Properties

Label 5733.2.a.br.1.1
Level $5733$
Weight $2$
Character 5733.1
Self dual yes
Analytic conductor $45.778$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5733 = 3^{2} \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5733.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(45.7782354788\)
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: \(\Z[a_1, \ldots, a_{5}]\)
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 \(\chi\) \(=\) 5733.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.18322 q^{2} +2.76645 q^{4} -2.11065 q^{5} -1.67333 q^{8} +O(q^{10})\) \(q-2.18322 q^{2} +2.76645 q^{4} -2.11065 q^{5} -1.67333 q^{8} +4.60802 q^{10} +5.76889 q^{11} +1.00000 q^{13} -1.87966 q^{16} -1.64082 q^{17} -2.67077 q^{19} -5.83901 q^{20} -12.5948 q^{22} -6.42469 q^{23} -0.545141 q^{25} -2.18322 q^{26} +6.04973 q^{29} +5.12202 q^{31} +7.45036 q^{32} +3.58227 q^{34} +5.74772 q^{37} +5.83087 q^{38} +3.53181 q^{40} +7.14100 q^{41} -4.47061 q^{43} +15.9593 q^{44} +14.0265 q^{46} -11.7910 q^{47} +1.19016 q^{50} +2.76645 q^{52} -3.44959 q^{53} -12.1761 q^{55} -13.2079 q^{58} -13.1805 q^{59} +6.24666 q^{61} -11.1825 q^{62} -12.5065 q^{64} -2.11065 q^{65} +7.74216 q^{67} -4.53925 q^{68} -13.6372 q^{71} -15.5041 q^{73} -12.5485 q^{74} -7.38854 q^{76} +1.12214 q^{79} +3.96731 q^{80} -15.5904 q^{82} -4.96925 q^{83} +3.46321 q^{85} +9.76032 q^{86} -9.65324 q^{88} -1.14630 q^{89} -17.7736 q^{92} +25.7423 q^{94} +5.63707 q^{95} -6.97223 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 4 q^{4} - 6 q^{5} + O(q^{10}) \) \( 6 q + 4 q^{4} - 6 q^{5} + 4 q^{10} - 4 q^{11} + 6 q^{13} - 16 q^{17} + 2 q^{19} - 16 q^{20} - 12 q^{22} + 6 q^{23} - 4 q^{25} + 6 q^{29} + 6 q^{31} + 20 q^{32} - 8 q^{38} + 4 q^{40} + 8 q^{41} + 2 q^{43} + 4 q^{44} + 8 q^{46} - 30 q^{47} - 8 q^{50} + 4 q^{52} + 14 q^{53} - 8 q^{55} - 8 q^{58} - 24 q^{59} - 28 q^{62} - 20 q^{64} - 6 q^{65} + 16 q^{67} - 28 q^{68} - 8 q^{71} - 6 q^{73} + 12 q^{74} - 16 q^{76} - 22 q^{79} + 28 q^{80} - 40 q^{82} - 50 q^{83} - 8 q^{85} + 16 q^{86} - 44 q^{88} - 26 q^{89} - 20 q^{92} - 32 q^{94} + 6 q^{95} - 14 q^{97} + O(q^{100}) \)

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 \(0.780686\pi\)
\(3\) 0 0
\(4\) 2.76645 1.38322
\(5\) −2.11065 −0.943913 −0.471957 0.881622i \(-0.656452\pi\)
−0.471957 + 0.881622i \(0.656452\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −1.67333 −0.591610
\(9\) 0 0
\(10\) 4.60802 1.45718
\(11\) 5.76889 1.73939 0.869694 0.493592i \(-0.164316\pi\)
0.869694 + 0.493592i \(0.164316\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) 0 0
\(16\) −1.87966 −0.469915
\(17\) −1.64082 −0.397958 −0.198979 0.980004i \(-0.563763\pi\)
−0.198979 + 0.980004i \(0.563763\pi\)
\(18\) 0 0
\(19\) −2.67077 −0.612716 −0.306358 0.951916i \(-0.599110\pi\)
−0.306358 + 0.951916i \(0.599110\pi\)
\(20\) −5.83901 −1.30564
\(21\) 0 0
\(22\) −12.5948 −2.68521
\(23\) −6.42469 −1.33964 −0.669820 0.742523i \(-0.733629\pi\)
−0.669820 + 0.742523i \(0.733629\pi\)
\(24\) 0 0
\(25\) −0.545141 −0.109028
\(26\) −2.18322 −0.428165
\(27\) 0 0
\(28\) 0 0
\(29\) 6.04973 1.12341 0.561704 0.827338i \(-0.310146\pi\)
0.561704 + 0.827338i \(0.310146\pi\)
\(30\) 0 0
\(31\) 5.12202 0.919942 0.459971 0.887934i \(-0.347860\pi\)
0.459971 + 0.887934i \(0.347860\pi\)
\(32\) 7.45036 1.31705
\(33\) 0 0
\(34\) 3.58227 0.614355
\(35\) 0 0
\(36\) 0 0
\(37\) 5.74772 0.944919 0.472460 0.881352i \(-0.343366\pi\)
0.472460 + 0.881352i \(0.343366\pi\)
\(38\) 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 0
\(46\) 14.0265 2.06810
\(47\) −11.7910 −1.71989 −0.859947 0.510384i \(-0.829503\pi\)
−0.859947 + 0.510384i \(0.829503\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 1.19016 0.168314
\(51\) 0 0
\(52\) 2.76645 0.383637
\(53\) −3.44959 −0.473838 −0.236919 0.971529i \(-0.576138\pi\)
−0.236919 + 0.971529i \(0.576138\pi\)
\(54\) 0 0
\(55\) −12.1761 −1.64183
\(56\) 0 0
\(57\) 0 0
\(58\) −13.2079 −1.73428
\(59\) −13.1805 −1.71596 −0.857981 0.513682i \(-0.828281\pi\)
−0.857981 + 0.513682i \(0.828281\pi\)
\(60\) 0 0
\(61\) 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
\(65\) −2.11065 −0.261794
\(66\) 0 0
\(67\) 7.74216 0.945856 0.472928 0.881101i \(-0.343197\pi\)
0.472928 + 0.881101i \(0.343197\pi\)
\(68\) −4.53925 −0.550465
\(69\) 0 0
\(70\) 0 0
\(71\) −13.6372 −1.61844 −0.809221 0.587504i \(-0.800111\pi\)
−0.809221 + 0.587504i \(0.800111\pi\)
\(72\) 0 0
\(73\) −15.5041 −1.81462 −0.907308 0.420467i \(-0.861866\pi\)
−0.907308 + 0.420467i \(0.861866\pi\)
\(74\) −12.5485 −1.45874
\(75\) 0 0
\(76\) −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
\(81\) 0 0
\(82\) −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
\(87\) 0 0
\(88\) −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 0
\(91\) 0 0
\(92\) −17.7736 −1.85302
\(93\) 0 0
\(94\) 25.7423 2.65512
\(95\) 5.63707 0.578351
\(96\) 0 0
\(97\) −6.97223 −0.707923 −0.353961 0.935260i \(-0.615166\pi\)
−0.353961 + 0.935260i \(0.615166\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −1.50810 −0.150810
\(101\) 6.49919 0.646694 0.323347 0.946280i \(-0.395192\pi\)
0.323347 + 0.946280i \(0.395192\pi\)
\(102\) 0 0
\(103\) −0.578048 −0.0569568 −0.0284784 0.999594i \(-0.509066\pi\)
−0.0284784 + 0.999594i \(0.509066\pi\)
\(104\) −1.67333 −0.164083
\(105\) 0 0
\(106\) 7.53122 0.731497
\(107\) 16.2573 1.57165 0.785824 0.618450i \(-0.212239\pi\)
0.785824 + 0.618450i \(0.212239\pi\)
\(108\) 0 0
\(109\) 1.78191 0.170676 0.0853378 0.996352i \(-0.472803\pi\)
0.0853378 + 0.996352i \(0.472803\pi\)
\(110\) 26.5832 2.53461
\(111\) 0 0
\(112\) 0 0
\(113\) 7.52215 0.707624 0.353812 0.935316i \(-0.384885\pi\)
0.353812 + 0.935316i \(0.384885\pi\)
\(114\) 0 0
\(115\) 13.5603 1.26450
\(116\) 16.7363 1.55392
\(117\) 0 0
\(118\) 28.7760 2.64905
\(119\) 0 0
\(120\) 0 0
\(121\) 22.2801 2.02547
\(122\) −13.6378 −1.23471
\(123\) 0 0
\(124\) 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
\(129\) 0 0
\(130\) 4.60802 0.404150
\(131\) −9.69703 −0.847234 −0.423617 0.905842i \(-0.639240\pi\)
−0.423617 + 0.905842i \(0.639240\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −16.9028 −1.46018
\(135\) 0 0
\(136\) 2.74563 0.235436
\(137\) 15.4493 1.31993 0.659963 0.751298i \(-0.270572\pi\)
0.659963 + 0.751298i \(0.270572\pi\)
\(138\) 0 0
\(139\) 3.84912 0.326478 0.163239 0.986587i \(-0.447806\pi\)
0.163239 + 0.986587i \(0.447806\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 29.7731 2.49850
\(143\) 5.76889 0.482419
\(144\) 0 0
\(145\) −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 \(0.486432\pi\)
\(150\) 0 0
\(151\) −4.34336 −0.353458 −0.176729 0.984260i \(-0.556552\pi\)
−0.176729 + 0.984260i \(0.556552\pi\)
\(152\) 4.46906 0.362489
\(153\) 0 0
\(154\) 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
\(159\) 0 0
\(160\) −15.7251 −1.24318
\(161\) 0 0
\(162\) 0 0
\(163\) 6.79919 0.532553 0.266277 0.963897i \(-0.414207\pi\)
0.266277 + 0.963897i \(0.414207\pi\)
\(164\) 19.7552 1.54262
\(165\) 0 0
\(166\) 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
\(169\) 1.00000 0.0769231
\(170\) −7.56094 −0.579898
\(171\) 0 0
\(172\) −12.3677 −0.943029
\(173\) −25.1178 −1.90967 −0.954837 0.297131i \(-0.903970\pi\)
−0.954837 + 0.297131i \(0.903970\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −10.8436 −0.817364
\(177\) 0 0
\(178\) 2.50262 0.187579
\(179\) 1.89527 0.141659 0.0708294 0.997488i \(-0.477435\pi\)
0.0708294 + 0.997488i \(0.477435\pi\)
\(180\) 0 0
\(181\) 11.3595 0.844347 0.422174 0.906515i \(-0.361267\pi\)
0.422174 + 0.906515i \(0.361267\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 10.7506 0.792544
\(185\) −12.1314 −0.891922
\(186\) 0 0
\(187\) −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 \(-0.177774\pi\)
0.848055 + 0.529909i \(0.177774\pi\)
\(192\) 0 0
\(193\) −1.85210 −0.133317 −0.0666584 0.997776i \(-0.521234\pi\)
−0.0666584 + 0.997776i \(0.521234\pi\)
\(194\) 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 \(0.385626\pi\)
\(198\) 0 0
\(199\) 12.2492 0.868324 0.434162 0.900835i \(-0.357044\pi\)
0.434162 + 0.900835i \(0.357044\pi\)
\(200\) 0.912199 0.0645022
\(201\) 0 0
\(202\) −14.1892 −0.998347
\(203\) 0 0
\(204\) 0 0
\(205\) −15.0722 −1.05269
\(206\) 1.26201 0.0879281
\(207\) 0 0
\(208\) −1.87966 −0.130331
\(209\) −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
\(213\) 0 0
\(214\) −35.4932 −2.42626
\(215\) 9.43591 0.643523
\(216\) 0 0
\(217\) 0 0
\(218\) −3.89029 −0.263484
\(219\) 0 0
\(220\) −33.6847 −2.27102
\(221\) −1.64082 −0.110374
\(222\) 0 0
\(223\) 8.30577 0.556196 0.278098 0.960553i \(-0.410296\pi\)
0.278098 + 0.960553i \(0.410296\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −16.4225 −1.09241
\(227\) −13.2917 −0.882198 −0.441099 0.897458i \(-0.645411\pi\)
−0.441099 + 0.897458i \(0.645411\pi\)
\(228\) 0 0
\(229\) 17.9207 1.18423 0.592115 0.805853i \(-0.298293\pi\)
0.592115 + 0.805853i \(0.298293\pi\)
\(230\) −29.6051 −1.95210
\(231\) 0 0
\(232\) −10.1232 −0.664619
\(233\) −6.29968 −0.412705 −0.206353 0.978478i \(-0.566159\pi\)
−0.206353 + 0.978478i \(0.566159\pi\)
\(234\) 0 0
\(235\) 24.8867 1.62343
\(236\) −36.4633 −2.37356
\(237\) 0 0
\(238\) 0 0
\(239\) −9.41783 −0.609189 −0.304594 0.952482i \(-0.598521\pi\)
−0.304594 + 0.952482i \(0.598521\pi\)
\(240\) 0 0
\(241\) −19.9493 −1.28505 −0.642524 0.766266i \(-0.722113\pi\)
−0.642524 + 0.766266i \(0.722113\pi\)
\(242\) −48.6424 −3.12685
\(243\) 0 0
\(244\) 17.2811 1.10631
\(245\) 0 0
\(246\) 0 0
\(247\) −2.67077 −0.169937
\(248\) −8.57080 −0.544246
\(249\) 0 0
\(250\) −25.5521 −1.61606
\(251\) −1.22202 −0.0771330 −0.0385665 0.999256i \(-0.512279\pi\)
−0.0385665 + 0.999256i \(0.512279\pi\)
\(252\) 0 0
\(253\) −37.0633 −2.33015
\(254\) 42.1484 2.64463
\(255\) 0 0
\(256\) −2.06691 −0.129182
\(257\) 22.4240 1.39877 0.699386 0.714745i \(-0.253457\pi\)
0.699386 + 0.714745i \(0.253457\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −5.83901 −0.362120
\(261\) 0 0
\(262\) 21.1707 1.30793
\(263\) −8.99048 −0.554377 −0.277188 0.960816i \(-0.589403\pi\)
−0.277188 + 0.960816i \(0.589403\pi\)
\(264\) 0 0
\(265\) 7.28090 0.447262
\(266\) 0 0
\(267\) 0 0
\(268\) 21.4183 1.30833
\(269\) 22.2054 1.35389 0.676945 0.736034i \(-0.263303\pi\)
0.676945 + 0.736034i \(0.263303\pi\)
\(270\) 0 0
\(271\) −10.8423 −0.658624 −0.329312 0.944221i \(-0.606817\pi\)
−0.329312 + 0.944221i \(0.606817\pi\)
\(272\) 3.08419 0.187006
\(273\) 0 0
\(274\) −33.7293 −2.03766
\(275\) −3.14486 −0.189642
\(276\) 0 0
\(277\) 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 0
\(280\) 0 0
\(281\) −26.9071 −1.60514 −0.802572 0.596556i \(-0.796535\pi\)
−0.802572 + 0.596556i \(0.796535\pi\)
\(282\) 0 0
\(283\) −3.49717 −0.207885 −0.103943 0.994583i \(-0.533146\pi\)
−0.103943 + 0.994583i \(0.533146\pi\)
\(284\) −37.7267 −2.23867
\(285\) 0 0
\(286\) −12.5948 −0.744744
\(287\) 0 0
\(288\) 0 0
\(289\) −14.3077 −0.841630
\(290\) 27.8773 1.63701
\(291\) 0 0
\(292\) −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
\(297\) 0 0
\(298\) −2.27116 −0.131565
\(299\) −6.42469 −0.371549
\(300\) 0 0
\(301\) 0 0
\(302\) 9.48252 0.545657
\(303\) 0 0
\(304\) 5.02013 0.287924
\(305\) −13.1845 −0.754944
\(306\) 0 0
\(307\) 4.25772 0.243001 0.121501 0.992591i \(-0.461229\pi\)
0.121501 + 0.992591i \(0.461229\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 23.6024 1.34052
\(311\) 6.94043 0.393556 0.196778 0.980448i \(-0.436952\pi\)
0.196778 + 0.980448i \(0.436952\pi\)
\(312\) 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 \(0.533615\pi\)
\(318\) 0 0
\(319\) 34.9003 1.95404
\(320\) 26.3968 1.47563
\(321\) 0 0
\(322\) 0 0
\(323\) 4.38225 0.243835
\(324\) 0 0
\(325\) −0.545141 −0.0302390
\(326\) −14.8441 −0.822140
\(327\) 0 0
\(328\) −11.9492 −0.659785
\(329\) 0 0
\(330\) 0 0
\(331\) −19.9812 −1.09827 −0.549134 0.835734i \(-0.685042\pi\)
−0.549134 + 0.835734i \(0.685042\pi\)
\(332\) −13.7472 −0.754474
\(333\) 0 0
\(334\) 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 \(0.332019\pi\)
\(338\) −2.18322 −0.118751
\(339\) 0 0
\(340\) 9.58078 0.519591
\(341\) 29.5484 1.60013
\(342\) 0 0
\(343\) 0 0
\(344\) 7.48078 0.403337
\(345\) 0 0
\(346\) 54.8377 2.94810
\(347\) 0.796100 0.0427369 0.0213684 0.999772i \(-0.493198\pi\)
0.0213684 + 0.999772i \(0.493198\pi\)
\(348\) 0 0
\(349\) −11.3725 −0.608754 −0.304377 0.952552i \(-0.598448\pi\)
−0.304377 + 0.952552i \(0.598448\pi\)
\(350\) 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 \(0.721703\pi\)
\(354\) 0 0
\(355\) 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 0
\(361\) −11.8670 −0.624579
\(362\) −24.8003 −1.30348
\(363\) 0 0
\(364\) 0 0
\(365\) 32.7238 1.71284
\(366\) 0 0
\(367\) −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 0
\(370\) 26.4856 1.37692
\(371\) 0 0
\(372\) 0 0
\(373\) −16.0581 −0.831457 −0.415728 0.909489i \(-0.636473\pi\)
−0.415728 + 0.909489i \(0.636473\pi\)
\(374\) 20.6658 1.06860
\(375\) 0 0
\(376\) 19.7302 1.01751
\(377\) 6.04973 0.311577
\(378\) 0 0
\(379\) −5.36895 −0.275785 −0.137892 0.990447i \(-0.544033\pi\)
−0.137892 + 0.990447i \(0.544033\pi\)
\(380\) 15.5947 0.799989
\(381\) 0 0
\(382\) −51.1762 −2.61840
\(383\) −34.5075 −1.76325 −0.881625 0.471951i \(-0.843550\pi\)
−0.881625 + 0.471951i \(0.843550\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 4.04353 0.205810
\(387\) 0 0
\(388\) −19.2883 −0.979216
\(389\) −22.9210 −1.16214 −0.581070 0.813854i \(-0.697366\pi\)
−0.581070 + 0.813854i \(0.697366\pi\)
\(390\) 0 0
\(391\) 10.5418 0.533120
\(392\) 0 0
\(393\) 0 0
\(394\) −21.5502 −1.08568
\(395\) −2.36845 −0.119170
\(396\) 0 0
\(397\) 10.8294 0.543510 0.271755 0.962366i \(-0.412396\pi\)
0.271755 + 0.962366i \(0.412396\pi\)
\(398\) −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 \(0.886745\pi\)
\(402\) 0 0
\(403\) 5.12202 0.255146
\(404\) 17.9797 0.894523
\(405\) 0 0
\(406\) 0 0
\(407\) 33.1580 1.64358
\(408\) 0 0
\(409\) −20.4727 −1.01231 −0.506155 0.862443i \(-0.668934\pi\)
−0.506155 + 0.862443i \(0.668934\pi\)
\(410\) 32.9059 1.62511
\(411\) 0 0
\(412\) −1.59914 −0.0787840
\(413\) 0 0
\(414\) 0 0
\(415\) 10.4884 0.514854
\(416\) 7.45036 0.365284
\(417\) 0 0
\(418\) 33.6377 1.64527
\(419\) 15.4980 0.757127 0.378564 0.925575i \(-0.376418\pi\)
0.378564 + 0.925575i \(0.376418\pi\)
\(420\) 0 0
\(421\) 17.9390 0.874293 0.437147 0.899390i \(-0.355989\pi\)
0.437147 + 0.899390i \(0.355989\pi\)
\(422\) 1.61536 0.0786346
\(423\) 0 0
\(424\) 5.77229 0.280327
\(425\) 0.894480 0.0433886
\(426\) 0 0
\(427\) 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 \(0.672749\pi\)
\(432\) 0 0
\(433\) −14.9365 −0.717800 −0.358900 0.933376i \(-0.616848\pi\)
−0.358900 + 0.933376i \(0.616848\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 4.92955 0.236083
\(437\) 17.1589 0.820819
\(438\) 0 0
\(439\) −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
\(442\) 3.58227 0.170391
\(443\) 13.6534 0.648694 0.324347 0.945938i \(-0.394855\pi\)
0.324347 + 0.945938i \(0.394855\pi\)
\(444\) 0 0
\(445\) 2.41943 0.114692
\(446\) −18.1333 −0.858638
\(447\) 0 0
\(448\) 0 0
\(449\) 8.72412 0.411717 0.205858 0.978582i \(-0.434001\pi\)
0.205858 + 0.978582i \(0.434001\pi\)
\(450\) 0 0
\(451\) 41.1957 1.93983
\(452\) 20.8096 0.978803
\(453\) 0 0
\(454\) 29.0186 1.36191
\(455\) 0 0
\(456\) 0 0
\(457\) −30.4659 −1.42514 −0.712568 0.701603i \(-0.752468\pi\)
−0.712568 + 0.701603i \(0.752468\pi\)
\(458\) −39.1247 −1.82818
\(459\) 0 0
\(460\) 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 \(-0.340929\pi\)
0.479193 + 0.877709i \(0.340929\pi\)
\(464\) −11.3714 −0.527906
\(465\) 0 0
\(466\) 13.7536 0.637122
\(467\) −1.67149 −0.0773474 −0.0386737 0.999252i \(-0.512313\pi\)
−0.0386737 + 0.999252i \(0.512313\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −54.3332 −2.50620
\(471\) 0 0
\(472\) 22.0553 1.01518
\(473\) −25.7905 −1.18585
\(474\) 0 0
\(475\) 1.45595 0.0668034
\(476\) 0 0
\(477\) 0 0
\(478\) 20.5612 0.940447
\(479\) −10.6439 −0.486333 −0.243166 0.969985i \(-0.578186\pi\)
−0.243166 + 0.969985i \(0.578186\pi\)
\(480\) 0 0
\(481\) 5.74772 0.262073
\(482\) 43.5537 1.98382
\(483\) 0 0
\(484\) 61.6369 2.80168
\(485\) 14.7160 0.668217
\(486\) 0 0
\(487\) −31.8312 −1.44241 −0.721206 0.692721i \(-0.756412\pi\)
−0.721206 + 0.692721i \(0.756412\pi\)
\(488\) −10.4527 −0.473171
\(489\) 0 0
\(490\) 0 0
\(491\) −6.87077 −0.310074 −0.155037 0.987909i \(-0.549550\pi\)
−0.155037 + 0.987909i \(0.549550\pi\)
\(492\) 0 0
\(493\) −9.92654 −0.447069
\(494\) 5.83087 0.262343
\(495\) 0 0
\(496\) −9.62765 −0.432294
\(497\) 0 0
\(498\) 0 0
\(499\) 0.344335 0.0154145 0.00770727 0.999970i \(-0.497547\pi\)
0.00770727 + 0.999970i \(0.497547\pi\)
\(500\) 32.3782 1.44800
\(501\) 0 0
\(502\) 2.66793 0.119076
\(503\) −30.0808 −1.34123 −0.670617 0.741803i \(-0.733971\pi\)
−0.670617 + 0.741803i \(0.733971\pi\)
\(504\) 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 \(0.764339\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −20.2947 −0.896908
\(513\) 0 0
\(514\) −48.9565 −2.15938
\(515\) 1.22006 0.0537622
\(516\) 0 0
\(517\) −68.0210 −2.99156
\(518\) 0 0
\(519\) 0 0
\(520\) 3.53181 0.154880
\(521\) −32.7726 −1.43579 −0.717897 0.696149i \(-0.754895\pi\)
−0.717897 + 0.696149i \(0.754895\pi\)
\(522\) 0 0
\(523\) 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
\(528\) 0 0
\(529\) 18.2766 0.794636
\(530\) −15.8958 −0.690469
\(531\) 0 0
\(532\) 0 0
\(533\) 7.14100 0.309311
\(534\) 0 0
\(535\) −34.3134 −1.48350
\(536\) −12.9552 −0.559578
\(537\) 0 0
\(538\) −48.4794 −2.09009
\(539\) 0 0
\(540\) 0 0
\(541\) 13.1630 0.565920 0.282960 0.959132i \(-0.408684\pi\)
0.282960 + 0.959132i \(0.408684\pi\)
\(542\) 23.6712 1.01676
\(543\) 0 0
\(544\) −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 0
\(550\) 6.86592 0.292764
\(551\) −16.1574 −0.688330
\(552\) 0 0
\(553\) 0 0
\(554\) −12.8554 −0.546174
\(555\) 0 0
\(556\) 10.6484 0.451592
\(557\) 4.61284 0.195452 0.0977262 0.995213i \(-0.468843\pi\)
0.0977262 + 0.995213i \(0.468843\pi\)
\(558\) 0 0
\(559\) −4.47061 −0.189087
\(560\) 0 0
\(561\) 0 0
\(562\) 58.7441 2.47797
\(563\) −40.2805 −1.69762 −0.848811 0.528697i \(-0.822681\pi\)
−0.848811 + 0.528697i \(0.822681\pi\)
\(564\) 0 0
\(565\) −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 \(-0.548215\pi\)
−0.150892 + 0.988550i \(0.548215\pi\)
\(570\) 0 0
\(571\) 11.3820 0.476323 0.238161 0.971226i \(-0.423455\pi\)
0.238161 + 0.971226i \(0.423455\pi\)
\(572\) 15.9593 0.667294
\(573\) 0 0
\(574\) 0 0
\(575\) 3.50236 0.146059
\(576\) 0 0
\(577\) −35.6257 −1.48312 −0.741560 0.670887i \(-0.765914\pi\)
−0.741560 + 0.670887i \(0.765914\pi\)
\(578\) 31.2369 1.29928
\(579\) 0 0
\(580\) −35.3245 −1.46677
\(581\) 0 0
\(582\) 0 0
\(583\) −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
\(591\) 0 0
\(592\) −10.8038 −0.444032
\(593\) 24.1313 0.990953 0.495476 0.868621i \(-0.334994\pi\)
0.495476 + 0.868621i \(0.334994\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 2.87788 0.117883
\(597\) 0 0
\(598\) 14.0265 0.573587
\(599\) 48.5607 1.98414 0.992068 0.125703i \(-0.0401185\pi\)
0.992068 + 0.125703i \(0.0401185\pi\)
\(600\) 0 0
\(601\) −33.2069 −1.35454 −0.677270 0.735735i \(-0.736837\pi\)
−0.677270 + 0.735735i \(0.736837\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −12.0157 −0.488911
\(605\) −47.0257 −1.91186
\(606\) 0 0
\(607\) −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
\(611\) −11.7910 −0.477013
\(612\) 0 0
\(613\) −22.7991 −0.920847 −0.460423 0.887700i \(-0.652302\pi\)
−0.460423 + 0.887700i \(0.652302\pi\)
\(614\) −9.29554 −0.375138
\(615\) 0 0
\(616\) 0 0
\(617\) 42.5433 1.71273 0.856364 0.516373i \(-0.172718\pi\)
0.856364 + 0.516373i \(0.172718\pi\)
\(618\) 0 0
\(619\) −44.1285 −1.77367 −0.886836 0.462084i \(-0.847102\pi\)
−0.886836 + 0.462084i \(0.847102\pi\)
\(620\) −29.9075 −1.20112
\(621\) 0 0
\(622\) −15.1525 −0.607559
\(623\) 0 0
\(624\) 0 0
\(625\) −21.9771 −0.879085
\(626\) 21.7514 0.869361
\(627\) 0 0
\(628\) 0.929528 0.0370922
\(629\) −9.43098 −0.376038
\(630\) 0 0
\(631\) 9.09226 0.361957 0.180979 0.983487i \(-0.442074\pi\)
0.180979 + 0.983487i \(0.442074\pi\)
\(632\) −1.87771 −0.0746912
\(633\) 0 0
\(634\) 8.19461 0.325449
\(635\) 40.7475 1.61701
\(636\) 0 0
\(637\) 0 0
\(638\) −76.1950 −3.01659
\(639\) 0 0
\(640\) −26.1797 −1.03485
\(641\) 22.6642 0.895181 0.447591 0.894239i \(-0.352282\pi\)
0.447591 + 0.894239i \(0.352282\pi\)
\(642\) 0 0
\(643\) 24.3792 0.961422 0.480711 0.876879i \(-0.340379\pi\)
0.480711 + 0.876879i \(0.340379\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −9.56742 −0.376425
\(647\) 37.0364 1.45605 0.728026 0.685549i \(-0.240438\pi\)
0.728026 + 0.685549i \(0.240438\pi\)
\(648\) 0 0
\(649\) −76.0372 −2.98472
\(650\) 1.19016 0.0466820
\(651\) 0 0
\(652\) 18.8096 0.736641
\(653\) −18.7341 −0.733122 −0.366561 0.930394i \(-0.619465\pi\)
−0.366561 + 0.930394i \(0.619465\pi\)
\(654\) 0 0
\(655\) 20.4671 0.799715
\(656\) −13.4227 −0.524067
\(657\) 0 0
\(658\) 0 0
\(659\) 28.5206 1.11100 0.555502 0.831515i \(-0.312526\pi\)
0.555502 + 0.831515i \(0.312526\pi\)
\(660\) 0 0
\(661\) −19.9758 −0.776968 −0.388484 0.921455i \(-0.627001\pi\)
−0.388484 + 0.921455i \(0.627001\pi\)
\(662\) 43.6234 1.69547
\(663\) 0 0
\(664\) 8.31517 0.322691
\(665\) 0 0
\(666\) 0 0
\(667\) −38.8677 −1.50496
\(668\) −1.84978 −0.0715702
\(669\) 0 0
\(670\) 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
\(675\) 0 0
\(676\) 2.76645 0.106402
\(677\) 38.3396 1.47351 0.736755 0.676159i \(-0.236357\pi\)
0.736755 + 0.676159i \(0.236357\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −5.79507 −0.222231
\(681\) 0 0
\(682\) −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 0
\(685\) −32.6082 −1.24589
\(686\) 0 0
\(687\) 0 0
\(688\) 8.40322 0.320370
\(689\) −3.44959 −0.131419
\(690\) 0 0
\(691\) −40.9185 −1.55661 −0.778307 0.627884i \(-0.783921\pi\)
−0.778307 + 0.627884i \(0.783921\pi\)
\(692\) −69.4872 −2.64151
\(693\) 0 0
\(694\) −1.73806 −0.0659759
\(695\) −8.12415 −0.308167
\(696\) 0 0
\(697\) −11.7171 −0.443817
\(698\) 24.8286 0.939776
\(699\) 0 0
\(700\) 0 0
\(701\) 0.762896 0.0288142 0.0144071 0.999896i \(-0.495414\pi\)
0.0144071 + 0.999896i \(0.495414\pi\)
\(702\) 0 0
\(703\) −15.3508 −0.578967
\(704\) −72.1484 −2.71920
\(705\) 0 0
\(706\) 52.6304 1.98077
\(707\) 0 0
\(708\) 0 0
\(709\) 2.65281 0.0996284 0.0498142 0.998759i \(-0.484137\pi\)
0.0498142 + 0.998759i \(0.484137\pi\)
\(710\) −62.8407 −2.35837
\(711\) 0 0
\(712\) 1.91813 0.0718848
\(713\) −32.9074 −1.23239
\(714\) 0 0
\(715\) −12.1761 −0.455362
\(716\) 5.24315 0.195946
\(717\) 0 0
\(718\) 19.8058 0.739145
\(719\) −11.0443 −0.411883 −0.205942 0.978564i \(-0.566026\pi\)
−0.205942 + 0.978564i \(0.566026\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 25.9083 0.964206
\(723\) 0 0
\(724\) 31.4255 1.16792
\(725\) −3.29796 −0.122483
\(726\) 0 0
\(727\) 3.46566 0.128534 0.0642672 0.997933i \(-0.479529\pi\)
0.0642672 + 0.997933i \(0.479529\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −71.4432 −2.64423
\(731\) 7.33547 0.271312
\(732\) 0 0
\(733\) −29.0332 −1.07236 −0.536182 0.844102i \(-0.680134\pi\)
−0.536182 + 0.844102i \(0.680134\pi\)
\(734\) 68.8822 2.54249
\(735\) 0 0
\(736\) −47.8663 −1.76437
\(737\) 44.6637 1.64521
\(738\) 0 0
\(739\) 47.5522 1.74924 0.874618 0.484813i \(-0.161112\pi\)
0.874618 + 0.484813i \(0.161112\pi\)
\(740\) −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 \(0.574314\pi\)
\(744\) 0 0
\(745\) −2.19567 −0.0804432
\(746\) 35.0584 1.28358
\(747\) 0 0
\(748\) −26.1864 −0.957471
\(749\) 0 0
\(750\) 0 0
\(751\) −7.70781 −0.281262 −0.140631 0.990062i \(-0.544913\pi\)
−0.140631 + 0.990062i \(0.544913\pi\)
\(752\) 22.1631 0.808204
\(753\) 0 0
\(754\) −13.2079 −0.481003
\(755\) 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
\(759\) 0 0
\(760\) −9.43265 −0.342158
\(761\) 41.3296 1.49820 0.749098 0.662459i \(-0.230487\pi\)
0.749098 + 0.662459i \(0.230487\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 64.8475 2.34610
\(765\) 0 0
\(766\) 75.3374 2.72205
\(767\) −13.1805 −0.475922
\(768\) 0 0
\(769\) 36.7414 1.32493 0.662464 0.749094i \(-0.269511\pi\)
0.662464 + 0.749094i \(0.269511\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −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 0
\(775\) −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
\(783\) 0 0
\(784\) 0 0
\(785\) −0.709180 −0.0253117
\(786\) 0 0
\(787\) 36.0134 1.28374 0.641870 0.766814i \(-0.278159\pi\)
0.641870 + 0.766814i \(0.278159\pi\)
\(788\) 27.3071 0.972776
\(789\) 0 0
\(790\) 5.17085 0.183971
\(791\) 0 0
\(792\) 0 0
\(793\) 6.24666 0.221825
\(794\) −23.6429 −0.839054
\(795\) 0 0
\(796\) 33.8868 1.20109
\(797\) 18.8155 0.666478 0.333239 0.942842i \(-0.391858\pi\)
0.333239 + 0.942842i \(0.391858\pi\)
\(798\) 0 0
\(799\) 19.3469 0.684445
\(800\) −4.06150 −0.143596
\(801\) 0 0
\(802\) 81.9614 2.89416
\(803\) −89.4414 −3.15632
\(804\) 0 0
\(805\) 0 0
\(806\) −11.1825 −0.393886
\(807\) 0 0
\(808\) −10.8753 −0.382591
\(809\) 47.3322 1.66411 0.832056 0.554692i \(-0.187164\pi\)
0.832056 + 0.554692i \(0.187164\pi\)
\(810\) 0 0
\(811\) 19.2555 0.676153 0.338077 0.941119i \(-0.390224\pi\)
0.338077 + 0.941119i \(0.390224\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −72.3912 −2.53731
\(815\) −14.3507 −0.502684
\(816\) 0 0
\(817\) 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 \(0.757915\pi\)
\(822\) 0 0
\(823\) 24.5627 0.856201 0.428100 0.903731i \(-0.359183\pi\)
0.428100 + 0.903731i \(0.359183\pi\)
\(824\) 0.967262 0.0336962
\(825\) 0 0
\(826\) 0 0
\(827\) 0.580957 0.0202019 0.0101009 0.999949i \(-0.496785\pi\)
0.0101009 + 0.999949i \(0.496785\pi\)
\(828\) 0 0
\(829\) 14.6630 0.509268 0.254634 0.967038i \(-0.418045\pi\)
0.254634 + 0.967038i \(0.418045\pi\)
\(830\) −22.8984 −0.794815
\(831\) 0 0
\(832\) −12.5065 −0.433583
\(833\) 0 0
\(834\) 0 0
\(835\) 1.41129 0.0488396
\(836\) −42.6237 −1.47417
\(837\) 0 0
\(838\) −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
\(843\) 0 0
\(844\) −2.04689 −0.0704569
\(845\) −2.11065 −0.0726087
\(846\) 0 0
\(847\) 0 0
\(848\) 6.48406 0.222664
\(849\) 0 0
\(850\) −1.95285 −0.0669820
\(851\) −36.9273 −1.26585
\(852\) 0 0
\(853\) −0.587340 −0.0201101 −0.0100551 0.999949i \(-0.503201\pi\)
−0.0100551 + 0.999949i \(0.503201\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −27.2037 −0.929803
\(857\) −31.1680 −1.06468 −0.532340 0.846531i \(-0.678687\pi\)
−0.532340 + 0.846531i \(0.678687\pi\)
\(858\) 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 \(0.830566\pi\)
\(864\) 0 0
\(865\) 53.0150 1.80257
\(866\) 32.6096 1.10812
\(867\) 0 0
\(868\) 0 0
\(869\) 6.47352 0.219599
\(870\) 0 0
\(871\) 7.74216 0.262333
\(872\) −2.98171 −0.100973
\(873\) 0 0
\(874\) −37.4615 −1.26716
\(875\) 0 0
\(876\) 0 0
\(877\) 11.9495 0.403505 0.201753 0.979436i \(-0.435336\pi\)
0.201753 + 0.979436i \(0.435336\pi\)
\(878\) 7.84480 0.264749
\(879\) 0 0
\(880\) 22.8870 0.771520
\(881\) −54.6400 −1.84087 −0.920434 0.390898i \(-0.872165\pi\)
−0.920434 + 0.390898i \(0.872165\pi\)
\(882\) 0 0
\(883\) −26.7256 −0.899388 −0.449694 0.893183i \(-0.648467\pi\)
−0.449694 + 0.893183i \(0.648467\pi\)
\(884\) −4.53925 −0.152671
\(885\) 0 0
\(886\) −29.8084 −1.00143
\(887\) 5.82278 0.195510 0.0977549 0.995211i \(-0.468834\pi\)
0.0977549 + 0.995211i \(0.468834\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −5.28215 −0.177058
\(891\) 0 0
\(892\) 22.9775 0.769343
\(893\) 31.4910 1.05381
\(894\) 0 0
\(895\) −4.00025 −0.133714
\(896\) 0 0
\(897\) 0 0
\(898\) −19.0467 −0.635595
\(899\) 30.9868 1.03347
\(900\) 0 0
\(901\) 5.66017 0.188568
\(902\) −89.9392 −2.99465
\(903\) 0 0
\(904\) −12.5870 −0.418638
\(905\) −23.9760 −0.796990
\(906\) 0 0
\(907\) −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 0
\(910\) 0 0
\(911\) −5.94815 −0.197071 −0.0985355 0.995134i \(-0.531416\pi\)
−0.0985355 + 0.995134i \(0.531416\pi\)
\(912\) 0 0
\(913\) −28.6671 −0.948742
\(914\) 66.5138 2.20008
\(915\) 0 0
\(916\) 49.5766 1.63806
\(917\) 0 0
\(918\) 0 0
\(919\) 13.6480 0.450207 0.225103 0.974335i \(-0.427728\pi\)
0.225103 + 0.974335i \(0.427728\pi\)
\(920\) −22.6908 −0.748093
\(921\) 0 0
\(922\) 11.4019 0.375503
\(923\) −13.6372 −0.448875
\(924\) 0 0
\(925\) −3.13332 −0.103023
\(926\) −45.0224 −1.47953
\(927\) 0 0
\(928\) 45.0727 1.47958
\(929\) −36.3062 −1.19117 −0.595585 0.803293i \(-0.703080\pi\)
−0.595585 + 0.803293i \(0.703080\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −17.4277 −0.570864
\(933\) 0 0
\(934\) 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
\(939\) 0 0
\(940\) 68.8478 2.24557
\(941\) 31.9163 1.04044 0.520222 0.854031i \(-0.325849\pi\)
0.520222 + 0.854031i \(0.325849\pi\)
\(942\) 0 0
\(943\) −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 \(0.622836\pi\)
\(948\) 0 0
\(949\) −15.5041 −0.503284
\(950\) −3.17865 −0.103129
\(951\) 0 0
\(952\) 0 0
\(953\) −25.7340 −0.833607 −0.416803 0.908997i \(-0.636850\pi\)
−0.416803 + 0.908997i \(0.636850\pi\)
\(954\) 0 0
\(955\) −49.4752 −1.60098
\(956\) −26.0539 −0.842645
\(957\) 0 0
\(958\) 23.2380 0.750786
\(959\) 0 0
\(960\) 0 0
\(961\) −4.76494 −0.153708
\(962\) −12.5485 −0.404581
\(963\) 0 0
\(964\) −55.1887 −1.77751
\(965\) 3.90913 0.125839
\(966\) 0 0
\(967\) 16.2544 0.522706 0.261353 0.965243i \(-0.415831\pi\)
0.261353 + 0.965243i \(0.415831\pi\)
\(968\) −37.2819 −1.19829
\(969\) 0 0
\(970\) −32.1282 −1.03157
\(971\) −39.7465 −1.27553 −0.637763 0.770232i \(-0.720140\pi\)
−0.637763 + 0.770232i \(0.720140\pi\)
\(972\) 0 0
\(973\) 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 \(0.362408\pi\)
\(978\) 0 0
\(979\) −6.61286 −0.211348
\(980\) 0 0
\(981\) 0 0
\(982\) 15.0004 0.478682
\(983\) −12.2915 −0.392039 −0.196020 0.980600i \(-0.562802\pi\)
−0.196020 + 0.980600i \(0.562802\pi\)
\(984\) 0 0
\(985\) −20.8339 −0.663823
\(986\) 21.6718 0.690171
\(987\) 0 0
\(988\) −7.38854 −0.235061
\(989\) 28.7223 0.913315
\(990\) 0 0
\(991\) 11.4681 0.364298 0.182149 0.983271i \(-0.441695\pi\)
0.182149 + 0.983271i \(0.441695\pi\)
\(992\) 38.1609 1.21161
\(993\) 0 0
\(994\) 0 0
\(995\) −25.8539 −0.819623
\(996\) 0 0
\(997\) −5.30214 −0.167920 −0.0839602 0.996469i \(-0.526757\pi\)
−0.0839602 + 0.996469i \(0.526757\pi\)
\(998\) −0.751758 −0.0237965
\(999\) 0 0
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 5733.2.a.br.1.1 6
3.2 odd 2 637.2.a.n.1.6 yes 6
7.6 odd 2 5733.2.a.bu.1.1 6
21.2 odd 6 637.2.e.n.508.1 12
21.5 even 6 637.2.e.o.508.1 12
21.11 odd 6 637.2.e.n.79.1 12
21.17 even 6 637.2.e.o.79.1 12
21.20 even 2 637.2.a.m.1.6 6
39.38 odd 2 8281.2.a.cd.1.1 6
273.272 even 2 8281.2.a.cc.1.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
637.2.a.m.1.6 6 21.20 even 2
637.2.a.n.1.6 yes 6 3.2 odd 2
637.2.e.n.79.1 12 21.11 odd 6
637.2.e.n.508.1 12 21.2 odd 6
637.2.e.o.79.1 12 21.17 even 6
637.2.e.o.508.1 12 21.5 even 6
5733.2.a.br.1.1 6 1.1 even 1 trivial
5733.2.a.bu.1.1 6 7.6 odd 2
8281.2.a.cc.1.1 6 273.272 even 2
8281.2.a.cd.1.1 6 39.38 odd 2