Properties

Label 5733.2.a.bu.1.1
Level $5733$
Weight $2$
Character 5733.1
Self dual yes
Analytic conductor $45.778$
Analytic rank $0$
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: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.4507648.1
Defining polynomial: \( x^{6} - 2x^{5} - 5x^{4} + 8x^{3} + 7x^{2} - 6x - 1 \) Copy content Toggle raw display
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

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

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 \(-0.343548\pi\)
0.471957 + 0.881622i \(0.343548\pi\)
\(6\) 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 \(-0.436237\pi\)
0.198979 + 0.980004i \(0.436237\pi\)
\(18\) 0 0
\(19\) 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 \(-0.733629\pi\)
−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 \(-0.652140\pi\)
−0.459971 + 0.887934i \(0.652140\pi\)
\(32\) 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 \(-0.688285\pi\)
−0.557619 + 0.830097i \(0.688285\pi\)
\(42\) 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 \(-0.170497\pi\)
0.859947 + 0.510384i \(0.170497\pi\)
\(48\) 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 \(-0.171719\pi\)
0.857981 + 0.513682i \(0.171719\pi\)
\(60\) 0 0
\(61\) −6.24666 −0.799803 −0.399901 0.916558i \(-0.630956\pi\)
−0.399901 + 0.916558i \(0.630956\pi\)
\(62\) 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 \(-0.138134\pi\)
0.907308 + 0.420467i \(0.138134\pi\)
\(74\) −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 \(-0.412076\pi\)
0.272723 + 0.962093i \(0.412076\pi\)
\(84\) 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 \(-0.480650\pi\)
0.0607535 + 0.998153i \(0.480650\pi\)
\(90\) 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 \(-0.384834\pi\)
0.353961 + 0.935260i \(0.384834\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −1.50810 −0.150810
\(101\) −6.49919 −0.646694 −0.323347 0.946280i \(-0.604808\pi\)
−0.323347 + 0.946280i \(0.604808\pi\)
\(102\) 0 0
\(103\) 0.578048 0.0569568 0.0284784 0.999594i \(-0.490934\pi\)
0.0284784 + 0.999594i \(0.490934\pi\)
\(104\) 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 \(-0.360760\pi\)
0.423617 + 0.905842i \(0.360760\pi\)
\(132\) 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 \(-0.552194\pi\)
−0.163239 + 0.986587i \(0.552194\pi\)
\(140\) 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 \(-0.504268\pi\)
−0.0134079 + 0.999910i \(0.504268\pi\)
\(158\) −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 \(-0.491764\pi\)
0.0258708 + 0.999665i \(0.491764\pi\)
\(168\) 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 \(-0.0960298\pi\)
0.954837 + 0.297131i \(0.0960298\pi\)
\(174\) 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 \(-0.638733\pi\)
−0.422174 + 0.906515i \(0.638733\pi\)
\(182\) 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 \(-0.642956\pi\)
−0.434162 + 0.900835i \(0.642956\pi\)
\(200\) 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 \(-0.589704\pi\)
−0.278098 + 0.960553i \(0.589704\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −16.4225 −1.09241
\(227\) 13.2917 0.882198 0.441099 0.897458i \(-0.354589\pi\)
0.441099 + 0.897458i \(0.354589\pi\)
\(228\) 0 0
\(229\) −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 \(-0.566159\pi\)
−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 \(-0.277887\pi\)
0.642524 + 0.766266i \(0.277887\pi\)
\(242\) −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 \(-0.487721\pi\)
0.0385665 + 0.999256i \(0.487721\pi\)
\(252\) 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 \(-0.746543\pi\)
−0.699386 + 0.714745i \(0.746543\pi\)
\(258\) 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 \(-0.736697\pi\)
−0.676945 + 0.736034i \(0.736697\pi\)
\(270\) 0 0
\(271\) 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
\(276\) 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 \(-0.466854\pi\)
0.103943 + 0.994583i \(0.466854\pi\)
\(284\) −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 \(-0.490653\pi\)
0.0293589 + 0.999569i \(0.490653\pi\)
\(294\) 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 \(-0.538771\pi\)
−0.121501 + 0.992591i \(0.538771\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 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 \(-0.409145\pi\)
0.281571 + 0.959540i \(0.409145\pi\)
\(314\) 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 \(-0.401552\pi\)
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 \(-0.278297\pi\)
0.641537 + 0.767092i \(0.278297\pi\)
\(354\) 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 \(-0.192037\pi\)
0.823467 + 0.567364i \(0.192037\pi\)
\(368\) 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 \(-0.156450\pi\)
0.881625 + 0.471951i \(0.156450\pi\)
\(384\) 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 \(-0.587604\pi\)
−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 \(0.886745\pi\)
\(402\) 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 \(-0.331066\pi\)
0.506155 + 0.862443i \(0.331066\pi\)
\(410\) 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 \(-0.623582\pi\)
−0.378564 + 0.925575i \(0.623582\pi\)
\(420\) 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 \(-0.383152\pi\)
0.358900 + 0.933376i \(0.383152\pi\)
\(434\) 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 \(-0.472672\pi\)
0.0857476 + 0.996317i \(0.472672\pi\)
\(440\) −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 \(-0.461191\pi\)
0.121619 + 0.992577i \(0.461191\pi\)
\(462\) 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 \(-0.487687\pi\)
0.0386737 + 0.999252i \(0.487687\pi\)
\(468\) 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 \(-0.421814\pi\)
0.243166 + 0.969985i \(0.421814\pi\)
\(480\) 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 \(-0.266029\pi\)
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 \(-0.235661\pi\)
0.738232 + 0.674547i \(0.235661\pi\)
\(510\) 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 \(-0.245105\pi\)
0.717897 + 0.696149i \(0.245105\pi\)
\(522\) 0 0
\(523\) −25.2616 −1.10461 −0.552307 0.833641i \(-0.686252\pi\)
−0.552307 + 0.833641i \(0.686252\pi\)
\(524\) 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 \(-0.177319\pi\)
0.848811 + 0.528697i \(0.177319\pi\)
\(564\) 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 \(-0.234086\pi\)
0.741560 + 0.670887i \(0.234086\pi\)
\(578\) 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 \(-0.472721\pi\)
0.0855938 + 0.996330i \(0.472721\pi\)
\(588\) 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 \(-0.665006\pi\)
−0.495476 + 0.868621i \(0.665006\pi\)
\(594\) 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 \(-0.263163\pi\)
0.677270 + 0.735735i \(0.263163\pi\)
\(602\) 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 \(-0.354160\pi\)
0.442308 + 0.896863i \(0.354160\pi\)
\(608\) 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 \(-0.152898\pi\)
0.886836 + 0.462084i \(0.152898\pi\)
\(620\) −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 \(-0.659621\pi\)
−0.480711 + 0.876879i \(0.659621\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −9.56742 −0.376425
\(647\) −37.0364 −1.45605 −0.728026 0.685549i \(-0.759562\pi\)
−0.728026 + 0.685549i \(0.759562\pi\)
\(648\) 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 \(-0.372999\pi\)
0.388484 + 0.921455i \(0.372999\pi\)
\(662\) 43.6234 1.69547
\(663\) 0 0
\(664\) −8.31517 −0.322691
\(665\) 0 0
\(666\) 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 \(-0.763643\pi\)
−0.736755 + 0.676159i \(0.763643\pi\)
\(678\) 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 \(-0.216079\pi\)
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
\(696\) 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 \(-0.433974\pi\)
0.205942 + 0.978564i \(0.433974\pi\)
\(720\) 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 \(-0.520471\pi\)
−0.0642672 + 0.997933i \(0.520471\pi\)
\(728\) 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 \(-0.319866\pi\)
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
\(738\) 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 \(-0.769513\pi\)
−0.749098 + 0.662459i \(0.769513\pi\)
\(762\) 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 \(-0.730489\pi\)
−0.662464 + 0.749094i \(0.730489\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −5.12373 −0.184407
\(773\) −23.9672 −0.862040 −0.431020 0.902342i \(-0.641846\pi\)
−0.431020 + 0.902342i \(0.641846\pi\)
\(774\) 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 \(-0.721841\pi\)
−0.641870 + 0.766814i \(0.721841\pi\)
\(788\) 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 \(-0.608142\pi\)
−0.333239 + 0.942842i \(0.608142\pi\)
\(798\) 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 \(-0.609776\pi\)
−0.338077 + 0.941119i \(0.609776\pi\)
\(812\) 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 \(-0.581955\pi\)
−0.254634 + 0.967038i \(0.581955\pi\)
\(830\) −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 \(-0.354210\pi\)
0.442166 + 0.896933i \(0.354210\pi\)
\(840\) 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 \(-0.496799\pi\)
0.0100551 + 0.999949i \(0.496799\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −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 \(-0.372866\pi\)
0.388867 + 0.921294i \(0.372866\pi\)
\(860\) −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 \(-0.127835\pi\)
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 \(0.648467\pi\)
\(884\) −4.53925 −0.152671
\(885\) 0 0
\(886\) −29.8084 −1.00143
\(887\) −5.82278 −0.195510 −0.0977549 0.995211i \(-0.531166\pi\)
−0.0977549 + 0.995211i \(0.531166\pi\)
\(888\) 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 \(-0.296920\pi\)
0.595585 + 0.803293i \(0.296920\pi\)
\(930\) 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 \(-0.649677\pi\)
−0.453087 + 0.891466i \(0.649677\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 68.8478 2.24557
\(941\) −31.9163 −1.04044 −0.520222 0.854031i \(-0.674151\pi\)
−0.520222 + 0.854031i \(0.674151\pi\)
\(942\) 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 \(-0.279860\pi\)
0.637763 + 0.770232i \(0.279860\pi\)
\(972\) 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 \(-0.437198\pi\)
0.196020 + 0.980600i \(0.437198\pi\)
\(984\) 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 \(-0.473243\pi\)
0.0839602 + 0.996469i \(0.473243\pi\)
\(998\) −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.bu.1.1 6
3.2 odd 2 637.2.a.m.1.6 6
7.6 odd 2 5733.2.a.br.1.1 6
21.2 odd 6 637.2.e.o.508.1 12
21.5 even 6 637.2.e.n.508.1 12
21.11 odd 6 637.2.e.o.79.1 12
21.17 even 6 637.2.e.n.79.1 12
21.20 even 2 637.2.a.n.1.6 yes 6
39.38 odd 2 8281.2.a.cc.1.1 6
273.272 even 2 8281.2.a.cd.1.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
637.2.a.m.1.6 6 3.2 odd 2
637.2.a.n.1.6 yes 6 21.20 even 2
637.2.e.n.79.1 12 21.17 even 6
637.2.e.n.508.1 12 21.5 even 6
637.2.e.o.79.1 12 21.11 odd 6
637.2.e.o.508.1 12 21.2 odd 6
5733.2.a.br.1.1 6 7.6 odd 2
5733.2.a.bu.1.1 6 1.1 even 1 trivial
8281.2.a.cc.1.1 6 39.38 odd 2
8281.2.a.cd.1.1 6 273.272 even 2