Properties

Label 6864.2.a.ca.1.1
Level $6864$
Weight $2$
Character 6864.1
Self dual yes
Analytic conductor $54.809$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6864 = 2^{4} \cdot 3 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6864.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(54.8093159474\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.70164.1
Defining polynomial: \(x^{4} - x^{3} - 10 x^{2} + 10 x - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3432)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-3.18590\) of defining polynomial
Character \(\chi\) \(=\) 6864.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} -3.18590 q^{5} -1.10556 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -3.18590 q^{5} -1.10556 q^{7} +1.00000 q^{9} -1.00000 q^{11} -1.00000 q^{13} -3.18590 q^{15} -6.29146 q^{17} -3.74404 q^{19} -1.10556 q^{21} -3.26624 q^{23} +5.14997 q^{25} +1.00000 q^{27} +3.60255 q^{29} +0.336304 q^{31} -1.00000 q^{33} +3.52221 q^{35} +6.37180 q^{37} -1.00000 q^{39} +0.894440 q^{41} -8.85808 q^{43} -3.18590 q^{45} +8.21112 q^{47} -5.77774 q^{49} -6.29146 q^{51} +5.32739 q^{53} +3.18590 q^{55} -3.74404 q^{57} +9.01028 q^{59} -14.9384 q^{61} -1.10556 q^{63} +3.18590 q^{65} -10.6021 q^{67} -3.26624 q^{69} +5.32739 q^{71} -5.03370 q^{73} +5.14997 q^{75} +1.10556 q^{77} -9.54699 q^{79} +1.00000 q^{81} +5.11627 q^{83} +20.0440 q^{85} +3.60255 q^{87} +3.66370 q^{89} +1.10556 q^{91} +0.336304 q^{93} +11.9281 q^{95} +8.13926 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{3} + q^{5} + q^{7} + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{3} + q^{5} + q^{7} + 4q^{9} - 4q^{11} - 4q^{13} + q^{15} - 6q^{17} + q^{21} + 9q^{23} + q^{25} + 4q^{27} - q^{29} + 8q^{31} - 4q^{33} + 7q^{35} - 2q^{37} - 4q^{39} + 9q^{41} + 5q^{43} + q^{45} + 22q^{47} + 9q^{49} - 6q^{51} + 8q^{53} - q^{55} - q^{59} - 11q^{61} + q^{63} - q^{65} + 13q^{67} + 9q^{69} + 8q^{71} - 3q^{73} + q^{75} - q^{77} + 6q^{79} + 4q^{81} + 18q^{83} + 26q^{85} - q^{87} + 8q^{89} - q^{91} + 8q^{93} + 36q^{95} + 10q^{97} - 4q^{99} + O(q^{100}) \)

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
\(5\) −3.18590 −1.42478 −0.712389 0.701784i \(-0.752387\pi\)
−0.712389 + 0.701784i \(0.752387\pi\)
\(6\) 0 0
\(7\) −1.10556 −0.417862 −0.208931 0.977930i \(-0.566998\pi\)
−0.208931 + 0.977930i \(0.566998\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) −3.18590 −0.822596
\(16\) 0 0
\(17\) −6.29146 −1.52590 −0.762952 0.646455i \(-0.776251\pi\)
−0.762952 + 0.646455i \(0.776251\pi\)
\(18\) 0 0
\(19\) −3.74404 −0.858941 −0.429471 0.903081i \(-0.641300\pi\)
−0.429471 + 0.903081i \(0.641300\pi\)
\(20\) 0 0
\(21\) −1.10556 −0.241253
\(22\) 0 0
\(23\) −3.26624 −0.681059 −0.340529 0.940234i \(-0.610606\pi\)
−0.340529 + 0.940234i \(0.610606\pi\)
\(24\) 0 0
\(25\) 5.14997 1.02999
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 3.60255 0.668976 0.334488 0.942400i \(-0.391437\pi\)
0.334488 + 0.942400i \(0.391437\pi\)
\(30\) 0 0
\(31\) 0.336304 0.0604019 0.0302010 0.999544i \(-0.490385\pi\)
0.0302010 + 0.999544i \(0.490385\pi\)
\(32\) 0 0
\(33\) −1.00000 −0.174078
\(34\) 0 0
\(35\) 3.52221 0.595361
\(36\) 0 0
\(37\) 6.37180 1.04752 0.523759 0.851866i \(-0.324529\pi\)
0.523759 + 0.851866i \(0.324529\pi\)
\(38\) 0 0
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) 0.894440 0.139688 0.0698440 0.997558i \(-0.477750\pi\)
0.0698440 + 0.997558i \(0.477750\pi\)
\(42\) 0 0
\(43\) −8.85808 −1.35084 −0.675422 0.737431i \(-0.736039\pi\)
−0.675422 + 0.737431i \(0.736039\pi\)
\(44\) 0 0
\(45\) −3.18590 −0.474926
\(46\) 0 0
\(47\) 8.21112 1.19771 0.598857 0.800856i \(-0.295622\pi\)
0.598857 + 0.800856i \(0.295622\pi\)
\(48\) 0 0
\(49\) −5.77774 −0.825391
\(50\) 0 0
\(51\) −6.29146 −0.880981
\(52\) 0 0
\(53\) 5.32739 0.731774 0.365887 0.930659i \(-0.380766\pi\)
0.365887 + 0.930659i \(0.380766\pi\)
\(54\) 0 0
\(55\) 3.18590 0.429587
\(56\) 0 0
\(57\) −3.74404 −0.495910
\(58\) 0 0
\(59\) 9.01028 1.17304 0.586519 0.809935i \(-0.300498\pi\)
0.586519 + 0.809935i \(0.300498\pi\)
\(60\) 0 0
\(61\) −14.9384 −1.91267 −0.956334 0.292275i \(-0.905588\pi\)
−0.956334 + 0.292275i \(0.905588\pi\)
\(62\) 0 0
\(63\) −1.10556 −0.139287
\(64\) 0 0
\(65\) 3.18590 0.395163
\(66\) 0 0
\(67\) −10.6021 −1.29525 −0.647627 0.761957i \(-0.724239\pi\)
−0.647627 + 0.761957i \(0.724239\pi\)
\(68\) 0 0
\(69\) −3.26624 −0.393210
\(70\) 0 0
\(71\) 5.32739 0.632245 0.316123 0.948718i \(-0.397619\pi\)
0.316123 + 0.948718i \(0.397619\pi\)
\(72\) 0 0
\(73\) −5.03370 −0.589150 −0.294575 0.955628i \(-0.595178\pi\)
−0.294575 + 0.955628i \(0.595178\pi\)
\(74\) 0 0
\(75\) 5.14997 0.594667
\(76\) 0 0
\(77\) 1.10556 0.125990
\(78\) 0 0
\(79\) −9.54699 −1.07412 −0.537060 0.843544i \(-0.680465\pi\)
−0.537060 + 0.843544i \(0.680465\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 5.11627 0.561584 0.280792 0.959769i \(-0.409403\pi\)
0.280792 + 0.959769i \(0.409403\pi\)
\(84\) 0 0
\(85\) 20.0440 2.17408
\(86\) 0 0
\(87\) 3.60255 0.386234
\(88\) 0 0
\(89\) 3.66370 0.388351 0.194176 0.980967i \(-0.437797\pi\)
0.194176 + 0.980967i \(0.437797\pi\)
\(90\) 0 0
\(91\) 1.10556 0.115894
\(92\) 0 0
\(93\) 0.336304 0.0348731
\(94\) 0 0
\(95\) 11.9281 1.22380
\(96\) 0 0
\(97\) 8.13926 0.826417 0.413208 0.910637i \(-0.364408\pi\)
0.413208 + 0.910637i \(0.364408\pi\)
\(98\) 0 0
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) 18.5914 1.84991 0.924957 0.380072i \(-0.124101\pi\)
0.924957 + 0.380072i \(0.124101\pi\)
\(102\) 0 0
\(103\) 18.9384 1.86606 0.933029 0.359801i \(-0.117155\pi\)
0.933029 + 0.359801i \(0.117155\pi\)
\(104\) 0 0
\(105\) 3.52221 0.343732
\(106\) 0 0
\(107\) 16.0547 1.55207 0.776033 0.630692i \(-0.217229\pi\)
0.776033 + 0.630692i \(0.217229\pi\)
\(108\) 0 0
\(109\) −6.41665 −0.614603 −0.307302 0.951612i \(-0.599426\pi\)
−0.307302 + 0.951612i \(0.599426\pi\)
\(110\) 0 0
\(111\) 6.37180 0.604785
\(112\) 0 0
\(113\) −0.477794 −0.0449471 −0.0224736 0.999747i \(-0.507154\pi\)
−0.0224736 + 0.999747i \(0.507154\pi\)
\(114\) 0 0
\(115\) 10.4059 0.970358
\(116\) 0 0
\(117\) −1.00000 −0.0924500
\(118\) 0 0
\(119\) 6.95559 0.637618
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 0.894440 0.0806489
\(124\) 0 0
\(125\) −0.477794 −0.0427352
\(126\) 0 0
\(127\) 7.40773 0.657330 0.328665 0.944447i \(-0.393401\pi\)
0.328665 + 0.944447i \(0.393401\pi\)
\(128\) 0 0
\(129\) −8.85808 −0.779910
\(130\) 0 0
\(131\) 2.19481 0.191762 0.0958809 0.995393i \(-0.469433\pi\)
0.0958809 + 0.995393i \(0.469433\pi\)
\(132\) 0 0
\(133\) 4.13926 0.358919
\(134\) 0 0
\(135\) −3.18590 −0.274199
\(136\) 0 0
\(137\) −20.5466 −1.75541 −0.877706 0.479200i \(-0.840927\pi\)
−0.877706 + 0.479200i \(0.840927\pi\)
\(138\) 0 0
\(139\) −15.1248 −1.28286 −0.641432 0.767180i \(-0.721660\pi\)
−0.641432 + 0.767180i \(0.721660\pi\)
\(140\) 0 0
\(141\) 8.21112 0.691501
\(142\) 0 0
\(143\) 1.00000 0.0836242
\(144\) 0 0
\(145\) −11.4774 −0.953143
\(146\) 0 0
\(147\) −5.77774 −0.476540
\(148\) 0 0
\(149\) −6.99957 −0.573427 −0.286714 0.958016i \(-0.592563\pi\)
−0.286714 + 0.958016i \(0.592563\pi\)
\(150\) 0 0
\(151\) 5.95516 0.484624 0.242312 0.970198i \(-0.422094\pi\)
0.242312 + 0.970198i \(0.422094\pi\)
\(152\) 0 0
\(153\) −6.29146 −0.508635
\(154\) 0 0
\(155\) −1.07143 −0.0860594
\(156\) 0 0
\(157\) 2.91075 0.232303 0.116151 0.993232i \(-0.462944\pi\)
0.116151 + 0.993232i \(0.462944\pi\)
\(158\) 0 0
\(159\) 5.32739 0.422490
\(160\) 0 0
\(161\) 3.61103 0.284589
\(162\) 0 0
\(163\) 6.06963 0.475410 0.237705 0.971337i \(-0.423605\pi\)
0.237705 + 0.971337i \(0.423605\pi\)
\(164\) 0 0
\(165\) 3.18590 0.248022
\(166\) 0 0
\(167\) 20.8496 1.61339 0.806695 0.590968i \(-0.201254\pi\)
0.806695 + 0.590968i \(0.201254\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) −3.74404 −0.286314
\(172\) 0 0
\(173\) −7.81367 −0.594062 −0.297031 0.954868i \(-0.595996\pi\)
−0.297031 + 0.954868i \(0.595996\pi\)
\(174\) 0 0
\(175\) −5.69360 −0.430396
\(176\) 0 0
\(177\) 9.01028 0.677254
\(178\) 0 0
\(179\) 6.67820 0.499152 0.249576 0.968355i \(-0.419709\pi\)
0.249576 + 0.968355i \(0.419709\pi\)
\(180\) 0 0
\(181\) −12.9996 −0.966250 −0.483125 0.875551i \(-0.660498\pi\)
−0.483125 + 0.875551i \(0.660498\pi\)
\(182\) 0 0
\(183\) −14.9384 −1.10428
\(184\) 0 0
\(185\) −20.2999 −1.49248
\(186\) 0 0
\(187\) 6.29146 0.460077
\(188\) 0 0
\(189\) −1.10556 −0.0804177
\(190\) 0 0
\(191\) 13.8436 1.00169 0.500843 0.865538i \(-0.333023\pi\)
0.500843 + 0.865538i \(0.333023\pi\)
\(192\) 0 0
\(193\) 21.2277 1.52800 0.764000 0.645216i \(-0.223233\pi\)
0.764000 + 0.645216i \(0.223233\pi\)
\(194\) 0 0
\(195\) 3.18590 0.228147
\(196\) 0 0
\(197\) −23.9935 −1.70947 −0.854735 0.519065i \(-0.826280\pi\)
−0.854735 + 0.519065i \(0.826280\pi\)
\(198\) 0 0
\(199\) 25.9828 1.84187 0.920937 0.389711i \(-0.127425\pi\)
0.920937 + 0.389711i \(0.127425\pi\)
\(200\) 0 0
\(201\) −10.6021 −0.747816
\(202\) 0 0
\(203\) −3.98283 −0.279540
\(204\) 0 0
\(205\) −2.84960 −0.199025
\(206\) 0 0
\(207\) −3.26624 −0.227020
\(208\) 0 0
\(209\) 3.74404 0.258981
\(210\) 0 0
\(211\) 9.26401 0.637761 0.318880 0.947795i \(-0.396693\pi\)
0.318880 + 0.947795i \(0.396693\pi\)
\(212\) 0 0
\(213\) 5.32739 0.365027
\(214\) 0 0
\(215\) 28.2210 1.92465
\(216\) 0 0
\(217\) −0.371804 −0.0252397
\(218\) 0 0
\(219\) −5.03370 −0.340146
\(220\) 0 0
\(221\) 6.29146 0.423210
\(222\) 0 0
\(223\) 17.7525 1.18880 0.594398 0.804171i \(-0.297390\pi\)
0.594398 + 0.804171i \(0.297390\pi\)
\(224\) 0 0
\(225\) 5.14997 0.343331
\(226\) 0 0
\(227\) −10.7884 −0.716055 −0.358027 0.933711i \(-0.616551\pi\)
−0.358027 + 0.933711i \(0.616551\pi\)
\(228\) 0 0
\(229\) 3.57264 0.236087 0.118043 0.993008i \(-0.462338\pi\)
0.118043 + 0.993008i \(0.462338\pi\)
\(230\) 0 0
\(231\) 1.10556 0.0727405
\(232\) 0 0
\(233\) −7.70854 −0.505003 −0.252502 0.967596i \(-0.581253\pi\)
−0.252502 + 0.967596i \(0.581253\pi\)
\(234\) 0 0
\(235\) −26.1598 −1.70648
\(236\) 0 0
\(237\) −9.54699 −0.620144
\(238\) 0 0
\(239\) 9.42693 0.609777 0.304889 0.952388i \(-0.401381\pi\)
0.304889 + 0.952388i \(0.401381\pi\)
\(240\) 0 0
\(241\) −15.8385 −1.02024 −0.510122 0.860102i \(-0.670400\pi\)
−0.510122 + 0.860102i \(0.670400\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 18.4073 1.17600
\(246\) 0 0
\(247\) 3.74404 0.238227
\(248\) 0 0
\(249\) 5.11627 0.324231
\(250\) 0 0
\(251\) −3.62734 −0.228955 −0.114478 0.993426i \(-0.536519\pi\)
−0.114478 + 0.993426i \(0.536519\pi\)
\(252\) 0 0
\(253\) 3.26624 0.205347
\(254\) 0 0
\(255\) 20.0440 1.25520
\(256\) 0 0
\(257\) −15.6829 −0.978272 −0.489136 0.872208i \(-0.662688\pi\)
−0.489136 + 0.872208i \(0.662688\pi\)
\(258\) 0 0
\(259\) −7.04441 −0.437718
\(260\) 0 0
\(261\) 3.60255 0.222992
\(262\) 0 0
\(263\) −1.41622 −0.0873276 −0.0436638 0.999046i \(-0.513903\pi\)
−0.0436638 + 0.999046i \(0.513903\pi\)
\(264\) 0 0
\(265\) −16.9726 −1.04262
\(266\) 0 0
\(267\) 3.66370 0.224215
\(268\) 0 0
\(269\) 6.97255 0.425124 0.212562 0.977148i \(-0.431819\pi\)
0.212562 + 0.977148i \(0.431819\pi\)
\(270\) 0 0
\(271\) −3.46106 −0.210244 −0.105122 0.994459i \(-0.533523\pi\)
−0.105122 + 0.994459i \(0.533523\pi\)
\(272\) 0 0
\(273\) 1.10556 0.0669115
\(274\) 0 0
\(275\) −5.14997 −0.310555
\(276\) 0 0
\(277\) −19.9828 −1.20065 −0.600326 0.799755i \(-0.704963\pi\)
−0.600326 + 0.799755i \(0.704963\pi\)
\(278\) 0 0
\(279\) 0.336304 0.0201340
\(280\) 0 0
\(281\) 10.7543 0.641549 0.320774 0.947156i \(-0.396057\pi\)
0.320774 + 0.947156i \(0.396057\pi\)
\(282\) 0 0
\(283\) 1.35218 0.0803788 0.0401894 0.999192i \(-0.487204\pi\)
0.0401894 + 0.999192i \(0.487204\pi\)
\(284\) 0 0
\(285\) 11.9281 0.706562
\(286\) 0 0
\(287\) −0.988857 −0.0583704
\(288\) 0 0
\(289\) 22.5825 1.32838
\(290\) 0 0
\(291\) 8.13926 0.477132
\(292\) 0 0
\(293\) −9.05001 −0.528707 −0.264353 0.964426i \(-0.585159\pi\)
−0.264353 + 0.964426i \(0.585159\pi\)
\(294\) 0 0
\(295\) −28.7059 −1.67132
\(296\) 0 0
\(297\) −1.00000 −0.0580259
\(298\) 0 0
\(299\) 3.26624 0.188892
\(300\) 0 0
\(301\) 9.79314 0.564467
\(302\) 0 0
\(303\) 18.5914 1.06805
\(304\) 0 0
\(305\) 47.5923 2.72513
\(306\) 0 0
\(307\) 26.9987 1.54090 0.770449 0.637502i \(-0.220032\pi\)
0.770449 + 0.637502i \(0.220032\pi\)
\(308\) 0 0
\(309\) 18.9384 1.07737
\(310\) 0 0
\(311\) −8.92771 −0.506244 −0.253122 0.967434i \(-0.581457\pi\)
−0.253122 + 0.967434i \(0.581457\pi\)
\(312\) 0 0
\(313\) 2.07811 0.117462 0.0587309 0.998274i \(-0.481295\pi\)
0.0587309 + 0.998274i \(0.481295\pi\)
\(314\) 0 0
\(315\) 3.52221 0.198454
\(316\) 0 0
\(317\) 8.49187 0.476951 0.238475 0.971149i \(-0.423352\pi\)
0.238475 + 0.971149i \(0.423352\pi\)
\(318\) 0 0
\(319\) −3.60255 −0.201704
\(320\) 0 0
\(321\) 16.0547 0.896086
\(322\) 0 0
\(323\) 23.5555 1.31066
\(324\) 0 0
\(325\) −5.14997 −0.285669
\(326\) 0 0
\(327\) −6.41665 −0.354841
\(328\) 0 0
\(329\) −9.07789 −0.500480
\(330\) 0 0
\(331\) −29.6287 −1.62854 −0.814270 0.580486i \(-0.802863\pi\)
−0.814270 + 0.580486i \(0.802863\pi\)
\(332\) 0 0
\(333\) 6.37180 0.349173
\(334\) 0 0
\(335\) 33.7773 1.84545
\(336\) 0 0
\(337\) 27.8385 1.51646 0.758229 0.651989i \(-0.226065\pi\)
0.758229 + 0.651989i \(0.226065\pi\)
\(338\) 0 0
\(339\) −0.477794 −0.0259502
\(340\) 0 0
\(341\) −0.336304 −0.0182119
\(342\) 0 0
\(343\) 14.1266 0.762762
\(344\) 0 0
\(345\) 10.4059 0.560237
\(346\) 0 0
\(347\) 8.69317 0.466674 0.233337 0.972396i \(-0.425035\pi\)
0.233337 + 0.972396i \(0.425035\pi\)
\(348\) 0 0
\(349\) 6.28298 0.336320 0.168160 0.985760i \(-0.446217\pi\)
0.168160 + 0.985760i \(0.446217\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) 0 0
\(353\) 10.6689 0.567846 0.283923 0.958847i \(-0.408364\pi\)
0.283923 + 0.958847i \(0.408364\pi\)
\(354\) 0 0
\(355\) −16.9726 −0.900809
\(356\) 0 0
\(357\) 6.95559 0.368129
\(358\) 0 0
\(359\) 9.08413 0.479442 0.239721 0.970842i \(-0.422944\pi\)
0.239721 + 0.970842i \(0.422944\pi\)
\(360\) 0 0
\(361\) −4.98218 −0.262220
\(362\) 0 0
\(363\) 1.00000 0.0524864
\(364\) 0 0
\(365\) 16.0369 0.839408
\(366\) 0 0
\(367\) −18.5821 −0.969976 −0.484988 0.874521i \(-0.661176\pi\)
−0.484988 + 0.874521i \(0.661176\pi\)
\(368\) 0 0
\(369\) 0.894440 0.0465627
\(370\) 0 0
\(371\) −5.88975 −0.305781
\(372\) 0 0
\(373\) −22.4769 −1.16381 −0.581906 0.813256i \(-0.697693\pi\)
−0.581906 + 0.813256i \(0.697693\pi\)
\(374\) 0 0
\(375\) −0.477794 −0.0246732
\(376\) 0 0
\(377\) −3.60255 −0.185541
\(378\) 0 0
\(379\) −4.85183 −0.249222 −0.124611 0.992206i \(-0.539768\pi\)
−0.124611 + 0.992206i \(0.539768\pi\)
\(380\) 0 0
\(381\) 7.40773 0.379510
\(382\) 0 0
\(383\) 21.7162 1.10964 0.554822 0.831969i \(-0.312786\pi\)
0.554822 + 0.831969i \(0.312786\pi\)
\(384\) 0 0
\(385\) −3.52221 −0.179508
\(386\) 0 0
\(387\) −8.85808 −0.450281
\(388\) 0 0
\(389\) −1.64876 −0.0835955 −0.0417977 0.999126i \(-0.513309\pi\)
−0.0417977 + 0.999126i \(0.513309\pi\)
\(390\) 0 0
\(391\) 20.5494 1.03923
\(392\) 0 0
\(393\) 2.19481 0.110714
\(394\) 0 0
\(395\) 30.4158 1.53038
\(396\) 0 0
\(397\) 20.0547 1.00652 0.503258 0.864136i \(-0.332134\pi\)
0.503258 + 0.864136i \(0.332134\pi\)
\(398\) 0 0
\(399\) 4.13926 0.207222
\(400\) 0 0
\(401\) 14.2680 0.712512 0.356256 0.934388i \(-0.384053\pi\)
0.356256 + 0.934388i \(0.384053\pi\)
\(402\) 0 0
\(403\) −0.336304 −0.0167525
\(404\) 0 0
\(405\) −3.18590 −0.158309
\(406\) 0 0
\(407\) −6.37180 −0.315839
\(408\) 0 0
\(409\) 39.4979 1.95305 0.976523 0.215411i \(-0.0691092\pi\)
0.976523 + 0.215411i \(0.0691092\pi\)
\(410\) 0 0
\(411\) −20.5466 −1.01349
\(412\) 0 0
\(413\) −9.96141 −0.490169
\(414\) 0 0
\(415\) −16.2999 −0.800133
\(416\) 0 0
\(417\) −15.1248 −0.740662
\(418\) 0 0
\(419\) −18.3439 −0.896159 −0.448080 0.893994i \(-0.647892\pi\)
−0.448080 + 0.893994i \(0.647892\pi\)
\(420\) 0 0
\(421\) 28.0154 1.36539 0.682695 0.730704i \(-0.260808\pi\)
0.682695 + 0.730704i \(0.260808\pi\)
\(422\) 0 0
\(423\) 8.21112 0.399238
\(424\) 0 0
\(425\) −32.4008 −1.57167
\(426\) 0 0
\(427\) 16.5153 0.799232
\(428\) 0 0
\(429\) 1.00000 0.0482805
\(430\) 0 0
\(431\) 18.7445 0.902889 0.451445 0.892299i \(-0.350909\pi\)
0.451445 + 0.892299i \(0.350909\pi\)
\(432\) 0 0
\(433\) −32.9170 −1.58189 −0.790945 0.611887i \(-0.790411\pi\)
−0.790945 + 0.611887i \(0.790411\pi\)
\(434\) 0 0
\(435\) −11.4774 −0.550297
\(436\) 0 0
\(437\) 12.2289 0.584990
\(438\) 0 0
\(439\) 24.1804 1.15407 0.577033 0.816721i \(-0.304211\pi\)
0.577033 + 0.816721i \(0.304211\pi\)
\(440\) 0 0
\(441\) −5.77774 −0.275130
\(442\) 0 0
\(443\) −8.74361 −0.415421 −0.207711 0.978190i \(-0.566601\pi\)
−0.207711 + 0.978190i \(0.566601\pi\)
\(444\) 0 0
\(445\) −11.6722 −0.553314
\(446\) 0 0
\(447\) −6.99957 −0.331068
\(448\) 0 0
\(449\) 30.7969 1.45340 0.726699 0.686957i \(-0.241054\pi\)
0.726699 + 0.686957i \(0.241054\pi\)
\(450\) 0 0
\(451\) −0.894440 −0.0421175
\(452\) 0 0
\(453\) 5.95516 0.279798
\(454\) 0 0
\(455\) −3.52221 −0.165124
\(456\) 0 0
\(457\) −21.6820 −1.01424 −0.507121 0.861875i \(-0.669290\pi\)
−0.507121 + 0.861875i \(0.669290\pi\)
\(458\) 0 0
\(459\) −6.29146 −0.293660
\(460\) 0 0
\(461\) 38.2542 1.78168 0.890839 0.454320i \(-0.150118\pi\)
0.890839 + 0.454320i \(0.150118\pi\)
\(462\) 0 0
\(463\) 6.64071 0.308620 0.154310 0.988022i \(-0.450685\pi\)
0.154310 + 0.988022i \(0.450685\pi\)
\(464\) 0 0
\(465\) −1.07143 −0.0496864
\(466\) 0 0
\(467\) −12.4158 −0.574534 −0.287267 0.957851i \(-0.592747\pi\)
−0.287267 + 0.957851i \(0.592747\pi\)
\(468\) 0 0
\(469\) 11.7213 0.541238
\(470\) 0 0
\(471\) 2.91075 0.134120
\(472\) 0 0
\(473\) 8.85808 0.407295
\(474\) 0 0
\(475\) −19.2817 −0.884705
\(476\) 0 0
\(477\) 5.32739 0.243925
\(478\) 0 0
\(479\) −6.34990 −0.290134 −0.145067 0.989422i \(-0.546340\pi\)
−0.145067 + 0.989422i \(0.546340\pi\)
\(480\) 0 0
\(481\) −6.37180 −0.290529
\(482\) 0 0
\(483\) 3.61103 0.164307
\(484\) 0 0
\(485\) −25.9309 −1.17746
\(486\) 0 0
\(487\) −16.9201 −0.766722 −0.383361 0.923599i \(-0.625234\pi\)
−0.383361 + 0.923599i \(0.625234\pi\)
\(488\) 0 0
\(489\) 6.06963 0.274478
\(490\) 0 0
\(491\) −11.6231 −0.524542 −0.262271 0.964994i \(-0.584471\pi\)
−0.262271 + 0.964994i \(0.584471\pi\)
\(492\) 0 0
\(493\) −22.6653 −1.02079
\(494\) 0 0
\(495\) 3.18590 0.143196
\(496\) 0 0
\(497\) −5.88975 −0.264191
\(498\) 0 0
\(499\) 8.35261 0.373914 0.186957 0.982368i \(-0.440137\pi\)
0.186957 + 0.982368i \(0.440137\pi\)
\(500\) 0 0
\(501\) 20.8496 0.931491
\(502\) 0 0
\(503\) −1.23857 −0.0552251 −0.0276125 0.999619i \(-0.508790\pi\)
−0.0276125 + 0.999619i \(0.508790\pi\)
\(504\) 0 0
\(505\) −59.2304 −2.63572
\(506\) 0 0
\(507\) 1.00000 0.0444116
\(508\) 0 0
\(509\) 39.0576 1.73120 0.865599 0.500737i \(-0.166938\pi\)
0.865599 + 0.500737i \(0.166938\pi\)
\(510\) 0 0
\(511\) 5.56506 0.246184
\(512\) 0 0
\(513\) −3.74404 −0.165303
\(514\) 0 0
\(515\) −60.3360 −2.65872
\(516\) 0 0
\(517\) −8.21112 −0.361125
\(518\) 0 0
\(519\) −7.81367 −0.342982
\(520\) 0 0
\(521\) −7.82661 −0.342890 −0.171445 0.985194i \(-0.554844\pi\)
−0.171445 + 0.985194i \(0.554844\pi\)
\(522\) 0 0
\(523\) −13.1033 −0.572969 −0.286484 0.958085i \(-0.592487\pi\)
−0.286484 + 0.958085i \(0.592487\pi\)
\(524\) 0 0
\(525\) −5.69360 −0.248489
\(526\) 0 0
\(527\) −2.11584 −0.0921675
\(528\) 0 0
\(529\) −12.3317 −0.536159
\(530\) 0 0
\(531\) 9.01028 0.391013
\(532\) 0 0
\(533\) −0.894440 −0.0387425
\(534\) 0 0
\(535\) −51.1487 −2.21135
\(536\) 0 0
\(537\) 6.67820 0.288186
\(538\) 0 0
\(539\) 5.77774 0.248865
\(540\) 0 0
\(541\) 11.9451 0.513560 0.256780 0.966470i \(-0.417338\pi\)
0.256780 + 0.966470i \(0.417338\pi\)
\(542\) 0 0
\(543\) −12.9996 −0.557865
\(544\) 0 0
\(545\) 20.4428 0.875674
\(546\) 0 0
\(547\) 27.2504 1.16515 0.582573 0.812779i \(-0.302046\pi\)
0.582573 + 0.812779i \(0.302046\pi\)
\(548\) 0 0
\(549\) −14.9384 −0.637556
\(550\) 0 0
\(551\) −13.4881 −0.574611
\(552\) 0 0
\(553\) 10.5548 0.448835
\(554\) 0 0
\(555\) −20.2999 −0.861685
\(556\) 0 0
\(557\) −25.2987 −1.07194 −0.535969 0.844238i \(-0.680054\pi\)
−0.535969 + 0.844238i \(0.680054\pi\)
\(558\) 0 0
\(559\) 8.85808 0.374657
\(560\) 0 0
\(561\) 6.29146 0.265626
\(562\) 0 0
\(563\) 37.6778 1.58793 0.793964 0.607964i \(-0.208014\pi\)
0.793964 + 0.607964i \(0.208014\pi\)
\(564\) 0 0
\(565\) 1.52221 0.0640397
\(566\) 0 0
\(567\) −1.10556 −0.0464292
\(568\) 0 0
\(569\) −15.5854 −0.653373 −0.326687 0.945133i \(-0.605932\pi\)
−0.326687 + 0.945133i \(0.605932\pi\)
\(570\) 0 0
\(571\) −4.18719 −0.175229 −0.0876143 0.996154i \(-0.527924\pi\)
−0.0876143 + 0.996154i \(0.527924\pi\)
\(572\) 0 0
\(573\) 13.8436 0.578324
\(574\) 0 0
\(575\) −16.8211 −0.701487
\(576\) 0 0
\(577\) 33.8760 1.41028 0.705138 0.709070i \(-0.250885\pi\)
0.705138 + 0.709070i \(0.250885\pi\)
\(578\) 0 0
\(579\) 21.2277 0.882191
\(580\) 0 0
\(581\) −5.65635 −0.234665
\(582\) 0 0
\(583\) −5.32739 −0.220638
\(584\) 0 0
\(585\) 3.18590 0.131721
\(586\) 0 0
\(587\) −17.7539 −0.732781 −0.366391 0.930461i \(-0.619407\pi\)
−0.366391 + 0.930461i \(0.619407\pi\)
\(588\) 0 0
\(589\) −1.25913 −0.0518817
\(590\) 0 0
\(591\) −23.9935 −0.986963
\(592\) 0 0
\(593\) −12.6278 −0.518560 −0.259280 0.965802i \(-0.583485\pi\)
−0.259280 + 0.965802i \(0.583485\pi\)
\(594\) 0 0
\(595\) −22.1598 −0.908464
\(596\) 0 0
\(597\) 25.9828 1.06341
\(598\) 0 0
\(599\) 27.0777 1.10636 0.553182 0.833060i \(-0.313413\pi\)
0.553182 + 0.833060i \(0.313413\pi\)
\(600\) 0 0
\(601\) 11.2770 0.459997 0.229998 0.973191i \(-0.426128\pi\)
0.229998 + 0.973191i \(0.426128\pi\)
\(602\) 0 0
\(603\) −10.6021 −0.431752
\(604\) 0 0
\(605\) −3.18590 −0.129525
\(606\) 0 0
\(607\) 11.5854 0.470236 0.235118 0.971967i \(-0.424452\pi\)
0.235118 + 0.971967i \(0.424452\pi\)
\(608\) 0 0
\(609\) −3.98283 −0.161393
\(610\) 0 0
\(611\) −8.21112 −0.332186
\(612\) 0 0
\(613\) −20.5167 −0.828660 −0.414330 0.910127i \(-0.635984\pi\)
−0.414330 + 0.910127i \(0.635984\pi\)
\(614\) 0 0
\(615\) −2.84960 −0.114907
\(616\) 0 0
\(617\) 40.8175 1.64325 0.821625 0.570028i \(-0.193068\pi\)
0.821625 + 0.570028i \(0.193068\pi\)
\(618\) 0 0
\(619\) 22.7619 0.914880 0.457440 0.889241i \(-0.348767\pi\)
0.457440 + 0.889241i \(0.348767\pi\)
\(620\) 0 0
\(621\) −3.26624 −0.131070
\(622\) 0 0
\(623\) −4.05044 −0.162277
\(624\) 0 0
\(625\) −24.2277 −0.969106
\(626\) 0 0
\(627\) 3.74404 0.149522
\(628\) 0 0
\(629\) −40.0880 −1.59841
\(630\) 0 0
\(631\) 26.2167 1.04367 0.521836 0.853046i \(-0.325247\pi\)
0.521836 + 0.853046i \(0.325247\pi\)
\(632\) 0 0
\(633\) 9.26401 0.368211
\(634\) 0 0
\(635\) −23.6003 −0.936550
\(636\) 0 0
\(637\) 5.77774 0.228922
\(638\) 0 0
\(639\) 5.32739 0.210748
\(640\) 0 0
\(641\) −28.7947 −1.13732 −0.568661 0.822572i \(-0.692538\pi\)
−0.568661 + 0.822572i \(0.692538\pi\)
\(642\) 0 0
\(643\) −48.5251 −1.91364 −0.956822 0.290673i \(-0.906121\pi\)
−0.956822 + 0.290673i \(0.906121\pi\)
\(644\) 0 0
\(645\) 28.2210 1.11120
\(646\) 0 0
\(647\) 8.72665 0.343080 0.171540 0.985177i \(-0.445126\pi\)
0.171540 + 0.985177i \(0.445126\pi\)
\(648\) 0 0
\(649\) −9.01028 −0.353685
\(650\) 0 0
\(651\) −0.371804 −0.0145721
\(652\) 0 0
\(653\) −43.5881 −1.70573 −0.852867 0.522128i \(-0.825138\pi\)
−0.852867 + 0.522128i \(0.825138\pi\)
\(654\) 0 0
\(655\) −6.99246 −0.273218
\(656\) 0 0
\(657\) −5.03370 −0.196383
\(658\) 0 0
\(659\) −27.9665 −1.08942 −0.544711 0.838624i \(-0.683360\pi\)
−0.544711 + 0.838624i \(0.683360\pi\)
\(660\) 0 0
\(661\) −21.4154 −0.832961 −0.416480 0.909145i \(-0.636737\pi\)
−0.416480 + 0.909145i \(0.636737\pi\)
\(662\) 0 0
\(663\) 6.29146 0.244340
\(664\) 0 0
\(665\) −13.1873 −0.511381
\(666\) 0 0
\(667\) −11.7668 −0.455612
\(668\) 0 0
\(669\) 17.7525 0.686352
\(670\) 0 0
\(671\) 14.9384 0.576691
\(672\) 0 0
\(673\) −36.2486 −1.39728 −0.698641 0.715472i \(-0.746212\pi\)
−0.698641 + 0.715472i \(0.746212\pi\)
\(674\) 0 0
\(675\) 5.14997 0.198222
\(676\) 0 0
\(677\) 38.1469 1.46610 0.733052 0.680173i \(-0.238095\pi\)
0.733052 + 0.680173i \(0.238095\pi\)
\(678\) 0 0
\(679\) −8.99844 −0.345328
\(680\) 0 0
\(681\) −10.7884 −0.413414
\(682\) 0 0
\(683\) −2.58804 −0.0990287 −0.0495143 0.998773i \(-0.515767\pi\)
−0.0495143 + 0.998773i \(0.515767\pi\)
\(684\) 0 0
\(685\) 65.4593 2.50107
\(686\) 0 0
\(687\) 3.57264 0.136305
\(688\) 0 0
\(689\) −5.32739 −0.202957
\(690\) 0 0
\(691\) 3.55345 0.135180 0.0675898 0.997713i \(-0.478469\pi\)
0.0675898 + 0.997713i \(0.478469\pi\)
\(692\) 0 0
\(693\) 1.10556 0.0419968
\(694\) 0 0
\(695\) 48.1860 1.82780
\(696\) 0 0
\(697\) −5.62734 −0.213151
\(698\) 0 0
\(699\) −7.70854 −0.291564
\(700\) 0 0
\(701\) −27.9342 −1.05506 −0.527531 0.849536i \(-0.676882\pi\)
−0.527531 + 0.849536i \(0.676882\pi\)
\(702\) 0 0
\(703\) −23.8563 −0.899757
\(704\) 0 0
\(705\) −26.1598 −0.985236
\(706\) 0 0
\(707\) −20.5539 −0.773009
\(708\) 0 0
\(709\) −43.1701 −1.62129 −0.810644 0.585540i \(-0.800883\pi\)
−0.810644 + 0.585540i \(0.800883\pi\)
\(710\) 0 0
\(711\) −9.54699 −0.358040
\(712\) 0 0
\(713\) −1.09845 −0.0411373
\(714\) 0 0
\(715\) −3.18590 −0.119146
\(716\) 0 0
\(717\) 9.42693 0.352055
\(718\) 0 0
\(719\) 16.4385 0.613054 0.306527 0.951862i \(-0.400833\pi\)
0.306527 + 0.951862i \(0.400833\pi\)
\(720\) 0 0
\(721\) −20.9376 −0.779755
\(722\) 0 0
\(723\) −15.8385 −0.589038
\(724\) 0 0
\(725\) 18.5530 0.689042
\(726\) 0 0
\(727\) −50.8137 −1.88458 −0.942289 0.334801i \(-0.891331\pi\)
−0.942289 + 0.334801i \(0.891331\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 55.7303 2.06126
\(732\) 0 0
\(733\) −15.8842 −0.586695 −0.293347 0.956006i \(-0.594769\pi\)
−0.293347 + 0.956006i \(0.594769\pi\)
\(734\) 0 0
\(735\) 18.4073 0.678964
\(736\) 0 0
\(737\) 10.6021 0.390534
\(738\) 0 0
\(739\) −35.7994 −1.31690 −0.658451 0.752624i \(-0.728788\pi\)
−0.658451 + 0.752624i \(0.728788\pi\)
\(740\) 0 0
\(741\) 3.74404 0.137541
\(742\) 0 0
\(743\) −4.19842 −0.154025 −0.0770125 0.997030i \(-0.524538\pi\)
−0.0770125 + 0.997030i \(0.524538\pi\)
\(744\) 0 0
\(745\) 22.2999 0.817007
\(746\) 0 0
\(747\) 5.11627 0.187195
\(748\) 0 0
\(749\) −17.7494 −0.648550
\(750\) 0 0
\(751\) 45.6428 1.66553 0.832764 0.553628i \(-0.186757\pi\)
0.832764 + 0.553628i \(0.186757\pi\)
\(752\) 0 0
\(753\) −3.62734 −0.132187
\(754\) 0 0
\(755\) −18.9726 −0.690482
\(756\) 0 0
\(757\) 16.8987 0.614194 0.307097 0.951678i \(-0.400642\pi\)
0.307097 + 0.951678i \(0.400642\pi\)
\(758\) 0 0
\(759\) 3.26624 0.118557
\(760\) 0 0
\(761\) 42.1995 1.52973 0.764866 0.644189i \(-0.222805\pi\)
0.764866 + 0.644189i \(0.222805\pi\)
\(762\) 0 0
\(763\) 7.09399 0.256820
\(764\) 0 0
\(765\) 20.0440 0.724692
\(766\) 0 0
\(767\) −9.01028 −0.325342
\(768\) 0 0
\(769\) 40.3836 1.45627 0.728136 0.685433i \(-0.240387\pi\)
0.728136 + 0.685433i \(0.240387\pi\)
\(770\) 0 0
\(771\) −15.6829 −0.564805
\(772\) 0 0
\(773\) −15.3790 −0.553144 −0.276572 0.960993i \(-0.589198\pi\)
−0.276572 + 0.960993i \(0.589198\pi\)
\(774\) 0 0
\(775\) 1.73195 0.0622136
\(776\) 0 0
\(777\) −7.04441 −0.252717
\(778\) 0 0
\(779\) −3.34882 −0.119984
\(780\) 0 0
\(781\) −5.32739 −0.190629
\(782\) 0 0
\(783\) 3.60255 0.128745
\(784\) 0 0
\(785\) −9.27335 −0.330980
\(786\) 0 0
\(787\) 11.2535 0.401145 0.200573 0.979679i \(-0.435720\pi\)
0.200573 + 0.979679i \(0.435720\pi\)
\(788\) 0 0
\(789\) −1.41622 −0.0504186
\(790\) 0 0
\(791\) 0.528230 0.0187817
\(792\) 0 0
\(793\) 14.9384 0.530479
\(794\) 0 0
\(795\) −16.9726 −0.601954
\(796\) 0 0
\(797\) −37.3770 −1.32396 −0.661980 0.749521i \(-0.730284\pi\)
−0.661980 + 0.749521i \(0.730284\pi\)
\(798\) 0 0
\(799\) −51.6599 −1.82760
\(800\) 0 0
\(801\) 3.66370 0.129450
\(802\) 0 0
\(803\) 5.03370 0.177635
\(804\) 0 0
\(805\) −11.5044 −0.405476
\(806\) 0 0
\(807\) 6.97255 0.245445
\(808\) 0 0
\(809\) −1.96165 −0.0689679 −0.0344839 0.999405i \(-0.510979\pi\)
−0.0344839 + 0.999405i \(0.510979\pi\)
\(810\) 0 0
\(811\) −42.4542 −1.49077 −0.745384 0.666636i \(-0.767734\pi\)
−0.745384 + 0.666636i \(0.767734\pi\)
\(812\) 0 0
\(813\) −3.46106 −0.121385
\(814\) 0 0
\(815\) −19.3372 −0.677354
\(816\) 0 0
\(817\) 33.1650 1.16030
\(818\) 0 0
\(819\) 1.10556 0.0386314
\(820\) 0 0
\(821\) −37.3714 −1.30427 −0.652135 0.758103i \(-0.726126\pi\)
−0.652135 + 0.758103i \(0.726126\pi\)
\(822\) 0 0
\(823\) −44.9206 −1.56583 −0.782917 0.622126i \(-0.786269\pi\)
−0.782917 + 0.622126i \(0.786269\pi\)
\(824\) 0 0
\(825\) −5.14997 −0.179299
\(826\) 0 0
\(827\) 52.1860 1.81468 0.907342 0.420393i \(-0.138108\pi\)
0.907342 + 0.420393i \(0.138108\pi\)
\(828\) 0 0
\(829\) 36.2308 1.25835 0.629174 0.777264i \(-0.283393\pi\)
0.629174 + 0.777264i \(0.283393\pi\)
\(830\) 0 0
\(831\) −19.9828 −0.693197
\(832\) 0 0
\(833\) 36.3504 1.25947
\(834\) 0 0
\(835\) −66.4248 −2.29872
\(836\) 0 0
\(837\) 0.336304 0.0116244
\(838\) 0 0
\(839\) −3.73398 −0.128911 −0.0644557 0.997921i \(-0.520531\pi\)
−0.0644557 + 0.997921i \(0.520531\pi\)
\(840\) 0 0
\(841\) −16.0216 −0.552471
\(842\) 0 0
\(843\) 10.7543 0.370398
\(844\) 0 0
\(845\) −3.18590 −0.109598
\(846\) 0 0
\(847\) −1.10556 −0.0379875
\(848\) 0 0
\(849\) 1.35218 0.0464067
\(850\) 0 0
\(851\) −20.8119 −0.713422
\(852\) 0 0
\(853\) 3.46579 0.118666 0.0593332 0.998238i \(-0.481103\pi\)
0.0593332 + 0.998238i \(0.481103\pi\)
\(854\) 0 0
\(855\) 11.9281 0.407934
\(856\) 0 0
\(857\) 0.996683 0.0340460 0.0170230 0.999855i \(-0.494581\pi\)
0.0170230 + 0.999855i \(0.494581\pi\)
\(858\) 0 0
\(859\) 40.1590 1.37021 0.685103 0.728446i \(-0.259757\pi\)
0.685103 + 0.728446i \(0.259757\pi\)
\(860\) 0 0
\(861\) −0.988857 −0.0337002
\(862\) 0 0
\(863\) 24.3044 0.827332 0.413666 0.910429i \(-0.364248\pi\)
0.413666 + 0.910429i \(0.364248\pi\)
\(864\) 0 0
\(865\) 24.8936 0.846407
\(866\) 0 0
\(867\) 22.5825 0.766942
\(868\) 0 0
\(869\) 9.54699 0.323860
\(870\) 0 0
\(871\) 10.6021 0.359239
\(872\) 0 0
\(873\) 8.13926 0.275472
\(874\) 0 0
\(875\) 0.528230 0.0178574
\(876\) 0 0
\(877\) 33.5377 1.13249 0.566243 0.824238i \(-0.308396\pi\)
0.566243 + 0.824238i \(0.308396\pi\)
\(878\) 0 0
\(879\) −9.05001 −0.305249
\(880\) 0 0
\(881\) −5.10357 −0.171944 −0.0859718 0.996298i \(-0.527399\pi\)
−0.0859718 + 0.996298i \(0.527399\pi\)
\(882\) 0 0
\(883\) 44.6196 1.50157 0.750784 0.660547i \(-0.229676\pi\)
0.750784 + 0.660547i \(0.229676\pi\)
\(884\) 0 0
\(885\) −28.7059 −0.964937
\(886\) 0 0
\(887\) −50.4419 −1.69367 −0.846837 0.531852i \(-0.821496\pi\)
−0.846837 + 0.531852i \(0.821496\pi\)
\(888\) 0 0
\(889\) −8.18969 −0.274674
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) 0 0
\(893\) −30.7427 −1.02877
\(894\) 0 0
\(895\) −21.2761 −0.711181
\(896\) 0 0
\(897\) 3.26624 0.109057
\(898\) 0 0
\(899\) 1.21155 0.0404075
\(900\) 0 0
\(901\) −33.5171 −1.11662
\(902\) 0 0
\(903\) 9.79314 0.325895
\(904\) 0 0
\(905\) 41.4154 1.37669
\(906\) 0 0
\(907\) 41.5939 1.38110 0.690551 0.723284i \(-0.257368\pi\)
0.690551 + 0.723284i \(0.257368\pi\)
\(908\) 0 0
\(909\) 18.5914 0.616638
\(910\) 0 0
\(911\) 10.8155 0.358332 0.179166 0.983819i \(-0.442660\pi\)
0.179166 + 0.983819i \(0.442660\pi\)
\(912\) 0 0
\(913\) −5.11627 −0.169324
\(914\) 0 0
\(915\) 47.5923 1.57335
\(916\) 0 0
\(917\) −2.42650 −0.0801300
\(918\) 0 0
\(919\) −12.4231 −0.409801 −0.204901 0.978783i \(-0.565687\pi\)
−0.204901 + 0.978783i \(0.565687\pi\)
\(920\) 0 0
\(921\) 26.9987 0.889638
\(922\) 0 0
\(923\) −5.32739 −0.175353
\(924\) 0 0
\(925\) 32.8146 1.07894
\(926\) 0 0
\(927\) 18.9384 0.622019
\(928\) 0 0
\(929\) −17.4481 −0.572454 −0.286227 0.958162i \(-0.592401\pi\)
−0.286227 + 0.958162i \(0.592401\pi\)
\(930\) 0 0
\(931\) 21.6321 0.708962
\(932\) 0 0
\(933\) −8.92771 −0.292280
\(934\) 0 0
\(935\) −20.0440 −0.655508
\(936\) 0 0
\(937\) 15.4582 0.504998 0.252499 0.967597i \(-0.418748\pi\)
0.252499 + 0.967597i \(0.418748\pi\)
\(938\) 0 0
\(939\) 2.07811 0.0678166
\(940\) 0 0
\(941\) 11.7432 0.382817 0.191408 0.981510i \(-0.438695\pi\)
0.191408 + 0.981510i \(0.438695\pi\)
\(942\) 0 0
\(943\) −2.92146 −0.0951358
\(944\) 0 0
\(945\) 3.52221 0.114577
\(946\) 0 0
\(947\) 37.4238 1.21611 0.608055 0.793895i \(-0.291950\pi\)
0.608055 + 0.793895i \(0.291950\pi\)
\(948\) 0 0
\(949\) 5.03370 0.163401
\(950\) 0 0
\(951\) 8.49187 0.275368
\(952\) 0 0
\(953\) 2.71010 0.0877887 0.0438944 0.999036i \(-0.486024\pi\)
0.0438944 + 0.999036i \(0.486024\pi\)
\(954\) 0 0
\(955\) −44.1043 −1.42718
\(956\) 0 0
\(957\) −3.60255 −0.116454
\(958\) 0 0
\(959\) 22.7155 0.733520
\(960\) 0 0
\(961\) −30.8869 −0.996352
\(962\) 0 0
\(963\) 16.0547 0.517355
\(964\) 0 0
\(965\) −67.6292 −2.17706
\(966\) 0 0
\(967\) −23.7559 −0.763938 −0.381969 0.924175i \(-0.624754\pi\)
−0.381969 + 0.924175i \(0.624754\pi\)
\(968\) 0 0
\(969\) 23.5555 0.756711
\(970\) 0 0
\(971\) −18.6286 −0.597821 −0.298911 0.954281i \(-0.596623\pi\)
−0.298911 + 0.954281i \(0.596623\pi\)
\(972\) 0 0
\(973\) 16.7213 0.536061
\(974\) 0 0
\(975\) −5.14997 −0.164931
\(976\) 0 0
\(977\) −28.7287 −0.919112 −0.459556 0.888149i \(-0.651991\pi\)
−0.459556 + 0.888149i \(0.651991\pi\)
\(978\) 0 0
\(979\) −3.66370 −0.117092
\(980\) 0 0
\(981\) −6.41665 −0.204868
\(982\) 0 0
\(983\) −25.6863 −0.819265 −0.409633 0.912251i \(-0.634343\pi\)
−0.409633 + 0.912251i \(0.634343\pi\)
\(984\) 0 0
\(985\) 76.4411 2.43562
\(986\) 0 0
\(987\) −9.07789 −0.288952
\(988\) 0 0
\(989\) 28.9326 0.920005
\(990\) 0 0
\(991\) 8.58804 0.272808 0.136404 0.990653i \(-0.456445\pi\)
0.136404 + 0.990653i \(0.456445\pi\)
\(992\) 0 0
\(993\) −29.6287 −0.940239
\(994\) 0 0
\(995\) −82.7788 −2.62426
\(996\) 0 0
\(997\) 12.8104 0.405708 0.202854 0.979209i \(-0.434978\pi\)
0.202854 + 0.979209i \(0.434978\pi\)
\(998\) 0 0
\(999\) 6.37180 0.201595
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 6864.2.a.ca.1.1 4
4.3 odd 2 3432.2.a.r.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3432.2.a.r.1.1 4 4.3 odd 2
6864.2.a.ca.1.1 4 1.1 even 1 trivial