Properties

Label 350.2.a.f.1.1
Level 350
Weight 2
Character 350.1
Self dual yes
Analytic conductor 2.795
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 350.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.79476407074\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 14)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(0\)
Character \(\chi\) = 350.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} +2.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} +2.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{12} +4.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} -6.00000 q^{17} +1.00000 q^{18} +2.00000 q^{19} -2.00000 q^{21} +2.00000 q^{24} +4.00000 q^{26} -4.00000 q^{27} -1.00000 q^{28} -6.00000 q^{29} -4.00000 q^{31} +1.00000 q^{32} -6.00000 q^{34} +1.00000 q^{36} -2.00000 q^{37} +2.00000 q^{38} +8.00000 q^{39} +6.00000 q^{41} -2.00000 q^{42} -8.00000 q^{43} +12.0000 q^{47} +2.00000 q^{48} +1.00000 q^{49} -12.0000 q^{51} +4.00000 q^{52} -6.00000 q^{53} -4.00000 q^{54} -1.00000 q^{56} +4.00000 q^{57} -6.00000 q^{58} -6.00000 q^{59} +8.00000 q^{61} -4.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} +4.00000 q^{67} -6.00000 q^{68} +1.00000 q^{72} -2.00000 q^{73} -2.00000 q^{74} +2.00000 q^{76} +8.00000 q^{78} +8.00000 q^{79} -11.0000 q^{81} +6.00000 q^{82} +6.00000 q^{83} -2.00000 q^{84} -8.00000 q^{86} -12.0000 q^{87} -6.00000 q^{89} -4.00000 q^{91} -8.00000 q^{93} +12.0000 q^{94} +2.00000 q^{96} +10.0000 q^{97} +1.00000 q^{98} +O(q^{100})\)

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
\(3\) 2.00000 1.15470 0.577350 0.816497i \(-0.304087\pi\)
0.577350 + 0.816497i \(0.304087\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 2.00000 0.816497
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 2.00000 0.577350
\(13\) 4.00000 1.10940 0.554700 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −6.00000 −1.45521 −0.727607 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 1.00000 0.235702
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 0 0
\(21\) −2.00000 −0.436436
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 2.00000 0.408248
\(25\) 0 0
\(26\) 4.00000 0.784465
\(27\) −4.00000 −0.769800
\(28\) −1.00000 −0.188982
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) −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
\(34\) −6.00000 −1.02899
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 2.00000 0.324443
\(39\) 8.00000 1.28103
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) −2.00000 −0.308607
\(43\) −8.00000 −1.21999 −0.609994 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 12.0000 1.75038 0.875190 0.483779i \(-0.160736\pi\)
0.875190 + 0.483779i \(0.160736\pi\)
\(48\) 2.00000 0.288675
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −12.0000 −1.68034
\(52\) 4.00000 0.554700
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) −4.00000 −0.544331
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) 4.00000 0.529813
\(58\) −6.00000 −0.787839
\(59\) −6.00000 −0.781133 −0.390567 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 0 0
\(61\) 8.00000 1.02430 0.512148 0.858898i \(-0.328850\pi\)
0.512148 + 0.858898i \(0.328850\pi\)
\(62\) −4.00000 −0.508001
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) −6.00000 −0.727607
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 1.00000 0.117851
\(73\) −2.00000 −0.234082 −0.117041 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) −2.00000 −0.232495
\(75\) 0 0
\(76\) 2.00000 0.229416
\(77\) 0 0
\(78\) 8.00000 0.905822
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 6.00000 0.662589
\(83\) 6.00000 0.658586 0.329293 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) −2.00000 −0.218218
\(85\) 0 0
\(86\) −8.00000 −0.862662
\(87\) −12.0000 −1.28654
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) −4.00000 −0.419314
\(92\) 0 0
\(93\) −8.00000 −0.829561
\(94\) 12.0000 1.23771
\(95\) 0 0
\(96\) 2.00000 0.204124
\(97\) 10.0000 1.01535 0.507673 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) −12.0000 −1.18818
\(103\) 4.00000 0.394132 0.197066 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) 4.00000 0.392232
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) −4.00000 −0.384900
\(109\) 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
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 4.00000 0.374634
\(115\) 0 0
\(116\) −6.00000 −0.557086
\(117\) 4.00000 0.369800
\(118\) −6.00000 −0.552345
\(119\) 6.00000 0.550019
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 8.00000 0.724286
\(123\) 12.0000 1.08200
\(124\) −4.00000 −0.359211
\(125\) 0 0
\(126\) −1.00000 −0.0890871
\(127\) 16.0000 1.41977 0.709885 0.704317i \(-0.248747\pi\)
0.709885 + 0.704317i \(0.248747\pi\)
\(128\) 1.00000 0.0883883
\(129\) −16.0000 −1.40872
\(130\) 0 0
\(131\) 18.0000 1.57267 0.786334 0.617802i \(-0.211977\pi\)
0.786334 + 0.617802i \(0.211977\pi\)
\(132\) 0 0
\(133\) −2.00000 −0.173422
\(134\) 4.00000 0.345547
\(135\) 0 0
\(136\) −6.00000 −0.514496
\(137\) −18.0000 −1.53784 −0.768922 0.639343i \(-0.779207\pi\)
−0.768922 + 0.639343i \(0.779207\pi\)
\(138\) 0 0
\(139\) 14.0000 1.18746 0.593732 0.804663i \(-0.297654\pi\)
0.593732 + 0.804663i \(0.297654\pi\)
\(140\) 0 0
\(141\) 24.0000 2.02116
\(142\) 0 0
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −2.00000 −0.165521
\(147\) 2.00000 0.164957
\(148\) −2.00000 −0.164399
\(149\) −18.0000 −1.47462 −0.737309 0.675556i \(-0.763904\pi\)
−0.737309 + 0.675556i \(0.763904\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 2.00000 0.162221
\(153\) −6.00000 −0.485071
\(154\) 0 0
\(155\) 0 0
\(156\) 8.00000 0.640513
\(157\) 4.00000 0.319235 0.159617 0.987179i \(-0.448974\pi\)
0.159617 + 0.987179i \(0.448974\pi\)
\(158\) 8.00000 0.636446
\(159\) −12.0000 −0.951662
\(160\) 0 0
\(161\) 0 0
\(162\) −11.0000 −0.864242
\(163\) 16.0000 1.25322 0.626608 0.779334i \(-0.284443\pi\)
0.626608 + 0.779334i \(0.284443\pi\)
\(164\) 6.00000 0.468521
\(165\) 0 0
\(166\) 6.00000 0.465690
\(167\) 12.0000 0.928588 0.464294 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) −2.00000 −0.154303
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) 2.00000 0.152944
\(172\) −8.00000 −0.609994
\(173\) 12.0000 0.912343 0.456172 0.889892i \(-0.349220\pi\)
0.456172 + 0.889892i \(0.349220\pi\)
\(174\) −12.0000 −0.909718
\(175\) 0 0
\(176\) 0 0
\(177\) −12.0000 −0.901975
\(178\) −6.00000 −0.449719
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) 20.0000 1.48659 0.743294 0.668965i \(-0.233262\pi\)
0.743294 + 0.668965i \(0.233262\pi\)
\(182\) −4.00000 −0.296500
\(183\) 16.0000 1.18275
\(184\) 0 0
\(185\) 0 0
\(186\) −8.00000 −0.586588
\(187\) 0 0
\(188\) 12.0000 0.875190
\(189\) 4.00000 0.290957
\(190\) 0 0
\(191\) 24.0000 1.73658 0.868290 0.496058i \(-0.165220\pi\)
0.868290 + 0.496058i \(0.165220\pi\)
\(192\) 2.00000 0.144338
\(193\) −14.0000 −1.00774 −0.503871 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) 10.0000 0.717958
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 18.0000 1.28245 0.641223 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) 0 0
\(199\) 20.0000 1.41776 0.708881 0.705328i \(-0.249200\pi\)
0.708881 + 0.705328i \(0.249200\pi\)
\(200\) 0 0
\(201\) 8.00000 0.564276
\(202\) 0 0
\(203\) 6.00000 0.421117
\(204\) −12.0000 −0.840168
\(205\) 0 0
\(206\) 4.00000 0.278693
\(207\) 0 0
\(208\) 4.00000 0.277350
\(209\) 0 0
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) −6.00000 −0.412082
\(213\) 0 0
\(214\) −12.0000 −0.820303
\(215\) 0 0
\(216\) −4.00000 −0.272166
\(217\) 4.00000 0.271538
\(218\) 2.00000 0.135457
\(219\) −4.00000 −0.270295
\(220\) 0 0
\(221\) −24.0000 −1.61441
\(222\) −4.00000 −0.268462
\(223\) −8.00000 −0.535720 −0.267860 0.963458i \(-0.586316\pi\)
−0.267860 + 0.963458i \(0.586316\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) −6.00000 −0.399114
\(227\) −18.0000 −1.19470 −0.597351 0.801980i \(-0.703780\pi\)
−0.597351 + 0.801980i \(0.703780\pi\)
\(228\) 4.00000 0.264906
\(229\) −4.00000 −0.264327 −0.132164 0.991228i \(-0.542192\pi\)
−0.132164 + 0.991228i \(0.542192\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −6.00000 −0.393919
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 4.00000 0.261488
\(235\) 0 0
\(236\) −6.00000 −0.390567
\(237\) 16.0000 1.03931
\(238\) 6.00000 0.388922
\(239\) 24.0000 1.55243 0.776215 0.630468i \(-0.217137\pi\)
0.776215 + 0.630468i \(0.217137\pi\)
\(240\) 0 0
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) −11.0000 −0.707107
\(243\) −10.0000 −0.641500
\(244\) 8.00000 0.512148
\(245\) 0 0
\(246\) 12.0000 0.765092
\(247\) 8.00000 0.509028
\(248\) −4.00000 −0.254000
\(249\) 12.0000 0.760469
\(250\) 0 0
\(251\) −18.0000 −1.13615 −0.568075 0.822977i \(-0.692312\pi\)
−0.568075 + 0.822977i \(0.692312\pi\)
\(252\) −1.00000 −0.0629941
\(253\) 0 0
\(254\) 16.0000 1.00393
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −18.0000 −1.12281 −0.561405 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) −16.0000 −0.996116
\(259\) 2.00000 0.124274
\(260\) 0 0
\(261\) −6.00000 −0.371391
\(262\) 18.0000 1.11204
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −2.00000 −0.122628
\(267\) −12.0000 −0.734388
\(268\) 4.00000 0.244339
\(269\) −12.0000 −0.731653 −0.365826 0.930683i \(-0.619214\pi\)
−0.365826 + 0.930683i \(0.619214\pi\)
\(270\) 0 0
\(271\) −16.0000 −0.971931 −0.485965 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) −6.00000 −0.363803
\(273\) −8.00000 −0.484182
\(274\) −18.0000 −1.08742
\(275\) 0 0
\(276\) 0 0
\(277\) 10.0000 0.600842 0.300421 0.953807i \(-0.402873\pi\)
0.300421 + 0.953807i \(0.402873\pi\)
\(278\) 14.0000 0.839664
\(279\) −4.00000 −0.239474
\(280\) 0 0
\(281\) −6.00000 −0.357930 −0.178965 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 24.0000 1.42918
\(283\) 22.0000 1.30776 0.653882 0.756596i \(-0.273139\pi\)
0.653882 + 0.756596i \(0.273139\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −6.00000 −0.354169
\(288\) 1.00000 0.0589256
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 20.0000 1.17242
\(292\) −2.00000 −0.117041
\(293\) −24.0000 −1.40209 −0.701047 0.713115i \(-0.747284\pi\)
−0.701047 + 0.713115i \(0.747284\pi\)
\(294\) 2.00000 0.116642
\(295\) 0 0
\(296\) −2.00000 −0.116248
\(297\) 0 0
\(298\) −18.0000 −1.04271
\(299\) 0 0
\(300\) 0 0
\(301\) 8.00000 0.461112
\(302\) 8.00000 0.460348
\(303\) 0 0
\(304\) 2.00000 0.114708
\(305\) 0 0
\(306\) −6.00000 −0.342997
\(307\) −2.00000 −0.114146 −0.0570730 0.998370i \(-0.518177\pi\)
−0.0570730 + 0.998370i \(0.518177\pi\)
\(308\) 0 0
\(309\) 8.00000 0.455104
\(310\) 0 0
\(311\) −24.0000 −1.36092 −0.680458 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) 8.00000 0.452911
\(313\) 10.0000 0.565233 0.282617 0.959233i \(-0.408798\pi\)
0.282617 + 0.959233i \(0.408798\pi\)
\(314\) 4.00000 0.225733
\(315\) 0 0
\(316\) 8.00000 0.450035
\(317\) −6.00000 −0.336994 −0.168497 0.985702i \(-0.553891\pi\)
−0.168497 + 0.985702i \(0.553891\pi\)
\(318\) −12.0000 −0.672927
\(319\) 0 0
\(320\) 0 0
\(321\) −24.0000 −1.33955
\(322\) 0 0
\(323\) −12.0000 −0.667698
\(324\) −11.0000 −0.611111
\(325\) 0 0
\(326\) 16.0000 0.886158
\(327\) 4.00000 0.221201
\(328\) 6.00000 0.331295
\(329\) −12.0000 −0.661581
\(330\) 0 0
\(331\) 8.00000 0.439720 0.219860 0.975531i \(-0.429440\pi\)
0.219860 + 0.975531i \(0.429440\pi\)
\(332\) 6.00000 0.329293
\(333\) −2.00000 −0.109599
\(334\) 12.0000 0.656611
\(335\) 0 0
\(336\) −2.00000 −0.109109
\(337\) −14.0000 −0.762629 −0.381314 0.924445i \(-0.624528\pi\)
−0.381314 + 0.924445i \(0.624528\pi\)
\(338\) 3.00000 0.163178
\(339\) −12.0000 −0.651751
\(340\) 0 0
\(341\) 0 0
\(342\) 2.00000 0.108148
\(343\) −1.00000 −0.0539949
\(344\) −8.00000 −0.431331
\(345\) 0 0
\(346\) 12.0000 0.645124
\(347\) 24.0000 1.28839 0.644194 0.764862i \(-0.277193\pi\)
0.644194 + 0.764862i \(0.277193\pi\)
\(348\) −12.0000 −0.643268
\(349\) −28.0000 −1.49881 −0.749403 0.662114i \(-0.769659\pi\)
−0.749403 + 0.662114i \(0.769659\pi\)
\(350\) 0 0
\(351\) −16.0000 −0.854017
\(352\) 0 0
\(353\) −18.0000 −0.958043 −0.479022 0.877803i \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) −12.0000 −0.637793
\(355\) 0 0
\(356\) −6.00000 −0.317999
\(357\) 12.0000 0.635107
\(358\) −12.0000 −0.634220
\(359\) −24.0000 −1.26667 −0.633336 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 20.0000 1.05118
\(363\) −22.0000 −1.15470
\(364\) −4.00000 −0.209657
\(365\) 0 0
\(366\) 16.0000 0.836333
\(367\) −8.00000 −0.417597 −0.208798 0.977959i \(-0.566955\pi\)
−0.208798 + 0.977959i \(0.566955\pi\)
\(368\) 0 0
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) 6.00000 0.311504
\(372\) −8.00000 −0.414781
\(373\) −14.0000 −0.724893 −0.362446 0.932005i \(-0.618058\pi\)
−0.362446 + 0.932005i \(0.618058\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 12.0000 0.618853
\(377\) −24.0000 −1.23606
\(378\) 4.00000 0.205738
\(379\) −16.0000 −0.821865 −0.410932 0.911666i \(-0.634797\pi\)
−0.410932 + 0.911666i \(0.634797\pi\)
\(380\) 0 0
\(381\) 32.0000 1.63941
\(382\) 24.0000 1.22795
\(383\) −36.0000 −1.83951 −0.919757 0.392488i \(-0.871614\pi\)
−0.919757 + 0.392488i \(0.871614\pi\)
\(384\) 2.00000 0.102062
\(385\) 0 0
\(386\) −14.0000 −0.712581
\(387\) −8.00000 −0.406663
\(388\) 10.0000 0.507673
\(389\) 18.0000 0.912636 0.456318 0.889817i \(-0.349168\pi\)
0.456318 + 0.889817i \(0.349168\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 1.00000 0.0505076
\(393\) 36.0000 1.81596
\(394\) 18.0000 0.906827
\(395\) 0 0
\(396\) 0 0
\(397\) −20.0000 −1.00377 −0.501886 0.864934i \(-0.667360\pi\)
−0.501886 + 0.864934i \(0.667360\pi\)
\(398\) 20.0000 1.00251
\(399\) −4.00000 −0.200250
\(400\) 0 0
\(401\) −18.0000 −0.898877 −0.449439 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) 8.00000 0.399004
\(403\) −16.0000 −0.797017
\(404\) 0 0
\(405\) 0 0
\(406\) 6.00000 0.297775
\(407\) 0 0
\(408\) −12.0000 −0.594089
\(409\) 14.0000 0.692255 0.346128 0.938187i \(-0.387496\pi\)
0.346128 + 0.938187i \(0.387496\pi\)
\(410\) 0 0
\(411\) −36.0000 −1.77575
\(412\) 4.00000 0.197066
\(413\) 6.00000 0.295241
\(414\) 0 0
\(415\) 0 0
\(416\) 4.00000 0.196116
\(417\) 28.0000 1.37117
\(418\) 0 0
\(419\) 6.00000 0.293119 0.146560 0.989202i \(-0.453180\pi\)
0.146560 + 0.989202i \(0.453180\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) −4.00000 −0.194717
\(423\) 12.0000 0.583460
\(424\) −6.00000 −0.291386
\(425\) 0 0
\(426\) 0 0
\(427\) −8.00000 −0.387147
\(428\) −12.0000 −0.580042
\(429\) 0 0
\(430\) 0 0
\(431\) 24.0000 1.15604 0.578020 0.816023i \(-0.303826\pi\)
0.578020 + 0.816023i \(0.303826\pi\)
\(432\) −4.00000 −0.192450
\(433\) 34.0000 1.63394 0.816968 0.576683i \(-0.195653\pi\)
0.816968 + 0.576683i \(0.195653\pi\)
\(434\) 4.00000 0.192006
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) 0 0
\(438\) −4.00000 −0.191127
\(439\) 8.00000 0.381819 0.190910 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) 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
\(447\) −36.0000 −1.70274
\(448\) −1.00000 −0.0472456
\(449\) 18.0000 0.849473 0.424736 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −6.00000 −0.282216
\(453\) 16.0000 0.751746
\(454\) −18.0000 −0.844782
\(455\) 0 0
\(456\) 4.00000 0.187317
\(457\) 10.0000 0.467780 0.233890 0.972263i \(-0.424854\pi\)
0.233890 + 0.972263i \(0.424854\pi\)
\(458\) −4.00000 −0.186908
\(459\) 24.0000 1.12022
\(460\) 0 0
\(461\) 12.0000 0.558896 0.279448 0.960161i \(-0.409849\pi\)
0.279448 + 0.960161i \(0.409849\pi\)
\(462\) 0 0
\(463\) −32.0000 −1.48717 −0.743583 0.668644i \(-0.766875\pi\)
−0.743583 + 0.668644i \(0.766875\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) 6.00000 0.277945
\(467\) 6.00000 0.277647 0.138823 0.990317i \(-0.455668\pi\)
0.138823 + 0.990317i \(0.455668\pi\)
\(468\) 4.00000 0.184900
\(469\) −4.00000 −0.184703
\(470\) 0 0
\(471\) 8.00000 0.368621
\(472\) −6.00000 −0.276172
\(473\) 0 0
\(474\) 16.0000 0.734904
\(475\) 0 0
\(476\) 6.00000 0.275010
\(477\) −6.00000 −0.274721
\(478\) 24.0000 1.09773
\(479\) −36.0000 −1.64488 −0.822441 0.568850i \(-0.807388\pi\)
−0.822441 + 0.568850i \(0.807388\pi\)
\(480\) 0 0
\(481\) −8.00000 −0.364769
\(482\) −10.0000 −0.455488
\(483\) 0 0
\(484\) −11.0000 −0.500000
\(485\) 0 0
\(486\) −10.0000 −0.453609
\(487\) 16.0000 0.725029 0.362515 0.931978i \(-0.381918\pi\)
0.362515 + 0.931978i \(0.381918\pi\)
\(488\) 8.00000 0.362143
\(489\) 32.0000 1.44709
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 12.0000 0.541002
\(493\) 36.0000 1.62136
\(494\) 8.00000 0.359937
\(495\) 0 0
\(496\) −4.00000 −0.179605
\(497\) 0 0
\(498\) 12.0000 0.537733
\(499\) −4.00000 −0.179065 −0.0895323 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) 0 0
\(501\) 24.0000 1.07224
\(502\) −18.0000 −0.803379
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) −1.00000 −0.0445435
\(505\) 0 0
\(506\) 0 0
\(507\) 6.00000 0.266469
\(508\) 16.0000 0.709885
\(509\) 36.0000 1.59567 0.797836 0.602875i \(-0.205978\pi\)
0.797836 + 0.602875i \(0.205978\pi\)
\(510\) 0 0
\(511\) 2.00000 0.0884748
\(512\) 1.00000 0.0441942
\(513\) −8.00000 −0.353209
\(514\) −18.0000 −0.793946
\(515\) 0 0
\(516\) −16.0000 −0.704361
\(517\) 0 0
\(518\) 2.00000 0.0878750
\(519\) 24.0000 1.05348
\(520\) 0 0
\(521\) 6.00000 0.262865 0.131432 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) −6.00000 −0.262613
\(523\) −2.00000 −0.0874539 −0.0437269 0.999044i \(-0.513923\pi\)
−0.0437269 + 0.999044i \(0.513923\pi\)
\(524\) 18.0000 0.786334
\(525\) 0 0
\(526\) 0 0
\(527\) 24.0000 1.04546
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) −6.00000 −0.260378
\(532\) −2.00000 −0.0867110
\(533\) 24.0000 1.03956
\(534\) −12.0000 −0.519291
\(535\) 0 0
\(536\) 4.00000 0.172774
\(537\) −24.0000 −1.03568
\(538\) −12.0000 −0.517357
\(539\) 0 0
\(540\) 0 0
\(541\) 38.0000 1.63375 0.816874 0.576816i \(-0.195705\pi\)
0.816874 + 0.576816i \(0.195705\pi\)
\(542\) −16.0000 −0.687259
\(543\) 40.0000 1.71656
\(544\) −6.00000 −0.257248
\(545\) 0 0
\(546\) −8.00000 −0.342368
\(547\) −8.00000 −0.342055 −0.171028 0.985266i \(-0.554709\pi\)
−0.171028 + 0.985266i \(0.554709\pi\)
\(548\) −18.0000 −0.768922
\(549\) 8.00000 0.341432
\(550\) 0 0
\(551\) −12.0000 −0.511217
\(552\) 0 0
\(553\) −8.00000 −0.340195
\(554\) 10.0000 0.424859
\(555\) 0 0
\(556\) 14.0000 0.593732
\(557\) −6.00000 −0.254228 −0.127114 0.991888i \(-0.540571\pi\)
−0.127114 + 0.991888i \(0.540571\pi\)
\(558\) −4.00000 −0.169334
\(559\) −32.0000 −1.35346
\(560\) 0 0
\(561\) 0 0
\(562\) −6.00000 −0.253095
\(563\) −30.0000 −1.26435 −0.632175 0.774826i \(-0.717837\pi\)
−0.632175 + 0.774826i \(0.717837\pi\)
\(564\) 24.0000 1.01058
\(565\) 0 0
\(566\) 22.0000 0.924729
\(567\) 11.0000 0.461957
\(568\) 0 0
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 0 0
\(571\) 32.0000 1.33916 0.669579 0.742741i \(-0.266474\pi\)
0.669579 + 0.742741i \(0.266474\pi\)
\(572\) 0 0
\(573\) 48.0000 2.00523
\(574\) −6.00000 −0.250435
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −2.00000 −0.0832611 −0.0416305 0.999133i \(-0.513255\pi\)
−0.0416305 + 0.999133i \(0.513255\pi\)
\(578\) 19.0000 0.790296
\(579\) −28.0000 −1.16364
\(580\) 0 0
\(581\) −6.00000 −0.248922
\(582\) 20.0000 0.829027
\(583\) 0 0
\(584\) −2.00000 −0.0827606
\(585\) 0 0
\(586\) −24.0000 −0.991431
\(587\) 42.0000 1.73353 0.866763 0.498721i \(-0.166197\pi\)
0.866763 + 0.498721i \(0.166197\pi\)
\(588\) 2.00000 0.0824786
\(589\) −8.00000 −0.329634
\(590\) 0 0
\(591\) 36.0000 1.48084
\(592\) −2.00000 −0.0821995
\(593\) 6.00000 0.246390 0.123195 0.992382i \(-0.460686\pi\)
0.123195 + 0.992382i \(0.460686\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −18.0000 −0.737309
\(597\) 40.0000 1.63709
\(598\) 0 0
\(599\) −24.0000 −0.980613 −0.490307 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 0 0
\(601\) 26.0000 1.06056 0.530281 0.847822i \(-0.322086\pi\)
0.530281 + 0.847822i \(0.322086\pi\)
\(602\) 8.00000 0.326056
\(603\) 4.00000 0.162893
\(604\) 8.00000 0.325515
\(605\) 0 0
\(606\) 0 0
\(607\) −32.0000 −1.29884 −0.649420 0.760430i \(-0.724988\pi\)
−0.649420 + 0.760430i \(0.724988\pi\)
\(608\) 2.00000 0.0811107
\(609\) 12.0000 0.486265
\(610\) 0 0
\(611\) 48.0000 1.94187
\(612\) −6.00000 −0.242536
\(613\) −2.00000 −0.0807792 −0.0403896 0.999184i \(-0.512860\pi\)
−0.0403896 + 0.999184i \(0.512860\pi\)
\(614\) −2.00000 −0.0807134
\(615\) 0 0
\(616\) 0 0
\(617\) −6.00000 −0.241551 −0.120775 0.992680i \(-0.538538\pi\)
−0.120775 + 0.992680i \(0.538538\pi\)
\(618\) 8.00000 0.321807
\(619\) 26.0000 1.04503 0.522514 0.852631i \(-0.324994\pi\)
0.522514 + 0.852631i \(0.324994\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −24.0000 −0.962312
\(623\) 6.00000 0.240385
\(624\) 8.00000 0.320256
\(625\) 0 0
\(626\) 10.0000 0.399680
\(627\) 0 0
\(628\) 4.00000 0.159617
\(629\) 12.0000 0.478471
\(630\) 0 0
\(631\) −16.0000 −0.636950 −0.318475 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) 8.00000 0.318223
\(633\) −8.00000 −0.317971
\(634\) −6.00000 −0.238290
\(635\) 0 0
\(636\) −12.0000 −0.475831
\(637\) 4.00000 0.158486
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −18.0000 −0.710957 −0.355479 0.934684i \(-0.615682\pi\)
−0.355479 + 0.934684i \(0.615682\pi\)
\(642\) −24.0000 −0.947204
\(643\) −14.0000 −0.552106 −0.276053 0.961142i \(-0.589027\pi\)
−0.276053 + 0.961142i \(0.589027\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −12.0000 −0.472134
\(647\) 12.0000 0.471769 0.235884 0.971781i \(-0.424201\pi\)
0.235884 + 0.971781i \(0.424201\pi\)
\(648\) −11.0000 −0.432121
\(649\) 0 0
\(650\) 0 0
\(651\) 8.00000 0.313545
\(652\) 16.0000 0.626608
\(653\) −18.0000 −0.704394 −0.352197 0.935926i \(-0.614565\pi\)
−0.352197 + 0.935926i \(0.614565\pi\)
\(654\) 4.00000 0.156412
\(655\) 0 0
\(656\) 6.00000 0.234261
\(657\) −2.00000 −0.0780274
\(658\) −12.0000 −0.467809
\(659\) −24.0000 −0.934907 −0.467454 0.884018i \(-0.654829\pi\)
−0.467454 + 0.884018i \(0.654829\pi\)
\(660\) 0 0
\(661\) −40.0000 −1.55582 −0.777910 0.628376i \(-0.783720\pi\)
−0.777910 + 0.628376i \(0.783720\pi\)
\(662\) 8.00000 0.310929
\(663\) −48.0000 −1.86417
\(664\) 6.00000 0.232845
\(665\) 0 0
\(666\) −2.00000 −0.0774984
\(667\) 0 0
\(668\) 12.0000 0.464294
\(669\) −16.0000 −0.618596
\(670\) 0 0
\(671\) 0 0
\(672\) −2.00000 −0.0771517
\(673\) −26.0000 −1.00223 −0.501113 0.865382i \(-0.667076\pi\)
−0.501113 + 0.865382i \(0.667076\pi\)
\(674\) −14.0000 −0.539260
\(675\) 0 0
\(676\) 3.00000 0.115385
\(677\) 12.0000 0.461197 0.230599 0.973049i \(-0.425932\pi\)
0.230599 + 0.973049i \(0.425932\pi\)
\(678\) −12.0000 −0.460857
\(679\) −10.0000 −0.383765
\(680\) 0 0
\(681\) −36.0000 −1.37952
\(682\) 0 0
\(683\) 12.0000 0.459167 0.229584 0.973289i \(-0.426264\pi\)
0.229584 + 0.973289i \(0.426264\pi\)
\(684\) 2.00000 0.0764719
\(685\) 0 0
\(686\) −1.00000 −0.0381802
\(687\) −8.00000 −0.305219
\(688\) −8.00000 −0.304997
\(689\) −24.0000 −0.914327
\(690\) 0 0
\(691\) −46.0000 −1.74992 −0.874961 0.484193i \(-0.839113\pi\)
−0.874961 + 0.484193i \(0.839113\pi\)
\(692\) 12.0000 0.456172
\(693\) 0 0
\(694\) 24.0000 0.911028
\(695\) 0 0
\(696\) −12.0000 −0.454859
\(697\) −36.0000 −1.36360
\(698\) −28.0000 −1.05982
\(699\) 12.0000 0.453882
\(700\) 0 0
\(701\) 18.0000 0.679851 0.339925 0.940452i \(-0.389598\pi\)
0.339925 + 0.940452i \(0.389598\pi\)
\(702\) −16.0000 −0.603881
\(703\) −4.00000 −0.150863
\(704\) 0 0
\(705\) 0 0
\(706\) −18.0000 −0.677439
\(707\) 0 0
\(708\) −12.0000 −0.450988
\(709\) −46.0000 −1.72757 −0.863783 0.503864i \(-0.831911\pi\)
−0.863783 + 0.503864i \(0.831911\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) −6.00000 −0.224860
\(713\) 0 0
\(714\) 12.0000 0.449089
\(715\) 0 0
\(716\) −12.0000 −0.448461
\(717\) 48.0000 1.79259
\(718\) −24.0000 −0.895672
\(719\) 12.0000 0.447524 0.223762 0.974644i \(-0.428166\pi\)
0.223762 + 0.974644i \(0.428166\pi\)
\(720\) 0 0
\(721\) −4.00000 −0.148968
\(722\) −15.0000 −0.558242
\(723\) −20.0000 −0.743808
\(724\) 20.0000 0.743294
\(725\) 0 0
\(726\) −22.0000 −0.816497
\(727\) −44.0000 −1.63187 −0.815935 0.578144i \(-0.803777\pi\)
−0.815935 + 0.578144i \(0.803777\pi\)
\(728\) −4.00000 −0.148250
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) 48.0000 1.77534
\(732\) 16.0000 0.591377
\(733\) 40.0000 1.47743 0.738717 0.674016i \(-0.235432\pi\)
0.738717 + 0.674016i \(0.235432\pi\)
\(734\) −8.00000 −0.295285
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 6.00000 0.220863
\(739\) −16.0000 −0.588570 −0.294285 0.955718i \(-0.595081\pi\)
−0.294285 + 0.955718i \(0.595081\pi\)
\(740\) 0 0
\(741\) 16.0000 0.587775
\(742\) 6.00000 0.220267
\(743\) −24.0000 −0.880475 −0.440237 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) −8.00000 −0.293294
\(745\) 0 0
\(746\) −14.0000 −0.512576
\(747\) 6.00000 0.219529
\(748\) 0 0
\(749\) 12.0000 0.438470
\(750\) 0 0
\(751\) −40.0000 −1.45962 −0.729810 0.683650i \(-0.760392\pi\)
−0.729810 + 0.683650i \(0.760392\pi\)
\(752\) 12.0000 0.437595
\(753\) −36.0000 −1.31191
\(754\) −24.0000 −0.874028
\(755\) 0 0
\(756\) 4.00000 0.145479
\(757\) −2.00000 −0.0726912 −0.0363456 0.999339i \(-0.511572\pi\)
−0.0363456 + 0.999339i \(0.511572\pi\)
\(758\) −16.0000 −0.581146
\(759\) 0 0
\(760\) 0 0
\(761\) −18.0000 −0.652499 −0.326250 0.945284i \(-0.605785\pi\)
−0.326250 + 0.945284i \(0.605785\pi\)
\(762\) 32.0000 1.15924
\(763\) −2.00000 −0.0724049
\(764\) 24.0000 0.868290
\(765\) 0 0
\(766\) −36.0000 −1.30073
\(767\) −24.0000 −0.866590
\(768\) 2.00000 0.0721688
\(769\) 14.0000 0.504853 0.252426 0.967616i \(-0.418771\pi\)
0.252426 + 0.967616i \(0.418771\pi\)
\(770\) 0 0
\(771\) −36.0000 −1.29651
\(772\) −14.0000 −0.503871
\(773\) −24.0000 −0.863220 −0.431610 0.902060i \(-0.642054\pi\)
−0.431610 + 0.902060i \(0.642054\pi\)
\(774\) −8.00000 −0.287554
\(775\) 0 0
\(776\) 10.0000 0.358979
\(777\) 4.00000 0.143499
\(778\) 18.0000 0.645331
\(779\) 12.0000 0.429945
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 24.0000 0.857690
\(784\) 1.00000 0.0357143
\(785\) 0 0
\(786\) 36.0000 1.28408
\(787\) 22.0000 0.784215 0.392108 0.919919i \(-0.371746\pi\)
0.392108 + 0.919919i \(0.371746\pi\)
\(788\) 18.0000 0.641223
\(789\) 0 0
\(790\) 0 0
\(791\) 6.00000 0.213335
\(792\) 0 0
\(793\) 32.0000 1.13635
\(794\) −20.0000 −0.709773
\(795\) 0 0
\(796\) 20.0000 0.708881
\(797\) 12.0000 0.425062 0.212531 0.977154i \(-0.431829\pi\)
0.212531 + 0.977154i \(0.431829\pi\)
\(798\) −4.00000 −0.141598
\(799\) −72.0000 −2.54718
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) −18.0000 −0.635602
\(803\) 0 0
\(804\) 8.00000 0.282138
\(805\) 0 0
\(806\) −16.0000 −0.563576
\(807\) −24.0000 −0.844840
\(808\) 0 0
\(809\) 6.00000 0.210949 0.105474 0.994422i \(-0.466364\pi\)
0.105474 + 0.994422i \(0.466364\pi\)
\(810\) 0 0
\(811\) 2.00000 0.0702295 0.0351147 0.999383i \(-0.488820\pi\)
0.0351147 + 0.999383i \(0.488820\pi\)
\(812\) 6.00000 0.210559
\(813\) −32.0000 −1.12229
\(814\) 0 0
\(815\) 0 0
\(816\) −12.0000 −0.420084
\(817\) −16.0000 −0.559769
\(818\) 14.0000 0.489499
\(819\) −4.00000 −0.139771
\(820\) 0 0
\(821\) 6.00000 0.209401 0.104701 0.994504i \(-0.466612\pi\)
0.104701 + 0.994504i \(0.466612\pi\)
\(822\) −36.0000 −1.25564
\(823\) 40.0000 1.39431 0.697156 0.716919i \(-0.254448\pi\)
0.697156 + 0.716919i \(0.254448\pi\)
\(824\) 4.00000 0.139347
\(825\) 0 0
\(826\) 6.00000 0.208767
\(827\) 36.0000 1.25184 0.625921 0.779886i \(-0.284723\pi\)
0.625921 + 0.779886i \(0.284723\pi\)
\(828\) 0 0
\(829\) 56.0000 1.94496 0.972480 0.232986i \(-0.0748495\pi\)
0.972480 + 0.232986i \(0.0748495\pi\)
\(830\) 0 0
\(831\) 20.0000 0.693792
\(832\) 4.00000 0.138675
\(833\) −6.00000 −0.207888
\(834\) 28.0000 0.969561
\(835\) 0 0
\(836\) 0 0
\(837\) 16.0000 0.553041
\(838\) 6.00000 0.207267
\(839\) 12.0000 0.414286 0.207143 0.978311i \(-0.433583\pi\)
0.207143 + 0.978311i \(0.433583\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) −10.0000 −0.344623
\(843\) −12.0000 −0.413302
\(844\) −4.00000 −0.137686
\(845\) 0 0
\(846\) 12.0000 0.412568
\(847\) 11.0000 0.377964
\(848\) −6.00000 −0.206041
\(849\) 44.0000 1.51008
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −44.0000 −1.50653 −0.753266 0.657716i \(-0.771523\pi\)
−0.753266 + 0.657716i \(0.771523\pi\)
\(854\) −8.00000 −0.273754
\(855\) 0 0
\(856\) −12.0000 −0.410152
\(857\) 18.0000 0.614868 0.307434 0.951569i \(-0.400530\pi\)
0.307434 + 0.951569i \(0.400530\pi\)
\(858\) 0 0
\(859\) 14.0000 0.477674 0.238837 0.971060i \(-0.423234\pi\)
0.238837 + 0.971060i \(0.423234\pi\)
\(860\) 0 0
\(861\) −12.0000 −0.408959
\(862\) 24.0000 0.817443
\(863\) 24.0000 0.816970 0.408485 0.912765i \(-0.366057\pi\)
0.408485 + 0.912765i \(0.366057\pi\)
\(864\) −4.00000 −0.136083
\(865\) 0 0
\(866\) 34.0000 1.15537
\(867\) 38.0000 1.29055
\(868\) 4.00000 0.135769
\(869\) 0 0
\(870\) 0 0
\(871\) 16.0000 0.542139
\(872\) 2.00000 0.0677285
\(873\) 10.0000 0.338449
\(874\) 0 0
\(875\) 0 0
\(876\) −4.00000 −0.135147
\(877\) 22.0000 0.742887 0.371444 0.928456i \(-0.378863\pi\)
0.371444 + 0.928456i \(0.378863\pi\)
\(878\) 8.00000 0.269987
\(879\) −48.0000 −1.61900
\(880\) 0 0
\(881\) −54.0000 −1.81931 −0.909653 0.415369i \(-0.863653\pi\)
−0.909653 + 0.415369i \(0.863653\pi\)
\(882\) 1.00000 0.0336718
\(883\) −20.0000 −0.673054 −0.336527 0.941674i \(-0.609252\pi\)
−0.336527 + 0.941674i \(0.609252\pi\)
\(884\) −24.0000 −0.807207
\(885\) 0 0
\(886\) 12.0000 0.403148
\(887\) 36.0000 1.20876 0.604381 0.796696i \(-0.293421\pi\)
0.604381 + 0.796696i \(0.293421\pi\)
\(888\) −4.00000 −0.134231
\(889\) −16.0000 −0.536623
\(890\) 0 0
\(891\) 0 0
\(892\) −8.00000 −0.267860
\(893\) 24.0000 0.803129
\(894\) −36.0000 −1.20402
\(895\) 0 0
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) 18.0000 0.600668
\(899\) 24.0000 0.800445
\(900\) 0 0
\(901\) 36.0000 1.19933
\(902\) 0 0
\(903\) 16.0000 0.532447
\(904\) −6.00000 −0.199557
\(905\) 0 0
\(906\) 16.0000 0.531564
\(907\) −44.0000 −1.46100 −0.730498 0.682915i \(-0.760712\pi\)
−0.730498 + 0.682915i \(0.760712\pi\)
\(908\) −18.0000 −0.597351
\(909\) 0 0
\(910\) 0 0
\(911\) 48.0000 1.59031 0.795155 0.606406i \(-0.207389\pi\)
0.795155 + 0.606406i \(0.207389\pi\)
\(912\) 4.00000 0.132453
\(913\) 0 0
\(914\) 10.0000 0.330771
\(915\) 0 0
\(916\) −4.00000 −0.132164
\(917\) −18.0000 −0.594412
\(918\) 24.0000 0.792118
\(919\) 56.0000 1.84727 0.923635 0.383274i \(-0.125203\pi\)
0.923635 + 0.383274i \(0.125203\pi\)
\(920\) 0 0
\(921\) −4.00000 −0.131804
\(922\) 12.0000 0.395199
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) −32.0000 −1.05159
\(927\) 4.00000 0.131377
\(928\) −6.00000 −0.196960
\(929\) 6.00000 0.196854 0.0984268 0.995144i \(-0.468619\pi\)
0.0984268 + 0.995144i \(0.468619\pi\)
\(930\) 0 0
\(931\) 2.00000 0.0655474
\(932\) 6.00000 0.196537
\(933\) −48.0000 −1.57145
\(934\) 6.00000 0.196326
\(935\) 0 0
\(936\) 4.00000 0.130744
\(937\) −2.00000 −0.0653372 −0.0326686 0.999466i \(-0.510401\pi\)
−0.0326686 + 0.999466i \(0.510401\pi\)
\(938\) −4.00000 −0.130605
\(939\) 20.0000 0.652675
\(940\) 0 0
\(941\) −24.0000 −0.782378 −0.391189 0.920310i \(-0.627936\pi\)
−0.391189 + 0.920310i \(0.627936\pi\)
\(942\) 8.00000 0.260654
\(943\) 0 0
\(944\) −6.00000 −0.195283
\(945\) 0 0
\(946\) 0 0
\(947\) −24.0000 −0.779895 −0.389948 0.920837i \(-0.627507\pi\)
−0.389948 + 0.920837i \(0.627507\pi\)
\(948\) 16.0000 0.519656
\(949\) −8.00000 −0.259691
\(950\) 0 0
\(951\) −12.0000 −0.389127
\(952\) 6.00000 0.194461
\(953\) 54.0000 1.74923 0.874616 0.484817i \(-0.161114\pi\)
0.874616 + 0.484817i \(0.161114\pi\)
\(954\) −6.00000 −0.194257
\(955\) 0 0
\(956\) 24.0000 0.776215
\(957\) 0 0
\(958\) −36.0000 −1.16311
\(959\) 18.0000 0.581250
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) −8.00000 −0.257930
\(963\) −12.0000 −0.386695
\(964\) −10.0000 −0.322078
\(965\) 0 0
\(966\) 0 0
\(967\) −32.0000 −1.02905 −0.514525 0.857475i \(-0.672032\pi\)
−0.514525 + 0.857475i \(0.672032\pi\)
\(968\) −11.0000 −0.353553
\(969\) −24.0000 −0.770991
\(970\) 0 0
\(971\) −6.00000 −0.192549 −0.0962746 0.995355i \(-0.530693\pi\)
−0.0962746 + 0.995355i \(0.530693\pi\)
\(972\) −10.0000 −0.320750
\(973\) −14.0000 −0.448819
\(974\) 16.0000 0.512673
\(975\) 0 0
\(976\) 8.00000 0.256074
\(977\) 6.00000 0.191957 0.0959785 0.995383i \(-0.469402\pi\)
0.0959785 + 0.995383i \(0.469402\pi\)
\(978\) 32.0000 1.02325
\(979\) 0 0
\(980\) 0 0
\(981\) 2.00000 0.0638551
\(982\) −12.0000 −0.382935
\(983\) 36.0000 1.14822 0.574111 0.818778i \(-0.305348\pi\)
0.574111 + 0.818778i \(0.305348\pi\)
\(984\) 12.0000 0.382546
\(985\) 0 0
\(986\) 36.0000 1.14647
\(987\) −24.0000 −0.763928
\(988\) 8.00000 0.254514
\(989\) 0 0
\(990\) 0 0
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) −4.00000 −0.127000
\(993\) 16.0000 0.507745
\(994\) 0 0
\(995\) 0 0
\(996\) 12.0000 0.380235
\(997\) −8.00000 −0.253363 −0.126681 0.991943i \(-0.540433\pi\)
−0.126681 + 0.991943i \(0.540433\pi\)
\(998\) −4.00000 −0.126618
\(999\) 8.00000 0.253109
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 350.2.a.f.1.1 1
3.2 odd 2 3150.2.a.i.1.1 1
4.3 odd 2 2800.2.a.g.1.1 1
5.2 odd 4 350.2.c.d.99.2 2
5.3 odd 4 350.2.c.d.99.1 2
5.4 even 2 14.2.a.a.1.1 1
7.6 odd 2 2450.2.a.t.1.1 1
15.2 even 4 3150.2.g.j.2899.1 2
15.8 even 4 3150.2.g.j.2899.2 2
15.14 odd 2 126.2.a.b.1.1 1
20.3 even 4 2800.2.g.h.449.1 2
20.7 even 4 2800.2.g.h.449.2 2
20.19 odd 2 112.2.a.c.1.1 1
35.4 even 6 98.2.c.b.79.1 2
35.9 even 6 98.2.c.b.67.1 2
35.13 even 4 2450.2.c.c.99.1 2
35.19 odd 6 98.2.c.a.67.1 2
35.24 odd 6 98.2.c.a.79.1 2
35.27 even 4 2450.2.c.c.99.2 2
35.34 odd 2 98.2.a.a.1.1 1
40.19 odd 2 448.2.a.a.1.1 1
40.29 even 2 448.2.a.g.1.1 1
45.4 even 6 1134.2.f.l.379.1 2
45.14 odd 6 1134.2.f.f.379.1 2
45.29 odd 6 1134.2.f.f.757.1 2
45.34 even 6 1134.2.f.l.757.1 2
55.54 odd 2 1694.2.a.e.1.1 1
60.59 even 2 1008.2.a.h.1.1 1
65.34 odd 4 2366.2.d.b.337.1 2
65.44 odd 4 2366.2.d.b.337.2 2
65.64 even 2 2366.2.a.j.1.1 1
80.19 odd 4 1792.2.b.g.897.2 2
80.29 even 4 1792.2.b.c.897.1 2
80.59 odd 4 1792.2.b.g.897.1 2
80.69 even 4 1792.2.b.c.897.2 2
85.84 even 2 4046.2.a.f.1.1 1
95.94 odd 2 5054.2.a.c.1.1 1
105.44 odd 6 882.2.g.c.361.1 2
105.59 even 6 882.2.g.d.667.1 2
105.74 odd 6 882.2.g.c.667.1 2
105.89 even 6 882.2.g.d.361.1 2
105.104 even 2 882.2.a.i.1.1 1
115.114 odd 2 7406.2.a.a.1.1 1
120.29 odd 2 4032.2.a.w.1.1 1
120.59 even 2 4032.2.a.r.1.1 1
140.19 even 6 784.2.i.i.753.1 2
140.39 odd 6 784.2.i.c.177.1 2
140.59 even 6 784.2.i.i.177.1 2
140.79 odd 6 784.2.i.c.753.1 2
140.139 even 2 784.2.a.b.1.1 1
280.69 odd 2 3136.2.a.e.1.1 1
280.139 even 2 3136.2.a.z.1.1 1
420.419 odd 2 7056.2.a.bd.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
14.2.a.a.1.1 1 5.4 even 2
98.2.a.a.1.1 1 35.34 odd 2
98.2.c.a.67.1 2 35.19 odd 6
98.2.c.a.79.1 2 35.24 odd 6
98.2.c.b.67.1 2 35.9 even 6
98.2.c.b.79.1 2 35.4 even 6
112.2.a.c.1.1 1 20.19 odd 2
126.2.a.b.1.1 1 15.14 odd 2
350.2.a.f.1.1 1 1.1 even 1 trivial
350.2.c.d.99.1 2 5.3 odd 4
350.2.c.d.99.2 2 5.2 odd 4
448.2.a.a.1.1 1 40.19 odd 2
448.2.a.g.1.1 1 40.29 even 2
784.2.a.b.1.1 1 140.139 even 2
784.2.i.c.177.1 2 140.39 odd 6
784.2.i.c.753.1 2 140.79 odd 6
784.2.i.i.177.1 2 140.59 even 6
784.2.i.i.753.1 2 140.19 even 6
882.2.a.i.1.1 1 105.104 even 2
882.2.g.c.361.1 2 105.44 odd 6
882.2.g.c.667.1 2 105.74 odd 6
882.2.g.d.361.1 2 105.89 even 6
882.2.g.d.667.1 2 105.59 even 6
1008.2.a.h.1.1 1 60.59 even 2
1134.2.f.f.379.1 2 45.14 odd 6
1134.2.f.f.757.1 2 45.29 odd 6
1134.2.f.l.379.1 2 45.4 even 6
1134.2.f.l.757.1 2 45.34 even 6
1694.2.a.e.1.1 1 55.54 odd 2
1792.2.b.c.897.1 2 80.29 even 4
1792.2.b.c.897.2 2 80.69 even 4
1792.2.b.g.897.1 2 80.59 odd 4
1792.2.b.g.897.2 2 80.19 odd 4
2366.2.a.j.1.1 1 65.64 even 2
2366.2.d.b.337.1 2 65.34 odd 4
2366.2.d.b.337.2 2 65.44 odd 4
2450.2.a.t.1.1 1 7.6 odd 2
2450.2.c.c.99.1 2 35.13 even 4
2450.2.c.c.99.2 2 35.27 even 4
2800.2.a.g.1.1 1 4.3 odd 2
2800.2.g.h.449.1 2 20.3 even 4
2800.2.g.h.449.2 2 20.7 even 4
3136.2.a.e.1.1 1 280.69 odd 2
3136.2.a.z.1.1 1 280.139 even 2
3150.2.a.i.1.1 1 3.2 odd 2
3150.2.g.j.2899.1 2 15.2 even 4
3150.2.g.j.2899.2 2 15.8 even 4
4032.2.a.r.1.1 1 120.59 even 2
4032.2.a.w.1.1 1 120.29 odd 2
4046.2.a.f.1.1 1 85.84 even 2
5054.2.a.c.1.1 1 95.94 odd 2
7056.2.a.bd.1.1 1 420.419 odd 2
7406.2.a.a.1.1 1 115.114 odd 2