Properties

Label 6240.2.a.bo.1.2
Level $6240$
Weight $2$
Character 6240.1
Self dual yes
Analytic conductor $49.827$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6240 = 2^{5} \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6240.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(49.8266508613\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
Defining polynomial: \( x^{2} - x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-2.37228\) of defining polynomial
Character \(\chi\) \(=\) 6240.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} -1.00000 q^{5} +1.37228 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -1.00000 q^{5} +1.37228 q^{7} +1.00000 q^{9} -1.37228 q^{11} -1.00000 q^{13} -1.00000 q^{15} -5.37228 q^{17} -2.00000 q^{19} +1.37228 q^{21} +3.37228 q^{23} +1.00000 q^{25} +1.00000 q^{27} +8.74456 q^{29} -4.74456 q^{31} -1.37228 q^{33} -1.37228 q^{35} -1.37228 q^{37} -1.00000 q^{39} +5.37228 q^{41} -6.74456 q^{43} -1.00000 q^{45} -8.74456 q^{47} -5.11684 q^{49} -5.37228 q^{51} -13.3723 q^{53} +1.37228 q^{55} -2.00000 q^{57} +8.74456 q^{59} +8.11684 q^{61} +1.37228 q^{63} +1.00000 q^{65} +12.7446 q^{67} +3.37228 q^{69} -1.37228 q^{71} -6.00000 q^{73} +1.00000 q^{75} -1.88316 q^{77} -15.3723 q^{79} +1.00000 q^{81} -3.25544 q^{83} +5.37228 q^{85} +8.74456 q^{87} +4.11684 q^{89} -1.37228 q^{91} -4.74456 q^{93} +2.00000 q^{95} -8.11684 q^{97} -1.37228 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 2 q^{5} - 3 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} - 2 q^{5} - 3 q^{7} + 2 q^{9} + 3 q^{11} - 2 q^{13} - 2 q^{15} - 5 q^{17} - 4 q^{19} - 3 q^{21} + q^{23} + 2 q^{25} + 2 q^{27} + 6 q^{29} + 2 q^{31} + 3 q^{33} + 3 q^{35} + 3 q^{37} - 2 q^{39} + 5 q^{41} - 2 q^{43} - 2 q^{45} - 6 q^{47} + 7 q^{49} - 5 q^{51} - 21 q^{53} - 3 q^{55} - 4 q^{57} + 6 q^{59} - q^{61} - 3 q^{63} + 2 q^{65} + 14 q^{67} + q^{69} + 3 q^{71} - 12 q^{73} + 2 q^{75} - 21 q^{77} - 25 q^{79} + 2 q^{81} - 18 q^{83} + 5 q^{85} + 6 q^{87} - 9 q^{89} + 3 q^{91} + 2 q^{93} + 4 q^{95} + q^{97} + 3 q^{99}+O(q^{100}) \) 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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 1.37228 0.518674 0.259337 0.965787i \(-0.416496\pi\)
0.259337 + 0.965787i \(0.416496\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.37228 −0.413758 −0.206879 0.978366i \(-0.566331\pi\)
−0.206879 + 0.978366i \(0.566331\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) −5.37228 −1.30297 −0.651485 0.758662i \(-0.725854\pi\)
−0.651485 + 0.758662i \(0.725854\pi\)
\(18\) 0 0
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) 1.37228 0.299456
\(22\) 0 0
\(23\) 3.37228 0.703169 0.351585 0.936156i \(-0.385643\pi\)
0.351585 + 0.936156i \(0.385643\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 8.74456 1.62382 0.811912 0.583779i \(-0.198427\pi\)
0.811912 + 0.583779i \(0.198427\pi\)
\(30\) 0 0
\(31\) −4.74456 −0.852149 −0.426074 0.904688i \(-0.640104\pi\)
−0.426074 + 0.904688i \(0.640104\pi\)
\(32\) 0 0
\(33\) −1.37228 −0.238884
\(34\) 0 0
\(35\) −1.37228 −0.231958
\(36\) 0 0
\(37\) −1.37228 −0.225602 −0.112801 0.993618i \(-0.535982\pi\)
−0.112801 + 0.993618i \(0.535982\pi\)
\(38\) 0 0
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) 5.37228 0.839009 0.419505 0.907753i \(-0.362204\pi\)
0.419505 + 0.907753i \(0.362204\pi\)
\(42\) 0 0
\(43\) −6.74456 −1.02854 −0.514268 0.857629i \(-0.671936\pi\)
−0.514268 + 0.857629i \(0.671936\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) −8.74456 −1.27553 −0.637763 0.770233i \(-0.720140\pi\)
−0.637763 + 0.770233i \(0.720140\pi\)
\(48\) 0 0
\(49\) −5.11684 −0.730978
\(50\) 0 0
\(51\) −5.37228 −0.752270
\(52\) 0 0
\(53\) −13.3723 −1.83682 −0.918412 0.395625i \(-0.870528\pi\)
−0.918412 + 0.395625i \(0.870528\pi\)
\(54\) 0 0
\(55\) 1.37228 0.185038
\(56\) 0 0
\(57\) −2.00000 −0.264906
\(58\) 0 0
\(59\) 8.74456 1.13845 0.569223 0.822183i \(-0.307244\pi\)
0.569223 + 0.822183i \(0.307244\pi\)
\(60\) 0 0
\(61\) 8.11684 1.03926 0.519628 0.854393i \(-0.326071\pi\)
0.519628 + 0.854393i \(0.326071\pi\)
\(62\) 0 0
\(63\) 1.37228 0.172891
\(64\) 0 0
\(65\) 1.00000 0.124035
\(66\) 0 0
\(67\) 12.7446 1.55700 0.778498 0.627647i \(-0.215982\pi\)
0.778498 + 0.627647i \(0.215982\pi\)
\(68\) 0 0
\(69\) 3.37228 0.405975
\(70\) 0 0
\(71\) −1.37228 −0.162860 −0.0814299 0.996679i \(-0.525949\pi\)
−0.0814299 + 0.996679i \(0.525949\pi\)
\(72\) 0 0
\(73\) −6.00000 −0.702247 −0.351123 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) −1.88316 −0.214606
\(78\) 0 0
\(79\) −15.3723 −1.72952 −0.864758 0.502188i \(-0.832528\pi\)
−0.864758 + 0.502188i \(0.832528\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −3.25544 −0.357331 −0.178665 0.983910i \(-0.557178\pi\)
−0.178665 + 0.983910i \(0.557178\pi\)
\(84\) 0 0
\(85\) 5.37228 0.582706
\(86\) 0 0
\(87\) 8.74456 0.937516
\(88\) 0 0
\(89\) 4.11684 0.436385 0.218192 0.975906i \(-0.429984\pi\)
0.218192 + 0.975906i \(0.429984\pi\)
\(90\) 0 0
\(91\) −1.37228 −0.143854
\(92\) 0 0
\(93\) −4.74456 −0.491988
\(94\) 0 0
\(95\) 2.00000 0.205196
\(96\) 0 0
\(97\) −8.11684 −0.824141 −0.412070 0.911152i \(-0.635194\pi\)
−0.412070 + 0.911152i \(0.635194\pi\)
\(98\) 0 0
\(99\) −1.37228 −0.137919
\(100\) 0 0
\(101\) −16.7446 −1.66615 −0.833073 0.553163i \(-0.813421\pi\)
−0.833073 + 0.553163i \(0.813421\pi\)
\(102\) 0 0
\(103\) −8.00000 −0.788263 −0.394132 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) 0 0
\(105\) −1.37228 −0.133921
\(106\) 0 0
\(107\) 7.37228 0.712705 0.356353 0.934352i \(-0.384020\pi\)
0.356353 + 0.934352i \(0.384020\pi\)
\(108\) 0 0
\(109\) −20.7446 −1.98697 −0.993484 0.113968i \(-0.963644\pi\)
−0.993484 + 0.113968i \(0.963644\pi\)
\(110\) 0 0
\(111\) −1.37228 −0.130251
\(112\) 0 0
\(113\) −2.00000 −0.188144 −0.0940721 0.995565i \(-0.529988\pi\)
−0.0940721 + 0.995565i \(0.529988\pi\)
\(114\) 0 0
\(115\) −3.37228 −0.314467
\(116\) 0 0
\(117\) −1.00000 −0.0924500
\(118\) 0 0
\(119\) −7.37228 −0.675816
\(120\) 0 0
\(121\) −9.11684 −0.828804
\(122\) 0 0
\(123\) 5.37228 0.484402
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −16.0000 −1.41977 −0.709885 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) 0 0
\(129\) −6.74456 −0.593826
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 0 0
\(133\) −2.74456 −0.237984
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) 15.4891 1.32333 0.661663 0.749802i \(-0.269851\pi\)
0.661663 + 0.749802i \(0.269851\pi\)
\(138\) 0 0
\(139\) −14.1168 −1.19738 −0.598688 0.800983i \(-0.704311\pi\)
−0.598688 + 0.800983i \(0.704311\pi\)
\(140\) 0 0
\(141\) −8.74456 −0.736425
\(142\) 0 0
\(143\) 1.37228 0.114756
\(144\) 0 0
\(145\) −8.74456 −0.726196
\(146\) 0 0
\(147\) −5.11684 −0.422030
\(148\) 0 0
\(149\) 4.11684 0.337265 0.168632 0.985679i \(-0.446065\pi\)
0.168632 + 0.985679i \(0.446065\pi\)
\(150\) 0 0
\(151\) −14.2337 −1.15832 −0.579161 0.815214i \(-0.696620\pi\)
−0.579161 + 0.815214i \(0.696620\pi\)
\(152\) 0 0
\(153\) −5.37228 −0.434323
\(154\) 0 0
\(155\) 4.74456 0.381092
\(156\) 0 0
\(157\) 2.00000 0.159617 0.0798087 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) 0 0
\(159\) −13.3723 −1.06049
\(160\) 0 0
\(161\) 4.62772 0.364715
\(162\) 0 0
\(163\) 10.8614 0.850731 0.425366 0.905022i \(-0.360146\pi\)
0.425366 + 0.905022i \(0.360146\pi\)
\(164\) 0 0
\(165\) 1.37228 0.106832
\(166\) 0 0
\(167\) −8.74456 −0.676675 −0.338337 0.941025i \(-0.609864\pi\)
−0.338337 + 0.941025i \(0.609864\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) −2.00000 −0.152944
\(172\) 0 0
\(173\) 11.4891 0.873502 0.436751 0.899582i \(-0.356129\pi\)
0.436751 + 0.899582i \(0.356129\pi\)
\(174\) 0 0
\(175\) 1.37228 0.103735
\(176\) 0 0
\(177\) 8.74456 0.657282
\(178\) 0 0
\(179\) 14.7446 1.10206 0.551030 0.834485i \(-0.314235\pi\)
0.551030 + 0.834485i \(0.314235\pi\)
\(180\) 0 0
\(181\) 12.1168 0.900638 0.450319 0.892868i \(-0.351310\pi\)
0.450319 + 0.892868i \(0.351310\pi\)
\(182\) 0 0
\(183\) 8.11684 0.600014
\(184\) 0 0
\(185\) 1.37228 0.100892
\(186\) 0 0
\(187\) 7.37228 0.539115
\(188\) 0 0
\(189\) 1.37228 0.0998188
\(190\) 0 0
\(191\) −2.74456 −0.198590 −0.0992948 0.995058i \(-0.531659\pi\)
−0.0992948 + 0.995058i \(0.531659\pi\)
\(192\) 0 0
\(193\) −3.88316 −0.279516 −0.139758 0.990186i \(-0.544632\pi\)
−0.139758 + 0.990186i \(0.544632\pi\)
\(194\) 0 0
\(195\) 1.00000 0.0716115
\(196\) 0 0
\(197\) −19.4891 −1.38854 −0.694271 0.719713i \(-0.744273\pi\)
−0.694271 + 0.719713i \(0.744273\pi\)
\(198\) 0 0
\(199\) 8.00000 0.567105 0.283552 0.958957i \(-0.408487\pi\)
0.283552 + 0.958957i \(0.408487\pi\)
\(200\) 0 0
\(201\) 12.7446 0.898932
\(202\) 0 0
\(203\) 12.0000 0.842235
\(204\) 0 0
\(205\) −5.37228 −0.375216
\(206\) 0 0
\(207\) 3.37228 0.234390
\(208\) 0 0
\(209\) 2.74456 0.189845
\(210\) 0 0
\(211\) 1.48913 0.102516 0.0512578 0.998685i \(-0.483677\pi\)
0.0512578 + 0.998685i \(0.483677\pi\)
\(212\) 0 0
\(213\) −1.37228 −0.0940272
\(214\) 0 0
\(215\) 6.74456 0.459975
\(216\) 0 0
\(217\) −6.51087 −0.441987
\(218\) 0 0
\(219\) −6.00000 −0.405442
\(220\) 0 0
\(221\) 5.37228 0.361379
\(222\) 0 0
\(223\) 14.2337 0.953158 0.476579 0.879132i \(-0.341877\pi\)
0.476579 + 0.879132i \(0.341877\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 14.0000 0.929213 0.464606 0.885517i \(-0.346196\pi\)
0.464606 + 0.885517i \(0.346196\pi\)
\(228\) 0 0
\(229\) −10.2337 −0.676261 −0.338131 0.941099i \(-0.609795\pi\)
−0.338131 + 0.941099i \(0.609795\pi\)
\(230\) 0 0
\(231\) −1.88316 −0.123903
\(232\) 0 0
\(233\) −10.6277 −0.696245 −0.348122 0.937449i \(-0.613181\pi\)
−0.348122 + 0.937449i \(0.613181\pi\)
\(234\) 0 0
\(235\) 8.74456 0.570432
\(236\) 0 0
\(237\) −15.3723 −0.998537
\(238\) 0 0
\(239\) 25.6060 1.65631 0.828156 0.560497i \(-0.189390\pi\)
0.828156 + 0.560497i \(0.189390\pi\)
\(240\) 0 0
\(241\) −4.74456 −0.305624 −0.152812 0.988255i \(-0.548833\pi\)
−0.152812 + 0.988255i \(0.548833\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 5.11684 0.326903
\(246\) 0 0
\(247\) 2.00000 0.127257
\(248\) 0 0
\(249\) −3.25544 −0.206305
\(250\) 0 0
\(251\) −26.9783 −1.70285 −0.851426 0.524475i \(-0.824262\pi\)
−0.851426 + 0.524475i \(0.824262\pi\)
\(252\) 0 0
\(253\) −4.62772 −0.290942
\(254\) 0 0
\(255\) 5.37228 0.336425
\(256\) 0 0
\(257\) −14.0000 −0.873296 −0.436648 0.899632i \(-0.643834\pi\)
−0.436648 + 0.899632i \(0.643834\pi\)
\(258\) 0 0
\(259\) −1.88316 −0.117014
\(260\) 0 0
\(261\) 8.74456 0.541275
\(262\) 0 0
\(263\) −25.4891 −1.57173 −0.785863 0.618400i \(-0.787781\pi\)
−0.785863 + 0.618400i \(0.787781\pi\)
\(264\) 0 0
\(265\) 13.3723 0.821453
\(266\) 0 0
\(267\) 4.11684 0.251947
\(268\) 0 0
\(269\) −31.7228 −1.93417 −0.967087 0.254446i \(-0.918107\pi\)
−0.967087 + 0.254446i \(0.918107\pi\)
\(270\) 0 0
\(271\) 19.4891 1.18388 0.591940 0.805982i \(-0.298362\pi\)
0.591940 + 0.805982i \(0.298362\pi\)
\(272\) 0 0
\(273\) −1.37228 −0.0830542
\(274\) 0 0
\(275\) −1.37228 −0.0827517
\(276\) 0 0
\(277\) −32.7446 −1.96743 −0.983715 0.179735i \(-0.942476\pi\)
−0.983715 + 0.179735i \(0.942476\pi\)
\(278\) 0 0
\(279\) −4.74456 −0.284050
\(280\) 0 0
\(281\) −19.4891 −1.16262 −0.581312 0.813681i \(-0.697460\pi\)
−0.581312 + 0.813681i \(0.697460\pi\)
\(282\) 0 0
\(283\) −2.51087 −0.149256 −0.0746280 0.997211i \(-0.523777\pi\)
−0.0746280 + 0.997211i \(0.523777\pi\)
\(284\) 0 0
\(285\) 2.00000 0.118470
\(286\) 0 0
\(287\) 7.37228 0.435172
\(288\) 0 0
\(289\) 11.8614 0.697730
\(290\) 0 0
\(291\) −8.11684 −0.475818
\(292\) 0 0
\(293\) 8.74456 0.510863 0.255431 0.966827i \(-0.417783\pi\)
0.255431 + 0.966827i \(0.417783\pi\)
\(294\) 0 0
\(295\) −8.74456 −0.509128
\(296\) 0 0
\(297\) −1.37228 −0.0796278
\(298\) 0 0
\(299\) −3.37228 −0.195024
\(300\) 0 0
\(301\) −9.25544 −0.533475
\(302\) 0 0
\(303\) −16.7446 −0.961950
\(304\) 0 0
\(305\) −8.11684 −0.464769
\(306\) 0 0
\(307\) −8.11684 −0.463253 −0.231626 0.972805i \(-0.574405\pi\)
−0.231626 + 0.972805i \(0.574405\pi\)
\(308\) 0 0
\(309\) −8.00000 −0.455104
\(310\) 0 0
\(311\) 25.7228 1.45861 0.729303 0.684190i \(-0.239844\pi\)
0.729303 + 0.684190i \(0.239844\pi\)
\(312\) 0 0
\(313\) 16.9783 0.959667 0.479834 0.877359i \(-0.340697\pi\)
0.479834 + 0.877359i \(0.340697\pi\)
\(314\) 0 0
\(315\) −1.37228 −0.0773193
\(316\) 0 0
\(317\) −6.00000 −0.336994 −0.168497 0.985702i \(-0.553891\pi\)
−0.168497 + 0.985702i \(0.553891\pi\)
\(318\) 0 0
\(319\) −12.0000 −0.671871
\(320\) 0 0
\(321\) 7.37228 0.411481
\(322\) 0 0
\(323\) 10.7446 0.597843
\(324\) 0 0
\(325\) −1.00000 −0.0554700
\(326\) 0 0
\(327\) −20.7446 −1.14718
\(328\) 0 0
\(329\) −12.0000 −0.661581
\(330\) 0 0
\(331\) 22.0000 1.20923 0.604615 0.796518i \(-0.293327\pi\)
0.604615 + 0.796518i \(0.293327\pi\)
\(332\) 0 0
\(333\) −1.37228 −0.0752006
\(334\) 0 0
\(335\) −12.7446 −0.696310
\(336\) 0 0
\(337\) −26.0000 −1.41631 −0.708155 0.706057i \(-0.750472\pi\)
−0.708155 + 0.706057i \(0.750472\pi\)
\(338\) 0 0
\(339\) −2.00000 −0.108625
\(340\) 0 0
\(341\) 6.51087 0.352584
\(342\) 0 0
\(343\) −16.6277 −0.897812
\(344\) 0 0
\(345\) −3.37228 −0.181558
\(346\) 0 0
\(347\) 17.8832 0.960018 0.480009 0.877264i \(-0.340633\pi\)
0.480009 + 0.877264i \(0.340633\pi\)
\(348\) 0 0
\(349\) −24.9783 −1.33706 −0.668528 0.743687i \(-0.733075\pi\)
−0.668528 + 0.743687i \(0.733075\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) 0 0
\(353\) −20.7446 −1.10412 −0.552061 0.833804i \(-0.686158\pi\)
−0.552061 + 0.833804i \(0.686158\pi\)
\(354\) 0 0
\(355\) 1.37228 0.0728331
\(356\) 0 0
\(357\) −7.37228 −0.390183
\(358\) 0 0
\(359\) 20.7446 1.09486 0.547428 0.836853i \(-0.315607\pi\)
0.547428 + 0.836853i \(0.315607\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) −9.11684 −0.478510
\(364\) 0 0
\(365\) 6.00000 0.314054
\(366\) 0 0
\(367\) 13.2554 0.691928 0.345964 0.938248i \(-0.387552\pi\)
0.345964 + 0.938248i \(0.387552\pi\)
\(368\) 0 0
\(369\) 5.37228 0.279670
\(370\) 0 0
\(371\) −18.3505 −0.952712
\(372\) 0 0
\(373\) 7.48913 0.387772 0.193886 0.981024i \(-0.437891\pi\)
0.193886 + 0.981024i \(0.437891\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) −8.74456 −0.450368
\(378\) 0 0
\(379\) 26.2337 1.34753 0.673767 0.738944i \(-0.264675\pi\)
0.673767 + 0.738944i \(0.264675\pi\)
\(380\) 0 0
\(381\) −16.0000 −0.819705
\(382\) 0 0
\(383\) 4.51087 0.230495 0.115247 0.993337i \(-0.463234\pi\)
0.115247 + 0.993337i \(0.463234\pi\)
\(384\) 0 0
\(385\) 1.88316 0.0959745
\(386\) 0 0
\(387\) −6.74456 −0.342845
\(388\) 0 0
\(389\) −28.7446 −1.45741 −0.728704 0.684829i \(-0.759877\pi\)
−0.728704 + 0.684829i \(0.759877\pi\)
\(390\) 0 0
\(391\) −18.1168 −0.916208
\(392\) 0 0
\(393\) 12.0000 0.605320
\(394\) 0 0
\(395\) 15.3723 0.773463
\(396\) 0 0
\(397\) 25.6060 1.28513 0.642563 0.766233i \(-0.277871\pi\)
0.642563 + 0.766233i \(0.277871\pi\)
\(398\) 0 0
\(399\) −2.74456 −0.137400
\(400\) 0 0
\(401\) −11.4891 −0.573740 −0.286870 0.957970i \(-0.592615\pi\)
−0.286870 + 0.957970i \(0.592615\pi\)
\(402\) 0 0
\(403\) 4.74456 0.236343
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) 1.88316 0.0933446
\(408\) 0 0
\(409\) 6.23369 0.308236 0.154118 0.988052i \(-0.450746\pi\)
0.154118 + 0.988052i \(0.450746\pi\)
\(410\) 0 0
\(411\) 15.4891 0.764022
\(412\) 0 0
\(413\) 12.0000 0.590481
\(414\) 0 0
\(415\) 3.25544 0.159803
\(416\) 0 0
\(417\) −14.1168 −0.691305
\(418\) 0 0
\(419\) 10.7446 0.524906 0.262453 0.964945i \(-0.415468\pi\)
0.262453 + 0.964945i \(0.415468\pi\)
\(420\) 0 0
\(421\) 24.7446 1.20598 0.602988 0.797750i \(-0.293977\pi\)
0.602988 + 0.797750i \(0.293977\pi\)
\(422\) 0 0
\(423\) −8.74456 −0.425175
\(424\) 0 0
\(425\) −5.37228 −0.260594
\(426\) 0 0
\(427\) 11.1386 0.539034
\(428\) 0 0
\(429\) 1.37228 0.0662544
\(430\) 0 0
\(431\) 14.2337 0.685613 0.342806 0.939406i \(-0.388623\pi\)
0.342806 + 0.939406i \(0.388623\pi\)
\(432\) 0 0
\(433\) −0.510875 −0.0245511 −0.0122755 0.999925i \(-0.503908\pi\)
−0.0122755 + 0.999925i \(0.503908\pi\)
\(434\) 0 0
\(435\) −8.74456 −0.419270
\(436\) 0 0
\(437\) −6.74456 −0.322636
\(438\) 0 0
\(439\) −10.1168 −0.482851 −0.241425 0.970419i \(-0.577615\pi\)
−0.241425 + 0.970419i \(0.577615\pi\)
\(440\) 0 0
\(441\) −5.11684 −0.243659
\(442\) 0 0
\(443\) 12.8614 0.611064 0.305532 0.952182i \(-0.401166\pi\)
0.305532 + 0.952182i \(0.401166\pi\)
\(444\) 0 0
\(445\) −4.11684 −0.195157
\(446\) 0 0
\(447\) 4.11684 0.194720
\(448\) 0 0
\(449\) −16.1168 −0.760601 −0.380300 0.924863i \(-0.624179\pi\)
−0.380300 + 0.924863i \(0.624179\pi\)
\(450\) 0 0
\(451\) −7.37228 −0.347147
\(452\) 0 0
\(453\) −14.2337 −0.668757
\(454\) 0 0
\(455\) 1.37228 0.0643335
\(456\) 0 0
\(457\) −2.62772 −0.122919 −0.0614597 0.998110i \(-0.519576\pi\)
−0.0614597 + 0.998110i \(0.519576\pi\)
\(458\) 0 0
\(459\) −5.37228 −0.250757
\(460\) 0 0
\(461\) 18.8614 0.878463 0.439232 0.898374i \(-0.355251\pi\)
0.439232 + 0.898374i \(0.355251\pi\)
\(462\) 0 0
\(463\) −41.6060 −1.93359 −0.966797 0.255547i \(-0.917745\pi\)
−0.966797 + 0.255547i \(0.917745\pi\)
\(464\) 0 0
\(465\) 4.74456 0.220024
\(466\) 0 0
\(467\) 3.37228 0.156051 0.0780253 0.996951i \(-0.475139\pi\)
0.0780253 + 0.996951i \(0.475139\pi\)
\(468\) 0 0
\(469\) 17.4891 0.807573
\(470\) 0 0
\(471\) 2.00000 0.0921551
\(472\) 0 0
\(473\) 9.25544 0.425565
\(474\) 0 0
\(475\) −2.00000 −0.0917663
\(476\) 0 0
\(477\) −13.3723 −0.612275
\(478\) 0 0
\(479\) −23.0951 −1.05524 −0.527621 0.849480i \(-0.676916\pi\)
−0.527621 + 0.849480i \(0.676916\pi\)
\(480\) 0 0
\(481\) 1.37228 0.0625706
\(482\) 0 0
\(483\) 4.62772 0.210568
\(484\) 0 0
\(485\) 8.11684 0.368567
\(486\) 0 0
\(487\) −41.8397 −1.89594 −0.947968 0.318366i \(-0.896866\pi\)
−0.947968 + 0.318366i \(0.896866\pi\)
\(488\) 0 0
\(489\) 10.8614 0.491170
\(490\) 0 0
\(491\) −6.74456 −0.304378 −0.152189 0.988351i \(-0.548632\pi\)
−0.152189 + 0.988351i \(0.548632\pi\)
\(492\) 0 0
\(493\) −46.9783 −2.11579
\(494\) 0 0
\(495\) 1.37228 0.0616795
\(496\) 0 0
\(497\) −1.88316 −0.0844711
\(498\) 0 0
\(499\) −36.9783 −1.65537 −0.827687 0.561190i \(-0.810344\pi\)
−0.827687 + 0.561190i \(0.810344\pi\)
\(500\) 0 0
\(501\) −8.74456 −0.390678
\(502\) 0 0
\(503\) 41.4891 1.84991 0.924954 0.380078i \(-0.124103\pi\)
0.924954 + 0.380078i \(0.124103\pi\)
\(504\) 0 0
\(505\) 16.7446 0.745123
\(506\) 0 0
\(507\) 1.00000 0.0444116
\(508\) 0 0
\(509\) −19.8832 −0.881305 −0.440653 0.897678i \(-0.645253\pi\)
−0.440653 + 0.897678i \(0.645253\pi\)
\(510\) 0 0
\(511\) −8.23369 −0.364237
\(512\) 0 0
\(513\) −2.00000 −0.0883022
\(514\) 0 0
\(515\) 8.00000 0.352522
\(516\) 0 0
\(517\) 12.0000 0.527759
\(518\) 0 0
\(519\) 11.4891 0.504317
\(520\) 0 0
\(521\) 8.51087 0.372868 0.186434 0.982467i \(-0.440307\pi\)
0.186434 + 0.982467i \(0.440307\pi\)
\(522\) 0 0
\(523\) −13.7228 −0.600057 −0.300028 0.953930i \(-0.596996\pi\)
−0.300028 + 0.953930i \(0.596996\pi\)
\(524\) 0 0
\(525\) 1.37228 0.0598913
\(526\) 0 0
\(527\) 25.4891 1.11032
\(528\) 0 0
\(529\) −11.6277 −0.505553
\(530\) 0 0
\(531\) 8.74456 0.379482
\(532\) 0 0
\(533\) −5.37228 −0.232699
\(534\) 0 0
\(535\) −7.37228 −0.318732
\(536\) 0 0
\(537\) 14.7446 0.636275
\(538\) 0 0
\(539\) 7.02175 0.302448
\(540\) 0 0
\(541\) 7.48913 0.321983 0.160991 0.986956i \(-0.448531\pi\)
0.160991 + 0.986956i \(0.448531\pi\)
\(542\) 0 0
\(543\) 12.1168 0.519984
\(544\) 0 0
\(545\) 20.7446 0.888599
\(546\) 0 0
\(547\) −12.0000 −0.513083 −0.256541 0.966533i \(-0.582583\pi\)
−0.256541 + 0.966533i \(0.582583\pi\)
\(548\) 0 0
\(549\) 8.11684 0.346418
\(550\) 0 0
\(551\) −17.4891 −0.745062
\(552\) 0 0
\(553\) −21.0951 −0.897055
\(554\) 0 0
\(555\) 1.37228 0.0582501
\(556\) 0 0
\(557\) 3.25544 0.137937 0.0689687 0.997619i \(-0.478029\pi\)
0.0689687 + 0.997619i \(0.478029\pi\)
\(558\) 0 0
\(559\) 6.74456 0.285265
\(560\) 0 0
\(561\) 7.37228 0.311258
\(562\) 0 0
\(563\) 13.8832 0.585105 0.292553 0.956249i \(-0.405495\pi\)
0.292553 + 0.956249i \(0.405495\pi\)
\(564\) 0 0
\(565\) 2.00000 0.0841406
\(566\) 0 0
\(567\) 1.37228 0.0576304
\(568\) 0 0
\(569\) 36.9783 1.55021 0.775104 0.631833i \(-0.217697\pi\)
0.775104 + 0.631833i \(0.217697\pi\)
\(570\) 0 0
\(571\) 23.6060 0.987879 0.493940 0.869496i \(-0.335556\pi\)
0.493940 + 0.869496i \(0.335556\pi\)
\(572\) 0 0
\(573\) −2.74456 −0.114656
\(574\) 0 0
\(575\) 3.37228 0.140634
\(576\) 0 0
\(577\) 18.8614 0.785211 0.392605 0.919707i \(-0.371574\pi\)
0.392605 + 0.919707i \(0.371574\pi\)
\(578\) 0 0
\(579\) −3.88316 −0.161378
\(580\) 0 0
\(581\) −4.46738 −0.185338
\(582\) 0 0
\(583\) 18.3505 0.760001
\(584\) 0 0
\(585\) 1.00000 0.0413449
\(586\) 0 0
\(587\) 43.4891 1.79499 0.897494 0.441026i \(-0.145385\pi\)
0.897494 + 0.441026i \(0.145385\pi\)
\(588\) 0 0
\(589\) 9.48913 0.390993
\(590\) 0 0
\(591\) −19.4891 −0.801675
\(592\) 0 0
\(593\) 27.7228 1.13844 0.569220 0.822185i \(-0.307245\pi\)
0.569220 + 0.822185i \(0.307245\pi\)
\(594\) 0 0
\(595\) 7.37228 0.302234
\(596\) 0 0
\(597\) 8.00000 0.327418
\(598\) 0 0
\(599\) −20.0000 −0.817178 −0.408589 0.912719i \(-0.633979\pi\)
−0.408589 + 0.912719i \(0.633979\pi\)
\(600\) 0 0
\(601\) 14.8614 0.606209 0.303105 0.952957i \(-0.401977\pi\)
0.303105 + 0.952957i \(0.401977\pi\)
\(602\) 0 0
\(603\) 12.7446 0.518999
\(604\) 0 0
\(605\) 9.11684 0.370652
\(606\) 0 0
\(607\) 13.7228 0.556992 0.278496 0.960437i \(-0.410164\pi\)
0.278496 + 0.960437i \(0.410164\pi\)
\(608\) 0 0
\(609\) 12.0000 0.486265
\(610\) 0 0
\(611\) 8.74456 0.353767
\(612\) 0 0
\(613\) 48.3505 1.95286 0.976430 0.215835i \(-0.0692474\pi\)
0.976430 + 0.215835i \(0.0692474\pi\)
\(614\) 0 0
\(615\) −5.37228 −0.216631
\(616\) 0 0
\(617\) −46.4674 −1.87071 −0.935353 0.353716i \(-0.884918\pi\)
−0.935353 + 0.353716i \(0.884918\pi\)
\(618\) 0 0
\(619\) −12.7446 −0.512247 −0.256124 0.966644i \(-0.582445\pi\)
−0.256124 + 0.966644i \(0.582445\pi\)
\(620\) 0 0
\(621\) 3.37228 0.135325
\(622\) 0 0
\(623\) 5.64947 0.226341
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 2.74456 0.109607
\(628\) 0 0
\(629\) 7.37228 0.293952
\(630\) 0 0
\(631\) 1.76631 0.0703158 0.0351579 0.999382i \(-0.488807\pi\)
0.0351579 + 0.999382i \(0.488807\pi\)
\(632\) 0 0
\(633\) 1.48913 0.0591874
\(634\) 0 0
\(635\) 16.0000 0.634941
\(636\) 0 0
\(637\) 5.11684 0.202737
\(638\) 0 0
\(639\) −1.37228 −0.0542866
\(640\) 0 0
\(641\) −4.97825 −0.196629 −0.0983145 0.995155i \(-0.531345\pi\)
−0.0983145 + 0.995155i \(0.531345\pi\)
\(642\) 0 0
\(643\) −20.1168 −0.793331 −0.396665 0.917963i \(-0.629833\pi\)
−0.396665 + 0.917963i \(0.629833\pi\)
\(644\) 0 0
\(645\) 6.74456 0.265567
\(646\) 0 0
\(647\) 15.6060 0.613534 0.306767 0.951785i \(-0.400753\pi\)
0.306767 + 0.951785i \(0.400753\pi\)
\(648\) 0 0
\(649\) −12.0000 −0.471041
\(650\) 0 0
\(651\) −6.51087 −0.255181
\(652\) 0 0
\(653\) −27.4891 −1.07573 −0.537866 0.843030i \(-0.680769\pi\)
−0.537866 + 0.843030i \(0.680769\pi\)
\(654\) 0 0
\(655\) −12.0000 −0.468879
\(656\) 0 0
\(657\) −6.00000 −0.234082
\(658\) 0 0
\(659\) 22.9783 0.895106 0.447553 0.894258i \(-0.352296\pi\)
0.447553 + 0.894258i \(0.352296\pi\)
\(660\) 0 0
\(661\) 39.4891 1.53595 0.767974 0.640480i \(-0.221265\pi\)
0.767974 + 0.640480i \(0.221265\pi\)
\(662\) 0 0
\(663\) 5.37228 0.208642
\(664\) 0 0
\(665\) 2.74456 0.106430
\(666\) 0 0
\(667\) 29.4891 1.14182
\(668\) 0 0
\(669\) 14.2337 0.550306
\(670\) 0 0
\(671\) −11.1386 −0.430001
\(672\) 0 0
\(673\) −30.2337 −1.16542 −0.582712 0.812679i \(-0.698008\pi\)
−0.582712 + 0.812679i \(0.698008\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) −26.8614 −1.03237 −0.516184 0.856478i \(-0.672648\pi\)
−0.516184 + 0.856478i \(0.672648\pi\)
\(678\) 0 0
\(679\) −11.1386 −0.427460
\(680\) 0 0
\(681\) 14.0000 0.536481
\(682\) 0 0
\(683\) 2.00000 0.0765279 0.0382639 0.999268i \(-0.487817\pi\)
0.0382639 + 0.999268i \(0.487817\pi\)
\(684\) 0 0
\(685\) −15.4891 −0.591809
\(686\) 0 0
\(687\) −10.2337 −0.390440
\(688\) 0 0
\(689\) 13.3723 0.509443
\(690\) 0 0
\(691\) 16.9783 0.645883 0.322942 0.946419i \(-0.395328\pi\)
0.322942 + 0.946419i \(0.395328\pi\)
\(692\) 0 0
\(693\) −1.88316 −0.0715352
\(694\) 0 0
\(695\) 14.1168 0.535482
\(696\) 0 0
\(697\) −28.8614 −1.09320
\(698\) 0 0
\(699\) −10.6277 −0.401977
\(700\) 0 0
\(701\) 6.23369 0.235443 0.117722 0.993047i \(-0.462441\pi\)
0.117722 + 0.993047i \(0.462441\pi\)
\(702\) 0 0
\(703\) 2.74456 0.103513
\(704\) 0 0
\(705\) 8.74456 0.329339
\(706\) 0 0
\(707\) −22.9783 −0.864186
\(708\) 0 0
\(709\) −45.2119 −1.69797 −0.848985 0.528417i \(-0.822786\pi\)
−0.848985 + 0.528417i \(0.822786\pi\)
\(710\) 0 0
\(711\) −15.3723 −0.576506
\(712\) 0 0
\(713\) −16.0000 −0.599205
\(714\) 0 0
\(715\) −1.37228 −0.0513204
\(716\) 0 0
\(717\) 25.6060 0.956272
\(718\) 0 0
\(719\) −48.4674 −1.80753 −0.903764 0.428031i \(-0.859207\pi\)
−0.903764 + 0.428031i \(0.859207\pi\)
\(720\) 0 0
\(721\) −10.9783 −0.408851
\(722\) 0 0
\(723\) −4.74456 −0.176452
\(724\) 0 0
\(725\) 8.74456 0.324765
\(726\) 0 0
\(727\) 4.23369 0.157019 0.0785094 0.996913i \(-0.474984\pi\)
0.0785094 + 0.996913i \(0.474984\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 36.2337 1.34015
\(732\) 0 0
\(733\) 33.6060 1.24126 0.620632 0.784102i \(-0.286876\pi\)
0.620632 + 0.784102i \(0.286876\pi\)
\(734\) 0 0
\(735\) 5.11684 0.188738
\(736\) 0 0
\(737\) −17.4891 −0.644220
\(738\) 0 0
\(739\) −43.9565 −1.61697 −0.808483 0.588520i \(-0.799711\pi\)
−0.808483 + 0.588520i \(0.799711\pi\)
\(740\) 0 0
\(741\) 2.00000 0.0734718
\(742\) 0 0
\(743\) −38.0000 −1.39408 −0.697042 0.717030i \(-0.745501\pi\)
−0.697042 + 0.717030i \(0.745501\pi\)
\(744\) 0 0
\(745\) −4.11684 −0.150829
\(746\) 0 0
\(747\) −3.25544 −0.119110
\(748\) 0 0
\(749\) 10.1168 0.369661
\(750\) 0 0
\(751\) 35.8397 1.30781 0.653904 0.756578i \(-0.273130\pi\)
0.653904 + 0.756578i \(0.273130\pi\)
\(752\) 0 0
\(753\) −26.9783 −0.983142
\(754\) 0 0
\(755\) 14.2337 0.518017
\(756\) 0 0
\(757\) −8.74456 −0.317827 −0.158913 0.987293i \(-0.550799\pi\)
−0.158913 + 0.987293i \(0.550799\pi\)
\(758\) 0 0
\(759\) −4.62772 −0.167976
\(760\) 0 0
\(761\) 31.4891 1.14148 0.570740 0.821131i \(-0.306656\pi\)
0.570740 + 0.821131i \(0.306656\pi\)
\(762\) 0 0
\(763\) −28.4674 −1.03059
\(764\) 0 0
\(765\) 5.37228 0.194235
\(766\) 0 0
\(767\) −8.74456 −0.315748
\(768\) 0 0
\(769\) −51.9565 −1.87360 −0.936800 0.349866i \(-0.886227\pi\)
−0.936800 + 0.349866i \(0.886227\pi\)
\(770\) 0 0
\(771\) −14.0000 −0.504198
\(772\) 0 0
\(773\) −7.72281 −0.277770 −0.138885 0.990308i \(-0.544352\pi\)
−0.138885 + 0.990308i \(0.544352\pi\)
\(774\) 0 0
\(775\) −4.74456 −0.170430
\(776\) 0 0
\(777\) −1.88316 −0.0675578
\(778\) 0 0
\(779\) −10.7446 −0.384964
\(780\) 0 0
\(781\) 1.88316 0.0673846
\(782\) 0 0
\(783\) 8.74456 0.312505
\(784\) 0 0
\(785\) −2.00000 −0.0713831
\(786\) 0 0
\(787\) 11.2554 0.401213 0.200607 0.979672i \(-0.435709\pi\)
0.200607 + 0.979672i \(0.435709\pi\)
\(788\) 0 0
\(789\) −25.4891 −0.907437
\(790\) 0 0
\(791\) −2.74456 −0.0975854
\(792\) 0 0
\(793\) −8.11684 −0.288238
\(794\) 0 0
\(795\) 13.3723 0.474266
\(796\) 0 0
\(797\) −27.0951 −0.959757 −0.479879 0.877335i \(-0.659319\pi\)
−0.479879 + 0.877335i \(0.659319\pi\)
\(798\) 0 0
\(799\) 46.9783 1.66197
\(800\) 0 0
\(801\) 4.11684 0.145462
\(802\) 0 0
\(803\) 8.23369 0.290561
\(804\) 0 0
\(805\) −4.62772 −0.163106
\(806\) 0 0
\(807\) −31.7228 −1.11670
\(808\) 0 0
\(809\) −4.97825 −0.175026 −0.0875130 0.996163i \(-0.527892\pi\)
−0.0875130 + 0.996163i \(0.527892\pi\)
\(810\) 0 0
\(811\) 0.978251 0.0343510 0.0171755 0.999852i \(-0.494533\pi\)
0.0171755 + 0.999852i \(0.494533\pi\)
\(812\) 0 0
\(813\) 19.4891 0.683513
\(814\) 0 0
\(815\) −10.8614 −0.380458
\(816\) 0 0
\(817\) 13.4891 0.471925
\(818\) 0 0
\(819\) −1.37228 −0.0479514
\(820\) 0 0
\(821\) 21.3723 0.745898 0.372949 0.927852i \(-0.378347\pi\)
0.372949 + 0.927852i \(0.378347\pi\)
\(822\) 0 0
\(823\) −34.9783 −1.21927 −0.609633 0.792684i \(-0.708683\pi\)
−0.609633 + 0.792684i \(0.708683\pi\)
\(824\) 0 0
\(825\) −1.37228 −0.0477767
\(826\) 0 0
\(827\) 48.7446 1.69501 0.847507 0.530784i \(-0.178102\pi\)
0.847507 + 0.530784i \(0.178102\pi\)
\(828\) 0 0
\(829\) 27.4891 0.954737 0.477368 0.878703i \(-0.341591\pi\)
0.477368 + 0.878703i \(0.341591\pi\)
\(830\) 0 0
\(831\) −32.7446 −1.13590
\(832\) 0 0
\(833\) 27.4891 0.952442
\(834\) 0 0
\(835\) 8.74456 0.302618
\(836\) 0 0
\(837\) −4.74456 −0.163996
\(838\) 0 0
\(839\) 32.5842 1.12493 0.562466 0.826820i \(-0.309853\pi\)
0.562466 + 0.826820i \(0.309853\pi\)
\(840\) 0 0
\(841\) 47.4674 1.63681
\(842\) 0 0
\(843\) −19.4891 −0.671241
\(844\) 0 0
\(845\) −1.00000 −0.0344010
\(846\) 0 0
\(847\) −12.5109 −0.429879
\(848\) 0 0
\(849\) −2.51087 −0.0861730
\(850\) 0 0
\(851\) −4.62772 −0.158636
\(852\) 0 0
\(853\) 31.8832 1.09166 0.545829 0.837896i \(-0.316215\pi\)
0.545829 + 0.837896i \(0.316215\pi\)
\(854\) 0 0
\(855\) 2.00000 0.0683986
\(856\) 0 0
\(857\) 45.8397 1.56585 0.782926 0.622114i \(-0.213726\pi\)
0.782926 + 0.622114i \(0.213726\pi\)
\(858\) 0 0
\(859\) 50.3505 1.71794 0.858969 0.512028i \(-0.171105\pi\)
0.858969 + 0.512028i \(0.171105\pi\)
\(860\) 0 0
\(861\) 7.37228 0.251247
\(862\) 0 0
\(863\) −11.4891 −0.391094 −0.195547 0.980694i \(-0.562648\pi\)
−0.195547 + 0.980694i \(0.562648\pi\)
\(864\) 0 0
\(865\) −11.4891 −0.390642
\(866\) 0 0
\(867\) 11.8614 0.402834
\(868\) 0 0
\(869\) 21.0951 0.715602
\(870\) 0 0
\(871\) −12.7446 −0.431833
\(872\) 0 0
\(873\) −8.11684 −0.274714
\(874\) 0 0
\(875\) −1.37228 −0.0463916
\(876\) 0 0
\(877\) 7.48913 0.252890 0.126445 0.991974i \(-0.459643\pi\)
0.126445 + 0.991974i \(0.459643\pi\)
\(878\) 0 0
\(879\) 8.74456 0.294947
\(880\) 0 0
\(881\) 35.7228 1.20353 0.601766 0.798672i \(-0.294464\pi\)
0.601766 + 0.798672i \(0.294464\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(884\) 0 0
\(885\) −8.74456 −0.293945
\(886\) 0 0
\(887\) −12.3940 −0.416151 −0.208075 0.978113i \(-0.566720\pi\)
−0.208075 + 0.978113i \(0.566720\pi\)
\(888\) 0 0
\(889\) −21.9565 −0.736397
\(890\) 0 0
\(891\) −1.37228 −0.0459732
\(892\) 0 0
\(893\) 17.4891 0.585251
\(894\) 0 0
\(895\) −14.7446 −0.492856
\(896\) 0 0
\(897\) −3.37228 −0.112597
\(898\) 0 0
\(899\) −41.4891 −1.38374
\(900\) 0 0
\(901\) 71.8397 2.39333
\(902\) 0 0
\(903\) −9.25544 −0.308002
\(904\) 0 0
\(905\) −12.1168 −0.402778
\(906\) 0 0
\(907\) 22.7446 0.755221 0.377610 0.925965i \(-0.376746\pi\)
0.377610 + 0.925965i \(0.376746\pi\)
\(908\) 0 0
\(909\) −16.7446 −0.555382
\(910\) 0 0
\(911\) 29.7228 0.984761 0.492380 0.870380i \(-0.336127\pi\)
0.492380 + 0.870380i \(0.336127\pi\)
\(912\) 0 0
\(913\) 4.46738 0.147849
\(914\) 0 0
\(915\) −8.11684 −0.268335
\(916\) 0 0
\(917\) 16.4674 0.543801
\(918\) 0 0
\(919\) 14.1168 0.465672 0.232836 0.972516i \(-0.425200\pi\)
0.232836 + 0.972516i \(0.425200\pi\)
\(920\) 0 0
\(921\) −8.11684 −0.267459
\(922\) 0 0
\(923\) 1.37228 0.0451692
\(924\) 0 0
\(925\) −1.37228 −0.0451203
\(926\) 0 0
\(927\) −8.00000 −0.262754
\(928\) 0 0
\(929\) −4.35053 −0.142736 −0.0713682 0.997450i \(-0.522737\pi\)
−0.0713682 + 0.997450i \(0.522737\pi\)
\(930\) 0 0
\(931\) 10.2337 0.335396
\(932\) 0 0
\(933\) 25.7228 0.842127
\(934\) 0 0
\(935\) −7.37228 −0.241099
\(936\) 0 0
\(937\) −28.7446 −0.939044 −0.469522 0.882921i \(-0.655574\pi\)
−0.469522 + 0.882921i \(0.655574\pi\)
\(938\) 0 0
\(939\) 16.9783 0.554064
\(940\) 0 0
\(941\) 49.6060 1.61711 0.808554 0.588422i \(-0.200250\pi\)
0.808554 + 0.588422i \(0.200250\pi\)
\(942\) 0 0
\(943\) 18.1168 0.589966
\(944\) 0 0
\(945\) −1.37228 −0.0446403
\(946\) 0 0
\(947\) 3.72281 0.120975 0.0604876 0.998169i \(-0.480734\pi\)
0.0604876 + 0.998169i \(0.480734\pi\)
\(948\) 0 0
\(949\) 6.00000 0.194768
\(950\) 0 0
\(951\) −6.00000 −0.194563
\(952\) 0 0
\(953\) −39.3288 −1.27398 −0.636992 0.770870i \(-0.719822\pi\)
−0.636992 + 0.770870i \(0.719822\pi\)
\(954\) 0 0
\(955\) 2.74456 0.0888120
\(956\) 0 0
\(957\) −12.0000 −0.387905
\(958\) 0 0
\(959\) 21.2554 0.686374
\(960\) 0 0
\(961\) −8.48913 −0.273843
\(962\) 0 0
\(963\) 7.37228 0.237568
\(964\) 0 0
\(965\) 3.88316 0.125003
\(966\) 0 0
\(967\) −28.7446 −0.924363 −0.462181 0.886785i \(-0.652933\pi\)
−0.462181 + 0.886785i \(0.652933\pi\)
\(968\) 0 0
\(969\) 10.7446 0.345165
\(970\) 0 0
\(971\) 51.2119 1.64347 0.821735 0.569870i \(-0.193007\pi\)
0.821735 + 0.569870i \(0.193007\pi\)
\(972\) 0 0
\(973\) −19.3723 −0.621047
\(974\) 0 0
\(975\) −1.00000 −0.0320256
\(976\) 0 0
\(977\) −22.0000 −0.703842 −0.351921 0.936030i \(-0.614471\pi\)
−0.351921 + 0.936030i \(0.614471\pi\)
\(978\) 0 0
\(979\) −5.64947 −0.180558
\(980\) 0 0
\(981\) −20.7446 −0.662323
\(982\) 0 0
\(983\) 17.7663 0.566657 0.283329 0.959023i \(-0.408561\pi\)
0.283329 + 0.959023i \(0.408561\pi\)
\(984\) 0 0
\(985\) 19.4891 0.620975
\(986\) 0 0
\(987\) −12.0000 −0.381964
\(988\) 0 0
\(989\) −22.7446 −0.723235
\(990\) 0 0
\(991\) −11.1386 −0.353829 −0.176915 0.984226i \(-0.556612\pi\)
−0.176915 + 0.984226i \(0.556612\pi\)
\(992\) 0 0
\(993\) 22.0000 0.698149
\(994\) 0 0
\(995\) −8.00000 −0.253617
\(996\) 0 0
\(997\) −3.72281 −0.117903 −0.0589513 0.998261i \(-0.518776\pi\)
−0.0589513 + 0.998261i \(0.518776\pi\)
\(998\) 0 0
\(999\) −1.37228 −0.0434171
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 6240.2.a.bo.1.2 yes 2
4.3 odd 2 6240.2.a.bk.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6240.2.a.bk.1.1 2 4.3 odd 2
6240.2.a.bo.1.2 yes 2 1.1 even 1 trivial