Properties

Label 4400.2.a.bu.1.2
Level $4400$
Weight $2$
Character 4400.1
Self dual yes
Analytic conductor $35.134$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4400 = 2^{4} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4400.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(35.1341768894\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{21}) \)
Defining polynomial: \( x^{2} - x - 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1100)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.79129\) of defining polynomial
Character \(\chi\) \(=\) 4400.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.79129 q^{3} +0.208712 q^{7} +4.79129 q^{9} +O(q^{10})\) \(q+2.79129 q^{3} +0.208712 q^{7} +4.79129 q^{9} +1.00000 q^{11} -1.00000 q^{13} +0.791288 q^{17} -6.58258 q^{19} +0.582576 q^{21} +3.79129 q^{23} +5.00000 q^{27} +6.79129 q^{29} +8.58258 q^{31} +2.79129 q^{33} -2.58258 q^{37} -2.79129 q^{39} -1.41742 q^{41} +10.0000 q^{43} +1.41742 q^{47} -6.95644 q^{49} +2.20871 q^{51} +11.3739 q^{53} -18.3739 q^{57} +10.5826 q^{59} +4.20871 q^{61} +1.00000 q^{63} +4.00000 q^{67} +10.5826 q^{69} +10.7477 q^{71} -7.79129 q^{73} +0.208712 q^{77} +15.5390 q^{79} -0.417424 q^{81} +9.95644 q^{83} +18.9564 q^{87} -0.791288 q^{89} -0.208712 q^{91} +23.9564 q^{93} -6.20871 q^{97} +4.79129 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} + 5 q^{7} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{3} + 5 q^{7} + 5 q^{9} + 2 q^{11} - 2 q^{13} - 3 q^{17} - 4 q^{19} - 8 q^{21} + 3 q^{23} + 10 q^{27} + 9 q^{29} + 8 q^{31} + q^{33} + 4 q^{37} - q^{39} - 12 q^{41} + 20 q^{43} + 12 q^{47} + 9 q^{49} + 9 q^{51} + 9 q^{53} - 23 q^{57} + 12 q^{59} + 13 q^{61} + 2 q^{63} + 8 q^{67} + 12 q^{69} - 6 q^{71} - 11 q^{73} + 5 q^{77} - q^{79} - 10 q^{81} - 3 q^{83} + 15 q^{87} + 3 q^{89} - 5 q^{91} + 25 q^{93} - 17 q^{97} + 5 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) 2.79129 1.61155 0.805775 0.592221i \(-0.201749\pi\)
0.805775 + 0.592221i \(0.201749\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0.208712 0.0788858 0.0394429 0.999222i \(-0.487442\pi\)
0.0394429 + 0.999222i \(0.487442\pi\)
\(8\) 0 0
\(9\) 4.79129 1.59710
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) −1.00000 −0.277350 −0.138675 0.990338i \(-0.544284\pi\)
−0.138675 + 0.990338i \(0.544284\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0.791288 0.191915 0.0959577 0.995385i \(-0.469409\pi\)
0.0959577 + 0.995385i \(0.469409\pi\)
\(18\) 0 0
\(19\) −6.58258 −1.51015 −0.755073 0.655640i \(-0.772399\pi\)
−0.755073 + 0.655640i \(0.772399\pi\)
\(20\) 0 0
\(21\) 0.582576 0.127128
\(22\) 0 0
\(23\) 3.79129 0.790538 0.395269 0.918565i \(-0.370651\pi\)
0.395269 + 0.918565i \(0.370651\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 5.00000 0.962250
\(28\) 0 0
\(29\) 6.79129 1.26111 0.630555 0.776144i \(-0.282827\pi\)
0.630555 + 0.776144i \(0.282827\pi\)
\(30\) 0 0
\(31\) 8.58258 1.54148 0.770738 0.637152i \(-0.219888\pi\)
0.770738 + 0.637152i \(0.219888\pi\)
\(32\) 0 0
\(33\) 2.79129 0.485901
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −2.58258 −0.424573 −0.212286 0.977207i \(-0.568091\pi\)
−0.212286 + 0.977207i \(0.568091\pi\)
\(38\) 0 0
\(39\) −2.79129 −0.446964
\(40\) 0 0
\(41\) −1.41742 −0.221364 −0.110682 0.993856i \(-0.535304\pi\)
−0.110682 + 0.993856i \(0.535304\pi\)
\(42\) 0 0
\(43\) 10.0000 1.52499 0.762493 0.646997i \(-0.223975\pi\)
0.762493 + 0.646997i \(0.223975\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.41742 0.206753 0.103376 0.994642i \(-0.467035\pi\)
0.103376 + 0.994642i \(0.467035\pi\)
\(48\) 0 0
\(49\) −6.95644 −0.993777
\(50\) 0 0
\(51\) 2.20871 0.309282
\(52\) 0 0
\(53\) 11.3739 1.56232 0.781160 0.624331i \(-0.214628\pi\)
0.781160 + 0.624331i \(0.214628\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −18.3739 −2.43368
\(58\) 0 0
\(59\) 10.5826 1.37773 0.688867 0.724888i \(-0.258108\pi\)
0.688867 + 0.724888i \(0.258108\pi\)
\(60\) 0 0
\(61\) 4.20871 0.538870 0.269435 0.963019i \(-0.413163\pi\)
0.269435 + 0.963019i \(0.413163\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 0 0
\(69\) 10.5826 1.27399
\(70\) 0 0
\(71\) 10.7477 1.27552 0.637760 0.770235i \(-0.279861\pi\)
0.637760 + 0.770235i \(0.279861\pi\)
\(72\) 0 0
\(73\) −7.79129 −0.911901 −0.455951 0.890005i \(-0.650701\pi\)
−0.455951 + 0.890005i \(0.650701\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0.208712 0.0237850
\(78\) 0 0
\(79\) 15.5390 1.74828 0.874138 0.485678i \(-0.161427\pi\)
0.874138 + 0.485678i \(0.161427\pi\)
\(80\) 0 0
\(81\) −0.417424 −0.0463805
\(82\) 0 0
\(83\) 9.95644 1.09286 0.546431 0.837504i \(-0.315986\pi\)
0.546431 + 0.837504i \(0.315986\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 18.9564 2.03234
\(88\) 0 0
\(89\) −0.791288 −0.0838763 −0.0419382 0.999120i \(-0.513353\pi\)
−0.0419382 + 0.999120i \(0.513353\pi\)
\(90\) 0 0
\(91\) −0.208712 −0.0218790
\(92\) 0 0
\(93\) 23.9564 2.48417
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −6.20871 −0.630399 −0.315200 0.949025i \(-0.602071\pi\)
−0.315200 + 0.949025i \(0.602071\pi\)
\(98\) 0 0
\(99\) 4.79129 0.481543
\(100\) 0 0
\(101\) −17.3739 −1.72876 −0.864382 0.502836i \(-0.832290\pi\)
−0.864382 + 0.502836i \(0.832290\pi\)
\(102\) 0 0
\(103\) −5.95644 −0.586905 −0.293453 0.955974i \(-0.594804\pi\)
−0.293453 + 0.955974i \(0.594804\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −19.7477 −1.90908 −0.954542 0.298075i \(-0.903655\pi\)
−0.954542 + 0.298075i \(0.903655\pi\)
\(108\) 0 0
\(109\) −4.79129 −0.458922 −0.229461 0.973318i \(-0.573696\pi\)
−0.229461 + 0.973318i \(0.573696\pi\)
\(110\) 0 0
\(111\) −7.20871 −0.684221
\(112\) 0 0
\(113\) 1.41742 0.133340 0.0666700 0.997775i \(-0.478763\pi\)
0.0666700 + 0.997775i \(0.478763\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −4.79129 −0.442955
\(118\) 0 0
\(119\) 0.165151 0.0151394
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) −3.95644 −0.356740
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −16.3739 −1.45295 −0.726473 0.687195i \(-0.758842\pi\)
−0.726473 + 0.687195i \(0.758842\pi\)
\(128\) 0 0
\(129\) 27.9129 2.45759
\(130\) 0 0
\(131\) 0.626136 0.0547058 0.0273529 0.999626i \(-0.491292\pi\)
0.0273529 + 0.999626i \(0.491292\pi\)
\(132\) 0 0
\(133\) −1.37386 −0.119129
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −11.5390 −0.985845 −0.492922 0.870073i \(-0.664071\pi\)
−0.492922 + 0.870073i \(0.664071\pi\)
\(138\) 0 0
\(139\) −9.74773 −0.826791 −0.413396 0.910551i \(-0.635657\pi\)
−0.413396 + 0.910551i \(0.635657\pi\)
\(140\) 0 0
\(141\) 3.95644 0.333192
\(142\) 0 0
\(143\) −1.00000 −0.0836242
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −19.4174 −1.60152
\(148\) 0 0
\(149\) 3.16515 0.259299 0.129650 0.991560i \(-0.458615\pi\)
0.129650 + 0.991560i \(0.458615\pi\)
\(150\) 0 0
\(151\) −5.00000 −0.406894 −0.203447 0.979086i \(-0.565214\pi\)
−0.203447 + 0.979086i \(0.565214\pi\)
\(152\) 0 0
\(153\) 3.79129 0.306507
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 17.1652 1.36993 0.684964 0.728577i \(-0.259818\pi\)
0.684964 + 0.728577i \(0.259818\pi\)
\(158\) 0 0
\(159\) 31.7477 2.51776
\(160\) 0 0
\(161\) 0.791288 0.0623622
\(162\) 0 0
\(163\) 21.3739 1.67413 0.837065 0.547103i \(-0.184270\pi\)
0.837065 + 0.547103i \(0.184270\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 16.5826 1.28320 0.641599 0.767040i \(-0.278271\pi\)
0.641599 + 0.767040i \(0.278271\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) −31.5390 −2.41185
\(172\) 0 0
\(173\) −13.7477 −1.04522 −0.522610 0.852572i \(-0.675042\pi\)
−0.522610 + 0.852572i \(0.675042\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 29.5390 2.22029
\(178\) 0 0
\(179\) −22.1216 −1.65345 −0.826723 0.562610i \(-0.809797\pi\)
−0.826723 + 0.562610i \(0.809797\pi\)
\(180\) 0 0
\(181\) −3.37386 −0.250777 −0.125389 0.992108i \(-0.540018\pi\)
−0.125389 + 0.992108i \(0.540018\pi\)
\(182\) 0 0
\(183\) 11.7477 0.868417
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0.791288 0.0578647
\(188\) 0 0
\(189\) 1.04356 0.0759079
\(190\) 0 0
\(191\) −17.2087 −1.24518 −0.622589 0.782549i \(-0.713919\pi\)
−0.622589 + 0.782549i \(0.713919\pi\)
\(192\) 0 0
\(193\) −7.16515 −0.515759 −0.257879 0.966177i \(-0.583024\pi\)
−0.257879 + 0.966177i \(0.583024\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 26.5390 1.89083 0.945413 0.325874i \(-0.105658\pi\)
0.945413 + 0.325874i \(0.105658\pi\)
\(198\) 0 0
\(199\) 1.62614 0.115274 0.0576369 0.998338i \(-0.481643\pi\)
0.0576369 + 0.998338i \(0.481643\pi\)
\(200\) 0 0
\(201\) 11.1652 0.787529
\(202\) 0 0
\(203\) 1.41742 0.0994837
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 18.1652 1.26257
\(208\) 0 0
\(209\) −6.58258 −0.455326
\(210\) 0 0
\(211\) −5.00000 −0.344214 −0.172107 0.985078i \(-0.555058\pi\)
−0.172107 + 0.985078i \(0.555058\pi\)
\(212\) 0 0
\(213\) 30.0000 2.05557
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 1.79129 0.121601
\(218\) 0 0
\(219\) −21.7477 −1.46958
\(220\) 0 0
\(221\) −0.791288 −0.0532278
\(222\) 0 0
\(223\) 20.5826 1.37831 0.689156 0.724613i \(-0.257982\pi\)
0.689156 + 0.724613i \(0.257982\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −11.5390 −0.765871 −0.382936 0.923775i \(-0.625087\pi\)
−0.382936 + 0.923775i \(0.625087\pi\)
\(228\) 0 0
\(229\) −7.62614 −0.503949 −0.251975 0.967734i \(-0.581080\pi\)
−0.251975 + 0.967734i \(0.581080\pi\)
\(230\) 0 0
\(231\) 0.582576 0.0383307
\(232\) 0 0
\(233\) 15.7913 1.03452 0.517261 0.855828i \(-0.326952\pi\)
0.517261 + 0.855828i \(0.326952\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 43.3739 2.81744
\(238\) 0 0
\(239\) 2.20871 0.142870 0.0714349 0.997445i \(-0.477242\pi\)
0.0714349 + 0.997445i \(0.477242\pi\)
\(240\) 0 0
\(241\) −23.1216 −1.48939 −0.744696 0.667404i \(-0.767406\pi\)
−0.744696 + 0.667404i \(0.767406\pi\)
\(242\) 0 0
\(243\) −16.1652 −1.03699
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 6.58258 0.418839
\(248\) 0 0
\(249\) 27.7913 1.76120
\(250\) 0 0
\(251\) 11.5390 0.728336 0.364168 0.931333i \(-0.381353\pi\)
0.364168 + 0.931333i \(0.381353\pi\)
\(252\) 0 0
\(253\) 3.79129 0.238356
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 18.0000 1.12281 0.561405 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) 0 0
\(259\) −0.539015 −0.0334928
\(260\) 0 0
\(261\) 32.5390 2.01411
\(262\) 0 0
\(263\) −24.1652 −1.49009 −0.745044 0.667016i \(-0.767571\pi\)
−0.745044 + 0.667016i \(0.767571\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −2.20871 −0.135171
\(268\) 0 0
\(269\) −20.7042 −1.26236 −0.631178 0.775638i \(-0.717428\pi\)
−0.631178 + 0.775638i \(0.717428\pi\)
\(270\) 0 0
\(271\) −15.7477 −0.956606 −0.478303 0.878195i \(-0.658748\pi\)
−0.478303 + 0.878195i \(0.658748\pi\)
\(272\) 0 0
\(273\) −0.582576 −0.0352591
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −8.58258 −0.515677 −0.257838 0.966188i \(-0.583010\pi\)
−0.257838 + 0.966188i \(0.583010\pi\)
\(278\) 0 0
\(279\) 41.1216 2.46189
\(280\) 0 0
\(281\) −13.7477 −0.820121 −0.410060 0.912058i \(-0.634492\pi\)
−0.410060 + 0.912058i \(0.634492\pi\)
\(282\) 0 0
\(283\) 30.7042 1.82517 0.912587 0.408883i \(-0.134082\pi\)
0.912587 + 0.408883i \(0.134082\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −0.295834 −0.0174625
\(288\) 0 0
\(289\) −16.3739 −0.963168
\(290\) 0 0
\(291\) −17.3303 −1.01592
\(292\) 0 0
\(293\) −2.83485 −0.165614 −0.0828068 0.996566i \(-0.526388\pi\)
−0.0828068 + 0.996566i \(0.526388\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 5.00000 0.290129
\(298\) 0 0
\(299\) −3.79129 −0.219256
\(300\) 0 0
\(301\) 2.08712 0.120300
\(302\) 0 0
\(303\) −48.4955 −2.78599
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 27.2087 1.55288 0.776442 0.630189i \(-0.217023\pi\)
0.776442 + 0.630189i \(0.217023\pi\)
\(308\) 0 0
\(309\) −16.6261 −0.945828
\(310\) 0 0
\(311\) 5.83485 0.330864 0.165432 0.986221i \(-0.447098\pi\)
0.165432 + 0.986221i \(0.447098\pi\)
\(312\) 0 0
\(313\) 6.74773 0.381404 0.190702 0.981648i \(-0.438924\pi\)
0.190702 + 0.981648i \(0.438924\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −8.37386 −0.470323 −0.235162 0.971956i \(-0.575562\pi\)
−0.235162 + 0.971956i \(0.575562\pi\)
\(318\) 0 0
\(319\) 6.79129 0.380239
\(320\) 0 0
\(321\) −55.1216 −3.07659
\(322\) 0 0
\(323\) −5.20871 −0.289820
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −13.3739 −0.739576
\(328\) 0 0
\(329\) 0.295834 0.0163098
\(330\) 0 0
\(331\) −17.3303 −0.952560 −0.476280 0.879294i \(-0.658015\pi\)
−0.476280 + 0.879294i \(0.658015\pi\)
\(332\) 0 0
\(333\) −12.3739 −0.678084
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −22.0000 −1.19842 −0.599208 0.800593i \(-0.704518\pi\)
−0.599208 + 0.800593i \(0.704518\pi\)
\(338\) 0 0
\(339\) 3.95644 0.214884
\(340\) 0 0
\(341\) 8.58258 0.464773
\(342\) 0 0
\(343\) −2.91288 −0.157281
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −12.6261 −0.677807 −0.338903 0.940821i \(-0.610056\pi\)
−0.338903 + 0.940821i \(0.610056\pi\)
\(348\) 0 0
\(349\) −14.5826 −0.780587 −0.390294 0.920690i \(-0.627627\pi\)
−0.390294 + 0.920690i \(0.627627\pi\)
\(350\) 0 0
\(351\) −5.00000 −0.266880
\(352\) 0 0
\(353\) 18.1652 0.966833 0.483417 0.875390i \(-0.339396\pi\)
0.483417 + 0.875390i \(0.339396\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0.460985 0.0243979
\(358\) 0 0
\(359\) 6.00000 0.316668 0.158334 0.987386i \(-0.449388\pi\)
0.158334 + 0.987386i \(0.449388\pi\)
\(360\) 0 0
\(361\) 24.3303 1.28054
\(362\) 0 0
\(363\) 2.79129 0.146505
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −8.62614 −0.450281 −0.225140 0.974326i \(-0.572284\pi\)
−0.225140 + 0.974326i \(0.572284\pi\)
\(368\) 0 0
\(369\) −6.79129 −0.353540
\(370\) 0 0
\(371\) 2.37386 0.123245
\(372\) 0 0
\(373\) −19.3303 −1.00089 −0.500443 0.865770i \(-0.666829\pi\)
−0.500443 + 0.865770i \(0.666829\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −6.79129 −0.349769
\(378\) 0 0
\(379\) 31.0000 1.59236 0.796182 0.605058i \(-0.206850\pi\)
0.796182 + 0.605058i \(0.206850\pi\)
\(380\) 0 0
\(381\) −45.7042 −2.34150
\(382\) 0 0
\(383\) 18.0000 0.919757 0.459879 0.887982i \(-0.347893\pi\)
0.459879 + 0.887982i \(0.347893\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 47.9129 2.43555
\(388\) 0 0
\(389\) −12.0000 −0.608424 −0.304212 0.952604i \(-0.598393\pi\)
−0.304212 + 0.952604i \(0.598393\pi\)
\(390\) 0 0
\(391\) 3.00000 0.151717
\(392\) 0 0
\(393\) 1.74773 0.0881612
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 7.20871 0.361795 0.180897 0.983502i \(-0.442100\pi\)
0.180897 + 0.983502i \(0.442100\pi\)
\(398\) 0 0
\(399\) −3.83485 −0.191983
\(400\) 0 0
\(401\) −14.8348 −0.740817 −0.370408 0.928869i \(-0.620782\pi\)
−0.370408 + 0.928869i \(0.620782\pi\)
\(402\) 0 0
\(403\) −8.58258 −0.427529
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −2.58258 −0.128014
\(408\) 0 0
\(409\) 17.0000 0.840596 0.420298 0.907386i \(-0.361926\pi\)
0.420298 + 0.907386i \(0.361926\pi\)
\(410\) 0 0
\(411\) −32.2087 −1.58874
\(412\) 0 0
\(413\) 2.20871 0.108684
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −27.2087 −1.33242
\(418\) 0 0
\(419\) −27.0000 −1.31904 −0.659518 0.751689i \(-0.729240\pi\)
−0.659518 + 0.751689i \(0.729240\pi\)
\(420\) 0 0
\(421\) 25.3739 1.23665 0.618323 0.785924i \(-0.287812\pi\)
0.618323 + 0.785924i \(0.287812\pi\)
\(422\) 0 0
\(423\) 6.79129 0.330204
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0.878409 0.0425092
\(428\) 0 0
\(429\) −2.79129 −0.134765
\(430\) 0 0
\(431\) −10.9129 −0.525655 −0.262827 0.964843i \(-0.584655\pi\)
−0.262827 + 0.964843i \(0.584655\pi\)
\(432\) 0 0
\(433\) −13.3303 −0.640613 −0.320307 0.947314i \(-0.603786\pi\)
−0.320307 + 0.947314i \(0.603786\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −24.9564 −1.19383
\(438\) 0 0
\(439\) −11.9564 −0.570650 −0.285325 0.958431i \(-0.592101\pi\)
−0.285325 + 0.958431i \(0.592101\pi\)
\(440\) 0 0
\(441\) −33.3303 −1.58716
\(442\) 0 0
\(443\) −22.4174 −1.06508 −0.532542 0.846404i \(-0.678763\pi\)
−0.532542 + 0.846404i \(0.678763\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 8.83485 0.417874
\(448\) 0 0
\(449\) 26.2087 1.23687 0.618433 0.785838i \(-0.287768\pi\)
0.618433 + 0.785838i \(0.287768\pi\)
\(450\) 0 0
\(451\) −1.41742 −0.0667439
\(452\) 0 0
\(453\) −13.9564 −0.655731
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 13.3739 0.625603 0.312801 0.949819i \(-0.398733\pi\)
0.312801 + 0.949819i \(0.398733\pi\)
\(458\) 0 0
\(459\) 3.95644 0.184671
\(460\) 0 0
\(461\) −24.4955 −1.14087 −0.570434 0.821344i \(-0.693225\pi\)
−0.570434 + 0.821344i \(0.693225\pi\)
\(462\) 0 0
\(463\) 2.25227 0.104672 0.0523360 0.998630i \(-0.483333\pi\)
0.0523360 + 0.998630i \(0.483333\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −10.5826 −0.489703 −0.244852 0.969561i \(-0.578739\pi\)
−0.244852 + 0.969561i \(0.578739\pi\)
\(468\) 0 0
\(469\) 0.834849 0.0385497
\(470\) 0 0
\(471\) 47.9129 2.20771
\(472\) 0 0
\(473\) 10.0000 0.459800
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 54.4955 2.49518
\(478\) 0 0
\(479\) −0.165151 −0.00754596 −0.00377298 0.999993i \(-0.501201\pi\)
−0.00377298 + 0.999993i \(0.501201\pi\)
\(480\) 0 0
\(481\) 2.58258 0.117755
\(482\) 0 0
\(483\) 2.20871 0.100500
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 29.7477 1.34800 0.673999 0.738732i \(-0.264575\pi\)
0.673999 + 0.738732i \(0.264575\pi\)
\(488\) 0 0
\(489\) 59.6606 2.69795
\(490\) 0 0
\(491\) 7.41742 0.334744 0.167372 0.985894i \(-0.446472\pi\)
0.167372 + 0.985894i \(0.446472\pi\)
\(492\) 0 0
\(493\) 5.37386 0.242027
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 2.24318 0.100620
\(498\) 0 0
\(499\) −14.9564 −0.669542 −0.334771 0.942299i \(-0.608659\pi\)
−0.334771 + 0.942299i \(0.608659\pi\)
\(500\) 0 0
\(501\) 46.2867 2.06794
\(502\) 0 0
\(503\) −16.4174 −0.732017 −0.366008 0.930612i \(-0.619276\pi\)
−0.366008 + 0.930612i \(0.619276\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −33.4955 −1.48759
\(508\) 0 0
\(509\) 10.1216 0.448632 0.224316 0.974517i \(-0.427985\pi\)
0.224316 + 0.974517i \(0.427985\pi\)
\(510\) 0 0
\(511\) −1.62614 −0.0719360
\(512\) 0 0
\(513\) −32.9129 −1.45314
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 1.41742 0.0623382
\(518\) 0 0
\(519\) −38.3739 −1.68443
\(520\) 0 0
\(521\) 1.58258 0.0693339 0.0346670 0.999399i \(-0.488963\pi\)
0.0346670 + 0.999399i \(0.488963\pi\)
\(522\) 0 0
\(523\) 25.1652 1.10040 0.550198 0.835034i \(-0.314552\pi\)
0.550198 + 0.835034i \(0.314552\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 6.79129 0.295833
\(528\) 0 0
\(529\) −8.62614 −0.375049
\(530\) 0 0
\(531\) 50.7042 2.20037
\(532\) 0 0
\(533\) 1.41742 0.0613955
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −61.7477 −2.66461
\(538\) 0 0
\(539\) −6.95644 −0.299635
\(540\) 0 0
\(541\) −10.6261 −0.456853 −0.228427 0.973561i \(-0.573358\pi\)
−0.228427 + 0.973561i \(0.573358\pi\)
\(542\) 0 0
\(543\) −9.41742 −0.404140
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −7.37386 −0.315284 −0.157642 0.987496i \(-0.550389\pi\)
−0.157642 + 0.987496i \(0.550389\pi\)
\(548\) 0 0
\(549\) 20.1652 0.860628
\(550\) 0 0
\(551\) −44.7042 −1.90446
\(552\) 0 0
\(553\) 3.24318 0.137914
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −45.6606 −1.93470 −0.967351 0.253441i \(-0.918438\pi\)
−0.967351 + 0.253441i \(0.918438\pi\)
\(558\) 0 0
\(559\) −10.0000 −0.422955
\(560\) 0 0
\(561\) 2.20871 0.0932519
\(562\) 0 0
\(563\) 40.1216 1.69092 0.845462 0.534036i \(-0.179325\pi\)
0.845462 + 0.534036i \(0.179325\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −0.0871215 −0.00365876
\(568\) 0 0
\(569\) 18.9564 0.794695 0.397348 0.917668i \(-0.369931\pi\)
0.397348 + 0.917668i \(0.369931\pi\)
\(570\) 0 0
\(571\) −27.2867 −1.14191 −0.570957 0.820980i \(-0.693428\pi\)
−0.570957 + 0.820980i \(0.693428\pi\)
\(572\) 0 0
\(573\) −48.0345 −2.00667
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 11.9564 0.497753 0.248877 0.968535i \(-0.419939\pi\)
0.248877 + 0.968535i \(0.419939\pi\)
\(578\) 0 0
\(579\) −20.0000 −0.831172
\(580\) 0 0
\(581\) 2.07803 0.0862112
\(582\) 0 0
\(583\) 11.3739 0.471057
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −15.9564 −0.658593 −0.329296 0.944227i \(-0.606812\pi\)
−0.329296 + 0.944227i \(0.606812\pi\)
\(588\) 0 0
\(589\) −56.4955 −2.32785
\(590\) 0 0
\(591\) 74.0780 3.04716
\(592\) 0 0
\(593\) −22.5826 −0.927355 −0.463678 0.886004i \(-0.653470\pi\)
−0.463678 + 0.886004i \(0.653470\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 4.53901 0.185770
\(598\) 0 0
\(599\) −44.7042 −1.82656 −0.913281 0.407329i \(-0.866460\pi\)
−0.913281 + 0.407329i \(0.866460\pi\)
\(600\) 0 0
\(601\) 28.5390 1.16413 0.582065 0.813142i \(-0.302245\pi\)
0.582065 + 0.813142i \(0.302245\pi\)
\(602\) 0 0
\(603\) 19.1652 0.780465
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 34.0000 1.38002 0.690009 0.723801i \(-0.257607\pi\)
0.690009 + 0.723801i \(0.257607\pi\)
\(608\) 0 0
\(609\) 3.95644 0.160323
\(610\) 0 0
\(611\) −1.41742 −0.0573428
\(612\) 0 0
\(613\) 33.4519 1.35111 0.675555 0.737310i \(-0.263904\pi\)
0.675555 + 0.737310i \(0.263904\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −28.7477 −1.15734 −0.578670 0.815562i \(-0.696428\pi\)
−0.578670 + 0.815562i \(0.696428\pi\)
\(618\) 0 0
\(619\) −6.25227 −0.251300 −0.125650 0.992075i \(-0.540102\pi\)
−0.125650 + 0.992075i \(0.540102\pi\)
\(620\) 0 0
\(621\) 18.9564 0.760696
\(622\) 0 0
\(623\) −0.165151 −0.00661665
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −18.3739 −0.733781
\(628\) 0 0
\(629\) −2.04356 −0.0814821
\(630\) 0 0
\(631\) −18.1216 −0.721409 −0.360705 0.932680i \(-0.617464\pi\)
−0.360705 + 0.932680i \(0.617464\pi\)
\(632\) 0 0
\(633\) −13.9564 −0.554719
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 6.95644 0.275624
\(638\) 0 0
\(639\) 51.4955 2.03713
\(640\) 0 0
\(641\) −23.8348 −0.941420 −0.470710 0.882288i \(-0.656002\pi\)
−0.470710 + 0.882288i \(0.656002\pi\)
\(642\) 0 0
\(643\) 43.4955 1.71529 0.857647 0.514239i \(-0.171926\pi\)
0.857647 + 0.514239i \(0.171926\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −21.3303 −0.838581 −0.419290 0.907852i \(-0.637721\pi\)
−0.419290 + 0.907852i \(0.637721\pi\)
\(648\) 0 0
\(649\) 10.5826 0.415402
\(650\) 0 0
\(651\) 5.00000 0.195965
\(652\) 0 0
\(653\) 26.7042 1.04501 0.522507 0.852635i \(-0.324997\pi\)
0.522507 + 0.852635i \(0.324997\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −37.3303 −1.45639
\(658\) 0 0
\(659\) −0.460985 −0.0179574 −0.00897871 0.999960i \(-0.502858\pi\)
−0.00897871 + 0.999960i \(0.502858\pi\)
\(660\) 0 0
\(661\) 30.5826 1.18952 0.594762 0.803902i \(-0.297246\pi\)
0.594762 + 0.803902i \(0.297246\pi\)
\(662\) 0 0
\(663\) −2.20871 −0.0857793
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 25.7477 0.996956
\(668\) 0 0
\(669\) 57.4519 2.22122
\(670\) 0 0
\(671\) 4.20871 0.162476
\(672\) 0 0
\(673\) −38.9129 −1.49998 −0.749991 0.661448i \(-0.769942\pi\)
−0.749991 + 0.661448i \(0.769942\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −1.58258 −0.0608233 −0.0304117 0.999537i \(-0.509682\pi\)
−0.0304117 + 0.999537i \(0.509682\pi\)
\(678\) 0 0
\(679\) −1.29583 −0.0497295
\(680\) 0 0
\(681\) −32.2087 −1.23424
\(682\) 0 0
\(683\) −34.9129 −1.33590 −0.667952 0.744204i \(-0.732829\pi\)
−0.667952 + 0.744204i \(0.732829\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −21.2867 −0.812140
\(688\) 0 0
\(689\) −11.3739 −0.433310
\(690\) 0 0
\(691\) 32.1216 1.22196 0.610981 0.791645i \(-0.290775\pi\)
0.610981 + 0.791645i \(0.290775\pi\)
\(692\) 0 0
\(693\) 1.00000 0.0379869
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −1.12159 −0.0424833
\(698\) 0 0
\(699\) 44.0780 1.66718
\(700\) 0 0
\(701\) 20.8348 0.786921 0.393461 0.919341i \(-0.371278\pi\)
0.393461 + 0.919341i \(0.371278\pi\)
\(702\) 0 0
\(703\) 17.0000 0.641167
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −3.62614 −0.136375
\(708\) 0 0
\(709\) 29.4955 1.10773 0.553863 0.832608i \(-0.313153\pi\)
0.553863 + 0.832608i \(0.313153\pi\)
\(710\) 0 0
\(711\) 74.4519 2.79216
\(712\) 0 0
\(713\) 32.5390 1.21860
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 6.16515 0.230242
\(718\) 0 0
\(719\) −6.49545 −0.242240 −0.121120 0.992638i \(-0.538649\pi\)
−0.121120 + 0.992638i \(0.538649\pi\)
\(720\) 0 0
\(721\) −1.24318 −0.0462985
\(722\) 0 0
\(723\) −64.5390 −2.40023
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 23.1216 0.857532 0.428766 0.903416i \(-0.358948\pi\)
0.428766 + 0.903416i \(0.358948\pi\)
\(728\) 0 0
\(729\) −43.8693 −1.62479
\(730\) 0 0
\(731\) 7.91288 0.292668
\(732\) 0 0
\(733\) 31.2432 1.15399 0.576997 0.816747i \(-0.304225\pi\)
0.576997 + 0.816747i \(0.304225\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 4.00000 0.147342
\(738\) 0 0
\(739\) −42.1216 −1.54947 −0.774734 0.632287i \(-0.782116\pi\)
−0.774734 + 0.632287i \(0.782116\pi\)
\(740\) 0 0
\(741\) 18.3739 0.674981
\(742\) 0 0
\(743\) −39.7913 −1.45980 −0.729900 0.683554i \(-0.760434\pi\)
−0.729900 + 0.683554i \(0.760434\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 47.7042 1.74540
\(748\) 0 0
\(749\) −4.12159 −0.150600
\(750\) 0 0
\(751\) −13.2087 −0.481993 −0.240996 0.970526i \(-0.577474\pi\)
−0.240996 + 0.970526i \(0.577474\pi\)
\(752\) 0 0
\(753\) 32.2087 1.17375
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −7.33030 −0.266424 −0.133212 0.991088i \(-0.542529\pi\)
−0.133212 + 0.991088i \(0.542529\pi\)
\(758\) 0 0
\(759\) 10.5826 0.384123
\(760\) 0 0
\(761\) 30.3303 1.09947 0.549736 0.835338i \(-0.314728\pi\)
0.549736 + 0.835338i \(0.314728\pi\)
\(762\) 0 0
\(763\) −1.00000 −0.0362024
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −10.5826 −0.382115
\(768\) 0 0
\(769\) −41.7477 −1.50546 −0.752731 0.658328i \(-0.771264\pi\)
−0.752731 + 0.658328i \(0.771264\pi\)
\(770\) 0 0
\(771\) 50.2432 1.80946
\(772\) 0 0
\(773\) 26.2087 0.942662 0.471331 0.881956i \(-0.343774\pi\)
0.471331 + 0.881956i \(0.343774\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −1.50455 −0.0539753
\(778\) 0 0
\(779\) 9.33030 0.334293
\(780\) 0 0
\(781\) 10.7477 0.384584
\(782\) 0 0
\(783\) 33.9564 1.21350
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 17.7477 0.632638 0.316319 0.948653i \(-0.397553\pi\)
0.316319 + 0.948653i \(0.397553\pi\)
\(788\) 0 0
\(789\) −67.4519 −2.40135
\(790\) 0 0
\(791\) 0.295834 0.0105186
\(792\) 0 0
\(793\) −4.20871 −0.149456
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 34.2867 1.21450 0.607249 0.794511i \(-0.292273\pi\)
0.607249 + 0.794511i \(0.292273\pi\)
\(798\) 0 0
\(799\) 1.12159 0.0396790
\(800\) 0 0
\(801\) −3.79129 −0.133959
\(802\) 0 0
\(803\) −7.79129 −0.274949
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −57.7913 −2.03435
\(808\) 0 0
\(809\) −45.4955 −1.59953 −0.799767 0.600310i \(-0.795044\pi\)
−0.799767 + 0.600310i \(0.795044\pi\)
\(810\) 0 0
\(811\) 55.6606 1.95451 0.977254 0.212072i \(-0.0680210\pi\)
0.977254 + 0.212072i \(0.0680210\pi\)
\(812\) 0 0
\(813\) −43.9564 −1.54162
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −65.8258 −2.30295
\(818\) 0 0
\(819\) −1.00000 −0.0349428
\(820\) 0 0
\(821\) 48.6606 1.69827 0.849133 0.528178i \(-0.177125\pi\)
0.849133 + 0.528178i \(0.177125\pi\)
\(822\) 0 0
\(823\) −8.00000 −0.278862 −0.139431 0.990232i \(-0.544527\pi\)
−0.139431 + 0.990232i \(0.544527\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 10.4174 0.362249 0.181125 0.983460i \(-0.442026\pi\)
0.181125 + 0.983460i \(0.442026\pi\)
\(828\) 0 0
\(829\) −37.9564 −1.31828 −0.659141 0.752020i \(-0.729080\pi\)
−0.659141 + 0.752020i \(0.729080\pi\)
\(830\) 0 0
\(831\) −23.9564 −0.831040
\(832\) 0 0
\(833\) −5.50455 −0.190721
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 42.9129 1.48329
\(838\) 0 0
\(839\) 23.7042 0.818359 0.409179 0.912454i \(-0.365815\pi\)
0.409179 + 0.912454i \(0.365815\pi\)
\(840\) 0 0
\(841\) 17.1216 0.590400
\(842\) 0 0
\(843\) −38.3739 −1.32167
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0.208712 0.00717143
\(848\) 0 0
\(849\) 85.7042 2.94136
\(850\) 0 0
\(851\) −9.79129 −0.335641
\(852\) 0 0
\(853\) 3.12159 0.106881 0.0534406 0.998571i \(-0.482981\pi\)
0.0534406 + 0.998571i \(0.482981\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −30.4955 −1.04170 −0.520852 0.853647i \(-0.674386\pi\)
−0.520852 + 0.853647i \(0.674386\pi\)
\(858\) 0 0
\(859\) 2.08712 0.0712117 0.0356058 0.999366i \(-0.488664\pi\)
0.0356058 + 0.999366i \(0.488664\pi\)
\(860\) 0 0
\(861\) −0.825757 −0.0281417
\(862\) 0 0
\(863\) 27.3303 0.930334 0.465167 0.885223i \(-0.345994\pi\)
0.465167 + 0.885223i \(0.345994\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −45.7042 −1.55219
\(868\) 0 0
\(869\) 15.5390 0.527125
\(870\) 0 0
\(871\) −4.00000 −0.135535
\(872\) 0 0
\(873\) −29.7477 −1.00681
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −16.6606 −0.562589 −0.281294 0.959622i \(-0.590764\pi\)
−0.281294 + 0.959622i \(0.590764\pi\)
\(878\) 0 0
\(879\) −7.91288 −0.266895
\(880\) 0 0
\(881\) 1.87841 0.0632852 0.0316426 0.999499i \(-0.489926\pi\)
0.0316426 + 0.999499i \(0.489926\pi\)
\(882\) 0 0
\(883\) −26.1652 −0.880527 −0.440264 0.897869i \(-0.645115\pi\)
−0.440264 + 0.897869i \(0.645115\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 39.3303 1.32058 0.660291 0.751010i \(-0.270433\pi\)
0.660291 + 0.751010i \(0.270433\pi\)
\(888\) 0 0
\(889\) −3.41742 −0.114617
\(890\) 0 0
\(891\) −0.417424 −0.0139842
\(892\) 0 0
\(893\) −9.33030 −0.312227
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −10.5826 −0.353342
\(898\) 0 0
\(899\) 58.2867 1.94397
\(900\) 0 0
\(901\) 9.00000 0.299833
\(902\) 0 0
\(903\) 5.82576 0.193869
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −32.6606 −1.08448 −0.542239 0.840224i \(-0.682423\pi\)
−0.542239 + 0.840224i \(0.682423\pi\)
\(908\) 0 0
\(909\) −83.2432 −2.76100
\(910\) 0 0
\(911\) 12.3303 0.408521 0.204261 0.978917i \(-0.434521\pi\)
0.204261 + 0.978917i \(0.434521\pi\)
\(912\) 0 0
\(913\) 9.95644 0.329510
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0.130682 0.00431551
\(918\) 0 0
\(919\) −28.8348 −0.951174 −0.475587 0.879669i \(-0.657764\pi\)
−0.475587 + 0.879669i \(0.657764\pi\)
\(920\) 0 0
\(921\) 75.9473 2.50255
\(922\) 0 0
\(923\) −10.7477 −0.353766
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −28.5390 −0.937344
\(928\) 0 0
\(929\) 13.7477 0.451048 0.225524 0.974238i \(-0.427591\pi\)
0.225524 + 0.974238i \(0.427591\pi\)
\(930\) 0 0
\(931\) 45.7913 1.50075
\(932\) 0 0
\(933\) 16.2867 0.533204
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 35.1652 1.14880 0.574398 0.818576i \(-0.305236\pi\)
0.574398 + 0.818576i \(0.305236\pi\)
\(938\) 0 0
\(939\) 18.8348 0.614652
\(940\) 0 0
\(941\) −50.0780 −1.63250 −0.816249 0.577701i \(-0.803950\pi\)
−0.816249 + 0.577701i \(0.803950\pi\)
\(942\) 0 0
\(943\) −5.37386 −0.174997
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 4.91288 0.159647 0.0798235 0.996809i \(-0.474564\pi\)
0.0798235 + 0.996809i \(0.474564\pi\)
\(948\) 0 0
\(949\) 7.79129 0.252916
\(950\) 0 0
\(951\) −23.3739 −0.757949
\(952\) 0 0
\(953\) −42.0000 −1.36051 −0.680257 0.732974i \(-0.738132\pi\)
−0.680257 + 0.732974i \(0.738132\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 18.9564 0.612775
\(958\) 0 0
\(959\) −2.40833 −0.0777691
\(960\) 0 0
\(961\) 42.6606 1.37615
\(962\) 0 0
\(963\) −94.6170 −3.04899
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −43.5390 −1.40012 −0.700060 0.714084i \(-0.746844\pi\)
−0.700060 + 0.714084i \(0.746844\pi\)
\(968\) 0 0
\(969\) −14.5390 −0.467060
\(970\) 0 0
\(971\) −18.9564 −0.608341 −0.304171 0.952618i \(-0.598379\pi\)
−0.304171 + 0.952618i \(0.598379\pi\)
\(972\) 0 0
\(973\) −2.03447 −0.0652221
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −25.7477 −0.823743 −0.411871 0.911242i \(-0.635125\pi\)
−0.411871 + 0.911242i \(0.635125\pi\)
\(978\) 0 0
\(979\) −0.791288 −0.0252897
\(980\) 0 0
\(981\) −22.9564 −0.732943
\(982\) 0 0
\(983\) 48.6606 1.55203 0.776016 0.630713i \(-0.217237\pi\)
0.776016 + 0.630713i \(0.217237\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0.825757 0.0262841
\(988\) 0 0
\(989\) 37.9129 1.20556
\(990\) 0 0
\(991\) −55.8693 −1.77475 −0.887374 0.461051i \(-0.847473\pi\)
−0.887374 + 0.461051i \(0.847473\pi\)
\(992\) 0 0
\(993\) −48.3739 −1.53510
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 40.3739 1.27865 0.639327 0.768935i \(-0.279213\pi\)
0.639327 + 0.768935i \(0.279213\pi\)
\(998\) 0 0
\(999\) −12.9129 −0.408545
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 4400.2.a.bu.1.2 2
4.3 odd 2 1100.2.a.g.1.1 2
5.2 odd 4 4400.2.b.s.4049.1 4
5.3 odd 4 4400.2.b.s.4049.4 4
5.4 even 2 4400.2.a.bi.1.1 2
12.11 even 2 9900.2.a.bh.1.2 2
20.3 even 4 1100.2.b.d.749.1 4
20.7 even 4 1100.2.b.d.749.4 4
20.19 odd 2 1100.2.a.h.1.2 yes 2
60.23 odd 4 9900.2.c.x.5149.3 4
60.47 odd 4 9900.2.c.x.5149.2 4
60.59 even 2 9900.2.a.bz.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1100.2.a.g.1.1 2 4.3 odd 2
1100.2.a.h.1.2 yes 2 20.19 odd 2
1100.2.b.d.749.1 4 20.3 even 4
1100.2.b.d.749.4 4 20.7 even 4
4400.2.a.bi.1.1 2 5.4 even 2
4400.2.a.bu.1.2 2 1.1 even 1 trivial
4400.2.b.s.4049.1 4 5.2 odd 4
4400.2.b.s.4049.4 4 5.3 odd 4
9900.2.a.bh.1.2 2 12.11 even 2
9900.2.a.bz.1.1 2 60.59 even 2
9900.2.c.x.5149.2 4 60.47 odd 4
9900.2.c.x.5149.3 4 60.23 odd 4