Properties

Label 3330.2.a.p.1.1
Level $3330$
Weight $2$
Character 3330.1
Self dual yes
Analytic conductor $26.590$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3330.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.5901838731\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 370)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3330.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} -1.00000 q^{10} -3.00000 q^{11} +1.00000 q^{14} +1.00000 q^{16} -3.00000 q^{17} -6.00000 q^{19} -1.00000 q^{20} -3.00000 q^{22} -2.00000 q^{23} +1.00000 q^{25} +1.00000 q^{28} +3.00000 q^{29} +3.00000 q^{31} +1.00000 q^{32} -3.00000 q^{34} -1.00000 q^{35} -1.00000 q^{37} -6.00000 q^{38} -1.00000 q^{40} -3.00000 q^{41} -1.00000 q^{43} -3.00000 q^{44} -2.00000 q^{46} -4.00000 q^{47} -6.00000 q^{49} +1.00000 q^{50} -13.0000 q^{53} +3.00000 q^{55} +1.00000 q^{56} +3.00000 q^{58} -15.0000 q^{61} +3.00000 q^{62} +1.00000 q^{64} -3.00000 q^{68} -1.00000 q^{70} +2.00000 q^{71} -1.00000 q^{74} -6.00000 q^{76} -3.00000 q^{77} -8.00000 q^{79} -1.00000 q^{80} -3.00000 q^{82} +4.00000 q^{83} +3.00000 q^{85} -1.00000 q^{86} -3.00000 q^{88} +18.0000 q^{89} -2.00000 q^{92} -4.00000 q^{94} +6.00000 q^{95} -7.00000 q^{97} -6.00000 q^{98} +O(q^{100})\)

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
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 1.00000 0.377964 0.188982 0.981981i \(-0.439481\pi\)
0.188982 + 0.981981i \(0.439481\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −1.00000 −0.316228
\(11\) −3.00000 −0.904534 −0.452267 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −3.00000 −0.727607 −0.363803 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 0 0
\(19\) −6.00000 −1.37649 −0.688247 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) −3.00000 −0.639602
\(23\) −2.00000 −0.417029 −0.208514 0.978019i \(-0.566863\pi\)
−0.208514 + 0.978019i \(0.566863\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 1.00000 0.188982
\(29\) 3.00000 0.557086 0.278543 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) 0 0
\(31\) 3.00000 0.538816 0.269408 0.963026i \(-0.413172\pi\)
0.269408 + 0.963026i \(0.413172\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −3.00000 −0.514496
\(35\) −1.00000 −0.169031
\(36\) 0 0
\(37\) −1.00000 −0.164399
\(38\) −6.00000 −0.973329
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) −3.00000 −0.468521 −0.234261 0.972174i \(-0.575267\pi\)
−0.234261 + 0.972174i \(0.575267\pi\)
\(42\) 0 0
\(43\) −1.00000 −0.152499 −0.0762493 0.997089i \(-0.524294\pi\)
−0.0762493 + 0.997089i \(0.524294\pi\)
\(44\) −3.00000 −0.452267
\(45\) 0 0
\(46\) −2.00000 −0.294884
\(47\) −4.00000 −0.583460 −0.291730 0.956501i \(-0.594231\pi\)
−0.291730 + 0.956501i \(0.594231\pi\)
\(48\) 0 0
\(49\) −6.00000 −0.857143
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) 0 0
\(53\) −13.0000 −1.78569 −0.892844 0.450367i \(-0.851293\pi\)
−0.892844 + 0.450367i \(0.851293\pi\)
\(54\) 0 0
\(55\) 3.00000 0.404520
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) 3.00000 0.393919
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −15.0000 −1.92055 −0.960277 0.279050i \(-0.909981\pi\)
−0.960277 + 0.279050i \(0.909981\pi\)
\(62\) 3.00000 0.381000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) −3.00000 −0.363803
\(69\) 0 0
\(70\) −1.00000 −0.119523
\(71\) 2.00000 0.237356 0.118678 0.992933i \(-0.462134\pi\)
0.118678 + 0.992933i \(0.462134\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) −1.00000 −0.116248
\(75\) 0 0
\(76\) −6.00000 −0.688247
\(77\) −3.00000 −0.341882
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) −3.00000 −0.331295
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 0 0
\(85\) 3.00000 0.325396
\(86\) −1.00000 −0.107833
\(87\) 0 0
\(88\) −3.00000 −0.319801
\(89\) 18.0000 1.90800 0.953998 0.299813i \(-0.0969242\pi\)
0.953998 + 0.299813i \(0.0969242\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −2.00000 −0.208514
\(93\) 0 0
\(94\) −4.00000 −0.412568
\(95\) 6.00000 0.615587
\(96\) 0 0
\(97\) −7.00000 −0.710742 −0.355371 0.934725i \(-0.615646\pi\)
−0.355371 + 0.934725i \(0.615646\pi\)
\(98\) −6.00000 −0.606092
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 10.0000 0.995037 0.497519 0.867453i \(-0.334245\pi\)
0.497519 + 0.867453i \(0.334245\pi\)
\(102\) 0 0
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −13.0000 −1.26267
\(107\) −2.00000 −0.193347 −0.0966736 0.995316i \(-0.530820\pi\)
−0.0966736 + 0.995316i \(0.530820\pi\)
\(108\) 0 0
\(109\) −3.00000 −0.287348 −0.143674 0.989625i \(-0.545892\pi\)
−0.143674 + 0.989625i \(0.545892\pi\)
\(110\) 3.00000 0.286039
\(111\) 0 0
\(112\) 1.00000 0.0944911
\(113\) 7.00000 0.658505 0.329252 0.944242i \(-0.393203\pi\)
0.329252 + 0.944242i \(0.393203\pi\)
\(114\) 0 0
\(115\) 2.00000 0.186501
\(116\) 3.00000 0.278543
\(117\) 0 0
\(118\) 0 0
\(119\) −3.00000 −0.275010
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) −15.0000 −1.35804
\(123\) 0 0
\(124\) 3.00000 0.269408
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −4.00000 −0.354943 −0.177471 0.984126i \(-0.556792\pi\)
−0.177471 + 0.984126i \(0.556792\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 0 0
\(133\) −6.00000 −0.520266
\(134\) 0 0
\(135\) 0 0
\(136\) −3.00000 −0.257248
\(137\) −8.00000 −0.683486 −0.341743 0.939793i \(-0.611017\pi\)
−0.341743 + 0.939793i \(0.611017\pi\)
\(138\) 0 0
\(139\) 3.00000 0.254457 0.127228 0.991873i \(-0.459392\pi\)
0.127228 + 0.991873i \(0.459392\pi\)
\(140\) −1.00000 −0.0845154
\(141\) 0 0
\(142\) 2.00000 0.167836
\(143\) 0 0
\(144\) 0 0
\(145\) −3.00000 −0.249136
\(146\) 0 0
\(147\) 0 0
\(148\) −1.00000 −0.0821995
\(149\) −18.0000 −1.47462 −0.737309 0.675556i \(-0.763904\pi\)
−0.737309 + 0.675556i \(0.763904\pi\)
\(150\) 0 0
\(151\) 16.0000 1.30206 0.651031 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) −6.00000 −0.486664
\(153\) 0 0
\(154\) −3.00000 −0.241747
\(155\) −3.00000 −0.240966
\(156\) 0 0
\(157\) −3.00000 −0.239426 −0.119713 0.992809i \(-0.538197\pi\)
−0.119713 + 0.992809i \(0.538197\pi\)
\(158\) −8.00000 −0.636446
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) −2.00000 −0.157622
\(162\) 0 0
\(163\) −5.00000 −0.391630 −0.195815 0.980641i \(-0.562735\pi\)
−0.195815 + 0.980641i \(0.562735\pi\)
\(164\) −3.00000 −0.234261
\(165\) 0 0
\(166\) 4.00000 0.310460
\(167\) −12.0000 −0.928588 −0.464294 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) 3.00000 0.230089
\(171\) 0 0
\(172\) −1.00000 −0.0762493
\(173\) −9.00000 −0.684257 −0.342129 0.939653i \(-0.611148\pi\)
−0.342129 + 0.939653i \(0.611148\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) −3.00000 −0.226134
\(177\) 0 0
\(178\) 18.0000 1.34916
\(179\) −24.0000 −1.79384 −0.896922 0.442189i \(-0.854202\pi\)
−0.896922 + 0.442189i \(0.854202\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −2.00000 −0.147442
\(185\) 1.00000 0.0735215
\(186\) 0 0
\(187\) 9.00000 0.658145
\(188\) −4.00000 −0.291730
\(189\) 0 0
\(190\) 6.00000 0.435286
\(191\) −21.0000 −1.51951 −0.759753 0.650211i \(-0.774680\pi\)
−0.759753 + 0.650211i \(0.774680\pi\)
\(192\) 0 0
\(193\) −10.0000 −0.719816 −0.359908 0.932988i \(-0.617192\pi\)
−0.359908 + 0.932988i \(0.617192\pi\)
\(194\) −7.00000 −0.502571
\(195\) 0 0
\(196\) −6.00000 −0.428571
\(197\) 18.0000 1.28245 0.641223 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) 0 0
\(199\) 8.00000 0.567105 0.283552 0.958957i \(-0.408487\pi\)
0.283552 + 0.958957i \(0.408487\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0 0
\(202\) 10.0000 0.703598
\(203\) 3.00000 0.210559
\(204\) 0 0
\(205\) 3.00000 0.209529
\(206\) 8.00000 0.557386
\(207\) 0 0
\(208\) 0 0
\(209\) 18.0000 1.24509
\(210\) 0 0
\(211\) −3.00000 −0.206529 −0.103264 0.994654i \(-0.532929\pi\)
−0.103264 + 0.994654i \(0.532929\pi\)
\(212\) −13.0000 −0.892844
\(213\) 0 0
\(214\) −2.00000 −0.136717
\(215\) 1.00000 0.0681994
\(216\) 0 0
\(217\) 3.00000 0.203653
\(218\) −3.00000 −0.203186
\(219\) 0 0
\(220\) 3.00000 0.202260
\(221\) 0 0
\(222\) 0 0
\(223\) 23.0000 1.54019 0.770097 0.637927i \(-0.220208\pi\)
0.770097 + 0.637927i \(0.220208\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) 7.00000 0.465633
\(227\) −13.0000 −0.862840 −0.431420 0.902151i \(-0.641987\pi\)
−0.431420 + 0.902151i \(0.641987\pi\)
\(228\) 0 0
\(229\) −6.00000 −0.396491 −0.198246 0.980152i \(-0.563524\pi\)
−0.198246 + 0.980152i \(0.563524\pi\)
\(230\) 2.00000 0.131876
\(231\) 0 0
\(232\) 3.00000 0.196960
\(233\) −4.00000 −0.262049 −0.131024 0.991379i \(-0.541827\pi\)
−0.131024 + 0.991379i \(0.541827\pi\)
\(234\) 0 0
\(235\) 4.00000 0.260931
\(236\) 0 0
\(237\) 0 0
\(238\) −3.00000 −0.194461
\(239\) −9.00000 −0.582162 −0.291081 0.956698i \(-0.594015\pi\)
−0.291081 + 0.956698i \(0.594015\pi\)
\(240\) 0 0
\(241\) 24.0000 1.54598 0.772988 0.634421i \(-0.218761\pi\)
0.772988 + 0.634421i \(0.218761\pi\)
\(242\) −2.00000 −0.128565
\(243\) 0 0
\(244\) −15.0000 −0.960277
\(245\) 6.00000 0.383326
\(246\) 0 0
\(247\) 0 0
\(248\) 3.00000 0.190500
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) −6.00000 −0.378717 −0.189358 0.981908i \(-0.560641\pi\)
−0.189358 + 0.981908i \(0.560641\pi\)
\(252\) 0 0
\(253\) 6.00000 0.377217
\(254\) −4.00000 −0.250982
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 22.0000 1.37232 0.686161 0.727450i \(-0.259294\pi\)
0.686161 + 0.727450i \(0.259294\pi\)
\(258\) 0 0
\(259\) −1.00000 −0.0621370
\(260\) 0 0
\(261\) 0 0
\(262\) 12.0000 0.741362
\(263\) 1.00000 0.0616626 0.0308313 0.999525i \(-0.490185\pi\)
0.0308313 + 0.999525i \(0.490185\pi\)
\(264\) 0 0
\(265\) 13.0000 0.798584
\(266\) −6.00000 −0.367884
\(267\) 0 0
\(268\) 0 0
\(269\) 10.0000 0.609711 0.304855 0.952399i \(-0.401392\pi\)
0.304855 + 0.952399i \(0.401392\pi\)
\(270\) 0 0
\(271\) −10.0000 −0.607457 −0.303728 0.952759i \(-0.598232\pi\)
−0.303728 + 0.952759i \(0.598232\pi\)
\(272\) −3.00000 −0.181902
\(273\) 0 0
\(274\) −8.00000 −0.483298
\(275\) −3.00000 −0.180907
\(276\) 0 0
\(277\) −20.0000 −1.20168 −0.600842 0.799368i \(-0.705168\pi\)
−0.600842 + 0.799368i \(0.705168\pi\)
\(278\) 3.00000 0.179928
\(279\) 0 0
\(280\) −1.00000 −0.0597614
\(281\) 24.0000 1.43172 0.715860 0.698244i \(-0.246035\pi\)
0.715860 + 0.698244i \(0.246035\pi\)
\(282\) 0 0
\(283\) −20.0000 −1.18888 −0.594438 0.804141i \(-0.702626\pi\)
−0.594438 + 0.804141i \(0.702626\pi\)
\(284\) 2.00000 0.118678
\(285\) 0 0
\(286\) 0 0
\(287\) −3.00000 −0.177084
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) −3.00000 −0.176166
\(291\) 0 0
\(292\) 0 0
\(293\) −3.00000 −0.175262 −0.0876309 0.996153i \(-0.527930\pi\)
−0.0876309 + 0.996153i \(0.527930\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −1.00000 −0.0581238
\(297\) 0 0
\(298\) −18.0000 −1.04271
\(299\) 0 0
\(300\) 0 0
\(301\) −1.00000 −0.0576390
\(302\) 16.0000 0.920697
\(303\) 0 0
\(304\) −6.00000 −0.344124
\(305\) 15.0000 0.858898
\(306\) 0 0
\(307\) 34.0000 1.94048 0.970241 0.242140i \(-0.0778494\pi\)
0.970241 + 0.242140i \(0.0778494\pi\)
\(308\) −3.00000 −0.170941
\(309\) 0 0
\(310\) −3.00000 −0.170389
\(311\) −25.0000 −1.41762 −0.708810 0.705399i \(-0.750768\pi\)
−0.708810 + 0.705399i \(0.750768\pi\)
\(312\) 0 0
\(313\) 26.0000 1.46961 0.734803 0.678280i \(-0.237274\pi\)
0.734803 + 0.678280i \(0.237274\pi\)
\(314\) −3.00000 −0.169300
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) −21.0000 −1.17948 −0.589739 0.807594i \(-0.700769\pi\)
−0.589739 + 0.807594i \(0.700769\pi\)
\(318\) 0 0
\(319\) −9.00000 −0.503903
\(320\) −1.00000 −0.0559017
\(321\) 0 0
\(322\) −2.00000 −0.111456
\(323\) 18.0000 1.00155
\(324\) 0 0
\(325\) 0 0
\(326\) −5.00000 −0.276924
\(327\) 0 0
\(328\) −3.00000 −0.165647
\(329\) −4.00000 −0.220527
\(330\) 0 0
\(331\) 30.0000 1.64895 0.824475 0.565899i \(-0.191471\pi\)
0.824475 + 0.565899i \(0.191471\pi\)
\(332\) 4.00000 0.219529
\(333\) 0 0
\(334\) −12.0000 −0.656611
\(335\) 0 0
\(336\) 0 0
\(337\) 12.0000 0.653682 0.326841 0.945079i \(-0.394016\pi\)
0.326841 + 0.945079i \(0.394016\pi\)
\(338\) −13.0000 −0.707107
\(339\) 0 0
\(340\) 3.00000 0.162698
\(341\) −9.00000 −0.487377
\(342\) 0 0
\(343\) −13.0000 −0.701934
\(344\) −1.00000 −0.0539164
\(345\) 0 0
\(346\) −9.00000 −0.483843
\(347\) 28.0000 1.50312 0.751559 0.659665i \(-0.229302\pi\)
0.751559 + 0.659665i \(0.229302\pi\)
\(348\) 0 0
\(349\) −22.0000 −1.17763 −0.588817 0.808267i \(-0.700406\pi\)
−0.588817 + 0.808267i \(0.700406\pi\)
\(350\) 1.00000 0.0534522
\(351\) 0 0
\(352\) −3.00000 −0.159901
\(353\) −19.0000 −1.01127 −0.505634 0.862748i \(-0.668741\pi\)
−0.505634 + 0.862748i \(0.668741\pi\)
\(354\) 0 0
\(355\) −2.00000 −0.106149
\(356\) 18.0000 0.953998
\(357\) 0 0
\(358\) −24.0000 −1.26844
\(359\) −16.0000 −0.844448 −0.422224 0.906492i \(-0.638750\pi\)
−0.422224 + 0.906492i \(0.638750\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) 2.00000 0.105118
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 13.0000 0.678594 0.339297 0.940679i \(-0.389811\pi\)
0.339297 + 0.940679i \(0.389811\pi\)
\(368\) −2.00000 −0.104257
\(369\) 0 0
\(370\) 1.00000 0.0519875
\(371\) −13.0000 −0.674926
\(372\) 0 0
\(373\) 22.0000 1.13912 0.569558 0.821951i \(-0.307114\pi\)
0.569558 + 0.821951i \(0.307114\pi\)
\(374\) 9.00000 0.465379
\(375\) 0 0
\(376\) −4.00000 −0.206284
\(377\) 0 0
\(378\) 0 0
\(379\) 16.0000 0.821865 0.410932 0.911666i \(-0.365203\pi\)
0.410932 + 0.911666i \(0.365203\pi\)
\(380\) 6.00000 0.307794
\(381\) 0 0
\(382\) −21.0000 −1.07445
\(383\) 12.0000 0.613171 0.306586 0.951843i \(-0.400813\pi\)
0.306586 + 0.951843i \(0.400813\pi\)
\(384\) 0 0
\(385\) 3.00000 0.152894
\(386\) −10.0000 −0.508987
\(387\) 0 0
\(388\) −7.00000 −0.355371
\(389\) −15.0000 −0.760530 −0.380265 0.924878i \(-0.624167\pi\)
−0.380265 + 0.924878i \(0.624167\pi\)
\(390\) 0 0
\(391\) 6.00000 0.303433
\(392\) −6.00000 −0.303046
\(393\) 0 0
\(394\) 18.0000 0.906827
\(395\) 8.00000 0.402524
\(396\) 0 0
\(397\) 18.0000 0.903394 0.451697 0.892171i \(-0.350819\pi\)
0.451697 + 0.892171i \(0.350819\pi\)
\(398\) 8.00000 0.401004
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −34.0000 −1.69788 −0.848939 0.528490i \(-0.822758\pi\)
−0.848939 + 0.528490i \(0.822758\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 10.0000 0.497519
\(405\) 0 0
\(406\) 3.00000 0.148888
\(407\) 3.00000 0.148704
\(408\) 0 0
\(409\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(410\) 3.00000 0.148159
\(411\) 0 0
\(412\) 8.00000 0.394132
\(413\) 0 0
\(414\) 0 0
\(415\) −4.00000 −0.196352
\(416\) 0 0
\(417\) 0 0
\(418\) 18.0000 0.880409
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) 34.0000 1.65706 0.828529 0.559946i \(-0.189178\pi\)
0.828529 + 0.559946i \(0.189178\pi\)
\(422\) −3.00000 −0.146038
\(423\) 0 0
\(424\) −13.0000 −0.631336
\(425\) −3.00000 −0.145521
\(426\) 0 0
\(427\) −15.0000 −0.725901
\(428\) −2.00000 −0.0966736
\(429\) 0 0
\(430\) 1.00000 0.0482243
\(431\) 31.0000 1.49322 0.746609 0.665263i \(-0.231681\pi\)
0.746609 + 0.665263i \(0.231681\pi\)
\(432\) 0 0
\(433\) −22.0000 −1.05725 −0.528626 0.848855i \(-0.677293\pi\)
−0.528626 + 0.848855i \(0.677293\pi\)
\(434\) 3.00000 0.144005
\(435\) 0 0
\(436\) −3.00000 −0.143674
\(437\) 12.0000 0.574038
\(438\) 0 0
\(439\) 37.0000 1.76591 0.882957 0.469454i \(-0.155549\pi\)
0.882957 + 0.469454i \(0.155549\pi\)
\(440\) 3.00000 0.143019
\(441\) 0 0
\(442\) 0 0
\(443\) 34.0000 1.61539 0.807694 0.589601i \(-0.200715\pi\)
0.807694 + 0.589601i \(0.200715\pi\)
\(444\) 0 0
\(445\) −18.0000 −0.853282
\(446\) 23.0000 1.08908
\(447\) 0 0
\(448\) 1.00000 0.0472456
\(449\) 10.0000 0.471929 0.235965 0.971762i \(-0.424175\pi\)
0.235965 + 0.971762i \(0.424175\pi\)
\(450\) 0 0
\(451\) 9.00000 0.423793
\(452\) 7.00000 0.329252
\(453\) 0 0
\(454\) −13.0000 −0.610120
\(455\) 0 0
\(456\) 0 0
\(457\) 9.00000 0.421002 0.210501 0.977594i \(-0.432490\pi\)
0.210501 + 0.977594i \(0.432490\pi\)
\(458\) −6.00000 −0.280362
\(459\) 0 0
\(460\) 2.00000 0.0932505
\(461\) 1.00000 0.0465746 0.0232873 0.999729i \(-0.492587\pi\)
0.0232873 + 0.999729i \(0.492587\pi\)
\(462\) 0 0
\(463\) 12.0000 0.557687 0.278844 0.960337i \(-0.410049\pi\)
0.278844 + 0.960337i \(0.410049\pi\)
\(464\) 3.00000 0.139272
\(465\) 0 0
\(466\) −4.00000 −0.185296
\(467\) 37.0000 1.71216 0.856078 0.516847i \(-0.172894\pi\)
0.856078 + 0.516847i \(0.172894\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 4.00000 0.184506
\(471\) 0 0
\(472\) 0 0
\(473\) 3.00000 0.137940
\(474\) 0 0
\(475\) −6.00000 −0.275299
\(476\) −3.00000 −0.137505
\(477\) 0 0
\(478\) −9.00000 −0.411650
\(479\) 8.00000 0.365529 0.182765 0.983157i \(-0.441495\pi\)
0.182765 + 0.983157i \(0.441495\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 24.0000 1.09317
\(483\) 0 0
\(484\) −2.00000 −0.0909091
\(485\) 7.00000 0.317854
\(486\) 0 0
\(487\) 34.0000 1.54069 0.770344 0.637629i \(-0.220085\pi\)
0.770344 + 0.637629i \(0.220085\pi\)
\(488\) −15.0000 −0.679018
\(489\) 0 0
\(490\) 6.00000 0.271052
\(491\) 20.0000 0.902587 0.451294 0.892375i \(-0.350963\pi\)
0.451294 + 0.892375i \(0.350963\pi\)
\(492\) 0 0
\(493\) −9.00000 −0.405340
\(494\) 0 0
\(495\) 0 0
\(496\) 3.00000 0.134704
\(497\) 2.00000 0.0897123
\(498\) 0 0
\(499\) −22.0000 −0.984855 −0.492428 0.870353i \(-0.663890\pi\)
−0.492428 + 0.870353i \(0.663890\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) −6.00000 −0.267793
\(503\) 24.0000 1.07011 0.535054 0.844818i \(-0.320291\pi\)
0.535054 + 0.844818i \(0.320291\pi\)
\(504\) 0 0
\(505\) −10.0000 −0.444994
\(506\) 6.00000 0.266733
\(507\) 0 0
\(508\) −4.00000 −0.177471
\(509\) −36.0000 −1.59567 −0.797836 0.602875i \(-0.794022\pi\)
−0.797836 + 0.602875i \(0.794022\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 22.0000 0.970378
\(515\) −8.00000 −0.352522
\(516\) 0 0
\(517\) 12.0000 0.527759
\(518\) −1.00000 −0.0439375
\(519\) 0 0
\(520\) 0 0
\(521\) 23.0000 1.00765 0.503824 0.863806i \(-0.331926\pi\)
0.503824 + 0.863806i \(0.331926\pi\)
\(522\) 0 0
\(523\) −44.0000 −1.92399 −0.961993 0.273075i \(-0.911959\pi\)
−0.961993 + 0.273075i \(0.911959\pi\)
\(524\) 12.0000 0.524222
\(525\) 0 0
\(526\) 1.00000 0.0436021
\(527\) −9.00000 −0.392046
\(528\) 0 0
\(529\) −19.0000 −0.826087
\(530\) 13.0000 0.564684
\(531\) 0 0
\(532\) −6.00000 −0.260133
\(533\) 0 0
\(534\) 0 0
\(535\) 2.00000 0.0864675
\(536\) 0 0
\(537\) 0 0
\(538\) 10.0000 0.431131
\(539\) 18.0000 0.775315
\(540\) 0 0
\(541\) −2.00000 −0.0859867 −0.0429934 0.999075i \(-0.513689\pi\)
−0.0429934 + 0.999075i \(0.513689\pi\)
\(542\) −10.0000 −0.429537
\(543\) 0 0
\(544\) −3.00000 −0.128624
\(545\) 3.00000 0.128506
\(546\) 0 0
\(547\) 23.0000 0.983409 0.491704 0.870762i \(-0.336374\pi\)
0.491704 + 0.870762i \(0.336374\pi\)
\(548\) −8.00000 −0.341743
\(549\) 0 0
\(550\) −3.00000 −0.127920
\(551\) −18.0000 −0.766826
\(552\) 0 0
\(553\) −8.00000 −0.340195
\(554\) −20.0000 −0.849719
\(555\) 0 0
\(556\) 3.00000 0.127228
\(557\) 40.0000 1.69485 0.847427 0.530912i \(-0.178150\pi\)
0.847427 + 0.530912i \(0.178150\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) 24.0000 1.01238
\(563\) 11.0000 0.463595 0.231797 0.972764i \(-0.425539\pi\)
0.231797 + 0.972764i \(0.425539\pi\)
\(564\) 0 0
\(565\) −7.00000 −0.294492
\(566\) −20.0000 −0.840663
\(567\) 0 0
\(568\) 2.00000 0.0839181
\(569\) −10.0000 −0.419222 −0.209611 0.977785i \(-0.567220\pi\)
−0.209611 + 0.977785i \(0.567220\pi\)
\(570\) 0 0
\(571\) −31.0000 −1.29731 −0.648655 0.761083i \(-0.724668\pi\)
−0.648655 + 0.761083i \(0.724668\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −3.00000 −0.125218
\(575\) −2.00000 −0.0834058
\(576\) 0 0
\(577\) 30.0000 1.24892 0.624458 0.781058i \(-0.285320\pi\)
0.624458 + 0.781058i \(0.285320\pi\)
\(578\) −8.00000 −0.332756
\(579\) 0 0
\(580\) −3.00000 −0.124568
\(581\) 4.00000 0.165948
\(582\) 0 0
\(583\) 39.0000 1.61521
\(584\) 0 0
\(585\) 0 0
\(586\) −3.00000 −0.123929
\(587\) −35.0000 −1.44460 −0.722302 0.691577i \(-0.756916\pi\)
−0.722302 + 0.691577i \(0.756916\pi\)
\(588\) 0 0
\(589\) −18.0000 −0.741677
\(590\) 0 0
\(591\) 0 0
\(592\) −1.00000 −0.0410997
\(593\) −12.0000 −0.492781 −0.246390 0.969171i \(-0.579245\pi\)
−0.246390 + 0.969171i \(0.579245\pi\)
\(594\) 0 0
\(595\) 3.00000 0.122988
\(596\) −18.0000 −0.737309
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 21.0000 0.856608 0.428304 0.903635i \(-0.359111\pi\)
0.428304 + 0.903635i \(0.359111\pi\)
\(602\) −1.00000 −0.0407570
\(603\) 0 0
\(604\) 16.0000 0.651031
\(605\) 2.00000 0.0813116
\(606\) 0 0
\(607\) 22.0000 0.892952 0.446476 0.894795i \(-0.352679\pi\)
0.446476 + 0.894795i \(0.352679\pi\)
\(608\) −6.00000 −0.243332
\(609\) 0 0
\(610\) 15.0000 0.607332
\(611\) 0 0
\(612\) 0 0
\(613\) 11.0000 0.444286 0.222143 0.975014i \(-0.428695\pi\)
0.222143 + 0.975014i \(0.428695\pi\)
\(614\) 34.0000 1.37213
\(615\) 0 0
\(616\) −3.00000 −0.120873
\(617\) −36.0000 −1.44931 −0.724653 0.689114i \(-0.758000\pi\)
−0.724653 + 0.689114i \(0.758000\pi\)
\(618\) 0 0
\(619\) −37.0000 −1.48716 −0.743578 0.668649i \(-0.766873\pi\)
−0.743578 + 0.668649i \(0.766873\pi\)
\(620\) −3.00000 −0.120483
\(621\) 0 0
\(622\) −25.0000 −1.00241
\(623\) 18.0000 0.721155
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 26.0000 1.03917
\(627\) 0 0
\(628\) −3.00000 −0.119713
\(629\) 3.00000 0.119618
\(630\) 0 0
\(631\) −41.0000 −1.63218 −0.816092 0.577922i \(-0.803864\pi\)
−0.816092 + 0.577922i \(0.803864\pi\)
\(632\) −8.00000 −0.318223
\(633\) 0 0
\(634\) −21.0000 −0.834017
\(635\) 4.00000 0.158735
\(636\) 0 0
\(637\) 0 0
\(638\) −9.00000 −0.356313
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) 33.0000 1.30342 0.651711 0.758468i \(-0.274052\pi\)
0.651711 + 0.758468i \(0.274052\pi\)
\(642\) 0 0
\(643\) 11.0000 0.433798 0.216899 0.976194i \(-0.430406\pi\)
0.216899 + 0.976194i \(0.430406\pi\)
\(644\) −2.00000 −0.0788110
\(645\) 0 0
\(646\) 18.0000 0.708201
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) −5.00000 −0.195815
\(653\) −30.0000 −1.17399 −0.586995 0.809590i \(-0.699689\pi\)
−0.586995 + 0.809590i \(0.699689\pi\)
\(654\) 0 0
\(655\) −12.0000 −0.468879
\(656\) −3.00000 −0.117130
\(657\) 0 0
\(658\) −4.00000 −0.155936
\(659\) −36.0000 −1.40236 −0.701180 0.712984i \(-0.747343\pi\)
−0.701180 + 0.712984i \(0.747343\pi\)
\(660\) 0 0
\(661\) −13.0000 −0.505641 −0.252821 0.967513i \(-0.581358\pi\)
−0.252821 + 0.967513i \(0.581358\pi\)
\(662\) 30.0000 1.16598
\(663\) 0 0
\(664\) 4.00000 0.155230
\(665\) 6.00000 0.232670
\(666\) 0 0
\(667\) −6.00000 −0.232321
\(668\) −12.0000 −0.464294
\(669\) 0 0
\(670\) 0 0
\(671\) 45.0000 1.73721
\(672\) 0 0
\(673\) −34.0000 −1.31060 −0.655302 0.755367i \(-0.727459\pi\)
−0.655302 + 0.755367i \(0.727459\pi\)
\(674\) 12.0000 0.462223
\(675\) 0 0
\(676\) −13.0000 −0.500000
\(677\) −34.0000 −1.30673 −0.653363 0.757045i \(-0.726642\pi\)
−0.653363 + 0.757045i \(0.726642\pi\)
\(678\) 0 0
\(679\) −7.00000 −0.268635
\(680\) 3.00000 0.115045
\(681\) 0 0
\(682\) −9.00000 −0.344628
\(683\) 31.0000 1.18618 0.593091 0.805135i \(-0.297907\pi\)
0.593091 + 0.805135i \(0.297907\pi\)
\(684\) 0 0
\(685\) 8.00000 0.305664
\(686\) −13.0000 −0.496342
\(687\) 0 0
\(688\) −1.00000 −0.0381246
\(689\) 0 0
\(690\) 0 0
\(691\) 5.00000 0.190209 0.0951045 0.995467i \(-0.469681\pi\)
0.0951045 + 0.995467i \(0.469681\pi\)
\(692\) −9.00000 −0.342129
\(693\) 0 0
\(694\) 28.0000 1.06287
\(695\) −3.00000 −0.113796
\(696\) 0 0
\(697\) 9.00000 0.340899
\(698\) −22.0000 −0.832712
\(699\) 0 0
\(700\) 1.00000 0.0377964
\(701\) 22.0000 0.830929 0.415464 0.909610i \(-0.363619\pi\)
0.415464 + 0.909610i \(0.363619\pi\)
\(702\) 0 0
\(703\) 6.00000 0.226294
\(704\) −3.00000 −0.113067
\(705\) 0 0
\(706\) −19.0000 −0.715074
\(707\) 10.0000 0.376089
\(708\) 0 0
\(709\) −19.0000 −0.713560 −0.356780 0.934188i \(-0.616125\pi\)
−0.356780 + 0.934188i \(0.616125\pi\)
\(710\) −2.00000 −0.0750587
\(711\) 0 0
\(712\) 18.0000 0.674579
\(713\) −6.00000 −0.224702
\(714\) 0 0
\(715\) 0 0
\(716\) −24.0000 −0.896922
\(717\) 0 0
\(718\) −16.0000 −0.597115
\(719\) 18.0000 0.671287 0.335643 0.941989i \(-0.391046\pi\)
0.335643 + 0.941989i \(0.391046\pi\)
\(720\) 0 0
\(721\) 8.00000 0.297936
\(722\) 17.0000 0.632674
\(723\) 0 0
\(724\) 2.00000 0.0743294
\(725\) 3.00000 0.111417
\(726\) 0 0
\(727\) 52.0000 1.92857 0.964287 0.264861i \(-0.0853260\pi\)
0.964287 + 0.264861i \(0.0853260\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 3.00000 0.110959
\(732\) 0 0
\(733\) −41.0000 −1.51437 −0.757185 0.653201i \(-0.773426\pi\)
−0.757185 + 0.653201i \(0.773426\pi\)
\(734\) 13.0000 0.479839
\(735\) 0 0
\(736\) −2.00000 −0.0737210
\(737\) 0 0
\(738\) 0 0
\(739\) 27.0000 0.993211 0.496606 0.867976i \(-0.334580\pi\)
0.496606 + 0.867976i \(0.334580\pi\)
\(740\) 1.00000 0.0367607
\(741\) 0 0
\(742\) −13.0000 −0.477245
\(743\) −3.00000 −0.110059 −0.0550297 0.998485i \(-0.517525\pi\)
−0.0550297 + 0.998485i \(0.517525\pi\)
\(744\) 0 0
\(745\) 18.0000 0.659469
\(746\) 22.0000 0.805477
\(747\) 0 0
\(748\) 9.00000 0.329073
\(749\) −2.00000 −0.0730784
\(750\) 0 0
\(751\) −22.0000 −0.802791 −0.401396 0.915905i \(-0.631475\pi\)
−0.401396 + 0.915905i \(0.631475\pi\)
\(752\) −4.00000 −0.145865
\(753\) 0 0
\(754\) 0 0
\(755\) −16.0000 −0.582300
\(756\) 0 0
\(757\) −50.0000 −1.81728 −0.908640 0.417579i \(-0.862879\pi\)
−0.908640 + 0.417579i \(0.862879\pi\)
\(758\) 16.0000 0.581146
\(759\) 0 0
\(760\) 6.00000 0.217643
\(761\) 21.0000 0.761249 0.380625 0.924730i \(-0.375709\pi\)
0.380625 + 0.924730i \(0.375709\pi\)
\(762\) 0 0
\(763\) −3.00000 −0.108607
\(764\) −21.0000 −0.759753
\(765\) 0 0
\(766\) 12.0000 0.433578
\(767\) 0 0
\(768\) 0 0
\(769\) 8.00000 0.288487 0.144244 0.989542i \(-0.453925\pi\)
0.144244 + 0.989542i \(0.453925\pi\)
\(770\) 3.00000 0.108112
\(771\) 0 0
\(772\) −10.0000 −0.359908
\(773\) 41.0000 1.47467 0.737334 0.675529i \(-0.236085\pi\)
0.737334 + 0.675529i \(0.236085\pi\)
\(774\) 0 0
\(775\) 3.00000 0.107763
\(776\) −7.00000 −0.251285
\(777\) 0 0
\(778\) −15.0000 −0.537776
\(779\) 18.0000 0.644917
\(780\) 0 0
\(781\) −6.00000 −0.214697
\(782\) 6.00000 0.214560
\(783\) 0 0
\(784\) −6.00000 −0.214286
\(785\) 3.00000 0.107075
\(786\) 0 0
\(787\) −10.0000 −0.356462 −0.178231 0.983989i \(-0.557037\pi\)
−0.178231 + 0.983989i \(0.557037\pi\)
\(788\) 18.0000 0.641223
\(789\) 0 0
\(790\) 8.00000 0.284627
\(791\) 7.00000 0.248891
\(792\) 0 0
\(793\) 0 0
\(794\) 18.0000 0.638796
\(795\) 0 0
\(796\) 8.00000 0.283552
\(797\) 26.0000 0.920967 0.460484 0.887668i \(-0.347676\pi\)
0.460484 + 0.887668i \(0.347676\pi\)
\(798\) 0 0
\(799\) 12.0000 0.424529
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) −34.0000 −1.20058
\(803\) 0 0
\(804\) 0 0
\(805\) 2.00000 0.0704907
\(806\) 0 0
\(807\) 0 0
\(808\) 10.0000 0.351799
\(809\) −34.0000 −1.19538 −0.597688 0.801729i \(-0.703914\pi\)
−0.597688 + 0.801729i \(0.703914\pi\)
\(810\) 0 0
\(811\) −16.0000 −0.561836 −0.280918 0.959732i \(-0.590639\pi\)
−0.280918 + 0.959732i \(0.590639\pi\)
\(812\) 3.00000 0.105279
\(813\) 0 0
\(814\) 3.00000 0.105150
\(815\) 5.00000 0.175142
\(816\) 0 0
\(817\) 6.00000 0.209913
\(818\) 0 0
\(819\) 0 0
\(820\) 3.00000 0.104765
\(821\) −8.00000 −0.279202 −0.139601 0.990208i \(-0.544582\pi\)
−0.139601 + 0.990208i \(0.544582\pi\)
\(822\) 0 0
\(823\) 24.0000 0.836587 0.418294 0.908312i \(-0.362628\pi\)
0.418294 + 0.908312i \(0.362628\pi\)
\(824\) 8.00000 0.278693
\(825\) 0 0
\(826\) 0 0
\(827\) 11.0000 0.382507 0.191254 0.981541i \(-0.438745\pi\)
0.191254 + 0.981541i \(0.438745\pi\)
\(828\) 0 0
\(829\) −39.0000 −1.35453 −0.677263 0.735741i \(-0.736834\pi\)
−0.677263 + 0.735741i \(0.736834\pi\)
\(830\) −4.00000 −0.138842
\(831\) 0 0
\(832\) 0 0
\(833\) 18.0000 0.623663
\(834\) 0 0
\(835\) 12.0000 0.415277
\(836\) 18.0000 0.622543
\(837\) 0 0
\(838\) −12.0000 −0.414533
\(839\) −36.0000 −1.24286 −0.621429 0.783470i \(-0.713448\pi\)
−0.621429 + 0.783470i \(0.713448\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 34.0000 1.17172
\(843\) 0 0
\(844\) −3.00000 −0.103264
\(845\) 13.0000 0.447214
\(846\) 0 0
\(847\) −2.00000 −0.0687208
\(848\) −13.0000 −0.446422
\(849\) 0 0
\(850\) −3.00000 −0.102899
\(851\) 2.00000 0.0685591
\(852\) 0 0
\(853\) 22.0000 0.753266 0.376633 0.926363i \(-0.377082\pi\)
0.376633 + 0.926363i \(0.377082\pi\)
\(854\) −15.0000 −0.513289
\(855\) 0 0
\(856\) −2.00000 −0.0683586
\(857\) −33.0000 −1.12726 −0.563629 0.826028i \(-0.690595\pi\)
−0.563629 + 0.826028i \(0.690595\pi\)
\(858\) 0 0
\(859\) −36.0000 −1.22830 −0.614152 0.789188i \(-0.710502\pi\)
−0.614152 + 0.789188i \(0.710502\pi\)
\(860\) 1.00000 0.0340997
\(861\) 0 0
\(862\) 31.0000 1.05586
\(863\) −33.0000 −1.12333 −0.561667 0.827364i \(-0.689840\pi\)
−0.561667 + 0.827364i \(0.689840\pi\)
\(864\) 0 0
\(865\) 9.00000 0.306009
\(866\) −22.0000 −0.747590
\(867\) 0 0
\(868\) 3.00000 0.101827
\(869\) 24.0000 0.814144
\(870\) 0 0
\(871\) 0 0
\(872\) −3.00000 −0.101593
\(873\) 0 0
\(874\) 12.0000 0.405906
\(875\) −1.00000 −0.0338062
\(876\) 0 0
\(877\) −43.0000 −1.45201 −0.726003 0.687691i \(-0.758624\pi\)
−0.726003 + 0.687691i \(0.758624\pi\)
\(878\) 37.0000 1.24869
\(879\) 0 0
\(880\) 3.00000 0.101130
\(881\) 9.00000 0.303218 0.151609 0.988441i \(-0.451555\pi\)
0.151609 + 0.988441i \(0.451555\pi\)
\(882\) 0 0
\(883\) 29.0000 0.975928 0.487964 0.872864i \(-0.337740\pi\)
0.487964 + 0.872864i \(0.337740\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 34.0000 1.14225
\(887\) 33.0000 1.10803 0.554016 0.832506i \(-0.313095\pi\)
0.554016 + 0.832506i \(0.313095\pi\)
\(888\) 0 0
\(889\) −4.00000 −0.134156
\(890\) −18.0000 −0.603361
\(891\) 0 0
\(892\) 23.0000 0.770097
\(893\) 24.0000 0.803129
\(894\) 0 0
\(895\) 24.0000 0.802232
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) 10.0000 0.333704
\(899\) 9.00000 0.300167
\(900\) 0 0
\(901\) 39.0000 1.29928
\(902\) 9.00000 0.299667
\(903\) 0 0
\(904\) 7.00000 0.232817
\(905\) −2.00000 −0.0664822
\(906\) 0 0
\(907\) −12.0000 −0.398453 −0.199227 0.979953i \(-0.563843\pi\)
−0.199227 + 0.979953i \(0.563843\pi\)
\(908\) −13.0000 −0.431420
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) −12.0000 −0.397142
\(914\) 9.00000 0.297694
\(915\) 0 0
\(916\) −6.00000 −0.198246
\(917\) 12.0000 0.396275
\(918\) 0 0
\(919\) 32.0000 1.05558 0.527791 0.849374i \(-0.323020\pi\)
0.527791 + 0.849374i \(0.323020\pi\)
\(920\) 2.00000 0.0659380
\(921\) 0 0
\(922\) 1.00000 0.0329332
\(923\) 0 0
\(924\) 0 0
\(925\) −1.00000 −0.0328798
\(926\) 12.0000 0.394344
\(927\) 0 0
\(928\) 3.00000 0.0984798
\(929\) 3.00000 0.0984268 0.0492134 0.998788i \(-0.484329\pi\)
0.0492134 + 0.998788i \(0.484329\pi\)
\(930\) 0 0
\(931\) 36.0000 1.17985
\(932\) −4.00000 −0.131024
\(933\) 0 0
\(934\) 37.0000 1.21068
\(935\) −9.00000 −0.294331
\(936\) 0 0
\(937\) −14.0000 −0.457360 −0.228680 0.973502i \(-0.573441\pi\)
−0.228680 + 0.973502i \(0.573441\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 4.00000 0.130466
\(941\) 6.00000 0.195594 0.0977972 0.995206i \(-0.468820\pi\)
0.0977972 + 0.995206i \(0.468820\pi\)
\(942\) 0 0
\(943\) 6.00000 0.195387
\(944\) 0 0
\(945\) 0 0
\(946\) 3.00000 0.0975384
\(947\) −9.00000 −0.292461 −0.146230 0.989251i \(-0.546714\pi\)
−0.146230 + 0.989251i \(0.546714\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −6.00000 −0.194666
\(951\) 0 0
\(952\) −3.00000 −0.0972306
\(953\) −36.0000 −1.16615 −0.583077 0.812417i \(-0.698151\pi\)
−0.583077 + 0.812417i \(0.698151\pi\)
\(954\) 0 0
\(955\) 21.0000 0.679544
\(956\) −9.00000 −0.291081
\(957\) 0 0
\(958\) 8.00000 0.258468
\(959\) −8.00000 −0.258333
\(960\) 0 0
\(961\) −22.0000 −0.709677
\(962\) 0 0
\(963\) 0 0
\(964\) 24.0000 0.772988
\(965\) 10.0000 0.321911
\(966\) 0 0
\(967\) 48.0000 1.54358 0.771788 0.635880i \(-0.219363\pi\)
0.771788 + 0.635880i \(0.219363\pi\)
\(968\) −2.00000 −0.0642824
\(969\) 0 0
\(970\) 7.00000 0.224756
\(971\) −9.00000 −0.288824 −0.144412 0.989518i \(-0.546129\pi\)
−0.144412 + 0.989518i \(0.546129\pi\)
\(972\) 0 0
\(973\) 3.00000 0.0961756
\(974\) 34.0000 1.08943
\(975\) 0 0
\(976\) −15.0000 −0.480138
\(977\) −35.0000 −1.11975 −0.559875 0.828577i \(-0.689151\pi\)
−0.559875 + 0.828577i \(0.689151\pi\)
\(978\) 0 0
\(979\) −54.0000 −1.72585
\(980\) 6.00000 0.191663
\(981\) 0 0
\(982\) 20.0000 0.638226
\(983\) 27.0000 0.861166 0.430583 0.902551i \(-0.358308\pi\)
0.430583 + 0.902551i \(0.358308\pi\)
\(984\) 0 0
\(985\) −18.0000 −0.573528
\(986\) −9.00000 −0.286618
\(987\) 0 0
\(988\) 0 0
\(989\) 2.00000 0.0635963
\(990\) 0 0
\(991\) 33.0000 1.04828 0.524140 0.851632i \(-0.324387\pi\)
0.524140 + 0.851632i \(0.324387\pi\)
\(992\) 3.00000 0.0952501
\(993\) 0 0
\(994\) 2.00000 0.0634361
\(995\) −8.00000 −0.253617
\(996\) 0 0
\(997\) −12.0000 −0.380044 −0.190022 0.981780i \(-0.560856\pi\)
−0.190022 + 0.981780i \(0.560856\pi\)
\(998\) −22.0000 −0.696398
\(999\) 0 0
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 3330.2.a.p.1.1 1
3.2 odd 2 370.2.a.c.1.1 1
12.11 even 2 2960.2.a.c.1.1 1
15.2 even 4 1850.2.b.c.149.1 2
15.8 even 4 1850.2.b.c.149.2 2
15.14 odd 2 1850.2.a.i.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.a.c.1.1 1 3.2 odd 2
1850.2.a.i.1.1 1 15.14 odd 2
1850.2.b.c.149.1 2 15.2 even 4
1850.2.b.c.149.2 2 15.8 even 4
2960.2.a.c.1.1 1 12.11 even 2
3330.2.a.p.1.1 1 1.1 even 1 trivial