Properties

Label 6160.2.a.t.1.2
Level $6160$
Weight $2$
Character 6160.1
Self dual yes
Analytic conductor $49.188$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6160 = 2^{4} \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6160.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(49.1878476451\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \( x^{2} - 3 \) 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 770)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 6160.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.732051 q^{3} -1.00000 q^{5} -1.00000 q^{7} -2.46410 q^{9} +O(q^{10})\) \(q+0.732051 q^{3} -1.00000 q^{5} -1.00000 q^{7} -2.46410 q^{9} -1.00000 q^{11} +5.46410 q^{13} -0.732051 q^{15} +3.46410 q^{17} -3.26795 q^{19} -0.732051 q^{21} -2.19615 q^{23} +1.00000 q^{25} -4.00000 q^{27} -1.26795 q^{29} -2.00000 q^{31} -0.732051 q^{33} +1.00000 q^{35} -2.73205 q^{37} +4.00000 q^{39} +8.19615 q^{41} -2.00000 q^{43} +2.46410 q^{45} +6.92820 q^{47} +1.00000 q^{49} +2.53590 q^{51} -10.7321 q^{53} +1.00000 q^{55} -2.39230 q^{57} +6.92820 q^{59} +8.92820 q^{61} +2.46410 q^{63} -5.46410 q^{65} +4.00000 q^{67} -1.60770 q^{69} -2.53590 q^{71} +6.39230 q^{73} +0.732051 q^{75} +1.00000 q^{77} +1.80385 q^{79} +4.46410 q^{81} -4.39230 q^{83} -3.46410 q^{85} -0.928203 q^{87} -3.46410 q^{89} -5.46410 q^{91} -1.46410 q^{93} +3.26795 q^{95} -16.5885 q^{97} +2.46410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9} - 2 q^{11} + 4 q^{13} + 2 q^{15} - 10 q^{19} + 2 q^{21} + 6 q^{23} + 2 q^{25} - 8 q^{27} - 6 q^{29} - 4 q^{31} + 2 q^{33} + 2 q^{35} - 2 q^{37} + 8 q^{39} + 6 q^{41} - 4 q^{43} - 2 q^{45} + 2 q^{49} + 12 q^{51} - 18 q^{53} + 2 q^{55} + 16 q^{57} + 4 q^{61} - 2 q^{63} - 4 q^{65} + 8 q^{67} - 24 q^{69} - 12 q^{71} - 8 q^{73} - 2 q^{75} + 2 q^{77} + 14 q^{79} + 2 q^{81} + 12 q^{83} + 12 q^{87} - 4 q^{91} + 4 q^{93} + 10 q^{95} - 2 q^{97} - 2 q^{99}+O(q^{100}) \) 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
\(3\) 0.732051 0.422650 0.211325 0.977416i \(-0.432222\pi\)
0.211325 + 0.977416i \(0.432222\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) −2.46410 −0.821367
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 5.46410 1.51547 0.757735 0.652563i \(-0.226306\pi\)
0.757735 + 0.652563i \(0.226306\pi\)
\(14\) 0 0
\(15\) −0.732051 −0.189015
\(16\) 0 0
\(17\) 3.46410 0.840168 0.420084 0.907485i \(-0.362001\pi\)
0.420084 + 0.907485i \(0.362001\pi\)
\(18\) 0 0
\(19\) −3.26795 −0.749719 −0.374859 0.927082i \(-0.622309\pi\)
−0.374859 + 0.927082i \(0.622309\pi\)
\(20\) 0 0
\(21\) −0.732051 −0.159747
\(22\) 0 0
\(23\) −2.19615 −0.457929 −0.228965 0.973435i \(-0.573534\pi\)
−0.228965 + 0.973435i \(0.573534\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −4.00000 −0.769800
\(28\) 0 0
\(29\) −1.26795 −0.235452 −0.117726 0.993046i \(-0.537560\pi\)
−0.117726 + 0.993046i \(0.537560\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 0 0
\(33\) −0.732051 −0.127434
\(34\) 0 0
\(35\) 1.00000 0.169031
\(36\) 0 0
\(37\) −2.73205 −0.449146 −0.224573 0.974457i \(-0.572099\pi\)
−0.224573 + 0.974457i \(0.572099\pi\)
\(38\) 0 0
\(39\) 4.00000 0.640513
\(40\) 0 0
\(41\) 8.19615 1.28002 0.640012 0.768365i \(-0.278929\pi\)
0.640012 + 0.768365i \(0.278929\pi\)
\(42\) 0 0
\(43\) −2.00000 −0.304997 −0.152499 0.988304i \(-0.548732\pi\)
−0.152499 + 0.988304i \(0.548732\pi\)
\(44\) 0 0
\(45\) 2.46410 0.367327
\(46\) 0 0
\(47\) 6.92820 1.01058 0.505291 0.862949i \(-0.331385\pi\)
0.505291 + 0.862949i \(0.331385\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 2.53590 0.355097
\(52\) 0 0
\(53\) −10.7321 −1.47416 −0.737080 0.675805i \(-0.763796\pi\)
−0.737080 + 0.675805i \(0.763796\pi\)
\(54\) 0 0
\(55\) 1.00000 0.134840
\(56\) 0 0
\(57\) −2.39230 −0.316869
\(58\) 0 0
\(59\) 6.92820 0.901975 0.450988 0.892530i \(-0.351072\pi\)
0.450988 + 0.892530i \(0.351072\pi\)
\(60\) 0 0
\(61\) 8.92820 1.14314 0.571570 0.820554i \(-0.306335\pi\)
0.571570 + 0.820554i \(0.306335\pi\)
\(62\) 0 0
\(63\) 2.46410 0.310448
\(64\) 0 0
\(65\) −5.46410 −0.677738
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 0 0
\(69\) −1.60770 −0.193544
\(70\) 0 0
\(71\) −2.53590 −0.300956 −0.150478 0.988613i \(-0.548081\pi\)
−0.150478 + 0.988613i \(0.548081\pi\)
\(72\) 0 0
\(73\) 6.39230 0.748163 0.374081 0.927396i \(-0.377958\pi\)
0.374081 + 0.927396i \(0.377958\pi\)
\(74\) 0 0
\(75\) 0.732051 0.0845299
\(76\) 0 0
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) 1.80385 0.202949 0.101474 0.994838i \(-0.467644\pi\)
0.101474 + 0.994838i \(0.467644\pi\)
\(80\) 0 0
\(81\) 4.46410 0.496011
\(82\) 0 0
\(83\) −4.39230 −0.482118 −0.241059 0.970510i \(-0.577495\pi\)
−0.241059 + 0.970510i \(0.577495\pi\)
\(84\) 0 0
\(85\) −3.46410 −0.375735
\(86\) 0 0
\(87\) −0.928203 −0.0995138
\(88\) 0 0
\(89\) −3.46410 −0.367194 −0.183597 0.983002i \(-0.558774\pi\)
−0.183597 + 0.983002i \(0.558774\pi\)
\(90\) 0 0
\(91\) −5.46410 −0.572793
\(92\) 0 0
\(93\) −1.46410 −0.151820
\(94\) 0 0
\(95\) 3.26795 0.335285
\(96\) 0 0
\(97\) −16.5885 −1.68430 −0.842151 0.539241i \(-0.818711\pi\)
−0.842151 + 0.539241i \(0.818711\pi\)
\(98\) 0 0
\(99\) 2.46410 0.247652
\(100\) 0 0
\(101\) 19.8564 1.97579 0.987893 0.155136i \(-0.0495815\pi\)
0.987893 + 0.155136i \(0.0495815\pi\)
\(102\) 0 0
\(103\) 8.39230 0.826918 0.413459 0.910523i \(-0.364320\pi\)
0.413459 + 0.910523i \(0.364320\pi\)
\(104\) 0 0
\(105\) 0.732051 0.0714408
\(106\) 0 0
\(107\) 19.8564 1.91959 0.959796 0.280700i \(-0.0905665\pi\)
0.959796 + 0.280700i \(0.0905665\pi\)
\(108\) 0 0
\(109\) 1.66025 0.159023 0.0795117 0.996834i \(-0.474664\pi\)
0.0795117 + 0.996834i \(0.474664\pi\)
\(110\) 0 0
\(111\) −2.00000 −0.189832
\(112\) 0 0
\(113\) −19.8564 −1.86793 −0.933967 0.357360i \(-0.883677\pi\)
−0.933967 + 0.357360i \(0.883677\pi\)
\(114\) 0 0
\(115\) 2.19615 0.204792
\(116\) 0 0
\(117\) −13.4641 −1.24476
\(118\) 0 0
\(119\) −3.46410 −0.317554
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 6.00000 0.541002
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −14.9282 −1.32466 −0.662332 0.749211i \(-0.730433\pi\)
−0.662332 + 0.749211i \(0.730433\pi\)
\(128\) 0 0
\(129\) −1.46410 −0.128907
\(130\) 0 0
\(131\) 11.6603 1.01876 0.509381 0.860541i \(-0.329875\pi\)
0.509381 + 0.860541i \(0.329875\pi\)
\(132\) 0 0
\(133\) 3.26795 0.283367
\(134\) 0 0
\(135\) 4.00000 0.344265
\(136\) 0 0
\(137\) 12.9282 1.10453 0.552265 0.833668i \(-0.313763\pi\)
0.552265 + 0.833668i \(0.313763\pi\)
\(138\) 0 0
\(139\) 3.66025 0.310459 0.155229 0.987878i \(-0.450388\pi\)
0.155229 + 0.987878i \(0.450388\pi\)
\(140\) 0 0
\(141\) 5.07180 0.427122
\(142\) 0 0
\(143\) −5.46410 −0.456931
\(144\) 0 0
\(145\) 1.26795 0.105297
\(146\) 0 0
\(147\) 0.732051 0.0603785
\(148\) 0 0
\(149\) −10.7321 −0.879204 −0.439602 0.898193i \(-0.644880\pi\)
−0.439602 + 0.898193i \(0.644880\pi\)
\(150\) 0 0
\(151\) 13.1244 1.06804 0.534022 0.845470i \(-0.320680\pi\)
0.534022 + 0.845470i \(0.320680\pi\)
\(152\) 0 0
\(153\) −8.53590 −0.690086
\(154\) 0 0
\(155\) 2.00000 0.160644
\(156\) 0 0
\(157\) 11.4641 0.914935 0.457467 0.889226i \(-0.348757\pi\)
0.457467 + 0.889226i \(0.348757\pi\)
\(158\) 0 0
\(159\) −7.85641 −0.623054
\(160\) 0 0
\(161\) 2.19615 0.173081
\(162\) 0 0
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) 0 0
\(165\) 0.732051 0.0569901
\(166\) 0 0
\(167\) 13.8564 1.07224 0.536120 0.844141i \(-0.319889\pi\)
0.536120 + 0.844141i \(0.319889\pi\)
\(168\) 0 0
\(169\) 16.8564 1.29665
\(170\) 0 0
\(171\) 8.05256 0.615795
\(172\) 0 0
\(173\) −12.9282 −0.982913 −0.491457 0.870902i \(-0.663535\pi\)
−0.491457 + 0.870902i \(0.663535\pi\)
\(174\) 0 0
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) 5.07180 0.381220
\(178\) 0 0
\(179\) 6.00000 0.448461 0.224231 0.974536i \(-0.428013\pi\)
0.224231 + 0.974536i \(0.428013\pi\)
\(180\) 0 0
\(181\) 14.0000 1.04061 0.520306 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) 0 0
\(183\) 6.53590 0.483148
\(184\) 0 0
\(185\) 2.73205 0.200864
\(186\) 0 0
\(187\) −3.46410 −0.253320
\(188\) 0 0
\(189\) 4.00000 0.290957
\(190\) 0 0
\(191\) 12.0000 0.868290 0.434145 0.900843i \(-0.357051\pi\)
0.434145 + 0.900843i \(0.357051\pi\)
\(192\) 0 0
\(193\) −8.39230 −0.604091 −0.302046 0.953293i \(-0.597670\pi\)
−0.302046 + 0.953293i \(0.597670\pi\)
\(194\) 0 0
\(195\) −4.00000 −0.286446
\(196\) 0 0
\(197\) 24.2487 1.72765 0.863825 0.503793i \(-0.168062\pi\)
0.863825 + 0.503793i \(0.168062\pi\)
\(198\) 0 0
\(199\) 10.9282 0.774680 0.387340 0.921937i \(-0.373394\pi\)
0.387340 + 0.921937i \(0.373394\pi\)
\(200\) 0 0
\(201\) 2.92820 0.206540
\(202\) 0 0
\(203\) 1.26795 0.0889926
\(204\) 0 0
\(205\) −8.19615 −0.572444
\(206\) 0 0
\(207\) 5.41154 0.376128
\(208\) 0 0
\(209\) 3.26795 0.226049
\(210\) 0 0
\(211\) −13.0718 −0.899900 −0.449950 0.893054i \(-0.648558\pi\)
−0.449950 + 0.893054i \(0.648558\pi\)
\(212\) 0 0
\(213\) −1.85641 −0.127199
\(214\) 0 0
\(215\) 2.00000 0.136399
\(216\) 0 0
\(217\) 2.00000 0.135769
\(218\) 0 0
\(219\) 4.67949 0.316211
\(220\) 0 0
\(221\) 18.9282 1.27325
\(222\) 0 0
\(223\) 18.5359 1.24126 0.620628 0.784105i \(-0.286878\pi\)
0.620628 + 0.784105i \(0.286878\pi\)
\(224\) 0 0
\(225\) −2.46410 −0.164273
\(226\) 0 0
\(227\) 6.92820 0.459841 0.229920 0.973209i \(-0.426153\pi\)
0.229920 + 0.973209i \(0.426153\pi\)
\(228\) 0 0
\(229\) 3.60770 0.238403 0.119202 0.992870i \(-0.461967\pi\)
0.119202 + 0.992870i \(0.461967\pi\)
\(230\) 0 0
\(231\) 0.732051 0.0481654
\(232\) 0 0
\(233\) 19.8564 1.30084 0.650418 0.759576i \(-0.274594\pi\)
0.650418 + 0.759576i \(0.274594\pi\)
\(234\) 0 0
\(235\) −6.92820 −0.451946
\(236\) 0 0
\(237\) 1.32051 0.0857762
\(238\) 0 0
\(239\) −4.73205 −0.306091 −0.153045 0.988219i \(-0.548908\pi\)
−0.153045 + 0.988219i \(0.548908\pi\)
\(240\) 0 0
\(241\) 6.73205 0.433650 0.216825 0.976211i \(-0.430430\pi\)
0.216825 + 0.976211i \(0.430430\pi\)
\(242\) 0 0
\(243\) 15.2679 0.979439
\(244\) 0 0
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) −17.8564 −1.13618
\(248\) 0 0
\(249\) −3.21539 −0.203767
\(250\) 0 0
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 0 0
\(253\) 2.19615 0.138071
\(254\) 0 0
\(255\) −2.53590 −0.158804
\(256\) 0 0
\(257\) −6.33975 −0.395462 −0.197731 0.980256i \(-0.563357\pi\)
−0.197731 + 0.980256i \(0.563357\pi\)
\(258\) 0 0
\(259\) 2.73205 0.169761
\(260\) 0 0
\(261\) 3.12436 0.193393
\(262\) 0 0
\(263\) 24.0000 1.47990 0.739952 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) 0 0
\(265\) 10.7321 0.659265
\(266\) 0 0
\(267\) −2.53590 −0.155194
\(268\) 0 0
\(269\) 7.60770 0.463849 0.231925 0.972734i \(-0.425498\pi\)
0.231925 + 0.972734i \(0.425498\pi\)
\(270\) 0 0
\(271\) 20.3923 1.23874 0.619372 0.785098i \(-0.287387\pi\)
0.619372 + 0.785098i \(0.287387\pi\)
\(272\) 0 0
\(273\) −4.00000 −0.242091
\(274\) 0 0
\(275\) −1.00000 −0.0603023
\(276\) 0 0
\(277\) 20.9282 1.25745 0.628727 0.777626i \(-0.283576\pi\)
0.628727 + 0.777626i \(0.283576\pi\)
\(278\) 0 0
\(279\) 4.92820 0.295044
\(280\) 0 0
\(281\) 1.60770 0.0959071 0.0479535 0.998850i \(-0.484730\pi\)
0.0479535 + 0.998850i \(0.484730\pi\)
\(282\) 0 0
\(283\) −23.7128 −1.40958 −0.704790 0.709416i \(-0.748959\pi\)
−0.704790 + 0.709416i \(0.748959\pi\)
\(284\) 0 0
\(285\) 2.39230 0.141708
\(286\) 0 0
\(287\) −8.19615 −0.483804
\(288\) 0 0
\(289\) −5.00000 −0.294118
\(290\) 0 0
\(291\) −12.1436 −0.711870
\(292\) 0 0
\(293\) 14.5359 0.849196 0.424598 0.905382i \(-0.360415\pi\)
0.424598 + 0.905382i \(0.360415\pi\)
\(294\) 0 0
\(295\) −6.92820 −0.403376
\(296\) 0 0
\(297\) 4.00000 0.232104
\(298\) 0 0
\(299\) −12.0000 −0.693978
\(300\) 0 0
\(301\) 2.00000 0.115278
\(302\) 0 0
\(303\) 14.5359 0.835066
\(304\) 0 0
\(305\) −8.92820 −0.511227
\(306\) 0 0
\(307\) −3.60770 −0.205902 −0.102951 0.994686i \(-0.532828\pi\)
−0.102951 + 0.994686i \(0.532828\pi\)
\(308\) 0 0
\(309\) 6.14359 0.349497
\(310\) 0 0
\(311\) −11.0718 −0.627824 −0.313912 0.949452i \(-0.601640\pi\)
−0.313912 + 0.949452i \(0.601640\pi\)
\(312\) 0 0
\(313\) 11.8038 0.667193 0.333596 0.942716i \(-0.391738\pi\)
0.333596 + 0.942716i \(0.391738\pi\)
\(314\) 0 0
\(315\) −2.46410 −0.138836
\(316\) 0 0
\(317\) −0.588457 −0.0330511 −0.0165255 0.999863i \(-0.505260\pi\)
−0.0165255 + 0.999863i \(0.505260\pi\)
\(318\) 0 0
\(319\) 1.26795 0.0709915
\(320\) 0 0
\(321\) 14.5359 0.811315
\(322\) 0 0
\(323\) −11.3205 −0.629890
\(324\) 0 0
\(325\) 5.46410 0.303094
\(326\) 0 0
\(327\) 1.21539 0.0672112
\(328\) 0 0
\(329\) −6.92820 −0.381964
\(330\) 0 0
\(331\) −22.7846 −1.25236 −0.626178 0.779680i \(-0.715382\pi\)
−0.626178 + 0.779680i \(0.715382\pi\)
\(332\) 0 0
\(333\) 6.73205 0.368914
\(334\) 0 0
\(335\) −4.00000 −0.218543
\(336\) 0 0
\(337\) −18.7846 −1.02326 −0.511631 0.859205i \(-0.670959\pi\)
−0.511631 + 0.859205i \(0.670959\pi\)
\(338\) 0 0
\(339\) −14.5359 −0.789482
\(340\) 0 0
\(341\) 2.00000 0.108306
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 1.60770 0.0865554
\(346\) 0 0
\(347\) 36.9282 1.98241 0.991205 0.132336i \(-0.0422478\pi\)
0.991205 + 0.132336i \(0.0422478\pi\)
\(348\) 0 0
\(349\) −26.3923 −1.41275 −0.706374 0.707839i \(-0.749670\pi\)
−0.706374 + 0.707839i \(0.749670\pi\)
\(350\) 0 0
\(351\) −21.8564 −1.16661
\(352\) 0 0
\(353\) 3.80385 0.202458 0.101229 0.994863i \(-0.467722\pi\)
0.101229 + 0.994863i \(0.467722\pi\)
\(354\) 0 0
\(355\) 2.53590 0.134592
\(356\) 0 0
\(357\) −2.53590 −0.134214
\(358\) 0 0
\(359\) −4.05256 −0.213886 −0.106943 0.994265i \(-0.534106\pi\)
−0.106943 + 0.994265i \(0.534106\pi\)
\(360\) 0 0
\(361\) −8.32051 −0.437921
\(362\) 0 0
\(363\) 0.732051 0.0384227
\(364\) 0 0
\(365\) −6.39230 −0.334589
\(366\) 0 0
\(367\) 8.39230 0.438075 0.219037 0.975716i \(-0.429708\pi\)
0.219037 + 0.975716i \(0.429708\pi\)
\(368\) 0 0
\(369\) −20.1962 −1.05137
\(370\) 0 0
\(371\) 10.7321 0.557180
\(372\) 0 0
\(373\) −2.39230 −0.123869 −0.0619344 0.998080i \(-0.519727\pi\)
−0.0619344 + 0.998080i \(0.519727\pi\)
\(374\) 0 0
\(375\) −0.732051 −0.0378029
\(376\) 0 0
\(377\) −6.92820 −0.356821
\(378\) 0 0
\(379\) −33.8564 −1.73909 −0.869543 0.493857i \(-0.835587\pi\)
−0.869543 + 0.493857i \(0.835587\pi\)
\(380\) 0 0
\(381\) −10.9282 −0.559869
\(382\) 0 0
\(383\) 26.5359 1.35592 0.677961 0.735098i \(-0.262864\pi\)
0.677961 + 0.735098i \(0.262864\pi\)
\(384\) 0 0
\(385\) −1.00000 −0.0509647
\(386\) 0 0
\(387\) 4.92820 0.250515
\(388\) 0 0
\(389\) −22.3923 −1.13533 −0.567667 0.823258i \(-0.692154\pi\)
−0.567667 + 0.823258i \(0.692154\pi\)
\(390\) 0 0
\(391\) −7.60770 −0.384738
\(392\) 0 0
\(393\) 8.53590 0.430579
\(394\) 0 0
\(395\) −1.80385 −0.0907614
\(396\) 0 0
\(397\) 9.60770 0.482196 0.241098 0.970501i \(-0.422492\pi\)
0.241098 + 0.970501i \(0.422492\pi\)
\(398\) 0 0
\(399\) 2.39230 0.119765
\(400\) 0 0
\(401\) 9.46410 0.472615 0.236307 0.971678i \(-0.424063\pi\)
0.236307 + 0.971678i \(0.424063\pi\)
\(402\) 0 0
\(403\) −10.9282 −0.544373
\(404\) 0 0
\(405\) −4.46410 −0.221823
\(406\) 0 0
\(407\) 2.73205 0.135423
\(408\) 0 0
\(409\) 35.1244 1.73679 0.868394 0.495875i \(-0.165153\pi\)
0.868394 + 0.495875i \(0.165153\pi\)
\(410\) 0 0
\(411\) 9.46410 0.466830
\(412\) 0 0
\(413\) −6.92820 −0.340915
\(414\) 0 0
\(415\) 4.39230 0.215610
\(416\) 0 0
\(417\) 2.67949 0.131215
\(418\) 0 0
\(419\) −17.0718 −0.834012 −0.417006 0.908904i \(-0.636921\pi\)
−0.417006 + 0.908904i \(0.636921\pi\)
\(420\) 0 0
\(421\) −8.14359 −0.396894 −0.198447 0.980112i \(-0.563590\pi\)
−0.198447 + 0.980112i \(0.563590\pi\)
\(422\) 0 0
\(423\) −17.0718 −0.830059
\(424\) 0 0
\(425\) 3.46410 0.168034
\(426\) 0 0
\(427\) −8.92820 −0.432066
\(428\) 0 0
\(429\) −4.00000 −0.193122
\(430\) 0 0
\(431\) 21.1244 1.01752 0.508762 0.860907i \(-0.330103\pi\)
0.508762 + 0.860907i \(0.330103\pi\)
\(432\) 0 0
\(433\) −7.80385 −0.375029 −0.187514 0.982262i \(-0.560043\pi\)
−0.187514 + 0.982262i \(0.560043\pi\)
\(434\) 0 0
\(435\) 0.928203 0.0445039
\(436\) 0 0
\(437\) 7.17691 0.343318
\(438\) 0 0
\(439\) 22.2487 1.06187 0.530937 0.847412i \(-0.321840\pi\)
0.530937 + 0.847412i \(0.321840\pi\)
\(440\) 0 0
\(441\) −2.46410 −0.117338
\(442\) 0 0
\(443\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) 0 0
\(445\) 3.46410 0.164214
\(446\) 0 0
\(447\) −7.85641 −0.371595
\(448\) 0 0
\(449\) 19.6077 0.925344 0.462672 0.886529i \(-0.346891\pi\)
0.462672 + 0.886529i \(0.346891\pi\)
\(450\) 0 0
\(451\) −8.19615 −0.385942
\(452\) 0 0
\(453\) 9.60770 0.451409
\(454\) 0 0
\(455\) 5.46410 0.256161
\(456\) 0 0
\(457\) 2.00000 0.0935561 0.0467780 0.998905i \(-0.485105\pi\)
0.0467780 + 0.998905i \(0.485105\pi\)
\(458\) 0 0
\(459\) −13.8564 −0.646762
\(460\) 0 0
\(461\) 1.60770 0.0748778 0.0374389 0.999299i \(-0.488080\pi\)
0.0374389 + 0.999299i \(0.488080\pi\)
\(462\) 0 0
\(463\) 17.5167 0.814068 0.407034 0.913413i \(-0.366563\pi\)
0.407034 + 0.913413i \(0.366563\pi\)
\(464\) 0 0
\(465\) 1.46410 0.0678961
\(466\) 0 0
\(467\) −4.05256 −0.187530 −0.0937650 0.995594i \(-0.529890\pi\)
−0.0937650 + 0.995594i \(0.529890\pi\)
\(468\) 0 0
\(469\) −4.00000 −0.184703
\(470\) 0 0
\(471\) 8.39230 0.386697
\(472\) 0 0
\(473\) 2.00000 0.0919601
\(474\) 0 0
\(475\) −3.26795 −0.149944
\(476\) 0 0
\(477\) 26.4449 1.21083
\(478\) 0 0
\(479\) −8.78461 −0.401379 −0.200690 0.979655i \(-0.564318\pi\)
−0.200690 + 0.979655i \(0.564318\pi\)
\(480\) 0 0
\(481\) −14.9282 −0.680667
\(482\) 0 0
\(483\) 1.60770 0.0731527
\(484\) 0 0
\(485\) 16.5885 0.753243
\(486\) 0 0
\(487\) 6.19615 0.280774 0.140387 0.990097i \(-0.455165\pi\)
0.140387 + 0.990097i \(0.455165\pi\)
\(488\) 0 0
\(489\) 2.92820 0.132418
\(490\) 0 0
\(491\) −27.7128 −1.25066 −0.625331 0.780360i \(-0.715036\pi\)
−0.625331 + 0.780360i \(0.715036\pi\)
\(492\) 0 0
\(493\) −4.39230 −0.197819
\(494\) 0 0
\(495\) −2.46410 −0.110753
\(496\) 0 0
\(497\) 2.53590 0.113751
\(498\) 0 0
\(499\) −39.8564 −1.78422 −0.892109 0.451820i \(-0.850775\pi\)
−0.892109 + 0.451820i \(0.850775\pi\)
\(500\) 0 0
\(501\) 10.1436 0.453182
\(502\) 0 0
\(503\) 32.7846 1.46179 0.730897 0.682488i \(-0.239102\pi\)
0.730897 + 0.682488i \(0.239102\pi\)
\(504\) 0 0
\(505\) −19.8564 −0.883598
\(506\) 0 0
\(507\) 12.3397 0.548027
\(508\) 0 0
\(509\) 24.9282 1.10492 0.552462 0.833538i \(-0.313689\pi\)
0.552462 + 0.833538i \(0.313689\pi\)
\(510\) 0 0
\(511\) −6.39230 −0.282779
\(512\) 0 0
\(513\) 13.0718 0.577134
\(514\) 0 0
\(515\) −8.39230 −0.369809
\(516\) 0 0
\(517\) −6.92820 −0.304702
\(518\) 0 0
\(519\) −9.46410 −0.415428
\(520\) 0 0
\(521\) −10.3923 −0.455295 −0.227648 0.973744i \(-0.573103\pi\)
−0.227648 + 0.973744i \(0.573103\pi\)
\(522\) 0 0
\(523\) −33.1769 −1.45073 −0.725363 0.688367i \(-0.758328\pi\)
−0.725363 + 0.688367i \(0.758328\pi\)
\(524\) 0 0
\(525\) −0.732051 −0.0319493
\(526\) 0 0
\(527\) −6.92820 −0.301797
\(528\) 0 0
\(529\) −18.1769 −0.790301
\(530\) 0 0
\(531\) −17.0718 −0.740853
\(532\) 0 0
\(533\) 44.7846 1.93984
\(534\) 0 0
\(535\) −19.8564 −0.858467
\(536\) 0 0
\(537\) 4.39230 0.189542
\(538\) 0 0
\(539\) −1.00000 −0.0430730
\(540\) 0 0
\(541\) −17.2679 −0.742407 −0.371204 0.928552i \(-0.621055\pi\)
−0.371204 + 0.928552i \(0.621055\pi\)
\(542\) 0 0
\(543\) 10.2487 0.439814
\(544\) 0 0
\(545\) −1.66025 −0.0711175
\(546\) 0 0
\(547\) 12.7846 0.546630 0.273315 0.961925i \(-0.411880\pi\)
0.273315 + 0.961925i \(0.411880\pi\)
\(548\) 0 0
\(549\) −22.0000 −0.938937
\(550\) 0 0
\(551\) 4.14359 0.176523
\(552\) 0 0
\(553\) −1.80385 −0.0767074
\(554\) 0 0
\(555\) 2.00000 0.0848953
\(556\) 0 0
\(557\) −46.3923 −1.96571 −0.982853 0.184393i \(-0.940968\pi\)
−0.982853 + 0.184393i \(0.940968\pi\)
\(558\) 0 0
\(559\) −10.9282 −0.462214
\(560\) 0 0
\(561\) −2.53590 −0.107066
\(562\) 0 0
\(563\) −18.9282 −0.797729 −0.398864 0.917010i \(-0.630596\pi\)
−0.398864 + 0.917010i \(0.630596\pi\)
\(564\) 0 0
\(565\) 19.8564 0.835365
\(566\) 0 0
\(567\) −4.46410 −0.187475
\(568\) 0 0
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 0 0
\(571\) −24.3923 −1.02079 −0.510393 0.859941i \(-0.670500\pi\)
−0.510393 + 0.859941i \(0.670500\pi\)
\(572\) 0 0
\(573\) 8.78461 0.366982
\(574\) 0 0
\(575\) −2.19615 −0.0915859
\(576\) 0 0
\(577\) −3.41154 −0.142024 −0.0710122 0.997475i \(-0.522623\pi\)
−0.0710122 + 0.997475i \(0.522623\pi\)
\(578\) 0 0
\(579\) −6.14359 −0.255319
\(580\) 0 0
\(581\) 4.39230 0.182224
\(582\) 0 0
\(583\) 10.7321 0.444476
\(584\) 0 0
\(585\) 13.4641 0.556672
\(586\) 0 0
\(587\) 42.5885 1.75781 0.878907 0.476993i \(-0.158273\pi\)
0.878907 + 0.476993i \(0.158273\pi\)
\(588\) 0 0
\(589\) 6.53590 0.269307
\(590\) 0 0
\(591\) 17.7513 0.730190
\(592\) 0 0
\(593\) −48.2487 −1.98134 −0.990669 0.136293i \(-0.956481\pi\)
−0.990669 + 0.136293i \(0.956481\pi\)
\(594\) 0 0
\(595\) 3.46410 0.142014
\(596\) 0 0
\(597\) 8.00000 0.327418
\(598\) 0 0
\(599\) 25.1769 1.02870 0.514350 0.857580i \(-0.328033\pi\)
0.514350 + 0.857580i \(0.328033\pi\)
\(600\) 0 0
\(601\) 39.5167 1.61192 0.805959 0.591971i \(-0.201650\pi\)
0.805959 + 0.591971i \(0.201650\pi\)
\(602\) 0 0
\(603\) −9.85641 −0.401384
\(604\) 0 0
\(605\) −1.00000 −0.0406558
\(606\) 0 0
\(607\) −7.07180 −0.287035 −0.143518 0.989648i \(-0.545841\pi\)
−0.143518 + 0.989648i \(0.545841\pi\)
\(608\) 0 0
\(609\) 0.928203 0.0376127
\(610\) 0 0
\(611\) 37.8564 1.53151
\(612\) 0 0
\(613\) 27.1769 1.09767 0.548833 0.835932i \(-0.315072\pi\)
0.548833 + 0.835932i \(0.315072\pi\)
\(614\) 0 0
\(615\) −6.00000 −0.241943
\(616\) 0 0
\(617\) −17.3205 −0.697297 −0.348649 0.937253i \(-0.613359\pi\)
−0.348649 + 0.937253i \(0.613359\pi\)
\(618\) 0 0
\(619\) 12.7846 0.513857 0.256928 0.966430i \(-0.417290\pi\)
0.256928 + 0.966430i \(0.417290\pi\)
\(620\) 0 0
\(621\) 8.78461 0.352514
\(622\) 0 0
\(623\) 3.46410 0.138786
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 2.39230 0.0955395
\(628\) 0 0
\(629\) −9.46410 −0.377358
\(630\) 0 0
\(631\) 33.0718 1.31657 0.658284 0.752770i \(-0.271283\pi\)
0.658284 + 0.752770i \(0.271283\pi\)
\(632\) 0 0
\(633\) −9.56922 −0.380342
\(634\) 0 0
\(635\) 14.9282 0.592408
\(636\) 0 0
\(637\) 5.46410 0.216496
\(638\) 0 0
\(639\) 6.24871 0.247195
\(640\) 0 0
\(641\) 47.5692 1.87887 0.939436 0.342725i \(-0.111350\pi\)
0.939436 + 0.342725i \(0.111350\pi\)
\(642\) 0 0
\(643\) −36.7321 −1.44857 −0.724285 0.689500i \(-0.757830\pi\)
−0.724285 + 0.689500i \(0.757830\pi\)
\(644\) 0 0
\(645\) 1.46410 0.0576489
\(646\) 0 0
\(647\) 33.4641 1.31561 0.657805 0.753188i \(-0.271485\pi\)
0.657805 + 0.753188i \(0.271485\pi\)
\(648\) 0 0
\(649\) −6.92820 −0.271956
\(650\) 0 0
\(651\) 1.46410 0.0573827
\(652\) 0 0
\(653\) −5.66025 −0.221503 −0.110751 0.993848i \(-0.535326\pi\)
−0.110751 + 0.993848i \(0.535326\pi\)
\(654\) 0 0
\(655\) −11.6603 −0.455604
\(656\) 0 0
\(657\) −15.7513 −0.614516
\(658\) 0 0
\(659\) 18.2487 0.710869 0.355434 0.934701i \(-0.384333\pi\)
0.355434 + 0.934701i \(0.384333\pi\)
\(660\) 0 0
\(661\) −46.0000 −1.78919 −0.894596 0.446875i \(-0.852537\pi\)
−0.894596 + 0.446875i \(0.852537\pi\)
\(662\) 0 0
\(663\) 13.8564 0.538138
\(664\) 0 0
\(665\) −3.26795 −0.126726
\(666\) 0 0
\(667\) 2.78461 0.107821
\(668\) 0 0
\(669\) 13.5692 0.524616
\(670\) 0 0
\(671\) −8.92820 −0.344669
\(672\) 0 0
\(673\) 2.00000 0.0770943 0.0385472 0.999257i \(-0.487727\pi\)
0.0385472 + 0.999257i \(0.487727\pi\)
\(674\) 0 0
\(675\) −4.00000 −0.153960
\(676\) 0 0
\(677\) −31.8564 −1.22434 −0.612171 0.790726i \(-0.709703\pi\)
−0.612171 + 0.790726i \(0.709703\pi\)
\(678\) 0 0
\(679\) 16.5885 0.636607
\(680\) 0 0
\(681\) 5.07180 0.194352
\(682\) 0 0
\(683\) 42.2487 1.61660 0.808301 0.588769i \(-0.200387\pi\)
0.808301 + 0.588769i \(0.200387\pi\)
\(684\) 0 0
\(685\) −12.9282 −0.493961
\(686\) 0 0
\(687\) 2.64102 0.100761
\(688\) 0 0
\(689\) −58.6410 −2.23404
\(690\) 0 0
\(691\) 6.53590 0.248637 0.124319 0.992242i \(-0.460325\pi\)
0.124319 + 0.992242i \(0.460325\pi\)
\(692\) 0 0
\(693\) −2.46410 −0.0936035
\(694\) 0 0
\(695\) −3.66025 −0.138841
\(696\) 0 0
\(697\) 28.3923 1.07544
\(698\) 0 0
\(699\) 14.5359 0.549798
\(700\) 0 0
\(701\) −23.9090 −0.903029 −0.451515 0.892264i \(-0.649116\pi\)
−0.451515 + 0.892264i \(0.649116\pi\)
\(702\) 0 0
\(703\) 8.92820 0.336734
\(704\) 0 0
\(705\) −5.07180 −0.191015
\(706\) 0 0
\(707\) −19.8564 −0.746777
\(708\) 0 0
\(709\) −9.32051 −0.350039 −0.175020 0.984565i \(-0.555999\pi\)
−0.175020 + 0.984565i \(0.555999\pi\)
\(710\) 0 0
\(711\) −4.44486 −0.166695
\(712\) 0 0
\(713\) 4.39230 0.164493
\(714\) 0 0
\(715\) 5.46410 0.204346
\(716\) 0 0
\(717\) −3.46410 −0.129369
\(718\) 0 0
\(719\) −27.7128 −1.03351 −0.516757 0.856132i \(-0.672861\pi\)
−0.516757 + 0.856132i \(0.672861\pi\)
\(720\) 0 0
\(721\) −8.39230 −0.312546
\(722\) 0 0
\(723\) 4.92820 0.183282
\(724\) 0 0
\(725\) −1.26795 −0.0470905
\(726\) 0 0
\(727\) 4.00000 0.148352 0.0741759 0.997245i \(-0.476367\pi\)
0.0741759 + 0.997245i \(0.476367\pi\)
\(728\) 0 0
\(729\) −2.21539 −0.0820515
\(730\) 0 0
\(731\) −6.92820 −0.256249
\(732\) 0 0
\(733\) 26.0000 0.960332 0.480166 0.877178i \(-0.340576\pi\)
0.480166 + 0.877178i \(0.340576\pi\)
\(734\) 0 0
\(735\) −0.732051 −0.0270021
\(736\) 0 0
\(737\) −4.00000 −0.147342
\(738\) 0 0
\(739\) 43.7128 1.60800 0.804001 0.594628i \(-0.202701\pi\)
0.804001 + 0.594628i \(0.202701\pi\)
\(740\) 0 0
\(741\) −13.0718 −0.480204
\(742\) 0 0
\(743\) −8.78461 −0.322276 −0.161138 0.986932i \(-0.551516\pi\)
−0.161138 + 0.986932i \(0.551516\pi\)
\(744\) 0 0
\(745\) 10.7321 0.393192
\(746\) 0 0
\(747\) 10.8231 0.395996
\(748\) 0 0
\(749\) −19.8564 −0.725537
\(750\) 0 0
\(751\) 32.3923 1.18201 0.591006 0.806667i \(-0.298731\pi\)
0.591006 + 0.806667i \(0.298731\pi\)
\(752\) 0 0
\(753\) −8.78461 −0.320129
\(754\) 0 0
\(755\) −13.1244 −0.477644
\(756\) 0 0
\(757\) 29.3731 1.06758 0.533791 0.845616i \(-0.320767\pi\)
0.533791 + 0.845616i \(0.320767\pi\)
\(758\) 0 0
\(759\) 1.60770 0.0583556
\(760\) 0 0
\(761\) −29.6603 −1.07518 −0.537592 0.843205i \(-0.680666\pi\)
−0.537592 + 0.843205i \(0.680666\pi\)
\(762\) 0 0
\(763\) −1.66025 −0.0601052
\(764\) 0 0
\(765\) 8.53590 0.308616
\(766\) 0 0
\(767\) 37.8564 1.36692
\(768\) 0 0
\(769\) −7.80385 −0.281414 −0.140707 0.990051i \(-0.544938\pi\)
−0.140707 + 0.990051i \(0.544938\pi\)
\(770\) 0 0
\(771\) −4.64102 −0.167142
\(772\) 0 0
\(773\) 12.9282 0.464995 0.232498 0.972597i \(-0.425310\pi\)
0.232498 + 0.972597i \(0.425310\pi\)
\(774\) 0 0
\(775\) −2.00000 −0.0718421
\(776\) 0 0
\(777\) 2.00000 0.0717496
\(778\) 0 0
\(779\) −26.7846 −0.959658
\(780\) 0 0
\(781\) 2.53590 0.0907416
\(782\) 0 0
\(783\) 5.07180 0.181251
\(784\) 0 0
\(785\) −11.4641 −0.409171
\(786\) 0 0
\(787\) −45.8564 −1.63460 −0.817302 0.576209i \(-0.804531\pi\)
−0.817302 + 0.576209i \(0.804531\pi\)
\(788\) 0 0
\(789\) 17.5692 0.625481
\(790\) 0 0
\(791\) 19.8564 0.706013
\(792\) 0 0
\(793\) 48.7846 1.73239
\(794\) 0 0
\(795\) 7.85641 0.278638
\(796\) 0 0
\(797\) −46.3923 −1.64330 −0.821650 0.569993i \(-0.806946\pi\)
−0.821650 + 0.569993i \(0.806946\pi\)
\(798\) 0 0
\(799\) 24.0000 0.849059
\(800\) 0 0
\(801\) 8.53590 0.301601
\(802\) 0 0
\(803\) −6.39230 −0.225580
\(804\) 0 0
\(805\) −2.19615 −0.0774042
\(806\) 0 0
\(807\) 5.56922 0.196046
\(808\) 0 0
\(809\) 31.1769 1.09612 0.548061 0.836438i \(-0.315366\pi\)
0.548061 + 0.836438i \(0.315366\pi\)
\(810\) 0 0
\(811\) 18.8756 0.662814 0.331407 0.943488i \(-0.392477\pi\)
0.331407 + 0.943488i \(0.392477\pi\)
\(812\) 0 0
\(813\) 14.9282 0.523555
\(814\) 0 0
\(815\) −4.00000 −0.140114
\(816\) 0 0
\(817\) 6.53590 0.228662
\(818\) 0 0
\(819\) 13.4641 0.470474
\(820\) 0 0
\(821\) −47.9090 −1.67203 −0.836017 0.548703i \(-0.815122\pi\)
−0.836017 + 0.548703i \(0.815122\pi\)
\(822\) 0 0
\(823\) 25.1244 0.875780 0.437890 0.899029i \(-0.355726\pi\)
0.437890 + 0.899029i \(0.355726\pi\)
\(824\) 0 0
\(825\) −0.732051 −0.0254867
\(826\) 0 0
\(827\) −1.85641 −0.0645536 −0.0322768 0.999479i \(-0.510276\pi\)
−0.0322768 + 0.999479i \(0.510276\pi\)
\(828\) 0 0
\(829\) −46.2487 −1.60628 −0.803142 0.595788i \(-0.796840\pi\)
−0.803142 + 0.595788i \(0.796840\pi\)
\(830\) 0 0
\(831\) 15.3205 0.531463
\(832\) 0 0
\(833\) 3.46410 0.120024
\(834\) 0 0
\(835\) −13.8564 −0.479521
\(836\) 0 0
\(837\) 8.00000 0.276520
\(838\) 0 0
\(839\) 4.14359 0.143053 0.0715264 0.997439i \(-0.477213\pi\)
0.0715264 + 0.997439i \(0.477213\pi\)
\(840\) 0 0
\(841\) −27.3923 −0.944562
\(842\) 0 0
\(843\) 1.17691 0.0405351
\(844\) 0 0
\(845\) −16.8564 −0.579878
\(846\) 0 0
\(847\) −1.00000 −0.0343604
\(848\) 0 0
\(849\) −17.3590 −0.595759
\(850\) 0 0
\(851\) 6.00000 0.205677
\(852\) 0 0
\(853\) −13.2154 −0.452486 −0.226243 0.974071i \(-0.572644\pi\)
−0.226243 + 0.974071i \(0.572644\pi\)
\(854\) 0 0
\(855\) −8.05256 −0.275392
\(856\) 0 0
\(857\) −20.5359 −0.701493 −0.350746 0.936470i \(-0.614072\pi\)
−0.350746 + 0.936470i \(0.614072\pi\)
\(858\) 0 0
\(859\) −28.7846 −0.982118 −0.491059 0.871126i \(-0.663390\pi\)
−0.491059 + 0.871126i \(0.663390\pi\)
\(860\) 0 0
\(861\) −6.00000 −0.204479
\(862\) 0 0
\(863\) −37.5167 −1.27708 −0.638541 0.769588i \(-0.720462\pi\)
−0.638541 + 0.769588i \(0.720462\pi\)
\(864\) 0 0
\(865\) 12.9282 0.439572
\(866\) 0 0
\(867\) −3.66025 −0.124309
\(868\) 0 0
\(869\) −1.80385 −0.0611913
\(870\) 0 0
\(871\) 21.8564 0.740576
\(872\) 0 0
\(873\) 40.8756 1.38343
\(874\) 0 0
\(875\) 1.00000 0.0338062
\(876\) 0 0
\(877\) 13.3205 0.449802 0.224901 0.974382i \(-0.427794\pi\)
0.224901 + 0.974382i \(0.427794\pi\)
\(878\) 0 0
\(879\) 10.6410 0.358913
\(880\) 0 0
\(881\) 24.9282 0.839853 0.419926 0.907558i \(-0.362056\pi\)
0.419926 + 0.907558i \(0.362056\pi\)
\(882\) 0 0
\(883\) 56.3923 1.89775 0.948876 0.315649i \(-0.102222\pi\)
0.948876 + 0.315649i \(0.102222\pi\)
\(884\) 0 0
\(885\) −5.07180 −0.170487
\(886\) 0 0
\(887\) 13.8564 0.465253 0.232626 0.972566i \(-0.425268\pi\)
0.232626 + 0.972566i \(0.425268\pi\)
\(888\) 0 0
\(889\) 14.9282 0.500676
\(890\) 0 0
\(891\) −4.46410 −0.149553
\(892\) 0 0
\(893\) −22.6410 −0.757653
\(894\) 0 0
\(895\) −6.00000 −0.200558
\(896\) 0 0
\(897\) −8.78461 −0.293310
\(898\) 0 0
\(899\) 2.53590 0.0845769
\(900\) 0 0
\(901\) −37.1769 −1.23854
\(902\) 0 0
\(903\) 1.46410 0.0487223
\(904\) 0 0
\(905\) −14.0000 −0.465376
\(906\) 0 0
\(907\) −59.0333 −1.96017 −0.980085 0.198580i \(-0.936367\pi\)
−0.980085 + 0.198580i \(0.936367\pi\)
\(908\) 0 0
\(909\) −48.9282 −1.62285
\(910\) 0 0
\(911\) −2.53590 −0.0840181 −0.0420090 0.999117i \(-0.513376\pi\)
−0.0420090 + 0.999117i \(0.513376\pi\)
\(912\) 0 0
\(913\) 4.39230 0.145364
\(914\) 0 0
\(915\) −6.53590 −0.216070
\(916\) 0 0
\(917\) −11.6603 −0.385056
\(918\) 0 0
\(919\) −47.3731 −1.56269 −0.781347 0.624097i \(-0.785467\pi\)
−0.781347 + 0.624097i \(0.785467\pi\)
\(920\) 0 0
\(921\) −2.64102 −0.0870244
\(922\) 0 0
\(923\) −13.8564 −0.456089
\(924\) 0 0
\(925\) −2.73205 −0.0898293
\(926\) 0 0
\(927\) −20.6795 −0.679204
\(928\) 0 0
\(929\) −46.3923 −1.52208 −0.761041 0.648704i \(-0.775311\pi\)
−0.761041 + 0.648704i \(0.775311\pi\)
\(930\) 0 0
\(931\) −3.26795 −0.107103
\(932\) 0 0
\(933\) −8.10512 −0.265350
\(934\) 0 0
\(935\) 3.46410 0.113288
\(936\) 0 0
\(937\) −42.7846 −1.39771 −0.698856 0.715262i \(-0.746307\pi\)
−0.698856 + 0.715262i \(0.746307\pi\)
\(938\) 0 0
\(939\) 8.64102 0.281989
\(940\) 0 0
\(941\) 26.7846 0.873153 0.436577 0.899667i \(-0.356191\pi\)
0.436577 + 0.899667i \(0.356191\pi\)
\(942\) 0 0
\(943\) −18.0000 −0.586161
\(944\) 0 0
\(945\) −4.00000 −0.130120
\(946\) 0 0
\(947\) −12.6795 −0.412028 −0.206014 0.978549i \(-0.566049\pi\)
−0.206014 + 0.978549i \(0.566049\pi\)
\(948\) 0 0
\(949\) 34.9282 1.13382
\(950\) 0 0
\(951\) −0.430781 −0.0139690
\(952\) 0 0
\(953\) 33.4641 1.08401 0.542004 0.840376i \(-0.317666\pi\)
0.542004 + 0.840376i \(0.317666\pi\)
\(954\) 0 0
\(955\) −12.0000 −0.388311
\(956\) 0 0
\(957\) 0.928203 0.0300045
\(958\) 0 0
\(959\) −12.9282 −0.417473
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 0 0
\(963\) −48.9282 −1.57669
\(964\) 0 0
\(965\) 8.39230 0.270158
\(966\) 0 0
\(967\) −13.0718 −0.420361 −0.210180 0.977663i \(-0.567405\pi\)
−0.210180 + 0.977663i \(0.567405\pi\)
\(968\) 0 0
\(969\) −8.28719 −0.266223
\(970\) 0 0
\(971\) −29.0718 −0.932958 −0.466479 0.884532i \(-0.654478\pi\)
−0.466479 + 0.884532i \(0.654478\pi\)
\(972\) 0 0
\(973\) −3.66025 −0.117342
\(974\) 0 0
\(975\) 4.00000 0.128103
\(976\) 0 0
\(977\) 21.7128 0.694654 0.347327 0.937744i \(-0.387089\pi\)
0.347327 + 0.937744i \(0.387089\pi\)
\(978\) 0 0
\(979\) 3.46410 0.110713
\(980\) 0 0
\(981\) −4.09103 −0.130617
\(982\) 0 0
\(983\) −25.1769 −0.803019 −0.401509 0.915855i \(-0.631514\pi\)
−0.401509 + 0.915855i \(0.631514\pi\)
\(984\) 0 0
\(985\) −24.2487 −0.772628
\(986\) 0 0
\(987\) −5.07180 −0.161437
\(988\) 0 0
\(989\) 4.39230 0.139667
\(990\) 0 0
\(991\) 4.00000 0.127064 0.0635321 0.997980i \(-0.479763\pi\)
0.0635321 + 0.997980i \(0.479763\pi\)
\(992\) 0 0
\(993\) −16.6795 −0.529308
\(994\) 0 0
\(995\) −10.9282 −0.346447
\(996\) 0 0
\(997\) −37.7128 −1.19438 −0.597188 0.802101i \(-0.703716\pi\)
−0.597188 + 0.802101i \(0.703716\pi\)
\(998\) 0 0
\(999\) 10.9282 0.345753
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 6160.2.a.t.1.2 2
4.3 odd 2 770.2.a.j.1.1 2
12.11 even 2 6930.2.a.bv.1.2 2
20.3 even 4 3850.2.c.x.1849.1 4
20.7 even 4 3850.2.c.x.1849.4 4
20.19 odd 2 3850.2.a.bd.1.2 2
28.27 even 2 5390.2.a.bs.1.2 2
44.43 even 2 8470.2.a.br.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.a.j.1.1 2 4.3 odd 2
3850.2.a.bd.1.2 2 20.19 odd 2
3850.2.c.x.1849.1 4 20.3 even 4
3850.2.c.x.1849.4 4 20.7 even 4
5390.2.a.bs.1.2 2 28.27 even 2
6160.2.a.t.1.2 2 1.1 even 1 trivial
6930.2.a.bv.1.2 2 12.11 even 2
8470.2.a.br.1.1 2 44.43 even 2