Properties

Label 2070.4.a.bj.1.4
Level $2070$
Weight $4$
Character 2070.1
Self dual yes
Analytic conductor $122.134$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2070.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(122.133953712\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \( x^{4} - 68x^{2} - 111x + 342 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 230)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-3.74869\) of defining polynomial
Character \(\chi\) \(=\) 2070.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.00000 q^{2} +4.00000 q^{4} +5.00000 q^{5} +29.3684 q^{7} +8.00000 q^{8} +O(q^{10})\) \(q+2.00000 q^{2} +4.00000 q^{4} +5.00000 q^{5} +29.3684 q^{7} +8.00000 q^{8} +10.0000 q^{10} +38.1645 q^{11} -22.5396 q^{13} +58.7368 q^{14} +16.0000 q^{16} +104.049 q^{17} +142.000 q^{19} +20.0000 q^{20} +76.3291 q^{22} +23.0000 q^{23} +25.0000 q^{25} -45.0792 q^{26} +117.474 q^{28} -241.429 q^{29} +99.2706 q^{31} +32.0000 q^{32} +208.097 q^{34} +146.842 q^{35} +59.9452 q^{37} +284.000 q^{38} +40.0000 q^{40} -249.248 q^{41} -163.863 q^{43} +152.658 q^{44} +46.0000 q^{46} +205.591 q^{47} +519.502 q^{49} +50.0000 q^{50} -90.1584 q^{52} -491.274 q^{53} +190.823 q^{55} +234.947 q^{56} -482.858 q^{58} -433.734 q^{59} +660.902 q^{61} +198.541 q^{62} +64.0000 q^{64} -112.698 q^{65} -323.564 q^{67} +416.195 q^{68} +293.684 q^{70} -893.243 q^{71} +196.273 q^{73} +119.890 q^{74} +567.999 q^{76} +1120.83 q^{77} -500.211 q^{79} +80.0000 q^{80} -498.496 q^{82} -800.944 q^{83} +520.243 q^{85} -327.726 q^{86} +305.316 q^{88} +729.016 q^{89} -661.952 q^{91} +92.0000 q^{92} +411.181 q^{94} +709.999 q^{95} +1139.96 q^{97} +1039.00 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{2} + 16 q^{4} + 20 q^{5} - q^{7} + 32 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{2} + 16 q^{4} + 20 q^{5} - q^{7} + 32 q^{8} + 40 q^{10} + 39 q^{11} - 20 q^{13} - 2 q^{14} + 64 q^{16} + 23 q^{17} + 53 q^{19} + 80 q^{20} + 78 q^{22} + 92 q^{23} + 100 q^{25} - 40 q^{26} - 4 q^{28} - 161 q^{29} + 388 q^{31} + 128 q^{32} + 46 q^{34} - 5 q^{35} + 466 q^{37} + 106 q^{38} + 160 q^{40} - 484 q^{41} + 894 q^{43} + 156 q^{44} + 184 q^{46} + 265 q^{47} + 1643 q^{49} + 200 q^{50} - 80 q^{52} - 576 q^{53} + 195 q^{55} - 8 q^{56} - 322 q^{58} + 94 q^{59} + 1153 q^{61} + 776 q^{62} + 256 q^{64} - 100 q^{65} - 1472 q^{67} + 92 q^{68} - 10 q^{70} - 200 q^{71} + 1147 q^{73} + 932 q^{74} + 212 q^{76} + 2176 q^{77} - 908 q^{79} + 320 q^{80} - 968 q^{82} + 1048 q^{83} + 115 q^{85} + 1788 q^{86} + 312 q^{88} + 1784 q^{89} + 2329 q^{91} + 368 q^{92} + 530 q^{94} + 265 q^{95} - 2047 q^{97} + 3286 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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
\(5\) 5.00000 0.447214
\(6\) 0 0
\(7\) 29.3684 1.58574 0.792872 0.609388i \(-0.208585\pi\)
0.792872 + 0.609388i \(0.208585\pi\)
\(8\) 8.00000 0.353553
\(9\) 0 0
\(10\) 10.0000 0.316228
\(11\) 38.1645 1.04609 0.523047 0.852304i \(-0.324795\pi\)
0.523047 + 0.852304i \(0.324795\pi\)
\(12\) 0 0
\(13\) −22.5396 −0.480874 −0.240437 0.970665i \(-0.577291\pi\)
−0.240437 + 0.970665i \(0.577291\pi\)
\(14\) 58.7368 1.12129
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) 104.049 1.48444 0.742221 0.670155i \(-0.233772\pi\)
0.742221 + 0.670155i \(0.233772\pi\)
\(18\) 0 0
\(19\) 142.000 1.71458 0.857289 0.514835i \(-0.172147\pi\)
0.857289 + 0.514835i \(0.172147\pi\)
\(20\) 20.0000 0.223607
\(21\) 0 0
\(22\) 76.3291 0.739701
\(23\) 23.0000 0.208514
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) −45.0792 −0.340029
\(27\) 0 0
\(28\) 117.474 0.792872
\(29\) −241.429 −1.54594 −0.772969 0.634444i \(-0.781229\pi\)
−0.772969 + 0.634444i \(0.781229\pi\)
\(30\) 0 0
\(31\) 99.2706 0.575146 0.287573 0.957759i \(-0.407152\pi\)
0.287573 + 0.957759i \(0.407152\pi\)
\(32\) 32.0000 0.176777
\(33\) 0 0
\(34\) 208.097 1.04966
\(35\) 146.842 0.709166
\(36\) 0 0
\(37\) 59.9452 0.266349 0.133175 0.991093i \(-0.457483\pi\)
0.133175 + 0.991093i \(0.457483\pi\)
\(38\) 284.000 1.21239
\(39\) 0 0
\(40\) 40.0000 0.158114
\(41\) −249.248 −0.949414 −0.474707 0.880144i \(-0.657446\pi\)
−0.474707 + 0.880144i \(0.657446\pi\)
\(42\) 0 0
\(43\) −163.863 −0.581136 −0.290568 0.956854i \(-0.593844\pi\)
−0.290568 + 0.956854i \(0.593844\pi\)
\(44\) 152.658 0.523047
\(45\) 0 0
\(46\) 46.0000 0.147442
\(47\) 205.591 0.638052 0.319026 0.947746i \(-0.396644\pi\)
0.319026 + 0.947746i \(0.396644\pi\)
\(48\) 0 0
\(49\) 519.502 1.51458
\(50\) 50.0000 0.141421
\(51\) 0 0
\(52\) −90.1584 −0.240437
\(53\) −491.274 −1.27324 −0.636620 0.771178i \(-0.719668\pi\)
−0.636620 + 0.771178i \(0.719668\pi\)
\(54\) 0 0
\(55\) 190.823 0.467828
\(56\) 234.947 0.560645
\(57\) 0 0
\(58\) −482.858 −1.09314
\(59\) −433.734 −0.957074 −0.478537 0.878067i \(-0.658833\pi\)
−0.478537 + 0.878067i \(0.658833\pi\)
\(60\) 0 0
\(61\) 660.902 1.38721 0.693605 0.720355i \(-0.256021\pi\)
0.693605 + 0.720355i \(0.256021\pi\)
\(62\) 198.541 0.406690
\(63\) 0 0
\(64\) 64.0000 0.125000
\(65\) −112.698 −0.215053
\(66\) 0 0
\(67\) −323.564 −0.589995 −0.294997 0.955498i \(-0.595319\pi\)
−0.294997 + 0.955498i \(0.595319\pi\)
\(68\) 416.195 0.742221
\(69\) 0 0
\(70\) 293.684 0.501456
\(71\) −893.243 −1.49308 −0.746538 0.665342i \(-0.768286\pi\)
−0.746538 + 0.665342i \(0.768286\pi\)
\(72\) 0 0
\(73\) 196.273 0.314685 0.157343 0.987544i \(-0.449707\pi\)
0.157343 + 0.987544i \(0.449707\pi\)
\(74\) 119.890 0.188337
\(75\) 0 0
\(76\) 567.999 0.857289
\(77\) 1120.83 1.65884
\(78\) 0 0
\(79\) −500.211 −0.712381 −0.356191 0.934413i \(-0.615925\pi\)
−0.356191 + 0.934413i \(0.615925\pi\)
\(80\) 80.0000 0.111803
\(81\) 0 0
\(82\) −498.496 −0.671337
\(83\) −800.944 −1.05922 −0.529609 0.848242i \(-0.677661\pi\)
−0.529609 + 0.848242i \(0.677661\pi\)
\(84\) 0 0
\(85\) 520.243 0.663863
\(86\) −327.726 −0.410925
\(87\) 0 0
\(88\) 305.316 0.369850
\(89\) 729.016 0.868264 0.434132 0.900849i \(-0.357055\pi\)
0.434132 + 0.900849i \(0.357055\pi\)
\(90\) 0 0
\(91\) −661.952 −0.762543
\(92\) 92.0000 0.104257
\(93\) 0 0
\(94\) 411.181 0.451171
\(95\) 709.999 0.766783
\(96\) 0 0
\(97\) 1139.96 1.19325 0.596626 0.802520i \(-0.296508\pi\)
0.596626 + 0.802520i \(0.296508\pi\)
\(98\) 1039.00 1.07097
\(99\) 0 0
\(100\) 100.000 0.100000
\(101\) 1669.38 1.64465 0.822326 0.569017i \(-0.192676\pi\)
0.822326 + 0.569017i \(0.192676\pi\)
\(102\) 0 0
\(103\) −1110.63 −1.06246 −0.531229 0.847228i \(-0.678270\pi\)
−0.531229 + 0.847228i \(0.678270\pi\)
\(104\) −180.317 −0.170015
\(105\) 0 0
\(106\) −982.549 −0.900317
\(107\) −39.1892 −0.0354071 −0.0177036 0.999843i \(-0.505636\pi\)
−0.0177036 + 0.999843i \(0.505636\pi\)
\(108\) 0 0
\(109\) 807.545 0.709622 0.354811 0.934938i \(-0.384545\pi\)
0.354811 + 0.934938i \(0.384545\pi\)
\(110\) 381.645 0.330804
\(111\) 0 0
\(112\) 469.894 0.396436
\(113\) −1066.58 −0.887923 −0.443961 0.896046i \(-0.646427\pi\)
−0.443961 + 0.896046i \(0.646427\pi\)
\(114\) 0 0
\(115\) 115.000 0.0932505
\(116\) −965.715 −0.772969
\(117\) 0 0
\(118\) −867.468 −0.676753
\(119\) 3055.74 2.35395
\(120\) 0 0
\(121\) 125.532 0.0943139
\(122\) 1321.80 0.980906
\(123\) 0 0
\(124\) 397.082 0.287573
\(125\) 125.000 0.0894427
\(126\) 0 0
\(127\) −641.707 −0.448364 −0.224182 0.974547i \(-0.571971\pi\)
−0.224182 + 0.974547i \(0.571971\pi\)
\(128\) 128.000 0.0883883
\(129\) 0 0
\(130\) −225.396 −0.152066
\(131\) −2389.92 −1.59396 −0.796979 0.604007i \(-0.793570\pi\)
−0.796979 + 0.604007i \(0.793570\pi\)
\(132\) 0 0
\(133\) 4170.31 2.71888
\(134\) −647.128 −0.417189
\(135\) 0 0
\(136\) 832.390 0.524830
\(137\) −899.291 −0.560814 −0.280407 0.959881i \(-0.590470\pi\)
−0.280407 + 0.959881i \(0.590470\pi\)
\(138\) 0 0
\(139\) 309.341 0.188762 0.0943811 0.995536i \(-0.469913\pi\)
0.0943811 + 0.995536i \(0.469913\pi\)
\(140\) 587.368 0.354583
\(141\) 0 0
\(142\) −1786.49 −1.05576
\(143\) −860.214 −0.503040
\(144\) 0 0
\(145\) −1207.14 −0.691364
\(146\) 392.546 0.222516
\(147\) 0 0
\(148\) 239.781 0.133175
\(149\) −2560.87 −1.40802 −0.704010 0.710190i \(-0.748609\pi\)
−0.704010 + 0.710190i \(0.748609\pi\)
\(150\) 0 0
\(151\) −2463.15 −1.32747 −0.663737 0.747966i \(-0.731031\pi\)
−0.663737 + 0.747966i \(0.731031\pi\)
\(152\) 1136.00 0.606195
\(153\) 0 0
\(154\) 2241.66 1.17298
\(155\) 496.353 0.257213
\(156\) 0 0
\(157\) −566.720 −0.288084 −0.144042 0.989572i \(-0.546010\pi\)
−0.144042 + 0.989572i \(0.546010\pi\)
\(158\) −1000.42 −0.503730
\(159\) 0 0
\(160\) 160.000 0.0790569
\(161\) 675.473 0.330650
\(162\) 0 0
\(163\) 960.902 0.461740 0.230870 0.972985i \(-0.425843\pi\)
0.230870 + 0.972985i \(0.425843\pi\)
\(164\) −996.992 −0.474707
\(165\) 0 0
\(166\) −1601.89 −0.748980
\(167\) 4224.70 1.95759 0.978794 0.204847i \(-0.0656696\pi\)
0.978794 + 0.204847i \(0.0656696\pi\)
\(168\) 0 0
\(169\) −1688.97 −0.768760
\(170\) 1040.49 0.469422
\(171\) 0 0
\(172\) −655.451 −0.290568
\(173\) −3448.98 −1.51573 −0.757865 0.652412i \(-0.773757\pi\)
−0.757865 + 0.652412i \(0.773757\pi\)
\(174\) 0 0
\(175\) 734.210 0.317149
\(176\) 610.633 0.261524
\(177\) 0 0
\(178\) 1458.03 0.613955
\(179\) −1457.82 −0.608728 −0.304364 0.952556i \(-0.598444\pi\)
−0.304364 + 0.952556i \(0.598444\pi\)
\(180\) 0 0
\(181\) −3634.97 −1.49274 −0.746368 0.665534i \(-0.768204\pi\)
−0.746368 + 0.665534i \(0.768204\pi\)
\(182\) −1323.90 −0.539199
\(183\) 0 0
\(184\) 184.000 0.0737210
\(185\) 299.726 0.119115
\(186\) 0 0
\(187\) 3970.97 1.55287
\(188\) 822.362 0.319026
\(189\) 0 0
\(190\) 1420.00 0.542197
\(191\) −448.811 −0.170025 −0.0850127 0.996380i \(-0.527093\pi\)
−0.0850127 + 0.996380i \(0.527093\pi\)
\(192\) 0 0
\(193\) 4259.35 1.58857 0.794286 0.607544i \(-0.207845\pi\)
0.794286 + 0.607544i \(0.207845\pi\)
\(194\) 2279.92 0.843756
\(195\) 0 0
\(196\) 2078.01 0.757292
\(197\) 1767.37 0.639189 0.319594 0.947554i \(-0.396453\pi\)
0.319594 + 0.947554i \(0.396453\pi\)
\(198\) 0 0
\(199\) 1004.29 0.357750 0.178875 0.983872i \(-0.442754\pi\)
0.178875 + 0.983872i \(0.442754\pi\)
\(200\) 200.000 0.0707107
\(201\) 0 0
\(202\) 3338.76 1.16294
\(203\) −7090.37 −2.45146
\(204\) 0 0
\(205\) −1246.24 −0.424591
\(206\) −2221.25 −0.751272
\(207\) 0 0
\(208\) −360.634 −0.120218
\(209\) 5419.36 1.79361
\(210\) 0 0
\(211\) 5395.17 1.76028 0.880140 0.474714i \(-0.157449\pi\)
0.880140 + 0.474714i \(0.157449\pi\)
\(212\) −1965.10 −0.636620
\(213\) 0 0
\(214\) −78.3783 −0.0250366
\(215\) −819.314 −0.259892
\(216\) 0 0
\(217\) 2915.42 0.912034
\(218\) 1615.09 0.501779
\(219\) 0 0
\(220\) 763.291 0.233914
\(221\) −2345.22 −0.713830
\(222\) 0 0
\(223\) −1504.79 −0.451876 −0.225938 0.974142i \(-0.572545\pi\)
−0.225938 + 0.974142i \(0.572545\pi\)
\(224\) 939.788 0.280323
\(225\) 0 0
\(226\) −2133.16 −0.627856
\(227\) 1779.44 0.520288 0.260144 0.965570i \(-0.416230\pi\)
0.260144 + 0.965570i \(0.416230\pi\)
\(228\) 0 0
\(229\) 3976.16 1.14739 0.573694 0.819070i \(-0.305510\pi\)
0.573694 + 0.819070i \(0.305510\pi\)
\(230\) 230.000 0.0659380
\(231\) 0 0
\(232\) −1931.43 −0.546572
\(233\) 2311.12 0.649815 0.324907 0.945746i \(-0.394667\pi\)
0.324907 + 0.945746i \(0.394667\pi\)
\(234\) 0 0
\(235\) 1027.95 0.285346
\(236\) −1734.94 −0.478537
\(237\) 0 0
\(238\) 6111.49 1.66449
\(239\) −824.267 −0.223085 −0.111543 0.993760i \(-0.535579\pi\)
−0.111543 + 0.993760i \(0.535579\pi\)
\(240\) 0 0
\(241\) 3641.83 0.973406 0.486703 0.873567i \(-0.338199\pi\)
0.486703 + 0.873567i \(0.338199\pi\)
\(242\) 251.064 0.0666900
\(243\) 0 0
\(244\) 2643.61 0.693605
\(245\) 2597.51 0.677343
\(246\) 0 0
\(247\) −3200.62 −0.824496
\(248\) 794.165 0.203345
\(249\) 0 0
\(250\) 250.000 0.0632456
\(251\) 7767.51 1.95331 0.976655 0.214813i \(-0.0689143\pi\)
0.976655 + 0.214813i \(0.0689143\pi\)
\(252\) 0 0
\(253\) 877.784 0.218126
\(254\) −1283.41 −0.317042
\(255\) 0 0
\(256\) 256.000 0.0625000
\(257\) 1501.17 0.364359 0.182179 0.983265i \(-0.441685\pi\)
0.182179 + 0.983265i \(0.441685\pi\)
\(258\) 0 0
\(259\) 1760.49 0.422362
\(260\) −450.792 −0.107527
\(261\) 0 0
\(262\) −4779.84 −1.12710
\(263\) 5430.98 1.27334 0.636671 0.771136i \(-0.280311\pi\)
0.636671 + 0.771136i \(0.280311\pi\)
\(264\) 0 0
\(265\) −2456.37 −0.569410
\(266\) 8340.61 1.92254
\(267\) 0 0
\(268\) −1294.26 −0.294997
\(269\) 5014.20 1.13651 0.568255 0.822852i \(-0.307619\pi\)
0.568255 + 0.822852i \(0.307619\pi\)
\(270\) 0 0
\(271\) 7215.54 1.61739 0.808695 0.588228i \(-0.200174\pi\)
0.808695 + 0.588228i \(0.200174\pi\)
\(272\) 1664.78 0.371111
\(273\) 0 0
\(274\) −1798.58 −0.396556
\(275\) 954.113 0.209219
\(276\) 0 0
\(277\) −7439.60 −1.61373 −0.806863 0.590738i \(-0.798837\pi\)
−0.806863 + 0.590738i \(0.798837\pi\)
\(278\) 618.681 0.133475
\(279\) 0 0
\(280\) 1174.74 0.250728
\(281\) 2605.59 0.553155 0.276578 0.960992i \(-0.410800\pi\)
0.276578 + 0.960992i \(0.410800\pi\)
\(282\) 0 0
\(283\) −7162.57 −1.50449 −0.752245 0.658884i \(-0.771029\pi\)
−0.752245 + 0.658884i \(0.771029\pi\)
\(284\) −3572.97 −0.746538
\(285\) 0 0
\(286\) −1720.43 −0.355703
\(287\) −7320.01 −1.50553
\(288\) 0 0
\(289\) 5913.13 1.20357
\(290\) −2414.29 −0.488868
\(291\) 0 0
\(292\) 785.093 0.157343
\(293\) −9310.55 −1.85641 −0.928205 0.372069i \(-0.878649\pi\)
−0.928205 + 0.372069i \(0.878649\pi\)
\(294\) 0 0
\(295\) −2168.67 −0.428016
\(296\) 479.562 0.0941687
\(297\) 0 0
\(298\) −5121.74 −0.995620
\(299\) −518.411 −0.100269
\(300\) 0 0
\(301\) −4812.39 −0.921533
\(302\) −4926.31 −0.938666
\(303\) 0 0
\(304\) 2272.00 0.428645
\(305\) 3304.51 0.620380
\(306\) 0 0
\(307\) 3359.07 0.624470 0.312235 0.950005i \(-0.398922\pi\)
0.312235 + 0.950005i \(0.398922\pi\)
\(308\) 4483.32 0.829419
\(309\) 0 0
\(310\) 992.706 0.181877
\(311\) 5509.20 1.00450 0.502248 0.864724i \(-0.332507\pi\)
0.502248 + 0.864724i \(0.332507\pi\)
\(312\) 0 0
\(313\) 2401.42 0.433661 0.216831 0.976209i \(-0.430428\pi\)
0.216831 + 0.976209i \(0.430428\pi\)
\(314\) −1133.44 −0.203706
\(315\) 0 0
\(316\) −2000.84 −0.356191
\(317\) 1057.36 0.187341 0.0936704 0.995603i \(-0.470140\pi\)
0.0936704 + 0.995603i \(0.470140\pi\)
\(318\) 0 0
\(319\) −9214.02 −1.61720
\(320\) 320.000 0.0559017
\(321\) 0 0
\(322\) 1350.95 0.233805
\(323\) 14774.9 2.54519
\(324\) 0 0
\(325\) −563.490 −0.0961748
\(326\) 1921.80 0.326500
\(327\) 0 0
\(328\) −1993.98 −0.335669
\(329\) 6037.86 1.01179
\(330\) 0 0
\(331\) −9680.21 −1.60747 −0.803735 0.594987i \(-0.797157\pi\)
−0.803735 + 0.594987i \(0.797157\pi\)
\(332\) −3203.78 −0.529609
\(333\) 0 0
\(334\) 8449.40 1.38422
\(335\) −1617.82 −0.263854
\(336\) 0 0
\(337\) 467.694 0.0755991 0.0377996 0.999285i \(-0.487965\pi\)
0.0377996 + 0.999285i \(0.487965\pi\)
\(338\) −3377.93 −0.543596
\(339\) 0 0
\(340\) 2080.97 0.331931
\(341\) 3788.62 0.601657
\(342\) 0 0
\(343\) 5183.59 0.815998
\(344\) −1310.90 −0.205463
\(345\) 0 0
\(346\) −6897.96 −1.07178
\(347\) −3371.55 −0.521597 −0.260798 0.965393i \(-0.583986\pi\)
−0.260798 + 0.965393i \(0.583986\pi\)
\(348\) 0 0
\(349\) 10958.7 1.68082 0.840412 0.541947i \(-0.182313\pi\)
0.840412 + 0.541947i \(0.182313\pi\)
\(350\) 1468.42 0.224258
\(351\) 0 0
\(352\) 1221.27 0.184925
\(353\) 1584.76 0.238947 0.119474 0.992837i \(-0.461879\pi\)
0.119474 + 0.992837i \(0.461879\pi\)
\(354\) 0 0
\(355\) −4466.21 −0.667724
\(356\) 2916.06 0.434132
\(357\) 0 0
\(358\) −2915.63 −0.430436
\(359\) −5130.67 −0.754280 −0.377140 0.926156i \(-0.623092\pi\)
−0.377140 + 0.926156i \(0.623092\pi\)
\(360\) 0 0
\(361\) 13304.9 1.93978
\(362\) −7269.94 −1.05552
\(363\) 0 0
\(364\) −2647.81 −0.381271
\(365\) 981.366 0.140732
\(366\) 0 0
\(367\) 2811.27 0.399856 0.199928 0.979811i \(-0.435929\pi\)
0.199928 + 0.979811i \(0.435929\pi\)
\(368\) 368.000 0.0521286
\(369\) 0 0
\(370\) 599.452 0.0842271
\(371\) −14427.9 −2.01903
\(372\) 0 0
\(373\) −1952.42 −0.271026 −0.135513 0.990776i \(-0.543268\pi\)
−0.135513 + 0.990776i \(0.543268\pi\)
\(374\) 7941.94 1.09804
\(375\) 0 0
\(376\) 1644.72 0.225586
\(377\) 5441.71 0.743401
\(378\) 0 0
\(379\) 9609.27 1.30236 0.651181 0.758923i \(-0.274274\pi\)
0.651181 + 0.758923i \(0.274274\pi\)
\(380\) 2840.00 0.383391
\(381\) 0 0
\(382\) −897.623 −0.120226
\(383\) −5027.44 −0.670732 −0.335366 0.942088i \(-0.608860\pi\)
−0.335366 + 0.942088i \(0.608860\pi\)
\(384\) 0 0
\(385\) 5604.15 0.741855
\(386\) 8518.70 1.12329
\(387\) 0 0
\(388\) 4559.84 0.596626
\(389\) −5892.29 −0.767997 −0.383999 0.923334i \(-0.625453\pi\)
−0.383999 + 0.923334i \(0.625453\pi\)
\(390\) 0 0
\(391\) 2393.12 0.309528
\(392\) 4156.02 0.535486
\(393\) 0 0
\(394\) 3534.75 0.451975
\(395\) −2501.05 −0.318587
\(396\) 0 0
\(397\) −2454.41 −0.310285 −0.155143 0.987892i \(-0.549584\pi\)
−0.155143 + 0.987892i \(0.549584\pi\)
\(398\) 2008.58 0.252967
\(399\) 0 0
\(400\) 400.000 0.0500000
\(401\) −9458.39 −1.17788 −0.588939 0.808177i \(-0.700454\pi\)
−0.588939 + 0.808177i \(0.700454\pi\)
\(402\) 0 0
\(403\) −2237.52 −0.276573
\(404\) 6677.53 0.822326
\(405\) 0 0
\(406\) −14180.7 −1.73345
\(407\) 2287.78 0.278627
\(408\) 0 0
\(409\) −5857.25 −0.708123 −0.354062 0.935222i \(-0.615200\pi\)
−0.354062 + 0.935222i \(0.615200\pi\)
\(410\) −2492.48 −0.300231
\(411\) 0 0
\(412\) −4442.51 −0.531229
\(413\) −12738.1 −1.51767
\(414\) 0 0
\(415\) −4004.72 −0.473696
\(416\) −721.267 −0.0850073
\(417\) 0 0
\(418\) 10838.7 1.26827
\(419\) 5252.61 0.612426 0.306213 0.951963i \(-0.400938\pi\)
0.306213 + 0.951963i \(0.400938\pi\)
\(420\) 0 0
\(421\) −203.517 −0.0235602 −0.0117801 0.999931i \(-0.503750\pi\)
−0.0117801 + 0.999931i \(0.503750\pi\)
\(422\) 10790.3 1.24471
\(423\) 0 0
\(424\) −3930.20 −0.450158
\(425\) 2601.22 0.296888
\(426\) 0 0
\(427\) 19409.6 2.19976
\(428\) −156.757 −0.0177036
\(429\) 0 0
\(430\) −1638.63 −0.183771
\(431\) −3107.72 −0.347317 −0.173659 0.984806i \(-0.555559\pi\)
−0.173659 + 0.984806i \(0.555559\pi\)
\(432\) 0 0
\(433\) −5713.46 −0.634114 −0.317057 0.948406i \(-0.602695\pi\)
−0.317057 + 0.948406i \(0.602695\pi\)
\(434\) 5830.83 0.644905
\(435\) 0 0
\(436\) 3230.18 0.354811
\(437\) 3266.00 0.357514
\(438\) 0 0
\(439\) −4587.14 −0.498706 −0.249353 0.968413i \(-0.580218\pi\)
−0.249353 + 0.968413i \(0.580218\pi\)
\(440\) 1526.58 0.165402
\(441\) 0 0
\(442\) −4690.43 −0.504754
\(443\) −5268.01 −0.564990 −0.282495 0.959269i \(-0.591162\pi\)
−0.282495 + 0.959269i \(0.591162\pi\)
\(444\) 0 0
\(445\) 3645.08 0.388299
\(446\) −3009.58 −0.319525
\(447\) 0 0
\(448\) 1879.58 0.198218
\(449\) −16866.8 −1.77282 −0.886409 0.462902i \(-0.846808\pi\)
−0.886409 + 0.462902i \(0.846808\pi\)
\(450\) 0 0
\(451\) −9512.43 −0.993177
\(452\) −4266.31 −0.443961
\(453\) 0 0
\(454\) 3558.87 0.367899
\(455\) −3309.76 −0.341020
\(456\) 0 0
\(457\) 9124.25 0.933948 0.466974 0.884271i \(-0.345344\pi\)
0.466974 + 0.884271i \(0.345344\pi\)
\(458\) 7952.31 0.811326
\(459\) 0 0
\(460\) 460.000 0.0466252
\(461\) −6011.92 −0.607382 −0.303691 0.952771i \(-0.598219\pi\)
−0.303691 + 0.952771i \(0.598219\pi\)
\(462\) 0 0
\(463\) −8584.09 −0.861634 −0.430817 0.902439i \(-0.641775\pi\)
−0.430817 + 0.902439i \(0.641775\pi\)
\(464\) −3862.86 −0.386484
\(465\) 0 0
\(466\) 4622.25 0.459488
\(467\) −3954.09 −0.391806 −0.195903 0.980623i \(-0.562764\pi\)
−0.195903 + 0.980623i \(0.562764\pi\)
\(468\) 0 0
\(469\) −9502.56 −0.935581
\(470\) 2055.91 0.201770
\(471\) 0 0
\(472\) −3469.87 −0.338377
\(473\) −6253.75 −0.607923
\(474\) 0 0
\(475\) 3549.99 0.342916
\(476\) 12223.0 1.17697
\(477\) 0 0
\(478\) −1648.53 −0.157745
\(479\) 95.3377 0.00909413 0.00454707 0.999990i \(-0.498553\pi\)
0.00454707 + 0.999990i \(0.498553\pi\)
\(480\) 0 0
\(481\) −1351.14 −0.128080
\(482\) 7283.66 0.688302
\(483\) 0 0
\(484\) 502.127 0.0471570
\(485\) 5699.80 0.533638
\(486\) 0 0
\(487\) 13915.2 1.29478 0.647391 0.762158i \(-0.275860\pi\)
0.647391 + 0.762158i \(0.275860\pi\)
\(488\) 5287.22 0.490453
\(489\) 0 0
\(490\) 5195.02 0.478954
\(491\) −6291.33 −0.578256 −0.289128 0.957290i \(-0.593365\pi\)
−0.289128 + 0.957290i \(0.593365\pi\)
\(492\) 0 0
\(493\) −25120.3 −2.29486
\(494\) −6401.24 −0.583007
\(495\) 0 0
\(496\) 1588.33 0.143786
\(497\) −26233.1 −2.36764
\(498\) 0 0
\(499\) −638.332 −0.0572658 −0.0286329 0.999590i \(-0.509115\pi\)
−0.0286329 + 0.999590i \(0.509115\pi\)
\(500\) 500.000 0.0447214
\(501\) 0 0
\(502\) 15535.0 1.38120
\(503\) −15063.4 −1.33527 −0.667637 0.744487i \(-0.732694\pi\)
−0.667637 + 0.744487i \(0.732694\pi\)
\(504\) 0 0
\(505\) 8346.91 0.735510
\(506\) 1755.57 0.154238
\(507\) 0 0
\(508\) −2566.83 −0.224182
\(509\) 13623.1 1.18631 0.593157 0.805087i \(-0.297881\pi\)
0.593157 + 0.805087i \(0.297881\pi\)
\(510\) 0 0
\(511\) 5764.23 0.499010
\(512\) 512.000 0.0441942
\(513\) 0 0
\(514\) 3002.33 0.257640
\(515\) −5553.13 −0.475146
\(516\) 0 0
\(517\) 7846.27 0.667463
\(518\) 3520.99 0.298655
\(519\) 0 0
\(520\) −901.584 −0.0760328
\(521\) 21659.4 1.82133 0.910666 0.413143i \(-0.135569\pi\)
0.910666 + 0.413143i \(0.135569\pi\)
\(522\) 0 0
\(523\) −3813.85 −0.318868 −0.159434 0.987209i \(-0.550967\pi\)
−0.159434 + 0.987209i \(0.550967\pi\)
\(524\) −9559.69 −0.796979
\(525\) 0 0
\(526\) 10862.0 0.900388
\(527\) 10329.0 0.853771
\(528\) 0 0
\(529\) 529.000 0.0434783
\(530\) −4912.74 −0.402634
\(531\) 0 0
\(532\) 16681.2 1.35944
\(533\) 5617.95 0.456549
\(534\) 0 0
\(535\) −195.946 −0.0158345
\(536\) −2588.51 −0.208595
\(537\) 0 0
\(538\) 10028.4 0.803634
\(539\) 19826.6 1.58440
\(540\) 0 0
\(541\) 9727.32 0.773031 0.386516 0.922283i \(-0.373679\pi\)
0.386516 + 0.922283i \(0.373679\pi\)
\(542\) 14431.1 1.14367
\(543\) 0 0
\(544\) 3329.56 0.262415
\(545\) 4037.73 0.317353
\(546\) 0 0
\(547\) 782.647 0.0611766 0.0305883 0.999532i \(-0.490262\pi\)
0.0305883 + 0.999532i \(0.490262\pi\)
\(548\) −3597.16 −0.280407
\(549\) 0 0
\(550\) 1908.23 0.147940
\(551\) −34282.8 −2.65063
\(552\) 0 0
\(553\) −14690.4 −1.12965
\(554\) −14879.2 −1.14108
\(555\) 0 0
\(556\) 1237.36 0.0943811
\(557\) −3472.87 −0.264184 −0.132092 0.991237i \(-0.542169\pi\)
−0.132092 + 0.991237i \(0.542169\pi\)
\(558\) 0 0
\(559\) 3693.40 0.279453
\(560\) 2349.47 0.177292
\(561\) 0 0
\(562\) 5211.18 0.391140
\(563\) −7544.75 −0.564784 −0.282392 0.959299i \(-0.591128\pi\)
−0.282392 + 0.959299i \(0.591128\pi\)
\(564\) 0 0
\(565\) −5332.89 −0.397091
\(566\) −14325.1 −1.06383
\(567\) 0 0
\(568\) −7145.94 −0.527882
\(569\) −13222.5 −0.974193 −0.487096 0.873348i \(-0.661944\pi\)
−0.487096 + 0.873348i \(0.661944\pi\)
\(570\) 0 0
\(571\) 4069.49 0.298254 0.149127 0.988818i \(-0.452354\pi\)
0.149127 + 0.988818i \(0.452354\pi\)
\(572\) −3440.85 −0.251520
\(573\) 0 0
\(574\) −14640.0 −1.06457
\(575\) 575.000 0.0417029
\(576\) 0 0
\(577\) 846.627 0.0610841 0.0305421 0.999533i \(-0.490277\pi\)
0.0305421 + 0.999533i \(0.490277\pi\)
\(578\) 11826.3 0.851051
\(579\) 0 0
\(580\) −4828.58 −0.345682
\(581\) −23522.4 −1.67965
\(582\) 0 0
\(583\) −18749.3 −1.33193
\(584\) 1570.19 0.111258
\(585\) 0 0
\(586\) −18621.1 −1.31268
\(587\) −4702.77 −0.330672 −0.165336 0.986237i \(-0.552871\pi\)
−0.165336 + 0.986237i \(0.552871\pi\)
\(588\) 0 0
\(589\) 14096.4 0.986133
\(590\) −4337.34 −0.302653
\(591\) 0 0
\(592\) 959.123 0.0665874
\(593\) 14016.5 0.970640 0.485320 0.874337i \(-0.338703\pi\)
0.485320 + 0.874337i \(0.338703\pi\)
\(594\) 0 0
\(595\) 15278.7 1.05272
\(596\) −10243.5 −0.704010
\(597\) 0 0
\(598\) −1036.82 −0.0709010
\(599\) −12885.0 −0.878910 −0.439455 0.898265i \(-0.644828\pi\)
−0.439455 + 0.898265i \(0.644828\pi\)
\(600\) 0 0
\(601\) −4753.41 −0.322622 −0.161311 0.986904i \(-0.551572\pi\)
−0.161311 + 0.986904i \(0.551572\pi\)
\(602\) −9624.77 −0.651622
\(603\) 0 0
\(604\) −9852.61 −0.663737
\(605\) 627.659 0.0421785
\(606\) 0 0
\(607\) −6800.06 −0.454705 −0.227352 0.973813i \(-0.573007\pi\)
−0.227352 + 0.973813i \(0.573007\pi\)
\(608\) 4543.99 0.303097
\(609\) 0 0
\(610\) 6609.02 0.438675
\(611\) −4633.93 −0.306823
\(612\) 0 0
\(613\) 20842.3 1.37327 0.686634 0.727003i \(-0.259087\pi\)
0.686634 + 0.727003i \(0.259087\pi\)
\(614\) 6718.14 0.441567
\(615\) 0 0
\(616\) 8966.65 0.586488
\(617\) −18917.7 −1.23436 −0.617179 0.786823i \(-0.711724\pi\)
−0.617179 + 0.786823i \(0.711724\pi\)
\(618\) 0 0
\(619\) −28044.8 −1.82103 −0.910513 0.413480i \(-0.864313\pi\)
−0.910513 + 0.413480i \(0.864313\pi\)
\(620\) 1985.41 0.128607
\(621\) 0 0
\(622\) 11018.4 0.710285
\(623\) 21410.0 1.37684
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 4802.83 0.306645
\(627\) 0 0
\(628\) −2266.88 −0.144042
\(629\) 6237.22 0.395380
\(630\) 0 0
\(631\) −25796.2 −1.62746 −0.813732 0.581241i \(-0.802567\pi\)
−0.813732 + 0.581241i \(0.802567\pi\)
\(632\) −4001.69 −0.251865
\(633\) 0 0
\(634\) 2114.71 0.132470
\(635\) −3208.54 −0.200515
\(636\) 0 0
\(637\) −11709.4 −0.728324
\(638\) −18428.0 −1.14353
\(639\) 0 0
\(640\) 640.000 0.0395285
\(641\) 26482.3 1.63181 0.815903 0.578188i \(-0.196240\pi\)
0.815903 + 0.578188i \(0.196240\pi\)
\(642\) 0 0
\(643\) 30458.1 1.86804 0.934020 0.357219i \(-0.116275\pi\)
0.934020 + 0.357219i \(0.116275\pi\)
\(644\) 2701.89 0.165325
\(645\) 0 0
\(646\) 29549.8 1.79972
\(647\) 6746.24 0.409926 0.204963 0.978770i \(-0.434293\pi\)
0.204963 + 0.978770i \(0.434293\pi\)
\(648\) 0 0
\(649\) −16553.3 −1.00119
\(650\) −1126.98 −0.0680058
\(651\) 0 0
\(652\) 3843.61 0.230870
\(653\) −12976.2 −0.777640 −0.388820 0.921314i \(-0.627117\pi\)
−0.388820 + 0.921314i \(0.627117\pi\)
\(654\) 0 0
\(655\) −11949.6 −0.712840
\(656\) −3987.97 −0.237354
\(657\) 0 0
\(658\) 12075.7 0.715442
\(659\) 6538.56 0.386504 0.193252 0.981149i \(-0.438097\pi\)
0.193252 + 0.981149i \(0.438097\pi\)
\(660\) 0 0
\(661\) −25177.1 −1.48151 −0.740753 0.671777i \(-0.765531\pi\)
−0.740753 + 0.671777i \(0.765531\pi\)
\(662\) −19360.4 −1.13665
\(663\) 0 0
\(664\) −6407.55 −0.374490
\(665\) 20851.5 1.21592
\(666\) 0 0
\(667\) −5552.86 −0.322350
\(668\) 16898.8 0.978794
\(669\) 0 0
\(670\) −3235.64 −0.186573
\(671\) 25223.0 1.45115
\(672\) 0 0
\(673\) 18339.3 1.05041 0.525207 0.850975i \(-0.323988\pi\)
0.525207 + 0.850975i \(0.323988\pi\)
\(674\) 935.388 0.0534567
\(675\) 0 0
\(676\) −6755.86 −0.384380
\(677\) −31876.9 −1.80965 −0.904823 0.425789i \(-0.859997\pi\)
−0.904823 + 0.425789i \(0.859997\pi\)
\(678\) 0 0
\(679\) 33478.8 1.89219
\(680\) 4161.95 0.234711
\(681\) 0 0
\(682\) 7577.23 0.425436
\(683\) 28843.7 1.61592 0.807959 0.589239i \(-0.200572\pi\)
0.807959 + 0.589239i \(0.200572\pi\)
\(684\) 0 0
\(685\) −4496.45 −0.250804
\(686\) 10367.2 0.576998
\(687\) 0 0
\(688\) −2621.80 −0.145284
\(689\) 11073.1 0.612268
\(690\) 0 0
\(691\) −18327.7 −1.00900 −0.504499 0.863413i \(-0.668323\pi\)
−0.504499 + 0.863413i \(0.668323\pi\)
\(692\) −13795.9 −0.757865
\(693\) 0 0
\(694\) −6743.09 −0.368825
\(695\) 1546.70 0.0844170
\(696\) 0 0
\(697\) −25933.9 −1.40935
\(698\) 21917.5 1.18852
\(699\) 0 0
\(700\) 2936.84 0.158574
\(701\) 8329.48 0.448788 0.224394 0.974499i \(-0.427960\pi\)
0.224394 + 0.974499i \(0.427960\pi\)
\(702\) 0 0
\(703\) 8512.20 0.456677
\(704\) 2442.53 0.130762
\(705\) 0 0
\(706\) 3169.52 0.168961
\(707\) 49027.1 2.60800
\(708\) 0 0
\(709\) −6169.85 −0.326818 −0.163409 0.986558i \(-0.552249\pi\)
−0.163409 + 0.986558i \(0.552249\pi\)
\(710\) −8932.43 −0.472152
\(711\) 0 0
\(712\) 5832.12 0.306978
\(713\) 2283.22 0.119926
\(714\) 0 0
\(715\) −4301.07 −0.224966
\(716\) −5831.27 −0.304364
\(717\) 0 0
\(718\) −10261.3 −0.533356
\(719\) −28740.8 −1.49075 −0.745377 0.666644i \(-0.767730\pi\)
−0.745377 + 0.666644i \(0.767730\pi\)
\(720\) 0 0
\(721\) −32617.3 −1.68479
\(722\) 26609.9 1.37163
\(723\) 0 0
\(724\) −14539.9 −0.746368
\(725\) −6035.72 −0.309188
\(726\) 0 0
\(727\) 11174.0 0.570041 0.285021 0.958521i \(-0.408000\pi\)
0.285021 + 0.958521i \(0.408000\pi\)
\(728\) −5295.62 −0.269600
\(729\) 0 0
\(730\) 1962.73 0.0995123
\(731\) −17049.7 −0.862663
\(732\) 0 0
\(733\) −23560.7 −1.18722 −0.593612 0.804751i \(-0.702299\pi\)
−0.593612 + 0.804751i \(0.702299\pi\)
\(734\) 5622.54 0.282741
\(735\) 0 0
\(736\) 736.000 0.0368605
\(737\) −12348.7 −0.617191
\(738\) 0 0
\(739\) 22713.6 1.13063 0.565313 0.824877i \(-0.308755\pi\)
0.565313 + 0.824877i \(0.308755\pi\)
\(740\) 1198.90 0.0595575
\(741\) 0 0
\(742\) −28855.9 −1.42767
\(743\) 15882.7 0.784224 0.392112 0.919917i \(-0.371745\pi\)
0.392112 + 0.919917i \(0.371745\pi\)
\(744\) 0 0
\(745\) −12804.4 −0.629685
\(746\) −3904.84 −0.191644
\(747\) 0 0
\(748\) 15883.9 0.776433
\(749\) −1150.92 −0.0561466
\(750\) 0 0
\(751\) −30250.6 −1.46985 −0.734927 0.678146i \(-0.762784\pi\)
−0.734927 + 0.678146i \(0.762784\pi\)
\(752\) 3289.45 0.159513
\(753\) 0 0
\(754\) 10883.4 0.525664
\(755\) −12315.8 −0.593664
\(756\) 0 0
\(757\) −9611.05 −0.461453 −0.230726 0.973019i \(-0.574110\pi\)
−0.230726 + 0.973019i \(0.574110\pi\)
\(758\) 19218.5 0.920908
\(759\) 0 0
\(760\) 5679.99 0.271099
\(761\) −5892.64 −0.280694 −0.140347 0.990102i \(-0.544822\pi\)
−0.140347 + 0.990102i \(0.544822\pi\)
\(762\) 0 0
\(763\) 23716.3 1.12528
\(764\) −1795.25 −0.0850127
\(765\) 0 0
\(766\) −10054.9 −0.474279
\(767\) 9776.19 0.460232
\(768\) 0 0
\(769\) 1683.72 0.0789553 0.0394777 0.999220i \(-0.487431\pi\)
0.0394777 + 0.999220i \(0.487431\pi\)
\(770\) 11208.3 0.524571
\(771\) 0 0
\(772\) 17037.4 0.794286
\(773\) −19880.8 −0.925048 −0.462524 0.886607i \(-0.653056\pi\)
−0.462524 + 0.886607i \(0.653056\pi\)
\(774\) 0 0
\(775\) 2481.76 0.115029
\(776\) 9119.68 0.421878
\(777\) 0 0
\(778\) −11784.6 −0.543056
\(779\) −35393.2 −1.62785
\(780\) 0 0
\(781\) −34090.2 −1.56190
\(782\) 4786.24 0.218869
\(783\) 0 0
\(784\) 8312.04 0.378646
\(785\) −2833.60 −0.128835
\(786\) 0 0
\(787\) −7622.64 −0.345258 −0.172629 0.984987i \(-0.555226\pi\)
−0.172629 + 0.984987i \(0.555226\pi\)
\(788\) 7069.50 0.319594
\(789\) 0 0
\(790\) −5002.11 −0.225275
\(791\) −31323.7 −1.40802
\(792\) 0 0
\(793\) −14896.5 −0.667074
\(794\) −4908.82 −0.219405
\(795\) 0 0
\(796\) 4017.16 0.178875
\(797\) 22380.9 0.994693 0.497347 0.867552i \(-0.334308\pi\)
0.497347 + 0.867552i \(0.334308\pi\)
\(798\) 0 0
\(799\) 21391.4 0.947152
\(800\) 800.000 0.0353553
\(801\) 0 0
\(802\) −18916.8 −0.832886
\(803\) 7490.67 0.329191
\(804\) 0 0
\(805\) 3377.36 0.147871
\(806\) −4475.04 −0.195566
\(807\) 0 0
\(808\) 13355.1 0.581472
\(809\) −8117.18 −0.352762 −0.176381 0.984322i \(-0.556439\pi\)
−0.176381 + 0.984322i \(0.556439\pi\)
\(810\) 0 0
\(811\) 7221.05 0.312658 0.156329 0.987705i \(-0.450034\pi\)
0.156329 + 0.987705i \(0.450034\pi\)
\(812\) −28361.5 −1.22573
\(813\) 0 0
\(814\) 4575.56 0.197019
\(815\) 4804.51 0.206497
\(816\) 0 0
\(817\) −23268.5 −0.996403
\(818\) −11714.5 −0.500719
\(819\) 0 0
\(820\) −4984.96 −0.212296
\(821\) −34803.3 −1.47947 −0.739735 0.672899i \(-0.765049\pi\)
−0.739735 + 0.672899i \(0.765049\pi\)
\(822\) 0 0
\(823\) 490.883 0.0207911 0.0103956 0.999946i \(-0.496691\pi\)
0.0103956 + 0.999946i \(0.496691\pi\)
\(824\) −8885.01 −0.375636
\(825\) 0 0
\(826\) −25476.1 −1.07316
\(827\) −30952.0 −1.30146 −0.650729 0.759310i \(-0.725537\pi\)
−0.650729 + 0.759310i \(0.725537\pi\)
\(828\) 0 0
\(829\) 8587.63 0.359784 0.179892 0.983686i \(-0.442425\pi\)
0.179892 + 0.983686i \(0.442425\pi\)
\(830\) −8009.44 −0.334954
\(831\) 0 0
\(832\) −1442.53 −0.0601092
\(833\) 54053.5 2.24831
\(834\) 0 0
\(835\) 21123.5 0.875460
\(836\) 21677.4 0.896806
\(837\) 0 0
\(838\) 10505.2 0.433051
\(839\) −3077.94 −0.126653 −0.0633267 0.997993i \(-0.520171\pi\)
−0.0633267 + 0.997993i \(0.520171\pi\)
\(840\) 0 0
\(841\) 33898.9 1.38992
\(842\) −407.035 −0.0166596
\(843\) 0 0
\(844\) 21580.7 0.880140
\(845\) −8444.83 −0.343800
\(846\) 0 0
\(847\) 3686.67 0.149558
\(848\) −7860.39 −0.318310
\(849\) 0 0
\(850\) 5202.43 0.209932
\(851\) 1378.74 0.0555377
\(852\) 0 0
\(853\) −9193.14 −0.369012 −0.184506 0.982831i \(-0.559068\pi\)
−0.184506 + 0.982831i \(0.559068\pi\)
\(854\) 38819.3 1.55547
\(855\) 0 0
\(856\) −313.513 −0.0125183
\(857\) −21445.8 −0.854813 −0.427407 0.904059i \(-0.640573\pi\)
−0.427407 + 0.904059i \(0.640573\pi\)
\(858\) 0 0
\(859\) −40276.7 −1.59979 −0.799897 0.600137i \(-0.795113\pi\)
−0.799897 + 0.600137i \(0.795113\pi\)
\(860\) −3277.26 −0.129946
\(861\) 0 0
\(862\) −6215.44 −0.245590
\(863\) −43713.4 −1.72424 −0.862120 0.506703i \(-0.830864\pi\)
−0.862120 + 0.506703i \(0.830864\pi\)
\(864\) 0 0
\(865\) −17244.9 −0.677855
\(866\) −11426.9 −0.448386
\(867\) 0 0
\(868\) 11661.7 0.456017
\(869\) −19090.3 −0.745218
\(870\) 0 0
\(871\) 7293.01 0.283713
\(872\) 6460.36 0.250889
\(873\) 0 0
\(874\) 6531.99 0.252801
\(875\) 3671.05 0.141833
\(876\) 0 0
\(877\) 24740.5 0.952596 0.476298 0.879284i \(-0.341978\pi\)
0.476298 + 0.879284i \(0.341978\pi\)
\(878\) −9174.27 −0.352639
\(879\) 0 0
\(880\) 3053.16 0.116957
\(881\) 44027.2 1.68367 0.841835 0.539735i \(-0.181476\pi\)
0.841835 + 0.539735i \(0.181476\pi\)
\(882\) 0 0
\(883\) 30245.0 1.15269 0.576344 0.817207i \(-0.304479\pi\)
0.576344 + 0.817207i \(0.304479\pi\)
\(884\) −9380.87 −0.356915
\(885\) 0 0
\(886\) −10536.0 −0.399509
\(887\) 7317.82 0.277010 0.138505 0.990362i \(-0.455770\pi\)
0.138505 + 0.990362i \(0.455770\pi\)
\(888\) 0 0
\(889\) −18845.9 −0.710991
\(890\) 7290.16 0.274569
\(891\) 0 0
\(892\) −6019.17 −0.225938
\(893\) 29193.8 1.09399
\(894\) 0 0
\(895\) −7289.08 −0.272232
\(896\) 3759.15 0.140161
\(897\) 0 0
\(898\) −33733.7 −1.25357
\(899\) −23966.8 −0.889140
\(900\) 0 0
\(901\) −51116.5 −1.89005
\(902\) −19024.9 −0.702282
\(903\) 0 0
\(904\) −8532.63 −0.313928
\(905\) −18174.8 −0.667572
\(906\) 0 0
\(907\) −35436.0 −1.29728 −0.648640 0.761096i \(-0.724662\pi\)
−0.648640 + 0.761096i \(0.724662\pi\)
\(908\) 7117.75 0.260144
\(909\) 0 0
\(910\) −6619.52 −0.241137
\(911\) 11359.1 0.413112 0.206556 0.978435i \(-0.433775\pi\)
0.206556 + 0.978435i \(0.433775\pi\)
\(912\) 0 0
\(913\) −30567.7 −1.10804
\(914\) 18248.5 0.660401
\(915\) 0 0
\(916\) 15904.6 0.573694
\(917\) −70188.2 −2.52761
\(918\) 0 0
\(919\) 40517.5 1.45435 0.727175 0.686452i \(-0.240833\pi\)
0.727175 + 0.686452i \(0.240833\pi\)
\(920\) 920.000 0.0329690
\(921\) 0 0
\(922\) −12023.8 −0.429484
\(923\) 20133.3 0.717982
\(924\) 0 0
\(925\) 1498.63 0.0532699
\(926\) −17168.2 −0.609267
\(927\) 0 0
\(928\) −7725.72 −0.273286
\(929\) −16242.2 −0.573614 −0.286807 0.957988i \(-0.592594\pi\)
−0.286807 + 0.957988i \(0.592594\pi\)
\(930\) 0 0
\(931\) 73769.2 2.59687
\(932\) 9244.50 0.324907
\(933\) 0 0
\(934\) −7908.19 −0.277049
\(935\) 19854.9 0.694463
\(936\) 0 0
\(937\) −47445.0 −1.65417 −0.827087 0.562074i \(-0.810004\pi\)
−0.827087 + 0.562074i \(0.810004\pi\)
\(938\) −19005.1 −0.661556
\(939\) 0 0
\(940\) 4111.81 0.142673
\(941\) 48063.5 1.66506 0.832532 0.553976i \(-0.186890\pi\)
0.832532 + 0.553976i \(0.186890\pi\)
\(942\) 0 0
\(943\) −5732.70 −0.197967
\(944\) −6939.74 −0.239268
\(945\) 0 0
\(946\) −12507.5 −0.429867
\(947\) −13745.4 −0.471663 −0.235831 0.971794i \(-0.575781\pi\)
−0.235831 + 0.971794i \(0.575781\pi\)
\(948\) 0 0
\(949\) −4423.92 −0.151324
\(950\) 7099.99 0.242478
\(951\) 0 0
\(952\) 24445.9 0.832245
\(953\) 13999.1 0.475839 0.237920 0.971285i \(-0.423535\pi\)
0.237920 + 0.971285i \(0.423535\pi\)
\(954\) 0 0
\(955\) −2244.06 −0.0760377
\(956\) −3297.07 −0.111543
\(957\) 0 0
\(958\) 190.675 0.00643052
\(959\) −26410.7 −0.889308
\(960\) 0 0
\(961\) −19936.4 −0.669207
\(962\) −2702.28 −0.0905666
\(963\) 0 0
\(964\) 14567.3 0.486703
\(965\) 21296.7 0.710431
\(966\) 0 0
\(967\) −800.187 −0.0266104 −0.0133052 0.999911i \(-0.504235\pi\)
−0.0133052 + 0.999911i \(0.504235\pi\)
\(968\) 1004.25 0.0333450
\(969\) 0 0
\(970\) 11399.6 0.377339
\(971\) 11363.5 0.375563 0.187782 0.982211i \(-0.439870\pi\)
0.187782 + 0.982211i \(0.439870\pi\)
\(972\) 0 0
\(973\) 9084.84 0.299328
\(974\) 27830.4 0.915549
\(975\) 0 0
\(976\) 10574.4 0.346803
\(977\) 17444.0 0.571220 0.285610 0.958346i \(-0.407804\pi\)
0.285610 + 0.958346i \(0.407804\pi\)
\(978\) 0 0
\(979\) 27822.5 0.908286
\(980\) 10390.0 0.338671
\(981\) 0 0
\(982\) −12582.7 −0.408889
\(983\) 54032.7 1.75318 0.876590 0.481237i \(-0.159812\pi\)
0.876590 + 0.481237i \(0.159812\pi\)
\(984\) 0 0
\(985\) 8836.87 0.285854
\(986\) −50240.7 −1.62271
\(987\) 0 0
\(988\) −12802.5 −0.412248
\(989\) −3768.84 −0.121175
\(990\) 0 0
\(991\) 28869.3 0.925392 0.462696 0.886517i \(-0.346882\pi\)
0.462696 + 0.886517i \(0.346882\pi\)
\(992\) 3176.66 0.101672
\(993\) 0 0
\(994\) −52466.2 −1.67417
\(995\) 5021.45 0.159991
\(996\) 0 0
\(997\) 3358.78 0.106694 0.0533469 0.998576i \(-0.483011\pi\)
0.0533469 + 0.998576i \(0.483011\pi\)
\(998\) −1276.66 −0.0404931
\(999\) 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 2070.4.a.bj.1.4 4
3.2 odd 2 230.4.a.h.1.2 4
12.11 even 2 1840.4.a.m.1.3 4
15.2 even 4 1150.4.b.n.599.3 8
15.8 even 4 1150.4.b.n.599.6 8
15.14 odd 2 1150.4.a.p.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.4.a.h.1.2 4 3.2 odd 2
1150.4.a.p.1.3 4 15.14 odd 2
1150.4.b.n.599.3 8 15.2 even 4
1150.4.b.n.599.6 8 15.8 even 4
1840.4.a.m.1.3 4 12.11 even 2
2070.4.a.bj.1.4 4 1.1 even 1 trivial