Properties

Label 6762.2.a.ci.1.3
Level $6762$
Weight $2$
Character 6762.1
Self dual yes
Analytic conductor $53.995$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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 1.3
Root \(0.814115\) of defining polynomial
Character \(\chi\) \(=\) 6762.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.585786 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.585786 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -0.585786 q^{10} -3.93089 q^{11} -1.00000 q^{12} -2.37177 q^{13} +0.585786 q^{15} +1.00000 q^{16} +4.45666 q^{17} +1.00000 q^{18} -1.56821 q^{19} -0.585786 q^{20} -3.93089 q^{22} +1.00000 q^{23} -1.00000 q^{24} -4.65685 q^{25} -2.37177 q^{26} -1.00000 q^{27} +10.6053 q^{29} +0.585786 q^{30} +0.431794 q^{31} +1.00000 q^{32} +3.93089 q^{33} +4.45666 q^{34} +1.00000 q^{36} +4.45666 q^{37} -1.56821 q^{38} +2.37177 q^{39} -0.585786 q^{40} -9.27600 q^{41} +5.71312 q^{43} -3.93089 q^{44} -0.585786 q^{45} +1.00000 q^{46} +7.42999 q^{47} -1.00000 q^{48} -4.65685 q^{50} -4.45666 q^{51} -2.37177 q^{52} +7.40513 q^{53} -1.00000 q^{54} +2.30266 q^{55} +1.56821 q^{57} +10.6053 q^{58} -12.6053 q^{59} +0.585786 q^{60} -15.2760 q^{61} +0.431794 q^{62} +1.00000 q^{64} +1.38935 q^{65} +3.93089 q^{66} -4.11351 q^{67} +4.45666 q^{68} -1.00000 q^{69} -11.5591 q^{71} +1.00000 q^{72} -10.5453 q^{73} +4.45666 q^{74} +4.65685 q^{75} -1.56821 q^{76} +2.37177 q^{78} +3.21598 q^{79} -0.585786 q^{80} +1.00000 q^{81} -9.27600 q^{82} +7.34511 q^{83} -2.61065 q^{85} +5.71312 q^{86} -10.6053 q^{87} -3.93089 q^{88} -16.7465 q^{89} -0.585786 q^{90} +1.00000 q^{92} -0.431794 q^{93} +7.42999 q^{94} +0.918634 q^{95} -1.00000 q^{96} -12.9419 q^{97} -3.93089 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{2} - 4q^{3} + 4q^{4} - 8q^{5} - 4q^{6} + 4q^{8} + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{2} - 4q^{3} + 4q^{4} - 8q^{5} - 4q^{6} + 4q^{8} + 4q^{9} - 8q^{10} + 4q^{11} - 4q^{12} - 12q^{13} + 8q^{15} + 4q^{16} + 4q^{17} + 4q^{18} - 4q^{19} - 8q^{20} + 4q^{22} + 4q^{23} - 4q^{24} + 4q^{25} - 12q^{26} - 4q^{27} + 8q^{29} + 8q^{30} + 4q^{31} + 4q^{32} - 4q^{33} + 4q^{34} + 4q^{36} + 4q^{37} - 4q^{38} + 12q^{39} - 8q^{40} + 8q^{41} + 4q^{43} + 4q^{44} - 8q^{45} + 4q^{46} - 12q^{47} - 4q^{48} + 4q^{50} - 4q^{51} - 12q^{52} + 4q^{53} - 4q^{54} - 8q^{55} + 4q^{57} + 8q^{58} - 16q^{59} + 8q^{60} - 16q^{61} + 4q^{62} + 4q^{64} + 16q^{65} - 4q^{66} + 20q^{67} + 4q^{68} - 4q^{69} - 24q^{71} + 4q^{72} - 8q^{73} + 4q^{74} - 4q^{75} - 4q^{76} + 12q^{78} - 32q^{79} - 8q^{80} + 4q^{81} + 8q^{82} + 4q^{83} + 4q^{86} - 8q^{87} + 4q^{88} - 20q^{89} - 8q^{90} + 4q^{92} - 4q^{93} - 12q^{94} - 4q^{96} - 4q^{97} + 4q^{99} + O(q^{100}) \)

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
\(11\) −3.93089 −1.18521 −0.592604 0.805494i \(-0.701900\pi\)
−0.592604 + 0.805494i \(0.701900\pi\)
\(12\) −1.00000 −0.288675
\(13\) −2.37177 −0.657811 −0.328905 0.944363i \(-0.606680\pi\)
−0.328905 + 0.944363i \(0.606680\pi\)
\(14\) 0 0
\(15\) 0.585786 0.151249
\(16\) 1.00000 0.250000
\(17\) 4.45666 1.08090 0.540449 0.841377i \(-0.318254\pi\)
0.540449 + 0.841377i \(0.318254\pi\)
\(18\) 1.00000 0.235702
\(19\) −1.56821 −0.359771 −0.179886 0.983688i \(-0.557573\pi\)
−0.179886 + 0.983688i \(0.557573\pi\)
\(20\) −0.585786 −0.130986
\(21\) 0 0
\(22\) −3.93089 −0.838069
\(23\) 1.00000 0.208514
\(24\) −1.00000 −0.204124
\(25\) −4.65685 −0.931371
\(26\) −2.37177 −0.465143
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 10.6053 1.96936 0.984680 0.174371i \(-0.0557893\pi\)
0.984680 + 0.174371i \(0.0557893\pi\)
\(30\) 0.585786 0.106949
\(31\) 0.431794 0.0775525 0.0387762 0.999248i \(-0.487654\pi\)
0.0387762 + 0.999248i \(0.487654\pi\)
\(32\) 1.00000 0.176777
\(33\) 3.93089 0.684281
\(34\) 4.45666 0.764310
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 4.45666 0.732670 0.366335 0.930483i \(-0.380612\pi\)
0.366335 + 0.930483i \(0.380612\pi\)
\(38\) −1.56821 −0.254397
\(39\) 2.37177 0.379787
\(40\) −0.585786 −0.0926210
\(41\) −9.27600 −1.44867 −0.724334 0.689449i \(-0.757853\pi\)
−0.724334 + 0.689449i \(0.757853\pi\)
\(42\) 0 0
\(43\) 5.71312 0.871242 0.435621 0.900130i \(-0.356529\pi\)
0.435621 + 0.900130i \(0.356529\pi\)
\(44\) −3.93089 −0.592604
\(45\) −0.585786 −0.0873239
\(46\) 1.00000 0.147442
\(47\) 7.42999 1.08378 0.541888 0.840451i \(-0.317710\pi\)
0.541888 + 0.840451i \(0.317710\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) −4.65685 −0.658579
\(51\) −4.45666 −0.624057
\(52\) −2.37177 −0.328905
\(53\) 7.40513 1.01717 0.508586 0.861011i \(-0.330168\pi\)
0.508586 + 0.861011i \(0.330168\pi\)
\(54\) −1.00000 −0.136083
\(55\) 2.30266 0.310491
\(56\) 0 0
\(57\) 1.56821 0.207714
\(58\) 10.6053 1.39255
\(59\) −12.6053 −1.64107 −0.820537 0.571593i \(-0.806325\pi\)
−0.820537 + 0.571593i \(0.806325\pi\)
\(60\) 0.585786 0.0756247
\(61\) −15.2760 −1.95589 −0.977946 0.208859i \(-0.933025\pi\)
−0.977946 + 0.208859i \(0.933025\pi\)
\(62\) 0.431794 0.0548379
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 1.38935 0.172328
\(66\) 3.93089 0.483860
\(67\) −4.11351 −0.502545 −0.251273 0.967916i \(-0.580849\pi\)
−0.251273 + 0.967916i \(0.580849\pi\)
\(68\) 4.45666 0.540449
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −11.5591 −1.37182 −0.685908 0.727689i \(-0.740595\pi\)
−0.685908 + 0.727689i \(0.740595\pi\)
\(72\) 1.00000 0.117851
\(73\) −10.5453 −1.23423 −0.617117 0.786871i \(-0.711700\pi\)
−0.617117 + 0.786871i \(0.711700\pi\)
\(74\) 4.45666 0.518076
\(75\) 4.65685 0.537727
\(76\) −1.56821 −0.179886
\(77\) 0 0
\(78\) 2.37177 0.268550
\(79\) 3.21598 0.361826 0.180913 0.983499i \(-0.442095\pi\)
0.180913 + 0.983499i \(0.442095\pi\)
\(80\) −0.585786 −0.0654929
\(81\) 1.00000 0.111111
\(82\) −9.27600 −1.02436
\(83\) 7.34511 0.806230 0.403115 0.915149i \(-0.367927\pi\)
0.403115 + 0.915149i \(0.367927\pi\)
\(84\) 0 0
\(85\) −2.61065 −0.283165
\(86\) 5.71312 0.616061
\(87\) −10.6053 −1.13701
\(88\) −3.93089 −0.419035
\(89\) −16.7465 −1.77512 −0.887561 0.460689i \(-0.847602\pi\)
−0.887561 + 0.460689i \(0.847602\pi\)
\(90\) −0.585786 −0.0617473
\(91\) 0 0
\(92\) 1.00000 0.104257
\(93\) −0.431794 −0.0447749
\(94\) 7.42999 0.766345
\(95\) 0.918634 0.0942498
\(96\) −1.00000 −0.102062
\(97\) −12.9419 −1.31405 −0.657027 0.753867i \(-0.728186\pi\)
−0.657027 + 0.753867i \(0.728186\pi\)
\(98\) 0 0
\(99\) −3.93089 −0.395070
\(100\) −4.65685 −0.465685
\(101\) −7.71312 −0.767484 −0.383742 0.923440i \(-0.625365\pi\)
−0.383742 + 0.923440i \(0.625365\pi\)
\(102\) −4.45666 −0.441275
\(103\) 5.77690 0.569215 0.284607 0.958644i \(-0.408137\pi\)
0.284607 + 0.958644i \(0.408137\pi\)
\(104\) −2.37177 −0.232571
\(105\) 0 0
\(106\) 7.40513 0.719250
\(107\) 13.8957 1.34335 0.671676 0.740845i \(-0.265575\pi\)
0.671676 + 0.740845i \(0.265575\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −14.9419 −1.43118 −0.715589 0.698522i \(-0.753842\pi\)
−0.715589 + 0.698522i \(0.753842\pi\)
\(110\) 2.30266 0.219550
\(111\) −4.45666 −0.423007
\(112\) 0 0
\(113\) −16.3027 −1.53363 −0.766813 0.641871i \(-0.778159\pi\)
−0.766813 + 0.641871i \(0.778159\pi\)
\(114\) 1.56821 0.146876
\(115\) −0.585786 −0.0546249
\(116\) 10.6053 0.984680
\(117\) −2.37177 −0.219270
\(118\) −12.6053 −1.16041
\(119\) 0 0
\(120\) 0.585786 0.0534747
\(121\) 4.45192 0.404720
\(122\) −15.2760 −1.38302
\(123\) 9.27600 0.836389
\(124\) 0.431794 0.0387762
\(125\) 5.65685 0.505964
\(126\) 0 0
\(127\) 13.0778 1.16046 0.580232 0.814451i \(-0.302962\pi\)
0.580232 + 0.814451i \(0.302962\pi\)
\(128\) 1.00000 0.0883883
\(129\) −5.71312 −0.503012
\(130\) 1.38935 0.121854
\(131\) 9.96484 0.870632 0.435316 0.900278i \(-0.356637\pi\)
0.435316 + 0.900278i \(0.356637\pi\)
\(132\) 3.93089 0.342140
\(133\) 0 0
\(134\) −4.11351 −0.355353
\(135\) 0.585786 0.0504165
\(136\) 4.45666 0.382155
\(137\) −18.9133 −1.61587 −0.807937 0.589269i \(-0.799416\pi\)
−0.807937 + 0.589269i \(0.799416\pi\)
\(138\) −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
\(141\) −7.42999 −0.625718
\(142\) −11.5591 −0.970020
\(143\) 9.32318 0.779643
\(144\) 1.00000 0.0833333
\(145\) −6.21246 −0.515916
\(146\) −10.5453 −0.872736
\(147\) 0 0
\(148\) 4.45666 0.366335
\(149\) −13.1469 −1.07703 −0.538517 0.842615i \(-0.681015\pi\)
−0.538517 + 0.842615i \(0.681015\pi\)
\(150\) 4.65685 0.380231
\(151\) 17.3857 1.41483 0.707416 0.706797i \(-0.249861\pi\)
0.707416 + 0.706797i \(0.249861\pi\)
\(152\) −1.56821 −0.127198
\(153\) 4.45666 0.360299
\(154\) 0 0
\(155\) −0.252939 −0.0203166
\(156\) 2.37177 0.189894
\(157\) −7.00850 −0.559339 −0.279669 0.960096i \(-0.590225\pi\)
−0.279669 + 0.960096i \(0.590225\pi\)
\(158\) 3.21598 0.255849
\(159\) −7.40513 −0.587265
\(160\) −0.585786 −0.0463105
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) 7.95952 0.623438 0.311719 0.950174i \(-0.399095\pi\)
0.311719 + 0.950174i \(0.399095\pi\)
\(164\) −9.27600 −0.724334
\(165\) −2.30266 −0.179262
\(166\) 7.34511 0.570091
\(167\) −6.03140 −0.466724 −0.233362 0.972390i \(-0.574973\pi\)
−0.233362 + 0.972390i \(0.574973\pi\)
\(168\) 0 0
\(169\) −7.37470 −0.567285
\(170\) −2.61065 −0.200228
\(171\) −1.56821 −0.119924
\(172\) 5.71312 0.435621
\(173\) 18.7188 1.42317 0.711583 0.702602i \(-0.247978\pi\)
0.711583 + 0.702602i \(0.247978\pi\)
\(174\) −10.6053 −0.803988
\(175\) 0 0
\(176\) −3.93089 −0.296302
\(177\) 12.6053 0.947474
\(178\) −16.7465 −1.25520
\(179\) −25.6884 −1.92004 −0.960021 0.279928i \(-0.909689\pi\)
−0.960021 + 0.279928i \(0.909689\pi\)
\(180\) −0.585786 −0.0436619
\(181\) 6.10443 0.453739 0.226869 0.973925i \(-0.427151\pi\)
0.226869 + 0.973925i \(0.427151\pi\)
\(182\) 0 0
\(183\) 15.2760 1.12923
\(184\) 1.00000 0.0737210
\(185\) −2.61065 −0.191939
\(186\) −0.431794 −0.0316607
\(187\) −17.5186 −1.28109
\(188\) 7.42999 0.541888
\(189\) 0 0
\(190\) 0.918634 0.0666447
\(191\) −12.8156 −0.927303 −0.463651 0.886018i \(-0.653461\pi\)
−0.463651 + 0.886018i \(0.653461\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 9.15341 0.658877 0.329438 0.944177i \(-0.393141\pi\)
0.329438 + 0.944177i \(0.393141\pi\)
\(194\) −12.9419 −0.929177
\(195\) −1.38935 −0.0994935
\(196\) 0 0
\(197\) 19.4209 1.38368 0.691841 0.722050i \(-0.256800\pi\)
0.691841 + 0.722050i \(0.256800\pi\)
\(198\) −3.93089 −0.279356
\(199\) 25.6387 1.81748 0.908739 0.417364i \(-0.137046\pi\)
0.908739 + 0.417364i \(0.137046\pi\)
\(200\) −4.65685 −0.329289
\(201\) 4.11351 0.290145
\(202\) −7.71312 −0.542693
\(203\) 0 0
\(204\) −4.45666 −0.312028
\(205\) 5.43376 0.379510
\(206\) 5.77690 0.402496
\(207\) 1.00000 0.0695048
\(208\) −2.37177 −0.164453
\(209\) 6.16445 0.426404
\(210\) 0 0
\(211\) −3.90227 −0.268643 −0.134322 0.990938i \(-0.542886\pi\)
−0.134322 + 0.990938i \(0.542886\pi\)
\(212\) 7.40513 0.508586
\(213\) 11.5591 0.792018
\(214\) 13.8957 0.949893
\(215\) −3.34667 −0.228241
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) −14.9419 −1.01200
\(219\) 10.5453 0.712586
\(220\) 2.30266 0.155246
\(221\) −10.5702 −0.711026
\(222\) −4.45666 −0.299111
\(223\) 4.79703 0.321233 0.160616 0.987017i \(-0.448652\pi\)
0.160616 + 0.987017i \(0.448652\pi\)
\(224\) 0 0
\(225\) −4.65685 −0.310457
\(226\) −16.3027 −1.08444
\(227\) −26.4006 −1.75227 −0.876133 0.482069i \(-0.839885\pi\)
−0.876133 + 0.482069i \(0.839885\pi\)
\(228\) 1.56821 0.103857
\(229\) 2.91077 0.192349 0.0961744 0.995364i \(-0.469339\pi\)
0.0961744 + 0.995364i \(0.469339\pi\)
\(230\) −0.585786 −0.0386256
\(231\) 0 0
\(232\) 10.6053 0.696274
\(233\) −6.54808 −0.428979 −0.214489 0.976726i \(-0.568809\pi\)
−0.214489 + 0.976726i \(0.568809\pi\)
\(234\) −2.37177 −0.155048
\(235\) −4.35239 −0.283919
\(236\) −12.6053 −0.820537
\(237\) −3.21598 −0.208900
\(238\) 0 0
\(239\) −28.4724 −1.84173 −0.920864 0.389883i \(-0.872515\pi\)
−0.920864 + 0.389883i \(0.872515\pi\)
\(240\) 0.585786 0.0378124
\(241\) 21.6673 1.39571 0.697857 0.716237i \(-0.254137\pi\)
0.697857 + 0.716237i \(0.254137\pi\)
\(242\) 4.45192 0.286180
\(243\) −1.00000 −0.0641500
\(244\) −15.2760 −0.977946
\(245\) 0 0
\(246\) 9.27600 0.591416
\(247\) 3.71943 0.236661
\(248\) 0.431794 0.0274189
\(249\) −7.34511 −0.465477
\(250\) 5.65685 0.357771
\(251\) −5.08292 −0.320831 −0.160416 0.987050i \(-0.551283\pi\)
−0.160416 + 0.987050i \(0.551283\pi\)
\(252\) 0 0
\(253\) −3.93089 −0.247133
\(254\) 13.0778 0.820572
\(255\) 2.61065 0.163485
\(256\) 1.00000 0.0625000
\(257\) −16.8480 −1.05095 −0.525474 0.850810i \(-0.676112\pi\)
−0.525474 + 0.850810i \(0.676112\pi\)
\(258\) −5.71312 −0.355683
\(259\) 0 0
\(260\) 1.38935 0.0861639
\(261\) 10.6053 0.656453
\(262\) 9.96484 0.615630
\(263\) −31.3137 −1.93089 −0.965443 0.260614i \(-0.916075\pi\)
−0.965443 + 0.260614i \(0.916075\pi\)
\(264\) 3.93089 0.241930
\(265\) −4.33782 −0.266470
\(266\) 0 0
\(267\) 16.7465 1.02487
\(268\) −4.11351 −0.251273
\(269\) −13.6358 −0.831387 −0.415693 0.909505i \(-0.636461\pi\)
−0.415693 + 0.909505i \(0.636461\pi\)
\(270\) 0.585786 0.0356498
\(271\) 14.9042 0.905367 0.452684 0.891671i \(-0.350467\pi\)
0.452684 + 0.891671i \(0.350467\pi\)
\(272\) 4.45666 0.270224
\(273\) 0 0
\(274\) −18.9133 −1.14260
\(275\) 18.3056 1.10387
\(276\) −1.00000 −0.0601929
\(277\) −23.9542 −1.43927 −0.719634 0.694353i \(-0.755690\pi\)
−0.719634 + 0.694353i \(0.755690\pi\)
\(278\) −10.8284 −0.649446
\(279\) 0.431794 0.0258508
\(280\) 0 0
\(281\) −3.46139 −0.206489 −0.103245 0.994656i \(-0.532922\pi\)
−0.103245 + 0.994656i \(0.532922\pi\)
\(282\) −7.42999 −0.442450
\(283\) −23.0464 −1.36996 −0.684982 0.728560i \(-0.740190\pi\)
−0.684982 + 0.728560i \(0.740190\pi\)
\(284\) −11.5591 −0.685908
\(285\) −0.918634 −0.0544152
\(286\) 9.32318 0.551291
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) 2.86179 0.168340
\(290\) −6.21246 −0.364808
\(291\) 12.9419 0.758670
\(292\) −10.5453 −0.617117
\(293\) −13.2813 −0.775903 −0.387952 0.921680i \(-0.626817\pi\)
−0.387952 + 0.921680i \(0.626817\pi\)
\(294\) 0 0
\(295\) 7.38403 0.429915
\(296\) 4.45666 0.259038
\(297\) 3.93089 0.228094
\(298\) −13.1469 −0.761578
\(299\) −2.37177 −0.137163
\(300\) 4.65685 0.268864
\(301\) 0 0
\(302\) 17.3857 1.00044
\(303\) 7.71312 0.443107
\(304\) −1.56821 −0.0899428
\(305\) 8.94847 0.512388
\(306\) 4.45666 0.254770
\(307\) 2.94095 0.167849 0.0839244 0.996472i \(-0.473255\pi\)
0.0839244 + 0.996472i \(0.473255\pi\)
\(308\) 0 0
\(309\) −5.77690 −0.328636
\(310\) −0.252939 −0.0143660
\(311\) −25.9206 −1.46982 −0.734911 0.678164i \(-0.762776\pi\)
−0.734911 + 0.678164i \(0.762776\pi\)
\(312\) 2.37177 0.134275
\(313\) 13.6101 0.769286 0.384643 0.923065i \(-0.374325\pi\)
0.384643 + 0.923065i \(0.374325\pi\)
\(314\) −7.00850 −0.395512
\(315\) 0 0
\(316\) 3.21598 0.180913
\(317\) 22.3248 1.25388 0.626942 0.779066i \(-0.284306\pi\)
0.626942 + 0.779066i \(0.284306\pi\)
\(318\) −7.40513 −0.415259
\(319\) −41.6884 −2.33410
\(320\) −0.585786 −0.0327465
\(321\) −13.8957 −0.775584
\(322\) 0 0
\(323\) −6.98896 −0.388876
\(324\) 1.00000 0.0555556
\(325\) 11.0450 0.612666
\(326\) 7.95952 0.440837
\(327\) 14.9419 0.826291
\(328\) −9.27600 −0.512181
\(329\) 0 0
\(330\) −2.30266 −0.126757
\(331\) 17.6569 0.970508 0.485254 0.874373i \(-0.338727\pi\)
0.485254 + 0.874373i \(0.338727\pi\)
\(332\) 7.34511 0.403115
\(333\) 4.45666 0.244223
\(334\) −6.03140 −0.330023
\(335\) 2.40964 0.131653
\(336\) 0 0
\(337\) −3.25114 −0.177101 −0.0885504 0.996072i \(-0.528223\pi\)
−0.0885504 + 0.996072i \(0.528223\pi\)
\(338\) −7.37470 −0.401131
\(339\) 16.3027 0.885439
\(340\) −2.61065 −0.141582
\(341\) −1.69734 −0.0919159
\(342\) −1.56821 −0.0847989
\(343\) 0 0
\(344\) 5.71312 0.308031
\(345\) 0.585786 0.0315377
\(346\) 18.7188 1.00633
\(347\) −16.5755 −0.889819 −0.444909 0.895576i \(-0.646764\pi\)
−0.444909 + 0.895576i \(0.646764\pi\)
\(348\) −10.6053 −0.568505
\(349\) −24.4882 −1.31082 −0.655412 0.755272i \(-0.727505\pi\)
−0.655412 + 0.755272i \(0.727505\pi\)
\(350\) 0 0
\(351\) 2.37177 0.126596
\(352\) −3.93089 −0.209517
\(353\) 20.1765 1.07388 0.536942 0.843619i \(-0.319579\pi\)
0.536942 + 0.843619i \(0.319579\pi\)
\(354\) 12.6053 0.669966
\(355\) 6.77118 0.359377
\(356\) −16.7465 −0.887561
\(357\) 0 0
\(358\) −25.6884 −1.35767
\(359\) 7.95952 0.420087 0.210044 0.977692i \(-0.432639\pi\)
0.210044 + 0.977692i \(0.432639\pi\)
\(360\) −0.585786 −0.0308737
\(361\) −16.5407 −0.870565
\(362\) 6.10443 0.320842
\(363\) −4.45192 −0.233665
\(364\) 0 0
\(365\) 6.17730 0.323334
\(366\) 15.2760 0.798489
\(367\) 17.3986 0.908199 0.454100 0.890951i \(-0.349961\pi\)
0.454100 + 0.890951i \(0.349961\pi\)
\(368\) 1.00000 0.0521286
\(369\) −9.27600 −0.482889
\(370\) −2.61065 −0.135721
\(371\) 0 0
\(372\) −0.431794 −0.0223875
\(373\) 19.7427 1.02224 0.511120 0.859509i \(-0.329231\pi\)
0.511120 + 0.859509i \(0.329231\pi\)
\(374\) −17.5186 −0.905867
\(375\) −5.65685 −0.292119
\(376\) 7.42999 0.383173
\(377\) −25.1534 −1.29547
\(378\) 0 0
\(379\) −5.03967 −0.258870 −0.129435 0.991588i \(-0.541316\pi\)
−0.129435 + 0.991588i \(0.541316\pi\)
\(380\) 0.918634 0.0471249
\(381\) −13.0778 −0.669994
\(382\) −12.8156 −0.655702
\(383\) 34.6017 1.76807 0.884033 0.467425i \(-0.154818\pi\)
0.884033 + 0.467425i \(0.154818\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 9.15341 0.465896
\(387\) 5.71312 0.290414
\(388\) −12.9419 −0.657027
\(389\) 11.6598 0.591175 0.295587 0.955316i \(-0.404485\pi\)
0.295587 + 0.955316i \(0.404485\pi\)
\(390\) −1.38935 −0.0703525
\(391\) 4.45666 0.225383
\(392\) 0 0
\(393\) −9.96484 −0.502660
\(394\) 19.4209 0.978411
\(395\) −1.88388 −0.0947881
\(396\) −3.93089 −0.197535
\(397\) −8.80372 −0.441846 −0.220923 0.975291i \(-0.570907\pi\)
−0.220923 + 0.975291i \(0.570907\pi\)
\(398\) 25.6387 1.28515
\(399\) 0 0
\(400\) −4.65685 −0.232843
\(401\) 12.9800 0.648192 0.324096 0.946024i \(-0.394940\pi\)
0.324096 + 0.946024i \(0.394940\pi\)
\(402\) 4.11351 0.205163
\(403\) −1.02412 −0.0510149
\(404\) −7.71312 −0.383742
\(405\) −0.585786 −0.0291080
\(406\) 0 0
\(407\) −17.5186 −0.868367
\(408\) −4.45666 −0.220637
\(409\) −18.9957 −0.939274 −0.469637 0.882859i \(-0.655615\pi\)
−0.469637 + 0.882859i \(0.655615\pi\)
\(410\) 5.43376 0.268354
\(411\) 18.9133 0.932925
\(412\) 5.77690 0.284607
\(413\) 0 0
\(414\) 1.00000 0.0491473
\(415\) −4.30266 −0.211209
\(416\) −2.37177 −0.116286
\(417\) 10.8284 0.530270
\(418\) 6.16445 0.301513
\(419\) 19.2566 0.940747 0.470374 0.882467i \(-0.344119\pi\)
0.470374 + 0.882467i \(0.344119\pi\)
\(420\) 0 0
\(421\) 11.1078 0.541360 0.270680 0.962669i \(-0.412751\pi\)
0.270680 + 0.962669i \(0.412751\pi\)
\(422\) −3.90227 −0.189959
\(423\) 7.42999 0.361259
\(424\) 7.40513 0.359625
\(425\) −20.7540 −1.00672
\(426\) 11.5591 0.560041
\(427\) 0 0
\(428\) 13.8957 0.671676
\(429\) −9.32318 −0.450127
\(430\) −3.34667 −0.161391
\(431\) −14.7656 −0.711235 −0.355618 0.934632i \(-0.615729\pi\)
−0.355618 + 0.934632i \(0.615729\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −5.51571 −0.265068 −0.132534 0.991178i \(-0.542311\pi\)
−0.132534 + 0.991178i \(0.542311\pi\)
\(434\) 0 0
\(435\) 6.21246 0.297865
\(436\) −14.9419 −0.715589
\(437\) −1.56821 −0.0750175
\(438\) 10.5453 0.503874
\(439\) −4.98912 −0.238117 −0.119059 0.992887i \(-0.537988\pi\)
−0.119059 + 0.992887i \(0.537988\pi\)
\(440\) 2.30266 0.109775
\(441\) 0 0
\(442\) −10.5702 −0.502772
\(443\) 10.7382 0.510188 0.255094 0.966916i \(-0.417894\pi\)
0.255094 + 0.966916i \(0.417894\pi\)
\(444\) −4.45666 −0.211504
\(445\) 9.80986 0.465032
\(446\) 4.79703 0.227146
\(447\) 13.1469 0.621826
\(448\) 0 0
\(449\) −30.0924 −1.42015 −0.710074 0.704127i \(-0.751339\pi\)
−0.710074 + 0.704127i \(0.751339\pi\)
\(450\) −4.65685 −0.219526
\(451\) 36.4630 1.71697
\(452\) −16.3027 −0.766813
\(453\) −17.3857 −0.816854
\(454\) −26.4006 −1.23904
\(455\) 0 0
\(456\) 1.56821 0.0734380
\(457\) −21.8618 −1.02265 −0.511326 0.859387i \(-0.670845\pi\)
−0.511326 + 0.859387i \(0.670845\pi\)
\(458\) 2.91077 0.136011
\(459\) −4.45666 −0.208019
\(460\) −0.585786 −0.0273124
\(461\) 20.9238 0.974517 0.487259 0.873258i \(-0.337997\pi\)
0.487259 + 0.873258i \(0.337997\pi\)
\(462\) 0 0
\(463\) 3.73422 0.173544 0.0867718 0.996228i \(-0.472345\pi\)
0.0867718 + 0.996228i \(0.472345\pi\)
\(464\) 10.6053 0.492340
\(465\) 0.252939 0.0117298
\(466\) −6.54808 −0.303334
\(467\) −12.9042 −0.597137 −0.298568 0.954388i \(-0.596509\pi\)
−0.298568 + 0.954388i \(0.596509\pi\)
\(468\) −2.37177 −0.109635
\(469\) 0 0
\(470\) −4.35239 −0.200761
\(471\) 7.00850 0.322934
\(472\) −12.6053 −0.580207
\(473\) −22.4576 −1.03260
\(474\) −3.21598 −0.147715
\(475\) 7.30291 0.335080
\(476\) 0 0
\(477\) 7.40513 0.339058
\(478\) −28.4724 −1.30230
\(479\) 35.0298 1.60055 0.800275 0.599633i \(-0.204687\pi\)
0.800275 + 0.599633i \(0.204687\pi\)
\(480\) 0.585786 0.0267374
\(481\) −10.5702 −0.481958
\(482\) 21.6673 0.986919
\(483\) 0 0
\(484\) 4.45192 0.202360
\(485\) 7.58121 0.344245
\(486\) −1.00000 −0.0453609
\(487\) 0.195463 0.00885729 0.00442864 0.999990i \(-0.498590\pi\)
0.00442864 + 0.999990i \(0.498590\pi\)
\(488\) −15.2760 −0.691512
\(489\) −7.95952 −0.359942
\(490\) 0 0
\(491\) 18.0519 0.814672 0.407336 0.913278i \(-0.366458\pi\)
0.407336 + 0.913278i \(0.366458\pi\)
\(492\) 9.27600 0.418194
\(493\) 47.2643 2.12868
\(494\) 3.71943 0.167345
\(495\) 2.30266 0.103497
\(496\) 0.431794 0.0193881
\(497\) 0 0
\(498\) −7.34511 −0.329142
\(499\) 12.4057 0.555356 0.277678 0.960674i \(-0.410435\pi\)
0.277678 + 0.960674i \(0.410435\pi\)
\(500\) 5.65685 0.252982
\(501\) 6.03140 0.269463
\(502\) −5.08292 −0.226862
\(503\) 2.84659 0.126923 0.0634617 0.997984i \(-0.479786\pi\)
0.0634617 + 0.997984i \(0.479786\pi\)
\(504\) 0 0
\(505\) 4.51824 0.201059
\(506\) −3.93089 −0.174750
\(507\) 7.37470 0.327522
\(508\) 13.0778 0.580232
\(509\) 7.26692 0.322100 0.161050 0.986946i \(-0.448512\pi\)
0.161050 + 0.986946i \(0.448512\pi\)
\(510\) 2.61065 0.115601
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 1.56821 0.0692380
\(514\) −16.8480 −0.743132
\(515\) −3.38403 −0.149118
\(516\) −5.71312 −0.251506
\(517\) −29.2065 −1.28450
\(518\) 0 0
\(519\) −18.7188 −0.821666
\(520\) 1.38935 0.0609271
\(521\) 5.08015 0.222565 0.111283 0.993789i \(-0.464504\pi\)
0.111283 + 0.993789i \(0.464504\pi\)
\(522\) 10.6053 0.464183
\(523\) −1.82647 −0.0798658 −0.0399329 0.999202i \(-0.512714\pi\)
−0.0399329 + 0.999202i \(0.512714\pi\)
\(524\) 9.96484 0.435316
\(525\) 0 0
\(526\) −31.3137 −1.36534
\(527\) 1.92436 0.0838263
\(528\) 3.93089 0.171070
\(529\) 1.00000 0.0434783
\(530\) −4.33782 −0.188423
\(531\) −12.6053 −0.547025
\(532\) 0 0
\(533\) 22.0005 0.952949
\(534\) 16.7465 0.724691
\(535\) −8.13993 −0.351920
\(536\) −4.11351 −0.177677
\(537\) 25.6884 1.10854
\(538\) −13.6358 −0.587879
\(539\) 0 0
\(540\) 0.585786 0.0252082
\(541\) −24.5445 −1.05525 −0.527625 0.849478i \(-0.676917\pi\)
−0.527625 + 0.849478i \(0.676917\pi\)
\(542\) 14.9042 0.640191
\(543\) −6.10443 −0.261966
\(544\) 4.45666 0.191078
\(545\) 8.75278 0.374928
\(546\) 0 0
\(547\) 18.6827 0.798814 0.399407 0.916774i \(-0.369216\pi\)
0.399407 + 0.916774i \(0.369216\pi\)
\(548\) −18.9133 −0.807937
\(549\) −15.2760 −0.651964
\(550\) 18.3056 0.780553
\(551\) −16.6313 −0.708519
\(552\) −1.00000 −0.0425628
\(553\) 0 0
\(554\) −23.9542 −1.01772
\(555\) 2.61065 0.110816
\(556\) −10.8284 −0.459228
\(557\) 3.05446 0.129422 0.0647108 0.997904i \(-0.479388\pi\)
0.0647108 + 0.997904i \(0.479388\pi\)
\(558\) 0.431794 0.0182793
\(559\) −13.5502 −0.573112
\(560\) 0 0
\(561\) 17.5186 0.739638
\(562\) −3.46139 −0.146010
\(563\) −32.2626 −1.35971 −0.679853 0.733349i \(-0.737956\pi\)
−0.679853 + 0.733349i \(0.737956\pi\)
\(564\) −7.42999 −0.312859
\(565\) 9.54988 0.401766
\(566\) −23.0464 −0.968711
\(567\) 0 0
\(568\) −11.5591 −0.485010
\(569\) −1.96844 −0.0825214 −0.0412607 0.999148i \(-0.513137\pi\)
−0.0412607 + 0.999148i \(0.513137\pi\)
\(570\) −0.918634 −0.0384773
\(571\) 13.0971 0.548098 0.274049 0.961716i \(-0.411637\pi\)
0.274049 + 0.961716i \(0.411637\pi\)
\(572\) 9.32318 0.389822
\(573\) 12.8156 0.535379
\(574\) 0 0
\(575\) −4.65685 −0.194204
\(576\) 1.00000 0.0416667
\(577\) 6.57100 0.273554 0.136777 0.990602i \(-0.456326\pi\)
0.136777 + 0.990602i \(0.456326\pi\)
\(578\) 2.86179 0.119035
\(579\) −9.15341 −0.380403
\(580\) −6.21246 −0.257958
\(581\) 0 0
\(582\) 12.9419 0.536461
\(583\) −29.1088 −1.20556
\(584\) −10.5453 −0.436368
\(585\) 1.38935 0.0574426
\(586\) −13.2813 −0.548646
\(587\) −24.0134 −0.991139 −0.495569 0.868568i \(-0.665041\pi\)
−0.495569 + 0.868568i \(0.665041\pi\)
\(588\) 0 0
\(589\) −0.677142 −0.0279011
\(590\) 7.38403 0.303996
\(591\) −19.4209 −0.798869
\(592\) 4.45666 0.183167
\(593\) −35.8205 −1.47097 −0.735485 0.677541i \(-0.763046\pi\)
−0.735485 + 0.677541i \(0.763046\pi\)
\(594\) 3.93089 0.161287
\(595\) 0 0
\(596\) −13.1469 −0.538517
\(597\) −25.6387 −1.04932
\(598\) −2.37177 −0.0969889
\(599\) 25.5502 1.04395 0.521976 0.852960i \(-0.325195\pi\)
0.521976 + 0.852960i \(0.325195\pi\)
\(600\) 4.65685 0.190115
\(601\) −47.1194 −1.92204 −0.961020 0.276479i \(-0.910833\pi\)
−0.961020 + 0.276479i \(0.910833\pi\)
\(602\) 0 0
\(603\) −4.11351 −0.167515
\(604\) 17.3857 0.707416
\(605\) −2.60788 −0.106025
\(606\) 7.71312 0.313324
\(607\) −37.8197 −1.53506 −0.767528 0.641015i \(-0.778513\pi\)
−0.767528 + 0.641015i \(0.778513\pi\)
\(608\) −1.56821 −0.0635992
\(609\) 0 0
\(610\) 8.94847 0.362313
\(611\) −17.6222 −0.712919
\(612\) 4.45666 0.180150
\(613\) 29.4549 1.18967 0.594835 0.803848i \(-0.297217\pi\)
0.594835 + 0.803848i \(0.297217\pi\)
\(614\) 2.94095 0.118687
\(615\) −5.43376 −0.219110
\(616\) 0 0
\(617\) −31.8266 −1.28129 −0.640646 0.767837i \(-0.721333\pi\)
−0.640646 + 0.767837i \(0.721333\pi\)
\(618\) −5.77690 −0.232381
\(619\) 40.0183 1.60847 0.804236 0.594310i \(-0.202575\pi\)
0.804236 + 0.594310i \(0.202575\pi\)
\(620\) −0.252939 −0.0101583
\(621\) −1.00000 −0.0401286
\(622\) −25.9206 −1.03932
\(623\) 0 0
\(624\) 2.37177 0.0949468
\(625\) 19.9706 0.798823
\(626\) 13.6101 0.543967
\(627\) −6.16445 −0.246184
\(628\) −7.00850 −0.279669
\(629\) 19.8618 0.791941
\(630\) 0 0
\(631\) −46.5356 −1.85255 −0.926276 0.376847i \(-0.877008\pi\)
−0.926276 + 0.376847i \(0.877008\pi\)
\(632\) 3.21598 0.127925
\(633\) 3.90227 0.155101
\(634\) 22.3248 0.886629
\(635\) −7.66078 −0.304009
\(636\) −7.40513 −0.293633
\(637\) 0 0
\(638\) −41.6884 −1.65046
\(639\) −11.5591 −0.457272
\(640\) −0.585786 −0.0231552
\(641\) −24.3747 −0.962743 −0.481371 0.876517i \(-0.659861\pi\)
−0.481371 + 0.876517i \(0.659861\pi\)
\(642\) −13.8957 −0.548421
\(643\) 12.0886 0.476730 0.238365 0.971176i \(-0.423389\pi\)
0.238365 + 0.971176i \(0.423389\pi\)
\(644\) 0 0
\(645\) 3.34667 0.131775
\(646\) −6.98896 −0.274977
\(647\) −17.9652 −0.706286 −0.353143 0.935569i \(-0.614887\pi\)
−0.353143 + 0.935569i \(0.614887\pi\)
\(648\) 1.00000 0.0392837
\(649\) 49.5502 1.94502
\(650\) 11.0450 0.433220
\(651\) 0 0
\(652\) 7.95952 0.311719
\(653\) −2.98363 −0.116759 −0.0583793 0.998294i \(-0.518593\pi\)
−0.0583793 + 0.998294i \(0.518593\pi\)
\(654\) 14.9419 0.584276
\(655\) −5.83727 −0.228081
\(656\) −9.27600 −0.362167
\(657\) −10.5453 −0.411412
\(658\) 0 0
\(659\) −12.5806 −0.490072 −0.245036 0.969514i \(-0.578800\pi\)
−0.245036 + 0.969514i \(0.578800\pi\)
\(660\) −2.30266 −0.0896311
\(661\) 38.5194 1.49823 0.749115 0.662440i \(-0.230479\pi\)
0.749115 + 0.662440i \(0.230479\pi\)
\(662\) 17.6569 0.686253
\(663\) 10.5702 0.410511
\(664\) 7.34511 0.285045
\(665\) 0 0
\(666\) 4.45666 0.172692
\(667\) 10.6053 0.410640
\(668\) −6.03140 −0.233362
\(669\) −4.79703 −0.185464
\(670\) 2.40964 0.0930925
\(671\) 60.0483 2.31814
\(672\) 0 0
\(673\) 13.3840 0.515916 0.257958 0.966156i \(-0.416950\pi\)
0.257958 + 0.966156i \(0.416950\pi\)
\(674\) −3.25114 −0.125229
\(675\) 4.65685 0.179242
\(676\) −7.37470 −0.283642
\(677\) 4.41601 0.169721 0.0848606 0.996393i \(-0.472955\pi\)
0.0848606 + 0.996393i \(0.472955\pi\)
\(678\) 16.3027 0.626100
\(679\) 0 0
\(680\) −2.61065 −0.100114
\(681\) 26.4006 1.01167
\(682\) −1.69734 −0.0649944
\(683\) −39.9101 −1.52712 −0.763559 0.645738i \(-0.776550\pi\)
−0.763559 + 0.645738i \(0.776550\pi\)
\(684\) −1.56821 −0.0599619
\(685\) 11.0792 0.423313
\(686\) 0 0
\(687\) −2.91077 −0.111053
\(688\) 5.71312 0.217810
\(689\) −17.5633 −0.669107
\(690\) 0.585786 0.0223005
\(691\) −31.1937 −1.18666 −0.593331 0.804958i \(-0.702188\pi\)
−0.593331 + 0.804958i \(0.702188\pi\)
\(692\) 18.7188 0.711583
\(693\) 0 0
\(694\) −16.5755 −0.629197
\(695\) 6.34315 0.240609
\(696\) −10.6053 −0.401994
\(697\) −41.3399 −1.56586
\(698\) −24.4882 −0.926893
\(699\) 6.54808 0.247671
\(700\) 0 0
\(701\) −14.7761 −0.558085 −0.279043 0.960279i \(-0.590017\pi\)
−0.279043 + 0.960279i \(0.590017\pi\)
\(702\) 2.37177 0.0895167
\(703\) −6.98896 −0.263593
\(704\) −3.93089 −0.148151
\(705\) 4.35239 0.163920
\(706\) 20.1765 0.759351
\(707\) 0 0
\(708\) 12.6053 0.473737
\(709\) 32.2928 1.21278 0.606390 0.795167i \(-0.292617\pi\)
0.606390 + 0.795167i \(0.292617\pi\)
\(710\) 6.77118 0.254118
\(711\) 3.21598 0.120609
\(712\) −16.7465 −0.627601
\(713\) 0.431794 0.0161708
\(714\) 0 0
\(715\) −5.46139 −0.204244
\(716\) −25.6884 −0.960021
\(717\) 28.4724 1.06332
\(718\) 7.95952 0.297047
\(719\) 25.7327 0.959666 0.479833 0.877360i \(-0.340697\pi\)
0.479833 + 0.877360i \(0.340697\pi\)
\(720\) −0.585786 −0.0218310
\(721\) 0 0
\(722\) −16.5407 −0.615582
\(723\) −21.6673 −0.805816
\(724\) 6.10443 0.226869
\(725\) −49.3875 −1.83420
\(726\) −4.45192 −0.165226
\(727\) 0.623495 0.0231241 0.0115621 0.999933i \(-0.496320\pi\)
0.0115621 + 0.999933i \(0.496320\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 6.17730 0.228632
\(731\) 25.4614 0.941724
\(732\) 15.2760 0.564617
\(733\) −32.7449 −1.20946 −0.604731 0.796430i \(-0.706719\pi\)
−0.604731 + 0.796430i \(0.706719\pi\)
\(734\) 17.3986 0.642194
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) 16.1698 0.595621
\(738\) −9.27600 −0.341454
\(739\) 34.8270 1.28113 0.640566 0.767903i \(-0.278700\pi\)
0.640566 + 0.767903i \(0.278700\pi\)
\(740\) −2.61065 −0.0959694
\(741\) −3.71943 −0.136636
\(742\) 0 0
\(743\) 7.38575 0.270957 0.135478 0.990780i \(-0.456743\pi\)
0.135478 + 0.990780i \(0.456743\pi\)
\(744\) −0.431794 −0.0158303
\(745\) 7.70126 0.282152
\(746\) 19.7427 0.722833
\(747\) 7.34511 0.268743
\(748\) −17.5186 −0.640545
\(749\) 0 0
\(750\) −5.65685 −0.206559
\(751\) 26.9800 0.984515 0.492258 0.870450i \(-0.336172\pi\)
0.492258 + 0.870450i \(0.336172\pi\)
\(752\) 7.42999 0.270944
\(753\) 5.08292 0.185232
\(754\) −25.1534 −0.916033
\(755\) −10.1843 −0.370646
\(756\) 0 0
\(757\) 0.577876 0.0210033 0.0105016 0.999945i \(-0.496657\pi\)
0.0105016 + 0.999945i \(0.496657\pi\)
\(758\) −5.03967 −0.183049
\(759\) 3.93089 0.142682
\(760\) 0.918634 0.0333224
\(761\) −7.36981 −0.267155 −0.133578 0.991038i \(-0.542647\pi\)
−0.133578 + 0.991038i \(0.542647\pi\)
\(762\) −13.0778 −0.473757
\(763\) 0 0
\(764\) −12.8156 −0.463651
\(765\) −2.61065 −0.0943882
\(766\) 34.6017 1.25021
\(767\) 29.8969 1.07952
\(768\) −1.00000 −0.0360844
\(769\) −15.0011 −0.540955 −0.270477 0.962726i \(-0.587182\pi\)
−0.270477 + 0.962726i \(0.587182\pi\)
\(770\) 0 0
\(771\) 16.8480 0.606765
\(772\) 9.15341 0.329438
\(773\) 15.5013 0.557543 0.278772 0.960357i \(-0.410073\pi\)
0.278772 + 0.960357i \(0.410073\pi\)
\(774\) 5.71312 0.205354
\(775\) −2.01080 −0.0722301
\(776\) −12.9419 −0.464588
\(777\) 0 0
\(778\) 11.6598 0.418024
\(779\) 14.5467 0.521189
\(780\) −1.38935 −0.0497467
\(781\) 45.4377 1.62589
\(782\) 4.45666 0.159370
\(783\) −10.6053 −0.379004
\(784\) 0 0
\(785\) 4.10548 0.146531
\(786\) −9.96484 −0.355434
\(787\) 19.3767 0.690704 0.345352 0.938473i \(-0.387760\pi\)
0.345352 + 0.938473i \(0.387760\pi\)
\(788\) 19.4209 0.691841
\(789\) 31.3137 1.11480
\(790\) −1.88388 −0.0670253
\(791\) 0 0
\(792\) −3.93089 −0.139678
\(793\) 36.2312 1.28661
\(794\) −8.80372 −0.312432
\(795\) 4.33782 0.153847
\(796\) 25.6387 0.908739
\(797\) 50.6436 1.79389 0.896944 0.442144i \(-0.145782\pi\)
0.896944 + 0.442144i \(0.145782\pi\)
\(798\) 0 0
\(799\) 33.1129 1.17145
\(800\) −4.65685 −0.164645
\(801\) −16.7465 −0.591708
\(802\) 12.9800 0.458341
\(803\) 41.4525 1.46283
\(804\) 4.11351 0.145072
\(805\) 0 0
\(806\) −1.02412 −0.0360730
\(807\) 13.6358 0.480001
\(808\) −7.71312 −0.271346
\(809\) −14.7530 −0.518688 −0.259344 0.965785i \(-0.583506\pi\)
−0.259344 + 0.965785i \(0.583506\pi\)
\(810\) −0.585786 −0.0205824
\(811\) −1.24699 −0.0437877 −0.0218939 0.999760i \(-0.506970\pi\)
−0.0218939 + 0.999760i \(0.506970\pi\)
\(812\) 0 0
\(813\) −14.9042 −0.522714
\(814\) −17.5186 −0.614028
\(815\) −4.66258 −0.163323
\(816\) −4.45666 −0.156014
\(817\) −8.95934 −0.313448
\(818\) −18.9957 −0.664167
\(819\) 0 0
\(820\) 5.43376 0.189755
\(821\) 28.9652 1.01089 0.505447 0.862858i \(-0.331328\pi\)
0.505447 + 0.862858i \(0.331328\pi\)
\(822\) 18.9133 0.659678
\(823\) 30.3820 1.05905 0.529525 0.848294i \(-0.322370\pi\)
0.529525 + 0.848294i \(0.322370\pi\)
\(824\) 5.77690 0.201248
\(825\) −18.3056 −0.637319
\(826\) 0 0
\(827\) 30.6286 1.06506 0.532531 0.846411i \(-0.321241\pi\)
0.532531 + 0.846411i \(0.321241\pi\)
\(828\) 1.00000 0.0347524
\(829\) 5.36487 0.186330 0.0931649 0.995651i \(-0.470302\pi\)
0.0931649 + 0.995651i \(0.470302\pi\)
\(830\) −4.30266 −0.149348
\(831\) 23.9542 0.830962
\(832\) −2.37177 −0.0822264
\(833\) 0 0
\(834\) 10.8284 0.374958
\(835\) 3.53311 0.122268
\(836\) 6.16445 0.213202
\(837\) −0.431794 −0.0149250
\(838\) 19.2566 0.665209
\(839\) 14.2306 0.491296 0.245648 0.969359i \(-0.420999\pi\)
0.245648 + 0.969359i \(0.420999\pi\)
\(840\) 0 0
\(841\) 83.4730 2.87838
\(842\) 11.1078 0.382800
\(843\) 3.46139 0.119217
\(844\) −3.90227 −0.134322
\(845\) 4.32000 0.148613
\(846\) 7.42999 0.255448
\(847\) 0 0
\(848\) 7.40513 0.254293
\(849\) 23.0464 0.790949
\(850\) −20.7540 −0.711856
\(851\) 4.45666 0.152772
\(852\) 11.5591 0.396009
\(853\) −6.59879 −0.225938 −0.112969 0.993598i \(-0.536036\pi\)
−0.112969 + 0.993598i \(0.536036\pi\)
\(854\) 0 0
\(855\) 0.918634 0.0314166
\(856\) 13.8957 0.474946
\(857\) −13.2280 −0.451860 −0.225930 0.974144i \(-0.572542\pi\)
−0.225930 + 0.974144i \(0.572542\pi\)
\(858\) −9.32318 −0.318288
\(859\) 38.3692 1.30914 0.654569 0.756002i \(-0.272850\pi\)
0.654569 + 0.756002i \(0.272850\pi\)
\(860\) −3.34667 −0.114120
\(861\) 0 0
\(862\) −14.7656 −0.502919
\(863\) −55.3143 −1.88292 −0.941460 0.337126i \(-0.890545\pi\)
−0.941460 + 0.337126i \(0.890545\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −10.9652 −0.372829
\(866\) −5.51571 −0.187431
\(867\) −2.86179 −0.0971914
\(868\) 0 0
\(869\) −12.6417 −0.428839
\(870\) 6.21246 0.210622
\(871\) 9.75630 0.330580
\(872\) −14.9419 −0.505998
\(873\) −12.9419 −0.438018
\(874\) −1.56821 −0.0530454
\(875\) 0 0
\(876\) 10.5453 0.356293
\(877\) −29.0099 −0.979594 −0.489797 0.871837i \(-0.662929\pi\)
−0.489797 + 0.871837i \(0.662929\pi\)
\(878\) −4.98912 −0.168374
\(879\) 13.2813 0.447968
\(880\) 2.30266 0.0776228
\(881\) 56.4179 1.90077 0.950384 0.311081i \(-0.100691\pi\)
0.950384 + 0.311081i \(0.100691\pi\)
\(882\) 0 0
\(883\) 21.9690 0.739315 0.369658 0.929168i \(-0.379475\pi\)
0.369658 + 0.929168i \(0.379475\pi\)
\(884\) −10.5702 −0.355513
\(885\) −7.38403 −0.248211
\(886\) 10.7382 0.360758
\(887\) 12.7844 0.429258 0.214629 0.976696i \(-0.431146\pi\)
0.214629 + 0.976696i \(0.431146\pi\)
\(888\) −4.45666 −0.149556
\(889\) 0 0
\(890\) 9.80986 0.328827
\(891\) −3.93089 −0.131690
\(892\) 4.79703 0.160616
\(893\) −11.6518 −0.389911
\(894\) 13.1469 0.439697
\(895\) 15.0479 0.502997
\(896\) 0 0
\(897\) 2.37177 0.0791911
\(898\) −30.0924 −1.00420
\(899\) 4.57932 0.152729
\(900\) −4.65685 −0.155228
\(901\) 33.0021 1.09946
\(902\) 36.4630 1.21408
\(903\) 0 0
\(904\) −16.3027 −0.542219
\(905\) −3.57589 −0.118867
\(906\) −17.3857 −0.577603
\(907\) 27.8349 0.924242 0.462121 0.886817i \(-0.347089\pi\)
0.462121 + 0.886817i \(0.347089\pi\)
\(908\) −26.4006 −0.876133
\(909\) −7.71312 −0.255828
\(910\) 0 0
\(911\) 48.2181 1.59754 0.798768 0.601639i \(-0.205485\pi\)
0.798768 + 0.601639i \(0.205485\pi\)
\(912\) 1.56821 0.0519285
\(913\) −28.8728 −0.955551
\(914\) −21.8618 −0.723124
\(915\) −8.94847 −0.295827
\(916\) 2.91077 0.0961744
\(917\) 0 0
\(918\) −4.45666 −0.147092
\(919\) −30.0941 −0.992714 −0.496357 0.868119i \(-0.665329\pi\)
−0.496357 + 0.868119i \(0.665329\pi\)
\(920\) −0.585786 −0.0193128
\(921\) −2.94095 −0.0969076
\(922\) 20.9238 0.689088
\(923\) 27.4156 0.902395
\(924\) 0 0
\(925\) −20.7540 −0.682387
\(926\) 3.73422 0.122714
\(927\) 5.77690 0.189738
\(928\) 10.6053 0.348137
\(929\) −7.98438 −0.261959 −0.130980 0.991385i \(-0.541812\pi\)
−0.130980 + 0.991385i \(0.541812\pi\)
\(930\) 0.252939 0.00829420
\(931\) 0 0
\(932\) −6.54808 −0.214489
\(933\) 25.9206 0.848602
\(934\) −12.9042 −0.422239
\(935\) 10.2622 0.335609
\(936\) −2.37177 −0.0775238
\(937\) 17.4146 0.568910 0.284455 0.958689i \(-0.408187\pi\)
0.284455 + 0.958689i \(0.408187\pi\)
\(938\) 0 0
\(939\) −13.6101 −0.444148
\(940\) −4.35239 −0.141959
\(941\) −36.8310 −1.20066 −0.600328 0.799754i \(-0.704963\pi\)
−0.600328 + 0.799754i \(0.704963\pi\)
\(942\) 7.00850 0.228349
\(943\) −9.27600 −0.302068
\(944\) −12.6053 −0.410268
\(945\) 0 0
\(946\) −22.4576 −0.730161
\(947\) −9.61637 −0.312490 −0.156245 0.987718i \(-0.549939\pi\)
−0.156245 + 0.987718i \(0.549939\pi\)
\(948\) −3.21598 −0.104450
\(949\) 25.0110 0.811893
\(950\) 7.30291 0.236938
\(951\) −22.3248 −0.723930
\(952\) 0 0
\(953\) −34.5964 −1.12069 −0.560344 0.828260i \(-0.689331\pi\)
−0.560344 + 0.828260i \(0.689331\pi\)
\(954\) 7.40513 0.239750
\(955\) 7.50719 0.242927
\(956\) −28.4724 −0.920864
\(957\) 41.6884 1.34759
\(958\) 35.0298 1.13176
\(959\) 0 0
\(960\) 0.585786 0.0189062
\(961\) −30.8136 −0.993986
\(962\) −10.5702 −0.340796
\(963\) 13.8957 0.447784
\(964\) 21.6673 0.697857
\(965\) −5.36194 −0.172607
\(966\) 0 0
\(967\) −29.1146 −0.936264 −0.468132 0.883659i \(-0.655073\pi\)
−0.468132 + 0.883659i \(0.655073\pi\)
\(968\) 4.45192 0.143090
\(969\) 6.98896 0.224518
\(970\) 7.58121 0.243418
\(971\) 20.7601 0.666225 0.333112 0.942887i \(-0.391901\pi\)
0.333112 + 0.942887i \(0.391901\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) 0.195463 0.00626305
\(975\) −11.0450 −0.353723
\(976\) −15.2760 −0.488973
\(977\) 47.8582 1.53112 0.765559 0.643365i \(-0.222462\pi\)
0.765559 + 0.643365i \(0.222462\pi\)
\(978\) −7.95952 −0.254517
\(979\) 65.8286 2.10389
\(980\) 0 0
\(981\) −14.9419 −0.477059
\(982\) 18.0519 0.576060
\(983\) −47.2959 −1.50850 −0.754252 0.656585i \(-0.772000\pi\)
−0.754252 + 0.656585i \(0.772000\pi\)
\(984\) 9.27600 0.295708
\(985\) −11.3765 −0.362486
\(986\) 47.2643 1.50520
\(987\) 0 0
\(988\) 3.71943 0.118331
\(989\) 5.71312 0.181666
\(990\) 2.30266 0.0731835
\(991\) −19.3530 −0.614769 −0.307384 0.951585i \(-0.599454\pi\)
−0.307384 + 0.951585i \(0.599454\pi\)
\(992\) 0.431794 0.0137095
\(993\) −17.6569 −0.560323
\(994\) 0 0
\(995\) −15.0188 −0.476128
\(996\) −7.34511 −0.232739
\(997\) 32.4957 1.02915 0.514575 0.857445i \(-0.327950\pi\)
0.514575 + 0.857445i \(0.327950\pi\)
\(998\) 12.4057 0.392696
\(999\) −4.45666 −0.141002
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.1.3 4
7.6 odd 2 6762.2.a.ct.1.1 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6762.2.a.ci.1.3 4 1.1 even 1 trivial
6762.2.a.ct.1.1 yes 4 7.6 odd 2