Properties

Label 567.2.a.d.1.3
Level $567$
Weight $2$
Character 567.1
Self dual yes
Analytic conductor $4.528$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 567 = 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 567.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.52751779461\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.321.1
Defining polynomial: \(x^{3} - x^{2} - 4 x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 63)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.69963\) of defining polynomial
Character \(\chi\) \(=\) 567.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.69963 q^{2} +0.888736 q^{4} -3.58836 q^{5} -1.00000 q^{7} -1.88874 q^{8} +O(q^{10})\) \(q+1.69963 q^{2} +0.888736 q^{4} -3.58836 q^{5} -1.00000 q^{7} -1.88874 q^{8} -6.09888 q^{10} -2.81089 q^{11} +1.00000 q^{13} -1.69963 q^{14} -4.98762 q^{16} -4.11126 q^{17} -0.888736 q^{19} -3.18911 q^{20} -4.77747 q^{22} +5.87636 q^{23} +7.87636 q^{25} +1.69963 q^{26} -0.888736 q^{28} -1.69963 q^{29} -6.98762 q^{31} -4.69963 q^{32} -6.98762 q^{34} +3.58836 q^{35} +4.76509 q^{37} -1.51052 q^{38} +6.77747 q^{40} -5.41164 q^{41} +5.21015 q^{43} -2.49814 q^{44} +9.98762 q^{46} -2.66621 q^{47} +1.00000 q^{49} +13.3869 q^{50} +0.888736 q^{52} -0.123644 q^{53} +10.0865 q^{55} +1.88874 q^{56} -2.88874 q^{58} -8.87636 q^{59} +3.87636 q^{61} -11.8764 q^{62} +1.98762 q^{64} -3.58836 q^{65} +12.3090 q^{67} -3.65383 q^{68} +6.09888 q^{70} -2.87636 q^{71} -10.6414 q^{73} +8.09888 q^{74} -0.789851 q^{76} +2.81089 q^{77} -7.08650 q^{79} +17.8974 q^{80} -9.19777 q^{82} -4.11126 q^{83} +14.7527 q^{85} +8.85532 q^{86} +5.30903 q^{88} +9.60940 q^{89} -1.00000 q^{91} +5.22253 q^{92} -4.53156 q^{94} +3.18911 q^{95} +7.32141 q^{97} +1.69963 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - q^{2} + 3q^{4} - 5q^{5} - 3q^{7} - 6q^{8} + O(q^{10}) \) \( 3q - q^{2} + 3q^{4} - 5q^{5} - 3q^{7} - 6q^{8} - 2q^{11} + 3q^{13} + q^{14} + 3q^{16} - 12q^{17} - 3q^{19} - 16q^{20} - 15q^{22} + 6q^{25} - q^{26} - 3q^{28} + q^{29} - 3q^{31} - 8q^{32} - 3q^{34} + 5q^{35} - 3q^{37} + 8q^{38} + 21q^{40} - 22q^{41} - 3q^{43} + 23q^{44} + 12q^{46} - 9q^{47} + 3q^{49} + 10q^{50} + 3q^{52} - 18q^{53} - 6q^{55} + 6q^{56} - 9q^{58} - 9q^{59} - 6q^{61} - 18q^{62} - 12q^{64} - 5q^{65} + 6q^{68} + 9q^{71} + 3q^{73} + 6q^{74} - 21q^{76} + 2q^{77} + 15q^{79} + 11q^{80} + 9q^{82} - 12q^{83} + 9q^{85} + 34q^{86} - 21q^{88} - 2q^{89} - 3q^{91} + 15q^{92} + 24q^{94} + 16q^{95} + 3q^{97} - q^{98} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.69963 1.20182 0.600909 0.799317i \(-0.294805\pi\)
0.600909 + 0.799317i \(0.294805\pi\)
\(3\) 0 0
\(4\) 0.888736 0.444368
\(5\) −3.58836 −1.60477 −0.802383 0.596810i \(-0.796435\pi\)
−0.802383 + 0.596810i \(0.796435\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) −1.88874 −0.667769
\(9\) 0 0
\(10\) −6.09888 −1.92864
\(11\) −2.81089 −0.847516 −0.423758 0.905775i \(-0.639289\pi\)
−0.423758 + 0.905775i \(0.639289\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350 0.138675 0.990338i \(-0.455716\pi\)
0.138675 + 0.990338i \(0.455716\pi\)
\(14\) −1.69963 −0.454245
\(15\) 0 0
\(16\) −4.98762 −1.24691
\(17\) −4.11126 −0.997128 −0.498564 0.866853i \(-0.666139\pi\)
−0.498564 + 0.866853i \(0.666139\pi\)
\(18\) 0 0
\(19\) −0.888736 −0.203890 −0.101945 0.994790i \(-0.532507\pi\)
−0.101945 + 0.994790i \(0.532507\pi\)
\(20\) −3.18911 −0.713106
\(21\) 0 0
\(22\) −4.77747 −1.01856
\(23\) 5.87636 1.22530 0.612652 0.790352i \(-0.290103\pi\)
0.612652 + 0.790352i \(0.290103\pi\)
\(24\) 0 0
\(25\) 7.87636 1.57527
\(26\) 1.69963 0.333325
\(27\) 0 0
\(28\) −0.888736 −0.167955
\(29\) −1.69963 −0.315613 −0.157807 0.987470i \(-0.550442\pi\)
−0.157807 + 0.987470i \(0.550442\pi\)
\(30\) 0 0
\(31\) −6.98762 −1.25501 −0.627507 0.778611i \(-0.715925\pi\)
−0.627507 + 0.778611i \(0.715925\pi\)
\(32\) −4.69963 −0.830785
\(33\) 0 0
\(34\) −6.98762 −1.19837
\(35\) 3.58836 0.606544
\(36\) 0 0
\(37\) 4.76509 0.783376 0.391688 0.920098i \(-0.371891\pi\)
0.391688 + 0.920098i \(0.371891\pi\)
\(38\) −1.51052 −0.245039
\(39\) 0 0
\(40\) 6.77747 1.07161
\(41\) −5.41164 −0.845156 −0.422578 0.906327i \(-0.638875\pi\)
−0.422578 + 0.906327i \(0.638875\pi\)
\(42\) 0 0
\(43\) 5.21015 0.794540 0.397270 0.917702i \(-0.369958\pi\)
0.397270 + 0.917702i \(0.369958\pi\)
\(44\) −2.49814 −0.376609
\(45\) 0 0
\(46\) 9.98762 1.47259
\(47\) −2.66621 −0.388906 −0.194453 0.980912i \(-0.562293\pi\)
−0.194453 + 0.980912i \(0.562293\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 13.3869 1.89319
\(51\) 0 0
\(52\) 0.888736 0.123245
\(53\) −0.123644 −0.0169838 −0.00849190 0.999964i \(-0.502703\pi\)
−0.00849190 + 0.999964i \(0.502703\pi\)
\(54\) 0 0
\(55\) 10.0865 1.36006
\(56\) 1.88874 0.252393
\(57\) 0 0
\(58\) −2.88874 −0.379310
\(59\) −8.87636 −1.15560 −0.577802 0.816177i \(-0.696089\pi\)
−0.577802 + 0.816177i \(0.696089\pi\)
\(60\) 0 0
\(61\) 3.87636 0.496317 0.248158 0.968720i \(-0.420175\pi\)
0.248158 + 0.968720i \(0.420175\pi\)
\(62\) −11.8764 −1.50830
\(63\) 0 0
\(64\) 1.98762 0.248453
\(65\) −3.58836 −0.445082
\(66\) 0 0
\(67\) 12.3090 1.50379 0.751894 0.659284i \(-0.229141\pi\)
0.751894 + 0.659284i \(0.229141\pi\)
\(68\) −3.65383 −0.443092
\(69\) 0 0
\(70\) 6.09888 0.728956
\(71\) −2.87636 −0.341361 −0.170680 0.985326i \(-0.554597\pi\)
−0.170680 + 0.985326i \(0.554597\pi\)
\(72\) 0 0
\(73\) −10.6414 −1.24549 −0.622744 0.782426i \(-0.713982\pi\)
−0.622744 + 0.782426i \(0.713982\pi\)
\(74\) 8.09888 0.941476
\(75\) 0 0
\(76\) −0.789851 −0.0906022
\(77\) 2.81089 0.320331
\(78\) 0 0
\(79\) −7.08650 −0.797294 −0.398647 0.917104i \(-0.630520\pi\)
−0.398647 + 0.917104i \(0.630520\pi\)
\(80\) 17.8974 2.00099
\(81\) 0 0
\(82\) −9.19777 −1.01572
\(83\) −4.11126 −0.451270 −0.225635 0.974212i \(-0.572446\pi\)
−0.225635 + 0.974212i \(0.572446\pi\)
\(84\) 0 0
\(85\) 14.7527 1.60016
\(86\) 8.85532 0.954893
\(87\) 0 0
\(88\) 5.30903 0.565945
\(89\) 9.60940 1.01859 0.509297 0.860591i \(-0.329905\pi\)
0.509297 + 0.860591i \(0.329905\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) 5.22253 0.544486
\(93\) 0 0
\(94\) −4.53156 −0.467395
\(95\) 3.18911 0.327196
\(96\) 0 0
\(97\) 7.32141 0.743377 0.371688 0.928358i \(-0.378779\pi\)
0.371688 + 0.928358i \(0.378779\pi\)
\(98\) 1.69963 0.171688
\(99\) 0 0
\(100\) 7.00000 0.700000
\(101\) 3.46472 0.344753 0.172376 0.985031i \(-0.444856\pi\)
0.172376 + 0.985031i \(0.444856\pi\)
\(102\) 0 0
\(103\) −15.8764 −1.56434 −0.782172 0.623063i \(-0.785888\pi\)
−0.782172 + 0.623063i \(0.785888\pi\)
\(104\) −1.88874 −0.185206
\(105\) 0 0
\(106\) −0.210149 −0.0204114
\(107\) −5.35346 −0.517538 −0.258769 0.965939i \(-0.583317\pi\)
−0.258769 + 0.965939i \(0.583317\pi\)
\(108\) 0 0
\(109\) −18.8640 −1.80684 −0.903421 0.428755i \(-0.858952\pi\)
−0.903421 + 0.428755i \(0.858952\pi\)
\(110\) 17.1433 1.63455
\(111\) 0 0
\(112\) 4.98762 0.471286
\(113\) −18.5512 −1.74515 −0.872576 0.488478i \(-0.837552\pi\)
−0.872576 + 0.488478i \(0.837552\pi\)
\(114\) 0 0
\(115\) −21.0865 −1.96633
\(116\) −1.51052 −0.140248
\(117\) 0 0
\(118\) −15.0865 −1.38883
\(119\) 4.11126 0.376879
\(120\) 0 0
\(121\) −3.09888 −0.281717
\(122\) 6.58836 0.596482
\(123\) 0 0
\(124\) −6.21015 −0.557688
\(125\) −10.3214 −0.923175
\(126\) 0 0
\(127\) 9.98762 0.886258 0.443129 0.896458i \(-0.353868\pi\)
0.443129 + 0.896458i \(0.353868\pi\)
\(128\) 12.7775 1.12938
\(129\) 0 0
\(130\) −6.09888 −0.534908
\(131\) 16.0531 1.40256 0.701282 0.712884i \(-0.252611\pi\)
0.701282 + 0.712884i \(0.252611\pi\)
\(132\) 0 0
\(133\) 0.888736 0.0770632
\(134\) 20.9208 1.80728
\(135\) 0 0
\(136\) 7.76509 0.665851
\(137\) −12.9876 −1.10961 −0.554804 0.831981i \(-0.687207\pi\)
−0.554804 + 0.831981i \(0.687207\pi\)
\(138\) 0 0
\(139\) 1.11126 0.0942562 0.0471281 0.998889i \(-0.484993\pi\)
0.0471281 + 0.998889i \(0.484993\pi\)
\(140\) 3.18911 0.269529
\(141\) 0 0
\(142\) −4.88874 −0.410254
\(143\) −2.81089 −0.235059
\(144\) 0 0
\(145\) 6.09888 0.506485
\(146\) −18.0865 −1.49685
\(147\) 0 0
\(148\) 4.23491 0.348107
\(149\) 8.43268 0.690832 0.345416 0.938450i \(-0.387738\pi\)
0.345416 + 0.938450i \(0.387738\pi\)
\(150\) 0 0
\(151\) −14.8516 −1.20861 −0.604303 0.796755i \(-0.706548\pi\)
−0.604303 + 0.796755i \(0.706548\pi\)
\(152\) 1.67859 0.136151
\(153\) 0 0
\(154\) 4.77747 0.384980
\(155\) 25.0741 2.01400
\(156\) 0 0
\(157\) 2.88874 0.230546 0.115273 0.993334i \(-0.463226\pi\)
0.115273 + 0.993334i \(0.463226\pi\)
\(158\) −12.0444 −0.958203
\(159\) 0 0
\(160\) 16.8640 1.33321
\(161\) −5.87636 −0.463122
\(162\) 0 0
\(163\) −10.3090 −0.807466 −0.403733 0.914877i \(-0.632287\pi\)
−0.403733 + 0.914877i \(0.632287\pi\)
\(164\) −4.80951 −0.375560
\(165\) 0 0
\(166\) −6.98762 −0.542345
\(167\) 12.1520 0.940348 0.470174 0.882574i \(-0.344191\pi\)
0.470174 + 0.882574i \(0.344191\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 25.0741 1.92310
\(171\) 0 0
\(172\) 4.63045 0.353068
\(173\) −6.60940 −0.502504 −0.251252 0.967922i \(-0.580842\pi\)
−0.251252 + 0.967922i \(0.580842\pi\)
\(174\) 0 0
\(175\) −7.87636 −0.595397
\(176\) 14.0197 1.05677
\(177\) 0 0
\(178\) 16.3324 1.22417
\(179\) −3.84294 −0.287234 −0.143617 0.989633i \(-0.545873\pi\)
−0.143617 + 0.989633i \(0.545873\pi\)
\(180\) 0 0
\(181\) 18.5426 1.37826 0.689129 0.724639i \(-0.257993\pi\)
0.689129 + 0.724639i \(0.257993\pi\)
\(182\) −1.69963 −0.125985
\(183\) 0 0
\(184\) −11.0989 −0.818221
\(185\) −17.0989 −1.25713
\(186\) 0 0
\(187\) 11.5563 0.845082
\(188\) −2.36955 −0.172818
\(189\) 0 0
\(190\) 5.42030 0.393230
\(191\) 4.63416 0.335316 0.167658 0.985845i \(-0.446380\pi\)
0.167658 + 0.985845i \(0.446380\pi\)
\(192\) 0 0
\(193\) −25.2967 −1.82089 −0.910446 0.413627i \(-0.864262\pi\)
−0.910446 + 0.413627i \(0.864262\pi\)
\(194\) 12.4437 0.893404
\(195\) 0 0
\(196\) 0.888736 0.0634811
\(197\) 10.7207 0.763816 0.381908 0.924200i \(-0.375267\pi\)
0.381908 + 0.924200i \(0.375267\pi\)
\(198\) 0 0
\(199\) −8.76647 −0.621439 −0.310719 0.950502i \(-0.600570\pi\)
−0.310719 + 0.950502i \(0.600570\pi\)
\(200\) −14.8764 −1.05192
\(201\) 0 0
\(202\) 5.88874 0.414330
\(203\) 1.69963 0.119291
\(204\) 0 0
\(205\) 19.4189 1.35628
\(206\) −26.9839 −1.88006
\(207\) 0 0
\(208\) −4.98762 −0.345829
\(209\) 2.49814 0.172800
\(210\) 0 0
\(211\) 10.5302 0.724928 0.362464 0.931998i \(-0.381936\pi\)
0.362464 + 0.931998i \(0.381936\pi\)
\(212\) −0.109887 −0.00754705
\(213\) 0 0
\(214\) −9.09888 −0.621987
\(215\) −18.6959 −1.27505
\(216\) 0 0
\(217\) 6.98762 0.474351
\(218\) −32.0617 −2.17150
\(219\) 0 0
\(220\) 8.96424 0.604369
\(221\) −4.11126 −0.276554
\(222\) 0 0
\(223\) −5.66758 −0.379530 −0.189765 0.981830i \(-0.560773\pi\)
−0.189765 + 0.981830i \(0.560773\pi\)
\(224\) 4.69963 0.314007
\(225\) 0 0
\(226\) −31.5302 −2.09736
\(227\) −11.0989 −0.736659 −0.368329 0.929695i \(-0.620070\pi\)
−0.368329 + 0.929695i \(0.620070\pi\)
\(228\) 0 0
\(229\) 19.6428 1.29803 0.649017 0.760774i \(-0.275180\pi\)
0.649017 + 0.760774i \(0.275180\pi\)
\(230\) −35.8392 −2.36317
\(231\) 0 0
\(232\) 3.21015 0.210757
\(233\) 8.96286 0.587177 0.293588 0.955932i \(-0.405151\pi\)
0.293588 + 0.955932i \(0.405151\pi\)
\(234\) 0 0
\(235\) 9.56732 0.624103
\(236\) −7.88874 −0.513513
\(237\) 0 0
\(238\) 6.98762 0.452940
\(239\) 11.2225 0.725925 0.362963 0.931804i \(-0.381765\pi\)
0.362963 + 0.931804i \(0.381765\pi\)
\(240\) 0 0
\(241\) −6.98624 −0.450023 −0.225012 0.974356i \(-0.572242\pi\)
−0.225012 + 0.974356i \(0.572242\pi\)
\(242\) −5.26695 −0.338572
\(243\) 0 0
\(244\) 3.44506 0.220547
\(245\) −3.58836 −0.229252
\(246\) 0 0
\(247\) −0.888736 −0.0565489
\(248\) 13.1978 0.838059
\(249\) 0 0
\(250\) −17.5426 −1.10949
\(251\) 4.62041 0.291638 0.145819 0.989311i \(-0.453418\pi\)
0.145819 + 0.989311i \(0.453418\pi\)
\(252\) 0 0
\(253\) −16.5178 −1.03847
\(254\) 16.9752 1.06512
\(255\) 0 0
\(256\) 17.7417 1.10886
\(257\) −1.42402 −0.0888277 −0.0444138 0.999013i \(-0.514142\pi\)
−0.0444138 + 0.999013i \(0.514142\pi\)
\(258\) 0 0
\(259\) −4.76509 −0.296088
\(260\) −3.18911 −0.197780
\(261\) 0 0
\(262\) 27.2843 1.68563
\(263\) 16.2632 1.00283 0.501417 0.865206i \(-0.332812\pi\)
0.501417 + 0.865206i \(0.332812\pi\)
\(264\) 0 0
\(265\) 0.443679 0.0272550
\(266\) 1.51052 0.0926160
\(267\) 0 0
\(268\) 10.9395 0.668235
\(269\) −18.6538 −1.13734 −0.568672 0.822564i \(-0.692543\pi\)
−0.568672 + 0.822564i \(0.692543\pi\)
\(270\) 0 0
\(271\) 3.96286 0.240727 0.120363 0.992730i \(-0.461594\pi\)
0.120363 + 0.992730i \(0.461594\pi\)
\(272\) 20.5054 1.24332
\(273\) 0 0
\(274\) −22.0741 −1.33355
\(275\) −22.1396 −1.33507
\(276\) 0 0
\(277\) −2.33379 −0.140224 −0.0701120 0.997539i \(-0.522336\pi\)
−0.0701120 + 0.997539i \(0.522336\pi\)
\(278\) 1.88874 0.113279
\(279\) 0 0
\(280\) −6.77747 −0.405031
\(281\) 27.9949 1.67004 0.835018 0.550223i \(-0.185457\pi\)
0.835018 + 0.550223i \(0.185457\pi\)
\(282\) 0 0
\(283\) 10.3200 0.613462 0.306731 0.951796i \(-0.400765\pi\)
0.306731 + 0.951796i \(0.400765\pi\)
\(284\) −2.55632 −0.151690
\(285\) 0 0
\(286\) −4.77747 −0.282498
\(287\) 5.41164 0.319439
\(288\) 0 0
\(289\) −0.0975070 −0.00573571
\(290\) 10.3658 0.608703
\(291\) 0 0
\(292\) −9.45744 −0.553455
\(293\) −30.6959 −1.79327 −0.896637 0.442766i \(-0.853997\pi\)
−0.896637 + 0.442766i \(0.853997\pi\)
\(294\) 0 0
\(295\) 31.8516 1.85447
\(296\) −9.00000 −0.523114
\(297\) 0 0
\(298\) 14.3324 0.830255
\(299\) 5.87636 0.339838
\(300\) 0 0
\(301\) −5.21015 −0.300308
\(302\) −25.2422 −1.45252
\(303\) 0 0
\(304\) 4.43268 0.254231
\(305\) −13.9098 −0.796471
\(306\) 0 0
\(307\) −11.4437 −0.653125 −0.326563 0.945176i \(-0.605890\pi\)
−0.326563 + 0.945176i \(0.605890\pi\)
\(308\) 2.49814 0.142345
\(309\) 0 0
\(310\) 42.6167 2.42047
\(311\) 11.9629 0.678352 0.339176 0.940723i \(-0.389852\pi\)
0.339176 + 0.940723i \(0.389852\pi\)
\(312\) 0 0
\(313\) −13.5439 −0.765549 −0.382774 0.923842i \(-0.625031\pi\)
−0.382774 + 0.923842i \(0.625031\pi\)
\(314\) 4.90978 0.277075
\(315\) 0 0
\(316\) −6.29803 −0.354292
\(317\) −29.9629 −1.68288 −0.841441 0.540349i \(-0.818292\pi\)
−0.841441 + 0.540349i \(0.818292\pi\)
\(318\) 0 0
\(319\) 4.77747 0.267487
\(320\) −7.13231 −0.398708
\(321\) 0 0
\(322\) −9.98762 −0.556588
\(323\) 3.65383 0.203304
\(324\) 0 0
\(325\) 7.87636 0.436902
\(326\) −17.5215 −0.970427
\(327\) 0 0
\(328\) 10.2212 0.564369
\(329\) 2.66621 0.146993
\(330\) 0 0
\(331\) 2.08650 0.114685 0.0573423 0.998355i \(-0.481737\pi\)
0.0573423 + 0.998355i \(0.481737\pi\)
\(332\) −3.65383 −0.200530
\(333\) 0 0
\(334\) 20.6538 1.13013
\(335\) −44.1693 −2.41323
\(336\) 0 0
\(337\) −16.2088 −0.882948 −0.441474 0.897274i \(-0.645544\pi\)
−0.441474 + 0.897274i \(0.645544\pi\)
\(338\) −20.3955 −1.10937
\(339\) 0 0
\(340\) 13.1113 0.711058
\(341\) 19.6414 1.06364
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) −9.84059 −0.530569
\(345\) 0 0
\(346\) −11.2335 −0.603918
\(347\) 11.2670 0.604842 0.302421 0.953175i \(-0.402205\pi\)
0.302421 + 0.953175i \(0.402205\pi\)
\(348\) 0 0
\(349\) −0.197769 −0.0105863 −0.00529316 0.999986i \(-0.501685\pi\)
−0.00529316 + 0.999986i \(0.501685\pi\)
\(350\) −13.3869 −0.715559
\(351\) 0 0
\(352\) 13.2101 0.704103
\(353\) −12.5019 −0.665407 −0.332703 0.943032i \(-0.607961\pi\)
−0.332703 + 0.943032i \(0.607961\pi\)
\(354\) 0 0
\(355\) 10.3214 0.547804
\(356\) 8.54022 0.452631
\(357\) 0 0
\(358\) −6.53156 −0.345204
\(359\) 20.0197 1.05660 0.528299 0.849059i \(-0.322830\pi\)
0.528299 + 0.849059i \(0.322830\pi\)
\(360\) 0 0
\(361\) −18.2101 −0.958429
\(362\) 31.5155 1.65642
\(363\) 0 0
\(364\) −0.888736 −0.0465824
\(365\) 38.1854 1.99871
\(366\) 0 0
\(367\) 30.0727 1.56978 0.784892 0.619632i \(-0.212718\pi\)
0.784892 + 0.619632i \(0.212718\pi\)
\(368\) −29.3090 −1.52784
\(369\) 0 0
\(370\) −29.0617 −1.51085
\(371\) 0.123644 0.00641927
\(372\) 0 0
\(373\) 7.01238 0.363087 0.181544 0.983383i \(-0.441891\pi\)
0.181544 + 0.983383i \(0.441891\pi\)
\(374\) 19.6414 1.01564
\(375\) 0 0
\(376\) 5.03576 0.259700
\(377\) −1.69963 −0.0875353
\(378\) 0 0
\(379\) −19.0741 −0.979772 −0.489886 0.871787i \(-0.662962\pi\)
−0.489886 + 0.871787i \(0.662962\pi\)
\(380\) 2.83427 0.145395
\(381\) 0 0
\(382\) 7.87636 0.402989
\(383\) 3.21015 0.164031 0.0820155 0.996631i \(-0.473864\pi\)
0.0820155 + 0.996631i \(0.473864\pi\)
\(384\) 0 0
\(385\) −10.0865 −0.514056
\(386\) −42.9949 −2.18838
\(387\) 0 0
\(388\) 6.50680 0.330333
\(389\) 5.13602 0.260407 0.130203 0.991487i \(-0.458437\pi\)
0.130203 + 0.991487i \(0.458437\pi\)
\(390\) 0 0
\(391\) −24.1593 −1.22179
\(392\) −1.88874 −0.0953956
\(393\) 0 0
\(394\) 18.2212 0.917968
\(395\) 25.4290 1.27947
\(396\) 0 0
\(397\) 22.9381 1.15123 0.575615 0.817721i \(-0.304763\pi\)
0.575615 + 0.817721i \(0.304763\pi\)
\(398\) −14.8997 −0.746856
\(399\) 0 0
\(400\) −39.2843 −1.96421
\(401\) −18.2101 −0.909371 −0.454686 0.890652i \(-0.650248\pi\)
−0.454686 + 0.890652i \(0.650248\pi\)
\(402\) 0 0
\(403\) −6.98762 −0.348078
\(404\) 3.07922 0.153197
\(405\) 0 0
\(406\) 2.88874 0.143366
\(407\) −13.3942 −0.663924
\(408\) 0 0
\(409\) −15.3324 −0.758139 −0.379070 0.925368i \(-0.623756\pi\)
−0.379070 + 0.925368i \(0.623756\pi\)
\(410\) 33.0049 1.63000
\(411\) 0 0
\(412\) −14.1099 −0.695144
\(413\) 8.87636 0.436777
\(414\) 0 0
\(415\) 14.7527 0.724182
\(416\) −4.69963 −0.230418
\(417\) 0 0
\(418\) 4.24591 0.207674
\(419\) 10.5687 0.516315 0.258157 0.966103i \(-0.416885\pi\)
0.258157 + 0.966103i \(0.416885\pi\)
\(420\) 0 0
\(421\) −36.1716 −1.76290 −0.881449 0.472280i \(-0.843431\pi\)
−0.881449 + 0.472280i \(0.843431\pi\)
\(422\) 17.8974 0.871232
\(423\) 0 0
\(424\) 0.233531 0.0113412
\(425\) −32.3818 −1.57075
\(426\) 0 0
\(427\) −3.87636 −0.187590
\(428\) −4.75781 −0.229977
\(429\) 0 0
\(430\) −31.7761 −1.53238
\(431\) 35.0989 1.69065 0.845327 0.534249i \(-0.179406\pi\)
0.845327 + 0.534249i \(0.179406\pi\)
\(432\) 0 0
\(433\) −41.1730 −1.97865 −0.989324 0.145731i \(-0.953447\pi\)
−0.989324 + 0.145731i \(0.953447\pi\)
\(434\) 11.8764 0.570083
\(435\) 0 0
\(436\) −16.7651 −0.802902
\(437\) −5.22253 −0.249827
\(438\) 0 0
\(439\) −4.67859 −0.223297 −0.111648 0.993748i \(-0.535613\pi\)
−0.111648 + 0.993748i \(0.535613\pi\)
\(440\) −19.0507 −0.908209
\(441\) 0 0
\(442\) −6.98762 −0.332367
\(443\) 30.1730 1.43356 0.716781 0.697298i \(-0.245615\pi\)
0.716781 + 0.697298i \(0.245615\pi\)
\(444\) 0 0
\(445\) −34.4820 −1.63461
\(446\) −9.63279 −0.456126
\(447\) 0 0
\(448\) −1.98762 −0.0939062
\(449\) 0.333792 0.0157526 0.00787632 0.999969i \(-0.497493\pi\)
0.00787632 + 0.999969i \(0.497493\pi\)
\(450\) 0 0
\(451\) 15.2115 0.716283
\(452\) −16.4871 −0.775490
\(453\) 0 0
\(454\) −18.8640 −0.885330
\(455\) 3.58836 0.168225
\(456\) 0 0
\(457\) −19.3090 −0.903238 −0.451619 0.892211i \(-0.649153\pi\)
−0.451619 + 0.892211i \(0.649153\pi\)
\(458\) 33.3855 1.56000
\(459\) 0 0
\(460\) −18.7403 −0.873773
\(461\) 39.1075 1.82142 0.910710 0.413046i \(-0.135535\pi\)
0.910710 + 0.413046i \(0.135535\pi\)
\(462\) 0 0
\(463\) 21.8764 1.01668 0.508340 0.861156i \(-0.330259\pi\)
0.508340 + 0.861156i \(0.330259\pi\)
\(464\) 8.47710 0.393539
\(465\) 0 0
\(466\) 15.2335 0.705680
\(467\) −12.3200 −0.570103 −0.285052 0.958512i \(-0.592011\pi\)
−0.285052 + 0.958512i \(0.592011\pi\)
\(468\) 0 0
\(469\) −12.3090 −0.568378
\(470\) 16.2609 0.750059
\(471\) 0 0
\(472\) 16.7651 0.771676
\(473\) −14.6452 −0.673385
\(474\) 0 0
\(475\) −7.00000 −0.321182
\(476\) 3.65383 0.167473
\(477\) 0 0
\(478\) 19.0741 0.872430
\(479\) 13.4895 0.616350 0.308175 0.951330i \(-0.400282\pi\)
0.308175 + 0.951330i \(0.400282\pi\)
\(480\) 0 0
\(481\) 4.76509 0.217269
\(482\) −11.8740 −0.540847
\(483\) 0 0
\(484\) −2.75409 −0.125186
\(485\) −26.2719 −1.19295
\(486\) 0 0
\(487\) 7.54394 0.341849 0.170924 0.985284i \(-0.445325\pi\)
0.170924 + 0.985284i \(0.445325\pi\)
\(488\) −7.32141 −0.331425
\(489\) 0 0
\(490\) −6.09888 −0.275520
\(491\) −16.1396 −0.728369 −0.364185 0.931327i \(-0.618652\pi\)
−0.364185 + 0.931327i \(0.618652\pi\)
\(492\) 0 0
\(493\) 6.98762 0.314707
\(494\) −1.51052 −0.0679615
\(495\) 0 0
\(496\) 34.8516 1.56488
\(497\) 2.87636 0.129022
\(498\) 0 0
\(499\) −30.8654 −1.38172 −0.690862 0.722987i \(-0.742769\pi\)
−0.690862 + 0.722987i \(0.742769\pi\)
\(500\) −9.17301 −0.410229
\(501\) 0 0
\(502\) 7.85297 0.350495
\(503\) 24.6304 1.09822 0.549109 0.835751i \(-0.314967\pi\)
0.549109 + 0.835751i \(0.314967\pi\)
\(504\) 0 0
\(505\) −12.4327 −0.553247
\(506\) −28.0741 −1.24805
\(507\) 0 0
\(508\) 8.87636 0.393825
\(509\) 13.5897 0.602355 0.301177 0.953568i \(-0.402620\pi\)
0.301177 + 0.953568i \(0.402620\pi\)
\(510\) 0 0
\(511\) 10.6414 0.470750
\(512\) 4.59937 0.203265
\(513\) 0 0
\(514\) −2.42030 −0.106755
\(515\) 56.9701 2.51040
\(516\) 0 0
\(517\) 7.49442 0.329604
\(518\) −8.09888 −0.355845
\(519\) 0 0
\(520\) 6.77747 0.297212
\(521\) −39.1730 −1.71620 −0.858100 0.513482i \(-0.828355\pi\)
−0.858100 + 0.513482i \(0.828355\pi\)
\(522\) 0 0
\(523\) 19.1236 0.836219 0.418109 0.908397i \(-0.362693\pi\)
0.418109 + 0.908397i \(0.362693\pi\)
\(524\) 14.2670 0.623255
\(525\) 0 0
\(526\) 27.6414 1.20522
\(527\) 28.7280 1.25141
\(528\) 0 0
\(529\) 11.5316 0.501372
\(530\) 0.754090 0.0327556
\(531\) 0 0
\(532\) 0.789851 0.0342444
\(533\) −5.41164 −0.234404
\(534\) 0 0
\(535\) 19.2101 0.830527
\(536\) −23.2485 −1.00418
\(537\) 0 0
\(538\) −31.7046 −1.36688
\(539\) −2.81089 −0.121074
\(540\) 0 0
\(541\) 2.53018 0.108781 0.0543906 0.998520i \(-0.482678\pi\)
0.0543906 + 0.998520i \(0.482678\pi\)
\(542\) 6.73539 0.289310
\(543\) 0 0
\(544\) 19.3214 0.828399
\(545\) 67.6908 2.89956
\(546\) 0 0
\(547\) 17.8516 0.763279 0.381640 0.924311i \(-0.375360\pi\)
0.381640 + 0.924311i \(0.375360\pi\)
\(548\) −11.5426 −0.493074
\(549\) 0 0
\(550\) −37.6291 −1.60451
\(551\) 1.51052 0.0643503
\(552\) 0 0
\(553\) 7.08650 0.301349
\(554\) −3.96658 −0.168524
\(555\) 0 0
\(556\) 0.987620 0.0418844
\(557\) 41.3607 1.75251 0.876255 0.481847i \(-0.160034\pi\)
0.876255 + 0.481847i \(0.160034\pi\)
\(558\) 0 0
\(559\) 5.21015 0.220366
\(560\) −17.8974 −0.756303
\(561\) 0 0
\(562\) 47.5809 2.00708
\(563\) −20.7366 −0.873944 −0.436972 0.899475i \(-0.643949\pi\)
−0.436972 + 0.899475i \(0.643949\pi\)
\(564\) 0 0
\(565\) 66.5685 2.80056
\(566\) 17.5402 0.737271
\(567\) 0 0
\(568\) 5.43268 0.227950
\(569\) −0.268329 −0.0112489 −0.00562446 0.999984i \(-0.501790\pi\)
−0.00562446 + 0.999984i \(0.501790\pi\)
\(570\) 0 0
\(571\) 35.9367 1.50391 0.751953 0.659217i \(-0.229112\pi\)
0.751953 + 0.659217i \(0.229112\pi\)
\(572\) −2.49814 −0.104453
\(573\) 0 0
\(574\) 9.19777 0.383907
\(575\) 46.2843 1.93019
\(576\) 0 0
\(577\) 5.43130 0.226108 0.113054 0.993589i \(-0.463937\pi\)
0.113054 + 0.993589i \(0.463937\pi\)
\(578\) −0.165726 −0.00689328
\(579\) 0 0
\(580\) 5.42030 0.225066
\(581\) 4.11126 0.170564
\(582\) 0 0
\(583\) 0.347550 0.0143940
\(584\) 20.0989 0.831698
\(585\) 0 0
\(586\) −52.1716 −2.15519
\(587\) 35.1643 1.45139 0.725694 0.688018i \(-0.241519\pi\)
0.725694 + 0.688018i \(0.241519\pi\)
\(588\) 0 0
\(589\) 6.21015 0.255885
\(590\) 54.1359 2.22874
\(591\) 0 0
\(592\) −23.7665 −0.976796
\(593\) −33.5068 −1.37596 −0.687980 0.725730i \(-0.741502\pi\)
−0.687980 + 0.725730i \(0.741502\pi\)
\(594\) 0 0
\(595\) −14.7527 −0.604802
\(596\) 7.49442 0.306983
\(597\) 0 0
\(598\) 9.98762 0.408424
\(599\) 6.24729 0.255257 0.127629 0.991822i \(-0.459263\pi\)
0.127629 + 0.991822i \(0.459263\pi\)
\(600\) 0 0
\(601\) 22.4079 0.914038 0.457019 0.889457i \(-0.348917\pi\)
0.457019 + 0.889457i \(0.348917\pi\)
\(602\) −8.85532 −0.360916
\(603\) 0 0
\(604\) −13.1991 −0.537066
\(605\) 11.1199 0.452089
\(606\) 0 0
\(607\) 14.9505 0.606821 0.303411 0.952860i \(-0.401875\pi\)
0.303411 + 0.952860i \(0.401875\pi\)
\(608\) 4.17673 0.169389
\(609\) 0 0
\(610\) −23.6414 −0.957214
\(611\) −2.66621 −0.107863
\(612\) 0 0
\(613\) 35.1978 1.42162 0.710812 0.703382i \(-0.248328\pi\)
0.710812 + 0.703382i \(0.248328\pi\)
\(614\) −19.4500 −0.784938
\(615\) 0 0
\(616\) −5.30903 −0.213907
\(617\) −2.01238 −0.0810154 −0.0405077 0.999179i \(-0.512898\pi\)
−0.0405077 + 0.999179i \(0.512898\pi\)
\(618\) 0 0
\(619\) 39.3818 1.58289 0.791444 0.611242i \(-0.209330\pi\)
0.791444 + 0.611242i \(0.209330\pi\)
\(620\) 22.2843 0.894958
\(621\) 0 0
\(622\) 20.3324 0.815256
\(623\) −9.60940 −0.384993
\(624\) 0 0
\(625\) −2.34479 −0.0937918
\(626\) −23.0197 −0.920051
\(627\) 0 0
\(628\) 2.56732 0.102447
\(629\) −19.5906 −0.781126
\(630\) 0 0
\(631\) 44.3832 1.76687 0.883433 0.468558i \(-0.155226\pi\)
0.883433 + 0.468558i \(0.155226\pi\)
\(632\) 13.3845 0.532408
\(633\) 0 0
\(634\) −50.9257 −2.02252
\(635\) −35.8392 −1.42224
\(636\) 0 0
\(637\) 1.00000 0.0396214
\(638\) 8.11993 0.321471
\(639\) 0 0
\(640\) −45.8502 −1.81239
\(641\) −14.9862 −0.591921 −0.295961 0.955200i \(-0.595640\pi\)
−0.295961 + 0.955200i \(0.595640\pi\)
\(642\) 0 0
\(643\) −10.6538 −0.420146 −0.210073 0.977686i \(-0.567370\pi\)
−0.210073 + 0.977686i \(0.567370\pi\)
\(644\) −5.22253 −0.205796
\(645\) 0 0
\(646\) 6.21015 0.244335
\(647\) 2.12955 0.0837213 0.0418606 0.999123i \(-0.486671\pi\)
0.0418606 + 0.999123i \(0.486671\pi\)
\(648\) 0 0
\(649\) 24.9505 0.979392
\(650\) 13.3869 0.525076
\(651\) 0 0
\(652\) −9.16201 −0.358812
\(653\) 11.1716 0.437180 0.218590 0.975817i \(-0.429854\pi\)
0.218590 + 0.975817i \(0.429854\pi\)
\(654\) 0 0
\(655\) −57.6043 −2.25079
\(656\) 26.9912 1.05383
\(657\) 0 0
\(658\) 4.53156 0.176659
\(659\) −11.3090 −0.440537 −0.220269 0.975439i \(-0.570693\pi\)
−0.220269 + 0.975439i \(0.570693\pi\)
\(660\) 0 0
\(661\) 32.3570 1.25854 0.629271 0.777186i \(-0.283354\pi\)
0.629271 + 0.777186i \(0.283354\pi\)
\(662\) 3.54628 0.137830
\(663\) 0 0
\(664\) 7.76509 0.301344
\(665\) −3.18911 −0.123668
\(666\) 0 0
\(667\) −9.98762 −0.386722
\(668\) 10.7999 0.417860
\(669\) 0 0
\(670\) −75.0714 −2.90026
\(671\) −10.8960 −0.420636
\(672\) 0 0
\(673\) −24.1606 −0.931324 −0.465662 0.884963i \(-0.654184\pi\)
−0.465662 + 0.884963i \(0.654184\pi\)
\(674\) −27.5489 −1.06114
\(675\) 0 0
\(676\) −10.6648 −0.410186
\(677\) 25.0741 0.963677 0.481838 0.876260i \(-0.339969\pi\)
0.481838 + 0.876260i \(0.339969\pi\)
\(678\) 0 0
\(679\) −7.32141 −0.280970
\(680\) −27.8640 −1.06853
\(681\) 0 0
\(682\) 33.3832 1.27831
\(683\) −47.6784 −1.82436 −0.912182 0.409785i \(-0.865604\pi\)
−0.912182 + 0.409785i \(0.865604\pi\)
\(684\) 0 0
\(685\) 46.6043 1.78066
\(686\) −1.69963 −0.0648921
\(687\) 0 0
\(688\) −25.9862 −0.990716
\(689\) −0.123644 −0.00471046
\(690\) 0 0
\(691\) −24.6800 −0.938870 −0.469435 0.882967i \(-0.655542\pi\)
−0.469435 + 0.882967i \(0.655542\pi\)
\(692\) −5.87402 −0.223297
\(693\) 0 0
\(694\) 19.1496 0.726910
\(695\) −3.98762 −0.151259
\(696\) 0 0
\(697\) 22.2487 0.842728
\(698\) −0.336134 −0.0127228
\(699\) 0 0
\(700\) −7.00000 −0.264575
\(701\) −29.6784 −1.12094 −0.560469 0.828175i \(-0.689379\pi\)
−0.560469 + 0.828175i \(0.689379\pi\)
\(702\) 0 0
\(703\) −4.23491 −0.159723
\(704\) −5.58699 −0.210567
\(705\) 0 0
\(706\) −21.2485 −0.799698
\(707\) −3.46472 −0.130304
\(708\) 0 0
\(709\) −29.2581 −1.09881 −0.549406 0.835555i \(-0.685146\pi\)
−0.549406 + 0.835555i \(0.685146\pi\)
\(710\) 17.5426 0.658361
\(711\) 0 0
\(712\) −18.1496 −0.680186
\(713\) −41.0617 −1.53777
\(714\) 0 0
\(715\) 10.0865 0.377214
\(716\) −3.41535 −0.127638
\(717\) 0 0
\(718\) 34.0260 1.26984
\(719\) 1.07413 0.0400581 0.0200291 0.999799i \(-0.493624\pi\)
0.0200291 + 0.999799i \(0.493624\pi\)
\(720\) 0 0
\(721\) 15.8764 0.591266
\(722\) −30.9505 −1.15186
\(723\) 0 0
\(724\) 16.4794 0.612454
\(725\) −13.3869 −0.497176
\(726\) 0 0
\(727\) 25.4327 0.943246 0.471623 0.881800i \(-0.343668\pi\)
0.471623 + 0.881800i \(0.343668\pi\)
\(728\) 1.88874 0.0700012
\(729\) 0 0
\(730\) 64.9010 2.40209
\(731\) −21.4203 −0.792258
\(732\) 0 0
\(733\) −11.3955 −0.420904 −0.210452 0.977604i \(-0.567494\pi\)
−0.210452 + 0.977604i \(0.567494\pi\)
\(734\) 51.1125 1.88660
\(735\) 0 0
\(736\) −27.6167 −1.01796
\(737\) −34.5994 −1.27448
\(738\) 0 0
\(739\) −29.9395 −1.10134 −0.550671 0.834723i \(-0.685628\pi\)
−0.550671 + 0.834723i \(0.685628\pi\)
\(740\) −15.1964 −0.558630
\(741\) 0 0
\(742\) 0.210149 0.00771480
\(743\) −19.0014 −0.697093 −0.348546 0.937292i \(-0.613325\pi\)
−0.348546 + 0.937292i \(0.613325\pi\)
\(744\) 0 0
\(745\) −30.2595 −1.10862
\(746\) 11.9184 0.436365
\(747\) 0 0
\(748\) 10.2705 0.375527
\(749\) 5.35346 0.195611
\(750\) 0 0
\(751\) 0.0261368 0.000953747 0 0.000476873 1.00000i \(-0.499848\pi\)
0.000476873 1.00000i \(0.499848\pi\)
\(752\) 13.2980 0.484929
\(753\) 0 0
\(754\) −2.88874 −0.105202
\(755\) 53.2929 1.93953
\(756\) 0 0
\(757\) −13.6910 −0.497607 −0.248803 0.968554i \(-0.580037\pi\)
−0.248803 + 0.968554i \(0.580037\pi\)
\(758\) −32.4189 −1.17751
\(759\) 0 0
\(760\) −6.02338 −0.218491
\(761\) −14.6428 −0.530802 −0.265401 0.964138i \(-0.585504\pi\)
−0.265401 + 0.964138i \(0.585504\pi\)
\(762\) 0 0
\(763\) 18.8640 0.682922
\(764\) 4.11855 0.149004
\(765\) 0 0
\(766\) 5.45606 0.197135
\(767\) −8.87636 −0.320507
\(768\) 0 0
\(769\) 49.1345 1.77184 0.885918 0.463843i \(-0.153530\pi\)
0.885918 + 0.463843i \(0.153530\pi\)
\(770\) −17.1433 −0.617802
\(771\) 0 0
\(772\) −22.4820 −0.809146
\(773\) 12.4413 0.447484 0.223742 0.974648i \(-0.428173\pi\)
0.223742 + 0.974648i \(0.428173\pi\)
\(774\) 0 0
\(775\) −55.0370 −1.97699
\(776\) −13.8282 −0.496404
\(777\) 0 0
\(778\) 8.72933 0.312962
\(779\) 4.80951 0.172319
\(780\) 0 0
\(781\) 8.08513 0.289309
\(782\) −41.0617 −1.46837
\(783\) 0 0
\(784\) −4.98762 −0.178129
\(785\) −10.3658 −0.369973
\(786\) 0 0
\(787\) −32.9133 −1.17323 −0.586617 0.809865i \(-0.699541\pi\)
−0.586617 + 0.809865i \(0.699541\pi\)
\(788\) 9.52784 0.339415
\(789\) 0 0
\(790\) 43.2198 1.53769
\(791\) 18.5512 0.659606
\(792\) 0 0
\(793\) 3.87636 0.137653
\(794\) 38.9862 1.38357
\(795\) 0 0
\(796\) −7.79108 −0.276147
\(797\) 26.3979 0.935061 0.467530 0.883977i \(-0.345144\pi\)
0.467530 + 0.883977i \(0.345144\pi\)
\(798\) 0 0
\(799\) 10.9615 0.387789
\(800\) −37.0159 −1.30871
\(801\) 0 0
\(802\) −30.9505 −1.09290
\(803\) 29.9120 1.05557
\(804\) 0 0
\(805\) 21.0865 0.743202
\(806\) −11.8764 −0.418327
\(807\) 0 0
\(808\) −6.54394 −0.230215
\(809\) 35.5919 1.25135 0.625673 0.780086i \(-0.284825\pi\)
0.625673 + 0.780086i \(0.284825\pi\)
\(810\) 0 0
\(811\) −37.8268 −1.32828 −0.664140 0.747608i \(-0.731202\pi\)
−0.664140 + 0.747608i \(0.731202\pi\)
\(812\) 1.51052 0.0530089
\(813\) 0 0
\(814\) −22.7651 −0.797916
\(815\) 36.9926 1.29579
\(816\) 0 0
\(817\) −4.63045 −0.161999
\(818\) −26.0594 −0.911146
\(819\) 0 0
\(820\) 17.2583 0.602686
\(821\) −18.3128 −0.639119 −0.319560 0.947566i \(-0.603535\pi\)
−0.319560 + 0.947566i \(0.603535\pi\)
\(822\) 0 0
\(823\) −36.0000 −1.25488 −0.627441 0.778664i \(-0.715897\pi\)
−0.627441 + 0.778664i \(0.715897\pi\)
\(824\) 29.9862 1.04462
\(825\) 0 0
\(826\) 15.0865 0.524927
\(827\) −28.2115 −0.981011 −0.490505 0.871438i \(-0.663188\pi\)
−0.490505 + 0.871438i \(0.663188\pi\)
\(828\) 0 0
\(829\) −11.2843 −0.391919 −0.195960 0.980612i \(-0.562782\pi\)
−0.195960 + 0.980612i \(0.562782\pi\)
\(830\) 25.0741 0.870336
\(831\) 0 0
\(832\) 1.98762 0.0689083
\(833\) −4.11126 −0.142447
\(834\) 0 0
\(835\) −43.6057 −1.50904
\(836\) 2.22019 0.0767868
\(837\) 0 0
\(838\) 17.9629 0.620517
\(839\) −2.04305 −0.0705338 −0.0352669 0.999378i \(-0.511228\pi\)
−0.0352669 + 0.999378i \(0.511228\pi\)
\(840\) 0 0
\(841\) −26.1113 −0.900388
\(842\) −61.4783 −2.11868
\(843\) 0 0
\(844\) 9.35855 0.322135
\(845\) 43.0604 1.48132
\(846\) 0 0
\(847\) 3.09888 0.106479
\(848\) 0.616689 0.0211772
\(849\) 0 0
\(850\) −55.0370 −1.88775
\(851\) 28.0014 0.959875
\(852\) 0 0
\(853\) −48.5919 −1.66376 −0.831878 0.554959i \(-0.812734\pi\)
−0.831878 + 0.554959i \(0.812734\pi\)
\(854\) −6.58836 −0.225449
\(855\) 0 0
\(856\) 10.1113 0.345596
\(857\) −44.8974 −1.53367 −0.766833 0.641847i \(-0.778169\pi\)
−0.766833 + 0.641847i \(0.778169\pi\)
\(858\) 0 0
\(859\) 29.8131 1.01721 0.508605 0.861000i \(-0.330162\pi\)
0.508605 + 0.861000i \(0.330162\pi\)
\(860\) −16.6157 −0.566592
\(861\) 0 0
\(862\) 59.6551 2.03186
\(863\) −42.2595 −1.43853 −0.719265 0.694736i \(-0.755521\pi\)
−0.719265 + 0.694736i \(0.755521\pi\)
\(864\) 0 0
\(865\) 23.7170 0.806401
\(866\) −69.9788 −2.37798
\(867\) 0 0
\(868\) 6.21015 0.210786
\(869\) 19.9194 0.675719
\(870\) 0 0
\(871\) 12.3090 0.417076
\(872\) 35.6291 1.20655
\(873\) 0 0
\(874\) −8.87636 −0.300247
\(875\) 10.3214 0.348927
\(876\) 0 0
\(877\) −30.5316 −1.03098 −0.515489 0.856896i \(-0.672390\pi\)
−0.515489 + 0.856896i \(0.672390\pi\)
\(878\) −7.95186 −0.268362
\(879\) 0 0
\(880\) −50.3077 −1.69587
\(881\) −13.4079 −0.451724 −0.225862 0.974159i \(-0.572520\pi\)
−0.225862 + 0.974159i \(0.572520\pi\)
\(882\) 0 0
\(883\) −14.1250 −0.475345 −0.237672 0.971345i \(-0.576384\pi\)
−0.237672 + 0.971345i \(0.576384\pi\)
\(884\) −3.65383 −0.122892
\(885\) 0 0
\(886\) 51.2829 1.72288
\(887\) −39.9432 −1.34116 −0.670581 0.741837i \(-0.733955\pi\)
−0.670581 + 0.741837i \(0.733955\pi\)
\(888\) 0 0
\(889\) −9.98762 −0.334974
\(890\) −58.6067 −1.96450
\(891\) 0 0
\(892\) −5.03699 −0.168651
\(893\) 2.36955 0.0792941
\(894\) 0 0
\(895\) 13.7899 0.460944
\(896\) −12.7775 −0.426865
\(897\) 0 0
\(898\) 0.567323 0.0189318
\(899\) 11.8764 0.396099
\(900\) 0 0
\(901\) 0.508333 0.0169350
\(902\) 25.8539 0.860842
\(903\) 0 0
\(904\) 35.0384 1.16536
\(905\) −66.5375 −2.21178
\(906\) 0 0
\(907\) 41.4203 1.37534 0.687669 0.726024i \(-0.258634\pi\)
0.687669 + 0.726024i \(0.258634\pi\)
\(908\) −9.86398 −0.327348
\(909\) 0 0
\(910\) 6.09888 0.202176
\(911\) 1.78847 0.0592548 0.0296274 0.999561i \(-0.490568\pi\)
0.0296274 + 0.999561i \(0.490568\pi\)
\(912\) 0 0
\(913\) 11.5563 0.382458
\(914\) −32.8182 −1.08553
\(915\) 0 0
\(916\) 17.4573 0.576805
\(917\) −16.0531 −0.530120
\(918\) 0 0
\(919\) −57.4683 −1.89570 −0.947852 0.318711i \(-0.896750\pi\)
−0.947852 + 0.318711i \(0.896750\pi\)
\(920\) 39.8268 1.31305
\(921\) 0 0
\(922\) 66.4683 2.18902
\(923\) −2.87636 −0.0946764
\(924\) 0 0
\(925\) 37.5316 1.23403
\(926\) 37.1817 1.22187
\(927\) 0 0
\(928\) 7.98762 0.262206
\(929\) 34.7352 1.13963 0.569813 0.821774i \(-0.307016\pi\)
0.569813 + 0.821774i \(0.307016\pi\)
\(930\) 0 0
\(931\) −0.888736 −0.0291271
\(932\) 7.96562 0.260922
\(933\) 0 0
\(934\) −20.9395 −0.685161
\(935\) −41.4683 −1.35616
\(936\) 0 0
\(937\) 11.6662 0.381118 0.190559 0.981676i \(-0.438970\pi\)
0.190559 + 0.981676i \(0.438970\pi\)
\(938\) −20.9208 −0.683088
\(939\) 0 0
\(940\) 8.50282 0.277332
\(941\) −50.3374 −1.64095 −0.820475 0.571682i \(-0.806291\pi\)
−0.820475 + 0.571682i \(0.806291\pi\)
\(942\) 0 0
\(943\) −31.8007 −1.03557
\(944\) 44.2719 1.44093
\(945\) 0 0
\(946\) −24.8913 −0.809287
\(947\) −32.3883 −1.05248 −0.526238 0.850337i \(-0.676398\pi\)
−0.526238 + 0.850337i \(0.676398\pi\)
\(948\) 0 0
\(949\) −10.6414 −0.345436
\(950\) −11.8974 −0.386003
\(951\) 0 0
\(952\) −7.76509 −0.251668
\(953\) −12.5367 −0.406102 −0.203051 0.979168i \(-0.565086\pi\)
−0.203051 + 0.979168i \(0.565086\pi\)
\(954\) 0 0
\(955\) −16.6291 −0.538104
\(956\) 9.97386 0.322578
\(957\) 0 0
\(958\) 22.9271 0.740741
\(959\) 12.9876 0.419392
\(960\) 0 0
\(961\) 17.8268 0.575059
\(962\) 8.09888 0.261119
\(963\) 0 0
\(964\) −6.20892 −0.199976
\(965\) 90.7736 2.92211
\(966\) 0 0
\(967\) −57.9875 −1.86475 −0.932376 0.361491i \(-0.882268\pi\)
−0.932376 + 0.361491i \(0.882268\pi\)
\(968\) 5.85297 0.188122
\(969\) 0 0
\(970\) −44.6525 −1.43370
\(971\) 28.0370 0.899750 0.449875 0.893092i \(-0.351469\pi\)
0.449875 + 0.893092i \(0.351469\pi\)
\(972\) 0 0
\(973\) −1.11126 −0.0356255
\(974\) 12.8219 0.410840
\(975\) 0 0
\(976\) −19.3338 −0.618860
\(977\) 9.84059 0.314829 0.157414 0.987533i \(-0.449684\pi\)
0.157414 + 0.987533i \(0.449684\pi\)
\(978\) 0 0
\(979\) −27.0110 −0.863275
\(980\) −3.18911 −0.101872
\(981\) 0 0
\(982\) −27.4313 −0.875368
\(983\) −48.6894 −1.55295 −0.776476 0.630147i \(-0.782995\pi\)
−0.776476 + 0.630147i \(0.782995\pi\)
\(984\) 0 0
\(985\) −38.4697 −1.22575
\(986\) 11.8764 0.378220
\(987\) 0 0
\(988\) −0.789851 −0.0251285
\(989\) 30.6167 0.973554
\(990\) 0 0
\(991\) 2.86398 0.0909772 0.0454886 0.998965i \(-0.485516\pi\)
0.0454886 + 0.998965i \(0.485516\pi\)
\(992\) 32.8392 1.04265
\(993\) 0 0
\(994\) 4.88874 0.155061
\(995\) 31.4573 0.997263
\(996\) 0 0
\(997\) −50.8406 −1.61014 −0.805069 0.593181i \(-0.797872\pi\)
−0.805069 + 0.593181i \(0.797872\pi\)
\(998\) −52.4596 −1.66058
\(999\) 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 567.2.a.d.1.3 3
3.2 odd 2 567.2.a.g.1.1 3
4.3 odd 2 9072.2.a.bq.1.1 3
7.6 odd 2 3969.2.a.m.1.3 3
9.2 odd 6 189.2.f.a.64.3 6
9.4 even 3 63.2.f.b.43.1 yes 6
9.5 odd 6 189.2.f.a.127.3 6
9.7 even 3 63.2.f.b.22.1 6
12.11 even 2 9072.2.a.cd.1.3 3
21.20 even 2 3969.2.a.p.1.1 3
36.7 odd 6 1008.2.r.k.337.1 6
36.11 even 6 3024.2.r.g.1009.1 6
36.23 even 6 3024.2.r.g.2017.1 6
36.31 odd 6 1008.2.r.k.673.1 6
63.2 odd 6 1323.2.g.c.361.3 6
63.4 even 3 441.2.g.e.79.1 6
63.5 even 6 1323.2.h.e.802.1 6
63.11 odd 6 1323.2.h.d.226.1 6
63.13 odd 6 441.2.f.d.295.1 6
63.16 even 3 441.2.g.e.67.1 6
63.20 even 6 1323.2.f.c.442.3 6
63.23 odd 6 1323.2.h.d.802.1 6
63.25 even 3 441.2.h.c.373.3 6
63.31 odd 6 441.2.g.d.79.1 6
63.32 odd 6 1323.2.g.c.667.3 6
63.34 odd 6 441.2.f.d.148.1 6
63.38 even 6 1323.2.h.e.226.1 6
63.40 odd 6 441.2.h.b.214.3 6
63.41 even 6 1323.2.f.c.883.3 6
63.47 even 6 1323.2.g.b.361.3 6
63.52 odd 6 441.2.h.b.373.3 6
63.58 even 3 441.2.h.c.214.3 6
63.59 even 6 1323.2.g.b.667.3 6
63.61 odd 6 441.2.g.d.67.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.2.f.b.22.1 6 9.7 even 3
63.2.f.b.43.1 yes 6 9.4 even 3
189.2.f.a.64.3 6 9.2 odd 6
189.2.f.a.127.3 6 9.5 odd 6
441.2.f.d.148.1 6 63.34 odd 6
441.2.f.d.295.1 6 63.13 odd 6
441.2.g.d.67.1 6 63.61 odd 6
441.2.g.d.79.1 6 63.31 odd 6
441.2.g.e.67.1 6 63.16 even 3
441.2.g.e.79.1 6 63.4 even 3
441.2.h.b.214.3 6 63.40 odd 6
441.2.h.b.373.3 6 63.52 odd 6
441.2.h.c.214.3 6 63.58 even 3
441.2.h.c.373.3 6 63.25 even 3
567.2.a.d.1.3 3 1.1 even 1 trivial
567.2.a.g.1.1 3 3.2 odd 2
1008.2.r.k.337.1 6 36.7 odd 6
1008.2.r.k.673.1 6 36.31 odd 6
1323.2.f.c.442.3 6 63.20 even 6
1323.2.f.c.883.3 6 63.41 even 6
1323.2.g.b.361.3 6 63.47 even 6
1323.2.g.b.667.3 6 63.59 even 6
1323.2.g.c.361.3 6 63.2 odd 6
1323.2.g.c.667.3 6 63.32 odd 6
1323.2.h.d.226.1 6 63.11 odd 6
1323.2.h.d.802.1 6 63.23 odd 6
1323.2.h.e.226.1 6 63.38 even 6
1323.2.h.e.802.1 6 63.5 even 6
3024.2.r.g.1009.1 6 36.11 even 6
3024.2.r.g.2017.1 6 36.23 even 6
3969.2.a.m.1.3 3 7.6 odd 2
3969.2.a.p.1.1 3 21.20 even 2
9072.2.a.bq.1.1 3 4.3 odd 2
9072.2.a.cd.1.3 3 12.11 even 2