Properties

Label 1152.2.c.a.1151.3
Level $1152$
Weight $2$
Character 1152.1151
Analytic conductor $9.199$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1152 = 2^{7} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1152.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.19876631285\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1151.3
Root \(0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 1152.1151
Dual form 1152.2.c.a.1151.2

$q$-expansion

\(f(q)\) \(=\) \(q+0.585786i q^{5} -0.828427i q^{7} +O(q^{10})\) \(q+0.585786i q^{5} -0.828427i q^{7} -2.82843 q^{11} -2.82843 q^{13} -2.58579i q^{17} -5.65685i q^{19} -6.82843 q^{23} +4.65685 q^{25} +3.41421i q^{29} -8.82843i q^{31} +0.485281 q^{35} -7.65685 q^{37} -5.41421i q^{41} -1.65685i q^{43} +4.48528 q^{47} +6.31371 q^{49} +9.07107i q^{53} -1.65685i q^{55} -13.6569 q^{59} -3.65685 q^{61} -1.65685i q^{65} -12.0000i q^{67} -12.4853 q^{71} +4.00000 q^{73} +2.34315i q^{77} -10.4853i q^{79} +10.8284 q^{83} +1.51472 q^{85} -3.75736i q^{89} +2.34315i q^{91} +3.31371 q^{95} +2.34315 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q - 16q^{23} - 4q^{25} - 32q^{35} - 8q^{37} - 16q^{47} - 20q^{49} - 32q^{59} + 8q^{61} - 16q^{71} + 16q^{73} + 32q^{83} + 40q^{85} - 32q^{95} + 32q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(641\) \(901\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 0.585786i 0.261972i 0.991384 + 0.130986i \(0.0418142\pi\)
−0.991384 + 0.130986i \(0.958186\pi\)
\(6\) 0 0
\(7\) − 0.828427i − 0.313116i −0.987669 0.156558i \(-0.949960\pi\)
0.987669 0.156558i \(-0.0500398\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −2.82843 −0.852803 −0.426401 0.904534i \(-0.640219\pi\)
−0.426401 + 0.904534i \(0.640219\pi\)
\(12\) 0 0
\(13\) −2.82843 −0.784465 −0.392232 0.919866i \(-0.628297\pi\)
−0.392232 + 0.919866i \(0.628297\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 2.58579i − 0.627145i −0.949564 0.313573i \(-0.898474\pi\)
0.949564 0.313573i \(-0.101526\pi\)
\(18\) 0 0
\(19\) − 5.65685i − 1.29777i −0.760886 0.648886i \(-0.775235\pi\)
0.760886 0.648886i \(-0.224765\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −6.82843 −1.42383 −0.711913 0.702268i \(-0.752171\pi\)
−0.711913 + 0.702268i \(0.752171\pi\)
\(24\) 0 0
\(25\) 4.65685 0.931371
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 3.41421i 0.634004i 0.948425 + 0.317002i \(0.102676\pi\)
−0.948425 + 0.317002i \(0.897324\pi\)
\(30\) 0 0
\(31\) − 8.82843i − 1.58563i −0.609461 0.792816i \(-0.708614\pi\)
0.609461 0.792816i \(-0.291386\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0.485281 0.0820275
\(36\) 0 0
\(37\) −7.65685 −1.25878 −0.629390 0.777090i \(-0.716695\pi\)
−0.629390 + 0.777090i \(0.716695\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 5.41421i − 0.845558i −0.906233 0.422779i \(-0.861055\pi\)
0.906233 0.422779i \(-0.138945\pi\)
\(42\) 0 0
\(43\) − 1.65685i − 0.252668i −0.991988 0.126334i \(-0.959679\pi\)
0.991988 0.126334i \(-0.0403211\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.48528 0.654246 0.327123 0.944982i \(-0.393921\pi\)
0.327123 + 0.944982i \(0.393921\pi\)
\(48\) 0 0
\(49\) 6.31371 0.901958
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 9.07107i 1.24601i 0.782219 + 0.623003i \(0.214088\pi\)
−0.782219 + 0.623003i \(0.785912\pi\)
\(54\) 0 0
\(55\) − 1.65685i − 0.223410i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −13.6569 −1.77797 −0.888985 0.457935i \(-0.848589\pi\)
−0.888985 + 0.457935i \(0.848589\pi\)
\(60\) 0 0
\(61\) −3.65685 −0.468212 −0.234106 0.972211i \(-0.575216\pi\)
−0.234106 + 0.972211i \(0.575216\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 1.65685i − 0.205507i
\(66\) 0 0
\(67\) − 12.0000i − 1.46603i −0.680211 0.733017i \(-0.738112\pi\)
0.680211 0.733017i \(-0.261888\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −12.4853 −1.48173 −0.740865 0.671654i \(-0.765584\pi\)
−0.740865 + 0.671654i \(0.765584\pi\)
\(72\) 0 0
\(73\) 4.00000 0.468165 0.234082 0.972217i \(-0.424791\pi\)
0.234082 + 0.972217i \(0.424791\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.34315i 0.267026i
\(78\) 0 0
\(79\) − 10.4853i − 1.17969i −0.807518 0.589843i \(-0.799190\pi\)
0.807518 0.589843i \(-0.200810\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 10.8284 1.18857 0.594287 0.804253i \(-0.297434\pi\)
0.594287 + 0.804253i \(0.297434\pi\)
\(84\) 0 0
\(85\) 1.51472 0.164294
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 3.75736i − 0.398279i −0.979971 0.199140i \(-0.936185\pi\)
0.979971 0.199140i \(-0.0638147\pi\)
\(90\) 0 0
\(91\) 2.34315i 0.245628i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 3.31371 0.339979
\(96\) 0 0
\(97\) 2.34315 0.237910 0.118955 0.992900i \(-0.462046\pi\)
0.118955 + 0.992900i \(0.462046\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 14.7279i − 1.46548i −0.680507 0.732742i \(-0.738240\pi\)
0.680507 0.732742i \(-0.261760\pi\)
\(102\) 0 0
\(103\) 8.82843i 0.869891i 0.900457 + 0.434945i \(0.143232\pi\)
−0.900457 + 0.434945i \(0.856768\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −11.3137 −1.09374 −0.546869 0.837218i \(-0.684180\pi\)
−0.546869 + 0.837218i \(0.684180\pi\)
\(108\) 0 0
\(109\) −5.17157 −0.495347 −0.247673 0.968844i \(-0.579666\pi\)
−0.247673 + 0.968844i \(0.579666\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 3.75736i − 0.353463i −0.984259 0.176731i \(-0.943448\pi\)
0.984259 0.176731i \(-0.0565524\pi\)
\(114\) 0 0
\(115\) − 4.00000i − 0.373002i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −2.14214 −0.196369
\(120\) 0 0
\(121\) −3.00000 −0.272727
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 5.65685i 0.505964i
\(126\) 0 0
\(127\) − 0.828427i − 0.0735110i −0.999324 0.0367555i \(-0.988298\pi\)
0.999324 0.0367555i \(-0.0117023\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 5.65685 0.494242 0.247121 0.968985i \(-0.420516\pi\)
0.247121 + 0.968985i \(0.420516\pi\)
\(132\) 0 0
\(133\) −4.68629 −0.406353
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 13.4142i 1.14605i 0.819537 + 0.573027i \(0.194231\pi\)
−0.819537 + 0.573027i \(0.805769\pi\)
\(138\) 0 0
\(139\) 12.0000i 1.01783i 0.860818 + 0.508913i \(0.169953\pi\)
−0.860818 + 0.508913i \(0.830047\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 8.00000 0.668994
\(144\) 0 0
\(145\) −2.00000 −0.166091
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 18.7279i 1.53425i 0.641497 + 0.767126i \(0.278314\pi\)
−0.641497 + 0.767126i \(0.721686\pi\)
\(150\) 0 0
\(151\) 10.4853i 0.853280i 0.904422 + 0.426640i \(0.140303\pi\)
−0.904422 + 0.426640i \(0.859697\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 5.17157 0.415391
\(156\) 0 0
\(157\) 11.6569 0.930318 0.465159 0.885227i \(-0.345997\pi\)
0.465159 + 0.885227i \(0.345997\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 5.65685i 0.445823i
\(162\) 0 0
\(163\) 9.65685i 0.756383i 0.925727 + 0.378192i \(0.123454\pi\)
−0.925727 + 0.378192i \(0.876546\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −5.65685 −0.437741 −0.218870 0.975754i \(-0.570237\pi\)
−0.218870 + 0.975754i \(0.570237\pi\)
\(168\) 0 0
\(169\) −5.00000 −0.384615
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 23.4142i − 1.78015i −0.455814 0.890075i \(-0.650652\pi\)
0.455814 0.890075i \(-0.349348\pi\)
\(174\) 0 0
\(175\) − 3.85786i − 0.291627i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 3.31371 0.247678 0.123839 0.992302i \(-0.460479\pi\)
0.123839 + 0.992302i \(0.460479\pi\)
\(180\) 0 0
\(181\) 22.1421 1.64581 0.822906 0.568178i \(-0.192351\pi\)
0.822906 + 0.568178i \(0.192351\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) − 4.48528i − 0.329764i
\(186\) 0 0
\(187\) 7.31371i 0.534831i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 21.6569 1.56703 0.783517 0.621370i \(-0.213423\pi\)
0.783517 + 0.621370i \(0.213423\pi\)
\(192\) 0 0
\(193\) −17.3137 −1.24627 −0.623134 0.782115i \(-0.714141\pi\)
−0.623134 + 0.782115i \(0.714141\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 20.3848i − 1.45236i −0.687507 0.726178i \(-0.741295\pi\)
0.687507 0.726178i \(-0.258705\pi\)
\(198\) 0 0
\(199\) 0.828427i 0.0587256i 0.999569 + 0.0293628i \(0.00934782\pi\)
−0.999569 + 0.0293628i \(0.990652\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 2.82843 0.198517
\(204\) 0 0
\(205\) 3.17157 0.221512
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 16.0000i 1.10674i
\(210\) 0 0
\(211\) − 0.686292i − 0.0472463i −0.999721 0.0236231i \(-0.992480\pi\)
0.999721 0.0236231i \(-0.00752017\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0.970563 0.0661918
\(216\) 0 0
\(217\) −7.31371 −0.496487
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 7.31371i 0.491973i
\(222\) 0 0
\(223\) 21.7990i 1.45977i 0.683571 + 0.729884i \(0.260426\pi\)
−0.683571 + 0.729884i \(0.739574\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 21.1716 1.40521 0.702603 0.711582i \(-0.252021\pi\)
0.702603 + 0.711582i \(0.252021\pi\)
\(228\) 0 0
\(229\) −10.8284 −0.715563 −0.357781 0.933805i \(-0.616467\pi\)
−0.357781 + 0.933805i \(0.616467\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 1.89949i − 0.124440i −0.998062 0.0622200i \(-0.980182\pi\)
0.998062 0.0622200i \(-0.0198181\pi\)
\(234\) 0 0
\(235\) 2.62742i 0.171394i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −8.00000 −0.517477 −0.258738 0.965947i \(-0.583307\pi\)
−0.258738 + 0.965947i \(0.583307\pi\)
\(240\) 0 0
\(241\) −12.9706 −0.835507 −0.417754 0.908560i \(-0.637183\pi\)
−0.417754 + 0.908560i \(0.637183\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 3.69848i 0.236288i
\(246\) 0 0
\(247\) 16.0000i 1.01806i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 7.51472 0.474325 0.237162 0.971470i \(-0.423783\pi\)
0.237162 + 0.971470i \(0.423783\pi\)
\(252\) 0 0
\(253\) 19.3137 1.21424
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 9.89949i − 0.617514i −0.951141 0.308757i \(-0.900087\pi\)
0.951141 0.308757i \(-0.0999129\pi\)
\(258\) 0 0
\(259\) 6.34315i 0.394144i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 29.6569 1.82872 0.914360 0.404902i \(-0.132694\pi\)
0.914360 + 0.404902i \(0.132694\pi\)
\(264\) 0 0
\(265\) −5.31371 −0.326419
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 4.38478i − 0.267345i −0.991026 0.133672i \(-0.957323\pi\)
0.991026 0.133672i \(-0.0426769\pi\)
\(270\) 0 0
\(271\) 18.4853i 1.12290i 0.827510 + 0.561450i \(0.189756\pi\)
−0.827510 + 0.561450i \(0.810244\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −13.1716 −0.794276
\(276\) 0 0
\(277\) 26.8284 1.61196 0.805982 0.591940i \(-0.201638\pi\)
0.805982 + 0.591940i \(0.201638\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 25.8995i 1.54503i 0.634994 + 0.772517i \(0.281003\pi\)
−0.634994 + 0.772517i \(0.718997\pi\)
\(282\) 0 0
\(283\) 19.3137i 1.14808i 0.818827 + 0.574040i \(0.194625\pi\)
−0.818827 + 0.574040i \(0.805375\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −4.48528 −0.264758
\(288\) 0 0
\(289\) 10.3137 0.606689
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 24.8701i − 1.45292i −0.687206 0.726462i \(-0.741163\pi\)
0.687206 0.726462i \(-0.258837\pi\)
\(294\) 0 0
\(295\) − 8.00000i − 0.465778i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 19.3137 1.11694
\(300\) 0 0
\(301\) −1.37258 −0.0791144
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) − 2.14214i − 0.122658i
\(306\) 0 0
\(307\) 23.3137i 1.33058i 0.746583 + 0.665292i \(0.231693\pi\)
−0.746583 + 0.665292i \(0.768307\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 2.34315 0.132868 0.0664338 0.997791i \(-0.478838\pi\)
0.0664338 + 0.997791i \(0.478838\pi\)
\(312\) 0 0
\(313\) −13.3137 −0.752535 −0.376268 0.926511i \(-0.622793\pi\)
−0.376268 + 0.926511i \(0.622793\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 5.75736i − 0.323366i −0.986843 0.161683i \(-0.948308\pi\)
0.986843 0.161683i \(-0.0516921\pi\)
\(318\) 0 0
\(319\) − 9.65685i − 0.540680i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −14.6274 −0.813891
\(324\) 0 0
\(325\) −13.1716 −0.730627
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) − 3.71573i − 0.204855i
\(330\) 0 0
\(331\) − 4.00000i − 0.219860i −0.993939 0.109930i \(-0.964937\pi\)
0.993939 0.109930i \(-0.0350627\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 7.02944 0.384059
\(336\) 0 0
\(337\) 0.686292 0.0373847 0.0186923 0.999825i \(-0.494050\pi\)
0.0186923 + 0.999825i \(0.494050\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 24.9706i 1.35223i
\(342\) 0 0
\(343\) − 11.0294i − 0.595534i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0.485281 0.0260513 0.0130256 0.999915i \(-0.495854\pi\)
0.0130256 + 0.999915i \(0.495854\pi\)
\(348\) 0 0
\(349\) −0.343146 −0.0183682 −0.00918409 0.999958i \(-0.502923\pi\)
−0.00918409 + 0.999958i \(0.502923\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 34.8701i 1.85595i 0.372648 + 0.927973i \(0.378450\pi\)
−0.372648 + 0.927973i \(0.621550\pi\)
\(354\) 0 0
\(355\) − 7.31371i − 0.388171i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −3.51472 −0.185500 −0.0927499 0.995689i \(-0.529566\pi\)
−0.0927499 + 0.995689i \(0.529566\pi\)
\(360\) 0 0
\(361\) −13.0000 −0.684211
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 2.34315i 0.122646i
\(366\) 0 0
\(367\) 15.1716i 0.791950i 0.918261 + 0.395975i \(0.129593\pi\)
−0.918261 + 0.395975i \(0.870407\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 7.51472 0.390145
\(372\) 0 0
\(373\) 18.9706 0.982259 0.491129 0.871087i \(-0.336584\pi\)
0.491129 + 0.871087i \(0.336584\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 9.65685i − 0.497353i
\(378\) 0 0
\(379\) − 36.2843i − 1.86380i −0.362718 0.931899i \(-0.618151\pi\)
0.362718 0.931899i \(-0.381849\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −30.6274 −1.56499 −0.782494 0.622658i \(-0.786053\pi\)
−0.782494 + 0.622658i \(0.786053\pi\)
\(384\) 0 0
\(385\) −1.37258 −0.0699533
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) − 33.5563i − 1.70137i −0.525672 0.850687i \(-0.676186\pi\)
0.525672 0.850687i \(-0.323814\pi\)
\(390\) 0 0
\(391\) 17.6569i 0.892946i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 6.14214 0.309044
\(396\) 0 0
\(397\) −14.9706 −0.751351 −0.375676 0.926751i \(-0.622589\pi\)
−0.375676 + 0.926751i \(0.622589\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) − 10.5858i − 0.528629i −0.964437 0.264314i \(-0.914854\pi\)
0.964437 0.264314i \(-0.0851457\pi\)
\(402\) 0 0
\(403\) 24.9706i 1.24387i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 21.6569 1.07349
\(408\) 0 0
\(409\) 21.6569 1.07086 0.535431 0.844579i \(-0.320149\pi\)
0.535431 + 0.844579i \(0.320149\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 11.3137i 0.556711i
\(414\) 0 0
\(415\) 6.34315i 0.311373i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −27.7990 −1.35807 −0.679035 0.734106i \(-0.737601\pi\)
−0.679035 + 0.734106i \(0.737601\pi\)
\(420\) 0 0
\(421\) −24.4853 −1.19334 −0.596670 0.802487i \(-0.703510\pi\)
−0.596670 + 0.802487i \(0.703510\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) − 12.0416i − 0.584105i
\(426\) 0 0
\(427\) 3.02944i 0.146605i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −11.5147 −0.554644 −0.277322 0.960777i \(-0.589447\pi\)
−0.277322 + 0.960777i \(0.589447\pi\)
\(432\) 0 0
\(433\) −16.6274 −0.799063 −0.399531 0.916720i \(-0.630827\pi\)
−0.399531 + 0.916720i \(0.630827\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 38.6274i 1.84780i
\(438\) 0 0
\(439\) − 36.1421i − 1.72497i −0.506083 0.862485i \(-0.668907\pi\)
0.506083 0.862485i \(-0.331093\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 33.4558 1.58954 0.794768 0.606914i \(-0.207593\pi\)
0.794768 + 0.606914i \(0.207593\pi\)
\(444\) 0 0
\(445\) 2.20101 0.104338
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) − 40.7279i − 1.92207i −0.276428 0.961035i \(-0.589151\pi\)
0.276428 0.961035i \(-0.410849\pi\)
\(450\) 0 0
\(451\) 15.3137i 0.721094i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −1.37258 −0.0643477
\(456\) 0 0
\(457\) 24.9706 1.16807 0.584037 0.811727i \(-0.301472\pi\)
0.584037 + 0.811727i \(0.301472\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 27.8995i − 1.29941i −0.760187 0.649705i \(-0.774893\pi\)
0.760187 0.649705i \(-0.225107\pi\)
\(462\) 0 0
\(463\) − 5.79899i − 0.269502i −0.990879 0.134751i \(-0.956977\pi\)
0.990879 0.134751i \(-0.0430234\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −22.1421 −1.02462 −0.512308 0.858802i \(-0.671209\pi\)
−0.512308 + 0.858802i \(0.671209\pi\)
\(468\) 0 0
\(469\) −9.94113 −0.459039
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 4.68629i 0.215476i
\(474\) 0 0
\(475\) − 26.3431i − 1.20871i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −14.8284 −0.677528 −0.338764 0.940871i \(-0.610009\pi\)
−0.338764 + 0.940871i \(0.610009\pi\)
\(480\) 0 0
\(481\) 21.6569 0.987468
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 1.37258i 0.0623258i
\(486\) 0 0
\(487\) − 23.1716i − 1.05000i −0.851101 0.525002i \(-0.824065\pi\)
0.851101 0.525002i \(-0.175935\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −12.6863 −0.572524 −0.286262 0.958151i \(-0.592413\pi\)
−0.286262 + 0.958151i \(0.592413\pi\)
\(492\) 0 0
\(493\) 8.82843 0.397612
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 10.3431i 0.463953i
\(498\) 0 0
\(499\) − 38.6274i − 1.72920i −0.502460 0.864600i \(-0.667572\pi\)
0.502460 0.864600i \(-0.332428\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 23.7990 1.06114 0.530572 0.847640i \(-0.321977\pi\)
0.530572 + 0.847640i \(0.321977\pi\)
\(504\) 0 0
\(505\) 8.62742 0.383915
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 38.7279i 1.71658i 0.513161 + 0.858292i \(0.328474\pi\)
−0.513161 + 0.858292i \(0.671526\pi\)
\(510\) 0 0
\(511\) − 3.31371i − 0.146590i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −5.17157 −0.227887
\(516\) 0 0
\(517\) −12.6863 −0.557942
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 11.0711i 0.485032i 0.970147 + 0.242516i \(0.0779727\pi\)
−0.970147 + 0.242516i \(0.922027\pi\)
\(522\) 0 0
\(523\) − 34.3431i − 1.50172i −0.660461 0.750860i \(-0.729639\pi\)
0.660461 0.750860i \(-0.270361\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −22.8284 −0.994422
\(528\) 0 0
\(529\) 23.6274 1.02728
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 15.3137i 0.663310i
\(534\) 0 0
\(535\) − 6.62742i − 0.286528i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −17.8579 −0.769193
\(540\) 0 0
\(541\) −3.79899 −0.163331 −0.0816657 0.996660i \(-0.526024\pi\)
−0.0816657 + 0.996660i \(0.526024\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 3.02944i − 0.129767i
\(546\) 0 0
\(547\) − 1.65685i − 0.0708420i −0.999372 0.0354210i \(-0.988723\pi\)
0.999372 0.0354210i \(-0.0112772\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 19.3137 0.822792
\(552\) 0 0
\(553\) −8.68629 −0.369379
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 5.07107i 0.214868i 0.994212 + 0.107434i \(0.0342634\pi\)
−0.994212 + 0.107434i \(0.965737\pi\)
\(558\) 0 0
\(559\) 4.68629i 0.198209i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −25.4558 −1.07284 −0.536418 0.843952i \(-0.680223\pi\)
−0.536418 + 0.843952i \(0.680223\pi\)
\(564\) 0 0
\(565\) 2.20101 0.0925972
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) − 21.8995i − 0.918075i −0.888417 0.459037i \(-0.848194\pi\)
0.888417 0.459037i \(-0.151806\pi\)
\(570\) 0 0
\(571\) 24.0000i 1.00437i 0.864761 + 0.502184i \(0.167470\pi\)
−0.864761 + 0.502184i \(0.832530\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −31.7990 −1.32611
\(576\) 0 0
\(577\) −6.68629 −0.278354 −0.139177 0.990268i \(-0.544446\pi\)
−0.139177 + 0.990268i \(0.544446\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) − 8.97056i − 0.372162i
\(582\) 0 0
\(583\) − 25.6569i − 1.06260i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 18.3431 0.757103 0.378551 0.925580i \(-0.376422\pi\)
0.378551 + 0.925580i \(0.376422\pi\)
\(588\) 0 0
\(589\) −49.9411 −2.05779
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 0.928932i − 0.0381467i −0.999818 0.0190733i \(-0.993928\pi\)
0.999818 0.0190733i \(-0.00607160\pi\)
\(594\) 0 0
\(595\) − 1.25483i − 0.0514432i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −27.5147 −1.12422 −0.562110 0.827062i \(-0.690010\pi\)
−0.562110 + 0.827062i \(0.690010\pi\)
\(600\) 0 0
\(601\) −1.31371 −0.0535873 −0.0267936 0.999641i \(-0.508530\pi\)
−0.0267936 + 0.999641i \(0.508530\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 1.75736i − 0.0714468i
\(606\) 0 0
\(607\) − 24.8284i − 1.00775i −0.863775 0.503877i \(-0.831906\pi\)
0.863775 0.503877i \(-0.168094\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −12.6863 −0.513232
\(612\) 0 0
\(613\) −0.343146 −0.0138595 −0.00692976 0.999976i \(-0.502206\pi\)
−0.00692976 + 0.999976i \(0.502206\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 33.4142i − 1.34521i −0.740004 0.672603i \(-0.765176\pi\)
0.740004 0.672603i \(-0.234824\pi\)
\(618\) 0 0
\(619\) − 33.9411i − 1.36421i −0.731255 0.682105i \(-0.761065\pi\)
0.731255 0.682105i \(-0.238935\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −3.11270 −0.124708
\(624\) 0 0
\(625\) 19.9706 0.798823
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 19.7990i 0.789437i
\(630\) 0 0
\(631\) − 18.4853i − 0.735887i −0.929848 0.367944i \(-0.880062\pi\)
0.929848 0.367944i \(-0.119938\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0.485281 0.0192578
\(636\) 0 0
\(637\) −17.8579 −0.707554
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) − 14.8701i − 0.587332i −0.955908 0.293666i \(-0.905125\pi\)
0.955908 0.293666i \(-0.0948753\pi\)
\(642\) 0 0
\(643\) 9.65685i 0.380829i 0.981704 + 0.190415i \(0.0609832\pi\)
−0.981704 + 0.190415i \(0.939017\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −15.7990 −0.621122 −0.310561 0.950553i \(-0.600517\pi\)
−0.310561 + 0.950553i \(0.600517\pi\)
\(648\) 0 0
\(649\) 38.6274 1.51626
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 5.27208i 0.206312i 0.994665 + 0.103156i \(0.0328941\pi\)
−0.994665 + 0.103156i \(0.967106\pi\)
\(654\) 0 0
\(655\) 3.31371i 0.129477i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −8.97056 −0.349444 −0.174722 0.984618i \(-0.555903\pi\)
−0.174722 + 0.984618i \(0.555903\pi\)
\(660\) 0 0
\(661\) 3.65685 0.142235 0.0711176 0.997468i \(-0.477343\pi\)
0.0711176 + 0.997468i \(0.477343\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 2.74517i − 0.106453i
\(666\) 0 0
\(667\) − 23.3137i − 0.902710i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 10.3431 0.399293
\(672\) 0 0
\(673\) 18.0000 0.693849 0.346925 0.937893i \(-0.387226\pi\)
0.346925 + 0.937893i \(0.387226\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 18.7279i − 0.719773i −0.932996 0.359886i \(-0.882815\pi\)
0.932996 0.359886i \(-0.117185\pi\)
\(678\) 0 0
\(679\) − 1.94113i − 0.0744936i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 22.1421 0.847245 0.423623 0.905839i \(-0.360758\pi\)
0.423623 + 0.905839i \(0.360758\pi\)
\(684\) 0 0
\(685\) −7.85786 −0.300234
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 25.6569i − 0.977448i
\(690\) 0 0
\(691\) 3.02944i 0.115245i 0.998338 + 0.0576226i \(0.0183520\pi\)
−0.998338 + 0.0576226i \(0.981648\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −7.02944 −0.266642
\(696\) 0 0
\(697\) −14.0000 −0.530288
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 13.7574i 0.519608i 0.965661 + 0.259804i \(0.0836580\pi\)
−0.965661 + 0.259804i \(0.916342\pi\)
\(702\) 0 0
\(703\) 43.3137i 1.63361i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −12.2010 −0.458866
\(708\) 0 0
\(709\) −48.0833 −1.80580 −0.902902 0.429846i \(-0.858568\pi\)
−0.902902 + 0.429846i \(0.858568\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 60.2843i 2.25766i
\(714\) 0 0
\(715\) 4.68629i 0.175257i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −28.4853 −1.06232 −0.531161 0.847271i \(-0.678244\pi\)
−0.531161 + 0.847271i \(0.678244\pi\)
\(720\) 0 0
\(721\) 7.31371 0.272377
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 15.8995i 0.590492i
\(726\) 0 0
\(727\) 8.82843i 0.327428i 0.986508 + 0.163714i \(0.0523475\pi\)
−0.986508 + 0.163714i \(0.947653\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −4.28427 −0.158459
\(732\) 0 0
\(733\) 24.4853 0.904385 0.452192 0.891920i \(-0.350642\pi\)
0.452192 + 0.891920i \(0.350642\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 33.9411i 1.25024i
\(738\) 0 0
\(739\) − 8.00000i − 0.294285i −0.989115 0.147142i \(-0.952992\pi\)
0.989115 0.147142i \(-0.0470076\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 16.9706 0.622590 0.311295 0.950313i \(-0.399237\pi\)
0.311295 + 0.950313i \(0.399237\pi\)
\(744\) 0 0
\(745\) −10.9706 −0.401930
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 9.37258i 0.342467i
\(750\) 0 0
\(751\) − 40.8284i − 1.48985i −0.667148 0.744925i \(-0.732485\pi\)
0.667148 0.744925i \(-0.267515\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −6.14214 −0.223535
\(756\) 0 0
\(757\) 0.485281 0.0176379 0.00881893 0.999961i \(-0.497193\pi\)
0.00881893 + 0.999961i \(0.497193\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) − 25.6985i − 0.931569i −0.884898 0.465785i \(-0.845772\pi\)
0.884898 0.465785i \(-0.154228\pi\)
\(762\) 0 0
\(763\) 4.28427i 0.155101i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 38.6274 1.39476
\(768\) 0 0
\(769\) −21.3137 −0.768592 −0.384296 0.923210i \(-0.625556\pi\)
−0.384296 + 0.923210i \(0.625556\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 12.5858i − 0.452679i −0.974048 0.226340i \(-0.927324\pi\)
0.974048 0.226340i \(-0.0726759\pi\)
\(774\) 0 0
\(775\) − 41.1127i − 1.47681i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −30.6274 −1.09734
\(780\) 0 0
\(781\) 35.3137 1.26362
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 6.82843i 0.243717i
\(786\) 0 0
\(787\) − 26.3431i − 0.939032i −0.882924 0.469516i \(-0.844428\pi\)
0.882924 0.469516i \(-0.155572\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −3.11270 −0.110675
\(792\) 0 0
\(793\) 10.3431 0.367296
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 13.0711i − 0.463001i −0.972835 0.231500i \(-0.925637\pi\)
0.972835 0.231500i \(-0.0743635\pi\)
\(798\) 0 0
\(799\) − 11.5980i − 0.410307i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −11.3137 −0.399252
\(804\) 0 0
\(805\) −3.31371 −0.116793
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) − 42.6690i − 1.50016i −0.661345 0.750082i \(-0.730014\pi\)
0.661345 0.750082i \(-0.269986\pi\)
\(810\) 0 0
\(811\) 31.5980i 1.10956i 0.831999 + 0.554778i \(0.187197\pi\)
−0.831999 + 0.554778i \(0.812803\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −5.65685 −0.198151
\(816\) 0 0
\(817\) −9.37258 −0.327905
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 11.4142i − 0.398359i −0.979963 0.199179i \(-0.936172\pi\)
0.979963 0.199179i \(-0.0638276\pi\)
\(822\) 0 0
\(823\) 52.4264i 1.82747i 0.406311 + 0.913735i \(0.366815\pi\)
−0.406311 + 0.913735i \(0.633185\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −15.0294 −0.522625 −0.261312 0.965254i \(-0.584155\pi\)
−0.261312 + 0.965254i \(0.584155\pi\)
\(828\) 0 0
\(829\) 4.20101 0.145907 0.0729536 0.997335i \(-0.476758\pi\)
0.0729536 + 0.997335i \(0.476758\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 16.3259i − 0.565659i
\(834\) 0 0
\(835\) − 3.31371i − 0.114676i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −3.51472 −0.121342 −0.0606708 0.998158i \(-0.519324\pi\)
−0.0606708 + 0.998158i \(0.519324\pi\)
\(840\) 0 0
\(841\) 17.3431 0.598040
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) − 2.92893i − 0.100758i
\(846\) 0 0
\(847\) 2.48528i 0.0853953i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 52.2843 1.79228
\(852\) 0 0
\(853\) 42.2843 1.44779 0.723893 0.689912i \(-0.242351\pi\)
0.723893 + 0.689912i \(0.242351\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 21.8995i 0.748072i 0.927414 + 0.374036i \(0.122026\pi\)
−0.927414 + 0.374036i \(0.877974\pi\)
\(858\) 0 0
\(859\) − 1.65685i − 0.0565311i −0.999600 0.0282656i \(-0.991002\pi\)
0.999600 0.0282656i \(-0.00899841\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0.970563 0.0330383 0.0165192 0.999864i \(-0.494742\pi\)
0.0165192 + 0.999864i \(0.494742\pi\)
\(864\) 0 0
\(865\) 13.7157 0.466349
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 29.6569i 1.00604i
\(870\) 0 0
\(871\) 33.9411i 1.15005i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 4.68629 0.158426
\(876\) 0 0
\(877\) −6.97056 −0.235379 −0.117690 0.993050i \(-0.537549\pi\)
−0.117690 + 0.993050i \(0.537549\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 12.7279i 0.428815i 0.976744 + 0.214407i \(0.0687820\pi\)
−0.976744 + 0.214407i \(0.931218\pi\)
\(882\) 0 0
\(883\) 40.2843i 1.35567i 0.735212 + 0.677837i \(0.237082\pi\)
−0.735212 + 0.677837i \(0.762918\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −2.34315 −0.0786751 −0.0393376 0.999226i \(-0.512525\pi\)
−0.0393376 + 0.999226i \(0.512525\pi\)
\(888\) 0 0
\(889\) −0.686292 −0.0230175
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 25.3726i − 0.849061i
\(894\) 0 0
\(895\) 1.94113i 0.0648847i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 30.1421 1.00530
\(900\) 0 0
\(901\) 23.4558 0.781427
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 12.9706i 0.431156i
\(906\) 0 0
\(907\) 14.3431i 0.476256i 0.971234 + 0.238128i \(0.0765338\pi\)
−0.971234 + 0.238128i \(0.923466\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 23.5980 0.781836 0.390918 0.920426i \(-0.372158\pi\)
0.390918 + 0.920426i \(0.372158\pi\)
\(912\) 0 0
\(913\) −30.6274 −1.01362
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 4.68629i − 0.154755i
\(918\) 0 0
\(919\) − 23.4558i − 0.773737i −0.922135 0.386868i \(-0.873557\pi\)
0.922135 0.386868i \(-0.126443\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 35.3137 1.16236
\(924\) 0 0
\(925\) −35.6569 −1.17239
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 16.7279i 0.548825i 0.961612 + 0.274413i \(0.0884834\pi\)
−0.961612 + 0.274413i \(0.911517\pi\)
\(930\) 0 0
\(931\) − 35.7157i − 1.17054i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −4.28427 −0.140111
\(936\) 0 0
\(937\) −56.6274 −1.84994 −0.924969 0.380044i \(-0.875909\pi\)
−0.924969 + 0.380044i \(0.875909\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 21.2721i − 0.693450i −0.937967 0.346725i \(-0.887294\pi\)
0.937967 0.346725i \(-0.112706\pi\)
\(942\) 0 0
\(943\) 36.9706i 1.20393i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 31.5980 1.02680 0.513398 0.858151i \(-0.328386\pi\)
0.513398 + 0.858151i \(0.328386\pi\)
\(948\) 0 0
\(949\) −11.3137 −0.367259
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 4.44365i 0.143944i 0.997407 + 0.0719720i \(0.0229292\pi\)
−0.997407 + 0.0719720i \(0.977071\pi\)
\(954\) 0 0
\(955\) 12.6863i 0.410519i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 11.1127 0.358848
\(960\) 0 0
\(961\) −46.9411 −1.51423
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 10.1421i − 0.326487i
\(966\) 0 0
\(967\) 46.0833i 1.48194i 0.671539 + 0.740969i \(0.265633\pi\)
−0.671539 + 0.740969i \(0.734367\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −21.1716 −0.679428 −0.339714 0.940529i \(-0.610330\pi\)
−0.339714 + 0.940529i \(0.610330\pi\)
\(972\) 0 0
\(973\) 9.94113 0.318698
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 12.5269i 0.400771i 0.979717 + 0.200386i \(0.0642195\pi\)
−0.979717 + 0.200386i \(0.935780\pi\)
\(978\) 0 0
\(979\) 10.6274i 0.339654i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 34.3431 1.09538 0.547688 0.836683i \(-0.315508\pi\)
0.547688 + 0.836683i \(0.315508\pi\)
\(984\) 0 0
\(985\) 11.9411 0.380476
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 11.3137i 0.359755i
\(990\) 0 0
\(991\) 13.7990i 0.438339i 0.975687 + 0.219170i \(0.0703348\pi\)
−0.975687 + 0.219170i \(0.929665\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −0.485281 −0.0153845
\(996\) 0 0
\(997\) 17.5980 0.557334 0.278667 0.960388i \(-0.410107\pi\)
0.278667 + 0.960388i \(0.410107\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1152.2.c.a.1151.3 yes 4
3.2 odd 2 1152.2.c.d.1151.2 yes 4
4.3 odd 2 1152.2.c.d.1151.3 yes 4
8.3 odd 2 1152.2.c.c.1151.2 yes 4
8.5 even 2 1152.2.c.b.1151.2 yes 4
12.11 even 2 inner 1152.2.c.a.1151.2 4
16.3 odd 4 2304.2.f.g.1151.1 4
16.5 even 4 2304.2.f.b.1151.4 4
16.11 odd 4 2304.2.f.a.1151.3 4
16.13 even 4 2304.2.f.h.1151.2 4
24.5 odd 2 1152.2.c.c.1151.3 yes 4
24.11 even 2 1152.2.c.b.1151.3 yes 4
48.5 odd 4 2304.2.f.g.1151.2 4
48.11 even 4 2304.2.f.h.1151.1 4
48.29 odd 4 2304.2.f.a.1151.4 4
48.35 even 4 2304.2.f.b.1151.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1152.2.c.a.1151.2 4 12.11 even 2 inner
1152.2.c.a.1151.3 yes 4 1.1 even 1 trivial
1152.2.c.b.1151.2 yes 4 8.5 even 2
1152.2.c.b.1151.3 yes 4 24.11 even 2
1152.2.c.c.1151.2 yes 4 8.3 odd 2
1152.2.c.c.1151.3 yes 4 24.5 odd 2
1152.2.c.d.1151.2 yes 4 3.2 odd 2
1152.2.c.d.1151.3 yes 4 4.3 odd 2
2304.2.f.a.1151.3 4 16.11 odd 4
2304.2.f.a.1151.4 4 48.29 odd 4
2304.2.f.b.1151.3 4 48.35 even 4
2304.2.f.b.1151.4 4 16.5 even 4
2304.2.f.g.1151.1 4 16.3 odd 4
2304.2.f.g.1151.2 4 48.5 odd 4
2304.2.f.h.1151.1 4 48.11 even 4
2304.2.f.h.1151.2 4 16.13 even 4