Properties

Label 507.4.a.j.1.3
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 1.3
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 \(-0.424599\pi\)
0.234671 + 0.972075i \(0.424599\pi\)
\(3\) −3.00000 −0.577350
\(4\) −6.23774 −0.779717
\(5\) 15.4241 1.37957 0.689785 0.724014i \(-0.257705\pi\)
0.689785 + 0.724014i \(0.257705\pi\)
\(6\) −3.98251 −0.270975
\(7\) −7.96501 −0.430070 −0.215035 0.976606i \(-0.568987\pi\)
−0.215035 + 0.976606i \(0.568987\pi\)
\(8\) −18.9006 −0.835297
\(9\) 9.00000 0.333333
\(10\) 20.4755 0.647491
\(11\) 12.7691 0.350002 0.175001 0.984568i \(-0.444007\pi\)
0.175001 + 0.984568i \(0.444007\pi\)
\(12\) 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 \(-0.670587\pi\)
−0.510628 + 0.859802i \(0.670587\pi\)
\(20\) −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 \(-0.609517\pi\)
−0.337309 + 0.941394i \(0.609517\pi\)
\(32\) 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 \(-0.914271\pi\)
−0.963951 + 0.266081i \(0.914271\pi\)
\(38\) −112.279 −0.479319
\(39\) 0 0
\(40\) −291.525 −1.15235
\(41\) −205.823 −0.784003 −0.392002 0.919965i \(-0.628217\pi\)
−0.392002 + 0.919965i \(0.628217\pi\)
\(42\) 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 \(-0.228152\pi\)
0.753937 + 0.656947i \(0.228152\pi\)
\(48\) −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 \(-0.434130\pi\)
0.205462 + 0.978665i \(0.434130\pi\)
\(60\) 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 \(-0.495926\pi\)
0.0127972 + 0.999918i \(0.495926\pi\)
\(68\) 336.838 0.600700
\(69\) −368.559 −0.643033
\(70\) −163.087 −0.278467
\(71\) −346.789 −0.579665 −0.289833 0.957077i \(-0.593600\pi\)
−0.289833 + 0.957077i \(0.593600\pi\)
\(72\) −170.106 −0.278432
\(73\) −832.900 −1.33539 −0.667695 0.744435i \(-0.732719\pi\)
−0.667695 + 0.744435i \(0.732719\pi\)
\(74\) −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 \(-0.377263\pi\)
0.376106 + 0.926577i \(0.377263\pi\)
\(84\) −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 \(-0.545012\pi\)
−0.140939 + 0.990018i \(0.545012\pi\)
\(90\) 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 \(-0.733434\pi\)
−0.669365 + 0.742934i \(0.733434\pi\)
\(98\) −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 \(-0.451220\pi\)
0.152648 + 0.988281i \(0.451220\pi\)
\(110\) 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 \(-0.206203\pi\)
0.797410 + 0.603438i \(0.206203\pi\)
\(138\) −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 \(-0.376443\pi\)
0.378491 + 0.925605i \(0.376443\pi\)
\(150\) −449.632 −0.244749
\(151\) 2733.47 1.47316 0.736579 0.676352i \(-0.236440\pi\)
0.736579 + 0.676352i \(0.236440\pi\)
\(152\) 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 \(-0.743561\pi\)
−0.692660 + 0.721264i \(0.743561\pi\)
\(164\) 1283.87 0.611301
\(165\) −590.853 −0.278775
\(166\) 755.079 0.353045
\(167\) 1153.90 0.534679 0.267340 0.963602i \(-0.413855\pi\)
0.267340 + 0.963602i \(0.413855\pi\)
\(168\) −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 \(-0.513603\pi\)
−0.0427212 + 0.999087i \(0.513603\pi\)
\(194\) −1697.80 −0.628323
\(195\) 0 0
\(196\) 1743.81 0.635501
\(197\) −228.335 −0.0825798 −0.0412899 0.999147i \(-0.513147\pi\)
−0.0412899 + 0.999147i \(0.513147\pi\)
\(198\) 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 \(-0.493745\pi\)
0.0196490 + 0.999807i \(0.493745\pi\)
\(224\) −1466.69 −0.437489
\(225\) 1016.12 0.301072
\(226\) 875.761 0.257764
\(227\) 4325.19 1.26464 0.632319 0.774708i \(-0.282103\pi\)
0.632319 + 0.774708i \(0.282103\pi\)
\(228\) −1582.75 −0.459738
\(229\) −2621.57 −0.756499 −0.378250 0.925704i \(-0.623474\pi\)
−0.378250 + 0.925704i \(0.623474\pi\)
\(230\) 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 \(-0.138956\pi\)
0.906219 + 0.422809i \(0.138956\pi\)
\(240\) −1148.07 −0.308783
\(241\) 2301.47 0.615148 0.307574 0.951524i \(-0.400483\pi\)
0.307574 + 0.951524i \(0.400483\pi\)
\(242\) −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 \(-0.613578\pi\)
−0.349293 + 0.937014i \(0.613578\pi\)
\(272\) −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 \(-0.252569\pi\)
0.701377 + 0.712791i \(0.252569\pi\)
\(282\) −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 \(-0.320999\pi\)
0.533175 + 0.846005i \(0.320999\pi\)
\(294\) 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 \(-0.362550\pi\)
0.418516 + 0.908210i \(0.362550\pi\)
\(308\) 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 \(-0.570091\pi\)
−0.218423 + 0.975854i \(0.570091\pi\)
\(318\) 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 \(-0.389014\pi\)
0.341650 + 0.939827i \(0.389014\pi\)
\(332\) −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 \(-0.490924\pi\)
0.0285083 + 0.999594i \(0.490924\pi\)
\(350\) −1193.78 −0.182314
\(351\) 0 0
\(352\) 2351.32 0.356039
\(353\) −7539.10 −1.13673 −0.568365 0.822776i \(-0.692424\pi\)
−0.568365 + 0.822776i \(0.692424\pi\)
\(354\) −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 \(-0.886033\pi\)
−0.936586 + 0.350437i \(0.886033\pi\)
\(360\) −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 \(-0.586974\pi\)
−0.269850 + 0.962902i \(0.586974\pi\)
\(380\) 8137.50 1.09854
\(381\) −825.057 −0.110942
\(382\) 5606.47 0.750921
\(383\) −8638.43 −1.15249 −0.576244 0.817278i \(-0.695482\pi\)
−0.576244 + 0.817278i \(0.695482\pi\)
\(384\) 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 \(-0.405614\pi\)
0.292197 + 0.956358i \(0.405614\pi\)
\(398\) 3901.55 0.491375
\(399\) −2021.03 −0.253579
\(400\) 2801.24 0.350155
\(401\) 138.075 0.0171949 0.00859743 0.999963i \(-0.497263\pi\)
0.00859743 + 0.999963i \(0.497263\pi\)
\(402\) −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 \(-0.476801\pi\)
0.0728186 + 0.997345i \(0.476801\pi\)
\(410\) −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 \(-0.194941\pi\)
0.818256 + 0.574854i \(0.194941\pi\)
\(422\) −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 \(-0.459338\pi\)
0.127396 + 0.991852i \(0.459338\pi\)
\(432\) −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 \(-0.669914\pi\)
−0.508808 + 0.860880i \(0.669914\pi\)
\(450\) 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 \(-0.442044\pi\)
0.181070 + 0.983470i \(0.442044\pi\)
\(458\) −3480.14 −0.355057
\(459\) 1458.00 0.148265
\(460\) −11819.8 −1.19805
\(461\) 15074.9 1.52302 0.761508 0.648156i \(-0.224459\pi\)
0.761508 + 0.648156i \(0.224459\pi\)
\(462\) 405.044 0.0407886
\(463\) 11070.0 1.11116 0.555580 0.831463i \(-0.312496\pi\)
0.555580 + 0.831463i \(0.312496\pi\)
\(464\) 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 \(-0.698452\pi\)
−0.583843 + 0.811866i \(0.698452\pi\)
\(480\) −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 \(-0.285490\pi\)
0.624041 + 0.781392i \(0.285490\pi\)
\(488\) 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 \(-0.526669\pi\)
−0.0836852 + 0.996492i \(0.526669\pi\)
\(500\) 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 \(-0.557537\pi\)
−0.179775 + 0.983708i \(0.557537\pi\)
\(510\) 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 \(-0.687799\pi\)
−0.556351 + 0.830948i \(0.687799\pi\)
\(542\) −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 \(-0.352764\pi\)
0.446236 + 0.894915i \(0.352764\pi\)
\(558\) −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 \(-0.590154\pi\)
−0.279455 + 0.960159i \(0.590154\pi\)
\(578\) −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 \(-0.468991\pi\)
0.0972635 + 0.995259i \(0.468991\pi\)
\(588\) −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 \(-0.484114\pi\)
0.0498871 + 0.998755i \(0.484114\pi\)
\(594\) −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 \(-0.353407\pi\)
0.444428 + 0.895814i \(0.353407\pi\)
\(614\) 5977.02 0.392855
\(615\) 9523.88 0.624455
\(616\) 1922.30 0.125733
\(617\) −7470.76 −0.487458 −0.243729 0.969843i \(-0.578371\pi\)
−0.243729 + 0.969843i \(0.578371\pi\)
\(618\) −5998.71 −0.390458
\(619\) −24806.9 −1.61078 −0.805389 0.592746i \(-0.798044\pi\)
−0.805389 + 0.592746i \(0.798044\pi\)
\(620\) 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 \(-0.503156\pi\)
−0.00991554 + 0.999951i \(0.503156\pi\)
\(632\) 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 \(-0.553741\pi\)
−0.168032 + 0.985781i \(0.553741\pi\)
\(644\) 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 \(-0.270502\pi\)
0.660127 + 0.751154i \(0.270502\pi\)
\(662\) 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 \(-0.387391\pi\)
0.346437 + 0.938073i \(0.387391\pi\)
\(684\) 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 \(-0.509210\pi\)
−0.0289302 + 0.999581i \(0.509210\pi\)
\(692\) 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 \(-0.240712\pi\)
0.727436 + 0.686176i \(0.240712\pi\)
\(710\) −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 \(-0.328945\pi\)
0.511893 + 0.859049i \(0.328945\pi\)
\(734\) 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 \(-0.767532\pi\)
−0.744962 + 0.667107i \(0.767532\pi\)
\(740\) 41745.9 2.07380
\(741\) 0 0
\(742\) 7132.49 0.352887
\(743\) 24376.4 1.20361 0.601805 0.798643i \(-0.294448\pi\)
0.601805 + 0.798643i \(0.294448\pi\)
\(744\) −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 \(-0.501640\pi\)
−0.00515226 + 0.999987i \(0.501640\pi\)
\(762\) −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 \(-0.351241\pi\)
0.450512 + 0.892770i \(0.351241\pi\)
\(770\) −2082.47 −0.0974638
\(771\) −2654.29 −0.123984
\(772\) 1429.01 0.0666209
\(773\) 33175.6 1.54365 0.771826 0.635834i \(-0.219344\pi\)
0.771826 + 0.635834i \(0.219344\pi\)
\(774\) −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 \(-0.609468\pi\)
−0.337165 + 0.941446i \(0.609468\pi\)
\(788\) 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 \(-0.306960\pi\)
0.569956 + 0.821675i \(0.306960\pi\)
\(812\) 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 \(-0.761409\pi\)
−0.731992 + 0.681314i \(0.761409\pi\)
\(822\) −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 \(-0.485726\pi\)
0.0448292 + 0.998995i \(0.485726\pi\)
\(828\) −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 \(-0.529845\pi\)
−0.0936242 + 0.995608i \(0.529845\pi\)
\(840\) −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 \(-0.420846\pi\)
0.246114 + 0.969241i \(0.420846\pi\)
\(854\) 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 \(-0.627776\pi\)
−0.390725 + 0.920507i \(0.627776\pi\)
\(864\) −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 \(-0.664866\pi\)
−0.495093 + 0.868840i \(0.664866\pi\)
\(878\) −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 \(-0.697236\pi\)
−0.580738 + 0.814090i \(0.697236\pi\)
\(930\) 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 \(-0.433041\pi\)
0.208810 + 0.977956i \(0.433041\pi\)
\(942\) −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 \(-0.387220\pi\)
0.346942 + 0.937886i \(0.387220\pi\)
\(948\) −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 \(-0.558992\pi\)
−0.184270 + 0.982876i \(0.558992\pi\)
\(968\) 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 \(-0.409373\pi\)
0.280883 + 0.959742i \(0.409373\pi\)
\(978\) 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 \(-0.714207\pi\)
−0.623296 + 0.781986i \(0.714207\pi\)
\(984\) −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.1.3 4
3.2 odd 2 1521.4.a.x.1.2 4
13.5 odd 4 39.4.b.a.25.2 4
13.8 odd 4 39.4.b.a.25.3 yes 4
13.12 even 2 inner 507.4.a.j.1.2 4
39.5 even 4 117.4.b.d.64.3 4
39.8 even 4 117.4.b.d.64.2 4
39.38 odd 2 1521.4.a.x.1.3 4
52.31 even 4 624.4.c.e.337.1 4
52.47 even 4 624.4.c.e.337.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.b.a.25.2 4 13.5 odd 4
39.4.b.a.25.3 yes 4 13.8 odd 4
117.4.b.d.64.2 4 39.8 even 4
117.4.b.d.64.3 4 39.5 even 4
507.4.a.j.1.2 4 13.12 even 2 inner
507.4.a.j.1.3 4 1.1 even 1 trivial
624.4.c.e.337.1 4 52.31 even 4
624.4.c.e.337.4 4 52.47 even 4
1521.4.a.x.1.2 4 3.2 odd 2
1521.4.a.x.1.3 4 39.38 odd 2