Properties

Label 1849.4.a.j.1.8
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 1.8
Character \(\chi\) \(=\) 1849.1

$q$-expansion

\(f(q)\) \(=\) \(q-4.15048 q^{2} +4.65912 q^{3} +9.22650 q^{4} +19.9023 q^{5} -19.3376 q^{6} -27.8552 q^{7} -5.09055 q^{8} -5.29261 q^{9} +O(q^{10})\) \(q-4.15048 q^{2} +4.65912 q^{3} +9.22650 q^{4} +19.9023 q^{5} -19.3376 q^{6} -27.8552 q^{7} -5.09055 q^{8} -5.29261 q^{9} -82.6043 q^{10} -0.733230 q^{11} +42.9873 q^{12} +67.6888 q^{13} +115.613 q^{14} +92.7274 q^{15} -52.6837 q^{16} -45.0640 q^{17} +21.9669 q^{18} -96.4948 q^{19} +183.629 q^{20} -129.781 q^{21} +3.04326 q^{22} -76.2235 q^{23} -23.7175 q^{24} +271.103 q^{25} -280.941 q^{26} -150.455 q^{27} -257.006 q^{28} +182.776 q^{29} -384.863 q^{30} -37.9792 q^{31} +259.387 q^{32} -3.41621 q^{33} +187.037 q^{34} -554.384 q^{35} -48.8323 q^{36} -95.8006 q^{37} +400.500 q^{38} +315.370 q^{39} -101.314 q^{40} +33.5766 q^{41} +538.653 q^{42} -6.76515 q^{44} -105.335 q^{45} +316.364 q^{46} +533.353 q^{47} -245.460 q^{48} +432.913 q^{49} -1125.21 q^{50} -209.958 q^{51} +624.530 q^{52} -0.134055 q^{53} +624.461 q^{54} -14.5930 q^{55} +141.798 q^{56} -449.581 q^{57} -758.607 q^{58} -110.795 q^{59} +855.549 q^{60} -533.253 q^{61} +157.632 q^{62} +147.427 q^{63} -655.112 q^{64} +1347.16 q^{65} +14.1789 q^{66} -411.431 q^{67} -415.783 q^{68} -355.134 q^{69} +2300.96 q^{70} +216.076 q^{71} +26.9423 q^{72} +136.316 q^{73} +397.619 q^{74} +1263.10 q^{75} -890.309 q^{76} +20.4243 q^{77} -1308.94 q^{78} +81.5918 q^{79} -1048.53 q^{80} -558.088 q^{81} -139.359 q^{82} -926.163 q^{83} -1197.42 q^{84} -896.879 q^{85} +851.573 q^{87} +3.73254 q^{88} +1125.82 q^{89} +437.193 q^{90} -1885.49 q^{91} -703.276 q^{92} -176.950 q^{93} -2213.67 q^{94} -1920.47 q^{95} +1208.52 q^{96} -1512.02 q^{97} -1796.80 q^{98} +3.88071 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 50q + 10q^{2} + 2q^{3} + 186q^{4} - 8q^{5} - 51q^{6} + 6q^{7} + 138q^{8} + 360q^{9} + O(q^{10}) \) \( 50q + 10q^{2} + 2q^{3} + 186q^{4} - 8q^{5} - 51q^{6} + 6q^{7} + 138q^{8} + 360q^{9} - 137q^{10} - 252q^{11} + 48q^{12} - 192q^{13} - 272q^{14} - 314q^{15} + 542q^{16} - 236q^{17} + 386q^{18} - 12q^{19} - 108q^{20} - 408q^{21} - 1235q^{22} - 630q^{23} - 613q^{24} + 1098q^{25} - 1493q^{26} - 10q^{27} + 242q^{28} - 208q^{29} + 48q^{30} - 932q^{31} + 1124q^{32} - 254q^{33} - 765q^{34} - 1452q^{35} + 747q^{36} + 90q^{37} - 1213q^{38} + 1610q^{39} - 1693q^{40} - 1354q^{41} + 16q^{42} - 2704q^{44} - 4508q^{45} - 233q^{46} - 3484q^{47} + 376q^{48} + 1324q^{49} + 408q^{50} - 4054q^{51} - 2176q^{52} - 726q^{53} - 6497q^{54} + 3288q^{55} - 7097q^{56} - 870q^{57} + 275q^{58} - 4370q^{59} - 3891q^{60} - 1172q^{61} + 1546q^{62} + 3686q^{63} + 606q^{64} - 2610q^{65} - 4697q^{66} - 344q^{67} - 3221q^{68} - 136q^{69} + 1310q^{70} - 162q^{71} + 5814q^{72} - 746q^{73} - 4332q^{74} - 236q^{75} - 1338q^{76} - 2024q^{77} - 2782q^{78} - 2656q^{79} + 5713q^{80} - 86q^{81} + 4168q^{82} - 3514q^{83} - 4269q^{84} + 7558q^{85} - 10278q^{87} - 11692q^{88} - 2640q^{89} - 8286q^{90} + 5946q^{91} - 4271q^{92} + 2q^{93} - 9062q^{94} - 12140q^{95} - 700q^{96} - 3864q^{97} + 2826q^{98} - 8174q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.15048 −1.46742 −0.733708 0.679465i \(-0.762212\pi\)
−0.733708 + 0.679465i \(0.762212\pi\)
\(3\) 4.65912 0.896648 0.448324 0.893871i \(-0.352021\pi\)
0.448324 + 0.893871i \(0.352021\pi\)
\(4\) 9.22650 1.15331
\(5\) 19.9023 1.78012 0.890060 0.455844i \(-0.150662\pi\)
0.890060 + 0.455844i \(0.150662\pi\)
\(6\) −19.3376 −1.31576
\(7\) −27.8552 −1.50404 −0.752020 0.659140i \(-0.770921\pi\)
−0.752020 + 0.659140i \(0.770921\pi\)
\(8\) −5.09055 −0.224973
\(9\) −5.29261 −0.196023
\(10\) −82.6043 −2.61218
\(11\) −0.733230 −0.0200979 −0.0100490 0.999950i \(-0.503199\pi\)
−0.0100490 + 0.999950i \(0.503199\pi\)
\(12\) 42.9873 1.03411
\(13\) 67.6888 1.44411 0.722057 0.691834i \(-0.243197\pi\)
0.722057 + 0.691834i \(0.243197\pi\)
\(14\) 115.613 2.20705
\(15\) 92.7274 1.59614
\(16\) −52.6837 −0.823183
\(17\) −45.0640 −0.642919 −0.321460 0.946923i \(-0.604173\pi\)
−0.321460 + 0.946923i \(0.604173\pi\)
\(18\) 21.9669 0.287647
\(19\) −96.4948 −1.16513 −0.582564 0.812785i \(-0.697950\pi\)
−0.582564 + 0.812785i \(0.697950\pi\)
\(20\) 183.629 2.05303
\(21\) −129.781 −1.34859
\(22\) 3.04326 0.0294920
\(23\) −76.2235 −0.691031 −0.345515 0.938413i \(-0.612296\pi\)
−0.345515 + 0.938413i \(0.612296\pi\)
\(24\) −23.7175 −0.201721
\(25\) 271.103 2.16883
\(26\) −280.941 −2.11912
\(27\) −150.455 −1.07241
\(28\) −257.006 −1.73463
\(29\) 182.776 1.17036 0.585182 0.810902i \(-0.301023\pi\)
0.585182 + 0.810902i \(0.301023\pi\)
\(30\) −384.863 −2.34220
\(31\) −37.9792 −0.220041 −0.110020 0.993929i \(-0.535092\pi\)
−0.110020 + 0.993929i \(0.535092\pi\)
\(32\) 259.387 1.43293
\(33\) −3.41621 −0.0180208
\(34\) 187.037 0.943430
\(35\) −554.384 −2.67737
\(36\) −48.8323 −0.226075
\(37\) −95.8006 −0.425663 −0.212831 0.977089i \(-0.568269\pi\)
−0.212831 + 0.977089i \(0.568269\pi\)
\(38\) 400.500 1.70973
\(39\) 315.370 1.29486
\(40\) −101.314 −0.400478
\(41\) 33.5766 0.127897 0.0639485 0.997953i \(-0.479631\pi\)
0.0639485 + 0.997953i \(0.479631\pi\)
\(42\) 538.653 1.97895
\(43\) 0 0
\(44\) −6.76515 −0.0231792
\(45\) −105.335 −0.348944
\(46\) 316.364 1.01403
\(47\) 533.353 1.65527 0.827634 0.561268i \(-0.189686\pi\)
0.827634 + 0.561268i \(0.189686\pi\)
\(48\) −245.460 −0.738106
\(49\) 432.913 1.26214
\(50\) −1125.21 −3.18257
\(51\) −209.958 −0.576472
\(52\) 624.530 1.66551
\(53\) −0.134055 −0.000347430 0 −0.000173715 1.00000i \(-0.500055\pi\)
−0.000173715 1.00000i \(0.500055\pi\)
\(54\) 624.461 1.57367
\(55\) −14.5930 −0.0357767
\(56\) 141.798 0.338368
\(57\) −449.581 −1.04471
\(58\) −758.607 −1.71741
\(59\) −110.795 −0.244480 −0.122240 0.992501i \(-0.539008\pi\)
−0.122240 + 0.992501i \(0.539008\pi\)
\(60\) 855.549 1.84085
\(61\) −533.253 −1.11928 −0.559640 0.828736i \(-0.689061\pi\)
−0.559640 + 0.828736i \(0.689061\pi\)
\(62\) 157.632 0.322891
\(63\) 147.427 0.294826
\(64\) −655.112 −1.27952
\(65\) 1347.16 2.57070
\(66\) 14.1789 0.0264440
\(67\) −411.431 −0.750214 −0.375107 0.926982i \(-0.622394\pi\)
−0.375107 + 0.926982i \(0.622394\pi\)
\(68\) −415.783 −0.741486
\(69\) −355.134 −0.619611
\(70\) 2300.96 3.92882
\(71\) 216.076 0.361176 0.180588 0.983559i \(-0.442200\pi\)
0.180588 + 0.983559i \(0.442200\pi\)
\(72\) 26.9423 0.0440997
\(73\) 136.316 0.218557 0.109278 0.994011i \(-0.465146\pi\)
0.109278 + 0.994011i \(0.465146\pi\)
\(74\) 397.619 0.624625
\(75\) 1263.10 1.94467
\(76\) −890.309 −1.34376
\(77\) 20.4243 0.0302281
\(78\) −1308.94 −1.90010
\(79\) 81.5918 0.116200 0.0581000 0.998311i \(-0.481496\pi\)
0.0581000 + 0.998311i \(0.481496\pi\)
\(80\) −1048.53 −1.46536
\(81\) −558.088 −0.765552
\(82\) −139.359 −0.187678
\(83\) −926.163 −1.22481 −0.612407 0.790542i \(-0.709799\pi\)
−0.612407 + 0.790542i \(0.709799\pi\)
\(84\) −1197.42 −1.55535
\(85\) −896.879 −1.14447
\(86\) 0 0
\(87\) 851.573 1.04941
\(88\) 3.73254 0.00452148
\(89\) 1125.82 1.34086 0.670429 0.741974i \(-0.266110\pi\)
0.670429 + 0.741974i \(0.266110\pi\)
\(90\) 437.193 0.512046
\(91\) −1885.49 −2.17201
\(92\) −703.276 −0.796974
\(93\) −176.950 −0.197299
\(94\) −2213.67 −2.42897
\(95\) −1920.47 −2.07407
\(96\) 1208.52 1.28483
\(97\) −1512.02 −1.58270 −0.791352 0.611361i \(-0.790622\pi\)
−0.791352 + 0.611361i \(0.790622\pi\)
\(98\) −1796.80 −1.85208
\(99\) 3.88071 0.00393965
\(100\) 2501.33 2.50133
\(101\) −1324.79 −1.30517 −0.652583 0.757717i \(-0.726315\pi\)
−0.652583 + 0.757717i \(0.726315\pi\)
\(102\) 871.429 0.845925
\(103\) −1115.34 −1.06696 −0.533482 0.845811i \(-0.679117\pi\)
−0.533482 + 0.845811i \(0.679117\pi\)
\(104\) −344.573 −0.324886
\(105\) −2582.94 −2.40066
\(106\) 0.556391 0.000509825 0
\(107\) −647.960 −0.585427 −0.292713 0.956200i \(-0.594558\pi\)
−0.292713 + 0.956200i \(0.594558\pi\)
\(108\) −1388.17 −1.23682
\(109\) −2058.34 −1.80875 −0.904374 0.426740i \(-0.859662\pi\)
−0.904374 + 0.426740i \(0.859662\pi\)
\(110\) 60.5680 0.0524994
\(111\) −446.346 −0.381670
\(112\) 1467.52 1.23810
\(113\) −911.539 −0.758853 −0.379426 0.925222i \(-0.623879\pi\)
−0.379426 + 0.925222i \(0.623879\pi\)
\(114\) 1865.98 1.53302
\(115\) −1517.03 −1.23012
\(116\) 1686.38 1.34980
\(117\) −358.250 −0.283079
\(118\) 459.854 0.358754
\(119\) 1255.27 0.966976
\(120\) −472.033 −0.359088
\(121\) −1330.46 −0.999596
\(122\) 2213.26 1.64245
\(123\) 156.437 0.114679
\(124\) −350.415 −0.253776
\(125\) 2907.79 2.08065
\(126\) −611.893 −0.432633
\(127\) −375.272 −0.262205 −0.131102 0.991369i \(-0.541852\pi\)
−0.131102 + 0.991369i \(0.541852\pi\)
\(128\) 643.932 0.444657
\(129\) 0 0
\(130\) −5591.38 −3.77228
\(131\) 2701.28 1.80161 0.900807 0.434219i \(-0.142976\pi\)
0.900807 + 0.434219i \(0.142976\pi\)
\(132\) −31.5196 −0.0207836
\(133\) 2687.88 1.75240
\(134\) 1707.64 1.10088
\(135\) −2994.41 −1.90902
\(136\) 229.400 0.144639
\(137\) 2022.29 1.26114 0.630568 0.776134i \(-0.282822\pi\)
0.630568 + 0.776134i \(0.282822\pi\)
\(138\) 1473.98 0.909228
\(139\) −564.430 −0.344419 −0.172210 0.985060i \(-0.555091\pi\)
−0.172210 + 0.985060i \(0.555091\pi\)
\(140\) −5115.02 −3.08785
\(141\) 2484.96 1.48419
\(142\) −896.820 −0.529996
\(143\) −49.6314 −0.0290237
\(144\) 278.835 0.161363
\(145\) 3637.66 2.08339
\(146\) −565.779 −0.320714
\(147\) 2017.00 1.13169
\(148\) −883.904 −0.490922
\(149\) 754.273 0.414714 0.207357 0.978265i \(-0.433514\pi\)
0.207357 + 0.978265i \(0.433514\pi\)
\(150\) −5242.48 −2.85364
\(151\) −2555.27 −1.37712 −0.688560 0.725179i \(-0.741757\pi\)
−0.688560 + 0.725179i \(0.741757\pi\)
\(152\) 491.211 0.262122
\(153\) 238.506 0.126027
\(154\) −84.7707 −0.0443572
\(155\) −755.875 −0.391699
\(156\) 2909.76 1.49338
\(157\) 1811.53 0.920864 0.460432 0.887695i \(-0.347695\pi\)
0.460432 + 0.887695i \(0.347695\pi\)
\(158\) −338.645 −0.170514
\(159\) −0.624576 −0.000311523 0
\(160\) 5162.41 2.55078
\(161\) 2123.22 1.03934
\(162\) 2316.33 1.12338
\(163\) −2865.26 −1.37684 −0.688419 0.725313i \(-0.741695\pi\)
−0.688419 + 0.725313i \(0.741695\pi\)
\(164\) 309.794 0.147505
\(165\) −67.9905 −0.0320791
\(166\) 3844.02 1.79731
\(167\) 14.5741 0.00675318 0.00337659 0.999994i \(-0.498925\pi\)
0.00337659 + 0.999994i \(0.498925\pi\)
\(168\) 660.655 0.303397
\(169\) 2384.77 1.08547
\(170\) 3722.48 1.67942
\(171\) 510.710 0.228392
\(172\) 0 0
\(173\) 1036.88 0.455678 0.227839 0.973699i \(-0.426834\pi\)
0.227839 + 0.973699i \(0.426834\pi\)
\(174\) −3534.44 −1.53991
\(175\) −7551.64 −3.26200
\(176\) 38.6293 0.0165443
\(177\) −516.209 −0.219213
\(178\) −4672.68 −1.96760
\(179\) −1266.66 −0.528909 −0.264455 0.964398i \(-0.585192\pi\)
−0.264455 + 0.964398i \(0.585192\pi\)
\(180\) −971.877 −0.402441
\(181\) −60.5440 −0.0248630 −0.0124315 0.999923i \(-0.503957\pi\)
−0.0124315 + 0.999923i \(0.503957\pi\)
\(182\) 7825.67 3.18724
\(183\) −2484.49 −1.00360
\(184\) 388.019 0.155463
\(185\) −1906.66 −0.757731
\(186\) 734.426 0.289520
\(187\) 33.0423 0.0129213
\(188\) 4920.98 1.90904
\(189\) 4190.96 1.61295
\(190\) 7970.89 3.04352
\(191\) −2237.34 −0.847581 −0.423791 0.905760i \(-0.639301\pi\)
−0.423791 + 0.905760i \(0.639301\pi\)
\(192\) −3052.24 −1.14728
\(193\) −1150.33 −0.429029 −0.214515 0.976721i \(-0.568817\pi\)
−0.214515 + 0.976721i \(0.568817\pi\)
\(194\) 6275.60 2.32249
\(195\) 6276.60 2.30501
\(196\) 3994.27 1.45564
\(197\) 3565.84 1.28962 0.644811 0.764342i \(-0.276936\pi\)
0.644811 + 0.764342i \(0.276936\pi\)
\(198\) −16.1068 −0.00578111
\(199\) 2313.80 0.824226 0.412113 0.911133i \(-0.364791\pi\)
0.412113 + 0.911133i \(0.364791\pi\)
\(200\) −1380.06 −0.487926
\(201\) −1916.91 −0.672678
\(202\) 5498.52 1.91522
\(203\) −5091.26 −1.76028
\(204\) −1937.18 −0.664852
\(205\) 668.252 0.227672
\(206\) 4629.18 1.56568
\(207\) 403.422 0.135458
\(208\) −3566.10 −1.18877
\(209\) 70.7529 0.0234167
\(210\) 10720.5 3.52277
\(211\) −1691.58 −0.551912 −0.275956 0.961170i \(-0.588994\pi\)
−0.275956 + 0.961170i \(0.588994\pi\)
\(212\) −1.23685 −0.000400695 0
\(213\) 1006.72 0.323848
\(214\) 2689.34 0.859065
\(215\) 0 0
\(216\) 765.899 0.241263
\(217\) 1057.92 0.330950
\(218\) 8543.12 2.65419
\(219\) 635.115 0.195968
\(220\) −134.642 −0.0412617
\(221\) −3050.33 −0.928448
\(222\) 1852.55 0.560068
\(223\) 406.066 0.121938 0.0609691 0.998140i \(-0.480581\pi\)
0.0609691 + 0.998140i \(0.480581\pi\)
\(224\) −7225.29 −2.15518
\(225\) −1434.84 −0.425139
\(226\) 3783.32 1.11355
\(227\) −4463.18 −1.30499 −0.652493 0.757795i \(-0.726277\pi\)
−0.652493 + 0.757795i \(0.726277\pi\)
\(228\) −4148.06 −1.20488
\(229\) −5509.43 −1.58984 −0.794920 0.606714i \(-0.792487\pi\)
−0.794920 + 0.606714i \(0.792487\pi\)
\(230\) 6296.39 1.80509
\(231\) 95.1592 0.0271040
\(232\) −930.428 −0.263300
\(233\) −4222.96 −1.18736 −0.593680 0.804701i \(-0.702326\pi\)
−0.593680 + 0.804701i \(0.702326\pi\)
\(234\) 1486.91 0.415395
\(235\) 10615.0 2.94657
\(236\) −1022.25 −0.281962
\(237\) 380.146 0.104190
\(238\) −5209.96 −1.41896
\(239\) −876.821 −0.237309 −0.118654 0.992936i \(-0.537858\pi\)
−0.118654 + 0.992936i \(0.537858\pi\)
\(240\) −4885.22 −1.31392
\(241\) −3169.86 −0.847257 −0.423628 0.905836i \(-0.639244\pi\)
−0.423628 + 0.905836i \(0.639244\pi\)
\(242\) 5522.06 1.46682
\(243\) 1462.09 0.385980
\(244\) −4920.06 −1.29088
\(245\) 8615.99 2.24676
\(246\) −649.290 −0.168281
\(247\) −6531.61 −1.68258
\(248\) 193.335 0.0495031
\(249\) −4315.10 −1.09823
\(250\) −12068.7 −3.05318
\(251\) 3990.59 1.00352 0.501761 0.865006i \(-0.332686\pi\)
0.501761 + 0.865006i \(0.332686\pi\)
\(252\) 1360.23 0.340027
\(253\) 55.8894 0.0138883
\(254\) 1557.56 0.384763
\(255\) −4178.67 −1.02619
\(256\) 2568.27 0.627018
\(257\) 4425.70 1.07419 0.537096 0.843521i \(-0.319521\pi\)
0.537096 + 0.843521i \(0.319521\pi\)
\(258\) 0 0
\(259\) 2668.55 0.640214
\(260\) 12429.6 2.96481
\(261\) −967.361 −0.229418
\(262\) −11211.6 −2.64372
\(263\) 72.8494 0.0170802 0.00854009 0.999964i \(-0.497282\pi\)
0.00854009 + 0.999964i \(0.497282\pi\)
\(264\) 17.3904 0.00405418
\(265\) −2.66800 −0.000618467 0
\(266\) −11156.0 −2.57150
\(267\) 5245.32 1.20228
\(268\) −3796.07 −0.865231
\(269\) 3884.39 0.880430 0.440215 0.897892i \(-0.354902\pi\)
0.440215 + 0.897892i \(0.354902\pi\)
\(270\) 12428.2 2.80133
\(271\) 6716.63 1.50556 0.752779 0.658273i \(-0.228713\pi\)
0.752779 + 0.658273i \(0.228713\pi\)
\(272\) 2374.14 0.529240
\(273\) −8784.70 −1.94752
\(274\) −8393.46 −1.85061
\(275\) −198.781 −0.0435889
\(276\) −3276.65 −0.714605
\(277\) 2043.39 0.443233 0.221616 0.975134i \(-0.428867\pi\)
0.221616 + 0.975134i \(0.428867\pi\)
\(278\) 2342.66 0.505407
\(279\) 201.009 0.0431330
\(280\) 2822.12 0.602335
\(281\) −2091.71 −0.444060 −0.222030 0.975040i \(-0.571268\pi\)
−0.222030 + 0.975040i \(0.571268\pi\)
\(282\) −10313.8 −2.17793
\(283\) −4108.63 −0.863013 −0.431507 0.902110i \(-0.642018\pi\)
−0.431507 + 0.902110i \(0.642018\pi\)
\(284\) 1993.63 0.416549
\(285\) −8947.71 −1.85971
\(286\) 205.994 0.0425899
\(287\) −935.283 −0.192362
\(288\) −1372.84 −0.280886
\(289\) −2882.24 −0.586655
\(290\) −15098.1 −3.05720
\(291\) −7044.67 −1.41913
\(292\) 1257.72 0.252064
\(293\) −771.427 −0.153813 −0.0769066 0.997038i \(-0.524504\pi\)
−0.0769066 + 0.997038i \(0.524504\pi\)
\(294\) −8371.50 −1.66067
\(295\) −2205.09 −0.435204
\(296\) 487.678 0.0957624
\(297\) 110.318 0.0215532
\(298\) −3130.60 −0.608559
\(299\) −5159.48 −0.997927
\(300\) 11654.0 2.24281
\(301\) 0 0
\(302\) 10605.6 2.02081
\(303\) −6172.36 −1.17027
\(304\) 5083.71 0.959114
\(305\) −10613.0 −1.99245
\(306\) −989.916 −0.184934
\(307\) −6628.00 −1.23218 −0.616092 0.787675i \(-0.711285\pi\)
−0.616092 + 0.787675i \(0.711285\pi\)
\(308\) 188.445 0.0348624
\(309\) −5196.49 −0.956692
\(310\) 3137.24 0.574785
\(311\) 4804.84 0.876069 0.438035 0.898958i \(-0.355675\pi\)
0.438035 + 0.898958i \(0.355675\pi\)
\(312\) −1605.41 −0.291308
\(313\) 969.070 0.175000 0.0875001 0.996165i \(-0.472112\pi\)
0.0875001 + 0.996165i \(0.472112\pi\)
\(314\) −7518.71 −1.35129
\(315\) 2934.14 0.524826
\(316\) 752.806 0.134015
\(317\) −3279.49 −0.581055 −0.290528 0.956867i \(-0.593831\pi\)
−0.290528 + 0.956867i \(0.593831\pi\)
\(318\) 2.59229 0.000457133 0
\(319\) −134.017 −0.0235219
\(320\) −13038.3 −2.27769
\(321\) −3018.92 −0.524921
\(322\) −8812.40 −1.52514
\(323\) 4348.44 0.749083
\(324\) −5149.19 −0.882921
\(325\) 18350.6 3.13203
\(326\) 11892.2 2.02040
\(327\) −9590.07 −1.62181
\(328\) −170.923 −0.0287733
\(329\) −14856.7 −2.48959
\(330\) 282.193 0.0470734
\(331\) −6633.82 −1.10159 −0.550797 0.834639i \(-0.685676\pi\)
−0.550797 + 0.834639i \(0.685676\pi\)
\(332\) −8545.24 −1.41259
\(333\) 507.036 0.0834396
\(334\) −60.4897 −0.00990973
\(335\) −8188.45 −1.33547
\(336\) 6837.34 1.11014
\(337\) −4061.37 −0.656489 −0.328244 0.944593i \(-0.606457\pi\)
−0.328244 + 0.944593i \(0.606457\pi\)
\(338\) −9897.93 −1.59283
\(339\) −4246.97 −0.680424
\(340\) −8275.05 −1.31993
\(341\) 27.8475 0.00442236
\(342\) −2119.69 −0.335146
\(343\) −2504.56 −0.394267
\(344\) 0 0
\(345\) −7068.01 −1.10298
\(346\) −4303.54 −0.668669
\(347\) −7916.21 −1.22468 −0.612341 0.790594i \(-0.709772\pi\)
−0.612341 + 0.790594i \(0.709772\pi\)
\(348\) 7857.04 1.21029
\(349\) 12863.0 1.97290 0.986448 0.164077i \(-0.0524645\pi\)
0.986448 + 0.164077i \(0.0524645\pi\)
\(350\) 31342.9 4.78672
\(351\) −10184.1 −1.54868
\(352\) −190.191 −0.0287988
\(353\) −7949.47 −1.19861 −0.599303 0.800522i \(-0.704555\pi\)
−0.599303 + 0.800522i \(0.704555\pi\)
\(354\) 2142.52 0.321676
\(355\) 4300.42 0.642937
\(356\) 10387.3 1.54643
\(357\) 5848.44 0.867037
\(358\) 5257.26 0.776130
\(359\) −3114.11 −0.457817 −0.228909 0.973448i \(-0.573516\pi\)
−0.228909 + 0.973448i \(0.573516\pi\)
\(360\) 536.215 0.0785028
\(361\) 2452.25 0.357523
\(362\) 251.287 0.0364843
\(363\) −6198.78 −0.896286
\(364\) −17396.4 −2.50500
\(365\) 2713.02 0.389057
\(366\) 10311.8 1.47270
\(367\) −4292.76 −0.610573 −0.305287 0.952260i \(-0.598752\pi\)
−0.305287 + 0.952260i \(0.598752\pi\)
\(368\) 4015.74 0.568845
\(369\) −177.708 −0.0250707
\(370\) 7913.54 1.11191
\(371\) 3.73412 0.000522549 0
\(372\) −1632.62 −0.227547
\(373\) 2868.12 0.398138 0.199069 0.979985i \(-0.436208\pi\)
0.199069 + 0.979985i \(0.436208\pi\)
\(374\) −137.141 −0.0189610
\(375\) 13547.8 1.86561
\(376\) −2715.06 −0.372390
\(377\) 12371.9 1.69014
\(378\) −17394.5 −2.36687
\(379\) 5154.61 0.698613 0.349307 0.937009i \(-0.386417\pi\)
0.349307 + 0.937009i \(0.386417\pi\)
\(380\) −17719.2 −2.39205
\(381\) −1748.44 −0.235105
\(382\) 9286.03 1.24376
\(383\) −4471.13 −0.596512 −0.298256 0.954486i \(-0.596405\pi\)
−0.298256 + 0.954486i \(0.596405\pi\)
\(384\) 3000.16 0.398701
\(385\) 406.491 0.0538096
\(386\) 4774.43 0.629565
\(387\) 0 0
\(388\) −13950.6 −1.82535
\(389\) 8980.63 1.17053 0.585265 0.810842i \(-0.300990\pi\)
0.585265 + 0.810842i \(0.300990\pi\)
\(390\) −26050.9 −3.38241
\(391\) 3434.94 0.444277
\(392\) −2203.77 −0.283946
\(393\) 12585.6 1.61541
\(394\) −14800.0 −1.89241
\(395\) 1623.87 0.206850
\(396\) 35.8053 0.00454365
\(397\) 6091.78 0.770120 0.385060 0.922892i \(-0.374181\pi\)
0.385060 + 0.922892i \(0.374181\pi\)
\(398\) −9603.39 −1.20948
\(399\) 12523.2 1.57129
\(400\) −14282.7 −1.78534
\(401\) −6404.00 −0.797507 −0.398753 0.917058i \(-0.630557\pi\)
−0.398753 + 0.917058i \(0.630557\pi\)
\(402\) 7956.09 0.987099
\(403\) −2570.76 −0.317764
\(404\) −12223.2 −1.50526
\(405\) −11107.3 −1.36277
\(406\) 21131.2 2.58306
\(407\) 70.2439 0.00855494
\(408\) 1068.80 0.129690
\(409\) −12843.8 −1.55277 −0.776386 0.630258i \(-0.782949\pi\)
−0.776386 + 0.630258i \(0.782949\pi\)
\(410\) −2773.57 −0.334090
\(411\) 9422.07 1.13079
\(412\) −10290.6 −1.23054
\(413\) 3086.23 0.367708
\(414\) −1674.39 −0.198773
\(415\) −18432.8 −2.18032
\(416\) 17557.6 2.06931
\(417\) −2629.75 −0.308823
\(418\) −293.659 −0.0343620
\(419\) 8367.04 0.975552 0.487776 0.872969i \(-0.337808\pi\)
0.487776 + 0.872969i \(0.337808\pi\)
\(420\) −23831.5 −2.76871
\(421\) 2176.45 0.251956 0.125978 0.992033i \(-0.459793\pi\)
0.125978 + 0.992033i \(0.459793\pi\)
\(422\) 7020.88 0.809884
\(423\) −2822.83 −0.324470
\(424\) 0.682411 7.81623e−5 0
\(425\) −12217.0 −1.39438
\(426\) −4178.39 −0.475220
\(427\) 14853.9 1.68344
\(428\) −5978.40 −0.675179
\(429\) −231.239 −0.0260240
\(430\) 0 0
\(431\) 2491.13 0.278408 0.139204 0.990264i \(-0.455546\pi\)
0.139204 + 0.990264i \(0.455546\pi\)
\(432\) 7926.54 0.882791
\(433\) 15117.3 1.67781 0.838906 0.544277i \(-0.183196\pi\)
0.838906 + 0.544277i \(0.183196\pi\)
\(434\) −4390.87 −0.485642
\(435\) 16948.3 1.86807
\(436\) −18991.3 −2.08605
\(437\) 7355.18 0.805139
\(438\) −2636.03 −0.287567
\(439\) 2947.73 0.320473 0.160237 0.987079i \(-0.448774\pi\)
0.160237 + 0.987079i \(0.448774\pi\)
\(440\) 74.2864 0.00804878
\(441\) −2291.24 −0.247408
\(442\) 12660.3 1.36242
\(443\) −6749.23 −0.723850 −0.361925 0.932207i \(-0.617880\pi\)
−0.361925 + 0.932207i \(0.617880\pi\)
\(444\) −4118.21 −0.440184
\(445\) 22406.4 2.38689
\(446\) −1685.37 −0.178934
\(447\) 3514.25 0.371853
\(448\) 18248.3 1.92444
\(449\) 5120.14 0.538162 0.269081 0.963118i \(-0.413280\pi\)
0.269081 + 0.963118i \(0.413280\pi\)
\(450\) 5955.29 0.623856
\(451\) −24.6194 −0.00257047
\(452\) −8410.31 −0.875194
\(453\) −11905.3 −1.23479
\(454\) 18524.4 1.91496
\(455\) −37525.6 −3.86643
\(456\) 2288.61 0.235031
\(457\) −11669.3 −1.19446 −0.597230 0.802070i \(-0.703732\pi\)
−0.597230 + 0.802070i \(0.703732\pi\)
\(458\) 22866.8 2.33296
\(459\) 6780.11 0.689474
\(460\) −13996.8 −1.41871
\(461\) −12772.3 −1.29038 −0.645188 0.764024i \(-0.723221\pi\)
−0.645188 + 0.764024i \(0.723221\pi\)
\(462\) −394.957 −0.0397728
\(463\) −6707.16 −0.673235 −0.336618 0.941641i \(-0.609283\pi\)
−0.336618 + 0.941641i \(0.609283\pi\)
\(464\) −9629.30 −0.963425
\(465\) −3521.71 −0.351216
\(466\) 17527.3 1.74235
\(467\) −13188.3 −1.30681 −0.653407 0.757007i \(-0.726661\pi\)
−0.653407 + 0.757007i \(0.726661\pi\)
\(468\) −3305.40 −0.326479
\(469\) 11460.5 1.12835
\(470\) −44057.3 −4.32385
\(471\) 8440.12 0.825691
\(472\) 564.009 0.0550013
\(473\) 0 0
\(474\) −1577.79 −0.152891
\(475\) −26160.0 −2.52696
\(476\) 11581.7 1.11523
\(477\) 0.709499 6.81042e−5 0
\(478\) 3639.23 0.348231
\(479\) 2983.64 0.284606 0.142303 0.989823i \(-0.454549\pi\)
0.142303 + 0.989823i \(0.454549\pi\)
\(480\) 24052.3 2.28715
\(481\) −6484.62 −0.614706
\(482\) 13156.5 1.24328
\(483\) 9892.35 0.931920
\(484\) −12275.5 −1.15285
\(485\) −30092.7 −2.81740
\(486\) −6068.38 −0.566394
\(487\) 903.861 0.0841024 0.0420512 0.999115i \(-0.486611\pi\)
0.0420512 + 0.999115i \(0.486611\pi\)
\(488\) 2714.55 0.251807
\(489\) −13349.6 −1.23454
\(490\) −35760.5 −3.29693
\(491\) −14745.6 −1.35531 −0.677656 0.735379i \(-0.737004\pi\)
−0.677656 + 0.735379i \(0.737004\pi\)
\(492\) 1443.37 0.132260
\(493\) −8236.60 −0.752450
\(494\) 27109.3 2.46904
\(495\) 77.2351 0.00701305
\(496\) 2000.89 0.181134
\(497\) −6018.85 −0.543224
\(498\) 17909.8 1.61156
\(499\) −846.074 −0.0759028 −0.0379514 0.999280i \(-0.512083\pi\)
−0.0379514 + 0.999280i \(0.512083\pi\)
\(500\) 26828.8 2.39964
\(501\) 67.9026 0.00605522
\(502\) −16562.9 −1.47258
\(503\) −239.890 −0.0212647 −0.0106324 0.999943i \(-0.503384\pi\)
−0.0106324 + 0.999943i \(0.503384\pi\)
\(504\) −750.484 −0.0663278
\(505\) −26366.5 −2.32335
\(506\) −231.968 −0.0203799
\(507\) 11110.9 0.973280
\(508\) −3462.44 −0.302404
\(509\) 5624.56 0.489792 0.244896 0.969549i \(-0.421246\pi\)
0.244896 + 0.969549i \(0.421246\pi\)
\(510\) 17343.5 1.50585
\(511\) −3797.13 −0.328718
\(512\) −15811.0 −1.36475
\(513\) 14518.1 1.24950
\(514\) −18368.8 −1.57629
\(515\) −22197.8 −1.89932
\(516\) 0 0
\(517\) −391.071 −0.0332675
\(518\) −11075.8 −0.939461
\(519\) 4830.93 0.408582
\(520\) −6857.80 −0.578336
\(521\) 12416.2 1.04408 0.522038 0.852922i \(-0.325172\pi\)
0.522038 + 0.852922i \(0.325172\pi\)
\(522\) 4015.01 0.336652
\(523\) 8470.79 0.708225 0.354112 0.935203i \(-0.384783\pi\)
0.354112 + 0.935203i \(0.384783\pi\)
\(524\) 24923.3 2.07782
\(525\) −35184.0 −2.92487
\(526\) −302.360 −0.0250637
\(527\) 1711.49 0.141468
\(528\) 179.979 0.0148344
\(529\) −6356.97 −0.522477
\(530\) 11.0735 0.000907549 0
\(531\) 586.397 0.0479237
\(532\) 24799.8 2.02106
\(533\) 2272.76 0.184698
\(534\) −21770.6 −1.76424
\(535\) −12895.9 −1.04213
\(536\) 2094.41 0.168778
\(537\) −5901.53 −0.474245
\(538\) −16122.1 −1.29196
\(539\) −317.425 −0.0253664
\(540\) −27627.9 −2.20170
\(541\) 1109.59 0.0881792 0.0440896 0.999028i \(-0.485961\pi\)
0.0440896 + 0.999028i \(0.485961\pi\)
\(542\) −27877.3 −2.20928
\(543\) −282.082 −0.0222933
\(544\) −11689.0 −0.921255
\(545\) −40965.9 −3.21979
\(546\) 36460.7 2.85783
\(547\) −13123.6 −1.02582 −0.512912 0.858441i \(-0.671433\pi\)
−0.512912 + 0.858441i \(0.671433\pi\)
\(548\) 18658.6 1.45448
\(549\) 2822.30 0.219404
\(550\) 825.037 0.0639631
\(551\) −17636.9 −1.36362
\(552\) 1807.83 0.139395
\(553\) −2272.76 −0.174769
\(554\) −8481.06 −0.650407
\(555\) −8883.34 −0.679417
\(556\) −5207.71 −0.397223
\(557\) 16074.4 1.22279 0.611395 0.791325i \(-0.290608\pi\)
0.611395 + 0.791325i \(0.290608\pi\)
\(558\) −834.285 −0.0632941
\(559\) 0 0
\(560\) 29207.0 2.20397
\(561\) 153.948 0.0115859
\(562\) 8681.60 0.651621
\(563\) −13329.8 −0.997839 −0.498919 0.866648i \(-0.666270\pi\)
−0.498919 + 0.866648i \(0.666270\pi\)
\(564\) 22927.4 1.71174
\(565\) −18141.8 −1.35085
\(566\) 17052.8 1.26640
\(567\) 15545.7 1.15142
\(568\) −1099.95 −0.0812548
\(569\) −17313.5 −1.27561 −0.637803 0.770199i \(-0.720157\pi\)
−0.637803 + 0.770199i \(0.720157\pi\)
\(570\) 37137.3 2.72897
\(571\) −8514.71 −0.624045 −0.312022 0.950075i \(-0.601006\pi\)
−0.312022 + 0.950075i \(0.601006\pi\)
\(572\) −457.924 −0.0334734
\(573\) −10424.0 −0.759982
\(574\) 3881.87 0.282276
\(575\) −20664.4 −1.49872
\(576\) 3467.26 0.250814
\(577\) 23224.4 1.67564 0.837820 0.545947i \(-0.183830\pi\)
0.837820 + 0.545947i \(0.183830\pi\)
\(578\) 11962.7 0.860868
\(579\) −5359.53 −0.384688
\(580\) 33562.9 2.40280
\(581\) 25798.5 1.84217
\(582\) 29238.8 2.08245
\(583\) 0.0982928 6.98263e−6 0
\(584\) −693.926 −0.0491693
\(585\) −7130.02 −0.503915
\(586\) 3201.79 0.225708
\(587\) 15872.8 1.11608 0.558042 0.829813i \(-0.311553\pi\)
0.558042 + 0.829813i \(0.311553\pi\)
\(588\) 18609.8 1.30520
\(589\) 3664.79 0.256376
\(590\) 9152.18 0.638626
\(591\) 16613.7 1.15634
\(592\) 5047.13 0.350399
\(593\) 21818.4 1.51092 0.755461 0.655194i \(-0.227413\pi\)
0.755461 + 0.655194i \(0.227413\pi\)
\(594\) −457.874 −0.0316276
\(595\) 24982.8 1.72133
\(596\) 6959.30 0.478295
\(597\) 10780.3 0.739041
\(598\) 21414.3 1.46437
\(599\) −10835.2 −0.739087 −0.369544 0.929213i \(-0.620486\pi\)
−0.369544 + 0.929213i \(0.620486\pi\)
\(600\) −6429.88 −0.437498
\(601\) −7125.29 −0.483605 −0.241803 0.970325i \(-0.577739\pi\)
−0.241803 + 0.970325i \(0.577739\pi\)
\(602\) 0 0
\(603\) 2177.55 0.147059
\(604\) −23576.2 −1.58825
\(605\) −26479.3 −1.77940
\(606\) 25618.3 1.71728
\(607\) −10703.4 −0.715710 −0.357855 0.933777i \(-0.616492\pi\)
−0.357855 + 0.933777i \(0.616492\pi\)
\(608\) −25029.5 −1.66954
\(609\) −23720.8 −1.57835
\(610\) 44049.0 2.92376
\(611\) 36102.0 2.39040
\(612\) 2200.58 0.145348
\(613\) −3774.44 −0.248692 −0.124346 0.992239i \(-0.539683\pi\)
−0.124346 + 0.992239i \(0.539683\pi\)
\(614\) 27509.4 1.80813
\(615\) 3113.47 0.204142
\(616\) −103.971 −0.00680049
\(617\) −19188.0 −1.25199 −0.625997 0.779826i \(-0.715308\pi\)
−0.625997 + 0.779826i \(0.715308\pi\)
\(618\) 21567.9 1.40387
\(619\) −19479.3 −1.26485 −0.632424 0.774622i \(-0.717940\pi\)
−0.632424 + 0.774622i \(0.717940\pi\)
\(620\) −6974.07 −0.451751
\(621\) 11468.2 0.741069
\(622\) −19942.4 −1.28556
\(623\) −31359.9 −2.01671
\(624\) −16614.9 −1.06591
\(625\) 23984.0 1.53498
\(626\) −4022.11 −0.256798
\(627\) 329.646 0.0209965
\(628\) 16714.0 1.06204
\(629\) 4317.16 0.273667
\(630\) −12178.1 −0.770138
\(631\) 7063.66 0.445642 0.222821 0.974859i \(-0.428473\pi\)
0.222821 + 0.974859i \(0.428473\pi\)
\(632\) −415.347 −0.0261418
\(633\) −7881.28 −0.494870
\(634\) 13611.5 0.852650
\(635\) −7468.79 −0.466756
\(636\) −5.76265 −0.000359283 0
\(637\) 29303.4 1.82267
\(638\) 556.234 0.0345165
\(639\) −1143.61 −0.0707988
\(640\) 12815.8 0.791543
\(641\) −8274.88 −0.509888 −0.254944 0.966956i \(-0.582057\pi\)
−0.254944 + 0.966956i \(0.582057\pi\)
\(642\) 12530.0 0.770278
\(643\) 18880.9 1.15799 0.578996 0.815331i \(-0.303445\pi\)
0.578996 + 0.815331i \(0.303445\pi\)
\(644\) 19589.9 1.19868
\(645\) 0 0
\(646\) −18048.1 −1.09922
\(647\) 13808.8 0.839075 0.419537 0.907738i \(-0.362192\pi\)
0.419537 + 0.907738i \(0.362192\pi\)
\(648\) 2840.97 0.172228
\(649\) 81.2386 0.00491355
\(650\) −76164.0 −4.59599
\(651\) 4928.97 0.296746
\(652\) −26436.3 −1.58792
\(653\) −30854.7 −1.84906 −0.924530 0.381109i \(-0.875542\pi\)
−0.924530 + 0.381109i \(0.875542\pi\)
\(654\) 39803.4 2.37987
\(655\) 53761.7 3.20709
\(656\) −1768.94 −0.105283
\(657\) −721.471 −0.0428421
\(658\) 61662.4 3.65327
\(659\) −11484.2 −0.678848 −0.339424 0.940633i \(-0.610232\pi\)
−0.339424 + 0.940633i \(0.610232\pi\)
\(660\) −627.314 −0.0369972
\(661\) 6233.04 0.366773 0.183387 0.983041i \(-0.441294\pi\)
0.183387 + 0.983041i \(0.441294\pi\)
\(662\) 27533.5 1.61650
\(663\) −14211.8 −0.832491
\(664\) 4714.68 0.275550
\(665\) 53495.2 3.11948
\(666\) −2104.44 −0.122441
\(667\) −13931.8 −0.808758
\(668\) 134.468 0.00778852
\(669\) 1891.91 0.109336
\(670\) 33986.0 1.95969
\(671\) 390.997 0.0224952
\(672\) −33663.5 −1.93244
\(673\) 27119.3 1.55330 0.776651 0.629931i \(-0.216917\pi\)
0.776651 + 0.629931i \(0.216917\pi\)
\(674\) 16856.6 0.963343
\(675\) −40788.9 −2.32587
\(676\) 22003.0 1.25188
\(677\) −22288.2 −1.26530 −0.632649 0.774439i \(-0.718032\pi\)
−0.632649 + 0.774439i \(0.718032\pi\)
\(678\) 17627.0 0.998465
\(679\) 42117.6 2.38045
\(680\) 4565.60 0.257475
\(681\) −20794.5 −1.17011
\(682\) −115.580 −0.00648945
\(683\) 2868.96 0.160729 0.0803644 0.996766i \(-0.474392\pi\)
0.0803644 + 0.996766i \(0.474392\pi\)
\(684\) 4712.06 0.263407
\(685\) 40248.2 2.24497
\(686\) 10395.1 0.578554
\(687\) −25669.1 −1.42553
\(688\) 0 0
\(689\) −9.07398 −0.000501729 0
\(690\) 29335.6 1.61853
\(691\) −32355.3 −1.78126 −0.890631 0.454726i \(-0.849737\pi\)
−0.890631 + 0.454726i \(0.849737\pi\)
\(692\) 9566.74 0.525539
\(693\) −108.098 −0.00592540
\(694\) 32856.1 1.79712
\(695\) −11233.5 −0.613108
\(696\) −4334.97 −0.236087
\(697\) −1513.09 −0.0822274
\(698\) −53387.6 −2.89506
\(699\) −19675.2 −1.06464
\(700\) −69675.2 −3.76211
\(701\) −5159.16 −0.277973 −0.138986 0.990294i \(-0.544384\pi\)
−0.138986 + 0.990294i \(0.544384\pi\)
\(702\) 42269.0 2.27256
\(703\) 9244.26 0.495952
\(704\) 480.348 0.0257156
\(705\) 49456.5 2.64204
\(706\) 32994.1 1.75885
\(707\) 36902.4 1.96302
\(708\) −4762.80 −0.252821
\(709\) 20468.6 1.08422 0.542112 0.840306i \(-0.317625\pi\)
0.542112 + 0.840306i \(0.317625\pi\)
\(710\) −17848.8 −0.943457
\(711\) −431.834 −0.0227778
\(712\) −5731.03 −0.301656
\(713\) 2894.91 0.152055
\(714\) −24273.8 −1.27230
\(715\) −987.782 −0.0516657
\(716\) −11686.9 −0.609997
\(717\) −4085.21 −0.212782
\(718\) 12925.0 0.671809
\(719\) −16111.8 −0.835702 −0.417851 0.908515i \(-0.637217\pi\)
−0.417851 + 0.908515i \(0.637217\pi\)
\(720\) 5549.46 0.287245
\(721\) 31068.0 1.60476
\(722\) −10178.0 −0.524635
\(723\) −14768.8 −0.759691
\(724\) −558.609 −0.0286748
\(725\) 49551.1 2.53832
\(726\) 25727.9 1.31522
\(727\) 9681.53 0.493904 0.246952 0.969028i \(-0.420571\pi\)
0.246952 + 0.969028i \(0.420571\pi\)
\(728\) 9598.15 0.488642
\(729\) 21880.4 1.11164
\(730\) −11260.3 −0.570909
\(731\) 0 0
\(732\) −22923.1 −1.15746
\(733\) 18580.8 0.936288 0.468144 0.883652i \(-0.344923\pi\)
0.468144 + 0.883652i \(0.344923\pi\)
\(734\) 17817.0 0.895966
\(735\) 40142.9 2.01455
\(736\) −19771.4 −0.990196
\(737\) 301.674 0.0150778
\(738\) 737.573 0.0367892
\(739\) −27864.0 −1.38700 −0.693502 0.720455i \(-0.743933\pi\)
−0.693502 + 0.720455i \(0.743933\pi\)
\(740\) −17591.8 −0.873900
\(741\) −30431.6 −1.50868
\(742\) −15.4984 −0.000766798 0
\(743\) 9787.89 0.483288 0.241644 0.970365i \(-0.422313\pi\)
0.241644 + 0.970365i \(0.422313\pi\)
\(744\) 900.770 0.0443869
\(745\) 15011.8 0.738241
\(746\) −11904.1 −0.584235
\(747\) 4901.82 0.240091
\(748\) 304.864 0.0149023
\(749\) 18049.1 0.880505
\(750\) −56229.7 −2.73763
\(751\) 19515.4 0.948239 0.474119 0.880461i \(-0.342766\pi\)
0.474119 + 0.880461i \(0.342766\pi\)
\(752\) −28099.1 −1.36259
\(753\) 18592.6 0.899805
\(754\) −51349.2 −2.48014
\(755\) −50855.9 −2.45144
\(756\) 38667.9 1.86023
\(757\) 23470.3 1.12687 0.563436 0.826160i \(-0.309479\pi\)
0.563436 + 0.826160i \(0.309479\pi\)
\(758\) −21394.1 −1.02516
\(759\) 260.395 0.0124529
\(760\) 9776.26 0.466608
\(761\) −24891.6 −1.18570 −0.592851 0.805312i \(-0.701998\pi\)
−0.592851 + 0.805312i \(0.701998\pi\)
\(762\) 7256.85 0.344997
\(763\) 57335.6 2.72043
\(764\) −20642.8 −0.977526
\(765\) 4746.83 0.224343
\(766\) 18557.4 0.875332
\(767\) −7499.60 −0.353057
\(768\) 11965.9 0.562215
\(769\) −32481.7 −1.52317 −0.761586 0.648064i \(-0.775579\pi\)
−0.761586 + 0.648064i \(0.775579\pi\)
\(770\) −1687.13 −0.0789612
\(771\) 20619.8 0.963172
\(772\) −10613.5 −0.494805
\(773\) 1476.22 0.0686879 0.0343440 0.999410i \(-0.489066\pi\)
0.0343440 + 0.999410i \(0.489066\pi\)
\(774\) 0 0
\(775\) −10296.3 −0.477230
\(776\) 7697.00 0.356065
\(777\) 12433.1 0.574047
\(778\) −37274.0 −1.71766
\(779\) −3239.96 −0.149016
\(780\) 57911.0 2.65839
\(781\) −158.434 −0.00725890
\(782\) −14256.6 −0.651939
\(783\) −27499.5 −1.25511
\(784\) −22807.5 −1.03897
\(785\) 36053.6 1.63925
\(786\) −52236.1 −2.37049
\(787\) 7837.58 0.354993 0.177496 0.984121i \(-0.443200\pi\)
0.177496 + 0.984121i \(0.443200\pi\)
\(788\) 32900.2 1.48734
\(789\) 339.414 0.0153149
\(790\) −6739.83 −0.303535
\(791\) 25391.1 1.14135
\(792\) −19.7549 −0.000886313 0
\(793\) −36095.2 −1.61637
\(794\) −25283.8 −1.13009
\(795\) −12.4305 −0.000554547 0
\(796\) 21348.3 0.950590
\(797\) 10321.0 0.458706 0.229353 0.973343i \(-0.426339\pi\)
0.229353 + 0.973343i \(0.426339\pi\)
\(798\) −51977.2 −2.30573
\(799\) −24035.0 −1.06420
\(800\) 70320.7 3.10777
\(801\) −5958.52 −0.262839
\(802\) 26579.7 1.17028
\(803\) −99.9514 −0.00439254
\(804\) −17686.3 −0.775807
\(805\) 42257.1 1.85015
\(806\) 10669.9 0.466292
\(807\) 18097.8 0.789435
\(808\) 6743.92 0.293626
\(809\) 30215.4 1.31312 0.656561 0.754273i \(-0.272010\pi\)
0.656561 + 0.754273i \(0.272010\pi\)
\(810\) 46100.4 1.99976
\(811\) −43448.1 −1.88122 −0.940611 0.339487i \(-0.889747\pi\)
−0.940611 + 0.339487i \(0.889747\pi\)
\(812\) −46974.5 −2.03015
\(813\) 31293.6 1.34996
\(814\) −291.546 −0.0125537
\(815\) −57025.4 −2.45094
\(816\) 11061.4 0.474542
\(817\) 0 0
\(818\) 53307.8 2.27856
\(819\) 9979.15 0.425763
\(820\) 6165.63 0.262577
\(821\) 28087.5 1.19398 0.596991 0.802248i \(-0.296363\pi\)
0.596991 + 0.802248i \(0.296363\pi\)
\(822\) −39106.1 −1.65935
\(823\) −44076.2 −1.86683 −0.933413 0.358803i \(-0.883185\pi\)
−0.933413 + 0.358803i \(0.883185\pi\)
\(824\) 5677.67 0.240038
\(825\) −926.144 −0.0390839
\(826\) −12809.3 −0.539581
\(827\) 6095.24 0.256291 0.128145 0.991755i \(-0.459098\pi\)
0.128145 + 0.991755i \(0.459098\pi\)
\(828\) 3722.17 0.156225
\(829\) 8358.62 0.350189 0.175095 0.984552i \(-0.443977\pi\)
0.175095 + 0.984552i \(0.443977\pi\)
\(830\) 76505.0 3.19943
\(831\) 9520.40 0.397424
\(832\) −44343.7 −1.84777
\(833\) −19508.8 −0.811453
\(834\) 10914.7 0.453172
\(835\) 290.059 0.0120215
\(836\) 652.802 0.0270067
\(837\) 5714.16 0.235974
\(838\) −34727.2 −1.43154
\(839\) 12358.3 0.508530 0.254265 0.967135i \(-0.418166\pi\)
0.254265 + 0.967135i \(0.418166\pi\)
\(840\) 13148.6 0.540082
\(841\) 9017.93 0.369754
\(842\) −9033.31 −0.369725
\(843\) −9745.52 −0.398165
\(844\) −15607.4 −0.636526
\(845\) 47462.4 1.93226
\(846\) 11716.1 0.476133
\(847\) 37060.3 1.50343
\(848\) 7.06249 0.000285999 0
\(849\) −19142.6 −0.773819
\(850\) 50706.4 2.04614
\(851\) 7302.26 0.294146
\(852\) 9288.54 0.373498
\(853\) −17084.9 −0.685786 −0.342893 0.939374i \(-0.611407\pi\)
−0.342893 + 0.939374i \(0.611407\pi\)
\(854\) −61650.8 −2.47031
\(855\) 10164.3 0.406564
\(856\) 3298.47 0.131705
\(857\) −16684.6 −0.665037 −0.332519 0.943097i \(-0.607898\pi\)
−0.332519 + 0.943097i \(0.607898\pi\)
\(858\) 959.752 0.0381881
\(859\) −48813.4 −1.93887 −0.969436 0.245343i \(-0.921099\pi\)
−0.969436 + 0.245343i \(0.921099\pi\)
\(860\) 0 0
\(861\) −4357.59 −0.172481
\(862\) −10339.4 −0.408540
\(863\) 38260.4 1.50915 0.754577 0.656212i \(-0.227842\pi\)
0.754577 + 0.656212i \(0.227842\pi\)
\(864\) −39026.1 −1.53669
\(865\) 20636.3 0.811161
\(866\) −62744.2 −2.46205
\(867\) −13428.7 −0.526023
\(868\) 9760.88 0.381689
\(869\) −59.8256 −0.00233538
\(870\) −70343.6 −2.74123
\(871\) −27849.3 −1.08339
\(872\) 10478.1 0.406919
\(873\) 8002.53 0.310246
\(874\) −30527.5 −1.18147
\(875\) −80997.3 −3.12938
\(876\) 5859.88 0.226013
\(877\) 29665.4 1.14222 0.571112 0.820872i \(-0.306512\pi\)
0.571112 + 0.820872i \(0.306512\pi\)
\(878\) −12234.5 −0.470268
\(879\) −3594.17 −0.137916
\(880\) 768.814 0.0294508
\(881\) 3377.77 0.129171 0.0645857 0.997912i \(-0.479427\pi\)
0.0645857 + 0.997912i \(0.479427\pi\)
\(882\) 9509.77 0.363050
\(883\) 32125.9 1.22437 0.612186 0.790713i \(-0.290290\pi\)
0.612186 + 0.790713i \(0.290290\pi\)
\(884\) −28143.8 −1.07079
\(885\) −10273.8 −0.390225
\(886\) 28012.6 1.06219
\(887\) 25106.0 0.950368 0.475184 0.879886i \(-0.342382\pi\)
0.475184 + 0.879886i \(0.342382\pi\)
\(888\) 2272.15 0.0858652
\(889\) 10453.3 0.394366
\(890\) −92997.3 −3.50256
\(891\) 409.207 0.0153860
\(892\) 3746.57 0.140633
\(893\) −51465.8 −1.92860
\(894\) −14585.8 −0.545663
\(895\) −25209.5 −0.941522
\(896\) −17936.9 −0.668782
\(897\) −24038.6 −0.894789
\(898\) −21251.1 −0.789707
\(899\) −6941.67 −0.257528
\(900\) −13238.6 −0.490318
\(901\) 6.04103 0.000223370 0
\(902\) 102.182 0.00377194
\(903\) 0 0
\(904\) 4640.23 0.170721
\(905\) −1204.97 −0.0442591
\(906\) 49412.8 1.81195
\(907\) 17495.5 0.640496 0.320248 0.947334i \(-0.396234\pi\)
0.320248 + 0.947334i \(0.396234\pi\)
\(908\) −41179.5 −1.50506
\(909\) 7011.61 0.255842
\(910\) 155749. 5.67366
\(911\) 44170.3 1.60639 0.803197 0.595713i \(-0.203130\pi\)
0.803197 + 0.595713i \(0.203130\pi\)
\(912\) 23685.6 0.859987
\(913\) 679.091 0.0246162
\(914\) 48433.3 1.75277
\(915\) −49447.2 −1.78653
\(916\) −50832.7 −1.83358
\(917\) −75244.6 −2.70970
\(918\) −28140.7 −1.01174
\(919\) 2965.31 0.106438 0.0532191 0.998583i \(-0.483052\pi\)
0.0532191 + 0.998583i \(0.483052\pi\)
\(920\) 7722.50 0.276743
\(921\) −30880.7 −1.10483
\(922\) 53011.0 1.89352
\(923\) 14625.9 0.521580
\(924\) 877.986 0.0312593
\(925\) −25971.8 −0.923188
\(926\) 27837.9 0.987917
\(927\) 5903.05 0.209149
\(928\) 47409.7 1.67705
\(929\) 34759.3 1.22757 0.613786 0.789472i \(-0.289646\pi\)
0.613786 + 0.789472i \(0.289646\pi\)
\(930\) 14616.8 0.515380
\(931\) −41773.9 −1.47055
\(932\) −38963.1 −1.36940
\(933\) 22386.3 0.785526
\(934\) 54737.8 1.91764
\(935\) 657.619 0.0230015
\(936\) 1823.69 0.0636850
\(937\) −27998.7 −0.976178 −0.488089 0.872794i \(-0.662306\pi\)
−0.488089 + 0.872794i \(0.662306\pi\)
\(938\) −47566.6 −1.65576
\(939\) 4515.01 0.156914
\(940\) 97939.1 3.39832
\(941\) −23821.3 −0.825242 −0.412621 0.910903i \(-0.635387\pi\)
−0.412621 + 0.910903i \(0.635387\pi\)
\(942\) −35030.6 −1.21163
\(943\) −2559.32 −0.0883808
\(944\) 5837.12 0.201252
\(945\) 83409.9 2.87124
\(946\) 0 0
\(947\) −15304.0 −0.525145 −0.262573 0.964912i \(-0.584571\pi\)
−0.262573 + 0.964912i \(0.584571\pi\)
\(948\) 3507.41 0.120164
\(949\) 9227.09 0.315621
\(950\) 108577. 3.70810
\(951\) −15279.5 −0.521002
\(952\) −6390.00 −0.217543
\(953\) −528.850 −0.0179760 −0.00898800 0.999960i \(-0.502861\pi\)
−0.00898800 + 0.999960i \(0.502861\pi\)
\(954\) −2.94476 −9.99373e−5 0
\(955\) −44528.2 −1.50880
\(956\) −8089.98 −0.273691
\(957\) −624.399 −0.0210909
\(958\) −12383.6 −0.417635
\(959\) −56331.2 −1.89680
\(960\) −60746.8 −2.04229
\(961\) −28348.6 −0.951582
\(962\) 26914.3 0.902029
\(963\) 3429.40 0.114757
\(964\) −29246.7 −0.977151
\(965\) −22894.3 −0.763724
\(966\) −41058.0 −1.36752
\(967\) 22602.1 0.751639 0.375819 0.926693i \(-0.377361\pi\)
0.375819 + 0.926693i \(0.377361\pi\)
\(968\) 6772.78 0.224882
\(969\) 20259.9 0.671663
\(970\) 124899. 4.13430
\(971\) 23322.2 0.770797 0.385398 0.922750i \(-0.374064\pi\)
0.385398 + 0.922750i \(0.374064\pi\)
\(972\) 13490.0 0.445156
\(973\) 15722.3 0.518021
\(974\) −3751.46 −0.123413
\(975\) 85497.8 2.80833
\(976\) 28093.8 0.921373
\(977\) 27151.1 0.889090 0.444545 0.895757i \(-0.353365\pi\)
0.444545 + 0.895757i \(0.353365\pi\)
\(978\) 55407.2 1.81158
\(979\) −825.483 −0.0269485
\(980\) 79495.4 2.59121
\(981\) 10894.0 0.354556
\(982\) 61201.2 1.98881
\(983\) 16794.4 0.544922 0.272461 0.962167i \(-0.412163\pi\)
0.272461 + 0.962167i \(0.412163\pi\)
\(984\) −796.351 −0.0257995
\(985\) 70968.6 2.29568
\(986\) 34185.9 1.10416
\(987\) −69219.0 −2.23229
\(988\) −60263.9 −1.94054
\(989\) 0 0
\(990\) −320.563 −0.0102911
\(991\) −6415.98 −0.205661 −0.102831 0.994699i \(-0.532790\pi\)
−0.102831 + 0.994699i \(0.532790\pi\)
\(992\) −9851.32 −0.315302
\(993\) −30907.7 −0.987742
\(994\) 24981.1 0.797136
\(995\) 46050.1 1.46722
\(996\) −39813.3 −1.26660
\(997\) 38417.5 1.22036 0.610178 0.792264i \(-0.291098\pi\)
0.610178 + 0.792264i \(0.291098\pi\)
\(998\) 3511.62 0.111381
\(999\) 14413.7 0.456486
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.1.8 yes 50
43.42 odd 2 1849.4.a.i.1.43 50
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1849.4.a.i.1.43 50 43.42 odd 2
1849.4.a.j.1.8 yes 50 1.1 even 1 trivial