Properties

Label 5445.2.a.bl.1.3
Level $5445$
Weight $2$
Character 5445.1
Self dual yes
Analytic conductor $43.479$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 5445 = 3^{2} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5445.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(43.4785439006\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{3}, \sqrt{7})\)
Defining polynomial: \(x^{4} - 5 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1815)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-2.18890\) of defining polynomial
Character \(\chi\) \(=\) 5445.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.456850 q^{2} -1.79129 q^{4} +1.00000 q^{5} -4.37780 q^{7} -1.73205 q^{8} +O(q^{10})\) \(q+0.456850 q^{2} -1.79129 q^{4} +1.00000 q^{5} -4.37780 q^{7} -1.73205 q^{8} +0.456850 q^{10} -0.913701 q^{13} -2.00000 q^{14} +2.79129 q^{16} +1.73205 q^{17} +3.46410 q^{19} -1.79129 q^{20} +0.582576 q^{23} +1.00000 q^{25} -0.417424 q^{26} +7.84190 q^{28} +9.66930 q^{29} -8.58258 q^{31} +4.73930 q^{32} +0.791288 q^{34} -4.37780 q^{35} +7.58258 q^{37} +1.58258 q^{38} -1.73205 q^{40} +3.46410 q^{41} +9.66930 q^{43} +0.266150 q^{46} -3.41742 q^{47} +12.1652 q^{49} +0.456850 q^{50} +1.63670 q^{52} -12.1652 q^{53} +7.58258 q^{56} +4.41742 q^{58} -13.5826 q^{59} +3.55945 q^{61} -3.92095 q^{62} -3.41742 q^{64} -0.913701 q^{65} -4.41742 q^{67} -3.10260 q^{68} -2.00000 q^{70} -8.00000 q^{71} -5.10080 q^{73} +3.46410 q^{74} -6.20520 q^{76} -0.818350 q^{79} +2.79129 q^{80} +1.58258 q^{82} -15.8745 q^{83} +1.73205 q^{85} +4.41742 q^{86} -14.7477 q^{89} +4.00000 q^{91} -1.04356 q^{92} -1.56125 q^{94} +3.46410 q^{95} +4.41742 q^{97} +5.55765 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{4} + 4q^{5} + O(q^{10}) \) \( 4q + 2q^{4} + 4q^{5} - 8q^{14} + 2q^{16} + 2q^{20} - 16q^{23} + 4q^{25} - 20q^{26} - 16q^{31} - 6q^{34} + 12q^{37} - 12q^{38} - 32q^{47} + 12q^{49} - 12q^{53} + 12q^{56} + 36q^{58} - 36q^{59} - 32q^{64} - 36q^{67} - 8q^{70} - 32q^{71} + 2q^{80} - 12q^{82} + 36q^{86} - 4q^{89} + 16q^{91} - 50q^{92} + 36q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.456850 0.323042 0.161521 0.986869i \(-0.448360\pi\)
0.161521 + 0.986869i \(0.448360\pi\)
\(3\) 0 0
\(4\) −1.79129 −0.895644
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −4.37780 −1.65465 −0.827327 0.561721i \(-0.810140\pi\)
−0.827327 + 0.561721i \(0.810140\pi\)
\(8\) −1.73205 −0.612372
\(9\) 0 0
\(10\) 0.456850 0.144469
\(11\) 0 0
\(12\) 0 0
\(13\) −0.913701 −0.253415 −0.126707 0.991940i \(-0.540441\pi\)
−0.126707 + 0.991940i \(0.540441\pi\)
\(14\) −2.00000 −0.534522
\(15\) 0 0
\(16\) 2.79129 0.697822
\(17\) 1.73205 0.420084 0.210042 0.977692i \(-0.432640\pi\)
0.210042 + 0.977692i \(0.432640\pi\)
\(18\) 0 0
\(19\) 3.46410 0.794719 0.397360 0.917663i \(-0.369927\pi\)
0.397360 + 0.917663i \(0.369927\pi\)
\(20\) −1.79129 −0.400544
\(21\) 0 0
\(22\) 0 0
\(23\) 0.582576 0.121475 0.0607377 0.998154i \(-0.480655\pi\)
0.0607377 + 0.998154i \(0.480655\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −0.417424 −0.0818636
\(27\) 0 0
\(28\) 7.84190 1.48198
\(29\) 9.66930 1.79554 0.897772 0.440460i \(-0.145185\pi\)
0.897772 + 0.440460i \(0.145185\pi\)
\(30\) 0 0
\(31\) −8.58258 −1.54148 −0.770738 0.637152i \(-0.780112\pi\)
−0.770738 + 0.637152i \(0.780112\pi\)
\(32\) 4.73930 0.837798
\(33\) 0 0
\(34\) 0.791288 0.135705
\(35\) −4.37780 −0.739984
\(36\) 0 0
\(37\) 7.58258 1.24657 0.623284 0.781996i \(-0.285798\pi\)
0.623284 + 0.781996i \(0.285798\pi\)
\(38\) 1.58258 0.256728
\(39\) 0 0
\(40\) −1.73205 −0.273861
\(41\) 3.46410 0.541002 0.270501 0.962720i \(-0.412811\pi\)
0.270501 + 0.962720i \(0.412811\pi\)
\(42\) 0 0
\(43\) 9.66930 1.47456 0.737278 0.675590i \(-0.236111\pi\)
0.737278 + 0.675590i \(0.236111\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0.266150 0.0392417
\(47\) −3.41742 −0.498483 −0.249241 0.968441i \(-0.580181\pi\)
−0.249241 + 0.968441i \(0.580181\pi\)
\(48\) 0 0
\(49\) 12.1652 1.73788
\(50\) 0.456850 0.0646084
\(51\) 0 0
\(52\) 1.63670 0.226970
\(53\) −12.1652 −1.67101 −0.835506 0.549481i \(-0.814825\pi\)
−0.835506 + 0.549481i \(0.814825\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 7.58258 1.01326
\(57\) 0 0
\(58\) 4.41742 0.580036
\(59\) −13.5826 −1.76830 −0.884150 0.467202i \(-0.845262\pi\)
−0.884150 + 0.467202i \(0.845262\pi\)
\(60\) 0 0
\(61\) 3.55945 0.455741 0.227871 0.973691i \(-0.426824\pi\)
0.227871 + 0.973691i \(0.426824\pi\)
\(62\) −3.92095 −0.497961
\(63\) 0 0
\(64\) −3.41742 −0.427178
\(65\) −0.913701 −0.113331
\(66\) 0 0
\(67\) −4.41742 −0.539674 −0.269837 0.962906i \(-0.586970\pi\)
−0.269837 + 0.962906i \(0.586970\pi\)
\(68\) −3.10260 −0.376246
\(69\) 0 0
\(70\) −2.00000 −0.239046
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) −5.10080 −0.597004 −0.298502 0.954409i \(-0.596487\pi\)
−0.298502 + 0.954409i \(0.596487\pi\)
\(74\) 3.46410 0.402694
\(75\) 0 0
\(76\) −6.20520 −0.711786
\(77\) 0 0
\(78\) 0 0
\(79\) −0.818350 −0.0920716 −0.0460358 0.998940i \(-0.514659\pi\)
−0.0460358 + 0.998940i \(0.514659\pi\)
\(80\) 2.79129 0.312075
\(81\) 0 0
\(82\) 1.58258 0.174766
\(83\) −15.8745 −1.74245 −0.871227 0.490881i \(-0.836675\pi\)
−0.871227 + 0.490881i \(0.836675\pi\)
\(84\) 0 0
\(85\) 1.73205 0.187867
\(86\) 4.41742 0.476343
\(87\) 0 0
\(88\) 0 0
\(89\) −14.7477 −1.56326 −0.781628 0.623745i \(-0.785610\pi\)
−0.781628 + 0.623745i \(0.785610\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) −1.04356 −0.108799
\(93\) 0 0
\(94\) −1.56125 −0.161031
\(95\) 3.46410 0.355409
\(96\) 0 0
\(97\) 4.41742 0.448521 0.224261 0.974529i \(-0.428003\pi\)
0.224261 + 0.974529i \(0.428003\pi\)
\(98\) 5.55765 0.561408
\(99\) 0 0
\(100\) −1.79129 −0.179129
\(101\) 1.82740 0.181833 0.0909166 0.995859i \(-0.471020\pi\)
0.0909166 + 0.995859i \(0.471020\pi\)
\(102\) 0 0
\(103\) 7.16515 0.706003 0.353002 0.935623i \(-0.385161\pi\)
0.353002 + 0.935623i \(0.385161\pi\)
\(104\) 1.58258 0.155184
\(105\) 0 0
\(106\) −5.55765 −0.539807
\(107\) −13.0381 −1.26044 −0.630218 0.776418i \(-0.717035\pi\)
−0.630218 + 0.776418i \(0.717035\pi\)
\(108\) 0 0
\(109\) 6.92820 0.663602 0.331801 0.943349i \(-0.392344\pi\)
0.331801 + 0.943349i \(0.392344\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −12.2197 −1.15465
\(113\) −9.00000 −0.846649 −0.423324 0.905978i \(-0.639137\pi\)
−0.423324 + 0.905978i \(0.639137\pi\)
\(114\) 0 0
\(115\) 0.582576 0.0543255
\(116\) −17.3205 −1.60817
\(117\) 0 0
\(118\) −6.20520 −0.571235
\(119\) −7.58258 −0.695094
\(120\) 0 0
\(121\) 0 0
\(122\) 1.62614 0.147223
\(123\) 0 0
\(124\) 15.3739 1.38061
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −7.84190 −0.695856 −0.347928 0.937521i \(-0.613115\pi\)
−0.347928 + 0.937521i \(0.613115\pi\)
\(128\) −11.0399 −0.975795
\(129\) 0 0
\(130\) −0.417424 −0.0366105
\(131\) 20.9753 1.83262 0.916311 0.400468i \(-0.131152\pi\)
0.916311 + 0.400468i \(0.131152\pi\)
\(132\) 0 0
\(133\) −15.1652 −1.31499
\(134\) −2.01810 −0.174337
\(135\) 0 0
\(136\) −3.00000 −0.257248
\(137\) 19.3303 1.65150 0.825750 0.564037i \(-0.190752\pi\)
0.825750 + 0.564037i \(0.190752\pi\)
\(138\) 0 0
\(139\) −7.74655 −0.657054 −0.328527 0.944495i \(-0.606552\pi\)
−0.328527 + 0.944495i \(0.606552\pi\)
\(140\) 7.84190 0.662762
\(141\) 0 0
\(142\) −3.65480 −0.306704
\(143\) 0 0
\(144\) 0 0
\(145\) 9.66930 0.802992
\(146\) −2.33030 −0.192857
\(147\) 0 0
\(148\) −13.5826 −1.11648
\(149\) −2.74110 −0.224560 −0.112280 0.993677i \(-0.535815\pi\)
−0.112280 + 0.993677i \(0.535815\pi\)
\(150\) 0 0
\(151\) −7.74655 −0.630406 −0.315203 0.949024i \(-0.602073\pi\)
−0.315203 + 0.949024i \(0.602073\pi\)
\(152\) −6.00000 −0.486664
\(153\) 0 0
\(154\) 0 0
\(155\) −8.58258 −0.689369
\(156\) 0 0
\(157\) 10.7477 0.857762 0.428881 0.903361i \(-0.358908\pi\)
0.428881 + 0.903361i \(0.358908\pi\)
\(158\) −0.373864 −0.0297430
\(159\) 0 0
\(160\) 4.73930 0.374675
\(161\) −2.55040 −0.201000
\(162\) 0 0
\(163\) −20.7477 −1.62509 −0.812544 0.582900i \(-0.801918\pi\)
−0.812544 + 0.582900i \(0.801918\pi\)
\(164\) −6.20520 −0.484545
\(165\) 0 0
\(166\) −7.25227 −0.562886
\(167\) 4.47315 0.346143 0.173071 0.984909i \(-0.444631\pi\)
0.173071 + 0.984909i \(0.444631\pi\)
\(168\) 0 0
\(169\) −12.1652 −0.935781
\(170\) 0.791288 0.0606890
\(171\) 0 0
\(172\) −17.3205 −1.32068
\(173\) −6.92820 −0.526742 −0.263371 0.964695i \(-0.584834\pi\)
−0.263371 + 0.964695i \(0.584834\pi\)
\(174\) 0 0
\(175\) −4.37780 −0.330931
\(176\) 0 0
\(177\) 0 0
\(178\) −6.73750 −0.504997
\(179\) −7.58258 −0.566748 −0.283374 0.959009i \(-0.591454\pi\)
−0.283374 + 0.959009i \(0.591454\pi\)
\(180\) 0 0
\(181\) −5.16515 −0.383923 −0.191961 0.981402i \(-0.561485\pi\)
−0.191961 + 0.981402i \(0.561485\pi\)
\(182\) 1.82740 0.135456
\(183\) 0 0
\(184\) −1.00905 −0.0743882
\(185\) 7.58258 0.557482
\(186\) 0 0
\(187\) 0 0
\(188\) 6.12159 0.446463
\(189\) 0 0
\(190\) 1.58258 0.114812
\(191\) 22.7477 1.64597 0.822984 0.568065i \(-0.192308\pi\)
0.822984 + 0.568065i \(0.192308\pi\)
\(192\) 0 0
\(193\) −16.4068 −1.18099 −0.590494 0.807042i \(-0.701067\pi\)
−0.590494 + 0.807042i \(0.701067\pi\)
\(194\) 2.01810 0.144891
\(195\) 0 0
\(196\) −21.7913 −1.55652
\(197\) −12.4104 −0.884205 −0.442102 0.896965i \(-0.645767\pi\)
−0.442102 + 0.896965i \(0.645767\pi\)
\(198\) 0 0
\(199\) −15.7477 −1.11633 −0.558163 0.829731i \(-0.688494\pi\)
−0.558163 + 0.829731i \(0.688494\pi\)
\(200\) −1.73205 −0.122474
\(201\) 0 0
\(202\) 0.834849 0.0587397
\(203\) −42.3303 −2.97100
\(204\) 0 0
\(205\) 3.46410 0.241943
\(206\) 3.27340 0.228069
\(207\) 0 0
\(208\) −2.55040 −0.176838
\(209\) 0 0
\(210\) 0 0
\(211\) 23.4304 1.61301 0.806506 0.591226i \(-0.201356\pi\)
0.806506 + 0.591226i \(0.201356\pi\)
\(212\) 21.7913 1.49663
\(213\) 0 0
\(214\) −5.95644 −0.407174
\(215\) 9.66930 0.659441
\(216\) 0 0
\(217\) 37.5728 2.55061
\(218\) 3.16515 0.214371
\(219\) 0 0
\(220\) 0 0
\(221\) −1.58258 −0.106456
\(222\) 0 0
\(223\) −16.3303 −1.09356 −0.546779 0.837277i \(-0.684146\pi\)
−0.546779 + 0.837277i \(0.684146\pi\)
\(224\) −20.7477 −1.38627
\(225\) 0 0
\(226\) −4.11165 −0.273503
\(227\) −20.1570 −1.33786 −0.668932 0.743323i \(-0.733248\pi\)
−0.668932 + 0.743323i \(0.733248\pi\)
\(228\) 0 0
\(229\) 17.0000 1.12339 0.561696 0.827344i \(-0.310149\pi\)
0.561696 + 0.827344i \(0.310149\pi\)
\(230\) 0.266150 0.0175494
\(231\) 0 0
\(232\) −16.7477 −1.09954
\(233\) −14.1425 −0.926503 −0.463252 0.886227i \(-0.653317\pi\)
−0.463252 + 0.886227i \(0.653317\pi\)
\(234\) 0 0
\(235\) −3.41742 −0.222928
\(236\) 24.3303 1.58377
\(237\) 0 0
\(238\) −3.46410 −0.224544
\(239\) 11.1153 0.718989 0.359495 0.933147i \(-0.382949\pi\)
0.359495 + 0.933147i \(0.382949\pi\)
\(240\) 0 0
\(241\) 26.1715 1.68585 0.842926 0.538029i \(-0.180831\pi\)
0.842926 + 0.538029i \(0.180831\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −6.37600 −0.408182
\(245\) 12.1652 0.777203
\(246\) 0 0
\(247\) −3.16515 −0.201394
\(248\) 14.8655 0.943957
\(249\) 0 0
\(250\) 0.456850 0.0288937
\(251\) 5.58258 0.352369 0.176185 0.984357i \(-0.443624\pi\)
0.176185 + 0.984357i \(0.443624\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −3.58258 −0.224791
\(255\) 0 0
\(256\) 1.79129 0.111955
\(257\) −5.83485 −0.363968 −0.181984 0.983302i \(-0.558252\pi\)
−0.181984 + 0.983302i \(0.558252\pi\)
\(258\) 0 0
\(259\) −33.1950 −2.06264
\(260\) 1.63670 0.101504
\(261\) 0 0
\(262\) 9.58258 0.592014
\(263\) −11.4014 −0.703038 −0.351519 0.936181i \(-0.614335\pi\)
−0.351519 + 0.936181i \(0.614335\pi\)
\(264\) 0 0
\(265\) −12.1652 −0.747299
\(266\) −6.92820 −0.424795
\(267\) 0 0
\(268\) 7.91288 0.483356
\(269\) 8.83485 0.538670 0.269335 0.963047i \(-0.413196\pi\)
0.269335 + 0.963047i \(0.413196\pi\)
\(270\) 0 0
\(271\) −16.6929 −1.01402 −0.507009 0.861940i \(-0.669249\pi\)
−0.507009 + 0.861940i \(0.669249\pi\)
\(272\) 4.83465 0.293144
\(273\) 0 0
\(274\) 8.83105 0.533503
\(275\) 0 0
\(276\) 0 0
\(277\) 1.82740 0.109798 0.0548989 0.998492i \(-0.482516\pi\)
0.0548989 + 0.998492i \(0.482516\pi\)
\(278\) −3.53901 −0.212256
\(279\) 0 0
\(280\) 7.58258 0.453146
\(281\) −23.7164 −1.41480 −0.707401 0.706812i \(-0.750133\pi\)
−0.707401 + 0.706812i \(0.750133\pi\)
\(282\) 0 0
\(283\) 10.7737 0.640430 0.320215 0.947345i \(-0.396245\pi\)
0.320215 + 0.947345i \(0.396245\pi\)
\(284\) 14.3303 0.850347
\(285\) 0 0
\(286\) 0 0
\(287\) −15.1652 −0.895171
\(288\) 0 0
\(289\) −14.0000 −0.823529
\(290\) 4.41742 0.259400
\(291\) 0 0
\(292\) 9.13701 0.534703
\(293\) 1.73205 0.101187 0.0505937 0.998719i \(-0.483889\pi\)
0.0505937 + 0.998719i \(0.483889\pi\)
\(294\) 0 0
\(295\) −13.5826 −0.790808
\(296\) −13.1334 −0.763364
\(297\) 0 0
\(298\) −1.25227 −0.0725422
\(299\) −0.532300 −0.0307837
\(300\) 0 0
\(301\) −42.3303 −2.43988
\(302\) −3.53901 −0.203647
\(303\) 0 0
\(304\) 9.66930 0.554573
\(305\) 3.55945 0.203814
\(306\) 0 0
\(307\) 15.4931 0.884238 0.442119 0.896956i \(-0.354227\pi\)
0.442119 + 0.896956i \(0.354227\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −3.92095 −0.222695
\(311\) 11.5826 0.656788 0.328394 0.944541i \(-0.393493\pi\)
0.328394 + 0.944541i \(0.393493\pi\)
\(312\) 0 0
\(313\) 5.58258 0.315546 0.157773 0.987475i \(-0.449569\pi\)
0.157773 + 0.987475i \(0.449569\pi\)
\(314\) 4.91010 0.277093
\(315\) 0 0
\(316\) 1.46590 0.0824634
\(317\) −30.1652 −1.69424 −0.847122 0.531399i \(-0.821667\pi\)
−0.847122 + 0.531399i \(0.821667\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −3.41742 −0.191040
\(321\) 0 0
\(322\) −1.16515 −0.0649313
\(323\) 6.00000 0.333849
\(324\) 0 0
\(325\) −0.913701 −0.0506830
\(326\) −9.47860 −0.524971
\(327\) 0 0
\(328\) −6.00000 −0.331295
\(329\) 14.9608 0.824816
\(330\) 0 0
\(331\) 27.7477 1.52515 0.762577 0.646898i \(-0.223934\pi\)
0.762577 + 0.646898i \(0.223934\pi\)
\(332\) 28.4358 1.56062
\(333\) 0 0
\(334\) 2.04356 0.111819
\(335\) −4.41742 −0.241350
\(336\) 0 0
\(337\) −0.190700 −0.0103881 −0.00519406 0.999987i \(-0.501653\pi\)
−0.00519406 + 0.999987i \(0.501653\pi\)
\(338\) −5.55765 −0.302296
\(339\) 0 0
\(340\) −3.10260 −0.168262
\(341\) 0 0
\(342\) 0 0
\(343\) −22.6120 −1.22093
\(344\) −16.7477 −0.902977
\(345\) 0 0
\(346\) −3.16515 −0.170160
\(347\) 21.9844 1.18018 0.590091 0.807337i \(-0.299092\pi\)
0.590091 + 0.807337i \(0.299092\pi\)
\(348\) 0 0
\(349\) 25.9808 1.39072 0.695359 0.718662i \(-0.255245\pi\)
0.695359 + 0.718662i \(0.255245\pi\)
\(350\) −2.00000 −0.106904
\(351\) 0 0
\(352\) 0 0
\(353\) −17.8348 −0.949253 −0.474627 0.880187i \(-0.657417\pi\)
−0.474627 + 0.880187i \(0.657417\pi\)
\(354\) 0 0
\(355\) −8.00000 −0.424596
\(356\) 26.4174 1.40012
\(357\) 0 0
\(358\) −3.46410 −0.183083
\(359\) 8.75560 0.462103 0.231052 0.972942i \(-0.425783\pi\)
0.231052 + 0.972942i \(0.425783\pi\)
\(360\) 0 0
\(361\) −7.00000 −0.368421
\(362\) −2.35970 −0.124023
\(363\) 0 0
\(364\) −7.16515 −0.375556
\(365\) −5.10080 −0.266988
\(366\) 0 0
\(367\) −1.25227 −0.0653681 −0.0326841 0.999466i \(-0.510406\pi\)
−0.0326841 + 0.999466i \(0.510406\pi\)
\(368\) 1.62614 0.0847682
\(369\) 0 0
\(370\) 3.46410 0.180090
\(371\) 53.2566 2.76495
\(372\) 0 0
\(373\) 13.8564 0.717458 0.358729 0.933442i \(-0.383210\pi\)
0.358729 + 0.933442i \(0.383210\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 5.91915 0.305257
\(377\) −8.83485 −0.455018
\(378\) 0 0
\(379\) −15.7477 −0.808906 −0.404453 0.914559i \(-0.632538\pi\)
−0.404453 + 0.914559i \(0.632538\pi\)
\(380\) −6.20520 −0.318320
\(381\) 0 0
\(382\) 10.3923 0.531717
\(383\) −6.33030 −0.323463 −0.161732 0.986835i \(-0.551708\pi\)
−0.161732 + 0.986835i \(0.551708\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −7.49545 −0.381509
\(387\) 0 0
\(388\) −7.91288 −0.401716
\(389\) −4.74773 −0.240719 −0.120360 0.992730i \(-0.538405\pi\)
−0.120360 + 0.992730i \(0.538405\pi\)
\(390\) 0 0
\(391\) 1.00905 0.0510299
\(392\) −21.0707 −1.06423
\(393\) 0 0
\(394\) −5.66970 −0.285635
\(395\) −0.818350 −0.0411757
\(396\) 0 0
\(397\) 32.3303 1.62261 0.811306 0.584622i \(-0.198757\pi\)
0.811306 + 0.584622i \(0.198757\pi\)
\(398\) −7.19435 −0.360620
\(399\) 0 0
\(400\) 2.79129 0.139564
\(401\) −14.3303 −0.715621 −0.357811 0.933794i \(-0.616477\pi\)
−0.357811 + 0.933794i \(0.616477\pi\)
\(402\) 0 0
\(403\) 7.84190 0.390633
\(404\) −3.27340 −0.162858
\(405\) 0 0
\(406\) −19.3386 −0.959759
\(407\) 0 0
\(408\) 0 0
\(409\) −19.4340 −0.960947 −0.480474 0.877009i \(-0.659535\pi\)
−0.480474 + 0.877009i \(0.659535\pi\)
\(410\) 1.58258 0.0781578
\(411\) 0 0
\(412\) −12.8348 −0.632328
\(413\) 59.4618 2.92593
\(414\) 0 0
\(415\) −15.8745 −0.779249
\(416\) −4.33030 −0.212311
\(417\) 0 0
\(418\) 0 0
\(419\) 4.33030 0.211549 0.105775 0.994390i \(-0.466268\pi\)
0.105775 + 0.994390i \(0.466268\pi\)
\(420\) 0 0
\(421\) −39.3303 −1.91684 −0.958421 0.285359i \(-0.907887\pi\)
−0.958421 + 0.285359i \(0.907887\pi\)
\(422\) 10.7042 0.521071
\(423\) 0 0
\(424\) 21.0707 1.02328
\(425\) 1.73205 0.0840168
\(426\) 0 0
\(427\) −15.5826 −0.754094
\(428\) 23.3549 1.12890
\(429\) 0 0
\(430\) 4.41742 0.213027
\(431\) −4.18710 −0.201686 −0.100843 0.994902i \(-0.532154\pi\)
−0.100843 + 0.994902i \(0.532154\pi\)
\(432\) 0 0
\(433\) −13.5826 −0.652737 −0.326368 0.945243i \(-0.605825\pi\)
−0.326368 + 0.945243i \(0.605825\pi\)
\(434\) 17.1652 0.823954
\(435\) 0 0
\(436\) −12.4104 −0.594351
\(437\) 2.01810 0.0965389
\(438\) 0 0
\(439\) 4.28245 0.204390 0.102195 0.994764i \(-0.467413\pi\)
0.102195 + 0.994764i \(0.467413\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −0.723000 −0.0343896
\(443\) 11.1652 0.530472 0.265236 0.964183i \(-0.414550\pi\)
0.265236 + 0.964183i \(0.414550\pi\)
\(444\) 0 0
\(445\) −14.7477 −0.699109
\(446\) −7.46050 −0.353265
\(447\) 0 0
\(448\) 14.9608 0.706832
\(449\) −21.1652 −0.998845 −0.499423 0.866358i \(-0.666454\pi\)
−0.499423 + 0.866358i \(0.666454\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 16.1216 0.758296
\(453\) 0 0
\(454\) −9.20871 −0.432186
\(455\) 4.00000 0.187523
\(456\) 0 0
\(457\) −33.0043 −1.54388 −0.771938 0.635697i \(-0.780713\pi\)
−0.771938 + 0.635697i \(0.780713\pi\)
\(458\) 7.76645 0.362903
\(459\) 0 0
\(460\) −1.04356 −0.0486563
\(461\) 18.4249 0.858134 0.429067 0.903273i \(-0.358842\pi\)
0.429067 + 0.903273i \(0.358842\pi\)
\(462\) 0 0
\(463\) 19.4955 0.906031 0.453015 0.891503i \(-0.350348\pi\)
0.453015 + 0.891503i \(0.350348\pi\)
\(464\) 26.9898 1.25297
\(465\) 0 0
\(466\) −6.46099 −0.299299
\(467\) 12.5826 0.582252 0.291126 0.956685i \(-0.405970\pi\)
0.291126 + 0.956685i \(0.405970\pi\)
\(468\) 0 0
\(469\) 19.3386 0.892974
\(470\) −1.56125 −0.0720151
\(471\) 0 0
\(472\) 23.5257 1.08286
\(473\) 0 0
\(474\) 0 0
\(475\) 3.46410 0.158944
\(476\) 13.5826 0.622556
\(477\) 0 0
\(478\) 5.07803 0.232264
\(479\) −30.8353 −1.40890 −0.704451 0.709753i \(-0.748807\pi\)
−0.704451 + 0.709753i \(0.748807\pi\)
\(480\) 0 0
\(481\) −6.92820 −0.315899
\(482\) 11.9564 0.544601
\(483\) 0 0
\(484\) 0 0
\(485\) 4.41742 0.200585
\(486\) 0 0
\(487\) −5.58258 −0.252971 −0.126485 0.991968i \(-0.540370\pi\)
−0.126485 + 0.991968i \(0.540370\pi\)
\(488\) −6.16515 −0.279083
\(489\) 0 0
\(490\) 5.55765 0.251069
\(491\) −27.7128 −1.25066 −0.625331 0.780360i \(-0.715036\pi\)
−0.625331 + 0.780360i \(0.715036\pi\)
\(492\) 0 0
\(493\) 16.7477 0.754280
\(494\) −1.44600 −0.0650586
\(495\) 0 0
\(496\) −23.9564 −1.07568
\(497\) 35.0224 1.57097
\(498\) 0 0
\(499\) −10.3303 −0.462448 −0.231224 0.972901i \(-0.574273\pi\)
−0.231224 + 0.972901i \(0.574273\pi\)
\(500\) −1.79129 −0.0801088
\(501\) 0 0
\(502\) 2.55040 0.113830
\(503\) −6.10985 −0.272425 −0.136212 0.990680i \(-0.543493\pi\)
−0.136212 + 0.990680i \(0.543493\pi\)
\(504\) 0 0
\(505\) 1.82740 0.0813183
\(506\) 0 0
\(507\) 0 0
\(508\) 14.0471 0.623240
\(509\) 26.7477 1.18557 0.592786 0.805360i \(-0.298028\pi\)
0.592786 + 0.805360i \(0.298028\pi\)
\(510\) 0 0
\(511\) 22.3303 0.987834
\(512\) 22.8981 1.01196
\(513\) 0 0
\(514\) −2.66565 −0.117577
\(515\) 7.16515 0.315734
\(516\) 0 0
\(517\) 0 0
\(518\) −15.1652 −0.666318
\(519\) 0 0
\(520\) 1.58258 0.0694005
\(521\) −36.7477 −1.60995 −0.804974 0.593311i \(-0.797821\pi\)
−0.804974 + 0.593311i \(0.797821\pi\)
\(522\) 0 0
\(523\) 21.8890 0.957140 0.478570 0.878050i \(-0.341155\pi\)
0.478570 + 0.878050i \(0.341155\pi\)
\(524\) −37.5728 −1.64138
\(525\) 0 0
\(526\) −5.20871 −0.227111
\(527\) −14.8655 −0.647549
\(528\) 0 0
\(529\) −22.6606 −0.985244
\(530\) −5.55765 −0.241409
\(531\) 0 0
\(532\) 27.1652 1.17776
\(533\) −3.16515 −0.137098
\(534\) 0 0
\(535\) −13.0381 −0.563684
\(536\) 7.65120 0.330482
\(537\) 0 0
\(538\) 4.03620 0.174013
\(539\) 0 0
\(540\) 0 0
\(541\) 22.9934 0.988564 0.494282 0.869302i \(-0.335431\pi\)
0.494282 + 0.869302i \(0.335431\pi\)
\(542\) −7.62614 −0.327571
\(543\) 0 0
\(544\) 8.20871 0.351946
\(545\) 6.92820 0.296772
\(546\) 0 0
\(547\) −41.0369 −1.75461 −0.877306 0.479931i \(-0.840662\pi\)
−0.877306 + 0.479931i \(0.840662\pi\)
\(548\) −34.6261 −1.47916
\(549\) 0 0
\(550\) 0 0
\(551\) 33.4955 1.42695
\(552\) 0 0
\(553\) 3.58258 0.152347
\(554\) 0.834849 0.0354693
\(555\) 0 0
\(556\) 13.8763 0.588487
\(557\) −8.46955 −0.358867 −0.179433 0.983770i \(-0.557426\pi\)
−0.179433 + 0.983770i \(0.557426\pi\)
\(558\) 0 0
\(559\) −8.83485 −0.373674
\(560\) −12.2197 −0.516377
\(561\) 0 0
\(562\) −10.8348 −0.457041
\(563\) −43.5873 −1.83699 −0.918493 0.395437i \(-0.870593\pi\)
−0.918493 + 0.395437i \(0.870593\pi\)
\(564\) 0 0
\(565\) −9.00000 −0.378633
\(566\) 4.92197 0.206886
\(567\) 0 0
\(568\) 13.8564 0.581402
\(569\) −37.3821 −1.56714 −0.783570 0.621304i \(-0.786603\pi\)
−0.783570 + 0.621304i \(0.786603\pi\)
\(570\) 0 0
\(571\) −0.627650 −0.0262663 −0.0131332 0.999914i \(-0.504181\pi\)
−0.0131332 + 0.999914i \(0.504181\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −6.92820 −0.289178
\(575\) 0.582576 0.0242951
\(576\) 0 0
\(577\) 20.0000 0.832611 0.416305 0.909225i \(-0.363325\pi\)
0.416305 + 0.909225i \(0.363325\pi\)
\(578\) −6.39590 −0.266035
\(579\) 0 0
\(580\) −17.3205 −0.719195
\(581\) 69.4955 2.88316
\(582\) 0 0
\(583\) 0 0
\(584\) 8.83485 0.365589
\(585\) 0 0
\(586\) 0.791288 0.0326878
\(587\) −7.74773 −0.319783 −0.159891 0.987135i \(-0.551114\pi\)
−0.159891 + 0.987135i \(0.551114\pi\)
\(588\) 0 0
\(589\) −29.7309 −1.22504
\(590\) −6.20520 −0.255464
\(591\) 0 0
\(592\) 21.1652 0.869882
\(593\) −27.7128 −1.13803 −0.569014 0.822328i \(-0.692675\pi\)
−0.569014 + 0.822328i \(0.692675\pi\)
\(594\) 0 0
\(595\) −7.58258 −0.310855
\(596\) 4.91010 0.201126
\(597\) 0 0
\(598\) −0.243181 −0.00994442
\(599\) −6.74773 −0.275705 −0.137852 0.990453i \(-0.544020\pi\)
−0.137852 + 0.990453i \(0.544020\pi\)
\(600\) 0 0
\(601\) −1.44600 −0.0589836 −0.0294918 0.999565i \(-0.509389\pi\)
−0.0294918 + 0.999565i \(0.509389\pi\)
\(602\) −19.3386 −0.788183
\(603\) 0 0
\(604\) 13.8763 0.564619
\(605\) 0 0
\(606\) 0 0
\(607\) −25.5438 −1.03679 −0.518396 0.855141i \(-0.673471\pi\)
−0.518396 + 0.855141i \(0.673471\pi\)
\(608\) 16.4174 0.665814
\(609\) 0 0
\(610\) 1.62614 0.0658403
\(611\) 3.12250 0.126323
\(612\) 0 0
\(613\) −1.82740 −0.0738080 −0.0369040 0.999319i \(-0.511750\pi\)
−0.0369040 + 0.999319i \(0.511750\pi\)
\(614\) 7.07803 0.285646
\(615\) 0 0
\(616\) 0 0
\(617\) 6.00000 0.241551 0.120775 0.992680i \(-0.461462\pi\)
0.120775 + 0.992680i \(0.461462\pi\)
\(618\) 0 0
\(619\) 13.4955 0.542428 0.271214 0.962519i \(-0.412575\pi\)
0.271214 + 0.962519i \(0.412575\pi\)
\(620\) 15.3739 0.617429
\(621\) 0 0
\(622\) 5.29150 0.212170
\(623\) 64.5626 2.58665
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 2.55040 0.101935
\(627\) 0 0
\(628\) −19.2523 −0.768249
\(629\) 13.1334 0.523663
\(630\) 0 0
\(631\) −21.4174 −0.852614 −0.426307 0.904578i \(-0.640186\pi\)
−0.426307 + 0.904578i \(0.640186\pi\)
\(632\) 1.41742 0.0563821
\(633\) 0 0
\(634\) −13.7810 −0.547312
\(635\) −7.84190 −0.311196
\(636\) 0 0
\(637\) −11.1153 −0.440404
\(638\) 0 0
\(639\) 0 0
\(640\) −11.0399 −0.436389
\(641\) 6.74773 0.266519 0.133260 0.991081i \(-0.457456\pi\)
0.133260 + 0.991081i \(0.457456\pi\)
\(642\) 0 0
\(643\) −39.4955 −1.55755 −0.778774 0.627304i \(-0.784158\pi\)
−0.778774 + 0.627304i \(0.784158\pi\)
\(644\) 4.56850 0.180024
\(645\) 0 0
\(646\) 2.74110 0.107847
\(647\) −29.7477 −1.16950 −0.584752 0.811212i \(-0.698808\pi\)
−0.584752 + 0.811212i \(0.698808\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −0.417424 −0.0163727
\(651\) 0 0
\(652\) 37.1652 1.45550
\(653\) 10.0000 0.391330 0.195665 0.980671i \(-0.437313\pi\)
0.195665 + 0.980671i \(0.437313\pi\)
\(654\) 0 0
\(655\) 20.9753 0.819573
\(656\) 9.66930 0.377523
\(657\) 0 0
\(658\) 6.83485 0.266450
\(659\) 18.2342 0.710304 0.355152 0.934809i \(-0.384429\pi\)
0.355152 + 0.934809i \(0.384429\pi\)
\(660\) 0 0
\(661\) 22.0000 0.855701 0.427850 0.903850i \(-0.359271\pi\)
0.427850 + 0.903850i \(0.359271\pi\)
\(662\) 12.6766 0.492688
\(663\) 0 0
\(664\) 27.4955 1.06703
\(665\) −15.1652 −0.588079
\(666\) 0 0
\(667\) 5.63310 0.218115
\(668\) −8.01270 −0.310021
\(669\) 0 0
\(670\) −2.01810 −0.0779661
\(671\) 0 0
\(672\) 0 0
\(673\) −11.3060 −0.435814 −0.217907 0.975969i \(-0.569923\pi\)
−0.217907 + 0.975969i \(0.569923\pi\)
\(674\) −0.0871215 −0.00335580
\(675\) 0 0
\(676\) 21.7913 0.838126
\(677\) −12.0290 −0.462312 −0.231156 0.972917i \(-0.574251\pi\)
−0.231156 + 0.972917i \(0.574251\pi\)
\(678\) 0 0
\(679\) −19.3386 −0.742148
\(680\) −3.00000 −0.115045
\(681\) 0 0
\(682\) 0 0
\(683\) −46.3303 −1.77278 −0.886390 0.462939i \(-0.846795\pi\)
−0.886390 + 0.462939i \(0.846795\pi\)
\(684\) 0 0
\(685\) 19.3303 0.738573
\(686\) −10.3303 −0.394413
\(687\) 0 0
\(688\) 26.9898 1.02898
\(689\) 11.1153 0.423459
\(690\) 0 0
\(691\) −0.582576 −0.0221622 −0.0110811 0.999939i \(-0.503527\pi\)
−0.0110811 + 0.999939i \(0.503527\pi\)
\(692\) 12.4104 0.471773
\(693\) 0 0
\(694\) 10.0436 0.381248
\(695\) −7.74655 −0.293844
\(696\) 0 0
\(697\) 6.00000 0.227266
\(698\) 11.8693 0.449260
\(699\) 0 0
\(700\) 7.84190 0.296396
\(701\) −21.8890 −0.826737 −0.413368 0.910564i \(-0.635648\pi\)
−0.413368 + 0.910564i \(0.635648\pi\)
\(702\) 0 0
\(703\) 26.2668 0.990672
\(704\) 0 0
\(705\) 0 0
\(706\) −8.14786 −0.306649
\(707\) −8.00000 −0.300871
\(708\) 0 0
\(709\) −45.0000 −1.69001 −0.845005 0.534758i \(-0.820403\pi\)
−0.845005 + 0.534758i \(0.820403\pi\)
\(710\) −3.65480 −0.137162
\(711\) 0 0
\(712\) 25.5438 0.957295
\(713\) −5.00000 −0.187251
\(714\) 0 0
\(715\) 0 0
\(716\) 13.5826 0.507605
\(717\) 0 0
\(718\) 4.00000 0.149279
\(719\) 16.3303 0.609018 0.304509 0.952510i \(-0.401508\pi\)
0.304509 + 0.952510i \(0.401508\pi\)
\(720\) 0 0
\(721\) −31.3676 −1.16819
\(722\) −3.19795 −0.119015
\(723\) 0 0
\(724\) 9.25227 0.343858
\(725\) 9.66930 0.359109
\(726\) 0 0
\(727\) −5.16515 −0.191565 −0.0957824 0.995402i \(-0.530535\pi\)
−0.0957824 + 0.995402i \(0.530535\pi\)
\(728\) −6.92820 −0.256776
\(729\) 0 0
\(730\) −2.33030 −0.0862484
\(731\) 16.7477 0.619437
\(732\) 0 0
\(733\) 13.6657 0.504754 0.252377 0.967629i \(-0.418788\pi\)
0.252377 + 0.967629i \(0.418788\pi\)
\(734\) −0.572101 −0.0211166
\(735\) 0 0
\(736\) 2.76100 0.101772
\(737\) 0 0
\(738\) 0 0
\(739\) 21.9844 0.808708 0.404354 0.914603i \(-0.367496\pi\)
0.404354 + 0.914603i \(0.367496\pi\)
\(740\) −13.5826 −0.499305
\(741\) 0 0
\(742\) 24.3303 0.893194
\(743\) −12.6567 −0.464328 −0.232164 0.972677i \(-0.574581\pi\)
−0.232164 + 0.972677i \(0.574581\pi\)
\(744\) 0 0
\(745\) −2.74110 −0.100426
\(746\) 6.33030 0.231769
\(747\) 0 0
\(748\) 0 0
\(749\) 57.0780 2.08559
\(750\) 0 0
\(751\) 20.0780 0.732658 0.366329 0.930485i \(-0.380615\pi\)
0.366329 + 0.930485i \(0.380615\pi\)
\(752\) −9.53901 −0.347852
\(753\) 0 0
\(754\) −4.03620 −0.146990
\(755\) −7.74655 −0.281926
\(756\) 0 0
\(757\) 6.83485 0.248417 0.124208 0.992256i \(-0.460361\pi\)
0.124208 + 0.992256i \(0.460361\pi\)
\(758\) −7.19435 −0.261311
\(759\) 0 0
\(760\) −6.00000 −0.217643
\(761\) 14.7701 0.535416 0.267708 0.963500i \(-0.413734\pi\)
0.267708 + 0.963500i \(0.413734\pi\)
\(762\) 0 0
\(763\) −30.3303 −1.09803
\(764\) −40.7477 −1.47420
\(765\) 0 0
\(766\) −2.89200 −0.104492
\(767\) 12.4104 0.448114
\(768\) 0 0
\(769\) −45.5101 −1.64114 −0.820568 0.571550i \(-0.806343\pi\)
−0.820568 + 0.571550i \(0.806343\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 29.3893 1.05774
\(773\) −48.4955 −1.74426 −0.872130 0.489274i \(-0.837262\pi\)
−0.872130 + 0.489274i \(0.837262\pi\)
\(774\) 0 0
\(775\) −8.58258 −0.308295
\(776\) −7.65120 −0.274662
\(777\) 0 0
\(778\) −2.16900 −0.0777624
\(779\) 12.0000 0.429945
\(780\) 0 0
\(781\) 0 0
\(782\) 0.460985 0.0164848
\(783\) 0 0
\(784\) 33.9564 1.21273
\(785\) 10.7477 0.383603
\(786\) 0 0
\(787\) −14.2378 −0.507523 −0.253762 0.967267i \(-0.581668\pi\)
−0.253762 + 0.967267i \(0.581668\pi\)
\(788\) 22.2306 0.791933
\(789\) 0 0
\(790\) −0.373864 −0.0133015
\(791\) 39.4002 1.40091
\(792\) 0 0
\(793\) −3.25227 −0.115492
\(794\) 14.7701 0.524171
\(795\) 0 0
\(796\) 28.2087 0.999831
\(797\) 4.33030 0.153387 0.0766936 0.997055i \(-0.475564\pi\)
0.0766936 + 0.997055i \(0.475564\pi\)
\(798\) 0 0
\(799\) −5.91915 −0.209405
\(800\) 4.73930 0.167560
\(801\) 0 0
\(802\) −6.54680 −0.231176
\(803\) 0 0
\(804\) 0 0
\(805\) −2.55040 −0.0898898
\(806\) 3.58258 0.126191
\(807\) 0 0
\(808\) −3.16515 −0.111350
\(809\) −8.75560 −0.307831 −0.153915 0.988084i \(-0.549188\pi\)
−0.153915 + 0.988084i \(0.549188\pi\)
\(810\) 0 0
\(811\) −54.7980 −1.92422 −0.962109 0.272667i \(-0.912094\pi\)
−0.962109 + 0.272667i \(0.912094\pi\)
\(812\) 75.8258 2.66096
\(813\) 0 0
\(814\) 0 0
\(815\) −20.7477 −0.726761
\(816\) 0 0
\(817\) 33.4955 1.17186
\(818\) −8.87841 −0.310426
\(819\) 0 0
\(820\) −6.20520 −0.216695
\(821\) 18.7665 0.654956 0.327478 0.944859i \(-0.393801\pi\)
0.327478 + 0.944859i \(0.393801\pi\)
\(822\) 0 0
\(823\) −19.1652 −0.668055 −0.334028 0.942563i \(-0.608408\pi\)
−0.334028 + 0.942563i \(0.608408\pi\)
\(824\) −12.4104 −0.432337
\(825\) 0 0
\(826\) 27.1652 0.945197
\(827\) −5.67290 −0.197266 −0.0986331 0.995124i \(-0.531447\pi\)
−0.0986331 + 0.995124i \(0.531447\pi\)
\(828\) 0 0
\(829\) 19.0000 0.659897 0.329949 0.943999i \(-0.392969\pi\)
0.329949 + 0.943999i \(0.392969\pi\)
\(830\) −7.25227 −0.251730
\(831\) 0 0
\(832\) 3.12250 0.108253
\(833\) 21.0707 0.730055
\(834\) 0 0
\(835\) 4.47315 0.154800
\(836\) 0 0
\(837\) 0 0
\(838\) 1.97830 0.0683392
\(839\) 12.0000 0.414286 0.207143 0.978311i \(-0.433583\pi\)
0.207143 + 0.978311i \(0.433583\pi\)
\(840\) 0 0
\(841\) 64.4955 2.22398
\(842\) −17.9681 −0.619220
\(843\) 0 0
\(844\) −41.9705 −1.44468
\(845\) −12.1652 −0.418494
\(846\) 0 0
\(847\) 0 0
\(848\) −33.9564 −1.16607
\(849\) 0 0
\(850\) 0.791288 0.0271409
\(851\) 4.41742 0.151427
\(852\) 0 0
\(853\) 21.3567 0.731240 0.365620 0.930764i \(-0.380857\pi\)
0.365620 + 0.930764i \(0.380857\pi\)
\(854\) −7.11890 −0.243604
\(855\) 0 0
\(856\) 22.5826 0.771857
\(857\) −6.83285 −0.233406 −0.116703 0.993167i \(-0.537233\pi\)
−0.116703 + 0.993167i \(0.537233\pi\)
\(858\) 0 0
\(859\) 48.6606 1.66028 0.830139 0.557556i \(-0.188261\pi\)
0.830139 + 0.557556i \(0.188261\pi\)
\(860\) −17.3205 −0.590624
\(861\) 0 0
\(862\) −1.91288 −0.0651529
\(863\) −36.0000 −1.22545 −0.612727 0.790295i \(-0.709928\pi\)
−0.612727 + 0.790295i \(0.709928\pi\)
\(864\) 0 0
\(865\) −6.92820 −0.235566
\(866\) −6.20520 −0.210861
\(867\) 0 0
\(868\) −67.3037 −2.28444
\(869\) 0 0
\(870\) 0 0
\(871\) 4.03620 0.136762
\(872\) −12.0000 −0.406371
\(873\) 0 0
\(874\) 0.921970 0.0311861
\(875\) −4.37780 −0.147997
\(876\) 0 0
\(877\) 20.5939 0.695407 0.347703 0.937605i \(-0.386962\pi\)
0.347703 + 0.937605i \(0.386962\pi\)
\(878\) 1.95644 0.0660266
\(879\) 0 0
\(880\) 0 0
\(881\) 29.4955 0.993727 0.496864 0.867829i \(-0.334485\pi\)
0.496864 + 0.867829i \(0.334485\pi\)
\(882\) 0 0
\(883\) 34.2432 1.15237 0.576187 0.817318i \(-0.304540\pi\)
0.576187 + 0.817318i \(0.304540\pi\)
\(884\) 2.83485 0.0953463
\(885\) 0 0
\(886\) 5.10080 0.171365
\(887\) 29.7309 0.998266 0.499133 0.866525i \(-0.333652\pi\)
0.499133 + 0.866525i \(0.333652\pi\)
\(888\) 0 0
\(889\) 34.3303 1.15140
\(890\) −6.73750 −0.225842
\(891\) 0 0
\(892\) 29.2523 0.979439
\(893\) −11.8383 −0.396154
\(894\) 0 0
\(895\) −7.58258 −0.253458
\(896\) 48.3303 1.61460
\(897\) 0 0
\(898\) −9.66930 −0.322669
\(899\) −82.9875 −2.76779
\(900\) 0 0
\(901\) −21.0707 −0.701965
\(902\) 0 0
\(903\) 0 0
\(904\) 15.5885 0.518464
\(905\) −5.16515 −0.171695
\(906\) 0 0
\(907\) 10.8348 0.359765 0.179883 0.983688i \(-0.442428\pi\)
0.179883 + 0.983688i \(0.442428\pi\)
\(908\) 36.1069 1.19825
\(909\) 0 0
\(910\) 1.82740 0.0605778
\(911\) 17.2523 0.571593 0.285797 0.958290i \(-0.407742\pi\)
0.285797 + 0.958290i \(0.407742\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −15.0780 −0.498737
\(915\) 0 0
\(916\) −30.4519 −1.00616
\(917\) −91.8258 −3.03235
\(918\) 0 0
\(919\) 6.73750 0.222250 0.111125 0.993806i \(-0.464555\pi\)
0.111125 + 0.993806i \(0.464555\pi\)
\(920\) −1.00905 −0.0332674
\(921\) 0 0
\(922\) 8.41742 0.277213
\(923\) 7.30960 0.240599
\(924\) 0 0
\(925\) 7.58258 0.249314
\(926\) 8.90650 0.292686
\(927\) 0 0
\(928\) 45.8258 1.50430
\(929\) −20.3303 −0.667016 −0.333508 0.942747i \(-0.608232\pi\)
−0.333508 + 0.942747i \(0.608232\pi\)
\(930\) 0 0
\(931\) 42.1413 1.38113
\(932\) 25.3332 0.829817
\(933\) 0 0
\(934\) 5.74835 0.188092
\(935\) 0 0
\(936\) 0 0
\(937\) −9.86001 −0.322112 −0.161056 0.986945i \(-0.551490\pi\)
−0.161056 + 0.986945i \(0.551490\pi\)
\(938\) 8.83485 0.288468
\(939\) 0 0
\(940\) 6.12159 0.199664
\(941\) −41.0369 −1.33777 −0.668883 0.743368i \(-0.733227\pi\)
−0.668883 + 0.743368i \(0.733227\pi\)
\(942\) 0 0
\(943\) 2.01810 0.0657184
\(944\) −37.9129 −1.23396
\(945\) 0 0
\(946\) 0 0
\(947\) 24.9129 0.809560 0.404780 0.914414i \(-0.367348\pi\)
0.404780 + 0.914414i \(0.367348\pi\)
\(948\) 0 0
\(949\) 4.66061 0.151290
\(950\) 1.58258 0.0513455
\(951\) 0 0
\(952\) 13.1334 0.425656
\(953\) 20.7846 0.673280 0.336640 0.941634i \(-0.390710\pi\)
0.336640 + 0.941634i \(0.390710\pi\)
\(954\) 0 0
\(955\) 22.7477 0.736099
\(956\) −19.9107 −0.643958
\(957\) 0 0
\(958\) −14.0871 −0.455134
\(959\) −84.6242 −2.73266
\(960\) 0 0
\(961\) 42.6606 1.37615
\(962\) −3.16515 −0.102049
\(963\) 0 0
\(964\) −46.8806 −1.50992
\(965\) −16.4068 −0.528154
\(966\) 0 0
\(967\) 4.71940 0.151766 0.0758829 0.997117i \(-0.475822\pi\)
0.0758829 + 0.997117i \(0.475822\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 2.01810 0.0647973
\(971\) −25.5826 −0.820984 −0.410492 0.911864i \(-0.634643\pi\)
−0.410492 + 0.911864i \(0.634643\pi\)
\(972\) 0 0
\(973\) 33.9129 1.08720
\(974\) −2.55040 −0.0817201
\(975\) 0 0
\(976\) 9.93545 0.318026
\(977\) −12.1652 −0.389198 −0.194599 0.980883i \(-0.562340\pi\)
−0.194599 + 0.980883i \(0.562340\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −21.7913 −0.696097
\(981\) 0 0
\(982\) −12.6606 −0.404016
\(983\) −21.4174 −0.683110 −0.341555 0.939862i \(-0.610954\pi\)
−0.341555 + 0.939862i \(0.610954\pi\)
\(984\) 0 0
\(985\) −12.4104 −0.395428
\(986\) 7.65120 0.243664
\(987\) 0 0
\(988\) 5.66970 0.180377
\(989\) 5.63310 0.179122
\(990\) 0 0
\(991\) −37.2432 −1.18307 −0.591534 0.806280i \(-0.701478\pi\)
−0.591534 + 0.806280i \(0.701478\pi\)
\(992\) −40.6754 −1.29145
\(993\) 0 0
\(994\) 16.0000 0.507489
\(995\) −15.7477 −0.499237
\(996\) 0 0
\(997\) 41.4183 1.31173 0.655866 0.754878i \(-0.272304\pi\)
0.655866 + 0.754878i \(0.272304\pi\)
\(998\) −4.71940 −0.149390
\(999\) 0 0
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 5445.2.a.bl.1.3 4
3.2 odd 2 1815.2.a.t.1.2 4
11.10 odd 2 inner 5445.2.a.bl.1.2 4
15.14 odd 2 9075.2.a.cu.1.3 4
33.32 even 2 1815.2.a.t.1.3 yes 4
165.164 even 2 9075.2.a.cu.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1815.2.a.t.1.2 4 3.2 odd 2
1815.2.a.t.1.3 yes 4 33.32 even 2
5445.2.a.bl.1.2 4 11.10 odd 2 inner
5445.2.a.bl.1.3 4 1.1 even 1 trivial
9075.2.a.cu.1.2 4 165.164 even 2
9075.2.a.cu.1.3 4 15.14 odd 2