Properties

Label 256.2.b.c.129.2
Level $256$
Weight $2$
Character 256.129
Analytic conductor $2.044$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 256 = 2^{8} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 256.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.04417029174\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 128)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 129.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 256.129
Dual form 256.2.b.c.129.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.00000i q^{3} -2.00000i q^{5} +4.00000 q^{7} -1.00000 q^{9} +O(q^{10})\) \(q+2.00000i q^{3} -2.00000i q^{5} +4.00000 q^{7} -1.00000 q^{9} +2.00000i q^{11} +2.00000i q^{13} +4.00000 q^{15} -2.00000 q^{17} +2.00000i q^{19} +8.00000i q^{21} -4.00000 q^{23} +1.00000 q^{25} +4.00000i q^{27} -6.00000i q^{29} -4.00000 q^{33} -8.00000i q^{35} -10.0000i q^{37} -4.00000 q^{39} +6.00000 q^{41} -6.00000i q^{43} +2.00000i q^{45} -8.00000 q^{47} +9.00000 q^{49} -4.00000i q^{51} +6.00000i q^{53} +4.00000 q^{55} -4.00000 q^{57} -14.0000i q^{59} +2.00000i q^{61} -4.00000 q^{63} +4.00000 q^{65} +10.0000i q^{67} -8.00000i q^{69} -12.0000 q^{71} -14.0000 q^{73} +2.00000i q^{75} +8.00000i q^{77} -8.00000 q^{79} -11.0000 q^{81} -6.00000i q^{83} +4.00000i q^{85} +12.0000 q^{87} +2.00000 q^{89} +8.00000i q^{91} +4.00000 q^{95} -2.00000 q^{97} -2.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 8q^{7} - 2q^{9} + O(q^{10}) \) \( 2q + 8q^{7} - 2q^{9} + 8q^{15} - 4q^{17} - 8q^{23} + 2q^{25} - 8q^{33} - 8q^{39} + 12q^{41} - 16q^{47} + 18q^{49} + 8q^{55} - 8q^{57} - 8q^{63} + 8q^{65} - 24q^{71} - 28q^{73} - 16q^{79} - 22q^{81} + 24q^{87} + 4q^{89} + 8q^{95} - 4q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/256\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(255\)
\(\chi(n)\) \(-1\) \(1\)

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
\(3\) 2.00000i 1.15470i 0.816497 + 0.577350i \(0.195913\pi\)
−0.816497 + 0.577350i \(0.804087\pi\)
\(4\) 0 0
\(5\) − 2.00000i − 0.894427i −0.894427 0.447214i \(-0.852416\pi\)
0.894427 0.447214i \(-0.147584\pi\)
\(6\) 0 0
\(7\) 4.00000 1.51186 0.755929 0.654654i \(-0.227186\pi\)
0.755929 + 0.654654i \(0.227186\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 2.00000i 0.603023i 0.953463 + 0.301511i \(0.0974911\pi\)
−0.953463 + 0.301511i \(0.902509\pi\)
\(12\) 0 0
\(13\) 2.00000i 0.554700i 0.960769 + 0.277350i \(0.0894562\pi\)
−0.960769 + 0.277350i \(0.910544\pi\)
\(14\) 0 0
\(15\) 4.00000 1.03280
\(16\) 0 0
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) 2.00000i 0.458831i 0.973329 + 0.229416i \(0.0736815\pi\)
−0.973329 + 0.229416i \(0.926318\pi\)
\(20\) 0 0
\(21\) 8.00000i 1.74574i
\(22\) 0 0
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 4.00000i 0.769800i
\(28\) 0 0
\(29\) − 6.00000i − 1.11417i −0.830455 0.557086i \(-0.811919\pi\)
0.830455 0.557086i \(-0.188081\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) −4.00000 −0.696311
\(34\) 0 0
\(35\) − 8.00000i − 1.35225i
\(36\) 0 0
\(37\) − 10.0000i − 1.64399i −0.569495 0.821995i \(-0.692861\pi\)
0.569495 0.821995i \(-0.307139\pi\)
\(38\) 0 0
\(39\) −4.00000 −0.640513
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) − 6.00000i − 0.914991i −0.889212 0.457496i \(-0.848747\pi\)
0.889212 0.457496i \(-0.151253\pi\)
\(44\) 0 0
\(45\) 2.00000i 0.298142i
\(46\) 0 0
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 0 0
\(49\) 9.00000 1.28571
\(50\) 0 0
\(51\) − 4.00000i − 0.560112i
\(52\) 0 0
\(53\) 6.00000i 0.824163i 0.911147 + 0.412082i \(0.135198\pi\)
−0.911147 + 0.412082i \(0.864802\pi\)
\(54\) 0 0
\(55\) 4.00000 0.539360
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) 0 0
\(59\) − 14.0000i − 1.82264i −0.411693 0.911322i \(-0.635063\pi\)
0.411693 0.911322i \(-0.364937\pi\)
\(60\) 0 0
\(61\) 2.00000i 0.256074i 0.991769 + 0.128037i \(0.0408676\pi\)
−0.991769 + 0.128037i \(0.959132\pi\)
\(62\) 0 0
\(63\) −4.00000 −0.503953
\(64\) 0 0
\(65\) 4.00000 0.496139
\(66\) 0 0
\(67\) 10.0000i 1.22169i 0.791748 + 0.610847i \(0.209171\pi\)
−0.791748 + 0.610847i \(0.790829\pi\)
\(68\) 0 0
\(69\) − 8.00000i − 0.963087i
\(70\) 0 0
\(71\) −12.0000 −1.42414 −0.712069 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 0 0
\(73\) −14.0000 −1.63858 −0.819288 0.573382i \(-0.805631\pi\)
−0.819288 + 0.573382i \(0.805631\pi\)
\(74\) 0 0
\(75\) 2.00000i 0.230940i
\(76\) 0 0
\(77\) 8.00000i 0.911685i
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) − 6.00000i − 0.658586i −0.944228 0.329293i \(-0.893190\pi\)
0.944228 0.329293i \(-0.106810\pi\)
\(84\) 0 0
\(85\) 4.00000i 0.433861i
\(86\) 0 0
\(87\) 12.0000 1.28654
\(88\) 0 0
\(89\) 2.00000 0.212000 0.106000 0.994366i \(-0.466196\pi\)
0.106000 + 0.994366i \(0.466196\pi\)
\(90\) 0 0
\(91\) 8.00000i 0.838628i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 4.00000 0.410391
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 0 0
\(99\) − 2.00000i − 0.201008i
\(100\) 0 0
\(101\) 6.00000i 0.597022i 0.954406 + 0.298511i \(0.0964900\pi\)
−0.954406 + 0.298511i \(0.903510\pi\)
\(102\) 0 0
\(103\) 4.00000 0.394132 0.197066 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) 0 0
\(105\) 16.0000 1.56144
\(106\) 0 0
\(107\) 2.00000i 0.193347i 0.995316 + 0.0966736i \(0.0308203\pi\)
−0.995316 + 0.0966736i \(0.969180\pi\)
\(108\) 0 0
\(109\) − 6.00000i − 0.574696i −0.957826 0.287348i \(-0.907226\pi\)
0.957826 0.287348i \(-0.0927736\pi\)
\(110\) 0 0
\(111\) 20.0000 1.89832
\(112\) 0 0
\(113\) 2.00000 0.188144 0.0940721 0.995565i \(-0.470012\pi\)
0.0940721 + 0.995565i \(0.470012\pi\)
\(114\) 0 0
\(115\) 8.00000i 0.746004i
\(116\) 0 0
\(117\) − 2.00000i − 0.184900i
\(118\) 0 0
\(119\) −8.00000 −0.733359
\(120\) 0 0
\(121\) 7.00000 0.636364
\(122\) 0 0
\(123\) 12.0000i 1.08200i
\(124\) 0 0
\(125\) − 12.0000i − 1.07331i
\(126\) 0 0
\(127\) 16.0000 1.41977 0.709885 0.704317i \(-0.248747\pi\)
0.709885 + 0.704317i \(0.248747\pi\)
\(128\) 0 0
\(129\) 12.0000 1.05654
\(130\) 0 0
\(131\) − 6.00000i − 0.524222i −0.965038 0.262111i \(-0.915581\pi\)
0.965038 0.262111i \(-0.0844187\pi\)
\(132\) 0 0
\(133\) 8.00000i 0.693688i
\(134\) 0 0
\(135\) 8.00000 0.688530
\(136\) 0 0
\(137\) −10.0000 −0.854358 −0.427179 0.904167i \(-0.640493\pi\)
−0.427179 + 0.904167i \(0.640493\pi\)
\(138\) 0 0
\(139\) 10.0000i 0.848189i 0.905618 + 0.424094i \(0.139408\pi\)
−0.905618 + 0.424094i \(0.860592\pi\)
\(140\) 0 0
\(141\) − 16.0000i − 1.34744i
\(142\) 0 0
\(143\) −4.00000 −0.334497
\(144\) 0 0
\(145\) −12.0000 −0.996546
\(146\) 0 0
\(147\) 18.0000i 1.48461i
\(148\) 0 0
\(149\) − 18.0000i − 1.47462i −0.675556 0.737309i \(-0.736096\pi\)
0.675556 0.737309i \(-0.263904\pi\)
\(150\) 0 0
\(151\) −4.00000 −0.325515 −0.162758 0.986666i \(-0.552039\pi\)
−0.162758 + 0.986666i \(0.552039\pi\)
\(152\) 0 0
\(153\) 2.00000 0.161690
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 18.0000i 1.43656i 0.695756 + 0.718278i \(0.255069\pi\)
−0.695756 + 0.718278i \(0.744931\pi\)
\(158\) 0 0
\(159\) −12.0000 −0.951662
\(160\) 0 0
\(161\) −16.0000 −1.26098
\(162\) 0 0
\(163\) 2.00000i 0.156652i 0.996928 + 0.0783260i \(0.0249575\pi\)
−0.996928 + 0.0783260i \(0.975042\pi\)
\(164\) 0 0
\(165\) 8.00000i 0.622799i
\(166\) 0 0
\(167\) 20.0000 1.54765 0.773823 0.633402i \(-0.218342\pi\)
0.773823 + 0.633402i \(0.218342\pi\)
\(168\) 0 0
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) − 2.00000i − 0.152944i
\(172\) 0 0
\(173\) 18.0000i 1.36851i 0.729241 + 0.684257i \(0.239873\pi\)
−0.729241 + 0.684257i \(0.760127\pi\)
\(174\) 0 0
\(175\) 4.00000 0.302372
\(176\) 0 0
\(177\) 28.0000 2.10461
\(178\) 0 0
\(179\) − 6.00000i − 0.448461i −0.974536 0.224231i \(-0.928013\pi\)
0.974536 0.224231i \(-0.0719869\pi\)
\(180\) 0 0
\(181\) − 2.00000i − 0.148659i −0.997234 0.0743294i \(-0.976318\pi\)
0.997234 0.0743294i \(-0.0236816\pi\)
\(182\) 0 0
\(183\) −4.00000 −0.295689
\(184\) 0 0
\(185\) −20.0000 −1.47043
\(186\) 0 0
\(187\) − 4.00000i − 0.292509i
\(188\) 0 0
\(189\) 16.0000i 1.16383i
\(190\) 0 0
\(191\) −16.0000 −1.15772 −0.578860 0.815427i \(-0.696502\pi\)
−0.578860 + 0.815427i \(0.696502\pi\)
\(192\) 0 0
\(193\) −2.00000 −0.143963 −0.0719816 0.997406i \(-0.522932\pi\)
−0.0719816 + 0.997406i \(0.522932\pi\)
\(194\) 0 0
\(195\) 8.00000i 0.572892i
\(196\) 0 0
\(197\) 14.0000i 0.997459i 0.866758 + 0.498729i \(0.166200\pi\)
−0.866758 + 0.498729i \(0.833800\pi\)
\(198\) 0 0
\(199\) 4.00000 0.283552 0.141776 0.989899i \(-0.454719\pi\)
0.141776 + 0.989899i \(0.454719\pi\)
\(200\) 0 0
\(201\) −20.0000 −1.41069
\(202\) 0 0
\(203\) − 24.0000i − 1.68447i
\(204\) 0 0
\(205\) − 12.0000i − 0.838116i
\(206\) 0 0
\(207\) 4.00000 0.278019
\(208\) 0 0
\(209\) −4.00000 −0.276686
\(210\) 0 0
\(211\) − 22.0000i − 1.51454i −0.653101 0.757271i \(-0.726532\pi\)
0.653101 0.757271i \(-0.273468\pi\)
\(212\) 0 0
\(213\) − 24.0000i − 1.64445i
\(214\) 0 0
\(215\) −12.0000 −0.818393
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) − 28.0000i − 1.89206i
\(220\) 0 0
\(221\) − 4.00000i − 0.269069i
\(222\) 0 0
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) −1.00000 −0.0666667
\(226\) 0 0
\(227\) 18.0000i 1.19470i 0.801980 + 0.597351i \(0.203780\pi\)
−0.801980 + 0.597351i \(0.796220\pi\)
\(228\) 0 0
\(229\) 14.0000i 0.925146i 0.886581 + 0.462573i \(0.153074\pi\)
−0.886581 + 0.462573i \(0.846926\pi\)
\(230\) 0 0
\(231\) −16.0000 −1.05272
\(232\) 0 0
\(233\) 18.0000 1.17922 0.589610 0.807688i \(-0.299282\pi\)
0.589610 + 0.807688i \(0.299282\pi\)
\(234\) 0 0
\(235\) 16.0000i 1.04372i
\(236\) 0 0
\(237\) − 16.0000i − 1.03931i
\(238\) 0 0
\(239\) −24.0000 −1.55243 −0.776215 0.630468i \(-0.782863\pi\)
−0.776215 + 0.630468i \(0.782863\pi\)
\(240\) 0 0
\(241\) −2.00000 −0.128831 −0.0644157 0.997923i \(-0.520518\pi\)
−0.0644157 + 0.997923i \(0.520518\pi\)
\(242\) 0 0
\(243\) − 10.0000i − 0.641500i
\(244\) 0 0
\(245\) − 18.0000i − 1.14998i
\(246\) 0 0
\(247\) −4.00000 −0.254514
\(248\) 0 0
\(249\) 12.0000 0.760469
\(250\) 0 0
\(251\) 18.0000i 1.13615i 0.822977 + 0.568075i \(0.192312\pi\)
−0.822977 + 0.568075i \(0.807688\pi\)
\(252\) 0 0
\(253\) − 8.00000i − 0.502956i
\(254\) 0 0
\(255\) −8.00000 −0.500979
\(256\) 0 0
\(257\) 18.0000 1.12281 0.561405 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) 0 0
\(259\) − 40.0000i − 2.48548i
\(260\) 0 0
\(261\) 6.00000i 0.371391i
\(262\) 0 0
\(263\) −12.0000 −0.739952 −0.369976 0.929041i \(-0.620634\pi\)
−0.369976 + 0.929041i \(0.620634\pi\)
\(264\) 0 0
\(265\) 12.0000 0.737154
\(266\) 0 0
\(267\) 4.00000i 0.244796i
\(268\) 0 0
\(269\) 10.0000i 0.609711i 0.952399 + 0.304855i \(0.0986081\pi\)
−0.952399 + 0.304855i \(0.901392\pi\)
\(270\) 0 0
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) 0 0
\(273\) −16.0000 −0.968364
\(274\) 0 0
\(275\) 2.00000i 0.120605i
\(276\) 0 0
\(277\) 6.00000i 0.360505i 0.983620 + 0.180253i \(0.0576915\pi\)
−0.983620 + 0.180253i \(0.942309\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 18.0000 1.07379 0.536895 0.843649i \(-0.319597\pi\)
0.536895 + 0.843649i \(0.319597\pi\)
\(282\) 0 0
\(283\) − 6.00000i − 0.356663i −0.983970 0.178331i \(-0.942930\pi\)
0.983970 0.178331i \(-0.0570699\pi\)
\(284\) 0 0
\(285\) 8.00000i 0.473879i
\(286\) 0 0
\(287\) 24.0000 1.41668
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) − 4.00000i − 0.234484i
\(292\) 0 0
\(293\) 14.0000i 0.817889i 0.912559 + 0.408944i \(0.134103\pi\)
−0.912559 + 0.408944i \(0.865897\pi\)
\(294\) 0 0
\(295\) −28.0000 −1.63022
\(296\) 0 0
\(297\) −8.00000 −0.464207
\(298\) 0 0
\(299\) − 8.00000i − 0.462652i
\(300\) 0 0
\(301\) − 24.0000i − 1.38334i
\(302\) 0 0
\(303\) −12.0000 −0.689382
\(304\) 0 0
\(305\) 4.00000 0.229039
\(306\) 0 0
\(307\) 18.0000i 1.02731i 0.857996 + 0.513657i \(0.171710\pi\)
−0.857996 + 0.513657i \(0.828290\pi\)
\(308\) 0 0
\(309\) 8.00000i 0.455104i
\(310\) 0 0
\(311\) 28.0000 1.58773 0.793867 0.608091i \(-0.208065\pi\)
0.793867 + 0.608091i \(0.208065\pi\)
\(312\) 0 0
\(313\) −10.0000 −0.565233 −0.282617 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) 0 0
\(315\) 8.00000i 0.450749i
\(316\) 0 0
\(317\) − 6.00000i − 0.336994i −0.985702 0.168497i \(-0.946109\pi\)
0.985702 0.168497i \(-0.0538913\pi\)
\(318\) 0 0
\(319\) 12.0000 0.671871
\(320\) 0 0
\(321\) −4.00000 −0.223258
\(322\) 0 0
\(323\) − 4.00000i − 0.222566i
\(324\) 0 0
\(325\) 2.00000i 0.110940i
\(326\) 0 0
\(327\) 12.0000 0.663602
\(328\) 0 0
\(329\) −32.0000 −1.76422
\(330\) 0 0
\(331\) − 14.0000i − 0.769510i −0.923019 0.384755i \(-0.874286\pi\)
0.923019 0.384755i \(-0.125714\pi\)
\(332\) 0 0
\(333\) 10.0000i 0.547997i
\(334\) 0 0
\(335\) 20.0000 1.09272
\(336\) 0 0
\(337\) −14.0000 −0.762629 −0.381314 0.924445i \(-0.624528\pi\)
−0.381314 + 0.924445i \(0.624528\pi\)
\(338\) 0 0
\(339\) 4.00000i 0.217250i
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 8.00000 0.431959
\(344\) 0 0
\(345\) −16.0000 −0.861411
\(346\) 0 0
\(347\) 18.0000i 0.966291i 0.875540 + 0.483145i \(0.160506\pi\)
−0.875540 + 0.483145i \(0.839494\pi\)
\(348\) 0 0
\(349\) 10.0000i 0.535288i 0.963518 + 0.267644i \(0.0862451\pi\)
−0.963518 + 0.267644i \(0.913755\pi\)
\(350\) 0 0
\(351\) −8.00000 −0.427008
\(352\) 0 0
\(353\) 18.0000 0.958043 0.479022 0.877803i \(-0.340992\pi\)
0.479022 + 0.877803i \(0.340992\pi\)
\(354\) 0 0
\(355\) 24.0000i 1.27379i
\(356\) 0 0
\(357\) − 16.0000i − 0.846810i
\(358\) 0 0
\(359\) 4.00000 0.211112 0.105556 0.994413i \(-0.466338\pi\)
0.105556 + 0.994413i \(0.466338\pi\)
\(360\) 0 0
\(361\) 15.0000 0.789474
\(362\) 0 0
\(363\) 14.0000i 0.734809i
\(364\) 0 0
\(365\) 28.0000i 1.46559i
\(366\) 0 0
\(367\) −8.00000 −0.417597 −0.208798 0.977959i \(-0.566955\pi\)
−0.208798 + 0.977959i \(0.566955\pi\)
\(368\) 0 0
\(369\) −6.00000 −0.312348
\(370\) 0 0
\(371\) 24.0000i 1.24602i
\(372\) 0 0
\(373\) − 10.0000i − 0.517780i −0.965907 0.258890i \(-0.916643\pi\)
0.965907 0.258890i \(-0.0833568\pi\)
\(374\) 0 0
\(375\) 24.0000 1.23935
\(376\) 0 0
\(377\) 12.0000 0.618031
\(378\) 0 0
\(379\) 2.00000i 0.102733i 0.998680 + 0.0513665i \(0.0163577\pi\)
−0.998680 + 0.0513665i \(0.983642\pi\)
\(380\) 0 0
\(381\) 32.0000i 1.63941i
\(382\) 0 0
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 16.0000 0.815436
\(386\) 0 0
\(387\) 6.00000i 0.304997i
\(388\) 0 0
\(389\) − 10.0000i − 0.507020i −0.967333 0.253510i \(-0.918415\pi\)
0.967333 0.253510i \(-0.0815851\pi\)
\(390\) 0 0
\(391\) 8.00000 0.404577
\(392\) 0 0
\(393\) 12.0000 0.605320
\(394\) 0 0
\(395\) 16.0000i 0.805047i
\(396\) 0 0
\(397\) − 6.00000i − 0.301131i −0.988600 0.150566i \(-0.951890\pi\)
0.988600 0.150566i \(-0.0481095\pi\)
\(398\) 0 0
\(399\) −16.0000 −0.801002
\(400\) 0 0
\(401\) 30.0000 1.49813 0.749064 0.662497i \(-0.230503\pi\)
0.749064 + 0.662497i \(0.230503\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 22.0000i 1.09319i
\(406\) 0 0
\(407\) 20.0000 0.991363
\(408\) 0 0
\(409\) −10.0000 −0.494468 −0.247234 0.968956i \(-0.579522\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) 0 0
\(411\) − 20.0000i − 0.986527i
\(412\) 0 0
\(413\) − 56.0000i − 2.75558i
\(414\) 0 0
\(415\) −12.0000 −0.589057
\(416\) 0 0
\(417\) −20.0000 −0.979404
\(418\) 0 0
\(419\) 26.0000i 1.27018i 0.772437 + 0.635092i \(0.219038\pi\)
−0.772437 + 0.635092i \(0.780962\pi\)
\(420\) 0 0
\(421\) − 34.0000i − 1.65706i −0.559946 0.828529i \(-0.689178\pi\)
0.559946 0.828529i \(-0.310822\pi\)
\(422\) 0 0
\(423\) 8.00000 0.388973
\(424\) 0 0
\(425\) −2.00000 −0.0970143
\(426\) 0 0
\(427\) 8.00000i 0.387147i
\(428\) 0 0
\(429\) − 8.00000i − 0.386244i
\(430\) 0 0
\(431\) 40.0000 1.92673 0.963366 0.268190i \(-0.0864254\pi\)
0.963366 + 0.268190i \(0.0864254\pi\)
\(432\) 0 0
\(433\) 30.0000 1.44171 0.720854 0.693087i \(-0.243750\pi\)
0.720854 + 0.693087i \(0.243750\pi\)
\(434\) 0 0
\(435\) − 24.0000i − 1.15071i
\(436\) 0 0
\(437\) − 8.00000i − 0.382692i
\(438\) 0 0
\(439\) −36.0000 −1.71819 −0.859093 0.511819i \(-0.828972\pi\)
−0.859093 + 0.511819i \(0.828972\pi\)
\(440\) 0 0
\(441\) −9.00000 −0.428571
\(442\) 0 0
\(443\) − 6.00000i − 0.285069i −0.989790 0.142534i \(-0.954475\pi\)
0.989790 0.142534i \(-0.0455251\pi\)
\(444\) 0 0
\(445\) − 4.00000i − 0.189618i
\(446\) 0 0
\(447\) 36.0000 1.70274
\(448\) 0 0
\(449\) −34.0000 −1.60456 −0.802280 0.596948i \(-0.796380\pi\)
−0.802280 + 0.596948i \(0.796380\pi\)
\(450\) 0 0
\(451\) 12.0000i 0.565058i
\(452\) 0 0
\(453\) − 8.00000i − 0.375873i
\(454\) 0 0
\(455\) 16.0000 0.750092
\(456\) 0 0
\(457\) 6.00000 0.280668 0.140334 0.990104i \(-0.455182\pi\)
0.140334 + 0.990104i \(0.455182\pi\)
\(458\) 0 0
\(459\) − 8.00000i − 0.373408i
\(460\) 0 0
\(461\) 10.0000i 0.465746i 0.972507 + 0.232873i \(0.0748127\pi\)
−0.972507 + 0.232873i \(0.925187\pi\)
\(462\) 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
\(467\) − 14.0000i − 0.647843i −0.946084 0.323921i \(-0.894999\pi\)
0.946084 0.323921i \(-0.105001\pi\)
\(468\) 0 0
\(469\) 40.0000i 1.84703i
\(470\) 0 0
\(471\) −36.0000 −1.65879
\(472\) 0 0
\(473\) 12.0000 0.551761
\(474\) 0 0
\(475\) 2.00000i 0.0917663i
\(476\) 0 0
\(477\) − 6.00000i − 0.274721i
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 20.0000 0.911922
\(482\) 0 0
\(483\) − 32.0000i − 1.45605i
\(484\) 0 0
\(485\) 4.00000i 0.181631i
\(486\) 0 0
\(487\) 20.0000 0.906287 0.453143 0.891438i \(-0.350303\pi\)
0.453143 + 0.891438i \(0.350303\pi\)
\(488\) 0 0
\(489\) −4.00000 −0.180886
\(490\) 0 0
\(491\) 10.0000i 0.451294i 0.974209 + 0.225647i \(0.0724495\pi\)
−0.974209 + 0.225647i \(0.927550\pi\)
\(492\) 0 0
\(493\) 12.0000i 0.540453i
\(494\) 0 0
\(495\) −4.00000 −0.179787
\(496\) 0 0
\(497\) −48.0000 −2.15309
\(498\) 0 0
\(499\) − 22.0000i − 0.984855i −0.870353 0.492428i \(-0.836110\pi\)
0.870353 0.492428i \(-0.163890\pi\)
\(500\) 0 0
\(501\) 40.0000i 1.78707i
\(502\) 0 0
\(503\) −20.0000 −0.891756 −0.445878 0.895094i \(-0.647108\pi\)
−0.445878 + 0.895094i \(0.647108\pi\)
\(504\) 0 0
\(505\) 12.0000 0.533993
\(506\) 0 0
\(507\) 18.0000i 0.799408i
\(508\) 0 0
\(509\) − 14.0000i − 0.620539i −0.950649 0.310270i \(-0.899581\pi\)
0.950649 0.310270i \(-0.100419\pi\)
\(510\) 0 0
\(511\) −56.0000 −2.47729
\(512\) 0 0
\(513\) −8.00000 −0.353209
\(514\) 0 0
\(515\) − 8.00000i − 0.352522i
\(516\) 0 0
\(517\) − 16.0000i − 0.703679i
\(518\) 0 0
\(519\) −36.0000 −1.58022
\(520\) 0 0
\(521\) 22.0000 0.963837 0.481919 0.876216i \(-0.339940\pi\)
0.481919 + 0.876216i \(0.339940\pi\)
\(522\) 0 0
\(523\) − 14.0000i − 0.612177i −0.952003 0.306089i \(-0.900980\pi\)
0.952003 0.306089i \(-0.0990204\pi\)
\(524\) 0 0
\(525\) 8.00000i 0.349149i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 14.0000i 0.607548i
\(532\) 0 0
\(533\) 12.0000i 0.519778i
\(534\) 0 0
\(535\) 4.00000 0.172935
\(536\) 0 0
\(537\) 12.0000 0.517838
\(538\) 0 0
\(539\) 18.0000i 0.775315i
\(540\) 0 0
\(541\) 34.0000i 1.46177i 0.682498 + 0.730887i \(0.260893\pi\)
−0.682498 + 0.730887i \(0.739107\pi\)
\(542\) 0 0
\(543\) 4.00000 0.171656
\(544\) 0 0
\(545\) −12.0000 −0.514024
\(546\) 0 0
\(547\) − 38.0000i − 1.62476i −0.583127 0.812381i \(-0.698171\pi\)
0.583127 0.812381i \(-0.301829\pi\)
\(548\) 0 0
\(549\) − 2.00000i − 0.0853579i
\(550\) 0 0
\(551\) 12.0000 0.511217
\(552\) 0 0
\(553\) −32.0000 −1.36078
\(554\) 0 0
\(555\) − 40.0000i − 1.69791i
\(556\) 0 0
\(557\) 2.00000i 0.0847427i 0.999102 + 0.0423714i \(0.0134913\pi\)
−0.999102 + 0.0423714i \(0.986509\pi\)
\(558\) 0 0
\(559\) 12.0000 0.507546
\(560\) 0 0
\(561\) 8.00000 0.337760
\(562\) 0 0
\(563\) 18.0000i 0.758610i 0.925272 + 0.379305i \(0.123837\pi\)
−0.925272 + 0.379305i \(0.876163\pi\)
\(564\) 0 0
\(565\) − 4.00000i − 0.168281i
\(566\) 0 0
\(567\) −44.0000 −1.84783
\(568\) 0 0
\(569\) −26.0000 −1.08998 −0.544988 0.838444i \(-0.683466\pi\)
−0.544988 + 0.838444i \(0.683466\pi\)
\(570\) 0 0
\(571\) − 38.0000i − 1.59025i −0.606445 0.795125i \(-0.707405\pi\)
0.606445 0.795125i \(-0.292595\pi\)
\(572\) 0 0
\(573\) − 32.0000i − 1.33682i
\(574\) 0 0
\(575\) −4.00000 −0.166812
\(576\) 0 0
\(577\) 2.00000 0.0832611 0.0416305 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) 0 0
\(579\) − 4.00000i − 0.166234i
\(580\) 0 0
\(581\) − 24.0000i − 0.995688i
\(582\) 0 0
\(583\) −12.0000 −0.496989
\(584\) 0 0
\(585\) −4.00000 −0.165380
\(586\) 0 0
\(587\) 34.0000i 1.40333i 0.712507 + 0.701665i \(0.247560\pi\)
−0.712507 + 0.701665i \(0.752440\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) −28.0000 −1.15177
\(592\) 0 0
\(593\) 18.0000 0.739171 0.369586 0.929197i \(-0.379500\pi\)
0.369586 + 0.929197i \(0.379500\pi\)
\(594\) 0 0
\(595\) 16.0000i 0.655936i
\(596\) 0 0
\(597\) 8.00000i 0.327418i
\(598\) 0 0
\(599\) 12.0000 0.490307 0.245153 0.969484i \(-0.421162\pi\)
0.245153 + 0.969484i \(0.421162\pi\)
\(600\) 0 0
\(601\) −30.0000 −1.22373 −0.611863 0.790964i \(-0.709580\pi\)
−0.611863 + 0.790964i \(0.709580\pi\)
\(602\) 0 0
\(603\) − 10.0000i − 0.407231i
\(604\) 0 0
\(605\) − 14.0000i − 0.569181i
\(606\) 0 0
\(607\) −16.0000 −0.649420 −0.324710 0.945814i \(-0.605267\pi\)
−0.324710 + 0.945814i \(0.605267\pi\)
\(608\) 0 0
\(609\) 48.0000 1.94506
\(610\) 0 0
\(611\) − 16.0000i − 0.647291i
\(612\) 0 0
\(613\) − 34.0000i − 1.37325i −0.727013 0.686624i \(-0.759092\pi\)
0.727013 0.686624i \(-0.240908\pi\)
\(614\) 0 0
\(615\) 24.0000 0.967773
\(616\) 0 0
\(617\) 2.00000 0.0805170 0.0402585 0.999189i \(-0.487182\pi\)
0.0402585 + 0.999189i \(0.487182\pi\)
\(618\) 0 0
\(619\) − 46.0000i − 1.84890i −0.381308 0.924448i \(-0.624526\pi\)
0.381308 0.924448i \(-0.375474\pi\)
\(620\) 0 0
\(621\) − 16.0000i − 0.642058i
\(622\) 0 0
\(623\) 8.00000 0.320513
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 0 0
\(627\) − 8.00000i − 0.319489i
\(628\) 0 0
\(629\) 20.0000i 0.797452i
\(630\) 0 0
\(631\) 44.0000 1.75161 0.875806 0.482663i \(-0.160330\pi\)
0.875806 + 0.482663i \(0.160330\pi\)
\(632\) 0 0
\(633\) 44.0000 1.74884
\(634\) 0 0
\(635\) − 32.0000i − 1.26988i
\(636\) 0 0
\(637\) 18.0000i 0.713186i
\(638\) 0 0
\(639\) 12.0000 0.474713
\(640\) 0 0
\(641\) 30.0000 1.18493 0.592464 0.805597i \(-0.298155\pi\)
0.592464 + 0.805597i \(0.298155\pi\)
\(642\) 0 0
\(643\) 42.0000i 1.65632i 0.560493 + 0.828159i \(0.310612\pi\)
−0.560493 + 0.828159i \(0.689388\pi\)
\(644\) 0 0
\(645\) − 24.0000i − 0.944999i
\(646\) 0 0
\(647\) −12.0000 −0.471769 −0.235884 0.971781i \(-0.575799\pi\)
−0.235884 + 0.971781i \(0.575799\pi\)
\(648\) 0 0
\(649\) 28.0000 1.09910
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 42.0000i 1.64359i 0.569785 + 0.821794i \(0.307026\pi\)
−0.569785 + 0.821794i \(0.692974\pi\)
\(654\) 0 0
\(655\) −12.0000 −0.468879
\(656\) 0 0
\(657\) 14.0000 0.546192
\(658\) 0 0
\(659\) − 6.00000i − 0.233727i −0.993148 0.116863i \(-0.962716\pi\)
0.993148 0.116863i \(-0.0372840\pi\)
\(660\) 0 0
\(661\) − 34.0000i − 1.32245i −0.750189 0.661223i \(-0.770038\pi\)
0.750189 0.661223i \(-0.229962\pi\)
\(662\) 0 0
\(663\) 8.00000 0.310694
\(664\) 0 0
\(665\) 16.0000 0.620453
\(666\) 0 0
\(667\) 24.0000i 0.929284i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −4.00000 −0.154418
\(672\) 0 0
\(673\) −2.00000 −0.0770943 −0.0385472 0.999257i \(-0.512273\pi\)
−0.0385472 + 0.999257i \(0.512273\pi\)
\(674\) 0 0
\(675\) 4.00000i 0.153960i
\(676\) 0 0
\(677\) 22.0000i 0.845529i 0.906240 + 0.422764i \(0.138940\pi\)
−0.906240 + 0.422764i \(0.861060\pi\)
\(678\) 0 0
\(679\) −8.00000 −0.307012
\(680\) 0 0
\(681\) −36.0000 −1.37952
\(682\) 0 0
\(683\) 42.0000i 1.60709i 0.595247 + 0.803543i \(0.297054\pi\)
−0.595247 + 0.803543i \(0.702946\pi\)
\(684\) 0 0
\(685\) 20.0000i 0.764161i
\(686\) 0 0
\(687\) −28.0000 −1.06827
\(688\) 0 0
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) − 6.00000i − 0.228251i −0.993466 0.114125i \(-0.963593\pi\)
0.993466 0.114125i \(-0.0364066\pi\)
\(692\) 0 0
\(693\) − 8.00000i − 0.303895i
\(694\) 0 0
\(695\) 20.0000 0.758643
\(696\) 0 0
\(697\) −12.0000 −0.454532
\(698\) 0 0
\(699\) 36.0000i 1.36165i
\(700\) 0 0
\(701\) 2.00000i 0.0755390i 0.999286 + 0.0377695i \(0.0120253\pi\)
−0.999286 + 0.0377695i \(0.987975\pi\)
\(702\) 0 0
\(703\) 20.0000 0.754314
\(704\) 0 0
\(705\) −32.0000 −1.20519
\(706\) 0 0
\(707\) 24.0000i 0.902613i
\(708\) 0 0
\(709\) 22.0000i 0.826227i 0.910679 + 0.413114i \(0.135559\pi\)
−0.910679 + 0.413114i \(0.864441\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 8.00000i 0.299183i
\(716\) 0 0
\(717\) − 48.0000i − 1.79259i
\(718\) 0 0
\(719\) 24.0000 0.895049 0.447524 0.894272i \(-0.352306\pi\)
0.447524 + 0.894272i \(0.352306\pi\)
\(720\) 0 0
\(721\) 16.0000 0.595871
\(722\) 0 0
\(723\) − 4.00000i − 0.148762i
\(724\) 0 0
\(725\) − 6.00000i − 0.222834i
\(726\) 0 0
\(727\) 12.0000 0.445055 0.222528 0.974926i \(-0.428569\pi\)
0.222528 + 0.974926i \(0.428569\pi\)
\(728\) 0 0
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) 12.0000i 0.443836i
\(732\) 0 0
\(733\) − 6.00000i − 0.221615i −0.993842 0.110808i \(-0.964656\pi\)
0.993842 0.110808i \(-0.0353437\pi\)
\(734\) 0 0
\(735\) 36.0000 1.32788
\(736\) 0 0
\(737\) −20.0000 −0.736709
\(738\) 0 0
\(739\) 18.0000i 0.662141i 0.943606 + 0.331070i \(0.107410\pi\)
−0.943606 + 0.331070i \(0.892590\pi\)
\(740\) 0 0
\(741\) − 8.00000i − 0.293887i
\(742\) 0 0
\(743\) −44.0000 −1.61420 −0.807102 0.590412i \(-0.798965\pi\)
−0.807102 + 0.590412i \(0.798965\pi\)
\(744\) 0 0
\(745\) −36.0000 −1.31894
\(746\) 0 0
\(747\) 6.00000i 0.219529i
\(748\) 0 0
\(749\) 8.00000i 0.292314i
\(750\) 0 0
\(751\) 8.00000 0.291924 0.145962 0.989290i \(-0.453372\pi\)
0.145962 + 0.989290i \(0.453372\pi\)
\(752\) 0 0
\(753\) −36.0000 −1.31191
\(754\) 0 0
\(755\) 8.00000i 0.291150i
\(756\) 0 0
\(757\) 46.0000i 1.67190i 0.548807 + 0.835949i \(0.315082\pi\)
−0.548807 + 0.835949i \(0.684918\pi\)
\(758\) 0 0
\(759\) 16.0000 0.580763
\(760\) 0 0
\(761\) −10.0000 −0.362500 −0.181250 0.983437i \(-0.558014\pi\)
−0.181250 + 0.983437i \(0.558014\pi\)
\(762\) 0 0
\(763\) − 24.0000i − 0.868858i
\(764\) 0 0
\(765\) − 4.00000i − 0.144620i
\(766\) 0 0
\(767\) 28.0000 1.01102
\(768\) 0 0
\(769\) −34.0000 −1.22607 −0.613036 0.790055i \(-0.710052\pi\)
−0.613036 + 0.790055i \(0.710052\pi\)
\(770\) 0 0
\(771\) 36.0000i 1.29651i
\(772\) 0 0
\(773\) 54.0000i 1.94225i 0.238581 + 0.971123i \(0.423318\pi\)
−0.238581 + 0.971123i \(0.576682\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 80.0000 2.86998
\(778\) 0 0
\(779\) 12.0000i 0.429945i
\(780\) 0 0
\(781\) − 24.0000i − 0.858788i
\(782\) 0 0
\(783\) 24.0000 0.857690
\(784\) 0 0
\(785\) 36.0000 1.28490
\(786\) 0 0
\(787\) − 22.0000i − 0.784215i −0.919919 0.392108i \(-0.871746\pi\)
0.919919 0.392108i \(-0.128254\pi\)
\(788\) 0 0
\(789\) − 24.0000i − 0.854423i
\(790\) 0 0
\(791\) 8.00000 0.284447
\(792\) 0 0
\(793\) −4.00000 −0.142044
\(794\) 0 0
\(795\) 24.0000i 0.851192i
\(796\) 0 0
\(797\) 18.0000i 0.637593i 0.947823 + 0.318796i \(0.103279\pi\)
−0.947823 + 0.318796i \(0.896721\pi\)
\(798\) 0 0
\(799\) 16.0000 0.566039
\(800\) 0 0
\(801\) −2.00000 −0.0706665
\(802\) 0 0
\(803\) − 28.0000i − 0.988099i
\(804\) 0 0
\(805\) 32.0000i 1.12785i
\(806\) 0 0
\(807\) −20.0000 −0.704033
\(808\) 0 0
\(809\) 6.00000 0.210949 0.105474 0.994422i \(-0.466364\pi\)
0.105474 + 0.994422i \(0.466364\pi\)
\(810\) 0 0
\(811\) 18.0000i 0.632065i 0.948748 + 0.316033i \(0.102351\pi\)
−0.948748 + 0.316033i \(0.897649\pi\)
\(812\) 0 0
\(813\) 16.0000i 0.561144i
\(814\) 0 0
\(815\) 4.00000 0.140114
\(816\) 0 0
\(817\) 12.0000 0.419827
\(818\) 0 0
\(819\) − 8.00000i − 0.279543i
\(820\) 0 0
\(821\) − 10.0000i − 0.349002i −0.984657 0.174501i \(-0.944169\pi\)
0.984657 0.174501i \(-0.0558313\pi\)
\(822\) 0 0
\(823\) 28.0000 0.976019 0.488009 0.872838i \(-0.337723\pi\)
0.488009 + 0.872838i \(0.337723\pi\)
\(824\) 0 0
\(825\) −4.00000 −0.139262
\(826\) 0 0
\(827\) − 22.0000i − 0.765015i −0.923952 0.382507i \(-0.875061\pi\)
0.923952 0.382507i \(-0.124939\pi\)
\(828\) 0 0
\(829\) − 14.0000i − 0.486240i −0.969996 0.243120i \(-0.921829\pi\)
0.969996 0.243120i \(-0.0781709\pi\)
\(830\) 0 0
\(831\) −12.0000 −0.416275
\(832\) 0 0
\(833\) −18.0000 −0.623663
\(834\) 0 0
\(835\) − 40.0000i − 1.38426i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 36.0000 1.24286 0.621429 0.783470i \(-0.286552\pi\)
0.621429 + 0.783470i \(0.286552\pi\)
\(840\) 0 0
\(841\) −7.00000 −0.241379
\(842\) 0 0
\(843\) 36.0000i 1.23991i
\(844\) 0 0
\(845\) − 18.0000i − 0.619219i
\(846\) 0 0
\(847\) 28.0000 0.962091
\(848\) 0 0
\(849\) 12.0000 0.411839
\(850\) 0 0
\(851\) 40.0000i 1.37118i
\(852\) 0 0
\(853\) − 26.0000i − 0.890223i −0.895475 0.445112i \(-0.853164\pi\)
0.895475 0.445112i \(-0.146836\pi\)
\(854\) 0 0
\(855\) −4.00000 −0.136797
\(856\) 0 0
\(857\) 6.00000 0.204956 0.102478 0.994735i \(-0.467323\pi\)
0.102478 + 0.994735i \(0.467323\pi\)
\(858\) 0 0
\(859\) 50.0000i 1.70598i 0.521929 + 0.852989i \(0.325213\pi\)
−0.521929 + 0.852989i \(0.674787\pi\)
\(860\) 0 0
\(861\) 48.0000i 1.63584i
\(862\) 0 0
\(863\) −32.0000 −1.08929 −0.544646 0.838666i \(-0.683336\pi\)
−0.544646 + 0.838666i \(0.683336\pi\)
\(864\) 0 0
\(865\) 36.0000 1.22404
\(866\) 0 0
\(867\) − 26.0000i − 0.883006i
\(868\) 0 0
\(869\) − 16.0000i − 0.542763i
\(870\) 0 0
\(871\) −20.0000 −0.677674
\(872\) 0 0
\(873\) 2.00000 0.0676897
\(874\) 0 0
\(875\) − 48.0000i − 1.62270i
\(876\) 0 0
\(877\) − 22.0000i − 0.742887i −0.928456 0.371444i \(-0.878863\pi\)
0.928456 0.371444i \(-0.121137\pi\)
\(878\) 0 0
\(879\) −28.0000 −0.944417
\(880\) 0 0
\(881\) −46.0000 −1.54978 −0.774890 0.632096i \(-0.782195\pi\)
−0.774890 + 0.632096i \(0.782195\pi\)
\(882\) 0 0
\(883\) 34.0000i 1.14419i 0.820187 + 0.572096i \(0.193869\pi\)
−0.820187 + 0.572096i \(0.806131\pi\)
\(884\) 0 0
\(885\) − 56.0000i − 1.88242i
\(886\) 0 0
\(887\) −36.0000 −1.20876 −0.604381 0.796696i \(-0.706579\pi\)
−0.604381 + 0.796696i \(0.706579\pi\)
\(888\) 0 0
\(889\) 64.0000 2.14649
\(890\) 0 0
\(891\) − 22.0000i − 0.737028i
\(892\) 0 0
\(893\) − 16.0000i − 0.535420i
\(894\) 0 0
\(895\) −12.0000 −0.401116
\(896\) 0 0
\(897\) 16.0000 0.534224
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) − 12.0000i − 0.399778i
\(902\) 0 0
\(903\) 48.0000 1.59734
\(904\) 0 0
\(905\) −4.00000 −0.132964
\(906\) 0 0
\(907\) − 38.0000i − 1.26177i −0.775877 0.630885i \(-0.782692\pi\)
0.775877 0.630885i \(-0.217308\pi\)
\(908\) 0 0
\(909\) − 6.00000i − 0.199007i
\(910\) 0 0
\(911\) −24.0000 −0.795155 −0.397578 0.917568i \(-0.630149\pi\)
−0.397578 + 0.917568i \(0.630149\pi\)
\(912\) 0 0
\(913\) 12.0000 0.397142
\(914\) 0 0
\(915\) 8.00000i 0.264472i
\(916\) 0 0
\(917\) − 24.0000i − 0.792550i
\(918\) 0 0
\(919\) −36.0000 −1.18753 −0.593765 0.804638i \(-0.702359\pi\)
−0.593765 + 0.804638i \(0.702359\pi\)
\(920\) 0 0
\(921\) −36.0000 −1.18624
\(922\) 0 0
\(923\) − 24.0000i − 0.789970i
\(924\) 0 0
\(925\) − 10.0000i − 0.328798i
\(926\) 0 0
\(927\) −4.00000 −0.131377
\(928\) 0 0
\(929\) 30.0000 0.984268 0.492134 0.870519i \(-0.336217\pi\)
0.492134 + 0.870519i \(0.336217\pi\)
\(930\) 0 0
\(931\) 18.0000i 0.589926i
\(932\) 0 0
\(933\) 56.0000i 1.83336i
\(934\) 0 0
\(935\) −8.00000 −0.261628
\(936\) 0 0
\(937\) −46.0000 −1.50275 −0.751377 0.659873i \(-0.770610\pi\)
−0.751377 + 0.659873i \(0.770610\pi\)
\(938\) 0 0
\(939\) − 20.0000i − 0.652675i
\(940\) 0 0
\(941\) − 38.0000i − 1.23876i −0.785090 0.619382i \(-0.787383\pi\)
0.785090 0.619382i \(-0.212617\pi\)
\(942\) 0 0
\(943\) −24.0000 −0.781548
\(944\) 0 0
\(945\) 32.0000 1.04096
\(946\) 0 0
\(947\) − 14.0000i − 0.454939i −0.973785 0.227469i \(-0.926955\pi\)
0.973785 0.227469i \(-0.0730452\pi\)
\(948\) 0 0
\(949\) − 28.0000i − 0.908918i
\(950\) 0 0
\(951\) 12.0000 0.389127
\(952\) 0 0
\(953\) −58.0000 −1.87880 −0.939402 0.342817i \(-0.888619\pi\)
−0.939402 + 0.342817i \(0.888619\pi\)
\(954\) 0 0
\(955\) 32.0000i 1.03550i
\(956\) 0 0
\(957\) 24.0000i 0.775810i
\(958\) 0 0
\(959\) −40.0000 −1.29167
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) − 2.00000i − 0.0644491i
\(964\) 0 0
\(965\) 4.00000i 0.128765i
\(966\) 0 0
\(967\) −28.0000 −0.900419 −0.450210 0.892923i \(-0.648651\pi\)
−0.450210 + 0.892923i \(0.648651\pi\)
\(968\) 0 0
\(969\) 8.00000 0.256997
\(970\) 0 0
\(971\) − 38.0000i − 1.21948i −0.792602 0.609739i \(-0.791274\pi\)
0.792602 0.609739i \(-0.208726\pi\)
\(972\) 0 0
\(973\) 40.0000i 1.28234i
\(974\) 0 0
\(975\) −4.00000 −0.128103
\(976\) 0 0
\(977\) 30.0000 0.959785 0.479893 0.877327i \(-0.340676\pi\)
0.479893 + 0.877327i \(0.340676\pi\)
\(978\) 0 0
\(979\) 4.00000i 0.127841i
\(980\) 0 0
\(981\) 6.00000i 0.191565i
\(982\) 0 0
\(983\) −20.0000 −0.637901 −0.318950 0.947771i \(-0.603330\pi\)
−0.318950 + 0.947771i \(0.603330\pi\)
\(984\) 0 0
\(985\) 28.0000 0.892154
\(986\) 0 0
\(987\) − 64.0000i − 2.03714i
\(988\) 0 0
\(989\) 24.0000i 0.763156i
\(990\) 0 0
\(991\) 16.0000 0.508257 0.254128 0.967170i \(-0.418211\pi\)
0.254128 + 0.967170i \(0.418211\pi\)
\(992\) 0 0
\(993\) 28.0000 0.888553
\(994\) 0 0
\(995\) − 8.00000i − 0.253617i
\(996\) 0 0
\(997\) 54.0000i 1.71020i 0.518465 + 0.855099i \(0.326503\pi\)
−0.518465 + 0.855099i \(0.673497\pi\)
\(998\) 0 0
\(999\) 40.0000 1.26554
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 256.2.b.c.129.2 2
3.2 odd 2 2304.2.d.r.1153.2 2
4.3 odd 2 256.2.b.a.129.1 2
8.3 odd 2 256.2.b.a.129.2 2
8.5 even 2 inner 256.2.b.c.129.1 2
12.11 even 2 2304.2.d.b.1153.2 2
16.3 odd 4 128.2.a.c.1.1 yes 1
16.5 even 4 128.2.a.d.1.1 yes 1
16.11 odd 4 128.2.a.b.1.1 yes 1
16.13 even 4 128.2.a.a.1.1 1
24.5 odd 2 2304.2.d.r.1153.1 2
24.11 even 2 2304.2.d.b.1153.1 2
32.3 odd 8 1024.2.e.m.769.2 4
32.5 even 8 1024.2.e.i.257.1 4
32.11 odd 8 1024.2.e.m.257.1 4
32.13 even 8 1024.2.e.i.769.2 4
32.19 odd 8 1024.2.e.m.769.1 4
32.21 even 8 1024.2.e.i.257.2 4
32.27 odd 8 1024.2.e.m.257.2 4
32.29 even 8 1024.2.e.i.769.1 4
48.5 odd 4 1152.2.a.c.1.1 1
48.11 even 4 1152.2.a.h.1.1 1
48.29 odd 4 1152.2.a.m.1.1 1
48.35 even 4 1152.2.a.r.1.1 1
80.3 even 4 3200.2.c.f.2049.2 2
80.13 odd 4 3200.2.c.l.2049.1 2
80.19 odd 4 3200.2.a.e.1.1 1
80.27 even 4 3200.2.c.k.2049.2 2
80.29 even 4 3200.2.a.x.1.1 1
80.37 odd 4 3200.2.c.e.2049.1 2
80.43 even 4 3200.2.c.k.2049.1 2
80.53 odd 4 3200.2.c.e.2049.2 2
80.59 odd 4 3200.2.a.u.1.1 1
80.67 even 4 3200.2.c.f.2049.1 2
80.69 even 4 3200.2.a.h.1.1 1
80.77 odd 4 3200.2.c.l.2049.2 2
112.13 odd 4 6272.2.a.h.1.1 1
112.27 even 4 6272.2.a.g.1.1 1
112.69 odd 4 6272.2.a.a.1.1 1
112.83 even 4 6272.2.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
128.2.a.a.1.1 1 16.13 even 4
128.2.a.b.1.1 yes 1 16.11 odd 4
128.2.a.c.1.1 yes 1 16.3 odd 4
128.2.a.d.1.1 yes 1 16.5 even 4
256.2.b.a.129.1 2 4.3 odd 2
256.2.b.a.129.2 2 8.3 odd 2
256.2.b.c.129.1 2 8.5 even 2 inner
256.2.b.c.129.2 2 1.1 even 1 trivial
1024.2.e.i.257.1 4 32.5 even 8
1024.2.e.i.257.2 4 32.21 even 8
1024.2.e.i.769.1 4 32.29 even 8
1024.2.e.i.769.2 4 32.13 even 8
1024.2.e.m.257.1 4 32.11 odd 8
1024.2.e.m.257.2 4 32.27 odd 8
1024.2.e.m.769.1 4 32.19 odd 8
1024.2.e.m.769.2 4 32.3 odd 8
1152.2.a.c.1.1 1 48.5 odd 4
1152.2.a.h.1.1 1 48.11 even 4
1152.2.a.m.1.1 1 48.29 odd 4
1152.2.a.r.1.1 1 48.35 even 4
2304.2.d.b.1153.1 2 24.11 even 2
2304.2.d.b.1153.2 2 12.11 even 2
2304.2.d.r.1153.1 2 24.5 odd 2
2304.2.d.r.1153.2 2 3.2 odd 2
3200.2.a.e.1.1 1 80.19 odd 4
3200.2.a.h.1.1 1 80.69 even 4
3200.2.a.u.1.1 1 80.59 odd 4
3200.2.a.x.1.1 1 80.29 even 4
3200.2.c.e.2049.1 2 80.37 odd 4
3200.2.c.e.2049.2 2 80.53 odd 4
3200.2.c.f.2049.1 2 80.67 even 4
3200.2.c.f.2049.2 2 80.3 even 4
3200.2.c.k.2049.1 2 80.43 even 4
3200.2.c.k.2049.2 2 80.27 even 4
3200.2.c.l.2049.1 2 80.13 odd 4
3200.2.c.l.2049.2 2 80.77 odd 4
6272.2.a.a.1.1 1 112.69 odd 4
6272.2.a.b.1.1 1 112.83 even 4
6272.2.a.g.1.1 1 112.27 even 4
6272.2.a.h.1.1 1 112.13 odd 4