Properties

Label 4830.2.a.u.1.1
Level $4830$
Weight $2$
Character 4830.1
Self dual yes
Analytic conductor $38.568$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 4830 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4830.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(38.5677441763\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 4830.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -1.00000 q^{12} +6.00000 q^{13} +1.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} +4.00000 q^{17} +1.00000 q^{18} +4.00000 q^{19} -1.00000 q^{20} -1.00000 q^{21} +1.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} +6.00000 q^{26} -1.00000 q^{27} +1.00000 q^{28} -8.00000 q^{29} +1.00000 q^{30} -4.00000 q^{31} +1.00000 q^{32} +4.00000 q^{34} -1.00000 q^{35} +1.00000 q^{36} +2.00000 q^{37} +4.00000 q^{38} -6.00000 q^{39} -1.00000 q^{40} -1.00000 q^{42} +4.00000 q^{43} -1.00000 q^{45} +1.00000 q^{46} -4.00000 q^{47} -1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} -4.00000 q^{51} +6.00000 q^{52} -1.00000 q^{54} +1.00000 q^{56} -4.00000 q^{57} -8.00000 q^{58} -8.00000 q^{59} +1.00000 q^{60} -6.00000 q^{61} -4.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} -6.00000 q^{65} +4.00000 q^{67} +4.00000 q^{68} -1.00000 q^{69} -1.00000 q^{70} +12.0000 q^{71} +1.00000 q^{72} -4.00000 q^{73} +2.00000 q^{74} -1.00000 q^{75} +4.00000 q^{76} -6.00000 q^{78} +12.0000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +4.00000 q^{83} -1.00000 q^{84} -4.00000 q^{85} +4.00000 q^{86} +8.00000 q^{87} -6.00000 q^{89} -1.00000 q^{90} +6.00000 q^{91} +1.00000 q^{92} +4.00000 q^{93} -4.00000 q^{94} -4.00000 q^{95} -1.00000 q^{96} +6.00000 q^{97} +1.00000 q^{98} +O(q^{100})\)

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
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) −1.00000 −0.408248
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) −1.00000 −0.288675
\(13\) 6.00000 1.66410 0.832050 0.554700i \(-0.187167\pi\)
0.832050 + 0.554700i \(0.187167\pi\)
\(14\) 1.00000 0.267261
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 1.00000 0.235702
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) −1.00000 −0.223607
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) 6.00000 1.17670
\(27\) −1.00000 −0.192450
\(28\) 1.00000 0.188982
\(29\) −8.00000 −1.48556 −0.742781 0.669534i \(-0.766494\pi\)
−0.742781 + 0.669534i \(0.766494\pi\)
\(30\) 1.00000 0.182574
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 4.00000 0.685994
\(35\) −1.00000 −0.169031
\(36\) 1.00000 0.166667
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 4.00000 0.648886
\(39\) −6.00000 −0.960769
\(40\) −1.00000 −0.158114
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) −1.00000 −0.154303
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 1.00000 0.147442
\(47\) −4.00000 −0.583460 −0.291730 0.956501i \(-0.594231\pi\)
−0.291730 + 0.956501i \(0.594231\pi\)
\(48\) −1.00000 −0.144338
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) −4.00000 −0.560112
\(52\) 6.00000 0.832050
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) −4.00000 −0.529813
\(58\) −8.00000 −1.05045
\(59\) −8.00000 −1.04151 −0.520756 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) 1.00000 0.129099
\(61\) −6.00000 −0.768221 −0.384111 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) −4.00000 −0.508001
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) −6.00000 −0.744208
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 4.00000 0.485071
\(69\) −1.00000 −0.120386
\(70\) −1.00000 −0.119523
\(71\) 12.0000 1.42414 0.712069 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 1.00000 0.117851
\(73\) −4.00000 −0.468165 −0.234082 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) 2.00000 0.232495
\(75\) −1.00000 −0.115470
\(76\) 4.00000 0.458831
\(77\) 0 0
\(78\) −6.00000 −0.679366
\(79\) 12.0000 1.35011 0.675053 0.737769i \(-0.264121\pi\)
0.675053 + 0.737769i \(0.264121\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) −1.00000 −0.109109
\(85\) −4.00000 −0.433861
\(86\) 4.00000 0.431331
\(87\) 8.00000 0.857690
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) −1.00000 −0.105409
\(91\) 6.00000 0.628971
\(92\) 1.00000 0.104257
\(93\) 4.00000 0.414781
\(94\) −4.00000 −0.412568
\(95\) −4.00000 −0.410391
\(96\) −1.00000 −0.102062
\(97\) 6.00000 0.609208 0.304604 0.952479i \(-0.401476\pi\)
0.304604 + 0.952479i \(0.401476\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 2.00000 0.199007 0.0995037 0.995037i \(-0.468274\pi\)
0.0995037 + 0.995037i \(0.468274\pi\)
\(102\) −4.00000 −0.396059
\(103\) 4.00000 0.394132 0.197066 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) 6.00000 0.588348
\(105\) 1.00000 0.0975900
\(106\) 0 0
\(107\) 14.0000 1.35343 0.676716 0.736245i \(-0.263403\pi\)
0.676716 + 0.736245i \(0.263403\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 4.00000 0.383131 0.191565 0.981480i \(-0.438644\pi\)
0.191565 + 0.981480i \(0.438644\pi\)
\(110\) 0 0
\(111\) −2.00000 −0.189832
\(112\) 1.00000 0.0944911
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) −4.00000 −0.374634
\(115\) −1.00000 −0.0932505
\(116\) −8.00000 −0.742781
\(117\) 6.00000 0.554700
\(118\) −8.00000 −0.736460
\(119\) 4.00000 0.366679
\(120\) 1.00000 0.0912871
\(121\) −11.0000 −1.00000
\(122\) −6.00000 −0.543214
\(123\) 0 0
\(124\) −4.00000 −0.359211
\(125\) −1.00000 −0.0894427
\(126\) 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
\(129\) −4.00000 −0.352180
\(130\) −6.00000 −0.526235
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 4.00000 0.346844
\(134\) 4.00000 0.345547
\(135\) 1.00000 0.0860663
\(136\) 4.00000 0.342997
\(137\) 2.00000 0.170872 0.0854358 0.996344i \(-0.472772\pi\)
0.0854358 + 0.996344i \(0.472772\pi\)
\(138\) −1.00000 −0.0851257
\(139\) 20.0000 1.69638 0.848189 0.529694i \(-0.177693\pi\)
0.848189 + 0.529694i \(0.177693\pi\)
\(140\) −1.00000 −0.0845154
\(141\) 4.00000 0.336861
\(142\) 12.0000 1.00702
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 8.00000 0.664364
\(146\) −4.00000 −0.331042
\(147\) −1.00000 −0.0824786
\(148\) 2.00000 0.164399
\(149\) 10.0000 0.819232 0.409616 0.912258i \(-0.365663\pi\)
0.409616 + 0.912258i \(0.365663\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 16.0000 1.30206 0.651031 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) 4.00000 0.324443
\(153\) 4.00000 0.323381
\(154\) 0 0
\(155\) 4.00000 0.321288
\(156\) −6.00000 −0.480384
\(157\) −2.00000 −0.159617 −0.0798087 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) 12.0000 0.954669
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) 1.00000 0.0788110
\(162\) 1.00000 0.0785674
\(163\) −2.00000 −0.156652 −0.0783260 0.996928i \(-0.524958\pi\)
−0.0783260 + 0.996928i \(0.524958\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 4.00000 0.310460
\(167\) −16.0000 −1.23812 −0.619059 0.785345i \(-0.712486\pi\)
−0.619059 + 0.785345i \(0.712486\pi\)
\(168\) −1.00000 −0.0771517
\(169\) 23.0000 1.76923
\(170\) −4.00000 −0.306786
\(171\) 4.00000 0.305888
\(172\) 4.00000 0.304997
\(173\) 10.0000 0.760286 0.380143 0.924928i \(-0.375875\pi\)
0.380143 + 0.924928i \(0.375875\pi\)
\(174\) 8.00000 0.606478
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) 8.00000 0.601317
\(178\) −6.00000 −0.449719
\(179\) 10.0000 0.747435 0.373718 0.927543i \(-0.378083\pi\)
0.373718 + 0.927543i \(0.378083\pi\)
\(180\) −1.00000 −0.0745356
\(181\) 14.0000 1.04061 0.520306 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) 6.00000 0.444750
\(183\) 6.00000 0.443533
\(184\) 1.00000 0.0737210
\(185\) −2.00000 −0.147043
\(186\) 4.00000 0.293294
\(187\) 0 0
\(188\) −4.00000 −0.291730
\(189\) −1.00000 −0.0727393
\(190\) −4.00000 −0.290191
\(191\) 8.00000 0.578860 0.289430 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −10.0000 −0.719816 −0.359908 0.932988i \(-0.617192\pi\)
−0.359908 + 0.932988i \(0.617192\pi\)
\(194\) 6.00000 0.430775
\(195\) 6.00000 0.429669
\(196\) 1.00000 0.0714286
\(197\) −2.00000 −0.142494 −0.0712470 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) 0 0
\(199\) −14.0000 −0.992434 −0.496217 0.868199i \(-0.665278\pi\)
−0.496217 + 0.868199i \(0.665278\pi\)
\(200\) 1.00000 0.0707107
\(201\) −4.00000 −0.282138
\(202\) 2.00000 0.140720
\(203\) −8.00000 −0.561490
\(204\) −4.00000 −0.280056
\(205\) 0 0
\(206\) 4.00000 0.278693
\(207\) 1.00000 0.0695048
\(208\) 6.00000 0.416025
\(209\) 0 0
\(210\) 1.00000 0.0690066
\(211\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(212\) 0 0
\(213\) −12.0000 −0.822226
\(214\) 14.0000 0.957020
\(215\) −4.00000 −0.272798
\(216\) −1.00000 −0.0680414
\(217\) −4.00000 −0.271538
\(218\) 4.00000 0.270914
\(219\) 4.00000 0.270295
\(220\) 0 0
\(221\) 24.0000 1.61441
\(222\) −2.00000 −0.134231
\(223\) −2.00000 −0.133930 −0.0669650 0.997755i \(-0.521332\pi\)
−0.0669650 + 0.997755i \(0.521332\pi\)
\(224\) 1.00000 0.0668153
\(225\) 1.00000 0.0666667
\(226\) −6.00000 −0.399114
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) −4.00000 −0.264906
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) −1.00000 −0.0659380
\(231\) 0 0
\(232\) −8.00000 −0.525226
\(233\) 22.0000 1.44127 0.720634 0.693316i \(-0.243851\pi\)
0.720634 + 0.693316i \(0.243851\pi\)
\(234\) 6.00000 0.392232
\(235\) 4.00000 0.260931
\(236\) −8.00000 −0.520756
\(237\) −12.0000 −0.779484
\(238\) 4.00000 0.259281
\(239\) 12.0000 0.776215 0.388108 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) 1.00000 0.0645497
\(241\) 4.00000 0.257663 0.128831 0.991667i \(-0.458877\pi\)
0.128831 + 0.991667i \(0.458877\pi\)
\(242\) −11.0000 −0.707107
\(243\) −1.00000 −0.0641500
\(244\) −6.00000 −0.384111
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) 24.0000 1.52708
\(248\) −4.00000 −0.254000
\(249\) −4.00000 −0.253490
\(250\) −1.00000 −0.0632456
\(251\) −28.0000 −1.76734 −0.883672 0.468106i \(-0.844936\pi\)
−0.883672 + 0.468106i \(0.844936\pi\)
\(252\) 1.00000 0.0629941
\(253\) 0 0
\(254\) 8.00000 0.501965
\(255\) 4.00000 0.250490
\(256\) 1.00000 0.0625000
\(257\) 6.00000 0.374270 0.187135 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) −4.00000 −0.249029
\(259\) 2.00000 0.124274
\(260\) −6.00000 −0.372104
\(261\) −8.00000 −0.495188
\(262\) 0 0
\(263\) −24.0000 −1.47990 −0.739952 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 4.00000 0.245256
\(267\) 6.00000 0.367194
\(268\) 4.00000 0.244339
\(269\) −10.0000 −0.609711 −0.304855 0.952399i \(-0.598608\pi\)
−0.304855 + 0.952399i \(0.598608\pi\)
\(270\) 1.00000 0.0608581
\(271\) −4.00000 −0.242983 −0.121491 0.992592i \(-0.538768\pi\)
−0.121491 + 0.992592i \(0.538768\pi\)
\(272\) 4.00000 0.242536
\(273\) −6.00000 −0.363137
\(274\) 2.00000 0.120824
\(275\) 0 0
\(276\) −1.00000 −0.0601929
\(277\) 24.0000 1.44202 0.721010 0.692925i \(-0.243678\pi\)
0.721010 + 0.692925i \(0.243678\pi\)
\(278\) 20.0000 1.19952
\(279\) −4.00000 −0.239474
\(280\) −1.00000 −0.0597614
\(281\) −26.0000 −1.55103 −0.775515 0.631329i \(-0.782510\pi\)
−0.775515 + 0.631329i \(0.782510\pi\)
\(282\) 4.00000 0.238197
\(283\) −20.0000 −1.18888 −0.594438 0.804141i \(-0.702626\pi\)
−0.594438 + 0.804141i \(0.702626\pi\)
\(284\) 12.0000 0.712069
\(285\) 4.00000 0.236940
\(286\) 0 0
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) −1.00000 −0.0588235
\(290\) 8.00000 0.469776
\(291\) −6.00000 −0.351726
\(292\) −4.00000 −0.234082
\(293\) −14.0000 −0.817889 −0.408944 0.912559i \(-0.634103\pi\)
−0.408944 + 0.912559i \(0.634103\pi\)
\(294\) −1.00000 −0.0583212
\(295\) 8.00000 0.465778
\(296\) 2.00000 0.116248
\(297\) 0 0
\(298\) 10.0000 0.579284
\(299\) 6.00000 0.346989
\(300\) −1.00000 −0.0577350
\(301\) 4.00000 0.230556
\(302\) 16.0000 0.920697
\(303\) −2.00000 −0.114897
\(304\) 4.00000 0.229416
\(305\) 6.00000 0.343559
\(306\) 4.00000 0.228665
\(307\) −16.0000 −0.913168 −0.456584 0.889680i \(-0.650927\pi\)
−0.456584 + 0.889680i \(0.650927\pi\)
\(308\) 0 0
\(309\) −4.00000 −0.227552
\(310\) 4.00000 0.227185
\(311\) −10.0000 −0.567048 −0.283524 0.958965i \(-0.591504\pi\)
−0.283524 + 0.958965i \(0.591504\pi\)
\(312\) −6.00000 −0.339683
\(313\) −14.0000 −0.791327 −0.395663 0.918396i \(-0.629485\pi\)
−0.395663 + 0.918396i \(0.629485\pi\)
\(314\) −2.00000 −0.112867
\(315\) −1.00000 −0.0563436
\(316\) 12.0000 0.675053
\(317\) 10.0000 0.561656 0.280828 0.959758i \(-0.409391\pi\)
0.280828 + 0.959758i \(0.409391\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) −14.0000 −0.781404
\(322\) 1.00000 0.0557278
\(323\) 16.0000 0.890264
\(324\) 1.00000 0.0555556
\(325\) 6.00000 0.332820
\(326\) −2.00000 −0.110770
\(327\) −4.00000 −0.221201
\(328\) 0 0
\(329\) −4.00000 −0.220527
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) 4.00000 0.219529
\(333\) 2.00000 0.109599
\(334\) −16.0000 −0.875481
\(335\) −4.00000 −0.218543
\(336\) −1.00000 −0.0545545
\(337\) −18.0000 −0.980522 −0.490261 0.871576i \(-0.663099\pi\)
−0.490261 + 0.871576i \(0.663099\pi\)
\(338\) 23.0000 1.25104
\(339\) 6.00000 0.325875
\(340\) −4.00000 −0.216930
\(341\) 0 0
\(342\) 4.00000 0.216295
\(343\) 1.00000 0.0539949
\(344\) 4.00000 0.215666
\(345\) 1.00000 0.0538382
\(346\) 10.0000 0.537603
\(347\) −12.0000 −0.644194 −0.322097 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) 8.00000 0.428845
\(349\) 10.0000 0.535288 0.267644 0.963518i \(-0.413755\pi\)
0.267644 + 0.963518i \(0.413755\pi\)
\(350\) 1.00000 0.0534522
\(351\) −6.00000 −0.320256
\(352\) 0 0
\(353\) 6.00000 0.319348 0.159674 0.987170i \(-0.448956\pi\)
0.159674 + 0.987170i \(0.448956\pi\)
\(354\) 8.00000 0.425195
\(355\) −12.0000 −0.636894
\(356\) −6.00000 −0.317999
\(357\) −4.00000 −0.211702
\(358\) 10.0000 0.528516
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −3.00000 −0.157895
\(362\) 14.0000 0.735824
\(363\) 11.0000 0.577350
\(364\) 6.00000 0.314485
\(365\) 4.00000 0.209370
\(366\) 6.00000 0.313625
\(367\) −8.00000 −0.417597 −0.208798 0.977959i \(-0.566955\pi\)
−0.208798 + 0.977959i \(0.566955\pi\)
\(368\) 1.00000 0.0521286
\(369\) 0 0
\(370\) −2.00000 −0.103975
\(371\) 0 0
\(372\) 4.00000 0.207390
\(373\) −10.0000 −0.517780 −0.258890 0.965907i \(-0.583357\pi\)
−0.258890 + 0.965907i \(0.583357\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) −4.00000 −0.206284
\(377\) −48.0000 −2.47213
\(378\) −1.00000 −0.0514344
\(379\) 30.0000 1.54100 0.770498 0.637442i \(-0.220007\pi\)
0.770498 + 0.637442i \(0.220007\pi\)
\(380\) −4.00000 −0.205196
\(381\) −8.00000 −0.409852
\(382\) 8.00000 0.409316
\(383\) 10.0000 0.510976 0.255488 0.966812i \(-0.417764\pi\)
0.255488 + 0.966812i \(0.417764\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −10.0000 −0.508987
\(387\) 4.00000 0.203331
\(388\) 6.00000 0.304604
\(389\) 30.0000 1.52106 0.760530 0.649303i \(-0.224939\pi\)
0.760530 + 0.649303i \(0.224939\pi\)
\(390\) 6.00000 0.303822
\(391\) 4.00000 0.202289
\(392\) 1.00000 0.0505076
\(393\) 0 0
\(394\) −2.00000 −0.100759
\(395\) −12.0000 −0.603786
\(396\) 0 0
\(397\) −2.00000 −0.100377 −0.0501886 0.998740i \(-0.515982\pi\)
−0.0501886 + 0.998740i \(0.515982\pi\)
\(398\) −14.0000 −0.701757
\(399\) −4.00000 −0.200250
\(400\) 1.00000 0.0500000
\(401\) −30.0000 −1.49813 −0.749064 0.662497i \(-0.769497\pi\)
−0.749064 + 0.662497i \(0.769497\pi\)
\(402\) −4.00000 −0.199502
\(403\) −24.0000 −1.19553
\(404\) 2.00000 0.0995037
\(405\) −1.00000 −0.0496904
\(406\) −8.00000 −0.397033
\(407\) 0 0
\(408\) −4.00000 −0.198030
\(409\) −14.0000 −0.692255 −0.346128 0.938187i \(-0.612504\pi\)
−0.346128 + 0.938187i \(0.612504\pi\)
\(410\) 0 0
\(411\) −2.00000 −0.0986527
\(412\) 4.00000 0.197066
\(413\) −8.00000 −0.393654
\(414\) 1.00000 0.0491473
\(415\) −4.00000 −0.196352
\(416\) 6.00000 0.294174
\(417\) −20.0000 −0.979404
\(418\) 0 0
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 1.00000 0.0487950
\(421\) −16.0000 −0.779792 −0.389896 0.920859i \(-0.627489\pi\)
−0.389896 + 0.920859i \(0.627489\pi\)
\(422\) 0 0
\(423\) −4.00000 −0.194487
\(424\) 0 0
\(425\) 4.00000 0.194029
\(426\) −12.0000 −0.581402
\(427\) −6.00000 −0.290360
\(428\) 14.0000 0.676716
\(429\) 0 0
\(430\) −4.00000 −0.192897
\(431\) −16.0000 −0.770693 −0.385346 0.922772i \(-0.625918\pi\)
−0.385346 + 0.922772i \(0.625918\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 14.0000 0.672797 0.336399 0.941720i \(-0.390791\pi\)
0.336399 + 0.941720i \(0.390791\pi\)
\(434\) −4.00000 −0.192006
\(435\) −8.00000 −0.383571
\(436\) 4.00000 0.191565
\(437\) 4.00000 0.191346
\(438\) 4.00000 0.191127
\(439\) 36.0000 1.71819 0.859093 0.511819i \(-0.171028\pi\)
0.859093 + 0.511819i \(0.171028\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 24.0000 1.14156
\(443\) −20.0000 −0.950229 −0.475114 0.879924i \(-0.657593\pi\)
−0.475114 + 0.879924i \(0.657593\pi\)
\(444\) −2.00000 −0.0949158
\(445\) 6.00000 0.284427
\(446\) −2.00000 −0.0947027
\(447\) −10.0000 −0.472984
\(448\) 1.00000 0.0472456
\(449\) 6.00000 0.283158 0.141579 0.989927i \(-0.454782\pi\)
0.141579 + 0.989927i \(0.454782\pi\)
\(450\) 1.00000 0.0471405
\(451\) 0 0
\(452\) −6.00000 −0.282216
\(453\) −16.0000 −0.751746
\(454\) 0 0
\(455\) −6.00000 −0.281284
\(456\) −4.00000 −0.187317
\(457\) 6.00000 0.280668 0.140334 0.990104i \(-0.455182\pi\)
0.140334 + 0.990104i \(0.455182\pi\)
\(458\) −10.0000 −0.467269
\(459\) −4.00000 −0.186704
\(460\) −1.00000 −0.0466252
\(461\) 30.0000 1.39724 0.698620 0.715493i \(-0.253798\pi\)
0.698620 + 0.715493i \(0.253798\pi\)
\(462\) 0 0
\(463\) 32.0000 1.48717 0.743583 0.668644i \(-0.233125\pi\)
0.743583 + 0.668644i \(0.233125\pi\)
\(464\) −8.00000 −0.371391
\(465\) −4.00000 −0.185496
\(466\) 22.0000 1.01913
\(467\) −8.00000 −0.370196 −0.185098 0.982720i \(-0.559260\pi\)
−0.185098 + 0.982720i \(0.559260\pi\)
\(468\) 6.00000 0.277350
\(469\) 4.00000 0.184703
\(470\) 4.00000 0.184506
\(471\) 2.00000 0.0921551
\(472\) −8.00000 −0.368230
\(473\) 0 0
\(474\) −12.0000 −0.551178
\(475\) 4.00000 0.183533
\(476\) 4.00000 0.183340
\(477\) 0 0
\(478\) 12.0000 0.548867
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 1.00000 0.0456435
\(481\) 12.0000 0.547153
\(482\) 4.00000 0.182195
\(483\) −1.00000 −0.0455016
\(484\) −11.0000 −0.500000
\(485\) −6.00000 −0.272446
\(486\) −1.00000 −0.0453609
\(487\) 20.0000 0.906287 0.453143 0.891438i \(-0.350303\pi\)
0.453143 + 0.891438i \(0.350303\pi\)
\(488\) −6.00000 −0.271607
\(489\) 2.00000 0.0904431
\(490\) −1.00000 −0.0451754
\(491\) 22.0000 0.992846 0.496423 0.868081i \(-0.334646\pi\)
0.496423 + 0.868081i \(0.334646\pi\)
\(492\) 0 0
\(493\) −32.0000 −1.44121
\(494\) 24.0000 1.07981
\(495\) 0 0
\(496\) −4.00000 −0.179605
\(497\) 12.0000 0.538274
\(498\) −4.00000 −0.179244
\(499\) 32.0000 1.43252 0.716258 0.697835i \(-0.245853\pi\)
0.716258 + 0.697835i \(0.245853\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 16.0000 0.714827
\(502\) −28.0000 −1.24970
\(503\) 2.00000 0.0891756 0.0445878 0.999005i \(-0.485803\pi\)
0.0445878 + 0.999005i \(0.485803\pi\)
\(504\) 1.00000 0.0445435
\(505\) −2.00000 −0.0889988
\(506\) 0 0
\(507\) −23.0000 −1.02147
\(508\) 8.00000 0.354943
\(509\) −2.00000 −0.0886484 −0.0443242 0.999017i \(-0.514113\pi\)
−0.0443242 + 0.999017i \(0.514113\pi\)
\(510\) 4.00000 0.177123
\(511\) −4.00000 −0.176950
\(512\) 1.00000 0.0441942
\(513\) −4.00000 −0.176604
\(514\) 6.00000 0.264649
\(515\) −4.00000 −0.176261
\(516\) −4.00000 −0.176090
\(517\) 0 0
\(518\) 2.00000 0.0878750
\(519\) −10.0000 −0.438951
\(520\) −6.00000 −0.263117
\(521\) −18.0000 −0.788594 −0.394297 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) −8.00000 −0.350150
\(523\) 20.0000 0.874539 0.437269 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) 0 0
\(525\) −1.00000 −0.0436436
\(526\) −24.0000 −1.04645
\(527\) −16.0000 −0.696971
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −8.00000 −0.347170
\(532\) 4.00000 0.173422
\(533\) 0 0
\(534\) 6.00000 0.259645
\(535\) −14.0000 −0.605273
\(536\) 4.00000 0.172774
\(537\) −10.0000 −0.431532
\(538\) −10.0000 −0.431131
\(539\) 0 0
\(540\) 1.00000 0.0430331
\(541\) −10.0000 −0.429934 −0.214967 0.976621i \(-0.568964\pi\)
−0.214967 + 0.976621i \(0.568964\pi\)
\(542\) −4.00000 −0.171815
\(543\) −14.0000 −0.600798
\(544\) 4.00000 0.171499
\(545\) −4.00000 −0.171341
\(546\) −6.00000 −0.256776
\(547\) −6.00000 −0.256541 −0.128271 0.991739i \(-0.540943\pi\)
−0.128271 + 0.991739i \(0.540943\pi\)
\(548\) 2.00000 0.0854358
\(549\) −6.00000 −0.256074
\(550\) 0 0
\(551\) −32.0000 −1.36325
\(552\) −1.00000 −0.0425628
\(553\) 12.0000 0.510292
\(554\) 24.0000 1.01966
\(555\) 2.00000 0.0848953
\(556\) 20.0000 0.848189
\(557\) −28.0000 −1.18640 −0.593199 0.805056i \(-0.702135\pi\)
−0.593199 + 0.805056i \(0.702135\pi\)
\(558\) −4.00000 −0.169334
\(559\) 24.0000 1.01509
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) −26.0000 −1.09674
\(563\) −32.0000 −1.34864 −0.674320 0.738440i \(-0.735563\pi\)
−0.674320 + 0.738440i \(0.735563\pi\)
\(564\) 4.00000 0.168430
\(565\) 6.00000 0.252422
\(566\) −20.0000 −0.840663
\(567\) 1.00000 0.0419961
\(568\) 12.0000 0.503509
\(569\) −22.0000 −0.922288 −0.461144 0.887325i \(-0.652561\pi\)
−0.461144 + 0.887325i \(0.652561\pi\)
\(570\) 4.00000 0.167542
\(571\) −18.0000 −0.753277 −0.376638 0.926360i \(-0.622920\pi\)
−0.376638 + 0.926360i \(0.622920\pi\)
\(572\) 0 0
\(573\) −8.00000 −0.334205
\(574\) 0 0
\(575\) 1.00000 0.0417029
\(576\) 1.00000 0.0416667
\(577\) −8.00000 −0.333044 −0.166522 0.986038i \(-0.553254\pi\)
−0.166522 + 0.986038i \(0.553254\pi\)
\(578\) −1.00000 −0.0415945
\(579\) 10.0000 0.415586
\(580\) 8.00000 0.332182
\(581\) 4.00000 0.165948
\(582\) −6.00000 −0.248708
\(583\) 0 0
\(584\) −4.00000 −0.165521
\(585\) −6.00000 −0.248069
\(586\) −14.0000 −0.578335
\(587\) 28.0000 1.15568 0.577842 0.816149i \(-0.303895\pi\)
0.577842 + 0.816149i \(0.303895\pi\)
\(588\) −1.00000 −0.0412393
\(589\) −16.0000 −0.659269
\(590\) 8.00000 0.329355
\(591\) 2.00000 0.0822690
\(592\) 2.00000 0.0821995
\(593\) 14.0000 0.574911 0.287456 0.957794i \(-0.407191\pi\)
0.287456 + 0.957794i \(0.407191\pi\)
\(594\) 0 0
\(595\) −4.00000 −0.163984
\(596\) 10.0000 0.409616
\(597\) 14.0000 0.572982
\(598\) 6.00000 0.245358
\(599\) −16.0000 −0.653742 −0.326871 0.945069i \(-0.605994\pi\)
−0.326871 + 0.945069i \(0.605994\pi\)
\(600\) −1.00000 −0.0408248
\(601\) 10.0000 0.407909 0.203954 0.978980i \(-0.434621\pi\)
0.203954 + 0.978980i \(0.434621\pi\)
\(602\) 4.00000 0.163028
\(603\) 4.00000 0.162893
\(604\) 16.0000 0.651031
\(605\) 11.0000 0.447214
\(606\) −2.00000 −0.0812444
\(607\) −38.0000 −1.54237 −0.771186 0.636610i \(-0.780336\pi\)
−0.771186 + 0.636610i \(0.780336\pi\)
\(608\) 4.00000 0.162221
\(609\) 8.00000 0.324176
\(610\) 6.00000 0.242933
\(611\) −24.0000 −0.970936
\(612\) 4.00000 0.161690
\(613\) −34.0000 −1.37325 −0.686624 0.727013i \(-0.740908\pi\)
−0.686624 + 0.727013i \(0.740908\pi\)
\(614\) −16.0000 −0.645707
\(615\) 0 0
\(616\) 0 0
\(617\) 22.0000 0.885687 0.442843 0.896599i \(-0.353970\pi\)
0.442843 + 0.896599i \(0.353970\pi\)
\(618\) −4.00000 −0.160904
\(619\) 12.0000 0.482321 0.241160 0.970485i \(-0.422472\pi\)
0.241160 + 0.970485i \(0.422472\pi\)
\(620\) 4.00000 0.160644
\(621\) −1.00000 −0.0401286
\(622\) −10.0000 −0.400963
\(623\) −6.00000 −0.240385
\(624\) −6.00000 −0.240192
\(625\) 1.00000 0.0400000
\(626\) −14.0000 −0.559553
\(627\) 0 0
\(628\) −2.00000 −0.0798087
\(629\) 8.00000 0.318981
\(630\) −1.00000 −0.0398410
\(631\) 4.00000 0.159237 0.0796187 0.996825i \(-0.474630\pi\)
0.0796187 + 0.996825i \(0.474630\pi\)
\(632\) 12.0000 0.477334
\(633\) 0 0
\(634\) 10.0000 0.397151
\(635\) −8.00000 −0.317470
\(636\) 0 0
\(637\) 6.00000 0.237729
\(638\) 0 0
\(639\) 12.0000 0.474713
\(640\) −1.00000 −0.0395285
\(641\) −34.0000 −1.34292 −0.671460 0.741041i \(-0.734332\pi\)
−0.671460 + 0.741041i \(0.734332\pi\)
\(642\) −14.0000 −0.552536
\(643\) −28.0000 −1.10421 −0.552106 0.833774i \(-0.686176\pi\)
−0.552106 + 0.833774i \(0.686176\pi\)
\(644\) 1.00000 0.0394055
\(645\) 4.00000 0.157500
\(646\) 16.0000 0.629512
\(647\) 40.0000 1.57256 0.786281 0.617869i \(-0.212004\pi\)
0.786281 + 0.617869i \(0.212004\pi\)
\(648\) 1.00000 0.0392837
\(649\) 0 0
\(650\) 6.00000 0.235339
\(651\) 4.00000 0.156772
\(652\) −2.00000 −0.0783260
\(653\) −10.0000 −0.391330 −0.195665 0.980671i \(-0.562687\pi\)
−0.195665 + 0.980671i \(0.562687\pi\)
\(654\) −4.00000 −0.156412
\(655\) 0 0
\(656\) 0 0
\(657\) −4.00000 −0.156055
\(658\) −4.00000 −0.155936
\(659\) 44.0000 1.71400 0.856998 0.515319i \(-0.172327\pi\)
0.856998 + 0.515319i \(0.172327\pi\)
\(660\) 0 0
\(661\) −46.0000 −1.78919 −0.894596 0.446875i \(-0.852537\pi\)
−0.894596 + 0.446875i \(0.852537\pi\)
\(662\) 20.0000 0.777322
\(663\) −24.0000 −0.932083
\(664\) 4.00000 0.155230
\(665\) −4.00000 −0.155113
\(666\) 2.00000 0.0774984
\(667\) −8.00000 −0.309761
\(668\) −16.0000 −0.619059
\(669\) 2.00000 0.0773245
\(670\) −4.00000 −0.154533
\(671\) 0 0
\(672\) −1.00000 −0.0385758
\(673\) −6.00000 −0.231283 −0.115642 0.993291i \(-0.536892\pi\)
−0.115642 + 0.993291i \(0.536892\pi\)
\(674\) −18.0000 −0.693334
\(675\) −1.00000 −0.0384900
\(676\) 23.0000 0.884615
\(677\) −42.0000 −1.61419 −0.807096 0.590421i \(-0.798962\pi\)
−0.807096 + 0.590421i \(0.798962\pi\)
\(678\) 6.00000 0.230429
\(679\) 6.00000 0.230259
\(680\) −4.00000 −0.153393
\(681\) 0 0
\(682\) 0 0
\(683\) 4.00000 0.153056 0.0765279 0.997067i \(-0.475617\pi\)
0.0765279 + 0.997067i \(0.475617\pi\)
\(684\) 4.00000 0.152944
\(685\) −2.00000 −0.0764161
\(686\) 1.00000 0.0381802
\(687\) 10.0000 0.381524
\(688\) 4.00000 0.152499
\(689\) 0 0
\(690\) 1.00000 0.0380693
\(691\) −20.0000 −0.760836 −0.380418 0.924815i \(-0.624220\pi\)
−0.380418 + 0.924815i \(0.624220\pi\)
\(692\) 10.0000 0.380143
\(693\) 0 0
\(694\) −12.0000 −0.455514
\(695\) −20.0000 −0.758643
\(696\) 8.00000 0.303239
\(697\) 0 0
\(698\) 10.0000 0.378506
\(699\) −22.0000 −0.832116
\(700\) 1.00000 0.0377964
\(701\) −10.0000 −0.377695 −0.188847 0.982006i \(-0.560475\pi\)
−0.188847 + 0.982006i \(0.560475\pi\)
\(702\) −6.00000 −0.226455
\(703\) 8.00000 0.301726
\(704\) 0 0
\(705\) −4.00000 −0.150649
\(706\) 6.00000 0.225813
\(707\) 2.00000 0.0752177
\(708\) 8.00000 0.300658
\(709\) −4.00000 −0.150223 −0.0751116 0.997175i \(-0.523931\pi\)
−0.0751116 + 0.997175i \(0.523931\pi\)
\(710\) −12.0000 −0.450352
\(711\) 12.0000 0.450035
\(712\) −6.00000 −0.224860
\(713\) −4.00000 −0.149801
\(714\) −4.00000 −0.149696
\(715\) 0 0
\(716\) 10.0000 0.373718
\(717\) −12.0000 −0.448148
\(718\) 24.0000 0.895672
\(719\) −26.0000 −0.969636 −0.484818 0.874615i \(-0.661114\pi\)
−0.484818 + 0.874615i \(0.661114\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 4.00000 0.148968
\(722\) −3.00000 −0.111648
\(723\) −4.00000 −0.148762
\(724\) 14.0000 0.520306
\(725\) −8.00000 −0.297113
\(726\) 11.0000 0.408248
\(727\) −28.0000 −1.03846 −0.519231 0.854634i \(-0.673782\pi\)
−0.519231 + 0.854634i \(0.673782\pi\)
\(728\) 6.00000 0.222375
\(729\) 1.00000 0.0370370
\(730\) 4.00000 0.148047
\(731\) 16.0000 0.591781
\(732\) 6.00000 0.221766
\(733\) 34.0000 1.25582 0.627909 0.778287i \(-0.283911\pi\)
0.627909 + 0.778287i \(0.283911\pi\)
\(734\) −8.00000 −0.295285
\(735\) 1.00000 0.0368856
\(736\) 1.00000 0.0368605
\(737\) 0 0
\(738\) 0 0
\(739\) −16.0000 −0.588570 −0.294285 0.955718i \(-0.595081\pi\)
−0.294285 + 0.955718i \(0.595081\pi\)
\(740\) −2.00000 −0.0735215
\(741\) −24.0000 −0.881662
\(742\) 0 0
\(743\) −24.0000 −0.880475 −0.440237 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) 4.00000 0.146647
\(745\) −10.0000 −0.366372
\(746\) −10.0000 −0.366126
\(747\) 4.00000 0.146352
\(748\) 0 0
\(749\) 14.0000 0.511549
\(750\) 1.00000 0.0365148
\(751\) 24.0000 0.875772 0.437886 0.899030i \(-0.355727\pi\)
0.437886 + 0.899030i \(0.355727\pi\)
\(752\) −4.00000 −0.145865
\(753\) 28.0000 1.02038
\(754\) −48.0000 −1.74806
\(755\) −16.0000 −0.582300
\(756\) −1.00000 −0.0363696
\(757\) 26.0000 0.944986 0.472493 0.881334i \(-0.343354\pi\)
0.472493 + 0.881334i \(0.343354\pi\)
\(758\) 30.0000 1.08965
\(759\) 0 0
\(760\) −4.00000 −0.145095
\(761\) 16.0000 0.580000 0.290000 0.957027i \(-0.406345\pi\)
0.290000 + 0.957027i \(0.406345\pi\)
\(762\) −8.00000 −0.289809
\(763\) 4.00000 0.144810
\(764\) 8.00000 0.289430
\(765\) −4.00000 −0.144620
\(766\) 10.0000 0.361315
\(767\) −48.0000 −1.73318
\(768\) −1.00000 −0.0360844
\(769\) 16.0000 0.576975 0.288487 0.957484i \(-0.406848\pi\)
0.288487 + 0.957484i \(0.406848\pi\)
\(770\) 0 0
\(771\) −6.00000 −0.216085
\(772\) −10.0000 −0.359908
\(773\) −30.0000 −1.07903 −0.539513 0.841978i \(-0.681391\pi\)
−0.539513 + 0.841978i \(0.681391\pi\)
\(774\) 4.00000 0.143777
\(775\) −4.00000 −0.143684
\(776\) 6.00000 0.215387
\(777\) −2.00000 −0.0717496
\(778\) 30.0000 1.07555
\(779\) 0 0
\(780\) 6.00000 0.214834
\(781\) 0 0
\(782\) 4.00000 0.143040
\(783\) 8.00000 0.285897
\(784\) 1.00000 0.0357143
\(785\) 2.00000 0.0713831
\(786\) 0 0
\(787\) −28.0000 −0.998092 −0.499046 0.866575i \(-0.666316\pi\)
−0.499046 + 0.866575i \(0.666316\pi\)
\(788\) −2.00000 −0.0712470
\(789\) 24.0000 0.854423
\(790\) −12.0000 −0.426941
\(791\) −6.00000 −0.213335
\(792\) 0 0
\(793\) −36.0000 −1.27840
\(794\) −2.00000 −0.0709773
\(795\) 0 0
\(796\) −14.0000 −0.496217
\(797\) 22.0000 0.779280 0.389640 0.920967i \(-0.372599\pi\)
0.389640 + 0.920967i \(0.372599\pi\)
\(798\) −4.00000 −0.141598
\(799\) −16.0000 −0.566039
\(800\) 1.00000 0.0353553
\(801\) −6.00000 −0.212000
\(802\) −30.0000 −1.05934
\(803\) 0 0
\(804\) −4.00000 −0.141069
\(805\) −1.00000 −0.0352454
\(806\) −24.0000 −0.845364
\(807\) 10.0000 0.352017
\(808\) 2.00000 0.0703598
\(809\) −50.0000 −1.75791 −0.878953 0.476908i \(-0.841757\pi\)
−0.878953 + 0.476908i \(0.841757\pi\)
\(810\) −1.00000 −0.0351364
\(811\) 4.00000 0.140459 0.0702295 0.997531i \(-0.477627\pi\)
0.0702295 + 0.997531i \(0.477627\pi\)
\(812\) −8.00000 −0.280745
\(813\) 4.00000 0.140286
\(814\) 0 0
\(815\) 2.00000 0.0700569
\(816\) −4.00000 −0.140028
\(817\) 16.0000 0.559769
\(818\) −14.0000 −0.489499
\(819\) 6.00000 0.209657
\(820\) 0 0
\(821\) 24.0000 0.837606 0.418803 0.908077i \(-0.362450\pi\)
0.418803 + 0.908077i \(0.362450\pi\)
\(822\) −2.00000 −0.0697580
\(823\) −16.0000 −0.557725 −0.278862 0.960331i \(-0.589957\pi\)
−0.278862 + 0.960331i \(0.589957\pi\)
\(824\) 4.00000 0.139347
\(825\) 0 0
\(826\) −8.00000 −0.278356
\(827\) −22.0000 −0.765015 −0.382507 0.923952i \(-0.624939\pi\)
−0.382507 + 0.923952i \(0.624939\pi\)
\(828\) 1.00000 0.0347524
\(829\) 14.0000 0.486240 0.243120 0.969996i \(-0.421829\pi\)
0.243120 + 0.969996i \(0.421829\pi\)
\(830\) −4.00000 −0.138842
\(831\) −24.0000 −0.832551
\(832\) 6.00000 0.208013
\(833\) 4.00000 0.138592
\(834\) −20.0000 −0.692543
\(835\) 16.0000 0.553703
\(836\) 0 0
\(837\) 4.00000 0.138260
\(838\) 12.0000 0.414533
\(839\) 12.0000 0.414286 0.207143 0.978311i \(-0.433583\pi\)
0.207143 + 0.978311i \(0.433583\pi\)
\(840\) 1.00000 0.0345033
\(841\) 35.0000 1.20690
\(842\) −16.0000 −0.551396
\(843\) 26.0000 0.895488
\(844\) 0 0
\(845\) −23.0000 −0.791224
\(846\) −4.00000 −0.137523
\(847\) −11.0000 −0.377964
\(848\) 0 0
\(849\) 20.0000 0.686398
\(850\) 4.00000 0.137199
\(851\) 2.00000 0.0685591
\(852\) −12.0000 −0.411113
\(853\) −34.0000 −1.16414 −0.582069 0.813139i \(-0.697757\pi\)
−0.582069 + 0.813139i \(0.697757\pi\)
\(854\) −6.00000 −0.205316
\(855\) −4.00000 −0.136797
\(856\) 14.0000 0.478510
\(857\) 42.0000 1.43469 0.717346 0.696717i \(-0.245357\pi\)
0.717346 + 0.696717i \(0.245357\pi\)
\(858\) 0 0
\(859\) −20.0000 −0.682391 −0.341196 0.939992i \(-0.610832\pi\)
−0.341196 + 0.939992i \(0.610832\pi\)
\(860\) −4.00000 −0.136399
\(861\) 0 0
\(862\) −16.0000 −0.544962
\(863\) −48.0000 −1.63394 −0.816970 0.576681i \(-0.804348\pi\)
−0.816970 + 0.576681i \(0.804348\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −10.0000 −0.340010
\(866\) 14.0000 0.475739
\(867\) 1.00000 0.0339618
\(868\) −4.00000 −0.135769
\(869\) 0 0
\(870\) −8.00000 −0.271225
\(871\) 24.0000 0.813209
\(872\) 4.00000 0.135457
\(873\) 6.00000 0.203069
\(874\) 4.00000 0.135302
\(875\) −1.00000 −0.0338062
\(876\) 4.00000 0.135147
\(877\) 4.00000 0.135070 0.0675352 0.997717i \(-0.478487\pi\)
0.0675352 + 0.997717i \(0.478487\pi\)
\(878\) 36.0000 1.21494
\(879\) 14.0000 0.472208
\(880\) 0 0
\(881\) −38.0000 −1.28025 −0.640126 0.768270i \(-0.721118\pi\)
−0.640126 + 0.768270i \(0.721118\pi\)
\(882\) 1.00000 0.0336718
\(883\) 38.0000 1.27880 0.639401 0.768874i \(-0.279182\pi\)
0.639401 + 0.768874i \(0.279182\pi\)
\(884\) 24.0000 0.807207
\(885\) −8.00000 −0.268917
\(886\) −20.0000 −0.671913
\(887\) −20.0000 −0.671534 −0.335767 0.941945i \(-0.608996\pi\)
−0.335767 + 0.941945i \(0.608996\pi\)
\(888\) −2.00000 −0.0671156
\(889\) 8.00000 0.268311
\(890\) 6.00000 0.201120
\(891\) 0 0
\(892\) −2.00000 −0.0669650
\(893\) −16.0000 −0.535420
\(894\) −10.0000 −0.334450
\(895\) −10.0000 −0.334263
\(896\) 1.00000 0.0334077
\(897\) −6.00000 −0.200334
\(898\) 6.00000 0.200223
\(899\) 32.0000 1.06726
\(900\) 1.00000 0.0333333
\(901\) 0 0
\(902\) 0 0
\(903\) −4.00000 −0.133112
\(904\) −6.00000 −0.199557
\(905\) −14.0000 −0.465376
\(906\) −16.0000 −0.531564
\(907\) −44.0000 −1.46100 −0.730498 0.682915i \(-0.760712\pi\)
−0.730498 + 0.682915i \(0.760712\pi\)
\(908\) 0 0
\(909\) 2.00000 0.0663358
\(910\) −6.00000 −0.198898
\(911\) −40.0000 −1.32526 −0.662630 0.748947i \(-0.730560\pi\)
−0.662630 + 0.748947i \(0.730560\pi\)
\(912\) −4.00000 −0.132453
\(913\) 0 0
\(914\) 6.00000 0.198462
\(915\) −6.00000 −0.198354
\(916\) −10.0000 −0.330409
\(917\) 0 0
\(918\) −4.00000 −0.132020
\(919\) 40.0000 1.31948 0.659739 0.751495i \(-0.270667\pi\)
0.659739 + 0.751495i \(0.270667\pi\)
\(920\) −1.00000 −0.0329690
\(921\) 16.0000 0.527218
\(922\) 30.0000 0.987997
\(923\) 72.0000 2.36991
\(924\) 0 0
\(925\) 2.00000 0.0657596
\(926\) 32.0000 1.05159
\(927\) 4.00000 0.131377
\(928\) −8.00000 −0.262613
\(929\) −4.00000 −0.131236 −0.0656179 0.997845i \(-0.520902\pi\)
−0.0656179 + 0.997845i \(0.520902\pi\)
\(930\) −4.00000 −0.131165
\(931\) 4.00000 0.131095
\(932\) 22.0000 0.720634
\(933\) 10.0000 0.327385
\(934\) −8.00000 −0.261768
\(935\) 0 0
\(936\) 6.00000 0.196116
\(937\) −22.0000 −0.718709 −0.359354 0.933201i \(-0.617003\pi\)
−0.359354 + 0.933201i \(0.617003\pi\)
\(938\) 4.00000 0.130605
\(939\) 14.0000 0.456873
\(940\) 4.00000 0.130466
\(941\) 22.0000 0.717180 0.358590 0.933495i \(-0.383258\pi\)
0.358590 + 0.933495i \(0.383258\pi\)
\(942\) 2.00000 0.0651635
\(943\) 0 0
\(944\) −8.00000 −0.260378
\(945\) 1.00000 0.0325300
\(946\) 0 0
\(947\) −16.0000 −0.519930 −0.259965 0.965618i \(-0.583711\pi\)
−0.259965 + 0.965618i \(0.583711\pi\)
\(948\) −12.0000 −0.389742
\(949\) −24.0000 −0.779073
\(950\) 4.00000 0.129777
\(951\) −10.0000 −0.324272
\(952\) 4.00000 0.129641
\(953\) 6.00000 0.194359 0.0971795 0.995267i \(-0.469018\pi\)
0.0971795 + 0.995267i \(0.469018\pi\)
\(954\) 0 0
\(955\) −8.00000 −0.258874
\(956\) 12.0000 0.388108
\(957\) 0 0
\(958\) 24.0000 0.775405
\(959\) 2.00000 0.0645834
\(960\) 1.00000 0.0322749
\(961\) −15.0000 −0.483871
\(962\) 12.0000 0.386896
\(963\) 14.0000 0.451144
\(964\) 4.00000 0.128831
\(965\) 10.0000 0.321911
\(966\) −1.00000 −0.0321745
\(967\) 4.00000 0.128631 0.0643157 0.997930i \(-0.479514\pi\)
0.0643157 + 0.997930i \(0.479514\pi\)
\(968\) −11.0000 −0.353553
\(969\) −16.0000 −0.513994
\(970\) −6.00000 −0.192648
\(971\) 28.0000 0.898563 0.449281 0.893390i \(-0.351680\pi\)
0.449281 + 0.893390i \(0.351680\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 20.0000 0.641171
\(974\) 20.0000 0.640841
\(975\) −6.00000 −0.192154
\(976\) −6.00000 −0.192055
\(977\) −6.00000 −0.191957 −0.0959785 0.995383i \(-0.530598\pi\)
−0.0959785 + 0.995383i \(0.530598\pi\)
\(978\) 2.00000 0.0639529
\(979\) 0 0
\(980\) −1.00000 −0.0319438
\(981\) 4.00000 0.127710
\(982\) 22.0000 0.702048
\(983\) −54.0000 −1.72233 −0.861166 0.508323i \(-0.830265\pi\)
−0.861166 + 0.508323i \(0.830265\pi\)
\(984\) 0 0
\(985\) 2.00000 0.0637253
\(986\) −32.0000 −1.01909
\(987\) 4.00000 0.127321
\(988\) 24.0000 0.763542
\(989\) 4.00000 0.127193
\(990\) 0 0
\(991\) 16.0000 0.508257 0.254128 0.967170i \(-0.418211\pi\)
0.254128 + 0.967170i \(0.418211\pi\)
\(992\) −4.00000 −0.127000
\(993\) −20.0000 −0.634681
\(994\) 12.0000 0.380617
\(995\) 14.0000 0.443830
\(996\) −4.00000 −0.126745
\(997\) −14.0000 −0.443384 −0.221692 0.975117i \(-0.571158\pi\)
−0.221692 + 0.975117i \(0.571158\pi\)
\(998\) 32.0000 1.01294
\(999\) −2.00000 −0.0632772
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 4830.2.a.u.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4830.2.a.u.1.1 1 1.1 even 1 trivial