Properties

Label 6762.2.a.ci.1.1
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 1.1
Root \(-0.360409\) of defining polynomial
Character \(\chi\) \(=\) 6762.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.41421 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.41421 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -3.41421 q^{10} -0.298573 q^{11} -1.00000 q^{12} -4.72082 q^{13} +3.41421 q^{15} +1.00000 q^{16} -3.54925 q^{17} +1.00000 q^{18} +7.98285 q^{19} -3.41421 q^{20} -0.298573 q^{22} +1.00000 q^{23} -1.00000 q^{24} +6.65685 q^{25} -4.72082 q^{26} -1.00000 q^{27} +8.03878 q^{29} +3.41421 q^{30} +9.98285 q^{31} +1.00000 q^{32} +0.298573 q^{33} -3.54925 q^{34} +1.00000 q^{36} -3.54925 q^{37} +7.98285 q^{38} +4.72082 q^{39} -3.41421 q^{40} +0.817068 q^{41} -6.99088 q^{43} -0.298573 q^{44} -3.41421 q^{45} +1.00000 q^{46} -9.38571 q^{47} -1.00000 q^{48} +6.65685 q^{50} +3.54925 q^{51} -4.72082 q^{52} +8.14639 q^{53} -1.00000 q^{54} +1.01939 q^{55} -7.98285 q^{57} +8.03878 q^{58} -10.0388 q^{59} +3.41421 q^{60} -5.18293 q^{61} +9.98285 q^{62} +1.00000 q^{64} +16.1179 q^{65} +0.298573 q^{66} +15.2061 q^{67} -3.54925 q^{68} -1.00000 q^{69} -5.57775 q^{71} +1.00000 q^{72} -0.776751 q^{73} -3.54925 q^{74} -6.65685 q^{75} +7.98285 q^{76} +4.72082 q^{78} -14.0791 q^{79} -3.41421 q^{80} +1.00000 q^{81} +0.817068 q^{82} +0.884359 q^{83} +12.1179 q^{85} -6.99088 q^{86} -8.03878 q^{87} -0.298573 q^{88} -2.43469 q^{89} -3.41421 q^{90} +1.00000 q^{92} -9.98285 q^{93} -9.38571 q^{94} -27.2552 q^{95} -1.00000 q^{96} +12.0345 q^{97} -0.298573 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{2} - 4q^{3} + 4q^{4} - 8q^{5} - 4q^{6} + 4q^{8} + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{2} - 4q^{3} + 4q^{4} - 8q^{5} - 4q^{6} + 4q^{8} + 4q^{9} - 8q^{10} + 4q^{11} - 4q^{12} - 12q^{13} + 8q^{15} + 4q^{16} + 4q^{17} + 4q^{18} - 4q^{19} - 8q^{20} + 4q^{22} + 4q^{23} - 4q^{24} + 4q^{25} - 12q^{26} - 4q^{27} + 8q^{29} + 8q^{30} + 4q^{31} + 4q^{32} - 4q^{33} + 4q^{34} + 4q^{36} + 4q^{37} - 4q^{38} + 12q^{39} - 8q^{40} + 8q^{41} + 4q^{43} + 4q^{44} - 8q^{45} + 4q^{46} - 12q^{47} - 4q^{48} + 4q^{50} - 4q^{51} - 12q^{52} + 4q^{53} - 4q^{54} - 8q^{55} + 4q^{57} + 8q^{58} - 16q^{59} + 8q^{60} - 16q^{61} + 4q^{62} + 4q^{64} + 16q^{65} - 4q^{66} + 20q^{67} + 4q^{68} - 4q^{69} - 24q^{71} + 4q^{72} - 8q^{73} + 4q^{74} - 4q^{75} - 4q^{76} + 12q^{78} - 32q^{79} - 8q^{80} + 4q^{81} + 8q^{82} + 4q^{83} + 4q^{86} - 8q^{87} + 4q^{88} - 20q^{89} - 8q^{90} + 4q^{92} - 4q^{93} - 12q^{94} - 4q^{96} - 4q^{97} + 4q^{99} + O(q^{100}) \)

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
\(5\) −3.41421 −1.52688 −0.763441 0.645877i \(-0.776492\pi\)
−0.763441 + 0.645877i \(0.776492\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −3.41421 −1.07967
\(11\) −0.298573 −0.0900231 −0.0450116 0.998986i \(-0.514332\pi\)
−0.0450116 + 0.998986i \(0.514332\pi\)
\(12\) −1.00000 −0.288675
\(13\) −4.72082 −1.30932 −0.654660 0.755924i \(-0.727188\pi\)
−0.654660 + 0.755924i \(0.727188\pi\)
\(14\) 0 0
\(15\) 3.41421 0.881546
\(16\) 1.00000 0.250000
\(17\) −3.54925 −0.860819 −0.430409 0.902634i \(-0.641631\pi\)
−0.430409 + 0.902634i \(0.641631\pi\)
\(18\) 1.00000 0.235702
\(19\) 7.98285 1.83139 0.915696 0.401872i \(-0.131640\pi\)
0.915696 + 0.401872i \(0.131640\pi\)
\(20\) −3.41421 −0.763441
\(21\) 0 0
\(22\) −0.298573 −0.0636559
\(23\) 1.00000 0.208514
\(24\) −1.00000 −0.204124
\(25\) 6.65685 1.33137
\(26\) −4.72082 −0.925829
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 8.03878 1.49276 0.746382 0.665517i \(-0.231789\pi\)
0.746382 + 0.665517i \(0.231789\pi\)
\(30\) 3.41421 0.623347
\(31\) 9.98285 1.79297 0.896486 0.443071i \(-0.146111\pi\)
0.896486 + 0.443071i \(0.146111\pi\)
\(32\) 1.00000 0.176777
\(33\) 0.298573 0.0519749
\(34\) −3.54925 −0.608691
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −3.54925 −0.583492 −0.291746 0.956496i \(-0.594236\pi\)
−0.291746 + 0.956496i \(0.594236\pi\)
\(38\) 7.98285 1.29499
\(39\) 4.72082 0.755936
\(40\) −3.41421 −0.539835
\(41\) 0.817068 0.127605 0.0638023 0.997963i \(-0.479677\pi\)
0.0638023 + 0.997963i \(0.479677\pi\)
\(42\) 0 0
\(43\) −6.99088 −1.06610 −0.533050 0.846084i \(-0.678954\pi\)
−0.533050 + 0.846084i \(0.678954\pi\)
\(44\) −0.298573 −0.0450116
\(45\) −3.41421 −0.508961
\(46\) 1.00000 0.147442
\(47\) −9.38571 −1.36905 −0.684523 0.728991i \(-0.739989\pi\)
−0.684523 + 0.728991i \(0.739989\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) 6.65685 0.941421
\(51\) 3.54925 0.496994
\(52\) −4.72082 −0.654660
\(53\) 8.14639 1.11899 0.559496 0.828833i \(-0.310995\pi\)
0.559496 + 0.828833i \(0.310995\pi\)
\(54\) −1.00000 −0.136083
\(55\) 1.01939 0.137455
\(56\) 0 0
\(57\) −7.98285 −1.05735
\(58\) 8.03878 1.05554
\(59\) −10.0388 −1.30694 −0.653469 0.756953i \(-0.726687\pi\)
−0.653469 + 0.756953i \(0.726687\pi\)
\(60\) 3.41421 0.440773
\(61\) −5.18293 −0.663606 −0.331803 0.943349i \(-0.607657\pi\)
−0.331803 + 0.943349i \(0.607657\pi\)
\(62\) 9.98285 1.26782
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 16.1179 1.99918
\(66\) 0.298573 0.0367518
\(67\) 15.2061 1.85772 0.928860 0.370430i \(-0.120790\pi\)
0.928860 + 0.370430i \(0.120790\pi\)
\(68\) −3.54925 −0.430409
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −5.57775 −0.661958 −0.330979 0.943638i \(-0.607379\pi\)
−0.330979 + 0.943638i \(0.607379\pi\)
\(72\) 1.00000 0.117851
\(73\) −0.776751 −0.0909118 −0.0454559 0.998966i \(-0.514474\pi\)
−0.0454559 + 0.998966i \(0.514474\pi\)
\(74\) −3.54925 −0.412591
\(75\) −6.65685 −0.768667
\(76\) 7.98285 0.915696
\(77\) 0 0
\(78\) 4.72082 0.534527
\(79\) −14.0791 −1.58402 −0.792011 0.610506i \(-0.790966\pi\)
−0.792011 + 0.610506i \(0.790966\pi\)
\(80\) −3.41421 −0.381721
\(81\) 1.00000 0.111111
\(82\) 0.817068 0.0902300
\(83\) 0.884359 0.0970710 0.0485355 0.998821i \(-0.484545\pi\)
0.0485355 + 0.998821i \(0.484545\pi\)
\(84\) 0 0
\(85\) 12.1179 1.31437
\(86\) −6.99088 −0.753846
\(87\) −8.03878 −0.861848
\(88\) −0.298573 −0.0318280
\(89\) −2.43469 −0.258077 −0.129038 0.991640i \(-0.541189\pi\)
−0.129038 + 0.991640i \(0.541189\pi\)
\(90\) −3.41421 −0.359890
\(91\) 0 0
\(92\) 1.00000 0.104257
\(93\) −9.98285 −1.03517
\(94\) −9.38571 −0.968062
\(95\) −27.2552 −2.79632
\(96\) −1.00000 −0.102062
\(97\) 12.0345 1.22192 0.610961 0.791661i \(-0.290783\pi\)
0.610961 + 0.791661i \(0.290783\pi\)
\(98\) 0 0
\(99\) −0.298573 −0.0300077
\(100\) 6.65685 0.665685
\(101\) 4.99088 0.496611 0.248306 0.968682i \(-0.420126\pi\)
0.248306 + 0.968682i \(0.420126\pi\)
\(102\) 3.54925 0.351428
\(103\) 8.86721 0.873712 0.436856 0.899531i \(-0.356092\pi\)
0.436856 + 0.899531i \(0.356092\pi\)
\(104\) −4.72082 −0.462914
\(105\) 0 0
\(106\) 8.14639 0.791247
\(107\) −14.4956 −1.40134 −0.700669 0.713486i \(-0.747115\pi\)
−0.700669 + 0.713486i \(0.747115\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 10.0345 0.961133 0.480567 0.876958i \(-0.340431\pi\)
0.480567 + 0.876958i \(0.340431\pi\)
\(110\) 1.01939 0.0971952
\(111\) 3.54925 0.336880
\(112\) 0 0
\(113\) −15.0194 −1.41291 −0.706453 0.707760i \(-0.749706\pi\)
−0.706453 + 0.707760i \(0.749706\pi\)
\(114\) −7.98285 −0.747662
\(115\) −3.41421 −0.318377
\(116\) 8.03878 0.746382
\(117\) −4.72082 −0.436440
\(118\) −10.0388 −0.924145
\(119\) 0 0
\(120\) 3.41421 0.311674
\(121\) −10.9109 −0.991896
\(122\) −5.18293 −0.469241
\(123\) −0.817068 −0.0736725
\(124\) 9.98285 0.896486
\(125\) −5.65685 −0.505964
\(126\) 0 0
\(127\) −11.4820 −1.01886 −0.509429 0.860512i \(-0.670143\pi\)
−0.509429 + 0.860512i \(0.670143\pi\)
\(128\) 1.00000 0.0883883
\(129\) 6.99088 0.615513
\(130\) 16.1179 1.41363
\(131\) −14.7941 −1.29257 −0.646285 0.763097i \(-0.723678\pi\)
−0.646285 + 0.763097i \(0.723678\pi\)
\(132\) 0.298573 0.0259874
\(133\) 0 0
\(134\) 15.2061 1.31361
\(135\) 3.41421 0.293849
\(136\) −3.54925 −0.304345
\(137\) −2.90151 −0.247893 −0.123946 0.992289i \(-0.539555\pi\)
−0.123946 + 0.992289i \(0.539555\pi\)
\(138\) −1.00000 −0.0851257
\(139\) −5.17157 −0.438647 −0.219324 0.975652i \(-0.570385\pi\)
−0.219324 + 0.975652i \(0.570385\pi\)
\(140\) 0 0
\(141\) 9.38571 0.790419
\(142\) −5.57775 −0.468075
\(143\) 1.40951 0.117869
\(144\) 1.00000 0.0833333
\(145\) −27.4461 −2.27928
\(146\) −0.776751 −0.0642844
\(147\) 0 0
\(148\) −3.54925 −0.291746
\(149\) 7.78053 0.637406 0.318703 0.947855i \(-0.396753\pi\)
0.318703 + 0.947855i \(0.396753\pi\)
\(150\) −6.65685 −0.543530
\(151\) −20.6192 −1.67797 −0.838985 0.544155i \(-0.816850\pi\)
−0.838985 + 0.544155i \(0.816850\pi\)
\(152\) 7.98285 0.647495
\(153\) −3.54925 −0.286940
\(154\) 0 0
\(155\) −34.0836 −2.73766
\(156\) 4.72082 0.377968
\(157\) −22.9577 −1.83222 −0.916111 0.400925i \(-0.868689\pi\)
−0.916111 + 0.400925i \(0.868689\pi\)
\(158\) −14.0791 −1.12007
\(159\) −8.14639 −0.646051
\(160\) −3.41421 −0.269917
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) −4.63746 −0.363234 −0.181617 0.983369i \(-0.558133\pi\)
−0.181617 + 0.983369i \(0.558133\pi\)
\(164\) 0.817068 0.0638023
\(165\) −1.01939 −0.0793595
\(166\) 0.884359 0.0686396
\(167\) −22.1981 −1.71774 −0.858869 0.512195i \(-0.828832\pi\)
−0.858869 + 0.512195i \(0.828832\pi\)
\(168\) 0 0
\(169\) 9.28613 0.714318
\(170\) 12.1179 0.929399
\(171\) 7.98285 0.610464
\(172\) −6.99088 −0.533050
\(173\) −3.16732 −0.240807 −0.120403 0.992725i \(-0.538419\pi\)
−0.120403 + 0.992725i \(0.538419\pi\)
\(174\) −8.03878 −0.609419
\(175\) 0 0
\(176\) −0.298573 −0.0225058
\(177\) 10.0388 0.754561
\(178\) −2.43469 −0.182488
\(179\) 13.5998 1.01650 0.508250 0.861210i \(-0.330293\pi\)
0.508250 + 0.861210i \(0.330293\pi\)
\(180\) −3.41421 −0.254480
\(181\) −9.64549 −0.716944 −0.358472 0.933540i \(-0.616702\pi\)
−0.358472 + 0.933540i \(0.616702\pi\)
\(182\) 0 0
\(183\) 5.18293 0.383133
\(184\) 1.00000 0.0737210
\(185\) 12.1179 0.890924
\(186\) −9.98285 −0.731978
\(187\) 1.05971 0.0774936
\(188\) −9.38571 −0.684523
\(189\) 0 0
\(190\) −27.2552 −1.97730
\(191\) −2.13612 −0.154564 −0.0772820 0.997009i \(-0.524624\pi\)
−0.0772820 + 0.997009i \(0.524624\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 21.9496 1.57997 0.789985 0.613127i \(-0.210088\pi\)
0.789985 + 0.613127i \(0.210088\pi\)
\(194\) 12.0345 0.864029
\(195\) −16.1179 −1.15423
\(196\) 0 0
\(197\) 6.17490 0.439943 0.219972 0.975506i \(-0.429403\pi\)
0.219972 + 0.975506i \(0.429403\pi\)
\(198\) −0.298573 −0.0212186
\(199\) 21.4644 1.52157 0.760784 0.649005i \(-0.224815\pi\)
0.760784 + 0.649005i \(0.224815\pi\)
\(200\) 6.65685 0.470711
\(201\) −15.2061 −1.07256
\(202\) 4.99088 0.351157
\(203\) 0 0
\(204\) 3.54925 0.248497
\(205\) −2.78964 −0.194837
\(206\) 8.86721 0.617808
\(207\) 1.00000 0.0695048
\(208\) −4.72082 −0.327330
\(209\) −2.38346 −0.164868
\(210\) 0 0
\(211\) −9.23461 −0.635737 −0.317868 0.948135i \(-0.602967\pi\)
−0.317868 + 0.948135i \(0.602967\pi\)
\(212\) 8.14639 0.559496
\(213\) 5.57775 0.382181
\(214\) −14.4956 −0.990896
\(215\) 23.8684 1.62781
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) 10.0345 0.679624
\(219\) 0.776751 0.0524880
\(220\) 1.01939 0.0687274
\(221\) 16.7553 1.12709
\(222\) 3.54925 0.238210
\(223\) −17.0265 −1.14018 −0.570089 0.821583i \(-0.693091\pi\)
−0.570089 + 0.821583i \(0.693091\pi\)
\(224\) 0 0
\(225\) 6.65685 0.443790
\(226\) −15.0194 −0.999075
\(227\) 24.3563 1.61658 0.808291 0.588783i \(-0.200393\pi\)
0.808291 + 0.588783i \(0.200393\pi\)
\(228\) −7.98285 −0.528677
\(229\) 24.1923 1.59867 0.799335 0.600885i \(-0.205185\pi\)
0.799335 + 0.600885i \(0.205185\pi\)
\(230\) −3.41421 −0.225127
\(231\) 0 0
\(232\) 8.03878 0.527772
\(233\) −21.9109 −1.43543 −0.717714 0.696338i \(-0.754811\pi\)
−0.717714 + 0.696338i \(0.754811\pi\)
\(234\) −4.72082 −0.308610
\(235\) 32.0448 2.09037
\(236\) −10.0388 −0.653469
\(237\) 14.0791 0.914536
\(238\) 0 0
\(239\) −6.47926 −0.419109 −0.209554 0.977797i \(-0.567201\pi\)
−0.209554 + 0.977797i \(0.567201\pi\)
\(240\) 3.41421 0.220387
\(241\) 8.52832 0.549357 0.274679 0.961536i \(-0.411429\pi\)
0.274679 + 0.961536i \(0.411429\pi\)
\(242\) −10.9109 −0.701376
\(243\) −1.00000 −0.0641500
\(244\) −5.18293 −0.331803
\(245\) 0 0
\(246\) −0.817068 −0.0520943
\(247\) −37.6856 −2.39788
\(248\) 9.98285 0.633912
\(249\) −0.884359 −0.0560440
\(250\) −5.65685 −0.357771
\(251\) −12.5024 −0.789146 −0.394573 0.918865i \(-0.629108\pi\)
−0.394573 + 0.918865i \(0.629108\pi\)
\(252\) 0 0
\(253\) −0.298573 −0.0187711
\(254\) −11.4820 −0.720442
\(255\) −12.1179 −0.758851
\(256\) 1.00000 0.0625000
\(257\) −5.79614 −0.361553 −0.180777 0.983524i \(-0.557861\pi\)
−0.180777 + 0.983524i \(0.557861\pi\)
\(258\) 6.99088 0.435233
\(259\) 0 0
\(260\) 16.1179 0.999589
\(261\) 8.03878 0.497588
\(262\) −14.7941 −0.913984
\(263\) −8.68629 −0.535620 −0.267810 0.963472i \(-0.586300\pi\)
−0.267810 + 0.963472i \(0.586300\pi\)
\(264\) 0.298573 0.0183759
\(265\) −27.8135 −1.70857
\(266\) 0 0
\(267\) 2.43469 0.149001
\(268\) 15.2061 0.928860
\(269\) −28.4713 −1.73593 −0.867963 0.496629i \(-0.834571\pi\)
−0.867963 + 0.496629i \(0.834571\pi\)
\(270\) 3.41421 0.207782
\(271\) 2.46211 0.149563 0.0747814 0.997200i \(-0.476174\pi\)
0.0747814 + 0.997200i \(0.476174\pi\)
\(272\) −3.54925 −0.215205
\(273\) 0 0
\(274\) −2.90151 −0.175287
\(275\) −1.98756 −0.119854
\(276\) −1.00000 −0.0601929
\(277\) −23.5192 −1.41313 −0.706566 0.707647i \(-0.749757\pi\)
−0.706566 + 0.707647i \(0.749757\pi\)
\(278\) −5.17157 −0.310170
\(279\) 9.98285 0.597658
\(280\) 0 0
\(281\) −2.81236 −0.167771 −0.0838857 0.996475i \(-0.526733\pi\)
−0.0838857 + 0.996475i \(0.526733\pi\)
\(282\) 9.38571 0.558911
\(283\) 17.6800 1.05097 0.525484 0.850803i \(-0.323884\pi\)
0.525484 + 0.850803i \(0.323884\pi\)
\(284\) −5.57775 −0.330979
\(285\) 27.2552 1.61446
\(286\) 1.40951 0.0833460
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) −4.40285 −0.258991
\(290\) −27.4461 −1.61169
\(291\) −12.0345 −0.705476
\(292\) −0.776751 −0.0454559
\(293\) 8.97373 0.524251 0.262126 0.965034i \(-0.415577\pi\)
0.262126 + 0.965034i \(0.415577\pi\)
\(294\) 0 0
\(295\) 34.2745 1.99554
\(296\) −3.54925 −0.206296
\(297\) 0.298573 0.0173250
\(298\) 7.78053 0.450714
\(299\) −4.72082 −0.273012
\(300\) −6.65685 −0.384334
\(301\) 0 0
\(302\) −20.6192 −1.18650
\(303\) −4.99088 −0.286719
\(304\) 7.98285 0.457848
\(305\) 17.6956 1.01325
\(306\) −3.54925 −0.202897
\(307\) −5.49648 −0.313701 −0.156850 0.987622i \(-0.550134\pi\)
−0.156850 + 0.987622i \(0.550134\pi\)
\(308\) 0 0
\(309\) −8.86721 −0.504438
\(310\) −34.0836 −1.93582
\(311\) 20.0277 1.13566 0.567832 0.823144i \(-0.307782\pi\)
0.567832 + 0.823144i \(0.307782\pi\)
\(312\) 4.72082 0.267264
\(313\) 18.4004 1.04005 0.520026 0.854151i \(-0.325922\pi\)
0.520026 + 0.854151i \(0.325922\pi\)
\(314\) −22.9577 −1.29558
\(315\) 0 0
\(316\) −14.0791 −0.792011
\(317\) −21.6468 −1.21581 −0.607903 0.794011i \(-0.707989\pi\)
−0.607903 + 0.794011i \(0.707989\pi\)
\(318\) −8.14639 −0.456827
\(319\) −2.40016 −0.134383
\(320\) −3.41421 −0.190860
\(321\) 14.4956 0.809063
\(322\) 0 0
\(323\) −28.3331 −1.57650
\(324\) 1.00000 0.0555556
\(325\) −31.4258 −1.74319
\(326\) −4.63746 −0.256845
\(327\) −10.0345 −0.554911
\(328\) 0.817068 0.0451150
\(329\) 0 0
\(330\) −1.01939 −0.0561157
\(331\) 6.34315 0.348651 0.174325 0.984688i \(-0.444226\pi\)
0.174325 + 0.984688i \(0.444226\pi\)
\(332\) 0.884359 0.0485355
\(333\) −3.54925 −0.194497
\(334\) −22.1981 −1.21462
\(335\) −51.9169 −2.83652
\(336\) 0 0
\(337\) −10.7150 −0.583685 −0.291842 0.956466i \(-0.594268\pi\)
−0.291842 + 0.956466i \(0.594268\pi\)
\(338\) 9.28613 0.505099
\(339\) 15.0194 0.815741
\(340\) 12.1179 0.657184
\(341\) −2.98061 −0.161409
\(342\) 7.98285 0.431663
\(343\) 0 0
\(344\) −6.99088 −0.376923
\(345\) 3.41421 0.183815
\(346\) −3.16732 −0.170276
\(347\) 22.9120 1.22998 0.614991 0.788534i \(-0.289160\pi\)
0.614991 + 0.788534i \(0.289160\pi\)
\(348\) −8.03878 −0.430924
\(349\) 11.4922 0.615165 0.307582 0.951521i \(-0.400480\pi\)
0.307582 + 0.951521i \(0.400480\pi\)
\(350\) 0 0
\(351\) 4.72082 0.251979
\(352\) −0.298573 −0.0159140
\(353\) −10.9510 −0.582864 −0.291432 0.956592i \(-0.594132\pi\)
−0.291432 + 0.956592i \(0.594132\pi\)
\(354\) 10.0388 0.533555
\(355\) 19.0436 1.01073
\(356\) −2.43469 −0.129038
\(357\) 0 0
\(358\) 13.5998 0.718774
\(359\) −4.63746 −0.244756 −0.122378 0.992484i \(-0.539052\pi\)
−0.122378 + 0.992484i \(0.539052\pi\)
\(360\) −3.41421 −0.179945
\(361\) 44.7259 2.35400
\(362\) −9.64549 −0.506956
\(363\) 10.9109 0.572671
\(364\) 0 0
\(365\) 2.65199 0.138812
\(366\) 5.18293 0.270916
\(367\) −15.5838 −0.813466 −0.406733 0.913547i \(-0.633332\pi\)
−0.406733 + 0.913547i \(0.633332\pi\)
\(368\) 1.00000 0.0521286
\(369\) 0.817068 0.0425349
\(370\) 12.1179 0.629979
\(371\) 0 0
\(372\) −9.98285 −0.517587
\(373\) −18.4650 −0.956079 −0.478040 0.878338i \(-0.658653\pi\)
−0.478040 + 0.878338i \(0.658653\pi\)
\(374\) 1.05971 0.0547962
\(375\) 5.65685 0.292119
\(376\) −9.38571 −0.484031
\(377\) −37.9496 −1.95451
\(378\) 0 0
\(379\) 25.2691 1.29799 0.648994 0.760793i \(-0.275190\pi\)
0.648994 + 0.760793i \(0.275190\pi\)
\(380\) −27.2552 −1.39816
\(381\) 11.4820 0.588238
\(382\) −2.13612 −0.109293
\(383\) −20.6983 −1.05764 −0.528818 0.848735i \(-0.677364\pi\)
−0.528818 + 0.848735i \(0.677364\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 21.9496 1.11721
\(387\) −6.99088 −0.355367
\(388\) 12.0345 0.610961
\(389\) −18.6638 −0.946292 −0.473146 0.880984i \(-0.656882\pi\)
−0.473146 + 0.880984i \(0.656882\pi\)
\(390\) −16.1179 −0.816161
\(391\) −3.54925 −0.179493
\(392\) 0 0
\(393\) 14.7941 0.746265
\(394\) 6.17490 0.311087
\(395\) 48.0691 2.41862
\(396\) −0.298573 −0.0150039
\(397\) 23.4374 1.17629 0.588144 0.808756i \(-0.299859\pi\)
0.588144 + 0.808756i \(0.299859\pi\)
\(398\) 21.4644 1.07591
\(399\) 0 0
\(400\) 6.65685 0.332843
\(401\) −6.24735 −0.311978 −0.155989 0.987759i \(-0.549856\pi\)
−0.155989 + 0.987759i \(0.549856\pi\)
\(402\) −15.2061 −0.758411
\(403\) −47.1272 −2.34757
\(404\) 4.99088 0.248306
\(405\) −3.41421 −0.169654
\(406\) 0 0
\(407\) 1.05971 0.0525278
\(408\) 3.54925 0.175714
\(409\) −29.9222 −1.47956 −0.739779 0.672850i \(-0.765070\pi\)
−0.739779 + 0.672850i \(0.765070\pi\)
\(410\) −2.78964 −0.137771
\(411\) 2.90151 0.143121
\(412\) 8.86721 0.436856
\(413\) 0 0
\(414\) 1.00000 0.0491473
\(415\) −3.01939 −0.148216
\(416\) −4.72082 −0.231457
\(417\) 5.17157 0.253253
\(418\) −2.38346 −0.116579
\(419\) −29.5827 −1.44521 −0.722604 0.691262i \(-0.757055\pi\)
−0.722604 + 0.691262i \(0.757055\pi\)
\(420\) 0 0
\(421\) 0.970334 0.0472912 0.0236456 0.999720i \(-0.492473\pi\)
0.0236456 + 0.999720i \(0.492473\pi\)
\(422\) −9.23461 −0.449534
\(423\) −9.38571 −0.456349
\(424\) 8.14639 0.395624
\(425\) −23.6268 −1.14607
\(426\) 5.57775 0.270243
\(427\) 0 0
\(428\) −14.4956 −0.700669
\(429\) −1.40951 −0.0680517
\(430\) 23.8684 1.15103
\(431\) 23.2246 1.11869 0.559344 0.828936i \(-0.311053\pi\)
0.559344 + 0.828936i \(0.311053\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −5.94724 −0.285806 −0.142903 0.989737i \(-0.545644\pi\)
−0.142903 + 0.989737i \(0.545644\pi\)
\(434\) 0 0
\(435\) 27.4461 1.31594
\(436\) 10.0345 0.480567
\(437\) 7.98285 0.381872
\(438\) 0.776751 0.0371146
\(439\) 17.8080 0.849927 0.424964 0.905210i \(-0.360287\pi\)
0.424964 + 0.905210i \(0.360287\pi\)
\(440\) 1.01939 0.0485976
\(441\) 0 0
\(442\) 16.7553 0.796971
\(443\) 27.5983 1.31123 0.655617 0.755093i \(-0.272408\pi\)
0.655617 + 0.755093i \(0.272408\pi\)
\(444\) 3.54925 0.168440
\(445\) 8.31255 0.394053
\(446\) −17.0265 −0.806228
\(447\) −7.78053 −0.368006
\(448\) 0 0
\(449\) −36.9221 −1.74246 −0.871230 0.490875i \(-0.836677\pi\)
−0.871230 + 0.490875i \(0.836677\pi\)
\(450\) 6.65685 0.313807
\(451\) −0.243954 −0.0114874
\(452\) −15.0194 −0.706453
\(453\) 20.6192 0.968776
\(454\) 24.3563 1.14310
\(455\) 0 0
\(456\) −7.98285 −0.373831
\(457\) −14.5971 −0.682826 −0.341413 0.939913i \(-0.610905\pi\)
−0.341413 + 0.939913i \(0.610905\pi\)
\(458\) 24.1923 1.13043
\(459\) 3.54925 0.165665
\(460\) −3.41421 −0.159189
\(461\) 3.08668 0.143761 0.0718806 0.997413i \(-0.477100\pi\)
0.0718806 + 0.997413i \(0.477100\pi\)
\(462\) 0 0
\(463\) −35.1190 −1.63212 −0.816060 0.577967i \(-0.803846\pi\)
−0.816060 + 0.577967i \(0.803846\pi\)
\(464\) 8.03878 0.373191
\(465\) 34.0836 1.58059
\(466\) −21.9109 −1.01500
\(467\) −0.462114 −0.0213841 −0.0106920 0.999943i \(-0.503403\pi\)
−0.0106920 + 0.999943i \(0.503403\pi\)
\(468\) −4.72082 −0.218220
\(469\) 0 0
\(470\) 32.0448 1.47812
\(471\) 22.9577 1.05783
\(472\) −10.0388 −0.462072
\(473\) 2.08729 0.0959736
\(474\) 14.0791 0.646674
\(475\) 53.1407 2.43826
\(476\) 0 0
\(477\) 8.14639 0.372998
\(478\) −6.47926 −0.296355
\(479\) −19.3115 −0.882367 −0.441183 0.897417i \(-0.645441\pi\)
−0.441183 + 0.897417i \(0.645441\pi\)
\(480\) 3.41421 0.155837
\(481\) 16.7553 0.763978
\(482\) 8.52832 0.388454
\(483\) 0 0
\(484\) −10.9109 −0.495948
\(485\) −41.0884 −1.86573
\(486\) −1.00000 −0.0453609
\(487\) −10.4692 −0.474406 −0.237203 0.971460i \(-0.576231\pi\)
−0.237203 + 0.971460i \(0.576231\pi\)
\(488\) −5.18293 −0.234620
\(489\) 4.63746 0.209713
\(490\) 0 0
\(491\) 12.2846 0.554396 0.277198 0.960813i \(-0.410594\pi\)
0.277198 + 0.960813i \(0.410594\pi\)
\(492\) −0.817068 −0.0368363
\(493\) −28.5316 −1.28500
\(494\) −37.6856 −1.69555
\(495\) 1.01939 0.0458182
\(496\) 9.98285 0.448243
\(497\) 0 0
\(498\) −0.884359 −0.0396291
\(499\) −6.37188 −0.285245 −0.142622 0.989777i \(-0.545553\pi\)
−0.142622 + 0.989777i \(0.545553\pi\)
\(500\) −5.65685 −0.252982
\(501\) 22.1981 0.991737
\(502\) −12.5024 −0.558011
\(503\) −9.94964 −0.443632 −0.221816 0.975088i \(-0.571198\pi\)
−0.221816 + 0.975088i \(0.571198\pi\)
\(504\) 0 0
\(505\) −17.0399 −0.758267
\(506\) −0.298573 −0.0132732
\(507\) −9.28613 −0.412411
\(508\) −11.4820 −0.509429
\(509\) 0.743537 0.0329567 0.0164784 0.999864i \(-0.494755\pi\)
0.0164784 + 0.999864i \(0.494755\pi\)
\(510\) −12.1179 −0.536589
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) −7.98285 −0.352451
\(514\) −5.79614 −0.255657
\(515\) −30.2745 −1.33406
\(516\) 6.99088 0.307756
\(517\) 2.80232 0.123246
\(518\) 0 0
\(519\) 3.16732 0.139030
\(520\) 16.1179 0.706816
\(521\) −12.6317 −0.553404 −0.276702 0.960956i \(-0.589241\pi\)
−0.276702 + 0.960956i \(0.589241\pi\)
\(522\) 8.03878 0.351848
\(523\) −13.9441 −0.609732 −0.304866 0.952395i \(-0.598612\pi\)
−0.304866 + 0.952395i \(0.598612\pi\)
\(524\) −14.7941 −0.646285
\(525\) 0 0
\(526\) −8.68629 −0.378740
\(527\) −35.4316 −1.54342
\(528\) 0.298573 0.0129937
\(529\) 1.00000 0.0434783
\(530\) −27.8135 −1.20814
\(531\) −10.0388 −0.435646
\(532\) 0 0
\(533\) −3.85723 −0.167075
\(534\) 2.43469 0.105359
\(535\) 49.4909 2.13968
\(536\) 15.2061 0.656803
\(537\) −13.5998 −0.586876
\(538\) −28.4713 −1.22749
\(539\) 0 0
\(540\) 3.41421 0.146924
\(541\) 12.8263 0.551444 0.275722 0.961237i \(-0.411083\pi\)
0.275722 + 0.961237i \(0.411083\pi\)
\(542\) 2.46211 0.105757
\(543\) 9.64549 0.413928
\(544\) −3.54925 −0.152173
\(545\) −34.2600 −1.46754
\(546\) 0 0
\(547\) −11.4234 −0.488429 −0.244215 0.969721i \(-0.578530\pi\)
−0.244215 + 0.969721i \(0.578530\pi\)
\(548\) −2.90151 −0.123946
\(549\) −5.18293 −0.221202
\(550\) −1.98756 −0.0847497
\(551\) 64.1724 2.73384
\(552\) −1.00000 −0.0425628
\(553\) 0 0
\(554\) −23.5192 −0.999235
\(555\) −12.1179 −0.514375
\(556\) −5.17157 −0.219324
\(557\) −24.7026 −1.04668 −0.523341 0.852123i \(-0.675315\pi\)
−0.523341 + 0.852123i \(0.675315\pi\)
\(558\) 9.98285 0.422608
\(559\) 33.0027 1.39587
\(560\) 0 0
\(561\) −1.05971 −0.0447409
\(562\) −2.81236 −0.118632
\(563\) −36.6658 −1.54528 −0.772639 0.634846i \(-0.781063\pi\)
−0.772639 + 0.634846i \(0.781063\pi\)
\(564\) 9.38571 0.395209
\(565\) 51.2794 2.15734
\(566\) 17.6800 0.743147
\(567\) 0 0
\(568\) −5.57775 −0.234037
\(569\) −29.9430 −1.25528 −0.627638 0.778506i \(-0.715978\pi\)
−0.627638 + 0.778506i \(0.715978\pi\)
\(570\) 27.2552 1.14159
\(571\) 27.2837 1.14179 0.570893 0.821025i \(-0.306597\pi\)
0.570893 + 0.821025i \(0.306597\pi\)
\(572\) 1.40951 0.0589345
\(573\) 2.13612 0.0892376
\(574\) 0 0
\(575\) 6.65685 0.277610
\(576\) 1.00000 0.0416667
\(577\) 6.84766 0.285072 0.142536 0.989790i \(-0.454474\pi\)
0.142536 + 0.989790i \(0.454474\pi\)
\(578\) −4.40285 −0.183135
\(579\) −21.9496 −0.912196
\(580\) −27.4461 −1.13964
\(581\) 0 0
\(582\) −12.0345 −0.498847
\(583\) −2.43229 −0.100735
\(584\) −0.776751 −0.0321422
\(585\) 16.1179 0.666392
\(586\) 8.97373 0.370702
\(587\) −3.17823 −0.131179 −0.0655897 0.997847i \(-0.520893\pi\)
−0.0655897 + 0.997847i \(0.520893\pi\)
\(588\) 0 0
\(589\) 79.6916 3.28364
\(590\) 34.2745 1.41106
\(591\) −6.17490 −0.254001
\(592\) −3.54925 −0.145873
\(593\) 11.6433 0.478134 0.239067 0.971003i \(-0.423158\pi\)
0.239067 + 0.971003i \(0.423158\pi\)
\(594\) 0.298573 0.0122506
\(595\) 0 0
\(596\) 7.78053 0.318703
\(597\) −21.4644 −0.878478
\(598\) −4.72082 −0.193049
\(599\) −21.0027 −0.858147 −0.429073 0.903270i \(-0.641160\pi\)
−0.429073 + 0.903270i \(0.641160\pi\)
\(600\) −6.65685 −0.271765
\(601\) 25.5236 1.04113 0.520564 0.853822i \(-0.325722\pi\)
0.520564 + 0.853822i \(0.325722\pi\)
\(602\) 0 0
\(603\) 15.2061 0.619240
\(604\) −20.6192 −0.838985
\(605\) 37.2520 1.51451
\(606\) −4.99088 −0.202741
\(607\) −38.7124 −1.57129 −0.785644 0.618679i \(-0.787668\pi\)
−0.785644 + 0.618679i \(0.787668\pi\)
\(608\) 7.98285 0.323747
\(609\) 0 0
\(610\) 17.6956 0.716475
\(611\) 44.3082 1.79252
\(612\) −3.54925 −0.143470
\(613\) −4.91780 −0.198628 −0.0993141 0.995056i \(-0.531665\pi\)
−0.0993141 + 0.995056i \(0.531665\pi\)
\(614\) −5.49648 −0.221820
\(615\) 2.78964 0.112489
\(616\) 0 0
\(617\) 0.196983 0.00793024 0.00396512 0.999992i \(-0.498738\pi\)
0.00396512 + 0.999992i \(0.498738\pi\)
\(618\) −8.86721 −0.356692
\(619\) −11.2623 −0.452668 −0.226334 0.974050i \(-0.572674\pi\)
−0.226334 + 0.974050i \(0.572674\pi\)
\(620\) −34.0836 −1.36883
\(621\) −1.00000 −0.0401286
\(622\) 20.0277 0.803036
\(623\) 0 0
\(624\) 4.72082 0.188984
\(625\) −13.9706 −0.558823
\(626\) 18.4004 0.735428
\(627\) 2.38346 0.0951863
\(628\) −22.9577 −0.916111
\(629\) 12.5971 0.502281
\(630\) 0 0
\(631\) 31.4067 1.25028 0.625141 0.780512i \(-0.285042\pi\)
0.625141 + 0.780512i \(0.285042\pi\)
\(632\) −14.0791 −0.560037
\(633\) 9.23461 0.367043
\(634\) −21.6468 −0.859705
\(635\) 39.2018 1.55568
\(636\) −8.14639 −0.323025
\(637\) 0 0
\(638\) −2.40016 −0.0950233
\(639\) −5.57775 −0.220653
\(640\) −3.41421 −0.134959
\(641\) −7.71387 −0.304680 −0.152340 0.988328i \(-0.548681\pi\)
−0.152340 + 0.988328i \(0.548681\pi\)
\(642\) 14.4956 0.572094
\(643\) 10.3260 0.407218 0.203609 0.979052i \(-0.434733\pi\)
0.203609 + 0.979052i \(0.434733\pi\)
\(644\) 0 0
\(645\) −23.8684 −0.939816
\(646\) −28.3331 −1.11475
\(647\) −11.4897 −0.451706 −0.225853 0.974161i \(-0.572517\pi\)
−0.225853 + 0.974161i \(0.572517\pi\)
\(648\) 1.00000 0.0392837
\(649\) 2.99731 0.117655
\(650\) −31.4258 −1.23262
\(651\) 0 0
\(652\) −4.63746 −0.181617
\(653\) −36.4898 −1.42795 −0.713977 0.700169i \(-0.753108\pi\)
−0.713977 + 0.700169i \(0.753108\pi\)
\(654\) −10.0345 −0.392381
\(655\) 50.5103 1.97360
\(656\) 0.817068 0.0319011
\(657\) −0.776751 −0.0303039
\(658\) 0 0
\(659\) 16.5702 0.645482 0.322741 0.946487i \(-0.395396\pi\)
0.322741 + 0.946487i \(0.395396\pi\)
\(660\) −1.01939 −0.0396798
\(661\) −43.7190 −1.70047 −0.850237 0.526401i \(-0.823541\pi\)
−0.850237 + 0.526401i \(0.823541\pi\)
\(662\) 6.34315 0.246533
\(663\) −16.7553 −0.650724
\(664\) 0.884359 0.0343198
\(665\) 0 0
\(666\) −3.54925 −0.137530
\(667\) 8.03878 0.311263
\(668\) −22.1981 −0.858869
\(669\) 17.0265 0.658282
\(670\) −51.9169 −2.00572
\(671\) 1.54748 0.0597399
\(672\) 0 0
\(673\) 40.2745 1.55247 0.776235 0.630444i \(-0.217127\pi\)
0.776235 + 0.630444i \(0.217127\pi\)
\(674\) −10.7150 −0.412727
\(675\) −6.65685 −0.256222
\(676\) 9.28613 0.357159
\(677\) 27.9543 1.07437 0.537186 0.843464i \(-0.319487\pi\)
0.537186 + 0.843464i \(0.319487\pi\)
\(678\) 15.0194 0.576816
\(679\) 0 0
\(680\) 12.1179 0.464700
\(681\) −24.3563 −0.933334
\(682\) −2.98061 −0.114133
\(683\) 25.8554 0.989328 0.494664 0.869084i \(-0.335291\pi\)
0.494664 + 0.869084i \(0.335291\pi\)
\(684\) 7.98285 0.305232
\(685\) 9.90637 0.378503
\(686\) 0 0
\(687\) −24.1923 −0.922993
\(688\) −6.99088 −0.266525
\(689\) −38.4576 −1.46512
\(690\) 3.41421 0.129977
\(691\) 5.83777 0.222079 0.111040 0.993816i \(-0.464582\pi\)
0.111040 + 0.993816i \(0.464582\pi\)
\(692\) −3.16732 −0.120403
\(693\) 0 0
\(694\) 22.9120 0.869728
\(695\) 17.6569 0.669763
\(696\) −8.03878 −0.304709
\(697\) −2.89997 −0.109844
\(698\) 11.4922 0.434987
\(699\) 21.9109 0.828745
\(700\) 0 0
\(701\) 25.0394 0.945725 0.472862 0.881136i \(-0.343221\pi\)
0.472862 + 0.881136i \(0.343221\pi\)
\(702\) 4.72082 0.178176
\(703\) −28.3331 −1.06860
\(704\) −0.298573 −0.0112529
\(705\) −32.0448 −1.20688
\(706\) −10.9510 −0.412147
\(707\) 0 0
\(708\) 10.0388 0.377281
\(709\) 6.97699 0.262026 0.131013 0.991381i \(-0.458177\pi\)
0.131013 + 0.991381i \(0.458177\pi\)
\(710\) 19.0436 0.714695
\(711\) −14.0791 −0.528008
\(712\) −2.43469 −0.0912439
\(713\) 9.98285 0.373861
\(714\) 0 0
\(715\) −4.81236 −0.179972
\(716\) 13.5998 0.508250
\(717\) 6.47926 0.241972
\(718\) −4.63746 −0.173069
\(719\) 7.63369 0.284688 0.142344 0.989817i \(-0.454536\pi\)
0.142344 + 0.989817i \(0.454536\pi\)
\(720\) −3.41421 −0.127240
\(721\) 0 0
\(722\) 44.7259 1.66453
\(723\) −8.52832 −0.317172
\(724\) −9.64549 −0.358472
\(725\) 53.5130 1.98742
\(726\) 10.9109 0.404940
\(727\) −9.08243 −0.336849 −0.168424 0.985715i \(-0.553868\pi\)
−0.168424 + 0.985715i \(0.553868\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 2.65199 0.0981547
\(731\) 24.8124 0.917718
\(732\) 5.18293 0.191567
\(733\) −39.1874 −1.44742 −0.723710 0.690104i \(-0.757565\pi\)
−0.723710 + 0.690104i \(0.757565\pi\)
\(734\) −15.5838 −0.575208
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) −4.54013 −0.167238
\(738\) 0.817068 0.0300767
\(739\) 5.78325 0.212740 0.106370 0.994327i \(-0.466077\pi\)
0.106370 + 0.994327i \(0.466077\pi\)
\(740\) 12.1179 0.445462
\(741\) 37.6856 1.38441
\(742\) 0 0
\(743\) −30.6192 −1.12331 −0.561655 0.827371i \(-0.689835\pi\)
−0.561655 + 0.827371i \(0.689835\pi\)
\(744\) −9.98285 −0.365989
\(745\) −26.5644 −0.973244
\(746\) −18.4650 −0.676050
\(747\) 0.884359 0.0323570
\(748\) 1.05971 0.0387468
\(749\) 0 0
\(750\) 5.65685 0.206559
\(751\) 7.75265 0.282898 0.141449 0.989946i \(-0.454824\pi\)
0.141449 + 0.989946i \(0.454824\pi\)
\(752\) −9.38571 −0.342261
\(753\) 12.5024 0.455614
\(754\) −37.9496 −1.38204
\(755\) 70.3985 2.56206
\(756\) 0 0
\(757\) −32.0617 −1.16530 −0.582652 0.812722i \(-0.697985\pi\)
−0.582652 + 0.812722i \(0.697985\pi\)
\(758\) 25.2691 0.917816
\(759\) 0.298573 0.0108375
\(760\) −27.2552 −0.988648
\(761\) −27.4933 −0.996632 −0.498316 0.866996i \(-0.666048\pi\)
−0.498316 + 0.866996i \(0.666048\pi\)
\(762\) 11.4820 0.415947
\(763\) 0 0
\(764\) −2.13612 −0.0772820
\(765\) 12.1179 0.438123
\(766\) −20.6983 −0.747861
\(767\) 47.3913 1.71120
\(768\) −1.00000 −0.0360844
\(769\) 30.3755 1.09537 0.547684 0.836685i \(-0.315509\pi\)
0.547684 + 0.836685i \(0.315509\pi\)
\(770\) 0 0
\(771\) 5.79614 0.208743
\(772\) 21.9496 0.789985
\(773\) 31.6645 1.13889 0.569447 0.822028i \(-0.307158\pi\)
0.569447 + 0.822028i \(0.307158\pi\)
\(774\) −6.99088 −0.251282
\(775\) 66.4544 2.38711
\(776\) 12.0345 0.432014
\(777\) 0 0
\(778\) −18.6638 −0.669130
\(779\) 6.52253 0.233694
\(780\) −16.1179 −0.577113
\(781\) 1.66537 0.0595915
\(782\) −3.54925 −0.126921
\(783\) −8.03878 −0.287283
\(784\) 0 0
\(785\) 78.3824 2.79759
\(786\) 14.7941 0.527689
\(787\) −15.0586 −0.536782 −0.268391 0.963310i \(-0.586492\pi\)
−0.268391 + 0.963310i \(0.586492\pi\)
\(788\) 6.17490 0.219972
\(789\) 8.68629 0.309240
\(790\) 48.0691 1.71022
\(791\) 0 0
\(792\) −0.298573 −0.0106093
\(793\) 24.4677 0.868873
\(794\) 23.4374 0.831762
\(795\) 27.8135 0.986444
\(796\) 21.4644 0.760784
\(797\) 9.68491 0.343057 0.171529 0.985179i \(-0.445129\pi\)
0.171529 + 0.985179i \(0.445129\pi\)
\(798\) 0 0
\(799\) 33.3122 1.17850
\(800\) 6.65685 0.235355
\(801\) −2.43469 −0.0860255
\(802\) −6.24735 −0.220601
\(803\) 0.231917 0.00818416
\(804\) −15.2061 −0.536278
\(805\) 0 0
\(806\) −47.1272 −1.65999
\(807\) 28.4713 1.00224
\(808\) 4.99088 0.175579
\(809\) −34.1649 −1.20117 −0.600586 0.799560i \(-0.705066\pi\)
−0.600586 + 0.799560i \(0.705066\pi\)
\(810\) −3.41421 −0.119963
\(811\) 18.1649 0.637854 0.318927 0.947779i \(-0.396677\pi\)
0.318927 + 0.947779i \(0.396677\pi\)
\(812\) 0 0
\(813\) −2.46211 −0.0863501
\(814\) 1.05971 0.0371428
\(815\) 15.8333 0.554616
\(816\) 3.54925 0.124248
\(817\) −55.8072 −1.95245
\(818\) −29.9222 −1.04621
\(819\) 0 0
\(820\) −2.78964 −0.0974186
\(821\) 7.18610 0.250797 0.125398 0.992106i \(-0.459979\pi\)
0.125398 + 0.992106i \(0.459979\pi\)
\(822\) 2.90151 0.101202
\(823\) −31.5189 −1.09868 −0.549340 0.835599i \(-0.685121\pi\)
−0.549340 + 0.835599i \(0.685121\pi\)
\(824\) 8.86721 0.308904
\(825\) 1.98756 0.0691978
\(826\) 0 0
\(827\) 31.2594 1.08700 0.543498 0.839410i \(-0.317099\pi\)
0.543498 + 0.839410i \(0.317099\pi\)
\(828\) 1.00000 0.0347524
\(829\) 55.9338 1.94266 0.971331 0.237733i \(-0.0764042\pi\)
0.971331 + 0.237733i \(0.0764042\pi\)
\(830\) −3.01939 −0.104805
\(831\) 23.5192 0.815872
\(832\) −4.72082 −0.163665
\(833\) 0 0
\(834\) 5.17157 0.179077
\(835\) 75.7889 2.62278
\(836\) −2.38346 −0.0824338
\(837\) −9.98285 −0.345058
\(838\) −29.5827 −1.02192
\(839\) 28.3249 0.977885 0.488942 0.872316i \(-0.337383\pi\)
0.488942 + 0.872316i \(0.337383\pi\)
\(840\) 0 0
\(841\) 35.6220 1.22835
\(842\) 0.970334 0.0334399
\(843\) 2.81236 0.0968629
\(844\) −9.23461 −0.317868
\(845\) −31.7048 −1.09068
\(846\) −9.38571 −0.322687
\(847\) 0 0
\(848\) 8.14639 0.279748
\(849\) −17.6800 −0.606777
\(850\) −23.6268 −0.810393
\(851\) −3.54925 −0.121667
\(852\) 5.57775 0.191091
\(853\) 29.6914 1.01661 0.508307 0.861176i \(-0.330272\pi\)
0.508307 + 0.861176i \(0.330272\pi\)
\(854\) 0 0
\(855\) −27.2552 −0.932107
\(856\) −14.4956 −0.495448
\(857\) 26.6467 0.910232 0.455116 0.890432i \(-0.349598\pi\)
0.455116 + 0.890432i \(0.349598\pi\)
\(858\) −1.40951 −0.0481198
\(859\) −28.5543 −0.974261 −0.487131 0.873329i \(-0.661956\pi\)
−0.487131 + 0.873329i \(0.661956\pi\)
\(860\) 23.8684 0.813905
\(861\) 0 0
\(862\) 23.2246 0.791032
\(863\) −6.82906 −0.232464 −0.116232 0.993222i \(-0.537082\pi\)
−0.116232 + 0.993222i \(0.537082\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 10.8139 0.367684
\(866\) −5.94724 −0.202095
\(867\) 4.40285 0.149529
\(868\) 0 0
\(869\) 4.20364 0.142599
\(870\) 27.4461 0.930511
\(871\) −71.7852 −2.43235
\(872\) 10.0345 0.339812
\(873\) 12.0345 0.407307
\(874\) 7.98285 0.270024
\(875\) 0 0
\(876\) 0.776751 0.0262440
\(877\) −46.7034 −1.57706 −0.788532 0.614994i \(-0.789158\pi\)
−0.788532 + 0.614994i \(0.789158\pi\)
\(878\) 17.8080 0.600989
\(879\) −8.97373 −0.302677
\(880\) 1.01939 0.0343637
\(881\) −29.0805 −0.979746 −0.489873 0.871794i \(-0.662957\pi\)
−0.489873 + 0.871794i \(0.662957\pi\)
\(882\) 0 0
\(883\) 24.0858 0.810550 0.405275 0.914195i \(-0.367176\pi\)
0.405275 + 0.914195i \(0.367176\pi\)
\(884\) 16.7553 0.563543
\(885\) −34.2745 −1.15213
\(886\) 27.5983 0.927183
\(887\) 48.3629 1.62387 0.811934 0.583749i \(-0.198415\pi\)
0.811934 + 0.583749i \(0.198415\pi\)
\(888\) 3.54925 0.119105
\(889\) 0 0
\(890\) 8.31255 0.278637
\(891\) −0.298573 −0.0100026
\(892\) −17.0265 −0.570089
\(893\) −74.9247 −2.50726
\(894\) −7.78053 −0.260220
\(895\) −46.4327 −1.55208
\(896\) 0 0
\(897\) 4.72082 0.157624
\(898\) −36.9221 −1.23211
\(899\) 80.2500 2.67649
\(900\) 6.65685 0.221895
\(901\) −28.9135 −0.963250
\(902\) −0.243954 −0.00812279
\(903\) 0 0
\(904\) −15.0194 −0.499538
\(905\) 32.9318 1.09469
\(906\) 20.6192 0.685028
\(907\) −35.3606 −1.17413 −0.587065 0.809540i \(-0.699717\pi\)
−0.587065 + 0.809540i \(0.699717\pi\)
\(908\) 24.3563 0.808291
\(909\) 4.99088 0.165537
\(910\) 0 0
\(911\) −30.9926 −1.02683 −0.513416 0.858140i \(-0.671620\pi\)
−0.513416 + 0.858140i \(0.671620\pi\)
\(912\) −7.98285 −0.264339
\(913\) −0.264046 −0.00873864
\(914\) −14.5971 −0.482831
\(915\) −17.6956 −0.585000
\(916\) 24.1923 0.799335
\(917\) 0 0
\(918\) 3.54925 0.117143
\(919\) 27.9717 0.922702 0.461351 0.887218i \(-0.347365\pi\)
0.461351 + 0.887218i \(0.347365\pi\)
\(920\) −3.41421 −0.112563
\(921\) 5.49648 0.181115
\(922\) 3.08668 0.101654
\(923\) 26.3316 0.866714
\(924\) 0 0
\(925\) −23.6268 −0.776845
\(926\) −35.1190 −1.15408
\(927\) 8.86721 0.291237
\(928\) 8.03878 0.263886
\(929\) 22.1696 0.727360 0.363680 0.931524i \(-0.381520\pi\)
0.363680 + 0.931524i \(0.381520\pi\)
\(930\) 34.0836 1.11764
\(931\) 0 0
\(932\) −21.9109 −0.717714
\(933\) −20.0277 −0.655676
\(934\) −0.462114 −0.0151208
\(935\) −3.61807 −0.118324
\(936\) −4.72082 −0.154305
\(937\) 32.8696 1.07380 0.536902 0.843645i \(-0.319595\pi\)
0.536902 + 0.843645i \(0.319595\pi\)
\(938\) 0 0
\(939\) −18.4004 −0.600474
\(940\) 32.0448 1.04519
\(941\) 6.11920 0.199480 0.0997401 0.995014i \(-0.468199\pi\)
0.0997401 + 0.995014i \(0.468199\pi\)
\(942\) 22.9577 0.748001
\(943\) 0.817068 0.0266074
\(944\) −10.0388 −0.326735
\(945\) 0 0
\(946\) 2.08729 0.0678636
\(947\) 14.2943 0.464503 0.232251 0.972656i \(-0.425391\pi\)
0.232251 + 0.972656i \(0.425391\pi\)
\(948\) 14.0791 0.457268
\(949\) 3.66690 0.119033
\(950\) 53.1407 1.72411
\(951\) 21.6468 0.701946
\(952\) 0 0
\(953\) 8.54166 0.276692 0.138346 0.990384i \(-0.455821\pi\)
0.138346 + 0.990384i \(0.455821\pi\)
\(954\) 8.14639 0.263749
\(955\) 7.29316 0.236001
\(956\) −6.47926 −0.209554
\(957\) 2.40016 0.0775862
\(958\) −19.3115 −0.623928
\(959\) 0 0
\(960\) 3.41421 0.110193
\(961\) 68.6573 2.21475
\(962\) 16.7553 0.540214
\(963\) −14.4956 −0.467113
\(964\) 8.52832 0.274679
\(965\) −74.9407 −2.41243
\(966\) 0 0
\(967\) 35.5816 1.14423 0.572114 0.820174i \(-0.306124\pi\)
0.572114 + 0.820174i \(0.306124\pi\)
\(968\) −10.9109 −0.350688
\(969\) 28.3331 0.910190
\(970\) −41.0884 −1.31927
\(971\) 39.0732 1.25392 0.626959 0.779053i \(-0.284300\pi\)
0.626959 + 0.779053i \(0.284300\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) −10.4692 −0.335455
\(975\) 31.4258 1.00643
\(976\) −5.18293 −0.165902
\(977\) −12.1400 −0.388392 −0.194196 0.980963i \(-0.562210\pi\)
−0.194196 + 0.980963i \(0.562210\pi\)
\(978\) 4.63746 0.148290
\(979\) 0.726932 0.0232329
\(980\) 0 0
\(981\) 10.0345 0.320378
\(982\) 12.2846 0.392017
\(983\) 56.4746 1.80126 0.900630 0.434587i \(-0.143106\pi\)
0.900630 + 0.434587i \(0.143106\pi\)
\(984\) −0.817068 −0.0260472
\(985\) −21.0824 −0.671742
\(986\) −28.5316 −0.908632
\(987\) 0 0
\(988\) −37.6856 −1.19894
\(989\) −6.99088 −0.222297
\(990\) 1.01939 0.0323984
\(991\) −48.3603 −1.53622 −0.768108 0.640321i \(-0.778802\pi\)
−0.768108 + 0.640321i \(0.778802\pi\)
\(992\) 9.98285 0.316956
\(993\) −6.34315 −0.201294
\(994\) 0 0
\(995\) −73.2839 −2.32326
\(996\) −0.884359 −0.0280220
\(997\) 13.6999 0.433880 0.216940 0.976185i \(-0.430392\pi\)
0.216940 + 0.976185i \(0.430392\pi\)
\(998\) −6.37188 −0.201698
\(999\) 3.54925 0.112293
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.1.1 4
7.6 odd 2 6762.2.a.ct.1.3 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6762.2.a.ci.1.1 4 1.1 even 1 trivial
6762.2.a.ct.1.3 yes 4 7.6 odd 2