Properties

Label 4410.2.a.bt.1.1
Level $4410$
Weight $2$
Character 4410.1
Self dual yes
Analytic conductor $35.214$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4410 = 2 \cdot 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4410.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(35.2140272914\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 490)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 4410.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{8} -1.00000 q^{10} -4.82843 q^{11} +0.828427 q^{13} +1.00000 q^{16} -5.41421 q^{17} +3.41421 q^{19} -1.00000 q^{20} -4.82843 q^{22} +6.82843 q^{23} +1.00000 q^{25} +0.828427 q^{26} -0.828427 q^{29} +2.82843 q^{31} +1.00000 q^{32} -5.41421 q^{34} +3.65685 q^{37} +3.41421 q^{38} -1.00000 q^{40} -11.0711 q^{41} -3.17157 q^{43} -4.82843 q^{44} +6.82843 q^{46} -10.8284 q^{47} +1.00000 q^{50} +0.828427 q^{52} +10.4853 q^{53} +4.82843 q^{55} -0.828427 q^{58} -11.4142 q^{59} -13.3137 q^{61} +2.82843 q^{62} +1.00000 q^{64} -0.828427 q^{65} -9.65685 q^{67} -5.41421 q^{68} -12.4853 q^{71} -6.58579 q^{73} +3.65685 q^{74} +3.41421 q^{76} -1.17157 q^{79} -1.00000 q^{80} -11.0711 q^{82} -6.24264 q^{83} +5.41421 q^{85} -3.17157 q^{86} -4.82843 q^{88} +12.7279 q^{89} +6.82843 q^{92} -10.8284 q^{94} -3.41421 q^{95} -16.2426 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{4} - 2q^{5} + 2q^{8} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{4} - 2q^{5} + 2q^{8} - 2q^{10} - 4q^{11} - 4q^{13} + 2q^{16} - 8q^{17} + 4q^{19} - 2q^{20} - 4q^{22} + 8q^{23} + 2q^{25} - 4q^{26} + 4q^{29} + 2q^{32} - 8q^{34} - 4q^{37} + 4q^{38} - 2q^{40} - 8q^{41} - 12q^{43} - 4q^{44} + 8q^{46} - 16q^{47} + 2q^{50} - 4q^{52} + 4q^{53} + 4q^{55} + 4q^{58} - 20q^{59} - 4q^{61} + 2q^{64} + 4q^{65} - 8q^{67} - 8q^{68} - 8q^{71} - 16q^{73} - 4q^{74} + 4q^{76} - 8q^{79} - 2q^{80} - 8q^{82} - 4q^{83} + 8q^{85} - 12q^{86} - 4q^{88} + 8q^{92} - 16q^{94} - 4q^{95} - 24q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −1.00000 −0.316228
\(11\) −4.82843 −1.45583 −0.727913 0.685670i \(-0.759509\pi\)
−0.727913 + 0.685670i \(0.759509\pi\)
\(12\) 0 0
\(13\) 0.828427 0.229764 0.114882 0.993379i \(-0.463351\pi\)
0.114882 + 0.993379i \(0.463351\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −5.41421 −1.31314 −0.656570 0.754265i \(-0.727993\pi\)
−0.656570 + 0.754265i \(0.727993\pi\)
\(18\) 0 0
\(19\) 3.41421 0.783274 0.391637 0.920120i \(-0.371909\pi\)
0.391637 + 0.920120i \(0.371909\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) −4.82843 −1.02942
\(23\) 6.82843 1.42383 0.711913 0.702268i \(-0.247829\pi\)
0.711913 + 0.702268i \(0.247829\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0.828427 0.162468
\(27\) 0 0
\(28\) 0 0
\(29\) −0.828427 −0.153835 −0.0769175 0.997037i \(-0.524508\pi\)
−0.0769175 + 0.997037i \(0.524508\pi\)
\(30\) 0 0
\(31\) 2.82843 0.508001 0.254000 0.967204i \(-0.418254\pi\)
0.254000 + 0.967204i \(0.418254\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −5.41421 −0.928530
\(35\) 0 0
\(36\) 0 0
\(37\) 3.65685 0.601183 0.300592 0.953753i \(-0.402816\pi\)
0.300592 + 0.953753i \(0.402816\pi\)
\(38\) 3.41421 0.553859
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) −11.0711 −1.72901 −0.864505 0.502624i \(-0.832368\pi\)
−0.864505 + 0.502624i \(0.832368\pi\)
\(42\) 0 0
\(43\) −3.17157 −0.483660 −0.241830 0.970319i \(-0.577748\pi\)
−0.241830 + 0.970319i \(0.577748\pi\)
\(44\) −4.82843 −0.727913
\(45\) 0 0
\(46\) 6.82843 1.00680
\(47\) −10.8284 −1.57949 −0.789744 0.613436i \(-0.789787\pi\)
−0.789744 + 0.613436i \(0.789787\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) 0.828427 0.114882
\(53\) 10.4853 1.44026 0.720132 0.693837i \(-0.244081\pi\)
0.720132 + 0.693837i \(0.244081\pi\)
\(54\) 0 0
\(55\) 4.82843 0.651065
\(56\) 0 0
\(57\) 0 0
\(58\) −0.828427 −0.108778
\(59\) −11.4142 −1.48600 −0.743002 0.669289i \(-0.766599\pi\)
−0.743002 + 0.669289i \(0.766599\pi\)
\(60\) 0 0
\(61\) −13.3137 −1.70465 −0.852323 0.523016i \(-0.824807\pi\)
−0.852323 + 0.523016i \(0.824807\pi\)
\(62\) 2.82843 0.359211
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −0.828427 −0.102754
\(66\) 0 0
\(67\) −9.65685 −1.17977 −0.589886 0.807486i \(-0.700827\pi\)
−0.589886 + 0.807486i \(0.700827\pi\)
\(68\) −5.41421 −0.656570
\(69\) 0 0
\(70\) 0 0
\(71\) −12.4853 −1.48173 −0.740865 0.671654i \(-0.765584\pi\)
−0.740865 + 0.671654i \(0.765584\pi\)
\(72\) 0 0
\(73\) −6.58579 −0.770808 −0.385404 0.922748i \(-0.625938\pi\)
−0.385404 + 0.922748i \(0.625938\pi\)
\(74\) 3.65685 0.425101
\(75\) 0 0
\(76\) 3.41421 0.391637
\(77\) 0 0
\(78\) 0 0
\(79\) −1.17157 −0.131812 −0.0659061 0.997826i \(-0.520994\pi\)
−0.0659061 + 0.997826i \(0.520994\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) −11.0711 −1.22259
\(83\) −6.24264 −0.685219 −0.342609 0.939478i \(-0.611311\pi\)
−0.342609 + 0.939478i \(0.611311\pi\)
\(84\) 0 0
\(85\) 5.41421 0.587254
\(86\) −3.17157 −0.341999
\(87\) 0 0
\(88\) −4.82843 −0.514712
\(89\) 12.7279 1.34916 0.674579 0.738203i \(-0.264325\pi\)
0.674579 + 0.738203i \(0.264325\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 6.82843 0.711913
\(93\) 0 0
\(94\) −10.8284 −1.11687
\(95\) −3.41421 −0.350291
\(96\) 0 0
\(97\) −16.2426 −1.64919 −0.824595 0.565723i \(-0.808597\pi\)
−0.824595 + 0.565723i \(0.808597\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 9.31371 0.926749 0.463374 0.886163i \(-0.346639\pi\)
0.463374 + 0.886163i \(0.346639\pi\)
\(102\) 0 0
\(103\) −9.17157 −0.903702 −0.451851 0.892093i \(-0.649236\pi\)
−0.451851 + 0.892093i \(0.649236\pi\)
\(104\) 0.828427 0.0812340
\(105\) 0 0
\(106\) 10.4853 1.01842
\(107\) 1.65685 0.160174 0.0800871 0.996788i \(-0.474480\pi\)
0.0800871 + 0.996788i \(0.474480\pi\)
\(108\) 0 0
\(109\) −14.4853 −1.38744 −0.693719 0.720246i \(-0.744029\pi\)
−0.693719 + 0.720246i \(0.744029\pi\)
\(110\) 4.82843 0.460372
\(111\) 0 0
\(112\) 0 0
\(113\) −7.31371 −0.688016 −0.344008 0.938967i \(-0.611785\pi\)
−0.344008 + 0.938967i \(0.611785\pi\)
\(114\) 0 0
\(115\) −6.82843 −0.636754
\(116\) −0.828427 −0.0769175
\(117\) 0 0
\(118\) −11.4142 −1.05076
\(119\) 0 0
\(120\) 0 0
\(121\) 12.3137 1.11943
\(122\) −13.3137 −1.20537
\(123\) 0 0
\(124\) 2.82843 0.254000
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −2.82843 −0.250982 −0.125491 0.992095i \(-0.540051\pi\)
−0.125491 + 0.992095i \(0.540051\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) −0.828427 −0.0726579
\(131\) −2.24264 −0.195940 −0.0979702 0.995189i \(-0.531235\pi\)
−0.0979702 + 0.995189i \(0.531235\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −9.65685 −0.834225
\(135\) 0 0
\(136\) −5.41421 −0.464265
\(137\) 16.0000 1.36697 0.683486 0.729964i \(-0.260463\pi\)
0.683486 + 0.729964i \(0.260463\pi\)
\(138\) 0 0
\(139\) 0.100505 0.00852473 0.00426236 0.999991i \(-0.498643\pi\)
0.00426236 + 0.999991i \(0.498643\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −12.4853 −1.04774
\(143\) −4.00000 −0.334497
\(144\) 0 0
\(145\) 0.828427 0.0687971
\(146\) −6.58579 −0.545044
\(147\) 0 0
\(148\) 3.65685 0.300592
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) 11.3137 0.920697 0.460348 0.887738i \(-0.347725\pi\)
0.460348 + 0.887738i \(0.347725\pi\)
\(152\) 3.41421 0.276929
\(153\) 0 0
\(154\) 0 0
\(155\) −2.82843 −0.227185
\(156\) 0 0
\(157\) −10.4853 −0.836817 −0.418408 0.908259i \(-0.637412\pi\)
−0.418408 + 0.908259i \(0.637412\pi\)
\(158\) −1.17157 −0.0932053
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) 0 0
\(162\) 0 0
\(163\) 8.14214 0.637741 0.318871 0.947798i \(-0.396696\pi\)
0.318871 + 0.947798i \(0.396696\pi\)
\(164\) −11.0711 −0.864505
\(165\) 0 0
\(166\) −6.24264 −0.484523
\(167\) 23.7990 1.84162 0.920811 0.390010i \(-0.127529\pi\)
0.920811 + 0.390010i \(0.127529\pi\)
\(168\) 0 0
\(169\) −12.3137 −0.947208
\(170\) 5.41421 0.415251
\(171\) 0 0
\(172\) −3.17157 −0.241830
\(173\) 3.17157 0.241130 0.120565 0.992705i \(-0.461529\pi\)
0.120565 + 0.992705i \(0.461529\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −4.82843 −0.363956
\(177\) 0 0
\(178\) 12.7279 0.953998
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 0 0
\(181\) 14.4853 1.07668 0.538341 0.842727i \(-0.319051\pi\)
0.538341 + 0.842727i \(0.319051\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 6.82843 0.503398
\(185\) −3.65685 −0.268857
\(186\) 0 0
\(187\) 26.1421 1.91170
\(188\) −10.8284 −0.789744
\(189\) 0 0
\(190\) −3.41421 −0.247693
\(191\) 18.1421 1.31272 0.656359 0.754448i \(-0.272096\pi\)
0.656359 + 0.754448i \(0.272096\pi\)
\(192\) 0 0
\(193\) −5.65685 −0.407189 −0.203595 0.979055i \(-0.565262\pi\)
−0.203595 + 0.979055i \(0.565262\pi\)
\(194\) −16.2426 −1.16615
\(195\) 0 0
\(196\) 0 0
\(197\) −13.7990 −0.983137 −0.491569 0.870839i \(-0.663576\pi\)
−0.491569 + 0.870839i \(0.663576\pi\)
\(198\) 0 0
\(199\) −0.485281 −0.0344007 −0.0172003 0.999852i \(-0.505475\pi\)
−0.0172003 + 0.999852i \(0.505475\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0 0
\(202\) 9.31371 0.655310
\(203\) 0 0
\(204\) 0 0
\(205\) 11.0711 0.773237
\(206\) −9.17157 −0.639014
\(207\) 0 0
\(208\) 0.828427 0.0574411
\(209\) −16.4853 −1.14031
\(210\) 0 0
\(211\) −26.6274 −1.83311 −0.916553 0.399912i \(-0.869041\pi\)
−0.916553 + 0.399912i \(0.869041\pi\)
\(212\) 10.4853 0.720132
\(213\) 0 0
\(214\) 1.65685 0.113260
\(215\) 3.17157 0.216299
\(216\) 0 0
\(217\) 0 0
\(218\) −14.4853 −0.981067
\(219\) 0 0
\(220\) 4.82843 0.325532
\(221\) −4.48528 −0.301713
\(222\) 0 0
\(223\) 15.3137 1.02548 0.512741 0.858543i \(-0.328630\pi\)
0.512741 + 0.858543i \(0.328630\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −7.31371 −0.486501
\(227\) −9.75736 −0.647619 −0.323809 0.946122i \(-0.604964\pi\)
−0.323809 + 0.946122i \(0.604964\pi\)
\(228\) 0 0
\(229\) 12.1421 0.802375 0.401187 0.915996i \(-0.368598\pi\)
0.401187 + 0.915996i \(0.368598\pi\)
\(230\) −6.82843 −0.450253
\(231\) 0 0
\(232\) −0.828427 −0.0543889
\(233\) −0.686292 −0.0449605 −0.0224802 0.999747i \(-0.507156\pi\)
−0.0224802 + 0.999747i \(0.507156\pi\)
\(234\) 0 0
\(235\) 10.8284 0.706369
\(236\) −11.4142 −0.743002
\(237\) 0 0
\(238\) 0 0
\(239\) 9.65685 0.624650 0.312325 0.949975i \(-0.398892\pi\)
0.312325 + 0.949975i \(0.398892\pi\)
\(240\) 0 0
\(241\) −10.5858 −0.681890 −0.340945 0.940083i \(-0.610747\pi\)
−0.340945 + 0.940083i \(0.610747\pi\)
\(242\) 12.3137 0.791555
\(243\) 0 0
\(244\) −13.3137 −0.852323
\(245\) 0 0
\(246\) 0 0
\(247\) 2.82843 0.179969
\(248\) 2.82843 0.179605
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) −3.41421 −0.215503 −0.107752 0.994178i \(-0.534365\pi\)
−0.107752 + 0.994178i \(0.534365\pi\)
\(252\) 0 0
\(253\) −32.9706 −2.07284
\(254\) −2.82843 −0.177471
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −9.89949 −0.617514 −0.308757 0.951141i \(-0.599913\pi\)
−0.308757 + 0.951141i \(0.599913\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −0.828427 −0.0513769
\(261\) 0 0
\(262\) −2.24264 −0.138551
\(263\) −28.0000 −1.72655 −0.863277 0.504730i \(-0.831592\pi\)
−0.863277 + 0.504730i \(0.831592\pi\)
\(264\) 0 0
\(265\) −10.4853 −0.644106
\(266\) 0 0
\(267\) 0 0
\(268\) −9.65685 −0.589886
\(269\) −1.51472 −0.0923540 −0.0461770 0.998933i \(-0.514704\pi\)
−0.0461770 + 0.998933i \(0.514704\pi\)
\(270\) 0 0
\(271\) 12.0000 0.728948 0.364474 0.931214i \(-0.381249\pi\)
0.364474 + 0.931214i \(0.381249\pi\)
\(272\) −5.41421 −0.328285
\(273\) 0 0
\(274\) 16.0000 0.966595
\(275\) −4.82843 −0.291165
\(276\) 0 0
\(277\) 20.1421 1.21022 0.605112 0.796140i \(-0.293128\pi\)
0.605112 + 0.796140i \(0.293128\pi\)
\(278\) 0.100505 0.00602789
\(279\) 0 0
\(280\) 0 0
\(281\) −8.00000 −0.477240 −0.238620 0.971113i \(-0.576695\pi\)
−0.238620 + 0.971113i \(0.576695\pi\)
\(282\) 0 0
\(283\) 6.24264 0.371086 0.185543 0.982636i \(-0.440596\pi\)
0.185543 + 0.982636i \(0.440596\pi\)
\(284\) −12.4853 −0.740865
\(285\) 0 0
\(286\) −4.00000 −0.236525
\(287\) 0 0
\(288\) 0 0
\(289\) 12.3137 0.724336
\(290\) 0.828427 0.0486469
\(291\) 0 0
\(292\) −6.58579 −0.385404
\(293\) 19.6569 1.14837 0.574183 0.818727i \(-0.305320\pi\)
0.574183 + 0.818727i \(0.305320\pi\)
\(294\) 0 0
\(295\) 11.4142 0.664561
\(296\) 3.65685 0.212550
\(297\) 0 0
\(298\) 6.00000 0.347571
\(299\) 5.65685 0.327144
\(300\) 0 0
\(301\) 0 0
\(302\) 11.3137 0.651031
\(303\) 0 0
\(304\) 3.41421 0.195819
\(305\) 13.3137 0.762341
\(306\) 0 0
\(307\) 29.0711 1.65917 0.829587 0.558378i \(-0.188576\pi\)
0.829587 + 0.558378i \(0.188576\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −2.82843 −0.160644
\(311\) −4.00000 −0.226819 −0.113410 0.993548i \(-0.536177\pi\)
−0.113410 + 0.993548i \(0.536177\pi\)
\(312\) 0 0
\(313\) −22.3848 −1.26526 −0.632631 0.774453i \(-0.718025\pi\)
−0.632631 + 0.774453i \(0.718025\pi\)
\(314\) −10.4853 −0.591719
\(315\) 0 0
\(316\) −1.17157 −0.0659061
\(317\) 6.48528 0.364250 0.182125 0.983275i \(-0.441702\pi\)
0.182125 + 0.983275i \(0.441702\pi\)
\(318\) 0 0
\(319\) 4.00000 0.223957
\(320\) −1.00000 −0.0559017
\(321\) 0 0
\(322\) 0 0
\(323\) −18.4853 −1.02855
\(324\) 0 0
\(325\) 0.828427 0.0459529
\(326\) 8.14214 0.450951
\(327\) 0 0
\(328\) −11.0711 −0.611297
\(329\) 0 0
\(330\) 0 0
\(331\) −5.79899 −0.318741 −0.159371 0.987219i \(-0.550946\pi\)
−0.159371 + 0.987219i \(0.550946\pi\)
\(332\) −6.24264 −0.342609
\(333\) 0 0
\(334\) 23.7990 1.30222
\(335\) 9.65685 0.527610
\(336\) 0 0
\(337\) 6.00000 0.326841 0.163420 0.986557i \(-0.447747\pi\)
0.163420 + 0.986557i \(0.447747\pi\)
\(338\) −12.3137 −0.669777
\(339\) 0 0
\(340\) 5.41421 0.293627
\(341\) −13.6569 −0.739560
\(342\) 0 0
\(343\) 0 0
\(344\) −3.17157 −0.171000
\(345\) 0 0
\(346\) 3.17157 0.170505
\(347\) 8.82843 0.473935 0.236967 0.971518i \(-0.423847\pi\)
0.236967 + 0.971518i \(0.423847\pi\)
\(348\) 0 0
\(349\) −14.4853 −0.775379 −0.387690 0.921790i \(-0.626727\pi\)
−0.387690 + 0.921790i \(0.626727\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −4.82843 −0.257356
\(353\) −34.3848 −1.83012 −0.915058 0.403321i \(-0.867856\pi\)
−0.915058 + 0.403321i \(0.867856\pi\)
\(354\) 0 0
\(355\) 12.4853 0.662650
\(356\) 12.7279 0.674579
\(357\) 0 0
\(358\) −4.00000 −0.211407
\(359\) 28.2843 1.49279 0.746393 0.665505i \(-0.231784\pi\)
0.746393 + 0.665505i \(0.231784\pi\)
\(360\) 0 0
\(361\) −7.34315 −0.386481
\(362\) 14.4853 0.761329
\(363\) 0 0
\(364\) 0 0
\(365\) 6.58579 0.344716
\(366\) 0 0
\(367\) −8.97056 −0.468260 −0.234130 0.972205i \(-0.575224\pi\)
−0.234130 + 0.972205i \(0.575224\pi\)
\(368\) 6.82843 0.355956
\(369\) 0 0
\(370\) −3.65685 −0.190111
\(371\) 0 0
\(372\) 0 0
\(373\) −13.5147 −0.699766 −0.349883 0.936793i \(-0.613779\pi\)
−0.349883 + 0.936793i \(0.613779\pi\)
\(374\) 26.1421 1.35178
\(375\) 0 0
\(376\) −10.8284 −0.558433
\(377\) −0.686292 −0.0353458
\(378\) 0 0
\(379\) 17.5147 0.899671 0.449835 0.893112i \(-0.351483\pi\)
0.449835 + 0.893112i \(0.351483\pi\)
\(380\) −3.41421 −0.175145
\(381\) 0 0
\(382\) 18.1421 0.928232
\(383\) 15.5147 0.792765 0.396383 0.918085i \(-0.370265\pi\)
0.396383 + 0.918085i \(0.370265\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −5.65685 −0.287926
\(387\) 0 0
\(388\) −16.2426 −0.824595
\(389\) 0.142136 0.00720656 0.00360328 0.999994i \(-0.498853\pi\)
0.00360328 + 0.999994i \(0.498853\pi\)
\(390\) 0 0
\(391\) −36.9706 −1.86968
\(392\) 0 0
\(393\) 0 0
\(394\) −13.7990 −0.695183
\(395\) 1.17157 0.0589482
\(396\) 0 0
\(397\) −5.79899 −0.291043 −0.145521 0.989355i \(-0.546486\pi\)
−0.145521 + 0.989355i \(0.546486\pi\)
\(398\) −0.485281 −0.0243250
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 6.00000 0.299626 0.149813 0.988714i \(-0.452133\pi\)
0.149813 + 0.988714i \(0.452133\pi\)
\(402\) 0 0
\(403\) 2.34315 0.116720
\(404\) 9.31371 0.463374
\(405\) 0 0
\(406\) 0 0
\(407\) −17.6569 −0.875218
\(408\) 0 0
\(409\) −13.4142 −0.663290 −0.331645 0.943404i \(-0.607604\pi\)
−0.331645 + 0.943404i \(0.607604\pi\)
\(410\) 11.0711 0.546761
\(411\) 0 0
\(412\) −9.17157 −0.451851
\(413\) 0 0
\(414\) 0 0
\(415\) 6.24264 0.306439
\(416\) 0.828427 0.0406170
\(417\) 0 0
\(418\) −16.4853 −0.806321
\(419\) −32.8701 −1.60581 −0.802904 0.596109i \(-0.796713\pi\)
−0.802904 + 0.596109i \(0.796713\pi\)
\(420\) 0 0
\(421\) −5.31371 −0.258974 −0.129487 0.991581i \(-0.541333\pi\)
−0.129487 + 0.991581i \(0.541333\pi\)
\(422\) −26.6274 −1.29620
\(423\) 0 0
\(424\) 10.4853 0.509210
\(425\) −5.41421 −0.262628
\(426\) 0 0
\(427\) 0 0
\(428\) 1.65685 0.0800871
\(429\) 0 0
\(430\) 3.17157 0.152947
\(431\) 33.6569 1.62119 0.810597 0.585605i \(-0.199143\pi\)
0.810597 + 0.585605i \(0.199143\pi\)
\(432\) 0 0
\(433\) 13.4142 0.644646 0.322323 0.946630i \(-0.395536\pi\)
0.322323 + 0.946630i \(0.395536\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −14.4853 −0.693719
\(437\) 23.3137 1.11525
\(438\) 0 0
\(439\) 8.97056 0.428142 0.214071 0.976818i \(-0.431328\pi\)
0.214071 + 0.976818i \(0.431328\pi\)
\(440\) 4.82843 0.230186
\(441\) 0 0
\(442\) −4.48528 −0.213343
\(443\) −36.9706 −1.75652 −0.878262 0.478179i \(-0.841297\pi\)
−0.878262 + 0.478179i \(0.841297\pi\)
\(444\) 0 0
\(445\) −12.7279 −0.603361
\(446\) 15.3137 0.725125
\(447\) 0 0
\(448\) 0 0
\(449\) −28.6274 −1.35101 −0.675506 0.737355i \(-0.736075\pi\)
−0.675506 + 0.737355i \(0.736075\pi\)
\(450\) 0 0
\(451\) 53.4558 2.51714
\(452\) −7.31371 −0.344008
\(453\) 0 0
\(454\) −9.75736 −0.457936
\(455\) 0 0
\(456\) 0 0
\(457\) 10.3431 0.483832 0.241916 0.970297i \(-0.422224\pi\)
0.241916 + 0.970297i \(0.422224\pi\)
\(458\) 12.1421 0.567365
\(459\) 0 0
\(460\) −6.82843 −0.318377
\(461\) 7.17157 0.334013 0.167007 0.985956i \(-0.446590\pi\)
0.167007 + 0.985956i \(0.446590\pi\)
\(462\) 0 0
\(463\) 16.9706 0.788689 0.394344 0.918963i \(-0.370972\pi\)
0.394344 + 0.918963i \(0.370972\pi\)
\(464\) −0.828427 −0.0384588
\(465\) 0 0
\(466\) −0.686292 −0.0317918
\(467\) −3.89949 −0.180447 −0.0902236 0.995922i \(-0.528758\pi\)
−0.0902236 + 0.995922i \(0.528758\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 10.8284 0.499478
\(471\) 0 0
\(472\) −11.4142 −0.525382
\(473\) 15.3137 0.704125
\(474\) 0 0
\(475\) 3.41421 0.156655
\(476\) 0 0
\(477\) 0 0
\(478\) 9.65685 0.441694
\(479\) 22.8284 1.04306 0.521529 0.853234i \(-0.325362\pi\)
0.521529 + 0.853234i \(0.325362\pi\)
\(480\) 0 0
\(481\) 3.02944 0.138130
\(482\) −10.5858 −0.482169
\(483\) 0 0
\(484\) 12.3137 0.559714
\(485\) 16.2426 0.737540
\(486\) 0 0
\(487\) −7.79899 −0.353406 −0.176703 0.984264i \(-0.556543\pi\)
−0.176703 + 0.984264i \(0.556543\pi\)
\(488\) −13.3137 −0.602683
\(489\) 0 0
\(490\) 0 0
\(491\) 24.2843 1.09593 0.547967 0.836500i \(-0.315402\pi\)
0.547967 + 0.836500i \(0.315402\pi\)
\(492\) 0 0
\(493\) 4.48528 0.202007
\(494\) 2.82843 0.127257
\(495\) 0 0
\(496\) 2.82843 0.127000
\(497\) 0 0
\(498\) 0 0
\(499\) 41.6569 1.86482 0.932408 0.361406i \(-0.117703\pi\)
0.932408 + 0.361406i \(0.117703\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) −3.41421 −0.152384
\(503\) −6.34315 −0.282827 −0.141413 0.989951i \(-0.545165\pi\)
−0.141413 + 0.989951i \(0.545165\pi\)
\(504\) 0 0
\(505\) −9.31371 −0.414455
\(506\) −32.9706 −1.46572
\(507\) 0 0
\(508\) −2.82843 −0.125491
\(509\) 33.7990 1.49811 0.749057 0.662506i \(-0.230507\pi\)
0.749057 + 0.662506i \(0.230507\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −9.89949 −0.436648
\(515\) 9.17157 0.404148
\(516\) 0 0
\(517\) 52.2843 2.29946
\(518\) 0 0
\(519\) 0 0
\(520\) −0.828427 −0.0363289
\(521\) −4.92893 −0.215940 −0.107970 0.994154i \(-0.534435\pi\)
−0.107970 + 0.994154i \(0.534435\pi\)
\(522\) 0 0
\(523\) 4.10051 0.179303 0.0896513 0.995973i \(-0.471425\pi\)
0.0896513 + 0.995973i \(0.471425\pi\)
\(524\) −2.24264 −0.0979702
\(525\) 0 0
\(526\) −28.0000 −1.22086
\(527\) −15.3137 −0.667076
\(528\) 0 0
\(529\) 23.6274 1.02728
\(530\) −10.4853 −0.455452
\(531\) 0 0
\(532\) 0 0
\(533\) −9.17157 −0.397265
\(534\) 0 0
\(535\) −1.65685 −0.0716321
\(536\) −9.65685 −0.417113
\(537\) 0 0
\(538\) −1.51472 −0.0653042
\(539\) 0 0
\(540\) 0 0
\(541\) 18.9706 0.815608 0.407804 0.913069i \(-0.366295\pi\)
0.407804 + 0.913069i \(0.366295\pi\)
\(542\) 12.0000 0.515444
\(543\) 0 0
\(544\) −5.41421 −0.232132
\(545\) 14.4853 0.620481
\(546\) 0 0
\(547\) 6.48528 0.277291 0.138645 0.990342i \(-0.455725\pi\)
0.138645 + 0.990342i \(0.455725\pi\)
\(548\) 16.0000 0.683486
\(549\) 0 0
\(550\) −4.82843 −0.205885
\(551\) −2.82843 −0.120495
\(552\) 0 0
\(553\) 0 0
\(554\) 20.1421 0.855757
\(555\) 0 0
\(556\) 0.100505 0.00426236
\(557\) −20.8284 −0.882529 −0.441264 0.897377i \(-0.645470\pi\)
−0.441264 + 0.897377i \(0.645470\pi\)
\(558\) 0 0
\(559\) −2.62742 −0.111128
\(560\) 0 0
\(561\) 0 0
\(562\) −8.00000 −0.337460
\(563\) 39.4142 1.66111 0.830556 0.556936i \(-0.188023\pi\)
0.830556 + 0.556936i \(0.188023\pi\)
\(564\) 0 0
\(565\) 7.31371 0.307690
\(566\) 6.24264 0.262398
\(567\) 0 0
\(568\) −12.4853 −0.523871
\(569\) 6.68629 0.280304 0.140152 0.990130i \(-0.455241\pi\)
0.140152 + 0.990130i \(0.455241\pi\)
\(570\) 0 0
\(571\) −41.7990 −1.74923 −0.874617 0.484815i \(-0.838887\pi\)
−0.874617 + 0.484815i \(0.838887\pi\)
\(572\) −4.00000 −0.167248
\(573\) 0 0
\(574\) 0 0
\(575\) 6.82843 0.284765
\(576\) 0 0
\(577\) 25.8995 1.07821 0.539105 0.842239i \(-0.318763\pi\)
0.539105 + 0.842239i \(0.318763\pi\)
\(578\) 12.3137 0.512183
\(579\) 0 0
\(580\) 0.828427 0.0343986
\(581\) 0 0
\(582\) 0 0
\(583\) −50.6274 −2.09677
\(584\) −6.58579 −0.272522
\(585\) 0 0
\(586\) 19.6569 0.812017
\(587\) −2.92893 −0.120890 −0.0604450 0.998172i \(-0.519252\pi\)
−0.0604450 + 0.998172i \(0.519252\pi\)
\(588\) 0 0
\(589\) 9.65685 0.397904
\(590\) 11.4142 0.469916
\(591\) 0 0
\(592\) 3.65685 0.150296
\(593\) −28.7279 −1.17971 −0.589857 0.807508i \(-0.700816\pi\)
−0.589857 + 0.807508i \(0.700816\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 6.00000 0.245770
\(597\) 0 0
\(598\) 5.65685 0.231326
\(599\) 5.17157 0.211305 0.105652 0.994403i \(-0.466307\pi\)
0.105652 + 0.994403i \(0.466307\pi\)
\(600\) 0 0
\(601\) 9.41421 0.384014 0.192007 0.981394i \(-0.438500\pi\)
0.192007 + 0.981394i \(0.438500\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 11.3137 0.460348
\(605\) −12.3137 −0.500623
\(606\) 0 0
\(607\) −40.2843 −1.63509 −0.817544 0.575866i \(-0.804665\pi\)
−0.817544 + 0.575866i \(0.804665\pi\)
\(608\) 3.41421 0.138465
\(609\) 0 0
\(610\) 13.3137 0.539056
\(611\) −8.97056 −0.362910
\(612\) 0 0
\(613\) −23.6569 −0.955491 −0.477746 0.878498i \(-0.658546\pi\)
−0.477746 + 0.878498i \(0.658546\pi\)
\(614\) 29.0711 1.17321
\(615\) 0 0
\(616\) 0 0
\(617\) 10.6863 0.430214 0.215107 0.976590i \(-0.430990\pi\)
0.215107 + 0.976590i \(0.430990\pi\)
\(618\) 0 0
\(619\) 14.9289 0.600044 0.300022 0.953932i \(-0.403006\pi\)
0.300022 + 0.953932i \(0.403006\pi\)
\(620\) −2.82843 −0.113592
\(621\) 0 0
\(622\) −4.00000 −0.160385
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −22.3848 −0.894676
\(627\) 0 0
\(628\) −10.4853 −0.418408
\(629\) −19.7990 −0.789437
\(630\) 0 0
\(631\) 4.48528 0.178556 0.0892781 0.996007i \(-0.471544\pi\)
0.0892781 + 0.996007i \(0.471544\pi\)
\(632\) −1.17157 −0.0466027
\(633\) 0 0
\(634\) 6.48528 0.257563
\(635\) 2.82843 0.112243
\(636\) 0 0
\(637\) 0 0
\(638\) 4.00000 0.158362
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) −20.6274 −0.814734 −0.407367 0.913265i \(-0.633553\pi\)
−0.407367 + 0.913265i \(0.633553\pi\)
\(642\) 0 0
\(643\) 47.2132 1.86191 0.930953 0.365138i \(-0.118978\pi\)
0.930953 + 0.365138i \(0.118978\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −18.4853 −0.727294
\(647\) −39.1127 −1.53768 −0.768839 0.639442i \(-0.779165\pi\)
−0.768839 + 0.639442i \(0.779165\pi\)
\(648\) 0 0
\(649\) 55.1127 2.16336
\(650\) 0.828427 0.0324936
\(651\) 0 0
\(652\) 8.14214 0.318871
\(653\) −15.6569 −0.612700 −0.306350 0.951919i \(-0.599108\pi\)
−0.306350 + 0.951919i \(0.599108\pi\)
\(654\) 0 0
\(655\) 2.24264 0.0876272
\(656\) −11.0711 −0.432253
\(657\) 0 0
\(658\) 0 0
\(659\) 32.8284 1.27881 0.639407 0.768868i \(-0.279180\pi\)
0.639407 + 0.768868i \(0.279180\pi\)
\(660\) 0 0
\(661\) 18.2843 0.711176 0.355588 0.934643i \(-0.384281\pi\)
0.355588 + 0.934643i \(0.384281\pi\)
\(662\) −5.79899 −0.225384
\(663\) 0 0
\(664\) −6.24264 −0.242261
\(665\) 0 0
\(666\) 0 0
\(667\) −5.65685 −0.219034
\(668\) 23.7990 0.920811
\(669\) 0 0
\(670\) 9.65685 0.373077
\(671\) 64.2843 2.48167
\(672\) 0 0
\(673\) −48.0000 −1.85026 −0.925132 0.379646i \(-0.876046\pi\)
−0.925132 + 0.379646i \(0.876046\pi\)
\(674\) 6.00000 0.231111
\(675\) 0 0
\(676\) −12.3137 −0.473604
\(677\) 11.4558 0.440284 0.220142 0.975468i \(-0.429348\pi\)
0.220142 + 0.975468i \(0.429348\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 5.41421 0.207626
\(681\) 0 0
\(682\) −13.6569 −0.522948
\(683\) −22.3431 −0.854937 −0.427468 0.904030i \(-0.640594\pi\)
−0.427468 + 0.904030i \(0.640594\pi\)
\(684\) 0 0
\(685\) −16.0000 −0.611329
\(686\) 0 0
\(687\) 0 0
\(688\) −3.17157 −0.120915
\(689\) 8.68629 0.330921
\(690\) 0 0
\(691\) 10.2426 0.389648 0.194824 0.980838i \(-0.437586\pi\)
0.194824 + 0.980838i \(0.437586\pi\)
\(692\) 3.17157 0.120565
\(693\) 0 0
\(694\) 8.82843 0.335123
\(695\) −0.100505 −0.00381237
\(696\) 0 0
\(697\) 59.9411 2.27043
\(698\) −14.4853 −0.548276
\(699\) 0 0
\(700\) 0 0
\(701\) 14.4853 0.547102 0.273551 0.961858i \(-0.411802\pi\)
0.273551 + 0.961858i \(0.411802\pi\)
\(702\) 0 0
\(703\) 12.4853 0.470891
\(704\) −4.82843 −0.181978
\(705\) 0 0
\(706\) −34.3848 −1.29409
\(707\) 0 0
\(708\) 0 0
\(709\) −17.1127 −0.642681 −0.321340 0.946964i \(-0.604133\pi\)
−0.321340 + 0.946964i \(0.604133\pi\)
\(710\) 12.4853 0.468564
\(711\) 0 0
\(712\) 12.7279 0.476999
\(713\) 19.3137 0.723304
\(714\) 0 0
\(715\) 4.00000 0.149592
\(716\) −4.00000 −0.149487
\(717\) 0 0
\(718\) 28.2843 1.05556
\(719\) 9.45584 0.352643 0.176322 0.984333i \(-0.443580\pi\)
0.176322 + 0.984333i \(0.443580\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −7.34315 −0.273284
\(723\) 0 0
\(724\) 14.4853 0.538341
\(725\) −0.828427 −0.0307670
\(726\) 0 0
\(727\) −20.4853 −0.759757 −0.379879 0.925036i \(-0.624034\pi\)
−0.379879 + 0.925036i \(0.624034\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 6.58579 0.243751
\(731\) 17.1716 0.635114
\(732\) 0 0
\(733\) 34.0000 1.25582 0.627909 0.778287i \(-0.283911\pi\)
0.627909 + 0.778287i \(0.283911\pi\)
\(734\) −8.97056 −0.331110
\(735\) 0 0
\(736\) 6.82843 0.251699
\(737\) 46.6274 1.71754
\(738\) 0 0
\(739\) 8.82843 0.324759 0.162379 0.986728i \(-0.448083\pi\)
0.162379 + 0.986728i \(0.448083\pi\)
\(740\) −3.65685 −0.134429
\(741\) 0 0
\(742\) 0 0
\(743\) −12.2010 −0.447612 −0.223806 0.974634i \(-0.571848\pi\)
−0.223806 + 0.974634i \(0.571848\pi\)
\(744\) 0 0
\(745\) −6.00000 −0.219823
\(746\) −13.5147 −0.494809
\(747\) 0 0
\(748\) 26.1421 0.955851
\(749\) 0 0
\(750\) 0 0
\(751\) −16.6863 −0.608891 −0.304446 0.952530i \(-0.598471\pi\)
−0.304446 + 0.952530i \(0.598471\pi\)
\(752\) −10.8284 −0.394872
\(753\) 0 0
\(754\) −0.686292 −0.0249933
\(755\) −11.3137 −0.411748
\(756\) 0 0
\(757\) −7.65685 −0.278293 −0.139147 0.990272i \(-0.544436\pi\)
−0.139147 + 0.990272i \(0.544436\pi\)
\(758\) 17.5147 0.636163
\(759\) 0 0
\(760\) −3.41421 −0.123847
\(761\) 14.3848 0.521448 0.260724 0.965413i \(-0.416039\pi\)
0.260724 + 0.965413i \(0.416039\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 18.1421 0.656359
\(765\) 0 0
\(766\) 15.5147 0.560570
\(767\) −9.45584 −0.341431
\(768\) 0 0
\(769\) −11.5563 −0.416733 −0.208366 0.978051i \(-0.566815\pi\)
−0.208366 + 0.978051i \(0.566815\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −5.65685 −0.203595
\(773\) 2.00000 0.0719350 0.0359675 0.999353i \(-0.488549\pi\)
0.0359675 + 0.999353i \(0.488549\pi\)
\(774\) 0 0
\(775\) 2.82843 0.101600
\(776\) −16.2426 −0.583077
\(777\) 0 0
\(778\) 0.142136 0.00509581
\(779\) −37.7990 −1.35429
\(780\) 0 0
\(781\) 60.2843 2.15714
\(782\) −36.9706 −1.32206
\(783\) 0 0
\(784\) 0 0
\(785\) 10.4853 0.374236
\(786\) 0 0
\(787\) −26.7279 −0.952748 −0.476374 0.879243i \(-0.658049\pi\)
−0.476374 + 0.879243i \(0.658049\pi\)
\(788\) −13.7990 −0.491569
\(789\) 0 0
\(790\) 1.17157 0.0416827
\(791\) 0 0
\(792\) 0 0
\(793\) −11.0294 −0.391667
\(794\) −5.79899 −0.205798
\(795\) 0 0
\(796\) −0.485281 −0.0172003
\(797\) 2.20101 0.0779638 0.0389819 0.999240i \(-0.487589\pi\)
0.0389819 + 0.999240i \(0.487589\pi\)
\(798\) 0 0
\(799\) 58.6274 2.07409
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) 6.00000 0.211867
\(803\) 31.7990 1.12216
\(804\) 0 0
\(805\) 0 0
\(806\) 2.34315 0.0825338
\(807\) 0 0
\(808\) 9.31371 0.327655
\(809\) −36.9706 −1.29982 −0.649908 0.760013i \(-0.725193\pi\)
−0.649908 + 0.760013i \(0.725193\pi\)
\(810\) 0 0
\(811\) 35.4142 1.24356 0.621781 0.783191i \(-0.286410\pi\)
0.621781 + 0.783191i \(0.286410\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −17.6569 −0.618872
\(815\) −8.14214 −0.285207
\(816\) 0 0
\(817\) −10.8284 −0.378839
\(818\) −13.4142 −0.469017
\(819\) 0 0
\(820\) 11.0711 0.386618
\(821\) 5.31371 0.185450 0.0927249 0.995692i \(-0.470442\pi\)
0.0927249 + 0.995692i \(0.470442\pi\)
\(822\) 0 0
\(823\) −36.2843 −1.26479 −0.632395 0.774646i \(-0.717928\pi\)
−0.632395 + 0.774646i \(0.717928\pi\)
\(824\) −9.17157 −0.319507
\(825\) 0 0
\(826\) 0 0
\(827\) 50.6274 1.76049 0.880244 0.474522i \(-0.157379\pi\)
0.880244 + 0.474522i \(0.157379\pi\)
\(828\) 0 0
\(829\) −38.9706 −1.35350 −0.676752 0.736211i \(-0.736613\pi\)
−0.676752 + 0.736211i \(0.736613\pi\)
\(830\) 6.24264 0.216685
\(831\) 0 0
\(832\) 0.828427 0.0287205
\(833\) 0 0
\(834\) 0 0
\(835\) −23.7990 −0.823598
\(836\) −16.4853 −0.570155
\(837\) 0 0
\(838\) −32.8701 −1.13548
\(839\) 13.8579 0.478427 0.239213 0.970967i \(-0.423110\pi\)
0.239213 + 0.970967i \(0.423110\pi\)
\(840\) 0 0
\(841\) −28.3137 −0.976335
\(842\) −5.31371 −0.183122
\(843\) 0 0
\(844\) −26.6274 −0.916553
\(845\) 12.3137 0.423604
\(846\) 0 0
\(847\) 0 0
\(848\) 10.4853 0.360066
\(849\) 0 0
\(850\) −5.41421 −0.185706
\(851\) 24.9706 0.855980
\(852\) 0 0
\(853\) −48.8284 −1.67185 −0.835927 0.548841i \(-0.815069\pi\)
−0.835927 + 0.548841i \(0.815069\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 1.65685 0.0566301
\(857\) 19.0711 0.651455 0.325728 0.945464i \(-0.394391\pi\)
0.325728 + 0.945464i \(0.394391\pi\)
\(858\) 0 0
\(859\) 35.2132 1.20146 0.600729 0.799452i \(-0.294877\pi\)
0.600729 + 0.799452i \(0.294877\pi\)
\(860\) 3.17157 0.108150
\(861\) 0 0
\(862\) 33.6569 1.14636
\(863\) −28.9706 −0.986169 −0.493085 0.869981i \(-0.664131\pi\)
−0.493085 + 0.869981i \(0.664131\pi\)
\(864\) 0 0
\(865\) −3.17157 −0.107837
\(866\) 13.4142 0.455834
\(867\) 0 0
\(868\) 0 0
\(869\) 5.65685 0.191896
\(870\) 0 0
\(871\) −8.00000 −0.271070
\(872\) −14.4853 −0.490534
\(873\) 0 0
\(874\) 23.3137 0.788598
\(875\) 0 0
\(876\) 0 0
\(877\) 26.2843 0.887557 0.443778 0.896137i \(-0.353638\pi\)
0.443778 + 0.896137i \(0.353638\pi\)
\(878\) 8.97056 0.302742
\(879\) 0 0
\(880\) 4.82843 0.162766
\(881\) −34.3848 −1.15845 −0.579226 0.815167i \(-0.696645\pi\)
−0.579226 + 0.815167i \(0.696645\pi\)
\(882\) 0 0
\(883\) −30.3431 −1.02113 −0.510564 0.859840i \(-0.670563\pi\)
−0.510564 + 0.859840i \(0.670563\pi\)
\(884\) −4.48528 −0.150856
\(885\) 0 0
\(886\) −36.9706 −1.24205
\(887\) −7.11270 −0.238821 −0.119411 0.992845i \(-0.538100\pi\)
−0.119411 + 0.992845i \(0.538100\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −12.7279 −0.426641
\(891\) 0 0
\(892\) 15.3137 0.512741
\(893\) −36.9706 −1.23717
\(894\) 0 0
\(895\) 4.00000 0.133705
\(896\) 0 0
\(897\) 0 0
\(898\) −28.6274 −0.955309
\(899\) −2.34315 −0.0781483
\(900\) 0 0
\(901\) −56.7696 −1.89127
\(902\) 53.4558 1.77988
\(903\) 0 0
\(904\) −7.31371 −0.243250
\(905\) −14.4853 −0.481507
\(906\) 0 0
\(907\) −56.2843 −1.86889 −0.934444 0.356109i \(-0.884103\pi\)
−0.934444 + 0.356109i \(0.884103\pi\)
\(908\) −9.75736 −0.323809
\(909\) 0 0
\(910\) 0 0
\(911\) 20.2843 0.672048 0.336024 0.941853i \(-0.390918\pi\)
0.336024 + 0.941853i \(0.390918\pi\)
\(912\) 0 0
\(913\) 30.1421 0.997559
\(914\) 10.3431 0.342121
\(915\) 0 0
\(916\) 12.1421 0.401187
\(917\) 0 0
\(918\) 0 0
\(919\) 32.4853 1.07159 0.535795 0.844348i \(-0.320012\pi\)
0.535795 + 0.844348i \(0.320012\pi\)
\(920\) −6.82843 −0.225127
\(921\) 0 0
\(922\) 7.17157 0.236183
\(923\) −10.3431 −0.340449
\(924\) 0 0
\(925\) 3.65685 0.120237
\(926\) 16.9706 0.557687
\(927\) 0 0
\(928\) −0.828427 −0.0271945
\(929\) −25.2132 −0.827218 −0.413609 0.910455i \(-0.635732\pi\)
−0.413609 + 0.910455i \(0.635732\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −0.686292 −0.0224802
\(933\) 0 0
\(934\) −3.89949 −0.127595
\(935\) −26.1421 −0.854939
\(936\) 0 0
\(937\) 11.7574 0.384096 0.192048 0.981386i \(-0.438487\pi\)
0.192048 + 0.981386i \(0.438487\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 10.8284 0.353184
\(941\) 50.0000 1.62995 0.814977 0.579494i \(-0.196750\pi\)
0.814977 + 0.579494i \(0.196750\pi\)
\(942\) 0 0
\(943\) −75.5980 −2.46181
\(944\) −11.4142 −0.371501
\(945\) 0 0
\(946\) 15.3137 0.497892
\(947\) −0.828427 −0.0269203 −0.0134601 0.999909i \(-0.504285\pi\)
−0.0134601 + 0.999909i \(0.504285\pi\)
\(948\) 0 0
\(949\) −5.45584 −0.177104
\(950\) 3.41421 0.110772
\(951\) 0 0
\(952\) 0 0
\(953\) −11.6569 −0.377603 −0.188801 0.982015i \(-0.560460\pi\)
−0.188801 + 0.982015i \(0.560460\pi\)
\(954\) 0 0
\(955\) −18.1421 −0.587066
\(956\) 9.65685 0.312325
\(957\) 0 0
\(958\) 22.8284 0.737553
\(959\) 0 0
\(960\) 0 0
\(961\) −23.0000 −0.741935
\(962\) 3.02944 0.0976730
\(963\) 0 0
\(964\) −10.5858 −0.340945
\(965\) 5.65685 0.182101
\(966\) 0 0
\(967\) 13.4558 0.432711 0.216355 0.976315i \(-0.430583\pi\)
0.216355 + 0.976315i \(0.430583\pi\)
\(968\) 12.3137 0.395778
\(969\) 0 0
\(970\) 16.2426 0.521520
\(971\) −37.3553 −1.19879 −0.599395 0.800453i \(-0.704592\pi\)
−0.599395 + 0.800453i \(0.704592\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −7.79899 −0.249896
\(975\) 0 0
\(976\) −13.3137 −0.426161
\(977\) 35.3137 1.12979 0.564893 0.825164i \(-0.308918\pi\)
0.564893 + 0.825164i \(0.308918\pi\)
\(978\) 0 0
\(979\) −61.4558 −1.96414
\(980\) 0 0
\(981\) 0 0
\(982\) 24.2843 0.774942
\(983\) 51.7990 1.65213 0.826066 0.563574i \(-0.190574\pi\)
0.826066 + 0.563574i \(0.190574\pi\)
\(984\) 0 0
\(985\) 13.7990 0.439672
\(986\) 4.48528 0.142840
\(987\) 0 0
\(988\) 2.82843 0.0899843
\(989\) −21.6569 −0.688648
\(990\) 0 0
\(991\) −28.7696 −0.913895 −0.456947 0.889494i \(-0.651057\pi\)
−0.456947 + 0.889494i \(0.651057\pi\)
\(992\) 2.82843 0.0898027
\(993\) 0 0
\(994\) 0 0
\(995\) 0.485281 0.0153845
\(996\) 0 0
\(997\) 38.2843 1.21248 0.606238 0.795284i \(-0.292678\pi\)
0.606238 + 0.795284i \(0.292678\pi\)
\(998\) 41.6569 1.31862
\(999\) 0 0
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 4410.2.a.bt.1.1 2
3.2 odd 2 490.2.a.m.1.1 yes 2
7.6 odd 2 4410.2.a.by.1.1 2
12.11 even 2 3920.2.a.bm.1.2 2
15.2 even 4 2450.2.c.t.99.2 4
15.8 even 4 2450.2.c.t.99.3 4
15.14 odd 2 2450.2.a.bn.1.2 2
21.2 odd 6 490.2.e.i.361.2 4
21.5 even 6 490.2.e.j.361.1 4
21.11 odd 6 490.2.e.i.471.2 4
21.17 even 6 490.2.e.j.471.1 4
21.20 even 2 490.2.a.l.1.2 2
84.83 odd 2 3920.2.a.ca.1.1 2
105.62 odd 4 2450.2.c.w.99.1 4
105.83 odd 4 2450.2.c.w.99.4 4
105.104 even 2 2450.2.a.bs.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
490.2.a.l.1.2 2 21.20 even 2
490.2.a.m.1.1 yes 2 3.2 odd 2
490.2.e.i.361.2 4 21.2 odd 6
490.2.e.i.471.2 4 21.11 odd 6
490.2.e.j.361.1 4 21.5 even 6
490.2.e.j.471.1 4 21.17 even 6
2450.2.a.bn.1.2 2 15.14 odd 2
2450.2.a.bs.1.1 2 105.104 even 2
2450.2.c.t.99.2 4 15.2 even 4
2450.2.c.t.99.3 4 15.8 even 4
2450.2.c.w.99.1 4 105.62 odd 4
2450.2.c.w.99.4 4 105.83 odd 4
3920.2.a.bm.1.2 2 12.11 even 2
3920.2.a.ca.1.1 2 84.83 odd 2
4410.2.a.bt.1.1 2 1.1 even 1 trivial
4410.2.a.by.1.1 2 7.6 odd 2