Properties

Label 1792.2.a.u.1.4
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 1.4
Root \(1.84776\) of defining polynomial
Character \(\chi\) \(=\) 1792.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.61313 q^{3} -2.61313 q^{5} -1.00000 q^{7} +3.82843 q^{9} +O(q^{10})\) \(q+2.61313 q^{3} -2.61313 q^{5} -1.00000 q^{7} +3.82843 q^{9} -2.16478 q^{11} -0.448342 q^{13} -6.82843 q^{15} -7.65685 q^{17} +4.77791 q^{19} -2.61313 q^{21} -6.82843 q^{23} +1.82843 q^{25} +2.16478 q^{27} -9.55582 q^{29} +5.65685 q^{31} -5.65685 q^{33} +2.61313 q^{35} -5.22625 q^{37} -1.17157 q^{39} +3.65685 q^{41} +2.16478 q^{43} -10.0042 q^{45} -8.00000 q^{47} +1.00000 q^{49} -20.0083 q^{51} +10.4525 q^{53} +5.65685 q^{55} +12.4853 q^{57} +0.448342 q^{59} +12.1689 q^{61} -3.82843 q^{63} +1.17157 q^{65} +3.06147 q^{67} -17.8435 q^{69} -2.34315 q^{71} -11.6569 q^{73} +4.77791 q^{75} +2.16478 q^{77} -2.34315 q^{79} -5.82843 q^{81} -13.0656 q^{83} +20.0083 q^{85} -24.9706 q^{87} +2.00000 q^{89} +0.448342 q^{91} +14.7821 q^{93} -12.4853 q^{95} -10.0000 q^{97} -8.28772 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{7} + 4q^{9} + O(q^{10}) \) \( 4q - 4q^{7} + 4q^{9} - 16q^{15} - 8q^{17} - 16q^{23} - 4q^{25} - 16q^{39} - 8q^{41} - 32q^{47} + 4q^{49} + 16q^{57} - 4q^{63} + 16q^{65} - 32q^{71} - 24q^{73} - 32q^{79} - 12q^{81} - 32q^{87} + 8q^{89} - 16q^{95} - 40q^{97} + O(q^{100}) \)

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
\(3\) 2.61313 1.50869 0.754344 0.656479i \(-0.227955\pi\)
0.754344 + 0.656479i \(0.227955\pi\)
\(4\) 0 0
\(5\) −2.61313 −1.16863 −0.584313 0.811529i \(-0.698636\pi\)
−0.584313 + 0.811529i \(0.698636\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 3.82843 1.27614
\(10\) 0 0
\(11\) −2.16478 −0.652707 −0.326354 0.945248i \(-0.605820\pi\)
−0.326354 + 0.945248i \(0.605820\pi\)
\(12\) 0 0
\(13\) −0.448342 −0.124348 −0.0621738 0.998065i \(-0.519803\pi\)
−0.0621738 + 0.998065i \(0.519803\pi\)
\(14\) 0 0
\(15\) −6.82843 −1.76309
\(16\) 0 0
\(17\) −7.65685 −1.85706 −0.928530 0.371257i \(-0.878927\pi\)
−0.928530 + 0.371257i \(0.878927\pi\)
\(18\) 0 0
\(19\) 4.77791 1.09613 0.548064 0.836436i \(-0.315365\pi\)
0.548064 + 0.836436i \(0.315365\pi\)
\(20\) 0 0
\(21\) −2.61313 −0.570231
\(22\) 0 0
\(23\) −6.82843 −1.42383 −0.711913 0.702268i \(-0.752171\pi\)
−0.711913 + 0.702268i \(0.752171\pi\)
\(24\) 0 0
\(25\) 1.82843 0.365685
\(26\) 0 0
\(27\) 2.16478 0.416613
\(28\) 0 0
\(29\) −9.55582 −1.77447 −0.887236 0.461316i \(-0.847377\pi\)
−0.887236 + 0.461316i \(0.847377\pi\)
\(30\) 0 0
\(31\) 5.65685 1.01600 0.508001 0.861357i \(-0.330385\pi\)
0.508001 + 0.861357i \(0.330385\pi\)
\(32\) 0 0
\(33\) −5.65685 −0.984732
\(34\) 0 0
\(35\) 2.61313 0.441699
\(36\) 0 0
\(37\) −5.22625 −0.859191 −0.429595 0.903022i \(-0.641344\pi\)
−0.429595 + 0.903022i \(0.641344\pi\)
\(38\) 0 0
\(39\) −1.17157 −0.187602
\(40\) 0 0
\(41\) 3.65685 0.571105 0.285552 0.958363i \(-0.407823\pi\)
0.285552 + 0.958363i \(0.407823\pi\)
\(42\) 0 0
\(43\) 2.16478 0.330127 0.165063 0.986283i \(-0.447217\pi\)
0.165063 + 0.986283i \(0.447217\pi\)
\(44\) 0 0
\(45\) −10.0042 −1.49133
\(46\) 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
\(51\) −20.0083 −2.80173
\(52\) 0 0
\(53\) 10.4525 1.43576 0.717881 0.696166i \(-0.245112\pi\)
0.717881 + 0.696166i \(0.245112\pi\)
\(54\) 0 0
\(55\) 5.65685 0.762770
\(56\) 0 0
\(57\) 12.4853 1.65372
\(58\) 0 0
\(59\) 0.448342 0.0583691 0.0291845 0.999574i \(-0.490709\pi\)
0.0291845 + 0.999574i \(0.490709\pi\)
\(60\) 0 0
\(61\) 12.1689 1.55807 0.779037 0.626978i \(-0.215708\pi\)
0.779037 + 0.626978i \(0.215708\pi\)
\(62\) 0 0
\(63\) −3.82843 −0.482336
\(64\) 0 0
\(65\) 1.17157 0.145316
\(66\) 0 0
\(67\) 3.06147 0.374018 0.187009 0.982358i \(-0.440121\pi\)
0.187009 + 0.982358i \(0.440121\pi\)
\(68\) 0 0
\(69\) −17.8435 −2.14811
\(70\) 0 0
\(71\) −2.34315 −0.278080 −0.139040 0.990287i \(-0.544402\pi\)
−0.139040 + 0.990287i \(0.544402\pi\)
\(72\) 0 0
\(73\) −11.6569 −1.36433 −0.682166 0.731198i \(-0.738962\pi\)
−0.682166 + 0.731198i \(0.738962\pi\)
\(74\) 0 0
\(75\) 4.77791 0.551706
\(76\) 0 0
\(77\) 2.16478 0.246700
\(78\) 0 0
\(79\) −2.34315 −0.263624 −0.131812 0.991275i \(-0.542080\pi\)
−0.131812 + 0.991275i \(0.542080\pi\)
\(80\) 0 0
\(81\) −5.82843 −0.647603
\(82\) 0 0
\(83\) −13.0656 −1.43414 −0.717070 0.697002i \(-0.754517\pi\)
−0.717070 + 0.697002i \(0.754517\pi\)
\(84\) 0 0
\(85\) 20.0083 2.17021
\(86\) 0 0
\(87\) −24.9706 −2.67713
\(88\) 0 0
\(89\) 2.00000 0.212000 0.106000 0.994366i \(-0.466196\pi\)
0.106000 + 0.994366i \(0.466196\pi\)
\(90\) 0 0
\(91\) 0.448342 0.0469990
\(92\) 0 0
\(93\) 14.7821 1.53283
\(94\) 0 0
\(95\) −12.4853 −1.28096
\(96\) 0 0
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 0 0
\(99\) −8.28772 −0.832947
\(100\) 0 0
\(101\) 4.77791 0.475420 0.237710 0.971336i \(-0.423603\pi\)
0.237710 + 0.971336i \(0.423603\pi\)
\(102\) 0 0
\(103\) −13.6569 −1.34565 −0.672825 0.739802i \(-0.734919\pi\)
−0.672825 + 0.739802i \(0.734919\pi\)
\(104\) 0 0
\(105\) 6.82843 0.666386
\(106\) 0 0
\(107\) −13.5140 −1.30644 −0.653222 0.757166i \(-0.726583\pi\)
−0.653222 + 0.757166i \(0.726583\pi\)
\(108\) 0 0
\(109\) 11.3492 1.08705 0.543527 0.839391i \(-0.317088\pi\)
0.543527 + 0.839391i \(0.317088\pi\)
\(110\) 0 0
\(111\) −13.6569 −1.29625
\(112\) 0 0
\(113\) 8.82843 0.830509 0.415254 0.909705i \(-0.363693\pi\)
0.415254 + 0.909705i \(0.363693\pi\)
\(114\) 0 0
\(115\) 17.8435 1.66392
\(116\) 0 0
\(117\) −1.71644 −0.158685
\(118\) 0 0
\(119\) 7.65685 0.701903
\(120\) 0 0
\(121\) −6.31371 −0.573973
\(122\) 0 0
\(123\) 9.55582 0.861619
\(124\) 0 0
\(125\) 8.28772 0.741276
\(126\) 0 0
\(127\) −4.48528 −0.398004 −0.199002 0.979999i \(-0.563770\pi\)
−0.199002 + 0.979999i \(0.563770\pi\)
\(128\) 0 0
\(129\) 5.65685 0.498058
\(130\) 0 0
\(131\) 18.2919 1.59817 0.799085 0.601219i \(-0.205318\pi\)
0.799085 + 0.601219i \(0.205318\pi\)
\(132\) 0 0
\(133\) −4.77791 −0.414297
\(134\) 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
\(139\) −19.5600 −1.65906 −0.829528 0.558465i \(-0.811390\pi\)
−0.829528 + 0.558465i \(0.811390\pi\)
\(140\) 0 0
\(141\) −20.9050 −1.76052
\(142\) 0 0
\(143\) 0.970563 0.0811625
\(144\) 0 0
\(145\) 24.9706 2.07369
\(146\) 0 0
\(147\) 2.61313 0.215527
\(148\) 0 0
\(149\) 16.5754 1.35791 0.678956 0.734179i \(-0.262433\pi\)
0.678956 + 0.734179i \(0.262433\pi\)
\(150\) 0 0
\(151\) 18.1421 1.47639 0.738193 0.674590i \(-0.235679\pi\)
0.738193 + 0.674590i \(0.235679\pi\)
\(152\) 0 0
\(153\) −29.3137 −2.36987
\(154\) 0 0
\(155\) −14.7821 −1.18732
\(156\) 0 0
\(157\) −4.77791 −0.381319 −0.190659 0.981656i \(-0.561063\pi\)
−0.190659 + 0.981656i \(0.561063\pi\)
\(158\) 0 0
\(159\) 27.3137 2.16612
\(160\) 0 0
\(161\) 6.82843 0.538155
\(162\) 0 0
\(163\) −3.95815 −0.310026 −0.155013 0.987912i \(-0.549542\pi\)
−0.155013 + 0.987912i \(0.549542\pi\)
\(164\) 0 0
\(165\) 14.7821 1.15078
\(166\) 0 0
\(167\) −16.0000 −1.23812 −0.619059 0.785345i \(-0.712486\pi\)
−0.619059 + 0.785345i \(0.712486\pi\)
\(168\) 0 0
\(169\) −12.7990 −0.984538
\(170\) 0 0
\(171\) 18.2919 1.39882
\(172\) 0 0
\(173\) 16.1271 1.22612 0.613060 0.790036i \(-0.289938\pi\)
0.613060 + 0.790036i \(0.289938\pi\)
\(174\) 0 0
\(175\) −1.82843 −0.138216
\(176\) 0 0
\(177\) 1.17157 0.0880608
\(178\) 0 0
\(179\) 22.1731 1.65730 0.828648 0.559770i \(-0.189111\pi\)
0.828648 + 0.559770i \(0.189111\pi\)
\(180\) 0 0
\(181\) 2.61313 0.194232 0.0971161 0.995273i \(-0.469038\pi\)
0.0971161 + 0.995273i \(0.469038\pi\)
\(182\) 0 0
\(183\) 31.7990 2.35065
\(184\) 0 0
\(185\) 13.6569 1.00407
\(186\) 0 0
\(187\) 16.5754 1.21212
\(188\) 0 0
\(189\) −2.16478 −0.157465
\(190\) 0 0
\(191\) 19.3137 1.39749 0.698745 0.715370i \(-0.253742\pi\)
0.698745 + 0.715370i \(0.253742\pi\)
\(192\) 0 0
\(193\) 13.7990 0.993273 0.496637 0.867959i \(-0.334568\pi\)
0.496637 + 0.867959i \(0.334568\pi\)
\(194\) 0 0
\(195\) 3.06147 0.219236
\(196\) 0 0
\(197\) −6.12293 −0.436241 −0.218121 0.975922i \(-0.569993\pi\)
−0.218121 + 0.975922i \(0.569993\pi\)
\(198\) 0 0
\(199\) −8.97056 −0.635906 −0.317953 0.948106i \(-0.602995\pi\)
−0.317953 + 0.948106i \(0.602995\pi\)
\(200\) 0 0
\(201\) 8.00000 0.564276
\(202\) 0 0
\(203\) 9.55582 0.670687
\(204\) 0 0
\(205\) −9.55582 −0.667407
\(206\) 0 0
\(207\) −26.1421 −1.81700
\(208\) 0 0
\(209\) −10.3431 −0.715450
\(210\) 0 0
\(211\) 9.18440 0.632280 0.316140 0.948712i \(-0.397613\pi\)
0.316140 + 0.948712i \(0.397613\pi\)
\(212\) 0 0
\(213\) −6.12293 −0.419537
\(214\) 0 0
\(215\) −5.65685 −0.385794
\(216\) 0 0
\(217\) −5.65685 −0.384012
\(218\) 0 0
\(219\) −30.4608 −2.05835
\(220\) 0 0
\(221\) 3.43289 0.230921
\(222\) 0 0
\(223\) −8.97056 −0.600713 −0.300357 0.953827i \(-0.597106\pi\)
−0.300357 + 0.953827i \(0.597106\pi\)
\(224\) 0 0
\(225\) 7.00000 0.466667
\(226\) 0 0
\(227\) −14.3337 −0.951363 −0.475682 0.879618i \(-0.657799\pi\)
−0.475682 + 0.879618i \(0.657799\pi\)
\(228\) 0 0
\(229\) 5.67459 0.374988 0.187494 0.982266i \(-0.439964\pi\)
0.187494 + 0.982266i \(0.439964\pi\)
\(230\) 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
\(235\) 20.9050 1.36369
\(236\) 0 0
\(237\) −6.12293 −0.397727
\(238\) 0 0
\(239\) −18.1421 −1.17352 −0.586759 0.809762i \(-0.699596\pi\)
−0.586759 + 0.809762i \(0.699596\pi\)
\(240\) 0 0
\(241\) 1.31371 0.0846234 0.0423117 0.999104i \(-0.486528\pi\)
0.0423117 + 0.999104i \(0.486528\pi\)
\(242\) 0 0
\(243\) −21.7248 −1.39364
\(244\) 0 0
\(245\) −2.61313 −0.166946
\(246\) 0 0
\(247\) −2.14214 −0.136301
\(248\) 0 0
\(249\) −34.1421 −2.16367
\(250\) 0 0
\(251\) 0.819760 0.0517428 0.0258714 0.999665i \(-0.491764\pi\)
0.0258714 + 0.999665i \(0.491764\pi\)
\(252\) 0 0
\(253\) 14.7821 0.929341
\(254\) 0 0
\(255\) 52.2843 3.27417
\(256\) 0 0
\(257\) −9.31371 −0.580973 −0.290487 0.956879i \(-0.593817\pi\)
−0.290487 + 0.956879i \(0.593817\pi\)
\(258\) 0 0
\(259\) 5.22625 0.324743
\(260\) 0 0
\(261\) −36.5838 −2.26448
\(262\) 0 0
\(263\) −2.34315 −0.144485 −0.0722423 0.997387i \(-0.523015\pi\)
−0.0722423 + 0.997387i \(0.523015\pi\)
\(264\) 0 0
\(265\) −27.3137 −1.67787
\(266\) 0 0
\(267\) 5.22625 0.319841
\(268\) 0 0
\(269\) 8.21080 0.500621 0.250311 0.968166i \(-0.419467\pi\)
0.250311 + 0.968166i \(0.419467\pi\)
\(270\) 0 0
\(271\) −2.34315 −0.142336 −0.0711680 0.997464i \(-0.522673\pi\)
−0.0711680 + 0.997464i \(0.522673\pi\)
\(272\) 0 0
\(273\) 1.17157 0.0709068
\(274\) 0 0
\(275\) −3.95815 −0.238685
\(276\) 0 0
\(277\) −14.7821 −0.888169 −0.444084 0.895985i \(-0.646471\pi\)
−0.444084 + 0.895985i \(0.646471\pi\)
\(278\) 0 0
\(279\) 21.6569 1.29656
\(280\) 0 0
\(281\) 2.00000 0.119310 0.0596550 0.998219i \(-0.481000\pi\)
0.0596550 + 0.998219i \(0.481000\pi\)
\(282\) 0 0
\(283\) 20.8281 1.23810 0.619051 0.785351i \(-0.287518\pi\)
0.619051 + 0.785351i \(0.287518\pi\)
\(284\) 0 0
\(285\) −32.6256 −1.93257
\(286\) 0 0
\(287\) −3.65685 −0.215857
\(288\) 0 0
\(289\) 41.6274 2.44867
\(290\) 0 0
\(291\) −26.1313 −1.53184
\(292\) 0 0
\(293\) 10.9008 0.636834 0.318417 0.947951i \(-0.396849\pi\)
0.318417 + 0.947951i \(0.396849\pi\)
\(294\) 0 0
\(295\) −1.17157 −0.0682116
\(296\) 0 0
\(297\) −4.68629 −0.271926
\(298\) 0 0
\(299\) 3.06147 0.177049
\(300\) 0 0
\(301\) −2.16478 −0.124776
\(302\) 0 0
\(303\) 12.4853 0.717261
\(304\) 0 0
\(305\) −31.7990 −1.82080
\(306\) 0 0
\(307\) −8.73606 −0.498593 −0.249297 0.968427i \(-0.580199\pi\)
−0.249297 + 0.968427i \(0.580199\pi\)
\(308\) 0 0
\(309\) −35.6871 −2.03017
\(310\) 0 0
\(311\) −14.6274 −0.829445 −0.414722 0.909948i \(-0.636121\pi\)
−0.414722 + 0.909948i \(0.636121\pi\)
\(312\) 0 0
\(313\) 8.34315 0.471582 0.235791 0.971804i \(-0.424232\pi\)
0.235791 + 0.971804i \(0.424232\pi\)
\(314\) 0 0
\(315\) 10.0042 0.563671
\(316\) 0 0
\(317\) −8.65914 −0.486346 −0.243173 0.969983i \(-0.578188\pi\)
−0.243173 + 0.969983i \(0.578188\pi\)
\(318\) 0 0
\(319\) 20.6863 1.15821
\(320\) 0 0
\(321\) −35.3137 −1.97102
\(322\) 0 0
\(323\) −36.5838 −2.03558
\(324\) 0 0
\(325\) −0.819760 −0.0454721
\(326\) 0 0
\(327\) 29.6569 1.64003
\(328\) 0 0
\(329\) 8.00000 0.441054
\(330\) 0 0
\(331\) −18.7402 −1.03006 −0.515028 0.857173i \(-0.672218\pi\)
−0.515028 + 0.857173i \(0.672218\pi\)
\(332\) 0 0
\(333\) −20.0083 −1.09645
\(334\) 0 0
\(335\) −8.00000 −0.437087
\(336\) 0 0
\(337\) 11.1716 0.608554 0.304277 0.952584i \(-0.401585\pi\)
0.304277 + 0.952584i \(0.401585\pi\)
\(338\) 0 0
\(339\) 23.0698 1.25298
\(340\) 0 0
\(341\) −12.2459 −0.663151
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 46.6274 2.51034
\(346\) 0 0
\(347\) −12.6173 −0.677332 −0.338666 0.940907i \(-0.609976\pi\)
−0.338666 + 0.940907i \(0.609976\pi\)
\(348\) 0 0
\(349\) −22.6215 −1.21090 −0.605449 0.795884i \(-0.707007\pi\)
−0.605449 + 0.795884i \(0.707007\pi\)
\(350\) 0 0
\(351\) −0.970563 −0.0518048
\(352\) 0 0
\(353\) 15.6569 0.833330 0.416665 0.909060i \(-0.363199\pi\)
0.416665 + 0.909060i \(0.363199\pi\)
\(354\) 0 0
\(355\) 6.12293 0.324972
\(356\) 0 0
\(357\) 20.0083 1.05895
\(358\) 0 0
\(359\) −6.82843 −0.360391 −0.180195 0.983631i \(-0.557673\pi\)
−0.180195 + 0.983631i \(0.557673\pi\)
\(360\) 0 0
\(361\) 3.82843 0.201496
\(362\) 0 0
\(363\) −16.4985 −0.865947
\(364\) 0 0
\(365\) 30.4608 1.59439
\(366\) 0 0
\(367\) 24.9706 1.30345 0.651726 0.758454i \(-0.274045\pi\)
0.651726 + 0.758454i \(0.274045\pi\)
\(368\) 0 0
\(369\) 14.0000 0.728811
\(370\) 0 0
\(371\) −10.4525 −0.542667
\(372\) 0 0
\(373\) 33.8937 1.75495 0.877475 0.479622i \(-0.159226\pi\)
0.877475 + 0.479622i \(0.159226\pi\)
\(374\) 0 0
\(375\) 21.6569 1.11836
\(376\) 0 0
\(377\) 4.28427 0.220651
\(378\) 0 0
\(379\) −23.0698 −1.18502 −0.592508 0.805565i \(-0.701862\pi\)
−0.592508 + 0.805565i \(0.701862\pi\)
\(380\) 0 0
\(381\) −11.7206 −0.600465
\(382\) 0 0
\(383\) −19.3137 −0.986884 −0.493442 0.869779i \(-0.664262\pi\)
−0.493442 + 0.869779i \(0.664262\pi\)
\(384\) 0 0
\(385\) −5.65685 −0.288300
\(386\) 0 0
\(387\) 8.28772 0.421288
\(388\) 0 0
\(389\) −20.0083 −1.01446 −0.507231 0.861810i \(-0.669331\pi\)
−0.507231 + 0.861810i \(0.669331\pi\)
\(390\) 0 0
\(391\) 52.2843 2.64413
\(392\) 0 0
\(393\) 47.7990 2.41114
\(394\) 0 0
\(395\) 6.12293 0.308078
\(396\) 0 0
\(397\) −18.2919 −0.918043 −0.459022 0.888425i \(-0.651800\pi\)
−0.459022 + 0.888425i \(0.651800\pi\)
\(398\) 0 0
\(399\) −12.4853 −0.625046
\(400\) 0 0
\(401\) 9.51472 0.475142 0.237571 0.971370i \(-0.423649\pi\)
0.237571 + 0.971370i \(0.423649\pi\)
\(402\) 0 0
\(403\) −2.53620 −0.126337
\(404\) 0 0
\(405\) 15.2304 0.756805
\(406\) 0 0
\(407\) 11.3137 0.560800
\(408\) 0 0
\(409\) −5.31371 −0.262746 −0.131373 0.991333i \(-0.541939\pi\)
−0.131373 + 0.991333i \(0.541939\pi\)
\(410\) 0 0
\(411\) −5.22625 −0.257792
\(412\) 0 0
\(413\) −0.448342 −0.0220614
\(414\) 0 0
\(415\) 34.1421 1.67597
\(416\) 0 0
\(417\) −51.1127 −2.50300
\(418\) 0 0
\(419\) 13.0656 0.638298 0.319149 0.947705i \(-0.396603\pi\)
0.319149 + 0.947705i \(0.396603\pi\)
\(420\) 0 0
\(421\) −2.53620 −0.123607 −0.0618035 0.998088i \(-0.519685\pi\)
−0.0618035 + 0.998088i \(0.519685\pi\)
\(422\) 0 0
\(423\) −30.6274 −1.48916
\(424\) 0 0
\(425\) −14.0000 −0.679100
\(426\) 0 0
\(427\) −12.1689 −0.588897
\(428\) 0 0
\(429\) 2.53620 0.122449
\(430\) 0 0
\(431\) 6.82843 0.328914 0.164457 0.986384i \(-0.447413\pi\)
0.164457 + 0.986384i \(0.447413\pi\)
\(432\) 0 0
\(433\) −12.3431 −0.593174 −0.296587 0.955006i \(-0.595848\pi\)
−0.296587 + 0.955006i \(0.595848\pi\)
\(434\) 0 0
\(435\) 65.2512 3.12856
\(436\) 0 0
\(437\) −32.6256 −1.56069
\(438\) 0 0
\(439\) −0.970563 −0.0463224 −0.0231612 0.999732i \(-0.507373\pi\)
−0.0231612 + 0.999732i \(0.507373\pi\)
\(440\) 0 0
\(441\) 3.82843 0.182306
\(442\) 0 0
\(443\) 9.18440 0.436364 0.218182 0.975908i \(-0.429987\pi\)
0.218182 + 0.975908i \(0.429987\pi\)
\(444\) 0 0
\(445\) −5.22625 −0.247748
\(446\) 0 0
\(447\) 43.3137 2.04867
\(448\) 0 0
\(449\) −24.6274 −1.16224 −0.581120 0.813818i \(-0.697385\pi\)
−0.581120 + 0.813818i \(0.697385\pi\)
\(450\) 0 0
\(451\) −7.91630 −0.372764
\(452\) 0 0
\(453\) 47.4077 2.22741
\(454\) 0 0
\(455\) −1.17157 −0.0549242
\(456\) 0 0
\(457\) 10.4853 0.490481 0.245240 0.969462i \(-0.421133\pi\)
0.245240 + 0.969462i \(0.421133\pi\)
\(458\) 0 0
\(459\) −16.5754 −0.773675
\(460\) 0 0
\(461\) −10.3756 −0.483239 −0.241619 0.970371i \(-0.577679\pi\)
−0.241619 + 0.970371i \(0.577679\pi\)
\(462\) 0 0
\(463\) −8.00000 −0.371792 −0.185896 0.982569i \(-0.559519\pi\)
−0.185896 + 0.982569i \(0.559519\pi\)
\(464\) 0 0
\(465\) −38.6274 −1.79130
\(466\) 0 0
\(467\) −1.34502 −0.0622403 −0.0311202 0.999516i \(-0.509907\pi\)
−0.0311202 + 0.999516i \(0.509907\pi\)
\(468\) 0 0
\(469\) −3.06147 −0.141365
\(470\) 0 0
\(471\) −12.4853 −0.575291
\(472\) 0 0
\(473\) −4.68629 −0.215476
\(474\) 0 0
\(475\) 8.73606 0.400838
\(476\) 0 0
\(477\) 40.0166 1.83224
\(478\) 0 0
\(479\) 5.65685 0.258468 0.129234 0.991614i \(-0.458748\pi\)
0.129234 + 0.991614i \(0.458748\pi\)
\(480\) 0 0
\(481\) 2.34315 0.106838
\(482\) 0 0
\(483\) 17.8435 0.811909
\(484\) 0 0
\(485\) 26.1313 1.18656
\(486\) 0 0
\(487\) 31.7990 1.44095 0.720475 0.693481i \(-0.243924\pi\)
0.720475 + 0.693481i \(0.243924\pi\)
\(488\) 0 0
\(489\) −10.3431 −0.467733
\(490\) 0 0
\(491\) 32.6256 1.47237 0.736187 0.676779i \(-0.236625\pi\)
0.736187 + 0.676779i \(0.236625\pi\)
\(492\) 0 0
\(493\) 73.1675 3.29530
\(494\) 0 0
\(495\) 21.6569 0.973403
\(496\) 0 0
\(497\) 2.34315 0.105104
\(498\) 0 0
\(499\) −11.7206 −0.524686 −0.262343 0.964975i \(-0.584495\pi\)
−0.262343 + 0.964975i \(0.584495\pi\)
\(500\) 0 0
\(501\) −41.8100 −1.86793
\(502\) 0 0
\(503\) −5.65685 −0.252227 −0.126113 0.992016i \(-0.540250\pi\)
−0.126113 + 0.992016i \(0.540250\pi\)
\(504\) 0 0
\(505\) −12.4853 −0.555588
\(506\) 0 0
\(507\) −33.4454 −1.48536
\(508\) 0 0
\(509\) −3.50981 −0.155570 −0.0777848 0.996970i \(-0.524785\pi\)
−0.0777848 + 0.996970i \(0.524785\pi\)
\(510\) 0 0
\(511\) 11.6569 0.515669
\(512\) 0 0
\(513\) 10.3431 0.456661
\(514\) 0 0
\(515\) 35.6871 1.57256
\(516\) 0 0
\(517\) 17.3183 0.761657
\(518\) 0 0
\(519\) 42.1421 1.84983
\(520\) 0 0
\(521\) −14.6863 −0.643418 −0.321709 0.946839i \(-0.604257\pi\)
−0.321709 + 0.946839i \(0.604257\pi\)
\(522\) 0 0
\(523\) −7.46796 −0.326551 −0.163276 0.986581i \(-0.552206\pi\)
−0.163276 + 0.986581i \(0.552206\pi\)
\(524\) 0 0
\(525\) −4.77791 −0.208525
\(526\) 0 0
\(527\) −43.3137 −1.88677
\(528\) 0 0
\(529\) 23.6274 1.02728
\(530\) 0 0
\(531\) 1.71644 0.0744873
\(532\) 0 0
\(533\) −1.63952 −0.0710155
\(534\) 0 0
\(535\) 35.3137 1.52674
\(536\) 0 0
\(537\) 57.9411 2.50034
\(538\) 0 0
\(539\) −2.16478 −0.0932439
\(540\) 0 0
\(541\) −31.3575 −1.34816 −0.674082 0.738656i \(-0.735461\pi\)
−0.674082 + 0.738656i \(0.735461\pi\)
\(542\) 0 0
\(543\) 6.82843 0.293036
\(544\) 0 0
\(545\) −29.6569 −1.27036
\(546\) 0 0
\(547\) −43.9748 −1.88023 −0.940113 0.340862i \(-0.889281\pi\)
−0.940113 + 0.340862i \(0.889281\pi\)
\(548\) 0 0
\(549\) 46.5879 1.98832
\(550\) 0 0
\(551\) −45.6569 −1.94505
\(552\) 0 0
\(553\) 2.34315 0.0996407
\(554\) 0 0
\(555\) 35.6871 1.51483
\(556\) 0 0
\(557\) 14.7821 0.626337 0.313168 0.949698i \(-0.398610\pi\)
0.313168 + 0.949698i \(0.398610\pi\)
\(558\) 0 0
\(559\) −0.970563 −0.0410504
\(560\) 0 0
\(561\) 43.3137 1.82871
\(562\) 0 0
\(563\) −17.7666 −0.748774 −0.374387 0.927273i \(-0.622147\pi\)
−0.374387 + 0.927273i \(0.622147\pi\)
\(564\) 0 0
\(565\) −23.0698 −0.970553
\(566\) 0 0
\(567\) 5.82843 0.244771
\(568\) 0 0
\(569\) 17.5147 0.734255 0.367128 0.930171i \(-0.380341\pi\)
0.367128 + 0.930171i \(0.380341\pi\)
\(570\) 0 0
\(571\) 18.7402 0.784254 0.392127 0.919911i \(-0.371739\pi\)
0.392127 + 0.919911i \(0.371739\pi\)
\(572\) 0 0
\(573\) 50.4692 2.10838
\(574\) 0 0
\(575\) −12.4853 −0.520672
\(576\) 0 0
\(577\) −22.9706 −0.956277 −0.478139 0.878284i \(-0.658688\pi\)
−0.478139 + 0.878284i \(0.658688\pi\)
\(578\) 0 0
\(579\) 36.0585 1.49854
\(580\) 0 0
\(581\) 13.0656 0.542054
\(582\) 0 0
\(583\) −22.6274 −0.937132
\(584\) 0 0
\(585\) 4.48528 0.185444
\(586\) 0 0
\(587\) 33.4454 1.38044 0.690219 0.723600i \(-0.257514\pi\)
0.690219 + 0.723600i \(0.257514\pi\)
\(588\) 0 0
\(589\) 27.0279 1.11367
\(590\) 0 0
\(591\) −16.0000 −0.658152
\(592\) 0 0
\(593\) −25.3137 −1.03951 −0.519755 0.854316i \(-0.673977\pi\)
−0.519755 + 0.854316i \(0.673977\pi\)
\(594\) 0 0
\(595\) −20.0083 −0.820261
\(596\) 0 0
\(597\) −23.4412 −0.959385
\(598\) 0 0
\(599\) −8.00000 −0.326871 −0.163436 0.986554i \(-0.552258\pi\)
−0.163436 + 0.986554i \(0.552258\pi\)
\(600\) 0 0
\(601\) −41.3137 −1.68522 −0.842611 0.538523i \(-0.818982\pi\)
−0.842611 + 0.538523i \(0.818982\pi\)
\(602\) 0 0
\(603\) 11.7206 0.477300
\(604\) 0 0
\(605\) 16.4985 0.670760
\(606\) 0 0
\(607\) 27.3137 1.10863 0.554315 0.832307i \(-0.312980\pi\)
0.554315 + 0.832307i \(0.312980\pi\)
\(608\) 0 0
\(609\) 24.9706 1.01186
\(610\) 0 0
\(611\) 3.58673 0.145104
\(612\) 0 0
\(613\) 26.1313 1.05543 0.527716 0.849421i \(-0.323049\pi\)
0.527716 + 0.849421i \(0.323049\pi\)
\(614\) 0 0
\(615\) −24.9706 −1.00691
\(616\) 0 0
\(617\) 41.7990 1.68276 0.841382 0.540441i \(-0.181743\pi\)
0.841382 + 0.540441i \(0.181743\pi\)
\(618\) 0 0
\(619\) −33.9706 −1.36540 −0.682698 0.730701i \(-0.739193\pi\)
−0.682698 + 0.730701i \(0.739193\pi\)
\(620\) 0 0
\(621\) −14.7821 −0.593184
\(622\) 0 0
\(623\) −2.00000 −0.0801283
\(624\) 0 0
\(625\) −30.7990 −1.23196
\(626\) 0 0
\(627\) −27.0279 −1.07939
\(628\) 0 0
\(629\) 40.0166 1.59557
\(630\) 0 0
\(631\) −30.6274 −1.21926 −0.609629 0.792687i \(-0.708682\pi\)
−0.609629 + 0.792687i \(0.708682\pi\)
\(632\) 0 0
\(633\) 24.0000 0.953914
\(634\) 0 0
\(635\) 11.7206 0.465118
\(636\) 0 0
\(637\) −0.448342 −0.0177639
\(638\) 0 0
\(639\) −8.97056 −0.354870
\(640\) 0 0
\(641\) 27.4558 1.08444 0.542220 0.840236i \(-0.317584\pi\)
0.542220 + 0.840236i \(0.317584\pi\)
\(642\) 0 0
\(643\) −2.98454 −0.117699 −0.0588495 0.998267i \(-0.518743\pi\)
−0.0588495 + 0.998267i \(0.518743\pi\)
\(644\) 0 0
\(645\) −14.7821 −0.582044
\(646\) 0 0
\(647\) −29.6569 −1.16593 −0.582966 0.812497i \(-0.698108\pi\)
−0.582966 + 0.812497i \(0.698108\pi\)
\(648\) 0 0
\(649\) −0.970563 −0.0380979
\(650\) 0 0
\(651\) −14.7821 −0.579355
\(652\) 0 0
\(653\) −22.5445 −0.882236 −0.441118 0.897449i \(-0.645418\pi\)
−0.441118 + 0.897449i \(0.645418\pi\)
\(654\) 0 0
\(655\) −47.7990 −1.86766
\(656\) 0 0
\(657\) −44.6274 −1.74108
\(658\) 0 0
\(659\) 10.0811 0.392703 0.196352 0.980534i \(-0.437091\pi\)
0.196352 + 0.980534i \(0.437091\pi\)
\(660\) 0 0
\(661\) 0.448342 0.0174385 0.00871923 0.999962i \(-0.497225\pi\)
0.00871923 + 0.999962i \(0.497225\pi\)
\(662\) 0 0
\(663\) 8.97056 0.348388
\(664\) 0 0
\(665\) 12.4853 0.484158
\(666\) 0 0
\(667\) 65.2512 2.52654
\(668\) 0 0
\(669\) −23.4412 −0.906290
\(670\) 0 0
\(671\) −26.3431 −1.01697
\(672\) 0 0
\(673\) 28.6274 1.10351 0.551753 0.834008i \(-0.313959\pi\)
0.551753 + 0.834008i \(0.313959\pi\)
\(674\) 0 0
\(675\) 3.95815 0.152349
\(676\) 0 0
\(677\) 8.73606 0.335754 0.167877 0.985808i \(-0.446309\pi\)
0.167877 + 0.985808i \(0.446309\pi\)
\(678\) 0 0
\(679\) 10.0000 0.383765
\(680\) 0 0
\(681\) −37.4558 −1.43531
\(682\) 0 0
\(683\) −5.59767 −0.214189 −0.107094 0.994249i \(-0.534155\pi\)
−0.107094 + 0.994249i \(0.534155\pi\)
\(684\) 0 0
\(685\) 5.22625 0.199685
\(686\) 0 0
\(687\) 14.8284 0.565740
\(688\) 0 0
\(689\) −4.68629 −0.178533
\(690\) 0 0
\(691\) −35.7640 −1.36053 −0.680263 0.732968i \(-0.738135\pi\)
−0.680263 + 0.732968i \(0.738135\pi\)
\(692\) 0 0
\(693\) 8.28772 0.314824
\(694\) 0 0
\(695\) 51.1127 1.93882
\(696\) 0 0
\(697\) −28.0000 −1.06058
\(698\) 0 0
\(699\) 26.1313 0.988375
\(700\) 0 0
\(701\) −40.9133 −1.54528 −0.772638 0.634847i \(-0.781063\pi\)
−0.772638 + 0.634847i \(0.781063\pi\)
\(702\) 0 0
\(703\) −24.9706 −0.941783
\(704\) 0 0
\(705\) 54.6274 2.05739
\(706\) 0 0
\(707\) −4.77791 −0.179692
\(708\) 0 0
\(709\) −17.4721 −0.656179 −0.328090 0.944647i \(-0.606405\pi\)
−0.328090 + 0.944647i \(0.606405\pi\)
\(710\) 0 0
\(711\) −8.97056 −0.336422
\(712\) 0 0
\(713\) −38.6274 −1.44661
\(714\) 0 0
\(715\) −2.53620 −0.0948486
\(716\) 0 0
\(717\) −47.4077 −1.77047
\(718\) 0 0
\(719\) 3.31371 0.123580 0.0617902 0.998089i \(-0.480319\pi\)
0.0617902 + 0.998089i \(0.480319\pi\)
\(720\) 0 0
\(721\) 13.6569 0.508608
\(722\) 0 0
\(723\) 3.43289 0.127670
\(724\) 0 0
\(725\) −17.4721 −0.648898
\(726\) 0 0
\(727\) −20.6863 −0.767212 −0.383606 0.923497i \(-0.625318\pi\)
−0.383606 + 0.923497i \(0.625318\pi\)
\(728\) 0 0
\(729\) −39.2843 −1.45497
\(730\) 0 0
\(731\) −16.5754 −0.613065
\(732\) 0 0
\(733\) −21.7248 −0.802423 −0.401211 0.915986i \(-0.631411\pi\)
−0.401211 + 0.915986i \(0.631411\pi\)
\(734\) 0 0
\(735\) −6.82843 −0.251870
\(736\) 0 0
\(737\) −6.62742 −0.244124
\(738\) 0 0
\(739\) −24.8632 −0.914606 −0.457303 0.889311i \(-0.651185\pi\)
−0.457303 + 0.889311i \(0.651185\pi\)
\(740\) 0 0
\(741\) −5.59767 −0.205636
\(742\) 0 0
\(743\) −27.5147 −1.00942 −0.504709 0.863290i \(-0.668400\pi\)
−0.504709 + 0.863290i \(0.668400\pi\)
\(744\) 0 0
\(745\) −43.3137 −1.58689
\(746\) 0 0
\(747\) −50.0208 −1.83017
\(748\) 0 0
\(749\) 13.5140 0.493790
\(750\) 0 0
\(751\) 45.4558 1.65871 0.829354 0.558724i \(-0.188709\pi\)
0.829354 + 0.558724i \(0.188709\pi\)
\(752\) 0 0
\(753\) 2.14214 0.0780638
\(754\) 0 0
\(755\) −47.4077 −1.72534
\(756\) 0 0
\(757\) 22.5445 0.819395 0.409697 0.912221i \(-0.365634\pi\)
0.409697 + 0.912221i \(0.365634\pi\)
\(758\) 0 0
\(759\) 38.6274 1.40209
\(760\) 0 0
\(761\) −18.9706 −0.687682 −0.343841 0.939028i \(-0.611728\pi\)
−0.343841 + 0.939028i \(0.611728\pi\)
\(762\) 0 0
\(763\) −11.3492 −0.410868
\(764\) 0 0
\(765\) 76.6004 2.76949
\(766\) 0 0
\(767\) −0.201010 −0.00725806
\(768\) 0 0
\(769\) 35.2548 1.27132 0.635661 0.771968i \(-0.280728\pi\)
0.635661 + 0.771968i \(0.280728\pi\)
\(770\) 0 0
\(771\) −24.3379 −0.876508
\(772\) 0 0
\(773\) −8.73606 −0.314214 −0.157107 0.987582i \(-0.550217\pi\)
−0.157107 + 0.987582i \(0.550217\pi\)
\(774\) 0 0
\(775\) 10.3431 0.371537
\(776\) 0 0
\(777\) 13.6569 0.489937
\(778\) 0 0
\(779\) 17.4721 0.626004
\(780\) 0 0
\(781\) 5.07241 0.181505
\(782\) 0 0
\(783\) −20.6863 −0.739268
\(784\) 0 0
\(785\) 12.4853 0.445619
\(786\) 0 0
\(787\) −7.46796 −0.266204 −0.133102 0.991102i \(-0.542494\pi\)
−0.133102 + 0.991102i \(0.542494\pi\)
\(788\) 0 0
\(789\) −6.12293 −0.217982
\(790\) 0 0
\(791\) −8.82843 −0.313903
\(792\) 0 0
\(793\) −5.45584 −0.193743
\(794\) 0 0
\(795\) −71.3742 −2.53138
\(796\) 0 0
\(797\) −27.4763 −0.973260 −0.486630 0.873608i \(-0.661774\pi\)
−0.486630 + 0.873608i \(0.661774\pi\)
\(798\) 0 0
\(799\) 61.2548 2.16704
\(800\) 0 0
\(801\) 7.65685 0.270542
\(802\) 0 0
\(803\) 25.2346 0.890509
\(804\) 0 0
\(805\) −17.8435 −0.628902
\(806\) 0 0
\(807\) 21.4558 0.755281
\(808\) 0 0
\(809\) 31.1716 1.09593 0.547967 0.836500i \(-0.315402\pi\)
0.547967 + 0.836500i \(0.315402\pi\)
\(810\) 0 0
\(811\) −20.9819 −0.736775 −0.368388 0.929672i \(-0.620090\pi\)
−0.368388 + 0.929672i \(0.620090\pi\)
\(812\) 0 0
\(813\) −6.12293 −0.214741
\(814\) 0 0
\(815\) 10.3431 0.362305
\(816\) 0 0
\(817\) 10.3431 0.361861
\(818\) 0 0
\(819\) 1.71644 0.0599774
\(820\) 0 0
\(821\) 8.65914 0.302206 0.151103 0.988518i \(-0.451717\pi\)
0.151103 + 0.988518i \(0.451717\pi\)
\(822\) 0 0
\(823\) −40.9706 −1.42814 −0.714072 0.700072i \(-0.753151\pi\)
−0.714072 + 0.700072i \(0.753151\pi\)
\(824\) 0 0
\(825\) −10.3431 −0.360102
\(826\) 0 0
\(827\) −10.9778 −0.381734 −0.190867 0.981616i \(-0.561130\pi\)
−0.190867 + 0.981616i \(0.561130\pi\)
\(828\) 0 0
\(829\) −6.19986 −0.215330 −0.107665 0.994187i \(-0.534337\pi\)
−0.107665 + 0.994187i \(0.534337\pi\)
\(830\) 0 0
\(831\) −38.6274 −1.33997
\(832\) 0 0
\(833\) −7.65685 −0.265294
\(834\) 0 0
\(835\) 41.8100 1.44690
\(836\) 0 0
\(837\) 12.2459 0.423279
\(838\) 0 0
\(839\) −36.2843 −1.25267 −0.626336 0.779553i \(-0.715446\pi\)
−0.626336 + 0.779553i \(0.715446\pi\)
\(840\) 0 0
\(841\) 62.3137 2.14875
\(842\) 0 0
\(843\) 5.22625 0.180002
\(844\) 0 0
\(845\) 33.4454 1.15056
\(846\) 0 0
\(847\) 6.31371 0.216942
\(848\) 0 0
\(849\) 54.4264 1.86791
\(850\) 0 0
\(851\) 35.6871 1.22334
\(852\) 0 0
\(853\) −47.3308 −1.62057 −0.810287 0.586033i \(-0.800689\pi\)
−0.810287 + 0.586033i \(0.800689\pi\)
\(854\) 0 0
\(855\) −47.7990 −1.63469
\(856\) 0 0
\(857\) −0.627417 −0.0214322 −0.0107161 0.999943i \(-0.503411\pi\)
−0.0107161 + 0.999943i \(0.503411\pi\)
\(858\) 0 0
\(859\) −3.50981 −0.119753 −0.0598766 0.998206i \(-0.519071\pi\)
−0.0598766 + 0.998206i \(0.519071\pi\)
\(860\) 0 0
\(861\) −9.55582 −0.325661
\(862\) 0 0
\(863\) −51.3137 −1.74674 −0.873369 0.487058i \(-0.838070\pi\)
−0.873369 + 0.487058i \(0.838070\pi\)
\(864\) 0 0
\(865\) −42.1421 −1.43288
\(866\) 0 0
\(867\) 108.778 3.69428
\(868\) 0 0
\(869\) 5.07241 0.172070
\(870\) 0 0
\(871\) −1.37258 −0.0465082
\(872\) 0 0
\(873\) −38.2843 −1.29573
\(874\) 0 0
\(875\) −8.28772 −0.280176
\(876\) 0 0
\(877\) −32.2542 −1.08915 −0.544573 0.838713i \(-0.683308\pi\)
−0.544573 + 0.838713i \(0.683308\pi\)
\(878\) 0 0
\(879\) 28.4853 0.960785
\(880\) 0 0
\(881\) 10.9706 0.369608 0.184804 0.982775i \(-0.440835\pi\)
0.184804 + 0.982775i \(0.440835\pi\)
\(882\) 0 0
\(883\) 51.7373 1.74110 0.870549 0.492082i \(-0.163764\pi\)
0.870549 + 0.492082i \(0.163764\pi\)
\(884\) 0 0
\(885\) −3.06147 −0.102910
\(886\) 0 0
\(887\) 22.6274 0.759754 0.379877 0.925037i \(-0.375966\pi\)
0.379877 + 0.925037i \(0.375966\pi\)
\(888\) 0 0
\(889\) 4.48528 0.150432
\(890\) 0 0
\(891\) 12.6173 0.422695
\(892\) 0 0
\(893\) −38.2233 −1.27909
\(894\) 0 0
\(895\) −57.9411 −1.93676
\(896\) 0 0
\(897\) 8.00000 0.267112
\(898\) 0 0
\(899\) −54.0559 −1.80286
\(900\) 0 0
\(901\) −80.0333 −2.66630
\(902\) 0 0
\(903\) −5.65685 −0.188248
\(904\) 0 0
\(905\) −6.82843 −0.226985
\(906\) 0 0
\(907\) 1.26810 0.0421066 0.0210533 0.999778i \(-0.493298\pi\)
0.0210533 + 0.999778i \(0.493298\pi\)
\(908\) 0 0
\(909\) 18.2919 0.606703
\(910\) 0 0
\(911\) −38.8284 −1.28644 −0.643222 0.765680i \(-0.722403\pi\)
−0.643222 + 0.765680i \(0.722403\pi\)
\(912\) 0 0
\(913\) 28.2843 0.936073
\(914\) 0 0
\(915\) −83.0948 −2.74703
\(916\) 0 0
\(917\) −18.2919 −0.604051
\(918\) 0 0
\(919\) −2.34315 −0.0772932 −0.0386466 0.999253i \(-0.512305\pi\)
−0.0386466 + 0.999253i \(0.512305\pi\)
\(920\) 0 0
\(921\) −22.8284 −0.752222
\(922\) 0 0
\(923\) 1.05053 0.0345786
\(924\) 0 0
\(925\) −9.55582 −0.314193
\(926\) 0 0
\(927\) −52.2843 −1.71724
\(928\) 0 0
\(929\) 10.2843 0.337416 0.168708 0.985666i \(-0.446041\pi\)
0.168708 + 0.985666i \(0.446041\pi\)
\(930\) 0 0
\(931\) 4.77791 0.156590
\(932\) 0 0
\(933\) −38.2233 −1.25137
\(934\) 0 0
\(935\) −43.3137 −1.41651
\(936\) 0 0
\(937\) 38.2843 1.25069 0.625346 0.780347i \(-0.284958\pi\)
0.625346 + 0.780347i \(0.284958\pi\)
\(938\) 0 0
\(939\) 21.8017 0.711471
\(940\) 0 0
\(941\) −37.7749 −1.23143 −0.615714 0.787970i \(-0.711132\pi\)
−0.615714 + 0.787970i \(0.711132\pi\)
\(942\) 0 0
\(943\) −24.9706 −0.813153
\(944\) 0 0
\(945\) 5.65685 0.184017
\(946\) 0 0
\(947\) 48.3044 1.56968 0.784841 0.619698i \(-0.212745\pi\)
0.784841 + 0.619698i \(0.212745\pi\)
\(948\) 0 0
\(949\) 5.22625 0.169651
\(950\) 0 0
\(951\) −22.6274 −0.733744
\(952\) 0 0
\(953\) −37.3137 −1.20871 −0.604355 0.796715i \(-0.706569\pi\)
−0.604355 + 0.796715i \(0.706569\pi\)
\(954\) 0 0
\(955\) −50.4692 −1.63314
\(956\) 0 0
\(957\) 54.0559 1.74738
\(958\) 0 0
\(959\) 2.00000 0.0645834
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) 0 0
\(963\) −51.7373 −1.66721
\(964\) 0 0
\(965\) −36.0585 −1.16076
\(966\) 0 0
\(967\) 29.8579 0.960164 0.480082 0.877224i \(-0.340607\pi\)
0.480082 + 0.877224i \(0.340607\pi\)
\(968\) 0 0
\(969\) −95.5980 −3.07105
\(970\) 0 0
\(971\) −52.5570 −1.68663 −0.843317 0.537416i \(-0.819401\pi\)
−0.843317 + 0.537416i \(0.819401\pi\)
\(972\) 0 0
\(973\) 19.5600 0.627064
\(974\) 0 0
\(975\) −2.14214 −0.0686032
\(976\) 0 0
\(977\) 30.0000 0.959785 0.479893 0.877327i \(-0.340676\pi\)
0.479893 + 0.877327i \(0.340676\pi\)
\(978\) 0 0
\(979\) −4.32957 −0.138374
\(980\) 0 0
\(981\) 43.4495 1.38724
\(982\) 0 0
\(983\) −23.0294 −0.734525 −0.367262 0.930117i \(-0.619705\pi\)
−0.367262 + 0.930117i \(0.619705\pi\)
\(984\) 0 0
\(985\) 16.0000 0.509802
\(986\) 0 0
\(987\) 20.9050 0.665414
\(988\) 0 0
\(989\) −14.7821 −0.470043
\(990\) 0 0
\(991\) −29.6569 −0.942081 −0.471041 0.882112i \(-0.656121\pi\)
−0.471041 + 0.882112i \(0.656121\pi\)
\(992\) 0 0
\(993\) −48.9706 −1.55403
\(994\) 0 0
\(995\) 23.4412 0.743136
\(996\) 0 0
\(997\) −0.0769232 −0.00243618 −0.00121809 0.999999i \(-0.500388\pi\)
−0.00121809 + 0.999999i \(0.500388\pi\)
\(998\) 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.1.4 4
4.3 odd 2 1792.2.a.w.1.1 4
8.3 odd 2 1792.2.a.w.1.4 4
8.5 even 2 inner 1792.2.a.u.1.1 4
16.3 odd 4 896.2.b.f.449.1 4
16.5 even 4 896.2.b.h.449.1 yes 4
16.11 odd 4 896.2.b.f.449.4 yes 4
16.13 even 4 896.2.b.h.449.4 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
896.2.b.f.449.1 4 16.3 odd 4
896.2.b.f.449.4 yes 4 16.11 odd 4
896.2.b.h.449.1 yes 4 16.5 even 4
896.2.b.h.449.4 yes 4 16.13 even 4
1792.2.a.u.1.1 4 8.5 even 2 inner
1792.2.a.u.1.4 4 1.1 even 1 trivial
1792.2.a.w.1.1 4 4.3 odd 2
1792.2.a.w.1.4 4 8.3 odd 2