Properties

Label 1849.4.a.j.1.44
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 1.44
Character \(\chi\) \(=\) 1849.1

$q$-expansion

\(f(q)\) \(=\) \(q+4.78088 q^{2} -1.15757 q^{3} +14.8568 q^{4} +21.3015 q^{5} -5.53419 q^{6} -24.0536 q^{7} +32.7814 q^{8} -25.6600 q^{9} +O(q^{10})\) \(q+4.78088 q^{2} -1.15757 q^{3} +14.8568 q^{4} +21.3015 q^{5} -5.53419 q^{6} -24.0536 q^{7} +32.7814 q^{8} -25.6600 q^{9} +101.840 q^{10} +17.5943 q^{11} -17.1977 q^{12} -56.9423 q^{13} -114.997 q^{14} -24.6580 q^{15} +37.8697 q^{16} -74.3144 q^{17} -122.677 q^{18} -100.891 q^{19} +316.472 q^{20} +27.8436 q^{21} +84.1161 q^{22} -13.9922 q^{23} -37.9467 q^{24} +328.755 q^{25} -272.234 q^{26} +60.9576 q^{27} -357.359 q^{28} -87.1198 q^{29} -117.887 q^{30} -127.229 q^{31} -81.2011 q^{32} -20.3666 q^{33} -355.288 q^{34} -512.378 q^{35} -381.226 q^{36} -125.930 q^{37} -482.347 q^{38} +65.9146 q^{39} +698.294 q^{40} -82.7414 q^{41} +133.117 q^{42} +261.394 q^{44} -546.598 q^{45} -66.8948 q^{46} -71.5573 q^{47} -43.8367 q^{48} +235.575 q^{49} +1571.74 q^{50} +86.0239 q^{51} -845.980 q^{52} +152.062 q^{53} +291.431 q^{54} +374.785 q^{55} -788.511 q^{56} +116.788 q^{57} -416.509 q^{58} +319.919 q^{59} -366.338 q^{60} +353.020 q^{61} -608.268 q^{62} +617.216 q^{63} -691.170 q^{64} -1212.96 q^{65} -97.3701 q^{66} +37.2004 q^{67} -1104.07 q^{68} +16.1969 q^{69} -2449.62 q^{70} +781.921 q^{71} -841.173 q^{72} -1171.74 q^{73} -602.056 q^{74} -380.556 q^{75} -1498.91 q^{76} -423.206 q^{77} +315.130 q^{78} +841.963 q^{79} +806.682 q^{80} +622.259 q^{81} -395.577 q^{82} -1113.00 q^{83} +413.667 q^{84} -1583.01 q^{85} +100.847 q^{87} +576.766 q^{88} -308.616 q^{89} -2613.22 q^{90} +1369.67 q^{91} -207.879 q^{92} +147.277 q^{93} -342.106 q^{94} -2149.13 q^{95} +93.9957 q^{96} +1108.36 q^{97} +1126.25 q^{98} -451.470 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 50q + 10q^{2} + 2q^{3} + 186q^{4} - 8q^{5} - 51q^{6} + 6q^{7} + 138q^{8} + 360q^{9} + O(q^{10}) \) \( 50q + 10q^{2} + 2q^{3} + 186q^{4} - 8q^{5} - 51q^{6} + 6q^{7} + 138q^{8} + 360q^{9} - 137q^{10} - 252q^{11} + 48q^{12} - 192q^{13} - 272q^{14} - 314q^{15} + 542q^{16} - 236q^{17} + 386q^{18} - 12q^{19} - 108q^{20} - 408q^{21} - 1235q^{22} - 630q^{23} - 613q^{24} + 1098q^{25} - 1493q^{26} - 10q^{27} + 242q^{28} - 208q^{29} + 48q^{30} - 932q^{31} + 1124q^{32} - 254q^{33} - 765q^{34} - 1452q^{35} + 747q^{36} + 90q^{37} - 1213q^{38} + 1610q^{39} - 1693q^{40} - 1354q^{41} + 16q^{42} - 2704q^{44} - 4508q^{45} - 233q^{46} - 3484q^{47} + 376q^{48} + 1324q^{49} + 408q^{50} - 4054q^{51} - 2176q^{52} - 726q^{53} - 6497q^{54} + 3288q^{55} - 7097q^{56} - 870q^{57} + 275q^{58} - 4370q^{59} - 3891q^{60} - 1172q^{61} + 1546q^{62} + 3686q^{63} + 606q^{64} - 2610q^{65} - 4697q^{66} - 344q^{67} - 3221q^{68} - 136q^{69} + 1310q^{70} - 162q^{71} + 5814q^{72} - 746q^{73} - 4332q^{74} - 236q^{75} - 1338q^{76} - 2024q^{77} - 2782q^{78} - 2656q^{79} + 5713q^{80} - 86q^{81} + 4168q^{82} - 3514q^{83} - 4269q^{84} + 7558q^{85} - 10278q^{87} - 11692q^{88} - 2640q^{89} - 8286q^{90} + 5946q^{91} - 4271q^{92} + 2q^{93} - 9062q^{94} - 12140q^{95} - 700q^{96} - 3864q^{97} + 2826q^{98} - 8174q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.78088 1.69030 0.845148 0.534533i \(-0.179513\pi\)
0.845148 + 0.534533i \(0.179513\pi\)
\(3\) −1.15757 −0.222774 −0.111387 0.993777i \(-0.535529\pi\)
−0.111387 + 0.993777i \(0.535529\pi\)
\(4\) 14.8568 1.85710
\(5\) 21.3015 1.90527 0.952633 0.304122i \(-0.0983630\pi\)
0.952633 + 0.304122i \(0.0983630\pi\)
\(6\) −5.53419 −0.376554
\(7\) −24.0536 −1.29877 −0.649386 0.760459i \(-0.724974\pi\)
−0.649386 + 0.760459i \(0.724974\pi\)
\(8\) 32.7814 1.44875
\(9\) −25.6600 −0.950372
\(10\) 101.840 3.22046
\(11\) 17.5943 0.482262 0.241131 0.970493i \(-0.422482\pi\)
0.241131 + 0.970493i \(0.422482\pi\)
\(12\) −17.1977 −0.413713
\(13\) −56.9423 −1.21484 −0.607422 0.794380i \(-0.707796\pi\)
−0.607422 + 0.794380i \(0.707796\pi\)
\(14\) −114.997 −2.19531
\(15\) −24.6580 −0.424444
\(16\) 37.8697 0.591714
\(17\) −74.3144 −1.06023 −0.530114 0.847926i \(-0.677851\pi\)
−0.530114 + 0.847926i \(0.677851\pi\)
\(18\) −122.677 −1.60641
\(19\) −100.891 −1.21821 −0.609104 0.793090i \(-0.708471\pi\)
−0.609104 + 0.793090i \(0.708471\pi\)
\(20\) 316.472 3.53827
\(21\) 27.8436 0.289332
\(22\) 84.1161 0.815164
\(23\) −13.9922 −0.126851 −0.0634254 0.997987i \(-0.520202\pi\)
−0.0634254 + 0.997987i \(0.520202\pi\)
\(24\) −37.9467 −0.322743
\(25\) 328.755 2.63004
\(26\) −272.234 −2.05344
\(27\) 60.9576 0.434492
\(28\) −357.359 −2.41195
\(29\) −87.1198 −0.557853 −0.278927 0.960312i \(-0.589979\pi\)
−0.278927 + 0.960312i \(0.589979\pi\)
\(30\) −117.887 −0.717435
\(31\) −127.229 −0.737131 −0.368566 0.929602i \(-0.620151\pi\)
−0.368566 + 0.929602i \(0.620151\pi\)
\(32\) −81.2011 −0.448577
\(33\) −20.3666 −0.107435
\(34\) −355.288 −1.79210
\(35\) −512.378 −2.47451
\(36\) −381.226 −1.76493
\(37\) −125.930 −0.559535 −0.279767 0.960068i \(-0.590257\pi\)
−0.279767 + 0.960068i \(0.590257\pi\)
\(38\) −482.347 −2.05913
\(39\) 65.9146 0.270635
\(40\) 698.294 2.76025
\(41\) −82.7414 −0.315172 −0.157586 0.987505i \(-0.550371\pi\)
−0.157586 + 0.987505i \(0.550371\pi\)
\(42\) 133.117 0.489057
\(43\) 0 0
\(44\) 261.394 0.895607
\(45\) −546.598 −1.81071
\(46\) −66.8948 −0.214415
\(47\) −71.5573 −0.222079 −0.111039 0.993816i \(-0.535418\pi\)
−0.111039 + 0.993816i \(0.535418\pi\)
\(48\) −43.8367 −0.131818
\(49\) 235.575 0.686807
\(50\) 1571.74 4.44554
\(51\) 86.0239 0.236191
\(52\) −845.980 −2.25608
\(53\) 152.062 0.394101 0.197050 0.980393i \(-0.436864\pi\)
0.197050 + 0.980393i \(0.436864\pi\)
\(54\) 291.431 0.734420
\(55\) 374.785 0.918837
\(56\) −788.511 −1.88159
\(57\) 116.788 0.271385
\(58\) −416.509 −0.942936
\(59\) 319.919 0.705931 0.352965 0.935636i \(-0.385173\pi\)
0.352965 + 0.935636i \(0.385173\pi\)
\(60\) −366.338 −0.788234
\(61\) 353.020 0.740977 0.370488 0.928837i \(-0.379190\pi\)
0.370488 + 0.928837i \(0.379190\pi\)
\(62\) −608.268 −1.24597
\(63\) 617.216 1.23432
\(64\) −691.170 −1.34994
\(65\) −1212.96 −2.31460
\(66\) −97.3701 −0.181597
\(67\) 37.2004 0.0678320 0.0339160 0.999425i \(-0.489202\pi\)
0.0339160 + 0.999425i \(0.489202\pi\)
\(68\) −1104.07 −1.96895
\(69\) 16.1969 0.0282591
\(70\) −2449.62 −4.18264
\(71\) 781.921 1.30700 0.653500 0.756927i \(-0.273300\pi\)
0.653500 + 0.756927i \(0.273300\pi\)
\(72\) −841.173 −1.37685
\(73\) −1171.74 −1.87866 −0.939331 0.343013i \(-0.888552\pi\)
−0.939331 + 0.343013i \(0.888552\pi\)
\(74\) −602.056 −0.945779
\(75\) −380.556 −0.585905
\(76\) −1498.91 −2.26233
\(77\) −423.206 −0.626347
\(78\) 315.130 0.457454
\(79\) 841.963 1.19909 0.599546 0.800340i \(-0.295348\pi\)
0.599546 + 0.800340i \(0.295348\pi\)
\(80\) 806.682 1.12737
\(81\) 622.259 0.853578
\(82\) −395.577 −0.532733
\(83\) −1113.00 −1.47190 −0.735950 0.677036i \(-0.763264\pi\)
−0.735950 + 0.677036i \(0.763264\pi\)
\(84\) 413.667 0.537319
\(85\) −1583.01 −2.02002
\(86\) 0 0
\(87\) 100.847 0.124275
\(88\) 576.766 0.698675
\(89\) −308.616 −0.367564 −0.183782 0.982967i \(-0.558834\pi\)
−0.183782 + 0.982967i \(0.558834\pi\)
\(90\) −2613.22 −3.06064
\(91\) 1369.67 1.57780
\(92\) −207.879 −0.235574
\(93\) 147.277 0.164214
\(94\) −342.106 −0.375379
\(95\) −2149.13 −2.32101
\(96\) 93.9957 0.0999312
\(97\) 1108.36 1.16017 0.580085 0.814556i \(-0.303019\pi\)
0.580085 + 0.814556i \(0.303019\pi\)
\(98\) 1126.25 1.16091
\(99\) −451.470 −0.458328
\(100\) 4884.24 4.88424
\(101\) −774.120 −0.762652 −0.381326 0.924441i \(-0.624532\pi\)
−0.381326 + 0.924441i \(0.624532\pi\)
\(102\) 411.270 0.399233
\(103\) 317.377 0.303612 0.151806 0.988410i \(-0.451491\pi\)
0.151806 + 0.988410i \(0.451491\pi\)
\(104\) −1866.65 −1.76000
\(105\) 593.112 0.551255
\(106\) 726.990 0.666146
\(107\) 648.591 0.585997 0.292998 0.956113i \(-0.405347\pi\)
0.292998 + 0.956113i \(0.405347\pi\)
\(108\) 905.633 0.806894
\(109\) 575.465 0.505684 0.252842 0.967508i \(-0.418635\pi\)
0.252842 + 0.967508i \(0.418635\pi\)
\(110\) 1791.80 1.55311
\(111\) 145.773 0.124650
\(112\) −910.902 −0.768501
\(113\) −509.868 −0.424463 −0.212232 0.977219i \(-0.568073\pi\)
−0.212232 + 0.977219i \(0.568073\pi\)
\(114\) 558.349 0.458721
\(115\) −298.055 −0.241685
\(116\) −1294.32 −1.03599
\(117\) 1461.14 1.15455
\(118\) 1529.49 1.19323
\(119\) 1787.53 1.37699
\(120\) −808.323 −0.614912
\(121\) −1021.44 −0.767424
\(122\) 1687.75 1.25247
\(123\) 95.7788 0.0702121
\(124\) −1890.22 −1.36892
\(125\) 4340.29 3.10566
\(126\) 2950.83 2.08636
\(127\) 2623.19 1.83284 0.916421 0.400217i \(-0.131065\pi\)
0.916421 + 0.400217i \(0.131065\pi\)
\(128\) −2654.79 −1.83322
\(129\) 0 0
\(130\) −5799.01 −3.91236
\(131\) −2604.53 −1.73709 −0.868544 0.495611i \(-0.834944\pi\)
−0.868544 + 0.495611i \(0.834944\pi\)
\(132\) −302.582 −0.199518
\(133\) 2426.79 1.58217
\(134\) 177.850 0.114656
\(135\) 1298.49 0.827823
\(136\) −2436.13 −1.53600
\(137\) 665.268 0.414873 0.207437 0.978248i \(-0.433488\pi\)
0.207437 + 0.978248i \(0.433488\pi\)
\(138\) 77.4353 0.0477662
\(139\) −876.154 −0.534636 −0.267318 0.963608i \(-0.586137\pi\)
−0.267318 + 0.963608i \(0.586137\pi\)
\(140\) −7612.29 −4.59540
\(141\) 82.8324 0.0494734
\(142\) 3738.27 2.20921
\(143\) −1001.86 −0.585872
\(144\) −971.738 −0.562348
\(145\) −1855.78 −1.06286
\(146\) −5601.96 −3.17549
\(147\) −272.694 −0.153003
\(148\) −1870.92 −1.03911
\(149\) 1157.27 0.636289 0.318144 0.948042i \(-0.396940\pi\)
0.318144 + 0.948042i \(0.396940\pi\)
\(150\) −1819.39 −0.990352
\(151\) 1380.36 0.743922 0.371961 0.928248i \(-0.378686\pi\)
0.371961 + 0.928248i \(0.378686\pi\)
\(152\) −3307.35 −1.76488
\(153\) 1906.91 1.00761
\(154\) −2023.29 −1.05871
\(155\) −2710.18 −1.40443
\(156\) 979.279 0.502596
\(157\) 2859.38 1.45352 0.726762 0.686890i \(-0.241024\pi\)
0.726762 + 0.686890i \(0.241024\pi\)
\(158\) 4025.32 2.02682
\(159\) −176.022 −0.0877954
\(160\) −1729.71 −0.854658
\(161\) 336.562 0.164750
\(162\) 2974.94 1.44280
\(163\) 1990.78 0.956627 0.478314 0.878189i \(-0.341248\pi\)
0.478314 + 0.878189i \(0.341248\pi\)
\(164\) −1229.27 −0.585305
\(165\) −433.839 −0.204693
\(166\) −5321.12 −2.48794
\(167\) −1753.53 −0.812527 −0.406263 0.913756i \(-0.633168\pi\)
−0.406263 + 0.913756i \(0.633168\pi\)
\(168\) 912.754 0.419170
\(169\) 1045.43 0.475844
\(170\) −7568.17 −3.41443
\(171\) 2588.86 1.15775
\(172\) 0 0
\(173\) 1857.37 0.816263 0.408131 0.912923i \(-0.366180\pi\)
0.408131 + 0.912923i \(0.366180\pi\)
\(174\) 482.137 0.210062
\(175\) −7907.73 −3.41582
\(176\) 666.290 0.285361
\(177\) −370.328 −0.157263
\(178\) −1475.45 −0.621291
\(179\) −3915.43 −1.63493 −0.817467 0.575975i \(-0.804623\pi\)
−0.817467 + 0.575975i \(0.804623\pi\)
\(180\) −8120.69 −3.36267
\(181\) 3009.44 1.23586 0.617928 0.786234i \(-0.287972\pi\)
0.617928 + 0.786234i \(0.287972\pi\)
\(182\) 6548.21 2.66695
\(183\) −408.645 −0.165070
\(184\) −458.683 −0.183775
\(185\) −2682.50 −1.06606
\(186\) 704.111 0.277570
\(187\) −1307.51 −0.511308
\(188\) −1063.11 −0.412422
\(189\) −1466.25 −0.564306
\(190\) −10274.7 −3.92319
\(191\) −393.305 −0.148998 −0.0744989 0.997221i \(-0.523736\pi\)
−0.0744989 + 0.997221i \(0.523736\pi\)
\(192\) 800.076 0.300732
\(193\) 1329.16 0.495725 0.247862 0.968795i \(-0.420272\pi\)
0.247862 + 0.968795i \(0.420272\pi\)
\(194\) 5298.91 1.96103
\(195\) 1404.08 0.515633
\(196\) 3499.88 1.27547
\(197\) −533.591 −0.192978 −0.0964892 0.995334i \(-0.530761\pi\)
−0.0964892 + 0.995334i \(0.530761\pi\)
\(198\) −2158.42 −0.774709
\(199\) −2186.74 −0.778964 −0.389482 0.921034i \(-0.627346\pi\)
−0.389482 + 0.921034i \(0.627346\pi\)
\(200\) 10777.1 3.81026
\(201\) −43.0619 −0.0151112
\(202\) −3700.97 −1.28911
\(203\) 2095.54 0.724524
\(204\) 1278.04 0.438630
\(205\) −1762.52 −0.600486
\(206\) 1517.34 0.513195
\(207\) 359.040 0.120555
\(208\) −2156.39 −0.718840
\(209\) −1775.10 −0.587495
\(210\) 2835.60 0.931784
\(211\) −5062.24 −1.65165 −0.825827 0.563923i \(-0.809291\pi\)
−0.825827 + 0.563923i \(0.809291\pi\)
\(212\) 2259.15 0.731883
\(213\) −905.127 −0.291165
\(214\) 3100.83 0.990508
\(215\) 0 0
\(216\) 1998.28 0.629470
\(217\) 3060.32 0.957365
\(218\) 2751.23 0.854755
\(219\) 1356.37 0.418517
\(220\) 5568.10 1.70637
\(221\) 4231.63 1.28801
\(222\) 696.921 0.210695
\(223\) −5423.05 −1.62850 −0.814248 0.580518i \(-0.802850\pi\)
−0.814248 + 0.580518i \(0.802850\pi\)
\(224\) 1953.18 0.582599
\(225\) −8435.87 −2.49952
\(226\) −2437.62 −0.717468
\(227\) −1368.39 −0.400103 −0.200052 0.979785i \(-0.564111\pi\)
−0.200052 + 0.979785i \(0.564111\pi\)
\(228\) 1735.09 0.503988
\(229\) 3079.65 0.888685 0.444342 0.895857i \(-0.353437\pi\)
0.444342 + 0.895857i \(0.353437\pi\)
\(230\) −1424.96 −0.408518
\(231\) 489.889 0.139534
\(232\) −2855.91 −0.808189
\(233\) −1874.11 −0.526939 −0.263469 0.964668i \(-0.584867\pi\)
−0.263469 + 0.964668i \(0.584867\pi\)
\(234\) 6985.54 1.95153
\(235\) −1524.28 −0.423119
\(236\) 4752.97 1.31098
\(237\) −974.629 −0.267126
\(238\) 8545.95 2.32753
\(239\) −2112.85 −0.571837 −0.285919 0.958254i \(-0.592299\pi\)
−0.285919 + 0.958254i \(0.592299\pi\)
\(240\) −933.789 −0.251149
\(241\) 242.170 0.0647285 0.0323642 0.999476i \(-0.489696\pi\)
0.0323642 + 0.999476i \(0.489696\pi\)
\(242\) −4883.38 −1.29717
\(243\) −2366.16 −0.624647
\(244\) 5244.74 1.37607
\(245\) 5018.10 1.30855
\(246\) 457.907 0.118679
\(247\) 5744.96 1.47993
\(248\) −4170.76 −1.06792
\(249\) 1288.37 0.327901
\(250\) 20750.4 5.24948
\(251\) −2026.35 −0.509569 −0.254784 0.966998i \(-0.582005\pi\)
−0.254784 + 0.966998i \(0.582005\pi\)
\(252\) 9169.84 2.29224
\(253\) −246.182 −0.0611753
\(254\) 12541.2 3.09804
\(255\) 1832.44 0.450007
\(256\) −7162.86 −1.74875
\(257\) −2709.06 −0.657534 −0.328767 0.944411i \(-0.606633\pi\)
−0.328767 + 0.944411i \(0.606633\pi\)
\(258\) 0 0
\(259\) 3029.07 0.726707
\(260\) −18020.7 −4.29844
\(261\) 2235.50 0.530168
\(262\) −12451.9 −2.93619
\(263\) −1614.38 −0.378507 −0.189253 0.981928i \(-0.560607\pi\)
−0.189253 + 0.981928i \(0.560607\pi\)
\(264\) −667.645 −0.155647
\(265\) 3239.15 0.750867
\(266\) 11602.2 2.67434
\(267\) 357.243 0.0818837
\(268\) 552.678 0.125971
\(269\) −3571.10 −0.809420 −0.404710 0.914445i \(-0.632627\pi\)
−0.404710 + 0.914445i \(0.632627\pi\)
\(270\) 6207.92 1.39927
\(271\) −7251.79 −1.62552 −0.812758 0.582602i \(-0.802035\pi\)
−0.812758 + 0.582602i \(0.802035\pi\)
\(272\) −2814.26 −0.627352
\(273\) −1585.48 −0.351494
\(274\) 3180.56 0.701258
\(275\) 5784.21 1.26837
\(276\) 240.634 0.0524799
\(277\) 4897.85 1.06239 0.531197 0.847248i \(-0.321742\pi\)
0.531197 + 0.847248i \(0.321742\pi\)
\(278\) −4188.78 −0.903693
\(279\) 3264.71 0.700549
\(280\) −16796.5 −3.58493
\(281\) −3332.97 −0.707573 −0.353787 0.935326i \(-0.615106\pi\)
−0.353787 + 0.935326i \(0.615106\pi\)
\(282\) 396.011 0.0836246
\(283\) −3141.03 −0.659770 −0.329885 0.944021i \(-0.607010\pi\)
−0.329885 + 0.944021i \(0.607010\pi\)
\(284\) 11616.8 2.42723
\(285\) 2487.76 0.517061
\(286\) −4789.77 −0.990297
\(287\) 1990.23 0.409336
\(288\) 2083.62 0.426315
\(289\) 609.628 0.124085
\(290\) −8872.28 −1.79654
\(291\) −1283.00 −0.258456
\(292\) −17408.3 −3.48886
\(293\) −4342.75 −0.865891 −0.432945 0.901420i \(-0.642526\pi\)
−0.432945 + 0.901420i \(0.642526\pi\)
\(294\) −1303.71 −0.258620
\(295\) 6814.76 1.34499
\(296\) −4128.17 −0.810625
\(297\) 1072.50 0.209539
\(298\) 5532.75 1.07552
\(299\) 796.747 0.154104
\(300\) −5653.84 −1.08808
\(301\) 0 0
\(302\) 6599.34 1.25745
\(303\) 896.096 0.169899
\(304\) −3820.70 −0.720830
\(305\) 7519.87 1.41176
\(306\) 9116.70 1.70316
\(307\) −8180.43 −1.52079 −0.760394 0.649462i \(-0.774994\pi\)
−0.760394 + 0.649462i \(0.774994\pi\)
\(308\) −6287.47 −1.16319
\(309\) −367.385 −0.0676370
\(310\) −12957.0 −2.37390
\(311\) −4932.66 −0.899375 −0.449687 0.893186i \(-0.648465\pi\)
−0.449687 + 0.893186i \(0.648465\pi\)
\(312\) 2160.77 0.392083
\(313\) 4008.91 0.723952 0.361976 0.932187i \(-0.382102\pi\)
0.361976 + 0.932187i \(0.382102\pi\)
\(314\) 13670.3 2.45688
\(315\) 13147.6 2.35170
\(316\) 12508.9 2.22683
\(317\) 1568.08 0.277831 0.138915 0.990304i \(-0.455638\pi\)
0.138915 + 0.990304i \(0.455638\pi\)
\(318\) −841.540 −0.148400
\(319\) −1532.81 −0.269031
\(320\) −14723.0 −2.57200
\(321\) −750.788 −0.130545
\(322\) 1609.06 0.278477
\(323\) 7497.64 1.29158
\(324\) 9244.76 1.58518
\(325\) −18720.1 −3.19509
\(326\) 9517.69 1.61698
\(327\) −666.140 −0.112653
\(328\) −2712.38 −0.456604
\(329\) 1721.21 0.288429
\(330\) −2074.13 −0.345991
\(331\) −2987.88 −0.496160 −0.248080 0.968740i \(-0.579800\pi\)
−0.248080 + 0.968740i \(0.579800\pi\)
\(332\) −16535.6 −2.73346
\(333\) 3231.37 0.531766
\(334\) −8383.39 −1.37341
\(335\) 792.424 0.129238
\(336\) 1054.43 0.171202
\(337\) −5177.88 −0.836965 −0.418483 0.908225i \(-0.637438\pi\)
−0.418483 + 0.908225i \(0.637438\pi\)
\(338\) 4998.07 0.804317
\(339\) 590.207 0.0945594
\(340\) −23518.4 −3.75137
\(341\) −2238.51 −0.355490
\(342\) 12377.0 1.95694
\(343\) 2583.96 0.406766
\(344\) 0 0
\(345\) 345.018 0.0538411
\(346\) 8879.87 1.37973
\(347\) 8237.38 1.27437 0.637184 0.770712i \(-0.280099\pi\)
0.637184 + 0.770712i \(0.280099\pi\)
\(348\) 1498.26 0.230791
\(349\) −2750.91 −0.421928 −0.210964 0.977494i \(-0.567660\pi\)
−0.210964 + 0.977494i \(0.567660\pi\)
\(350\) −37805.9 −5.77374
\(351\) −3471.07 −0.527840
\(352\) −1428.67 −0.216331
\(353\) −109.272 −0.0164758 −0.00823789 0.999966i \(-0.502622\pi\)
−0.00823789 + 0.999966i \(0.502622\pi\)
\(354\) −1770.49 −0.265821
\(355\) 16656.1 2.49018
\(356\) −4585.03 −0.682602
\(357\) −2069.18 −0.306759
\(358\) −18719.2 −2.76352
\(359\) −11539.3 −1.69643 −0.848216 0.529650i \(-0.822323\pi\)
−0.848216 + 0.529650i \(0.822323\pi\)
\(360\) −17918.3 −2.62326
\(361\) 3319.96 0.484030
\(362\) 14387.8 2.08896
\(363\) 1182.39 0.170962
\(364\) 20348.8 2.93013
\(365\) −24959.9 −3.57935
\(366\) −1953.68 −0.279018
\(367\) −13092.1 −1.86213 −0.931064 0.364857i \(-0.881118\pi\)
−0.931064 + 0.364857i \(0.881118\pi\)
\(368\) −529.879 −0.0750594
\(369\) 2123.15 0.299530
\(370\) −12824.7 −1.80196
\(371\) −3657.64 −0.511847
\(372\) 2188.06 0.304961
\(373\) 9972.25 1.38430 0.692150 0.721754i \(-0.256664\pi\)
0.692150 + 0.721754i \(0.256664\pi\)
\(374\) −6251.04 −0.864261
\(375\) −5024.18 −0.691860
\(376\) −2345.75 −0.321736
\(377\) 4960.80 0.677704
\(378\) −7009.95 −0.953843
\(379\) 1702.20 0.230702 0.115351 0.993325i \(-0.463201\pi\)
0.115351 + 0.993325i \(0.463201\pi\)
\(380\) −31929.1 −4.31034
\(381\) −3036.52 −0.408309
\(382\) −1880.34 −0.251850
\(383\) −3035.45 −0.404972 −0.202486 0.979285i \(-0.564902\pi\)
−0.202486 + 0.979285i \(0.564902\pi\)
\(384\) 3073.10 0.408394
\(385\) −9014.92 −1.19336
\(386\) 6354.54 0.837921
\(387\) 0 0
\(388\) 16466.6 2.15455
\(389\) 9435.36 1.22980 0.614900 0.788605i \(-0.289196\pi\)
0.614900 + 0.788605i \(0.289196\pi\)
\(390\) 6712.74 0.871571
\(391\) 1039.82 0.134491
\(392\) 7722.47 0.995010
\(393\) 3014.92 0.386978
\(394\) −2551.03 −0.326191
\(395\) 17935.1 2.28459
\(396\) −6707.39 −0.851159
\(397\) 4943.82 0.624995 0.312498 0.949919i \(-0.398834\pi\)
0.312498 + 0.949919i \(0.398834\pi\)
\(398\) −10454.5 −1.31668
\(399\) −2809.17 −0.352467
\(400\) 12449.8 1.55623
\(401\) −10139.0 −1.26264 −0.631318 0.775524i \(-0.717486\pi\)
−0.631318 + 0.775524i \(0.717486\pi\)
\(402\) −205.874 −0.0255424
\(403\) 7244.74 0.895499
\(404\) −11500.9 −1.41632
\(405\) 13255.1 1.62629
\(406\) 10018.5 1.22466
\(407\) −2215.65 −0.269842
\(408\) 2819.99 0.342182
\(409\) −740.411 −0.0895134 −0.0447567 0.998998i \(-0.514251\pi\)
−0.0447567 + 0.998998i \(0.514251\pi\)
\(410\) −8426.39 −1.01500
\(411\) −770.092 −0.0924230
\(412\) 4715.20 0.563838
\(413\) −7695.20 −0.916843
\(414\) 1716.52 0.203774
\(415\) −23708.6 −2.80436
\(416\) 4623.78 0.544950
\(417\) 1014.21 0.119103
\(418\) −8486.55 −0.993040
\(419\) 12031.8 1.40284 0.701421 0.712747i \(-0.252549\pi\)
0.701421 + 0.712747i \(0.252549\pi\)
\(420\) 8811.74 1.02374
\(421\) −1281.44 −0.148346 −0.0741731 0.997245i \(-0.523632\pi\)
−0.0741731 + 0.997245i \(0.523632\pi\)
\(422\) −24202.0 −2.79178
\(423\) 1836.16 0.211057
\(424\) 4984.81 0.570952
\(425\) −24431.2 −2.78844
\(426\) −4327.30 −0.492156
\(427\) −8491.40 −0.962360
\(428\) 9635.98 1.08825
\(429\) 1159.72 0.130517
\(430\) 0 0
\(431\) −5537.45 −0.618862 −0.309431 0.950922i \(-0.600139\pi\)
−0.309431 + 0.950922i \(0.600139\pi\)
\(432\) 2308.44 0.257095
\(433\) 10109.8 1.12205 0.561023 0.827800i \(-0.310408\pi\)
0.561023 + 0.827800i \(0.310408\pi\)
\(434\) 14631.0 1.61823
\(435\) 2148.20 0.236777
\(436\) 8549.56 0.939104
\(437\) 1411.68 0.154531
\(438\) 6484.65 0.707417
\(439\) 13005.7 1.41396 0.706981 0.707233i \(-0.250057\pi\)
0.706981 + 0.707233i \(0.250057\pi\)
\(440\) 12286.0 1.33116
\(441\) −6044.86 −0.652722
\(442\) 20230.9 2.17712
\(443\) 12636.5 1.35526 0.677630 0.735403i \(-0.263007\pi\)
0.677630 + 0.735403i \(0.263007\pi\)
\(444\) 2165.71 0.231487
\(445\) −6573.98 −0.700307
\(446\) −25926.9 −2.75264
\(447\) −1339.61 −0.141749
\(448\) 16625.1 1.75326
\(449\) 1602.63 0.168447 0.0842237 0.996447i \(-0.473159\pi\)
0.0842237 + 0.996447i \(0.473159\pi\)
\(450\) −40330.8 −4.22492
\(451\) −1455.78 −0.151995
\(452\) −7575.00 −0.788269
\(453\) −1597.86 −0.165726
\(454\) −6542.12 −0.676292
\(455\) 29176.0 3.00614
\(456\) 3828.48 0.393168
\(457\) −12386.5 −1.26786 −0.633932 0.773389i \(-0.718560\pi\)
−0.633932 + 0.773389i \(0.718560\pi\)
\(458\) 14723.4 1.50214
\(459\) −4530.02 −0.460661
\(460\) −4428.13 −0.448832
\(461\) 7814.88 0.789534 0.394767 0.918781i \(-0.370825\pi\)
0.394767 + 0.918781i \(0.370825\pi\)
\(462\) 2342.10 0.235854
\(463\) −7828.20 −0.785761 −0.392881 0.919589i \(-0.628521\pi\)
−0.392881 + 0.919589i \(0.628521\pi\)
\(464\) −3299.20 −0.330089
\(465\) 3137.22 0.312871
\(466\) −8959.87 −0.890682
\(467\) −372.814 −0.0369417 −0.0184709 0.999829i \(-0.505880\pi\)
−0.0184709 + 0.999829i \(0.505880\pi\)
\(468\) 21707.9 2.14412
\(469\) −894.802 −0.0880983
\(470\) −7287.39 −0.715196
\(471\) −3309.92 −0.323807
\(472\) 10487.4 1.02272
\(473\) 0 0
\(474\) −4659.58 −0.451522
\(475\) −33168.4 −3.20393
\(476\) 26556.9 2.55721
\(477\) −3901.92 −0.374542
\(478\) −10101.3 −0.966574
\(479\) 6085.36 0.580474 0.290237 0.956955i \(-0.406266\pi\)
0.290237 + 0.956955i \(0.406266\pi\)
\(480\) 2002.25 0.190396
\(481\) 7170.75 0.679747
\(482\) 1157.79 0.109410
\(483\) −389.593 −0.0367021
\(484\) −15175.3 −1.42518
\(485\) 23609.7 2.21043
\(486\) −11312.3 −1.05584
\(487\) −1655.22 −0.154015 −0.0770075 0.997031i \(-0.524537\pi\)
−0.0770075 + 0.997031i \(0.524537\pi\)
\(488\) 11572.5 1.07349
\(489\) −2304.47 −0.213112
\(490\) 23990.9 2.21184
\(491\) −14763.9 −1.35700 −0.678498 0.734602i \(-0.737369\pi\)
−0.678498 + 0.734602i \(0.737369\pi\)
\(492\) 1422.97 0.130391
\(493\) 6474.25 0.591452
\(494\) 27465.9 2.50152
\(495\) −9617.00 −0.873236
\(496\) −4818.14 −0.436171
\(497\) −18808.0 −1.69749
\(498\) 6159.55 0.554249
\(499\) 5889.43 0.528350 0.264175 0.964475i \(-0.414900\pi\)
0.264175 + 0.964475i \(0.414900\pi\)
\(500\) 64482.8 5.76751
\(501\) 2029.83 0.181010
\(502\) −9687.71 −0.861322
\(503\) 18376.1 1.62893 0.814463 0.580215i \(-0.197032\pi\)
0.814463 + 0.580215i \(0.197032\pi\)
\(504\) 20233.2 1.78821
\(505\) −16489.9 −1.45305
\(506\) −1176.97 −0.103404
\(507\) −1210.16 −0.106006
\(508\) 38972.2 3.40376
\(509\) 8355.89 0.727639 0.363820 0.931469i \(-0.381472\pi\)
0.363820 + 0.931469i \(0.381472\pi\)
\(510\) 8760.67 0.760645
\(511\) 28184.6 2.43995
\(512\) −13006.4 −1.12267
\(513\) −6150.06 −0.529302
\(514\) −12951.7 −1.11143
\(515\) 6760.61 0.578463
\(516\) 0 0
\(517\) −1259.00 −0.107100
\(518\) 14481.6 1.22835
\(519\) −2150.04 −0.181842
\(520\) −39762.5 −3.35327
\(521\) 19044.7 1.60146 0.800732 0.599022i \(-0.204444\pi\)
0.800732 + 0.599022i \(0.204444\pi\)
\(522\) 10687.6 0.896140
\(523\) −5907.61 −0.493923 −0.246961 0.969025i \(-0.579432\pi\)
−0.246961 + 0.969025i \(0.579432\pi\)
\(524\) −38694.9 −3.22594
\(525\) 9153.74 0.760956
\(526\) −7718.18 −0.639788
\(527\) 9454.97 0.781528
\(528\) −771.276 −0.0635710
\(529\) −11971.2 −0.983909
\(530\) 15486.0 1.26919
\(531\) −8209.14 −0.670897
\(532\) 36054.2 2.93825
\(533\) 4711.49 0.382884
\(534\) 1707.94 0.138408
\(535\) 13816.0 1.11648
\(536\) 1219.48 0.0982715
\(537\) 4532.38 0.364221
\(538\) −17073.0 −1.36816
\(539\) 4144.77 0.331221
\(540\) 19291.4 1.53735
\(541\) −22011.7 −1.74927 −0.874636 0.484780i \(-0.838900\pi\)
−0.874636 + 0.484780i \(0.838900\pi\)
\(542\) −34669.9 −2.74760
\(543\) −3483.63 −0.275317
\(544\) 6034.41 0.475594
\(545\) 12258.3 0.963463
\(546\) −7579.99 −0.594128
\(547\) −19182.2 −1.49940 −0.749700 0.661777i \(-0.769802\pi\)
−0.749700 + 0.661777i \(0.769802\pi\)
\(548\) 9883.74 0.770460
\(549\) −9058.51 −0.704204
\(550\) 27653.6 2.14391
\(551\) 8789.59 0.679581
\(552\) 530.957 0.0409403
\(553\) −20252.2 −1.55735
\(554\) 23416.0 1.79576
\(555\) 3105.18 0.237491
\(556\) −13016.8 −0.992872
\(557\) 12163.8 0.925306 0.462653 0.886539i \(-0.346898\pi\)
0.462653 + 0.886539i \(0.346898\pi\)
\(558\) 15608.2 1.18413
\(559\) 0 0
\(560\) −19403.6 −1.46420
\(561\) 1513.53 0.113906
\(562\) −15934.5 −1.19601
\(563\) 9583.99 0.717437 0.358718 0.933446i \(-0.383214\pi\)
0.358718 + 0.933446i \(0.383214\pi\)
\(564\) 1230.62 0.0918769
\(565\) −10861.0 −0.808715
\(566\) −15016.9 −1.11521
\(567\) −14967.5 −1.10860
\(568\) 25632.5 1.89351
\(569\) 7956.37 0.586201 0.293100 0.956082i \(-0.405313\pi\)
0.293100 + 0.956082i \(0.405313\pi\)
\(570\) 11893.7 0.873985
\(571\) 4971.46 0.364360 0.182180 0.983265i \(-0.441685\pi\)
0.182180 + 0.983265i \(0.441685\pi\)
\(572\) −14884.4 −1.08802
\(573\) 455.277 0.0331928
\(574\) 9515.03 0.691899
\(575\) −4600.00 −0.333623
\(576\) 17735.4 1.28295
\(577\) 23376.7 1.68663 0.843315 0.537420i \(-0.180601\pi\)
0.843315 + 0.537420i \(0.180601\pi\)
\(578\) 2914.56 0.209740
\(579\) −1538.59 −0.110435
\(580\) −27571.0 −1.97383
\(581\) 26771.6 1.91166
\(582\) −6133.85 −0.436866
\(583\) 2675.42 0.190060
\(584\) −38411.4 −2.72171
\(585\) 31124.6 2.19973
\(586\) −20762.1 −1.46361
\(587\) −13790.5 −0.969670 −0.484835 0.874606i \(-0.661120\pi\)
−0.484835 + 0.874606i \(0.661120\pi\)
\(588\) −4051.35 −0.284141
\(589\) 12836.3 0.897979
\(590\) 32580.5 2.27342
\(591\) 617.667 0.0429906
\(592\) −4768.93 −0.331084
\(593\) 22432.3 1.55343 0.776716 0.629851i \(-0.216884\pi\)
0.776716 + 0.629851i \(0.216884\pi\)
\(594\) 5127.51 0.354182
\(595\) 38077.1 2.62354
\(596\) 17193.3 1.18165
\(597\) 2531.30 0.173533
\(598\) 3809.15 0.260481
\(599\) 2181.92 0.148833 0.0744165 0.997227i \(-0.476291\pi\)
0.0744165 + 0.997227i \(0.476291\pi\)
\(600\) −12475.2 −0.848828
\(601\) −10415.4 −0.706907 −0.353453 0.935452i \(-0.614993\pi\)
−0.353453 + 0.935452i \(0.614993\pi\)
\(602\) 0 0
\(603\) −954.562 −0.0644657
\(604\) 20507.7 1.38154
\(605\) −21758.3 −1.46215
\(606\) 4284.12 0.287179
\(607\) 6163.42 0.412134 0.206067 0.978538i \(-0.433933\pi\)
0.206067 + 0.978538i \(0.433933\pi\)
\(608\) 8192.44 0.546460
\(609\) −2425.73 −0.161405
\(610\) 35951.6 2.38629
\(611\) 4074.64 0.269791
\(612\) 28330.5 1.87123
\(613\) 6705.56 0.441819 0.220910 0.975294i \(-0.429097\pi\)
0.220910 + 0.975294i \(0.429097\pi\)
\(614\) −39109.6 −2.57058
\(615\) 2040.24 0.133773
\(616\) −13873.3 −0.907420
\(617\) −1560.14 −0.101797 −0.0508985 0.998704i \(-0.516209\pi\)
−0.0508985 + 0.998704i \(0.516209\pi\)
\(618\) −1756.42 −0.114326
\(619\) −17652.2 −1.14621 −0.573104 0.819483i \(-0.694261\pi\)
−0.573104 + 0.819483i \(0.694261\pi\)
\(620\) −40264.5 −2.60817
\(621\) −852.929 −0.0551157
\(622\) −23582.4 −1.52021
\(623\) 7423.31 0.477381
\(624\) 2496.17 0.160139
\(625\) 51360.5 3.28707
\(626\) 19166.1 1.22369
\(627\) 2054.80 0.130879
\(628\) 42481.1 2.69933
\(629\) 9358.42 0.593235
\(630\) 62857.2 3.97507
\(631\) 31138.2 1.96449 0.982245 0.187601i \(-0.0600712\pi\)
0.982245 + 0.187601i \(0.0600712\pi\)
\(632\) 27600.7 1.73718
\(633\) 5859.89 0.367946
\(634\) 7496.81 0.469616
\(635\) 55878.0 3.49205
\(636\) −2615.12 −0.163045
\(637\) −13414.2 −0.834363
\(638\) −7328.18 −0.454742
\(639\) −20064.1 −1.24214
\(640\) −56551.1 −3.49278
\(641\) 17820.0 1.09805 0.549023 0.835807i \(-0.315000\pi\)
0.549023 + 0.835807i \(0.315000\pi\)
\(642\) −3589.43 −0.220659
\(643\) −3961.12 −0.242941 −0.121471 0.992595i \(-0.538761\pi\)
−0.121471 + 0.992595i \(0.538761\pi\)
\(644\) 5000.23 0.305957
\(645\) 0 0
\(646\) 35845.3 2.18315
\(647\) 1807.07 0.109804 0.0549019 0.998492i \(-0.482515\pi\)
0.0549019 + 0.998492i \(0.482515\pi\)
\(648\) 20398.5 1.23662
\(649\) 5628.75 0.340443
\(650\) −89498.4 −5.40064
\(651\) −3542.53 −0.213276
\(652\) 29576.6 1.77655
\(653\) −20608.3 −1.23502 −0.617509 0.786564i \(-0.711858\pi\)
−0.617509 + 0.786564i \(0.711858\pi\)
\(654\) −3184.73 −0.190417
\(655\) −55480.4 −3.30962
\(656\) −3133.39 −0.186492
\(657\) 30067.0 1.78543
\(658\) 8228.89 0.487531
\(659\) −5125.22 −0.302959 −0.151480 0.988460i \(-0.548404\pi\)
−0.151480 + 0.988460i \(0.548404\pi\)
\(660\) −6445.45 −0.380135
\(661\) −19830.2 −1.16688 −0.583439 0.812157i \(-0.698293\pi\)
−0.583439 + 0.812157i \(0.698293\pi\)
\(662\) −14284.7 −0.838656
\(663\) −4898.40 −0.286935
\(664\) −36485.7 −2.13241
\(665\) 51694.2 3.01446
\(666\) 15448.8 0.898841
\(667\) 1219.00 0.0707641
\(668\) −26051.8 −1.50894
\(669\) 6277.55 0.362786
\(670\) 3788.48 0.218451
\(671\) 6211.14 0.357345
\(672\) −2260.93 −0.129788
\(673\) −20904.7 −1.19735 −0.598675 0.800992i \(-0.704306\pi\)
−0.598675 + 0.800992i \(0.704306\pi\)
\(674\) −24754.8 −1.41472
\(675\) 20040.1 1.14273
\(676\) 15531.7 0.883689
\(677\) −8266.41 −0.469282 −0.234641 0.972082i \(-0.575392\pi\)
−0.234641 + 0.972082i \(0.575392\pi\)
\(678\) 2821.70 0.159833
\(679\) −26659.9 −1.50680
\(680\) −51893.3 −2.92650
\(681\) 1584.01 0.0891326
\(682\) −10702.0 −0.600883
\(683\) −27099.4 −1.51820 −0.759098 0.650976i \(-0.774360\pi\)
−0.759098 + 0.650976i \(0.774360\pi\)
\(684\) 38462.2 2.15006
\(685\) 14171.2 0.790444
\(686\) 12353.6 0.687555
\(687\) −3564.90 −0.197976
\(688\) 0 0
\(689\) −8658.77 −0.478770
\(690\) 1649.49 0.0910073
\(691\) −18247.5 −1.00458 −0.502292 0.864698i \(-0.667510\pi\)
−0.502292 + 0.864698i \(0.667510\pi\)
\(692\) 27594.6 1.51588
\(693\) 10859.5 0.595263
\(694\) 39381.9 2.15406
\(695\) −18663.4 −1.01862
\(696\) 3305.91 0.180043
\(697\) 6148.88 0.334154
\(698\) −13151.8 −0.713183
\(699\) 2169.40 0.117388
\(700\) −117483. −6.34351
\(701\) 27373.3 1.47486 0.737430 0.675424i \(-0.236039\pi\)
0.737430 + 0.675424i \(0.236039\pi\)
\(702\) −16594.7 −0.892205
\(703\) 12705.2 0.681629
\(704\) −12160.6 −0.651025
\(705\) 1764.46 0.0942599
\(706\) −522.415 −0.0278489
\(707\) 18620.4 0.990510
\(708\) −5501.88 −0.292053
\(709\) −12752.9 −0.675524 −0.337762 0.941232i \(-0.609670\pi\)
−0.337762 + 0.941232i \(0.609670\pi\)
\(710\) 79630.8 4.20914
\(711\) −21604.8 −1.13958
\(712\) −10116.9 −0.532507
\(713\) 1780.21 0.0935057
\(714\) −9892.51 −0.518513
\(715\) −21341.1 −1.11624
\(716\) −58170.7 −3.03623
\(717\) 2445.77 0.127391
\(718\) −55167.8 −2.86747
\(719\) −26662.5 −1.38295 −0.691477 0.722398i \(-0.743040\pi\)
−0.691477 + 0.722398i \(0.743040\pi\)
\(720\) −20699.5 −1.07142
\(721\) −7634.05 −0.394323
\(722\) 15872.3 0.818154
\(723\) −280.328 −0.0144198
\(724\) 44710.6 2.29511
\(725\) −28641.1 −1.46718
\(726\) 5652.85 0.288976
\(727\) 27767.2 1.41655 0.708273 0.705939i \(-0.249475\pi\)
0.708273 + 0.705939i \(0.249475\pi\)
\(728\) 44899.6 2.28584
\(729\) −14062.0 −0.714423
\(730\) −119330. −6.05016
\(731\) 0 0
\(732\) −6071.14 −0.306552
\(733\) 5819.92 0.293265 0.146633 0.989191i \(-0.453156\pi\)
0.146633 + 0.989191i \(0.453156\pi\)
\(734\) −62591.6 −3.14754
\(735\) −5808.79 −0.291511
\(736\) 1136.18 0.0569023
\(737\) 654.514 0.0327128
\(738\) 10150.5 0.506295
\(739\) −4469.69 −0.222490 −0.111245 0.993793i \(-0.535484\pi\)
−0.111245 + 0.993793i \(0.535484\pi\)
\(740\) −39853.4 −1.97978
\(741\) −6650.18 −0.329690
\(742\) −17486.7 −0.865172
\(743\) −31260.1 −1.54350 −0.771752 0.635924i \(-0.780619\pi\)
−0.771752 + 0.635924i \(0.780619\pi\)
\(744\) 4827.94 0.237904
\(745\) 24651.6 1.21230
\(746\) 47676.1 2.33987
\(747\) 28559.6 1.39885
\(748\) −19425.4 −0.949548
\(749\) −15600.9 −0.761076
\(750\) −24020.0 −1.16945
\(751\) 9943.65 0.483154 0.241577 0.970382i \(-0.422335\pi\)
0.241577 + 0.970382i \(0.422335\pi\)
\(752\) −2709.85 −0.131407
\(753\) 2345.63 0.113519
\(754\) 23717.0 1.14552
\(755\) 29403.8 1.41737
\(756\) −21783.7 −1.04797
\(757\) −11664.7 −0.560056 −0.280028 0.959992i \(-0.590344\pi\)
−0.280028 + 0.959992i \(0.590344\pi\)
\(758\) 8138.02 0.389955
\(759\) 284.973 0.0136283
\(760\) −70451.5 −3.36256
\(761\) −23261.8 −1.10807 −0.554033 0.832495i \(-0.686912\pi\)
−0.554033 + 0.832495i \(0.686912\pi\)
\(762\) −14517.2 −0.690163
\(763\) −13842.0 −0.656768
\(764\) −5843.25 −0.276703
\(765\) 40620.1 1.91977
\(766\) −14512.1 −0.684522
\(767\) −18216.9 −0.857595
\(768\) 8291.50 0.389575
\(769\) −12725.0 −0.596717 −0.298358 0.954454i \(-0.596439\pi\)
−0.298358 + 0.954454i \(0.596439\pi\)
\(770\) −43099.2 −2.01713
\(771\) 3135.92 0.146482
\(772\) 19747.0 0.920609
\(773\) 31896.0 1.48411 0.742056 0.670337i \(-0.233851\pi\)
0.742056 + 0.670337i \(0.233851\pi\)
\(774\) 0 0
\(775\) −41827.3 −1.93868
\(776\) 36333.5 1.68079
\(777\) −3506.35 −0.161892
\(778\) 45109.3 2.07872
\(779\) 8347.85 0.383945
\(780\) 20860.1 0.957580
\(781\) 13757.3 0.630316
\(782\) 4971.25 0.227329
\(783\) −5310.61 −0.242383
\(784\) 8921.14 0.406393
\(785\) 60909.1 2.76935
\(786\) 14413.9 0.654107
\(787\) −3554.63 −0.161003 −0.0805013 0.996755i \(-0.525652\pi\)
−0.0805013 + 0.996755i \(0.525652\pi\)
\(788\) −7927.44 −0.358380
\(789\) 1868.76 0.0843214
\(790\) 85745.5 3.86163
\(791\) 12264.1 0.551281
\(792\) −14799.8 −0.664001
\(793\) −20101.8 −0.900171
\(794\) 23635.8 1.05643
\(795\) −3749.54 −0.167274
\(796\) −32487.9 −1.44661
\(797\) 15944.6 0.708640 0.354320 0.935124i \(-0.384712\pi\)
0.354320 + 0.935124i \(0.384712\pi\)
\(798\) −13430.3 −0.595773
\(799\) 5317.73 0.235454
\(800\) −26695.3 −1.17977
\(801\) 7919.09 0.349322
\(802\) −48473.2 −2.13423
\(803\) −20616.0 −0.906006
\(804\) −639.762 −0.0280630
\(805\) 7169.28 0.313893
\(806\) 34636.2 1.51366
\(807\) 4133.79 0.180318
\(808\) −25376.7 −1.10489
\(809\) 12758.5 0.554467 0.277234 0.960803i \(-0.410582\pi\)
0.277234 + 0.960803i \(0.410582\pi\)
\(810\) 63370.8 2.74892
\(811\) 33232.7 1.43891 0.719457 0.694537i \(-0.244391\pi\)
0.719457 + 0.694537i \(0.244391\pi\)
\(812\) 31133.0 1.34551
\(813\) 8394.44 0.362123
\(814\) −10592.8 −0.456113
\(815\) 42406.7 1.82263
\(816\) 3257.70 0.139758
\(817\) 0 0
\(818\) −3539.81 −0.151304
\(819\) −35145.7 −1.49950
\(820\) −26185.4 −1.11516
\(821\) 1216.94 0.0517314 0.0258657 0.999665i \(-0.491766\pi\)
0.0258657 + 0.999665i \(0.491766\pi\)
\(822\) −3681.72 −0.156222
\(823\) 4475.87 0.189574 0.0947869 0.995498i \(-0.469783\pi\)
0.0947869 + 0.995498i \(0.469783\pi\)
\(824\) 10404.1 0.439858
\(825\) −6695.61 −0.282559
\(826\) −36789.8 −1.54973
\(827\) 38299.6 1.61041 0.805205 0.592997i \(-0.202055\pi\)
0.805205 + 0.592997i \(0.202055\pi\)
\(828\) 5334.17 0.223883
\(829\) 14373.6 0.602192 0.301096 0.953594i \(-0.402648\pi\)
0.301096 + 0.953594i \(0.402648\pi\)
\(830\) −113348. −4.74020
\(831\) −5669.59 −0.236674
\(832\) 39356.8 1.63997
\(833\) −17506.6 −0.728172
\(834\) 4848.80 0.201319
\(835\) −37352.8 −1.54808
\(836\) −26372.3 −1.09104
\(837\) −7755.59 −0.320278
\(838\) 57522.5 2.37122
\(839\) 20770.8 0.854691 0.427346 0.904088i \(-0.359449\pi\)
0.427346 + 0.904088i \(0.359449\pi\)
\(840\) 19443.1 0.798630
\(841\) −16799.1 −0.688800
\(842\) −6126.43 −0.250749
\(843\) 3858.13 0.157629
\(844\) −75208.6 −3.06728
\(845\) 22269.2 0.906609
\(846\) 8778.47 0.356749
\(847\) 24569.3 0.996708
\(848\) 5758.54 0.233195
\(849\) 3635.96 0.146980
\(850\) −116803. −4.71329
\(851\) 1762.04 0.0709774
\(852\) −13447.3 −0.540723
\(853\) 21536.0 0.864455 0.432228 0.901765i \(-0.357728\pi\)
0.432228 + 0.901765i \(0.357728\pi\)
\(854\) −40596.3 −1.62667
\(855\) 55146.7 2.20582
\(856\) 21261.7 0.848962
\(857\) 34255.5 1.36540 0.682699 0.730700i \(-0.260806\pi\)
0.682699 + 0.730700i \(0.260806\pi\)
\(858\) 5544.48 0.220612
\(859\) 39691.7 1.57656 0.788279 0.615317i \(-0.210972\pi\)
0.788279 + 0.615317i \(0.210972\pi\)
\(860\) 0 0
\(861\) −2303.82 −0.0911894
\(862\) −26473.9 −1.04606
\(863\) 10554.4 0.416310 0.208155 0.978096i \(-0.433254\pi\)
0.208155 + 0.978096i \(0.433254\pi\)
\(864\) −4949.82 −0.194903
\(865\) 39564.9 1.55520
\(866\) 48333.7 1.89659
\(867\) −705.686 −0.0276429
\(868\) 45466.5 1.77792
\(869\) 14813.7 0.578276
\(870\) 10270.3 0.400224
\(871\) −2118.27 −0.0824053
\(872\) 18864.6 0.732608
\(873\) −28440.5 −1.10259
\(874\) 6749.08 0.261202
\(875\) −104400. −4.03354
\(876\) 20151.3 0.777227
\(877\) −2628.97 −0.101224 −0.0506122 0.998718i \(-0.516117\pi\)
−0.0506122 + 0.998718i \(0.516117\pi\)
\(878\) 62178.7 2.39001
\(879\) 5027.02 0.192898
\(880\) 14193.0 0.543688
\(881\) −44531.4 −1.70295 −0.851476 0.524394i \(-0.824292\pi\)
−0.851476 + 0.524394i \(0.824292\pi\)
\(882\) −28899.7 −1.10329
\(883\) 12638.0 0.481657 0.240828 0.970568i \(-0.422581\pi\)
0.240828 + 0.970568i \(0.422581\pi\)
\(884\) 62868.5 2.39196
\(885\) −7888.55 −0.299628
\(886\) 60413.8 2.29079
\(887\) −47366.5 −1.79302 −0.896511 0.443022i \(-0.853906\pi\)
−0.896511 + 0.443022i \(0.853906\pi\)
\(888\) 4778.63 0.180586
\(889\) −63097.2 −2.38044
\(890\) −31429.4 −1.18373
\(891\) 10948.2 0.411648
\(892\) −80569.1 −3.02427
\(893\) 7219.47 0.270538
\(894\) −6404.53 −0.239597
\(895\) −83404.7 −3.11499
\(896\) 63857.2 2.38094
\(897\) −922.288 −0.0343303
\(898\) 7661.99 0.284726
\(899\) 11084.2 0.411211
\(900\) −125330. −4.64184
\(901\) −11300.4 −0.417837
\(902\) −6959.89 −0.256917
\(903\) 0 0
\(904\) −16714.2 −0.614940
\(905\) 64105.7 2.35464
\(906\) −7639.18 −0.280127
\(907\) 3162.85 0.115789 0.0578946 0.998323i \(-0.481561\pi\)
0.0578946 + 0.998323i \(0.481561\pi\)
\(908\) −20329.9 −0.743031
\(909\) 19863.9 0.724802
\(910\) 139487. 5.08126
\(911\) −13963.5 −0.507829 −0.253914 0.967227i \(-0.581718\pi\)
−0.253914 + 0.967227i \(0.581718\pi\)
\(912\) 4422.72 0.160582
\(913\) −19582.4 −0.709840
\(914\) −59218.1 −2.14306
\(915\) −8704.75 −0.314503
\(916\) 45753.7 1.65037
\(917\) 62648.2 2.25608
\(918\) −21657.5 −0.778653
\(919\) 5006.75 0.179714 0.0898572 0.995955i \(-0.471359\pi\)
0.0898572 + 0.995955i \(0.471359\pi\)
\(920\) −9770.65 −0.350140
\(921\) 9469.40 0.338792
\(922\) 37362.0 1.33455
\(923\) −44524.4 −1.58780
\(924\) 7278.18 0.259128
\(925\) −41400.1 −1.47160
\(926\) −37425.7 −1.32817
\(927\) −8143.91 −0.288545
\(928\) 7074.22 0.250240
\(929\) −35356.7 −1.24867 −0.624336 0.781156i \(-0.714630\pi\)
−0.624336 + 0.781156i \(0.714630\pi\)
\(930\) 14998.6 0.528844
\(931\) −23767.3 −0.836673
\(932\) −27843.2 −0.978577
\(933\) 5709.89 0.200357
\(934\) −1782.38 −0.0624425
\(935\) −27851.9 −0.974177
\(936\) 47898.3 1.67266
\(937\) −42579.5 −1.48454 −0.742269 0.670103i \(-0.766250\pi\)
−0.742269 + 0.670103i \(0.766250\pi\)
\(938\) −4277.94 −0.148912
\(939\) −4640.58 −0.161278
\(940\) −22645.9 −0.785774
\(941\) −24475.5 −0.847905 −0.423952 0.905685i \(-0.639358\pi\)
−0.423952 + 0.905685i \(0.639358\pi\)
\(942\) −15824.3 −0.547330
\(943\) 1157.73 0.0399798
\(944\) 12115.2 0.417709
\(945\) −31233.3 −1.07515
\(946\) 0 0
\(947\) 5042.58 0.173032 0.0865162 0.996250i \(-0.472427\pi\)
0.0865162 + 0.996250i \(0.472427\pi\)
\(948\) −14479.9 −0.496080
\(949\) 66721.8 2.28228
\(950\) −158574. −5.41560
\(951\) −1815.16 −0.0618934
\(952\) 58597.7 1.99492
\(953\) 22662.7 0.770323 0.385161 0.922849i \(-0.374146\pi\)
0.385161 + 0.922849i \(0.374146\pi\)
\(954\) −18654.6 −0.633087
\(955\) −8378.00 −0.283880
\(956\) −31390.2 −1.06196
\(957\) 1774.33 0.0599331
\(958\) 29093.4 0.981173
\(959\) −16002.1 −0.538826
\(960\) 17042.8 0.572974
\(961\) −13603.7 −0.456638
\(962\) 34282.5 1.14897
\(963\) −16642.9 −0.556915
\(964\) 3597.87 0.120207
\(965\) 28313.1 0.944488
\(966\) −1862.60 −0.0620373
\(967\) 20319.2 0.675720 0.337860 0.941196i \(-0.390297\pi\)
0.337860 + 0.941196i \(0.390297\pi\)
\(968\) −33484.3 −1.11180
\(969\) −8679.03 −0.287730
\(970\) 112875. 3.73628
\(971\) 2458.32 0.0812474 0.0406237 0.999175i \(-0.487066\pi\)
0.0406237 + 0.999175i \(0.487066\pi\)
\(972\) −35153.5 −1.16003
\(973\) 21074.6 0.694370
\(974\) −7913.41 −0.260331
\(975\) 21669.8 0.711782
\(976\) 13368.8 0.438446
\(977\) 56845.4 1.86146 0.930729 0.365709i \(-0.119174\pi\)
0.930729 + 0.365709i \(0.119174\pi\)
\(978\) −11017.4 −0.360222
\(979\) −5429.87 −0.177262
\(980\) 74552.8 2.43010
\(981\) −14766.5 −0.480588
\(982\) −70584.4 −2.29372
\(983\) 9749.84 0.316350 0.158175 0.987411i \(-0.449439\pi\)
0.158175 + 0.987411i \(0.449439\pi\)
\(984\) 3139.77 0.101720
\(985\) −11366.3 −0.367675
\(986\) 30952.6 0.999728
\(987\) −1992.42 −0.0642546
\(988\) 85351.6 2.74838
\(989\) 0 0
\(990\) −45977.7 −1.47603
\(991\) −44926.5 −1.44010 −0.720050 0.693922i \(-0.755881\pi\)
−0.720050 + 0.693922i \(0.755881\pi\)
\(992\) 10331.2 0.330660
\(993\) 3458.68 0.110531
\(994\) −89918.7 −2.86926
\(995\) −46580.9 −1.48413
\(996\) 19141.1 0.608944
\(997\) −1536.04 −0.0487934 −0.0243967 0.999702i \(-0.507766\pi\)
−0.0243967 + 0.999702i \(0.507766\pi\)
\(998\) 28156.6 0.893068
\(999\) −7676.39 −0.243113
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.1.44 yes 50
43.42 odd 2 1849.4.a.i.1.7 50
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1849.4.a.i.1.7 50 43.42 odd 2
1849.4.a.j.1.44 yes 50 1.1 even 1 trivial