Properties

Label 8004.2.a.i.1.14
Level $8004$
Weight $2$
Character 8004.1
Self dual yes
Analytic conductor $63.912$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8004.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.9122617778\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 5 x^{15} - 43 x^{14} + 234 x^{13} + 634 x^{12} - 4048 x^{11} - 3483 x^{10} + 32512 x^{9} - 137 x^{8} - 121665 x^{7} + 60168 x^{6} + 172218 x^{5} - 138024 x^{4} - 17844 x^{3} + 14352 x^{2} + 72 x - 208\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.14
Root \(3.56414\) of defining polynomial
Character \(\chi\) \(=\) 8004.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +3.56414 q^{5} -0.955621 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +3.56414 q^{5} -0.955621 q^{7} +1.00000 q^{9} -5.80108 q^{11} +2.78668 q^{13} -3.56414 q^{15} -4.80496 q^{17} -2.83599 q^{19} +0.955621 q^{21} +1.00000 q^{23} +7.70306 q^{25} -1.00000 q^{27} +1.00000 q^{29} +1.35740 q^{31} +5.80108 q^{33} -3.40596 q^{35} -2.67980 q^{37} -2.78668 q^{39} +10.9216 q^{41} -5.88946 q^{43} +3.56414 q^{45} -0.527133 q^{47} -6.08679 q^{49} +4.80496 q^{51} -7.00804 q^{53} -20.6758 q^{55} +2.83599 q^{57} +2.89886 q^{59} +8.42802 q^{61} -0.955621 q^{63} +9.93209 q^{65} +11.0194 q^{67} -1.00000 q^{69} +12.6124 q^{71} +5.58199 q^{73} -7.70306 q^{75} +5.54363 q^{77} -9.15554 q^{79} +1.00000 q^{81} +5.98706 q^{83} -17.1255 q^{85} -1.00000 q^{87} +8.05698 q^{89} -2.66300 q^{91} -1.35740 q^{93} -10.1079 q^{95} +8.87633 q^{97} -5.80108 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 16q^{3} + 5q^{5} - 4q^{7} + 16q^{9} + O(q^{10}) \) \( 16q - 16q^{3} + 5q^{5} - 4q^{7} + 16q^{9} + 5q^{11} + 8q^{13} - 5q^{15} + 7q^{17} + q^{19} + 4q^{21} + 16q^{23} + 31q^{25} - 16q^{27} + 16q^{29} - 2q^{31} - 5q^{33} + 5q^{35} + 14q^{37} - 8q^{39} - q^{41} - 13q^{43} + 5q^{45} - 4q^{47} + 30q^{49} - 7q^{51} + 19q^{53} - 37q^{55} - q^{57} + 12q^{59} + 21q^{61} - 4q^{63} + 26q^{65} - 11q^{67} - 16q^{69} + 7q^{71} - 13q^{73} - 31q^{75} + 4q^{77} - 18q^{79} + 16q^{81} + 25q^{83} + 48q^{85} - 16q^{87} + 12q^{89} - 11q^{91} + 2q^{93} - q^{95} + 5q^{97} + 5q^{99} + O(q^{100}) \)

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
\(5\) 3.56414 1.59393 0.796965 0.604025i \(-0.206437\pi\)
0.796965 + 0.604025i \(0.206437\pi\)
\(6\) 0 0
\(7\) −0.955621 −0.361191 −0.180595 0.983557i \(-0.557802\pi\)
−0.180595 + 0.983557i \(0.557802\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −5.80108 −1.74909 −0.874546 0.484943i \(-0.838840\pi\)
−0.874546 + 0.484943i \(0.838840\pi\)
\(12\) 0 0
\(13\) 2.78668 0.772885 0.386442 0.922314i \(-0.373704\pi\)
0.386442 + 0.922314i \(0.373704\pi\)
\(14\) 0 0
\(15\) −3.56414 −0.920256
\(16\) 0 0
\(17\) −4.80496 −1.16537 −0.582687 0.812697i \(-0.697999\pi\)
−0.582687 + 0.812697i \(0.697999\pi\)
\(18\) 0 0
\(19\) −2.83599 −0.650621 −0.325311 0.945607i \(-0.605469\pi\)
−0.325311 + 0.945607i \(0.605469\pi\)
\(20\) 0 0
\(21\) 0.955621 0.208533
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 7.70306 1.54061
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) 1.35740 0.243796 0.121898 0.992543i \(-0.461102\pi\)
0.121898 + 0.992543i \(0.461102\pi\)
\(32\) 0 0
\(33\) 5.80108 1.00984
\(34\) 0 0
\(35\) −3.40596 −0.575713
\(36\) 0 0
\(37\) −2.67980 −0.440556 −0.220278 0.975437i \(-0.570697\pi\)
−0.220278 + 0.975437i \(0.570697\pi\)
\(38\) 0 0
\(39\) −2.78668 −0.446225
\(40\) 0 0
\(41\) 10.9216 1.70566 0.852832 0.522186i \(-0.174883\pi\)
0.852832 + 0.522186i \(0.174883\pi\)
\(42\) 0 0
\(43\) −5.88946 −0.898134 −0.449067 0.893498i \(-0.648243\pi\)
−0.449067 + 0.893498i \(0.648243\pi\)
\(44\) 0 0
\(45\) 3.56414 0.531310
\(46\) 0 0
\(47\) −0.527133 −0.0768903 −0.0384452 0.999261i \(-0.512240\pi\)
−0.0384452 + 0.999261i \(0.512240\pi\)
\(48\) 0 0
\(49\) −6.08679 −0.869541
\(50\) 0 0
\(51\) 4.80496 0.672829
\(52\) 0 0
\(53\) −7.00804 −0.962628 −0.481314 0.876548i \(-0.659840\pi\)
−0.481314 + 0.876548i \(0.659840\pi\)
\(54\) 0 0
\(55\) −20.6758 −2.78793
\(56\) 0 0
\(57\) 2.83599 0.375636
\(58\) 0 0
\(59\) 2.89886 0.377400 0.188700 0.982035i \(-0.439573\pi\)
0.188700 + 0.982035i \(0.439573\pi\)
\(60\) 0 0
\(61\) 8.42802 1.07910 0.539549 0.841954i \(-0.318595\pi\)
0.539549 + 0.841954i \(0.318595\pi\)
\(62\) 0 0
\(63\) −0.955621 −0.120397
\(64\) 0 0
\(65\) 9.93209 1.23192
\(66\) 0 0
\(67\) 11.0194 1.34624 0.673120 0.739534i \(-0.264954\pi\)
0.673120 + 0.739534i \(0.264954\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 12.6124 1.49681 0.748407 0.663240i \(-0.230819\pi\)
0.748407 + 0.663240i \(0.230819\pi\)
\(72\) 0 0
\(73\) 5.58199 0.653322 0.326661 0.945142i \(-0.394076\pi\)
0.326661 + 0.945142i \(0.394076\pi\)
\(74\) 0 0
\(75\) −7.70306 −0.889473
\(76\) 0 0
\(77\) 5.54363 0.631755
\(78\) 0 0
\(79\) −9.15554 −1.03008 −0.515039 0.857167i \(-0.672223\pi\)
−0.515039 + 0.857167i \(0.672223\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 5.98706 0.657166 0.328583 0.944475i \(-0.393429\pi\)
0.328583 + 0.944475i \(0.393429\pi\)
\(84\) 0 0
\(85\) −17.1255 −1.85752
\(86\) 0 0
\(87\) −1.00000 −0.107211
\(88\) 0 0
\(89\) 8.05698 0.854038 0.427019 0.904243i \(-0.359564\pi\)
0.427019 + 0.904243i \(0.359564\pi\)
\(90\) 0 0
\(91\) −2.66300 −0.279159
\(92\) 0 0
\(93\) −1.35740 −0.140756
\(94\) 0 0
\(95\) −10.1079 −1.03704
\(96\) 0 0
\(97\) 8.87633 0.901255 0.450627 0.892712i \(-0.351200\pi\)
0.450627 + 0.892712i \(0.351200\pi\)
\(98\) 0 0
\(99\) −5.80108 −0.583030
\(100\) 0 0
\(101\) 15.8122 1.57337 0.786687 0.617352i \(-0.211795\pi\)
0.786687 + 0.617352i \(0.211795\pi\)
\(102\) 0 0
\(103\) 6.02264 0.593429 0.296714 0.954966i \(-0.404109\pi\)
0.296714 + 0.954966i \(0.404109\pi\)
\(104\) 0 0
\(105\) 3.40596 0.332388
\(106\) 0 0
\(107\) 10.9137 1.05506 0.527531 0.849536i \(-0.323118\pi\)
0.527531 + 0.849536i \(0.323118\pi\)
\(108\) 0 0
\(109\) 7.69560 0.737104 0.368552 0.929607i \(-0.379854\pi\)
0.368552 + 0.929607i \(0.379854\pi\)
\(110\) 0 0
\(111\) 2.67980 0.254355
\(112\) 0 0
\(113\) −9.48154 −0.891948 −0.445974 0.895046i \(-0.647143\pi\)
−0.445974 + 0.895046i \(0.647143\pi\)
\(114\) 0 0
\(115\) 3.56414 0.332357
\(116\) 0 0
\(117\) 2.78668 0.257628
\(118\) 0 0
\(119\) 4.59172 0.420922
\(120\) 0 0
\(121\) 22.6525 2.05932
\(122\) 0 0
\(123\) −10.9216 −0.984765
\(124\) 0 0
\(125\) 9.63409 0.861699
\(126\) 0 0
\(127\) −3.02443 −0.268375 −0.134188 0.990956i \(-0.542842\pi\)
−0.134188 + 0.990956i \(0.542842\pi\)
\(128\) 0 0
\(129\) 5.88946 0.518538
\(130\) 0 0
\(131\) 8.68706 0.758992 0.379496 0.925193i \(-0.376097\pi\)
0.379496 + 0.925193i \(0.376097\pi\)
\(132\) 0 0
\(133\) 2.71013 0.234998
\(134\) 0 0
\(135\) −3.56414 −0.306752
\(136\) 0 0
\(137\) 3.01854 0.257891 0.128946 0.991652i \(-0.458841\pi\)
0.128946 + 0.991652i \(0.458841\pi\)
\(138\) 0 0
\(139\) −5.16341 −0.437955 −0.218978 0.975730i \(-0.570272\pi\)
−0.218978 + 0.975730i \(0.570272\pi\)
\(140\) 0 0
\(141\) 0.527133 0.0443926
\(142\) 0 0
\(143\) −16.1657 −1.35185
\(144\) 0 0
\(145\) 3.56414 0.295985
\(146\) 0 0
\(147\) 6.08679 0.502030
\(148\) 0 0
\(149\) 3.58064 0.293338 0.146669 0.989186i \(-0.453145\pi\)
0.146669 + 0.989186i \(0.453145\pi\)
\(150\) 0 0
\(151\) 2.96648 0.241408 0.120704 0.992689i \(-0.461485\pi\)
0.120704 + 0.992689i \(0.461485\pi\)
\(152\) 0 0
\(153\) −4.80496 −0.388458
\(154\) 0 0
\(155\) 4.83796 0.388594
\(156\) 0 0
\(157\) 18.2771 1.45867 0.729334 0.684158i \(-0.239830\pi\)
0.729334 + 0.684158i \(0.239830\pi\)
\(158\) 0 0
\(159\) 7.00804 0.555773
\(160\) 0 0
\(161\) −0.955621 −0.0753134
\(162\) 0 0
\(163\) 7.06758 0.553575 0.276788 0.960931i \(-0.410730\pi\)
0.276788 + 0.960931i \(0.410730\pi\)
\(164\) 0 0
\(165\) 20.6758 1.60961
\(166\) 0 0
\(167\) −0.993252 −0.0768602 −0.0384301 0.999261i \(-0.512236\pi\)
−0.0384301 + 0.999261i \(0.512236\pi\)
\(168\) 0 0
\(169\) −5.23444 −0.402649
\(170\) 0 0
\(171\) −2.83599 −0.216874
\(172\) 0 0
\(173\) 10.1738 0.773502 0.386751 0.922184i \(-0.373597\pi\)
0.386751 + 0.922184i \(0.373597\pi\)
\(174\) 0 0
\(175\) −7.36121 −0.556455
\(176\) 0 0
\(177\) −2.89886 −0.217892
\(178\) 0 0
\(179\) 13.2458 0.990036 0.495018 0.868883i \(-0.335161\pi\)
0.495018 + 0.868883i \(0.335161\pi\)
\(180\) 0 0
\(181\) −6.67164 −0.495899 −0.247950 0.968773i \(-0.579757\pi\)
−0.247950 + 0.968773i \(0.579757\pi\)
\(182\) 0 0
\(183\) −8.42802 −0.623017
\(184\) 0 0
\(185\) −9.55117 −0.702216
\(186\) 0 0
\(187\) 27.8740 2.03835
\(188\) 0 0
\(189\) 0.955621 0.0695112
\(190\) 0 0
\(191\) −8.22295 −0.594991 −0.297496 0.954723i \(-0.596151\pi\)
−0.297496 + 0.954723i \(0.596151\pi\)
\(192\) 0 0
\(193\) −9.08660 −0.654068 −0.327034 0.945013i \(-0.606049\pi\)
−0.327034 + 0.945013i \(0.606049\pi\)
\(194\) 0 0
\(195\) −9.93209 −0.711252
\(196\) 0 0
\(197\) 9.97613 0.710770 0.355385 0.934720i \(-0.384350\pi\)
0.355385 + 0.934720i \(0.384350\pi\)
\(198\) 0 0
\(199\) 12.4190 0.880361 0.440180 0.897909i \(-0.354914\pi\)
0.440180 + 0.897909i \(0.354914\pi\)
\(200\) 0 0
\(201\) −11.0194 −0.777252
\(202\) 0 0
\(203\) −0.955621 −0.0670714
\(204\) 0 0
\(205\) 38.9260 2.71871
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) 16.4518 1.13800
\(210\) 0 0
\(211\) −20.1635 −1.38811 −0.694057 0.719920i \(-0.744178\pi\)
−0.694057 + 0.719920i \(0.744178\pi\)
\(212\) 0 0
\(213\) −12.6124 −0.864186
\(214\) 0 0
\(215\) −20.9908 −1.43156
\(216\) 0 0
\(217\) −1.29716 −0.0880569
\(218\) 0 0
\(219\) −5.58199 −0.377196
\(220\) 0 0
\(221\) −13.3899 −0.900700
\(222\) 0 0
\(223\) 21.7862 1.45891 0.729455 0.684028i \(-0.239774\pi\)
0.729455 + 0.684028i \(0.239774\pi\)
\(224\) 0 0
\(225\) 7.70306 0.513538
\(226\) 0 0
\(227\) 0.0750304 0.00497994 0.00248997 0.999997i \(-0.499207\pi\)
0.00248997 + 0.999997i \(0.499207\pi\)
\(228\) 0 0
\(229\) 15.9524 1.05417 0.527084 0.849813i \(-0.323285\pi\)
0.527084 + 0.849813i \(0.323285\pi\)
\(230\) 0 0
\(231\) −5.54363 −0.364744
\(232\) 0 0
\(233\) −13.0981 −0.858082 −0.429041 0.903285i \(-0.641148\pi\)
−0.429041 + 0.903285i \(0.641148\pi\)
\(234\) 0 0
\(235\) −1.87877 −0.122558
\(236\) 0 0
\(237\) 9.15554 0.594716
\(238\) 0 0
\(239\) −27.9484 −1.80783 −0.903916 0.427710i \(-0.859321\pi\)
−0.903916 + 0.427710i \(0.859321\pi\)
\(240\) 0 0
\(241\) −23.4964 −1.51353 −0.756767 0.653685i \(-0.773222\pi\)
−0.756767 + 0.653685i \(0.773222\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −21.6941 −1.38599
\(246\) 0 0
\(247\) −7.90299 −0.502855
\(248\) 0 0
\(249\) −5.98706 −0.379415
\(250\) 0 0
\(251\) 6.46207 0.407882 0.203941 0.978983i \(-0.434625\pi\)
0.203941 + 0.978983i \(0.434625\pi\)
\(252\) 0 0
\(253\) −5.80108 −0.364711
\(254\) 0 0
\(255\) 17.1255 1.07244
\(256\) 0 0
\(257\) −9.92325 −0.618995 −0.309498 0.950900i \(-0.600161\pi\)
−0.309498 + 0.950900i \(0.600161\pi\)
\(258\) 0 0
\(259\) 2.56087 0.159125
\(260\) 0 0
\(261\) 1.00000 0.0618984
\(262\) 0 0
\(263\) 5.08891 0.313795 0.156898 0.987615i \(-0.449851\pi\)
0.156898 + 0.987615i \(0.449851\pi\)
\(264\) 0 0
\(265\) −24.9776 −1.53436
\(266\) 0 0
\(267\) −8.05698 −0.493079
\(268\) 0 0
\(269\) −4.38799 −0.267540 −0.133770 0.991012i \(-0.542708\pi\)
−0.133770 + 0.991012i \(0.542708\pi\)
\(270\) 0 0
\(271\) 10.9204 0.663367 0.331684 0.943391i \(-0.392383\pi\)
0.331684 + 0.943391i \(0.392383\pi\)
\(272\) 0 0
\(273\) 2.66300 0.161172
\(274\) 0 0
\(275\) −44.6861 −2.69467
\(276\) 0 0
\(277\) 1.80890 0.108686 0.0543432 0.998522i \(-0.482693\pi\)
0.0543432 + 0.998522i \(0.482693\pi\)
\(278\) 0 0
\(279\) 1.35740 0.0812654
\(280\) 0 0
\(281\) 23.3353 1.39207 0.696035 0.718008i \(-0.254946\pi\)
0.696035 + 0.718008i \(0.254946\pi\)
\(282\) 0 0
\(283\) −25.4070 −1.51029 −0.755144 0.655559i \(-0.772433\pi\)
−0.755144 + 0.655559i \(0.772433\pi\)
\(284\) 0 0
\(285\) 10.1079 0.598738
\(286\) 0 0
\(287\) −10.4369 −0.616070
\(288\) 0 0
\(289\) 6.08765 0.358097
\(290\) 0 0
\(291\) −8.87633 −0.520340
\(292\) 0 0
\(293\) −23.0688 −1.34769 −0.673846 0.738872i \(-0.735359\pi\)
−0.673846 + 0.738872i \(0.735359\pi\)
\(294\) 0 0
\(295\) 10.3319 0.601549
\(296\) 0 0
\(297\) 5.80108 0.336613
\(298\) 0 0
\(299\) 2.78668 0.161158
\(300\) 0 0
\(301\) 5.62809 0.324397
\(302\) 0 0
\(303\) −15.8122 −0.908388
\(304\) 0 0
\(305\) 30.0386 1.72001
\(306\) 0 0
\(307\) −22.2938 −1.27237 −0.636187 0.771535i \(-0.719489\pi\)
−0.636187 + 0.771535i \(0.719489\pi\)
\(308\) 0 0
\(309\) −6.02264 −0.342616
\(310\) 0 0
\(311\) 11.0508 0.626635 0.313317 0.949648i \(-0.398560\pi\)
0.313317 + 0.949648i \(0.398560\pi\)
\(312\) 0 0
\(313\) −2.95462 −0.167005 −0.0835024 0.996508i \(-0.526611\pi\)
−0.0835024 + 0.996508i \(0.526611\pi\)
\(314\) 0 0
\(315\) −3.40596 −0.191904
\(316\) 0 0
\(317\) 3.09880 0.174046 0.0870230 0.996206i \(-0.472265\pi\)
0.0870230 + 0.996206i \(0.472265\pi\)
\(318\) 0 0
\(319\) −5.80108 −0.324798
\(320\) 0 0
\(321\) −10.9137 −0.609141
\(322\) 0 0
\(323\) 13.6268 0.758217
\(324\) 0 0
\(325\) 21.4659 1.19072
\(326\) 0 0
\(327\) −7.69560 −0.425567
\(328\) 0 0
\(329\) 0.503739 0.0277721
\(330\) 0 0
\(331\) −16.6986 −0.917838 −0.458919 0.888478i \(-0.651763\pi\)
−0.458919 + 0.888478i \(0.651763\pi\)
\(332\) 0 0
\(333\) −2.67980 −0.146852
\(334\) 0 0
\(335\) 39.2748 2.14581
\(336\) 0 0
\(337\) 13.8811 0.756151 0.378075 0.925775i \(-0.376586\pi\)
0.378075 + 0.925775i \(0.376586\pi\)
\(338\) 0 0
\(339\) 9.48154 0.514966
\(340\) 0 0
\(341\) −7.87438 −0.426422
\(342\) 0 0
\(343\) 12.5060 0.675261
\(344\) 0 0
\(345\) −3.56414 −0.191887
\(346\) 0 0
\(347\) −15.9087 −0.854026 −0.427013 0.904245i \(-0.640434\pi\)
−0.427013 + 0.904245i \(0.640434\pi\)
\(348\) 0 0
\(349\) 19.4711 1.04227 0.521133 0.853476i \(-0.325510\pi\)
0.521133 + 0.853476i \(0.325510\pi\)
\(350\) 0 0
\(351\) −2.78668 −0.148742
\(352\) 0 0
\(353\) 11.5300 0.613677 0.306839 0.951762i \(-0.400729\pi\)
0.306839 + 0.951762i \(0.400729\pi\)
\(354\) 0 0
\(355\) 44.9522 2.38582
\(356\) 0 0
\(357\) −4.59172 −0.243020
\(358\) 0 0
\(359\) 2.90372 0.153253 0.0766263 0.997060i \(-0.475585\pi\)
0.0766263 + 0.997060i \(0.475585\pi\)
\(360\) 0 0
\(361\) −10.9571 −0.576692
\(362\) 0 0
\(363\) −22.6525 −1.18895
\(364\) 0 0
\(365\) 19.8950 1.04135
\(366\) 0 0
\(367\) 24.7354 1.29118 0.645588 0.763686i \(-0.276613\pi\)
0.645588 + 0.763686i \(0.276613\pi\)
\(368\) 0 0
\(369\) 10.9216 0.568554
\(370\) 0 0
\(371\) 6.69702 0.347692
\(372\) 0 0
\(373\) 18.0826 0.936283 0.468142 0.883654i \(-0.344924\pi\)
0.468142 + 0.883654i \(0.344924\pi\)
\(374\) 0 0
\(375\) −9.63409 −0.497502
\(376\) 0 0
\(377\) 2.78668 0.143521
\(378\) 0 0
\(379\) −16.0998 −0.826990 −0.413495 0.910506i \(-0.635692\pi\)
−0.413495 + 0.910506i \(0.635692\pi\)
\(380\) 0 0
\(381\) 3.02443 0.154946
\(382\) 0 0
\(383\) −13.8440 −0.707395 −0.353697 0.935360i \(-0.615076\pi\)
−0.353697 + 0.935360i \(0.615076\pi\)
\(384\) 0 0
\(385\) 19.7583 1.00697
\(386\) 0 0
\(387\) −5.88946 −0.299378
\(388\) 0 0
\(389\) 1.10286 0.0559171 0.0279585 0.999609i \(-0.491099\pi\)
0.0279585 + 0.999609i \(0.491099\pi\)
\(390\) 0 0
\(391\) −4.80496 −0.242997
\(392\) 0 0
\(393\) −8.68706 −0.438204
\(394\) 0 0
\(395\) −32.6316 −1.64187
\(396\) 0 0
\(397\) 15.7315 0.789544 0.394772 0.918779i \(-0.370824\pi\)
0.394772 + 0.918779i \(0.370824\pi\)
\(398\) 0 0
\(399\) −2.71013 −0.135676
\(400\) 0 0
\(401\) 19.8224 0.989882 0.494941 0.868926i \(-0.335190\pi\)
0.494941 + 0.868926i \(0.335190\pi\)
\(402\) 0 0
\(403\) 3.78263 0.188426
\(404\) 0 0
\(405\) 3.56414 0.177103
\(406\) 0 0
\(407\) 15.5457 0.770573
\(408\) 0 0
\(409\) 17.2715 0.854022 0.427011 0.904246i \(-0.359567\pi\)
0.427011 + 0.904246i \(0.359567\pi\)
\(410\) 0 0
\(411\) −3.01854 −0.148894
\(412\) 0 0
\(413\) −2.77021 −0.136313
\(414\) 0 0
\(415\) 21.3387 1.04748
\(416\) 0 0
\(417\) 5.16341 0.252853
\(418\) 0 0
\(419\) −8.01063 −0.391345 −0.195672 0.980669i \(-0.562689\pi\)
−0.195672 + 0.980669i \(0.562689\pi\)
\(420\) 0 0
\(421\) −30.3538 −1.47935 −0.739676 0.672963i \(-0.765021\pi\)
−0.739676 + 0.672963i \(0.765021\pi\)
\(422\) 0 0
\(423\) −0.527133 −0.0256301
\(424\) 0 0
\(425\) −37.0129 −1.79539
\(426\) 0 0
\(427\) −8.05399 −0.389760
\(428\) 0 0
\(429\) 16.1657 0.780489
\(430\) 0 0
\(431\) 9.19842 0.443072 0.221536 0.975152i \(-0.428893\pi\)
0.221536 + 0.975152i \(0.428893\pi\)
\(432\) 0 0
\(433\) −0.834370 −0.0400973 −0.0200486 0.999799i \(-0.506382\pi\)
−0.0200486 + 0.999799i \(0.506382\pi\)
\(434\) 0 0
\(435\) −3.56414 −0.170887
\(436\) 0 0
\(437\) −2.83599 −0.135664
\(438\) 0 0
\(439\) 22.2395 1.06143 0.530716 0.847550i \(-0.321923\pi\)
0.530716 + 0.847550i \(0.321923\pi\)
\(440\) 0 0
\(441\) −6.08679 −0.289847
\(442\) 0 0
\(443\) 13.1992 0.627115 0.313558 0.949569i \(-0.398479\pi\)
0.313558 + 0.949569i \(0.398479\pi\)
\(444\) 0 0
\(445\) 28.7162 1.36128
\(446\) 0 0
\(447\) −3.58064 −0.169359
\(448\) 0 0
\(449\) −16.8450 −0.794964 −0.397482 0.917610i \(-0.630116\pi\)
−0.397482 + 0.917610i \(0.630116\pi\)
\(450\) 0 0
\(451\) −63.3569 −2.98336
\(452\) 0 0
\(453\) −2.96648 −0.139377
\(454\) 0 0
\(455\) −9.49131 −0.444960
\(456\) 0 0
\(457\) −41.8557 −1.95792 −0.978962 0.204041i \(-0.934593\pi\)
−0.978962 + 0.204041i \(0.934593\pi\)
\(458\) 0 0
\(459\) 4.80496 0.224276
\(460\) 0 0
\(461\) 11.4646 0.533961 0.266981 0.963702i \(-0.413974\pi\)
0.266981 + 0.963702i \(0.413974\pi\)
\(462\) 0 0
\(463\) 9.04554 0.420382 0.210191 0.977660i \(-0.432591\pi\)
0.210191 + 0.977660i \(0.432591\pi\)
\(464\) 0 0
\(465\) −4.83796 −0.224355
\(466\) 0 0
\(467\) 9.38461 0.434268 0.217134 0.976142i \(-0.430329\pi\)
0.217134 + 0.976142i \(0.430329\pi\)
\(468\) 0 0
\(469\) −10.5304 −0.486249
\(470\) 0 0
\(471\) −18.2771 −0.842162
\(472\) 0 0
\(473\) 34.1652 1.57092
\(474\) 0 0
\(475\) −21.8458 −1.00236
\(476\) 0 0
\(477\) −7.00804 −0.320876
\(478\) 0 0
\(479\) −1.45125 −0.0663091 −0.0331546 0.999450i \(-0.510555\pi\)
−0.0331546 + 0.999450i \(0.510555\pi\)
\(480\) 0 0
\(481\) −7.46773 −0.340499
\(482\) 0 0
\(483\) 0.955621 0.0434822
\(484\) 0 0
\(485\) 31.6364 1.43654
\(486\) 0 0
\(487\) 21.0211 0.952555 0.476277 0.879295i \(-0.341986\pi\)
0.476277 + 0.879295i \(0.341986\pi\)
\(488\) 0 0
\(489\) −7.06758 −0.319607
\(490\) 0 0
\(491\) −23.5710 −1.06374 −0.531871 0.846825i \(-0.678511\pi\)
−0.531871 + 0.846825i \(0.678511\pi\)
\(492\) 0 0
\(493\) −4.80496 −0.216405
\(494\) 0 0
\(495\) −20.6758 −0.929310
\(496\) 0 0
\(497\) −12.0527 −0.540635
\(498\) 0 0
\(499\) 25.1831 1.12735 0.563676 0.825996i \(-0.309387\pi\)
0.563676 + 0.825996i \(0.309387\pi\)
\(500\) 0 0
\(501\) 0.993252 0.0443752
\(502\) 0 0
\(503\) 39.2545 1.75027 0.875136 0.483877i \(-0.160772\pi\)
0.875136 + 0.483877i \(0.160772\pi\)
\(504\) 0 0
\(505\) 56.3569 2.50785
\(506\) 0 0
\(507\) 5.23444 0.232469
\(508\) 0 0
\(509\) 29.5193 1.30842 0.654209 0.756314i \(-0.273002\pi\)
0.654209 + 0.756314i \(0.273002\pi\)
\(510\) 0 0
\(511\) −5.33426 −0.235974
\(512\) 0 0
\(513\) 2.83599 0.125212
\(514\) 0 0
\(515\) 21.4655 0.945884
\(516\) 0 0
\(517\) 3.05794 0.134488
\(518\) 0 0
\(519\) −10.1738 −0.446582
\(520\) 0 0
\(521\) −0.834372 −0.0365545 −0.0182773 0.999833i \(-0.505818\pi\)
−0.0182773 + 0.999833i \(0.505818\pi\)
\(522\) 0 0
\(523\) −35.1951 −1.53898 −0.769488 0.638662i \(-0.779488\pi\)
−0.769488 + 0.638662i \(0.779488\pi\)
\(524\) 0 0
\(525\) 7.36121 0.321269
\(526\) 0 0
\(527\) −6.52225 −0.284114
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 2.89886 0.125800
\(532\) 0 0
\(533\) 30.4349 1.31828
\(534\) 0 0
\(535\) 38.8977 1.68170
\(536\) 0 0
\(537\) −13.2458 −0.571597
\(538\) 0 0
\(539\) 35.3099 1.52091
\(540\) 0 0
\(541\) 26.0125 1.11836 0.559181 0.829045i \(-0.311116\pi\)
0.559181 + 0.829045i \(0.311116\pi\)
\(542\) 0 0
\(543\) 6.67164 0.286308
\(544\) 0 0
\(545\) 27.4281 1.17489
\(546\) 0 0
\(547\) 25.3017 1.08182 0.540910 0.841080i \(-0.318080\pi\)
0.540910 + 0.841080i \(0.318080\pi\)
\(548\) 0 0
\(549\) 8.42802 0.359699
\(550\) 0 0
\(551\) −2.83599 −0.120817
\(552\) 0 0
\(553\) 8.74922 0.372055
\(554\) 0 0
\(555\) 9.55117 0.405425
\(556\) 0 0
\(557\) −23.6524 −1.00219 −0.501093 0.865393i \(-0.667069\pi\)
−0.501093 + 0.865393i \(0.667069\pi\)
\(558\) 0 0
\(559\) −16.4120 −0.694154
\(560\) 0 0
\(561\) −27.8740 −1.17684
\(562\) 0 0
\(563\) −3.37852 −0.142388 −0.0711939 0.997462i \(-0.522681\pi\)
−0.0711939 + 0.997462i \(0.522681\pi\)
\(564\) 0 0
\(565\) −33.7935 −1.42170
\(566\) 0 0
\(567\) −0.955621 −0.0401323
\(568\) 0 0
\(569\) −2.95456 −0.123862 −0.0619308 0.998080i \(-0.519726\pi\)
−0.0619308 + 0.998080i \(0.519726\pi\)
\(570\) 0 0
\(571\) −3.06701 −0.128350 −0.0641752 0.997939i \(-0.520442\pi\)
−0.0641752 + 0.997939i \(0.520442\pi\)
\(572\) 0 0
\(573\) 8.22295 0.343518
\(574\) 0 0
\(575\) 7.70306 0.321240
\(576\) 0 0
\(577\) −23.4116 −0.974637 −0.487319 0.873224i \(-0.662025\pi\)
−0.487319 + 0.873224i \(0.662025\pi\)
\(578\) 0 0
\(579\) 9.08660 0.377626
\(580\) 0 0
\(581\) −5.72136 −0.237362
\(582\) 0 0
\(583\) 40.6542 1.68372
\(584\) 0 0
\(585\) 9.93209 0.410641
\(586\) 0 0
\(587\) 31.7728 1.31140 0.655702 0.755019i \(-0.272373\pi\)
0.655702 + 0.755019i \(0.272373\pi\)
\(588\) 0 0
\(589\) −3.84957 −0.158619
\(590\) 0 0
\(591\) −9.97613 −0.410363
\(592\) 0 0
\(593\) 29.9630 1.23043 0.615216 0.788358i \(-0.289069\pi\)
0.615216 + 0.788358i \(0.289069\pi\)
\(594\) 0 0
\(595\) 16.3655 0.670921
\(596\) 0 0
\(597\) −12.4190 −0.508277
\(598\) 0 0
\(599\) 15.7292 0.642677 0.321339 0.946964i \(-0.395867\pi\)
0.321339 + 0.946964i \(0.395867\pi\)
\(600\) 0 0
\(601\) 6.85702 0.279703 0.139852 0.990172i \(-0.455337\pi\)
0.139852 + 0.990172i \(0.455337\pi\)
\(602\) 0 0
\(603\) 11.0194 0.448746
\(604\) 0 0
\(605\) 80.7366 3.28241
\(606\) 0 0
\(607\) −7.61547 −0.309102 −0.154551 0.987985i \(-0.549393\pi\)
−0.154551 + 0.987985i \(0.549393\pi\)
\(608\) 0 0
\(609\) 0.955621 0.0387237
\(610\) 0 0
\(611\) −1.46895 −0.0594274
\(612\) 0 0
\(613\) −29.1497 −1.17734 −0.588672 0.808372i \(-0.700349\pi\)
−0.588672 + 0.808372i \(0.700349\pi\)
\(614\) 0 0
\(615\) −38.9260 −1.56965
\(616\) 0 0
\(617\) −25.8141 −1.03924 −0.519618 0.854399i \(-0.673926\pi\)
−0.519618 + 0.854399i \(0.673926\pi\)
\(618\) 0 0
\(619\) 7.90003 0.317529 0.158764 0.987316i \(-0.449249\pi\)
0.158764 + 0.987316i \(0.449249\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) −7.69941 −0.308470
\(624\) 0 0
\(625\) −4.17812 −0.167125
\(626\) 0 0
\(627\) −16.4518 −0.657022
\(628\) 0 0
\(629\) 12.8763 0.513413
\(630\) 0 0
\(631\) −0.734933 −0.0292572 −0.0146286 0.999893i \(-0.504657\pi\)
−0.0146286 + 0.999893i \(0.504657\pi\)
\(632\) 0 0
\(633\) 20.1635 0.801428
\(634\) 0 0
\(635\) −10.7795 −0.427771
\(636\) 0 0
\(637\) −16.9619 −0.672055
\(638\) 0 0
\(639\) 12.6124 0.498938
\(640\) 0 0
\(641\) −1.91285 −0.0755529 −0.0377765 0.999286i \(-0.512027\pi\)
−0.0377765 + 0.999286i \(0.512027\pi\)
\(642\) 0 0
\(643\) −35.0815 −1.38348 −0.691739 0.722148i \(-0.743155\pi\)
−0.691739 + 0.722148i \(0.743155\pi\)
\(644\) 0 0
\(645\) 20.9908 0.826513
\(646\) 0 0
\(647\) 48.3884 1.90234 0.951172 0.308661i \(-0.0998807\pi\)
0.951172 + 0.308661i \(0.0998807\pi\)
\(648\) 0 0
\(649\) −16.8165 −0.660106
\(650\) 0 0
\(651\) 1.29716 0.0508397
\(652\) 0 0
\(653\) −28.9349 −1.13231 −0.566154 0.824299i \(-0.691569\pi\)
−0.566154 + 0.824299i \(0.691569\pi\)
\(654\) 0 0
\(655\) 30.9618 1.20978
\(656\) 0 0
\(657\) 5.58199 0.217774
\(658\) 0 0
\(659\) 13.9584 0.543740 0.271870 0.962334i \(-0.412358\pi\)
0.271870 + 0.962334i \(0.412358\pi\)
\(660\) 0 0
\(661\) −1.51563 −0.0589514 −0.0294757 0.999565i \(-0.509384\pi\)
−0.0294757 + 0.999565i \(0.509384\pi\)
\(662\) 0 0
\(663\) 13.3899 0.520019
\(664\) 0 0
\(665\) 9.65928 0.374571
\(666\) 0 0
\(667\) 1.00000 0.0387202
\(668\) 0 0
\(669\) −21.7862 −0.842303
\(670\) 0 0
\(671\) −48.8916 −1.88744
\(672\) 0 0
\(673\) 17.6349 0.679777 0.339888 0.940466i \(-0.389611\pi\)
0.339888 + 0.940466i \(0.389611\pi\)
\(674\) 0 0
\(675\) −7.70306 −0.296491
\(676\) 0 0
\(677\) 4.50314 0.173070 0.0865348 0.996249i \(-0.472421\pi\)
0.0865348 + 0.996249i \(0.472421\pi\)
\(678\) 0 0
\(679\) −8.48240 −0.325525
\(680\) 0 0
\(681\) −0.0750304 −0.00287517
\(682\) 0 0
\(683\) 26.3201 1.00711 0.503556 0.863963i \(-0.332025\pi\)
0.503556 + 0.863963i \(0.332025\pi\)
\(684\) 0 0
\(685\) 10.7585 0.411061
\(686\) 0 0
\(687\) −15.9524 −0.608624
\(688\) 0 0
\(689\) −19.5291 −0.744000
\(690\) 0 0
\(691\) −7.65638 −0.291262 −0.145631 0.989339i \(-0.546521\pi\)
−0.145631 + 0.989339i \(0.546521\pi\)
\(692\) 0 0
\(693\) 5.54363 0.210585
\(694\) 0 0
\(695\) −18.4031 −0.698070
\(696\) 0 0
\(697\) −52.4777 −1.98774
\(698\) 0 0
\(699\) 13.0981 0.495414
\(700\) 0 0
\(701\) 36.4761 1.37768 0.688842 0.724912i \(-0.258120\pi\)
0.688842 + 0.724912i \(0.258120\pi\)
\(702\) 0 0
\(703\) 7.59989 0.286635
\(704\) 0 0
\(705\) 1.87877 0.0707588
\(706\) 0 0
\(707\) −15.1105 −0.568288
\(708\) 0 0
\(709\) −7.48088 −0.280950 −0.140475 0.990084i \(-0.544863\pi\)
−0.140475 + 0.990084i \(0.544863\pi\)
\(710\) 0 0
\(711\) −9.15554 −0.343360
\(712\) 0 0
\(713\) 1.35740 0.0508350
\(714\) 0 0
\(715\) −57.6169 −2.15475
\(716\) 0 0
\(717\) 27.9484 1.04375
\(718\) 0 0
\(719\) −0.709976 −0.0264776 −0.0132388 0.999912i \(-0.504214\pi\)
−0.0132388 + 0.999912i \(0.504214\pi\)
\(720\) 0 0
\(721\) −5.75536 −0.214341
\(722\) 0 0
\(723\) 23.4964 0.873839
\(724\) 0 0
\(725\) 7.70306 0.286085
\(726\) 0 0
\(727\) −1.81386 −0.0672724 −0.0336362 0.999434i \(-0.510709\pi\)
−0.0336362 + 0.999434i \(0.510709\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 28.2986 1.04666
\(732\) 0 0
\(733\) 38.4328 1.41955 0.709774 0.704430i \(-0.248797\pi\)
0.709774 + 0.704430i \(0.248797\pi\)
\(734\) 0 0
\(735\) 21.6941 0.800201
\(736\) 0 0
\(737\) −63.9247 −2.35469
\(738\) 0 0
\(739\) 9.80458 0.360667 0.180334 0.983605i \(-0.442282\pi\)
0.180334 + 0.983605i \(0.442282\pi\)
\(740\) 0 0
\(741\) 7.90299 0.290324
\(742\) 0 0
\(743\) −34.2477 −1.25643 −0.628214 0.778041i \(-0.716214\pi\)
−0.628214 + 0.778041i \(0.716214\pi\)
\(744\) 0 0
\(745\) 12.7619 0.467560
\(746\) 0 0
\(747\) 5.98706 0.219055
\(748\) 0 0
\(749\) −10.4293 −0.381079
\(750\) 0 0
\(751\) −27.2508 −0.994398 −0.497199 0.867637i \(-0.665638\pi\)
−0.497199 + 0.867637i \(0.665638\pi\)
\(752\) 0 0
\(753\) −6.46207 −0.235491
\(754\) 0 0
\(755\) 10.5729 0.384788
\(756\) 0 0
\(757\) −2.11360 −0.0768202 −0.0384101 0.999262i \(-0.512229\pi\)
−0.0384101 + 0.999262i \(0.512229\pi\)
\(758\) 0 0
\(759\) 5.80108 0.210566
\(760\) 0 0
\(761\) −34.5154 −1.25118 −0.625590 0.780152i \(-0.715142\pi\)
−0.625590 + 0.780152i \(0.715142\pi\)
\(762\) 0 0
\(763\) −7.35407 −0.266235
\(764\) 0 0
\(765\) −17.1255 −0.619175
\(766\) 0 0
\(767\) 8.07819 0.291687
\(768\) 0 0
\(769\) −48.9236 −1.76423 −0.882115 0.471035i \(-0.843881\pi\)
−0.882115 + 0.471035i \(0.843881\pi\)
\(770\) 0 0
\(771\) 9.92325 0.357377
\(772\) 0 0
\(773\) 25.5483 0.918909 0.459454 0.888201i \(-0.348045\pi\)
0.459454 + 0.888201i \(0.348045\pi\)
\(774\) 0 0
\(775\) 10.4561 0.375596
\(776\) 0 0
\(777\) −2.56087 −0.0918707
\(778\) 0 0
\(779\) −30.9735 −1.10974
\(780\) 0 0
\(781\) −73.1654 −2.61806
\(782\) 0 0
\(783\) −1.00000 −0.0357371
\(784\) 0 0
\(785\) 65.1419 2.32501
\(786\) 0 0
\(787\) 12.2817 0.437794 0.218897 0.975748i \(-0.429754\pi\)
0.218897 + 0.975748i \(0.429754\pi\)
\(788\) 0 0
\(789\) −5.08891 −0.181170
\(790\) 0 0
\(791\) 9.06075 0.322163
\(792\) 0 0
\(793\) 23.4862 0.834018
\(794\) 0 0
\(795\) 24.9776 0.885864
\(796\) 0 0
\(797\) 5.91892 0.209659 0.104829 0.994490i \(-0.466570\pi\)
0.104829 + 0.994490i \(0.466570\pi\)
\(798\) 0 0
\(799\) 2.53286 0.0896060
\(800\) 0 0
\(801\) 8.05698 0.284679
\(802\) 0 0
\(803\) −32.3815 −1.14272
\(804\) 0 0
\(805\) −3.40596 −0.120044
\(806\) 0 0
\(807\) 4.38799 0.154465
\(808\) 0 0
\(809\) −17.4302 −0.612813 −0.306406 0.951901i \(-0.599127\pi\)
−0.306406 + 0.951901i \(0.599127\pi\)
\(810\) 0 0
\(811\) 44.5700 1.56506 0.782532 0.622611i \(-0.213928\pi\)
0.782532 + 0.622611i \(0.213928\pi\)
\(812\) 0 0
\(813\) −10.9204 −0.382995
\(814\) 0 0
\(815\) 25.1898 0.882360
\(816\) 0 0
\(817\) 16.7025 0.584345
\(818\) 0 0
\(819\) −2.66300 −0.0930529
\(820\) 0 0
\(821\) 18.6106 0.649513 0.324757 0.945798i \(-0.394718\pi\)
0.324757 + 0.945798i \(0.394718\pi\)
\(822\) 0 0
\(823\) −39.8324 −1.38847 −0.694235 0.719749i \(-0.744257\pi\)
−0.694235 + 0.719749i \(0.744257\pi\)
\(824\) 0 0
\(825\) 44.6861 1.55577
\(826\) 0 0
\(827\) −28.0395 −0.975030 −0.487515 0.873115i \(-0.662097\pi\)
−0.487515 + 0.873115i \(0.662097\pi\)
\(828\) 0 0
\(829\) 35.5688 1.23535 0.617677 0.786432i \(-0.288074\pi\)
0.617677 + 0.786432i \(0.288074\pi\)
\(830\) 0 0
\(831\) −1.80890 −0.0627502
\(832\) 0 0
\(833\) 29.2468 1.01334
\(834\) 0 0
\(835\) −3.54009 −0.122510
\(836\) 0 0
\(837\) −1.35740 −0.0469186
\(838\) 0 0
\(839\) 51.4430 1.77601 0.888005 0.459834i \(-0.152091\pi\)
0.888005 + 0.459834i \(0.152091\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 0 0
\(843\) −23.3353 −0.803712
\(844\) 0 0
\(845\) −18.6562 −0.641794
\(846\) 0 0
\(847\) −21.6472 −0.743807
\(848\) 0 0
\(849\) 25.4070 0.871965
\(850\) 0 0
\(851\) −2.67980 −0.0918623
\(852\) 0 0
\(853\) −37.6496 −1.28910 −0.644550 0.764562i \(-0.722955\pi\)
−0.644550 + 0.764562i \(0.722955\pi\)
\(854\) 0 0
\(855\) −10.1079 −0.345682
\(856\) 0 0
\(857\) 33.4469 1.14252 0.571262 0.820768i \(-0.306454\pi\)
0.571262 + 0.820768i \(0.306454\pi\)
\(858\) 0 0
\(859\) 5.19282 0.177177 0.0885884 0.996068i \(-0.471764\pi\)
0.0885884 + 0.996068i \(0.471764\pi\)
\(860\) 0 0
\(861\) 10.4369 0.355688
\(862\) 0 0
\(863\) 28.6534 0.975374 0.487687 0.873019i \(-0.337841\pi\)
0.487687 + 0.873019i \(0.337841\pi\)
\(864\) 0 0
\(865\) 36.2609 1.23291
\(866\) 0 0
\(867\) −6.08765 −0.206747
\(868\) 0 0
\(869\) 53.1120 1.80170
\(870\) 0 0
\(871\) 30.7076 1.04049
\(872\) 0 0
\(873\) 8.87633 0.300418
\(874\) 0 0
\(875\) −9.20653 −0.311238
\(876\) 0 0
\(877\) 1.06764 0.0360517 0.0180259 0.999838i \(-0.494262\pi\)
0.0180259 + 0.999838i \(0.494262\pi\)
\(878\) 0 0
\(879\) 23.0688 0.778091
\(880\) 0 0
\(881\) 28.4201 0.957499 0.478749 0.877952i \(-0.341090\pi\)
0.478749 + 0.877952i \(0.341090\pi\)
\(882\) 0 0
\(883\) −9.84307 −0.331246 −0.165623 0.986189i \(-0.552963\pi\)
−0.165623 + 0.986189i \(0.552963\pi\)
\(884\) 0 0
\(885\) −10.3319 −0.347304
\(886\) 0 0
\(887\) 42.3392 1.42161 0.710806 0.703388i \(-0.248330\pi\)
0.710806 + 0.703388i \(0.248330\pi\)
\(888\) 0 0
\(889\) 2.89021 0.0969346
\(890\) 0 0
\(891\) −5.80108 −0.194343
\(892\) 0 0
\(893\) 1.49495 0.0500265
\(894\) 0 0
\(895\) 47.2097 1.57805
\(896\) 0 0
\(897\) −2.78668 −0.0930444
\(898\) 0 0
\(899\) 1.35740 0.0452718
\(900\) 0 0
\(901\) 33.6733 1.12182
\(902\) 0 0
\(903\) −5.62809 −0.187291
\(904\) 0 0
\(905\) −23.7786 −0.790429
\(906\) 0 0
\(907\) 21.7926 0.723610 0.361805 0.932254i \(-0.382161\pi\)
0.361805 + 0.932254i \(0.382161\pi\)
\(908\) 0 0
\(909\) 15.8122 0.524458
\(910\) 0 0
\(911\) −32.6889 −1.08303 −0.541517 0.840690i \(-0.682150\pi\)
−0.541517 + 0.840690i \(0.682150\pi\)
\(912\) 0 0
\(913\) −34.7314 −1.14944
\(914\) 0 0
\(915\) −30.0386 −0.993046
\(916\) 0 0
\(917\) −8.30153 −0.274141
\(918\) 0 0
\(919\) 6.57313 0.216828 0.108414 0.994106i \(-0.465423\pi\)
0.108414 + 0.994106i \(0.465423\pi\)
\(920\) 0 0
\(921\) 22.2938 0.734606
\(922\) 0 0
\(923\) 35.1466 1.15687
\(924\) 0 0
\(925\) −20.6427 −0.678727
\(926\) 0 0
\(927\) 6.02264 0.197810
\(928\) 0 0
\(929\) −20.7768 −0.681664 −0.340832 0.940124i \(-0.610709\pi\)
−0.340832 + 0.940124i \(0.610709\pi\)
\(930\) 0 0
\(931\) 17.2621 0.565742
\(932\) 0 0
\(933\) −11.0508 −0.361788
\(934\) 0 0
\(935\) 99.3466 3.24898
\(936\) 0 0
\(937\) −24.7579 −0.808805 −0.404403 0.914581i \(-0.632521\pi\)
−0.404403 + 0.914581i \(0.632521\pi\)
\(938\) 0 0
\(939\) 2.95462 0.0964203
\(940\) 0 0
\(941\) 6.82478 0.222481 0.111241 0.993793i \(-0.464518\pi\)
0.111241 + 0.993793i \(0.464518\pi\)
\(942\) 0 0
\(943\) 10.9216 0.355655
\(944\) 0 0
\(945\) 3.40596 0.110796
\(946\) 0 0
\(947\) 38.4065 1.24804 0.624022 0.781407i \(-0.285497\pi\)
0.624022 + 0.781407i \(0.285497\pi\)
\(948\) 0 0
\(949\) 15.5552 0.504943
\(950\) 0 0
\(951\) −3.09880 −0.100485
\(952\) 0 0
\(953\) −37.7845 −1.22396 −0.611981 0.790873i \(-0.709627\pi\)
−0.611981 + 0.790873i \(0.709627\pi\)
\(954\) 0 0
\(955\) −29.3077 −0.948375
\(956\) 0 0
\(957\) 5.80108 0.187522
\(958\) 0 0
\(959\) −2.88458 −0.0931480
\(960\) 0 0
\(961\) −29.1575 −0.940563
\(962\) 0 0
\(963\) 10.9137 0.351687
\(964\) 0 0
\(965\) −32.3859 −1.04254
\(966\) 0 0
\(967\) 40.8040 1.31217 0.656085 0.754687i \(-0.272211\pi\)
0.656085 + 0.754687i \(0.272211\pi\)
\(968\) 0 0
\(969\) −13.6268 −0.437757
\(970\) 0 0
\(971\) −27.4923 −0.882271 −0.441136 0.897440i \(-0.645424\pi\)
−0.441136 + 0.897440i \(0.645424\pi\)
\(972\) 0 0
\(973\) 4.93426 0.158185
\(974\) 0 0
\(975\) −21.4659 −0.687460
\(976\) 0 0
\(977\) 6.64545 0.212607 0.106303 0.994334i \(-0.466099\pi\)
0.106303 + 0.994334i \(0.466099\pi\)
\(978\) 0 0
\(979\) −46.7392 −1.49379
\(980\) 0 0
\(981\) 7.69560 0.245701
\(982\) 0 0
\(983\) −16.1250 −0.514307 −0.257153 0.966371i \(-0.582785\pi\)
−0.257153 + 0.966371i \(0.582785\pi\)
\(984\) 0 0
\(985\) 35.5563 1.13292
\(986\) 0 0
\(987\) −0.503739 −0.0160342
\(988\) 0 0
\(989\) −5.88946 −0.187274
\(990\) 0 0
\(991\) −31.9866 −1.01609 −0.508044 0.861331i \(-0.669631\pi\)
−0.508044 + 0.861331i \(0.669631\pi\)
\(992\) 0 0
\(993\) 16.6986 0.529914
\(994\) 0 0
\(995\) 44.2631 1.40323
\(996\) 0 0
\(997\) −57.6602 −1.82612 −0.913058 0.407829i \(-0.866286\pi\)
−0.913058 + 0.407829i \(0.866286\pi\)
\(998\) 0 0
\(999\) 2.67980 0.0847851
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 8004.2.a.i.1.14 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8004.2.a.i.1.14 16 1.1 even 1 trivial