 11.0999 0.380052 0.190026 0.981779i \(-0.439143\pi\)
0.190026 + 0.981779i \(0.439143\pi\)
\(854\) 0 0
\(855\) 9.69545 0.331577
\(856\) 0 0
\(857\) −0.658909 −0.0225079 −0.0112539 0.999937i \(-0.503582\pi\)
−0.0112539 + 0.999937i \(0.503582\pi\)
\(858\) 0 0
\(859\) −20.9062 −0.713310 −0.356655 0.934236i \(-0.616083\pi\)
−0.356655 + 0.934236i \(0.616083\pi\)
\(860\) 0 0
\(861\) −5.76698 −0.196538
\(862\) 0 0
\(863\) 45.9133 1.56291 0.781453 0.623964i \(-0.214479\pi\)
0.781453 + 0.623964i \(0.214479\pi\)
\(864\) 0 0
\(865\) 55.7945 1.89707
\(866\) 0 0
\(867\) −16.4573 −0.558918
\(868\) 0 0
\(869\) 31.5193 1.06922
\(870\) 0 0
\(871\) 35.0384 1.18723
\(872\) 0 0
\(873\) 9.62123 0.325629
\(874\) 0 0
\(875\) −0.220612 −0.00745806
\(876\) 0 0
\(877\) 23.1757 0.782587 0.391293 0.920266i \(-0.372028\pi\)
0.391293 + 0.920266i \(0.372028\pi\)
\(878\) 0 0
\(879\) 0.965119 0.0325526
\(880\) 0 0
\(881\) 33.5672 1.13091 0.565454 0.824780i \(-0.308701\pi\)
0.565454 + 0.824780i \(0.308701\pi\)
\(882\) 0 0
\(883\) 22.5629 0.759301 0.379651 0.925130i \(-0.376044\pi\)
0.379651 + 0.925130i \(0.376044\pi\)
\(884\) 0 0
\(885\) −9.83465 −0.330588
\(886\) 0 0
\(887\) 18.5003 0.621181 0.310590 0.950544i \(-0.399473\pi\)
0.310590 + 0.950544i \(0.399473\pi\)
\(888\) 0 0
\(889\) 2.97832 0.0998896
\(890\) 0 0
\(891\) −2.35582 −0.0789230
\(892\) 0 0
\(893\) −9.69885 −0.324560
\(894\) 0 0
\(895\) −68.0336 −2.27411
\(896\) 0 0
\(897\) 8.40305 0.280570
\(898\) 0 0
\(899\) −20.3422 −0.678450
\(900\) 0 0
\(901\) −1.10606 −0.0368484
\(902\) 0 0
\(903\) −3.23557 −0.107673
\(904\) 0 0
\(905\) −33.1276 −1.10120
\(906\) 0 0
\(907\) 33.1224 1.09981 0.549905 0.835227i \(-0.314664\pi\)
0.549905 + 0.835227i \(0.314664\pi\)
\(908\) 0 0
\(909\) −0.374583 −0.0124241
\(910\) 0 0
\(911\) 49.6212 1.64402 0.822012 0.569470i \(-0.192851\pi\)
0.822012 + 0.569470i \(0.192851\pi\)
\(912\) 0 0
\(913\) −19.5626 −0.647427
\(914\) 0 0
\(915\) −43.9637 −1.45340
\(916\) 0 0
\(917\) −5.51332 −0.182066
\(918\) 0 0
\(919\) −7.74469 −0.255474 −0.127737 0.991808i \(-0.540771\pi\)
−0.127737 + 0.991808i \(0.540771\pi\)
\(920\) 0 0
\(921\) 10.8989 0.359129
\(922\) 0 0
\(923\) 39.2637 1.29238
\(924\) 0 0
\(925\) 36.5474 1.20167
\(926\) 0 0
\(927\) −10.0604 −0.330427
\(928\) 0 0
\(929\) −28.7929 −0.944663 −0.472332 0.881421i \(-0.656587\pi\)
−0.472332 + 0.881421i \(0.656587\pi\)
\(930\) 0 0
\(931\) −20.0537 −0.657235
\(932\) 0 0
\(933\) 23.2877 0.762405
\(934\) 0 0
\(935\) 5.51740 0.180438
\(936\) 0 0
\(937\) 16.6507 0.543953 0.271977 0.962304i \(-0.412323\pi\)
0.271977 + 0.962304i \(0.412323\pi\)
\(938\) 0 0
\(939\) 26.2329 0.856079
\(940\) 0 0
\(941\) 36.4130 1.18703 0.593515 0.804823i \(-0.297740\pi\)
0.593515 + 0.804823i \(0.297740\pi\)
\(942\) 0 0
\(943\) 12.8695 0.419089
\(944\) 0 0
\(945\) −2.07132 −0.0673800
\(946\) 0 0
\(947\) −34.7734 −1.12998 −0.564992 0.825097i \(-0.691121\pi\)
−0.564992 + 0.825097i \(0.691121\pi\)
\(948\) 0 0
\(949\) −5.83511 −0.189416
\(950\) 0 0
\(951\) 27.5938 0.894790
\(952\) 0 0
\(953\) −33.4050 −1.08209 −0.541046 0.840993i \(-0.681972\pi\)
−0.541046 + 0.840993i \(0.681972\pi\)
\(954\) 0 0
\(955\) −32.9983 −1.06780
\(956\) 0 0
\(957\) −8.47103 −0.273830
\(958\) 0 0
\(959\) 5.96344 0.192570
\(960\) 0 0
\(961\) 1.00428 0.0323962
\(962\) 0 0
\(963\) 14.8116 0.477296
\(964\) 0 0
\(965\) −63.3850 −2.04043
\(966\) 0 0
\(967\) 40.8553 1.31382 0.656910 0.753969i \(-0.271863\pi\)
0.656910 + 0.753969i \(0.271863\pi\)
\(968\) 0 0
\(969\) −2.24677 −0.0721767
\(970\) 0 0
\(971\) 32.6557 1.04797 0.523986 0.851727i \(-0.324444\pi\)
0.523986 + 0.851727i \(0.324444\pi\)
\(972\) 0 0
\(973\) −3.22300 −0.103325
\(974\) 0 0
\(975\) 29.5121 0.945145
\(976\) 0 0
\(977\) 26.0370 0.832996 0.416498 0.909137i \(-0.363257\pi\)
0.416498 + 0.909137i \(0.363257\pi\)
\(978\) 0 0
\(979\) 27.2459 0.870782
\(980\) 0 0
\(981\) 15.4949 0.494713
\(982\) 0 0
\(983\) −50.8084 −1.62054 −0.810269 0.586059i \(-0.800679\pi\)
−0.810269 + 0.586059i \(0.800679\pi\)
\(984\) 0 0
\(985\) 68.4368 2.18058
\(986\) 0 0
\(987\) 2.07205 0.0659540
\(988\) 0 0
\(989\) 7.22045 0.229597
\(990\) 0 0
\(991\) 14.1305 0.448871 0.224436 0.974489i \(-0.427946\pi\)
0.224436 + 0.974489i \(0.427946\pi\)
\(992\) 0 0
\(993\) −11.6057 −0.368295
\(994\) 0 0
\(995\) −5.01772 −0.159073
\(996\) 0 0
\(997\) −50.7523 −1.60734 −0.803671 0.595074i \(-0.797123\pi\)
−0.803671 + 0.595074i \(0.797123\pi\)
\(998\) 0 0
\(999\) 7.15702 0.226438
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.be.1.8 11
4.3 odd 2 4008.2.a.k.1.8 11
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4008.2.a.k.1.8 11 4.3 odd 2
8016.2.a.be.1.8 11 1.1 even 1 trivial