Properties

Label 3900.2.h.b.1249.1
Level $3900$
Weight $2$
Character 3900.1249
Analytic conductor $31.142$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3900.h (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(31.1416567883\)
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: \( 1 \)
Twist minimal: no (minimal twist has level 156)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1249.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 3900.1249
Dual form 3900.2.h.b.1249.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{3} +2.00000i q^{7} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{3} +2.00000i q^{7} -1.00000 q^{9} -4.00000 q^{11} +1.00000i q^{13} -2.00000i q^{17} +2.00000 q^{19} +2.00000 q^{21} +1.00000i q^{27} +6.00000 q^{29} -10.0000 q^{31} +4.00000i q^{33} -10.0000i q^{37} +1.00000 q^{39} +8.00000 q^{41} +4.00000i q^{43} +4.00000i q^{47} +3.00000 q^{49} -2.00000 q^{51} -10.0000i q^{53} -2.00000i q^{57} +8.00000 q^{59} -14.0000 q^{61} -2.00000i q^{63} -2.00000i q^{67} +16.0000 q^{71} -10.0000i q^{73} -8.00000i q^{77} +16.0000 q^{79} +1.00000 q^{81} -6.00000i q^{87} +4.00000 q^{89} -2.00000 q^{91} +10.0000i q^{93} +2.00000i q^{97} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{9} - 8q^{11} + 4q^{19} + 4q^{21} + 12q^{29} - 20q^{31} + 2q^{39} + 16q^{41} + 6q^{49} - 4q^{51} + 16q^{59} - 28q^{61} + 32q^{71} + 32q^{79} + 2q^{81} + 8q^{89} - 4q^{91} + 8q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3900\mathbb{Z}\right)^\times\).

\(n\) \(301\) \(1301\) \(1951\) \(3277\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(3\) − 1.00000i − 0.577350i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 2.00000i 0.755929i 0.925820 + 0.377964i \(0.123376\pi\)
−0.925820 + 0.377964i \(0.876624\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 0 0
\(13\) 1.00000i 0.277350i
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 2.00000i − 0.485071i −0.970143 0.242536i \(-0.922021\pi\)
0.970143 0.242536i \(-0.0779791\pi\)
\(18\) 0 0
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) −10.0000 −1.79605 −0.898027 0.439941i \(-0.854999\pi\)
−0.898027 + 0.439941i \(0.854999\pi\)
\(32\) 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
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) 8.00000 1.24939 0.624695 0.780869i \(-0.285223\pi\)
0.624695 + 0.780869i \(0.285223\pi\)
\(42\) 0 0
\(43\) 4.00000i 0.609994i 0.952353 + 0.304997i \(0.0986555\pi\)
−0.952353 + 0.304997i \(0.901344\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.00000i 0.583460i 0.956501 + 0.291730i \(0.0942309\pi\)
−0.956501 + 0.291730i \(0.905769\pi\)
\(48\) 0 0
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) −2.00000 −0.280056
\(52\) 0 0
\(53\) − 10.0000i − 1.37361i −0.726844 0.686803i \(-0.759014\pi\)
0.726844 0.686803i \(-0.240986\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 2.00000i − 0.264906i
\(58\) 0 0
\(59\) 8.00000 1.04151 0.520756 0.853706i \(-0.325650\pi\)
0.520756 + 0.853706i \(0.325650\pi\)
\(60\) 0 0
\(61\) −14.0000 −1.79252 −0.896258 0.443533i \(-0.853725\pi\)
−0.896258 + 0.443533i \(0.853725\pi\)
\(62\) 0 0
\(63\) − 2.00000i − 0.251976i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 2.00000i − 0.244339i −0.992509 0.122169i \(-0.961015\pi\)
0.992509 0.122169i \(-0.0389851\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 16.0000 1.89885 0.949425 0.313993i \(-0.101667\pi\)
0.949425 + 0.313993i \(0.101667\pi\)
\(72\) 0 0
\(73\) − 10.0000i − 1.17041i −0.810885 0.585206i \(-0.801014\pi\)
0.810885 0.585206i \(-0.198986\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 8.00000i − 0.911685i
\(78\) 0 0
\(79\) 16.0000 1.80014 0.900070 0.435745i \(-0.143515\pi\)
0.900070 + 0.435745i \(0.143515\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 6.00000i − 0.643268i
\(88\) 0 0
\(89\) 4.00000 0.423999 0.212000 0.977270i \(-0.432002\pi\)
0.212000 + 0.977270i \(0.432002\pi\)
\(90\) 0 0
\(91\) −2.00000 −0.209657
\(92\) 0 0
\(93\) 10.0000i 1.03695i
\(94\) 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
\(99\) 4.00000 0.402015
\(100\) 0 0
\(101\) 10.0000 0.995037 0.497519 0.867453i \(-0.334245\pi\)
0.497519 + 0.867453i \(0.334245\pi\)
\(102\) 0 0
\(103\) − 8.00000i − 0.788263i −0.919054 0.394132i \(-0.871045\pi\)
0.919054 0.394132i \(-0.128955\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 12.0000i − 1.16008i −0.814587 0.580042i \(-0.803036\pi\)
0.814587 0.580042i \(-0.196964\pi\)
\(108\) 0 0
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) −10.0000 −0.949158
\(112\) 0 0
\(113\) 6.00000i 0.564433i 0.959351 + 0.282216i \(0.0910696\pi\)
−0.959351 + 0.282216i \(0.908930\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 1.00000i − 0.0924500i
\(118\) 0 0
\(119\) 4.00000 0.366679
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 0 0
\(123\) − 8.00000i − 0.721336i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 12.0000i − 1.06483i −0.846484 0.532414i \(-0.821285\pi\)
0.846484 0.532414i \(-0.178715\pi\)
\(128\) 0 0
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) 4.00000 0.349482 0.174741 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) 0 0
\(133\) 4.00000i 0.346844i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 8.00000i − 0.683486i −0.939793 0.341743i \(-0.888983\pi\)
0.939793 0.341743i \(-0.111017\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 4.00000 0.336861
\(142\) 0 0
\(143\) − 4.00000i − 0.334497i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 3.00000i − 0.247436i
\(148\) 0 0
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) 18.0000 1.46482 0.732410 0.680864i \(-0.238396\pi\)
0.732410 + 0.680864i \(0.238396\pi\)
\(152\) 0 0
\(153\) 2.00000i 0.161690i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 2.00000i − 0.159617i −0.996810 0.0798087i \(-0.974569\pi\)
0.996810 0.0798087i \(-0.0254309\pi\)
\(158\) 0 0
\(159\) −10.0000 −0.793052
\(160\) 0 0
\(161\) 0 0
\(162\) 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
\(167\) 12.0000i 0.928588i 0.885681 + 0.464294i \(0.153692\pi\)
−0.885681 + 0.464294i \(0.846308\pi\)
\(168\) 0 0
\(169\) −1.00000 −0.0769231
\(170\) 0 0
\(171\) −2.00000 −0.152944
\(172\) 0 0
\(173\) 2.00000i 0.152057i 0.997106 + 0.0760286i \(0.0242240\pi\)
−0.997106 + 0.0760286i \(0.975776\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 8.00000i − 0.601317i
\(178\) 0 0
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 0 0
\(183\) 14.0000i 1.03491i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 8.00000i 0.585018i
\(188\) 0 0
\(189\) −2.00000 −0.145479
\(190\) 0 0
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) 0 0
\(193\) − 14.0000i − 1.00774i −0.863779 0.503871i \(-0.831909\pi\)
0.863779 0.503871i \(-0.168091\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 12.0000i 0.854965i 0.904024 + 0.427482i \(0.140599\pi\)
−0.904024 + 0.427482i \(0.859401\pi\)
\(198\) 0 0
\(199\) −4.00000 −0.283552 −0.141776 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) 0 0
\(201\) −2.00000 −0.141069
\(202\) 0 0
\(203\) 12.0000i 0.842235i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −8.00000 −0.553372
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) 0 0
\(213\) − 16.0000i − 1.09630i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 20.0000i − 1.35769i
\(218\) 0 0
\(219\) −10.0000 −0.675737
\(220\) 0 0
\(221\) 2.00000 0.134535
\(222\) 0 0
\(223\) 14.0000i 0.937509i 0.883328 + 0.468755i \(0.155297\pi\)
−0.883328 + 0.468755i \(0.844703\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 4.00000i 0.265489i 0.991150 + 0.132745i \(0.0423790\pi\)
−0.991150 + 0.132745i \(0.957621\pi\)
\(228\) 0 0
\(229\) 2.00000 0.132164 0.0660819 0.997814i \(-0.478950\pi\)
0.0660819 + 0.997814i \(0.478950\pi\)
\(230\) 0 0
\(231\) −8.00000 −0.526361
\(232\) 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
\(243\) − 1.00000i − 0.0641500i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 2.00000i 0.127257i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −20.0000 −1.26239 −0.631194 0.775625i \(-0.717435\pi\)
−0.631194 + 0.775625i \(0.717435\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 30.0000i − 1.87135i −0.352865 0.935674i \(-0.614792\pi\)
0.352865 0.935674i \(-0.385208\pi\)
\(258\) 0 0
\(259\) 20.0000 1.24274
\(260\) 0 0
\(261\) −6.00000 −0.371391
\(262\) 0 0
\(263\) − 16.0000i − 0.986602i −0.869859 0.493301i \(-0.835790\pi\)
0.869859 0.493301i \(-0.164210\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 4.00000i − 0.244796i
\(268\) 0 0
\(269\) 14.0000 0.853595 0.426798 0.904347i \(-0.359642\pi\)
0.426798 + 0.904347i \(0.359642\pi\)
\(270\) 0 0
\(271\) −10.0000 −0.607457 −0.303728 0.952759i \(-0.598232\pi\)
−0.303728 + 0.952759i \(0.598232\pi\)
\(272\) 0 0
\(273\) 2.00000i 0.121046i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 22.0000i − 1.32185i −0.750451 0.660926i \(-0.770164\pi\)
0.750451 0.660926i \(-0.229836\pi\)
\(278\) 0 0
\(279\) 10.0000 0.598684
\(280\) 0 0
\(281\) −8.00000 −0.477240 −0.238620 0.971113i \(-0.576695\pi\)
−0.238620 + 0.971113i \(0.576695\pi\)
\(282\) 0 0
\(283\) − 16.0000i − 0.951101i −0.879688 0.475551i \(-0.842249\pi\)
0.879688 0.475551i \(-0.157751\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 16.0000i 0.944450i
\(288\) 0 0
\(289\) 13.0000 0.764706
\(290\) 0 0
\(291\) 2.00000 0.117242
\(292\) 0 0
\(293\) 8.00000i 0.467365i 0.972313 + 0.233682i \(0.0750776\pi\)
−0.972313 + 0.233682i \(0.924922\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 4.00000i − 0.232104i
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −8.00000 −0.461112
\(302\) 0 0
\(303\) − 10.0000i − 0.574485i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 22.0000i − 1.25561i −0.778372 0.627803i \(-0.783954\pi\)
0.778372 0.627803i \(-0.216046\pi\)
\(308\) 0 0
\(309\) −8.00000 −0.455104
\(310\) 0 0
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) 0 0
\(313\) − 6.00000i − 0.339140i −0.985518 0.169570i \(-0.945762\pi\)
0.985518 0.169570i \(-0.0542379\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 12.0000i 0.673987i 0.941507 + 0.336994i \(0.109410\pi\)
−0.941507 + 0.336994i \(0.890590\pi\)
\(318\) 0 0
\(319\) −24.0000 −1.34374
\(320\) 0 0
\(321\) −12.0000 −0.669775
\(322\) 0 0
\(323\) − 4.00000i − 0.222566i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 2.00000i − 0.110600i
\(328\) 0 0
\(329\) −8.00000 −0.441054
\(330\) 0 0
\(331\) 2.00000 0.109930 0.0549650 0.998488i \(-0.482495\pi\)
0.0549650 + 0.998488i \(0.482495\pi\)
\(332\) 0 0
\(333\) 10.0000i 0.547997i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 30.0000i 1.63420i 0.576493 + 0.817102i \(0.304421\pi\)
−0.576493 + 0.817102i \(0.695579\pi\)
\(338\) 0 0
\(339\) 6.00000 0.325875
\(340\) 0 0
\(341\) 40.0000 2.16612
\(342\) 0 0
\(343\) 20.0000i 1.07990i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 12.0000i − 0.644194i −0.946707 0.322097i \(-0.895612\pi\)
0.946707 0.322097i \(-0.104388\pi\)
\(348\) 0 0
\(349\) −18.0000 −0.963518 −0.481759 0.876304i \(-0.660002\pi\)
−0.481759 + 0.876304i \(0.660002\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) 0 0
\(353\) − 16.0000i − 0.851594i −0.904819 0.425797i \(-0.859994\pi\)
0.904819 0.425797i \(-0.140006\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 4.00000i − 0.211702i
\(358\) 0 0
\(359\) 12.0000 0.633336 0.316668 0.948536i \(-0.397436\pi\)
0.316668 + 0.948536i \(0.397436\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) − 5.00000i − 0.262432i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 4.00000i − 0.208798i −0.994535 0.104399i \(-0.966708\pi\)
0.994535 0.104399i \(-0.0332919\pi\)
\(368\) 0 0
\(369\) −8.00000 −0.416463
\(370\) 0 0
\(371\) 20.0000 1.03835
\(372\) 0 0
\(373\) − 18.0000i − 0.932005i −0.884783 0.466002i \(-0.845694\pi\)
0.884783 0.466002i \(-0.154306\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 6.00000i 0.309016i
\(378\) 0 0
\(379\) −6.00000 −0.308199 −0.154100 0.988055i \(-0.549248\pi\)
−0.154100 + 0.988055i \(0.549248\pi\)
\(380\) 0 0
\(381\) −12.0000 −0.614779
\(382\) 0 0
\(383\) 12.0000i 0.613171i 0.951843 + 0.306586i \(0.0991866\pi\)
−0.951843 + 0.306586i \(0.900813\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 4.00000i − 0.203331i
\(388\) 0 0
\(389\) 26.0000 1.31825 0.659126 0.752032i \(-0.270926\pi\)
0.659126 + 0.752032i \(0.270926\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) − 4.00000i − 0.201773i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 18.0000i − 0.903394i −0.892171 0.451697i \(-0.850819\pi\)
0.892171 0.451697i \(-0.149181\pi\)
\(398\) 0 0
\(399\) 4.00000 0.200250
\(400\) 0 0
\(401\) −12.0000 −0.599251 −0.299626 0.954057i \(-0.596862\pi\)
−0.299626 + 0.954057i \(0.596862\pi\)
\(402\) 0 0
\(403\) − 10.0000i − 0.498135i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 40.0000i 1.98273i
\(408\) 0 0
\(409\) −14.0000 −0.692255 −0.346128 0.938187i \(-0.612504\pi\)
−0.346128 + 0.938187i \(0.612504\pi\)
\(410\) 0 0
\(411\) −8.00000 −0.394611
\(412\) 0 0
\(413\) 16.0000i 0.787309i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 20.0000 0.977064 0.488532 0.872546i \(-0.337533\pi\)
0.488532 + 0.872546i \(0.337533\pi\)
\(420\) 0 0
\(421\) −2.00000 −0.0974740 −0.0487370 0.998812i \(-0.515520\pi\)
−0.0487370 + 0.998812i \(0.515520\pi\)
\(422\) 0 0
\(423\) − 4.00000i − 0.194487i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 28.0000i − 1.35501i
\(428\) 0 0
\(429\) −4.00000 −0.193122
\(430\) 0 0
\(431\) −20.0000 −0.963366 −0.481683 0.876346i \(-0.659974\pi\)
−0.481683 + 0.876346i \(0.659974\pi\)
\(432\) 0 0
\(433\) − 18.0000i − 0.865025i −0.901628 0.432512i \(-0.857627\pi\)
0.901628 0.432512i \(-0.142373\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −16.0000 −0.763638 −0.381819 0.924237i \(-0.624702\pi\)
−0.381819 + 0.924237i \(0.624702\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 0 0
\(443\) − 4.00000i − 0.190046i −0.995475 0.0950229i \(-0.969708\pi\)
0.995475 0.0950229i \(-0.0302924\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −12.0000 −0.566315 −0.283158 0.959073i \(-0.591382\pi\)
−0.283158 + 0.959073i \(0.591382\pi\)
\(450\) 0 0
\(451\) −32.0000 −1.50682
\(452\) 0 0
\(453\) − 18.0000i − 0.845714i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 18.0000i 0.842004i 0.907060 + 0.421002i \(0.138322\pi\)
−0.907060 + 0.421002i \(0.861678\pi\)
\(458\) 0 0
\(459\) 2.00000 0.0933520
\(460\) 0 0
\(461\) 12.0000 0.558896 0.279448 0.960161i \(-0.409849\pi\)
0.279448 + 0.960161i \(0.409849\pi\)
\(462\) 0 0
\(463\) 22.0000i 1.02243i 0.859454 + 0.511213i \(0.170804\pi\)
−0.859454 + 0.511213i \(0.829196\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 28.0000i 1.29569i 0.761774 + 0.647843i \(0.224329\pi\)
−0.761774 + 0.647843i \(0.775671\pi\)
\(468\) 0 0
\(469\) 4.00000 0.184703
\(470\) 0 0
\(471\) −2.00000 −0.0921551
\(472\) 0 0
\(473\) − 16.0000i − 0.735681i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 10.0000i 0.457869i
\(478\) 0 0
\(479\) 16.0000 0.731059 0.365529 0.930800i \(-0.380888\pi\)
0.365529 + 0.930800i \(0.380888\pi\)
\(480\) 0 0
\(481\) 10.0000 0.455961
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 26.0000i 1.17817i 0.808070 + 0.589086i \(0.200512\pi\)
−0.808070 + 0.589086i \(0.799488\pi\)
\(488\) 0 0
\(489\) 2.00000 0.0904431
\(490\) 0 0
\(491\) 12.0000 0.541552 0.270776 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) 0 0
\(493\) − 12.0000i − 0.540453i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 32.0000i 1.43540i
\(498\) 0 0
\(499\) 6.00000 0.268597 0.134298 0.990941i \(-0.457122\pi\)
0.134298 + 0.990941i \(0.457122\pi\)
\(500\) 0 0
\(501\) 12.0000 0.536120
\(502\) 0 0
\(503\) 16.0000i 0.713405i 0.934218 + 0.356702i \(0.116099\pi\)
−0.934218 + 0.356702i \(0.883901\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 1.00000i 0.0444116i
\(508\) 0 0
\(509\) −12.0000 −0.531891 −0.265945 0.963988i \(-0.585684\pi\)
−0.265945 + 0.963988i \(0.585684\pi\)
\(510\) 0 0
\(511\) 20.0000 0.884748
\(512\) 0 0
\(513\) 2.00000i 0.0883022i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 16.0000i − 0.703679i
\(518\) 0 0
\(519\) 2.00000 0.0877903
\(520\) 0 0
\(521\) −2.00000 −0.0876216 −0.0438108 0.999040i \(-0.513950\pi\)
−0.0438108 + 0.999040i \(0.513950\pi\)
\(522\) 0 0
\(523\) 16.0000i 0.699631i 0.936819 + 0.349816i \(0.113756\pi\)
−0.936819 + 0.349816i \(0.886244\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 20.0000i 0.871214i
\(528\) 0 0
\(529\) 23.0000 1.00000
\(530\) 0 0
\(531\) −8.00000 −0.347170
\(532\) 0 0
\(533\) 8.00000i 0.346518i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 12.0000i 0.517838i
\(538\) 0 0
\(539\) −12.0000 −0.516877
\(540\) 0 0
\(541\) 38.0000 1.63375 0.816874 0.576816i \(-0.195705\pi\)
0.816874 + 0.576816i \(0.195705\pi\)
\(542\) 0 0
\(543\) − 2.00000i − 0.0858282i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 32.0000i − 1.36822i −0.729378 0.684111i \(-0.760191\pi\)
0.729378 0.684111i \(-0.239809\pi\)
\(548\) 0 0
\(549\) 14.0000 0.597505
\(550\) 0 0
\(551\) 12.0000 0.511217
\(552\) 0 0
\(553\) 32.0000i 1.36078i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 24.0000i − 1.01691i −0.861088 0.508456i \(-0.830216\pi\)
0.861088 0.508456i \(-0.169784\pi\)
\(558\) 0 0
\(559\) −4.00000 −0.169182
\(560\) 0 0
\(561\) 8.00000 0.337760
\(562\) 0 0
\(563\) 28.0000i 1.18006i 0.807382 + 0.590030i \(0.200884\pi\)
−0.807382 + 0.590030i \(0.799116\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 2.00000i 0.0839921i
\(568\) 0 0
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 0 0
\(571\) −8.00000 −0.334790 −0.167395 0.985890i \(-0.553535\pi\)
−0.167395 + 0.985890i \(0.553535\pi\)
\(572\) 0 0
\(573\) 8.00000i 0.334205i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 30.0000i 1.24892i 0.781058 + 0.624458i \(0.214680\pi\)
−0.781058 + 0.624458i \(0.785320\pi\)
\(578\) 0 0
\(579\) −14.0000 −0.581820
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 40.0000i 1.65663i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 8.00000i 0.330195i 0.986277 + 0.165098i \(0.0527939\pi\)
−0.986277 + 0.165098i \(0.947206\pi\)
\(588\) 0 0
\(589\) −20.0000 −0.824086
\(590\) 0 0
\(591\) 12.0000 0.493614
\(592\) 0 0
\(593\) − 36.0000i − 1.47834i −0.673517 0.739171i \(-0.735217\pi\)
0.673517 0.739171i \(-0.264783\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 4.00000i 0.163709i
\(598\) 0 0
\(599\) −24.0000 −0.980613 −0.490307 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 0 0
\(601\) −22.0000 −0.897399 −0.448699 0.893683i \(-0.648113\pi\)
−0.448699 + 0.893683i \(0.648113\pi\)
\(602\) 0 0
\(603\) 2.00000i 0.0814463i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 4.00000i 0.162355i 0.996700 + 0.0811775i \(0.0258681\pi\)
−0.996700 + 0.0811775i \(0.974132\pi\)
\(608\) 0 0
\(609\) 12.0000 0.486265
\(610\) 0 0
\(611\) −4.00000 −0.161823
\(612\) 0 0
\(613\) − 14.0000i − 0.565455i −0.959200 0.282727i \(-0.908761\pi\)
0.959200 0.282727i \(-0.0912392\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 48.0000i 1.93241i 0.257780 + 0.966204i \(0.417009\pi\)
−0.257780 + 0.966204i \(0.582991\pi\)
\(618\) 0 0
\(619\) −14.0000 −0.562708 −0.281354 0.959604i \(-0.590783\pi\)
−0.281354 + 0.959604i \(0.590783\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 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
\(631\) −30.0000 −1.19428 −0.597141 0.802137i \(-0.703697\pi\)
−0.597141 + 0.802137i \(0.703697\pi\)
\(632\) 0 0
\(633\) 12.0000i 0.476957i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 3.00000i 0.118864i
\(638\) 0 0
\(639\) −16.0000 −0.632950
\(640\) 0 0
\(641\) 26.0000 1.02694 0.513469 0.858108i \(-0.328360\pi\)
0.513469 + 0.858108i \(0.328360\pi\)
\(642\) 0 0
\(643\) 26.0000i 1.02534i 0.858586 + 0.512670i \(0.171344\pi\)
−0.858586 + 0.512670i \(0.828656\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 24.0000i 0.943537i 0.881722 + 0.471769i \(0.156384\pi\)
−0.881722 + 0.471769i \(0.843616\pi\)
\(648\) 0 0
\(649\) −32.0000 −1.25611
\(650\) 0 0
\(651\) −20.0000 −0.783862
\(652\) 0 0
\(653\) − 46.0000i − 1.80012i −0.435767 0.900060i \(-0.643523\pi\)
0.435767 0.900060i \(-0.356477\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 10.0000i 0.390137i
\(658\) 0 0
\(659\) 12.0000 0.467454 0.233727 0.972302i \(-0.424908\pi\)
0.233727 + 0.972302i \(0.424908\pi\)
\(660\) 0 0
\(661\) 10.0000 0.388955 0.194477 0.980907i \(-0.437699\pi\)
0.194477 + 0.980907i \(0.437699\pi\)
\(662\) 0 0
\(663\) − 2.00000i − 0.0776736i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 14.0000 0.541271
\(670\) 0 0
\(671\) 56.0000 2.16186
\(672\) 0 0
\(673\) 22.0000i 0.848038i 0.905653 + 0.424019i \(0.139381\pi\)
−0.905653 + 0.424019i \(0.860619\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 38.0000i − 1.46046i −0.683202 0.730229i \(-0.739413\pi\)
0.683202 0.730229i \(-0.260587\pi\)
\(678\) 0 0
\(679\) −4.00000 −0.153506
\(680\) 0 0
\(681\) 4.00000 0.153280
\(682\) 0 0
\(683\) 20.0000i 0.765279i 0.923898 + 0.382639i \(0.124985\pi\)
−0.923898 + 0.382639i \(0.875015\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 2.00000i − 0.0763048i
\(688\) 0 0
\(689\) 10.0000 0.380970
\(690\) 0 0
\(691\) 10.0000 0.380418 0.190209 0.981744i \(-0.439083\pi\)
0.190209 + 0.981744i \(0.439083\pi\)
\(692\) 0 0
\(693\) 8.00000i 0.303895i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 16.0000i − 0.606043i
\(698\) 0 0
\(699\) −18.0000 −0.680823
\(700\) 0 0
\(701\) −2.00000 −0.0755390 −0.0377695 0.999286i \(-0.512025\pi\)
−0.0377695 + 0.999286i \(0.512025\pi\)
\(702\) 0 0
\(703\) − 20.0000i − 0.754314i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 20.0000i 0.752177i
\(708\) 0 0
\(709\) −6.00000 −0.225335 −0.112667 0.993633i \(-0.535939\pi\)
−0.112667 + 0.993633i \(0.535939\pi\)
\(710\) 0 0
\(711\) −16.0000 −0.600047
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 24.0000i − 0.896296i
\(718\) 0 0
\(719\) 48.0000 1.79010 0.895049 0.445968i \(-0.147140\pi\)
0.895049 + 0.445968i \(0.147140\pi\)
\(720\) 0 0
\(721\) 16.0000 0.595871
\(722\) 0 0
\(723\) 2.00000i 0.0743808i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 44.0000i − 1.63187i −0.578144 0.815935i \(-0.696223\pi\)
0.578144 0.815935i \(-0.303777\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 8.00000 0.295891
\(732\) 0 0
\(733\) 22.0000i 0.812589i 0.913742 + 0.406294i \(0.133179\pi\)
−0.913742 + 0.406294i \(0.866821\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 8.00000i 0.294684i
\(738\) 0 0
\(739\) −22.0000 −0.809283 −0.404642 0.914475i \(-0.632604\pi\)
−0.404642 + 0.914475i \(0.632604\pi\)
\(740\) 0 0
\(741\) 2.00000 0.0734718
\(742\) 0 0
\(743\) 48.0000i 1.76095i 0.474093 + 0.880475i \(0.342776\pi\)
−0.474093 + 0.880475i \(0.657224\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 24.0000 0.876941
\(750\) 0 0
\(751\) 8.00000 0.291924 0.145962 0.989290i \(-0.453372\pi\)
0.145962 + 0.989290i \(0.453372\pi\)
\(752\) 0 0
\(753\) 20.0000i 0.728841i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 18.0000i − 0.654221i −0.944986 0.327111i \(-0.893925\pi\)
0.944986 0.327111i \(-0.106075\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 40.0000 1.45000 0.724999 0.688749i \(-0.241840\pi\)
0.724999 + 0.688749i \(0.241840\pi\)
\(762\) 0 0
\(763\) 4.00000i 0.144810i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 8.00000i 0.288863i
\(768\) 0 0
\(769\) 22.0000 0.793340 0.396670 0.917961i \(-0.370166\pi\)
0.396670 + 0.917961i \(0.370166\pi\)
\(770\) 0 0
\(771\) −30.0000 −1.08042
\(772\) 0 0
\(773\) − 48.0000i − 1.72644i −0.504828 0.863220i \(-0.668444\pi\)
0.504828 0.863220i \(-0.331556\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 20.0000i − 0.717496i
\(778\) 0 0
\(779\) 16.0000 0.573259
\(780\) 0 0
\(781\) −64.0000 −2.29010
\(782\) 0 0
\(783\) 6.00000i 0.214423i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 50.0000i 1.78231i 0.453701 + 0.891154i \(0.350103\pi\)
−0.453701 + 0.891154i \(0.649897\pi\)
\(788\) 0 0
\(789\) −16.0000 −0.569615
\(790\) 0 0
\(791\) −12.0000 −0.426671
\(792\) 0 0
\(793\) − 14.0000i − 0.497155i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 26.0000i 0.920967i 0.887668 + 0.460484i \(0.152324\pi\)
−0.887668 + 0.460484i \(0.847676\pi\)
\(798\) 0 0
\(799\) 8.00000 0.283020
\(800\) 0 0
\(801\) −4.00000 −0.141333
\(802\) 0 0
\(803\) 40.0000i 1.41157i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 14.0000i − 0.492823i
\(808\) 0 0
\(809\) −38.0000 −1.33601 −0.668004 0.744157i \(-0.732851\pi\)
−0.668004 + 0.744157i \(0.732851\pi\)
\(810\) 0 0
\(811\) 6.00000 0.210688 0.105344 0.994436i \(-0.466406\pi\)
0.105344 + 0.994436i \(0.466406\pi\)
\(812\) 0 0
\(813\) 10.0000i 0.350715i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 8.00000i 0.279885i
\(818\) 0 0
\(819\) 2.00000 0.0698857
\(820\) 0 0
\(821\) 24.0000 0.837606 0.418803 0.908077i \(-0.362450\pi\)
0.418803 + 0.908077i \(0.362450\pi\)
\(822\) 0 0
\(823\) 32.0000i 1.11545i 0.830026 + 0.557725i \(0.188326\pi\)
−0.830026 + 0.557725i \(0.811674\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 20.0000i 0.695468i 0.937593 + 0.347734i \(0.113049\pi\)
−0.937593 + 0.347734i \(0.886951\pi\)
\(828\) 0 0
\(829\) 14.0000 0.486240 0.243120 0.969996i \(-0.421829\pi\)
0.243120 + 0.969996i \(0.421829\pi\)
\(830\) 0 0
\(831\) −22.0000 −0.763172
\(832\) 0 0
\(833\) − 6.00000i − 0.207888i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 10.0000i − 0.345651i
\(838\) 0 0
\(839\) 16.0000 0.552381 0.276191 0.961103i \(-0.410928\pi\)
0.276191 + 0.961103i \(0.410928\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) 8.00000i 0.275535i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 10.0000i 0.343604i
\(848\) 0 0
\(849\) −16.0000 −0.549119
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) − 14.0000i − 0.479351i −0.970853 0.239675i \(-0.922959\pi\)
0.970853 0.239675i \(-0.0770410\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 18.0000i 0.614868i 0.951569 + 0.307434i \(0.0994704\pi\)
−0.951569 + 0.307434i \(0.900530\pi\)
\(858\) 0 0
\(859\) −52.0000 −1.77422 −0.887109 0.461561i \(-0.847290\pi\)
−0.887109 + 0.461561i \(0.847290\pi\)
\(860\) 0 0
\(861\) 16.0000 0.545279
\(862\) 0 0
\(863\) 4.00000i 0.136162i 0.997680 + 0.0680808i \(0.0216876\pi\)
−0.997680 + 0.0680808i \(0.978312\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 13.0000i − 0.441503i
\(868\) 0 0
\(869\) −64.0000 −2.17105
\(870\) 0 0
\(871\) 2.00000 0.0677674
\(872\) 0 0
\(873\) − 2.00000i − 0.0676897i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 10.0000i − 0.337676i −0.985644 0.168838i \(-0.945999\pi\)
0.985644 0.168838i \(-0.0540015\pi\)
\(878\) 0 0
\(879\) 8.00000 0.269833
\(880\) 0 0
\(881\) 14.0000 0.471672 0.235836 0.971793i \(-0.424217\pi\)
0.235836 + 0.971793i \(0.424217\pi\)
\(882\) 0 0
\(883\) − 32.0000i − 1.07689i −0.842662 0.538443i \(-0.819013\pi\)
0.842662 0.538443i \(-0.180987\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 48.0000i − 1.61168i −0.592132 0.805841i \(-0.701714\pi\)
0.592132 0.805841i \(-0.298286\pi\)
\(888\) 0 0
\(889\) 24.0000 0.804934
\(890\) 0 0
\(891\) −4.00000 −0.134005
\(892\) 0 0
\(893\) 8.00000i 0.267710i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −60.0000 −2.00111
\(900\) 0 0
\(901\) −20.0000 −0.666297
\(902\) 0 0
\(903\) 8.00000i 0.266223i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 56.0000i 1.85945i 0.368255 + 0.929725i \(0.379955\pi\)
−0.368255 + 0.929725i \(0.620045\pi\)
\(908\) 0 0
\(909\) −10.0000 −0.331679
\(910\) 0 0
\(911\) −56.0000 −1.85536 −0.927681 0.373373i \(-0.878201\pi\)
−0.927681 + 0.373373i \(0.878201\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 8.00000i 0.264183i
\(918\) 0 0
\(919\) −56.0000 −1.84727 −0.923635 0.383274i \(-0.874797\pi\)
−0.923635 + 0.383274i \(0.874797\pi\)
\(920\) 0 0
\(921\) −22.0000 −0.724925
\(922\) 0 0
\(923\) 16.0000i 0.526646i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 8.00000i 0.262754i
\(928\) 0 0
\(929\) −16.0000 −0.524943 −0.262471 0.964940i \(-0.584538\pi\)
−0.262471 + 0.964940i \(0.584538\pi\)
\(930\) 0 0
\(931\) 6.00000 0.196642
\(932\) 0 0
\(933\) − 24.0000i − 0.785725i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 34.0000i 1.11073i 0.831606 + 0.555366i \(0.187422\pi\)
−0.831606 + 0.555366i \(0.812578\pi\)
\(938\) 0 0
\(939\) −6.00000 −0.195803
\(940\) 0 0
\(941\) 8.00000 0.260793 0.130396 0.991462i \(-0.458375\pi\)
0.130396 + 0.991462i \(0.458375\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 8.00000i − 0.259965i −0.991516 0.129983i \(-0.958508\pi\)
0.991516 0.129983i \(-0.0414921\pi\)
\(948\) 0 0
\(949\) 10.0000 0.324614
\(950\) 0 0
\(951\) 12.0000 0.389127
\(952\) 0 0
\(953\) − 42.0000i − 1.36051i −0.732974 0.680257i \(-0.761868\pi\)
0.732974 0.680257i \(-0.238132\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 24.0000i 0.775810i
\(958\) 0 0
\(959\) 16.0000 0.516667
\(960\) 0 0
\(961\) 69.0000 2.22581
\(962\) 0 0
\(963\) 12.0000i 0.386695i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 22.0000i − 0.707472i −0.935345 0.353736i \(-0.884911\pi\)
0.935345 0.353736i \(-0.115089\pi\)
\(968\) 0 0
\(969\) −4.00000 −0.128499
\(970\) 0 0
\(971\) −28.0000 −0.898563 −0.449281 0.893390i \(-0.648320\pi\)
−0.449281 + 0.893390i \(0.648320\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 12.0000i 0.383914i 0.981403 + 0.191957i \(0.0614834\pi\)
−0.981403 + 0.191957i \(0.938517\pi\)
\(978\) 0 0
\(979\) −16.0000 −0.511362
\(980\) 0 0
\(981\) −2.00000 −0.0638551
\(982\) 0 0
\(983\) 24.0000i 0.765481i 0.923856 + 0.382741i \(0.125020\pi\)
−0.923856 + 0.382741i \(0.874980\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 8.00000i 0.254643i
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −36.0000 −1.14358 −0.571789 0.820401i \(-0.693750\pi\)
−0.571789 + 0.820401i \(0.693750\pi\)
\(992\) 0 0
\(993\) − 2.00000i − 0.0634681i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 22.0000i − 0.696747i −0.937356 0.348373i \(-0.886734\pi\)
0.937356 0.348373i \(-0.113266\pi\)
\(998\) 0 0
\(999\) 10.0000 0.316386
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 3900.2.h.b.1249.1 2
5.2 odd 4 156.2.a.a.1.1 1
5.3 odd 4 3900.2.a.m.1.1 1
5.4 even 2 inner 3900.2.h.b.1249.2 2
15.2 even 4 468.2.a.d.1.1 1
20.7 even 4 624.2.a.e.1.1 1
35.27 even 4 7644.2.a.k.1.1 1
40.27 even 4 2496.2.a.o.1.1 1
40.37 odd 4 2496.2.a.bc.1.1 1
45.2 even 12 4212.2.i.b.1405.1 2
45.7 odd 12 4212.2.i.l.1405.1 2
45.22 odd 12 4212.2.i.l.2809.1 2
45.32 even 12 4212.2.i.b.2809.1 2
60.47 odd 4 1872.2.a.s.1.1 1
65.2 even 12 2028.2.q.h.1837.2 4
65.7 even 12 2028.2.q.h.361.1 4
65.12 odd 4 2028.2.a.c.1.1 1
65.17 odd 12 2028.2.i.g.2005.1 2
65.22 odd 12 2028.2.i.e.2005.1 2
65.32 even 12 2028.2.q.h.361.2 4
65.37 even 12 2028.2.q.h.1837.1 4
65.42 odd 12 2028.2.i.e.529.1 2
65.47 even 4 2028.2.b.a.337.1 2
65.57 even 4 2028.2.b.a.337.2 2
65.62 odd 12 2028.2.i.g.529.1 2
120.77 even 4 7488.2.a.c.1.1 1
120.107 odd 4 7488.2.a.d.1.1 1
195.47 odd 4 6084.2.b.j.4393.2 2
195.77 even 4 6084.2.a.b.1.1 1
195.122 odd 4 6084.2.b.j.4393.1 2
260.207 even 4 8112.2.a.bi.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
156.2.a.a.1.1 1 5.2 odd 4
468.2.a.d.1.1 1 15.2 even 4
624.2.a.e.1.1 1 20.7 even 4
1872.2.a.s.1.1 1 60.47 odd 4
2028.2.a.c.1.1 1 65.12 odd 4
2028.2.b.a.337.1 2 65.47 even 4
2028.2.b.a.337.2 2 65.57 even 4
2028.2.i.e.529.1 2 65.42 odd 12
2028.2.i.e.2005.1 2 65.22 odd 12
2028.2.i.g.529.1 2 65.62 odd 12
2028.2.i.g.2005.1 2 65.17 odd 12
2028.2.q.h.361.1 4 65.7 even 12
2028.2.q.h.361.2 4 65.32 even 12
2028.2.q.h.1837.1 4 65.37 even 12
2028.2.q.h.1837.2 4 65.2 even 12
2496.2.a.o.1.1 1 40.27 even 4
2496.2.a.bc.1.1 1 40.37 odd 4
3900.2.a.m.1.1 1 5.3 odd 4
3900.2.h.b.1249.1 2 1.1 even 1 trivial
3900.2.h.b.1249.2 2 5.4 even 2 inner
4212.2.i.b.1405.1 2 45.2 even 12
4212.2.i.b.2809.1 2 45.32 even 12
4212.2.i.l.1405.1 2 45.7 odd 12
4212.2.i.l.2809.1 2 45.22 odd 12
6084.2.a.b.1.1 1 195.77 even 4
6084.2.b.j.4393.1 2 195.122 odd 4
6084.2.b.j.4393.2 2 195.47 odd 4
7488.2.a.c.1.1 1 120.77 even 4
7488.2.a.d.1.1 1 120.107 odd 4
7644.2.a.k.1.1 1 35.27 even 4
8112.2.a.bi.1.1 1 260.207 even 4