Properties

Label 4560.2.a.bp.1.2
Level $4560$
Weight $2$
Character 4560.1
Self dual yes
Analytic conductor $36.412$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4560.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(36.4117833217\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{7}) \)
Defining polynomial: \(x^{2} - 7\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2280)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.64575\) of defining polynomial
Character \(\chi\) \(=\) 4560.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +1.00000 q^{5} +3.64575 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +1.00000 q^{5} +3.64575 q^{7} +1.00000 q^{9} +5.64575 q^{11} -1.64575 q^{13} +1.00000 q^{15} +1.00000 q^{19} +3.64575 q^{21} -2.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} +4.35425 q^{29} +2.00000 q^{31} +5.64575 q^{33} +3.64575 q^{35} +8.93725 q^{37} -1.64575 q^{39} -3.64575 q^{41} -10.9373 q^{43} +1.00000 q^{45} -6.00000 q^{47} +6.29150 q^{49} +4.00000 q^{53} +5.64575 q^{55} +1.00000 q^{57} +7.29150 q^{59} +0.708497 q^{61} +3.64575 q^{63} -1.64575 q^{65} -4.00000 q^{67} -2.00000 q^{69} -7.29150 q^{71} -2.00000 q^{73} +1.00000 q^{75} +20.5830 q^{77} +8.00000 q^{79} +1.00000 q^{81} -15.8745 q^{83} +4.35425 q^{87} +14.2288 q^{89} -6.00000 q^{91} +2.00000 q^{93} +1.00000 q^{95} -8.22876 q^{97} +5.64575 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9} + O(q^{10}) \) \( 2 q + 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9} + 6 q^{11} + 2 q^{13} + 2 q^{15} + 2 q^{19} + 2 q^{21} - 4 q^{23} + 2 q^{25} + 2 q^{27} + 14 q^{29} + 4 q^{31} + 6 q^{33} + 2 q^{35} + 2 q^{37} + 2 q^{39} - 2 q^{41} - 6 q^{43} + 2 q^{45} - 12 q^{47} + 2 q^{49} + 8 q^{53} + 6 q^{55} + 2 q^{57} + 4 q^{59} + 12 q^{61} + 2 q^{63} + 2 q^{65} - 8 q^{67} - 4 q^{69} - 4 q^{71} - 4 q^{73} + 2 q^{75} + 20 q^{77} + 16 q^{79} + 2 q^{81} + 14 q^{87} + 2 q^{89} - 12 q^{91} + 4 q^{93} + 2 q^{95} + 10 q^{97} + 6 q^{99} + O(q^{100}) \)

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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 3.64575 1.37796 0.688982 0.724778i \(-0.258058\pi\)
0.688982 + 0.724778i \(0.258058\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 5.64575 1.70226 0.851129 0.524957i \(-0.175918\pi\)
0.851129 + 0.524957i \(0.175918\pi\)
\(12\) 0 0
\(13\) −1.64575 −0.456449 −0.228225 0.973609i \(-0.573292\pi\)
−0.228225 + 0.973609i \(0.573292\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 3.64575 0.795568
\(22\) 0 0
\(23\) −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
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 4.35425 0.808564 0.404282 0.914634i \(-0.367521\pi\)
0.404282 + 0.914634i \(0.367521\pi\)
\(30\) 0 0
\(31\) 2.00000 0.359211 0.179605 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) 0 0
\(33\) 5.64575 0.982799
\(34\) 0 0
\(35\) 3.64575 0.616244
\(36\) 0 0
\(37\) 8.93725 1.46928 0.734638 0.678460i \(-0.237352\pi\)
0.734638 + 0.678460i \(0.237352\pi\)
\(38\) 0 0
\(39\) −1.64575 −0.263531
\(40\) 0 0
\(41\) −3.64575 −0.569371 −0.284685 0.958621i \(-0.591889\pi\)
−0.284685 + 0.958621i \(0.591889\pi\)
\(42\) 0 0
\(43\) −10.9373 −1.66792 −0.833958 0.551828i \(-0.813930\pi\)
−0.833958 + 0.551828i \(0.813930\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) −6.00000 −0.875190 −0.437595 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) 0 0
\(49\) 6.29150 0.898786
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 4.00000 0.549442 0.274721 0.961524i \(-0.411414\pi\)
0.274721 + 0.961524i \(0.411414\pi\)
\(54\) 0 0
\(55\) 5.64575 0.761273
\(56\) 0 0
\(57\) 1.00000 0.132453
\(58\) 0 0
\(59\) 7.29150 0.949273 0.474636 0.880182i \(-0.342580\pi\)
0.474636 + 0.880182i \(0.342580\pi\)
\(60\) 0 0
\(61\) 0.708497 0.0907138 0.0453569 0.998971i \(-0.485557\pi\)
0.0453569 + 0.998971i \(0.485557\pi\)
\(62\) 0 0
\(63\) 3.64575 0.459321
\(64\) 0 0
\(65\) −1.64575 −0.204130
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 0 0
\(69\) −2.00000 −0.240772
\(70\) 0 0
\(71\) −7.29150 −0.865342 −0.432671 0.901552i \(-0.642429\pi\)
−0.432671 + 0.901552i \(0.642429\pi\)
\(72\) 0 0
\(73\) −2.00000 −0.234082 −0.117041 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) 20.5830 2.34565
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −15.8745 −1.74245 −0.871227 0.490881i \(-0.836675\pi\)
−0.871227 + 0.490881i \(0.836675\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 4.35425 0.466824
\(88\) 0 0
\(89\) 14.2288 1.50825 0.754123 0.656734i \(-0.228062\pi\)
0.754123 + 0.656734i \(0.228062\pi\)
\(90\) 0 0
\(91\) −6.00000 −0.628971
\(92\) 0 0
\(93\) 2.00000 0.207390
\(94\) 0 0
\(95\) 1.00000 0.102598
\(96\) 0 0
\(97\) −8.22876 −0.835504 −0.417752 0.908561i \(-0.637182\pi\)
−0.417752 + 0.908561i \(0.637182\pi\)
\(98\) 0 0
\(99\) 5.64575 0.567419
\(100\) 0 0
\(101\) −15.8745 −1.57957 −0.789786 0.613382i \(-0.789809\pi\)
−0.789786 + 0.613382i \(0.789809\pi\)
\(102\) 0 0
\(103\) 6.58301 0.648643 0.324321 0.945947i \(-0.394864\pi\)
0.324321 + 0.945947i \(0.394864\pi\)
\(104\) 0 0
\(105\) 3.64575 0.355789
\(106\) 0 0
\(107\) 6.58301 0.636403 0.318202 0.948023i \(-0.396921\pi\)
0.318202 + 0.948023i \(0.396921\pi\)
\(108\) 0 0
\(109\) −5.29150 −0.506834 −0.253417 0.967357i \(-0.581554\pi\)
−0.253417 + 0.967357i \(0.581554\pi\)
\(110\) 0 0
\(111\) 8.93725 0.848287
\(112\) 0 0
\(113\) −13.8745 −1.30520 −0.652602 0.757701i \(-0.726323\pi\)
−0.652602 + 0.757701i \(0.726323\pi\)
\(114\) 0 0
\(115\) −2.00000 −0.186501
\(116\) 0 0
\(117\) −1.64575 −0.152150
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 20.8745 1.89768
\(122\) 0 0
\(123\) −3.64575 −0.328726
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −18.5830 −1.64898 −0.824488 0.565880i \(-0.808536\pi\)
−0.824488 + 0.565880i \(0.808536\pi\)
\(128\) 0 0
\(129\) −10.9373 −0.962972
\(130\) 0 0
\(131\) −2.35425 −0.205692 −0.102846 0.994697i \(-0.532795\pi\)
−0.102846 + 0.994697i \(0.532795\pi\)
\(132\) 0 0
\(133\) 3.64575 0.316127
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) 20.5830 1.75852 0.879262 0.476338i \(-0.158036\pi\)
0.879262 + 0.476338i \(0.158036\pi\)
\(138\) 0 0
\(139\) −13.8745 −1.17682 −0.588410 0.808563i \(-0.700246\pi\)
−0.588410 + 0.808563i \(0.700246\pi\)
\(140\) 0 0
\(141\) −6.00000 −0.505291
\(142\) 0 0
\(143\) −9.29150 −0.776994
\(144\) 0 0
\(145\) 4.35425 0.361601
\(146\) 0 0
\(147\) 6.29150 0.518914
\(148\) 0 0
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) −6.00000 −0.488273 −0.244137 0.969741i \(-0.578505\pi\)
−0.244137 + 0.969741i \(0.578505\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 2.00000 0.160644
\(156\) 0 0
\(157\) −15.8745 −1.26692 −0.633462 0.773774i \(-0.718367\pi\)
−0.633462 + 0.773774i \(0.718367\pi\)
\(158\) 0 0
\(159\) 4.00000 0.317221
\(160\) 0 0
\(161\) −7.29150 −0.574651
\(162\) 0 0
\(163\) −4.35425 −0.341051 −0.170526 0.985353i \(-0.554547\pi\)
−0.170526 + 0.985353i \(0.554547\pi\)
\(164\) 0 0
\(165\) 5.64575 0.439521
\(166\) 0 0
\(167\) −9.29150 −0.718998 −0.359499 0.933145i \(-0.617052\pi\)
−0.359499 + 0.933145i \(0.617052\pi\)
\(168\) 0 0
\(169\) −10.2915 −0.791654
\(170\) 0 0
\(171\) 1.00000 0.0764719
\(172\) 0 0
\(173\) 19.2915 1.46671 0.733353 0.679848i \(-0.237954\pi\)
0.733353 + 0.679848i \(0.237954\pi\)
\(174\) 0 0
\(175\) 3.64575 0.275593
\(176\) 0 0
\(177\) 7.29150 0.548063
\(178\) 0 0
\(179\) −3.29150 −0.246018 −0.123009 0.992406i \(-0.539254\pi\)
−0.123009 + 0.992406i \(0.539254\pi\)
\(180\) 0 0
\(181\) 5.29150 0.393314 0.196657 0.980472i \(-0.436991\pi\)
0.196657 + 0.980472i \(0.436991\pi\)
\(182\) 0 0
\(183\) 0.708497 0.0523736
\(184\) 0 0
\(185\) 8.93725 0.657080
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 3.64575 0.265189
\(190\) 0 0
\(191\) 4.93725 0.357247 0.178624 0.983917i \(-0.442836\pi\)
0.178624 + 0.983917i \(0.442836\pi\)
\(192\) 0 0
\(193\) −12.9373 −0.931244 −0.465622 0.884984i \(-0.654169\pi\)
−0.465622 + 0.884984i \(0.654169\pi\)
\(194\) 0 0
\(195\) −1.64575 −0.117855
\(196\) 0 0
\(197\) 24.0000 1.70993 0.854965 0.518686i \(-0.173579\pi\)
0.854965 + 0.518686i \(0.173579\pi\)
\(198\) 0 0
\(199\) 12.7085 0.900881 0.450441 0.892806i \(-0.351267\pi\)
0.450441 + 0.892806i \(0.351267\pi\)
\(200\) 0 0
\(201\) −4.00000 −0.282138
\(202\) 0 0
\(203\) 15.8745 1.11417
\(204\) 0 0
\(205\) −3.64575 −0.254630
\(206\) 0 0
\(207\) −2.00000 −0.139010
\(208\) 0 0
\(209\) 5.64575 0.390525
\(210\) 0 0
\(211\) −10.5830 −0.728564 −0.364282 0.931289i \(-0.618686\pi\)
−0.364282 + 0.931289i \(0.618686\pi\)
\(212\) 0 0
\(213\) −7.29150 −0.499606
\(214\) 0 0
\(215\) −10.9373 −0.745915
\(216\) 0 0
\(217\) 7.29150 0.494979
\(218\) 0 0
\(219\) −2.00000 −0.135147
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 22.5830 1.51227 0.756135 0.654416i \(-0.227085\pi\)
0.756135 + 0.654416i \(0.227085\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −14.7085 −0.976237 −0.488119 0.872777i \(-0.662317\pi\)
−0.488119 + 0.872777i \(0.662317\pi\)
\(228\) 0 0
\(229\) −3.29150 −0.217509 −0.108754 0.994069i \(-0.534686\pi\)
−0.108754 + 0.994069i \(0.534686\pi\)
\(230\) 0 0
\(231\) 20.5830 1.35426
\(232\) 0 0
\(233\) 22.0000 1.44127 0.720634 0.693316i \(-0.243851\pi\)
0.720634 + 0.693316i \(0.243851\pi\)
\(234\) 0 0
\(235\) −6.00000 −0.391397
\(236\) 0 0
\(237\) 8.00000 0.519656
\(238\) 0 0
\(239\) 28.9373 1.87180 0.935898 0.352272i \(-0.114591\pi\)
0.935898 + 0.352272i \(0.114591\pi\)
\(240\) 0 0
\(241\) 16.5830 1.06821 0.534103 0.845420i \(-0.320650\pi\)
0.534103 + 0.845420i \(0.320650\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 6.29150 0.401949
\(246\) 0 0
\(247\) −1.64575 −0.104717
\(248\) 0 0
\(249\) −15.8745 −1.00601
\(250\) 0 0
\(251\) −20.2288 −1.27683 −0.638414 0.769693i \(-0.720409\pi\)
−0.638414 + 0.769693i \(0.720409\pi\)
\(252\) 0 0
\(253\) −11.2915 −0.709891
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 21.8745 1.36449 0.682247 0.731122i \(-0.261003\pi\)
0.682247 + 0.731122i \(0.261003\pi\)
\(258\) 0 0
\(259\) 32.5830 2.02461
\(260\) 0 0
\(261\) 4.35425 0.269521
\(262\) 0 0
\(263\) −9.29150 −0.572939 −0.286469 0.958089i \(-0.592482\pi\)
−0.286469 + 0.958089i \(0.592482\pi\)
\(264\) 0 0
\(265\) 4.00000 0.245718
\(266\) 0 0
\(267\) 14.2288 0.870786
\(268\) 0 0
\(269\) −10.9373 −0.666856 −0.333428 0.942776i \(-0.608205\pi\)
−0.333428 + 0.942776i \(0.608205\pi\)
\(270\) 0 0
\(271\) −23.2915 −1.41486 −0.707429 0.706784i \(-0.750145\pi\)
−0.707429 + 0.706784i \(0.750145\pi\)
\(272\) 0 0
\(273\) −6.00000 −0.363137
\(274\) 0 0
\(275\) 5.64575 0.340452
\(276\) 0 0
\(277\) −3.41699 −0.205307 −0.102654 0.994717i \(-0.532733\pi\)
−0.102654 + 0.994717i \(0.532733\pi\)
\(278\) 0 0
\(279\) 2.00000 0.119737
\(280\) 0 0
\(281\) 10.9373 0.652462 0.326231 0.945290i \(-0.394221\pi\)
0.326231 + 0.945290i \(0.394221\pi\)
\(282\) 0 0
\(283\) 30.2288 1.79691 0.898457 0.439062i \(-0.144689\pi\)
0.898457 + 0.439062i \(0.144689\pi\)
\(284\) 0 0
\(285\) 1.00000 0.0592349
\(286\) 0 0
\(287\) −13.2915 −0.784573
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) −8.22876 −0.482378
\(292\) 0 0
\(293\) 13.8745 0.810557 0.405279 0.914193i \(-0.367174\pi\)
0.405279 + 0.914193i \(0.367174\pi\)
\(294\) 0 0
\(295\) 7.29150 0.424528
\(296\) 0 0
\(297\) 5.64575 0.327600
\(298\) 0 0
\(299\) 3.29150 0.190353
\(300\) 0 0
\(301\) −39.8745 −2.29833
\(302\) 0 0
\(303\) −15.8745 −0.911967
\(304\) 0 0
\(305\) 0.708497 0.0405684
\(306\) 0 0
\(307\) −23.2915 −1.32932 −0.664658 0.747148i \(-0.731423\pi\)
−0.664658 + 0.747148i \(0.731423\pi\)
\(308\) 0 0
\(309\) 6.58301 0.374494
\(310\) 0 0
\(311\) 9.64575 0.546960 0.273480 0.961878i \(-0.411825\pi\)
0.273480 + 0.961878i \(0.411825\pi\)
\(312\) 0 0
\(313\) −29.2915 −1.65565 −0.827827 0.560984i \(-0.810423\pi\)
−0.827827 + 0.560984i \(0.810423\pi\)
\(314\) 0 0
\(315\) 3.64575 0.205415
\(316\) 0 0
\(317\) 1.41699 0.0795864 0.0397932 0.999208i \(-0.487330\pi\)
0.0397932 + 0.999208i \(0.487330\pi\)
\(318\) 0 0
\(319\) 24.5830 1.37638
\(320\) 0 0
\(321\) 6.58301 0.367428
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −1.64575 −0.0912899
\(326\) 0 0
\(327\) −5.29150 −0.292621
\(328\) 0 0
\(329\) −21.8745 −1.20598
\(330\) 0 0
\(331\) 4.58301 0.251905 0.125952 0.992036i \(-0.459801\pi\)
0.125952 + 0.992036i \(0.459801\pi\)
\(332\) 0 0
\(333\) 8.93725 0.489758
\(334\) 0 0
\(335\) −4.00000 −0.218543
\(336\) 0 0
\(337\) 4.22876 0.230355 0.115178 0.993345i \(-0.463256\pi\)
0.115178 + 0.993345i \(0.463256\pi\)
\(338\) 0 0
\(339\) −13.8745 −0.753560
\(340\) 0 0
\(341\) 11.2915 0.611469
\(342\) 0 0
\(343\) −2.58301 −0.139469
\(344\) 0 0
\(345\) −2.00000 −0.107676
\(346\) 0 0
\(347\) 1.29150 0.0693315 0.0346657 0.999399i \(-0.488963\pi\)
0.0346657 + 0.999399i \(0.488963\pi\)
\(348\) 0 0
\(349\) 15.1660 0.811818 0.405909 0.913914i \(-0.366955\pi\)
0.405909 + 0.913914i \(0.366955\pi\)
\(350\) 0 0
\(351\) −1.64575 −0.0878437
\(352\) 0 0
\(353\) 7.16601 0.381408 0.190704 0.981648i \(-0.438923\pi\)
0.190704 + 0.981648i \(0.438923\pi\)
\(354\) 0 0
\(355\) −7.29150 −0.386993
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −16.2288 −0.856521 −0.428261 0.903655i \(-0.640873\pi\)
−0.428261 + 0.903655i \(0.640873\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 20.8745 1.09563
\(364\) 0 0
\(365\) −2.00000 −0.104685
\(366\) 0 0
\(367\) 26.2288 1.36913 0.684565 0.728952i \(-0.259992\pi\)
0.684565 + 0.728952i \(0.259992\pi\)
\(368\) 0 0
\(369\) −3.64575 −0.189790
\(370\) 0 0
\(371\) 14.5830 0.757112
\(372\) 0 0
\(373\) −13.6458 −0.706550 −0.353275 0.935519i \(-0.614932\pi\)
−0.353275 + 0.935519i \(0.614932\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) −7.16601 −0.369068
\(378\) 0 0
\(379\) −10.0000 −0.513665 −0.256833 0.966456i \(-0.582679\pi\)
−0.256833 + 0.966456i \(0.582679\pi\)
\(380\) 0 0
\(381\) −18.5830 −0.952036
\(382\) 0 0
\(383\) 26.5830 1.35833 0.679164 0.733987i \(-0.262342\pi\)
0.679164 + 0.733987i \(0.262342\pi\)
\(384\) 0 0
\(385\) 20.5830 1.04901
\(386\) 0 0
\(387\) −10.9373 −0.555972
\(388\) 0 0
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) −2.35425 −0.118756
\(394\) 0 0
\(395\) 8.00000 0.402524
\(396\) 0 0
\(397\) −14.0000 −0.702640 −0.351320 0.936255i \(-0.614267\pi\)
−0.351320 + 0.936255i \(0.614267\pi\)
\(398\) 0 0
\(399\) 3.64575 0.182516
\(400\) 0 0
\(401\) 11.6458 0.581561 0.290781 0.956790i \(-0.406085\pi\)
0.290781 + 0.956790i \(0.406085\pi\)
\(402\) 0 0
\(403\) −3.29150 −0.163961
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) 50.4575 2.50109
\(408\) 0 0
\(409\) −17.2915 −0.855010 −0.427505 0.904013i \(-0.640607\pi\)
−0.427505 + 0.904013i \(0.640607\pi\)
\(410\) 0 0
\(411\) 20.5830 1.01528
\(412\) 0 0
\(413\) 26.5830 1.30806
\(414\) 0 0
\(415\) −15.8745 −0.779249
\(416\) 0 0
\(417\) −13.8745 −0.679438
\(418\) 0 0
\(419\) 23.0627 1.12669 0.563344 0.826222i \(-0.309514\pi\)
0.563344 + 0.826222i \(0.309514\pi\)
\(420\) 0 0
\(421\) 20.5830 1.00315 0.501577 0.865113i \(-0.332753\pi\)
0.501577 + 0.865113i \(0.332753\pi\)
\(422\) 0 0
\(423\) −6.00000 −0.291730
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 2.58301 0.125000
\(428\) 0 0
\(429\) −9.29150 −0.448598
\(430\) 0 0
\(431\) −20.4575 −0.985404 −0.492702 0.870198i \(-0.663991\pi\)
−0.492702 + 0.870198i \(0.663991\pi\)
\(432\) 0 0
\(433\) 0.228757 0.0109933 0.00549667 0.999985i \(-0.498250\pi\)
0.00549667 + 0.999985i \(0.498250\pi\)
\(434\) 0 0
\(435\) 4.35425 0.208770
\(436\) 0 0
\(437\) −2.00000 −0.0956730
\(438\) 0 0
\(439\) 37.1660 1.77384 0.886918 0.461926i \(-0.152841\pi\)
0.886918 + 0.461926i \(0.152841\pi\)
\(440\) 0 0
\(441\) 6.29150 0.299595
\(442\) 0 0
\(443\) 12.5830 0.597837 0.298918 0.954279i \(-0.403374\pi\)
0.298918 + 0.954279i \(0.403374\pi\)
\(444\) 0 0
\(445\) 14.2288 0.674508
\(446\) 0 0
\(447\) 6.00000 0.283790
\(448\) 0 0
\(449\) −7.64575 −0.360825 −0.180413 0.983591i \(-0.557743\pi\)
−0.180413 + 0.983591i \(0.557743\pi\)
\(450\) 0 0
\(451\) −20.5830 −0.969216
\(452\) 0 0
\(453\) −6.00000 −0.281905
\(454\) 0 0
\(455\) −6.00000 −0.281284
\(456\) 0 0
\(457\) −1.29150 −0.0604139 −0.0302070 0.999544i \(-0.509617\pi\)
−0.0302070 + 0.999544i \(0.509617\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −6.00000 −0.279448 −0.139724 0.990190i \(-0.544622\pi\)
−0.139724 + 0.990190i \(0.544622\pi\)
\(462\) 0 0
\(463\) 34.2288 1.59075 0.795373 0.606121i \(-0.207275\pi\)
0.795373 + 0.606121i \(0.207275\pi\)
\(464\) 0 0
\(465\) 2.00000 0.0927478
\(466\) 0 0
\(467\) −20.5830 −0.952468 −0.476234 0.879319i \(-0.657998\pi\)
−0.476234 + 0.879319i \(0.657998\pi\)
\(468\) 0 0
\(469\) −14.5830 −0.673381
\(470\) 0 0
\(471\) −15.8745 −0.731459
\(472\) 0 0
\(473\) −61.7490 −2.83922
\(474\) 0 0
\(475\) 1.00000 0.0458831
\(476\) 0 0
\(477\) 4.00000 0.183147
\(478\) 0 0
\(479\) −23.5203 −1.07467 −0.537334 0.843370i \(-0.680569\pi\)
−0.537334 + 0.843370i \(0.680569\pi\)
\(480\) 0 0
\(481\) −14.7085 −0.670650
\(482\) 0 0
\(483\) −7.29150 −0.331775
\(484\) 0 0
\(485\) −8.22876 −0.373649
\(486\) 0 0
\(487\) 7.29150 0.330410 0.165205 0.986259i \(-0.447172\pi\)
0.165205 + 0.986259i \(0.447172\pi\)
\(488\) 0 0
\(489\) −4.35425 −0.196906
\(490\) 0 0
\(491\) −14.8118 −0.668445 −0.334223 0.942494i \(-0.608474\pi\)
−0.334223 + 0.942494i \(0.608474\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 5.64575 0.253758
\(496\) 0 0
\(497\) −26.5830 −1.19241
\(498\) 0 0
\(499\) −8.70850 −0.389846 −0.194923 0.980819i \(-0.562446\pi\)
−0.194923 + 0.980819i \(0.562446\pi\)
\(500\) 0 0
\(501\) −9.29150 −0.415114
\(502\) 0 0
\(503\) −9.29150 −0.414288 −0.207144 0.978311i \(-0.566417\pi\)
−0.207144 + 0.978311i \(0.566417\pi\)
\(504\) 0 0
\(505\) −15.8745 −0.706406
\(506\) 0 0
\(507\) −10.2915 −0.457062
\(508\) 0 0
\(509\) 39.6458 1.75727 0.878634 0.477497i \(-0.158456\pi\)
0.878634 + 0.477497i \(0.158456\pi\)
\(510\) 0 0
\(511\) −7.29150 −0.322557
\(512\) 0 0
\(513\) 1.00000 0.0441511
\(514\) 0 0
\(515\) 6.58301 0.290082
\(516\) 0 0
\(517\) −33.8745 −1.48980
\(518\) 0 0
\(519\) 19.2915 0.846803
\(520\) 0 0
\(521\) −20.8118 −0.911780 −0.455890 0.890036i \(-0.650679\pi\)
−0.455890 + 0.890036i \(0.650679\pi\)
\(522\) 0 0
\(523\) −37.8745 −1.65614 −0.828068 0.560627i \(-0.810560\pi\)
−0.828068 + 0.560627i \(0.810560\pi\)
\(524\) 0 0
\(525\) 3.64575 0.159114
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −19.0000 −0.826087
\(530\) 0 0
\(531\) 7.29150 0.316424
\(532\) 0 0
\(533\) 6.00000 0.259889
\(534\) 0 0
\(535\) 6.58301 0.284608
\(536\) 0 0
\(537\) −3.29150 −0.142039
\(538\) 0 0
\(539\) 35.5203 1.52997
\(540\) 0 0
\(541\) −26.0000 −1.11783 −0.558914 0.829226i \(-0.688782\pi\)
−0.558914 + 0.829226i \(0.688782\pi\)
\(542\) 0 0
\(543\) 5.29150 0.227080
\(544\) 0 0
\(545\) −5.29150 −0.226663
\(546\) 0 0
\(547\) −12.4575 −0.532645 −0.266322 0.963884i \(-0.585809\pi\)
−0.266322 + 0.963884i \(0.585809\pi\)
\(548\) 0 0
\(549\) 0.708497 0.0302379
\(550\) 0 0
\(551\) 4.35425 0.185497
\(552\) 0 0
\(553\) 29.1660 1.24026
\(554\) 0 0
\(555\) 8.93725 0.379365
\(556\) 0 0
\(557\) −10.0000 −0.423714 −0.211857 0.977301i \(-0.567951\pi\)
−0.211857 + 0.977301i \(0.567951\pi\)
\(558\) 0 0
\(559\) 18.0000 0.761319
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −27.8745 −1.17477 −0.587385 0.809307i \(-0.699843\pi\)
−0.587385 + 0.809307i \(0.699843\pi\)
\(564\) 0 0
\(565\) −13.8745 −0.583705
\(566\) 0 0
\(567\) 3.64575 0.153107
\(568\) 0 0
\(569\) 37.5203 1.57293 0.786466 0.617634i \(-0.211909\pi\)
0.786466 + 0.617634i \(0.211909\pi\)
\(570\) 0 0
\(571\) 17.8745 0.748025 0.374012 0.927424i \(-0.377982\pi\)
0.374012 + 0.927424i \(0.377982\pi\)
\(572\) 0 0
\(573\) 4.93725 0.206257
\(574\) 0 0
\(575\) −2.00000 −0.0834058
\(576\) 0 0
\(577\) −3.87451 −0.161298 −0.0806489 0.996743i \(-0.525699\pi\)
−0.0806489 + 0.996743i \(0.525699\pi\)
\(578\) 0 0
\(579\) −12.9373 −0.537654
\(580\) 0 0
\(581\) −57.8745 −2.40104
\(582\) 0 0
\(583\) 22.5830 0.935293
\(584\) 0 0
\(585\) −1.64575 −0.0680434
\(586\) 0 0
\(587\) −15.4170 −0.636327 −0.318164 0.948036i \(-0.603066\pi\)
−0.318164 + 0.948036i \(0.603066\pi\)
\(588\) 0 0
\(589\) 2.00000 0.0824086
\(590\) 0 0
\(591\) 24.0000 0.987228
\(592\) 0 0
\(593\) −36.5830 −1.50228 −0.751142 0.660141i \(-0.770497\pi\)
−0.751142 + 0.660141i \(0.770497\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 12.7085 0.520124
\(598\) 0 0
\(599\) 26.5830 1.08615 0.543076 0.839683i \(-0.317260\pi\)
0.543076 + 0.839683i \(0.317260\pi\)
\(600\) 0 0
\(601\) 15.8745 0.647535 0.323767 0.946137i \(-0.395050\pi\)
0.323767 + 0.946137i \(0.395050\pi\)
\(602\) 0 0
\(603\) −4.00000 −0.162893
\(604\) 0 0
\(605\) 20.8745 0.848669
\(606\) 0 0
\(607\) −40.4575 −1.64212 −0.821060 0.570842i \(-0.806617\pi\)
−0.821060 + 0.570842i \(0.806617\pi\)
\(608\) 0 0
\(609\) 15.8745 0.643268
\(610\) 0 0
\(611\) 9.87451 0.399480
\(612\) 0 0
\(613\) 38.4575 1.55328 0.776642 0.629942i \(-0.216921\pi\)
0.776642 + 0.629942i \(0.216921\pi\)
\(614\) 0 0
\(615\) −3.64575 −0.147011
\(616\) 0 0
\(617\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) 0 0
\(619\) 6.12549 0.246204 0.123102 0.992394i \(-0.460716\pi\)
0.123102 + 0.992394i \(0.460716\pi\)
\(620\) 0 0
\(621\) −2.00000 −0.0802572
\(622\) 0 0
\(623\) 51.8745 2.07831
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 5.64575 0.225470
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 5.16601 0.205656 0.102828 0.994699i \(-0.467211\pi\)
0.102828 + 0.994699i \(0.467211\pi\)
\(632\) 0 0
\(633\) −10.5830 −0.420637
\(634\) 0 0
\(635\) −18.5830 −0.737444
\(636\) 0 0
\(637\) −10.3542 −0.410250
\(638\) 0 0
\(639\) −7.29150 −0.288447
\(640\) 0 0
\(641\) 6.22876 0.246021 0.123011 0.992405i \(-0.460745\pi\)
0.123011 + 0.992405i \(0.460745\pi\)
\(642\) 0 0
\(643\) −29.0627 −1.14612 −0.573061 0.819512i \(-0.694244\pi\)
−0.573061 + 0.819512i \(0.694244\pi\)
\(644\) 0 0
\(645\) −10.9373 −0.430654
\(646\) 0 0
\(647\) 14.7085 0.578251 0.289125 0.957291i \(-0.406636\pi\)
0.289125 + 0.957291i \(0.406636\pi\)
\(648\) 0 0
\(649\) 41.1660 1.61591
\(650\) 0 0
\(651\) 7.29150 0.285777
\(652\) 0 0
\(653\) −4.00000 −0.156532 −0.0782660 0.996933i \(-0.524938\pi\)
−0.0782660 + 0.996933i \(0.524938\pi\)
\(654\) 0 0
\(655\) −2.35425 −0.0919881
\(656\) 0 0
\(657\) −2.00000 −0.0780274
\(658\) 0 0
\(659\) 29.1660 1.13615 0.568073 0.822978i \(-0.307689\pi\)
0.568073 + 0.822978i \(0.307689\pi\)
\(660\) 0 0
\(661\) 5.29150 0.205816 0.102908 0.994691i \(-0.467185\pi\)
0.102908 + 0.994691i \(0.467185\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 3.64575 0.141376
\(666\) 0 0
\(667\) −8.70850 −0.337194
\(668\) 0 0
\(669\) 22.5830 0.873109
\(670\) 0 0
\(671\) 4.00000 0.154418
\(672\) 0 0
\(673\) −27.5203 −1.06083 −0.530414 0.847739i \(-0.677964\pi\)
−0.530414 + 0.847739i \(0.677964\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) −4.00000 −0.153732 −0.0768662 0.997041i \(-0.524491\pi\)
−0.0768662 + 0.997041i \(0.524491\pi\)
\(678\) 0 0
\(679\) −30.0000 −1.15129
\(680\) 0 0
\(681\) −14.7085 −0.563631
\(682\) 0 0
\(683\) −8.00000 −0.306111 −0.153056 0.988218i \(-0.548911\pi\)
−0.153056 + 0.988218i \(0.548911\pi\)
\(684\) 0 0
\(685\) 20.5830 0.786436
\(686\) 0 0
\(687\) −3.29150 −0.125579
\(688\) 0 0
\(689\) −6.58301 −0.250793
\(690\) 0 0
\(691\) −20.7085 −0.787788 −0.393894 0.919156i \(-0.628872\pi\)
−0.393894 + 0.919156i \(0.628872\pi\)
\(692\) 0 0
\(693\) 20.5830 0.781884
\(694\) 0 0
\(695\) −13.8745 −0.526290
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 22.0000 0.832116
\(700\) 0 0
\(701\) −10.0000 −0.377695 −0.188847 0.982006i \(-0.560475\pi\)
−0.188847 + 0.982006i \(0.560475\pi\)
\(702\) 0 0
\(703\) 8.93725 0.337075
\(704\) 0 0
\(705\) −6.00000 −0.225973
\(706\) 0 0
\(707\) −57.8745 −2.17659
\(708\) 0 0
\(709\) 20.4575 0.768298 0.384149 0.923271i \(-0.374495\pi\)
0.384149 + 0.923271i \(0.374495\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) 0 0
\(713\) −4.00000 −0.149801
\(714\) 0 0
\(715\) −9.29150 −0.347482
\(716\) 0 0
\(717\) 28.9373 1.08068
\(718\) 0 0
\(719\) 2.35425 0.0877987 0.0438993 0.999036i \(-0.486022\pi\)
0.0438993 + 0.999036i \(0.486022\pi\)
\(720\) 0 0
\(721\) 24.0000 0.893807
\(722\) 0 0
\(723\) 16.5830 0.616729
\(724\) 0 0
\(725\) 4.35425 0.161713
\(726\) 0 0
\(727\) −49.9778 −1.85357 −0.926786 0.375589i \(-0.877441\pi\)
−0.926786 + 0.375589i \(0.877441\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −11.4170 −0.421696 −0.210848 0.977519i \(-0.567623\pi\)
−0.210848 + 0.977519i \(0.567623\pi\)
\(734\) 0 0
\(735\) 6.29150 0.232066
\(736\) 0 0
\(737\) −22.5830 −0.831856
\(738\) 0 0
\(739\) −10.5830 −0.389302 −0.194651 0.980873i \(-0.562357\pi\)
−0.194651 + 0.980873i \(0.562357\pi\)
\(740\) 0 0
\(741\) −1.64575 −0.0604582
\(742\) 0 0
\(743\) −31.7490 −1.16476 −0.582379 0.812917i \(-0.697878\pi\)
−0.582379 + 0.812917i \(0.697878\pi\)
\(744\) 0 0
\(745\) 6.00000 0.219823
\(746\) 0 0
\(747\) −15.8745 −0.580818
\(748\) 0 0
\(749\) 24.0000 0.876941
\(750\) 0 0
\(751\) 48.3320 1.76366 0.881830 0.471567i \(-0.156311\pi\)
0.881830 + 0.471567i \(0.156311\pi\)
\(752\) 0 0
\(753\) −20.2288 −0.737177
\(754\) 0 0
\(755\) −6.00000 −0.218362
\(756\) 0 0
\(757\) 11.8745 0.431586 0.215793 0.976439i \(-0.430766\pi\)
0.215793 + 0.976439i \(0.430766\pi\)
\(758\) 0 0
\(759\) −11.2915 −0.409856
\(760\) 0 0
\(761\) 31.1660 1.12977 0.564883 0.825171i \(-0.308921\pi\)
0.564883 + 0.825171i \(0.308921\pi\)
\(762\) 0 0
\(763\) −19.2915 −0.698399
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −12.0000 −0.433295
\(768\) 0 0
\(769\) −41.7490 −1.50551 −0.752754 0.658302i \(-0.771275\pi\)
−0.752754 + 0.658302i \(0.771275\pi\)
\(770\) 0 0
\(771\) 21.8745 0.787791
\(772\) 0 0
\(773\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(774\) 0 0
\(775\) 2.00000 0.0718421
\(776\) 0 0
\(777\) 32.5830 1.16891
\(778\) 0 0
\(779\) −3.64575 −0.130623
\(780\) 0 0
\(781\) −41.1660 −1.47304
\(782\) 0 0
\(783\) 4.35425 0.155608
\(784\) 0 0
\(785\) −15.8745 −0.566585
\(786\) 0 0
\(787\) −0.708497 −0.0252552 −0.0126276 0.999920i \(-0.504020\pi\)
−0.0126276 + 0.999920i \(0.504020\pi\)
\(788\) 0 0
\(789\) −9.29150 −0.330786
\(790\) 0 0
\(791\) −50.5830 −1.79852
\(792\) 0 0
\(793\) −1.16601 −0.0414062
\(794\) 0 0
\(795\) 4.00000 0.141865
\(796\) 0 0
\(797\) 28.7085 1.01691 0.508454 0.861089i \(-0.330217\pi\)
0.508454 + 0.861089i \(0.330217\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 14.2288 0.502748
\(802\) 0 0
\(803\) −11.2915 −0.398468
\(804\) 0 0
\(805\) −7.29150 −0.256992
\(806\) 0 0
\(807\) −10.9373 −0.385010
\(808\) 0 0
\(809\) 12.5830 0.442395 0.221197 0.975229i \(-0.429003\pi\)
0.221197 + 0.975229i \(0.429003\pi\)
\(810\) 0 0
\(811\) −26.5830 −0.933456 −0.466728 0.884401i \(-0.654567\pi\)
−0.466728 + 0.884401i \(0.654567\pi\)
\(812\) 0 0
\(813\) −23.2915 −0.816869
\(814\) 0 0
\(815\) −4.35425 −0.152523
\(816\) 0 0
\(817\) −10.9373 −0.382646
\(818\) 0 0
\(819\) −6.00000 −0.209657
\(820\) 0 0
\(821\) 25.2915 0.882680 0.441340 0.897340i \(-0.354503\pi\)
0.441340 + 0.897340i \(0.354503\pi\)
\(822\) 0 0
\(823\) −57.2693 −1.99628 −0.998141 0.0609517i \(-0.980586\pi\)
−0.998141 + 0.0609517i \(0.980586\pi\)
\(824\) 0 0
\(825\) 5.64575 0.196560
\(826\) 0 0
\(827\) −33.2915 −1.15766 −0.578829 0.815449i \(-0.696490\pi\)
−0.578829 + 0.815449i \(0.696490\pi\)
\(828\) 0 0
\(829\) 3.87451 0.134567 0.0672836 0.997734i \(-0.478567\pi\)
0.0672836 + 0.997734i \(0.478567\pi\)
\(830\) 0 0
\(831\) −3.41699 −0.118534
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −9.29150 −0.321546
\(836\) 0 0
\(837\) 2.00000 0.0691301
\(838\) 0 0
\(839\) 39.2915 1.35649 0.678247 0.734834i \(-0.262740\pi\)
0.678247 + 0.734834i \(0.262740\pi\)
\(840\) 0 0
\(841\) −10.0405 −0.346225
\(842\) 0 0
\(843\) 10.9373 0.376699
\(844\) 0 0
\(845\) −10.2915 −0.354038
\(846\) 0 0
\(847\) 76.1033 2.61494
\(848\) 0 0
\(849\) 30.2288 1.03745
\(850\) 0 0
\(851\) −17.8745 −0.612730
\(852\) 0 0
\(853\) −34.7085 −1.18840 −0.594198 0.804319i \(-0.702531\pi\)
−0.594198 + 0.804319i \(0.702531\pi\)
\(854\) 0 0
\(855\) 1.00000 0.0341993
\(856\) 0 0
\(857\) 39.7490 1.35780 0.678900 0.734231i \(-0.262457\pi\)
0.678900 + 0.734231i \(0.262457\pi\)
\(858\) 0 0
\(859\) −45.1660 −1.54104 −0.770522 0.637413i \(-0.780004\pi\)
−0.770522 + 0.637413i \(0.780004\pi\)
\(860\) 0 0
\(861\) −13.2915 −0.452973
\(862\) 0 0
\(863\) 41.1660 1.40131 0.700654 0.713502i \(-0.252892\pi\)
0.700654 + 0.713502i \(0.252892\pi\)
\(864\) 0 0
\(865\) 19.2915 0.655931
\(866\) 0 0
\(867\) −17.0000 −0.577350
\(868\) 0 0
\(869\) 45.1660 1.53215
\(870\) 0 0
\(871\) 6.58301 0.223057
\(872\) 0 0
\(873\) −8.22876 −0.278501
\(874\) 0 0
\(875\) 3.64575 0.123249
\(876\) 0 0
\(877\) 30.8118 1.04044 0.520220 0.854033i \(-0.325850\pi\)
0.520220 + 0.854033i \(0.325850\pi\)
\(878\) 0 0
\(879\) 13.8745 0.467976
\(880\) 0 0
\(881\) −49.2915 −1.66067 −0.830336 0.557262i \(-0.811852\pi\)
−0.830336 + 0.557262i \(0.811852\pi\)
\(882\) 0 0
\(883\) −20.3542 −0.684975 −0.342488 0.939522i \(-0.611269\pi\)
−0.342488 + 0.939522i \(0.611269\pi\)
\(884\) 0 0
\(885\) 7.29150 0.245101
\(886\) 0 0
\(887\) 19.7490 0.663107 0.331554 0.943436i \(-0.392427\pi\)
0.331554 + 0.943436i \(0.392427\pi\)
\(888\) 0 0
\(889\) −67.7490 −2.27223
\(890\) 0 0
\(891\) 5.64575 0.189140
\(892\) 0 0
\(893\) −6.00000 −0.200782
\(894\) 0 0
\(895\) −3.29150 −0.110023
\(896\) 0 0
\(897\) 3.29150 0.109900
\(898\) 0 0
\(899\) 8.70850 0.290445
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) −39.8745 −1.32694
\(904\) 0 0
\(905\) 5.29150 0.175895
\(906\) 0 0
\(907\) −22.1255 −0.734665 −0.367332 0.930090i \(-0.619729\pi\)
−0.367332 + 0.930090i \(0.619729\pi\)
\(908\) 0 0
\(909\) −15.8745 −0.526524
\(910\) 0 0
\(911\) 39.7490 1.31694 0.658472 0.752605i \(-0.271203\pi\)
0.658472 + 0.752605i \(0.271203\pi\)
\(912\) 0 0
\(913\) −89.6235 −2.96611
\(914\) 0 0
\(915\) 0.708497 0.0234222
\(916\) 0 0
\(917\) −8.58301 −0.283436
\(918\) 0 0
\(919\) 2.58301 0.0852055 0.0426027 0.999092i \(-0.486435\pi\)
0.0426027 + 0.999092i \(0.486435\pi\)
\(920\) 0 0
\(921\) −23.2915 −0.767481
\(922\) 0 0
\(923\) 12.0000 0.394985
\(924\) 0 0
\(925\) 8.93725 0.293855
\(926\) 0 0
\(927\) 6.58301 0.216214
\(928\) 0 0
\(929\) 15.8745 0.520826 0.260413 0.965497i \(-0.416141\pi\)
0.260413 + 0.965497i \(0.416141\pi\)
\(930\) 0 0
\(931\) 6.29150 0.206196
\(932\) 0 0
\(933\) 9.64575 0.315788
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 6.70850 0.219157 0.109579 0.993978i \(-0.465050\pi\)
0.109579 + 0.993978i \(0.465050\pi\)
\(938\) 0 0
\(939\) −29.2915 −0.955892
\(940\) 0 0
\(941\) 10.2288 0.333448 0.166724 0.986004i \(-0.446681\pi\)
0.166724 + 0.986004i \(0.446681\pi\)
\(942\) 0 0
\(943\) 7.29150 0.237444
\(944\) 0 0
\(945\) 3.64575 0.118596
\(946\) 0 0
\(947\) −37.2915 −1.21181 −0.605906 0.795537i \(-0.707189\pi\)
−0.605906 + 0.795537i \(0.707189\pi\)
\(948\) 0 0
\(949\) 3.29150 0.106847
\(950\) 0 0
\(951\) 1.41699 0.0459492
\(952\) 0 0
\(953\) −39.2915 −1.27278 −0.636388 0.771369i \(-0.719572\pi\)
−0.636388 + 0.771369i \(0.719572\pi\)
\(954\) 0 0
\(955\) 4.93725 0.159766
\(956\) 0 0
\(957\) 24.5830 0.794656
\(958\) 0 0
\(959\) 75.0405 2.42318
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 0 0
\(963\) 6.58301 0.212134
\(964\) 0 0
\(965\) −12.9373 −0.416465
\(966\) 0 0
\(967\) −14.4797 −0.465637 −0.232819 0.972520i \(-0.574795\pi\)
−0.232819 + 0.972520i \(0.574795\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −51.7490 −1.66070 −0.830352 0.557239i \(-0.811861\pi\)
−0.830352 + 0.557239i \(0.811861\pi\)
\(972\) 0 0
\(973\) −50.5830 −1.62162
\(974\) 0 0
\(975\) −1.64575 −0.0527062
\(976\) 0 0
\(977\) 34.3320 1.09838 0.549189 0.835698i \(-0.314937\pi\)
0.549189 + 0.835698i \(0.314937\pi\)
\(978\) 0 0
\(979\) 80.3320 2.56742
\(980\) 0 0
\(981\) −5.29150 −0.168945
\(982\) 0 0
\(983\) −22.7085 −0.724289 −0.362144 0.932122i \(-0.617955\pi\)
−0.362144 + 0.932122i \(0.617955\pi\)
\(984\) 0 0
\(985\) 24.0000 0.764704
\(986\) 0 0
\(987\) −21.8745 −0.696273
\(988\) 0 0
\(989\) 21.8745 0.695569
\(990\) 0 0
\(991\) −59.7490 −1.89799 −0.948995 0.315291i \(-0.897898\pi\)
−0.948995 + 0.315291i \(0.897898\pi\)
\(992\) 0 0
\(993\) 4.58301 0.145437
\(994\) 0 0
\(995\) 12.7085 0.402886
\(996\) 0 0
\(997\) −42.7085 −1.35259 −0.676296 0.736630i \(-0.736416\pi\)
−0.676296 + 0.736630i \(0.736416\pi\)
\(998\) 0 0
\(999\) 8.93725 0.282762
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 4560.2.a.bp.1.2 2
4.3 odd 2 2280.2.a.n.1.1 2
12.11 even 2 6840.2.a.w.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2280.2.a.n.1.1 2 4.3 odd 2
4560.2.a.bp.1.2 2 1.1 even 1 trivial
6840.2.a.w.1.1 2 12.11 even 2