Properties

Label 9075.2.a.n.1.1
Level $9075$
Weight $2$
Character 9075.1
Self dual yes
Analytic conductor $72.464$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 9075 = 3 \cdot 5^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9075.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(72.4642398343\)
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 \(\chi\) \(=\) 9075.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000 q^{6} +2.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000 q^{6} +2.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +1.00000 q^{12} -4.00000 q^{13} +2.00000 q^{14} -1.00000 q^{16} +6.00000 q^{17} +1.00000 q^{18} -6.00000 q^{19} -2.00000 q^{21} -4.00000 q^{23} +3.00000 q^{24} -4.00000 q^{26} -1.00000 q^{27} -2.00000 q^{28} +6.00000 q^{29} +8.00000 q^{31} +5.00000 q^{32} +6.00000 q^{34} -1.00000 q^{36} +6.00000 q^{37} -6.00000 q^{38} +4.00000 q^{39} -6.00000 q^{41} -2.00000 q^{42} +6.00000 q^{43} -4.00000 q^{46} -8.00000 q^{47} +1.00000 q^{48} -3.00000 q^{49} -6.00000 q^{51} +4.00000 q^{52} -6.00000 q^{53} -1.00000 q^{54} -6.00000 q^{56} +6.00000 q^{57} +6.00000 q^{58} +4.00000 q^{61} +8.00000 q^{62} +2.00000 q^{63} +7.00000 q^{64} -12.0000 q^{67} -6.00000 q^{68} +4.00000 q^{69} +8.00000 q^{71} -3.00000 q^{72} -16.0000 q^{73} +6.00000 q^{74} +6.00000 q^{76} +4.00000 q^{78} +2.00000 q^{79} +1.00000 q^{81} -6.00000 q^{82} +2.00000 q^{84} +6.00000 q^{86} -6.00000 q^{87} +10.0000 q^{89} -8.00000 q^{91} +4.00000 q^{92} -8.00000 q^{93} -8.00000 q^{94} -5.00000 q^{96} +6.00000 q^{97} -3.00000 q^{98} +O(q^{100})\)

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 \(-0.384973\pi\)
0.353553 + 0.935414i \(0.384973\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) 2.00000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) −3.00000 −1.06066
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0
\(12\) 1.00000 0.288675
\(13\) −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 \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 1.00000 0.235702
\(19\) −6.00000 −1.37649 −0.688247 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) 0 0
\(21\) −2.00000 −0.436436
\(22\) 0 0
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 3.00000 0.612372
\(25\) 0 0
\(26\) −4.00000 −0.784465
\(27\) −1.00000 −0.192450
\(28\) −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
\(35\) 0 0
\(36\) −1.00000 −0.166667
\(37\) 6.00000 0.986394 0.493197 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) −6.00000 −0.973329
\(39\) 4.00000 0.640513
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) −2.00000 −0.308607
\(43\) 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 \(0.698301\pi\)
\(48\) 1.00000 0.144338
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) −6.00000 −0.840168
\(52\) 4.00000 0.554700
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) −6.00000 −0.801784
\(57\) 6.00000 0.794719
\(58\) 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 \(-0.417571\pi\)
0.256074 + 0.966657i \(0.417571\pi\)
\(62\) 8.00000 1.01600
\(63\) 2.00000 0.251976
\(64\) 7.00000 0.875000
\(65\) 0 0
\(66\) 0 0
\(67\) −12.0000 −1.46603 −0.733017 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) −6.00000 −0.727607
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) −3.00000 −0.353553
\(73\) −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
\(78\) 4.00000 0.452911
\(79\) 2.00000 0.225018 0.112509 0.993651i \(-0.464111\pi\)
0.112509 + 0.993651i \(0.464111\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −6.00000 −0.662589
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 2.00000 0.218218
\(85\) 0 0
\(86\) 6.00000 0.646997
\(87\) −6.00000 −0.643268
\(88\) 0 0
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) −8.00000 −0.838628
\(92\) 4.00000 0.417029
\(93\) −8.00000 −0.829561
\(94\) −8.00000 −0.825137
\(95\) 0 0
\(96\) −5.00000 −0.510310
\(97\) 6.00000 0.609208 0.304604 0.952479i \(-0.401476\pi\)
0.304604 + 0.952479i \(0.401476\pi\)
\(98\) −3.00000 −0.303046
\(99\) 0 0
\(100\) 0 0
\(101\) 14.0000 1.39305 0.696526 0.717532i \(-0.254728\pi\)
0.696526 + 0.717532i \(0.254728\pi\)
\(102\) −6.00000 −0.594089
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 12.0000 1.17670
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) −8.00000 −0.773389 −0.386695 0.922208i \(-0.626383\pi\)
−0.386695 + 0.922208i \(0.626383\pi\)
\(108\) 1.00000 0.0962250
\(109\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(110\) 0 0
\(111\) −6.00000 −0.569495
\(112\) −2.00000 −0.188982
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 6.00000 0.561951
\(115\) 0 0
\(116\) −6.00000 −0.557086
\(117\) −4.00000 −0.369800
\(118\) 0 0
\(119\) 12.0000 1.10004
\(120\) 0 0
\(121\) 0 0
\(122\) 4.00000 0.362143
\(123\) 6.00000 0.541002
\(124\) −8.00000 −0.718421
\(125\) 0 0
\(126\) 2.00000 0.178174
\(127\) 2.00000 0.177471 0.0887357 0.996055i \(-0.471717\pi\)
0.0887357 + 0.996055i \(0.471717\pi\)
\(128\) −3.00000 −0.265165
\(129\) −6.00000 −0.528271
\(130\) 0 0
\(131\) 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 \(0.704058\pi\)
\(138\) 4.00000 0.340503
\(139\) −22.0000 −1.86602 −0.933008 0.359856i \(-0.882826\pi\)
−0.933008 + 0.359856i \(0.882826\pi\)
\(140\) 0 0
\(141\) 8.00000 0.673722
\(142\) 8.00000 0.671345
\(143\) 0 0
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) −16.0000 −1.32417
\(147\) 3.00000 0.247436
\(148\) −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 \(-0.474068\pi\)
0.0813788 + 0.996683i \(0.474068\pi\)
\(152\) 18.0000 1.45999
\(153\) 6.00000 0.485071
\(154\) 0 0
\(155\) 0 0
\(156\) −4.00000 −0.320256
\(157\) −18.0000 −1.43656 −0.718278 0.695756i \(-0.755069\pi\)
−0.718278 + 0.695756i \(0.755069\pi\)
\(158\) 2.00000 0.159111
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) −8.00000 −0.630488
\(162\) 1.00000 0.0785674
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) 6.00000 0.468521
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 6.00000 0.462910
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) −6.00000 −0.458831
\(172\) −6.00000 −0.457496
\(173\) −18.0000 −1.36851 −0.684257 0.729241i \(-0.739873\pi\)
−0.684257 + 0.729241i \(0.739873\pi\)
\(174\) −6.00000 −0.454859
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 10.0000 0.749532
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 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
\(183\) −4.00000 −0.295689
\(184\) 12.0000 0.884652
\(185\) 0 0
\(186\) −8.00000 −0.586588
\(187\) 0 0
\(188\) 8.00000 0.583460
\(189\) −2.00000 −0.145479
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) −7.00000 −0.505181
\(193\) −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 \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 0 0
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 0 0
\(201\) 12.0000 0.846415
\(202\) 14.0000 0.985037
\(203\) 12.0000 0.842235
\(204\) 6.00000 0.420084
\(205\) 0 0
\(206\) 8.00000 0.557386
\(207\) −4.00000 −0.278019
\(208\) 4.00000 0.277350
\(209\) 0 0
\(210\) 0 0
\(211\) −22.0000 −1.51454 −0.757271 0.653101i \(-0.773468\pi\)
−0.757271 + 0.653101i \(0.773468\pi\)
\(212\) 6.00000 0.412082
\(213\) −8.00000 −0.548151
\(214\) −8.00000 −0.546869
\(215\) 0 0
\(216\) 3.00000 0.204124
\(217\) 16.0000 1.08615
\(218\) 0 0
\(219\) 16.0000 1.08118
\(220\) 0 0
\(221\) −24.0000 −1.61441
\(222\) −6.00000 −0.402694
\(223\) 16.0000 1.07144 0.535720 0.844396i \(-0.320040\pi\)
0.535720 + 0.844396i \(0.320040\pi\)
\(224\) 10.0000 0.668153
\(225\) 0 0
\(226\) −6.00000 −0.399114
\(227\) −16.0000 −1.06196 −0.530979 0.847385i \(-0.678176\pi\)
−0.530979 + 0.847385i \(0.678176\pi\)
\(228\) −6.00000 −0.397360
\(229\) −22.0000 −1.45380 −0.726900 0.686743i \(-0.759040\pi\)
−0.726900 + 0.686743i \(0.759040\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −18.0000 −1.18176
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) −4.00000 −0.261488
\(235\) 0 0
\(236\) 0 0
\(237\) −2.00000 −0.129914
\(238\) 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 \(-0.277208\pi\)
0.644157 + 0.764894i \(0.277208\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) −4.00000 −0.256074
\(245\) 0 0
\(246\) 6.00000 0.382546
\(247\) 24.0000 1.52708
\(248\) −24.0000 −1.52400
\(249\) 0 0
\(250\) 0 0
\(251\) 20.0000 1.26239 0.631194 0.775625i \(-0.282565\pi\)
0.631194 + 0.775625i \(0.282565\pi\)
\(252\) −2.00000 −0.125988
\(253\) 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 \(0.689739\pi\)
\(258\) −6.00000 −0.373544
\(259\) 12.0000 0.745644
\(260\) 0 0
\(261\) 6.00000 0.371391
\(262\) 16.0000 0.988483
\(263\) 24.0000 1.47990 0.739952 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −12.0000 −0.735767
\(267\) −10.0000 −0.611990
\(268\) 12.0000 0.733017
\(269\) 6.00000 0.365826 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) 0 0
\(271\) 14.0000 0.850439 0.425220 0.905090i \(-0.360197\pi\)
0.425220 + 0.905090i \(0.360197\pi\)
\(272\) −6.00000 −0.363803
\(273\) 8.00000 0.484182
\(274\) −14.0000 −0.845771
\(275\) 0 0
\(276\) −4.00000 −0.240772
\(277\) −4.00000 −0.240337 −0.120168 0.992754i \(-0.538343\pi\)
−0.120168 + 0.992754i \(0.538343\pi\)
\(278\) −22.0000 −1.31947
\(279\) 8.00000 0.478947
\(280\) 0 0
\(281\) −22.0000 −1.31241 −0.656205 0.754583i \(-0.727839\pi\)
−0.656205 + 0.754583i \(0.727839\pi\)
\(282\) 8.00000 0.476393
\(283\) 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
\(288\) 5.00000 0.294628
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) −6.00000 −0.351726
\(292\) 16.0000 0.936329
\(293\) 18.0000 1.05157 0.525786 0.850617i \(-0.323771\pi\)
0.525786 + 0.850617i \(0.323771\pi\)
\(294\) 3.00000 0.174964
\(295\) 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
\(303\) −14.0000 −0.804279
\(304\) 6.00000 0.344124
\(305\) 0 0
\(306\) 6.00000 0.342997
\(307\) −14.0000 −0.799022 −0.399511 0.916728i \(-0.630820\pi\)
−0.399511 + 0.916728i \(0.630820\pi\)
\(308\) 0 0
\(309\) −8.00000 −0.455104
\(310\) 0 0
\(311\) −28.0000 −1.58773 −0.793867 0.608091i \(-0.791935\pi\)
−0.793867 + 0.608091i \(0.791935\pi\)
\(312\) −12.0000 −0.679366
\(313\) 26.0000 1.46961 0.734803 0.678280i \(-0.237274\pi\)
0.734803 + 0.678280i \(0.237274\pi\)
\(314\) −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 \(0.181092\pi\)
\(318\) 6.00000 0.336463
\(319\) 0 0
\(320\) 0 0
\(321\) 8.00000 0.446516
\(322\) −8.00000 −0.445823
\(323\) −36.0000 −2.00309
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) 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
\(333\) 6.00000 0.328798
\(334\) 0 0
\(335\) 0 0
\(336\) 2.00000 0.109109
\(337\) 8.00000 0.435788 0.217894 0.975972i \(-0.430081\pi\)
0.217894 + 0.975972i \(0.430081\pi\)
\(338\) 3.00000 0.163178
\(339\) 6.00000 0.325875
\(340\) 0 0
\(341\) 0 0
\(342\) −6.00000 −0.324443
\(343\) −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 \(-0.431112\pi\)
0.214731 + 0.976673i \(0.431112\pi\)
\(348\) 6.00000 0.321634
\(349\) 36.0000 1.92704 0.963518 0.267644i \(-0.0862451\pi\)
0.963518 + 0.267644i \(0.0862451\pi\)
\(350\) 0 0
\(351\) 4.00000 0.213504
\(352\) 0 0
\(353\) −18.0000 −0.958043 −0.479022 0.877803i \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −10.0000 −0.529999
\(357\) −12.0000 −0.635107
\(358\) 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
\(365\) 0 0
\(366\) −4.00000 −0.209083
\(367\) −24.0000 −1.25279 −0.626395 0.779506i \(-0.715470\pi\)
−0.626395 + 0.779506i \(0.715470\pi\)
\(368\) 4.00000 0.208514
\(369\) −6.00000 −0.312348
\(370\) 0 0
\(371\) −12.0000 −0.623009
\(372\) 8.00000 0.414781
\(373\) −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
\(378\) −2.00000 −0.102869
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 0 0
\(381\) −2.00000 −0.102463
\(382\) 0 0
\(383\) −36.0000 −1.83951 −0.919757 0.392488i \(-0.871614\pi\)
−0.919757 + 0.392488i \(0.871614\pi\)
\(384\) 3.00000 0.153093
\(385\) 0 0
\(386\) −20.0000 −1.01797
\(387\) 6.00000 0.304997
\(388\) −6.00000 −0.304604
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) −24.0000 −1.21373
\(392\) 9.00000 0.454569
\(393\) −16.0000 −0.807093
\(394\) −18.0000 −0.906827
\(395\) 0 0
\(396\) 0 0
\(397\) −14.0000 −0.702640 −0.351320 0.936255i \(-0.614267\pi\)
−0.351320 + 0.936255i \(0.614267\pi\)
\(398\) 16.0000 0.802008
\(399\) 12.0000 0.600751
\(400\) 0 0
\(401\) −10.0000 −0.499376 −0.249688 0.968326i \(-0.580328\pi\)
−0.249688 + 0.968326i \(0.580328\pi\)
\(402\) 12.0000 0.598506
\(403\) −32.0000 −1.59403
\(404\) −14.0000 −0.696526
\(405\) 0 0
\(406\) 12.0000 0.595550
\(407\) 0 0
\(408\) 18.0000 0.891133
\(409\) −4.00000 −0.197787 −0.0988936 0.995098i \(-0.531530\pi\)
−0.0988936 + 0.995098i \(0.531530\pi\)
\(410\) 0 0
\(411\) 14.0000 0.690569
\(412\) −8.00000 −0.394132
\(413\) 0 0
\(414\) −4.00000 −0.196589
\(415\) 0 0
\(416\) −20.0000 −0.980581
\(417\) 22.0000 1.07734
\(418\) 0 0
\(419\) −28.0000 −1.36789 −0.683945 0.729534i \(-0.739737\pi\)
−0.683945 + 0.729534i \(0.739737\pi\)
\(420\) 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
\(423\) −8.00000 −0.388973
\(424\) 18.0000 0.874157
\(425\) 0 0
\(426\) −8.00000 −0.387601
\(427\) 8.00000 0.387147
\(428\) 8.00000 0.386695
\(429\) 0 0
\(430\) 0 0
\(431\) −12.0000 −0.578020 −0.289010 0.957326i \(-0.593326\pi\)
−0.289010 + 0.957326i \(0.593326\pi\)
\(432\) 1.00000 0.0481125
\(433\) −18.0000 −0.865025 −0.432512 0.901628i \(-0.642373\pi\)
−0.432512 + 0.901628i \(0.642373\pi\)
\(434\) 16.0000 0.768025
\(435\) 0 0
\(436\) 0 0
\(437\) 24.0000 1.14808
\(438\) 16.0000 0.764510
\(439\) −26.0000 −1.24091 −0.620456 0.784241i \(-0.713053\pi\)
−0.620456 + 0.784241i \(0.713053\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) −24.0000 −1.14156
\(443\) −16.0000 −0.760183 −0.380091 0.924949i \(-0.624107\pi\)
−0.380091 + 0.924949i \(0.624107\pi\)
\(444\) 6.00000 0.284747
\(445\) 0 0
\(446\) 16.0000 0.757622
\(447\) 6.00000 0.283790
\(448\) 14.0000 0.661438
\(449\) −30.0000 −1.41579 −0.707894 0.706319i \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 6.00000 0.282216
\(453\) −2.00000 −0.0939682
\(454\) −16.0000 −0.750917
\(455\) 0 0
\(456\) −18.0000 −0.842927
\(457\) −8.00000 −0.374224 −0.187112 0.982339i \(-0.559913\pi\)
−0.187112 + 0.982339i \(0.559913\pi\)
\(458\) −22.0000 −1.02799
\(459\) −6.00000 −0.280056
\(460\) 0 0
\(461\) −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 \(-0.440481\pi\)
0.185896 + 0.982569i \(0.440481\pi\)
\(464\) −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 \(0.620709\pi\)
\(468\) 4.00000 0.184900
\(469\) −24.0000 −1.10822
\(470\) 0 0
\(471\) 18.0000 0.829396
\(472\) 0 0
\(473\) 0 0
\(474\) −2.00000 −0.0918630
\(475\) 0 0
\(476\) −12.0000 −0.550019
\(477\) −6.00000 −0.274721
\(478\) −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
\(483\) 8.00000 0.364013
\(484\) 0 0
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) 16.0000 0.725029 0.362515 0.931978i \(-0.381918\pi\)
0.362515 + 0.931978i \(0.381918\pi\)
\(488\) −12.0000 −0.543214
\(489\) −4.00000 −0.180886
\(490\) 0 0
\(491\) −36.0000 −1.62466 −0.812329 0.583200i \(-0.801800\pi\)
−0.812329 + 0.583200i \(0.801800\pi\)
\(492\) −6.00000 −0.270501
\(493\) 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 \(0.528537\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 20.0000 0.892644
\(503\) 16.0000 0.713405 0.356702 0.934218i \(-0.383901\pi\)
0.356702 + 0.934218i \(0.383901\pi\)
\(504\) −6.00000 −0.267261
\(505\) 0 0
\(506\) 0 0
\(507\) −3.00000 −0.133235
\(508\) −2.00000 −0.0887357
\(509\) 14.0000 0.620539 0.310270 0.950649i \(-0.399581\pi\)
0.310270 + 0.950649i \(0.399581\pi\)
\(510\) 0 0
\(511\) −32.0000 −1.41560
\(512\) −11.0000 −0.486136
\(513\) 6.00000 0.264906
\(514\) −18.0000 −0.793946
\(515\) 0 0
\(516\) 6.00000 0.264135
\(517\) 0 0
\(518\) 12.0000 0.527250
\(519\) 18.0000 0.790112
\(520\) 0 0
\(521\) −10.0000 −0.438108 −0.219054 0.975713i \(-0.570297\pi\)
−0.219054 + 0.975713i \(0.570297\pi\)
\(522\) 6.00000 0.262613
\(523\) 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
\(530\) 0 0
\(531\) 0 0
\(532\) 12.0000 0.520266
\(533\) 24.0000 1.03956
\(534\) −10.0000 −0.432742
\(535\) 0 0
\(536\) 36.0000 1.55496
\(537\) −12.0000 −0.517838
\(538\) 6.00000 0.258678
\(539\) 0 0
\(540\) 0 0
\(541\) 32.0000 1.37579 0.687894 0.725811i \(-0.258536\pi\)
0.687894 + 0.725811i \(0.258536\pi\)
\(542\) 14.0000 0.601351
\(543\) 26.0000 1.11577
\(544\) 30.0000 1.28624
\(545\) 0 0
\(546\) 8.00000 0.342368
\(547\) −34.0000 −1.45374 −0.726868 0.686778i \(-0.759025\pi\)
−0.726868 + 0.686778i \(0.759025\pi\)
\(548\) 14.0000 0.598050
\(549\) 4.00000 0.170716
\(550\) 0 0
\(551\) −36.0000 −1.53365
\(552\) −12.0000 −0.510754
\(553\) 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 \(-0.486509\pi\)
0.0423714 + 0.999102i \(0.486509\pi\)
\(558\) 8.00000 0.338667
\(559\) −24.0000 −1.01509
\(560\) 0 0
\(561\) 0 0
\(562\) −22.0000 −0.928014
\(563\) −12.0000 −0.505740 −0.252870 0.967500i \(-0.581374\pi\)
−0.252870 + 0.967500i \(0.581374\pi\)
\(564\) −8.00000 −0.336861
\(565\) 0 0
\(566\) 14.0000 0.588464
\(567\) 2.00000 0.0839921
\(568\) −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 \(-0.683103\pi\)
−0.544033 + 0.839064i \(0.683103\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −12.0000 −0.500870
\(575\) 0 0
\(576\) 7.00000 0.291667
\(577\) −14.0000 −0.582828 −0.291414 0.956597i \(-0.594126\pi\)
−0.291414 + 0.956597i \(0.594126\pi\)
\(578\) 19.0000 0.790296
\(579\) 20.0000 0.831172
\(580\) 0 0
\(581\) 0 0
\(582\) −6.00000 −0.248708
\(583\) 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 \(0.5\pi\)
\(588\) −3.00000 −0.123718
\(589\) −48.0000 −1.97781
\(590\) 0 0
\(591\) 18.0000 0.740421
\(592\) −6.00000 −0.246598
\(593\) −6.00000 −0.246390 −0.123195 0.992382i \(-0.539314\pi\)
−0.123195 + 0.992382i \(0.539314\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 6.00000 0.245770
\(597\) −16.0000 −0.654836
\(598\) 16.0000 0.654289
\(599\) −4.00000 −0.163436 −0.0817178 0.996656i \(-0.526041\pi\)
−0.0817178 + 0.996656i \(0.526041\pi\)
\(600\) 0 0
\(601\) 24.0000 0.978980 0.489490 0.872009i \(-0.337183\pi\)
0.489490 + 0.872009i \(0.337183\pi\)
\(602\) 12.0000 0.489083
\(603\) −12.0000 −0.488678
\(604\) −2.00000 −0.0813788
\(605\) 0 0
\(606\) −14.0000 −0.568711
\(607\) 42.0000 1.70473 0.852364 0.522949i \(-0.175168\pi\)
0.852364 + 0.522949i \(0.175168\pi\)
\(608\) −30.0000 −1.21666
\(609\) −12.0000 −0.486265
\(610\) 0 0
\(611\) 32.0000 1.29458
\(612\) −6.00000 −0.242536
\(613\) −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 \(0.706378\pi\)
\(618\) −8.00000 −0.321807
\(619\) 4.00000 0.160774 0.0803868 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) 0 0
\(621\) 4.00000 0.160514
\(622\) −28.0000 −1.12270
\(623\) 20.0000 0.801283
\(624\) −4.00000 −0.160128
\(625\) 0 0
\(626\) 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
\(633\) 22.0000 0.874421
\(634\) 30.0000 1.19145
\(635\) 0 0
\(636\) −6.00000 −0.237915
\(637\) 12.0000 0.475457
\(638\) 0 0
\(639\) 8.00000 0.316475
\(640\) 0 0
\(641\) −38.0000 −1.50091 −0.750455 0.660922i \(-0.770166\pi\)
−0.750455 + 0.660922i \(0.770166\pi\)
\(642\) 8.00000 0.315735
\(643\) −12.0000 −0.473234 −0.236617 0.971603i \(-0.576039\pi\)
−0.236617 + 0.971603i \(0.576039\pi\)
\(644\) 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 \(0.371389\pi\)
\(648\) −3.00000 −0.117851
\(649\) 0 0
\(650\) 0 0
\(651\) −16.0000 −0.627089
\(652\) −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
\(655\) 0 0
\(656\) 6.00000 0.234261
\(657\) −16.0000 −0.624219
\(658\) −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
\(663\) 24.0000 0.932083
\(664\) 0 0
\(665\) 0 0
\(666\) 6.00000 0.232495
\(667\) −24.0000 −0.929284
\(668\) 0 0
\(669\) −16.0000 −0.618596
\(670\) 0 0
\(671\) 0 0
\(672\) −10.0000 −0.385758
\(673\) −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 \(-0.512237\pi\)
−0.0384331 + 0.999261i \(0.512237\pi\)
\(678\) 6.00000 0.230429
\(679\) 12.0000 0.460518
\(680\) 0 0
\(681\) 16.0000 0.613121
\(682\) 0 0
\(683\) −28.0000 −1.07139 −0.535695 0.844411i \(-0.679950\pi\)
−0.535695 + 0.844411i \(0.679950\pi\)
\(684\) 6.00000 0.229416
\(685\) 0 0
\(686\) −20.0000 −0.763604
\(687\) 22.0000 0.839352
\(688\) −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
\(695\) 0 0
\(696\) 18.0000 0.682288
\(697\) −36.0000 −1.36360
\(698\) 36.0000 1.36262
\(699\) 6.00000 0.226941
\(700\) 0 0
\(701\) −14.0000 −0.528773 −0.264386 0.964417i \(-0.585169\pi\)
−0.264386 + 0.964417i \(0.585169\pi\)
\(702\) 4.00000 0.150970
\(703\) −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 \(0.535939\pi\)
\(710\) 0 0
\(711\) 2.00000 0.0750059
\(712\) −30.0000 −1.12430
\(713\) −32.0000 −1.19841
\(714\) −12.0000 −0.449089
\(715\) 0 0
\(716\) −12.0000 −0.448461
\(717\) 12.0000 0.448148
\(718\) 8.00000 0.298557
\(719\) −40.0000 −1.49175 −0.745874 0.666087i \(-0.767968\pi\)
−0.745874 + 0.666087i \(0.767968\pi\)
\(720\) 0 0
\(721\) 16.0000 0.595871
\(722\) 17.0000 0.632674
\(723\) −20.0000 −0.743808
\(724\) 26.0000 0.966282
\(725\) 0 0
\(726\) 0 0
\(727\) 8.00000 0.296704 0.148352 0.988935i \(-0.452603\pi\)
0.148352 + 0.988935i \(0.452603\pi\)
\(728\) 24.0000 0.889499
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 36.0000 1.33151
\(732\) 4.00000 0.147844
\(733\) 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
\(738\) −6.00000 −0.220863
\(739\) −34.0000 −1.25071 −0.625355 0.780340i \(-0.715046\pi\)
−0.625355 + 0.780340i \(0.715046\pi\)
\(740\) 0 0
\(741\) −24.0000 −0.881662
\(742\) −12.0000 −0.440534
\(743\) 20.0000 0.733729 0.366864 0.930274i \(-0.380431\pi\)
0.366864 + 0.930274i \(0.380431\pi\)
\(744\) 24.0000 0.879883
\(745\) 0 0
\(746\) −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
\(753\) −20.0000 −0.728841
\(754\) −24.0000 −0.874028
\(755\) 0 0
\(756\) 2.00000 0.0727393
\(757\) −10.0000 −0.363456 −0.181728 0.983349i \(-0.558169\pi\)
−0.181728 + 0.983349i \(0.558169\pi\)
\(758\) −20.0000 −0.726433
\(759\) 0 0
\(760\) 0 0
\(761\) −50.0000 −1.81250 −0.906249 0.422744i \(-0.861067\pi\)
−0.906249 + 0.422744i \(0.861067\pi\)
\(762\) −2.00000 −0.0724524
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) −36.0000 −1.30073
\(767\) 0 0
\(768\) 17.0000 0.613435
\(769\) 40.0000 1.44244 0.721218 0.692708i \(-0.243582\pi\)
0.721218 + 0.692708i \(0.243582\pi\)
\(770\) 0 0
\(771\) 18.0000 0.648254
\(772\) 20.0000 0.719816
\(773\) −54.0000 −1.94225 −0.971123 0.238581i \(-0.923318\pi\)
−0.971123 + 0.238581i \(0.923318\pi\)
\(774\) 6.00000 0.215666
\(775\) 0 0
\(776\) −18.0000 −0.646162
\(777\) −12.0000 −0.430498
\(778\) −6.00000 −0.215110
\(779\) 36.0000 1.28983
\(780\) 0 0
\(781\) 0 0
\(782\) −24.0000 −0.858238
\(783\) −6.00000 −0.214423
\(784\) 3.00000 0.107143
\(785\) 0 0
\(786\) −16.0000 −0.570701
\(787\) 22.0000 0.784215 0.392108 0.919919i \(-0.371746\pi\)
0.392108 + 0.919919i \(0.371746\pi\)
\(788\) 18.0000 0.641223
\(789\) −24.0000 −0.854423
\(790\) 0 0
\(791\) −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 \(0.652324\pi\)
\(798\) 12.0000 0.424795
\(799\) −48.0000 −1.69812
\(800\) 0 0
\(801\) 10.0000 0.353333
\(802\) −10.0000 −0.353112
\(803\) 0 0
\(804\) −12.0000 −0.423207
\(805\) 0 0
\(806\) −32.0000 −1.12715
\(807\) −6.00000 −0.211210
\(808\) −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 \(-0.158965\pi\)
0.877869 + 0.478901i \(0.158965\pi\)
\(812\) −12.0000 −0.421117
\(813\) −14.0000 −0.491001
\(814\) 0 0
\(815\) 0 0
\(816\) 6.00000 0.210042
\(817\) −36.0000 −1.25948
\(818\) −4.00000 −0.139857
\(819\) −8.00000 −0.279543
\(820\) 0 0
\(821\) 30.0000 1.04701 0.523504 0.852023i \(-0.324625\pi\)
0.523504 + 0.852023i \(0.324625\pi\)
\(822\) 14.0000 0.488306
\(823\) −8.00000 −0.278862 −0.139431 0.990232i \(-0.544527\pi\)
−0.139431 + 0.990232i \(0.544527\pi\)
\(824\) −24.0000 −0.836080
\(825\) 0 0
\(826\) 0 0
\(827\) −8.00000 −0.278187 −0.139094 0.990279i \(-0.544419\pi\)
−0.139094 + 0.990279i \(0.544419\pi\)
\(828\) 4.00000 0.139010
\(829\) −2.00000 −0.0694629 −0.0347314 0.999397i \(-0.511058\pi\)
−0.0347314 + 0.999397i \(0.511058\pi\)
\(830\) 0 0
\(831\) 4.00000 0.138758
\(832\) −28.0000 −0.970725
\(833\) −18.0000 −0.623663
\(834\) 22.0000 0.761798
\(835\) 0 0
\(836\) 0 0
\(837\) −8.00000 −0.276520
\(838\) −28.0000 −0.967244
\(839\) 12.0000 0.414286 0.207143 0.978311i \(-0.433583\pi\)
0.207143 + 0.978311i \(0.433583\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) −6.00000 −0.206774
\(843\) 22.0000 0.757720
\(844\) 22.0000 0.757271
\(845\) 0 0
\(846\) −8.00000 −0.275046
\(847\) 0 0
\(848\) 6.00000 0.206041
\(849\) −14.0000 −0.480479
\(850\) 0 0
\(851\) −24.0000 −0.822709
\(852\) 8.00000 0.274075
\(853\) −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 \(-0.377384\pi\)
0.375753 + 0.926720i \(0.377384\pi\)
\(858\) 0 0
\(859\) −36.0000 −1.22830 −0.614152 0.789188i \(-0.710502\pi\)
−0.614152 + 0.789188i \(0.710502\pi\)
\(860\) 0 0
\(861\) 12.0000 0.408959
\(862\) −12.0000 −0.408722
\(863\) 36.0000 1.22545 0.612727 0.790295i \(-0.290072\pi\)
0.612727 + 0.790295i \(0.290072\pi\)
\(864\) −5.00000 −0.170103
\(865\) 0 0
\(866\) −18.0000 −0.611665
\(867\) −19.0000 −0.645274
\(868\) −16.0000 −0.543075
\(869\) 0 0
\(870\) 0 0
\(871\) 48.0000 1.62642
\(872\) 0 0
\(873\) 6.00000 0.203069
\(874\) 24.0000 0.811812
\(875\) 0 0
\(876\) −16.0000 −0.540590
\(877\) 32.0000 1.08056 0.540282 0.841484i \(-0.318318\pi\)
0.540282 + 0.841484i \(0.318318\pi\)
\(878\) −26.0000 −0.877457
\(879\) −18.0000 −0.607125
\(880\) 0 0
\(881\) 46.0000 1.54978 0.774890 0.632096i \(-0.217805\pi\)
0.774890 + 0.632096i \(0.217805\pi\)
\(882\) −3.00000 −0.101015
\(883\) 52.0000 1.74994 0.874970 0.484178i \(-0.160881\pi\)
0.874970 + 0.484178i \(0.160881\pi\)
\(884\) 24.0000 0.807207
\(885\) 0 0
\(886\) −16.0000 −0.537531
\(887\) −12.0000 −0.402921 −0.201460 0.979497i \(-0.564569\pi\)
−0.201460 + 0.979497i \(0.564569\pi\)
\(888\) 18.0000 0.604040
\(889\) 4.00000 0.134156
\(890\) 0 0
\(891\) 0 0
\(892\) −16.0000 −0.535720
\(893\) 48.0000 1.60626
\(894\) 6.00000 0.200670
\(895\) 0 0
\(896\) −6.00000 −0.200446
\(897\) −16.0000 −0.534224
\(898\) −30.0000 −1.00111
\(899\) 48.0000 1.60089
\(900\) 0 0
\(901\) −36.0000 −1.19933
\(902\) 0 0
\(903\) −12.0000 −0.399335
\(904\) 18.0000 0.598671
\(905\) 0 0
\(906\) −2.00000 −0.0664455
\(907\) −20.0000 −0.664089 −0.332045 0.943264i \(-0.607738\pi\)
−0.332045 + 0.943264i \(0.607738\pi\)
\(908\) 16.0000 0.530979
\(909\) 14.0000 0.464351
\(910\) 0 0
\(911\) 32.0000 1.06021 0.530104 0.847933i \(-0.322153\pi\)
0.530104 + 0.847933i \(0.322153\pi\)
\(912\) −6.00000 −0.198680
\(913\) 0 0
\(914\) −8.00000 −0.264616
\(915\) 0 0
\(916\) 22.0000 0.726900
\(917\) 32.0000 1.05673
\(918\) −6.00000 −0.198030
\(919\) 22.0000 0.725713 0.362857 0.931845i \(-0.381802\pi\)
0.362857 + 0.931845i \(0.381802\pi\)
\(920\) 0 0
\(921\) 14.0000 0.461316
\(922\) −22.0000 −0.724531
\(923\) −32.0000 −1.05329
\(924\) 0 0
\(925\) 0 0
\(926\) 8.00000 0.262896
\(927\) 8.00000 0.262754
\(928\) 30.0000 0.984798
\(929\) 14.0000 0.459325 0.229663 0.973270i \(-0.426238\pi\)
0.229663 + 0.973270i \(0.426238\pi\)
\(930\) 0 0
\(931\) 18.0000 0.589926
\(932\) 6.00000 0.196537
\(933\) 28.0000 0.916679
\(934\) −16.0000 −0.523536
\(935\) 0 0
\(936\) 12.0000 0.392232
\(937\) 20.0000 0.653372 0.326686 0.945133i \(-0.394068\pi\)
0.326686 + 0.945133i \(0.394068\pi\)
\(938\) −24.0000 −0.783628
\(939\) −26.0000 −0.848478
\(940\) 0 0
\(941\) 38.0000 1.23876 0.619382 0.785090i \(-0.287383\pi\)
0.619382 + 0.785090i \(0.287383\pi\)
\(942\) 18.0000 0.586472
\(943\) 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 \(0.520702\pi\)
\(948\) 2.00000 0.0649570
\(949\) 64.0000 2.07753
\(950\) 0 0
\(951\) −30.0000 −0.972817
\(952\) −36.0000 −1.16677
\(953\) 6.00000 0.194359 0.0971795 0.995267i \(-0.469018\pi\)
0.0971795 + 0.995267i \(0.469018\pi\)
\(954\) −6.00000 −0.194257
\(955\) 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
\(963\) −8.00000 −0.257796
\(964\) −20.0000 −0.644157
\(965\) 0 0
\(966\) 8.00000 0.257396
\(967\) −46.0000 −1.47926 −0.739630 0.673014i \(-0.765000\pi\)
−0.739630 + 0.673014i \(0.765000\pi\)
\(968\) 0 0
\(969\) 36.0000 1.15649
\(970\) 0 0
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 1.00000 0.0320750
\(973\) −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 \(0.659324\pi\)
\(978\) −4.00000 −0.127906
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) −36.0000 −1.14881
\(983\) 4.00000 0.127580 0.0637901 0.997963i \(-0.479681\pi\)
0.0637901 + 0.997963i \(0.479681\pi\)
\(984\) −18.0000 −0.573819
\(985\) 0 0
\(986\) 36.0000 1.14647
\(987\) 16.0000 0.509286
\(988\) −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
\(993\) 28.0000 0.888553
\(994\) 16.0000 0.507489
\(995\) 0 0
\(996\) 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
\(999\) −6.00000 −0.189832
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 9075.2.a.n.1.1 1
5.4 even 2 1815.2.a.a.1.1 1
11.10 odd 2 9075.2.a.d.1.1 1
15.14 odd 2 5445.2.a.j.1.1 1
55.54 odd 2 1815.2.a.e.1.1 yes 1
165.164 even 2 5445.2.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1815.2.a.a.1.1 1 5.4 even 2
1815.2.a.e.1.1 yes 1 55.54 odd 2
5445.2.a.e.1.1 1 165.164 even 2
5445.2.a.j.1.1 1 15.14 odd 2
9075.2.a.d.1.1 1 11.10 odd 2
9075.2.a.n.1.1 1 1.1 even 1 trivial