Properties

Label 8470.2.a.c.1.1
Level $8470$
Weight $2$
Character 8470.1
Self dual yes
Analytic conductor $67.633$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8470.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(67.6332905120\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 770)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +2.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +2.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -2.00000 q^{12} +4.00000 q^{13} +1.00000 q^{14} +2.00000 q^{15} +1.00000 q^{16} -1.00000 q^{18} +4.00000 q^{19} -1.00000 q^{20} +2.00000 q^{21} +2.00000 q^{24} +1.00000 q^{25} -4.00000 q^{26} +4.00000 q^{27} -1.00000 q^{28} +6.00000 q^{29} -2.00000 q^{30} -10.0000 q^{31} -1.00000 q^{32} +1.00000 q^{35} +1.00000 q^{36} +2.00000 q^{37} -4.00000 q^{38} -8.00000 q^{39} +1.00000 q^{40} +12.0000 q^{41} -2.00000 q^{42} +4.00000 q^{43} -1.00000 q^{45} +6.00000 q^{47} -2.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} +4.00000 q^{52} -6.00000 q^{53} -4.00000 q^{54} +1.00000 q^{56} -8.00000 q^{57} -6.00000 q^{58} -6.00000 q^{59} +2.00000 q^{60} +4.00000 q^{61} +10.0000 q^{62} -1.00000 q^{63} +1.00000 q^{64} -4.00000 q^{65} -4.00000 q^{67} -1.00000 q^{70} +12.0000 q^{71} -1.00000 q^{72} +4.00000 q^{73} -2.00000 q^{74} -2.00000 q^{75} +4.00000 q^{76} +8.00000 q^{78} -8.00000 q^{79} -1.00000 q^{80} -11.0000 q^{81} -12.0000 q^{82} -12.0000 q^{83} +2.00000 q^{84} -4.00000 q^{86} -12.0000 q^{87} +18.0000 q^{89} +1.00000 q^{90} -4.00000 q^{91} +20.0000 q^{93} -6.00000 q^{94} -4.00000 q^{95} +2.00000 q^{96} -10.0000 q^{97} -1.00000 q^{98} +O(q^{100})\)

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 \(-0.695913\pi\)
−0.577350 + 0.816497i \(0.695913\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 2.00000 0.816497
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) 0 0
\(12\) −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
\(15\) 2.00000 0.516398
\(16\) 1.00000 0.250000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) −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
\(21\) 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
\(25\) 1.00000 0.200000
\(26\) −4.00000 −0.784465
\(27\) 4.00000 0.769800
\(28\) −1.00000 −0.188982
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) −2.00000 −0.365148
\(31\) −10.0000 −1.79605 −0.898027 0.439941i \(-0.854999\pi\)
−0.898027 + 0.439941i \(0.854999\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 0 0
\(35\) 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
\(39\) −8.00000 −1.28103
\(40\) 1.00000 0.158114
\(41\) 12.0000 1.87409 0.937043 0.349215i \(-0.113552\pi\)
0.937043 + 0.349215i \(0.113552\pi\)
\(42\) −2.00000 −0.308607
\(43\) 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
\(46\) 0 0
\(47\) 6.00000 0.875190 0.437595 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) −2.00000 −0.288675
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) 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
\(57\) −8.00000 −1.05963
\(58\) −6.00000 −0.787839
\(59\) −6.00000 −0.781133 −0.390567 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 2.00000 0.258199
\(61\) 4.00000 0.512148 0.256074 0.966657i \(-0.417571\pi\)
0.256074 + 0.966657i \(0.417571\pi\)
\(62\) 10.0000 1.27000
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) −4.00000 −0.496139
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) −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 \(-0.424791\pi\)
0.234082 + 0.972217i \(0.424791\pi\)
\(74\) −2.00000 −0.232495
\(75\) −2.00000 −0.230940
\(76\) 4.00000 0.458831
\(77\) 0 0
\(78\) 8.00000 0.905822
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) −1.00000 −0.111803
\(81\) −11.0000 −1.22222
\(82\) −12.0000 −1.32518
\(83\) −12.0000 −1.31717 −0.658586 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 2.00000 0.218218
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) −12.0000 −1.28654
\(88\) 0 0
\(89\) 18.0000 1.90800 0.953998 0.299813i \(-0.0969242\pi\)
0.953998 + 0.299813i \(0.0969242\pi\)
\(90\) 1.00000 0.105409
\(91\) −4.00000 −0.419314
\(92\) 0 0
\(93\) 20.0000 2.07390
\(94\) −6.00000 −0.618853
\(95\) −4.00000 −0.410391
\(96\) 2.00000 0.204124
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 14.0000 1.37946 0.689730 0.724066i \(-0.257729\pi\)
0.689730 + 0.724066i \(0.257729\pi\)
\(104\) −4.00000 −0.392232
\(105\) −2.00000 −0.195180
\(106\) 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 \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 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 \(0.591070\pi\)
\(114\) 8.00000 0.749269
\(115\) 0 0
\(116\) 6.00000 0.557086
\(117\) 4.00000 0.369800
\(118\) 6.00000 0.552345
\(119\) 0 0
\(120\) −2.00000 −0.182574
\(121\) 0 0
\(122\) −4.00000 −0.362143
\(123\) −24.0000 −2.16401
\(124\) −10.0000 −0.898027
\(125\) −1.00000 −0.0894427
\(126\) 1.00000 0.0890871
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −8.00000 −0.704361
\(130\) 4.00000 0.350823
\(131\) 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
\(135\) −4.00000 −0.344265
\(136\) 0 0
\(137\) 6.00000 0.512615 0.256307 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) 0 0
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 1.00000 0.0845154
\(141\) −12.0000 −1.01058
\(142\) −12.0000 −1.00702
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −6.00000 −0.498273
\(146\) −4.00000 −0.331042
\(147\) −2.00000 −0.164957
\(148\) 2.00000 0.164399
\(149\) 18.0000 1.47462 0.737309 0.675556i \(-0.236096\pi\)
0.737309 + 0.675556i \(0.236096\pi\)
\(150\) 2.00000 0.163299
\(151\) 16.0000 1.30206 0.651031 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) −4.00000 −0.324443
\(153\) 0 0
\(154\) 0 0
\(155\) 10.0000 0.803219
\(156\) −8.00000 −0.640513
\(157\) −22.0000 −1.75579 −0.877896 0.478852i \(-0.841053\pi\)
−0.877896 + 0.478852i \(0.841053\pi\)
\(158\) 8.00000 0.636446
\(159\) 12.0000 0.951662
\(160\) 1.00000 0.0790569
\(161\) 0 0
\(162\) 11.0000 0.864242
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 12.0000 0.937043
\(165\) 0 0
\(166\) 12.0000 0.931381
\(167\) −24.0000 −1.85718 −0.928588 0.371113i \(-0.878976\pi\)
−0.928588 + 0.371113i \(0.878976\pi\)
\(168\) −2.00000 −0.154303
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) 4.00000 0.305888
\(172\) 4.00000 0.304997
\(173\) −12.0000 −0.912343 −0.456172 0.889892i \(-0.650780\pi\)
−0.456172 + 0.889892i \(0.650780\pi\)
\(174\) 12.0000 0.909718
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) 12.0000 0.901975
\(178\) −18.0000 −1.34916
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) −1.00000 −0.0745356
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 4.00000 0.296500
\(183\) −8.00000 −0.591377
\(184\) 0 0
\(185\) −2.00000 −0.147043
\(186\) −20.0000 −1.46647
\(187\) 0 0
\(188\) 6.00000 0.437595
\(189\) −4.00000 −0.290957
\(190\) 4.00000 0.290191
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) −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
\(195\) 8.00000 0.572892
\(196\) 1.00000 0.0714286
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 0 0
\(199\) 2.00000 0.141776 0.0708881 0.997484i \(-0.477417\pi\)
0.0708881 + 0.997484i \(0.477417\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 8.00000 0.564276
\(202\) 0 0
\(203\) −6.00000 −0.421117
\(204\) 0 0
\(205\) −12.0000 −0.838116
\(206\) −14.0000 −0.975426
\(207\) 0 0
\(208\) 4.00000 0.277350
\(209\) 0 0
\(210\) 2.00000 0.138013
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) −6.00000 −0.412082
\(213\) −24.0000 −1.64445
\(214\) 12.0000 0.820303
\(215\) −4.00000 −0.272798
\(216\) −4.00000 −0.272166
\(217\) 10.0000 0.678844
\(218\) 2.00000 0.135457
\(219\) −8.00000 −0.540590
\(220\) 0 0
\(221\) 0 0
\(222\) 4.00000 0.268462
\(223\) −10.0000 −0.669650 −0.334825 0.942280i \(-0.608677\pi\)
−0.334825 + 0.942280i \(0.608677\pi\)
\(224\) 1.00000 0.0668153
\(225\) 1.00000 0.0666667
\(226\) 6.00000 0.399114
\(227\) 24.0000 1.59294 0.796468 0.604681i \(-0.206699\pi\)
0.796468 + 0.604681i \(0.206699\pi\)
\(228\) −8.00000 −0.529813
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 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
\(235\) −6.00000 −0.391397
\(236\) −6.00000 −0.390567
\(237\) 16.0000 1.03931
\(238\) 0 0
\(239\) 24.0000 1.55243 0.776215 0.630468i \(-0.217137\pi\)
0.776215 + 0.630468i \(0.217137\pi\)
\(240\) 2.00000 0.129099
\(241\) 4.00000 0.257663 0.128831 0.991667i \(-0.458877\pi\)
0.128831 + 0.991667i \(0.458877\pi\)
\(242\) 0 0
\(243\) 10.0000 0.641500
\(244\) 4.00000 0.256074
\(245\) −1.00000 −0.0638877
\(246\) 24.0000 1.53018
\(247\) 16.0000 1.01806
\(248\) 10.0000 0.635001
\(249\) 24.0000 1.52094
\(250\) 1.00000 0.0632456
\(251\) −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
\(254\) 8.00000 0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 18.0000 1.12281 0.561405 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) 8.00000 0.498058
\(259\) −2.00000 −0.124274
\(260\) −4.00000 −0.248069
\(261\) 6.00000 0.371391
\(262\) 0 0
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 6.00000 0.368577
\(266\) 4.00000 0.245256
\(267\) −36.0000 −2.20316
\(268\) −4.00000 −0.244339
\(269\) 6.00000 0.365826 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) 4.00000 0.243432
\(271\) −20.0000 −1.21491 −0.607457 0.794353i \(-0.707810\pi\)
−0.607457 + 0.794353i \(0.707810\pi\)
\(272\) 0 0
\(273\) 8.00000 0.484182
\(274\) −6.00000 −0.362473
\(275\) 0 0
\(276\) 0 0
\(277\) 22.0000 1.32185 0.660926 0.750451i \(-0.270164\pi\)
0.660926 + 0.750451i \(0.270164\pi\)
\(278\) −4.00000 −0.239904
\(279\) −10.0000 −0.598684
\(280\) −1.00000 −0.0597614
\(281\) 18.0000 1.07379 0.536895 0.843649i \(-0.319597\pi\)
0.536895 + 0.843649i \(0.319597\pi\)
\(282\) 12.0000 0.714590
\(283\) 16.0000 0.951101 0.475551 0.879688i \(-0.342249\pi\)
0.475551 + 0.879688i \(0.342249\pi\)
\(284\) 12.0000 0.712069
\(285\) 8.00000 0.473879
\(286\) 0 0
\(287\) −12.0000 −0.708338
\(288\) −1.00000 −0.0589256
\(289\) −17.0000 −1.00000
\(290\) 6.00000 0.352332
\(291\) 20.0000 1.17242
\(292\) 4.00000 0.234082
\(293\) −24.0000 −1.40209 −0.701047 0.713115i \(-0.747284\pi\)
−0.701047 + 0.713115i \(0.747284\pi\)
\(294\) 2.00000 0.116642
\(295\) 6.00000 0.349334
\(296\) −2.00000 −0.116248
\(297\) 0 0
\(298\) −18.0000 −1.04271
\(299\) 0 0
\(300\) −2.00000 −0.115470
\(301\) −4.00000 −0.230556
\(302\) −16.0000 −0.920697
\(303\) 0 0
\(304\) 4.00000 0.229416
\(305\) −4.00000 −0.229039
\(306\) 0 0
\(307\) −32.0000 −1.82634 −0.913168 0.407583i \(-0.866372\pi\)
−0.913168 + 0.407583i \(0.866372\pi\)
\(308\) 0 0
\(309\) −28.0000 −1.59286
\(310\) −10.0000 −0.567962
\(311\) 18.0000 1.02069 0.510343 0.859971i \(-0.329518\pi\)
0.510343 + 0.859971i \(0.329518\pi\)
\(312\) 8.00000 0.452911
\(313\) 2.00000 0.113047 0.0565233 0.998401i \(-0.481998\pi\)
0.0565233 + 0.998401i \(0.481998\pi\)
\(314\) 22.0000 1.24153
\(315\) 1.00000 0.0563436
\(316\) −8.00000 −0.450035
\(317\) 6.00000 0.336994 0.168497 0.985702i \(-0.446109\pi\)
0.168497 + 0.985702i \(0.446109\pi\)
\(318\) −12.0000 −0.672927
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) 24.0000 1.33955
\(322\) 0 0
\(323\) 0 0
\(324\) −11.0000 −0.611111
\(325\) 4.00000 0.221880
\(326\) 4.00000 0.221540
\(327\) 4.00000 0.221201
\(328\) −12.0000 −0.662589
\(329\) −6.00000 −0.330791
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) −12.0000 −0.658586
\(333\) 2.00000 0.109599
\(334\) 24.0000 1.31322
\(335\) 4.00000 0.218543
\(336\) 2.00000 0.109109
\(337\) 22.0000 1.19842 0.599208 0.800593i \(-0.295482\pi\)
0.599208 + 0.800593i \(0.295482\pi\)
\(338\) −3.00000 −0.163178
\(339\) 12.0000 0.651751
\(340\) 0 0
\(341\) 0 0
\(342\) −4.00000 −0.216295
\(343\) −1.00000 −0.0539949
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) 12.0000 0.645124
\(347\) −12.0000 −0.644194 −0.322097 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) −12.0000 −0.643268
\(349\) −20.0000 −1.07058 −0.535288 0.844670i \(-0.679797\pi\)
−0.535288 + 0.844670i \(0.679797\pi\)
\(350\) 1.00000 0.0534522
\(351\) 16.0000 0.854017
\(352\) 0 0
\(353\) −6.00000 −0.319348 −0.159674 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) −12.0000 −0.637793
\(355\) −12.0000 −0.636894
\(356\) 18.0000 0.953998
\(357\) 0 0
\(358\) −12.0000 −0.634220
\(359\) 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
\(362\) 10.0000 0.525588
\(363\) 0 0
\(364\) −4.00000 −0.209657
\(365\) −4.00000 −0.209370
\(366\) 8.00000 0.418167
\(367\) −34.0000 −1.77479 −0.887393 0.461014i \(-0.847486\pi\)
−0.887393 + 0.461014i \(0.847486\pi\)
\(368\) 0 0
\(369\) 12.0000 0.624695
\(370\) 2.00000 0.103975
\(371\) 6.00000 0.311504
\(372\) 20.0000 1.03695
\(373\) 22.0000 1.13912 0.569558 0.821951i \(-0.307114\pi\)
0.569558 + 0.821951i \(0.307114\pi\)
\(374\) 0 0
\(375\) 2.00000 0.103280
\(376\) −6.00000 −0.309426
\(377\) 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 \(0.634797\pi\)
\(380\) −4.00000 −0.205196
\(381\) 16.0000 0.819705
\(382\) 0 0
\(383\) −6.00000 −0.306586 −0.153293 0.988181i \(-0.548988\pi\)
−0.153293 + 0.988181i \(0.548988\pi\)
\(384\) 2.00000 0.102062
\(385\) 0 0
\(386\) 14.0000 0.712581
\(387\) 4.00000 0.203331
\(388\) −10.0000 −0.507673
\(389\) 30.0000 1.52106 0.760530 0.649303i \(-0.224939\pi\)
0.760530 + 0.649303i \(0.224939\pi\)
\(390\) −8.00000 −0.405096
\(391\) 0 0
\(392\) −1.00000 −0.0505076
\(393\) 0 0
\(394\) −6.00000 −0.302276
\(395\) 8.00000 0.402524
\(396\) 0 0
\(397\) 2.00000 0.100377 0.0501886 0.998740i \(-0.484018\pi\)
0.0501886 + 0.998740i \(0.484018\pi\)
\(398\) −2.00000 −0.100251
\(399\) 8.00000 0.400501
\(400\) 1.00000 0.0500000
\(401\) −6.00000 −0.299626 −0.149813 0.988714i \(-0.547867\pi\)
−0.149813 + 0.988714i \(0.547867\pi\)
\(402\) −8.00000 −0.399004
\(403\) −40.0000 −1.99254
\(404\) 0 0
\(405\) 11.0000 0.546594
\(406\) 6.00000 0.297775
\(407\) 0 0
\(408\) 0 0
\(409\) −32.0000 −1.58230 −0.791149 0.611623i \(-0.790517\pi\)
−0.791149 + 0.611623i \(0.790517\pi\)
\(410\) 12.0000 0.592638
\(411\) −12.0000 −0.591916
\(412\) 14.0000 0.689730
\(413\) 6.00000 0.295241
\(414\) 0 0
\(415\) 12.0000 0.589057
\(416\) −4.00000 −0.196116
\(417\) −8.00000 −0.391762
\(418\) 0 0
\(419\) −6.00000 −0.293119 −0.146560 0.989202i \(-0.546820\pi\)
−0.146560 + 0.989202i \(0.546820\pi\)
\(420\) −2.00000 −0.0975900
\(421\) 26.0000 1.26716 0.633581 0.773676i \(-0.281584\pi\)
0.633581 + 0.773676i \(0.281584\pi\)
\(422\) −4.00000 −0.194717
\(423\) 6.00000 0.291730
\(424\) 6.00000 0.291386
\(425\) 0 0
\(426\) 24.0000 1.16280
\(427\) −4.00000 −0.193574
\(428\) −12.0000 −0.580042
\(429\) 0 0
\(430\) 4.00000 0.192897
\(431\) −24.0000 −1.15604 −0.578020 0.816023i \(-0.696174\pi\)
−0.578020 + 0.816023i \(0.696174\pi\)
\(432\) 4.00000 0.192450
\(433\) 14.0000 0.672797 0.336399 0.941720i \(-0.390791\pi\)
0.336399 + 0.941720i \(0.390791\pi\)
\(434\) −10.0000 −0.480015
\(435\) 12.0000 0.575356
\(436\) −2.00000 −0.0957826
\(437\) 0 0
\(438\) 8.00000 0.382255
\(439\) 28.0000 1.33637 0.668184 0.743996i \(-0.267072\pi\)
0.668184 + 0.743996i \(0.267072\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −24.0000 −1.14027 −0.570137 0.821549i \(-0.693110\pi\)
−0.570137 + 0.821549i \(0.693110\pi\)
\(444\) −4.00000 −0.189832
\(445\) −18.0000 −0.853282
\(446\) 10.0000 0.473514
\(447\) −36.0000 −1.70274
\(448\) −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
\(453\) −32.0000 −1.50349
\(454\) −24.0000 −1.12638
\(455\) 4.00000 0.187523
\(456\) 8.00000 0.374634
\(457\) 10.0000 0.467780 0.233890 0.972263i \(-0.424854\pi\)
0.233890 + 0.972263i \(0.424854\pi\)
\(458\) 10.0000 0.467269
\(459\) 0 0
\(460\) 0 0
\(461\) 36.0000 1.67669 0.838344 0.545142i \(-0.183524\pi\)
0.838344 + 0.545142i \(0.183524\pi\)
\(462\) 0 0
\(463\) −40.0000 −1.85896 −0.929479 0.368875i \(-0.879743\pi\)
−0.929479 + 0.368875i \(0.879743\pi\)
\(464\) 6.00000 0.278543
\(465\) −20.0000 −0.927478
\(466\) −6.00000 −0.277945
\(467\) 42.0000 1.94353 0.971764 0.235954i \(-0.0758216\pi\)
0.971764 + 0.235954i \(0.0758216\pi\)
\(468\) 4.00000 0.184900
\(469\) 4.00000 0.184703
\(470\) 6.00000 0.276759
\(471\) 44.0000 2.02741
\(472\) 6.00000 0.276172
\(473\) 0 0
\(474\) −16.0000 −0.734904
\(475\) 4.00000 0.183533
\(476\) 0 0
\(477\) −6.00000 −0.274721
\(478\) −24.0000 −1.09773
\(479\) −12.0000 −0.548294 −0.274147 0.961688i \(-0.588395\pi\)
−0.274147 + 0.961688i \(0.588395\pi\)
\(480\) −2.00000 −0.0912871
\(481\) 8.00000 0.364769
\(482\) −4.00000 −0.182195
\(483\) 0 0
\(484\) 0 0
\(485\) 10.0000 0.454077
\(486\) −10.0000 −0.453609
\(487\) 32.0000 1.45006 0.725029 0.688718i \(-0.241826\pi\)
0.725029 + 0.688718i \(0.241826\pi\)
\(488\) −4.00000 −0.181071
\(489\) 8.00000 0.361773
\(490\) 1.00000 0.0451754
\(491\) 12.0000 0.541552 0.270776 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) −24.0000 −1.08200
\(493\) 0 0
\(494\) −16.0000 −0.719874
\(495\) 0 0
\(496\) −10.0000 −0.449013
\(497\) −12.0000 −0.538274
\(498\) −24.0000 −1.07547
\(499\) −40.0000 −1.79065 −0.895323 0.445418i \(-0.853055\pi\)
−0.895323 + 0.445418i \(0.853055\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 48.0000 2.14448
\(502\) 18.0000 0.803379
\(503\) 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
\(506\) 0 0
\(507\) −6.00000 −0.266469
\(508\) −8.00000 −0.354943
\(509\) 6.00000 0.265945 0.132973 0.991120i \(-0.457548\pi\)
0.132973 + 0.991120i \(0.457548\pi\)
\(510\) 0 0
\(511\) −4.00000 −0.176950
\(512\) −1.00000 −0.0441942
\(513\) 16.0000 0.706417
\(514\) −18.0000 −0.793946
\(515\) −14.0000 −0.616914
\(516\) −8.00000 −0.352180
\(517\) 0 0
\(518\) 2.00000 0.0878750
\(519\) 24.0000 1.05348
\(520\) 4.00000 0.175412
\(521\) −42.0000 −1.84005 −0.920027 0.391856i \(-0.871833\pi\)
−0.920027 + 0.391856i \(0.871833\pi\)
\(522\) −6.00000 −0.262613
\(523\) 16.0000 0.699631 0.349816 0.936819i \(-0.386244\pi\)
0.349816 + 0.936819i \(0.386244\pi\)
\(524\) 0 0
\(525\) 2.00000 0.0872872
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) −6.00000 −0.260623
\(531\) −6.00000 −0.260378
\(532\) −4.00000 −0.173422
\(533\) 48.0000 2.07911
\(534\) 36.0000 1.55787
\(535\) 12.0000 0.518805
\(536\) 4.00000 0.172774
\(537\) −24.0000 −1.03568
\(538\) −6.00000 −0.258678
\(539\) 0 0
\(540\) −4.00000 −0.172133
\(541\) 10.0000 0.429934 0.214967 0.976621i \(-0.431036\pi\)
0.214967 + 0.976621i \(0.431036\pi\)
\(542\) 20.0000 0.859074
\(543\) 20.0000 0.858282
\(544\) 0 0
\(545\) 2.00000 0.0856706
\(546\) −8.00000 −0.342368
\(547\) 28.0000 1.19719 0.598597 0.801050i \(-0.295725\pi\)
0.598597 + 0.801050i \(0.295725\pi\)
\(548\) 6.00000 0.256307
\(549\) 4.00000 0.170716
\(550\) 0 0
\(551\) 24.0000 1.02243
\(552\) 0 0
\(553\) 8.00000 0.340195
\(554\) −22.0000 −0.934690
\(555\) 4.00000 0.169791
\(556\) 4.00000 0.169638
\(557\) 6.00000 0.254228 0.127114 0.991888i \(-0.459429\pi\)
0.127114 + 0.991888i \(0.459429\pi\)
\(558\) 10.0000 0.423334
\(559\) 16.0000 0.676728
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) −18.0000 −0.759284
\(563\) 12.0000 0.505740 0.252870 0.967500i \(-0.418626\pi\)
0.252870 + 0.967500i \(0.418626\pi\)
\(564\) −12.0000 −0.505291
\(565\) 6.00000 0.252422
\(566\) −16.0000 −0.672530
\(567\) 11.0000 0.461957
\(568\) −12.0000 −0.503509
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) −8.00000 −0.335083
\(571\) 4.00000 0.167395 0.0836974 0.996491i \(-0.473327\pi\)
0.0836974 + 0.996491i \(0.473327\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 12.0000 0.500870
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 38.0000 1.58196 0.790980 0.611842i \(-0.209571\pi\)
0.790980 + 0.611842i \(0.209571\pi\)
\(578\) 17.0000 0.707107
\(579\) 28.0000 1.16364
\(580\) −6.00000 −0.249136
\(581\) 12.0000 0.497844
\(582\) −20.0000 −0.829027
\(583\) 0 0
\(584\) −4.00000 −0.165521
\(585\) −4.00000 −0.165380
\(586\) 24.0000 0.991431
\(587\) 18.0000 0.742940 0.371470 0.928445i \(-0.378854\pi\)
0.371470 + 0.928445i \(0.378854\pi\)
\(588\) −2.00000 −0.0824786
\(589\) −40.0000 −1.64817
\(590\) −6.00000 −0.247016
\(591\) −12.0000 −0.493614
\(592\) 2.00000 0.0821995
\(593\) 12.0000 0.492781 0.246390 0.969171i \(-0.420755\pi\)
0.246390 + 0.969171i \(0.420755\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 18.0000 0.737309
\(597\) −4.00000 −0.163709
\(598\) 0 0
\(599\) 36.0000 1.47092 0.735460 0.677568i \(-0.236966\pi\)
0.735460 + 0.677568i \(0.236966\pi\)
\(600\) 2.00000 0.0816497
\(601\) 4.00000 0.163163 0.0815817 0.996667i \(-0.474003\pi\)
0.0815817 + 0.996667i \(0.474003\pi\)
\(602\) 4.00000 0.163028
\(603\) −4.00000 −0.162893
\(604\) 16.0000 0.651031
\(605\) 0 0
\(606\) 0 0
\(607\) 40.0000 1.62355 0.811775 0.583970i \(-0.198502\pi\)
0.811775 + 0.583970i \(0.198502\pi\)
\(608\) −4.00000 −0.162221
\(609\) 12.0000 0.486265
\(610\) 4.00000 0.161955
\(611\) 24.0000 0.970936
\(612\) 0 0
\(613\) 10.0000 0.403896 0.201948 0.979396i \(-0.435273\pi\)
0.201948 + 0.979396i \(0.435273\pi\)
\(614\) 32.0000 1.29141
\(615\) 24.0000 0.967773
\(616\) 0 0
\(617\) 6.00000 0.241551 0.120775 0.992680i \(-0.461462\pi\)
0.120775 + 0.992680i \(0.461462\pi\)
\(618\) 28.0000 1.12633
\(619\) −10.0000 −0.401934 −0.200967 0.979598i \(-0.564408\pi\)
−0.200967 + 0.979598i \(0.564408\pi\)
\(620\) 10.0000 0.401610
\(621\) 0 0
\(622\) −18.0000 −0.721734
\(623\) −18.0000 −0.721155
\(624\) −8.00000 −0.320256
\(625\) 1.00000 0.0400000
\(626\) −2.00000 −0.0799361
\(627\) 0 0
\(628\) −22.0000 −0.877896
\(629\) 0 0
\(630\) −1.00000 −0.0398410
\(631\) 8.00000 0.318475 0.159237 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) 8.00000 0.318223
\(633\) −8.00000 −0.317971
\(634\) −6.00000 −0.238290
\(635\) 8.00000 0.317470
\(636\) 12.0000 0.475831
\(637\) 4.00000 0.158486
\(638\) 0 0
\(639\) 12.0000 0.474713
\(640\) 1.00000 0.0395285
\(641\) 30.0000 1.18493 0.592464 0.805597i \(-0.298155\pi\)
0.592464 + 0.805597i \(0.298155\pi\)
\(642\) −24.0000 −0.947204
\(643\) 2.00000 0.0788723 0.0394362 0.999222i \(-0.487444\pi\)
0.0394362 + 0.999222i \(0.487444\pi\)
\(644\) 0 0
\(645\) 8.00000 0.315000
\(646\) 0 0
\(647\) −6.00000 −0.235884 −0.117942 0.993020i \(-0.537630\pi\)
−0.117942 + 0.993020i \(0.537630\pi\)
\(648\) 11.0000 0.432121
\(649\) 0 0
\(650\) −4.00000 −0.156893
\(651\) −20.0000 −0.783862
\(652\) −4.00000 −0.156652
\(653\) −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
\(656\) 12.0000 0.468521
\(657\) 4.00000 0.156055
\(658\) 6.00000 0.233904
\(659\) −12.0000 −0.467454 −0.233727 0.972302i \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) 0 0
\(661\) 26.0000 1.01128 0.505641 0.862744i \(-0.331256\pi\)
0.505641 + 0.862744i \(0.331256\pi\)
\(662\) −20.0000 −0.777322
\(663\) 0 0
\(664\) 12.0000 0.465690
\(665\) 4.00000 0.155113
\(666\) −2.00000 −0.0774984
\(667\) 0 0
\(668\) −24.0000 −0.928588
\(669\) 20.0000 0.773245
\(670\) −4.00000 −0.154533
\(671\) 0 0
\(672\) −2.00000 −0.0771517
\(673\) −2.00000 −0.0770943 −0.0385472 0.999257i \(-0.512273\pi\)
−0.0385472 + 0.999257i \(0.512273\pi\)
\(674\) −22.0000 −0.847408
\(675\) 4.00000 0.153960
\(676\) 3.00000 0.115385
\(677\) −12.0000 −0.461197 −0.230599 0.973049i \(-0.574068\pi\)
−0.230599 + 0.973049i \(0.574068\pi\)
\(678\) −12.0000 −0.460857
\(679\) 10.0000 0.383765
\(680\) 0 0
\(681\) −48.0000 −1.83936
\(682\) 0 0
\(683\) −24.0000 −0.918334 −0.459167 0.888350i \(-0.651852\pi\)
−0.459167 + 0.888350i \(0.651852\pi\)
\(684\) 4.00000 0.152944
\(685\) −6.00000 −0.229248
\(686\) 1.00000 0.0381802
\(687\) 20.0000 0.763048
\(688\) 4.00000 0.152499
\(689\) −24.0000 −0.914327
\(690\) 0 0
\(691\) 2.00000 0.0760836 0.0380418 0.999276i \(-0.487888\pi\)
0.0380418 + 0.999276i \(0.487888\pi\)
\(692\) −12.0000 −0.456172
\(693\) 0 0
\(694\) 12.0000 0.455514
\(695\) −4.00000 −0.151729
\(696\) 12.0000 0.454859
\(697\) 0 0
\(698\) 20.0000 0.757011
\(699\) −12.0000 −0.453882
\(700\) −1.00000 −0.0377964
\(701\) 30.0000 1.13308 0.566542 0.824033i \(-0.308281\pi\)
0.566542 + 0.824033i \(0.308281\pi\)
\(702\) −16.0000 −0.603881
\(703\) 8.00000 0.301726
\(704\) 0 0
\(705\) 12.0000 0.451946
\(706\) 6.00000 0.225813
\(707\) 0 0
\(708\) 12.0000 0.450988
\(709\) 38.0000 1.42712 0.713560 0.700594i \(-0.247082\pi\)
0.713560 + 0.700594i \(0.247082\pi\)
\(710\) 12.0000 0.450352
\(711\) −8.00000 −0.300023
\(712\) −18.0000 −0.674579
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 12.0000 0.448461
\(717\) −48.0000 −1.79259
\(718\) −24.0000 −0.895672
\(719\) 30.0000 1.11881 0.559406 0.828894i \(-0.311029\pi\)
0.559406 + 0.828894i \(0.311029\pi\)
\(720\) −1.00000 −0.0372678
\(721\) −14.0000 −0.521387
\(722\) 3.00000 0.111648
\(723\) −8.00000 −0.297523
\(724\) −10.0000 −0.371647
\(725\) 6.00000 0.222834
\(726\) 0 0
\(727\) 2.00000 0.0741759 0.0370879 0.999312i \(-0.488192\pi\)
0.0370879 + 0.999312i \(0.488192\pi\)
\(728\) 4.00000 0.148250
\(729\) 13.0000 0.481481
\(730\) 4.00000 0.148047
\(731\) 0 0
\(732\) −8.00000 −0.295689
\(733\) 40.0000 1.47743 0.738717 0.674016i \(-0.235432\pi\)
0.738717 + 0.674016i \(0.235432\pi\)
\(734\) 34.0000 1.25496
\(735\) 2.00000 0.0737711
\(736\) 0 0
\(737\) 0 0
\(738\) −12.0000 −0.441726
\(739\) 4.00000 0.147142 0.0735712 0.997290i \(-0.476560\pi\)
0.0735712 + 0.997290i \(0.476560\pi\)
\(740\) −2.00000 −0.0735215
\(741\) −32.0000 −1.17555
\(742\) −6.00000 −0.220267
\(743\) −24.0000 −0.880475 −0.440237 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) −20.0000 −0.733236
\(745\) −18.0000 −0.659469
\(746\) −22.0000 −0.805477
\(747\) −12.0000 −0.439057
\(748\) 0 0
\(749\) 12.0000 0.438470
\(750\) −2.00000 −0.0730297
\(751\) −28.0000 −1.02173 −0.510867 0.859660i \(-0.670676\pi\)
−0.510867 + 0.859660i \(0.670676\pi\)
\(752\) 6.00000 0.218797
\(753\) 36.0000 1.31191
\(754\) −24.0000 −0.874028
\(755\) −16.0000 −0.582300
\(756\) −4.00000 −0.145479
\(757\) 38.0000 1.38113 0.690567 0.723269i \(-0.257361\pi\)
0.690567 + 0.723269i \(0.257361\pi\)
\(758\) 16.0000 0.581146
\(759\) 0 0
\(760\) 4.00000 0.145095
\(761\) −24.0000 −0.869999 −0.435000 0.900431i \(-0.643252\pi\)
−0.435000 + 0.900431i \(0.643252\pi\)
\(762\) −16.0000 −0.579619
\(763\) 2.00000 0.0724049
\(764\) 0 0
\(765\) 0 0
\(766\) 6.00000 0.216789
\(767\) −24.0000 −0.866590
\(768\) −2.00000 −0.0721688
\(769\) 16.0000 0.576975 0.288487 0.957484i \(-0.406848\pi\)
0.288487 + 0.957484i \(0.406848\pi\)
\(770\) 0 0
\(771\) −36.0000 −1.29651
\(772\) −14.0000 −0.503871
\(773\) −18.0000 −0.647415 −0.323708 0.946157i \(-0.604929\pi\)
−0.323708 + 0.946157i \(0.604929\pi\)
\(774\) −4.00000 −0.143777
\(775\) −10.0000 −0.359211
\(776\) 10.0000 0.358979
\(777\) 4.00000 0.143499
\(778\) −30.0000 −1.07555
\(779\) 48.0000 1.71978
\(780\) 8.00000 0.286446
\(781\) 0 0
\(782\) 0 0
\(783\) 24.0000 0.857690
\(784\) 1.00000 0.0357143
\(785\) 22.0000 0.785214
\(786\) 0 0
\(787\) −44.0000 −1.56843 −0.784215 0.620489i \(-0.786934\pi\)
−0.784215 + 0.620489i \(0.786934\pi\)
\(788\) 6.00000 0.213741
\(789\) 0 0
\(790\) −8.00000 −0.284627
\(791\) 6.00000 0.213335
\(792\) 0 0
\(793\) 16.0000 0.568177
\(794\) −2.00000 −0.0709773
\(795\) −12.0000 −0.425596
\(796\) 2.00000 0.0708881
\(797\) 42.0000 1.48772 0.743858 0.668338i \(-0.232994\pi\)
0.743858 + 0.668338i \(0.232994\pi\)
\(798\) −8.00000 −0.283197
\(799\) 0 0
\(800\) −1.00000 −0.0353553
\(801\) 18.0000 0.635999
\(802\) 6.00000 0.211867
\(803\) 0 0
\(804\) 8.00000 0.282138
\(805\) 0 0
\(806\) 40.0000 1.40894
\(807\) −12.0000 −0.422420
\(808\) 0 0
\(809\) 30.0000 1.05474 0.527372 0.849635i \(-0.323177\pi\)
0.527372 + 0.849635i \(0.323177\pi\)
\(810\) −11.0000 −0.386501
\(811\) 16.0000 0.561836 0.280918 0.959732i \(-0.409361\pi\)
0.280918 + 0.959732i \(0.409361\pi\)
\(812\) −6.00000 −0.210559
\(813\) 40.0000 1.40286
\(814\) 0 0
\(815\) 4.00000 0.140114
\(816\) 0 0
\(817\) 16.0000 0.559769
\(818\) 32.0000 1.11885
\(819\) −4.00000 −0.139771
\(820\) −12.0000 −0.419058
\(821\) −30.0000 −1.04701 −0.523504 0.852023i \(-0.675375\pi\)
−0.523504 + 0.852023i \(0.675375\pi\)
\(822\) 12.0000 0.418548
\(823\) −4.00000 −0.139431 −0.0697156 0.997567i \(-0.522209\pi\)
−0.0697156 + 0.997567i \(0.522209\pi\)
\(824\) −14.0000 −0.487713
\(825\) 0 0
\(826\) −6.00000 −0.208767
\(827\) 12.0000 0.417281 0.208640 0.977992i \(-0.433096\pi\)
0.208640 + 0.977992i \(0.433096\pi\)
\(828\) 0 0
\(829\) 26.0000 0.903017 0.451509 0.892267i \(-0.350886\pi\)
0.451509 + 0.892267i \(0.350886\pi\)
\(830\) −12.0000 −0.416526
\(831\) −44.0000 −1.52634
\(832\) 4.00000 0.138675
\(833\) 0 0
\(834\) 8.00000 0.277017
\(835\) 24.0000 0.830554
\(836\) 0 0
\(837\) −40.0000 −1.38260
\(838\) 6.00000 0.207267
\(839\) 30.0000 1.03572 0.517858 0.855467i \(-0.326730\pi\)
0.517858 + 0.855467i \(0.326730\pi\)
\(840\) 2.00000 0.0690066
\(841\) 7.00000 0.241379
\(842\) −26.0000 −0.896019
\(843\) −36.0000 −1.23991
\(844\) 4.00000 0.137686
\(845\) −3.00000 −0.103203
\(846\) −6.00000 −0.206284
\(847\) 0 0
\(848\) −6.00000 −0.206041
\(849\) −32.0000 −1.09824
\(850\) 0 0
\(851\) 0 0
\(852\) −24.0000 −0.822226
\(853\) 4.00000 0.136957 0.0684787 0.997653i \(-0.478185\pi\)
0.0684787 + 0.997653i \(0.478185\pi\)
\(854\) 4.00000 0.136877
\(855\) −4.00000 −0.136797
\(856\) 12.0000 0.410152
\(857\) 12.0000 0.409912 0.204956 0.978771i \(-0.434295\pi\)
0.204956 + 0.978771i \(0.434295\pi\)
\(858\) 0 0
\(859\) −10.0000 −0.341196 −0.170598 0.985341i \(-0.554570\pi\)
−0.170598 + 0.985341i \(0.554570\pi\)
\(860\) −4.00000 −0.136399
\(861\) 24.0000 0.817918
\(862\) 24.0000 0.817443
\(863\) 12.0000 0.408485 0.204242 0.978920i \(-0.434527\pi\)
0.204242 + 0.978920i \(0.434527\pi\)
\(864\) −4.00000 −0.136083
\(865\) 12.0000 0.408012
\(866\) −14.0000 −0.475739
\(867\) 34.0000 1.15470
\(868\) 10.0000 0.339422
\(869\) 0 0
\(870\) −12.0000 −0.406838
\(871\) −16.0000 −0.542139
\(872\) 2.00000 0.0677285
\(873\) −10.0000 −0.338449
\(874\) 0 0
\(875\) 1.00000 0.0338062
\(876\) −8.00000 −0.270295
\(877\) 34.0000 1.14810 0.574049 0.818821i \(-0.305372\pi\)
0.574049 + 0.818821i \(0.305372\pi\)
\(878\) −28.0000 −0.944954
\(879\) 48.0000 1.61900
\(880\) 0 0
\(881\) 6.00000 0.202145 0.101073 0.994879i \(-0.467773\pi\)
0.101073 + 0.994879i \(0.467773\pi\)
\(882\) −1.00000 −0.0336718
\(883\) −16.0000 −0.538443 −0.269221 0.963078i \(-0.586766\pi\)
−0.269221 + 0.963078i \(0.586766\pi\)
\(884\) 0 0
\(885\) −12.0000 −0.403376
\(886\) 24.0000 0.806296
\(887\) −12.0000 −0.402921 −0.201460 0.979497i \(-0.564569\pi\)
−0.201460 + 0.979497i \(0.564569\pi\)
\(888\) 4.00000 0.134231
\(889\) 8.00000 0.268311
\(890\) 18.0000 0.603361
\(891\) 0 0
\(892\) −10.0000 −0.334825
\(893\) 24.0000 0.803129
\(894\) 36.0000 1.20402
\(895\) −12.0000 −0.401116
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) −6.00000 −0.200223
\(899\) −60.0000 −2.00111
\(900\) 1.00000 0.0333333
\(901\) 0 0
\(902\) 0 0
\(903\) 8.00000 0.266223
\(904\) 6.00000 0.199557
\(905\) 10.0000 0.332411
\(906\) 32.0000 1.06313
\(907\) −28.0000 −0.929725 −0.464862 0.885383i \(-0.653896\pi\)
−0.464862 + 0.885383i \(0.653896\pi\)
\(908\) 24.0000 0.796468
\(909\) 0 0
\(910\) −4.00000 −0.132599
\(911\) 36.0000 1.19273 0.596367 0.802712i \(-0.296610\pi\)
0.596367 + 0.802712i \(0.296610\pi\)
\(912\) −8.00000 −0.264906
\(913\) 0 0
\(914\) −10.0000 −0.330771
\(915\) 8.00000 0.264472
\(916\) −10.0000 −0.330409
\(917\) 0 0
\(918\) 0 0
\(919\) 16.0000 0.527791 0.263896 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) 0 0
\(921\) 64.0000 2.10887
\(922\) −36.0000 −1.18560
\(923\) 48.0000 1.57994
\(924\) 0 0
\(925\) 2.00000 0.0657596
\(926\) 40.0000 1.31448
\(927\) 14.0000 0.459820
\(928\) −6.00000 −0.196960
\(929\) −42.0000 −1.37798 −0.688988 0.724773i \(-0.741945\pi\)
−0.688988 + 0.724773i \(0.741945\pi\)
\(930\) 20.0000 0.655826
\(931\) 4.00000 0.131095
\(932\) 6.00000 0.196537
\(933\) −36.0000 −1.17859
\(934\) −42.0000 −1.37428
\(935\) 0 0
\(936\) −4.00000 −0.130744
\(937\) 52.0000 1.69877 0.849383 0.527777i \(-0.176974\pi\)
0.849383 + 0.527777i \(0.176974\pi\)
\(938\) −4.00000 −0.130605
\(939\) −4.00000 −0.130535
\(940\) −6.00000 −0.195698
\(941\) 36.0000 1.17357 0.586783 0.809744i \(-0.300394\pi\)
0.586783 + 0.809744i \(0.300394\pi\)
\(942\) −44.0000 −1.43360
\(943\) 0 0
\(944\) −6.00000 −0.195283
\(945\) 4.00000 0.130120
\(946\) 0 0
\(947\) 48.0000 1.55979 0.779895 0.625910i \(-0.215272\pi\)
0.779895 + 0.625910i \(0.215272\pi\)
\(948\) 16.0000 0.519656
\(949\) 16.0000 0.519382
\(950\) −4.00000 −0.129777
\(951\) −12.0000 −0.389127
\(952\) 0 0
\(953\) 18.0000 0.583077 0.291539 0.956559i \(-0.405833\pi\)
0.291539 + 0.956559i \(0.405833\pi\)
\(954\) 6.00000 0.194257
\(955\) 0 0
\(956\) 24.0000 0.776215
\(957\) 0 0
\(958\) 12.0000 0.387702
\(959\) −6.00000 −0.193750
\(960\) 2.00000 0.0645497
\(961\) 69.0000 2.22581
\(962\) −8.00000 −0.257930
\(963\) −12.0000 −0.386695
\(964\) 4.00000 0.128831
\(965\) 14.0000 0.450676
\(966\) 0 0
\(967\) −32.0000 −1.02905 −0.514525 0.857475i \(-0.672032\pi\)
−0.514525 + 0.857475i \(0.672032\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −10.0000 −0.321081
\(971\) 30.0000 0.962746 0.481373 0.876516i \(-0.340138\pi\)
0.481373 + 0.876516i \(0.340138\pi\)
\(972\) 10.0000 0.320750
\(973\) −4.00000 −0.128234
\(974\) −32.0000 −1.02535
\(975\) −8.00000 −0.256205
\(976\) 4.00000 0.128037
\(977\) 18.0000 0.575871 0.287936 0.957650i \(-0.407031\pi\)
0.287936 + 0.957650i \(0.407031\pi\)
\(978\) −8.00000 −0.255812
\(979\) 0 0
\(980\) −1.00000 −0.0319438
\(981\) −2.00000 −0.0638551
\(982\) −12.0000 −0.382935
\(983\) −42.0000 −1.33959 −0.669796 0.742545i \(-0.733618\pi\)
−0.669796 + 0.742545i \(0.733618\pi\)
\(984\) 24.0000 0.765092
\(985\) −6.00000 −0.191176
\(986\) 0 0
\(987\) 12.0000 0.381964
\(988\) 16.0000 0.509028
\(989\) 0 0
\(990\) 0 0
\(991\) 20.0000 0.635321 0.317660 0.948205i \(-0.397103\pi\)
0.317660 + 0.948205i \(0.397103\pi\)
\(992\) 10.0000 0.317500
\(993\) −40.0000 −1.26936
\(994\) 12.0000 0.380617
\(995\) −2.00000 −0.0634043
\(996\) 24.0000 0.760469
\(997\) 52.0000 1.64686 0.823428 0.567420i \(-0.192059\pi\)
0.823428 + 0.567420i \(0.192059\pi\)
\(998\) 40.0000 1.26618
\(999\) 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 8470.2.a.c.1.1 1
11.10 odd 2 770.2.a.f.1.1 1
33.32 even 2 6930.2.a.o.1.1 1
44.43 even 2 6160.2.a.j.1.1 1
55.32 even 4 3850.2.c.b.1849.2 2
55.43 even 4 3850.2.c.b.1849.1 2
55.54 odd 2 3850.2.a.k.1.1 1
77.76 even 2 5390.2.a.bj.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.a.f.1.1 1 11.10 odd 2
3850.2.a.k.1.1 1 55.54 odd 2
3850.2.c.b.1849.1 2 55.43 even 4
3850.2.c.b.1849.2 2 55.32 even 4
5390.2.a.bj.1.1 1 77.76 even 2
6160.2.a.j.1.1 1 44.43 even 2
6930.2.a.o.1.1 1 33.32 even 2
8470.2.a.c.1.1 1 1.1 even 1 trivial