Properties

Label 3381.2.a.f.1.1
Level $3381$
Weight $2$
Character 3381.1
Self dual yes
Analytic conductor $26.997$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3381 = 3 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3381.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.9974209234\)
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 \(\chi\) \(=\) 3381.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -2.00000 q^{4} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -2.00000 q^{4} +1.00000 q^{9} +5.00000 q^{11} +2.00000 q^{12} -4.00000 q^{13} +4.00000 q^{16} -2.00000 q^{17} -3.00000 q^{19} +1.00000 q^{23} -5.00000 q^{25} -1.00000 q^{27} -8.00000 q^{29} +10.0000 q^{31} -5.00000 q^{33} -2.00000 q^{36} +4.00000 q^{37} +4.00000 q^{39} -7.00000 q^{41} +10.0000 q^{43} -10.0000 q^{44} +3.00000 q^{47} -4.00000 q^{48} +2.00000 q^{51} +8.00000 q^{52} +3.00000 q^{53} +3.00000 q^{57} +13.0000 q^{59} +9.00000 q^{61} -8.00000 q^{64} +8.00000 q^{67} +4.00000 q^{68} -1.00000 q^{69} -6.00000 q^{71} -8.00000 q^{73} +5.00000 q^{75} +6.00000 q^{76} +1.00000 q^{81} -18.0000 q^{83} +8.00000 q^{87} -14.0000 q^{89} -2.00000 q^{92} -10.0000 q^{93} -18.0000 q^{97} +5.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(3\) −1.00000 −0.577350
\(4\) −2.00000 −1.00000
\(5\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 5.00000 1.50756 0.753778 0.657129i \(-0.228229\pi\)
0.753778 + 0.657129i \(0.228229\pi\)
\(12\) 2.00000 0.577350
\(13\) −4.00000 −1.10940 −0.554700 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 4.00000 1.00000
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) −3.00000 −0.688247 −0.344124 0.938924i \(-0.611824\pi\)
−0.344124 + 0.938924i \(0.611824\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) −5.00000 −1.00000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −8.00000 −1.48556 −0.742781 0.669534i \(-0.766494\pi\)
−0.742781 + 0.669534i \(0.766494\pi\)
\(30\) 0 0
\(31\) 10.0000 1.79605 0.898027 0.439941i \(-0.145001\pi\)
0.898027 + 0.439941i \(0.145001\pi\)
\(32\) 0 0
\(33\) −5.00000 −0.870388
\(34\) 0 0
\(35\) 0 0
\(36\) −2.00000 −0.333333
\(37\) 4.00000 0.657596 0.328798 0.944400i \(-0.393356\pi\)
0.328798 + 0.944400i \(0.393356\pi\)
\(38\) 0 0
\(39\) 4.00000 0.640513
\(40\) 0 0
\(41\) −7.00000 −1.09322 −0.546608 0.837389i \(-0.684081\pi\)
−0.546608 + 0.837389i \(0.684081\pi\)
\(42\) 0 0
\(43\) 10.0000 1.52499 0.762493 0.646997i \(-0.223975\pi\)
0.762493 + 0.646997i \(0.223975\pi\)
\(44\) −10.0000 −1.50756
\(45\) 0 0
\(46\) 0 0
\(47\) 3.00000 0.437595 0.218797 0.975770i \(-0.429787\pi\)
0.218797 + 0.975770i \(0.429787\pi\)
\(48\) −4.00000 −0.577350
\(49\) 0 0
\(50\) 0 0
\(51\) 2.00000 0.280056
\(52\) 8.00000 1.10940
\(53\) 3.00000 0.412082 0.206041 0.978543i \(-0.433942\pi\)
0.206041 + 0.978543i \(0.433942\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 3.00000 0.397360
\(58\) 0 0
\(59\) 13.0000 1.69246 0.846228 0.532821i \(-0.178868\pi\)
0.846228 + 0.532821i \(0.178868\pi\)
\(60\) 0 0
\(61\) 9.00000 1.15233 0.576166 0.817333i \(-0.304548\pi\)
0.576166 + 0.817333i \(0.304548\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) 4.00000 0.485071
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −6.00000 −0.712069 −0.356034 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 0 0
\(73\) −8.00000 −0.936329 −0.468165 0.883641i \(-0.655085\pi\)
−0.468165 + 0.883641i \(0.655085\pi\)
\(74\) 0 0
\(75\) 5.00000 0.577350
\(76\) 6.00000 0.688247
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −18.0000 −1.97576 −0.987878 0.155230i \(-0.950388\pi\)
−0.987878 + 0.155230i \(0.950388\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 8.00000 0.857690
\(88\) 0 0
\(89\) −14.0000 −1.48400 −0.741999 0.670402i \(-0.766122\pi\)
−0.741999 + 0.670402i \(0.766122\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −2.00000 −0.208514
\(93\) −10.0000 −1.03695
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −18.0000 −1.82762 −0.913812 0.406138i \(-0.866875\pi\)
−0.913812 + 0.406138i \(0.866875\pi\)
\(98\) 0 0
\(99\) 5.00000 0.502519
\(100\) 10.0000 1.00000
\(101\) 5.00000 0.497519 0.248759 0.968565i \(-0.419977\pi\)
0.248759 + 0.968565i \(0.419977\pi\)
\(102\) 0 0
\(103\) −13.0000 −1.28093 −0.640464 0.767988i \(-0.721258\pi\)
−0.640464 + 0.767988i \(0.721258\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 2.00000 0.192450
\(109\) −16.0000 −1.53252 −0.766261 0.642529i \(-0.777885\pi\)
−0.766261 + 0.642529i \(0.777885\pi\)
\(110\) 0 0
\(111\) −4.00000 −0.379663
\(112\) 0 0
\(113\) 10.0000 0.940721 0.470360 0.882474i \(-0.344124\pi\)
0.470360 + 0.882474i \(0.344124\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 16.0000 1.48556
\(117\) −4.00000 −0.369800
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) 0 0
\(123\) 7.00000 0.631169
\(124\) −20.0000 −1.79605
\(125\) 0 0
\(126\) 0 0
\(127\) 1.00000 0.0887357 0.0443678 0.999015i \(-0.485873\pi\)
0.0443678 + 0.999015i \(0.485873\pi\)
\(128\) 0 0
\(129\) −10.0000 −0.880451
\(130\) 0 0
\(131\) −21.0000 −1.83478 −0.917389 0.397991i \(-0.869707\pi\)
−0.917389 + 0.397991i \(0.869707\pi\)
\(132\) 10.0000 0.870388
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −9.00000 −0.768922 −0.384461 0.923141i \(-0.625613\pi\)
−0.384461 + 0.923141i \(0.625613\pi\)
\(138\) 0 0
\(139\) −14.0000 −1.18746 −0.593732 0.804663i \(-0.702346\pi\)
−0.593732 + 0.804663i \(0.702346\pi\)
\(140\) 0 0
\(141\) −3.00000 −0.252646
\(142\) 0 0
\(143\) −20.0000 −1.67248
\(144\) 4.00000 0.333333
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) −8.00000 −0.657596
\(149\) −11.0000 −0.901155 −0.450578 0.892737i \(-0.648782\pi\)
−0.450578 + 0.892737i \(0.648782\pi\)
\(150\) 0 0
\(151\) −19.0000 −1.54620 −0.773099 0.634285i \(-0.781294\pi\)
−0.773099 + 0.634285i \(0.781294\pi\)
\(152\) 0 0
\(153\) −2.00000 −0.161690
\(154\) 0 0
\(155\) 0 0
\(156\) −8.00000 −0.640513
\(157\) −19.0000 −1.51637 −0.758183 0.652042i \(-0.773912\pi\)
−0.758183 + 0.652042i \(0.773912\pi\)
\(158\) 0 0
\(159\) −3.00000 −0.237915
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 1.00000 0.0783260 0.0391630 0.999233i \(-0.487531\pi\)
0.0391630 + 0.999233i \(0.487531\pi\)
\(164\) 14.0000 1.09322
\(165\) 0 0
\(166\) 0 0
\(167\) 19.0000 1.47026 0.735132 0.677924i \(-0.237120\pi\)
0.735132 + 0.677924i \(0.237120\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) −3.00000 −0.229416
\(172\) −20.0000 −1.52499
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 20.0000 1.50756
\(177\) −13.0000 −0.977140
\(178\) 0 0
\(179\) −6.00000 −0.448461 −0.224231 0.974536i \(-0.571987\pi\)
−0.224231 + 0.974536i \(0.571987\pi\)
\(180\) 0 0
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 0 0
\(183\) −9.00000 −0.665299
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −10.0000 −0.731272
\(188\) −6.00000 −0.437595
\(189\) 0 0
\(190\) 0 0
\(191\) −13.0000 −0.940647 −0.470323 0.882494i \(-0.655863\pi\)
−0.470323 + 0.882494i \(0.655863\pi\)
\(192\) 8.00000 0.577350
\(193\) 9.00000 0.647834 0.323917 0.946085i \(-0.395000\pi\)
0.323917 + 0.946085i \(0.395000\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 18.0000 1.28245 0.641223 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) 0 0
\(199\) 1.00000 0.0708881 0.0354441 0.999372i \(-0.488715\pi\)
0.0354441 + 0.999372i \(0.488715\pi\)
\(200\) 0 0
\(201\) −8.00000 −0.564276
\(202\) 0 0
\(203\) 0 0
\(204\) −4.00000 −0.280056
\(205\) 0 0
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) −16.0000 −1.10940
\(209\) −15.0000 −1.03757
\(210\) 0 0
\(211\) −23.0000 −1.58339 −0.791693 0.610920i \(-0.790800\pi\)
−0.791693 + 0.610920i \(0.790800\pi\)
\(212\) −6.00000 −0.412082
\(213\) 6.00000 0.411113
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 8.00000 0.540590
\(220\) 0 0
\(221\) 8.00000 0.538138
\(222\) 0 0
\(223\) −2.00000 −0.133930 −0.0669650 0.997755i \(-0.521332\pi\)
−0.0669650 + 0.997755i \(0.521332\pi\)
\(224\) 0 0
\(225\) −5.00000 −0.333333
\(226\) 0 0
\(227\) 12.0000 0.796468 0.398234 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) −6.00000 −0.397360
\(229\) 13.0000 0.859064 0.429532 0.903052i \(-0.358679\pi\)
0.429532 + 0.903052i \(0.358679\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 16.0000 1.04819 0.524097 0.851658i \(-0.324403\pi\)
0.524097 + 0.851658i \(0.324403\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −26.0000 −1.69246
\(237\) 0 0
\(238\) 0 0
\(239\) −10.0000 −0.646846 −0.323423 0.946254i \(-0.604834\pi\)
−0.323423 + 0.946254i \(0.604834\pi\)
\(240\) 0 0
\(241\) 7.00000 0.450910 0.225455 0.974254i \(-0.427613\pi\)
0.225455 + 0.974254i \(0.427613\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) −18.0000 −1.15233
\(245\) 0 0
\(246\) 0 0
\(247\) 12.0000 0.763542
\(248\) 0 0
\(249\) 18.0000 1.14070
\(250\) 0 0
\(251\) −2.00000 −0.126239 −0.0631194 0.998006i \(-0.520105\pi\)
−0.0631194 + 0.998006i \(0.520105\pi\)
\(252\) 0 0
\(253\) 5.00000 0.314347
\(254\) 0 0
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) 13.0000 0.810918 0.405459 0.914113i \(-0.367112\pi\)
0.405459 + 0.914113i \(0.367112\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −8.00000 −0.495188
\(262\) 0 0
\(263\) −7.00000 −0.431638 −0.215819 0.976433i \(-0.569242\pi\)
−0.215819 + 0.976433i \(0.569242\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 14.0000 0.856786
\(268\) −16.0000 −0.977356
\(269\) −18.0000 −1.09748 −0.548740 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) 0 0
\(271\) −6.00000 −0.364474 −0.182237 0.983255i \(-0.558334\pi\)
−0.182237 + 0.983255i \(0.558334\pi\)
\(272\) −8.00000 −0.485071
\(273\) 0 0
\(274\) 0 0
\(275\) −25.0000 −1.50756
\(276\) 2.00000 0.120386
\(277\) −21.0000 −1.26177 −0.630884 0.775877i \(-0.717308\pi\)
−0.630884 + 0.775877i \(0.717308\pi\)
\(278\) 0 0
\(279\) 10.0000 0.598684
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) 20.0000 1.18888 0.594438 0.804141i \(-0.297374\pi\)
0.594438 + 0.804141i \(0.297374\pi\)
\(284\) 12.0000 0.712069
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 18.0000 1.05518
\(292\) 16.0000 0.936329
\(293\) −22.0000 −1.28525 −0.642627 0.766179i \(-0.722155\pi\)
−0.642627 + 0.766179i \(0.722155\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −5.00000 −0.290129
\(298\) 0 0
\(299\) −4.00000 −0.231326
\(300\) −10.0000 −0.577350
\(301\) 0 0
\(302\) 0 0
\(303\) −5.00000 −0.287242
\(304\) −12.0000 −0.688247
\(305\) 0 0
\(306\) 0 0
\(307\) 22.0000 1.25561 0.627803 0.778372i \(-0.283954\pi\)
0.627803 + 0.778372i \(0.283954\pi\)
\(308\) 0 0
\(309\) 13.0000 0.739544
\(310\) 0 0
\(311\) −27.0000 −1.53103 −0.765515 0.643418i \(-0.777516\pi\)
−0.765515 + 0.643418i \(0.777516\pi\)
\(312\) 0 0
\(313\) 13.0000 0.734803 0.367402 0.930062i \(-0.380247\pi\)
0.367402 + 0.930062i \(0.380247\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 2.00000 0.112331 0.0561656 0.998421i \(-0.482113\pi\)
0.0561656 + 0.998421i \(0.482113\pi\)
\(318\) 0 0
\(319\) −40.0000 −2.23957
\(320\) 0 0
\(321\) −12.0000 −0.669775
\(322\) 0 0
\(323\) 6.00000 0.333849
\(324\) −2.00000 −0.111111
\(325\) 20.0000 1.10940
\(326\) 0 0
\(327\) 16.0000 0.884802
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 1.00000 0.0549650 0.0274825 0.999622i \(-0.491251\pi\)
0.0274825 + 0.999622i \(0.491251\pi\)
\(332\) 36.0000 1.97576
\(333\) 4.00000 0.219199
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −22.0000 −1.19842 −0.599208 0.800593i \(-0.704518\pi\)
−0.599208 + 0.800593i \(0.704518\pi\)
\(338\) 0 0
\(339\) −10.0000 −0.543125
\(340\) 0 0
\(341\) 50.0000 2.70765
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 14.0000 0.751559 0.375780 0.926709i \(-0.377375\pi\)
0.375780 + 0.926709i \(0.377375\pi\)
\(348\) −16.0000 −0.857690
\(349\) −10.0000 −0.535288 −0.267644 0.963518i \(-0.586245\pi\)
−0.267644 + 0.963518i \(0.586245\pi\)
\(350\) 0 0
\(351\) 4.00000 0.213504
\(352\) 0 0
\(353\) 18.0000 0.958043 0.479022 0.877803i \(-0.340992\pi\)
0.479022 + 0.877803i \(0.340992\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 28.0000 1.48400
\(357\) 0 0
\(358\) 0 0
\(359\) 32.0000 1.68890 0.844448 0.535638i \(-0.179929\pi\)
0.844448 + 0.535638i \(0.179929\pi\)
\(360\) 0 0
\(361\) −10.0000 −0.526316
\(362\) 0 0
\(363\) −14.0000 −0.734809
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −23.0000 −1.20059 −0.600295 0.799779i \(-0.704950\pi\)
−0.600295 + 0.799779i \(0.704950\pi\)
\(368\) 4.00000 0.208514
\(369\) −7.00000 −0.364405
\(370\) 0 0
\(371\) 0 0
\(372\) 20.0000 1.03695
\(373\) −4.00000 −0.207112 −0.103556 0.994624i \(-0.533022\pi\)
−0.103556 + 0.994624i \(0.533022\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 32.0000 1.64808
\(378\) 0 0
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 0 0
\(381\) −1.00000 −0.0512316
\(382\) 0 0
\(383\) −18.0000 −0.919757 −0.459879 0.887982i \(-0.652107\pi\)
−0.459879 + 0.887982i \(0.652107\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 10.0000 0.508329
\(388\) 36.0000 1.82762
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) −2.00000 −0.101144
\(392\) 0 0
\(393\) 21.0000 1.05931
\(394\) 0 0
\(395\) 0 0
\(396\) −10.0000 −0.502519
\(397\) 14.0000 0.702640 0.351320 0.936255i \(-0.385733\pi\)
0.351320 + 0.936255i \(0.385733\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −20.0000 −1.00000
\(401\) 25.0000 1.24844 0.624220 0.781248i \(-0.285417\pi\)
0.624220 + 0.781248i \(0.285417\pi\)
\(402\) 0 0
\(403\) −40.0000 −1.99254
\(404\) −10.0000 −0.497519
\(405\) 0 0
\(406\) 0 0
\(407\) 20.0000 0.991363
\(408\) 0 0
\(409\) 12.0000 0.593362 0.296681 0.954977i \(-0.404120\pi\)
0.296681 + 0.954977i \(0.404120\pi\)
\(410\) 0 0
\(411\) 9.00000 0.443937
\(412\) 26.0000 1.28093
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 14.0000 0.685583
\(418\) 0 0
\(419\) −28.0000 −1.36789 −0.683945 0.729534i \(-0.739737\pi\)
−0.683945 + 0.729534i \(0.739737\pi\)
\(420\) 0 0
\(421\) −6.00000 −0.292422 −0.146211 0.989253i \(-0.546708\pi\)
−0.146211 + 0.989253i \(0.546708\pi\)
\(422\) 0 0
\(423\) 3.00000 0.145865
\(424\) 0 0
\(425\) 10.0000 0.485071
\(426\) 0 0
\(427\) 0 0
\(428\) −24.0000 −1.16008
\(429\) 20.0000 0.965609
\(430\) 0 0
\(431\) 5.00000 0.240842 0.120421 0.992723i \(-0.461576\pi\)
0.120421 + 0.992723i \(0.461576\pi\)
\(432\) −4.00000 −0.192450
\(433\) −21.0000 −1.00920 −0.504598 0.863355i \(-0.668359\pi\)
−0.504598 + 0.863355i \(0.668359\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 32.0000 1.53252
\(437\) −3.00000 −0.143509
\(438\) 0 0
\(439\) −8.00000 −0.381819 −0.190910 0.981608i \(-0.561144\pi\)
−0.190910 + 0.981608i \(0.561144\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −28.0000 −1.33032 −0.665160 0.746701i \(-0.731637\pi\)
−0.665160 + 0.746701i \(0.731637\pi\)
\(444\) 8.00000 0.379663
\(445\) 0 0
\(446\) 0 0
\(447\) 11.0000 0.520282
\(448\) 0 0
\(449\) 8.00000 0.377543 0.188772 0.982021i \(-0.439549\pi\)
0.188772 + 0.982021i \(0.439549\pi\)
\(450\) 0 0
\(451\) −35.0000 −1.64809
\(452\) −20.0000 −0.940721
\(453\) 19.0000 0.892698
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −22.0000 −1.02912 −0.514558 0.857455i \(-0.672044\pi\)
−0.514558 + 0.857455i \(0.672044\pi\)
\(458\) 0 0
\(459\) 2.00000 0.0933520
\(460\) 0 0
\(461\) −18.0000 −0.838344 −0.419172 0.907907i \(-0.637680\pi\)
−0.419172 + 0.907907i \(0.637680\pi\)
\(462\) 0 0
\(463\) 13.0000 0.604161 0.302081 0.953282i \(-0.402319\pi\)
0.302081 + 0.953282i \(0.402319\pi\)
\(464\) −32.0000 −1.48556
\(465\) 0 0
\(466\) 0 0
\(467\) 12.0000 0.555294 0.277647 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) 8.00000 0.369800
\(469\) 0 0
\(470\) 0 0
\(471\) 19.0000 0.875474
\(472\) 0 0
\(473\) 50.0000 2.29900
\(474\) 0 0
\(475\) 15.0000 0.688247
\(476\) 0 0
\(477\) 3.00000 0.137361
\(478\) 0 0
\(479\) −32.0000 −1.46212 −0.731059 0.682315i \(-0.760973\pi\)
−0.731059 + 0.682315i \(0.760973\pi\)
\(480\) 0 0
\(481\) −16.0000 −0.729537
\(482\) 0 0
\(483\) 0 0
\(484\) −28.0000 −1.27273
\(485\) 0 0
\(486\) 0 0
\(487\) 40.0000 1.81257 0.906287 0.422664i \(-0.138905\pi\)
0.906287 + 0.422664i \(0.138905\pi\)
\(488\) 0 0
\(489\) −1.00000 −0.0452216
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) −14.0000 −0.631169
\(493\) 16.0000 0.720604
\(494\) 0 0
\(495\) 0 0
\(496\) 40.0000 1.79605
\(497\) 0 0
\(498\) 0 0
\(499\) 28.0000 1.25345 0.626726 0.779240i \(-0.284395\pi\)
0.626726 + 0.779240i \(0.284395\pi\)
\(500\) 0 0
\(501\) −19.0000 −0.848857
\(502\) 0 0
\(503\) −6.00000 −0.267527 −0.133763 0.991013i \(-0.542706\pi\)
−0.133763 + 0.991013i \(0.542706\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −3.00000 −0.133235
\(508\) −2.00000 −0.0887357
\(509\) 1.00000 0.0443242 0.0221621 0.999754i \(-0.492945\pi\)
0.0221621 + 0.999754i \(0.492945\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 3.00000 0.132453
\(514\) 0 0
\(515\) 0 0
\(516\) 20.0000 0.880451
\(517\) 15.0000 0.659699
\(518\) 0 0
\(519\) 6.00000 0.263371
\(520\) 0 0
\(521\) −20.0000 −0.876216 −0.438108 0.898922i \(-0.644351\pi\)
−0.438108 + 0.898922i \(0.644351\pi\)
\(522\) 0 0
\(523\) −7.00000 −0.306089 −0.153044 0.988219i \(-0.548908\pi\)
−0.153044 + 0.988219i \(0.548908\pi\)
\(524\) 42.0000 1.83478
\(525\) 0 0
\(526\) 0 0
\(527\) −20.0000 −0.871214
\(528\) −20.0000 −0.870388
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 13.0000 0.564152
\(532\) 0 0
\(533\) 28.0000 1.21281
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 6.00000 0.258919
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −25.0000 −1.07483 −0.537417 0.843317i \(-0.680600\pi\)
−0.537417 + 0.843317i \(0.680600\pi\)
\(542\) 0 0
\(543\) 2.00000 0.0858282
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −28.0000 −1.19719 −0.598597 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) 18.0000 0.768922
\(549\) 9.00000 0.384111
\(550\) 0 0
\(551\) 24.0000 1.02243
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 28.0000 1.18746
\(557\) 18.0000 0.762684 0.381342 0.924434i \(-0.375462\pi\)
0.381342 + 0.924434i \(0.375462\pi\)
\(558\) 0 0
\(559\) −40.0000 −1.69182
\(560\) 0 0
\(561\) 10.0000 0.422200
\(562\) 0 0
\(563\) 36.0000 1.51722 0.758610 0.651546i \(-0.225879\pi\)
0.758610 + 0.651546i \(0.225879\pi\)
\(564\) 6.00000 0.252646
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −19.0000 −0.796521 −0.398261 0.917272i \(-0.630386\pi\)
−0.398261 + 0.917272i \(0.630386\pi\)
\(570\) 0 0
\(571\) −10.0000 −0.418487 −0.209243 0.977864i \(-0.567100\pi\)
−0.209243 + 0.977864i \(0.567100\pi\)
\(572\) 40.0000 1.67248
\(573\) 13.0000 0.543083
\(574\) 0 0
\(575\) −5.00000 −0.208514
\(576\) −8.00000 −0.333333
\(577\) −38.0000 −1.58196 −0.790980 0.611842i \(-0.790429\pi\)
−0.790980 + 0.611842i \(0.790429\pi\)
\(578\) 0 0
\(579\) −9.00000 −0.374027
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 15.0000 0.621237
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 3.00000 0.123823 0.0619116 0.998082i \(-0.480280\pi\)
0.0619116 + 0.998082i \(0.480280\pi\)
\(588\) 0 0
\(589\) −30.0000 −1.23613
\(590\) 0 0
\(591\) −18.0000 −0.740421
\(592\) 16.0000 0.657596
\(593\) 6.00000 0.246390 0.123195 0.992382i \(-0.460686\pi\)
0.123195 + 0.992382i \(0.460686\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 22.0000 0.901155
\(597\) −1.00000 −0.0409273
\(598\) 0 0
\(599\) 2.00000 0.0817178 0.0408589 0.999165i \(-0.486991\pi\)
0.0408589 + 0.999165i \(0.486991\pi\)
\(600\) 0 0
\(601\) 34.0000 1.38689 0.693444 0.720510i \(-0.256092\pi\)
0.693444 + 0.720510i \(0.256092\pi\)
\(602\) 0 0
\(603\) 8.00000 0.325785
\(604\) 38.0000 1.54620
\(605\) 0 0
\(606\) 0 0
\(607\) −4.00000 −0.162355 −0.0811775 0.996700i \(-0.525868\pi\)
−0.0811775 + 0.996700i \(0.525868\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −12.0000 −0.485468
\(612\) 4.00000 0.161690
\(613\) −24.0000 −0.969351 −0.484675 0.874694i \(-0.661062\pi\)
−0.484675 + 0.874694i \(0.661062\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −22.0000 −0.885687 −0.442843 0.896599i \(-0.646030\pi\)
−0.442843 + 0.896599i \(0.646030\pi\)
\(618\) 0 0
\(619\) −28.0000 −1.12542 −0.562708 0.826656i \(-0.690240\pi\)
−0.562708 + 0.826656i \(0.690240\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) 0 0
\(624\) 16.0000 0.640513
\(625\) 25.0000 1.00000
\(626\) 0 0
\(627\) 15.0000 0.599042
\(628\) 38.0000 1.51637
\(629\) −8.00000 −0.318981
\(630\) 0 0
\(631\) −12.0000 −0.477712 −0.238856 0.971055i \(-0.576772\pi\)
−0.238856 + 0.971055i \(0.576772\pi\)
\(632\) 0 0
\(633\) 23.0000 0.914168
\(634\) 0 0
\(635\) 0 0
\(636\) 6.00000 0.237915
\(637\) 0 0
\(638\) 0 0
\(639\) −6.00000 −0.237356
\(640\) 0 0
\(641\) 41.0000 1.61940 0.809701 0.586842i \(-0.199629\pi\)
0.809701 + 0.586842i \(0.199629\pi\)
\(642\) 0 0
\(643\) −29.0000 −1.14365 −0.571824 0.820376i \(-0.693764\pi\)
−0.571824 + 0.820376i \(0.693764\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 16.0000 0.629025 0.314512 0.949253i \(-0.398159\pi\)
0.314512 + 0.949253i \(0.398159\pi\)
\(648\) 0 0
\(649\) 65.0000 2.55147
\(650\) 0 0
\(651\) 0 0
\(652\) −2.00000 −0.0783260
\(653\) 30.0000 1.17399 0.586995 0.809590i \(-0.300311\pi\)
0.586995 + 0.809590i \(0.300311\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −28.0000 −1.09322
\(657\) −8.00000 −0.312110
\(658\) 0 0
\(659\) −28.0000 −1.09073 −0.545363 0.838200i \(-0.683608\pi\)
−0.545363 + 0.838200i \(0.683608\pi\)
\(660\) 0 0
\(661\) −31.0000 −1.20576 −0.602880 0.797832i \(-0.705980\pi\)
−0.602880 + 0.797832i \(0.705980\pi\)
\(662\) 0 0
\(663\) −8.00000 −0.310694
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −8.00000 −0.309761
\(668\) −38.0000 −1.47026
\(669\) 2.00000 0.0773245
\(670\) 0 0
\(671\) 45.0000 1.73721
\(672\) 0 0
\(673\) −7.00000 −0.269830 −0.134915 0.990857i \(-0.543076\pi\)
−0.134915 + 0.990857i \(0.543076\pi\)
\(674\) 0 0
\(675\) 5.00000 0.192450
\(676\) −6.00000 −0.230769
\(677\) −22.0000 −0.845529 −0.422764 0.906240i \(-0.638940\pi\)
−0.422764 + 0.906240i \(0.638940\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −12.0000 −0.459841
\(682\) 0 0
\(683\) −4.00000 −0.153056 −0.0765279 0.997067i \(-0.524383\pi\)
−0.0765279 + 0.997067i \(0.524383\pi\)
\(684\) 6.00000 0.229416
\(685\) 0 0
\(686\) 0 0
\(687\) −13.0000 −0.495981
\(688\) 40.0000 1.52499
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) 48.0000 1.82601 0.913003 0.407953i \(-0.133757\pi\)
0.913003 + 0.407953i \(0.133757\pi\)
\(692\) 12.0000 0.456172
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 14.0000 0.530288
\(698\) 0 0
\(699\) −16.0000 −0.605176
\(700\) 0 0
\(701\) 27.0000 1.01978 0.509888 0.860241i \(-0.329687\pi\)
0.509888 + 0.860241i \(0.329687\pi\)
\(702\) 0 0
\(703\) −12.0000 −0.452589
\(704\) −40.0000 −1.50756
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 26.0000 0.977140
\(709\) −38.0000 −1.42712 −0.713560 0.700594i \(-0.752918\pi\)
−0.713560 + 0.700594i \(0.752918\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 10.0000 0.374503
\(714\) 0 0
\(715\) 0 0
\(716\) 12.0000 0.448461
\(717\) 10.0000 0.373457
\(718\) 0 0
\(719\) 44.0000 1.64092 0.820462 0.571702i \(-0.193717\pi\)
0.820462 + 0.571702i \(0.193717\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −7.00000 −0.260333
\(724\) 4.00000 0.148659
\(725\) 40.0000 1.48556
\(726\) 0 0
\(727\) 27.0000 1.00137 0.500687 0.865628i \(-0.333081\pi\)
0.500687 + 0.865628i \(0.333081\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −20.0000 −0.739727
\(732\) 18.0000 0.665299
\(733\) 14.0000 0.517102 0.258551 0.965998i \(-0.416755\pi\)
0.258551 + 0.965998i \(0.416755\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 40.0000 1.47342
\(738\) 0 0
\(739\) −20.0000 −0.735712 −0.367856 0.929883i \(-0.619908\pi\)
−0.367856 + 0.929883i \(0.619908\pi\)
\(740\) 0 0
\(741\) −12.0000 −0.440831
\(742\) 0 0
\(743\) 41.0000 1.50414 0.752072 0.659081i \(-0.229055\pi\)
0.752072 + 0.659081i \(0.229055\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −18.0000 −0.658586
\(748\) 20.0000 0.731272
\(749\) 0 0
\(750\) 0 0
\(751\) −10.0000 −0.364905 −0.182453 0.983215i \(-0.558404\pi\)
−0.182453 + 0.983215i \(0.558404\pi\)
\(752\) 12.0000 0.437595
\(753\) 2.00000 0.0728841
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −46.0000 −1.67190 −0.835949 0.548807i \(-0.815082\pi\)
−0.835949 + 0.548807i \(0.815082\pi\)
\(758\) 0 0
\(759\) −5.00000 −0.181489
\(760\) 0 0
\(761\) −9.00000 −0.326250 −0.163125 0.986605i \(-0.552157\pi\)
−0.163125 + 0.986605i \(0.552157\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 26.0000 0.940647
\(765\) 0 0
\(766\) 0 0
\(767\) −52.0000 −1.87761
\(768\) −16.0000 −0.577350
\(769\) 34.0000 1.22607 0.613036 0.790055i \(-0.289948\pi\)
0.613036 + 0.790055i \(0.289948\pi\)
\(770\) 0 0
\(771\) −13.0000 −0.468184
\(772\) −18.0000 −0.647834
\(773\) 18.0000 0.647415 0.323708 0.946157i \(-0.395071\pi\)
0.323708 + 0.946157i \(0.395071\pi\)
\(774\) 0 0
\(775\) −50.0000 −1.79605
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 21.0000 0.752403
\(780\) 0 0
\(781\) −30.0000 −1.07348
\(782\) 0 0
\(783\) 8.00000 0.285897
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −1.00000 −0.0356462 −0.0178231 0.999841i \(-0.505674\pi\)
−0.0178231 + 0.999841i \(0.505674\pi\)
\(788\) −36.0000 −1.28245
\(789\) 7.00000 0.249207
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −36.0000 −1.27840
\(794\) 0 0
\(795\) 0 0
\(796\) −2.00000 −0.0708881
\(797\) 8.00000 0.283375 0.141687 0.989911i \(-0.454747\pi\)
0.141687 + 0.989911i \(0.454747\pi\)
\(798\) 0 0
\(799\) −6.00000 −0.212265
\(800\) 0 0
\(801\) −14.0000 −0.494666
\(802\) 0 0
\(803\) −40.0000 −1.41157
\(804\) 16.0000 0.564276
\(805\) 0 0
\(806\) 0 0
\(807\) 18.0000 0.633630
\(808\) 0 0
\(809\) 4.00000 0.140633 0.0703163 0.997525i \(-0.477599\pi\)
0.0703163 + 0.997525i \(0.477599\pi\)
\(810\) 0 0
\(811\) 12.0000 0.421377 0.210688 0.977553i \(-0.432429\pi\)
0.210688 + 0.977553i \(0.432429\pi\)
\(812\) 0 0
\(813\) 6.00000 0.210429
\(814\) 0 0
\(815\) 0 0
\(816\) 8.00000 0.280056
\(817\) −30.0000 −1.04957
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −32.0000 −1.11681 −0.558404 0.829569i \(-0.688586\pi\)
−0.558404 + 0.829569i \(0.688586\pi\)
\(822\) 0 0
\(823\) −9.00000 −0.313720 −0.156860 0.987621i \(-0.550137\pi\)
−0.156860 + 0.987621i \(0.550137\pi\)
\(824\) 0 0
\(825\) 25.0000 0.870388
\(826\) 0 0
\(827\) −3.00000 −0.104320 −0.0521601 0.998639i \(-0.516611\pi\)
−0.0521601 + 0.998639i \(0.516611\pi\)
\(828\) −2.00000 −0.0695048
\(829\) 6.00000 0.208389 0.104194 0.994557i \(-0.466774\pi\)
0.104194 + 0.994557i \(0.466774\pi\)
\(830\) 0 0
\(831\) 21.0000 0.728482
\(832\) 32.0000 1.10940
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 30.0000 1.03757
\(837\) −10.0000 −0.345651
\(838\) 0 0
\(839\) 44.0000 1.51905 0.759524 0.650479i \(-0.225432\pi\)
0.759524 + 0.650479i \(0.225432\pi\)
\(840\) 0 0
\(841\) 35.0000 1.20690
\(842\) 0 0
\(843\) −6.00000 −0.206651
\(844\) 46.0000 1.58339
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 12.0000 0.412082
\(849\) −20.0000 −0.686398
\(850\) 0 0
\(851\) 4.00000 0.137118
\(852\) −12.0000 −0.411113
\(853\) −4.00000 −0.136957 −0.0684787 0.997653i \(-0.521815\pi\)
−0.0684787 + 0.997653i \(0.521815\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −7.00000 −0.239115 −0.119558 0.992827i \(-0.538148\pi\)
−0.119558 + 0.992827i \(0.538148\pi\)
\(858\) 0 0
\(859\) 6.00000 0.204717 0.102359 0.994748i \(-0.467361\pi\)
0.102359 + 0.994748i \(0.467361\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 52.0000 1.77010 0.885050 0.465495i \(-0.154124\pi\)
0.885050 + 0.465495i \(0.154124\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 13.0000 0.441503
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −32.0000 −1.08428
\(872\) 0 0
\(873\) −18.0000 −0.609208
\(874\) 0 0
\(875\) 0 0
\(876\) −16.0000 −0.540590
\(877\) −49.0000 −1.65461 −0.827306 0.561751i \(-0.810128\pi\)
−0.827306 + 0.561751i \(0.810128\pi\)
\(878\) 0 0
\(879\) 22.0000 0.742042
\(880\) 0 0
\(881\) 12.0000 0.404290 0.202145 0.979356i \(-0.435209\pi\)
0.202145 + 0.979356i \(0.435209\pi\)
\(882\) 0 0
\(883\) 12.0000 0.403832 0.201916 0.979403i \(-0.435283\pi\)
0.201916 + 0.979403i \(0.435283\pi\)
\(884\) −16.0000 −0.538138
\(885\) 0 0
\(886\) 0 0
\(887\) 20.0000 0.671534 0.335767 0.941945i \(-0.391004\pi\)
0.335767 + 0.941945i \(0.391004\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 5.00000 0.167506
\(892\) 4.00000 0.133930
\(893\) −9.00000 −0.301174
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 4.00000 0.133556
\(898\) 0 0
\(899\) −80.0000 −2.66815
\(900\) 10.0000 0.333333
\(901\) −6.00000 −0.199889
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −52.0000 −1.72663 −0.863316 0.504664i \(-0.831616\pi\)
−0.863316 + 0.504664i \(0.831616\pi\)
\(908\) −24.0000 −0.796468
\(909\) 5.00000 0.165840
\(910\) 0 0
\(911\) 48.0000 1.59031 0.795155 0.606406i \(-0.207389\pi\)
0.795155 + 0.606406i \(0.207389\pi\)
\(912\) 12.0000 0.397360
\(913\) −90.0000 −2.97857
\(914\) 0 0
\(915\) 0 0
\(916\) −26.0000 −0.859064
\(917\) 0 0
\(918\) 0 0
\(919\) 10.0000 0.329870 0.164935 0.986304i \(-0.447259\pi\)
0.164935 + 0.986304i \(0.447259\pi\)
\(920\) 0 0
\(921\) −22.0000 −0.724925
\(922\) 0 0
\(923\) 24.0000 0.789970
\(924\) 0 0
\(925\) −20.0000 −0.657596
\(926\) 0 0
\(927\) −13.0000 −0.426976
\(928\) 0 0
\(929\) −6.00000 −0.196854 −0.0984268 0.995144i \(-0.531381\pi\)
−0.0984268 + 0.995144i \(0.531381\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −32.0000 −1.04819
\(933\) 27.0000 0.883940
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 31.0000 1.01273 0.506363 0.862320i \(-0.330990\pi\)
0.506363 + 0.862320i \(0.330990\pi\)
\(938\) 0 0
\(939\) −13.0000 −0.424239
\(940\) 0 0
\(941\) −6.00000 −0.195594 −0.0977972 0.995206i \(-0.531180\pi\)
−0.0977972 + 0.995206i \(0.531180\pi\)
\(942\) 0 0
\(943\) −7.00000 −0.227951
\(944\) 52.0000 1.69246
\(945\) 0 0
\(946\) 0 0
\(947\) −24.0000 −0.779895 −0.389948 0.920837i \(-0.627507\pi\)
−0.389948 + 0.920837i \(0.627507\pi\)
\(948\) 0 0
\(949\) 32.0000 1.03876
\(950\) 0 0
\(951\) −2.00000 −0.0648544
\(952\) 0 0
\(953\) −33.0000 −1.06897 −0.534487 0.845176i \(-0.679495\pi\)
−0.534487 + 0.845176i \(0.679495\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 20.0000 0.646846
\(957\) 40.0000 1.29302
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 69.0000 2.22581
\(962\) 0 0
\(963\) 12.0000 0.386695
\(964\) −14.0000 −0.450910
\(965\) 0 0
\(966\) 0 0
\(967\) 40.0000 1.28631 0.643157 0.765735i \(-0.277624\pi\)
0.643157 + 0.765735i \(0.277624\pi\)
\(968\) 0 0
\(969\) −6.00000 −0.192748
\(970\) 0 0
\(971\) 40.0000 1.28366 0.641831 0.766846i \(-0.278175\pi\)
0.641831 + 0.766846i \(0.278175\pi\)
\(972\) 2.00000 0.0641500
\(973\) 0 0
\(974\) 0 0
\(975\) −20.0000 −0.640513
\(976\) 36.0000 1.15233
\(977\) −29.0000 −0.927792 −0.463896 0.885890i \(-0.653549\pi\)
−0.463896 + 0.885890i \(0.653549\pi\)
\(978\) 0 0
\(979\) −70.0000 −2.23721
\(980\) 0 0
\(981\) −16.0000 −0.510841
\(982\) 0 0
\(983\) −36.0000 −1.14822 −0.574111 0.818778i \(-0.694652\pi\)
−0.574111 + 0.818778i \(0.694652\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) −24.0000 −0.763542
\(989\) 10.0000 0.317982
\(990\) 0 0
\(991\) −19.0000 −0.603555 −0.301777 0.953378i \(-0.597580\pi\)
−0.301777 + 0.953378i \(0.597580\pi\)
\(992\) 0 0
\(993\) −1.00000 −0.0317340
\(994\) 0 0
\(995\) 0 0
\(996\) −36.0000 −1.14070
\(997\) 50.0000 1.58352 0.791758 0.610835i \(-0.209166\pi\)
0.791758 + 0.610835i \(0.209166\pi\)
\(998\) 0 0
\(999\) −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 3381.2.a.f.1.1 1
7.6 odd 2 3381.2.a.i.1.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3381.2.a.f.1.1 1 1.1 even 1 trivial
3381.2.a.i.1.1 yes 1 7.6 odd 2