Properties

Label 882.4.a.w.1.2
Level $882$
Weight $4$
Character 882.1
Self dual yes
Analytic conductor $52.040$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 882.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(52.0396846251\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{22}) \)
Defining polynomial: \(x^{2} - 22\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 98)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(4.69042\) of defining polynomial
Character \(\chi\) \(=\) 882.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.00000 q^{2} +4.00000 q^{4} +9.38083 q^{5} -8.00000 q^{8} +O(q^{10})\) \(q-2.00000 q^{2} +4.00000 q^{4} +9.38083 q^{5} -8.00000 q^{8} -18.7617 q^{10} -20.0000 q^{11} -65.6658 q^{13} +16.0000 q^{16} +56.2850 q^{17} -9.38083 q^{19} +37.5233 q^{20} +40.0000 q^{22} -48.0000 q^{23} -37.0000 q^{25} +131.332 q^{26} +166.000 q^{29} +206.378 q^{31} -32.0000 q^{32} -112.570 q^{34} -78.0000 q^{37} +18.7617 q^{38} -75.0467 q^{40} +393.995 q^{41} +436.000 q^{43} -80.0000 q^{44} +96.0000 q^{46} +206.378 q^{47} +74.0000 q^{50} -262.663 q^{52} -62.0000 q^{53} -187.617 q^{55} -332.000 q^{58} -666.039 q^{59} -272.044 q^{61} -412.757 q^{62} +64.0000 q^{64} -616.000 q^{65} +580.000 q^{67} +225.140 q^{68} +544.000 q^{71} +600.373 q^{73} +156.000 q^{74} -37.5233 q^{76} -680.000 q^{79} +150.093 q^{80} -787.990 q^{82} +196.997 q^{83} +528.000 q^{85} -872.000 q^{86} +160.000 q^{88} -1500.93 q^{89} -192.000 q^{92} -412.757 q^{94} -88.0000 q^{95} +656.658 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{2} + 8q^{4} - 16q^{8} + O(q^{10}) \) \( 2q - 4q^{2} + 8q^{4} - 16q^{8} - 40q^{11} + 32q^{16} + 80q^{22} - 96q^{23} - 74q^{25} + 332q^{29} - 64q^{32} - 156q^{37} + 872q^{43} - 160q^{44} + 192q^{46} + 148q^{50} - 124q^{53} - 664q^{58} + 128q^{64} - 1232q^{65} + 1160q^{67} + 1088q^{71} + 312q^{74} - 1360q^{79} + 1056q^{85} - 1744q^{86} + 320q^{88} - 384q^{92} - 176q^{95} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −0.707107
\(3\) 0 0
\(4\) 4.00000 0.500000
\(5\) 9.38083 0.839047 0.419524 0.907744i \(-0.362197\pi\)
0.419524 + 0.907744i \(0.362197\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −8.00000 −0.353553
\(9\) 0 0
\(10\) −18.7617 −0.593296
\(11\) −20.0000 −0.548202 −0.274101 0.961701i \(-0.588380\pi\)
−0.274101 + 0.961701i \(0.588380\pi\)
\(12\) 0 0
\(13\) −65.6658 −1.40096 −0.700478 0.713674i \(-0.747030\pi\)
−0.700478 + 0.713674i \(0.747030\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) 56.2850 0.803007 0.401503 0.915858i \(-0.368488\pi\)
0.401503 + 0.915858i \(0.368488\pi\)
\(18\) 0 0
\(19\) −9.38083 −0.113269 −0.0566345 0.998395i \(-0.518037\pi\)
−0.0566345 + 0.998395i \(0.518037\pi\)
\(20\) 37.5233 0.419524
\(21\) 0 0
\(22\) 40.0000 0.387638
\(23\) −48.0000 −0.435161 −0.217580 0.976042i \(-0.569816\pi\)
−0.217580 + 0.976042i \(0.569816\pi\)
\(24\) 0 0
\(25\) −37.0000 −0.296000
\(26\) 131.332 0.990625
\(27\) 0 0
\(28\) 0 0
\(29\) 166.000 1.06295 0.531473 0.847075i \(-0.321639\pi\)
0.531473 + 0.847075i \(0.321639\pi\)
\(30\) 0 0
\(31\) 206.378 1.19570 0.597849 0.801609i \(-0.296022\pi\)
0.597849 + 0.801609i \(0.296022\pi\)
\(32\) −32.0000 −0.176777
\(33\) 0 0
\(34\) −112.570 −0.567812
\(35\) 0 0
\(36\) 0 0
\(37\) −78.0000 −0.346571 −0.173285 0.984872i \(-0.555438\pi\)
−0.173285 + 0.984872i \(0.555438\pi\)
\(38\) 18.7617 0.0800933
\(39\) 0 0
\(40\) −75.0467 −0.296648
\(41\) 393.995 1.50077 0.750386 0.661000i \(-0.229868\pi\)
0.750386 + 0.661000i \(0.229868\pi\)
\(42\) 0 0
\(43\) 436.000 1.54626 0.773132 0.634245i \(-0.218689\pi\)
0.773132 + 0.634245i \(0.218689\pi\)
\(44\) −80.0000 −0.274101
\(45\) 0 0
\(46\) 96.0000 0.307705
\(47\) 206.378 0.640497 0.320249 0.947334i \(-0.396234\pi\)
0.320249 + 0.947334i \(0.396234\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 74.0000 0.209304
\(51\) 0 0
\(52\) −262.663 −0.700478
\(53\) −62.0000 −0.160686 −0.0803430 0.996767i \(-0.525602\pi\)
−0.0803430 + 0.996767i \(0.525602\pi\)
\(54\) 0 0
\(55\) −187.617 −0.459968
\(56\) 0 0
\(57\) 0 0
\(58\) −332.000 −0.751616
\(59\) −666.039 −1.46968 −0.734838 0.678243i \(-0.762742\pi\)
−0.734838 + 0.678243i \(0.762742\pi\)
\(60\) 0 0
\(61\) −272.044 −0.571011 −0.285506 0.958377i \(-0.592162\pi\)
−0.285506 + 0.958377i \(0.592162\pi\)
\(62\) −412.757 −0.845486
\(63\) 0 0
\(64\) 64.0000 0.125000
\(65\) −616.000 −1.17547
\(66\) 0 0
\(67\) 580.000 1.05759 0.528793 0.848751i \(-0.322645\pi\)
0.528793 + 0.848751i \(0.322645\pi\)
\(68\) 225.140 0.401503
\(69\) 0 0
\(70\) 0 0
\(71\) 544.000 0.909309 0.454654 0.890668i \(-0.349763\pi\)
0.454654 + 0.890668i \(0.349763\pi\)
\(72\) 0 0
\(73\) 600.373 0.962580 0.481290 0.876561i \(-0.340168\pi\)
0.481290 + 0.876561i \(0.340168\pi\)
\(74\) 156.000 0.245063
\(75\) 0 0
\(76\) −37.5233 −0.0566345
\(77\) 0 0
\(78\) 0 0
\(79\) −680.000 −0.968430 −0.484215 0.874949i \(-0.660895\pi\)
−0.484215 + 0.874949i \(0.660895\pi\)
\(80\) 150.093 0.209762
\(81\) 0 0
\(82\) −787.990 −1.06121
\(83\) 196.997 0.260521 0.130261 0.991480i \(-0.458419\pi\)
0.130261 + 0.991480i \(0.458419\pi\)
\(84\) 0 0
\(85\) 528.000 0.673760
\(86\) −872.000 −1.09337
\(87\) 0 0
\(88\) 160.000 0.193819
\(89\) −1500.93 −1.78762 −0.893812 0.448441i \(-0.851979\pi\)
−0.893812 + 0.448441i \(0.851979\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −192.000 −0.217580
\(93\) 0 0
\(94\) −412.757 −0.452900
\(95\) −88.0000 −0.0950380
\(96\) 0 0
\(97\) 656.658 0.687356 0.343678 0.939088i \(-0.388327\pi\)
0.343678 + 0.939088i \(0.388327\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −148.000 −0.148000
\(101\) 121.951 0.120144 0.0600721 0.998194i \(-0.480867\pi\)
0.0600721 + 0.998194i \(0.480867\pi\)
\(102\) 0 0
\(103\) 1369.60 1.31020 0.655101 0.755541i \(-0.272626\pi\)
0.655101 + 0.755541i \(0.272626\pi\)
\(104\) 525.327 0.495313
\(105\) 0 0
\(106\) 124.000 0.113622
\(107\) 260.000 0.234908 0.117454 0.993078i \(-0.462527\pi\)
0.117454 + 0.993078i \(0.462527\pi\)
\(108\) 0 0
\(109\) 1882.00 1.65379 0.826894 0.562358i \(-0.190106\pi\)
0.826894 + 0.562358i \(0.190106\pi\)
\(110\) 375.233 0.325246
\(111\) 0 0
\(112\) 0 0
\(113\) 1286.00 1.07059 0.535295 0.844665i \(-0.320200\pi\)
0.535295 + 0.844665i \(0.320200\pi\)
\(114\) 0 0
\(115\) −450.280 −0.365120
\(116\) 664.000 0.531473
\(117\) 0 0
\(118\) 1332.08 1.03922
\(119\) 0 0
\(120\) 0 0
\(121\) −931.000 −0.699474
\(122\) 544.088 0.403766
\(123\) 0 0
\(124\) 825.513 0.597849
\(125\) −1519.69 −1.08741
\(126\) 0 0
\(127\) 2312.00 1.61541 0.807704 0.589588i \(-0.200710\pi\)
0.807704 + 0.589588i \(0.200710\pi\)
\(128\) −128.000 −0.0883883
\(129\) 0 0
\(130\) 1232.00 0.831181
\(131\) −253.282 −0.168927 −0.0844633 0.996427i \(-0.526918\pi\)
−0.0844633 + 0.996427i \(0.526918\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −1160.00 −0.747826
\(135\) 0 0
\(136\) −450.280 −0.283906
\(137\) 1114.00 0.694711 0.347356 0.937733i \(-0.387080\pi\)
0.347356 + 0.937733i \(0.387080\pi\)
\(138\) 0 0
\(139\) −1378.98 −0.841466 −0.420733 0.907185i \(-0.638227\pi\)
−0.420733 + 0.907185i \(0.638227\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −1088.00 −0.642978
\(143\) 1313.32 0.768007
\(144\) 0 0
\(145\) 1557.22 0.891862
\(146\) −1200.75 −0.680647
\(147\) 0 0
\(148\) −312.000 −0.173285
\(149\) 946.000 0.520130 0.260065 0.965591i \(-0.416256\pi\)
0.260065 + 0.965591i \(0.416256\pi\)
\(150\) 0 0
\(151\) 832.000 0.448392 0.224196 0.974544i \(-0.428024\pi\)
0.224196 + 0.974544i \(0.428024\pi\)
\(152\) 75.0467 0.0400466
\(153\) 0 0
\(154\) 0 0
\(155\) 1936.00 1.00325
\(156\) 0 0
\(157\) −2879.92 −1.46396 −0.731982 0.681324i \(-0.761404\pi\)
−0.731982 + 0.681324i \(0.761404\pi\)
\(158\) 1360.00 0.684783
\(159\) 0 0
\(160\) −300.187 −0.148324
\(161\) 0 0
\(162\) 0 0
\(163\) 636.000 0.305616 0.152808 0.988256i \(-0.451168\pi\)
0.152808 + 0.988256i \(0.451168\pi\)
\(164\) 1575.98 0.750386
\(165\) 0 0
\(166\) −393.995 −0.184216
\(167\) −656.658 −0.304274 −0.152137 0.988359i \(-0.548615\pi\)
−0.152137 + 0.988359i \(0.548615\pi\)
\(168\) 0 0
\(169\) 2115.00 0.962676
\(170\) −1056.00 −0.476421
\(171\) 0 0
\(172\) 1744.00 0.773132
\(173\) 666.039 0.292705 0.146353 0.989232i \(-0.453247\pi\)
0.146353 + 0.989232i \(0.453247\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −320.000 −0.137051
\(177\) 0 0
\(178\) 3001.87 1.26404
\(179\) 3228.00 1.34789 0.673944 0.738782i \(-0.264599\pi\)
0.673944 + 0.738782i \(0.264599\pi\)
\(180\) 0 0
\(181\) 2823.63 1.15955 0.579776 0.814776i \(-0.303140\pi\)
0.579776 + 0.814776i \(0.303140\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 384.000 0.153852
\(185\) −731.705 −0.290789
\(186\) 0 0
\(187\) −1125.70 −0.440210
\(188\) 825.513 0.320249
\(189\) 0 0
\(190\) 176.000 0.0672020
\(191\) 2136.00 0.809191 0.404596 0.914496i \(-0.367412\pi\)
0.404596 + 0.914496i \(0.367412\pi\)
\(192\) 0 0
\(193\) 1658.00 0.618370 0.309185 0.951002i \(-0.399944\pi\)
0.309185 + 0.951002i \(0.399944\pi\)
\(194\) −1313.32 −0.486034
\(195\) 0 0
\(196\) 0 0
\(197\) 978.000 0.353704 0.176852 0.984237i \(-0.443409\pi\)
0.176852 + 0.984237i \(0.443409\pi\)
\(198\) 0 0
\(199\) 4934.32 1.75771 0.878855 0.477088i \(-0.158308\pi\)
0.878855 + 0.477088i \(0.158308\pi\)
\(200\) 296.000 0.104652
\(201\) 0 0
\(202\) −243.902 −0.0849547
\(203\) 0 0
\(204\) 0 0
\(205\) 3696.00 1.25922
\(206\) −2739.20 −0.926453
\(207\) 0 0
\(208\) −1050.65 −0.350239
\(209\) 187.617 0.0620943
\(210\) 0 0
\(211\) 1556.00 0.507675 0.253838 0.967247i \(-0.418307\pi\)
0.253838 + 0.967247i \(0.418307\pi\)
\(212\) −248.000 −0.0803430
\(213\) 0 0
\(214\) −520.000 −0.166105
\(215\) 4090.04 1.29739
\(216\) 0 0
\(217\) 0 0
\(218\) −3764.00 −1.16940
\(219\) 0 0
\(220\) −750.467 −0.229984
\(221\) −3696.00 −1.12498
\(222\) 0 0
\(223\) −2889.30 −0.867630 −0.433815 0.901002i \(-0.642833\pi\)
−0.433815 + 0.901002i \(0.642833\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −2572.00 −0.757022
\(227\) −1979.36 −0.578742 −0.289371 0.957217i \(-0.593446\pi\)
−0.289371 + 0.957217i \(0.593446\pi\)
\(228\) 0 0
\(229\) −2767.35 −0.798565 −0.399282 0.916828i \(-0.630741\pi\)
−0.399282 + 0.916828i \(0.630741\pi\)
\(230\) 900.560 0.258179
\(231\) 0 0
\(232\) −1328.00 −0.375808
\(233\) 6490.00 1.82478 0.912391 0.409321i \(-0.134234\pi\)
0.912391 + 0.409321i \(0.134234\pi\)
\(234\) 0 0
\(235\) 1936.00 0.537407
\(236\) −2664.16 −0.734838
\(237\) 0 0
\(238\) 0 0
\(239\) 4296.00 1.16270 0.581350 0.813654i \(-0.302525\pi\)
0.581350 + 0.813654i \(0.302525\pi\)
\(240\) 0 0
\(241\) −4521.56 −1.20854 −0.604272 0.796778i \(-0.706536\pi\)
−0.604272 + 0.796778i \(0.706536\pi\)
\(242\) 1862.00 0.494603
\(243\) 0 0
\(244\) −1088.18 −0.285506
\(245\) 0 0
\(246\) 0 0
\(247\) 616.000 0.158685
\(248\) −1651.03 −0.422743
\(249\) 0 0
\(250\) 3039.39 0.768911
\(251\) 5581.59 1.40361 0.701807 0.712367i \(-0.252377\pi\)
0.701807 + 0.712367i \(0.252377\pi\)
\(252\) 0 0
\(253\) 960.000 0.238556
\(254\) −4624.00 −1.14227
\(255\) 0 0
\(256\) 256.000 0.0625000
\(257\) −1500.93 −0.364302 −0.182151 0.983271i \(-0.558306\pi\)
−0.182151 + 0.983271i \(0.558306\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −2464.00 −0.587734
\(261\) 0 0
\(262\) 506.565 0.119449
\(263\) 400.000 0.0937835 0.0468917 0.998900i \(-0.485068\pi\)
0.0468917 + 0.998900i \(0.485068\pi\)
\(264\) 0 0
\(265\) −581.612 −0.134823
\(266\) 0 0
\(267\) 0 0
\(268\) 2320.00 0.528793
\(269\) −272.044 −0.0616610 −0.0308305 0.999525i \(-0.509815\pi\)
−0.0308305 + 0.999525i \(0.509815\pi\)
\(270\) 0 0
\(271\) −6904.29 −1.54762 −0.773812 0.633416i \(-0.781652\pi\)
−0.773812 + 0.633416i \(0.781652\pi\)
\(272\) 900.560 0.200752
\(273\) 0 0
\(274\) −2228.00 −0.491235
\(275\) 740.000 0.162268
\(276\) 0 0
\(277\) −6770.00 −1.46848 −0.734242 0.678888i \(-0.762462\pi\)
−0.734242 + 0.678888i \(0.762462\pi\)
\(278\) 2757.96 0.595006
\(279\) 0 0
\(280\) 0 0
\(281\) −1878.00 −0.398691 −0.199345 0.979929i \(-0.563882\pi\)
−0.199345 + 0.979929i \(0.563882\pi\)
\(282\) 0 0
\(283\) −384.614 −0.0807878 −0.0403939 0.999184i \(-0.512861\pi\)
−0.0403939 + 0.999184i \(0.512861\pi\)
\(284\) 2176.00 0.454654
\(285\) 0 0
\(286\) −2626.63 −0.543063
\(287\) 0 0
\(288\) 0 0
\(289\) −1745.00 −0.355180
\(290\) −3114.44 −0.630641
\(291\) 0 0
\(292\) 2401.49 0.481290
\(293\) −3742.95 −0.746299 −0.373149 0.927771i \(-0.621722\pi\)
−0.373149 + 0.927771i \(0.621722\pi\)
\(294\) 0 0
\(295\) −6248.00 −1.23313
\(296\) 624.000 0.122531
\(297\) 0 0
\(298\) −1892.00 −0.367787
\(299\) 3151.96 0.609641
\(300\) 0 0
\(301\) 0 0
\(302\) −1664.00 −0.317061
\(303\) 0 0
\(304\) −150.093 −0.0283172
\(305\) −2552.00 −0.479105
\(306\) 0 0
\(307\) −722.324 −0.134284 −0.0671420 0.997743i \(-0.521388\pi\)
−0.0671420 + 0.997743i \(0.521388\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −3872.00 −0.709403
\(311\) 7279.53 1.32728 0.663640 0.748052i \(-0.269011\pi\)
0.663640 + 0.748052i \(0.269011\pi\)
\(312\) 0 0
\(313\) −1519.69 −0.274435 −0.137218 0.990541i \(-0.543816\pi\)
−0.137218 + 0.990541i \(0.543816\pi\)
\(314\) 5759.83 1.03518
\(315\) 0 0
\(316\) −2720.00 −0.484215
\(317\) −2358.00 −0.417787 −0.208893 0.977938i \(-0.566986\pi\)
−0.208893 + 0.977938i \(0.566986\pi\)
\(318\) 0 0
\(319\) −3320.00 −0.582709
\(320\) 600.373 0.104881
\(321\) 0 0
\(322\) 0 0
\(323\) −528.000 −0.0909557
\(324\) 0 0
\(325\) 2429.64 0.414683
\(326\) −1272.00 −0.216103
\(327\) 0 0
\(328\) −3151.96 −0.530603
\(329\) 0 0
\(330\) 0 0
\(331\) 2372.00 0.393888 0.196944 0.980415i \(-0.436898\pi\)
0.196944 + 0.980415i \(0.436898\pi\)
\(332\) 787.990 0.130261
\(333\) 0 0
\(334\) 1313.32 0.215154
\(335\) 5440.88 0.887365
\(336\) 0 0
\(337\) −250.000 −0.0404106 −0.0202053 0.999796i \(-0.506432\pi\)
−0.0202053 + 0.999796i \(0.506432\pi\)
\(338\) −4230.00 −0.680715
\(339\) 0 0
\(340\) 2112.00 0.336880
\(341\) −4127.57 −0.655485
\(342\) 0 0
\(343\) 0 0
\(344\) −3488.00 −0.546687
\(345\) 0 0
\(346\) −1332.08 −0.206974
\(347\) −9540.00 −1.47589 −0.737945 0.674861i \(-0.764204\pi\)
−0.737945 + 0.674861i \(0.764204\pi\)
\(348\) 0 0
\(349\) 5712.93 0.876235 0.438117 0.898918i \(-0.355645\pi\)
0.438117 + 0.898918i \(0.355645\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 640.000 0.0969094
\(353\) 4390.23 0.661950 0.330975 0.943640i \(-0.392622\pi\)
0.330975 + 0.943640i \(0.392622\pi\)
\(354\) 0 0
\(355\) 5103.17 0.762953
\(356\) −6003.73 −0.893812
\(357\) 0 0
\(358\) −6456.00 −0.953101
\(359\) −1840.00 −0.270506 −0.135253 0.990811i \(-0.543185\pi\)
−0.135253 + 0.990811i \(0.543185\pi\)
\(360\) 0 0
\(361\) −6771.00 −0.987170
\(362\) −5647.26 −0.819927
\(363\) 0 0
\(364\) 0 0
\(365\) 5632.00 0.807650
\(366\) 0 0
\(367\) 2964.34 0.421628 0.210814 0.977526i \(-0.432389\pi\)
0.210814 + 0.977526i \(0.432389\pi\)
\(368\) −768.000 −0.108790
\(369\) 0 0
\(370\) 1463.41 0.205619
\(371\) 0 0
\(372\) 0 0
\(373\) 3982.00 0.552762 0.276381 0.961048i \(-0.410865\pi\)
0.276381 + 0.961048i \(0.410865\pi\)
\(374\) 2251.40 0.311276
\(375\) 0 0
\(376\) −1651.03 −0.226450
\(377\) −10900.5 −1.48914
\(378\) 0 0
\(379\) 2676.00 0.362683 0.181342 0.983420i \(-0.441956\pi\)
0.181342 + 0.983420i \(0.441956\pi\)
\(380\) −352.000 −0.0475190
\(381\) 0 0
\(382\) −4272.00 −0.572185
\(383\) 7035.62 0.938652 0.469326 0.883025i \(-0.344497\pi\)
0.469326 + 0.883025i \(0.344497\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −3316.00 −0.437254
\(387\) 0 0
\(388\) 2626.63 0.343678
\(389\) −8658.00 −1.12848 −0.564239 0.825611i \(-0.690830\pi\)
−0.564239 + 0.825611i \(0.690830\pi\)
\(390\) 0 0
\(391\) −2701.68 −0.349437
\(392\) 0 0
\(393\) 0 0
\(394\) −1956.00 −0.250106
\(395\) −6378.97 −0.812558
\(396\) 0 0
\(397\) 9052.50 1.14441 0.572207 0.820109i \(-0.306088\pi\)
0.572207 + 0.820109i \(0.306088\pi\)
\(398\) −9868.63 −1.24289
\(399\) 0 0
\(400\) −592.000 −0.0740000
\(401\) 5706.00 0.710584 0.355292 0.934755i \(-0.384381\pi\)
0.355292 + 0.934755i \(0.384381\pi\)
\(402\) 0 0
\(403\) −13552.0 −1.67512
\(404\) 487.803 0.0600721
\(405\) 0 0
\(406\) 0 0
\(407\) 1560.00 0.189991
\(408\) 0 0
\(409\) −2420.25 −0.292601 −0.146301 0.989240i \(-0.546737\pi\)
−0.146301 + 0.989240i \(0.546737\pi\)
\(410\) −7392.00 −0.890402
\(411\) 0 0
\(412\) 5478.41 0.655101
\(413\) 0 0
\(414\) 0 0
\(415\) 1848.00 0.218590
\(416\) 2101.31 0.247656
\(417\) 0 0
\(418\) −375.233 −0.0439073
\(419\) −1510.31 −0.176095 −0.0880473 0.996116i \(-0.528063\pi\)
−0.0880473 + 0.996116i \(0.528063\pi\)
\(420\) 0 0
\(421\) −16770.0 −1.94138 −0.970689 0.240341i \(-0.922741\pi\)
−0.970689 + 0.240341i \(0.922741\pi\)
\(422\) −3112.00 −0.358981
\(423\) 0 0
\(424\) 496.000 0.0568111
\(425\) −2082.54 −0.237690
\(426\) 0 0
\(427\) 0 0
\(428\) 1040.00 0.117454
\(429\) 0 0
\(430\) −8180.09 −0.917392
\(431\) −1336.00 −0.149311 −0.0746553 0.997209i \(-0.523786\pi\)
−0.0746553 + 0.997209i \(0.523786\pi\)
\(432\) 0 0
\(433\) 11163.2 1.23896 0.619479 0.785013i \(-0.287344\pi\)
0.619479 + 0.785013i \(0.287344\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 7528.00 0.826894
\(437\) 450.280 0.0492902
\(438\) 0 0
\(439\) 3602.24 0.391630 0.195815 0.980641i \(-0.437265\pi\)
0.195815 + 0.980641i \(0.437265\pi\)
\(440\) 1500.93 0.162623
\(441\) 0 0
\(442\) 7392.00 0.795479
\(443\) −6348.00 −0.680818 −0.340409 0.940277i \(-0.610566\pi\)
−0.340409 + 0.940277i \(0.610566\pi\)
\(444\) 0 0
\(445\) −14080.0 −1.49990
\(446\) 5778.59 0.613507
\(447\) 0 0
\(448\) 0 0
\(449\) −7170.00 −0.753615 −0.376808 0.926292i \(-0.622978\pi\)
−0.376808 + 0.926292i \(0.622978\pi\)
\(450\) 0 0
\(451\) −7879.90 −0.822727
\(452\) 5144.00 0.535295
\(453\) 0 0
\(454\) 3958.71 0.409232
\(455\) 0 0
\(456\) 0 0
\(457\) 6866.00 0.702796 0.351398 0.936226i \(-0.385706\pi\)
0.351398 + 0.936226i \(0.385706\pi\)
\(458\) 5534.69 0.564671
\(459\) 0 0
\(460\) −1801.12 −0.182560
\(461\) 1378.98 0.139318 0.0696590 0.997571i \(-0.477809\pi\)
0.0696590 + 0.997571i \(0.477809\pi\)
\(462\) 0 0
\(463\) 2648.00 0.265795 0.132897 0.991130i \(-0.457572\pi\)
0.132897 + 0.991130i \(0.457572\pi\)
\(464\) 2656.00 0.265736
\(465\) 0 0
\(466\) −12980.0 −1.29032
\(467\) −12335.8 −1.22234 −0.611170 0.791500i \(-0.709301\pi\)
−0.611170 + 0.791500i \(0.709301\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −3872.00 −0.380004
\(471\) 0 0
\(472\) 5328.31 0.519609
\(473\) −8720.00 −0.847666
\(474\) 0 0
\(475\) 347.091 0.0335276
\(476\) 0 0
\(477\) 0 0
\(478\) −8592.00 −0.822153
\(479\) −13339.5 −1.27244 −0.636221 0.771507i \(-0.719503\pi\)
−0.636221 + 0.771507i \(0.719503\pi\)
\(480\) 0 0
\(481\) 5121.93 0.485530
\(482\) 9043.12 0.854570
\(483\) 0 0
\(484\) −3724.00 −0.349737
\(485\) 6160.00 0.576724
\(486\) 0 0
\(487\) 13936.0 1.29672 0.648358 0.761336i \(-0.275456\pi\)
0.648358 + 0.761336i \(0.275456\pi\)
\(488\) 2176.35 0.201883
\(489\) 0 0
\(490\) 0 0
\(491\) 12276.0 1.12833 0.564163 0.825663i \(-0.309199\pi\)
0.564163 + 0.825663i \(0.309199\pi\)
\(492\) 0 0
\(493\) 9343.31 0.853553
\(494\) −1232.00 −0.112207
\(495\) 0 0
\(496\) 3302.05 0.298924
\(497\) 0 0
\(498\) 0 0
\(499\) −2220.00 −0.199160 −0.0995800 0.995030i \(-0.531750\pi\)
−0.0995800 + 0.995030i \(0.531750\pi\)
\(500\) −6078.78 −0.543703
\(501\) 0 0
\(502\) −11163.2 −0.992505
\(503\) −11294.5 −1.00119 −0.500594 0.865682i \(-0.666885\pi\)
−0.500594 + 0.865682i \(0.666885\pi\)
\(504\) 0 0
\(505\) 1144.00 0.100807
\(506\) −1920.00 −0.168685
\(507\) 0 0
\(508\) 9248.00 0.807704
\(509\) −15881.7 −1.38300 −0.691499 0.722377i \(-0.743049\pi\)
−0.691499 + 0.722377i \(0.743049\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −512.000 −0.0441942
\(513\) 0 0
\(514\) 3001.87 0.257600
\(515\) 12848.0 1.09932
\(516\) 0 0
\(517\) −4127.57 −0.351122
\(518\) 0 0
\(519\) 0 0
\(520\) 4928.00 0.415591
\(521\) 11613.5 0.976575 0.488287 0.872683i \(-0.337622\pi\)
0.488287 + 0.872683i \(0.337622\pi\)
\(522\) 0 0
\(523\) −12617.2 −1.05490 −0.527450 0.849586i \(-0.676852\pi\)
−0.527450 + 0.849586i \(0.676852\pi\)
\(524\) −1013.13 −0.0844633
\(525\) 0 0
\(526\) −800.000 −0.0663149
\(527\) 11616.0 0.960154
\(528\) 0 0
\(529\) −9863.00 −0.810635
\(530\) 1163.22 0.0953343
\(531\) 0 0
\(532\) 0 0
\(533\) −25872.0 −2.10252
\(534\) 0 0
\(535\) 2439.02 0.197099
\(536\) −4640.00 −0.373913
\(537\) 0 0
\(538\) 544.088 0.0436009
\(539\) 0 0
\(540\) 0 0
\(541\) 1798.00 0.142887 0.0714437 0.997445i \(-0.477239\pi\)
0.0714437 + 0.997445i \(0.477239\pi\)
\(542\) 13808.6 1.09433
\(543\) 0 0
\(544\) −1801.12 −0.141953
\(545\) 17654.7 1.38761
\(546\) 0 0
\(547\) 1276.00 0.0997401 0.0498700 0.998756i \(-0.484119\pi\)
0.0498700 + 0.998756i \(0.484119\pi\)
\(548\) 4456.00 0.347356
\(549\) 0 0
\(550\) −1480.00 −0.114741
\(551\) −1557.22 −0.120399
\(552\) 0 0
\(553\) 0 0
\(554\) 13540.0 1.03837
\(555\) 0 0
\(556\) −5515.93 −0.420733
\(557\) −2694.00 −0.204934 −0.102467 0.994736i \(-0.532674\pi\)
−0.102467 + 0.994736i \(0.532674\pi\)
\(558\) 0 0
\(559\) −28630.3 −2.16625
\(560\) 0 0
\(561\) 0 0
\(562\) 3756.00 0.281917
\(563\) −15769.2 −1.18045 −0.590223 0.807240i \(-0.700960\pi\)
−0.590223 + 0.807240i \(0.700960\pi\)
\(564\) 0 0
\(565\) 12063.7 0.898276
\(566\) 769.228 0.0571256
\(567\) 0 0
\(568\) −4352.00 −0.321489
\(569\) −12606.0 −0.928772 −0.464386 0.885633i \(-0.653725\pi\)
−0.464386 + 0.885633i \(0.653725\pi\)
\(570\) 0 0
\(571\) 6852.00 0.502184 0.251092 0.967963i \(-0.419210\pi\)
0.251092 + 0.967963i \(0.419210\pi\)
\(572\) 5253.27 0.384004
\(573\) 0 0
\(574\) 0 0
\(575\) 1776.00 0.128808
\(576\) 0 0
\(577\) −14371.4 −1.03690 −0.518449 0.855108i \(-0.673491\pi\)
−0.518449 + 0.855108i \(0.673491\pi\)
\(578\) 3490.00 0.251150
\(579\) 0 0
\(580\) 6228.87 0.445931
\(581\) 0 0
\(582\) 0 0
\(583\) 1240.00 0.0880884
\(584\) −4802.99 −0.340324
\(585\) 0 0
\(586\) 7485.90 0.527713
\(587\) 18977.4 1.33438 0.667191 0.744887i \(-0.267497\pi\)
0.667191 + 0.744887i \(0.267497\pi\)
\(588\) 0 0
\(589\) −1936.00 −0.135435
\(590\) 12496.0 0.871953
\(591\) 0 0
\(592\) −1248.00 −0.0866427
\(593\) 8217.61 0.569067 0.284534 0.958666i \(-0.408161\pi\)
0.284534 + 0.958666i \(0.408161\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 3784.00 0.260065
\(597\) 0 0
\(598\) −6303.92 −0.431081
\(599\) 19104.0 1.30312 0.651559 0.758598i \(-0.274115\pi\)
0.651559 + 0.758598i \(0.274115\pi\)
\(600\) 0 0
\(601\) −21538.4 −1.46185 −0.730923 0.682460i \(-0.760910\pi\)
−0.730923 + 0.682460i \(0.760910\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 3328.00 0.224196
\(605\) −8733.55 −0.586892
\(606\) 0 0
\(607\) −13733.5 −0.918331 −0.459166 0.888351i \(-0.651852\pi\)
−0.459166 + 0.888351i \(0.651852\pi\)
\(608\) 300.187 0.0200233
\(609\) 0 0
\(610\) 5104.00 0.338779
\(611\) −13552.0 −0.897308
\(612\) 0 0
\(613\) 28034.0 1.84712 0.923558 0.383458i \(-0.125267\pi\)
0.923558 + 0.383458i \(0.125267\pi\)
\(614\) 1444.65 0.0949532
\(615\) 0 0
\(616\) 0 0
\(617\) 8258.00 0.538824 0.269412 0.963025i \(-0.413171\pi\)
0.269412 + 0.963025i \(0.413171\pi\)
\(618\) 0 0
\(619\) 5131.31 0.333191 0.166595 0.986025i \(-0.446723\pi\)
0.166595 + 0.986025i \(0.446723\pi\)
\(620\) 7744.00 0.501623
\(621\) 0 0
\(622\) −14559.1 −0.938529
\(623\) 0 0
\(624\) 0 0
\(625\) −9631.00 −0.616384
\(626\) 3039.39 0.194055
\(627\) 0 0
\(628\) −11519.7 −0.731982
\(629\) −4390.23 −0.278299
\(630\) 0 0
\(631\) 912.000 0.0575375 0.0287687 0.999586i \(-0.490841\pi\)
0.0287687 + 0.999586i \(0.490841\pi\)
\(632\) 5440.00 0.342392
\(633\) 0 0
\(634\) 4716.00 0.295420
\(635\) 21688.5 1.35540
\(636\) 0 0
\(637\) 0 0
\(638\) 6640.00 0.412038
\(639\) 0 0
\(640\) −1200.75 −0.0741620
\(641\) 890.000 0.0548407 0.0274203 0.999624i \(-0.491271\pi\)
0.0274203 + 0.999624i \(0.491271\pi\)
\(642\) 0 0
\(643\) 29352.6 1.80024 0.900120 0.435642i \(-0.143479\pi\)
0.900120 + 0.435642i \(0.143479\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 1056.00 0.0643154
\(647\) 11876.1 0.721637 0.360818 0.932636i \(-0.382497\pi\)
0.360818 + 0.932636i \(0.382497\pi\)
\(648\) 0 0
\(649\) 13320.8 0.805680
\(650\) −4859.27 −0.293225
\(651\) 0 0
\(652\) 2544.00 0.152808
\(653\) 21526.0 1.29001 0.645006 0.764178i \(-0.276855\pi\)
0.645006 + 0.764178i \(0.276855\pi\)
\(654\) 0 0
\(655\) −2376.00 −0.141737
\(656\) 6303.92 0.375193
\(657\) 0 0
\(658\) 0 0
\(659\) −23452.0 −1.38628 −0.693141 0.720802i \(-0.743774\pi\)
−0.693141 + 0.720802i \(0.743774\pi\)
\(660\) 0 0
\(661\) 26669.7 1.56934 0.784668 0.619916i \(-0.212833\pi\)
0.784668 + 0.619916i \(0.212833\pi\)
\(662\) −4744.00 −0.278521
\(663\) 0 0
\(664\) −1575.98 −0.0921082
\(665\) 0 0
\(666\) 0 0
\(667\) −7968.00 −0.462552
\(668\) −2626.63 −0.152137
\(669\) 0 0
\(670\) −10881.8 −0.627462
\(671\) 5440.88 0.313030
\(672\) 0 0
\(673\) −13858.0 −0.793739 −0.396870 0.917875i \(-0.629904\pi\)
−0.396870 + 0.917875i \(0.629904\pi\)
\(674\) 500.000 0.0285746
\(675\) 0 0
\(676\) 8460.00 0.481338
\(677\) 32448.3 1.84208 0.921041 0.389466i \(-0.127340\pi\)
0.921041 + 0.389466i \(0.127340\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −4224.00 −0.238210
\(681\) 0 0
\(682\) 8255.13 0.463498
\(683\) 27812.0 1.55812 0.779060 0.626949i \(-0.215696\pi\)
0.779060 + 0.626949i \(0.215696\pi\)
\(684\) 0 0
\(685\) 10450.2 0.582895
\(686\) 0 0
\(687\) 0 0
\(688\) 6976.00 0.386566
\(689\) 4071.28 0.225114
\(690\) 0 0
\(691\) 1303.94 0.0717859 0.0358929 0.999356i \(-0.488572\pi\)
0.0358929 + 0.999356i \(0.488572\pi\)
\(692\) 2664.16 0.146353
\(693\) 0 0
\(694\) 19080.0 1.04361
\(695\) −12936.0 −0.706029
\(696\) 0 0
\(697\) 22176.0 1.20513
\(698\) −11425.9 −0.619592
\(699\) 0 0
\(700\) 0 0
\(701\) −22906.0 −1.23416 −0.617081 0.786900i \(-0.711685\pi\)
−0.617081 + 0.786900i \(0.711685\pi\)
\(702\) 0 0
\(703\) 731.705 0.0392557
\(704\) −1280.00 −0.0685253
\(705\) 0 0
\(706\) −8780.46 −0.468069
\(707\) 0 0
\(708\) 0 0
\(709\) −15086.0 −0.799107 −0.399553 0.916710i \(-0.630835\pi\)
−0.399553 + 0.916710i \(0.630835\pi\)
\(710\) −10206.3 −0.539489
\(711\) 0 0
\(712\) 12007.5 0.632021
\(713\) −9906.16 −0.520321
\(714\) 0 0
\(715\) 12320.0 0.644394
\(716\) 12912.0 0.673944
\(717\) 0 0
\(718\) 3680.00 0.191276
\(719\) −20544.0 −1.06559 −0.532797 0.846243i \(-0.678859\pi\)
−0.532797 + 0.846243i \(0.678859\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 13542.0 0.698035
\(723\) 0 0
\(724\) 11294.5 0.579776
\(725\) −6142.00 −0.314632
\(726\) 0 0
\(727\) 7223.24 0.368494 0.184247 0.982880i \(-0.441015\pi\)
0.184247 + 0.982880i \(0.441015\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −11264.0 −0.571095
\(731\) 24540.3 1.24166
\(732\) 0 0
\(733\) −29427.7 −1.48286 −0.741430 0.671031i \(-0.765852\pi\)
−0.741430 + 0.671031i \(0.765852\pi\)
\(734\) −5928.69 −0.298136
\(735\) 0 0
\(736\) 1536.00 0.0769262
\(737\) −11600.0 −0.579771
\(738\) 0 0
\(739\) 32668.0 1.62613 0.813066 0.582171i \(-0.197797\pi\)
0.813066 + 0.582171i \(0.197797\pi\)
\(740\) −2926.82 −0.145395
\(741\) 0 0
\(742\) 0 0
\(743\) 37056.0 1.82968 0.914840 0.403816i \(-0.132316\pi\)
0.914840 + 0.403816i \(0.132316\pi\)
\(744\) 0 0
\(745\) 8874.27 0.436413
\(746\) −7964.00 −0.390862
\(747\) 0 0
\(748\) −4502.80 −0.220105
\(749\) 0 0
\(750\) 0 0
\(751\) −19608.0 −0.952738 −0.476369 0.879246i \(-0.658047\pi\)
−0.476369 + 0.879246i \(0.658047\pi\)
\(752\) 3302.05 0.160124
\(753\) 0 0
\(754\) 21801.1 1.05298
\(755\) 7804.85 0.376222
\(756\) 0 0
\(757\) 19378.0 0.930390 0.465195 0.885208i \(-0.345984\pi\)
0.465195 + 0.885208i \(0.345984\pi\)
\(758\) −5352.00 −0.256456
\(759\) 0 0
\(760\) 704.000 0.0336010
\(761\) −13977.4 −0.665810 −0.332905 0.942960i \(-0.608029\pi\)
−0.332905 + 0.942960i \(0.608029\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 8544.00 0.404596
\(765\) 0 0
\(766\) −14071.2 −0.663727
\(767\) 43736.0 2.05895
\(768\) 0 0
\(769\) −8536.56 −0.400307 −0.200154 0.979765i \(-0.564144\pi\)
−0.200154 + 0.979765i \(0.564144\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 6632.00 0.309185
\(773\) −29296.3 −1.36315 −0.681576 0.731748i \(-0.738705\pi\)
−0.681576 + 0.731748i \(0.738705\pi\)
\(774\) 0 0
\(775\) −7636.00 −0.353927
\(776\) −5253.27 −0.243017
\(777\) 0 0
\(778\) 17316.0 0.797955
\(779\) −3696.00 −0.169991
\(780\) 0 0
\(781\) −10880.0 −0.498485
\(782\) 5403.36 0.247089
\(783\) 0 0
\(784\) 0 0
\(785\) −27016.0 −1.22833
\(786\) 0 0
\(787\) −13780.4 −0.624167 −0.312084 0.950055i \(-0.601027\pi\)
−0.312084 + 0.950055i \(0.601027\pi\)
\(788\) 3912.00 0.176852
\(789\) 0 0
\(790\) 12757.9 0.574566
\(791\) 0 0
\(792\) 0 0
\(793\) 17864.0 0.799961
\(794\) −18105.0 −0.809222
\(795\) 0 0
\(796\) 19737.3 0.878855
\(797\) −34868.6 −1.54970 −0.774848 0.632148i \(-0.782174\pi\)
−0.774848 + 0.632148i \(0.782174\pi\)
\(798\) 0 0
\(799\) 11616.0 0.514324
\(800\) 1184.00 0.0523259
\(801\) 0 0
\(802\) −11412.0 −0.502459
\(803\) −12007.5 −0.527689
\(804\) 0 0
\(805\) 0 0
\(806\) 27104.0 1.18449
\(807\) 0 0
\(808\) −975.606 −0.0424774
\(809\) −14034.0 −0.609900 −0.304950 0.952368i \(-0.598640\pi\)
−0.304950 + 0.952368i \(0.598640\pi\)
\(810\) 0 0
\(811\) 6632.25 0.287164 0.143582 0.989638i \(-0.454138\pi\)
0.143582 + 0.989638i \(0.454138\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −3120.00 −0.134344
\(815\) 5966.21 0.256426
\(816\) 0 0
\(817\) −4090.04 −0.175144
\(818\) 4840.51 0.206900
\(819\) 0 0
\(820\) 14784.0 0.629609
\(821\) −28622.0 −1.21670 −0.608352 0.793667i \(-0.708169\pi\)
−0.608352 + 0.793667i \(0.708169\pi\)
\(822\) 0 0
\(823\) 24688.0 1.04565 0.522825 0.852440i \(-0.324878\pi\)
0.522825 + 0.852440i \(0.324878\pi\)
\(824\) −10956.8 −0.463226
\(825\) 0 0
\(826\) 0 0
\(827\) 30756.0 1.29322 0.646609 0.762822i \(-0.276187\pi\)
0.646609 + 0.762822i \(0.276187\pi\)
\(828\) 0 0
\(829\) −23236.3 −0.973499 −0.486750 0.873542i \(-0.661818\pi\)
−0.486750 + 0.873542i \(0.661818\pi\)
\(830\) −3696.00 −0.154566
\(831\) 0 0
\(832\) −4202.61 −0.175119
\(833\) 0 0
\(834\) 0 0
\(835\) −6160.00 −0.255300
\(836\) 750.467 0.0310472
\(837\) 0 0
\(838\) 3020.63 0.124518
\(839\) −24033.7 −0.988957 −0.494479 0.869190i \(-0.664641\pi\)
−0.494479 + 0.869190i \(0.664641\pi\)
\(840\) 0 0
\(841\) 3167.00 0.129854
\(842\) 33540.0 1.37276
\(843\) 0 0
\(844\) 6224.00 0.253838
\(845\) 19840.5 0.807731
\(846\) 0 0
\(847\) 0 0
\(848\) −992.000 −0.0401715
\(849\) 0 0
\(850\) 4165.09 0.168072
\(851\) 3744.00 0.150814
\(852\) 0 0
\(853\) 23574.0 0.946260 0.473130 0.880993i \(-0.343124\pi\)
0.473130 + 0.880993i \(0.343124\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −2080.00 −0.0830525
\(857\) 24484.0 0.975912 0.487956 0.872868i \(-0.337743\pi\)
0.487956 + 0.872868i \(0.337743\pi\)
\(858\) 0 0
\(859\) 32954.9 1.30897 0.654485 0.756075i \(-0.272885\pi\)
0.654485 + 0.756075i \(0.272885\pi\)
\(860\) 16360.2 0.648694
\(861\) 0 0
\(862\) 2672.00 0.105579
\(863\) −40872.0 −1.61217 −0.806083 0.591803i \(-0.798416\pi\)
−0.806083 + 0.591803i \(0.798416\pi\)
\(864\) 0 0
\(865\) 6248.00 0.245593
\(866\) −22326.4 −0.876075
\(867\) 0 0
\(868\) 0 0
\(869\) 13600.0 0.530896
\(870\) 0 0
\(871\) −38086.2 −1.48163
\(872\) −15056.0 −0.584702
\(873\) 0 0
\(874\) −900.560 −0.0348534
\(875\) 0 0
\(876\) 0 0
\(877\) −12006.0 −0.462273 −0.231137 0.972921i \(-0.574244\pi\)
−0.231137 + 0.972921i \(0.574244\pi\)
\(878\) −7204.48 −0.276924
\(879\) 0 0
\(880\) −3001.87 −0.114992
\(881\) 35722.2 1.36607 0.683037 0.730383i \(-0.260659\pi\)
0.683037 + 0.730383i \(0.260659\pi\)
\(882\) 0 0
\(883\) 19588.0 0.746533 0.373267 0.927724i \(-0.378238\pi\)
0.373267 + 0.927724i \(0.378238\pi\)
\(884\) −14784.0 −0.562488
\(885\) 0 0
\(886\) 12696.0 0.481411
\(887\) −40243.8 −1.52340 −0.761699 0.647931i \(-0.775634\pi\)
−0.761699 + 0.647931i \(0.775634\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 28160.0 1.06059
\(891\) 0 0
\(892\) −11557.2 −0.433815
\(893\) −1936.00 −0.0725485
\(894\) 0 0
\(895\) 30281.3 1.13094
\(896\) 0 0
\(897\) 0 0
\(898\) 14340.0 0.532886
\(899\) 34258.8 1.27096
\(900\) 0 0
\(901\) −3489.67 −0.129032
\(902\) 15759.8 0.581756
\(903\) 0 0
\(904\) −10288.0 −0.378511
\(905\) 26488.0 0.972918
\(906\) 0 0
\(907\) 15868.0 0.580913 0.290457 0.956888i \(-0.406193\pi\)
0.290457 + 0.956888i \(0.406193\pi\)
\(908\) −7917.42 −0.289371
\(909\) 0 0
\(910\) 0 0
\(911\) −39832.0 −1.44862 −0.724310 0.689474i \(-0.757842\pi\)
−0.724310 + 0.689474i \(0.757842\pi\)
\(912\) 0 0
\(913\) −3939.95 −0.142818
\(914\) −13732.0 −0.496952
\(915\) 0 0
\(916\) −11069.4 −0.399282
\(917\) 0 0
\(918\) 0 0
\(919\) −30528.0 −1.09578 −0.547892 0.836549i \(-0.684570\pi\)
−0.547892 + 0.836549i \(0.684570\pi\)
\(920\) 3602.24 0.129089
\(921\) 0 0
\(922\) −2757.96 −0.0985127
\(923\) −35722.2 −1.27390
\(924\) 0 0
\(925\) 2886.00 0.102585
\(926\) −5296.00 −0.187945
\(927\) 0 0
\(928\) −5312.00 −0.187904
\(929\) −16604.1 −0.586396 −0.293198 0.956052i \(-0.594720\pi\)
−0.293198 + 0.956052i \(0.594720\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 25960.0 0.912391
\(933\) 0 0
\(934\) 24671.6 0.864324
\(935\) −10560.0 −0.369357
\(936\) 0 0
\(937\) −29943.6 −1.04399 −0.521993 0.852950i \(-0.674811\pi\)
−0.521993 + 0.852950i \(0.674811\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 7744.00 0.268704
\(941\) 5375.22 0.186214 0.0931068 0.995656i \(-0.470320\pi\)
0.0931068 + 0.995656i \(0.470320\pi\)
\(942\) 0 0
\(943\) −18911.8 −0.653077
\(944\) −10656.6 −0.367419
\(945\) 0 0
\(946\) 17440.0 0.599390
\(947\) −45212.0 −1.55142 −0.775709 0.631091i \(-0.782607\pi\)
−0.775709 + 0.631091i \(0.782607\pi\)
\(948\) 0 0
\(949\) −39424.0 −1.34853
\(950\) −694.182 −0.0237076
\(951\) 0 0
\(952\) 0 0
\(953\) −34218.0 −1.16310 −0.581548 0.813512i \(-0.697553\pi\)
−0.581548 + 0.813512i \(0.697553\pi\)
\(954\) 0 0
\(955\) 20037.5 0.678950
\(956\) 17184.0 0.581350
\(957\) 0 0
\(958\) 26679.1 0.899752
\(959\) 0 0
\(960\) 0 0
\(961\) 12801.0 0.429694
\(962\) −10243.9 −0.343322
\(963\) 0 0
\(964\) −18086.2 −0.604272
\(965\) 15553.4 0.518842
\(966\) 0 0
\(967\) 14464.0 0.481004 0.240502 0.970649i \(-0.422688\pi\)
0.240502 + 0.970649i \(0.422688\pi\)
\(968\) 7448.00 0.247301
\(969\) 0 0
\(970\) −12320.0 −0.407806
\(971\) 37832.9 1.25038 0.625188 0.780474i \(-0.285022\pi\)
0.625188 + 0.780474i \(0.285022\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −27872.0 −0.916916
\(975\) 0 0
\(976\) −4352.71 −0.142753
\(977\) −42062.0 −1.37736 −0.688681 0.725065i \(-0.741810\pi\)
−0.688681 + 0.725065i \(0.741810\pi\)
\(978\) 0 0
\(979\) 30018.7 0.979980
\(980\) 0 0
\(981\) 0 0
\(982\) −24552.0 −0.797847
\(983\) −43020.5 −1.39587 −0.697935 0.716161i \(-0.745898\pi\)
−0.697935 + 0.716161i \(0.745898\pi\)
\(984\) 0 0
\(985\) 9174.45 0.296774
\(986\) −18686.6 −0.603553
\(987\) 0 0
\(988\) 2464.00 0.0793424
\(989\) −20928.0 −0.672873
\(990\) 0 0
\(991\) 21272.0 0.681864 0.340932 0.940088i \(-0.389257\pi\)
0.340932 + 0.940088i \(0.389257\pi\)
\(992\) −6604.11 −0.211372
\(993\) 0 0
\(994\) 0 0
\(995\) 46288.0 1.47480
\(996\) 0 0
\(997\) −121.951 −0.00387384 −0.00193692 0.999998i \(-0.500617\pi\)
−0.00193692 + 0.999998i \(0.500617\pi\)
\(998\) 4440.00 0.140827
\(999\) 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 882.4.a.w.1.2 2
3.2 odd 2 98.4.a.h.1.2 yes 2
7.2 even 3 882.4.g.bi.361.1 4
7.3 odd 6 882.4.g.bi.667.2 4
7.4 even 3 882.4.g.bi.667.1 4
7.5 odd 6 882.4.g.bi.361.2 4
7.6 odd 2 inner 882.4.a.w.1.1 2
12.11 even 2 784.4.a.z.1.1 2
15.14 odd 2 2450.4.a.bs.1.1 2
21.2 odd 6 98.4.c.g.67.1 4
21.5 even 6 98.4.c.g.67.2 4
21.11 odd 6 98.4.c.g.79.1 4
21.17 even 6 98.4.c.g.79.2 4
21.20 even 2 98.4.a.h.1.1 2
84.83 odd 2 784.4.a.z.1.2 2
105.104 even 2 2450.4.a.bs.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
98.4.a.h.1.1 2 21.20 even 2
98.4.a.h.1.2 yes 2 3.2 odd 2
98.4.c.g.67.1 4 21.2 odd 6
98.4.c.g.67.2 4 21.5 even 6
98.4.c.g.79.1 4 21.11 odd 6
98.4.c.g.79.2 4 21.17 even 6
784.4.a.z.1.1 2 12.11 even 2
784.4.a.z.1.2 2 84.83 odd 2
882.4.a.w.1.1 2 7.6 odd 2 inner
882.4.a.w.1.2 2 1.1 even 1 trivial
882.4.g.bi.361.1 4 7.2 even 3
882.4.g.bi.361.2 4 7.5 odd 6
882.4.g.bi.667.1 4 7.4 even 3
882.4.g.bi.667.2 4 7.3 odd 6
2450.4.a.bs.1.1 2 15.14 odd 2
2450.4.a.bs.1.2 2 105.104 even 2