Properties

Label 462.4.a.d.1.1
Level $462$
Weight $4$
Character 462.1
Self dual yes
Analytic conductor $27.259$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 462 = 2 \cdot 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 462.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(27.2588824227\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 462.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -7.00000 q^{5} -6.00000 q^{6} +7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -7.00000 q^{5} -6.00000 q^{6} +7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +14.0000 q^{10} +11.0000 q^{11} +12.0000 q^{12} -67.0000 q^{13} -14.0000 q^{14} -21.0000 q^{15} +16.0000 q^{16} +30.0000 q^{17} -18.0000 q^{18} -7.00000 q^{19} -28.0000 q^{20} +21.0000 q^{21} -22.0000 q^{22} +28.0000 q^{23} -24.0000 q^{24} -76.0000 q^{25} +134.000 q^{26} +27.0000 q^{27} +28.0000 q^{28} +121.000 q^{29} +42.0000 q^{30} -310.000 q^{31} -32.0000 q^{32} +33.0000 q^{33} -60.0000 q^{34} -49.0000 q^{35} +36.0000 q^{36} -71.0000 q^{37} +14.0000 q^{38} -201.000 q^{39} +56.0000 q^{40} -180.000 q^{41} -42.0000 q^{42} -108.000 q^{43} +44.0000 q^{44} -63.0000 q^{45} -56.0000 q^{46} +71.0000 q^{47} +48.0000 q^{48} +49.0000 q^{49} +152.000 q^{50} +90.0000 q^{51} -268.000 q^{52} +128.000 q^{53} -54.0000 q^{54} -77.0000 q^{55} -56.0000 q^{56} -21.0000 q^{57} -242.000 q^{58} -429.000 q^{59} -84.0000 q^{60} +22.0000 q^{61} +620.000 q^{62} +63.0000 q^{63} +64.0000 q^{64} +469.000 q^{65} -66.0000 q^{66} -803.000 q^{67} +120.000 q^{68} +84.0000 q^{69} +98.0000 q^{70} +468.000 q^{71} -72.0000 q^{72} -117.000 q^{73} +142.000 q^{74} -228.000 q^{75} -28.0000 q^{76} +77.0000 q^{77} +402.000 q^{78} -96.0000 q^{79} -112.000 q^{80} +81.0000 q^{81} +360.000 q^{82} -1122.00 q^{83} +84.0000 q^{84} -210.000 q^{85} +216.000 q^{86} +363.000 q^{87} -88.0000 q^{88} -1146.00 q^{89} +126.000 q^{90} -469.000 q^{91} +112.000 q^{92} -930.000 q^{93} -142.000 q^{94} +49.0000 q^{95} -96.0000 q^{96} -92.0000 q^{97} -98.0000 q^{98} +99.0000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −0.707107
\(3\) 3.00000 0.577350
\(4\) 4.00000 0.500000
\(5\) −7.00000 −0.626099 −0.313050 0.949737i \(-0.601351\pi\)
−0.313050 + 0.949737i \(0.601351\pi\)
\(6\) −6.00000 −0.408248
\(7\) 7.00000 0.377964
\(8\) −8.00000 −0.353553
\(9\) 9.00000 0.333333
\(10\) 14.0000 0.442719
\(11\) 11.0000 0.301511
\(12\) 12.0000 0.288675
\(13\) −67.0000 −1.42942 −0.714710 0.699421i \(-0.753441\pi\)
−0.714710 + 0.699421i \(0.753441\pi\)
\(14\) −14.0000 −0.267261
\(15\) −21.0000 −0.361478
\(16\) 16.0000 0.250000
\(17\) 30.0000 0.428004 0.214002 0.976833i \(-0.431350\pi\)
0.214002 + 0.976833i \(0.431350\pi\)
\(18\) −18.0000 −0.235702
\(19\) −7.00000 −0.0845216 −0.0422608 0.999107i \(-0.513456\pi\)
−0.0422608 + 0.999107i \(0.513456\pi\)
\(20\) −28.0000 −0.313050
\(21\) 21.0000 0.218218
\(22\) −22.0000 −0.213201
\(23\) 28.0000 0.253844 0.126922 0.991913i \(-0.459490\pi\)
0.126922 + 0.991913i \(0.459490\pi\)
\(24\) −24.0000 −0.204124
\(25\) −76.0000 −0.608000
\(26\) 134.000 1.01075
\(27\) 27.0000 0.192450
\(28\) 28.0000 0.188982
\(29\) 121.000 0.774798 0.387399 0.921912i \(-0.373374\pi\)
0.387399 + 0.921912i \(0.373374\pi\)
\(30\) 42.0000 0.255604
\(31\) −310.000 −1.79605 −0.898027 0.439941i \(-0.854999\pi\)
−0.898027 + 0.439941i \(0.854999\pi\)
\(32\) −32.0000 −0.176777
\(33\) 33.0000 0.174078
\(34\) −60.0000 −0.302645
\(35\) −49.0000 −0.236643
\(36\) 36.0000 0.166667
\(37\) −71.0000 −0.315468 −0.157734 0.987482i \(-0.550419\pi\)
−0.157734 + 0.987482i \(0.550419\pi\)
\(38\) 14.0000 0.0597658
\(39\) −201.000 −0.825276
\(40\) 56.0000 0.221359
\(41\) −180.000 −0.685641 −0.342820 0.939401i \(-0.611382\pi\)
−0.342820 + 0.939401i \(0.611382\pi\)
\(42\) −42.0000 −0.154303
\(43\) −108.000 −0.383020 −0.191510 0.981491i \(-0.561338\pi\)
−0.191510 + 0.981491i \(0.561338\pi\)
\(44\) 44.0000 0.150756
\(45\) −63.0000 −0.208700
\(46\) −56.0000 −0.179495
\(47\) 71.0000 0.220349 0.110175 0.993912i \(-0.464859\pi\)
0.110175 + 0.993912i \(0.464859\pi\)
\(48\) 48.0000 0.144338
\(49\) 49.0000 0.142857
\(50\) 152.000 0.429921
\(51\) 90.0000 0.247108
\(52\) −268.000 −0.714710
\(53\) 128.000 0.331739 0.165869 0.986148i \(-0.446957\pi\)
0.165869 + 0.986148i \(0.446957\pi\)
\(54\) −54.0000 −0.136083
\(55\) −77.0000 −0.188776
\(56\) −56.0000 −0.133631
\(57\) −21.0000 −0.0487986
\(58\) −242.000 −0.547865
\(59\) −429.000 −0.946628 −0.473314 0.880894i \(-0.656942\pi\)
−0.473314 + 0.880894i \(0.656942\pi\)
\(60\) −84.0000 −0.180739
\(61\) 22.0000 0.0461772 0.0230886 0.999733i \(-0.492650\pi\)
0.0230886 + 0.999733i \(0.492650\pi\)
\(62\) 620.000 1.27000
\(63\) 63.0000 0.125988
\(64\) 64.0000 0.125000
\(65\) 469.000 0.894958
\(66\) −66.0000 −0.123091
\(67\) −803.000 −1.46421 −0.732105 0.681192i \(-0.761462\pi\)
−0.732105 + 0.681192i \(0.761462\pi\)
\(68\) 120.000 0.214002
\(69\) 84.0000 0.146557
\(70\) 98.0000 0.167332
\(71\) 468.000 0.782273 0.391136 0.920333i \(-0.372082\pi\)
0.391136 + 0.920333i \(0.372082\pi\)
\(72\) −72.0000 −0.117851
\(73\) −117.000 −0.187586 −0.0937932 0.995592i \(-0.529899\pi\)
−0.0937932 + 0.995592i \(0.529899\pi\)
\(74\) 142.000 0.223070
\(75\) −228.000 −0.351029
\(76\) −28.0000 −0.0422608
\(77\) 77.0000 0.113961
\(78\) 402.000 0.583558
\(79\) −96.0000 −0.136720 −0.0683598 0.997661i \(-0.521777\pi\)
−0.0683598 + 0.997661i \(0.521777\pi\)
\(80\) −112.000 −0.156525
\(81\) 81.0000 0.111111
\(82\) 360.000 0.484821
\(83\) −1122.00 −1.48380 −0.741901 0.670510i \(-0.766075\pi\)
−0.741901 + 0.670510i \(0.766075\pi\)
\(84\) 84.0000 0.109109
\(85\) −210.000 −0.267973
\(86\) 216.000 0.270836
\(87\) 363.000 0.447330
\(88\) −88.0000 −0.106600
\(89\) −1146.00 −1.36490 −0.682448 0.730934i \(-0.739085\pi\)
−0.682448 + 0.730934i \(0.739085\pi\)
\(90\) 126.000 0.147573
\(91\) −469.000 −0.540270
\(92\) 112.000 0.126922
\(93\) −930.000 −1.03695
\(94\) −142.000 −0.155810
\(95\) 49.0000 0.0529189
\(96\) −96.0000 −0.102062
\(97\) −92.0000 −0.0963009 −0.0481504 0.998840i \(-0.515333\pi\)
−0.0481504 + 0.998840i \(0.515333\pi\)
\(98\) −98.0000 −0.101015
\(99\) 99.0000 0.100504
\(100\) −304.000 −0.304000
\(101\) −202.000 −0.199007 −0.0995037 0.995037i \(-0.531726\pi\)
−0.0995037 + 0.995037i \(0.531726\pi\)
\(102\) −180.000 −0.174732
\(103\) −1798.00 −1.72002 −0.860011 0.510276i \(-0.829543\pi\)
−0.860011 + 0.510276i \(0.829543\pi\)
\(104\) 536.000 0.505376
\(105\) −147.000 −0.136626
\(106\) −256.000 −0.234575
\(107\) −1431.00 −1.29290 −0.646449 0.762958i \(-0.723747\pi\)
−0.646449 + 0.762958i \(0.723747\pi\)
\(108\) 108.000 0.0962250
\(109\) −278.000 −0.244290 −0.122145 0.992512i \(-0.538977\pi\)
−0.122145 + 0.992512i \(0.538977\pi\)
\(110\) 154.000 0.133485
\(111\) −213.000 −0.182136
\(112\) 112.000 0.0944911
\(113\) −2322.00 −1.93306 −0.966528 0.256560i \(-0.917411\pi\)
−0.966528 + 0.256560i \(0.917411\pi\)
\(114\) 42.0000 0.0345058
\(115\) −196.000 −0.158931
\(116\) 484.000 0.387399
\(117\) −603.000 −0.476473
\(118\) 858.000 0.669367
\(119\) 210.000 0.161770
\(120\) 168.000 0.127802
\(121\) 121.000 0.0909091
\(122\) −44.0000 −0.0326522
\(123\) −540.000 −0.395855
\(124\) −1240.00 −0.898027
\(125\) 1407.00 1.00677
\(126\) −126.000 −0.0890871
\(127\) 2254.00 1.57488 0.787442 0.616389i \(-0.211405\pi\)
0.787442 + 0.616389i \(0.211405\pi\)
\(128\) −128.000 −0.0883883
\(129\) −324.000 −0.221137
\(130\) −938.000 −0.632831
\(131\) 2196.00 1.46462 0.732311 0.680971i \(-0.238442\pi\)
0.732311 + 0.680971i \(0.238442\pi\)
\(132\) 132.000 0.0870388
\(133\) −49.0000 −0.0319462
\(134\) 1606.00 1.03535
\(135\) −189.000 −0.120493
\(136\) −240.000 −0.151322
\(137\) 766.000 0.477692 0.238846 0.971057i \(-0.423231\pi\)
0.238846 + 0.971057i \(0.423231\pi\)
\(138\) −168.000 −0.103631
\(139\) −1080.00 −0.659024 −0.329512 0.944151i \(-0.606884\pi\)
−0.329512 + 0.944151i \(0.606884\pi\)
\(140\) −196.000 −0.118322
\(141\) 213.000 0.127219
\(142\) −936.000 −0.553151
\(143\) −737.000 −0.430986
\(144\) 144.000 0.0833333
\(145\) −847.000 −0.485100
\(146\) 234.000 0.132644
\(147\) 147.000 0.0824786
\(148\) −284.000 −0.157734
\(149\) 1373.00 0.754903 0.377451 0.926029i \(-0.376800\pi\)
0.377451 + 0.926029i \(0.376800\pi\)
\(150\) 456.000 0.248215
\(151\) −3210.00 −1.72997 −0.864987 0.501794i \(-0.832673\pi\)
−0.864987 + 0.501794i \(0.832673\pi\)
\(152\) 56.0000 0.0298829
\(153\) 270.000 0.142668
\(154\) −154.000 −0.0805823
\(155\) 2170.00 1.12451
\(156\) −804.000 −0.412638
\(157\) 452.000 0.229768 0.114884 0.993379i \(-0.463350\pi\)
0.114884 + 0.993379i \(0.463350\pi\)
\(158\) 192.000 0.0966753
\(159\) 384.000 0.191529
\(160\) 224.000 0.110680
\(161\) 196.000 0.0959439
\(162\) −162.000 −0.0785674
\(163\) −11.0000 −0.00528581 −0.00264290 0.999997i \(-0.500841\pi\)
−0.00264290 + 0.999997i \(0.500841\pi\)
\(164\) −720.000 −0.342820
\(165\) −231.000 −0.108990
\(166\) 2244.00 1.04921
\(167\) 504.000 0.233537 0.116769 0.993159i \(-0.462746\pi\)
0.116769 + 0.993159i \(0.462746\pi\)
\(168\) −168.000 −0.0771517
\(169\) 2292.00 1.04324
\(170\) 420.000 0.189485
\(171\) −63.0000 −0.0281739
\(172\) −432.000 −0.191510
\(173\) 1688.00 0.741828 0.370914 0.928667i \(-0.379044\pi\)
0.370914 + 0.928667i \(0.379044\pi\)
\(174\) −726.000 −0.316310
\(175\) −532.000 −0.229802
\(176\) 176.000 0.0753778
\(177\) −1287.00 −0.546536
\(178\) 2292.00 0.965127
\(179\) 460.000 0.192078 0.0960391 0.995378i \(-0.469383\pi\)
0.0960391 + 0.995378i \(0.469383\pi\)
\(180\) −252.000 −0.104350
\(181\) 2296.00 0.942875 0.471437 0.881900i \(-0.343735\pi\)
0.471437 + 0.881900i \(0.343735\pi\)
\(182\) 938.000 0.382028
\(183\) 66.0000 0.0266604
\(184\) −224.000 −0.0897473
\(185\) 497.000 0.197514
\(186\) 1860.00 0.733236
\(187\) 330.000 0.129048
\(188\) 284.000 0.110175
\(189\) 189.000 0.0727393
\(190\) −98.0000 −0.0374193
\(191\) 878.000 0.332617 0.166309 0.986074i \(-0.446815\pi\)
0.166309 + 0.986074i \(0.446815\pi\)
\(192\) 192.000 0.0721688
\(193\) −2268.00 −0.845877 −0.422938 0.906158i \(-0.639001\pi\)
−0.422938 + 0.906158i \(0.639001\pi\)
\(194\) 184.000 0.0680950
\(195\) 1407.00 0.516704
\(196\) 196.000 0.0714286
\(197\) 5298.00 1.91608 0.958038 0.286642i \(-0.0925390\pi\)
0.958038 + 0.286642i \(0.0925390\pi\)
\(198\) −198.000 −0.0710669
\(199\) 1044.00 0.371895 0.185948 0.982560i \(-0.440464\pi\)
0.185948 + 0.982560i \(0.440464\pi\)
\(200\) 608.000 0.214960
\(201\) −2409.00 −0.845362
\(202\) 404.000 0.140720
\(203\) 847.000 0.292846
\(204\) 360.000 0.123554
\(205\) 1260.00 0.429279
\(206\) 3596.00 1.21624
\(207\) 252.000 0.0846145
\(208\) −1072.00 −0.357355
\(209\) −77.0000 −0.0254842
\(210\) 294.000 0.0966092
\(211\) 4566.00 1.48975 0.744873 0.667206i \(-0.232510\pi\)
0.744873 + 0.667206i \(0.232510\pi\)
\(212\) 512.000 0.165869
\(213\) 1404.00 0.451646
\(214\) 2862.00 0.914216
\(215\) 756.000 0.239808
\(216\) −216.000 −0.0680414
\(217\) −2170.00 −0.678844
\(218\) 556.000 0.172739
\(219\) −351.000 −0.108303
\(220\) −308.000 −0.0943880
\(221\) −2010.00 −0.611797
\(222\) 426.000 0.128789
\(223\) 198.000 0.0594577 0.0297288 0.999558i \(-0.490536\pi\)
0.0297288 + 0.999558i \(0.490536\pi\)
\(224\) −224.000 −0.0668153
\(225\) −684.000 −0.202667
\(226\) 4644.00 1.36688
\(227\) 3762.00 1.09997 0.549984 0.835175i \(-0.314634\pi\)
0.549984 + 0.835175i \(0.314634\pi\)
\(228\) −84.0000 −0.0243993
\(229\) −704.000 −0.203151 −0.101576 0.994828i \(-0.532388\pi\)
−0.101576 + 0.994828i \(0.532388\pi\)
\(230\) 392.000 0.112381
\(231\) 231.000 0.0657952
\(232\) −968.000 −0.273932
\(233\) 2210.00 0.621382 0.310691 0.950511i \(-0.399440\pi\)
0.310691 + 0.950511i \(0.399440\pi\)
\(234\) 1206.00 0.336917
\(235\) −497.000 −0.137960
\(236\) −1716.00 −0.473314
\(237\) −288.000 −0.0789351
\(238\) −420.000 −0.114389
\(239\) 2451.00 0.663356 0.331678 0.943393i \(-0.392385\pi\)
0.331678 + 0.943393i \(0.392385\pi\)
\(240\) −336.000 −0.0903696
\(241\) 25.0000 0.00668212 0.00334106 0.999994i \(-0.498937\pi\)
0.00334106 + 0.999994i \(0.498937\pi\)
\(242\) −242.000 −0.0642824
\(243\) 243.000 0.0641500
\(244\) 88.0000 0.0230886
\(245\) −343.000 −0.0894427
\(246\) 1080.00 0.279912
\(247\) 469.000 0.120817
\(248\) 2480.00 0.635001
\(249\) −3366.00 −0.856673
\(250\) −2814.00 −0.711892
\(251\) −5267.00 −1.32450 −0.662251 0.749282i \(-0.730399\pi\)
−0.662251 + 0.749282i \(0.730399\pi\)
\(252\) 252.000 0.0629941
\(253\) 308.000 0.0765367
\(254\) −4508.00 −1.11361
\(255\) −630.000 −0.154714
\(256\) 256.000 0.0625000
\(257\) −1623.00 −0.393930 −0.196965 0.980411i \(-0.563109\pi\)
−0.196965 + 0.980411i \(0.563109\pi\)
\(258\) 648.000 0.156367
\(259\) −497.000 −0.119236
\(260\) 1876.00 0.447479
\(261\) 1089.00 0.258266
\(262\) −4392.00 −1.03564
\(263\) −2475.00 −0.580285 −0.290143 0.956983i \(-0.593703\pi\)
−0.290143 + 0.956983i \(0.593703\pi\)
\(264\) −264.000 −0.0615457
\(265\) −896.000 −0.207701
\(266\) 98.0000 0.0225893
\(267\) −3438.00 −0.788023
\(268\) −3212.00 −0.732105
\(269\) 1746.00 0.395745 0.197873 0.980228i \(-0.436597\pi\)
0.197873 + 0.980228i \(0.436597\pi\)
\(270\) 378.000 0.0852013
\(271\) 1653.00 0.370526 0.185263 0.982689i \(-0.440686\pi\)
0.185263 + 0.982689i \(0.440686\pi\)
\(272\) 480.000 0.107001
\(273\) −1407.00 −0.311925
\(274\) −1532.00 −0.337779
\(275\) −836.000 −0.183319
\(276\) 336.000 0.0732783
\(277\) −2056.00 −0.445968 −0.222984 0.974822i \(-0.571580\pi\)
−0.222984 + 0.974822i \(0.571580\pi\)
\(278\) 2160.00 0.466001
\(279\) −2790.00 −0.598684
\(280\) 392.000 0.0836660
\(281\) 837.000 0.177691 0.0888456 0.996045i \(-0.471682\pi\)
0.0888456 + 0.996045i \(0.471682\pi\)
\(282\) −426.000 −0.0899572
\(283\) 8969.00 1.88393 0.941964 0.335713i \(-0.108977\pi\)
0.941964 + 0.335713i \(0.108977\pi\)
\(284\) 1872.00 0.391136
\(285\) 147.000 0.0305527
\(286\) 1474.00 0.304753
\(287\) −1260.00 −0.259148
\(288\) −288.000 −0.0589256
\(289\) −4013.00 −0.816813
\(290\) 1694.00 0.343018
\(291\) −276.000 −0.0555993
\(292\) −468.000 −0.0937932
\(293\) 5120.00 1.02087 0.510433 0.859918i \(-0.329485\pi\)
0.510433 + 0.859918i \(0.329485\pi\)
\(294\) −294.000 −0.0583212
\(295\) 3003.00 0.592683
\(296\) 568.000 0.111535
\(297\) 297.000 0.0580259
\(298\) −2746.00 −0.533797
\(299\) −1876.00 −0.362849
\(300\) −912.000 −0.175514
\(301\) −756.000 −0.144768
\(302\) 6420.00 1.22328
\(303\) −606.000 −0.114897
\(304\) −112.000 −0.0211304
\(305\) −154.000 −0.0289115
\(306\) −540.000 −0.100882
\(307\) −2892.00 −0.537639 −0.268819 0.963191i \(-0.586634\pi\)
−0.268819 + 0.963191i \(0.586634\pi\)
\(308\) 308.000 0.0569803
\(309\) −5394.00 −0.993055
\(310\) −4340.00 −0.795147
\(311\) 3208.00 0.584916 0.292458 0.956278i \(-0.405527\pi\)
0.292458 + 0.956278i \(0.405527\pi\)
\(312\) 1608.00 0.291779
\(313\) 3550.00 0.641079 0.320540 0.947235i \(-0.396136\pi\)
0.320540 + 0.947235i \(0.396136\pi\)
\(314\) −904.000 −0.162470
\(315\) −441.000 −0.0788811
\(316\) −384.000 −0.0683598
\(317\) −5524.00 −0.978734 −0.489367 0.872078i \(-0.662772\pi\)
−0.489367 + 0.872078i \(0.662772\pi\)
\(318\) −768.000 −0.135432
\(319\) 1331.00 0.233610
\(320\) −448.000 −0.0782624
\(321\) −4293.00 −0.746454
\(322\) −392.000 −0.0678426
\(323\) −210.000 −0.0361756
\(324\) 324.000 0.0555556
\(325\) 5092.00 0.869087
\(326\) 22.0000 0.00373763
\(327\) −834.000 −0.141041
\(328\) 1440.00 0.242411
\(329\) 497.000 0.0832842
\(330\) 462.000 0.0770675
\(331\) −2144.00 −0.356027 −0.178013 0.984028i \(-0.556967\pi\)
−0.178013 + 0.984028i \(0.556967\pi\)
\(332\) −4488.00 −0.741901
\(333\) −639.000 −0.105156
\(334\) −1008.00 −0.165136
\(335\) 5621.00 0.916740
\(336\) 336.000 0.0545545
\(337\) −884.000 −0.142892 −0.0714459 0.997444i \(-0.522761\pi\)
−0.0714459 + 0.997444i \(0.522761\pi\)
\(338\) −4584.00 −0.737683
\(339\) −6966.00 −1.11605
\(340\) −840.000 −0.133986
\(341\) −3410.00 −0.541530
\(342\) 126.000 0.0199219
\(343\) 343.000 0.0539949
\(344\) 864.000 0.135418
\(345\) −588.000 −0.0917590
\(346\) −3376.00 −0.524552
\(347\) −4040.00 −0.625010 −0.312505 0.949916i \(-0.601168\pi\)
−0.312505 + 0.949916i \(0.601168\pi\)
\(348\) 1452.00 0.223665
\(349\) 1225.00 0.187888 0.0939438 0.995578i \(-0.470053\pi\)
0.0939438 + 0.995578i \(0.470053\pi\)
\(350\) 1064.00 0.162495
\(351\) −1809.00 −0.275092
\(352\) −352.000 −0.0533002
\(353\) 2483.00 0.374382 0.187191 0.982324i \(-0.440062\pi\)
0.187191 + 0.982324i \(0.440062\pi\)
\(354\) 2574.00 0.386459
\(355\) −3276.00 −0.489780
\(356\) −4584.00 −0.682448
\(357\) 630.000 0.0933981
\(358\) −920.000 −0.135820
\(359\) 7804.00 1.14730 0.573648 0.819102i \(-0.305528\pi\)
0.573648 + 0.819102i \(0.305528\pi\)
\(360\) 504.000 0.0737865
\(361\) −6810.00 −0.992856
\(362\) −4592.00 −0.666713
\(363\) 363.000 0.0524864
\(364\) −1876.00 −0.270135
\(365\) 819.000 0.117448
\(366\) −132.000 −0.0188518
\(367\) 3074.00 0.437225 0.218612 0.975812i \(-0.429847\pi\)
0.218612 + 0.975812i \(0.429847\pi\)
\(368\) 448.000 0.0634609
\(369\) −1620.00 −0.228547
\(370\) −994.000 −0.139664
\(371\) 896.000 0.125385
\(372\) −3720.00 −0.518476
\(373\) 674.000 0.0935614 0.0467807 0.998905i \(-0.485104\pi\)
0.0467807 + 0.998905i \(0.485104\pi\)
\(374\) −660.000 −0.0912508
\(375\) 4221.00 0.581257
\(376\) −568.000 −0.0779052
\(377\) −8107.00 −1.10751
\(378\) −378.000 −0.0514344
\(379\) 12051.0 1.63329 0.816647 0.577138i \(-0.195830\pi\)
0.816647 + 0.577138i \(0.195830\pi\)
\(380\) 196.000 0.0264594
\(381\) 6762.00 0.909259
\(382\) −1756.00 −0.235196
\(383\) −12664.0 −1.68956 −0.844778 0.535116i \(-0.820268\pi\)
−0.844778 + 0.535116i \(0.820268\pi\)
\(384\) −384.000 −0.0510310
\(385\) −539.000 −0.0713506
\(386\) 4536.00 0.598125
\(387\) −972.000 −0.127673
\(388\) −368.000 −0.0481504
\(389\) 264.000 0.0344096 0.0172048 0.999852i \(-0.494523\pi\)
0.0172048 + 0.999852i \(0.494523\pi\)
\(390\) −2814.00 −0.365365
\(391\) 840.000 0.108646
\(392\) −392.000 −0.0505076
\(393\) 6588.00 0.845600
\(394\) −10596.0 −1.35487
\(395\) 672.000 0.0856000
\(396\) 396.000 0.0502519
\(397\) −14394.0 −1.81968 −0.909842 0.414956i \(-0.863797\pi\)
−0.909842 + 0.414956i \(0.863797\pi\)
\(398\) −2088.00 −0.262970
\(399\) −147.000 −0.0184441
\(400\) −1216.00 −0.152000
\(401\) −12304.0 −1.53225 −0.766125 0.642691i \(-0.777818\pi\)
−0.766125 + 0.642691i \(0.777818\pi\)
\(402\) 4818.00 0.597761
\(403\) 20770.0 2.56731
\(404\) −808.000 −0.0995037
\(405\) −567.000 −0.0695666
\(406\) −1694.00 −0.207073
\(407\) −781.000 −0.0951173
\(408\) −720.000 −0.0873660
\(409\) −3154.00 −0.381309 −0.190654 0.981657i \(-0.561061\pi\)
−0.190654 + 0.981657i \(0.561061\pi\)
\(410\) −2520.00 −0.303546
\(411\) 2298.00 0.275796
\(412\) −7192.00 −0.860011
\(413\) −3003.00 −0.357792
\(414\) −504.000 −0.0598315
\(415\) 7854.00 0.929006
\(416\) 2144.00 0.252688
\(417\) −3240.00 −0.380488
\(418\) 154.000 0.0180201
\(419\) −4635.00 −0.540417 −0.270208 0.962802i \(-0.587093\pi\)
−0.270208 + 0.962802i \(0.587093\pi\)
\(420\) −588.000 −0.0683130
\(421\) −7265.00 −0.841032 −0.420516 0.907285i \(-0.638151\pi\)
−0.420516 + 0.907285i \(0.638151\pi\)
\(422\) −9132.00 −1.05341
\(423\) 639.000 0.0734497
\(424\) −1024.00 −0.117287
\(425\) −2280.00 −0.260226
\(426\) −2808.00 −0.319362
\(427\) 154.000 0.0174534
\(428\) −5724.00 −0.646449
\(429\) −2211.00 −0.248830
\(430\) −1512.00 −0.169570
\(431\) 7589.00 0.848142 0.424071 0.905629i \(-0.360601\pi\)
0.424071 + 0.905629i \(0.360601\pi\)
\(432\) 432.000 0.0481125
\(433\) 5164.00 0.573132 0.286566 0.958061i \(-0.407486\pi\)
0.286566 + 0.958061i \(0.407486\pi\)
\(434\) 4340.00 0.480015
\(435\) −2541.00 −0.280073
\(436\) −1112.00 −0.122145
\(437\) −196.000 −0.0214553
\(438\) 702.000 0.0765819
\(439\) 14221.0 1.54608 0.773042 0.634354i \(-0.218734\pi\)
0.773042 + 0.634354i \(0.218734\pi\)
\(440\) 616.000 0.0667424
\(441\) 441.000 0.0476190
\(442\) 4020.00 0.432606
\(443\) 6924.00 0.742594 0.371297 0.928514i \(-0.378913\pi\)
0.371297 + 0.928514i \(0.378913\pi\)
\(444\) −852.000 −0.0910679
\(445\) 8022.00 0.854560
\(446\) −396.000 −0.0420429
\(447\) 4119.00 0.435843
\(448\) 448.000 0.0472456
\(449\) 2524.00 0.265289 0.132645 0.991164i \(-0.457653\pi\)
0.132645 + 0.991164i \(0.457653\pi\)
\(450\) 1368.00 0.143307
\(451\) −1980.00 −0.206729
\(452\) −9288.00 −0.966528
\(453\) −9630.00 −0.998801
\(454\) −7524.00 −0.777795
\(455\) 3283.00 0.338262
\(456\) 168.000 0.0172529
\(457\) 1928.00 0.197348 0.0986740 0.995120i \(-0.468540\pi\)
0.0986740 + 0.995120i \(0.468540\pi\)
\(458\) 1408.00 0.143650
\(459\) 810.000 0.0823694
\(460\) −784.000 −0.0794656
\(461\) −12852.0 −1.29843 −0.649216 0.760604i \(-0.724903\pi\)
−0.649216 + 0.760604i \(0.724903\pi\)
\(462\) −462.000 −0.0465242
\(463\) 4741.00 0.475881 0.237941 0.971280i \(-0.423528\pi\)
0.237941 + 0.971280i \(0.423528\pi\)
\(464\) 1936.00 0.193699
\(465\) 6510.00 0.649234
\(466\) −4420.00 −0.439383
\(467\) −6809.00 −0.674696 −0.337348 0.941380i \(-0.609530\pi\)
−0.337348 + 0.941380i \(0.609530\pi\)
\(468\) −2412.00 −0.238237
\(469\) −5621.00 −0.553419
\(470\) 994.000 0.0975528
\(471\) 1356.00 0.132656
\(472\) 3432.00 0.334683
\(473\) −1188.00 −0.115485
\(474\) 576.000 0.0558155
\(475\) 532.000 0.0513891
\(476\) 840.000 0.0808852
\(477\) 1152.00 0.110580
\(478\) −4902.00 −0.469063
\(479\) 1734.00 0.165404 0.0827020 0.996574i \(-0.473645\pi\)
0.0827020 + 0.996574i \(0.473645\pi\)
\(480\) 672.000 0.0639010
\(481\) 4757.00 0.450937
\(482\) −50.0000 −0.00472497
\(483\) 588.000 0.0553932
\(484\) 484.000 0.0454545
\(485\) 644.000 0.0602939
\(486\) −486.000 −0.0453609
\(487\) −7368.00 −0.685577 −0.342788 0.939413i \(-0.611371\pi\)
−0.342788 + 0.939413i \(0.611371\pi\)
\(488\) −176.000 −0.0163261
\(489\) −33.0000 −0.00305176
\(490\) 686.000 0.0632456
\(491\) 13201.0 1.21335 0.606673 0.794952i \(-0.292504\pi\)
0.606673 + 0.794952i \(0.292504\pi\)
\(492\) −2160.00 −0.197927
\(493\) 3630.00 0.331617
\(494\) −938.000 −0.0854304
\(495\) −693.000 −0.0629253
\(496\) −4960.00 −0.449013
\(497\) 3276.00 0.295671
\(498\) 6732.00 0.605759
\(499\) 16539.0 1.48374 0.741871 0.670543i \(-0.233939\pi\)
0.741871 + 0.670543i \(0.233939\pi\)
\(500\) 5628.00 0.503384
\(501\) 1512.00 0.134833
\(502\) 10534.0 0.936565
\(503\) −14780.0 −1.31015 −0.655077 0.755562i \(-0.727364\pi\)
−0.655077 + 0.755562i \(0.727364\pi\)
\(504\) −504.000 −0.0445435
\(505\) 1414.00 0.124598
\(506\) −616.000 −0.0541196
\(507\) 6876.00 0.602315
\(508\) 9016.00 0.787442
\(509\) −11766.0 −1.02459 −0.512297 0.858808i \(-0.671205\pi\)
−0.512297 + 0.858808i \(0.671205\pi\)
\(510\) 1260.00 0.109399
\(511\) −819.000 −0.0709010
\(512\) −512.000 −0.0441942
\(513\) −189.000 −0.0162662
\(514\) 3246.00 0.278550
\(515\) 12586.0 1.07690
\(516\) −1296.00 −0.110568
\(517\) 781.000 0.0664378
\(518\) 994.000 0.0843125
\(519\) 5064.00 0.428295
\(520\) −3752.00 −0.316416
\(521\) −13503.0 −1.13546 −0.567732 0.823213i \(-0.692179\pi\)
−0.567732 + 0.823213i \(0.692179\pi\)
\(522\) −2178.00 −0.182622
\(523\) −18681.0 −1.56188 −0.780940 0.624606i \(-0.785259\pi\)
−0.780940 + 0.624606i \(0.785259\pi\)
\(524\) 8784.00 0.732311
\(525\) −1596.00 −0.132676
\(526\) 4950.00 0.410324
\(527\) −9300.00 −0.768718
\(528\) 528.000 0.0435194
\(529\) −11383.0 −0.935563
\(530\) 1792.00 0.146867
\(531\) −3861.00 −0.315543
\(532\) −196.000 −0.0159731
\(533\) 12060.0 0.980069
\(534\) 6876.00 0.557217
\(535\) 10017.0 0.809482
\(536\) 6424.00 0.517676
\(537\) 1380.00 0.110896
\(538\) −3492.00 −0.279834
\(539\) 539.000 0.0430730
\(540\) −756.000 −0.0602464
\(541\) −2000.00 −0.158940 −0.0794702 0.996837i \(-0.525323\pi\)
−0.0794702 + 0.996837i \(0.525323\pi\)
\(542\) −3306.00 −0.262002
\(543\) 6888.00 0.544369
\(544\) −960.000 −0.0756611
\(545\) 1946.00 0.152950
\(546\) 2814.00 0.220564
\(547\) 19368.0 1.51392 0.756961 0.653459i \(-0.226683\pi\)
0.756961 + 0.653459i \(0.226683\pi\)
\(548\) 3064.00 0.238846
\(549\) 198.000 0.0153924
\(550\) 1672.00 0.129626
\(551\) −847.000 −0.0654871
\(552\) −672.000 −0.0518156
\(553\) −672.000 −0.0516751
\(554\) 4112.00 0.315347
\(555\) 1491.00 0.114035
\(556\) −4320.00 −0.329512
\(557\) 4669.00 0.355174 0.177587 0.984105i \(-0.443171\pi\)
0.177587 + 0.984105i \(0.443171\pi\)
\(558\) 5580.00 0.423334
\(559\) 7236.00 0.547496
\(560\) −784.000 −0.0591608
\(561\) 990.000 0.0745059
\(562\) −1674.00 −0.125647
\(563\) −9324.00 −0.697975 −0.348987 0.937127i \(-0.613474\pi\)
−0.348987 + 0.937127i \(0.613474\pi\)
\(564\) 852.000 0.0636093
\(565\) 16254.0 1.21028
\(566\) −17938.0 −1.33214
\(567\) 567.000 0.0419961
\(568\) −3744.00 −0.276575
\(569\) 25646.0 1.88952 0.944759 0.327765i \(-0.106295\pi\)
0.944759 + 0.327765i \(0.106295\pi\)
\(570\) −294.000 −0.0216040
\(571\) −12928.0 −0.947496 −0.473748 0.880661i \(-0.657099\pi\)
−0.473748 + 0.880661i \(0.657099\pi\)
\(572\) −2948.00 −0.215493
\(573\) 2634.00 0.192037
\(574\) 2520.00 0.183245
\(575\) −2128.00 −0.154337
\(576\) 576.000 0.0416667
\(577\) −14862.0 −1.07229 −0.536147 0.844125i \(-0.680121\pi\)
−0.536147 + 0.844125i \(0.680121\pi\)
\(578\) 8026.00 0.577574
\(579\) −6804.00 −0.488367
\(580\) −3388.00 −0.242550
\(581\) −7854.00 −0.560824
\(582\) 552.000 0.0393147
\(583\) 1408.00 0.100023
\(584\) 936.000 0.0663218
\(585\) 4221.00 0.298319
\(586\) −10240.0 −0.721861
\(587\) −13383.0 −0.941015 −0.470507 0.882396i \(-0.655929\pi\)
−0.470507 + 0.882396i \(0.655929\pi\)
\(588\) 588.000 0.0412393
\(589\) 2170.00 0.151805
\(590\) −6006.00 −0.419090
\(591\) 15894.0 1.10625
\(592\) −1136.00 −0.0788671
\(593\) −10152.0 −0.703023 −0.351512 0.936184i \(-0.614332\pi\)
−0.351512 + 0.936184i \(0.614332\pi\)
\(594\) −594.000 −0.0410305
\(595\) −1470.00 −0.101284
\(596\) 5492.00 0.377451
\(597\) 3132.00 0.214714
\(598\) 3752.00 0.256573
\(599\) 16592.0 1.13177 0.565885 0.824484i \(-0.308534\pi\)
0.565885 + 0.824484i \(0.308534\pi\)
\(600\) 1824.00 0.124107
\(601\) −1001.00 −0.0679395 −0.0339698 0.999423i \(-0.510815\pi\)
−0.0339698 + 0.999423i \(0.510815\pi\)
\(602\) 1512.00 0.102366
\(603\) −7227.00 −0.488070
\(604\) −12840.0 −0.864987
\(605\) −847.000 −0.0569181
\(606\) 1212.00 0.0812444
\(607\) −18427.0 −1.23217 −0.616086 0.787679i \(-0.711283\pi\)
−0.616086 + 0.787679i \(0.711283\pi\)
\(608\) 224.000 0.0149414
\(609\) 2541.00 0.169075
\(610\) 308.000 0.0204435
\(611\) −4757.00 −0.314972
\(612\) 1080.00 0.0713340
\(613\) 5606.00 0.369371 0.184685 0.982798i \(-0.440873\pi\)
0.184685 + 0.982798i \(0.440873\pi\)
\(614\) 5784.00 0.380168
\(615\) 3780.00 0.247844
\(616\) −616.000 −0.0402911
\(617\) 6378.00 0.416157 0.208078 0.978112i \(-0.433279\pi\)
0.208078 + 0.978112i \(0.433279\pi\)
\(618\) 10788.0 0.702196
\(619\) −4070.00 −0.264276 −0.132138 0.991231i \(-0.542184\pi\)
−0.132138 + 0.991231i \(0.542184\pi\)
\(620\) 8680.00 0.562254
\(621\) 756.000 0.0488522
\(622\) −6416.00 −0.413598
\(623\) −8022.00 −0.515882
\(624\) −3216.00 −0.206319
\(625\) −349.000 −0.0223360
\(626\) −7100.00 −0.453312
\(627\) −231.000 −0.0147133
\(628\) 1808.00 0.114884
\(629\) −2130.00 −0.135022
\(630\) 882.000 0.0557773
\(631\) 23252.0 1.46695 0.733477 0.679715i \(-0.237896\pi\)
0.733477 + 0.679715i \(0.237896\pi\)
\(632\) 768.000 0.0483377
\(633\) 13698.0 0.860105
\(634\) 11048.0 0.692070
\(635\) −15778.0 −0.986033
\(636\) 1536.00 0.0957647
\(637\) −3283.00 −0.204203
\(638\) −2662.00 −0.165187
\(639\) 4212.00 0.260758
\(640\) 896.000 0.0553399
\(641\) 3758.00 0.231563 0.115782 0.993275i \(-0.463063\pi\)
0.115782 + 0.993275i \(0.463063\pi\)
\(642\) 8586.00 0.527823
\(643\) 13068.0 0.801480 0.400740 0.916192i \(-0.368753\pi\)
0.400740 + 0.916192i \(0.368753\pi\)
\(644\) 784.000 0.0479719
\(645\) 2268.00 0.138453
\(646\) 420.000 0.0255800
\(647\) −31027.0 −1.88531 −0.942656 0.333765i \(-0.891681\pi\)
−0.942656 + 0.333765i \(0.891681\pi\)
\(648\) −648.000 −0.0392837
\(649\) −4719.00 −0.285419
\(650\) −10184.0 −0.614537
\(651\) −6510.00 −0.391931
\(652\) −44.0000 −0.00264290
\(653\) −18456.0 −1.10603 −0.553016 0.833171i \(-0.686523\pi\)
−0.553016 + 0.833171i \(0.686523\pi\)
\(654\) 1668.00 0.0997308
\(655\) −15372.0 −0.916998
\(656\) −2880.00 −0.171410
\(657\) −1053.00 −0.0625288
\(658\) −994.000 −0.0588908
\(659\) 11141.0 0.658561 0.329281 0.944232i \(-0.393194\pi\)
0.329281 + 0.944232i \(0.393194\pi\)
\(660\) −924.000 −0.0544949
\(661\) 5172.00 0.304338 0.152169 0.988354i \(-0.451374\pi\)
0.152169 + 0.988354i \(0.451374\pi\)
\(662\) 4288.00 0.251749
\(663\) −6030.00 −0.353221
\(664\) 8976.00 0.524603
\(665\) 343.000 0.0200015
\(666\) 1278.00 0.0743566
\(667\) 3388.00 0.196677
\(668\) 2016.00 0.116769
\(669\) 594.000 0.0343279
\(670\) −11242.0 −0.648233
\(671\) 242.000 0.0139230
\(672\) −672.000 −0.0385758
\(673\) −10460.0 −0.599113 −0.299557 0.954078i \(-0.596839\pi\)
−0.299557 + 0.954078i \(0.596839\pi\)
\(674\) 1768.00 0.101040
\(675\) −2052.00 −0.117010
\(676\) 9168.00 0.521620
\(677\) −15546.0 −0.882543 −0.441271 0.897374i \(-0.645472\pi\)
−0.441271 + 0.897374i \(0.645472\pi\)
\(678\) 13932.0 0.789167
\(679\) −644.000 −0.0363983
\(680\) 1680.00 0.0947427
\(681\) 11286.0 0.635067
\(682\) 6820.00 0.382920
\(683\) 31138.0 1.74445 0.872227 0.489101i \(-0.162675\pi\)
0.872227 + 0.489101i \(0.162675\pi\)
\(684\) −252.000 −0.0140869
\(685\) −5362.00 −0.299082
\(686\) −686.000 −0.0381802
\(687\) −2112.00 −0.117289
\(688\) −1728.00 −0.0957549
\(689\) −8576.00 −0.474194
\(690\) 1176.00 0.0648834
\(691\) 21574.0 1.18772 0.593859 0.804569i \(-0.297604\pi\)
0.593859 + 0.804569i \(0.297604\pi\)
\(692\) 6752.00 0.370914
\(693\) 693.000 0.0379869
\(694\) 8080.00 0.441949
\(695\) 7560.00 0.412615
\(696\) −2904.00 −0.158155
\(697\) −5400.00 −0.293457
\(698\) −2450.00 −0.132857
\(699\) 6630.00 0.358755
\(700\) −2128.00 −0.114901
\(701\) 2074.00 0.111746 0.0558730 0.998438i \(-0.482206\pi\)
0.0558730 + 0.998438i \(0.482206\pi\)
\(702\) 3618.00 0.194519
\(703\) 497.000 0.0266639
\(704\) 704.000 0.0376889
\(705\) −1491.00 −0.0796515
\(706\) −4966.00 −0.264728
\(707\) −1414.00 −0.0752177
\(708\) −5148.00 −0.273268
\(709\) −6829.00 −0.361733 −0.180866 0.983508i \(-0.557890\pi\)
−0.180866 + 0.983508i \(0.557890\pi\)
\(710\) 6552.00 0.346327
\(711\) −864.000 −0.0455732
\(712\) 9168.00 0.482564
\(713\) −8680.00 −0.455917
\(714\) −1260.00 −0.0660425
\(715\) 5159.00 0.269840
\(716\) 1840.00 0.0960391
\(717\) 7353.00 0.382989
\(718\) −15608.0 −0.811261
\(719\) 6921.00 0.358984 0.179492 0.983759i \(-0.442555\pi\)
0.179492 + 0.983759i \(0.442555\pi\)
\(720\) −1008.00 −0.0521749
\(721\) −12586.0 −0.650107
\(722\) 13620.0 0.702055
\(723\) 75.0000 0.00385793
\(724\) 9184.00 0.471437
\(725\) −9196.00 −0.471077
\(726\) −726.000 −0.0371135
\(727\) 24064.0 1.22763 0.613813 0.789451i \(-0.289635\pi\)
0.613813 + 0.789451i \(0.289635\pi\)
\(728\) 3752.00 0.191014
\(729\) 729.000 0.0370370
\(730\) −1638.00 −0.0830481
\(731\) −3240.00 −0.163934
\(732\) 264.000 0.0133302
\(733\) 30262.0 1.52490 0.762451 0.647047i \(-0.223996\pi\)
0.762451 + 0.647047i \(0.223996\pi\)
\(734\) −6148.00 −0.309165
\(735\) −1029.00 −0.0516398
\(736\) −896.000 −0.0448736
\(737\) −8833.00 −0.441476
\(738\) 3240.00 0.161607
\(739\) 24024.0 1.19586 0.597928 0.801550i \(-0.295991\pi\)
0.597928 + 0.801550i \(0.295991\pi\)
\(740\) 1988.00 0.0987572
\(741\) 1407.00 0.0697536
\(742\) −1792.00 −0.0886609
\(743\) −227.000 −0.0112084 −0.00560419 0.999984i \(-0.501784\pi\)
−0.00560419 + 0.999984i \(0.501784\pi\)
\(744\) 7440.00 0.366618
\(745\) −9611.00 −0.472644
\(746\) −1348.00 −0.0661579
\(747\) −10098.0 −0.494600
\(748\) 1320.00 0.0645240
\(749\) −10017.0 −0.488669
\(750\) −8442.00 −0.411011
\(751\) −23435.0 −1.13869 −0.569344 0.822099i \(-0.692803\pi\)
−0.569344 + 0.822099i \(0.692803\pi\)
\(752\) 1136.00 0.0550873
\(753\) −15801.0 −0.764702
\(754\) 16214.0 0.783129
\(755\) 22470.0 1.08314
\(756\) 756.000 0.0363696
\(757\) 1609.00 0.0772524 0.0386262 0.999254i \(-0.487702\pi\)
0.0386262 + 0.999254i \(0.487702\pi\)
\(758\) −24102.0 −1.15491
\(759\) 924.000 0.0441885
\(760\) −392.000 −0.0187097
\(761\) 6976.00 0.332299 0.166150 0.986101i \(-0.446867\pi\)
0.166150 + 0.986101i \(0.446867\pi\)
\(762\) −13524.0 −0.642943
\(763\) −1946.00 −0.0923328
\(764\) 3512.00 0.166309
\(765\) −1890.00 −0.0893243
\(766\) 25328.0 1.19470
\(767\) 28743.0 1.35313
\(768\) 768.000 0.0360844
\(769\) −36079.0 −1.69186 −0.845931 0.533292i \(-0.820955\pi\)
−0.845931 + 0.533292i \(0.820955\pi\)
\(770\) 1078.00 0.0504525
\(771\) −4869.00 −0.227435
\(772\) −9072.00 −0.422938
\(773\) −23507.0 −1.09377 −0.546887 0.837206i \(-0.684187\pi\)
−0.546887 + 0.837206i \(0.684187\pi\)
\(774\) 1944.00 0.0902786
\(775\) 23560.0 1.09200
\(776\) 736.000 0.0340475
\(777\) −1491.00 −0.0688408
\(778\) −528.000 −0.0243313
\(779\) 1260.00 0.0579515
\(780\) 5628.00 0.258352
\(781\) 5148.00 0.235864
\(782\) −1680.00 −0.0768244
\(783\) 3267.00 0.149110
\(784\) 784.000 0.0357143
\(785\) −3164.00 −0.143857
\(786\) −13176.0 −0.597929
\(787\) −35145.0 −1.59185 −0.795924 0.605397i \(-0.793014\pi\)
−0.795924 + 0.605397i \(0.793014\pi\)
\(788\) 21192.0 0.958038
\(789\) −7425.00 −0.335028
\(790\) −1344.00 −0.0605283
\(791\) −16254.0 −0.730627
\(792\) −792.000 −0.0355335
\(793\) −1474.00 −0.0660067
\(794\) 28788.0 1.28671
\(795\) −2688.00 −0.119916
\(796\) 4176.00 0.185948
\(797\) 34155.0 1.51798 0.758991 0.651101i \(-0.225692\pi\)
0.758991 + 0.651101i \(0.225692\pi\)
\(798\) 294.000 0.0130420
\(799\) 2130.00 0.0943104
\(800\) 2432.00 0.107480
\(801\) −10314.0 −0.454965
\(802\) 24608.0 1.08346
\(803\) −1287.00 −0.0565595
\(804\) −9636.00 −0.422681
\(805\) −1372.00 −0.0600704
\(806\) −41540.0 −1.81536
\(807\) 5238.00 0.228484
\(808\) 1616.00 0.0703598
\(809\) −3439.00 −0.149455 −0.0747273 0.997204i \(-0.523809\pi\)
−0.0747273 + 0.997204i \(0.523809\pi\)
\(810\) 1134.00 0.0491910
\(811\) −1625.00 −0.0703594 −0.0351797 0.999381i \(-0.511200\pi\)
−0.0351797 + 0.999381i \(0.511200\pi\)
\(812\) 3388.00 0.146423
\(813\) 4959.00 0.213923
\(814\) 1562.00 0.0672581
\(815\) 77.0000 0.00330944
\(816\) 1440.00 0.0617771
\(817\) 756.000 0.0323734
\(818\) 6308.00 0.269626
\(819\) −4221.00 −0.180090
\(820\) 5040.00 0.214640
\(821\) 31531.0 1.34036 0.670182 0.742196i \(-0.266216\pi\)
0.670182 + 0.742196i \(0.266216\pi\)
\(822\) −4596.00 −0.195017
\(823\) −31003.0 −1.31312 −0.656559 0.754274i \(-0.727989\pi\)
−0.656559 + 0.754274i \(0.727989\pi\)
\(824\) 14384.0 0.608119
\(825\) −2508.00 −0.105839
\(826\) 6006.00 0.252997
\(827\) −15015.0 −0.631345 −0.315673 0.948868i \(-0.602230\pi\)
−0.315673 + 0.948868i \(0.602230\pi\)
\(828\) 1008.00 0.0423073
\(829\) 12280.0 0.514478 0.257239 0.966348i \(-0.417187\pi\)
0.257239 + 0.966348i \(0.417187\pi\)
\(830\) −15708.0 −0.656907
\(831\) −6168.00 −0.257480
\(832\) −4288.00 −0.178677
\(833\) 1470.00 0.0611434
\(834\) 6480.00 0.269046
\(835\) −3528.00 −0.146217
\(836\) −308.000 −0.0127421
\(837\) −8370.00 −0.345651
\(838\) 9270.00 0.382132
\(839\) 26599.0 1.09452 0.547258 0.836964i \(-0.315672\pi\)
0.547258 + 0.836964i \(0.315672\pi\)
\(840\) 1176.00 0.0483046
\(841\) −9748.00 −0.399688
\(842\) 14530.0 0.594699
\(843\) 2511.00 0.102590
\(844\) 18264.0 0.744873
\(845\) −16044.0 −0.653172
\(846\) −1278.00 −0.0519368
\(847\) 847.000 0.0343604
\(848\) 2048.00 0.0829347
\(849\) 26907.0 1.08769
\(850\) 4560.00 0.184008
\(851\) −1988.00 −0.0800796
\(852\) 5616.00 0.225823
\(853\) −3798.00 −0.152451 −0.0762257 0.997091i \(-0.524287\pi\)
−0.0762257 + 0.997091i \(0.524287\pi\)
\(854\) −308.000 −0.0123414
\(855\) 441.000 0.0176396
\(856\) 11448.0 0.457108
\(857\) −38726.0 −1.54359 −0.771794 0.635873i \(-0.780640\pi\)
−0.771794 + 0.635873i \(0.780640\pi\)
\(858\) 4422.00 0.175949
\(859\) 3296.00 0.130917 0.0654587 0.997855i \(-0.479149\pi\)
0.0654587 + 0.997855i \(0.479149\pi\)
\(860\) 3024.00 0.119904
\(861\) −3780.00 −0.149619
\(862\) −15178.0 −0.599727
\(863\) 13488.0 0.532024 0.266012 0.963970i \(-0.414294\pi\)
0.266012 + 0.963970i \(0.414294\pi\)
\(864\) −864.000 −0.0340207
\(865\) −11816.0 −0.464458
\(866\) −10328.0 −0.405265
\(867\) −12039.0 −0.471587
\(868\) −8680.00 −0.339422
\(869\) −1056.00 −0.0412225
\(870\) 5082.00 0.198041
\(871\) 53801.0 2.09297
\(872\) 2224.00 0.0863694
\(873\) −828.000 −0.0321003
\(874\) 392.000 0.0151712
\(875\) 9849.00 0.380522
\(876\) −1404.00 −0.0541516
\(877\) −17144.0 −0.660105 −0.330052 0.943963i \(-0.607066\pi\)
−0.330052 + 0.943963i \(0.607066\pi\)
\(878\) −28442.0 −1.09325
\(879\) 15360.0 0.589397
\(880\) −1232.00 −0.0471940
\(881\) 11993.0 0.458632 0.229316 0.973352i \(-0.426351\pi\)
0.229316 + 0.973352i \(0.426351\pi\)
\(882\) −882.000 −0.0336718
\(883\) 10483.0 0.399526 0.199763 0.979844i \(-0.435983\pi\)
0.199763 + 0.979844i \(0.435983\pi\)
\(884\) −8040.00 −0.305899
\(885\) 9009.00 0.342186
\(886\) −13848.0 −0.525093
\(887\) 4562.00 0.172691 0.0863455 0.996265i \(-0.472481\pi\)
0.0863455 + 0.996265i \(0.472481\pi\)
\(888\) 1704.00 0.0643947
\(889\) 15778.0 0.595250
\(890\) −16044.0 −0.604265
\(891\) 891.000 0.0335013
\(892\) 792.000 0.0297288
\(893\) −497.000 −0.0186243
\(894\) −8238.00 −0.308188
\(895\) −3220.00 −0.120260
\(896\) −896.000 −0.0334077
\(897\) −5628.00 −0.209491
\(898\) −5048.00 −0.187588
\(899\) −37510.0 −1.39158
\(900\) −2736.00 −0.101333
\(901\) 3840.00 0.141986
\(902\) 3960.00 0.146179
\(903\) −2268.00 −0.0835817
\(904\) 18576.0 0.683439
\(905\) −16072.0 −0.590333
\(906\) 19260.0 0.706259
\(907\) 18556.0 0.679318 0.339659 0.940549i \(-0.389688\pi\)
0.339659 + 0.940549i \(0.389688\pi\)
\(908\) 15048.0 0.549984
\(909\) −1818.00 −0.0663358
\(910\) −6566.00 −0.239188
\(911\) 5826.00 0.211881 0.105941 0.994372i \(-0.466215\pi\)
0.105941 + 0.994372i \(0.466215\pi\)
\(912\) −336.000 −0.0121996
\(913\) −12342.0 −0.447383
\(914\) −3856.00 −0.139546
\(915\) −462.000 −0.0166921
\(916\) −2816.00 −0.101576
\(917\) 15372.0 0.553575
\(918\) −1620.00 −0.0582440
\(919\) 41492.0 1.48933 0.744665 0.667438i \(-0.232609\pi\)
0.744665 + 0.667438i \(0.232609\pi\)
\(920\) 1568.00 0.0561907
\(921\) −8676.00 −0.310406
\(922\) 25704.0 0.918130
\(923\) −31356.0 −1.11820
\(924\) 924.000 0.0328976
\(925\) 5396.00 0.191805
\(926\) −9482.00 −0.336499
\(927\) −16182.0 −0.573340
\(928\) −3872.00 −0.136966
\(929\) −15381.0 −0.543202 −0.271601 0.962410i \(-0.587553\pi\)
−0.271601 + 0.962410i \(0.587553\pi\)
\(930\) −13020.0 −0.459078
\(931\) −343.000 −0.0120745
\(932\) 8840.00 0.310691
\(933\) 9624.00 0.337702
\(934\) 13618.0 0.477082
\(935\) −2310.00 −0.0807969
\(936\) 4824.00 0.168459
\(937\) −55054.0 −1.91946 −0.959731 0.280921i \(-0.909360\pi\)
−0.959731 + 0.280921i \(0.909360\pi\)
\(938\) 11242.0 0.391327
\(939\) 10650.0 0.370127
\(940\) −1988.00 −0.0689802
\(941\) −3892.00 −0.134831 −0.0674153 0.997725i \(-0.521475\pi\)
−0.0674153 + 0.997725i \(0.521475\pi\)
\(942\) −2712.00 −0.0938023
\(943\) −5040.00 −0.174046
\(944\) −6864.00 −0.236657
\(945\) −1323.00 −0.0455420
\(946\) 2376.00 0.0816601
\(947\) −36158.0 −1.24074 −0.620368 0.784311i \(-0.713017\pi\)
−0.620368 + 0.784311i \(0.713017\pi\)
\(948\) −1152.00 −0.0394675
\(949\) 7839.00 0.268140
\(950\) −1064.00 −0.0363376
\(951\) −16572.0 −0.565072
\(952\) −1680.00 −0.0571944
\(953\) −23959.0 −0.814384 −0.407192 0.913343i \(-0.633492\pi\)
−0.407192 + 0.913343i \(0.633492\pi\)
\(954\) −2304.00 −0.0781916
\(955\) −6146.00 −0.208251
\(956\) 9804.00 0.331678
\(957\) 3993.00 0.134875
\(958\) −3468.00 −0.116958
\(959\) 5362.00 0.180551
\(960\) −1344.00 −0.0451848
\(961\) 66309.0 2.22581
\(962\) −9514.00 −0.318860
\(963\) −12879.0 −0.430966
\(964\) 100.000 0.00334106
\(965\) 15876.0 0.529603
\(966\) −1176.00 −0.0391689
\(967\) −56106.0 −1.86582 −0.932910 0.360110i \(-0.882739\pi\)
−0.932910 + 0.360110i \(0.882739\pi\)
\(968\) −968.000 −0.0321412
\(969\) −630.000 −0.0208860
\(970\) −1288.00 −0.0426342
\(971\) −35079.0 −1.15936 −0.579680 0.814844i \(-0.696822\pi\)
−0.579680 + 0.814844i \(0.696822\pi\)
\(972\) 972.000 0.0320750
\(973\) −7560.00 −0.249088
\(974\) 14736.0 0.484776
\(975\) 15276.0 0.501768
\(976\) 352.000 0.0115443
\(977\) −26184.0 −0.857421 −0.428711 0.903442i \(-0.641032\pi\)
−0.428711 + 0.903442i \(0.641032\pi\)
\(978\) 66.0000 0.00215792
\(979\) −12606.0 −0.411532
\(980\) −1372.00 −0.0447214
\(981\) −2502.00 −0.0814299
\(982\) −26402.0 −0.857965
\(983\) −8584.00 −0.278522 −0.139261 0.990256i \(-0.544473\pi\)
−0.139261 + 0.990256i \(0.544473\pi\)
\(984\) 4320.00 0.139956
\(985\) −37086.0 −1.19965
\(986\) −7260.00 −0.234488
\(987\) 1491.00 0.0480841
\(988\) 1876.00 0.0604084
\(989\) −3024.00 −0.0972271
\(990\) 1386.00 0.0444949
\(991\) −51331.0 −1.64539 −0.822696 0.568482i \(-0.807531\pi\)
−0.822696 + 0.568482i \(0.807531\pi\)
\(992\) 9920.00 0.317500
\(993\) −6432.00 −0.205552
\(994\) −6552.00 −0.209071
\(995\) −7308.00 −0.232843
\(996\) −13464.0 −0.428337
\(997\) −38542.0 −1.22431 −0.612155 0.790738i \(-0.709697\pi\)
−0.612155 + 0.790738i \(0.709697\pi\)
\(998\) −33078.0 −1.04916
\(999\) −1917.00 −0.0607119
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 462.4.a.d.1.1 1
3.2 odd 2 1386.4.a.l.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
462.4.a.d.1.1 1 1.1 even 1 trivial
1386.4.a.l.1.1 1 3.2 odd 2