Properties

Label 5445.2.a.s.1.1
Level $5445$
Weight $2$
Character 5445.1
Self dual yes
Analytic conductor $43.479$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5445 = 3^{2} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5445.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(43.4785439006\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 5445.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.73205 q^{2} +1.00000 q^{4} +1.00000 q^{5} -2.00000 q^{7} +1.73205 q^{8} +O(q^{10})\) \(q-1.73205 q^{2} +1.00000 q^{4} +1.00000 q^{5} -2.00000 q^{7} +1.73205 q^{8} -1.73205 q^{10} -5.46410 q^{13} +3.46410 q^{14} -5.00000 q^{16} -5.46410 q^{19} +1.00000 q^{20} -6.92820 q^{23} +1.00000 q^{25} +9.46410 q^{26} -2.00000 q^{28} -3.46410 q^{29} -10.9282 q^{31} +5.19615 q^{32} -2.00000 q^{35} -4.92820 q^{37} +9.46410 q^{38} +1.73205 q^{40} +3.46410 q^{41} +4.92820 q^{43} +12.0000 q^{46} +6.92820 q^{47} -3.00000 q^{49} -1.73205 q^{50} -5.46410 q^{52} -0.928203 q^{53} -3.46410 q^{56} +6.00000 q^{58} +6.92820 q^{59} -2.00000 q^{61} +18.9282 q^{62} +1.00000 q^{64} -5.46410 q^{65} +8.00000 q^{67} +3.46410 q^{70} -13.8564 q^{71} +8.39230 q^{73} +8.53590 q^{74} -5.46410 q^{76} +6.53590 q^{79} -5.00000 q^{80} -6.00000 q^{82} +8.53590 q^{83} -8.53590 q^{86} -0.928203 q^{89} +10.9282 q^{91} -6.92820 q^{92} -12.0000 q^{94} -5.46410 q^{95} -10.0000 q^{97} +5.19615 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{4} + 2q^{5} - 4q^{7} + O(q^{10}) \) \( 2q + 2q^{4} + 2q^{5} - 4q^{7} - 4q^{13} - 10q^{16} - 4q^{19} + 2q^{20} + 2q^{25} + 12q^{26} - 4q^{28} - 8q^{31} - 4q^{35} + 4q^{37} + 12q^{38} - 4q^{43} + 24q^{46} - 6q^{49} - 4q^{52} + 12q^{53} + 12q^{58} - 4q^{61} + 24q^{62} + 2q^{64} - 4q^{65} + 16q^{67} - 4q^{73} + 24q^{74} - 4q^{76} + 20q^{79} - 10q^{80} - 12q^{82} + 24q^{83} - 24q^{86} + 12q^{89} + 8q^{91} - 24q^{94} - 4q^{95} - 20q^{97} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.73205 −1.22474 −0.612372 0.790569i \(-0.709785\pi\)
−0.612372 + 0.790569i \(0.709785\pi\)
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −2.00000 −0.755929 −0.377964 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 1.73205 0.612372
\(9\) 0 0
\(10\) −1.73205 −0.547723
\(11\) 0 0
\(12\) 0 0
\(13\) −5.46410 −1.51547 −0.757735 0.652563i \(-0.773694\pi\)
−0.757735 + 0.652563i \(0.773694\pi\)
\(14\) 3.46410 0.925820
\(15\) 0 0
\(16\) −5.00000 −1.25000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) −5.46410 −1.25355 −0.626775 0.779200i \(-0.715626\pi\)
−0.626775 + 0.779200i \(0.715626\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) 0 0
\(23\) −6.92820 −1.44463 −0.722315 0.691564i \(-0.756922\pi\)
−0.722315 + 0.691564i \(0.756922\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 9.46410 1.85606
\(27\) 0 0
\(28\) −2.00000 −0.377964
\(29\) −3.46410 −0.643268 −0.321634 0.946864i \(-0.604232\pi\)
−0.321634 + 0.946864i \(0.604232\pi\)
\(30\) 0 0
\(31\) −10.9282 −1.96276 −0.981382 0.192068i \(-0.938481\pi\)
−0.981382 + 0.192068i \(0.938481\pi\)
\(32\) 5.19615 0.918559
\(33\) 0 0
\(34\) 0 0
\(35\) −2.00000 −0.338062
\(36\) 0 0
\(37\) −4.92820 −0.810192 −0.405096 0.914274i \(-0.632762\pi\)
−0.405096 + 0.914274i \(0.632762\pi\)
\(38\) 9.46410 1.53528
\(39\) 0 0
\(40\) 1.73205 0.273861
\(41\) 3.46410 0.541002 0.270501 0.962720i \(-0.412811\pi\)
0.270501 + 0.962720i \(0.412811\pi\)
\(42\) 0 0
\(43\) 4.92820 0.751544 0.375772 0.926712i \(-0.377378\pi\)
0.375772 + 0.926712i \(0.377378\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 12.0000 1.76930
\(47\) 6.92820 1.01058 0.505291 0.862949i \(-0.331385\pi\)
0.505291 + 0.862949i \(0.331385\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) −1.73205 −0.244949
\(51\) 0 0
\(52\) −5.46410 −0.757735
\(53\) −0.928203 −0.127499 −0.0637493 0.997966i \(-0.520306\pi\)
−0.0637493 + 0.997966i \(0.520306\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −3.46410 −0.462910
\(57\) 0 0
\(58\) 6.00000 0.787839
\(59\) 6.92820 0.901975 0.450988 0.892530i \(-0.351072\pi\)
0.450988 + 0.892530i \(0.351072\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 18.9282 2.40388
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −5.46410 −0.677738
\(66\) 0 0
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 3.46410 0.414039
\(71\) −13.8564 −1.64445 −0.822226 0.569160i \(-0.807268\pi\)
−0.822226 + 0.569160i \(0.807268\pi\)
\(72\) 0 0
\(73\) 8.39230 0.982245 0.491122 0.871091i \(-0.336587\pi\)
0.491122 + 0.871091i \(0.336587\pi\)
\(74\) 8.53590 0.992278
\(75\) 0 0
\(76\) −5.46410 −0.626775
\(77\) 0 0
\(78\) 0 0
\(79\) 6.53590 0.735346 0.367673 0.929955i \(-0.380155\pi\)
0.367673 + 0.929955i \(0.380155\pi\)
\(80\) −5.00000 −0.559017
\(81\) 0 0
\(82\) −6.00000 −0.662589
\(83\) 8.53590 0.936937 0.468468 0.883480i \(-0.344806\pi\)
0.468468 + 0.883480i \(0.344806\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −8.53590 −0.920450
\(87\) 0 0
\(88\) 0 0
\(89\) −0.928203 −0.0983893 −0.0491947 0.998789i \(-0.515665\pi\)
−0.0491947 + 0.998789i \(0.515665\pi\)
\(90\) 0 0
\(91\) 10.9282 1.14559
\(92\) −6.92820 −0.722315
\(93\) 0 0
\(94\) −12.0000 −1.23771
\(95\) −5.46410 −0.560605
\(96\) 0 0
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 5.19615 0.524891
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −10.3923 −1.03407 −0.517036 0.855963i \(-0.672965\pi\)
−0.517036 + 0.855963i \(0.672965\pi\)
\(102\) 0 0
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) −9.46410 −0.928032
\(105\) 0 0
\(106\) 1.60770 0.156153
\(107\) 8.53590 0.825196 0.412598 0.910913i \(-0.364621\pi\)
0.412598 + 0.910913i \(0.364621\pi\)
\(108\) 0 0
\(109\) 10.0000 0.957826 0.478913 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 10.0000 0.944911
\(113\) 12.9282 1.21618 0.608092 0.793867i \(-0.291935\pi\)
0.608092 + 0.793867i \(0.291935\pi\)
\(114\) 0 0
\(115\) −6.92820 −0.646058
\(116\) −3.46410 −0.321634
\(117\) 0 0
\(118\) −12.0000 −1.10469
\(119\) 0 0
\(120\) 0 0
\(121\) 0 0
\(122\) 3.46410 0.313625
\(123\) 0 0
\(124\) −10.9282 −0.981382
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −8.92820 −0.792250 −0.396125 0.918197i \(-0.629645\pi\)
−0.396125 + 0.918197i \(0.629645\pi\)
\(128\) −12.1244 −1.07165
\(129\) 0 0
\(130\) 9.46410 0.830057
\(131\) 18.9282 1.65376 0.826882 0.562375i \(-0.190112\pi\)
0.826882 + 0.562375i \(0.190112\pi\)
\(132\) 0 0
\(133\) 10.9282 0.947595
\(134\) −13.8564 −1.19701
\(135\) 0 0
\(136\) 0 0
\(137\) 18.0000 1.53784 0.768922 0.639343i \(-0.220793\pi\)
0.768922 + 0.639343i \(0.220793\pi\)
\(138\) 0 0
\(139\) −12.3923 −1.05110 −0.525551 0.850762i \(-0.676141\pi\)
−0.525551 + 0.850762i \(0.676141\pi\)
\(140\) −2.00000 −0.169031
\(141\) 0 0
\(142\) 24.0000 2.01404
\(143\) 0 0
\(144\) 0 0
\(145\) −3.46410 −0.287678
\(146\) −14.5359 −1.20300
\(147\) 0 0
\(148\) −4.92820 −0.405096
\(149\) −15.4641 −1.26687 −0.633434 0.773796i \(-0.718355\pi\)
−0.633434 + 0.773796i \(0.718355\pi\)
\(150\) 0 0
\(151\) 20.3923 1.65950 0.829751 0.558134i \(-0.188482\pi\)
0.829751 + 0.558134i \(0.188482\pi\)
\(152\) −9.46410 −0.767640
\(153\) 0 0
\(154\) 0 0
\(155\) −10.9282 −0.877774
\(156\) 0 0
\(157\) −3.07180 −0.245156 −0.122578 0.992459i \(-0.539116\pi\)
−0.122578 + 0.992459i \(0.539116\pi\)
\(158\) −11.3205 −0.900611
\(159\) 0 0
\(160\) 5.19615 0.410792
\(161\) 13.8564 1.09204
\(162\) 0 0
\(163\) 9.85641 0.772013 0.386007 0.922496i \(-0.373854\pi\)
0.386007 + 0.922496i \(0.373854\pi\)
\(164\) 3.46410 0.270501
\(165\) 0 0
\(166\) −14.7846 −1.14751
\(167\) −10.3923 −0.804181 −0.402090 0.915600i \(-0.631716\pi\)
−0.402090 + 0.915600i \(0.631716\pi\)
\(168\) 0 0
\(169\) 16.8564 1.29665
\(170\) 0 0
\(171\) 0 0
\(172\) 4.92820 0.375772
\(173\) −12.0000 −0.912343 −0.456172 0.889892i \(-0.650780\pi\)
−0.456172 + 0.889892i \(0.650780\pi\)
\(174\) 0 0
\(175\) −2.00000 −0.151186
\(176\) 0 0
\(177\) 0 0
\(178\) 1.60770 0.120502
\(179\) −6.92820 −0.517838 −0.258919 0.965899i \(-0.583366\pi\)
−0.258919 + 0.965899i \(0.583366\pi\)
\(180\) 0 0
\(181\) 15.8564 1.17860 0.589299 0.807915i \(-0.299404\pi\)
0.589299 + 0.807915i \(0.299404\pi\)
\(182\) −18.9282 −1.40305
\(183\) 0 0
\(184\) −12.0000 −0.884652
\(185\) −4.92820 −0.362329
\(186\) 0 0
\(187\) 0 0
\(188\) 6.92820 0.505291
\(189\) 0 0
\(190\) 9.46410 0.686598
\(191\) 18.9282 1.36960 0.684798 0.728733i \(-0.259890\pi\)
0.684798 + 0.728733i \(0.259890\pi\)
\(192\) 0 0
\(193\) −24.3923 −1.75580 −0.877898 0.478847i \(-0.841055\pi\)
−0.877898 + 0.478847i \(0.841055\pi\)
\(194\) 17.3205 1.24354
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) −12.0000 −0.854965 −0.427482 0.904024i \(-0.640599\pi\)
−0.427482 + 0.904024i \(0.640599\pi\)
\(198\) 0 0
\(199\) −24.7846 −1.75693 −0.878467 0.477803i \(-0.841433\pi\)
−0.878467 + 0.477803i \(0.841433\pi\)
\(200\) 1.73205 0.122474
\(201\) 0 0
\(202\) 18.0000 1.26648
\(203\) 6.92820 0.486265
\(204\) 0 0
\(205\) 3.46410 0.241943
\(206\) −13.8564 −0.965422
\(207\) 0 0
\(208\) 27.3205 1.89434
\(209\) 0 0
\(210\) 0 0
\(211\) 8.39230 0.577750 0.288875 0.957367i \(-0.406719\pi\)
0.288875 + 0.957367i \(0.406719\pi\)
\(212\) −0.928203 −0.0637493
\(213\) 0 0
\(214\) −14.7846 −1.01066
\(215\) 4.92820 0.336101
\(216\) 0 0
\(217\) 21.8564 1.48371
\(218\) −17.3205 −1.17309
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 9.85641 0.660034 0.330017 0.943975i \(-0.392946\pi\)
0.330017 + 0.943975i \(0.392946\pi\)
\(224\) −10.3923 −0.694365
\(225\) 0 0
\(226\) −22.3923 −1.48951
\(227\) 15.4641 1.02639 0.513194 0.858272i \(-0.328462\pi\)
0.513194 + 0.858272i \(0.328462\pi\)
\(228\) 0 0
\(229\) −23.8564 −1.57648 −0.788238 0.615371i \(-0.789006\pi\)
−0.788238 + 0.615371i \(0.789006\pi\)
\(230\) 12.0000 0.791257
\(231\) 0 0
\(232\) −6.00000 −0.393919
\(233\) 12.0000 0.786146 0.393073 0.919507i \(-0.371412\pi\)
0.393073 + 0.919507i \(0.371412\pi\)
\(234\) 0 0
\(235\) 6.92820 0.451946
\(236\) 6.92820 0.450988
\(237\) 0 0
\(238\) 0 0
\(239\) −12.0000 −0.776215 −0.388108 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 0 0
\(241\) −0.143594 −0.00924967 −0.00462484 0.999989i \(-0.501472\pi\)
−0.00462484 + 0.999989i \(0.501472\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −2.00000 −0.128037
\(245\) −3.00000 −0.191663
\(246\) 0 0
\(247\) 29.8564 1.89972
\(248\) −18.9282 −1.20194
\(249\) 0 0
\(250\) −1.73205 −0.109545
\(251\) 1.85641 0.117175 0.0585877 0.998282i \(-0.481340\pi\)
0.0585877 + 0.998282i \(0.481340\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 15.4641 0.970304
\(255\) 0 0
\(256\) 19.0000 1.18750
\(257\) 19.8564 1.23861 0.619304 0.785151i \(-0.287415\pi\)
0.619304 + 0.785151i \(0.287415\pi\)
\(258\) 0 0
\(259\) 9.85641 0.612447
\(260\) −5.46410 −0.338869
\(261\) 0 0
\(262\) −32.7846 −2.02544
\(263\) 20.5359 1.26630 0.633149 0.774030i \(-0.281762\pi\)
0.633149 + 0.774030i \(0.281762\pi\)
\(264\) 0 0
\(265\) −0.928203 −0.0570191
\(266\) −18.9282 −1.16056
\(267\) 0 0
\(268\) 8.00000 0.488678
\(269\) −19.8564 −1.21067 −0.605333 0.795972i \(-0.706960\pi\)
−0.605333 + 0.795972i \(0.706960\pi\)
\(270\) 0 0
\(271\) 11.6077 0.705117 0.352559 0.935790i \(-0.385312\pi\)
0.352559 + 0.935790i \(0.385312\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) −31.1769 −1.88347
\(275\) 0 0
\(276\) 0 0
\(277\) −29.4641 −1.77033 −0.885163 0.465281i \(-0.845953\pi\)
−0.885163 + 0.465281i \(0.845953\pi\)
\(278\) 21.4641 1.28733
\(279\) 0 0
\(280\) −3.46410 −0.207020
\(281\) 3.46410 0.206651 0.103325 0.994648i \(-0.467052\pi\)
0.103325 + 0.994648i \(0.467052\pi\)
\(282\) 0 0
\(283\) 4.92820 0.292951 0.146476 0.989214i \(-0.453207\pi\)
0.146476 + 0.989214i \(0.453207\pi\)
\(284\) −13.8564 −0.822226
\(285\) 0 0
\(286\) 0 0
\(287\) −6.92820 −0.408959
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 6.00000 0.352332
\(291\) 0 0
\(292\) 8.39230 0.491122
\(293\) −13.8564 −0.809500 −0.404750 0.914427i \(-0.632641\pi\)
−0.404750 + 0.914427i \(0.632641\pi\)
\(294\) 0 0
\(295\) 6.92820 0.403376
\(296\) −8.53590 −0.496139
\(297\) 0 0
\(298\) 26.7846 1.55159
\(299\) 37.8564 2.18929
\(300\) 0 0
\(301\) −9.85641 −0.568114
\(302\) −35.3205 −2.03247
\(303\) 0 0
\(304\) 27.3205 1.56694
\(305\) −2.00000 −0.114520
\(306\) 0 0
\(307\) −14.0000 −0.799022 −0.399511 0.916728i \(-0.630820\pi\)
−0.399511 + 0.916728i \(0.630820\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 18.9282 1.07505
\(311\) −5.07180 −0.287595 −0.143798 0.989607i \(-0.545931\pi\)
−0.143798 + 0.989607i \(0.545931\pi\)
\(312\) 0 0
\(313\) 20.9282 1.18293 0.591466 0.806330i \(-0.298549\pi\)
0.591466 + 0.806330i \(0.298549\pi\)
\(314\) 5.32051 0.300254
\(315\) 0 0
\(316\) 6.53590 0.367673
\(317\) 24.9282 1.40011 0.700054 0.714090i \(-0.253159\pi\)
0.700054 + 0.714090i \(0.253159\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 1.00000 0.0559017
\(321\) 0 0
\(322\) −24.0000 −1.33747
\(323\) 0 0
\(324\) 0 0
\(325\) −5.46410 −0.303094
\(326\) −17.0718 −0.945519
\(327\) 0 0
\(328\) 6.00000 0.331295
\(329\) −13.8564 −0.763928
\(330\) 0 0
\(331\) 9.85641 0.541757 0.270879 0.962614i \(-0.412686\pi\)
0.270879 + 0.962614i \(0.412686\pi\)
\(332\) 8.53590 0.468468
\(333\) 0 0
\(334\) 18.0000 0.984916
\(335\) 8.00000 0.437087
\(336\) 0 0
\(337\) −33.1769 −1.80726 −0.903631 0.428312i \(-0.859108\pi\)
−0.903631 + 0.428312i \(0.859108\pi\)
\(338\) −29.1962 −1.58806
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 20.0000 1.07990
\(344\) 8.53590 0.460225
\(345\) 0 0
\(346\) 20.7846 1.11739
\(347\) 22.3923 1.20208 0.601041 0.799218i \(-0.294753\pi\)
0.601041 + 0.799218i \(0.294753\pi\)
\(348\) 0 0
\(349\) 8.14359 0.435917 0.217958 0.975958i \(-0.430060\pi\)
0.217958 + 0.975958i \(0.430060\pi\)
\(350\) 3.46410 0.185164
\(351\) 0 0
\(352\) 0 0
\(353\) −12.9282 −0.688099 −0.344049 0.938952i \(-0.611799\pi\)
−0.344049 + 0.938952i \(0.611799\pi\)
\(354\) 0 0
\(355\) −13.8564 −0.735422
\(356\) −0.928203 −0.0491947
\(357\) 0 0
\(358\) 12.0000 0.634220
\(359\) −20.7846 −1.09697 −0.548485 0.836160i \(-0.684795\pi\)
−0.548485 + 0.836160i \(0.684795\pi\)
\(360\) 0 0
\(361\) 10.8564 0.571390
\(362\) −27.4641 −1.44348
\(363\) 0 0
\(364\) 10.9282 0.572793
\(365\) 8.39230 0.439273
\(366\) 0 0
\(367\) 20.0000 1.04399 0.521996 0.852948i \(-0.325188\pi\)
0.521996 + 0.852948i \(0.325188\pi\)
\(368\) 34.6410 1.80579
\(369\) 0 0
\(370\) 8.53590 0.443760
\(371\) 1.85641 0.0963798
\(372\) 0 0
\(373\) 20.3923 1.05587 0.527937 0.849284i \(-0.322966\pi\)
0.527937 + 0.849284i \(0.322966\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 12.0000 0.618853
\(377\) 18.9282 0.974852
\(378\) 0 0
\(379\) −17.8564 −0.917222 −0.458611 0.888637i \(-0.651653\pi\)
−0.458611 + 0.888637i \(0.651653\pi\)
\(380\) −5.46410 −0.280302
\(381\) 0 0
\(382\) −32.7846 −1.67741
\(383\) 13.8564 0.708029 0.354015 0.935240i \(-0.384816\pi\)
0.354015 + 0.935240i \(0.384816\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 42.2487 2.15040
\(387\) 0 0
\(388\) −10.0000 −0.507673
\(389\) −11.0718 −0.561362 −0.280681 0.959801i \(-0.590560\pi\)
−0.280681 + 0.959801i \(0.590560\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −5.19615 −0.262445
\(393\) 0 0
\(394\) 20.7846 1.04711
\(395\) 6.53590 0.328857
\(396\) 0 0
\(397\) 2.00000 0.100377 0.0501886 0.998740i \(-0.484018\pi\)
0.0501886 + 0.998740i \(0.484018\pi\)
\(398\) 42.9282 2.15180
\(399\) 0 0
\(400\) −5.00000 −0.250000
\(401\) 7.85641 0.392330 0.196165 0.980571i \(-0.437151\pi\)
0.196165 + 0.980571i \(0.437151\pi\)
\(402\) 0 0
\(403\) 59.7128 2.97451
\(404\) −10.3923 −0.517036
\(405\) 0 0
\(406\) −12.0000 −0.595550
\(407\) 0 0
\(408\) 0 0
\(409\) 6.78461 0.335477 0.167739 0.985831i \(-0.446354\pi\)
0.167739 + 0.985831i \(0.446354\pi\)
\(410\) −6.00000 −0.296319
\(411\) 0 0
\(412\) 8.00000 0.394132
\(413\) −13.8564 −0.681829
\(414\) 0 0
\(415\) 8.53590 0.419011
\(416\) −28.3923 −1.39205
\(417\) 0 0
\(418\) 0 0
\(419\) −30.9282 −1.51094 −0.755471 0.655182i \(-0.772592\pi\)
−0.755471 + 0.655182i \(0.772592\pi\)
\(420\) 0 0
\(421\) 2.00000 0.0974740 0.0487370 0.998812i \(-0.484480\pi\)
0.0487370 + 0.998812i \(0.484480\pi\)
\(422\) −14.5359 −0.707596
\(423\) 0 0
\(424\) −1.60770 −0.0780766
\(425\) 0 0
\(426\) 0 0
\(427\) 4.00000 0.193574
\(428\) 8.53590 0.412598
\(429\) 0 0
\(430\) −8.53590 −0.411638
\(431\) 8.78461 0.423140 0.211570 0.977363i \(-0.432142\pi\)
0.211570 + 0.977363i \(0.432142\pi\)
\(432\) 0 0
\(433\) 0.143594 0.00690067 0.00345033 0.999994i \(-0.498902\pi\)
0.00345033 + 0.999994i \(0.498902\pi\)
\(434\) −37.8564 −1.81717
\(435\) 0 0
\(436\) 10.0000 0.478913
\(437\) 37.8564 1.81092
\(438\) 0 0
\(439\) −33.1769 −1.58345 −0.791724 0.610879i \(-0.790816\pi\)
−0.791724 + 0.610879i \(0.790816\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −12.0000 −0.570137 −0.285069 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) 0 0
\(445\) −0.928203 −0.0440011
\(446\) −17.0718 −0.808373
\(447\) 0 0
\(448\) −2.00000 −0.0944911
\(449\) 26.7846 1.26404 0.632022 0.774950i \(-0.282225\pi\)
0.632022 + 0.774950i \(0.282225\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 12.9282 0.608092
\(453\) 0 0
\(454\) −26.7846 −1.25706
\(455\) 10.9282 0.512322
\(456\) 0 0
\(457\) −12.3923 −0.579688 −0.289844 0.957074i \(-0.593603\pi\)
−0.289844 + 0.957074i \(0.593603\pi\)
\(458\) 41.3205 1.93078
\(459\) 0 0
\(460\) −6.92820 −0.323029
\(461\) 36.2487 1.68827 0.844135 0.536130i \(-0.180114\pi\)
0.844135 + 0.536130i \(0.180114\pi\)
\(462\) 0 0
\(463\) −28.0000 −1.30127 −0.650635 0.759390i \(-0.725497\pi\)
−0.650635 + 0.759390i \(0.725497\pi\)
\(464\) 17.3205 0.804084
\(465\) 0 0
\(466\) −20.7846 −0.962828
\(467\) −5.07180 −0.234695 −0.117347 0.993091i \(-0.537439\pi\)
−0.117347 + 0.993091i \(0.537439\pi\)
\(468\) 0 0
\(469\) −16.0000 −0.738811
\(470\) −12.0000 −0.553519
\(471\) 0 0
\(472\) 12.0000 0.552345
\(473\) 0 0
\(474\) 0 0
\(475\) −5.46410 −0.250710
\(476\) 0 0
\(477\) 0 0
\(478\) 20.7846 0.950666
\(479\) 12.0000 0.548294 0.274147 0.961688i \(-0.411605\pi\)
0.274147 + 0.961688i \(0.411605\pi\)
\(480\) 0 0
\(481\) 26.9282 1.22782
\(482\) 0.248711 0.0113285
\(483\) 0 0
\(484\) 0 0
\(485\) −10.0000 −0.454077
\(486\) 0 0
\(487\) −31.7128 −1.43704 −0.718522 0.695504i \(-0.755181\pi\)
−0.718522 + 0.695504i \(0.755181\pi\)
\(488\) −3.46410 −0.156813
\(489\) 0 0
\(490\) 5.19615 0.234738
\(491\) −30.9282 −1.39577 −0.697885 0.716210i \(-0.745875\pi\)
−0.697885 + 0.716210i \(0.745875\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) −51.7128 −2.32667
\(495\) 0 0
\(496\) 54.6410 2.45345
\(497\) 27.7128 1.24309
\(498\) 0 0
\(499\) 28.7846 1.28858 0.644288 0.764783i \(-0.277154\pi\)
0.644288 + 0.764783i \(0.277154\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) −3.21539 −0.143510
\(503\) 31.1769 1.39011 0.695055 0.718957i \(-0.255380\pi\)
0.695055 + 0.718957i \(0.255380\pi\)
\(504\) 0 0
\(505\) −10.3923 −0.462451
\(506\) 0 0
\(507\) 0 0
\(508\) −8.92820 −0.396125
\(509\) −19.8564 −0.880120 −0.440060 0.897968i \(-0.645043\pi\)
−0.440060 + 0.897968i \(0.645043\pi\)
\(510\) 0 0
\(511\) −16.7846 −0.742507
\(512\) −8.66025 −0.382733
\(513\) 0 0
\(514\) −34.3923 −1.51698
\(515\) 8.00000 0.352522
\(516\) 0 0
\(517\) 0 0
\(518\) −17.0718 −0.750092
\(519\) 0 0
\(520\) −9.46410 −0.415028
\(521\) −6.00000 −0.262865 −0.131432 0.991325i \(-0.541958\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) 0 0
\(523\) 22.0000 0.961993 0.480996 0.876723i \(-0.340275\pi\)
0.480996 + 0.876723i \(0.340275\pi\)
\(524\) 18.9282 0.826882
\(525\) 0 0
\(526\) −35.5692 −1.55089
\(527\) 0 0
\(528\) 0 0
\(529\) 25.0000 1.08696
\(530\) 1.60770 0.0698338
\(531\) 0 0
\(532\) 10.9282 0.473798
\(533\) −18.9282 −0.819871
\(534\) 0 0
\(535\) 8.53590 0.369039
\(536\) 13.8564 0.598506
\(537\) 0 0
\(538\) 34.3923 1.48276
\(539\) 0 0
\(540\) 0 0
\(541\) −27.8564 −1.19764 −0.598820 0.800883i \(-0.704364\pi\)
−0.598820 + 0.800883i \(0.704364\pi\)
\(542\) −20.1051 −0.863589
\(543\) 0 0
\(544\) 0 0
\(545\) 10.0000 0.428353
\(546\) 0 0
\(547\) −2.00000 −0.0855138 −0.0427569 0.999086i \(-0.513614\pi\)
−0.0427569 + 0.999086i \(0.513614\pi\)
\(548\) 18.0000 0.768922
\(549\) 0 0
\(550\) 0 0
\(551\) 18.9282 0.806369
\(552\) 0 0
\(553\) −13.0718 −0.555869
\(554\) 51.0333 2.16820
\(555\) 0 0
\(556\) −12.3923 −0.525551
\(557\) 3.21539 0.136240 0.0681202 0.997677i \(-0.478300\pi\)
0.0681202 + 0.997677i \(0.478300\pi\)
\(558\) 0 0
\(559\) −26.9282 −1.13894
\(560\) 10.0000 0.422577
\(561\) 0 0
\(562\) −6.00000 −0.253095
\(563\) 10.3923 0.437983 0.218992 0.975727i \(-0.429723\pi\)
0.218992 + 0.975727i \(0.429723\pi\)
\(564\) 0 0
\(565\) 12.9282 0.543894
\(566\) −8.53590 −0.358791
\(567\) 0 0
\(568\) −24.0000 −1.00702
\(569\) 5.32051 0.223047 0.111524 0.993762i \(-0.464427\pi\)
0.111524 + 0.993762i \(0.464427\pi\)
\(570\) 0 0
\(571\) −3.60770 −0.150977 −0.0754887 0.997147i \(-0.524052\pi\)
−0.0754887 + 0.997147i \(0.524052\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 12.0000 0.500870
\(575\) −6.92820 −0.288926
\(576\) 0 0
\(577\) −18.7846 −0.782014 −0.391007 0.920388i \(-0.627873\pi\)
−0.391007 + 0.920388i \(0.627873\pi\)
\(578\) 29.4449 1.22474
\(579\) 0 0
\(580\) −3.46410 −0.143839
\(581\) −17.0718 −0.708257
\(582\) 0 0
\(583\) 0 0
\(584\) 14.5359 0.601500
\(585\) 0 0
\(586\) 24.0000 0.991431
\(587\) 18.9282 0.781251 0.390625 0.920550i \(-0.372259\pi\)
0.390625 + 0.920550i \(0.372259\pi\)
\(588\) 0 0
\(589\) 59.7128 2.46042
\(590\) −12.0000 −0.494032
\(591\) 0 0
\(592\) 24.6410 1.01274
\(593\) 8.78461 0.360741 0.180370 0.983599i \(-0.442270\pi\)
0.180370 + 0.983599i \(0.442270\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −15.4641 −0.633434
\(597\) 0 0
\(598\) −65.5692 −2.68132
\(599\) 37.8564 1.54677 0.773385 0.633936i \(-0.218562\pi\)
0.773385 + 0.633936i \(0.218562\pi\)
\(600\) 0 0
\(601\) 32.6410 1.33145 0.665727 0.746195i \(-0.268121\pi\)
0.665727 + 0.746195i \(0.268121\pi\)
\(602\) 17.0718 0.695794
\(603\) 0 0
\(604\) 20.3923 0.829751
\(605\) 0 0
\(606\) 0 0
\(607\) 18.7846 0.762444 0.381222 0.924484i \(-0.375503\pi\)
0.381222 + 0.924484i \(0.375503\pi\)
\(608\) −28.3923 −1.15146
\(609\) 0 0
\(610\) 3.46410 0.140257
\(611\) −37.8564 −1.53151
\(612\) 0 0
\(613\) 20.3923 0.823637 0.411819 0.911266i \(-0.364894\pi\)
0.411819 + 0.911266i \(0.364894\pi\)
\(614\) 24.2487 0.978598
\(615\) 0 0
\(616\) 0 0
\(617\) 36.9282 1.48667 0.743337 0.668917i \(-0.233242\pi\)
0.743337 + 0.668917i \(0.233242\pi\)
\(618\) 0 0
\(619\) 20.0000 0.803868 0.401934 0.915669i \(-0.368338\pi\)
0.401934 + 0.915669i \(0.368338\pi\)
\(620\) −10.9282 −0.438887
\(621\) 0 0
\(622\) 8.78461 0.352231
\(623\) 1.85641 0.0743754
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −36.2487 −1.44879
\(627\) 0 0
\(628\) −3.07180 −0.122578
\(629\) 0 0
\(630\) 0 0
\(631\) −21.0718 −0.838855 −0.419427 0.907789i \(-0.637769\pi\)
−0.419427 + 0.907789i \(0.637769\pi\)
\(632\) 11.3205 0.450306
\(633\) 0 0
\(634\) −43.1769 −1.71477
\(635\) −8.92820 −0.354305
\(636\) 0 0
\(637\) 16.3923 0.649487
\(638\) 0 0
\(639\) 0 0
\(640\) −12.1244 −0.479257
\(641\) 12.9282 0.510633 0.255317 0.966857i \(-0.417820\pi\)
0.255317 + 0.966857i \(0.417820\pi\)
\(642\) 0 0
\(643\) 37.5692 1.48159 0.740793 0.671734i \(-0.234450\pi\)
0.740793 + 0.671734i \(0.234450\pi\)
\(644\) 13.8564 0.546019
\(645\) 0 0
\(646\) 0 0
\(647\) −27.7128 −1.08950 −0.544752 0.838597i \(-0.683376\pi\)
−0.544752 + 0.838597i \(0.683376\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 9.46410 0.371213
\(651\) 0 0
\(652\) 9.85641 0.386007
\(653\) −19.8564 −0.777041 −0.388521 0.921440i \(-0.627014\pi\)
−0.388521 + 0.921440i \(0.627014\pi\)
\(654\) 0 0
\(655\) 18.9282 0.739586
\(656\) −17.3205 −0.676252
\(657\) 0 0
\(658\) 24.0000 0.935617
\(659\) −15.7128 −0.612084 −0.306042 0.952018i \(-0.599005\pi\)
−0.306042 + 0.952018i \(0.599005\pi\)
\(660\) 0 0
\(661\) 14.0000 0.544537 0.272268 0.962221i \(-0.412226\pi\)
0.272268 + 0.962221i \(0.412226\pi\)
\(662\) −17.0718 −0.663514
\(663\) 0 0
\(664\) 14.7846 0.573754
\(665\) 10.9282 0.423778
\(666\) 0 0
\(667\) 24.0000 0.929284
\(668\) −10.3923 −0.402090
\(669\) 0 0
\(670\) −13.8564 −0.535320
\(671\) 0 0
\(672\) 0 0
\(673\) 3.32051 0.127996 0.0639981 0.997950i \(-0.479615\pi\)
0.0639981 + 0.997950i \(0.479615\pi\)
\(674\) 57.4641 2.21343
\(675\) 0 0
\(676\) 16.8564 0.648323
\(677\) 8.78461 0.337620 0.168810 0.985649i \(-0.446008\pi\)
0.168810 + 0.985649i \(0.446008\pi\)
\(678\) 0 0
\(679\) 20.0000 0.767530
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 32.7846 1.25447 0.627234 0.778831i \(-0.284187\pi\)
0.627234 + 0.778831i \(0.284187\pi\)
\(684\) 0 0
\(685\) 18.0000 0.687745
\(686\) −34.6410 −1.32260
\(687\) 0 0
\(688\) −24.6410 −0.939430
\(689\) 5.07180 0.193220
\(690\) 0 0
\(691\) 47.7128 1.81508 0.907540 0.419965i \(-0.137958\pi\)
0.907540 + 0.419965i \(0.137958\pi\)
\(692\) −12.0000 −0.456172
\(693\) 0 0
\(694\) −38.7846 −1.47224
\(695\) −12.3923 −0.470067
\(696\) 0 0
\(697\) 0 0
\(698\) −14.1051 −0.533887
\(699\) 0 0
\(700\) −2.00000 −0.0755929
\(701\) 39.4641 1.49054 0.745269 0.666764i \(-0.232321\pi\)
0.745269 + 0.666764i \(0.232321\pi\)
\(702\) 0 0
\(703\) 26.9282 1.01562
\(704\) 0 0
\(705\) 0 0
\(706\) 22.3923 0.842746
\(707\) 20.7846 0.781686
\(708\) 0 0
\(709\) −11.8564 −0.445277 −0.222638 0.974901i \(-0.571467\pi\)
−0.222638 + 0.974901i \(0.571467\pi\)
\(710\) 24.0000 0.900704
\(711\) 0 0
\(712\) −1.60770 −0.0602509
\(713\) 75.7128 2.83547
\(714\) 0 0
\(715\) 0 0
\(716\) −6.92820 −0.258919
\(717\) 0 0
\(718\) 36.0000 1.34351
\(719\) 5.07180 0.189146 0.0945731 0.995518i \(-0.469851\pi\)
0.0945731 + 0.995518i \(0.469851\pi\)
\(720\) 0 0
\(721\) −16.0000 −0.595871
\(722\) −18.8038 −0.699807
\(723\) 0 0
\(724\) 15.8564 0.589299
\(725\) −3.46410 −0.128654
\(726\) 0 0
\(727\) 8.00000 0.296704 0.148352 0.988935i \(-0.452603\pi\)
0.148352 + 0.988935i \(0.452603\pi\)
\(728\) 18.9282 0.701526
\(729\) 0 0
\(730\) −14.5359 −0.537998
\(731\) 0 0
\(732\) 0 0
\(733\) −53.9615 −1.99311 −0.996557 0.0829082i \(-0.973579\pi\)
−0.996557 + 0.0829082i \(0.973579\pi\)
\(734\) −34.6410 −1.27862
\(735\) 0 0
\(736\) −36.0000 −1.32698
\(737\) 0 0
\(738\) 0 0
\(739\) −17.4641 −0.642427 −0.321214 0.947007i \(-0.604091\pi\)
−0.321214 + 0.947007i \(0.604091\pi\)
\(740\) −4.92820 −0.181164
\(741\) 0 0
\(742\) −3.21539 −0.118041
\(743\) 25.6077 0.939455 0.469728 0.882811i \(-0.344352\pi\)
0.469728 + 0.882811i \(0.344352\pi\)
\(744\) 0 0
\(745\) −15.4641 −0.566561
\(746\) −35.3205 −1.29318
\(747\) 0 0
\(748\) 0 0
\(749\) −17.0718 −0.623790
\(750\) 0 0
\(751\) 26.9282 0.982624 0.491312 0.870984i \(-0.336517\pi\)
0.491312 + 0.870984i \(0.336517\pi\)
\(752\) −34.6410 −1.26323
\(753\) 0 0
\(754\) −32.7846 −1.19395
\(755\) 20.3923 0.742152
\(756\) 0 0
\(757\) 34.7846 1.26427 0.632134 0.774859i \(-0.282179\pi\)
0.632134 + 0.774859i \(0.282179\pi\)
\(758\) 30.9282 1.12336
\(759\) 0 0
\(760\) −9.46410 −0.343299
\(761\) −32.5359 −1.17943 −0.589713 0.807613i \(-0.700759\pi\)
−0.589713 + 0.807613i \(0.700759\pi\)
\(762\) 0 0
\(763\) −20.0000 −0.724049
\(764\) 18.9282 0.684798
\(765\) 0 0
\(766\) −24.0000 −0.867155
\(767\) −37.8564 −1.36692
\(768\) 0 0
\(769\) −50.4974 −1.82098 −0.910492 0.413527i \(-0.864297\pi\)
−0.910492 + 0.413527i \(0.864297\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −24.3923 −0.877898
\(773\) 4.14359 0.149035 0.0745174 0.997220i \(-0.476258\pi\)
0.0745174 + 0.997220i \(0.476258\pi\)
\(774\) 0 0
\(775\) −10.9282 −0.392553
\(776\) −17.3205 −0.621770
\(777\) 0 0
\(778\) 19.1769 0.687526
\(779\) −18.9282 −0.678173
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 15.0000 0.535714
\(785\) −3.07180 −0.109637
\(786\) 0 0
\(787\) −22.7846 −0.812184 −0.406092 0.913832i \(-0.633109\pi\)
−0.406092 + 0.913832i \(0.633109\pi\)
\(788\) −12.0000 −0.427482
\(789\) 0 0
\(790\) −11.3205 −0.402766
\(791\) −25.8564 −0.919348
\(792\) 0 0
\(793\) 10.9282 0.388072
\(794\) −3.46410 −0.122936
\(795\) 0 0
\(796\) −24.7846 −0.878467
\(797\) 52.6410 1.86464 0.932320 0.361634i \(-0.117781\pi\)
0.932320 + 0.361634i \(0.117781\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 5.19615 0.183712
\(801\) 0 0
\(802\) −13.6077 −0.480504
\(803\) 0 0
\(804\) 0 0
\(805\) 13.8564 0.488374
\(806\) −103.426 −3.64301
\(807\) 0 0
\(808\) −18.0000 −0.633238
\(809\) −15.4641 −0.543689 −0.271844 0.962341i \(-0.587634\pi\)
−0.271844 + 0.962341i \(0.587634\pi\)
\(810\) 0 0
\(811\) −12.3923 −0.435153 −0.217576 0.976043i \(-0.569815\pi\)
−0.217576 + 0.976043i \(0.569815\pi\)
\(812\) 6.92820 0.243132
\(813\) 0 0
\(814\) 0 0
\(815\) 9.85641 0.345255
\(816\) 0 0
\(817\) −26.9282 −0.942099
\(818\) −11.7513 −0.410874
\(819\) 0 0
\(820\) 3.46410 0.120972
\(821\) 20.5359 0.716708 0.358354 0.933586i \(-0.383338\pi\)
0.358354 + 0.933586i \(0.383338\pi\)
\(822\) 0 0
\(823\) −33.5692 −1.17015 −0.585075 0.810979i \(-0.698935\pi\)
−0.585075 + 0.810979i \(0.698935\pi\)
\(824\) 13.8564 0.482711
\(825\) 0 0
\(826\) 24.0000 0.835067
\(827\) 22.3923 0.778657 0.389328 0.921099i \(-0.372707\pi\)
0.389328 + 0.921099i \(0.372707\pi\)
\(828\) 0 0
\(829\) 29.7128 1.03197 0.515984 0.856598i \(-0.327426\pi\)
0.515984 + 0.856598i \(0.327426\pi\)
\(830\) −14.7846 −0.513181
\(831\) 0 0
\(832\) −5.46410 −0.189434
\(833\) 0 0
\(834\) 0 0
\(835\) −10.3923 −0.359641
\(836\) 0 0
\(837\) 0 0
\(838\) 53.5692 1.85052
\(839\) −56.7846 −1.96042 −0.980211 0.197954i \(-0.936570\pi\)
−0.980211 + 0.197954i \(0.936570\pi\)
\(840\) 0 0
\(841\) −17.0000 −0.586207
\(842\) −3.46410 −0.119381
\(843\) 0 0
\(844\) 8.39230 0.288875
\(845\) 16.8564 0.579878
\(846\) 0 0
\(847\) 0 0
\(848\) 4.64102 0.159373
\(849\) 0 0
\(850\) 0 0
\(851\) 34.1436 1.17043
\(852\) 0 0
\(853\) −3.60770 −0.123525 −0.0617626 0.998091i \(-0.519672\pi\)
−0.0617626 + 0.998091i \(0.519672\pi\)
\(854\) −6.92820 −0.237078
\(855\) 0 0
\(856\) 14.7846 0.505328
\(857\) −37.8564 −1.29315 −0.646575 0.762850i \(-0.723799\pi\)
−0.646575 + 0.762850i \(0.723799\pi\)
\(858\) 0 0
\(859\) −7.71281 −0.263158 −0.131579 0.991306i \(-0.542005\pi\)
−0.131579 + 0.991306i \(0.542005\pi\)
\(860\) 4.92820 0.168050
\(861\) 0 0
\(862\) −15.2154 −0.518238
\(863\) 37.8564 1.28865 0.644324 0.764753i \(-0.277139\pi\)
0.644324 + 0.764753i \(0.277139\pi\)
\(864\) 0 0
\(865\) −12.0000 −0.408012
\(866\) −0.248711 −0.00845155
\(867\) 0 0
\(868\) 21.8564 0.741855
\(869\) 0 0
\(870\) 0 0
\(871\) −43.7128 −1.48115
\(872\) 17.3205 0.586546
\(873\) 0 0
\(874\) −65.5692 −2.21791
\(875\) −2.00000 −0.0676123
\(876\) 0 0
\(877\) 34.2487 1.15650 0.578248 0.815861i \(-0.303736\pi\)
0.578248 + 0.815861i \(0.303736\pi\)
\(878\) 57.4641 1.93932
\(879\) 0 0
\(880\) 0 0
\(881\) 0.928203 0.0312720 0.0156360 0.999878i \(-0.495023\pi\)
0.0156360 + 0.999878i \(0.495023\pi\)
\(882\) 0 0
\(883\) −4.00000 −0.134611 −0.0673054 0.997732i \(-0.521440\pi\)
−0.0673054 + 0.997732i \(0.521440\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 20.7846 0.698273
\(887\) 12.2487 0.411271 0.205636 0.978629i \(-0.434074\pi\)
0.205636 + 0.978629i \(0.434074\pi\)
\(888\) 0 0
\(889\) 17.8564 0.598885
\(890\) 1.60770 0.0538901
\(891\) 0 0
\(892\) 9.85641 0.330017
\(893\) −37.8564 −1.26682
\(894\) 0 0
\(895\) −6.92820 −0.231584
\(896\) 24.2487 0.810093
\(897\) 0 0
\(898\) −46.3923 −1.54813
\(899\) 37.8564 1.26258
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 22.3923 0.744757
\(905\) 15.8564 0.527085
\(906\) 0 0
\(907\) 18.1436 0.602448 0.301224 0.953553i \(-0.402605\pi\)
0.301224 + 0.953553i \(0.402605\pi\)
\(908\) 15.4641 0.513194
\(909\) 0 0
\(910\) −18.9282 −0.627464
\(911\) −18.9282 −0.627119 −0.313560 0.949568i \(-0.601522\pi\)
−0.313560 + 0.949568i \(0.601522\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 21.4641 0.709969
\(915\) 0 0
\(916\) −23.8564 −0.788238
\(917\) −37.8564 −1.25013
\(918\) 0 0
\(919\) 32.3923 1.06852 0.534262 0.845319i \(-0.320590\pi\)
0.534262 + 0.845319i \(0.320590\pi\)
\(920\) −12.0000 −0.395628
\(921\) 0 0
\(922\) −62.7846 −2.06770
\(923\) 75.7128 2.49212
\(924\) 0 0
\(925\) −4.92820 −0.162038
\(926\) 48.4974 1.59372
\(927\) 0 0
\(928\) −18.0000 −0.590879
\(929\) −2.78461 −0.0913601 −0.0456800 0.998956i \(-0.514545\pi\)
−0.0456800 + 0.998956i \(0.514545\pi\)
\(930\) 0 0
\(931\) 16.3923 0.537236
\(932\) 12.0000 0.393073
\(933\) 0 0
\(934\) 8.78461 0.287441
\(935\) 0 0
\(936\) 0 0
\(937\) 20.3923 0.666188 0.333094 0.942894i \(-0.391907\pi\)
0.333094 + 0.942894i \(0.391907\pi\)
\(938\) 27.7128 0.904855
\(939\) 0 0
\(940\) 6.92820 0.225973
\(941\) 27.4641 0.895304 0.447652 0.894208i \(-0.352260\pi\)
0.447652 + 0.894208i \(0.352260\pi\)
\(942\) 0 0
\(943\) −24.0000 −0.781548
\(944\) −34.6410 −1.12747
\(945\) 0 0
\(946\) 0 0
\(947\) 18.9282 0.615084 0.307542 0.951535i \(-0.400494\pi\)
0.307542 + 0.951535i \(0.400494\pi\)
\(948\) 0 0
\(949\) −45.8564 −1.48856
\(950\) 9.46410 0.307056
\(951\) 0 0
\(952\) 0 0
\(953\) −3.21539 −0.104157 −0.0520784 0.998643i \(-0.516585\pi\)
−0.0520784 + 0.998643i \(0.516585\pi\)
\(954\) 0 0
\(955\) 18.9282 0.612502
\(956\) −12.0000 −0.388108
\(957\) 0 0
\(958\) −20.7846 −0.671520
\(959\) −36.0000 −1.16250
\(960\) 0 0
\(961\) 88.4256 2.85244
\(962\) −46.6410 −1.50377
\(963\) 0 0
\(964\) −0.143594 −0.00462484
\(965\) −24.3923 −0.785216
\(966\) 0 0
\(967\) −22.7846 −0.732704 −0.366352 0.930476i \(-0.619393\pi\)
−0.366352 + 0.930476i \(0.619393\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 17.3205 0.556128
\(971\) −25.8564 −0.829772 −0.414886 0.909873i \(-0.636178\pi\)
−0.414886 + 0.909873i \(0.636178\pi\)
\(972\) 0 0
\(973\) 24.7846 0.794558
\(974\) 54.9282 1.76001
\(975\) 0 0
\(976\) 10.0000 0.320092
\(977\) −47.5692 −1.52187 −0.760937 0.648826i \(-0.775260\pi\)
−0.760937 + 0.648826i \(0.775260\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −3.00000 −0.0958315
\(981\) 0 0
\(982\) 53.5692 1.70946
\(983\) −24.0000 −0.765481 −0.382741 0.923856i \(-0.625020\pi\)
−0.382741 + 0.923856i \(0.625020\pi\)
\(984\) 0 0
\(985\) −12.0000 −0.382352
\(986\) 0 0
\(987\) 0 0
\(988\) 29.8564 0.949859
\(989\) −34.1436 −1.08570
\(990\) 0 0
\(991\) −7.21539 −0.229204 −0.114602 0.993411i \(-0.536559\pi\)
−0.114602 + 0.993411i \(0.536559\pi\)
\(992\) −56.7846 −1.80291
\(993\) 0 0
\(994\) −48.0000 −1.52247
\(995\) −24.7846 −0.785725
\(996\) 0 0
\(997\) −27.6077 −0.874344 −0.437172 0.899378i \(-0.644020\pi\)
−0.437172 + 0.899378i \(0.644020\pi\)
\(998\) −49.8564 −1.57818
\(999\) 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 5445.2.a.s.1.1 2
3.2 odd 2 1815.2.a.i.1.2 2
11.10 odd 2 495.2.a.c.1.2 2
15.14 odd 2 9075.2.a.bh.1.1 2
33.32 even 2 165.2.a.b.1.1 2
44.43 even 2 7920.2.a.bz.1.2 2
55.32 even 4 2475.2.c.n.199.4 4
55.43 even 4 2475.2.c.n.199.1 4
55.54 odd 2 2475.2.a.r.1.1 2
132.131 odd 2 2640.2.a.x.1.2 2
165.32 odd 4 825.2.c.c.199.1 4
165.98 odd 4 825.2.c.c.199.4 4
165.164 even 2 825.2.a.e.1.2 2
231.230 odd 2 8085.2.a.bd.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.a.b.1.1 2 33.32 even 2
495.2.a.c.1.2 2 11.10 odd 2
825.2.a.e.1.2 2 165.164 even 2
825.2.c.c.199.1 4 165.32 odd 4
825.2.c.c.199.4 4 165.98 odd 4
1815.2.a.i.1.2 2 3.2 odd 2
2475.2.a.r.1.1 2 55.54 odd 2
2475.2.c.n.199.1 4 55.43 even 4
2475.2.c.n.199.4 4 55.32 even 4
2640.2.a.x.1.2 2 132.131 odd 2
5445.2.a.s.1.1 2 1.1 even 1 trivial
7920.2.a.bz.1.2 2 44.43 even 2
8085.2.a.bd.1.1 2 231.230 odd 2
9075.2.a.bh.1.1 2 15.14 odd 2