Properties

Label 1849.4.a.i.1.17
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 1.17
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 \(-0.647743\pi\)
−0.447661 + 0.894203i \(0.647743\pi\)
\(3\) 8.01322 1.54214 0.771072 0.636747i \(-0.219721\pi\)
0.771072 + 0.636747i \(0.219721\pi\)
\(4\) −1.58719 −0.198399
\(5\) 20.7136 1.85268 0.926340 0.376687i \(-0.122937\pi\)
0.926340 + 0.376687i \(0.122937\pi\)
\(6\) −20.2923 −1.38072
\(7\) 3.06093 0.165275 0.0826374 0.996580i \(-0.473666\pi\)
0.0826374 + 0.996580i \(0.473666\pi\)
\(8\) 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 \(-0.806235\pi\)
−0.820375 + 0.571826i \(0.806235\pi\)
\(20\) −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 \(-0.515804\pi\)
−0.0496285 + 0.998768i \(0.515804\pi\)
\(30\) −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 \(-0.663559\pi\)
−0.491522 + 0.870865i \(0.663559\pi\)
\(38\) 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 \(-0.482704\pi\)
0.0543103 + 0.998524i \(0.482704\pi\)
\(62\) 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 \(-0.633646\pi\)
−0.407634 + 0.913145i \(0.633646\pi\)
\(72\) 903.431 1.47875
\(73\) −722.667 −1.15865 −0.579327 0.815095i \(-0.696685\pi\)
−0.579327 + 0.815095i \(0.696685\pi\)
\(74\) 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 \(-0.563556\pi\)
−0.198343 + 0.980133i \(0.563556\pi\)
\(90\) −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 \(-0.262279\pi\)
0.679311 + 0.733850i \(0.262279\pi\)
\(114\) 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 \(-0.785333\pi\)
−0.781083 + 0.624427i \(0.785333\pi\)
\(132\) 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 \(-0.374410\pi\)
0.384394 + 0.923169i \(0.374410\pi\)
\(138\) 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 \(-0.256141\pi\)
0.693334 + 0.720616i \(0.256141\pi\)
\(150\) −6169.94 −3.35849
\(151\) −999.113 −0.538455 −0.269227 0.963077i \(-0.586768\pi\)
−0.269227 + 0.963077i \(0.586768\pi\)
\(152\) −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 \(-0.597989\pi\)
−0.303001 + 0.952990i \(0.597989\pi\)
\(158\) 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 \(-0.508736\pi\)
−0.0274424 + 0.999623i \(0.508736\pi\)
\(164\) −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 \(-0.386562\pi\)
0.348881 + 0.937167i \(0.386562\pi\)
\(180\) −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 \(-0.495197\pi\)
0.0150890 + 0.999886i \(0.495197\pi\)
\(192\) 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 \(-0.511922\pi\)
−0.0374450 + 0.999299i \(0.511922\pi\)
\(200\) 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 \(-0.544678\pi\)
−0.139901 + 0.990166i \(0.544678\pi\)
\(212\) −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 \(-0.322058\pi\)
0.530356 + 0.847775i \(0.322058\pi\)
\(224\) −216.374 −0.0645405
\(225\) 11314.3 3.35240
\(226\) −4132.76 −1.21640
\(227\) 5188.68 1.51711 0.758557 0.651607i \(-0.225905\pi\)
0.758557 + 0.651607i \(0.225905\pi\)
\(228\) 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 \(-0.569709\pi\)
−0.217250 + 0.976116i \(0.569709\pi\)
\(234\) 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 \(-0.270365\pi\)
0.660452 + 0.750868i \(0.270365\pi\)
\(242\) −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 \(-0.242936\pi\)
0.722624 + 0.691242i \(0.242936\pi\)
\(258\) 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 \(-0.501718\pi\)
−0.00539706 + 0.999985i \(0.501718\pi\)
\(264\) −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 \(-0.762743\pi\)
−0.734840 + 0.678240i \(0.762743\pi\)
\(278\) 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 \(-0.708124\pi\)
−0.608238 + 0.793754i \(0.708124\pi\)
\(314\) 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 \(-0.651872\pi\)
−0.459222 + 0.888322i \(0.651872\pi\)
\(332\) 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 \(-0.724076\pi\)
−0.647237 + 0.762289i \(0.724076\pi\)
\(348\) 197.149 0.0303687
\(349\) 2243.31 0.344074 0.172037 0.985090i \(-0.444965\pi\)
0.172037 + 0.985090i \(0.444965\pi\)
\(350\) −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 \(-0.839228\pi\)
−0.875136 + 0.483876i \(0.839228\pi\)
\(374\) −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 \(-0.756797\pi\)
−0.722043 + 0.691849i \(0.756797\pi\)
\(384\) −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 \(-0.634051\pi\)
−0.408795 + 0.912626i \(0.634051\pi\)
\(390\) 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 \(-0.317557\pi\)
0.542292 + 0.840190i \(0.317557\pi\)
\(410\) −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 \(-0.275687\pi\)
0.647806 + 0.761805i \(0.275687\pi\)
\(420\) −806.393 −0.0936856
\(421\) −1138.09 −0.131751 −0.0658754 0.997828i \(-0.520984\pi\)
−0.0658754 + 0.997828i \(0.520984\pi\)
\(422\) 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 \(-0.651425\pi\)
−0.457975 + 0.888965i \(0.651425\pi\)
\(434\) 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 \(-0.666731\pi\)
−0.500174 + 0.865925i \(0.666731\pi\)
\(450\) −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 \(-0.693868\pi\)
−0.572093 + 0.820189i \(0.693868\pi\)
\(458\) −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 \(-0.540807\pi\)
−0.127847 + 0.991794i \(0.540807\pi\)
\(464\) 756.187 0.0756575
\(465\) −5805.92 −0.579017
\(466\) 3913.34 0.389017
\(467\) −7822.43 −0.775115 −0.387558 0.921845i \(-0.626681\pi\)
−0.387558 + 0.921845i \(0.626681\pi\)
\(468\) 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 \(-0.797633\pi\)
−0.804624 + 0.593784i \(0.797633\pi\)
\(492\) −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 \(-0.522296\pi\)
−0.0699888 + 0.997548i \(0.522296\pi\)
\(500\) −5886.65 −0.526518
\(501\) 24894.2 2.21994
\(502\) −11306.4 −1.00523
\(503\) 7420.52 0.657783 0.328891 0.944368i \(-0.393325\pi\)
0.328891 + 0.944368i \(0.393325\pi\)
\(504\) 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 \(-0.315304\pi\)
0.548225 + 0.836331i \(0.315304\pi\)
\(522\) 1460.70 0.122477
\(523\) −18176.0 −1.51966 −0.759829 0.650123i \(-0.774717\pi\)
−0.759829 + 0.650123i \(0.774717\pi\)
\(524\) 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 \(-0.526486\pi\)
−0.0831110 + 0.996540i \(0.526486\pi\)
\(572\) −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 \(-0.361119\pi\)
0.422595 + 0.906319i \(0.361119\pi\)
\(578\) 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 \(-0.660404\pi\)
−0.482865 + 0.875695i \(0.660404\pi\)
\(588\) 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 \(-0.883924\pi\)
−0.934244 + 0.356634i \(0.883924\pi\)
\(594\) 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 \(-0.746737\pi\)
−0.699820 + 0.714319i \(0.746737\pi\)
\(602\) 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 \(-0.545071\pi\)
−0.141121 + 0.989992i \(0.545071\pi\)
\(608\) 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 \(-0.271314\pi\)
0.658209 + 0.752835i \(0.271314\pi\)
\(632\) −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 \(-0.276763\pi\)
0.645226 + 0.763992i \(0.276763\pi\)
\(642\) −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 \(-0.334275\pi\)
0.497437 + 0.867500i \(0.334275\pi\)
\(648\) −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 \(-0.535355\pi\)
−0.110843 + 0.993838i \(0.535355\pi\)
\(654\) 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 \(-0.458149\pi\)
0.131101 + 0.991369i \(0.458149\pi\)
\(674\) −6925.30 −0.395776
\(675\) 24880.2 1.41873
\(676\) 2385.28 0.135713
\(677\) 12344.6 0.700801 0.350401 0.936600i \(-0.386045\pi\)
0.350401 + 0.936600i \(0.386045\pi\)
\(678\) −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 \(-0.401211\pi\)
0.305396 + 0.952226i \(0.401211\pi\)
\(692\) 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 \(-0.358615\pi\)
0.429713 + 0.902966i \(0.358615\pi\)
\(728\) −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 \(-0.614042\pi\)
−0.350658 + 0.936503i \(0.614042\pi\)
\(734\) −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 \(-0.419927\pi\)
0.248912 + 0.968526i \(0.419927\pi\)
\(740\) 7273.79 0.361338
\(741\) 28688.8 1.42228
\(742\) −3163.81 −0.156533
\(743\) −5356.46 −0.264481 −0.132240 0.991218i \(-0.542217\pi\)
−0.132240 + 0.991218i \(0.542217\pi\)
\(744\) −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 \(-0.238223\pi\)
0.732779 + 0.680467i \(0.238223\pi\)
\(752\) 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 \(-0.169961\pi\)
0.860805 + 0.508935i \(0.169961\pi\)
\(758\) −4205.24 −0.201505
\(759\) 37047.7 1.77173
\(760\) −68335.1 −3.26155
\(761\) 3433.69 0.163563 0.0817814 0.996650i \(-0.473939\pi\)
0.0817814 + 0.996650i \(0.473939\pi\)
\(762\) −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 \(-0.129602\pi\)
0.918251 + 0.395999i \(0.129602\pi\)
\(774\) 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 \(-0.204565\pi\)
0.800504 + 0.599328i \(0.204565\pi\)
\(812\) 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 \(-0.170718\pi\)
0.859591 + 0.510983i \(0.170718\pi\)
\(830\) 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 \(-0.301629\pi\)
0.583636 + 0.812015i \(0.301629\pi\)
\(840\) 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 \(-0.486365\pi\)
0.0428221 + 0.999083i \(0.486365\pi\)
\(860\) 0 0
\(861\) 5070.66 0.200706
\(862\) 18842.3 0.744516
\(863\) 20877.3 0.823489 0.411744 0.911299i \(-0.364920\pi\)
0.411744 + 0.911299i \(0.364920\pi\)
\(864\) −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 \(-0.507946\pi\)
−0.0249592 + 0.999688i \(0.507946\pi\)
\(888\) −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 \(-0.560725\pi\)
−0.189618 + 0.981858i \(0.560725\pi\)
\(912\) 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 \(-0.773924\pi\)
−0.758207 + 0.652014i \(0.773924\pi\)
\(930\) 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 \(-0.601372\pi\)
−0.313112 + 0.949716i \(0.601372\pi\)
\(938\) −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 \(-0.733701\pi\)
−0.669989 + 0.742371i \(0.733701\pi\)
\(954\) −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 \(-0.576638\pi\)
−0.238445 + 0.971156i \(0.576638\pi\)
\(984\) 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 \(-0.745450\pi\)
−0.696927 + 0.717143i \(0.745450\pi\)
\(992\) 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 \(-0.691968\pi\)
−0.567187 + 0.823589i \(0.691968\pi\)
\(998\) 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.i.1.17 50
43.42 odd 2 1849.4.a.j.1.34 yes 50
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1849.4.a.i.1.17 50 1.1 even 1 trivial
1849.4.a.j.1.34 yes 50 43.42 odd 2