Properties

Label 8550.2.a.bv.1.2
Level $8550$
Weight $2$
Character 8550.1
Self dual yes
Analytic conductor $68.272$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8550.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(68.2720937282\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
Defining polynomial: \(x^{2} - 6\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2850)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.44949\) of defining polynomial
Character \(\chi\) \(=\) 8550.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +0.449490 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +0.449490 q^{7} +1.00000 q^{8} +1.44949 q^{11} -2.44949 q^{13} +0.449490 q^{14} +1.00000 q^{16} -0.449490 q^{17} +1.00000 q^{19} +1.44949 q^{22} -1.00000 q^{23} -2.44949 q^{26} +0.449490 q^{28} -10.3485 q^{29} -3.00000 q^{31} +1.00000 q^{32} -0.449490 q^{34} -11.7980 q^{37} +1.00000 q^{38} -8.89898 q^{41} -2.44949 q^{43} +1.44949 q^{44} -1.00000 q^{46} +11.7980 q^{47} -6.79796 q^{49} -2.44949 q^{52} -2.55051 q^{53} +0.449490 q^{56} -10.3485 q^{58} -1.55051 q^{59} -4.55051 q^{61} -3.00000 q^{62} +1.00000 q^{64} +9.24745 q^{67} -0.449490 q^{68} +6.44949 q^{71} +1.00000 q^{73} -11.7980 q^{74} +1.00000 q^{76} +0.651531 q^{77} -5.00000 q^{79} -8.89898 q^{82} -6.34847 q^{83} -2.44949 q^{86} +1.44949 q^{88} -11.8990 q^{89} -1.10102 q^{91} -1.00000 q^{92} +11.7980 q^{94} -6.44949 q^{97} -6.79796 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{4} - 4q^{7} + 2q^{8} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{4} - 4q^{7} + 2q^{8} - 2q^{11} - 4q^{14} + 2q^{16} + 4q^{17} + 2q^{19} - 2q^{22} - 2q^{23} - 4q^{28} - 6q^{29} - 6q^{31} + 2q^{32} + 4q^{34} - 4q^{37} + 2q^{38} - 8q^{41} - 2q^{44} - 2q^{46} + 4q^{47} + 6q^{49} - 10q^{53} - 4q^{56} - 6q^{58} - 8q^{59} - 14q^{61} - 6q^{62} + 2q^{64} - 6q^{67} + 4q^{68} + 8q^{71} + 2q^{73} - 4q^{74} + 2q^{76} + 16q^{77} - 10q^{79} - 8q^{82} + 2q^{83} - 2q^{88} - 14q^{89} - 12q^{91} - 2q^{92} + 4q^{94} - 8q^{97} + 6q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) 0.449490 0.169891 0.0849456 0.996386i \(-0.472928\pi\)
0.0849456 + 0.996386i \(0.472928\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) 1.44949 0.437038 0.218519 0.975833i \(-0.429878\pi\)
0.218519 + 0.975833i \(0.429878\pi\)
\(12\) 0 0
\(13\) −2.44949 −0.679366 −0.339683 0.940540i \(-0.610320\pi\)
−0.339683 + 0.940540i \(0.610320\pi\)
\(14\) 0.449490 0.120131
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −0.449490 −0.109017 −0.0545086 0.998513i \(-0.517359\pi\)
−0.0545086 + 0.998513i \(0.517359\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) 1.44949 0.309032
\(23\) −1.00000 −0.208514 −0.104257 0.994550i \(-0.533247\pi\)
−0.104257 + 0.994550i \(0.533247\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −2.44949 −0.480384
\(27\) 0 0
\(28\) 0.449490 0.0849456
\(29\) −10.3485 −1.92166 −0.960831 0.277134i \(-0.910615\pi\)
−0.960831 + 0.277134i \(0.910615\pi\)
\(30\) 0 0
\(31\) −3.00000 −0.538816 −0.269408 0.963026i \(-0.586828\pi\)
−0.269408 + 0.963026i \(0.586828\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −0.449490 −0.0770869
\(35\) 0 0
\(36\) 0 0
\(37\) −11.7980 −1.93957 −0.969786 0.243956i \(-0.921555\pi\)
−0.969786 + 0.243956i \(0.921555\pi\)
\(38\) 1.00000 0.162221
\(39\) 0 0
\(40\) 0 0
\(41\) −8.89898 −1.38979 −0.694894 0.719113i \(-0.744549\pi\)
−0.694894 + 0.719113i \(0.744549\pi\)
\(42\) 0 0
\(43\) −2.44949 −0.373544 −0.186772 0.982403i \(-0.559803\pi\)
−0.186772 + 0.982403i \(0.559803\pi\)
\(44\) 1.44949 0.218519
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) 11.7980 1.72091 0.860455 0.509527i \(-0.170180\pi\)
0.860455 + 0.509527i \(0.170180\pi\)
\(48\) 0 0
\(49\) −6.79796 −0.971137
\(50\) 0 0
\(51\) 0 0
\(52\) −2.44949 −0.339683
\(53\) −2.55051 −0.350340 −0.175170 0.984538i \(-0.556047\pi\)
−0.175170 + 0.984538i \(0.556047\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0.449490 0.0600656
\(57\) 0 0
\(58\) −10.3485 −1.35882
\(59\) −1.55051 −0.201859 −0.100930 0.994894i \(-0.532182\pi\)
−0.100930 + 0.994894i \(0.532182\pi\)
\(60\) 0 0
\(61\) −4.55051 −0.582633 −0.291317 0.956627i \(-0.594093\pi\)
−0.291317 + 0.956627i \(0.594093\pi\)
\(62\) −3.00000 −0.381000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 9.24745 1.12976 0.564878 0.825175i \(-0.308923\pi\)
0.564878 + 0.825175i \(0.308923\pi\)
\(68\) −0.449490 −0.0545086
\(69\) 0 0
\(70\) 0 0
\(71\) 6.44949 0.765414 0.382707 0.923870i \(-0.374992\pi\)
0.382707 + 0.923870i \(0.374992\pi\)
\(72\) 0 0
\(73\) 1.00000 0.117041 0.0585206 0.998286i \(-0.481362\pi\)
0.0585206 + 0.998286i \(0.481362\pi\)
\(74\) −11.7980 −1.37148
\(75\) 0 0
\(76\) 1.00000 0.114708
\(77\) 0.651531 0.0742488
\(78\) 0 0
\(79\) −5.00000 −0.562544 −0.281272 0.959628i \(-0.590756\pi\)
−0.281272 + 0.959628i \(0.590756\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −8.89898 −0.982728
\(83\) −6.34847 −0.696835 −0.348418 0.937339i \(-0.613281\pi\)
−0.348418 + 0.937339i \(0.613281\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −2.44949 −0.264135
\(87\) 0 0
\(88\) 1.44949 0.154516
\(89\) −11.8990 −1.26129 −0.630645 0.776072i \(-0.717209\pi\)
−0.630645 + 0.776072i \(0.717209\pi\)
\(90\) 0 0
\(91\) −1.10102 −0.115418
\(92\) −1.00000 −0.104257
\(93\) 0 0
\(94\) 11.7980 1.21687
\(95\) 0 0
\(96\) 0 0
\(97\) −6.44949 −0.654846 −0.327423 0.944878i \(-0.606180\pi\)
−0.327423 + 0.944878i \(0.606180\pi\)
\(98\) −6.79796 −0.686698
\(99\) 0 0
\(100\) 0 0
\(101\) 6.44949 0.641748 0.320874 0.947122i \(-0.396023\pi\)
0.320874 + 0.947122i \(0.396023\pi\)
\(102\) 0 0
\(103\) 7.89898 0.778310 0.389155 0.921172i \(-0.372767\pi\)
0.389155 + 0.921172i \(0.372767\pi\)
\(104\) −2.44949 −0.240192
\(105\) 0 0
\(106\) −2.55051 −0.247727
\(107\) 2.65153 0.256333 0.128167 0.991753i \(-0.459091\pi\)
0.128167 + 0.991753i \(0.459091\pi\)
\(108\) 0 0
\(109\) 6.89898 0.660802 0.330401 0.943841i \(-0.392816\pi\)
0.330401 + 0.943841i \(0.392816\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0.449490 0.0424728
\(113\) 2.79796 0.263210 0.131605 0.991302i \(-0.457987\pi\)
0.131605 + 0.991302i \(0.457987\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −10.3485 −0.960831
\(117\) 0 0
\(118\) −1.55051 −0.142736
\(119\) −0.202041 −0.0185211
\(120\) 0 0
\(121\) −8.89898 −0.808998
\(122\) −4.55051 −0.411984
\(123\) 0 0
\(124\) −3.00000 −0.269408
\(125\) 0 0
\(126\) 0 0
\(127\) −9.89898 −0.878392 −0.439196 0.898391i \(-0.644737\pi\)
−0.439196 + 0.898391i \(0.644737\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) 15.2474 1.33218 0.666088 0.745873i \(-0.267968\pi\)
0.666088 + 0.745873i \(0.267968\pi\)
\(132\) 0 0
\(133\) 0.449490 0.0389757
\(134\) 9.24745 0.798858
\(135\) 0 0
\(136\) −0.449490 −0.0385434
\(137\) 4.89898 0.418548 0.209274 0.977857i \(-0.432890\pi\)
0.209274 + 0.977857i \(0.432890\pi\)
\(138\) 0 0
\(139\) 2.24745 0.190626 0.0953131 0.995447i \(-0.469615\pi\)
0.0953131 + 0.995447i \(0.469615\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 6.44949 0.541229
\(143\) −3.55051 −0.296909
\(144\) 0 0
\(145\) 0 0
\(146\) 1.00000 0.0827606
\(147\) 0 0
\(148\) −11.7980 −0.969786
\(149\) 13.7980 1.13037 0.565186 0.824963i \(-0.308804\pi\)
0.565186 + 0.824963i \(0.308804\pi\)
\(150\) 0 0
\(151\) −4.20204 −0.341957 −0.170979 0.985275i \(-0.554693\pi\)
−0.170979 + 0.985275i \(0.554693\pi\)
\(152\) 1.00000 0.0811107
\(153\) 0 0
\(154\) 0.651531 0.0525018
\(155\) 0 0
\(156\) 0 0
\(157\) −4.89898 −0.390981 −0.195491 0.980706i \(-0.562630\pi\)
−0.195491 + 0.980706i \(0.562630\pi\)
\(158\) −5.00000 −0.397779
\(159\) 0 0
\(160\) 0 0
\(161\) −0.449490 −0.0354248
\(162\) 0 0
\(163\) −0.202041 −0.0158251 −0.00791254 0.999969i \(-0.502519\pi\)
−0.00791254 + 0.999969i \(0.502519\pi\)
\(164\) −8.89898 −0.694894
\(165\) 0 0
\(166\) −6.34847 −0.492737
\(167\) 18.0000 1.39288 0.696441 0.717614i \(-0.254766\pi\)
0.696441 + 0.717614i \(0.254766\pi\)
\(168\) 0 0
\(169\) −7.00000 −0.538462
\(170\) 0 0
\(171\) 0 0
\(172\) −2.44949 −0.186772
\(173\) −16.3485 −1.24295 −0.621476 0.783434i \(-0.713466\pi\)
−0.621476 + 0.783434i \(0.713466\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 1.44949 0.109259
\(177\) 0 0
\(178\) −11.8990 −0.891866
\(179\) −6.20204 −0.463562 −0.231781 0.972768i \(-0.574455\pi\)
−0.231781 + 0.972768i \(0.574455\pi\)
\(180\) 0 0
\(181\) −9.55051 −0.709884 −0.354942 0.934888i \(-0.615499\pi\)
−0.354942 + 0.934888i \(0.615499\pi\)
\(182\) −1.10102 −0.0816131
\(183\) 0 0
\(184\) −1.00000 −0.0737210
\(185\) 0 0
\(186\) 0 0
\(187\) −0.651531 −0.0476446
\(188\) 11.7980 0.860455
\(189\) 0 0
\(190\) 0 0
\(191\) 9.89898 0.716265 0.358133 0.933671i \(-0.383414\pi\)
0.358133 + 0.933671i \(0.383414\pi\)
\(192\) 0 0
\(193\) 0.651531 0.0468982 0.0234491 0.999725i \(-0.492535\pi\)
0.0234491 + 0.999725i \(0.492535\pi\)
\(194\) −6.44949 −0.463046
\(195\) 0 0
\(196\) −6.79796 −0.485568
\(197\) 12.6515 0.901384 0.450692 0.892679i \(-0.351177\pi\)
0.450692 + 0.892679i \(0.351177\pi\)
\(198\) 0 0
\(199\) −10.0000 −0.708881 −0.354441 0.935079i \(-0.615329\pi\)
−0.354441 + 0.935079i \(0.615329\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 6.44949 0.453785
\(203\) −4.65153 −0.326473
\(204\) 0 0
\(205\) 0 0
\(206\) 7.89898 0.550348
\(207\) 0 0
\(208\) −2.44949 −0.169842
\(209\) 1.44949 0.100263
\(210\) 0 0
\(211\) −18.3485 −1.26316 −0.631580 0.775310i \(-0.717593\pi\)
−0.631580 + 0.775310i \(0.717593\pi\)
\(212\) −2.55051 −0.175170
\(213\) 0 0
\(214\) 2.65153 0.181255
\(215\) 0 0
\(216\) 0 0
\(217\) −1.34847 −0.0915401
\(218\) 6.89898 0.467258
\(219\) 0 0
\(220\) 0 0
\(221\) 1.10102 0.0740627
\(222\) 0 0
\(223\) −25.8990 −1.73432 −0.867162 0.498026i \(-0.834058\pi\)
−0.867162 + 0.498026i \(0.834058\pi\)
\(224\) 0.449490 0.0300328
\(225\) 0 0
\(226\) 2.79796 0.186117
\(227\) −9.59592 −0.636903 −0.318452 0.947939i \(-0.603163\pi\)
−0.318452 + 0.947939i \(0.603163\pi\)
\(228\) 0 0
\(229\) −17.2474 −1.13974 −0.569872 0.821734i \(-0.693007\pi\)
−0.569872 + 0.821734i \(0.693007\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −10.3485 −0.679410
\(233\) −18.2474 −1.19543 −0.597715 0.801709i \(-0.703925\pi\)
−0.597715 + 0.801709i \(0.703925\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −1.55051 −0.100930
\(237\) 0 0
\(238\) −0.202041 −0.0130964
\(239\) −20.6969 −1.33877 −0.669387 0.742914i \(-0.733443\pi\)
−0.669387 + 0.742914i \(0.733443\pi\)
\(240\) 0 0
\(241\) 0.449490 0.0289542 0.0144771 0.999895i \(-0.495392\pi\)
0.0144771 + 0.999895i \(0.495392\pi\)
\(242\) −8.89898 −0.572048
\(243\) 0 0
\(244\) −4.55051 −0.291317
\(245\) 0 0
\(246\) 0 0
\(247\) −2.44949 −0.155857
\(248\) −3.00000 −0.190500
\(249\) 0 0
\(250\) 0 0
\(251\) 18.0000 1.13615 0.568075 0.822977i \(-0.307688\pi\)
0.568075 + 0.822977i \(0.307688\pi\)
\(252\) 0 0
\(253\) −1.44949 −0.0911286
\(254\) −9.89898 −0.621117
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 27.4949 1.71508 0.857542 0.514413i \(-0.171990\pi\)
0.857542 + 0.514413i \(0.171990\pi\)
\(258\) 0 0
\(259\) −5.30306 −0.329516
\(260\) 0 0
\(261\) 0 0
\(262\) 15.2474 0.941991
\(263\) 12.7980 0.789156 0.394578 0.918862i \(-0.370891\pi\)
0.394578 + 0.918862i \(0.370891\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0.449490 0.0275600
\(267\) 0 0
\(268\) 9.24745 0.564878
\(269\) −6.89898 −0.420638 −0.210319 0.977633i \(-0.567450\pi\)
−0.210319 + 0.977633i \(0.567450\pi\)
\(270\) 0 0
\(271\) −30.9444 −1.87974 −0.939869 0.341536i \(-0.889053\pi\)
−0.939869 + 0.341536i \(0.889053\pi\)
\(272\) −0.449490 −0.0272543
\(273\) 0 0
\(274\) 4.89898 0.295958
\(275\) 0 0
\(276\) 0 0
\(277\) 23.0454 1.38466 0.692332 0.721579i \(-0.256583\pi\)
0.692332 + 0.721579i \(0.256583\pi\)
\(278\) 2.24745 0.134793
\(279\) 0 0
\(280\) 0 0
\(281\) −7.00000 −0.417585 −0.208792 0.977960i \(-0.566953\pi\)
−0.208792 + 0.977960i \(0.566953\pi\)
\(282\) 0 0
\(283\) −10.2020 −0.606448 −0.303224 0.952919i \(-0.598063\pi\)
−0.303224 + 0.952919i \(0.598063\pi\)
\(284\) 6.44949 0.382707
\(285\) 0 0
\(286\) −3.55051 −0.209946
\(287\) −4.00000 −0.236113
\(288\) 0 0
\(289\) −16.7980 −0.988115
\(290\) 0 0
\(291\) 0 0
\(292\) 1.00000 0.0585206
\(293\) 1.24745 0.0728767 0.0364384 0.999336i \(-0.488399\pi\)
0.0364384 + 0.999336i \(0.488399\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −11.7980 −0.685742
\(297\) 0 0
\(298\) 13.7980 0.799294
\(299\) 2.44949 0.141658
\(300\) 0 0
\(301\) −1.10102 −0.0634618
\(302\) −4.20204 −0.241800
\(303\) 0 0
\(304\) 1.00000 0.0573539
\(305\) 0 0
\(306\) 0 0
\(307\) 1.65153 0.0942578 0.0471289 0.998889i \(-0.484993\pi\)
0.0471289 + 0.998889i \(0.484993\pi\)
\(308\) 0.651531 0.0371244
\(309\) 0 0
\(310\) 0 0
\(311\) 4.20204 0.238276 0.119138 0.992878i \(-0.461987\pi\)
0.119138 + 0.992878i \(0.461987\pi\)
\(312\) 0 0
\(313\) −22.1010 −1.24922 −0.624612 0.780935i \(-0.714743\pi\)
−0.624612 + 0.780935i \(0.714743\pi\)
\(314\) −4.89898 −0.276465
\(315\) 0 0
\(316\) −5.00000 −0.281272
\(317\) −9.24745 −0.519388 −0.259694 0.965691i \(-0.583622\pi\)
−0.259694 + 0.965691i \(0.583622\pi\)
\(318\) 0 0
\(319\) −15.0000 −0.839839
\(320\) 0 0
\(321\) 0 0
\(322\) −0.449490 −0.0250491
\(323\) −0.449490 −0.0250103
\(324\) 0 0
\(325\) 0 0
\(326\) −0.202041 −0.0111900
\(327\) 0 0
\(328\) −8.89898 −0.491364
\(329\) 5.30306 0.292367
\(330\) 0 0
\(331\) −7.65153 −0.420566 −0.210283 0.977641i \(-0.567439\pi\)
−0.210283 + 0.977641i \(0.567439\pi\)
\(332\) −6.34847 −0.348418
\(333\) 0 0
\(334\) 18.0000 0.984916
\(335\) 0 0
\(336\) 0 0
\(337\) 15.1010 0.822605 0.411303 0.911499i \(-0.365074\pi\)
0.411303 + 0.911499i \(0.365074\pi\)
\(338\) −7.00000 −0.380750
\(339\) 0 0
\(340\) 0 0
\(341\) −4.34847 −0.235483
\(342\) 0 0
\(343\) −6.20204 −0.334879
\(344\) −2.44949 −0.132068
\(345\) 0 0
\(346\) −16.3485 −0.878899
\(347\) 15.5959 0.837233 0.418616 0.908163i \(-0.362515\pi\)
0.418616 + 0.908163i \(0.362515\pi\)
\(348\) 0 0
\(349\) 27.9444 1.49583 0.747914 0.663795i \(-0.231055\pi\)
0.747914 + 0.663795i \(0.231055\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 1.44949 0.0772581
\(353\) 6.24745 0.332518 0.166259 0.986082i \(-0.446831\pi\)
0.166259 + 0.986082i \(0.446831\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −11.8990 −0.630645
\(357\) 0 0
\(358\) −6.20204 −0.327788
\(359\) 13.1010 0.691445 0.345723 0.938337i \(-0.387634\pi\)
0.345723 + 0.938337i \(0.387634\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −9.55051 −0.501964
\(363\) 0 0
\(364\) −1.10102 −0.0577092
\(365\) 0 0
\(366\) 0 0
\(367\) 10.4495 0.545459 0.272729 0.962091i \(-0.412074\pi\)
0.272729 + 0.962091i \(0.412074\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 0 0
\(370\) 0 0
\(371\) −1.14643 −0.0595196
\(372\) 0 0
\(373\) 17.5505 0.908731 0.454365 0.890815i \(-0.349866\pi\)
0.454365 + 0.890815i \(0.349866\pi\)
\(374\) −0.651531 −0.0336899
\(375\) 0 0
\(376\) 11.7980 0.608433
\(377\) 25.3485 1.30551
\(378\) 0 0
\(379\) −30.6969 −1.57680 −0.788398 0.615166i \(-0.789089\pi\)
−0.788398 + 0.615166i \(0.789089\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 9.89898 0.506476
\(383\) −8.24745 −0.421425 −0.210712 0.977548i \(-0.567578\pi\)
−0.210712 + 0.977548i \(0.567578\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0.651531 0.0331620
\(387\) 0 0
\(388\) −6.44949 −0.327423
\(389\) 17.5959 0.892148 0.446074 0.894996i \(-0.352822\pi\)
0.446074 + 0.894996i \(0.352822\pi\)
\(390\) 0 0
\(391\) 0.449490 0.0227317
\(392\) −6.79796 −0.343349
\(393\) 0 0
\(394\) 12.6515 0.637375
\(395\) 0 0
\(396\) 0 0
\(397\) 29.9444 1.50287 0.751433 0.659810i \(-0.229363\pi\)
0.751433 + 0.659810i \(0.229363\pi\)
\(398\) −10.0000 −0.501255
\(399\) 0 0
\(400\) 0 0
\(401\) −30.7980 −1.53798 −0.768988 0.639263i \(-0.779240\pi\)
−0.768988 + 0.639263i \(0.779240\pi\)
\(402\) 0 0
\(403\) 7.34847 0.366053
\(404\) 6.44949 0.320874
\(405\) 0 0
\(406\) −4.65153 −0.230852
\(407\) −17.1010 −0.847666
\(408\) 0 0
\(409\) −13.1010 −0.647804 −0.323902 0.946091i \(-0.604995\pi\)
−0.323902 + 0.946091i \(0.604995\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 7.89898 0.389155
\(413\) −0.696938 −0.0342941
\(414\) 0 0
\(415\) 0 0
\(416\) −2.44949 −0.120096
\(417\) 0 0
\(418\) 1.44949 0.0708969
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 22.0000 1.07221 0.536107 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) −18.3485 −0.893190
\(423\) 0 0
\(424\) −2.55051 −0.123864
\(425\) 0 0
\(426\) 0 0
\(427\) −2.04541 −0.0989842
\(428\) 2.65153 0.128167
\(429\) 0 0
\(430\) 0 0
\(431\) 14.8990 0.717659 0.358829 0.933403i \(-0.383176\pi\)
0.358829 + 0.933403i \(0.383176\pi\)
\(432\) 0 0
\(433\) −19.3485 −0.929828 −0.464914 0.885356i \(-0.653915\pi\)
−0.464914 + 0.885356i \(0.653915\pi\)
\(434\) −1.34847 −0.0647286
\(435\) 0 0
\(436\) 6.89898 0.330401
\(437\) −1.00000 −0.0478365
\(438\) 0 0
\(439\) −21.8990 −1.04518 −0.522591 0.852584i \(-0.675034\pi\)
−0.522591 + 0.852584i \(0.675034\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 1.10102 0.0523702
\(443\) −3.24745 −0.154291 −0.0771455 0.997020i \(-0.524581\pi\)
−0.0771455 + 0.997020i \(0.524581\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −25.8990 −1.22635
\(447\) 0 0
\(448\) 0.449490 0.0212364
\(449\) 15.0000 0.707894 0.353947 0.935266i \(-0.384839\pi\)
0.353947 + 0.935266i \(0.384839\pi\)
\(450\) 0 0
\(451\) −12.8990 −0.607389
\(452\) 2.79796 0.131605
\(453\) 0 0
\(454\) −9.59592 −0.450359
\(455\) 0 0
\(456\) 0 0
\(457\) 15.1010 0.706396 0.353198 0.935549i \(-0.385094\pi\)
0.353198 + 0.935549i \(0.385094\pi\)
\(458\) −17.2474 −0.805920
\(459\) 0 0
\(460\) 0 0
\(461\) −12.0000 −0.558896 −0.279448 0.960161i \(-0.590151\pi\)
−0.279448 + 0.960161i \(0.590151\pi\)
\(462\) 0 0
\(463\) −14.6969 −0.683025 −0.341512 0.939877i \(-0.610939\pi\)
−0.341512 + 0.939877i \(0.610939\pi\)
\(464\) −10.3485 −0.480416
\(465\) 0 0
\(466\) −18.2474 −0.845297
\(467\) 8.34847 0.386321 0.193161 0.981167i \(-0.438126\pi\)
0.193161 + 0.981167i \(0.438126\pi\)
\(468\) 0 0
\(469\) 4.15663 0.191935
\(470\) 0 0
\(471\) 0 0
\(472\) −1.55051 −0.0713680
\(473\) −3.55051 −0.163253
\(474\) 0 0
\(475\) 0 0
\(476\) −0.202041 −0.00926054
\(477\) 0 0
\(478\) −20.6969 −0.946656
\(479\) 22.5959 1.03243 0.516217 0.856458i \(-0.327340\pi\)
0.516217 + 0.856458i \(0.327340\pi\)
\(480\) 0 0
\(481\) 28.8990 1.31768
\(482\) 0.449490 0.0204737
\(483\) 0 0
\(484\) −8.89898 −0.404499
\(485\) 0 0
\(486\) 0 0
\(487\) 32.0000 1.45006 0.725029 0.688718i \(-0.241826\pi\)
0.725029 + 0.688718i \(0.241826\pi\)
\(488\) −4.55051 −0.205992
\(489\) 0 0
\(490\) 0 0
\(491\) −4.40408 −0.198753 −0.0993767 0.995050i \(-0.531685\pi\)
−0.0993767 + 0.995050i \(0.531685\pi\)
\(492\) 0 0
\(493\) 4.65153 0.209494
\(494\) −2.44949 −0.110208
\(495\) 0 0
\(496\) −3.00000 −0.134704
\(497\) 2.89898 0.130037
\(498\) 0 0
\(499\) 39.8434 1.78363 0.891817 0.452396i \(-0.149431\pi\)
0.891817 + 0.452396i \(0.149431\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 18.0000 0.803379
\(503\) 10.2020 0.454887 0.227443 0.973791i \(-0.426963\pi\)
0.227443 + 0.973791i \(0.426963\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −1.44949 −0.0644377
\(507\) 0 0
\(508\) −9.89898 −0.439196
\(509\) −4.14643 −0.183787 −0.0918936 0.995769i \(-0.529292\pi\)
−0.0918936 + 0.995769i \(0.529292\pi\)
\(510\) 0 0
\(511\) 0.449490 0.0198843
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 27.4949 1.21275
\(515\) 0 0
\(516\) 0 0
\(517\) 17.1010 0.752102
\(518\) −5.30306 −0.233003
\(519\) 0 0
\(520\) 0 0
\(521\) 13.6969 0.600074 0.300037 0.953928i \(-0.403001\pi\)
0.300037 + 0.953928i \(0.403001\pi\)
\(522\) 0 0
\(523\) −25.3939 −1.11040 −0.555198 0.831718i \(-0.687358\pi\)
−0.555198 + 0.831718i \(0.687358\pi\)
\(524\) 15.2474 0.666088
\(525\) 0 0
\(526\) 12.7980 0.558018
\(527\) 1.34847 0.0587402
\(528\) 0 0
\(529\) −22.0000 −0.956522
\(530\) 0 0
\(531\) 0 0
\(532\) 0.449490 0.0194879
\(533\) 21.7980 0.944174
\(534\) 0 0
\(535\) 0 0
\(536\) 9.24745 0.399429
\(537\) 0 0
\(538\) −6.89898 −0.297436
\(539\) −9.85357 −0.424423
\(540\) 0 0
\(541\) −12.1464 −0.522216 −0.261108 0.965310i \(-0.584088\pi\)
−0.261108 + 0.965310i \(0.584088\pi\)
\(542\) −30.9444 −1.32918
\(543\) 0 0
\(544\) −0.449490 −0.0192717
\(545\) 0 0
\(546\) 0 0
\(547\) −36.6413 −1.56667 −0.783335 0.621600i \(-0.786483\pi\)
−0.783335 + 0.621600i \(0.786483\pi\)
\(548\) 4.89898 0.209274
\(549\) 0 0
\(550\) 0 0
\(551\) −10.3485 −0.440860
\(552\) 0 0
\(553\) −2.24745 −0.0955712
\(554\) 23.0454 0.979106
\(555\) 0 0
\(556\) 2.24745 0.0953131
\(557\) 12.6515 0.536063 0.268031 0.963410i \(-0.413627\pi\)
0.268031 + 0.963410i \(0.413627\pi\)
\(558\) 0 0
\(559\) 6.00000 0.253773
\(560\) 0 0
\(561\) 0 0
\(562\) −7.00000 −0.295277
\(563\) −18.2474 −0.769038 −0.384519 0.923117i \(-0.625633\pi\)
−0.384519 + 0.923117i \(0.625633\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −10.2020 −0.428824
\(567\) 0 0
\(568\) 6.44949 0.270615
\(569\) −45.1918 −1.89454 −0.947270 0.320436i \(-0.896171\pi\)
−0.947270 + 0.320436i \(0.896171\pi\)
\(570\) 0 0
\(571\) 14.2474 0.596237 0.298119 0.954529i \(-0.403641\pi\)
0.298119 + 0.954529i \(0.403641\pi\)
\(572\) −3.55051 −0.148454
\(573\) 0 0
\(574\) −4.00000 −0.166957
\(575\) 0 0
\(576\) 0 0
\(577\) −9.89898 −0.412100 −0.206050 0.978541i \(-0.566061\pi\)
−0.206050 + 0.978541i \(0.566061\pi\)
\(578\) −16.7980 −0.698703
\(579\) 0 0
\(580\) 0 0
\(581\) −2.85357 −0.118386
\(582\) 0 0
\(583\) −3.69694 −0.153112
\(584\) 1.00000 0.0413803
\(585\) 0 0
\(586\) 1.24745 0.0515316
\(587\) −28.5505 −1.17841 −0.589203 0.807985i \(-0.700558\pi\)
−0.589203 + 0.807985i \(0.700558\pi\)
\(588\) 0 0
\(589\) −3.00000 −0.123613
\(590\) 0 0
\(591\) 0 0
\(592\) −11.7980 −0.484893
\(593\) −40.4949 −1.66293 −0.831463 0.555580i \(-0.812496\pi\)
−0.831463 + 0.555580i \(0.812496\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 13.7980 0.565186
\(597\) 0 0
\(598\) 2.44949 0.100167
\(599\) −11.5505 −0.471941 −0.235971 0.971760i \(-0.575827\pi\)
−0.235971 + 0.971760i \(0.575827\pi\)
\(600\) 0 0
\(601\) −17.1464 −0.699417 −0.349709 0.936858i \(-0.613719\pi\)
−0.349709 + 0.936858i \(0.613719\pi\)
\(602\) −1.10102 −0.0448742
\(603\) 0 0
\(604\) −4.20204 −0.170979
\(605\) 0 0
\(606\) 0 0
\(607\) 32.1918 1.30663 0.653313 0.757088i \(-0.273379\pi\)
0.653313 + 0.757088i \(0.273379\pi\)
\(608\) 1.00000 0.0405554
\(609\) 0 0
\(610\) 0 0
\(611\) −28.8990 −1.16913
\(612\) 0 0
\(613\) 41.1918 1.66372 0.831861 0.554984i \(-0.187275\pi\)
0.831861 + 0.554984i \(0.187275\pi\)
\(614\) 1.65153 0.0666504
\(615\) 0 0
\(616\) 0.651531 0.0262509
\(617\) −40.2929 −1.62213 −0.811065 0.584957i \(-0.801112\pi\)
−0.811065 + 0.584957i \(0.801112\pi\)
\(618\) 0 0
\(619\) 10.8536 0.436242 0.218121 0.975922i \(-0.430007\pi\)
0.218121 + 0.975922i \(0.430007\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 4.20204 0.168486
\(623\) −5.34847 −0.214282
\(624\) 0 0
\(625\) 0 0
\(626\) −22.1010 −0.883334
\(627\) 0 0
\(628\) −4.89898 −0.195491
\(629\) 5.30306 0.211447
\(630\) 0 0
\(631\) −28.0000 −1.11466 −0.557331 0.830290i \(-0.688175\pi\)
−0.557331 + 0.830290i \(0.688175\pi\)
\(632\) −5.00000 −0.198889
\(633\) 0 0
\(634\) −9.24745 −0.367263
\(635\) 0 0
\(636\) 0 0
\(637\) 16.6515 0.659758
\(638\) −15.0000 −0.593856
\(639\) 0 0
\(640\) 0 0
\(641\) −18.2020 −0.718937 −0.359469 0.933157i \(-0.617042\pi\)
−0.359469 + 0.933157i \(0.617042\pi\)
\(642\) 0 0
\(643\) 19.7980 0.780755 0.390378 0.920655i \(-0.372344\pi\)
0.390378 + 0.920655i \(0.372344\pi\)
\(644\) −0.449490 −0.0177124
\(645\) 0 0
\(646\) −0.449490 −0.0176849
\(647\) 16.1010 0.632996 0.316498 0.948593i \(-0.397493\pi\)
0.316498 + 0.948593i \(0.397493\pi\)
\(648\) 0 0
\(649\) −2.24745 −0.0882201
\(650\) 0 0
\(651\) 0 0
\(652\) −0.202041 −0.00791254
\(653\) 14.6969 0.575136 0.287568 0.957760i \(-0.407153\pi\)
0.287568 + 0.957760i \(0.407153\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −8.89898 −0.347447
\(657\) 0 0
\(658\) 5.30306 0.206735
\(659\) −26.0454 −1.01459 −0.507293 0.861774i \(-0.669354\pi\)
−0.507293 + 0.861774i \(0.669354\pi\)
\(660\) 0 0
\(661\) −15.5959 −0.606611 −0.303305 0.952893i \(-0.598090\pi\)
−0.303305 + 0.952893i \(0.598090\pi\)
\(662\) −7.65153 −0.297385
\(663\) 0 0
\(664\) −6.34847 −0.246368
\(665\) 0 0
\(666\) 0 0
\(667\) 10.3485 0.400694
\(668\) 18.0000 0.696441
\(669\) 0 0
\(670\) 0 0
\(671\) −6.59592 −0.254633
\(672\) 0 0
\(673\) 30.6515 1.18153 0.590765 0.806844i \(-0.298826\pi\)
0.590765 + 0.806844i \(0.298826\pi\)
\(674\) 15.1010 0.581670
\(675\) 0 0
\(676\) −7.00000 −0.269231
\(677\) 7.65153 0.294072 0.147036 0.989131i \(-0.453027\pi\)
0.147036 + 0.989131i \(0.453027\pi\)
\(678\) 0 0
\(679\) −2.89898 −0.111253
\(680\) 0 0
\(681\) 0 0
\(682\) −4.34847 −0.166511
\(683\) 47.6413 1.82294 0.911472 0.411361i \(-0.134947\pi\)
0.911472 + 0.411361i \(0.134947\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −6.20204 −0.236795
\(687\) 0 0
\(688\) −2.44949 −0.0933859
\(689\) 6.24745 0.238009
\(690\) 0 0
\(691\) 34.2474 1.30283 0.651417 0.758720i \(-0.274175\pi\)
0.651417 + 0.758720i \(0.274175\pi\)
\(692\) −16.3485 −0.621476
\(693\) 0 0
\(694\) 15.5959 0.592013
\(695\) 0 0
\(696\) 0 0
\(697\) 4.00000 0.151511
\(698\) 27.9444 1.05771
\(699\) 0 0
\(700\) 0 0
\(701\) −16.4949 −0.623004 −0.311502 0.950246i \(-0.600832\pi\)
−0.311502 + 0.950246i \(0.600832\pi\)
\(702\) 0 0
\(703\) −11.7980 −0.444968
\(704\) 1.44949 0.0546297
\(705\) 0 0
\(706\) 6.24745 0.235126
\(707\) 2.89898 0.109027
\(708\) 0 0
\(709\) −37.2474 −1.39886 −0.699429 0.714702i \(-0.746562\pi\)
−0.699429 + 0.714702i \(0.746562\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −11.8990 −0.445933
\(713\) 3.00000 0.112351
\(714\) 0 0
\(715\) 0 0
\(716\) −6.20204 −0.231781
\(717\) 0 0
\(718\) 13.1010 0.488926
\(719\) 38.7980 1.44692 0.723460 0.690366i \(-0.242551\pi\)
0.723460 + 0.690366i \(0.242551\pi\)
\(720\) 0 0
\(721\) 3.55051 0.132228
\(722\) 1.00000 0.0372161
\(723\) 0 0
\(724\) −9.55051 −0.354942
\(725\) 0 0
\(726\) 0 0
\(727\) 26.4949 0.982641 0.491321 0.870979i \(-0.336514\pi\)
0.491321 + 0.870979i \(0.336514\pi\)
\(728\) −1.10102 −0.0408065
\(729\) 0 0
\(730\) 0 0
\(731\) 1.10102 0.0407227
\(732\) 0 0
\(733\) 30.1464 1.11348 0.556742 0.830686i \(-0.312051\pi\)
0.556742 + 0.830686i \(0.312051\pi\)
\(734\) 10.4495 0.385698
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) 13.4041 0.493746
\(738\) 0 0
\(739\) 23.1010 0.849785 0.424892 0.905244i \(-0.360312\pi\)
0.424892 + 0.905244i \(0.360312\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −1.14643 −0.0420867
\(743\) 5.39388 0.197882 0.0989411 0.995093i \(-0.468454\pi\)
0.0989411 + 0.995093i \(0.468454\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 17.5505 0.642570
\(747\) 0 0
\(748\) −0.651531 −0.0238223
\(749\) 1.19184 0.0435487
\(750\) 0 0
\(751\) −4.20204 −0.153335 −0.0766673 0.997057i \(-0.524428\pi\)
−0.0766673 + 0.997057i \(0.524428\pi\)
\(752\) 11.7980 0.430227
\(753\) 0 0
\(754\) 25.3485 0.923137
\(755\) 0 0
\(756\) 0 0
\(757\) −13.8536 −0.503517 −0.251758 0.967790i \(-0.581009\pi\)
−0.251758 + 0.967790i \(0.581009\pi\)
\(758\) −30.6969 −1.11496
\(759\) 0 0
\(760\) 0 0
\(761\) −36.6515 −1.32862 −0.664308 0.747459i \(-0.731274\pi\)
−0.664308 + 0.747459i \(0.731274\pi\)
\(762\) 0 0
\(763\) 3.10102 0.112264
\(764\) 9.89898 0.358133
\(765\) 0 0
\(766\) −8.24745 −0.297992
\(767\) 3.79796 0.137136
\(768\) 0 0
\(769\) 1.89898 0.0684790 0.0342395 0.999414i \(-0.489099\pi\)
0.0342395 + 0.999414i \(0.489099\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0.651531 0.0234491
\(773\) −30.4949 −1.09683 −0.548413 0.836208i \(-0.684768\pi\)
−0.548413 + 0.836208i \(0.684768\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −6.44949 −0.231523
\(777\) 0 0
\(778\) 17.5959 0.630844
\(779\) −8.89898 −0.318839
\(780\) 0 0
\(781\) 9.34847 0.334515
\(782\) 0.449490 0.0160737
\(783\) 0 0
\(784\) −6.79796 −0.242784
\(785\) 0 0
\(786\) 0 0
\(787\) 28.5505 1.01772 0.508858 0.860851i \(-0.330068\pi\)
0.508858 + 0.860851i \(0.330068\pi\)
\(788\) 12.6515 0.450692
\(789\) 0 0
\(790\) 0 0
\(791\) 1.25765 0.0447170
\(792\) 0 0
\(793\) 11.1464 0.395821
\(794\) 29.9444 1.06269
\(795\) 0 0
\(796\) −10.0000 −0.354441
\(797\) 52.4949 1.85946 0.929732 0.368236i \(-0.120038\pi\)
0.929732 + 0.368236i \(0.120038\pi\)
\(798\) 0 0
\(799\) −5.30306 −0.187609
\(800\) 0 0
\(801\) 0 0
\(802\) −30.7980 −1.08751
\(803\) 1.44949 0.0511514
\(804\) 0 0
\(805\) 0 0
\(806\) 7.34847 0.258839
\(807\) 0 0
\(808\) 6.44949 0.226892
\(809\) 8.44949 0.297068 0.148534 0.988907i \(-0.452545\pi\)
0.148534 + 0.988907i \(0.452545\pi\)
\(810\) 0 0
\(811\) −14.5505 −0.510938 −0.255469 0.966817i \(-0.582230\pi\)
−0.255469 + 0.966817i \(0.582230\pi\)
\(812\) −4.65153 −0.163237
\(813\) 0 0
\(814\) −17.1010 −0.599390
\(815\) 0 0
\(816\) 0 0
\(817\) −2.44949 −0.0856968
\(818\) −13.1010 −0.458066
\(819\) 0 0
\(820\) 0 0
\(821\) −11.1464 −0.389013 −0.194507 0.980901i \(-0.562311\pi\)
−0.194507 + 0.980901i \(0.562311\pi\)
\(822\) 0 0
\(823\) 12.2020 0.425336 0.212668 0.977124i \(-0.431785\pi\)
0.212668 + 0.977124i \(0.431785\pi\)
\(824\) 7.89898 0.275174
\(825\) 0 0
\(826\) −0.696938 −0.0242496
\(827\) −24.2474 −0.843166 −0.421583 0.906790i \(-0.638525\pi\)
−0.421583 + 0.906790i \(0.638525\pi\)
\(828\) 0 0
\(829\) 43.6413 1.51573 0.757863 0.652414i \(-0.226244\pi\)
0.757863 + 0.652414i \(0.226244\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −2.44949 −0.0849208
\(833\) 3.05561 0.105871
\(834\) 0 0
\(835\) 0 0
\(836\) 1.44949 0.0501317
\(837\) 0 0
\(838\) 0 0
\(839\) −25.3485 −0.875126 −0.437563 0.899188i \(-0.644158\pi\)
−0.437563 + 0.899188i \(0.644158\pi\)
\(840\) 0 0
\(841\) 78.0908 2.69279
\(842\) 22.0000 0.758170
\(843\) 0 0
\(844\) −18.3485 −0.631580
\(845\) 0 0
\(846\) 0 0
\(847\) −4.00000 −0.137442
\(848\) −2.55051 −0.0875849
\(849\) 0 0
\(850\) 0 0
\(851\) 11.7980 0.404429
\(852\) 0 0
\(853\) −10.8990 −0.373174 −0.186587 0.982438i \(-0.559743\pi\)
−0.186587 + 0.982438i \(0.559743\pi\)
\(854\) −2.04541 −0.0699924
\(855\) 0 0
\(856\) 2.65153 0.0906275
\(857\) 12.4949 0.426818 0.213409 0.976963i \(-0.431543\pi\)
0.213409 + 0.976963i \(0.431543\pi\)
\(858\) 0 0
\(859\) −23.7980 −0.811976 −0.405988 0.913878i \(-0.633073\pi\)
−0.405988 + 0.913878i \(0.633073\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 14.8990 0.507461
\(863\) −46.6969 −1.58958 −0.794791 0.606883i \(-0.792420\pi\)
−0.794791 + 0.606883i \(0.792420\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −19.3485 −0.657488
\(867\) 0 0
\(868\) −1.34847 −0.0457700
\(869\) −7.24745 −0.245853
\(870\) 0 0
\(871\) −22.6515 −0.767518
\(872\) 6.89898 0.233629
\(873\) 0 0
\(874\) −1.00000 −0.0338255
\(875\) 0 0
\(876\) 0 0
\(877\) 18.8990 0.638173 0.319087 0.947726i \(-0.396624\pi\)
0.319087 + 0.947726i \(0.396624\pi\)
\(878\) −21.8990 −0.739055
\(879\) 0 0
\(880\) 0 0
\(881\) −21.3031 −0.717718 −0.358859 0.933392i \(-0.616834\pi\)
−0.358859 + 0.933392i \(0.616834\pi\)
\(882\) 0 0
\(883\) −49.3485 −1.66071 −0.830354 0.557236i \(-0.811862\pi\)
−0.830354 + 0.557236i \(0.811862\pi\)
\(884\) 1.10102 0.0370313
\(885\) 0 0
\(886\) −3.24745 −0.109100
\(887\) 4.89898 0.164492 0.0822458 0.996612i \(-0.473791\pi\)
0.0822458 + 0.996612i \(0.473791\pi\)
\(888\) 0 0
\(889\) −4.44949 −0.149231
\(890\) 0 0
\(891\) 0 0
\(892\) −25.8990 −0.867162
\(893\) 11.7980 0.394804
\(894\) 0 0
\(895\) 0 0
\(896\) 0.449490 0.0150164
\(897\) 0 0
\(898\) 15.0000 0.500556
\(899\) 31.0454 1.03542
\(900\) 0 0
\(901\) 1.14643 0.0381931
\(902\) −12.8990 −0.429489
\(903\) 0 0
\(904\) 2.79796 0.0930587
\(905\) 0 0
\(906\) 0 0
\(907\) 22.0000 0.730498 0.365249 0.930910i \(-0.380984\pi\)
0.365249 + 0.930910i \(0.380984\pi\)
\(908\) −9.59592 −0.318452
\(909\) 0 0
\(910\) 0 0
\(911\) 42.4949 1.40792 0.703959 0.710240i \(-0.251414\pi\)
0.703959 + 0.710240i \(0.251414\pi\)
\(912\) 0 0
\(913\) −9.20204 −0.304543
\(914\) 15.1010 0.499497
\(915\) 0 0
\(916\) −17.2474 −0.569872
\(917\) 6.85357 0.226325
\(918\) 0 0
\(919\) −39.8434 −1.31431 −0.657156 0.753755i \(-0.728241\pi\)
−0.657156 + 0.753755i \(0.728241\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −12.0000 −0.395199
\(923\) −15.7980 −0.519996
\(924\) 0 0
\(925\) 0 0
\(926\) −14.6969 −0.482971
\(927\) 0 0
\(928\) −10.3485 −0.339705
\(929\) −7.59592 −0.249214 −0.124607 0.992206i \(-0.539767\pi\)
−0.124607 + 0.992206i \(0.539767\pi\)
\(930\) 0 0
\(931\) −6.79796 −0.222794
\(932\) −18.2474 −0.597715
\(933\) 0 0
\(934\) 8.34847 0.273170
\(935\) 0 0
\(936\) 0 0
\(937\) −39.3939 −1.28694 −0.643471 0.765471i \(-0.722506\pi\)
−0.643471 + 0.765471i \(0.722506\pi\)
\(938\) 4.15663 0.135719
\(939\) 0 0
\(940\) 0 0
\(941\) 46.6413 1.52046 0.760232 0.649652i \(-0.225085\pi\)
0.760232 + 0.649652i \(0.225085\pi\)
\(942\) 0 0
\(943\) 8.89898 0.289791
\(944\) −1.55051 −0.0504648
\(945\) 0 0
\(946\) −3.55051 −0.115437
\(947\) 8.00000 0.259965 0.129983 0.991516i \(-0.458508\pi\)
0.129983 + 0.991516i \(0.458508\pi\)
\(948\) 0 0
\(949\) −2.44949 −0.0795138
\(950\) 0 0
\(951\) 0 0
\(952\) −0.202041 −0.00654819
\(953\) 33.4949 1.08501 0.542503 0.840054i \(-0.317477\pi\)
0.542503 + 0.840054i \(0.317477\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −20.6969 −0.669387
\(957\) 0 0
\(958\) 22.5959 0.730041
\(959\) 2.20204 0.0711076
\(960\) 0 0
\(961\) −22.0000 −0.709677
\(962\) 28.8990 0.931740
\(963\) 0 0
\(964\) 0.449490 0.0144771
\(965\) 0 0
\(966\) 0 0
\(967\) 25.1010 0.807194 0.403597 0.914937i \(-0.367760\pi\)
0.403597 + 0.914937i \(0.367760\pi\)
\(968\) −8.89898 −0.286024
\(969\) 0 0
\(970\) 0 0
\(971\) 41.6413 1.33633 0.668167 0.744011i \(-0.267079\pi\)
0.668167 + 0.744011i \(0.267079\pi\)
\(972\) 0 0
\(973\) 1.01021 0.0323857
\(974\) 32.0000 1.02535
\(975\) 0 0
\(976\) −4.55051 −0.145658
\(977\) −19.5959 −0.626929 −0.313464 0.949600i \(-0.601490\pi\)
−0.313464 + 0.949600i \(0.601490\pi\)
\(978\) 0 0
\(979\) −17.2474 −0.551231
\(980\) 0 0
\(981\) 0 0
\(982\) −4.40408 −0.140540
\(983\) 45.5505 1.45284 0.726418 0.687253i \(-0.241184\pi\)
0.726418 + 0.687253i \(0.241184\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 4.65153 0.148135
\(987\) 0 0
\(988\) −2.44949 −0.0779287
\(989\) 2.44949 0.0778892
\(990\) 0 0
\(991\) 53.8990 1.71216 0.856079 0.516845i \(-0.172894\pi\)
0.856079 + 0.516845i \(0.172894\pi\)
\(992\) −3.00000 −0.0952501
\(993\) 0 0
\(994\) 2.89898 0.0919500
\(995\) 0 0
\(996\) 0 0
\(997\) −34.5505 −1.09423 −0.547113 0.837059i \(-0.684273\pi\)
−0.547113 + 0.837059i \(0.684273\pi\)
\(998\) 39.8434 1.26122
\(999\) 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 8550.2.a.bv.1.2 2
3.2 odd 2 2850.2.a.bc.1.2 2
5.4 even 2 8550.2.a.bu.1.1 2
15.2 even 4 2850.2.d.w.799.2 4
15.8 even 4 2850.2.d.w.799.3 4
15.14 odd 2 2850.2.a.bj.1.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2850.2.a.bc.1.2 2 3.2 odd 2
2850.2.a.bj.1.1 yes 2 15.14 odd 2
2850.2.d.w.799.2 4 15.2 even 4
2850.2.d.w.799.3 4 15.8 even 4
8550.2.a.bu.1.1 2 5.4 even 2
8550.2.a.bv.1.2 2 1.1 even 1 trivial