Properties

Label 6008.2.a.b.1.31
Level 6008
Weight 2
Character 6008.1
Self dual yes
Analytic conductor 47.974
Analytic rank 1
Dimension 44
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6008 = 2^{3} \cdot 751 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6008.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(47.9741215344\)
Analytic rank: \(1\)
Dimension: \(44\)
Coefficient ring index: multiple of None
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.31
Character \(\chi\) = 6008.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.16559 q^{3} -2.46634 q^{5} -4.75856 q^{7} -1.64140 q^{9} +O(q^{10})\) \(q+1.16559 q^{3} -2.46634 q^{5} -4.75856 q^{7} -1.64140 q^{9} -2.24706 q^{11} +5.94250 q^{13} -2.87474 q^{15} +3.75546 q^{17} +4.54795 q^{19} -5.54653 q^{21} +4.61898 q^{23} +1.08282 q^{25} -5.40997 q^{27} +8.83422 q^{29} +0.341568 q^{31} -2.61915 q^{33} +11.7362 q^{35} +3.19121 q^{37} +6.92651 q^{39} +3.24052 q^{41} -6.43907 q^{43} +4.04825 q^{45} -11.5124 q^{47} +15.6439 q^{49} +4.37733 q^{51} -7.92955 q^{53} +5.54202 q^{55} +5.30105 q^{57} -5.05486 q^{59} -0.698184 q^{61} +7.81069 q^{63} -14.6562 q^{65} -7.34445 q^{67} +5.38384 q^{69} -3.69800 q^{71} -13.6948 q^{73} +1.26213 q^{75} +10.6928 q^{77} +5.53147 q^{79} -1.38161 q^{81} -5.90215 q^{83} -9.26224 q^{85} +10.2971 q^{87} -5.58741 q^{89} -28.2777 q^{91} +0.398128 q^{93} -11.2168 q^{95} +0.699211 q^{97} +3.68833 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 44q - 14q^{3} + 7q^{5} - 20q^{7} + 38q^{9} + O(q^{10}) \) \( 44q - 14q^{3} + 7q^{5} - 20q^{7} + 38q^{9} - 19q^{11} - 10q^{13} - 17q^{15} - 16q^{17} - 25q^{19} + 16q^{21} - 29q^{23} + 29q^{25} - 50q^{27} + 35q^{29} - 49q^{31} - 28q^{33} - 37q^{35} - 30q^{37} - 28q^{39} - 14q^{41} - 35q^{43} + 6q^{45} - 45q^{47} + 20q^{49} - 17q^{51} + 18q^{53} - 53q^{55} - 31q^{57} - 57q^{59} + 27q^{61} - 77q^{63} - 21q^{65} - 56q^{67} + 36q^{69} - 52q^{71} - 68q^{73} - 77q^{75} + 37q^{77} - 55q^{79} + 28q^{81} - 51q^{83} - 16q^{85} - 67q^{87} - 21q^{89} - 51q^{91} - 14q^{93} - 56q^{95} - 67q^{97} - 58q^{99} + O(q^{100}) \)

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 0
\(3\) 1.16559 0.672954 0.336477 0.941692i \(-0.390765\pi\)
0.336477 + 0.941692i \(0.390765\pi\)
\(4\) 0 0
\(5\) −2.46634 −1.10298 −0.551490 0.834182i \(-0.685940\pi\)
−0.551490 + 0.834182i \(0.685940\pi\)
\(6\) 0 0
\(7\) −4.75856 −1.79856 −0.899282 0.437368i \(-0.855911\pi\)
−0.899282 + 0.437368i \(0.855911\pi\)
\(8\) 0 0
\(9\) −1.64140 −0.547133
\(10\) 0 0
\(11\) −2.24706 −0.677515 −0.338757 0.940874i \(-0.610007\pi\)
−0.338757 + 0.940874i \(0.610007\pi\)
\(12\) 0 0
\(13\) 5.94250 1.64815 0.824076 0.566479i \(-0.191695\pi\)
0.824076 + 0.566479i \(0.191695\pi\)
\(14\) 0 0
\(15\) −2.87474 −0.742254
\(16\) 0 0
\(17\) 3.75546 0.910834 0.455417 0.890278i \(-0.349490\pi\)
0.455417 + 0.890278i \(0.349490\pi\)
\(18\) 0 0
\(19\) 4.54795 1.04337 0.521686 0.853138i \(-0.325303\pi\)
0.521686 + 0.853138i \(0.325303\pi\)
\(20\) 0 0
\(21\) −5.54653 −1.21035
\(22\) 0 0
\(23\) 4.61898 0.963124 0.481562 0.876412i \(-0.340070\pi\)
0.481562 + 0.876412i \(0.340070\pi\)
\(24\) 0 0
\(25\) 1.08282 0.216564
\(26\) 0 0
\(27\) −5.40997 −1.04115
\(28\) 0 0
\(29\) 8.83422 1.64047 0.820237 0.572024i \(-0.193842\pi\)
0.820237 + 0.572024i \(0.193842\pi\)
\(30\) 0 0
\(31\) 0.341568 0.0613474 0.0306737 0.999529i \(-0.490235\pi\)
0.0306737 + 0.999529i \(0.490235\pi\)
\(32\) 0 0
\(33\) −2.61915 −0.455936
\(34\) 0 0
\(35\) 11.7362 1.98378
\(36\) 0 0
\(37\) 3.19121 0.524632 0.262316 0.964982i \(-0.415514\pi\)
0.262316 + 0.964982i \(0.415514\pi\)
\(38\) 0 0
\(39\) 6.92651 1.10913
\(40\) 0 0
\(41\) 3.24052 0.506084 0.253042 0.967455i \(-0.418569\pi\)
0.253042 + 0.967455i \(0.418569\pi\)
\(42\) 0 0
\(43\) −6.43907 −0.981949 −0.490974 0.871174i \(-0.663359\pi\)
−0.490974 + 0.871174i \(0.663359\pi\)
\(44\) 0 0
\(45\) 4.04825 0.603477
\(46\) 0 0
\(47\) −11.5124 −1.67925 −0.839625 0.543166i \(-0.817225\pi\)
−0.839625 + 0.543166i \(0.817225\pi\)
\(48\) 0 0
\(49\) 15.6439 2.23484
\(50\) 0 0
\(51\) 4.37733 0.612949
\(52\) 0 0
\(53\) −7.92955 −1.08921 −0.544604 0.838693i \(-0.683320\pi\)
−0.544604 + 0.838693i \(0.683320\pi\)
\(54\) 0 0
\(55\) 5.54202 0.747285
\(56\) 0 0
\(57\) 5.30105 0.702141
\(58\) 0 0
\(59\) −5.05486 −0.658086 −0.329043 0.944315i \(-0.606726\pi\)
−0.329043 + 0.944315i \(0.606726\pi\)
\(60\) 0 0
\(61\) −0.698184 −0.0893933 −0.0446967 0.999001i \(-0.514232\pi\)
−0.0446967 + 0.999001i \(0.514232\pi\)
\(62\) 0 0
\(63\) 7.81069 0.984055
\(64\) 0 0
\(65\) −14.6562 −1.81788
\(66\) 0 0
\(67\) −7.34445 −0.897268 −0.448634 0.893716i \(-0.648089\pi\)
−0.448634 + 0.893716i \(0.648089\pi\)
\(68\) 0 0
\(69\) 5.38384 0.648138
\(70\) 0 0
\(71\) −3.69800 −0.438872 −0.219436 0.975627i \(-0.570422\pi\)
−0.219436 + 0.975627i \(0.570422\pi\)
\(72\) 0 0
\(73\) −13.6948 −1.60285 −0.801426 0.598093i \(-0.795925\pi\)
−0.801426 + 0.598093i \(0.795925\pi\)
\(74\) 0 0
\(75\) 1.26213 0.145738
\(76\) 0 0
\(77\) 10.6928 1.21855
\(78\) 0 0
\(79\) 5.53147 0.622339 0.311170 0.950354i \(-0.399279\pi\)
0.311170 + 0.950354i \(0.399279\pi\)
\(80\) 0 0
\(81\) −1.38161 −0.153512
\(82\) 0 0
\(83\) −5.90215 −0.647845 −0.323923 0.946084i \(-0.605002\pi\)
−0.323923 + 0.946084i \(0.605002\pi\)
\(84\) 0 0
\(85\) −9.26224 −1.00463
\(86\) 0 0
\(87\) 10.2971 1.10396
\(88\) 0 0
\(89\) −5.58741 −0.592265 −0.296132 0.955147i \(-0.595697\pi\)
−0.296132 + 0.955147i \(0.595697\pi\)
\(90\) 0 0
\(91\) −28.2777 −2.96431
\(92\) 0 0
\(93\) 0.398128 0.0412840
\(94\) 0 0
\(95\) −11.2168 −1.15082
\(96\) 0 0
\(97\) 0.699211 0.0709941 0.0354970 0.999370i \(-0.488699\pi\)
0.0354970 + 0.999370i \(0.488699\pi\)
\(98\) 0 0
\(99\) 3.68833 0.370691
\(100\) 0 0
\(101\) 3.04083 0.302574 0.151287 0.988490i \(-0.451658\pi\)
0.151287 + 0.988490i \(0.451658\pi\)
\(102\) 0 0
\(103\) 0.202037 0.0199073 0.00995366 0.999950i \(-0.496832\pi\)
0.00995366 + 0.999950i \(0.496832\pi\)
\(104\) 0 0
\(105\) 13.6796 1.33499
\(106\) 0 0
\(107\) 12.3407 1.19302 0.596509 0.802606i \(-0.296554\pi\)
0.596509 + 0.802606i \(0.296554\pi\)
\(108\) 0 0
\(109\) −12.4684 −1.19426 −0.597129 0.802145i \(-0.703692\pi\)
−0.597129 + 0.802145i \(0.703692\pi\)
\(110\) 0 0
\(111\) 3.71965 0.353053
\(112\) 0 0
\(113\) 15.6516 1.47238 0.736191 0.676774i \(-0.236623\pi\)
0.736191 + 0.676774i \(0.236623\pi\)
\(114\) 0 0
\(115\) −11.3920 −1.06231
\(116\) 0 0
\(117\) −9.75401 −0.901759
\(118\) 0 0
\(119\) −17.8706 −1.63819
\(120\) 0 0
\(121\) −5.95071 −0.540973
\(122\) 0 0
\(123\) 3.77712 0.340571
\(124\) 0 0
\(125\) 9.66109 0.864114
\(126\) 0 0
\(127\) 2.11641 0.187801 0.0939007 0.995582i \(-0.470066\pi\)
0.0939007 + 0.995582i \(0.470066\pi\)
\(128\) 0 0
\(129\) −7.50532 −0.660806
\(130\) 0 0
\(131\) −13.9660 −1.22022 −0.610109 0.792318i \(-0.708874\pi\)
−0.610109 + 0.792318i \(0.708874\pi\)
\(132\) 0 0
\(133\) −21.6417 −1.87657
\(134\) 0 0
\(135\) 13.3428 1.14837
\(136\) 0 0
\(137\) 6.06697 0.518336 0.259168 0.965832i \(-0.416552\pi\)
0.259168 + 0.965832i \(0.416552\pi\)
\(138\) 0 0
\(139\) 3.26072 0.276571 0.138285 0.990392i \(-0.455841\pi\)
0.138285 + 0.990392i \(0.455841\pi\)
\(140\) 0 0
\(141\) −13.4187 −1.13006
\(142\) 0 0
\(143\) −13.3532 −1.11665
\(144\) 0 0
\(145\) −21.7882 −1.80941
\(146\) 0 0
\(147\) 18.2343 1.50394
\(148\) 0 0
\(149\) 13.6511 1.11834 0.559171 0.829052i \(-0.311120\pi\)
0.559171 + 0.829052i \(0.311120\pi\)
\(150\) 0 0
\(151\) 12.8281 1.04394 0.521970 0.852964i \(-0.325197\pi\)
0.521970 + 0.852964i \(0.325197\pi\)
\(152\) 0 0
\(153\) −6.16422 −0.498347
\(154\) 0 0
\(155\) −0.842421 −0.0676649
\(156\) 0 0
\(157\) −10.7555 −0.858382 −0.429191 0.903214i \(-0.641201\pi\)
−0.429191 + 0.903214i \(0.641201\pi\)
\(158\) 0 0
\(159\) −9.24261 −0.732986
\(160\) 0 0
\(161\) −21.9797 −1.73224
\(162\) 0 0
\(163\) −23.3765 −1.83099 −0.915494 0.402331i \(-0.868200\pi\)
−0.915494 + 0.402331i \(0.868200\pi\)
\(164\) 0 0
\(165\) 6.45972 0.502888
\(166\) 0 0
\(167\) −3.54470 −0.274297 −0.137149 0.990550i \(-0.543794\pi\)
−0.137149 + 0.990550i \(0.543794\pi\)
\(168\) 0 0
\(169\) 22.3133 1.71640
\(170\) 0 0
\(171\) −7.46501 −0.570863
\(172\) 0 0
\(173\) 0.564088 0.0428868 0.0214434 0.999770i \(-0.493174\pi\)
0.0214434 + 0.999770i \(0.493174\pi\)
\(174\) 0 0
\(175\) −5.15266 −0.389505
\(176\) 0 0
\(177\) −5.89189 −0.442862
\(178\) 0 0
\(179\) −15.1013 −1.12873 −0.564363 0.825526i \(-0.690878\pi\)
−0.564363 + 0.825526i \(0.690878\pi\)
\(180\) 0 0
\(181\) −3.76067 −0.279528 −0.139764 0.990185i \(-0.544634\pi\)
−0.139764 + 0.990185i \(0.544634\pi\)
\(182\) 0 0
\(183\) −0.813797 −0.0601576
\(184\) 0 0
\(185\) −7.87061 −0.578659
\(186\) 0 0
\(187\) −8.43876 −0.617103
\(188\) 0 0
\(189\) 25.7436 1.87257
\(190\) 0 0
\(191\) −4.28313 −0.309916 −0.154958 0.987921i \(-0.549524\pi\)
−0.154958 + 0.987921i \(0.549524\pi\)
\(192\) 0 0
\(193\) −20.7407 −1.49295 −0.746476 0.665413i \(-0.768256\pi\)
−0.746476 + 0.665413i \(0.768256\pi\)
\(194\) 0 0
\(195\) −17.0831 −1.22335
\(196\) 0 0
\(197\) −12.5117 −0.891418 −0.445709 0.895178i \(-0.647048\pi\)
−0.445709 + 0.895178i \(0.647048\pi\)
\(198\) 0 0
\(199\) 19.4335 1.37760 0.688801 0.724951i \(-0.258138\pi\)
0.688801 + 0.724951i \(0.258138\pi\)
\(200\) 0 0
\(201\) −8.56062 −0.603820
\(202\) 0 0
\(203\) −42.0381 −2.95050
\(204\) 0 0
\(205\) −7.99221 −0.558200
\(206\) 0 0
\(207\) −7.58159 −0.526957
\(208\) 0 0
\(209\) −10.2195 −0.706900
\(210\) 0 0
\(211\) 12.3049 0.847104 0.423552 0.905872i \(-0.360783\pi\)
0.423552 + 0.905872i \(0.360783\pi\)
\(212\) 0 0
\(213\) −4.31035 −0.295340
\(214\) 0 0
\(215\) 15.8809 1.08307
\(216\) 0 0
\(217\) −1.62537 −0.110337
\(218\) 0 0
\(219\) −15.9625 −1.07865
\(220\) 0 0
\(221\) 22.3168 1.50119
\(222\) 0 0
\(223\) 12.7500 0.853802 0.426901 0.904298i \(-0.359605\pi\)
0.426901 + 0.904298i \(0.359605\pi\)
\(224\) 0 0
\(225\) −1.77734 −0.118489
\(226\) 0 0
\(227\) 2.47776 0.164455 0.0822275 0.996614i \(-0.473797\pi\)
0.0822275 + 0.996614i \(0.473797\pi\)
\(228\) 0 0
\(229\) −28.7123 −1.89736 −0.948681 0.316235i \(-0.897581\pi\)
−0.948681 + 0.316235i \(0.897581\pi\)
\(230\) 0 0
\(231\) 12.4634 0.820031
\(232\) 0 0
\(233\) −21.8851 −1.43374 −0.716872 0.697205i \(-0.754427\pi\)
−0.716872 + 0.697205i \(0.754427\pi\)
\(234\) 0 0
\(235\) 28.3934 1.85218
\(236\) 0 0
\(237\) 6.44743 0.418805
\(238\) 0 0
\(239\) −19.2567 −1.24561 −0.622807 0.782376i \(-0.714008\pi\)
−0.622807 + 0.782376i \(0.714008\pi\)
\(240\) 0 0
\(241\) −12.6860 −0.817178 −0.408589 0.912718i \(-0.633979\pi\)
−0.408589 + 0.912718i \(0.633979\pi\)
\(242\) 0 0
\(243\) 14.6195 0.937843
\(244\) 0 0
\(245\) −38.5830 −2.46498
\(246\) 0 0
\(247\) 27.0262 1.71964
\(248\) 0 0
\(249\) −6.87949 −0.435970
\(250\) 0 0
\(251\) 16.3196 1.03008 0.515041 0.857165i \(-0.327777\pi\)
0.515041 + 0.857165i \(0.327777\pi\)
\(252\) 0 0
\(253\) −10.3791 −0.652531
\(254\) 0 0
\(255\) −10.7960 −0.676070
\(256\) 0 0
\(257\) −3.18924 −0.198939 −0.0994697 0.995041i \(-0.531715\pi\)
−0.0994697 + 0.995041i \(0.531715\pi\)
\(258\) 0 0
\(259\) −15.1856 −0.943585
\(260\) 0 0
\(261\) −14.5005 −0.897558
\(262\) 0 0
\(263\) −15.6005 −0.961969 −0.480984 0.876729i \(-0.659721\pi\)
−0.480984 + 0.876729i \(0.659721\pi\)
\(264\) 0 0
\(265\) 19.5569 1.20137
\(266\) 0 0
\(267\) −6.51263 −0.398567
\(268\) 0 0
\(269\) −0.540868 −0.0329773 −0.0164887 0.999864i \(-0.505249\pi\)
−0.0164887 + 0.999864i \(0.505249\pi\)
\(270\) 0 0
\(271\) 7.35835 0.446988 0.223494 0.974705i \(-0.428254\pi\)
0.223494 + 0.974705i \(0.428254\pi\)
\(272\) 0 0
\(273\) −32.9602 −1.99484
\(274\) 0 0
\(275\) −2.43317 −0.146725
\(276\) 0 0
\(277\) 17.9367 1.07771 0.538856 0.842398i \(-0.318857\pi\)
0.538856 + 0.842398i \(0.318857\pi\)
\(278\) 0 0
\(279\) −0.560649 −0.0335652
\(280\) 0 0
\(281\) −17.8856 −1.06696 −0.533482 0.845811i \(-0.679117\pi\)
−0.533482 + 0.845811i \(0.679117\pi\)
\(282\) 0 0
\(283\) −18.2714 −1.08612 −0.543060 0.839694i \(-0.682734\pi\)
−0.543060 + 0.839694i \(0.682734\pi\)
\(284\) 0 0
\(285\) −13.0742 −0.774447
\(286\) 0 0
\(287\) −15.4202 −0.910225
\(288\) 0 0
\(289\) −2.89650 −0.170382
\(290\) 0 0
\(291\) 0.814993 0.0477757
\(292\) 0 0
\(293\) −5.38306 −0.314482 −0.157241 0.987560i \(-0.550260\pi\)
−0.157241 + 0.987560i \(0.550260\pi\)
\(294\) 0 0
\(295\) 12.4670 0.725856
\(296\) 0 0
\(297\) 12.1565 0.705394
\(298\) 0 0
\(299\) 27.4483 1.58737
\(300\) 0 0
\(301\) 30.6407 1.76610
\(302\) 0 0
\(303\) 3.54436 0.203618
\(304\) 0 0
\(305\) 1.72196 0.0985990
\(306\) 0 0
\(307\) −28.1274 −1.60532 −0.802659 0.596438i \(-0.796582\pi\)
−0.802659 + 0.596438i \(0.796582\pi\)
\(308\) 0 0
\(309\) 0.235493 0.0133967
\(310\) 0 0
\(311\) 25.9310 1.47041 0.735207 0.677843i \(-0.237085\pi\)
0.735207 + 0.677843i \(0.237085\pi\)
\(312\) 0 0
\(313\) 0.634096 0.0358412 0.0179206 0.999839i \(-0.494295\pi\)
0.0179206 + 0.999839i \(0.494295\pi\)
\(314\) 0 0
\(315\) −19.2638 −1.08539
\(316\) 0 0
\(317\) 23.9684 1.34620 0.673100 0.739552i \(-0.264962\pi\)
0.673100 + 0.739552i \(0.264962\pi\)
\(318\) 0 0
\(319\) −19.8511 −1.11145
\(320\) 0 0
\(321\) 14.3842 0.802847
\(322\) 0 0
\(323\) 17.0797 0.950338
\(324\) 0 0
\(325\) 6.43466 0.356931
\(326\) 0 0
\(327\) −14.5331 −0.803681
\(328\) 0 0
\(329\) 54.7822 3.02024
\(330\) 0 0
\(331\) 14.7751 0.812113 0.406057 0.913848i \(-0.366904\pi\)
0.406057 + 0.913848i \(0.366904\pi\)
\(332\) 0 0
\(333\) −5.23806 −0.287044
\(334\) 0 0
\(335\) 18.1139 0.989668
\(336\) 0 0
\(337\) 0.964632 0.0525468 0.0262734 0.999655i \(-0.491636\pi\)
0.0262734 + 0.999655i \(0.491636\pi\)
\(338\) 0 0
\(339\) 18.2434 0.990845
\(340\) 0 0
\(341\) −0.767524 −0.0415638
\(342\) 0 0
\(343\) −41.1322 −2.22093
\(344\) 0 0
\(345\) −13.2784 −0.714883
\(346\) 0 0
\(347\) −27.5766 −1.48039 −0.740196 0.672391i \(-0.765267\pi\)
−0.740196 + 0.672391i \(0.765267\pi\)
\(348\) 0 0
\(349\) −11.5461 −0.618051 −0.309025 0.951054i \(-0.600003\pi\)
−0.309025 + 0.951054i \(0.600003\pi\)
\(350\) 0 0
\(351\) −32.1487 −1.71597
\(352\) 0 0
\(353\) 14.8283 0.789232 0.394616 0.918846i \(-0.370878\pi\)
0.394616 + 0.918846i \(0.370878\pi\)
\(354\) 0 0
\(355\) 9.12051 0.484067
\(356\) 0 0
\(357\) −20.8298 −1.10243
\(358\) 0 0
\(359\) 24.3537 1.28534 0.642670 0.766143i \(-0.277827\pi\)
0.642670 + 0.766143i \(0.277827\pi\)
\(360\) 0 0
\(361\) 1.68387 0.0886247
\(362\) 0 0
\(363\) −6.93609 −0.364050
\(364\) 0 0
\(365\) 33.7760 1.76791
\(366\) 0 0
\(367\) −8.60265 −0.449055 −0.224527 0.974468i \(-0.572084\pi\)
−0.224527 + 0.974468i \(0.572084\pi\)
\(368\) 0 0
\(369\) −5.31899 −0.276895
\(370\) 0 0
\(371\) 37.7332 1.95901
\(372\) 0 0
\(373\) 20.7019 1.07190 0.535952 0.844249i \(-0.319953\pi\)
0.535952 + 0.844249i \(0.319953\pi\)
\(374\) 0 0
\(375\) 11.2609 0.581509
\(376\) 0 0
\(377\) 52.4973 2.70375
\(378\) 0 0
\(379\) −19.5638 −1.00493 −0.502463 0.864599i \(-0.667573\pi\)
−0.502463 + 0.864599i \(0.667573\pi\)
\(380\) 0 0
\(381\) 2.46687 0.126382
\(382\) 0 0
\(383\) 7.85034 0.401133 0.200567 0.979680i \(-0.435722\pi\)
0.200567 + 0.979680i \(0.435722\pi\)
\(384\) 0 0
\(385\) −26.3720 −1.34404
\(386\) 0 0
\(387\) 10.5691 0.537257
\(388\) 0 0
\(389\) −0.112618 −0.00570995 −0.00285497 0.999996i \(-0.500909\pi\)
−0.00285497 + 0.999996i \(0.500909\pi\)
\(390\) 0 0
\(391\) 17.3464 0.877246
\(392\) 0 0
\(393\) −16.2787 −0.821150
\(394\) 0 0
\(395\) −13.6425 −0.686427
\(396\) 0 0
\(397\) 11.8685 0.595664 0.297832 0.954618i \(-0.403736\pi\)
0.297832 + 0.954618i \(0.403736\pi\)
\(398\) 0 0
\(399\) −25.2253 −1.26285
\(400\) 0 0
\(401\) 9.57030 0.477918 0.238959 0.971030i \(-0.423194\pi\)
0.238959 + 0.971030i \(0.423194\pi\)
\(402\) 0 0
\(403\) 2.02976 0.101110
\(404\) 0 0
\(405\) 3.40751 0.169321
\(406\) 0 0
\(407\) −7.17086 −0.355446
\(408\) 0 0
\(409\) 12.7693 0.631402 0.315701 0.948859i \(-0.397760\pi\)
0.315701 + 0.948859i \(0.397760\pi\)
\(410\) 0 0
\(411\) 7.07160 0.348816
\(412\) 0 0
\(413\) 24.0538 1.18361
\(414\) 0 0
\(415\) 14.5567 0.714560
\(416\) 0 0
\(417\) 3.80067 0.186119
\(418\) 0 0
\(419\) −0.697466 −0.0340735 −0.0170367 0.999855i \(-0.505423\pi\)
−0.0170367 + 0.999855i \(0.505423\pi\)
\(420\) 0 0
\(421\) −32.5433 −1.58606 −0.793031 0.609181i \(-0.791498\pi\)
−0.793031 + 0.609181i \(0.791498\pi\)
\(422\) 0 0
\(423\) 18.8964 0.918774
\(424\) 0 0
\(425\) 4.06649 0.197254
\(426\) 0 0
\(427\) 3.32235 0.160780
\(428\) 0 0
\(429\) −15.5643 −0.751452
\(430\) 0 0
\(431\) −28.0235 −1.34984 −0.674922 0.737889i \(-0.735823\pi\)
−0.674922 + 0.737889i \(0.735823\pi\)
\(432\) 0 0
\(433\) 25.8263 1.24113 0.620566 0.784154i \(-0.286903\pi\)
0.620566 + 0.784154i \(0.286903\pi\)
\(434\) 0 0
\(435\) −25.3961 −1.21765
\(436\) 0 0
\(437\) 21.0069 1.00490
\(438\) 0 0
\(439\) −4.36867 −0.208505 −0.104253 0.994551i \(-0.533245\pi\)
−0.104253 + 0.994551i \(0.533245\pi\)
\(440\) 0 0
\(441\) −25.6778 −1.22275
\(442\) 0 0
\(443\) −28.3633 −1.34758 −0.673790 0.738923i \(-0.735335\pi\)
−0.673790 + 0.738923i \(0.735335\pi\)
\(444\) 0 0
\(445\) 13.7804 0.653256
\(446\) 0 0
\(447\) 15.9116 0.752592
\(448\) 0 0
\(449\) 9.30972 0.439353 0.219676 0.975573i \(-0.429500\pi\)
0.219676 + 0.975573i \(0.429500\pi\)
\(450\) 0 0
\(451\) −7.28165 −0.342880
\(452\) 0 0
\(453\) 14.9524 0.702523
\(454\) 0 0
\(455\) 69.7423 3.26957
\(456\) 0 0
\(457\) 12.9193 0.604337 0.302169 0.953254i \(-0.402289\pi\)
0.302169 + 0.953254i \(0.402289\pi\)
\(458\) 0 0
\(459\) −20.3169 −0.948314
\(460\) 0 0
\(461\) −23.2926 −1.08484 −0.542422 0.840106i \(-0.682493\pi\)
−0.542422 + 0.840106i \(0.682493\pi\)
\(462\) 0 0
\(463\) −16.5028 −0.766950 −0.383475 0.923551i \(-0.625273\pi\)
−0.383475 + 0.923551i \(0.625273\pi\)
\(464\) 0 0
\(465\) −0.981918 −0.0455354
\(466\) 0 0
\(467\) 26.6096 1.23134 0.615672 0.788003i \(-0.288885\pi\)
0.615672 + 0.788003i \(0.288885\pi\)
\(468\) 0 0
\(469\) 34.9490 1.61379
\(470\) 0 0
\(471\) −12.5365 −0.577652
\(472\) 0 0
\(473\) 14.4690 0.665285
\(474\) 0 0
\(475\) 4.92462 0.225957
\(476\) 0 0
\(477\) 13.0156 0.595942
\(478\) 0 0
\(479\) 1.33450 0.0609749 0.0304874 0.999535i \(-0.490294\pi\)
0.0304874 + 0.999535i \(0.490294\pi\)
\(480\) 0 0
\(481\) 18.9638 0.864674
\(482\) 0 0
\(483\) −25.6193 −1.16572
\(484\) 0 0
\(485\) −1.72449 −0.0783050
\(486\) 0 0
\(487\) −13.7797 −0.624419 −0.312209 0.950013i \(-0.601069\pi\)
−0.312209 + 0.950013i \(0.601069\pi\)
\(488\) 0 0
\(489\) −27.2474 −1.23217
\(490\) 0 0
\(491\) −19.6459 −0.886607 −0.443303 0.896372i \(-0.646194\pi\)
−0.443303 + 0.896372i \(0.646194\pi\)
\(492\) 0 0
\(493\) 33.1766 1.49420
\(494\) 0 0
\(495\) −9.09666 −0.408865
\(496\) 0 0
\(497\) 17.5971 0.789339
\(498\) 0 0
\(499\) 3.03872 0.136032 0.0680159 0.997684i \(-0.478333\pi\)
0.0680159 + 0.997684i \(0.478333\pi\)
\(500\) 0 0
\(501\) −4.13167 −0.184589
\(502\) 0 0
\(503\) −16.8119 −0.749606 −0.374803 0.927104i \(-0.622290\pi\)
−0.374803 + 0.927104i \(0.622290\pi\)
\(504\) 0 0
\(505\) −7.49971 −0.333733
\(506\) 0 0
\(507\) 26.0081 1.15506
\(508\) 0 0
\(509\) 32.0706 1.42150 0.710751 0.703444i \(-0.248355\pi\)
0.710751 + 0.703444i \(0.248355\pi\)
\(510\) 0 0
\(511\) 65.1674 2.88284
\(512\) 0 0
\(513\) −24.6043 −1.08631
\(514\) 0 0
\(515\) −0.498292 −0.0219574
\(516\) 0 0
\(517\) 25.8690 1.13772
\(518\) 0 0
\(519\) 0.657495 0.0288608
\(520\) 0 0
\(521\) −21.5125 −0.942481 −0.471241 0.882005i \(-0.656194\pi\)
−0.471241 + 0.882005i \(0.656194\pi\)
\(522\) 0 0
\(523\) −39.6958 −1.73578 −0.867888 0.496759i \(-0.834523\pi\)
−0.867888 + 0.496759i \(0.834523\pi\)
\(524\) 0 0
\(525\) −6.00589 −0.262119
\(526\) 0 0
\(527\) 1.28275 0.0558773
\(528\) 0 0
\(529\) −1.66501 −0.0723917
\(530\) 0 0
\(531\) 8.29704 0.360061
\(532\) 0 0
\(533\) 19.2568 0.834103
\(534\) 0 0
\(535\) −30.4363 −1.31588
\(536\) 0 0
\(537\) −17.6020 −0.759581
\(538\) 0 0
\(539\) −35.1527 −1.51413
\(540\) 0 0
\(541\) −15.0193 −0.645728 −0.322864 0.946445i \(-0.604646\pi\)
−0.322864 + 0.946445i \(0.604646\pi\)
\(542\) 0 0
\(543\) −4.38339 −0.188109
\(544\) 0 0
\(545\) 30.7513 1.31724
\(546\) 0 0
\(547\) −24.3162 −1.03969 −0.519844 0.854261i \(-0.674010\pi\)
−0.519844 + 0.854261i \(0.674010\pi\)
\(548\) 0 0
\(549\) 1.14600 0.0489101
\(550\) 0 0
\(551\) 40.1776 1.71162
\(552\) 0 0
\(553\) −26.3218 −1.11932
\(554\) 0 0
\(555\) −9.17391 −0.389411
\(556\) 0 0
\(557\) 44.9748 1.90564 0.952821 0.303533i \(-0.0981662\pi\)
0.952821 + 0.303533i \(0.0981662\pi\)
\(558\) 0 0
\(559\) −38.2641 −1.61840
\(560\) 0 0
\(561\) −9.83614 −0.415282
\(562\) 0 0
\(563\) 2.37978 0.100296 0.0501478 0.998742i \(-0.484031\pi\)
0.0501478 + 0.998742i \(0.484031\pi\)
\(564\) 0 0
\(565\) −38.6022 −1.62401
\(566\) 0 0
\(567\) 6.57446 0.276102
\(568\) 0 0
\(569\) 11.7376 0.492064 0.246032 0.969262i \(-0.420873\pi\)
0.246032 + 0.969262i \(0.420873\pi\)
\(570\) 0 0
\(571\) −5.38516 −0.225362 −0.112681 0.993631i \(-0.535944\pi\)
−0.112681 + 0.993631i \(0.535944\pi\)
\(572\) 0 0
\(573\) −4.99237 −0.208559
\(574\) 0 0
\(575\) 5.00153 0.208578
\(576\) 0 0
\(577\) −29.4400 −1.22560 −0.612801 0.790237i \(-0.709957\pi\)
−0.612801 + 0.790237i \(0.709957\pi\)
\(578\) 0 0
\(579\) −24.1752 −1.00469
\(580\) 0 0
\(581\) 28.0857 1.16519
\(582\) 0 0
\(583\) 17.8182 0.737954
\(584\) 0 0
\(585\) 24.0567 0.994621
\(586\) 0 0
\(587\) 13.2316 0.546128 0.273064 0.961996i \(-0.411963\pi\)
0.273064 + 0.961996i \(0.411963\pi\)
\(588\) 0 0
\(589\) 1.55343 0.0640081
\(590\) 0 0
\(591\) −14.5835 −0.599883
\(592\) 0 0
\(593\) −14.2681 −0.585922 −0.292961 0.956124i \(-0.594641\pi\)
−0.292961 + 0.956124i \(0.594641\pi\)
\(594\) 0 0
\(595\) 44.0749 1.80689
\(596\) 0 0
\(597\) 22.6514 0.927062
\(598\) 0 0
\(599\) −16.8467 −0.688339 −0.344169 0.938908i \(-0.611839\pi\)
−0.344169 + 0.938908i \(0.611839\pi\)
\(600\) 0 0
\(601\) −22.9982 −0.938118 −0.469059 0.883167i \(-0.655407\pi\)
−0.469059 + 0.883167i \(0.655407\pi\)
\(602\) 0 0
\(603\) 12.0552 0.490925
\(604\) 0 0
\(605\) 14.6765 0.596683
\(606\) 0 0
\(607\) −34.4746 −1.39928 −0.699641 0.714495i \(-0.746657\pi\)
−0.699641 + 0.714495i \(0.746657\pi\)
\(608\) 0 0
\(609\) −48.9992 −1.98555
\(610\) 0 0
\(611\) −68.4122 −2.76766
\(612\) 0 0
\(613\) −16.1589 −0.652653 −0.326326 0.945257i \(-0.605811\pi\)
−0.326326 + 0.945257i \(0.605811\pi\)
\(614\) 0 0
\(615\) −9.31565 −0.375643
\(616\) 0 0
\(617\) 27.7521 1.11726 0.558628 0.829419i \(-0.311328\pi\)
0.558628 + 0.829419i \(0.311328\pi\)
\(618\) 0 0
\(619\) −38.2780 −1.53852 −0.769261 0.638935i \(-0.779375\pi\)
−0.769261 + 0.638935i \(0.779375\pi\)
\(620\) 0 0
\(621\) −24.9886 −1.00276
\(622\) 0 0
\(623\) 26.5880 1.06523
\(624\) 0 0
\(625\) −29.2416 −1.16966
\(626\) 0 0
\(627\) −11.9118 −0.475711
\(628\) 0 0
\(629\) 11.9845 0.477853
\(630\) 0 0
\(631\) 26.4958 1.05478 0.527392 0.849622i \(-0.323170\pi\)
0.527392 + 0.849622i \(0.323170\pi\)
\(632\) 0 0
\(633\) 14.3425 0.570062
\(634\) 0 0
\(635\) −5.21979 −0.207141
\(636\) 0 0
\(637\) 92.9635 3.68335
\(638\) 0 0
\(639\) 6.06989 0.240121
\(640\) 0 0
\(641\) −19.3438 −0.764036 −0.382018 0.924155i \(-0.624771\pi\)
−0.382018 + 0.924155i \(0.624771\pi\)
\(642\) 0 0
\(643\) −31.2614 −1.23283 −0.616414 0.787422i \(-0.711415\pi\)
−0.616414 + 0.787422i \(0.711415\pi\)
\(644\) 0 0
\(645\) 18.5106 0.728856
\(646\) 0 0
\(647\) 11.0023 0.432547 0.216273 0.976333i \(-0.430610\pi\)
0.216273 + 0.976333i \(0.430610\pi\)
\(648\) 0 0
\(649\) 11.3586 0.445863
\(650\) 0 0
\(651\) −1.89451 −0.0742519
\(652\) 0 0
\(653\) 2.27287 0.0889442 0.0444721 0.999011i \(-0.485839\pi\)
0.0444721 + 0.999011i \(0.485839\pi\)
\(654\) 0 0
\(655\) 34.4450 1.34588
\(656\) 0 0
\(657\) 22.4786 0.876974
\(658\) 0 0
\(659\) 23.6347 0.920676 0.460338 0.887744i \(-0.347728\pi\)
0.460338 + 0.887744i \(0.347728\pi\)
\(660\) 0 0
\(661\) 32.3110 1.25675 0.628377 0.777909i \(-0.283720\pi\)
0.628377 + 0.777909i \(0.283720\pi\)
\(662\) 0 0
\(663\) 26.0123 1.01023
\(664\) 0 0
\(665\) 53.3757 2.06982
\(666\) 0 0
\(667\) 40.8051 1.57998
\(668\) 0 0
\(669\) 14.8612 0.574569
\(670\) 0 0
\(671\) 1.56886 0.0605653
\(672\) 0 0
\(673\) −14.2022 −0.547453 −0.273726 0.961808i \(-0.588256\pi\)
−0.273726 + 0.961808i \(0.588256\pi\)
\(674\) 0 0
\(675\) −5.85803 −0.225476
\(676\) 0 0
\(677\) 49.7955 1.91380 0.956899 0.290422i \(-0.0937958\pi\)
0.956899 + 0.290422i \(0.0937958\pi\)
\(678\) 0 0
\(679\) −3.32723 −0.127687
\(680\) 0 0
\(681\) 2.88806 0.110671
\(682\) 0 0
\(683\) −17.6202 −0.674220 −0.337110 0.941465i \(-0.609449\pi\)
−0.337110 + 0.941465i \(0.609449\pi\)
\(684\) 0 0
\(685\) −14.9632 −0.571714
\(686\) 0 0
\(687\) −33.4668 −1.27684
\(688\) 0 0
\(689\) −47.1213 −1.79518
\(690\) 0 0
\(691\) 41.6482 1.58437 0.792186 0.610279i \(-0.208943\pi\)
0.792186 + 0.610279i \(0.208943\pi\)
\(692\) 0 0
\(693\) −17.5511 −0.666712
\(694\) 0 0
\(695\) −8.04204 −0.305052
\(696\) 0 0
\(697\) 12.1697 0.460958
\(698\) 0 0
\(699\) −25.5091 −0.964843
\(700\) 0 0
\(701\) −16.8494 −0.636392 −0.318196 0.948025i \(-0.603077\pi\)
−0.318196 + 0.948025i \(0.603077\pi\)
\(702\) 0 0
\(703\) 14.5135 0.547387
\(704\) 0 0
\(705\) 33.0950 1.24643
\(706\) 0 0
\(707\) −14.4699 −0.544198
\(708\) 0 0
\(709\) 31.5679 1.18556 0.592779 0.805365i \(-0.298031\pi\)
0.592779 + 0.805365i \(0.298031\pi\)
\(710\) 0 0
\(711\) −9.07935 −0.340502
\(712\) 0 0
\(713\) 1.57770 0.0590851
\(714\) 0 0
\(715\) 32.9334 1.23164
\(716\) 0 0
\(717\) −22.4454 −0.838240
\(718\) 0 0
\(719\) −28.4117 −1.05958 −0.529788 0.848130i \(-0.677729\pi\)
−0.529788 + 0.848130i \(0.677729\pi\)
\(720\) 0 0
\(721\) −0.961406 −0.0358046
\(722\) 0 0
\(723\) −14.7867 −0.549923
\(724\) 0 0
\(725\) 9.56588 0.355268
\(726\) 0 0
\(727\) −0.740528 −0.0274647 −0.0137323 0.999906i \(-0.504371\pi\)
−0.0137323 + 0.999906i \(0.504371\pi\)
\(728\) 0 0
\(729\) 21.1852 0.784637
\(730\) 0 0
\(731\) −24.1817 −0.894392
\(732\) 0 0
\(733\) 48.4611 1.78995 0.894976 0.446113i \(-0.147192\pi\)
0.894976 + 0.446113i \(0.147192\pi\)
\(734\) 0 0
\(735\) −44.9720 −1.65882
\(736\) 0 0
\(737\) 16.5035 0.607912
\(738\) 0 0
\(739\) −43.6186 −1.60454 −0.802268 0.596964i \(-0.796374\pi\)
−0.802268 + 0.596964i \(0.796374\pi\)
\(740\) 0 0
\(741\) 31.5015 1.15723
\(742\) 0 0
\(743\) 9.11821 0.334515 0.167257 0.985913i \(-0.446509\pi\)
0.167257 + 0.985913i \(0.446509\pi\)
\(744\) 0 0
\(745\) −33.6682 −1.23351
\(746\) 0 0
\(747\) 9.68778 0.354458
\(748\) 0 0
\(749\) −58.7238 −2.14572
\(750\) 0 0
\(751\) −1.00000 −0.0364905
\(752\) 0 0
\(753\) 19.0219 0.693198
\(754\) 0 0
\(755\) −31.6385 −1.15144
\(756\) 0 0
\(757\) −40.6682 −1.47811 −0.739055 0.673645i \(-0.764728\pi\)
−0.739055 + 0.673645i \(0.764728\pi\)
\(758\) 0 0
\(759\) −12.0978 −0.439123
\(760\) 0 0
\(761\) 8.68141 0.314701 0.157350 0.987543i \(-0.449705\pi\)
0.157350 + 0.987543i \(0.449705\pi\)
\(762\) 0 0
\(763\) 59.3317 2.14795
\(764\) 0 0
\(765\) 15.2030 0.549667
\(766\) 0 0
\(767\) −30.0385 −1.08463
\(768\) 0 0
\(769\) −22.3304 −0.805254 −0.402627 0.915364i \(-0.631903\pi\)
−0.402627 + 0.915364i \(0.631903\pi\)
\(770\) 0 0
\(771\) −3.71735 −0.133877
\(772\) 0 0
\(773\) −49.6069 −1.78424 −0.892119 0.451801i \(-0.850782\pi\)
−0.892119 + 0.451801i \(0.850782\pi\)
\(774\) 0 0
\(775\) 0.369857 0.0132856
\(776\) 0 0
\(777\) −17.7002 −0.634989
\(778\) 0 0
\(779\) 14.7377 0.528034
\(780\) 0 0
\(781\) 8.30964 0.297342
\(782\) 0 0
\(783\) −47.7929 −1.70798
\(784\) 0 0
\(785\) 26.5267 0.946778
\(786\) 0 0
\(787\) −11.5200 −0.410643 −0.205321 0.978695i \(-0.565824\pi\)
−0.205321 + 0.978695i \(0.565824\pi\)
\(788\) 0 0
\(789\) −18.1838 −0.647361
\(790\) 0 0
\(791\) −74.4791 −2.64817
\(792\) 0 0
\(793\) −4.14896 −0.147334
\(794\) 0 0
\(795\) 22.7954 0.808469
\(796\) 0 0
\(797\) 12.9665 0.459296 0.229648 0.973274i \(-0.426243\pi\)
0.229648 + 0.973274i \(0.426243\pi\)
\(798\) 0 0
\(799\) −43.2342 −1.52952
\(800\) 0 0
\(801\) 9.17118 0.324048
\(802\) 0 0
\(803\) 30.7730 1.08596
\(804\) 0 0
\(805\) 54.2093 1.91063
\(806\) 0 0
\(807\) −0.630431 −0.0221922
\(808\) 0 0
\(809\) −44.5479 −1.56622 −0.783110 0.621883i \(-0.786368\pi\)
−0.783110 + 0.621883i \(0.786368\pi\)
\(810\) 0 0
\(811\) −3.82599 −0.134349 −0.0671744 0.997741i \(-0.521398\pi\)
−0.0671744 + 0.997741i \(0.521398\pi\)
\(812\) 0 0
\(813\) 8.57682 0.300802
\(814\) 0 0
\(815\) 57.6543 2.01954
\(816\) 0 0
\(817\) −29.2846 −1.02454
\(818\) 0 0
\(819\) 46.4150 1.62187
\(820\) 0 0
\(821\) −15.0631 −0.525707 −0.262853 0.964836i \(-0.584664\pi\)
−0.262853 + 0.964836i \(0.584664\pi\)
\(822\) 0 0
\(823\) −31.9424 −1.11344 −0.556722 0.830699i \(-0.687941\pi\)
−0.556722 + 0.830699i \(0.687941\pi\)
\(824\) 0 0
\(825\) −2.83607 −0.0987394
\(826\) 0 0
\(827\) −56.4566 −1.96319 −0.981595 0.190974i \(-0.938835\pi\)
−0.981595 + 0.190974i \(0.938835\pi\)
\(828\) 0 0
\(829\) −35.8891 −1.24648 −0.623241 0.782030i \(-0.714184\pi\)
−0.623241 + 0.782030i \(0.714184\pi\)
\(830\) 0 0
\(831\) 20.9068 0.725250
\(832\) 0 0
\(833\) 58.7499 2.03556
\(834\) 0 0
\(835\) 8.74244 0.302544
\(836\) 0 0
\(837\) −1.84787 −0.0638718
\(838\) 0 0
\(839\) −22.2955 −0.769727 −0.384864 0.922973i \(-0.625752\pi\)
−0.384864 + 0.922973i \(0.625752\pi\)
\(840\) 0 0
\(841\) 49.0435 1.69116
\(842\) 0 0
\(843\) −20.8473 −0.718017
\(844\) 0 0
\(845\) −55.0320 −1.89316
\(846\) 0 0
\(847\) 28.3168 0.972976
\(848\) 0 0
\(849\) −21.2969 −0.730908
\(850\) 0 0
\(851\) 14.7402 0.505286
\(852\) 0 0
\(853\) 33.0277 1.13085 0.565423 0.824801i \(-0.308713\pi\)
0.565423 + 0.824801i \(0.308713\pi\)
\(854\) 0 0
\(855\) 18.4112 0.629651
\(856\) 0 0
\(857\) −43.4712 −1.48495 −0.742474 0.669875i \(-0.766348\pi\)
−0.742474 + 0.669875i \(0.766348\pi\)
\(858\) 0 0
\(859\) 21.5741 0.736100 0.368050 0.929806i \(-0.380026\pi\)
0.368050 + 0.929806i \(0.380026\pi\)
\(860\) 0 0
\(861\) −17.9736 −0.612539
\(862\) 0 0
\(863\) 28.7269 0.977874 0.488937 0.872319i \(-0.337385\pi\)
0.488937 + 0.872319i \(0.337385\pi\)
\(864\) 0 0
\(865\) −1.39123 −0.0473033
\(866\) 0 0
\(867\) −3.37613 −0.114659
\(868\) 0 0
\(869\) −12.4296 −0.421644
\(870\) 0 0
\(871\) −43.6444 −1.47883
\(872\) 0 0
\(873\) −1.14768 −0.0388432
\(874\) 0 0
\(875\) −45.9728 −1.55416
\(876\) 0 0
\(877\) −43.7197 −1.47631 −0.738154 0.674632i \(-0.764302\pi\)
−0.738154 + 0.674632i \(0.764302\pi\)
\(878\) 0 0
\(879\) −6.27444 −0.211632
\(880\) 0 0
\(881\) 2.03101 0.0684263 0.0342132 0.999415i \(-0.489107\pi\)
0.0342132 + 0.999415i \(0.489107\pi\)
\(882\) 0 0
\(883\) 29.5237 0.993551 0.496776 0.867879i \(-0.334517\pi\)
0.496776 + 0.867879i \(0.334517\pi\)
\(884\) 0 0
\(885\) 14.5314 0.488468
\(886\) 0 0
\(887\) 46.8644 1.57355 0.786776 0.617238i \(-0.211749\pi\)
0.786776 + 0.617238i \(0.211749\pi\)
\(888\) 0 0
\(889\) −10.0711 −0.337773
\(890\) 0 0
\(891\) 3.10456 0.104007
\(892\) 0 0
\(893\) −52.3577 −1.75208
\(894\) 0 0
\(895\) 37.2450 1.24496
\(896\) 0 0
\(897\) 31.9934 1.06823
\(898\) 0 0
\(899\) 3.01749 0.100639
\(900\) 0 0
\(901\) −29.7791 −0.992087
\(902\) 0 0
\(903\) 35.7145 1.18850
\(904\) 0 0
\(905\) 9.27507 0.308314
\(906\) 0 0
\(907\) 8.64308 0.286989 0.143494 0.989651i \(-0.454166\pi\)
0.143494 + 0.989651i \(0.454166\pi\)
\(908\) 0 0
\(909\) −4.99121 −0.165548
\(910\) 0 0
\(911\) 5.72164 0.189566 0.0947832 0.995498i \(-0.469784\pi\)
0.0947832 + 0.995498i \(0.469784\pi\)
\(912\) 0 0
\(913\) 13.2625 0.438925
\(914\) 0 0
\(915\) 2.00710 0.0663526
\(916\) 0 0
\(917\) 66.4581 2.19464
\(918\) 0 0
\(919\) 16.9977 0.560703 0.280351 0.959897i \(-0.409549\pi\)
0.280351 + 0.959897i \(0.409549\pi\)
\(920\) 0 0
\(921\) −32.7851 −1.08030
\(922\) 0 0
\(923\) −21.9753 −0.723327
\(924\) 0 0
\(925\) 3.45551 0.113617
\(926\) 0 0
\(927\) −0.331624 −0.0108920
\(928\) 0 0
\(929\) −20.1225 −0.660199 −0.330099 0.943946i \(-0.607082\pi\)
−0.330099 + 0.943946i \(0.607082\pi\)
\(930\) 0 0
\(931\) 71.1475 2.33176
\(932\) 0 0
\(933\) 30.2249 0.989520
\(934\) 0 0
\(935\) 20.8128 0.680652
\(936\) 0 0
\(937\) 38.4577 1.25636 0.628179 0.778069i \(-0.283801\pi\)
0.628179 + 0.778069i \(0.283801\pi\)
\(938\) 0 0
\(939\) 0.739096 0.0241195
\(940\) 0 0
\(941\) 38.6646 1.26043 0.630215 0.776421i \(-0.282967\pi\)
0.630215 + 0.776421i \(0.282967\pi\)
\(942\) 0 0
\(943\) 14.9679 0.487422
\(944\) 0 0
\(945\) −63.4925 −2.06541
\(946\) 0 0
\(947\) 28.9184 0.939722 0.469861 0.882740i \(-0.344304\pi\)
0.469861 + 0.882740i \(0.344304\pi\)
\(948\) 0 0
\(949\) −81.3812 −2.64174
\(950\) 0 0
\(951\) 27.9373 0.905930
\(952\) 0 0
\(953\) 11.0740 0.358721 0.179361 0.983783i \(-0.442597\pi\)
0.179361 + 0.983783i \(0.442597\pi\)
\(954\) 0 0
\(955\) 10.5636 0.341831
\(956\) 0 0
\(957\) −23.1382 −0.747952
\(958\) 0 0
\(959\) −28.8700 −0.932261
\(960\) 0 0
\(961\) −30.8833 −0.996236
\(962\) 0 0
\(963\) −20.2560 −0.652740
\(964\) 0 0
\(965\) 51.1537 1.64669
\(966\) 0 0
\(967\) −19.9193 −0.640562 −0.320281 0.947323i \(-0.603777\pi\)
−0.320281 + 0.947323i \(0.603777\pi\)
\(968\) 0 0
\(969\) 19.9079 0.639534
\(970\) 0 0
\(971\) −32.3540 −1.03829 −0.519144 0.854687i \(-0.673749\pi\)
−0.519144 + 0.854687i \(0.673749\pi\)
\(972\) 0 0
\(973\) −15.5163 −0.497431
\(974\) 0 0
\(975\) 7.50017 0.240198
\(976\) 0 0
\(977\) 20.7839 0.664935 0.332468 0.943115i \(-0.392119\pi\)
0.332468 + 0.943115i \(0.392119\pi\)
\(978\) 0 0
\(979\) 12.5553 0.401268
\(980\) 0 0
\(981\) 20.4657 0.653418
\(982\) 0 0
\(983\) −56.6633 −1.80728 −0.903639 0.428295i \(-0.859114\pi\)
−0.903639 + 0.428295i \(0.859114\pi\)
\(984\) 0 0
\(985\) 30.8580 0.983216
\(986\) 0 0
\(987\) 63.8536 2.03248
\(988\) 0 0
\(989\) −29.7419 −0.945739
\(990\) 0 0
\(991\) −47.7786 −1.51774 −0.758869 0.651244i \(-0.774248\pi\)
−0.758869 + 0.651244i \(0.774248\pi\)
\(992\) 0 0
\(993\) 17.2217 0.546515
\(994\) 0 0
\(995\) −47.9295 −1.51947
\(996\) 0 0
\(997\) 48.4167 1.53337 0.766686 0.642022i \(-0.221904\pi\)
0.766686 + 0.642022i \(0.221904\pi\)
\(998\) 0 0
\(999\) −17.2644 −0.546221
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 6008.2.a.b.1.31 44
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6008.2.a.b.1.31 44 1.1 even 1 trivial