Properties

Label 6720.2.a.cr.1.2
Level $6720$
Weight $2$
Character 6720.1
Self dual yes
Analytic conductor $53.659$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6720 = 2^{6} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6720.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(53.6594701583\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 3360)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 6720.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +2.82843 q^{11} -0.828427 q^{13} -1.00000 q^{15} +7.65685 q^{17} +2.82843 q^{19} +1.00000 q^{21} -4.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} +7.65685 q^{29} -2.82843 q^{31} -2.82843 q^{33} -1.00000 q^{35} +11.6569 q^{37} +0.828427 q^{39} -7.65685 q^{41} -1.65685 q^{43} +1.00000 q^{45} +11.3137 q^{47} +1.00000 q^{49} -7.65685 q^{51} -10.4853 q^{53} +2.82843 q^{55} -2.82843 q^{57} -9.31371 q^{61} -1.00000 q^{63} -0.828427 q^{65} +4.00000 q^{67} +4.00000 q^{69} -10.8284 q^{71} -4.82843 q^{73} -1.00000 q^{75} -2.82843 q^{77} -15.3137 q^{79} +1.00000 q^{81} +17.6569 q^{83} +7.65685 q^{85} -7.65685 q^{87} +11.6569 q^{89} +0.828427 q^{91} +2.82843 q^{93} +2.82843 q^{95} -7.17157 q^{97} +2.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + 2q^{5} - 2q^{7} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} + 2q^{5} - 2q^{7} + 2q^{9} + 4q^{13} - 2q^{15} + 4q^{17} + 2q^{21} - 8q^{23} + 2q^{25} - 2q^{27} + 4q^{29} - 2q^{35} + 12q^{37} - 4q^{39} - 4q^{41} + 8q^{43} + 2q^{45} + 2q^{49} - 4q^{51} - 4q^{53} + 4q^{61} - 2q^{63} + 4q^{65} + 8q^{67} + 8q^{69} - 16q^{71} - 4q^{73} - 2q^{75} - 8q^{79} + 2q^{81} + 24q^{83} + 4q^{85} - 4q^{87} + 12q^{89} - 4q^{91} - 20q^{97} + O(q^{100}) \)

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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.82843 0.852803 0.426401 0.904534i \(-0.359781\pi\)
0.426401 + 0.904534i \(0.359781\pi\)
\(12\) 0 0
\(13\) −0.828427 −0.229764 −0.114882 0.993379i \(-0.536649\pi\)
−0.114882 + 0.993379i \(0.536649\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) 7.65685 1.85706 0.928530 0.371257i \(-0.121073\pi\)
0.928530 + 0.371257i \(0.121073\pi\)
\(18\) 0 0
\(19\) 2.82843 0.648886 0.324443 0.945905i \(-0.394823\pi\)
0.324443 + 0.945905i \(0.394823\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 7.65685 1.42184 0.710921 0.703272i \(-0.248278\pi\)
0.710921 + 0.703272i \(0.248278\pi\)
\(30\) 0 0
\(31\) −2.82843 −0.508001 −0.254000 0.967204i \(-0.581746\pi\)
−0.254000 + 0.967204i \(0.581746\pi\)
\(32\) 0 0
\(33\) −2.82843 −0.492366
\(34\) 0 0
\(35\) −1.00000 −0.169031
\(36\) 0 0
\(37\) 11.6569 1.91638 0.958188 0.286141i \(-0.0923726\pi\)
0.958188 + 0.286141i \(0.0923726\pi\)
\(38\) 0 0
\(39\) 0.828427 0.132655
\(40\) 0 0
\(41\) −7.65685 −1.19580 −0.597900 0.801571i \(-0.703998\pi\)
−0.597900 + 0.801571i \(0.703998\pi\)
\(42\) 0 0
\(43\) −1.65685 −0.252668 −0.126334 0.991988i \(-0.540321\pi\)
−0.126334 + 0.991988i \(0.540321\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) 11.3137 1.65027 0.825137 0.564933i \(-0.191098\pi\)
0.825137 + 0.564933i \(0.191098\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −7.65685 −1.07217
\(52\) 0 0
\(53\) −10.4853 −1.44026 −0.720132 0.693837i \(-0.755919\pi\)
−0.720132 + 0.693837i \(0.755919\pi\)
\(54\) 0 0
\(55\) 2.82843 0.381385
\(56\) 0 0
\(57\) −2.82843 −0.374634
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −9.31371 −1.19250 −0.596249 0.802799i \(-0.703343\pi\)
−0.596249 + 0.802799i \(0.703343\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) −0.828427 −0.102754
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 0 0
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) −10.8284 −1.28510 −0.642549 0.766245i \(-0.722123\pi\)
−0.642549 + 0.766245i \(0.722123\pi\)
\(72\) 0 0
\(73\) −4.82843 −0.565125 −0.282562 0.959249i \(-0.591184\pi\)
−0.282562 + 0.959249i \(0.591184\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) −2.82843 −0.322329
\(78\) 0 0
\(79\) −15.3137 −1.72293 −0.861463 0.507820i \(-0.830452\pi\)
−0.861463 + 0.507820i \(0.830452\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 17.6569 1.93809 0.969046 0.246881i \(-0.0794057\pi\)
0.969046 + 0.246881i \(0.0794057\pi\)
\(84\) 0 0
\(85\) 7.65685 0.830502
\(86\) 0 0
\(87\) −7.65685 −0.820901
\(88\) 0 0
\(89\) 11.6569 1.23562 0.617812 0.786326i \(-0.288019\pi\)
0.617812 + 0.786326i \(0.288019\pi\)
\(90\) 0 0
\(91\) 0.828427 0.0868428
\(92\) 0 0
\(93\) 2.82843 0.293294
\(94\) 0 0
\(95\) 2.82843 0.290191
\(96\) 0 0
\(97\) −7.17157 −0.728163 −0.364081 0.931367i \(-0.618617\pi\)
−0.364081 + 0.931367i \(0.618617\pi\)
\(98\) 0 0
\(99\) 2.82843 0.284268
\(100\) 0 0
\(101\) 17.3137 1.72278 0.861389 0.507946i \(-0.169595\pi\)
0.861389 + 0.507946i \(0.169595\pi\)
\(102\) 0 0
\(103\) −8.00000 −0.788263 −0.394132 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) 0 0
\(105\) 1.00000 0.0975900
\(106\) 0 0
\(107\) 8.00000 0.773389 0.386695 0.922208i \(-0.373617\pi\)
0.386695 + 0.922208i \(0.373617\pi\)
\(108\) 0 0
\(109\) 13.3137 1.27522 0.637611 0.770358i \(-0.279923\pi\)
0.637611 + 0.770358i \(0.279923\pi\)
\(110\) 0 0
\(111\) −11.6569 −1.10642
\(112\) 0 0
\(113\) 20.8284 1.95937 0.979687 0.200534i \(-0.0642676\pi\)
0.979687 + 0.200534i \(0.0642676\pi\)
\(114\) 0 0
\(115\) −4.00000 −0.373002
\(116\) 0 0
\(117\) −0.828427 −0.0765881
\(118\) 0 0
\(119\) −7.65685 −0.701903
\(120\) 0 0
\(121\) −3.00000 −0.272727
\(122\) 0 0
\(123\) 7.65685 0.690395
\(124\) 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
\(129\) 1.65685 0.145878
\(130\) 0 0
\(131\) −5.65685 −0.494242 −0.247121 0.968985i \(-0.579484\pi\)
−0.247121 + 0.968985i \(0.579484\pi\)
\(132\) 0 0
\(133\) −2.82843 −0.245256
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) −0.828427 −0.0707773 −0.0353887 0.999374i \(-0.511267\pi\)
−0.0353887 + 0.999374i \(0.511267\pi\)
\(138\) 0 0
\(139\) −19.7990 −1.67933 −0.839664 0.543106i \(-0.817248\pi\)
−0.839664 + 0.543106i \(0.817248\pi\)
\(140\) 0 0
\(141\) −11.3137 −0.952786
\(142\) 0 0
\(143\) −2.34315 −0.195944
\(144\) 0 0
\(145\) 7.65685 0.635867
\(146\) 0 0
\(147\) −1.00000 −0.0824786
\(148\) 0 0
\(149\) −17.3137 −1.41839 −0.709197 0.705010i \(-0.750942\pi\)
−0.709197 + 0.705010i \(0.750942\pi\)
\(150\) 0 0
\(151\) 1.65685 0.134833 0.0674164 0.997725i \(-0.478524\pi\)
0.0674164 + 0.997725i \(0.478524\pi\)
\(152\) 0 0
\(153\) 7.65685 0.619020
\(154\) 0 0
\(155\) −2.82843 −0.227185
\(156\) 0 0
\(157\) 12.8284 1.02382 0.511910 0.859039i \(-0.328938\pi\)
0.511910 + 0.859039i \(0.328938\pi\)
\(158\) 0 0
\(159\) 10.4853 0.831537
\(160\) 0 0
\(161\) 4.00000 0.315244
\(162\) 0 0
\(163\) 17.6569 1.38299 0.691496 0.722380i \(-0.256952\pi\)
0.691496 + 0.722380i \(0.256952\pi\)
\(164\) 0 0
\(165\) −2.82843 −0.220193
\(166\) 0 0
\(167\) −24.9706 −1.93228 −0.966140 0.258018i \(-0.916931\pi\)
−0.966140 + 0.258018i \(0.916931\pi\)
\(168\) 0 0
\(169\) −12.3137 −0.947208
\(170\) 0 0
\(171\) 2.82843 0.216295
\(172\) 0 0
\(173\) 14.9706 1.13819 0.569095 0.822272i \(-0.307294\pi\)
0.569095 + 0.822272i \(0.307294\pi\)
\(174\) 0 0
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −5.17157 −0.386542 −0.193271 0.981145i \(-0.561910\pi\)
−0.193271 + 0.981145i \(0.561910\pi\)
\(180\) 0 0
\(181\) 12.3431 0.917459 0.458729 0.888576i \(-0.348305\pi\)
0.458729 + 0.888576i \(0.348305\pi\)
\(182\) 0 0
\(183\) 9.31371 0.688489
\(184\) 0 0
\(185\) 11.6569 0.857029
\(186\) 0 0
\(187\) 21.6569 1.58371
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) −13.1716 −0.953062 −0.476531 0.879158i \(-0.658106\pi\)
−0.476531 + 0.879158i \(0.658106\pi\)
\(192\) 0 0
\(193\) 10.0000 0.719816 0.359908 0.932988i \(-0.382808\pi\)
0.359908 + 0.932988i \(0.382808\pi\)
\(194\) 0 0
\(195\) 0.828427 0.0593249
\(196\) 0 0
\(197\) −12.8284 −0.913988 −0.456994 0.889470i \(-0.651074\pi\)
−0.456994 + 0.889470i \(0.651074\pi\)
\(198\) 0 0
\(199\) 8.48528 0.601506 0.300753 0.953702i \(-0.402762\pi\)
0.300753 + 0.953702i \(0.402762\pi\)
\(200\) 0 0
\(201\) −4.00000 −0.282138
\(202\) 0 0
\(203\) −7.65685 −0.537406
\(204\) 0 0
\(205\) −7.65685 −0.534778
\(206\) 0 0
\(207\) −4.00000 −0.278019
\(208\) 0 0
\(209\) 8.00000 0.553372
\(210\) 0 0
\(211\) 20.0000 1.37686 0.688428 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) 0 0
\(213\) 10.8284 0.741952
\(214\) 0 0
\(215\) −1.65685 −0.112997
\(216\) 0 0
\(217\) 2.82843 0.192006
\(218\) 0 0
\(219\) 4.82843 0.326275
\(220\) 0 0
\(221\) −6.34315 −0.426686
\(222\) 0 0
\(223\) 16.9706 1.13643 0.568216 0.822879i \(-0.307634\pi\)
0.568216 + 0.822879i \(0.307634\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 6.34315 0.421009 0.210505 0.977593i \(-0.432489\pi\)
0.210505 + 0.977593i \(0.432489\pi\)
\(228\) 0 0
\(229\) −2.68629 −0.177515 −0.0887576 0.996053i \(-0.528290\pi\)
−0.0887576 + 0.996053i \(0.528290\pi\)
\(230\) 0 0
\(231\) 2.82843 0.186097
\(232\) 0 0
\(233\) 28.8284 1.88861 0.944307 0.329067i \(-0.106734\pi\)
0.944307 + 0.329067i \(0.106734\pi\)
\(234\) 0 0
\(235\) 11.3137 0.738025
\(236\) 0 0
\(237\) 15.3137 0.994732
\(238\) 0 0
\(239\) 13.1716 0.851998 0.425999 0.904724i \(-0.359923\pi\)
0.425999 + 0.904724i \(0.359923\pi\)
\(240\) 0 0
\(241\) −3.65685 −0.235559 −0.117779 0.993040i \(-0.537578\pi\)
−0.117779 + 0.993040i \(0.537578\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 1.00000 0.0638877
\(246\) 0 0
\(247\) −2.34315 −0.149091
\(248\) 0 0
\(249\) −17.6569 −1.11896
\(250\) 0 0
\(251\) −11.3137 −0.714115 −0.357057 0.934082i \(-0.616220\pi\)
−0.357057 + 0.934082i \(0.616220\pi\)
\(252\) 0 0
\(253\) −11.3137 −0.711287
\(254\) 0 0
\(255\) −7.65685 −0.479491
\(256\) 0 0
\(257\) 6.68629 0.417079 0.208540 0.978014i \(-0.433129\pi\)
0.208540 + 0.978014i \(0.433129\pi\)
\(258\) 0 0
\(259\) −11.6569 −0.724322
\(260\) 0 0
\(261\) 7.65685 0.473947
\(262\) 0 0
\(263\) −20.9706 −1.29310 −0.646550 0.762871i \(-0.723789\pi\)
−0.646550 + 0.762871i \(0.723789\pi\)
\(264\) 0 0
\(265\) −10.4853 −0.644106
\(266\) 0 0
\(267\) −11.6569 −0.713388
\(268\) 0 0
\(269\) −5.31371 −0.323983 −0.161991 0.986792i \(-0.551792\pi\)
−0.161991 + 0.986792i \(0.551792\pi\)
\(270\) 0 0
\(271\) 10.8284 0.657780 0.328890 0.944368i \(-0.393325\pi\)
0.328890 + 0.944368i \(0.393325\pi\)
\(272\) 0 0
\(273\) −0.828427 −0.0501387
\(274\) 0 0
\(275\) 2.82843 0.170561
\(276\) 0 0
\(277\) −2.97056 −0.178484 −0.0892419 0.996010i \(-0.528444\pi\)
−0.0892419 + 0.996010i \(0.528444\pi\)
\(278\) 0 0
\(279\) −2.82843 −0.169334
\(280\) 0 0
\(281\) 18.9706 1.13169 0.565844 0.824512i \(-0.308550\pi\)
0.565844 + 0.824512i \(0.308550\pi\)
\(282\) 0 0
\(283\) 15.3137 0.910305 0.455153 0.890413i \(-0.349585\pi\)
0.455153 + 0.890413i \(0.349585\pi\)
\(284\) 0 0
\(285\) −2.82843 −0.167542
\(286\) 0 0
\(287\) 7.65685 0.451970
\(288\) 0 0
\(289\) 41.6274 2.44867
\(290\) 0 0
\(291\) 7.17157 0.420405
\(292\) 0 0
\(293\) 19.6569 1.14837 0.574183 0.818727i \(-0.305320\pi\)
0.574183 + 0.818727i \(0.305320\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −2.82843 −0.164122
\(298\) 0 0
\(299\) 3.31371 0.191637
\(300\) 0 0
\(301\) 1.65685 0.0954995
\(302\) 0 0
\(303\) −17.3137 −0.994647
\(304\) 0 0
\(305\) −9.31371 −0.533301
\(306\) 0 0
\(307\) 17.6569 1.00773 0.503865 0.863782i \(-0.331911\pi\)
0.503865 + 0.863782i \(0.331911\pi\)
\(308\) 0 0
\(309\) 8.00000 0.455104
\(310\) 0 0
\(311\) −8.00000 −0.453638 −0.226819 0.973937i \(-0.572833\pi\)
−0.226819 + 0.973937i \(0.572833\pi\)
\(312\) 0 0
\(313\) 0.828427 0.0468255 0.0234127 0.999726i \(-0.492547\pi\)
0.0234127 + 0.999726i \(0.492547\pi\)
\(314\) 0 0
\(315\) −1.00000 −0.0563436
\(316\) 0 0
\(317\) −7.17157 −0.402796 −0.201398 0.979510i \(-0.564548\pi\)
−0.201398 + 0.979510i \(0.564548\pi\)
\(318\) 0 0
\(319\) 21.6569 1.21255
\(320\) 0 0
\(321\) −8.00000 −0.446516
\(322\) 0 0
\(323\) 21.6569 1.20502
\(324\) 0 0
\(325\) −0.828427 −0.0459529
\(326\) 0 0
\(327\) −13.3137 −0.736250
\(328\) 0 0
\(329\) −11.3137 −0.623745
\(330\) 0 0
\(331\) 4.00000 0.219860 0.109930 0.993939i \(-0.464937\pi\)
0.109930 + 0.993939i \(0.464937\pi\)
\(332\) 0 0
\(333\) 11.6569 0.638792
\(334\) 0 0
\(335\) 4.00000 0.218543
\(336\) 0 0
\(337\) −1.31371 −0.0715623 −0.0357811 0.999360i \(-0.511392\pi\)
−0.0357811 + 0.999360i \(0.511392\pi\)
\(338\) 0 0
\(339\) −20.8284 −1.13124
\(340\) 0 0
\(341\) −8.00000 −0.433224
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 4.00000 0.215353
\(346\) 0 0
\(347\) −29.6569 −1.59206 −0.796032 0.605255i \(-0.793071\pi\)
−0.796032 + 0.605255i \(0.793071\pi\)
\(348\) 0 0
\(349\) 6.68629 0.357909 0.178954 0.983857i \(-0.442729\pi\)
0.178954 + 0.983857i \(0.442729\pi\)
\(350\) 0 0
\(351\) 0.828427 0.0442182
\(352\) 0 0
\(353\) −6.97056 −0.371006 −0.185503 0.982644i \(-0.559391\pi\)
−0.185503 + 0.982644i \(0.559391\pi\)
\(354\) 0 0
\(355\) −10.8284 −0.574713
\(356\) 0 0
\(357\) 7.65685 0.405244
\(358\) 0 0
\(359\) 33.4558 1.76573 0.882866 0.469625i \(-0.155611\pi\)
0.882866 + 0.469625i \(0.155611\pi\)
\(360\) 0 0
\(361\) −11.0000 −0.578947
\(362\) 0 0
\(363\) 3.00000 0.157459
\(364\) 0 0
\(365\) −4.82843 −0.252731
\(366\) 0 0
\(367\) 29.6569 1.54808 0.774038 0.633140i \(-0.218234\pi\)
0.774038 + 0.633140i \(0.218234\pi\)
\(368\) 0 0
\(369\) −7.65685 −0.398600
\(370\) 0 0
\(371\) 10.4853 0.544369
\(372\) 0 0
\(373\) −10.0000 −0.517780 −0.258890 0.965907i \(-0.583357\pi\)
−0.258890 + 0.965907i \(0.583357\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) −6.34315 −0.326689
\(378\) 0 0
\(379\) 1.65685 0.0851069 0.0425534 0.999094i \(-0.486451\pi\)
0.0425534 + 0.999094i \(0.486451\pi\)
\(380\) 0 0
\(381\) 5.65685 0.289809
\(382\) 0 0
\(383\) −28.2843 −1.44526 −0.722629 0.691236i \(-0.757067\pi\)
−0.722629 + 0.691236i \(0.757067\pi\)
\(384\) 0 0
\(385\) −2.82843 −0.144150
\(386\) 0 0
\(387\) −1.65685 −0.0842226
\(388\) 0 0
\(389\) −9.31371 −0.472224 −0.236112 0.971726i \(-0.575873\pi\)
−0.236112 + 0.971726i \(0.575873\pi\)
\(390\) 0 0
\(391\) −30.6274 −1.54890
\(392\) 0 0
\(393\) 5.65685 0.285351
\(394\) 0 0
\(395\) −15.3137 −0.770516
\(396\) 0 0
\(397\) 2.48528 0.124733 0.0623663 0.998053i \(-0.480135\pi\)
0.0623663 + 0.998053i \(0.480135\pi\)
\(398\) 0 0
\(399\) 2.82843 0.141598
\(400\) 0 0
\(401\) −1.31371 −0.0656035 −0.0328017 0.999462i \(-0.510443\pi\)
−0.0328017 + 0.999462i \(0.510443\pi\)
\(402\) 0 0
\(403\) 2.34315 0.116720
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) 32.9706 1.63429
\(408\) 0 0
\(409\) 13.3137 0.658321 0.329160 0.944274i \(-0.393234\pi\)
0.329160 + 0.944274i \(0.393234\pi\)
\(410\) 0 0
\(411\) 0.828427 0.0408633
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 17.6569 0.866741
\(416\) 0 0
\(417\) 19.7990 0.969561
\(418\) 0 0
\(419\) 16.9706 0.829066 0.414533 0.910034i \(-0.363945\pi\)
0.414533 + 0.910034i \(0.363945\pi\)
\(420\) 0 0
\(421\) 21.3137 1.03877 0.519383 0.854541i \(-0.326162\pi\)
0.519383 + 0.854541i \(0.326162\pi\)
\(422\) 0 0
\(423\) 11.3137 0.550091
\(424\) 0 0
\(425\) 7.65685 0.371412
\(426\) 0 0
\(427\) 9.31371 0.450722
\(428\) 0 0
\(429\) 2.34315 0.113128
\(430\) 0 0
\(431\) 0.485281 0.0233752 0.0116876 0.999932i \(-0.496280\pi\)
0.0116876 + 0.999932i \(0.496280\pi\)
\(432\) 0 0
\(433\) 13.5147 0.649476 0.324738 0.945804i \(-0.394724\pi\)
0.324738 + 0.945804i \(0.394724\pi\)
\(434\) 0 0
\(435\) −7.65685 −0.367118
\(436\) 0 0
\(437\) −11.3137 −0.541208
\(438\) 0 0
\(439\) 13.1716 0.628645 0.314322 0.949316i \(-0.398223\pi\)
0.314322 + 0.949316i \(0.398223\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −24.9706 −1.18639 −0.593194 0.805060i \(-0.702133\pi\)
−0.593194 + 0.805060i \(0.702133\pi\)
\(444\) 0 0
\(445\) 11.6569 0.552588
\(446\) 0 0
\(447\) 17.3137 0.818910
\(448\) 0 0
\(449\) 2.00000 0.0943858 0.0471929 0.998886i \(-0.484972\pi\)
0.0471929 + 0.998886i \(0.484972\pi\)
\(450\) 0 0
\(451\) −21.6569 −1.01978
\(452\) 0 0
\(453\) −1.65685 −0.0778458
\(454\) 0 0
\(455\) 0.828427 0.0388373
\(456\) 0 0
\(457\) −8.34315 −0.390276 −0.195138 0.980776i \(-0.562515\pi\)
−0.195138 + 0.980776i \(0.562515\pi\)
\(458\) 0 0
\(459\) −7.65685 −0.357391
\(460\) 0 0
\(461\) −16.6274 −0.774416 −0.387208 0.921992i \(-0.626560\pi\)
−0.387208 + 0.921992i \(0.626560\pi\)
\(462\) 0 0
\(463\) −20.2843 −0.942690 −0.471345 0.881949i \(-0.656231\pi\)
−0.471345 + 0.881949i \(0.656231\pi\)
\(464\) 0 0
\(465\) 2.82843 0.131165
\(466\) 0 0
\(467\) −28.9706 −1.34060 −0.670299 0.742091i \(-0.733834\pi\)
−0.670299 + 0.742091i \(0.733834\pi\)
\(468\) 0 0
\(469\) −4.00000 −0.184703
\(470\) 0 0
\(471\) −12.8284 −0.591103
\(472\) 0 0
\(473\) −4.68629 −0.215476
\(474\) 0 0
\(475\) 2.82843 0.129777
\(476\) 0 0
\(477\) −10.4853 −0.480088
\(478\) 0 0
\(479\) −3.31371 −0.151407 −0.0757036 0.997130i \(-0.524120\pi\)
−0.0757036 + 0.997130i \(0.524120\pi\)
\(480\) 0 0
\(481\) −9.65685 −0.440315
\(482\) 0 0
\(483\) −4.00000 −0.182006
\(484\) 0 0
\(485\) −7.17157 −0.325644
\(486\) 0 0
\(487\) 5.65685 0.256337 0.128168 0.991752i \(-0.459090\pi\)
0.128168 + 0.991752i \(0.459090\pi\)
\(488\) 0 0
\(489\) −17.6569 −0.798471
\(490\) 0 0
\(491\) 33.4558 1.50984 0.754921 0.655816i \(-0.227675\pi\)
0.754921 + 0.655816i \(0.227675\pi\)
\(492\) 0 0
\(493\) 58.6274 2.64045
\(494\) 0 0
\(495\) 2.82843 0.127128
\(496\) 0 0
\(497\) 10.8284 0.485721
\(498\) 0 0
\(499\) −6.34315 −0.283958 −0.141979 0.989870i \(-0.545347\pi\)
−0.141979 + 0.989870i \(0.545347\pi\)
\(500\) 0 0
\(501\) 24.9706 1.11560
\(502\) 0 0
\(503\) 39.5980 1.76559 0.882793 0.469762i \(-0.155660\pi\)
0.882793 + 0.469762i \(0.155660\pi\)
\(504\) 0 0
\(505\) 17.3137 0.770450
\(506\) 0 0
\(507\) 12.3137 0.546871
\(508\) 0 0
\(509\) −21.3137 −0.944714 −0.472357 0.881407i \(-0.656597\pi\)
−0.472357 + 0.881407i \(0.656597\pi\)
\(510\) 0 0
\(511\) 4.82843 0.213597
\(512\) 0 0
\(513\) −2.82843 −0.124878
\(514\) 0 0
\(515\) −8.00000 −0.352522
\(516\) 0 0
\(517\) 32.0000 1.40736
\(518\) 0 0
\(519\) −14.9706 −0.657135
\(520\) 0 0
\(521\) 14.9706 0.655872 0.327936 0.944700i \(-0.393647\pi\)
0.327936 + 0.944700i \(0.393647\pi\)
\(522\) 0 0
\(523\) 22.3431 0.976998 0.488499 0.872565i \(-0.337545\pi\)
0.488499 + 0.872565i \(0.337545\pi\)
\(524\) 0 0
\(525\) 1.00000 0.0436436
\(526\) 0 0
\(527\) −21.6569 −0.943387
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 6.34315 0.274752
\(534\) 0 0
\(535\) 8.00000 0.345870
\(536\) 0 0
\(537\) 5.17157 0.223170
\(538\) 0 0
\(539\) 2.82843 0.121829
\(540\) 0 0
\(541\) 0.627417 0.0269748 0.0134874 0.999909i \(-0.495707\pi\)
0.0134874 + 0.999909i \(0.495707\pi\)
\(542\) 0 0
\(543\) −12.3431 −0.529695
\(544\) 0 0
\(545\) 13.3137 0.570297
\(546\) 0 0
\(547\) 20.0000 0.855138 0.427569 0.903983i \(-0.359370\pi\)
0.427569 + 0.903983i \(0.359370\pi\)
\(548\) 0 0
\(549\) −9.31371 −0.397499
\(550\) 0 0
\(551\) 21.6569 0.922613
\(552\) 0 0
\(553\) 15.3137 0.651205
\(554\) 0 0
\(555\) −11.6569 −0.494806
\(556\) 0 0
\(557\) −15.1716 −0.642840 −0.321420 0.946937i \(-0.604160\pi\)
−0.321420 + 0.946937i \(0.604160\pi\)
\(558\) 0 0
\(559\) 1.37258 0.0580541
\(560\) 0 0
\(561\) −21.6569 −0.914353
\(562\) 0 0
\(563\) 0.686292 0.0289237 0.0144619 0.999895i \(-0.495396\pi\)
0.0144619 + 0.999895i \(0.495396\pi\)
\(564\) 0 0
\(565\) 20.8284 0.876259
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) −20.6274 −0.864746 −0.432373 0.901695i \(-0.642324\pi\)
−0.432373 + 0.901695i \(0.642324\pi\)
\(570\) 0 0
\(571\) −35.5980 −1.48973 −0.744865 0.667216i \(-0.767486\pi\)
−0.744865 + 0.667216i \(0.767486\pi\)
\(572\) 0 0
\(573\) 13.1716 0.550250
\(574\) 0 0
\(575\) −4.00000 −0.166812
\(576\) 0 0
\(577\) 0.828427 0.0344879 0.0172439 0.999851i \(-0.494511\pi\)
0.0172439 + 0.999851i \(0.494511\pi\)
\(578\) 0 0
\(579\) −10.0000 −0.415586
\(580\) 0 0
\(581\) −17.6569 −0.732530
\(582\) 0 0
\(583\) −29.6569 −1.22826
\(584\) 0 0
\(585\) −0.828427 −0.0342512
\(586\) 0 0
\(587\) 15.3137 0.632064 0.316032 0.948748i \(-0.397649\pi\)
0.316032 + 0.948748i \(0.397649\pi\)
\(588\) 0 0
\(589\) −8.00000 −0.329634
\(590\) 0 0
\(591\) 12.8284 0.527691
\(592\) 0 0
\(593\) 39.6569 1.62851 0.814256 0.580506i \(-0.197145\pi\)
0.814256 + 0.580506i \(0.197145\pi\)
\(594\) 0 0
\(595\) −7.65685 −0.313900
\(596\) 0 0
\(597\) −8.48528 −0.347279
\(598\) 0 0
\(599\) 15.5147 0.633914 0.316957 0.948440i \(-0.397339\pi\)
0.316957 + 0.948440i \(0.397339\pi\)
\(600\) 0 0
\(601\) −24.3431 −0.992978 −0.496489 0.868043i \(-0.665378\pi\)
−0.496489 + 0.868043i \(0.665378\pi\)
\(602\) 0 0
\(603\) 4.00000 0.162893
\(604\) 0 0
\(605\) −3.00000 −0.121967
\(606\) 0 0
\(607\) 18.3431 0.744525 0.372263 0.928127i \(-0.378582\pi\)
0.372263 + 0.928127i \(0.378582\pi\)
\(608\) 0 0
\(609\) 7.65685 0.310271
\(610\) 0 0
\(611\) −9.37258 −0.379174
\(612\) 0 0
\(613\) 27.6569 1.11705 0.558525 0.829488i \(-0.311368\pi\)
0.558525 + 0.829488i \(0.311368\pi\)
\(614\) 0 0
\(615\) 7.65685 0.308754
\(616\) 0 0
\(617\) 17.5147 0.705116 0.352558 0.935790i \(-0.385312\pi\)
0.352558 + 0.935790i \(0.385312\pi\)
\(618\) 0 0
\(619\) 9.45584 0.380062 0.190031 0.981778i \(-0.439141\pi\)
0.190031 + 0.981778i \(0.439141\pi\)
\(620\) 0 0
\(621\) 4.00000 0.160514
\(622\) 0 0
\(623\) −11.6569 −0.467022
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −8.00000 −0.319489
\(628\) 0 0
\(629\) 89.2548 3.55882
\(630\) 0 0
\(631\) 26.6274 1.06002 0.530010 0.847991i \(-0.322188\pi\)
0.530010 + 0.847991i \(0.322188\pi\)
\(632\) 0 0
\(633\) −20.0000 −0.794929
\(634\) 0 0
\(635\) −5.65685 −0.224485
\(636\) 0 0
\(637\) −0.828427 −0.0328235
\(638\) 0 0
\(639\) −10.8284 −0.428366
\(640\) 0 0
\(641\) −0.343146 −0.0135534 −0.00677672 0.999977i \(-0.502157\pi\)
−0.00677672 + 0.999977i \(0.502157\pi\)
\(642\) 0 0
\(643\) 33.6569 1.32730 0.663648 0.748045i \(-0.269007\pi\)
0.663648 + 0.748045i \(0.269007\pi\)
\(644\) 0 0
\(645\) 1.65685 0.0652386
\(646\) 0 0
\(647\) −24.9706 −0.981694 −0.490847 0.871246i \(-0.663313\pi\)
−0.490847 + 0.871246i \(0.663313\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −2.82843 −0.110855
\(652\) 0 0
\(653\) −9.51472 −0.372340 −0.186170 0.982518i \(-0.559607\pi\)
−0.186170 + 0.982518i \(0.559607\pi\)
\(654\) 0 0
\(655\) −5.65685 −0.221032
\(656\) 0 0
\(657\) −4.82843 −0.188375
\(658\) 0 0
\(659\) −36.7696 −1.43234 −0.716169 0.697927i \(-0.754106\pi\)
−0.716169 + 0.697927i \(0.754106\pi\)
\(660\) 0 0
\(661\) −6.00000 −0.233373 −0.116686 0.993169i \(-0.537227\pi\)
−0.116686 + 0.993169i \(0.537227\pi\)
\(662\) 0 0
\(663\) 6.34315 0.246347
\(664\) 0 0
\(665\) −2.82843 −0.109682
\(666\) 0 0
\(667\) −30.6274 −1.18590
\(668\) 0 0
\(669\) −16.9706 −0.656120
\(670\) 0 0
\(671\) −26.3431 −1.01697
\(672\) 0 0
\(673\) 28.3431 1.09255 0.546274 0.837607i \(-0.316046\pi\)
0.546274 + 0.837607i \(0.316046\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) 6.97056 0.267900 0.133950 0.990988i \(-0.457234\pi\)
0.133950 + 0.990988i \(0.457234\pi\)
\(678\) 0 0
\(679\) 7.17157 0.275220
\(680\) 0 0
\(681\) −6.34315 −0.243070
\(682\) 0 0
\(683\) −10.3431 −0.395769 −0.197885 0.980225i \(-0.563407\pi\)
−0.197885 + 0.980225i \(0.563407\pi\)
\(684\) 0 0
\(685\) −0.828427 −0.0316526
\(686\) 0 0
\(687\) 2.68629 0.102488
\(688\) 0 0
\(689\) 8.68629 0.330921
\(690\) 0 0
\(691\) 51.7990 1.97053 0.985263 0.171045i \(-0.0547143\pi\)
0.985263 + 0.171045i \(0.0547143\pi\)
\(692\) 0 0
\(693\) −2.82843 −0.107443
\(694\) 0 0
\(695\) −19.7990 −0.751018
\(696\) 0 0
\(697\) −58.6274 −2.22067
\(698\) 0 0
\(699\) −28.8284 −1.09039
\(700\) 0 0
\(701\) −13.0294 −0.492115 −0.246058 0.969255i \(-0.579135\pi\)
−0.246058 + 0.969255i \(0.579135\pi\)
\(702\) 0 0
\(703\) 32.9706 1.24351
\(704\) 0 0
\(705\) −11.3137 −0.426099
\(706\) 0 0
\(707\) −17.3137 −0.651149
\(708\) 0 0
\(709\) 5.31371 0.199561 0.0997803 0.995009i \(-0.468186\pi\)
0.0997803 + 0.995009i \(0.468186\pi\)
\(710\) 0 0
\(711\) −15.3137 −0.574309
\(712\) 0 0
\(713\) 11.3137 0.423702
\(714\) 0 0
\(715\) −2.34315 −0.0876287
\(716\) 0 0
\(717\) −13.1716 −0.491901
\(718\) 0 0
\(719\) 37.6569 1.40436 0.702182 0.711998i \(-0.252209\pi\)
0.702182 + 0.711998i \(0.252209\pi\)
\(720\) 0 0
\(721\) 8.00000 0.297936
\(722\) 0 0
\(723\) 3.65685 0.136000
\(724\) 0 0
\(725\) 7.65685 0.284368
\(726\) 0 0
\(727\) 37.6569 1.39662 0.698308 0.715798i \(-0.253937\pi\)
0.698308 + 0.715798i \(0.253937\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −12.6863 −0.469219
\(732\) 0 0
\(733\) −42.7696 −1.57973 −0.789865 0.613281i \(-0.789849\pi\)
−0.789865 + 0.613281i \(0.789849\pi\)
\(734\) 0 0
\(735\) −1.00000 −0.0368856
\(736\) 0 0
\(737\) 11.3137 0.416746
\(738\) 0 0
\(739\) −25.6569 −0.943803 −0.471901 0.881651i \(-0.656432\pi\)
−0.471901 + 0.881651i \(0.656432\pi\)
\(740\) 0 0
\(741\) 2.34315 0.0860776
\(742\) 0 0
\(743\) −14.3431 −0.526199 −0.263099 0.964769i \(-0.584745\pi\)
−0.263099 + 0.964769i \(0.584745\pi\)
\(744\) 0 0
\(745\) −17.3137 −0.634325
\(746\) 0 0
\(747\) 17.6569 0.646031
\(748\) 0 0
\(749\) −8.00000 −0.292314
\(750\) 0 0
\(751\) 20.0000 0.729810 0.364905 0.931045i \(-0.381101\pi\)
0.364905 + 0.931045i \(0.381101\pi\)
\(752\) 0 0
\(753\) 11.3137 0.412294
\(754\) 0 0
\(755\) 1.65685 0.0602991
\(756\) 0 0
\(757\) 22.0000 0.799604 0.399802 0.916602i \(-0.369079\pi\)
0.399802 + 0.916602i \(0.369079\pi\)
\(758\) 0 0
\(759\) 11.3137 0.410662
\(760\) 0 0
\(761\) 3.65685 0.132561 0.0662804 0.997801i \(-0.478887\pi\)
0.0662804 + 0.997801i \(0.478887\pi\)
\(762\) 0 0
\(763\) −13.3137 −0.481989
\(764\) 0 0
\(765\) 7.65685 0.276834
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −22.9706 −0.828340 −0.414170 0.910200i \(-0.635928\pi\)
−0.414170 + 0.910200i \(0.635928\pi\)
\(770\) 0 0
\(771\) −6.68629 −0.240801
\(772\) 0 0
\(773\) −2.97056 −0.106844 −0.0534219 0.998572i \(-0.517013\pi\)
−0.0534219 + 0.998572i \(0.517013\pi\)
\(774\) 0 0
\(775\) −2.82843 −0.101600
\(776\) 0 0
\(777\) 11.6569 0.418187
\(778\) 0 0
\(779\) −21.6569 −0.775937
\(780\) 0 0
\(781\) −30.6274 −1.09594
\(782\) 0 0
\(783\) −7.65685 −0.273634
\(784\) 0 0
\(785\) 12.8284 0.457866
\(786\) 0 0
\(787\) 30.3431 1.08162 0.540808 0.841146i \(-0.318118\pi\)
0.540808 + 0.841146i \(0.318118\pi\)
\(788\) 0 0
\(789\) 20.9706 0.746572
\(790\) 0 0
\(791\) −20.8284 −0.740574
\(792\) 0 0
\(793\) 7.71573 0.273994
\(794\) 0 0
\(795\) 10.4853 0.371875
\(796\) 0 0
\(797\) −1.02944 −0.0364645 −0.0182323 0.999834i \(-0.505804\pi\)
−0.0182323 + 0.999834i \(0.505804\pi\)
\(798\) 0 0
\(799\) 86.6274 3.06466
\(800\) 0 0
\(801\) 11.6569 0.411875
\(802\) 0 0
\(803\) −13.6569 −0.481940
\(804\) 0 0
\(805\) 4.00000 0.140981
\(806\) 0 0
\(807\) 5.31371 0.187051
\(808\) 0 0
\(809\) −38.9706 −1.37013 −0.685066 0.728481i \(-0.740227\pi\)
−0.685066 + 0.728481i \(0.740227\pi\)
\(810\) 0 0
\(811\) 1.45584 0.0511216 0.0255608 0.999673i \(-0.491863\pi\)
0.0255608 + 0.999673i \(0.491863\pi\)
\(812\) 0 0
\(813\) −10.8284 −0.379770
\(814\) 0 0
\(815\) 17.6569 0.618493
\(816\) 0 0
\(817\) −4.68629 −0.163953
\(818\) 0 0
\(819\) 0.828427 0.0289476
\(820\) 0 0
\(821\) −10.2843 −0.358924 −0.179462 0.983765i \(-0.557436\pi\)
−0.179462 + 0.983765i \(0.557436\pi\)
\(822\) 0 0
\(823\) −20.2843 −0.707065 −0.353533 0.935422i \(-0.615020\pi\)
−0.353533 + 0.935422i \(0.615020\pi\)
\(824\) 0 0
\(825\) −2.82843 −0.0984732
\(826\) 0 0
\(827\) 45.2548 1.57366 0.786832 0.617167i \(-0.211720\pi\)
0.786832 + 0.617167i \(0.211720\pi\)
\(828\) 0 0
\(829\) −34.2843 −1.19074 −0.595371 0.803451i \(-0.702995\pi\)
−0.595371 + 0.803451i \(0.702995\pi\)
\(830\) 0 0
\(831\) 2.97056 0.103048
\(832\) 0 0
\(833\) 7.65685 0.265294
\(834\) 0 0
\(835\) −24.9706 −0.864142
\(836\) 0 0
\(837\) 2.82843 0.0977647
\(838\) 0 0
\(839\) 48.9706 1.69065 0.845326 0.534251i \(-0.179406\pi\)
0.845326 + 0.534251i \(0.179406\pi\)
\(840\) 0 0
\(841\) 29.6274 1.02164
\(842\) 0 0
\(843\) −18.9706 −0.653381
\(844\) 0 0
\(845\) −12.3137 −0.423604
\(846\) 0 0
\(847\) 3.00000 0.103081
\(848\) 0 0
\(849\) −15.3137 −0.525565
\(850\) 0 0
\(851\) −46.6274 −1.59837
\(852\) 0 0
\(853\) 31.1716 1.06729 0.533647 0.845707i \(-0.320821\pi\)
0.533647 + 0.845707i \(0.320821\pi\)
\(854\) 0 0
\(855\) 2.82843 0.0967302
\(856\) 0 0
\(857\) −23.9411 −0.817813 −0.408907 0.912576i \(-0.634090\pi\)
−0.408907 + 0.912576i \(0.634090\pi\)
\(858\) 0 0
\(859\) −17.4558 −0.595586 −0.297793 0.954631i \(-0.596250\pi\)
−0.297793 + 0.954631i \(0.596250\pi\)
\(860\) 0 0
\(861\) −7.65685 −0.260945
\(862\) 0 0
\(863\) −36.0000 −1.22545 −0.612727 0.790295i \(-0.709928\pi\)
−0.612727 + 0.790295i \(0.709928\pi\)
\(864\) 0 0
\(865\) 14.9706 0.509014
\(866\) 0 0
\(867\) −41.6274 −1.41374
\(868\) 0 0
\(869\) −43.3137 −1.46932
\(870\) 0 0
\(871\) −3.31371 −0.112281
\(872\) 0 0
\(873\) −7.17157 −0.242721
\(874\) 0 0
\(875\) −1.00000 −0.0338062
\(876\) 0 0
\(877\) −54.2843 −1.83305 −0.916525 0.399978i \(-0.869018\pi\)
−0.916525 + 0.399978i \(0.869018\pi\)
\(878\) 0 0
\(879\) −19.6569 −0.663009
\(880\) 0 0
\(881\) −42.9706 −1.44772 −0.723858 0.689949i \(-0.757633\pi\)
−0.723858 + 0.689949i \(0.757633\pi\)
\(882\) 0 0
\(883\) −18.6274 −0.626862 −0.313431 0.949611i \(-0.601479\pi\)
−0.313431 + 0.949611i \(0.601479\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 12.6863 0.425964 0.212982 0.977056i \(-0.431682\pi\)
0.212982 + 0.977056i \(0.431682\pi\)
\(888\) 0 0
\(889\) 5.65685 0.189725
\(890\) 0 0
\(891\) 2.82843 0.0947559
\(892\) 0 0
\(893\) 32.0000 1.07084
\(894\) 0 0
\(895\) −5.17157 −0.172867
\(896\) 0 0
\(897\) −3.31371 −0.110642
\(898\) 0 0
\(899\) −21.6569 −0.722297
\(900\) 0 0
\(901\) −80.2843 −2.67466
\(902\) 0 0
\(903\) −1.65685 −0.0551367
\(904\) 0 0
\(905\) 12.3431 0.410300
\(906\) 0 0
\(907\) 10.6274 0.352878 0.176439 0.984312i \(-0.443542\pi\)
0.176439 + 0.984312i \(0.443542\pi\)
\(908\) 0 0
\(909\) 17.3137 0.574259
\(910\) 0 0
\(911\) −16.4853 −0.546182 −0.273091 0.961988i \(-0.588046\pi\)
−0.273091 + 0.961988i \(0.588046\pi\)
\(912\) 0 0
\(913\) 49.9411 1.65281
\(914\) 0 0
\(915\) 9.31371 0.307902
\(916\) 0 0
\(917\) 5.65685 0.186806
\(918\) 0 0
\(919\) 23.3137 0.769048 0.384524 0.923115i \(-0.374366\pi\)
0.384524 + 0.923115i \(0.374366\pi\)
\(920\) 0 0
\(921\) −17.6569 −0.581813
\(922\) 0 0
\(923\) 8.97056 0.295270
\(924\) 0 0
\(925\) 11.6569 0.383275
\(926\) 0 0
\(927\) −8.00000 −0.262754
\(928\) 0 0
\(929\) 34.2843 1.12483 0.562415 0.826855i \(-0.309872\pi\)
0.562415 + 0.826855i \(0.309872\pi\)
\(930\) 0 0
\(931\) 2.82843 0.0926980
\(932\) 0 0
\(933\) 8.00000 0.261908
\(934\) 0 0
\(935\) 21.6569 0.708255
\(936\) 0 0
\(937\) −41.5147 −1.35623 −0.678113 0.734957i \(-0.737202\pi\)
−0.678113 + 0.734957i \(0.737202\pi\)
\(938\) 0 0
\(939\) −0.828427 −0.0270347
\(940\) 0 0
\(941\) −34.0000 −1.10837 −0.554184 0.832394i \(-0.686970\pi\)
−0.554184 + 0.832394i \(0.686970\pi\)
\(942\) 0 0
\(943\) 30.6274 0.997366
\(944\) 0 0
\(945\) 1.00000 0.0325300
\(946\) 0 0
\(947\) 50.9117 1.65441 0.827204 0.561902i \(-0.189930\pi\)
0.827204 + 0.561902i \(0.189930\pi\)
\(948\) 0 0
\(949\) 4.00000 0.129845
\(950\) 0 0
\(951\) 7.17157 0.232554
\(952\) 0 0
\(953\) −56.4264 −1.82783 −0.913915 0.405905i \(-0.866956\pi\)
−0.913915 + 0.405905i \(0.866956\pi\)
\(954\) 0 0
\(955\) −13.1716 −0.426222
\(956\) 0 0
\(957\) −21.6569 −0.700067
\(958\) 0 0
\(959\) 0.828427 0.0267513
\(960\) 0 0
\(961\) −23.0000 −0.741935
\(962\) 0 0
\(963\) 8.00000 0.257796
\(964\) 0 0
\(965\) 10.0000 0.321911
\(966\) 0 0
\(967\) 41.9411 1.34874 0.674368 0.738396i \(-0.264416\pi\)
0.674368 + 0.738396i \(0.264416\pi\)
\(968\) 0 0
\(969\) −21.6569 −0.695718
\(970\) 0 0
\(971\) 8.00000 0.256732 0.128366 0.991727i \(-0.459027\pi\)
0.128366 + 0.991727i \(0.459027\pi\)
\(972\) 0 0
\(973\) 19.7990 0.634726
\(974\) 0 0
\(975\) 0.828427 0.0265309
\(976\) 0 0
\(977\) −7.85786 −0.251395 −0.125698 0.992069i \(-0.540117\pi\)
−0.125698 + 0.992069i \(0.540117\pi\)
\(978\) 0 0
\(979\) 32.9706 1.05374
\(980\) 0 0
\(981\) 13.3137 0.425074
\(982\) 0 0
\(983\) −51.3137 −1.63665 −0.818327 0.574754i \(-0.805098\pi\)
−0.818327 + 0.574754i \(0.805098\pi\)
\(984\) 0 0
\(985\) −12.8284 −0.408748
\(986\) 0 0
\(987\) 11.3137 0.360119
\(988\) 0 0
\(989\) 6.62742 0.210740
\(990\) 0 0
\(991\) −53.9411 −1.71350 −0.856748 0.515735i \(-0.827519\pi\)
−0.856748 + 0.515735i \(0.827519\pi\)
\(992\) 0 0
\(993\) −4.00000 −0.126936
\(994\) 0 0
\(995\) 8.48528 0.269002
\(996\) 0 0
\(997\) −4.14214 −0.131183 −0.0655914 0.997847i \(-0.520893\pi\)
−0.0655914 + 0.997847i \(0.520893\pi\)
\(998\) 0 0
\(999\) −11.6569 −0.368807
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 6720.2.a.cr.1.2 2
4.3 odd 2 6720.2.a.cy.1.1 2
8.3 odd 2 3360.2.a.bc.1.2 2
8.5 even 2 3360.2.a.be.1.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3360.2.a.bc.1.2 2 8.3 odd 2
3360.2.a.be.1.1 yes 2 8.5 even 2
6720.2.a.cr.1.2 2 1.1 even 1 trivial
6720.2.a.cy.1.1 2 4.3 odd 2