Properties

Label 464.4.a.f.1.2
Level $464$
Weight $4$
Character 464.1
Self dual yes
Analytic conductor $27.377$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 464 = 2^{4} \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 464.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(27.3768862427\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 29)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 464.1

$q$-expansion

\(f(q)\) \(=\) \(q+9.24264 q^{3} +0.656854 q^{5} -6.14214 q^{7} +58.4264 q^{9} +O(q^{10})\) \(q+9.24264 q^{3} +0.656854 q^{5} -6.14214 q^{7} +58.4264 q^{9} +65.3259 q^{11} -49.7696 q^{13} +6.07107 q^{15} +55.4558 q^{17} +64.7452 q^{19} -56.7696 q^{21} -93.8823 q^{23} -124.569 q^{25} +290.463 q^{27} +29.0000 q^{29} +236.095 q^{31} +603.784 q^{33} -4.03449 q^{35} +76.8040 q^{37} -460.002 q^{39} +215.161 q^{41} -80.8305 q^{43} +38.3776 q^{45} +357.742 q^{47} -305.274 q^{49} +512.558 q^{51} +328.466 q^{53} +42.9096 q^{55} +598.416 q^{57} +99.2750 q^{59} -725.730 q^{61} -358.863 q^{63} -32.6913 q^{65} -844.479 q^{67} -867.720 q^{69} +378.083 q^{71} -581.097 q^{73} -1151.34 q^{75} -401.241 q^{77} +353.247 q^{79} +1107.13 q^{81} -696.510 q^{83} +36.4264 q^{85} +268.037 q^{87} +1118.22 q^{89} +305.691 q^{91} +2182.15 q^{93} +42.5281 q^{95} -805.415 q^{97} +3816.76 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 10 q^{3} - 10 q^{5} + 16 q^{7} + 32 q^{9} + O(q^{10}) \) \( 2 q + 10 q^{3} - 10 q^{5} + 16 q^{7} + 32 q^{9} + 26 q^{11} - 26 q^{13} - 2 q^{15} + 60 q^{17} + 220 q^{19} - 40 q^{21} - 52 q^{23} - 136 q^{25} + 250 q^{27} + 58 q^{29} + 294 q^{31} + 574 q^{33} - 240 q^{35} + 312 q^{37} - 442 q^{39} + 40 q^{41} + 322 q^{43} + 320 q^{45} + 130 q^{47} - 158 q^{49} + 516 q^{51} + 1002 q^{53} + 462 q^{55} + 716 q^{57} + 900 q^{59} - 948 q^{61} - 944 q^{63} - 286 q^{65} - 320 q^{67} - 836 q^{69} + 660 q^{71} + 648 q^{73} - 1160 q^{75} - 1272 q^{77} - 258 q^{79} + 1790 q^{81} - 1212 q^{83} - 12 q^{85} + 290 q^{87} + 760 q^{89} + 832 q^{91} + 2226 q^{93} - 1612 q^{95} + 24 q^{97} + 4856 q^{99} + O(q^{100}) \)

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
\(3\) 9.24264 1.77875 0.889374 0.457181i \(-0.151141\pi\)
0.889374 + 0.457181i \(0.151141\pi\)
\(4\) 0 0
\(5\) 0.656854 0.0587508 0.0293754 0.999568i \(-0.490648\pi\)
0.0293754 + 0.999568i \(0.490648\pi\)
\(6\) 0 0
\(7\) −6.14214 −0.331644 −0.165822 0.986156i \(-0.553028\pi\)
−0.165822 + 0.986156i \(0.553028\pi\)
\(8\) 0 0
\(9\) 58.4264 2.16394
\(10\) 0 0
\(11\) 65.3259 1.79059 0.895295 0.445473i \(-0.146964\pi\)
0.895295 + 0.445473i \(0.146964\pi\)
\(12\) 0 0
\(13\) −49.7696 −1.06181 −0.530907 0.847430i \(-0.678149\pi\)
−0.530907 + 0.847430i \(0.678149\pi\)
\(14\) 0 0
\(15\) 6.07107 0.104503
\(16\) 0 0
\(17\) 55.4558 0.791178 0.395589 0.918428i \(-0.370541\pi\)
0.395589 + 0.918428i \(0.370541\pi\)
\(18\) 0 0
\(19\) 64.7452 0.781766 0.390883 0.920440i \(-0.372170\pi\)
0.390883 + 0.920440i \(0.372170\pi\)
\(20\) 0 0
\(21\) −56.7696 −0.589911
\(22\) 0 0
\(23\) −93.8823 −0.851122 −0.425561 0.904930i \(-0.639923\pi\)
−0.425561 + 0.904930i \(0.639923\pi\)
\(24\) 0 0
\(25\) −124.569 −0.996548
\(26\) 0 0
\(27\) 290.463 2.07036
\(28\) 0 0
\(29\) 29.0000 0.185695
\(30\) 0 0
\(31\) 236.095 1.36787 0.683935 0.729543i \(-0.260267\pi\)
0.683935 + 0.729543i \(0.260267\pi\)
\(32\) 0 0
\(33\) 603.784 3.18501
\(34\) 0 0
\(35\) −4.03449 −0.0194844
\(36\) 0 0
\(37\) 76.8040 0.341257 0.170628 0.985335i \(-0.445420\pi\)
0.170628 + 0.985335i \(0.445420\pi\)
\(38\) 0 0
\(39\) −460.002 −1.88870
\(40\) 0 0
\(41\) 215.161 0.819575 0.409788 0.912181i \(-0.365603\pi\)
0.409788 + 0.912181i \(0.365603\pi\)
\(42\) 0 0
\(43\) −80.8305 −0.286664 −0.143332 0.989675i \(-0.545782\pi\)
−0.143332 + 0.989675i \(0.545782\pi\)
\(44\) 0 0
\(45\) 38.3776 0.127133
\(46\) 0 0
\(47\) 357.742 1.11026 0.555128 0.831765i \(-0.312669\pi\)
0.555128 + 0.831765i \(0.312669\pi\)
\(48\) 0 0
\(49\) −305.274 −0.890012
\(50\) 0 0
\(51\) 512.558 1.40730
\(52\) 0 0
\(53\) 328.466 0.851288 0.425644 0.904891i \(-0.360048\pi\)
0.425644 + 0.904891i \(0.360048\pi\)
\(54\) 0 0
\(55\) 42.9096 0.105199
\(56\) 0 0
\(57\) 598.416 1.39056
\(58\) 0 0
\(59\) 99.2750 0.219059 0.109530 0.993984i \(-0.465066\pi\)
0.109530 + 0.993984i \(0.465066\pi\)
\(60\) 0 0
\(61\) −725.730 −1.52328 −0.761641 0.647999i \(-0.775606\pi\)
−0.761641 + 0.647999i \(0.775606\pi\)
\(62\) 0 0
\(63\) −358.863 −0.717658
\(64\) 0 0
\(65\) −32.6913 −0.0623825
\(66\) 0 0
\(67\) −844.479 −1.53984 −0.769922 0.638138i \(-0.779705\pi\)
−0.769922 + 0.638138i \(0.779705\pi\)
\(68\) 0 0
\(69\) −867.720 −1.51393
\(70\) 0 0
\(71\) 378.083 0.631975 0.315988 0.948763i \(-0.397664\pi\)
0.315988 + 0.948763i \(0.397664\pi\)
\(72\) 0 0
\(73\) −581.097 −0.931674 −0.465837 0.884870i \(-0.654247\pi\)
−0.465837 + 0.884870i \(0.654247\pi\)
\(74\) 0 0
\(75\) −1151.34 −1.77261
\(76\) 0 0
\(77\) −401.241 −0.593839
\(78\) 0 0
\(79\) 353.247 0.503081 0.251540 0.967847i \(-0.419063\pi\)
0.251540 + 0.967847i \(0.419063\pi\)
\(80\) 0 0
\(81\) 1107.13 1.51870
\(82\) 0 0
\(83\) −696.510 −0.921107 −0.460553 0.887632i \(-0.652349\pi\)
−0.460553 + 0.887632i \(0.652349\pi\)
\(84\) 0 0
\(85\) 36.4264 0.0464823
\(86\) 0 0
\(87\) 268.037 0.330305
\(88\) 0 0
\(89\) 1118.22 1.33181 0.665905 0.746037i \(-0.268046\pi\)
0.665905 + 0.746037i \(0.268046\pi\)
\(90\) 0 0
\(91\) 305.691 0.352145
\(92\) 0 0
\(93\) 2182.15 2.43310
\(94\) 0 0
\(95\) 42.5281 0.0459294
\(96\) 0 0
\(97\) −805.415 −0.843068 −0.421534 0.906813i \(-0.638508\pi\)
−0.421534 + 0.906813i \(0.638508\pi\)
\(98\) 0 0
\(99\) 3816.76 3.87473
\(100\) 0 0
\(101\) −1373.99 −1.35363 −0.676817 0.736151i \(-0.736641\pi\)
−0.676817 + 0.736151i \(0.736641\pi\)
\(102\) 0 0
\(103\) 634.672 0.607147 0.303573 0.952808i \(-0.401820\pi\)
0.303573 + 0.952808i \(0.401820\pi\)
\(104\) 0 0
\(105\) −37.2893 −0.0346578
\(106\) 0 0
\(107\) 180.956 0.163493 0.0817463 0.996653i \(-0.473950\pi\)
0.0817463 + 0.996653i \(0.473950\pi\)
\(108\) 0 0
\(109\) 1038.14 0.912251 0.456125 0.889916i \(-0.349237\pi\)
0.456125 + 0.889916i \(0.349237\pi\)
\(110\) 0 0
\(111\) 709.872 0.607010
\(112\) 0 0
\(113\) −184.765 −0.153816 −0.0769082 0.997038i \(-0.524505\pi\)
−0.0769082 + 0.997038i \(0.524505\pi\)
\(114\) 0 0
\(115\) −61.6670 −0.0500041
\(116\) 0 0
\(117\) −2907.86 −2.29770
\(118\) 0 0
\(119\) −340.617 −0.262389
\(120\) 0 0
\(121\) 2936.47 2.20622
\(122\) 0 0
\(123\) 1988.66 1.45782
\(124\) 0 0
\(125\) −163.930 −0.117299
\(126\) 0 0
\(127\) −1999.58 −1.39712 −0.698558 0.715554i \(-0.746175\pi\)
−0.698558 + 0.715554i \(0.746175\pi\)
\(128\) 0 0
\(129\) −747.087 −0.509902
\(130\) 0 0
\(131\) −561.468 −0.374471 −0.187236 0.982315i \(-0.559953\pi\)
−0.187236 + 0.982315i \(0.559953\pi\)
\(132\) 0 0
\(133\) −397.674 −0.259268
\(134\) 0 0
\(135\) 190.792 0.121635
\(136\) 0 0
\(137\) −250.489 −0.156210 −0.0781050 0.996945i \(-0.524887\pi\)
−0.0781050 + 0.996945i \(0.524887\pi\)
\(138\) 0 0
\(139\) 242.244 0.147819 0.0739096 0.997265i \(-0.476452\pi\)
0.0739096 + 0.997265i \(0.476452\pi\)
\(140\) 0 0
\(141\) 3306.48 1.97487
\(142\) 0 0
\(143\) −3251.24 −1.90128
\(144\) 0 0
\(145\) 19.0488 0.0109098
\(146\) 0 0
\(147\) −2821.54 −1.58311
\(148\) 0 0
\(149\) −1632.63 −0.897652 −0.448826 0.893619i \(-0.648158\pi\)
−0.448826 + 0.893619i \(0.648158\pi\)
\(150\) 0 0
\(151\) 121.582 0.0655245 0.0327623 0.999463i \(-0.489570\pi\)
0.0327623 + 0.999463i \(0.489570\pi\)
\(152\) 0 0
\(153\) 3240.09 1.71206
\(154\) 0 0
\(155\) 155.080 0.0803635
\(156\) 0 0
\(157\) −753.163 −0.382860 −0.191430 0.981506i \(-0.561312\pi\)
−0.191430 + 0.981506i \(0.561312\pi\)
\(158\) 0 0
\(159\) 3035.89 1.51423
\(160\) 0 0
\(161\) 576.638 0.282270
\(162\) 0 0
\(163\) −537.917 −0.258484 −0.129242 0.991613i \(-0.541254\pi\)
−0.129242 + 0.991613i \(0.541254\pi\)
\(164\) 0 0
\(165\) 396.598 0.187122
\(166\) 0 0
\(167\) −484.613 −0.224554 −0.112277 0.993677i \(-0.535814\pi\)
−0.112277 + 0.993677i \(0.535814\pi\)
\(168\) 0 0
\(169\) 280.008 0.127450
\(170\) 0 0
\(171\) 3782.83 1.69170
\(172\) 0 0
\(173\) −3269.70 −1.43694 −0.718469 0.695559i \(-0.755157\pi\)
−0.718469 + 0.695559i \(0.755157\pi\)
\(174\) 0 0
\(175\) 765.117 0.330499
\(176\) 0 0
\(177\) 917.563 0.389651
\(178\) 0 0
\(179\) 562.267 0.234781 0.117390 0.993086i \(-0.462547\pi\)
0.117390 + 0.993086i \(0.462547\pi\)
\(180\) 0 0
\(181\) −1507.32 −0.618998 −0.309499 0.950900i \(-0.600161\pi\)
−0.309499 + 0.950900i \(0.600161\pi\)
\(182\) 0 0
\(183\) −6707.66 −2.70953
\(184\) 0 0
\(185\) 50.4491 0.0200491
\(186\) 0 0
\(187\) 3622.70 1.41668
\(188\) 0 0
\(189\) −1784.06 −0.686622
\(190\) 0 0
\(191\) −4532.00 −1.71688 −0.858439 0.512915i \(-0.828566\pi\)
−0.858439 + 0.512915i \(0.828566\pi\)
\(192\) 0 0
\(193\) −2935.17 −1.09471 −0.547353 0.836902i \(-0.684364\pi\)
−0.547353 + 0.836902i \(0.684364\pi\)
\(194\) 0 0
\(195\) −302.154 −0.110963
\(196\) 0 0
\(197\) 2682.20 0.970043 0.485022 0.874502i \(-0.338812\pi\)
0.485022 + 0.874502i \(0.338812\pi\)
\(198\) 0 0
\(199\) 648.376 0.230966 0.115483 0.993309i \(-0.463158\pi\)
0.115483 + 0.993309i \(0.463158\pi\)
\(200\) 0 0
\(201\) −7805.22 −2.73899
\(202\) 0 0
\(203\) −178.122 −0.0615848
\(204\) 0 0
\(205\) 141.330 0.0481507
\(206\) 0 0
\(207\) −5485.20 −1.84178
\(208\) 0 0
\(209\) 4229.54 1.39982
\(210\) 0 0
\(211\) 4949.57 1.61489 0.807446 0.589941i \(-0.200849\pi\)
0.807446 + 0.589941i \(0.200849\pi\)
\(212\) 0 0
\(213\) 3494.49 1.12412
\(214\) 0 0
\(215\) −53.0939 −0.0168417
\(216\) 0 0
\(217\) −1450.13 −0.453646
\(218\) 0 0
\(219\) −5370.87 −1.65721
\(220\) 0 0
\(221\) −2760.01 −0.840084
\(222\) 0 0
\(223\) −2216.94 −0.665729 −0.332864 0.942975i \(-0.608015\pi\)
−0.332864 + 0.942975i \(0.608015\pi\)
\(224\) 0 0
\(225\) −7278.09 −2.15647
\(226\) 0 0
\(227\) −4546.09 −1.32923 −0.664613 0.747187i \(-0.731404\pi\)
−0.664613 + 0.747187i \(0.731404\pi\)
\(228\) 0 0
\(229\) 3339.05 0.963539 0.481770 0.876298i \(-0.339994\pi\)
0.481770 + 0.876298i \(0.339994\pi\)
\(230\) 0 0
\(231\) −3708.52 −1.05629
\(232\) 0 0
\(233\) −3995.35 −1.12336 −0.561682 0.827353i \(-0.689846\pi\)
−0.561682 + 0.827353i \(0.689846\pi\)
\(234\) 0 0
\(235\) 234.984 0.0652285
\(236\) 0 0
\(237\) 3264.93 0.894853
\(238\) 0 0
\(239\) 1400.04 0.378915 0.189458 0.981889i \(-0.439327\pi\)
0.189458 + 0.981889i \(0.439327\pi\)
\(240\) 0 0
\(241\) −2040.94 −0.545513 −0.272756 0.962083i \(-0.587935\pi\)
−0.272756 + 0.962083i \(0.587935\pi\)
\(242\) 0 0
\(243\) 2390.32 0.631026
\(244\) 0 0
\(245\) −200.521 −0.0522890
\(246\) 0 0
\(247\) −3222.34 −0.830091
\(248\) 0 0
\(249\) −6437.59 −1.63842
\(250\) 0 0
\(251\) −802.648 −0.201843 −0.100922 0.994894i \(-0.532179\pi\)
−0.100922 + 0.994894i \(0.532179\pi\)
\(252\) 0 0
\(253\) −6132.94 −1.52401
\(254\) 0 0
\(255\) 336.676 0.0826803
\(256\) 0 0
\(257\) −4464.10 −1.08351 −0.541756 0.840536i \(-0.682240\pi\)
−0.541756 + 0.840536i \(0.682240\pi\)
\(258\) 0 0
\(259\) −471.741 −0.113176
\(260\) 0 0
\(261\) 1694.37 0.401834
\(262\) 0 0
\(263\) 3815.21 0.894509 0.447255 0.894407i \(-0.352402\pi\)
0.447255 + 0.894407i \(0.352402\pi\)
\(264\) 0 0
\(265\) 215.754 0.0500139
\(266\) 0 0
\(267\) 10335.3 2.36895
\(268\) 0 0
\(269\) 4523.98 1.02540 0.512699 0.858569i \(-0.328646\pi\)
0.512699 + 0.858569i \(0.328646\pi\)
\(270\) 0 0
\(271\) −3962.65 −0.888242 −0.444121 0.895967i \(-0.646484\pi\)
−0.444121 + 0.895967i \(0.646484\pi\)
\(272\) 0 0
\(273\) 2825.40 0.626376
\(274\) 0 0
\(275\) −8137.55 −1.78441
\(276\) 0 0
\(277\) 2217.59 0.481019 0.240509 0.970647i \(-0.422685\pi\)
0.240509 + 0.970647i \(0.422685\pi\)
\(278\) 0 0
\(279\) 13794.2 2.95999
\(280\) 0 0
\(281\) −2562.96 −0.544105 −0.272053 0.962282i \(-0.587702\pi\)
−0.272053 + 0.962282i \(0.587702\pi\)
\(282\) 0 0
\(283\) 3869.29 0.812741 0.406370 0.913708i \(-0.366794\pi\)
0.406370 + 0.913708i \(0.366794\pi\)
\(284\) 0 0
\(285\) 393.072 0.0816968
\(286\) 0 0
\(287\) −1321.55 −0.271807
\(288\) 0 0
\(289\) −1837.65 −0.374038
\(290\) 0 0
\(291\) −7444.17 −1.49960
\(292\) 0 0
\(293\) −3883.83 −0.774388 −0.387194 0.921998i \(-0.626556\pi\)
−0.387194 + 0.921998i \(0.626556\pi\)
\(294\) 0 0
\(295\) 65.2092 0.0128699
\(296\) 0 0
\(297\) 18974.8 3.70716
\(298\) 0 0
\(299\) 4672.48 0.903734
\(300\) 0 0
\(301\) 496.472 0.0950703
\(302\) 0 0
\(303\) −12699.3 −2.40777
\(304\) 0 0
\(305\) −476.699 −0.0894941
\(306\) 0 0
\(307\) 403.210 0.0749590 0.0374795 0.999297i \(-0.488067\pi\)
0.0374795 + 0.999297i \(0.488067\pi\)
\(308\) 0 0
\(309\) 5866.05 1.07996
\(310\) 0 0
\(311\) 4838.71 0.882244 0.441122 0.897447i \(-0.354581\pi\)
0.441122 + 0.897447i \(0.354581\pi\)
\(312\) 0 0
\(313\) −8544.28 −1.54298 −0.771488 0.636244i \(-0.780487\pi\)
−0.771488 + 0.636244i \(0.780487\pi\)
\(314\) 0 0
\(315\) −235.721 −0.0421630
\(316\) 0 0
\(317\) −1773.06 −0.314148 −0.157074 0.987587i \(-0.550206\pi\)
−0.157074 + 0.987587i \(0.550206\pi\)
\(318\) 0 0
\(319\) 1894.45 0.332504
\(320\) 0 0
\(321\) 1672.51 0.290812
\(322\) 0 0
\(323\) 3590.50 0.618516
\(324\) 0 0
\(325\) 6199.72 1.05815
\(326\) 0 0
\(327\) 9595.11 1.62266
\(328\) 0 0
\(329\) −2197.30 −0.368210
\(330\) 0 0
\(331\) −801.875 −0.133157 −0.0665786 0.997781i \(-0.521208\pi\)
−0.0665786 + 0.997781i \(0.521208\pi\)
\(332\) 0 0
\(333\) 4487.38 0.738460
\(334\) 0 0
\(335\) −554.700 −0.0904671
\(336\) 0 0
\(337\) 8193.23 1.32437 0.662186 0.749339i \(-0.269629\pi\)
0.662186 + 0.749339i \(0.269629\pi\)
\(338\) 0 0
\(339\) −1707.72 −0.273601
\(340\) 0 0
\(341\) 15423.1 2.44930
\(342\) 0 0
\(343\) 3981.79 0.626811
\(344\) 0 0
\(345\) −569.966 −0.0889447
\(346\) 0 0
\(347\) 10914.9 1.68860 0.844301 0.535869i \(-0.180016\pi\)
0.844301 + 0.535869i \(0.180016\pi\)
\(348\) 0 0
\(349\) 6697.83 1.02730 0.513649 0.858001i \(-0.328294\pi\)
0.513649 + 0.858001i \(0.328294\pi\)
\(350\) 0 0
\(351\) −14456.2 −2.19833
\(352\) 0 0
\(353\) −3764.83 −0.567654 −0.283827 0.958875i \(-0.591604\pi\)
−0.283827 + 0.958875i \(0.591604\pi\)
\(354\) 0 0
\(355\) 248.346 0.0371291
\(356\) 0 0
\(357\) −3148.20 −0.466724
\(358\) 0 0
\(359\) −6577.13 −0.966930 −0.483465 0.875364i \(-0.660622\pi\)
−0.483465 + 0.875364i \(0.660622\pi\)
\(360\) 0 0
\(361\) −2667.06 −0.388841
\(362\) 0 0
\(363\) 27140.8 3.92430
\(364\) 0 0
\(365\) −381.696 −0.0547366
\(366\) 0 0
\(367\) −2274.27 −0.323477 −0.161738 0.986834i \(-0.551710\pi\)
−0.161738 + 0.986834i \(0.551710\pi\)
\(368\) 0 0
\(369\) 12571.1 1.77351
\(370\) 0 0
\(371\) −2017.48 −0.282325
\(372\) 0 0
\(373\) 1284.94 0.178369 0.0891844 0.996015i \(-0.471574\pi\)
0.0891844 + 0.996015i \(0.471574\pi\)
\(374\) 0 0
\(375\) −1515.15 −0.208645
\(376\) 0 0
\(377\) −1443.32 −0.197174
\(378\) 0 0
\(379\) −174.785 −0.0236890 −0.0118445 0.999930i \(-0.503770\pi\)
−0.0118445 + 0.999930i \(0.503770\pi\)
\(380\) 0 0
\(381\) −18481.4 −2.48512
\(382\) 0 0
\(383\) 5558.62 0.741599 0.370799 0.928713i \(-0.379084\pi\)
0.370799 + 0.928713i \(0.379084\pi\)
\(384\) 0 0
\(385\) −263.557 −0.0348885
\(386\) 0 0
\(387\) −4722.64 −0.620323
\(388\) 0 0
\(389\) −2556.05 −0.333154 −0.166577 0.986028i \(-0.553271\pi\)
−0.166577 + 0.986028i \(0.553271\pi\)
\(390\) 0 0
\(391\) −5206.32 −0.673388
\(392\) 0 0
\(393\) −5189.45 −0.666089
\(394\) 0 0
\(395\) 232.032 0.0295564
\(396\) 0 0
\(397\) 5927.27 0.749323 0.374662 0.927162i \(-0.377759\pi\)
0.374662 + 0.927162i \(0.377759\pi\)
\(398\) 0 0
\(399\) −3675.55 −0.461173
\(400\) 0 0
\(401\) 4747.99 0.591280 0.295640 0.955299i \(-0.404467\pi\)
0.295640 + 0.955299i \(0.404467\pi\)
\(402\) 0 0
\(403\) −11750.4 −1.45243
\(404\) 0 0
\(405\) 727.224 0.0892249
\(406\) 0 0
\(407\) 5017.29 0.611052
\(408\) 0 0
\(409\) 5200.19 0.628686 0.314343 0.949309i \(-0.398216\pi\)
0.314343 + 0.949309i \(0.398216\pi\)
\(410\) 0 0
\(411\) −2315.18 −0.277858
\(412\) 0 0
\(413\) −609.761 −0.0726498
\(414\) 0 0
\(415\) −457.505 −0.0541158
\(416\) 0 0
\(417\) 2238.97 0.262933
\(418\) 0 0
\(419\) 6425.59 0.749189 0.374595 0.927189i \(-0.377782\pi\)
0.374595 + 0.927189i \(0.377782\pi\)
\(420\) 0 0
\(421\) 10037.6 1.16201 0.581003 0.813902i \(-0.302661\pi\)
0.581003 + 0.813902i \(0.302661\pi\)
\(422\) 0 0
\(423\) 20901.6 2.40253
\(424\) 0 0
\(425\) −6908.05 −0.788447
\(426\) 0 0
\(427\) 4457.53 0.505188
\(428\) 0 0
\(429\) −30050.1 −3.38189
\(430\) 0 0
\(431\) −16646.8 −1.86044 −0.930218 0.367006i \(-0.880383\pi\)
−0.930218 + 0.367006i \(0.880383\pi\)
\(432\) 0 0
\(433\) 15089.1 1.67468 0.837340 0.546682i \(-0.184109\pi\)
0.837340 + 0.546682i \(0.184109\pi\)
\(434\) 0 0
\(435\) 176.061 0.0194057
\(436\) 0 0
\(437\) −6078.42 −0.665378
\(438\) 0 0
\(439\) 3777.24 0.410656 0.205328 0.978693i \(-0.434174\pi\)
0.205328 + 0.978693i \(0.434174\pi\)
\(440\) 0 0
\(441\) −17836.1 −1.92593
\(442\) 0 0
\(443\) −7992.65 −0.857206 −0.428603 0.903493i \(-0.640994\pi\)
−0.428603 + 0.903493i \(0.640994\pi\)
\(444\) 0 0
\(445\) 734.507 0.0782449
\(446\) 0 0
\(447\) −15089.8 −1.59670
\(448\) 0 0
\(449\) 6433.54 0.676209 0.338104 0.941109i \(-0.390214\pi\)
0.338104 + 0.941109i \(0.390214\pi\)
\(450\) 0 0
\(451\) 14055.6 1.46752
\(452\) 0 0
\(453\) 1123.74 0.116552
\(454\) 0 0
\(455\) 200.795 0.0206888
\(456\) 0 0
\(457\) 6975.18 0.713972 0.356986 0.934110i \(-0.383804\pi\)
0.356986 + 0.934110i \(0.383804\pi\)
\(458\) 0 0
\(459\) 16107.9 1.63802
\(460\) 0 0
\(461\) −14758.9 −1.49109 −0.745543 0.666458i \(-0.767810\pi\)
−0.745543 + 0.666458i \(0.767810\pi\)
\(462\) 0 0
\(463\) 18951.2 1.90224 0.951121 0.308818i \(-0.0999334\pi\)
0.951121 + 0.308818i \(0.0999334\pi\)
\(464\) 0 0
\(465\) 1433.35 0.142946
\(466\) 0 0
\(467\) −12442.4 −1.23290 −0.616449 0.787395i \(-0.711429\pi\)
−0.616449 + 0.787395i \(0.711429\pi\)
\(468\) 0 0
\(469\) 5186.91 0.510680
\(470\) 0 0
\(471\) −6961.22 −0.681010
\(472\) 0 0
\(473\) −5280.33 −0.513297
\(474\) 0 0
\(475\) −8065.21 −0.779068
\(476\) 0 0
\(477\) 19191.1 1.84214
\(478\) 0 0
\(479\) −12947.5 −1.23504 −0.617522 0.786554i \(-0.711863\pi\)
−0.617522 + 0.786554i \(0.711863\pi\)
\(480\) 0 0
\(481\) −3822.50 −0.362352
\(482\) 0 0
\(483\) 5329.65 0.502086
\(484\) 0 0
\(485\) −529.041 −0.0495309
\(486\) 0 0
\(487\) −9844.72 −0.916030 −0.458015 0.888944i \(-0.651439\pi\)
−0.458015 + 0.888944i \(0.651439\pi\)
\(488\) 0 0
\(489\) −4971.77 −0.459778
\(490\) 0 0
\(491\) 6809.50 0.625883 0.312941 0.949772i \(-0.398686\pi\)
0.312941 + 0.949772i \(0.398686\pi\)
\(492\) 0 0
\(493\) 1608.22 0.146918
\(494\) 0 0
\(495\) 2507.05 0.227644
\(496\) 0 0
\(497\) −2322.24 −0.209591
\(498\) 0 0
\(499\) 15953.8 1.43124 0.715622 0.698488i \(-0.246143\pi\)
0.715622 + 0.698488i \(0.246143\pi\)
\(500\) 0 0
\(501\) −4479.11 −0.399424
\(502\) 0 0
\(503\) 14582.7 1.29267 0.646334 0.763054i \(-0.276301\pi\)
0.646334 + 0.763054i \(0.276301\pi\)
\(504\) 0 0
\(505\) −902.511 −0.0795272
\(506\) 0 0
\(507\) 2588.02 0.226702
\(508\) 0 0
\(509\) 20906.4 1.82055 0.910273 0.414008i \(-0.135871\pi\)
0.910273 + 0.414008i \(0.135871\pi\)
\(510\) 0 0
\(511\) 3569.17 0.308984
\(512\) 0 0
\(513\) 18806.1 1.61854
\(514\) 0 0
\(515\) 416.887 0.0356704
\(516\) 0 0
\(517\) 23369.8 1.98802
\(518\) 0 0
\(519\) −30220.6 −2.55595
\(520\) 0 0
\(521\) −15131.7 −1.27242 −0.636212 0.771515i \(-0.719500\pi\)
−0.636212 + 0.771515i \(0.719500\pi\)
\(522\) 0 0
\(523\) −12146.9 −1.01558 −0.507790 0.861481i \(-0.669537\pi\)
−0.507790 + 0.861481i \(0.669537\pi\)
\(524\) 0 0
\(525\) 7071.70 0.587875
\(526\) 0 0
\(527\) 13092.9 1.08223
\(528\) 0 0
\(529\) −3353.12 −0.275592
\(530\) 0 0
\(531\) 5800.28 0.474032
\(532\) 0 0
\(533\) −10708.5 −0.870237
\(534\) 0 0
\(535\) 118.862 0.00960532
\(536\) 0 0
\(537\) 5196.83 0.417616
\(538\) 0 0
\(539\) −19942.3 −1.59365
\(540\) 0 0
\(541\) −22291.8 −1.77153 −0.885767 0.464130i \(-0.846367\pi\)
−0.885767 + 0.464130i \(0.846367\pi\)
\(542\) 0 0
\(543\) −13931.7 −1.10104
\(544\) 0 0
\(545\) 681.904 0.0535955
\(546\) 0 0
\(547\) 15439.4 1.20684 0.603421 0.797423i \(-0.293804\pi\)
0.603421 + 0.797423i \(0.293804\pi\)
\(548\) 0 0
\(549\) −42401.8 −3.29629
\(550\) 0 0
\(551\) 1877.61 0.145170
\(552\) 0 0
\(553\) −2169.69 −0.166844
\(554\) 0 0
\(555\) 466.283 0.0356623
\(556\) 0 0
\(557\) 2336.99 0.177776 0.0888881 0.996042i \(-0.471669\pi\)
0.0888881 + 0.996042i \(0.471669\pi\)
\(558\) 0 0
\(559\) 4022.90 0.304384
\(560\) 0 0
\(561\) 33483.3 2.51991
\(562\) 0 0
\(563\) −19833.3 −1.48468 −0.742340 0.670023i \(-0.766284\pi\)
−0.742340 + 0.670023i \(0.766284\pi\)
\(564\) 0 0
\(565\) −121.364 −0.00903685
\(566\) 0 0
\(567\) −6800.16 −0.503668
\(568\) 0 0
\(569\) 11063.7 0.815141 0.407571 0.913174i \(-0.366376\pi\)
0.407571 + 0.913174i \(0.366376\pi\)
\(570\) 0 0
\(571\) 665.827 0.0487986 0.0243993 0.999702i \(-0.492233\pi\)
0.0243993 + 0.999702i \(0.492233\pi\)
\(572\) 0 0
\(573\) −41887.6 −3.05389
\(574\) 0 0
\(575\) 11694.8 0.848184
\(576\) 0 0
\(577\) −7165.21 −0.516970 −0.258485 0.966015i \(-0.583223\pi\)
−0.258485 + 0.966015i \(0.583223\pi\)
\(578\) 0 0
\(579\) −27128.7 −1.94720
\(580\) 0 0
\(581\) 4278.06 0.305480
\(582\) 0 0
\(583\) 21457.3 1.52431
\(584\) 0 0
\(585\) −1910.04 −0.134992
\(586\) 0 0
\(587\) 10375.2 0.729525 0.364763 0.931101i \(-0.381150\pi\)
0.364763 + 0.931101i \(0.381150\pi\)
\(588\) 0 0
\(589\) 15286.0 1.06936
\(590\) 0 0
\(591\) 24790.6 1.72546
\(592\) 0 0
\(593\) 18931.5 1.31100 0.655501 0.755194i \(-0.272458\pi\)
0.655501 + 0.755194i \(0.272458\pi\)
\(594\) 0 0
\(595\) −223.736 −0.0154156
\(596\) 0 0
\(597\) 5992.71 0.410829
\(598\) 0 0
\(599\) 12244.2 0.835199 0.417600 0.908631i \(-0.362871\pi\)
0.417600 + 0.908631i \(0.362871\pi\)
\(600\) 0 0
\(601\) 15596.9 1.05859 0.529293 0.848439i \(-0.322457\pi\)
0.529293 + 0.848439i \(0.322457\pi\)
\(602\) 0 0
\(603\) −49339.9 −3.33213
\(604\) 0 0
\(605\) 1928.84 0.129617
\(606\) 0 0
\(607\) 10155.5 0.679076 0.339538 0.940592i \(-0.389729\pi\)
0.339538 + 0.940592i \(0.389729\pi\)
\(608\) 0 0
\(609\) −1646.32 −0.109544
\(610\) 0 0
\(611\) −17804.7 −1.17889
\(612\) 0 0
\(613\) −6227.67 −0.410331 −0.205166 0.978727i \(-0.565773\pi\)
−0.205166 + 0.978727i \(0.565773\pi\)
\(614\) 0 0
\(615\) 1306.26 0.0856479
\(616\) 0 0
\(617\) 14357.5 0.936808 0.468404 0.883514i \(-0.344829\pi\)
0.468404 + 0.883514i \(0.344829\pi\)
\(618\) 0 0
\(619\) 13220.6 0.858453 0.429227 0.903197i \(-0.358786\pi\)
0.429227 + 0.903197i \(0.358786\pi\)
\(620\) 0 0
\(621\) −27269.3 −1.76213
\(622\) 0 0
\(623\) −6868.26 −0.441687
\(624\) 0 0
\(625\) 15463.4 0.989657
\(626\) 0 0
\(627\) 39092.1 2.48993
\(628\) 0 0
\(629\) 4259.23 0.269995
\(630\) 0 0
\(631\) −1828.97 −0.115389 −0.0576943 0.998334i \(-0.518375\pi\)
−0.0576943 + 0.998334i \(0.518375\pi\)
\(632\) 0 0
\(633\) 45747.1 2.87249
\(634\) 0 0
\(635\) −1313.43 −0.0820817
\(636\) 0 0
\(637\) 15193.4 0.945028
\(638\) 0 0
\(639\) 22090.0 1.36756
\(640\) 0 0
\(641\) −22644.1 −1.39530 −0.697651 0.716437i \(-0.745772\pi\)
−0.697651 + 0.716437i \(0.745772\pi\)
\(642\) 0 0
\(643\) −22728.4 −1.39397 −0.696983 0.717088i \(-0.745475\pi\)
−0.696983 + 0.717088i \(0.745475\pi\)
\(644\) 0 0
\(645\) −490.728 −0.0299572
\(646\) 0 0
\(647\) 5844.85 0.355154 0.177577 0.984107i \(-0.443174\pi\)
0.177577 + 0.984107i \(0.443174\pi\)
\(648\) 0 0
\(649\) 6485.23 0.392246
\(650\) 0 0
\(651\) −13403.0 −0.806922
\(652\) 0 0
\(653\) 15174.1 0.909355 0.454677 0.890656i \(-0.349755\pi\)
0.454677 + 0.890656i \(0.349755\pi\)
\(654\) 0 0
\(655\) −368.803 −0.0220005
\(656\) 0 0
\(657\) −33951.4 −2.01609
\(658\) 0 0
\(659\) 27857.0 1.64667 0.823333 0.567558i \(-0.192112\pi\)
0.823333 + 0.567558i \(0.192112\pi\)
\(660\) 0 0
\(661\) −4966.64 −0.292254 −0.146127 0.989266i \(-0.546681\pi\)
−0.146127 + 0.989266i \(0.546681\pi\)
\(662\) 0 0
\(663\) −25509.8 −1.49430
\(664\) 0 0
\(665\) −261.214 −0.0152322
\(666\) 0 0
\(667\) −2722.59 −0.158049
\(668\) 0 0
\(669\) −20490.4 −1.18416
\(670\) 0 0
\(671\) −47409.0 −2.72758
\(672\) 0 0
\(673\) −2338.02 −0.133914 −0.0669569 0.997756i \(-0.521329\pi\)
−0.0669569 + 0.997756i \(0.521329\pi\)
\(674\) 0 0
\(675\) −36182.6 −2.06321
\(676\) 0 0
\(677\) 6342.30 0.360051 0.180025 0.983662i \(-0.442382\pi\)
0.180025 + 0.983662i \(0.442382\pi\)
\(678\) 0 0
\(679\) 4946.97 0.279598
\(680\) 0 0
\(681\) −42017.9 −2.36436
\(682\) 0 0
\(683\) 30366.6 1.70124 0.850620 0.525780i \(-0.176227\pi\)
0.850620 + 0.525780i \(0.176227\pi\)
\(684\) 0 0
\(685\) −164.535 −0.00917746
\(686\) 0 0
\(687\) 30861.6 1.71389
\(688\) 0 0
\(689\) −16347.6 −0.903910
\(690\) 0 0
\(691\) 11826.5 0.651089 0.325545 0.945527i \(-0.394452\pi\)
0.325545 + 0.945527i \(0.394452\pi\)
\(692\) 0 0
\(693\) −23443.0 −1.28503
\(694\) 0 0
\(695\) 159.119 0.00868450
\(696\) 0 0
\(697\) 11932.0 0.648429
\(698\) 0 0
\(699\) −36927.6 −1.99818
\(700\) 0 0
\(701\) −2776.33 −0.149587 −0.0747936 0.997199i \(-0.523830\pi\)
−0.0747936 + 0.997199i \(0.523830\pi\)
\(702\) 0 0
\(703\) 4972.69 0.266783
\(704\) 0 0
\(705\) 2171.88 0.116025
\(706\) 0 0
\(707\) 8439.23 0.448925
\(708\) 0 0
\(709\) −15962.7 −0.845543 −0.422772 0.906236i \(-0.638943\pi\)
−0.422772 + 0.906236i \(0.638943\pi\)
\(710\) 0 0
\(711\) 20638.9 1.08864
\(712\) 0 0
\(713\) −22165.2 −1.16422
\(714\) 0 0
\(715\) −2135.59 −0.111702
\(716\) 0 0
\(717\) 12940.0 0.673994
\(718\) 0 0
\(719\) 20832.9 1.08058 0.540289 0.841480i \(-0.318315\pi\)
0.540289 + 0.841480i \(0.318315\pi\)
\(720\) 0 0
\(721\) −3898.24 −0.201357
\(722\) 0 0
\(723\) −18863.7 −0.970329
\(724\) 0 0
\(725\) −3612.49 −0.185054
\(726\) 0 0
\(727\) −4452.04 −0.227121 −0.113561 0.993531i \(-0.536226\pi\)
−0.113561 + 0.993531i \(0.536226\pi\)
\(728\) 0 0
\(729\) −7799.67 −0.396264
\(730\) 0 0
\(731\) −4482.52 −0.226802
\(732\) 0 0
\(733\) −12107.2 −0.610082 −0.305041 0.952339i \(-0.598670\pi\)
−0.305041 + 0.952339i \(0.598670\pi\)
\(734\) 0 0
\(735\) −1853.34 −0.0930088
\(736\) 0 0
\(737\) −55166.4 −2.75723
\(738\) 0 0
\(739\) 4506.27 0.224311 0.112156 0.993691i \(-0.464225\pi\)
0.112156 + 0.993691i \(0.464225\pi\)
\(740\) 0 0
\(741\) −29782.9 −1.47652
\(742\) 0 0
\(743\) −1177.19 −0.0581253 −0.0290626 0.999578i \(-0.509252\pi\)
−0.0290626 + 0.999578i \(0.509252\pi\)
\(744\) 0 0
\(745\) −1072.40 −0.0527378
\(746\) 0 0
\(747\) −40694.6 −1.99322
\(748\) 0 0
\(749\) −1111.46 −0.0542214
\(750\) 0 0
\(751\) −27631.6 −1.34260 −0.671300 0.741186i \(-0.734264\pi\)
−0.671300 + 0.741186i \(0.734264\pi\)
\(752\) 0 0
\(753\) −7418.59 −0.359028
\(754\) 0 0
\(755\) 79.8616 0.00384962
\(756\) 0 0
\(757\) 11336.6 0.544300 0.272150 0.962255i \(-0.412265\pi\)
0.272150 + 0.962255i \(0.412265\pi\)
\(758\) 0 0
\(759\) −56684.6 −2.71083
\(760\) 0 0
\(761\) 4356.58 0.207524 0.103762 0.994602i \(-0.466912\pi\)
0.103762 + 0.994602i \(0.466912\pi\)
\(762\) 0 0
\(763\) −6376.37 −0.302543
\(764\) 0 0
\(765\) 2128.26 0.100585
\(766\) 0 0
\(767\) −4940.87 −0.232601
\(768\) 0 0
\(769\) −21718.1 −1.01843 −0.509217 0.860638i \(-0.670065\pi\)
−0.509217 + 0.860638i \(0.670065\pi\)
\(770\) 0 0
\(771\) −41260.0 −1.92729
\(772\) 0 0
\(773\) 22688.4 1.05568 0.527842 0.849343i \(-0.323001\pi\)
0.527842 + 0.849343i \(0.323001\pi\)
\(774\) 0 0
\(775\) −29410.1 −1.36315
\(776\) 0 0
\(777\) −4360.13 −0.201311
\(778\) 0 0
\(779\) 13930.7 0.640716
\(780\) 0 0
\(781\) 24698.6 1.13161
\(782\) 0 0
\(783\) 8423.43 0.384456
\(784\) 0 0
\(785\) −494.718 −0.0224933
\(786\) 0 0
\(787\) 32890.9 1.48975 0.744875 0.667204i \(-0.232509\pi\)
0.744875 + 0.667204i \(0.232509\pi\)
\(788\) 0 0
\(789\) 35262.6 1.59111
\(790\) 0 0
\(791\) 1134.85 0.0510123
\(792\) 0 0
\(793\) 36119.3 1.61744
\(794\) 0 0
\(795\) 1994.14 0.0889620
\(796\) 0 0
\(797\) 30404.1 1.35128 0.675638 0.737233i \(-0.263868\pi\)
0.675638 + 0.737233i \(0.263868\pi\)
\(798\) 0 0
\(799\) 19838.9 0.878410
\(800\) 0 0
\(801\) 65333.5 2.88196
\(802\) 0 0
\(803\) −37960.7 −1.66825
\(804\) 0 0
\(805\) 378.767 0.0165836
\(806\) 0 0
\(807\) 41813.5 1.82392
\(808\) 0 0
\(809\) −37889.3 −1.64662 −0.823311 0.567591i \(-0.807876\pi\)
−0.823311 + 0.567591i \(0.807876\pi\)
\(810\) 0 0
\(811\) 8123.23 0.351720 0.175860 0.984415i \(-0.443729\pi\)
0.175860 + 0.984415i \(0.443729\pi\)
\(812\) 0 0
\(813\) −36625.3 −1.57996
\(814\) 0 0
\(815\) −353.333 −0.0151862
\(816\) 0 0
\(817\) −5233.39 −0.224104
\(818\) 0 0
\(819\) 17860.4 0.762020
\(820\) 0 0
\(821\) 13226.8 0.562264 0.281132 0.959669i \(-0.409290\pi\)
0.281132 + 0.959669i \(0.409290\pi\)
\(822\) 0 0
\(823\) 29575.2 1.25265 0.626323 0.779563i \(-0.284559\pi\)
0.626323 + 0.779563i \(0.284559\pi\)
\(824\) 0 0
\(825\) −75212.5 −3.17401
\(826\) 0 0
\(827\) 36661.2 1.54152 0.770758 0.637128i \(-0.219878\pi\)
0.770758 + 0.637128i \(0.219878\pi\)
\(828\) 0 0
\(829\) 11277.0 0.472455 0.236228 0.971698i \(-0.424089\pi\)
0.236228 + 0.971698i \(0.424089\pi\)
\(830\) 0 0
\(831\) 20496.4 0.855611
\(832\) 0 0
\(833\) −16929.2 −0.704158
\(834\) 0 0
\(835\) −318.320 −0.0131927
\(836\) 0 0
\(837\) 68577.0 2.83198
\(838\) 0 0
\(839\) 18965.7 0.780417 0.390208 0.920727i \(-0.372403\pi\)
0.390208 + 0.920727i \(0.372403\pi\)
\(840\) 0 0
\(841\) 841.000 0.0344828
\(842\) 0 0
\(843\) −23688.5 −0.967825
\(844\) 0 0
\(845\) 183.925 0.00748781
\(846\) 0 0
\(847\) −18036.2 −0.731679
\(848\) 0 0
\(849\) 35762.5 1.44566
\(850\) 0 0
\(851\) −7210.54 −0.290451
\(852\) 0 0
\(853\) 8067.23 0.323818 0.161909 0.986806i \(-0.448235\pi\)
0.161909 + 0.986806i \(0.448235\pi\)
\(854\) 0 0
\(855\) 2484.77 0.0993886
\(856\) 0 0
\(857\) 15281.7 0.609118 0.304559 0.952493i \(-0.401491\pi\)
0.304559 + 0.952493i \(0.401491\pi\)
\(858\) 0 0
\(859\) 36789.1 1.46127 0.730634 0.682770i \(-0.239225\pi\)
0.730634 + 0.682770i \(0.239225\pi\)
\(860\) 0 0
\(861\) −12214.6 −0.483476
\(862\) 0 0
\(863\) −40907.8 −1.61358 −0.806788 0.590840i \(-0.798796\pi\)
−0.806788 + 0.590840i \(0.798796\pi\)
\(864\) 0 0
\(865\) −2147.71 −0.0844213
\(866\) 0 0
\(867\) −16984.7 −0.665319
\(868\) 0 0
\(869\) 23076.2 0.900812
\(870\) 0 0
\(871\) 42029.4 1.63503
\(872\) 0 0
\(873\) −47057.5 −1.82435
\(874\) 0 0
\(875\) 1006.88 0.0389015
\(876\) 0 0
\(877\) 2391.60 0.0920852 0.0460426 0.998939i \(-0.485339\pi\)
0.0460426 + 0.998939i \(0.485339\pi\)
\(878\) 0 0
\(879\) −35896.8 −1.37744
\(880\) 0 0
\(881\) 5487.72 0.209859 0.104930 0.994480i \(-0.466538\pi\)
0.104930 + 0.994480i \(0.466538\pi\)
\(882\) 0 0
\(883\) −170.008 −0.00647931 −0.00323966 0.999995i \(-0.501031\pi\)
−0.00323966 + 0.999995i \(0.501031\pi\)
\(884\) 0 0
\(885\) 602.705 0.0228923
\(886\) 0 0
\(887\) −25867.3 −0.979188 −0.489594 0.871950i \(-0.662855\pi\)
−0.489594 + 0.871950i \(0.662855\pi\)
\(888\) 0 0
\(889\) 12281.7 0.463345
\(890\) 0 0
\(891\) 72324.4 2.71937
\(892\) 0 0
\(893\) 23162.1 0.867961
\(894\) 0 0
\(895\) 369.327 0.0137936
\(896\) 0 0
\(897\) 43186.0 1.60751
\(898\) 0 0
\(899\) 6846.77 0.254007
\(900\) 0 0
\(901\) 18215.4 0.673520
\(902\) 0 0
\(903\) 4588.71 0.169106
\(904\) 0 0
\(905\) −990.093 −0.0363666
\(906\) 0 0
\(907\) −11411.5 −0.417765 −0.208882 0.977941i \(-0.566983\pi\)
−0.208882 + 0.977941i \(0.566983\pi\)
\(908\) 0 0
\(909\) −80277.3 −2.92919
\(910\) 0 0
\(911\) 38718.5 1.40813 0.704063 0.710138i \(-0.251367\pi\)
0.704063 + 0.710138i \(0.251367\pi\)
\(912\) 0 0
\(913\) −45500.1 −1.64933
\(914\) 0 0
\(915\) −4405.96 −0.159187
\(916\) 0 0
\(917\) 3448.62 0.124191
\(918\) 0 0
\(919\) 48465.3 1.73963 0.869817 0.493374i \(-0.164237\pi\)
0.869817 + 0.493374i \(0.164237\pi\)
\(920\) 0 0
\(921\) 3726.72 0.133333
\(922\) 0 0
\(923\) −18817.0 −0.671040
\(924\) 0 0
\(925\) −9567.37 −0.340079
\(926\) 0 0
\(927\) 37081.6 1.31383
\(928\) 0 0
\(929\) 10560.1 0.372946 0.186473 0.982460i \(-0.440294\pi\)
0.186473 + 0.982460i \(0.440294\pi\)
\(930\) 0 0
\(931\) −19765.0 −0.695782
\(932\) 0 0
\(933\) 44722.4 1.56929
\(934\) 0 0
\(935\) 2379.59 0.0832309
\(936\) 0 0
\(937\) −23025.0 −0.802769 −0.401384 0.915910i \(-0.631471\pi\)
−0.401384 + 0.915910i \(0.631471\pi\)
\(938\) 0 0
\(939\) −78971.7 −2.74456
\(940\) 0 0
\(941\) 40778.1 1.41268 0.706338 0.707874i \(-0.250346\pi\)
0.706338 + 0.707874i \(0.250346\pi\)
\(942\) 0 0
\(943\) −20199.8 −0.697558
\(944\) 0 0
\(945\) −1171.87 −0.0403396
\(946\) 0 0
\(947\) −36129.8 −1.23977 −0.619884 0.784693i \(-0.712821\pi\)
−0.619884 + 0.784693i \(0.712821\pi\)
\(948\) 0 0
\(949\) 28920.9 0.989265
\(950\) 0 0
\(951\) −16387.8 −0.558790
\(952\) 0 0
\(953\) 20831.4 0.708075 0.354037 0.935231i \(-0.384809\pi\)
0.354037 + 0.935231i \(0.384809\pi\)
\(954\) 0 0
\(955\) −2976.86 −0.100868
\(956\) 0 0
\(957\) 17509.7 0.591441
\(958\) 0 0
\(959\) 1538.54 0.0518061
\(960\) 0 0
\(961\) 25950.1 0.871071
\(962\) 0 0
\(963\) 10572.6 0.353788
\(964\) 0 0
\(965\) −1927.98 −0.0643149
\(966\) 0 0
\(967\) −49242.5 −1.63757 −0.818785 0.574100i \(-0.805352\pi\)
−0.818785 + 0.574100i \(0.805352\pi\)
\(968\) 0 0
\(969\) 33185.7 1.10018
\(970\) 0 0
\(971\) 2352.05 0.0777351 0.0388675 0.999244i \(-0.487625\pi\)
0.0388675 + 0.999244i \(0.487625\pi\)
\(972\) 0 0
\(973\) −1487.89 −0.0490233
\(974\) 0 0
\(975\) 57301.8 1.88218
\(976\) 0 0
\(977\) 18768.3 0.614588 0.307294 0.951615i \(-0.400577\pi\)
0.307294 + 0.951615i \(0.400577\pi\)
\(978\) 0 0
\(979\) 73048.7 2.38473
\(980\) 0 0
\(981\) 60654.5 1.97406
\(982\) 0 0
\(983\) 49014.5 1.59036 0.795179 0.606375i \(-0.207377\pi\)
0.795179 + 0.606375i \(0.207377\pi\)
\(984\) 0 0
\(985\) 1761.81 0.0569908
\(986\) 0 0
\(987\) −20308.9 −0.654953
\(988\) 0 0
\(989\) 7588.55 0.243986
\(990\) 0 0
\(991\) 48860.6 1.56620 0.783102 0.621893i \(-0.213636\pi\)
0.783102 + 0.621893i \(0.213636\pi\)
\(992\) 0 0
\(993\) −7411.44 −0.236853
\(994\) 0 0
\(995\) 425.888 0.0135694
\(996\) 0 0
\(997\) 2934.57 0.0932184 0.0466092 0.998913i \(-0.485158\pi\)
0.0466092 + 0.998913i \(0.485158\pi\)
\(998\) 0 0
\(999\) 22308.7 0.706524
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 464.4.a.f.1.2 2
4.3 odd 2 29.4.a.a.1.2 2
8.3 odd 2 1856.4.a.n.1.2 2
8.5 even 2 1856.4.a.h.1.1 2
12.11 even 2 261.4.a.b.1.1 2
20.19 odd 2 725.4.a.b.1.1 2
28.27 even 2 1421.4.a.c.1.2 2
116.115 odd 2 841.4.a.a.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
29.4.a.a.1.2 2 4.3 odd 2
261.4.a.b.1.1 2 12.11 even 2
464.4.a.f.1.2 2 1.1 even 1 trivial
725.4.a.b.1.1 2 20.19 odd 2
841.4.a.a.1.1 2 116.115 odd 2
1421.4.a.c.1.2 2 28.27 even 2
1856.4.a.h.1.1 2 8.5 even 2
1856.4.a.n.1.2 2 8.3 odd 2