Properties

Label 6762.2.a.ci.1.4
Level $6762$
Weight $2$
Character 6762.1
Self dual yes
Analytic conductor $53.995$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6762 = 2 \cdot 3 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6762.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(53.9948418468\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.2624.1
Defining polynomial: \(x^{4} - 2 x^{3} - 3 x^{2} + 2 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-1.22833\) of defining polynomial
Character \(\chi\) \(=\) 6762.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.585786 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.585786 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -0.585786 q^{10} +5.93089 q^{11} -1.00000 q^{12} -6.45666 q^{13} +0.585786 q^{15} +1.00000 q^{16} +0.371771 q^{17} +1.00000 q^{18} -3.26022 q^{19} -0.585786 q^{20} +5.93089 q^{22} +1.00000 q^{23} -1.00000 q^{24} -4.65685 q^{25} -6.45666 q^{26} -1.00000 q^{27} -0.948474 q^{29} +0.585786 q^{30} -1.26022 q^{31} +1.00000 q^{32} -5.93089 q^{33} +0.371771 q^{34} +1.00000 q^{36} +0.371771 q^{37} -3.26022 q^{38} +6.45666 q^{39} -0.585786 q^{40} +10.4476 q^{41} -6.54154 q^{43} +5.93089 q^{44} -0.585786 q^{45} +1.00000 q^{46} -10.6016 q^{47} -1.00000 q^{48} -4.65685 q^{50} -0.371771 q^{51} -6.45666 q^{52} -8.23356 q^{53} -1.00000 q^{54} -3.47424 q^{55} +3.26022 q^{57} -0.948474 q^{58} -1.05153 q^{59} +0.585786 q^{60} +4.44757 q^{61} -1.26022 q^{62} +1.00000 q^{64} +3.78222 q^{65} -5.93089 q^{66} -0.0286249 q^{67} +0.371771 q^{68} -1.00000 q^{69} +2.38755 q^{71} +1.00000 q^{72} -4.76840 q^{73} +0.371771 q^{74} +4.65685 q^{75} -3.26022 q^{76} +6.45666 q^{78} -10.7307 q^{79} -0.585786 q^{80} +1.00000 q^{81} +10.4476 q^{82} -2.51668 q^{83} -0.217778 q^{85} -6.54154 q^{86} +0.948474 q^{87} +5.93089 q^{88} +15.2318 q^{89} -0.585786 q^{90} +1.00000 q^{92} +1.26022 q^{93} -10.6016 q^{94} +1.90979 q^{95} -1.00000 q^{96} -8.85705 q^{97} +5.93089 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{2} - 4q^{3} + 4q^{4} - 8q^{5} - 4q^{6} + 4q^{8} + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{2} - 4q^{3} + 4q^{4} - 8q^{5} - 4q^{6} + 4q^{8} + 4q^{9} - 8q^{10} + 4q^{11} - 4q^{12} - 12q^{13} + 8q^{15} + 4q^{16} + 4q^{17} + 4q^{18} - 4q^{19} - 8q^{20} + 4q^{22} + 4q^{23} - 4q^{24} + 4q^{25} - 12q^{26} - 4q^{27} + 8q^{29} + 8q^{30} + 4q^{31} + 4q^{32} - 4q^{33} + 4q^{34} + 4q^{36} + 4q^{37} - 4q^{38} + 12q^{39} - 8q^{40} + 8q^{41} + 4q^{43} + 4q^{44} - 8q^{45} + 4q^{46} - 12q^{47} - 4q^{48} + 4q^{50} - 4q^{51} - 12q^{52} + 4q^{53} - 4q^{54} - 8q^{55} + 4q^{57} + 8q^{58} - 16q^{59} + 8q^{60} - 16q^{61} + 4q^{62} + 4q^{64} + 16q^{65} - 4q^{66} + 20q^{67} + 4q^{68} - 4q^{69} - 24q^{71} + 4q^{72} - 8q^{73} + 4q^{74} - 4q^{75} - 4q^{76} + 12q^{78} - 32q^{79} - 8q^{80} + 4q^{81} + 8q^{82} + 4q^{83} + 4q^{86} - 8q^{87} + 4q^{88} - 20q^{89} - 8q^{90} + 4q^{92} - 4q^{93} - 12q^{94} - 4q^{96} - 4q^{97} + 4q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) −0.585786 −0.261972 −0.130986 0.991384i \(-0.541814\pi\)
−0.130986 + 0.991384i \(0.541814\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −0.585786 −0.185242
\(11\) 5.93089 1.78823 0.894116 0.447836i \(-0.147805\pi\)
0.894116 + 0.447836i \(0.147805\pi\)
\(12\) −1.00000 −0.288675
\(13\) −6.45666 −1.79075 −0.895377 0.445309i \(-0.853094\pi\)
−0.895377 + 0.445309i \(0.853094\pi\)
\(14\) 0 0
\(15\) 0.585786 0.151249
\(16\) 1.00000 0.250000
\(17\) 0.371771 0.0901676 0.0450838 0.998983i \(-0.485645\pi\)
0.0450838 + 0.998983i \(0.485645\pi\)
\(18\) 1.00000 0.235702
\(19\) −3.26022 −0.747946 −0.373973 0.927440i \(-0.622005\pi\)
−0.373973 + 0.927440i \(0.622005\pi\)
\(20\) −0.585786 −0.130986
\(21\) 0 0
\(22\) 5.93089 1.26447
\(23\) 1.00000 0.208514
\(24\) −1.00000 −0.204124
\(25\) −4.65685 −0.931371
\(26\) −6.45666 −1.26625
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −0.948474 −0.176127 −0.0880636 0.996115i \(-0.528068\pi\)
−0.0880636 + 0.996115i \(0.528068\pi\)
\(30\) 0.585786 0.106949
\(31\) −1.26022 −0.226342 −0.113171 0.993576i \(-0.536101\pi\)
−0.113171 + 0.993576i \(0.536101\pi\)
\(32\) 1.00000 0.176777
\(33\) −5.93089 −1.03244
\(34\) 0.371771 0.0637581
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 0.371771 0.0611187 0.0305594 0.999533i \(-0.490271\pi\)
0.0305594 + 0.999533i \(0.490271\pi\)
\(38\) −3.26022 −0.528878
\(39\) 6.45666 1.03389
\(40\) −0.585786 −0.0926210
\(41\) 10.4476 1.63164 0.815818 0.578308i \(-0.196287\pi\)
0.815818 + 0.578308i \(0.196287\pi\)
\(42\) 0 0
\(43\) −6.54154 −0.997576 −0.498788 0.866724i \(-0.666221\pi\)
−0.498788 + 0.866724i \(0.666221\pi\)
\(44\) 5.93089 0.894116
\(45\) −0.585786 −0.0873239
\(46\) 1.00000 0.147442
\(47\) −10.6016 −1.54640 −0.773199 0.634164i \(-0.781344\pi\)
−0.773199 + 0.634164i \(0.781344\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) −4.65685 −0.658579
\(51\) −0.371771 −0.0520583
\(52\) −6.45666 −0.895377
\(53\) −8.23356 −1.13097 −0.565483 0.824760i \(-0.691310\pi\)
−0.565483 + 0.824760i \(0.691310\pi\)
\(54\) −1.00000 −0.136083
\(55\) −3.47424 −0.468466
\(56\) 0 0
\(57\) 3.26022 0.431827
\(58\) −0.948474 −0.124541
\(59\) −1.05153 −0.136897 −0.0684485 0.997655i \(-0.521805\pi\)
−0.0684485 + 0.997655i \(0.521805\pi\)
\(60\) 0.585786 0.0756247
\(61\) 4.44757 0.569453 0.284727 0.958609i \(-0.408097\pi\)
0.284727 + 0.958609i \(0.408097\pi\)
\(62\) −1.26022 −0.160048
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 3.78222 0.469127
\(66\) −5.93089 −0.730042
\(67\) −0.0286249 −0.00349709 −0.00174855 0.999998i \(-0.500557\pi\)
−0.00174855 + 0.999998i \(0.500557\pi\)
\(68\) 0.371771 0.0450838
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 2.38755 0.283350 0.141675 0.989913i \(-0.454751\pi\)
0.141675 + 0.989913i \(0.454751\pi\)
\(72\) 1.00000 0.117851
\(73\) −4.76840 −0.558099 −0.279050 0.960277i \(-0.590019\pi\)
−0.279050 + 0.960277i \(0.590019\pi\)
\(74\) 0.371771 0.0432175
\(75\) 4.65685 0.537727
\(76\) −3.26022 −0.373973
\(77\) 0 0
\(78\) 6.45666 0.731072
\(79\) −10.7307 −1.20730 −0.603649 0.797250i \(-0.706287\pi\)
−0.603649 + 0.797250i \(0.706287\pi\)
\(80\) −0.585786 −0.0654929
\(81\) 1.00000 0.111111
\(82\) 10.4476 1.15374
\(83\) −2.51668 −0.276241 −0.138121 0.990415i \(-0.544106\pi\)
−0.138121 + 0.990415i \(0.544106\pi\)
\(84\) 0 0
\(85\) −0.217778 −0.0236214
\(86\) −6.54154 −0.705393
\(87\) 0.948474 0.101687
\(88\) 5.93089 0.632235
\(89\) 15.2318 1.61456 0.807281 0.590167i \(-0.200938\pi\)
0.807281 + 0.590167i \(0.200938\pi\)
\(90\) −0.585786 −0.0617473
\(91\) 0 0
\(92\) 1.00000 0.104257
\(93\) 1.26022 0.130679
\(94\) −10.6016 −1.09347
\(95\) 1.90979 0.195941
\(96\) −1.00000 −0.102062
\(97\) −8.85705 −0.899297 −0.449649 0.893205i \(-0.648451\pi\)
−0.449649 + 0.893205i \(0.648451\pi\)
\(98\) 0 0
\(99\) 5.93089 0.596077
\(100\) −4.65685 −0.465685
\(101\) 4.54154 0.451900 0.225950 0.974139i \(-0.427451\pi\)
0.225950 + 0.974139i \(0.427451\pi\)
\(102\) −0.371771 −0.0368108
\(103\) −5.77690 −0.569215 −0.284607 0.958644i \(-0.591863\pi\)
−0.284607 + 0.958644i \(0.591863\pi\)
\(104\) −6.45666 −0.633127
\(105\) 0 0
\(106\) −8.23356 −0.799714
\(107\) 7.41798 0.717123 0.358561 0.933506i \(-0.383267\pi\)
0.358561 + 0.933506i \(0.383267\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −10.8571 −1.03992 −0.519958 0.854192i \(-0.674053\pi\)
−0.519958 + 0.854192i \(0.674053\pi\)
\(110\) −3.47424 −0.331255
\(111\) −0.371771 −0.0352869
\(112\) 0 0
\(113\) −10.5258 −0.990181 −0.495090 0.868842i \(-0.664865\pi\)
−0.495090 + 0.868842i \(0.664865\pi\)
\(114\) 3.26022 0.305348
\(115\) −0.585786 −0.0546249
\(116\) −0.948474 −0.0880636
\(117\) −6.45666 −0.596918
\(118\) −1.05153 −0.0968008
\(119\) 0 0
\(120\) 0.585786 0.0534747
\(121\) 24.1755 2.19777
\(122\) 4.44757 0.402664
\(123\) −10.4476 −0.942026
\(124\) −1.26022 −0.113171
\(125\) 5.65685 0.505964
\(126\) 0 0
\(127\) −20.5925 −1.82729 −0.913644 0.406516i \(-0.866744\pi\)
−0.913644 + 0.406516i \(0.866744\pi\)
\(128\) 1.00000 0.0883883
\(129\) 6.54154 0.575951
\(130\) 3.78222 0.331723
\(131\) 13.3489 1.16630 0.583148 0.812366i \(-0.301821\pi\)
0.583148 + 0.812366i \(0.301821\pi\)
\(132\) −5.93089 −0.516218
\(133\) 0 0
\(134\) −0.0286249 −0.00247282
\(135\) 0.585786 0.0504165
\(136\) 0.371771 0.0318791
\(137\) −10.7435 −0.917883 −0.458941 0.888467i \(-0.651771\pi\)
−0.458941 + 0.888467i \(0.651771\pi\)
\(138\) −1.00000 −0.0851257
\(139\) −10.8284 −0.918455 −0.459228 0.888319i \(-0.651874\pi\)
−0.459228 + 0.888319i \(0.651874\pi\)
\(140\) 0 0
\(141\) 10.6016 0.892813
\(142\) 2.38755 0.200359
\(143\) −38.2937 −3.20228
\(144\) 1.00000 0.0833333
\(145\) 0.555603 0.0461403
\(146\) −4.76840 −0.394636
\(147\) 0 0
\(148\) 0.371771 0.0305594
\(149\) 10.6616 0.873431 0.436716 0.899600i \(-0.356142\pi\)
0.436716 + 0.899600i \(0.356142\pi\)
\(150\) 4.65685 0.380231
\(151\) −12.9005 −1.04983 −0.524913 0.851156i \(-0.675902\pi\)
−0.524913 + 0.851156i \(0.675902\pi\)
\(152\) −3.26022 −0.264439
\(153\) 0.371771 0.0300559
\(154\) 0 0
\(155\) 0.738220 0.0592953
\(156\) 6.45666 0.516946
\(157\) 10.3222 0.823802 0.411901 0.911229i \(-0.364865\pi\)
0.411901 + 0.911229i \(0.364865\pi\)
\(158\) −10.7307 −0.853688
\(159\) 8.23356 0.652964
\(160\) −0.585786 −0.0463105
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) 2.18262 0.170956 0.0854779 0.996340i \(-0.472758\pi\)
0.0854779 + 0.996340i \(0.472758\pi\)
\(164\) 10.4476 0.815818
\(165\) 3.47424 0.270469
\(166\) −2.51668 −0.195332
\(167\) 3.83039 0.296404 0.148202 0.988957i \(-0.452651\pi\)
0.148202 + 0.988957i \(0.452651\pi\)
\(168\) 0 0
\(169\) 28.6884 2.20680
\(170\) −0.217778 −0.0167028
\(171\) −3.26022 −0.249315
\(172\) −6.54154 −0.498788
\(173\) 3.08015 0.234180 0.117090 0.993121i \(-0.462643\pi\)
0.117090 + 0.993121i \(0.462643\pi\)
\(174\) 0.948474 0.0719036
\(175\) 0 0
\(176\) 5.93089 0.447058
\(177\) 1.05153 0.0790376
\(178\) 15.2318 1.14167
\(179\) 10.3747 0.775442 0.387721 0.921777i \(-0.373262\pi\)
0.387721 + 0.921777i \(0.373262\pi\)
\(180\) −0.585786 −0.0436619
\(181\) −13.6191 −1.01230 −0.506152 0.862445i \(-0.668932\pi\)
−0.506152 + 0.862445i \(0.668932\pi\)
\(182\) 0 0
\(183\) −4.44757 −0.328774
\(184\) 1.00000 0.0737210
\(185\) −0.217778 −0.0160114
\(186\) 1.26022 0.0924039
\(187\) 2.20493 0.161241
\(188\) −10.6016 −0.773199
\(189\) 0 0
\(190\) 1.90979 0.138551
\(191\) 9.30086 0.672987 0.336493 0.941686i \(-0.390759\pi\)
0.336493 + 0.941686i \(0.390759\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −22.1240 −1.59252 −0.796259 0.604956i \(-0.793191\pi\)
−0.796259 + 0.604956i \(0.793191\pi\)
\(194\) −8.85705 −0.635899
\(195\) −3.78222 −0.270851
\(196\) 0 0
\(197\) −14.2493 −1.01522 −0.507612 0.861586i \(-0.669471\pi\)
−0.507612 + 0.861586i \(0.669471\pi\)
\(198\) 5.93089 0.421490
\(199\) −5.63869 −0.399716 −0.199858 0.979825i \(-0.564048\pi\)
−0.199858 + 0.979825i \(0.564048\pi\)
\(200\) −4.65685 −0.329289
\(201\) 0.0286249 0.00201905
\(202\) 4.54154 0.319542
\(203\) 0 0
\(204\) −0.371771 −0.0260292
\(205\) −6.12005 −0.427443
\(206\) −5.77690 −0.402496
\(207\) 1.00000 0.0695048
\(208\) −6.45666 −0.447689
\(209\) −19.3360 −1.33750
\(210\) 0 0
\(211\) 10.0444 0.691485 0.345743 0.938329i \(-0.387627\pi\)
0.345743 + 0.938329i \(0.387627\pi\)
\(212\) −8.23356 −0.565483
\(213\) −2.38755 −0.163592
\(214\) 7.41798 0.507082
\(215\) 3.83195 0.261337
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) −10.8571 −0.735332
\(219\) 4.76840 0.322219
\(220\) −3.47424 −0.234233
\(221\) −2.40040 −0.161468
\(222\) −0.371771 −0.0249516
\(223\) 14.6588 0.981627 0.490813 0.871265i \(-0.336700\pi\)
0.490813 + 0.871265i \(0.336700\pi\)
\(224\) 0 0
\(225\) −4.65685 −0.310457
\(226\) −10.5258 −0.700163
\(227\) −8.36900 −0.555470 −0.277735 0.960658i \(-0.589584\pi\)
−0.277735 + 0.960658i \(0.589584\pi\)
\(228\) 3.26022 0.215913
\(229\) −28.3666 −1.87452 −0.937259 0.348633i \(-0.886646\pi\)
−0.937259 + 0.348633i \(0.886646\pi\)
\(230\) −0.585786 −0.0386256
\(231\) 0 0
\(232\) −0.948474 −0.0622703
\(233\) 13.1755 0.863155 0.431578 0.902076i \(-0.357957\pi\)
0.431578 + 0.902076i \(0.357957\pi\)
\(234\) −6.45666 −0.422085
\(235\) 6.21025 0.405112
\(236\) −1.05153 −0.0684485
\(237\) 10.7307 0.697034
\(238\) 0 0
\(239\) −6.35599 −0.411135 −0.205567 0.978643i \(-0.565904\pi\)
−0.205567 + 0.978643i \(0.565904\pi\)
\(240\) 0.585786 0.0378124
\(241\) −5.52518 −0.355908 −0.177954 0.984039i \(-0.556948\pi\)
−0.177954 + 0.984039i \(0.556948\pi\)
\(242\) 24.1755 1.55406
\(243\) −1.00000 −0.0641500
\(244\) 4.44757 0.284727
\(245\) 0 0
\(246\) −10.4476 −0.666113
\(247\) 21.0501 1.33939
\(248\) −1.26022 −0.0800241
\(249\) 2.51668 0.159488
\(250\) 5.65685 0.357771
\(251\) −6.77494 −0.427630 −0.213815 0.976874i \(-0.568589\pi\)
−0.213815 + 0.976874i \(0.568589\pi\)
\(252\) 0 0
\(253\) 5.93089 0.372872
\(254\) −20.5925 −1.29209
\(255\) 0.217778 0.0136378
\(256\) 1.00000 0.0625000
\(257\) −5.29417 −0.330241 −0.165121 0.986273i \(-0.552801\pi\)
−0.165121 + 0.986273i \(0.552801\pi\)
\(258\) 6.54154 0.407259
\(259\) 0 0
\(260\) 3.78222 0.234563
\(261\) −0.948474 −0.0587090
\(262\) 13.3489 0.824696
\(263\) −31.3137 −1.93089 −0.965443 0.260614i \(-0.916075\pi\)
−0.965443 + 0.260614i \(0.916075\pi\)
\(264\) −5.93089 −0.365021
\(265\) 4.82311 0.296281
\(266\) 0 0
\(267\) −15.2318 −0.932168
\(268\) −0.0286249 −0.00174855
\(269\) −22.5064 −1.37224 −0.686119 0.727489i \(-0.740687\pi\)
−0.686119 + 0.727489i \(0.740687\pi\)
\(270\) 0.585786 0.0356498
\(271\) −8.90423 −0.540893 −0.270447 0.962735i \(-0.587171\pi\)
−0.270447 + 0.962735i \(0.587171\pi\)
\(272\) 0.371771 0.0225419
\(273\) 0 0
\(274\) −10.7435 −0.649041
\(275\) −27.6193 −1.66551
\(276\) −1.00000 −0.0601929
\(277\) −9.01637 −0.541741 −0.270870 0.962616i \(-0.587312\pi\)
−0.270870 + 0.962616i \(0.587312\pi\)
\(278\) −10.8284 −0.649446
\(279\) −1.26022 −0.0754475
\(280\) 0 0
\(281\) 24.4320 1.45749 0.728744 0.684786i \(-0.240104\pi\)
0.728744 + 0.684786i \(0.240104\pi\)
\(282\) 10.6016 0.631314
\(283\) 0.762094 0.0453018 0.0226509 0.999743i \(-0.492789\pi\)
0.0226509 + 0.999743i \(0.492789\pi\)
\(284\) 2.38755 0.141675
\(285\) −1.90979 −0.113126
\(286\) −38.2937 −2.26436
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) −16.8618 −0.991870
\(290\) 0.555603 0.0326261
\(291\) 8.85705 0.519210
\(292\) −4.76840 −0.279050
\(293\) −2.71868 −0.158827 −0.0794135 0.996842i \(-0.525305\pi\)
−0.0794135 + 0.996842i \(0.525305\pi\)
\(294\) 0 0
\(295\) 0.615970 0.0358632
\(296\) 0.371771 0.0216087
\(297\) −5.93089 −0.344145
\(298\) 10.6616 0.617609
\(299\) −6.45666 −0.373398
\(300\) 4.65685 0.268864
\(301\) 0 0
\(302\) −12.9005 −0.742338
\(303\) −4.54154 −0.260905
\(304\) −3.26022 −0.186987
\(305\) −2.60533 −0.149181
\(306\) 0.371771 0.0212527
\(307\) −21.5684 −1.23097 −0.615486 0.788148i \(-0.711040\pi\)
−0.615486 + 0.788148i \(0.711040\pi\)
\(308\) 0 0
\(309\) 5.77690 0.328636
\(310\) 0.738220 0.0419281
\(311\) −17.0500 −0.966815 −0.483407 0.875395i \(-0.660601\pi\)
−0.483407 + 0.875395i \(0.660601\pi\)
\(312\) 6.45666 0.365536
\(313\) −21.7522 −1.22951 −0.614753 0.788719i \(-0.710744\pi\)
−0.614753 + 0.788719i \(0.710744\pi\)
\(314\) 10.3222 0.582516
\(315\) 0 0
\(316\) −10.7307 −0.603649
\(317\) 28.1017 1.57835 0.789173 0.614171i \(-0.210509\pi\)
0.789173 + 0.614171i \(0.210509\pi\)
\(318\) 8.23356 0.461715
\(319\) −5.62530 −0.314956
\(320\) −0.585786 −0.0327465
\(321\) −7.41798 −0.414031
\(322\) 0 0
\(323\) −1.21205 −0.0674405
\(324\) 1.00000 0.0555556
\(325\) 30.0677 1.66786
\(326\) 2.18262 0.120884
\(327\) 10.8571 0.600396
\(328\) 10.4476 0.576871
\(329\) 0 0
\(330\) 3.47424 0.191250
\(331\) 17.6569 0.970508 0.485254 0.874373i \(-0.338727\pi\)
0.485254 + 0.874373i \(0.338727\pi\)
\(332\) −2.51668 −0.138121
\(333\) 0.371771 0.0203729
\(334\) 3.83039 0.209590
\(335\) 0.0167681 0.000916139 0
\(336\) 0 0
\(337\) 14.0796 0.766963 0.383481 0.923549i \(-0.374725\pi\)
0.383481 + 0.923549i \(0.374725\pi\)
\(338\) 28.6884 1.56044
\(339\) 10.5258 0.571681
\(340\) −0.217778 −0.0118107
\(341\) −7.47424 −0.404753
\(342\) −3.26022 −0.176293
\(343\) 0 0
\(344\) −6.54154 −0.352696
\(345\) 0.585786 0.0315377
\(346\) 3.08015 0.165590
\(347\) −17.5666 −0.943027 −0.471514 0.881859i \(-0.656292\pi\)
−0.471514 + 0.881859i \(0.656292\pi\)
\(348\) 0.948474 0.0508435
\(349\) 15.6598 0.838249 0.419125 0.907929i \(-0.362337\pi\)
0.419125 + 0.907929i \(0.362337\pi\)
\(350\) 0 0
\(351\) 6.45666 0.344631
\(352\) 5.93089 0.316118
\(353\) −29.8333 −1.58787 −0.793934 0.608004i \(-0.791970\pi\)
−0.793934 + 0.608004i \(0.791970\pi\)
\(354\) 1.05153 0.0558880
\(355\) −1.39859 −0.0742297
\(356\) 15.2318 0.807281
\(357\) 0 0
\(358\) 10.3747 0.548320
\(359\) 2.18262 0.115194 0.0575971 0.998340i \(-0.481656\pi\)
0.0575971 + 0.998340i \(0.481656\pi\)
\(360\) −0.585786 −0.0308737
\(361\) −8.37096 −0.440577
\(362\) −13.6191 −0.715806
\(363\) −24.1755 −1.26888
\(364\) 0 0
\(365\) 2.79327 0.146206
\(366\) −4.44757 −0.232478
\(367\) 9.22882 0.481741 0.240870 0.970557i \(-0.422567\pi\)
0.240870 + 0.970557i \(0.422567\pi\)
\(368\) 1.00000 0.0521286
\(369\) 10.4476 0.543879
\(370\) −0.217778 −0.0113218
\(371\) 0 0
\(372\) 1.26022 0.0653394
\(373\) 31.9974 1.65676 0.828381 0.560165i \(-0.189262\pi\)
0.828381 + 0.560165i \(0.189262\pi\)
\(374\) 2.20493 0.114014
\(375\) −5.65685 −0.292119
\(376\) −10.6016 −0.546734
\(377\) 6.12397 0.315400
\(378\) 0 0
\(379\) −14.9015 −0.765436 −0.382718 0.923865i \(-0.625012\pi\)
−0.382718 + 0.923865i \(0.625012\pi\)
\(380\) 1.90979 0.0979703
\(381\) 20.5925 1.05498
\(382\) 9.30086 0.475874
\(383\) −9.63116 −0.492129 −0.246065 0.969253i \(-0.579138\pi\)
−0.246065 + 0.969253i \(0.579138\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −22.1240 −1.12608
\(387\) −6.54154 −0.332525
\(388\) −8.85705 −0.449649
\(389\) −28.4882 −1.44441 −0.722205 0.691679i \(-0.756871\pi\)
−0.722205 + 0.691679i \(0.756871\pi\)
\(390\) −3.78222 −0.191520
\(391\) 0.371771 0.0188013
\(392\) 0 0
\(393\) −13.3489 −0.673361
\(394\) −14.2493 −0.717871
\(395\) 6.28590 0.316278
\(396\) 5.93089 0.298039
\(397\) 15.0047 0.753066 0.376533 0.926403i \(-0.377116\pi\)
0.376533 + 0.926403i \(0.377116\pi\)
\(398\) −5.63869 −0.282642
\(399\) 0 0
\(400\) −4.65685 −0.232843
\(401\) −34.6369 −1.72968 −0.864842 0.502045i \(-0.832581\pi\)
−0.864842 + 0.502045i \(0.832581\pi\)
\(402\) 0.0286249 0.00142768
\(403\) 8.13681 0.405324
\(404\) 4.54154 0.225950
\(405\) −0.585786 −0.0291080
\(406\) 0 0
\(407\) 2.20493 0.109294
\(408\) −0.371771 −0.0184054
\(409\) 20.4515 1.01126 0.505631 0.862750i \(-0.331260\pi\)
0.505631 + 0.862750i \(0.331260\pi\)
\(410\) −6.12005 −0.302248
\(411\) 10.7435 0.529940
\(412\) −5.77690 −0.284607
\(413\) 0 0
\(414\) 1.00000 0.0491473
\(415\) 1.47424 0.0723674
\(416\) −6.45666 −0.316564
\(417\) 10.8284 0.530270
\(418\) −19.3360 −0.945756
\(419\) −15.1145 −0.738391 −0.369195 0.929352i \(-0.620367\pi\)
−0.369195 + 0.929352i \(0.620367\pi\)
\(420\) 0 0
\(421\) 10.4069 0.507203 0.253601 0.967309i \(-0.418385\pi\)
0.253601 + 0.967309i \(0.418385\pi\)
\(422\) 10.0444 0.488954
\(423\) −10.6016 −0.515466
\(424\) −8.23356 −0.399857
\(425\) −1.73128 −0.0839795
\(426\) −2.38755 −0.115677
\(427\) 0 0
\(428\) 7.41798 0.358561
\(429\) 38.2937 1.84884
\(430\) 3.83195 0.184793
\(431\) −34.4892 −1.66129 −0.830643 0.556805i \(-0.812027\pi\)
−0.830643 + 0.556805i \(0.812027\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −25.9401 −1.24660 −0.623302 0.781981i \(-0.714209\pi\)
−0.623302 + 0.781981i \(0.714209\pi\)
\(434\) 0 0
\(435\) −0.555603 −0.0266391
\(436\) −10.8571 −0.519958
\(437\) −3.26022 −0.155958
\(438\) 4.76840 0.227843
\(439\) 26.9891 1.28812 0.644060 0.764975i \(-0.277249\pi\)
0.644060 + 0.764975i \(0.277249\pi\)
\(440\) −3.47424 −0.165628
\(441\) 0 0
\(442\) −2.40040 −0.114175
\(443\) 9.74706 0.463097 0.231548 0.972823i \(-0.425621\pi\)
0.231548 + 0.972823i \(0.425621\pi\)
\(444\) −0.371771 −0.0176435
\(445\) −8.92256 −0.422970
\(446\) 14.6588 0.694115
\(447\) −10.6616 −0.504276
\(448\) 0 0
\(449\) −34.8782 −1.64600 −0.823001 0.568040i \(-0.807702\pi\)
−0.823001 + 0.568040i \(0.807702\pi\)
\(450\) −4.65685 −0.219526
\(451\) 61.9634 2.91774
\(452\) −10.5258 −0.495090
\(453\) 12.9005 0.606117
\(454\) −8.36900 −0.392777
\(455\) 0 0
\(456\) 3.26022 0.152674
\(457\) −2.13821 −0.100021 −0.0500107 0.998749i \(-0.515926\pi\)
−0.0500107 + 0.998749i \(0.515926\pi\)
\(458\) −28.3666 −1.32548
\(459\) −0.371771 −0.0173528
\(460\) −0.585786 −0.0273124
\(461\) −14.4385 −0.672468 −0.336234 0.941779i \(-0.609153\pi\)
−0.336234 + 0.941779i \(0.609153\pi\)
\(462\) 0 0
\(463\) −17.3911 −0.808232 −0.404116 0.914708i \(-0.632421\pi\)
−0.404116 + 0.914708i \(0.632421\pi\)
\(464\) −0.948474 −0.0440318
\(465\) −0.738220 −0.0342342
\(466\) 13.1755 0.610343
\(467\) 10.9042 0.504588 0.252294 0.967651i \(-0.418815\pi\)
0.252294 + 0.967651i \(0.418815\pi\)
\(468\) −6.45666 −0.298459
\(469\) 0 0
\(470\) 6.21025 0.286458
\(471\) −10.3222 −0.475622
\(472\) −1.05153 −0.0484004
\(473\) −38.7972 −1.78390
\(474\) 10.7307 0.492877
\(475\) 15.1824 0.696615
\(476\) 0 0
\(477\) −8.23356 −0.376989
\(478\) −6.35599 −0.290716
\(479\) −17.3729 −0.793788 −0.396894 0.917864i \(-0.629912\pi\)
−0.396894 + 0.917864i \(0.629912\pi\)
\(480\) 0.585786 0.0267374
\(481\) −2.40040 −0.109449
\(482\) −5.52518 −0.251665
\(483\) 0 0
\(484\) 24.1755 1.09889
\(485\) 5.18834 0.235590
\(486\) −1.00000 −0.0453609
\(487\) 28.0888 1.27283 0.636413 0.771349i \(-0.280418\pi\)
0.636413 + 0.771349i \(0.280418\pi\)
\(488\) 4.44757 0.201332
\(489\) −2.18262 −0.0987014
\(490\) 0 0
\(491\) 17.0608 0.769942 0.384971 0.922929i \(-0.374211\pi\)
0.384971 + 0.922929i \(0.374211\pi\)
\(492\) −10.4476 −0.471013
\(493\) −0.352615 −0.0158810
\(494\) 21.0501 0.947090
\(495\) −3.47424 −0.156155
\(496\) −1.26022 −0.0565856
\(497\) 0 0
\(498\) 2.51668 0.112775
\(499\) 29.7364 1.33118 0.665592 0.746315i \(-0.268179\pi\)
0.665592 + 0.746315i \(0.268179\pi\)
\(500\) 5.65685 0.252982
\(501\) −3.83039 −0.171129
\(502\) −6.77494 −0.302380
\(503\) 34.1240 1.52151 0.760756 0.649038i \(-0.224828\pi\)
0.760756 + 0.649038i \(0.224828\pi\)
\(504\) 0 0
\(505\) −2.66037 −0.118385
\(506\) 5.93089 0.263660
\(507\) −28.6884 −1.27410
\(508\) −20.5925 −0.913644
\(509\) −28.0953 −1.24530 −0.622652 0.782499i \(-0.713945\pi\)
−0.622652 + 0.782499i \(0.713945\pi\)
\(510\) 0.217778 0.00964338
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 3.26022 0.143942
\(514\) −5.29417 −0.233516
\(515\) 3.38403 0.149118
\(516\) 6.54154 0.287975
\(517\) −62.8768 −2.76532
\(518\) 0 0
\(519\) −3.08015 −0.135204
\(520\) 3.78222 0.165861
\(521\) 20.7188 0.907709 0.453854 0.891076i \(-0.350049\pi\)
0.453854 + 0.891076i \(0.350049\pi\)
\(522\) −0.948474 −0.0415136
\(523\) −11.6883 −0.511092 −0.255546 0.966797i \(-0.582255\pi\)
−0.255546 + 0.966797i \(0.582255\pi\)
\(524\) 13.3489 0.583148
\(525\) 0 0
\(526\) −31.3137 −1.36534
\(527\) −0.468513 −0.0204088
\(528\) −5.93089 −0.258109
\(529\) 1.00000 0.0434783
\(530\) 4.82311 0.209502
\(531\) −1.05153 −0.0456324
\(532\) 0 0
\(533\) −67.4564 −2.92186
\(534\) −15.2318 −0.659143
\(535\) −4.34535 −0.187866
\(536\) −0.0286249 −0.00123641
\(537\) −10.3747 −0.447702
\(538\) −22.5064 −0.970319
\(539\) 0 0
\(540\) 0.585786 0.0252082
\(541\) 27.8582 1.19772 0.598858 0.800855i \(-0.295621\pi\)
0.598858 + 0.800855i \(0.295621\pi\)
\(542\) −8.90423 −0.382469
\(543\) 13.6191 0.584453
\(544\) 0.371771 0.0159395
\(545\) 6.35991 0.272429
\(546\) 0 0
\(547\) −13.9964 −0.598443 −0.299221 0.954184i \(-0.596727\pi\)
−0.299221 + 0.954184i \(0.596727\pi\)
\(548\) −10.7435 −0.458941
\(549\) 4.44757 0.189818
\(550\) −27.6193 −1.17769
\(551\) 3.09223 0.131734
\(552\) −1.00000 −0.0425628
\(553\) 0 0
\(554\) −9.01637 −0.383069
\(555\) 0.217778 0.00924417
\(556\) −10.8284 −0.459228
\(557\) −25.5397 −1.08215 −0.541077 0.840973i \(-0.681983\pi\)
−0.541077 + 0.840973i \(0.681983\pi\)
\(558\) −1.26022 −0.0533494
\(559\) 42.2365 1.78641
\(560\) 0 0
\(561\) −2.20493 −0.0930923
\(562\) 24.4320 1.03060
\(563\) 42.5468 1.79314 0.896568 0.442907i \(-0.146053\pi\)
0.896568 + 0.442907i \(0.146053\pi\)
\(564\) 10.6016 0.446406
\(565\) 6.16585 0.259399
\(566\) 0.762094 0.0320332
\(567\) 0 0
\(568\) 2.38755 0.100179
\(569\) −38.0316 −1.59437 −0.797183 0.603738i \(-0.793677\pi\)
−0.797183 + 0.603738i \(0.793677\pi\)
\(570\) −1.90979 −0.0799924
\(571\) −5.92557 −0.247977 −0.123989 0.992284i \(-0.539569\pi\)
−0.123989 + 0.992284i \(0.539569\pi\)
\(572\) −38.2937 −1.60114
\(573\) −9.30086 −0.388549
\(574\) 0 0
\(575\) −4.65685 −0.194204
\(576\) 1.00000 0.0416667
\(577\) 45.0270 1.87450 0.937249 0.348661i \(-0.113364\pi\)
0.937249 + 0.348661i \(0.113364\pi\)
\(578\) −16.8618 −0.701358
\(579\) 22.1240 0.919441
\(580\) 0.555603 0.0230702
\(581\) 0 0
\(582\) 8.85705 0.367137
\(583\) −48.8323 −2.02243
\(584\) −4.76840 −0.197318
\(585\) 3.78222 0.156376
\(586\) −2.71868 −0.112308
\(587\) 43.3271 1.78830 0.894150 0.447767i \(-0.147781\pi\)
0.894150 + 0.447767i \(0.147781\pi\)
\(588\) 0 0
\(589\) 4.10860 0.169292
\(590\) 0.615970 0.0253591
\(591\) 14.2493 0.586139
\(592\) 0.371771 0.0152797
\(593\) 36.3058 1.49090 0.745449 0.666562i \(-0.232235\pi\)
0.745449 + 0.666562i \(0.232235\pi\)
\(594\) −5.93089 −0.243347
\(595\) 0 0
\(596\) 10.6616 0.436716
\(597\) 5.63869 0.230776
\(598\) −6.45666 −0.264032
\(599\) −30.2365 −1.23543 −0.617715 0.786402i \(-0.711941\pi\)
−0.617715 + 0.786402i \(0.711941\pi\)
\(600\) 4.65685 0.190115
\(601\) −13.4491 −0.548602 −0.274301 0.961644i \(-0.588446\pi\)
−0.274301 + 0.961644i \(0.588446\pi\)
\(602\) 0 0
\(603\) −0.0286249 −0.00116570
\(604\) −12.9005 −0.524913
\(605\) −14.1617 −0.575754
\(606\) −4.54154 −0.184488
\(607\) −9.63610 −0.391117 −0.195559 0.980692i \(-0.562652\pi\)
−0.195559 + 0.980692i \(0.562652\pi\)
\(608\) −3.26022 −0.132219
\(609\) 0 0
\(610\) −2.60533 −0.105487
\(611\) 68.4507 2.76922
\(612\) 0.371771 0.0150279
\(613\) 9.03043 0.364735 0.182368 0.983230i \(-0.441624\pi\)
0.182368 + 0.983230i \(0.441624\pi\)
\(614\) −21.5684 −0.870428
\(615\) 6.12005 0.246784
\(616\) 0 0
\(617\) −15.4871 −0.623486 −0.311743 0.950166i \(-0.600913\pi\)
−0.311743 + 0.950166i \(0.600913\pi\)
\(618\) 5.77690 0.232381
\(619\) 45.0944 1.81250 0.906248 0.422747i \(-0.138934\pi\)
0.906248 + 0.422747i \(0.138934\pi\)
\(620\) 0.738220 0.0296476
\(621\) −1.00000 −0.0401286
\(622\) −17.0500 −0.683641
\(623\) 0 0
\(624\) 6.45666 0.258473
\(625\) 19.9706 0.798823
\(626\) −21.7522 −0.869393
\(627\) 19.3360 0.772206
\(628\) 10.3222 0.411901
\(629\) 0.138213 0.00551093
\(630\) 0 0
\(631\) 47.7071 1.89919 0.949595 0.313478i \(-0.101494\pi\)
0.949595 + 0.313478i \(0.101494\pi\)
\(632\) −10.7307 −0.426844
\(633\) −10.0444 −0.399229
\(634\) 28.1017 1.11606
\(635\) 12.0628 0.478697
\(636\) 8.23356 0.326482
\(637\) 0 0
\(638\) −5.62530 −0.222708
\(639\) 2.38755 0.0944500
\(640\) −0.585786 −0.0231552
\(641\) 11.6884 0.461665 0.230832 0.972994i \(-0.425855\pi\)
0.230832 + 0.972994i \(0.425855\pi\)
\(642\) −7.41798 −0.292764
\(643\) 10.3966 0.410003 0.205002 0.978762i \(-0.434280\pi\)
0.205002 + 0.978762i \(0.434280\pi\)
\(644\) 0 0
\(645\) −3.83195 −0.150883
\(646\) −1.21205 −0.0476877
\(647\) 41.9064 1.64751 0.823754 0.566947i \(-0.191876\pi\)
0.823754 + 0.566947i \(0.191876\pi\)
\(648\) 1.00000 0.0392837
\(649\) −6.23649 −0.244804
\(650\) 30.0677 1.17935
\(651\) 0 0
\(652\) 2.18262 0.0854779
\(653\) 11.9542 0.467804 0.233902 0.972260i \(-0.424851\pi\)
0.233902 + 0.972260i \(0.424851\pi\)
\(654\) 10.8571 0.424544
\(655\) −7.81959 −0.305537
\(656\) 10.4476 0.407909
\(657\) −4.76840 −0.186033
\(658\) 0 0
\(659\) 22.7816 0.887447 0.443723 0.896164i \(-0.353657\pi\)
0.443723 + 0.896164i \(0.353657\pi\)
\(660\) 3.47424 0.135234
\(661\) 25.5639 0.994319 0.497160 0.867659i \(-0.334376\pi\)
0.497160 + 0.867659i \(0.334376\pi\)
\(662\) 17.6569 0.686253
\(663\) 2.40040 0.0932236
\(664\) −2.51668 −0.0976661
\(665\) 0 0
\(666\) 0.371771 0.0144058
\(667\) −0.948474 −0.0367250
\(668\) 3.83039 0.148202
\(669\) −14.6588 −0.566743
\(670\) 0.0167681 0.000647808 0
\(671\) 26.3781 1.01831
\(672\) 0 0
\(673\) 6.61597 0.255027 0.127513 0.991837i \(-0.459300\pi\)
0.127513 + 0.991837i \(0.459300\pi\)
\(674\) 14.0796 0.542325
\(675\) 4.65685 0.179242
\(676\) 28.6884 1.10340
\(677\) 20.7556 0.797701 0.398850 0.917016i \(-0.369409\pi\)
0.398850 + 0.917016i \(0.369409\pi\)
\(678\) 10.5258 0.404239
\(679\) 0 0
\(680\) −0.217778 −0.00835141
\(681\) 8.36900 0.320701
\(682\) −7.47424 −0.286203
\(683\) 13.4837 0.515940 0.257970 0.966153i \(-0.416946\pi\)
0.257970 + 0.966153i \(0.416946\pi\)
\(684\) −3.26022 −0.124658
\(685\) 6.29342 0.240459
\(686\) 0 0
\(687\) 28.3666 1.08225
\(688\) −6.54154 −0.249394
\(689\) 53.1613 2.02528
\(690\) 0.585786 0.0223005
\(691\) −42.7475 −1.62619 −0.813095 0.582131i \(-0.802219\pi\)
−0.813095 + 0.582131i \(0.802219\pi\)
\(692\) 3.08015 0.117090
\(693\) 0 0
\(694\) −17.5666 −0.666821
\(695\) 6.34315 0.240609
\(696\) 0.948474 0.0359518
\(697\) 3.88410 0.147121
\(698\) 15.6598 0.592732
\(699\) −13.1755 −0.498343
\(700\) 0 0
\(701\) −7.30717 −0.275988 −0.137994 0.990433i \(-0.544065\pi\)
−0.137994 + 0.990433i \(0.544065\pi\)
\(702\) 6.45666 0.243691
\(703\) −1.21205 −0.0457135
\(704\) 5.93089 0.223529
\(705\) −6.21025 −0.233892
\(706\) −29.8333 −1.12279
\(707\) 0 0
\(708\) 1.05153 0.0395188
\(709\) −35.7486 −1.34257 −0.671283 0.741201i \(-0.734257\pi\)
−0.671283 + 0.741201i \(0.734257\pi\)
\(710\) −1.39859 −0.0524883
\(711\) −10.7307 −0.402433
\(712\) 15.2318 0.570834
\(713\) −1.26022 −0.0471957
\(714\) 0 0
\(715\) 22.4320 0.838908
\(716\) 10.3747 0.387721
\(717\) 6.35599 0.237369
\(718\) 2.18262 0.0814546
\(719\) 1.92420 0.0717605 0.0358802 0.999356i \(-0.488577\pi\)
0.0358802 + 0.999356i \(0.488577\pi\)
\(720\) −0.585786 −0.0218310
\(721\) 0 0
\(722\) −8.37096 −0.311535
\(723\) 5.52518 0.205484
\(724\) −13.6191 −0.506152
\(725\) 4.41690 0.164040
\(726\) −24.1755 −0.897237
\(727\) 20.3471 0.754631 0.377315 0.926085i \(-0.376847\pi\)
0.377315 + 0.926085i \(0.376847\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 2.79327 0.103383
\(731\) −2.43195 −0.0899491
\(732\) −4.44757 −0.164387
\(733\) 1.91649 0.0707871 0.0353936 0.999373i \(-0.488732\pi\)
0.0353936 + 0.999373i \(0.488732\pi\)
\(734\) 9.22882 0.340642
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) −0.169771 −0.00625361
\(738\) 10.4476 0.384580
\(739\) 5.94252 0.218599 0.109300 0.994009i \(-0.465139\pi\)
0.109300 + 0.994009i \(0.465139\pi\)
\(740\) −0.217778 −0.00800569
\(741\) −21.0501 −0.773296
\(742\) 0 0
\(743\) −22.9005 −0.840137 −0.420068 0.907492i \(-0.637994\pi\)
−0.420068 + 0.907492i \(0.637994\pi\)
\(744\) 1.26022 0.0462019
\(745\) −6.24541 −0.228814
\(746\) 31.9974 1.17151
\(747\) −2.51668 −0.0920805
\(748\) 2.20493 0.0806203
\(749\) 0 0
\(750\) −5.65685 −0.206559
\(751\) −20.6369 −0.753051 −0.376525 0.926406i \(-0.622881\pi\)
−0.376525 + 0.926406i \(0.622881\pi\)
\(752\) −10.6016 −0.386599
\(753\) 6.77494 0.246893
\(754\) 6.12397 0.223022
\(755\) 7.55692 0.275024
\(756\) 0 0
\(757\) 50.8780 1.84919 0.924596 0.380950i \(-0.124403\pi\)
0.924596 + 0.380950i \(0.124403\pi\)
\(758\) −14.9015 −0.541245
\(759\) −5.93089 −0.215278
\(760\) 1.90979 0.0692755
\(761\) −21.3165 −0.772722 −0.386361 0.922348i \(-0.626268\pi\)
−0.386361 + 0.922348i \(0.626268\pi\)
\(762\) 20.5925 0.745987
\(763\) 0 0
\(764\) 9.30086 0.336493
\(765\) −0.217778 −0.00787379
\(766\) −9.63116 −0.347988
\(767\) 6.78934 0.245149
\(768\) −1.00000 −0.0360844
\(769\) 41.4864 1.49604 0.748019 0.663677i \(-0.231005\pi\)
0.748019 + 0.663677i \(0.231005\pi\)
\(770\) 0 0
\(771\) 5.29417 0.190665
\(772\) −22.1240 −0.796259
\(773\) 11.1261 0.400178 0.200089 0.979778i \(-0.435877\pi\)
0.200089 + 0.979778i \(0.435877\pi\)
\(774\) −6.54154 −0.235131
\(775\) 5.86867 0.210809
\(776\) −8.85705 −0.317950
\(777\) 0 0
\(778\) −28.4882 −1.02135
\(779\) −34.0614 −1.22038
\(780\) −3.78222 −0.135425
\(781\) 14.1603 0.506695
\(782\) 0.371771 0.0132945
\(783\) 0.948474 0.0338957
\(784\) 0 0
\(785\) −6.04661 −0.215813
\(786\) −13.3489 −0.476138
\(787\) −26.5482 −0.946342 −0.473171 0.880970i \(-0.656891\pi\)
−0.473171 + 0.880970i \(0.656891\pi\)
\(788\) −14.2493 −0.507612
\(789\) 31.3137 1.11480
\(790\) 6.28590 0.223642
\(791\) 0 0
\(792\) 5.93089 0.210745
\(793\) −28.7165 −1.01975
\(794\) 15.0047 0.532498
\(795\) −4.82311 −0.171058
\(796\) −5.63869 −0.199858
\(797\) −30.6436 −1.08545 −0.542726 0.839910i \(-0.682608\pi\)
−0.542726 + 0.839910i \(0.682608\pi\)
\(798\) 0 0
\(799\) −3.94135 −0.139435
\(800\) −4.65685 −0.164645
\(801\) 15.2318 0.538188
\(802\) −34.6369 −1.22307
\(803\) −28.2809 −0.998011
\(804\) 0.0286249 0.00100952
\(805\) 0 0
\(806\) 8.13681 0.286607
\(807\) 22.5064 0.792262
\(808\) 4.54154 0.159771
\(809\) 24.6941 0.868200 0.434100 0.900865i \(-0.357067\pi\)
0.434100 + 0.900865i \(0.357067\pi\)
\(810\) −0.585786 −0.0205824
\(811\) −40.6941 −1.42896 −0.714482 0.699654i \(-0.753338\pi\)
−0.714482 + 0.699654i \(0.753338\pi\)
\(812\) 0 0
\(813\) 8.90423 0.312285
\(814\) 2.20493 0.0772828
\(815\) −1.27855 −0.0447856
\(816\) −0.371771 −0.0130146
\(817\) 21.3269 0.746133
\(818\) 20.4515 0.715070
\(819\) 0 0
\(820\) −6.12005 −0.213721
\(821\) 19.8043 0.691175 0.345588 0.938386i \(-0.387680\pi\)
0.345588 + 0.938386i \(0.387680\pi\)
\(822\) 10.7435 0.374724
\(823\) 44.3287 1.54520 0.772600 0.634893i \(-0.218956\pi\)
0.772600 + 0.634893i \(0.218956\pi\)
\(824\) −5.77690 −0.201248
\(825\) 27.6193 0.961581
\(826\) 0 0
\(827\) 13.9988 0.486785 0.243393 0.969928i \(-0.421740\pi\)
0.243393 + 0.969928i \(0.421740\pi\)
\(828\) 1.00000 0.0347524
\(829\) −53.1050 −1.84441 −0.922206 0.386698i \(-0.873615\pi\)
−0.922206 + 0.386698i \(0.873615\pi\)
\(830\) 1.47424 0.0511715
\(831\) 9.01637 0.312774
\(832\) −6.45666 −0.223844
\(833\) 0 0
\(834\) 10.8284 0.374958
\(835\) −2.24379 −0.0776495
\(836\) −19.3360 −0.668750
\(837\) 1.26022 0.0435596
\(838\) −15.1145 −0.522121
\(839\) 38.7399 1.33745 0.668726 0.743509i \(-0.266840\pi\)
0.668726 + 0.743509i \(0.266840\pi\)
\(840\) 0 0
\(841\) −28.1004 −0.968979
\(842\) 10.4069 0.358647
\(843\) −24.4320 −0.841481
\(844\) 10.0444 0.345743
\(845\) −16.8053 −0.578119
\(846\) −10.6016 −0.364489
\(847\) 0 0
\(848\) −8.23356 −0.282742
\(849\) −0.762094 −0.0261550
\(850\) −1.73128 −0.0593825
\(851\) 0.371771 0.0127441
\(852\) −2.38755 −0.0817961
\(853\) −2.51391 −0.0860745 −0.0430373 0.999073i \(-0.513703\pi\)
−0.0430373 + 0.999073i \(0.513703\pi\)
\(854\) 0 0
\(855\) 1.90979 0.0653136
\(856\) 7.41798 0.253541
\(857\) 25.2280 0.861772 0.430886 0.902406i \(-0.358201\pi\)
0.430886 + 0.902406i \(0.358201\pi\)
\(858\) 38.2937 1.30733
\(859\) 30.1994 1.03039 0.515195 0.857073i \(-0.327720\pi\)
0.515195 + 0.857073i \(0.327720\pi\)
\(860\) 3.83195 0.130668
\(861\) 0 0
\(862\) −34.4892 −1.17471
\(863\) 34.1427 1.16223 0.581115 0.813821i \(-0.302617\pi\)
0.581115 + 0.813821i \(0.302617\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −1.80431 −0.0613484
\(866\) −25.9401 −0.881482
\(867\) 16.8618 0.572656
\(868\) 0 0
\(869\) −63.6426 −2.15893
\(870\) −0.555603 −0.0188367
\(871\) 0.184821 0.00626243
\(872\) −10.8571 −0.367666
\(873\) −8.85705 −0.299766
\(874\) −3.26022 −0.110279
\(875\) 0 0
\(876\) 4.76840 0.161109
\(877\) 31.1520 1.05193 0.525964 0.850507i \(-0.323705\pi\)
0.525964 + 0.850507i \(0.323705\pi\)
\(878\) 26.9891 0.910839
\(879\) 2.71868 0.0916988
\(880\) −3.47424 −0.117117
\(881\) 23.0380 0.776168 0.388084 0.921624i \(-0.373137\pi\)
0.388084 + 0.921624i \(0.373137\pi\)
\(882\) 0 0
\(883\) −31.4248 −1.05753 −0.528765 0.848768i \(-0.677345\pi\)
−0.528765 + 0.848768i \(0.677345\pi\)
\(884\) −2.40040 −0.0807340
\(885\) −0.615970 −0.0207056
\(886\) 9.74706 0.327459
\(887\) −36.5245 −1.22637 −0.613187 0.789938i \(-0.710113\pi\)
−0.613187 + 0.789938i \(0.710113\pi\)
\(888\) −0.371771 −0.0124758
\(889\) 0 0
\(890\) −8.92256 −0.299085
\(891\) 5.93089 0.198692
\(892\) 14.6588 0.490813
\(893\) 34.5634 1.15662
\(894\) −10.6616 −0.356577
\(895\) −6.07736 −0.203144
\(896\) 0 0
\(897\) 6.45666 0.215581
\(898\) −34.8782 −1.16390
\(899\) 1.19529 0.0398650
\(900\) −4.65685 −0.155228
\(901\) −3.06100 −0.101977
\(902\) 61.9634 2.06316
\(903\) 0 0
\(904\) −10.5258 −0.350082
\(905\) 7.97791 0.265195
\(906\) 12.9005 0.428589
\(907\) −19.4917 −0.647212 −0.323606 0.946192i \(-0.604895\pi\)
−0.323606 + 0.946192i \(0.604895\pi\)
\(908\) −8.36900 −0.277735
\(909\) 4.54154 0.150633
\(910\) 0 0
\(911\) −1.79169 −0.0593614 −0.0296807 0.999559i \(-0.509449\pi\)
−0.0296807 + 0.999559i \(0.509449\pi\)
\(912\) 3.26022 0.107957
\(913\) −14.9262 −0.493984
\(914\) −2.13821 −0.0707258
\(915\) 2.60533 0.0861295
\(916\) −28.3666 −0.937259
\(917\) 0 0
\(918\) −0.371771 −0.0122703
\(919\) −11.3617 −0.374788 −0.187394 0.982285i \(-0.560004\pi\)
−0.187394 + 0.982285i \(0.560004\pi\)
\(920\) −0.585786 −0.0193128
\(921\) 21.5684 0.710702
\(922\) −14.4385 −0.475506
\(923\) −15.4156 −0.507410
\(924\) 0 0
\(925\) −1.73128 −0.0569242
\(926\) −17.3911 −0.571506
\(927\) −5.77690 −0.189738
\(928\) −0.948474 −0.0311352
\(929\) 0.185391 0.00608247 0.00304124 0.999995i \(-0.499032\pi\)
0.00304124 + 0.999995i \(0.499032\pi\)
\(930\) −0.738220 −0.0242072
\(931\) 0 0
\(932\) 13.1755 0.431578
\(933\) 17.0500 0.558191
\(934\) 10.9042 0.356797
\(935\) −1.29162 −0.0422405
\(936\) −6.45666 −0.211042
\(937\) −45.8410 −1.49756 −0.748780 0.662818i \(-0.769360\pi\)
−0.748780 + 0.662818i \(0.769360\pi\)
\(938\) 0 0
\(939\) 21.7522 0.709856
\(940\) 6.21025 0.202556
\(941\) −59.9386 −1.95394 −0.976971 0.213373i \(-0.931555\pi\)
−0.976971 + 0.213373i \(0.931555\pi\)
\(942\) −10.3222 −0.336316
\(943\) 10.4476 0.340220
\(944\) −1.05153 −0.0342243
\(945\) 0 0
\(946\) −38.7972 −1.26141
\(947\) −3.83947 −0.124766 −0.0623830 0.998052i \(-0.519870\pi\)
−0.0623830 + 0.998052i \(0.519870\pi\)
\(948\) 10.7307 0.348517
\(949\) 30.7879 0.999419
\(950\) 15.1824 0.492581
\(951\) −28.1017 −0.911259
\(952\) 0 0
\(953\) 18.7974 0.608908 0.304454 0.952527i \(-0.401526\pi\)
0.304454 + 0.952527i \(0.401526\pi\)
\(954\) −8.23356 −0.266571
\(955\) −5.44832 −0.176303
\(956\) −6.35599 −0.205567
\(957\) 5.62530 0.181840
\(958\) −17.3729 −0.561293
\(959\) 0 0
\(960\) 0.585786 0.0189062
\(961\) −29.4118 −0.948769
\(962\) −2.40040 −0.0773919
\(963\) 7.41798 0.239041
\(964\) −5.52518 −0.177954
\(965\) 12.9599 0.417195
\(966\) 0 0
\(967\) 31.4578 1.01161 0.505807 0.862647i \(-0.331195\pi\)
0.505807 + 0.862647i \(0.331195\pi\)
\(968\) 24.1755 0.777030
\(969\) 1.21205 0.0389368
\(970\) 5.18834 0.166588
\(971\) 57.5241 1.84604 0.923018 0.384756i \(-0.125714\pi\)
0.923018 + 0.384756i \(0.125714\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) 28.0888 0.900023
\(975\) −30.0677 −0.962937
\(976\) 4.44757 0.142363
\(977\) −4.54448 −0.145391 −0.0726953 0.997354i \(-0.523160\pi\)
−0.0726953 + 0.997354i \(0.523160\pi\)
\(978\) −2.18262 −0.0697924
\(979\) 90.3379 2.88721
\(980\) 0 0
\(981\) −10.8571 −0.346639
\(982\) 17.0608 0.544431
\(983\) 36.3842 1.16047 0.580237 0.814447i \(-0.302960\pi\)
0.580237 + 0.814447i \(0.302960\pi\)
\(984\) −10.4476 −0.333056
\(985\) 8.34707 0.265960
\(986\) −0.352615 −0.0112295
\(987\) 0 0
\(988\) 21.0501 0.669694
\(989\) −6.54154 −0.208009
\(990\) −3.47424 −0.110418
\(991\) 40.8089 1.29634 0.648168 0.761497i \(-0.275535\pi\)
0.648168 + 0.761497i \(0.275535\pi\)
\(992\) −1.26022 −0.0400121
\(993\) −17.6569 −0.560323
\(994\) 0 0
\(995\) 3.30307 0.104714
\(996\) 2.51668 0.0797440
\(997\) 5.30325 0.167956 0.0839778 0.996468i \(-0.473238\pi\)
0.0839778 + 0.996468i \(0.473238\pi\)
\(998\) 29.7364 0.941290
\(999\) −0.371771 −0.0117623
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 6762.2.a.ci.1.4 4
7.6 odd 2 6762.2.a.ct.1.2 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6762.2.a.ci.1.4 4 1.1 even 1 trivial
6762.2.a.ct.1.2 yes 4 7.6 odd 2