Properties

Label 1849.4.a.g.1.25
Level $1849$
Weight $4$
Character 1849.1
Self dual yes
Analytic conductor $109.095$
Analytic rank $1$
Dimension $30$
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: \(30\)
Twist minimal: no (minimal twist has level 43)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.25
Character \(\chi\) \(=\) 1849.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.72135 q^{2} -5.79603 q^{3} +5.84847 q^{4} +14.8139 q^{5} -21.5691 q^{6} +12.2090 q^{7} -8.00659 q^{8} +6.59399 q^{9} +O(q^{10})\) \(q+3.72135 q^{2} -5.79603 q^{3} +5.84847 q^{4} +14.8139 q^{5} -21.5691 q^{6} +12.2090 q^{7} -8.00659 q^{8} +6.59399 q^{9} +55.1279 q^{10} +2.05254 q^{11} -33.8979 q^{12} +9.53050 q^{13} +45.4339 q^{14} -85.8621 q^{15} -76.5831 q^{16} -74.1396 q^{17} +24.5386 q^{18} +103.100 q^{19} +86.6389 q^{20} -70.7635 q^{21} +7.63821 q^{22} -146.983 q^{23} +46.4064 q^{24} +94.4528 q^{25} +35.4663 q^{26} +118.274 q^{27} +71.4038 q^{28} +102.214 q^{29} -319.523 q^{30} -308.185 q^{31} -220.940 q^{32} -11.8966 q^{33} -275.900 q^{34} +180.863 q^{35} +38.5648 q^{36} -434.968 q^{37} +383.670 q^{38} -55.2391 q^{39} -118.609 q^{40} +183.943 q^{41} -263.336 q^{42} +12.0042 q^{44} +97.6829 q^{45} -546.975 q^{46} +348.672 q^{47} +443.878 q^{48} -193.941 q^{49} +351.492 q^{50} +429.716 q^{51} +55.7389 q^{52} -199.807 q^{53} +440.139 q^{54} +30.4062 q^{55} -97.7522 q^{56} -597.568 q^{57} +380.375 q^{58} -156.936 q^{59} -502.162 q^{60} +261.234 q^{61} -1146.86 q^{62} +80.5057 q^{63} -209.532 q^{64} +141.184 q^{65} -44.2713 q^{66} -43.3366 q^{67} -433.604 q^{68} +851.917 q^{69} +673.055 q^{70} +654.799 q^{71} -52.7953 q^{72} -868.625 q^{73} -1618.67 q^{74} -547.452 q^{75} +602.975 q^{76} +25.0593 q^{77} -205.564 q^{78} -1354.05 q^{79} -1134.50 q^{80} -863.557 q^{81} +684.515 q^{82} +166.764 q^{83} -413.859 q^{84} -1098.30 q^{85} -592.436 q^{87} -16.4338 q^{88} +1189.13 q^{89} +363.513 q^{90} +116.357 q^{91} -859.625 q^{92} +1786.25 q^{93} +1297.53 q^{94} +1527.31 q^{95} +1280.58 q^{96} -398.059 q^{97} -721.724 q^{98} +13.5344 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 30 q - 6 q^{2} - 2 q^{3} + 114 q^{4} - 27 q^{5} + 8 q^{6} - 48 q^{7} - 90 q^{8} + 216 q^{9} + O(q^{10}) \) \( 30 q - 6 q^{2} - 2 q^{3} + 114 q^{4} - 27 q^{5} + 8 q^{6} - 48 q^{7} - 90 q^{8} + 216 q^{9} - 27 q^{10} + 80 q^{11} + 36 q^{12} - 13 q^{13} + 36 q^{14} + 16 q^{15} + 318 q^{16} + 66 q^{17} - 80 q^{18} - 254 q^{19} - 312 q^{20} - 548 q^{21} - 305 q^{22} - 105 q^{23} + 123 q^{24} + 523 q^{25} - 549 q^{26} + 10 q^{27} - 578 q^{28} - 793 q^{29} - 1560 q^{30} - 359 q^{31} - 676 q^{32} - 208 q^{33} - 1007 q^{34} - 514 q^{35} + 776 q^{36} - 510 q^{37} - 2066 q^{38} - 898 q^{39} - 1248 q^{40} - 270 q^{41} + 915 q^{42} + 3256 q^{44} - 807 q^{45} - 1960 q^{46} + 1421 q^{47} + 632 q^{48} + 386 q^{49} + 141 q^{50} - 209 q^{51} + 2825 q^{52} - 21 q^{53} + 2368 q^{54} - 2258 q^{55} + 2521 q^{56} - 1723 q^{57} - 347 q^{58} + 1752 q^{59} + 2711 q^{60} - 1759 q^{61} - 395 q^{62} - 2204 q^{63} + 222 q^{64} - 1151 q^{65} + 160 q^{66} - 3001 q^{67} + 1921 q^{68} - 1660 q^{69} - 1597 q^{70} - 727 q^{71} - 9100 q^{72} - 4623 q^{73} - 2649 q^{74} - 1027 q^{75} - 874 q^{76} - 3556 q^{77} - 4979 q^{78} + 546 q^{79} - 5809 q^{80} - 410 q^{81} + 4397 q^{82} - 492 q^{83} - 10611 q^{84} + 1723 q^{85} + 5937 q^{87} - 3974 q^{88} - 5218 q^{89} + 10492 q^{90} - 1104 q^{91} + 1060 q^{92} - 1997 q^{93} + 2134 q^{94} + 6346 q^{95} - 11984 q^{96} + 2590 q^{97} - 6270 q^{98} - 2693 q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.72135 1.31570 0.657849 0.753150i \(-0.271467\pi\)
0.657849 + 0.753150i \(0.271467\pi\)
\(3\) −5.79603 −1.11545 −0.557723 0.830027i \(-0.688325\pi\)
−0.557723 + 0.830027i \(0.688325\pi\)
\(4\) 5.84847 0.731059
\(5\) 14.8139 1.32500 0.662500 0.749062i \(-0.269496\pi\)
0.662500 + 0.749062i \(0.269496\pi\)
\(6\) −21.5691 −1.46759
\(7\) 12.2090 0.659222 0.329611 0.944117i \(-0.393082\pi\)
0.329611 + 0.944117i \(0.393082\pi\)
\(8\) −8.00659 −0.353845
\(9\) 6.59399 0.244222
\(10\) 55.1279 1.74330
\(11\) 2.05254 0.0562603 0.0281301 0.999604i \(-0.491045\pi\)
0.0281301 + 0.999604i \(0.491045\pi\)
\(12\) −33.8979 −0.815458
\(13\) 9.53050 0.203330 0.101665 0.994819i \(-0.467583\pi\)
0.101665 + 0.994819i \(0.467583\pi\)
\(14\) 45.4339 0.867337
\(15\) −85.8621 −1.47797
\(16\) −76.5831 −1.19661
\(17\) −74.1396 −1.05774 −0.528868 0.848704i \(-0.677383\pi\)
−0.528868 + 0.848704i \(0.677383\pi\)
\(18\) 24.5386 0.321322
\(19\) 103.100 1.24488 0.622439 0.782669i \(-0.286142\pi\)
0.622439 + 0.782669i \(0.286142\pi\)
\(20\) 86.6389 0.968653
\(21\) −70.7635 −0.735327
\(22\) 7.63821 0.0740215
\(23\) −146.983 −1.33252 −0.666261 0.745718i \(-0.732107\pi\)
−0.666261 + 0.745718i \(0.732107\pi\)
\(24\) 46.4064 0.394695
\(25\) 94.4528 0.755623
\(26\) 35.4663 0.267520
\(27\) 118.274 0.843031
\(28\) 71.4038 0.481930
\(29\) 102.214 0.654506 0.327253 0.944937i \(-0.393877\pi\)
0.327253 + 0.944937i \(0.393877\pi\)
\(30\) −319.523 −1.94456
\(31\) −308.185 −1.78554 −0.892768 0.450516i \(-0.851240\pi\)
−0.892768 + 0.450516i \(0.851240\pi\)
\(32\) −220.940 −1.22053
\(33\) −11.8966 −0.0627553
\(34\) −275.900 −1.39166
\(35\) 180.863 0.873469
\(36\) 38.5648 0.178541
\(37\) −434.968 −1.93266 −0.966329 0.257312i \(-0.917163\pi\)
−0.966329 + 0.257312i \(0.917163\pi\)
\(38\) 383.670 1.63788
\(39\) −55.2391 −0.226803
\(40\) −118.609 −0.468844
\(41\) 183.943 0.700659 0.350329 0.936627i \(-0.386070\pi\)
0.350329 + 0.936627i \(0.386070\pi\)
\(42\) −263.336 −0.967468
\(43\) 0 0
\(44\) 12.0042 0.0411296
\(45\) 97.6829 0.323594
\(46\) −546.975 −1.75320
\(47\) 348.672 1.08211 0.541053 0.840988i \(-0.318026\pi\)
0.541053 + 0.840988i \(0.318026\pi\)
\(48\) 443.878 1.33476
\(49\) −193.941 −0.565426
\(50\) 351.492 0.994171
\(51\) 429.716 1.17985
\(52\) 55.7389 0.148646
\(53\) −199.807 −0.517842 −0.258921 0.965898i \(-0.583367\pi\)
−0.258921 + 0.965898i \(0.583367\pi\)
\(54\) 440.139 1.10917
\(55\) 30.4062 0.0745448
\(56\) −97.7522 −0.233262
\(57\) −597.568 −1.38859
\(58\) 380.375 0.861132
\(59\) −156.936 −0.346294 −0.173147 0.984896i \(-0.555394\pi\)
−0.173147 + 0.984896i \(0.555394\pi\)
\(60\) −502.162 −1.08048
\(61\) 261.234 0.548321 0.274160 0.961684i \(-0.411600\pi\)
0.274160 + 0.961684i \(0.411600\pi\)
\(62\) −1146.86 −2.34923
\(63\) 80.5057 0.160996
\(64\) −209.532 −0.409242
\(65\) 141.184 0.269411
\(66\) −44.2713 −0.0825670
\(67\) −43.3366 −0.0790209 −0.0395105 0.999219i \(-0.512580\pi\)
−0.0395105 + 0.999219i \(0.512580\pi\)
\(68\) −433.604 −0.773267
\(69\) 851.917 1.48636
\(70\) 673.055 1.14922
\(71\) 654.799 1.09451 0.547256 0.836965i \(-0.315672\pi\)
0.547256 + 0.836965i \(0.315672\pi\)
\(72\) −52.7953 −0.0864165
\(73\) −868.625 −1.39267 −0.696335 0.717717i \(-0.745187\pi\)
−0.696335 + 0.717717i \(0.745187\pi\)
\(74\) −1618.67 −2.54279
\(75\) −547.452 −0.842857
\(76\) 602.975 0.910079
\(77\) 25.0593 0.0370880
\(78\) −205.564 −0.298404
\(79\) −1354.05 −1.92838 −0.964190 0.265213i \(-0.914558\pi\)
−0.964190 + 0.265213i \(0.914558\pi\)
\(80\) −1134.50 −1.58551
\(81\) −863.557 −1.18458
\(82\) 684.515 0.921855
\(83\) 166.764 0.220539 0.110270 0.993902i \(-0.464829\pi\)
0.110270 + 0.993902i \(0.464829\pi\)
\(84\) −413.859 −0.537568
\(85\) −1098.30 −1.40150
\(86\) 0 0
\(87\) −592.436 −0.730067
\(88\) −16.4338 −0.0199074
\(89\) 1189.13 1.41627 0.708134 0.706078i \(-0.249537\pi\)
0.708134 + 0.706078i \(0.249537\pi\)
\(90\) 363.513 0.425751
\(91\) 116.357 0.134039
\(92\) −859.625 −0.974153
\(93\) 1786.25 1.99167
\(94\) 1297.53 1.42373
\(95\) 1527.31 1.64946
\(96\) 1280.58 1.36144
\(97\) −398.059 −0.416668 −0.208334 0.978058i \(-0.566804\pi\)
−0.208334 + 0.978058i \(0.566804\pi\)
\(98\) −721.724 −0.743930
\(99\) 13.5344 0.0137400
\(100\) 552.405 0.552405
\(101\) 592.846 0.584063 0.292032 0.956409i \(-0.405669\pi\)
0.292032 + 0.956409i \(0.405669\pi\)
\(102\) 1599.12 1.55232
\(103\) 644.170 0.616233 0.308116 0.951349i \(-0.400301\pi\)
0.308116 + 0.951349i \(0.400301\pi\)
\(104\) −76.3068 −0.0719471
\(105\) −1048.29 −0.974308
\(106\) −743.553 −0.681323
\(107\) 215.065 0.194309 0.0971547 0.995269i \(-0.469026\pi\)
0.0971547 + 0.995269i \(0.469026\pi\)
\(108\) 691.722 0.616305
\(109\) −1496.74 −1.31524 −0.657621 0.753349i \(-0.728437\pi\)
−0.657621 + 0.753349i \(0.728437\pi\)
\(110\) 113.152 0.0980784
\(111\) 2521.09 2.15578
\(112\) −935.001 −0.788833
\(113\) 436.886 0.363706 0.181853 0.983326i \(-0.441791\pi\)
0.181853 + 0.983326i \(0.441791\pi\)
\(114\) −2223.76 −1.82697
\(115\) −2177.39 −1.76559
\(116\) 597.797 0.478483
\(117\) 62.8440 0.0496575
\(118\) −584.016 −0.455619
\(119\) −905.168 −0.697283
\(120\) 687.462 0.522970
\(121\) −1326.79 −0.996835
\(122\) 972.143 0.721424
\(123\) −1066.14 −0.781547
\(124\) −1802.41 −1.30533
\(125\) −452.524 −0.323800
\(126\) 299.590 0.211822
\(127\) −1556.68 −1.08766 −0.543830 0.839196i \(-0.683026\pi\)
−0.543830 + 0.839196i \(0.683026\pi\)
\(128\) 987.780 0.682096
\(129\) 0 0
\(130\) 525.396 0.354464
\(131\) 278.172 0.185527 0.0927633 0.995688i \(-0.470430\pi\)
0.0927633 + 0.995688i \(0.470430\pi\)
\(132\) −69.5768 −0.0458779
\(133\) 1258.74 0.820651
\(134\) −161.271 −0.103968
\(135\) 1752.10 1.11701
\(136\) 593.606 0.374274
\(137\) 1770.68 1.10423 0.552114 0.833769i \(-0.313821\pi\)
0.552114 + 0.833769i \(0.313821\pi\)
\(138\) 3170.28 1.95560
\(139\) −1023.01 −0.624249 −0.312125 0.950041i \(-0.601041\pi\)
−0.312125 + 0.950041i \(0.601041\pi\)
\(140\) 1057.77 0.638557
\(141\) −2020.91 −1.20703
\(142\) 2436.74 1.44005
\(143\) 19.5617 0.0114394
\(144\) −504.988 −0.292239
\(145\) 1514.19 0.867220
\(146\) −3232.46 −1.83233
\(147\) 1124.09 0.630703
\(148\) −2543.90 −1.41289
\(149\) −2224.57 −1.22311 −0.611556 0.791201i \(-0.709456\pi\)
−0.611556 + 0.791201i \(0.709456\pi\)
\(150\) −2037.26 −1.10894
\(151\) 753.417 0.406041 0.203020 0.979174i \(-0.434924\pi\)
0.203020 + 0.979174i \(0.434924\pi\)
\(152\) −825.476 −0.440493
\(153\) −488.876 −0.258322
\(154\) 93.2547 0.0487966
\(155\) −4565.43 −2.36583
\(156\) −323.064 −0.165807
\(157\) −297.824 −0.151395 −0.0756973 0.997131i \(-0.524118\pi\)
−0.0756973 + 0.997131i \(0.524118\pi\)
\(158\) −5038.88 −2.53716
\(159\) 1158.09 0.577625
\(160\) −3273.00 −1.61721
\(161\) −1794.51 −0.878429
\(162\) −3213.60 −1.55855
\(163\) −2993.34 −1.43838 −0.719191 0.694812i \(-0.755487\pi\)
−0.719191 + 0.694812i \(0.755487\pi\)
\(164\) 1075.78 0.512223
\(165\) −176.235 −0.0831508
\(166\) 620.589 0.290163
\(167\) −300.196 −0.139101 −0.0695504 0.997578i \(-0.522156\pi\)
−0.0695504 + 0.997578i \(0.522156\pi\)
\(168\) 566.575 0.260192
\(169\) −2106.17 −0.958657
\(170\) −4087.16 −1.84395
\(171\) 679.837 0.304026
\(172\) 0 0
\(173\) −1441.64 −0.633559 −0.316780 0.948499i \(-0.602602\pi\)
−0.316780 + 0.948499i \(0.602602\pi\)
\(174\) −2204.66 −0.960547
\(175\) 1153.17 0.498123
\(176\) −157.190 −0.0673217
\(177\) 909.608 0.386273
\(178\) 4425.19 1.86338
\(179\) −4500.30 −1.87915 −0.939576 0.342340i \(-0.888781\pi\)
−0.939576 + 0.342340i \(0.888781\pi\)
\(180\) 571.296 0.236566
\(181\) 3180.09 1.30593 0.652967 0.757386i \(-0.273524\pi\)
0.652967 + 0.757386i \(0.273524\pi\)
\(182\) 433.007 0.176355
\(183\) −1514.12 −0.611623
\(184\) 1176.83 0.471506
\(185\) −6443.59 −2.56077
\(186\) 6647.27 2.62044
\(187\) −152.174 −0.0595085
\(188\) 2039.20 0.791084
\(189\) 1444.00 0.555744
\(190\) 5683.67 2.17019
\(191\) −3654.44 −1.38443 −0.692215 0.721691i \(-0.743365\pi\)
−0.692215 + 0.721691i \(0.743365\pi\)
\(192\) 1214.45 0.456487
\(193\) 2259.01 0.842522 0.421261 0.906939i \(-0.361588\pi\)
0.421261 + 0.906939i \(0.361588\pi\)
\(194\) −1481.32 −0.548209
\(195\) −818.308 −0.300514
\(196\) −1134.26 −0.413360
\(197\) 350.103 0.126618 0.0633091 0.997994i \(-0.479835\pi\)
0.0633091 + 0.997994i \(0.479835\pi\)
\(198\) 50.3663 0.0180777
\(199\) −3583.53 −1.27653 −0.638265 0.769817i \(-0.720348\pi\)
−0.638265 + 0.769817i \(0.720348\pi\)
\(200\) −756.245 −0.267373
\(201\) 251.180 0.0881437
\(202\) 2206.19 0.768451
\(203\) 1247.93 0.431465
\(204\) 2513.18 0.862539
\(205\) 2724.91 0.928372
\(206\) 2397.18 0.810776
\(207\) −969.202 −0.325431
\(208\) −729.875 −0.243307
\(209\) 211.616 0.0700371
\(210\) −3901.05 −1.28189
\(211\) −653.378 −0.213177 −0.106589 0.994303i \(-0.533993\pi\)
−0.106589 + 0.994303i \(0.533993\pi\)
\(212\) −1168.57 −0.378573
\(213\) −3795.24 −1.22087
\(214\) 800.332 0.255652
\(215\) 0 0
\(216\) −946.971 −0.298302
\(217\) −3762.62 −1.17707
\(218\) −5569.88 −1.73046
\(219\) 5034.58 1.55345
\(220\) 177.830 0.0544967
\(221\) −706.588 −0.215069
\(222\) 9381.86 2.83635
\(223\) −304.058 −0.0913061 −0.0456530 0.998957i \(-0.514537\pi\)
−0.0456530 + 0.998957i \(0.514537\pi\)
\(224\) −2697.45 −0.804603
\(225\) 622.821 0.184539
\(226\) 1625.81 0.478527
\(227\) 1310.85 0.383277 0.191638 0.981466i \(-0.438620\pi\)
0.191638 + 0.981466i \(0.438620\pi\)
\(228\) −3494.86 −1.01514
\(229\) −4790.56 −1.38240 −0.691199 0.722664i \(-0.742917\pi\)
−0.691199 + 0.722664i \(0.742917\pi\)
\(230\) −8102.85 −2.32298
\(231\) −145.245 −0.0413697
\(232\) −818.386 −0.231594
\(233\) −374.148 −0.105199 −0.0525993 0.998616i \(-0.516751\pi\)
−0.0525993 + 0.998616i \(0.516751\pi\)
\(234\) 233.865 0.0653342
\(235\) 5165.20 1.43379
\(236\) −917.838 −0.253162
\(237\) 7848.09 2.15101
\(238\) −3368.45 −0.917413
\(239\) −4096.57 −1.10872 −0.554362 0.832276i \(-0.687038\pi\)
−0.554362 + 0.832276i \(0.687038\pi\)
\(240\) 6575.59 1.76855
\(241\) 6255.68 1.67205 0.836024 0.548693i \(-0.184874\pi\)
0.836024 + 0.548693i \(0.184874\pi\)
\(242\) −4937.44 −1.31153
\(243\) 1811.81 0.478303
\(244\) 1527.82 0.400855
\(245\) −2873.03 −0.749189
\(246\) −3967.47 −1.02828
\(247\) 982.590 0.253120
\(248\) 2467.51 0.631802
\(249\) −966.572 −0.246000
\(250\) −1684.00 −0.426022
\(251\) −404.137 −0.101629 −0.0508146 0.998708i \(-0.516182\pi\)
−0.0508146 + 0.998708i \(0.516182\pi\)
\(252\) 470.836 0.117698
\(253\) −301.687 −0.0749681
\(254\) −5792.95 −1.43103
\(255\) 6365.78 1.56330
\(256\) 5352.13 1.30667
\(257\) 2899.07 0.703655 0.351827 0.936065i \(-0.385560\pi\)
0.351827 + 0.936065i \(0.385560\pi\)
\(258\) 0 0
\(259\) −5310.51 −1.27405
\(260\) 825.712 0.196956
\(261\) 673.998 0.159845
\(262\) 1035.18 0.244097
\(263\) −5855.49 −1.37287 −0.686435 0.727191i \(-0.740825\pi\)
−0.686435 + 0.727191i \(0.740825\pi\)
\(264\) 95.2509 0.0222056
\(265\) −2959.93 −0.686140
\(266\) 4684.21 1.07973
\(267\) −6892.26 −1.57977
\(268\) −253.453 −0.0577690
\(269\) 6624.95 1.50160 0.750800 0.660530i \(-0.229668\pi\)
0.750800 + 0.660530i \(0.229668\pi\)
\(270\) 6520.19 1.46965
\(271\) −3887.82 −0.871470 −0.435735 0.900075i \(-0.643512\pi\)
−0.435735 + 0.900075i \(0.643512\pi\)
\(272\) 5677.85 1.26570
\(273\) −674.412 −0.149514
\(274\) 6589.32 1.45283
\(275\) 193.868 0.0425115
\(276\) 4982.41 1.08662
\(277\) −1269.13 −0.275287 −0.137644 0.990482i \(-0.543953\pi\)
−0.137644 + 0.990482i \(0.543953\pi\)
\(278\) −3806.98 −0.821323
\(279\) −2032.17 −0.436067
\(280\) −1448.09 −0.309072
\(281\) −1292.70 −0.274434 −0.137217 0.990541i \(-0.543816\pi\)
−0.137217 + 0.990541i \(0.543816\pi\)
\(282\) −7520.53 −1.58809
\(283\) −291.953 −0.0613243 −0.0306622 0.999530i \(-0.509762\pi\)
−0.0306622 + 0.999530i \(0.509762\pi\)
\(284\) 3829.58 0.800153
\(285\) −8852.34 −1.83989
\(286\) 72.7960 0.0150508
\(287\) 2245.75 0.461890
\(288\) −1456.88 −0.298081
\(289\) 583.686 0.118804
\(290\) 5634.85 1.14100
\(291\) 2307.17 0.464771
\(292\) −5080.13 −1.01812
\(293\) 4659.82 0.929111 0.464555 0.885544i \(-0.346214\pi\)
0.464555 + 0.885544i \(0.346214\pi\)
\(294\) 4183.13 0.829814
\(295\) −2324.85 −0.458840
\(296\) 3482.61 0.683860
\(297\) 242.762 0.0474291
\(298\) −8278.40 −1.60924
\(299\) −1400.82 −0.270941
\(300\) −3201.76 −0.616178
\(301\) 0 0
\(302\) 2803.73 0.534227
\(303\) −3436.16 −0.651492
\(304\) −7895.69 −1.48963
\(305\) 3869.90 0.726524
\(306\) −1819.28 −0.339873
\(307\) 4104.01 0.762959 0.381479 0.924377i \(-0.375415\pi\)
0.381479 + 0.924377i \(0.375415\pi\)
\(308\) 146.559 0.0271135
\(309\) −3733.63 −0.687375
\(310\) −16989.6 −3.11272
\(311\) 5349.77 0.975426 0.487713 0.873004i \(-0.337831\pi\)
0.487713 + 0.873004i \(0.337831\pi\)
\(312\) 442.277 0.0802531
\(313\) 2013.28 0.363569 0.181784 0.983338i \(-0.441813\pi\)
0.181784 + 0.983338i \(0.441813\pi\)
\(314\) −1108.31 −0.199189
\(315\) 1192.61 0.213320
\(316\) −7919.10 −1.40976
\(317\) −4266.99 −0.756019 −0.378010 0.925802i \(-0.623391\pi\)
−0.378010 + 0.925802i \(0.623391\pi\)
\(318\) 4309.66 0.759980
\(319\) 209.798 0.0368227
\(320\) −3103.99 −0.542245
\(321\) −1246.52 −0.216742
\(322\) −6678.00 −1.15575
\(323\) −7643.77 −1.31675
\(324\) −5050.49 −0.865996
\(325\) 900.182 0.153640
\(326\) −11139.3 −1.89248
\(327\) 8675.13 1.46708
\(328\) −1472.75 −0.247924
\(329\) 4256.92 0.713349
\(330\) −655.833 −0.109401
\(331\) −815.461 −0.135413 −0.0677066 0.997705i \(-0.521568\pi\)
−0.0677066 + 0.997705i \(0.521568\pi\)
\(332\) 975.317 0.161227
\(333\) −2868.17 −0.471997
\(334\) −1117.13 −0.183014
\(335\) −641.985 −0.104703
\(336\) 5419.29 0.879901
\(337\) 8384.84 1.35534 0.677672 0.735364i \(-0.262989\pi\)
0.677672 + 0.735364i \(0.262989\pi\)
\(338\) −7837.80 −1.26130
\(339\) −2532.20 −0.405695
\(340\) −6423.38 −1.02458
\(341\) −632.561 −0.100455
\(342\) 2529.91 0.400006
\(343\) −6555.50 −1.03196
\(344\) 0 0
\(345\) 12620.2 1.96942
\(346\) −5364.85 −0.833572
\(347\) −7627.48 −1.18001 −0.590006 0.807399i \(-0.700875\pi\)
−0.590006 + 0.807399i \(0.700875\pi\)
\(348\) −3464.85 −0.533722
\(349\) 1526.25 0.234093 0.117046 0.993126i \(-0.462657\pi\)
0.117046 + 0.993126i \(0.462657\pi\)
\(350\) 4291.36 0.655379
\(351\) 1127.21 0.171413
\(352\) −453.488 −0.0686676
\(353\) 11239.7 1.69471 0.847353 0.531031i \(-0.178195\pi\)
0.847353 + 0.531031i \(0.178195\pi\)
\(354\) 3384.97 0.508218
\(355\) 9700.15 1.45023
\(356\) 6954.62 1.03538
\(357\) 5246.38 0.777782
\(358\) −16747.2 −2.47240
\(359\) 3923.43 0.576799 0.288400 0.957510i \(-0.406877\pi\)
0.288400 + 0.957510i \(0.406877\pi\)
\(360\) −782.107 −0.114502
\(361\) 3770.52 0.549719
\(362\) 11834.2 1.71821
\(363\) 7690.10 1.11192
\(364\) 680.514 0.0979907
\(365\) −12867.8 −1.84529
\(366\) −5634.57 −0.804710
\(367\) 1953.59 0.277866 0.138933 0.990302i \(-0.455633\pi\)
0.138933 + 0.990302i \(0.455633\pi\)
\(368\) 11256.4 1.59451
\(369\) 1212.91 0.171116
\(370\) −23978.9 −3.36920
\(371\) −2439.44 −0.341373
\(372\) 10446.8 1.45603
\(373\) 2889.05 0.401043 0.200522 0.979689i \(-0.435736\pi\)
0.200522 + 0.979689i \(0.435736\pi\)
\(374\) −566.294 −0.0782952
\(375\) 2622.84 0.361181
\(376\) −2791.67 −0.382898
\(377\) 974.151 0.133080
\(378\) 5373.64 0.731191
\(379\) 2202.12 0.298457 0.149229 0.988803i \(-0.452321\pi\)
0.149229 + 0.988803i \(0.452321\pi\)
\(380\) 8932.44 1.20585
\(381\) 9022.55 1.21323
\(382\) −13599.5 −1.82149
\(383\) −13559.5 −1.80903 −0.904515 0.426442i \(-0.859767\pi\)
−0.904515 + 0.426442i \(0.859767\pi\)
\(384\) −5725.21 −0.760842
\(385\) 371.228 0.0491416
\(386\) 8406.56 1.10850
\(387\) 0 0
\(388\) −2328.04 −0.304609
\(389\) −6665.93 −0.868833 −0.434417 0.900712i \(-0.643045\pi\)
−0.434417 + 0.900712i \(0.643045\pi\)
\(390\) −3045.21 −0.395386
\(391\) 10897.2 1.40946
\(392\) 1552.81 0.200073
\(393\) −1612.29 −0.206945
\(394\) 1302.86 0.166591
\(395\) −20058.7 −2.55510
\(396\) 79.1556 0.0100447
\(397\) 2959.76 0.374172 0.187086 0.982344i \(-0.440096\pi\)
0.187086 + 0.982344i \(0.440096\pi\)
\(398\) −13335.6 −1.67953
\(399\) −7295.69 −0.915392
\(400\) −7233.50 −0.904187
\(401\) 516.898 0.0643707 0.0321854 0.999482i \(-0.489753\pi\)
0.0321854 + 0.999482i \(0.489753\pi\)
\(402\) 934.730 0.115970
\(403\) −2937.15 −0.363052
\(404\) 3467.25 0.426985
\(405\) −12792.7 −1.56956
\(406\) 4643.98 0.567677
\(407\) −892.788 −0.108732
\(408\) −3440.56 −0.417483
\(409\) −10919.9 −1.32018 −0.660091 0.751186i \(-0.729482\pi\)
−0.660091 + 0.751186i \(0.729482\pi\)
\(410\) 10140.4 1.22146
\(411\) −10262.9 −1.23171
\(412\) 3767.41 0.450503
\(413\) −1916.03 −0.228285
\(414\) −3606.74 −0.428169
\(415\) 2470.44 0.292214
\(416\) −2105.67 −0.248171
\(417\) 5929.40 0.696317
\(418\) 787.497 0.0921477
\(419\) 9649.13 1.12504 0.562519 0.826784i \(-0.309832\pi\)
0.562519 + 0.826784i \(0.309832\pi\)
\(420\) −6130.88 −0.712277
\(421\) −14758.4 −1.70850 −0.854250 0.519862i \(-0.825983\pi\)
−0.854250 + 0.519862i \(0.825983\pi\)
\(422\) −2431.45 −0.280477
\(423\) 2299.14 0.264274
\(424\) 1599.77 0.183236
\(425\) −7002.70 −0.799249
\(426\) −14123.4 −1.60630
\(427\) 3189.39 0.361465
\(428\) 1257.80 0.142052
\(429\) −113.380 −0.0127600
\(430\) 0 0
\(431\) 11145.4 1.24560 0.622799 0.782382i \(-0.285995\pi\)
0.622799 + 0.782382i \(0.285995\pi\)
\(432\) −9057.79 −1.00878
\(433\) 14713.2 1.63296 0.816479 0.577375i \(-0.195923\pi\)
0.816479 + 0.577375i \(0.195923\pi\)
\(434\) −14002.0 −1.54866
\(435\) −8776.32 −0.967338
\(436\) −8753.62 −0.961519
\(437\) −15153.9 −1.65883
\(438\) 18735.5 2.04387
\(439\) −7024.07 −0.763646 −0.381823 0.924235i \(-0.624704\pi\)
−0.381823 + 0.924235i \(0.624704\pi\)
\(440\) −243.450 −0.0263773
\(441\) −1278.85 −0.138089
\(442\) −2629.46 −0.282966
\(443\) 6017.83 0.645408 0.322704 0.946500i \(-0.395408\pi\)
0.322704 + 0.946500i \(0.395408\pi\)
\(444\) 14744.5 1.57600
\(445\) 17615.8 1.87656
\(446\) −1131.51 −0.120131
\(447\) 12893.7 1.36432
\(448\) −2558.17 −0.269781
\(449\) 11364.2 1.19446 0.597229 0.802071i \(-0.296268\pi\)
0.597229 + 0.802071i \(0.296268\pi\)
\(450\) 2317.74 0.242798
\(451\) 377.549 0.0394192
\(452\) 2555.12 0.265890
\(453\) −4366.83 −0.452917
\(454\) 4878.12 0.504276
\(455\) 1723.71 0.177602
\(456\) 4784.49 0.491347
\(457\) 9100.15 0.931482 0.465741 0.884921i \(-0.345788\pi\)
0.465741 + 0.884921i \(0.345788\pi\)
\(458\) −17827.4 −1.81882
\(459\) −8768.78 −0.891703
\(460\) −12734.4 −1.29075
\(461\) −5401.64 −0.545725 −0.272863 0.962053i \(-0.587970\pi\)
−0.272863 + 0.962053i \(0.587970\pi\)
\(462\) −540.507 −0.0544300
\(463\) 11462.7 1.15057 0.575286 0.817953i \(-0.304891\pi\)
0.575286 + 0.817953i \(0.304891\pi\)
\(464\) −7827.88 −0.783190
\(465\) 26461.4 2.63896
\(466\) −1392.34 −0.138410
\(467\) 5843.46 0.579022 0.289511 0.957175i \(-0.406507\pi\)
0.289511 + 0.957175i \(0.406507\pi\)
\(468\) 367.541 0.0363026
\(469\) −529.094 −0.0520924
\(470\) 19221.6 1.88643
\(471\) 1726.20 0.168873
\(472\) 1256.53 0.122534
\(473\) 0 0
\(474\) 29205.5 2.83007
\(475\) 9738.05 0.940658
\(476\) −5293.85 −0.509755
\(477\) −1317.53 −0.126468
\(478\) −15244.8 −1.45874
\(479\) −7333.33 −0.699516 −0.349758 0.936840i \(-0.613736\pi\)
−0.349758 + 0.936840i \(0.613736\pi\)
\(480\) 18970.4 1.80391
\(481\) −4145.46 −0.392966
\(482\) 23279.6 2.19991
\(483\) 10401.0 0.979840
\(484\) −7759.68 −0.728745
\(485\) −5896.83 −0.552085
\(486\) 6742.38 0.629301
\(487\) −9898.48 −0.921032 −0.460516 0.887651i \(-0.652336\pi\)
−0.460516 + 0.887651i \(0.652336\pi\)
\(488\) −2091.59 −0.194020
\(489\) 17349.5 1.60444
\(490\) −10691.6 −0.985706
\(491\) −1867.36 −0.171635 −0.0858176 0.996311i \(-0.527350\pi\)
−0.0858176 + 0.996311i \(0.527350\pi\)
\(492\) −6235.27 −0.571357
\(493\) −7578.12 −0.692295
\(494\) 3656.57 0.333030
\(495\) 200.498 0.0182055
\(496\) 23601.8 2.13659
\(497\) 7994.42 0.721526
\(498\) −3596.95 −0.323661
\(499\) 8422.05 0.755556 0.377778 0.925896i \(-0.376688\pi\)
0.377778 + 0.925896i \(0.376688\pi\)
\(500\) −2646.57 −0.236717
\(501\) 1739.94 0.155160
\(502\) −1503.94 −0.133713
\(503\) 13588.9 1.20457 0.602286 0.798280i \(-0.294257\pi\)
0.602286 + 0.798280i \(0.294257\pi\)
\(504\) −644.576 −0.0569677
\(505\) 8782.39 0.773884
\(506\) −1122.69 −0.0986353
\(507\) 12207.4 1.06933
\(508\) −9104.18 −0.795143
\(509\) −1287.12 −0.112084 −0.0560420 0.998428i \(-0.517848\pi\)
−0.0560420 + 0.998428i \(0.517848\pi\)
\(510\) 23689.3 2.05683
\(511\) −10605.0 −0.918079
\(512\) 12014.9 1.03709
\(513\) 12194.0 1.04947
\(514\) 10788.5 0.925797
\(515\) 9542.70 0.816508
\(516\) 0 0
\(517\) 715.662 0.0608796
\(518\) −19762.3 −1.67626
\(519\) 8355.78 0.706701
\(520\) −1130.40 −0.0953298
\(521\) 18347.4 1.54283 0.771415 0.636333i \(-0.219549\pi\)
0.771415 + 0.636333i \(0.219549\pi\)
\(522\) 2508.19 0.210307
\(523\) 10556.8 0.882635 0.441318 0.897351i \(-0.354511\pi\)
0.441318 + 0.897351i \(0.354511\pi\)
\(524\) 1626.88 0.135631
\(525\) −6683.82 −0.555630
\(526\) −21790.3 −1.80628
\(527\) 22848.7 1.88863
\(528\) 911.076 0.0750938
\(529\) 9436.94 0.775617
\(530\) −11015.0 −0.902753
\(531\) −1034.84 −0.0845726
\(532\) 7361.70 0.599944
\(533\) 1753.06 0.142465
\(534\) −25648.5 −2.07850
\(535\) 3185.96 0.257460
\(536\) 346.978 0.0279611
\(537\) 26083.9 2.09609
\(538\) 24653.8 1.97565
\(539\) −398.071 −0.0318110
\(540\) 10247.1 0.816604
\(541\) −18649.2 −1.48205 −0.741027 0.671475i \(-0.765661\pi\)
−0.741027 + 0.671475i \(0.765661\pi\)
\(542\) −14468.0 −1.14659
\(543\) −18431.9 −1.45670
\(544\) 16380.4 1.29100
\(545\) −22172.6 −1.74269
\(546\) −2509.72 −0.196715
\(547\) 12594.9 0.984498 0.492249 0.870454i \(-0.336175\pi\)
0.492249 + 0.870454i \(0.336175\pi\)
\(548\) 10355.8 0.807256
\(549\) 1722.57 0.133912
\(550\) 721.451 0.0559323
\(551\) 10538.2 0.814780
\(552\) −6820.95 −0.525940
\(553\) −16531.5 −1.27123
\(554\) −4722.87 −0.362194
\(555\) 37347.3 2.85640
\(556\) −5983.05 −0.456363
\(557\) −2619.40 −0.199259 −0.0996297 0.995025i \(-0.531766\pi\)
−0.0996297 + 0.995025i \(0.531766\pi\)
\(558\) −7562.41 −0.573732
\(559\) 0 0
\(560\) −13851.0 −1.04520
\(561\) 882.007 0.0663786
\(562\) −4810.58 −0.361071
\(563\) 3061.94 0.229211 0.114605 0.993411i \(-0.463440\pi\)
0.114605 + 0.993411i \(0.463440\pi\)
\(564\) −11819.3 −0.882412
\(565\) 6472.00 0.481910
\(566\) −1086.46 −0.0806842
\(567\) −10543.1 −0.780900
\(568\) −5242.71 −0.387287
\(569\) −7153.09 −0.527017 −0.263509 0.964657i \(-0.584880\pi\)
−0.263509 + 0.964657i \(0.584880\pi\)
\(570\) −32942.7 −2.42073
\(571\) −439.404 −0.0322040 −0.0161020 0.999870i \(-0.505126\pi\)
−0.0161020 + 0.999870i \(0.505126\pi\)
\(572\) 114.406 0.00836286
\(573\) 21181.3 1.54426
\(574\) 8357.22 0.607707
\(575\) −13882.9 −1.00688
\(576\) −1381.65 −0.0999457
\(577\) 12722.8 0.917951 0.458975 0.888449i \(-0.348217\pi\)
0.458975 + 0.888449i \(0.348217\pi\)
\(578\) 2172.10 0.156311
\(579\) −13093.3 −0.939789
\(580\) 8855.72 0.633989
\(581\) 2036.02 0.145384
\(582\) 8585.78 0.611498
\(583\) −410.112 −0.0291339
\(584\) 6954.73 0.492789
\(585\) 930.967 0.0657961
\(586\) 17340.8 1.22243
\(587\) −1310.20 −0.0921256 −0.0460628 0.998939i \(-0.514667\pi\)
−0.0460628 + 0.998939i \(0.514667\pi\)
\(588\) 6574.21 0.461081
\(589\) −31773.7 −2.22277
\(590\) −8651.57 −0.603694
\(591\) −2029.21 −0.141236
\(592\) 33311.2 2.31264
\(593\) −8523.59 −0.590256 −0.295128 0.955458i \(-0.595362\pi\)
−0.295128 + 0.955458i \(0.595362\pi\)
\(594\) 903.401 0.0624024
\(595\) −13409.1 −0.923899
\(596\) −13010.3 −0.894167
\(597\) 20770.2 1.42390
\(598\) −5212.94 −0.356477
\(599\) −21107.4 −1.43977 −0.719887 0.694091i \(-0.755806\pi\)
−0.719887 + 0.694091i \(0.755806\pi\)
\(600\) 4383.22 0.298240
\(601\) 27867.5 1.89141 0.945705 0.325026i \(-0.105373\pi\)
0.945705 + 0.325026i \(0.105373\pi\)
\(602\) 0 0
\(603\) −285.761 −0.0192986
\(604\) 4406.34 0.296840
\(605\) −19654.9 −1.32081
\(606\) −12787.2 −0.857166
\(607\) 21396.3 1.43072 0.715361 0.698755i \(-0.246262\pi\)
0.715361 + 0.698755i \(0.246262\pi\)
\(608\) −22778.8 −1.51942
\(609\) −7233.03 −0.481276
\(610\) 14401.3 0.955886
\(611\) 3323.02 0.220024
\(612\) −2859.18 −0.188849
\(613\) 21983.4 1.44845 0.724226 0.689563i \(-0.242197\pi\)
0.724226 + 0.689563i \(0.242197\pi\)
\(614\) 15272.5 1.00382
\(615\) −15793.7 −1.03555
\(616\) −200.640 −0.0131234
\(617\) 11090.6 0.723647 0.361824 0.932247i \(-0.382154\pi\)
0.361824 + 0.932247i \(0.382154\pi\)
\(618\) −13894.2 −0.904377
\(619\) −14015.0 −0.910030 −0.455015 0.890484i \(-0.650366\pi\)
−0.455015 + 0.890484i \(0.650366\pi\)
\(620\) −26700.8 −1.72957
\(621\) −17384.2 −1.12336
\(622\) 19908.4 1.28337
\(623\) 14518.1 0.933636
\(624\) 4230.38 0.271395
\(625\) −18510.3 −1.18466
\(626\) 7492.11 0.478346
\(627\) −1226.53 −0.0781227
\(628\) −1741.82 −0.110678
\(629\) 32248.4 2.04424
\(630\) 4438.11 0.280665
\(631\) 20579.0 1.29832 0.649158 0.760654i \(-0.275122\pi\)
0.649158 + 0.760654i \(0.275122\pi\)
\(632\) 10841.3 0.682347
\(633\) 3787.00 0.237788
\(634\) −15879.0 −0.994693
\(635\) −23060.5 −1.44115
\(636\) 6773.05 0.422278
\(637\) −1848.36 −0.114968
\(638\) 780.733 0.0484475
\(639\) 4317.74 0.267304
\(640\) 14632.9 0.903777
\(641\) 21587.1 1.33017 0.665087 0.746766i \(-0.268395\pi\)
0.665087 + 0.746766i \(0.268395\pi\)
\(642\) −4638.75 −0.285166
\(643\) 6576.06 0.403320 0.201660 0.979456i \(-0.435366\pi\)
0.201660 + 0.979456i \(0.435366\pi\)
\(644\) −10495.1 −0.642183
\(645\) 0 0
\(646\) −28445.2 −1.73245
\(647\) −10311.6 −0.626571 −0.313285 0.949659i \(-0.601430\pi\)
−0.313285 + 0.949659i \(0.601430\pi\)
\(648\) 6914.15 0.419156
\(649\) −322.118 −0.0194826
\(650\) 3349.90 0.202144
\(651\) 21808.3 1.31295
\(652\) −17506.5 −1.05154
\(653\) −10297.6 −0.617116 −0.308558 0.951206i \(-0.599846\pi\)
−0.308558 + 0.951206i \(0.599846\pi\)
\(654\) 32283.2 1.93024
\(655\) 4120.82 0.245823
\(656\) −14086.9 −0.838416
\(657\) −5727.70 −0.340120
\(658\) 15841.5 0.938551
\(659\) 12986.6 0.767658 0.383829 0.923404i \(-0.374605\pi\)
0.383829 + 0.923404i \(0.374605\pi\)
\(660\) −1030.71 −0.0607881
\(661\) 6720.12 0.395435 0.197717 0.980259i \(-0.436647\pi\)
0.197717 + 0.980259i \(0.436647\pi\)
\(662\) −3034.62 −0.178163
\(663\) 4095.40 0.239898
\(664\) −1335.21 −0.0780367
\(665\) 18646.9 1.08736
\(666\) −10673.5 −0.621005
\(667\) −15023.7 −0.872145
\(668\) −1755.69 −0.101691
\(669\) 1762.33 0.101847
\(670\) −2389.05 −0.137757
\(671\) 536.192 0.0308487
\(672\) 15634.5 0.897492
\(673\) 5865.01 0.335928 0.167964 0.985793i \(-0.446281\pi\)
0.167964 + 0.985793i \(0.446281\pi\)
\(674\) 31202.9 1.78322
\(675\) 11171.3 0.637013
\(676\) −12317.9 −0.700835
\(677\) −22538.1 −1.27948 −0.639740 0.768591i \(-0.720958\pi\)
−0.639740 + 0.768591i \(0.720958\pi\)
\(678\) −9423.23 −0.533771
\(679\) −4859.89 −0.274677
\(680\) 8793.64 0.495913
\(681\) −7597.70 −0.427525
\(682\) −2353.98 −0.132168
\(683\) 15142.2 0.848318 0.424159 0.905588i \(-0.360570\pi\)
0.424159 + 0.905588i \(0.360570\pi\)
\(684\) 3976.01 0.222261
\(685\) 26230.7 1.46310
\(686\) −24395.3 −1.35775
\(687\) 27766.3 1.54199
\(688\) 0 0
\(689\) −1904.26 −0.105293
\(690\) 46964.4 2.59117
\(691\) 9967.42 0.548739 0.274369 0.961624i \(-0.411531\pi\)
0.274369 + 0.961624i \(0.411531\pi\)
\(692\) −8431.38 −0.463169
\(693\) 165.241 0.00905770
\(694\) −28384.5 −1.55254
\(695\) −15154.8 −0.827129
\(696\) 4743.39 0.258330
\(697\) −13637.4 −0.741111
\(698\) 5679.72 0.307995
\(699\) 2168.58 0.117343
\(700\) 6744.29 0.364158
\(701\) 14945.7 0.805267 0.402634 0.915361i \(-0.368095\pi\)
0.402634 + 0.915361i \(0.368095\pi\)
\(702\) 4194.74 0.225528
\(703\) −44845.0 −2.40592
\(704\) −430.071 −0.0230240
\(705\) −29937.7 −1.59932
\(706\) 41827.0 2.22972
\(707\) 7238.04 0.385027
\(708\) 5319.82 0.282389
\(709\) −16453.5 −0.871541 −0.435770 0.900058i \(-0.643524\pi\)
−0.435770 + 0.900058i \(0.643524\pi\)
\(710\) 36097.7 1.90806
\(711\) −8928.55 −0.470952
\(712\) −9520.91 −0.501139
\(713\) 45297.9 2.37927
\(714\) 19523.6 1.02333
\(715\) 289.786 0.0151572
\(716\) −26319.9 −1.37377
\(717\) 23743.8 1.23672
\(718\) 14600.5 0.758893
\(719\) 17776.5 0.922048 0.461024 0.887388i \(-0.347482\pi\)
0.461024 + 0.887388i \(0.347482\pi\)
\(720\) −7480.86 −0.387216
\(721\) 7864.65 0.406234
\(722\) 14031.5 0.723264
\(723\) −36258.1 −1.86508
\(724\) 18598.7 0.954716
\(725\) 9654.41 0.494560
\(726\) 28617.6 1.46295
\(727\) 782.314 0.0399098 0.0199549 0.999801i \(-0.493648\pi\)
0.0199549 + 0.999801i \(0.493648\pi\)
\(728\) −931.627 −0.0474291
\(729\) 12814.7 0.651056
\(730\) −47885.5 −2.42784
\(731\) 0 0
\(732\) −8855.29 −0.447132
\(733\) 38475.2 1.93876 0.969381 0.245560i \(-0.0789720\pi\)
0.969381 + 0.245560i \(0.0789720\pi\)
\(734\) 7270.02 0.365588
\(735\) 16652.2 0.835681
\(736\) 32474.4 1.62639
\(737\) −88.9499 −0.00444574
\(738\) 4513.68 0.225137
\(739\) −12300.1 −0.612266 −0.306133 0.951989i \(-0.599035\pi\)
−0.306133 + 0.951989i \(0.599035\pi\)
\(740\) −37685.2 −1.87207
\(741\) −5695.12 −0.282342
\(742\) −9078.02 −0.449143
\(743\) −38280.9 −1.89016 −0.945080 0.326840i \(-0.894016\pi\)
−0.945080 + 0.326840i \(0.894016\pi\)
\(744\) −14301.8 −0.704742
\(745\) −32954.6 −1.62062
\(746\) 10751.2 0.527652
\(747\) 1099.64 0.0538605
\(748\) −889.987 −0.0435042
\(749\) 2625.72 0.128093
\(750\) 9760.53 0.475205
\(751\) −12564.6 −0.610506 −0.305253 0.952271i \(-0.598741\pi\)
−0.305253 + 0.952271i \(0.598741\pi\)
\(752\) −26702.4 −1.29486
\(753\) 2342.39 0.113362
\(754\) 3625.16 0.175094
\(755\) 11161.1 0.538004
\(756\) 8445.21 0.406282
\(757\) 12863.3 0.617603 0.308801 0.951127i \(-0.400072\pi\)
0.308801 + 0.951127i \(0.400072\pi\)
\(758\) 8194.86 0.392679
\(759\) 1748.59 0.0836229
\(760\) −12228.6 −0.583653
\(761\) 3597.42 0.171362 0.0856810 0.996323i \(-0.472693\pi\)
0.0856810 + 0.996323i \(0.472693\pi\)
\(762\) 33576.1 1.59624
\(763\) −18273.6 −0.867036
\(764\) −21372.9 −1.01210
\(765\) −7242.18 −0.342276
\(766\) −50459.7 −2.38014
\(767\) −1495.68 −0.0704119
\(768\) −31021.1 −1.45752
\(769\) 18986.1 0.890318 0.445159 0.895452i \(-0.353147\pi\)
0.445159 + 0.895452i \(0.353147\pi\)
\(770\) 1381.47 0.0646555
\(771\) −16803.1 −0.784890
\(772\) 13211.7 0.615934
\(773\) −23805.7 −1.10767 −0.553836 0.832626i \(-0.686836\pi\)
−0.553836 + 0.832626i \(0.686836\pi\)
\(774\) 0 0
\(775\) −29108.9 −1.34919
\(776\) 3187.10 0.147436
\(777\) 30779.9 1.42114
\(778\) −24806.3 −1.14312
\(779\) 18964.4 0.872234
\(780\) −4785.85 −0.219694
\(781\) 1344.00 0.0615775
\(782\) 40552.5 1.85442
\(783\) 12089.3 0.551769
\(784\) 14852.6 0.676596
\(785\) −4411.95 −0.200598
\(786\) −5999.91 −0.272277
\(787\) 5921.07 0.268187 0.134094 0.990969i \(-0.457188\pi\)
0.134094 + 0.990969i \(0.457188\pi\)
\(788\) 2047.57 0.0925655
\(789\) 33938.6 1.53136
\(790\) −74645.7 −3.36174
\(791\) 5333.92 0.239763
\(792\) −108.364 −0.00486182
\(793\) 2489.69 0.111490
\(794\) 11014.3 0.492297
\(795\) 17155.9 0.765353
\(796\) −20958.2 −0.933219
\(797\) −31492.1 −1.39963 −0.699817 0.714322i \(-0.746735\pi\)
−0.699817 + 0.714322i \(0.746735\pi\)
\(798\) −27149.9 −1.20438
\(799\) −25850.4 −1.14458
\(800\) −20868.4 −0.922263
\(801\) 7841.13 0.345884
\(802\) 1923.56 0.0846924
\(803\) −1782.89 −0.0783520
\(804\) 1469.02 0.0644382
\(805\) −26583.7 −1.16392
\(806\) −10930.2 −0.477667
\(807\) −38398.4 −1.67495
\(808\) −4746.68 −0.206668
\(809\) −8959.91 −0.389386 −0.194693 0.980864i \(-0.562371\pi\)
−0.194693 + 0.980864i \(0.562371\pi\)
\(810\) −47606.1 −2.06507
\(811\) −15879.3 −0.687543 −0.343772 0.939053i \(-0.611705\pi\)
−0.343772 + 0.939053i \(0.611705\pi\)
\(812\) 7298.48 0.315427
\(813\) 22533.9 0.972079
\(814\) −3322.38 −0.143058
\(815\) −44343.1 −1.90586
\(816\) −32909.0 −1.41182
\(817\) 0 0
\(818\) −40636.8 −1.73696
\(819\) 767.260 0.0327353
\(820\) 15936.6 0.678695
\(821\) 1143.25 0.0485990 0.0242995 0.999705i \(-0.492264\pi\)
0.0242995 + 0.999705i \(0.492264\pi\)
\(822\) −38191.9 −1.62055
\(823\) 6089.64 0.257924 0.128962 0.991650i \(-0.458835\pi\)
0.128962 + 0.991650i \(0.458835\pi\)
\(824\) −5157.61 −0.218051
\(825\) −1123.66 −0.0474194
\(826\) −7130.23 −0.300354
\(827\) −347.605 −0.0146160 −0.00730798 0.999973i \(-0.502326\pi\)
−0.00730798 + 0.999973i \(0.502326\pi\)
\(828\) −5668.35 −0.237909
\(829\) 10417.2 0.436437 0.218218 0.975900i \(-0.429976\pi\)
0.218218 + 0.975900i \(0.429976\pi\)
\(830\) 9193.37 0.384466
\(831\) 7355.91 0.307068
\(832\) −1996.94 −0.0832109
\(833\) 14378.7 0.598071
\(834\) 22065.4 0.916142
\(835\) −4447.08 −0.184308
\(836\) 1237.63 0.0512013
\(837\) −36450.2 −1.50526
\(838\) 35907.8 1.48021
\(839\) −27147.7 −1.11710 −0.558548 0.829472i \(-0.688641\pi\)
−0.558548 + 0.829472i \(0.688641\pi\)
\(840\) 8393.20 0.344754
\(841\) −13941.3 −0.571621
\(842\) −54921.1 −2.24787
\(843\) 7492.51 0.306116
\(844\) −3821.26 −0.155845
\(845\) −31200.7 −1.27022
\(846\) 8555.90 0.347705
\(847\) −16198.7 −0.657136
\(848\) 15301.9 0.619656
\(849\) 1692.17 0.0684040
\(850\) −26059.5 −1.05157
\(851\) 63932.8 2.57531
\(852\) −22196.3 −0.892528
\(853\) −28796.5 −1.15589 −0.577945 0.816076i \(-0.696145\pi\)
−0.577945 + 0.816076i \(0.696145\pi\)
\(854\) 11868.9 0.475579
\(855\) 10071.1 0.402834
\(856\) −1721.94 −0.0687553
\(857\) 183.644 0.00731990 0.00365995 0.999993i \(-0.498835\pi\)
0.00365995 + 0.999993i \(0.498835\pi\)
\(858\) −421.928 −0.0167883
\(859\) −29667.8 −1.17841 −0.589203 0.807985i \(-0.700558\pi\)
−0.589203 + 0.807985i \(0.700558\pi\)
\(860\) 0 0
\(861\) −13016.4 −0.515213
\(862\) 41475.8 1.63883
\(863\) 37230.7 1.46854 0.734268 0.678860i \(-0.237526\pi\)
0.734268 + 0.678860i \(0.237526\pi\)
\(864\) −26131.5 −1.02895
\(865\) −21356.3 −0.839465
\(866\) 54753.0 2.14848
\(867\) −3383.06 −0.132520
\(868\) −22005.6 −0.860504
\(869\) −2779.23 −0.108491
\(870\) −32659.8 −1.27272
\(871\) −413.019 −0.0160673
\(872\) 11983.8 0.465391
\(873\) −2624.80 −0.101759
\(874\) −56392.9 −2.18251
\(875\) −5524.85 −0.213456
\(876\) 29444.6 1.13566
\(877\) −9915.97 −0.381800 −0.190900 0.981610i \(-0.561141\pi\)
−0.190900 + 0.981610i \(0.561141\pi\)
\(878\) −26139.1 −1.00473
\(879\) −27008.4 −1.03637
\(880\) −2328.60 −0.0892012
\(881\) 31781.6 1.21538 0.607691 0.794174i \(-0.292096\pi\)
0.607691 + 0.794174i \(0.292096\pi\)
\(882\) −4759.04 −0.181684
\(883\) −30414.3 −1.15914 −0.579572 0.814921i \(-0.696780\pi\)
−0.579572 + 0.814921i \(0.696780\pi\)
\(884\) −4132.46 −0.157228
\(885\) 13474.9 0.511812
\(886\) 22394.5 0.849161
\(887\) −32756.2 −1.23996 −0.619981 0.784617i \(-0.712860\pi\)
−0.619981 + 0.784617i \(0.712860\pi\)
\(888\) −20185.3 −0.762810
\(889\) −19005.4 −0.717009
\(890\) 65554.5 2.46898
\(891\) −1772.48 −0.0666447
\(892\) −1778.28 −0.0667501
\(893\) 35947.9 1.34709
\(894\) 47981.9 1.79503
\(895\) −66667.2 −2.48988
\(896\) 12059.8 0.449653
\(897\) 8119.19 0.302221
\(898\) 42290.4 1.57155
\(899\) −31500.8 −1.16865
\(900\) 3642.55 0.134909
\(901\) 14813.6 0.547740
\(902\) 1404.99 0.0518638
\(903\) 0 0
\(904\) −3497.97 −0.128695
\(905\) 47109.7 1.73036
\(906\) −16250.5 −0.595902
\(907\) 11089.6 0.405981 0.202991 0.979181i \(-0.434934\pi\)
0.202991 + 0.979181i \(0.434934\pi\)
\(908\) 7666.45 0.280198
\(909\) 3909.22 0.142641
\(910\) 6414.55 0.233670
\(911\) −27247.3 −0.990938 −0.495469 0.868626i \(-0.665004\pi\)
−0.495469 + 0.868626i \(0.665004\pi\)
\(912\) 45763.7 1.66161
\(913\) 342.290 0.0124076
\(914\) 33864.9 1.22555
\(915\) −22430.1 −0.810399
\(916\) −28017.5 −1.01062
\(917\) 3396.19 0.122303
\(918\) −32631.8 −1.17321
\(919\) −3098.95 −0.111235 −0.0556174 0.998452i \(-0.517713\pi\)
−0.0556174 + 0.998452i \(0.517713\pi\)
\(920\) 17433.5 0.624745
\(921\) −23787.0 −0.851040
\(922\) −20101.4 −0.718009
\(923\) 6240.56 0.222547
\(924\) −849.460 −0.0302437
\(925\) −41084.0 −1.46036
\(926\) 42656.6 1.51380
\(927\) 4247.65 0.150497
\(928\) −22583.2 −0.798847
\(929\) −15691.2 −0.554158 −0.277079 0.960847i \(-0.589366\pi\)
−0.277079 + 0.960847i \(0.589366\pi\)
\(930\) 98472.2 3.47208
\(931\) −19995.3 −0.703886
\(932\) −2188.20 −0.0769064
\(933\) −31007.4 −1.08804
\(934\) 21745.6 0.761817
\(935\) −2254.30 −0.0788487
\(936\) −503.166 −0.0175710
\(937\) 30261.1 1.05505 0.527527 0.849538i \(-0.323119\pi\)
0.527527 + 0.849538i \(0.323119\pi\)
\(938\) −1968.95 −0.0685378
\(939\) −11669.0 −0.405542
\(940\) 30208.6 1.04819
\(941\) −51902.6 −1.79806 −0.899032 0.437884i \(-0.855728\pi\)
−0.899032 + 0.437884i \(0.855728\pi\)
\(942\) 6423.79 0.222185
\(943\) −27036.4 −0.933644
\(944\) 12018.7 0.414380
\(945\) 21391.4 0.736361
\(946\) 0 0
\(947\) −18308.0 −0.628225 −0.314112 0.949386i \(-0.601707\pi\)
−0.314112 + 0.949386i \(0.601707\pi\)
\(948\) 45899.3 1.57251
\(949\) −8278.43 −0.283171
\(950\) 36238.7 1.23762
\(951\) 24731.6 0.843299
\(952\) 7247.31 0.246730
\(953\) 1809.03 0.0614904 0.0307452 0.999527i \(-0.490212\pi\)
0.0307452 + 0.999527i \(0.490212\pi\)
\(954\) −4902.98 −0.166394
\(955\) −54136.7 −1.83437
\(956\) −23958.7 −0.810543
\(957\) −1216.00 −0.0410738
\(958\) −27289.9 −0.920351
\(959\) 21618.1 0.727931
\(960\) 17990.8 0.604845
\(961\) 65186.9 2.18814
\(962\) −15426.7 −0.517025
\(963\) 1418.13 0.0474546
\(964\) 36586.2 1.22237
\(965\) 33464.8 1.11634
\(966\) 38705.9 1.28917
\(967\) −9570.15 −0.318258 −0.159129 0.987258i \(-0.550869\pi\)
−0.159129 + 0.987258i \(0.550869\pi\)
\(968\) 10623.0 0.352725
\(969\) 44303.5 1.46877
\(970\) −21944.2 −0.726377
\(971\) 24689.1 0.815973 0.407987 0.912988i \(-0.366231\pi\)
0.407987 + 0.912988i \(0.366231\pi\)
\(972\) 10596.3 0.349668
\(973\) −12489.9 −0.411519
\(974\) −36835.7 −1.21180
\(975\) −5217.49 −0.171378
\(976\) −20006.1 −0.656127
\(977\) −57659.0 −1.88810 −0.944051 0.329801i \(-0.893018\pi\)
−0.944051 + 0.329801i \(0.893018\pi\)
\(978\) 64563.6 2.11096
\(979\) 2440.74 0.0796797
\(980\) −16802.9 −0.547702
\(981\) −9869.46 −0.321210
\(982\) −6949.11 −0.225820
\(983\) −22540.1 −0.731351 −0.365676 0.930742i \(-0.619162\pi\)
−0.365676 + 0.930742i \(0.619162\pi\)
\(984\) 8536.12 0.276546
\(985\) 5186.40 0.167769
\(986\) −28200.9 −0.910850
\(987\) −24673.3 −0.795703
\(988\) 5746.65 0.185046
\(989\) 0 0
\(990\) 746.123 0.0239529
\(991\) −19732.0 −0.632499 −0.316249 0.948676i \(-0.602424\pi\)
−0.316249 + 0.948676i \(0.602424\pi\)
\(992\) 68090.4 2.17931
\(993\) 4726.44 0.151046
\(994\) 29750.1 0.949310
\(995\) −53086.1 −1.69140
\(996\) −5652.97 −0.179841
\(997\) 34498.1 1.09585 0.547927 0.836526i \(-0.315417\pi\)
0.547927 + 0.836526i \(0.315417\pi\)
\(998\) 31341.4 0.994084
\(999\) −51445.4 −1.62929
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.g.1.25 30
43.2 odd 14 43.4.e.a.4.2 60
43.22 odd 14 43.4.e.a.11.2 yes 60
43.42 odd 2 1849.4.a.h.1.6 30
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.4.e.a.4.2 60 43.2 odd 14
43.4.e.a.11.2 yes 60 43.22 odd 14
1849.4.a.g.1.25 30 1.1 even 1 trivial
1849.4.a.h.1.6 30 43.42 odd 2