Properties

Label 1849.4.a.i.1.47
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 1.47
Character \(\chi\) \(=\) 1849.1

$q$-expansion

\(f(q)\) \(=\) \(q+4.75171 q^{2} -7.02039 q^{3} +14.5788 q^{4} -4.22415 q^{5} -33.3589 q^{6} -11.1787 q^{7} +31.2605 q^{8} +22.2859 q^{9} +O(q^{10})\) \(q+4.75171 q^{2} -7.02039 q^{3} +14.5788 q^{4} -4.22415 q^{5} -33.3589 q^{6} -11.1787 q^{7} +31.2605 q^{8} +22.2859 q^{9} -20.0719 q^{10} -43.0911 q^{11} -102.349 q^{12} +62.9713 q^{13} -53.1182 q^{14} +29.6552 q^{15} +31.9107 q^{16} +136.464 q^{17} +105.896 q^{18} +108.392 q^{19} -61.5829 q^{20} +78.4792 q^{21} -204.757 q^{22} -142.408 q^{23} -219.461 q^{24} -107.157 q^{25} +299.222 q^{26} +33.0947 q^{27} -162.973 q^{28} +149.025 q^{29} +140.913 q^{30} -29.2265 q^{31} -98.4535 q^{32} +302.516 q^{33} +648.439 q^{34} +47.2206 q^{35} +324.902 q^{36} +69.6966 q^{37} +515.047 q^{38} -442.084 q^{39} -132.049 q^{40} +128.292 q^{41} +372.911 q^{42} -628.216 q^{44} -94.1390 q^{45} -676.684 q^{46} -95.1118 q^{47} -224.026 q^{48} -218.036 q^{49} -509.177 q^{50} -958.032 q^{51} +918.046 q^{52} -416.499 q^{53} +157.256 q^{54} +182.023 q^{55} -349.453 q^{56} -760.953 q^{57} +708.123 q^{58} -644.210 q^{59} +432.336 q^{60} -228.604 q^{61} -138.876 q^{62} -249.129 q^{63} -723.109 q^{64} -266.000 q^{65} +1437.47 q^{66} -877.104 q^{67} +1989.48 q^{68} +999.763 q^{69} +224.379 q^{70} -258.534 q^{71} +696.669 q^{72} -396.446 q^{73} +331.179 q^{74} +752.281 q^{75} +1580.22 q^{76} +481.704 q^{77} -2100.65 q^{78} +916.890 q^{79} -134.796 q^{80} -834.058 q^{81} +609.605 q^{82} -66.9221 q^{83} +1144.13 q^{84} -576.445 q^{85} -1046.21 q^{87} -1347.05 q^{88} +365.655 q^{89} -447.322 q^{90} -703.941 q^{91} -2076.14 q^{92} +205.182 q^{93} -451.944 q^{94} -457.863 q^{95} +691.182 q^{96} +1014.49 q^{97} -1036.04 q^{98} -960.325 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 50q - 10q^{2} - 2q^{3} + 186q^{4} + 8q^{5} - 51q^{6} - 6q^{7} - 138q^{8} + 360q^{9} + O(q^{10}) \) \( 50q - 10q^{2} - 2q^{3} + 186q^{4} + 8q^{5} - 51q^{6} - 6q^{7} - 138q^{8} + 360q^{9} - 137q^{10} - 252q^{11} - 48q^{12} - 192q^{13} - 272q^{14} - 314q^{15} + 542q^{16} - 236q^{17} - 386q^{18} + 12q^{19} + 108q^{20} - 408q^{21} + 1235q^{22} - 630q^{23} - 613q^{24} + 1098q^{25} + 1493q^{26} + 10q^{27} - 242q^{28} + 208q^{29} - 48q^{30} - 932q^{31} - 1124q^{32} + 254q^{33} + 765q^{34} - 1452q^{35} + 747q^{36} - 90q^{37} - 1213q^{38} - 1610q^{39} - 1693q^{40} - 1354q^{41} - 16q^{42} - 2704q^{44} + 4508q^{45} + 233q^{46} - 3484q^{47} - 376q^{48} + 1324q^{49} - 408q^{50} + 4054q^{51} - 2176q^{52} - 726q^{53} - 6497q^{54} - 3288q^{55} - 7097q^{56} - 870q^{57} + 275q^{58} - 4370q^{59} - 3891q^{60} + 1172q^{61} - 1546q^{62} - 3686q^{63} + 606q^{64} + 2610q^{65} - 4697q^{66} - 344q^{67} - 3221q^{68} + 136q^{69} - 1310q^{70} + 162q^{71} - 5814q^{72} + 746q^{73} - 4332q^{74} + 236q^{75} + 1338q^{76} + 2024q^{77} - 2782q^{78} - 2656q^{79} - 5713q^{80} - 86q^{81} - 4168q^{82} - 3514q^{83} - 4269q^{84} - 7558q^{85} - 10278q^{87} + 11692q^{88} + 2640q^{89} - 8286q^{90} - 5946q^{91} - 4271q^{92} - 2q^{93} + 9062q^{94} - 12140q^{95} - 700q^{96} - 3864q^{97} - 2826q^{98} - 8174q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.75171 1.67998 0.839992 0.542598i \(-0.182559\pi\)
0.839992 + 0.542598i \(0.182559\pi\)
\(3\) −7.02039 −1.35108 −0.675538 0.737325i \(-0.736088\pi\)
−0.675538 + 0.737325i \(0.736088\pi\)
\(4\) 14.5788 1.82235
\(5\) −4.22415 −0.377819 −0.188910 0.981994i \(-0.560495\pi\)
−0.188910 + 0.981994i \(0.560495\pi\)
\(6\) −33.3589 −2.26979
\(7\) −11.1787 −0.603595 −0.301798 0.953372i \(-0.597587\pi\)
−0.301798 + 0.953372i \(0.597587\pi\)
\(8\) 31.2605 1.38153
\(9\) 22.2859 0.825405
\(10\) −20.0719 −0.634730
\(11\) −43.0911 −1.18113 −0.590566 0.806989i \(-0.701095\pi\)
−0.590566 + 0.806989i \(0.701095\pi\)
\(12\) −102.349 −2.46213
\(13\) 62.9713 1.34347 0.671735 0.740792i \(-0.265549\pi\)
0.671735 + 0.740792i \(0.265549\pi\)
\(14\) −53.1182 −1.01403
\(15\) 29.6552 0.510462
\(16\) 31.9107 0.498605
\(17\) 136.464 1.94691 0.973454 0.228884i \(-0.0735077\pi\)
0.973454 + 0.228884i \(0.0735077\pi\)
\(18\) 105.896 1.38667
\(19\) 108.392 1.30878 0.654389 0.756158i \(-0.272926\pi\)
0.654389 + 0.756158i \(0.272926\pi\)
\(20\) −61.5829 −0.688518
\(21\) 78.4792 0.815503
\(22\) −204.757 −1.98428
\(23\) −142.408 −1.29105 −0.645526 0.763738i \(-0.723362\pi\)
−0.645526 + 0.763738i \(0.723362\pi\)
\(24\) −219.461 −1.86655
\(25\) −107.157 −0.857253
\(26\) 299.222 2.25701
\(27\) 33.0947 0.235892
\(28\) −162.973 −1.09996
\(29\) 149.025 0.954249 0.477124 0.878836i \(-0.341679\pi\)
0.477124 + 0.878836i \(0.341679\pi\)
\(30\) 140.913 0.857568
\(31\) −29.2265 −0.169330 −0.0846652 0.996409i \(-0.526982\pi\)
−0.0846652 + 0.996409i \(0.526982\pi\)
\(32\) −98.4535 −0.543884
\(33\) 302.516 1.59580
\(34\) 648.439 3.27077
\(35\) 47.2206 0.228050
\(36\) 324.902 1.50417
\(37\) 69.6966 0.309677 0.154839 0.987940i \(-0.450514\pi\)
0.154839 + 0.987940i \(0.450514\pi\)
\(38\) 515.047 2.19873
\(39\) −442.084 −1.81513
\(40\) −132.049 −0.521969
\(41\) 128.292 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(42\) 372.911 1.37003
\(43\) 0 0
\(44\) −628.216 −2.15243
\(45\) −94.1390 −0.311854
\(46\) −676.684 −2.16895
\(47\) −95.1118 −0.295180 −0.147590 0.989049i \(-0.547152\pi\)
−0.147590 + 0.989049i \(0.547152\pi\)
\(48\) −224.026 −0.673653
\(49\) −218.036 −0.635673
\(50\) −509.177 −1.44017
\(51\) −958.032 −2.63042
\(52\) 918.046 2.44827
\(53\) −416.499 −1.07944 −0.539722 0.841843i \(-0.681471\pi\)
−0.539722 + 0.841843i \(0.681471\pi\)
\(54\) 157.256 0.396294
\(55\) 182.023 0.446254
\(56\) −349.453 −0.833887
\(57\) −760.953 −1.76826
\(58\) 708.123 1.60312
\(59\) −644.210 −1.42151 −0.710754 0.703440i \(-0.751646\pi\)
−0.710754 + 0.703440i \(0.751646\pi\)
\(60\) 432.336 0.930240
\(61\) −228.604 −0.479832 −0.239916 0.970794i \(-0.577120\pi\)
−0.239916 + 0.970794i \(0.577120\pi\)
\(62\) −138.876 −0.284472
\(63\) −249.129 −0.498210
\(64\) −723.109 −1.41232
\(65\) −266.000 −0.507589
\(66\) 1437.47 2.68092
\(67\) −877.104 −1.59933 −0.799667 0.600444i \(-0.794990\pi\)
−0.799667 + 0.600444i \(0.794990\pi\)
\(68\) 1989.48 3.54794
\(69\) 999.763 1.74431
\(70\) 224.379 0.383120
\(71\) −258.534 −0.432145 −0.216073 0.976377i \(-0.569325\pi\)
−0.216073 + 0.976377i \(0.569325\pi\)
\(72\) 696.669 1.14032
\(73\) −396.446 −0.635624 −0.317812 0.948154i \(-0.602948\pi\)
−0.317812 + 0.948154i \(0.602948\pi\)
\(74\) 331.179 0.520253
\(75\) 752.281 1.15821
\(76\) 1580.22 2.38505
\(77\) 481.704 0.712926
\(78\) −2100.65 −3.04939
\(79\) 916.890 1.30580 0.652900 0.757444i \(-0.273552\pi\)
0.652900 + 0.757444i \(0.273552\pi\)
\(80\) −134.796 −0.188382
\(81\) −834.058 −1.14411
\(82\) 609.605 0.820971
\(83\) −66.9221 −0.0885019 −0.0442510 0.999020i \(-0.514090\pi\)
−0.0442510 + 0.999020i \(0.514090\pi\)
\(84\) 1144.13 1.48613
\(85\) −576.445 −0.735579
\(86\) 0 0
\(87\) −1046.21 −1.28926
\(88\) −1347.05 −1.63177
\(89\) 365.655 0.435498 0.217749 0.976005i \(-0.430129\pi\)
0.217749 + 0.976005i \(0.430129\pi\)
\(90\) −447.322 −0.523909
\(91\) −703.941 −0.810912
\(92\) −2076.14 −2.35275
\(93\) 205.182 0.228778
\(94\) −451.944 −0.495899
\(95\) −457.863 −0.494481
\(96\) 691.182 0.734828
\(97\) 1014.49 1.06192 0.530958 0.847398i \(-0.321832\pi\)
0.530958 + 0.847398i \(0.321832\pi\)
\(98\) −1036.04 −1.06792
\(99\) −960.325 −0.974912
\(100\) −1562.21 −1.56221
\(101\) −1840.07 −1.81281 −0.906406 0.422408i \(-0.861185\pi\)
−0.906406 + 0.422408i \(0.861185\pi\)
\(102\) −4552.30 −4.41906
\(103\) −1091.88 −1.04452 −0.522261 0.852786i \(-0.674911\pi\)
−0.522261 + 0.852786i \(0.674911\pi\)
\(104\) 1968.52 1.85605
\(105\) −331.508 −0.308113
\(106\) −1979.09 −1.81345
\(107\) −110.329 −0.0996812 −0.0498406 0.998757i \(-0.515871\pi\)
−0.0498406 + 0.998757i \(0.515871\pi\)
\(108\) 482.480 0.429876
\(109\) −743.706 −0.653524 −0.326762 0.945107i \(-0.605958\pi\)
−0.326762 + 0.945107i \(0.605958\pi\)
\(110\) 864.921 0.749700
\(111\) −489.298 −0.418397
\(112\) −356.722 −0.300956
\(113\) 1133.22 0.943402 0.471701 0.881759i \(-0.343640\pi\)
0.471701 + 0.881759i \(0.343640\pi\)
\(114\) −3615.83 −2.97064
\(115\) 601.554 0.487784
\(116\) 2172.60 1.73897
\(117\) 1403.37 1.10891
\(118\) −3061.10 −2.38811
\(119\) −1525.50 −1.17514
\(120\) 927.036 0.705220
\(121\) 525.842 0.395073
\(122\) −1086.26 −0.806110
\(123\) −900.657 −0.660240
\(124\) −426.087 −0.308579
\(125\) 980.663 0.701706
\(126\) −1183.79 −0.836986
\(127\) 220.566 0.154111 0.0770555 0.997027i \(-0.475448\pi\)
0.0770555 + 0.997027i \(0.475448\pi\)
\(128\) −2648.38 −1.82879
\(129\) 0 0
\(130\) −1263.96 −0.852741
\(131\) 1305.39 0.870628 0.435314 0.900279i \(-0.356637\pi\)
0.435314 + 0.900279i \(0.356637\pi\)
\(132\) 4410.32 2.90810
\(133\) −1211.68 −0.789972
\(134\) −4167.75 −2.68686
\(135\) −139.797 −0.0891243
\(136\) 4265.94 2.68972
\(137\) 2023.98 1.26219 0.631096 0.775705i \(-0.282605\pi\)
0.631096 + 0.775705i \(0.282605\pi\)
\(138\) 4750.59 2.93041
\(139\) 324.650 0.198104 0.0990521 0.995082i \(-0.468419\pi\)
0.0990521 + 0.995082i \(0.468419\pi\)
\(140\) 688.420 0.415586
\(141\) 667.722 0.398811
\(142\) −1228.48 −0.725997
\(143\) −2713.50 −1.58682
\(144\) 711.160 0.411551
\(145\) −629.502 −0.360533
\(146\) −1883.80 −1.06784
\(147\) 1530.70 0.858842
\(148\) 1016.09 0.564340
\(149\) −1524.69 −0.838308 −0.419154 0.907915i \(-0.637673\pi\)
−0.419154 + 0.907915i \(0.637673\pi\)
\(150\) 3574.63 1.94578
\(151\) −3148.37 −1.69676 −0.848380 0.529388i \(-0.822421\pi\)
−0.848380 + 0.529388i \(0.822421\pi\)
\(152\) 3388.38 1.80812
\(153\) 3041.23 1.60699
\(154\) 2288.92 1.19770
\(155\) 123.457 0.0639762
\(156\) −6445.04 −3.30780
\(157\) −1808.52 −0.919335 −0.459667 0.888091i \(-0.652031\pi\)
−0.459667 + 0.888091i \(0.652031\pi\)
\(158\) 4356.80 2.19372
\(159\) 2923.99 1.45841
\(160\) 415.882 0.205490
\(161\) 1591.95 0.779273
\(162\) −3963.20 −1.92209
\(163\) −668.967 −0.321457 −0.160729 0.986999i \(-0.551384\pi\)
−0.160729 + 0.986999i \(0.551384\pi\)
\(164\) 1870.34 0.890541
\(165\) −1277.87 −0.602923
\(166\) −317.995 −0.148682
\(167\) −2632.15 −1.21965 −0.609825 0.792536i \(-0.708760\pi\)
−0.609825 + 0.792536i \(0.708760\pi\)
\(168\) 2453.30 1.12664
\(169\) 1768.39 0.804911
\(170\) −2739.10 −1.23576
\(171\) 2415.61 1.08027
\(172\) 0 0
\(173\) 2225.67 0.978119 0.489060 0.872250i \(-0.337340\pi\)
0.489060 + 0.872250i \(0.337340\pi\)
\(174\) −4971.30 −2.16594
\(175\) 1197.88 0.517434
\(176\) −1375.07 −0.588918
\(177\) 4522.61 1.92057
\(178\) 1737.49 0.731630
\(179\) −1594.43 −0.665774 −0.332887 0.942967i \(-0.608023\pi\)
−0.332887 + 0.942967i \(0.608023\pi\)
\(180\) −1372.43 −0.568306
\(181\) 3965.16 1.62833 0.814166 0.580632i \(-0.197194\pi\)
0.814166 + 0.580632i \(0.197194\pi\)
\(182\) −3344.92 −1.36232
\(183\) 1604.89 0.648289
\(184\) −4451.76 −1.78363
\(185\) −294.409 −0.117002
\(186\) 974.965 0.384344
\(187\) −5880.39 −2.29955
\(188\) −1386.61 −0.537922
\(189\) −369.957 −0.142383
\(190\) −2175.63 −0.830721
\(191\) 2496.73 0.945848 0.472924 0.881103i \(-0.343199\pi\)
0.472924 + 0.881103i \(0.343199\pi\)
\(192\) 5076.51 1.90815
\(193\) −2258.43 −0.842307 −0.421154 0.906989i \(-0.638375\pi\)
−0.421154 + 0.906989i \(0.638375\pi\)
\(194\) 4820.56 1.78400
\(195\) 1867.43 0.685790
\(196\) −3178.70 −1.15842
\(197\) −1867.33 −0.675338 −0.337669 0.941265i \(-0.609638\pi\)
−0.337669 + 0.941265i \(0.609638\pi\)
\(198\) −4563.19 −1.63784
\(199\) −3008.91 −1.07184 −0.535920 0.844269i \(-0.680035\pi\)
−0.535920 + 0.844269i \(0.680035\pi\)
\(200\) −3349.77 −1.18432
\(201\) 6157.62 2.16082
\(202\) −8743.50 −3.04550
\(203\) −1665.91 −0.575980
\(204\) −13966.9 −4.79354
\(205\) −541.922 −0.184632
\(206\) −5188.28 −1.75478
\(207\) −3173.70 −1.06564
\(208\) 2009.46 0.669861
\(209\) −4670.72 −1.54584
\(210\) −1575.23 −0.517624
\(211\) −253.754 −0.0827923 −0.0413961 0.999143i \(-0.513181\pi\)
−0.0413961 + 0.999143i \(0.513181\pi\)
\(212\) −6072.05 −1.96712
\(213\) 1815.01 0.583861
\(214\) −524.251 −0.167463
\(215\) 0 0
\(216\) 1034.56 0.325892
\(217\) 326.716 0.102207
\(218\) −3533.88 −1.09791
\(219\) 2783.21 0.858776
\(220\) 2653.68 0.813231
\(221\) 8593.33 2.61561
\(222\) −2325.00 −0.702901
\(223\) −2211.05 −0.663960 −0.331980 0.943286i \(-0.607717\pi\)
−0.331980 + 0.943286i \(0.607717\pi\)
\(224\) 1100.59 0.328286
\(225\) −2388.08 −0.707580
\(226\) 5384.74 1.58490
\(227\) −4508.16 −1.31814 −0.659068 0.752084i \(-0.729049\pi\)
−0.659068 + 0.752084i \(0.729049\pi\)
\(228\) −11093.8 −3.22238
\(229\) 4592.16 1.32515 0.662573 0.748997i \(-0.269464\pi\)
0.662573 + 0.748997i \(0.269464\pi\)
\(230\) 2858.41 0.819470
\(231\) −3381.75 −0.963217
\(232\) 4658.59 1.31833
\(233\) −5822.36 −1.63706 −0.818531 0.574463i \(-0.805211\pi\)
−0.818531 + 0.574463i \(0.805211\pi\)
\(234\) 6668.44 1.86295
\(235\) 401.766 0.111525
\(236\) −9391.80 −2.59048
\(237\) −6436.93 −1.76423
\(238\) −7248.73 −1.97422
\(239\) −2551.34 −0.690512 −0.345256 0.938509i \(-0.612208\pi\)
−0.345256 + 0.938509i \(0.612208\pi\)
\(240\) 946.318 0.254519
\(241\) 2606.50 0.696679 0.348339 0.937369i \(-0.386746\pi\)
0.348339 + 0.937369i \(0.386746\pi\)
\(242\) 2498.65 0.663716
\(243\) 4961.86 1.30989
\(244\) −3332.77 −0.874421
\(245\) 921.015 0.240169
\(246\) −4279.67 −1.10919
\(247\) 6825.57 1.75830
\(248\) −913.636 −0.233935
\(249\) 469.820 0.119573
\(250\) 4659.83 1.17885
\(251\) −936.255 −0.235442 −0.117721 0.993047i \(-0.537559\pi\)
−0.117721 + 0.993047i \(0.537559\pi\)
\(252\) −3631.99 −0.907913
\(253\) 6136.53 1.52490
\(254\) 1048.07 0.258904
\(255\) 4046.87 0.993822
\(256\) −6799.46 −1.66003
\(257\) −6041.18 −1.46630 −0.733149 0.680068i \(-0.761950\pi\)
−0.733149 + 0.680068i \(0.761950\pi\)
\(258\) 0 0
\(259\) −779.121 −0.186920
\(260\) −3877.96 −0.925003
\(261\) 3321.16 0.787641
\(262\) 6202.83 1.46264
\(263\) 5628.61 1.31968 0.659839 0.751407i \(-0.270625\pi\)
0.659839 + 0.751407i \(0.270625\pi\)
\(264\) 9456.82 2.20465
\(265\) 1759.35 0.407835
\(266\) −5757.57 −1.32714
\(267\) −2567.04 −0.588391
\(268\) −12787.1 −2.91454
\(269\) −4773.16 −1.08188 −0.540938 0.841062i \(-0.681931\pi\)
−0.540938 + 0.841062i \(0.681931\pi\)
\(270\) −664.274 −0.149727
\(271\) 2753.55 0.617219 0.308609 0.951189i \(-0.400136\pi\)
0.308609 + 0.951189i \(0.400136\pi\)
\(272\) 4354.67 0.970738
\(273\) 4941.94 1.09560
\(274\) 9617.38 2.12046
\(275\) 4617.49 1.01253
\(276\) 14575.3 3.17874
\(277\) 1587.11 0.344260 0.172130 0.985074i \(-0.444935\pi\)
0.172130 + 0.985074i \(0.444935\pi\)
\(278\) 1542.65 0.332812
\(279\) −651.340 −0.139766
\(280\) 1476.14 0.315058
\(281\) 3096.65 0.657405 0.328702 0.944434i \(-0.393389\pi\)
0.328702 + 0.944434i \(0.393389\pi\)
\(282\) 3172.82 0.669996
\(283\) 2960.70 0.621891 0.310946 0.950428i \(-0.399354\pi\)
0.310946 + 0.950428i \(0.399354\pi\)
\(284\) −3769.11 −0.787519
\(285\) 3214.38 0.668081
\(286\) −12893.8 −2.66583
\(287\) −1434.14 −0.294964
\(288\) −2194.13 −0.448924
\(289\) 13709.5 2.79045
\(290\) −2991.22 −0.605690
\(291\) −7122.12 −1.43473
\(292\) −5779.71 −1.15833
\(293\) 4920.02 0.980992 0.490496 0.871443i \(-0.336816\pi\)
0.490496 + 0.871443i \(0.336816\pi\)
\(294\) 7273.43 1.44284
\(295\) 2721.24 0.537073
\(296\) 2178.75 0.427829
\(297\) −1426.08 −0.278619
\(298\) −7244.91 −1.40834
\(299\) −8967.65 −1.73449
\(300\) 10967.4 2.11067
\(301\) 0 0
\(302\) −14960.1 −2.85053
\(303\) 12918.0 2.44925
\(304\) 3458.86 0.652563
\(305\) 965.657 0.181290
\(306\) 14451.1 2.69971
\(307\) −9438.56 −1.75468 −0.877340 0.479869i \(-0.840684\pi\)
−0.877340 + 0.479869i \(0.840684\pi\)
\(308\) 7022.66 1.29920
\(309\) 7665.40 1.41123
\(310\) 586.633 0.107479
\(311\) −3537.86 −0.645059 −0.322530 0.946559i \(-0.604533\pi\)
−0.322530 + 0.946559i \(0.604533\pi\)
\(312\) −13819.8 −2.50766
\(313\) 1481.13 0.267471 0.133736 0.991017i \(-0.457303\pi\)
0.133736 + 0.991017i \(0.457303\pi\)
\(314\) −8593.57 −1.54447
\(315\) 1052.36 0.188233
\(316\) 13367.1 2.37962
\(317\) −732.044 −0.129702 −0.0648512 0.997895i \(-0.520657\pi\)
−0.0648512 + 0.997895i \(0.520657\pi\)
\(318\) 13894.0 2.45011
\(319\) −6421.64 −1.12709
\(320\) 3054.52 0.533602
\(321\) 774.552 0.134677
\(322\) 7564.48 1.30917
\(323\) 14791.6 2.54807
\(324\) −12159.5 −2.08497
\(325\) −6747.79 −1.15169
\(326\) −3178.74 −0.540043
\(327\) 5221.11 0.882961
\(328\) 4010.46 0.675124
\(329\) 1063.23 0.178170
\(330\) −6072.09 −1.01290
\(331\) −417.381 −0.0693092 −0.0346546 0.999399i \(-0.511033\pi\)
−0.0346546 + 0.999399i \(0.511033\pi\)
\(332\) −975.643 −0.161281
\(333\) 1553.25 0.255609
\(334\) −12507.2 −2.04899
\(335\) 3705.02 0.604259
\(336\) 2504.33 0.406614
\(337\) −2917.52 −0.471594 −0.235797 0.971802i \(-0.575770\pi\)
−0.235797 + 0.971802i \(0.575770\pi\)
\(338\) 8402.88 1.35224
\(339\) −7955.65 −1.27461
\(340\) −8403.86 −1.34048
\(341\) 1259.40 0.200001
\(342\) 11478.3 1.81484
\(343\) 6271.67 0.987284
\(344\) 0 0
\(345\) −4223.14 −0.659033
\(346\) 10575.8 1.64323
\(347\) −2114.38 −0.327106 −0.163553 0.986535i \(-0.552295\pi\)
−0.163553 + 0.986535i \(0.552295\pi\)
\(348\) −15252.5 −2.34948
\(349\) −2209.08 −0.338823 −0.169412 0.985545i \(-0.554187\pi\)
−0.169412 + 0.985545i \(0.554187\pi\)
\(350\) 5691.96 0.869281
\(351\) 2084.02 0.316913
\(352\) 4242.47 0.642398
\(353\) −9680.14 −1.45955 −0.729776 0.683686i \(-0.760376\pi\)
−0.729776 + 0.683686i \(0.760376\pi\)
\(354\) 21490.1 3.22652
\(355\) 1092.08 0.163273
\(356\) 5330.80 0.793629
\(357\) 10709.6 1.58771
\(358\) −7576.30 −1.11849
\(359\) −6934.75 −1.01951 −0.509753 0.860321i \(-0.670263\pi\)
−0.509753 + 0.860321i \(0.670263\pi\)
\(360\) −2942.83 −0.430836
\(361\) 4889.77 0.712899
\(362\) 18841.3 2.73557
\(363\) −3691.62 −0.533773
\(364\) −10262.6 −1.47776
\(365\) 1674.65 0.240151
\(366\) 7625.98 1.08912
\(367\) −2151.01 −0.305945 −0.152972 0.988230i \(-0.548885\pi\)
−0.152972 + 0.988230i \(0.548885\pi\)
\(368\) −4544.35 −0.643725
\(369\) 2859.10 0.403357
\(370\) −1398.95 −0.196562
\(371\) 4655.94 0.651548
\(372\) 2991.30 0.416913
\(373\) −2182.22 −0.302925 −0.151463 0.988463i \(-0.548398\pi\)
−0.151463 + 0.988463i \(0.548398\pi\)
\(374\) −27941.9 −3.86322
\(375\) −6884.64 −0.948057
\(376\) −2973.24 −0.407801
\(377\) 9384.29 1.28200
\(378\) −1757.93 −0.239201
\(379\) −1718.17 −0.232866 −0.116433 0.993199i \(-0.537146\pi\)
−0.116433 + 0.993199i \(0.537146\pi\)
\(380\) −6675.08 −0.901117
\(381\) −1548.46 −0.208216
\(382\) 11863.7 1.58901
\(383\) 435.777 0.0581388 0.0290694 0.999577i \(-0.490746\pi\)
0.0290694 + 0.999577i \(0.490746\pi\)
\(384\) 18592.6 2.47084
\(385\) −2034.79 −0.269357
\(386\) −10731.4 −1.41506
\(387\) 0 0
\(388\) 14790.0 1.93518
\(389\) −2870.76 −0.374173 −0.187087 0.982343i \(-0.559904\pi\)
−0.187087 + 0.982343i \(0.559904\pi\)
\(390\) 8873.47 1.15212
\(391\) −19433.6 −2.51356
\(392\) −6815.91 −0.878202
\(393\) −9164.34 −1.17628
\(394\) −8873.01 −1.13456
\(395\) −3873.08 −0.493356
\(396\) −14000.4 −1.77663
\(397\) −9081.90 −1.14813 −0.574065 0.818810i \(-0.694634\pi\)
−0.574065 + 0.818810i \(0.694634\pi\)
\(398\) −14297.5 −1.80067
\(399\) 8506.50 1.06731
\(400\) −3419.44 −0.427430
\(401\) 4802.23 0.598035 0.299017 0.954248i \(-0.403341\pi\)
0.299017 + 0.954248i \(0.403341\pi\)
\(402\) 29259.2 3.63014
\(403\) −1840.43 −0.227490
\(404\) −26826.0 −3.30357
\(405\) 3523.18 0.432267
\(406\) −7915.93 −0.967638
\(407\) −3003.30 −0.365770
\(408\) −29948.6 −3.63401
\(409\) 3318.18 0.401158 0.200579 0.979678i \(-0.435718\pi\)
0.200579 + 0.979678i \(0.435718\pi\)
\(410\) −2575.06 −0.310178
\(411\) −14209.1 −1.70532
\(412\) −15918.2 −1.90348
\(413\) 7201.46 0.858016
\(414\) −15080.5 −1.79026
\(415\) 282.689 0.0334377
\(416\) −6199.75 −0.730691
\(417\) −2279.17 −0.267654
\(418\) −22193.9 −2.59699
\(419\) 9133.20 1.06488 0.532442 0.846467i \(-0.321275\pi\)
0.532442 + 0.846467i \(0.321275\pi\)
\(420\) −4832.98 −0.561488
\(421\) −12874.9 −1.49046 −0.745232 0.666806i \(-0.767661\pi\)
−0.745232 + 0.666806i \(0.767661\pi\)
\(422\) −1205.77 −0.139090
\(423\) −2119.65 −0.243643
\(424\) −13020.0 −1.49129
\(425\) −14623.0 −1.66899
\(426\) 8624.40 0.980877
\(427\) 2555.51 0.289624
\(428\) −1608.46 −0.181654
\(429\) 19049.9 2.14391
\(430\) 0 0
\(431\) 5433.33 0.607226 0.303613 0.952795i \(-0.401807\pi\)
0.303613 + 0.952795i \(0.401807\pi\)
\(432\) 1056.07 0.117617
\(433\) 8137.92 0.903195 0.451598 0.892222i \(-0.350854\pi\)
0.451598 + 0.892222i \(0.350854\pi\)
\(434\) 1552.46 0.171706
\(435\) 4419.35 0.487108
\(436\) −10842.3 −1.19095
\(437\) −15435.9 −1.68970
\(438\) 13225.0 1.44273
\(439\) 2828.77 0.307540 0.153770 0.988107i \(-0.450859\pi\)
0.153770 + 0.988107i \(0.450859\pi\)
\(440\) 5690.13 0.616515
\(441\) −4859.13 −0.524687
\(442\) 40833.1 4.39419
\(443\) −1724.64 −0.184966 −0.0924830 0.995714i \(-0.529480\pi\)
−0.0924830 + 0.995714i \(0.529480\pi\)
\(444\) −7133.37 −0.762466
\(445\) −1544.58 −0.164540
\(446\) −10506.3 −1.11544
\(447\) 10704.0 1.13262
\(448\) 8083.44 0.852471
\(449\) −18393.7 −1.93331 −0.966653 0.256090i \(-0.917566\pi\)
−0.966653 + 0.256090i \(0.917566\pi\)
\(450\) −11347.5 −1.18872
\(451\) −5528.22 −0.577193
\(452\) 16521.0 1.71921
\(453\) 22102.8 2.29245
\(454\) −21421.5 −2.21445
\(455\) 2973.55 0.306378
\(456\) −23787.8 −2.44290
\(457\) −2439.34 −0.249689 −0.124844 0.992176i \(-0.539843\pi\)
−0.124844 + 0.992176i \(0.539843\pi\)
\(458\) 21820.6 2.22622
\(459\) 4516.24 0.459259
\(460\) 8769.92 0.888913
\(461\) 11301.0 1.14173 0.570866 0.821043i \(-0.306608\pi\)
0.570866 + 0.821043i \(0.306608\pi\)
\(462\) −16069.1 −1.61819
\(463\) 10842.6 1.08833 0.544165 0.838979i \(-0.316847\pi\)
0.544165 + 0.838979i \(0.316847\pi\)
\(464\) 4755.49 0.475793
\(465\) −866.718 −0.0864367
\(466\) −27666.2 −2.75024
\(467\) 6330.48 0.627280 0.313640 0.949542i \(-0.398452\pi\)
0.313640 + 0.949542i \(0.398452\pi\)
\(468\) 20459.5 2.02081
\(469\) 9804.92 0.965350
\(470\) 1909.08 0.187360
\(471\) 12696.5 1.24209
\(472\) −20138.3 −1.96386
\(473\) 0 0
\(474\) −30586.4 −2.96389
\(475\) −11614.9 −1.12195
\(476\) −22239.9 −2.14152
\(477\) −9282.07 −0.890979
\(478\) −12123.2 −1.16005
\(479\) −8678.52 −0.827832 −0.413916 0.910315i \(-0.635839\pi\)
−0.413916 + 0.910315i \(0.635839\pi\)
\(480\) −2919.65 −0.277632
\(481\) 4388.89 0.416042
\(482\) 12385.4 1.17041
\(483\) −11176.1 −1.05286
\(484\) 7666.14 0.719960
\(485\) −4285.35 −0.401212
\(486\) 23577.3 2.20059
\(487\) 11021.8 1.02556 0.512778 0.858521i \(-0.328616\pi\)
0.512778 + 0.858521i \(0.328616\pi\)
\(488\) −7146.28 −0.662903
\(489\) 4696.41 0.434313
\(490\) 4376.40 0.403481
\(491\) 5786.44 0.531850 0.265925 0.963994i \(-0.414323\pi\)
0.265925 + 0.963994i \(0.414323\pi\)
\(492\) −13130.5 −1.20319
\(493\) 20336.5 1.85783
\(494\) 32433.2 2.95392
\(495\) 4056.55 0.368340
\(496\) −932.640 −0.0844289
\(497\) 2890.08 0.260841
\(498\) 2232.45 0.200880
\(499\) −5756.39 −0.516415 −0.258208 0.966089i \(-0.583132\pi\)
−0.258208 + 0.966089i \(0.583132\pi\)
\(500\) 14296.9 1.27875
\(501\) 18478.7 1.64784
\(502\) −4448.82 −0.395539
\(503\) −3112.75 −0.275926 −0.137963 0.990437i \(-0.544055\pi\)
−0.137963 + 0.990437i \(0.544055\pi\)
\(504\) −7787.89 −0.688294
\(505\) 7772.73 0.684915
\(506\) 29159.0 2.56181
\(507\) −12414.8 −1.08750
\(508\) 3215.59 0.280844
\(509\) −17859.5 −1.55522 −0.777611 0.628746i \(-0.783569\pi\)
−0.777611 + 0.628746i \(0.783569\pi\)
\(510\) 19229.6 1.66961
\(511\) 4431.77 0.383660
\(512\) −11122.1 −0.960022
\(513\) 3587.19 0.308730
\(514\) −28706.0 −2.46336
\(515\) 4612.24 0.394640
\(516\) 0 0
\(517\) 4098.47 0.348647
\(518\) −3702.16 −0.314022
\(519\) −15625.1 −1.32151
\(520\) −8315.30 −0.701250
\(521\) 13837.7 1.16361 0.581803 0.813329i \(-0.302347\pi\)
0.581803 + 0.813329i \(0.302347\pi\)
\(522\) 15781.2 1.32323
\(523\) 21318.7 1.78241 0.891204 0.453602i \(-0.149861\pi\)
0.891204 + 0.453602i \(0.149861\pi\)
\(524\) 19031.0 1.58659
\(525\) −8409.56 −0.699092
\(526\) 26745.6 2.21704
\(527\) −3988.37 −0.329670
\(528\) 9653.52 0.795673
\(529\) 8113.15 0.666816
\(530\) 8359.95 0.685156
\(531\) −14356.8 −1.17332
\(532\) −17664.9 −1.43960
\(533\) 8078.69 0.656524
\(534\) −12197.8 −0.988488
\(535\) 466.045 0.0376615
\(536\) −27418.7 −2.20953
\(537\) 11193.6 0.899511
\(538\) −22680.7 −1.81754
\(539\) 9395.40 0.750813
\(540\) −2038.07 −0.162416
\(541\) 4358.16 0.346343 0.173172 0.984892i \(-0.444598\pi\)
0.173172 + 0.984892i \(0.444598\pi\)
\(542\) 13084.1 1.03692
\(543\) −27837.0 −2.20000
\(544\) −13435.4 −1.05889
\(545\) 3141.52 0.246914
\(546\) 23482.7 1.84060
\(547\) −17185.3 −1.34331 −0.671657 0.740862i \(-0.734417\pi\)
−0.671657 + 0.740862i \(0.734417\pi\)
\(548\) 29507.2 2.30015
\(549\) −5094.65 −0.396055
\(550\) 21941.0 1.70103
\(551\) 16153.1 1.24890
\(552\) 31253.1 2.40982
\(553\) −10249.7 −0.788175
\(554\) 7541.47 0.578351
\(555\) 2066.87 0.158079
\(556\) 4733.01 0.361015
\(557\) −2037.15 −0.154967 −0.0774836 0.996994i \(-0.524689\pi\)
−0.0774836 + 0.996994i \(0.524689\pi\)
\(558\) −3094.98 −0.234805
\(559\) 0 0
\(560\) 1506.84 0.113707
\(561\) 41282.7 3.10687
\(562\) 14714.4 1.10443
\(563\) −4997.46 −0.374099 −0.187049 0.982351i \(-0.559892\pi\)
−0.187049 + 0.982351i \(0.559892\pi\)
\(564\) 9734.58 0.726773
\(565\) −4786.89 −0.356435
\(566\) 14068.4 1.04477
\(567\) 9323.71 0.690581
\(568\) −8081.90 −0.597023
\(569\) −14634.1 −1.07819 −0.539096 0.842244i \(-0.681234\pi\)
−0.539096 + 0.842244i \(0.681234\pi\)
\(570\) 15273.8 1.12237
\(571\) −24429.4 −1.79044 −0.895219 0.445626i \(-0.852981\pi\)
−0.895219 + 0.445626i \(0.852981\pi\)
\(572\) −39559.6 −2.89173
\(573\) −17528.0 −1.27791
\(574\) −6814.62 −0.495534
\(575\) 15260.0 1.10676
\(576\) −16115.1 −1.16574
\(577\) 7423.87 0.535632 0.267816 0.963470i \(-0.413698\pi\)
0.267816 + 0.963470i \(0.413698\pi\)
\(578\) 65143.5 4.68791
\(579\) 15855.1 1.13802
\(580\) −9177.38 −0.657017
\(581\) 748.105 0.0534193
\(582\) −33842.3 −2.41032
\(583\) 17947.4 1.27497
\(584\) −12393.1 −0.878135
\(585\) −5928.06 −0.418966
\(586\) 23378.5 1.64805
\(587\) 3113.36 0.218913 0.109457 0.993992i \(-0.465089\pi\)
0.109457 + 0.993992i \(0.465089\pi\)
\(588\) 22315.7 1.56511
\(589\) −3167.92 −0.221616
\(590\) 12930.5 0.902275
\(591\) 13109.4 0.912433
\(592\) 2224.07 0.154407
\(593\) 950.921 0.0658510 0.0329255 0.999458i \(-0.489518\pi\)
0.0329255 + 0.999458i \(0.489518\pi\)
\(594\) −6776.35 −0.468076
\(595\) 6443.93 0.443992
\(596\) −22228.2 −1.52769
\(597\) 21123.8 1.44814
\(598\) −42611.7 −2.91392
\(599\) −20463.9 −1.39588 −0.697941 0.716155i \(-0.745900\pi\)
−0.697941 + 0.716155i \(0.745900\pi\)
\(600\) 23516.7 1.60011
\(601\) 11078.2 0.751899 0.375949 0.926640i \(-0.377317\pi\)
0.375949 + 0.926640i \(0.377317\pi\)
\(602\) 0 0
\(603\) −19547.1 −1.32010
\(604\) −45899.4 −3.09209
\(605\) −2221.23 −0.149266
\(606\) 61382.8 4.11470
\(607\) −14486.3 −0.968670 −0.484335 0.874883i \(-0.660938\pi\)
−0.484335 + 0.874883i \(0.660938\pi\)
\(608\) −10671.5 −0.711823
\(609\) 11695.3 0.778192
\(610\) 4588.52 0.304564
\(611\) −5989.32 −0.396566
\(612\) 44337.5 2.92849
\(613\) 12062.1 0.794753 0.397376 0.917656i \(-0.369921\pi\)
0.397376 + 0.917656i \(0.369921\pi\)
\(614\) −44849.3 −2.94784
\(615\) 3804.51 0.249451
\(616\) 15058.3 0.984930
\(617\) 10781.2 0.703458 0.351729 0.936102i \(-0.385594\pi\)
0.351729 + 0.936102i \(0.385594\pi\)
\(618\) 36423.8 2.37084
\(619\) 24086.1 1.56398 0.781990 0.623291i \(-0.214205\pi\)
0.781990 + 0.623291i \(0.214205\pi\)
\(620\) 1799.86 0.116587
\(621\) −4712.96 −0.304548
\(622\) −16810.9 −1.08369
\(623\) −4087.56 −0.262865
\(624\) −14107.2 −0.905032
\(625\) 9252.11 0.592135
\(626\) 7037.91 0.449347
\(627\) 32790.3 2.08855
\(628\) −26366.0 −1.67535
\(629\) 9511.10 0.602913
\(630\) 5000.49 0.316229
\(631\) −213.204 −0.0134509 −0.00672545 0.999977i \(-0.502141\pi\)
−0.00672545 + 0.999977i \(0.502141\pi\)
\(632\) 28662.5 1.80401
\(633\) 1781.46 0.111859
\(634\) −3478.46 −0.217898
\(635\) −931.705 −0.0582261
\(636\) 42628.2 2.65773
\(637\) −13730.0 −0.854007
\(638\) −30513.8 −1.89350
\(639\) −5761.66 −0.356695
\(640\) 11187.1 0.690953
\(641\) 19848.9 1.22306 0.611532 0.791219i \(-0.290553\pi\)
0.611532 + 0.791219i \(0.290553\pi\)
\(642\) 3680.45 0.226255
\(643\) 7733.15 0.474286 0.237143 0.971475i \(-0.423789\pi\)
0.237143 + 0.971475i \(0.423789\pi\)
\(644\) 23208.7 1.42011
\(645\) 0 0
\(646\) 70285.4 4.28072
\(647\) 8440.19 0.512856 0.256428 0.966563i \(-0.417454\pi\)
0.256428 + 0.966563i \(0.417454\pi\)
\(648\) −26073.1 −1.58063
\(649\) 27759.7 1.67899
\(650\) −32063.6 −1.93483
\(651\) −2293.67 −0.138089
\(652\) −9752.72 −0.585807
\(653\) −11887.4 −0.712390 −0.356195 0.934412i \(-0.615926\pi\)
−0.356195 + 0.934412i \(0.615926\pi\)
\(654\) 24809.2 1.48336
\(655\) −5514.15 −0.328940
\(656\) 4093.88 0.243657
\(657\) −8835.18 −0.524647
\(658\) 5052.17 0.299322
\(659\) 16141.8 0.954164 0.477082 0.878859i \(-0.341694\pi\)
0.477082 + 0.878859i \(0.341694\pi\)
\(660\) −18629.8 −1.09874
\(661\) −12054.7 −0.709338 −0.354669 0.934992i \(-0.615407\pi\)
−0.354669 + 0.934992i \(0.615407\pi\)
\(662\) −1983.28 −0.116438
\(663\) −60328.6 −3.53389
\(664\) −2092.02 −0.122268
\(665\) 5118.33 0.298467
\(666\) 7380.62 0.429419
\(667\) −21222.4 −1.23198
\(668\) −38373.5 −2.22263
\(669\) 15522.5 0.897060
\(670\) 17605.2 1.01515
\(671\) 9850.80 0.566745
\(672\) −7726.55 −0.443539
\(673\) 15294.5 0.876017 0.438008 0.898971i \(-0.355684\pi\)
0.438008 + 0.898971i \(0.355684\pi\)
\(674\) −13863.2 −0.792271
\(675\) −3546.31 −0.202219
\(676\) 25781.0 1.46683
\(677\) −10935.3 −0.620797 −0.310398 0.950607i \(-0.600462\pi\)
−0.310398 + 0.950607i \(0.600462\pi\)
\(678\) −37803.0 −2.14132
\(679\) −11340.7 −0.640967
\(680\) −18020.0 −1.01623
\(681\) 31649.0 1.78090
\(682\) 5984.32 0.335999
\(683\) −27098.4 −1.51814 −0.759070 0.651009i \(-0.774346\pi\)
−0.759070 + 0.651009i \(0.774346\pi\)
\(684\) 35216.7 1.96863
\(685\) −8549.59 −0.476880
\(686\) 29801.2 1.65862
\(687\) −32238.8 −1.79037
\(688\) 0 0
\(689\) −26227.5 −1.45020
\(690\) −20067.2 −1.10717
\(691\) 15040.8 0.828046 0.414023 0.910266i \(-0.364123\pi\)
0.414023 + 0.910266i \(0.364123\pi\)
\(692\) 32447.6 1.78247
\(693\) 10735.2 0.588452
\(694\) −10046.9 −0.549532
\(695\) −1371.37 −0.0748476
\(696\) −32705.1 −1.78116
\(697\) 17507.2 0.951410
\(698\) −10496.9 −0.569218
\(699\) 40875.2 2.21179
\(700\) 17463.6 0.942945
\(701\) −7504.62 −0.404344 −0.202172 0.979350i \(-0.564800\pi\)
−0.202172 + 0.979350i \(0.564800\pi\)
\(702\) 9902.65 0.532409
\(703\) 7554.54 0.405299
\(704\) 31159.5 1.66814
\(705\) −2820.56 −0.150678
\(706\) −45997.3 −2.45203
\(707\) 20569.7 1.09421
\(708\) 65934.2 3.49994
\(709\) 28975.7 1.53484 0.767421 0.641143i \(-0.221539\pi\)
0.767421 + 0.641143i \(0.221539\pi\)
\(710\) 5189.27 0.274296
\(711\) 20433.7 1.07781
\(712\) 11430.6 0.601655
\(713\) 4162.10 0.218614
\(714\) 50888.9 2.66733
\(715\) 11462.2 0.599529
\(716\) −23244.9 −1.21327
\(717\) 17911.4 0.932934
\(718\) −32952.0 −1.71275
\(719\) 13465.7 0.698448 0.349224 0.937039i \(-0.386445\pi\)
0.349224 + 0.937039i \(0.386445\pi\)
\(720\) −3004.04 −0.155492
\(721\) 12205.8 0.630468
\(722\) 23234.8 1.19766
\(723\) −18298.7 −0.941265
\(724\) 57807.3 2.96739
\(725\) −15969.0 −0.818032
\(726\) −17541.5 −0.896731
\(727\) 7095.20 0.361962 0.180981 0.983487i \(-0.442073\pi\)
0.180981 + 0.983487i \(0.442073\pi\)
\(728\) −22005.5 −1.12030
\(729\) −12314.6 −0.625648
\(730\) 7957.45 0.403450
\(731\) 0 0
\(732\) 23397.3 1.18141
\(733\) −11709.6 −0.590048 −0.295024 0.955490i \(-0.595328\pi\)
−0.295024 + 0.955490i \(0.595328\pi\)
\(734\) −10221.0 −0.513982
\(735\) −6465.89 −0.324487
\(736\) 14020.6 0.702182
\(737\) 37795.4 1.88902
\(738\) 13585.6 0.677633
\(739\) 33371.8 1.66117 0.830584 0.556894i \(-0.188007\pi\)
0.830584 + 0.556894i \(0.188007\pi\)
\(740\) −4292.12 −0.213218
\(741\) −47918.2 −2.37560
\(742\) 22123.7 1.09459
\(743\) −11255.4 −0.555745 −0.277873 0.960618i \(-0.589629\pi\)
−0.277873 + 0.960618i \(0.589629\pi\)
\(744\) 6414.09 0.316064
\(745\) 6440.53 0.316729
\(746\) −10369.3 −0.508909
\(747\) −1491.42 −0.0730499
\(748\) −85729.0 −4.19059
\(749\) 1233.34 0.0601671
\(750\) −32713.9 −1.59272
\(751\) 28744.1 1.39665 0.698327 0.715779i \(-0.253928\pi\)
0.698327 + 0.715779i \(0.253928\pi\)
\(752\) −3035.09 −0.147178
\(753\) 6572.88 0.318100
\(754\) 44591.5 2.15375
\(755\) 13299.2 0.641068
\(756\) −5393.52 −0.259471
\(757\) 963.643 0.0462671 0.0231336 0.999732i \(-0.492636\pi\)
0.0231336 + 0.999732i \(0.492636\pi\)
\(758\) −8164.24 −0.391212
\(759\) −43080.9 −2.06026
\(760\) −14313.0 −0.683142
\(761\) 18973.6 0.903799 0.451899 0.892069i \(-0.350747\pi\)
0.451899 + 0.892069i \(0.350747\pi\)
\(762\) −7357.85 −0.349799
\(763\) 8313.70 0.394464
\(764\) 36399.3 1.72366
\(765\) −12846.6 −0.607150
\(766\) 2070.69 0.0976723
\(767\) −40566.8 −1.90975
\(768\) 47734.9 2.24282
\(769\) −24386.0 −1.14354 −0.571770 0.820414i \(-0.693743\pi\)
−0.571770 + 0.820414i \(0.693743\pi\)
\(770\) −9668.74 −0.452516
\(771\) 42411.5 1.98108
\(772\) −32925.2 −1.53498
\(773\) −6779.29 −0.315439 −0.157719 0.987484i \(-0.550414\pi\)
−0.157719 + 0.987484i \(0.550414\pi\)
\(774\) 0 0
\(775\) 3131.82 0.145159
\(776\) 31713.5 1.46707
\(777\) 5469.74 0.252543
\(778\) −13641.0 −0.628605
\(779\) 13905.7 0.639570
\(780\) 27224.8 1.24975
\(781\) 11140.5 0.510421
\(782\) −92343.1 −4.22274
\(783\) 4931.92 0.225099
\(784\) −6957.68 −0.316950
\(785\) 7639.45 0.347342
\(786\) −43546.3 −1.97614
\(787\) 24019.6 1.08794 0.543968 0.839106i \(-0.316921\pi\)
0.543968 + 0.839106i \(0.316921\pi\)
\(788\) −27223.4 −1.23070
\(789\) −39515.1 −1.78298
\(790\) −18403.8 −0.828831
\(791\) −12668.0 −0.569433
\(792\) −30020.2 −1.34687
\(793\) −14395.5 −0.644640
\(794\) −43154.6 −1.92884
\(795\) −12351.4 −0.551016
\(796\) −43866.3 −1.95327
\(797\) 13544.4 0.601966 0.300983 0.953629i \(-0.402685\pi\)
0.300983 + 0.953629i \(0.402685\pi\)
\(798\) 40420.4 1.79307
\(799\) −12979.4 −0.574689
\(800\) 10549.9 0.466246
\(801\) 8148.96 0.359462
\(802\) 22818.8 1.00469
\(803\) 17083.3 0.750756
\(804\) 89770.6 3.93777
\(805\) −6724.62 −0.294424
\(806\) −8745.22 −0.382180
\(807\) 33509.5 1.46170
\(808\) −57521.6 −2.50446
\(809\) −26389.6 −1.14686 −0.573430 0.819255i \(-0.694388\pi\)
−0.573430 + 0.819255i \(0.694388\pi\)
\(810\) 16741.1 0.726202
\(811\) −21060.8 −0.911892 −0.455946 0.890007i \(-0.650699\pi\)
−0.455946 + 0.890007i \(0.650699\pi\)
\(812\) −24286.9 −1.04964
\(813\) −19331.0 −0.833909
\(814\) −14270.8 −0.614487
\(815\) 2825.81 0.121453
\(816\) −30571.5 −1.31154
\(817\) 0 0
\(818\) 15767.0 0.673939
\(819\) −15688.0 −0.669331
\(820\) −7900.57 −0.336463
\(821\) 612.058 0.0260182 0.0130091 0.999915i \(-0.495859\pi\)
0.0130091 + 0.999915i \(0.495859\pi\)
\(822\) −67517.8 −2.86491
\(823\) −21153.4 −0.895944 −0.447972 0.894048i \(-0.647854\pi\)
−0.447972 + 0.894048i \(0.647854\pi\)
\(824\) −34132.6 −1.44304
\(825\) −32416.6 −1.36800
\(826\) 34219.3 1.44145
\(827\) 22165.3 0.931999 0.465999 0.884785i \(-0.345695\pi\)
0.465999 + 0.884785i \(0.345695\pi\)
\(828\) −46268.7 −1.94197
\(829\) 42815.7 1.79379 0.896895 0.442243i \(-0.145817\pi\)
0.896895 + 0.442243i \(0.145817\pi\)
\(830\) 1343.26 0.0561748
\(831\) −11142.1 −0.465121
\(832\) −45535.1 −1.89741
\(833\) −29754.1 −1.23760
\(834\) −10830.0 −0.449654
\(835\) 11118.6 0.460807
\(836\) −68093.4 −2.81706
\(837\) −967.242 −0.0399436
\(838\) 43398.4 1.78899
\(839\) −28252.1 −1.16254 −0.581270 0.813711i \(-0.697444\pi\)
−0.581270 + 0.813711i \(0.697444\pi\)
\(840\) −10363.1 −0.425668
\(841\) −2180.61 −0.0894097
\(842\) −61177.9 −2.50396
\(843\) −21739.7 −0.888203
\(844\) −3699.43 −0.150876
\(845\) −7469.94 −0.304111
\(846\) −10072.0 −0.409317
\(847\) −5878.25 −0.238464
\(848\) −13290.8 −0.538217
\(849\) −20785.3 −0.840222
\(850\) −69484.5 −2.80388
\(851\) −9925.39 −0.399810
\(852\) 26460.6 1.06400
\(853\) 17130.3 0.687609 0.343804 0.939041i \(-0.388284\pi\)
0.343804 + 0.939041i \(0.388284\pi\)
\(854\) 12143.0 0.486564
\(855\) −10203.9 −0.408147
\(856\) −3448.94 −0.137713
\(857\) −36446.6 −1.45273 −0.726366 0.687309i \(-0.758792\pi\)
−0.726366 + 0.687309i \(0.758792\pi\)
\(858\) 90519.5 3.60173
\(859\) −34690.3 −1.37790 −0.688951 0.724808i \(-0.741929\pi\)
−0.688951 + 0.724808i \(0.741929\pi\)
\(860\) 0 0
\(861\) 10068.2 0.398518
\(862\) 25817.6 1.02013
\(863\) −32454.0 −1.28012 −0.640062 0.768323i \(-0.721091\pi\)
−0.640062 + 0.768323i \(0.721091\pi\)
\(864\) −3258.28 −0.128298
\(865\) −9401.56 −0.369552
\(866\) 38669.1 1.51735
\(867\) −96245.9 −3.77011
\(868\) 4763.12 0.186257
\(869\) −39509.8 −1.54232
\(870\) 20999.5 0.818333
\(871\) −55232.4 −2.14866
\(872\) −23248.6 −0.902865
\(873\) 22608.8 0.876510
\(874\) −73347.0 −2.83867
\(875\) −10962.6 −0.423546
\(876\) 40575.8 1.56499
\(877\) 26856.7 1.03408 0.517038 0.855962i \(-0.327034\pi\)
0.517038 + 0.855962i \(0.327034\pi\)
\(878\) 13441.5 0.516662
\(879\) −34540.5 −1.32539
\(880\) 5808.49 0.222505
\(881\) −17343.7 −0.663250 −0.331625 0.943411i \(-0.607597\pi\)
−0.331625 + 0.943411i \(0.607597\pi\)
\(882\) −23089.2 −0.881466
\(883\) 8366.36 0.318857 0.159428 0.987210i \(-0.449035\pi\)
0.159428 + 0.987210i \(0.449035\pi\)
\(884\) 125280. 4.76656
\(885\) −19104.2 −0.725626
\(886\) −8194.98 −0.310740
\(887\) −13819.8 −0.523137 −0.261568 0.965185i \(-0.584240\pi\)
−0.261568 + 0.965185i \(0.584240\pi\)
\(888\) −15295.7 −0.578029
\(889\) −2465.66 −0.0930207
\(890\) −7339.40 −0.276424
\(891\) 35940.4 1.35135
\(892\) −32234.5 −1.20997
\(893\) −10309.3 −0.386326
\(894\) 50862.2 1.90278
\(895\) 6735.12 0.251542
\(896\) 29605.5 1.10385
\(897\) 62956.4 2.34343
\(898\) −87401.8 −3.24792
\(899\) −4355.48 −0.161583
\(900\) −34815.4 −1.28946
\(901\) −56837.2 −2.10158
\(902\) −26268.5 −0.969675
\(903\) 0 0
\(904\) 35425.1 1.30334
\(905\) −16749.4 −0.615215
\(906\) 105026. 3.85128
\(907\) −38464.8 −1.40816 −0.704082 0.710119i \(-0.748641\pi\)
−0.704082 + 0.710119i \(0.748641\pi\)
\(908\) −65723.4 −2.40210
\(909\) −41007.7 −1.49630
\(910\) 14129.4 0.514711
\(911\) 31120.3 1.13179 0.565896 0.824476i \(-0.308530\pi\)
0.565896 + 0.824476i \(0.308530\pi\)
\(912\) −24282.6 −0.881662
\(913\) 2883.75 0.104532
\(914\) −11591.1 −0.419473
\(915\) −6779.29 −0.244936
\(916\) 66948.1 2.41488
\(917\) −14592.6 −0.525507
\(918\) 21459.9 0.771548
\(919\) −30589.6 −1.09799 −0.548997 0.835824i \(-0.684990\pi\)
−0.548997 + 0.835824i \(0.684990\pi\)
\(920\) 18804.9 0.673890
\(921\) 66262.4 2.37071
\(922\) 53699.0 1.91809
\(923\) −16280.2 −0.580574
\(924\) −49301.9 −1.75532
\(925\) −7468.46 −0.265472
\(926\) 51520.7 1.82838
\(927\) −24333.5 −0.862153
\(928\) −14672.0 −0.519000
\(929\) 33705.0 1.19034 0.595170 0.803600i \(-0.297085\pi\)
0.595170 + 0.803600i \(0.297085\pi\)
\(930\) −4118.39 −0.145212
\(931\) −23633.3 −0.831954
\(932\) −84882.9 −2.98330
\(933\) 24837.1 0.871524
\(934\) 30080.6 1.05382
\(935\) 24839.6 0.868816
\(936\) 43870.2 1.53199
\(937\) −9240.51 −0.322171 −0.161086 0.986940i \(-0.551500\pi\)
−0.161086 + 0.986940i \(0.551500\pi\)
\(938\) 46590.2 1.62177
\(939\) −10398.1 −0.361374
\(940\) 5857.26 0.203237
\(941\) 2959.70 0.102533 0.0512665 0.998685i \(-0.483674\pi\)
0.0512665 + 0.998685i \(0.483674\pi\)
\(942\) 60330.2 2.08669
\(943\) −18269.8 −0.630908
\(944\) −20557.2 −0.708771
\(945\) 1562.75 0.0537950
\(946\) 0 0
\(947\) −8960.28 −0.307466 −0.153733 0.988112i \(-0.549130\pi\)
−0.153733 + 0.988112i \(0.549130\pi\)
\(948\) −93842.6 −3.21505
\(949\) −24964.8 −0.853942
\(950\) −55190.6 −1.88486
\(951\) 5139.23 0.175238
\(952\) −47687.8 −1.62350
\(953\) −29389.4 −0.998969 −0.499484 0.866323i \(-0.666477\pi\)
−0.499484 + 0.866323i \(0.666477\pi\)
\(954\) −44105.7 −1.49683
\(955\) −10546.5 −0.357359
\(956\) −37195.4 −1.25835
\(957\) 45082.4 1.52279
\(958\) −41237.9 −1.39075
\(959\) −22625.6 −0.761854
\(960\) −21443.9 −0.720936
\(961\) −28936.8 −0.971327
\(962\) 20854.8 0.698944
\(963\) −2458.78 −0.0822774
\(964\) 37999.6 1.26959
\(965\) 9539.94 0.318240
\(966\) −53105.6 −1.76878
\(967\) 26171.5 0.870341 0.435170 0.900348i \(-0.356688\pi\)
0.435170 + 0.900348i \(0.356688\pi\)
\(968\) 16438.1 0.545806
\(969\) −103843. −3.44263
\(970\) −20362.8 −0.674030
\(971\) 27259.4 0.900922 0.450461 0.892796i \(-0.351260\pi\)
0.450461 + 0.892796i \(0.351260\pi\)
\(972\) 72337.8 2.38708
\(973\) −3629.18 −0.119575
\(974\) 52372.5 1.72292
\(975\) 47372.2 1.55602
\(976\) −7294.92 −0.239247
\(977\) 19534.3 0.639669 0.319834 0.947473i \(-0.396373\pi\)
0.319834 + 0.947473i \(0.396373\pi\)
\(978\) 22316.0 0.729639
\(979\) −15756.5 −0.514381
\(980\) 13427.3 0.437672
\(981\) −16574.2 −0.539422
\(982\) 27495.5 0.893500
\(983\) 48920.6 1.58731 0.793655 0.608368i \(-0.208176\pi\)
0.793655 + 0.608368i \(0.208176\pi\)
\(984\) −28155.0 −0.912143
\(985\) 7887.87 0.255156
\(986\) 96633.4 3.12113
\(987\) −7464.29 −0.240720
\(988\) 99508.6 3.20424
\(989\) 0 0
\(990\) 19275.6 0.618806
\(991\) 47979.4 1.53796 0.768979 0.639274i \(-0.220765\pi\)
0.768979 + 0.639274i \(0.220765\pi\)
\(992\) 2877.45 0.0920960
\(993\) 2930.18 0.0936420
\(994\) 13732.8 0.438209
\(995\) 12710.1 0.404962
\(996\) 6849.40 0.217903
\(997\) −19997.6 −0.635235 −0.317617 0.948219i \(-0.602883\pi\)
−0.317617 + 0.948219i \(0.602883\pi\)
\(998\) −27352.7 −0.867570
\(999\) 2306.59 0.0730502
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.1.47 50
43.42 odd 2 1849.4.a.j.1.4 yes 50
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1849.4.a.i.1.47 50 1.1 even 1 trivial
1849.4.a.j.1.4 yes 50 43.42 odd 2