Properties

Label 4608.2.a.f.1.2
Level $4608$
Weight $2$
Character 4608.1
Self dual yes
Analytic conductor $36.795$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4608 = 2^{9} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4608.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(36.7950652514\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 512)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 4608.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.82843 q^{5} -4.00000 q^{7} +O(q^{10})\) \(q+2.82843 q^{5} -4.00000 q^{7} -1.41421 q^{11} +2.82843 q^{13} +4.00000 q^{17} -7.07107 q^{19} +4.00000 q^{23} +3.00000 q^{25} -8.48528 q^{29} -8.00000 q^{31} -11.3137 q^{35} +2.82843 q^{37} -2.00000 q^{41} +4.24264 q^{43} +9.00000 q^{49} -2.82843 q^{53} -4.00000 q^{55} -4.24264 q^{59} -8.48528 q^{61} +8.00000 q^{65} +4.24264 q^{67} -4.00000 q^{71} -4.00000 q^{73} +5.65685 q^{77} +8.00000 q^{79} +9.89949 q^{83} +11.3137 q^{85} -12.0000 q^{89} -11.3137 q^{91} -20.0000 q^{95} -4.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 8q^{7} + O(q^{10}) \) \( 2q - 8q^{7} + 8q^{17} + 8q^{23} + 6q^{25} - 16q^{31} - 4q^{41} + 18q^{49} - 8q^{55} + 16q^{65} - 8q^{71} - 8q^{73} + 16q^{79} - 24q^{89} - 40q^{95} - 8q^{97} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 2.82843 1.26491 0.632456 0.774597i \(-0.282047\pi\)
0.632456 + 0.774597i \(0.282047\pi\)
\(6\) 0 0
\(7\) −4.00000 −1.51186 −0.755929 0.654654i \(-0.772814\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.41421 −0.426401 −0.213201 0.977008i \(-0.568389\pi\)
−0.213201 + 0.977008i \(0.568389\pi\)
\(12\) 0 0
\(13\) 2.82843 0.784465 0.392232 0.919866i \(-0.371703\pi\)
0.392232 + 0.919866i \(0.371703\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 0 0
\(19\) −7.07107 −1.62221 −0.811107 0.584898i \(-0.801135\pi\)
−0.811107 + 0.584898i \(0.801135\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) 3.00000 0.600000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −8.48528 −1.57568 −0.787839 0.615882i \(-0.788800\pi\)
−0.787839 + 0.615882i \(0.788800\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −11.3137 −1.91237
\(36\) 0 0
\(37\) 2.82843 0.464991 0.232495 0.972598i \(-0.425311\pi\)
0.232495 + 0.972598i \(0.425311\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) 4.24264 0.646997 0.323498 0.946229i \(-0.395141\pi\)
0.323498 + 0.946229i \(0.395141\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 9.00000 1.28571
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −2.82843 −0.388514 −0.194257 0.980951i \(-0.562230\pi\)
−0.194257 + 0.980951i \(0.562230\pi\)
\(54\) 0 0
\(55\) −4.00000 −0.539360
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −4.24264 −0.552345 −0.276172 0.961108i \(-0.589066\pi\)
−0.276172 + 0.961108i \(0.589066\pi\)
\(60\) 0 0
\(61\) −8.48528 −1.08643 −0.543214 0.839594i \(-0.682793\pi\)
−0.543214 + 0.839594i \(0.682793\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 8.00000 0.992278
\(66\) 0 0
\(67\) 4.24264 0.518321 0.259161 0.965834i \(-0.416554\pi\)
0.259161 + 0.965834i \(0.416554\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −4.00000 −0.474713 −0.237356 0.971423i \(-0.576281\pi\)
−0.237356 + 0.971423i \(0.576281\pi\)
\(72\) 0 0
\(73\) −4.00000 −0.468165 −0.234082 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 5.65685 0.644658
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 9.89949 1.08661 0.543305 0.839535i \(-0.317173\pi\)
0.543305 + 0.839535i \(0.317173\pi\)
\(84\) 0 0
\(85\) 11.3137 1.22714
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −12.0000 −1.27200 −0.635999 0.771690i \(-0.719412\pi\)
−0.635999 + 0.771690i \(0.719412\pi\)
\(90\) 0 0
\(91\) −11.3137 −1.18600
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −20.0000 −2.05196
\(96\) 0 0
\(97\) −4.00000 −0.406138 −0.203069 0.979164i \(-0.565092\pi\)
−0.203069 + 0.979164i \(0.565092\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −2.82843 −0.281439 −0.140720 0.990050i \(-0.544942\pi\)
−0.140720 + 0.990050i \(0.544942\pi\)
\(102\) 0 0
\(103\) −12.0000 −1.18240 −0.591198 0.806527i \(-0.701345\pi\)
−0.591198 + 0.806527i \(0.701345\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 9.89949 0.957020 0.478510 0.878082i \(-0.341177\pi\)
0.478510 + 0.878082i \(0.341177\pi\)
\(108\) 0 0
\(109\) −14.1421 −1.35457 −0.677285 0.735720i \(-0.736844\pi\)
−0.677285 + 0.735720i \(0.736844\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −14.0000 −1.31701 −0.658505 0.752577i \(-0.728811\pi\)
−0.658505 + 0.752577i \(0.728811\pi\)
\(114\) 0 0
\(115\) 11.3137 1.05501
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −16.0000 −1.46672
\(120\) 0 0
\(121\) −9.00000 −0.818182
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −5.65685 −0.505964
\(126\) 0 0
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 7.07107 0.617802 0.308901 0.951094i \(-0.400039\pi\)
0.308901 + 0.951094i \(0.400039\pi\)
\(132\) 0 0
\(133\) 28.2843 2.45256
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 18.0000 1.53784 0.768922 0.639343i \(-0.220793\pi\)
0.768922 + 0.639343i \(0.220793\pi\)
\(138\) 0 0
\(139\) 4.24264 0.359856 0.179928 0.983680i \(-0.442414\pi\)
0.179928 + 0.983680i \(0.442414\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −4.00000 −0.334497
\(144\) 0 0
\(145\) −24.0000 −1.99309
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 2.82843 0.231714 0.115857 0.993266i \(-0.463039\pi\)
0.115857 + 0.993266i \(0.463039\pi\)
\(150\) 0 0
\(151\) −4.00000 −0.325515 −0.162758 0.986666i \(-0.552039\pi\)
−0.162758 + 0.986666i \(0.552039\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −22.6274 −1.81748
\(156\) 0 0
\(157\) −8.48528 −0.677199 −0.338600 0.940931i \(-0.609953\pi\)
−0.338600 + 0.940931i \(0.609953\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −16.0000 −1.26098
\(162\) 0 0
\(163\) −9.89949 −0.775388 −0.387694 0.921788i \(-0.626728\pi\)
−0.387694 + 0.921788i \(0.626728\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −12.0000 −0.928588 −0.464294 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 0 0
\(169\) −5.00000 −0.384615
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −14.1421 −1.07521 −0.537603 0.843198i \(-0.680670\pi\)
−0.537603 + 0.843198i \(0.680670\pi\)
\(174\) 0 0
\(175\) −12.0000 −0.907115
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −1.41421 −0.105703 −0.0528516 0.998602i \(-0.516831\pi\)
−0.0528516 + 0.998602i \(0.516831\pi\)
\(180\) 0 0
\(181\) 8.48528 0.630706 0.315353 0.948974i \(-0.397877\pi\)
0.315353 + 0.948974i \(0.397877\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 8.00000 0.588172
\(186\) 0 0
\(187\) −5.65685 −0.413670
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 8.00000 0.578860 0.289430 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) 0 0
\(193\) −4.00000 −0.287926 −0.143963 0.989583i \(-0.545985\pi\)
−0.143963 + 0.989583i \(0.545985\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 14.1421 1.00759 0.503793 0.863825i \(-0.331938\pi\)
0.503793 + 0.863825i \(0.331938\pi\)
\(198\) 0 0
\(199\) −4.00000 −0.283552 −0.141776 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 33.9411 2.38220
\(204\) 0 0
\(205\) −5.65685 −0.395092
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 10.0000 0.691714
\(210\) 0 0
\(211\) 24.0416 1.65509 0.827547 0.561396i \(-0.189736\pi\)
0.827547 + 0.561396i \(0.189736\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 12.0000 0.818393
\(216\) 0 0
\(217\) 32.0000 2.17230
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 11.3137 0.761042
\(222\) 0 0
\(223\) −24.0000 −1.60716 −0.803579 0.595198i \(-0.797074\pi\)
−0.803579 + 0.595198i \(0.797074\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 21.2132 1.40797 0.703985 0.710215i \(-0.251402\pi\)
0.703985 + 0.710215i \(0.251402\pi\)
\(228\) 0 0
\(229\) −25.4558 −1.68217 −0.841085 0.540903i \(-0.818082\pi\)
−0.841085 + 0.540903i \(0.818082\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −12.0000 −0.786146 −0.393073 0.919507i \(-0.628588\pi\)
−0.393073 + 0.919507i \(0.628588\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 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
\(241\) 12.0000 0.772988 0.386494 0.922292i \(-0.373686\pi\)
0.386494 + 0.922292i \(0.373686\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 25.4558 1.62631
\(246\) 0 0
\(247\) −20.0000 −1.27257
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −4.24264 −0.267793 −0.133897 0.990995i \(-0.542749\pi\)
−0.133897 + 0.990995i \(0.542749\pi\)
\(252\) 0 0
\(253\) −5.65685 −0.355643
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 2.00000 0.124757 0.0623783 0.998053i \(-0.480131\pi\)
0.0623783 + 0.998053i \(0.480131\pi\)
\(258\) 0 0
\(259\) −11.3137 −0.703000
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 28.0000 1.72655 0.863277 0.504730i \(-0.168408\pi\)
0.863277 + 0.504730i \(0.168408\pi\)
\(264\) 0 0
\(265\) −8.00000 −0.491436
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −19.7990 −1.20717 −0.603583 0.797300i \(-0.706261\pi\)
−0.603583 + 0.797300i \(0.706261\pi\)
\(270\) 0 0
\(271\) −32.0000 −1.94386 −0.971931 0.235267i \(-0.924404\pi\)
−0.971931 + 0.235267i \(0.924404\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −4.24264 −0.255841
\(276\) 0 0
\(277\) −8.48528 −0.509831 −0.254916 0.966963i \(-0.582048\pi\)
−0.254916 + 0.966963i \(0.582048\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 20.0000 1.19310 0.596550 0.802576i \(-0.296538\pi\)
0.596550 + 0.802576i \(0.296538\pi\)
\(282\) 0 0
\(283\) 24.0416 1.42913 0.714563 0.699571i \(-0.246625\pi\)
0.714563 + 0.699571i \(0.246625\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 8.00000 0.472225
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −8.48528 −0.495715 −0.247858 0.968796i \(-0.579727\pi\)
−0.247858 + 0.968796i \(0.579727\pi\)
\(294\) 0 0
\(295\) −12.0000 −0.698667
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 11.3137 0.654289
\(300\) 0 0
\(301\) −16.9706 −0.978167
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −24.0000 −1.37424
\(306\) 0 0
\(307\) −7.07107 −0.403567 −0.201784 0.979430i \(-0.564674\pi\)
−0.201784 + 0.979430i \(0.564674\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −20.0000 −1.13410 −0.567048 0.823685i \(-0.691915\pi\)
−0.567048 + 0.823685i \(0.691915\pi\)
\(312\) 0 0
\(313\) −14.0000 −0.791327 −0.395663 0.918396i \(-0.629485\pi\)
−0.395663 + 0.918396i \(0.629485\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 2.82843 0.158860 0.0794301 0.996840i \(-0.474690\pi\)
0.0794301 + 0.996840i \(0.474690\pi\)
\(318\) 0 0
\(319\) 12.0000 0.671871
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −28.2843 −1.57378
\(324\) 0 0
\(325\) 8.48528 0.470679
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 12.7279 0.699590 0.349795 0.936826i \(-0.386251\pi\)
0.349795 + 0.936826i \(0.386251\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 12.0000 0.655630
\(336\) 0 0
\(337\) 30.0000 1.63420 0.817102 0.576493i \(-0.195579\pi\)
0.817102 + 0.576493i \(0.195579\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 11.3137 0.612672
\(342\) 0 0
\(343\) −8.00000 −0.431959
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −15.5563 −0.835109 −0.417554 0.908652i \(-0.637113\pi\)
−0.417554 + 0.908652i \(0.637113\pi\)
\(348\) 0 0
\(349\) 19.7990 1.05982 0.529908 0.848055i \(-0.322227\pi\)
0.529908 + 0.848055i \(0.322227\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 14.0000 0.745145 0.372572 0.928003i \(-0.378476\pi\)
0.372572 + 0.928003i \(0.378476\pi\)
\(354\) 0 0
\(355\) −11.3137 −0.600469
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −36.0000 −1.90001 −0.950004 0.312239i \(-0.898921\pi\)
−0.950004 + 0.312239i \(0.898921\pi\)
\(360\) 0 0
\(361\) 31.0000 1.63158
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −11.3137 −0.592187
\(366\) 0 0
\(367\) −16.0000 −0.835193 −0.417597 0.908633i \(-0.637127\pi\)
−0.417597 + 0.908633i \(0.637127\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 11.3137 0.587378
\(372\) 0 0
\(373\) 36.7696 1.90386 0.951928 0.306323i \(-0.0990988\pi\)
0.951928 + 0.306323i \(0.0990988\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −24.0000 −1.23606
\(378\) 0 0
\(379\) −29.6985 −1.52551 −0.762754 0.646688i \(-0.776153\pi\)
−0.762754 + 0.646688i \(0.776153\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 16.0000 0.817562 0.408781 0.912633i \(-0.365954\pi\)
0.408781 + 0.912633i \(0.365954\pi\)
\(384\) 0 0
\(385\) 16.0000 0.815436
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 19.7990 1.00385 0.501924 0.864912i \(-0.332626\pi\)
0.501924 + 0.864912i \(0.332626\pi\)
\(390\) 0 0
\(391\) 16.0000 0.809155
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 22.6274 1.13851
\(396\) 0 0
\(397\) −2.82843 −0.141955 −0.0709773 0.997478i \(-0.522612\pi\)
−0.0709773 + 0.997478i \(0.522612\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 36.0000 1.79775 0.898877 0.438201i \(-0.144384\pi\)
0.898877 + 0.438201i \(0.144384\pi\)
\(402\) 0 0
\(403\) −22.6274 −1.12715
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −4.00000 −0.198273
\(408\) 0 0
\(409\) −14.0000 −0.692255 −0.346128 0.938187i \(-0.612504\pi\)
−0.346128 + 0.938187i \(0.612504\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 16.9706 0.835067
\(414\) 0 0
\(415\) 28.0000 1.37447
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −26.8701 −1.31269 −0.656344 0.754462i \(-0.727898\pi\)
−0.656344 + 0.754462i \(0.727898\pi\)
\(420\) 0 0
\(421\) 8.48528 0.413547 0.206774 0.978389i \(-0.433704\pi\)
0.206774 + 0.978389i \(0.433704\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 12.0000 0.582086
\(426\) 0 0
\(427\) 33.9411 1.64253
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −24.0000 −1.15604 −0.578020 0.816023i \(-0.696174\pi\)
−0.578020 + 0.816023i \(0.696174\pi\)
\(432\) 0 0
\(433\) −20.0000 −0.961139 −0.480569 0.876957i \(-0.659570\pi\)
−0.480569 + 0.876957i \(0.659570\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −28.2843 −1.35302
\(438\) 0 0
\(439\) 4.00000 0.190910 0.0954548 0.995434i \(-0.469569\pi\)
0.0954548 + 0.995434i \(0.469569\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −35.3553 −1.67978 −0.839891 0.542754i \(-0.817381\pi\)
−0.839891 + 0.542754i \(0.817381\pi\)
\(444\) 0 0
\(445\) −33.9411 −1.60896
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 20.0000 0.943858 0.471929 0.881636i \(-0.343558\pi\)
0.471929 + 0.881636i \(0.343558\pi\)
\(450\) 0 0
\(451\) 2.82843 0.133185
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −32.0000 −1.50018
\(456\) 0 0
\(457\) 2.00000 0.0935561 0.0467780 0.998905i \(-0.485105\pi\)
0.0467780 + 0.998905i \(0.485105\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 14.1421 0.658665 0.329332 0.944214i \(-0.393176\pi\)
0.329332 + 0.944214i \(0.393176\pi\)
\(462\) 0 0
\(463\) −8.00000 −0.371792 −0.185896 0.982569i \(-0.559519\pi\)
−0.185896 + 0.982569i \(0.559519\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 18.3848 0.850746 0.425373 0.905018i \(-0.360143\pi\)
0.425373 + 0.905018i \(0.360143\pi\)
\(468\) 0 0
\(469\) −16.9706 −0.783628
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −6.00000 −0.275880
\(474\) 0 0
\(475\) −21.2132 −0.973329
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −24.0000 −1.09659 −0.548294 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 0 0
\(481\) 8.00000 0.364769
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −11.3137 −0.513729
\(486\) 0 0
\(487\) 12.0000 0.543772 0.271886 0.962329i \(-0.412353\pi\)
0.271886 + 0.962329i \(0.412353\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −15.5563 −0.702048 −0.351024 0.936366i \(-0.614166\pi\)
−0.351024 + 0.936366i \(0.614166\pi\)
\(492\) 0 0
\(493\) −33.9411 −1.52863
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 16.0000 0.717698
\(498\) 0 0
\(499\) 12.7279 0.569780 0.284890 0.958560i \(-0.408043\pi\)
0.284890 + 0.958560i \(0.408043\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 20.0000 0.891756 0.445878 0.895094i \(-0.352892\pi\)
0.445878 + 0.895094i \(0.352892\pi\)
\(504\) 0 0
\(505\) −8.00000 −0.355995
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −2.82843 −0.125368 −0.0626839 0.998033i \(-0.519966\pi\)
−0.0626839 + 0.998033i \(0.519966\pi\)
\(510\) 0 0
\(511\) 16.0000 0.707798
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −33.9411 −1.49562
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −14.0000 −0.613351 −0.306676 0.951814i \(-0.599217\pi\)
−0.306676 + 0.951814i \(0.599217\pi\)
\(522\) 0 0
\(523\) −9.89949 −0.432875 −0.216437 0.976297i \(-0.569444\pi\)
−0.216437 + 0.976297i \(0.569444\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −32.0000 −1.39394
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −5.65685 −0.245026
\(534\) 0 0
\(535\) 28.0000 1.21055
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −12.7279 −0.548230
\(540\) 0 0
\(541\) −8.48528 −0.364811 −0.182405 0.983223i \(-0.558388\pi\)
−0.182405 + 0.983223i \(0.558388\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −40.0000 −1.71341
\(546\) 0 0
\(547\) 26.8701 1.14888 0.574440 0.818546i \(-0.305220\pi\)
0.574440 + 0.818546i \(0.305220\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 60.0000 2.55609
\(552\) 0 0
\(553\) −32.0000 −1.36078
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 31.1127 1.31829 0.659144 0.752017i \(-0.270919\pi\)
0.659144 + 0.752017i \(0.270919\pi\)
\(558\) 0 0
\(559\) 12.0000 0.507546
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 7.07107 0.298010 0.149005 0.988836i \(-0.452393\pi\)
0.149005 + 0.988836i \(0.452393\pi\)
\(564\) 0 0
\(565\) −39.5980 −1.66590
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −30.0000 −1.25767 −0.628833 0.777541i \(-0.716467\pi\)
−0.628833 + 0.777541i \(0.716467\pi\)
\(570\) 0 0
\(571\) −32.5269 −1.36121 −0.680604 0.732651i \(-0.738283\pi\)
−0.680604 + 0.732651i \(0.738283\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 12.0000 0.500435
\(576\) 0 0
\(577\) −18.0000 −0.749350 −0.374675 0.927156i \(-0.622246\pi\)
−0.374675 + 0.927156i \(0.622246\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −39.5980 −1.64280
\(582\) 0 0
\(583\) 4.00000 0.165663
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −1.41421 −0.0583708 −0.0291854 0.999574i \(-0.509291\pi\)
−0.0291854 + 0.999574i \(0.509291\pi\)
\(588\) 0 0
\(589\) 56.5685 2.33087
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −18.0000 −0.739171 −0.369586 0.929197i \(-0.620500\pi\)
−0.369586 + 0.929197i \(0.620500\pi\)
\(594\) 0 0
\(595\) −45.2548 −1.85527
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −36.0000 −1.47092 −0.735460 0.677568i \(-0.763034\pi\)
−0.735460 + 0.677568i \(0.763034\pi\)
\(600\) 0 0
\(601\) −4.00000 −0.163163 −0.0815817 0.996667i \(-0.525997\pi\)
−0.0815817 + 0.996667i \(0.525997\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −25.4558 −1.03493
\(606\) 0 0
\(607\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 8.48528 0.342717 0.171359 0.985209i \(-0.445184\pi\)
0.171359 + 0.985209i \(0.445184\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −28.0000 −1.12724 −0.563619 0.826035i \(-0.690591\pi\)
−0.563619 + 0.826035i \(0.690591\pi\)
\(618\) 0 0
\(619\) 1.41421 0.0568420 0.0284210 0.999596i \(-0.490952\pi\)
0.0284210 + 0.999596i \(0.490952\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 48.0000 1.92308
\(624\) 0 0
\(625\) −31.0000 −1.24000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 11.3137 0.451107
\(630\) 0 0
\(631\) −4.00000 −0.159237 −0.0796187 0.996825i \(-0.525370\pi\)
−0.0796187 + 0.996825i \(0.525370\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −22.6274 −0.897942
\(636\) 0 0
\(637\) 25.4558 1.00860
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 4.00000 0.157991 0.0789953 0.996875i \(-0.474829\pi\)
0.0789953 + 0.996875i \(0.474829\pi\)
\(642\) 0 0
\(643\) 4.24264 0.167313 0.0836567 0.996495i \(-0.473340\pi\)
0.0836567 + 0.996495i \(0.473340\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 12.0000 0.471769 0.235884 0.971781i \(-0.424201\pi\)
0.235884 + 0.971781i \(0.424201\pi\)
\(648\) 0 0
\(649\) 6.00000 0.235521
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 48.0833 1.88164 0.940822 0.338902i \(-0.110055\pi\)
0.940822 + 0.338902i \(0.110055\pi\)
\(654\) 0 0
\(655\) 20.0000 0.781465
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 32.5269 1.26707 0.633534 0.773715i \(-0.281604\pi\)
0.633534 + 0.773715i \(0.281604\pi\)
\(660\) 0 0
\(661\) 31.1127 1.21014 0.605072 0.796171i \(-0.293144\pi\)
0.605072 + 0.796171i \(0.293144\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 80.0000 3.10227
\(666\) 0 0
\(667\) −33.9411 −1.31421
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 12.0000 0.463255
\(672\) 0 0
\(673\) −20.0000 −0.770943 −0.385472 0.922720i \(-0.625961\pi\)
−0.385472 + 0.922720i \(0.625961\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 42.4264 1.63058 0.815290 0.579053i \(-0.196578\pi\)
0.815290 + 0.579053i \(0.196578\pi\)
\(678\) 0 0
\(679\) 16.0000 0.614024
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 29.6985 1.13638 0.568190 0.822897i \(-0.307644\pi\)
0.568190 + 0.822897i \(0.307644\pi\)
\(684\) 0 0
\(685\) 50.9117 1.94524
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −8.00000 −0.304776
\(690\) 0 0
\(691\) 46.6690 1.77537 0.887687 0.460447i \(-0.152311\pi\)
0.887687 + 0.460447i \(0.152311\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 12.0000 0.455186
\(696\) 0 0
\(697\) −8.00000 −0.303022
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −48.0833 −1.81608 −0.908040 0.418884i \(-0.862421\pi\)
−0.908040 + 0.418884i \(0.862421\pi\)
\(702\) 0 0
\(703\) −20.0000 −0.754314
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 11.3137 0.425496
\(708\) 0 0
\(709\) −8.48528 −0.318671 −0.159336 0.987224i \(-0.550935\pi\)
−0.159336 + 0.987224i \(0.550935\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −32.0000 −1.19841
\(714\) 0 0
\(715\) −11.3137 −0.423109
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 24.0000 0.895049 0.447524 0.894272i \(-0.352306\pi\)
0.447524 + 0.894272i \(0.352306\pi\)
\(720\) 0 0
\(721\) 48.0000 1.78761
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −25.4558 −0.945406
\(726\) 0 0
\(727\) −28.0000 −1.03846 −0.519231 0.854634i \(-0.673782\pi\)
−0.519231 + 0.854634i \(0.673782\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 16.9706 0.627679
\(732\) 0 0
\(733\) 19.7990 0.731292 0.365646 0.930754i \(-0.380848\pi\)
0.365646 + 0.930754i \(0.380848\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −6.00000 −0.221013
\(738\) 0 0
\(739\) 24.0416 0.884386 0.442193 0.896920i \(-0.354201\pi\)
0.442193 + 0.896920i \(0.354201\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 4.00000 0.146746 0.0733729 0.997305i \(-0.476624\pi\)
0.0733729 + 0.997305i \(0.476624\pi\)
\(744\) 0 0
\(745\) 8.00000 0.293097
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −39.5980 −1.44688
\(750\) 0 0
\(751\) 40.0000 1.45962 0.729810 0.683650i \(-0.239608\pi\)
0.729810 + 0.683650i \(0.239608\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −11.3137 −0.411748
\(756\) 0 0
\(757\) −25.4558 −0.925208 −0.462604 0.886565i \(-0.653085\pi\)
−0.462604 + 0.886565i \(0.653085\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −18.0000 −0.652499 −0.326250 0.945284i \(-0.605785\pi\)
−0.326250 + 0.945284i \(0.605785\pi\)
\(762\) 0 0
\(763\) 56.5685 2.04792
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −12.0000 −0.433295
\(768\) 0 0
\(769\) 44.0000 1.58668 0.793340 0.608778i \(-0.208340\pi\)
0.793340 + 0.608778i \(0.208340\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 8.48528 0.305194 0.152597 0.988288i \(-0.451236\pi\)
0.152597 + 0.988288i \(0.451236\pi\)
\(774\) 0 0
\(775\) −24.0000 −0.862105
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 14.1421 0.506695
\(780\) 0 0
\(781\) 5.65685 0.202418
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −24.0000 −0.856597
\(786\) 0 0
\(787\) 46.6690 1.66357 0.831786 0.555097i \(-0.187319\pi\)
0.831786 + 0.555097i \(0.187319\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 56.0000 1.99113
\(792\) 0 0
\(793\) −24.0000 −0.852265
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 19.7990 0.701316 0.350658 0.936504i \(-0.385958\pi\)
0.350658 + 0.936504i \(0.385958\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 5.65685 0.199626
\(804\) 0 0
\(805\) −45.2548 −1.59502
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 30.0000 1.05474 0.527372 0.849635i \(-0.323177\pi\)
0.527372 + 0.849635i \(0.323177\pi\)
\(810\) 0 0
\(811\) 24.0416 0.844216 0.422108 0.906546i \(-0.361290\pi\)
0.422108 + 0.906546i \(0.361290\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −28.0000 −0.980797
\(816\) 0 0
\(817\) −30.0000 −1.04957
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 8.48528 0.296138 0.148069 0.988977i \(-0.452694\pi\)
0.148069 + 0.988977i \(0.452694\pi\)
\(822\) 0 0
\(823\) 4.00000 0.139431 0.0697156 0.997567i \(-0.477791\pi\)
0.0697156 + 0.997567i \(0.477791\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 21.2132 0.737655 0.368828 0.929498i \(-0.379759\pi\)
0.368828 + 0.929498i \(0.379759\pi\)
\(828\) 0 0
\(829\) 25.4558 0.884118 0.442059 0.896986i \(-0.354248\pi\)
0.442059 + 0.896986i \(0.354248\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 36.0000 1.24733
\(834\) 0 0
\(835\) −33.9411 −1.17458
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 20.0000 0.690477 0.345238 0.938515i \(-0.387798\pi\)
0.345238 + 0.938515i \(0.387798\pi\)
\(840\) 0 0
\(841\) 43.0000 1.48276
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −14.1421 −0.486504
\(846\) 0 0
\(847\) 36.0000 1.23697
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 11.3137 0.387829
\(852\) 0 0
\(853\) −42.4264 −1.45265 −0.726326 0.687350i \(-0.758774\pi\)
−0.726326 + 0.687350i \(0.758774\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 34.0000 1.16142 0.580709 0.814111i \(-0.302775\pi\)
0.580709 + 0.814111i \(0.302775\pi\)
\(858\) 0 0
\(859\) 38.1838 1.30281 0.651407 0.758729i \(-0.274179\pi\)
0.651407 + 0.758729i \(0.274179\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 24.0000 0.816970 0.408485 0.912765i \(-0.366057\pi\)
0.408485 + 0.912765i \(0.366057\pi\)
\(864\) 0 0
\(865\) −40.0000 −1.36004
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −11.3137 −0.383791
\(870\) 0 0
\(871\) 12.0000 0.406604
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 22.6274 0.764946
\(876\) 0 0
\(877\) 8.48528 0.286528 0.143264 0.989685i \(-0.454240\pi\)
0.143264 + 0.989685i \(0.454240\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 34.0000 1.14549 0.572745 0.819734i \(-0.305879\pi\)
0.572745 + 0.819734i \(0.305879\pi\)
\(882\) 0 0
\(883\) −41.0122 −1.38017 −0.690085 0.723728i \(-0.742427\pi\)
−0.690085 + 0.723728i \(0.742427\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 28.0000 0.940148 0.470074 0.882627i \(-0.344227\pi\)
0.470074 + 0.882627i \(0.344227\pi\)
\(888\) 0 0
\(889\) 32.0000 1.07325
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) −4.00000 −0.133705
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 67.8823 2.26400
\(900\) 0 0
\(901\) −11.3137 −0.376914
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 24.0000 0.797787
\(906\) 0 0
\(907\) −7.07107 −0.234791 −0.117395 0.993085i \(-0.537455\pi\)
−0.117395 + 0.993085i \(0.537455\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 48.0000 1.59031 0.795155 0.606406i \(-0.207389\pi\)
0.795155 + 0.606406i \(0.207389\pi\)
\(912\) 0 0
\(913\) −14.0000 −0.463332
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −28.2843 −0.934029
\(918\) 0 0
\(919\) 44.0000 1.45143 0.725713 0.687998i \(-0.241510\pi\)
0.725713 + 0.687998i \(0.241510\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −11.3137 −0.372395
\(924\) 0 0
\(925\) 8.48528 0.278994
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −60.0000 −1.96854 −0.984268 0.176682i \(-0.943464\pi\)
−0.984268 + 0.176682i \(0.943464\pi\)
\(930\) 0 0
\(931\) −63.6396 −2.08570
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −16.0000 −0.523256
\(936\) 0 0
\(937\) −20.0000 −0.653372 −0.326686 0.945133i \(-0.605932\pi\)
−0.326686 + 0.945133i \(0.605932\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 36.7696 1.19865 0.599327 0.800505i \(-0.295435\pi\)
0.599327 + 0.800505i \(0.295435\pi\)
\(942\) 0 0
\(943\) −8.00000 −0.260516
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −60.8112 −1.97610 −0.988049 0.154140i \(-0.950739\pi\)
−0.988049 + 0.154140i \(0.950739\pi\)
\(948\) 0 0
\(949\) −11.3137 −0.367259
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −30.0000 −0.971795 −0.485898 0.874016i \(-0.661507\pi\)
−0.485898 + 0.874016i \(0.661507\pi\)
\(954\) 0 0
\(955\) 22.6274 0.732206
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −72.0000 −2.32500
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −11.3137 −0.364201
\(966\) 0 0
\(967\) −20.0000 −0.643157 −0.321578 0.946883i \(-0.604213\pi\)
−0.321578 + 0.946883i \(0.604213\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −15.5563 −0.499227 −0.249614 0.968346i \(-0.580304\pi\)
−0.249614 + 0.968346i \(0.580304\pi\)
\(972\) 0 0
\(973\) −16.9706 −0.544051
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −12.0000 −0.383914 −0.191957 0.981403i \(-0.561483\pi\)
−0.191957 + 0.981403i \(0.561483\pi\)
\(978\) 0 0
\(979\) 16.9706 0.542382
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −36.0000 −1.14822 −0.574111 0.818778i \(-0.694652\pi\)
−0.574111 + 0.818778i \(0.694652\pi\)
\(984\) 0 0
\(985\) 40.0000 1.27451
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 16.9706 0.539633
\(990\) 0 0
\(991\) 32.0000 1.01651 0.508257 0.861206i \(-0.330290\pi\)
0.508257 + 0.861206i \(0.330290\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −11.3137 −0.358669
\(996\) 0 0
\(997\) 14.1421 0.447886 0.223943 0.974602i \(-0.428107\pi\)
0.223943 + 0.974602i \(0.428107\pi\)
\(998\) 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 4608.2.a.f.1.2 2
3.2 odd 2 512.2.a.c.1.2 yes 2
4.3 odd 2 4608.2.a.m.1.2 2
8.3 odd 2 4608.2.a.m.1.1 2
8.5 even 2 inner 4608.2.a.f.1.1 2
12.11 even 2 512.2.a.d.1.1 yes 2
16.3 odd 4 4608.2.d.a.2305.1 2
16.5 even 4 4608.2.d.b.2305.2 2
16.11 odd 4 4608.2.d.a.2305.2 2
16.13 even 4 4608.2.d.b.2305.1 2
24.5 odd 2 512.2.a.c.1.1 2
24.11 even 2 512.2.a.d.1.2 yes 2
48.5 odd 4 512.2.b.b.257.1 2
48.11 even 4 512.2.b.a.257.2 2
48.29 odd 4 512.2.b.b.257.2 2
48.35 even 4 512.2.b.a.257.1 2
96.5 odd 8 1024.2.e.b.769.1 2
96.11 even 8 1024.2.e.c.769.1 2
96.29 odd 8 1024.2.e.e.257.1 2
96.35 even 8 1024.2.e.c.257.1 2
96.53 odd 8 1024.2.e.e.769.1 2
96.59 even 8 1024.2.e.d.769.1 2
96.77 odd 8 1024.2.e.b.257.1 2
96.83 even 8 1024.2.e.d.257.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
512.2.a.c.1.1 2 24.5 odd 2
512.2.a.c.1.2 yes 2 3.2 odd 2
512.2.a.d.1.1 yes 2 12.11 even 2
512.2.a.d.1.2 yes 2 24.11 even 2
512.2.b.a.257.1 2 48.35 even 4
512.2.b.a.257.2 2 48.11 even 4
512.2.b.b.257.1 2 48.5 odd 4
512.2.b.b.257.2 2 48.29 odd 4
1024.2.e.b.257.1 2 96.77 odd 8
1024.2.e.b.769.1 2 96.5 odd 8
1024.2.e.c.257.1 2 96.35 even 8
1024.2.e.c.769.1 2 96.11 even 8
1024.2.e.d.257.1 2 96.83 even 8
1024.2.e.d.769.1 2 96.59 even 8
1024.2.e.e.257.1 2 96.29 odd 8
1024.2.e.e.769.1 2 96.53 odd 8
4608.2.a.f.1.1 2 8.5 even 2 inner
4608.2.a.f.1.2 2 1.1 even 1 trivial
4608.2.a.m.1.1 2 8.3 odd 2
4608.2.a.m.1.2 2 4.3 odd 2
4608.2.d.a.2305.1 2 16.3 odd 4
4608.2.d.a.2305.2 2 16.11 odd 4
4608.2.d.b.2305.1 2 16.13 even 4
4608.2.d.b.2305.2 2 16.5 even 4