Properties

Label 966.4.a.b.1.1
Level $966$
Weight $4$
Character 966.1
Self dual yes
Analytic conductor $56.996$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 966 = 2 \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 966.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(56.9958450655\)
Analytic rank: \(0\)
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 \(\chi\) \(=\) 966.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -9.00000 q^{5} -6.00000 q^{6} +7.00000 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -9.00000 q^{5} -6.00000 q^{6} +7.00000 q^{7} +8.00000 q^{8} +9.00000 q^{9} -18.0000 q^{10} -12.0000 q^{11} -12.0000 q^{12} +11.0000 q^{13} +14.0000 q^{14} +27.0000 q^{15} +16.0000 q^{16} +96.0000 q^{17} +18.0000 q^{18} -40.0000 q^{19} -36.0000 q^{20} -21.0000 q^{21} -24.0000 q^{22} -23.0000 q^{23} -24.0000 q^{24} -44.0000 q^{25} +22.0000 q^{26} -27.0000 q^{27} +28.0000 q^{28} -231.000 q^{29} +54.0000 q^{30} -94.0000 q^{31} +32.0000 q^{32} +36.0000 q^{33} +192.000 q^{34} -63.0000 q^{35} +36.0000 q^{36} +47.0000 q^{37} -80.0000 q^{38} -33.0000 q^{39} -72.0000 q^{40} -27.0000 q^{41} -42.0000 q^{42} +479.000 q^{43} -48.0000 q^{44} -81.0000 q^{45} -46.0000 q^{46} +423.000 q^{47} -48.0000 q^{48} +49.0000 q^{49} -88.0000 q^{50} -288.000 q^{51} +44.0000 q^{52} +516.000 q^{53} -54.0000 q^{54} +108.000 q^{55} +56.0000 q^{56} +120.000 q^{57} -462.000 q^{58} +882.000 q^{59} +108.000 q^{60} +842.000 q^{61} -188.000 q^{62} +63.0000 q^{63} +64.0000 q^{64} -99.0000 q^{65} +72.0000 q^{66} -844.000 q^{67} +384.000 q^{68} +69.0000 q^{69} -126.000 q^{70} +654.000 q^{71} +72.0000 q^{72} -496.000 q^{73} +94.0000 q^{74} +132.000 q^{75} -160.000 q^{76} -84.0000 q^{77} -66.0000 q^{78} +260.000 q^{79} -144.000 q^{80} +81.0000 q^{81} -54.0000 q^{82} -156.000 q^{83} -84.0000 q^{84} -864.000 q^{85} +958.000 q^{86} +693.000 q^{87} -96.0000 q^{88} -414.000 q^{89} -162.000 q^{90} +77.0000 q^{91} -92.0000 q^{92} +282.000 q^{93} +846.000 q^{94} +360.000 q^{95} -96.0000 q^{96} +1343.00 q^{97} +98.0000 q^{98} -108.000 q^{99} +O(q^{100})\)

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
\(5\) −9.00000 −0.804984 −0.402492 0.915423i \(-0.631856\pi\)
−0.402492 + 0.915423i \(0.631856\pi\)
\(6\) −6.00000 −0.408248
\(7\) 7.00000 0.377964
\(8\) 8.00000 0.353553
\(9\) 9.00000 0.333333
\(10\) −18.0000 −0.569210
\(11\) −12.0000 −0.328921 −0.164461 0.986384i \(-0.552588\pi\)
−0.164461 + 0.986384i \(0.552588\pi\)
\(12\) −12.0000 −0.288675
\(13\) 11.0000 0.234681 0.117340 0.993092i \(-0.462563\pi\)
0.117340 + 0.993092i \(0.462563\pi\)
\(14\) 14.0000 0.267261
\(15\) 27.0000 0.464758
\(16\) 16.0000 0.250000
\(17\) 96.0000 1.36961 0.684806 0.728725i \(-0.259887\pi\)
0.684806 + 0.728725i \(0.259887\pi\)
\(18\) 18.0000 0.235702
\(19\) −40.0000 −0.482980 −0.241490 0.970403i \(-0.577636\pi\)
−0.241490 + 0.970403i \(0.577636\pi\)
\(20\) −36.0000 −0.402492
\(21\) −21.0000 −0.218218
\(22\) −24.0000 −0.232583
\(23\) −23.0000 −0.208514
\(24\) −24.0000 −0.204124
\(25\) −44.0000 −0.352000
\(26\) 22.0000 0.165944
\(27\) −27.0000 −0.192450
\(28\) 28.0000 0.188982
\(29\) −231.000 −1.47916 −0.739580 0.673069i \(-0.764976\pi\)
−0.739580 + 0.673069i \(0.764976\pi\)
\(30\) 54.0000 0.328634
\(31\) −94.0000 −0.544610 −0.272305 0.962211i \(-0.587786\pi\)
−0.272305 + 0.962211i \(0.587786\pi\)
\(32\) 32.0000 0.176777
\(33\) 36.0000 0.189903
\(34\) 192.000 0.968463
\(35\) −63.0000 −0.304256
\(36\) 36.0000 0.166667
\(37\) 47.0000 0.208831 0.104416 0.994534i \(-0.466703\pi\)
0.104416 + 0.994534i \(0.466703\pi\)
\(38\) −80.0000 −0.341519
\(39\) −33.0000 −0.135493
\(40\) −72.0000 −0.284605
\(41\) −27.0000 −0.102846 −0.0514231 0.998677i \(-0.516376\pi\)
−0.0514231 + 0.998677i \(0.516376\pi\)
\(42\) −42.0000 −0.154303
\(43\) 479.000 1.69876 0.849382 0.527779i \(-0.176975\pi\)
0.849382 + 0.527779i \(0.176975\pi\)
\(44\) −48.0000 −0.164461
\(45\) −81.0000 −0.268328
\(46\) −46.0000 −0.147442
\(47\) 423.000 1.31278 0.656392 0.754420i \(-0.272082\pi\)
0.656392 + 0.754420i \(0.272082\pi\)
\(48\) −48.0000 −0.144338
\(49\) 49.0000 0.142857
\(50\) −88.0000 −0.248902
\(51\) −288.000 −0.790746
\(52\) 44.0000 0.117340
\(53\) 516.000 1.33732 0.668661 0.743568i \(-0.266868\pi\)
0.668661 + 0.743568i \(0.266868\pi\)
\(54\) −54.0000 −0.136083
\(55\) 108.000 0.264777
\(56\) 56.0000 0.133631
\(57\) 120.000 0.278849
\(58\) −462.000 −1.04592
\(59\) 882.000 1.94621 0.973107 0.230354i \(-0.0739883\pi\)
0.973107 + 0.230354i \(0.0739883\pi\)
\(60\) 108.000 0.232379
\(61\) 842.000 1.76733 0.883664 0.468121i \(-0.155069\pi\)
0.883664 + 0.468121i \(0.155069\pi\)
\(62\) −188.000 −0.385097
\(63\) 63.0000 0.125988
\(64\) 64.0000 0.125000
\(65\) −99.0000 −0.188914
\(66\) 72.0000 0.134282
\(67\) −844.000 −1.53897 −0.769485 0.638665i \(-0.779487\pi\)
−0.769485 + 0.638665i \(0.779487\pi\)
\(68\) 384.000 0.684806
\(69\) 69.0000 0.120386
\(70\) −126.000 −0.215141
\(71\) 654.000 1.09318 0.546588 0.837402i \(-0.315926\pi\)
0.546588 + 0.837402i \(0.315926\pi\)
\(72\) 72.0000 0.117851
\(73\) −496.000 −0.795238 −0.397619 0.917551i \(-0.630163\pi\)
−0.397619 + 0.917551i \(0.630163\pi\)
\(74\) 94.0000 0.147666
\(75\) 132.000 0.203227
\(76\) −160.000 −0.241490
\(77\) −84.0000 −0.124321
\(78\) −66.0000 −0.0958081
\(79\) 260.000 0.370282 0.185141 0.982712i \(-0.440726\pi\)
0.185141 + 0.982712i \(0.440726\pi\)
\(80\) −144.000 −0.201246
\(81\) 81.0000 0.111111
\(82\) −54.0000 −0.0727232
\(83\) −156.000 −0.206304 −0.103152 0.994666i \(-0.532893\pi\)
−0.103152 + 0.994666i \(0.532893\pi\)
\(84\) −84.0000 −0.109109
\(85\) −864.000 −1.10252
\(86\) 958.000 1.20121
\(87\) 693.000 0.853993
\(88\) −96.0000 −0.116291
\(89\) −414.000 −0.493078 −0.246539 0.969133i \(-0.579293\pi\)
−0.246539 + 0.969133i \(0.579293\pi\)
\(90\) −162.000 −0.189737
\(91\) 77.0000 0.0887010
\(92\) −92.0000 −0.104257
\(93\) 282.000 0.314431
\(94\) 846.000 0.928279
\(95\) 360.000 0.388792
\(96\) −96.0000 −0.102062
\(97\) 1343.00 1.40578 0.702892 0.711297i \(-0.251892\pi\)
0.702892 + 0.711297i \(0.251892\pi\)
\(98\) 98.0000 0.101015
\(99\) −108.000 −0.109640
\(100\) −176.000 −0.176000
\(101\) 270.000 0.266000 0.133000 0.991116i \(-0.457539\pi\)
0.133000 + 0.991116i \(0.457539\pi\)
\(102\) −576.000 −0.559142
\(103\) 449.000 0.429527 0.214764 0.976666i \(-0.431102\pi\)
0.214764 + 0.976666i \(0.431102\pi\)
\(104\) 88.0000 0.0829722
\(105\) 189.000 0.175662
\(106\) 1032.00 0.945629
\(107\) 276.000 0.249364 0.124682 0.992197i \(-0.460209\pi\)
0.124682 + 0.992197i \(0.460209\pi\)
\(108\) −108.000 −0.0962250
\(109\) −589.000 −0.517578 −0.258789 0.965934i \(-0.583323\pi\)
−0.258789 + 0.965934i \(0.583323\pi\)
\(110\) 216.000 0.187225
\(111\) −141.000 −0.120569
\(112\) 112.000 0.0944911
\(113\) 1779.00 1.48101 0.740505 0.672050i \(-0.234586\pi\)
0.740505 + 0.672050i \(0.234586\pi\)
\(114\) 240.000 0.197176
\(115\) 207.000 0.167851
\(116\) −924.000 −0.739580
\(117\) 99.0000 0.0782270
\(118\) 1764.00 1.37618
\(119\) 672.000 0.517665
\(120\) 216.000 0.164317
\(121\) −1187.00 −0.891811
\(122\) 1684.00 1.24969
\(123\) 81.0000 0.0593782
\(124\) −376.000 −0.272305
\(125\) 1521.00 1.08834
\(126\) 126.000 0.0890871
\(127\) 773.000 0.540100 0.270050 0.962846i \(-0.412960\pi\)
0.270050 + 0.962846i \(0.412960\pi\)
\(128\) 128.000 0.0883883
\(129\) −1437.00 −0.980781
\(130\) −198.000 −0.133583
\(131\) 570.000 0.380161 0.190081 0.981768i \(-0.439125\pi\)
0.190081 + 0.981768i \(0.439125\pi\)
\(132\) 144.000 0.0949514
\(133\) −280.000 −0.182549
\(134\) −1688.00 −1.08822
\(135\) 243.000 0.154919
\(136\) 768.000 0.484231
\(137\) 2355.00 1.46862 0.734311 0.678813i \(-0.237505\pi\)
0.734311 + 0.678813i \(0.237505\pi\)
\(138\) 138.000 0.0851257
\(139\) 2333.00 1.42361 0.711807 0.702375i \(-0.247877\pi\)
0.711807 + 0.702375i \(0.247877\pi\)
\(140\) −252.000 −0.152128
\(141\) −1269.00 −0.757937
\(142\) 1308.00 0.772992
\(143\) −132.000 −0.0771916
\(144\) 144.000 0.0833333
\(145\) 2079.00 1.19070
\(146\) −992.000 −0.562319
\(147\) −147.000 −0.0824786
\(148\) 188.000 0.104416
\(149\) −2220.00 −1.22060 −0.610300 0.792170i \(-0.708951\pi\)
−0.610300 + 0.792170i \(0.708951\pi\)
\(150\) 264.000 0.143703
\(151\) −2827.00 −1.52356 −0.761781 0.647834i \(-0.775675\pi\)
−0.761781 + 0.647834i \(0.775675\pi\)
\(152\) −320.000 −0.170759
\(153\) 864.000 0.456538
\(154\) −168.000 −0.0879080
\(155\) 846.000 0.438402
\(156\) −132.000 −0.0677465
\(157\) 1928.00 0.980071 0.490036 0.871702i \(-0.336984\pi\)
0.490036 + 0.871702i \(0.336984\pi\)
\(158\) 520.000 0.261829
\(159\) −1548.00 −0.772103
\(160\) −288.000 −0.142302
\(161\) −161.000 −0.0788110
\(162\) 162.000 0.0785674
\(163\) −1996.00 −0.959134 −0.479567 0.877505i \(-0.659206\pi\)
−0.479567 + 0.877505i \(0.659206\pi\)
\(164\) −108.000 −0.0514231
\(165\) −324.000 −0.152869
\(166\) −312.000 −0.145879
\(167\) 1140.00 0.528239 0.264119 0.964490i \(-0.414919\pi\)
0.264119 + 0.964490i \(0.414919\pi\)
\(168\) −168.000 −0.0771517
\(169\) −2076.00 −0.944925
\(170\) −1728.00 −0.779597
\(171\) −360.000 −0.160993
\(172\) 1916.00 0.849382
\(173\) 3618.00 1.59001 0.795004 0.606604i \(-0.207469\pi\)
0.795004 + 0.606604i \(0.207469\pi\)
\(174\) 1386.00 0.603864
\(175\) −308.000 −0.133043
\(176\) −192.000 −0.0822304
\(177\) −2646.00 −1.12365
\(178\) −828.000 −0.348659
\(179\) 939.000 0.392090 0.196045 0.980595i \(-0.437190\pi\)
0.196045 + 0.980595i \(0.437190\pi\)
\(180\) −324.000 −0.134164
\(181\) −3598.00 −1.47755 −0.738777 0.673950i \(-0.764596\pi\)
−0.738777 + 0.673950i \(0.764596\pi\)
\(182\) 154.000 0.0627211
\(183\) −2526.00 −1.02037
\(184\) −184.000 −0.0737210
\(185\) −423.000 −0.168106
\(186\) 564.000 0.222336
\(187\) −1152.00 −0.450495
\(188\) 1692.00 0.656392
\(189\) −189.000 −0.0727393
\(190\) 720.000 0.274917
\(191\) −5004.00 −1.89569 −0.947845 0.318732i \(-0.896743\pi\)
−0.947845 + 0.318732i \(0.896743\pi\)
\(192\) −192.000 −0.0721688
\(193\) 3095.00 1.15432 0.577158 0.816633i \(-0.304162\pi\)
0.577158 + 0.816633i \(0.304162\pi\)
\(194\) 2686.00 0.994039
\(195\) 297.000 0.109070
\(196\) 196.000 0.0714286
\(197\) −621.000 −0.224591 −0.112295 0.993675i \(-0.535820\pi\)
−0.112295 + 0.993675i \(0.535820\pi\)
\(198\) −216.000 −0.0775275
\(199\) 3191.00 1.13670 0.568352 0.822786i \(-0.307581\pi\)
0.568352 + 0.822786i \(0.307581\pi\)
\(200\) −352.000 −0.124451
\(201\) 2532.00 0.888525
\(202\) 540.000 0.188090
\(203\) −1617.00 −0.559070
\(204\) −1152.00 −0.395373
\(205\) 243.000 0.0827895
\(206\) 898.000 0.303721
\(207\) −207.000 −0.0695048
\(208\) 176.000 0.0586702
\(209\) 480.000 0.158863
\(210\) 378.000 0.124212
\(211\) 80.0000 0.0261016 0.0130508 0.999915i \(-0.495846\pi\)
0.0130508 + 0.999915i \(0.495846\pi\)
\(212\) 2064.00 0.668661
\(213\) −1962.00 −0.631146
\(214\) 552.000 0.176327
\(215\) −4311.00 −1.36748
\(216\) −216.000 −0.0680414
\(217\) −658.000 −0.205843
\(218\) −1178.00 −0.365983
\(219\) 1488.00 0.459131
\(220\) 432.000 0.132388
\(221\) 1056.00 0.321422
\(222\) −282.000 −0.0852550
\(223\) 2606.00 0.782559 0.391280 0.920272i \(-0.372033\pi\)
0.391280 + 0.920272i \(0.372033\pi\)
\(224\) 224.000 0.0668153
\(225\) −396.000 −0.117333
\(226\) 3558.00 1.04723
\(227\) −3699.00 −1.08155 −0.540774 0.841168i \(-0.681868\pi\)
−0.540774 + 0.841168i \(0.681868\pi\)
\(228\) 480.000 0.139424
\(229\) −6712.00 −1.93686 −0.968431 0.249281i \(-0.919806\pi\)
−0.968431 + 0.249281i \(0.919806\pi\)
\(230\) 414.000 0.118688
\(231\) 252.000 0.0717765
\(232\) −1848.00 −0.522962
\(233\) 5190.00 1.45926 0.729631 0.683841i \(-0.239692\pi\)
0.729631 + 0.683841i \(0.239692\pi\)
\(234\) 198.000 0.0553148
\(235\) −3807.00 −1.05677
\(236\) 3528.00 0.973107
\(237\) −780.000 −0.213782
\(238\) 1344.00 0.366044
\(239\) 4800.00 1.29911 0.649553 0.760317i \(-0.274956\pi\)
0.649553 + 0.760317i \(0.274956\pi\)
\(240\) 432.000 0.116190
\(241\) 7229.00 1.93220 0.966101 0.258163i \(-0.0831173\pi\)
0.966101 + 0.258163i \(0.0831173\pi\)
\(242\) −2374.00 −0.630605
\(243\) −243.000 −0.0641500
\(244\) 3368.00 0.883664
\(245\) −441.000 −0.114998
\(246\) 162.000 0.0419868
\(247\) −440.000 −0.113346
\(248\) −752.000 −0.192549
\(249\) 468.000 0.119110
\(250\) 3042.00 0.769572
\(251\) −123.000 −0.0309310 −0.0154655 0.999880i \(-0.504923\pi\)
−0.0154655 + 0.999880i \(0.504923\pi\)
\(252\) 252.000 0.0629941
\(253\) 276.000 0.0685849
\(254\) 1546.00 0.381908
\(255\) 2592.00 0.636539
\(256\) 256.000 0.0625000
\(257\) 5886.00 1.42863 0.714316 0.699823i \(-0.246738\pi\)
0.714316 + 0.699823i \(0.246738\pi\)
\(258\) −2874.00 −0.693517
\(259\) 329.000 0.0789308
\(260\) −396.000 −0.0944572
\(261\) −2079.00 −0.493053
\(262\) 1140.00 0.268815
\(263\) 357.000 0.0837018 0.0418509 0.999124i \(-0.486675\pi\)
0.0418509 + 0.999124i \(0.486675\pi\)
\(264\) 288.000 0.0671408
\(265\) −4644.00 −1.07652
\(266\) −560.000 −0.129082
\(267\) 1242.00 0.284679
\(268\) −3376.00 −0.769485
\(269\) −8418.00 −1.90801 −0.954005 0.299792i \(-0.903083\pi\)
−0.954005 + 0.299792i \(0.903083\pi\)
\(270\) 486.000 0.109545
\(271\) 4172.00 0.935170 0.467585 0.883948i \(-0.345124\pi\)
0.467585 + 0.883948i \(0.345124\pi\)
\(272\) 1536.00 0.342403
\(273\) −231.000 −0.0512116
\(274\) 4710.00 1.03847
\(275\) 528.000 0.115780
\(276\) 276.000 0.0601929
\(277\) −760.000 −0.164852 −0.0824259 0.996597i \(-0.526267\pi\)
−0.0824259 + 0.996597i \(0.526267\pi\)
\(278\) 4666.00 1.00665
\(279\) −846.000 −0.181537
\(280\) −504.000 −0.107571
\(281\) −9339.00 −1.98263 −0.991313 0.131522i \(-0.958014\pi\)
−0.991313 + 0.131522i \(0.958014\pi\)
\(282\) −2538.00 −0.535942
\(283\) −6262.00 −1.31533 −0.657663 0.753312i \(-0.728455\pi\)
−0.657663 + 0.753312i \(0.728455\pi\)
\(284\) 2616.00 0.546588
\(285\) −1080.00 −0.224469
\(286\) −264.000 −0.0545827
\(287\) −189.000 −0.0388722
\(288\) 288.000 0.0589256
\(289\) 4303.00 0.875840
\(290\) 4158.00 0.841952
\(291\) −4029.00 −0.811629
\(292\) −1984.00 −0.397619
\(293\) −5754.00 −1.14728 −0.573639 0.819108i \(-0.694469\pi\)
−0.573639 + 0.819108i \(0.694469\pi\)
\(294\) −294.000 −0.0583212
\(295\) −7938.00 −1.56667
\(296\) 376.000 0.0738330
\(297\) 324.000 0.0633010
\(298\) −4440.00 −0.863095
\(299\) −253.000 −0.0489343
\(300\) 528.000 0.101614
\(301\) 3353.00 0.642072
\(302\) −5654.00 −1.07732
\(303\) −810.000 −0.153575
\(304\) −640.000 −0.120745
\(305\) −7578.00 −1.42267
\(306\) 1728.00 0.322821
\(307\) 3971.00 0.738231 0.369116 0.929384i \(-0.379661\pi\)
0.369116 + 0.929384i \(0.379661\pi\)
\(308\) −336.000 −0.0621603
\(309\) −1347.00 −0.247988
\(310\) 1692.00 0.309997
\(311\) 6156.00 1.12243 0.561213 0.827671i \(-0.310335\pi\)
0.561213 + 0.827671i \(0.310335\pi\)
\(312\) −264.000 −0.0479040
\(313\) −4210.00 −0.760266 −0.380133 0.924932i \(-0.624122\pi\)
−0.380133 + 0.924932i \(0.624122\pi\)
\(314\) 3856.00 0.693015
\(315\) −567.000 −0.101419
\(316\) 1040.00 0.185141
\(317\) −1329.00 −0.235470 −0.117735 0.993045i \(-0.537563\pi\)
−0.117735 + 0.993045i \(0.537563\pi\)
\(318\) −3096.00 −0.545959
\(319\) 2772.00 0.486527
\(320\) −576.000 −0.100623
\(321\) −828.000 −0.143970
\(322\) −322.000 −0.0557278
\(323\) −3840.00 −0.661496
\(324\) 324.000 0.0555556
\(325\) −484.000 −0.0826077
\(326\) −3992.00 −0.678210
\(327\) 1767.00 0.298824
\(328\) −216.000 −0.0363616
\(329\) 2961.00 0.496186
\(330\) −648.000 −0.108095
\(331\) −7444.00 −1.23613 −0.618065 0.786127i \(-0.712083\pi\)
−0.618065 + 0.786127i \(0.712083\pi\)
\(332\) −624.000 −0.103152
\(333\) 423.000 0.0696104
\(334\) 2280.00 0.373521
\(335\) 7596.00 1.23885
\(336\) −336.000 −0.0545545
\(337\) 9596.00 1.55112 0.775560 0.631274i \(-0.217468\pi\)
0.775560 + 0.631274i \(0.217468\pi\)
\(338\) −4152.00 −0.668163
\(339\) −5337.00 −0.855062
\(340\) −3456.00 −0.551259
\(341\) 1128.00 0.179134
\(342\) −720.000 −0.113840
\(343\) 343.000 0.0539949
\(344\) 3832.00 0.600603
\(345\) −621.000 −0.0969087
\(346\) 7236.00 1.12431
\(347\) 2079.00 0.321633 0.160816 0.986984i \(-0.448587\pi\)
0.160816 + 0.986984i \(0.448587\pi\)
\(348\) 2772.00 0.426997
\(349\) −1582.00 −0.242643 −0.121322 0.992613i \(-0.538713\pi\)
−0.121322 + 0.992613i \(0.538713\pi\)
\(350\) −616.000 −0.0940760
\(351\) −297.000 −0.0451644
\(352\) −384.000 −0.0581456
\(353\) 4191.00 0.631911 0.315955 0.948774i \(-0.397675\pi\)
0.315955 + 0.948774i \(0.397675\pi\)
\(354\) −5292.00 −0.794538
\(355\) −5886.00 −0.879990
\(356\) −1656.00 −0.246539
\(357\) −2016.00 −0.298874
\(358\) 1878.00 0.277250
\(359\) −11721.0 −1.72315 −0.861575 0.507631i \(-0.830521\pi\)
−0.861575 + 0.507631i \(0.830521\pi\)
\(360\) −648.000 −0.0948683
\(361\) −5259.00 −0.766730
\(362\) −7196.00 −1.04479
\(363\) 3561.00 0.514887
\(364\) 308.000 0.0443505
\(365\) 4464.00 0.640155
\(366\) −5052.00 −0.721509
\(367\) 9899.00 1.40797 0.703983 0.710217i \(-0.251403\pi\)
0.703983 + 0.710217i \(0.251403\pi\)
\(368\) −368.000 −0.0521286
\(369\) −243.000 −0.0342820
\(370\) −846.000 −0.118869
\(371\) 3612.00 0.505460
\(372\) 1128.00 0.157215
\(373\) −8170.00 −1.13412 −0.567060 0.823677i \(-0.691919\pi\)
−0.567060 + 0.823677i \(0.691919\pi\)
\(374\) −2304.00 −0.318548
\(375\) −4563.00 −0.628353
\(376\) 3384.00 0.464140
\(377\) −2541.00 −0.347130
\(378\) −378.000 −0.0514344
\(379\) −6163.00 −0.835282 −0.417641 0.908612i \(-0.637143\pi\)
−0.417641 + 0.908612i \(0.637143\pi\)
\(380\) 1440.00 0.194396
\(381\) −2319.00 −0.311827
\(382\) −10008.0 −1.34046
\(383\) −9168.00 −1.22314 −0.611570 0.791190i \(-0.709462\pi\)
−0.611570 + 0.791190i \(0.709462\pi\)
\(384\) −384.000 −0.0510310
\(385\) 756.000 0.100076
\(386\) 6190.00 0.816225
\(387\) 4311.00 0.566254
\(388\) 5372.00 0.702892
\(389\) 6834.00 0.890739 0.445370 0.895347i \(-0.353072\pi\)
0.445370 + 0.895347i \(0.353072\pi\)
\(390\) 594.000 0.0771240
\(391\) −2208.00 −0.285584
\(392\) 392.000 0.0505076
\(393\) −1710.00 −0.219486
\(394\) −1242.00 −0.158810
\(395\) −2340.00 −0.298071
\(396\) −432.000 −0.0548202
\(397\) −5698.00 −0.720339 −0.360169 0.932887i \(-0.617281\pi\)
−0.360169 + 0.932887i \(0.617281\pi\)
\(398\) 6382.00 0.803771
\(399\) 840.000 0.105395
\(400\) −704.000 −0.0880000
\(401\) −942.000 −0.117310 −0.0586549 0.998278i \(-0.518681\pi\)
−0.0586549 + 0.998278i \(0.518681\pi\)
\(402\) 5064.00 0.628282
\(403\) −1034.00 −0.127809
\(404\) 1080.00 0.133000
\(405\) −729.000 −0.0894427
\(406\) −3234.00 −0.395322
\(407\) −564.000 −0.0686890
\(408\) −2304.00 −0.279571
\(409\) −12976.0 −1.56876 −0.784379 0.620282i \(-0.787018\pi\)
−0.784379 + 0.620282i \(0.787018\pi\)
\(410\) 486.000 0.0585410
\(411\) −7065.00 −0.847909
\(412\) 1796.00 0.214764
\(413\) 6174.00 0.735600
\(414\) −414.000 −0.0491473
\(415\) 1404.00 0.166071
\(416\) 352.000 0.0414861
\(417\) −6999.00 −0.821924
\(418\) 960.000 0.112333
\(419\) 36.0000 0.00419741 0.00209871 0.999998i \(-0.499332\pi\)
0.00209871 + 0.999998i \(0.499332\pi\)
\(420\) 756.000 0.0878310
\(421\) 659.000 0.0762891 0.0381445 0.999272i \(-0.487855\pi\)
0.0381445 + 0.999272i \(0.487855\pi\)
\(422\) 160.000 0.0184566
\(423\) 3807.00 0.437595
\(424\) 4128.00 0.472815
\(425\) −4224.00 −0.482104
\(426\) −3924.00 −0.446287
\(427\) 5894.00 0.667987
\(428\) 1104.00 0.124682
\(429\) 396.000 0.0445666
\(430\) −8622.00 −0.966953
\(431\) 15573.0 1.74043 0.870215 0.492673i \(-0.163980\pi\)
0.870215 + 0.492673i \(0.163980\pi\)
\(432\) −432.000 −0.0481125
\(433\) −15727.0 −1.74548 −0.872738 0.488188i \(-0.837658\pi\)
−0.872738 + 0.488188i \(0.837658\pi\)
\(434\) −1316.00 −0.145553
\(435\) −6237.00 −0.687451
\(436\) −2356.00 −0.258789
\(437\) 920.000 0.100708
\(438\) 2976.00 0.324655
\(439\) 11666.0 1.26831 0.634155 0.773206i \(-0.281348\pi\)
0.634155 + 0.773206i \(0.281348\pi\)
\(440\) 864.000 0.0936127
\(441\) 441.000 0.0476190
\(442\) 2112.00 0.227280
\(443\) 14457.0 1.55050 0.775251 0.631653i \(-0.217623\pi\)
0.775251 + 0.631653i \(0.217623\pi\)
\(444\) −564.000 −0.0602844
\(445\) 3726.00 0.396920
\(446\) 5212.00 0.553353
\(447\) 6660.00 0.704714
\(448\) 448.000 0.0472456
\(449\) 5886.00 0.618658 0.309329 0.950955i \(-0.399896\pi\)
0.309329 + 0.950955i \(0.399896\pi\)
\(450\) −792.000 −0.0829672
\(451\) 324.000 0.0338283
\(452\) 7116.00 0.740505
\(453\) 8481.00 0.879629
\(454\) −7398.00 −0.764769
\(455\) −693.000 −0.0714029
\(456\) 960.000 0.0985880
\(457\) −7540.00 −0.771786 −0.385893 0.922543i \(-0.626107\pi\)
−0.385893 + 0.922543i \(0.626107\pi\)
\(458\) −13424.0 −1.36957
\(459\) −2592.00 −0.263582
\(460\) 828.000 0.0839254
\(461\) 1266.00 0.127903 0.0639517 0.997953i \(-0.479630\pi\)
0.0639517 + 0.997953i \(0.479630\pi\)
\(462\) 504.000 0.0507537
\(463\) −12265.0 −1.23111 −0.615554 0.788095i \(-0.711068\pi\)
−0.615554 + 0.788095i \(0.711068\pi\)
\(464\) −3696.00 −0.369790
\(465\) −2538.00 −0.253112
\(466\) 10380.0 1.03185
\(467\) 10491.0 1.03954 0.519770 0.854306i \(-0.326017\pi\)
0.519770 + 0.854306i \(0.326017\pi\)
\(468\) 396.000 0.0391135
\(469\) −5908.00 −0.581676
\(470\) −7614.00 −0.747250
\(471\) −5784.00 −0.565844
\(472\) 7056.00 0.688091
\(473\) −5748.00 −0.558760
\(474\) −1560.00 −0.151167
\(475\) 1760.00 0.170009
\(476\) 2688.00 0.258833
\(477\) 4644.00 0.445774
\(478\) 9600.00 0.918606
\(479\) 10062.0 0.959801 0.479900 0.877323i \(-0.340673\pi\)
0.479900 + 0.877323i \(0.340673\pi\)
\(480\) 864.000 0.0821584
\(481\) 517.000 0.0490087
\(482\) 14458.0 1.36627
\(483\) 483.000 0.0455016
\(484\) −4748.00 −0.445905
\(485\) −12087.0 −1.13163
\(486\) −486.000 −0.0453609
\(487\) 13967.0 1.29960 0.649800 0.760105i \(-0.274853\pi\)
0.649800 + 0.760105i \(0.274853\pi\)
\(488\) 6736.00 0.624845
\(489\) 5988.00 0.553756
\(490\) −882.000 −0.0813157
\(491\) 9732.00 0.894499 0.447250 0.894409i \(-0.352404\pi\)
0.447250 + 0.894409i \(0.352404\pi\)
\(492\) 324.000 0.0296891
\(493\) −22176.0 −2.02588
\(494\) −880.000 −0.0801479
\(495\) 972.000 0.0882589
\(496\) −1504.00 −0.136152
\(497\) 4578.00 0.413182
\(498\) 936.000 0.0842232
\(499\) −6190.00 −0.555316 −0.277658 0.960680i \(-0.589558\pi\)
−0.277658 + 0.960680i \(0.589558\pi\)
\(500\) 6084.00 0.544170
\(501\) −3420.00 −0.304979
\(502\) −246.000 −0.0218715
\(503\) 10026.0 0.888742 0.444371 0.895843i \(-0.353427\pi\)
0.444371 + 0.895843i \(0.353427\pi\)
\(504\) 504.000 0.0445435
\(505\) −2430.00 −0.214126
\(506\) 552.000 0.0484968
\(507\) 6228.00 0.545553
\(508\) 3092.00 0.270050
\(509\) −14640.0 −1.27487 −0.637433 0.770506i \(-0.720004\pi\)
−0.637433 + 0.770506i \(0.720004\pi\)
\(510\) 5184.00 0.450101
\(511\) −3472.00 −0.300572
\(512\) 512.000 0.0441942
\(513\) 1080.00 0.0929496
\(514\) 11772.0 1.01020
\(515\) −4041.00 −0.345763
\(516\) −5748.00 −0.490391
\(517\) −5076.00 −0.431803
\(518\) 658.000 0.0558125
\(519\) −10854.0 −0.917992
\(520\) −792.000 −0.0667913
\(521\) 11682.0 0.982337 0.491169 0.871065i \(-0.336570\pi\)
0.491169 + 0.871065i \(0.336570\pi\)
\(522\) −4158.00 −0.348641
\(523\) −20254.0 −1.69339 −0.846697 0.532075i \(-0.821413\pi\)
−0.846697 + 0.532075i \(0.821413\pi\)
\(524\) 2280.00 0.190081
\(525\) 924.000 0.0768127
\(526\) 714.000 0.0591861
\(527\) −9024.00 −0.745904
\(528\) 576.000 0.0474757
\(529\) 529.000 0.0434783
\(530\) −9288.00 −0.761217
\(531\) 7938.00 0.648738
\(532\) −1120.00 −0.0912747
\(533\) −297.000 −0.0241360
\(534\) 2484.00 0.201298
\(535\) −2484.00 −0.200734
\(536\) −6752.00 −0.544108
\(537\) −2817.00 −0.226373
\(538\) −16836.0 −1.34917
\(539\) −588.000 −0.0469888
\(540\) 972.000 0.0774597
\(541\) −6712.00 −0.533404 −0.266702 0.963779i \(-0.585934\pi\)
−0.266702 + 0.963779i \(0.585934\pi\)
\(542\) 8344.00 0.661265
\(543\) 10794.0 0.853066
\(544\) 3072.00 0.242116
\(545\) 5301.00 0.416642
\(546\) −462.000 −0.0362120
\(547\) 10658.0 0.833095 0.416548 0.909114i \(-0.363240\pi\)
0.416548 + 0.909114i \(0.363240\pi\)
\(548\) 9420.00 0.734311
\(549\) 7578.00 0.589110
\(550\) 1056.00 0.0818691
\(551\) 9240.00 0.714405
\(552\) 552.000 0.0425628
\(553\) 1820.00 0.139953
\(554\) −1520.00 −0.116568
\(555\) 1269.00 0.0970559
\(556\) 9332.00 0.711807
\(557\) 12972.0 0.986789 0.493394 0.869806i \(-0.335756\pi\)
0.493394 + 0.869806i \(0.335756\pi\)
\(558\) −1692.00 −0.128366
\(559\) 5269.00 0.398667
\(560\) −1008.00 −0.0760639
\(561\) 3456.00 0.260093
\(562\) −18678.0 −1.40193
\(563\) 16329.0 1.22235 0.611177 0.791494i \(-0.290696\pi\)
0.611177 + 0.791494i \(0.290696\pi\)
\(564\) −5076.00 −0.378968
\(565\) −16011.0 −1.19219
\(566\) −12524.0 −0.930076
\(567\) 567.000 0.0419961
\(568\) 5232.00 0.386496
\(569\) −2823.00 −0.207990 −0.103995 0.994578i \(-0.533163\pi\)
−0.103995 + 0.994578i \(0.533163\pi\)
\(570\) −2160.00 −0.158724
\(571\) −21580.0 −1.58160 −0.790801 0.612073i \(-0.790336\pi\)
−0.790801 + 0.612073i \(0.790336\pi\)
\(572\) −528.000 −0.0385958
\(573\) 15012.0 1.09448
\(574\) −378.000 −0.0274868
\(575\) 1012.00 0.0733971
\(576\) 576.000 0.0416667
\(577\) 11354.0 0.819191 0.409595 0.912267i \(-0.365670\pi\)
0.409595 + 0.912267i \(0.365670\pi\)
\(578\) 8606.00 0.619312
\(579\) −9285.00 −0.666445
\(580\) 8316.00 0.595350
\(581\) −1092.00 −0.0779755
\(582\) −8058.00 −0.573909
\(583\) −6192.00 −0.439874
\(584\) −3968.00 −0.281159
\(585\) −891.000 −0.0629715
\(586\) −11508.0 −0.811248
\(587\) 16872.0 1.18634 0.593170 0.805077i \(-0.297876\pi\)
0.593170 + 0.805077i \(0.297876\pi\)
\(588\) −588.000 −0.0412393
\(589\) 3760.00 0.263036
\(590\) −15876.0 −1.10780
\(591\) 1863.00 0.129668
\(592\) 752.000 0.0522078
\(593\) −19527.0 −1.35224 −0.676120 0.736792i \(-0.736340\pi\)
−0.676120 + 0.736792i \(0.736340\pi\)
\(594\) 648.000 0.0447605
\(595\) −6048.00 −0.416712
\(596\) −8880.00 −0.610300
\(597\) −9573.00 −0.656276
\(598\) −506.000 −0.0346018
\(599\) 14076.0 0.960150 0.480075 0.877227i \(-0.340609\pi\)
0.480075 + 0.877227i \(0.340609\pi\)
\(600\) 1056.00 0.0718517
\(601\) −7264.00 −0.493020 −0.246510 0.969140i \(-0.579284\pi\)
−0.246510 + 0.969140i \(0.579284\pi\)
\(602\) 6706.00 0.454014
\(603\) −7596.00 −0.512990
\(604\) −11308.0 −0.761781
\(605\) 10683.0 0.717894
\(606\) −1620.00 −0.108594
\(607\) −18394.0 −1.22997 −0.614983 0.788540i \(-0.710837\pi\)
−0.614983 + 0.788540i \(0.710837\pi\)
\(608\) −1280.00 −0.0853797
\(609\) 4851.00 0.322779
\(610\) −15156.0 −1.00598
\(611\) 4653.00 0.308085
\(612\) 3456.00 0.228269
\(613\) −8527.00 −0.561831 −0.280915 0.959733i \(-0.590638\pi\)
−0.280915 + 0.959733i \(0.590638\pi\)
\(614\) 7942.00 0.522008
\(615\) −729.000 −0.0477986
\(616\) −672.000 −0.0439540
\(617\) −17706.0 −1.15529 −0.577647 0.816286i \(-0.696029\pi\)
−0.577647 + 0.816286i \(0.696029\pi\)
\(618\) −2694.00 −0.175354
\(619\) 7238.00 0.469983 0.234992 0.971997i \(-0.424494\pi\)
0.234992 + 0.971997i \(0.424494\pi\)
\(620\) 3384.00 0.219201
\(621\) 621.000 0.0401286
\(622\) 12312.0 0.793676
\(623\) −2898.00 −0.186366
\(624\) −528.000 −0.0338733
\(625\) −8189.00 −0.524096
\(626\) −8420.00 −0.537589
\(627\) −1440.00 −0.0917194
\(628\) 7712.00 0.490036
\(629\) 4512.00 0.286018
\(630\) −1134.00 −0.0717137
\(631\) 21656.0 1.36626 0.683131 0.730296i \(-0.260618\pi\)
0.683131 + 0.730296i \(0.260618\pi\)
\(632\) 2080.00 0.130914
\(633\) −240.000 −0.0150697
\(634\) −2658.00 −0.166503
\(635\) −6957.00 −0.434772
\(636\) −6192.00 −0.386052
\(637\) 539.000 0.0335258
\(638\) 5544.00 0.344027
\(639\) 5886.00 0.364392
\(640\) −1152.00 −0.0711512
\(641\) −18027.0 −1.11080 −0.555401 0.831583i \(-0.687435\pi\)
−0.555401 + 0.831583i \(0.687435\pi\)
\(642\) −1656.00 −0.101802
\(643\) −14218.0 −0.872011 −0.436006 0.899944i \(-0.643607\pi\)
−0.436006 + 0.899944i \(0.643607\pi\)
\(644\) −644.000 −0.0394055
\(645\) 12933.0 0.789514
\(646\) −7680.00 −0.467749
\(647\) −18840.0 −1.14479 −0.572393 0.819979i \(-0.693985\pi\)
−0.572393 + 0.819979i \(0.693985\pi\)
\(648\) 648.000 0.0392837
\(649\) −10584.0 −0.640152
\(650\) −968.000 −0.0584124
\(651\) 1974.00 0.118844
\(652\) −7984.00 −0.479567
\(653\) −7611.00 −0.456112 −0.228056 0.973648i \(-0.573237\pi\)
−0.228056 + 0.973648i \(0.573237\pi\)
\(654\) 3534.00 0.211300
\(655\) −5130.00 −0.306024
\(656\) −432.000 −0.0257115
\(657\) −4464.00 −0.265079
\(658\) 5922.00 0.350857
\(659\) 30456.0 1.80030 0.900150 0.435581i \(-0.143457\pi\)
0.900150 + 0.435581i \(0.143457\pi\)
\(660\) −1296.00 −0.0764344
\(661\) 25298.0 1.48862 0.744310 0.667834i \(-0.232778\pi\)
0.744310 + 0.667834i \(0.232778\pi\)
\(662\) −14888.0 −0.874076
\(663\) −3168.00 −0.185573
\(664\) −1248.00 −0.0729394
\(665\) 2520.00 0.146949
\(666\) 846.000 0.0492220
\(667\) 5313.00 0.308426
\(668\) 4560.00 0.264119
\(669\) −7818.00 −0.451811
\(670\) 15192.0 0.875997
\(671\) −10104.0 −0.581312
\(672\) −672.000 −0.0385758
\(673\) 23555.0 1.34915 0.674575 0.738206i \(-0.264327\pi\)
0.674575 + 0.738206i \(0.264327\pi\)
\(674\) 19192.0 1.09681
\(675\) 1188.00 0.0677424
\(676\) −8304.00 −0.472462
\(677\) −15954.0 −0.905705 −0.452852 0.891586i \(-0.649594\pi\)
−0.452852 + 0.891586i \(0.649594\pi\)
\(678\) −10674.0 −0.604620
\(679\) 9401.00 0.531336
\(680\) −6912.00 −0.389799
\(681\) 11097.0 0.624432
\(682\) 2256.00 0.126667
\(683\) −10284.0 −0.576144 −0.288072 0.957609i \(-0.593014\pi\)
−0.288072 + 0.957609i \(0.593014\pi\)
\(684\) −1440.00 −0.0804967
\(685\) −21195.0 −1.18222
\(686\) 686.000 0.0381802
\(687\) 20136.0 1.11825
\(688\) 7664.00 0.424691
\(689\) 5676.00 0.313844
\(690\) −1242.00 −0.0685248
\(691\) −193.000 −0.0106253 −0.00531264 0.999986i \(-0.501691\pi\)
−0.00531264 + 0.999986i \(0.501691\pi\)
\(692\) 14472.0 0.795004
\(693\) −756.000 −0.0414402
\(694\) 4158.00 0.227429
\(695\) −20997.0 −1.14599
\(696\) 5544.00 0.301932
\(697\) −2592.00 −0.140859
\(698\) −3164.00 −0.171575
\(699\) −15570.0 −0.842506
\(700\) −1232.00 −0.0665217
\(701\) −6840.00 −0.368535 −0.184268 0.982876i \(-0.558991\pi\)
−0.184268 + 0.982876i \(0.558991\pi\)
\(702\) −594.000 −0.0319360
\(703\) −1880.00 −0.100861
\(704\) −768.000 −0.0411152
\(705\) 11421.0 0.610127
\(706\) 8382.00 0.446828
\(707\) 1890.00 0.100539
\(708\) −10584.0 −0.561824
\(709\) −10354.0 −0.548452 −0.274226 0.961665i \(-0.588422\pi\)
−0.274226 + 0.961665i \(0.588422\pi\)
\(710\) −11772.0 −0.622247
\(711\) 2340.00 0.123427
\(712\) −3312.00 −0.174329
\(713\) 2162.00 0.113559
\(714\) −4032.00 −0.211336
\(715\) 1188.00 0.0621380
\(716\) 3756.00 0.196045
\(717\) −14400.0 −0.750039
\(718\) −23442.0 −1.21845
\(719\) 9111.00 0.472577 0.236289 0.971683i \(-0.424069\pi\)
0.236289 + 0.971683i \(0.424069\pi\)
\(720\) −1296.00 −0.0670820
\(721\) 3143.00 0.162346
\(722\) −10518.0 −0.542160
\(723\) −21687.0 −1.11556
\(724\) −14392.0 −0.738777
\(725\) 10164.0 0.520664
\(726\) 7122.00 0.364080
\(727\) −6676.00 −0.340577 −0.170288 0.985394i \(-0.554470\pi\)
−0.170288 + 0.985394i \(0.554470\pi\)
\(728\) 616.000 0.0313605
\(729\) 729.000 0.0370370
\(730\) 8928.00 0.452658
\(731\) 45984.0 2.32665
\(732\) −10104.0 −0.510184
\(733\) 116.000 0.00584524 0.00292262 0.999996i \(-0.499070\pi\)
0.00292262 + 0.999996i \(0.499070\pi\)
\(734\) 19798.0 0.995582
\(735\) 1323.00 0.0663940
\(736\) −736.000 −0.0368605
\(737\) 10128.0 0.506200
\(738\) −486.000 −0.0242411
\(739\) 314.000 0.0156301 0.00781507 0.999969i \(-0.497512\pi\)
0.00781507 + 0.999969i \(0.497512\pi\)
\(740\) −1692.00 −0.0840529
\(741\) 1320.00 0.0654405
\(742\) 7224.00 0.357414
\(743\) 9744.00 0.481121 0.240560 0.970634i \(-0.422669\pi\)
0.240560 + 0.970634i \(0.422669\pi\)
\(744\) 2256.00 0.111168
\(745\) 19980.0 0.982565
\(746\) −16340.0 −0.801944
\(747\) −1404.00 −0.0687680
\(748\) −4608.00 −0.225248
\(749\) 1932.00 0.0942507
\(750\) −9126.00 −0.444313
\(751\) −8476.00 −0.411842 −0.205921 0.978569i \(-0.566019\pi\)
−0.205921 + 0.978569i \(0.566019\pi\)
\(752\) 6768.00 0.328196
\(753\) 369.000 0.0178580
\(754\) −5082.00 −0.245458
\(755\) 25443.0 1.22644
\(756\) −756.000 −0.0363696
\(757\) −25918.0 −1.24439 −0.622197 0.782861i \(-0.713760\pi\)
−0.622197 + 0.782861i \(0.713760\pi\)
\(758\) −12326.0 −0.590634
\(759\) −828.000 −0.0395975
\(760\) 2880.00 0.137459
\(761\) 12534.0 0.597053 0.298526 0.954401i \(-0.403505\pi\)
0.298526 + 0.954401i \(0.403505\pi\)
\(762\) −4638.00 −0.220495
\(763\) −4123.00 −0.195626
\(764\) −20016.0 −0.947845
\(765\) −7776.00 −0.367506
\(766\) −18336.0 −0.864891
\(767\) 9702.00 0.456739
\(768\) −768.000 −0.0360844
\(769\) 9179.00 0.430433 0.215217 0.976566i \(-0.430954\pi\)
0.215217 + 0.976566i \(0.430954\pi\)
\(770\) 1512.00 0.0707645
\(771\) −17658.0 −0.824821
\(772\) 12380.0 0.577158
\(773\) −23121.0 −1.07581 −0.537907 0.843004i \(-0.680785\pi\)
−0.537907 + 0.843004i \(0.680785\pi\)
\(774\) 8622.00 0.400402
\(775\) 4136.00 0.191703
\(776\) 10744.0 0.497019
\(777\) −987.000 −0.0455707
\(778\) 13668.0 0.629848
\(779\) 1080.00 0.0496727
\(780\) 1188.00 0.0545349
\(781\) −7848.00 −0.359569
\(782\) −4416.00 −0.201938
\(783\) 6237.00 0.284664
\(784\) 784.000 0.0357143
\(785\) −17352.0 −0.788942
\(786\) −3420.00 −0.155200
\(787\) 16436.0 0.744447 0.372224 0.928143i \(-0.378595\pi\)
0.372224 + 0.928143i \(0.378595\pi\)
\(788\) −2484.00 −0.112295
\(789\) −1071.00 −0.0483252
\(790\) −4680.00 −0.210768
\(791\) 12453.0 0.559770
\(792\) −864.000 −0.0387638
\(793\) 9262.00 0.414758
\(794\) −11396.0 −0.509356
\(795\) 13932.0 0.621531
\(796\) 12764.0 0.568352
\(797\) −18669.0 −0.829724 −0.414862 0.909884i \(-0.636170\pi\)
−0.414862 + 0.909884i \(0.636170\pi\)
\(798\) 1680.00 0.0745255
\(799\) 40608.0 1.79801
\(800\) −1408.00 −0.0622254
\(801\) −3726.00 −0.164359
\(802\) −1884.00 −0.0829506
\(803\) 5952.00 0.261571
\(804\) 10128.0 0.444262
\(805\) 1449.00 0.0634417
\(806\) −2068.00 −0.0903749
\(807\) 25254.0 1.10159
\(808\) 2160.00 0.0940452
\(809\) −20616.0 −0.895946 −0.447973 0.894047i \(-0.647854\pi\)
−0.447973 + 0.894047i \(0.647854\pi\)
\(810\) −1458.00 −0.0632456
\(811\) −16075.0 −0.696017 −0.348008 0.937491i \(-0.613142\pi\)
−0.348008 + 0.937491i \(0.613142\pi\)
\(812\) −6468.00 −0.279535
\(813\) −12516.0 −0.539920
\(814\) −1128.00 −0.0485705
\(815\) 17964.0 0.772088
\(816\) −4608.00 −0.197687
\(817\) −19160.0 −0.820469
\(818\) −25952.0 −1.10928
\(819\) 693.000 0.0295670
\(820\) 972.000 0.0413948
\(821\) −17010.0 −0.723085 −0.361543 0.932356i \(-0.617750\pi\)
−0.361543 + 0.932356i \(0.617750\pi\)
\(822\) −14130.0 −0.599562
\(823\) −1699.00 −0.0719604 −0.0359802 0.999353i \(-0.511455\pi\)
−0.0359802 + 0.999353i \(0.511455\pi\)
\(824\) 3592.00 0.151861
\(825\) −1584.00 −0.0668458
\(826\) 12348.0 0.520148
\(827\) 35106.0 1.47612 0.738062 0.674732i \(-0.235741\pi\)
0.738062 + 0.674732i \(0.235741\pi\)
\(828\) −828.000 −0.0347524
\(829\) −15070.0 −0.631366 −0.315683 0.948865i \(-0.602234\pi\)
−0.315683 + 0.948865i \(0.602234\pi\)
\(830\) 2808.00 0.117430
\(831\) 2280.00 0.0951773
\(832\) 704.000 0.0293351
\(833\) 4704.00 0.195659
\(834\) −13998.0 −0.581188
\(835\) −10260.0 −0.425224
\(836\) 1920.00 0.0794313
\(837\) 2538.00 0.104810
\(838\) 72.0000 0.00296802
\(839\) 14082.0 0.579457 0.289729 0.957109i \(-0.406435\pi\)
0.289729 + 0.957109i \(0.406435\pi\)
\(840\) 1512.00 0.0621059
\(841\) 28972.0 1.18791
\(842\) 1318.00 0.0539445
\(843\) 28017.0 1.14467
\(844\) 320.000 0.0130508
\(845\) 18684.0 0.760650
\(846\) 7614.00 0.309426
\(847\) −8309.00 −0.337073
\(848\) 8256.00 0.334330
\(849\) 18786.0 0.759404
\(850\) −8448.00 −0.340899
\(851\) −1081.00 −0.0435443
\(852\) −7848.00 −0.315573
\(853\) 42737.0 1.71546 0.857730 0.514101i \(-0.171874\pi\)
0.857730 + 0.514101i \(0.171874\pi\)
\(854\) 11788.0 0.472338
\(855\) 3240.00 0.129597
\(856\) 2208.00 0.0881634
\(857\) 28773.0 1.14687 0.573435 0.819251i \(-0.305611\pi\)
0.573435 + 0.819251i \(0.305611\pi\)
\(858\) 792.000 0.0315133
\(859\) −4471.00 −0.177589 −0.0887943 0.996050i \(-0.528301\pi\)
−0.0887943 + 0.996050i \(0.528301\pi\)
\(860\) −17244.0 −0.683739
\(861\) 567.000 0.0224429
\(862\) 31146.0 1.23067
\(863\) 27672.0 1.09150 0.545751 0.837948i \(-0.316245\pi\)
0.545751 + 0.837948i \(0.316245\pi\)
\(864\) −864.000 −0.0340207
\(865\) −32562.0 −1.27993
\(866\) −31454.0 −1.23424
\(867\) −12909.0 −0.505666
\(868\) −2632.00 −0.102922
\(869\) −3120.00 −0.121794
\(870\) −12474.0 −0.486101
\(871\) −9284.00 −0.361167
\(872\) −4712.00 −0.182991
\(873\) 12087.0 0.468594
\(874\) 1840.00 0.0712116
\(875\) 10647.0 0.411353
\(876\) 5952.00 0.229566
\(877\) 1316.00 0.0506707 0.0253353 0.999679i \(-0.491935\pi\)
0.0253353 + 0.999679i \(0.491935\pi\)
\(878\) 23332.0 0.896830
\(879\) 17262.0 0.662381
\(880\) 1728.00 0.0661942
\(881\) 34950.0 1.33654 0.668272 0.743917i \(-0.267034\pi\)
0.668272 + 0.743917i \(0.267034\pi\)
\(882\) 882.000 0.0336718
\(883\) 15746.0 0.600108 0.300054 0.953922i \(-0.402995\pi\)
0.300054 + 0.953922i \(0.402995\pi\)
\(884\) 4224.00 0.160711
\(885\) 23814.0 0.904518
\(886\) 28914.0 1.09637
\(887\) −9240.00 −0.349773 −0.174887 0.984589i \(-0.555956\pi\)
−0.174887 + 0.984589i \(0.555956\pi\)
\(888\) −1128.00 −0.0426275
\(889\) 5411.00 0.204138
\(890\) 7452.00 0.280665
\(891\) −972.000 −0.0365468
\(892\) 10424.0 0.391280
\(893\) −16920.0 −0.634050
\(894\) 13320.0 0.498308
\(895\) −8451.00 −0.315627
\(896\) 896.000 0.0334077
\(897\) 759.000 0.0282523
\(898\) 11772.0 0.437457
\(899\) 21714.0 0.805564
\(900\) −1584.00 −0.0586667
\(901\) 49536.0 1.83161
\(902\) 648.000 0.0239202
\(903\) −10059.0 −0.370701
\(904\) 14232.0 0.523616
\(905\) 32382.0 1.18941
\(906\) 16962.0 0.621992
\(907\) −11695.0 −0.428143 −0.214072 0.976818i \(-0.568673\pi\)
−0.214072 + 0.976818i \(0.568673\pi\)
\(908\) −14796.0 −0.540774
\(909\) 2430.00 0.0886667
\(910\) −1386.00 −0.0504895
\(911\) 48495.0 1.76368 0.881839 0.471550i \(-0.156305\pi\)
0.881839 + 0.471550i \(0.156305\pi\)
\(912\) 1920.00 0.0697122
\(913\) 1872.00 0.0678578
\(914\) −15080.0 −0.545735
\(915\) 22734.0 0.821380
\(916\) −26848.0 −0.968431
\(917\) 3990.00 0.143687
\(918\) −5184.00 −0.186381
\(919\) 24788.0 0.889750 0.444875 0.895593i \(-0.353248\pi\)
0.444875 + 0.895593i \(0.353248\pi\)
\(920\) 1656.00 0.0593442
\(921\) −11913.0 −0.426218
\(922\) 2532.00 0.0904414
\(923\) 7194.00 0.256548
\(924\) 1008.00 0.0358883
\(925\) −2068.00 −0.0735086
\(926\) −24530.0 −0.870525
\(927\) 4041.00 0.143176
\(928\) −7392.00 −0.261481
\(929\) −32859.0 −1.16046 −0.580231 0.814452i \(-0.697038\pi\)
−0.580231 + 0.814452i \(0.697038\pi\)
\(930\) −5076.00 −0.178977
\(931\) −1960.00 −0.0689972
\(932\) 20760.0 0.729631
\(933\) −18468.0 −0.648033
\(934\) 20982.0 0.735066
\(935\) 10368.0 0.362642
\(936\) 792.000 0.0276574
\(937\) −45817.0 −1.59741 −0.798707 0.601721i \(-0.794482\pi\)
−0.798707 + 0.601721i \(0.794482\pi\)
\(938\) −11816.0 −0.411307
\(939\) 12630.0 0.438940
\(940\) −15228.0 −0.528386
\(941\) −10737.0 −0.371962 −0.185981 0.982553i \(-0.559546\pi\)
−0.185981 + 0.982553i \(0.559546\pi\)
\(942\) −11568.0 −0.400112
\(943\) 621.000 0.0214449
\(944\) 14112.0 0.486553
\(945\) 1701.00 0.0585540
\(946\) −11496.0 −0.395103
\(947\) 54225.0 1.86069 0.930346 0.366682i \(-0.119506\pi\)
0.930346 + 0.366682i \(0.119506\pi\)
\(948\) −3120.00 −0.106891
\(949\) −5456.00 −0.186627
\(950\) 3520.00 0.120215
\(951\) 3987.00 0.135949
\(952\) 5376.00 0.183022
\(953\) −4050.00 −0.137663 −0.0688313 0.997628i \(-0.521927\pi\)
−0.0688313 + 0.997628i \(0.521927\pi\)
\(954\) 9288.00 0.315210
\(955\) 45036.0 1.52600
\(956\) 19200.0 0.649553
\(957\) −8316.00 −0.280897
\(958\) 20124.0 0.678682
\(959\) 16485.0 0.555087
\(960\) 1728.00 0.0580948
\(961\) −20955.0 −0.703400
\(962\) 1034.00 0.0346544
\(963\) 2484.00 0.0831213
\(964\) 28916.0 0.966101
\(965\) −27855.0 −0.929206
\(966\) 966.000 0.0321745
\(967\) −35452.0 −1.17897 −0.589483 0.807781i \(-0.700668\pi\)
−0.589483 + 0.807781i \(0.700668\pi\)
\(968\) −9496.00 −0.315303
\(969\) 11520.0 0.381915
\(970\) −24174.0 −0.800186
\(971\) −48048.0 −1.58799 −0.793993 0.607927i \(-0.792001\pi\)
−0.793993 + 0.607927i \(0.792001\pi\)
\(972\) −972.000 −0.0320750
\(973\) 16331.0 0.538076
\(974\) 27934.0 0.918956
\(975\) 1452.00 0.0476936
\(976\) 13472.0 0.441832
\(977\) 8391.00 0.274772 0.137386 0.990518i \(-0.456130\pi\)
0.137386 + 0.990518i \(0.456130\pi\)
\(978\) 11976.0 0.391565
\(979\) 4968.00 0.162184
\(980\) −1764.00 −0.0574989
\(981\) −5301.00 −0.172526
\(982\) 19464.0 0.632506
\(983\) −2322.00 −0.0753411 −0.0376705 0.999290i \(-0.511994\pi\)
−0.0376705 + 0.999290i \(0.511994\pi\)
\(984\) 648.000 0.0209934
\(985\) 5589.00 0.180792
\(986\) −44352.0 −1.43251
\(987\) −8883.00 −0.286473
\(988\) −1760.00 −0.0566731
\(989\) −11017.0 −0.354217
\(990\) 1944.00 0.0624085
\(991\) 6476.00 0.207585 0.103793 0.994599i \(-0.466902\pi\)
0.103793 + 0.994599i \(0.466902\pi\)
\(992\) −3008.00 −0.0962743
\(993\) 22332.0 0.713680
\(994\) 9156.00 0.292164
\(995\) −28719.0 −0.915029
\(996\) 1872.00 0.0595548
\(997\) 61238.0 1.94526 0.972631 0.232354i \(-0.0746427\pi\)
0.972631 + 0.232354i \(0.0746427\pi\)
\(998\) −12380.0 −0.392667
\(999\) −1269.00 −0.0401896
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 966.4.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
966.4.a.b.1.1 1 1.1 even 1 trivial