Properties

Label 1050.2.a.d.1.1
Level $1050$
Weight $2$
Character 1050.1
Self dual yes
Analytic conductor $8.384$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1050.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.38429221223\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 210)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1050.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{11} -1.00000 q^{12} -6.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} -4.00000 q^{17} -1.00000 q^{18} -6.00000 q^{19} -1.00000 q^{21} -2.00000 q^{22} +8.00000 q^{23} +1.00000 q^{24} +6.00000 q^{26} -1.00000 q^{27} +1.00000 q^{28} +6.00000 q^{29} -2.00000 q^{31} -1.00000 q^{32} -2.00000 q^{33} +4.00000 q^{34} +1.00000 q^{36} -4.00000 q^{37} +6.00000 q^{38} +6.00000 q^{39} +2.00000 q^{41} +1.00000 q^{42} -4.00000 q^{43} +2.00000 q^{44} -8.00000 q^{46} -8.00000 q^{47} -1.00000 q^{48} +1.00000 q^{49} +4.00000 q^{51} -6.00000 q^{52} -6.00000 q^{53} +1.00000 q^{54} -1.00000 q^{56} +6.00000 q^{57} -6.00000 q^{58} -8.00000 q^{59} -10.0000 q^{61} +2.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} +2.00000 q^{66} -8.00000 q^{67} -4.00000 q^{68} -8.00000 q^{69} -6.00000 q^{71} -1.00000 q^{72} +14.0000 q^{73} +4.00000 q^{74} -6.00000 q^{76} +2.00000 q^{77} -6.00000 q^{78} -12.0000 q^{79} +1.00000 q^{81} -2.00000 q^{82} +8.00000 q^{83} -1.00000 q^{84} +4.00000 q^{86} -6.00000 q^{87} -2.00000 q^{88} -10.0000 q^{89} -6.00000 q^{91} +8.00000 q^{92} +2.00000 q^{93} +8.00000 q^{94} +1.00000 q^{96} +10.0000 q^{97} -1.00000 q^{98} +2.00000 q^{99} +O(q^{100})\)

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
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) −1.00000 −0.288675
\(13\) −6.00000 −1.66410 −0.832050 0.554700i \(-0.812833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −4.00000 −0.970143 −0.485071 0.874475i \(-0.661206\pi\)
−0.485071 + 0.874475i \(0.661206\pi\)
\(18\) −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
\(21\) −1.00000 −0.218218
\(22\) −2.00000 −0.426401
\(23\) 8.00000 1.66812 0.834058 0.551677i \(-0.186012\pi\)
0.834058 + 0.551677i \(0.186012\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) 6.00000 1.17670
\(27\) −1.00000 −0.192450
\(28\) 1.00000 0.188982
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) −1.00000 −0.176777
\(33\) −2.00000 −0.348155
\(34\) 4.00000 0.685994
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −4.00000 −0.657596 −0.328798 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 6.00000 0.973329
\(39\) 6.00000 0.960769
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 1.00000 0.154303
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 2.00000 0.301511
\(45\) 0 0
\(46\) −8.00000 −1.17954
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) −1.00000 −0.144338
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 4.00000 0.560112
\(52\) −6.00000 −0.832050
\(53\) −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
\(56\) −1.00000 −0.133631
\(57\) 6.00000 0.794719
\(58\) −6.00000 −0.787839
\(59\) −8.00000 −1.04151 −0.520756 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 2.00000 0.254000
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 2.00000 0.246183
\(67\) −8.00000 −0.977356 −0.488678 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) −4.00000 −0.485071
\(69\) −8.00000 −0.963087
\(70\) 0 0
\(71\) −6.00000 −0.712069 −0.356034 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) −1.00000 −0.117851
\(73\) 14.0000 1.63858 0.819288 0.573382i \(-0.194369\pi\)
0.819288 + 0.573382i \(0.194369\pi\)
\(74\) 4.00000 0.464991
\(75\) 0 0
\(76\) −6.00000 −0.688247
\(77\) 2.00000 0.227921
\(78\) −6.00000 −0.679366
\(79\) −12.0000 −1.35011 −0.675053 0.737769i \(-0.735879\pi\)
−0.675053 + 0.737769i \(0.735879\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −2.00000 −0.220863
\(83\) 8.00000 0.878114 0.439057 0.898459i \(-0.355313\pi\)
0.439057 + 0.898459i \(0.355313\pi\)
\(84\) −1.00000 −0.109109
\(85\) 0 0
\(86\) 4.00000 0.431331
\(87\) −6.00000 −0.643268
\(88\) −2.00000 −0.213201
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) −6.00000 −0.628971
\(92\) 8.00000 0.834058
\(93\) 2.00000 0.207390
\(94\) 8.00000 0.825137
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) 10.0000 1.01535 0.507673 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) −1.00000 −0.101015
\(99\) 2.00000 0.201008
\(100\) 0 0
\(101\) 10.0000 0.995037 0.497519 0.867453i \(-0.334245\pi\)
0.497519 + 0.867453i \(0.334245\pi\)
\(102\) −4.00000 −0.396059
\(103\) −8.00000 −0.788263 −0.394132 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) 6.00000 0.588348
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −14.0000 −1.34096 −0.670478 0.741929i \(-0.733911\pi\)
−0.670478 + 0.741929i \(0.733911\pi\)
\(110\) 0 0
\(111\) 4.00000 0.379663
\(112\) 1.00000 0.0944911
\(113\) −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
\(117\) −6.00000 −0.554700
\(118\) 8.00000 0.736460
\(119\) −4.00000 −0.366679
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 10.0000 0.905357
\(123\) −2.00000 −0.180334
\(124\) −2.00000 −0.179605
\(125\) 0 0
\(126\) −1.00000 −0.0890871
\(127\) −4.00000 −0.354943 −0.177471 0.984126i \(-0.556792\pi\)
−0.177471 + 0.984126i \(0.556792\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) −2.00000 −0.174078
\(133\) −6.00000 −0.520266
\(134\) 8.00000 0.691095
\(135\) 0 0
\(136\) 4.00000 0.342997
\(137\) 6.00000 0.512615 0.256307 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) 8.00000 0.681005
\(139\) −14.0000 −1.18746 −0.593732 0.804663i \(-0.702346\pi\)
−0.593732 + 0.804663i \(0.702346\pi\)
\(140\) 0 0
\(141\) 8.00000 0.673722
\(142\) 6.00000 0.503509
\(143\) −12.0000 −1.00349
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −14.0000 −1.15865
\(147\) −1.00000 −0.0824786
\(148\) −4.00000 −0.328798
\(149\) 10.0000 0.819232 0.409616 0.912258i \(-0.365663\pi\)
0.409616 + 0.912258i \(0.365663\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 6.00000 0.486664
\(153\) −4.00000 −0.323381
\(154\) −2.00000 −0.161165
\(155\) 0 0
\(156\) 6.00000 0.480384
\(157\) 22.0000 1.75579 0.877896 0.478852i \(-0.158947\pi\)
0.877896 + 0.478852i \(0.158947\pi\)
\(158\) 12.0000 0.954669
\(159\) 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 \(0.449930\pi\)
\(164\) 2.00000 0.156174
\(165\) 0 0
\(166\) −8.00000 −0.620920
\(167\) 12.0000 0.928588 0.464294 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 1.00000 0.0771517
\(169\) 23.0000 1.76923
\(170\) 0 0
\(171\) −6.00000 −0.458831
\(172\) −4.00000 −0.304997
\(173\) −8.00000 −0.608229 −0.304114 0.952636i \(-0.598361\pi\)
−0.304114 + 0.952636i \(0.598361\pi\)
\(174\) 6.00000 0.454859
\(175\) 0 0
\(176\) 2.00000 0.150756
\(177\) 8.00000 0.601317
\(178\) 10.0000 0.749532
\(179\) 10.0000 0.747435 0.373718 0.927543i \(-0.378083\pi\)
0.373718 + 0.927543i \(0.378083\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 6.00000 0.444750
\(183\) 10.0000 0.739221
\(184\) −8.00000 −0.589768
\(185\) 0 0
\(186\) −2.00000 −0.146647
\(187\) −8.00000 −0.585018
\(188\) −8.00000 −0.583460
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −18.0000 −1.30243 −0.651217 0.758891i \(-0.725741\pi\)
−0.651217 + 0.758891i \(0.725741\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 8.00000 0.575853 0.287926 0.957653i \(-0.407034\pi\)
0.287926 + 0.957653i \(0.407034\pi\)
\(194\) −10.0000 −0.717958
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −2.00000 −0.142494 −0.0712470 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) −2.00000 −0.142134
\(199\) 6.00000 0.425329 0.212664 0.977125i \(-0.431786\pi\)
0.212664 + 0.977125i \(0.431786\pi\)
\(200\) 0 0
\(201\) 8.00000 0.564276
\(202\) −10.0000 −0.703598
\(203\) 6.00000 0.421117
\(204\) 4.00000 0.280056
\(205\) 0 0
\(206\) 8.00000 0.557386
\(207\) 8.00000 0.556038
\(208\) −6.00000 −0.416025
\(209\) −12.0000 −0.830057
\(210\) 0 0
\(211\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(212\) −6.00000 −0.412082
\(213\) 6.00000 0.411113
\(214\) 12.0000 0.820303
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) −2.00000 −0.135769
\(218\) 14.0000 0.948200
\(219\) −14.0000 −0.946032
\(220\) 0 0
\(221\) 24.0000 1.61441
\(222\) −4.00000 −0.268462
\(223\) 8.00000 0.535720 0.267860 0.963458i \(-0.413684\pi\)
0.267860 + 0.963458i \(0.413684\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) 6.00000 0.399114
\(227\) 8.00000 0.530979 0.265489 0.964114i \(-0.414466\pi\)
0.265489 + 0.964114i \(0.414466\pi\)
\(228\) 6.00000 0.397360
\(229\) 14.0000 0.925146 0.462573 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) 0 0
\(231\) −2.00000 −0.131590
\(232\) −6.00000 −0.393919
\(233\) 10.0000 0.655122 0.327561 0.944830i \(-0.393773\pi\)
0.327561 + 0.944830i \(0.393773\pi\)
\(234\) 6.00000 0.392232
\(235\) 0 0
\(236\) −8.00000 −0.520756
\(237\) 12.0000 0.779484
\(238\) 4.00000 0.259281
\(239\) 26.0000 1.68180 0.840900 0.541190i \(-0.182026\pi\)
0.840900 + 0.541190i \(0.182026\pi\)
\(240\) 0 0
\(241\) −26.0000 −1.67481 −0.837404 0.546585i \(-0.815928\pi\)
−0.837404 + 0.546585i \(0.815928\pi\)
\(242\) 7.00000 0.449977
\(243\) −1.00000 −0.0641500
\(244\) −10.0000 −0.640184
\(245\) 0 0
\(246\) 2.00000 0.127515
\(247\) 36.0000 2.29063
\(248\) 2.00000 0.127000
\(249\) −8.00000 −0.506979
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 1.00000 0.0629941
\(253\) 16.0000 1.00591
\(254\) 4.00000 0.250982
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) −4.00000 −0.249029
\(259\) −4.00000 −0.248548
\(260\) 0 0
\(261\) 6.00000 0.371391
\(262\) 12.0000 0.741362
\(263\) −16.0000 −0.986602 −0.493301 0.869859i \(-0.664210\pi\)
−0.493301 + 0.869859i \(0.664210\pi\)
\(264\) 2.00000 0.123091
\(265\) 0 0
\(266\) 6.00000 0.367884
\(267\) 10.0000 0.611990
\(268\) −8.00000 −0.488678
\(269\) 18.0000 1.09748 0.548740 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) 0 0
\(271\) 10.0000 0.607457 0.303728 0.952759i \(-0.401768\pi\)
0.303728 + 0.952759i \(0.401768\pi\)
\(272\) −4.00000 −0.242536
\(273\) 6.00000 0.363137
\(274\) −6.00000 −0.362473
\(275\) 0 0
\(276\) −8.00000 −0.481543
\(277\) −28.0000 −1.68236 −0.841178 0.540758i \(-0.818138\pi\)
−0.841178 + 0.540758i \(0.818138\pi\)
\(278\) 14.0000 0.839664
\(279\) −2.00000 −0.119737
\(280\) 0 0
\(281\) 30.0000 1.78965 0.894825 0.446417i \(-0.147300\pi\)
0.894825 + 0.446417i \(0.147300\pi\)
\(282\) −8.00000 −0.476393
\(283\) −20.0000 −1.18888 −0.594438 0.804141i \(-0.702626\pi\)
−0.594438 + 0.804141i \(0.702626\pi\)
\(284\) −6.00000 −0.356034
\(285\) 0 0
\(286\) 12.0000 0.709575
\(287\) 2.00000 0.118056
\(288\) −1.00000 −0.0589256
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) −10.0000 −0.586210
\(292\) 14.0000 0.819288
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 1.00000 0.0583212
\(295\) 0 0
\(296\) 4.00000 0.232495
\(297\) −2.00000 −0.116052
\(298\) −10.0000 −0.579284
\(299\) −48.0000 −2.77591
\(300\) 0 0
\(301\) −4.00000 −0.230556
\(302\) 8.00000 0.460348
\(303\) −10.0000 −0.574485
\(304\) −6.00000 −0.344124
\(305\) 0 0
\(306\) 4.00000 0.228665
\(307\) 12.0000 0.684876 0.342438 0.939540i \(-0.388747\pi\)
0.342438 + 0.939540i \(0.388747\pi\)
\(308\) 2.00000 0.113961
\(309\) 8.00000 0.455104
\(310\) 0 0
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) −6.00000 −0.339683
\(313\) 2.00000 0.113047 0.0565233 0.998401i \(-0.481998\pi\)
0.0565233 + 0.998401i \(0.481998\pi\)
\(314\) −22.0000 −1.24153
\(315\) 0 0
\(316\) −12.0000 −0.675053
\(317\) 2.00000 0.112331 0.0561656 0.998421i \(-0.482113\pi\)
0.0561656 + 0.998421i \(0.482113\pi\)
\(318\) −6.00000 −0.336463
\(319\) 12.0000 0.671871
\(320\) 0 0
\(321\) 12.0000 0.669775
\(322\) −8.00000 −0.445823
\(323\) 24.0000 1.33540
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −4.00000 −0.221540
\(327\) 14.0000 0.774202
\(328\) −2.00000 −0.110432
\(329\) −8.00000 −0.441054
\(330\) 0 0
\(331\) −8.00000 −0.439720 −0.219860 0.975531i \(-0.570560\pi\)
−0.219860 + 0.975531i \(0.570560\pi\)
\(332\) 8.00000 0.439057
\(333\) −4.00000 −0.219199
\(334\) −12.0000 −0.656611
\(335\) 0 0
\(336\) −1.00000 −0.0545545
\(337\) 8.00000 0.435788 0.217894 0.975972i \(-0.430081\pi\)
0.217894 + 0.975972i \(0.430081\pi\)
\(338\) −23.0000 −1.25104
\(339\) 6.00000 0.325875
\(340\) 0 0
\(341\) −4.00000 −0.216612
\(342\) 6.00000 0.324443
\(343\) 1.00000 0.0539949
\(344\) 4.00000 0.215666
\(345\) 0 0
\(346\) 8.00000 0.430083
\(347\) −36.0000 −1.93258 −0.966291 0.257454i \(-0.917117\pi\)
−0.966291 + 0.257454i \(0.917117\pi\)
\(348\) −6.00000 −0.321634
\(349\) −26.0000 −1.39175 −0.695874 0.718164i \(-0.744983\pi\)
−0.695874 + 0.718164i \(0.744983\pi\)
\(350\) 0 0
\(351\) 6.00000 0.320256
\(352\) −2.00000 −0.106600
\(353\) −20.0000 −1.06449 −0.532246 0.846590i \(-0.678652\pi\)
−0.532246 + 0.846590i \(0.678652\pi\)
\(354\) −8.00000 −0.425195
\(355\) 0 0
\(356\) −10.0000 −0.529999
\(357\) 4.00000 0.211702
\(358\) −10.0000 −0.528516
\(359\) 6.00000 0.316668 0.158334 0.987386i \(-0.449388\pi\)
0.158334 + 0.987386i \(0.449388\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) −2.00000 −0.105118
\(363\) 7.00000 0.367405
\(364\) −6.00000 −0.314485
\(365\) 0 0
\(366\) −10.0000 −0.522708
\(367\) 32.0000 1.67039 0.835193 0.549957i \(-0.185356\pi\)
0.835193 + 0.549957i \(0.185356\pi\)
\(368\) 8.00000 0.417029
\(369\) 2.00000 0.104116
\(370\) 0 0
\(371\) −6.00000 −0.311504
\(372\) 2.00000 0.103695
\(373\) 24.0000 1.24267 0.621336 0.783544i \(-0.286590\pi\)
0.621336 + 0.783544i \(0.286590\pi\)
\(374\) 8.00000 0.413670
\(375\) 0 0
\(376\) 8.00000 0.412568
\(377\) −36.0000 −1.85409
\(378\) 1.00000 0.0514344
\(379\) 20.0000 1.02733 0.513665 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 0 0
\(381\) 4.00000 0.204926
\(382\) 18.0000 0.920960
\(383\) −20.0000 −1.02195 −0.510976 0.859595i \(-0.670716\pi\)
−0.510976 + 0.859595i \(0.670716\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −8.00000 −0.407189
\(387\) −4.00000 −0.203331
\(388\) 10.0000 0.507673
\(389\) 2.00000 0.101404 0.0507020 0.998714i \(-0.483854\pi\)
0.0507020 + 0.998714i \(0.483854\pi\)
\(390\) 0 0
\(391\) −32.0000 −1.61831
\(392\) −1.00000 −0.0505076
\(393\) 12.0000 0.605320
\(394\) 2.00000 0.100759
\(395\) 0 0
\(396\) 2.00000 0.100504
\(397\) 2.00000 0.100377 0.0501886 0.998740i \(-0.484018\pi\)
0.0501886 + 0.998740i \(0.484018\pi\)
\(398\) −6.00000 −0.300753
\(399\) 6.00000 0.300376
\(400\) 0 0
\(401\) −14.0000 −0.699127 −0.349563 0.936913i \(-0.613670\pi\)
−0.349563 + 0.936913i \(0.613670\pi\)
\(402\) −8.00000 −0.399004
\(403\) 12.0000 0.597763
\(404\) 10.0000 0.497519
\(405\) 0 0
\(406\) −6.00000 −0.297775
\(407\) −8.00000 −0.396545
\(408\) −4.00000 −0.198030
\(409\) 26.0000 1.28562 0.642809 0.766027i \(-0.277769\pi\)
0.642809 + 0.766027i \(0.277769\pi\)
\(410\) 0 0
\(411\) −6.00000 −0.295958
\(412\) −8.00000 −0.394132
\(413\) −8.00000 −0.393654
\(414\) −8.00000 −0.393179
\(415\) 0 0
\(416\) 6.00000 0.294174
\(417\) 14.0000 0.685583
\(418\) 12.0000 0.586939
\(419\) 20.0000 0.977064 0.488532 0.872546i \(-0.337533\pi\)
0.488532 + 0.872546i \(0.337533\pi\)
\(420\) 0 0
\(421\) 34.0000 1.65706 0.828529 0.559946i \(-0.189178\pi\)
0.828529 + 0.559946i \(0.189178\pi\)
\(422\) 0 0
\(423\) −8.00000 −0.388973
\(424\) 6.00000 0.291386
\(425\) 0 0
\(426\) −6.00000 −0.290701
\(427\) −10.0000 −0.483934
\(428\) −12.0000 −0.580042
\(429\) 12.0000 0.579365
\(430\) 0 0
\(431\) 2.00000 0.0963366 0.0481683 0.998839i \(-0.484662\pi\)
0.0481683 + 0.998839i \(0.484662\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 26.0000 1.24948 0.624740 0.780833i \(-0.285205\pi\)
0.624740 + 0.780833i \(0.285205\pi\)
\(434\) 2.00000 0.0960031
\(435\) 0 0
\(436\) −14.0000 −0.670478
\(437\) −48.0000 −2.29615
\(438\) 14.0000 0.668946
\(439\) −18.0000 −0.859093 −0.429547 0.903045i \(-0.641327\pi\)
−0.429547 + 0.903045i \(0.641327\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) −24.0000 −1.14156
\(443\) 12.0000 0.570137 0.285069 0.958507i \(-0.407984\pi\)
0.285069 + 0.958507i \(0.407984\pi\)
\(444\) 4.00000 0.189832
\(445\) 0 0
\(446\) −8.00000 −0.378811
\(447\) −10.0000 −0.472984
\(448\) 1.00000 0.0472456
\(449\) −6.00000 −0.283158 −0.141579 0.989927i \(-0.545218\pi\)
−0.141579 + 0.989927i \(0.545218\pi\)
\(450\) 0 0
\(451\) 4.00000 0.188353
\(452\) −6.00000 −0.282216
\(453\) 8.00000 0.375873
\(454\) −8.00000 −0.375459
\(455\) 0 0
\(456\) −6.00000 −0.280976
\(457\) −28.0000 −1.30978 −0.654892 0.755722i \(-0.727286\pi\)
−0.654892 + 0.755722i \(0.727286\pi\)
\(458\) −14.0000 −0.654177
\(459\) 4.00000 0.186704
\(460\) 0 0
\(461\) −18.0000 −0.838344 −0.419172 0.907907i \(-0.637680\pi\)
−0.419172 + 0.907907i \(0.637680\pi\)
\(462\) 2.00000 0.0930484
\(463\) −36.0000 −1.67306 −0.836531 0.547920i \(-0.815420\pi\)
−0.836531 + 0.547920i \(0.815420\pi\)
\(464\) 6.00000 0.278543
\(465\) 0 0
\(466\) −10.0000 −0.463241
\(467\) 24.0000 1.11059 0.555294 0.831654i \(-0.312606\pi\)
0.555294 + 0.831654i \(0.312606\pi\)
\(468\) −6.00000 −0.277350
\(469\) −8.00000 −0.369406
\(470\) 0 0
\(471\) −22.0000 −1.01371
\(472\) 8.00000 0.368230
\(473\) −8.00000 −0.367840
\(474\) −12.0000 −0.551178
\(475\) 0 0
\(476\) −4.00000 −0.183340
\(477\) −6.00000 −0.274721
\(478\) −26.0000 −1.18921
\(479\) 8.00000 0.365529 0.182765 0.983157i \(-0.441495\pi\)
0.182765 + 0.983157i \(0.441495\pi\)
\(480\) 0 0
\(481\) 24.0000 1.09431
\(482\) 26.0000 1.18427
\(483\) −8.00000 −0.364013
\(484\) −7.00000 −0.318182
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) −12.0000 −0.543772 −0.271886 0.962329i \(-0.587647\pi\)
−0.271886 + 0.962329i \(0.587647\pi\)
\(488\) 10.0000 0.452679
\(489\) −4.00000 −0.180886
\(490\) 0 0
\(491\) −30.0000 −1.35388 −0.676941 0.736038i \(-0.736695\pi\)
−0.676941 + 0.736038i \(0.736695\pi\)
\(492\) −2.00000 −0.0901670
\(493\) −24.0000 −1.08091
\(494\) −36.0000 −1.61972
\(495\) 0 0
\(496\) −2.00000 −0.0898027
\(497\) −6.00000 −0.269137
\(498\) 8.00000 0.358489
\(499\) −12.0000 −0.537194 −0.268597 0.963253i \(-0.586560\pi\)
−0.268597 + 0.963253i \(0.586560\pi\)
\(500\) 0 0
\(501\) −12.0000 −0.536120
\(502\) 0 0
\(503\) 12.0000 0.535054 0.267527 0.963550i \(-0.413794\pi\)
0.267527 + 0.963550i \(0.413794\pi\)
\(504\) −1.00000 −0.0445435
\(505\) 0 0
\(506\) −16.0000 −0.711287
\(507\) −23.0000 −1.02147
\(508\) −4.00000 −0.177471
\(509\) 30.0000 1.32973 0.664863 0.746965i \(-0.268490\pi\)
0.664863 + 0.746965i \(0.268490\pi\)
\(510\) 0 0
\(511\) 14.0000 0.619324
\(512\) −1.00000 −0.0441942
\(513\) 6.00000 0.264906
\(514\) 0 0
\(515\) 0 0
\(516\) 4.00000 0.176090
\(517\) −16.0000 −0.703679
\(518\) 4.00000 0.175750
\(519\) 8.00000 0.351161
\(520\) 0 0
\(521\) −26.0000 −1.13908 −0.569540 0.821963i \(-0.692879\pi\)
−0.569540 + 0.821963i \(0.692879\pi\)
\(522\) −6.00000 −0.262613
\(523\) 28.0000 1.22435 0.612177 0.790721i \(-0.290294\pi\)
0.612177 + 0.790721i \(0.290294\pi\)
\(524\) −12.0000 −0.524222
\(525\) 0 0
\(526\) 16.0000 0.697633
\(527\) 8.00000 0.348485
\(528\) −2.00000 −0.0870388
\(529\) 41.0000 1.78261
\(530\) 0 0
\(531\) −8.00000 −0.347170
\(532\) −6.00000 −0.260133
\(533\) −12.0000 −0.519778
\(534\) −10.0000 −0.432742
\(535\) 0 0
\(536\) 8.00000 0.345547
\(537\) −10.0000 −0.431532
\(538\) −18.0000 −0.776035
\(539\) 2.00000 0.0861461
\(540\) 0 0
\(541\) 22.0000 0.945854 0.472927 0.881102i \(-0.343197\pi\)
0.472927 + 0.881102i \(0.343197\pi\)
\(542\) −10.0000 −0.429537
\(543\) −2.00000 −0.0858282
\(544\) 4.00000 0.171499
\(545\) 0 0
\(546\) −6.00000 −0.256776
\(547\) −40.0000 −1.71028 −0.855138 0.518400i \(-0.826528\pi\)
−0.855138 + 0.518400i \(0.826528\pi\)
\(548\) 6.00000 0.256307
\(549\) −10.0000 −0.426790
\(550\) 0 0
\(551\) −36.0000 −1.53365
\(552\) 8.00000 0.340503
\(553\) −12.0000 −0.510292
\(554\) 28.0000 1.18961
\(555\) 0 0
\(556\) −14.0000 −0.593732
\(557\) 42.0000 1.77960 0.889799 0.456354i \(-0.150845\pi\)
0.889799 + 0.456354i \(0.150845\pi\)
\(558\) 2.00000 0.0846668
\(559\) 24.0000 1.01509
\(560\) 0 0
\(561\) 8.00000 0.337760
\(562\) −30.0000 −1.26547
\(563\) 4.00000 0.168580 0.0842900 0.996441i \(-0.473138\pi\)
0.0842900 + 0.996441i \(0.473138\pi\)
\(564\) 8.00000 0.336861
\(565\) 0 0
\(566\) 20.0000 0.840663
\(567\) 1.00000 0.0419961
\(568\) 6.00000 0.251754
\(569\) −46.0000 −1.92842 −0.964210 0.265139i \(-0.914582\pi\)
−0.964210 + 0.265139i \(0.914582\pi\)
\(570\) 0 0
\(571\) 4.00000 0.167395 0.0836974 0.996491i \(-0.473327\pi\)
0.0836974 + 0.996491i \(0.473327\pi\)
\(572\) −12.0000 −0.501745
\(573\) 18.0000 0.751961
\(574\) −2.00000 −0.0834784
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 26.0000 1.08239 0.541197 0.840896i \(-0.317971\pi\)
0.541197 + 0.840896i \(0.317971\pi\)
\(578\) 1.00000 0.0415945
\(579\) −8.00000 −0.332469
\(580\) 0 0
\(581\) 8.00000 0.331896
\(582\) 10.0000 0.414513
\(583\) −12.0000 −0.496989
\(584\) −14.0000 −0.579324
\(585\) 0 0
\(586\) 0 0
\(587\) 12.0000 0.495293 0.247647 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) −1.00000 −0.0412393
\(589\) 12.0000 0.494451
\(590\) 0 0
\(591\) 2.00000 0.0822690
\(592\) −4.00000 −0.164399
\(593\) −12.0000 −0.492781 −0.246390 0.969171i \(-0.579245\pi\)
−0.246390 + 0.969171i \(0.579245\pi\)
\(594\) 2.00000 0.0820610
\(595\) 0 0
\(596\) 10.0000 0.409616
\(597\) −6.00000 −0.245564
\(598\) 48.0000 1.96287
\(599\) 30.0000 1.22577 0.612883 0.790173i \(-0.290010\pi\)
0.612883 + 0.790173i \(0.290010\pi\)
\(600\) 0 0
\(601\) 38.0000 1.55005 0.775026 0.631929i \(-0.217737\pi\)
0.775026 + 0.631929i \(0.217737\pi\)
\(602\) 4.00000 0.163028
\(603\) −8.00000 −0.325785
\(604\) −8.00000 −0.325515
\(605\) 0 0
\(606\) 10.0000 0.406222
\(607\) −8.00000 −0.324710 −0.162355 0.986732i \(-0.551909\pi\)
−0.162355 + 0.986732i \(0.551909\pi\)
\(608\) 6.00000 0.243332
\(609\) −6.00000 −0.243132
\(610\) 0 0
\(611\) 48.0000 1.94187
\(612\) −4.00000 −0.161690
\(613\) −28.0000 −1.13091 −0.565455 0.824779i \(-0.691299\pi\)
−0.565455 + 0.824779i \(0.691299\pi\)
\(614\) −12.0000 −0.484281
\(615\) 0 0
\(616\) −2.00000 −0.0805823
\(617\) 26.0000 1.04672 0.523360 0.852111i \(-0.324678\pi\)
0.523360 + 0.852111i \(0.324678\pi\)
\(618\) −8.00000 −0.321807
\(619\) −22.0000 −0.884255 −0.442127 0.896952i \(-0.645776\pi\)
−0.442127 + 0.896952i \(0.645776\pi\)
\(620\) 0 0
\(621\) −8.00000 −0.321029
\(622\) −24.0000 −0.962312
\(623\) −10.0000 −0.400642
\(624\) 6.00000 0.240192
\(625\) 0 0
\(626\) −2.00000 −0.0799361
\(627\) 12.0000 0.479234
\(628\) 22.0000 0.877896
\(629\) 16.0000 0.637962
\(630\) 0 0
\(631\) −4.00000 −0.159237 −0.0796187 0.996825i \(-0.525370\pi\)
−0.0796187 + 0.996825i \(0.525370\pi\)
\(632\) 12.0000 0.477334
\(633\) 0 0
\(634\) −2.00000 −0.0794301
\(635\) 0 0
\(636\) 6.00000 0.237915
\(637\) −6.00000 −0.237729
\(638\) −12.0000 −0.475085
\(639\) −6.00000 −0.237356
\(640\) 0 0
\(641\) 14.0000 0.552967 0.276483 0.961019i \(-0.410831\pi\)
0.276483 + 0.961019i \(0.410831\pi\)
\(642\) −12.0000 −0.473602
\(643\) −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
\(646\) −24.0000 −0.944267
\(647\) −36.0000 −1.41531 −0.707653 0.706560i \(-0.750246\pi\)
−0.707653 + 0.706560i \(0.750246\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −16.0000 −0.628055
\(650\) 0 0
\(651\) 2.00000 0.0783862
\(652\) 4.00000 0.156652
\(653\) −34.0000 −1.33052 −0.665261 0.746611i \(-0.731680\pi\)
−0.665261 + 0.746611i \(0.731680\pi\)
\(654\) −14.0000 −0.547443
\(655\) 0 0
\(656\) 2.00000 0.0780869
\(657\) 14.0000 0.546192
\(658\) 8.00000 0.311872
\(659\) −26.0000 −1.01282 −0.506408 0.862294i \(-0.669027\pi\)
−0.506408 + 0.862294i \(0.669027\pi\)
\(660\) 0 0
\(661\) 38.0000 1.47803 0.739014 0.673690i \(-0.235292\pi\)
0.739014 + 0.673690i \(0.235292\pi\)
\(662\) 8.00000 0.310929
\(663\) −24.0000 −0.932083
\(664\) −8.00000 −0.310460
\(665\) 0 0
\(666\) 4.00000 0.154997
\(667\) 48.0000 1.85857
\(668\) 12.0000 0.464294
\(669\) −8.00000 −0.309298
\(670\) 0 0
\(671\) −20.0000 −0.772091
\(672\) 1.00000 0.0385758
\(673\) −12.0000 −0.462566 −0.231283 0.972887i \(-0.574292\pi\)
−0.231283 + 0.972887i \(0.574292\pi\)
\(674\) −8.00000 −0.308148
\(675\) 0 0
\(676\) 23.0000 0.884615
\(677\) −40.0000 −1.53732 −0.768662 0.639655i \(-0.779077\pi\)
−0.768662 + 0.639655i \(0.779077\pi\)
\(678\) −6.00000 −0.230429
\(679\) 10.0000 0.383765
\(680\) 0 0
\(681\) −8.00000 −0.306561
\(682\) 4.00000 0.153168
\(683\) −4.00000 −0.153056 −0.0765279 0.997067i \(-0.524383\pi\)
−0.0765279 + 0.997067i \(0.524383\pi\)
\(684\) −6.00000 −0.229416
\(685\) 0 0
\(686\) −1.00000 −0.0381802
\(687\) −14.0000 −0.534133
\(688\) −4.00000 −0.152499
\(689\) 36.0000 1.37149
\(690\) 0 0
\(691\) −2.00000 −0.0760836 −0.0380418 0.999276i \(-0.512112\pi\)
−0.0380418 + 0.999276i \(0.512112\pi\)
\(692\) −8.00000 −0.304114
\(693\) 2.00000 0.0759737
\(694\) 36.0000 1.36654
\(695\) 0 0
\(696\) 6.00000 0.227429
\(697\) −8.00000 −0.303022
\(698\) 26.0000 0.984115
\(699\) −10.0000 −0.378235
\(700\) 0 0
\(701\) −34.0000 −1.28416 −0.642081 0.766637i \(-0.721929\pi\)
−0.642081 + 0.766637i \(0.721929\pi\)
\(702\) −6.00000 −0.226455
\(703\) 24.0000 0.905177
\(704\) 2.00000 0.0753778
\(705\) 0 0
\(706\) 20.0000 0.752710
\(707\) 10.0000 0.376089
\(708\) 8.00000 0.300658
\(709\) −6.00000 −0.225335 −0.112667 0.993633i \(-0.535939\pi\)
−0.112667 + 0.993633i \(0.535939\pi\)
\(710\) 0 0
\(711\) −12.0000 −0.450035
\(712\) 10.0000 0.374766
\(713\) −16.0000 −0.599205
\(714\) −4.00000 −0.149696
\(715\) 0 0
\(716\) 10.0000 0.373718
\(717\) −26.0000 −0.970988
\(718\) −6.00000 −0.223918
\(719\) −4.00000 −0.149175 −0.0745874 0.997214i \(-0.523764\pi\)
−0.0745874 + 0.997214i \(0.523764\pi\)
\(720\) 0 0
\(721\) −8.00000 −0.297936
\(722\) −17.0000 −0.632674
\(723\) 26.0000 0.966950
\(724\) 2.00000 0.0743294
\(725\) 0 0
\(726\) −7.00000 −0.259794
\(727\) −24.0000 −0.890111 −0.445055 0.895503i \(-0.646816\pi\)
−0.445055 + 0.895503i \(0.646816\pi\)
\(728\) 6.00000 0.222375
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 16.0000 0.591781
\(732\) 10.0000 0.369611
\(733\) −14.0000 −0.517102 −0.258551 0.965998i \(-0.583245\pi\)
−0.258551 + 0.965998i \(0.583245\pi\)
\(734\) −32.0000 −1.18114
\(735\) 0 0
\(736\) −8.00000 −0.294884
\(737\) −16.0000 −0.589368
\(738\) −2.00000 −0.0736210
\(739\) −44.0000 −1.61857 −0.809283 0.587419i \(-0.800144\pi\)
−0.809283 + 0.587419i \(0.800144\pi\)
\(740\) 0 0
\(741\) −36.0000 −1.32249
\(742\) 6.00000 0.220267
\(743\) 48.0000 1.76095 0.880475 0.474093i \(-0.157224\pi\)
0.880475 + 0.474093i \(0.157224\pi\)
\(744\) −2.00000 −0.0733236
\(745\) 0 0
\(746\) −24.0000 −0.878702
\(747\) 8.00000 0.292705
\(748\) −8.00000 −0.292509
\(749\) −12.0000 −0.438470
\(750\) 0 0
\(751\) 20.0000 0.729810 0.364905 0.931045i \(-0.381101\pi\)
0.364905 + 0.931045i \(0.381101\pi\)
\(752\) −8.00000 −0.291730
\(753\) 0 0
\(754\) 36.0000 1.31104
\(755\) 0 0
\(756\) −1.00000 −0.0363696
\(757\) 32.0000 1.16306 0.581530 0.813525i \(-0.302454\pi\)
0.581530 + 0.813525i \(0.302454\pi\)
\(758\) −20.0000 −0.726433
\(759\) −16.0000 −0.580763
\(760\) 0 0
\(761\) −6.00000 −0.217500 −0.108750 0.994069i \(-0.534685\pi\)
−0.108750 + 0.994069i \(0.534685\pi\)
\(762\) −4.00000 −0.144905
\(763\) −14.0000 −0.506834
\(764\) −18.0000 −0.651217
\(765\) 0 0
\(766\) 20.0000 0.722629
\(767\) 48.0000 1.73318
\(768\) −1.00000 −0.0360844
\(769\) 30.0000 1.08183 0.540914 0.841078i \(-0.318079\pi\)
0.540914 + 0.841078i \(0.318079\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 8.00000 0.287926
\(773\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(774\) 4.00000 0.143777
\(775\) 0 0
\(776\) −10.0000 −0.358979
\(777\) 4.00000 0.143499
\(778\) −2.00000 −0.0717035
\(779\) −12.0000 −0.429945
\(780\) 0 0
\(781\) −12.0000 −0.429394
\(782\) 32.0000 1.14432
\(783\) −6.00000 −0.214423
\(784\) 1.00000 0.0357143
\(785\) 0 0
\(786\) −12.0000 −0.428026
\(787\) −28.0000 −0.998092 −0.499046 0.866575i \(-0.666316\pi\)
−0.499046 + 0.866575i \(0.666316\pi\)
\(788\) −2.00000 −0.0712470
\(789\) 16.0000 0.569615
\(790\) 0 0
\(791\) −6.00000 −0.213335
\(792\) −2.00000 −0.0710669
\(793\) 60.0000 2.13066
\(794\) −2.00000 −0.0709773
\(795\) 0 0
\(796\) 6.00000 0.212664
\(797\) 8.00000 0.283375 0.141687 0.989911i \(-0.454747\pi\)
0.141687 + 0.989911i \(0.454747\pi\)
\(798\) −6.00000 −0.212398
\(799\) 32.0000 1.13208
\(800\) 0 0
\(801\) −10.0000 −0.353333
\(802\) 14.0000 0.494357
\(803\) 28.0000 0.988099
\(804\) 8.00000 0.282138
\(805\) 0 0
\(806\) −12.0000 −0.422682
\(807\) −18.0000 −0.633630
\(808\) −10.0000 −0.351799
\(809\) 30.0000 1.05474 0.527372 0.849635i \(-0.323177\pi\)
0.527372 + 0.849635i \(0.323177\pi\)
\(810\) 0 0
\(811\) −14.0000 −0.491606 −0.245803 0.969320i \(-0.579052\pi\)
−0.245803 + 0.969320i \(0.579052\pi\)
\(812\) 6.00000 0.210559
\(813\) −10.0000 −0.350715
\(814\) 8.00000 0.280400
\(815\) 0 0
\(816\) 4.00000 0.140028
\(817\) 24.0000 0.839654
\(818\) −26.0000 −0.909069
\(819\) −6.00000 −0.209657
\(820\) 0 0
\(821\) 22.0000 0.767805 0.383903 0.923374i \(-0.374580\pi\)
0.383903 + 0.923374i \(0.374580\pi\)
\(822\) 6.00000 0.209274
\(823\) −12.0000 −0.418294 −0.209147 0.977884i \(-0.567069\pi\)
−0.209147 + 0.977884i \(0.567069\pi\)
\(824\) 8.00000 0.278693
\(825\) 0 0
\(826\) 8.00000 0.278356
\(827\) 20.0000 0.695468 0.347734 0.937593i \(-0.386951\pi\)
0.347734 + 0.937593i \(0.386951\pi\)
\(828\) 8.00000 0.278019
\(829\) 18.0000 0.625166 0.312583 0.949890i \(-0.398806\pi\)
0.312583 + 0.949890i \(0.398806\pi\)
\(830\) 0 0
\(831\) 28.0000 0.971309
\(832\) −6.00000 −0.208013
\(833\) −4.00000 −0.138592
\(834\) −14.0000 −0.484780
\(835\) 0 0
\(836\) −12.0000 −0.415029
\(837\) 2.00000 0.0691301
\(838\) −20.0000 −0.690889
\(839\) −20.0000 −0.690477 −0.345238 0.938515i \(-0.612202\pi\)
−0.345238 + 0.938515i \(0.612202\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) −34.0000 −1.17172
\(843\) −30.0000 −1.03325
\(844\) 0 0
\(845\) 0 0
\(846\) 8.00000 0.275046
\(847\) −7.00000 −0.240523
\(848\) −6.00000 −0.206041
\(849\) 20.0000 0.686398
\(850\) 0 0
\(851\) −32.0000 −1.09695
\(852\) 6.00000 0.205557
\(853\) −38.0000 −1.30110 −0.650548 0.759465i \(-0.725461\pi\)
−0.650548 + 0.759465i \(0.725461\pi\)
\(854\) 10.0000 0.342193
\(855\) 0 0
\(856\) 12.0000 0.410152
\(857\) −24.0000 −0.819824 −0.409912 0.912125i \(-0.634441\pi\)
−0.409912 + 0.912125i \(0.634441\pi\)
\(858\) −12.0000 −0.409673
\(859\) −10.0000 −0.341196 −0.170598 0.985341i \(-0.554570\pi\)
−0.170598 + 0.985341i \(0.554570\pi\)
\(860\) 0 0
\(861\) −2.00000 −0.0681598
\(862\) −2.00000 −0.0681203
\(863\) 48.0000 1.63394 0.816970 0.576681i \(-0.195652\pi\)
0.816970 + 0.576681i \(0.195652\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) −26.0000 −0.883516
\(867\) 1.00000 0.0339618
\(868\) −2.00000 −0.0678844
\(869\) −24.0000 −0.814144
\(870\) 0 0
\(871\) 48.0000 1.62642
\(872\) 14.0000 0.474100
\(873\) 10.0000 0.338449
\(874\) 48.0000 1.62362
\(875\) 0 0
\(876\) −14.0000 −0.473016
\(877\) 28.0000 0.945493 0.472746 0.881199i \(-0.343263\pi\)
0.472746 + 0.881199i \(0.343263\pi\)
\(878\) 18.0000 0.607471
\(879\) 0 0
\(880\) 0 0
\(881\) 30.0000 1.01073 0.505363 0.862907i \(-0.331359\pi\)
0.505363 + 0.862907i \(0.331359\pi\)
\(882\) −1.00000 −0.0336718
\(883\) −24.0000 −0.807664 −0.403832 0.914833i \(-0.632322\pi\)
−0.403832 + 0.914833i \(0.632322\pi\)
\(884\) 24.0000 0.807207
\(885\) 0 0
\(886\) −12.0000 −0.403148
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) −4.00000 −0.134231
\(889\) −4.00000 −0.134156
\(890\) 0 0
\(891\) 2.00000 0.0670025
\(892\) 8.00000 0.267860
\(893\) 48.0000 1.60626
\(894\) 10.0000 0.334450
\(895\) 0 0
\(896\) −1.00000 −0.0334077
\(897\) 48.0000 1.60267
\(898\) 6.00000 0.200223
\(899\) −12.0000 −0.400222
\(900\) 0 0
\(901\) 24.0000 0.799556
\(902\) −4.00000 −0.133185
\(903\) 4.00000 0.133112
\(904\) 6.00000 0.199557
\(905\) 0 0
\(906\) −8.00000 −0.265782
\(907\) −24.0000 −0.796907 −0.398453 0.917189i \(-0.630453\pi\)
−0.398453 + 0.917189i \(0.630453\pi\)
\(908\) 8.00000 0.265489
\(909\) 10.0000 0.331679
\(910\) 0 0
\(911\) 6.00000 0.198789 0.0993944 0.995048i \(-0.468309\pi\)
0.0993944 + 0.995048i \(0.468309\pi\)
\(912\) 6.00000 0.198680
\(913\) 16.0000 0.529523
\(914\) 28.0000 0.926158
\(915\) 0 0
\(916\) 14.0000 0.462573
\(917\) −12.0000 −0.396275
\(918\) −4.00000 −0.132020
\(919\) 60.0000 1.97922 0.989609 0.143787i \(-0.0459280\pi\)
0.989609 + 0.143787i \(0.0459280\pi\)
\(920\) 0 0
\(921\) −12.0000 −0.395413
\(922\) 18.0000 0.592798
\(923\) 36.0000 1.18495
\(924\) −2.00000 −0.0657952
\(925\) 0 0
\(926\) 36.0000 1.18303
\(927\) −8.00000 −0.262754
\(928\) −6.00000 −0.196960
\(929\) −30.0000 −0.984268 −0.492134 0.870519i \(-0.663783\pi\)
−0.492134 + 0.870519i \(0.663783\pi\)
\(930\) 0 0
\(931\) −6.00000 −0.196642
\(932\) 10.0000 0.327561
\(933\) −24.0000 −0.785725
\(934\) −24.0000 −0.785304
\(935\) 0 0
\(936\) 6.00000 0.196116
\(937\) −26.0000 −0.849383 −0.424691 0.905338i \(-0.639617\pi\)
−0.424691 + 0.905338i \(0.639617\pi\)
\(938\) 8.00000 0.261209
\(939\) −2.00000 −0.0652675
\(940\) 0 0
\(941\) −10.0000 −0.325991 −0.162995 0.986627i \(-0.552116\pi\)
−0.162995 + 0.986627i \(0.552116\pi\)
\(942\) 22.0000 0.716799
\(943\) 16.0000 0.521032
\(944\) −8.00000 −0.260378
\(945\) 0 0
\(946\) 8.00000 0.260102
\(947\) 12.0000 0.389948 0.194974 0.980808i \(-0.437538\pi\)
0.194974 + 0.980808i \(0.437538\pi\)
\(948\) 12.0000 0.389742
\(949\) −84.0000 −2.72676
\(950\) 0 0
\(951\) −2.00000 −0.0648544
\(952\) 4.00000 0.129641
\(953\) 10.0000 0.323932 0.161966 0.986796i \(-0.448217\pi\)
0.161966 + 0.986796i \(0.448217\pi\)
\(954\) 6.00000 0.194257
\(955\) 0 0
\(956\) 26.0000 0.840900
\(957\) −12.0000 −0.387905
\(958\) −8.00000 −0.258468
\(959\) 6.00000 0.193750
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) −24.0000 −0.773791
\(963\) −12.0000 −0.386695
\(964\) −26.0000 −0.837404
\(965\) 0 0
\(966\) 8.00000 0.257396
\(967\) −56.0000 −1.80084 −0.900419 0.435023i \(-0.856740\pi\)
−0.900419 + 0.435023i \(0.856740\pi\)
\(968\) 7.00000 0.224989
\(969\) −24.0000 −0.770991
\(970\) 0 0
\(971\) −24.0000 −0.770197 −0.385098 0.922876i \(-0.625832\pi\)
−0.385098 + 0.922876i \(0.625832\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −14.0000 −0.448819
\(974\) 12.0000 0.384505
\(975\) 0 0
\(976\) −10.0000 −0.320092
\(977\) 18.0000 0.575871 0.287936 0.957650i \(-0.407031\pi\)
0.287936 + 0.957650i \(0.407031\pi\)
\(978\) 4.00000 0.127906
\(979\) −20.0000 −0.639203
\(980\) 0 0
\(981\) −14.0000 −0.446986
\(982\) 30.0000 0.957338
\(983\) 16.0000 0.510321 0.255160 0.966899i \(-0.417872\pi\)
0.255160 + 0.966899i \(0.417872\pi\)
\(984\) 2.00000 0.0637577
\(985\) 0 0
\(986\) 24.0000 0.764316
\(987\) 8.00000 0.254643
\(988\) 36.0000 1.14531
\(989\) −32.0000 −1.01754
\(990\) 0 0
\(991\) 4.00000 0.127064 0.0635321 0.997980i \(-0.479763\pi\)
0.0635321 + 0.997980i \(0.479763\pi\)
\(992\) 2.00000 0.0635001
\(993\) 8.00000 0.253872
\(994\) 6.00000 0.190308
\(995\) 0 0
\(996\) −8.00000 −0.253490
\(997\) 42.0000 1.33015 0.665077 0.746775i \(-0.268399\pi\)
0.665077 + 0.746775i \(0.268399\pi\)
\(998\) 12.0000 0.379853
\(999\) 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 1050.2.a.d.1.1 1
3.2 odd 2 3150.2.a.bk.1.1 1
4.3 odd 2 8400.2.a.bp.1.1 1
5.2 odd 4 210.2.g.b.169.1 2
5.3 odd 4 210.2.g.b.169.2 yes 2
5.4 even 2 1050.2.a.p.1.1 1
7.6 odd 2 7350.2.a.bk.1.1 1
15.2 even 4 630.2.g.c.379.2 2
15.8 even 4 630.2.g.c.379.1 2
15.14 odd 2 3150.2.a.d.1.1 1
20.3 even 4 1680.2.t.e.1009.2 2
20.7 even 4 1680.2.t.e.1009.1 2
20.19 odd 2 8400.2.a.w.1.1 1
35.2 odd 12 1470.2.n.b.949.1 4
35.3 even 12 1470.2.n.f.79.1 4
35.12 even 12 1470.2.n.f.949.1 4
35.13 even 4 1470.2.g.b.589.2 2
35.17 even 12 1470.2.n.f.79.2 4
35.18 odd 12 1470.2.n.b.79.1 4
35.23 odd 12 1470.2.n.b.949.2 4
35.27 even 4 1470.2.g.b.589.1 2
35.32 odd 12 1470.2.n.b.79.2 4
35.33 even 12 1470.2.n.f.949.2 4
35.34 odd 2 7350.2.a.bz.1.1 1
60.23 odd 4 5040.2.t.h.1009.2 2
60.47 odd 4 5040.2.t.h.1009.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
210.2.g.b.169.1 2 5.2 odd 4
210.2.g.b.169.2 yes 2 5.3 odd 4
630.2.g.c.379.1 2 15.8 even 4
630.2.g.c.379.2 2 15.2 even 4
1050.2.a.d.1.1 1 1.1 even 1 trivial
1050.2.a.p.1.1 1 5.4 even 2
1470.2.g.b.589.1 2 35.27 even 4
1470.2.g.b.589.2 2 35.13 even 4
1470.2.n.b.79.1 4 35.18 odd 12
1470.2.n.b.79.2 4 35.32 odd 12
1470.2.n.b.949.1 4 35.2 odd 12
1470.2.n.b.949.2 4 35.23 odd 12
1470.2.n.f.79.1 4 35.3 even 12
1470.2.n.f.79.2 4 35.17 even 12
1470.2.n.f.949.1 4 35.12 even 12
1470.2.n.f.949.2 4 35.33 even 12
1680.2.t.e.1009.1 2 20.7 even 4
1680.2.t.e.1009.2 2 20.3 even 4
3150.2.a.d.1.1 1 15.14 odd 2
3150.2.a.bk.1.1 1 3.2 odd 2
5040.2.t.h.1009.1 2 60.47 odd 4
5040.2.t.h.1009.2 2 60.23 odd 4
7350.2.a.bk.1.1 1 7.6 odd 2
7350.2.a.bz.1.1 1 35.34 odd 2
8400.2.a.w.1.1 1 20.19 odd 2
8400.2.a.bp.1.1 1 4.3 odd 2