Properties

Label 6762.2.a.ct.1.2
Level $6762$
Weight $2$
Character 6762.1
Self dual yes
Analytic conductor $53.995$
Analytic rank $0$
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: \(0\)
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 1.2
Root \(-1.22833\) of defining polynomial
Character \(\chi\) \(=\) 6762.1

$q$-expansion

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 0.585786 0.261972 0.130986 0.991384i \(-0.458186\pi\)
0.130986 + 0.991384i \(0.458186\pi\)
\(6\) 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 \(-0.146906\pi\)
0.895377 + 0.445309i \(0.146906\pi\)
\(14\) 0 0
\(15\) 0.585786 0.151249
\(16\) 1.00000 0.250000
\(17\) −0.371771 −0.0901676 −0.0450838 0.998983i \(-0.514355\pi\)
−0.0450838 + 0.998983i \(0.514355\pi\)
\(18\) 1.00000 0.235702
\(19\) 3.26022 0.747946 0.373973 0.927440i \(-0.377995\pi\)
0.373973 + 0.927440i \(0.377995\pi\)
\(20\) 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 \(-0.463899\pi\)
0.113171 + 0.993576i \(0.463899\pi\)
\(32\) 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 \(-0.803713\pi\)
−0.815818 + 0.578308i \(0.803713\pi\)
\(42\) 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 \(-0.218656\pi\)
0.773199 + 0.634164i \(0.218656\pi\)
\(48\) 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 \(-0.478195\pi\)
0.0684485 + 0.997655i \(0.478195\pi\)
\(60\) 0.585786 0.0756247
\(61\) −4.44757 −0.569453 −0.284727 0.958609i \(-0.591903\pi\)
−0.284727 + 0.958609i \(0.591903\pi\)
\(62\) 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 \(-0.409981\pi\)
0.279050 + 0.960277i \(0.409981\pi\)
\(74\) 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 \(-0.455894\pi\)
0.138121 + 0.990415i \(0.455894\pi\)
\(84\) 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 \(-0.799062\pi\)
−0.807281 + 0.590167i \(0.799062\pi\)
\(90\) 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 \(-0.351549\pi\)
0.449649 + 0.893205i \(0.351549\pi\)
\(98\) 0 0
\(99\) 5.93089 0.596077
\(100\) −4.65685 −0.465685
\(101\) −4.54154 −0.451900 −0.225950 0.974139i \(-0.572549\pi\)
−0.225950 + 0.974139i \(0.572549\pi\)
\(102\) −0.371771 −0.0368108
\(103\) 5.77690 0.569215 0.284607 0.958644i \(-0.408137\pi\)
0.284607 + 0.958644i \(0.408137\pi\)
\(104\) 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 \(-0.698179\pi\)
−0.583148 + 0.812366i \(0.698179\pi\)
\(132\) 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 \(-0.348126\pi\)
0.459228 + 0.888319i \(0.348126\pi\)
\(140\) 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 \(-0.635135\pi\)
−0.411901 + 0.911229i \(0.635135\pi\)
\(158\) −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 \(-0.547349\pi\)
−0.148202 + 0.988957i \(0.547349\pi\)
\(168\) 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 \(-0.537357\pi\)
−0.117090 + 0.993121i \(0.537357\pi\)
\(174\) −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 \(-0.331068\pi\)
0.506152 + 0.862445i \(0.331068\pi\)
\(182\) 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 \(-0.435952\pi\)
0.199858 + 0.979825i \(0.435952\pi\)
\(200\) −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 \(-0.663300\pi\)
−0.490813 + 0.871265i \(0.663300\pi\)
\(224\) 0 0
\(225\) −4.65685 −0.310457
\(226\) −10.5258 −0.700163
\(227\) 8.36900 0.555470 0.277735 0.960658i \(-0.410416\pi\)
0.277735 + 0.960658i \(0.410416\pi\)
\(228\) 3.26022 0.215913
\(229\) 28.3666 1.87452 0.937259 0.348633i \(-0.113354\pi\)
0.937259 + 0.348633i \(0.113354\pi\)
\(230\) 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 \(-0.443052\pi\)
0.177954 + 0.984039i \(0.443052\pi\)
\(242\) 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 \(-0.431411\pi\)
0.213815 + 0.976874i \(0.431411\pi\)
\(252\) 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 \(-0.447199\pi\)
0.165121 + 0.986273i \(0.447199\pi\)
\(258\) −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 \(-0.259313\pi\)
0.686119 + 0.727489i \(0.259313\pi\)
\(270\) 0.585786 0.0356498
\(271\) 8.90423 0.540893 0.270447 0.962735i \(-0.412829\pi\)
0.270447 + 0.962735i \(0.412829\pi\)
\(272\) −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 \(-0.507211\pi\)
−0.0226509 + 0.999743i \(0.507211\pi\)
\(284\) 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 \(-0.474695\pi\)
0.0794135 + 0.996842i \(0.474695\pi\)
\(294\) 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 \(-0.288960\pi\)
0.615486 + 0.788148i \(0.288960\pi\)
\(308\) 0 0
\(309\) 5.77690 0.328636
\(310\) 0.738220 0.0419281
\(311\) 17.0500 0.966815 0.483407 0.875395i \(-0.339399\pi\)
0.483407 + 0.875395i \(0.339399\pi\)
\(312\) 6.45666 0.365536
\(313\) 21.7522 1.22951 0.614753 0.788719i \(-0.289256\pi\)
0.614753 + 0.788719i \(0.289256\pi\)
\(314\) −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 \(-0.637663\pi\)
−0.419125 + 0.907929i \(0.637663\pi\)
\(350\) 0 0
\(351\) 6.45666 0.344631
\(352\) 5.93089 0.316118
\(353\) 29.8333 1.58787 0.793934 0.608004i \(-0.208030\pi\)
0.793934 + 0.608004i \(0.208030\pi\)
\(354\) 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 \(-0.577433\pi\)
−0.240870 + 0.970557i \(0.577433\pi\)
\(368\) 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 \(-0.420862\pi\)
0.246065 + 0.969253i \(0.420862\pi\)
\(384\) 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 \(-0.622884\pi\)
−0.376533 + 0.926403i \(0.622884\pi\)
\(398\) 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 \(-0.668740\pi\)
−0.505631 + 0.862750i \(0.668740\pi\)
\(410\) −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 \(-0.379633\pi\)
0.369195 + 0.929352i \(0.379633\pi\)
\(420\) 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 \(-0.285791\pi\)
0.623302 + 0.781981i \(0.285791\pi\)
\(434\) 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 \(-0.722751\pi\)
−0.644060 + 0.764975i \(0.722751\pi\)
\(440\) 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 \(-0.390847\pi\)
0.336234 + 0.941779i \(0.390847\pi\)
\(462\) 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 \(-0.581185\pi\)
−0.252294 + 0.967651i \(0.581185\pi\)
\(468\) 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 \(-0.370088\pi\)
0.396894 + 0.917864i \(0.370088\pi\)
\(480\) 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 \(-0.775172\pi\)
−0.760756 + 0.649038i \(0.775172\pi\)
\(504\) 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 \(-0.286055\pi\)
0.622652 + 0.782499i \(0.286055\pi\)
\(510\) −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 \(-0.649951\pi\)
−0.453854 + 0.891076i \(0.649951\pi\)
\(522\) −0.948474 −0.0415136
\(523\) 11.6883 0.511092 0.255546 0.966797i \(-0.417745\pi\)
0.255546 + 0.966797i \(0.417745\pi\)
\(524\) −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 \(-0.853947\pi\)
−0.896568 + 0.442907i \(0.853947\pi\)
\(564\) 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 \(-0.886636\pi\)
−0.937249 + 0.348661i \(0.886636\pi\)
\(578\) −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 \(-0.852219\pi\)
−0.894150 + 0.447767i \(0.852219\pi\)
\(588\) 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 \(-0.767765\pi\)
−0.745449 + 0.666562i \(0.767765\pi\)
\(594\) 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 \(-0.411554\pi\)
0.274301 + 0.961644i \(0.411554\pi\)
\(602\) 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 \(-0.437348\pi\)
0.195559 + 0.980692i \(0.437348\pi\)
\(608\) 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 \(-0.861066\pi\)
−0.906248 + 0.422747i \(0.861066\pi\)
\(620\) 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 \(-0.565720\pi\)
−0.205002 + 0.978762i \(0.565720\pi\)
\(644\) 0 0
\(645\) −3.83195 −0.150883
\(646\) −1.21205 −0.0476877
\(647\) −41.9064 −1.64751 −0.823754 0.566947i \(-0.808124\pi\)
−0.823754 + 0.566947i \(0.808124\pi\)
\(648\) 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 \(-0.665624\pi\)
−0.497160 + 0.867659i \(0.665624\pi\)
\(662\) 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 \(-0.630591\pi\)
−0.398850 + 0.917016i \(0.630591\pi\)
\(678\) −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 \(-0.197781\pi\)
0.813095 + 0.582131i \(0.197781\pi\)
\(692\) −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 \(-0.511423\pi\)
−0.0358802 + 0.999356i \(0.511423\pi\)
\(720\) 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 \(-0.623153\pi\)
−0.377315 + 0.926085i \(0.623153\pi\)
\(728\) 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 \(-0.511268\pi\)
−0.0353936 + 0.999373i \(0.511268\pi\)
\(734\) −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 \(-0.373732\pi\)
0.386361 + 0.922348i \(0.373732\pi\)
\(762\) −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 \(-0.768995\pi\)
−0.748019 + 0.663677i \(0.768995\pi\)
\(770\) 0 0
\(771\) 5.29417 0.190665
\(772\) −22.1240 −0.796259
\(773\) −11.1261 −0.400178 −0.200089 0.979778i \(-0.564123\pi\)
−0.200089 + 0.979778i \(0.564123\pi\)
\(774\) −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 \(-0.343109\pi\)
0.473171 + 0.880970i \(0.343109\pi\)
\(788\) −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 \(-0.317392\pi\)
0.542726 + 0.839910i \(0.317392\pi\)
\(798\) 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 \(-0.246662\pi\)
0.714482 + 0.699654i \(0.246662\pi\)
\(812\) 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 \(-0.126385\pi\)
0.922206 + 0.386698i \(0.126385\pi\)
\(830\) 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 \(-0.733160\pi\)
−0.668726 + 0.743509i \(0.733160\pi\)
\(840\) 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 \(-0.486297\pi\)
0.0430373 + 0.999073i \(0.486297\pi\)
\(854\) 0 0
\(855\) 1.90979 0.0653136
\(856\) 7.41798 0.253541
\(857\) −25.2280 −0.861772 −0.430886 0.902406i \(-0.641799\pi\)
−0.430886 + 0.902406i \(0.641799\pi\)
\(858\) 38.2937 1.30733
\(859\) −30.1994 −1.03039 −0.515195 0.857073i \(-0.672280\pi\)
−0.515195 + 0.857073i \(0.672280\pi\)
\(860\) −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 \(-0.626863\pi\)
−0.388084 + 0.921624i \(0.626863\pi\)
\(882\) 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 \(-0.289887\pi\)
0.613187 + 0.789938i \(0.289887\pi\)
\(888\) 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 \(-0.500968\pi\)
−0.00304124 + 0.999995i \(0.500968\pi\)
\(930\) 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 \(-0.230640\pi\)
0.748780 + 0.662818i \(0.230640\pi\)
\(938\) 0 0
\(939\) 21.7522 0.709856
\(940\) 6.21025 0.202556
\(941\) 59.9386 1.95394 0.976971 0.213373i \(-0.0684448\pi\)
0.976971 + 0.213373i \(0.0684448\pi\)
\(942\) −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 \(-0.874286\pi\)
−0.923018 + 0.384756i \(0.874286\pi\)
\(972\) 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 \(-0.697040\pi\)
−0.580237 + 0.814447i \(0.697040\pi\)
\(984\) −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 \(-0.526762\pi\)
−0.0839778 + 0.996468i \(0.526762\pi\)
\(998\) 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.ct.1.2 yes 4
7.6 odd 2 6762.2.a.ci.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6762.2.a.ci.1.4 4 7.6 odd 2
6762.2.a.ct.1.2 yes 4 1.1 even 1 trivial