Properties

Label 39.4.a.c.1.3
Level $39$
Weight $4$
Character 39.1
Self dual yes
Analytic conductor $2.301$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 39 = 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 39.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.30107449022\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.3144.1
Defining polynomial: \( x^{3} - x^{2} - 16x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-3.20905\) of defining polynomial
Character \(\chi\) \(=\) 39.1

$q$-expansion

\(f(q)\) \(=\) \(q+4.20905 q^{2} +3.00000 q^{3} +9.71610 q^{4} -11.4322 q^{5} +12.6271 q^{6} -11.2543 q^{7} +7.22315 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+4.20905 q^{2} +3.00000 q^{3} +9.71610 q^{4} -11.4322 q^{5} +12.6271 q^{6} -11.2543 q^{7} +7.22315 q^{8} +9.00000 q^{9} -48.1187 q^{10} +25.8785 q^{11} +29.1483 q^{12} +13.0000 q^{13} -47.3699 q^{14} -34.2966 q^{15} -47.3262 q^{16} -20.3276 q^{17} +37.8814 q^{18} +154.712 q^{19} -111.076 q^{20} -33.7629 q^{21} +108.924 q^{22} -180.418 q^{23} +21.6695 q^{24} +5.69520 q^{25} +54.7176 q^{26} +27.0000 q^{27} -109.348 q^{28} -20.4522 q^{29} -144.356 q^{30} +266.424 q^{31} -256.984 q^{32} +77.6355 q^{33} -85.5599 q^{34} +128.661 q^{35} +87.4449 q^{36} +115.984 q^{37} +651.190 q^{38} +39.0000 q^{39} -82.5765 q^{40} +391.184 q^{41} -142.110 q^{42} +151.407 q^{43} +251.438 q^{44} -102.890 q^{45} -759.390 q^{46} -467.365 q^{47} -141.979 q^{48} -216.341 q^{49} +23.9714 q^{50} -60.9828 q^{51} +126.309 q^{52} +79.9842 q^{53} +113.644 q^{54} -295.848 q^{55} -81.2915 q^{56} +464.136 q^{57} -86.0843 q^{58} -873.710 q^{59} -333.229 q^{60} -187.068 q^{61} +1121.39 q^{62} -101.289 q^{63} -703.047 q^{64} -148.619 q^{65} +326.772 q^{66} -609.204 q^{67} -197.505 q^{68} -541.255 q^{69} +541.542 q^{70} +248.038 q^{71} +65.0084 q^{72} +852.765 q^{73} +488.181 q^{74} +17.0856 q^{75} +1503.20 q^{76} -291.244 q^{77} +164.153 q^{78} -331.221 q^{79} +541.043 q^{80} +81.0000 q^{81} +1646.51 q^{82} -435.432 q^{83} -328.044 q^{84} +232.389 q^{85} +637.281 q^{86} -61.3566 q^{87} +186.924 q^{88} +259.233 q^{89} -433.068 q^{90} -146.306 q^{91} -1752.96 q^{92} +799.273 q^{93} -1967.16 q^{94} -1768.70 q^{95} -770.951 q^{96} +1225.17 q^{97} -910.589 q^{98} +232.907 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{2} + 9 q^{3} + 10 q^{4} + 4 q^{5} + 6 q^{6} + 30 q^{7} - 6 q^{8} + 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 2 q^{2} + 9 q^{3} + 10 q^{4} + 4 q^{5} + 6 q^{6} + 30 q^{7} - 6 q^{8} + 27 q^{9} - 4 q^{10} - 16 q^{11} + 30 q^{12} + 39 q^{13} - 176 q^{14} + 12 q^{15} - 110 q^{16} - 146 q^{17} + 18 q^{18} + 94 q^{19} - 244 q^{20} + 90 q^{21} - 56 q^{22} - 48 q^{23} - 18 q^{24} + 145 q^{25} + 26 q^{26} + 81 q^{27} + 80 q^{28} - 2 q^{29} - 12 q^{30} + 302 q^{31} + 154 q^{32} - 48 q^{33} + 164 q^{34} + 80 q^{35} + 90 q^{36} + 374 q^{37} + 312 q^{38} + 117 q^{39} - 516 q^{40} + 480 q^{41} - 528 q^{42} - 260 q^{43} + 712 q^{44} + 36 q^{45} - 1104 q^{46} - 24 q^{47} - 330 q^{48} + 447 q^{49} + 814 q^{50} - 438 q^{51} + 130 q^{52} - 678 q^{53} + 54 q^{54} - 1552 q^{55} + 96 q^{56} + 282 q^{57} - 628 q^{58} - 1788 q^{59} - 732 q^{60} + 230 q^{61} + 1952 q^{62} + 270 q^{63} - 750 q^{64} + 52 q^{65} - 168 q^{66} + 74 q^{67} - 460 q^{68} - 144 q^{69} + 1216 q^{70} - 948 q^{71} - 54 q^{72} - 222 q^{73} + 1724 q^{74} + 435 q^{75} + 2392 q^{76} + 112 q^{77} + 78 q^{78} - 24 q^{79} + 1100 q^{80} + 243 q^{81} + 564 q^{82} - 796 q^{83} + 240 q^{84} - 248 q^{85} + 1800 q^{86} - 6 q^{87} + 1608 q^{88} + 1436 q^{89} - 36 q^{90} + 390 q^{91} - 1296 q^{92} + 906 q^{93} - 1920 q^{94} - 4032 q^{95} + 462 q^{96} + 3242 q^{97} - 5070 q^{98} - 144 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.20905 1.48812 0.744062 0.668111i \(-0.232897\pi\)
0.744062 + 0.668111i \(0.232897\pi\)
\(3\) 3.00000 0.577350
\(4\) 9.71610 1.21451
\(5\) −11.4322 −1.02253 −0.511264 0.859424i \(-0.670822\pi\)
−0.511264 + 0.859424i \(0.670822\pi\)
\(6\) 12.6271 0.859169
\(7\) −11.2543 −0.607675 −0.303838 0.952724i \(-0.598268\pi\)
−0.303838 + 0.952724i \(0.598268\pi\)
\(8\) 7.22315 0.319221
\(9\) 9.00000 0.333333
\(10\) −48.1187 −1.52165
\(11\) 25.8785 0.709333 0.354666 0.934993i \(-0.384594\pi\)
0.354666 + 0.934993i \(0.384594\pi\)
\(12\) 29.1483 0.701199
\(13\) 13.0000 0.277350
\(14\) −47.3699 −0.904296
\(15\) −34.2966 −0.590356
\(16\) −47.3262 −0.739472
\(17\) −20.3276 −0.290010 −0.145005 0.989431i \(-0.546320\pi\)
−0.145005 + 0.989431i \(0.546320\pi\)
\(18\) 37.8814 0.496041
\(19\) 154.712 1.86807 0.934035 0.357181i \(-0.116262\pi\)
0.934035 + 0.357181i \(0.116262\pi\)
\(20\) −111.076 −1.24187
\(21\) −33.7629 −0.350841
\(22\) 108.924 1.05558
\(23\) −180.418 −1.63565 −0.817823 0.575471i \(-0.804819\pi\)
−0.817823 + 0.575471i \(0.804819\pi\)
\(24\) 21.6695 0.184302
\(25\) 5.69520 0.0455616
\(26\) 54.7176 0.412731
\(27\) 27.0000 0.192450
\(28\) −109.348 −0.738029
\(29\) −20.4522 −0.130961 −0.0654806 0.997854i \(-0.520858\pi\)
−0.0654806 + 0.997854i \(0.520858\pi\)
\(30\) −144.356 −0.878523
\(31\) 266.424 1.54359 0.771794 0.635873i \(-0.219360\pi\)
0.771794 + 0.635873i \(0.219360\pi\)
\(32\) −256.984 −1.41965
\(33\) 77.6355 0.409534
\(34\) −85.5599 −0.431571
\(35\) 128.661 0.621364
\(36\) 87.4449 0.404837
\(37\) 115.984 0.515340 0.257670 0.966233i \(-0.417045\pi\)
0.257670 + 0.966233i \(0.417045\pi\)
\(38\) 651.190 2.77992
\(39\) 39.0000 0.160128
\(40\) −82.5765 −0.326412
\(41\) 391.184 1.49006 0.745032 0.667029i \(-0.232434\pi\)
0.745032 + 0.667029i \(0.232434\pi\)
\(42\) −142.110 −0.522095
\(43\) 151.407 0.536963 0.268482 0.963285i \(-0.413478\pi\)
0.268482 + 0.963285i \(0.413478\pi\)
\(44\) 251.438 0.861494
\(45\) −102.890 −0.340842
\(46\) −759.390 −2.43404
\(47\) −467.365 −1.45047 −0.725236 0.688500i \(-0.758269\pi\)
−0.725236 + 0.688500i \(0.758269\pi\)
\(48\) −141.979 −0.426934
\(49\) −216.341 −0.630731
\(50\) 23.9714 0.0678012
\(51\) −60.9828 −0.167437
\(52\) 126.309 0.336845
\(53\) 79.9842 0.207296 0.103648 0.994614i \(-0.466949\pi\)
0.103648 + 0.994614i \(0.466949\pi\)
\(54\) 113.644 0.286390
\(55\) −295.848 −0.725312
\(56\) −81.2915 −0.193983
\(57\) 464.136 1.07853
\(58\) −86.0843 −0.194887
\(59\) −873.710 −1.92792 −0.963960 0.266045i \(-0.914283\pi\)
−0.963960 + 0.266045i \(0.914283\pi\)
\(60\) −333.229 −0.716995
\(61\) −187.068 −0.392649 −0.196325 0.980539i \(-0.562901\pi\)
−0.196325 + 0.980539i \(0.562901\pi\)
\(62\) 1121.39 2.29705
\(63\) −101.289 −0.202558
\(64\) −703.047 −1.37314
\(65\) −148.619 −0.283598
\(66\) 326.772 0.609437
\(67\) −609.204 −1.11084 −0.555418 0.831571i \(-0.687442\pi\)
−0.555418 + 0.831571i \(0.687442\pi\)
\(68\) −197.505 −0.352221
\(69\) −541.255 −0.944340
\(70\) 541.542 0.924667
\(71\) 248.038 0.414601 0.207301 0.978277i \(-0.433532\pi\)
0.207301 + 0.978277i \(0.433532\pi\)
\(72\) 65.0084 0.106407
\(73\) 852.765 1.36724 0.683621 0.729838i \(-0.260404\pi\)
0.683621 + 0.729838i \(0.260404\pi\)
\(74\) 488.181 0.766890
\(75\) 17.0856 0.0263050
\(76\) 1503.20 2.26880
\(77\) −291.244 −0.431044
\(78\) 164.153 0.238291
\(79\) −331.221 −0.471712 −0.235856 0.971788i \(-0.575789\pi\)
−0.235856 + 0.971788i \(0.575789\pi\)
\(80\) 541.043 0.756130
\(81\) 81.0000 0.111111
\(82\) 1646.51 2.21740
\(83\) −435.432 −0.575842 −0.287921 0.957654i \(-0.592964\pi\)
−0.287921 + 0.957654i \(0.592964\pi\)
\(84\) −328.044 −0.426101
\(85\) 232.389 0.296543
\(86\) 637.281 0.799067
\(87\) −61.3566 −0.0756105
\(88\) 186.924 0.226434
\(89\) 259.233 0.308749 0.154375 0.988012i \(-0.450664\pi\)
0.154375 + 0.988012i \(0.450664\pi\)
\(90\) −433.068 −0.507216
\(91\) −146.306 −0.168539
\(92\) −1752.96 −1.98651
\(93\) 799.273 0.891191
\(94\) −1967.16 −2.15848
\(95\) −1768.70 −1.91015
\(96\) −770.951 −0.819634
\(97\) 1225.17 1.28245 0.641223 0.767355i \(-0.278428\pi\)
0.641223 + 0.767355i \(0.278428\pi\)
\(98\) −910.589 −0.938606
\(99\) 232.907 0.236444
\(100\) 55.3351 0.0553351
\(101\) 645.416 0.635855 0.317927 0.948115i \(-0.397013\pi\)
0.317927 + 0.948115i \(0.397013\pi\)
\(102\) −256.680 −0.249167
\(103\) −511.137 −0.488969 −0.244484 0.969653i \(-0.578619\pi\)
−0.244484 + 0.969653i \(0.578619\pi\)
\(104\) 93.9010 0.0885360
\(105\) 385.984 0.358745
\(106\) 336.657 0.308482
\(107\) 608.195 0.549499 0.274750 0.961516i \(-0.411405\pi\)
0.274750 + 0.961516i \(0.411405\pi\)
\(108\) 262.335 0.233733
\(109\) −1300.04 −1.14239 −0.571197 0.820813i \(-0.693521\pi\)
−0.571197 + 0.820813i \(0.693521\pi\)
\(110\) −1245.24 −1.07935
\(111\) 347.951 0.297532
\(112\) 532.623 0.449359
\(113\) 42.1953 0.0351274 0.0175637 0.999846i \(-0.494409\pi\)
0.0175637 + 0.999846i \(0.494409\pi\)
\(114\) 1953.57 1.60499
\(115\) 2062.58 1.67249
\(116\) −198.716 −0.159054
\(117\) 117.000 0.0924500
\(118\) −3677.49 −2.86899
\(119\) 228.773 0.176232
\(120\) −247.729 −0.188454
\(121\) −661.303 −0.496847
\(122\) −787.378 −0.584311
\(123\) 1173.55 0.860289
\(124\) 2588.61 1.87471
\(125\) 1363.92 0.975939
\(126\) −426.329 −0.301432
\(127\) −311.018 −0.217310 −0.108655 0.994080i \(-0.534654\pi\)
−0.108655 + 0.994080i \(0.534654\pi\)
\(128\) −903.291 −0.623753
\(129\) 454.222 0.310016
\(130\) −625.543 −0.422029
\(131\) 2000.98 1.33456 0.667278 0.744809i \(-0.267459\pi\)
0.667278 + 0.744809i \(0.267459\pi\)
\(132\) 754.314 0.497384
\(133\) −1741.17 −1.13518
\(134\) −2564.17 −1.65306
\(135\) −308.669 −0.196785
\(136\) −146.829 −0.0925773
\(137\) 1038.53 0.647644 0.323822 0.946118i \(-0.395032\pi\)
0.323822 + 0.946118i \(0.395032\pi\)
\(138\) −2278.17 −1.40529
\(139\) −2858.46 −1.74426 −0.872128 0.489277i \(-0.837261\pi\)
−0.872128 + 0.489277i \(0.837261\pi\)
\(140\) 1250.09 0.754655
\(141\) −1402.09 −0.837430
\(142\) 1044.00 0.616978
\(143\) 336.421 0.196734
\(144\) −425.936 −0.246491
\(145\) 233.814 0.133911
\(146\) 3589.33 2.03462
\(147\) −649.022 −0.364153
\(148\) 1126.91 0.625887
\(149\) 743.479 0.408780 0.204390 0.978890i \(-0.434479\pi\)
0.204390 + 0.978890i \(0.434479\pi\)
\(150\) 71.9141 0.0391451
\(151\) 2277.24 1.22728 0.613640 0.789586i \(-0.289705\pi\)
0.613640 + 0.789586i \(0.289705\pi\)
\(152\) 1117.51 0.596328
\(153\) −182.948 −0.0966700
\(154\) −1225.86 −0.641447
\(155\) −3045.82 −1.57836
\(156\) 378.928 0.194478
\(157\) 3173.51 1.61321 0.806605 0.591091i \(-0.201303\pi\)
0.806605 + 0.591091i \(0.201303\pi\)
\(158\) −1394.12 −0.701966
\(159\) 239.953 0.119682
\(160\) 2937.89 1.45163
\(161\) 2030.48 0.993941
\(162\) 340.933 0.165347
\(163\) −2314.65 −1.11225 −0.556126 0.831098i \(-0.687713\pi\)
−0.556126 + 0.831098i \(0.687713\pi\)
\(164\) 3800.78 1.80970
\(165\) −887.545 −0.418759
\(166\) −1832.76 −0.856925
\(167\) −2665.65 −1.23517 −0.617587 0.786502i \(-0.711890\pi\)
−0.617587 + 0.786502i \(0.711890\pi\)
\(168\) −243.874 −0.111996
\(169\) 169.000 0.0769231
\(170\) 978.138 0.441293
\(171\) 1392.41 0.622690
\(172\) 1471.09 0.652148
\(173\) −165.243 −0.0726198 −0.0363099 0.999341i \(-0.511560\pi\)
−0.0363099 + 0.999341i \(0.511560\pi\)
\(174\) −258.253 −0.112518
\(175\) −64.0954 −0.0276866
\(176\) −1224.73 −0.524532
\(177\) −2621.13 −1.11309
\(178\) 1091.13 0.459457
\(179\) 712.339 0.297446 0.148723 0.988879i \(-0.452484\pi\)
0.148723 + 0.988879i \(0.452484\pi\)
\(180\) −999.688 −0.413957
\(181\) 2206.53 0.906133 0.453066 0.891477i \(-0.350330\pi\)
0.453066 + 0.891477i \(0.350330\pi\)
\(182\) −615.809 −0.250806
\(183\) −561.204 −0.226696
\(184\) −1303.19 −0.522132
\(185\) −1325.95 −0.526949
\(186\) 3364.18 1.32620
\(187\) −526.048 −0.205714
\(188\) −4540.96 −1.76162
\(189\) −303.866 −0.116947
\(190\) −7444.54 −2.84254
\(191\) 1470.64 0.557129 0.278565 0.960417i \(-0.410141\pi\)
0.278565 + 0.960417i \(0.410141\pi\)
\(192\) −2109.14 −0.792782
\(193\) 369.560 0.137832 0.0689158 0.997622i \(-0.478046\pi\)
0.0689158 + 0.997622i \(0.478046\pi\)
\(194\) 5156.80 1.90844
\(195\) −445.856 −0.163735
\(196\) −2101.99 −0.766031
\(197\) −4273.41 −1.54552 −0.772761 0.634697i \(-0.781125\pi\)
−0.772761 + 0.634697i \(0.781125\pi\)
\(198\) 980.315 0.351858
\(199\) 4154.31 1.47985 0.739927 0.672687i \(-0.234860\pi\)
0.739927 + 0.672687i \(0.234860\pi\)
\(200\) 41.1373 0.0145442
\(201\) −1827.61 −0.641342
\(202\) 2716.59 0.946230
\(203\) 230.175 0.0795819
\(204\) −592.515 −0.203355
\(205\) −4472.09 −1.52363
\(206\) −2151.40 −0.727646
\(207\) −1623.77 −0.545215
\(208\) −615.241 −0.205093
\(209\) 4003.71 1.32508
\(210\) 1624.63 0.533857
\(211\) 1231.59 0.401830 0.200915 0.979609i \(-0.435608\pi\)
0.200915 + 0.979609i \(0.435608\pi\)
\(212\) 777.134 0.251763
\(213\) 744.114 0.239370
\(214\) 2559.92 0.817723
\(215\) −1730.92 −0.549059
\(216\) 195.025 0.0614341
\(217\) −2998.42 −0.938000
\(218\) −5471.92 −1.70002
\(219\) 2558.30 0.789377
\(220\) −2874.49 −0.880901
\(221\) −264.259 −0.0804343
\(222\) 1464.54 0.442764
\(223\) −2187.24 −0.656809 −0.328404 0.944537i \(-0.606511\pi\)
−0.328404 + 0.944537i \(0.606511\pi\)
\(224\) 2892.17 0.862684
\(225\) 51.2568 0.0151872
\(226\) 177.602 0.0522739
\(227\) 4138.67 1.21010 0.605051 0.796187i \(-0.293153\pi\)
0.605051 + 0.796187i \(0.293153\pi\)
\(228\) 4509.59 1.30989
\(229\) −835.354 −0.241056 −0.120528 0.992710i \(-0.538459\pi\)
−0.120528 + 0.992710i \(0.538459\pi\)
\(230\) 8681.50 2.48887
\(231\) −873.733 −0.248863
\(232\) −147.729 −0.0418056
\(233\) 3685.51 1.03625 0.518124 0.855305i \(-0.326630\pi\)
0.518124 + 0.855305i \(0.326630\pi\)
\(234\) 492.459 0.137577
\(235\) 5343.01 1.48315
\(236\) −8489.05 −2.34148
\(237\) −993.662 −0.272343
\(238\) 962.917 0.262255
\(239\) 3026.21 0.819034 0.409517 0.912303i \(-0.365697\pi\)
0.409517 + 0.912303i \(0.365697\pi\)
\(240\) 1623.13 0.436552
\(241\) 3265.58 0.872839 0.436420 0.899743i \(-0.356246\pi\)
0.436420 + 0.899743i \(0.356246\pi\)
\(242\) −2783.46 −0.739370
\(243\) 243.000 0.0641500
\(244\) −1817.57 −0.476877
\(245\) 2473.25 0.644940
\(246\) 4939.53 1.28022
\(247\) 2011.25 0.518110
\(248\) 1924.42 0.492746
\(249\) −1306.30 −0.332463
\(250\) 5740.79 1.45232
\(251\) −6363.16 −1.60016 −0.800078 0.599897i \(-0.795208\pi\)
−0.800078 + 0.599897i \(0.795208\pi\)
\(252\) −984.131 −0.246010
\(253\) −4668.96 −1.16022
\(254\) −1309.09 −0.323385
\(255\) 697.168 0.171209
\(256\) 1822.38 0.444917
\(257\) −6085.36 −1.47702 −0.738511 0.674242i \(-0.764471\pi\)
−0.738511 + 0.674242i \(0.764471\pi\)
\(258\) 1911.84 0.461342
\(259\) −1305.31 −0.313159
\(260\) −1443.99 −0.344433
\(261\) −184.070 −0.0436538
\(262\) 8422.24 1.98598
\(263\) 123.227 0.0288916 0.0144458 0.999896i \(-0.495402\pi\)
0.0144458 + 0.999896i \(0.495402\pi\)
\(264\) 560.773 0.130732
\(265\) −914.395 −0.211965
\(266\) −7328.69 −1.68929
\(267\) 777.700 0.178256
\(268\) −5919.08 −1.34913
\(269\) −1935.79 −0.438763 −0.219381 0.975639i \(-0.570404\pi\)
−0.219381 + 0.975639i \(0.570404\pi\)
\(270\) −1299.20 −0.292841
\(271\) −4612.69 −1.03395 −0.516976 0.856000i \(-0.672942\pi\)
−0.516976 + 0.856000i \(0.672942\pi\)
\(272\) 962.028 0.214454
\(273\) −438.918 −0.0973059
\(274\) 4371.20 0.963774
\(275\) 147.383 0.0323183
\(276\) −5258.89 −1.14691
\(277\) −5834.30 −1.26552 −0.632761 0.774347i \(-0.718078\pi\)
−0.632761 + 0.774347i \(0.718078\pi\)
\(278\) −12031.4 −2.59567
\(279\) 2397.82 0.514529
\(280\) 929.341 0.198353
\(281\) 4691.91 0.996071 0.498036 0.867157i \(-0.334055\pi\)
0.498036 + 0.867157i \(0.334055\pi\)
\(282\) −5901.49 −1.24620
\(283\) 3465.60 0.727945 0.363973 0.931410i \(-0.381420\pi\)
0.363973 + 0.931410i \(0.381420\pi\)
\(284\) 2409.96 0.503539
\(285\) −5306.09 −1.10283
\(286\) 1416.01 0.292764
\(287\) −4402.50 −0.905475
\(288\) −2312.85 −0.473216
\(289\) −4499.79 −0.915894
\(290\) 984.133 0.199277
\(291\) 3675.51 0.740420
\(292\) 8285.55 1.66053
\(293\) 2677.31 0.533822 0.266911 0.963721i \(-0.413997\pi\)
0.266911 + 0.963721i \(0.413997\pi\)
\(294\) −2731.77 −0.541904
\(295\) 9988.43 1.97135
\(296\) 837.767 0.164507
\(297\) 698.720 0.136511
\(298\) 3129.34 0.608315
\(299\) −2345.44 −0.453646
\(300\) 166.005 0.0319477
\(301\) −1703.98 −0.326299
\(302\) 9585.02 1.82634
\(303\) 1936.25 0.367111
\(304\) −7321.93 −1.38139
\(305\) 2138.60 0.401494
\(306\) −770.039 −0.143857
\(307\) 471.915 0.0877316 0.0438658 0.999037i \(-0.486033\pi\)
0.0438658 + 0.999037i \(0.486033\pi\)
\(308\) −2829.76 −0.523508
\(309\) −1533.41 −0.282306
\(310\) −12820.0 −2.34880
\(311\) −1518.52 −0.276872 −0.138436 0.990371i \(-0.544207\pi\)
−0.138436 + 0.990371i \(0.544207\pi\)
\(312\) 281.703 0.0511163
\(313\) 4049.86 0.731348 0.365674 0.930743i \(-0.380839\pi\)
0.365674 + 0.930743i \(0.380839\pi\)
\(314\) 13357.5 2.40066
\(315\) 1157.95 0.207121
\(316\) −3218.17 −0.572900
\(317\) 3253.96 0.576532 0.288266 0.957550i \(-0.406921\pi\)
0.288266 + 0.957550i \(0.406921\pi\)
\(318\) 1009.97 0.178102
\(319\) −529.272 −0.0928951
\(320\) 8037.37 1.40407
\(321\) 1824.59 0.317254
\(322\) 8546.40 1.47911
\(323\) −3144.92 −0.541759
\(324\) 787.004 0.134946
\(325\) 74.0375 0.0126365
\(326\) −9742.46 −1.65517
\(327\) −3900.11 −0.659561
\(328\) 2825.58 0.475660
\(329\) 5259.86 0.881415
\(330\) −3735.72 −0.623165
\(331\) −3422.45 −0.568322 −0.284161 0.958777i \(-0.591715\pi\)
−0.284161 + 0.958777i \(0.591715\pi\)
\(332\) −4230.71 −0.699368
\(333\) 1043.85 0.171780
\(334\) −11219.8 −1.83809
\(335\) 6964.54 1.13586
\(336\) 1597.87 0.259437
\(337\) −9301.67 −1.50354 −0.751772 0.659423i \(-0.770801\pi\)
−0.751772 + 0.659423i \(0.770801\pi\)
\(338\) 711.329 0.114471
\(339\) 126.586 0.0202808
\(340\) 2257.92 0.360155
\(341\) 6894.66 1.09492
\(342\) 5860.71 0.926640
\(343\) 6294.99 0.990955
\(344\) 1093.64 0.171410
\(345\) 6187.74 0.965613
\(346\) −695.518 −0.108067
\(347\) 216.898 0.0335554 0.0167777 0.999859i \(-0.494659\pi\)
0.0167777 + 0.999859i \(0.494659\pi\)
\(348\) −596.147 −0.0918299
\(349\) −4809.84 −0.737721 −0.368861 0.929485i \(-0.620252\pi\)
−0.368861 + 0.929485i \(0.620252\pi\)
\(350\) −269.781 −0.0412011
\(351\) 351.000 0.0533761
\(352\) −6650.35 −1.00700
\(353\) −2859.64 −0.431170 −0.215585 0.976485i \(-0.569166\pi\)
−0.215585 + 0.976485i \(0.569166\pi\)
\(354\) −11032.5 −1.65641
\(355\) −2835.62 −0.423941
\(356\) 2518.74 0.374980
\(357\) 686.319 0.101747
\(358\) 2998.27 0.442636
\(359\) 3686.04 0.541899 0.270949 0.962594i \(-0.412662\pi\)
0.270949 + 0.962594i \(0.412662\pi\)
\(360\) −743.188 −0.108804
\(361\) 17076.8 2.48969
\(362\) 9287.39 1.34844
\(363\) −1983.91 −0.286855
\(364\) −1421.52 −0.204692
\(365\) −9748.98 −1.39804
\(366\) −2362.14 −0.337352
\(367\) −3470.59 −0.493633 −0.246816 0.969062i \(-0.579384\pi\)
−0.246816 + 0.969062i \(0.579384\pi\)
\(368\) 8538.52 1.20951
\(369\) 3520.65 0.496688
\(370\) −5580.98 −0.784166
\(371\) −900.166 −0.125968
\(372\) 7765.82 1.08236
\(373\) −11963.4 −1.66070 −0.830352 0.557240i \(-0.811860\pi\)
−0.830352 + 0.557240i \(0.811860\pi\)
\(374\) −2214.16 −0.306127
\(375\) 4091.75 0.563459
\(376\) −3375.85 −0.463021
\(377\) −265.879 −0.0363221
\(378\) −1278.99 −0.174032
\(379\) 345.604 0.0468403 0.0234202 0.999726i \(-0.492544\pi\)
0.0234202 + 0.999726i \(0.492544\pi\)
\(380\) −17184.8 −2.31990
\(381\) −933.055 −0.125464
\(382\) 6189.99 0.829078
\(383\) −3386.40 −0.451793 −0.225897 0.974151i \(-0.572531\pi\)
−0.225897 + 0.974151i \(0.572531\pi\)
\(384\) −2709.87 −0.360124
\(385\) 3329.56 0.440754
\(386\) 1555.49 0.205110
\(387\) 1362.67 0.178988
\(388\) 11903.9 1.55755
\(389\) −1629.88 −0.212438 −0.106219 0.994343i \(-0.533874\pi\)
−0.106219 + 0.994343i \(0.533874\pi\)
\(390\) −1876.63 −0.243659
\(391\) 3667.47 0.474353
\(392\) −1562.66 −0.201343
\(393\) 6002.95 0.770506
\(394\) −17987.0 −2.29993
\(395\) 3786.58 0.482338
\(396\) 2262.94 0.287165
\(397\) 7938.94 1.00364 0.501819 0.864973i \(-0.332664\pi\)
0.501819 + 0.864973i \(0.332664\pi\)
\(398\) 17485.7 2.20221
\(399\) −5223.52 −0.655396
\(400\) −269.532 −0.0336915
\(401\) 214.402 0.0267001 0.0133500 0.999911i \(-0.495750\pi\)
0.0133500 + 0.999911i \(0.495750\pi\)
\(402\) −7692.51 −0.954396
\(403\) 3463.52 0.428114
\(404\) 6270.93 0.772253
\(405\) −926.008 −0.113614
\(406\) 968.819 0.118428
\(407\) 3001.48 0.365548
\(408\) −440.488 −0.0534495
\(409\) −4783.73 −0.578338 −0.289169 0.957278i \(-0.593379\pi\)
−0.289169 + 0.957278i \(0.593379\pi\)
\(410\) −18823.2 −2.26735
\(411\) 3115.58 0.373917
\(412\) −4966.25 −0.593859
\(413\) 9832.99 1.17155
\(414\) −6834.51 −0.811347
\(415\) 4977.95 0.588815
\(416\) −3340.79 −0.393739
\(417\) −8575.39 −1.00705
\(418\) 16851.8 1.97189
\(419\) −9903.67 −1.15472 −0.577358 0.816491i \(-0.695916\pi\)
−0.577358 + 0.816491i \(0.695916\pi\)
\(420\) 3750.26 0.435700
\(421\) −12120.6 −1.40314 −0.701572 0.712598i \(-0.747518\pi\)
−0.701572 + 0.712598i \(0.747518\pi\)
\(422\) 5183.82 0.597973
\(423\) −4206.28 −0.483491
\(424\) 577.738 0.0661732
\(425\) −115.770 −0.0132133
\(426\) 3132.01 0.356213
\(427\) 2105.32 0.238603
\(428\) 5909.28 0.667374
\(429\) 1009.26 0.113584
\(430\) −7285.53 −0.817068
\(431\) −13672.6 −1.52805 −0.764023 0.645189i \(-0.776779\pi\)
−0.764023 + 0.645189i \(0.776779\pi\)
\(432\) −1277.81 −0.142311
\(433\) 7113.10 0.789455 0.394727 0.918798i \(-0.370839\pi\)
0.394727 + 0.918798i \(0.370839\pi\)
\(434\) −12620.5 −1.39586
\(435\) 701.441 0.0773138
\(436\) −12631.3 −1.38745
\(437\) −27912.9 −3.05550
\(438\) 10768.0 1.17469
\(439\) −6022.04 −0.654707 −0.327353 0.944902i \(-0.606157\pi\)
−0.327353 + 0.944902i \(0.606157\pi\)
\(440\) −2136.96 −0.231535
\(441\) −1947.07 −0.210244
\(442\) −1112.28 −0.119696
\(443\) −12994.4 −1.39364 −0.696821 0.717245i \(-0.745403\pi\)
−0.696821 + 0.717245i \(0.745403\pi\)
\(444\) 3380.72 0.361356
\(445\) −2963.61 −0.315704
\(446\) −9206.20 −0.977413
\(447\) 2230.44 0.236009
\(448\) 7912.30 0.834422
\(449\) 10984.3 1.15452 0.577260 0.816560i \(-0.304122\pi\)
0.577260 + 0.816560i \(0.304122\pi\)
\(450\) 215.742 0.0226004
\(451\) 10123.2 1.05695
\(452\) 409.973 0.0426627
\(453\) 6831.72 0.708570
\(454\) 17419.9 1.80078
\(455\) 1672.60 0.172335
\(456\) 3352.52 0.344290
\(457\) 9834.10 1.00661 0.503304 0.864109i \(-0.332118\pi\)
0.503304 + 0.864109i \(0.332118\pi\)
\(458\) −3516.05 −0.358721
\(459\) −548.845 −0.0558124
\(460\) 20040.2 2.03126
\(461\) 3401.42 0.343644 0.171822 0.985128i \(-0.445035\pi\)
0.171822 + 0.985128i \(0.445035\pi\)
\(462\) −3677.59 −0.370339
\(463\) 1739.42 0.174596 0.0872979 0.996182i \(-0.472177\pi\)
0.0872979 + 0.996182i \(0.472177\pi\)
\(464\) 967.925 0.0968422
\(465\) −9137.45 −0.911267
\(466\) 15512.5 1.54207
\(467\) −7958.82 −0.788630 −0.394315 0.918975i \(-0.629018\pi\)
−0.394315 + 0.918975i \(0.629018\pi\)
\(468\) 1136.78 0.112282
\(469\) 6856.16 0.675028
\(470\) 22489.0 2.20711
\(471\) 9520.54 0.931387
\(472\) −6310.94 −0.615433
\(473\) 3918.20 0.380886
\(474\) −4182.37 −0.405280
\(475\) 881.114 0.0851122
\(476\) 2222.78 0.214036
\(477\) 719.858 0.0690986
\(478\) 12737.5 1.21882
\(479\) 8431.98 0.804315 0.402158 0.915570i \(-0.368260\pi\)
0.402158 + 0.915570i \(0.368260\pi\)
\(480\) 8813.66 0.838097
\(481\) 1507.79 0.142930
\(482\) 13745.0 1.29889
\(483\) 6091.45 0.573852
\(484\) −6425.29 −0.603427
\(485\) −14006.4 −1.31133
\(486\) 1022.80 0.0954632
\(487\) −11684.7 −1.08723 −0.543617 0.839334i \(-0.682945\pi\)
−0.543617 + 0.839334i \(0.682945\pi\)
\(488\) −1351.22 −0.125342
\(489\) −6943.94 −0.642159
\(490\) 10410.0 0.959750
\(491\) −3954.70 −0.363489 −0.181745 0.983346i \(-0.558174\pi\)
−0.181745 + 0.983346i \(0.558174\pi\)
\(492\) 11402.3 1.04483
\(493\) 415.744 0.0379801
\(494\) 8465.47 0.771011
\(495\) −2662.63 −0.241771
\(496\) −12608.8 −1.14144
\(497\) −2791.49 −0.251943
\(498\) −5498.27 −0.494746
\(499\) −5690.37 −0.510493 −0.255246 0.966876i \(-0.582157\pi\)
−0.255246 + 0.966876i \(0.582157\pi\)
\(500\) 13251.9 1.18529
\(501\) −7996.95 −0.713128
\(502\) −26782.8 −2.38123
\(503\) 10859.1 0.962595 0.481298 0.876557i \(-0.340166\pi\)
0.481298 + 0.876557i \(0.340166\pi\)
\(504\) −731.623 −0.0646609
\(505\) −7378.53 −0.650178
\(506\) −19651.9 −1.72655
\(507\) 507.000 0.0444116
\(508\) −3021.88 −0.263926
\(509\) −18558.6 −1.61610 −0.808049 0.589115i \(-0.799476\pi\)
−0.808049 + 0.589115i \(0.799476\pi\)
\(510\) 2934.41 0.254780
\(511\) −9597.27 −0.830838
\(512\) 14896.8 1.28584
\(513\) 4177.22 0.359510
\(514\) −25613.6 −2.19799
\(515\) 5843.42 0.499984
\(516\) 4413.27 0.376518
\(517\) −12094.7 −1.02887
\(518\) −5494.13 −0.466020
\(519\) −495.730 −0.0419271
\(520\) −1073.49 −0.0905305
\(521\) 17297.5 1.45454 0.727271 0.686350i \(-0.240788\pi\)
0.727271 + 0.686350i \(0.240788\pi\)
\(522\) −774.759 −0.0649622
\(523\) −5016.11 −0.419386 −0.209693 0.977767i \(-0.567247\pi\)
−0.209693 + 0.977767i \(0.567247\pi\)
\(524\) 19441.8 1.62084
\(525\) −192.286 −0.0159849
\(526\) 518.667 0.0429942
\(527\) −5415.77 −0.447656
\(528\) −3674.19 −0.302839
\(529\) 20383.8 1.67533
\(530\) −3848.73 −0.315431
\(531\) −7863.39 −0.642640
\(532\) −16917.4 −1.37869
\(533\) 5085.39 0.413269
\(534\) 3273.38 0.265268
\(535\) −6953.01 −0.561878
\(536\) −4400.37 −0.354603
\(537\) 2137.02 0.171730
\(538\) −8147.83 −0.652933
\(539\) −5598.57 −0.447398
\(540\) −2999.06 −0.238998
\(541\) 17642.3 1.40204 0.701018 0.713144i \(-0.252729\pi\)
0.701018 + 0.713144i \(0.252729\pi\)
\(542\) −19415.0 −1.53865
\(543\) 6619.59 0.523156
\(544\) 5223.86 0.411712
\(545\) 14862.3 1.16813
\(546\) −1847.43 −0.144803
\(547\) −18414.9 −1.43943 −0.719713 0.694271i \(-0.755727\pi\)
−0.719713 + 0.694271i \(0.755727\pi\)
\(548\) 10090.4 0.786571
\(549\) −1683.61 −0.130883
\(550\) 620.343 0.0480937
\(551\) −3164.20 −0.244645
\(552\) −3909.57 −0.301453
\(553\) 3727.66 0.286648
\(554\) −24556.9 −1.88325
\(555\) −3977.84 −0.304234
\(556\) −27773.1 −2.11842
\(557\) 8179.15 0.622193 0.311096 0.950378i \(-0.399304\pi\)
0.311096 + 0.950378i \(0.399304\pi\)
\(558\) 10092.5 0.765683
\(559\) 1968.30 0.148927
\(560\) −6089.05 −0.459481
\(561\) −1578.14 −0.118769
\(562\) 19748.5 1.48228
\(563\) −1880.07 −0.140738 −0.0703690 0.997521i \(-0.522418\pi\)
−0.0703690 + 0.997521i \(0.522418\pi\)
\(564\) −13622.9 −1.01707
\(565\) −482.385 −0.0359187
\(566\) 14586.9 1.08327
\(567\) −911.598 −0.0675194
\(568\) 1791.62 0.132350
\(569\) 10118.3 0.745485 0.372743 0.927935i \(-0.378417\pi\)
0.372743 + 0.927935i \(0.378417\pi\)
\(570\) −22333.6 −1.64114
\(571\) 23428.9 1.71711 0.858555 0.512721i \(-0.171362\pi\)
0.858555 + 0.512721i \(0.171362\pi\)
\(572\) 3268.70 0.238935
\(573\) 4411.92 0.321659
\(574\) −18530.3 −1.34746
\(575\) −1027.52 −0.0745225
\(576\) −6327.42 −0.457713
\(577\) 20508.1 1.47966 0.739831 0.672793i \(-0.234906\pi\)
0.739831 + 0.672793i \(0.234906\pi\)
\(578\) −18939.8 −1.36296
\(579\) 1108.68 0.0795771
\(580\) 2271.76 0.162637
\(581\) 4900.49 0.349925
\(582\) 15470.4 1.10184
\(583\) 2069.87 0.147042
\(584\) 6159.65 0.436452
\(585\) −1337.57 −0.0945327
\(586\) 11268.9 0.794394
\(587\) −5968.43 −0.419665 −0.209833 0.977737i \(-0.567292\pi\)
−0.209833 + 0.977737i \(0.567292\pi\)
\(588\) −6305.97 −0.442268
\(589\) 41219.0 2.88353
\(590\) 42041.8 2.93361
\(591\) −12820.2 −0.892308
\(592\) −5489.06 −0.381080
\(593\) −14659.5 −1.01517 −0.507584 0.861602i \(-0.669461\pi\)
−0.507584 + 0.861602i \(0.669461\pi\)
\(594\) 2940.95 0.203146
\(595\) −2615.38 −0.180202
\(596\) 7223.72 0.496468
\(597\) 12462.9 0.854394
\(598\) −9872.07 −0.675082
\(599\) 23635.9 1.61225 0.806125 0.591746i \(-0.201561\pi\)
0.806125 + 0.591746i \(0.201561\pi\)
\(600\) 123.412 0.00839711
\(601\) −11527.0 −0.782356 −0.391178 0.920315i \(-0.627932\pi\)
−0.391178 + 0.920315i \(0.627932\pi\)
\(602\) −7172.15 −0.485573
\(603\) −5482.83 −0.370279
\(604\) 22125.9 1.49055
\(605\) 7560.15 0.508039
\(606\) 8149.77 0.546306
\(607\) −5098.56 −0.340930 −0.170465 0.985364i \(-0.554527\pi\)
−0.170465 + 0.985364i \(0.554527\pi\)
\(608\) −39758.4 −2.65200
\(609\) 690.525 0.0459466
\(610\) 9001.47 0.597473
\(611\) −6075.74 −0.402288
\(612\) −1777.55 −0.117407
\(613\) 1516.39 0.0999128 0.0499564 0.998751i \(-0.484092\pi\)
0.0499564 + 0.998751i \(0.484092\pi\)
\(614\) 1986.31 0.130556
\(615\) −13416.3 −0.879668
\(616\) −2103.70 −0.137598
\(617\) 18539.3 1.20966 0.604832 0.796353i \(-0.293240\pi\)
0.604832 + 0.796353i \(0.293240\pi\)
\(618\) −6454.20 −0.420107
\(619\) 25684.9 1.66779 0.833897 0.551920i \(-0.186105\pi\)
0.833897 + 0.551920i \(0.186105\pi\)
\(620\) −29593.5 −1.91694
\(621\) −4871.30 −0.314780
\(622\) −6391.51 −0.412020
\(623\) −2917.49 −0.187619
\(624\) −1845.72 −0.118410
\(625\) −16304.5 −1.04349
\(626\) 17046.1 1.08834
\(627\) 12011.1 0.765038
\(628\) 30834.2 1.95926
\(629\) −2357.67 −0.149454
\(630\) 4873.88 0.308222
\(631\) −22410.9 −1.41389 −0.706945 0.707269i \(-0.749927\pi\)
−0.706945 + 0.707269i \(0.749927\pi\)
\(632\) −2392.46 −0.150580
\(633\) 3694.77 0.231997
\(634\) 13696.1 0.857950
\(635\) 3555.62 0.222206
\(636\) 2331.40 0.145356
\(637\) −2812.43 −0.174933
\(638\) −2227.73 −0.138239
\(639\) 2232.34 0.138200
\(640\) 10326.6 0.637804
\(641\) 6827.81 0.420721 0.210361 0.977624i \(-0.432536\pi\)
0.210361 + 0.977624i \(0.432536\pi\)
\(642\) 7679.77 0.472113
\(643\) −23264.3 −1.42684 −0.713418 0.700738i \(-0.752854\pi\)
−0.713418 + 0.700738i \(0.752854\pi\)
\(644\) 19728.4 1.20715
\(645\) −5192.76 −0.316999
\(646\) −13237.1 −0.806204
\(647\) 14745.9 0.896014 0.448007 0.894030i \(-0.352134\pi\)
0.448007 + 0.894030i \(0.352134\pi\)
\(648\) 585.075 0.0354690
\(649\) −22610.3 −1.36754
\(650\) 311.628 0.0188047
\(651\) −8995.26 −0.541554
\(652\) −22489.3 −1.35084
\(653\) 10909.0 0.653755 0.326878 0.945067i \(-0.394003\pi\)
0.326878 + 0.945067i \(0.394003\pi\)
\(654\) −16415.8 −0.981509
\(655\) −22875.7 −1.36462
\(656\) −18513.2 −1.10186
\(657\) 7674.89 0.455747
\(658\) 22139.0 1.31166
\(659\) −4182.99 −0.247263 −0.123631 0.992328i \(-0.539454\pi\)
−0.123631 + 0.992328i \(0.539454\pi\)
\(660\) −8623.47 −0.508588
\(661\) 2224.23 0.130881 0.0654406 0.997856i \(-0.479155\pi\)
0.0654406 + 0.997856i \(0.479155\pi\)
\(662\) −14405.2 −0.845734
\(663\) −792.776 −0.0464387
\(664\) −3145.19 −0.183821
\(665\) 19905.4 1.16075
\(666\) 4393.63 0.255630
\(667\) 3689.95 0.214206
\(668\) −25899.7 −1.50013
\(669\) −6561.72 −0.379209
\(670\) 29314.1 1.69030
\(671\) −4841.04 −0.278519
\(672\) 8676.51 0.498071
\(673\) −24152.5 −1.38337 −0.691687 0.722197i \(-0.743132\pi\)
−0.691687 + 0.722197i \(0.743132\pi\)
\(674\) −39151.2 −2.23746
\(675\) 153.770 0.00876833
\(676\) 1642.02 0.0934240
\(677\) −15310.7 −0.869187 −0.434593 0.900627i \(-0.643108\pi\)
−0.434593 + 0.900627i \(0.643108\pi\)
\(678\) 532.806 0.0301804
\(679\) −13788.4 −0.779310
\(680\) 1678.58 0.0946628
\(681\) 12416.0 0.698652
\(682\) 29020.0 1.62937
\(683\) 11399.6 0.638646 0.319323 0.947646i \(-0.396545\pi\)
0.319323 + 0.947646i \(0.396545\pi\)
\(684\) 13528.8 0.756265
\(685\) −11872.6 −0.662233
\(686\) 26495.9 1.47466
\(687\) −2506.06 −0.139174
\(688\) −7165.54 −0.397069
\(689\) 1039.79 0.0574935
\(690\) 26044.5 1.43695
\(691\) −3323.23 −0.182955 −0.0914773 0.995807i \(-0.529159\pi\)
−0.0914773 + 0.995807i \(0.529159\pi\)
\(692\) −1605.52 −0.0881976
\(693\) −2621.20 −0.143681
\(694\) 912.936 0.0499345
\(695\) 32678.5 1.78355
\(696\) −443.188 −0.0241365
\(697\) −7951.82 −0.432133
\(698\) −20244.8 −1.09782
\(699\) 11056.5 0.598279
\(700\) −622.758 −0.0336257
\(701\) −12670.4 −0.682673 −0.341336 0.939941i \(-0.610880\pi\)
−0.341336 + 0.939941i \(0.610880\pi\)
\(702\) 1477.38 0.0794302
\(703\) 17944.0 0.962692
\(704\) −18193.8 −0.974012
\(705\) 16029.0 0.856295
\(706\) −12036.4 −0.641635
\(707\) −7263.71 −0.386393
\(708\) −25467.2 −1.35186
\(709\) 13075.2 0.692594 0.346297 0.938125i \(-0.387439\pi\)
0.346297 + 0.938125i \(0.387439\pi\)
\(710\) −11935.3 −0.630877
\(711\) −2980.99 −0.157237
\(712\) 1872.48 0.0985592
\(713\) −48067.8 −2.52476
\(714\) 2888.75 0.151413
\(715\) −3846.03 −0.201165
\(716\) 6921.16 0.361251
\(717\) 9078.62 0.472869
\(718\) 15514.7 0.806412
\(719\) −2988.41 −0.155005 −0.0775026 0.996992i \(-0.524695\pi\)
−0.0775026 + 0.996992i \(0.524695\pi\)
\(720\) 4869.38 0.252043
\(721\) 5752.48 0.297134
\(722\) 71877.0 3.70496
\(723\) 9796.73 0.503934
\(724\) 21438.9 1.10051
\(725\) −116.479 −0.00596680
\(726\) −8350.37 −0.426875
\(727\) −5507.46 −0.280963 −0.140482 0.990083i \(-0.544865\pi\)
−0.140482 + 0.990083i \(0.544865\pi\)
\(728\) −1056.79 −0.0538011
\(729\) 729.000 0.0370370
\(730\) −41033.9 −2.08046
\(731\) −3077.75 −0.155725
\(732\) −5452.71 −0.275325
\(733\) −36585.2 −1.84353 −0.921764 0.387751i \(-0.873252\pi\)
−0.921764 + 0.387751i \(0.873252\pi\)
\(734\) −14607.9 −0.734587
\(735\) 7419.75 0.372356
\(736\) 46364.6 2.32204
\(737\) −15765.3 −0.787953
\(738\) 14818.6 0.739133
\(739\) 6425.89 0.319865 0.159933 0.987128i \(-0.448872\pi\)
0.159933 + 0.987128i \(0.448872\pi\)
\(740\) −12883.0 −0.639986
\(741\) 6033.76 0.299131
\(742\) −3788.84 −0.187457
\(743\) 20411.0 1.00782 0.503908 0.863757i \(-0.331895\pi\)
0.503908 + 0.863757i \(0.331895\pi\)
\(744\) 5773.27 0.284487
\(745\) −8499.60 −0.417988
\(746\) −50354.6 −2.47133
\(747\) −3918.89 −0.191947
\(748\) −5111.13 −0.249842
\(749\) −6844.81 −0.333917
\(750\) 17222.4 0.838496
\(751\) −24259.5 −1.17875 −0.589375 0.807860i \(-0.700626\pi\)
−0.589375 + 0.807860i \(0.700626\pi\)
\(752\) 22118.6 1.07258
\(753\) −19089.5 −0.923850
\(754\) −1119.10 −0.0540518
\(755\) −26033.9 −1.25493
\(756\) −2952.39 −0.142034
\(757\) 9295.39 0.446297 0.223148 0.974785i \(-0.428367\pi\)
0.223148 + 0.974785i \(0.428367\pi\)
\(758\) 1454.66 0.0697042
\(759\) −14006.9 −0.669851
\(760\) −12775.6 −0.609761
\(761\) −21974.7 −1.04676 −0.523378 0.852101i \(-0.675328\pi\)
−0.523378 + 0.852101i \(0.675328\pi\)
\(762\) −3927.27 −0.186706
\(763\) 14631.0 0.694204
\(764\) 14288.9 0.676641
\(765\) 2091.50 0.0988476
\(766\) −14253.5 −0.672324
\(767\) −11358.2 −0.534709
\(768\) 5467.13 0.256873
\(769\) 22987.4 1.07795 0.538977 0.842320i \(-0.318811\pi\)
0.538977 + 0.842320i \(0.318811\pi\)
\(770\) 14014.3 0.655897
\(771\) −18256.1 −0.852759
\(772\) 3590.68 0.167398
\(773\) 31970.9 1.48760 0.743799 0.668404i \(-0.233022\pi\)
0.743799 + 0.668404i \(0.233022\pi\)
\(774\) 5735.53 0.266356
\(775\) 1517.34 0.0703283
\(776\) 8849.59 0.409384
\(777\) −3915.94 −0.180803
\(778\) −6860.25 −0.316133
\(779\) 60520.8 2.78354
\(780\) −4331.98 −0.198859
\(781\) 6418.85 0.294090
\(782\) 15436.6 0.705896
\(783\) −552.209 −0.0252035
\(784\) 10238.6 0.466408
\(785\) −36280.2 −1.64955
\(786\) 25266.7 1.14661
\(787\) 6087.26 0.275715 0.137857 0.990452i \(-0.455978\pi\)
0.137857 + 0.990452i \(0.455978\pi\)
\(788\) −41520.9 −1.87706
\(789\) 369.680 0.0166805
\(790\) 15937.9 0.717779
\(791\) −474.878 −0.0213460
\(792\) 1682.32 0.0754780
\(793\) −2431.88 −0.108901
\(794\) 33415.4 1.49354
\(795\) −2743.19 −0.122378
\(796\) 40363.6 1.79730
\(797\) 23080.0 1.02577 0.512883 0.858458i \(-0.328577\pi\)
0.512883 + 0.858458i \(0.328577\pi\)
\(798\) −21986.1 −0.975311
\(799\) 9500.41 0.420651
\(800\) −1463.57 −0.0646813
\(801\) 2333.10 0.102916
\(802\) 902.429 0.0397330
\(803\) 22068.3 0.969829
\(804\) −17757.3 −0.778918
\(805\) −23212.9 −1.01633
\(806\) 14578.1 0.637087
\(807\) −5807.37 −0.253320
\(808\) 4661.94 0.202978
\(809\) −32377.8 −1.40710 −0.703550 0.710646i \(-0.748403\pi\)
−0.703550 + 0.710646i \(0.748403\pi\)
\(810\) −3897.61 −0.169072
\(811\) 26352.8 1.14103 0.570513 0.821288i \(-0.306744\pi\)
0.570513 + 0.821288i \(0.306744\pi\)
\(812\) 2236.40 0.0966532
\(813\) −13838.1 −0.596952
\(814\) 12633.4 0.543980
\(815\) 26461.5 1.13731
\(816\) 2886.08 0.123815
\(817\) 23424.5 1.00308
\(818\) −20135.0 −0.860638
\(819\) −1316.75 −0.0561796
\(820\) −43451.3 −1.85047
\(821\) 35355.3 1.50294 0.751468 0.659770i \(-0.229346\pi\)
0.751468 + 0.659770i \(0.229346\pi\)
\(822\) 13113.6 0.556435
\(823\) −12663.3 −0.536347 −0.268173 0.963371i \(-0.586420\pi\)
−0.268173 + 0.963371i \(0.586420\pi\)
\(824\) −3692.02 −0.156089
\(825\) 442.149 0.0186590
\(826\) 41387.6 1.74341
\(827\) −16295.2 −0.685176 −0.342588 0.939486i \(-0.611303\pi\)
−0.342588 + 0.939486i \(0.611303\pi\)
\(828\) −15776.7 −0.662170
\(829\) 13638.9 0.571411 0.285705 0.958318i \(-0.407772\pi\)
0.285705 + 0.958318i \(0.407772\pi\)
\(830\) 20952.4 0.876229
\(831\) −17502.9 −0.730649
\(832\) −9139.61 −0.380840
\(833\) 4397.69 0.182918
\(834\) −36094.2 −1.49861
\(835\) 30474.2 1.26300
\(836\) 38900.5 1.60933
\(837\) 7193.46 0.297064
\(838\) −41685.0 −1.71836
\(839\) −1890.31 −0.0777838 −0.0388919 0.999243i \(-0.512383\pi\)
−0.0388919 + 0.999243i \(0.512383\pi\)
\(840\) 2788.02 0.114519
\(841\) −23970.7 −0.982849
\(842\) −51016.4 −2.08805
\(843\) 14075.7 0.575082
\(844\) 11966.3 0.488028
\(845\) −1932.04 −0.0786559
\(846\) −17704.5 −0.719494
\(847\) 7442.50 0.301921
\(848\) −3785.35 −0.153289
\(849\) 10396.8 0.420279
\(850\) −487.280 −0.0196630
\(851\) −20925.6 −0.842914
\(852\) 7229.89 0.290718
\(853\) 1620.21 0.0650351 0.0325175 0.999471i \(-0.489648\pi\)
0.0325175 + 0.999471i \(0.489648\pi\)
\(854\) 8861.39 0.355071
\(855\) −15918.3 −0.636718
\(856\) 4393.08 0.175412
\(857\) −14508.4 −0.578292 −0.289146 0.957285i \(-0.593371\pi\)
−0.289146 + 0.957285i \(0.593371\pi\)
\(858\) 4248.03 0.169027
\(859\) 29639.8 1.17730 0.588648 0.808389i \(-0.299660\pi\)
0.588648 + 0.808389i \(0.299660\pi\)
\(860\) −16817.8 −0.666839
\(861\) −13207.5 −0.522776
\(862\) −57548.8 −2.27392
\(863\) 21528.8 0.849186 0.424593 0.905384i \(-0.360417\pi\)
0.424593 + 0.905384i \(0.360417\pi\)
\(864\) −6938.56 −0.273211
\(865\) 1889.10 0.0742557
\(866\) 29939.4 1.17481
\(867\) −13499.4 −0.528792
\(868\) −29132.9 −1.13921
\(869\) −8571.50 −0.334601
\(870\) 2952.40 0.115053
\(871\) −7919.65 −0.308091
\(872\) −9390.36 −0.364676
\(873\) 11026.5 0.427482
\(874\) −117487. −4.54696
\(875\) −15349.9 −0.593054
\(876\) 24856.7 0.958708
\(877\) 14865.3 0.572366 0.286183 0.958175i \(-0.407613\pi\)
0.286183 + 0.958175i \(0.407613\pi\)
\(878\) −25347.1 −0.974285
\(879\) 8031.92 0.308202
\(880\) 14001.4 0.536348
\(881\) −21336.0 −0.815921 −0.407961 0.913000i \(-0.633760\pi\)
−0.407961 + 0.913000i \(0.633760\pi\)
\(882\) −8195.30 −0.312869
\(883\) 37538.2 1.43065 0.715323 0.698794i \(-0.246280\pi\)
0.715323 + 0.698794i \(0.246280\pi\)
\(884\) −2567.57 −0.0976884
\(885\) 29965.3 1.13816
\(886\) −54694.1 −2.07391
\(887\) 34575.0 1.30881 0.654406 0.756144i \(-0.272919\pi\)
0.654406 + 0.756144i \(0.272919\pi\)
\(888\) 2513.30 0.0949784
\(889\) 3500.29 0.132054
\(890\) −12474.0 −0.469807
\(891\) 2096.16 0.0788148
\(892\) −21251.4 −0.797703
\(893\) −72306.9 −2.70958
\(894\) 9388.02 0.351211
\(895\) −8143.61 −0.304146
\(896\) 10165.9 0.379039
\(897\) −7036.32 −0.261913
\(898\) 46233.3 1.71807
\(899\) −5448.96 −0.202150
\(900\) 498.016 0.0184450
\(901\) −1625.89 −0.0601178
\(902\) 42609.2 1.57287
\(903\) −5111.95 −0.188389
\(904\) 304.783 0.0112134
\(905\) −25225.5 −0.926545
\(906\) 28755.0 1.05444
\(907\) −10424.8 −0.381641 −0.190820 0.981625i \(-0.561115\pi\)
−0.190820 + 0.981625i \(0.561115\pi\)
\(908\) 40211.7 1.46968
\(909\) 5808.75 0.211952
\(910\) 7040.05 0.256456
\(911\) 10961.8 0.398661 0.199331 0.979932i \(-0.436123\pi\)
0.199331 + 0.979932i \(0.436123\pi\)
\(912\) −21965.8 −0.797543
\(913\) −11268.3 −0.408464
\(914\) 41392.2 1.49796
\(915\) 6415.80 0.231803
\(916\) −8116.38 −0.292765
\(917\) −22519.7 −0.810976
\(918\) −2310.12 −0.0830558
\(919\) −10779.2 −0.386914 −0.193457 0.981109i \(-0.561970\pi\)
−0.193457 + 0.981109i \(0.561970\pi\)
\(920\) 14898.3 0.533895
\(921\) 1415.74 0.0506519
\(922\) 14316.7 0.511384
\(923\) 3224.49 0.114990
\(924\) −8489.28 −0.302248
\(925\) 660.549 0.0234797
\(926\) 7321.32 0.259820
\(927\) −4600.23 −0.162990
\(928\) 5255.88 0.185919
\(929\) 5429.07 0.191735 0.0958675 0.995394i \(-0.469437\pi\)
0.0958675 + 0.995394i \(0.469437\pi\)
\(930\) −38460.0 −1.35608
\(931\) −33470.5 −1.17825
\(932\) 35808.8 1.25854
\(933\) −4555.55 −0.159852
\(934\) −33499.1 −1.17358
\(935\) 6013.88 0.210348
\(936\) 845.109 0.0295120
\(937\) −21300.1 −0.742631 −0.371315 0.928507i \(-0.621093\pi\)
−0.371315 + 0.928507i \(0.621093\pi\)
\(938\) 28857.9 1.00453
\(939\) 12149.6 0.422244
\(940\) 51913.2 1.80130
\(941\) 26851.2 0.930207 0.465103 0.885256i \(-0.346017\pi\)
0.465103 + 0.885256i \(0.346017\pi\)
\(942\) 40072.4 1.38602
\(943\) −70576.7 −2.43722
\(944\) 41349.4 1.42564
\(945\) 3473.86 0.119582
\(946\) 16491.9 0.566805
\(947\) −8021.68 −0.275258 −0.137629 0.990484i \(-0.543948\pi\)
−0.137629 + 0.990484i \(0.543948\pi\)
\(948\) −9654.52 −0.330764
\(949\) 11085.9 0.379204
\(950\) 3708.65 0.126658
\(951\) 9761.88 0.332861
\(952\) 1652.46 0.0562569
\(953\) 35715.0 1.21398 0.606990 0.794709i \(-0.292377\pi\)
0.606990 + 0.794709i \(0.292377\pi\)
\(954\) 3029.92 0.102827
\(955\) −16812.6 −0.569680
\(956\) 29402.9 0.994726
\(957\) −1587.82 −0.0536330
\(958\) 35490.6 1.19692
\(959\) −11687.9 −0.393557
\(960\) 24112.1 0.810641
\(961\) 41190.9 1.38266
\(962\) 6346.35 0.212697
\(963\) 5473.76 0.183166
\(964\) 31728.7 1.06007
\(965\) −4224.88 −0.140936
\(966\) 25639.2 0.853963
\(967\) 53338.8 1.77380 0.886898 0.461965i \(-0.152855\pi\)
0.886898 + 0.461965i \(0.152855\pi\)
\(968\) −4776.69 −0.158604
\(969\) −9434.77 −0.312785
\(970\) −58953.6 −1.95143
\(971\) 23112.9 0.763882 0.381941 0.924187i \(-0.375256\pi\)
0.381941 + 0.924187i \(0.375256\pi\)
\(972\) 2361.01 0.0779110
\(973\) 32170.0 1.05994
\(974\) −49181.3 −1.61794
\(975\) 222.113 0.00729569
\(976\) 8853.22 0.290353
\(977\) −52874.6 −1.73143 −0.865715 0.500538i \(-0.833136\pi\)
−0.865715 + 0.500538i \(0.833136\pi\)
\(978\) −29227.4 −0.955612
\(979\) 6708.57 0.219006
\(980\) 24030.4 0.783287
\(981\) −11700.3 −0.380798
\(982\) −16645.5 −0.540917
\(983\) 45173.1 1.46572 0.732858 0.680381i \(-0.238186\pi\)
0.732858 + 0.680381i \(0.238186\pi\)
\(984\) 8476.73 0.274622
\(985\) 48854.5 1.58034
\(986\) 1749.89 0.0565190
\(987\) 15779.6 0.508885
\(988\) 19541.6 0.629251
\(989\) −27316.7 −0.878281
\(990\) −11207.2 −0.359785
\(991\) 60485.6 1.93884 0.969418 0.245414i \(-0.0789239\pi\)
0.969418 + 0.245414i \(0.0789239\pi\)
\(992\) −68466.7 −2.19135
\(993\) −10267.3 −0.328121
\(994\) −11749.5 −0.374922
\(995\) −47492.8 −1.51319
\(996\) −12692.1 −0.403780
\(997\) −18108.1 −0.575214 −0.287607 0.957749i \(-0.592860\pi\)
−0.287607 + 0.957749i \(0.592860\pi\)
\(998\) −23951.1 −0.759677
\(999\) 3131.56 0.0991773
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 39.4.a.c.1.3 3
3.2 odd 2 117.4.a.f.1.1 3
4.3 odd 2 624.4.a.t.1.1 3
5.4 even 2 975.4.a.l.1.1 3
7.6 odd 2 1911.4.a.k.1.3 3
8.3 odd 2 2496.4.a.bp.1.3 3
8.5 even 2 2496.4.a.bl.1.3 3
12.11 even 2 1872.4.a.bk.1.3 3
13.5 odd 4 507.4.b.g.337.1 6
13.8 odd 4 507.4.b.g.337.6 6
13.12 even 2 507.4.a.h.1.1 3
39.38 odd 2 1521.4.a.u.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.a.c.1.3 3 1.1 even 1 trivial
117.4.a.f.1.1 3 3.2 odd 2
507.4.a.h.1.1 3 13.12 even 2
507.4.b.g.337.1 6 13.5 odd 4
507.4.b.g.337.6 6 13.8 odd 4
624.4.a.t.1.1 3 4.3 odd 2
975.4.a.l.1.1 3 5.4 even 2
1521.4.a.u.1.3 3 39.38 odd 2
1872.4.a.bk.1.3 3 12.11 even 2
1911.4.a.k.1.3 3 7.6 odd 2
2496.4.a.bl.1.3 3 8.5 even 2
2496.4.a.bp.1.3 3 8.3 odd 2