Properties

Label 507.4.a.j.1.2
Level $507$
Weight $4$
Character 507.1
Self dual yes
Analytic conductor $29.914$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 507.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(29.9139683729\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.5054412.1
Defining polynomial: \( x^{4} - 29x^{2} + 48 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 39)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.32750\) of defining polynomial
Character \(\chi\) \(=\) 507.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.32750 q^{2} -3.00000 q^{3} -6.23774 q^{4} -15.4241 q^{5} +3.98251 q^{6} +7.96501 q^{7} +18.9006 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-1.32750 q^{2} -3.00000 q^{3} -6.23774 q^{4} -15.4241 q^{5} +3.98251 q^{6} +7.96501 q^{7} +18.9006 q^{8} +9.00000 q^{9} +20.4755 q^{10} -12.7691 q^{11} +18.7132 q^{12} -10.5736 q^{14} +46.2722 q^{15} +24.8113 q^{16} -54.0000 q^{17} -11.9475 q^{18} +84.5794 q^{19} +96.2113 q^{20} -23.8950 q^{21} +16.9510 q^{22} +122.853 q^{23} -56.7019 q^{24} +112.902 q^{25} -27.0000 q^{27} -49.6837 q^{28} +140.853 q^{29} -61.4264 q^{30} +116.439 q^{31} -184.142 q^{32} +38.3072 q^{33} +71.6851 q^{34} -122.853 q^{35} -56.1397 q^{36} +433.898 q^{37} -112.279 q^{38} -291.525 q^{40} +205.823 q^{41} +31.7207 q^{42} -418.853 q^{43} +79.6501 q^{44} -138.817 q^{45} -163.087 q^{46} -485.861 q^{47} -74.4339 q^{48} -279.559 q^{49} -149.877 q^{50} +162.000 q^{51} -674.559 q^{53} +35.8425 q^{54} +196.951 q^{55} +150.544 q^{56} -253.738 q^{57} -186.982 q^{58} -186.226 q^{59} -288.634 q^{60} -671.902 q^{61} -154.574 q^{62} +71.6851 q^{63} +45.9584 q^{64} -50.8529 q^{66} -14.0364 q^{67} +336.838 q^{68} -368.559 q^{69} +163.087 q^{70} +346.789 q^{71} +170.106 q^{72} +832.900 q^{73} -576.000 q^{74} -338.706 q^{75} -527.584 q^{76} -101.706 q^{77} -335.608 q^{79} -382.691 q^{80} +81.0000 q^{81} -273.230 q^{82} -568.797 q^{83} +149.051 q^{84} +832.900 q^{85} +556.028 q^{86} -422.559 q^{87} -241.343 q^{88} +236.671 q^{89} +184.279 q^{90} -766.324 q^{92} -349.318 q^{93} +644.981 q^{94} -1304.56 q^{95} +552.426 q^{96} +1278.94 q^{97} +371.115 q^{98} -114.922 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{3} + 26 q^{4} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 12 q^{3} + 26 q^{4} + 36 q^{9} - 20 q^{10} - 78 q^{12} - 348 q^{14} + 354 q^{16} - 216 q^{17} - 136 q^{22} - 120 q^{23} + 44 q^{25} - 108 q^{27} - 48 q^{29} + 60 q^{30} + 120 q^{35} + 234 q^{36} + 468 q^{38} - 1268 q^{40} + 1044 q^{42} - 1064 q^{43} - 1062 q^{48} + 716 q^{49} + 648 q^{51} - 864 q^{53} + 584 q^{55} - 3372 q^{56} - 2280 q^{61} - 924 q^{62} + 1050 q^{64} + 408 q^{66} - 1404 q^{68} + 360 q^{69} - 2304 q^{74} - 132 q^{75} + 816 q^{77} + 288 q^{79} + 324 q^{81} + 28 q^{82} + 144 q^{87} - 2392 q^{88} - 180 q^{90} - 8568 q^{92} + 6656 q^{94} - 3384 q^{95}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.32750 −0.469343 −0.234671 0.972075i \(-0.575401\pi\)
−0.234671 + 0.972075i \(0.575401\pi\)
\(3\) −3.00000 −0.577350
\(4\) −6.23774 −0.779717
\(5\) −15.4241 −1.37957 −0.689785 0.724014i \(-0.742295\pi\)
−0.689785 + 0.724014i \(0.742295\pi\)
\(6\) 3.98251 0.270975
\(7\) 7.96501 0.430070 0.215035 0.976606i \(-0.431013\pi\)
0.215035 + 0.976606i \(0.431013\pi\)
\(8\) 18.9006 0.835297
\(9\) 9.00000 0.333333
\(10\) 20.4755 0.647491
\(11\) −12.7691 −0.350002 −0.175001 0.984568i \(-0.555993\pi\)
−0.175001 + 0.984568i \(0.555993\pi\)
\(12\) 18.7132 0.450170
\(13\) 0 0
\(14\) −10.5736 −0.201850
\(15\) 46.2722 0.796496
\(16\) 24.8113 0.387677
\(17\) −54.0000 −0.770407 −0.385204 0.922832i \(-0.625869\pi\)
−0.385204 + 0.922832i \(0.625869\pi\)
\(18\) −11.9475 −0.156448
\(19\) 84.5794 1.02126 0.510628 0.859802i \(-0.329413\pi\)
0.510628 + 0.859802i \(0.329413\pi\)
\(20\) 96.2113 1.07568
\(21\) −23.8950 −0.248301
\(22\) 16.9510 0.164271
\(23\) 122.853 1.11376 0.556882 0.830591i \(-0.311997\pi\)
0.556882 + 0.830591i \(0.311997\pi\)
\(24\) −56.7019 −0.482259
\(25\) 112.902 0.903215
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) −49.6837 −0.335333
\(29\) 140.853 0.901921 0.450961 0.892544i \(-0.351081\pi\)
0.450961 + 0.892544i \(0.351081\pi\)
\(30\) −61.4264 −0.373829
\(31\) 116.439 0.674617 0.337309 0.941394i \(-0.390483\pi\)
0.337309 + 0.941394i \(0.390483\pi\)
\(32\) −184.142 −1.01725
\(33\) 38.3072 0.202074
\(34\) 71.6851 0.361585
\(35\) −122.853 −0.593312
\(36\) −56.1397 −0.259906
\(37\) 433.898 1.92790 0.963951 0.266081i \(-0.0857289\pi\)
0.963951 + 0.266081i \(0.0857289\pi\)
\(38\) −112.279 −0.479319
\(39\) 0 0
\(40\) −291.525 −1.15235
\(41\) 205.823 0.784003 0.392002 0.919965i \(-0.371783\pi\)
0.392002 + 0.919965i \(0.371783\pi\)
\(42\) 31.7207 0.116538
\(43\) −418.853 −1.48545 −0.742726 0.669595i \(-0.766468\pi\)
−0.742726 + 0.669595i \(0.766468\pi\)
\(44\) 79.6501 0.272902
\(45\) −138.817 −0.459857
\(46\) −163.087 −0.522737
\(47\) −485.861 −1.50787 −0.753937 0.656947i \(-0.771848\pi\)
−0.753937 + 0.656947i \(0.771848\pi\)
\(48\) −74.4339 −0.223825
\(49\) −279.559 −0.815040
\(50\) −149.877 −0.423918
\(51\) 162.000 0.444795
\(52\) 0 0
\(53\) −674.559 −1.74826 −0.874130 0.485693i \(-0.838567\pi\)
−0.874130 + 0.485693i \(0.838567\pi\)
\(54\) 35.8425 0.0903251
\(55\) 196.951 0.482852
\(56\) 150.544 0.359236
\(57\) −253.738 −0.589622
\(58\) −186.982 −0.423310
\(59\) −186.226 −0.410925 −0.205462 0.978665i \(-0.565870\pi\)
−0.205462 + 0.978665i \(0.565870\pi\)
\(60\) −288.634 −0.621041
\(61\) −671.902 −1.41030 −0.705149 0.709059i \(-0.749120\pi\)
−0.705149 + 0.709059i \(0.749120\pi\)
\(62\) −154.574 −0.316627
\(63\) 71.6851 0.143357
\(64\) 45.9584 0.0897626
\(65\) 0 0
\(66\) −50.8529 −0.0948418
\(67\) −14.0364 −0.0255944 −0.0127972 0.999918i \(-0.504074\pi\)
−0.0127972 + 0.999918i \(0.504074\pi\)
\(68\) 336.838 0.600700
\(69\) −368.559 −0.643033
\(70\) 163.087 0.278467
\(71\) 346.789 0.579665 0.289833 0.957077i \(-0.406400\pi\)
0.289833 + 0.957077i \(0.406400\pi\)
\(72\) 170.106 0.278432
\(73\) 832.900 1.33539 0.667695 0.744435i \(-0.267281\pi\)
0.667695 + 0.744435i \(0.267281\pi\)
\(74\) −576.000 −0.904846
\(75\) −338.706 −0.521472
\(76\) −527.584 −0.796290
\(77\) −101.706 −0.150525
\(78\) 0 0
\(79\) −335.608 −0.477960 −0.238980 0.971025i \(-0.576813\pi\)
−0.238980 + 0.971025i \(0.576813\pi\)
\(80\) −382.691 −0.534827
\(81\) 81.0000 0.111111
\(82\) −273.230 −0.367966
\(83\) −568.797 −0.752212 −0.376106 0.926577i \(-0.622737\pi\)
−0.376106 + 0.926577i \(0.622737\pi\)
\(84\) 149.051 0.193605
\(85\) 832.900 1.06283
\(86\) 556.028 0.697186
\(87\) −422.559 −0.520725
\(88\) −241.343 −0.292355
\(89\) 236.671 0.281877 0.140939 0.990018i \(-0.454988\pi\)
0.140939 + 0.990018i \(0.454988\pi\)
\(90\) 184.279 0.215830
\(91\) 0 0
\(92\) −766.324 −0.868422
\(93\) −349.318 −0.389491
\(94\) 644.981 0.707710
\(95\) −1304.56 −1.40889
\(96\) 552.426 0.587310
\(97\) 1278.94 1.33873 0.669365 0.742934i \(-0.266566\pi\)
0.669365 + 0.742934i \(0.266566\pi\)
\(98\) 371.115 0.382533
\(99\) −114.922 −0.116667
\(100\) −704.253 −0.704253
\(101\) 632.264 0.622898 0.311449 0.950263i \(-0.399186\pi\)
0.311449 + 0.950263i \(0.399186\pi\)
\(102\) −215.055 −0.208761
\(103\) 1506.26 1.44094 0.720469 0.693487i \(-0.243927\pi\)
0.720469 + 0.693487i \(0.243927\pi\)
\(104\) 0 0
\(105\) 368.559 0.342549
\(106\) 895.478 0.820533
\(107\) −1268.56 −1.14613 −0.573066 0.819509i \(-0.694246\pi\)
−0.573066 + 0.819509i \(0.694246\pi\)
\(108\) 168.419 0.150057
\(109\) −347.425 −0.305296 −0.152648 0.988281i \(-0.548780\pi\)
−0.152648 + 0.988281i \(0.548780\pi\)
\(110\) −261.453 −0.226623
\(111\) −1301.69 −1.11307
\(112\) 197.622 0.166728
\(113\) 659.706 0.549203 0.274601 0.961558i \(-0.411454\pi\)
0.274601 + 0.961558i \(0.411454\pi\)
\(114\) 336.838 0.276735
\(115\) −1894.89 −1.53652
\(116\) −878.603 −0.703244
\(117\) 0 0
\(118\) 247.215 0.192865
\(119\) −430.111 −0.331329
\(120\) 874.574 0.665311
\(121\) −1167.95 −0.877499
\(122\) 891.951 0.661913
\(123\) −617.469 −0.452645
\(124\) −726.319 −0.526011
\(125\) 186.602 0.133521
\(126\) −95.1621 −0.0672834
\(127\) 275.019 0.192157 0.0960787 0.995374i \(-0.469370\pi\)
0.0960787 + 0.995374i \(0.469370\pi\)
\(128\) 1412.13 0.975121
\(129\) 1256.56 0.857626
\(130\) 0 0
\(131\) −1183.97 −0.789648 −0.394824 0.918757i \(-0.629195\pi\)
−0.394824 + 0.918757i \(0.629195\pi\)
\(132\) −238.950 −0.157560
\(133\) 673.676 0.439211
\(134\) 18.6334 0.0120125
\(135\) 416.450 0.265499
\(136\) −1020.63 −0.643519
\(137\) −2557.36 −1.59482 −0.797410 0.603438i \(-0.793797\pi\)
−0.797410 + 0.603438i \(0.793797\pi\)
\(138\) 489.262 0.301803
\(139\) 545.736 0.333012 0.166506 0.986040i \(-0.446751\pi\)
0.166506 + 0.986040i \(0.446751\pi\)
\(140\) 766.324 0.462616
\(141\) 1457.58 0.870571
\(142\) −460.362 −0.272062
\(143\) 0 0
\(144\) 223.302 0.129226
\(145\) −2172.52 −1.24426
\(146\) −1105.68 −0.626756
\(147\) 838.676 0.470563
\(148\) −2706.54 −1.50322
\(149\) −1376.78 −0.756981 −0.378491 0.925605i \(-0.623557\pi\)
−0.378491 + 0.925605i \(0.623557\pi\)
\(150\) 449.632 0.244749
\(151\) −2733.47 −1.47316 −0.736579 0.676352i \(-0.763560\pi\)
−0.736579 + 0.676352i \(0.763560\pi\)
\(152\) 1598.60 0.853052
\(153\) −486.000 −0.256802
\(154\) 135.015 0.0706479
\(155\) −1795.97 −0.930683
\(156\) 0 0
\(157\) 1029.97 0.523570 0.261785 0.965126i \(-0.415689\pi\)
0.261785 + 0.965126i \(0.415689\pi\)
\(158\) 445.520 0.224327
\(159\) 2023.68 1.00936
\(160\) 2840.22 1.40337
\(161\) 978.524 0.478997
\(162\) −107.528 −0.0521492
\(163\) 2882.91 1.38532 0.692660 0.721264i \(-0.256439\pi\)
0.692660 + 0.721264i \(0.256439\pi\)
\(164\) −1283.87 −0.611301
\(165\) −590.853 −0.278775
\(166\) 755.079 0.353045
\(167\) −1153.90 −0.534679 −0.267340 0.963602i \(-0.586145\pi\)
−0.267340 + 0.963602i \(0.586145\pi\)
\(168\) −451.631 −0.207405
\(169\) 0 0
\(170\) −1105.68 −0.498832
\(171\) 761.215 0.340418
\(172\) 2612.69 1.15823
\(173\) −1688.85 −0.742203 −0.371101 0.928592i \(-0.621020\pi\)
−0.371101 + 0.928592i \(0.621020\pi\)
\(174\) 560.947 0.244398
\(175\) 899.265 0.388446
\(176\) −316.817 −0.135687
\(177\) 558.678 0.237247
\(178\) −314.181 −0.132297
\(179\) −942.793 −0.393674 −0.196837 0.980436i \(-0.563067\pi\)
−0.196837 + 0.980436i \(0.563067\pi\)
\(180\) 865.902 0.358558
\(181\) 482.030 0.197950 0.0989751 0.995090i \(-0.468444\pi\)
0.0989751 + 0.995090i \(0.468444\pi\)
\(182\) 0 0
\(183\) 2015.71 0.814236
\(184\) 2322.00 0.930325
\(185\) −6692.47 −2.65968
\(186\) 463.721 0.182805
\(187\) 689.530 0.269644
\(188\) 3030.67 1.17572
\(189\) −215.055 −0.0827670
\(190\) 1731.80 0.661254
\(191\) 4223.32 1.59994 0.799971 0.600039i \(-0.204848\pi\)
0.799971 + 0.600039i \(0.204848\pi\)
\(192\) −137.875 −0.0518244
\(193\) 229.092 0.0854424 0.0427212 0.999087i \(-0.486397\pi\)
0.0427212 + 0.999087i \(0.486397\pi\)
\(194\) −1697.80 −0.628323
\(195\) 0 0
\(196\) 1743.81 0.635501
\(197\) 228.335 0.0825798 0.0412899 0.999147i \(-0.486853\pi\)
0.0412899 + 0.999147i \(0.486853\pi\)
\(198\) 152.559 0.0547569
\(199\) 2939.02 1.04694 0.523471 0.852043i \(-0.324637\pi\)
0.523471 + 0.852043i \(0.324637\pi\)
\(200\) 2133.92 0.754453
\(201\) 42.1093 0.0147769
\(202\) −839.332 −0.292352
\(203\) 1121.89 0.387889
\(204\) −1010.51 −0.346814
\(205\) −3174.63 −1.08159
\(206\) −1999.57 −0.676294
\(207\) 1105.68 0.371255
\(208\) 0 0
\(209\) −1080.00 −0.357441
\(210\) −489.262 −0.160773
\(211\) −1607.02 −0.524321 −0.262161 0.965024i \(-0.584435\pi\)
−0.262161 + 0.965024i \(0.584435\pi\)
\(212\) 4207.72 1.36315
\(213\) −1040.37 −0.334670
\(214\) 1684.01 0.537929
\(215\) 6460.42 2.04929
\(216\) −510.317 −0.160753
\(217\) 927.441 0.290133
\(218\) 461.207 0.143288
\(219\) −2498.70 −0.770988
\(220\) −1228.53 −0.376488
\(221\) 0 0
\(222\) 1728.00 0.522413
\(223\) −130.867 −0.0392981 −0.0196490 0.999807i \(-0.506255\pi\)
−0.0196490 + 0.999807i \(0.506255\pi\)
\(224\) −1466.69 −0.437489
\(225\) 1016.12 0.301072
\(226\) −875.761 −0.257764
\(227\) −4325.19 −1.26464 −0.632319 0.774708i \(-0.717897\pi\)
−0.632319 + 0.774708i \(0.717897\pi\)
\(228\) 1582.75 0.459738
\(229\) 2621.57 0.756499 0.378250 0.925704i \(-0.376526\pi\)
0.378250 + 0.925704i \(0.376526\pi\)
\(230\) 2515.47 0.721153
\(231\) 305.117 0.0869058
\(232\) 2662.21 0.753373
\(233\) −4643.12 −1.30550 −0.652748 0.757575i \(-0.726384\pi\)
−0.652748 + 0.757575i \(0.726384\pi\)
\(234\) 0 0
\(235\) 7493.95 2.08022
\(236\) 1161.63 0.320405
\(237\) 1006.82 0.275950
\(238\) 570.972 0.155507
\(239\) −6696.69 −1.81244 −0.906219 0.422809i \(-0.861044\pi\)
−0.906219 + 0.422809i \(0.861044\pi\)
\(240\) 1148.07 0.308783
\(241\) −2301.47 −0.615148 −0.307574 0.951524i \(-0.599517\pi\)
−0.307574 + 0.951524i \(0.599517\pi\)
\(242\) 1550.46 0.411848
\(243\) −243.000 −0.0641500
\(244\) 4191.15 1.09963
\(245\) 4311.93 1.12440
\(246\) 819.691 0.212445
\(247\) 0 0
\(248\) 2200.78 0.563506
\(249\) 1706.39 0.434290
\(250\) −247.714 −0.0626673
\(251\) 828.000 0.208219 0.104109 0.994566i \(-0.466801\pi\)
0.104109 + 0.994566i \(0.466801\pi\)
\(252\) −447.153 −0.111778
\(253\) −1568.72 −0.389820
\(254\) −365.088 −0.0901877
\(255\) −2498.70 −0.613626
\(256\) −2242.27 −0.547429
\(257\) 884.763 0.214747 0.107374 0.994219i \(-0.465756\pi\)
0.107374 + 0.994219i \(0.465756\pi\)
\(258\) −1668.08 −0.402521
\(259\) 3456.00 0.829133
\(260\) 0 0
\(261\) 1267.68 0.300640
\(262\) 1571.72 0.370616
\(263\) −8343.94 −1.95631 −0.978155 0.207878i \(-0.933344\pi\)
−0.978155 + 0.207878i \(0.933344\pi\)
\(264\) 724.030 0.168792
\(265\) 10404.4 2.41185
\(266\) −894.306 −0.206141
\(267\) −710.013 −0.162742
\(268\) 87.5556 0.0199564
\(269\) 2762.56 0.626157 0.313078 0.949727i \(-0.398640\pi\)
0.313078 + 0.949727i \(0.398640\pi\)
\(270\) −552.838 −0.124610
\(271\) 3116.54 0.698585 0.349293 0.937014i \(-0.386422\pi\)
0.349293 + 0.937014i \(0.386422\pi\)
\(272\) −1339.81 −0.298669
\(273\) 0 0
\(274\) 3394.91 0.748517
\(275\) −1441.65 −0.316127
\(276\) 2298.97 0.501384
\(277\) −502.060 −0.108902 −0.0544510 0.998516i \(-0.517341\pi\)
−0.0544510 + 0.998516i \(0.517341\pi\)
\(278\) −724.465 −0.156297
\(279\) 1047.96 0.224872
\(280\) −2322.00 −0.495592
\(281\) −6607.56 −1.40275 −0.701377 0.712791i \(-0.747431\pi\)
−0.701377 + 0.712791i \(0.747431\pi\)
\(282\) −1934.94 −0.408596
\(283\) −4368.98 −0.917699 −0.458850 0.888514i \(-0.651738\pi\)
−0.458850 + 0.888514i \(0.651738\pi\)
\(284\) −2163.18 −0.451975
\(285\) 3913.68 0.813425
\(286\) 0 0
\(287\) 1639.38 0.337176
\(288\) −1657.28 −0.339084
\(289\) −1997.00 −0.406473
\(290\) 2884.03 0.583986
\(291\) −3836.82 −0.772916
\(292\) −5195.41 −1.04123
\(293\) −5348.12 −1.06635 −0.533175 0.846005i \(-0.679001\pi\)
−0.533175 + 0.846005i \(0.679001\pi\)
\(294\) −1113.34 −0.220856
\(295\) 2872.36 0.566900
\(296\) 8200.94 1.61037
\(297\) 344.765 0.0673579
\(298\) 1827.68 0.355284
\(299\) 0 0
\(300\) 2112.76 0.406600
\(301\) −3336.17 −0.638849
\(302\) 3628.69 0.691416
\(303\) −1896.79 −0.359630
\(304\) 2098.53 0.395917
\(305\) 10363.5 1.94561
\(306\) 645.166 0.120528
\(307\) −4502.46 −0.837032 −0.418516 0.908210i \(-0.637450\pi\)
−0.418516 + 0.908210i \(0.637450\pi\)
\(308\) 634.414 0.117367
\(309\) −4518.79 −0.831926
\(310\) 2384.15 0.436809
\(311\) −7447.20 −1.35785 −0.678926 0.734207i \(-0.737554\pi\)
−0.678926 + 0.734207i \(0.737554\pi\)
\(312\) 0 0
\(313\) −6508.93 −1.17542 −0.587710 0.809072i \(-0.699970\pi\)
−0.587710 + 0.809072i \(0.699970\pi\)
\(314\) −1367.29 −0.245734
\(315\) −1105.68 −0.197771
\(316\) 2093.43 0.372673
\(317\) 2465.57 0.436846 0.218423 0.975854i \(-0.429909\pi\)
0.218423 + 0.975854i \(0.429909\pi\)
\(318\) −2686.43 −0.473735
\(319\) −1798.56 −0.315674
\(320\) −708.866 −0.123834
\(321\) 3805.68 0.661720
\(322\) −1298.99 −0.224814
\(323\) −4567.29 −0.786782
\(324\) −505.257 −0.0866353
\(325\) 0 0
\(326\) −3827.07 −0.650190
\(327\) 1042.27 0.176263
\(328\) 3890.18 0.654876
\(329\) −3869.89 −0.648491
\(330\) 784.358 0.130841
\(331\) −4114.84 −0.683300 −0.341650 0.939827i \(-0.610986\pi\)
−0.341650 + 0.939827i \(0.610986\pi\)
\(332\) 3548.01 0.586513
\(333\) 3905.08 0.642634
\(334\) 1531.80 0.250948
\(335\) 216.499 0.0353092
\(336\) −592.867 −0.0962605
\(337\) 4798.05 0.775568 0.387784 0.921750i \(-0.373241\pi\)
0.387784 + 0.921750i \(0.373241\pi\)
\(338\) 0 0
\(339\) −1979.12 −0.317082
\(340\) −5195.41 −0.828708
\(341\) −1486.82 −0.236117
\(342\) −1010.51 −0.159773
\(343\) −4958.69 −0.780594
\(344\) −7916.58 −1.24079
\(345\) 5684.67 0.887109
\(346\) 2241.96 0.348348
\(347\) −3314.76 −0.512812 −0.256406 0.966569i \(-0.582538\pi\)
−0.256406 + 0.966569i \(0.582538\pi\)
\(348\) 2635.81 0.406018
\(349\) −371.740 −0.0570166 −0.0285083 0.999594i \(-0.509076\pi\)
−0.0285083 + 0.999594i \(0.509076\pi\)
\(350\) −1193.78 −0.182314
\(351\) 0 0
\(352\) 2351.32 0.356039
\(353\) 7539.10 1.13673 0.568365 0.822776i \(-0.307576\pi\)
0.568365 + 0.822776i \(0.307576\pi\)
\(354\) −741.646 −0.111350
\(355\) −5348.89 −0.799689
\(356\) −1476.29 −0.219785
\(357\) 1290.33 0.191293
\(358\) 1251.56 0.184768
\(359\) 12741.5 1.87317 0.936586 0.350437i \(-0.113967\pi\)
0.936586 + 0.350437i \(0.113967\pi\)
\(360\) −2623.72 −0.384117
\(361\) 294.676 0.0429619
\(362\) −639.896 −0.0929065
\(363\) 3503.85 0.506624
\(364\) 0 0
\(365\) −12846.7 −1.84227
\(366\) −2675.85 −0.382156
\(367\) 7187.26 1.02227 0.511134 0.859501i \(-0.329226\pi\)
0.511134 + 0.859501i \(0.329226\pi\)
\(368\) 3048.14 0.431781
\(369\) 1852.41 0.261334
\(370\) 8884.26 1.24830
\(371\) −5372.87 −0.751874
\(372\) 2178.96 0.303693
\(373\) −2087.99 −0.289845 −0.144922 0.989443i \(-0.546293\pi\)
−0.144922 + 0.989443i \(0.546293\pi\)
\(374\) −915.352 −0.126555
\(375\) −559.805 −0.0770886
\(376\) −9183.07 −1.25952
\(377\) 0 0
\(378\) 285.486 0.0388461
\(379\) 3982.08 0.539699 0.269850 0.962902i \(-0.413026\pi\)
0.269850 + 0.962902i \(0.413026\pi\)
\(380\) 8137.50 1.09854
\(381\) −825.057 −0.110942
\(382\) −5606.47 −0.750921
\(383\) 8638.43 1.15249 0.576244 0.817278i \(-0.304518\pi\)
0.576244 + 0.817278i \(0.304518\pi\)
\(384\) −4236.38 −0.562987
\(385\) 1568.72 0.207660
\(386\) −304.120 −0.0401018
\(387\) −3769.68 −0.495151
\(388\) −7977.70 −1.04383
\(389\) −1275.74 −0.166279 −0.0831393 0.996538i \(-0.526495\pi\)
−0.0831393 + 0.996538i \(0.526495\pi\)
\(390\) 0 0
\(391\) −6634.06 −0.858053
\(392\) −5283.83 −0.680801
\(393\) 3551.91 0.455904
\(394\) −303.115 −0.0387582
\(395\) 5176.44 0.659379
\(396\) 716.851 0.0909675
\(397\) −4622.65 −0.584394 −0.292197 0.956358i \(-0.594386\pi\)
−0.292197 + 0.956358i \(0.594386\pi\)
\(398\) −3901.55 −0.491375
\(399\) −2021.03 −0.253579
\(400\) 2801.24 0.350155
\(401\) −138.075 −0.0171949 −0.00859743 0.999963i \(-0.502737\pi\)
−0.00859743 + 0.999963i \(0.502737\pi\)
\(402\) −55.9001 −0.00693544
\(403\) 0 0
\(404\) −3943.90 −0.485684
\(405\) −1249.35 −0.153286
\(406\) −1489.32 −0.182053
\(407\) −5540.47 −0.674769
\(408\) 3061.90 0.371536
\(409\) −1204.64 −0.145637 −0.0728186 0.997345i \(-0.523199\pi\)
−0.0728186 + 0.997345i \(0.523199\pi\)
\(410\) 4214.32 0.507635
\(411\) 7672.09 0.920770
\(412\) −9395.68 −1.12352
\(413\) −1483.29 −0.176726
\(414\) −1467.79 −0.174246
\(415\) 8773.16 1.03773
\(416\) 0 0
\(417\) −1637.21 −0.192265
\(418\) 1433.70 0.167762
\(419\) 5199.85 0.606275 0.303138 0.952947i \(-0.401966\pi\)
0.303138 + 0.952947i \(0.401966\pi\)
\(420\) −2298.97 −0.267091
\(421\) −14136.5 −1.63651 −0.818256 0.574854i \(-0.805059\pi\)
−0.818256 + 0.574854i \(0.805059\pi\)
\(422\) 2133.32 0.246086
\(423\) −4372.75 −0.502625
\(424\) −12749.6 −1.46032
\(425\) −6096.70 −0.695844
\(426\) 1381.09 0.157075
\(427\) −5351.71 −0.606527
\(428\) 7912.94 0.893660
\(429\) 0 0
\(430\) −8576.21 −0.961818
\(431\) −2279.83 −0.254793 −0.127396 0.991852i \(-0.540662\pi\)
−0.127396 + 0.991852i \(0.540662\pi\)
\(432\) −669.905 −0.0746084
\(433\) −13298.7 −1.47597 −0.737984 0.674819i \(-0.764222\pi\)
−0.737984 + 0.674819i \(0.764222\pi\)
\(434\) −1231.18 −0.136172
\(435\) 6517.57 0.718376
\(436\) 2167.14 0.238045
\(437\) 10390.8 1.13744
\(438\) 3317.03 0.361858
\(439\) −10452.3 −1.13635 −0.568177 0.822907i \(-0.692351\pi\)
−0.568177 + 0.822907i \(0.692351\pi\)
\(440\) 3722.50 0.403325
\(441\) −2516.03 −0.271680
\(442\) 0 0
\(443\) 5363.50 0.575232 0.287616 0.957746i \(-0.407137\pi\)
0.287616 + 0.957746i \(0.407137\pi\)
\(444\) 8119.62 0.867883
\(445\) −3650.43 −0.388870
\(446\) 173.726 0.0184443
\(447\) 4130.34 0.437043
\(448\) 366.059 0.0386042
\(449\) 9681.73 1.01762 0.508808 0.860880i \(-0.330086\pi\)
0.508808 + 0.860880i \(0.330086\pi\)
\(450\) −1348.90 −0.141306
\(451\) −2628.17 −0.274402
\(452\) −4115.07 −0.428223
\(453\) 8200.42 0.850528
\(454\) 5741.69 0.593548
\(455\) 0 0
\(456\) −4795.81 −0.492510
\(457\) −3537.94 −0.362139 −0.181070 0.983470i \(-0.557956\pi\)
−0.181070 + 0.983470i \(0.557956\pi\)
\(458\) −3480.14 −0.355057
\(459\) 1458.00 0.148265
\(460\) 11819.8 1.19805
\(461\) −15074.9 −1.52302 −0.761508 0.648156i \(-0.775541\pi\)
−0.761508 + 0.648156i \(0.775541\pi\)
\(462\) −405.044 −0.0407886
\(463\) −11070.0 −1.11116 −0.555580 0.831463i \(-0.687504\pi\)
−0.555580 + 0.831463i \(0.687504\pi\)
\(464\) 3494.74 0.349654
\(465\) 5387.91 0.537330
\(466\) 6163.75 0.612725
\(467\) 13252.8 1.31320 0.656600 0.754239i \(-0.271994\pi\)
0.656600 + 0.754239i \(0.271994\pi\)
\(468\) 0 0
\(469\) −111.800 −0.0110074
\(470\) −9948.23 −0.976335
\(471\) −3089.91 −0.302284
\(472\) −3519.79 −0.343244
\(473\) 5348.36 0.519911
\(474\) −1336.56 −0.129515
\(475\) 9549.18 0.922413
\(476\) 2682.92 0.258343
\(477\) −6071.03 −0.582753
\(478\) 8889.86 0.850654
\(479\) 12241.4 1.16769 0.583843 0.811866i \(-0.301548\pi\)
0.583843 + 0.811866i \(0.301548\pi\)
\(480\) −8520.66 −0.810236
\(481\) 0 0
\(482\) 3055.20 0.288715
\(483\) −2935.57 −0.276549
\(484\) 7285.37 0.684201
\(485\) −19726.5 −1.84687
\(486\) 322.583 0.0301084
\(487\) −13413.3 −1.24808 −0.624041 0.781392i \(-0.714510\pi\)
−0.624041 + 0.781392i \(0.714510\pi\)
\(488\) −12699.4 −1.17802
\(489\) −8648.74 −0.799815
\(490\) −5724.10 −0.527731
\(491\) −737.885 −0.0678214 −0.0339107 0.999425i \(-0.510796\pi\)
−0.0339107 + 0.999425i \(0.510796\pi\)
\(492\) 3851.61 0.352935
\(493\) −7606.06 −0.694847
\(494\) 0 0
\(495\) 1772.56 0.160951
\(496\) 2889.01 0.261533
\(497\) 2762.17 0.249297
\(498\) −2265.24 −0.203831
\(499\) 1865.65 0.167370 0.0836852 0.996492i \(-0.473331\pi\)
0.0836852 + 0.996492i \(0.473331\pi\)
\(500\) −1163.97 −0.104109
\(501\) 3461.70 0.308697
\(502\) −1099.17 −0.0977259
\(503\) −16632.0 −1.47432 −0.737161 0.675717i \(-0.763834\pi\)
−0.737161 + 0.675717i \(0.763834\pi\)
\(504\) 1354.89 0.119745
\(505\) −9752.09 −0.859331
\(506\) 2082.47 0.182959
\(507\) 0 0
\(508\) −1715.50 −0.149829
\(509\) 4128.91 0.359549 0.179775 0.983708i \(-0.442463\pi\)
0.179775 + 0.983708i \(0.442463\pi\)
\(510\) 3317.03 0.288001
\(511\) 6634.06 0.574312
\(512\) −8320.40 −0.718190
\(513\) −2283.64 −0.196541
\(514\) −1174.52 −0.100790
\(515\) −23232.7 −1.98788
\(516\) −7838.08 −0.668706
\(517\) 6203.99 0.527758
\(518\) −4587.85 −0.389147
\(519\) 5066.56 0.428511
\(520\) 0 0
\(521\) −988.234 −0.0831005 −0.0415502 0.999136i \(-0.513230\pi\)
−0.0415502 + 0.999136i \(0.513230\pi\)
\(522\) −1682.84 −0.141103
\(523\) −9441.62 −0.789394 −0.394697 0.918811i \(-0.629150\pi\)
−0.394697 + 0.918811i \(0.629150\pi\)
\(524\) 7385.30 0.615703
\(525\) −2697.79 −0.224269
\(526\) 11076.6 0.918180
\(527\) −6287.73 −0.519730
\(528\) 950.452 0.0783392
\(529\) 2925.83 0.240472
\(530\) −13811.9 −1.13198
\(531\) −1676.03 −0.136975
\(532\) −4202.21 −0.342461
\(533\) 0 0
\(534\) 942.544 0.0763817
\(535\) 19566.3 1.58117
\(536\) −265.297 −0.0213789
\(537\) 2828.38 0.227288
\(538\) −3667.30 −0.293882
\(539\) 3569.70 0.285265
\(540\) −2597.71 −0.207014
\(541\) 14001.5 1.11270 0.556351 0.830948i \(-0.312201\pi\)
0.556351 + 0.830948i \(0.312201\pi\)
\(542\) −4137.22 −0.327876
\(543\) −1446.09 −0.114287
\(544\) 9943.67 0.783697
\(545\) 5358.70 0.421177
\(546\) 0 0
\(547\) −4244.85 −0.331804 −0.165902 0.986142i \(-0.553053\pi\)
−0.165902 + 0.986142i \(0.553053\pi\)
\(548\) 15952.2 1.24351
\(549\) −6047.12 −0.470100
\(550\) 1913.80 0.148372
\(551\) 11913.3 0.921092
\(552\) −6965.99 −0.537123
\(553\) −2673.12 −0.205556
\(554\) 666.485 0.0511124
\(555\) 20077.4 1.53556
\(556\) −3404.16 −0.259655
\(557\) −11732.2 −0.892473 −0.446236 0.894915i \(-0.647236\pi\)
−0.446236 + 0.894915i \(0.647236\pi\)
\(558\) −1391.16 −0.105542
\(559\) 0 0
\(560\) −3048.14 −0.230013
\(561\) −2068.59 −0.155679
\(562\) 8771.54 0.658372
\(563\) −9941.80 −0.744222 −0.372111 0.928188i \(-0.621366\pi\)
−0.372111 + 0.928188i \(0.621366\pi\)
\(564\) −9092.02 −0.678800
\(565\) −10175.3 −0.757664
\(566\) 5799.83 0.430716
\(567\) 645.166 0.0477856
\(568\) 6554.52 0.484193
\(569\) 3690.77 0.271925 0.135962 0.990714i \(-0.456587\pi\)
0.135962 + 0.990714i \(0.456587\pi\)
\(570\) −5195.41 −0.381775
\(571\) −5685.09 −0.416661 −0.208331 0.978058i \(-0.566803\pi\)
−0.208331 + 0.978058i \(0.566803\pi\)
\(572\) 0 0
\(573\) −12670.0 −0.923727
\(574\) −2176.28 −0.158251
\(575\) 13870.3 1.00597
\(576\) 413.626 0.0299209
\(577\) 7746.50 0.558910 0.279455 0.960159i \(-0.409846\pi\)
0.279455 + 0.960159i \(0.409846\pi\)
\(578\) 2651.02 0.190775
\(579\) −687.275 −0.0493302
\(580\) 13551.6 0.970175
\(581\) −4530.47 −0.323504
\(582\) 5093.39 0.362762
\(583\) 8613.48 0.611894
\(584\) 15742.3 1.11545
\(585\) 0 0
\(586\) 7099.64 0.500484
\(587\) −2766.54 −0.194527 −0.0972635 0.995259i \(-0.531009\pi\)
−0.0972635 + 0.995259i \(0.531009\pi\)
\(588\) −5231.44 −0.366906
\(589\) 9848.38 0.688957
\(590\) −3813.07 −0.266070
\(591\) −685.006 −0.0476774
\(592\) 10765.6 0.747402
\(593\) −1440.79 −0.0997743 −0.0498871 0.998755i \(-0.515886\pi\)
−0.0498871 + 0.998755i \(0.515886\pi\)
\(594\) −457.676 −0.0316139
\(595\) 6634.06 0.457092
\(596\) 8587.99 0.590231
\(597\) −8817.06 −0.604452
\(598\) 0 0
\(599\) 23837.5 1.62600 0.813001 0.582263i \(-0.197832\pi\)
0.813001 + 0.582263i \(0.197832\pi\)
\(600\) −6401.75 −0.435584
\(601\) −6694.23 −0.454348 −0.227174 0.973854i \(-0.572949\pi\)
−0.227174 + 0.973854i \(0.572949\pi\)
\(602\) 4428.77 0.299839
\(603\) −126.328 −0.00853145
\(604\) 17050.7 1.14865
\(605\) 18014.6 1.21057
\(606\) 2518.00 0.168790
\(607\) 3330.50 0.222703 0.111352 0.993781i \(-0.464482\pi\)
0.111352 + 0.993781i \(0.464482\pi\)
\(608\) −15574.6 −1.03887
\(609\) −3365.68 −0.223948
\(610\) −13757.5 −0.913156
\(611\) 0 0
\(612\) 3031.54 0.200233
\(613\) −13490.3 −0.888857 −0.444428 0.895814i \(-0.646593\pi\)
−0.444428 + 0.895814i \(0.646593\pi\)
\(614\) 5977.02 0.392855
\(615\) 9523.88 0.624455
\(616\) −1922.30 −0.125733
\(617\) 7470.76 0.487458 0.243729 0.969843i \(-0.421629\pi\)
0.243729 + 0.969843i \(0.421629\pi\)
\(618\) 5998.71 0.390458
\(619\) 24806.9 1.61078 0.805389 0.592746i \(-0.201956\pi\)
0.805389 + 0.592746i \(0.201956\pi\)
\(620\) 11202.8 0.725669
\(621\) −3317.03 −0.214344
\(622\) 9886.17 0.637298
\(623\) 1885.09 0.121227
\(624\) 0 0
\(625\) −16990.9 −1.08742
\(626\) 8640.61 0.551675
\(627\) 3240.00 0.206369
\(628\) −6424.68 −0.408237
\(629\) −23430.5 −1.48527
\(630\) 1467.79 0.0928222
\(631\) 314.333 0.0198311 0.00991554 0.999951i \(-0.496844\pi\)
0.00991554 + 0.999951i \(0.496844\pi\)
\(632\) −6343.19 −0.399238
\(633\) 4821.06 0.302717
\(634\) −3273.05 −0.205031
\(635\) −4241.91 −0.265095
\(636\) −12623.2 −0.787014
\(637\) 0 0
\(638\) 2387.59 0.148159
\(639\) 3121.10 0.193222
\(640\) −21780.7 −1.34525
\(641\) 5550.41 0.342009 0.171005 0.985270i \(-0.445299\pi\)
0.171005 + 0.985270i \(0.445299\pi\)
\(642\) −5052.04 −0.310573
\(643\) 5479.48 0.336065 0.168032 0.985781i \(-0.446259\pi\)
0.168032 + 0.985781i \(0.446259\pi\)
\(644\) −6103.78 −0.373482
\(645\) −19381.2 −1.18316
\(646\) 6063.08 0.369271
\(647\) 4724.83 0.287098 0.143549 0.989643i \(-0.454149\pi\)
0.143549 + 0.989643i \(0.454149\pi\)
\(648\) 1530.95 0.0928108
\(649\) 2377.93 0.143824
\(650\) 0 0
\(651\) −2782.32 −0.167508
\(652\) −17982.9 −1.08016
\(653\) 3463.91 0.207585 0.103793 0.994599i \(-0.466902\pi\)
0.103793 + 0.994599i \(0.466902\pi\)
\(654\) −1383.62 −0.0827276
\(655\) 18261.6 1.08938
\(656\) 5106.73 0.303940
\(657\) 7496.10 0.445130
\(658\) 5137.28 0.304365
\(659\) −2606.35 −0.154065 −0.0770327 0.997029i \(-0.524545\pi\)
−0.0770327 + 0.997029i \(0.524545\pi\)
\(660\) 3685.59 0.217366
\(661\) −22436.7 −1.32025 −0.660127 0.751154i \(-0.729498\pi\)
−0.660127 + 0.751154i \(0.729498\pi\)
\(662\) 5462.46 0.320702
\(663\) 0 0
\(664\) −10750.6 −0.628321
\(665\) −10390.8 −0.605923
\(666\) −5184.00 −0.301615
\(667\) 17304.2 1.00453
\(668\) 7197.73 0.416899
\(669\) 392.600 0.0226888
\(670\) −287.403 −0.0165721
\(671\) 8579.56 0.493607
\(672\) 4400.08 0.252584
\(673\) −633.970 −0.0363117 −0.0181558 0.999835i \(-0.505779\pi\)
−0.0181558 + 0.999835i \(0.505779\pi\)
\(674\) −6369.42 −0.364007
\(675\) −3048.35 −0.173824
\(676\) 0 0
\(677\) 24457.4 1.38844 0.694221 0.719762i \(-0.255749\pi\)
0.694221 + 0.719762i \(0.255749\pi\)
\(678\) 2627.28 0.148820
\(679\) 10186.8 0.575747
\(680\) 15742.3 0.887780
\(681\) 12975.6 0.730139
\(682\) 1973.76 0.110820
\(683\) −12367.6 −0.692875 −0.346437 0.938073i \(-0.612609\pi\)
−0.346437 + 0.938073i \(0.612609\pi\)
\(684\) −4748.26 −0.265430
\(685\) 39445.0 2.20017
\(686\) 6582.66 0.366366
\(687\) −7864.71 −0.436765
\(688\) −10392.3 −0.575875
\(689\) 0 0
\(690\) −7546.41 −0.416358
\(691\) 1050.99 0.0578605 0.0289302 0.999581i \(-0.490790\pi\)
0.0289302 + 0.999581i \(0.490790\pi\)
\(692\) 10534.6 0.578709
\(693\) −915.352 −0.0501751
\(694\) 4400.35 0.240685
\(695\) −8417.46 −0.459414
\(696\) −7986.62 −0.434960
\(697\) −11114.4 −0.604002
\(698\) 493.486 0.0267603
\(699\) 13929.4 0.753729
\(700\) −5609.38 −0.302878
\(701\) −24294.1 −1.30895 −0.654476 0.756083i \(-0.727111\pi\)
−0.654476 + 0.756083i \(0.727111\pi\)
\(702\) 0 0
\(703\) 36698.8 1.96888
\(704\) −586.846 −0.0314170
\(705\) −22481.8 −1.20101
\(706\) −10008.2 −0.533516
\(707\) 5035.99 0.267890
\(708\) −3484.89 −0.184986
\(709\) −27465.9 −1.45487 −0.727436 0.686176i \(-0.759288\pi\)
−0.727436 + 0.686176i \(0.759288\pi\)
\(710\) 7100.66 0.375328
\(711\) −3020.47 −0.159320
\(712\) 4473.23 0.235451
\(713\) 14304.9 0.751365
\(714\) −1712.92 −0.0897820
\(715\) 0 0
\(716\) 5880.90 0.306955
\(717\) 20090.1 1.04641
\(718\) −16914.3 −0.879160
\(719\) 36433.5 1.88976 0.944882 0.327411i \(-0.106176\pi\)
0.944882 + 0.327411i \(0.106176\pi\)
\(720\) −3444.22 −0.178276
\(721\) 11997.4 0.619704
\(722\) −391.183 −0.0201639
\(723\) 6904.40 0.355156
\(724\) −3006.78 −0.154345
\(725\) 15902.6 0.814629
\(726\) −4651.37 −0.237780
\(727\) 551.608 0.0281403 0.0140701 0.999901i \(-0.495521\pi\)
0.0140701 + 0.999901i \(0.495521\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 17054.0 0.864654
\(731\) 22618.1 1.14440
\(732\) −12573.4 −0.634874
\(733\) −20317.2 −1.02379 −0.511893 0.859049i \(-0.671055\pi\)
−0.511893 + 0.859049i \(0.671055\pi\)
\(734\) −9541.10 −0.479794
\(735\) −12935.8 −0.649175
\(736\) −22622.4 −1.13298
\(737\) 179.232 0.00895807
\(738\) −2459.07 −0.122655
\(739\) 29931.6 1.48992 0.744962 0.667107i \(-0.232468\pi\)
0.744962 + 0.667107i \(0.232468\pi\)
\(740\) 41745.9 2.07380
\(741\) 0 0
\(742\) 7132.49 0.352887
\(743\) −24376.4 −1.20361 −0.601805 0.798643i \(-0.705552\pi\)
−0.601805 + 0.798643i \(0.705552\pi\)
\(744\) −6602.33 −0.325340
\(745\) 21235.5 1.04431
\(746\) 2771.81 0.136037
\(747\) −5119.17 −0.250737
\(748\) −4301.11 −0.210246
\(749\) −10104.1 −0.492917
\(750\) 743.142 0.0361810
\(751\) 22692.2 1.10260 0.551298 0.834308i \(-0.314133\pi\)
0.551298 + 0.834308i \(0.314133\pi\)
\(752\) −12054.8 −0.584567
\(753\) −2484.00 −0.120215
\(754\) 0 0
\(755\) 42161.3 2.03232
\(756\) 1341.46 0.0645349
\(757\) −33063.5 −1.58747 −0.793734 0.608265i \(-0.791866\pi\)
−0.793734 + 0.608265i \(0.791866\pi\)
\(758\) −5286.22 −0.253304
\(759\) 4706.15 0.225062
\(760\) −24657.0 −1.17685
\(761\) 216.324 0.0103045 0.00515226 0.999987i \(-0.498360\pi\)
0.00515226 + 0.999987i \(0.498360\pi\)
\(762\) 1095.27 0.0520699
\(763\) −2767.24 −0.131299
\(764\) −26344.0 −1.24750
\(765\) 7496.10 0.354277
\(766\) −11467.5 −0.540912
\(767\) 0 0
\(768\) 6726.80 0.316058
\(769\) −19214.4 −0.901025 −0.450512 0.892770i \(-0.648759\pi\)
−0.450512 + 0.892770i \(0.648759\pi\)
\(770\) −2082.47 −0.0974638
\(771\) −2654.29 −0.123984
\(772\) −1429.01 −0.0666209
\(773\) −33175.6 −1.54365 −0.771826 0.635834i \(-0.780656\pi\)
−0.771826 + 0.635834i \(0.780656\pi\)
\(774\) 5004.25 0.232395
\(775\) 13146.2 0.609325
\(776\) 24172.8 1.11824
\(777\) −10368.0 −0.478700
\(778\) 1693.54 0.0780416
\(779\) 17408.4 0.800667
\(780\) 0 0
\(781\) −4428.17 −0.202884
\(782\) 8806.72 0.402721
\(783\) −3803.03 −0.173575
\(784\) −6936.21 −0.315972
\(785\) −15886.3 −0.722302
\(786\) −4715.17 −0.213975
\(787\) 14887.9 0.674330 0.337165 0.941446i \(-0.390532\pi\)
0.337165 + 0.941446i \(0.390532\pi\)
\(788\) −1424.30 −0.0643889
\(789\) 25031.8 1.12948
\(790\) −6871.73 −0.309475
\(791\) 5254.56 0.236196
\(792\) −2172.09 −0.0974518
\(793\) 0 0
\(794\) 6136.58 0.274281
\(795\) −31213.3 −1.39248
\(796\) −18332.8 −0.816319
\(797\) 12954.5 0.575751 0.287876 0.957668i \(-0.407051\pi\)
0.287876 + 0.957668i \(0.407051\pi\)
\(798\) 2682.92 0.119015
\(799\) 26236.5 1.16168
\(800\) −20790.0 −0.918796
\(801\) 2130.04 0.0939591
\(802\) 183.295 0.00807028
\(803\) −10635.4 −0.467389
\(804\) −262.667 −0.0115218
\(805\) −15092.8 −0.660810
\(806\) 0 0
\(807\) −8287.68 −0.361512
\(808\) 11950.2 0.520305
\(809\) 8275.59 0.359647 0.179823 0.983699i \(-0.442447\pi\)
0.179823 + 0.983699i \(0.442447\pi\)
\(810\) 1658.51 0.0719435
\(811\) −26327.1 −1.13991 −0.569956 0.821675i \(-0.693040\pi\)
−0.569956 + 0.821675i \(0.693040\pi\)
\(812\) −6998.09 −0.302444
\(813\) −9349.63 −0.403328
\(814\) 7354.98 0.316698
\(815\) −44466.3 −1.91115
\(816\) 4019.43 0.172437
\(817\) −35426.3 −1.51703
\(818\) 1599.16 0.0683538
\(819\) 0 0
\(820\) 19802.5 0.843333
\(821\) 34439.0 1.46398 0.731992 0.681314i \(-0.238591\pi\)
0.731992 + 0.681314i \(0.238591\pi\)
\(822\) −10184.7 −0.432157
\(823\) −13870.5 −0.587479 −0.293739 0.955886i \(-0.594900\pi\)
−0.293739 + 0.955886i \(0.594900\pi\)
\(824\) 28469.3 1.20361
\(825\) 4324.96 0.182516
\(826\) 1969.07 0.0829453
\(827\) −2132.30 −0.0896583 −0.0448292 0.998995i \(-0.514274\pi\)
−0.0448292 + 0.998995i \(0.514274\pi\)
\(828\) −6896.92 −0.289474
\(829\) −6212.39 −0.260272 −0.130136 0.991496i \(-0.541541\pi\)
−0.130136 + 0.991496i \(0.541541\pi\)
\(830\) −11646.4 −0.487051
\(831\) 1506.18 0.0628746
\(832\) 0 0
\(833\) 15096.2 0.627913
\(834\) 2173.40 0.0902380
\(835\) 17797.8 0.737628
\(836\) 6736.76 0.278703
\(837\) −3143.87 −0.129830
\(838\) −6902.81 −0.284551
\(839\) 4550.52 0.187248 0.0936242 0.995608i \(-0.470155\pi\)
0.0936242 + 0.995608i \(0.470155\pi\)
\(840\) 6965.99 0.286130
\(841\) −4549.47 −0.186538
\(842\) 18766.2 0.768085
\(843\) 19822.7 0.809880
\(844\) 10024.2 0.408822
\(845\) 0 0
\(846\) 5804.83 0.235903
\(847\) −9302.74 −0.377386
\(848\) −16736.7 −0.677759
\(849\) 13106.9 0.529834
\(850\) 8093.38 0.326589
\(851\) 53305.6 2.14723
\(852\) 6489.53 0.260948
\(853\) −12262.8 −0.492228 −0.246114 0.969241i \(-0.579154\pi\)
−0.246114 + 0.969241i \(0.579154\pi\)
\(854\) 7104.40 0.284669
\(855\) −11741.0 −0.469631
\(856\) −23976.5 −0.957362
\(857\) −34949.1 −1.39304 −0.696521 0.717536i \(-0.745270\pi\)
−0.696521 + 0.717536i \(0.745270\pi\)
\(858\) 0 0
\(859\) −21762.1 −0.864394 −0.432197 0.901779i \(-0.642261\pi\)
−0.432197 + 0.901779i \(0.642261\pi\)
\(860\) −40298.4 −1.59786
\(861\) −4918.14 −0.194669
\(862\) 3026.48 0.119585
\(863\) 19811.5 0.781450 0.390725 0.920507i \(-0.372224\pi\)
0.390725 + 0.920507i \(0.372224\pi\)
\(864\) 4971.83 0.195770
\(865\) 26049.0 1.02392
\(866\) 17654.0 0.692735
\(867\) 5991.00 0.234677
\(868\) −5785.14 −0.226222
\(869\) 4285.40 0.167287
\(870\) −8652.09 −0.337165
\(871\) 0 0
\(872\) −6566.54 −0.255013
\(873\) 11510.5 0.446243
\(874\) −13793.8 −0.533848
\(875\) 1486.28 0.0574235
\(876\) 15586.2 0.601153
\(877\) 25716.8 0.990186 0.495093 0.868840i \(-0.335134\pi\)
0.495093 + 0.868840i \(0.335134\pi\)
\(878\) 13875.4 0.533339
\(879\) 16044.4 0.615658
\(880\) 4886.61 0.187190
\(881\) 34709.6 1.32735 0.663676 0.748020i \(-0.268995\pi\)
0.663676 + 0.748020i \(0.268995\pi\)
\(882\) 3340.03 0.127511
\(883\) −3848.68 −0.146680 −0.0733400 0.997307i \(-0.523366\pi\)
−0.0733400 + 0.997307i \(0.523366\pi\)
\(884\) 0 0
\(885\) −8617.09 −0.327300
\(886\) −7120.06 −0.269981
\(887\) 32804.8 1.24180 0.620900 0.783890i \(-0.286767\pi\)
0.620900 + 0.783890i \(0.286767\pi\)
\(888\) −24602.8 −0.929748
\(889\) 2190.53 0.0826412
\(890\) 4845.95 0.182513
\(891\) −1034.29 −0.0388891
\(892\) 816.311 0.0306414
\(893\) −41093.8 −1.53992
\(894\) −5483.03 −0.205123
\(895\) 14541.7 0.543101
\(896\) 11247.6 0.419371
\(897\) 0 0
\(898\) −12852.5 −0.477610
\(899\) 16400.8 0.608452
\(900\) −6338.27 −0.234751
\(901\) 36426.2 1.34687
\(902\) 3488.90 0.128789
\(903\) 10008.5 0.368840
\(904\) 12468.8 0.458748
\(905\) −7434.86 −0.273086
\(906\) −10886.1 −0.399189
\(907\) −22262.2 −0.814999 −0.407500 0.913205i \(-0.633599\pi\)
−0.407500 + 0.913205i \(0.633599\pi\)
\(908\) 26979.4 0.986060
\(909\) 5690.38 0.207633
\(910\) 0 0
\(911\) −13515.3 −0.491528 −0.245764 0.969330i \(-0.579039\pi\)
−0.245764 + 0.969330i \(0.579039\pi\)
\(912\) −6295.58 −0.228583
\(913\) 7263.01 0.263275
\(914\) 4696.62 0.169968
\(915\) −31090.4 −1.12330
\(916\) −16352.7 −0.589855
\(917\) −9430.33 −0.339604
\(918\) −1935.50 −0.0695871
\(919\) 26600.6 0.954811 0.477405 0.878683i \(-0.341577\pi\)
0.477405 + 0.878683i \(0.341577\pi\)
\(920\) −35814.6 −1.28345
\(921\) 13507.4 0.483260
\(922\) 20012.0 0.714816
\(923\) 0 0
\(924\) −1903.24 −0.0677620
\(925\) 48987.9 1.74131
\(926\) 14695.5 0.521515
\(927\) 13556.4 0.480313
\(928\) −25936.9 −0.917480
\(929\) 32887.7 1.16148 0.580738 0.814090i \(-0.302764\pi\)
0.580738 + 0.814090i \(0.302764\pi\)
\(930\) −7152.46 −0.252192
\(931\) −23644.9 −0.832363
\(932\) 28962.6 1.01792
\(933\) 22341.6 0.783956
\(934\) −17593.1 −0.616341
\(935\) −10635.4 −0.371993
\(936\) 0 0
\(937\) −9261.78 −0.322913 −0.161456 0.986880i \(-0.551619\pi\)
−0.161456 + 0.986880i \(0.551619\pi\)
\(938\) 148.415 0.00516623
\(939\) 19526.8 0.678629
\(940\) −46745.3 −1.62198
\(941\) −12054.9 −0.417619 −0.208810 0.977956i \(-0.566959\pi\)
−0.208810 + 0.977956i \(0.566959\pi\)
\(942\) 4101.86 0.141875
\(943\) 25285.9 0.873195
\(944\) −4620.51 −0.159306
\(945\) 3317.03 0.114183
\(946\) −7099.96 −0.244016
\(947\) −20221.4 −0.693885 −0.346942 0.937886i \(-0.612780\pi\)
−0.346942 + 0.937886i \(0.612780\pi\)
\(948\) −6280.30 −0.215163
\(949\) 0 0
\(950\) −12676.5 −0.432928
\(951\) −7396.72 −0.252213
\(952\) −8129.36 −0.276758
\(953\) −20331.5 −0.691083 −0.345542 0.938403i \(-0.612305\pi\)
−0.345542 + 0.938403i \(0.612305\pi\)
\(954\) 8059.30 0.273511
\(955\) −65140.8 −2.20723
\(956\) 41772.2 1.41319
\(957\) 5395.68 0.182254
\(958\) −16250.4 −0.548045
\(959\) −20369.4 −0.685885
\(960\) 2126.60 0.0714955
\(961\) −16232.9 −0.544891
\(962\) 0 0
\(963\) −11417.0 −0.382044
\(964\) 14356.0 0.479641
\(965\) −3533.53 −0.117874
\(966\) 3896.98 0.129796
\(967\) 11082.2 0.368541 0.184270 0.982876i \(-0.441008\pi\)
0.184270 + 0.982876i \(0.441008\pi\)
\(968\) −22075.0 −0.732973
\(969\) 13701.9 0.454249
\(970\) 26186.9 0.866816
\(971\) −36694.1 −1.21274 −0.606369 0.795184i \(-0.707374\pi\)
−0.606369 + 0.795184i \(0.707374\pi\)
\(972\) 1515.77 0.0500189
\(973\) 4346.79 0.143219
\(974\) 17806.2 0.585778
\(975\) 0 0
\(976\) −16670.8 −0.546740
\(977\) −17155.3 −0.561767 −0.280883 0.959742i \(-0.590627\pi\)
−0.280883 + 0.959742i \(0.590627\pi\)
\(978\) 11481.2 0.375387
\(979\) −3022.07 −0.0986575
\(980\) −26896.7 −0.876718
\(981\) −3126.82 −0.101765
\(982\) 979.544 0.0318315
\(983\) 38419.7 1.24659 0.623296 0.781986i \(-0.285793\pi\)
0.623296 + 0.781986i \(0.285793\pi\)
\(984\) −11670.5 −0.378093
\(985\) −3521.86 −0.113925
\(986\) 10097.1 0.326121
\(987\) 11609.7 0.374407
\(988\) 0 0
\(989\) −51457.3 −1.65445
\(990\) −2353.07 −0.0755410
\(991\) 51728.9 1.65815 0.829073 0.559140i \(-0.188869\pi\)
0.829073 + 0.559140i \(0.188869\pi\)
\(992\) −21441.4 −0.686255
\(993\) 12344.5 0.394503
\(994\) −3666.79 −0.117006
\(995\) −45331.6 −1.44433
\(996\) −10644.0 −0.338623
\(997\) −26846.8 −0.852805 −0.426403 0.904533i \(-0.640219\pi\)
−0.426403 + 0.904533i \(0.640219\pi\)
\(998\) −2476.65 −0.0785541
\(999\) −11715.2 −0.371025
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 507.4.a.j.1.2 4
3.2 odd 2 1521.4.a.x.1.3 4
13.5 odd 4 39.4.b.a.25.3 yes 4
13.8 odd 4 39.4.b.a.25.2 4
13.12 even 2 inner 507.4.a.j.1.3 4
39.5 even 4 117.4.b.d.64.2 4
39.8 even 4 117.4.b.d.64.3 4
39.38 odd 2 1521.4.a.x.1.2 4
52.31 even 4 624.4.c.e.337.4 4
52.47 even 4 624.4.c.e.337.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.b.a.25.2 4 13.8 odd 4
39.4.b.a.25.3 yes 4 13.5 odd 4
117.4.b.d.64.2 4 39.5 even 4
117.4.b.d.64.3 4 39.8 even 4
507.4.a.j.1.2 4 1.1 even 1 trivial
507.4.a.j.1.3 4 13.12 even 2 inner
624.4.c.e.337.1 4 52.47 even 4
624.4.c.e.337.4 4 52.31 even 4
1521.4.a.x.1.2 4 39.38 odd 2
1521.4.a.x.1.3 4 3.2 odd 2