Properties

Label 6840.2.a.bb.1.2
Level $6840$
Weight $2$
Character 6840.1
Self dual yes
Analytic conductor $54.618$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6840 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6840.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(54.6176749826\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2280)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 6840.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{5} +1.41421 q^{7} +O(q^{10})\) \(q+1.00000 q^{5} +1.41421 q^{7} +0.585786 q^{11} +0.585786 q^{13} +2.82843 q^{17} +1.00000 q^{19} +4.82843 q^{23} +1.00000 q^{25} +7.07107 q^{29} -4.82843 q^{31} +1.41421 q^{35} +6.24264 q^{37} -9.89949 q^{41} +11.0711 q^{43} +3.17157 q^{47} -5.00000 q^{49} -8.48528 q^{53} +0.585786 q^{55} +1.17157 q^{59} -1.65685 q^{61} +0.585786 q^{65} -11.3137 q^{67} +14.8284 q^{71} -11.6569 q^{73} +0.828427 q^{77} -13.6569 q^{79} +7.65685 q^{83} +2.82843 q^{85} +12.2426 q^{89} +0.828427 q^{91} +1.00000 q^{95} +11.8995 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} + O(q^{10}) \) \( 2 q + 2 q^{5} + 4 q^{11} + 4 q^{13} + 2 q^{19} + 4 q^{23} + 2 q^{25} - 4 q^{31} + 4 q^{37} + 8 q^{43} + 12 q^{47} - 10 q^{49} + 4 q^{55} + 8 q^{59} + 8 q^{61} + 4 q^{65} + 24 q^{71} - 12 q^{73} - 4 q^{77} - 16 q^{79} + 4 q^{83} + 16 q^{89} - 4 q^{91} + 2 q^{95} + 4 q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 1.41421 0.534522 0.267261 0.963624i \(-0.413881\pi\)
0.267261 + 0.963624i \(0.413881\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0.585786 0.176621 0.0883106 0.996093i \(-0.471853\pi\)
0.0883106 + 0.996093i \(0.471853\pi\)
\(12\) 0 0
\(13\) 0.585786 0.162468 0.0812340 0.996695i \(-0.474114\pi\)
0.0812340 + 0.996695i \(0.474114\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.82843 0.685994 0.342997 0.939336i \(-0.388558\pi\)
0.342997 + 0.939336i \(0.388558\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.82843 1.00680 0.503398 0.864054i \(-0.332083\pi\)
0.503398 + 0.864054i \(0.332083\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 7.07107 1.31306 0.656532 0.754298i \(-0.272023\pi\)
0.656532 + 0.754298i \(0.272023\pi\)
\(30\) 0 0
\(31\) −4.82843 −0.867211 −0.433606 0.901103i \(-0.642759\pi\)
−0.433606 + 0.901103i \(0.642759\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.41421 0.239046
\(36\) 0 0
\(37\) 6.24264 1.02628 0.513142 0.858304i \(-0.328481\pi\)
0.513142 + 0.858304i \(0.328481\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −9.89949 −1.54604 −0.773021 0.634381i \(-0.781255\pi\)
−0.773021 + 0.634381i \(0.781255\pi\)
\(42\) 0 0
\(43\) 11.0711 1.68832 0.844161 0.536090i \(-0.180099\pi\)
0.844161 + 0.536090i \(0.180099\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.17157 0.462621 0.231311 0.972880i \(-0.425699\pi\)
0.231311 + 0.972880i \(0.425699\pi\)
\(48\) 0 0
\(49\) −5.00000 −0.714286
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −8.48528 −1.16554 −0.582772 0.812636i \(-0.698032\pi\)
−0.582772 + 0.812636i \(0.698032\pi\)
\(54\) 0 0
\(55\) 0.585786 0.0789874
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 1.17157 0.152526 0.0762629 0.997088i \(-0.475701\pi\)
0.0762629 + 0.997088i \(0.475701\pi\)
\(60\) 0 0
\(61\) −1.65685 −0.212138 −0.106069 0.994359i \(-0.533827\pi\)
−0.106069 + 0.994359i \(0.533827\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0.585786 0.0726579
\(66\) 0 0
\(67\) −11.3137 −1.38219 −0.691095 0.722764i \(-0.742871\pi\)
−0.691095 + 0.722764i \(0.742871\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 14.8284 1.75981 0.879905 0.475149i \(-0.157606\pi\)
0.879905 + 0.475149i \(0.157606\pi\)
\(72\) 0 0
\(73\) −11.6569 −1.36433 −0.682166 0.731198i \(-0.738962\pi\)
−0.682166 + 0.731198i \(0.738962\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0.828427 0.0944080
\(78\) 0 0
\(79\) −13.6569 −1.53652 −0.768258 0.640140i \(-0.778876\pi\)
−0.768258 + 0.640140i \(0.778876\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 7.65685 0.840449 0.420224 0.907420i \(-0.361951\pi\)
0.420224 + 0.907420i \(0.361951\pi\)
\(84\) 0 0
\(85\) 2.82843 0.306786
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 12.2426 1.29772 0.648859 0.760909i \(-0.275247\pi\)
0.648859 + 0.760909i \(0.275247\pi\)
\(90\) 0 0
\(91\) 0.828427 0.0868428
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1.00000 0.102598
\(96\) 0 0
\(97\) 11.8995 1.20821 0.604105 0.796904i \(-0.293531\pi\)
0.604105 + 0.796904i \(0.293531\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0.828427 0.0824316 0.0412158 0.999150i \(-0.486877\pi\)
0.0412158 + 0.999150i \(0.486877\pi\)
\(102\) 0 0
\(103\) 3.31371 0.326509 0.163255 0.986584i \(-0.447801\pi\)
0.163255 + 0.986584i \(0.447801\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 13.6569 1.32026 0.660129 0.751152i \(-0.270502\pi\)
0.660129 + 0.751152i \(0.270502\pi\)
\(108\) 0 0
\(109\) 8.14214 0.779875 0.389938 0.920841i \(-0.372497\pi\)
0.389938 + 0.920841i \(0.372497\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 15.3137 1.44059 0.720296 0.693667i \(-0.244006\pi\)
0.720296 + 0.693667i \(0.244006\pi\)
\(114\) 0 0
\(115\) 4.82843 0.450253
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 4.00000 0.366679
\(120\) 0 0
\(121\) −10.6569 −0.968805
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −5.65685 −0.501965 −0.250982 0.967992i \(-0.580754\pi\)
−0.250982 + 0.967992i \(0.580754\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −6.72792 −0.587821 −0.293911 0.955833i \(-0.594957\pi\)
−0.293911 + 0.955833i \(0.594957\pi\)
\(132\) 0 0
\(133\) 1.41421 0.122628
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −15.6569 −1.33766 −0.668828 0.743417i \(-0.733204\pi\)
−0.668828 + 0.743417i \(0.733204\pi\)
\(138\) 0 0
\(139\) −2.82843 −0.239904 −0.119952 0.992780i \(-0.538274\pi\)
−0.119952 + 0.992780i \(0.538274\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0.343146 0.0286953
\(144\) 0 0
\(145\) 7.07107 0.587220
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 4.34315 0.355804 0.177902 0.984048i \(-0.443069\pi\)
0.177902 + 0.984048i \(0.443069\pi\)
\(150\) 0 0
\(151\) 0.828427 0.0674164 0.0337082 0.999432i \(-0.489268\pi\)
0.0337082 + 0.999432i \(0.489268\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −4.82843 −0.387829
\(156\) 0 0
\(157\) −4.14214 −0.330578 −0.165289 0.986245i \(-0.552856\pi\)
−0.165289 + 0.986245i \(0.552856\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 6.82843 0.538155
\(162\) 0 0
\(163\) −23.5563 −1.84508 −0.922538 0.385907i \(-0.873889\pi\)
−0.922538 + 0.385907i \(0.873889\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 17.3137 1.33977 0.669887 0.742463i \(-0.266342\pi\)
0.669887 + 0.742463i \(0.266342\pi\)
\(168\) 0 0
\(169\) −12.6569 −0.973604
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 11.3137 0.860165 0.430083 0.902790i \(-0.358484\pi\)
0.430083 + 0.902790i \(0.358484\pi\)
\(174\) 0 0
\(175\) 1.41421 0.106904
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 13.1716 0.984490 0.492245 0.870457i \(-0.336176\pi\)
0.492245 + 0.870457i \(0.336176\pi\)
\(180\) 0 0
\(181\) 20.1421 1.49715 0.748577 0.663048i \(-0.230738\pi\)
0.748577 + 0.663048i \(0.230738\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 6.24264 0.458968
\(186\) 0 0
\(187\) 1.65685 0.121161
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 13.7574 0.995448 0.497724 0.867336i \(-0.334169\pi\)
0.497724 + 0.867336i \(0.334169\pi\)
\(192\) 0 0
\(193\) 13.0711 0.940876 0.470438 0.882433i \(-0.344096\pi\)
0.470438 + 0.882433i \(0.344096\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −22.1421 −1.57756 −0.788781 0.614674i \(-0.789287\pi\)
−0.788781 + 0.614674i \(0.789287\pi\)
\(198\) 0 0
\(199\) −9.17157 −0.650156 −0.325078 0.945687i \(-0.605390\pi\)
−0.325078 + 0.945687i \(0.605390\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 10.0000 0.701862
\(204\) 0 0
\(205\) −9.89949 −0.691411
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0.585786 0.0405197
\(210\) 0 0
\(211\) 20.0000 1.37686 0.688428 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 11.0711 0.755041
\(216\) 0 0
\(217\) −6.82843 −0.463544
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 1.65685 0.111452
\(222\) 0 0
\(223\) 11.3137 0.757622 0.378811 0.925474i \(-0.376333\pi\)
0.378811 + 0.925474i \(0.376333\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −11.6569 −0.773693 −0.386846 0.922144i \(-0.626436\pi\)
−0.386846 + 0.922144i \(0.626436\pi\)
\(228\) 0 0
\(229\) −21.6569 −1.43113 −0.715563 0.698549i \(-0.753830\pi\)
−0.715563 + 0.698549i \(0.753830\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −15.6569 −1.02571 −0.512857 0.858474i \(-0.671413\pi\)
−0.512857 + 0.858474i \(0.671413\pi\)
\(234\) 0 0
\(235\) 3.17157 0.206891
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 17.7574 1.14863 0.574314 0.818635i \(-0.305269\pi\)
0.574314 + 0.818635i \(0.305269\pi\)
\(240\) 0 0
\(241\) 12.3431 0.795092 0.397546 0.917582i \(-0.369862\pi\)
0.397546 + 0.917582i \(0.369862\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −5.00000 −0.319438
\(246\) 0 0
\(247\) 0.585786 0.0372727
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 5.75736 0.363401 0.181701 0.983354i \(-0.441840\pi\)
0.181701 + 0.983354i \(0.441840\pi\)
\(252\) 0 0
\(253\) 2.82843 0.177822
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −12.0000 −0.748539 −0.374270 0.927320i \(-0.622107\pi\)
−0.374270 + 0.927320i \(0.622107\pi\)
\(258\) 0 0
\(259\) 8.82843 0.548572
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 2.00000 0.123325 0.0616626 0.998097i \(-0.480360\pi\)
0.0616626 + 0.998097i \(0.480360\pi\)
\(264\) 0 0
\(265\) −8.48528 −0.521247
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −14.3848 −0.877055 −0.438528 0.898718i \(-0.644500\pi\)
−0.438528 + 0.898718i \(0.644500\pi\)
\(270\) 0 0
\(271\) −2.82843 −0.171815 −0.0859074 0.996303i \(-0.527379\pi\)
−0.0859074 + 0.996303i \(0.527379\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0.585786 0.0353243
\(276\) 0 0
\(277\) −24.6274 −1.47972 −0.739859 0.672762i \(-0.765108\pi\)
−0.739859 + 0.672762i \(0.765108\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 30.3848 1.81260 0.906302 0.422631i \(-0.138893\pi\)
0.906302 + 0.422631i \(0.138893\pi\)
\(282\) 0 0
\(283\) 7.75736 0.461127 0.230564 0.973057i \(-0.425943\pi\)
0.230564 + 0.973057i \(0.425943\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −14.0000 −0.826394
\(288\) 0 0
\(289\) −9.00000 −0.529412
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −1.65685 −0.0967945 −0.0483972 0.998828i \(-0.515411\pi\)
−0.0483972 + 0.998828i \(0.515411\pi\)
\(294\) 0 0
\(295\) 1.17157 0.0682116
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 2.82843 0.163572
\(300\) 0 0
\(301\) 15.6569 0.902446
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −1.65685 −0.0948712
\(306\) 0 0
\(307\) 30.1421 1.72030 0.860151 0.510039i \(-0.170369\pi\)
0.860151 + 0.510039i \(0.170369\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 23.2132 1.31630 0.658150 0.752887i \(-0.271339\pi\)
0.658150 + 0.752887i \(0.271339\pi\)
\(312\) 0 0
\(313\) 25.7990 1.45825 0.729123 0.684383i \(-0.239928\pi\)
0.729123 + 0.684383i \(0.239928\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −26.1421 −1.46829 −0.734144 0.678993i \(-0.762416\pi\)
−0.734144 + 0.678993i \(0.762416\pi\)
\(318\) 0 0
\(319\) 4.14214 0.231915
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 2.82843 0.157378
\(324\) 0 0
\(325\) 0.585786 0.0324936
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 4.48528 0.247282
\(330\) 0 0
\(331\) −12.1421 −0.667392 −0.333696 0.942681i \(-0.608296\pi\)
−0.333696 + 0.942681i \(0.608296\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −11.3137 −0.618134
\(336\) 0 0
\(337\) 17.7574 0.967305 0.483652 0.875260i \(-0.339310\pi\)
0.483652 + 0.875260i \(0.339310\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −2.82843 −0.153168
\(342\) 0 0
\(343\) −16.9706 −0.916324
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −21.3137 −1.14418 −0.572090 0.820191i \(-0.693867\pi\)
−0.572090 + 0.820191i \(0.693867\pi\)
\(348\) 0 0
\(349\) −6.00000 −0.321173 −0.160586 0.987022i \(-0.551338\pi\)
−0.160586 + 0.987022i \(0.551338\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 6.68629 0.355875 0.177938 0.984042i \(-0.443057\pi\)
0.177938 + 0.984042i \(0.443057\pi\)
\(354\) 0 0
\(355\) 14.8284 0.787011
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −13.5563 −0.715477 −0.357738 0.933822i \(-0.616452\pi\)
−0.357738 + 0.933822i \(0.616452\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −11.6569 −0.610148
\(366\) 0 0
\(367\) 24.7279 1.29079 0.645394 0.763850i \(-0.276693\pi\)
0.645394 + 0.763850i \(0.276693\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −12.0000 −0.623009
\(372\) 0 0
\(373\) −17.0711 −0.883906 −0.441953 0.897038i \(-0.645714\pi\)
−0.441953 + 0.897038i \(0.645714\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 4.14214 0.213331
\(378\) 0 0
\(379\) 9.51472 0.488738 0.244369 0.969682i \(-0.421419\pi\)
0.244369 + 0.969682i \(0.421419\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 34.6274 1.76938 0.884689 0.466181i \(-0.154371\pi\)
0.884689 + 0.466181i \(0.154371\pi\)
\(384\) 0 0
\(385\) 0.828427 0.0422206
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 34.9706 1.77308 0.886539 0.462654i \(-0.153103\pi\)
0.886539 + 0.462654i \(0.153103\pi\)
\(390\) 0 0
\(391\) 13.6569 0.690657
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −13.6569 −0.687151
\(396\) 0 0
\(397\) 16.6274 0.834506 0.417253 0.908790i \(-0.362993\pi\)
0.417253 + 0.908790i \(0.362993\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −18.3848 −0.918092 −0.459046 0.888413i \(-0.651809\pi\)
−0.459046 + 0.888413i \(0.651809\pi\)
\(402\) 0 0
\(403\) −2.82843 −0.140894
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 3.65685 0.181264
\(408\) 0 0
\(409\) −4.82843 −0.238750 −0.119375 0.992849i \(-0.538089\pi\)
−0.119375 + 0.992849i \(0.538089\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 1.65685 0.0815285
\(414\) 0 0
\(415\) 7.65685 0.375860
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 1.55635 0.0760326 0.0380163 0.999277i \(-0.487896\pi\)
0.0380163 + 0.999277i \(0.487896\pi\)
\(420\) 0 0
\(421\) −16.6274 −0.810371 −0.405185 0.914235i \(-0.632793\pi\)
−0.405185 + 0.914235i \(0.632793\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 2.82843 0.137199
\(426\) 0 0
\(427\) −2.34315 −0.113393
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −6.14214 −0.295856 −0.147928 0.988998i \(-0.547260\pi\)
−0.147928 + 0.988998i \(0.547260\pi\)
\(432\) 0 0
\(433\) −14.2426 −0.684458 −0.342229 0.939617i \(-0.611182\pi\)
−0.342229 + 0.939617i \(0.611182\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 4.82843 0.230975
\(438\) 0 0
\(439\) 4.68629 0.223664 0.111832 0.993727i \(-0.464328\pi\)
0.111832 + 0.993727i \(0.464328\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −23.4558 −1.11442 −0.557210 0.830371i \(-0.688128\pi\)
−0.557210 + 0.830371i \(0.688128\pi\)
\(444\) 0 0
\(445\) 12.2426 0.580357
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 14.3848 0.678860 0.339430 0.940631i \(-0.389766\pi\)
0.339430 + 0.940631i \(0.389766\pi\)
\(450\) 0 0
\(451\) −5.79899 −0.273064
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0.828427 0.0388373
\(456\) 0 0
\(457\) 16.1421 0.755097 0.377549 0.925990i \(-0.376767\pi\)
0.377549 + 0.925990i \(0.376767\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −6.68629 −0.311412 −0.155706 0.987803i \(-0.549765\pi\)
−0.155706 + 0.987803i \(0.549765\pi\)
\(462\) 0 0
\(463\) 24.7279 1.14920 0.574602 0.818433i \(-0.305157\pi\)
0.574602 + 0.818433i \(0.305157\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 16.8284 0.778727 0.389363 0.921084i \(-0.372695\pi\)
0.389363 + 0.921084i \(0.372695\pi\)
\(468\) 0 0
\(469\) −16.0000 −0.738811
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 6.48528 0.298194
\(474\) 0 0
\(475\) 1.00000 0.0458831
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −5.75736 −0.263060 −0.131530 0.991312i \(-0.541989\pi\)
−0.131530 + 0.991312i \(0.541989\pi\)
\(480\) 0 0
\(481\) 3.65685 0.166738
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 11.8995 0.540328
\(486\) 0 0
\(487\) 23.7990 1.07844 0.539218 0.842166i \(-0.318720\pi\)
0.539218 + 0.842166i \(0.318720\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 12.3848 0.558917 0.279459 0.960158i \(-0.409845\pi\)
0.279459 + 0.960158i \(0.409845\pi\)
\(492\) 0 0
\(493\) 20.0000 0.900755
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 20.9706 0.940658
\(498\) 0 0
\(499\) 1.85786 0.0831694 0.0415847 0.999135i \(-0.486759\pi\)
0.0415847 + 0.999135i \(0.486759\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 6.68629 0.298127 0.149064 0.988828i \(-0.452374\pi\)
0.149064 + 0.988828i \(0.452374\pi\)
\(504\) 0 0
\(505\) 0.828427 0.0368645
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 4.92893 0.218471 0.109236 0.994016i \(-0.465160\pi\)
0.109236 + 0.994016i \(0.465160\pi\)
\(510\) 0 0
\(511\) −16.4853 −0.729266
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 3.31371 0.146019
\(516\) 0 0
\(517\) 1.85786 0.0817088
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −7.55635 −0.331050 −0.165525 0.986206i \(-0.552932\pi\)
−0.165525 + 0.986206i \(0.552932\pi\)
\(522\) 0 0
\(523\) 23.7990 1.04066 0.520329 0.853966i \(-0.325809\pi\)
0.520329 + 0.853966i \(0.325809\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −13.6569 −0.594902
\(528\) 0 0
\(529\) 0.313708 0.0136395
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −5.79899 −0.251182
\(534\) 0 0
\(535\) 13.6569 0.590437
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −2.92893 −0.126158
\(540\) 0 0
\(541\) −13.3137 −0.572401 −0.286201 0.958170i \(-0.592392\pi\)
−0.286201 + 0.958170i \(0.592392\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 8.14214 0.348771
\(546\) 0 0
\(547\) −8.48528 −0.362804 −0.181402 0.983409i \(-0.558064\pi\)
−0.181402 + 0.983409i \(0.558064\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 7.07107 0.301238
\(552\) 0 0
\(553\) −19.3137 −0.821302
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −12.3431 −0.522996 −0.261498 0.965204i \(-0.584216\pi\)
−0.261498 + 0.965204i \(0.584216\pi\)
\(558\) 0 0
\(559\) 6.48528 0.274298
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −11.6569 −0.491278 −0.245639 0.969361i \(-0.578998\pi\)
−0.245639 + 0.969361i \(0.578998\pi\)
\(564\) 0 0
\(565\) 15.3137 0.644253
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −34.1838 −1.43306 −0.716529 0.697557i \(-0.754270\pi\)
−0.716529 + 0.697557i \(0.754270\pi\)
\(570\) 0 0
\(571\) 26.1421 1.09401 0.547007 0.837128i \(-0.315767\pi\)
0.547007 + 0.837128i \(0.315767\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 4.82843 0.201359
\(576\) 0 0
\(577\) −26.4853 −1.10260 −0.551298 0.834308i \(-0.685867\pi\)
−0.551298 + 0.834308i \(0.685867\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 10.8284 0.449239
\(582\) 0 0
\(583\) −4.97056 −0.205860
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 23.4558 0.968126 0.484063 0.875033i \(-0.339160\pi\)
0.484063 + 0.875033i \(0.339160\pi\)
\(588\) 0 0
\(589\) −4.82843 −0.198952
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0.627417 0.0257649 0.0128825 0.999917i \(-0.495899\pi\)
0.0128825 + 0.999917i \(0.495899\pi\)
\(594\) 0 0
\(595\) 4.00000 0.163984
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 16.9706 0.693398 0.346699 0.937976i \(-0.387302\pi\)
0.346699 + 0.937976i \(0.387302\pi\)
\(600\) 0 0
\(601\) 45.7990 1.86818 0.934090 0.357038i \(-0.116213\pi\)
0.934090 + 0.357038i \(0.116213\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −10.6569 −0.433263
\(606\) 0 0
\(607\) −17.4558 −0.708511 −0.354255 0.935149i \(-0.615266\pi\)
−0.354255 + 0.935149i \(0.615266\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 1.85786 0.0751611
\(612\) 0 0
\(613\) −7.17157 −0.289657 −0.144829 0.989457i \(-0.546263\pi\)
−0.144829 + 0.989457i \(0.546263\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −47.1127 −1.89669 −0.948343 0.317247i \(-0.897242\pi\)
−0.948343 + 0.317247i \(0.897242\pi\)
\(618\) 0 0
\(619\) 20.4853 0.823373 0.411686 0.911326i \(-0.364940\pi\)
0.411686 + 0.911326i \(0.364940\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 17.3137 0.693659
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 17.6569 0.704025
\(630\) 0 0
\(631\) 20.2843 0.807504 0.403752 0.914868i \(-0.367706\pi\)
0.403752 + 0.914868i \(0.367706\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −5.65685 −0.224485
\(636\) 0 0
\(637\) −2.92893 −0.116049
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −31.0711 −1.22723 −0.613617 0.789604i \(-0.710286\pi\)
−0.613617 + 0.789604i \(0.710286\pi\)
\(642\) 0 0
\(643\) −43.0711 −1.69856 −0.849279 0.527945i \(-0.822963\pi\)
−0.849279 + 0.527945i \(0.822963\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −2.68629 −0.105609 −0.0528045 0.998605i \(-0.516816\pi\)
−0.0528045 + 0.998605i \(0.516816\pi\)
\(648\) 0 0
\(649\) 0.686292 0.0269393
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 18.1421 0.709957 0.354978 0.934875i \(-0.384488\pi\)
0.354978 + 0.934875i \(0.384488\pi\)
\(654\) 0 0
\(655\) −6.72792 −0.262882
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 4.97056 0.193626 0.0968128 0.995303i \(-0.469135\pi\)
0.0968128 + 0.995303i \(0.469135\pi\)
\(660\) 0 0
\(661\) 13.1127 0.510025 0.255012 0.966938i \(-0.417920\pi\)
0.255012 + 0.966938i \(0.417920\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 1.41421 0.0548408
\(666\) 0 0
\(667\) 34.1421 1.32199
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −0.970563 −0.0374682
\(672\) 0 0
\(673\) 5.75736 0.221930 0.110965 0.993824i \(-0.464606\pi\)
0.110965 + 0.993824i \(0.464606\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −34.8284 −1.33857 −0.669283 0.743008i \(-0.733398\pi\)
−0.669283 + 0.743008i \(0.733398\pi\)
\(678\) 0 0
\(679\) 16.8284 0.645816
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −12.6863 −0.485427 −0.242714 0.970098i \(-0.578038\pi\)
−0.242714 + 0.970098i \(0.578038\pi\)
\(684\) 0 0
\(685\) −15.6569 −0.598218
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −4.97056 −0.189363
\(690\) 0 0
\(691\) 47.7990 1.81836 0.909180 0.416404i \(-0.136710\pi\)
0.909180 + 0.416404i \(0.136710\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −2.82843 −0.107288
\(696\) 0 0
\(697\) −28.0000 −1.06058
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 10.9706 0.414352 0.207176 0.978304i \(-0.433573\pi\)
0.207176 + 0.978304i \(0.433573\pi\)
\(702\) 0 0
\(703\) 6.24264 0.235446
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 1.17157 0.0440615
\(708\) 0 0
\(709\) −20.0000 −0.751116 −0.375558 0.926799i \(-0.622549\pi\)
−0.375558 + 0.926799i \(0.622549\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −23.3137 −0.873105
\(714\) 0 0
\(715\) 0.343146 0.0128329
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −10.2426 −0.381986 −0.190993 0.981591i \(-0.561171\pi\)
−0.190993 + 0.981591i \(0.561171\pi\)
\(720\) 0 0
\(721\) 4.68629 0.174527
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 7.07107 0.262613
\(726\) 0 0
\(727\) 24.2426 0.899110 0.449555 0.893253i \(-0.351583\pi\)
0.449555 + 0.893253i \(0.351583\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 31.3137 1.15818
\(732\) 0 0
\(733\) −10.6863 −0.394707 −0.197354 0.980332i \(-0.563235\pi\)
−0.197354 + 0.980332i \(0.563235\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −6.62742 −0.244124
\(738\) 0 0
\(739\) 7.31371 0.269039 0.134520 0.990911i \(-0.457051\pi\)
0.134520 + 0.990911i \(0.457051\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −41.2548 −1.51349 −0.756747 0.653708i \(-0.773212\pi\)
−0.756747 + 0.653708i \(0.773212\pi\)
\(744\) 0 0
\(745\) 4.34315 0.159121
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 19.3137 0.705708
\(750\) 0 0
\(751\) −44.1421 −1.61077 −0.805385 0.592752i \(-0.798041\pi\)
−0.805385 + 0.592752i \(0.798041\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0.828427 0.0301496
\(756\) 0 0
\(757\) −26.4853 −0.962624 −0.481312 0.876549i \(-0.659840\pi\)
−0.481312 + 0.876549i \(0.659840\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 39.6569 1.43756 0.718780 0.695238i \(-0.244701\pi\)
0.718780 + 0.695238i \(0.244701\pi\)
\(762\) 0 0
\(763\) 11.5147 0.416861
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0.686292 0.0247805
\(768\) 0 0
\(769\) −34.2843 −1.23632 −0.618161 0.786051i \(-0.712122\pi\)
−0.618161 + 0.786051i \(0.712122\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −5.45584 −0.196233 −0.0981165 0.995175i \(-0.531282\pi\)
−0.0981165 + 0.995175i \(0.531282\pi\)
\(774\) 0 0
\(775\) −4.82843 −0.173442
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −9.89949 −0.354686
\(780\) 0 0
\(781\) 8.68629 0.310820
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −4.14214 −0.147839
\(786\) 0 0
\(787\) −6.82843 −0.243407 −0.121704 0.992566i \(-0.538836\pi\)
−0.121704 + 0.992566i \(0.538836\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 21.6569 0.770029
\(792\) 0 0
\(793\) −0.970563 −0.0344657
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −4.68629 −0.165997 −0.0829985 0.996550i \(-0.526450\pi\)
−0.0829985 + 0.996550i \(0.526450\pi\)
\(798\) 0 0
\(799\) 8.97056 0.317356
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −6.82843 −0.240970
\(804\) 0 0
\(805\) 6.82843 0.240670
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 53.3137 1.87441 0.937205 0.348779i \(-0.113404\pi\)
0.937205 + 0.348779i \(0.113404\pi\)
\(810\) 0 0
\(811\) 52.9706 1.86005 0.930024 0.367499i \(-0.119786\pi\)
0.930024 + 0.367499i \(0.119786\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −23.5563 −0.825143
\(816\) 0 0
\(817\) 11.0711 0.387328
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 21.7990 0.760790 0.380395 0.924824i \(-0.375788\pi\)
0.380395 + 0.924824i \(0.375788\pi\)
\(822\) 0 0
\(823\) −50.8701 −1.77322 −0.886609 0.462519i \(-0.846946\pi\)
−0.886609 + 0.462519i \(0.846946\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −7.65685 −0.266255 −0.133127 0.991099i \(-0.542502\pi\)
−0.133127 + 0.991099i \(0.542502\pi\)
\(828\) 0 0
\(829\) −0.544156 −0.0188993 −0.00944966 0.999955i \(-0.503008\pi\)
−0.00944966 + 0.999955i \(0.503008\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −14.1421 −0.489996
\(834\) 0 0
\(835\) 17.3137 0.599166
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −34.1421 −1.17872 −0.589359 0.807871i \(-0.700620\pi\)
−0.589359 + 0.807871i \(0.700620\pi\)
\(840\) 0 0
\(841\) 21.0000 0.724138
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −12.6569 −0.435409
\(846\) 0 0
\(847\) −15.0711 −0.517848
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 30.1421 1.03326
\(852\) 0 0
\(853\) 3.17157 0.108593 0.0542963 0.998525i \(-0.482708\pi\)
0.0542963 + 0.998525i \(0.482708\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 25.4558 0.869555 0.434778 0.900538i \(-0.356827\pi\)
0.434778 + 0.900538i \(0.356827\pi\)
\(858\) 0 0
\(859\) −8.00000 −0.272956 −0.136478 0.990643i \(-0.543578\pi\)
−0.136478 + 0.990643i \(0.543578\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 57.2548 1.94898 0.974489 0.224437i \(-0.0720543\pi\)
0.974489 + 0.224437i \(0.0720543\pi\)
\(864\) 0 0
\(865\) 11.3137 0.384678
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −8.00000 −0.271381
\(870\) 0 0
\(871\) −6.62742 −0.224561
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 1.41421 0.0478091
\(876\) 0 0
\(877\) 2.04163 0.0689410 0.0344705 0.999406i \(-0.489026\pi\)
0.0344705 + 0.999406i \(0.489026\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 17.5147 0.590086 0.295043 0.955484i \(-0.404666\pi\)
0.295043 + 0.955484i \(0.404666\pi\)
\(882\) 0 0
\(883\) −44.2426 −1.48888 −0.744442 0.667687i \(-0.767284\pi\)
−0.744442 + 0.667687i \(0.767284\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 10.3431 0.347289 0.173644 0.984808i \(-0.444446\pi\)
0.173644 + 0.984808i \(0.444446\pi\)
\(888\) 0 0
\(889\) −8.00000 −0.268311
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 3.17157 0.106133
\(894\) 0 0
\(895\) 13.1716 0.440277
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −34.1421 −1.13870
\(900\) 0 0
\(901\) −24.0000 −0.799556
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 20.1421 0.669547
\(906\) 0 0
\(907\) −20.7696 −0.689642 −0.344821 0.938669i \(-0.612060\pi\)
−0.344821 + 0.938669i \(0.612060\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −34.6274 −1.14726 −0.573629 0.819115i \(-0.694465\pi\)
−0.573629 + 0.819115i \(0.694465\pi\)
\(912\) 0 0
\(913\) 4.48528 0.148441
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −9.51472 −0.314204
\(918\) 0 0
\(919\) −51.5980 −1.70206 −0.851030 0.525117i \(-0.824022\pi\)
−0.851030 + 0.525117i \(0.824022\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 8.68629 0.285913
\(924\) 0 0
\(925\) 6.24264 0.205257
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 14.2010 0.465920 0.232960 0.972486i \(-0.425159\pi\)
0.232960 + 0.972486i \(0.425159\pi\)
\(930\) 0 0
\(931\) −5.00000 −0.163868
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 1.65685 0.0541849
\(936\) 0 0
\(937\) −29.7990 −0.973491 −0.486745 0.873544i \(-0.661816\pi\)
−0.486745 + 0.873544i \(0.661816\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −7.75736 −0.252883 −0.126441 0.991974i \(-0.540356\pi\)
−0.126441 + 0.991974i \(0.540356\pi\)
\(942\) 0 0
\(943\) −47.7990 −1.55655
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −38.2843 −1.24407 −0.622036 0.782989i \(-0.713694\pi\)
−0.622036 + 0.782989i \(0.713694\pi\)
\(948\) 0 0
\(949\) −6.82843 −0.221660
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −36.9706 −1.19759 −0.598797 0.800901i \(-0.704354\pi\)
−0.598797 + 0.800901i \(0.704354\pi\)
\(954\) 0 0
\(955\) 13.7574 0.445178
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −22.1421 −0.715007
\(960\) 0 0
\(961\) −7.68629 −0.247945
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 13.0711 0.420773
\(966\) 0 0
\(967\) −17.4142 −0.560003 −0.280002 0.960000i \(-0.590335\pi\)
−0.280002 + 0.960000i \(0.590335\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −44.0000 −1.41203 −0.706014 0.708198i \(-0.749508\pi\)
−0.706014 + 0.708198i \(0.749508\pi\)
\(972\) 0 0
\(973\) −4.00000 −0.128234
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −3.51472 −0.112446 −0.0562229 0.998418i \(-0.517906\pi\)
−0.0562229 + 0.998418i \(0.517906\pi\)
\(978\) 0 0
\(979\) 7.17157 0.229204
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 51.9411 1.65666 0.828332 0.560237i \(-0.189290\pi\)
0.828332 + 0.560237i \(0.189290\pi\)
\(984\) 0 0
\(985\) −22.1421 −0.705507
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 53.4558 1.69980
\(990\) 0 0
\(991\) −21.6569 −0.687953 −0.343976 0.938978i \(-0.611774\pi\)
−0.343976 + 0.938978i \(0.611774\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −9.17157 −0.290758
\(996\) 0 0
\(997\) −23.4558 −0.742854 −0.371427 0.928462i \(-0.621131\pi\)
−0.371427 + 0.928462i \(0.621131\pi\)
\(998\) 0 0
\(999\) 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 6840.2.a.bb.1.2 2
3.2 odd 2 2280.2.a.k.1.2 2
12.11 even 2 4560.2.a.bn.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2280.2.a.k.1.2 2 3.2 odd 2
4560.2.a.bn.1.1 2 12.11 even 2
6840.2.a.bb.1.2 2 1.1 even 1 trivial