Properties

Label 2450.4.a.bs.1.1
Level $2450$
Weight $4$
Character 2450.1
Self dual yes
Analytic conductor $144.555$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2450.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(144.554679514\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{22}) \)
Defining polynomial: \(x^{2} - 22\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 1.1
Root \(-4.69042\) of defining polynomial
Character \(\chi\) \(=\) 2450.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.00000 q^{2} -9.38083 q^{3} +4.00000 q^{4} +18.7617 q^{6} -8.00000 q^{8} +61.0000 q^{9} +O(q^{10})\) \(q-2.00000 q^{2} -9.38083 q^{3} +4.00000 q^{4} +18.7617 q^{6} -8.00000 q^{8} +61.0000 q^{9} +20.0000 q^{11} -37.5233 q^{12} +65.6658 q^{13} +16.0000 q^{16} +56.2850 q^{17} -122.000 q^{18} -9.38083 q^{19} -40.0000 q^{22} -48.0000 q^{23} +75.0467 q^{24} -131.332 q^{26} -318.948 q^{27} -166.000 q^{29} +206.378 q^{31} -32.0000 q^{32} -187.617 q^{33} -112.570 q^{34} +244.000 q^{36} +78.0000 q^{37} +18.7617 q^{38} -616.000 q^{39} -393.995 q^{41} -436.000 q^{43} +80.0000 q^{44} +96.0000 q^{46} +206.378 q^{47} -150.093 q^{48} -528.000 q^{51} +262.663 q^{52} -62.0000 q^{53} +637.897 q^{54} +88.0000 q^{57} +332.000 q^{58} +666.039 q^{59} -272.044 q^{61} -412.757 q^{62} +64.0000 q^{64} +375.233 q^{66} -580.000 q^{67} +225.140 q^{68} +450.280 q^{69} -544.000 q^{71} -488.000 q^{72} -600.373 q^{73} -156.000 q^{74} -37.5233 q^{76} +1232.00 q^{78} -680.000 q^{79} +1345.00 q^{81} +787.990 q^{82} +196.997 q^{83} +872.000 q^{86} +1557.22 q^{87} -160.000 q^{88} +1500.93 q^{89} -192.000 q^{92} -1936.00 q^{93} -412.757 q^{94} +300.187 q^{96} -656.658 q^{97} +1220.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{2} + 8q^{4} - 16q^{8} + 122q^{9} + O(q^{10}) \) \( 2q - 4q^{2} + 8q^{4} - 16q^{8} + 122q^{9} + 40q^{11} + 32q^{16} - 244q^{18} - 80q^{22} - 96q^{23} - 332q^{29} - 64q^{32} + 488q^{36} + 156q^{37} - 1232q^{39} - 872q^{43} + 160q^{44} + 192q^{46} - 1056q^{51} - 124q^{53} + 176q^{57} + 664q^{58} + 128q^{64} - 1160q^{67} - 1088q^{71} - 976q^{72} - 312q^{74} + 2464q^{78} - 1360q^{79} + 2690q^{81} + 1744q^{86} - 320q^{88} - 384q^{92} - 3872q^{93} + 2440q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −0.707107
\(3\) −9.38083 −1.80534 −0.902671 0.430331i \(-0.858397\pi\)
−0.902671 + 0.430331i \(0.858397\pi\)
\(4\) 4.00000 0.500000
\(5\) 0 0
\(6\) 18.7617 1.27657
\(7\) 0 0
\(8\) −8.00000 −0.353553
\(9\) 61.0000 2.25926
\(10\) 0 0
\(11\) 20.0000 0.548202 0.274101 0.961701i \(-0.411620\pi\)
0.274101 + 0.961701i \(0.411620\pi\)
\(12\) −37.5233 −0.902671
\(13\) 65.6658 1.40096 0.700478 0.713674i \(-0.252970\pi\)
0.700478 + 0.713674i \(0.252970\pi\)
\(14\) 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 \(0.368488\pi\)
\(18\) −122.000 −1.59754
\(19\) −9.38083 −0.113269 −0.0566345 0.998395i \(-0.518037\pi\)
−0.0566345 + 0.998395i \(0.518037\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −40.0000 −0.387638
\(23\) −48.0000 −0.435161 −0.217580 0.976042i \(-0.569816\pi\)
−0.217580 + 0.976042i \(0.569816\pi\)
\(24\) 75.0467 0.638285
\(25\) 0 0
\(26\) −131.332 −0.990625
\(27\) −318.948 −2.27339
\(28\) 0 0
\(29\) −166.000 −1.06295 −0.531473 0.847075i \(-0.678361\pi\)
−0.531473 + 0.847075i \(0.678361\pi\)
\(30\) 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
\(33\) −187.617 −0.989693
\(34\) −112.570 −0.567812
\(35\) 0 0
\(36\) 244.000 1.12963
\(37\) 78.0000 0.346571 0.173285 0.984872i \(-0.444562\pi\)
0.173285 + 0.984872i \(0.444562\pi\)
\(38\) 18.7617 0.0800933
\(39\) −616.000 −2.52920
\(40\) 0 0
\(41\) −393.995 −1.50077 −0.750386 0.661000i \(-0.770132\pi\)
−0.750386 + 0.661000i \(0.770132\pi\)
\(42\) 0 0
\(43\) −436.000 −1.54626 −0.773132 0.634245i \(-0.781311\pi\)
−0.773132 + 0.634245i \(0.781311\pi\)
\(44\) 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 \(0.396234\pi\)
\(48\) −150.093 −0.451335
\(49\) 0 0
\(50\) 0 0
\(51\) −528.000 −1.44970
\(52\) 262.663 0.700478
\(53\) −62.0000 −0.160686 −0.0803430 0.996767i \(-0.525602\pi\)
−0.0803430 + 0.996767i \(0.525602\pi\)
\(54\) 637.897 1.60753
\(55\) 0 0
\(56\) 0 0
\(57\) 88.0000 0.204489
\(58\) 332.000 0.751616
\(59\) 666.039 1.46968 0.734838 0.678243i \(-0.237258\pi\)
0.734838 + 0.678243i \(0.237258\pi\)
\(60\) 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
\(65\) 0 0
\(66\) 375.233 0.699819
\(67\) −580.000 −1.05759 −0.528793 0.848751i \(-0.677355\pi\)
−0.528793 + 0.848751i \(0.677355\pi\)
\(68\) 225.140 0.401503
\(69\) 450.280 0.785613
\(70\) 0 0
\(71\) −544.000 −0.909309 −0.454654 0.890668i \(-0.650237\pi\)
−0.454654 + 0.890668i \(0.650237\pi\)
\(72\) −488.000 −0.798769
\(73\) −600.373 −0.962580 −0.481290 0.876561i \(-0.659832\pi\)
−0.481290 + 0.876561i \(0.659832\pi\)
\(74\) −156.000 −0.245063
\(75\) 0 0
\(76\) −37.5233 −0.0566345
\(77\) 0 0
\(78\) 1232.00 1.78842
\(79\) −680.000 −0.968430 −0.484215 0.874949i \(-0.660895\pi\)
−0.484215 + 0.874949i \(0.660895\pi\)
\(80\) 0 0
\(81\) 1345.00 1.84499
\(82\) 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
\(85\) 0 0
\(86\) 872.000 1.09337
\(87\) 1557.22 1.91898
\(88\) −160.000 −0.193819
\(89\) 1500.93 1.78762 0.893812 0.448441i \(-0.148021\pi\)
0.893812 + 0.448441i \(0.148021\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −192.000 −0.217580
\(93\) −1936.00 −2.15864
\(94\) −412.757 −0.452900
\(95\) 0 0
\(96\) 300.187 0.319142
\(97\) −656.658 −0.687356 −0.343678 0.939088i \(-0.611673\pi\)
−0.343678 + 0.939088i \(0.611673\pi\)
\(98\) 0 0
\(99\) 1220.00 1.23853
\(100\) 0 0
\(101\) −121.951 −0.120144 −0.0600721 0.998194i \(-0.519133\pi\)
−0.0600721 + 0.998194i \(0.519133\pi\)
\(102\) 1056.00 1.02509
\(103\) −1369.60 −1.31020 −0.655101 0.755541i \(-0.727374\pi\)
−0.655101 + 0.755541i \(0.727374\pi\)
\(104\) −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 \(0.462527\pi\)
\(108\) −1275.79 −1.13670
\(109\) 1882.00 1.65379 0.826894 0.562358i \(-0.190106\pi\)
0.826894 + 0.562358i \(0.190106\pi\)
\(110\) 0 0
\(111\) −731.705 −0.625679
\(112\) 0 0
\(113\) 1286.00 1.07059 0.535295 0.844665i \(-0.320200\pi\)
0.535295 + 0.844665i \(0.320200\pi\)
\(114\) −176.000 −0.144596
\(115\) 0 0
\(116\) −664.000 −0.531473
\(117\) 4005.62 3.16512
\(118\) −1332.08 −1.03922
\(119\) 0 0
\(120\) 0 0
\(121\) −931.000 −0.699474
\(122\) 544.088 0.403766
\(123\) 3696.00 2.70941
\(124\) 825.513 0.597849
\(125\) 0 0
\(126\) 0 0
\(127\) −2312.00 −1.61541 −0.807704 0.589588i \(-0.799290\pi\)
−0.807704 + 0.589588i \(0.799290\pi\)
\(128\) −128.000 −0.0883883
\(129\) 4090.04 2.79154
\(130\) 0 0
\(131\) 253.282 0.168927 0.0844633 0.996427i \(-0.473082\pi\)
0.0844633 + 0.996427i \(0.473082\pi\)
\(132\) −750.467 −0.494846
\(133\) 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 \(0.387080\pi\)
\(138\) −900.560 −0.555513
\(139\) −1378.98 −0.841466 −0.420733 0.907185i \(-0.638227\pi\)
−0.420733 + 0.907185i \(0.638227\pi\)
\(140\) 0 0
\(141\) −1936.00 −1.15632
\(142\) 1088.00 0.642978
\(143\) 1313.32 0.768007
\(144\) 976.000 0.564815
\(145\) 0 0
\(146\) 1200.75 0.680647
\(147\) 0 0
\(148\) 312.000 0.173285
\(149\) −946.000 −0.520130 −0.260065 0.965591i \(-0.583744\pi\)
−0.260065 + 0.965591i \(0.583744\pi\)
\(150\) 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
\(153\) 3433.38 1.81420
\(154\) 0 0
\(155\) 0 0
\(156\) −2464.00 −1.26460
\(157\) 2879.92 1.46396 0.731982 0.681324i \(-0.238596\pi\)
0.731982 + 0.681324i \(0.238596\pi\)
\(158\) 1360.00 0.684783
\(159\) 581.612 0.290093
\(160\) 0 0
\(161\) 0 0
\(162\) −2690.00 −1.30461
\(163\) −636.000 −0.305616 −0.152808 0.988256i \(-0.548832\pi\)
−0.152808 + 0.988256i \(0.548832\pi\)
\(164\) −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
\(170\) 0 0
\(171\) −572.231 −0.255904
\(172\) −1744.00 −0.773132
\(173\) 666.039 0.292705 0.146353 0.989232i \(-0.453247\pi\)
0.146353 + 0.989232i \(0.453247\pi\)
\(174\) −3114.44 −1.35692
\(175\) 0 0
\(176\) 320.000 0.137051
\(177\) −6248.00 −2.65327
\(178\) −3001.87 −1.26404
\(179\) −3228.00 −1.34789 −0.673944 0.738782i \(-0.735401\pi\)
−0.673944 + 0.738782i \(0.735401\pi\)
\(180\) 0 0
\(181\) 2823.63 1.15955 0.579776 0.814776i \(-0.303140\pi\)
0.579776 + 0.814776i \(0.303140\pi\)
\(182\) 0 0
\(183\) 2552.00 1.03087
\(184\) 384.000 0.153852
\(185\) 0 0
\(186\) 3872.00 1.52639
\(187\) 1125.70 0.440210
\(188\) 825.513 0.320249
\(189\) 0 0
\(190\) 0 0
\(191\) −2136.00 −0.809191 −0.404596 0.914496i \(-0.632588\pi\)
−0.404596 + 0.914496i \(0.632588\pi\)
\(192\) −600.373 −0.225668
\(193\) −1658.00 −0.618370 −0.309185 0.951002i \(-0.600056\pi\)
−0.309185 + 0.951002i \(0.600056\pi\)
\(194\) 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 \(0.443409\pi\)
\(198\) −2440.00 −0.875774
\(199\) 4934.32 1.75771 0.878855 0.477088i \(-0.158308\pi\)
0.878855 + 0.477088i \(0.158308\pi\)
\(200\) 0 0
\(201\) 5440.88 1.90930
\(202\) 243.902 0.0849547
\(203\) 0 0
\(204\) −2112.00 −0.724851
\(205\) 0 0
\(206\) 2739.20 0.926453
\(207\) −2928.00 −0.983140
\(208\) 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
\(213\) 5103.17 1.64161
\(214\) −520.000 −0.166105
\(215\) 0 0
\(216\) 2551.59 0.803766
\(217\) 0 0
\(218\) −3764.00 −1.16940
\(219\) 5632.00 1.73779
\(220\) 0 0
\(221\) 3696.00 1.12498
\(222\) 1463.41 0.442422
\(223\) 2889.30 0.867630 0.433815 0.901002i \(-0.357167\pi\)
0.433815 + 0.901002i \(0.357167\pi\)
\(224\) 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 \(0.593446\pi\)
\(228\) 352.000 0.102245
\(229\) −2767.35 −0.798565 −0.399282 0.916828i \(-0.630741\pi\)
−0.399282 + 0.916828i \(0.630741\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 1328.00 0.375808
\(233\) 6490.00 1.82478 0.912391 0.409321i \(-0.134234\pi\)
0.912391 + 0.409321i \(0.134234\pi\)
\(234\) −8011.23 −2.23808
\(235\) 0 0
\(236\) 2664.16 0.734838
\(237\) 6378.97 1.74835
\(238\) 0 0
\(239\) −4296.00 −1.16270 −0.581350 0.813654i \(-0.697475\pi\)
−0.581350 + 0.813654i \(0.697475\pi\)
\(240\) 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
\(243\) −4005.62 −1.05745
\(244\) −1088.18 −0.285506
\(245\) 0 0
\(246\) −7392.00 −1.91584
\(247\) −616.000 −0.158685
\(248\) −1651.03 −0.422743
\(249\) −1848.00 −0.470330
\(250\) 0 0
\(251\) −5581.59 −1.40361 −0.701807 0.712367i \(-0.747623\pi\)
−0.701807 + 0.712367i \(0.747623\pi\)
\(252\) 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 \(0.558306\pi\)
\(258\) −8180.09 −1.97391
\(259\) 0 0
\(260\) 0 0
\(261\) −10126.0 −2.40147
\(262\) −506.565 −0.119449
\(263\) 400.000 0.0937835 0.0468917 0.998900i \(-0.485068\pi\)
0.0468917 + 0.998900i \(0.485068\pi\)
\(264\) 1500.93 0.349909
\(265\) 0 0
\(266\) 0 0
\(267\) −14080.0 −3.22727
\(268\) −2320.00 −0.528793
\(269\) 272.044 0.0616610 0.0308305 0.999525i \(-0.490185\pi\)
0.0308305 + 0.999525i \(0.490185\pi\)
\(270\) 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
\(275\) 0 0
\(276\) 1801.12 0.392807
\(277\) 6770.00 1.46848 0.734242 0.678888i \(-0.237538\pi\)
0.734242 + 0.678888i \(0.237538\pi\)
\(278\) 2757.96 0.595006
\(279\) 12589.1 2.70139
\(280\) 0 0
\(281\) 1878.00 0.398691 0.199345 0.979929i \(-0.436118\pi\)
0.199345 + 0.979929i \(0.436118\pi\)
\(282\) 3872.00 0.817639
\(283\) 384.614 0.0807878 0.0403939 0.999184i \(-0.487139\pi\)
0.0403939 + 0.999184i \(0.487139\pi\)
\(284\) −2176.00 −0.454654
\(285\) 0 0
\(286\) −2626.63 −0.543063
\(287\) 0 0
\(288\) −1952.00 −0.399384
\(289\) −1745.00 −0.355180
\(290\) 0 0
\(291\) 6160.00 1.24091
\(292\) −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
\(295\) 0 0
\(296\) −624.000 −0.122531
\(297\) −6378.97 −1.24628
\(298\) 1892.00 0.367787
\(299\) −3151.96 −0.609641
\(300\) 0 0
\(301\) 0 0
\(302\) −1664.00 −0.317061
\(303\) 1144.00 0.216901
\(304\) −150.093 −0.0283172
\(305\) 0 0
\(306\) −6866.77 −1.28283
\(307\) 722.324 0.134284 0.0671420 0.997743i \(-0.478612\pi\)
0.0671420 + 0.997743i \(0.478612\pi\)
\(308\) 0 0
\(309\) 12848.0 2.36536
\(310\) 0 0
\(311\) −7279.53 −1.32728 −0.663640 0.748052i \(-0.730989\pi\)
−0.663640 + 0.748052i \(0.730989\pi\)
\(312\) 4928.00 0.894209
\(313\) 1519.69 0.274435 0.137218 0.990541i \(-0.456184\pi\)
0.137218 + 0.990541i \(0.456184\pi\)
\(314\) −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 \(0.566986\pi\)
\(318\) −1163.22 −0.205127
\(319\) −3320.00 −0.582709
\(320\) 0 0
\(321\) −2439.02 −0.424089
\(322\) 0 0
\(323\) −528.000 −0.0909557
\(324\) 5380.00 0.922497
\(325\) 0 0
\(326\) 1272.00 0.216103
\(327\) −17654.7 −2.98565
\(328\) 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
\(333\) 4758.00 0.782993
\(334\) 1313.32 0.215154
\(335\) 0 0
\(336\) 0 0
\(337\) 250.000 0.0404106 0.0202053 0.999796i \(-0.493568\pi\)
0.0202053 + 0.999796i \(0.493568\pi\)
\(338\) −4230.00 −0.680715
\(339\) −12063.7 −1.93278
\(340\) 0 0
\(341\) 4127.57 0.655485
\(342\) 1144.46 0.180951
\(343\) 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 \(0.764204\pi\)
\(348\) 6228.87 0.959490
\(349\) 5712.93 0.876235 0.438117 0.898918i \(-0.355645\pi\)
0.438117 + 0.898918i \(0.355645\pi\)
\(350\) 0 0
\(351\) −20944.0 −3.18492
\(352\) −640.000 −0.0969094
\(353\) 4390.23 0.661950 0.330975 0.943640i \(-0.392622\pi\)
0.330975 + 0.943640i \(0.392622\pi\)
\(354\) 12496.0 1.87614
\(355\) 0 0
\(356\) 6003.73 0.893812
\(357\) 0 0
\(358\) 6456.00 0.953101
\(359\) 1840.00 0.270506 0.135253 0.990811i \(-0.456815\pi\)
0.135253 + 0.990811i \(0.456815\pi\)
\(360\) 0 0
\(361\) −6771.00 −0.987170
\(362\) −5647.26 −0.819927
\(363\) 8733.55 1.26279
\(364\) 0 0
\(365\) 0 0
\(366\) −5104.00 −0.728935
\(367\) −2964.34 −0.421628 −0.210814 0.977526i \(-0.567611\pi\)
−0.210814 + 0.977526i \(0.567611\pi\)
\(368\) −768.000 −0.108790
\(369\) −24033.7 −3.39063
\(370\) 0 0
\(371\) 0 0
\(372\) −7744.00 −1.07932
\(373\) −3982.00 −0.552762 −0.276381 0.961048i \(-0.589135\pi\)
−0.276381 + 0.961048i \(0.589135\pi\)
\(374\) −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 \(0.441956\pi\)
\(380\) 0 0
\(381\) 21688.5 2.91636
\(382\) 4272.00 0.572185
\(383\) 7035.62 0.938652 0.469326 0.883025i \(-0.344497\pi\)
0.469326 + 0.883025i \(0.344497\pi\)
\(384\) 1200.75 0.159571
\(385\) 0 0
\(386\) 3316.00 0.437254
\(387\) −26596.0 −3.49341
\(388\) −2626.63 −0.343678
\(389\) 8658.00 1.12848 0.564239 0.825611i \(-0.309170\pi\)
0.564239 + 0.825611i \(0.309170\pi\)
\(390\) 0 0
\(391\) −2701.68 −0.349437
\(392\) 0 0
\(393\) −2376.00 −0.304970
\(394\) −1956.00 −0.250106
\(395\) 0 0
\(396\) 4880.00 0.619266
\(397\) −9052.50 −1.14441 −0.572207 0.820109i \(-0.693912\pi\)
−0.572207 + 0.820109i \(0.693912\pi\)
\(398\) −9868.63 −1.24289
\(399\) 0 0
\(400\) 0 0
\(401\) −5706.00 −0.710584 −0.355292 0.934755i \(-0.615619\pi\)
−0.355292 + 0.934755i \(0.615619\pi\)
\(402\) −10881.8 −1.35008
\(403\) 13552.0 1.67512
\(404\) −487.803 −0.0600721
\(405\) 0 0
\(406\) 0 0
\(407\) 1560.00 0.189991
\(408\) 4224.00 0.512547
\(409\) −2420.25 −0.292601 −0.146301 0.989240i \(-0.546737\pi\)
−0.146301 + 0.989240i \(0.546737\pi\)
\(410\) 0 0
\(411\) −10450.2 −1.25419
\(412\) −5478.41 −0.655101
\(413\) 0 0
\(414\) 5856.00 0.695185
\(415\) 0 0
\(416\) −2101.31 −0.247656
\(417\) 12936.0 1.51913
\(418\) 375.233 0.0439073
\(419\) 1510.31 0.176095 0.0880473 0.996116i \(-0.471937\pi\)
0.0880473 + 0.996116i \(0.471937\pi\)
\(420\) 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
\(423\) 12589.1 1.44705
\(424\) 496.000 0.0568111
\(425\) 0 0
\(426\) −10206.3 −1.16080
\(427\) 0 0
\(428\) 1040.00 0.117454
\(429\) −12320.0 −1.38652
\(430\) 0 0
\(431\) 1336.00 0.149311 0.0746553 0.997209i \(-0.476214\pi\)
0.0746553 + 0.997209i \(0.476214\pi\)
\(432\) −5103.17 −0.568348
\(433\) −11163.2 −1.23896 −0.619479 0.785013i \(-0.712656\pi\)
−0.619479 + 0.785013i \(0.712656\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 7528.00 0.826894
\(437\) 450.280 0.0492902
\(438\) −11264.0 −1.22880
\(439\) 3602.24 0.391630 0.195815 0.980641i \(-0.437265\pi\)
0.195815 + 0.980641i \(0.437265\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −7392.00 −0.795479
\(443\) −6348.00 −0.680818 −0.340409 0.940277i \(-0.610566\pi\)
−0.340409 + 0.940277i \(0.610566\pi\)
\(444\) −2926.82 −0.312839
\(445\) 0 0
\(446\) −5778.59 −0.613507
\(447\) 8874.27 0.939012
\(448\) 0 0
\(449\) 7170.00 0.753615 0.376808 0.926292i \(-0.377022\pi\)
0.376808 + 0.926292i \(0.377022\pi\)
\(450\) 0 0
\(451\) −7879.90 −0.822727
\(452\) 5144.00 0.535295
\(453\) −7804.85 −0.809501
\(454\) 3958.71 0.409232
\(455\) 0 0
\(456\) −704.000 −0.0722979
\(457\) −6866.00 −0.702796 −0.351398 0.936226i \(-0.614294\pi\)
−0.351398 + 0.936226i \(0.614294\pi\)
\(458\) 5534.69 0.564671
\(459\) −17952.0 −1.82555
\(460\) 0 0
\(461\) −1378.98 −0.139318 −0.0696590 0.997571i \(-0.522191\pi\)
−0.0696590 + 0.997571i \(0.522191\pi\)
\(462\) 0 0
\(463\) −2648.00 −0.265795 −0.132897 0.991130i \(-0.542428\pi\)
−0.132897 + 0.991130i \(0.542428\pi\)
\(464\) −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 \(0.709301\pi\)
\(468\) 16022.5 1.58256
\(469\) 0 0
\(470\) 0 0
\(471\) −27016.0 −2.64295
\(472\) −5328.31 −0.519609
\(473\) −8720.00 −0.847666
\(474\) −12757.9 −1.23627
\(475\) 0 0
\(476\) 0 0
\(477\) −3782.00 −0.363031
\(478\) 8592.00 0.822153
\(479\) 13339.5 1.27244 0.636221 0.771507i \(-0.280497\pi\)
0.636221 + 0.771507i \(0.280497\pi\)
\(480\) 0 0
\(481\) 5121.93 0.485530
\(482\) 9043.12 0.854570
\(483\) 0 0
\(484\) −3724.00 −0.349737
\(485\) 0 0
\(486\) 8011.23 0.747730
\(487\) −13936.0 −1.29672 −0.648358 0.761336i \(-0.724544\pi\)
−0.648358 + 0.761336i \(0.724544\pi\)
\(488\) 2176.35 0.201883
\(489\) 5966.21 0.551741
\(490\) 0 0
\(491\) −12276.0 −1.12833 −0.564163 0.825663i \(-0.690801\pi\)
−0.564163 + 0.825663i \(0.690801\pi\)
\(492\) 14784.0 1.35470
\(493\) −9343.31 −0.853553
\(494\) 1232.00 0.112207
\(495\) 0 0
\(496\) 3302.05 0.298924
\(497\) 0 0
\(498\) 3696.00 0.332574
\(499\) −2220.00 −0.199160 −0.0995800 0.995030i \(-0.531750\pi\)
−0.0995800 + 0.995030i \(0.531750\pi\)
\(500\) 0 0
\(501\) 6160.00 0.549318
\(502\) 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
\(505\) 0 0
\(506\) 1920.00 0.168685
\(507\) −19840.5 −1.73796
\(508\) −9248.00 −0.807704
\(509\) 15881.7 1.38300 0.691499 0.722377i \(-0.256951\pi\)
0.691499 + 0.722377i \(0.256951\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −512.000 −0.0441942
\(513\) 2992.00 0.257505
\(514\) 3001.87 0.257600
\(515\) 0 0
\(516\) 16360.2 1.39577
\(517\) 4127.57 0.351122
\(518\) 0 0
\(519\) −6248.00 −0.528433
\(520\) 0 0
\(521\) −11613.5 −0.976575 −0.488287 0.872683i \(-0.662378\pi\)
−0.488287 + 0.872683i \(0.662378\pi\)
\(522\) 20252.0 1.69810
\(523\) 12617.2 1.05490 0.527450 0.849586i \(-0.323148\pi\)
0.527450 + 0.849586i \(0.323148\pi\)
\(524\) 1013.13 0.0844633
\(525\) 0 0
\(526\) −800.000 −0.0663149
\(527\) 11616.0 0.960154
\(528\) −3001.87 −0.247423
\(529\) −9863.00 −0.810635
\(530\) 0 0
\(531\) 40628.4 3.32038
\(532\) 0 0
\(533\) −25872.0 −2.10252
\(534\) 28160.0 2.28203
\(535\) 0 0
\(536\) 4640.00 0.373913
\(537\) 30281.3 2.43340
\(538\) −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
\(543\) −26488.0 −2.09339
\(544\) −1801.12 −0.141953
\(545\) 0 0
\(546\) 0 0
\(547\) −1276.00 −0.0997401 −0.0498700 0.998756i \(-0.515881\pi\)
−0.0498700 + 0.998756i \(0.515881\pi\)
\(548\) 4456.00 0.347356
\(549\) −16594.7 −1.29006
\(550\) 0 0
\(551\) 1557.22 0.120399
\(552\) −3602.24 −0.277756
\(553\) 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 \(0.532674\pi\)
\(558\) −25178.2 −1.91017
\(559\) −28630.3 −2.16625
\(560\) 0 0
\(561\) −10560.0 −0.794730
\(562\) −3756.00 −0.281917
\(563\) −15769.2 −1.18045 −0.590223 0.807240i \(-0.700960\pi\)
−0.590223 + 0.807240i \(0.700960\pi\)
\(564\) −7744.00 −0.578158
\(565\) 0 0
\(566\) −769.228 −0.0571256
\(567\) 0 0
\(568\) 4352.00 0.321489
\(569\) 12606.0 0.928772 0.464386 0.885633i \(-0.346275\pi\)
0.464386 + 0.885633i \(0.346275\pi\)
\(570\) 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
\(573\) 20037.5 1.46087
\(574\) 0 0
\(575\) 0 0
\(576\) 3904.00 0.282407
\(577\) 14371.4 1.03690 0.518449 0.855108i \(-0.326509\pi\)
0.518449 + 0.855108i \(0.326509\pi\)
\(578\) 3490.00 0.251150
\(579\) 15553.4 1.11637
\(580\) 0 0
\(581\) 0 0
\(582\) −12320.0 −0.877458
\(583\) −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
\(590\) 0 0
\(591\) −9174.45 −0.638556
\(592\) 1248.00 0.0866427
\(593\) 8217.61 0.569067 0.284534 0.958666i \(-0.408161\pi\)
0.284534 + 0.958666i \(0.408161\pi\)
\(594\) 12757.9 0.881253
\(595\) 0 0
\(596\) −3784.00 −0.260065
\(597\) −46288.0 −3.17327
\(598\) 6303.92 0.431081
\(599\) −19104.0 −1.30312 −0.651559 0.758598i \(-0.725885\pi\)
−0.651559 + 0.758598i \(0.725885\pi\)
\(600\) 0 0
\(601\) −21538.4 −1.46185 −0.730923 0.682460i \(-0.760910\pi\)
−0.730923 + 0.682460i \(0.760910\pi\)
\(602\) 0 0
\(603\) −35380.0 −2.38936
\(604\) 3328.00 0.224196
\(605\) 0 0
\(606\) −2288.00 −0.153372
\(607\) 13733.5 0.918331 0.459166 0.888351i \(-0.348148\pi\)
0.459166 + 0.888351i \(0.348148\pi\)
\(608\) 300.187 0.0200233
\(609\) 0 0
\(610\) 0 0
\(611\) 13552.0 0.897308
\(612\) 13733.5 0.907100
\(613\) −28034.0 −1.84712 −0.923558 0.383458i \(-0.874733\pi\)
−0.923558 + 0.383458i \(0.874733\pi\)
\(614\) −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 \(0.413171\pi\)
\(618\) −25696.0 −1.67256
\(619\) 5131.31 0.333191 0.166595 0.986025i \(-0.446723\pi\)
0.166595 + 0.986025i \(0.446723\pi\)
\(620\) 0 0
\(621\) 15309.5 0.989291
\(622\) 14559.1 0.938529
\(623\) 0 0
\(624\) −9856.00 −0.632301
\(625\) 0 0
\(626\) −3039.39 −0.194055
\(627\) 1760.00 0.112101
\(628\) 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
\(633\) −14596.6 −0.916527
\(634\) 4716.00 0.295420
\(635\) 0 0
\(636\) 2326.45 0.145047
\(637\) 0 0
\(638\) 6640.00 0.412038
\(639\) −33184.0 −2.05436
\(640\) 0 0
\(641\) −890.000 −0.0548407 −0.0274203 0.999624i \(-0.508729\pi\)
−0.0274203 + 0.999624i \(0.508729\pi\)
\(642\) 4878.03 0.299876
\(643\) −29352.6 −1.80024 −0.900120 0.435642i \(-0.856521\pi\)
−0.900120 + 0.435642i \(0.856521\pi\)
\(644\) 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 \(0.382497\pi\)
\(648\) −10760.0 −0.652304
\(649\) 13320.8 0.805680
\(650\) 0 0
\(651\) 0 0
\(652\) −2544.00 −0.152808
\(653\) 21526.0 1.29001 0.645006 0.764178i \(-0.276855\pi\)
0.645006 + 0.764178i \(0.276855\pi\)
\(654\) 35309.4 2.11118
\(655\) 0 0
\(656\) −6303.92 −0.375193
\(657\) −36622.8 −2.17472
\(658\) 0 0
\(659\) 23452.0 1.38628 0.693141 0.720802i \(-0.256226\pi\)
0.693141 + 0.720802i \(0.256226\pi\)
\(660\) 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
\(663\) −34671.6 −2.03097
\(664\) −1575.98 −0.0921082
\(665\) 0 0
\(666\) −9516.00 −0.553660
\(667\) 7968.00 0.462552
\(668\) −2626.63 −0.152137
\(669\) −27104.0 −1.56637
\(670\) 0 0
\(671\) −5440.88 −0.313030
\(672\) 0 0
\(673\) 13858.0 0.793739 0.396870 0.917875i \(-0.370096\pi\)
0.396870 + 0.917875i \(0.370096\pi\)
\(674\) −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 \(0.127340\pi\)
\(678\) 24127.5 1.36668
\(679\) 0 0
\(680\) 0 0
\(681\) 18568.0 1.04483
\(682\) −8255.13 −0.463498
\(683\) 27812.0 1.55812 0.779060 0.626949i \(-0.215696\pi\)
0.779060 + 0.626949i \(0.215696\pi\)
\(684\) −2288.92 −0.127952
\(685\) 0 0
\(686\) 0 0
\(687\) 25960.0 1.44168
\(688\) −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
\(695\) 0 0
\(696\) −12457.7 −0.678462
\(697\) −22176.0 −1.20513
\(698\) −11425.9 −0.619592
\(699\) −60881.6 −3.29435
\(700\) 0 0
\(701\) 22906.0 1.23416 0.617081 0.786900i \(-0.288315\pi\)
0.617081 + 0.786900i \(0.288315\pi\)
\(702\) 41888.0 2.25208
\(703\) −731.705 −0.0392557
\(704\) 1280.00 0.0685253
\(705\) 0 0
\(706\) −8780.46 −0.468069
\(707\) 0 0
\(708\) −24992.0 −1.32663
\(709\) −15086.0 −0.799107 −0.399553 0.916710i \(-0.630835\pi\)
−0.399553 + 0.916710i \(0.630835\pi\)
\(710\) 0 0
\(711\) −41480.0 −2.18793
\(712\) −12007.5 −0.632021
\(713\) −9906.16 −0.520321
\(714\) 0 0
\(715\) 0 0
\(716\) −12912.0 −0.673944
\(717\) 40300.1 2.09907
\(718\) −3680.00 −0.191276
\(719\) 20544.0 1.06559 0.532797 0.846243i \(-0.321141\pi\)
0.532797 + 0.846243i \(0.321141\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 13542.0 0.698035
\(723\) 42416.0 2.18184
\(724\) 11294.5 0.579776
\(725\) 0 0
\(726\) −17467.1 −0.892927
\(727\) −7223.24 −0.368494 −0.184247 0.982880i \(-0.558985\pi\)
−0.184247 + 0.982880i \(0.558985\pi\)
\(728\) 0 0
\(729\) 1261.00 0.0640654
\(730\) 0 0
\(731\) −24540.3 −1.24166
\(732\) 10208.0 0.515435
\(733\) 29427.7 1.48286 0.741430 0.671031i \(-0.234148\pi\)
0.741430 + 0.671031i \(0.234148\pi\)
\(734\) 5928.69 0.298136
\(735\) 0 0
\(736\) 1536.00 0.0769262
\(737\) −11600.0 −0.579771
\(738\) 48067.4 2.39754
\(739\) 32668.0 1.62613 0.813066 0.582171i \(-0.197797\pi\)
0.813066 + 0.582171i \(0.197797\pi\)
\(740\) 0 0
\(741\) 5778.59 0.286480
\(742\) 0 0
\(743\) 37056.0 1.82968 0.914840 0.403816i \(-0.132316\pi\)
0.914840 + 0.403816i \(0.132316\pi\)
\(744\) 15488.0 0.763196
\(745\) 0 0
\(746\) 7964.00 0.390862
\(747\) 12016.8 0.588586
\(748\) 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
\(753\) 52360.0 2.53400
\(754\) 21801.1 1.05298
\(755\) 0 0
\(756\) 0 0
\(757\) −19378.0 −0.930390 −0.465195 0.885208i \(-0.654016\pi\)
−0.465195 + 0.885208i \(0.654016\pi\)
\(758\) −5352.00 −0.256456
\(759\) 9005.60 0.430675
\(760\) 0 0
\(761\) 13977.4 0.665810 0.332905 0.942960i \(-0.391971\pi\)
0.332905 + 0.942960i \(0.391971\pi\)
\(762\) −43377.0 −2.06218
\(763\) 0 0
\(764\) −8544.00 −0.404596
\(765\) 0 0
\(766\) −14071.2 −0.663727
\(767\) 43736.0 2.05895
\(768\) −2401.49 −0.112834
\(769\) −8536.56 −0.400307 −0.200154 0.979765i \(-0.564144\pi\)
−0.200154 + 0.979765i \(0.564144\pi\)
\(770\) 0 0
\(771\) 14080.0 0.657690
\(772\) −6632.00 −0.309185
\(773\) −29296.3 −1.36315 −0.681576 0.731748i \(-0.738705\pi\)
−0.681576 + 0.731748i \(0.738705\pi\)
\(774\) 53192.0 2.47022
\(775\) 0 0
\(776\) 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
\(783\) 52945.4 2.41649
\(784\) 0 0
\(785\) 0 0
\(786\) 4752.00 0.215647
\(787\) 13780.4 0.624167 0.312084 0.950055i \(-0.398973\pi\)
0.312084 + 0.950055i \(0.398973\pi\)
\(788\) 3912.00 0.176852
\(789\) −3752.33 −0.169311
\(790\) 0 0
\(791\) 0 0
\(792\) −9760.00 −0.437887
\(793\) −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
\(800\) 0 0
\(801\) 91556.9 4.03871
\(802\) 11412.0 0.502459
\(803\) −12007.5 −0.527689
\(804\) 21763.5 0.954652
\(805\) 0 0
\(806\) −27104.0 −1.18449
\(807\) −2552.00 −0.111319
\(808\) 975.606 0.0424774
\(809\) 14034.0 0.609900 0.304950 0.952368i \(-0.401360\pi\)
0.304950 + 0.952368i \(0.401360\pi\)
\(810\) 0 0
\(811\) 6632.25 0.287164 0.143582 0.989638i \(-0.454138\pi\)
0.143582 + 0.989638i \(0.454138\pi\)
\(812\) 0 0
\(813\) 64768.0 2.79399
\(814\) −3120.00 −0.134344
\(815\) 0 0
\(816\) −8448.00 −0.362425
\(817\) 4090.04 0.175144
\(818\) 4840.51 0.206900
\(819\) 0 0
\(820\) 0 0
\(821\) 28622.0 1.21670 0.608352 0.793667i \(-0.291831\pi\)
0.608352 + 0.793667i \(0.291831\pi\)
\(822\) 20900.5 0.886847
\(823\) −24688.0 −1.04565 −0.522825 0.852440i \(-0.675122\pi\)
−0.522825 + 0.852440i \(0.675122\pi\)
\(824\) 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 \(0.276187\pi\)
\(828\) −11712.0 −0.491570
\(829\) −23236.3 −0.973499 −0.486750 0.873542i \(-0.661818\pi\)
−0.486750 + 0.873542i \(0.661818\pi\)
\(830\) 0 0
\(831\) −63508.2 −2.65111
\(832\) 4202.61 0.175119
\(833\) 0 0
\(834\) −25872.0 −1.07419
\(835\) 0 0
\(836\) −750.467 −0.0310472
\(837\) −65824.0 −2.71829
\(838\) −3020.63 −0.124518
\(839\) 24033.7 0.988957 0.494479 0.869190i \(-0.335359\pi\)
0.494479 + 0.869190i \(0.335359\pi\)
\(840\) 0 0
\(841\) 3167.00 0.129854
\(842\) 33540.0 1.37276
\(843\) −17617.2 −0.719773
\(844\) 6224.00 0.253838
\(845\) 0 0
\(846\) −25178.2 −1.02322
\(847\) 0 0
\(848\) −992.000 −0.0401715
\(849\) −3608.00 −0.145850
\(850\) 0 0
\(851\) −3744.00 −0.150814
\(852\) 20412.7 0.820807
\(853\) −23574.0 −0.946260 −0.473130 0.880993i \(-0.656876\pi\)
−0.473130 + 0.880993i \(0.656876\pi\)
\(854\) 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 \(0.337743\pi\)
\(858\) 24640.0 0.980415
\(859\) 32954.9 1.30897 0.654485 0.756075i \(-0.272885\pi\)
0.654485 + 0.756075i \(0.272885\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −2672.00 −0.105579
\(863\) −40872.0 −1.61217 −0.806083 0.591803i \(-0.798416\pi\)
−0.806083 + 0.591803i \(0.798416\pi\)
\(864\) 10206.3 0.401883
\(865\) 0 0
\(866\) 22326.4 0.876075
\(867\) 16369.6 0.641222
\(868\) 0 0
\(869\) −13600.0 −0.530896
\(870\) 0 0
\(871\) −38086.2 −1.48163
\(872\) −15056.0 −0.584702
\(873\) −40056.2 −1.55292
\(874\) −900.560 −0.0348534
\(875\) 0 0
\(876\) 22528.0 0.868893
\(877\) 12006.0 0.462273 0.231137 0.972921i \(-0.425756\pi\)
0.231137 + 0.972921i \(0.425756\pi\)
\(878\) −7204.48 −0.276924
\(879\) 35112.0 1.34732
\(880\) 0 0
\(881\) −35722.2 −1.36607 −0.683037 0.730383i \(-0.739341\pi\)
−0.683037 + 0.730383i \(0.739341\pi\)
\(882\) 0 0
\(883\) −19588.0 −0.746533 −0.373267 0.927724i \(-0.621762\pi\)
−0.373267 + 0.927724i \(0.621762\pi\)
\(884\) 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 \(0.775634\pi\)
\(888\) 5853.64 0.221211
\(889\) 0 0
\(890\) 0 0
\(891\) 26900.0 1.01143
\(892\) 11557.2 0.433815
\(893\) −1936.00 −0.0725485
\(894\) −17748.5 −0.663982
\(895\) 0 0
\(896\) 0 0
\(897\) 29568.0 1.10061
\(898\) −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
\(905\) 0 0
\(906\) 15609.7 0.572404
\(907\) −15868.0 −0.580913 −0.290457 0.956888i \(-0.593807\pi\)
−0.290457 + 0.956888i \(0.593807\pi\)
\(908\) −7917.42 −0.289371
\(909\) −7439.00 −0.271437
\(910\) 0 0
\(911\) 39832.0 1.44862 0.724310 0.689474i \(-0.242158\pi\)
0.724310 + 0.689474i \(0.242158\pi\)
\(912\) 1408.00 0.0511223
\(913\) 3939.95 0.142818
\(914\) 13732.0 0.496952
\(915\) 0 0
\(916\) −11069.4 −0.399282
\(917\) 0 0
\(918\) 35904.0 1.29086
\(919\) −30528.0 −1.09578 −0.547892 0.836549i \(-0.684570\pi\)
−0.547892 + 0.836549i \(0.684570\pi\)
\(920\) 0 0
\(921\) −6776.00 −0.242429
\(922\) 2757.96 0.0985127
\(923\) −35722.2 −1.27390
\(924\) 0 0
\(925\) 0 0
\(926\) 5296.00 0.187945
\(927\) −83545.7 −2.96009
\(928\) 5312.00 0.187904
\(929\) 16604.1 0.586396 0.293198 0.956052i \(-0.405280\pi\)
0.293198 + 0.956052i \(0.405280\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 25960.0 0.912391
\(933\) 68288.0 2.39619
\(934\) 24671.6 0.864324
\(935\) 0 0
\(936\) −32044.9 −1.11904
\(937\) 29943.6 1.04399 0.521993 0.852950i \(-0.325189\pi\)
0.521993 + 0.852950i \(0.325189\pi\)
\(938\) 0 0
\(939\) −14256.0 −0.495449
\(940\) 0 0
\(941\) −5375.22 −0.186214 −0.0931068 0.995656i \(-0.529680\pi\)
−0.0931068 + 0.995656i \(0.529680\pi\)
\(942\) 54032.0 1.86885
\(943\) 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 \(0.782607\pi\)
\(948\) 25515.9 0.874174
\(949\) −39424.0 −1.34853
\(950\) 0 0
\(951\) 22120.0 0.754248
\(952\) 0 0
\(953\) −34218.0 −1.16310 −0.581548 0.813512i \(-0.697553\pi\)
−0.581548 + 0.813512i \(0.697553\pi\)
\(954\) 7564.00 0.256702
\(955\) 0 0
\(956\) −17184.0 −0.581350
\(957\) 31144.4 1.05199
\(958\) −26679.1 −0.899752
\(959\) 0 0
\(960\) 0 0
\(961\) 12801.0 0.429694
\(962\) −10243.9 −0.343322
\(963\) 15860.0 0.530718
\(964\) −18086.2 −0.604272
\(965\) 0 0
\(966\) 0 0
\(967\) −14464.0 −0.481004 −0.240502 0.970649i \(-0.577312\pi\)
−0.240502 + 0.970649i \(0.577312\pi\)
\(968\) 7448.00 0.247301
\(969\) 4953.08 0.164206
\(970\) 0 0
\(971\) −37832.9 −1.25038 −0.625188 0.780474i \(-0.714978\pi\)
−0.625188 + 0.780474i \(0.714978\pi\)
\(972\) −16022.5 −0.528725
\(973\) 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 \(0.741810\pi\)
\(978\) −11932.4 −0.390140
\(979\) 30018.7 0.979980
\(980\) 0 0
\(981\) 114802. 3.73634
\(982\) 24552.0 0.797847
\(983\) −43020.5 −1.39587 −0.697935 0.716161i \(-0.745898\pi\)
−0.697935 + 0.716161i \(0.745898\pi\)
\(984\) −29568.0 −0.957920
\(985\) 0 0
\(986\) 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
\(993\) −22251.3 −0.711102
\(994\) 0 0
\(995\) 0 0
\(996\) −7392.00 −0.235165
\(997\) 121.951 0.00387384 0.00193692 0.999998i \(-0.499383\pi\)
0.00193692 + 0.999998i \(0.499383\pi\)
\(998\) 4440.00 0.140827
\(999\) −24878.0 −0.787892
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 2450.4.a.bs.1.1 2
5.4 even 2 98.4.a.h.1.2 yes 2
7.6 odd 2 inner 2450.4.a.bs.1.2 2
15.14 odd 2 882.4.a.w.1.2 2
20.19 odd 2 784.4.a.z.1.1 2
35.4 even 6 98.4.c.g.79.1 4
35.9 even 6 98.4.c.g.67.1 4
35.19 odd 6 98.4.c.g.67.2 4
35.24 odd 6 98.4.c.g.79.2 4
35.34 odd 2 98.4.a.h.1.1 2
105.44 odd 6 882.4.g.bi.361.1 4
105.59 even 6 882.4.g.bi.667.2 4
105.74 odd 6 882.4.g.bi.667.1 4
105.89 even 6 882.4.g.bi.361.2 4
105.104 even 2 882.4.a.w.1.1 2
140.139 even 2 784.4.a.z.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
98.4.a.h.1.1 2 35.34 odd 2
98.4.a.h.1.2 yes 2 5.4 even 2
98.4.c.g.67.1 4 35.9 even 6
98.4.c.g.67.2 4 35.19 odd 6
98.4.c.g.79.1 4 35.4 even 6
98.4.c.g.79.2 4 35.24 odd 6
784.4.a.z.1.1 2 20.19 odd 2
784.4.a.z.1.2 2 140.139 even 2
882.4.a.w.1.1 2 105.104 even 2
882.4.a.w.1.2 2 15.14 odd 2
882.4.g.bi.361.1 4 105.44 odd 6
882.4.g.bi.361.2 4 105.89 even 6
882.4.g.bi.667.1 4 105.74 odd 6
882.4.g.bi.667.2 4 105.59 even 6
2450.4.a.bs.1.1 2 1.1 even 1 trivial
2450.4.a.bs.1.2 2 7.6 odd 2 inner