Properties

Label 8624.2.a.cc.1.1
Level $8624$
Weight $2$
Character 8624.1
Self dual yes
Analytic conductor $68.863$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8624 = 2^{4} \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8624.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(68.8629867032\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 154)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 8624.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.414214 q^{3} -3.41421 q^{5} -2.82843 q^{9} +O(q^{10})\) \(q-0.414214 q^{3} -3.41421 q^{5} -2.82843 q^{9} -1.00000 q^{11} +1.82843 q^{13} +1.41421 q^{15} -7.65685 q^{17} +3.41421 q^{19} -2.24264 q^{23} +6.65685 q^{25} +2.41421 q^{27} -8.65685 q^{29} +4.00000 q^{31} +0.414214 q^{33} -6.58579 q^{37} -0.757359 q^{39} -2.58579 q^{41} -5.65685 q^{43} +9.65685 q^{45} -6.48528 q^{47} +3.17157 q^{51} -11.8995 q^{53} +3.41421 q^{55} -1.41421 q^{57} +8.41421 q^{59} +6.17157 q^{61} -6.24264 q^{65} -11.2426 q^{67} +0.928932 q^{69} -3.07107 q^{71} -6.58579 q^{73} -2.75736 q^{75} +4.75736 q^{79} +7.48528 q^{81} -16.1421 q^{83} +26.1421 q^{85} +3.58579 q^{87} -4.48528 q^{89} -1.65685 q^{93} -11.6569 q^{95} +1.82843 q^{97} +2.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 4 q^{5} + O(q^{10}) \) \( 2 q + 2 q^{3} - 4 q^{5} - 2 q^{11} - 2 q^{13} - 4 q^{17} + 4 q^{19} + 4 q^{23} + 2 q^{25} + 2 q^{27} - 6 q^{29} + 8 q^{31} - 2 q^{33} - 16 q^{37} - 10 q^{39} - 8 q^{41} + 8 q^{45} + 4 q^{47} + 12 q^{51} - 4 q^{53} + 4 q^{55} + 14 q^{59} + 18 q^{61} - 4 q^{65} - 14 q^{67} + 16 q^{69} + 8 q^{71} - 16 q^{73} - 14 q^{75} + 18 q^{79} - 2 q^{81} - 4 q^{83} + 24 q^{85} + 10 q^{87} + 8 q^{89} + 8 q^{93} - 12 q^{95} - 2 q^{97} + O(q^{100}) \)

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
\(3\) −0.414214 −0.239146 −0.119573 0.992825i \(-0.538153\pi\)
−0.119573 + 0.992825i \(0.538153\pi\)
\(4\) 0 0
\(5\) −3.41421 −1.52688 −0.763441 0.645877i \(-0.776492\pi\)
−0.763441 + 0.645877i \(0.776492\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −2.82843 −0.942809
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 1.82843 0.507114 0.253557 0.967320i \(-0.418399\pi\)
0.253557 + 0.967320i \(0.418399\pi\)
\(14\) 0 0
\(15\) 1.41421 0.365148
\(16\) 0 0
\(17\) −7.65685 −1.85706 −0.928530 0.371257i \(-0.878927\pi\)
−0.928530 + 0.371257i \(0.878927\pi\)
\(18\) 0 0
\(19\) 3.41421 0.783274 0.391637 0.920120i \(-0.371909\pi\)
0.391637 + 0.920120i \(0.371909\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −2.24264 −0.467623 −0.233811 0.972282i \(-0.575120\pi\)
−0.233811 + 0.972282i \(0.575120\pi\)
\(24\) 0 0
\(25\) 6.65685 1.33137
\(26\) 0 0
\(27\) 2.41421 0.464616
\(28\) 0 0
\(29\) −8.65685 −1.60754 −0.803769 0.594942i \(-0.797175\pi\)
−0.803769 + 0.594942i \(0.797175\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 0 0
\(33\) 0.414214 0.0721053
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −6.58579 −1.08270 −0.541348 0.840798i \(-0.682086\pi\)
−0.541348 + 0.840798i \(0.682086\pi\)
\(38\) 0 0
\(39\) −0.757359 −0.121275
\(40\) 0 0
\(41\) −2.58579 −0.403832 −0.201916 0.979403i \(-0.564717\pi\)
−0.201916 + 0.979403i \(0.564717\pi\)
\(42\) 0 0
\(43\) −5.65685 −0.862662 −0.431331 0.902194i \(-0.641956\pi\)
−0.431331 + 0.902194i \(0.641956\pi\)
\(44\) 0 0
\(45\) 9.65685 1.43956
\(46\) 0 0
\(47\) −6.48528 −0.945976 −0.472988 0.881069i \(-0.656825\pi\)
−0.472988 + 0.881069i \(0.656825\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 3.17157 0.444109
\(52\) 0 0
\(53\) −11.8995 −1.63452 −0.817261 0.576268i \(-0.804508\pi\)
−0.817261 + 0.576268i \(0.804508\pi\)
\(54\) 0 0
\(55\) 3.41421 0.460372
\(56\) 0 0
\(57\) −1.41421 −0.187317
\(58\) 0 0
\(59\) 8.41421 1.09544 0.547719 0.836663i \(-0.315496\pi\)
0.547719 + 0.836663i \(0.315496\pi\)
\(60\) 0 0
\(61\) 6.17157 0.790189 0.395094 0.918640i \(-0.370712\pi\)
0.395094 + 0.918640i \(0.370712\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −6.24264 −0.774304
\(66\) 0 0
\(67\) −11.2426 −1.37351 −0.686754 0.726890i \(-0.740965\pi\)
−0.686754 + 0.726890i \(0.740965\pi\)
\(68\) 0 0
\(69\) 0.928932 0.111830
\(70\) 0 0
\(71\) −3.07107 −0.364469 −0.182234 0.983255i \(-0.558333\pi\)
−0.182234 + 0.983255i \(0.558333\pi\)
\(72\) 0 0
\(73\) −6.58579 −0.770808 −0.385404 0.922748i \(-0.625938\pi\)
−0.385404 + 0.922748i \(0.625938\pi\)
\(74\) 0 0
\(75\) −2.75736 −0.318392
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 4.75736 0.535245 0.267622 0.963524i \(-0.413762\pi\)
0.267622 + 0.963524i \(0.413762\pi\)
\(80\) 0 0
\(81\) 7.48528 0.831698
\(82\) 0 0
\(83\) −16.1421 −1.77183 −0.885915 0.463848i \(-0.846468\pi\)
−0.885915 + 0.463848i \(0.846468\pi\)
\(84\) 0 0
\(85\) 26.1421 2.83551
\(86\) 0 0
\(87\) 3.58579 0.384437
\(88\) 0 0
\(89\) −4.48528 −0.475439 −0.237719 0.971334i \(-0.576400\pi\)
−0.237719 + 0.971334i \(0.576400\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −1.65685 −0.171808
\(94\) 0 0
\(95\) −11.6569 −1.19597
\(96\) 0 0
\(97\) 1.82843 0.185649 0.0928243 0.995683i \(-0.470411\pi\)
0.0928243 + 0.995683i \(0.470411\pi\)
\(98\) 0 0
\(99\) 2.82843 0.284268
\(100\) 0 0
\(101\) 11.8284 1.17697 0.588486 0.808507i \(-0.299724\pi\)
0.588486 + 0.808507i \(0.299724\pi\)
\(102\) 0 0
\(103\) −10.5858 −1.04305 −0.521524 0.853236i \(-0.674636\pi\)
−0.521524 + 0.853236i \(0.674636\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −11.0711 −1.07028 −0.535140 0.844763i \(-0.679741\pi\)
−0.535140 + 0.844763i \(0.679741\pi\)
\(108\) 0 0
\(109\) −0.485281 −0.0464815 −0.0232408 0.999730i \(-0.507398\pi\)
−0.0232408 + 0.999730i \(0.507398\pi\)
\(110\) 0 0
\(111\) 2.72792 0.258923
\(112\) 0 0
\(113\) −13.8284 −1.30087 −0.650434 0.759562i \(-0.725413\pi\)
−0.650434 + 0.759562i \(0.725413\pi\)
\(114\) 0 0
\(115\) 7.65685 0.714005
\(116\) 0 0
\(117\) −5.17157 −0.478112
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 1.07107 0.0965749
\(124\) 0 0
\(125\) −5.65685 −0.505964
\(126\) 0 0
\(127\) 9.72792 0.863213 0.431607 0.902062i \(-0.357947\pi\)
0.431607 + 0.902062i \(0.357947\pi\)
\(128\) 0 0
\(129\) 2.34315 0.206302
\(130\) 0 0
\(131\) 3.41421 0.298301 0.149151 0.988814i \(-0.452346\pi\)
0.149151 + 0.988814i \(0.452346\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −8.24264 −0.709414
\(136\) 0 0
\(137\) −5.34315 −0.456496 −0.228248 0.973603i \(-0.573300\pi\)
−0.228248 + 0.973603i \(0.573300\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 2.68629 0.226227
\(142\) 0 0
\(143\) −1.82843 −0.152901
\(144\) 0 0
\(145\) 29.5563 2.45452
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 6.34315 0.519651 0.259825 0.965656i \(-0.416335\pi\)
0.259825 + 0.965656i \(0.416335\pi\)
\(150\) 0 0
\(151\) −9.72792 −0.791647 −0.395824 0.918327i \(-0.629541\pi\)
−0.395824 + 0.918327i \(0.629541\pi\)
\(152\) 0 0
\(153\) 21.6569 1.75085
\(154\) 0 0
\(155\) −13.6569 −1.09694
\(156\) 0 0
\(157\) 6.34315 0.506238 0.253119 0.967435i \(-0.418544\pi\)
0.253119 + 0.967435i \(0.418544\pi\)
\(158\) 0 0
\(159\) 4.92893 0.390890
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 15.7279 1.23191 0.615953 0.787783i \(-0.288771\pi\)
0.615953 + 0.787783i \(0.288771\pi\)
\(164\) 0 0
\(165\) −1.41421 −0.110096
\(166\) 0 0
\(167\) 11.7279 0.907534 0.453767 0.891120i \(-0.350080\pi\)
0.453767 + 0.891120i \(0.350080\pi\)
\(168\) 0 0
\(169\) −9.65685 −0.742835
\(170\) 0 0
\(171\) −9.65685 −0.738478
\(172\) 0 0
\(173\) −4.17157 −0.317159 −0.158579 0.987346i \(-0.550691\pi\)
−0.158579 + 0.987346i \(0.550691\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −3.48528 −0.261970
\(178\) 0 0
\(179\) 18.8995 1.41261 0.706307 0.707905i \(-0.250360\pi\)
0.706307 + 0.707905i \(0.250360\pi\)
\(180\) 0 0
\(181\) 3.65685 0.271812 0.135906 0.990722i \(-0.456606\pi\)
0.135906 + 0.990722i \(0.456606\pi\)
\(182\) 0 0
\(183\) −2.55635 −0.188971
\(184\) 0 0
\(185\) 22.4853 1.65315
\(186\) 0 0
\(187\) 7.65685 0.559925
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 12.8284 0.928232 0.464116 0.885774i \(-0.346372\pi\)
0.464116 + 0.885774i \(0.346372\pi\)
\(192\) 0 0
\(193\) 2.10051 0.151198 0.0755988 0.997138i \(-0.475913\pi\)
0.0755988 + 0.997138i \(0.475913\pi\)
\(194\) 0 0
\(195\) 2.58579 0.185172
\(196\) 0 0
\(197\) −17.4853 −1.24577 −0.622887 0.782312i \(-0.714041\pi\)
−0.622887 + 0.782312i \(0.714041\pi\)
\(198\) 0 0
\(199\) −19.8995 −1.41064 −0.705319 0.708890i \(-0.749196\pi\)
−0.705319 + 0.708890i \(0.749196\pi\)
\(200\) 0 0
\(201\) 4.65685 0.328469
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 8.82843 0.616604
\(206\) 0 0
\(207\) 6.34315 0.440879
\(208\) 0 0
\(209\) −3.41421 −0.236166
\(210\) 0 0
\(211\) −4.58579 −0.315699 −0.157849 0.987463i \(-0.550456\pi\)
−0.157849 + 0.987463i \(0.550456\pi\)
\(212\) 0 0
\(213\) 1.27208 0.0871613
\(214\) 0 0
\(215\) 19.3137 1.31718
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 2.72792 0.184336
\(220\) 0 0
\(221\) −14.0000 −0.941742
\(222\) 0 0
\(223\) −11.4142 −0.764352 −0.382176 0.924089i \(-0.624825\pi\)
−0.382176 + 0.924089i \(0.624825\pi\)
\(224\) 0 0
\(225\) −18.8284 −1.25523
\(226\) 0 0
\(227\) 23.1716 1.53795 0.768976 0.639278i \(-0.220767\pi\)
0.768976 + 0.639278i \(0.220767\pi\)
\(228\) 0 0
\(229\) 0.686292 0.0453514 0.0226757 0.999743i \(-0.492781\pi\)
0.0226757 + 0.999743i \(0.492781\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −1.41421 −0.0926482 −0.0463241 0.998926i \(-0.514751\pi\)
−0.0463241 + 0.998926i \(0.514751\pi\)
\(234\) 0 0
\(235\) 22.1421 1.44439
\(236\) 0 0
\(237\) −1.97056 −0.128002
\(238\) 0 0
\(239\) 22.2132 1.43685 0.718426 0.695603i \(-0.244863\pi\)
0.718426 + 0.695603i \(0.244863\pi\)
\(240\) 0 0
\(241\) 3.75736 0.242033 0.121016 0.992651i \(-0.461385\pi\)
0.121016 + 0.992651i \(0.461385\pi\)
\(242\) 0 0
\(243\) −10.3431 −0.663513
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 6.24264 0.397210
\(248\) 0 0
\(249\) 6.68629 0.423727
\(250\) 0 0
\(251\) −2.14214 −0.135210 −0.0676052 0.997712i \(-0.521536\pi\)
−0.0676052 + 0.997712i \(0.521536\pi\)
\(252\) 0 0
\(253\) 2.24264 0.140994
\(254\) 0 0
\(255\) −10.8284 −0.678102
\(256\) 0 0
\(257\) 3.14214 0.196001 0.0980005 0.995186i \(-0.468755\pi\)
0.0980005 + 0.995186i \(0.468755\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 24.4853 1.51560
\(262\) 0 0
\(263\) −31.0416 −1.91411 −0.957054 0.289908i \(-0.906375\pi\)
−0.957054 + 0.289908i \(0.906375\pi\)
\(264\) 0 0
\(265\) 40.6274 2.49572
\(266\) 0 0
\(267\) 1.85786 0.113699
\(268\) 0 0
\(269\) 13.6569 0.832673 0.416337 0.909211i \(-0.363314\pi\)
0.416337 + 0.909211i \(0.363314\pi\)
\(270\) 0 0
\(271\) 26.5563 1.61318 0.806592 0.591109i \(-0.201310\pi\)
0.806592 + 0.591109i \(0.201310\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −6.65685 −0.401423
\(276\) 0 0
\(277\) −3.82843 −0.230028 −0.115014 0.993364i \(-0.536691\pi\)
−0.115014 + 0.993364i \(0.536691\pi\)
\(278\) 0 0
\(279\) −11.3137 −0.677334
\(280\) 0 0
\(281\) −16.7279 −0.997904 −0.498952 0.866630i \(-0.666282\pi\)
−0.498952 + 0.866630i \(0.666282\pi\)
\(282\) 0 0
\(283\) 20.5858 1.22370 0.611849 0.790975i \(-0.290426\pi\)
0.611849 + 0.790975i \(0.290426\pi\)
\(284\) 0 0
\(285\) 4.82843 0.286011
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 41.6274 2.44867
\(290\) 0 0
\(291\) −0.757359 −0.0443972
\(292\) 0 0
\(293\) −5.17157 −0.302127 −0.151063 0.988524i \(-0.548270\pi\)
−0.151063 + 0.988524i \(0.548270\pi\)
\(294\) 0 0
\(295\) −28.7279 −1.67260
\(296\) 0 0
\(297\) −2.41421 −0.140087
\(298\) 0 0
\(299\) −4.10051 −0.237138
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −4.89949 −0.281469
\(304\) 0 0
\(305\) −21.0711 −1.20653
\(306\) 0 0
\(307\) 9.89949 0.564994 0.282497 0.959268i \(-0.408837\pi\)
0.282497 + 0.959268i \(0.408837\pi\)
\(308\) 0 0
\(309\) 4.38478 0.249441
\(310\) 0 0
\(311\) 8.72792 0.494915 0.247458 0.968899i \(-0.420405\pi\)
0.247458 + 0.968899i \(0.420405\pi\)
\(312\) 0 0
\(313\) −9.34315 −0.528106 −0.264053 0.964508i \(-0.585059\pi\)
−0.264053 + 0.964508i \(0.585059\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 31.3137 1.75875 0.879377 0.476127i \(-0.157960\pi\)
0.879377 + 0.476127i \(0.157960\pi\)
\(318\) 0 0
\(319\) 8.65685 0.484691
\(320\) 0 0
\(321\) 4.58579 0.255954
\(322\) 0 0
\(323\) −26.1421 −1.45459
\(324\) 0 0
\(325\) 12.1716 0.675157
\(326\) 0 0
\(327\) 0.201010 0.0111159
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 9.92893 0.545743 0.272872 0.962050i \(-0.412027\pi\)
0.272872 + 0.962050i \(0.412027\pi\)
\(332\) 0 0
\(333\) 18.6274 1.02078
\(334\) 0 0
\(335\) 38.3848 2.09718
\(336\) 0 0
\(337\) −19.7574 −1.07625 −0.538126 0.842864i \(-0.680868\pi\)
−0.538126 + 0.842864i \(0.680868\pi\)
\(338\) 0 0
\(339\) 5.72792 0.311098
\(340\) 0 0
\(341\) −4.00000 −0.216612
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −3.17157 −0.170752
\(346\) 0 0
\(347\) −14.5858 −0.783006 −0.391503 0.920177i \(-0.628045\pi\)
−0.391503 + 0.920177i \(0.628045\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(350\) 0 0
\(351\) 4.41421 0.235613
\(352\) 0 0
\(353\) −25.3137 −1.34731 −0.673656 0.739045i \(-0.735277\pi\)
−0.673656 + 0.739045i \(0.735277\pi\)
\(354\) 0 0
\(355\) 10.4853 0.556501
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −10.7574 −0.567752 −0.283876 0.958861i \(-0.591620\pi\)
−0.283876 + 0.958861i \(0.591620\pi\)
\(360\) 0 0
\(361\) −7.34315 −0.386481
\(362\) 0 0
\(363\) −0.414214 −0.0217406
\(364\) 0 0
\(365\) 22.4853 1.17693
\(366\) 0 0
\(367\) −14.7279 −0.768791 −0.384396 0.923168i \(-0.625590\pi\)
−0.384396 + 0.923168i \(0.625590\pi\)
\(368\) 0 0
\(369\) 7.31371 0.380736
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 13.9706 0.723368 0.361684 0.932301i \(-0.382202\pi\)
0.361684 + 0.932301i \(0.382202\pi\)
\(374\) 0 0
\(375\) 2.34315 0.121000
\(376\) 0 0
\(377\) −15.8284 −0.815205
\(378\) 0 0
\(379\) 25.8701 1.32886 0.664428 0.747352i \(-0.268675\pi\)
0.664428 + 0.747352i \(0.268675\pi\)
\(380\) 0 0
\(381\) −4.02944 −0.206434
\(382\) 0 0
\(383\) −30.3848 −1.55259 −0.776295 0.630370i \(-0.782903\pi\)
−0.776295 + 0.630370i \(0.782903\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 16.0000 0.813326
\(388\) 0 0
\(389\) 2.72792 0.138311 0.0691556 0.997606i \(-0.477970\pi\)
0.0691556 + 0.997606i \(0.477970\pi\)
\(390\) 0 0
\(391\) 17.1716 0.868404
\(392\) 0 0
\(393\) −1.41421 −0.0713376
\(394\) 0 0
\(395\) −16.2426 −0.817256
\(396\) 0 0
\(397\) 22.0000 1.10415 0.552074 0.833795i \(-0.313837\pi\)
0.552074 + 0.833795i \(0.313837\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −4.31371 −0.215416 −0.107708 0.994183i \(-0.534351\pi\)
−0.107708 + 0.994183i \(0.534351\pi\)
\(402\) 0 0
\(403\) 7.31371 0.364322
\(404\) 0 0
\(405\) −25.5563 −1.26991
\(406\) 0 0
\(407\) 6.58579 0.326445
\(408\) 0 0
\(409\) −22.7279 −1.12382 −0.561912 0.827197i \(-0.689934\pi\)
−0.561912 + 0.827197i \(0.689934\pi\)
\(410\) 0 0
\(411\) 2.21320 0.109169
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 55.1127 2.70538
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −2.14214 −0.104650 −0.0523251 0.998630i \(-0.516663\pi\)
−0.0523251 + 0.998630i \(0.516663\pi\)
\(420\) 0 0
\(421\) 23.3137 1.13624 0.568120 0.822946i \(-0.307671\pi\)
0.568120 + 0.822946i \(0.307671\pi\)
\(422\) 0 0
\(423\) 18.3431 0.891874
\(424\) 0 0
\(425\) −50.9706 −2.47244
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0.757359 0.0365657
\(430\) 0 0
\(431\) 20.4142 0.983318 0.491659 0.870788i \(-0.336391\pi\)
0.491659 + 0.870788i \(0.336391\pi\)
\(432\) 0 0
\(433\) −2.14214 −0.102944 −0.0514722 0.998674i \(-0.516391\pi\)
−0.0514722 + 0.998674i \(0.516391\pi\)
\(434\) 0 0
\(435\) −12.2426 −0.586990
\(436\) 0 0
\(437\) −7.65685 −0.366277
\(438\) 0 0
\(439\) −9.38478 −0.447911 −0.223955 0.974599i \(-0.571897\pi\)
−0.223955 + 0.974599i \(0.571897\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 32.6274 1.55018 0.775088 0.631854i \(-0.217706\pi\)
0.775088 + 0.631854i \(0.217706\pi\)
\(444\) 0 0
\(445\) 15.3137 0.725939
\(446\) 0 0
\(447\) −2.62742 −0.124273
\(448\) 0 0
\(449\) 33.6569 1.58837 0.794183 0.607679i \(-0.207899\pi\)
0.794183 + 0.607679i \(0.207899\pi\)
\(450\) 0 0
\(451\) 2.58579 0.121760
\(452\) 0 0
\(453\) 4.02944 0.189319
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −0.343146 −0.0160517 −0.00802584 0.999968i \(-0.502555\pi\)
−0.00802584 + 0.999968i \(0.502555\pi\)
\(458\) 0 0
\(459\) −18.4853 −0.862819
\(460\) 0 0
\(461\) 14.3137 0.666656 0.333328 0.942811i \(-0.391828\pi\)
0.333328 + 0.942811i \(0.391828\pi\)
\(462\) 0 0
\(463\) −7.17157 −0.333291 −0.166646 0.986017i \(-0.553294\pi\)
−0.166646 + 0.986017i \(0.553294\pi\)
\(464\) 0 0
\(465\) 5.65685 0.262330
\(466\) 0 0
\(467\) 34.0000 1.57333 0.786666 0.617379i \(-0.211805\pi\)
0.786666 + 0.617379i \(0.211805\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −2.62742 −0.121065
\(472\) 0 0
\(473\) 5.65685 0.260102
\(474\) 0 0
\(475\) 22.7279 1.04283
\(476\) 0 0
\(477\) 33.6569 1.54104
\(478\) 0 0
\(479\) 26.0711 1.19122 0.595609 0.803275i \(-0.296911\pi\)
0.595609 + 0.803275i \(0.296911\pi\)
\(480\) 0 0
\(481\) −12.0416 −0.549051
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −6.24264 −0.283464
\(486\) 0 0
\(487\) −1.65685 −0.0750792 −0.0375396 0.999295i \(-0.511952\pi\)
−0.0375396 + 0.999295i \(0.511952\pi\)
\(488\) 0 0
\(489\) −6.51472 −0.294606
\(490\) 0 0
\(491\) −24.8284 −1.12049 −0.560246 0.828327i \(-0.689293\pi\)
−0.560246 + 0.828327i \(0.689293\pi\)
\(492\) 0 0
\(493\) 66.2843 2.98529
\(494\) 0 0
\(495\) −9.65685 −0.434043
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 6.14214 0.274960 0.137480 0.990505i \(-0.456100\pi\)
0.137480 + 0.990505i \(0.456100\pi\)
\(500\) 0 0
\(501\) −4.85786 −0.217033
\(502\) 0 0
\(503\) 38.2132 1.70384 0.851921 0.523670i \(-0.175437\pi\)
0.851921 + 0.523670i \(0.175437\pi\)
\(504\) 0 0
\(505\) −40.3848 −1.79710
\(506\) 0 0
\(507\) 4.00000 0.177646
\(508\) 0 0
\(509\) −13.3137 −0.590120 −0.295060 0.955479i \(-0.595340\pi\)
−0.295060 + 0.955479i \(0.595340\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 8.24264 0.363921
\(514\) 0 0
\(515\) 36.1421 1.59261
\(516\) 0 0
\(517\) 6.48528 0.285222
\(518\) 0 0
\(519\) 1.72792 0.0758474
\(520\) 0 0
\(521\) 28.2843 1.23916 0.619578 0.784935i \(-0.287304\pi\)
0.619578 + 0.784935i \(0.287304\pi\)
\(522\) 0 0
\(523\) −12.7279 −0.556553 −0.278277 0.960501i \(-0.589763\pi\)
−0.278277 + 0.960501i \(0.589763\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −30.6274 −1.33415
\(528\) 0 0
\(529\) −17.9706 −0.781329
\(530\) 0 0
\(531\) −23.7990 −1.03279
\(532\) 0 0
\(533\) −4.72792 −0.204789
\(534\) 0 0
\(535\) 37.7990 1.63419
\(536\) 0 0
\(537\) −7.82843 −0.337822
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −41.1421 −1.76884 −0.884419 0.466693i \(-0.845445\pi\)
−0.884419 + 0.466693i \(0.845445\pi\)
\(542\) 0 0
\(543\) −1.51472 −0.0650028
\(544\) 0 0
\(545\) 1.65685 0.0709718
\(546\) 0 0
\(547\) 18.8701 0.806825 0.403413 0.915018i \(-0.367824\pi\)
0.403413 + 0.915018i \(0.367824\pi\)
\(548\) 0 0
\(549\) −17.4558 −0.744997
\(550\) 0 0
\(551\) −29.5563 −1.25914
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −9.31371 −0.395345
\(556\) 0 0
\(557\) 24.4853 1.03747 0.518737 0.854934i \(-0.326402\pi\)
0.518737 + 0.854934i \(0.326402\pi\)
\(558\) 0 0
\(559\) −10.3431 −0.437468
\(560\) 0 0
\(561\) −3.17157 −0.133904
\(562\) 0 0
\(563\) 5.07107 0.213720 0.106860 0.994274i \(-0.465920\pi\)
0.106860 + 0.994274i \(0.465920\pi\)
\(564\) 0 0
\(565\) 47.2132 1.98627
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −4.00000 −0.167689 −0.0838444 0.996479i \(-0.526720\pi\)
−0.0838444 + 0.996479i \(0.526720\pi\)
\(570\) 0 0
\(571\) 10.3848 0.434589 0.217295 0.976106i \(-0.430277\pi\)
0.217295 + 0.976106i \(0.430277\pi\)
\(572\) 0 0
\(573\) −5.31371 −0.221983
\(574\) 0 0
\(575\) −14.9289 −0.622580
\(576\) 0 0
\(577\) −32.3137 −1.34524 −0.672619 0.739989i \(-0.734831\pi\)
−0.672619 + 0.739989i \(0.734831\pi\)
\(578\) 0 0
\(579\) −0.870058 −0.0361584
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 11.8995 0.492827
\(584\) 0 0
\(585\) 17.6569 0.730021
\(586\) 0 0
\(587\) 44.8995 1.85320 0.926600 0.376048i \(-0.122717\pi\)
0.926600 + 0.376048i \(0.122717\pi\)
\(588\) 0 0
\(589\) 13.6569 0.562721
\(590\) 0 0
\(591\) 7.24264 0.297922
\(592\) 0 0
\(593\) −35.6985 −1.46596 −0.732981 0.680250i \(-0.761871\pi\)
−0.732981 + 0.680250i \(0.761871\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 8.24264 0.337349
\(598\) 0 0
\(599\) 2.62742 0.107353 0.0536767 0.998558i \(-0.482906\pi\)
0.0536767 + 0.998558i \(0.482906\pi\)
\(600\) 0 0
\(601\) −35.9411 −1.46607 −0.733035 0.680191i \(-0.761897\pi\)
−0.733035 + 0.680191i \(0.761897\pi\)
\(602\) 0 0
\(603\) 31.7990 1.29495
\(604\) 0 0
\(605\) −3.41421 −0.138808
\(606\) 0 0
\(607\) 9.02944 0.366494 0.183247 0.983067i \(-0.441339\pi\)
0.183247 + 0.983067i \(0.441339\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −11.8579 −0.479718
\(612\) 0 0
\(613\) −16.6274 −0.671575 −0.335788 0.941938i \(-0.609002\pi\)
−0.335788 + 0.941938i \(0.609002\pi\)
\(614\) 0 0
\(615\) −3.65685 −0.147459
\(616\) 0 0
\(617\) −8.02944 −0.323253 −0.161626 0.986852i \(-0.551674\pi\)
−0.161626 + 0.986852i \(0.551674\pi\)
\(618\) 0 0
\(619\) −45.9411 −1.84653 −0.923265 0.384164i \(-0.874490\pi\)
−0.923265 + 0.384164i \(0.874490\pi\)
\(620\) 0 0
\(621\) −5.41421 −0.217265
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −13.9706 −0.558823
\(626\) 0 0
\(627\) 1.41421 0.0564782
\(628\) 0 0
\(629\) 50.4264 2.01063
\(630\) 0 0
\(631\) −48.7279 −1.93983 −0.969914 0.243448i \(-0.921722\pi\)
−0.969914 + 0.243448i \(0.921722\pi\)
\(632\) 0 0
\(633\) 1.89949 0.0754981
\(634\) 0 0
\(635\) −33.2132 −1.31803
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 8.68629 0.343624
\(640\) 0 0
\(641\) −41.2843 −1.63063 −0.815315 0.579017i \(-0.803436\pi\)
−0.815315 + 0.579017i \(0.803436\pi\)
\(642\) 0 0
\(643\) −4.41421 −0.174080 −0.0870398 0.996205i \(-0.527741\pi\)
−0.0870398 + 0.996205i \(0.527741\pi\)
\(644\) 0 0
\(645\) −8.00000 −0.315000
\(646\) 0 0
\(647\) 46.1838 1.81567 0.907836 0.419326i \(-0.137734\pi\)
0.907836 + 0.419326i \(0.137734\pi\)
\(648\) 0 0
\(649\) −8.41421 −0.330287
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 18.3848 0.719452 0.359726 0.933058i \(-0.382870\pi\)
0.359726 + 0.933058i \(0.382870\pi\)
\(654\) 0 0
\(655\) −11.6569 −0.455471
\(656\) 0 0
\(657\) 18.6274 0.726725
\(658\) 0 0
\(659\) 12.0000 0.467454 0.233727 0.972302i \(-0.424908\pi\)
0.233727 + 0.972302i \(0.424908\pi\)
\(660\) 0 0
\(661\) −14.9706 −0.582287 −0.291144 0.956679i \(-0.594036\pi\)
−0.291144 + 0.956679i \(0.594036\pi\)
\(662\) 0 0
\(663\) 5.79899 0.225214
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 19.4142 0.751721
\(668\) 0 0
\(669\) 4.72792 0.182792
\(670\) 0 0
\(671\) −6.17157 −0.238251
\(672\) 0 0
\(673\) −25.5563 −0.985125 −0.492562 0.870277i \(-0.663940\pi\)
−0.492562 + 0.870277i \(0.663940\pi\)
\(674\) 0 0
\(675\) 16.0711 0.618576
\(676\) 0 0
\(677\) −12.6863 −0.487574 −0.243787 0.969829i \(-0.578390\pi\)
−0.243787 + 0.969829i \(0.578390\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −9.59798 −0.367795
\(682\) 0 0
\(683\) −16.4142 −0.628072 −0.314036 0.949411i \(-0.601681\pi\)
−0.314036 + 0.949411i \(0.601681\pi\)
\(684\) 0 0
\(685\) 18.2426 0.697015
\(686\) 0 0
\(687\) −0.284271 −0.0108456
\(688\) 0 0
\(689\) −21.7574 −0.828889
\(690\) 0 0
\(691\) −13.9289 −0.529882 −0.264941 0.964265i \(-0.585352\pi\)
−0.264941 + 0.964265i \(0.585352\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 19.7990 0.749940
\(698\) 0 0
\(699\) 0.585786 0.0221565
\(700\) 0 0
\(701\) 36.1127 1.36396 0.681979 0.731372i \(-0.261120\pi\)
0.681979 + 0.731372i \(0.261120\pi\)
\(702\) 0 0
\(703\) −22.4853 −0.848048
\(704\) 0 0
\(705\) −9.17157 −0.345421
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 36.0416 1.35357 0.676786 0.736180i \(-0.263372\pi\)
0.676786 + 0.736180i \(0.263372\pi\)
\(710\) 0 0
\(711\) −13.4558 −0.504634
\(712\) 0 0
\(713\) −8.97056 −0.335950
\(714\) 0 0
\(715\) 6.24264 0.233462
\(716\) 0 0
\(717\) −9.20101 −0.343618
\(718\) 0 0
\(719\) 18.4853 0.689385 0.344692 0.938716i \(-0.387983\pi\)
0.344692 + 0.938716i \(0.387983\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −1.55635 −0.0578812
\(724\) 0 0
\(725\) −57.6274 −2.14023
\(726\) 0 0
\(727\) 48.4264 1.79604 0.898018 0.439959i \(-0.145007\pi\)
0.898018 + 0.439959i \(0.145007\pi\)
\(728\) 0 0
\(729\) −18.1716 −0.673021
\(730\) 0 0
\(731\) 43.3137 1.60202
\(732\) 0 0
\(733\) −13.0000 −0.480166 −0.240083 0.970752i \(-0.577175\pi\)
−0.240083 + 0.970752i \(0.577175\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 11.2426 0.414128
\(738\) 0 0
\(739\) −32.4264 −1.19282 −0.596412 0.802678i \(-0.703408\pi\)
−0.596412 + 0.802678i \(0.703408\pi\)
\(740\) 0 0
\(741\) −2.58579 −0.0949912
\(742\) 0 0
\(743\) −9.31371 −0.341687 −0.170843 0.985298i \(-0.554649\pi\)
−0.170843 + 0.985298i \(0.554649\pi\)
\(744\) 0 0
\(745\) −21.6569 −0.793446
\(746\) 0 0
\(747\) 45.6569 1.67050
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −34.3431 −1.25320 −0.626600 0.779341i \(-0.715554\pi\)
−0.626600 + 0.779341i \(0.715554\pi\)
\(752\) 0 0
\(753\) 0.887302 0.0323351
\(754\) 0 0
\(755\) 33.2132 1.20875
\(756\) 0 0
\(757\) 19.6569 0.714441 0.357220 0.934020i \(-0.383725\pi\)
0.357220 + 0.934020i \(0.383725\pi\)
\(758\) 0 0
\(759\) −0.928932 −0.0337181
\(760\) 0 0
\(761\) −2.97056 −0.107683 −0.0538414 0.998549i \(-0.517147\pi\)
−0.0538414 + 0.998549i \(0.517147\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −73.9411 −2.67335
\(766\) 0 0
\(767\) 15.3848 0.555512
\(768\) 0 0
\(769\) −10.9706 −0.395609 −0.197804 0.980242i \(-0.563381\pi\)
−0.197804 + 0.980242i \(0.563381\pi\)
\(770\) 0 0
\(771\) −1.30152 −0.0468729
\(772\) 0 0
\(773\) 45.9411 1.65239 0.826194 0.563386i \(-0.190502\pi\)
0.826194 + 0.563386i \(0.190502\pi\)
\(774\) 0 0
\(775\) 26.6274 0.956485
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −8.82843 −0.316311
\(780\) 0 0
\(781\) 3.07107 0.109891
\(782\) 0 0
\(783\) −20.8995 −0.746887
\(784\) 0 0
\(785\) −21.6569 −0.772966
\(786\) 0 0
\(787\) 17.5563 0.625816 0.312908 0.949783i \(-0.398697\pi\)
0.312908 + 0.949783i \(0.398697\pi\)
\(788\) 0 0
\(789\) 12.8579 0.457752
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 11.2843 0.400716
\(794\) 0 0
\(795\) −16.8284 −0.596843
\(796\) 0 0
\(797\) 3.65685 0.129532 0.0647662 0.997900i \(-0.479370\pi\)
0.0647662 + 0.997900i \(0.479370\pi\)
\(798\) 0 0
\(799\) 49.6569 1.75673
\(800\) 0 0
\(801\) 12.6863 0.448248
\(802\) 0 0
\(803\) 6.58579 0.232407
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −5.65685 −0.199131
\(808\) 0 0
\(809\) −1.37258 −0.0482574 −0.0241287 0.999709i \(-0.507681\pi\)
−0.0241287 + 0.999709i \(0.507681\pi\)
\(810\) 0 0
\(811\) −32.2843 −1.13365 −0.566827 0.823837i \(-0.691829\pi\)
−0.566827 + 0.823837i \(0.691829\pi\)
\(812\) 0 0
\(813\) −11.0000 −0.385787
\(814\) 0 0
\(815\) −53.6985 −1.88098
\(816\) 0 0
\(817\) −19.3137 −0.675701
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −12.5147 −0.436767 −0.218383 0.975863i \(-0.570078\pi\)
−0.218383 + 0.975863i \(0.570078\pi\)
\(822\) 0 0
\(823\) −7.45584 −0.259894 −0.129947 0.991521i \(-0.541481\pi\)
−0.129947 + 0.991521i \(0.541481\pi\)
\(824\) 0 0
\(825\) 2.75736 0.0959989
\(826\) 0 0
\(827\) −49.3553 −1.71625 −0.858127 0.513438i \(-0.828372\pi\)
−0.858127 + 0.513438i \(0.828372\pi\)
\(828\) 0 0
\(829\) 30.2426 1.05037 0.525185 0.850988i \(-0.323996\pi\)
0.525185 + 0.850988i \(0.323996\pi\)
\(830\) 0 0
\(831\) 1.58579 0.0550103
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −40.0416 −1.38570
\(836\) 0 0
\(837\) 9.65685 0.333790
\(838\) 0 0
\(839\) 1.51472 0.0522939 0.0261469 0.999658i \(-0.491676\pi\)
0.0261469 + 0.999658i \(0.491676\pi\)
\(840\) 0 0
\(841\) 45.9411 1.58418
\(842\) 0 0
\(843\) 6.92893 0.238645
\(844\) 0 0
\(845\) 32.9706 1.13422
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −8.52691 −0.292643
\(850\) 0 0
\(851\) 14.7696 0.506294
\(852\) 0 0
\(853\) 14.0000 0.479351 0.239675 0.970853i \(-0.422959\pi\)
0.239675 + 0.970853i \(0.422959\pi\)
\(854\) 0 0
\(855\) 32.9706 1.12757
\(856\) 0 0
\(857\) 10.6274 0.363026 0.181513 0.983389i \(-0.441901\pi\)
0.181513 + 0.983389i \(0.441901\pi\)
\(858\) 0 0
\(859\) −19.7279 −0.673108 −0.336554 0.941664i \(-0.609261\pi\)
−0.336554 + 0.941664i \(0.609261\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 7.21320 0.245540 0.122770 0.992435i \(-0.460822\pi\)
0.122770 + 0.992435i \(0.460822\pi\)
\(864\) 0 0
\(865\) 14.2426 0.484264
\(866\) 0 0
\(867\) −17.2426 −0.585591
\(868\) 0 0
\(869\) −4.75736 −0.161382
\(870\) 0 0
\(871\) −20.5563 −0.696525
\(872\) 0 0
\(873\) −5.17157 −0.175031
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −25.6274 −0.865376 −0.432688 0.901544i \(-0.642435\pi\)
−0.432688 + 0.901544i \(0.642435\pi\)
\(878\) 0 0
\(879\) 2.14214 0.0722524
\(880\) 0 0
\(881\) −44.4558 −1.49776 −0.748878 0.662708i \(-0.769407\pi\)
−0.748878 + 0.662708i \(0.769407\pi\)
\(882\) 0 0
\(883\) 9.72792 0.327371 0.163685 0.986513i \(-0.447662\pi\)
0.163685 + 0.986513i \(0.447662\pi\)
\(884\) 0 0
\(885\) 11.8995 0.399997
\(886\) 0 0
\(887\) 22.8995 0.768890 0.384445 0.923148i \(-0.374393\pi\)
0.384445 + 0.923148i \(0.374393\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −7.48528 −0.250766
\(892\) 0 0
\(893\) −22.1421 −0.740958
\(894\) 0 0
\(895\) −64.5269 −2.15690
\(896\) 0 0
\(897\) 1.69848 0.0567108
\(898\) 0 0
\(899\) −34.6274 −1.15489
\(900\) 0 0
\(901\) 91.1127 3.03540
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −12.4853 −0.415025
\(906\) 0 0
\(907\) −33.3137 −1.10616 −0.553082 0.833127i \(-0.686548\pi\)
−0.553082 + 0.833127i \(0.686548\pi\)
\(908\) 0 0
\(909\) −33.4558 −1.10966
\(910\) 0 0
\(911\) 13.5147 0.447763 0.223881 0.974616i \(-0.428127\pi\)
0.223881 + 0.974616i \(0.428127\pi\)
\(912\) 0 0
\(913\) 16.1421 0.534227
\(914\) 0 0
\(915\) 8.72792 0.288536
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 6.14214 0.202610 0.101305 0.994855i \(-0.467698\pi\)
0.101305 + 0.994855i \(0.467698\pi\)
\(920\) 0 0
\(921\) −4.10051 −0.135116
\(922\) 0 0
\(923\) −5.61522 −0.184827
\(924\) 0 0
\(925\) −43.8406 −1.44147
\(926\) 0 0
\(927\) 29.9411 0.983396
\(928\) 0 0
\(929\) 10.4558 0.343045 0.171523 0.985180i \(-0.445131\pi\)
0.171523 + 0.985180i \(0.445131\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −3.61522 −0.118357
\(934\) 0 0
\(935\) −26.1421 −0.854939
\(936\) 0 0
\(937\) −16.5858 −0.541834 −0.270917 0.962603i \(-0.587327\pi\)
−0.270917 + 0.962603i \(0.587327\pi\)
\(938\) 0 0
\(939\) 3.87006 0.126295
\(940\) 0 0
\(941\) 13.3431 0.434974 0.217487 0.976063i \(-0.430214\pi\)
0.217487 + 0.976063i \(0.430214\pi\)
\(942\) 0 0
\(943\) 5.79899 0.188841
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −29.1716 −0.947949 −0.473974 0.880539i \(-0.657181\pi\)
−0.473974 + 0.880539i \(0.657181\pi\)
\(948\) 0 0
\(949\) −12.0416 −0.390888
\(950\) 0 0
\(951\) −12.9706 −0.420599
\(952\) 0 0
\(953\) −25.5563 −0.827851 −0.413926 0.910311i \(-0.635843\pi\)
−0.413926 + 0.910311i \(0.635843\pi\)
\(954\) 0 0
\(955\) −43.7990 −1.41730
\(956\) 0 0
\(957\) −3.58579 −0.115912
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) 31.3137 1.00907
\(964\) 0 0
\(965\) −7.17157 −0.230861
\(966\) 0 0
\(967\) −20.2843 −0.652298 −0.326149 0.945318i \(-0.605751\pi\)
−0.326149 + 0.945318i \(0.605751\pi\)
\(968\) 0 0
\(969\) 10.8284 0.347859
\(970\) 0 0
\(971\) −47.7279 −1.53166 −0.765831 0.643042i \(-0.777672\pi\)
−0.765831 + 0.643042i \(0.777672\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −5.04163 −0.161461
\(976\) 0 0
\(977\) 40.0000 1.27971 0.639857 0.768494i \(-0.278994\pi\)
0.639857 + 0.768494i \(0.278994\pi\)
\(978\) 0 0
\(979\) 4.48528 0.143350
\(980\) 0 0
\(981\) 1.37258 0.0438232
\(982\) 0 0
\(983\) 52.4264 1.67214 0.836071 0.548621i \(-0.184847\pi\)
0.836071 + 0.548621i \(0.184847\pi\)
\(984\) 0 0
\(985\) 59.6985 1.90215
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 12.6863 0.403401
\(990\) 0 0
\(991\) 38.3848 1.21933 0.609666 0.792658i \(-0.291303\pi\)
0.609666 + 0.792658i \(0.291303\pi\)
\(992\) 0 0
\(993\) −4.11270 −0.130513
\(994\) 0 0
\(995\) 67.9411 2.15388
\(996\) 0 0
\(997\) 18.8284 0.596302 0.298151 0.954519i \(-0.403630\pi\)
0.298151 + 0.954519i \(0.403630\pi\)
\(998\) 0 0
\(999\) −15.8995 −0.503038
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 8624.2.a.cc.1.1 2
4.3 odd 2 1078.2.a.t.1.2 2
7.2 even 3 1232.2.q.f.529.2 4
7.4 even 3 1232.2.q.f.177.2 4
7.6 odd 2 8624.2.a.bh.1.2 2
12.11 even 2 9702.2.a.cx.1.2 2
28.3 even 6 1078.2.e.m.177.2 4
28.11 odd 6 154.2.e.e.23.1 4
28.19 even 6 1078.2.e.m.67.2 4
28.23 odd 6 154.2.e.e.67.1 yes 4
28.27 even 2 1078.2.a.x.1.1 2
84.11 even 6 1386.2.k.t.793.1 4
84.23 even 6 1386.2.k.t.991.1 4
84.83 odd 2 9702.2.a.ch.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
154.2.e.e.23.1 4 28.11 odd 6
154.2.e.e.67.1 yes 4 28.23 odd 6
1078.2.a.t.1.2 2 4.3 odd 2
1078.2.a.x.1.1 2 28.27 even 2
1078.2.e.m.67.2 4 28.19 even 6
1078.2.e.m.177.2 4 28.3 even 6
1232.2.q.f.177.2 4 7.4 even 3
1232.2.q.f.529.2 4 7.2 even 3
1386.2.k.t.793.1 4 84.11 even 6
1386.2.k.t.991.1 4 84.23 even 6
8624.2.a.bh.1.2 2 7.6 odd 2
8624.2.a.cc.1.1 2 1.1 even 1 trivial
9702.2.a.ch.1.1 2 84.83 odd 2
9702.2.a.cx.1.2 2 12.11 even 2