Properties

Label 4650.2.a.bc.1.1
Level $4650$
Weight $2$
Character 4650.1
Self dual yes
Analytic conductor $37.130$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4650.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(37.1304369399\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 930)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 4650.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -3.00000 q^{11} -1.00000 q^{12} -4.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +1.00000 q^{18} -1.00000 q^{19} -1.00000 q^{21} -3.00000 q^{22} +5.00000 q^{23} -1.00000 q^{24} -4.00000 q^{26} -1.00000 q^{27} +1.00000 q^{28} +2.00000 q^{29} -1.00000 q^{31} +1.00000 q^{32} +3.00000 q^{33} +1.00000 q^{36} +4.00000 q^{37} -1.00000 q^{38} +4.00000 q^{39} -10.0000 q^{41} -1.00000 q^{42} -5.00000 q^{43} -3.00000 q^{44} +5.00000 q^{46} -8.00000 q^{47} -1.00000 q^{48} -6.00000 q^{49} -4.00000 q^{52} +5.00000 q^{53} -1.00000 q^{54} +1.00000 q^{56} +1.00000 q^{57} +2.00000 q^{58} -6.00000 q^{59} -2.00000 q^{61} -1.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} +3.00000 q^{66} +2.00000 q^{67} -5.00000 q^{69} -5.00000 q^{71} +1.00000 q^{72} -7.00000 q^{73} +4.00000 q^{74} -1.00000 q^{76} -3.00000 q^{77} +4.00000 q^{78} +3.00000 q^{79} +1.00000 q^{81} -10.0000 q^{82} -2.00000 q^{83} -1.00000 q^{84} -5.00000 q^{86} -2.00000 q^{87} -3.00000 q^{88} +1.00000 q^{89} -4.00000 q^{91} +5.00000 q^{92} +1.00000 q^{93} -8.00000 q^{94} -1.00000 q^{96} -10.0000 q^{97} -6.00000 q^{98} -3.00000 q^{99} +O(q^{100})\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) 1.00000 0.377964 0.188982 0.981981i \(-0.439481\pi\)
0.188982 + 0.981981i \(0.439481\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −3.00000 −0.904534 −0.452267 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) −1.00000 −0.288675
\(13\) −4.00000 −1.10940 −0.554700 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 1.00000 0.235702
\(19\) −1.00000 −0.229416 −0.114708 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) −3.00000 −0.639602
\(23\) 5.00000 1.04257 0.521286 0.853382i \(-0.325452\pi\)
0.521286 + 0.853382i \(0.325452\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) −4.00000 −0.784465
\(27\) −1.00000 −0.192450
\(28\) 1.00000 0.188982
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605
\(32\) 1.00000 0.176777
\(33\) 3.00000 0.522233
\(34\) 0 0
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 4.00000 0.657596 0.328798 0.944400i \(-0.393356\pi\)
0.328798 + 0.944400i \(0.393356\pi\)
\(38\) −1.00000 −0.162221
\(39\) 4.00000 0.640513
\(40\) 0 0
\(41\) −10.0000 −1.56174 −0.780869 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) −1.00000 −0.154303
\(43\) −5.00000 −0.762493 −0.381246 0.924473i \(-0.624505\pi\)
−0.381246 + 0.924473i \(0.624505\pi\)
\(44\) −3.00000 −0.452267
\(45\) 0 0
\(46\) 5.00000 0.737210
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) −1.00000 −0.144338
\(49\) −6.00000 −0.857143
\(50\) 0 0
\(51\) 0 0
\(52\) −4.00000 −0.554700
\(53\) 5.00000 0.686803 0.343401 0.939189i \(-0.388421\pi\)
0.343401 + 0.939189i \(0.388421\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 1.00000 0.132453
\(58\) 2.00000 0.262613
\(59\) −6.00000 −0.781133 −0.390567 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) −1.00000 −0.127000
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 3.00000 0.369274
\(67\) 2.00000 0.244339 0.122169 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) 0 0
\(69\) −5.00000 −0.601929
\(70\) 0 0
\(71\) −5.00000 −0.593391 −0.296695 0.954972i \(-0.595885\pi\)
−0.296695 + 0.954972i \(0.595885\pi\)
\(72\) 1.00000 0.117851
\(73\) −7.00000 −0.819288 −0.409644 0.912245i \(-0.634347\pi\)
−0.409644 + 0.912245i \(0.634347\pi\)
\(74\) 4.00000 0.464991
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) −3.00000 −0.341882
\(78\) 4.00000 0.452911
\(79\) 3.00000 0.337526 0.168763 0.985657i \(-0.446023\pi\)
0.168763 + 0.985657i \(0.446023\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −10.0000 −1.10432
\(83\) −2.00000 −0.219529 −0.109764 0.993958i \(-0.535010\pi\)
−0.109764 + 0.993958i \(0.535010\pi\)
\(84\) −1.00000 −0.109109
\(85\) 0 0
\(86\) −5.00000 −0.539164
\(87\) −2.00000 −0.214423
\(88\) −3.00000 −0.319801
\(89\) 1.00000 0.106000 0.0529999 0.998595i \(-0.483122\pi\)
0.0529999 + 0.998595i \(0.483122\pi\)
\(90\) 0 0
\(91\) −4.00000 −0.419314
\(92\) 5.00000 0.521286
\(93\) 1.00000 0.103695
\(94\) −8.00000 −0.825137
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) −6.00000 −0.606092
\(99\) −3.00000 −0.301511
\(100\) 0 0
\(101\) −15.0000 −1.49256 −0.746278 0.665635i \(-0.768161\pi\)
−0.746278 + 0.665635i \(0.768161\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) −4.00000 −0.392232
\(105\) 0 0
\(106\) 5.00000 0.485643
\(107\) 19.0000 1.83680 0.918400 0.395654i \(-0.129482\pi\)
0.918400 + 0.395654i \(0.129482\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) −4.00000 −0.379663
\(112\) 1.00000 0.0944911
\(113\) −15.0000 −1.41108 −0.705541 0.708669i \(-0.749296\pi\)
−0.705541 + 0.708669i \(0.749296\pi\)
\(114\) 1.00000 0.0936586
\(115\) 0 0
\(116\) 2.00000 0.185695
\(117\) −4.00000 −0.369800
\(118\) −6.00000 −0.552345
\(119\) 0 0
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) −2.00000 −0.181071
\(123\) 10.0000 0.901670
\(124\) −1.00000 −0.0898027
\(125\) 0 0
\(126\) 1.00000 0.0890871
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) 1.00000 0.0883883
\(129\) 5.00000 0.440225
\(130\) 0 0
\(131\) −2.00000 −0.174741 −0.0873704 0.996176i \(-0.527846\pi\)
−0.0873704 + 0.996176i \(0.527846\pi\)
\(132\) 3.00000 0.261116
\(133\) −1.00000 −0.0867110
\(134\) 2.00000 0.172774
\(135\) 0 0
\(136\) 0 0
\(137\) 2.00000 0.170872 0.0854358 0.996344i \(-0.472772\pi\)
0.0854358 + 0.996344i \(0.472772\pi\)
\(138\) −5.00000 −0.425628
\(139\) −20.0000 −1.69638 −0.848189 0.529694i \(-0.822307\pi\)
−0.848189 + 0.529694i \(0.822307\pi\)
\(140\) 0 0
\(141\) 8.00000 0.673722
\(142\) −5.00000 −0.419591
\(143\) 12.0000 1.00349
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −7.00000 −0.579324
\(147\) 6.00000 0.494872
\(148\) 4.00000 0.328798
\(149\) 15.0000 1.22885 0.614424 0.788976i \(-0.289388\pi\)
0.614424 + 0.788976i \(0.289388\pi\)
\(150\) 0 0
\(151\) −16.0000 −1.30206 −0.651031 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) −1.00000 −0.0811107
\(153\) 0 0
\(154\) −3.00000 −0.241747
\(155\) 0 0
\(156\) 4.00000 0.320256
\(157\) 9.00000 0.718278 0.359139 0.933284i \(-0.383070\pi\)
0.359139 + 0.933284i \(0.383070\pi\)
\(158\) 3.00000 0.238667
\(159\) −5.00000 −0.396526
\(160\) 0 0
\(161\) 5.00000 0.394055
\(162\) 1.00000 0.0785674
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) −10.0000 −0.780869
\(165\) 0 0
\(166\) −2.00000 −0.155230
\(167\) 21.0000 1.62503 0.812514 0.582941i \(-0.198098\pi\)
0.812514 + 0.582941i \(0.198098\pi\)
\(168\) −1.00000 −0.0771517
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) −5.00000 −0.381246
\(173\) −2.00000 −0.152057 −0.0760286 0.997106i \(-0.524224\pi\)
−0.0760286 + 0.997106i \(0.524224\pi\)
\(174\) −2.00000 −0.151620
\(175\) 0 0
\(176\) −3.00000 −0.226134
\(177\) 6.00000 0.450988
\(178\) 1.00000 0.0749532
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) −5.00000 −0.371647 −0.185824 0.982583i \(-0.559495\pi\)
−0.185824 + 0.982583i \(0.559495\pi\)
\(182\) −4.00000 −0.296500
\(183\) 2.00000 0.147844
\(184\) 5.00000 0.368605
\(185\) 0 0
\(186\) 1.00000 0.0733236
\(187\) 0 0
\(188\) −8.00000 −0.583460
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −16.0000 −1.15772 −0.578860 0.815427i \(-0.696502\pi\)
−0.578860 + 0.815427i \(0.696502\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −10.0000 −0.719816 −0.359908 0.932988i \(-0.617192\pi\)
−0.359908 + 0.932988i \(0.617192\pi\)
\(194\) −10.0000 −0.717958
\(195\) 0 0
\(196\) −6.00000 −0.428571
\(197\) 18.0000 1.28245 0.641223 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) −3.00000 −0.213201
\(199\) −21.0000 −1.48865 −0.744325 0.667817i \(-0.767229\pi\)
−0.744325 + 0.667817i \(0.767229\pi\)
\(200\) 0 0
\(201\) −2.00000 −0.141069
\(202\) −15.0000 −1.05540
\(203\) 2.00000 0.140372
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 5.00000 0.347524
\(208\) −4.00000 −0.277350
\(209\) 3.00000 0.207514
\(210\) 0 0
\(211\) 7.00000 0.481900 0.240950 0.970538i \(-0.422541\pi\)
0.240950 + 0.970538i \(0.422541\pi\)
\(212\) 5.00000 0.343401
\(213\) 5.00000 0.342594
\(214\) 19.0000 1.29881
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) −1.00000 −0.0678844
\(218\) 2.00000 0.135457
\(219\) 7.00000 0.473016
\(220\) 0 0
\(221\) 0 0
\(222\) −4.00000 −0.268462
\(223\) −20.0000 −1.33930 −0.669650 0.742677i \(-0.733556\pi\)
−0.669650 + 0.742677i \(0.733556\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) −15.0000 −0.997785
\(227\) −9.00000 −0.597351 −0.298675 0.954355i \(-0.596545\pi\)
−0.298675 + 0.954355i \(0.596545\pi\)
\(228\) 1.00000 0.0662266
\(229\) 15.0000 0.991228 0.495614 0.868543i \(-0.334943\pi\)
0.495614 + 0.868543i \(0.334943\pi\)
\(230\) 0 0
\(231\) 3.00000 0.197386
\(232\) 2.00000 0.131306
\(233\) −21.0000 −1.37576 −0.687878 0.725826i \(-0.741458\pi\)
−0.687878 + 0.725826i \(0.741458\pi\)
\(234\) −4.00000 −0.261488
\(235\) 0 0
\(236\) −6.00000 −0.390567
\(237\) −3.00000 −0.194871
\(238\) 0 0
\(239\) −14.0000 −0.905585 −0.452792 0.891616i \(-0.649572\pi\)
−0.452792 + 0.891616i \(0.649572\pi\)
\(240\) 0 0
\(241\) −12.0000 −0.772988 −0.386494 0.922292i \(-0.626314\pi\)
−0.386494 + 0.922292i \(0.626314\pi\)
\(242\) −2.00000 −0.128565
\(243\) −1.00000 −0.0641500
\(244\) −2.00000 −0.128037
\(245\) 0 0
\(246\) 10.0000 0.637577
\(247\) 4.00000 0.254514
\(248\) −1.00000 −0.0635001
\(249\) 2.00000 0.126745
\(250\) 0 0
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 1.00000 0.0629941
\(253\) −15.0000 −0.943042
\(254\) 8.00000 0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 5.00000 0.311891 0.155946 0.987766i \(-0.450158\pi\)
0.155946 + 0.987766i \(0.450158\pi\)
\(258\) 5.00000 0.311286
\(259\) 4.00000 0.248548
\(260\) 0 0
\(261\) 2.00000 0.123797
\(262\) −2.00000 −0.123560
\(263\) 24.0000 1.47990 0.739952 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) 3.00000 0.184637
\(265\) 0 0
\(266\) −1.00000 −0.0613139
\(267\) −1.00000 −0.0611990
\(268\) 2.00000 0.122169
\(269\) 2.00000 0.121942 0.0609711 0.998140i \(-0.480580\pi\)
0.0609711 + 0.998140i \(0.480580\pi\)
\(270\) 0 0
\(271\) 9.00000 0.546711 0.273356 0.961913i \(-0.411866\pi\)
0.273356 + 0.961913i \(0.411866\pi\)
\(272\) 0 0
\(273\) 4.00000 0.242091
\(274\) 2.00000 0.120824
\(275\) 0 0
\(276\) −5.00000 −0.300965
\(277\) 2.00000 0.120168 0.0600842 0.998193i \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) −20.0000 −1.19952
\(279\) −1.00000 −0.0598684
\(280\) 0 0
\(281\) −32.0000 −1.90896 −0.954480 0.298275i \(-0.903589\pi\)
−0.954480 + 0.298275i \(0.903589\pi\)
\(282\) 8.00000 0.476393
\(283\) 14.0000 0.832214 0.416107 0.909316i \(-0.363394\pi\)
0.416107 + 0.909316i \(0.363394\pi\)
\(284\) −5.00000 −0.296695
\(285\) 0 0
\(286\) 12.0000 0.709575
\(287\) −10.0000 −0.590281
\(288\) 1.00000 0.0589256
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) 10.0000 0.586210
\(292\) −7.00000 −0.409644
\(293\) −18.0000 −1.05157 −0.525786 0.850617i \(-0.676229\pi\)
−0.525786 + 0.850617i \(0.676229\pi\)
\(294\) 6.00000 0.349927
\(295\) 0 0
\(296\) 4.00000 0.232495
\(297\) 3.00000 0.174078
\(298\) 15.0000 0.868927
\(299\) −20.0000 −1.15663
\(300\) 0 0
\(301\) −5.00000 −0.288195
\(302\) −16.0000 −0.920697
\(303\) 15.0000 0.861727
\(304\) −1.00000 −0.0573539
\(305\) 0 0
\(306\) 0 0
\(307\) −4.00000 −0.228292 −0.114146 0.993464i \(-0.536413\pi\)
−0.114146 + 0.993464i \(0.536413\pi\)
\(308\) −3.00000 −0.170941
\(309\) 0 0
\(310\) 0 0
\(311\) 16.0000 0.907277 0.453638 0.891186i \(-0.350126\pi\)
0.453638 + 0.891186i \(0.350126\pi\)
\(312\) 4.00000 0.226455
\(313\) −6.00000 −0.339140 −0.169570 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) 9.00000 0.507899
\(315\) 0 0
\(316\) 3.00000 0.168763
\(317\) 18.0000 1.01098 0.505490 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) −5.00000 −0.280386
\(319\) −6.00000 −0.335936
\(320\) 0 0
\(321\) −19.0000 −1.06048
\(322\) 5.00000 0.278639
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −4.00000 −0.221540
\(327\) −2.00000 −0.110600
\(328\) −10.0000 −0.552158
\(329\) −8.00000 −0.441054
\(330\) 0 0
\(331\) 8.00000 0.439720 0.219860 0.975531i \(-0.429440\pi\)
0.219860 + 0.975531i \(0.429440\pi\)
\(332\) −2.00000 −0.109764
\(333\) 4.00000 0.219199
\(334\) 21.0000 1.14907
\(335\) 0 0
\(336\) −1.00000 −0.0545545
\(337\) −34.0000 −1.85210 −0.926049 0.377403i \(-0.876817\pi\)
−0.926049 + 0.377403i \(0.876817\pi\)
\(338\) 3.00000 0.163178
\(339\) 15.0000 0.814688
\(340\) 0 0
\(341\) 3.00000 0.162459
\(342\) −1.00000 −0.0540738
\(343\) −13.0000 −0.701934
\(344\) −5.00000 −0.269582
\(345\) 0 0
\(346\) −2.00000 −0.107521
\(347\) −22.0000 −1.18102 −0.590511 0.807030i \(-0.701074\pi\)
−0.590511 + 0.807030i \(0.701074\pi\)
\(348\) −2.00000 −0.107211
\(349\) 20.0000 1.07058 0.535288 0.844670i \(-0.320203\pi\)
0.535288 + 0.844670i \(0.320203\pi\)
\(350\) 0 0
\(351\) 4.00000 0.213504
\(352\) −3.00000 −0.159901
\(353\) 24.0000 1.27739 0.638696 0.769460i \(-0.279474\pi\)
0.638696 + 0.769460i \(0.279474\pi\)
\(354\) 6.00000 0.318896
\(355\) 0 0
\(356\) 1.00000 0.0529999
\(357\) 0 0
\(358\) 0 0
\(359\) 3.00000 0.158334 0.0791670 0.996861i \(-0.474774\pi\)
0.0791670 + 0.996861i \(0.474774\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) −5.00000 −0.262794
\(363\) 2.00000 0.104973
\(364\) −4.00000 −0.209657
\(365\) 0 0
\(366\) 2.00000 0.104542
\(367\) 14.0000 0.730794 0.365397 0.930852i \(-0.380933\pi\)
0.365397 + 0.930852i \(0.380933\pi\)
\(368\) 5.00000 0.260643
\(369\) −10.0000 −0.520579
\(370\) 0 0
\(371\) 5.00000 0.259587
\(372\) 1.00000 0.0518476
\(373\) −11.0000 −0.569558 −0.284779 0.958593i \(-0.591920\pi\)
−0.284779 + 0.958593i \(0.591920\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −8.00000 −0.412568
\(377\) −8.00000 −0.412021
\(378\) −1.00000 −0.0514344
\(379\) −15.0000 −0.770498 −0.385249 0.922813i \(-0.625884\pi\)
−0.385249 + 0.922813i \(0.625884\pi\)
\(380\) 0 0
\(381\) −8.00000 −0.409852
\(382\) −16.0000 −0.818631
\(383\) 28.0000 1.43073 0.715367 0.698749i \(-0.246260\pi\)
0.715367 + 0.698749i \(0.246260\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −10.0000 −0.508987
\(387\) −5.00000 −0.254164
\(388\) −10.0000 −0.507673
\(389\) −28.0000 −1.41966 −0.709828 0.704375i \(-0.751227\pi\)
−0.709828 + 0.704375i \(0.751227\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −6.00000 −0.303046
\(393\) 2.00000 0.100887
\(394\) 18.0000 0.906827
\(395\) 0 0
\(396\) −3.00000 −0.150756
\(397\) −31.0000 −1.55585 −0.777923 0.628360i \(-0.783727\pi\)
−0.777923 + 0.628360i \(0.783727\pi\)
\(398\) −21.0000 −1.05263
\(399\) 1.00000 0.0500626
\(400\) 0 0
\(401\) 5.00000 0.249688 0.124844 0.992176i \(-0.460157\pi\)
0.124844 + 0.992176i \(0.460157\pi\)
\(402\) −2.00000 −0.0997509
\(403\) 4.00000 0.199254
\(404\) −15.0000 −0.746278
\(405\) 0 0
\(406\) 2.00000 0.0992583
\(407\) −12.0000 −0.594818
\(408\) 0 0
\(409\) 4.00000 0.197787 0.0988936 0.995098i \(-0.468470\pi\)
0.0988936 + 0.995098i \(0.468470\pi\)
\(410\) 0 0
\(411\) −2.00000 −0.0986527
\(412\) 0 0
\(413\) −6.00000 −0.295241
\(414\) 5.00000 0.245737
\(415\) 0 0
\(416\) −4.00000 −0.196116
\(417\) 20.0000 0.979404
\(418\) 3.00000 0.146735
\(419\) −18.0000 −0.879358 −0.439679 0.898155i \(-0.644908\pi\)
−0.439679 + 0.898155i \(0.644908\pi\)
\(420\) 0 0
\(421\) −14.0000 −0.682318 −0.341159 0.940006i \(-0.610819\pi\)
−0.341159 + 0.940006i \(0.610819\pi\)
\(422\) 7.00000 0.340755
\(423\) −8.00000 −0.388973
\(424\) 5.00000 0.242821
\(425\) 0 0
\(426\) 5.00000 0.242251
\(427\) −2.00000 −0.0967868
\(428\) 19.0000 0.918400
\(429\) −12.0000 −0.579365
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 19.0000 0.913082 0.456541 0.889702i \(-0.349088\pi\)
0.456541 + 0.889702i \(0.349088\pi\)
\(434\) −1.00000 −0.0480015
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) −5.00000 −0.239182
\(438\) 7.00000 0.334473
\(439\) 20.0000 0.954548 0.477274 0.878755i \(-0.341625\pi\)
0.477274 + 0.878755i \(0.341625\pi\)
\(440\) 0 0
\(441\) −6.00000 −0.285714
\(442\) 0 0
\(443\) 29.0000 1.37783 0.688916 0.724841i \(-0.258087\pi\)
0.688916 + 0.724841i \(0.258087\pi\)
\(444\) −4.00000 −0.189832
\(445\) 0 0
\(446\) −20.0000 −0.947027
\(447\) −15.0000 −0.709476
\(448\) 1.00000 0.0472456
\(449\) 2.00000 0.0943858 0.0471929 0.998886i \(-0.484972\pi\)
0.0471929 + 0.998886i \(0.484972\pi\)
\(450\) 0 0
\(451\) 30.0000 1.41264
\(452\) −15.0000 −0.705541
\(453\) 16.0000 0.751746
\(454\) −9.00000 −0.422391
\(455\) 0 0
\(456\) 1.00000 0.0468293
\(457\) 30.0000 1.40334 0.701670 0.712502i \(-0.252438\pi\)
0.701670 + 0.712502i \(0.252438\pi\)
\(458\) 15.0000 0.700904
\(459\) 0 0
\(460\) 0 0
\(461\) 16.0000 0.745194 0.372597 0.927993i \(-0.378467\pi\)
0.372597 + 0.927993i \(0.378467\pi\)
\(462\) 3.00000 0.139573
\(463\) −14.0000 −0.650635 −0.325318 0.945605i \(-0.605471\pi\)
−0.325318 + 0.945605i \(0.605471\pi\)
\(464\) 2.00000 0.0928477
\(465\) 0 0
\(466\) −21.0000 −0.972806
\(467\) 28.0000 1.29569 0.647843 0.761774i \(-0.275671\pi\)
0.647843 + 0.761774i \(0.275671\pi\)
\(468\) −4.00000 −0.184900
\(469\) 2.00000 0.0923514
\(470\) 0 0
\(471\) −9.00000 −0.414698
\(472\) −6.00000 −0.276172
\(473\) 15.0000 0.689701
\(474\) −3.00000 −0.137795
\(475\) 0 0
\(476\) 0 0
\(477\) 5.00000 0.228934
\(478\) −14.0000 −0.640345
\(479\) −25.0000 −1.14228 −0.571140 0.820853i \(-0.693499\pi\)
−0.571140 + 0.820853i \(0.693499\pi\)
\(480\) 0 0
\(481\) −16.0000 −0.729537
\(482\) −12.0000 −0.546585
\(483\) −5.00000 −0.227508
\(484\) −2.00000 −0.0909091
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) −4.00000 −0.181257 −0.0906287 0.995885i \(-0.528888\pi\)
−0.0906287 + 0.995885i \(0.528888\pi\)
\(488\) −2.00000 −0.0905357
\(489\) 4.00000 0.180886
\(490\) 0 0
\(491\) 33.0000 1.48927 0.744635 0.667472i \(-0.232624\pi\)
0.744635 + 0.667472i \(0.232624\pi\)
\(492\) 10.0000 0.450835
\(493\) 0 0
\(494\) 4.00000 0.179969
\(495\) 0 0
\(496\) −1.00000 −0.0449013
\(497\) −5.00000 −0.224281
\(498\) 2.00000 0.0896221
\(499\) 22.0000 0.984855 0.492428 0.870353i \(-0.336110\pi\)
0.492428 + 0.870353i \(0.336110\pi\)
\(500\) 0 0
\(501\) −21.0000 −0.938211
\(502\) 12.0000 0.535586
\(503\) 14.0000 0.624229 0.312115 0.950044i \(-0.398963\pi\)
0.312115 + 0.950044i \(0.398963\pi\)
\(504\) 1.00000 0.0445435
\(505\) 0 0
\(506\) −15.0000 −0.666831
\(507\) −3.00000 −0.133235
\(508\) 8.00000 0.354943
\(509\) 32.0000 1.41838 0.709188 0.705020i \(-0.249062\pi\)
0.709188 + 0.705020i \(0.249062\pi\)
\(510\) 0 0
\(511\) −7.00000 −0.309662
\(512\) 1.00000 0.0441942
\(513\) 1.00000 0.0441511
\(514\) 5.00000 0.220541
\(515\) 0 0
\(516\) 5.00000 0.220113
\(517\) 24.0000 1.05552
\(518\) 4.00000 0.175750
\(519\) 2.00000 0.0877903
\(520\) 0 0
\(521\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) 2.00000 0.0875376
\(523\) 19.0000 0.830812 0.415406 0.909636i \(-0.363640\pi\)
0.415406 + 0.909636i \(0.363640\pi\)
\(524\) −2.00000 −0.0873704
\(525\) 0 0
\(526\) 24.0000 1.04645
\(527\) 0 0
\(528\) 3.00000 0.130558
\(529\) 2.00000 0.0869565
\(530\) 0 0
\(531\) −6.00000 −0.260378
\(532\) −1.00000 −0.0433555
\(533\) 40.0000 1.73259
\(534\) −1.00000 −0.0432742
\(535\) 0 0
\(536\) 2.00000 0.0863868
\(537\) 0 0
\(538\) 2.00000 0.0862261
\(539\) 18.0000 0.775315
\(540\) 0 0
\(541\) 24.0000 1.03184 0.515920 0.856637i \(-0.327450\pi\)
0.515920 + 0.856637i \(0.327450\pi\)
\(542\) 9.00000 0.386583
\(543\) 5.00000 0.214571
\(544\) 0 0
\(545\) 0 0
\(546\) 4.00000 0.171184
\(547\) 28.0000 1.19719 0.598597 0.801050i \(-0.295725\pi\)
0.598597 + 0.801050i \(0.295725\pi\)
\(548\) 2.00000 0.0854358
\(549\) −2.00000 −0.0853579
\(550\) 0 0
\(551\) −2.00000 −0.0852029
\(552\) −5.00000 −0.212814
\(553\) 3.00000 0.127573
\(554\) 2.00000 0.0849719
\(555\) 0 0
\(556\) −20.0000 −0.848189
\(557\) 9.00000 0.381342 0.190671 0.981654i \(-0.438934\pi\)
0.190671 + 0.981654i \(0.438934\pi\)
\(558\) −1.00000 −0.0423334
\(559\) 20.0000 0.845910
\(560\) 0 0
\(561\) 0 0
\(562\) −32.0000 −1.34984
\(563\) 36.0000 1.51722 0.758610 0.651546i \(-0.225879\pi\)
0.758610 + 0.651546i \(0.225879\pi\)
\(564\) 8.00000 0.336861
\(565\) 0 0
\(566\) 14.0000 0.588464
\(567\) 1.00000 0.0419961
\(568\) −5.00000 −0.209795
\(569\) −39.0000 −1.63497 −0.817483 0.575953i \(-0.804631\pi\)
−0.817483 + 0.575953i \(0.804631\pi\)
\(570\) 0 0
\(571\) 26.0000 1.08807 0.544033 0.839064i \(-0.316897\pi\)
0.544033 + 0.839064i \(0.316897\pi\)
\(572\) 12.0000 0.501745
\(573\) 16.0000 0.668410
\(574\) −10.0000 −0.417392
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −34.0000 −1.41544 −0.707719 0.706494i \(-0.750276\pi\)
−0.707719 + 0.706494i \(0.750276\pi\)
\(578\) −17.0000 −0.707107
\(579\) 10.0000 0.415586
\(580\) 0 0
\(581\) −2.00000 −0.0829740
\(582\) 10.0000 0.414513
\(583\) −15.0000 −0.621237
\(584\) −7.00000 −0.289662
\(585\) 0 0
\(586\) −18.0000 −0.743573
\(587\) 6.00000 0.247647 0.123823 0.992304i \(-0.460484\pi\)
0.123823 + 0.992304i \(0.460484\pi\)
\(588\) 6.00000 0.247436
\(589\) 1.00000 0.0412043
\(590\) 0 0
\(591\) −18.0000 −0.740421
\(592\) 4.00000 0.164399
\(593\) 6.00000 0.246390 0.123195 0.992382i \(-0.460686\pi\)
0.123195 + 0.992382i \(0.460686\pi\)
\(594\) 3.00000 0.123091
\(595\) 0 0
\(596\) 15.0000 0.614424
\(597\) 21.0000 0.859473
\(598\) −20.0000 −0.817861
\(599\) 13.0000 0.531166 0.265583 0.964088i \(-0.414436\pi\)
0.265583 + 0.964088i \(0.414436\pi\)
\(600\) 0 0
\(601\) −44.0000 −1.79480 −0.897399 0.441221i \(-0.854546\pi\)
−0.897399 + 0.441221i \(0.854546\pi\)
\(602\) −5.00000 −0.203785
\(603\) 2.00000 0.0814463
\(604\) −16.0000 −0.651031
\(605\) 0 0
\(606\) 15.0000 0.609333
\(607\) −27.0000 −1.09590 −0.547948 0.836512i \(-0.684591\pi\)
−0.547948 + 0.836512i \(0.684591\pi\)
\(608\) −1.00000 −0.0405554
\(609\) −2.00000 −0.0810441
\(610\) 0 0
\(611\) 32.0000 1.29458
\(612\) 0 0
\(613\) 24.0000 0.969351 0.484675 0.874694i \(-0.338938\pi\)
0.484675 + 0.874694i \(0.338938\pi\)
\(614\) −4.00000 −0.161427
\(615\) 0 0
\(616\) −3.00000 −0.120873
\(617\) 5.00000 0.201292 0.100646 0.994922i \(-0.467909\pi\)
0.100646 + 0.994922i \(0.467909\pi\)
\(618\) 0 0
\(619\) −12.0000 −0.482321 −0.241160 0.970485i \(-0.577528\pi\)
−0.241160 + 0.970485i \(0.577528\pi\)
\(620\) 0 0
\(621\) −5.00000 −0.200643
\(622\) 16.0000 0.641542
\(623\) 1.00000 0.0400642
\(624\) 4.00000 0.160128
\(625\) 0 0
\(626\) −6.00000 −0.239808
\(627\) −3.00000 −0.119808
\(628\) 9.00000 0.359139
\(629\) 0 0
\(630\) 0 0
\(631\) 25.0000 0.995234 0.497617 0.867397i \(-0.334208\pi\)
0.497617 + 0.867397i \(0.334208\pi\)
\(632\) 3.00000 0.119334
\(633\) −7.00000 −0.278225
\(634\) 18.0000 0.714871
\(635\) 0 0
\(636\) −5.00000 −0.198263
\(637\) 24.0000 0.950915
\(638\) −6.00000 −0.237542
\(639\) −5.00000 −0.197797
\(640\) 0 0
\(641\) 50.0000 1.97488 0.987441 0.157991i \(-0.0505015\pi\)
0.987441 + 0.157991i \(0.0505015\pi\)
\(642\) −19.0000 −0.749870
\(643\) −7.00000 −0.276053 −0.138027 0.990429i \(-0.544076\pi\)
−0.138027 + 0.990429i \(0.544076\pi\)
\(644\) 5.00000 0.197028
\(645\) 0 0
\(646\) 0 0
\(647\) 7.00000 0.275198 0.137599 0.990488i \(-0.456061\pi\)
0.137599 + 0.990488i \(0.456061\pi\)
\(648\) 1.00000 0.0392837
\(649\) 18.0000 0.706562
\(650\) 0 0
\(651\) 1.00000 0.0391931
\(652\) −4.00000 −0.156652
\(653\) −42.0000 −1.64359 −0.821794 0.569785i \(-0.807026\pi\)
−0.821794 + 0.569785i \(0.807026\pi\)
\(654\) −2.00000 −0.0782062
\(655\) 0 0
\(656\) −10.0000 −0.390434
\(657\) −7.00000 −0.273096
\(658\) −8.00000 −0.311872
\(659\) 6.00000 0.233727 0.116863 0.993148i \(-0.462716\pi\)
0.116863 + 0.993148i \(0.462716\pi\)
\(660\) 0 0
\(661\) 36.0000 1.40024 0.700119 0.714026i \(-0.253130\pi\)
0.700119 + 0.714026i \(0.253130\pi\)
\(662\) 8.00000 0.310929
\(663\) 0 0
\(664\) −2.00000 −0.0776151
\(665\) 0 0
\(666\) 4.00000 0.154997
\(667\) 10.0000 0.387202
\(668\) 21.0000 0.812514
\(669\) 20.0000 0.773245
\(670\) 0 0
\(671\) 6.00000 0.231627
\(672\) −1.00000 −0.0385758
\(673\) −30.0000 −1.15642 −0.578208 0.815890i \(-0.696248\pi\)
−0.578208 + 0.815890i \(0.696248\pi\)
\(674\) −34.0000 −1.30963
\(675\) 0 0
\(676\) 3.00000 0.115385
\(677\) 13.0000 0.499631 0.249815 0.968294i \(-0.419630\pi\)
0.249815 + 0.968294i \(0.419630\pi\)
\(678\) 15.0000 0.576072
\(679\) −10.0000 −0.383765
\(680\) 0 0
\(681\) 9.00000 0.344881
\(682\) 3.00000 0.114876
\(683\) 35.0000 1.33924 0.669619 0.742705i \(-0.266457\pi\)
0.669619 + 0.742705i \(0.266457\pi\)
\(684\) −1.00000 −0.0382360
\(685\) 0 0
\(686\) −13.0000 −0.496342
\(687\) −15.0000 −0.572286
\(688\) −5.00000 −0.190623
\(689\) −20.0000 −0.761939
\(690\) 0 0
\(691\) −41.0000 −1.55971 −0.779857 0.625958i \(-0.784708\pi\)
−0.779857 + 0.625958i \(0.784708\pi\)
\(692\) −2.00000 −0.0760286
\(693\) −3.00000 −0.113961
\(694\) −22.0000 −0.835109
\(695\) 0 0
\(696\) −2.00000 −0.0758098
\(697\) 0 0
\(698\) 20.0000 0.757011
\(699\) 21.0000 0.794293
\(700\) 0 0
\(701\) −35.0000 −1.32193 −0.660966 0.750416i \(-0.729853\pi\)
−0.660966 + 0.750416i \(0.729853\pi\)
\(702\) 4.00000 0.150970
\(703\) −4.00000 −0.150863
\(704\) −3.00000 −0.113067
\(705\) 0 0
\(706\) 24.0000 0.903252
\(707\) −15.0000 −0.564133
\(708\) 6.00000 0.225494
\(709\) 5.00000 0.187779 0.0938895 0.995583i \(-0.470070\pi\)
0.0938895 + 0.995583i \(0.470070\pi\)
\(710\) 0 0
\(711\) 3.00000 0.112509
\(712\) 1.00000 0.0374766
\(713\) −5.00000 −0.187251
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 14.0000 0.522840
\(718\) 3.00000 0.111959
\(719\) 10.0000 0.372937 0.186469 0.982461i \(-0.440296\pi\)
0.186469 + 0.982461i \(0.440296\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −18.0000 −0.669891
\(723\) 12.0000 0.446285
\(724\) −5.00000 −0.185824
\(725\) 0 0
\(726\) 2.00000 0.0742270
\(727\) 49.0000 1.81731 0.908655 0.417548i \(-0.137111\pi\)
0.908655 + 0.417548i \(0.137111\pi\)
\(728\) −4.00000 −0.148250
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) 2.00000 0.0739221
\(733\) −22.0000 −0.812589 −0.406294 0.913742i \(-0.633179\pi\)
−0.406294 + 0.913742i \(0.633179\pi\)
\(734\) 14.0000 0.516749
\(735\) 0 0
\(736\) 5.00000 0.184302
\(737\) −6.00000 −0.221013
\(738\) −10.0000 −0.368105
\(739\) −18.0000 −0.662141 −0.331070 0.943606i \(-0.607410\pi\)
−0.331070 + 0.943606i \(0.607410\pi\)
\(740\) 0 0
\(741\) −4.00000 −0.146944
\(742\) 5.00000 0.183556
\(743\) 15.0000 0.550297 0.275148 0.961402i \(-0.411273\pi\)
0.275148 + 0.961402i \(0.411273\pi\)
\(744\) 1.00000 0.0366618
\(745\) 0 0
\(746\) −11.0000 −0.402739
\(747\) −2.00000 −0.0731762
\(748\) 0 0
\(749\) 19.0000 0.694245
\(750\) 0 0
\(751\) 22.0000 0.802791 0.401396 0.915905i \(-0.368525\pi\)
0.401396 + 0.915905i \(0.368525\pi\)
\(752\) −8.00000 −0.291730
\(753\) −12.0000 −0.437304
\(754\) −8.00000 −0.291343
\(755\) 0 0
\(756\) −1.00000 −0.0363696
\(757\) 20.0000 0.726912 0.363456 0.931611i \(-0.381597\pi\)
0.363456 + 0.931611i \(0.381597\pi\)
\(758\) −15.0000 −0.544825
\(759\) 15.0000 0.544466
\(760\) 0 0
\(761\) 21.0000 0.761249 0.380625 0.924730i \(-0.375709\pi\)
0.380625 + 0.924730i \(0.375709\pi\)
\(762\) −8.00000 −0.289809
\(763\) 2.00000 0.0724049
\(764\) −16.0000 −0.578860
\(765\) 0 0
\(766\) 28.0000 1.01168
\(767\) 24.0000 0.866590
\(768\) −1.00000 −0.0360844
\(769\) 45.0000 1.62274 0.811371 0.584532i \(-0.198722\pi\)
0.811371 + 0.584532i \(0.198722\pi\)
\(770\) 0 0
\(771\) −5.00000 −0.180071
\(772\) −10.0000 −0.359908
\(773\) −29.0000 −1.04306 −0.521529 0.853234i \(-0.674638\pi\)
−0.521529 + 0.853234i \(0.674638\pi\)
\(774\) −5.00000 −0.179721
\(775\) 0 0
\(776\) −10.0000 −0.358979
\(777\) −4.00000 −0.143499
\(778\) −28.0000 −1.00385
\(779\) 10.0000 0.358287
\(780\) 0 0
\(781\) 15.0000 0.536742
\(782\) 0 0
\(783\) −2.00000 −0.0714742
\(784\) −6.00000 −0.214286
\(785\) 0 0
\(786\) 2.00000 0.0713376
\(787\) 27.0000 0.962446 0.481223 0.876598i \(-0.340193\pi\)
0.481223 + 0.876598i \(0.340193\pi\)
\(788\) 18.0000 0.641223
\(789\) −24.0000 −0.854423
\(790\) 0 0
\(791\) −15.0000 −0.533339
\(792\) −3.00000 −0.106600
\(793\) 8.00000 0.284088
\(794\) −31.0000 −1.10015
\(795\) 0 0
\(796\) −21.0000 −0.744325
\(797\) −22.0000 −0.779280 −0.389640 0.920967i \(-0.627401\pi\)
−0.389640 + 0.920967i \(0.627401\pi\)
\(798\) 1.00000 0.0353996
\(799\) 0 0
\(800\) 0 0
\(801\) 1.00000 0.0353333
\(802\) 5.00000 0.176556
\(803\) 21.0000 0.741074
\(804\) −2.00000 −0.0705346
\(805\) 0 0
\(806\) 4.00000 0.140894
\(807\) −2.00000 −0.0704033
\(808\) −15.0000 −0.527698
\(809\) 9.00000 0.316423 0.158212 0.987405i \(-0.449427\pi\)
0.158212 + 0.987405i \(0.449427\pi\)
\(810\) 0 0
\(811\) 35.0000 1.22902 0.614508 0.788911i \(-0.289355\pi\)
0.614508 + 0.788911i \(0.289355\pi\)
\(812\) 2.00000 0.0701862
\(813\) −9.00000 −0.315644
\(814\) −12.0000 −0.420600
\(815\) 0 0
\(816\) 0 0
\(817\) 5.00000 0.174928
\(818\) 4.00000 0.139857
\(819\) −4.00000 −0.139771
\(820\) 0 0
\(821\) 18.0000 0.628204 0.314102 0.949389i \(-0.398297\pi\)
0.314102 + 0.949389i \(0.398297\pi\)
\(822\) −2.00000 −0.0697580
\(823\) −6.00000 −0.209147 −0.104573 0.994517i \(-0.533348\pi\)
−0.104573 + 0.994517i \(0.533348\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) −6.00000 −0.208767
\(827\) 14.0000 0.486828 0.243414 0.969923i \(-0.421733\pi\)
0.243414 + 0.969923i \(0.421733\pi\)
\(828\) 5.00000 0.173762
\(829\) −25.0000 −0.868286 −0.434143 0.900844i \(-0.642949\pi\)
−0.434143 + 0.900844i \(0.642949\pi\)
\(830\) 0 0
\(831\) −2.00000 −0.0693792
\(832\) −4.00000 −0.138675
\(833\) 0 0
\(834\) 20.0000 0.692543
\(835\) 0 0
\(836\) 3.00000 0.103757
\(837\) 1.00000 0.0345651
\(838\) −18.0000 −0.621800
\(839\) 39.0000 1.34643 0.673215 0.739447i \(-0.264913\pi\)
0.673215 + 0.739447i \(0.264913\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) −14.0000 −0.482472
\(843\) 32.0000 1.10214
\(844\) 7.00000 0.240950
\(845\) 0 0
\(846\) −8.00000 −0.275046
\(847\) −2.00000 −0.0687208
\(848\) 5.00000 0.171701
\(849\) −14.0000 −0.480479
\(850\) 0 0
\(851\) 20.0000 0.685591
\(852\) 5.00000 0.171297
\(853\) −55.0000 −1.88316 −0.941582 0.336784i \(-0.890661\pi\)
−0.941582 + 0.336784i \(0.890661\pi\)
\(854\) −2.00000 −0.0684386
\(855\) 0 0
\(856\) 19.0000 0.649407
\(857\) −22.0000 −0.751506 −0.375753 0.926720i \(-0.622616\pi\)
−0.375753 + 0.926720i \(0.622616\pi\)
\(858\) −12.0000 −0.409673
\(859\) −32.0000 −1.09183 −0.545913 0.837842i \(-0.683817\pi\)
−0.545913 + 0.837842i \(0.683817\pi\)
\(860\) 0 0
\(861\) 10.0000 0.340799
\(862\) 0 0
\(863\) −29.0000 −0.987171 −0.493586 0.869697i \(-0.664314\pi\)
−0.493586 + 0.869697i \(0.664314\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) 19.0000 0.645646
\(867\) 17.0000 0.577350
\(868\) −1.00000 −0.0339422
\(869\) −9.00000 −0.305304
\(870\) 0 0
\(871\) −8.00000 −0.271070
\(872\) 2.00000 0.0677285
\(873\) −10.0000 −0.338449
\(874\) −5.00000 −0.169128
\(875\) 0 0
\(876\) 7.00000 0.236508
\(877\) 42.0000 1.41824 0.709120 0.705088i \(-0.249093\pi\)
0.709120 + 0.705088i \(0.249093\pi\)
\(878\) 20.0000 0.674967
\(879\) 18.0000 0.607125
\(880\) 0 0
\(881\) 42.0000 1.41502 0.707508 0.706705i \(-0.249819\pi\)
0.707508 + 0.706705i \(0.249819\pi\)
\(882\) −6.00000 −0.202031
\(883\) −35.0000 −1.17784 −0.588922 0.808190i \(-0.700447\pi\)
−0.588922 + 0.808190i \(0.700447\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 29.0000 0.974274
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) −4.00000 −0.134231
\(889\) 8.00000 0.268311
\(890\) 0 0
\(891\) −3.00000 −0.100504
\(892\) −20.0000 −0.669650
\(893\) 8.00000 0.267710
\(894\) −15.0000 −0.501675
\(895\) 0 0
\(896\) 1.00000 0.0334077
\(897\) 20.0000 0.667781
\(898\) 2.00000 0.0667409
\(899\) −2.00000 −0.0667037
\(900\) 0 0
\(901\) 0 0
\(902\) 30.0000 0.998891
\(903\) 5.00000 0.166390
\(904\) −15.0000 −0.498893
\(905\) 0 0
\(906\) 16.0000 0.531564
\(907\) −36.0000 −1.19536 −0.597680 0.801735i \(-0.703911\pi\)
−0.597680 + 0.801735i \(0.703911\pi\)
\(908\) −9.00000 −0.298675
\(909\) −15.0000 −0.497519
\(910\) 0 0
\(911\) −46.0000 −1.52405 −0.762024 0.647549i \(-0.775794\pi\)
−0.762024 + 0.647549i \(0.775794\pi\)
\(912\) 1.00000 0.0331133
\(913\) 6.00000 0.198571
\(914\) 30.0000 0.992312
\(915\) 0 0
\(916\) 15.0000 0.495614
\(917\) −2.00000 −0.0660458
\(918\) 0 0
\(919\) 20.0000 0.659739 0.329870 0.944027i \(-0.392995\pi\)
0.329870 + 0.944027i \(0.392995\pi\)
\(920\) 0 0
\(921\) 4.00000 0.131804
\(922\) 16.0000 0.526932
\(923\) 20.0000 0.658308
\(924\) 3.00000 0.0986928
\(925\) 0 0
\(926\) −14.0000 −0.460069
\(927\) 0 0
\(928\) 2.00000 0.0656532
\(929\) −1.00000 −0.0328089 −0.0164045 0.999865i \(-0.505222\pi\)
−0.0164045 + 0.999865i \(0.505222\pi\)
\(930\) 0 0
\(931\) 6.00000 0.196642
\(932\) −21.0000 −0.687878
\(933\) −16.0000 −0.523816
\(934\) 28.0000 0.916188
\(935\) 0 0
\(936\) −4.00000 −0.130744
\(937\) −2.00000 −0.0653372 −0.0326686 0.999466i \(-0.510401\pi\)
−0.0326686 + 0.999466i \(0.510401\pi\)
\(938\) 2.00000 0.0653023
\(939\) 6.00000 0.195803
\(940\) 0 0
\(941\) 16.0000 0.521585 0.260793 0.965395i \(-0.416016\pi\)
0.260793 + 0.965395i \(0.416016\pi\)
\(942\) −9.00000 −0.293236
\(943\) −50.0000 −1.62822
\(944\) −6.00000 −0.195283
\(945\) 0 0
\(946\) 15.0000 0.487692
\(947\) 46.0000 1.49480 0.747400 0.664375i \(-0.231302\pi\)
0.747400 + 0.664375i \(0.231302\pi\)
\(948\) −3.00000 −0.0974355
\(949\) 28.0000 0.908918
\(950\) 0 0
\(951\) −18.0000 −0.583690
\(952\) 0 0
\(953\) −28.0000 −0.907009 −0.453504 0.891254i \(-0.649826\pi\)
−0.453504 + 0.891254i \(0.649826\pi\)
\(954\) 5.00000 0.161881
\(955\) 0 0
\(956\) −14.0000 −0.452792
\(957\) 6.00000 0.193952
\(958\) −25.0000 −0.807713
\(959\) 2.00000 0.0645834
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) −16.0000 −0.515861
\(963\) 19.0000 0.612266
\(964\) −12.0000 −0.386494
\(965\) 0 0
\(966\) −5.00000 −0.160872
\(967\) −8.00000 −0.257263 −0.128631 0.991692i \(-0.541058\pi\)
−0.128631 + 0.991692i \(0.541058\pi\)
\(968\) −2.00000 −0.0642824
\(969\) 0 0
\(970\) 0 0
\(971\) −18.0000 −0.577647 −0.288824 0.957382i \(-0.593264\pi\)
−0.288824 + 0.957382i \(0.593264\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −20.0000 −0.641171
\(974\) −4.00000 −0.128168
\(975\) 0 0
\(976\) −2.00000 −0.0640184
\(977\) 6.00000 0.191957 0.0959785 0.995383i \(-0.469402\pi\)
0.0959785 + 0.995383i \(0.469402\pi\)
\(978\) 4.00000 0.127906
\(979\) −3.00000 −0.0958804
\(980\) 0 0
\(981\) 2.00000 0.0638551
\(982\) 33.0000 1.05307
\(983\) −36.0000 −1.14822 −0.574111 0.818778i \(-0.694652\pi\)
−0.574111 + 0.818778i \(0.694652\pi\)
\(984\) 10.0000 0.318788
\(985\) 0 0
\(986\) 0 0
\(987\) 8.00000 0.254643
\(988\) 4.00000 0.127257
\(989\) −25.0000 −0.794954
\(990\) 0 0
\(991\) −53.0000 −1.68360 −0.841800 0.539789i \(-0.818504\pi\)
−0.841800 + 0.539789i \(0.818504\pi\)
\(992\) −1.00000 −0.0317500
\(993\) −8.00000 −0.253872
\(994\) −5.00000 −0.158590
\(995\) 0 0
\(996\) 2.00000 0.0633724
\(997\) 6.00000 0.190022 0.0950110 0.995476i \(-0.469711\pi\)
0.0950110 + 0.995476i \(0.469711\pi\)
\(998\) 22.0000 0.696398
\(999\) −4.00000 −0.126554
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 4650.2.a.bc.1.1 1
5.2 odd 4 930.2.d.a.559.2 yes 2
5.3 odd 4 930.2.d.a.559.1 2
5.4 even 2 4650.2.a.p.1.1 1
15.2 even 4 2790.2.d.f.559.1 2
15.8 even 4 2790.2.d.f.559.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.d.a.559.1 2 5.3 odd 4
930.2.d.a.559.2 yes 2 5.2 odd 4
2790.2.d.f.559.1 2 15.2 even 4
2790.2.d.f.559.2 2 15.8 even 4
4650.2.a.p.1.1 1 5.4 even 2
4650.2.a.bc.1.1 1 1.1 even 1 trivial