Properties

Label 1849.4.a.i.1.5
Level $1849$
Weight $4$
Character 1849.1
Self dual yes
Analytic conductor $109.095$
Analytic rank $1$
Dimension $50$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1849 = 43^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1849.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(109.094531601\)
Analytic rank: \(1\)
Dimension: \(50\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Character \(\chi\) \(=\) 1849.1

$q$-expansion

\(f(q)\) \(=\) \(q-4.85844 q^{2} -5.73018 q^{3} +15.6044 q^{4} +14.4569 q^{5} +27.8397 q^{6} -13.1859 q^{7} -36.9457 q^{8} +5.83494 q^{9} +O(q^{10})\) \(q-4.85844 q^{2} -5.73018 q^{3} +15.6044 q^{4} +14.4569 q^{5} +27.8397 q^{6} -13.1859 q^{7} -36.9457 q^{8} +5.83494 q^{9} -70.2380 q^{10} -56.6297 q^{11} -89.4162 q^{12} -68.5859 q^{13} +64.0631 q^{14} -82.8407 q^{15} +54.6629 q^{16} +23.3842 q^{17} -28.3487 q^{18} -119.755 q^{19} +225.592 q^{20} +75.5578 q^{21} +275.132 q^{22} -49.8515 q^{23} +211.705 q^{24} +84.0024 q^{25} +333.220 q^{26} +121.280 q^{27} -205.759 q^{28} +290.660 q^{29} +402.477 q^{30} +58.9050 q^{31} +29.9890 q^{32} +324.498 q^{33} -113.611 q^{34} -190.628 q^{35} +91.0510 q^{36} +147.330 q^{37} +581.820 q^{38} +393.009 q^{39} -534.121 q^{40} +216.626 q^{41} -367.093 q^{42} -883.674 q^{44} +84.3552 q^{45} +242.201 q^{46} -233.374 q^{47} -313.228 q^{48} -169.131 q^{49} -408.121 q^{50} -133.996 q^{51} -1070.24 q^{52} +712.940 q^{53} -589.229 q^{54} -818.690 q^{55} +487.164 q^{56} +686.215 q^{57} -1412.15 q^{58} +70.0677 q^{59} -1292.68 q^{60} +328.266 q^{61} -286.186 q^{62} -76.9392 q^{63} -583.003 q^{64} -991.541 q^{65} -1576.55 q^{66} +307.039 q^{67} +364.898 q^{68} +285.658 q^{69} +926.155 q^{70} +1135.63 q^{71} -215.576 q^{72} +203.220 q^{73} -715.792 q^{74} -481.349 q^{75} -1868.70 q^{76} +746.715 q^{77} -1909.41 q^{78} +255.296 q^{79} +790.257 q^{80} -852.497 q^{81} -1052.47 q^{82} +665.658 q^{83} +1179.04 q^{84} +338.064 q^{85} -1665.53 q^{87} +2092.22 q^{88} -834.098 q^{89} -409.835 q^{90} +904.369 q^{91} -777.905 q^{92} -337.536 q^{93} +1133.83 q^{94} -1731.28 q^{95} -171.842 q^{96} +817.923 q^{97} +821.713 q^{98} -330.431 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 50q - 10q^{2} - 2q^{3} + 186q^{4} + 8q^{5} - 51q^{6} - 6q^{7} - 138q^{8} + 360q^{9} + O(q^{10}) \) \( 50q - 10q^{2} - 2q^{3} + 186q^{4} + 8q^{5} - 51q^{6} - 6q^{7} - 138q^{8} + 360q^{9} - 137q^{10} - 252q^{11} - 48q^{12} - 192q^{13} - 272q^{14} - 314q^{15} + 542q^{16} - 236q^{17} - 386q^{18} + 12q^{19} + 108q^{20} - 408q^{21} + 1235q^{22} - 630q^{23} - 613q^{24} + 1098q^{25} + 1493q^{26} + 10q^{27} - 242q^{28} + 208q^{29} - 48q^{30} - 932q^{31} - 1124q^{32} + 254q^{33} + 765q^{34} - 1452q^{35} + 747q^{36} - 90q^{37} - 1213q^{38} - 1610q^{39} - 1693q^{40} - 1354q^{41} - 16q^{42} - 2704q^{44} + 4508q^{45} + 233q^{46} - 3484q^{47} - 376q^{48} + 1324q^{49} - 408q^{50} + 4054q^{51} - 2176q^{52} - 726q^{53} - 6497q^{54} - 3288q^{55} - 7097q^{56} - 870q^{57} + 275q^{58} - 4370q^{59} - 3891q^{60} + 1172q^{61} - 1546q^{62} - 3686q^{63} + 606q^{64} + 2610q^{65} - 4697q^{66} - 344q^{67} - 3221q^{68} + 136q^{69} - 1310q^{70} + 162q^{71} - 5814q^{72} + 746q^{73} - 4332q^{74} + 236q^{75} + 1338q^{76} + 2024q^{77} - 2782q^{78} - 2656q^{79} - 5713q^{80} - 86q^{81} - 4168q^{82} - 3514q^{83} - 4269q^{84} - 7558q^{85} - 10278q^{87} + 11692q^{88} + 2640q^{89} - 8286q^{90} - 5946q^{91} - 4271q^{92} - 2q^{93} + 9062q^{94} - 12140q^{95} - 700q^{96} - 3864q^{97} - 2826q^{98} - 8174q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.85844 −1.71772 −0.858859 0.512212i \(-0.828826\pi\)
−0.858859 + 0.512212i \(0.828826\pi\)
\(3\) −5.73018 −1.10277 −0.551387 0.834250i \(-0.685901\pi\)
−0.551387 + 0.834250i \(0.685901\pi\)
\(4\) 15.6044 1.95055
\(5\) 14.4569 1.29307 0.646533 0.762886i \(-0.276218\pi\)
0.646533 + 0.762886i \(0.276218\pi\)
\(6\) 27.8397 1.89425
\(7\) −13.1859 −0.711974 −0.355987 0.934491i \(-0.615855\pi\)
−0.355987 + 0.934491i \(0.615855\pi\)
\(8\) −36.9457 −1.63278
\(9\) 5.83494 0.216109
\(10\) −70.2380 −2.22112
\(11\) −56.6297 −1.55223 −0.776113 0.630594i \(-0.782811\pi\)
−0.776113 + 0.630594i \(0.782811\pi\)
\(12\) −89.4162 −2.15102
\(13\) −68.5859 −1.46325 −0.731627 0.681705i \(-0.761239\pi\)
−0.731627 + 0.681705i \(0.761239\pi\)
\(14\) 64.0631 1.22297
\(15\) −82.8407 −1.42596
\(16\) 54.6629 0.854108
\(17\) 23.3842 0.333618 0.166809 0.985989i \(-0.446654\pi\)
0.166809 + 0.985989i \(0.446654\pi\)
\(18\) −28.3487 −0.371214
\(19\) −119.755 −1.44598 −0.722989 0.690859i \(-0.757232\pi\)
−0.722989 + 0.690859i \(0.757232\pi\)
\(20\) 225.592 2.52220
\(21\) 75.5578 0.785146
\(22\) 275.132 2.66629
\(23\) −49.8515 −0.451946 −0.225973 0.974134i \(-0.572556\pi\)
−0.225973 + 0.974134i \(0.572556\pi\)
\(24\) 211.705 1.80059
\(25\) 84.0024 0.672019
\(26\) 333.220 2.51346
\(27\) 121.280 0.864454
\(28\) −205.759 −1.38874
\(29\) 290.660 1.86118 0.930590 0.366064i \(-0.119295\pi\)
0.930590 + 0.366064i \(0.119295\pi\)
\(30\) 402.477 2.44939
\(31\) 58.9050 0.341279 0.170639 0.985334i \(-0.445417\pi\)
0.170639 + 0.985334i \(0.445417\pi\)
\(32\) 29.9890 0.165667
\(33\) 324.498 1.71175
\(34\) −113.611 −0.573062
\(35\) −190.628 −0.920629
\(36\) 91.0510 0.421532
\(37\) 147.330 0.654617 0.327309 0.944918i \(-0.393858\pi\)
0.327309 + 0.944918i \(0.393858\pi\)
\(38\) 581.820 2.48378
\(39\) 393.009 1.61364
\(40\) −534.121 −2.11130
\(41\) 216.626 0.825154 0.412577 0.910923i \(-0.364629\pi\)
0.412577 + 0.910923i \(0.364629\pi\)
\(42\) −367.093 −1.34866
\(43\) 0 0
\(44\) −883.674 −3.02770
\(45\) 84.3552 0.279443
\(46\) 242.201 0.776316
\(47\) −233.374 −0.724277 −0.362139 0.932124i \(-0.617953\pi\)
−0.362139 + 0.932124i \(0.617953\pi\)
\(48\) −313.228 −0.941888
\(49\) −169.131 −0.493093
\(50\) −408.121 −1.15434
\(51\) −133.996 −0.367905
\(52\) −1070.24 −2.85416
\(53\) 712.940 1.84773 0.923866 0.382716i \(-0.125011\pi\)
0.923866 + 0.382716i \(0.125011\pi\)
\(54\) −589.229 −1.48489
\(55\) −818.690 −2.00713
\(56\) 487.164 1.16250
\(57\) 686.215 1.59459
\(58\) −1412.15 −3.19698
\(59\) 70.0677 0.154611 0.0773054 0.997007i \(-0.475368\pi\)
0.0773054 + 0.997007i \(0.475368\pi\)
\(60\) −1292.68 −2.78141
\(61\) 328.266 0.689019 0.344509 0.938783i \(-0.388045\pi\)
0.344509 + 0.938783i \(0.388045\pi\)
\(62\) −286.186 −0.586221
\(63\) −76.9392 −0.153864
\(64\) −583.003 −1.13868
\(65\) −991.541 −1.89208
\(66\) −1576.55 −2.94031
\(67\) 307.039 0.559863 0.279932 0.960020i \(-0.409688\pi\)
0.279932 + 0.960020i \(0.409688\pi\)
\(68\) 364.898 0.650740
\(69\) 285.658 0.498394
\(70\) 926.155 1.58138
\(71\) 1135.63 1.89823 0.949114 0.314933i \(-0.101982\pi\)
0.949114 + 0.314933i \(0.101982\pi\)
\(72\) −215.576 −0.352859
\(73\) 203.220 0.325824 0.162912 0.986641i \(-0.447911\pi\)
0.162912 + 0.986641i \(0.447911\pi\)
\(74\) −715.792 −1.12445
\(75\) −481.349 −0.741085
\(76\) −1868.70 −2.82046
\(77\) 746.715 1.10514
\(78\) −1909.41 −2.77177
\(79\) 255.296 0.363583 0.181791 0.983337i \(-0.441810\pi\)
0.181791 + 0.983337i \(0.441810\pi\)
\(80\) 790.257 1.10442
\(81\) −852.497 −1.16941
\(82\) −1052.47 −1.41738
\(83\) 665.658 0.880306 0.440153 0.897923i \(-0.354924\pi\)
0.440153 + 0.897923i \(0.354924\pi\)
\(84\) 1179.04 1.53147
\(85\) 338.064 0.431390
\(86\) 0 0
\(87\) −1665.53 −2.05246
\(88\) 2092.22 2.53445
\(89\) −834.098 −0.993419 −0.496709 0.867917i \(-0.665458\pi\)
−0.496709 + 0.867917i \(0.665458\pi\)
\(90\) −409.835 −0.480004
\(91\) 904.369 1.04180
\(92\) −777.905 −0.881545
\(93\) −337.536 −0.376353
\(94\) 1133.83 1.24410
\(95\) −1731.28 −1.86974
\(96\) −171.842 −0.182693
\(97\) 817.923 0.856160 0.428080 0.903741i \(-0.359190\pi\)
0.428080 + 0.903741i \(0.359190\pi\)
\(98\) 821.713 0.846995
\(99\) −330.431 −0.335450
\(100\) 1310.81 1.31081
\(101\) −642.462 −0.632944 −0.316472 0.948602i \(-0.602498\pi\)
−0.316472 + 0.948602i \(0.602498\pi\)
\(102\) 651.010 0.631957
\(103\) 76.9496 0.0736123 0.0368062 0.999322i \(-0.488282\pi\)
0.0368062 + 0.999322i \(0.488282\pi\)
\(104\) 2533.95 2.38918
\(105\) 1092.33 1.01524
\(106\) −3463.77 −3.17388
\(107\) −843.742 −0.762315 −0.381157 0.924510i \(-0.624474\pi\)
−0.381157 + 0.924510i \(0.624474\pi\)
\(108\) 1892.50 1.68616
\(109\) −1162.50 −1.02154 −0.510769 0.859718i \(-0.670639\pi\)
−0.510769 + 0.859718i \(0.670639\pi\)
\(110\) 3977.56 3.44768
\(111\) −844.225 −0.721894
\(112\) −720.782 −0.608103
\(113\) 884.391 0.736252 0.368126 0.929776i \(-0.379999\pi\)
0.368126 + 0.929776i \(0.379999\pi\)
\(114\) −3333.93 −2.73905
\(115\) −720.699 −0.584396
\(116\) 4535.58 3.63033
\(117\) −400.195 −0.316222
\(118\) −340.420 −0.265578
\(119\) −308.343 −0.237527
\(120\) 3060.61 2.32828
\(121\) 1875.92 1.40941
\(122\) −1594.86 −1.18354
\(123\) −1241.31 −0.909958
\(124\) 919.179 0.665683
\(125\) −592.699 −0.424101
\(126\) 373.804 0.264295
\(127\) −582.571 −0.407046 −0.203523 0.979070i \(-0.565239\pi\)
−0.203523 + 0.979070i \(0.565239\pi\)
\(128\) 2592.57 1.79026
\(129\) 0 0
\(130\) 4817.34 3.25007
\(131\) −903.044 −0.602285 −0.301142 0.953579i \(-0.597368\pi\)
−0.301142 + 0.953579i \(0.597368\pi\)
\(132\) 5063.61 3.33887
\(133\) 1579.08 1.02950
\(134\) −1491.73 −0.961687
\(135\) 1753.33 1.11780
\(136\) −863.947 −0.544727
\(137\) 2121.22 1.32283 0.661415 0.750020i \(-0.269956\pi\)
0.661415 + 0.750020i \(0.269956\pi\)
\(138\) −1387.85 −0.856100
\(139\) 1042.17 0.635939 0.317970 0.948101i \(-0.396999\pi\)
0.317970 + 0.948101i \(0.396999\pi\)
\(140\) −2974.64 −1.79574
\(141\) 1337.27 0.798713
\(142\) −5517.38 −3.26062
\(143\) 3884.00 2.27130
\(144\) 318.955 0.184580
\(145\) 4202.05 2.40663
\(146\) −987.334 −0.559673
\(147\) 969.151 0.543770
\(148\) 2298.99 1.27687
\(149\) −323.453 −0.177841 −0.0889205 0.996039i \(-0.528342\pi\)
−0.0889205 + 0.996039i \(0.528342\pi\)
\(150\) 2338.60 1.27297
\(151\) −864.578 −0.465949 −0.232975 0.972483i \(-0.574846\pi\)
−0.232975 + 0.972483i \(0.574846\pi\)
\(152\) 4424.42 2.36097
\(153\) 136.446 0.0720979
\(154\) −3627.87 −1.89833
\(155\) 851.584 0.441296
\(156\) 6132.69 3.14749
\(157\) −577.261 −0.293442 −0.146721 0.989178i \(-0.546872\pi\)
−0.146721 + 0.989178i \(0.546872\pi\)
\(158\) −1240.34 −0.624533
\(159\) −4085.27 −2.03763
\(160\) 433.548 0.214219
\(161\) 657.339 0.321774
\(162\) 4141.80 2.00871
\(163\) −1803.66 −0.866707 −0.433353 0.901224i \(-0.642670\pi\)
−0.433353 + 0.901224i \(0.642670\pi\)
\(164\) 3380.33 1.60951
\(165\) 4691.24 2.21341
\(166\) −3234.06 −1.51212
\(167\) −2552.51 −1.18275 −0.591374 0.806397i \(-0.701414\pi\)
−0.591374 + 0.806397i \(0.701414\pi\)
\(168\) −2791.53 −1.28197
\(169\) 2507.03 1.14111
\(170\) −1642.46 −0.741007
\(171\) −698.761 −0.312489
\(172\) 0 0
\(173\) −676.365 −0.297243 −0.148622 0.988894i \(-0.547484\pi\)
−0.148622 + 0.988894i \(0.547484\pi\)
\(174\) 8091.89 3.52554
\(175\) −1107.65 −0.478460
\(176\) −3095.54 −1.32577
\(177\) −401.500 −0.170501
\(178\) 4052.42 1.70641
\(179\) −1530.14 −0.638927 −0.319464 0.947598i \(-0.603503\pi\)
−0.319464 + 0.947598i \(0.603503\pi\)
\(180\) 1316.32 0.545069
\(181\) 2614.23 1.07356 0.536780 0.843722i \(-0.319641\pi\)
0.536780 + 0.843722i \(0.319641\pi\)
\(182\) −4393.82 −1.78952
\(183\) −1881.02 −0.759832
\(184\) 1841.80 0.737930
\(185\) 2129.93 0.846463
\(186\) 1639.90 0.646469
\(187\) −1324.24 −0.517851
\(188\) −3641.66 −1.41274
\(189\) −1599.18 −0.615469
\(190\) 8411.33 3.21169
\(191\) −3942.47 −1.49355 −0.746773 0.665079i \(-0.768398\pi\)
−0.746773 + 0.665079i \(0.768398\pi\)
\(192\) 3340.71 1.25570
\(193\) −80.0135 −0.0298420 −0.0149210 0.999889i \(-0.504750\pi\)
−0.0149210 + 0.999889i \(0.504750\pi\)
\(194\) −3973.83 −1.47064
\(195\) 5681.70 2.08654
\(196\) −2639.19 −0.961806
\(197\) 2777.66 1.00457 0.502284 0.864703i \(-0.332493\pi\)
0.502284 + 0.864703i \(0.332493\pi\)
\(198\) 1605.38 0.576208
\(199\) 2219.08 0.790484 0.395242 0.918577i \(-0.370661\pi\)
0.395242 + 0.918577i \(0.370661\pi\)
\(200\) −3103.53 −1.09726
\(201\) −1759.39 −0.617402
\(202\) 3121.36 1.08722
\(203\) −3832.62 −1.32511
\(204\) −2090.93 −0.717619
\(205\) 3131.75 1.06698
\(206\) −373.855 −0.126445
\(207\) −290.881 −0.0976696
\(208\) −3749.11 −1.24978
\(209\) 6781.66 2.24449
\(210\) −5307.03 −1.74390
\(211\) −3231.50 −1.05434 −0.527169 0.849760i \(-0.676747\pi\)
−0.527169 + 0.849760i \(0.676747\pi\)
\(212\) 11125.0 3.60410
\(213\) −6507.35 −2.09332
\(214\) 4099.27 1.30944
\(215\) 0 0
\(216\) −4480.76 −1.41147
\(217\) −776.717 −0.242982
\(218\) 5647.96 1.75471
\(219\) −1164.49 −0.359310
\(220\) −12775.2 −3.91502
\(221\) −1603.83 −0.488168
\(222\) 4101.61 1.24001
\(223\) 2987.09 0.896996 0.448498 0.893784i \(-0.351959\pi\)
0.448498 + 0.893784i \(0.351959\pi\)
\(224\) −395.433 −0.117951
\(225\) 490.149 0.145229
\(226\) −4296.76 −1.26467
\(227\) −2015.14 −0.589206 −0.294603 0.955620i \(-0.595187\pi\)
−0.294603 + 0.955620i \(0.595187\pi\)
\(228\) 10708.0 3.11033
\(229\) −4594.18 −1.32573 −0.662865 0.748739i \(-0.730659\pi\)
−0.662865 + 0.748739i \(0.730659\pi\)
\(230\) 3501.47 1.00383
\(231\) −4278.81 −1.21872
\(232\) −10738.6 −3.03890
\(233\) 919.717 0.258595 0.129298 0.991606i \(-0.458728\pi\)
0.129298 + 0.991606i \(0.458728\pi\)
\(234\) 1944.32 0.543181
\(235\) −3373.86 −0.936538
\(236\) 1093.37 0.301577
\(237\) −1462.89 −0.400950
\(238\) 1498.07 0.408005
\(239\) 1476.81 0.399695 0.199847 0.979827i \(-0.435955\pi\)
0.199847 + 0.979827i \(0.435955\pi\)
\(240\) −4528.32 −1.21792
\(241\) −5049.60 −1.34968 −0.674841 0.737963i \(-0.735788\pi\)
−0.674841 + 0.737963i \(0.735788\pi\)
\(242\) −9114.05 −2.42096
\(243\) 1610.41 0.425135
\(244\) 5122.41 1.34397
\(245\) −2445.11 −0.637602
\(246\) 6030.81 1.56305
\(247\) 8213.48 2.11583
\(248\) −2176.28 −0.557235
\(249\) −3814.34 −0.970778
\(250\) 2879.59 0.728486
\(251\) −292.300 −0.0735052 −0.0367526 0.999324i \(-0.511701\pi\)
−0.0367526 + 0.999324i \(0.511701\pi\)
\(252\) −1200.59 −0.300120
\(253\) 2823.08 0.701523
\(254\) 2830.39 0.699190
\(255\) −1937.17 −0.475726
\(256\) −7931.84 −1.93648
\(257\) −4.22507 −0.00102550 −0.000512749 1.00000i \(-0.500163\pi\)
−0.000512749 1.00000i \(0.500163\pi\)
\(258\) 0 0
\(259\) −1942.68 −0.466070
\(260\) −15472.4 −3.69061
\(261\) 1695.98 0.402217
\(262\) 4387.38 1.03456
\(263\) 8246.98 1.93358 0.966788 0.255580i \(-0.0822666\pi\)
0.966788 + 0.255580i \(0.0822666\pi\)
\(264\) −11988.8 −2.79492
\(265\) 10306.9 2.38924
\(266\) −7671.85 −1.76839
\(267\) 4779.53 1.09552
\(268\) 4791.18 1.09204
\(269\) 3136.78 0.710976 0.355488 0.934681i \(-0.384315\pi\)
0.355488 + 0.934681i \(0.384315\pi\)
\(270\) −8518.44 −1.92006
\(271\) −1246.03 −0.279303 −0.139652 0.990201i \(-0.544598\pi\)
−0.139652 + 0.990201i \(0.544598\pi\)
\(272\) 1278.25 0.284946
\(273\) −5182.20 −1.14887
\(274\) −10305.8 −2.27225
\(275\) −4757.03 −1.04313
\(276\) 4457.53 0.972145
\(277\) 1353.84 0.293662 0.146831 0.989162i \(-0.453093\pi\)
0.146831 + 0.989162i \(0.453093\pi\)
\(278\) −5063.31 −1.09236
\(279\) 343.707 0.0737534
\(280\) 7042.88 1.50319
\(281\) −5032.93 −1.06847 −0.534234 0.845337i \(-0.679400\pi\)
−0.534234 + 0.845337i \(0.679400\pi\)
\(282\) −6497.05 −1.37196
\(283\) −1540.76 −0.323636 −0.161818 0.986821i \(-0.551736\pi\)
−0.161818 + 0.986821i \(0.551736\pi\)
\(284\) 17720.8 3.70260
\(285\) 9920.55 2.06190
\(286\) −18870.2 −3.90146
\(287\) −2856.42 −0.587488
\(288\) 174.984 0.0358022
\(289\) −4366.18 −0.888699
\(290\) −20415.4 −4.13391
\(291\) −4686.85 −0.944150
\(292\) 3171.14 0.635537
\(293\) −2032.88 −0.405331 −0.202665 0.979248i \(-0.564960\pi\)
−0.202665 + 0.979248i \(0.564960\pi\)
\(294\) −4708.56 −0.934044
\(295\) 1012.96 0.199922
\(296\) −5443.19 −1.06885
\(297\) −6868.02 −1.34183
\(298\) 1571.48 0.305481
\(299\) 3419.11 0.661312
\(300\) −7511.17 −1.44553
\(301\) 0 0
\(302\) 4200.50 0.800369
\(303\) 3681.42 0.697994
\(304\) −6546.14 −1.23502
\(305\) 4745.71 0.890947
\(306\) −662.913 −0.123844
\(307\) −9698.57 −1.80302 −0.901509 0.432761i \(-0.857539\pi\)
−0.901509 + 0.432761i \(0.857539\pi\)
\(308\) 11652.1 2.15564
\(309\) −440.935 −0.0811777
\(310\) −4137.37 −0.758022
\(311\) −8337.39 −1.52016 −0.760081 0.649829i \(-0.774841\pi\)
−0.760081 + 0.649829i \(0.774841\pi\)
\(312\) −14520.0 −2.63472
\(313\) −9108.79 −1.64492 −0.822459 0.568825i \(-0.807398\pi\)
−0.822459 + 0.568825i \(0.807398\pi\)
\(314\) 2804.59 0.504051
\(315\) −1112.30 −0.198956
\(316\) 3983.75 0.709188
\(317\) 3842.07 0.680733 0.340366 0.940293i \(-0.389449\pi\)
0.340366 + 0.940293i \(0.389449\pi\)
\(318\) 19848.0 3.50007
\(319\) −16460.0 −2.88897
\(320\) −8428.43 −1.47239
\(321\) 4834.79 0.840660
\(322\) −3193.64 −0.552716
\(323\) −2800.37 −0.482405
\(324\) −13302.7 −2.28099
\(325\) −5761.38 −0.983335
\(326\) 8762.95 1.48876
\(327\) 6661.36 1.12653
\(328\) −8003.40 −1.34730
\(329\) 3077.25 0.515666
\(330\) −22792.1 −3.80201
\(331\) 2360.71 0.392014 0.196007 0.980603i \(-0.437202\pi\)
0.196007 + 0.980603i \(0.437202\pi\)
\(332\) 10387.2 1.71709
\(333\) 859.659 0.141469
\(334\) 12401.2 2.03163
\(335\) 4438.84 0.723940
\(336\) 4130.21 0.670599
\(337\) −66.5688 −0.0107603 −0.00538017 0.999986i \(-0.501713\pi\)
−0.00538017 + 0.999986i \(0.501713\pi\)
\(338\) −12180.2 −1.96011
\(339\) −5067.72 −0.811919
\(340\) 5275.29 0.841450
\(341\) −3335.77 −0.529742
\(342\) 3394.89 0.536768
\(343\) 6752.93 1.06304
\(344\) 0 0
\(345\) 4129.73 0.644456
\(346\) 3286.08 0.510580
\(347\) −2886.72 −0.446592 −0.223296 0.974751i \(-0.571682\pi\)
−0.223296 + 0.974751i \(0.571682\pi\)
\(348\) −25989.7 −4.00343
\(349\) −7743.50 −1.18768 −0.593840 0.804583i \(-0.702389\pi\)
−0.593840 + 0.804583i \(0.702389\pi\)
\(350\) 5381.45 0.821859
\(351\) −8318.07 −1.26492
\(352\) −1698.27 −0.257153
\(353\) −2857.45 −0.430840 −0.215420 0.976521i \(-0.569112\pi\)
−0.215420 + 0.976521i \(0.569112\pi\)
\(354\) 1950.67 0.292872
\(355\) 16417.7 2.45453
\(356\) −13015.6 −1.93772
\(357\) 1766.86 0.261939
\(358\) 7434.09 1.09750
\(359\) −2242.44 −0.329670 −0.164835 0.986321i \(-0.552709\pi\)
−0.164835 + 0.986321i \(0.552709\pi\)
\(360\) −3116.56 −0.456270
\(361\) 7482.16 1.09085
\(362\) −12701.1 −1.84407
\(363\) −10749.4 −1.55426
\(364\) 14112.2 2.03208
\(365\) 2937.94 0.421312
\(366\) 9138.84 1.30518
\(367\) 11678.8 1.66111 0.830557 0.556934i \(-0.188022\pi\)
0.830557 + 0.556934i \(0.188022\pi\)
\(368\) −2725.03 −0.386011
\(369\) 1264.00 0.178323
\(370\) −10348.1 −1.45398
\(371\) −9400.78 −1.31554
\(372\) −5267.06 −0.734097
\(373\) 4698.81 0.652266 0.326133 0.945324i \(-0.394254\pi\)
0.326133 + 0.945324i \(0.394254\pi\)
\(374\) 6433.75 0.889522
\(375\) 3396.27 0.467687
\(376\) 8622.15 1.18259
\(377\) −19935.2 −2.72338
\(378\) 7769.54 1.05720
\(379\) 9794.57 1.32748 0.663738 0.747965i \(-0.268969\pi\)
0.663738 + 0.747965i \(0.268969\pi\)
\(380\) −27015.7 −3.64704
\(381\) 3338.24 0.448879
\(382\) 19154.3 2.56549
\(383\) 4659.25 0.621610 0.310805 0.950474i \(-0.399401\pi\)
0.310805 + 0.950474i \(0.399401\pi\)
\(384\) −14855.9 −1.97425
\(385\) 10795.2 1.42902
\(386\) 388.741 0.0512601
\(387\) 0 0
\(388\) 12763.2 1.66999
\(389\) 15017.9 1.95743 0.978713 0.205236i \(-0.0657961\pi\)
0.978713 + 0.205236i \(0.0657961\pi\)
\(390\) −27604.2 −3.58409
\(391\) −1165.74 −0.150777
\(392\) 6248.66 0.805115
\(393\) 5174.60 0.664184
\(394\) −13495.1 −1.72557
\(395\) 3690.79 0.470137
\(396\) −5156.19 −0.654313
\(397\) 5330.02 0.673819 0.336909 0.941537i \(-0.390618\pi\)
0.336909 + 0.941537i \(0.390618\pi\)
\(398\) −10781.3 −1.35783
\(399\) −9048.39 −1.13530
\(400\) 4591.82 0.573977
\(401\) 1881.10 0.234259 0.117130 0.993117i \(-0.462631\pi\)
0.117130 + 0.993117i \(0.462631\pi\)
\(402\) 8547.89 1.06052
\(403\) −4040.05 −0.499378
\(404\) −10025.3 −1.23459
\(405\) −12324.5 −1.51212
\(406\) 18620.6 2.27617
\(407\) −8343.23 −1.01611
\(408\) 4950.57 0.600710
\(409\) 8907.66 1.07691 0.538454 0.842655i \(-0.319009\pi\)
0.538454 + 0.842655i \(0.319009\pi\)
\(410\) −15215.4 −1.83277
\(411\) −12154.9 −1.45878
\(412\) 1200.76 0.143585
\(413\) −923.908 −0.110079
\(414\) 1413.23 0.167769
\(415\) 9623.36 1.13829
\(416\) −2056.82 −0.242413
\(417\) −5971.81 −0.701297
\(418\) −32948.3 −3.85539
\(419\) −14129.0 −1.64736 −0.823682 0.567052i \(-0.808084\pi\)
−0.823682 + 0.567052i \(0.808084\pi\)
\(420\) 17045.2 1.98029
\(421\) 631.853 0.0731464 0.0365732 0.999331i \(-0.488356\pi\)
0.0365732 + 0.999331i \(0.488356\pi\)
\(422\) 15700.0 1.81106
\(423\) −1361.72 −0.156523
\(424\) −26340.0 −3.01695
\(425\) 1964.33 0.224198
\(426\) 31615.6 3.59572
\(427\) −4328.50 −0.490563
\(428\) −13166.1 −1.48694
\(429\) −22256.0 −2.50473
\(430\) 0 0
\(431\) −1391.25 −0.155485 −0.0777426 0.996973i \(-0.524771\pi\)
−0.0777426 + 0.996973i \(0.524771\pi\)
\(432\) 6629.50 0.738337
\(433\) −11215.1 −1.24472 −0.622359 0.782732i \(-0.713825\pi\)
−0.622359 + 0.782732i \(0.713825\pi\)
\(434\) 3773.63 0.417374
\(435\) −24078.5 −2.65396
\(436\) −18140.2 −1.99257
\(437\) 5969.95 0.653504
\(438\) 5657.60 0.617193
\(439\) 4321.01 0.469774 0.234887 0.972023i \(-0.424528\pi\)
0.234887 + 0.972023i \(0.424528\pi\)
\(440\) 30247.1 3.27721
\(441\) −986.870 −0.106562
\(442\) 7792.10 0.838535
\(443\) −10333.2 −1.10823 −0.554113 0.832441i \(-0.686943\pi\)
−0.554113 + 0.832441i \(0.686943\pi\)
\(444\) −13173.6 −1.40809
\(445\) −12058.5 −1.28456
\(446\) −14512.6 −1.54079
\(447\) 1853.44 0.196118
\(448\) 7687.44 0.810709
\(449\) 8737.55 0.918375 0.459188 0.888339i \(-0.348141\pi\)
0.459188 + 0.888339i \(0.348141\pi\)
\(450\) −2381.36 −0.249463
\(451\) −12267.5 −1.28083
\(452\) 13800.4 1.43610
\(453\) 4954.18 0.513836
\(454\) 9790.45 1.01209
\(455\) 13074.4 1.34711
\(456\) −25352.7 −2.60362
\(457\) 11206.6 1.14709 0.573545 0.819174i \(-0.305568\pi\)
0.573545 + 0.819174i \(0.305568\pi\)
\(458\) 22320.5 2.27723
\(459\) 2836.03 0.288398
\(460\) −11246.1 −1.13990
\(461\) −14525.7 −1.46752 −0.733762 0.679407i \(-0.762237\pi\)
−0.733762 + 0.679407i \(0.762237\pi\)
\(462\) 20788.4 2.09342
\(463\) 4930.50 0.494903 0.247451 0.968900i \(-0.420407\pi\)
0.247451 + 0.968900i \(0.420407\pi\)
\(464\) 15888.3 1.58965
\(465\) −4879.73 −0.486649
\(466\) −4468.39 −0.444193
\(467\) 6108.04 0.605238 0.302619 0.953112i \(-0.402139\pi\)
0.302619 + 0.953112i \(0.402139\pi\)
\(468\) −6244.81 −0.616809
\(469\) −4048.60 −0.398608
\(470\) 16391.7 1.60871
\(471\) 3307.81 0.323600
\(472\) −2588.70 −0.252446
\(473\) 0 0
\(474\) 7107.37 0.688718
\(475\) −10059.7 −0.971725
\(476\) −4811.52 −0.463310
\(477\) 4159.96 0.399311
\(478\) −7175.00 −0.686563
\(479\) −10162.4 −0.969382 −0.484691 0.874685i \(-0.661068\pi\)
−0.484691 + 0.874685i \(0.661068\pi\)
\(480\) −2484.31 −0.236235
\(481\) −10104.7 −0.957871
\(482\) 24533.2 2.31837
\(483\) −3766.67 −0.354843
\(484\) 29272.7 2.74912
\(485\) 11824.6 1.10707
\(486\) −7824.08 −0.730263
\(487\) 13884.5 1.29192 0.645961 0.763371i \(-0.276457\pi\)
0.645961 + 0.763371i \(0.276457\pi\)
\(488\) −12128.0 −1.12502
\(489\) 10335.3 0.955781
\(490\) 11879.4 1.09522
\(491\) 14658.8 1.34733 0.673667 0.739035i \(-0.264718\pi\)
0.673667 + 0.739035i \(0.264718\pi\)
\(492\) −19369.9 −1.77492
\(493\) 6796.86 0.620923
\(494\) −39904.7 −3.63441
\(495\) −4777.01 −0.433759
\(496\) 3219.92 0.291489
\(497\) −14974.3 −1.35149
\(498\) 18531.7 1.66752
\(499\) 15383.3 1.38006 0.690029 0.723782i \(-0.257598\pi\)
0.690029 + 0.723782i \(0.257598\pi\)
\(500\) −9248.73 −0.827232
\(501\) 14626.3 1.30430
\(502\) 1420.12 0.126261
\(503\) −12847.1 −1.13882 −0.569409 0.822054i \(-0.692828\pi\)
−0.569409 + 0.822054i \(0.692828\pi\)
\(504\) 2842.57 0.251227
\(505\) −9288.02 −0.818438
\(506\) −13715.7 −1.20502
\(507\) −14365.7 −1.25839
\(508\) −9090.69 −0.793965
\(509\) −14988.6 −1.30522 −0.652611 0.757693i \(-0.726326\pi\)
−0.652611 + 0.757693i \(0.726326\pi\)
\(510\) 9411.60 0.817162
\(511\) −2679.65 −0.231978
\(512\) 17795.8 1.53607
\(513\) −14523.8 −1.24998
\(514\) 20.5273 0.00176151
\(515\) 1112.45 0.0951856
\(516\) 0 0
\(517\) 13215.9 1.12424
\(518\) 9438.39 0.800577
\(519\) 3875.69 0.327792
\(520\) 36633.2 3.08937
\(521\) 10203.3 0.857997 0.428998 0.903305i \(-0.358867\pi\)
0.428998 + 0.903305i \(0.358867\pi\)
\(522\) −8239.83 −0.690896
\(523\) 5487.11 0.458766 0.229383 0.973336i \(-0.426329\pi\)
0.229383 + 0.973336i \(0.426329\pi\)
\(524\) −14091.5 −1.17479
\(525\) 6347.03 0.527633
\(526\) −40067.4 −3.32134
\(527\) 1377.45 0.113857
\(528\) 17738.0 1.46202
\(529\) −9681.83 −0.795745
\(530\) −50075.5 −4.10404
\(531\) 408.841 0.0334128
\(532\) 24640.6 2.00809
\(533\) −14857.5 −1.20741
\(534\) −23221.1 −1.88179
\(535\) −12197.9 −0.985723
\(536\) −11343.8 −0.914136
\(537\) 8767.97 0.704592
\(538\) −15239.8 −1.22126
\(539\) 9577.84 0.765393
\(540\) 27359.7 2.18032
\(541\) 13269.0 1.05449 0.527245 0.849713i \(-0.323225\pi\)
0.527245 + 0.849713i \(0.323225\pi\)
\(542\) 6053.78 0.479764
\(543\) −14980.0 −1.18389
\(544\) 701.269 0.0552696
\(545\) −16806.2 −1.32092
\(546\) 25177.4 1.97343
\(547\) 9377.89 0.733034 0.366517 0.930411i \(-0.380550\pi\)
0.366517 + 0.930411i \(0.380550\pi\)
\(548\) 33100.4 2.58025
\(549\) 1915.41 0.148903
\(550\) 23111.7 1.79180
\(551\) −34807.9 −2.69122
\(552\) −10553.8 −0.813770
\(553\) −3366.32 −0.258861
\(554\) −6577.55 −0.504429
\(555\) −12204.9 −0.933457
\(556\) 16262.4 1.24043
\(557\) 10526.2 0.800735 0.400368 0.916355i \(-0.368882\pi\)
0.400368 + 0.916355i \(0.368882\pi\)
\(558\) −1669.88 −0.126688
\(559\) 0 0
\(560\) −10420.3 −0.786317
\(561\) 7588.14 0.571072
\(562\) 24452.2 1.83533
\(563\) 20928.1 1.56663 0.783315 0.621625i \(-0.213527\pi\)
0.783315 + 0.621625i \(0.213527\pi\)
\(564\) 20867.4 1.55793
\(565\) 12785.6 0.952022
\(566\) 7485.71 0.555915
\(567\) 11241.0 0.832586
\(568\) −41956.5 −3.09940
\(569\) −16353.3 −1.20486 −0.602431 0.798171i \(-0.705801\pi\)
−0.602431 + 0.798171i \(0.705801\pi\)
\(570\) −48198.4 −3.54177
\(571\) −4027.42 −0.295171 −0.147585 0.989049i \(-0.547150\pi\)
−0.147585 + 0.989049i \(0.547150\pi\)
\(572\) 60607.6 4.43030
\(573\) 22591.1 1.64704
\(574\) 13877.7 1.00914
\(575\) −4187.65 −0.303716
\(576\) −3401.79 −0.246078
\(577\) 18455.6 1.33157 0.665786 0.746143i \(-0.268096\pi\)
0.665786 + 0.746143i \(0.268096\pi\)
\(578\) 21212.8 1.52653
\(579\) 458.492 0.0329089
\(580\) 65570.6 4.69426
\(581\) −8777.32 −0.626755
\(582\) 22770.8 1.62178
\(583\) −40373.5 −2.86810
\(584\) −7508.12 −0.532000
\(585\) −5785.58 −0.408896
\(586\) 9876.61 0.696244
\(587\) −12216.2 −0.858974 −0.429487 0.903073i \(-0.641306\pi\)
−0.429487 + 0.903073i \(0.641306\pi\)
\(588\) 15123.1 1.06065
\(589\) −7054.14 −0.493482
\(590\) −4921.42 −0.343409
\(591\) −15916.5 −1.10781
\(592\) 8053.47 0.559114
\(593\) 15656.9 1.08424 0.542118 0.840302i \(-0.317622\pi\)
0.542118 + 0.840302i \(0.317622\pi\)
\(594\) 33367.9 2.30488
\(595\) −4457.69 −0.307138
\(596\) −5047.31 −0.346889
\(597\) −12715.7 −0.871724
\(598\) −16611.5 −1.13595
\(599\) −7086.75 −0.483400 −0.241700 0.970351i \(-0.577705\pi\)
−0.241700 + 0.970351i \(0.577705\pi\)
\(600\) 17783.8 1.21003
\(601\) −4349.45 −0.295204 −0.147602 0.989047i \(-0.547155\pi\)
−0.147602 + 0.989047i \(0.547155\pi\)
\(602\) 0 0
\(603\) 1791.56 0.120991
\(604\) −13491.2 −0.908859
\(605\) 27120.0 1.82246
\(606\) −17886.0 −1.19896
\(607\) 11528.1 0.770860 0.385430 0.922737i \(-0.374053\pi\)
0.385430 + 0.922737i \(0.374053\pi\)
\(608\) −3591.32 −0.239551
\(609\) 21961.6 1.46130
\(610\) −23056.8 −1.53040
\(611\) 16006.1 1.05980
\(612\) 2129.16 0.140631
\(613\) 19355.5 1.27530 0.637652 0.770324i \(-0.279906\pi\)
0.637652 + 0.770324i \(0.279906\pi\)
\(614\) 47119.9 3.09708
\(615\) −17945.5 −1.17664
\(616\) −27587.9 −1.80446
\(617\) −13935.0 −0.909238 −0.454619 0.890686i \(-0.650225\pi\)
−0.454619 + 0.890686i \(0.650225\pi\)
\(618\) 2142.26 0.139440
\(619\) 15495.3 1.00615 0.503075 0.864243i \(-0.332202\pi\)
0.503075 + 0.864243i \(0.332202\pi\)
\(620\) 13288.5 0.860772
\(621\) −6045.97 −0.390687
\(622\) 40506.7 2.61121
\(623\) 10998.4 0.707288
\(624\) 21483.0 1.37822
\(625\) −19068.9 −1.22041
\(626\) 44254.5 2.82550
\(627\) −38860.1 −2.47516
\(628\) −9007.83 −0.572375
\(629\) 3445.19 0.218392
\(630\) 5404.06 0.341750
\(631\) 13393.6 0.844993 0.422496 0.906365i \(-0.361154\pi\)
0.422496 + 0.906365i \(0.361154\pi\)
\(632\) −9432.09 −0.593652
\(633\) 18517.0 1.16270
\(634\) −18666.5 −1.16931
\(635\) −8422.18 −0.526337
\(636\) −63748.3 −3.97451
\(637\) 11600.0 0.721521
\(638\) 79969.8 4.96244
\(639\) 6626.32 0.410224
\(640\) 37480.6 2.31492
\(641\) −2195.86 −0.135306 −0.0676530 0.997709i \(-0.521551\pi\)
−0.0676530 + 0.997709i \(0.521551\pi\)
\(642\) −23489.6 −1.44402
\(643\) −19987.0 −1.22583 −0.612915 0.790149i \(-0.710003\pi\)
−0.612915 + 0.790149i \(0.710003\pi\)
\(644\) 10257.4 0.627637
\(645\) 0 0
\(646\) 13605.4 0.828635
\(647\) −27112.0 −1.64743 −0.823713 0.567008i \(-0.808101\pi\)
−0.823713 + 0.567008i \(0.808101\pi\)
\(648\) 31496.1 1.90939
\(649\) −3967.91 −0.239991
\(650\) 27991.3 1.68909
\(651\) 4450.73 0.267954
\(652\) −28145.0 −1.69056
\(653\) −15361.6 −0.920591 −0.460296 0.887766i \(-0.652257\pi\)
−0.460296 + 0.887766i \(0.652257\pi\)
\(654\) −32363.8 −1.93505
\(655\) −13055.2 −0.778794
\(656\) 11841.4 0.704771
\(657\) 1185.78 0.0704134
\(658\) −14950.6 −0.885769
\(659\) −276.285 −0.0163316 −0.00816580 0.999967i \(-0.502599\pi\)
−0.00816580 + 0.999967i \(0.502599\pi\)
\(660\) 73204.2 4.31738
\(661\) 1668.69 0.0981916 0.0490958 0.998794i \(-0.484366\pi\)
0.0490958 + 0.998794i \(0.484366\pi\)
\(662\) −11469.4 −0.673369
\(663\) 9190.22 0.538339
\(664\) −24593.2 −1.43735
\(665\) 22828.6 1.33121
\(666\) −4176.60 −0.243003
\(667\) −14489.8 −0.841153
\(668\) −39830.4 −2.30701
\(669\) −17116.5 −0.989184
\(670\) −21565.9 −1.24352
\(671\) −18589.6 −1.06951
\(672\) 2265.90 0.130073
\(673\) 188.158 0.0107770 0.00538851 0.999985i \(-0.498285\pi\)
0.00538851 + 0.999985i \(0.498285\pi\)
\(674\) 323.421 0.0184832
\(675\) 10187.8 0.580930
\(676\) 39120.7 2.22580
\(677\) −15768.2 −0.895154 −0.447577 0.894245i \(-0.647713\pi\)
−0.447577 + 0.894245i \(0.647713\pi\)
\(678\) 24621.2 1.39465
\(679\) −10785.1 −0.609563
\(680\) −12490.0 −0.704367
\(681\) 11547.1 0.649761
\(682\) 16206.6 0.909947
\(683\) 7660.10 0.429144 0.214572 0.976708i \(-0.431164\pi\)
0.214572 + 0.976708i \(0.431164\pi\)
\(684\) −10903.8 −0.609526
\(685\) 30666.2 1.71051
\(686\) −32808.7 −1.82601
\(687\) 26325.5 1.46198
\(688\) 0 0
\(689\) −48897.6 −2.70370
\(690\) −20064.1 −1.10699
\(691\) 5645.69 0.310814 0.155407 0.987851i \(-0.450331\pi\)
0.155407 + 0.987851i \(0.450331\pi\)
\(692\) −10554.3 −0.579789
\(693\) 4357.04 0.238832
\(694\) 14025.0 0.767118
\(695\) 15066.5 0.822311
\(696\) 61534.3 3.35122
\(697\) 5065.64 0.275286
\(698\) 37621.3 2.04010
\(699\) −5270.14 −0.285172
\(700\) −17284.3 −0.933262
\(701\) −16807.3 −0.905568 −0.452784 0.891620i \(-0.649569\pi\)
−0.452784 + 0.891620i \(0.649569\pi\)
\(702\) 40412.8 2.17277
\(703\) −17643.4 −0.946562
\(704\) 33015.3 1.76749
\(705\) 19332.8 1.03279
\(706\) 13882.7 0.740062
\(707\) 8471.46 0.450640
\(708\) −6265.19 −0.332571
\(709\) 2473.18 0.131005 0.0655023 0.997852i \(-0.479135\pi\)
0.0655023 + 0.997852i \(0.479135\pi\)
\(710\) −79764.3 −4.21620
\(711\) 1489.64 0.0785735
\(712\) 30816.3 1.62204
\(713\) −2936.50 −0.154240
\(714\) −8584.18 −0.449937
\(715\) 56150.6 2.93694
\(716\) −23877.0 −1.24626
\(717\) −8462.39 −0.440773
\(718\) 10894.8 0.566280
\(719\) 1318.06 0.0683664 0.0341832 0.999416i \(-0.489117\pi\)
0.0341832 + 0.999416i \(0.489117\pi\)
\(720\) 4611.10 0.238675
\(721\) −1014.65 −0.0524100
\(722\) −36351.6 −1.87378
\(723\) 28935.1 1.48839
\(724\) 40793.6 2.09404
\(725\) 24416.1 1.25075
\(726\) 52225.1 2.66977
\(727\) −29579.7 −1.50901 −0.754505 0.656294i \(-0.772123\pi\)
−0.754505 + 0.656294i \(0.772123\pi\)
\(728\) −33412.6 −1.70103
\(729\) 13789.5 0.700578
\(730\) −14273.8 −0.723695
\(731\) 0 0
\(732\) −29352.3 −1.48209
\(733\) 20648.4 1.04047 0.520237 0.854022i \(-0.325844\pi\)
0.520237 + 0.854022i \(0.325844\pi\)
\(734\) −56740.8 −2.85332
\(735\) 14010.9 0.703131
\(736\) −1495.00 −0.0748727
\(737\) −17387.5 −0.869035
\(738\) −6141.07 −0.306309
\(739\) 412.419 0.0205292 0.0102646 0.999947i \(-0.496733\pi\)
0.0102646 + 0.999947i \(0.496733\pi\)
\(740\) 33236.4 1.65107
\(741\) −47064.7 −2.33328
\(742\) 45673.1 2.25972
\(743\) −5178.96 −0.255717 −0.127858 0.991792i \(-0.540810\pi\)
−0.127858 + 0.991792i \(0.540810\pi\)
\(744\) 12470.5 0.614504
\(745\) −4676.14 −0.229960
\(746\) −22828.9 −1.12041
\(747\) 3884.07 0.190242
\(748\) −20664.0 −1.01010
\(749\) 11125.5 0.542748
\(750\) −16500.6 −0.803355
\(751\) −34622.1 −1.68226 −0.841132 0.540831i \(-0.818110\pi\)
−0.841132 + 0.540831i \(0.818110\pi\)
\(752\) −12756.9 −0.618611
\(753\) 1674.93 0.0810596
\(754\) 96853.8 4.67800
\(755\) −12499.1 −0.602503
\(756\) −24954.4 −1.20051
\(757\) −18507.3 −0.888585 −0.444292 0.895882i \(-0.646545\pi\)
−0.444292 + 0.895882i \(0.646545\pi\)
\(758\) −47586.4 −2.28023
\(759\) −16176.7 −0.773620
\(760\) 63963.4 3.05289
\(761\) −20023.6 −0.953817 −0.476909 0.878953i \(-0.658243\pi\)
−0.476909 + 0.878953i \(0.658243\pi\)
\(762\) −16218.6 −0.771048
\(763\) 15328.7 0.727309
\(764\) −61520.0 −2.91324
\(765\) 1972.58 0.0932273
\(766\) −22636.7 −1.06775
\(767\) −4805.66 −0.226235
\(768\) 45450.8 2.13550
\(769\) −16391.2 −0.768636 −0.384318 0.923201i \(-0.625563\pi\)
−0.384318 + 0.923201i \(0.625563\pi\)
\(770\) −52447.8 −2.45466
\(771\) 24.2104 0.00113089
\(772\) −1248.57 −0.0582084
\(773\) 21244.9 0.988519 0.494260 0.869314i \(-0.335439\pi\)
0.494260 + 0.869314i \(0.335439\pi\)
\(774\) 0 0
\(775\) 4948.16 0.229346
\(776\) −30218.7 −1.39792
\(777\) 11131.9 0.513970
\(778\) −72963.6 −3.36230
\(779\) −25942.0 −1.19315
\(780\) 88659.8 4.06991
\(781\) −64310.2 −2.94648
\(782\) 5663.67 0.258993
\(783\) 35251.1 1.60890
\(784\) −9245.20 −0.421155
\(785\) −8345.41 −0.379440
\(786\) −25140.5 −1.14088
\(787\) 2206.07 0.0999212 0.0499606 0.998751i \(-0.484090\pi\)
0.0499606 + 0.998751i \(0.484090\pi\)
\(788\) 43343.8 1.95947
\(789\) −47256.6 −2.13230
\(790\) −17931.5 −0.807562
\(791\) −11661.5 −0.524192
\(792\) 12208.0 0.547717
\(793\) −22514.4 −1.00821
\(794\) −25895.6 −1.15743
\(795\) −59060.4 −2.63479
\(796\) 34627.5 1.54188
\(797\) −39524.6 −1.75663 −0.878314 0.478085i \(-0.841331\pi\)
−0.878314 + 0.478085i \(0.841331\pi\)
\(798\) 43961.1 1.95013
\(799\) −5457.26 −0.241632
\(800\) 2519.15 0.111332
\(801\) −4866.92 −0.214687
\(802\) −9139.23 −0.402391
\(803\) −11508.3 −0.505752
\(804\) −27454.3 −1.20428
\(805\) 9503.09 0.416075
\(806\) 19628.3 0.857790
\(807\) −17974.3 −0.784046
\(808\) 23736.2 1.03346
\(809\) −39225.2 −1.70468 −0.852339 0.522990i \(-0.824816\pi\)
−0.852339 + 0.522990i \(0.824816\pi\)
\(810\) 59877.7 2.59739
\(811\) −10801.0 −0.467664 −0.233832 0.972277i \(-0.575127\pi\)
−0.233832 + 0.972277i \(0.575127\pi\)
\(812\) −59805.9 −2.58470
\(813\) 7139.99 0.308008
\(814\) 40535.1 1.74540
\(815\) −26075.3 −1.12071
\(816\) −7324.60 −0.314231
\(817\) 0 0
\(818\) −43277.3 −1.84982
\(819\) 5276.94 0.225142
\(820\) 48869.1 2.08120
\(821\) 25868.4 1.09965 0.549826 0.835279i \(-0.314694\pi\)
0.549826 + 0.835279i \(0.314694\pi\)
\(822\) 59054.0 2.50577
\(823\) 45014.4 1.90657 0.953283 0.302079i \(-0.0976806\pi\)
0.953283 + 0.302079i \(0.0976806\pi\)
\(824\) −2842.96 −0.120193
\(825\) 27258.6 1.15033
\(826\) 4488.75 0.189084
\(827\) 46882.2 1.97129 0.985644 0.168839i \(-0.0540017\pi\)
0.985644 + 0.168839i \(0.0540017\pi\)
\(828\) −4539.03 −0.190510
\(829\) 26628.6 1.11562 0.557810 0.829969i \(-0.311642\pi\)
0.557810 + 0.829969i \(0.311642\pi\)
\(830\) −46754.5 −1.95527
\(831\) −7757.75 −0.323843
\(832\) 39985.8 1.66618
\(833\) −3955.00 −0.164505
\(834\) 29013.7 1.20463
\(835\) −36901.4 −1.52937
\(836\) 105824. 4.37799
\(837\) 7143.97 0.295020
\(838\) 68644.8 2.82971
\(839\) 23832.3 0.980670 0.490335 0.871534i \(-0.336874\pi\)
0.490335 + 0.871534i \(0.336874\pi\)
\(840\) −40357.0 −1.65768
\(841\) 60094.2 2.46399
\(842\) −3069.82 −0.125645
\(843\) 28839.6 1.17828
\(844\) −50425.7 −2.05654
\(845\) 36243.9 1.47553
\(846\) 6615.84 0.268862
\(847\) −24735.8 −1.00346
\(848\) 38971.4 1.57816
\(849\) 8828.85 0.356897
\(850\) −9543.58 −0.385108
\(851\) −7344.60 −0.295852
\(852\) −101543. −4.08313
\(853\) −8283.40 −0.332495 −0.166248 0.986084i \(-0.553165\pi\)
−0.166248 + 0.986084i \(0.553165\pi\)
\(854\) 21029.7 0.842649
\(855\) −10101.9 −0.404069
\(856\) 31172.6 1.24470
\(857\) −16912.0 −0.674098 −0.337049 0.941487i \(-0.609429\pi\)
−0.337049 + 0.941487i \(0.609429\pi\)
\(858\) 108129. 4.30242
\(859\) −35911.2 −1.42640 −0.713199 0.700962i \(-0.752754\pi\)
−0.713199 + 0.700962i \(0.752754\pi\)
\(860\) 0 0
\(861\) 16367.8 0.647866
\(862\) 6759.30 0.267080
\(863\) 12877.6 0.507949 0.253975 0.967211i \(-0.418262\pi\)
0.253975 + 0.967211i \(0.418262\pi\)
\(864\) 3637.05 0.143212
\(865\) −9778.15 −0.384355
\(866\) 54487.8 2.13807
\(867\) 25019.0 0.980033
\(868\) −12120.2 −0.473949
\(869\) −14457.3 −0.564363
\(870\) 116984. 4.55876
\(871\) −21058.6 −0.819222
\(872\) 42949.5 1.66795
\(873\) 4772.53 0.185024
\(874\) −29004.6 −1.12254
\(875\) 7815.29 0.301949
\(876\) −18171.2 −0.700853
\(877\) −40122.1 −1.54484 −0.772421 0.635110i \(-0.780955\pi\)
−0.772421 + 0.635110i \(0.780955\pi\)
\(878\) −20993.4 −0.806939
\(879\) 11648.7 0.446988
\(880\) −44752.0 −1.71431
\(881\) −32354.6 −1.23729 −0.618646 0.785670i \(-0.712318\pi\)
−0.618646 + 0.785670i \(0.712318\pi\)
\(882\) 4794.65 0.183043
\(883\) 2575.95 0.0981741 0.0490870 0.998795i \(-0.484369\pi\)
0.0490870 + 0.998795i \(0.484369\pi\)
\(884\) −25026.8 −0.952199
\(885\) −5804.46 −0.220469
\(886\) 50203.2 1.90362
\(887\) 171.714 0.00650009 0.00325004 0.999995i \(-0.498965\pi\)
0.00325004 + 0.999995i \(0.498965\pi\)
\(888\) 31190.5 1.17870
\(889\) 7681.75 0.289806
\(890\) 58585.4 2.20650
\(891\) 48276.6 1.81518
\(892\) 46611.8 1.74964
\(893\) 27947.5 1.04729
\(894\) −9004.85 −0.336876
\(895\) −22121.1 −0.826175
\(896\) −34185.5 −1.27462
\(897\) −19592.1 −0.729277
\(898\) −42450.9 −1.57751
\(899\) 17121.3 0.635181
\(900\) 7648.50 0.283278
\(901\) 16671.5 0.616437
\(902\) 59600.8 2.20010
\(903\) 0 0
\(904\) −32674.4 −1.20214
\(905\) 37793.7 1.38818
\(906\) −24069.6 −0.882626
\(907\) 6826.29 0.249904 0.124952 0.992163i \(-0.460122\pi\)
0.124952 + 0.992163i \(0.460122\pi\)
\(908\) −31445.2 −1.14928
\(909\) −3748.73 −0.136785
\(910\) −63521.1 −2.31396
\(911\) 2888.99 0.105067 0.0525337 0.998619i \(-0.483270\pi\)
0.0525337 + 0.998619i \(0.483270\pi\)
\(912\) 37510.5 1.36195
\(913\) −37696.0 −1.36643
\(914\) −54446.4 −1.97038
\(915\) −27193.8 −0.982512
\(916\) −71689.6 −2.58591
\(917\) 11907.5 0.428811
\(918\) −13778.7 −0.495386
\(919\) 16257.1 0.583539 0.291770 0.956489i \(-0.405756\pi\)
0.291770 + 0.956489i \(0.405756\pi\)
\(920\) 26626.7 0.954193
\(921\) 55574.5 1.98832
\(922\) 70572.1 2.52079
\(923\) −77888.0 −2.77759
\(924\) −66768.5 −2.37719
\(925\) 12376.0 0.439915
\(926\) −23954.6 −0.850103
\(927\) 448.996 0.0159083
\(928\) 8716.60 0.308337
\(929\) 46132.6 1.62924 0.814618 0.579998i \(-0.196947\pi\)
0.814618 + 0.579998i \(0.196947\pi\)
\(930\) 23707.9 0.835926
\(931\) 20254.2 0.713002
\(932\) 14351.7 0.504404
\(933\) 47774.7 1.67639
\(934\) −29675.5 −1.03963
\(935\) −19144.4 −0.669615
\(936\) 14785.5 0.516323
\(937\) −14113.4 −0.492066 −0.246033 0.969262i \(-0.579127\pi\)
−0.246033 + 0.969262i \(0.579127\pi\)
\(938\) 19669.9 0.684696
\(939\) 52195.0 1.81397
\(940\) −52647.2 −1.82677
\(941\) −27642.6 −0.957622 −0.478811 0.877918i \(-0.658932\pi\)
−0.478811 + 0.877918i \(0.658932\pi\)
\(942\) −16070.8 −0.555854
\(943\) −10799.1 −0.372925
\(944\) 3830.11 0.132054
\(945\) −23119.3 −0.795841
\(946\) 0 0
\(947\) 15987.3 0.548592 0.274296 0.961645i \(-0.411555\pi\)
0.274296 + 0.961645i \(0.411555\pi\)
\(948\) −22827.6 −0.782074
\(949\) −13938.0 −0.476763
\(950\) 48874.3 1.66915
\(951\) −22015.8 −0.750694
\(952\) 11391.9 0.387831
\(953\) 44654.8 1.51785 0.758925 0.651178i \(-0.225725\pi\)
0.758925 + 0.651178i \(0.225725\pi\)
\(954\) −20210.9 −0.685904
\(955\) −56996.0 −1.93125
\(956\) 23044.8 0.779626
\(957\) 94318.6 3.18588
\(958\) 49373.6 1.66512
\(959\) −27970.2 −0.941820
\(960\) 48296.4 1.62371
\(961\) −26321.2 −0.883529
\(962\) 49093.2 1.64535
\(963\) −4923.19 −0.164743
\(964\) −78796.2 −2.63263
\(965\) −1156.75 −0.0385876
\(966\) 18300.1 0.609521
\(967\) −27849.4 −0.926140 −0.463070 0.886322i \(-0.653252\pi\)
−0.463070 + 0.886322i \(0.653252\pi\)
\(968\) −69307.2 −2.30126
\(969\) 16046.6 0.531983
\(970\) −57449.3 −1.90164
\(971\) 25393.9 0.839268 0.419634 0.907694i \(-0.362159\pi\)
0.419634 + 0.907694i \(0.362159\pi\)
\(972\) 25129.6 0.829250
\(973\) −13742.0 −0.452772
\(974\) −67456.9 −2.21916
\(975\) 33013.7 1.08440
\(976\) 17944.0 0.588497
\(977\) −45241.2 −1.48147 −0.740734 0.671798i \(-0.765522\pi\)
−0.740734 + 0.671798i \(0.765522\pi\)
\(978\) −50213.3 −1.64176
\(979\) 47234.7 1.54201
\(980\) −38154.6 −1.24368
\(981\) −6783.14 −0.220764
\(982\) −71218.7 −2.31434
\(983\) 28920.1 0.938361 0.469180 0.883102i \(-0.344549\pi\)
0.469180 + 0.883102i \(0.344549\pi\)
\(984\) 45860.9 1.48577
\(985\) 40156.4 1.29897
\(986\) −33022.1 −1.06657
\(987\) −17633.2 −0.568663
\(988\) 128167. 4.12705
\(989\) 0 0
\(990\) 23208.8 0.745075
\(991\) 483.939 0.0155124 0.00775622 0.999970i \(-0.497531\pi\)
0.00775622 + 0.999970i \(0.497531\pi\)
\(992\) 1766.50 0.0565388
\(993\) −13527.3 −0.432302
\(994\) 72751.8 2.32148
\(995\) 32081.0 1.02215
\(996\) −59520.6 −1.89356
\(997\) 35131.0 1.11596 0.557978 0.829856i \(-0.311577\pi\)
0.557978 + 0.829856i \(0.311577\pi\)
\(998\) −74738.6 −2.37055
\(999\) 17868.1 0.565886
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 1849.4.a.i.1.5 50
43.42 odd 2 1849.4.a.j.1.46 yes 50
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1849.4.a.i.1.5 50 1.1 even 1 trivial
1849.4.a.j.1.46 yes 50 43.42 odd 2