Properties

Label 5445.2.a.e.1.1
Level $5445$
Weight $2$
Character 5445.1
Self dual yes
Analytic conductor $43.479$
Analytic rank $1$
Dimension $1$
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: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1815)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 5445.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{7} +3.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{7} +3.00000 q^{8} -1.00000 q^{10} -4.00000 q^{13} -2.00000 q^{14} -1.00000 q^{16} -6.00000 q^{17} +6.00000 q^{19} -1.00000 q^{20} -4.00000 q^{23} +1.00000 q^{25} +4.00000 q^{26} -2.00000 q^{28} +6.00000 q^{29} +8.00000 q^{31} -5.00000 q^{32} +6.00000 q^{34} +2.00000 q^{35} -6.00000 q^{37} -6.00000 q^{38} +3.00000 q^{40} -6.00000 q^{41} +6.00000 q^{43} +4.00000 q^{46} -8.00000 q^{47} -3.00000 q^{49} -1.00000 q^{50} +4.00000 q^{52} -6.00000 q^{53} +6.00000 q^{56} -6.00000 q^{58} -4.00000 q^{61} -8.00000 q^{62} +7.00000 q^{64} -4.00000 q^{65} +12.0000 q^{67} +6.00000 q^{68} -2.00000 q^{70} -8.00000 q^{71} -16.0000 q^{73} +6.00000 q^{74} -6.00000 q^{76} -2.00000 q^{79} -1.00000 q^{80} +6.00000 q^{82} -6.00000 q^{85} -6.00000 q^{86} -10.0000 q^{89} -8.00000 q^{91} +4.00000 q^{92} +8.00000 q^{94} +6.00000 q^{95} -6.00000 q^{97} +3.00000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107 −0.353553 0.935414i \(-0.615027\pi\)
−0.353553 + 0.935414i \(0.615027\pi\)
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 2.00000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 3.00000 1.06066
\(9\) 0 0
\(10\) −1.00000 −0.316228
\(11\) 0 0
\(12\) 0 0
\(13\) −4.00000 −1.10940 −0.554700 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) −2.00000 −0.534522
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) −6.00000 −1.45521 −0.727607 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 0 0
\(19\) 6.00000 1.37649 0.688247 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) 0 0
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 4.00000 0.784465
\(27\) 0 0
\(28\) −2.00000 −0.377964
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) −5.00000 −0.883883
\(33\) 0 0
\(34\) 6.00000 1.02899
\(35\) 2.00000 0.338062
\(36\) 0 0
\(37\) −6.00000 −0.986394 −0.493197 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) −6.00000 −0.973329
\(39\) 0 0
\(40\) 3.00000 0.474342
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 6.00000 0.914991 0.457496 0.889212i \(-0.348747\pi\)
0.457496 + 0.889212i \(0.348747\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 4.00000 0.589768
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) 4.00000 0.554700
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 6.00000 0.801784
\(57\) 0 0
\(58\) −6.00000 −0.787839
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −4.00000 −0.512148 −0.256074 0.966657i \(-0.582429\pi\)
−0.256074 + 0.966657i \(0.582429\pi\)
\(62\) −8.00000 −1.01600
\(63\) 0 0
\(64\) 7.00000 0.875000
\(65\) −4.00000 −0.496139
\(66\) 0 0
\(67\) 12.0000 1.46603 0.733017 0.680211i \(-0.238112\pi\)
0.733017 + 0.680211i \(0.238112\pi\)
\(68\) 6.00000 0.727607
\(69\) 0 0
\(70\) −2.00000 −0.239046
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) −16.0000 −1.87266 −0.936329 0.351123i \(-0.885800\pi\)
−0.936329 + 0.351123i \(0.885800\pi\)
\(74\) 6.00000 0.697486
\(75\) 0 0
\(76\) −6.00000 −0.688247
\(77\) 0 0
\(78\) 0 0
\(79\) −2.00000 −0.225018 −0.112509 0.993651i \(-0.535889\pi\)
−0.112509 + 0.993651i \(0.535889\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) 6.00000 0.662589
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) −6.00000 −0.650791
\(86\) −6.00000 −0.646997
\(87\) 0 0
\(88\) 0 0
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) −8.00000 −0.838628
\(92\) 4.00000 0.417029
\(93\) 0 0
\(94\) 8.00000 0.825137
\(95\) 6.00000 0.615587
\(96\) 0 0
\(97\) −6.00000 −0.609208 −0.304604 0.952479i \(-0.598524\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(98\) 3.00000 0.303046
\(99\) 0 0
\(100\) −1.00000 −0.100000
\(101\) 14.0000 1.39305 0.696526 0.717532i \(-0.254728\pi\)
0.696526 + 0.717532i \(0.254728\pi\)
\(102\) 0 0
\(103\) −8.00000 −0.788263 −0.394132 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) −12.0000 −1.17670
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) 8.00000 0.773389 0.386695 0.922208i \(-0.373617\pi\)
0.386695 + 0.922208i \(0.373617\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −2.00000 −0.188982
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) −4.00000 −0.373002
\(116\) −6.00000 −0.557086
\(117\) 0 0
\(118\) 0 0
\(119\) −12.0000 −1.10004
\(120\) 0 0
\(121\) 0 0
\(122\) 4.00000 0.362143
\(123\) 0 0
\(124\) −8.00000 −0.718421
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 2.00000 0.177471 0.0887357 0.996055i \(-0.471717\pi\)
0.0887357 + 0.996055i \(0.471717\pi\)
\(128\) 3.00000 0.265165
\(129\) 0 0
\(130\) 4.00000 0.350823
\(131\) 16.0000 1.39793 0.698963 0.715158i \(-0.253645\pi\)
0.698963 + 0.715158i \(0.253645\pi\)
\(132\) 0 0
\(133\) 12.0000 1.04053
\(134\) −12.0000 −1.03664
\(135\) 0 0
\(136\) −18.0000 −1.54349
\(137\) −14.0000 −1.19610 −0.598050 0.801459i \(-0.704058\pi\)
−0.598050 + 0.801459i \(0.704058\pi\)
\(138\) 0 0
\(139\) 22.0000 1.86602 0.933008 0.359856i \(-0.117174\pi\)
0.933008 + 0.359856i \(0.117174\pi\)
\(140\) −2.00000 −0.169031
\(141\) 0 0
\(142\) 8.00000 0.671345
\(143\) 0 0
\(144\) 0 0
\(145\) 6.00000 0.498273
\(146\) 16.0000 1.32417
\(147\) 0 0
\(148\) 6.00000 0.493197
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) −2.00000 −0.162758 −0.0813788 0.996683i \(-0.525932\pi\)
−0.0813788 + 0.996683i \(0.525932\pi\)
\(152\) 18.0000 1.45999
\(153\) 0 0
\(154\) 0 0
\(155\) 8.00000 0.642575
\(156\) 0 0
\(157\) 18.0000 1.43656 0.718278 0.695756i \(-0.244931\pi\)
0.718278 + 0.695756i \(0.244931\pi\)
\(158\) 2.00000 0.159111
\(159\) 0 0
\(160\) −5.00000 −0.395285
\(161\) −8.00000 −0.630488
\(162\) 0 0
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 6.00000 0.468521
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 6.00000 0.460179
\(171\) 0 0
\(172\) −6.00000 −0.457496
\(173\) 18.0000 1.36851 0.684257 0.729241i \(-0.260127\pi\)
0.684257 + 0.729241i \(0.260127\pi\)
\(174\) 0 0
\(175\) 2.00000 0.151186
\(176\) 0 0
\(177\) 0 0
\(178\) 10.0000 0.749532
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) −26.0000 −1.93256 −0.966282 0.257485i \(-0.917106\pi\)
−0.966282 + 0.257485i \(0.917106\pi\)
\(182\) 8.00000 0.592999
\(183\) 0 0
\(184\) −12.0000 −0.884652
\(185\) −6.00000 −0.441129
\(186\) 0 0
\(187\) 0 0
\(188\) 8.00000 0.583460
\(189\) 0 0
\(190\) −6.00000 −0.435286
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) −20.0000 −1.43963 −0.719816 0.694165i \(-0.755774\pi\)
−0.719816 + 0.694165i \(0.755774\pi\)
\(194\) 6.00000 0.430775
\(195\) 0 0
\(196\) 3.00000 0.214286
\(197\) 18.0000 1.28245 0.641223 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) 0 0
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 3.00000 0.212132
\(201\) 0 0
\(202\) −14.0000 −0.985037
\(203\) 12.0000 0.842235
\(204\) 0 0
\(205\) −6.00000 −0.419058
\(206\) 8.00000 0.557386
\(207\) 0 0
\(208\) 4.00000 0.277350
\(209\) 0 0
\(210\) 0 0
\(211\) 22.0000 1.51454 0.757271 0.653101i \(-0.226532\pi\)
0.757271 + 0.653101i \(0.226532\pi\)
\(212\) 6.00000 0.412082
\(213\) 0 0
\(214\) −8.00000 −0.546869
\(215\) 6.00000 0.409197
\(216\) 0 0
\(217\) 16.0000 1.08615
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 24.0000 1.61441
\(222\) 0 0
\(223\) −16.0000 −1.07144 −0.535720 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) −10.0000 −0.668153
\(225\) 0 0
\(226\) 6.00000 0.399114
\(227\) 16.0000 1.06196 0.530979 0.847385i \(-0.321824\pi\)
0.530979 + 0.847385i \(0.321824\pi\)
\(228\) 0 0
\(229\) −22.0000 −1.45380 −0.726900 0.686743i \(-0.759040\pi\)
−0.726900 + 0.686743i \(0.759040\pi\)
\(230\) 4.00000 0.263752
\(231\) 0 0
\(232\) 18.0000 1.18176
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) −8.00000 −0.521862
\(236\) 0 0
\(237\) 0 0
\(238\) 12.0000 0.777844
\(239\) −12.0000 −0.776215 −0.388108 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 0 0
\(241\) −20.0000 −1.28831 −0.644157 0.764894i \(-0.722792\pi\)
−0.644157 + 0.764894i \(0.722792\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 4.00000 0.256074
\(245\) −3.00000 −0.191663
\(246\) 0 0
\(247\) −24.0000 −1.52708
\(248\) 24.0000 1.52400
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) −20.0000 −1.26239 −0.631194 0.775625i \(-0.717435\pi\)
−0.631194 + 0.775625i \(0.717435\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −2.00000 −0.125491
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) −18.0000 −1.12281 −0.561405 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) 0 0
\(259\) −12.0000 −0.745644
\(260\) 4.00000 0.248069
\(261\) 0 0
\(262\) −16.0000 −0.988483
\(263\) −24.0000 −1.47990 −0.739952 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) 0 0
\(265\) −6.00000 −0.368577
\(266\) −12.0000 −0.735767
\(267\) 0 0
\(268\) −12.0000 −0.733017
\(269\) −6.00000 −0.365826 −0.182913 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 0 0
\(271\) −14.0000 −0.850439 −0.425220 0.905090i \(-0.639803\pi\)
−0.425220 + 0.905090i \(0.639803\pi\)
\(272\) 6.00000 0.363803
\(273\) 0 0
\(274\) 14.0000 0.845771
\(275\) 0 0
\(276\) 0 0
\(277\) −4.00000 −0.240337 −0.120168 0.992754i \(-0.538343\pi\)
−0.120168 + 0.992754i \(0.538343\pi\)
\(278\) −22.0000 −1.31947
\(279\) 0 0
\(280\) 6.00000 0.358569
\(281\) −22.0000 −1.31241 −0.656205 0.754583i \(-0.727839\pi\)
−0.656205 + 0.754583i \(0.727839\pi\)
\(282\) 0 0
\(283\) 14.0000 0.832214 0.416107 0.909316i \(-0.363394\pi\)
0.416107 + 0.909316i \(0.363394\pi\)
\(284\) 8.00000 0.474713
\(285\) 0 0
\(286\) 0 0
\(287\) −12.0000 −0.708338
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) −6.00000 −0.352332
\(291\) 0 0
\(292\) 16.0000 0.936329
\(293\) −18.0000 −1.05157 −0.525786 0.850617i \(-0.676229\pi\)
−0.525786 + 0.850617i \(0.676229\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −18.0000 −1.04623
\(297\) 0 0
\(298\) 6.00000 0.347571
\(299\) 16.0000 0.925304
\(300\) 0 0
\(301\) 12.0000 0.691669
\(302\) 2.00000 0.115087
\(303\) 0 0
\(304\) −6.00000 −0.344124
\(305\) −4.00000 −0.229039
\(306\) 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
\(310\) −8.00000 −0.454369
\(311\) 28.0000 1.58773 0.793867 0.608091i \(-0.208065\pi\)
0.793867 + 0.608091i \(0.208065\pi\)
\(312\) 0 0
\(313\) −26.0000 −1.46961 −0.734803 0.678280i \(-0.762726\pi\)
−0.734803 + 0.678280i \(0.762726\pi\)
\(314\) −18.0000 −1.01580
\(315\) 0 0
\(316\) 2.00000 0.112509
\(317\) 30.0000 1.68497 0.842484 0.538721i \(-0.181092\pi\)
0.842484 + 0.538721i \(0.181092\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 7.00000 0.391312
\(321\) 0 0
\(322\) 8.00000 0.445823
\(323\) −36.0000 −2.00309
\(324\) 0 0
\(325\) −4.00000 −0.221880
\(326\) 4.00000 0.221540
\(327\) 0 0
\(328\) −18.0000 −0.993884
\(329\) −16.0000 −0.882109
\(330\) 0 0
\(331\) −28.0000 −1.53902 −0.769510 0.638635i \(-0.779499\pi\)
−0.769510 + 0.638635i \(0.779499\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 12.0000 0.655630
\(336\) 0 0
\(337\) 8.00000 0.435788 0.217894 0.975972i \(-0.430081\pi\)
0.217894 + 0.975972i \(0.430081\pi\)
\(338\) −3.00000 −0.163178
\(339\) 0 0
\(340\) 6.00000 0.325396
\(341\) 0 0
\(342\) 0 0
\(343\) −20.0000 −1.07990
\(344\) 18.0000 0.970495
\(345\) 0 0
\(346\) −18.0000 −0.967686
\(347\) −8.00000 −0.429463 −0.214731 0.976673i \(-0.568888\pi\)
−0.214731 + 0.976673i \(0.568888\pi\)
\(348\) 0 0
\(349\) −36.0000 −1.92704 −0.963518 0.267644i \(-0.913755\pi\)
−0.963518 + 0.267644i \(0.913755\pi\)
\(350\) −2.00000 −0.106904
\(351\) 0 0
\(352\) 0 0
\(353\) −18.0000 −0.958043 −0.479022 0.877803i \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) 0 0
\(355\) −8.00000 −0.424596
\(356\) 10.0000 0.529999
\(357\) 0 0
\(358\) 12.0000 0.634220
\(359\) 8.00000 0.422224 0.211112 0.977462i \(-0.432292\pi\)
0.211112 + 0.977462i \(0.432292\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) 26.0000 1.36653
\(363\) 0 0
\(364\) 8.00000 0.419314
\(365\) −16.0000 −0.837478
\(366\) 0 0
\(367\) 24.0000 1.25279 0.626395 0.779506i \(-0.284530\pi\)
0.626395 + 0.779506i \(0.284530\pi\)
\(368\) 4.00000 0.208514
\(369\) 0 0
\(370\) 6.00000 0.311925
\(371\) −12.0000 −0.623009
\(372\) 0 0
\(373\) −24.0000 −1.24267 −0.621336 0.783544i \(-0.713410\pi\)
−0.621336 + 0.783544i \(0.713410\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −24.0000 −1.23771
\(377\) −24.0000 −1.23606
\(378\) 0 0
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) −6.00000 −0.307794
\(381\) 0 0
\(382\) 0 0
\(383\) −36.0000 −1.83951 −0.919757 0.392488i \(-0.871614\pi\)
−0.919757 + 0.392488i \(0.871614\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 20.0000 1.01797
\(387\) 0 0
\(388\) 6.00000 0.304604
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 0 0
\(391\) 24.0000 1.21373
\(392\) −9.00000 −0.454569
\(393\) 0 0
\(394\) −18.0000 −0.906827
\(395\) −2.00000 −0.100631
\(396\) 0 0
\(397\) 14.0000 0.702640 0.351320 0.936255i \(-0.385733\pi\)
0.351320 + 0.936255i \(0.385733\pi\)
\(398\) −16.0000 −0.802008
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) 10.0000 0.499376 0.249688 0.968326i \(-0.419672\pi\)
0.249688 + 0.968326i \(0.419672\pi\)
\(402\) 0 0
\(403\) −32.0000 −1.59403
\(404\) −14.0000 −0.696526
\(405\) 0 0
\(406\) −12.0000 −0.595550
\(407\) 0 0
\(408\) 0 0
\(409\) 4.00000 0.197787 0.0988936 0.995098i \(-0.468470\pi\)
0.0988936 + 0.995098i \(0.468470\pi\)
\(410\) 6.00000 0.296319
\(411\) 0 0
\(412\) 8.00000 0.394132
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 20.0000 0.980581
\(417\) 0 0
\(418\) 0 0
\(419\) 28.0000 1.36789 0.683945 0.729534i \(-0.260263\pi\)
0.683945 + 0.729534i \(0.260263\pi\)
\(420\) 0 0
\(421\) −6.00000 −0.292422 −0.146211 0.989253i \(-0.546708\pi\)
−0.146211 + 0.989253i \(0.546708\pi\)
\(422\) −22.0000 −1.07094
\(423\) 0 0
\(424\) −18.0000 −0.874157
\(425\) −6.00000 −0.291043
\(426\) 0 0
\(427\) −8.00000 −0.387147
\(428\) −8.00000 −0.386695
\(429\) 0 0
\(430\) −6.00000 −0.289346
\(431\) −12.0000 −0.578020 −0.289010 0.957326i \(-0.593326\pi\)
−0.289010 + 0.957326i \(0.593326\pi\)
\(432\) 0 0
\(433\) 18.0000 0.865025 0.432512 0.901628i \(-0.357627\pi\)
0.432512 + 0.901628i \(0.357627\pi\)
\(434\) −16.0000 −0.768025
\(435\) 0 0
\(436\) 0 0
\(437\) −24.0000 −1.14808
\(438\) 0 0
\(439\) 26.0000 1.24091 0.620456 0.784241i \(-0.286947\pi\)
0.620456 + 0.784241i \(0.286947\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −24.0000 −1.14156
\(443\) −16.0000 −0.760183 −0.380091 0.924949i \(-0.624107\pi\)
−0.380091 + 0.924949i \(0.624107\pi\)
\(444\) 0 0
\(445\) −10.0000 −0.474045
\(446\) 16.0000 0.757622
\(447\) 0 0
\(448\) 14.0000 0.661438
\(449\) 30.0000 1.41579 0.707894 0.706319i \(-0.249646\pi\)
0.707894 + 0.706319i \(0.249646\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 6.00000 0.282216
\(453\) 0 0
\(454\) −16.0000 −0.750917
\(455\) −8.00000 −0.375046
\(456\) 0 0
\(457\) −8.00000 −0.374224 −0.187112 0.982339i \(-0.559913\pi\)
−0.187112 + 0.982339i \(0.559913\pi\)
\(458\) 22.0000 1.02799
\(459\) 0 0
\(460\) 4.00000 0.186501
\(461\) −22.0000 −1.02464 −0.512321 0.858794i \(-0.671214\pi\)
−0.512321 + 0.858794i \(0.671214\pi\)
\(462\) 0 0
\(463\) −8.00000 −0.371792 −0.185896 0.982569i \(-0.559519\pi\)
−0.185896 + 0.982569i \(0.559519\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) −6.00000 −0.277945
\(467\) −16.0000 −0.740392 −0.370196 0.928954i \(-0.620709\pi\)
−0.370196 + 0.928954i \(0.620709\pi\)
\(468\) 0 0
\(469\) 24.0000 1.10822
\(470\) 8.00000 0.369012
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 6.00000 0.275299
\(476\) 12.0000 0.550019
\(477\) 0 0
\(478\) 12.0000 0.548867
\(479\) 28.0000 1.27935 0.639676 0.768644i \(-0.279068\pi\)
0.639676 + 0.768644i \(0.279068\pi\)
\(480\) 0 0
\(481\) 24.0000 1.09431
\(482\) 20.0000 0.910975
\(483\) 0 0
\(484\) 0 0
\(485\) −6.00000 −0.272446
\(486\) 0 0
\(487\) −16.0000 −0.725029 −0.362515 0.931978i \(-0.618082\pi\)
−0.362515 + 0.931978i \(0.618082\pi\)
\(488\) −12.0000 −0.543214
\(489\) 0 0
\(490\) 3.00000 0.135526
\(491\) −36.0000 −1.62466 −0.812329 0.583200i \(-0.801800\pi\)
−0.812329 + 0.583200i \(0.801800\pi\)
\(492\) 0 0
\(493\) −36.0000 −1.62136
\(494\) 24.0000 1.07981
\(495\) 0 0
\(496\) −8.00000 −0.359211
\(497\) −16.0000 −0.717698
\(498\) 0 0
\(499\) −4.00000 −0.179065 −0.0895323 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) 20.0000 0.892644
\(503\) −16.0000 −0.713405 −0.356702 0.934218i \(-0.616099\pi\)
−0.356702 + 0.934218i \(0.616099\pi\)
\(504\) 0 0
\(505\) 14.0000 0.622992
\(506\) 0 0
\(507\) 0 0
\(508\) −2.00000 −0.0887357
\(509\) −14.0000 −0.620539 −0.310270 0.950649i \(-0.600419\pi\)
−0.310270 + 0.950649i \(0.600419\pi\)
\(510\) 0 0
\(511\) −32.0000 −1.41560
\(512\) 11.0000 0.486136
\(513\) 0 0
\(514\) 18.0000 0.793946
\(515\) −8.00000 −0.352522
\(516\) 0 0
\(517\) 0 0
\(518\) 12.0000 0.527250
\(519\) 0 0
\(520\) −12.0000 −0.526235
\(521\) 10.0000 0.438108 0.219054 0.975713i \(-0.429703\pi\)
0.219054 + 0.975713i \(0.429703\pi\)
\(522\) 0 0
\(523\) 14.0000 0.612177 0.306089 0.952003i \(-0.400980\pi\)
0.306089 + 0.952003i \(0.400980\pi\)
\(524\) −16.0000 −0.698963
\(525\) 0 0
\(526\) 24.0000 1.04645
\(527\) −48.0000 −2.09091
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 6.00000 0.260623
\(531\) 0 0
\(532\) −12.0000 −0.520266
\(533\) 24.0000 1.03956
\(534\) 0 0
\(535\) 8.00000 0.345870
\(536\) 36.0000 1.55496
\(537\) 0 0
\(538\) 6.00000 0.258678
\(539\) 0 0
\(540\) 0 0
\(541\) −32.0000 −1.37579 −0.687894 0.725811i \(-0.741464\pi\)
−0.687894 + 0.725811i \(0.741464\pi\)
\(542\) 14.0000 0.601351
\(543\) 0 0
\(544\) 30.0000 1.28624
\(545\) 0 0
\(546\) 0 0
\(547\) −34.0000 −1.45374 −0.726868 0.686778i \(-0.759025\pi\)
−0.726868 + 0.686778i \(0.759025\pi\)
\(548\) 14.0000 0.598050
\(549\) 0 0
\(550\) 0 0
\(551\) 36.0000 1.53365
\(552\) 0 0
\(553\) −4.00000 −0.170097
\(554\) 4.00000 0.169944
\(555\) 0 0
\(556\) −22.0000 −0.933008
\(557\) −2.00000 −0.0847427 −0.0423714 0.999102i \(-0.513491\pi\)
−0.0423714 + 0.999102i \(0.513491\pi\)
\(558\) 0 0
\(559\) −24.0000 −1.01509
\(560\) −2.00000 −0.0845154
\(561\) 0 0
\(562\) 22.0000 0.928014
\(563\) 12.0000 0.505740 0.252870 0.967500i \(-0.418626\pi\)
0.252870 + 0.967500i \(0.418626\pi\)
\(564\) 0 0
\(565\) −6.00000 −0.252422
\(566\) −14.0000 −0.588464
\(567\) 0 0
\(568\) −24.0000 −1.00702
\(569\) −18.0000 −0.754599 −0.377300 0.926091i \(-0.623147\pi\)
−0.377300 + 0.926091i \(0.623147\pi\)
\(570\) 0 0
\(571\) 26.0000 1.08807 0.544033 0.839064i \(-0.316897\pi\)
0.544033 + 0.839064i \(0.316897\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 12.0000 0.500870
\(575\) −4.00000 −0.166812
\(576\) 0 0
\(577\) 14.0000 0.582828 0.291414 0.956597i \(-0.405874\pi\)
0.291414 + 0.956597i \(0.405874\pi\)
\(578\) −19.0000 −0.790296
\(579\) 0 0
\(580\) −6.00000 −0.249136
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) −48.0000 −1.98625
\(585\) 0 0
\(586\) 18.0000 0.743573
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 0 0
\(589\) 48.0000 1.97781
\(590\) 0 0
\(591\) 0 0
\(592\) 6.00000 0.246598
\(593\) 6.00000 0.246390 0.123195 0.992382i \(-0.460686\pi\)
0.123195 + 0.992382i \(0.460686\pi\)
\(594\) 0 0
\(595\) −12.0000 −0.491952
\(596\) 6.00000 0.245770
\(597\) 0 0
\(598\) −16.0000 −0.654289
\(599\) 4.00000 0.163436 0.0817178 0.996656i \(-0.473959\pi\)
0.0817178 + 0.996656i \(0.473959\pi\)
\(600\) 0 0
\(601\) −24.0000 −0.978980 −0.489490 0.872009i \(-0.662817\pi\)
−0.489490 + 0.872009i \(0.662817\pi\)
\(602\) −12.0000 −0.489083
\(603\) 0 0
\(604\) 2.00000 0.0813788
\(605\) 0 0
\(606\) 0 0
\(607\) 42.0000 1.70473 0.852364 0.522949i \(-0.175168\pi\)
0.852364 + 0.522949i \(0.175168\pi\)
\(608\) −30.0000 −1.21666
\(609\) 0 0
\(610\) 4.00000 0.161955
\(611\) 32.0000 1.29458
\(612\) 0 0
\(613\) −8.00000 −0.323117 −0.161558 0.986863i \(-0.551652\pi\)
−0.161558 + 0.986863i \(0.551652\pi\)
\(614\) 14.0000 0.564994
\(615\) 0 0
\(616\) 0 0
\(617\) −30.0000 −1.20775 −0.603877 0.797077i \(-0.706378\pi\)
−0.603877 + 0.797077i \(0.706378\pi\)
\(618\) 0 0
\(619\) 4.00000 0.160774 0.0803868 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) −8.00000 −0.321288
\(621\) 0 0
\(622\) −28.0000 −1.12270
\(623\) −20.0000 −0.801283
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 26.0000 1.03917
\(627\) 0 0
\(628\) −18.0000 −0.718278
\(629\) 36.0000 1.43541
\(630\) 0 0
\(631\) 16.0000 0.636950 0.318475 0.947931i \(-0.396829\pi\)
0.318475 + 0.947931i \(0.396829\pi\)
\(632\) −6.00000 −0.238667
\(633\) 0 0
\(634\) −30.0000 −1.19145
\(635\) 2.00000 0.0793676
\(636\) 0 0
\(637\) 12.0000 0.475457
\(638\) 0 0
\(639\) 0 0
\(640\) 3.00000 0.118585
\(641\) 38.0000 1.50091 0.750455 0.660922i \(-0.229834\pi\)
0.750455 + 0.660922i \(0.229834\pi\)
\(642\) 0 0
\(643\) 12.0000 0.473234 0.236617 0.971603i \(-0.423961\pi\)
0.236617 + 0.971603i \(0.423961\pi\)
\(644\) 8.00000 0.315244
\(645\) 0 0
\(646\) 36.0000 1.41640
\(647\) 20.0000 0.786281 0.393141 0.919478i \(-0.371389\pi\)
0.393141 + 0.919478i \(0.371389\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 4.00000 0.156893
\(651\) 0 0
\(652\) 4.00000 0.156652
\(653\) 10.0000 0.391330 0.195665 0.980671i \(-0.437313\pi\)
0.195665 + 0.980671i \(0.437313\pi\)
\(654\) 0 0
\(655\) 16.0000 0.625172
\(656\) 6.00000 0.234261
\(657\) 0 0
\(658\) 16.0000 0.623745
\(659\) 40.0000 1.55818 0.779089 0.626913i \(-0.215682\pi\)
0.779089 + 0.626913i \(0.215682\pi\)
\(660\) 0 0
\(661\) 10.0000 0.388955 0.194477 0.980907i \(-0.437699\pi\)
0.194477 + 0.980907i \(0.437699\pi\)
\(662\) 28.0000 1.08825
\(663\) 0 0
\(664\) 0 0
\(665\) 12.0000 0.465340
\(666\) 0 0
\(667\) −24.0000 −0.929284
\(668\) 0 0
\(669\) 0 0
\(670\) −12.0000 −0.463600
\(671\) 0 0
\(672\) 0 0
\(673\) −4.00000 −0.154189 −0.0770943 0.997024i \(-0.524564\pi\)
−0.0770943 + 0.997024i \(0.524564\pi\)
\(674\) −8.00000 −0.308148
\(675\) 0 0
\(676\) −3.00000 −0.115385
\(677\) 2.00000 0.0768662 0.0384331 0.999261i \(-0.487763\pi\)
0.0384331 + 0.999261i \(0.487763\pi\)
\(678\) 0 0
\(679\) −12.0000 −0.460518
\(680\) −18.0000 −0.690268
\(681\) 0 0
\(682\) 0 0
\(683\) −28.0000 −1.07139 −0.535695 0.844411i \(-0.679950\pi\)
−0.535695 + 0.844411i \(0.679950\pi\)
\(684\) 0 0
\(685\) −14.0000 −0.534913
\(686\) 20.0000 0.763604
\(687\) 0 0
\(688\) −6.00000 −0.228748
\(689\) 24.0000 0.914327
\(690\) 0 0
\(691\) 28.0000 1.06517 0.532585 0.846376i \(-0.321221\pi\)
0.532585 + 0.846376i \(0.321221\pi\)
\(692\) −18.0000 −0.684257
\(693\) 0 0
\(694\) 8.00000 0.303676
\(695\) 22.0000 0.834508
\(696\) 0 0
\(697\) 36.0000 1.36360
\(698\) 36.0000 1.36262
\(699\) 0 0
\(700\) −2.00000 −0.0755929
\(701\) −14.0000 −0.528773 −0.264386 0.964417i \(-0.585169\pi\)
−0.264386 + 0.964417i \(0.585169\pi\)
\(702\) 0 0
\(703\) −36.0000 −1.35777
\(704\) 0 0
\(705\) 0 0
\(706\) 18.0000 0.677439
\(707\) 28.0000 1.05305
\(708\) 0 0
\(709\) −6.00000 −0.225335 −0.112667 0.993633i \(-0.535939\pi\)
−0.112667 + 0.993633i \(0.535939\pi\)
\(710\) 8.00000 0.300235
\(711\) 0 0
\(712\) −30.0000 −1.12430
\(713\) −32.0000 −1.19841
\(714\) 0 0
\(715\) 0 0
\(716\) 12.0000 0.448461
\(717\) 0 0
\(718\) −8.00000 −0.298557
\(719\) 40.0000 1.49175 0.745874 0.666087i \(-0.232032\pi\)
0.745874 + 0.666087i \(0.232032\pi\)
\(720\) 0 0
\(721\) −16.0000 −0.595871
\(722\) −17.0000 −0.632674
\(723\) 0 0
\(724\) 26.0000 0.966282
\(725\) 6.00000 0.222834
\(726\) 0 0
\(727\) −8.00000 −0.296704 −0.148352 0.988935i \(-0.547397\pi\)
−0.148352 + 0.988935i \(0.547397\pi\)
\(728\) −24.0000 −0.889499
\(729\) 0 0
\(730\) 16.0000 0.592187
\(731\) −36.0000 −1.33151
\(732\) 0 0
\(733\) 44.0000 1.62518 0.812589 0.582838i \(-0.198058\pi\)
0.812589 + 0.582838i \(0.198058\pi\)
\(734\) −24.0000 −0.885856
\(735\) 0 0
\(736\) 20.0000 0.737210
\(737\) 0 0
\(738\) 0 0
\(739\) 34.0000 1.25071 0.625355 0.780340i \(-0.284954\pi\)
0.625355 + 0.780340i \(0.284954\pi\)
\(740\) 6.00000 0.220564
\(741\) 0 0
\(742\) 12.0000 0.440534
\(743\) −20.0000 −0.733729 −0.366864 0.930274i \(-0.619569\pi\)
−0.366864 + 0.930274i \(0.619569\pi\)
\(744\) 0 0
\(745\) −6.00000 −0.219823
\(746\) 24.0000 0.878702
\(747\) 0 0
\(748\) 0 0
\(749\) 16.0000 0.584627
\(750\) 0 0
\(751\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) 8.00000 0.291730
\(753\) 0 0
\(754\) 24.0000 0.874028
\(755\) −2.00000 −0.0727875
\(756\) 0 0
\(757\) 10.0000 0.363456 0.181728 0.983349i \(-0.441831\pi\)
0.181728 + 0.983349i \(0.441831\pi\)
\(758\) 20.0000 0.726433
\(759\) 0 0
\(760\) 18.0000 0.652929
\(761\) −50.0000 −1.81250 −0.906249 0.422744i \(-0.861067\pi\)
−0.906249 + 0.422744i \(0.861067\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 36.0000 1.30073
\(767\) 0 0
\(768\) 0 0
\(769\) −40.0000 −1.44244 −0.721218 0.692708i \(-0.756418\pi\)
−0.721218 + 0.692708i \(0.756418\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 20.0000 0.719816
\(773\) −54.0000 −1.94225 −0.971123 0.238581i \(-0.923318\pi\)
−0.971123 + 0.238581i \(0.923318\pi\)
\(774\) 0 0
\(775\) 8.00000 0.287368
\(776\) −18.0000 −0.646162
\(777\) 0 0
\(778\) −6.00000 −0.215110
\(779\) −36.0000 −1.28983
\(780\) 0 0
\(781\) 0 0
\(782\) −24.0000 −0.858238
\(783\) 0 0
\(784\) 3.00000 0.107143
\(785\) 18.0000 0.642448
\(786\) 0 0
\(787\) 22.0000 0.784215 0.392108 0.919919i \(-0.371746\pi\)
0.392108 + 0.919919i \(0.371746\pi\)
\(788\) −18.0000 −0.641223
\(789\) 0 0
\(790\) 2.00000 0.0711568
\(791\) −12.0000 −0.426671
\(792\) 0 0
\(793\) 16.0000 0.568177
\(794\) −14.0000 −0.496841
\(795\) 0 0
\(796\) −16.0000 −0.567105
\(797\) −26.0000 −0.920967 −0.460484 0.887668i \(-0.652324\pi\)
−0.460484 + 0.887668i \(0.652324\pi\)
\(798\) 0 0
\(799\) 48.0000 1.69812
\(800\) −5.00000 −0.176777
\(801\) 0 0
\(802\) −10.0000 −0.353112
\(803\) 0 0
\(804\) 0 0
\(805\) −8.00000 −0.281963
\(806\) 32.0000 1.12715
\(807\) 0 0
\(808\) 42.0000 1.47755
\(809\) 46.0000 1.61727 0.808637 0.588308i \(-0.200206\pi\)
0.808637 + 0.588308i \(0.200206\pi\)
\(810\) 0 0
\(811\) −50.0000 −1.75574 −0.877869 0.478901i \(-0.841035\pi\)
−0.877869 + 0.478901i \(0.841035\pi\)
\(812\) −12.0000 −0.421117
\(813\) 0 0
\(814\) 0 0
\(815\) −4.00000 −0.140114
\(816\) 0 0
\(817\) 36.0000 1.25948
\(818\) −4.00000 −0.139857
\(819\) 0 0
\(820\) 6.00000 0.209529
\(821\) 30.0000 1.04701 0.523504 0.852023i \(-0.324625\pi\)
0.523504 + 0.852023i \(0.324625\pi\)
\(822\) 0 0
\(823\) 8.00000 0.278862 0.139431 0.990232i \(-0.455473\pi\)
0.139431 + 0.990232i \(0.455473\pi\)
\(824\) −24.0000 −0.836080
\(825\) 0 0
\(826\) 0 0
\(827\) 8.00000 0.278187 0.139094 0.990279i \(-0.455581\pi\)
0.139094 + 0.990279i \(0.455581\pi\)
\(828\) 0 0
\(829\) −2.00000 −0.0694629 −0.0347314 0.999397i \(-0.511058\pi\)
−0.0347314 + 0.999397i \(0.511058\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −28.0000 −0.970725
\(833\) 18.0000 0.623663
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) −28.0000 −0.967244
\(839\) −12.0000 −0.414286 −0.207143 0.978311i \(-0.566417\pi\)
−0.207143 + 0.978311i \(0.566417\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 6.00000 0.206774
\(843\) 0 0
\(844\) −22.0000 −0.757271
\(845\) 3.00000 0.103203
\(846\) 0 0
\(847\) 0 0
\(848\) 6.00000 0.206041
\(849\) 0 0
\(850\) 6.00000 0.205798
\(851\) 24.0000 0.822709
\(852\) 0 0
\(853\) −24.0000 −0.821744 −0.410872 0.911693i \(-0.634776\pi\)
−0.410872 + 0.911693i \(0.634776\pi\)
\(854\) 8.00000 0.273754
\(855\) 0 0
\(856\) 24.0000 0.820303
\(857\) −22.0000 −0.751506 −0.375753 0.926720i \(-0.622616\pi\)
−0.375753 + 0.926720i \(0.622616\pi\)
\(858\) 0 0
\(859\) −36.0000 −1.22830 −0.614152 0.789188i \(-0.710502\pi\)
−0.614152 + 0.789188i \(0.710502\pi\)
\(860\) −6.00000 −0.204598
\(861\) 0 0
\(862\) 12.0000 0.408722
\(863\) 36.0000 1.22545 0.612727 0.790295i \(-0.290072\pi\)
0.612727 + 0.790295i \(0.290072\pi\)
\(864\) 0 0
\(865\) 18.0000 0.612018
\(866\) −18.0000 −0.611665
\(867\) 0 0
\(868\) −16.0000 −0.543075
\(869\) 0 0
\(870\) 0 0
\(871\) −48.0000 −1.62642
\(872\) 0 0
\(873\) 0 0
\(874\) 24.0000 0.811812
\(875\) 2.00000 0.0676123
\(876\) 0 0
\(877\) 32.0000 1.08056 0.540282 0.841484i \(-0.318318\pi\)
0.540282 + 0.841484i \(0.318318\pi\)
\(878\) −26.0000 −0.877457
\(879\) 0 0
\(880\) 0 0
\(881\) −46.0000 −1.54978 −0.774890 0.632096i \(-0.782195\pi\)
−0.774890 + 0.632096i \(0.782195\pi\)
\(882\) 0 0
\(883\) −52.0000 −1.74994 −0.874970 0.484178i \(-0.839119\pi\)
−0.874970 + 0.484178i \(0.839119\pi\)
\(884\) −24.0000 −0.807207
\(885\) 0 0
\(886\) 16.0000 0.537531
\(887\) 12.0000 0.402921 0.201460 0.979497i \(-0.435431\pi\)
0.201460 + 0.979497i \(0.435431\pi\)
\(888\) 0 0
\(889\) 4.00000 0.134156
\(890\) 10.0000 0.335201
\(891\) 0 0
\(892\) 16.0000 0.535720
\(893\) −48.0000 −1.60626
\(894\) 0 0
\(895\) −12.0000 −0.401116
\(896\) 6.00000 0.200446
\(897\) 0 0
\(898\) −30.0000 −1.00111
\(899\) 48.0000 1.60089
\(900\) 0 0
\(901\) 36.0000 1.19933
\(902\) 0 0
\(903\) 0 0
\(904\) −18.0000 −0.598671
\(905\) −26.0000 −0.864269
\(906\) 0 0
\(907\) 20.0000 0.664089 0.332045 0.943264i \(-0.392262\pi\)
0.332045 + 0.943264i \(0.392262\pi\)
\(908\) −16.0000 −0.530979
\(909\) 0 0
\(910\) 8.00000 0.265197
\(911\) −32.0000 −1.06021 −0.530104 0.847933i \(-0.677847\pi\)
−0.530104 + 0.847933i \(0.677847\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 8.00000 0.264616
\(915\) 0 0
\(916\) 22.0000 0.726900
\(917\) 32.0000 1.05673
\(918\) 0 0
\(919\) −22.0000 −0.725713 −0.362857 0.931845i \(-0.618198\pi\)
−0.362857 + 0.931845i \(0.618198\pi\)
\(920\) −12.0000 −0.395628
\(921\) 0 0
\(922\) 22.0000 0.724531
\(923\) 32.0000 1.05329
\(924\) 0 0
\(925\) −6.00000 −0.197279
\(926\) 8.00000 0.262896
\(927\) 0 0
\(928\) −30.0000 −0.984798
\(929\) −14.0000 −0.459325 −0.229663 0.973270i \(-0.573762\pi\)
−0.229663 + 0.973270i \(0.573762\pi\)
\(930\) 0 0
\(931\) −18.0000 −0.589926
\(932\) −6.00000 −0.196537
\(933\) 0 0
\(934\) 16.0000 0.523536
\(935\) 0 0
\(936\) 0 0
\(937\) 20.0000 0.653372 0.326686 0.945133i \(-0.394068\pi\)
0.326686 + 0.945133i \(0.394068\pi\)
\(938\) −24.0000 −0.783628
\(939\) 0 0
\(940\) 8.00000 0.260931
\(941\) 38.0000 1.23876 0.619382 0.785090i \(-0.287383\pi\)
0.619382 + 0.785090i \(0.287383\pi\)
\(942\) 0 0
\(943\) 24.0000 0.781548
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −4.00000 −0.129983 −0.0649913 0.997886i \(-0.520702\pi\)
−0.0649913 + 0.997886i \(0.520702\pi\)
\(948\) 0 0
\(949\) 64.0000 2.07753
\(950\) −6.00000 −0.194666
\(951\) 0 0
\(952\) −36.0000 −1.16677
\(953\) −6.00000 −0.194359 −0.0971795 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 12.0000 0.388108
\(957\) 0 0
\(958\) −28.0000 −0.904639
\(959\) −28.0000 −0.904167
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) −24.0000 −0.773791
\(963\) 0 0
\(964\) 20.0000 0.644157
\(965\) −20.0000 −0.643823
\(966\) 0 0
\(967\) −46.0000 −1.47926 −0.739630 0.673014i \(-0.765000\pi\)
−0.739630 + 0.673014i \(0.765000\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 6.00000 0.192648
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) 44.0000 1.41058
\(974\) 16.0000 0.512673
\(975\) 0 0
\(976\) 4.00000 0.128037
\(977\) −30.0000 −0.959785 −0.479893 0.877327i \(-0.659324\pi\)
−0.479893 + 0.877327i \(0.659324\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 3.00000 0.0958315
\(981\) 0 0
\(982\) 36.0000 1.14881
\(983\) 4.00000 0.127580 0.0637901 0.997963i \(-0.479681\pi\)
0.0637901 + 0.997963i \(0.479681\pi\)
\(984\) 0 0
\(985\) 18.0000 0.573528
\(986\) 36.0000 1.14647
\(987\) 0 0
\(988\) 24.0000 0.763542
\(989\) −24.0000 −0.763156
\(990\) 0 0
\(991\) −32.0000 −1.01651 −0.508257 0.861206i \(-0.669710\pi\)
−0.508257 + 0.861206i \(0.669710\pi\)
\(992\) −40.0000 −1.27000
\(993\) 0 0
\(994\) 16.0000 0.507489
\(995\) 16.0000 0.507234
\(996\) 0 0
\(997\) −24.0000 −0.760088 −0.380044 0.924968i \(-0.624091\pi\)
−0.380044 + 0.924968i \(0.624091\pi\)
\(998\) 4.00000 0.126618
\(999\) 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.e.1.1 1
3.2 odd 2 1815.2.a.e.1.1 yes 1
11.10 odd 2 5445.2.a.j.1.1 1
15.14 odd 2 9075.2.a.d.1.1 1
33.32 even 2 1815.2.a.a.1.1 1
165.164 even 2 9075.2.a.n.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1815.2.a.a.1.1 1 33.32 even 2
1815.2.a.e.1.1 yes 1 3.2 odd 2
5445.2.a.e.1.1 1 1.1 even 1 trivial
5445.2.a.j.1.1 1 11.10 odd 2
9075.2.a.d.1.1 1 15.14 odd 2
9075.2.a.n.1.1 1 165.164 even 2