Properties

Label 1792.2.a.n.1.2
Level $1792$
Weight $2$
Character 1792.1
Self dual yes
Analytic conductor $14.309$
Analytic rank $1$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 56)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 1792.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.41421 q^{3} +1.41421 q^{5} -1.00000 q^{7} -1.00000 q^{9} +O(q^{10})\) \(q+1.41421 q^{3} +1.41421 q^{5} -1.00000 q^{7} -1.00000 q^{9} -2.82843 q^{11} -4.24264 q^{13} +2.00000 q^{15} -6.00000 q^{17} -4.24264 q^{19} -1.41421 q^{21} +6.00000 q^{23} -3.00000 q^{25} -5.65685 q^{27} +2.82843 q^{29} -4.00000 q^{31} -4.00000 q^{33} -1.41421 q^{35} -8.48528 q^{37} -6.00000 q^{39} -6.00000 q^{41} +8.48528 q^{43} -1.41421 q^{45} +1.00000 q^{49} -8.48528 q^{51} +5.65685 q^{53} -4.00000 q^{55} -6.00000 q^{57} +1.41421 q^{59} +12.7279 q^{61} +1.00000 q^{63} -6.00000 q^{65} +8.48528 q^{69} -2.00000 q^{73} -4.24264 q^{75} +2.82843 q^{77} +8.00000 q^{79} -5.00000 q^{81} +15.5563 q^{83} -8.48528 q^{85} +4.00000 q^{87} -6.00000 q^{89} +4.24264 q^{91} -5.65685 q^{93} -6.00000 q^{95} -10.0000 q^{97} +2.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{7} - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{7} - 2q^{9} + 4q^{15} - 12q^{17} + 12q^{23} - 6q^{25} - 8q^{31} - 8q^{33} - 12q^{39} - 12q^{41} + 2q^{49} - 8q^{55} - 12q^{57} + 2q^{63} - 12q^{65} - 4q^{73} + 16q^{79} - 10q^{81} + 8q^{87} - 12q^{89} - 12q^{95} - 20q^{97} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.41421 0.816497 0.408248 0.912871i \(-0.366140\pi\)
0.408248 + 0.912871i \(0.366140\pi\)
\(4\) 0 0
\(5\) 1.41421 0.632456 0.316228 0.948683i \(-0.397584\pi\)
0.316228 + 0.948683i \(0.397584\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −2.82843 −0.852803 −0.426401 0.904534i \(-0.640219\pi\)
−0.426401 + 0.904534i \(0.640219\pi\)
\(12\) 0 0
\(13\) −4.24264 −1.17670 −0.588348 0.808608i \(-0.700222\pi\)
−0.588348 + 0.808608i \(0.700222\pi\)
\(14\) 0 0
\(15\) 2.00000 0.516398
\(16\) 0 0
\(17\) −6.00000 −1.45521 −0.727607 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 0 0
\(19\) −4.24264 −0.973329 −0.486664 0.873589i \(-0.661786\pi\)
−0.486664 + 0.873589i \(0.661786\pi\)
\(20\) 0 0
\(21\) −1.41421 −0.308607
\(22\) 0 0
\(23\) 6.00000 1.25109 0.625543 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) 0 0
\(25\) −3.00000 −0.600000
\(26\) 0 0
\(27\) −5.65685 −1.08866
\(28\) 0 0
\(29\) 2.82843 0.525226 0.262613 0.964901i \(-0.415416\pi\)
0.262613 + 0.964901i \(0.415416\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 0 0
\(33\) −4.00000 −0.696311
\(34\) 0 0
\(35\) −1.41421 −0.239046
\(36\) 0 0
\(37\) −8.48528 −1.39497 −0.697486 0.716599i \(-0.745698\pi\)
−0.697486 + 0.716599i \(0.745698\pi\)
\(38\) 0 0
\(39\) −6.00000 −0.960769
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 8.48528 1.29399 0.646997 0.762493i \(-0.276025\pi\)
0.646997 + 0.762493i \(0.276025\pi\)
\(44\) 0 0
\(45\) −1.41421 −0.210819
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −8.48528 −1.18818
\(52\) 0 0
\(53\) 5.65685 0.777029 0.388514 0.921443i \(-0.372988\pi\)
0.388514 + 0.921443i \(0.372988\pi\)
\(54\) 0 0
\(55\) −4.00000 −0.539360
\(56\) 0 0
\(57\) −6.00000 −0.794719
\(58\) 0 0
\(59\) 1.41421 0.184115 0.0920575 0.995754i \(-0.470656\pi\)
0.0920575 + 0.995754i \(0.470656\pi\)
\(60\) 0 0
\(61\) 12.7279 1.62964 0.814822 0.579712i \(-0.196835\pi\)
0.814822 + 0.579712i \(0.196835\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) −6.00000 −0.744208
\(66\) 0 0
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) 8.48528 1.02151
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −2.00000 −0.234082 −0.117041 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) 0 0
\(75\) −4.24264 −0.489898
\(76\) 0 0
\(77\) 2.82843 0.322329
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) −5.00000 −0.555556
\(82\) 0 0
\(83\) 15.5563 1.70753 0.853766 0.520658i \(-0.174313\pi\)
0.853766 + 0.520658i \(0.174313\pi\)
\(84\) 0 0
\(85\) −8.48528 −0.920358
\(86\) 0 0
\(87\) 4.00000 0.428845
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 4.24264 0.444750
\(92\) 0 0
\(93\) −5.65685 −0.586588
\(94\) 0 0
\(95\) −6.00000 −0.615587
\(96\) 0 0
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 0 0
\(99\) 2.82843 0.284268
\(100\) 0 0
\(101\) 9.89949 0.985037 0.492518 0.870302i \(-0.336076\pi\)
0.492518 + 0.870302i \(0.336076\pi\)
\(102\) 0 0
\(103\) 4.00000 0.394132 0.197066 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) 0 0
\(105\) −2.00000 −0.195180
\(106\) 0 0
\(107\) 5.65685 0.546869 0.273434 0.961891i \(-0.411840\pi\)
0.273434 + 0.961891i \(0.411840\pi\)
\(108\) 0 0
\(109\) −8.48528 −0.812743 −0.406371 0.913708i \(-0.633206\pi\)
−0.406371 + 0.913708i \(0.633206\pi\)
\(110\) 0 0
\(111\) −12.0000 −1.13899
\(112\) 0 0
\(113\) −12.0000 −1.12887 −0.564433 0.825479i \(-0.690905\pi\)
−0.564433 + 0.825479i \(0.690905\pi\)
\(114\) 0 0
\(115\) 8.48528 0.791257
\(116\) 0 0
\(117\) 4.24264 0.392232
\(118\) 0 0
\(119\) 6.00000 0.550019
\(120\) 0 0
\(121\) −3.00000 −0.272727
\(122\) 0 0
\(123\) −8.48528 −0.765092
\(124\) 0 0
\(125\) −11.3137 −1.01193
\(126\) 0 0
\(127\) 2.00000 0.177471 0.0887357 0.996055i \(-0.471717\pi\)
0.0887357 + 0.996055i \(0.471717\pi\)
\(128\) 0 0
\(129\) 12.0000 1.05654
\(130\) 0 0
\(131\) −1.41421 −0.123560 −0.0617802 0.998090i \(-0.519678\pi\)
−0.0617802 + 0.998090i \(0.519678\pi\)
\(132\) 0 0
\(133\) 4.24264 0.367884
\(134\) 0 0
\(135\) −8.00000 −0.688530
\(136\) 0 0
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 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
\(143\) 12.0000 1.00349
\(144\) 0 0
\(145\) 4.00000 0.332182
\(146\) 0 0
\(147\) 1.41421 0.116642
\(148\) 0 0
\(149\) −11.3137 −0.926855 −0.463428 0.886135i \(-0.653381\pi\)
−0.463428 + 0.886135i \(0.653381\pi\)
\(150\) 0 0
\(151\) 10.0000 0.813788 0.406894 0.913475i \(-0.366612\pi\)
0.406894 + 0.913475i \(0.366612\pi\)
\(152\) 0 0
\(153\) 6.00000 0.485071
\(154\) 0 0
\(155\) −5.65685 −0.454369
\(156\) 0 0
\(157\) 12.7279 1.01580 0.507899 0.861416i \(-0.330422\pi\)
0.507899 + 0.861416i \(0.330422\pi\)
\(158\) 0 0
\(159\) 8.00000 0.634441
\(160\) 0 0
\(161\) −6.00000 −0.472866
\(162\) 0 0
\(163\) 8.48528 0.664619 0.332309 0.943170i \(-0.392172\pi\)
0.332309 + 0.943170i \(0.392172\pi\)
\(164\) 0 0
\(165\) −5.65685 −0.440386
\(166\) 0 0
\(167\) −24.0000 −1.85718 −0.928588 0.371113i \(-0.878976\pi\)
−0.928588 + 0.371113i \(0.878976\pi\)
\(168\) 0 0
\(169\) 5.00000 0.384615
\(170\) 0 0
\(171\) 4.24264 0.324443
\(172\) 0 0
\(173\) −9.89949 −0.752645 −0.376322 0.926489i \(-0.622811\pi\)
−0.376322 + 0.926489i \(0.622811\pi\)
\(174\) 0 0
\(175\) 3.00000 0.226779
\(176\) 0 0
\(177\) 2.00000 0.150329
\(178\) 0 0
\(179\) −5.65685 −0.422813 −0.211407 0.977398i \(-0.567804\pi\)
−0.211407 + 0.977398i \(0.567804\pi\)
\(180\) 0 0
\(181\) −12.7279 −0.946059 −0.473029 0.881047i \(-0.656840\pi\)
−0.473029 + 0.881047i \(0.656840\pi\)
\(182\) 0 0
\(183\) 18.0000 1.33060
\(184\) 0 0
\(185\) −12.0000 −0.882258
\(186\) 0 0
\(187\) 16.9706 1.24101
\(188\) 0 0
\(189\) 5.65685 0.411476
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) −4.00000 −0.287926 −0.143963 0.989583i \(-0.545985\pi\)
−0.143963 + 0.989583i \(0.545985\pi\)
\(194\) 0 0
\(195\) −8.48528 −0.607644
\(196\) 0 0
\(197\) 5.65685 0.403034 0.201517 0.979485i \(-0.435413\pi\)
0.201517 + 0.979485i \(0.435413\pi\)
\(198\) 0 0
\(199\) −20.0000 −1.41776 −0.708881 0.705328i \(-0.750800\pi\)
−0.708881 + 0.705328i \(0.750800\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −2.82843 −0.198517
\(204\) 0 0
\(205\) −8.48528 −0.592638
\(206\) 0 0
\(207\) −6.00000 −0.417029
\(208\) 0 0
\(209\) 12.0000 0.830057
\(210\) 0 0
\(211\) −16.9706 −1.16830 −0.584151 0.811645i \(-0.698572\pi\)
−0.584151 + 0.811645i \(0.698572\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 12.0000 0.818393
\(216\) 0 0
\(217\) 4.00000 0.271538
\(218\) 0 0
\(219\) −2.82843 −0.191127
\(220\) 0 0
\(221\) 25.4558 1.71235
\(222\) 0 0
\(223\) −28.0000 −1.87502 −0.937509 0.347960i \(-0.886874\pi\)
−0.937509 + 0.347960i \(0.886874\pi\)
\(224\) 0 0
\(225\) 3.00000 0.200000
\(226\) 0 0
\(227\) −9.89949 −0.657053 −0.328526 0.944495i \(-0.606552\pi\)
−0.328526 + 0.944495i \(0.606552\pi\)
\(228\) 0 0
\(229\) −4.24264 −0.280362 −0.140181 0.990126i \(-0.544768\pi\)
−0.140181 + 0.990126i \(0.544768\pi\)
\(230\) 0 0
\(231\) 4.00000 0.263181
\(232\) 0 0
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 11.3137 0.734904
\(238\) 0 0
\(239\) 6.00000 0.388108 0.194054 0.980991i \(-0.437836\pi\)
0.194054 + 0.980991i \(0.437836\pi\)
\(240\) 0 0
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) 0 0
\(243\) 9.89949 0.635053
\(244\) 0 0
\(245\) 1.41421 0.0903508
\(246\) 0 0
\(247\) 18.0000 1.14531
\(248\) 0 0
\(249\) 22.0000 1.39419
\(250\) 0 0
\(251\) 18.3848 1.16044 0.580218 0.814461i \(-0.302967\pi\)
0.580218 + 0.814461i \(0.302967\pi\)
\(252\) 0 0
\(253\) −16.9706 −1.06693
\(254\) 0 0
\(255\) −12.0000 −0.751469
\(256\) 0 0
\(257\) −6.00000 −0.374270 −0.187135 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) 0 0
\(259\) 8.48528 0.527250
\(260\) 0 0
\(261\) −2.82843 −0.175075
\(262\) 0 0
\(263\) −24.0000 −1.47990 −0.739952 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) 0 0
\(265\) 8.00000 0.491436
\(266\) 0 0
\(267\) −8.48528 −0.519291
\(268\) 0 0
\(269\) 7.07107 0.431131 0.215565 0.976489i \(-0.430841\pi\)
0.215565 + 0.976489i \(0.430841\pi\)
\(270\) 0 0
\(271\) 20.0000 1.21491 0.607457 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) 0 0
\(273\) 6.00000 0.363137
\(274\) 0 0
\(275\) 8.48528 0.511682
\(276\) 0 0
\(277\) 16.9706 1.01966 0.509831 0.860274i \(-0.329708\pi\)
0.509831 + 0.860274i \(0.329708\pi\)
\(278\) 0 0
\(279\) 4.00000 0.239474
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) −12.7279 −0.756596 −0.378298 0.925684i \(-0.623491\pi\)
−0.378298 + 0.925684i \(0.623491\pi\)
\(284\) 0 0
\(285\) −8.48528 −0.502625
\(286\) 0 0
\(287\) 6.00000 0.354169
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) −14.1421 −0.829027
\(292\) 0 0
\(293\) −24.0416 −1.40453 −0.702264 0.711917i \(-0.747827\pi\)
−0.702264 + 0.711917i \(0.747827\pi\)
\(294\) 0 0
\(295\) 2.00000 0.116445
\(296\) 0 0
\(297\) 16.0000 0.928414
\(298\) 0 0
\(299\) −25.4558 −1.47215
\(300\) 0 0
\(301\) −8.48528 −0.489083
\(302\) 0 0
\(303\) 14.0000 0.804279
\(304\) 0 0
\(305\) 18.0000 1.03068
\(306\) 0 0
\(307\) −12.7279 −0.726421 −0.363210 0.931707i \(-0.618319\pi\)
−0.363210 + 0.931707i \(0.618319\pi\)
\(308\) 0 0
\(309\) 5.65685 0.321807
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 10.0000 0.565233 0.282617 0.959233i \(-0.408798\pi\)
0.282617 + 0.959233i \(0.408798\pi\)
\(314\) 0 0
\(315\) 1.41421 0.0796819
\(316\) 0 0
\(317\) −22.6274 −1.27088 −0.635441 0.772149i \(-0.719182\pi\)
−0.635441 + 0.772149i \(0.719182\pi\)
\(318\) 0 0
\(319\) −8.00000 −0.447914
\(320\) 0 0
\(321\) 8.00000 0.446516
\(322\) 0 0
\(323\) 25.4558 1.41640
\(324\) 0 0
\(325\) 12.7279 0.706018
\(326\) 0 0
\(327\) −12.0000 −0.663602
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −25.4558 −1.39918 −0.699590 0.714545i \(-0.746634\pi\)
−0.699590 + 0.714545i \(0.746634\pi\)
\(332\) 0 0
\(333\) 8.48528 0.464991
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 32.0000 1.74315 0.871576 0.490261i \(-0.163099\pi\)
0.871576 + 0.490261i \(0.163099\pi\)
\(338\) 0 0
\(339\) −16.9706 −0.921714
\(340\) 0 0
\(341\) 11.3137 0.612672
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 12.0000 0.646058
\(346\) 0 0
\(347\) 14.1421 0.759190 0.379595 0.925153i \(-0.376063\pi\)
0.379595 + 0.925153i \(0.376063\pi\)
\(348\) 0 0
\(349\) 4.24264 0.227103 0.113552 0.993532i \(-0.463777\pi\)
0.113552 + 0.993532i \(0.463777\pi\)
\(350\) 0 0
\(351\) 24.0000 1.28103
\(352\) 0 0
\(353\) 6.00000 0.319348 0.159674 0.987170i \(-0.448956\pi\)
0.159674 + 0.987170i \(0.448956\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 8.48528 0.449089
\(358\) 0 0
\(359\) 30.0000 1.58334 0.791670 0.610949i \(-0.209212\pi\)
0.791670 + 0.610949i \(0.209212\pi\)
\(360\) 0 0
\(361\) −1.00000 −0.0526316
\(362\) 0 0
\(363\) −4.24264 −0.222681
\(364\) 0 0
\(365\) −2.82843 −0.148047
\(366\) 0 0
\(367\) −28.0000 −1.46159 −0.730794 0.682598i \(-0.760850\pi\)
−0.730794 + 0.682598i \(0.760850\pi\)
\(368\) 0 0
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) −5.65685 −0.293689
\(372\) 0 0
\(373\) 33.9411 1.75740 0.878702 0.477370i \(-0.158410\pi\)
0.878702 + 0.477370i \(0.158410\pi\)
\(374\) 0 0
\(375\) −16.0000 −0.826236
\(376\) 0 0
\(377\) −12.0000 −0.618031
\(378\) 0 0
\(379\) −25.4558 −1.30758 −0.653789 0.756677i \(-0.726822\pi\)
−0.653789 + 0.756677i \(0.726822\pi\)
\(380\) 0 0
\(381\) 2.82843 0.144905
\(382\) 0 0
\(383\) −24.0000 −1.22634 −0.613171 0.789950i \(-0.710106\pi\)
−0.613171 + 0.789950i \(0.710106\pi\)
\(384\) 0 0
\(385\) 4.00000 0.203859
\(386\) 0 0
\(387\) −8.48528 −0.431331
\(388\) 0 0
\(389\) −2.82843 −0.143407 −0.0717035 0.997426i \(-0.522844\pi\)
−0.0717035 + 0.997426i \(0.522844\pi\)
\(390\) 0 0
\(391\) −36.0000 −1.82060
\(392\) 0 0
\(393\) −2.00000 −0.100887
\(394\) 0 0
\(395\) 11.3137 0.569254
\(396\) 0 0
\(397\) 21.2132 1.06466 0.532330 0.846537i \(-0.321317\pi\)
0.532330 + 0.846537i \(0.321317\pi\)
\(398\) 0 0
\(399\) 6.00000 0.300376
\(400\) 0 0
\(401\) −24.0000 −1.19850 −0.599251 0.800561i \(-0.704535\pi\)
−0.599251 + 0.800561i \(0.704535\pi\)
\(402\) 0 0
\(403\) 16.9706 0.845364
\(404\) 0 0
\(405\) −7.07107 −0.351364
\(406\) 0 0
\(407\) 24.0000 1.18964
\(408\) 0 0
\(409\) 22.0000 1.08783 0.543915 0.839140i \(-0.316941\pi\)
0.543915 + 0.839140i \(0.316941\pi\)
\(410\) 0 0
\(411\) −8.48528 −0.418548
\(412\) 0 0
\(413\) −1.41421 −0.0695889
\(414\) 0 0
\(415\) 22.0000 1.07994
\(416\) 0 0
\(417\) 6.00000 0.293821
\(418\) 0 0
\(419\) −26.8701 −1.31269 −0.656344 0.754462i \(-0.727898\pi\)
−0.656344 + 0.754462i \(0.727898\pi\)
\(420\) 0 0
\(421\) −16.9706 −0.827095 −0.413547 0.910483i \(-0.635710\pi\)
−0.413547 + 0.910483i \(0.635710\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 18.0000 0.873128
\(426\) 0 0
\(427\) −12.7279 −0.615947
\(428\) 0 0
\(429\) 16.9706 0.819346
\(430\) 0 0
\(431\) −6.00000 −0.289010 −0.144505 0.989504i \(-0.546159\pi\)
−0.144505 + 0.989504i \(0.546159\pi\)
\(432\) 0 0
\(433\) 2.00000 0.0961139 0.0480569 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) 0 0
\(435\) 5.65685 0.271225
\(436\) 0 0
\(437\) −25.4558 −1.21772
\(438\) 0 0
\(439\) 28.0000 1.33637 0.668184 0.743996i \(-0.267072\pi\)
0.668184 + 0.743996i \(0.267072\pi\)
\(440\) 0 0
\(441\) −1.00000 −0.0476190
\(442\) 0 0
\(443\) 22.6274 1.07506 0.537531 0.843244i \(-0.319357\pi\)
0.537531 + 0.843244i \(0.319357\pi\)
\(444\) 0 0
\(445\) −8.48528 −0.402241
\(446\) 0 0
\(447\) −16.0000 −0.756774
\(448\) 0 0
\(449\) −18.0000 −0.849473 −0.424736 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) 0 0
\(451\) 16.9706 0.799113
\(452\) 0 0
\(453\) 14.1421 0.664455
\(454\) 0 0
\(455\) 6.00000 0.281284
\(456\) 0 0
\(457\) 28.0000 1.30978 0.654892 0.755722i \(-0.272714\pi\)
0.654892 + 0.755722i \(0.272714\pi\)
\(458\) 0 0
\(459\) 33.9411 1.58424
\(460\) 0 0
\(461\) 15.5563 0.724531 0.362266 0.932075i \(-0.382003\pi\)
0.362266 + 0.932075i \(0.382003\pi\)
\(462\) 0 0
\(463\) 32.0000 1.48717 0.743583 0.668644i \(-0.233125\pi\)
0.743583 + 0.668644i \(0.233125\pi\)
\(464\) 0 0
\(465\) −8.00000 −0.370991
\(466\) 0 0
\(467\) 7.07107 0.327210 0.163605 0.986526i \(-0.447688\pi\)
0.163605 + 0.986526i \(0.447688\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 18.0000 0.829396
\(472\) 0 0
\(473\) −24.0000 −1.10352
\(474\) 0 0
\(475\) 12.7279 0.583997
\(476\) 0 0
\(477\) −5.65685 −0.259010
\(478\) 0 0
\(479\) −12.0000 −0.548294 −0.274147 0.961688i \(-0.588395\pi\)
−0.274147 + 0.961688i \(0.588395\pi\)
\(480\) 0 0
\(481\) 36.0000 1.64146
\(482\) 0 0
\(483\) −8.48528 −0.386094
\(484\) 0 0
\(485\) −14.1421 −0.642161
\(486\) 0 0
\(487\) −2.00000 −0.0906287 −0.0453143 0.998973i \(-0.514429\pi\)
−0.0453143 + 0.998973i \(0.514429\pi\)
\(488\) 0 0
\(489\) 12.0000 0.542659
\(490\) 0 0
\(491\) 39.5980 1.78703 0.893516 0.449032i \(-0.148231\pi\)
0.893516 + 0.449032i \(0.148231\pi\)
\(492\) 0 0
\(493\) −16.9706 −0.764316
\(494\) 0 0
\(495\) 4.00000 0.179787
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 16.9706 0.759707 0.379853 0.925047i \(-0.375974\pi\)
0.379853 + 0.925047i \(0.375974\pi\)
\(500\) 0 0
\(501\) −33.9411 −1.51638
\(502\) 0 0
\(503\) −36.0000 −1.60516 −0.802580 0.596544i \(-0.796540\pi\)
−0.802580 + 0.596544i \(0.796540\pi\)
\(504\) 0 0
\(505\) 14.0000 0.622992
\(506\) 0 0
\(507\) 7.07107 0.314037
\(508\) 0 0
\(509\) −1.41421 −0.0626839 −0.0313420 0.999509i \(-0.509978\pi\)
−0.0313420 + 0.999509i \(0.509978\pi\)
\(510\) 0 0
\(511\) 2.00000 0.0884748
\(512\) 0 0
\(513\) 24.0000 1.05963
\(514\) 0 0
\(515\) 5.65685 0.249271
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −14.0000 −0.614532
\(520\) 0 0
\(521\) 6.00000 0.262865 0.131432 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) 0 0
\(523\) 29.6985 1.29862 0.649312 0.760522i \(-0.275057\pi\)
0.649312 + 0.760522i \(0.275057\pi\)
\(524\) 0 0
\(525\) 4.24264 0.185164
\(526\) 0 0
\(527\) 24.0000 1.04546
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) −1.41421 −0.0613716
\(532\) 0 0
\(533\) 25.4558 1.10262
\(534\) 0 0
\(535\) 8.00000 0.345870
\(536\) 0 0
\(537\) −8.00000 −0.345225
\(538\) 0 0
\(539\) −2.82843 −0.121829
\(540\) 0 0
\(541\) −16.9706 −0.729621 −0.364811 0.931082i \(-0.618866\pi\)
−0.364811 + 0.931082i \(0.618866\pi\)
\(542\) 0 0
\(543\) −18.0000 −0.772454
\(544\) 0 0
\(545\) −12.0000 −0.514024
\(546\) 0 0
\(547\) −8.48528 −0.362804 −0.181402 0.983409i \(-0.558064\pi\)
−0.181402 + 0.983409i \(0.558064\pi\)
\(548\) 0 0
\(549\) −12.7279 −0.543214
\(550\) 0 0
\(551\) −12.0000 −0.511217
\(552\) 0 0
\(553\) −8.00000 −0.340195
\(554\) 0 0
\(555\) −16.9706 −0.720360
\(556\) 0 0
\(557\) −5.65685 −0.239689 −0.119844 0.992793i \(-0.538240\pi\)
−0.119844 + 0.992793i \(0.538240\pi\)
\(558\) 0 0
\(559\) −36.0000 −1.52264
\(560\) 0 0
\(561\) 24.0000 1.01328
\(562\) 0 0
\(563\) −1.41421 −0.0596020 −0.0298010 0.999556i \(-0.509487\pi\)
−0.0298010 + 0.999556i \(0.509487\pi\)
\(564\) 0 0
\(565\) −16.9706 −0.713957
\(566\) 0 0
\(567\) 5.00000 0.209980
\(568\) 0 0
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) −25.4558 −1.06529 −0.532647 0.846338i \(-0.678803\pi\)
−0.532647 + 0.846338i \(0.678803\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −18.0000 −0.750652
\(576\) 0 0
\(577\) 38.0000 1.58196 0.790980 0.611842i \(-0.209571\pi\)
0.790980 + 0.611842i \(0.209571\pi\)
\(578\) 0 0
\(579\) −5.65685 −0.235091
\(580\) 0 0
\(581\) −15.5563 −0.645386
\(582\) 0 0
\(583\) −16.0000 −0.662652
\(584\) 0 0
\(585\) 6.00000 0.248069
\(586\) 0 0
\(587\) −41.0122 −1.69275 −0.846377 0.532584i \(-0.821221\pi\)
−0.846377 + 0.532584i \(0.821221\pi\)
\(588\) 0 0
\(589\) 16.9706 0.699260
\(590\) 0 0
\(591\) 8.00000 0.329076
\(592\) 0 0
\(593\) −30.0000 −1.23195 −0.615976 0.787765i \(-0.711238\pi\)
−0.615976 + 0.787765i \(0.711238\pi\)
\(594\) 0 0
\(595\) 8.48528 0.347863
\(596\) 0 0
\(597\) −28.2843 −1.15760
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 10.0000 0.407909 0.203954 0.978980i \(-0.434621\pi\)
0.203954 + 0.978980i \(0.434621\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −4.24264 −0.172488
\(606\) 0 0
\(607\) 32.0000 1.29884 0.649420 0.760430i \(-0.275012\pi\)
0.649420 + 0.760430i \(0.275012\pi\)
\(608\) 0 0
\(609\) −4.00000 −0.162088
\(610\) 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
\(615\) −12.0000 −0.483887
\(616\) 0 0
\(617\) −24.0000 −0.966204 −0.483102 0.875564i \(-0.660490\pi\)
−0.483102 + 0.875564i \(0.660490\pi\)
\(618\) 0 0
\(619\) 4.24264 0.170526 0.0852631 0.996358i \(-0.472827\pi\)
0.0852631 + 0.996358i \(0.472827\pi\)
\(620\) 0 0
\(621\) −33.9411 −1.36201
\(622\) 0 0
\(623\) 6.00000 0.240385
\(624\) 0 0
\(625\) −1.00000 −0.0400000
\(626\) 0 0
\(627\) 16.9706 0.677739
\(628\) 0 0
\(629\) 50.9117 2.02998
\(630\) 0 0
\(631\) 16.0000 0.636950 0.318475 0.947931i \(-0.396829\pi\)
0.318475 + 0.947931i \(0.396829\pi\)
\(632\) 0 0
\(633\) −24.0000 −0.953914
\(634\) 0 0
\(635\) 2.82843 0.112243
\(636\) 0 0
\(637\) −4.24264 −0.168100
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −48.0000 −1.89589 −0.947943 0.318440i \(-0.896841\pi\)
−0.947943 + 0.318440i \(0.896841\pi\)
\(642\) 0 0
\(643\) 21.2132 0.836567 0.418284 0.908317i \(-0.362632\pi\)
0.418284 + 0.908317i \(0.362632\pi\)
\(644\) 0 0
\(645\) 16.9706 0.668215
\(646\) 0 0
\(647\) −12.0000 −0.471769 −0.235884 0.971781i \(-0.575799\pi\)
−0.235884 + 0.971781i \(0.575799\pi\)
\(648\) 0 0
\(649\) −4.00000 −0.157014
\(650\) 0 0
\(651\) 5.65685 0.221710
\(652\) 0 0
\(653\) 36.7696 1.43890 0.719452 0.694542i \(-0.244393\pi\)
0.719452 + 0.694542i \(0.244393\pi\)
\(654\) 0 0
\(655\) −2.00000 −0.0781465
\(656\) 0 0
\(657\) 2.00000 0.0780274
\(658\) 0 0
\(659\) 2.82843 0.110180 0.0550899 0.998481i \(-0.482455\pi\)
0.0550899 + 0.998481i \(0.482455\pi\)
\(660\) 0 0
\(661\) 38.1838 1.48518 0.742588 0.669748i \(-0.233598\pi\)
0.742588 + 0.669748i \(0.233598\pi\)
\(662\) 0 0
\(663\) 36.0000 1.39812
\(664\) 0 0
\(665\) 6.00000 0.232670
\(666\) 0 0
\(667\) 16.9706 0.657103
\(668\) 0 0
\(669\) −39.5980 −1.53095
\(670\) 0 0
\(671\) −36.0000 −1.38976
\(672\) 0 0
\(673\) −46.0000 −1.77317 −0.886585 0.462566i \(-0.846929\pi\)
−0.886585 + 0.462566i \(0.846929\pi\)
\(674\) 0 0
\(675\) 16.9706 0.653197
\(676\) 0 0
\(677\) 9.89949 0.380468 0.190234 0.981739i \(-0.439075\pi\)
0.190234 + 0.981739i \(0.439075\pi\)
\(678\) 0 0
\(679\) 10.0000 0.383765
\(680\) 0 0
\(681\) −14.0000 −0.536481
\(682\) 0 0
\(683\) 5.65685 0.216454 0.108227 0.994126i \(-0.465483\pi\)
0.108227 + 0.994126i \(0.465483\pi\)
\(684\) 0 0
\(685\) −8.48528 −0.324206
\(686\) 0 0
\(687\) −6.00000 −0.228914
\(688\) 0 0
\(689\) −24.0000 −0.914327
\(690\) 0 0
\(691\) −12.7279 −0.484193 −0.242096 0.970252i \(-0.577835\pi\)
−0.242096 + 0.970252i \(0.577835\pi\)
\(692\) 0 0
\(693\) −2.82843 −0.107443
\(694\) 0 0
\(695\) 6.00000 0.227593
\(696\) 0 0
\(697\) 36.0000 1.36360
\(698\) 0 0
\(699\) −8.48528 −0.320943
\(700\) 0 0
\(701\) 19.7990 0.747798 0.373899 0.927470i \(-0.378021\pi\)
0.373899 + 0.927470i \(0.378021\pi\)
\(702\) 0 0
\(703\) 36.0000 1.35777
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −9.89949 −0.372309
\(708\) 0 0
\(709\) 25.4558 0.956014 0.478007 0.878356i \(-0.341359\pi\)
0.478007 + 0.878356i \(0.341359\pi\)
\(710\) 0 0
\(711\) −8.00000 −0.300023
\(712\) 0 0
\(713\) −24.0000 −0.898807
\(714\) 0 0
\(715\) 16.9706 0.634663
\(716\) 0 0
\(717\) 8.48528 0.316889
\(718\) 0 0
\(719\) 24.0000 0.895049 0.447524 0.894272i \(-0.352306\pi\)
0.447524 + 0.894272i \(0.352306\pi\)
\(720\) 0 0
\(721\) −4.00000 −0.148968
\(722\) 0 0
\(723\) −14.1421 −0.525952
\(724\) 0 0
\(725\) −8.48528 −0.315135
\(726\) 0 0
\(727\) −8.00000 −0.296704 −0.148352 0.988935i \(-0.547397\pi\)
−0.148352 + 0.988935i \(0.547397\pi\)
\(728\) 0 0
\(729\) 29.0000 1.07407
\(730\) 0 0
\(731\) −50.9117 −1.88304
\(732\) 0 0
\(733\) 29.6985 1.09694 0.548469 0.836171i \(-0.315211\pi\)
0.548469 + 0.836171i \(0.315211\pi\)
\(734\) 0 0
\(735\) 2.00000 0.0737711
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 42.4264 1.56068 0.780340 0.625355i \(-0.215046\pi\)
0.780340 + 0.625355i \(0.215046\pi\)
\(740\) 0 0
\(741\) 25.4558 0.935144
\(742\) 0 0
\(743\) 30.0000 1.10059 0.550297 0.834969i \(-0.314515\pi\)
0.550297 + 0.834969i \(0.314515\pi\)
\(744\) 0 0
\(745\) −16.0000 −0.586195
\(746\) 0 0
\(747\) −15.5563 −0.569177
\(748\) 0 0
\(749\) −5.65685 −0.206697
\(750\) 0 0
\(751\) −22.0000 −0.802791 −0.401396 0.915905i \(-0.631475\pi\)
−0.401396 + 0.915905i \(0.631475\pi\)
\(752\) 0 0
\(753\) 26.0000 0.947493
\(754\) 0 0
\(755\) 14.1421 0.514685
\(756\) 0 0
\(757\) −25.4558 −0.925208 −0.462604 0.886565i \(-0.653085\pi\)
−0.462604 + 0.886565i \(0.653085\pi\)
\(758\) 0 0
\(759\) −24.0000 −0.871145
\(760\) 0 0
\(761\) −30.0000 −1.08750 −0.543750 0.839248i \(-0.682996\pi\)
−0.543750 + 0.839248i \(0.682996\pi\)
\(762\) 0 0
\(763\) 8.48528 0.307188
\(764\) 0 0
\(765\) 8.48528 0.306786
\(766\) 0 0
\(767\) −6.00000 −0.216647
\(768\) 0 0
\(769\) 14.0000 0.504853 0.252426 0.967616i \(-0.418771\pi\)
0.252426 + 0.967616i \(0.418771\pi\)
\(770\) 0 0
\(771\) −8.48528 −0.305590
\(772\) 0 0
\(773\) −32.5269 −1.16991 −0.584956 0.811065i \(-0.698888\pi\)
−0.584956 + 0.811065i \(0.698888\pi\)
\(774\) 0 0
\(775\) 12.0000 0.431053
\(776\) 0 0
\(777\) 12.0000 0.430498
\(778\) 0 0
\(779\) 25.4558 0.912050
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −16.0000 −0.571793
\(784\) 0 0
\(785\) 18.0000 0.642448
\(786\) 0 0
\(787\) −38.1838 −1.36110 −0.680552 0.732700i \(-0.738260\pi\)
−0.680552 + 0.732700i \(0.738260\pi\)
\(788\) 0 0
\(789\) −33.9411 −1.20834
\(790\) 0 0
\(791\) 12.0000 0.426671
\(792\) 0 0
\(793\) −54.0000 −1.91760
\(794\) 0 0
\(795\) 11.3137 0.401256
\(796\) 0 0
\(797\) −26.8701 −0.951786 −0.475893 0.879503i \(-0.657875\pi\)
−0.475893 + 0.879503i \(0.657875\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 6.00000 0.212000
\(802\) 0 0
\(803\) 5.65685 0.199626
\(804\) 0 0
\(805\) −8.48528 −0.299067
\(806\) 0 0
\(807\) 10.0000 0.352017
\(808\) 0 0
\(809\) −12.0000 −0.421898 −0.210949 0.977497i \(-0.567655\pi\)
−0.210949 + 0.977497i \(0.567655\pi\)
\(810\) 0 0
\(811\) −12.7279 −0.446938 −0.223469 0.974711i \(-0.571738\pi\)
−0.223469 + 0.974711i \(0.571738\pi\)
\(812\) 0 0
\(813\) 28.2843 0.991973
\(814\) 0 0
\(815\) 12.0000 0.420342
\(816\) 0 0
\(817\) −36.0000 −1.25948
\(818\) 0 0
\(819\) −4.24264 −0.148250
\(820\) 0 0
\(821\) 22.6274 0.789702 0.394851 0.918745i \(-0.370796\pi\)
0.394851 + 0.918745i \(0.370796\pi\)
\(822\) 0 0
\(823\) −32.0000 −1.11545 −0.557725 0.830026i \(-0.688326\pi\)
−0.557725 + 0.830026i \(0.688326\pi\)
\(824\) 0 0
\(825\) 12.0000 0.417786
\(826\) 0 0
\(827\) −45.2548 −1.57366 −0.786832 0.617167i \(-0.788280\pi\)
−0.786832 + 0.617167i \(0.788280\pi\)
\(828\) 0 0
\(829\) −21.2132 −0.736765 −0.368383 0.929674i \(-0.620088\pi\)
−0.368383 + 0.929674i \(0.620088\pi\)
\(830\) 0 0
\(831\) 24.0000 0.832551
\(832\) 0 0
\(833\) −6.00000 −0.207888
\(834\) 0 0
\(835\) −33.9411 −1.17458
\(836\) 0 0
\(837\) 22.6274 0.782118
\(838\) 0 0
\(839\) −12.0000 −0.414286 −0.207143 0.978311i \(-0.566417\pi\)
−0.207143 + 0.978311i \(0.566417\pi\)
\(840\) 0 0
\(841\) −21.0000 −0.724138
\(842\) 0 0
\(843\) 8.48528 0.292249
\(844\) 0 0
\(845\) 7.07107 0.243252
\(846\) 0 0
\(847\) 3.00000 0.103081
\(848\) 0 0
\(849\) −18.0000 −0.617758
\(850\) 0 0
\(851\) −50.9117 −1.74523
\(852\) 0 0
\(853\) −4.24264 −0.145265 −0.0726326 0.997359i \(-0.523140\pi\)
−0.0726326 + 0.997359i \(0.523140\pi\)
\(854\) 0 0
\(855\) 6.00000 0.205196
\(856\) 0 0
\(857\) −42.0000 −1.43469 −0.717346 0.696717i \(-0.754643\pi\)
−0.717346 + 0.696717i \(0.754643\pi\)
\(858\) 0 0
\(859\) 46.6690 1.59233 0.796164 0.605081i \(-0.206859\pi\)
0.796164 + 0.605081i \(0.206859\pi\)
\(860\) 0 0
\(861\) 8.48528 0.289178
\(862\) 0 0
\(863\) 24.0000 0.816970 0.408485 0.912765i \(-0.366057\pi\)
0.408485 + 0.912765i \(0.366057\pi\)
\(864\) 0 0
\(865\) −14.0000 −0.476014
\(866\) 0 0
\(867\) 26.8701 0.912555
\(868\) 0 0
\(869\) −22.6274 −0.767583
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 10.0000 0.338449
\(874\) 0 0
\(875\) 11.3137 0.382473
\(876\) 0 0
\(877\) 42.4264 1.43264 0.716319 0.697773i \(-0.245826\pi\)
0.716319 + 0.697773i \(0.245826\pi\)
\(878\) 0 0
\(879\) −34.0000 −1.14679
\(880\) 0 0
\(881\) −18.0000 −0.606435 −0.303218 0.952921i \(-0.598061\pi\)
−0.303218 + 0.952921i \(0.598061\pi\)
\(882\) 0 0
\(883\) −50.9117 −1.71331 −0.856657 0.515886i \(-0.827463\pi\)
−0.856657 + 0.515886i \(0.827463\pi\)
\(884\) 0 0
\(885\) 2.82843 0.0950765
\(886\) 0 0
\(887\) −48.0000 −1.61168 −0.805841 0.592132i \(-0.798286\pi\)
−0.805841 + 0.592132i \(0.798286\pi\)
\(888\) 0 0
\(889\) −2.00000 −0.0670778
\(890\) 0 0
\(891\) 14.1421 0.473779
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) −8.00000 −0.267411
\(896\) 0 0
\(897\) −36.0000 −1.20201
\(898\) 0 0
\(899\) −11.3137 −0.377333
\(900\) 0 0
\(901\) −33.9411 −1.13074
\(902\) 0 0
\(903\) −12.0000 −0.399335
\(904\) 0 0
\(905\) −18.0000 −0.598340
\(906\) 0 0
\(907\) −33.9411 −1.12700 −0.563498 0.826117i \(-0.690545\pi\)
−0.563498 + 0.826117i \(0.690545\pi\)
\(908\) 0 0
\(909\) −9.89949 −0.328346
\(910\) 0 0
\(911\) 30.0000 0.993944 0.496972 0.867766i \(-0.334445\pi\)
0.496972 + 0.867766i \(0.334445\pi\)
\(912\) 0 0
\(913\) −44.0000 −1.45619
\(914\) 0 0
\(915\) 25.4558 0.841544
\(916\) 0 0
\(917\) 1.41421 0.0467014
\(918\) 0 0
\(919\) 16.0000 0.527791 0.263896 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) 0 0
\(921\) −18.0000 −0.593120
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 25.4558 0.836983
\(926\) 0 0
\(927\) −4.00000 −0.131377
\(928\) 0 0
\(929\) 18.0000 0.590561 0.295280 0.955411i \(-0.404587\pi\)
0.295280 + 0.955411i \(0.404587\pi\)
\(930\) 0 0
\(931\) −4.24264 −0.139047
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 24.0000 0.784884
\(936\) 0 0
\(937\) −2.00000 −0.0653372 −0.0326686 0.999466i \(-0.510401\pi\)
−0.0326686 + 0.999466i \(0.510401\pi\)
\(938\) 0 0
\(939\) 14.1421 0.461511
\(940\) 0 0
\(941\) −52.3259 −1.70578 −0.852888 0.522094i \(-0.825151\pi\)
−0.852888 + 0.522094i \(0.825151\pi\)
\(942\) 0 0
\(943\) −36.0000 −1.17232
\(944\) 0 0
\(945\) 8.00000 0.260240
\(946\) 0 0
\(947\) 2.82843 0.0919115 0.0459558 0.998943i \(-0.485367\pi\)
0.0459558 + 0.998943i \(0.485367\pi\)
\(948\) 0 0
\(949\) 8.48528 0.275444
\(950\) 0 0
\(951\) −32.0000 −1.03767
\(952\) 0 0
\(953\) −18.0000 −0.583077 −0.291539 0.956559i \(-0.594167\pi\)
−0.291539 + 0.956559i \(0.594167\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −11.3137 −0.365720
\(958\) 0 0
\(959\) 6.00000 0.193750
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) −5.65685 −0.182290
\(964\) 0 0
\(965\) −5.65685 −0.182101
\(966\) 0 0
\(967\) 22.0000 0.707472 0.353736 0.935345i \(-0.384911\pi\)
0.353736 + 0.935345i \(0.384911\pi\)
\(968\) 0 0
\(969\) 36.0000 1.15649
\(970\) 0 0
\(971\) −32.5269 −1.04384 −0.521919 0.852995i \(-0.674784\pi\)
−0.521919 + 0.852995i \(0.674784\pi\)
\(972\) 0 0
\(973\) −4.24264 −0.136013
\(974\) 0 0
\(975\) 18.0000 0.576461
\(976\) 0 0
\(977\) −18.0000 −0.575871 −0.287936 0.957650i \(-0.592969\pi\)
−0.287936 + 0.957650i \(0.592969\pi\)
\(978\) 0 0
\(979\) 16.9706 0.542382
\(980\) 0 0
\(981\) 8.48528 0.270914
\(982\) 0 0
\(983\) 12.0000 0.382741 0.191370 0.981518i \(-0.438707\pi\)
0.191370 + 0.981518i \(0.438707\pi\)
\(984\) 0 0
\(985\) 8.00000 0.254901
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 50.9117 1.61890
\(990\) 0 0
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) 0 0
\(993\) −36.0000 −1.14243
\(994\) 0 0
\(995\) −28.2843 −0.896672
\(996\) 0 0
\(997\) −21.2132 −0.671829 −0.335914 0.941893i \(-0.609045\pi\)
−0.335914 + 0.941893i \(0.609045\pi\)
\(998\) 0 0
\(999\) 48.0000 1.51865
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.n.1.2 2
4.3 odd 2 1792.2.a.p.1.1 2
8.3 odd 2 1792.2.a.p.1.2 2
8.5 even 2 inner 1792.2.a.n.1.1 2
16.3 odd 4 224.2.b.a.113.1 2
16.5 even 4 56.2.b.a.29.1 2
16.11 odd 4 224.2.b.a.113.2 2
16.13 even 4 56.2.b.a.29.2 yes 2
48.5 odd 4 504.2.c.a.253.2 2
48.11 even 4 2016.2.c.a.1009.1 2
48.29 odd 4 504.2.c.a.253.1 2
48.35 even 4 2016.2.c.a.1009.2 2
112.3 even 12 1568.2.t.b.177.2 4
112.5 odd 12 392.2.p.b.165.1 4
112.11 odd 12 1568.2.t.c.177.2 4
112.13 odd 4 392.2.b.b.197.2 2
112.19 even 12 1568.2.t.b.753.1 4
112.27 even 4 1568.2.b.a.785.1 2
112.37 even 12 392.2.p.a.165.1 4
112.45 odd 12 392.2.p.b.373.1 4
112.51 odd 12 1568.2.t.c.753.2 4
112.53 even 12 392.2.p.a.373.2 4
112.59 even 12 1568.2.t.b.177.1 4
112.61 odd 12 392.2.p.b.165.2 4
112.67 odd 12 1568.2.t.c.177.1 4
112.69 odd 4 392.2.b.b.197.1 2
112.75 even 12 1568.2.t.b.753.2 4
112.83 even 4 1568.2.b.a.785.2 2
112.93 even 12 392.2.p.a.165.2 4
112.101 odd 12 392.2.p.b.373.2 4
112.107 odd 12 1568.2.t.c.753.1 4
112.109 even 12 392.2.p.a.373.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.2.b.a.29.1 2 16.5 even 4
56.2.b.a.29.2 yes 2 16.13 even 4
224.2.b.a.113.1 2 16.3 odd 4
224.2.b.a.113.2 2 16.11 odd 4
392.2.b.b.197.1 2 112.69 odd 4
392.2.b.b.197.2 2 112.13 odd 4
392.2.p.a.165.1 4 112.37 even 12
392.2.p.a.165.2 4 112.93 even 12
392.2.p.a.373.1 4 112.109 even 12
392.2.p.a.373.2 4 112.53 even 12
392.2.p.b.165.1 4 112.5 odd 12
392.2.p.b.165.2 4 112.61 odd 12
392.2.p.b.373.1 4 112.45 odd 12
392.2.p.b.373.2 4 112.101 odd 12
504.2.c.a.253.1 2 48.29 odd 4
504.2.c.a.253.2 2 48.5 odd 4
1568.2.b.a.785.1 2 112.27 even 4
1568.2.b.a.785.2 2 112.83 even 4
1568.2.t.b.177.1 4 112.59 even 12
1568.2.t.b.177.2 4 112.3 even 12
1568.2.t.b.753.1 4 112.19 even 12
1568.2.t.b.753.2 4 112.75 even 12
1568.2.t.c.177.1 4 112.67 odd 12
1568.2.t.c.177.2 4 112.11 odd 12
1568.2.t.c.753.1 4 112.107 odd 12
1568.2.t.c.753.2 4 112.51 odd 12
1792.2.a.n.1.1 2 8.5 even 2 inner
1792.2.a.n.1.2 2 1.1 even 1 trivial
1792.2.a.p.1.1 2 4.3 odd 2
1792.2.a.p.1.2 2 8.3 odd 2
2016.2.c.a.1009.1 2 48.11 even 4
2016.2.c.a.1009.2 2 48.35 even 4