Properties

Label 6864.2.a.bn.1.3
Level $6864$
Weight $2$
Character 6864.1
Self dual yes
Analytic conductor $54.809$
Analytic rank $0$
Dimension $3$
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: \(3\)
Coefficient field: 3.3.404.1
Defining polynomial: \(x^{3} - x^{2} - 5 x - 1\)
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 1.3
Root \(-0.210756\) of defining polynomial
Character \(\chi\) \(=\) 6864.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +1.74483 q^{5} +4.53407 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +1.74483 q^{5} +4.53407 q^{7} +1.00000 q^{9} -1.00000 q^{11} +1.00000 q^{13} -1.74483 q^{15} -1.63227 q^{17} -0.534070 q^{19} -4.53407 q^{21} +7.60221 q^{23} -1.95558 q^{25} -1.00000 q^{27} -1.21076 q^{29} +2.67669 q^{31} +1.00000 q^{33} +7.91116 q^{35} -9.91116 q^{37} -1.00000 q^{39} +3.37709 q^{41} -3.63227 q^{43} +1.74483 q^{45} +9.48965 q^{47} +13.5578 q^{49} +1.63227 q^{51} -3.57849 q^{53} -1.74483 q^{55} +0.534070 q^{57} +2.64663 q^{59} +12.5578 q^{61} +4.53407 q^{63} +1.74483 q^{65} +12.5878 q^{67} -7.60221 q^{69} -3.06814 q^{71} +4.95558 q^{73} +1.95558 q^{75} -4.53407 q^{77} +1.72110 q^{79} +1.00000 q^{81} -1.35337 q^{83} -2.84802 q^{85} +1.21076 q^{87} +6.39145 q^{89} +4.53407 q^{91} -2.67669 q^{93} -0.931860 q^{95} -2.93186 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{3} - 4q^{5} + 6q^{7} + 3q^{9} + O(q^{10}) \) \( 3q - 3q^{3} - 4q^{5} + 6q^{7} + 3q^{9} - 3q^{11} + 3q^{13} + 4q^{15} + 6q^{19} - 6q^{21} + 5q^{25} - 3q^{27} - 2q^{29} + 14q^{31} + 3q^{33} + 2q^{35} - 8q^{37} - 3q^{39} - 4q^{41} - 6q^{43} - 4q^{45} + 10q^{47} + 7q^{49} - 14q^{53} + 4q^{55} - 6q^{57} - 4q^{59} + 4q^{61} + 6q^{63} - 4q^{65} + 22q^{67} + 6q^{71} + 4q^{73} - 5q^{75} - 6q^{77} + 22q^{79} + 3q^{81} - 16q^{83} - 14q^{85} + 2q^{87} - 2q^{89} + 6q^{91} - 14q^{93} - 18q^{95} - 24q^{97} - 3q^{99} + O(q^{100}) \)

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
\(5\) 1.74483 0.780310 0.390155 0.920749i \(-0.372421\pi\)
0.390155 + 0.920749i \(0.372421\pi\)
\(6\) 0 0
\(7\) 4.53407 1.71372 0.856859 0.515551i \(-0.172413\pi\)
0.856859 + 0.515551i \(0.172413\pi\)
\(8\) 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
\(15\) −1.74483 −0.450512
\(16\) 0 0
\(17\) −1.63227 −0.395883 −0.197942 0.980214i \(-0.563426\pi\)
−0.197942 + 0.980214i \(0.563426\pi\)
\(18\) 0 0
\(19\) −0.534070 −0.122524 −0.0612621 0.998122i \(-0.519513\pi\)
−0.0612621 + 0.998122i \(0.519513\pi\)
\(20\) 0 0
\(21\) −4.53407 −0.989415
\(22\) 0 0
\(23\) 7.60221 1.58517 0.792585 0.609761i \(-0.208735\pi\)
0.792585 + 0.609761i \(0.208735\pi\)
\(24\) 0 0
\(25\) −1.95558 −0.391116
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −1.21076 −0.224832 −0.112416 0.993661i \(-0.535859\pi\)
−0.112416 + 0.993661i \(0.535859\pi\)
\(30\) 0 0
\(31\) 2.67669 0.480747 0.240373 0.970680i \(-0.422730\pi\)
0.240373 + 0.970680i \(0.422730\pi\)
\(32\) 0 0
\(33\) 1.00000 0.174078
\(34\) 0 0
\(35\) 7.91116 1.33723
\(36\) 0 0
\(37\) −9.91116 −1.62939 −0.814693 0.579893i \(-0.803094\pi\)
−0.814693 + 0.579893i \(0.803094\pi\)
\(38\) 0 0
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) 3.37709 0.527413 0.263707 0.964603i \(-0.415055\pi\)
0.263707 + 0.964603i \(0.415055\pi\)
\(42\) 0 0
\(43\) −3.63227 −0.553916 −0.276958 0.960882i \(-0.589326\pi\)
−0.276958 + 0.960882i \(0.589326\pi\)
\(44\) 0 0
\(45\) 1.74483 0.260103
\(46\) 0 0
\(47\) 9.48965 1.38421 0.692104 0.721798i \(-0.256684\pi\)
0.692104 + 0.721798i \(0.256684\pi\)
\(48\) 0 0
\(49\) 13.5578 1.93683
\(50\) 0 0
\(51\) 1.63227 0.228563
\(52\) 0 0
\(53\) −3.57849 −0.491543 −0.245772 0.969328i \(-0.579041\pi\)
−0.245772 + 0.969328i \(0.579041\pi\)
\(54\) 0 0
\(55\) −1.74483 −0.235272
\(56\) 0 0
\(57\) 0.534070 0.0707393
\(58\) 0 0
\(59\) 2.64663 0.344562 0.172281 0.985048i \(-0.444886\pi\)
0.172281 + 0.985048i \(0.444886\pi\)
\(60\) 0 0
\(61\) 12.5578 1.60786 0.803930 0.594724i \(-0.202738\pi\)
0.803930 + 0.594724i \(0.202738\pi\)
\(62\) 0 0
\(63\) 4.53407 0.571239
\(64\) 0 0
\(65\) 1.74483 0.216419
\(66\) 0 0
\(67\) 12.5878 1.53785 0.768925 0.639339i \(-0.220792\pi\)
0.768925 + 0.639339i \(0.220792\pi\)
\(68\) 0 0
\(69\) −7.60221 −0.915199
\(70\) 0 0
\(71\) −3.06814 −0.364121 −0.182061 0.983287i \(-0.558277\pi\)
−0.182061 + 0.983287i \(0.558277\pi\)
\(72\) 0 0
\(73\) 4.95558 0.580007 0.290003 0.957026i \(-0.406344\pi\)
0.290003 + 0.957026i \(0.406344\pi\)
\(74\) 0 0
\(75\) 1.95558 0.225811
\(76\) 0 0
\(77\) −4.53407 −0.516705
\(78\) 0 0
\(79\) 1.72110 0.193639 0.0968196 0.995302i \(-0.469133\pi\)
0.0968196 + 0.995302i \(0.469133\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −1.35337 −0.148552 −0.0742759 0.997238i \(-0.523665\pi\)
−0.0742759 + 0.997238i \(0.523665\pi\)
\(84\) 0 0
\(85\) −2.84802 −0.308911
\(86\) 0 0
\(87\) 1.21076 0.129807
\(88\) 0 0
\(89\) 6.39145 0.677493 0.338746 0.940878i \(-0.389997\pi\)
0.338746 + 0.940878i \(0.389997\pi\)
\(90\) 0 0
\(91\) 4.53407 0.475300
\(92\) 0 0
\(93\) −2.67669 −0.277559
\(94\) 0 0
\(95\) −0.931860 −0.0956068
\(96\) 0 0
\(97\) −2.93186 −0.297685 −0.148843 0.988861i \(-0.547555\pi\)
−0.148843 + 0.988861i \(0.547555\pi\)
\(98\) 0 0
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) −13.1219 −1.30568 −0.652840 0.757496i \(-0.726423\pi\)
−0.652840 + 0.757496i \(0.726423\pi\)
\(102\) 0 0
\(103\) 3.15698 0.311066 0.155533 0.987831i \(-0.450290\pi\)
0.155533 + 0.987831i \(0.450290\pi\)
\(104\) 0 0
\(105\) −7.91116 −0.772051
\(106\) 0 0
\(107\) 18.4690 1.78546 0.892731 0.450591i \(-0.148787\pi\)
0.892731 + 0.450591i \(0.148787\pi\)
\(108\) 0 0
\(109\) 6.86675 0.657715 0.328857 0.944380i \(-0.393336\pi\)
0.328857 + 0.944380i \(0.393336\pi\)
\(110\) 0 0
\(111\) 9.91116 0.940726
\(112\) 0 0
\(113\) −0.646629 −0.0608297 −0.0304149 0.999537i \(-0.509683\pi\)
−0.0304149 + 0.999537i \(0.509683\pi\)
\(114\) 0 0
\(115\) 13.2645 1.23692
\(116\) 0 0
\(117\) 1.00000 0.0924500
\(118\) 0 0
\(119\) −7.40082 −0.678432
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) −3.37709 −0.304502
\(124\) 0 0
\(125\) −12.1363 −1.08550
\(126\) 0 0
\(127\) 7.43587 0.659827 0.329914 0.944011i \(-0.392980\pi\)
0.329914 + 0.944011i \(0.392980\pi\)
\(128\) 0 0
\(129\) 3.63227 0.319803
\(130\) 0 0
\(131\) −15.2044 −1.32842 −0.664208 0.747548i \(-0.731231\pi\)
−0.664208 + 0.747548i \(0.731231\pi\)
\(132\) 0 0
\(133\) −2.42151 −0.209972
\(134\) 0 0
\(135\) −1.74483 −0.150171
\(136\) 0 0
\(137\) −16.5878 −1.41720 −0.708598 0.705613i \(-0.750672\pi\)
−0.708598 + 0.705613i \(0.750672\pi\)
\(138\) 0 0
\(139\) 3.21076 0.272333 0.136166 0.990686i \(-0.456522\pi\)
0.136166 + 0.990686i \(0.456522\pi\)
\(140\) 0 0
\(141\) −9.48965 −0.799173
\(142\) 0 0
\(143\) −1.00000 −0.0836242
\(144\) 0 0
\(145\) −2.11256 −0.175438
\(146\) 0 0
\(147\) −13.5578 −1.11823
\(148\) 0 0
\(149\) −10.7592 −0.881427 −0.440713 0.897648i \(-0.645275\pi\)
−0.440713 + 0.897648i \(0.645275\pi\)
\(150\) 0 0
\(151\) 0.486625 0.0396010 0.0198005 0.999804i \(-0.493697\pi\)
0.0198005 + 0.999804i \(0.493697\pi\)
\(152\) 0 0
\(153\) −1.63227 −0.131961
\(154\) 0 0
\(155\) 4.67035 0.375132
\(156\) 0 0
\(157\) 2.95558 0.235881 0.117941 0.993021i \(-0.462371\pi\)
0.117941 + 0.993021i \(0.462371\pi\)
\(158\) 0 0
\(159\) 3.57849 0.283793
\(160\) 0 0
\(161\) 34.4690 2.71653
\(162\) 0 0
\(163\) 23.8811 1.87051 0.935256 0.353971i \(-0.115169\pi\)
0.935256 + 0.353971i \(0.115169\pi\)
\(164\) 0 0
\(165\) 1.74483 0.135835
\(166\) 0 0
\(167\) −12.8905 −0.997494 −0.498747 0.866748i \(-0.666206\pi\)
−0.498747 + 0.866748i \(0.666206\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) −0.534070 −0.0408414
\(172\) 0 0
\(173\) −4.89680 −0.372297 −0.186149 0.982522i \(-0.559601\pi\)
−0.186149 + 0.982522i \(0.559601\pi\)
\(174\) 0 0
\(175\) −8.86675 −0.670263
\(176\) 0 0
\(177\) −2.64663 −0.198933
\(178\) 0 0
\(179\) 9.79861 0.732382 0.366191 0.930540i \(-0.380662\pi\)
0.366191 + 0.930540i \(0.380662\pi\)
\(180\) 0 0
\(181\) 8.02372 0.596399 0.298199 0.954504i \(-0.403614\pi\)
0.298199 + 0.954504i \(0.403614\pi\)
\(182\) 0 0
\(183\) −12.5578 −0.928299
\(184\) 0 0
\(185\) −17.2933 −1.27143
\(186\) 0 0
\(187\) 1.63227 0.119363
\(188\) 0 0
\(189\) −4.53407 −0.329805
\(190\) 0 0
\(191\) 12.3327 0.892361 0.446181 0.894943i \(-0.352784\pi\)
0.446181 + 0.894943i \(0.352784\pi\)
\(192\) 0 0
\(193\) 8.33768 0.600159 0.300080 0.953914i \(-0.402987\pi\)
0.300080 + 0.953914i \(0.402987\pi\)
\(194\) 0 0
\(195\) −1.74483 −0.124950
\(196\) 0 0
\(197\) −3.04442 −0.216906 −0.108453 0.994102i \(-0.534590\pi\)
−0.108453 + 0.994102i \(0.534590\pi\)
\(198\) 0 0
\(199\) 6.13628 0.434989 0.217495 0.976062i \(-0.430212\pi\)
0.217495 + 0.976062i \(0.430212\pi\)
\(200\) 0 0
\(201\) −12.5878 −0.887878
\(202\) 0 0
\(203\) −5.48965 −0.385298
\(204\) 0 0
\(205\) 5.89244 0.411546
\(206\) 0 0
\(207\) 7.60221 0.528390
\(208\) 0 0
\(209\) 0.534070 0.0369424
\(210\) 0 0
\(211\) 1.94622 0.133983 0.0669917 0.997754i \(-0.478660\pi\)
0.0669917 + 0.997754i \(0.478660\pi\)
\(212\) 0 0
\(213\) 3.06814 0.210226
\(214\) 0 0
\(215\) −6.33768 −0.432226
\(216\) 0 0
\(217\) 12.1363 0.823864
\(218\) 0 0
\(219\) −4.95558 −0.334867
\(220\) 0 0
\(221\) −1.63227 −0.109798
\(222\) 0 0
\(223\) −11.7923 −0.789669 −0.394834 0.918752i \(-0.629198\pi\)
−0.394834 + 0.918752i \(0.629198\pi\)
\(224\) 0 0
\(225\) −1.95558 −0.130372
\(226\) 0 0
\(227\) −22.6704 −1.50468 −0.752342 0.658773i \(-0.771076\pi\)
−0.752342 + 0.658773i \(0.771076\pi\)
\(228\) 0 0
\(229\) −8.51035 −0.562380 −0.281190 0.959652i \(-0.590729\pi\)
−0.281190 + 0.959652i \(0.590729\pi\)
\(230\) 0 0
\(231\) 4.53407 0.298320
\(232\) 0 0
\(233\) 21.4359 1.40431 0.702155 0.712024i \(-0.252221\pi\)
0.702155 + 0.712024i \(0.252221\pi\)
\(234\) 0 0
\(235\) 16.5578 1.08011
\(236\) 0 0
\(237\) −1.72110 −0.111798
\(238\) 0 0
\(239\) −7.91616 −0.512054 −0.256027 0.966670i \(-0.582414\pi\)
−0.256027 + 0.966670i \(0.582414\pi\)
\(240\) 0 0
\(241\) −24.6941 −1.59069 −0.795343 0.606160i \(-0.792709\pi\)
−0.795343 + 0.606160i \(0.792709\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 23.6560 1.51133
\(246\) 0 0
\(247\) −0.534070 −0.0339821
\(248\) 0 0
\(249\) 1.35337 0.0857664
\(250\) 0 0
\(251\) −29.6259 −1.86997 −0.934986 0.354684i \(-0.884588\pi\)
−0.934986 + 0.354684i \(0.884588\pi\)
\(252\) 0 0
\(253\) −7.60221 −0.477947
\(254\) 0 0
\(255\) 2.84802 0.178350
\(256\) 0 0
\(257\) 6.61791 0.412814 0.206407 0.978466i \(-0.433823\pi\)
0.206407 + 0.978466i \(0.433823\pi\)
\(258\) 0 0
\(259\) −44.9379 −2.79231
\(260\) 0 0
\(261\) −1.21076 −0.0749439
\(262\) 0 0
\(263\) −27.7622 −1.71189 −0.855946 0.517066i \(-0.827024\pi\)
−0.855946 + 0.517066i \(0.827024\pi\)
\(264\) 0 0
\(265\) −6.24384 −0.383556
\(266\) 0 0
\(267\) −6.39145 −0.391151
\(268\) 0 0
\(269\) 7.57849 0.462069 0.231034 0.972946i \(-0.425789\pi\)
0.231034 + 0.972946i \(0.425789\pi\)
\(270\) 0 0
\(271\) 19.1807 1.16514 0.582572 0.812779i \(-0.302046\pi\)
0.582572 + 0.812779i \(0.302046\pi\)
\(272\) 0 0
\(273\) −4.53407 −0.274414
\(274\) 0 0
\(275\) 1.95558 0.117926
\(276\) 0 0
\(277\) −13.2044 −0.793377 −0.396688 0.917953i \(-0.629841\pi\)
−0.396688 + 0.917953i \(0.629841\pi\)
\(278\) 0 0
\(279\) 2.67669 0.160249
\(280\) 0 0
\(281\) 19.7098 1.17579 0.587893 0.808939i \(-0.299958\pi\)
0.587893 + 0.808939i \(0.299958\pi\)
\(282\) 0 0
\(283\) 26.1299 1.55326 0.776632 0.629955i \(-0.216926\pi\)
0.776632 + 0.629955i \(0.216926\pi\)
\(284\) 0 0
\(285\) 0.931860 0.0551986
\(286\) 0 0
\(287\) 15.3120 0.903838
\(288\) 0 0
\(289\) −14.3357 −0.843277
\(290\) 0 0
\(291\) 2.93186 0.171869
\(292\) 0 0
\(293\) 5.97628 0.349138 0.174569 0.984645i \(-0.444147\pi\)
0.174569 + 0.984645i \(0.444147\pi\)
\(294\) 0 0
\(295\) 4.61791 0.268865
\(296\) 0 0
\(297\) 1.00000 0.0580259
\(298\) 0 0
\(299\) 7.60221 0.439647
\(300\) 0 0
\(301\) −16.4690 −0.949255
\(302\) 0 0
\(303\) 13.1219 0.753835
\(304\) 0 0
\(305\) 21.9112 1.25463
\(306\) 0 0
\(307\) 13.3771 0.763471 0.381736 0.924272i \(-0.375327\pi\)
0.381736 + 0.924272i \(0.375327\pi\)
\(308\) 0 0
\(309\) −3.15698 −0.179594
\(310\) 0 0
\(311\) −25.5608 −1.44942 −0.724711 0.689053i \(-0.758027\pi\)
−0.724711 + 0.689053i \(0.758027\pi\)
\(312\) 0 0
\(313\) 29.1994 1.65045 0.825224 0.564805i \(-0.191049\pi\)
0.825224 + 0.564805i \(0.191049\pi\)
\(314\) 0 0
\(315\) 7.91116 0.445744
\(316\) 0 0
\(317\) 9.94122 0.558355 0.279177 0.960240i \(-0.409938\pi\)
0.279177 + 0.960240i \(0.409938\pi\)
\(318\) 0 0
\(319\) 1.21076 0.0677893
\(320\) 0 0
\(321\) −18.4690 −1.03084
\(322\) 0 0
\(323\) 0.871746 0.0485052
\(324\) 0 0
\(325\) −1.95558 −0.108476
\(326\) 0 0
\(327\) −6.86675 −0.379732
\(328\) 0 0
\(329\) 43.0267 2.37214
\(330\) 0 0
\(331\) −6.87308 −0.377779 −0.188889 0.981998i \(-0.560489\pi\)
−0.188889 + 0.981998i \(0.560489\pi\)
\(332\) 0 0
\(333\) −9.91116 −0.543128
\(334\) 0 0
\(335\) 21.9636 1.20000
\(336\) 0 0
\(337\) 9.66732 0.526613 0.263306 0.964712i \(-0.415187\pi\)
0.263306 + 0.964712i \(0.415187\pi\)
\(338\) 0 0
\(339\) 0.646629 0.0351200
\(340\) 0 0
\(341\) −2.67669 −0.144951
\(342\) 0 0
\(343\) 29.7335 1.60546
\(344\) 0 0
\(345\) −13.2645 −0.714139
\(346\) 0 0
\(347\) 10.4502 0.560998 0.280499 0.959854i \(-0.409500\pi\)
0.280499 + 0.959854i \(0.409500\pi\)
\(348\) 0 0
\(349\) 1.44221 0.0771996 0.0385998 0.999255i \(-0.487710\pi\)
0.0385998 + 0.999255i \(0.487710\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) 0 0
\(353\) −16.7241 −0.890136 −0.445068 0.895497i \(-0.646820\pi\)
−0.445068 + 0.895497i \(0.646820\pi\)
\(354\) 0 0
\(355\) −5.35337 −0.284127
\(356\) 0 0
\(357\) 7.40082 0.391693
\(358\) 0 0
\(359\) 18.7178 0.987887 0.493944 0.869494i \(-0.335555\pi\)
0.493944 + 0.869494i \(0.335555\pi\)
\(360\) 0 0
\(361\) −18.7148 −0.984988
\(362\) 0 0
\(363\) −1.00000 −0.0524864
\(364\) 0 0
\(365\) 8.64663 0.452585
\(366\) 0 0
\(367\) −4.55779 −0.237915 −0.118957 0.992899i \(-0.537955\pi\)
−0.118957 + 0.992899i \(0.537955\pi\)
\(368\) 0 0
\(369\) 3.37709 0.175804
\(370\) 0 0
\(371\) −16.2251 −0.842366
\(372\) 0 0
\(373\) −29.3120 −1.51772 −0.758858 0.651256i \(-0.774243\pi\)
−0.758858 + 0.651256i \(0.774243\pi\)
\(374\) 0 0
\(375\) 12.1363 0.626715
\(376\) 0 0
\(377\) −1.21076 −0.0623571
\(378\) 0 0
\(379\) 19.9412 1.02431 0.512156 0.858893i \(-0.328847\pi\)
0.512156 + 0.858893i \(0.328847\pi\)
\(380\) 0 0
\(381\) −7.43587 −0.380951
\(382\) 0 0
\(383\) 15.2645 0.779981 0.389991 0.920819i \(-0.372478\pi\)
0.389991 + 0.920819i \(0.372478\pi\)
\(384\) 0 0
\(385\) −7.91116 −0.403190
\(386\) 0 0
\(387\) −3.63227 −0.184639
\(388\) 0 0
\(389\) −3.17570 −0.161014 −0.0805072 0.996754i \(-0.525654\pi\)
−0.0805072 + 0.996754i \(0.525654\pi\)
\(390\) 0 0
\(391\) −12.4088 −0.627542
\(392\) 0 0
\(393\) 15.2044 0.766962
\(394\) 0 0
\(395\) 3.00303 0.151099
\(396\) 0 0
\(397\) −9.91116 −0.497427 −0.248714 0.968577i \(-0.580008\pi\)
−0.248714 + 0.968577i \(0.580008\pi\)
\(398\) 0 0
\(399\) 2.42151 0.121227
\(400\) 0 0
\(401\) 19.1456 0.956088 0.478044 0.878336i \(-0.341346\pi\)
0.478044 + 0.878336i \(0.341346\pi\)
\(402\) 0 0
\(403\) 2.67669 0.133335
\(404\) 0 0
\(405\) 1.74483 0.0867011
\(406\) 0 0
\(407\) 9.91116 0.491278
\(408\) 0 0
\(409\) −5.18070 −0.256169 −0.128085 0.991763i \(-0.540883\pi\)
−0.128085 + 0.991763i \(0.540883\pi\)
\(410\) 0 0
\(411\) 16.5878 0.818218
\(412\) 0 0
\(413\) 12.0000 0.590481
\(414\) 0 0
\(415\) −2.36140 −0.115916
\(416\) 0 0
\(417\) −3.21076 −0.157231
\(418\) 0 0
\(419\) −14.0237 −0.685104 −0.342552 0.939499i \(-0.611291\pi\)
−0.342552 + 0.939499i \(0.611291\pi\)
\(420\) 0 0
\(421\) 27.0681 1.31922 0.659610 0.751608i \(-0.270721\pi\)
0.659610 + 0.751608i \(0.270721\pi\)
\(422\) 0 0
\(423\) 9.48965 0.461403
\(424\) 0 0
\(425\) 3.19203 0.154836
\(426\) 0 0
\(427\) 56.9379 2.75542
\(428\) 0 0
\(429\) 1.00000 0.0482805
\(430\) 0 0
\(431\) 35.2044 1.69574 0.847869 0.530206i \(-0.177886\pi\)
0.847869 + 0.530206i \(0.177886\pi\)
\(432\) 0 0
\(433\) 23.9586 1.15138 0.575689 0.817669i \(-0.304734\pi\)
0.575689 + 0.817669i \(0.304734\pi\)
\(434\) 0 0
\(435\) 2.11256 0.101289
\(436\) 0 0
\(437\) −4.06011 −0.194222
\(438\) 0 0
\(439\) −0.260174 −0.0124174 −0.00620870 0.999981i \(-0.501976\pi\)
−0.00620870 + 0.999981i \(0.501976\pi\)
\(440\) 0 0
\(441\) 13.5578 0.645609
\(442\) 0 0
\(443\) −15.8223 −0.751741 −0.375871 0.926672i \(-0.622656\pi\)
−0.375871 + 0.926672i \(0.622656\pi\)
\(444\) 0 0
\(445\) 11.1520 0.528654
\(446\) 0 0
\(447\) 10.7592 0.508892
\(448\) 0 0
\(449\) −26.3026 −1.24130 −0.620649 0.784089i \(-0.713131\pi\)
−0.620649 + 0.784089i \(0.713131\pi\)
\(450\) 0 0
\(451\) −3.37709 −0.159021
\(452\) 0 0
\(453\) −0.486625 −0.0228637
\(454\) 0 0
\(455\) 7.91116 0.370881
\(456\) 0 0
\(457\) 31.0681 1.45331 0.726653 0.687005i \(-0.241075\pi\)
0.726653 + 0.687005i \(0.241075\pi\)
\(458\) 0 0
\(459\) 1.63227 0.0761877
\(460\) 0 0
\(461\) 22.8066 1.06221 0.531105 0.847306i \(-0.321777\pi\)
0.531105 + 0.847306i \(0.321777\pi\)
\(462\) 0 0
\(463\) −11.9887 −0.557161 −0.278580 0.960413i \(-0.589864\pi\)
−0.278580 + 0.960413i \(0.589864\pi\)
\(464\) 0 0
\(465\) −4.67035 −0.216582
\(466\) 0 0
\(467\) −12.4927 −0.578092 −0.289046 0.957315i \(-0.593338\pi\)
−0.289046 + 0.957315i \(0.593338\pi\)
\(468\) 0 0
\(469\) 57.0742 2.63544
\(470\) 0 0
\(471\) −2.95558 −0.136186
\(472\) 0 0
\(473\) 3.63227 0.167012
\(474\) 0 0
\(475\) 1.04442 0.0479212
\(476\) 0 0
\(477\) −3.57849 −0.163848
\(478\) 0 0
\(479\) 8.48663 0.387764 0.193882 0.981025i \(-0.437892\pi\)
0.193882 + 0.981025i \(0.437892\pi\)
\(480\) 0 0
\(481\) −9.91116 −0.451910
\(482\) 0 0
\(483\) −34.4690 −1.56839
\(484\) 0 0
\(485\) −5.11559 −0.232287
\(486\) 0 0
\(487\) 35.1931 1.59475 0.797375 0.603483i \(-0.206221\pi\)
0.797375 + 0.603483i \(0.206221\pi\)
\(488\) 0 0
\(489\) −23.8811 −1.07994
\(490\) 0 0
\(491\) −4.08884 −0.184527 −0.0922633 0.995735i \(-0.529410\pi\)
−0.0922633 + 0.995735i \(0.529410\pi\)
\(492\) 0 0
\(493\) 1.97628 0.0890071
\(494\) 0 0
\(495\) −1.74483 −0.0784241
\(496\) 0 0
\(497\) −13.9112 −0.624001
\(498\) 0 0
\(499\) 9.92855 0.444463 0.222232 0.974994i \(-0.428666\pi\)
0.222232 + 0.974994i \(0.428666\pi\)
\(500\) 0 0
\(501\) 12.8905 0.575904
\(502\) 0 0
\(503\) −37.8411 −1.68725 −0.843625 0.536934i \(-0.819583\pi\)
−0.843625 + 0.536934i \(0.819583\pi\)
\(504\) 0 0
\(505\) −22.8955 −1.01883
\(506\) 0 0
\(507\) −1.00000 −0.0444116
\(508\) 0 0
\(509\) −10.3915 −0.460593 −0.230297 0.973120i \(-0.573970\pi\)
−0.230297 + 0.973120i \(0.573970\pi\)
\(510\) 0 0
\(511\) 22.4690 0.993968
\(512\) 0 0
\(513\) 0.534070 0.0235798
\(514\) 0 0
\(515\) 5.50837 0.242728
\(516\) 0 0
\(517\) −9.48965 −0.417354
\(518\) 0 0
\(519\) 4.89680 0.214946
\(520\) 0 0
\(521\) 30.9379 1.35541 0.677707 0.735332i \(-0.262974\pi\)
0.677707 + 0.735332i \(0.262974\pi\)
\(522\) 0 0
\(523\) −12.1901 −0.533034 −0.266517 0.963830i \(-0.585873\pi\)
−0.266517 + 0.963830i \(0.585873\pi\)
\(524\) 0 0
\(525\) 8.86675 0.386977
\(526\) 0 0
\(527\) −4.36907 −0.190320
\(528\) 0 0
\(529\) 34.7936 1.51277
\(530\) 0 0
\(531\) 2.64663 0.114854
\(532\) 0 0
\(533\) 3.37709 0.146278
\(534\) 0 0
\(535\) 32.2251 1.39321
\(536\) 0 0
\(537\) −9.79861 −0.422841
\(538\) 0 0
\(539\) −13.5578 −0.583975
\(540\) 0 0
\(541\) −37.8223 −1.62611 −0.813054 0.582188i \(-0.802197\pi\)
−0.813054 + 0.582188i \(0.802197\pi\)
\(542\) 0 0
\(543\) −8.02372 −0.344331
\(544\) 0 0
\(545\) 11.9813 0.513222
\(546\) 0 0
\(547\) −29.5622 −1.26399 −0.631993 0.774974i \(-0.717763\pi\)
−0.631993 + 0.774974i \(0.717763\pi\)
\(548\) 0 0
\(549\) 12.5578 0.535954
\(550\) 0 0
\(551\) 0.646629 0.0275473
\(552\) 0 0
\(553\) 7.80361 0.331843
\(554\) 0 0
\(555\) 17.2933 0.734058
\(556\) 0 0
\(557\) −11.4245 −0.484073 −0.242037 0.970267i \(-0.577815\pi\)
−0.242037 + 0.970267i \(0.577815\pi\)
\(558\) 0 0
\(559\) −3.63227 −0.153629
\(560\) 0 0
\(561\) −1.63227 −0.0689144
\(562\) 0 0
\(563\) −9.06814 −0.382177 −0.191088 0.981573i \(-0.561202\pi\)
−0.191088 + 0.981573i \(0.561202\pi\)
\(564\) 0 0
\(565\) −1.12825 −0.0474660
\(566\) 0 0
\(567\) 4.53407 0.190413
\(568\) 0 0
\(569\) 21.6323 0.906872 0.453436 0.891289i \(-0.350198\pi\)
0.453436 + 0.891289i \(0.350198\pi\)
\(570\) 0 0
\(571\) 44.9730 1.88206 0.941030 0.338323i \(-0.109860\pi\)
0.941030 + 0.338323i \(0.109860\pi\)
\(572\) 0 0
\(573\) −12.3327 −0.515205
\(574\) 0 0
\(575\) −14.8667 −0.619986
\(576\) 0 0
\(577\) 9.20442 0.383185 0.191593 0.981475i \(-0.438635\pi\)
0.191593 + 0.981475i \(0.438635\pi\)
\(578\) 0 0
\(579\) −8.33768 −0.346502
\(580\) 0 0
\(581\) −6.13628 −0.254576
\(582\) 0 0
\(583\) 3.57849 0.148206
\(584\) 0 0
\(585\) 1.74483 0.0721397
\(586\) 0 0
\(587\) 33.6259 1.38789 0.693945 0.720028i \(-0.255871\pi\)
0.693945 + 0.720028i \(0.255871\pi\)
\(588\) 0 0
\(589\) −1.42954 −0.0589031
\(590\) 0 0
\(591\) 3.04442 0.125231
\(592\) 0 0
\(593\) 20.3851 0.837117 0.418558 0.908190i \(-0.362536\pi\)
0.418558 + 0.908190i \(0.362536\pi\)
\(594\) 0 0
\(595\) −12.9131 −0.529387
\(596\) 0 0
\(597\) −6.13628 −0.251141
\(598\) 0 0
\(599\) 5.80361 0.237129 0.118564 0.992946i \(-0.462171\pi\)
0.118564 + 0.992946i \(0.462171\pi\)
\(600\) 0 0
\(601\) −30.0949 −1.22760 −0.613798 0.789463i \(-0.710359\pi\)
−0.613798 + 0.789463i \(0.710359\pi\)
\(602\) 0 0
\(603\) 12.5878 0.512617
\(604\) 0 0
\(605\) 1.74483 0.0709373
\(606\) 0 0
\(607\) 4.84936 0.196829 0.0984147 0.995145i \(-0.468623\pi\)
0.0984147 + 0.995145i \(0.468623\pi\)
\(608\) 0 0
\(609\) 5.48965 0.222452
\(610\) 0 0
\(611\) 9.48965 0.383910
\(612\) 0 0
\(613\) 35.7572 1.44422 0.722110 0.691778i \(-0.243172\pi\)
0.722110 + 0.691778i \(0.243172\pi\)
\(614\) 0 0
\(615\) −5.89244 −0.237606
\(616\) 0 0
\(617\) 6.63529 0.267127 0.133563 0.991040i \(-0.457358\pi\)
0.133563 + 0.991040i \(0.457358\pi\)
\(618\) 0 0
\(619\) −32.1062 −1.29046 −0.645229 0.763989i \(-0.723238\pi\)
−0.645229 + 0.763989i \(0.723238\pi\)
\(620\) 0 0
\(621\) −7.60221 −0.305066
\(622\) 0 0
\(623\) 28.9793 1.16103
\(624\) 0 0
\(625\) −11.3978 −0.455912
\(626\) 0 0
\(627\) −0.534070 −0.0213287
\(628\) 0 0
\(629\) 16.1777 0.645046
\(630\) 0 0
\(631\) 4.67669 0.186176 0.0930880 0.995658i \(-0.470326\pi\)
0.0930880 + 0.995658i \(0.470326\pi\)
\(632\) 0 0
\(633\) −1.94622 −0.0773553
\(634\) 0 0
\(635\) 12.9743 0.514870
\(636\) 0 0
\(637\) 13.5578 0.537179
\(638\) 0 0
\(639\) −3.06814 −0.121374
\(640\) 0 0
\(641\) 5.77488 0.228094 0.114047 0.993475i \(-0.463619\pi\)
0.114047 + 0.993475i \(0.463619\pi\)
\(642\) 0 0
\(643\) 36.0775 1.42276 0.711379 0.702809i \(-0.248071\pi\)
0.711379 + 0.702809i \(0.248071\pi\)
\(644\) 0 0
\(645\) 6.33768 0.249546
\(646\) 0 0
\(647\) −23.7048 −0.931931 −0.465965 0.884803i \(-0.654293\pi\)
−0.465965 + 0.884803i \(0.654293\pi\)
\(648\) 0 0
\(649\) −2.64663 −0.103889
\(650\) 0 0
\(651\) −12.1363 −0.475658
\(652\) 0 0
\(653\) −1.77488 −0.0694565 −0.0347283 0.999397i \(-0.511057\pi\)
−0.0347283 + 0.999397i \(0.511057\pi\)
\(654\) 0 0
\(655\) −26.5291 −1.03658
\(656\) 0 0
\(657\) 4.95558 0.193336
\(658\) 0 0
\(659\) −23.4008 −0.911566 −0.455783 0.890091i \(-0.650641\pi\)
−0.455783 + 0.890091i \(0.650641\pi\)
\(660\) 0 0
\(661\) 23.7936 0.925464 0.462732 0.886498i \(-0.346869\pi\)
0.462732 + 0.886498i \(0.346869\pi\)
\(662\) 0 0
\(663\) 1.63227 0.0633920
\(664\) 0 0
\(665\) −4.22512 −0.163843
\(666\) 0 0
\(667\) −9.20442 −0.356397
\(668\) 0 0
\(669\) 11.7923 0.455916
\(670\) 0 0
\(671\) −12.5578 −0.484788
\(672\) 0 0
\(673\) 22.9793 0.885787 0.442894 0.896574i \(-0.353952\pi\)
0.442894 + 0.896574i \(0.353952\pi\)
\(674\) 0 0
\(675\) 1.95558 0.0752704
\(676\) 0 0
\(677\) 40.8554 1.57020 0.785101 0.619368i \(-0.212611\pi\)
0.785101 + 0.619368i \(0.212611\pi\)
\(678\) 0 0
\(679\) −13.2933 −0.510148
\(680\) 0 0
\(681\) 22.6704 0.868730
\(682\) 0 0
\(683\) −7.24581 −0.277253 −0.138627 0.990345i \(-0.544269\pi\)
−0.138627 + 0.990345i \(0.544269\pi\)
\(684\) 0 0
\(685\) −28.9429 −1.10585
\(686\) 0 0
\(687\) 8.51035 0.324690
\(688\) 0 0
\(689\) −3.57849 −0.136330
\(690\) 0 0
\(691\) −34.5177 −1.31312 −0.656558 0.754275i \(-0.727988\pi\)
−0.656558 + 0.754275i \(0.727988\pi\)
\(692\) 0 0
\(693\) −4.53407 −0.172235
\(694\) 0 0
\(695\) 5.60221 0.212504
\(696\) 0 0
\(697\) −5.51232 −0.208794
\(698\) 0 0
\(699\) −21.4359 −0.810779
\(700\) 0 0
\(701\) −29.2168 −1.10350 −0.551752 0.834008i \(-0.686040\pi\)
−0.551752 + 0.834008i \(0.686040\pi\)
\(702\) 0 0
\(703\) 5.29326 0.199639
\(704\) 0 0
\(705\) −16.5578 −0.623603
\(706\) 0 0
\(707\) −59.4957 −2.23757
\(708\) 0 0
\(709\) 17.8510 0.670410 0.335205 0.942145i \(-0.391194\pi\)
0.335205 + 0.942145i \(0.391194\pi\)
\(710\) 0 0
\(711\) 1.72110 0.0645464
\(712\) 0 0
\(713\) 20.3487 0.762066
\(714\) 0 0
\(715\) −1.74483 −0.0652528
\(716\) 0 0
\(717\) 7.91616 0.295635
\(718\) 0 0
\(719\) −24.4502 −0.911840 −0.455920 0.890021i \(-0.650690\pi\)
−0.455920 + 0.890021i \(0.650690\pi\)
\(720\) 0 0
\(721\) 14.3140 0.533079
\(722\) 0 0
\(723\) 24.6941 0.918382
\(724\) 0 0
\(725\) 2.36773 0.0879354
\(726\) 0 0
\(727\) −9.00803 −0.334089 −0.167045 0.985949i \(-0.553422\pi\)
−0.167045 + 0.985949i \(0.553422\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 5.92883 0.219286
\(732\) 0 0
\(733\) 44.3851 1.63940 0.819701 0.572792i \(-0.194140\pi\)
0.819701 + 0.572792i \(0.194140\pi\)
\(734\) 0 0
\(735\) −23.6560 −0.872564
\(736\) 0 0
\(737\) −12.5878 −0.463679
\(738\) 0 0
\(739\) −40.4513 −1.48802 −0.744012 0.668166i \(-0.767080\pi\)
−0.744012 + 0.668166i \(0.767080\pi\)
\(740\) 0 0
\(741\) 0.534070 0.0196196
\(742\) 0 0
\(743\) −7.71477 −0.283027 −0.141514 0.989936i \(-0.545197\pi\)
−0.141514 + 0.989936i \(0.545197\pi\)
\(744\) 0 0
\(745\) −18.7729 −0.687786
\(746\) 0 0
\(747\) −1.35337 −0.0495173
\(748\) 0 0
\(749\) 83.7395 3.05978
\(750\) 0 0
\(751\) −17.1283 −0.625019 −0.312509 0.949915i \(-0.601170\pi\)
−0.312509 + 0.949915i \(0.601170\pi\)
\(752\) 0 0
\(753\) 29.6259 1.07963
\(754\) 0 0
\(755\) 0.849077 0.0309011
\(756\) 0 0
\(757\) −6.33768 −0.230347 −0.115173 0.993345i \(-0.536742\pi\)
−0.115173 + 0.993345i \(0.536742\pi\)
\(758\) 0 0
\(759\) 7.60221 0.275943
\(760\) 0 0
\(761\) −1.40582 −0.0509608 −0.0254804 0.999675i \(-0.508112\pi\)
−0.0254804 + 0.999675i \(0.508112\pi\)
\(762\) 0 0
\(763\) 31.1343 1.12714
\(764\) 0 0
\(765\) −2.84802 −0.102970
\(766\) 0 0
\(767\) 2.64663 0.0955642
\(768\) 0 0
\(769\) −14.2151 −0.512610 −0.256305 0.966596i \(-0.582505\pi\)
−0.256305 + 0.966596i \(0.582505\pi\)
\(770\) 0 0
\(771\) −6.61791 −0.238338
\(772\) 0 0
\(773\) −36.0174 −1.29546 −0.647728 0.761872i \(-0.724281\pi\)
−0.647728 + 0.761872i \(0.724281\pi\)
\(774\) 0 0
\(775\) −5.23448 −0.188028
\(776\) 0 0
\(777\) 44.9379 1.61214
\(778\) 0 0
\(779\) −1.80361 −0.0646209
\(780\) 0 0
\(781\) 3.06814 0.109787
\(782\) 0 0
\(783\) 1.21076 0.0432689
\(784\) 0 0
\(785\) 5.15698 0.184060
\(786\) 0 0
\(787\) 30.2676 1.07892 0.539461 0.842011i \(-0.318628\pi\)
0.539461 + 0.842011i \(0.318628\pi\)
\(788\) 0 0
\(789\) 27.7622 0.988361
\(790\) 0 0
\(791\) −2.93186 −0.104245
\(792\) 0 0
\(793\) 12.5578 0.445940
\(794\) 0 0
\(795\) 6.24384 0.221446
\(796\) 0 0
\(797\) −43.7809 −1.55080 −0.775400 0.631470i \(-0.782452\pi\)
−0.775400 + 0.631470i \(0.782452\pi\)
\(798\) 0 0
\(799\) −15.4897 −0.547985
\(800\) 0 0
\(801\) 6.39145 0.225831
\(802\) 0 0
\(803\) −4.95558 −0.174879
\(804\) 0 0
\(805\) 60.1423 2.11974
\(806\) 0 0
\(807\) −7.57849 −0.266775
\(808\) 0 0
\(809\) −33.6196 −1.18200 −0.591001 0.806671i \(-0.701267\pi\)
−0.591001 + 0.806671i \(0.701267\pi\)
\(810\) 0 0
\(811\) 24.6290 0.864840 0.432420 0.901672i \(-0.357660\pi\)
0.432420 + 0.901672i \(0.357660\pi\)
\(812\) 0 0
\(813\) −19.1807 −0.672696
\(814\) 0 0
\(815\) 41.6684 1.45958
\(816\) 0 0
\(817\) 1.93989 0.0678680
\(818\) 0 0
\(819\) 4.53407 0.158433
\(820\) 0 0
\(821\) −38.9142 −1.35811 −0.679057 0.734085i \(-0.737611\pi\)
−0.679057 + 0.734085i \(0.737611\pi\)
\(822\) 0 0
\(823\) −30.6941 −1.06993 −0.534964 0.844875i \(-0.679675\pi\)
−0.534964 + 0.844875i \(0.679675\pi\)
\(824\) 0 0
\(825\) −1.95558 −0.0680846
\(826\) 0 0
\(827\) 12.4038 0.431324 0.215662 0.976468i \(-0.430809\pi\)
0.215662 + 0.976468i \(0.430809\pi\)
\(828\) 0 0
\(829\) 37.4295 1.29998 0.649991 0.759942i \(-0.274773\pi\)
0.649991 + 0.759942i \(0.274773\pi\)
\(830\) 0 0
\(831\) 13.2044 0.458056
\(832\) 0 0
\(833\) −22.1299 −0.766757
\(834\) 0 0
\(835\) −22.4916 −0.778355
\(836\) 0 0
\(837\) −2.67669 −0.0925198
\(838\) 0 0
\(839\) −24.8905 −0.859314 −0.429657 0.902992i \(-0.641366\pi\)
−0.429657 + 0.902992i \(0.641366\pi\)
\(840\) 0 0
\(841\) −27.5341 −0.949451
\(842\) 0 0
\(843\) −19.7098 −0.678841
\(844\) 0 0
\(845\) 1.74483 0.0600238
\(846\) 0 0
\(847\) 4.53407 0.155792
\(848\) 0 0
\(849\) −26.1299 −0.896777
\(850\) 0 0
\(851\) −75.3468 −2.58285
\(852\) 0 0
\(853\) −27.1630 −0.930044 −0.465022 0.885299i \(-0.653954\pi\)
−0.465022 + 0.885299i \(0.653954\pi\)
\(854\) 0 0
\(855\) −0.931860 −0.0318689
\(856\) 0 0
\(857\) 15.0044 0.512539 0.256270 0.966605i \(-0.417507\pi\)
0.256270 + 0.966605i \(0.417507\pi\)
\(858\) 0 0
\(859\) −54.1236 −1.84667 −0.923337 0.383991i \(-0.874549\pi\)
−0.923337 + 0.383991i \(0.874549\pi\)
\(860\) 0 0
\(861\) −15.3120 −0.521831
\(862\) 0 0
\(863\) 26.2438 0.893351 0.446675 0.894696i \(-0.352608\pi\)
0.446675 + 0.894696i \(0.352608\pi\)
\(864\) 0 0
\(865\) −8.54407 −0.290507
\(866\) 0 0
\(867\) 14.3357 0.486866
\(868\) 0 0
\(869\) −1.72110 −0.0583844
\(870\) 0 0
\(871\) 12.5878 0.426523
\(872\) 0 0
\(873\) −2.93186 −0.0992284
\(874\) 0 0
\(875\) −55.0267 −1.86024
\(876\) 0 0
\(877\) 8.02872 0.271111 0.135555 0.990770i \(-0.456718\pi\)
0.135555 + 0.990770i \(0.456718\pi\)
\(878\) 0 0
\(879\) −5.97628 −0.201575
\(880\) 0 0
\(881\) −29.1944 −0.983585 −0.491793 0.870712i \(-0.663658\pi\)
−0.491793 + 0.870712i \(0.663658\pi\)
\(882\) 0 0
\(883\) 11.8223 0.397853 0.198927 0.980014i \(-0.436254\pi\)
0.198927 + 0.980014i \(0.436254\pi\)
\(884\) 0 0
\(885\) −4.61791 −0.155229
\(886\) 0 0
\(887\) 52.1236 1.75014 0.875070 0.483997i \(-0.160815\pi\)
0.875070 + 0.483997i \(0.160815\pi\)
\(888\) 0 0
\(889\) 33.7148 1.13076
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) 0 0
\(893\) −5.06814 −0.169599
\(894\) 0 0
\(895\) 17.0969 0.571485
\(896\) 0 0
\(897\) −7.60221 −0.253830
\(898\) 0 0
\(899\) −3.24081 −0.108087
\(900\) 0 0
\(901\) 5.84105 0.194594
\(902\) 0 0
\(903\) 16.4690 0.548053
\(904\) 0 0
\(905\) 14.0000 0.465376
\(906\) 0 0
\(907\) −56.6527 −1.88112 −0.940561 0.339626i \(-0.889700\pi\)
−0.940561 + 0.339626i \(0.889700\pi\)
\(908\) 0 0
\(909\) −13.1219 −0.435227
\(910\) 0 0
\(911\) 24.4276 0.809321 0.404661 0.914467i \(-0.367390\pi\)
0.404661 + 0.914467i \(0.367390\pi\)
\(912\) 0 0
\(913\) 1.35337 0.0447901
\(914\) 0 0
\(915\) −21.9112 −0.724361
\(916\) 0 0
\(917\) −68.9379 −2.27653
\(918\) 0 0
\(919\) −33.0144 −1.08904 −0.544522 0.838747i \(-0.683289\pi\)
−0.544522 + 0.838747i \(0.683289\pi\)
\(920\) 0 0
\(921\) −13.3771 −0.440790
\(922\) 0 0
\(923\) −3.06814 −0.100989
\(924\) 0 0
\(925\) 19.3821 0.637279
\(926\) 0 0
\(927\) 3.15698 0.103689
\(928\) 0 0
\(929\) −41.9286 −1.37563 −0.687816 0.725885i \(-0.741430\pi\)
−0.687816 + 0.725885i \(0.741430\pi\)
\(930\) 0 0
\(931\) −7.24081 −0.237308
\(932\) 0 0
\(933\) 25.5608 0.836824
\(934\) 0 0
\(935\) 2.84802 0.0931403
\(936\) 0 0
\(937\) −38.3614 −1.25321 −0.626606 0.779336i \(-0.715556\pi\)
−0.626606 + 0.779336i \(0.715556\pi\)
\(938\) 0 0
\(939\) −29.1994 −0.952887
\(940\) 0 0
\(941\) 32.2776 1.05222 0.526109 0.850417i \(-0.323650\pi\)
0.526109 + 0.850417i \(0.323650\pi\)
\(942\) 0 0
\(943\) 25.6734 0.836040
\(944\) 0 0
\(945\) −7.91116 −0.257350
\(946\) 0 0
\(947\) 0.243840 0.00792372 0.00396186 0.999992i \(-0.498739\pi\)
0.00396186 + 0.999992i \(0.498739\pi\)
\(948\) 0 0
\(949\) 4.95558 0.160865
\(950\) 0 0
\(951\) −9.94122 −0.322366
\(952\) 0 0
\(953\) 40.0725 1.29808 0.649038 0.760756i \(-0.275172\pi\)
0.649038 + 0.760756i \(0.275172\pi\)
\(954\) 0 0
\(955\) 21.5184 0.696318
\(956\) 0 0
\(957\) −1.21076 −0.0391382
\(958\) 0 0
\(959\) −75.2105 −2.42867
\(960\) 0 0
\(961\) −23.8354 −0.768882
\(962\) 0 0
\(963\) 18.4690 0.595154
\(964\) 0 0
\(965\) 14.5478 0.468310
\(966\) 0 0
\(967\) 51.5007 1.65615 0.828076 0.560617i \(-0.189436\pi\)
0.828076 + 0.560617i \(0.189436\pi\)
\(968\) 0 0
\(969\) −0.871746 −0.0280045
\(970\) 0 0
\(971\) −2.97430 −0.0954500 −0.0477250 0.998861i \(-0.515197\pi\)
−0.0477250 + 0.998861i \(0.515197\pi\)
\(972\) 0 0
\(973\) 14.5578 0.466701
\(974\) 0 0
\(975\) 1.95558 0.0626287
\(976\) 0 0
\(977\) −14.2966 −0.457388 −0.228694 0.973498i \(-0.573445\pi\)
−0.228694 + 0.973498i \(0.573445\pi\)
\(978\) 0 0
\(979\) −6.39145 −0.204272
\(980\) 0 0
\(981\) 6.86675 0.219238
\(982\) 0 0
\(983\) −4.61791 −0.147288 −0.0736442 0.997285i \(-0.523463\pi\)
−0.0736442 + 0.997285i \(0.523463\pi\)
\(984\) 0 0
\(985\) −5.31198 −0.169254
\(986\) 0 0
\(987\) −43.0267 −1.36956
\(988\) 0 0
\(989\) −27.6133 −0.878051
\(990\) 0 0
\(991\) 32.1550 1.02144 0.510719 0.859748i \(-0.329379\pi\)
0.510719 + 0.859748i \(0.329379\pi\)
\(992\) 0 0
\(993\) 6.87308 0.218111
\(994\) 0 0
\(995\) 10.7067 0.339427
\(996\) 0 0
\(997\) −50.6052 −1.60268 −0.801342 0.598207i \(-0.795880\pi\)
−0.801342 + 0.598207i \(0.795880\pi\)
\(998\) 0 0
\(999\) 9.91116 0.313575
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.bn.1.3 3
4.3 odd 2 3432.2.a.n.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3432.2.a.n.1.3 3 4.3 odd 2
6864.2.a.bn.1.3 3 1.1 even 1 trivial