Properties

Label 1792.2.a.u.1.3
Level $1792$
Weight $2$
Character 1792.1
Self dual yes
Analytic conductor $14.309$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1792 = 2^{8} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1792.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.3091920422\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{16})^+\)
Defining polynomial: \(x^{4} - 4 x^{2} + 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 896)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-0.765367\) of defining polynomial
Character \(\chi\) \(=\) 1792.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.08239 q^{3} -1.08239 q^{5} -1.00000 q^{7} -1.82843 q^{9} +O(q^{10})\) \(q+1.08239 q^{3} -1.08239 q^{5} -1.00000 q^{7} -1.82843 q^{9} +5.22625 q^{11} -6.30864 q^{13} -1.17157 q^{15} +3.65685 q^{17} -4.14386 q^{19} -1.08239 q^{21} -1.17157 q^{23} -3.82843 q^{25} -5.22625 q^{27} +8.28772 q^{29} -5.65685 q^{31} +5.65685 q^{33} +1.08239 q^{35} -2.16478 q^{37} -6.82843 q^{39} -7.65685 q^{41} -5.22625 q^{43} +1.97908 q^{45} -8.00000 q^{47} +1.00000 q^{49} +3.95815 q^{51} +4.32957 q^{53} -5.65685 q^{55} -4.48528 q^{57} +6.30864 q^{59} -7.20533 q^{61} +1.82843 q^{63} +6.82843 q^{65} +7.39104 q^{67} -1.26810 q^{69} -13.6569 q^{71} -0.343146 q^{73} -4.14386 q^{75} -5.22625 q^{77} -13.6569 q^{79} -0.171573 q^{81} -5.41196 q^{83} -3.95815 q^{85} +8.97056 q^{87} +2.00000 q^{89} +6.30864 q^{91} -6.12293 q^{93} +4.48528 q^{95} -10.0000 q^{97} -9.55582 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{7} + 4q^{9} + O(q^{10}) \) \( 4q - 4q^{7} + 4q^{9} - 16q^{15} - 8q^{17} - 16q^{23} - 4q^{25} - 16q^{39} - 8q^{41} - 32q^{47} + 4q^{49} + 16q^{57} - 4q^{63} + 16q^{65} - 32q^{71} - 24q^{73} - 32q^{79} - 12q^{81} - 32q^{87} + 8q^{89} - 16q^{95} - 40q^{97} + O(q^{100}) \)

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
\(3\) 1.08239 0.624919 0.312460 0.949931i \(-0.398847\pi\)
0.312460 + 0.949931i \(0.398847\pi\)
\(4\) 0 0
\(5\) −1.08239 −0.484061 −0.242030 0.970269i \(-0.577813\pi\)
−0.242030 + 0.970269i \(0.577813\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) −1.82843 −0.609476
\(10\) 0 0
\(11\) 5.22625 1.57577 0.787887 0.615820i \(-0.211175\pi\)
0.787887 + 0.615820i \(0.211175\pi\)
\(12\) 0 0
\(13\) −6.30864 −1.74970 −0.874852 0.484391i \(-0.839041\pi\)
−0.874852 + 0.484391i \(0.839041\pi\)
\(14\) 0 0
\(15\) −1.17157 −0.302499
\(16\) 0 0
\(17\) 3.65685 0.886917 0.443459 0.896295i \(-0.353751\pi\)
0.443459 + 0.896295i \(0.353751\pi\)
\(18\) 0 0
\(19\) −4.14386 −0.950667 −0.475333 0.879806i \(-0.657673\pi\)
−0.475333 + 0.879806i \(0.657673\pi\)
\(20\) 0 0
\(21\) −1.08239 −0.236197
\(22\) 0 0
\(23\) −1.17157 −0.244290 −0.122145 0.992512i \(-0.538977\pi\)
−0.122145 + 0.992512i \(0.538977\pi\)
\(24\) 0 0
\(25\) −3.82843 −0.765685
\(26\) 0 0
\(27\) −5.22625 −1.00579
\(28\) 0 0
\(29\) 8.28772 1.53899 0.769495 0.638652i \(-0.220508\pi\)
0.769495 + 0.638652i \(0.220508\pi\)
\(30\) 0 0
\(31\) −5.65685 −1.01600 −0.508001 0.861357i \(-0.669615\pi\)
−0.508001 + 0.861357i \(0.669615\pi\)
\(32\) 0 0
\(33\) 5.65685 0.984732
\(34\) 0 0
\(35\) 1.08239 0.182958
\(36\) 0 0
\(37\) −2.16478 −0.355888 −0.177944 0.984041i \(-0.556945\pi\)
−0.177944 + 0.984041i \(0.556945\pi\)
\(38\) 0 0
\(39\) −6.82843 −1.09342
\(40\) 0 0
\(41\) −7.65685 −1.19580 −0.597900 0.801571i \(-0.703998\pi\)
−0.597900 + 0.801571i \(0.703998\pi\)
\(42\) 0 0
\(43\) −5.22625 −0.796996 −0.398498 0.917169i \(-0.630468\pi\)
−0.398498 + 0.917169i \(0.630468\pi\)
\(44\) 0 0
\(45\) 1.97908 0.295023
\(46\) 0 0
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 3.95815 0.554252
\(52\) 0 0
\(53\) 4.32957 0.594712 0.297356 0.954767i \(-0.403895\pi\)
0.297356 + 0.954767i \(0.403895\pi\)
\(54\) 0 0
\(55\) −5.65685 −0.762770
\(56\) 0 0
\(57\) −4.48528 −0.594090
\(58\) 0 0
\(59\) 6.30864 0.821315 0.410658 0.911790i \(-0.365299\pi\)
0.410658 + 0.911790i \(0.365299\pi\)
\(60\) 0 0
\(61\) −7.20533 −0.922548 −0.461274 0.887258i \(-0.652607\pi\)
−0.461274 + 0.887258i \(0.652607\pi\)
\(62\) 0 0
\(63\) 1.82843 0.230360
\(64\) 0 0
\(65\) 6.82843 0.846962
\(66\) 0 0
\(67\) 7.39104 0.902959 0.451479 0.892282i \(-0.350896\pi\)
0.451479 + 0.892282i \(0.350896\pi\)
\(68\) 0 0
\(69\) −1.26810 −0.152661
\(70\) 0 0
\(71\) −13.6569 −1.62077 −0.810385 0.585897i \(-0.800742\pi\)
−0.810385 + 0.585897i \(0.800742\pi\)
\(72\) 0 0
\(73\) −0.343146 −0.0401622 −0.0200811 0.999798i \(-0.506392\pi\)
−0.0200811 + 0.999798i \(0.506392\pi\)
\(74\) 0 0
\(75\) −4.14386 −0.478492
\(76\) 0 0
\(77\) −5.22625 −0.595587
\(78\) 0 0
\(79\) −13.6569 −1.53652 −0.768258 0.640140i \(-0.778876\pi\)
−0.768258 + 0.640140i \(0.778876\pi\)
\(80\) 0 0
\(81\) −0.171573 −0.0190637
\(82\) 0 0
\(83\) −5.41196 −0.594040 −0.297020 0.954871i \(-0.595993\pi\)
−0.297020 + 0.954871i \(0.595993\pi\)
\(84\) 0 0
\(85\) −3.95815 −0.429322
\(86\) 0 0
\(87\) 8.97056 0.961745
\(88\) 0 0
\(89\) 2.00000 0.212000 0.106000 0.994366i \(-0.466196\pi\)
0.106000 + 0.994366i \(0.466196\pi\)
\(90\) 0 0
\(91\) 6.30864 0.661326
\(92\) 0 0
\(93\) −6.12293 −0.634919
\(94\) 0 0
\(95\) 4.48528 0.460180
\(96\) 0 0
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 0 0
\(99\) −9.55582 −0.960396
\(100\) 0 0
\(101\) −4.14386 −0.412329 −0.206165 0.978517i \(-0.566098\pi\)
−0.206165 + 0.978517i \(0.566098\pi\)
\(102\) 0 0
\(103\) −2.34315 −0.230877 −0.115439 0.993315i \(-0.536827\pi\)
−0.115439 + 0.993315i \(0.536827\pi\)
\(104\) 0 0
\(105\) 1.17157 0.114334
\(106\) 0 0
\(107\) −11.7206 −1.13307 −0.566537 0.824036i \(-0.691717\pi\)
−0.566537 + 0.824036i \(0.691717\pi\)
\(108\) 0 0
\(109\) 16.9469 1.62321 0.811607 0.584203i \(-0.198593\pi\)
0.811607 + 0.584203i \(0.198593\pi\)
\(110\) 0 0
\(111\) −2.34315 −0.222402
\(112\) 0 0
\(113\) 3.17157 0.298356 0.149178 0.988810i \(-0.452337\pi\)
0.149178 + 0.988810i \(0.452337\pi\)
\(114\) 0 0
\(115\) 1.26810 0.118251
\(116\) 0 0
\(117\) 11.5349 1.06640
\(118\) 0 0
\(119\) −3.65685 −0.335223
\(120\) 0 0
\(121\) 16.3137 1.48306
\(122\) 0 0
\(123\) −8.28772 −0.747278
\(124\) 0 0
\(125\) 9.55582 0.854699
\(126\) 0 0
\(127\) 12.4853 1.10789 0.553945 0.832553i \(-0.313122\pi\)
0.553945 + 0.832553i \(0.313122\pi\)
\(128\) 0 0
\(129\) −5.65685 −0.498058
\(130\) 0 0
\(131\) 7.57675 0.661983 0.330992 0.943634i \(-0.392617\pi\)
0.330992 + 0.943634i \(0.392617\pi\)
\(132\) 0 0
\(133\) 4.14386 0.359318
\(134\) 0 0
\(135\) 5.65685 0.486864
\(136\) 0 0
\(137\) −2.00000 −0.170872 −0.0854358 0.996344i \(-0.527228\pi\)
−0.0854358 + 0.996344i \(0.527228\pi\)
\(138\) 0 0
\(139\) 10.2668 0.870818 0.435409 0.900233i \(-0.356604\pi\)
0.435409 + 0.900233i \(0.356604\pi\)
\(140\) 0 0
\(141\) −8.65914 −0.729231
\(142\) 0 0
\(143\) −32.9706 −2.75714
\(144\) 0 0
\(145\) −8.97056 −0.744965
\(146\) 0 0
\(147\) 1.08239 0.0892742
\(148\) 0 0
\(149\) 19.1116 1.56569 0.782843 0.622219i \(-0.213769\pi\)
0.782843 + 0.622219i \(0.213769\pi\)
\(150\) 0 0
\(151\) −10.1421 −0.825355 −0.412678 0.910877i \(-0.635406\pi\)
−0.412678 + 0.910877i \(0.635406\pi\)
\(152\) 0 0
\(153\) −6.68629 −0.540555
\(154\) 0 0
\(155\) 6.12293 0.491806
\(156\) 0 0
\(157\) 4.14386 0.330716 0.165358 0.986234i \(-0.447122\pi\)
0.165358 + 0.986234i \(0.447122\pi\)
\(158\) 0 0
\(159\) 4.68629 0.371647
\(160\) 0 0
\(161\) 1.17157 0.0923329
\(162\) 0 0
\(163\) −20.0083 −1.56717 −0.783586 0.621283i \(-0.786612\pi\)
−0.783586 + 0.621283i \(0.786612\pi\)
\(164\) 0 0
\(165\) −6.12293 −0.476670
\(166\) 0 0
\(167\) −16.0000 −1.23812 −0.619059 0.785345i \(-0.712486\pi\)
−0.619059 + 0.785345i \(0.712486\pi\)
\(168\) 0 0
\(169\) 26.7990 2.06146
\(170\) 0 0
\(171\) 7.57675 0.579408
\(172\) 0 0
\(173\) 12.8030 0.973394 0.486697 0.873571i \(-0.338202\pi\)
0.486697 + 0.873571i \(0.338202\pi\)
\(174\) 0 0
\(175\) 3.82843 0.289402
\(176\) 0 0
\(177\) 6.82843 0.513256
\(178\) 0 0
\(179\) −9.18440 −0.686474 −0.343237 0.939249i \(-0.611523\pi\)
−0.343237 + 0.939249i \(0.611523\pi\)
\(180\) 0 0
\(181\) 1.08239 0.0804536 0.0402268 0.999191i \(-0.487192\pi\)
0.0402268 + 0.999191i \(0.487192\pi\)
\(182\) 0 0
\(183\) −7.79899 −0.576518
\(184\) 0 0
\(185\) 2.34315 0.172272
\(186\) 0 0
\(187\) 19.1116 1.39758
\(188\) 0 0
\(189\) 5.22625 0.380154
\(190\) 0 0
\(191\) −3.31371 −0.239772 −0.119886 0.992788i \(-0.538253\pi\)
−0.119886 + 0.992788i \(0.538253\pi\)
\(192\) 0 0
\(193\) −25.7990 −1.85705 −0.928526 0.371267i \(-0.878923\pi\)
−0.928526 + 0.371267i \(0.878923\pi\)
\(194\) 0 0
\(195\) 7.39104 0.529283
\(196\) 0 0
\(197\) −14.7821 −1.05318 −0.526590 0.850120i \(-0.676530\pi\)
−0.526590 + 0.850120i \(0.676530\pi\)
\(198\) 0 0
\(199\) 24.9706 1.77012 0.885058 0.465481i \(-0.154118\pi\)
0.885058 + 0.465481i \(0.154118\pi\)
\(200\) 0 0
\(201\) 8.00000 0.564276
\(202\) 0 0
\(203\) −8.28772 −0.581684
\(204\) 0 0
\(205\) 8.28772 0.578839
\(206\) 0 0
\(207\) 2.14214 0.148889
\(208\) 0 0
\(209\) −21.6569 −1.49804
\(210\) 0 0
\(211\) 22.1731 1.52646 0.763230 0.646127i \(-0.223612\pi\)
0.763230 + 0.646127i \(0.223612\pi\)
\(212\) 0 0
\(213\) −14.7821 −1.01285
\(214\) 0 0
\(215\) 5.65685 0.385794
\(216\) 0 0
\(217\) 5.65685 0.384012
\(218\) 0 0
\(219\) −0.371418 −0.0250981
\(220\) 0 0
\(221\) −23.0698 −1.55184
\(222\) 0 0
\(223\) 24.9706 1.67215 0.836076 0.548613i \(-0.184844\pi\)
0.836076 + 0.548613i \(0.184844\pi\)
\(224\) 0 0
\(225\) 7.00000 0.466667
\(226\) 0 0
\(227\) 12.4316 0.825113 0.412556 0.910932i \(-0.364636\pi\)
0.412556 + 0.910932i \(0.364636\pi\)
\(228\) 0 0
\(229\) 8.47343 0.559940 0.279970 0.960009i \(-0.409675\pi\)
0.279970 + 0.960009i \(0.409675\pi\)
\(230\) 0 0
\(231\) −5.65685 −0.372194
\(232\) 0 0
\(233\) 10.0000 0.655122 0.327561 0.944830i \(-0.393773\pi\)
0.327561 + 0.944830i \(0.393773\pi\)
\(234\) 0 0
\(235\) 8.65914 0.564860
\(236\) 0 0
\(237\) −14.7821 −0.960199
\(238\) 0 0
\(239\) 10.1421 0.656040 0.328020 0.944671i \(-0.393619\pi\)
0.328020 + 0.944671i \(0.393619\pi\)
\(240\) 0 0
\(241\) −21.3137 −1.37294 −0.686468 0.727160i \(-0.740840\pi\)
−0.686468 + 0.727160i \(0.740840\pi\)
\(242\) 0 0
\(243\) 15.4930 0.993879
\(244\) 0 0
\(245\) −1.08239 −0.0691515
\(246\) 0 0
\(247\) 26.1421 1.66338
\(248\) 0 0
\(249\) −5.85786 −0.371227
\(250\) 0 0
\(251\) −24.1522 −1.52447 −0.762236 0.647299i \(-0.775899\pi\)
−0.762236 + 0.647299i \(0.775899\pi\)
\(252\) 0 0
\(253\) −6.12293 −0.384946
\(254\) 0 0
\(255\) −4.28427 −0.268291
\(256\) 0 0
\(257\) 13.3137 0.830486 0.415243 0.909710i \(-0.363696\pi\)
0.415243 + 0.909710i \(0.363696\pi\)
\(258\) 0 0
\(259\) 2.16478 0.134513
\(260\) 0 0
\(261\) −15.1535 −0.937978
\(262\) 0 0
\(263\) −13.6569 −0.842118 −0.421059 0.907033i \(-0.638341\pi\)
−0.421059 + 0.907033i \(0.638341\pi\)
\(264\) 0 0
\(265\) −4.68629 −0.287877
\(266\) 0 0
\(267\) 2.16478 0.132483
\(268\) 0 0
\(269\) −27.2137 −1.65925 −0.829623 0.558324i \(-0.811445\pi\)
−0.829623 + 0.558324i \(0.811445\pi\)
\(270\) 0 0
\(271\) −13.6569 −0.829595 −0.414797 0.909914i \(-0.636148\pi\)
−0.414797 + 0.909914i \(0.636148\pi\)
\(272\) 0 0
\(273\) 6.82843 0.413275
\(274\) 0 0
\(275\) −20.0083 −1.20655
\(276\) 0 0
\(277\) 6.12293 0.367892 0.183946 0.982936i \(-0.441113\pi\)
0.183946 + 0.982936i \(0.441113\pi\)
\(278\) 0 0
\(279\) 10.3431 0.619228
\(280\) 0 0
\(281\) 2.00000 0.119310 0.0596550 0.998219i \(-0.481000\pi\)
0.0596550 + 0.998219i \(0.481000\pi\)
\(282\) 0 0
\(283\) −28.1103 −1.67099 −0.835493 0.549501i \(-0.814818\pi\)
−0.835493 + 0.549501i \(0.814818\pi\)
\(284\) 0 0
\(285\) 4.85483 0.287576
\(286\) 0 0
\(287\) 7.65685 0.451970
\(288\) 0 0
\(289\) −3.62742 −0.213377
\(290\) 0 0
\(291\) −10.8239 −0.634510
\(292\) 0 0
\(293\) 10.6382 0.621491 0.310746 0.950493i \(-0.399421\pi\)
0.310746 + 0.950493i \(0.399421\pi\)
\(294\) 0 0
\(295\) −6.82843 −0.397566
\(296\) 0 0
\(297\) −27.3137 −1.58490
\(298\) 0 0
\(299\) 7.39104 0.427435
\(300\) 0 0
\(301\) 5.22625 0.301236
\(302\) 0 0
\(303\) −4.48528 −0.257673
\(304\) 0 0
\(305\) 7.79899 0.446569
\(306\) 0 0
\(307\) −15.8645 −0.905433 −0.452716 0.891655i \(-0.649545\pi\)
−0.452716 + 0.891655i \(0.649545\pi\)
\(308\) 0 0
\(309\) −2.53620 −0.144280
\(310\) 0 0
\(311\) 30.6274 1.73672 0.868361 0.495933i \(-0.165174\pi\)
0.868361 + 0.495933i \(0.165174\pi\)
\(312\) 0 0
\(313\) 19.6569 1.11107 0.555536 0.831493i \(-0.312513\pi\)
0.555536 + 0.831493i \(0.312513\pi\)
\(314\) 0 0
\(315\) −1.97908 −0.111508
\(316\) 0 0
\(317\) 20.9050 1.17414 0.587071 0.809535i \(-0.300281\pi\)
0.587071 + 0.809535i \(0.300281\pi\)
\(318\) 0 0
\(319\) 43.3137 2.42510
\(320\) 0 0
\(321\) −12.6863 −0.708080
\(322\) 0 0
\(323\) −15.1535 −0.843163
\(324\) 0 0
\(325\) 24.1522 1.33972
\(326\) 0 0
\(327\) 18.3431 1.01438
\(328\) 0 0
\(329\) 8.00000 0.441054
\(330\) 0 0
\(331\) −13.8854 −0.763210 −0.381605 0.924325i \(-0.624629\pi\)
−0.381605 + 0.924325i \(0.624629\pi\)
\(332\) 0 0
\(333\) 3.95815 0.216905
\(334\) 0 0
\(335\) −8.00000 −0.437087
\(336\) 0 0
\(337\) 16.8284 0.916703 0.458351 0.888771i \(-0.348440\pi\)
0.458351 + 0.888771i \(0.348440\pi\)
\(338\) 0 0
\(339\) 3.43289 0.186449
\(340\) 0 0
\(341\) −29.5641 −1.60099
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 1.37258 0.0738974
\(346\) 0 0
\(347\) 0.896683 0.0481365 0.0240682 0.999710i \(-0.492338\pi\)
0.0240682 + 0.999710i \(0.492338\pi\)
\(348\) 0 0
\(349\) 2.87576 0.153936 0.0769679 0.997034i \(-0.475476\pi\)
0.0769679 + 0.997034i \(0.475476\pi\)
\(350\) 0 0
\(351\) 32.9706 1.75984
\(352\) 0 0
\(353\) 4.34315 0.231162 0.115581 0.993298i \(-0.463127\pi\)
0.115581 + 0.993298i \(0.463127\pi\)
\(354\) 0 0
\(355\) 14.7821 0.784551
\(356\) 0 0
\(357\) −3.95815 −0.209488
\(358\) 0 0
\(359\) −1.17157 −0.0618333 −0.0309166 0.999522i \(-0.509843\pi\)
−0.0309166 + 0.999522i \(0.509843\pi\)
\(360\) 0 0
\(361\) −1.82843 −0.0962330
\(362\) 0 0
\(363\) 17.6578 0.926796
\(364\) 0 0
\(365\) 0.371418 0.0194409
\(366\) 0 0
\(367\) −8.97056 −0.468260 −0.234130 0.972205i \(-0.575224\pi\)
−0.234130 + 0.972205i \(0.575224\pi\)
\(368\) 0 0
\(369\) 14.0000 0.728811
\(370\) 0 0
\(371\) −4.32957 −0.224780
\(372\) 0 0
\(373\) −22.6984 −1.17528 −0.587639 0.809124i \(-0.699942\pi\)
−0.587639 + 0.809124i \(0.699942\pi\)
\(374\) 0 0
\(375\) 10.3431 0.534118
\(376\) 0 0
\(377\) −52.2843 −2.69278
\(378\) 0 0
\(379\) −3.43289 −0.176335 −0.0881677 0.996106i \(-0.528101\pi\)
−0.0881677 + 0.996106i \(0.528101\pi\)
\(380\) 0 0
\(381\) 13.5140 0.692342
\(382\) 0 0
\(383\) 3.31371 0.169323 0.0846613 0.996410i \(-0.473019\pi\)
0.0846613 + 0.996410i \(0.473019\pi\)
\(384\) 0 0
\(385\) 5.65685 0.288300
\(386\) 0 0
\(387\) 9.55582 0.485750
\(388\) 0 0
\(389\) 3.95815 0.200686 0.100343 0.994953i \(-0.468006\pi\)
0.100343 + 0.994953i \(0.468006\pi\)
\(390\) 0 0
\(391\) −4.28427 −0.216665
\(392\) 0 0
\(393\) 8.20101 0.413686
\(394\) 0 0
\(395\) 14.7821 0.743767
\(396\) 0 0
\(397\) −7.57675 −0.380266 −0.190133 0.981758i \(-0.560892\pi\)
−0.190133 + 0.981758i \(0.560892\pi\)
\(398\) 0 0
\(399\) 4.48528 0.224545
\(400\) 0 0
\(401\) 26.4853 1.32261 0.661306 0.750116i \(-0.270003\pi\)
0.661306 + 0.750116i \(0.270003\pi\)
\(402\) 0 0
\(403\) 35.6871 1.77770
\(404\) 0 0
\(405\) 0.185709 0.00922796
\(406\) 0 0
\(407\) −11.3137 −0.560800
\(408\) 0 0
\(409\) 17.3137 0.856108 0.428054 0.903753i \(-0.359199\pi\)
0.428054 + 0.903753i \(0.359199\pi\)
\(410\) 0 0
\(411\) −2.16478 −0.106781
\(412\) 0 0
\(413\) −6.30864 −0.310428
\(414\) 0 0
\(415\) 5.85786 0.287551
\(416\) 0 0
\(417\) 11.1127 0.544191
\(418\) 0 0
\(419\) 5.41196 0.264392 0.132196 0.991224i \(-0.457797\pi\)
0.132196 + 0.991224i \(0.457797\pi\)
\(420\) 0 0
\(421\) 35.6871 1.73928 0.869641 0.493685i \(-0.164350\pi\)
0.869641 + 0.493685i \(0.164350\pi\)
\(422\) 0 0
\(423\) 14.6274 0.711209
\(424\) 0 0
\(425\) −14.0000 −0.679100
\(426\) 0 0
\(427\) 7.20533 0.348690
\(428\) 0 0
\(429\) −35.6871 −1.72299
\(430\) 0 0
\(431\) 1.17157 0.0564327 0.0282163 0.999602i \(-0.491017\pi\)
0.0282163 + 0.999602i \(0.491017\pi\)
\(432\) 0 0
\(433\) −23.6569 −1.13688 −0.568438 0.822726i \(-0.692452\pi\)
−0.568438 + 0.822726i \(0.692452\pi\)
\(434\) 0 0
\(435\) −9.70967 −0.465543
\(436\) 0 0
\(437\) 4.85483 0.232238
\(438\) 0 0
\(439\) 32.9706 1.57360 0.786800 0.617209i \(-0.211737\pi\)
0.786800 + 0.617209i \(0.211737\pi\)
\(440\) 0 0
\(441\) −1.82843 −0.0870680
\(442\) 0 0
\(443\) 22.1731 1.05348 0.526738 0.850028i \(-0.323415\pi\)
0.526738 + 0.850028i \(0.323415\pi\)
\(444\) 0 0
\(445\) −2.16478 −0.102621
\(446\) 0 0
\(447\) 20.6863 0.978428
\(448\) 0 0
\(449\) 20.6274 0.973468 0.486734 0.873550i \(-0.338188\pi\)
0.486734 + 0.873550i \(0.338188\pi\)
\(450\) 0 0
\(451\) −40.0166 −1.88431
\(452\) 0 0
\(453\) −10.9778 −0.515781
\(454\) 0 0
\(455\) −6.82843 −0.320122
\(456\) 0 0
\(457\) −6.48528 −0.303369 −0.151684 0.988429i \(-0.548470\pi\)
−0.151684 + 0.988429i \(0.548470\pi\)
\(458\) 0 0
\(459\) −19.1116 −0.892055
\(460\) 0 0
\(461\) 32.4399 1.51088 0.755438 0.655220i \(-0.227424\pi\)
0.755438 + 0.655220i \(0.227424\pi\)
\(462\) 0 0
\(463\) −8.00000 −0.371792 −0.185896 0.982569i \(-0.559519\pi\)
−0.185896 + 0.982569i \(0.559519\pi\)
\(464\) 0 0
\(465\) 6.62742 0.307339
\(466\) 0 0
\(467\) −18.9259 −0.875788 −0.437894 0.899027i \(-0.644275\pi\)
−0.437894 + 0.899027i \(0.644275\pi\)
\(468\) 0 0
\(469\) −7.39104 −0.341286
\(470\) 0 0
\(471\) 4.48528 0.206671
\(472\) 0 0
\(473\) −27.3137 −1.25589
\(474\) 0 0
\(475\) 15.8645 0.727912
\(476\) 0 0
\(477\) −7.91630 −0.362463
\(478\) 0 0
\(479\) −5.65685 −0.258468 −0.129234 0.991614i \(-0.541252\pi\)
−0.129234 + 0.991614i \(0.541252\pi\)
\(480\) 0 0
\(481\) 13.6569 0.622699
\(482\) 0 0
\(483\) 1.26810 0.0577006
\(484\) 0 0
\(485\) 10.8239 0.491489
\(486\) 0 0
\(487\) −7.79899 −0.353406 −0.176703 0.984264i \(-0.556543\pi\)
−0.176703 + 0.984264i \(0.556543\pi\)
\(488\) 0 0
\(489\) −21.6569 −0.979357
\(490\) 0 0
\(491\) −4.85483 −0.219096 −0.109548 0.993982i \(-0.534940\pi\)
−0.109548 + 0.993982i \(0.534940\pi\)
\(492\) 0 0
\(493\) 30.3070 1.36496
\(494\) 0 0
\(495\) 10.3431 0.464890
\(496\) 0 0
\(497\) 13.6569 0.612594
\(498\) 0 0
\(499\) 13.5140 0.604968 0.302484 0.953154i \(-0.402184\pi\)
0.302484 + 0.953154i \(0.402184\pi\)
\(500\) 0 0
\(501\) −17.3183 −0.773723
\(502\) 0 0
\(503\) 5.65685 0.252227 0.126113 0.992016i \(-0.459750\pi\)
0.126113 + 0.992016i \(0.459750\pi\)
\(504\) 0 0
\(505\) 4.48528 0.199592
\(506\) 0 0
\(507\) 29.0070 1.28825
\(508\) 0 0
\(509\) −13.6997 −0.607228 −0.303614 0.952795i \(-0.598193\pi\)
−0.303614 + 0.952795i \(0.598193\pi\)
\(510\) 0 0
\(511\) 0.343146 0.0151799
\(512\) 0 0
\(513\) 21.6569 0.956173
\(514\) 0 0
\(515\) 2.53620 0.111758
\(516\) 0 0
\(517\) −41.8100 −1.83880
\(518\) 0 0
\(519\) 13.8579 0.608293
\(520\) 0 0
\(521\) −37.3137 −1.63474 −0.817372 0.576111i \(-0.804570\pi\)
−0.817372 + 0.576111i \(0.804570\pi\)
\(522\) 0 0
\(523\) −33.7080 −1.47395 −0.736974 0.675921i \(-0.763746\pi\)
−0.736974 + 0.675921i \(0.763746\pi\)
\(524\) 0 0
\(525\) 4.14386 0.180853
\(526\) 0 0
\(527\) −20.6863 −0.901109
\(528\) 0 0
\(529\) −21.6274 −0.940322
\(530\) 0 0
\(531\) −11.5349 −0.500572
\(532\) 0 0
\(533\) 48.3044 2.09229
\(534\) 0 0
\(535\) 12.6863 0.548476
\(536\) 0 0
\(537\) −9.94113 −0.428991
\(538\) 0 0
\(539\) 5.22625 0.225111
\(540\) 0 0
\(541\) −12.9887 −0.558428 −0.279214 0.960229i \(-0.590074\pi\)
−0.279214 + 0.960229i \(0.590074\pi\)
\(542\) 0 0
\(543\) 1.17157 0.0502770
\(544\) 0 0
\(545\) −18.3431 −0.785734
\(546\) 0 0
\(547\) −12.0920 −0.517018 −0.258509 0.966009i \(-0.583231\pi\)
−0.258509 + 0.966009i \(0.583231\pi\)
\(548\) 0 0
\(549\) 13.1744 0.562270
\(550\) 0 0
\(551\) −34.3431 −1.46307
\(552\) 0 0
\(553\) 13.6569 0.580749
\(554\) 0 0
\(555\) 2.53620 0.107656
\(556\) 0 0
\(557\) −6.12293 −0.259437 −0.129719 0.991551i \(-0.541407\pi\)
−0.129719 + 0.991551i \(0.541407\pi\)
\(558\) 0 0
\(559\) 32.9706 1.39451
\(560\) 0 0
\(561\) 20.6863 0.873376
\(562\) 0 0
\(563\) 35.5014 1.49620 0.748102 0.663584i \(-0.230965\pi\)
0.748102 + 0.663584i \(0.230965\pi\)
\(564\) 0 0
\(565\) −3.43289 −0.144423
\(566\) 0 0
\(567\) 0.171573 0.00720538
\(568\) 0 0
\(569\) 34.4853 1.44570 0.722849 0.691006i \(-0.242832\pi\)
0.722849 + 0.691006i \(0.242832\pi\)
\(570\) 0 0
\(571\) 13.8854 0.581085 0.290543 0.956862i \(-0.406164\pi\)
0.290543 + 0.956862i \(0.406164\pi\)
\(572\) 0 0
\(573\) −3.58673 −0.149838
\(574\) 0 0
\(575\) 4.48528 0.187049
\(576\) 0 0
\(577\) 10.9706 0.456711 0.228355 0.973578i \(-0.426665\pi\)
0.228355 + 0.973578i \(0.426665\pi\)
\(578\) 0 0
\(579\) −27.9246 −1.16051
\(580\) 0 0
\(581\) 5.41196 0.224526
\(582\) 0 0
\(583\) 22.6274 0.937132
\(584\) 0 0
\(585\) −12.4853 −0.516203
\(586\) 0 0
\(587\) −29.0070 −1.19725 −0.598624 0.801030i \(-0.704286\pi\)
−0.598624 + 0.801030i \(0.704286\pi\)
\(588\) 0 0
\(589\) 23.4412 0.965878
\(590\) 0 0
\(591\) −16.0000 −0.658152
\(592\) 0 0
\(593\) −2.68629 −0.110313 −0.0551564 0.998478i \(-0.517566\pi\)
−0.0551564 + 0.998478i \(0.517566\pi\)
\(594\) 0 0
\(595\) 3.95815 0.162268
\(596\) 0 0
\(597\) 27.0279 1.10618
\(598\) 0 0
\(599\) −8.00000 −0.326871 −0.163436 0.986554i \(-0.552258\pi\)
−0.163436 + 0.986554i \(0.552258\pi\)
\(600\) 0 0
\(601\) −18.6863 −0.762230 −0.381115 0.924528i \(-0.624460\pi\)
−0.381115 + 0.924528i \(0.624460\pi\)
\(602\) 0 0
\(603\) −13.5140 −0.550331
\(604\) 0 0
\(605\) −17.6578 −0.717893
\(606\) 0 0
\(607\) 4.68629 0.190211 0.0951054 0.995467i \(-0.469681\pi\)
0.0951054 + 0.995467i \(0.469681\pi\)
\(608\) 0 0
\(609\) −8.97056 −0.363506
\(610\) 0 0
\(611\) 50.4692 2.04176
\(612\) 0 0
\(613\) 10.8239 0.437174 0.218587 0.975817i \(-0.429855\pi\)
0.218587 + 0.975817i \(0.429855\pi\)
\(614\) 0 0
\(615\) 8.97056 0.361728
\(616\) 0 0
\(617\) 2.20101 0.0886093 0.0443047 0.999018i \(-0.485893\pi\)
0.0443047 + 0.999018i \(0.485893\pi\)
\(618\) 0 0
\(619\) −14.0711 −0.565565 −0.282783 0.959184i \(-0.591257\pi\)
−0.282783 + 0.959184i \(0.591257\pi\)
\(620\) 0 0
\(621\) 6.12293 0.245705
\(622\) 0 0
\(623\) −2.00000 −0.0801283
\(624\) 0 0
\(625\) 8.79899 0.351960
\(626\) 0 0
\(627\) −23.4412 −0.936152
\(628\) 0 0
\(629\) −7.91630 −0.315644
\(630\) 0 0
\(631\) 14.6274 0.582308 0.291154 0.956676i \(-0.405961\pi\)
0.291154 + 0.956676i \(0.405961\pi\)
\(632\) 0 0
\(633\) 24.0000 0.953914
\(634\) 0 0
\(635\) −13.5140 −0.536286
\(636\) 0 0
\(637\) −6.30864 −0.249958
\(638\) 0 0
\(639\) 24.9706 0.987820
\(640\) 0 0
\(641\) −23.4558 −0.926450 −0.463225 0.886241i \(-0.653308\pi\)
−0.463225 + 0.886241i \(0.653308\pi\)
\(642\) 0 0
\(643\) 29.3784 1.15857 0.579286 0.815124i \(-0.303331\pi\)
0.579286 + 0.815124i \(0.303331\pi\)
\(644\) 0 0
\(645\) 6.12293 0.241090
\(646\) 0 0
\(647\) −18.3431 −0.721143 −0.360572 0.932731i \(-0.617418\pi\)
−0.360572 + 0.932731i \(0.617418\pi\)
\(648\) 0 0
\(649\) 32.9706 1.29421
\(650\) 0 0
\(651\) 6.12293 0.239977
\(652\) 0 0
\(653\) 39.6452 1.55144 0.775719 0.631079i \(-0.217387\pi\)
0.775719 + 0.631079i \(0.217387\pi\)
\(654\) 0 0
\(655\) −8.20101 −0.320440
\(656\) 0 0
\(657\) 0.627417 0.0244779
\(658\) 0 0
\(659\) 34.7904 1.35524 0.677621 0.735412i \(-0.263011\pi\)
0.677621 + 0.735412i \(0.263011\pi\)
\(660\) 0 0
\(661\) 6.30864 0.245378 0.122689 0.992445i \(-0.460848\pi\)
0.122689 + 0.992445i \(0.460848\pi\)
\(662\) 0 0
\(663\) −24.9706 −0.969776
\(664\) 0 0
\(665\) −4.48528 −0.173932
\(666\) 0 0
\(667\) −9.70967 −0.375960
\(668\) 0 0
\(669\) 27.0279 1.04496
\(670\) 0 0
\(671\) −37.6569 −1.45373
\(672\) 0 0
\(673\) −16.6274 −0.640940 −0.320470 0.947259i \(-0.603841\pi\)
−0.320470 + 0.947259i \(0.603841\pi\)
\(674\) 0 0
\(675\) 20.0083 0.770121
\(676\) 0 0
\(677\) 15.8645 0.609721 0.304860 0.952397i \(-0.401390\pi\)
0.304860 + 0.952397i \(0.401390\pi\)
\(678\) 0 0
\(679\) 10.0000 0.383765
\(680\) 0 0
\(681\) 13.4558 0.515629
\(682\) 0 0
\(683\) 28.2960 1.08272 0.541359 0.840792i \(-0.317910\pi\)
0.541359 + 0.840792i \(0.317910\pi\)
\(684\) 0 0
\(685\) 2.16478 0.0827122
\(686\) 0 0
\(687\) 9.17157 0.349917
\(688\) 0 0
\(689\) −27.3137 −1.04057
\(690\) 0 0
\(691\) −39.3057 −1.49526 −0.747629 0.664116i \(-0.768808\pi\)
−0.747629 + 0.664116i \(0.768808\pi\)
\(692\) 0 0
\(693\) 9.55582 0.362996
\(694\) 0 0
\(695\) −11.1127 −0.421529
\(696\) 0 0
\(697\) −28.0000 −1.06058
\(698\) 0 0
\(699\) 10.8239 0.409398
\(700\) 0 0
\(701\) −4.70099 −0.177554 −0.0887769 0.996052i \(-0.528296\pi\)
−0.0887769 + 0.996052i \(0.528296\pi\)
\(702\) 0 0
\(703\) 8.97056 0.338331
\(704\) 0 0
\(705\) 9.37258 0.352992
\(706\) 0 0
\(707\) 4.14386 0.155846
\(708\) 0 0
\(709\) −31.7289 −1.19160 −0.595802 0.803131i \(-0.703166\pi\)
−0.595802 + 0.803131i \(0.703166\pi\)
\(710\) 0 0
\(711\) 24.9706 0.936469
\(712\) 0 0
\(713\) 6.62742 0.248199
\(714\) 0 0
\(715\) 35.6871 1.33462
\(716\) 0 0
\(717\) 10.9778 0.409972
\(718\) 0 0
\(719\) −19.3137 −0.720280 −0.360140 0.932898i \(-0.617271\pi\)
−0.360140 + 0.932898i \(0.617271\pi\)
\(720\) 0 0
\(721\) 2.34315 0.0872633
\(722\) 0 0
\(723\) −23.0698 −0.857975
\(724\) 0 0
\(725\) −31.7289 −1.17838
\(726\) 0 0
\(727\) −43.3137 −1.60642 −0.803208 0.595698i \(-0.796875\pi\)
−0.803208 + 0.595698i \(0.796875\pi\)
\(728\) 0 0
\(729\) 17.2843 0.640158
\(730\) 0 0
\(731\) −19.1116 −0.706870
\(732\) 0 0
\(733\) 15.4930 0.572249 0.286124 0.958192i \(-0.407633\pi\)
0.286124 + 0.958192i \(0.407633\pi\)
\(734\) 0 0
\(735\) −1.17157 −0.0432141
\(736\) 0 0
\(737\) 38.6274 1.42286
\(738\) 0 0
\(739\) −28.6675 −1.05455 −0.527275 0.849695i \(-0.676786\pi\)
−0.527275 + 0.849695i \(0.676786\pi\)
\(740\) 0 0
\(741\) 28.2960 1.03948
\(742\) 0 0
\(743\) −44.4853 −1.63201 −0.816003 0.578047i \(-0.803815\pi\)
−0.816003 + 0.578047i \(0.803815\pi\)
\(744\) 0 0
\(745\) −20.6863 −0.757887
\(746\) 0 0
\(747\) 9.89538 0.362053
\(748\) 0 0
\(749\) 11.7206 0.428262
\(750\) 0 0
\(751\) −5.45584 −0.199087 −0.0995433 0.995033i \(-0.531738\pi\)
−0.0995433 + 0.995033i \(0.531738\pi\)
\(752\) 0 0
\(753\) −26.1421 −0.952672
\(754\) 0 0
\(755\) 10.9778 0.399522
\(756\) 0 0
\(757\) −39.6452 −1.44093 −0.720465 0.693491i \(-0.756072\pi\)
−0.720465 + 0.693491i \(0.756072\pi\)
\(758\) 0 0
\(759\) −6.62742 −0.240560
\(760\) 0 0
\(761\) 14.9706 0.542682 0.271341 0.962483i \(-0.412533\pi\)
0.271341 + 0.962483i \(0.412533\pi\)
\(762\) 0 0
\(763\) −16.9469 −0.613517
\(764\) 0 0
\(765\) 7.23719 0.261661
\(766\) 0 0
\(767\) −39.7990 −1.43706
\(768\) 0 0
\(769\) −55.2548 −1.99254 −0.996270 0.0862891i \(-0.972499\pi\)
−0.996270 + 0.0862891i \(0.972499\pi\)
\(770\) 0 0
\(771\) 14.4107 0.518987
\(772\) 0 0
\(773\) −15.8645 −0.570605 −0.285303 0.958438i \(-0.592094\pi\)
−0.285303 + 0.958438i \(0.592094\pi\)
\(774\) 0 0
\(775\) 21.6569 0.777937
\(776\) 0 0
\(777\) 2.34315 0.0840599
\(778\) 0 0
\(779\) 31.7289 1.13681
\(780\) 0 0
\(781\) −71.3742 −2.55397
\(782\) 0 0
\(783\) −43.3137 −1.54791
\(784\) 0 0
\(785\) −4.48528 −0.160087
\(786\) 0 0
\(787\) −33.7080 −1.20156 −0.600780 0.799414i \(-0.705143\pi\)
−0.600780 + 0.799414i \(0.705143\pi\)
\(788\) 0 0
\(789\) −14.7821 −0.526256
\(790\) 0 0
\(791\) −3.17157 −0.112768
\(792\) 0 0
\(793\) 45.4558 1.61418
\(794\) 0 0
\(795\) −5.07241 −0.179900
\(796\) 0 0
\(797\) −29.7499 −1.05379 −0.526897 0.849929i \(-0.676645\pi\)
−0.526897 + 0.849929i \(0.676645\pi\)
\(798\) 0 0
\(799\) −29.2548 −1.03496
\(800\) 0 0
\(801\) −3.65685 −0.129209
\(802\) 0 0
\(803\) −1.79337 −0.0632865
\(804\) 0 0
\(805\) −1.26810 −0.0446947
\(806\) 0 0
\(807\) −29.4558 −1.03689
\(808\) 0 0
\(809\) 36.8284 1.29482 0.647409 0.762143i \(-0.275852\pi\)
0.647409 + 0.762143i \(0.275852\pi\)
\(810\) 0 0
\(811\) −45.4286 −1.59521 −0.797607 0.603177i \(-0.793901\pi\)
−0.797607 + 0.603177i \(0.793901\pi\)
\(812\) 0 0
\(813\) −14.7821 −0.518430
\(814\) 0 0
\(815\) 21.6569 0.758607
\(816\) 0 0
\(817\) 21.6569 0.757677
\(818\) 0 0
\(819\) −11.5349 −0.403062
\(820\) 0 0
\(821\) −20.9050 −0.729590 −0.364795 0.931088i \(-0.618861\pi\)
−0.364795 + 0.931088i \(0.618861\pi\)
\(822\) 0 0
\(823\) −7.02944 −0.245031 −0.122515 0.992467i \(-0.539096\pi\)
−0.122515 + 0.992467i \(0.539096\pi\)
\(824\) 0 0
\(825\) −21.6569 −0.753995
\(826\) 0 0
\(827\) −47.4077 −1.64853 −0.824263 0.566207i \(-0.808410\pi\)
−0.824263 + 0.566207i \(0.808410\pi\)
\(828\) 0 0
\(829\) −51.5515 −1.79046 −0.895230 0.445605i \(-0.852989\pi\)
−0.895230 + 0.445605i \(0.852989\pi\)
\(830\) 0 0
\(831\) 6.62742 0.229903
\(832\) 0 0
\(833\) 3.65685 0.126702
\(834\) 0 0
\(835\) 17.3183 0.599324
\(836\) 0 0
\(837\) 29.5641 1.02189
\(838\) 0 0
\(839\) 20.2843 0.700291 0.350145 0.936695i \(-0.386132\pi\)
0.350145 + 0.936695i \(0.386132\pi\)
\(840\) 0 0
\(841\) 39.6863 1.36849
\(842\) 0 0
\(843\) 2.16478 0.0745591
\(844\) 0 0
\(845\) −29.0070 −0.997872
\(846\) 0 0
\(847\) −16.3137 −0.560546
\(848\) 0 0
\(849\) −30.4264 −1.04423
\(850\) 0 0
\(851\) 2.53620 0.0869399
\(852\) 0 0
\(853\) 47.7472 1.63483 0.817417 0.576046i \(-0.195405\pi\)
0.817417 + 0.576046i \(0.195405\pi\)
\(854\) 0 0
\(855\) −8.20101 −0.280469
\(856\) 0 0
\(857\) 44.6274 1.52444 0.762222 0.647316i \(-0.224109\pi\)
0.762222 + 0.647316i \(0.224109\pi\)
\(858\) 0 0
\(859\) −13.6997 −0.467427 −0.233714 0.972306i \(-0.575088\pi\)
−0.233714 + 0.972306i \(0.575088\pi\)
\(860\) 0 0
\(861\) 8.28772 0.282445
\(862\) 0 0
\(863\) −28.6863 −0.976493 −0.488246 0.872706i \(-0.662363\pi\)
−0.488246 + 0.872706i \(0.662363\pi\)
\(864\) 0 0
\(865\) −13.8579 −0.471182
\(866\) 0 0
\(867\) −3.92629 −0.133344
\(868\) 0 0
\(869\) −71.3742 −2.42120
\(870\) 0 0
\(871\) −46.6274 −1.57991
\(872\) 0 0
\(873\) 18.2843 0.618829
\(874\) 0 0
\(875\) −9.55582 −0.323046
\(876\) 0 0
\(877\) −25.6060 −0.864653 −0.432326 0.901717i \(-0.642307\pi\)
−0.432326 + 0.901717i \(0.642307\pi\)
\(878\) 0 0
\(879\) 11.5147 0.388382
\(880\) 0 0
\(881\) −22.9706 −0.773898 −0.386949 0.922101i \(-0.626471\pi\)
−0.386949 + 0.922101i \(0.626471\pi\)
\(882\) 0 0
\(883\) −21.4303 −0.721186 −0.360593 0.932723i \(-0.617426\pi\)
−0.360593 + 0.932723i \(0.617426\pi\)
\(884\) 0 0
\(885\) −7.39104 −0.248447
\(886\) 0 0
\(887\) −22.6274 −0.759754 −0.379877 0.925037i \(-0.624034\pi\)
−0.379877 + 0.925037i \(0.624034\pi\)
\(888\) 0 0
\(889\) −12.4853 −0.418743
\(890\) 0 0
\(891\) −0.896683 −0.0300400
\(892\) 0 0
\(893\) 33.1509 1.10935
\(894\) 0 0
\(895\) 9.94113 0.332295
\(896\) 0 0
\(897\) 8.00000 0.267112
\(898\) 0 0
\(899\) −46.8824 −1.56362
\(900\) 0 0
\(901\) 15.8326 0.527460
\(902\) 0 0
\(903\) 5.65685 0.188248
\(904\) 0 0
\(905\) −1.17157 −0.0389444
\(906\) 0 0
\(907\) −17.8435 −0.592485 −0.296243 0.955113i \(-0.595734\pi\)
−0.296243 + 0.955113i \(0.595734\pi\)
\(908\) 0 0
\(909\) 7.57675 0.251305
\(910\) 0 0
\(911\) −33.1716 −1.09902 −0.549512 0.835486i \(-0.685186\pi\)
−0.549512 + 0.835486i \(0.685186\pi\)
\(912\) 0 0
\(913\) −28.2843 −0.936073
\(914\) 0 0
\(915\) 8.44157 0.279070
\(916\) 0 0
\(917\) −7.57675 −0.250206
\(918\) 0 0
\(919\) −13.6569 −0.450498 −0.225249 0.974301i \(-0.572320\pi\)
−0.225249 + 0.974301i \(0.572320\pi\)
\(920\) 0 0
\(921\) −17.1716 −0.565823
\(922\) 0 0
\(923\) 86.1562 2.83587
\(924\) 0 0
\(925\) 8.28772 0.272499
\(926\) 0 0
\(927\) 4.28427 0.140714
\(928\) 0 0
\(929\) −46.2843 −1.51854 −0.759269 0.650777i \(-0.774443\pi\)
−0.759269 + 0.650777i \(0.774443\pi\)
\(930\) 0 0
\(931\) −4.14386 −0.135810
\(932\) 0 0
\(933\) 33.1509 1.08531
\(934\) 0 0
\(935\) −20.6863 −0.676514
\(936\) 0 0
\(937\) −18.2843 −0.597321 −0.298661 0.954359i \(-0.596540\pi\)
−0.298661 + 0.954359i \(0.596540\pi\)
\(938\) 0 0
\(939\) 21.2764 0.694330
\(940\) 0 0
\(941\) 39.4595 1.28634 0.643172 0.765722i \(-0.277618\pi\)
0.643172 + 0.765722i \(0.277618\pi\)
\(942\) 0 0
\(943\) 8.97056 0.292122
\(944\) 0 0
\(945\) −5.65685 −0.184017
\(946\) 0 0
\(947\) 1.63952 0.0532772 0.0266386 0.999645i \(-0.491520\pi\)
0.0266386 + 0.999645i \(0.491520\pi\)
\(948\) 0 0
\(949\) 2.16478 0.0702719
\(950\) 0 0
\(951\) 22.6274 0.733744
\(952\) 0 0
\(953\) −14.6863 −0.475736 −0.237868 0.971298i \(-0.576449\pi\)
−0.237868 + 0.971298i \(0.576449\pi\)
\(954\) 0 0
\(955\) 3.58673 0.116064
\(956\) 0 0
\(957\) 46.8824 1.51549
\(958\) 0 0
\(959\) 2.00000 0.0645834
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) 0 0
\(963\) 21.4303 0.690581
\(964\) 0 0
\(965\) 27.9246 0.898925
\(966\) 0 0
\(967\) 58.1421 1.86973 0.934863 0.355010i \(-0.115523\pi\)
0.934863 + 0.355010i \(0.115523\pi\)
\(968\) 0 0
\(969\) −16.4020 −0.526909
\(970\) 0 0
\(971\) 45.5825 1.46281 0.731405 0.681943i \(-0.238865\pi\)
0.731405 + 0.681943i \(0.238865\pi\)
\(972\) 0 0
\(973\) −10.2668 −0.329138
\(974\) 0 0
\(975\) 26.1421 0.837218
\(976\) 0 0
\(977\) 30.0000 0.959785 0.479893 0.877327i \(-0.340676\pi\)
0.479893 + 0.877327i \(0.340676\pi\)
\(978\) 0 0
\(979\) 10.4525 0.334063
\(980\) 0 0
\(981\) −30.9861 −0.989310
\(982\) 0 0
\(983\) −56.9706 −1.81708 −0.908539 0.417799i \(-0.862802\pi\)
−0.908539 + 0.417799i \(0.862802\pi\)
\(984\) 0 0
\(985\) 16.0000 0.509802
\(986\) 0 0
\(987\) 8.65914 0.275623
\(988\) 0 0
\(989\) 6.12293 0.194698
\(990\) 0 0
\(991\) −18.3431 −0.582689 −0.291345 0.956618i \(-0.594103\pi\)
−0.291345 + 0.956618i \(0.594103\pi\)
\(992\) 0 0
\(993\) −15.0294 −0.476945
\(994\) 0 0
\(995\) −27.0279 −0.856843
\(996\) 0 0
\(997\) −36.7695 −1.16450 −0.582250 0.813009i \(-0.697828\pi\)
−0.582250 + 0.813009i \(0.697828\pi\)
\(998\) 0 0
\(999\) 11.3137 0.357950
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 1792.2.a.u.1.3 4
4.3 odd 2 1792.2.a.w.1.2 4
8.3 odd 2 1792.2.a.w.1.3 4
8.5 even 2 inner 1792.2.a.u.1.2 4
16.3 odd 4 896.2.b.f.449.2 4
16.5 even 4 896.2.b.h.449.2 yes 4
16.11 odd 4 896.2.b.f.449.3 yes 4
16.13 even 4 896.2.b.h.449.3 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
896.2.b.f.449.2 4 16.3 odd 4
896.2.b.f.449.3 yes 4 16.11 odd 4
896.2.b.h.449.2 yes 4 16.5 even 4
896.2.b.h.449.3 yes 4 16.13 even 4
1792.2.a.u.1.2 4 8.5 even 2 inner
1792.2.a.u.1.3 4 1.1 even 1 trivial
1792.2.a.w.1.2 4 4.3 odd 2
1792.2.a.w.1.3 4 8.3 odd 2