Properties

Label 3330.2.a.i.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: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

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

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{8} -1.00000 q^{10} -2.00000 q^{11} -2.00000 q^{13} +1.00000 q^{16} +2.00000 q^{17} -8.00000 q^{19} +1.00000 q^{20} +2.00000 q^{22} +4.00000 q^{23} +1.00000 q^{25} +2.00000 q^{26} +10.0000 q^{29} +2.00000 q^{31} -1.00000 q^{32} -2.00000 q^{34} +1.00000 q^{37} +8.00000 q^{38} -1.00000 q^{40} -8.00000 q^{41} -6.00000 q^{43} -2.00000 q^{44} -4.00000 q^{46} -6.00000 q^{47} -7.00000 q^{49} -1.00000 q^{50} -2.00000 q^{52} -2.00000 q^{55} -10.0000 q^{58} +4.00000 q^{59} -4.00000 q^{61} -2.00000 q^{62} +1.00000 q^{64} -2.00000 q^{65} -16.0000 q^{67} +2.00000 q^{68} -4.00000 q^{71} +6.00000 q^{73} -1.00000 q^{74} -8.00000 q^{76} +10.0000 q^{79} +1.00000 q^{80} +8.00000 q^{82} -16.0000 q^{83} +2.00000 q^{85} +6.00000 q^{86} +2.00000 q^{88} -2.00000 q^{89} +4.00000 q^{92} +6.00000 q^{94} -8.00000 q^{95} -8.00000 q^{97} +7.00000 q^{98} +O(q^{100})\)

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
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) −1.00000 −0.316228
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) −8.00000 −1.83533 −0.917663 0.397360i \(-0.869927\pi\)
−0.917663 + 0.397360i \(0.869927\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) 2.00000 0.426401
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 2.00000 0.392232
\(27\) 0 0
\(28\) 0 0
\(29\) 10.0000 1.85695 0.928477 0.371391i \(-0.121119\pi\)
0.928477 + 0.371391i \(0.121119\pi\)
\(30\) 0 0
\(31\) 2.00000 0.359211 0.179605 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −2.00000 −0.342997
\(35\) 0 0
\(36\) 0 0
\(37\) 1.00000 0.164399
\(38\) 8.00000 1.29777
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) −8.00000 −1.24939 −0.624695 0.780869i \(-0.714777\pi\)
−0.624695 + 0.780869i \(0.714777\pi\)
\(42\) 0 0
\(43\) −6.00000 −0.914991 −0.457496 0.889212i \(-0.651253\pi\)
−0.457496 + 0.889212i \(0.651253\pi\)
\(44\) −2.00000 −0.301511
\(45\) 0 0
\(46\) −4.00000 −0.589768
\(47\) −6.00000 −0.875190 −0.437595 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) −2.00000 −0.277350
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) −2.00000 −0.269680
\(56\) 0 0
\(57\) 0 0
\(58\) −10.0000 −1.31306
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) −4.00000 −0.512148 −0.256074 0.966657i \(-0.582429\pi\)
−0.256074 + 0.966657i \(0.582429\pi\)
\(62\) −2.00000 −0.254000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −2.00000 −0.248069
\(66\) 0 0
\(67\) −16.0000 −1.95471 −0.977356 0.211604i \(-0.932131\pi\)
−0.977356 + 0.211604i \(0.932131\pi\)
\(68\) 2.00000 0.242536
\(69\) 0 0
\(70\) 0 0
\(71\) −4.00000 −0.474713 −0.237356 0.971423i \(-0.576281\pi\)
−0.237356 + 0.971423i \(0.576281\pi\)
\(72\) 0 0
\(73\) 6.00000 0.702247 0.351123 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) −1.00000 −0.116248
\(75\) 0 0
\(76\) −8.00000 −0.917663
\(77\) 0 0
\(78\) 0 0
\(79\) 10.0000 1.12509 0.562544 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) 1.00000 0.111803
\(81\) 0 0
\(82\) 8.00000 0.883452
\(83\) −16.0000 −1.75623 −0.878114 0.478451i \(-0.841198\pi\)
−0.878114 + 0.478451i \(0.841198\pi\)
\(84\) 0 0
\(85\) 2.00000 0.216930
\(86\) 6.00000 0.646997
\(87\) 0 0
\(88\) 2.00000 0.213201
\(89\) −2.00000 −0.212000 −0.106000 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 4.00000 0.417029
\(93\) 0 0
\(94\) 6.00000 0.618853
\(95\) −8.00000 −0.820783
\(96\) 0 0
\(97\) −8.00000 −0.812277 −0.406138 0.913812i \(-0.633125\pi\)
−0.406138 + 0.913812i \(0.633125\pi\)
\(98\) 7.00000 0.707107
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 18.0000 1.79107 0.895533 0.444994i \(-0.146794\pi\)
0.895533 + 0.444994i \(0.146794\pi\)
\(102\) 0 0
\(103\) −4.00000 −0.394132 −0.197066 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) 0 0
\(107\) 8.00000 0.773389 0.386695 0.922208i \(-0.373617\pi\)
0.386695 + 0.922208i \(0.373617\pi\)
\(108\) 0 0
\(109\) −4.00000 −0.383131 −0.191565 0.981480i \(-0.561356\pi\)
−0.191565 + 0.981480i \(0.561356\pi\)
\(110\) 2.00000 0.190693
\(111\) 0 0
\(112\) 0 0
\(113\) −2.00000 −0.188144 −0.0940721 0.995565i \(-0.529988\pi\)
−0.0940721 + 0.995565i \(0.529988\pi\)
\(114\) 0 0
\(115\) 4.00000 0.373002
\(116\) 10.0000 0.928477
\(117\) 0 0
\(118\) −4.00000 −0.368230
\(119\) 0 0
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 4.00000 0.362143
\(123\) 0 0
\(124\) 2.00000 0.179605
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 2.00000 0.175412
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 16.0000 1.38219
\(135\) 0 0
\(136\) −2.00000 −0.171499
\(137\) −22.0000 −1.87959 −0.939793 0.341743i \(-0.888983\pi\)
−0.939793 + 0.341743i \(0.888983\pi\)
\(138\) 0 0
\(139\) −12.0000 −1.01783 −0.508913 0.860818i \(-0.669953\pi\)
−0.508913 + 0.860818i \(0.669953\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 4.00000 0.335673
\(143\) 4.00000 0.334497
\(144\) 0 0
\(145\) 10.0000 0.830455
\(146\) −6.00000 −0.496564
\(147\) 0 0
\(148\) 1.00000 0.0821995
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 8.00000 0.648886
\(153\) 0 0
\(154\) 0 0
\(155\) 2.00000 0.160644
\(156\) 0 0
\(157\) −2.00000 −0.159617 −0.0798087 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) −10.0000 −0.795557
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) 0 0
\(162\) 0 0
\(163\) −18.0000 −1.40987 −0.704934 0.709273i \(-0.749024\pi\)
−0.704934 + 0.709273i \(0.749024\pi\)
\(164\) −8.00000 −0.624695
\(165\) 0 0
\(166\) 16.0000 1.24184
\(167\) −4.00000 −0.309529 −0.154765 0.987951i \(-0.549462\pi\)
−0.154765 + 0.987951i \(0.549462\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) −2.00000 −0.153393
\(171\) 0 0
\(172\) −6.00000 −0.457496
\(173\) −8.00000 −0.608229 −0.304114 0.952636i \(-0.598361\pi\)
−0.304114 + 0.952636i \(0.598361\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −2.00000 −0.150756
\(177\) 0 0
\(178\) 2.00000 0.149906
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 14.0000 1.04061 0.520306 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −4.00000 −0.294884
\(185\) 1.00000 0.0735215
\(186\) 0 0
\(187\) −4.00000 −0.292509
\(188\) −6.00000 −0.437595
\(189\) 0 0
\(190\) 8.00000 0.580381
\(191\) 8.00000 0.578860 0.289430 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) 0 0
\(193\) −4.00000 −0.287926 −0.143963 0.989583i \(-0.545985\pi\)
−0.143963 + 0.989583i \(0.545985\pi\)
\(194\) 8.00000 0.574367
\(195\) 0 0
\(196\) −7.00000 −0.500000
\(197\) 16.0000 1.13995 0.569976 0.821661i \(-0.306952\pi\)
0.569976 + 0.821661i \(0.306952\pi\)
\(198\) 0 0
\(199\) −26.0000 −1.84309 −0.921546 0.388270i \(-0.873073\pi\)
−0.921546 + 0.388270i \(0.873073\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 0 0
\(202\) −18.0000 −1.26648
\(203\) 0 0
\(204\) 0 0
\(205\) −8.00000 −0.558744
\(206\) 4.00000 0.278693
\(207\) 0 0
\(208\) −2.00000 −0.138675
\(209\) 16.0000 1.10674
\(210\) 0 0
\(211\) 20.0000 1.37686 0.688428 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) −8.00000 −0.546869
\(215\) −6.00000 −0.409197
\(216\) 0 0
\(217\) 0 0
\(218\) 4.00000 0.270914
\(219\) 0 0
\(220\) −2.00000 −0.134840
\(221\) −4.00000 −0.269069
\(222\) 0 0
\(223\) 24.0000 1.60716 0.803579 0.595198i \(-0.202926\pi\)
0.803579 + 0.595198i \(0.202926\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 2.00000 0.133038
\(227\) −28.0000 −1.85843 −0.929213 0.369546i \(-0.879513\pi\)
−0.929213 + 0.369546i \(0.879513\pi\)
\(228\) 0 0
\(229\) 26.0000 1.71813 0.859064 0.511868i \(-0.171046\pi\)
0.859064 + 0.511868i \(0.171046\pi\)
\(230\) −4.00000 −0.263752
\(231\) 0 0
\(232\) −10.0000 −0.656532
\(233\) 10.0000 0.655122 0.327561 0.944830i \(-0.393773\pi\)
0.327561 + 0.944830i \(0.393773\pi\)
\(234\) 0 0
\(235\) −6.00000 −0.391397
\(236\) 4.00000 0.260378
\(237\) 0 0
\(238\) 0 0
\(239\) −16.0000 −1.03495 −0.517477 0.855697i \(-0.673129\pi\)
−0.517477 + 0.855697i \(0.673129\pi\)
\(240\) 0 0
\(241\) −2.00000 −0.128831 −0.0644157 0.997923i \(-0.520518\pi\)
−0.0644157 + 0.997923i \(0.520518\pi\)
\(242\) 7.00000 0.449977
\(243\) 0 0
\(244\) −4.00000 −0.256074
\(245\) −7.00000 −0.447214
\(246\) 0 0
\(247\) 16.0000 1.01806
\(248\) −2.00000 −0.127000
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 0 0
\(253\) −8.00000 −0.502956
\(254\) 8.00000 0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −22.0000 −1.37232 −0.686161 0.727450i \(-0.740706\pi\)
−0.686161 + 0.727450i \(0.740706\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −2.00000 −0.124035
\(261\) 0 0
\(262\) −12.0000 −0.741362
\(263\) −10.0000 −0.616626 −0.308313 0.951285i \(-0.599764\pi\)
−0.308313 + 0.951285i \(0.599764\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −16.0000 −0.977356
\(269\) 10.0000 0.609711 0.304855 0.952399i \(-0.401392\pi\)
0.304855 + 0.952399i \(0.401392\pi\)
\(270\) 0 0
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) 2.00000 0.121268
\(273\) 0 0
\(274\) 22.0000 1.32907
\(275\) −2.00000 −0.120605
\(276\) 0 0
\(277\) −14.0000 −0.841178 −0.420589 0.907251i \(-0.638177\pi\)
−0.420589 + 0.907251i \(0.638177\pi\)
\(278\) 12.0000 0.719712
\(279\) 0 0
\(280\) 0 0
\(281\) −26.0000 −1.55103 −0.775515 0.631329i \(-0.782510\pi\)
−0.775515 + 0.631329i \(0.782510\pi\)
\(282\) 0 0
\(283\) 18.0000 1.06999 0.534994 0.844856i \(-0.320314\pi\)
0.534994 + 0.844856i \(0.320314\pi\)
\(284\) −4.00000 −0.237356
\(285\) 0 0
\(286\) −4.00000 −0.236525
\(287\) 0 0
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) −10.0000 −0.587220
\(291\) 0 0
\(292\) 6.00000 0.351123
\(293\) −12.0000 −0.701047 −0.350524 0.936554i \(-0.613996\pi\)
−0.350524 + 0.936554i \(0.613996\pi\)
\(294\) 0 0
\(295\) 4.00000 0.232889
\(296\) −1.00000 −0.0581238
\(297\) 0 0
\(298\) −6.00000 −0.347571
\(299\) −8.00000 −0.462652
\(300\) 0 0
\(301\) 0 0
\(302\) 8.00000 0.460348
\(303\) 0 0
\(304\) −8.00000 −0.458831
\(305\) −4.00000 −0.229039
\(306\) 0 0
\(307\) 12.0000 0.684876 0.342438 0.939540i \(-0.388747\pi\)
0.342438 + 0.939540i \(0.388747\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −2.00000 −0.113592
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) −28.0000 −1.58265 −0.791327 0.611393i \(-0.790609\pi\)
−0.791327 + 0.611393i \(0.790609\pi\)
\(314\) 2.00000 0.112867
\(315\) 0 0
\(316\) 10.0000 0.562544
\(317\) −24.0000 −1.34797 −0.673987 0.738743i \(-0.735420\pi\)
−0.673987 + 0.738743i \(0.735420\pi\)
\(318\) 0 0
\(319\) −20.0000 −1.11979
\(320\) 1.00000 0.0559017
\(321\) 0 0
\(322\) 0 0
\(323\) −16.0000 −0.890264
\(324\) 0 0
\(325\) −2.00000 −0.110940
\(326\) 18.0000 0.996928
\(327\) 0 0
\(328\) 8.00000 0.441726
\(329\) 0 0
\(330\) 0 0
\(331\) −12.0000 −0.659580 −0.329790 0.944054i \(-0.606978\pi\)
−0.329790 + 0.944054i \(0.606978\pi\)
\(332\) −16.0000 −0.878114
\(333\) 0 0
\(334\) 4.00000 0.218870
\(335\) −16.0000 −0.874173
\(336\) 0 0
\(337\) −34.0000 −1.85210 −0.926049 0.377403i \(-0.876817\pi\)
−0.926049 + 0.377403i \(0.876817\pi\)
\(338\) 9.00000 0.489535
\(339\) 0 0
\(340\) 2.00000 0.108465
\(341\) −4.00000 −0.216612
\(342\) 0 0
\(343\) 0 0
\(344\) 6.00000 0.323498
\(345\) 0 0
\(346\) 8.00000 0.430083
\(347\) 12.0000 0.644194 0.322097 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) 0 0
\(349\) 26.0000 1.39175 0.695874 0.718164i \(-0.255017\pi\)
0.695874 + 0.718164i \(0.255017\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 2.00000 0.106600
\(353\) 14.0000 0.745145 0.372572 0.928003i \(-0.378476\pi\)
0.372572 + 0.928003i \(0.378476\pi\)
\(354\) 0 0
\(355\) −4.00000 −0.212298
\(356\) −2.00000 −0.106000
\(357\) 0 0
\(358\) 0 0
\(359\) −32.0000 −1.68890 −0.844448 0.535638i \(-0.820071\pi\)
−0.844448 + 0.535638i \(0.820071\pi\)
\(360\) 0 0
\(361\) 45.0000 2.36842
\(362\) −14.0000 −0.735824
\(363\) 0 0
\(364\) 0 0
\(365\) 6.00000 0.314054
\(366\) 0 0
\(367\) −16.0000 −0.835193 −0.417597 0.908633i \(-0.637127\pi\)
−0.417597 + 0.908633i \(0.637127\pi\)
\(368\) 4.00000 0.208514
\(369\) 0 0
\(370\) −1.00000 −0.0519875
\(371\) 0 0
\(372\) 0 0
\(373\) 10.0000 0.517780 0.258890 0.965907i \(-0.416643\pi\)
0.258890 + 0.965907i \(0.416643\pi\)
\(374\) 4.00000 0.206835
\(375\) 0 0
\(376\) 6.00000 0.309426
\(377\) −20.0000 −1.03005
\(378\) 0 0
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) −8.00000 −0.410391
\(381\) 0 0
\(382\) −8.00000 −0.409316
\(383\) −8.00000 −0.408781 −0.204390 0.978889i \(-0.565521\pi\)
−0.204390 + 0.978889i \(0.565521\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 4.00000 0.203595
\(387\) 0 0
\(388\) −8.00000 −0.406138
\(389\) −10.0000 −0.507020 −0.253510 0.967333i \(-0.581585\pi\)
−0.253510 + 0.967333i \(0.581585\pi\)
\(390\) 0 0
\(391\) 8.00000 0.404577
\(392\) 7.00000 0.353553
\(393\) 0 0
\(394\) −16.0000 −0.806068
\(395\) 10.0000 0.503155
\(396\) 0 0
\(397\) 14.0000 0.702640 0.351320 0.936255i \(-0.385733\pi\)
0.351320 + 0.936255i \(0.385733\pi\)
\(398\) 26.0000 1.30326
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −14.0000 −0.699127 −0.349563 0.936913i \(-0.613670\pi\)
−0.349563 + 0.936913i \(0.613670\pi\)
\(402\) 0 0
\(403\) −4.00000 −0.199254
\(404\) 18.0000 0.895533
\(405\) 0 0
\(406\) 0 0
\(407\) −2.00000 −0.0991363
\(408\) 0 0
\(409\) 10.0000 0.494468 0.247234 0.968956i \(-0.420478\pi\)
0.247234 + 0.968956i \(0.420478\pi\)
\(410\) 8.00000 0.395092
\(411\) 0 0
\(412\) −4.00000 −0.197066
\(413\) 0 0
\(414\) 0 0
\(415\) −16.0000 −0.785409
\(416\) 2.00000 0.0980581
\(417\) 0 0
\(418\) −16.0000 −0.782586
\(419\) 2.00000 0.0977064 0.0488532 0.998806i \(-0.484443\pi\)
0.0488532 + 0.998806i \(0.484443\pi\)
\(420\) 0 0
\(421\) −32.0000 −1.55958 −0.779792 0.626038i \(-0.784675\pi\)
−0.779792 + 0.626038i \(0.784675\pi\)
\(422\) −20.0000 −0.973585
\(423\) 0 0
\(424\) 0 0
\(425\) 2.00000 0.0970143
\(426\) 0 0
\(427\) 0 0
\(428\) 8.00000 0.386695
\(429\) 0 0
\(430\) 6.00000 0.289346
\(431\) −24.0000 −1.15604 −0.578020 0.816023i \(-0.696174\pi\)
−0.578020 + 0.816023i \(0.696174\pi\)
\(432\) 0 0
\(433\) 14.0000 0.672797 0.336399 0.941720i \(-0.390791\pi\)
0.336399 + 0.941720i \(0.390791\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −4.00000 −0.191565
\(437\) −32.0000 −1.53077
\(438\) 0 0
\(439\) 6.00000 0.286364 0.143182 0.989696i \(-0.454267\pi\)
0.143182 + 0.989696i \(0.454267\pi\)
\(440\) 2.00000 0.0953463
\(441\) 0 0
\(442\) 4.00000 0.190261
\(443\) 24.0000 1.14027 0.570137 0.821549i \(-0.306890\pi\)
0.570137 + 0.821549i \(0.306890\pi\)
\(444\) 0 0
\(445\) −2.00000 −0.0948091
\(446\) −24.0000 −1.13643
\(447\) 0 0
\(448\) 0 0
\(449\) 30.0000 1.41579 0.707894 0.706319i \(-0.249646\pi\)
0.707894 + 0.706319i \(0.249646\pi\)
\(450\) 0 0
\(451\) 16.0000 0.753411
\(452\) −2.00000 −0.0940721
\(453\) 0 0
\(454\) 28.0000 1.31411
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(458\) −26.0000 −1.21490
\(459\) 0 0
\(460\) 4.00000 0.186501
\(461\) 18.0000 0.838344 0.419172 0.907907i \(-0.362320\pi\)
0.419172 + 0.907907i \(0.362320\pi\)
\(462\) 0 0
\(463\) 36.0000 1.67306 0.836531 0.547920i \(-0.184580\pi\)
0.836531 + 0.547920i \(0.184580\pi\)
\(464\) 10.0000 0.464238
\(465\) 0 0
\(466\) −10.0000 −0.463241
\(467\) −36.0000 −1.66588 −0.832941 0.553362i \(-0.813345\pi\)
−0.832941 + 0.553362i \(0.813345\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 6.00000 0.276759
\(471\) 0 0
\(472\) −4.00000 −0.184115
\(473\) 12.0000 0.551761
\(474\) 0 0
\(475\) −8.00000 −0.367065
\(476\) 0 0
\(477\) 0 0
\(478\) 16.0000 0.731823
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 0 0
\(481\) −2.00000 −0.0911922
\(482\) 2.00000 0.0910975
\(483\) 0 0
\(484\) −7.00000 −0.318182
\(485\) −8.00000 −0.363261
\(486\) 0 0
\(487\) −12.0000 −0.543772 −0.271886 0.962329i \(-0.587647\pi\)
−0.271886 + 0.962329i \(0.587647\pi\)
\(488\) 4.00000 0.181071
\(489\) 0 0
\(490\) 7.00000 0.316228
\(491\) −26.0000 −1.17336 −0.586682 0.809818i \(-0.699566\pi\)
−0.586682 + 0.809818i \(0.699566\pi\)
\(492\) 0 0
\(493\) 20.0000 0.900755
\(494\) −16.0000 −0.719874
\(495\) 0 0
\(496\) 2.00000 0.0898027
\(497\) 0 0
\(498\) 0 0
\(499\) −20.0000 −0.895323 −0.447661 0.894203i \(-0.647743\pi\)
−0.447661 + 0.894203i \(0.647743\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) −12.0000 −0.535586
\(503\) 12.0000 0.535054 0.267527 0.963550i \(-0.413794\pi\)
0.267527 + 0.963550i \(0.413794\pi\)
\(504\) 0 0
\(505\) 18.0000 0.800989
\(506\) 8.00000 0.355643
\(507\) 0 0
\(508\) −8.00000 −0.354943
\(509\) 22.0000 0.975133 0.487566 0.873086i \(-0.337885\pi\)
0.487566 + 0.873086i \(0.337885\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 22.0000 0.970378
\(515\) −4.00000 −0.176261
\(516\) 0 0
\(517\) 12.0000 0.527759
\(518\) 0 0
\(519\) 0 0
\(520\) 2.00000 0.0877058
\(521\) 36.0000 1.57719 0.788594 0.614914i \(-0.210809\pi\)
0.788594 + 0.614914i \(0.210809\pi\)
\(522\) 0 0
\(523\) 34.0000 1.48672 0.743358 0.668894i \(-0.233232\pi\)
0.743358 + 0.668894i \(0.233232\pi\)
\(524\) 12.0000 0.524222
\(525\) 0 0
\(526\) 10.0000 0.436021
\(527\) 4.00000 0.174243
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 16.0000 0.693037
\(534\) 0 0
\(535\) 8.00000 0.345870
\(536\) 16.0000 0.691095
\(537\) 0 0
\(538\) −10.0000 −0.431131
\(539\) 14.0000 0.603023
\(540\) 0 0
\(541\) 24.0000 1.03184 0.515920 0.856637i \(-0.327450\pi\)
0.515920 + 0.856637i \(0.327450\pi\)
\(542\) −8.00000 −0.343629
\(543\) 0 0
\(544\) −2.00000 −0.0857493
\(545\) −4.00000 −0.171341
\(546\) 0 0
\(547\) 26.0000 1.11168 0.555840 0.831289i \(-0.312397\pi\)
0.555840 + 0.831289i \(0.312397\pi\)
\(548\) −22.0000 −0.939793
\(549\) 0 0
\(550\) 2.00000 0.0852803
\(551\) −80.0000 −3.40811
\(552\) 0 0
\(553\) 0 0
\(554\) 14.0000 0.594803
\(555\) 0 0
\(556\) −12.0000 −0.508913
\(557\) 42.0000 1.77960 0.889799 0.456354i \(-0.150845\pi\)
0.889799 + 0.456354i \(0.150845\pi\)
\(558\) 0 0
\(559\) 12.0000 0.507546
\(560\) 0 0
\(561\) 0 0
\(562\) 26.0000 1.09674
\(563\) 12.0000 0.505740 0.252870 0.967500i \(-0.418626\pi\)
0.252870 + 0.967500i \(0.418626\pi\)
\(564\) 0 0
\(565\) −2.00000 −0.0841406
\(566\) −18.0000 −0.756596
\(567\) 0 0
\(568\) 4.00000 0.167836
\(569\) 14.0000 0.586911 0.293455 0.955973i \(-0.405195\pi\)
0.293455 + 0.955973i \(0.405195\pi\)
\(570\) 0 0
\(571\) −12.0000 −0.502184 −0.251092 0.967963i \(-0.580790\pi\)
−0.251092 + 0.967963i \(0.580790\pi\)
\(572\) 4.00000 0.167248
\(573\) 0 0
\(574\) 0 0
\(575\) 4.00000 0.166812
\(576\) 0 0
\(577\) −32.0000 −1.33218 −0.666089 0.745873i \(-0.732033\pi\)
−0.666089 + 0.745873i \(0.732033\pi\)
\(578\) 13.0000 0.540729
\(579\) 0 0
\(580\) 10.0000 0.415227
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) −6.00000 −0.248282
\(585\) 0 0
\(586\) 12.0000 0.495715
\(587\) −36.0000 −1.48588 −0.742940 0.669359i \(-0.766569\pi\)
−0.742940 + 0.669359i \(0.766569\pi\)
\(588\) 0 0
\(589\) −16.0000 −0.659269
\(590\) −4.00000 −0.164677
\(591\) 0 0
\(592\) 1.00000 0.0410997
\(593\) 22.0000 0.903432 0.451716 0.892162i \(-0.350812\pi\)
0.451716 + 0.892162i \(0.350812\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 6.00000 0.245770
\(597\) 0 0
\(598\) 8.00000 0.327144
\(599\) −24.0000 −0.980613 −0.490307 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 0 0
\(601\) 26.0000 1.06056 0.530281 0.847822i \(-0.322086\pi\)
0.530281 + 0.847822i \(0.322086\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −8.00000 −0.325515
\(605\) −7.00000 −0.284590
\(606\) 0 0
\(607\) 28.0000 1.13648 0.568242 0.822861i \(-0.307624\pi\)
0.568242 + 0.822861i \(0.307624\pi\)
\(608\) 8.00000 0.324443
\(609\) 0 0
\(610\) 4.00000 0.161955
\(611\) 12.0000 0.485468
\(612\) 0 0
\(613\) −14.0000 −0.565455 −0.282727 0.959200i \(-0.591239\pi\)
−0.282727 + 0.959200i \(0.591239\pi\)
\(614\) −12.0000 −0.484281
\(615\) 0 0
\(616\) 0 0
\(617\) −30.0000 −1.20775 −0.603877 0.797077i \(-0.706378\pi\)
−0.603877 + 0.797077i \(0.706378\pi\)
\(618\) 0 0
\(619\) −28.0000 −1.12542 −0.562708 0.826656i \(-0.690240\pi\)
−0.562708 + 0.826656i \(0.690240\pi\)
\(620\) 2.00000 0.0803219
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 28.0000 1.11911
\(627\) 0 0
\(628\) −2.00000 −0.0798087
\(629\) 2.00000 0.0797452
\(630\) 0 0
\(631\) −46.0000 −1.83123 −0.915616 0.402055i \(-0.868296\pi\)
−0.915616 + 0.402055i \(0.868296\pi\)
\(632\) −10.0000 −0.397779
\(633\) 0 0
\(634\) 24.0000 0.953162
\(635\) −8.00000 −0.317470
\(636\) 0 0
\(637\) 14.0000 0.554700
\(638\) 20.0000 0.791808
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) 8.00000 0.315981 0.157991 0.987441i \(-0.449498\pi\)
0.157991 + 0.987441i \(0.449498\pi\)
\(642\) 0 0
\(643\) 14.0000 0.552106 0.276053 0.961142i \(-0.410973\pi\)
0.276053 + 0.961142i \(0.410973\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 16.0000 0.629512
\(647\) −8.00000 −0.314512 −0.157256 0.987558i \(-0.550265\pi\)
−0.157256 + 0.987558i \(0.550265\pi\)
\(648\) 0 0
\(649\) −8.00000 −0.314027
\(650\) 2.00000 0.0784465
\(651\) 0 0
\(652\) −18.0000 −0.704934
\(653\) −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
\(656\) −8.00000 −0.312348
\(657\) 0 0
\(658\) 0 0
\(659\) 30.0000 1.16863 0.584317 0.811525i \(-0.301362\pi\)
0.584317 + 0.811525i \(0.301362\pi\)
\(660\) 0 0
\(661\) 16.0000 0.622328 0.311164 0.950356i \(-0.399281\pi\)
0.311164 + 0.950356i \(0.399281\pi\)
\(662\) 12.0000 0.466393
\(663\) 0 0
\(664\) 16.0000 0.620920
\(665\) 0 0
\(666\) 0 0
\(667\) 40.0000 1.54881
\(668\) −4.00000 −0.154765
\(669\) 0 0
\(670\) 16.0000 0.618134
\(671\) 8.00000 0.308837
\(672\) 0 0
\(673\) −38.0000 −1.46479 −0.732396 0.680879i \(-0.761598\pi\)
−0.732396 + 0.680879i \(0.761598\pi\)
\(674\) 34.0000 1.30963
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) 12.0000 0.461197 0.230599 0.973049i \(-0.425932\pi\)
0.230599 + 0.973049i \(0.425932\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −2.00000 −0.0766965
\(681\) 0 0
\(682\) 4.00000 0.153168
\(683\) 4.00000 0.153056 0.0765279 0.997067i \(-0.475617\pi\)
0.0765279 + 0.997067i \(0.475617\pi\)
\(684\) 0 0
\(685\) −22.0000 −0.840577
\(686\) 0 0
\(687\) 0 0
\(688\) −6.00000 −0.228748
\(689\) 0 0
\(690\) 0 0
\(691\) 28.0000 1.06517 0.532585 0.846376i \(-0.321221\pi\)
0.532585 + 0.846376i \(0.321221\pi\)
\(692\) −8.00000 −0.304114
\(693\) 0 0
\(694\) −12.0000 −0.455514
\(695\) −12.0000 −0.455186
\(696\) 0 0
\(697\) −16.0000 −0.606043
\(698\) −26.0000 −0.984115
\(699\) 0 0
\(700\) 0 0
\(701\) 50.0000 1.88847 0.944237 0.329267i \(-0.106802\pi\)
0.944237 + 0.329267i \(0.106802\pi\)
\(702\) 0 0
\(703\) −8.00000 −0.301726
\(704\) −2.00000 −0.0753778
\(705\) 0 0
\(706\) −14.0000 −0.526897
\(707\) 0 0
\(708\) 0 0
\(709\) −12.0000 −0.450669 −0.225335 0.974281i \(-0.572348\pi\)
−0.225335 + 0.974281i \(0.572348\pi\)
\(710\) 4.00000 0.150117
\(711\) 0 0
\(712\) 2.00000 0.0749532
\(713\) 8.00000 0.299602
\(714\) 0 0
\(715\) 4.00000 0.149592
\(716\) 0 0
\(717\) 0 0
\(718\) 32.0000 1.19423
\(719\) 4.00000 0.149175 0.0745874 0.997214i \(-0.476236\pi\)
0.0745874 + 0.997214i \(0.476236\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −45.0000 −1.67473
\(723\) 0 0
\(724\) 14.0000 0.520306
\(725\) 10.0000 0.371391
\(726\) 0 0
\(727\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −6.00000 −0.222070
\(731\) −12.0000 −0.443836
\(732\) 0 0
\(733\) 38.0000 1.40356 0.701781 0.712393i \(-0.252388\pi\)
0.701781 + 0.712393i \(0.252388\pi\)
\(734\) 16.0000 0.590571
\(735\) 0 0
\(736\) −4.00000 −0.147442
\(737\) 32.0000 1.17874
\(738\) 0 0
\(739\) 12.0000 0.441427 0.220714 0.975339i \(-0.429161\pi\)
0.220714 + 0.975339i \(0.429161\pi\)
\(740\) 1.00000 0.0367607
\(741\) 0 0
\(742\) 0 0
\(743\) −10.0000 −0.366864 −0.183432 0.983032i \(-0.558721\pi\)
−0.183432 + 0.983032i \(0.558721\pi\)
\(744\) 0 0
\(745\) 6.00000 0.219823
\(746\) −10.0000 −0.366126
\(747\) 0 0
\(748\) −4.00000 −0.146254
\(749\) 0 0
\(750\) 0 0
\(751\) 20.0000 0.729810 0.364905 0.931045i \(-0.381101\pi\)
0.364905 + 0.931045i \(0.381101\pi\)
\(752\) −6.00000 −0.218797
\(753\) 0 0
\(754\) 20.0000 0.728357
\(755\) −8.00000 −0.291150
\(756\) 0 0
\(757\) 10.0000 0.363456 0.181728 0.983349i \(-0.441831\pi\)
0.181728 + 0.983349i \(0.441831\pi\)
\(758\) 20.0000 0.726433
\(759\) 0 0
\(760\) 8.00000 0.290191
\(761\) 40.0000 1.45000 0.724999 0.688749i \(-0.241840\pi\)
0.724999 + 0.688749i \(0.241840\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 8.00000 0.289430
\(765\) 0 0
\(766\) 8.00000 0.289052
\(767\) −8.00000 −0.288863
\(768\) 0 0
\(769\) 2.00000 0.0721218 0.0360609 0.999350i \(-0.488519\pi\)
0.0360609 + 0.999350i \(0.488519\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −4.00000 −0.143963
\(773\) 36.0000 1.29483 0.647415 0.762138i \(-0.275850\pi\)
0.647415 + 0.762138i \(0.275850\pi\)
\(774\) 0 0
\(775\) 2.00000 0.0718421
\(776\) 8.00000 0.287183
\(777\) 0 0
\(778\) 10.0000 0.358517
\(779\) 64.0000 2.29304
\(780\) 0 0
\(781\) 8.00000 0.286263
\(782\) −8.00000 −0.286079
\(783\) 0 0
\(784\) −7.00000 −0.250000
\(785\) −2.00000 −0.0713831
\(786\) 0 0
\(787\) 32.0000 1.14068 0.570338 0.821410i \(-0.306812\pi\)
0.570338 + 0.821410i \(0.306812\pi\)
\(788\) 16.0000 0.569976
\(789\) 0 0
\(790\) −10.0000 −0.355784
\(791\) 0 0
\(792\) 0 0
\(793\) 8.00000 0.284088
\(794\) −14.0000 −0.496841
\(795\) 0 0
\(796\) −26.0000 −0.921546
\(797\) −34.0000 −1.20434 −0.602171 0.798367i \(-0.705697\pi\)
−0.602171 + 0.798367i \(0.705697\pi\)
\(798\) 0 0
\(799\) −12.0000 −0.424529
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) 14.0000 0.494357
\(803\) −12.0000 −0.423471
\(804\) 0 0
\(805\) 0 0
\(806\) 4.00000 0.140894
\(807\) 0 0
\(808\) −18.0000 −0.633238
\(809\) 2.00000 0.0703163 0.0351581 0.999382i \(-0.488807\pi\)
0.0351581 + 0.999382i \(0.488807\pi\)
\(810\) 0 0
\(811\) 44.0000 1.54505 0.772524 0.634985i \(-0.218994\pi\)
0.772524 + 0.634985i \(0.218994\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 2.00000 0.0701000
\(815\) −18.0000 −0.630512
\(816\) 0 0
\(817\) 48.0000 1.67931
\(818\) −10.0000 −0.349642
\(819\) 0 0
\(820\) −8.00000 −0.279372
\(821\) 30.0000 1.04701 0.523504 0.852023i \(-0.324625\pi\)
0.523504 + 0.852023i \(0.324625\pi\)
\(822\) 0 0
\(823\) −32.0000 −1.11545 −0.557725 0.830026i \(-0.688326\pi\)
−0.557725 + 0.830026i \(0.688326\pi\)
\(824\) 4.00000 0.139347
\(825\) 0 0
\(826\) 0 0
\(827\) −36.0000 −1.25184 −0.625921 0.779886i \(-0.715277\pi\)
−0.625921 + 0.779886i \(0.715277\pi\)
\(828\) 0 0
\(829\) 20.0000 0.694629 0.347314 0.937749i \(-0.387094\pi\)
0.347314 + 0.937749i \(0.387094\pi\)
\(830\) 16.0000 0.555368
\(831\) 0 0
\(832\) −2.00000 −0.0693375
\(833\) −14.0000 −0.485071
\(834\) 0 0
\(835\) −4.00000 −0.138426
\(836\) 16.0000 0.553372
\(837\) 0 0
\(838\) −2.00000 −0.0690889
\(839\) 12.0000 0.414286 0.207143 0.978311i \(-0.433583\pi\)
0.207143 + 0.978311i \(0.433583\pi\)
\(840\) 0 0
\(841\) 71.0000 2.44828
\(842\) 32.0000 1.10279
\(843\) 0 0
\(844\) 20.0000 0.688428
\(845\) −9.00000 −0.309609
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) −2.00000 −0.0685994
\(851\) 4.00000 0.137118
\(852\) 0 0
\(853\) 6.00000 0.205436 0.102718 0.994711i \(-0.467246\pi\)
0.102718 + 0.994711i \(0.467246\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −8.00000 −0.273434
\(857\) −42.0000 −1.43469 −0.717346 0.696717i \(-0.754643\pi\)
−0.717346 + 0.696717i \(0.754643\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(860\) −6.00000 −0.204598
\(861\) 0 0
\(862\) 24.0000 0.817443
\(863\) −18.0000 −0.612727 −0.306364 0.951915i \(-0.599112\pi\)
−0.306364 + 0.951915i \(0.599112\pi\)
\(864\) 0 0
\(865\) −8.00000 −0.272008
\(866\) −14.0000 −0.475739
\(867\) 0 0
\(868\) 0 0
\(869\) −20.0000 −0.678454
\(870\) 0 0
\(871\) 32.0000 1.08428
\(872\) 4.00000 0.135457
\(873\) 0 0
\(874\) 32.0000 1.08242
\(875\) 0 0
\(876\) 0 0
\(877\) −30.0000 −1.01303 −0.506514 0.862232i \(-0.669066\pi\)
−0.506514 + 0.862232i \(0.669066\pi\)
\(878\) −6.00000 −0.202490
\(879\) 0 0
\(880\) −2.00000 −0.0674200
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) 14.0000 0.471138 0.235569 0.971858i \(-0.424305\pi\)
0.235569 + 0.971858i \(0.424305\pi\)
\(884\) −4.00000 −0.134535
\(885\) 0 0
\(886\) −24.0000 −0.806296
\(887\) 30.0000 1.00730 0.503651 0.863907i \(-0.331990\pi\)
0.503651 + 0.863907i \(0.331990\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 2.00000 0.0670402
\(891\) 0 0
\(892\) 24.0000 0.803579
\(893\) 48.0000 1.60626
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) −30.0000 −1.00111
\(899\) 20.0000 0.667037
\(900\) 0 0
\(901\) 0 0
\(902\) −16.0000 −0.532742
\(903\) 0 0
\(904\) 2.00000 0.0665190
\(905\) 14.0000 0.465376
\(906\) 0 0
\(907\) −54.0000 −1.79304 −0.896520 0.443003i \(-0.853913\pi\)
−0.896520 + 0.443003i \(0.853913\pi\)
\(908\) −28.0000 −0.929213
\(909\) 0 0
\(910\) 0 0
\(911\) −16.0000 −0.530104 −0.265052 0.964234i \(-0.585389\pi\)
−0.265052 + 0.964234i \(0.585389\pi\)
\(912\) 0 0
\(913\) 32.0000 1.05905
\(914\) 0 0
\(915\) 0 0
\(916\) 26.0000 0.859064
\(917\) 0 0
\(918\) 0 0
\(919\) 42.0000 1.38545 0.692726 0.721201i \(-0.256409\pi\)
0.692726 + 0.721201i \(0.256409\pi\)
\(920\) −4.00000 −0.131876
\(921\) 0 0
\(922\) −18.0000 −0.592798
\(923\) 8.00000 0.263323
\(924\) 0 0
\(925\) 1.00000 0.0328798
\(926\) −36.0000 −1.18303
\(927\) 0 0
\(928\) −10.0000 −0.328266
\(929\) 28.0000 0.918650 0.459325 0.888268i \(-0.348091\pi\)
0.459325 + 0.888268i \(0.348091\pi\)
\(930\) 0 0
\(931\) 56.0000 1.83533
\(932\) 10.0000 0.327561
\(933\) 0 0
\(934\) 36.0000 1.17796
\(935\) −4.00000 −0.130814
\(936\) 0 0
\(937\) −34.0000 −1.11073 −0.555366 0.831606i \(-0.687422\pi\)
−0.555366 + 0.831606i \(0.687422\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −6.00000 −0.195698
\(941\) −18.0000 −0.586783 −0.293392 0.955992i \(-0.594784\pi\)
−0.293392 + 0.955992i \(0.594784\pi\)
\(942\) 0 0
\(943\) −32.0000 −1.04206
\(944\) 4.00000 0.130189
\(945\) 0 0
\(946\) −12.0000 −0.390154
\(947\) −44.0000 −1.42981 −0.714904 0.699223i \(-0.753530\pi\)
−0.714904 + 0.699223i \(0.753530\pi\)
\(948\) 0 0
\(949\) −12.0000 −0.389536
\(950\) 8.00000 0.259554
\(951\) 0 0
\(952\) 0 0
\(953\) 26.0000 0.842223 0.421111 0.907009i \(-0.361640\pi\)
0.421111 + 0.907009i \(0.361640\pi\)
\(954\) 0 0
\(955\) 8.00000 0.258874
\(956\) −16.0000 −0.517477
\(957\) 0 0
\(958\) −24.0000 −0.775405
\(959\) 0 0
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 2.00000 0.0644826
\(963\) 0 0
\(964\) −2.00000 −0.0644157
\(965\) −4.00000 −0.128765
\(966\) 0 0
\(967\) 32.0000 1.02905 0.514525 0.857475i \(-0.327968\pi\)
0.514525 + 0.857475i \(0.327968\pi\)
\(968\) 7.00000 0.224989
\(969\) 0 0
\(970\) 8.00000 0.256865
\(971\) 38.0000 1.21948 0.609739 0.792602i \(-0.291274\pi\)
0.609739 + 0.792602i \(0.291274\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 12.0000 0.384505
\(975\) 0 0
\(976\) −4.00000 −0.128037
\(977\) 62.0000 1.98356 0.991778 0.127971i \(-0.0408466\pi\)
0.991778 + 0.127971i \(0.0408466\pi\)
\(978\) 0 0
\(979\) 4.00000 0.127841
\(980\) −7.00000 −0.223607
\(981\) 0 0
\(982\) 26.0000 0.829693
\(983\) −18.0000 −0.574111 −0.287055 0.957914i \(-0.592676\pi\)
−0.287055 + 0.957914i \(0.592676\pi\)
\(984\) 0 0
\(985\) 16.0000 0.509802
\(986\) −20.0000 −0.636930
\(987\) 0 0
\(988\) 16.0000 0.509028
\(989\) −24.0000 −0.763156
\(990\) 0 0
\(991\) −38.0000 −1.20711 −0.603555 0.797321i \(-0.706250\pi\)
−0.603555 + 0.797321i \(0.706250\pi\)
\(992\) −2.00000 −0.0635001
\(993\) 0 0
\(994\) 0 0
\(995\) −26.0000 −0.824255
\(996\) 0 0
\(997\) −6.00000 −0.190022 −0.0950110 0.995476i \(-0.530289\pi\)
−0.0950110 + 0.995476i \(0.530289\pi\)
\(998\) 20.0000 0.633089
\(999\) 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.i.1.1 1
3.2 odd 2 3330.2.a.o.1.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3330.2.a.i.1.1 1 1.1 even 1 trivial
3330.2.a.o.1.1 yes 1 3.2 odd 2