Properties

Label 539.4.a.e.1.2
Level $539$
Weight $4$
Character 539.1
Self dual yes
Analytic conductor $31.802$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 539 = 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 539.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(31.8020294931\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 11)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 539.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.73205 q^{2} +7.92820 q^{3} -0.535898 q^{4} -14.8564 q^{5} +21.6603 q^{6} -23.3205 q^{8} +35.8564 q^{9} +O(q^{10})\) \(q+2.73205 q^{2} +7.92820 q^{3} -0.535898 q^{4} -14.8564 q^{5} +21.6603 q^{6} -23.3205 q^{8} +35.8564 q^{9} -40.5885 q^{10} -11.0000 q^{11} -4.24871 q^{12} -5.35898 q^{13} -117.785 q^{15} -59.4256 q^{16} +41.2154 q^{17} +97.9615 q^{18} -139.923 q^{19} +7.96152 q^{20} -30.0526 q^{22} -111.354 q^{23} -184.890 q^{24} +95.7128 q^{25} -14.6410 q^{26} +70.2154 q^{27} -24.9948 q^{29} -321.794 q^{30} -31.4974 q^{31} +24.2102 q^{32} -87.2102 q^{33} +112.603 q^{34} -19.2154 q^{36} +13.1436 q^{37} -382.277 q^{38} -42.4871 q^{39} +346.459 q^{40} -261.072 q^{41} -57.7128 q^{43} +5.89488 q^{44} -532.697 q^{45} -304.224 q^{46} +343.846 q^{47} -471.138 q^{48} +261.492 q^{50} +326.764 q^{51} +2.87187 q^{52} -342.995 q^{53} +191.832 q^{54} +163.420 q^{55} -1109.34 q^{57} -68.2872 q^{58} -88.3693 q^{59} +63.1206 q^{60} -738.697 q^{61} -86.0526 q^{62} +541.549 q^{64} +79.6152 q^{65} -238.263 q^{66} +342.359 q^{67} -22.0873 q^{68} -882.836 q^{69} -207.364 q^{71} -836.190 q^{72} +1010.60 q^{73} +35.9090 q^{74} +758.831 q^{75} +74.9845 q^{76} -116.077 q^{78} +1294.23 q^{79} +882.851 q^{80} -411.441 q^{81} -713.261 q^{82} -441.846 q^{83} -612.313 q^{85} -157.674 q^{86} -198.164 q^{87} +256.526 q^{88} +1489.11 q^{89} -1455.36 q^{90} +59.6743 q^{92} -249.718 q^{93} +939.405 q^{94} +2078.75 q^{95} +191.944 q^{96} -1346.42 q^{97} -394.420 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{3} - 8 q^{4} - 2 q^{5} + 26 q^{6} - 12 q^{8} + 44 q^{9} + O(q^{10}) \) \( 2 q + 2 q^{2} + 2 q^{3} - 8 q^{4} - 2 q^{5} + 26 q^{6} - 12 q^{8} + 44 q^{9} - 50 q^{10} - 22 q^{11} + 40 q^{12} - 80 q^{13} - 194 q^{15} - 8 q^{16} + 124 q^{17} + 92 q^{18} - 72 q^{19} - 88 q^{20} - 22 q^{22} - 98 q^{23} - 252 q^{24} + 136 q^{25} + 40 q^{26} + 182 q^{27} + 144 q^{29} - 266 q^{30} + 34 q^{31} - 104 q^{32} - 22 q^{33} + 52 q^{34} - 80 q^{36} + 54 q^{37} - 432 q^{38} + 400 q^{39} + 492 q^{40} - 536 q^{41} - 60 q^{43} + 88 q^{44} - 428 q^{45} - 314 q^{46} + 272 q^{47} - 776 q^{48} + 232 q^{50} - 164 q^{51} + 560 q^{52} - 492 q^{53} + 110 q^{54} + 22 q^{55} - 1512 q^{57} - 192 q^{58} - 634 q^{59} + 632 q^{60} - 840 q^{61} - 134 q^{62} + 224 q^{64} - 880 q^{65} - 286 q^{66} + 754 q^{67} - 640 q^{68} - 962 q^{69} - 678 q^{71} - 744 q^{72} + 400 q^{73} + 6 q^{74} + 520 q^{75} - 432 q^{76} - 440 q^{78} + 316 q^{79} + 1544 q^{80} - 1294 q^{81} - 512 q^{82} - 468 q^{83} + 452 q^{85} - 156 q^{86} - 1200 q^{87} + 132 q^{88} + 1842 q^{89} - 1532 q^{90} - 40 q^{92} - 638 q^{93} + 992 q^{94} + 2952 q^{95} + 952 q^{96} - 2194 q^{97} - 484 q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.73205 0.965926 0.482963 0.875641i \(-0.339561\pi\)
0.482963 + 0.875641i \(0.339561\pi\)
\(3\) 7.92820 1.52578 0.762892 0.646526i \(-0.223779\pi\)
0.762892 + 0.646526i \(0.223779\pi\)
\(4\) −0.535898 −0.0669873
\(5\) −14.8564 −1.32880 −0.664399 0.747378i \(-0.731312\pi\)
−0.664399 + 0.747378i \(0.731312\pi\)
\(6\) 21.6603 1.47379
\(7\) 0 0
\(8\) −23.3205 −1.03063
\(9\) 35.8564 1.32802
\(10\) −40.5885 −1.28352
\(11\) −11.0000 −0.301511
\(12\) −4.24871 −0.102208
\(13\) −5.35898 −0.114332 −0.0571659 0.998365i \(-0.518206\pi\)
−0.0571659 + 0.998365i \(0.518206\pi\)
\(14\) 0 0
\(15\) −117.785 −2.02746
\(16\) −59.4256 −0.928525
\(17\) 41.2154 0.588012 0.294006 0.955804i \(-0.405011\pi\)
0.294006 + 0.955804i \(0.405011\pi\)
\(18\) 97.9615 1.28276
\(19\) −139.923 −1.68950 −0.844751 0.535159i \(-0.820252\pi\)
−0.844751 + 0.535159i \(0.820252\pi\)
\(20\) 7.96152 0.0890125
\(21\) 0 0
\(22\) −30.0526 −0.291238
\(23\) −111.354 −1.00952 −0.504758 0.863261i \(-0.668418\pi\)
−0.504758 + 0.863261i \(0.668418\pi\)
\(24\) −184.890 −1.57252
\(25\) 95.7128 0.765703
\(26\) −14.6410 −0.110436
\(27\) 70.2154 0.500480
\(28\) 0 0
\(29\) −24.9948 −0.160049 −0.0800246 0.996793i \(-0.525500\pi\)
−0.0800246 + 0.996793i \(0.525500\pi\)
\(30\) −321.794 −1.95837
\(31\) −31.4974 −0.182487 −0.0912436 0.995829i \(-0.529084\pi\)
−0.0912436 + 0.995829i \(0.529084\pi\)
\(32\) 24.2102 0.133744
\(33\) −87.2102 −0.460041
\(34\) 112.603 0.567976
\(35\) 0 0
\(36\) −19.2154 −0.0889601
\(37\) 13.1436 0.0583998 0.0291999 0.999574i \(-0.490704\pi\)
0.0291999 + 0.999574i \(0.490704\pi\)
\(38\) −382.277 −1.63193
\(39\) −42.4871 −0.174446
\(40\) 346.459 1.36950
\(41\) −261.072 −0.994453 −0.497226 0.867621i \(-0.665648\pi\)
−0.497226 + 0.867621i \(0.665648\pi\)
\(42\) 0 0
\(43\) −57.7128 −0.204677 −0.102339 0.994750i \(-0.532633\pi\)
−0.102339 + 0.994750i \(0.532633\pi\)
\(44\) 5.89488 0.0201974
\(45\) −532.697 −1.76466
\(46\) −304.224 −0.975118
\(47\) 343.846 1.06713 0.533565 0.845759i \(-0.320852\pi\)
0.533565 + 0.845759i \(0.320852\pi\)
\(48\) −471.138 −1.41673
\(49\) 0 0
\(50\) 261.492 0.739612
\(51\) 326.764 0.897179
\(52\) 2.87187 0.00765879
\(53\) −342.995 −0.888943 −0.444471 0.895793i \(-0.646608\pi\)
−0.444471 + 0.895793i \(0.646608\pi\)
\(54\) 191.832 0.483426
\(55\) 163.420 0.400647
\(56\) 0 0
\(57\) −1109.34 −2.57782
\(58\) −68.2872 −0.154596
\(59\) −88.3693 −0.194995 −0.0974975 0.995236i \(-0.531084\pi\)
−0.0974975 + 0.995236i \(0.531084\pi\)
\(60\) 63.1206 0.135814
\(61\) −738.697 −1.55050 −0.775250 0.631654i \(-0.782376\pi\)
−0.775250 + 0.631654i \(0.782376\pi\)
\(62\) −86.0526 −0.176269
\(63\) 0 0
\(64\) 541.549 1.05771
\(65\) 79.6152 0.151924
\(66\) −238.263 −0.444365
\(67\) 342.359 0.624266 0.312133 0.950038i \(-0.398957\pi\)
0.312133 + 0.950038i \(0.398957\pi\)
\(68\) −22.0873 −0.0393893
\(69\) −882.836 −1.54030
\(70\) 0 0
\(71\) −207.364 −0.346614 −0.173307 0.984868i \(-0.555445\pi\)
−0.173307 + 0.984868i \(0.555445\pi\)
\(72\) −836.190 −1.36869
\(73\) 1010.60 1.62030 0.810149 0.586224i \(-0.199386\pi\)
0.810149 + 0.586224i \(0.199386\pi\)
\(74\) 35.9090 0.0564099
\(75\) 758.831 1.16830
\(76\) 74.9845 0.113175
\(77\) 0 0
\(78\) −116.077 −0.168502
\(79\) 1294.23 1.84319 0.921593 0.388157i \(-0.126888\pi\)
0.921593 + 0.388157i \(0.126888\pi\)
\(80\) 882.851 1.23382
\(81\) −411.441 −0.564391
\(82\) −713.261 −0.960568
\(83\) −441.846 −0.584324 −0.292162 0.956369i \(-0.594375\pi\)
−0.292162 + 0.956369i \(0.594375\pi\)
\(84\) 0 0
\(85\) −612.313 −0.781349
\(86\) −157.674 −0.197703
\(87\) −198.164 −0.244200
\(88\) 256.526 0.310747
\(89\) 1489.11 1.77355 0.886773 0.462205i \(-0.152942\pi\)
0.886773 + 0.462205i \(0.152942\pi\)
\(90\) −1455.36 −1.70453
\(91\) 0 0
\(92\) 59.6743 0.0676248
\(93\) −249.718 −0.278436
\(94\) 939.405 1.03077
\(95\) 2078.75 2.24501
\(96\) 191.944 0.204064
\(97\) −1346.42 −1.40936 −0.704679 0.709526i \(-0.748909\pi\)
−0.704679 + 0.709526i \(0.748909\pi\)
\(98\) 0 0
\(99\) −394.420 −0.400412
\(100\) −51.2923 −0.0512923
\(101\) 161.461 0.159069 0.0795347 0.996832i \(-0.474657\pi\)
0.0795347 + 0.996832i \(0.474657\pi\)
\(102\) 892.736 0.866608
\(103\) 34.7592 0.0332517 0.0166259 0.999862i \(-0.494708\pi\)
0.0166259 + 0.999862i \(0.494708\pi\)
\(104\) 124.974 0.117834
\(105\) 0 0
\(106\) −937.079 −0.858653
\(107\) 832.179 0.751867 0.375934 0.926647i \(-0.377322\pi\)
0.375934 + 0.926647i \(0.377322\pi\)
\(108\) −37.6283 −0.0335258
\(109\) 1044.26 0.917629 0.458815 0.888532i \(-0.348274\pi\)
0.458815 + 0.888532i \(0.348274\pi\)
\(110\) 446.473 0.386996
\(111\) 104.205 0.0891055
\(112\) 0 0
\(113\) 295.082 0.245654 0.122827 0.992428i \(-0.460804\pi\)
0.122827 + 0.992428i \(0.460804\pi\)
\(114\) −3030.77 −2.48998
\(115\) 1654.32 1.34144
\(116\) 13.3947 0.0107213
\(117\) −192.154 −0.151834
\(118\) −241.429 −0.188351
\(119\) 0 0
\(120\) 2746.80 2.08956
\(121\) 121.000 0.0909091
\(122\) −2018.16 −1.49767
\(123\) −2069.83 −1.51732
\(124\) 16.8794 0.0122243
\(125\) 435.102 0.311334
\(126\) 0 0
\(127\) −1317.60 −0.920618 −0.460309 0.887759i \(-0.652261\pi\)
−0.460309 + 0.887759i \(0.652261\pi\)
\(128\) 1285.86 0.887928
\(129\) −457.559 −0.312293
\(130\) 217.513 0.146747
\(131\) 1600.71 1.06759 0.533797 0.845612i \(-0.320765\pi\)
0.533797 + 0.845612i \(0.320765\pi\)
\(132\) 46.7358 0.0308169
\(133\) 0 0
\(134\) 935.342 0.602994
\(135\) −1043.15 −0.665036
\(136\) −961.164 −0.606023
\(137\) 1611.68 1.00507 0.502536 0.864556i \(-0.332400\pi\)
0.502536 + 0.864556i \(0.332400\pi\)
\(138\) −2411.95 −1.48782
\(139\) 31.8619 0.0194424 0.00972120 0.999953i \(-0.496906\pi\)
0.00972120 + 0.999953i \(0.496906\pi\)
\(140\) 0 0
\(141\) 2726.08 1.62821
\(142\) −566.529 −0.334803
\(143\) 58.9488 0.0344724
\(144\) −2130.79 −1.23310
\(145\) 371.334 0.212673
\(146\) 2761.01 1.56509
\(147\) 0 0
\(148\) −7.04363 −0.00391205
\(149\) −2428.34 −1.33515 −0.667576 0.744542i \(-0.732668\pi\)
−0.667576 + 0.744542i \(0.732668\pi\)
\(150\) 2073.16 1.12849
\(151\) −2576.68 −1.38866 −0.694328 0.719659i \(-0.744298\pi\)
−0.694328 + 0.719659i \(0.744298\pi\)
\(152\) 3263.08 1.74125
\(153\) 1477.84 0.780889
\(154\) 0 0
\(155\) 467.939 0.242489
\(156\) 22.7688 0.0116856
\(157\) −2475.94 −1.25861 −0.629305 0.777158i \(-0.716660\pi\)
−0.629305 + 0.777158i \(0.716660\pi\)
\(158\) 3535.89 1.78038
\(159\) −2719.33 −1.35633
\(160\) −359.677 −0.177719
\(161\) 0 0
\(162\) −1124.08 −0.545160
\(163\) −2725.11 −1.30949 −0.654745 0.755850i \(-0.727224\pi\)
−0.654745 + 0.755850i \(0.727224\pi\)
\(164\) 139.908 0.0666157
\(165\) 1295.63 0.611301
\(166\) −1207.15 −0.564414
\(167\) −2737.30 −1.26837 −0.634187 0.773180i \(-0.718665\pi\)
−0.634187 + 0.773180i \(0.718665\pi\)
\(168\) 0 0
\(169\) −2168.28 −0.986928
\(170\) −1672.87 −0.754725
\(171\) −5017.14 −2.24368
\(172\) 30.9282 0.0137108
\(173\) −2307.42 −1.01404 −0.507022 0.861933i \(-0.669254\pi\)
−0.507022 + 0.861933i \(0.669254\pi\)
\(174\) −541.395 −0.235879
\(175\) 0 0
\(176\) 653.682 0.279961
\(177\) −700.610 −0.297520
\(178\) 4068.33 1.71311
\(179\) −1312.15 −0.547905 −0.273953 0.961743i \(-0.588331\pi\)
−0.273953 + 0.961743i \(0.588331\pi\)
\(180\) 285.472 0.118210
\(181\) 803.174 0.329831 0.164916 0.986308i \(-0.447265\pi\)
0.164916 + 0.986308i \(0.447265\pi\)
\(182\) 0 0
\(183\) −5856.54 −2.36573
\(184\) 2596.83 1.04044
\(185\) −195.267 −0.0776015
\(186\) −682.242 −0.268949
\(187\) −453.369 −0.177292
\(188\) −184.267 −0.0714842
\(189\) 0 0
\(190\) 5679.26 2.16851
\(191\) 1718.25 0.650932 0.325466 0.945554i \(-0.394479\pi\)
0.325466 + 0.945554i \(0.394479\pi\)
\(192\) 4293.51 1.61384
\(193\) 1340.18 0.499837 0.249919 0.968267i \(-0.419596\pi\)
0.249919 + 0.968267i \(0.419596\pi\)
\(194\) −3678.48 −1.36134
\(195\) 631.206 0.231803
\(196\) 0 0
\(197\) −3518.33 −1.27244 −0.636220 0.771508i \(-0.719503\pi\)
−0.636220 + 0.771508i \(0.719503\pi\)
\(198\) −1077.58 −0.386768
\(199\) −823.692 −0.293417 −0.146709 0.989180i \(-0.546868\pi\)
−0.146709 + 0.989180i \(0.546868\pi\)
\(200\) −2232.07 −0.789156
\(201\) 2714.29 0.952494
\(202\) 441.121 0.153649
\(203\) 0 0
\(204\) −175.112 −0.0600996
\(205\) 3878.59 1.32143
\(206\) 94.9639 0.0321187
\(207\) −3992.75 −1.34065
\(208\) 318.461 0.106160
\(209\) 1539.15 0.509404
\(210\) 0 0
\(211\) −107.343 −0.0350228 −0.0175114 0.999847i \(-0.505574\pi\)
−0.0175114 + 0.999847i \(0.505574\pi\)
\(212\) 183.810 0.0595479
\(213\) −1644.03 −0.528858
\(214\) 2273.56 0.726248
\(215\) 857.405 0.271975
\(216\) −1637.46 −0.515810
\(217\) 0 0
\(218\) 2852.96 0.886362
\(219\) 8012.24 2.47222
\(220\) −87.5768 −0.0268383
\(221\) −220.873 −0.0672285
\(222\) 284.694 0.0860693
\(223\) 3933.68 1.18125 0.590625 0.806946i \(-0.298881\pi\)
0.590625 + 0.806946i \(0.298881\pi\)
\(224\) 0 0
\(225\) 3431.92 1.01686
\(226\) 806.178 0.237284
\(227\) 1771.90 0.518085 0.259042 0.965866i \(-0.416593\pi\)
0.259042 + 0.965866i \(0.416593\pi\)
\(228\) 594.493 0.172681
\(229\) −1915.37 −0.552713 −0.276356 0.961055i \(-0.589127\pi\)
−0.276356 + 0.961055i \(0.589127\pi\)
\(230\) 4519.68 1.29573
\(231\) 0 0
\(232\) 582.892 0.164952
\(233\) 4396.32 1.23610 0.618052 0.786137i \(-0.287922\pi\)
0.618052 + 0.786137i \(0.287922\pi\)
\(234\) −524.974 −0.146661
\(235\) −5108.32 −1.41800
\(236\) 47.3570 0.0130622
\(237\) 10260.9 2.81230
\(238\) 0 0
\(239\) −4084.49 −1.10546 −0.552728 0.833362i \(-0.686413\pi\)
−0.552728 + 0.833362i \(0.686413\pi\)
\(240\) 6999.42 1.88255
\(241\) −3908.58 −1.04471 −0.522353 0.852730i \(-0.674946\pi\)
−0.522353 + 0.852730i \(0.674946\pi\)
\(242\) 330.578 0.0878114
\(243\) −5157.80 −1.36162
\(244\) 395.867 0.103864
\(245\) 0 0
\(246\) −5654.88 −1.46562
\(247\) 749.845 0.193164
\(248\) 734.536 0.188077
\(249\) −3503.05 −0.891552
\(250\) 1188.72 0.300725
\(251\) −1094.89 −0.275335 −0.137667 0.990479i \(-0.543960\pi\)
−0.137667 + 0.990479i \(0.543960\pi\)
\(252\) 0 0
\(253\) 1224.89 0.304381
\(254\) −3599.76 −0.889249
\(255\) −4854.54 −1.19217
\(256\) −819.364 −0.200040
\(257\) −783.179 −0.190091 −0.0950454 0.995473i \(-0.530300\pi\)
−0.0950454 + 0.995473i \(0.530300\pi\)
\(258\) −1250.07 −0.301652
\(259\) 0 0
\(260\) −42.6657 −0.0101770
\(261\) −896.225 −0.212548
\(262\) 4373.23 1.03122
\(263\) 6180.06 1.44897 0.724484 0.689292i \(-0.242078\pi\)
0.724484 + 0.689292i \(0.242078\pi\)
\(264\) 2033.79 0.474132
\(265\) 5095.67 1.18122
\(266\) 0 0
\(267\) 11806.0 2.70605
\(268\) −183.470 −0.0418179
\(269\) −986.965 −0.223704 −0.111852 0.993725i \(-0.535678\pi\)
−0.111852 + 0.993725i \(0.535678\pi\)
\(270\) −2849.93 −0.642376
\(271\) −4576.99 −1.02595 −0.512975 0.858404i \(-0.671457\pi\)
−0.512975 + 0.858404i \(0.671457\pi\)
\(272\) −2449.25 −0.545984
\(273\) 0 0
\(274\) 4403.18 0.970825
\(275\) −1052.84 −0.230868
\(276\) 473.110 0.103181
\(277\) 567.836 0.123169 0.0615847 0.998102i \(-0.480385\pi\)
0.0615847 + 0.998102i \(0.480385\pi\)
\(278\) 87.0484 0.0187799
\(279\) −1129.38 −0.242346
\(280\) 0 0
\(281\) 5311.01 1.12750 0.563752 0.825944i \(-0.309357\pi\)
0.563752 + 0.825944i \(0.309357\pi\)
\(282\) 7447.79 1.57273
\(283\) 4728.44 0.993204 0.496602 0.867978i \(-0.334581\pi\)
0.496602 + 0.867978i \(0.334581\pi\)
\(284\) 111.126 0.0232187
\(285\) 16480.8 3.42539
\(286\) 161.051 0.0332977
\(287\) 0 0
\(288\) 868.092 0.177614
\(289\) −3214.29 −0.654242
\(290\) 1014.50 0.205426
\(291\) −10674.7 −2.15038
\(292\) −541.579 −0.108539
\(293\) −2328.92 −0.464358 −0.232179 0.972673i \(-0.574585\pi\)
−0.232179 + 0.972673i \(0.574585\pi\)
\(294\) 0 0
\(295\) 1312.85 0.259109
\(296\) −306.515 −0.0601886
\(297\) −772.369 −0.150900
\(298\) −6634.36 −1.28966
\(299\) 596.743 0.115420
\(300\) −406.656 −0.0782610
\(301\) 0 0
\(302\) −7039.61 −1.34134
\(303\) 1280.10 0.242705
\(304\) 8315.01 1.56875
\(305\) 10974.4 2.06030
\(306\) 4037.52 0.754280
\(307\) 1678.07 0.311962 0.155981 0.987760i \(-0.450146\pi\)
0.155981 + 0.987760i \(0.450146\pi\)
\(308\) 0 0
\(309\) 275.578 0.0507349
\(310\) 1278.43 0.234226
\(311\) −3572.71 −0.651413 −0.325707 0.945471i \(-0.605602\pi\)
−0.325707 + 0.945471i \(0.605602\pi\)
\(312\) 990.821 0.179789
\(313\) −7184.36 −1.29739 −0.648697 0.761047i \(-0.724686\pi\)
−0.648697 + 0.761047i \(0.724686\pi\)
\(314\) −6764.40 −1.21572
\(315\) 0 0
\(316\) −693.573 −0.123470
\(317\) −15.7077 −0.00278306 −0.00139153 0.999999i \(-0.500443\pi\)
−0.00139153 + 0.999999i \(0.500443\pi\)
\(318\) −7429.36 −1.31012
\(319\) 274.943 0.0482566
\(320\) −8045.47 −1.40549
\(321\) 6597.69 1.14719
\(322\) 0 0
\(323\) −5766.98 −0.993447
\(324\) 220.491 0.0378070
\(325\) −512.923 −0.0875442
\(326\) −7445.13 −1.26487
\(327\) 8279.08 1.40010
\(328\) 6088.33 1.02491
\(329\) 0 0
\(330\) 3539.73 0.590472
\(331\) −1318.95 −0.219022 −0.109511 0.993986i \(-0.534928\pi\)
−0.109511 + 0.993986i \(0.534928\pi\)
\(332\) 236.785 0.0391423
\(333\) 471.282 0.0775558
\(334\) −7478.43 −1.22515
\(335\) −5086.22 −0.829523
\(336\) 0 0
\(337\) −239.183 −0.0386621 −0.0193310 0.999813i \(-0.506154\pi\)
−0.0193310 + 0.999813i \(0.506154\pi\)
\(338\) −5923.85 −0.953299
\(339\) 2339.47 0.374816
\(340\) 328.137 0.0523404
\(341\) 346.472 0.0550220
\(342\) −13707.1 −2.16723
\(343\) 0 0
\(344\) 1345.89 0.210947
\(345\) 13115.8 2.04675
\(346\) −6303.98 −0.979491
\(347\) −5862.79 −0.907006 −0.453503 0.891255i \(-0.649826\pi\)
−0.453503 + 0.891255i \(0.649826\pi\)
\(348\) 106.196 0.0163583
\(349\) −3491.73 −0.535553 −0.267776 0.963481i \(-0.586289\pi\)
−0.267776 + 0.963481i \(0.586289\pi\)
\(350\) 0 0
\(351\) −376.283 −0.0572208
\(352\) −266.313 −0.0403253
\(353\) 10916.7 1.64600 0.822999 0.568043i \(-0.192299\pi\)
0.822999 + 0.568043i \(0.192299\pi\)
\(354\) −1914.10 −0.287382
\(355\) 3080.69 0.460580
\(356\) −798.013 −0.118805
\(357\) 0 0
\(358\) −3584.87 −0.529236
\(359\) −11500.7 −1.69077 −0.845384 0.534160i \(-0.820628\pi\)
−0.845384 + 0.534160i \(0.820628\pi\)
\(360\) 12422.8 1.81872
\(361\) 12719.5 1.85442
\(362\) 2194.31 0.318592
\(363\) 959.313 0.138708
\(364\) 0 0
\(365\) −15013.9 −2.15305
\(366\) −16000.4 −2.28512
\(367\) −6767.01 −0.962493 −0.481246 0.876585i \(-0.659816\pi\)
−0.481246 + 0.876585i \(0.659816\pi\)
\(368\) 6617.27 0.937362
\(369\) −9361.10 −1.32065
\(370\) −533.478 −0.0749573
\(371\) 0 0
\(372\) 133.823 0.0186517
\(373\) −5310.22 −0.737139 −0.368569 0.929600i \(-0.620152\pi\)
−0.368569 + 0.929600i \(0.620152\pi\)
\(374\) −1238.63 −0.171251
\(375\) 3449.58 0.475028
\(376\) −8018.67 −1.09982
\(377\) 133.947 0.0182987
\(378\) 0 0
\(379\) −838.267 −0.113612 −0.0568059 0.998385i \(-0.518092\pi\)
−0.0568059 + 0.998385i \(0.518092\pi\)
\(380\) −1114.00 −0.150387
\(381\) −10446.2 −1.40466
\(382\) 4694.34 0.628752
\(383\) 2832.16 0.377851 0.188925 0.981991i \(-0.439500\pi\)
0.188925 + 0.981991i \(0.439500\pi\)
\(384\) 10194.5 1.35479
\(385\) 0 0
\(386\) 3661.45 0.482806
\(387\) −2069.37 −0.271814
\(388\) 721.542 0.0944091
\(389\) 3111.25 0.405519 0.202759 0.979229i \(-0.435009\pi\)
0.202759 + 0.979229i \(0.435009\pi\)
\(390\) 1724.49 0.223905
\(391\) −4589.49 −0.593608
\(392\) 0 0
\(393\) 12690.8 1.62892
\(394\) −9612.25 −1.22908
\(395\) −19227.5 −2.44922
\(396\) 211.369 0.0268225
\(397\) −14208.7 −1.79626 −0.898131 0.439728i \(-0.855075\pi\)
−0.898131 + 0.439728i \(0.855075\pi\)
\(398\) −2250.37 −0.283419
\(399\) 0 0
\(400\) −5687.79 −0.710974
\(401\) −6261.68 −0.779784 −0.389892 0.920861i \(-0.627488\pi\)
−0.389892 + 0.920861i \(0.627488\pi\)
\(402\) 7415.58 0.920039
\(403\) 168.794 0.0208641
\(404\) −86.5269 −0.0106556
\(405\) 6112.54 0.749961
\(406\) 0 0
\(407\) −144.580 −0.0176082
\(408\) −7620.30 −0.924660
\(409\) 4192.50 0.506860 0.253430 0.967354i \(-0.418441\pi\)
0.253430 + 0.967354i \(0.418441\pi\)
\(410\) 10596.5 1.27640
\(411\) 12777.7 1.53352
\(412\) −18.6274 −0.00222744
\(413\) 0 0
\(414\) −10908.4 −1.29497
\(415\) 6564.25 0.776448
\(416\) −129.742 −0.0152912
\(417\) 252.608 0.0296649
\(418\) 4205.05 0.492047
\(419\) 9287.15 1.08283 0.541416 0.840755i \(-0.317888\pi\)
0.541416 + 0.840755i \(0.317888\pi\)
\(420\) 0 0
\(421\) 13146.0 1.52185 0.760923 0.648842i \(-0.224746\pi\)
0.760923 + 0.648842i \(0.224746\pi\)
\(422\) −293.267 −0.0338294
\(423\) 12329.1 1.41716
\(424\) 7998.81 0.916172
\(425\) 3944.84 0.450242
\(426\) −4491.56 −0.510838
\(427\) 0 0
\(428\) −445.964 −0.0503656
\(429\) 467.358 0.0525974
\(430\) 2342.47 0.262707
\(431\) 4909.67 0.548701 0.274351 0.961630i \(-0.411537\pi\)
0.274351 + 0.961630i \(0.411537\pi\)
\(432\) −4172.59 −0.464708
\(433\) 11743.3 1.30334 0.651671 0.758502i \(-0.274068\pi\)
0.651671 + 0.758502i \(0.274068\pi\)
\(434\) 0 0
\(435\) 2944.01 0.324493
\(436\) −559.615 −0.0614695
\(437\) 15581.0 1.70558
\(438\) 21889.8 2.38798
\(439\) 11824.2 1.28551 0.642754 0.766073i \(-0.277792\pi\)
0.642754 + 0.766073i \(0.277792\pi\)
\(440\) −3811.05 −0.412920
\(441\) 0 0
\(442\) −603.435 −0.0649377
\(443\) 10102.1 1.08344 0.541722 0.840558i \(-0.317772\pi\)
0.541722 + 0.840558i \(0.317772\pi\)
\(444\) −55.8433 −0.00596894
\(445\) −22122.9 −2.35668
\(446\) 10747.0 1.14100
\(447\) −19252.4 −2.03715
\(448\) 0 0
\(449\) −345.254 −0.0362885 −0.0181443 0.999835i \(-0.505776\pi\)
−0.0181443 + 0.999835i \(0.505776\pi\)
\(450\) 9376.17 0.982216
\(451\) 2871.79 0.299839
\(452\) −158.134 −0.0164557
\(453\) −20428.4 −2.11879
\(454\) 4840.93 0.500431
\(455\) 0 0
\(456\) 25870.3 2.65678
\(457\) −10567.1 −1.08164 −0.540821 0.841138i \(-0.681886\pi\)
−0.540821 + 0.841138i \(0.681886\pi\)
\(458\) −5232.89 −0.533879
\(459\) 2893.95 0.294288
\(460\) −886.546 −0.0898596
\(461\) −4733.96 −0.478270 −0.239135 0.970986i \(-0.576864\pi\)
−0.239135 + 0.970986i \(0.576864\pi\)
\(462\) 0 0
\(463\) 3431.20 0.344409 0.172204 0.985061i \(-0.444911\pi\)
0.172204 + 0.985061i \(0.444911\pi\)
\(464\) 1485.33 0.148610
\(465\) 3709.91 0.369985
\(466\) 12011.0 1.19399
\(467\) −5116.96 −0.507034 −0.253517 0.967331i \(-0.581587\pi\)
−0.253517 + 0.967331i \(0.581587\pi\)
\(468\) 102.975 0.0101710
\(469\) 0 0
\(470\) −13956.2 −1.36968
\(471\) −19629.8 −1.92037
\(472\) 2060.82 0.200968
\(473\) 634.841 0.0617125
\(474\) 28033.2 2.71648
\(475\) −13392.4 −1.29366
\(476\) 0 0
\(477\) −12298.6 −1.18053
\(478\) −11159.0 −1.06779
\(479\) −11566.9 −1.10335 −0.551675 0.834059i \(-0.686011\pi\)
−0.551675 + 0.834059i \(0.686011\pi\)
\(480\) −2851.59 −0.271160
\(481\) −70.4363 −0.00667696
\(482\) −10678.4 −1.00911
\(483\) 0 0
\(484\) −64.8437 −0.00608975
\(485\) 20002.9 1.87275
\(486\) −14091.4 −1.31522
\(487\) −18326.5 −1.70525 −0.852623 0.522527i \(-0.824990\pi\)
−0.852623 + 0.522527i \(0.824990\pi\)
\(488\) 17226.8 1.59799
\(489\) −21605.2 −1.99800
\(490\) 0 0
\(491\) −7617.58 −0.700156 −0.350078 0.936721i \(-0.613845\pi\)
−0.350078 + 0.936721i \(0.613845\pi\)
\(492\) 1109.22 0.101641
\(493\) −1030.17 −0.0941108
\(494\) 2048.62 0.186582
\(495\) 5859.67 0.532066
\(496\) 1871.75 0.169444
\(497\) 0 0
\(498\) −9570.50 −0.861173
\(499\) 12909.1 1.15810 0.579050 0.815292i \(-0.303424\pi\)
0.579050 + 0.815292i \(0.303424\pi\)
\(500\) −233.171 −0.0208554
\(501\) −21701.8 −1.93526
\(502\) −2991.30 −0.265953
\(503\) −10165.7 −0.901121 −0.450561 0.892746i \(-0.648776\pi\)
−0.450561 + 0.892746i \(0.648776\pi\)
\(504\) 0 0
\(505\) −2398.74 −0.211371
\(506\) 3346.47 0.294009
\(507\) −17190.6 −1.50584
\(508\) 706.102 0.0616697
\(509\) −6449.93 −0.561666 −0.280833 0.959757i \(-0.590611\pi\)
−0.280833 + 0.959757i \(0.590611\pi\)
\(510\) −13262.8 −1.15155
\(511\) 0 0
\(512\) −12525.4 −1.08115
\(513\) −9824.75 −0.845562
\(514\) −2139.68 −0.183614
\(515\) −516.397 −0.0441848
\(516\) 245.205 0.0209197
\(517\) −3782.31 −0.321752
\(518\) 0 0
\(519\) −18293.7 −1.54721
\(520\) −1856.67 −0.156577
\(521\) 19327.4 1.62524 0.812620 0.582794i \(-0.198041\pi\)
0.812620 + 0.582794i \(0.198041\pi\)
\(522\) −2448.53 −0.205305
\(523\) −6259.09 −0.523310 −0.261655 0.965161i \(-0.584268\pi\)
−0.261655 + 0.965161i \(0.584268\pi\)
\(524\) −857.819 −0.0715153
\(525\) 0 0
\(526\) 16884.2 1.39960
\(527\) −1298.18 −0.107305
\(528\) 5182.52 0.427160
\(529\) 232.675 0.0191235
\(530\) 13921.6 1.14098
\(531\) −3168.61 −0.258956
\(532\) 0 0
\(533\) 1399.08 0.113698
\(534\) 32254.6 2.61384
\(535\) −12363.2 −0.999079
\(536\) −7983.99 −0.643387
\(537\) −10403.0 −0.835985
\(538\) −2696.44 −0.216081
\(539\) 0 0
\(540\) 559.022 0.0445490
\(541\) −14008.2 −1.11323 −0.556616 0.830770i \(-0.687900\pi\)
−0.556616 + 0.830770i \(0.687900\pi\)
\(542\) −12504.6 −0.990991
\(543\) 6367.72 0.503251
\(544\) 997.834 0.0786430
\(545\) −15513.9 −1.21934
\(546\) 0 0
\(547\) −4949.45 −0.386879 −0.193440 0.981112i \(-0.561964\pi\)
−0.193440 + 0.981112i \(0.561964\pi\)
\(548\) −863.695 −0.0673270
\(549\) −26487.0 −2.05909
\(550\) −2876.41 −0.223001
\(551\) 3497.35 0.270404
\(552\) 20588.2 1.58748
\(553\) 0 0
\(554\) 1551.36 0.118973
\(555\) −1548.11 −0.118403
\(556\) −17.0748 −0.00130239
\(557\) −3801.58 −0.289188 −0.144594 0.989491i \(-0.546188\pi\)
−0.144594 + 0.989491i \(0.546188\pi\)
\(558\) −3085.54 −0.234088
\(559\) 309.282 0.0234011
\(560\) 0 0
\(561\) −3594.40 −0.270510
\(562\) 14510.0 1.08908
\(563\) 9900.11 0.741101 0.370551 0.928812i \(-0.379169\pi\)
0.370551 + 0.928812i \(0.379169\pi\)
\(564\) −1460.90 −0.109069
\(565\) −4383.85 −0.326425
\(566\) 12918.3 0.959361
\(567\) 0 0
\(568\) 4835.84 0.357231
\(569\) 5329.16 0.392636 0.196318 0.980540i \(-0.437102\pi\)
0.196318 + 0.980540i \(0.437102\pi\)
\(570\) 45026.3 3.30868
\(571\) −16962.6 −1.24319 −0.621597 0.783337i \(-0.713516\pi\)
−0.621597 + 0.783337i \(0.713516\pi\)
\(572\) −31.5906 −0.00230921
\(573\) 13622.6 0.993181
\(574\) 0 0
\(575\) −10658.0 −0.772989
\(576\) 19418.0 1.40466
\(577\) 15487.0 1.11738 0.558692 0.829375i \(-0.311303\pi\)
0.558692 + 0.829375i \(0.311303\pi\)
\(578\) −8781.61 −0.631949
\(579\) 10625.3 0.762643
\(580\) −198.997 −0.0142464
\(581\) 0 0
\(582\) −29163.7 −2.07710
\(583\) 3772.94 0.268026
\(584\) −23567.7 −1.66993
\(585\) 2854.72 0.201757
\(586\) −6362.72 −0.448535
\(587\) −11084.2 −0.779373 −0.389686 0.920948i \(-0.627417\pi\)
−0.389686 + 0.920948i \(0.627417\pi\)
\(588\) 0 0
\(589\) 4407.22 0.308313
\(590\) 3586.77 0.250280
\(591\) −27894.0 −1.94147
\(592\) −781.066 −0.0542257
\(593\) −4349.68 −0.301214 −0.150607 0.988594i \(-0.548123\pi\)
−0.150607 + 0.988594i \(0.548123\pi\)
\(594\) −2110.15 −0.145759
\(595\) 0 0
\(596\) 1301.34 0.0894382
\(597\) −6530.40 −0.447691
\(598\) 1630.33 0.111487
\(599\) 13183.9 0.899299 0.449650 0.893205i \(-0.351549\pi\)
0.449650 + 0.893205i \(0.351549\pi\)
\(600\) −17696.3 −1.20408
\(601\) 18765.0 1.27361 0.636806 0.771024i \(-0.280255\pi\)
0.636806 + 0.771024i \(0.280255\pi\)
\(602\) 0 0
\(603\) 12275.8 0.829034
\(604\) 1380.84 0.0930223
\(605\) −1797.63 −0.120800
\(606\) 3497.29 0.234435
\(607\) −21871.4 −1.46249 −0.731244 0.682116i \(-0.761060\pi\)
−0.731244 + 0.682116i \(0.761060\pi\)
\(608\) −3387.57 −0.225961
\(609\) 0 0
\(610\) 29982.6 1.99010
\(611\) −1842.67 −0.122007
\(612\) −791.970 −0.0523096
\(613\) −3527.85 −0.232445 −0.116222 0.993223i \(-0.537079\pi\)
−0.116222 + 0.993223i \(0.537079\pi\)
\(614\) 4584.56 0.301332
\(615\) 30750.2 2.01621
\(616\) 0 0
\(617\) −22728.1 −1.48298 −0.741490 0.670963i \(-0.765881\pi\)
−0.741490 + 0.670963i \(0.765881\pi\)
\(618\) 752.893 0.0490062
\(619\) 21443.3 1.39237 0.696187 0.717861i \(-0.254879\pi\)
0.696187 + 0.717861i \(0.254879\pi\)
\(620\) −250.767 −0.0162437
\(621\) −7818.75 −0.505243
\(622\) −9760.81 −0.629217
\(623\) 0 0
\(624\) 2524.82 0.161977
\(625\) −18428.2 −1.17940
\(626\) −19628.0 −1.25319
\(627\) 12202.7 0.777240
\(628\) 1326.85 0.0843109
\(629\) 541.718 0.0343398
\(630\) 0 0
\(631\) 21532.0 1.35844 0.679219 0.733936i \(-0.262319\pi\)
0.679219 + 0.733936i \(0.262319\pi\)
\(632\) −30182.0 −1.89964
\(633\) −851.038 −0.0534372
\(634\) −42.9141 −0.00268823
\(635\) 19574.9 1.22332
\(636\) 1457.29 0.0908572
\(637\) 0 0
\(638\) 751.159 0.0466123
\(639\) −7435.33 −0.460309
\(640\) −19103.2 −1.17988
\(641\) 20148.3 1.24151 0.620756 0.784004i \(-0.286826\pi\)
0.620756 + 0.784004i \(0.286826\pi\)
\(642\) 18025.2 1.10810
\(643\) −28869.7 −1.77062 −0.885310 0.465000i \(-0.846054\pi\)
−0.885310 + 0.465000i \(0.846054\pi\)
\(644\) 0 0
\(645\) 6797.68 0.414974
\(646\) −15755.7 −0.959597
\(647\) 1590.02 0.0966155 0.0483077 0.998833i \(-0.484617\pi\)
0.0483077 + 0.998833i \(0.484617\pi\)
\(648\) 9595.02 0.581679
\(649\) 972.062 0.0587932
\(650\) −1401.33 −0.0845612
\(651\) 0 0
\(652\) 1460.38 0.0877192
\(653\) 20028.1 1.20024 0.600122 0.799909i \(-0.295119\pi\)
0.600122 + 0.799909i \(0.295119\pi\)
\(654\) 22618.9 1.35240
\(655\) −23780.8 −1.41862
\(656\) 15514.4 0.923375
\(657\) 36236.5 2.15178
\(658\) 0 0
\(659\) −10520.7 −0.621897 −0.310948 0.950427i \(-0.600647\pi\)
−0.310948 + 0.950427i \(0.600647\pi\)
\(660\) −694.326 −0.0409494
\(661\) −3295.83 −0.193938 −0.0969690 0.995287i \(-0.530915\pi\)
−0.0969690 + 0.995287i \(0.530915\pi\)
\(662\) −3603.45 −0.211559
\(663\) −1751.12 −0.102576
\(664\) 10304.1 0.602222
\(665\) 0 0
\(666\) 1287.57 0.0749132
\(667\) 2783.27 0.161572
\(668\) 1466.91 0.0849649
\(669\) 31187.0 1.80233
\(670\) −13895.8 −0.801257
\(671\) 8125.67 0.467493
\(672\) 0 0
\(673\) −1187.64 −0.0680239 −0.0340119 0.999421i \(-0.510828\pi\)
−0.0340119 + 0.999421i \(0.510828\pi\)
\(674\) −653.460 −0.0373447
\(675\) 6720.51 0.383219
\(676\) 1161.98 0.0661117
\(677\) −13221.4 −0.750574 −0.375287 0.926909i \(-0.622456\pi\)
−0.375287 + 0.926909i \(0.622456\pi\)
\(678\) 6391.55 0.362044
\(679\) 0 0
\(680\) 14279.4 0.805282
\(681\) 14048.0 0.790485
\(682\) 946.578 0.0531471
\(683\) −13831.4 −0.774882 −0.387441 0.921894i \(-0.626641\pi\)
−0.387441 + 0.921894i \(0.626641\pi\)
\(684\) 2688.68 0.150298
\(685\) −23943.7 −1.33554
\(686\) 0 0
\(687\) −15185.4 −0.843320
\(688\) 3429.62 0.190048
\(689\) 1838.10 0.101635
\(690\) 35832.9 1.97701
\(691\) 9817.07 0.540462 0.270231 0.962796i \(-0.412900\pi\)
0.270231 + 0.962796i \(0.412900\pi\)
\(692\) 1236.54 0.0679280
\(693\) 0 0
\(694\) −16017.4 −0.876101
\(695\) −473.354 −0.0258350
\(696\) 4621.29 0.251680
\(697\) −10760.2 −0.584750
\(698\) −9539.58 −0.517304
\(699\) 34854.9 1.88603
\(700\) 0 0
\(701\) 29949.8 1.61368 0.806838 0.590773i \(-0.201177\pi\)
0.806838 + 0.590773i \(0.201177\pi\)
\(702\) −1028.02 −0.0552711
\(703\) −1839.09 −0.0986667
\(704\) −5957.03 −0.318912
\(705\) −40499.8 −2.16356
\(706\) 29825.0 1.58991
\(707\) 0 0
\(708\) 375.456 0.0199301
\(709\) 11307.5 0.598959 0.299479 0.954103i \(-0.403187\pi\)
0.299479 + 0.954103i \(0.403187\pi\)
\(710\) 8416.59 0.444886
\(711\) 46406.3 2.44778
\(712\) −34726.9 −1.82787
\(713\) 3507.36 0.184224
\(714\) 0 0
\(715\) −875.768 −0.0458068
\(716\) 703.181 0.0367027
\(717\) −32382.7 −1.68669
\(718\) −31420.6 −1.63316
\(719\) 32623.4 1.69214 0.846070 0.533071i \(-0.178962\pi\)
0.846070 + 0.533071i \(0.178962\pi\)
\(720\) 31655.9 1.63853
\(721\) 0 0
\(722\) 34750.2 1.79123
\(723\) −30988.0 −1.59399
\(724\) −430.420 −0.0220945
\(725\) −2392.33 −0.122550
\(726\) 2620.89 0.133981
\(727\) 502.545 0.0256373 0.0128187 0.999918i \(-0.495920\pi\)
0.0128187 + 0.999918i \(0.495920\pi\)
\(728\) 0 0
\(729\) −29783.2 −1.51314
\(730\) −41018.7 −2.07968
\(731\) −2378.66 −0.120353
\(732\) 3138.51 0.158474
\(733\) −8631.37 −0.434935 −0.217467 0.976068i \(-0.569780\pi\)
−0.217467 + 0.976068i \(0.569780\pi\)
\(734\) −18487.8 −0.929697
\(735\) 0 0
\(736\) −2695.90 −0.135017
\(737\) −3765.95 −0.188223
\(738\) −25575.0 −1.27565
\(739\) −18357.5 −0.913792 −0.456896 0.889520i \(-0.651039\pi\)
−0.456896 + 0.889520i \(0.651039\pi\)
\(740\) 104.643 0.00519832
\(741\) 5944.93 0.294726
\(742\) 0 0
\(743\) 11182.6 0.552155 0.276078 0.961135i \(-0.410965\pi\)
0.276078 + 0.961135i \(0.410965\pi\)
\(744\) 5823.55 0.286965
\(745\) 36076.4 1.77415
\(746\) −14507.8 −0.712021
\(747\) −15843.0 −0.775991
\(748\) 242.960 0.0118763
\(749\) 0 0
\(750\) 9424.43 0.458842
\(751\) 16733.4 0.813063 0.406531 0.913637i \(-0.366738\pi\)
0.406531 + 0.913637i \(0.366738\pi\)
\(752\) −20433.3 −0.990857
\(753\) −8680.53 −0.420101
\(754\) 365.950 0.0176752
\(755\) 38280.2 1.84524
\(756\) 0 0
\(757\) −24402.4 −1.17163 −0.585813 0.810446i \(-0.699225\pi\)
−0.585813 + 0.810446i \(0.699225\pi\)
\(758\) −2290.19 −0.109741
\(759\) 9711.19 0.464419
\(760\) −48477.6 −2.31377
\(761\) −8469.33 −0.403434 −0.201717 0.979444i \(-0.564652\pi\)
−0.201717 + 0.979444i \(0.564652\pi\)
\(762\) −28539.7 −1.35680
\(763\) 0 0
\(764\) −920.805 −0.0436042
\(765\) −21955.3 −1.03764
\(766\) 7737.62 0.364976
\(767\) 473.570 0.0222941
\(768\) −6496.08 −0.305218
\(769\) −32834.7 −1.53973 −0.769864 0.638208i \(-0.779676\pi\)
−0.769864 + 0.638208i \(0.779676\pi\)
\(770\) 0 0
\(771\) −6209.20 −0.290038
\(772\) −718.202 −0.0334827
\(773\) 35571.4 1.65513 0.827564 0.561371i \(-0.189726\pi\)
0.827564 + 0.561371i \(0.189726\pi\)
\(774\) −5653.64 −0.262553
\(775\) −3014.71 −0.139731
\(776\) 31399.1 1.45253
\(777\) 0 0
\(778\) 8500.11 0.391701
\(779\) 36530.0 1.68013
\(780\) −338.262 −0.0155279
\(781\) 2281.01 0.104508
\(782\) −12538.7 −0.573381
\(783\) −1755.02 −0.0801014
\(784\) 0 0
\(785\) 36783.6 1.67244
\(786\) 34671.8 1.57341
\(787\) −15729.6 −0.712452 −0.356226 0.934400i \(-0.615937\pi\)
−0.356226 + 0.934400i \(0.615937\pi\)
\(788\) 1885.47 0.0852373
\(789\) 48996.7 2.21081
\(790\) −52530.6 −2.36577
\(791\) 0 0
\(792\) 9198.09 0.412676
\(793\) 3958.67 0.177272
\(794\) −38819.0 −1.73506
\(795\) 40399.5 1.80229
\(796\) 441.415 0.0196552
\(797\) −7888.07 −0.350577 −0.175288 0.984517i \(-0.556086\pi\)
−0.175288 + 0.984517i \(0.556086\pi\)
\(798\) 0 0
\(799\) 14171.8 0.627485
\(800\) 2317.23 0.102408
\(801\) 53394.2 2.35530
\(802\) −17107.2 −0.753214
\(803\) −11116.6 −0.488538
\(804\) −1454.58 −0.0638050
\(805\) 0 0
\(806\) 461.154 0.0201532
\(807\) −7824.86 −0.341323
\(808\) −3765.36 −0.163942
\(809\) 5896.97 0.256275 0.128138 0.991756i \(-0.459100\pi\)
0.128138 + 0.991756i \(0.459100\pi\)
\(810\) 16699.8 0.724407
\(811\) −14197.9 −0.614744 −0.307372 0.951589i \(-0.599450\pi\)
−0.307372 + 0.951589i \(0.599450\pi\)
\(812\) 0 0
\(813\) −36287.3 −1.56538
\(814\) −394.999 −0.0170082
\(815\) 40485.3 1.74005
\(816\) −19418.2 −0.833053
\(817\) 8075.35 0.345803
\(818\) 11454.1 0.489589
\(819\) 0 0
\(820\) −2078.53 −0.0885188
\(821\) −19841.7 −0.843459 −0.421729 0.906722i \(-0.638577\pi\)
−0.421729 + 0.906722i \(0.638577\pi\)
\(822\) 34909.3 1.48127
\(823\) −28202.2 −1.19449 −0.597246 0.802058i \(-0.703738\pi\)
−0.597246 + 0.802058i \(0.703738\pi\)
\(824\) −810.602 −0.0342702
\(825\) −8347.14 −0.352255
\(826\) 0 0
\(827\) 34031.0 1.43092 0.715462 0.698651i \(-0.246216\pi\)
0.715462 + 0.698651i \(0.246216\pi\)
\(828\) 2139.71 0.0898067
\(829\) −4931.55 −0.206610 −0.103305 0.994650i \(-0.532942\pi\)
−0.103305 + 0.994650i \(0.532942\pi\)
\(830\) 17933.9 0.749992
\(831\) 4501.92 0.187930
\(832\) −2902.15 −0.120930
\(833\) 0 0
\(834\) 690.138 0.0286541
\(835\) 40666.4 1.68541
\(836\) −824.830 −0.0341236
\(837\) −2211.60 −0.0913312
\(838\) 25373.0 1.04594
\(839\) 38189.8 1.57146 0.785731 0.618568i \(-0.212287\pi\)
0.785731 + 0.618568i \(0.212287\pi\)
\(840\) 0 0
\(841\) −23764.3 −0.974384
\(842\) 35915.6 1.46999
\(843\) 42106.8 1.72033
\(844\) 57.5250 0.00234608
\(845\) 32212.9 1.31143
\(846\) 33683.7 1.36888
\(847\) 0 0
\(848\) 20382.7 0.825406
\(849\) 37488.0 1.51541
\(850\) 10777.5 0.434900
\(851\) −1463.59 −0.0589556
\(852\) 881.030 0.0354268
\(853\) −42966.8 −1.72469 −0.862343 0.506325i \(-0.831003\pi\)
−0.862343 + 0.506325i \(0.831003\pi\)
\(854\) 0 0
\(855\) 74536.6 2.98140
\(856\) −19406.8 −0.774898
\(857\) 17281.5 0.688828 0.344414 0.938818i \(-0.388078\pi\)
0.344414 + 0.938818i \(0.388078\pi\)
\(858\) 1276.85 0.0508052
\(859\) −9316.75 −0.370062 −0.185031 0.982733i \(-0.559239\pi\)
−0.185031 + 0.982733i \(0.559239\pi\)
\(860\) −459.482 −0.0182188
\(861\) 0 0
\(862\) 13413.5 0.530005
\(863\) −9647.65 −0.380544 −0.190272 0.981731i \(-0.560937\pi\)
−0.190272 + 0.981731i \(0.560937\pi\)
\(864\) 1699.93 0.0669361
\(865\) 34279.9 1.34746
\(866\) 32083.3 1.25893
\(867\) −25483.6 −0.998232
\(868\) 0 0
\(869\) −14236.5 −0.555742
\(870\) 8043.18 0.313436
\(871\) −1834.70 −0.0713735
\(872\) −24352.6 −0.945737
\(873\) −48277.6 −1.87165
\(874\) 42568.0 1.64746
\(875\) 0 0
\(876\) −4293.75 −0.165608
\(877\) 19728.7 0.759624 0.379812 0.925064i \(-0.375989\pi\)
0.379812 + 0.925064i \(0.375989\pi\)
\(878\) 32304.3 1.24171
\(879\) −18464.1 −0.708509
\(880\) −9711.36 −0.372011
\(881\) −19473.9 −0.744712 −0.372356 0.928090i \(-0.621450\pi\)
−0.372356 + 0.928090i \(0.621450\pi\)
\(882\) 0 0
\(883\) 49092.4 1.87100 0.935499 0.353329i \(-0.114950\pi\)
0.935499 + 0.353329i \(0.114950\pi\)
\(884\) 118.365 0.00450346
\(885\) 10408.5 0.395344
\(886\) 27599.5 1.04653
\(887\) −9292.86 −0.351774 −0.175887 0.984410i \(-0.556279\pi\)
−0.175887 + 0.984410i \(0.556279\pi\)
\(888\) −2430.12 −0.0918348
\(889\) 0 0
\(890\) −60440.8 −2.27638
\(891\) 4525.85 0.170170
\(892\) −2108.05 −0.0791288
\(893\) −48112.0 −1.80292
\(894\) −52598.5 −1.96774
\(895\) 19493.9 0.728055
\(896\) 0 0
\(897\) 4731.10 0.176106
\(898\) −943.252 −0.0350520
\(899\) 787.273 0.0292069
\(900\) −1839.16 −0.0681170
\(901\) −14136.7 −0.522709
\(902\) 7845.88 0.289622
\(903\) 0 0
\(904\) −6881.46 −0.253179
\(905\) −11932.3 −0.438279
\(906\) −55811.5 −2.04659
\(907\) 37688.7 1.37975 0.689875 0.723928i \(-0.257665\pi\)
0.689875 + 0.723928i \(0.257665\pi\)
\(908\) −949.559 −0.0347051
\(909\) 5789.42 0.211246
\(910\) 0 0
\(911\) 33049.6 1.20196 0.600979 0.799265i \(-0.294778\pi\)
0.600979 + 0.799265i \(0.294778\pi\)
\(912\) 65923.1 2.39357
\(913\) 4860.31 0.176180
\(914\) −28870.0 −1.04479
\(915\) 87007.2 3.14357
\(916\) 1026.44 0.0370247
\(917\) 0 0
\(918\) 7906.43 0.284260
\(919\) −23148.0 −0.830883 −0.415442 0.909620i \(-0.636373\pi\)
−0.415442 + 0.909620i \(0.636373\pi\)
\(920\) −38579.5 −1.38253
\(921\) 13304.1 0.475986
\(922\) −12933.4 −0.461973
\(923\) 1111.26 0.0396290
\(924\) 0 0
\(925\) 1258.01 0.0447169
\(926\) 9374.21 0.332673
\(927\) 1246.34 0.0441588
\(928\) −605.131 −0.0214056
\(929\) 23177.9 0.818561 0.409280 0.912409i \(-0.365780\pi\)
0.409280 + 0.912409i \(0.365780\pi\)
\(930\) 10135.7 0.357378
\(931\) 0 0
\(932\) −2355.98 −0.0828033
\(933\) −28325.1 −0.993916
\(934\) −13979.8 −0.489757
\(935\) 6735.44 0.235585
\(936\) 4481.13 0.156485
\(937\) 34574.7 1.20545 0.602724 0.797950i \(-0.294082\pi\)
0.602724 + 0.797950i \(0.294082\pi\)
\(938\) 0 0
\(939\) −56959.1 −1.97954
\(940\) 2737.54 0.0949880
\(941\) −41831.2 −1.44916 −0.724578 0.689192i \(-0.757966\pi\)
−0.724578 + 0.689192i \(0.757966\pi\)
\(942\) −53629.6 −1.85493
\(943\) 29071.3 1.00392
\(944\) 5251.40 0.181058
\(945\) 0 0
\(946\) 1734.42 0.0596097
\(947\) 27231.2 0.934419 0.467209 0.884147i \(-0.345259\pi\)
0.467209 + 0.884147i \(0.345259\pi\)
\(948\) −5498.79 −0.188389
\(949\) −5415.79 −0.185252
\(950\) −36588.8 −1.24958
\(951\) −124.534 −0.00424635
\(952\) 0 0
\(953\) 40939.4 1.39156 0.695781 0.718254i \(-0.255058\pi\)
0.695781 + 0.718254i \(0.255058\pi\)
\(954\) −33600.3 −1.14030
\(955\) −25527.0 −0.864956
\(956\) 2188.87 0.0740515
\(957\) 2179.81 0.0736292
\(958\) −31601.3 −1.06575
\(959\) 0 0
\(960\) −63786.1 −2.14447
\(961\) −28798.9 −0.966698
\(962\) −192.436 −0.00644945
\(963\) 29839.0 0.998491
\(964\) 2094.60 0.0699820
\(965\) −19910.3 −0.664182
\(966\) 0 0
\(967\) −46173.1 −1.53550 −0.767750 0.640750i \(-0.778624\pi\)
−0.767750 + 0.640750i \(0.778624\pi\)
\(968\) −2821.78 −0.0936937
\(969\) −45721.8 −1.51579
\(970\) 54648.9 1.80894
\(971\) 5153.91 0.170337 0.0851683 0.996367i \(-0.472857\pi\)
0.0851683 + 0.996367i \(0.472857\pi\)
\(972\) 2764.06 0.0912111
\(973\) 0 0
\(974\) −50069.0 −1.64714
\(975\) −4066.56 −0.133574
\(976\) 43897.6 1.43968
\(977\) 9692.13 0.317378 0.158689 0.987329i \(-0.449273\pi\)
0.158689 + 0.987329i \(0.449273\pi\)
\(978\) −59026.5 −1.92992
\(979\) −16380.2 −0.534744
\(980\) 0 0
\(981\) 37443.3 1.21863
\(982\) −20811.6 −0.676299
\(983\) −32915.7 −1.06800 −0.534002 0.845483i \(-0.679313\pi\)
−0.534002 + 0.845483i \(0.679313\pi\)
\(984\) 48269.5 1.56380
\(985\) 52269.7 1.69081
\(986\) −2814.48 −0.0909041
\(987\) 0 0
\(988\) −401.841 −0.0129395
\(989\) 6426.54 0.206625
\(990\) 16008.9 0.513936
\(991\) 29477.9 0.944901 0.472451 0.881357i \(-0.343370\pi\)
0.472451 + 0.881357i \(0.343370\pi\)
\(992\) −762.560 −0.0244066
\(993\) −10456.9 −0.334180
\(994\) 0 0
\(995\) 12237.1 0.389892
\(996\) 1877.28 0.0597227
\(997\) 31944.4 1.01473 0.507366 0.861731i \(-0.330619\pi\)
0.507366 + 0.861731i \(0.330619\pi\)
\(998\) 35268.4 1.11864
\(999\) 922.883 0.0292279
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 539.4.a.e.1.2 2
7.6 odd 2 11.4.a.a.1.2 2
21.20 even 2 99.4.a.c.1.1 2
28.27 even 2 176.4.a.i.1.2 2
35.13 even 4 275.4.b.c.199.1 4
35.27 even 4 275.4.b.c.199.4 4
35.34 odd 2 275.4.a.b.1.1 2
56.13 odd 2 704.4.a.p.1.2 2
56.27 even 2 704.4.a.n.1.1 2
77.6 even 10 121.4.c.f.3.2 8
77.13 even 10 121.4.c.f.81.2 8
77.20 odd 10 121.4.c.c.81.1 8
77.27 odd 10 121.4.c.c.3.1 8
77.41 even 10 121.4.c.f.9.1 8
77.48 odd 10 121.4.c.c.27.2 8
77.62 even 10 121.4.c.f.27.1 8
77.69 odd 10 121.4.c.c.9.2 8
77.76 even 2 121.4.a.c.1.1 2
84.83 odd 2 1584.4.a.bc.1.1 2
91.90 odd 2 1859.4.a.a.1.1 2
105.104 even 2 2475.4.a.q.1.2 2
231.230 odd 2 1089.4.a.v.1.2 2
308.307 odd 2 1936.4.a.w.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
11.4.a.a.1.2 2 7.6 odd 2
99.4.a.c.1.1 2 21.20 even 2
121.4.a.c.1.1 2 77.76 even 2
121.4.c.c.3.1 8 77.27 odd 10
121.4.c.c.9.2 8 77.69 odd 10
121.4.c.c.27.2 8 77.48 odd 10
121.4.c.c.81.1 8 77.20 odd 10
121.4.c.f.3.2 8 77.6 even 10
121.4.c.f.9.1 8 77.41 even 10
121.4.c.f.27.1 8 77.62 even 10
121.4.c.f.81.2 8 77.13 even 10
176.4.a.i.1.2 2 28.27 even 2
275.4.a.b.1.1 2 35.34 odd 2
275.4.b.c.199.1 4 35.13 even 4
275.4.b.c.199.4 4 35.27 even 4
539.4.a.e.1.2 2 1.1 even 1 trivial
704.4.a.n.1.1 2 56.27 even 2
704.4.a.p.1.2 2 56.13 odd 2
1089.4.a.v.1.2 2 231.230 odd 2
1584.4.a.bc.1.1 2 84.83 odd 2
1859.4.a.a.1.1 2 91.90 odd 2
1936.4.a.w.1.2 2 308.307 odd 2
2475.4.a.q.1.2 2 105.104 even 2