Properties

Label 8280.2.a.bt.1.3
Level $8280$
Weight $2$
Character 8280.1
Self dual yes
Analytic conductor $66.116$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8280 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8280.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(66.1161328736\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \( x^{6} - 2x^{5} - 16x^{4} + 26x^{3} + 52x^{2} - 48x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(3.58888\) of defining polynomial
Character \(\chi\) \(=\) 8280.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{5} -2.26189 q^{7} +O(q^{10})\) \(q-1.00000 q^{5} -2.26189 q^{7} +0.162910 q^{11} +7.01485 q^{13} -3.86140 q^{17} +1.83709 q^{19} -1.00000 q^{23} +1.00000 q^{25} -7.93835 q^{29} -5.15345 q^{31} +2.26189 q^{35} +6.51608 q^{37} +1.28436 q^{41} -5.17776 q^{43} +4.41710 q^{47} -1.88387 q^{49} -0.792589 q^{53} -0.162910 q^{55} +4.75296 q^{59} +13.0522 q^{61} -7.01485 q^{65} -1.99231 q^{67} -11.9307 q^{71} -11.9230 q^{73} -0.368483 q^{77} +9.21508 q^{79} -7.90634 q^{83} +3.86140 q^{85} +10.2915 q^{89} -15.8668 q^{91} -1.83709 q^{95} +0.560425 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{5} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{5} - 2 q^{7} + 4 q^{11} - 4 q^{17} + 8 q^{19} - 6 q^{23} + 6 q^{25} - 18 q^{29} - 8 q^{31} + 2 q^{35} + 6 q^{37} - 6 q^{41} + 8 q^{43} + 8 q^{47} + 10 q^{49} - 8 q^{53} - 4 q^{55} - 2 q^{59} - 8 q^{61} - 2 q^{67} - 2 q^{71} + 8 q^{73} - 16 q^{77} - 28 q^{79} - 24 q^{83} + 4 q^{85} - 4 q^{89} - 8 q^{91} - 8 q^{95} + 24 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) 0 0
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −2.26189 −0.854912 −0.427456 0.904036i \(-0.640590\pi\)
−0.427456 + 0.904036i \(0.640590\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0.162910 0.0491191 0.0245596 0.999698i \(-0.492182\pi\)
0.0245596 + 0.999698i \(0.492182\pi\)
\(12\) 0 0
\(13\) 7.01485 1.94557 0.972785 0.231711i \(-0.0744322\pi\)
0.972785 + 0.231711i \(0.0744322\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.86140 −0.936527 −0.468264 0.883589i \(-0.655120\pi\)
−0.468264 + 0.883589i \(0.655120\pi\)
\(18\) 0 0
\(19\) 1.83709 0.421457 0.210729 0.977545i \(-0.432416\pi\)
0.210729 + 0.977545i \(0.432416\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −7.93835 −1.47411 −0.737057 0.675831i \(-0.763785\pi\)
−0.737057 + 0.675831i \(0.763785\pi\)
\(30\) 0 0
\(31\) −5.15345 −0.925587 −0.462794 0.886466i \(-0.653153\pi\)
−0.462794 + 0.886466i \(0.653153\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.26189 0.382328
\(36\) 0 0
\(37\) 6.51608 1.07124 0.535618 0.844460i \(-0.320079\pi\)
0.535618 + 0.844460i \(0.320079\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 1.28436 0.200583 0.100292 0.994958i \(-0.468022\pi\)
0.100292 + 0.994958i \(0.468022\pi\)
\(42\) 0 0
\(43\) −5.17776 −0.789601 −0.394801 0.918767i \(-0.629186\pi\)
−0.394801 + 0.918767i \(0.629186\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.41710 0.644301 0.322150 0.946689i \(-0.395594\pi\)
0.322150 + 0.946689i \(0.395594\pi\)
\(48\) 0 0
\(49\) −1.88387 −0.269125
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −0.792589 −0.108870 −0.0544352 0.998517i \(-0.517336\pi\)
−0.0544352 + 0.998517i \(0.517336\pi\)
\(54\) 0 0
\(55\) −0.162910 −0.0219667
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 4.75296 0.618783 0.309392 0.950935i \(-0.399875\pi\)
0.309392 + 0.950935i \(0.399875\pi\)
\(60\) 0 0
\(61\) 13.0522 1.67116 0.835580 0.549369i \(-0.185132\pi\)
0.835580 + 0.549369i \(0.185132\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −7.01485 −0.870085
\(66\) 0 0
\(67\) −1.99231 −0.243399 −0.121700 0.992567i \(-0.538834\pi\)
−0.121700 + 0.992567i \(0.538834\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −11.9307 −1.41592 −0.707958 0.706254i \(-0.750384\pi\)
−0.707958 + 0.706254i \(0.750384\pi\)
\(72\) 0 0
\(73\) −11.9230 −1.39549 −0.697743 0.716348i \(-0.745812\pi\)
−0.697743 + 0.716348i \(0.745812\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −0.368483 −0.0419926
\(78\) 0 0
\(79\) 9.21508 1.03678 0.518389 0.855145i \(-0.326532\pi\)
0.518389 + 0.855145i \(0.326532\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −7.90634 −0.867834 −0.433917 0.900953i \(-0.642869\pi\)
−0.433917 + 0.900953i \(0.642869\pi\)
\(84\) 0 0
\(85\) 3.86140 0.418828
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 10.2915 1.09090 0.545449 0.838144i \(-0.316359\pi\)
0.545449 + 0.838144i \(0.316359\pi\)
\(90\) 0 0
\(91\) −15.8668 −1.66329
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1.83709 −0.188481
\(96\) 0 0
\(97\) 0.560425 0.0569025 0.0284513 0.999595i \(-0.490942\pi\)
0.0284513 + 0.999595i \(0.490942\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 15.9118 1.58328 0.791642 0.610985i \(-0.209227\pi\)
0.791642 + 0.610985i \(0.209227\pi\)
\(102\) 0 0
\(103\) −13.1588 −1.29658 −0.648290 0.761394i \(-0.724515\pi\)
−0.648290 + 0.761394i \(0.724515\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 14.4225 1.39428 0.697138 0.716937i \(-0.254457\pi\)
0.697138 + 0.716937i \(0.254457\pi\)
\(108\) 0 0
\(109\) 7.28443 0.697722 0.348861 0.937175i \(-0.386569\pi\)
0.348861 + 0.937175i \(0.386569\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −10.4414 −0.982246 −0.491123 0.871090i \(-0.663413\pi\)
−0.491123 + 0.871090i \(0.663413\pi\)
\(114\) 0 0
\(115\) 1.00000 0.0932505
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 8.73405 0.800649
\(120\) 0 0
\(121\) −10.9735 −0.997587
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −2.90590 −0.257857 −0.128929 0.991654i \(-0.541154\pi\)
−0.128929 + 0.991654i \(0.541154\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −13.9208 −1.21626 −0.608131 0.793837i \(-0.708080\pi\)
−0.608131 + 0.793837i \(0.708080\pi\)
\(132\) 0 0
\(133\) −4.15529 −0.360309
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 13.0095 1.11148 0.555739 0.831357i \(-0.312435\pi\)
0.555739 + 0.831357i \(0.312435\pi\)
\(138\) 0 0
\(139\) 16.9136 1.43459 0.717295 0.696769i \(-0.245380\pi\)
0.717295 + 0.696769i \(0.245380\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 1.14279 0.0955647
\(144\) 0 0
\(145\) 7.93835 0.659244
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −14.4698 −1.18541 −0.592707 0.805418i \(-0.701941\pi\)
−0.592707 + 0.805418i \(0.701941\pi\)
\(150\) 0 0
\(151\) −14.7780 −1.20261 −0.601307 0.799018i \(-0.705353\pi\)
−0.601307 + 0.799018i \(0.705353\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 5.15345 0.413935
\(156\) 0 0
\(157\) 7.32835 0.584866 0.292433 0.956286i \(-0.405535\pi\)
0.292433 + 0.956286i \(0.405535\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 2.26189 0.178262
\(162\) 0 0
\(163\) 20.3366 1.59289 0.796443 0.604714i \(-0.206713\pi\)
0.796443 + 0.604714i \(0.206713\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −9.87441 −0.764105 −0.382053 0.924141i \(-0.624783\pi\)
−0.382053 + 0.924141i \(0.624783\pi\)
\(168\) 0 0
\(169\) 36.2081 2.78524
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −6.99754 −0.532013 −0.266007 0.963971i \(-0.585704\pi\)
−0.266007 + 0.963971i \(0.585704\pi\)
\(174\) 0 0
\(175\) −2.26189 −0.170982
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −20.9166 −1.56338 −0.781691 0.623666i \(-0.785642\pi\)
−0.781691 + 0.623666i \(0.785642\pi\)
\(180\) 0 0
\(181\) −9.62991 −0.715785 −0.357893 0.933763i \(-0.616505\pi\)
−0.357893 + 0.933763i \(0.616505\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −6.51608 −0.479072
\(186\) 0 0
\(187\) −0.629060 −0.0460014
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 3.86679 0.279791 0.139896 0.990166i \(-0.455323\pi\)
0.139896 + 0.990166i \(0.455323\pi\)
\(192\) 0 0
\(193\) 2.45977 0.177058 0.0885290 0.996074i \(-0.471783\pi\)
0.0885290 + 0.996074i \(0.471783\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −16.5049 −1.17592 −0.587961 0.808889i \(-0.700069\pi\)
−0.587961 + 0.808889i \(0.700069\pi\)
\(198\) 0 0
\(199\) −13.1665 −0.933346 −0.466673 0.884430i \(-0.654547\pi\)
−0.466673 + 0.884430i \(0.654547\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 17.9556 1.26024
\(204\) 0 0
\(205\) −1.28436 −0.0897035
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0.299280 0.0207016
\(210\) 0 0
\(211\) 12.4538 0.857355 0.428678 0.903457i \(-0.358980\pi\)
0.428678 + 0.903457i \(0.358980\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 5.17776 0.353120
\(216\) 0 0
\(217\) 11.6565 0.791296
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −27.0871 −1.82208
\(222\) 0 0
\(223\) −17.5813 −1.17733 −0.588665 0.808377i \(-0.700346\pi\)
−0.588665 + 0.808377i \(0.700346\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −16.5634 −1.09935 −0.549677 0.835377i \(-0.685249\pi\)
−0.549677 + 0.835377i \(0.685249\pi\)
\(228\) 0 0
\(229\) −14.6273 −0.966600 −0.483300 0.875455i \(-0.660562\pi\)
−0.483300 + 0.875455i \(0.660562\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −0.414823 −0.0271759 −0.0135880 0.999908i \(-0.504325\pi\)
−0.0135880 + 0.999908i \(0.504325\pi\)
\(234\) 0 0
\(235\) −4.41710 −0.288140
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −24.6524 −1.59463 −0.797317 0.603561i \(-0.793748\pi\)
−0.797317 + 0.603561i \(0.793748\pi\)
\(240\) 0 0
\(241\) −11.3040 −0.728156 −0.364078 0.931369i \(-0.618616\pi\)
−0.364078 + 0.931369i \(0.618616\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 1.88387 0.120356
\(246\) 0 0
\(247\) 12.8869 0.819975
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −24.0297 −1.51674 −0.758371 0.651824i \(-0.774004\pi\)
−0.758371 + 0.651824i \(0.774004\pi\)
\(252\) 0 0
\(253\) −0.162910 −0.0102420
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −24.4622 −1.52591 −0.762955 0.646452i \(-0.776252\pi\)
−0.762955 + 0.646452i \(0.776252\pi\)
\(258\) 0 0
\(259\) −14.7386 −0.915814
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 3.90869 0.241020 0.120510 0.992712i \(-0.461547\pi\)
0.120510 + 0.992712i \(0.461547\pi\)
\(264\) 0 0
\(265\) 0.792589 0.0486883
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 16.8626 1.02813 0.514065 0.857751i \(-0.328139\pi\)
0.514065 + 0.857751i \(0.328139\pi\)
\(270\) 0 0
\(271\) −30.6128 −1.85959 −0.929796 0.368075i \(-0.880017\pi\)
−0.929796 + 0.368075i \(0.880017\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0.162910 0.00982383
\(276\) 0 0
\(277\) 14.4699 0.869411 0.434705 0.900573i \(-0.356852\pi\)
0.434705 + 0.900573i \(0.356852\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −9.92712 −0.592202 −0.296101 0.955157i \(-0.595687\pi\)
−0.296101 + 0.955157i \(0.595687\pi\)
\(282\) 0 0
\(283\) −13.5212 −0.803755 −0.401877 0.915694i \(-0.631642\pi\)
−0.401877 + 0.915694i \(0.631642\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −2.90507 −0.171481
\(288\) 0 0
\(289\) −2.08959 −0.122917
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −23.1655 −1.35334 −0.676671 0.736285i \(-0.736578\pi\)
−0.676671 + 0.736285i \(0.736578\pi\)
\(294\) 0 0
\(295\) −4.75296 −0.276728
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −7.01485 −0.405679
\(300\) 0 0
\(301\) 11.7115 0.675040
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −13.0522 −0.747365
\(306\) 0 0
\(307\) 32.6898 1.86571 0.932854 0.360255i \(-0.117310\pi\)
0.932854 + 0.360255i \(0.117310\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −34.2472 −1.94198 −0.970991 0.239114i \(-0.923143\pi\)
−0.970991 + 0.239114i \(0.923143\pi\)
\(312\) 0 0
\(313\) −26.6997 −1.50915 −0.754577 0.656211i \(-0.772158\pi\)
−0.754577 + 0.656211i \(0.772158\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 23.7359 1.33314 0.666569 0.745443i \(-0.267762\pi\)
0.666569 + 0.745443i \(0.267762\pi\)
\(318\) 0 0
\(319\) −1.29323 −0.0724072
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −7.09374 −0.394706
\(324\) 0 0
\(325\) 7.01485 0.389114
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −9.99098 −0.550821
\(330\) 0 0
\(331\) −5.10483 −0.280587 −0.140293 0.990110i \(-0.544805\pi\)
−0.140293 + 0.990110i \(0.544805\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 1.99231 0.108851
\(336\) 0 0
\(337\) 6.87848 0.374695 0.187347 0.982294i \(-0.440011\pi\)
0.187347 + 0.982294i \(0.440011\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −0.839547 −0.0454640
\(342\) 0 0
\(343\) 20.0943 1.08499
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −18.3555 −0.985376 −0.492688 0.870206i \(-0.663986\pi\)
−0.492688 + 0.870206i \(0.663986\pi\)
\(348\) 0 0
\(349\) −26.1280 −1.39860 −0.699300 0.714829i \(-0.746505\pi\)
−0.699300 + 0.714829i \(0.746505\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −0.215617 −0.0114762 −0.00573808 0.999984i \(-0.501826\pi\)
−0.00573808 + 0.999984i \(0.501826\pi\)
\(354\) 0 0
\(355\) 11.9307 0.633217
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −2.22691 −0.117532 −0.0587660 0.998272i \(-0.518717\pi\)
−0.0587660 + 0.998272i \(0.518717\pi\)
\(360\) 0 0
\(361\) −15.6251 −0.822374
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 11.9230 0.624080
\(366\) 0 0
\(367\) 20.3289 1.06116 0.530580 0.847635i \(-0.321974\pi\)
0.530580 + 0.847635i \(0.321974\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 1.79274 0.0930747
\(372\) 0 0
\(373\) 14.4704 0.749247 0.374623 0.927177i \(-0.377772\pi\)
0.374623 + 0.927177i \(0.377772\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −55.6863 −2.86799
\(378\) 0 0
\(379\) 10.7514 0.552264 0.276132 0.961120i \(-0.410947\pi\)
0.276132 + 0.961120i \(0.410947\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −16.3752 −0.836734 −0.418367 0.908278i \(-0.637397\pi\)
−0.418367 + 0.908278i \(0.637397\pi\)
\(384\) 0 0
\(385\) 0.368483 0.0187796
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −33.8461 −1.71606 −0.858032 0.513596i \(-0.828313\pi\)
−0.858032 + 0.513596i \(0.828313\pi\)
\(390\) 0 0
\(391\) 3.86140 0.195279
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −9.21508 −0.463661
\(396\) 0 0
\(397\) 13.6249 0.683812 0.341906 0.939734i \(-0.388928\pi\)
0.341906 + 0.939734i \(0.388928\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 14.2277 0.710495 0.355248 0.934772i \(-0.384397\pi\)
0.355248 + 0.934772i \(0.384397\pi\)
\(402\) 0 0
\(403\) −36.1507 −1.80079
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 1.06153 0.0526182
\(408\) 0 0
\(409\) −27.3581 −1.35277 −0.676385 0.736548i \(-0.736455\pi\)
−0.676385 + 0.736548i \(0.736455\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −10.7507 −0.529006
\(414\) 0 0
\(415\) 7.90634 0.388107
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 33.7129 1.64698 0.823490 0.567330i \(-0.192024\pi\)
0.823490 + 0.567330i \(0.192024\pi\)
\(420\) 0 0
\(421\) −21.6839 −1.05681 −0.528404 0.848993i \(-0.677209\pi\)
−0.528404 + 0.848993i \(0.677209\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −3.86140 −0.187305
\(426\) 0 0
\(427\) −29.5225 −1.42870
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 16.7550 0.807058 0.403529 0.914967i \(-0.367783\pi\)
0.403529 + 0.914967i \(0.367783\pi\)
\(432\) 0 0
\(433\) −5.18372 −0.249113 −0.124557 0.992212i \(-0.539751\pi\)
−0.124557 + 0.992212i \(0.539751\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −1.83709 −0.0878799
\(438\) 0 0
\(439\) −23.9780 −1.14441 −0.572204 0.820111i \(-0.693912\pi\)
−0.572204 + 0.820111i \(0.693912\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 21.0157 0.998484 0.499242 0.866463i \(-0.333612\pi\)
0.499242 + 0.866463i \(0.333612\pi\)
\(444\) 0 0
\(445\) −10.2915 −0.487865
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −0.756499 −0.0357014 −0.0178507 0.999841i \(-0.505682\pi\)
−0.0178507 + 0.999841i \(0.505682\pi\)
\(450\) 0 0
\(451\) 0.209234 0.00985246
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 15.8668 0.743847
\(456\) 0 0
\(457\) 40.0031 1.87127 0.935633 0.352975i \(-0.114830\pi\)
0.935633 + 0.352975i \(0.114830\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 8.97357 0.417941 0.208970 0.977922i \(-0.432989\pi\)
0.208970 + 0.977922i \(0.432989\pi\)
\(462\) 0 0
\(463\) 11.3218 0.526166 0.263083 0.964773i \(-0.415261\pi\)
0.263083 + 0.964773i \(0.415261\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −30.3146 −1.40279 −0.701396 0.712772i \(-0.747439\pi\)
−0.701396 + 0.712772i \(0.747439\pi\)
\(468\) 0 0
\(469\) 4.50637 0.208085
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −0.843508 −0.0387845
\(474\) 0 0
\(475\) 1.83709 0.0842915
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −24.5291 −1.12076 −0.560381 0.828235i \(-0.689345\pi\)
−0.560381 + 0.828235i \(0.689345\pi\)
\(480\) 0 0
\(481\) 45.7093 2.08417
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −0.560425 −0.0254476
\(486\) 0 0
\(487\) −10.8620 −0.492204 −0.246102 0.969244i \(-0.579150\pi\)
−0.246102 + 0.969244i \(0.579150\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −8.28328 −0.373819 −0.186910 0.982377i \(-0.559847\pi\)
−0.186910 + 0.982377i \(0.559847\pi\)
\(492\) 0 0
\(493\) 30.6531 1.38055
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 26.9859 1.21048
\(498\) 0 0
\(499\) −21.5719 −0.965693 −0.482847 0.875705i \(-0.660397\pi\)
−0.482847 + 0.875705i \(0.660397\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −10.1937 −0.454513 −0.227256 0.973835i \(-0.572976\pi\)
−0.227256 + 0.973835i \(0.572976\pi\)
\(504\) 0 0
\(505\) −15.9118 −0.708066
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 4.57107 0.202609 0.101304 0.994855i \(-0.467698\pi\)
0.101304 + 0.994855i \(0.467698\pi\)
\(510\) 0 0
\(511\) 26.9685 1.19302
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 13.1588 0.579848
\(516\) 0 0
\(517\) 0.719589 0.0316475
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 24.3266 1.06577 0.532883 0.846189i \(-0.321109\pi\)
0.532883 + 0.846189i \(0.321109\pi\)
\(522\) 0 0
\(523\) −15.8263 −0.692037 −0.346019 0.938228i \(-0.612467\pi\)
−0.346019 + 0.938228i \(0.612467\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 19.8995 0.866837
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 9.00958 0.390248
\(534\) 0 0
\(535\) −14.4225 −0.623539
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −0.306901 −0.0132192
\(540\) 0 0
\(541\) −29.7922 −1.28087 −0.640434 0.768013i \(-0.721246\pi\)
−0.640434 + 0.768013i \(0.721246\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −7.28443 −0.312031
\(546\) 0 0
\(547\) −4.87562 −0.208466 −0.104233 0.994553i \(-0.533239\pi\)
−0.104233 + 0.994553i \(0.533239\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −14.5835 −0.621276
\(552\) 0 0
\(553\) −20.8435 −0.886354
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −5.27495 −0.223507 −0.111753 0.993736i \(-0.535647\pi\)
−0.111753 + 0.993736i \(0.535647\pi\)
\(558\) 0 0
\(559\) −36.3212 −1.53622
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −8.32741 −0.350958 −0.175479 0.984483i \(-0.556147\pi\)
−0.175479 + 0.984483i \(0.556147\pi\)
\(564\) 0 0
\(565\) 10.4414 0.439274
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 19.2051 0.805120 0.402560 0.915394i \(-0.368120\pi\)
0.402560 + 0.915394i \(0.368120\pi\)
\(570\) 0 0
\(571\) 41.9511 1.75560 0.877799 0.479030i \(-0.159012\pi\)
0.877799 + 0.479030i \(0.159012\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −1.00000 −0.0417029
\(576\) 0 0
\(577\) 29.4610 1.22648 0.613238 0.789898i \(-0.289866\pi\)
0.613238 + 0.789898i \(0.289866\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 17.8832 0.741922
\(582\) 0 0
\(583\) −0.129120 −0.00534762
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 7.61488 0.314300 0.157150 0.987575i \(-0.449769\pi\)
0.157150 + 0.987575i \(0.449769\pi\)
\(588\) 0 0
\(589\) −9.46735 −0.390096
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 4.17421 0.171414 0.0857071 0.996320i \(-0.472685\pi\)
0.0857071 + 0.996320i \(0.472685\pi\)
\(594\) 0 0
\(595\) −8.73405 −0.358061
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 2.17716 0.0889562 0.0444781 0.999010i \(-0.485838\pi\)
0.0444781 + 0.999010i \(0.485838\pi\)
\(600\) 0 0
\(601\) 30.1253 1.22884 0.614418 0.788981i \(-0.289391\pi\)
0.614418 + 0.788981i \(0.289391\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 10.9735 0.446135
\(606\) 0 0
\(607\) −12.7789 −0.518680 −0.259340 0.965786i \(-0.583505\pi\)
−0.259340 + 0.965786i \(0.583505\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 30.9853 1.25353
\(612\) 0 0
\(613\) 13.6889 0.552888 0.276444 0.961030i \(-0.410844\pi\)
0.276444 + 0.961030i \(0.410844\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −3.20506 −0.129031 −0.0645155 0.997917i \(-0.520550\pi\)
−0.0645155 + 0.997917i \(0.520550\pi\)
\(618\) 0 0
\(619\) −23.2889 −0.936061 −0.468030 0.883712i \(-0.655036\pi\)
−0.468030 + 0.883712i \(0.655036\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −23.2782 −0.932623
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −25.1612 −1.00324
\(630\) 0 0
\(631\) −28.6706 −1.14136 −0.570679 0.821173i \(-0.693320\pi\)
−0.570679 + 0.821173i \(0.693320\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 2.90590 0.115317
\(636\) 0 0
\(637\) −13.2151 −0.523601
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 17.3621 0.685761 0.342880 0.939379i \(-0.388597\pi\)
0.342880 + 0.939379i \(0.388597\pi\)
\(642\) 0 0
\(643\) 27.2117 1.07313 0.536563 0.843860i \(-0.319722\pi\)
0.536563 + 0.843860i \(0.319722\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 32.7385 1.28708 0.643541 0.765412i \(-0.277465\pi\)
0.643541 + 0.765412i \(0.277465\pi\)
\(648\) 0 0
\(649\) 0.774304 0.0303941
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 6.56644 0.256964 0.128482 0.991712i \(-0.458989\pi\)
0.128482 + 0.991712i \(0.458989\pi\)
\(654\) 0 0
\(655\) 13.9208 0.543929
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −34.8822 −1.35882 −0.679409 0.733760i \(-0.737764\pi\)
−0.679409 + 0.733760i \(0.737764\pi\)
\(660\) 0 0
\(661\) 5.68921 0.221285 0.110642 0.993860i \(-0.464709\pi\)
0.110642 + 0.993860i \(0.464709\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 4.15529 0.161135
\(666\) 0 0
\(667\) 7.93835 0.307374
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 2.12633 0.0820859
\(672\) 0 0
\(673\) 6.24380 0.240681 0.120340 0.992733i \(-0.461601\pi\)
0.120340 + 0.992733i \(0.461601\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 17.1885 0.660607 0.330304 0.943875i \(-0.392849\pi\)
0.330304 + 0.943875i \(0.392849\pi\)
\(678\) 0 0
\(679\) −1.26762 −0.0486467
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −22.0085 −0.842132 −0.421066 0.907030i \(-0.638344\pi\)
−0.421066 + 0.907030i \(0.638344\pi\)
\(684\) 0 0
\(685\) −13.0095 −0.497068
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −5.55989 −0.211815
\(690\) 0 0
\(691\) −5.82031 −0.221415 −0.110708 0.993853i \(-0.535312\pi\)
−0.110708 + 0.993853i \(0.535312\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −16.9136 −0.641568
\(696\) 0 0
\(697\) −4.95942 −0.187851
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −33.9061 −1.28062 −0.640308 0.768118i \(-0.721193\pi\)
−0.640308 + 0.768118i \(0.721193\pi\)
\(702\) 0 0
\(703\) 11.9706 0.451481
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −35.9907 −1.35357
\(708\) 0 0
\(709\) 3.80820 0.143020 0.0715100 0.997440i \(-0.477218\pi\)
0.0715100 + 0.997440i \(0.477218\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 5.15345 0.192998
\(714\) 0 0
\(715\) −1.14279 −0.0427378
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 30.0024 1.11890 0.559449 0.828864i \(-0.311013\pi\)
0.559449 + 0.828864i \(0.311013\pi\)
\(720\) 0 0
\(721\) 29.7638 1.10846
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −7.93835 −0.294823
\(726\) 0 0
\(727\) −45.0419 −1.67051 −0.835257 0.549860i \(-0.814681\pi\)
−0.835257 + 0.549860i \(0.814681\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 19.9934 0.739483
\(732\) 0 0
\(733\) 35.2270 1.30114 0.650569 0.759447i \(-0.274530\pi\)
0.650569 + 0.759447i \(0.274530\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −0.324566 −0.0119556
\(738\) 0 0
\(739\) −31.9520 −1.17537 −0.587686 0.809089i \(-0.699961\pi\)
−0.587686 + 0.809089i \(0.699961\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −31.8142 −1.16715 −0.583575 0.812059i \(-0.698347\pi\)
−0.583575 + 0.812059i \(0.698347\pi\)
\(744\) 0 0
\(745\) 14.4698 0.530133
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −32.6220 −1.19198
\(750\) 0 0
\(751\) 1.82206 0.0664880 0.0332440 0.999447i \(-0.489416\pi\)
0.0332440 + 0.999447i \(0.489416\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 14.7780 0.537825
\(756\) 0 0
\(757\) 18.2276 0.662493 0.331246 0.943544i \(-0.392531\pi\)
0.331246 + 0.943544i \(0.392531\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −16.0469 −0.581701 −0.290851 0.956769i \(-0.593938\pi\)
−0.290851 + 0.956769i \(0.593938\pi\)
\(762\) 0 0
\(763\) −16.4765 −0.596491
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 33.3413 1.20389
\(768\) 0 0
\(769\) 15.9910 0.576650 0.288325 0.957533i \(-0.406902\pi\)
0.288325 + 0.957533i \(0.406902\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 30.1400 1.08406 0.542031 0.840359i \(-0.317656\pi\)
0.542031 + 0.840359i \(0.317656\pi\)
\(774\) 0 0
\(775\) −5.15345 −0.185117
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 2.35948 0.0845372
\(780\) 0 0
\(781\) −1.94363 −0.0695486
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −7.32835 −0.261560
\(786\) 0 0
\(787\) 24.4650 0.872083 0.436042 0.899926i \(-0.356380\pi\)
0.436042 + 0.899926i \(0.356380\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 23.6173 0.839734
\(792\) 0 0
\(793\) 91.5591 3.25136
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 11.5379 0.408694 0.204347 0.978898i \(-0.434493\pi\)
0.204347 + 0.978898i \(0.434493\pi\)
\(798\) 0 0
\(799\) −17.0562 −0.603405
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −1.94238 −0.0685450
\(804\) 0 0
\(805\) −2.26189 −0.0797210
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 33.1152 1.16427 0.582134 0.813093i \(-0.302218\pi\)
0.582134 + 0.813093i \(0.302218\pi\)
\(810\) 0 0
\(811\) −23.2159 −0.815221 −0.407611 0.913156i \(-0.633638\pi\)
−0.407611 + 0.913156i \(0.633638\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −20.3366 −0.712360
\(816\) 0 0
\(817\) −9.51201 −0.332783
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −25.2274 −0.880442 −0.440221 0.897889i \(-0.645100\pi\)
−0.440221 + 0.897889i \(0.645100\pi\)
\(822\) 0 0
\(823\) −30.6014 −1.06670 −0.533349 0.845895i \(-0.679067\pi\)
−0.533349 + 0.845895i \(0.679067\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 27.4393 0.954157 0.477078 0.878861i \(-0.341696\pi\)
0.477078 + 0.878861i \(0.341696\pi\)
\(828\) 0 0
\(829\) 34.5658 1.20052 0.600260 0.799805i \(-0.295064\pi\)
0.600260 + 0.799805i \(0.295064\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 7.27439 0.252043
\(834\) 0 0
\(835\) 9.87441 0.341718
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 40.8496 1.41028 0.705142 0.709067i \(-0.250883\pi\)
0.705142 + 0.709067i \(0.250883\pi\)
\(840\) 0 0
\(841\) 34.0173 1.17301
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −36.2081 −1.24560
\(846\) 0 0
\(847\) 24.8207 0.852850
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −6.51608 −0.223368
\(852\) 0 0
\(853\) −16.3456 −0.559661 −0.279830 0.960049i \(-0.590278\pi\)
−0.279830 + 0.960049i \(0.590278\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −56.9495 −1.94536 −0.972679 0.232156i \(-0.925422\pi\)
−0.972679 + 0.232156i \(0.925422\pi\)
\(858\) 0 0
\(859\) −40.5938 −1.38504 −0.692522 0.721397i \(-0.743500\pi\)
−0.692522 + 0.721397i \(0.743500\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −2.26093 −0.0769629 −0.0384815 0.999259i \(-0.512252\pi\)
−0.0384815 + 0.999259i \(0.512252\pi\)
\(864\) 0 0
\(865\) 6.99754 0.237924
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 1.50123 0.0509256
\(870\) 0 0
\(871\) −13.9757 −0.473550
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 2.26189 0.0764657
\(876\) 0 0
\(877\) 15.3501 0.518336 0.259168 0.965832i \(-0.416552\pi\)
0.259168 + 0.965832i \(0.416552\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 48.2365 1.62513 0.812564 0.582872i \(-0.198071\pi\)
0.812564 + 0.582872i \(0.198071\pi\)
\(882\) 0 0
\(883\) −11.9832 −0.403267 −0.201634 0.979461i \(-0.564625\pi\)
−0.201634 + 0.979461i \(0.564625\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −36.7597 −1.23427 −0.617134 0.786858i \(-0.711706\pi\)
−0.617134 + 0.786858i \(0.711706\pi\)
\(888\) 0 0
\(889\) 6.57282 0.220445
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 8.11462 0.271545
\(894\) 0 0
\(895\) 20.9166 0.699165
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 40.9099 1.36442
\(900\) 0 0
\(901\) 3.06050 0.101960
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 9.62991 0.320109
\(906\) 0 0
\(907\) 42.8633 1.42325 0.711627 0.702558i \(-0.247959\pi\)
0.711627 + 0.702558i \(0.247959\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −14.9800 −0.496310 −0.248155 0.968720i \(-0.579824\pi\)
−0.248155 + 0.968720i \(0.579824\pi\)
\(912\) 0 0
\(913\) −1.28802 −0.0426273
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 31.4872 1.03980
\(918\) 0 0
\(919\) −18.9236 −0.624231 −0.312115 0.950044i \(-0.601038\pi\)
−0.312115 + 0.950044i \(0.601038\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −83.6923 −2.75476
\(924\) 0 0
\(925\) 6.51608 0.214247
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 22.8137 0.748495 0.374247 0.927329i \(-0.377901\pi\)
0.374247 + 0.927329i \(0.377901\pi\)
\(930\) 0 0
\(931\) −3.46084 −0.113425
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0.629060 0.0205725
\(936\) 0 0
\(937\) 47.4243 1.54929 0.774643 0.632399i \(-0.217930\pi\)
0.774643 + 0.632399i \(0.217930\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 6.27642 0.204605 0.102303 0.994753i \(-0.467379\pi\)
0.102303 + 0.994753i \(0.467379\pi\)
\(942\) 0 0
\(943\) −1.28436 −0.0418244
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −30.4942 −0.990927 −0.495464 0.868629i \(-0.665002\pi\)
−0.495464 + 0.868629i \(0.665002\pi\)
\(948\) 0 0
\(949\) −83.6383 −2.71501
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 53.3375 1.72777 0.863885 0.503689i \(-0.168024\pi\)
0.863885 + 0.503689i \(0.168024\pi\)
\(954\) 0 0
\(955\) −3.86679 −0.125126
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −29.4260 −0.950216
\(960\) 0 0
\(961\) −4.44195 −0.143289
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −2.45977 −0.0791827
\(966\) 0 0
\(967\) 33.1227 1.06516 0.532578 0.846381i \(-0.321223\pi\)
0.532578 + 0.846381i \(0.321223\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −21.2383 −0.681571 −0.340786 0.940141i \(-0.610693\pi\)
−0.340786 + 0.940141i \(0.610693\pi\)
\(972\) 0 0
\(973\) −38.2566 −1.22645
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −38.2931 −1.22510 −0.612552 0.790430i \(-0.709857\pi\)
−0.612552 + 0.790430i \(0.709857\pi\)
\(978\) 0 0
\(979\) 1.67659 0.0535840
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 41.6546 1.32858 0.664288 0.747477i \(-0.268735\pi\)
0.664288 + 0.747477i \(0.268735\pi\)
\(984\) 0 0
\(985\) 16.5049 0.525888
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 5.17776 0.164643
\(990\) 0 0
\(991\) 47.1319 1.49719 0.748597 0.663025i \(-0.230728\pi\)
0.748597 + 0.663025i \(0.230728\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 13.1665 0.417405
\(996\) 0 0
\(997\) −5.83363 −0.184753 −0.0923764 0.995724i \(-0.529446\pi\)
−0.0923764 + 0.995724i \(0.529446\pi\)
\(998\) 0 0
\(999\) 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 8280.2.a.bt.1.3 6
3.2 odd 2 8280.2.a.bu.1.3 yes 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8280.2.a.bt.1.3 6 1.1 even 1 trivial
8280.2.a.bu.1.3 yes 6 3.2 odd 2