Properties

Label 8619.2.a.j.1.1
Level $8619$
Weight $2$
Character 8619.1
Self dual yes
Analytic conductor $68.823$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8619 = 3 \cdot 13^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8619.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(68.8230615021\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 663)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 8619.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{6} +2.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{6} +2.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +2.00000 q^{11} -1.00000 q^{12} +2.00000 q^{14} -1.00000 q^{16} +1.00000 q^{17} +1.00000 q^{18} +2.00000 q^{21} +2.00000 q^{22} -8.00000 q^{23} -3.00000 q^{24} -5.00000 q^{25} +1.00000 q^{27} -2.00000 q^{28} -6.00000 q^{29} -6.00000 q^{31} +5.00000 q^{32} +2.00000 q^{33} +1.00000 q^{34} -1.00000 q^{36} -4.00000 q^{37} +12.0000 q^{41} +2.00000 q^{42} -4.00000 q^{43} -2.00000 q^{44} -8.00000 q^{46} -1.00000 q^{48} -3.00000 q^{49} -5.00000 q^{50} +1.00000 q^{51} -6.00000 q^{53} +1.00000 q^{54} -6.00000 q^{56} -6.00000 q^{58} -2.00000 q^{61} -6.00000 q^{62} +2.00000 q^{63} +7.00000 q^{64} +2.00000 q^{66} -1.00000 q^{68} -8.00000 q^{69} -10.0000 q^{71} -3.00000 q^{72} +4.00000 q^{73} -4.00000 q^{74} -5.00000 q^{75} +4.00000 q^{77} +12.0000 q^{79} +1.00000 q^{81} +12.0000 q^{82} +4.00000 q^{83} -2.00000 q^{84} -4.00000 q^{86} -6.00000 q^{87} -6.00000 q^{88} +10.0000 q^{89} +8.00000 q^{92} -6.00000 q^{93} +5.00000 q^{96} -16.0000 q^{97} -3.00000 q^{98} +2.00000 q^{99} +O(q^{100})\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107 0.353553 0.935414i \(-0.384973\pi\)
0.353553 + 0.935414i \(0.384973\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.00000 −0.500000
\(5\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 1.00000 0.408248
\(7\) 2.00000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) −3.00000 −1.06066
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0 0
\(14\) 2.00000 0.534522
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) 1.00000 0.242536
\(18\) 1.00000 0.235702
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) 2.00000 0.426401
\(23\) −8.00000 −1.66812 −0.834058 0.551677i \(-0.813988\pi\)
−0.834058 + 0.551677i \(0.813988\pi\)
\(24\) −3.00000 −0.612372
\(25\) −5.00000 −1.00000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) −2.00000 −0.377964
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) −6.00000 −1.07763 −0.538816 0.842424i \(-0.681128\pi\)
−0.538816 + 0.842424i \(0.681128\pi\)
\(32\) 5.00000 0.883883
\(33\) 2.00000 0.348155
\(34\) 1.00000 0.171499
\(35\) 0 0
\(36\) −1.00000 −0.166667
\(37\) −4.00000 −0.657596 −0.328798 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 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 \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) −2.00000 −0.301511
\(45\) 0 0
\(46\) −8.00000 −1.17954
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −1.00000 −0.144338
\(49\) −3.00000 −0.428571
\(50\) −5.00000 −0.707107
\(51\) 1.00000 0.140028
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) −6.00000 −0.801784
\(57\) 0 0
\(58\) −6.00000 −0.787839
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) −6.00000 −0.762001
\(63\) 2.00000 0.251976
\(64\) 7.00000 0.875000
\(65\) 0 0
\(66\) 2.00000 0.246183
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) −1.00000 −0.121268
\(69\) −8.00000 −0.963087
\(70\) 0 0
\(71\) −10.0000 −1.18678 −0.593391 0.804914i \(-0.702211\pi\)
−0.593391 + 0.804914i \(0.702211\pi\)
\(72\) −3.00000 −0.353553
\(73\) 4.00000 0.468165 0.234082 0.972217i \(-0.424791\pi\)
0.234082 + 0.972217i \(0.424791\pi\)
\(74\) −4.00000 −0.464991
\(75\) −5.00000 −0.577350
\(76\) 0 0
\(77\) 4.00000 0.455842
\(78\) 0 0
\(79\) 12.0000 1.35011 0.675053 0.737769i \(-0.264121\pi\)
0.675053 + 0.737769i \(0.264121\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 12.0000 1.32518
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) −2.00000 −0.218218
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) −6.00000 −0.643268
\(88\) −6.00000 −0.639602
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 8.00000 0.834058
\(93\) −6.00000 −0.622171
\(94\) 0 0
\(95\) 0 0
\(96\) 5.00000 0.510310
\(97\) −16.0000 −1.62455 −0.812277 0.583272i \(-0.801772\pi\)
−0.812277 + 0.583272i \(0.801772\pi\)
\(98\) −3.00000 −0.303046
\(99\) 2.00000 0.201008
\(100\) 5.00000 0.500000
\(101\) −18.0000 −1.79107 −0.895533 0.444994i \(-0.853206\pi\)
−0.895533 + 0.444994i \(0.853206\pi\)
\(102\) 1.00000 0.0990148
\(103\) −16.0000 −1.57653 −0.788263 0.615338i \(-0.789020\pi\)
−0.788263 + 0.615338i \(0.789020\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) −8.00000 −0.773389 −0.386695 0.922208i \(-0.626383\pi\)
−0.386695 + 0.922208i \(0.626383\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(110\) 0 0
\(111\) −4.00000 −0.379663
\(112\) −2.00000 −0.188982
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 6.00000 0.557086
\(117\) 0 0
\(118\) 0 0
\(119\) 2.00000 0.183340
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) −2.00000 −0.181071
\(123\) 12.0000 1.08200
\(124\) 6.00000 0.538816
\(125\) 0 0
\(126\) 2.00000 0.178174
\(127\) 16.0000 1.41977 0.709885 0.704317i \(-0.248747\pi\)
0.709885 + 0.704317i \(0.248747\pi\)
\(128\) −3.00000 −0.265165
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) −2.00000 −0.174078
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) −3.00000 −0.257248
\(137\) 2.00000 0.170872 0.0854358 0.996344i \(-0.472772\pi\)
0.0854358 + 0.996344i \(0.472772\pi\)
\(138\) −8.00000 −0.681005
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −10.0000 −0.839181
\(143\) 0 0
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) 4.00000 0.331042
\(147\) −3.00000 −0.247436
\(148\) 4.00000 0.328798
\(149\) −14.0000 −1.14692 −0.573462 0.819232i \(-0.694400\pi\)
−0.573462 + 0.819232i \(0.694400\pi\)
\(150\) −5.00000 −0.408248
\(151\) −16.0000 −1.30206 −0.651031 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) 0 0
\(153\) 1.00000 0.0808452
\(154\) 4.00000 0.322329
\(155\) 0 0
\(156\) 0 0
\(157\) 18.0000 1.43656 0.718278 0.695756i \(-0.244931\pi\)
0.718278 + 0.695756i \(0.244931\pi\)
\(158\) 12.0000 0.954669
\(159\) −6.00000 −0.475831
\(160\) 0 0
\(161\) −16.0000 −1.26098
\(162\) 1.00000 0.0785674
\(163\) −2.00000 −0.156652 −0.0783260 0.996928i \(-0.524958\pi\)
−0.0783260 + 0.996928i \(0.524958\pi\)
\(164\) −12.0000 −0.937043
\(165\) 0 0
\(166\) 4.00000 0.310460
\(167\) −18.0000 −1.39288 −0.696441 0.717614i \(-0.745234\pi\)
−0.696441 + 0.717614i \(0.745234\pi\)
\(168\) −6.00000 −0.462910
\(169\) 0 0
\(170\) 0 0
\(171\) 0 0
\(172\) 4.00000 0.304997
\(173\) 2.00000 0.152057 0.0760286 0.997106i \(-0.475776\pi\)
0.0760286 + 0.997106i \(0.475776\pi\)
\(174\) −6.00000 −0.454859
\(175\) −10.0000 −0.755929
\(176\) −2.00000 −0.150756
\(177\) 0 0
\(178\) 10.0000 0.749532
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) 14.0000 1.04061 0.520306 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) 0 0
\(183\) −2.00000 −0.147844
\(184\) 24.0000 1.76930
\(185\) 0 0
\(186\) −6.00000 −0.439941
\(187\) 2.00000 0.146254
\(188\) 0 0
\(189\) 2.00000 0.145479
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 7.00000 0.505181
\(193\) 20.0000 1.43963 0.719816 0.694165i \(-0.244226\pi\)
0.719816 + 0.694165i \(0.244226\pi\)
\(194\) −16.0000 −1.14873
\(195\) 0 0
\(196\) 3.00000 0.214286
\(197\) −24.0000 −1.70993 −0.854965 0.518686i \(-0.826421\pi\)
−0.854965 + 0.518686i \(0.826421\pi\)
\(198\) 2.00000 0.142134
\(199\) −12.0000 −0.850657 −0.425329 0.905039i \(-0.639842\pi\)
−0.425329 + 0.905039i \(0.639842\pi\)
\(200\) 15.0000 1.06066
\(201\) 0 0
\(202\) −18.0000 −1.26648
\(203\) −12.0000 −0.842235
\(204\) −1.00000 −0.0700140
\(205\) 0 0
\(206\) −16.0000 −1.11477
\(207\) −8.00000 −0.556038
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 8.00000 0.550743 0.275371 0.961338i \(-0.411199\pi\)
0.275371 + 0.961338i \(0.411199\pi\)
\(212\) 6.00000 0.412082
\(213\) −10.0000 −0.685189
\(214\) −8.00000 −0.546869
\(215\) 0 0
\(216\) −3.00000 −0.204124
\(217\) −12.0000 −0.814613
\(218\) 0 0
\(219\) 4.00000 0.270295
\(220\) 0 0
\(221\) 0 0
\(222\) −4.00000 −0.268462
\(223\) 8.00000 0.535720 0.267860 0.963458i \(-0.413684\pi\)
0.267860 + 0.963458i \(0.413684\pi\)
\(224\) 10.0000 0.668153
\(225\) −5.00000 −0.333333
\(226\) 6.00000 0.399114
\(227\) 18.0000 1.19470 0.597351 0.801980i \(-0.296220\pi\)
0.597351 + 0.801980i \(0.296220\pi\)
\(228\) 0 0
\(229\) 10.0000 0.660819 0.330409 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) 0 0
\(231\) 4.00000 0.263181
\(232\) 18.0000 1.18176
\(233\) −18.0000 −1.17922 −0.589610 0.807688i \(-0.700718\pi\)
−0.589610 + 0.807688i \(0.700718\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 12.0000 0.779484
\(238\) 2.00000 0.129641
\(239\) 12.0000 0.776215 0.388108 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(242\) −7.00000 −0.449977
\(243\) 1.00000 0.0641500
\(244\) 2.00000 0.128037
\(245\) 0 0
\(246\) 12.0000 0.765092
\(247\) 0 0
\(248\) 18.0000 1.14300
\(249\) 4.00000 0.253490
\(250\) 0 0
\(251\) −28.0000 −1.76734 −0.883672 0.468106i \(-0.844936\pi\)
−0.883672 + 0.468106i \(0.844936\pi\)
\(252\) −2.00000 −0.125988
\(253\) −16.0000 −1.00591
\(254\) 16.0000 1.00393
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) −2.00000 −0.124757 −0.0623783 0.998053i \(-0.519869\pi\)
−0.0623783 + 0.998053i \(0.519869\pi\)
\(258\) −4.00000 −0.249029
\(259\) −8.00000 −0.497096
\(260\) 0 0
\(261\) −6.00000 −0.371391
\(262\) 12.0000 0.741362
\(263\) −16.0000 −0.986602 −0.493301 0.869859i \(-0.664210\pi\)
−0.493301 + 0.869859i \(0.664210\pi\)
\(264\) −6.00000 −0.369274
\(265\) 0 0
\(266\) 0 0
\(267\) 10.0000 0.611990
\(268\) 0 0
\(269\) −6.00000 −0.365826 −0.182913 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 0 0
\(271\) −28.0000 −1.70088 −0.850439 0.526073i \(-0.823664\pi\)
−0.850439 + 0.526073i \(0.823664\pi\)
\(272\) −1.00000 −0.0606339
\(273\) 0 0
\(274\) 2.00000 0.120824
\(275\) −10.0000 −0.603023
\(276\) 8.00000 0.481543
\(277\) −30.0000 −1.80253 −0.901263 0.433273i \(-0.857359\pi\)
−0.901263 + 0.433273i \(0.857359\pi\)
\(278\) 0 0
\(279\) −6.00000 −0.359211
\(280\) 0 0
\(281\) −6.00000 −0.357930 −0.178965 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 0 0
\(283\) 20.0000 1.18888 0.594438 0.804141i \(-0.297374\pi\)
0.594438 + 0.804141i \(0.297374\pi\)
\(284\) 10.0000 0.593391
\(285\) 0 0
\(286\) 0 0
\(287\) 24.0000 1.41668
\(288\) 5.00000 0.294628
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) −16.0000 −0.937937
\(292\) −4.00000 −0.234082
\(293\) −14.0000 −0.817889 −0.408944 0.912559i \(-0.634103\pi\)
−0.408944 + 0.912559i \(0.634103\pi\)
\(294\) −3.00000 −0.174964
\(295\) 0 0
\(296\) 12.0000 0.697486
\(297\) 2.00000 0.116052
\(298\) −14.0000 −0.810998
\(299\) 0 0
\(300\) 5.00000 0.288675
\(301\) −8.00000 −0.461112
\(302\) −16.0000 −0.920697
\(303\) −18.0000 −1.03407
\(304\) 0 0
\(305\) 0 0
\(306\) 1.00000 0.0571662
\(307\) 28.0000 1.59804 0.799022 0.601302i \(-0.205351\pi\)
0.799022 + 0.601302i \(0.205351\pi\)
\(308\) −4.00000 −0.227921
\(309\) −16.0000 −0.910208
\(310\) 0 0
\(311\) −4.00000 −0.226819 −0.113410 0.993548i \(-0.536177\pi\)
−0.113410 + 0.993548i \(0.536177\pi\)
\(312\) 0 0
\(313\) −10.0000 −0.565233 −0.282617 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) 18.0000 1.01580
\(315\) 0 0
\(316\) −12.0000 −0.675053
\(317\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(318\) −6.00000 −0.336463
\(319\) −12.0000 −0.671871
\(320\) 0 0
\(321\) −8.00000 −0.446516
\(322\) −16.0000 −0.891645
\(323\) 0 0
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) −2.00000 −0.110770
\(327\) 0 0
\(328\) −36.0000 −1.98777
\(329\) 0 0
\(330\) 0 0
\(331\) −28.0000 −1.53902 −0.769510 0.638635i \(-0.779499\pi\)
−0.769510 + 0.638635i \(0.779499\pi\)
\(332\) −4.00000 −0.219529
\(333\) −4.00000 −0.219199
\(334\) −18.0000 −0.984916
\(335\) 0 0
\(336\) −2.00000 −0.109109
\(337\) −14.0000 −0.762629 −0.381314 0.924445i \(-0.624528\pi\)
−0.381314 + 0.924445i \(0.624528\pi\)
\(338\) 0 0
\(339\) 6.00000 0.325875
\(340\) 0 0
\(341\) −12.0000 −0.649836
\(342\) 0 0
\(343\) −20.0000 −1.07990
\(344\) 12.0000 0.646997
\(345\) 0 0
\(346\) 2.00000 0.107521
\(347\) −24.0000 −1.28839 −0.644194 0.764862i \(-0.722807\pi\)
−0.644194 + 0.764862i \(0.722807\pi\)
\(348\) 6.00000 0.321634
\(349\) 14.0000 0.749403 0.374701 0.927146i \(-0.377745\pi\)
0.374701 + 0.927146i \(0.377745\pi\)
\(350\) −10.0000 −0.534522
\(351\) 0 0
\(352\) 10.0000 0.533002
\(353\) 6.00000 0.319348 0.159674 0.987170i \(-0.448956\pi\)
0.159674 + 0.987170i \(0.448956\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −10.0000 −0.529999
\(357\) 2.00000 0.105851
\(358\) 12.0000 0.634220
\(359\) −20.0000 −1.05556 −0.527780 0.849381i \(-0.676975\pi\)
−0.527780 + 0.849381i \(0.676975\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 14.0000 0.735824
\(363\) −7.00000 −0.367405
\(364\) 0 0
\(365\) 0 0
\(366\) −2.00000 −0.104542
\(367\) −24.0000 −1.25279 −0.626395 0.779506i \(-0.715470\pi\)
−0.626395 + 0.779506i \(0.715470\pi\)
\(368\) 8.00000 0.417029
\(369\) 12.0000 0.624695
\(370\) 0 0
\(371\) −12.0000 −0.623009
\(372\) 6.00000 0.311086
\(373\) −34.0000 −1.76045 −0.880227 0.474554i \(-0.842610\pi\)
−0.880227 + 0.474554i \(0.842610\pi\)
\(374\) 2.00000 0.103418
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 2.00000 0.102869
\(379\) 10.0000 0.513665 0.256833 0.966456i \(-0.417321\pi\)
0.256833 + 0.966456i \(0.417321\pi\)
\(380\) 0 0
\(381\) 16.0000 0.819705
\(382\) 0 0
\(383\) 28.0000 1.43073 0.715367 0.698749i \(-0.246260\pi\)
0.715367 + 0.698749i \(0.246260\pi\)
\(384\) −3.00000 −0.153093
\(385\) 0 0
\(386\) 20.0000 1.01797
\(387\) −4.00000 −0.203331
\(388\) 16.0000 0.812277
\(389\) 2.00000 0.101404 0.0507020 0.998714i \(-0.483854\pi\)
0.0507020 + 0.998714i \(0.483854\pi\)
\(390\) 0 0
\(391\) −8.00000 −0.404577
\(392\) 9.00000 0.454569
\(393\) 12.0000 0.605320
\(394\) −24.0000 −1.20910
\(395\) 0 0
\(396\) −2.00000 −0.100504
\(397\) 12.0000 0.602263 0.301131 0.953583i \(-0.402636\pi\)
0.301131 + 0.953583i \(0.402636\pi\)
\(398\) −12.0000 −0.601506
\(399\) 0 0
\(400\) 5.00000 0.250000
\(401\) −24.0000 −1.19850 −0.599251 0.800561i \(-0.704535\pi\)
−0.599251 + 0.800561i \(0.704535\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 18.0000 0.895533
\(405\) 0 0
\(406\) −12.0000 −0.595550
\(407\) −8.00000 −0.396545
\(408\) −3.00000 −0.148522
\(409\) 22.0000 1.08783 0.543915 0.839140i \(-0.316941\pi\)
0.543915 + 0.839140i \(0.316941\pi\)
\(410\) 0 0
\(411\) 2.00000 0.0986527
\(412\) 16.0000 0.788263
\(413\) 0 0
\(414\) −8.00000 −0.393179
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) −18.0000 −0.877266 −0.438633 0.898666i \(-0.644537\pi\)
−0.438633 + 0.898666i \(0.644537\pi\)
\(422\) 8.00000 0.389434
\(423\) 0 0
\(424\) 18.0000 0.874157
\(425\) −5.00000 −0.242536
\(426\) −10.0000 −0.484502
\(427\) −4.00000 −0.193574
\(428\) 8.00000 0.386695
\(429\) 0 0
\(430\) 0 0
\(431\) 30.0000 1.44505 0.722525 0.691345i \(-0.242982\pi\)
0.722525 + 0.691345i \(0.242982\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 34.0000 1.63394 0.816968 0.576683i \(-0.195653\pi\)
0.816968 + 0.576683i \(0.195653\pi\)
\(434\) −12.0000 −0.576018
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 4.00000 0.191127
\(439\) 40.0000 1.90910 0.954548 0.298057i \(-0.0963387\pi\)
0.954548 + 0.298057i \(0.0963387\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 0 0
\(443\) 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
\(447\) −14.0000 −0.662177
\(448\) 14.0000 0.661438
\(449\) −8.00000 −0.377543 −0.188772 0.982021i \(-0.560451\pi\)
−0.188772 + 0.982021i \(0.560451\pi\)
\(450\) −5.00000 −0.235702
\(451\) 24.0000 1.13012
\(452\) −6.00000 −0.282216
\(453\) −16.0000 −0.751746
\(454\) 18.0000 0.844782
\(455\) 0 0
\(456\) 0 0
\(457\) 30.0000 1.40334 0.701670 0.712502i \(-0.252438\pi\)
0.701670 + 0.712502i \(0.252438\pi\)
\(458\) 10.0000 0.467269
\(459\) 1.00000 0.0466760
\(460\) 0 0
\(461\) 14.0000 0.652045 0.326023 0.945362i \(-0.394291\pi\)
0.326023 + 0.945362i \(0.394291\pi\)
\(462\) 4.00000 0.186097
\(463\) 8.00000 0.371792 0.185896 0.982569i \(-0.440481\pi\)
0.185896 + 0.982569i \(0.440481\pi\)
\(464\) 6.00000 0.278543
\(465\) 0 0
\(466\) −18.0000 −0.833834
\(467\) 4.00000 0.185098 0.0925490 0.995708i \(-0.470499\pi\)
0.0925490 + 0.995708i \(0.470499\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 18.0000 0.829396
\(472\) 0 0
\(473\) −8.00000 −0.367840
\(474\) 12.0000 0.551178
\(475\) 0 0
\(476\) −2.00000 −0.0916698
\(477\) −6.00000 −0.274721
\(478\) 12.0000 0.548867
\(479\) 6.00000 0.274147 0.137073 0.990561i \(-0.456230\pi\)
0.137073 + 0.990561i \(0.456230\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) −16.0000 −0.728025
\(484\) 7.00000 0.318182
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) −26.0000 −1.17817 −0.589086 0.808070i \(-0.700512\pi\)
−0.589086 + 0.808070i \(0.700512\pi\)
\(488\) 6.00000 0.271607
\(489\) −2.00000 −0.0904431
\(490\) 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
\(493\) −6.00000 −0.270226
\(494\) 0 0
\(495\) 0 0
\(496\) 6.00000 0.269408
\(497\) −20.0000 −0.897123
\(498\) 4.00000 0.179244
\(499\) −34.0000 −1.52205 −0.761025 0.648723i \(-0.775303\pi\)
−0.761025 + 0.648723i \(0.775303\pi\)
\(500\) 0 0
\(501\) −18.0000 −0.804181
\(502\) −28.0000 −1.24970
\(503\) 8.00000 0.356702 0.178351 0.983967i \(-0.442924\pi\)
0.178351 + 0.983967i \(0.442924\pi\)
\(504\) −6.00000 −0.267261
\(505\) 0 0
\(506\) −16.0000 −0.711287
\(507\) 0 0
\(508\) −16.0000 −0.709885
\(509\) −30.0000 −1.32973 −0.664863 0.746965i \(-0.731510\pi\)
−0.664863 + 0.746965i \(0.731510\pi\)
\(510\) 0 0
\(511\) 8.00000 0.353899
\(512\) −11.0000 −0.486136
\(513\) 0 0
\(514\) −2.00000 −0.0882162
\(515\) 0 0
\(516\) 4.00000 0.176090
\(517\) 0 0
\(518\) −8.00000 −0.351500
\(519\) 2.00000 0.0877903
\(520\) 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
\(523\) −20.0000 −0.874539 −0.437269 0.899331i \(-0.644054\pi\)
−0.437269 + 0.899331i \(0.644054\pi\)
\(524\) −12.0000 −0.524222
\(525\) −10.0000 −0.436436
\(526\) −16.0000 −0.697633
\(527\) −6.00000 −0.261364
\(528\) −2.00000 −0.0870388
\(529\) 41.0000 1.78261
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 10.0000 0.432742
\(535\) 0 0
\(536\) 0 0
\(537\) 12.0000 0.517838
\(538\) −6.00000 −0.258678
\(539\) −6.00000 −0.258438
\(540\) 0 0
\(541\) 16.0000 0.687894 0.343947 0.938989i \(-0.388236\pi\)
0.343947 + 0.938989i \(0.388236\pi\)
\(542\) −28.0000 −1.20270
\(543\) 14.0000 0.600798
\(544\) 5.00000 0.214373
\(545\) 0 0
\(546\) 0 0
\(547\) 36.0000 1.53925 0.769624 0.638497i \(-0.220443\pi\)
0.769624 + 0.638497i \(0.220443\pi\)
\(548\) −2.00000 −0.0854358
\(549\) −2.00000 −0.0853579
\(550\) −10.0000 −0.426401
\(551\) 0 0
\(552\) 24.0000 1.02151
\(553\) 24.0000 1.02058
\(554\) −30.0000 −1.27458
\(555\) 0 0
\(556\) 0 0
\(557\) 22.0000 0.932170 0.466085 0.884740i \(-0.345664\pi\)
0.466085 + 0.884740i \(0.345664\pi\)
\(558\) −6.00000 −0.254000
\(559\) 0 0
\(560\) 0 0
\(561\) 2.00000 0.0844401
\(562\) −6.00000 −0.253095
\(563\) 4.00000 0.168580 0.0842900 0.996441i \(-0.473138\pi\)
0.0842900 + 0.996441i \(0.473138\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 20.0000 0.840663
\(567\) 2.00000 0.0839921
\(568\) 30.0000 1.25877
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 0 0
\(571\) 8.00000 0.334790 0.167395 0.985890i \(-0.446465\pi\)
0.167395 + 0.985890i \(0.446465\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 24.0000 1.00174
\(575\) 40.0000 1.66812
\(576\) 7.00000 0.291667
\(577\) 10.0000 0.416305 0.208153 0.978096i \(-0.433255\pi\)
0.208153 + 0.978096i \(0.433255\pi\)
\(578\) 1.00000 0.0415945
\(579\) 20.0000 0.831172
\(580\) 0 0
\(581\) 8.00000 0.331896
\(582\) −16.0000 −0.663221
\(583\) −12.0000 −0.496989
\(584\) −12.0000 −0.496564
\(585\) 0 0
\(586\) −14.0000 −0.578335
\(587\) 8.00000 0.330195 0.165098 0.986277i \(-0.447206\pi\)
0.165098 + 0.986277i \(0.447206\pi\)
\(588\) 3.00000 0.123718
\(589\) 0 0
\(590\) 0 0
\(591\) −24.0000 −0.987228
\(592\) 4.00000 0.164399
\(593\) 18.0000 0.739171 0.369586 0.929197i \(-0.379500\pi\)
0.369586 + 0.929197i \(0.379500\pi\)
\(594\) 2.00000 0.0820610
\(595\) 0 0
\(596\) 14.0000 0.573462
\(597\) −12.0000 −0.491127
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 15.0000 0.612372
\(601\) 30.0000 1.22373 0.611863 0.790964i \(-0.290420\pi\)
0.611863 + 0.790964i \(0.290420\pi\)
\(602\) −8.00000 −0.326056
\(603\) 0 0
\(604\) 16.0000 0.651031
\(605\) 0 0
\(606\) −18.0000 −0.731200
\(607\) −28.0000 −1.13648 −0.568242 0.822861i \(-0.692376\pi\)
−0.568242 + 0.822861i \(0.692376\pi\)
\(608\) 0 0
\(609\) −12.0000 −0.486265
\(610\) 0 0
\(611\) 0 0
\(612\) −1.00000 −0.0404226
\(613\) 18.0000 0.727013 0.363507 0.931592i \(-0.381579\pi\)
0.363507 + 0.931592i \(0.381579\pi\)
\(614\) 28.0000 1.12999
\(615\) 0 0
\(616\) −12.0000 −0.483494
\(617\) 12.0000 0.483102 0.241551 0.970388i \(-0.422344\pi\)
0.241551 + 0.970388i \(0.422344\pi\)
\(618\) −16.0000 −0.643614
\(619\) 26.0000 1.04503 0.522514 0.852631i \(-0.324994\pi\)
0.522514 + 0.852631i \(0.324994\pi\)
\(620\) 0 0
\(621\) −8.00000 −0.321029
\(622\) −4.00000 −0.160385
\(623\) 20.0000 0.801283
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) −10.0000 −0.399680
\(627\) 0 0
\(628\) −18.0000 −0.718278
\(629\) −4.00000 −0.159490
\(630\) 0 0
\(631\) −20.0000 −0.796187 −0.398094 0.917345i \(-0.630328\pi\)
−0.398094 + 0.917345i \(0.630328\pi\)
\(632\) −36.0000 −1.43200
\(633\) 8.00000 0.317971
\(634\) 0 0
\(635\) 0 0
\(636\) 6.00000 0.237915
\(637\) 0 0
\(638\) −12.0000 −0.475085
\(639\) −10.0000 −0.395594
\(640\) 0 0
\(641\) −38.0000 −1.50091 −0.750455 0.660922i \(-0.770166\pi\)
−0.750455 + 0.660922i \(0.770166\pi\)
\(642\) −8.00000 −0.315735
\(643\) 2.00000 0.0788723 0.0394362 0.999222i \(-0.487444\pi\)
0.0394362 + 0.999222i \(0.487444\pi\)
\(644\) 16.0000 0.630488
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) −3.00000 −0.117851
\(649\) 0 0
\(650\) 0 0
\(651\) −12.0000 −0.470317
\(652\) 2.00000 0.0783260
\(653\) −14.0000 −0.547862 −0.273931 0.961749i \(-0.588324\pi\)
−0.273931 + 0.961749i \(0.588324\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −12.0000 −0.468521
\(657\) 4.00000 0.156055
\(658\) 0 0
\(659\) −36.0000 −1.40236 −0.701180 0.712984i \(-0.747343\pi\)
−0.701180 + 0.712984i \(0.747343\pi\)
\(660\) 0 0
\(661\) −26.0000 −1.01128 −0.505641 0.862744i \(-0.668744\pi\)
−0.505641 + 0.862744i \(0.668744\pi\)
\(662\) −28.0000 −1.08825
\(663\) 0 0
\(664\) −12.0000 −0.465690
\(665\) 0 0
\(666\) −4.00000 −0.154997
\(667\) 48.0000 1.85857
\(668\) 18.0000 0.696441
\(669\) 8.00000 0.309298
\(670\) 0 0
\(671\) −4.00000 −0.154418
\(672\) 10.0000 0.385758
\(673\) 2.00000 0.0770943 0.0385472 0.999257i \(-0.487727\pi\)
0.0385472 + 0.999257i \(0.487727\pi\)
\(674\) −14.0000 −0.539260
\(675\) −5.00000 −0.192450
\(676\) 0 0
\(677\) −18.0000 −0.691796 −0.345898 0.938272i \(-0.612426\pi\)
−0.345898 + 0.938272i \(0.612426\pi\)
\(678\) 6.00000 0.230429
\(679\) −32.0000 −1.22805
\(680\) 0 0
\(681\) 18.0000 0.689761
\(682\) −12.0000 −0.459504
\(683\) −30.0000 −1.14792 −0.573959 0.818884i \(-0.694593\pi\)
−0.573959 + 0.818884i \(0.694593\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −20.0000 −0.763604
\(687\) 10.0000 0.381524
\(688\) 4.00000 0.152499
\(689\) 0 0
\(690\) 0 0
\(691\) 10.0000 0.380418 0.190209 0.981744i \(-0.439083\pi\)
0.190209 + 0.981744i \(0.439083\pi\)
\(692\) −2.00000 −0.0760286
\(693\) 4.00000 0.151947
\(694\) −24.0000 −0.911028
\(695\) 0 0
\(696\) 18.0000 0.682288
\(697\) 12.0000 0.454532
\(698\) 14.0000 0.529908
\(699\) −18.0000 −0.680823
\(700\) 10.0000 0.377964
\(701\) −2.00000 −0.0755390 −0.0377695 0.999286i \(-0.512025\pi\)
−0.0377695 + 0.999286i \(0.512025\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 14.0000 0.527645
\(705\) 0 0
\(706\) 6.00000 0.225813
\(707\) −36.0000 −1.35392
\(708\) 0 0
\(709\) 4.00000 0.150223 0.0751116 0.997175i \(-0.476069\pi\)
0.0751116 + 0.997175i \(0.476069\pi\)
\(710\) 0 0
\(711\) 12.0000 0.450035
\(712\) −30.0000 −1.12430
\(713\) 48.0000 1.79761
\(714\) 2.00000 0.0748481
\(715\) 0 0
\(716\) −12.0000 −0.448461
\(717\) 12.0000 0.448148
\(718\) −20.0000 −0.746393
\(719\) 48.0000 1.79010 0.895049 0.445968i \(-0.147140\pi\)
0.895049 + 0.445968i \(0.147140\pi\)
\(720\) 0 0
\(721\) −32.0000 −1.19174
\(722\) −19.0000 −0.707107
\(723\) 0 0
\(724\) −14.0000 −0.520306
\(725\) 30.0000 1.11417
\(726\) −7.00000 −0.259794
\(727\) 8.00000 0.296704 0.148352 0.988935i \(-0.452603\pi\)
0.148352 + 0.988935i \(0.452603\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −4.00000 −0.147945
\(732\) 2.00000 0.0739221
\(733\) 22.0000 0.812589 0.406294 0.913742i \(-0.366821\pi\)
0.406294 + 0.913742i \(0.366821\pi\)
\(734\) −24.0000 −0.885856
\(735\) 0 0
\(736\) −40.0000 −1.47442
\(737\) 0 0
\(738\) 12.0000 0.441726
\(739\) 4.00000 0.147142 0.0735712 0.997290i \(-0.476560\pi\)
0.0735712 + 0.997290i \(0.476560\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −12.0000 −0.440534
\(743\) −38.0000 −1.39408 −0.697042 0.717030i \(-0.745501\pi\)
−0.697042 + 0.717030i \(0.745501\pi\)
\(744\) 18.0000 0.659912
\(745\) 0 0
\(746\) −34.0000 −1.24483
\(747\) 4.00000 0.146352
\(748\) −2.00000 −0.0731272
\(749\) −16.0000 −0.584627
\(750\) 0 0
\(751\) 20.0000 0.729810 0.364905 0.931045i \(-0.381101\pi\)
0.364905 + 0.931045i \(0.381101\pi\)
\(752\) 0 0
\(753\) −28.0000 −1.02038
\(754\) 0 0
\(755\) 0 0
\(756\) −2.00000 −0.0727393
\(757\) −42.0000 −1.52652 −0.763258 0.646094i \(-0.776401\pi\)
−0.763258 + 0.646094i \(0.776401\pi\)
\(758\) 10.0000 0.363216
\(759\) −16.0000 −0.580763
\(760\) 0 0
\(761\) 18.0000 0.652499 0.326250 0.945284i \(-0.394215\pi\)
0.326250 + 0.945284i \(0.394215\pi\)
\(762\) 16.0000 0.579619
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 28.0000 1.01168
\(767\) 0 0
\(768\) −17.0000 −0.613435
\(769\) 26.0000 0.937584 0.468792 0.883309i \(-0.344689\pi\)
0.468792 + 0.883309i \(0.344689\pi\)
\(770\) 0 0
\(771\) −2.00000 −0.0720282
\(772\) −20.0000 −0.719816
\(773\) 42.0000 1.51064 0.755318 0.655359i \(-0.227483\pi\)
0.755318 + 0.655359i \(0.227483\pi\)
\(774\) −4.00000 −0.143777
\(775\) 30.0000 1.07763
\(776\) 48.0000 1.72310
\(777\) −8.00000 −0.286998
\(778\) 2.00000 0.0717035
\(779\) 0 0
\(780\) 0 0
\(781\) −20.0000 −0.715656
\(782\) −8.00000 −0.286079
\(783\) −6.00000 −0.214423
\(784\) 3.00000 0.107143
\(785\) 0 0
\(786\) 12.0000 0.428026
\(787\) −18.0000 −0.641631 −0.320815 0.947142i \(-0.603957\pi\)
−0.320815 + 0.947142i \(0.603957\pi\)
\(788\) 24.0000 0.854965
\(789\) −16.0000 −0.569615
\(790\) 0 0
\(791\) 12.0000 0.426671
\(792\) −6.00000 −0.213201
\(793\) 0 0
\(794\) 12.0000 0.425864
\(795\) 0 0
\(796\) 12.0000 0.425329
\(797\) 46.0000 1.62940 0.814702 0.579880i \(-0.196901\pi\)
0.814702 + 0.579880i \(0.196901\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −25.0000 −0.883883
\(801\) 10.0000 0.353333
\(802\) −24.0000 −0.847469
\(803\) 8.00000 0.282314
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −6.00000 −0.211210
\(808\) 54.0000 1.89971
\(809\) −50.0000 −1.75791 −0.878953 0.476908i \(-0.841757\pi\)
−0.878953 + 0.476908i \(0.841757\pi\)
\(810\) 0 0
\(811\) −50.0000 −1.75574 −0.877869 0.478901i \(-0.841035\pi\)
−0.877869 + 0.478901i \(0.841035\pi\)
\(812\) 12.0000 0.421117
\(813\) −28.0000 −0.982003
\(814\) −8.00000 −0.280400
\(815\) 0 0
\(816\) −1.00000 −0.0350070
\(817\) 0 0
\(818\) 22.0000 0.769212
\(819\) 0 0
\(820\) 0 0
\(821\) −36.0000 −1.25641 −0.628204 0.778048i \(-0.716210\pi\)
−0.628204 + 0.778048i \(0.716210\pi\)
\(822\) 2.00000 0.0697580
\(823\) −44.0000 −1.53374 −0.766872 0.641800i \(-0.778188\pi\)
−0.766872 + 0.641800i \(0.778188\pi\)
\(824\) 48.0000 1.67216
\(825\) −10.0000 −0.348155
\(826\) 0 0
\(827\) −22.0000 −0.765015 −0.382507 0.923952i \(-0.624939\pi\)
−0.382507 + 0.923952i \(0.624939\pi\)
\(828\) 8.00000 0.278019
\(829\) 34.0000 1.18087 0.590434 0.807086i \(-0.298956\pi\)
0.590434 + 0.807086i \(0.298956\pi\)
\(830\) 0 0
\(831\) −30.0000 −1.04069
\(832\) 0 0
\(833\) −3.00000 −0.103944
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −6.00000 −0.207390
\(838\) 0 0
\(839\) −18.0000 −0.621429 −0.310715 0.950503i \(-0.600568\pi\)
−0.310715 + 0.950503i \(0.600568\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) −18.0000 −0.620321
\(843\) −6.00000 −0.206651
\(844\) −8.00000 −0.275371
\(845\) 0 0
\(846\) 0 0
\(847\) −14.0000 −0.481046
\(848\) 6.00000 0.206041
\(849\) 20.0000 0.686398
\(850\) −5.00000 −0.171499
\(851\) 32.0000 1.09695
\(852\) 10.0000 0.342594
\(853\) 20.0000 0.684787 0.342393 0.939557i \(-0.388762\pi\)
0.342393 + 0.939557i \(0.388762\pi\)
\(854\) −4.00000 −0.136877
\(855\) 0 0
\(856\) 24.0000 0.820303
\(857\) −26.0000 −0.888143 −0.444072 0.895991i \(-0.646466\pi\)
−0.444072 + 0.895991i \(0.646466\pi\)
\(858\) 0 0
\(859\) −20.0000 −0.682391 −0.341196 0.939992i \(-0.610832\pi\)
−0.341196 + 0.939992i \(0.610832\pi\)
\(860\) 0 0
\(861\) 24.0000 0.817918
\(862\) 30.0000 1.02180
\(863\) 28.0000 0.953131 0.476566 0.879139i \(-0.341881\pi\)
0.476566 + 0.879139i \(0.341881\pi\)
\(864\) 5.00000 0.170103
\(865\) 0 0
\(866\) 34.0000 1.15537
\(867\) 1.00000 0.0339618
\(868\) 12.0000 0.407307
\(869\) 24.0000 0.814144
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −16.0000 −0.541518
\(874\) 0 0
\(875\) 0 0
\(876\) −4.00000 −0.135147
\(877\) 12.0000 0.405211 0.202606 0.979260i \(-0.435059\pi\)
0.202606 + 0.979260i \(0.435059\pi\)
\(878\) 40.0000 1.34993
\(879\) −14.0000 −0.472208
\(880\) 0 0
\(881\) 54.0000 1.81931 0.909653 0.415369i \(-0.136347\pi\)
0.909653 + 0.415369i \(0.136347\pi\)
\(882\) −3.00000 −0.101015
\(883\) 20.0000 0.673054 0.336527 0.941674i \(-0.390748\pi\)
0.336527 + 0.941674i \(0.390748\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 12.0000 0.403148
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 12.0000 0.402694
\(889\) 32.0000 1.07325
\(890\) 0 0
\(891\) 2.00000 0.0670025
\(892\) −8.00000 −0.267860
\(893\) 0 0
\(894\) −14.0000 −0.468230
\(895\) 0 0
\(896\) −6.00000 −0.200446
\(897\) 0 0
\(898\) −8.00000 −0.266963
\(899\) 36.0000 1.20067
\(900\) 5.00000 0.166667
\(901\) −6.00000 −0.199889
\(902\) 24.0000 0.799113
\(903\) −8.00000 −0.266223
\(904\) −18.0000 −0.598671
\(905\) 0 0
\(906\) −16.0000 −0.531564
\(907\) 52.0000 1.72663 0.863316 0.504664i \(-0.168384\pi\)
0.863316 + 0.504664i \(0.168384\pi\)
\(908\) −18.0000 −0.597351
\(909\) −18.0000 −0.597022
\(910\) 0 0
\(911\) 40.0000 1.32526 0.662630 0.748947i \(-0.269440\pi\)
0.662630 + 0.748947i \(0.269440\pi\)
\(912\) 0 0
\(913\) 8.00000 0.264761
\(914\) 30.0000 0.992312
\(915\) 0 0
\(916\) −10.0000 −0.330409
\(917\) 24.0000 0.792550
\(918\) 1.00000 0.0330049
\(919\) 16.0000 0.527791 0.263896 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) 0 0
\(921\) 28.0000 0.922631
\(922\) 14.0000 0.461065
\(923\) 0 0
\(924\) −4.00000 −0.131590
\(925\) 20.0000 0.657596
\(926\) 8.00000 0.262896
\(927\) −16.0000 −0.525509
\(928\) −30.0000 −0.984798
\(929\) 24.0000 0.787414 0.393707 0.919236i \(-0.371192\pi\)
0.393707 + 0.919236i \(0.371192\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 18.0000 0.589610
\(933\) −4.00000 −0.130954
\(934\) 4.00000 0.130884
\(935\) 0 0
\(936\) 0 0
\(937\) 26.0000 0.849383 0.424691 0.905338i \(-0.360383\pi\)
0.424691 + 0.905338i \(0.360383\pi\)
\(938\) 0 0
\(939\) −10.0000 −0.326338
\(940\) 0 0
\(941\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(942\) 18.0000 0.586472
\(943\) −96.0000 −3.12619
\(944\) 0 0
\(945\) 0 0
\(946\) −8.00000 −0.260102
\(947\) 30.0000 0.974869 0.487435 0.873160i \(-0.337933\pi\)
0.487435 + 0.873160i \(0.337933\pi\)
\(948\) −12.0000 −0.389742
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) −6.00000 −0.194461
\(953\) −30.0000 −0.971795 −0.485898 0.874016i \(-0.661507\pi\)
−0.485898 + 0.874016i \(0.661507\pi\)
\(954\) −6.00000 −0.194257
\(955\) 0 0
\(956\) −12.0000 −0.388108
\(957\) −12.0000 −0.387905
\(958\) 6.00000 0.193851
\(959\) 4.00000 0.129167
\(960\) 0 0
\(961\) 5.00000 0.161290
\(962\) 0 0
\(963\) −8.00000 −0.257796
\(964\) 0 0
\(965\) 0 0
\(966\) −16.0000 −0.514792
\(967\) 52.0000 1.67221 0.836104 0.548572i \(-0.184828\pi\)
0.836104 + 0.548572i \(0.184828\pi\)
\(968\) 21.0000 0.674966
\(969\) 0 0
\(970\) 0 0
\(971\) 12.0000 0.385098 0.192549 0.981287i \(-0.438325\pi\)
0.192549 + 0.981287i \(0.438325\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) −26.0000 −0.833094
\(975\) 0 0
\(976\) 2.00000 0.0640184
\(977\) 34.0000 1.08776 0.543878 0.839164i \(-0.316955\pi\)
0.543878 + 0.839164i \(0.316955\pi\)
\(978\) −2.00000 −0.0639529
\(979\) 20.0000 0.639203
\(980\) 0 0
\(981\) 0 0
\(982\) −12.0000 −0.382935
\(983\) 58.0000 1.84991 0.924956 0.380073i \(-0.124101\pi\)
0.924956 + 0.380073i \(0.124101\pi\)
\(984\) −36.0000 −1.14764
\(985\) 0 0
\(986\) −6.00000 −0.191079
\(987\) 0 0
\(988\) 0 0
\(989\) 32.0000 1.01754
\(990\) 0 0
\(991\) 4.00000 0.127064 0.0635321 0.997980i \(-0.479763\pi\)
0.0635321 + 0.997980i \(0.479763\pi\)
\(992\) −30.0000 −0.952501
\(993\) −28.0000 −0.888553
\(994\) −20.0000 −0.634361
\(995\) 0 0
\(996\) −4.00000 −0.126745
\(997\) −2.00000 −0.0633406 −0.0316703 0.999498i \(-0.510083\pi\)
−0.0316703 + 0.999498i \(0.510083\pi\)
\(998\) −34.0000 −1.07625
\(999\) −4.00000 −0.126554
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 8619.2.a.j.1.1 1
13.12 even 2 663.2.a.b.1.1 1
39.38 odd 2 1989.2.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
663.2.a.b.1.1 1 13.12 even 2
1989.2.a.d.1.1 1 39.38 odd 2
8619.2.a.j.1.1 1 1.1 even 1 trivial