Properties

Label 8281.2.a.cd.1.1
Level $8281$
Weight $2$
Character 8281.1
Self dual yes
Analytic conductor $66.124$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8281 = 7^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8281.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(66.1241179138\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.4507648.1
Defining polynomial: \(x^{6} - 2 x^{5} - 5 x^{4} + 8 x^{3} + 7 x^{2} - 6 x - 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 637)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(0.758419\) of defining polynomial
Character \(\chi\) \(=\) 8281.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.18322 q^{2} +1.75842 q^{3} +2.76645 q^{4} -2.11065 q^{5} -3.83901 q^{6} -1.67333 q^{8} +0.0920365 q^{9} +O(q^{10})\) \(q-2.18322 q^{2} +1.75842 q^{3} +2.76645 q^{4} -2.11065 q^{5} -3.83901 q^{6} -1.67333 q^{8} +0.0920365 q^{9} +4.60802 q^{10} +5.76889 q^{11} +4.86457 q^{12} -3.71141 q^{15} -1.87966 q^{16} +1.64082 q^{17} -0.200936 q^{18} +2.67077 q^{19} -5.83901 q^{20} -12.5948 q^{22} +6.42469 q^{23} -2.94241 q^{24} -0.545141 q^{25} -5.11342 q^{27} -6.04973 q^{29} +8.10283 q^{30} -5.12202 q^{31} +7.45036 q^{32} +10.1441 q^{33} -3.58227 q^{34} +0.254614 q^{36} -5.74772 q^{37} -5.83087 q^{38} +3.53181 q^{40} +7.14100 q^{41} -4.47061 q^{43} +15.9593 q^{44} -0.194257 q^{45} -14.0265 q^{46} -11.7910 q^{47} -3.30523 q^{48} +1.19016 q^{50} +2.88525 q^{51} +3.44959 q^{53} +11.1637 q^{54} -12.1761 q^{55} +4.69633 q^{57} +13.2079 q^{58} -13.1805 q^{59} -10.2674 q^{60} +6.24666 q^{61} +11.1825 q^{62} -12.5065 q^{64} -22.1469 q^{66} -7.74216 q^{67} +4.53925 q^{68} +11.2973 q^{69} -13.6372 q^{71} -0.154007 q^{72} +15.5041 q^{73} +12.5485 q^{74} -0.958586 q^{75} +7.38854 q^{76} +1.12214 q^{79} +3.96731 q^{80} -9.26764 q^{81} -15.5904 q^{82} -4.96925 q^{83} -3.46321 q^{85} +9.76032 q^{86} -10.6380 q^{87} -9.65324 q^{88} -1.14630 q^{89} +0.424106 q^{90} +17.7736 q^{92} -9.00665 q^{93} +25.7423 q^{94} -5.63707 q^{95} +13.1009 q^{96} +6.97223 q^{97} +0.530949 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 8q^{3} + 4q^{4} - 6q^{5} - 4q^{6} + 6q^{9} + O(q^{10}) \) \( 6q + 8q^{3} + 4q^{4} - 6q^{5} - 4q^{6} + 6q^{9} + 4q^{10} - 4q^{11} - 4q^{12} - 12q^{15} + 16q^{17} + 4q^{18} - 2q^{19} - 16q^{20} - 12q^{22} - 6q^{23} - 12q^{24} - 4q^{25} + 20q^{27} - 6q^{29} - 6q^{31} + 20q^{32} - 4q^{33} - 24q^{36} + 8q^{38} + 4q^{40} + 8q^{41} + 2q^{43} + 4q^{44} - 14q^{45} - 8q^{46} - 30q^{47} - 8q^{48} - 8q^{50} - 4q^{51} - 14q^{53} + 48q^{54} - 8q^{55} - 4q^{57} + 8q^{58} - 24q^{59} - 12q^{60} + 28q^{62} - 20q^{64} - 4q^{66} - 16q^{67} + 28q^{68} - 20q^{69} - 8q^{71} - 28q^{72} + 6q^{73} - 12q^{74} + 12q^{75} + 16q^{76} - 22q^{79} + 28q^{80} + 46q^{81} - 40q^{82} - 50q^{83} + 8q^{85} + 16q^{86} - 16q^{87} - 44q^{88} - 26q^{89} - 40q^{90} + 20q^{92} - 16q^{93} - 32q^{94} - 6q^{95} + 20q^{96} + 14q^{97} - 12q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.18322 −1.54377 −0.771885 0.635762i \(-0.780686\pi\)
−0.771885 + 0.635762i \(0.780686\pi\)
\(3\) 1.75842 1.01522 0.507612 0.861586i \(-0.330528\pi\)
0.507612 + 0.861586i \(0.330528\pi\)
\(4\) 2.76645 1.38322
\(5\) −2.11065 −0.943913 −0.471957 0.881622i \(-0.656452\pi\)
−0.471957 + 0.881622i \(0.656452\pi\)
\(6\) −3.83901 −1.56727
\(7\) 0 0
\(8\) −1.67333 −0.591610
\(9\) 0.0920365 0.0306788
\(10\) 4.60802 1.45718
\(11\) 5.76889 1.73939 0.869694 0.493592i \(-0.164316\pi\)
0.869694 + 0.493592i \(0.164316\pi\)
\(12\) 4.86457 1.40428
\(13\) 0 0
\(14\) 0 0
\(15\) −3.71141 −0.958283
\(16\) −1.87966 −0.469915
\(17\) 1.64082 0.397958 0.198979 0.980004i \(-0.436237\pi\)
0.198979 + 0.980004i \(0.436237\pi\)
\(18\) −0.200936 −0.0473611
\(19\) 2.67077 0.612716 0.306358 0.951916i \(-0.400890\pi\)
0.306358 + 0.951916i \(0.400890\pi\)
\(20\) −5.83901 −1.30564
\(21\) 0 0
\(22\) −12.5948 −2.68521
\(23\) 6.42469 1.33964 0.669820 0.742523i \(-0.266371\pi\)
0.669820 + 0.742523i \(0.266371\pi\)
\(24\) −2.94241 −0.600616
\(25\) −0.545141 −0.109028
\(26\) 0 0
\(27\) −5.11342 −0.984078
\(28\) 0 0
\(29\) −6.04973 −1.12341 −0.561704 0.827338i \(-0.689854\pi\)
−0.561704 + 0.827338i \(0.689854\pi\)
\(30\) 8.10283 1.47937
\(31\) −5.12202 −0.919942 −0.459971 0.887934i \(-0.652140\pi\)
−0.459971 + 0.887934i \(0.652140\pi\)
\(32\) 7.45036 1.31705
\(33\) 10.1441 1.76587
\(34\) −3.58227 −0.614355
\(35\) 0 0
\(36\) 0.254614 0.0424357
\(37\) −5.74772 −0.944919 −0.472460 0.881352i \(-0.656634\pi\)
−0.472460 + 0.881352i \(0.656634\pi\)
\(38\) −5.83087 −0.945892
\(39\) 0 0
\(40\) 3.53181 0.558428
\(41\) 7.14100 1.11524 0.557619 0.830097i \(-0.311715\pi\)
0.557619 + 0.830097i \(0.311715\pi\)
\(42\) 0 0
\(43\) −4.47061 −0.681761 −0.340881 0.940107i \(-0.610725\pi\)
−0.340881 + 0.940107i \(0.610725\pi\)
\(44\) 15.9593 2.40596
\(45\) −0.194257 −0.0289582
\(46\) −14.0265 −2.06810
\(47\) −11.7910 −1.71989 −0.859947 0.510384i \(-0.829503\pi\)
−0.859947 + 0.510384i \(0.829503\pi\)
\(48\) −3.30523 −0.477069
\(49\) 0 0
\(50\) 1.19016 0.168314
\(51\) 2.88525 0.404016
\(52\) 0 0
\(53\) 3.44959 0.473838 0.236919 0.971529i \(-0.423862\pi\)
0.236919 + 0.971529i \(0.423862\pi\)
\(54\) 11.1637 1.51919
\(55\) −12.1761 −1.64183
\(56\) 0 0
\(57\) 4.69633 0.622044
\(58\) 13.2079 1.73428
\(59\) −13.1805 −1.71596 −0.857981 0.513682i \(-0.828281\pi\)
−0.857981 + 0.513682i \(0.828281\pi\)
\(60\) −10.2674 −1.32552
\(61\) 6.24666 0.799803 0.399901 0.916558i \(-0.369044\pi\)
0.399901 + 0.916558i \(0.369044\pi\)
\(62\) 11.1825 1.42018
\(63\) 0 0
\(64\) −12.5065 −1.56331
\(65\) 0 0
\(66\) −22.1469 −2.72609
\(67\) −7.74216 −0.945856 −0.472928 0.881101i \(-0.656803\pi\)
−0.472928 + 0.881101i \(0.656803\pi\)
\(68\) 4.53925 0.550465
\(69\) 11.2973 1.36003
\(70\) 0 0
\(71\) −13.6372 −1.61844 −0.809221 0.587504i \(-0.800111\pi\)
−0.809221 + 0.587504i \(0.800111\pi\)
\(72\) −0.154007 −0.0181499
\(73\) 15.5041 1.81462 0.907308 0.420467i \(-0.138134\pi\)
0.907308 + 0.420467i \(0.138134\pi\)
\(74\) 12.5485 1.45874
\(75\) −0.958586 −0.110688
\(76\) 7.38854 0.847524
\(77\) 0 0
\(78\) 0 0
\(79\) 1.12214 0.126251 0.0631254 0.998006i \(-0.479893\pi\)
0.0631254 + 0.998006i \(0.479893\pi\)
\(80\) 3.96731 0.443559
\(81\) −9.26764 −1.02974
\(82\) −15.5904 −1.72167
\(83\) −4.96925 −0.545446 −0.272723 0.962093i \(-0.587924\pi\)
−0.272723 + 0.962093i \(0.587924\pi\)
\(84\) 0 0
\(85\) −3.46321 −0.375638
\(86\) 9.76032 1.05248
\(87\) −10.6380 −1.14051
\(88\) −9.65324 −1.02904
\(89\) −1.14630 −0.121507 −0.0607535 0.998153i \(-0.519350\pi\)
−0.0607535 + 0.998153i \(0.519350\pi\)
\(90\) 0.424106 0.0447047
\(91\) 0 0
\(92\) 17.7736 1.85302
\(93\) −9.00665 −0.933946
\(94\) 25.7423 2.65512
\(95\) −5.63707 −0.578351
\(96\) 13.1009 1.33710
\(97\) 6.97223 0.707923 0.353961 0.935260i \(-0.384834\pi\)
0.353961 + 0.935260i \(0.384834\pi\)
\(98\) 0 0
\(99\) 0.530949 0.0533624
\(100\) −1.50810 −0.150810
\(101\) −6.49919 −0.646694 −0.323347 0.946280i \(-0.604808\pi\)
−0.323347 + 0.946280i \(0.604808\pi\)
\(102\) −6.29914 −0.623708
\(103\) −0.578048 −0.0569568 −0.0284784 0.999594i \(-0.509066\pi\)
−0.0284784 + 0.999594i \(0.509066\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −7.53122 −0.731497
\(107\) −16.2573 −1.57165 −0.785824 0.618450i \(-0.787761\pi\)
−0.785824 + 0.618450i \(0.787761\pi\)
\(108\) −14.1460 −1.36120
\(109\) −1.78191 −0.170676 −0.0853378 0.996352i \(-0.527197\pi\)
−0.0853378 + 0.996352i \(0.527197\pi\)
\(110\) 26.5832 2.53461
\(111\) −10.1069 −0.959304
\(112\) 0 0
\(113\) −7.52215 −0.707624 −0.353812 0.935316i \(-0.615115\pi\)
−0.353812 + 0.935316i \(0.615115\pi\)
\(114\) −10.2531 −0.960292
\(115\) −13.5603 −1.26450
\(116\) −16.7363 −1.55392
\(117\) 0 0
\(118\) 28.7760 2.64905
\(119\) 0 0
\(120\) 6.21040 0.566930
\(121\) 22.2801 2.02547
\(122\) −13.6378 −1.23471
\(123\) 12.5569 1.13222
\(124\) −14.1698 −1.27249
\(125\) 11.7039 1.04683
\(126\) 0 0
\(127\) −19.3056 −1.71310 −0.856549 0.516067i \(-0.827396\pi\)
−0.856549 + 0.516067i \(0.827396\pi\)
\(128\) 12.4036 1.09634
\(129\) −7.86120 −0.692140
\(130\) 0 0
\(131\) 9.69703 0.847234 0.423617 0.905842i \(-0.360760\pi\)
0.423617 + 0.905842i \(0.360760\pi\)
\(132\) 28.0632 2.44259
\(133\) 0 0
\(134\) 16.9028 1.46018
\(135\) 10.7927 0.928884
\(136\) −2.74563 −0.235436
\(137\) 15.4493 1.31993 0.659963 0.751298i \(-0.270572\pi\)
0.659963 + 0.751298i \(0.270572\pi\)
\(138\) −24.6645 −2.09958
\(139\) 3.84912 0.326478 0.163239 0.986587i \(-0.447806\pi\)
0.163239 + 0.986587i \(0.447806\pi\)
\(140\) 0 0
\(141\) −20.7335 −1.74608
\(142\) 29.7731 2.49850
\(143\) 0 0
\(144\) −0.172997 −0.0144164
\(145\) 12.7689 1.06040
\(146\) −33.8488 −2.80135
\(147\) 0 0
\(148\) −15.9008 −1.30704
\(149\) 1.04028 0.0852231 0.0426115 0.999092i \(-0.486432\pi\)
0.0426115 + 0.999092i \(0.486432\pi\)
\(150\) 2.09280 0.170877
\(151\) 4.34336 0.353458 0.176729 0.984260i \(-0.443448\pi\)
0.176729 + 0.984260i \(0.443448\pi\)
\(152\) −4.46906 −0.362489
\(153\) 0.151016 0.0122089
\(154\) 0 0
\(155\) 10.8108 0.868345
\(156\) 0 0
\(157\) 0.336000 0.0268157 0.0134079 0.999910i \(-0.495732\pi\)
0.0134079 + 0.999910i \(0.495732\pi\)
\(158\) −2.44988 −0.194902
\(159\) 6.06583 0.481052
\(160\) −15.7251 −1.24318
\(161\) 0 0
\(162\) 20.2333 1.58968
\(163\) −6.79919 −0.532553 −0.266277 0.963897i \(-0.585793\pi\)
−0.266277 + 0.963897i \(0.585793\pi\)
\(164\) 19.7552 1.54262
\(165\) −21.4107 −1.66682
\(166\) 10.8490 0.842043
\(167\) −0.668649 −0.0517416 −0.0258708 0.999665i \(-0.508236\pi\)
−0.0258708 + 0.999665i \(0.508236\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 7.56094 0.579898
\(171\) 0.245808 0.0187974
\(172\) −12.3677 −0.943029
\(173\) 25.1178 1.90967 0.954837 0.297131i \(-0.0960298\pi\)
0.954837 + 0.297131i \(0.0960298\pi\)
\(174\) 23.2250 1.76068
\(175\) 0 0
\(176\) −10.8436 −0.817364
\(177\) −23.1769 −1.74208
\(178\) 2.50262 0.187579
\(179\) −1.89527 −0.141659 −0.0708294 0.997488i \(-0.522565\pi\)
−0.0708294 + 0.997488i \(0.522565\pi\)
\(180\) −0.537403 −0.0400556
\(181\) 11.3595 0.844347 0.422174 0.906515i \(-0.361267\pi\)
0.422174 + 0.906515i \(0.361267\pi\)
\(182\) 0 0
\(183\) 10.9842 0.811978
\(184\) −10.7506 −0.792544
\(185\) 12.1314 0.891922
\(186\) 19.6635 1.44180
\(187\) 9.46573 0.692203
\(188\) −32.6192 −2.37900
\(189\) 0 0
\(190\) 12.3070 0.892840
\(191\) −23.4407 −1.69611 −0.848055 0.529909i \(-0.822226\pi\)
−0.848055 + 0.529909i \(0.822226\pi\)
\(192\) −21.9916 −1.58711
\(193\) 1.85210 0.133317 0.0666584 0.997776i \(-0.478766\pi\)
0.0666584 + 0.997776i \(0.478766\pi\)
\(194\) −15.2219 −1.09287
\(195\) 0 0
\(196\) 0 0
\(197\) 9.87082 0.703267 0.351634 0.936138i \(-0.385626\pi\)
0.351634 + 0.936138i \(0.385626\pi\)
\(198\) −1.15918 −0.0823792
\(199\) 12.2492 0.868324 0.434162 0.900835i \(-0.357044\pi\)
0.434162 + 0.900835i \(0.357044\pi\)
\(200\) 0.912199 0.0645022
\(201\) −13.6140 −0.960255
\(202\) 14.1892 0.998347
\(203\) 0 0
\(204\) 7.98190 0.558845
\(205\) −15.0722 −1.05269
\(206\) 1.26201 0.0879281
\(207\) 0.591306 0.0410986
\(208\) 0 0
\(209\) 15.4074 1.06575
\(210\) 0 0
\(211\) −0.739899 −0.0509368 −0.0254684 0.999676i \(-0.508108\pi\)
−0.0254684 + 0.999676i \(0.508108\pi\)
\(212\) 9.54312 0.655424
\(213\) −23.9800 −1.64308
\(214\) 35.4932 2.42626
\(215\) 9.43591 0.643523
\(216\) 8.55641 0.582190
\(217\) 0 0
\(218\) 3.89029 0.263484
\(219\) 27.2627 1.84224
\(220\) −33.6847 −2.27102
\(221\) 0 0
\(222\) 22.0656 1.48094
\(223\) −8.30577 −0.556196 −0.278098 0.960553i \(-0.589704\pi\)
−0.278098 + 0.960553i \(0.589704\pi\)
\(224\) 0 0
\(225\) −0.0501729 −0.00334486
\(226\) 16.4225 1.09241
\(227\) −13.2917 −0.882198 −0.441099 0.897458i \(-0.645411\pi\)
−0.441099 + 0.897458i \(0.645411\pi\)
\(228\) 12.9921 0.860426
\(229\) −17.9207 −1.18423 −0.592115 0.805853i \(-0.701707\pi\)
−0.592115 + 0.805853i \(0.701707\pi\)
\(230\) 29.6051 1.95210
\(231\) 0 0
\(232\) 10.1232 0.664619
\(233\) 6.29968 0.412705 0.206353 0.978478i \(-0.433841\pi\)
0.206353 + 0.978478i \(0.433841\pi\)
\(234\) 0 0
\(235\) 24.8867 1.62343
\(236\) −36.4633 −2.37356
\(237\) 1.97320 0.128173
\(238\) 0 0
\(239\) −9.41783 −0.609189 −0.304594 0.952482i \(-0.598521\pi\)
−0.304594 + 0.952482i \(0.598521\pi\)
\(240\) 6.97619 0.450311
\(241\) 19.9493 1.28505 0.642524 0.766266i \(-0.277887\pi\)
0.642524 + 0.766266i \(0.277887\pi\)
\(242\) −48.6424 −3.12685
\(243\) −0.956137 −0.0613362
\(244\) 17.2811 1.10631
\(245\) 0 0
\(246\) −27.4144 −1.74788
\(247\) 0 0
\(248\) 8.57080 0.544246
\(249\) −8.73802 −0.553750
\(250\) −25.5521 −1.61606
\(251\) 1.22202 0.0771330 0.0385665 0.999256i \(-0.487721\pi\)
0.0385665 + 0.999256i \(0.487721\pi\)
\(252\) 0 0
\(253\) 37.0633 2.33015
\(254\) 42.1484 2.64463
\(255\) −6.08977 −0.381356
\(256\) −2.06691 −0.129182
\(257\) −22.4240 −1.39877 −0.699386 0.714745i \(-0.746543\pi\)
−0.699386 + 0.714745i \(0.746543\pi\)
\(258\) 17.1627 1.06850
\(259\) 0 0
\(260\) 0 0
\(261\) −0.556797 −0.0344648
\(262\) −21.1707 −1.30793
\(263\) 8.99048 0.554377 0.277188 0.960816i \(-0.410597\pi\)
0.277188 + 0.960816i \(0.410597\pi\)
\(264\) −16.9744 −1.04470
\(265\) −7.28090 −0.447262
\(266\) 0 0
\(267\) −2.01567 −0.123357
\(268\) −21.4183 −1.30833
\(269\) −22.2054 −1.35389 −0.676945 0.736034i \(-0.736697\pi\)
−0.676945 + 0.736034i \(0.736697\pi\)
\(270\) −23.5627 −1.43398
\(271\) 10.8423 0.658624 0.329312 0.944221i \(-0.393183\pi\)
0.329312 + 0.944221i \(0.393183\pi\)
\(272\) −3.08419 −0.187006
\(273\) 0 0
\(274\) −33.7293 −2.03766
\(275\) −3.14486 −0.189642
\(276\) 31.2534 1.88123
\(277\) 5.88828 0.353792 0.176896 0.984230i \(-0.443394\pi\)
0.176896 + 0.984230i \(0.443394\pi\)
\(278\) −8.40347 −0.504006
\(279\) −0.471413 −0.0282227
\(280\) 0 0
\(281\) −26.9071 −1.60514 −0.802572 0.596556i \(-0.796535\pi\)
−0.802572 + 0.596556i \(0.796535\pi\)
\(282\) 45.2658 2.69554
\(283\) −3.49717 −0.207885 −0.103943 0.994583i \(-0.533146\pi\)
−0.103943 + 0.994583i \(0.533146\pi\)
\(284\) −37.7267 −2.23867
\(285\) −9.91232 −0.587155
\(286\) 0 0
\(287\) 0 0
\(288\) 0.685705 0.0404056
\(289\) −14.3077 −0.841630
\(290\) −27.8773 −1.63701
\(291\) 12.2601 0.718700
\(292\) 42.8913 2.51002
\(293\) −1.00509 −0.0587179 −0.0293589 0.999569i \(-0.509347\pi\)
−0.0293589 + 0.999569i \(0.509347\pi\)
\(294\) 0 0
\(295\) 27.8196 1.61972
\(296\) 9.61780 0.559024
\(297\) −29.4988 −1.71169
\(298\) −2.27116 −0.131565
\(299\) 0 0
\(300\) −2.65188 −0.153106
\(301\) 0 0
\(302\) −9.48252 −0.545657
\(303\) −11.4283 −0.656539
\(304\) −5.02013 −0.287924
\(305\) −13.1845 −0.754944
\(306\) −0.329700 −0.0188477
\(307\) −4.25772 −0.243001 −0.121501 0.992591i \(-0.538771\pi\)
−0.121501 + 0.992591i \(0.538771\pi\)
\(308\) 0 0
\(309\) −1.01645 −0.0578238
\(310\) −23.6024 −1.34052
\(311\) −6.94043 −0.393556 −0.196778 0.980448i \(-0.563048\pi\)
−0.196778 + 0.980448i \(0.563048\pi\)
\(312\) 0 0
\(313\) −9.96299 −0.563142 −0.281571 0.959540i \(-0.590855\pi\)
−0.281571 + 0.959540i \(0.590855\pi\)
\(314\) −0.733563 −0.0413973
\(315\) 0 0
\(316\) 3.10435 0.174633
\(317\) −3.75345 −0.210815 −0.105407 0.994429i \(-0.533615\pi\)
−0.105407 + 0.994429i \(0.533615\pi\)
\(318\) −13.2430 −0.742633
\(319\) −34.9003 −1.95404
\(320\) 26.3968 1.47563
\(321\) −28.5871 −1.59557
\(322\) 0 0
\(323\) 4.38225 0.243835
\(324\) −25.6384 −1.42436
\(325\) 0 0
\(326\) 14.8441 0.822140
\(327\) −3.13334 −0.173274
\(328\) −11.9492 −0.659785
\(329\) 0 0
\(330\) 46.7444 2.57319
\(331\) 19.9812 1.09827 0.549134 0.835734i \(-0.314958\pi\)
0.549134 + 0.835734i \(0.314958\pi\)
\(332\) −13.7472 −0.754474
\(333\) −0.529000 −0.0289890
\(334\) 1.45981 0.0798771
\(335\) 16.3410 0.892805
\(336\) 0 0
\(337\) 18.4887 1.00714 0.503571 0.863954i \(-0.332019\pi\)
0.503571 + 0.863954i \(0.332019\pi\)
\(338\) 0 0
\(339\) −13.2271 −0.718397
\(340\) −9.58078 −0.519591
\(341\) −29.5484 −1.60013
\(342\) −0.536653 −0.0290189
\(343\) 0 0
\(344\) 7.48078 0.403337
\(345\) −23.8447 −1.28375
\(346\) −54.8377 −2.94810
\(347\) −0.796100 −0.0427369 −0.0213684 0.999772i \(-0.506802\pi\)
−0.0213684 + 0.999772i \(0.506802\pi\)
\(348\) −29.4294 −1.57758
\(349\) 11.3725 0.608754 0.304377 0.952552i \(-0.401552\pi\)
0.304377 + 0.952552i \(0.401552\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 42.9803 2.29086
\(353\) −24.1068 −1.28307 −0.641537 0.767092i \(-0.721703\pi\)
−0.641537 + 0.767092i \(0.721703\pi\)
\(354\) 50.6003 2.68938
\(355\) 28.7835 1.52767
\(356\) −3.17117 −0.168072
\(357\) 0 0
\(358\) 4.13778 0.218689
\(359\) −9.07182 −0.478792 −0.239396 0.970922i \(-0.576949\pi\)
−0.239396 + 0.970922i \(0.576949\pi\)
\(360\) 0.325056 0.0171319
\(361\) −11.8670 −0.624579
\(362\) −24.8003 −1.30348
\(363\) 39.1778 2.05630
\(364\) 0 0
\(365\) −32.7238 −1.71284
\(366\) −23.9810 −1.25351
\(367\) −31.5507 −1.64693 −0.823467 0.567364i \(-0.807963\pi\)
−0.823467 + 0.567364i \(0.807963\pi\)
\(368\) −12.0762 −0.629517
\(369\) 0.657233 0.0342142
\(370\) −26.4856 −1.37692
\(371\) 0 0
\(372\) −24.9164 −1.29186
\(373\) −16.0581 −0.831457 −0.415728 0.909489i \(-0.636473\pi\)
−0.415728 + 0.909489i \(0.636473\pi\)
\(374\) −20.6658 −1.06860
\(375\) 20.5803 1.06276
\(376\) 19.7302 1.01751
\(377\) 0 0
\(378\) 0 0
\(379\) 5.36895 0.275785 0.137892 0.990447i \(-0.455967\pi\)
0.137892 + 0.990447i \(0.455967\pi\)
\(380\) −15.5947 −0.799989
\(381\) −33.9474 −1.73918
\(382\) 51.1762 2.61840
\(383\) −34.5075 −1.76325 −0.881625 0.471951i \(-0.843550\pi\)
−0.881625 + 0.471951i \(0.843550\pi\)
\(384\) 21.8108 1.11303
\(385\) 0 0
\(386\) −4.04353 −0.205810
\(387\) −0.411459 −0.0209156
\(388\) 19.2883 0.979216
\(389\) 22.9210 1.16214 0.581070 0.813854i \(-0.302634\pi\)
0.581070 + 0.813854i \(0.302634\pi\)
\(390\) 0 0
\(391\) 10.5418 0.533120
\(392\) 0 0
\(393\) 17.0514 0.860131
\(394\) −21.5502 −1.08568
\(395\) −2.36845 −0.119170
\(396\) 1.46884 0.0738121
\(397\) −10.8294 −0.543510 −0.271755 0.962366i \(-0.587604\pi\)
−0.271755 + 0.962366i \(0.587604\pi\)
\(398\) −26.7427 −1.34049
\(399\) 0 0
\(400\) 1.02468 0.0512340
\(401\) −37.5415 −1.87473 −0.937367 0.348342i \(-0.886745\pi\)
−0.937367 + 0.348342i \(0.886745\pi\)
\(402\) 29.7223 1.48241
\(403\) 0 0
\(404\) −17.9797 −0.894523
\(405\) 19.5608 0.971983
\(406\) 0 0
\(407\) −33.1580 −1.64358
\(408\) −4.82797 −0.239020
\(409\) 20.4727 1.01231 0.506155 0.862443i \(-0.331066\pi\)
0.506155 + 0.862443i \(0.331066\pi\)
\(410\) 32.9059 1.62511
\(411\) 27.1664 1.34002
\(412\) −1.59914 −0.0787840
\(413\) 0 0
\(414\) −1.29095 −0.0634468
\(415\) 10.4884 0.514854
\(416\) 0 0
\(417\) 6.76836 0.331448
\(418\) −33.6377 −1.64527
\(419\) −15.4980 −0.757127 −0.378564 0.925575i \(-0.623582\pi\)
−0.378564 + 0.925575i \(0.623582\pi\)
\(420\) 0 0
\(421\) −17.9390 −0.874293 −0.437147 0.899390i \(-0.644011\pi\)
−0.437147 + 0.899390i \(0.644011\pi\)
\(422\) 1.61536 0.0786346
\(423\) −1.08520 −0.0527643
\(424\) −5.77229 −0.280327
\(425\) −0.894480 −0.0433886
\(426\) 52.3535 2.53654
\(427\) 0 0
\(428\) −44.9749 −2.17394
\(429\) 0 0
\(430\) −20.6007 −0.993452
\(431\) −21.4438 −1.03291 −0.516456 0.856313i \(-0.672749\pi\)
−0.516456 + 0.856313i \(0.672749\pi\)
\(432\) 9.61149 0.462433
\(433\) −14.9365 −0.717800 −0.358900 0.933376i \(-0.616848\pi\)
−0.358900 + 0.933376i \(0.616848\pi\)
\(434\) 0 0
\(435\) 22.4531 1.07654
\(436\) −4.92955 −0.236083
\(437\) 17.1589 0.820819
\(438\) −59.5204 −2.84400
\(439\) −3.59323 −0.171495 −0.0857476 0.996317i \(-0.527328\pi\)
−0.0857476 + 0.996317i \(0.527328\pi\)
\(440\) 20.3746 0.971323
\(441\) 0 0
\(442\) 0 0
\(443\) −13.6534 −0.648694 −0.324347 0.945938i \(-0.605145\pi\)
−0.324347 + 0.945938i \(0.605145\pi\)
\(444\) −27.9602 −1.32693
\(445\) 2.41943 0.114692
\(446\) 18.1333 0.858638
\(447\) 1.82925 0.0865205
\(448\) 0 0
\(449\) 8.72412 0.411717 0.205858 0.978582i \(-0.434001\pi\)
0.205858 + 0.978582i \(0.434001\pi\)
\(450\) 0.109538 0.00516369
\(451\) 41.1957 1.93983
\(452\) −20.8096 −0.978803
\(453\) 7.63745 0.358839
\(454\) 29.0186 1.36191
\(455\) 0 0
\(456\) −7.85848 −0.368007
\(457\) 30.4659 1.42514 0.712568 0.701603i \(-0.247532\pi\)
0.712568 + 0.701603i \(0.247532\pi\)
\(458\) 39.1247 1.82818
\(459\) −8.39021 −0.391621
\(460\) −37.5139 −1.74909
\(461\) −5.22253 −0.243237 −0.121619 0.992577i \(-0.538809\pi\)
−0.121619 + 0.992577i \(0.538809\pi\)
\(462\) 0 0
\(463\) −20.6220 −0.958386 −0.479193 0.877709i \(-0.659071\pi\)
−0.479193 + 0.877709i \(0.659071\pi\)
\(464\) 11.3714 0.527906
\(465\) 19.0099 0.881564
\(466\) −13.7536 −0.637122
\(467\) 1.67149 0.0773474 0.0386737 0.999252i \(-0.487687\pi\)
0.0386737 + 0.999252i \(0.487687\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −54.3332 −2.50620
\(471\) 0.590829 0.0272240
\(472\) 22.0553 1.01518
\(473\) −25.7905 −1.18585
\(474\) −4.30792 −0.197869
\(475\) −1.45595 −0.0668034
\(476\) 0 0
\(477\) 0.317489 0.0145368
\(478\) 20.5612 0.940447
\(479\) −10.6439 −0.486333 −0.243166 0.969985i \(-0.578186\pi\)
−0.243166 + 0.969985i \(0.578186\pi\)
\(480\) −27.6514 −1.26211
\(481\) 0 0
\(482\) −43.5537 −1.98382
\(483\) 0 0
\(484\) 61.6369 2.80168
\(485\) −14.7160 −0.668217
\(486\) 2.08746 0.0946890
\(487\) 31.8312 1.44241 0.721206 0.692721i \(-0.243588\pi\)
0.721206 + 0.692721i \(0.243588\pi\)
\(488\) −10.4527 −0.473171
\(489\) −11.9558 −0.540661
\(490\) 0 0
\(491\) 6.87077 0.310074 0.155037 0.987909i \(-0.450450\pi\)
0.155037 + 0.987909i \(0.450450\pi\)
\(492\) 34.7379 1.56611
\(493\) −9.92654 −0.447069
\(494\) 0 0
\(495\) −1.12065 −0.0503694
\(496\) 9.62765 0.432294
\(497\) 0 0
\(498\) 19.0770 0.854862
\(499\) −0.344335 −0.0154145 −0.00770727 0.999970i \(-0.502453\pi\)
−0.00770727 + 0.999970i \(0.502453\pi\)
\(500\) 32.3782 1.44800
\(501\) −1.17576 −0.0525293
\(502\) −2.66793 −0.119076
\(503\) 30.0808 1.34123 0.670617 0.741803i \(-0.266029\pi\)
0.670617 + 0.741803i \(0.266029\pi\)
\(504\) 0 0
\(505\) 13.7175 0.610423
\(506\) −80.9174 −3.59722
\(507\) 0 0
\(508\) −53.4080 −2.36960
\(509\) −33.3105 −1.47646 −0.738232 0.674547i \(-0.764339\pi\)
−0.738232 + 0.674547i \(0.764339\pi\)
\(510\) 13.2953 0.588726
\(511\) 0 0
\(512\) −20.2947 −0.896908
\(513\) −13.6568 −0.602960
\(514\) 48.9565 2.15938
\(515\) 1.22006 0.0537622
\(516\) −21.7476 −0.957385
\(517\) −68.0210 −2.99156
\(518\) 0 0
\(519\) 44.1677 1.93875
\(520\) 0 0
\(521\) 32.7726 1.43579 0.717897 0.696149i \(-0.245105\pi\)
0.717897 + 0.696149i \(0.245105\pi\)
\(522\) 1.21561 0.0532058
\(523\) 25.2616 1.10461 0.552307 0.833641i \(-0.313748\pi\)
0.552307 + 0.833641i \(0.313748\pi\)
\(524\) 26.8263 1.17191
\(525\) 0 0
\(526\) −19.6282 −0.855830
\(527\) −8.40432 −0.366098
\(528\) −19.0675 −0.829807
\(529\) 18.2766 0.794636
\(530\) 15.8958 0.690469
\(531\) −1.21309 −0.0526437
\(532\) 0 0
\(533\) 0 0
\(534\) 4.40065 0.190435
\(535\) 34.3134 1.48350
\(536\) 12.9552 0.559578
\(537\) −3.33267 −0.143815
\(538\) 48.4794 2.09009
\(539\) 0 0
\(540\) 29.8573 1.28485
\(541\) −13.1630 −0.565920 −0.282960 0.959132i \(-0.591316\pi\)
−0.282960 + 0.959132i \(0.591316\pi\)
\(542\) −23.6712 −1.01676
\(543\) 19.9748 0.857201
\(544\) 12.2247 0.524130
\(545\) 3.76099 0.161103
\(546\) 0 0
\(547\) −2.79349 −0.119441 −0.0597204 0.998215i \(-0.519021\pi\)
−0.0597204 + 0.998215i \(0.519021\pi\)
\(548\) 42.7398 1.82575
\(549\) 0.574921 0.0245370
\(550\) 6.86592 0.292764
\(551\) −16.1574 −0.688330
\(552\) −18.9040 −0.804610
\(553\) 0 0
\(554\) −12.8554 −0.546174
\(555\) 21.3322 0.905500
\(556\) 10.6484 0.451592
\(557\) 4.61284 0.195452 0.0977262 0.995213i \(-0.468843\pi\)
0.0977262 + 0.995213i \(0.468843\pi\)
\(558\) 1.02920 0.0435694
\(559\) 0 0
\(560\) 0 0
\(561\) 16.6447 0.702740
\(562\) 58.7441 2.47797
\(563\) 40.2805 1.69762 0.848811 0.528697i \(-0.177319\pi\)
0.848811 + 0.528697i \(0.177319\pi\)
\(564\) −57.3582 −2.41522
\(565\) 15.8767 0.667936
\(566\) 7.63510 0.320927
\(567\) 0 0
\(568\) 22.8195 0.957486
\(569\) 7.19869 0.301785 0.150892 0.988550i \(-0.451785\pi\)
0.150892 + 0.988550i \(0.451785\pi\)
\(570\) 21.6408 0.906432
\(571\) 11.3820 0.476323 0.238161 0.971226i \(-0.423455\pi\)
0.238161 + 0.971226i \(0.423455\pi\)
\(572\) 0 0
\(573\) −41.2186 −1.72193
\(574\) 0 0
\(575\) −3.50236 −0.146059
\(576\) −1.15105 −0.0479604
\(577\) 35.6257 1.48312 0.741560 0.670887i \(-0.234086\pi\)
0.741560 + 0.670887i \(0.234086\pi\)
\(578\) 31.2369 1.29928
\(579\) 3.25676 0.135346
\(580\) 35.3245 1.46677
\(581\) 0 0
\(582\) −26.7665 −1.10951
\(583\) 19.9003 0.824188
\(584\) −25.9434 −1.07354
\(585\) 0 0
\(586\) 2.19433 0.0906469
\(587\) −4.14755 −0.171188 −0.0855938 0.996330i \(-0.527279\pi\)
−0.0855938 + 0.996330i \(0.527279\pi\)
\(588\) 0 0
\(589\) −13.6797 −0.563663
\(590\) −60.7362 −2.50047
\(591\) 17.3570 0.713973
\(592\) 10.8038 0.444032
\(593\) 24.1313 0.990953 0.495476 0.868621i \(-0.334994\pi\)
0.495476 + 0.868621i \(0.334994\pi\)
\(594\) 64.4023 2.64246
\(595\) 0 0
\(596\) 2.87788 0.117883
\(597\) 21.5393 0.881543
\(598\) 0 0
\(599\) −48.5607 −1.98414 −0.992068 0.125703i \(-0.959881\pi\)
−0.992068 + 0.125703i \(0.959881\pi\)
\(600\) 1.60403 0.0654841
\(601\) −33.2069 −1.35454 −0.677270 0.735735i \(-0.736837\pi\)
−0.677270 + 0.735735i \(0.736837\pi\)
\(602\) 0 0
\(603\) −0.712562 −0.0290178
\(604\) 12.0157 0.488911
\(605\) −47.0257 −1.91186
\(606\) 24.9505 1.01354
\(607\) −21.7946 −0.884616 −0.442308 0.896863i \(-0.645840\pi\)
−0.442308 + 0.896863i \(0.645840\pi\)
\(608\) 19.8982 0.806978
\(609\) 0 0
\(610\) 28.7847 1.16546
\(611\) 0 0
\(612\) 0.417777 0.0168876
\(613\) 22.7991 0.920847 0.460423 0.887700i \(-0.347698\pi\)
0.460423 + 0.887700i \(0.347698\pi\)
\(614\) 9.29554 0.375138
\(615\) −26.5032 −1.06871
\(616\) 0 0
\(617\) 42.5433 1.71273 0.856364 0.516373i \(-0.172718\pi\)
0.856364 + 0.516373i \(0.172718\pi\)
\(618\) 2.21913 0.0892667
\(619\) 44.1285 1.77367 0.886836 0.462084i \(-0.152898\pi\)
0.886836 + 0.462084i \(0.152898\pi\)
\(620\) 29.9075 1.20112
\(621\) −32.8521 −1.31831
\(622\) 15.1525 0.607559
\(623\) 0 0
\(624\) 0 0
\(625\) −21.9771 −0.879085
\(626\) 21.7514 0.869361
\(627\) 27.0926 1.08197
\(628\) 0.929528 0.0370922
\(629\) −9.43098 −0.376038
\(630\) 0 0
\(631\) −9.09226 −0.361957 −0.180979 0.983487i \(-0.557926\pi\)
−0.180979 + 0.983487i \(0.557926\pi\)
\(632\) −1.87771 −0.0746912
\(633\) −1.30105 −0.0517122
\(634\) 8.19461 0.325449
\(635\) 40.7475 1.61701
\(636\) 16.7808 0.665402
\(637\) 0 0
\(638\) 76.1950 3.01659
\(639\) −1.25512 −0.0496519
\(640\) −26.1797 −1.03485
\(641\) −22.6642 −0.895181 −0.447591 0.894239i \(-0.647718\pi\)
−0.447591 + 0.894239i \(0.647718\pi\)
\(642\) 62.4119 2.46320
\(643\) −24.3792 −0.961422 −0.480711 0.876879i \(-0.659621\pi\)
−0.480711 + 0.876879i \(0.659621\pi\)
\(644\) 0 0
\(645\) 16.5923 0.653320
\(646\) −9.56742 −0.376425
\(647\) −37.0364 −1.45605 −0.728026 0.685549i \(-0.759562\pi\)
−0.728026 + 0.685549i \(0.759562\pi\)
\(648\) 15.5078 0.609203
\(649\) −76.0372 −2.98472
\(650\) 0 0
\(651\) 0 0
\(652\) −18.8096 −0.736641
\(653\) 18.7341 0.733122 0.366561 0.930394i \(-0.380535\pi\)
0.366561 + 0.930394i \(0.380535\pi\)
\(654\) 6.84077 0.267495
\(655\) −20.4671 −0.799715
\(656\) −13.4227 −0.524067
\(657\) 1.42694 0.0556703
\(658\) 0 0
\(659\) −28.5206 −1.11100 −0.555502 0.831515i \(-0.687474\pi\)
−0.555502 + 0.831515i \(0.687474\pi\)
\(660\) −59.2317 −2.30559
\(661\) 19.9758 0.776968 0.388484 0.921455i \(-0.372999\pi\)
0.388484 + 0.921455i \(0.372999\pi\)
\(662\) −43.6234 −1.69547
\(663\) 0 0
\(664\) 8.31517 0.322691
\(665\) 0 0
\(666\) 1.15492 0.0447524
\(667\) −38.8677 −1.50496
\(668\) −1.84978 −0.0715702
\(669\) −14.6050 −0.564663
\(670\) −35.6760 −1.37829
\(671\) 36.0363 1.39117
\(672\) 0 0
\(673\) −4.18849 −0.161454 −0.0807272 0.996736i \(-0.525724\pi\)
−0.0807272 + 0.996736i \(0.525724\pi\)
\(674\) −40.3649 −1.55480
\(675\) 2.78753 0.107292
\(676\) 0 0
\(677\) −38.3396 −1.47351 −0.736755 0.676159i \(-0.763643\pi\)
−0.736755 + 0.676159i \(0.763643\pi\)
\(678\) 28.8776 1.10904
\(679\) 0 0
\(680\) 5.79507 0.222231
\(681\) −23.3723 −0.895628
\(682\) 64.5106 2.47024
\(683\) 2.61207 0.0999482 0.0499741 0.998751i \(-0.484086\pi\)
0.0499741 + 0.998751i \(0.484086\pi\)
\(684\) 0.680016 0.0260010
\(685\) −32.6082 −1.24589
\(686\) 0 0
\(687\) −31.5120 −1.20226
\(688\) 8.40322 0.320370
\(689\) 0 0
\(690\) 52.0582 1.98182
\(691\) 40.9185 1.55661 0.778307 0.627884i \(-0.216079\pi\)
0.778307 + 0.627884i \(0.216079\pi\)
\(692\) 69.4872 2.64151
\(693\) 0 0
\(694\) 1.73806 0.0659759
\(695\) −8.12415 −0.308167
\(696\) 17.8008 0.674737
\(697\) 11.7171 0.443817
\(698\) −24.8286 −0.939776
\(699\) 11.0775 0.418988
\(700\) 0 0
\(701\) −0.762896 −0.0288142 −0.0144071 0.999896i \(-0.504586\pi\)
−0.0144071 + 0.999896i \(0.504586\pi\)
\(702\) 0 0
\(703\) −15.3508 −0.578967
\(704\) −72.1484 −2.71920
\(705\) 43.7613 1.64814
\(706\) 52.6304 1.98077
\(707\) 0 0
\(708\) −64.1178 −2.40969
\(709\) −2.65281 −0.0996284 −0.0498142 0.998759i \(-0.515863\pi\)
−0.0498142 + 0.998759i \(0.515863\pi\)
\(710\) −62.8407 −2.35837
\(711\) 0.103278 0.00387323
\(712\) 1.91813 0.0718848
\(713\) −32.9074 −1.23239
\(714\) 0 0
\(715\) 0 0
\(716\) −5.24315 −0.195946
\(717\) −16.5605 −0.618463
\(718\) 19.8058 0.739145
\(719\) 11.0443 0.411883 0.205942 0.978564i \(-0.433974\pi\)
0.205942 + 0.978564i \(0.433974\pi\)
\(720\) 0.365138 0.0136079
\(721\) 0 0
\(722\) 25.9083 0.964206
\(723\) 35.0792 1.30461
\(724\) 31.4255 1.16792
\(725\) 3.29796 0.122483
\(726\) −85.5338 −3.17446
\(727\) 3.46566 0.128534 0.0642672 0.997933i \(-0.479529\pi\)
0.0642672 + 0.997933i \(0.479529\pi\)
\(728\) 0 0
\(729\) 26.1216 0.967468
\(730\) 71.4432 2.64423
\(731\) −7.33547 −0.271312
\(732\) 30.3873 1.12315
\(733\) 29.0332 1.07236 0.536182 0.844102i \(-0.319866\pi\)
0.536182 + 0.844102i \(0.319866\pi\)
\(734\) 68.8822 2.54249
\(735\) 0 0
\(736\) 47.8663 1.76437
\(737\) −44.6637 −1.64521
\(738\) −1.43488 −0.0528188
\(739\) −47.5522 −1.74924 −0.874618 0.484813i \(-0.838888\pi\)
−0.874618 + 0.484813i \(0.838888\pi\)
\(740\) 33.5610 1.23373
\(741\) 0 0
\(742\) 0 0
\(743\) −12.6122 −0.462697 −0.231349 0.972871i \(-0.574314\pi\)
−0.231349 + 0.972871i \(0.574314\pi\)
\(744\) 15.0711 0.552532
\(745\) −2.19567 −0.0804432
\(746\) 35.0584 1.28358
\(747\) −0.457353 −0.0167337
\(748\) 26.1864 0.957471
\(749\) 0 0
\(750\) −44.9313 −1.64066
\(751\) −7.70781 −0.281262 −0.140631 0.990062i \(-0.544913\pi\)
−0.140631 + 0.990062i \(0.544913\pi\)
\(752\) 22.1631 0.808204
\(753\) 2.14882 0.0783073
\(754\) 0 0
\(755\) −9.16734 −0.333633
\(756\) 0 0
\(757\) 2.39454 0.0870311 0.0435156 0.999053i \(-0.486144\pi\)
0.0435156 + 0.999053i \(0.486144\pi\)
\(758\) −11.7216 −0.425748
\(759\) 65.1729 2.36563
\(760\) 9.43265 0.342158
\(761\) 41.3296 1.49820 0.749098 0.662459i \(-0.230487\pi\)
0.749098 + 0.662459i \(0.230487\pi\)
\(762\) 74.1146 2.68489
\(763\) 0 0
\(764\) −64.8475 −2.34610
\(765\) −0.318742 −0.0115241
\(766\) 75.3374 2.72205
\(767\) 0 0
\(768\) −3.63450 −0.131149
\(769\) −36.7414 −1.32493 −0.662464 0.749094i \(-0.730489\pi\)
−0.662464 + 0.749094i \(0.730489\pi\)
\(770\) 0 0
\(771\) −39.4308 −1.42007
\(772\) 5.12373 0.184407
\(773\) 23.9672 0.862040 0.431020 0.902342i \(-0.358154\pi\)
0.431020 + 0.902342i \(0.358154\pi\)
\(774\) 0.898306 0.0322889
\(775\) 2.79222 0.100300
\(776\) −11.6668 −0.418814
\(777\) 0 0
\(778\) −50.0415 −1.79408
\(779\) 19.0720 0.683324
\(780\) 0 0
\(781\) −78.6718 −2.81510
\(782\) −23.0150 −0.823015
\(783\) 30.9348 1.10552
\(784\) 0 0
\(785\) −0.709180 −0.0253117
\(786\) −37.2270 −1.32784
\(787\) −36.0134 −1.28374 −0.641870 0.766814i \(-0.721841\pi\)
−0.641870 + 0.766814i \(0.721841\pi\)
\(788\) 27.3071 0.972776
\(789\) 15.8090 0.562816
\(790\) 5.17085 0.183971
\(791\) 0 0
\(792\) −0.888450 −0.0315697
\(793\) 0 0
\(794\) 23.6429 0.839054
\(795\) −12.8029 −0.454071
\(796\) 33.8868 1.20109
\(797\) −18.8155 −0.666478 −0.333239 0.942842i \(-0.608142\pi\)
−0.333239 + 0.942842i \(0.608142\pi\)
\(798\) 0 0
\(799\) −19.3469 −0.684445
\(800\) −4.06150 −0.143596
\(801\) −0.105501 −0.00372770
\(802\) 81.9614 2.89416
\(803\) 89.4414 3.15632
\(804\) −37.6623 −1.32825
\(805\) 0 0
\(806\) 0 0
\(807\) −39.0465 −1.37450
\(808\) 10.8753 0.382591
\(809\) −47.3322 −1.66411 −0.832056 0.554692i \(-0.812836\pi\)
−0.832056 + 0.554692i \(0.812836\pi\)
\(810\) −42.7055 −1.50052
\(811\) −19.2555 −0.676153 −0.338077 0.941119i \(-0.609776\pi\)
−0.338077 + 0.941119i \(0.609776\pi\)
\(812\) 0 0
\(813\) 19.0653 0.668650
\(814\) 72.3912 2.53731
\(815\) 14.3507 0.502684
\(816\) −5.42329 −0.189853
\(817\) −11.9400 −0.417726
\(818\) −44.6964 −1.56277
\(819\) 0 0
\(820\) −41.6964 −1.45610
\(821\) −41.5165 −1.44894 −0.724468 0.689308i \(-0.757915\pi\)
−0.724468 + 0.689308i \(0.757915\pi\)
\(822\) −59.3102 −2.06868
\(823\) 24.5627 0.856201 0.428100 0.903731i \(-0.359183\pi\)
0.428100 + 0.903731i \(0.359183\pi\)
\(824\) 0.967262 0.0336962
\(825\) −5.52998 −0.192529
\(826\) 0 0
\(827\) 0.580957 0.0202019 0.0101009 0.999949i \(-0.496785\pi\)
0.0101009 + 0.999949i \(0.496785\pi\)
\(828\) 1.63582 0.0568486
\(829\) 14.6630 0.509268 0.254634 0.967038i \(-0.418045\pi\)
0.254634 + 0.967038i \(0.418045\pi\)
\(830\) −22.8984 −0.794815
\(831\) 10.3541 0.359178
\(832\) 0 0
\(833\) 0 0
\(834\) −14.7768 −0.511679
\(835\) 1.41129 0.0488396
\(836\) 42.6237 1.47417
\(837\) 26.1910 0.905294
\(838\) 33.8356 1.16883
\(839\) −25.6151 −0.884332 −0.442166 0.896933i \(-0.645790\pi\)
−0.442166 + 0.896933i \(0.645790\pi\)
\(840\) 0 0
\(841\) 7.59929 0.262044
\(842\) 39.1648 1.34971
\(843\) −47.3140 −1.62958
\(844\) −2.04689 −0.0704569
\(845\) 0 0
\(846\) 2.36924 0.0814560
\(847\) 0 0
\(848\) −6.48406 −0.222664
\(849\) −6.14949 −0.211050
\(850\) 1.95285 0.0669820
\(851\) −36.9273 −1.26585
\(852\) −66.3394 −2.27275
\(853\) 0.587340 0.0201101 0.0100551 0.999949i \(-0.496799\pi\)
0.0100551 + 0.999949i \(0.496799\pi\)
\(854\) 0 0
\(855\) −0.518816 −0.0177431
\(856\) 27.2037 0.929803
\(857\) 31.1680 1.06468 0.532340 0.846531i \(-0.321313\pi\)
0.532340 + 0.846531i \(0.321313\pi\)
\(858\) 0 0
\(859\) −22.7944 −0.777735 −0.388867 0.921294i \(-0.627134\pi\)
−0.388867 + 0.921294i \(0.627134\pi\)
\(860\) 26.1039 0.890137
\(861\) 0 0
\(862\) 46.8166 1.59458
\(863\) −50.6249 −1.72329 −0.861646 0.507510i \(-0.830566\pi\)
−0.861646 + 0.507510i \(0.830566\pi\)
\(864\) −38.0968 −1.29608
\(865\) −53.0150 −1.80257
\(866\) 32.6096 1.10812
\(867\) −25.1589 −0.854442
\(868\) 0 0
\(869\) 6.47352 0.219599
\(870\) −49.0200 −1.66193
\(871\) 0 0
\(872\) 2.98171 0.100973
\(873\) 0.641700 0.0217182
\(874\) −37.4615 −1.26716
\(875\) 0 0
\(876\) 75.4208 2.54823
\(877\) −11.9495 −0.403505 −0.201753 0.979436i \(-0.564664\pi\)
−0.201753 + 0.979436i \(0.564664\pi\)
\(878\) 7.84480 0.264749
\(879\) −1.76737 −0.0596118
\(880\) 22.8870 0.771520
\(881\) 54.6400 1.84087 0.920434 0.390898i \(-0.127835\pi\)
0.920434 + 0.390898i \(0.127835\pi\)
\(882\) 0 0
\(883\) −26.7256 −0.899388 −0.449694 0.893183i \(-0.648467\pi\)
−0.449694 + 0.893183i \(0.648467\pi\)
\(884\) 0 0
\(885\) 48.9185 1.64438
\(886\) 29.8084 1.00143
\(887\) −5.82278 −0.195510 −0.0977549 0.995211i \(-0.531166\pi\)
−0.0977549 + 0.995211i \(0.531166\pi\)
\(888\) 16.9121 0.567534
\(889\) 0 0
\(890\) −5.28215 −0.177058
\(891\) −53.4640 −1.79111
\(892\) −22.9775 −0.769343
\(893\) −31.4910 −1.05381
\(894\) −3.99365 −0.133568
\(895\) 4.00025 0.133714
\(896\) 0 0
\(897\) 0 0
\(898\) −19.0467 −0.635595
\(899\) 30.9868 1.03347
\(900\) −0.138801 −0.00462669
\(901\) 5.66017 0.188568
\(902\) −89.9392 −2.99465
\(903\) 0 0
\(904\) 12.5870 0.418638
\(905\) −23.9760 −0.796990
\(906\) −16.6742 −0.553964
\(907\) −54.4672 −1.80855 −0.904276 0.426948i \(-0.859589\pi\)
−0.904276 + 0.426948i \(0.859589\pi\)
\(908\) −36.7707 −1.22028
\(909\) −0.598163 −0.0198398
\(910\) 0 0
\(911\) 5.94815 0.197071 0.0985355 0.995134i \(-0.468584\pi\)
0.0985355 + 0.995134i \(0.468584\pi\)
\(912\) −8.82750 −0.292308
\(913\) −28.6671 −0.948742
\(914\) −66.5138 −2.20008
\(915\) −23.1839 −0.766437
\(916\) −49.5766 −1.63806
\(917\) 0 0
\(918\) 18.3177 0.604573
\(919\) 13.6480 0.450207 0.225103 0.974335i \(-0.427728\pi\)
0.225103 + 0.974335i \(0.427728\pi\)
\(920\) 22.6908 0.748093
\(921\) −7.48686 −0.246700
\(922\) 11.4019 0.375503
\(923\) 0 0
\(924\) 0 0
\(925\) 3.13332 0.103023
\(926\) 45.0224 1.47953
\(927\) −0.0532015 −0.00174737
\(928\) −45.0727 −1.47958
\(929\) −36.3062 −1.19117 −0.595585 0.803293i \(-0.703080\pi\)
−0.595585 + 0.803293i \(0.703080\pi\)
\(930\) −41.5028 −1.36093
\(931\) 0 0
\(932\) 17.4277 0.570864
\(933\) −12.2042 −0.399547
\(934\) −3.64923 −0.119406
\(935\) −19.9789 −0.653379
\(936\) 0 0
\(937\) 27.7384 0.906175 0.453087 0.891466i \(-0.350323\pi\)
0.453087 + 0.891466i \(0.350323\pi\)
\(938\) 0 0
\(939\) −17.5191 −0.571715
\(940\) 68.8478 2.24557
\(941\) 31.9163 1.04044 0.520222 0.854031i \(-0.325849\pi\)
0.520222 + 0.854031i \(0.325849\pi\)
\(942\) −1.28991 −0.0420275
\(943\) 45.8787 1.49402
\(944\) 24.7750 0.806356
\(945\) 0 0
\(946\) 56.3063 1.83067
\(947\) −23.1657 −0.752785 −0.376392 0.926460i \(-0.622836\pi\)
−0.376392 + 0.926460i \(0.622836\pi\)
\(948\) 5.45874 0.177292
\(949\) 0 0
\(950\) 3.17865 0.103129
\(951\) −6.60014 −0.214024
\(952\) 0 0
\(953\) 25.7340 0.833607 0.416803 0.908997i \(-0.363150\pi\)
0.416803 + 0.908997i \(0.363150\pi\)
\(954\) −0.693147 −0.0224415
\(955\) 49.4752 1.60098
\(956\) −26.0539 −0.842645
\(957\) −61.3693 −1.98379
\(958\) 23.2380 0.750786
\(959\) 0 0
\(960\) 46.4166 1.49809
\(961\) −4.76494 −0.153708
\(962\) 0 0
\(963\) −1.49626 −0.0482164
\(964\) 55.1887 1.77751
\(965\) −3.90913 −0.125839
\(966\) 0 0
\(967\) −16.2544 −0.522706 −0.261353 0.965243i \(-0.584169\pi\)
−0.261353 + 0.965243i \(0.584169\pi\)
\(968\) −37.2819 −1.19829
\(969\) 7.70584 0.247547
\(970\) 32.1282 1.03157
\(971\) 39.7465 1.27553 0.637763 0.770232i \(-0.279860\pi\)
0.637763 + 0.770232i \(0.279860\pi\)
\(972\) −2.64510 −0.0848418
\(973\) 0 0
\(974\) −69.4946 −2.22675
\(975\) 0 0
\(976\) −11.7416 −0.375839
\(977\) 26.1884 0.837842 0.418921 0.908023i \(-0.362408\pi\)
0.418921 + 0.908023i \(0.362408\pi\)
\(978\) 26.1022 0.834655
\(979\) −6.61286 −0.211348
\(980\) 0 0
\(981\) −0.164000 −0.00523613
\(982\) −15.0004 −0.478682
\(983\) −12.2915 −0.392039 −0.196020 0.980600i \(-0.562802\pi\)
−0.196020 + 0.980600i \(0.562802\pi\)
\(984\) −21.0117 −0.669830
\(985\) −20.8339 −0.663823
\(986\) 21.6718 0.690171
\(987\) 0 0
\(988\) 0 0
\(989\) −28.7223 −0.913315
\(990\) 2.44662 0.0777588
\(991\) 11.4681 0.364298 0.182149 0.983271i \(-0.441695\pi\)
0.182149 + 0.983271i \(0.441695\pi\)
\(992\) −38.1609 −1.21161
\(993\) 35.1354 1.11499
\(994\) 0 0
\(995\) −25.8539 −0.819623
\(996\) −24.1733 −0.765960
\(997\) −5.30214 −0.167920 −0.0839602 0.996469i \(-0.526757\pi\)
−0.0839602 + 0.996469i \(0.526757\pi\)
\(998\) 0.751758 0.0237965
\(999\) 29.3905 0.929874
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 8281.2.a.cd.1.1 6
7.6 odd 2 8281.2.a.cc.1.1 6
13.12 even 2 637.2.a.n.1.6 yes 6
39.38 odd 2 5733.2.a.br.1.1 6
91.12 odd 6 637.2.e.o.508.1 12
91.25 even 6 637.2.e.n.79.1 12
91.38 odd 6 637.2.e.o.79.1 12
91.51 even 6 637.2.e.n.508.1 12
91.90 odd 2 637.2.a.m.1.6 6
273.272 even 2 5733.2.a.bu.1.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
637.2.a.m.1.6 6 91.90 odd 2
637.2.a.n.1.6 yes 6 13.12 even 2
637.2.e.n.79.1 12 91.25 even 6
637.2.e.n.508.1 12 91.51 even 6
637.2.e.o.79.1 12 91.38 odd 6
637.2.e.o.508.1 12 91.12 odd 6
5733.2.a.br.1.1 6 39.38 odd 2
5733.2.a.bu.1.1 6 273.272 even 2
8281.2.a.cc.1.1 6 7.6 odd 2
8281.2.a.cd.1.1 6 1.1 even 1 trivial