Properties

Label 4600.2.e.r.4049.4
Level $4600$
Weight $2$
Character 4600.4049
Analytic conductor $36.731$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4600.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(36.7311849298\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.24681024.1
Defining polynomial: \( x^{6} + 12x^{4} + 36x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 920)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 4049.4
Root \(0.523976i\) of defining polynomial
Character \(\chi\) \(=\) 4600.4049
Dual form 4600.2.e.r.4049.3

$q$-expansion

\(f(q)\) \(=\) \(q+0.523976i q^{3} +0.476024i q^{7} +2.72545 q^{9} +O(q^{10})\) \(q+0.523976i q^{3} +0.476024i q^{7} +2.72545 q^{9} -1.67750 q^{11} -2.67750i q^{13} +1.67750i q^{17} -7.92692 q^{19} -0.249425 q^{21} -1.00000i q^{23} +3.00000i q^{27} +2.20147 q^{29} +6.77340 q^{31} -0.878968i q^{33} +4.00000i q^{37} +1.40294 q^{39} +5.97487 q^{41} +0.402945i q^{43} +1.79853i q^{47} +6.77340 q^{49} -0.878968 q^{51} +10.8059i q^{53} -4.15352i q^{57} -9.45090 q^{59} +6.32250 q^{61} +1.29738i q^{63} +14.7100i q^{67} +0.523976 q^{69} +9.97487 q^{71} +6.10557i q^{73} -0.798528i q^{77} +6.60442 q^{81} -4.49885i q^{83} +1.15352i q^{87} +3.59706 q^{89} +1.27455 q^{91} +3.54910i q^{93} -6.08044i q^{97} -4.57193 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{9} + 6 q^{11} - 6 q^{19} + 24 q^{21} - 6 q^{29} + 12 q^{31} - 30 q^{39} - 12 q^{41} + 12 q^{49} + 30 q^{51} - 12 q^{59} + 54 q^{61} + 12 q^{71} - 18 q^{81} + 60 q^{89} + 30 q^{91} - 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4600\mathbb{Z}\right)^\times\).

\(n\) \(1151\) \(1201\) \(2301\) \(2577\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(3\) 0.523976i 0.302518i 0.988494 + 0.151259i \(0.0483327\pi\)
−0.988494 + 0.151259i \(0.951667\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0.476024i 0.179920i 0.995945 + 0.0899600i \(0.0286739\pi\)
−0.995945 + 0.0899600i \(0.971326\pi\)
\(8\) 0 0
\(9\) 2.72545 0.908483
\(10\) 0 0
\(11\) −1.67750 −0.505784 −0.252892 0.967495i \(-0.581382\pi\)
−0.252892 + 0.967495i \(0.581382\pi\)
\(12\) 0 0
\(13\) − 2.67750i − 0.742604i −0.928512 0.371302i \(-0.878911\pi\)
0.928512 0.371302i \(-0.121089\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.67750i 0.406853i 0.979090 + 0.203426i \(0.0652077\pi\)
−0.979090 + 0.203426i \(0.934792\pi\)
\(18\) 0 0
\(19\) −7.92692 −1.81856 −0.909280 0.416184i \(-0.863367\pi\)
−0.909280 + 0.416184i \(0.863367\pi\)
\(20\) 0 0
\(21\) −0.249425 −0.0544290
\(22\) 0 0
\(23\) − 1.00000i − 0.208514i
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 3.00000i 0.577350i
\(28\) 0 0
\(29\) 2.20147 0.408803 0.204402 0.978887i \(-0.434475\pi\)
0.204402 + 0.978887i \(0.434475\pi\)
\(30\) 0 0
\(31\) 6.77340 1.21654 0.608269 0.793731i \(-0.291864\pi\)
0.608269 + 0.793731i \(0.291864\pi\)
\(32\) 0 0
\(33\) − 0.878968i − 0.153009i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 4.00000i 0.657596i 0.944400 + 0.328798i \(0.106644\pi\)
−0.944400 + 0.328798i \(0.893356\pi\)
\(38\) 0 0
\(39\) 1.40294 0.224651
\(40\) 0 0
\(41\) 5.97487 0.933119 0.466559 0.884490i \(-0.345493\pi\)
0.466559 + 0.884490i \(0.345493\pi\)
\(42\) 0 0
\(43\) 0.402945i 0.0614485i 0.999528 + 0.0307242i \(0.00978137\pi\)
−0.999528 + 0.0307242i \(0.990219\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.79853i 0.262342i 0.991360 + 0.131171i \(0.0418737\pi\)
−0.991360 + 0.131171i \(0.958126\pi\)
\(48\) 0 0
\(49\) 6.77340 0.967629
\(50\) 0 0
\(51\) −0.878968 −0.123080
\(52\) 0 0
\(53\) 10.8059i 1.48430i 0.670232 + 0.742152i \(0.266195\pi\)
−0.670232 + 0.742152i \(0.733805\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 4.15352i − 0.550147i
\(58\) 0 0
\(59\) −9.45090 −1.23040 −0.615201 0.788370i \(-0.710925\pi\)
−0.615201 + 0.788370i \(0.710925\pi\)
\(60\) 0 0
\(61\) 6.32250 0.809514 0.404757 0.914424i \(-0.367356\pi\)
0.404757 + 0.914424i \(0.367356\pi\)
\(62\) 0 0
\(63\) 1.29738i 0.163454i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 14.7100i 1.79711i 0.438860 + 0.898555i \(0.355382\pi\)
−0.438860 + 0.898555i \(0.644618\pi\)
\(68\) 0 0
\(69\) 0.523976 0.0630793
\(70\) 0 0
\(71\) 9.97487 1.18380 0.591900 0.806012i \(-0.298378\pi\)
0.591900 + 0.806012i \(0.298378\pi\)
\(72\) 0 0
\(73\) 6.10557i 0.714603i 0.933989 + 0.357301i \(0.116303\pi\)
−0.933989 + 0.357301i \(0.883697\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 0.798528i − 0.0910007i
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 6.60442 0.733824
\(82\) 0 0
\(83\) − 4.49885i − 0.493813i −0.969039 0.246906i \(-0.920586\pi\)
0.969039 0.246906i \(-0.0794141\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 1.15352i 0.123670i
\(88\) 0 0
\(89\) 3.59706 0.381287 0.190644 0.981659i \(-0.438943\pi\)
0.190644 + 0.981659i \(0.438943\pi\)
\(90\) 0 0
\(91\) 1.27455 0.133609
\(92\) 0 0
\(93\) 3.54910i 0.368025i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 6.08044i − 0.617375i −0.951163 0.308688i \(-0.900110\pi\)
0.951163 0.308688i \(-0.0998898\pi\)
\(98\) 0 0
\(99\) −4.57193 −0.459496
\(100\) 0 0
\(101\) 11.4509 1.13941 0.569703 0.821850i \(-0.307058\pi\)
0.569703 + 0.821850i \(0.307058\pi\)
\(102\) 0 0
\(103\) 2.32250i 0.228843i 0.993432 + 0.114422i \(0.0365015\pi\)
−0.993432 + 0.114422i \(0.963499\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 7.85384i − 0.759260i −0.925138 0.379630i \(-0.876051\pi\)
0.925138 0.379630i \(-0.123949\pi\)
\(108\) 0 0
\(109\) −13.9269 −1.33396 −0.666979 0.745077i \(-0.732413\pi\)
−0.666979 + 0.745077i \(0.732413\pi\)
\(110\) 0 0
\(111\) −2.09591 −0.198935
\(112\) 0 0
\(113\) 10.0000i 0.940721i 0.882474 + 0.470360i \(0.155876\pi\)
−0.882474 + 0.470360i \(0.844124\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 7.29738i − 0.674643i
\(118\) 0 0
\(119\) −0.798528 −0.0732009
\(120\) 0 0
\(121\) −8.18601 −0.744182
\(122\) 0 0
\(123\) 3.13069i 0.282285i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 11.1033i − 0.985256i −0.870240 0.492628i \(-0.836036\pi\)
0.870240 0.492628i \(-0.163964\pi\)
\(128\) 0 0
\(129\) −0.211133 −0.0185893
\(130\) 0 0
\(131\) 7.65237 0.668591 0.334295 0.942468i \(-0.391502\pi\)
0.334295 + 0.942468i \(0.391502\pi\)
\(132\) 0 0
\(133\) − 3.77340i − 0.327195i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 21.3778i − 1.82643i −0.407478 0.913215i \(-0.633592\pi\)
0.407478 0.913215i \(-0.366408\pi\)
\(138\) 0 0
\(139\) 8.60442 0.729817 0.364909 0.931043i \(-0.381100\pi\)
0.364909 + 0.931043i \(0.381100\pi\)
\(140\) 0 0
\(141\) −0.942386 −0.0793632
\(142\) 0 0
\(143\) 4.49149i 0.375597i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 3.54910i 0.292725i
\(148\) 0 0
\(149\) −4.97487 −0.407558 −0.203779 0.979017i \(-0.565322\pi\)
−0.203779 + 0.979017i \(0.565322\pi\)
\(150\) 0 0
\(151\) −0.926921 −0.0754318 −0.0377159 0.999289i \(-0.512008\pi\)
−0.0377159 + 0.999289i \(0.512008\pi\)
\(152\) 0 0
\(153\) 4.57193i 0.369619i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 22.3527i 1.78394i 0.452096 + 0.891970i \(0.350677\pi\)
−0.452096 + 0.891970i \(0.649323\pi\)
\(158\) 0 0
\(159\) −5.66203 −0.449028
\(160\) 0 0
\(161\) 0.476024 0.0375159
\(162\) 0 0
\(163\) 7.93658i 0.621641i 0.950469 + 0.310821i \(0.100604\pi\)
−0.950469 + 0.310821i \(0.899396\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 12.8059i 0.990949i 0.868622 + 0.495475i \(0.165006\pi\)
−0.868622 + 0.495475i \(0.834994\pi\)
\(168\) 0 0
\(169\) 5.83102 0.448540
\(170\) 0 0
\(171\) −21.6044 −1.65213
\(172\) 0 0
\(173\) 24.6752i 1.87602i 0.346608 + 0.938010i \(0.387334\pi\)
−0.346608 + 0.938010i \(0.612666\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 4.95205i − 0.372219i
\(178\) 0 0
\(179\) 13.1033 0.979384 0.489692 0.871895i \(-0.337109\pi\)
0.489692 + 0.871895i \(0.337109\pi\)
\(180\) 0 0
\(181\) 9.52398 0.707912 0.353956 0.935262i \(-0.384836\pi\)
0.353956 + 0.935262i \(0.384836\pi\)
\(182\) 0 0
\(183\) 3.31284i 0.244892i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 2.81399i − 0.205780i
\(188\) 0 0
\(189\) −1.42807 −0.103877
\(190\) 0 0
\(191\) 5.85384 0.423569 0.211785 0.977316i \(-0.432072\pi\)
0.211785 + 0.977316i \(0.432072\pi\)
\(192\) 0 0
\(193\) − 4.41261i − 0.317626i −0.987309 0.158813i \(-0.949233\pi\)
0.987309 0.158813i \(-0.0507667\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 11.4258i 0.814053i 0.913416 + 0.407026i \(0.133434\pi\)
−0.913416 + 0.407026i \(0.866566\pi\)
\(198\) 0 0
\(199\) 0.307039 0.0217654 0.0108827 0.999941i \(-0.496536\pi\)
0.0108827 + 0.999941i \(0.496536\pi\)
\(200\) 0 0
\(201\) −7.70768 −0.543658
\(202\) 0 0
\(203\) 1.04795i 0.0735519i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 2.72545i − 0.189432i
\(208\) 0 0
\(209\) 13.2974 0.919799
\(210\) 0 0
\(211\) 17.3047 1.19131 0.595654 0.803241i \(-0.296893\pi\)
0.595654 + 0.803241i \(0.296893\pi\)
\(212\) 0 0
\(213\) 5.22660i 0.358121i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 3.22430i 0.218880i
\(218\) 0 0
\(219\) −3.19917 −0.216180
\(220\) 0 0
\(221\) 4.49149 0.302130
\(222\) 0 0
\(223\) 1.90409i 0.127508i 0.997966 + 0.0637538i \(0.0203072\pi\)
−0.997966 + 0.0637538i \(0.979693\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 17.3550i 1.15189i 0.817488 + 0.575946i \(0.195366\pi\)
−0.817488 + 0.575946i \(0.804634\pi\)
\(228\) 0 0
\(229\) 6.19181 0.409166 0.204583 0.978849i \(-0.434416\pi\)
0.204583 + 0.978849i \(0.434416\pi\)
\(230\) 0 0
\(231\) 0.418410 0.0275293
\(232\) 0 0
\(233\) 15.7985i 1.03500i 0.855684 + 0.517498i \(0.173137\pi\)
−0.855684 + 0.517498i \(0.826863\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −6.34763 −0.410594 −0.205297 0.978700i \(-0.565816\pi\)
−0.205297 + 0.978700i \(0.565816\pi\)
\(240\) 0 0
\(241\) 5.69296 0.366716 0.183358 0.983046i \(-0.441303\pi\)
0.183358 + 0.983046i \(0.441303\pi\)
\(242\) 0 0
\(243\) 12.4606i 0.799345i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 21.2243i 1.35047i
\(248\) 0 0
\(249\) 2.35729 0.149387
\(250\) 0 0
\(251\) −28.9246 −1.82571 −0.912853 0.408288i \(-0.866126\pi\)
−0.912853 + 0.408288i \(0.866126\pi\)
\(252\) 0 0
\(253\) 1.67750i 0.105463i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 14.5085i − 0.905016i −0.891760 0.452508i \(-0.850529\pi\)
0.891760 0.452508i \(-0.149471\pi\)
\(258\) 0 0
\(259\) −1.90409 −0.118315
\(260\) 0 0
\(261\) 6.00000 0.371391
\(262\) 0 0
\(263\) 15.7734i 0.972630i 0.873784 + 0.486315i \(0.161659\pi\)
−0.873784 + 0.486315i \(0.838341\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 1.88477i 0.115346i
\(268\) 0 0
\(269\) 6.20147 0.378110 0.189055 0.981966i \(-0.439457\pi\)
0.189055 + 0.981966i \(0.439457\pi\)
\(270\) 0 0
\(271\) −25.5696 −1.55324 −0.776622 0.629967i \(-0.783069\pi\)
−0.776622 + 0.629967i \(0.783069\pi\)
\(272\) 0 0
\(273\) 0.667835i 0.0404192i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 7.03829i 0.422890i 0.977390 + 0.211445i \(0.0678169\pi\)
−0.977390 + 0.211445i \(0.932183\pi\)
\(278\) 0 0
\(279\) 18.4606 1.10520
\(280\) 0 0
\(281\) 29.0627 1.73373 0.866867 0.498540i \(-0.166130\pi\)
0.866867 + 0.498540i \(0.166130\pi\)
\(282\) 0 0
\(283\) 18.6141i 1.10649i 0.833018 + 0.553246i \(0.186611\pi\)
−0.833018 + 0.553246i \(0.813389\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 2.84418i 0.167887i
\(288\) 0 0
\(289\) 14.1860 0.834471
\(290\) 0 0
\(291\) 3.18601 0.186767
\(292\) 0 0
\(293\) − 23.4006i − 1.36708i −0.729913 0.683540i \(-0.760439\pi\)
0.729913 0.683540i \(-0.239561\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 5.03249i − 0.292015i
\(298\) 0 0
\(299\) −2.67750 −0.154844
\(300\) 0 0
\(301\) −0.191811 −0.0110558
\(302\) 0 0
\(303\) 6.00000i 0.344691i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 1.53134i − 0.0873981i −0.999045 0.0436990i \(-0.986086\pi\)
0.999045 0.0436990i \(-0.0139143\pi\)
\(308\) 0 0
\(309\) −1.21694 −0.0692291
\(310\) 0 0
\(311\) −7.55646 −0.428488 −0.214244 0.976780i \(-0.568729\pi\)
−0.214244 + 0.976780i \(0.568729\pi\)
\(312\) 0 0
\(313\) 2.32987i 0.131692i 0.997830 + 0.0658459i \(0.0209746\pi\)
−0.997830 + 0.0658459i \(0.979025\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 15.3778i − 0.863704i −0.901944 0.431852i \(-0.857860\pi\)
0.901944 0.431852i \(-0.142140\pi\)
\(318\) 0 0
\(319\) −3.69296 −0.206766
\(320\) 0 0
\(321\) 4.11523 0.229690
\(322\) 0 0
\(323\) − 13.2974i − 0.739886i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 7.29738i − 0.403546i
\(328\) 0 0
\(329\) −0.856142 −0.0472006
\(330\) 0 0
\(331\) 22.9115 1.25933 0.629664 0.776868i \(-0.283193\pi\)
0.629664 + 0.776868i \(0.283193\pi\)
\(332\) 0 0
\(333\) 10.9018i 0.597415i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 26.5410i 1.44578i 0.690963 + 0.722890i \(0.257187\pi\)
−0.690963 + 0.722890i \(0.742813\pi\)
\(338\) 0 0
\(339\) −5.23976 −0.284585
\(340\) 0 0
\(341\) −11.3624 −0.615306
\(342\) 0 0
\(343\) 6.55646i 0.354016i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 3.27455i − 0.175787i −0.996130 0.0878936i \(-0.971986\pi\)
0.996130 0.0878936i \(-0.0280135\pi\)
\(348\) 0 0
\(349\) −32.0553 −1.71588 −0.857941 0.513749i \(-0.828256\pi\)
−0.857941 + 0.513749i \(0.828256\pi\)
\(350\) 0 0
\(351\) 8.03249 0.428742
\(352\) 0 0
\(353\) − 11.1535i − 0.593642i −0.954933 0.296821i \(-0.904074\pi\)
0.954933 0.296821i \(-0.0959265\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 0.418410i − 0.0221446i
\(358\) 0 0
\(359\) −22.4029 −1.18238 −0.591191 0.806532i \(-0.701342\pi\)
−0.591191 + 0.806532i \(0.701342\pi\)
\(360\) 0 0
\(361\) 43.8361 2.30716
\(362\) 0 0
\(363\) − 4.28927i − 0.225129i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 14.2877i 0.745813i 0.927869 + 0.372906i \(0.121639\pi\)
−0.927869 + 0.372906i \(0.878361\pi\)
\(368\) 0 0
\(369\) 16.2842 0.847722
\(370\) 0 0
\(371\) −5.14386 −0.267056
\(372\) 0 0
\(373\) 1.23976i 0.0641925i 0.999485 + 0.0320963i \(0.0102183\pi\)
−0.999485 + 0.0320963i \(0.989782\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 5.89443i − 0.303579i
\(378\) 0 0
\(379\) −4.38748 −0.225370 −0.112685 0.993631i \(-0.535945\pi\)
−0.112685 + 0.993631i \(0.535945\pi\)
\(380\) 0 0
\(381\) 5.81785 0.298057
\(382\) 0 0
\(383\) − 18.4989i − 0.945247i −0.881264 0.472624i \(-0.843307\pi\)
0.881264 0.472624i \(-0.156693\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 1.09821i 0.0558249i
\(388\) 0 0
\(389\) −21.6775 −1.09909 −0.549546 0.835463i \(-0.685199\pi\)
−0.549546 + 0.835463i \(0.685199\pi\)
\(390\) 0 0
\(391\) 1.67750 0.0848346
\(392\) 0 0
\(393\) 4.00966i 0.202261i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 20.6849i − 1.03814i −0.854731 0.519072i \(-0.826278\pi\)
0.854731 0.519072i \(-0.173722\pi\)
\(398\) 0 0
\(399\) 1.97717 0.0989825
\(400\) 0 0
\(401\) −28.5638 −1.42641 −0.713205 0.700956i \(-0.752757\pi\)
−0.713205 + 0.700956i \(0.752757\pi\)
\(402\) 0 0
\(403\) − 18.1358i − 0.903406i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 6.70998i − 0.332602i
\(408\) 0 0
\(409\) −16.7734 −0.829391 −0.414696 0.909960i \(-0.636112\pi\)
−0.414696 + 0.909960i \(0.636112\pi\)
\(410\) 0 0
\(411\) 11.2015 0.552528
\(412\) 0 0
\(413\) − 4.49885i − 0.221374i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 4.50851i 0.220783i
\(418\) 0 0
\(419\) 4.59476 0.224469 0.112234 0.993682i \(-0.464199\pi\)
0.112234 + 0.993682i \(0.464199\pi\)
\(420\) 0 0
\(421\) 31.8310 1.55135 0.775674 0.631133i \(-0.217410\pi\)
0.775674 + 0.631133i \(0.217410\pi\)
\(422\) 0 0
\(423\) 4.90179i 0.238333i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 3.00966i 0.145648i
\(428\) 0 0
\(429\) −2.35343 −0.113625
\(430\) 0 0
\(431\) −16.0650 −0.773823 −0.386911 0.922117i \(-0.626458\pi\)
−0.386911 + 0.922117i \(0.626458\pi\)
\(432\) 0 0
\(433\) 7.68486i 0.369311i 0.982803 + 0.184655i \(0.0591169\pi\)
−0.982803 + 0.184655i \(0.940883\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 7.92692i 0.379196i
\(438\) 0 0
\(439\) −15.9822 −0.762790 −0.381395 0.924412i \(-0.624556\pi\)
−0.381395 + 0.924412i \(0.624556\pi\)
\(440\) 0 0
\(441\) 18.4606 0.879074
\(442\) 0 0
\(443\) 3.57929i 0.170057i 0.996379 + 0.0850286i \(0.0270982\pi\)
−0.996379 + 0.0850286i \(0.972902\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 2.60672i − 0.123293i
\(448\) 0 0
\(449\) 28.5217 1.34602 0.673011 0.739633i \(-0.265001\pi\)
0.673011 + 0.739633i \(0.265001\pi\)
\(450\) 0 0
\(451\) −10.0228 −0.471956
\(452\) 0 0
\(453\) − 0.485685i − 0.0228195i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 4.49885i − 0.210447i −0.994449 0.105224i \(-0.966444\pi\)
0.994449 0.105224i \(-0.0335559\pi\)
\(458\) 0 0
\(459\) −5.03249 −0.234896
\(460\) 0 0
\(461\) 2.29738 0.107000 0.0534998 0.998568i \(-0.482962\pi\)
0.0534998 + 0.998568i \(0.482962\pi\)
\(462\) 0 0
\(463\) 33.4966i 1.55672i 0.627820 + 0.778358i \(0.283947\pi\)
−0.627820 + 0.778358i \(0.716053\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 5.23976i − 0.242467i −0.992624 0.121234i \(-0.961315\pi\)
0.992624 0.121234i \(-0.0386850\pi\)
\(468\) 0 0
\(469\) −7.00230 −0.323336
\(470\) 0 0
\(471\) −11.7123 −0.539674
\(472\) 0 0
\(473\) − 0.675938i − 0.0310797i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 29.4509i 1.34846i
\(478\) 0 0
\(479\) 2.40294 0.109793 0.0548967 0.998492i \(-0.482517\pi\)
0.0548967 + 0.998492i \(0.482517\pi\)
\(480\) 0 0
\(481\) 10.7100 0.488333
\(482\) 0 0
\(483\) 0.249425i 0.0113492i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 25.8944i 1.17339i 0.809808 + 0.586694i \(0.199571\pi\)
−0.809808 + 0.586694i \(0.800429\pi\)
\(488\) 0 0
\(489\) −4.15858 −0.188058
\(490\) 0 0
\(491\) −0.489189 −0.0220768 −0.0110384 0.999939i \(-0.503514\pi\)
−0.0110384 + 0.999939i \(0.503514\pi\)
\(492\) 0 0
\(493\) 3.69296i 0.166323i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 4.74828i 0.212989i
\(498\) 0 0
\(499\) 38.4583 1.72163 0.860814 0.508920i \(-0.169955\pi\)
0.860814 + 0.508920i \(0.169955\pi\)
\(500\) 0 0
\(501\) −6.70998 −0.299780
\(502\) 0 0
\(503\) − 37.8960i − 1.68970i −0.535004 0.844849i \(-0.679690\pi\)
0.535004 0.844849i \(-0.320310\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 3.05531i 0.135691i
\(508\) 0 0
\(509\) −8.70032 −0.385635 −0.192818 0.981235i \(-0.561763\pi\)
−0.192818 + 0.981235i \(0.561763\pi\)
\(510\) 0 0
\(511\) −2.90639 −0.128571
\(512\) 0 0
\(513\) − 23.7808i − 1.04995i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 3.01702i − 0.132689i
\(518\) 0 0
\(519\) −12.9292 −0.567530
\(520\) 0 0
\(521\) 25.1129 1.10022 0.550109 0.835093i \(-0.314586\pi\)
0.550109 + 0.835093i \(0.314586\pi\)
\(522\) 0 0
\(523\) − 12.7100i − 0.555769i −0.960615 0.277884i \(-0.910367\pi\)
0.960615 0.277884i \(-0.0896332\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 11.3624i 0.494952i
\(528\) 0 0
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) −25.7579 −1.11780
\(532\) 0 0
\(533\) − 15.9977i − 0.692937i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 6.86580i 0.296281i
\(538\) 0 0
\(539\) −11.3624 −0.489411
\(540\) 0 0
\(541\) −0.251725 −0.0108225 −0.00541124 0.999985i \(-0.501722\pi\)
−0.00541124 + 0.999985i \(0.501722\pi\)
\(542\) 0 0
\(543\) 4.99034i 0.214156i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 24.3395i 1.04068i 0.853958 + 0.520342i \(0.174195\pi\)
−0.853958 + 0.520342i \(0.825805\pi\)
\(548\) 0 0
\(549\) 17.2317 0.735429
\(550\) 0 0
\(551\) −17.4509 −0.743433
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 23.4966i − 0.995581i −0.867297 0.497790i \(-0.834145\pi\)
0.867297 0.497790i \(-0.165855\pi\)
\(558\) 0 0
\(559\) 1.07888 0.0456319
\(560\) 0 0
\(561\) 1.47447 0.0622520
\(562\) 0 0
\(563\) 19.5159i 0.822496i 0.911523 + 0.411248i \(0.134907\pi\)
−0.911523 + 0.411248i \(0.865093\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 3.14386i 0.132030i
\(568\) 0 0
\(569\) 8.51817 0.357100 0.178550 0.983931i \(-0.442859\pi\)
0.178550 + 0.983931i \(0.442859\pi\)
\(570\) 0 0
\(571\) −27.4811 −1.15005 −0.575024 0.818137i \(-0.695007\pi\)
−0.575024 + 0.818137i \(0.695007\pi\)
\(572\) 0 0
\(573\) 3.06728i 0.128137i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 26.1512i − 1.08869i −0.838862 0.544345i \(-0.816778\pi\)
0.838862 0.544345i \(-0.183222\pi\)
\(578\) 0 0
\(579\) 2.31210 0.0960877
\(580\) 0 0
\(581\) 2.14156 0.0888468
\(582\) 0 0
\(583\) − 18.1268i − 0.750737i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 34.3276i − 1.41685i −0.705786 0.708425i \(-0.749406\pi\)
0.705786 0.708425i \(-0.250594\pi\)
\(588\) 0 0
\(589\) −53.6922 −2.21235
\(590\) 0 0
\(591\) −5.98683 −0.246265
\(592\) 0 0
\(593\) − 15.9497i − 0.654978i −0.944855 0.327489i \(-0.893798\pi\)
0.944855 0.327489i \(-0.106202\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0.160881i 0.00658443i
\(598\) 0 0
\(599\) −40.5940 −1.65863 −0.829313 0.558784i \(-0.811268\pi\)
−0.829313 + 0.558784i \(0.811268\pi\)
\(600\) 0 0
\(601\) −29.0302 −1.18417 −0.592083 0.805877i \(-0.701694\pi\)
−0.592083 + 0.805877i \(0.701694\pi\)
\(602\) 0 0
\(603\) 40.0913i 1.63264i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 16.9977i 0.689915i 0.938618 + 0.344958i \(0.112107\pi\)
−0.938618 + 0.344958i \(0.887893\pi\)
\(608\) 0 0
\(609\) −0.549103 −0.0222508
\(610\) 0 0
\(611\) 4.81555 0.194816
\(612\) 0 0
\(613\) − 30.3024i − 1.22390i −0.790895 0.611952i \(-0.790385\pi\)
0.790895 0.611952i \(-0.209615\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 6.13069i − 0.246812i −0.992356 0.123406i \(-0.960618\pi\)
0.992356 0.123406i \(-0.0393818\pi\)
\(618\) 0 0
\(619\) 33.7305 1.35574 0.677872 0.735180i \(-0.262902\pi\)
0.677872 + 0.735180i \(0.262902\pi\)
\(620\) 0 0
\(621\) 3.00000 0.120386
\(622\) 0 0
\(623\) 1.71228i 0.0686012i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 6.96751i 0.278256i
\(628\) 0 0
\(629\) −6.70998 −0.267545
\(630\) 0 0
\(631\) 4.85614 0.193320 0.0966600 0.995317i \(-0.469184\pi\)
0.0966600 + 0.995317i \(0.469184\pi\)
\(632\) 0 0
\(633\) 9.06728i 0.360392i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 18.1358i − 0.718565i
\(638\) 0 0
\(639\) 27.1860 1.07546
\(640\) 0 0
\(641\) 7.78887 0.307642 0.153821 0.988099i \(-0.450842\pi\)
0.153821 + 0.988099i \(0.450842\pi\)
\(642\) 0 0
\(643\) − 43.6427i − 1.72110i −0.509366 0.860550i \(-0.670120\pi\)
0.509366 0.860550i \(-0.329880\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 15.4606i − 0.607817i −0.952701 0.303909i \(-0.901708\pi\)
0.952701 0.303909i \(-0.0982918\pi\)
\(648\) 0 0
\(649\) 15.8538 0.622318
\(650\) 0 0
\(651\) −1.68946 −0.0662150
\(652\) 0 0
\(653\) 9.70613i 0.379830i 0.981801 + 0.189915i \(0.0608213\pi\)
−0.981801 + 0.189915i \(0.939179\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 16.6404i 0.649204i
\(658\) 0 0
\(659\) −50.7100 −1.97538 −0.987690 0.156422i \(-0.950004\pi\)
−0.987690 + 0.156422i \(0.950004\pi\)
\(660\) 0 0
\(661\) 33.0132 1.28406 0.642032 0.766678i \(-0.278092\pi\)
0.642032 + 0.766678i \(0.278092\pi\)
\(662\) 0 0
\(663\) 2.35343i 0.0913998i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 2.20147i − 0.0852413i
\(668\) 0 0
\(669\) −0.997701 −0.0385733
\(670\) 0 0
\(671\) −10.6060 −0.409439
\(672\) 0 0
\(673\) − 19.2494i − 0.742011i −0.928631 0.371005i \(-0.879013\pi\)
0.928631 0.371005i \(-0.120987\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 25.1941i − 0.968288i −0.874988 0.484144i \(-0.839131\pi\)
0.874988 0.484144i \(-0.160869\pi\)
\(678\) 0 0
\(679\) 2.89443 0.111078
\(680\) 0 0
\(681\) −9.09361 −0.348468
\(682\) 0 0
\(683\) − 9.48339i − 0.362872i −0.983403 0.181436i \(-0.941926\pi\)
0.983403 0.181436i \(-0.0580745\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 3.24436i 0.123780i
\(688\) 0 0
\(689\) 28.9327 1.10225
\(690\) 0 0
\(691\) −23.8538 −0.907443 −0.453721 0.891144i \(-0.649904\pi\)
−0.453721 + 0.891144i \(0.649904\pi\)
\(692\) 0 0
\(693\) − 2.17635i − 0.0826726i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 10.0228i 0.379642i
\(698\) 0 0
\(699\) −8.27806 −0.313105
\(700\) 0 0
\(701\) 19.8767 0.750731 0.375366 0.926877i \(-0.377517\pi\)
0.375366 + 0.926877i \(0.377517\pi\)
\(702\) 0 0
\(703\) − 31.7077i − 1.19588i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 5.45090i 0.205002i
\(708\) 0 0
\(709\) 6.72545 0.252580 0.126290 0.991993i \(-0.459693\pi\)
0.126290 + 0.991993i \(0.459693\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) − 6.77340i − 0.253666i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 3.32601i − 0.124212i
\(718\) 0 0
\(719\) 38.2294 1.42571 0.712857 0.701309i \(-0.247401\pi\)
0.712857 + 0.701309i \(0.247401\pi\)
\(720\) 0 0
\(721\) −1.10557 −0.0411735
\(722\) 0 0
\(723\) 2.98298i 0.110938i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 3.72315i − 0.138084i −0.997614 0.0690420i \(-0.978006\pi\)
0.997614 0.0690420i \(-0.0219942\pi\)
\(728\) 0 0
\(729\) 13.2842 0.492008
\(730\) 0 0
\(731\) −0.675938 −0.0250005
\(732\) 0 0
\(733\) 8.49885i 0.313912i 0.987606 + 0.156956i \(0.0501681\pi\)
−0.987606 + 0.156956i \(0.949832\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 24.6759i − 0.908950i
\(738\) 0 0
\(739\) 0.251725 0.00925984 0.00462992 0.999989i \(-0.498526\pi\)
0.00462992 + 0.999989i \(0.498526\pi\)
\(740\) 0 0
\(741\) −11.1210 −0.408541
\(742\) 0 0
\(743\) 29.7305i 1.09071i 0.838206 + 0.545353i \(0.183605\pi\)
−0.838206 + 0.545353i \(0.816395\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 12.2614i − 0.448621i
\(748\) 0 0
\(749\) 3.73861 0.136606
\(750\) 0 0
\(751\) 10.6597 0.388979 0.194490 0.980905i \(-0.437695\pi\)
0.194490 + 0.980905i \(0.437695\pi\)
\(752\) 0 0
\(753\) − 15.1558i − 0.552309i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 16.1918i − 0.588501i −0.955728 0.294251i \(-0.904930\pi\)
0.955728 0.294251i \(-0.0950701\pi\)
\(758\) 0 0
\(759\) −0.878968 −0.0319045
\(760\) 0 0
\(761\) −46.9343 −1.70137 −0.850683 0.525679i \(-0.823811\pi\)
−0.850683 + 0.525679i \(0.823811\pi\)
\(762\) 0 0
\(763\) − 6.62954i − 0.240006i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 25.3047i 0.913701i
\(768\) 0 0
\(769\) 35.9954 1.29803 0.649014 0.760777i \(-0.275182\pi\)
0.649014 + 0.760777i \(0.275182\pi\)
\(770\) 0 0
\(771\) 7.60212 0.273784
\(772\) 0 0
\(773\) 21.8229i 0.784916i 0.919770 + 0.392458i \(0.128375\pi\)
−0.919770 + 0.392458i \(0.871625\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 0.997701i − 0.0357923i
\(778\) 0 0
\(779\) −47.3624 −1.69693
\(780\) 0 0
\(781\) −16.7328 −0.598747
\(782\) 0 0
\(783\) 6.60442i 0.236023i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 7.16318i − 0.255340i −0.991817 0.127670i \(-0.959250\pi\)
0.991817 0.127670i \(-0.0407498\pi\)
\(788\) 0 0
\(789\) −8.26489 −0.294238
\(790\) 0 0
\(791\) −4.76024 −0.169255
\(792\) 0 0
\(793\) − 16.9285i − 0.601148i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 48.0604i − 1.70239i −0.524853 0.851193i \(-0.675880\pi\)
0.524853 0.851193i \(-0.324120\pi\)
\(798\) 0 0
\(799\) −3.01702 −0.106735
\(800\) 0 0
\(801\) 9.80359 0.346393
\(802\) 0 0
\(803\) − 10.2421i − 0.361435i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 3.24943i 0.114385i
\(808\) 0 0
\(809\) −0.0611183 −0.00214880 −0.00107440 0.999999i \(-0.500342\pi\)
−0.00107440 + 0.999999i \(0.500342\pi\)
\(810\) 0 0
\(811\) 25.9092 0.909794 0.454897 0.890544i \(-0.349676\pi\)
0.454897 + 0.890544i \(0.349676\pi\)
\(812\) 0 0
\(813\) − 13.3979i − 0.469884i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 3.19411i − 0.111748i
\(818\) 0 0
\(819\) 3.47372 0.121382
\(820\) 0 0
\(821\) −21.0936 −0.736172 −0.368086 0.929792i \(-0.619987\pi\)
−0.368086 + 0.929792i \(0.619987\pi\)
\(822\) 0 0
\(823\) 37.2641i 1.29895i 0.760384 + 0.649473i \(0.225011\pi\)
−0.760384 + 0.649473i \(0.774989\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 4.45320i − 0.154853i −0.996998 0.0774264i \(-0.975330\pi\)
0.996998 0.0774264i \(-0.0246703\pi\)
\(828\) 0 0
\(829\) 38.1300 1.32431 0.662154 0.749368i \(-0.269642\pi\)
0.662154 + 0.749368i \(0.269642\pi\)
\(830\) 0 0
\(831\) −3.68790 −0.127932
\(832\) 0 0
\(833\) 11.3624i 0.393682i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 20.3202i 0.702369i
\(838\) 0 0
\(839\) 7.15858 0.247142 0.123571 0.992336i \(-0.460565\pi\)
0.123571 + 0.992336i \(0.460565\pi\)
\(840\) 0 0
\(841\) −24.1535 −0.832880
\(842\) 0 0
\(843\) 15.2282i 0.524486i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 3.89673i − 0.133893i
\(848\) 0 0
\(849\) −9.75334 −0.334734
\(850\) 0 0
\(851\) 4.00000 0.137118
\(852\) 0 0
\(853\) 49.6199i 1.69895i 0.527627 + 0.849476i \(0.323082\pi\)
−0.527627 + 0.849476i \(0.676918\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 44.0360i − 1.50424i −0.659026 0.752120i \(-0.729031\pi\)
0.659026 0.752120i \(-0.270969\pi\)
\(858\) 0 0
\(859\) 49.5062 1.68913 0.844565 0.535453i \(-0.179859\pi\)
0.844565 + 0.535453i \(0.179859\pi\)
\(860\) 0 0
\(861\) −1.49028 −0.0507887
\(862\) 0 0
\(863\) − 20.7962i − 0.707912i −0.935262 0.353956i \(-0.884836\pi\)
0.935262 0.353956i \(-0.115164\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 7.43313i 0.252442i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 39.3859 1.33454
\(872\) 0 0
\(873\) − 16.5719i − 0.560875i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 5.42071i − 0.183044i −0.995803 0.0915222i \(-0.970827\pi\)
0.995803 0.0915222i \(-0.0291732\pi\)
\(878\) 0 0
\(879\) 12.2614 0.413566
\(880\) 0 0
\(881\) −45.8036 −1.54316 −0.771581 0.636131i \(-0.780534\pi\)
−0.771581 + 0.636131i \(0.780534\pi\)
\(882\) 0 0
\(883\) − 57.1860i − 1.92446i −0.272234 0.962231i \(-0.587762\pi\)
0.272234 0.962231i \(-0.412238\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 15.5371i − 0.521686i −0.965381 0.260843i \(-0.915999\pi\)
0.965381 0.260843i \(-0.0840005\pi\)
\(888\) 0 0
\(889\) 5.28542 0.177267
\(890\) 0 0
\(891\) −11.0789 −0.371157
\(892\) 0 0
\(893\) − 14.2568i − 0.477085i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 1.40294i − 0.0468430i
\(898\) 0 0
\(899\) 14.9115 0.497325
\(900\) 0 0
\(901\) −18.1268 −0.603892
\(902\) 0 0
\(903\) − 0.100505i − 0.00334458i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 32.8515i − 1.09082i −0.838170 0.545409i \(-0.816374\pi\)
0.838170 0.545409i \(-0.183626\pi\)
\(908\) 0 0
\(909\) 31.2088 1.03513
\(910\) 0 0
\(911\) 8.95205 0.296595 0.148297 0.988943i \(-0.452621\pi\)
0.148297 + 0.988943i \(0.452621\pi\)
\(912\) 0 0
\(913\) 7.54680i 0.249763i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 3.64271i 0.120293i
\(918\) 0 0
\(919\) 49.8229 1.64351 0.821753 0.569844i \(-0.192996\pi\)
0.821753 + 0.569844i \(0.192996\pi\)
\(920\) 0 0
\(921\) 0.802385 0.0264395
\(922\) 0 0
\(923\) − 26.7077i − 0.879094i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 6.32987i 0.207900i
\(928\) 0 0
\(929\) −25.0723 −0.822597 −0.411298 0.911501i \(-0.634925\pi\)
−0.411298 + 0.911501i \(0.634925\pi\)
\(930\) 0 0
\(931\) −53.6922 −1.75969
\(932\) 0 0
\(933\) − 3.95941i − 0.129625i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 21.3322i 0.696891i 0.937329 + 0.348446i \(0.113290\pi\)
−0.937329 + 0.348446i \(0.886710\pi\)
\(938\) 0 0
\(939\) −1.22079 −0.0398391
\(940\) 0 0
\(941\) 3.93428 0.128254 0.0641270 0.997942i \(-0.479574\pi\)
0.0641270 + 0.997942i \(0.479574\pi\)
\(942\) 0 0
\(943\) − 5.97487i − 0.194569i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 48.6420i 1.58065i 0.612687 + 0.790326i \(0.290089\pi\)
−0.612687 + 0.790326i \(0.709911\pi\)
\(948\) 0 0
\(949\) 16.3476 0.530667
\(950\) 0 0
\(951\) 8.05761 0.261286
\(952\) 0 0
\(953\) 15.5623i 0.504111i 0.967713 + 0.252056i \(0.0811066\pi\)
−0.967713 + 0.252056i \(0.918893\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) − 1.93502i − 0.0625505i
\(958\) 0 0
\(959\) 10.1763 0.328611
\(960\) 0 0
\(961\) 14.8790 0.479967
\(962\) 0 0
\(963\) − 21.4052i − 0.689774i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 6.80083i 0.218700i 0.994003 + 0.109350i \(0.0348769\pi\)
−0.994003 + 0.109350i \(0.965123\pi\)
\(968\) 0 0
\(969\) 6.96751 0.223829
\(970\) 0 0
\(971\) 6.46406 0.207442 0.103721 0.994606i \(-0.466925\pi\)
0.103721 + 0.994606i \(0.466925\pi\)
\(972\) 0 0
\(973\) 4.09591i 0.131309i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 9.18601i 0.293886i 0.989145 + 0.146943i \(0.0469435\pi\)
−0.989145 + 0.146943i \(0.953057\pi\)
\(978\) 0 0
\(979\) −6.03405 −0.192849
\(980\) 0 0
\(981\) −37.9571 −1.21188
\(982\) 0 0
\(983\) − 15.3705i − 0.490241i −0.969493 0.245121i \(-0.921172\pi\)
0.969493 0.245121i \(-0.0788276\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 0.448598i − 0.0142790i
\(988\) 0 0
\(989\) 0.402945 0.0128129
\(990\) 0 0
\(991\) −52.0109 −1.65218 −0.826090 0.563538i \(-0.809440\pi\)
−0.826090 + 0.563538i \(0.809440\pi\)
\(992\) 0 0
\(993\) 12.0051i 0.380969i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 15.7123i 0.497613i 0.968553 + 0.248807i \(0.0800383\pi\)
−0.968553 + 0.248807i \(0.919962\pi\)
\(998\) 0 0
\(999\) −12.0000 −0.379663
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 4600.2.e.r.4049.4 6
5.2 odd 4 4600.2.a.y.1.2 3
5.3 odd 4 920.2.a.g.1.2 3
5.4 even 2 inner 4600.2.e.r.4049.3 6
15.8 even 4 8280.2.a.bo.1.2 3
20.3 even 4 1840.2.a.t.1.2 3
20.7 even 4 9200.2.a.cd.1.2 3
40.3 even 4 7360.2.a.ca.1.2 3
40.13 odd 4 7360.2.a.cb.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.g.1.2 3 5.3 odd 4
1840.2.a.t.1.2 3 20.3 even 4
4600.2.a.y.1.2 3 5.2 odd 4
4600.2.e.r.4049.3 6 5.4 even 2 inner
4600.2.e.r.4049.4 6 1.1 even 1 trivial
7360.2.a.ca.1.2 3 40.3 even 4
7360.2.a.cb.1.2 3 40.13 odd 4
8280.2.a.bo.1.2 3 15.8 even 4
9200.2.a.cd.1.2 3 20.7 even 4