Properties

Label 3200.2.c.e.2049.2
Level $3200$
Weight $2$
Character 3200.2049
Analytic conductor $25.552$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3200 = 2^{7} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3200.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(25.5521286468\)
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 2049.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 3200.2049
Dual form 3200.2.c.e.2049.1

$q$-expansion

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3200\mathbb{Z}\right)^\times\).

\(n\) \(901\) \(1151\) \(2177\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2.00000i 1.15470i 0.816497 + 0.577350i \(0.195913\pi\)
−0.816497 + 0.577350i \(0.804087\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 4.00000i 1.51186i 0.654654 + 0.755929i \(0.272814\pi\)
−0.654654 + 0.755929i \(0.727186\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) 2.00000i 0.554700i 0.960769 + 0.277350i \(0.0894562\pi\)
−0.960769 + 0.277350i \(0.910544\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.00000i 0.485071i 0.970143 + 0.242536i \(0.0779791\pi\)
−0.970143 + 0.242536i \(0.922021\pi\)
\(18\) 0 0
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) −8.00000 −1.74574
\(22\) 0 0
\(23\) 4.00000i 0.834058i 0.908893 + 0.417029i \(0.136929\pi\)
−0.908893 + 0.417029i \(0.863071\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 4.00000i 0.769800i
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) − 4.00000i − 0.696311i
\(34\) 0 0
\(35\) 0 0
\(36\) 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 \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 6.00000i 0.914991i 0.889212 + 0.457496i \(0.151253\pi\)
−0.889212 + 0.457496i \(0.848747\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 8.00000i 1.16692i 0.812142 + 0.583460i \(0.198301\pi\)
−0.812142 + 0.583460i \(0.801699\pi\)
\(48\) 0 0
\(49\) −9.00000 −1.28571
\(50\) 0 0
\(51\) −4.00000 −0.560112
\(52\) 0 0
\(53\) − 6.00000i − 0.824163i −0.911147 0.412082i \(-0.864802\pi\)
0.911147 0.412082i \(-0.135198\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 4.00000i − 0.529813i
\(58\) 0 0
\(59\) −14.0000 −1.82264 −0.911322 0.411693i \(-0.864937\pi\)
−0.911322 + 0.411693i \(0.864937\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 0 0
\(63\) − 4.00000i − 0.503953i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 10.0000i − 1.22169i −0.791748 0.610847i \(-0.790829\pi\)
0.791748 0.610847i \(-0.209171\pi\)
\(68\) 0 0
\(69\) −8.00000 −0.963087
\(70\) 0 0
\(71\) 12.0000 1.42414 0.712069 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 0 0
\(73\) 14.0000i 1.63858i 0.573382 + 0.819288i \(0.305631\pi\)
−0.573382 + 0.819288i \(0.694369\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 8.00000i − 0.911685i
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) −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
\(85\) 0 0
\(86\) 0 0
\(87\) 12.0000i 1.28654i
\(88\) 0 0
\(89\) 2.00000 0.212000 0.106000 0.994366i \(-0.466196\pi\)
0.106000 + 0.994366i \(0.466196\pi\)
\(90\) 0 0
\(91\) −8.00000 −0.838628
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 2.00000i 0.203069i 0.994832 + 0.101535i \(0.0323753\pi\)
−0.994832 + 0.101535i \(0.967625\pi\)
\(98\) 0 0
\(99\) 2.00000 0.201008
\(100\) 0 0
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 0 0
\(103\) − 4.00000i − 0.394132i −0.980390 0.197066i \(-0.936859\pi\)
0.980390 0.197066i \(-0.0631413\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 2.00000i 0.193347i 0.995316 + 0.0966736i \(0.0308203\pi\)
−0.995316 + 0.0966736i \(0.969180\pi\)
\(108\) 0 0
\(109\) 6.00000 0.574696 0.287348 0.957826i \(-0.407226\pi\)
0.287348 + 0.957826i \(0.407226\pi\)
\(110\) 0 0
\(111\) 20.0000 1.89832
\(112\) 0 0
\(113\) 2.00000i 0.188144i 0.995565 + 0.0940721i \(0.0299884\pi\)
−0.995565 + 0.0940721i \(0.970012\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 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
\(125\) 0 0
\(126\) 0 0
\(127\) − 16.0000i − 1.41977i −0.704317 0.709885i \(-0.748747\pi\)
0.704317 0.709885i \(-0.251253\pi\)
\(128\) 0 0
\(129\) −12.0000 −1.05654
\(130\) 0 0
\(131\) −6.00000 −0.524222 −0.262111 0.965038i \(-0.584419\pi\)
−0.262111 + 0.965038i \(0.584419\pi\)
\(132\) 0 0
\(133\) − 8.00000i − 0.693688i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 10.0000i − 0.854358i −0.904167 0.427179i \(-0.859507\pi\)
0.904167 0.427179i \(-0.140493\pi\)
\(138\) 0 0
\(139\) 10.0000 0.848189 0.424094 0.905618i \(-0.360592\pi\)
0.424094 + 0.905618i \(0.360592\pi\)
\(140\) 0 0
\(141\) −16.0000 −1.34744
\(142\) 0 0
\(143\) − 4.00000i − 0.334497i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 18.0000i − 1.48461i
\(148\) 0 0
\(149\) −18.0000 −1.47462 −0.737309 0.675556i \(-0.763904\pi\)
−0.737309 + 0.675556i \(0.763904\pi\)
\(150\) 0 0
\(151\) 4.00000 0.325515 0.162758 0.986666i \(-0.447961\pi\)
0.162758 + 0.986666i \(0.447961\pi\)
\(152\) 0 0
\(153\) − 2.00000i − 0.161690i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 18.0000i − 1.43656i −0.695756 0.718278i \(-0.744931\pi\)
0.695756 0.718278i \(-0.255069\pi\)
\(158\) 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
\(165\) 0 0
\(166\) 0 0
\(167\) 20.0000i 1.54765i 0.633402 + 0.773823i \(0.281658\pi\)
−0.633402 + 0.773823i \(0.718342\pi\)
\(168\) 0 0
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) 2.00000 0.152944
\(172\) 0 0
\(173\) 18.0000i 1.36851i 0.729241 + 0.684257i \(0.239873\pi\)
−0.729241 + 0.684257i \(0.760127\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 28.0000i − 2.10461i
\(178\) 0 0
\(179\) 6.00000 0.448461 0.224231 0.974536i \(-0.428013\pi\)
0.224231 + 0.974536i \(0.428013\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 0 0
\(183\) 4.00000i 0.295689i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 4.00000i − 0.292509i
\(188\) 0 0
\(189\) −16.0000 −1.16383
\(190\) 0 0
\(191\) −16.0000 −1.15772 −0.578860 0.815427i \(-0.696502\pi\)
−0.578860 + 0.815427i \(0.696502\pi\)
\(192\) 0 0
\(193\) − 2.00000i − 0.143963i −0.997406 0.0719816i \(-0.977068\pi\)
0.997406 0.0719816i \(-0.0229323\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 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
\(205\) 0 0
\(206\) 0 0
\(207\) − 4.00000i − 0.278019i
\(208\) 0 0
\(209\) 4.00000 0.276686
\(210\) 0 0
\(211\) −22.0000 −1.51454 −0.757271 0.653101i \(-0.773468\pi\)
−0.757271 + 0.653101i \(0.773468\pi\)
\(212\) 0 0
\(213\) 24.0000i 1.64445i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −28.0000 −1.89206
\(220\) 0 0
\(221\) −4.00000 −0.269069
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 18.0000i − 1.19470i −0.801980 0.597351i \(-0.796220\pi\)
0.801980 0.597351i \(-0.203780\pi\)
\(228\) 0 0
\(229\) 14.0000 0.925146 0.462573 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) 0 0
\(231\) 16.0000 1.05272
\(232\) 0 0
\(233\) − 18.0000i − 1.17922i −0.807688 0.589610i \(-0.799282\pi\)
0.807688 0.589610i \(-0.200718\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 16.0000i 1.03931i
\(238\) 0 0
\(239\) 24.0000 1.55243 0.776215 0.630468i \(-0.217137\pi\)
0.776215 + 0.630468i \(0.217137\pi\)
\(240\) 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
\(245\) 0 0
\(246\) 0 0
\(247\) − 4.00000i − 0.254514i
\(248\) 0 0
\(249\) 12.0000 0.760469
\(250\) 0 0
\(251\) −18.0000 −1.13615 −0.568075 0.822977i \(-0.692312\pi\)
−0.568075 + 0.822977i \(0.692312\pi\)
\(252\) 0 0
\(253\) − 8.00000i − 0.502956i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 18.0000i − 1.12281i −0.827541 0.561405i \(-0.810261\pi\)
0.827541 0.561405i \(-0.189739\pi\)
\(258\) 0 0
\(259\) 40.0000 2.48548
\(260\) 0 0
\(261\) −6.00000 −0.371391
\(262\) 0 0
\(263\) 12.0000i 0.739952i 0.929041 + 0.369976i \(0.120634\pi\)
−0.929041 + 0.369976i \(0.879366\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 4.00000i 0.244796i
\(268\) 0 0
\(269\) −10.0000 −0.609711 −0.304855 0.952399i \(-0.598608\pi\)
−0.304855 + 0.952399i \(0.598608\pi\)
\(270\) 0 0
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) 0 0
\(273\) − 16.0000i − 0.968364i
\(274\) 0 0
\(275\) 0 0
\(276\) 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 \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) 0 0
\(283\) 6.00000i 0.356663i 0.983970 + 0.178331i \(0.0570699\pi\)
−0.983970 + 0.178331i \(0.942930\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 24.0000i − 1.41668i
\(288\) 0 0
\(289\) 13.0000 0.764706
\(290\) 0 0
\(291\) −4.00000 −0.234484
\(292\) 0 0
\(293\) − 14.0000i − 0.817889i −0.912559 0.408944i \(-0.865897\pi\)
0.912559 0.408944i \(-0.134103\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 8.00000i − 0.464207i
\(298\) 0 0
\(299\) −8.00000 −0.462652
\(300\) 0 0
\(301\) −24.0000 −1.38334
\(302\) 0 0
\(303\) − 12.0000i − 0.689382i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 18.0000i − 1.02731i −0.857996 0.513657i \(-0.828290\pi\)
0.857996 0.513657i \(-0.171710\pi\)
\(308\) 0 0
\(309\) 8.00000 0.455104
\(310\) 0 0
\(311\) −28.0000 −1.58773 −0.793867 0.608091i \(-0.791935\pi\)
−0.793867 + 0.608091i \(0.791935\pi\)
\(312\) 0 0
\(313\) 10.0000i 0.565233i 0.959233 + 0.282617i \(0.0912024\pi\)
−0.959233 + 0.282617i \(0.908798\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 6.00000i 0.336994i 0.985702 + 0.168497i \(0.0538913\pi\)
−0.985702 + 0.168497i \(0.946109\pi\)
\(318\) 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
\(325\) 0 0
\(326\) 0 0
\(327\) 12.0000i 0.663602i
\(328\) 0 0
\(329\) −32.0000 −1.76422
\(330\) 0 0
\(331\) 14.0000 0.769510 0.384755 0.923019i \(-0.374286\pi\)
0.384755 + 0.923019i \(0.374286\pi\)
\(332\) 0 0
\(333\) 10.0000i 0.547997i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 14.0000i 0.762629i 0.924445 + 0.381314i \(0.124528\pi\)
−0.924445 + 0.381314i \(0.875472\pi\)
\(338\) 0 0
\(339\) −4.00000 −0.217250
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) − 8.00000i − 0.431959i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 18.0000i 0.966291i 0.875540 + 0.483145i \(0.160506\pi\)
−0.875540 + 0.483145i \(0.839494\pi\)
\(348\) 0 0
\(349\) −10.0000 −0.535288 −0.267644 0.963518i \(-0.586245\pi\)
−0.267644 + 0.963518i \(0.586245\pi\)
\(350\) 0 0
\(351\) −8.00000 −0.427008
\(352\) 0 0
\(353\) 18.0000i 0.958043i 0.877803 + 0.479022i \(0.159008\pi\)
−0.877803 + 0.479022i \(0.840992\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 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
\(365\) 0 0
\(366\) 0 0
\(367\) 8.00000i 0.417597i 0.977959 + 0.208798i \(0.0669552\pi\)
−0.977959 + 0.208798i \(0.933045\pi\)
\(368\) 0 0
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) 24.0000 1.24602
\(372\) 0 0
\(373\) 10.0000i 0.517780i 0.965907 + 0.258890i \(0.0833568\pi\)
−0.965907 + 0.258890i \(0.916643\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 12.0000i 0.618031i
\(378\) 0 0
\(379\) 2.00000 0.102733 0.0513665 0.998680i \(-0.483642\pi\)
0.0513665 + 0.998680i \(0.483642\pi\)
\(380\) 0 0
\(381\) 32.0000 1.63941
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 6.00000i − 0.304997i
\(388\) 0 0
\(389\) −10.0000 −0.507020 −0.253510 0.967333i \(-0.581585\pi\)
−0.253510 + 0.967333i \(0.581585\pi\)
\(390\) 0 0
\(391\) −8.00000 −0.404577
\(392\) 0 0
\(393\) − 12.0000i − 0.605320i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 6.00000i 0.301131i 0.988600 + 0.150566i \(0.0481095\pi\)
−0.988600 + 0.150566i \(0.951890\pi\)
\(398\) 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
\(405\) 0 0
\(406\) 0 0
\(407\) 20.0000i 0.991363i
\(408\) 0 0
\(409\) −10.0000 −0.494468 −0.247234 0.968956i \(-0.579522\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) 0 0
\(411\) 20.0000 0.986527
\(412\) 0 0
\(413\) − 56.0000i − 2.75558i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 20.0000i 0.979404i
\(418\) 0 0
\(419\) −26.0000 −1.27018 −0.635092 0.772437i \(-0.719038\pi\)
−0.635092 + 0.772437i \(0.719038\pi\)
\(420\) 0 0
\(421\) 34.0000 1.65706 0.828529 0.559946i \(-0.189178\pi\)
0.828529 + 0.559946i \(0.189178\pi\)
\(422\) 0 0
\(423\) − 8.00000i − 0.388973i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 8.00000i 0.387147i
\(428\) 0 0
\(429\) 8.00000 0.386244
\(430\) 0 0
\(431\) 40.0000 1.92673 0.963366 0.268190i \(-0.0864254\pi\)
0.963366 + 0.268190i \(0.0864254\pi\)
\(432\) 0 0
\(433\) 30.0000i 1.44171i 0.693087 + 0.720854i \(0.256250\pi\)
−0.693087 + 0.720854i \(0.743750\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 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 \(0.0455251\pi\)
−0.989790 + 0.142534i \(0.954475\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 36.0000i − 1.70274i
\(448\) 0 0
\(449\) 34.0000 1.60456 0.802280 0.596948i \(-0.203620\pi\)
0.802280 + 0.596948i \(0.203620\pi\)
\(450\) 0 0
\(451\) 12.0000 0.565058
\(452\) 0 0
\(453\) 8.00000i 0.375873i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 6.00000i 0.280668i 0.990104 + 0.140334i \(0.0448177\pi\)
−0.990104 + 0.140334i \(0.955182\pi\)
\(458\) 0 0
\(459\) −8.00000 −0.373408
\(460\) 0 0
\(461\) 10.0000 0.465746 0.232873 0.972507i \(-0.425187\pi\)
0.232873 + 0.972507i \(0.425187\pi\)
\(462\) 0 0
\(463\) − 8.00000i − 0.371792i −0.982569 0.185896i \(-0.940481\pi\)
0.982569 0.185896i \(-0.0595187\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 14.0000i 0.647843i 0.946084 + 0.323921i \(0.105001\pi\)
−0.946084 + 0.323921i \(0.894999\pi\)
\(468\) 0 0
\(469\) 40.0000 1.84703
\(470\) 0 0
\(471\) 36.0000 1.65879
\(472\) 0 0
\(473\) − 12.0000i − 0.551761i
\(474\) 0 0
\(475\) 0 0
\(476\) 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
\(485\) 0 0
\(486\) 0 0
\(487\) 20.0000i 0.906287i 0.891438 + 0.453143i \(0.149697\pi\)
−0.891438 + 0.453143i \(0.850303\pi\)
\(488\) 0 0
\(489\) −4.00000 −0.180886
\(490\) 0 0
\(491\) −10.0000 −0.451294 −0.225647 0.974209i \(-0.572450\pi\)
−0.225647 + 0.974209i \(0.572450\pi\)
\(492\) 0 0
\(493\) 12.0000i 0.540453i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 48.0000i 2.15309i
\(498\) 0 0
\(499\) 22.0000 0.984855 0.492428 0.870353i \(-0.336110\pi\)
0.492428 + 0.870353i \(0.336110\pi\)
\(500\) 0 0
\(501\) −40.0000 −1.78707
\(502\) 0 0
\(503\) 20.0000i 0.891756i 0.895094 + 0.445878i \(0.147108\pi\)
−0.895094 + 0.445878i \(0.852892\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 18.0000i 0.799408i
\(508\) 0 0
\(509\) 14.0000 0.620539 0.310270 0.950649i \(-0.399581\pi\)
0.310270 + 0.950649i \(0.399581\pi\)
\(510\) 0 0
\(511\) −56.0000 −2.47729
\(512\) 0 0
\(513\) − 8.00000i − 0.353209i
\(514\) 0 0
\(515\) 0 0
\(516\) 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 \(-0.660060\pi\)
−0.481919 + 0.876216i \(0.660060\pi\)
\(522\) 0 0
\(523\) 14.0000i 0.612177i 0.952003 + 0.306089i \(0.0990204\pi\)
−0.952003 + 0.306089i \(0.900980\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 7.00000 0.304348
\(530\) 0 0
\(531\) 14.0000 0.607548
\(532\) 0 0
\(533\) − 12.0000i − 0.519778i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 12.0000i 0.517838i
\(538\) 0 0
\(539\) 18.0000 0.775315
\(540\) 0 0
\(541\) 34.0000 1.46177 0.730887 0.682498i \(-0.239107\pi\)
0.730887 + 0.682498i \(0.239107\pi\)
\(542\) 0 0
\(543\) 4.00000i 0.171656i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 38.0000i 1.62476i 0.583127 + 0.812381i \(0.301829\pi\)
−0.583127 + 0.812381i \(0.698171\pi\)
\(548\) 0 0
\(549\) −2.00000 −0.0853579
\(550\) 0 0
\(551\) −12.0000 −0.511217
\(552\) 0 0
\(553\) 32.0000i 1.36078i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 2.00000i − 0.0847427i −0.999102 0.0423714i \(-0.986509\pi\)
0.999102 0.0423714i \(-0.0134913\pi\)
\(558\) 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
\(565\) 0 0
\(566\) 0 0
\(567\) − 44.0000i − 1.84783i
\(568\) 0 0
\(569\) −26.0000 −1.08998 −0.544988 0.838444i \(-0.683466\pi\)
−0.544988 + 0.838444i \(0.683466\pi\)
\(570\) 0 0
\(571\) 38.0000 1.59025 0.795125 0.606445i \(-0.207405\pi\)
0.795125 + 0.606445i \(0.207405\pi\)
\(572\) 0 0
\(573\) − 32.0000i − 1.33682i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 2.00000i − 0.0832611i −0.999133 0.0416305i \(-0.986745\pi\)
0.999133 0.0416305i \(-0.0132552\pi\)
\(578\) 0 0
\(579\) 4.00000 0.166234
\(580\) 0 0
\(581\) 24.0000 0.995688
\(582\) 0 0
\(583\) 12.0000i 0.496989i
\(584\) 0 0
\(585\) 0 0
\(586\) 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
\(593\) 18.0000i 0.739171i 0.929197 + 0.369586i \(0.120500\pi\)
−0.929197 + 0.369586i \(0.879500\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 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 \(-0.290420\pi\)
0.611863 + 0.790964i \(0.290420\pi\)
\(602\) 0 0
\(603\) 10.0000i 0.407231i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 16.0000i 0.649420i 0.945814 + 0.324710i \(0.105267\pi\)
−0.945814 + 0.324710i \(0.894733\pi\)
\(608\) 0 0
\(609\) −48.0000 −1.94506
\(610\) 0 0
\(611\) −16.0000 −0.647291
\(612\) 0 0
\(613\) 34.0000i 1.37325i 0.727013 + 0.686624i \(0.240908\pi\)
−0.727013 + 0.686624i \(0.759092\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 2.00000i 0.0805170i 0.999189 + 0.0402585i \(0.0128181\pi\)
−0.999189 + 0.0402585i \(0.987182\pi\)
\(618\) 0 0
\(619\) −46.0000 −1.84890 −0.924448 0.381308i \(-0.875474\pi\)
−0.924448 + 0.381308i \(0.875474\pi\)
\(620\) 0 0
\(621\) −16.0000 −0.642058
\(622\) 0 0
\(623\) 8.00000i 0.320513i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 8.00000i 0.319489i
\(628\) 0 0
\(629\) 20.0000 0.797452
\(630\) 0 0
\(631\) −44.0000 −1.75161 −0.875806 0.482663i \(-0.839670\pi\)
−0.875806 + 0.482663i \(0.839670\pi\)
\(632\) 0 0
\(633\) − 44.0000i − 1.74884i
\(634\) 0 0
\(635\) 0 0
\(636\) 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
\(645\) 0 0
\(646\) 0 0
\(647\) − 12.0000i − 0.471769i −0.971781 0.235884i \(-0.924201\pi\)
0.971781 0.235884i \(-0.0757987\pi\)
\(648\) 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
\(655\) 0 0
\(656\) 0 0
\(657\) − 14.0000i − 0.546192i
\(658\) 0 0
\(659\) 6.00000 0.233727 0.116863 0.993148i \(-0.462716\pi\)
0.116863 + 0.993148i \(0.462716\pi\)
\(660\) 0 0
\(661\) 34.0000 1.32245 0.661223 0.750189i \(-0.270038\pi\)
0.661223 + 0.750189i \(0.270038\pi\)
\(662\) 0 0
\(663\) − 8.00000i − 0.310694i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 24.0000i 0.929284i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −4.00000 −0.154418
\(672\) 0 0
\(673\) − 2.00000i − 0.0770943i −0.999257 0.0385472i \(-0.987727\pi\)
0.999257 0.0385472i \(-0.0122730\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 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 \(-0.702946\pi\)
0.595247 0.803543i \(-0.297054\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 28.0000i 1.06827i
\(688\) 0 0
\(689\) 12.0000 0.457164
\(690\) 0 0
\(691\) −6.00000 −0.228251 −0.114125 0.993466i \(-0.536407\pi\)
−0.114125 + 0.993466i \(0.536407\pi\)
\(692\) 0 0
\(693\) 8.00000i 0.303895i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 12.0000i − 0.454532i
\(698\) 0 0
\(699\) 36.0000 1.36165
\(700\) 0 0
\(701\) 2.00000 0.0755390 0.0377695 0.999286i \(-0.487975\pi\)
0.0377695 + 0.999286i \(0.487975\pi\)
\(702\) 0 0
\(703\) 20.0000i 0.754314i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 24.0000i − 0.902613i
\(708\) 0 0
\(709\) 22.0000 0.826227 0.413114 0.910679i \(-0.364441\pi\)
0.413114 + 0.910679i \(0.364441\pi\)
\(710\) 0 0
\(711\) −8.00000 −0.300023
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 48.0000i 1.79259i
\(718\) 0 0
\(719\) −24.0000 −0.895049 −0.447524 0.894272i \(-0.647694\pi\)
−0.447524 + 0.894272i \(0.647694\pi\)
\(720\) 0 0
\(721\) 16.0000 0.595871
\(722\) 0 0
\(723\) − 4.00000i − 0.148762i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 12.0000i 0.445055i 0.974926 + 0.222528i \(0.0714308\pi\)
−0.974926 + 0.222528i \(0.928569\pi\)
\(728\) 0 0
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) −12.0000 −0.443836
\(732\) 0 0
\(733\) − 6.00000i − 0.221615i −0.993842 0.110808i \(-0.964656\pi\)
0.993842 0.110808i \(-0.0353437\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 20.0000i 0.736709i
\(738\) 0 0
\(739\) −18.0000 −0.662141 −0.331070 0.943606i \(-0.607410\pi\)
−0.331070 + 0.943606i \(0.607410\pi\)
\(740\) 0 0
\(741\) 8.00000 0.293887
\(742\) 0 0
\(743\) 44.0000i 1.61420i 0.590412 + 0.807102i \(0.298965\pi\)
−0.590412 + 0.807102i \(0.701035\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 6.00000i 0.219529i
\(748\) 0 0
\(749\) −8.00000 −0.292314
\(750\) 0 0
\(751\) 8.00000 0.291924 0.145962 0.989290i \(-0.453372\pi\)
0.145962 + 0.989290i \(0.453372\pi\)
\(752\) 0 0
\(753\) − 36.0000i − 1.31191i
\(754\) 0 0
\(755\) 0 0
\(756\) 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 \(-0.441986\pi\)
0.181250 + 0.983437i \(0.441986\pi\)
\(762\) 0 0
\(763\) 24.0000i 0.868858i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 28.0000i − 1.01102i
\(768\) 0 0
\(769\) 34.0000 1.22607 0.613036 0.790055i \(-0.289948\pi\)
0.613036 + 0.790055i \(0.289948\pi\)
\(770\) 0 0
\(771\) 36.0000 1.29651
\(772\) 0 0
\(773\) − 54.0000i − 1.94225i −0.238581 0.971123i \(-0.576682\pi\)
0.238581 0.971123i \(-0.423318\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 80.0000i 2.86998i
\(778\) 0 0
\(779\) 12.0000 0.429945
\(780\) 0 0
\(781\) −24.0000 −0.858788
\(782\) 0 0
\(783\) 24.0000i 0.857690i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 22.0000i 0.784215i 0.919919 + 0.392108i \(0.128254\pi\)
−0.919919 + 0.392108i \(0.871746\pi\)
\(788\) 0 0
\(789\) −24.0000 −0.854423
\(790\) 0 0
\(791\) −8.00000 −0.284447
\(792\) 0 0
\(793\) 4.00000i 0.142044i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 18.0000i − 0.637593i −0.947823 0.318796i \(-0.896721\pi\)
0.947823 0.318796i \(-0.103279\pi\)
\(798\) 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
\(805\) 0 0
\(806\) 0 0
\(807\) − 20.0000i − 0.704033i
\(808\) 0 0
\(809\) 6.00000 0.210949 0.105474 0.994422i \(-0.466364\pi\)
0.105474 + 0.994422i \(0.466364\pi\)
\(810\) 0 0
\(811\) −18.0000 −0.632065 −0.316033 0.948748i \(-0.602351\pi\)
−0.316033 + 0.948748i \(0.602351\pi\)
\(812\) 0 0
\(813\) 16.0000i 0.561144i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 12.0000i − 0.419827i
\(818\) 0 0
\(819\) 8.00000 0.279543
\(820\) 0 0
\(821\) 10.0000 0.349002 0.174501 0.984657i \(-0.444169\pi\)
0.174501 + 0.984657i \(0.444169\pi\)
\(822\) 0 0
\(823\) − 28.0000i − 0.976019i −0.872838 0.488009i \(-0.837723\pi\)
0.872838 0.488009i \(-0.162277\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 22.0000i − 0.765015i −0.923952 0.382507i \(-0.875061\pi\)
0.923952 0.382507i \(-0.124939\pi\)
\(828\) 0 0
\(829\) 14.0000 0.486240 0.243120 0.969996i \(-0.421829\pi\)
0.243120 + 0.969996i \(0.421829\pi\)
\(830\) 0 0
\(831\) −12.0000 −0.416275
\(832\) 0 0
\(833\) − 18.0000i − 0.623663i
\(834\) 0 0
\(835\) 0 0
\(836\) 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
\(845\) 0 0
\(846\) 0 0
\(847\) − 28.0000i − 0.962091i
\(848\) 0 0
\(849\) −12.0000 −0.411839
\(850\) 0 0
\(851\) 40.0000 1.37118
\(852\) 0 0
\(853\) 26.0000i 0.890223i 0.895475 + 0.445112i \(0.146836\pi\)
−0.895475 + 0.445112i \(0.853164\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 6.00000i 0.204956i 0.994735 + 0.102478i \(0.0326771\pi\)
−0.994735 + 0.102478i \(0.967323\pi\)
\(858\) 0 0
\(859\) 50.0000 1.70598 0.852989 0.521929i \(-0.174787\pi\)
0.852989 + 0.521929i \(0.174787\pi\)
\(860\) 0 0
\(861\) 48.0000 1.63584
\(862\) 0 0
\(863\) − 32.0000i − 1.08929i −0.838666 0.544646i \(-0.816664\pi\)
0.838666 0.544646i \(-0.183336\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 26.0000i 0.883006i
\(868\) 0 0
\(869\) −16.0000 −0.542763
\(870\) 0 0
\(871\) 20.0000 0.677674
\(872\) 0 0
\(873\) − 2.00000i − 0.0676897i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 22.0000i 0.742887i 0.928456 + 0.371444i \(0.121137\pi\)
−0.928456 + 0.371444i \(0.878863\pi\)
\(878\) 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
\(885\) 0 0
\(886\) 0 0
\(887\) − 36.0000i − 1.20876i −0.796696 0.604381i \(-0.793421\pi\)
0.796696 0.604381i \(-0.206579\pi\)
\(888\) 0 0
\(889\) 64.0000 2.14649
\(890\) 0 0
\(891\) 22.0000 0.737028
\(892\) 0 0
\(893\) − 16.0000i − 0.535420i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 16.0000i − 0.534224i
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 12.0000 0.399778
\(902\) 0 0
\(903\) − 48.0000i − 1.59734i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 38.0000i − 1.26177i −0.775877 0.630885i \(-0.782692\pi\)
0.775877 0.630885i \(-0.217308\pi\)
\(908\) 0 0
\(909\) 6.00000 0.199007
\(910\) 0 0
\(911\) −24.0000 −0.795155 −0.397578 0.917568i \(-0.630149\pi\)
−0.397578 + 0.917568i \(0.630149\pi\)
\(912\) 0 0
\(913\) 12.0000i 0.397142i
\(914\) 0 0
\(915\) 0 0
\(916\) 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
\(925\) 0 0
\(926\) 0 0
\(927\) 4.00000i 0.131377i
\(928\) 0 0
\(929\) −30.0000 −0.984268 −0.492134 0.870519i \(-0.663783\pi\)
−0.492134 + 0.870519i \(0.663783\pi\)
\(930\) 0 0
\(931\) 18.0000 0.589926
\(932\) 0 0
\(933\) − 56.0000i − 1.83336i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 46.0000i − 1.50275i −0.659873 0.751377i \(-0.729390\pi\)
0.659873 0.751377i \(-0.270610\pi\)
\(938\) 0 0
\(939\) −20.0000 −0.652675
\(940\) 0 0
\(941\) −38.0000 −1.23876 −0.619382 0.785090i \(-0.712617\pi\)
−0.619382 + 0.785090i \(0.712617\pi\)
\(942\) 0 0
\(943\) − 24.0000i − 0.781548i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 14.0000i 0.454939i 0.973785 + 0.227469i \(0.0730452\pi\)
−0.973785 + 0.227469i \(0.926955\pi\)
\(948\) 0 0
\(949\) −28.0000 −0.908918
\(950\) 0 0
\(951\) −12.0000 −0.389127
\(952\) 0 0
\(953\) 58.0000i 1.87880i 0.342817 + 0.939402i \(0.388619\pi\)
−0.342817 + 0.939402i \(0.611381\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 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
\(965\) 0 0
\(966\) 0 0
\(967\) − 28.0000i − 0.900419i −0.892923 0.450210i \(-0.851349\pi\)
0.892923 0.450210i \(-0.148651\pi\)
\(968\) 0 0
\(969\) 8.00000 0.256997
\(970\) 0 0
\(971\) 38.0000 1.21948 0.609739 0.792602i \(-0.291274\pi\)
0.609739 + 0.792602i \(0.291274\pi\)
\(972\) 0 0
\(973\) 40.0000i 1.28234i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 30.0000i − 0.959785i −0.877327 0.479893i \(-0.840676\pi\)
0.877327 0.479893i \(-0.159324\pi\)
\(978\) 0 0
\(979\) −4.00000 −0.127841
\(980\) 0 0
\(981\) −6.00000 −0.191565
\(982\) 0 0
\(983\) 20.0000i 0.637901i 0.947771 + 0.318950i \(0.103330\pi\)
−0.947771 + 0.318950i \(0.896670\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 64.0000i − 2.03714i
\(988\) 0 0
\(989\) −24.0000 −0.763156
\(990\) 0 0
\(991\) 16.0000 0.508257 0.254128 0.967170i \(-0.418211\pi\)
0.254128 + 0.967170i \(0.418211\pi\)
\(992\) 0 0
\(993\) 28.0000i 0.888553i
\(994\) 0 0
\(995\) 0 0
\(996\) 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 3200.2.c.e.2049.2 2
4.3 odd 2 3200.2.c.k.2049.1 2
5.2 odd 4 128.2.a.d.1.1 yes 1
5.3 odd 4 3200.2.a.h.1.1 1
5.4 even 2 inner 3200.2.c.e.2049.1 2
8.3 odd 2 3200.2.c.f.2049.2 2
8.5 even 2 3200.2.c.l.2049.1 2
15.2 even 4 1152.2.a.c.1.1 1
20.3 even 4 3200.2.a.u.1.1 1
20.7 even 4 128.2.a.b.1.1 yes 1
20.19 odd 2 3200.2.c.k.2049.2 2
35.27 even 4 6272.2.a.a.1.1 1
40.3 even 4 3200.2.a.e.1.1 1
40.13 odd 4 3200.2.a.x.1.1 1
40.19 odd 2 3200.2.c.f.2049.1 2
40.27 even 4 128.2.a.c.1.1 yes 1
40.29 even 2 3200.2.c.l.2049.2 2
40.37 odd 4 128.2.a.a.1.1 1
60.47 odd 4 1152.2.a.h.1.1 1
80.27 even 4 256.2.b.a.129.2 2
80.37 odd 4 256.2.b.c.129.1 2
80.67 even 4 256.2.b.a.129.1 2
80.77 odd 4 256.2.b.c.129.2 2
120.77 even 4 1152.2.a.m.1.1 1
120.107 odd 4 1152.2.a.r.1.1 1
140.27 odd 4 6272.2.a.g.1.1 1
160.27 even 8 1024.2.e.m.769.2 4
160.37 odd 8 1024.2.e.i.769.1 4
160.67 even 8 1024.2.e.m.257.1 4
160.77 odd 8 1024.2.e.i.257.1 4
160.107 even 8 1024.2.e.m.769.1 4
160.117 odd 8 1024.2.e.i.769.2 4
160.147 even 8 1024.2.e.m.257.2 4
160.157 odd 8 1024.2.e.i.257.2 4
240.77 even 4 2304.2.d.r.1153.2 2
240.107 odd 4 2304.2.d.b.1153.1 2
240.197 even 4 2304.2.d.r.1153.1 2
240.227 odd 4 2304.2.d.b.1153.2 2
280.27 odd 4 6272.2.a.b.1.1 1
280.237 even 4 6272.2.a.h.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
128.2.a.a.1.1 1 40.37 odd 4
128.2.a.b.1.1 yes 1 20.7 even 4
128.2.a.c.1.1 yes 1 40.27 even 4
128.2.a.d.1.1 yes 1 5.2 odd 4
256.2.b.a.129.1 2 80.67 even 4
256.2.b.a.129.2 2 80.27 even 4
256.2.b.c.129.1 2 80.37 odd 4
256.2.b.c.129.2 2 80.77 odd 4
1024.2.e.i.257.1 4 160.77 odd 8
1024.2.e.i.257.2 4 160.157 odd 8
1024.2.e.i.769.1 4 160.37 odd 8
1024.2.e.i.769.2 4 160.117 odd 8
1024.2.e.m.257.1 4 160.67 even 8
1024.2.e.m.257.2 4 160.147 even 8
1024.2.e.m.769.1 4 160.107 even 8
1024.2.e.m.769.2 4 160.27 even 8
1152.2.a.c.1.1 1 15.2 even 4
1152.2.a.h.1.1 1 60.47 odd 4
1152.2.a.m.1.1 1 120.77 even 4
1152.2.a.r.1.1 1 120.107 odd 4
2304.2.d.b.1153.1 2 240.107 odd 4
2304.2.d.b.1153.2 2 240.227 odd 4
2304.2.d.r.1153.1 2 240.197 even 4
2304.2.d.r.1153.2 2 240.77 even 4
3200.2.a.e.1.1 1 40.3 even 4
3200.2.a.h.1.1 1 5.3 odd 4
3200.2.a.u.1.1 1 20.3 even 4
3200.2.a.x.1.1 1 40.13 odd 4
3200.2.c.e.2049.1 2 5.4 even 2 inner
3200.2.c.e.2049.2 2 1.1 even 1 trivial
3200.2.c.f.2049.1 2 40.19 odd 2
3200.2.c.f.2049.2 2 8.3 odd 2
3200.2.c.k.2049.1 2 4.3 odd 2
3200.2.c.k.2049.2 2 20.19 odd 2
3200.2.c.l.2049.1 2 8.5 even 2
3200.2.c.l.2049.2 2 40.29 even 2
6272.2.a.a.1.1 1 35.27 even 4
6272.2.a.b.1.1 1 280.27 odd 4
6272.2.a.g.1.1 1 140.27 odd 4
6272.2.a.h.1.1 1 280.237 even 4