Properties

Label 1849.4.a.j.1.34
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 1.34
Character \(\chi\) \(=\) 1849.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.53235 q^{2} -8.01322 q^{3} -1.58719 q^{4} -20.7136 q^{5} -20.2923 q^{6} -3.06093 q^{7} -24.2781 q^{8} +37.2117 q^{9} +O(q^{10})\) \(q+2.53235 q^{2} -8.01322 q^{3} -1.58719 q^{4} -20.7136 q^{5} -20.2923 q^{6} -3.06093 q^{7} -24.2781 q^{8} +37.2117 q^{9} -52.4541 q^{10} -50.3240 q^{11} +12.7185 q^{12} -26.3471 q^{13} -7.75136 q^{14} +165.983 q^{15} -48.7833 q^{16} -23.1013 q^{17} +94.2331 q^{18} +135.885 q^{19} +32.8765 q^{20} +24.5279 q^{21} -127.438 q^{22} -91.8712 q^{23} +194.546 q^{24} +304.053 q^{25} -66.7201 q^{26} -81.8285 q^{27} +4.85830 q^{28} +15.5009 q^{29} +420.326 q^{30} -34.9791 q^{31} +70.6888 q^{32} +403.257 q^{33} -58.5007 q^{34} +63.4029 q^{35} -59.0622 q^{36} +221.246 q^{37} +344.110 q^{38} +211.125 q^{39} +502.888 q^{40} +206.730 q^{41} +62.1133 q^{42} +79.8739 q^{44} -770.788 q^{45} -232.650 q^{46} -600.224 q^{47} +390.911 q^{48} -333.631 q^{49} +769.970 q^{50} +185.116 q^{51} +41.8180 q^{52} +408.162 q^{53} -207.218 q^{54} +1042.39 q^{55} +74.3138 q^{56} -1088.88 q^{57} +39.2539 q^{58} -740.433 q^{59} -263.447 q^{60} -51.7496 q^{61} -88.5794 q^{62} -113.902 q^{63} +569.275 q^{64} +545.743 q^{65} +1021.19 q^{66} +334.159 q^{67} +36.6663 q^{68} +736.184 q^{69} +160.559 q^{70} +487.740 q^{71} -903.431 q^{72} +722.667 q^{73} +560.272 q^{74} -2436.45 q^{75} -215.677 q^{76} +154.038 q^{77} +534.643 q^{78} -171.428 q^{79} +1010.48 q^{80} -349.006 q^{81} +523.513 q^{82} -241.764 q^{83} -38.9306 q^{84} +478.511 q^{85} -124.213 q^{87} +1221.77 q^{88} +333.067 q^{89} -1951.91 q^{90} +80.6467 q^{91} +145.817 q^{92} +280.295 q^{93} -1519.98 q^{94} -2814.68 q^{95} -566.445 q^{96} +687.004 q^{97} -844.870 q^{98} -1872.64 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 50q + 10q^{2} + 2q^{3} + 186q^{4} - 8q^{5} - 51q^{6} + 6q^{7} + 138q^{8} + 360q^{9} + O(q^{10}) \) \( 50q + 10q^{2} + 2q^{3} + 186q^{4} - 8q^{5} - 51q^{6} + 6q^{7} + 138q^{8} + 360q^{9} - 137q^{10} - 252q^{11} + 48q^{12} - 192q^{13} - 272q^{14} - 314q^{15} + 542q^{16} - 236q^{17} + 386q^{18} - 12q^{19} - 108q^{20} - 408q^{21} - 1235q^{22} - 630q^{23} - 613q^{24} + 1098q^{25} - 1493q^{26} - 10q^{27} + 242q^{28} - 208q^{29} + 48q^{30} - 932q^{31} + 1124q^{32} - 254q^{33} - 765q^{34} - 1452q^{35} + 747q^{36} + 90q^{37} - 1213q^{38} + 1610q^{39} - 1693q^{40} - 1354q^{41} + 16q^{42} - 2704q^{44} - 4508q^{45} - 233q^{46} - 3484q^{47} + 376q^{48} + 1324q^{49} + 408q^{50} - 4054q^{51} - 2176q^{52} - 726q^{53} - 6497q^{54} + 3288q^{55} - 7097q^{56} - 870q^{57} + 275q^{58} - 4370q^{59} - 3891q^{60} - 1172q^{61} + 1546q^{62} + 3686q^{63} + 606q^{64} - 2610q^{65} - 4697q^{66} - 344q^{67} - 3221q^{68} - 136q^{69} + 1310q^{70} - 162q^{71} + 5814q^{72} - 746q^{73} - 4332q^{74} - 236q^{75} - 1338q^{76} - 2024q^{77} - 2782q^{78} - 2656q^{79} + 5713q^{80} - 86q^{81} + 4168q^{82} - 3514q^{83} - 4269q^{84} + 7558q^{85} - 10278q^{87} - 11692q^{88} - 2640q^{89} - 8286q^{90} + 5946q^{91} - 4271q^{92} + 2q^{93} - 9062q^{94} - 12140q^{95} - 700q^{96} - 3864q^{97} + 2826q^{98} - 8174q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.53235 0.895322 0.447661 0.894203i \(-0.352257\pi\)
0.447661 + 0.894203i \(0.352257\pi\)
\(3\) −8.01322 −1.54214 −0.771072 0.636747i \(-0.780279\pi\)
−0.771072 + 0.636747i \(0.780279\pi\)
\(4\) −1.58719 −0.198399
\(5\) −20.7136 −1.85268 −0.926340 0.376687i \(-0.877063\pi\)
−0.926340 + 0.376687i \(0.877063\pi\)
\(6\) −20.2923 −1.38072
\(7\) −3.06093 −0.165275 −0.0826374 0.996580i \(-0.526334\pi\)
−0.0826374 + 0.996580i \(0.526334\pi\)
\(8\) −24.2781 −1.07295
\(9\) 37.2117 1.37821
\(10\) −52.4541 −1.65875
\(11\) −50.3240 −1.37939 −0.689693 0.724102i \(-0.742255\pi\)
−0.689693 + 0.724102i \(0.742255\pi\)
\(12\) 12.7185 0.305961
\(13\) −26.3471 −0.562105 −0.281053 0.959692i \(-0.590684\pi\)
−0.281053 + 0.959692i \(0.590684\pi\)
\(14\) −7.75136 −0.147974
\(15\) 165.983 2.85710
\(16\) −48.7833 −0.762238
\(17\) −23.1013 −0.329582 −0.164791 0.986329i \(-0.552695\pi\)
−0.164791 + 0.986329i \(0.552695\pi\)
\(18\) 94.2331 1.23394
\(19\) 135.885 1.64075 0.820375 0.571826i \(-0.193765\pi\)
0.820375 + 0.571826i \(0.193765\pi\)
\(20\) 32.8765 0.367571
\(21\) 24.5279 0.254878
\(22\) −127.438 −1.23499
\(23\) −91.8712 −0.832890 −0.416445 0.909161i \(-0.636724\pi\)
−0.416445 + 0.909161i \(0.636724\pi\)
\(24\) 194.546 1.65465
\(25\) 304.053 2.43243
\(26\) −66.7201 −0.503265
\(27\) −81.8285 −0.583255
\(28\) 4.85830 0.0327904
\(29\) 15.5009 0.0992570 0.0496285 0.998768i \(-0.484196\pi\)
0.0496285 + 0.998768i \(0.484196\pi\)
\(30\) 420.326 2.55802
\(31\) −34.9791 −0.202659 −0.101330 0.994853i \(-0.532310\pi\)
−0.101330 + 0.994853i \(0.532310\pi\)
\(32\) 70.6888 0.390504
\(33\) 403.257 2.12721
\(34\) −58.5007 −0.295082
\(35\) 63.4029 0.306201
\(36\) −59.0622 −0.273436
\(37\) 221.246 0.983043 0.491522 0.870865i \(-0.336441\pi\)
0.491522 + 0.870865i \(0.336441\pi\)
\(38\) 344.110 1.46900
\(39\) 211.125 0.866848
\(40\) 502.888 1.98784
\(41\) 206.730 0.787459 0.393729 0.919226i \(-0.371185\pi\)
0.393729 + 0.919226i \(0.371185\pi\)
\(42\) 62.1133 0.228197
\(43\) 0 0
\(44\) 79.8739 0.273669
\(45\) −770.788 −2.55338
\(46\) −232.650 −0.745704
\(47\) −600.224 −1.86280 −0.931400 0.363997i \(-0.881412\pi\)
−0.931400 + 0.363997i \(0.881412\pi\)
\(48\) 390.911 1.17548
\(49\) −333.631 −0.972684
\(50\) 769.970 2.17780
\(51\) 185.116 0.508263
\(52\) 41.8180 0.111521
\(53\) 408.162 1.05784 0.528919 0.848672i \(-0.322598\pi\)
0.528919 + 0.848672i \(0.322598\pi\)
\(54\) −207.218 −0.522201
\(55\) 1042.39 2.55556
\(56\) 74.3138 0.177332
\(57\) −1088.88 −2.53027
\(58\) 39.2539 0.0888670
\(59\) −740.433 −1.63383 −0.816916 0.576756i \(-0.804318\pi\)
−0.816916 + 0.576756i \(0.804318\pi\)
\(60\) −263.447 −0.566847
\(61\) −51.7496 −0.108621 −0.0543103 0.998524i \(-0.517296\pi\)
−0.0543103 + 0.998524i \(0.517296\pi\)
\(62\) −88.5794 −0.181445
\(63\) −113.902 −0.227783
\(64\) 569.275 1.11187
\(65\) 545.743 1.04140
\(66\) 1021.19 1.90454
\(67\) 334.159 0.609314 0.304657 0.952462i \(-0.401458\pi\)
0.304657 + 0.952462i \(0.401458\pi\)
\(68\) 36.6663 0.0653888
\(69\) 736.184 1.28444
\(70\) 160.559 0.274149
\(71\) 487.740 0.815269 0.407634 0.913145i \(-0.366354\pi\)
0.407634 + 0.913145i \(0.366354\pi\)
\(72\) −903.431 −1.47875
\(73\) 722.667 1.15865 0.579327 0.815095i \(-0.303315\pi\)
0.579327 + 0.815095i \(0.303315\pi\)
\(74\) 560.272 0.880140
\(75\) −2436.45 −3.75115
\(76\) −215.677 −0.325524
\(77\) 154.038 0.227978
\(78\) 534.643 0.776107
\(79\) −171.428 −0.244141 −0.122071 0.992521i \(-0.538953\pi\)
−0.122071 + 0.992521i \(0.538953\pi\)
\(80\) 1010.48 1.41218
\(81\) −349.006 −0.478746
\(82\) 523.513 0.705029
\(83\) −241.764 −0.319724 −0.159862 0.987139i \(-0.551105\pi\)
−0.159862 + 0.987139i \(0.551105\pi\)
\(84\) −38.9306 −0.0505676
\(85\) 478.511 0.610610
\(86\) 0 0
\(87\) −124.213 −0.153069
\(88\) 1221.77 1.48002
\(89\) 333.067 0.396686 0.198343 0.980133i \(-0.436444\pi\)
0.198343 + 0.980133i \(0.436444\pi\)
\(90\) −1951.91 −2.28610
\(91\) 80.6467 0.0929019
\(92\) 145.817 0.165245
\(93\) 280.295 0.312530
\(94\) −1519.98 −1.66781
\(95\) −2814.68 −3.03979
\(96\) −566.445 −0.602214
\(97\) 687.004 0.719120 0.359560 0.933122i \(-0.382927\pi\)
0.359560 + 0.933122i \(0.382927\pi\)
\(98\) −844.870 −0.870865
\(99\) −1872.64 −1.90108
\(100\) −482.592 −0.482592
\(101\) 918.164 0.904561 0.452281 0.891876i \(-0.350611\pi\)
0.452281 + 0.891876i \(0.350611\pi\)
\(102\) 468.779 0.455059
\(103\) −348.289 −0.333184 −0.166592 0.986026i \(-0.553276\pi\)
−0.166592 + 0.986026i \(0.553276\pi\)
\(104\) 639.659 0.603112
\(105\) −508.062 −0.472207
\(106\) 1033.61 0.947105
\(107\) 830.251 0.750125 0.375063 0.927000i \(-0.377621\pi\)
0.375063 + 0.927000i \(0.377621\pi\)
\(108\) 129.878 0.115717
\(109\) −295.829 −0.259957 −0.129979 0.991517i \(-0.541491\pi\)
−0.129979 + 0.991517i \(0.541491\pi\)
\(110\) 2639.70 2.28805
\(111\) −1772.89 −1.51599
\(112\) 149.322 0.125979
\(113\) −1631.99 −1.35862 −0.679311 0.733850i \(-0.737721\pi\)
−0.679311 + 0.733850i \(0.737721\pi\)
\(114\) −2757.43 −2.26541
\(115\) 1902.98 1.54308
\(116\) −24.6030 −0.0196925
\(117\) −980.420 −0.774699
\(118\) −1875.04 −1.46281
\(119\) 70.7116 0.0544716
\(120\) −4029.75 −3.06554
\(121\) 1201.50 0.902706
\(122\) −131.048 −0.0972503
\(123\) −1656.57 −1.21438
\(124\) 55.5186 0.0402074
\(125\) −3708.84 −2.65383
\(126\) −288.441 −0.203939
\(127\) 243.422 0.170080 0.0850401 0.996378i \(-0.472898\pi\)
0.0850401 + 0.996378i \(0.472898\pi\)
\(128\) 876.094 0.604973
\(129\) 0 0
\(130\) 1382.01 0.932389
\(131\) 2342.26 1.56217 0.781083 0.624427i \(-0.214667\pi\)
0.781083 + 0.624427i \(0.214667\pi\)
\(132\) −640.047 −0.422038
\(133\) −415.936 −0.271175
\(134\) 846.208 0.545532
\(135\) 1694.96 1.08059
\(136\) 560.857 0.353626
\(137\) −1232.79 −0.768788 −0.384394 0.923169i \(-0.625590\pi\)
−0.384394 + 0.923169i \(0.625590\pi\)
\(138\) 1864.28 1.14998
\(139\) −310.063 −0.189203 −0.0946014 0.995515i \(-0.530158\pi\)
−0.0946014 + 0.995515i \(0.530158\pi\)
\(140\) −100.633 −0.0607502
\(141\) 4809.72 2.87271
\(142\) 1235.13 0.729928
\(143\) 1325.89 0.775360
\(144\) −1815.31 −1.05052
\(145\) −321.080 −0.183892
\(146\) 1830.05 1.03737
\(147\) 2673.46 1.50002
\(148\) −351.160 −0.195035
\(149\) −2522.04 −1.38667 −0.693334 0.720616i \(-0.743859\pi\)
−0.693334 + 0.720616i \(0.743859\pi\)
\(150\) −6169.94 −3.35849
\(151\) 999.113 0.538455 0.269227 0.963077i \(-0.413232\pi\)
0.269227 + 0.963077i \(0.413232\pi\)
\(152\) −3299.05 −1.76045
\(153\) −859.639 −0.454233
\(154\) 390.079 0.204113
\(155\) 724.543 0.375462
\(156\) −335.097 −0.171982
\(157\) 1192.13 0.606002 0.303001 0.952990i \(-0.402011\pi\)
0.303001 + 0.952990i \(0.402011\pi\)
\(158\) −434.116 −0.218585
\(159\) −3270.69 −1.63134
\(160\) −1464.22 −0.723480
\(161\) 281.212 0.137656
\(162\) −883.806 −0.428632
\(163\) 114.218 0.0548847 0.0274424 0.999623i \(-0.491264\pi\)
0.0274424 + 0.999623i \(0.491264\pi\)
\(164\) −328.121 −0.156231
\(165\) −8352.91 −3.94105
\(166\) −612.232 −0.286256
\(167\) 3106.64 1.43952 0.719758 0.694225i \(-0.244253\pi\)
0.719758 + 0.694225i \(0.244253\pi\)
\(168\) −595.493 −0.273472
\(169\) −1502.83 −0.684038
\(170\) 1211.76 0.546692
\(171\) 5056.53 2.26130
\(172\) 0 0
\(173\) −2342.86 −1.02962 −0.514810 0.857304i \(-0.672138\pi\)
−0.514810 + 0.857304i \(0.672138\pi\)
\(174\) −314.550 −0.137046
\(175\) −930.687 −0.402019
\(176\) 2454.97 1.05142
\(177\) 5933.25 2.51961
\(178\) 843.442 0.355161
\(179\) −1671.04 −0.697762 −0.348881 0.937167i \(-0.613438\pi\)
−0.348881 + 0.937167i \(0.613438\pi\)
\(180\) 1223.39 0.506590
\(181\) 2195.16 0.901465 0.450732 0.892659i \(-0.351163\pi\)
0.450732 + 0.892659i \(0.351163\pi\)
\(182\) 204.226 0.0831770
\(183\) 414.681 0.167509
\(184\) 2230.46 0.893652
\(185\) −4582.80 −1.82127
\(186\) 709.806 0.279814
\(187\) 1162.55 0.454621
\(188\) 952.672 0.369578
\(189\) 250.471 0.0963974
\(190\) −7127.75 −2.72159
\(191\) −79.6602 −0.0301781 −0.0150890 0.999886i \(-0.504803\pi\)
−0.0150890 + 0.999886i \(0.504803\pi\)
\(192\) −4561.73 −1.71466
\(193\) 1109.31 0.413730 0.206865 0.978369i \(-0.433674\pi\)
0.206865 + 0.978369i \(0.433674\pi\)
\(194\) 1739.73 0.643844
\(195\) −4373.16 −1.60599
\(196\) 529.537 0.192980
\(197\) 2392.69 0.865340 0.432670 0.901552i \(-0.357572\pi\)
0.432670 + 0.901552i \(0.357572\pi\)
\(198\) −4742.18 −1.70208
\(199\) 210.234 0.0748899 0.0374450 0.999299i \(-0.488078\pi\)
0.0374450 + 0.999299i \(0.488078\pi\)
\(200\) −7381.85 −2.60988
\(201\) −2677.69 −0.939650
\(202\) 2325.11 0.809873
\(203\) −47.4474 −0.0164047
\(204\) −293.815 −0.100839
\(205\) −4282.12 −1.45891
\(206\) −881.991 −0.298307
\(207\) −3418.68 −1.14790
\(208\) 1285.30 0.428458
\(209\) −6838.29 −2.26323
\(210\) −1286.59 −0.422777
\(211\) 857.579 0.279802 0.139901 0.990166i \(-0.455322\pi\)
0.139901 + 0.990166i \(0.455322\pi\)
\(212\) −647.833 −0.209874
\(213\) −3908.37 −1.25726
\(214\) 2102.49 0.671603
\(215\) 0 0
\(216\) 1986.64 0.625805
\(217\) 107.069 0.0334944
\(218\) −749.144 −0.232745
\(219\) −5790.89 −1.78681
\(220\) −1654.48 −0.507022
\(221\) 608.652 0.185260
\(222\) −4489.58 −1.35730
\(223\) −3532.28 −1.06071 −0.530356 0.847775i \(-0.677942\pi\)
−0.530356 + 0.847775i \(0.677942\pi\)
\(224\) −216.374 −0.0645405
\(225\) 11314.3 3.35240
\(226\) −4132.76 −1.21640
\(227\) −5188.68 −1.51711 −0.758557 0.651607i \(-0.774095\pi\)
−0.758557 + 0.651607i \(0.774095\pi\)
\(228\) 1728.26 0.502005
\(229\) 3482.36 1.00490 0.502448 0.864607i \(-0.332433\pi\)
0.502448 + 0.864607i \(0.332433\pi\)
\(230\) 4819.02 1.38155
\(231\) −1234.34 −0.351575
\(232\) −376.334 −0.106498
\(233\) 1545.34 0.434500 0.217250 0.976116i \(-0.430291\pi\)
0.217250 + 0.976116i \(0.430291\pi\)
\(234\) −2482.77 −0.693605
\(235\) 12432.8 3.45117
\(236\) 1175.21 0.324151
\(237\) 1373.69 0.376501
\(238\) 179.067 0.0487696
\(239\) −2562.59 −0.693558 −0.346779 0.937947i \(-0.612725\pi\)
−0.346779 + 0.937947i \(0.612725\pi\)
\(240\) −8097.17 −2.17779
\(241\) −4941.93 −1.32090 −0.660452 0.750868i \(-0.729635\pi\)
−0.660452 + 0.750868i \(0.729635\pi\)
\(242\) 3042.63 0.808212
\(243\) 5006.03 1.32155
\(244\) 82.1366 0.0215502
\(245\) 6910.69 1.80207
\(246\) −4195.03 −1.08726
\(247\) −3580.19 −0.922274
\(248\) 849.228 0.217444
\(249\) 1937.31 0.493060
\(250\) −9392.08 −2.37603
\(251\) 4464.77 1.12276 0.561382 0.827557i \(-0.310270\pi\)
0.561382 + 0.827557i \(0.310270\pi\)
\(252\) 180.785 0.0451921
\(253\) 4623.32 1.14888
\(254\) 616.430 0.152277
\(255\) −3834.42 −0.941649
\(256\) −2335.62 −0.570220
\(257\) −5954.45 −1.44525 −0.722624 0.691242i \(-0.757064\pi\)
−0.722624 + 0.691242i \(0.757064\pi\)
\(258\) 0 0
\(259\) −677.219 −0.162472
\(260\) −866.201 −0.206613
\(261\) 576.816 0.136797
\(262\) 5931.41 1.39864
\(263\) 46.0385 0.0107941 0.00539706 0.999985i \(-0.498282\pi\)
0.00539706 + 0.999985i \(0.498282\pi\)
\(264\) −9790.33 −2.28240
\(265\) −8454.51 −1.95984
\(266\) −1053.30 −0.242789
\(267\) −2668.94 −0.611747
\(268\) −530.375 −0.120887
\(269\) 924.553 0.209558 0.104779 0.994496i \(-0.466587\pi\)
0.104779 + 0.994496i \(0.466587\pi\)
\(270\) 4292.24 0.967472
\(271\) −5175.61 −1.16013 −0.580067 0.814569i \(-0.696974\pi\)
−0.580067 + 0.814569i \(0.696974\pi\)
\(272\) 1126.96 0.251220
\(273\) −646.239 −0.143268
\(274\) −3121.85 −0.688313
\(275\) −15301.2 −3.35526
\(276\) −1168.47 −0.254831
\(277\) 6775.52 1.46968 0.734840 0.678240i \(-0.237257\pi\)
0.734840 + 0.678240i \(0.237257\pi\)
\(278\) −785.189 −0.169397
\(279\) −1301.63 −0.279307
\(280\) −1539.31 −0.328540
\(281\) 2921.27 0.620172 0.310086 0.950709i \(-0.399642\pi\)
0.310086 + 0.950709i \(0.399642\pi\)
\(282\) 12179.9 2.57200
\(283\) −5154.70 −1.08274 −0.541369 0.840785i \(-0.682094\pi\)
−0.541369 + 0.840785i \(0.682094\pi\)
\(284\) −774.138 −0.161749
\(285\) 22554.6 4.68779
\(286\) 3357.62 0.694197
\(287\) −632.787 −0.130147
\(288\) 2630.45 0.538197
\(289\) −4379.33 −0.891376
\(290\) −813.089 −0.164642
\(291\) −5505.11 −1.10899
\(292\) −1147.01 −0.229876
\(293\) −5833.97 −1.16322 −0.581611 0.813467i \(-0.697578\pi\)
−0.581611 + 0.813467i \(0.697578\pi\)
\(294\) 6770.13 1.34300
\(295\) 15337.0 3.02697
\(296\) −5371.44 −1.05476
\(297\) 4117.93 0.804534
\(298\) −6386.69 −1.24151
\(299\) 2420.54 0.468172
\(300\) 3867.11 0.744226
\(301\) 0 0
\(302\) 2530.11 0.482090
\(303\) −7357.45 −1.39496
\(304\) −6628.93 −1.25064
\(305\) 1071.92 0.201239
\(306\) −2176.91 −0.406685
\(307\) 6732.61 1.25163 0.625815 0.779971i \(-0.284766\pi\)
0.625815 + 0.779971i \(0.284766\pi\)
\(308\) −244.489 −0.0452306
\(309\) 2790.92 0.513818
\(310\) 1834.80 0.336160
\(311\) 8757.73 1.59680 0.798401 0.602127i \(-0.205680\pi\)
0.798401 + 0.602127i \(0.205680\pi\)
\(312\) −5125.73 −0.930087
\(313\) 6736.28 1.21648 0.608238 0.793754i \(-0.291876\pi\)
0.608238 + 0.793754i \(0.291876\pi\)
\(314\) 3018.89 0.542567
\(315\) 2359.33 0.422010
\(316\) 272.090 0.0484374
\(317\) 8651.72 1.53290 0.766450 0.642304i \(-0.222021\pi\)
0.766450 + 0.642304i \(0.222021\pi\)
\(318\) −8282.55 −1.46057
\(319\) −780.069 −0.136914
\(320\) −11791.7 −2.05993
\(321\) −6652.98 −1.15680
\(322\) 712.127 0.123246
\(323\) −3139.13 −0.540762
\(324\) 553.941 0.0949830
\(325\) −8010.92 −1.36728
\(326\) 289.239 0.0491395
\(327\) 2370.55 0.400891
\(328\) −5019.02 −0.844906
\(329\) 1837.24 0.307874
\(330\) −21152.5 −3.52850
\(331\) 5530.88 0.918444 0.459222 0.888322i \(-0.348128\pi\)
0.459222 + 0.888322i \(0.348128\pi\)
\(332\) 383.727 0.0634330
\(333\) 8232.93 1.35484
\(334\) 7867.11 1.28883
\(335\) −6921.63 −1.12886
\(336\) −1196.55 −0.194278
\(337\) 2734.73 0.442049 0.221024 0.975268i \(-0.429060\pi\)
0.221024 + 0.975268i \(0.429060\pi\)
\(338\) −3805.70 −0.612434
\(339\) 13077.5 2.09519
\(340\) −759.491 −0.121145
\(341\) 1760.29 0.279545
\(342\) 12804.9 2.02459
\(343\) 2071.12 0.326035
\(344\) 0 0
\(345\) −15249.0 −2.37965
\(346\) −5932.95 −0.921842
\(347\) 8367.34 1.29447 0.647237 0.762289i \(-0.275924\pi\)
0.647237 + 0.762289i \(0.275924\pi\)
\(348\) 197.149 0.0303687
\(349\) −2243.31 −0.344074 −0.172037 0.985090i \(-0.555035\pi\)
−0.172037 + 0.985090i \(0.555035\pi\)
\(350\) −2356.83 −0.359936
\(351\) 2155.94 0.327851
\(352\) −3557.34 −0.538656
\(353\) 1573.11 0.237190 0.118595 0.992943i \(-0.462161\pi\)
0.118595 + 0.992943i \(0.462161\pi\)
\(354\) 15025.1 2.25586
\(355\) −10102.8 −1.51043
\(356\) −528.642 −0.0787022
\(357\) −566.627 −0.0840031
\(358\) −4231.66 −0.624722
\(359\) 5107.31 0.750846 0.375423 0.926854i \(-0.377497\pi\)
0.375423 + 0.926854i \(0.377497\pi\)
\(360\) 18713.3 2.73966
\(361\) 11605.8 1.69206
\(362\) 5558.92 0.807101
\(363\) −9627.90 −1.39210
\(364\) −128.002 −0.0184317
\(365\) −14969.0 −2.14662
\(366\) 1050.12 0.149974
\(367\) 3195.77 0.454545 0.227273 0.973831i \(-0.427019\pi\)
0.227273 + 0.973831i \(0.427019\pi\)
\(368\) 4481.78 0.634861
\(369\) 7692.77 1.08528
\(370\) −11605.3 −1.63062
\(371\) −1249.36 −0.174834
\(372\) −444.883 −0.0620057
\(373\) 12608.7 1.75027 0.875136 0.483876i \(-0.160772\pi\)
0.875136 + 0.483876i \(0.160772\pi\)
\(374\) 2943.99 0.407032
\(375\) 29719.7 4.09259
\(376\) 14572.3 1.99870
\(377\) −408.405 −0.0557929
\(378\) 634.282 0.0863067
\(379\) 1660.61 0.225065 0.112532 0.993648i \(-0.464104\pi\)
0.112532 + 0.993648i \(0.464104\pi\)
\(380\) 4467.44 0.603092
\(381\) −1950.59 −0.262288
\(382\) −201.728 −0.0270191
\(383\) 10824.1 1.44409 0.722043 0.691849i \(-0.243203\pi\)
0.722043 + 0.691849i \(0.243203\pi\)
\(384\) −7020.33 −0.932955
\(385\) −3190.69 −0.422370
\(386\) 2809.17 0.370422
\(387\) 0 0
\(388\) −1090.41 −0.142673
\(389\) 6272.78 0.817590 0.408795 0.912626i \(-0.365949\pi\)
0.408795 + 0.912626i \(0.365949\pi\)
\(390\) −11074.4 −1.43788
\(391\) 2122.35 0.274505
\(392\) 8099.94 1.04364
\(393\) −18769.0 −2.40909
\(394\) 6059.13 0.774758
\(395\) 3550.89 0.452316
\(396\) 2972.24 0.377174
\(397\) −2427.36 −0.306866 −0.153433 0.988159i \(-0.549033\pi\)
−0.153433 + 0.988159i \(0.549033\pi\)
\(398\) 532.387 0.0670506
\(399\) 3332.99 0.418191
\(400\) −14832.7 −1.85409
\(401\) 5002.47 0.622971 0.311486 0.950251i \(-0.399173\pi\)
0.311486 + 0.950251i \(0.399173\pi\)
\(402\) −6780.85 −0.841289
\(403\) 921.597 0.113916
\(404\) −1457.30 −0.179464
\(405\) 7229.17 0.886964
\(406\) −120.153 −0.0146875
\(407\) −11134.0 −1.35600
\(408\) −4494.27 −0.545342
\(409\) −8971.14 −1.08458 −0.542292 0.840190i \(-0.682443\pi\)
−0.542292 + 0.840190i \(0.682443\pi\)
\(410\) −10843.8 −1.30619
\(411\) 9878.58 1.18558
\(412\) 552.803 0.0661035
\(413\) 2266.41 0.270031
\(414\) −8657.31 −1.02774
\(415\) 5007.81 0.592346
\(416\) −1862.45 −0.219505
\(417\) 2484.60 0.291778
\(418\) −17317.0 −2.02632
\(419\) −11112.1 −1.29561 −0.647806 0.761805i \(-0.724313\pi\)
−0.647806 + 0.761805i \(0.724313\pi\)
\(420\) 806.393 0.0936856
\(421\) 1138.09 0.131751 0.0658754 0.997828i \(-0.479016\pi\)
0.0658754 + 0.997828i \(0.479016\pi\)
\(422\) 2171.69 0.250513
\(423\) −22335.3 −2.56733
\(424\) −9909.42 −1.13501
\(425\) −7024.03 −0.801684
\(426\) −9897.36 −1.12565
\(427\) 158.402 0.0179522
\(428\) −1317.77 −0.148824
\(429\) −10624.7 −1.19572
\(430\) 0 0
\(431\) −7440.65 −0.831563 −0.415781 0.909465i \(-0.636492\pi\)
−0.415781 + 0.909465i \(0.636492\pi\)
\(432\) 3991.86 0.444580
\(433\) 8252.84 0.915949 0.457975 0.888965i \(-0.348575\pi\)
0.457975 + 0.888965i \(0.348575\pi\)
\(434\) 271.135 0.0299883
\(435\) 2572.89 0.283587
\(436\) 469.539 0.0515753
\(437\) −12484.0 −1.36656
\(438\) −14664.6 −1.59977
\(439\) −11729.5 −1.27521 −0.637605 0.770364i \(-0.720075\pi\)
−0.637605 + 0.770364i \(0.720075\pi\)
\(440\) −25307.3 −2.74200
\(441\) −12415.0 −1.34056
\(442\) 1541.32 0.165867
\(443\) 1411.13 0.151343 0.0756714 0.997133i \(-0.475890\pi\)
0.0756714 + 0.997133i \(0.475890\pi\)
\(444\) 2813.92 0.300772
\(445\) −6899.01 −0.734932
\(446\) −8944.97 −0.949679
\(447\) 20209.7 2.13844
\(448\) −1742.51 −0.183763
\(449\) 9517.45 1.00035 0.500174 0.865925i \(-0.333269\pi\)
0.500174 + 0.865925i \(0.333269\pi\)
\(450\) 28651.9 3.00147
\(451\) −10403.5 −1.08621
\(452\) 2590.28 0.269550
\(453\) −8006.11 −0.830375
\(454\) −13139.6 −1.35830
\(455\) −1670.48 −0.172117
\(456\) 26436.0 2.71486
\(457\) 11178.2 1.14419 0.572093 0.820189i \(-0.306132\pi\)
0.572093 + 0.820189i \(0.306132\pi\)
\(458\) 8818.57 0.899705
\(459\) 1890.34 0.192230
\(460\) −3020.41 −0.306146
\(461\) −15694.1 −1.58557 −0.792786 0.609500i \(-0.791370\pi\)
−0.792786 + 0.609500i \(0.791370\pi\)
\(462\) −3125.79 −0.314772
\(463\) 2547.36 0.255693 0.127847 0.991794i \(-0.459193\pi\)
0.127847 + 0.991794i \(0.459193\pi\)
\(464\) −756.187 −0.0756575
\(465\) −5805.92 −0.579017
\(466\) 3913.34 0.389017
\(467\) 7822.43 0.775115 0.387558 0.921845i \(-0.373319\pi\)
0.387558 + 0.921845i \(0.373319\pi\)
\(468\) 1556.12 0.153700
\(469\) −1022.84 −0.100704
\(470\) 31484.2 3.08991
\(471\) −9552.80 −0.934543
\(472\) 17976.3 1.75302
\(473\) 0 0
\(474\) 3478.67 0.337089
\(475\) 41316.4 3.99100
\(476\) −112.233 −0.0108071
\(477\) 15188.4 1.45792
\(478\) −6489.39 −0.620957
\(479\) −14772.7 −1.40915 −0.704573 0.709632i \(-0.748861\pi\)
−0.704573 + 0.709632i \(0.748861\pi\)
\(480\) 11733.1 1.11571
\(481\) −5829.19 −0.552574
\(482\) −12514.7 −1.18263
\(483\) −2253.41 −0.212285
\(484\) −1907.02 −0.179096
\(485\) −14230.3 −1.33230
\(486\) 12677.0 1.18321
\(487\) 5286.95 0.491939 0.245970 0.969278i \(-0.420894\pi\)
0.245970 + 0.969278i \(0.420894\pi\)
\(488\) 1256.38 0.116545
\(489\) −915.251 −0.0846402
\(490\) 17500.3 1.61344
\(491\) 17508.4 1.60925 0.804624 0.593784i \(-0.202367\pi\)
0.804624 + 0.593784i \(0.202367\pi\)
\(492\) 2629.30 0.240931
\(493\) −358.092 −0.0327133
\(494\) −9066.29 −0.825732
\(495\) 38789.1 3.52210
\(496\) 1706.39 0.154474
\(497\) −1492.94 −0.134743
\(498\) 4905.95 0.441447
\(499\) 1560.31 0.139978 0.0699888 0.997548i \(-0.477704\pi\)
0.0699888 + 0.997548i \(0.477704\pi\)
\(500\) 5886.65 0.526518
\(501\) −24894.2 −2.21994
\(502\) 11306.4 1.00523
\(503\) −7420.52 −0.657783 −0.328891 0.944368i \(-0.606675\pi\)
−0.328891 + 0.944368i \(0.606675\pi\)
\(504\) 2765.34 0.244401
\(505\) −19018.5 −1.67586
\(506\) 11707.9 1.02861
\(507\) 12042.5 1.05489
\(508\) −386.358 −0.0337438
\(509\) 661.902 0.0576390 0.0288195 0.999585i \(-0.490825\pi\)
0.0288195 + 0.999585i \(0.490825\pi\)
\(510\) −9710.09 −0.843079
\(511\) −2212.03 −0.191496
\(512\) −12923.4 −1.11550
\(513\) −11119.3 −0.956976
\(514\) −15078.8 −1.29396
\(515\) 7214.33 0.617284
\(516\) 0 0
\(517\) 30205.6 2.56952
\(518\) −1714.96 −0.145465
\(519\) 18773.9 1.58782
\(520\) −13249.6 −1.11737
\(521\) −13039.0 −1.09645 −0.548225 0.836331i \(-0.684696\pi\)
−0.548225 + 0.836331i \(0.684696\pi\)
\(522\) 1460.70 0.122477
\(523\) 18176.0 1.51966 0.759829 0.650123i \(-0.225283\pi\)
0.759829 + 0.650123i \(0.225283\pi\)
\(524\) −3717.62 −0.309933
\(525\) 7457.80 0.619971
\(526\) 116.586 0.00966421
\(527\) 808.063 0.0667927
\(528\) −19672.2 −1.62144
\(529\) −3726.68 −0.306294
\(530\) −21409.8 −1.75468
\(531\) −27552.7 −2.25176
\(532\) 660.172 0.0538009
\(533\) −5446.74 −0.442635
\(534\) −6758.69 −0.547710
\(535\) −17197.5 −1.38974
\(536\) −8112.76 −0.653765
\(537\) 13390.4 1.07605
\(538\) 2341.29 0.187622
\(539\) 16789.6 1.34171
\(540\) −2690.23 −0.214388
\(541\) 3752.56 0.298216 0.149108 0.988821i \(-0.452360\pi\)
0.149108 + 0.988821i \(0.452360\pi\)
\(542\) −13106.5 −1.03869
\(543\) −17590.3 −1.39019
\(544\) −1633.00 −0.128703
\(545\) 6127.69 0.481617
\(546\) −1636.51 −0.128271
\(547\) 7414.69 0.579578 0.289789 0.957090i \(-0.406415\pi\)
0.289789 + 0.957090i \(0.406415\pi\)
\(548\) 1956.67 0.152527
\(549\) −1925.69 −0.149702
\(550\) −38747.9 −3.00403
\(551\) 2106.35 0.162856
\(552\) −17873.2 −1.37814
\(553\) 524.729 0.0403504
\(554\) 17158.0 1.31584
\(555\) 36723.0 2.80865
\(556\) 492.130 0.0375377
\(557\) −14232.7 −1.08269 −0.541345 0.840801i \(-0.682085\pi\)
−0.541345 + 0.840801i \(0.682085\pi\)
\(558\) −3296.19 −0.250069
\(559\) 0 0
\(560\) −3093.00 −0.233399
\(561\) −9315.77 −0.701091
\(562\) 7397.68 0.555253
\(563\) −20932.3 −1.56695 −0.783475 0.621423i \(-0.786555\pi\)
−0.783475 + 0.621423i \(0.786555\pi\)
\(564\) −7633.97 −0.569943
\(565\) 33804.3 2.51709
\(566\) −13053.5 −0.969399
\(567\) 1068.28 0.0791247
\(568\) −11841.4 −0.874745
\(569\) −3328.84 −0.245259 −0.122629 0.992453i \(-0.539133\pi\)
−0.122629 + 0.992453i \(0.539133\pi\)
\(570\) 57116.2 4.19708
\(571\) 2268.00 0.166222 0.0831110 0.996540i \(-0.473514\pi\)
0.0831110 + 0.996540i \(0.473514\pi\)
\(572\) −2104.45 −0.153831
\(573\) 638.335 0.0465389
\(574\) −1602.44 −0.116524
\(575\) −27933.7 −2.02594
\(576\) 21183.7 1.53238
\(577\) −11714.3 −0.845190 −0.422595 0.906319i \(-0.638881\pi\)
−0.422595 + 0.906319i \(0.638881\pi\)
\(578\) −11090.0 −0.798068
\(579\) −8889.16 −0.638032
\(580\) 509.617 0.0364840
\(581\) 740.024 0.0528423
\(582\) −13940.9 −0.992900
\(583\) −20540.3 −1.45917
\(584\) −17545.0 −1.24318
\(585\) 20308.0 1.43527
\(586\) −14773.7 −1.04146
\(587\) 13734.5 0.965730 0.482865 0.875695i \(-0.339596\pi\)
0.482865 + 0.875695i \(0.339596\pi\)
\(588\) −4243.30 −0.297603
\(589\) −4753.15 −0.332513
\(590\) 38838.7 2.71011
\(591\) −19173.1 −1.33448
\(592\) −10793.1 −0.749313
\(593\) 26981.9 1.86849 0.934244 0.356634i \(-0.116076\pi\)
0.934244 + 0.356634i \(0.116076\pi\)
\(594\) 10428.1 0.720317
\(595\) −1464.69 −0.100918
\(596\) 4002.97 0.275114
\(597\) −1684.65 −0.115491
\(598\) 6129.66 0.419164
\(599\) 2470.79 0.168537 0.0842685 0.996443i \(-0.473145\pi\)
0.0842685 + 0.996443i \(0.473145\pi\)
\(600\) 59152.4 4.02481
\(601\) 20621.9 1.39964 0.699820 0.714319i \(-0.253263\pi\)
0.699820 + 0.714319i \(0.253263\pi\)
\(602\) 0 0
\(603\) 12434.6 0.839762
\(604\) −1585.79 −0.106829
\(605\) −24887.4 −1.67243
\(606\) −18631.6 −1.24894
\(607\) 4220.90 0.282242 0.141121 0.989992i \(-0.454929\pi\)
0.141121 + 0.989992i \(0.454929\pi\)
\(608\) 9605.58 0.640720
\(609\) 380.206 0.0252984
\(610\) 2714.48 0.180174
\(611\) 15814.1 1.04709
\(612\) 1364.41 0.0901196
\(613\) 12018.1 0.791852 0.395926 0.918282i \(-0.370424\pi\)
0.395926 + 0.918282i \(0.370424\pi\)
\(614\) 17049.3 1.12061
\(615\) 34313.6 2.24985
\(616\) −3739.76 −0.244609
\(617\) −5171.02 −0.337403 −0.168701 0.985667i \(-0.553957\pi\)
−0.168701 + 0.985667i \(0.553957\pi\)
\(618\) 7067.59 0.460033
\(619\) 5055.56 0.328272 0.164136 0.986438i \(-0.447516\pi\)
0.164136 + 0.986438i \(0.447516\pi\)
\(620\) −1149.99 −0.0744915
\(621\) 7517.68 0.485788
\(622\) 22177.6 1.42965
\(623\) −1019.50 −0.0655621
\(624\) −10299.4 −0.660745
\(625\) 38816.7 2.48427
\(626\) 17058.6 1.08914
\(627\) 54796.8 3.49023
\(628\) −1892.14 −0.120230
\(629\) −5111.07 −0.323993
\(630\) 5974.65 0.377835
\(631\) −20865.9 −1.31642 −0.658209 0.752835i \(-0.728686\pi\)
−0.658209 + 0.752835i \(0.728686\pi\)
\(632\) 4161.95 0.261952
\(633\) −6871.97 −0.431495
\(634\) 21909.2 1.37244
\(635\) −5042.14 −0.315105
\(636\) 5191.23 0.323657
\(637\) 8790.20 0.546751
\(638\) −1975.41 −0.122582
\(639\) 18149.6 1.12361
\(640\) −18147.1 −1.12082
\(641\) −20942.5 −1.29045 −0.645226 0.763992i \(-0.723237\pi\)
−0.645226 + 0.763992i \(0.723237\pi\)
\(642\) −16847.7 −1.03571
\(643\) −1235.94 −0.0758020 −0.0379010 0.999281i \(-0.512067\pi\)
−0.0379010 + 0.999281i \(0.512067\pi\)
\(644\) −446.337 −0.0273108
\(645\) 0 0
\(646\) −7949.39 −0.484155
\(647\) −16372.8 −0.994873 −0.497437 0.867500i \(-0.665725\pi\)
−0.497437 + 0.867500i \(0.665725\pi\)
\(648\) 8473.22 0.513672
\(649\) 37261.5 2.25369
\(650\) −20286.5 −1.22415
\(651\) −857.964 −0.0516533
\(652\) −181.286 −0.0108891
\(653\) 3699.21 0.221686 0.110843 0.993838i \(-0.464645\pi\)
0.110843 + 0.993838i \(0.464645\pi\)
\(654\) 6003.06 0.358927
\(655\) −48516.5 −2.89420
\(656\) −10085.0 −0.600231
\(657\) 26891.6 1.59687
\(658\) 4652.55 0.275646
\(659\) −27786.0 −1.64247 −0.821237 0.570588i \(-0.806715\pi\)
−0.821237 + 0.570588i \(0.806715\pi\)
\(660\) 13257.7 0.781901
\(661\) 5798.03 0.341176 0.170588 0.985342i \(-0.445433\pi\)
0.170588 + 0.985342i \(0.445433\pi\)
\(662\) 14006.1 0.822302
\(663\) −4877.27 −0.285697
\(664\) 5869.59 0.343048
\(665\) 8615.53 0.502400
\(666\) 20848.7 1.21302
\(667\) −1424.09 −0.0826702
\(668\) −4930.85 −0.285599
\(669\) 28304.9 1.63577
\(670\) −17528.0 −1.01070
\(671\) 2604.24 0.149830
\(672\) 1733.85 0.0995308
\(673\) −4577.82 −0.262202 −0.131101 0.991369i \(-0.541851\pi\)
−0.131101 + 0.991369i \(0.541851\pi\)
\(674\) 6925.30 0.395776
\(675\) −24880.2 −1.41873
\(676\) 2385.28 0.135713
\(677\) −12344.6 −0.700801 −0.350401 0.936600i \(-0.613955\pi\)
−0.350401 + 0.936600i \(0.613955\pi\)
\(678\) 33116.7 1.87587
\(679\) −2102.87 −0.118852
\(680\) −11617.4 −0.655156
\(681\) 41578.0 2.33961
\(682\) 4457.66 0.250283
\(683\) 16504.5 0.924635 0.462318 0.886714i \(-0.347018\pi\)
0.462318 + 0.886714i \(0.347018\pi\)
\(684\) −8025.69 −0.448640
\(685\) 25535.4 1.42432
\(686\) 5244.81 0.291906
\(687\) −27905.0 −1.54969
\(688\) 0 0
\(689\) −10753.9 −0.594616
\(690\) −38615.9 −2.13055
\(691\) −11094.6 −0.610791 −0.305396 0.952226i \(-0.598789\pi\)
−0.305396 + 0.952226i \(0.598789\pi\)
\(692\) 3718.58 0.204276
\(693\) 5732.02 0.314201
\(694\) 21189.1 1.15897
\(695\) 6422.52 0.350533
\(696\) 3015.65 0.164236
\(697\) −4775.74 −0.259532
\(698\) −5680.86 −0.308057
\(699\) −12383.1 −0.670061
\(700\) 1477.18 0.0797603
\(701\) 3488.71 0.187970 0.0939850 0.995574i \(-0.470039\pi\)
0.0939850 + 0.995574i \(0.470039\pi\)
\(702\) 5459.60 0.293532
\(703\) 30064.1 1.61293
\(704\) −28648.2 −1.53369
\(705\) −99626.7 −5.32221
\(706\) 3983.66 0.212361
\(707\) −2810.44 −0.149501
\(708\) −9417.22 −0.499888
\(709\) 11358.7 0.601669 0.300835 0.953676i \(-0.402735\pi\)
0.300835 + 0.953676i \(0.402735\pi\)
\(710\) −25584.0 −1.35232
\(711\) −6379.12 −0.336478
\(712\) −8086.25 −0.425625
\(713\) 3213.57 0.168793
\(714\) −1434.90 −0.0752098
\(715\) −27464.0 −1.43650
\(716\) 2652.27 0.138436
\(717\) 20534.6 1.06957
\(718\) 12933.5 0.672249
\(719\) −34524.2 −1.79073 −0.895364 0.445335i \(-0.853085\pi\)
−0.895364 + 0.445335i \(0.853085\pi\)
\(720\) 37601.5 1.94629
\(721\) 1066.09 0.0550670
\(722\) 29390.1 1.51494
\(723\) 39600.8 2.03703
\(724\) −3484.15 −0.178850
\(725\) 4713.11 0.241435
\(726\) −24381.2 −1.24638
\(727\) −16846.5 −0.859426 −0.429713 0.902966i \(-0.641385\pi\)
−0.429713 + 0.902966i \(0.641385\pi\)
\(728\) −1957.95 −0.0996793
\(729\) −30691.3 −1.55928
\(730\) −37906.9 −1.92191
\(731\) 0 0
\(732\) −658.179 −0.0332336
\(733\) 13917.8 0.701317 0.350658 0.936503i \(-0.385958\pi\)
0.350658 + 0.936503i \(0.385958\pi\)
\(734\) 8092.82 0.406964
\(735\) −55376.9 −2.77906
\(736\) −6494.27 −0.325247
\(737\) −16816.2 −0.840479
\(738\) 19480.8 0.971678
\(739\) −10001.0 −0.497825 −0.248912 0.968526i \(-0.580073\pi\)
−0.248912 + 0.968526i \(0.580073\pi\)
\(740\) 7273.79 0.361338
\(741\) 28688.8 1.42228
\(742\) −3163.81 −0.156533
\(743\) 5356.46 0.264481 0.132240 0.991218i \(-0.457783\pi\)
0.132240 + 0.991218i \(0.457783\pi\)
\(744\) −6805.05 −0.335329
\(745\) 52240.5 2.56905
\(746\) 31929.6 1.56706
\(747\) −8996.45 −0.440647
\(748\) −1845.19 −0.0901964
\(749\) −2541.34 −0.123977
\(750\) 75260.8 3.66418
\(751\) −30162.2 −1.46556 −0.732779 0.680467i \(-0.761777\pi\)
−0.732779 + 0.680467i \(0.761777\pi\)
\(752\) 29280.9 1.41990
\(753\) −35777.2 −1.73146
\(754\) −1034.22 −0.0499526
\(755\) −20695.2 −0.997585
\(756\) −397.547 −0.0191252
\(757\) −35857.4 −1.72161 −0.860805 0.508935i \(-0.830039\pi\)
−0.860805 + 0.508935i \(0.830039\pi\)
\(758\) 4205.24 0.201505
\(759\) −37047.7 −1.77173
\(760\) 68335.1 3.26155
\(761\) −3433.69 −0.163563 −0.0817814 0.996650i \(-0.526061\pi\)
−0.0817814 + 0.996650i \(0.526061\pi\)
\(762\) −4939.59 −0.234832
\(763\) 905.514 0.0429644
\(764\) 126.436 0.00598731
\(765\) 17806.2 0.841549
\(766\) 27410.4 1.29292
\(767\) 19508.2 0.918386
\(768\) 18715.9 0.879362
\(769\) −2028.54 −0.0951248 −0.0475624 0.998868i \(-0.515145\pi\)
−0.0475624 + 0.998868i \(0.515145\pi\)
\(770\) −8079.94 −0.378157
\(771\) 47714.3 2.22878
\(772\) −1760.69 −0.0820839
\(773\) −39469.4 −1.83650 −0.918251 0.395999i \(-0.870398\pi\)
−0.918251 + 0.395999i \(0.870398\pi\)
\(774\) 0 0
\(775\) −10635.5 −0.492953
\(776\) −16679.2 −0.771582
\(777\) 5426.70 0.250556
\(778\) 15884.9 0.732006
\(779\) 28091.6 1.29202
\(780\) 6941.06 0.318628
\(781\) −24545.0 −1.12457
\(782\) 5374.53 0.245771
\(783\) −1268.42 −0.0578922
\(784\) 16275.6 0.741417
\(785\) −24693.3 −1.12273
\(786\) −47529.7 −2.15691
\(787\) −39757.2 −1.80075 −0.900376 0.435113i \(-0.856708\pi\)
−0.900376 + 0.435113i \(0.856708\pi\)
\(788\) −3797.66 −0.171683
\(789\) −368.916 −0.0166461
\(790\) 8992.10 0.404968
\(791\) 4995.40 0.224546
\(792\) 45464.2 2.03977
\(793\) 1363.45 0.0610562
\(794\) −6146.94 −0.274744
\(795\) 67747.8 3.02235
\(796\) −333.682 −0.0148581
\(797\) 9242.68 0.410781 0.205391 0.978680i \(-0.434154\pi\)
0.205391 + 0.978680i \(0.434154\pi\)
\(798\) 8440.30 0.374415
\(799\) 13866.0 0.613945
\(800\) 21493.2 0.949873
\(801\) 12394.0 0.546716
\(802\) 12668.0 0.557760
\(803\) −36367.5 −1.59823
\(804\) 4250.01 0.186426
\(805\) −5824.90 −0.255032
\(806\) 2333.81 0.101991
\(807\) −7408.65 −0.323168
\(808\) −22291.3 −0.970552
\(809\) −14493.7 −0.629877 −0.314939 0.949112i \(-0.601984\pi\)
−0.314939 + 0.949112i \(0.601984\pi\)
\(810\) 18306.8 0.794118
\(811\) −36976.4 −1.60101 −0.800504 0.599328i \(-0.795435\pi\)
−0.800504 + 0.599328i \(0.795435\pi\)
\(812\) 75.3082 0.00325468
\(813\) 41473.3 1.78909
\(814\) −28195.1 −1.21405
\(815\) −2365.86 −0.101684
\(816\) −9030.56 −0.387418
\(817\) 0 0
\(818\) −22718.1 −0.971051
\(819\) 3001.00 0.128038
\(820\) 6796.57 0.289447
\(821\) 41436.7 1.76145 0.880724 0.473629i \(-0.157056\pi\)
0.880724 + 0.473629i \(0.157056\pi\)
\(822\) 25016.0 1.06148
\(823\) −29097.3 −1.23241 −0.616203 0.787588i \(-0.711330\pi\)
−0.616203 + 0.787588i \(0.711330\pi\)
\(824\) 8455.82 0.357491
\(825\) 122612. 5.17429
\(826\) 5739.36 0.241765
\(827\) −16157.9 −0.679402 −0.339701 0.940534i \(-0.610326\pi\)
−0.339701 + 0.940534i \(0.610326\pi\)
\(828\) 5426.11 0.227742
\(829\) −41034.9 −1.71918 −0.859591 0.510983i \(-0.829282\pi\)
−0.859591 + 0.510983i \(0.829282\pi\)
\(830\) 12681.5 0.530340
\(831\) −54293.7 −2.26646
\(832\) −14998.7 −0.624985
\(833\) 7707.31 0.320579
\(834\) 6291.89 0.261235
\(835\) −64349.7 −2.66696
\(836\) 10853.7 0.449023
\(837\) 2862.28 0.118202
\(838\) −28139.7 −1.15999
\(839\) −28367.1 −1.16727 −0.583636 0.812015i \(-0.698371\pi\)
−0.583636 + 0.812015i \(0.698371\pi\)
\(840\) 12334.8 0.506656
\(841\) −24148.7 −0.990148
\(842\) 2882.04 0.117959
\(843\) −23408.8 −0.956395
\(844\) −1361.15 −0.0555125
\(845\) 31129.0 1.26730
\(846\) −56560.9 −2.29859
\(847\) −3677.72 −0.149195
\(848\) −19911.5 −0.806324
\(849\) 41305.7 1.66974
\(850\) −17787.3 −0.717765
\(851\) −20326.1 −0.818767
\(852\) 6203.34 0.249440
\(853\) −18754.4 −0.752801 −0.376400 0.926457i \(-0.622838\pi\)
−0.376400 + 0.926457i \(0.622838\pi\)
\(854\) 401.129 0.0160730
\(855\) −104739. −4.18947
\(856\) −20157.0 −0.804849
\(857\) −12782.0 −0.509481 −0.254741 0.967009i \(-0.581990\pi\)
−0.254741 + 0.967009i \(0.581990\pi\)
\(858\) −26905.4 −1.07055
\(859\) −2156.19 −0.0856442 −0.0428221 0.999083i \(-0.513635\pi\)
−0.0428221 + 0.999083i \(0.513635\pi\)
\(860\) 0 0
\(861\) 5070.66 0.200706
\(862\) −18842.3 −0.744516
\(863\) −20877.3 −0.823489 −0.411744 0.911299i \(-0.635080\pi\)
−0.411744 + 0.911299i \(0.635080\pi\)
\(864\) −5784.36 −0.227764
\(865\) 48529.1 1.90756
\(866\) 20899.1 0.820069
\(867\) 35092.5 1.37463
\(868\) −169.939 −0.00664527
\(869\) 8626.93 0.336765
\(870\) 6515.46 0.253902
\(871\) −8804.12 −0.342498
\(872\) 7182.19 0.278922
\(873\) 25564.6 0.991099
\(874\) −31613.8 −1.22351
\(875\) 11352.5 0.438611
\(876\) 9191.27 0.354502
\(877\) 21747.0 0.837336 0.418668 0.908139i \(-0.362497\pi\)
0.418668 + 0.908139i \(0.362497\pi\)
\(878\) −29703.1 −1.14172
\(879\) 46748.9 1.79386
\(880\) −50851.2 −1.94795
\(881\) 39665.9 1.51689 0.758444 0.651738i \(-0.225960\pi\)
0.758444 + 0.651738i \(0.225960\pi\)
\(882\) −31439.0 −1.20024
\(883\) −32675.0 −1.24530 −0.622652 0.782499i \(-0.713945\pi\)
−0.622652 + 0.782499i \(0.713945\pi\)
\(884\) −966.050 −0.0367554
\(885\) −122899. −4.66803
\(886\) 3573.48 0.135500
\(887\) 1318.70 0.0499183 0.0249592 0.999688i \(-0.492054\pi\)
0.0249592 + 0.999688i \(0.492054\pi\)
\(888\) 43042.5 1.62659
\(889\) −745.098 −0.0281100
\(890\) −17470.7 −0.658000
\(891\) 17563.4 0.660376
\(892\) 5606.41 0.210445
\(893\) −81561.6 −3.05639
\(894\) 51178.0 1.91459
\(895\) 34613.3 1.29273
\(896\) −2681.66 −0.0999867
\(897\) −19396.3 −0.721989
\(898\) 24101.5 0.895634
\(899\) −542.209 −0.0201153
\(900\) −17958.1 −0.665113
\(901\) −9429.08 −0.348644
\(902\) −26345.3 −0.972507
\(903\) 0 0
\(904\) 39621.6 1.45774
\(905\) −45469.7 −1.67013
\(906\) −20274.3 −0.743453
\(907\) −2202.10 −0.0806170 −0.0403085 0.999187i \(-0.512834\pi\)
−0.0403085 + 0.999187i \(0.512834\pi\)
\(908\) 8235.45 0.300994
\(909\) 34166.4 1.24668
\(910\) −4230.25 −0.154100
\(911\) 10427.7 0.379237 0.189618 0.981858i \(-0.439275\pi\)
0.189618 + 0.981858i \(0.439275\pi\)
\(912\) 53119.1 1.92867
\(913\) 12166.5 0.441022
\(914\) 28307.1 1.02441
\(915\) −8589.53 −0.310340
\(916\) −5527.19 −0.199371
\(917\) −7169.49 −0.258187
\(918\) 4787.02 0.172108
\(919\) −4793.05 −0.172044 −0.0860218 0.996293i \(-0.527415\pi\)
−0.0860218 + 0.996293i \(0.527415\pi\)
\(920\) −46200.9 −1.65565
\(921\) −53949.9 −1.93020
\(922\) −39743.1 −1.41960
\(923\) −12850.5 −0.458267
\(924\) 1959.14 0.0697522
\(925\) 67270.5 2.39118
\(926\) 6450.82 0.228928
\(927\) −12960.4 −0.459198
\(928\) 1095.74 0.0387603
\(929\) 42938.0 1.51641 0.758207 0.652014i \(-0.226076\pi\)
0.758207 + 0.652014i \(0.226076\pi\)
\(930\) −14702.6 −0.518407
\(931\) −45335.5 −1.59593
\(932\) −2452.75 −0.0862044
\(933\) −70177.6 −2.46250
\(934\) 19809.1 0.693977
\(935\) −24080.6 −0.842267
\(936\) 23802.8 0.831216
\(937\) 17961.4 0.626224 0.313112 0.949716i \(-0.398628\pi\)
0.313112 + 0.949716i \(0.398628\pi\)
\(938\) −2590.19 −0.0901626
\(939\) −53979.3 −1.87598
\(940\) −19733.3 −0.684711
\(941\) 1417.03 0.0490902 0.0245451 0.999699i \(-0.492186\pi\)
0.0245451 + 0.999699i \(0.492186\pi\)
\(942\) −24191.0 −0.836716
\(943\) −18992.5 −0.655867
\(944\) 36120.7 1.24537
\(945\) −5188.16 −0.178594
\(946\) 0 0
\(947\) −1240.24 −0.0425579 −0.0212789 0.999774i \(-0.506774\pi\)
−0.0212789 + 0.999774i \(0.506774\pi\)
\(948\) −2180.31 −0.0746975
\(949\) −19040.2 −0.651286
\(950\) 104628. 3.57323
\(951\) −69328.2 −2.36395
\(952\) −1716.75 −0.0584454
\(953\) 39421.9 1.33998 0.669989 0.742371i \(-0.266299\pi\)
0.669989 + 0.742371i \(0.266299\pi\)
\(954\) 38462.4 1.30531
\(955\) 1650.05 0.0559103
\(956\) 4067.33 0.137601
\(957\) 6250.87 0.211141
\(958\) −37409.6 −1.26164
\(959\) 3773.47 0.127061
\(960\) 94489.8 3.17671
\(961\) −28567.5 −0.958929
\(962\) −14761.5 −0.494731
\(963\) 30895.0 1.03383
\(964\) 7843.81 0.262067
\(965\) −22977.8 −0.766511
\(966\) −5706.43 −0.190063
\(967\) 13213.8 0.439428 0.219714 0.975564i \(-0.429488\pi\)
0.219714 + 0.975564i \(0.429488\pi\)
\(968\) −29170.2 −0.968561
\(969\) 25154.6 0.833933
\(970\) −36036.2 −1.19284
\(971\) 16647.1 0.550186 0.275093 0.961418i \(-0.411291\pi\)
0.275093 + 0.961418i \(0.411291\pi\)
\(972\) −7945.55 −0.262195
\(973\) 949.082 0.0312705
\(974\) 13388.4 0.440444
\(975\) 64193.3 2.10854
\(976\) 2524.51 0.0827948
\(977\) −3092.02 −0.101251 −0.0506256 0.998718i \(-0.516122\pi\)
−0.0506256 + 0.998718i \(0.516122\pi\)
\(978\) −2317.74 −0.0757802
\(979\) −16761.2 −0.547183
\(980\) −10968.6 −0.357530
\(981\) −11008.3 −0.358276
\(982\) 44337.3 1.44079
\(983\) 14697.7 0.476890 0.238445 0.971156i \(-0.423362\pi\)
0.238445 + 0.971156i \(0.423362\pi\)
\(984\) 40218.5 1.30297
\(985\) −49561.2 −1.60320
\(986\) −906.816 −0.0292889
\(987\) −14722.2 −0.474786
\(988\) 5682.45 0.182979
\(989\) 0 0
\(990\) 98227.7 3.15341
\(991\) 43483.8 1.39385 0.696927 0.717143i \(-0.254550\pi\)
0.696927 + 0.717143i \(0.254550\pi\)
\(992\) −2472.63 −0.0791392
\(993\) −44320.2 −1.41637
\(994\) −3780.65 −0.120639
\(995\) −4354.70 −0.138747
\(996\) −3074.89 −0.0978228
\(997\) 35710.8 1.13437 0.567187 0.823589i \(-0.308032\pi\)
0.567187 + 0.823589i \(0.308032\pi\)
\(998\) 3951.24 0.125325
\(999\) −18104.2 −0.573365
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.j.1.34 yes 50
43.42 odd 2 1849.4.a.i.1.17 50
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1849.4.a.i.1.17 50 43.42 odd 2
1849.4.a.j.1.34 yes 50 1.1 even 1 trivial