Properties

Label 2760.2.a.w.1.2
Level $2760$
Weight $2$
Character 2760.1
Self dual yes
Analytic conductor $22.039$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2760 = 2^{3} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2760.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(22.0387109579\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.20087896.1
Defining polynomial: \( x^{5} - x^{4} - 21x^{3} + 5x^{2} + 84x - 4 \) 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 1.2
Root \(2.27399\) of defining polynomial
Character \(\chi\) \(=\) 2760.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +1.00000 q^{5} -3.55148 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +1.00000 q^{5} -3.55148 q^{7} +1.00000 q^{9} -2.79304 q^{11} +3.73151 q^{13} -1.00000 q^{15} -7.83097 q^{17} +6.27949 q^{19} +3.55148 q^{21} -1.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} +2.75844 q^{29} -2.48995 q^{31} +2.79304 q^{33} -3.55148 q^{35} -1.55148 q^{37} -3.73151 q^{39} +5.78954 q^{41} -4.54798 q^{43} +1.00000 q^{45} -6.27949 q^{47} +5.61301 q^{49} +7.83097 q^{51} +8.89250 q^{53} -2.79304 q^{55} -6.27949 q^{57} +3.31342 q^{59} +11.2180 q^{61} -3.55148 q^{63} +3.73151 q^{65} +5.52105 q^{67} +1.00000 q^{69} +6.34452 q^{71} +15.3824 q^{73} -1.00000 q^{75} +9.91943 q^{77} -9.62051 q^{79} +1.00000 q^{81} +5.83097 q^{83} -7.83097 q^{85} -2.75844 q^{87} +0.390487 q^{89} -13.2524 q^{91} +2.48995 q^{93} +6.27949 q^{95} +0.961896 q^{97} -2.79304 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 5 q^{3} + 5 q^{5} - 4 q^{7} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 5 q^{3} + 5 q^{5} - 4 q^{7} + 5 q^{9} - 4 q^{11} + 4 q^{13} - 5 q^{15} + 10 q^{17} - 4 q^{19} + 4 q^{21} - 5 q^{23} + 5 q^{25} - 5 q^{27} + 10 q^{29} + 6 q^{31} + 4 q^{33} - 4 q^{35} + 6 q^{37} - 4 q^{39} + 12 q^{41} - 2 q^{43} + 5 q^{45} + 4 q^{47} + 19 q^{49} - 10 q^{51} - 4 q^{55} + 4 q^{57} + 6 q^{59} + 16 q^{61} - 4 q^{63} + 4 q^{65} - 4 q^{67} + 5 q^{69} + 8 q^{71} + 14 q^{73} - 5 q^{75} + 16 q^{77} + 18 q^{79} + 5 q^{81} - 20 q^{83} + 10 q^{85} - 10 q^{87} + 18 q^{89} + 20 q^{91} - 6 q^{93} - 4 q^{95} + 4 q^{97} - 4 q^{99}+O(q^{100}) \) 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
\(7\) −3.55148 −1.34233 −0.671167 0.741306i \(-0.734207\pi\)
−0.671167 + 0.741306i \(0.734207\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −2.79304 −0.842134 −0.421067 0.907030i \(-0.638344\pi\)
−0.421067 + 0.907030i \(0.638344\pi\)
\(12\) 0 0
\(13\) 3.73151 1.03493 0.517467 0.855703i \(-0.326875\pi\)
0.517467 + 0.855703i \(0.326875\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) −7.83097 −1.89929 −0.949644 0.313330i \(-0.898555\pi\)
−0.949644 + 0.313330i \(0.898555\pi\)
\(18\) 0 0
\(19\) 6.27949 1.44061 0.720307 0.693656i \(-0.244001\pi\)
0.720307 + 0.693656i \(0.244001\pi\)
\(20\) 0 0
\(21\) 3.55148 0.774997
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 2.75844 0.512229 0.256115 0.966646i \(-0.417558\pi\)
0.256115 + 0.966646i \(0.417558\pi\)
\(30\) 0 0
\(31\) −2.48995 −0.447208 −0.223604 0.974680i \(-0.571782\pi\)
−0.223604 + 0.974680i \(0.571782\pi\)
\(32\) 0 0
\(33\) 2.79304 0.486206
\(34\) 0 0
\(35\) −3.55148 −0.600310
\(36\) 0 0
\(37\) −1.55148 −0.255062 −0.127531 0.991835i \(-0.540705\pi\)
−0.127531 + 0.991835i \(0.540705\pi\)
\(38\) 0 0
\(39\) −3.73151 −0.597519
\(40\) 0 0
\(41\) 5.78954 0.904174 0.452087 0.891974i \(-0.350680\pi\)
0.452087 + 0.891974i \(0.350680\pi\)
\(42\) 0 0
\(43\) −4.54798 −0.693560 −0.346780 0.937946i \(-0.612725\pi\)
−0.346780 + 0.937946i \(0.612725\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) −6.27949 −0.915957 −0.457979 0.888963i \(-0.651426\pi\)
−0.457979 + 0.888963i \(0.651426\pi\)
\(48\) 0 0
\(49\) 5.61301 0.801859
\(50\) 0 0
\(51\) 7.83097 1.09655
\(52\) 0 0
\(53\) 8.89250 1.22148 0.610739 0.791832i \(-0.290872\pi\)
0.610739 + 0.791832i \(0.290872\pi\)
\(54\) 0 0
\(55\) −2.79304 −0.376614
\(56\) 0 0
\(57\) −6.27949 −0.831738
\(58\) 0 0
\(59\) 3.31342 0.431371 0.215685 0.976463i \(-0.430801\pi\)
0.215685 + 0.976463i \(0.430801\pi\)
\(60\) 0 0
\(61\) 11.2180 1.43631 0.718156 0.695882i \(-0.244986\pi\)
0.718156 + 0.695882i \(0.244986\pi\)
\(62\) 0 0
\(63\) −3.55148 −0.447444
\(64\) 0 0
\(65\) 3.73151 0.462837
\(66\) 0 0
\(67\) 5.52105 0.674503 0.337252 0.941415i \(-0.390503\pi\)
0.337252 + 0.941415i \(0.390503\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 6.34452 0.752956 0.376478 0.926426i \(-0.377135\pi\)
0.376478 + 0.926426i \(0.377135\pi\)
\(72\) 0 0
\(73\) 15.3824 1.80038 0.900190 0.435498i \(-0.143428\pi\)
0.900190 + 0.435498i \(0.143428\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) 9.91943 1.13042
\(78\) 0 0
\(79\) −9.62051 −1.08239 −0.541196 0.840897i \(-0.682028\pi\)
−0.541196 + 0.840897i \(0.682028\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 5.83097 0.640032 0.320016 0.947412i \(-0.396312\pi\)
0.320016 + 0.947412i \(0.396312\pi\)
\(84\) 0 0
\(85\) −7.83097 −0.849388
\(86\) 0 0
\(87\) −2.75844 −0.295736
\(88\) 0 0
\(89\) 0.390487 0.0413915 0.0206958 0.999786i \(-0.493412\pi\)
0.0206958 + 0.999786i \(0.493412\pi\)
\(90\) 0 0
\(91\) −13.2524 −1.38923
\(92\) 0 0
\(93\) 2.48995 0.258195
\(94\) 0 0
\(95\) 6.27949 0.644262
\(96\) 0 0
\(97\) 0.961896 0.0976657 0.0488329 0.998807i \(-0.484450\pi\)
0.0488329 + 0.998807i \(0.484450\pi\)
\(98\) 0 0
\(99\) −2.79304 −0.280711
\(100\) 0 0
\(101\) −16.9035 −1.68196 −0.840980 0.541066i \(-0.818021\pi\)
−0.840980 + 0.541066i \(0.818021\pi\)
\(102\) 0 0
\(103\) −6.04143 −0.595280 −0.297640 0.954678i \(-0.596199\pi\)
−0.297640 + 0.954678i \(0.596199\pi\)
\(104\) 0 0
\(105\) 3.55148 0.346589
\(106\) 0 0
\(107\) 12.3141 1.19045 0.595224 0.803560i \(-0.297063\pi\)
0.595224 + 0.803560i \(0.297063\pi\)
\(108\) 0 0
\(109\) 14.8579 1.42313 0.711564 0.702621i \(-0.247987\pi\)
0.711564 + 0.702621i \(0.247987\pi\)
\(110\) 0 0
\(111\) 1.55148 0.147260
\(112\) 0 0
\(113\) 3.21113 0.302077 0.151039 0.988528i \(-0.451738\pi\)
0.151039 + 0.988528i \(0.451738\pi\)
\(114\) 0 0
\(115\) −1.00000 −0.0932505
\(116\) 0 0
\(117\) 3.73151 0.344978
\(118\) 0 0
\(119\) 27.8115 2.54948
\(120\) 0 0
\(121\) −3.19892 −0.290811
\(122\) 0 0
\(123\) −5.78954 −0.522025
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 0.145425 0.0129043 0.00645217 0.999979i \(-0.497946\pi\)
0.00645217 + 0.999979i \(0.497946\pi\)
\(128\) 0 0
\(129\) 4.54798 0.400427
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) −22.3015 −1.93378
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) −6.16450 −0.526668 −0.263334 0.964705i \(-0.584822\pi\)
−0.263334 + 0.964705i \(0.584822\pi\)
\(138\) 0 0
\(139\) 14.0690 1.19332 0.596660 0.802494i \(-0.296494\pi\)
0.596660 + 0.802494i \(0.296494\pi\)
\(140\) 0 0
\(141\) 6.27949 0.528828
\(142\) 0 0
\(143\) −10.4223 −0.871553
\(144\) 0 0
\(145\) 2.75844 0.229076
\(146\) 0 0
\(147\) −5.61301 −0.462954
\(148\) 0 0
\(149\) 13.7964 1.13024 0.565121 0.825008i \(-0.308829\pi\)
0.565121 + 0.825008i \(0.308829\pi\)
\(150\) 0 0
\(151\) 16.1989 1.31825 0.659125 0.752034i \(-0.270927\pi\)
0.659125 + 0.752034i \(0.270927\pi\)
\(152\) 0 0
\(153\) −7.83097 −0.633096
\(154\) 0 0
\(155\) −2.48995 −0.199997
\(156\) 0 0
\(157\) −14.1105 −1.12614 −0.563068 0.826410i \(-0.690379\pi\)
−0.563068 + 0.826410i \(0.690379\pi\)
\(158\) 0 0
\(159\) −8.89250 −0.705221
\(160\) 0 0
\(161\) 3.55148 0.279896
\(162\) 0 0
\(163\) 21.2480 1.66427 0.832137 0.554571i \(-0.187118\pi\)
0.832137 + 0.554571i \(0.187118\pi\)
\(164\) 0 0
\(165\) 2.79304 0.217438
\(166\) 0 0
\(167\) 19.3895 1.50040 0.750200 0.661211i \(-0.229957\pi\)
0.750200 + 0.661211i \(0.229957\pi\)
\(168\) 0 0
\(169\) 0.924149 0.0710884
\(170\) 0 0
\(171\) 6.27949 0.480204
\(172\) 0 0
\(173\) −13.1030 −0.996200 −0.498100 0.867120i \(-0.665969\pi\)
−0.498100 + 0.867120i \(0.665969\pi\)
\(174\) 0 0
\(175\) −3.55148 −0.268467
\(176\) 0 0
\(177\) −3.31342 −0.249052
\(178\) 0 0
\(179\) −14.0759 −1.05208 −0.526039 0.850460i \(-0.676324\pi\)
−0.526039 + 0.850460i \(0.676324\pi\)
\(180\) 0 0
\(181\) 21.1374 1.57113 0.785565 0.618779i \(-0.212373\pi\)
0.785565 + 0.618779i \(0.212373\pi\)
\(182\) 0 0
\(183\) −11.2180 −0.829255
\(184\) 0 0
\(185\) −1.55148 −0.114067
\(186\) 0 0
\(187\) 21.8722 1.59945
\(188\) 0 0
\(189\) 3.55148 0.258332
\(190\) 0 0
\(191\) 2.81647 0.203793 0.101896 0.994795i \(-0.467509\pi\)
0.101896 + 0.994795i \(0.467509\pi\)
\(192\) 0 0
\(193\) 15.4560 1.11255 0.556274 0.830999i \(-0.312230\pi\)
0.556274 + 0.830999i \(0.312230\pi\)
\(194\) 0 0
\(195\) −3.73151 −0.267219
\(196\) 0 0
\(197\) 25.7311 1.83327 0.916634 0.399728i \(-0.130895\pi\)
0.916634 + 0.399728i \(0.130895\pi\)
\(198\) 0 0
\(199\) 5.95857 0.422392 0.211196 0.977444i \(-0.432264\pi\)
0.211196 + 0.977444i \(0.432264\pi\)
\(200\) 0 0
\(201\) −5.52105 −0.389425
\(202\) 0 0
\(203\) −9.79654 −0.687583
\(204\) 0 0
\(205\) 5.78954 0.404359
\(206\) 0 0
\(207\) −1.00000 −0.0695048
\(208\) 0 0
\(209\) −17.5389 −1.21319
\(210\) 0 0
\(211\) −6.06903 −0.417809 −0.208905 0.977936i \(-0.566990\pi\)
−0.208905 + 0.977936i \(0.566990\pi\)
\(212\) 0 0
\(213\) −6.34452 −0.434719
\(214\) 0 0
\(215\) −4.54798 −0.310170
\(216\) 0 0
\(217\) 8.84300 0.600302
\(218\) 0 0
\(219\) −15.3824 −1.03945
\(220\) 0 0
\(221\) −29.2213 −1.96564
\(222\) 0 0
\(223\) −1.57509 −0.105476 −0.0527379 0.998608i \(-0.516795\pi\)
−0.0527379 + 0.998608i \(0.516795\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −21.7034 −1.44050 −0.720251 0.693713i \(-0.755974\pi\)
−0.720251 + 0.693713i \(0.755974\pi\)
\(228\) 0 0
\(229\) 23.2136 1.53400 0.766999 0.641649i \(-0.221749\pi\)
0.766999 + 0.641649i \(0.221749\pi\)
\(230\) 0 0
\(231\) −9.91943 −0.652651
\(232\) 0 0
\(233\) −18.1989 −1.19225 −0.596125 0.802891i \(-0.703294\pi\)
−0.596125 + 0.802891i \(0.703294\pi\)
\(234\) 0 0
\(235\) −6.27949 −0.409629
\(236\) 0 0
\(237\) 9.62051 0.624919
\(238\) 0 0
\(239\) 21.9236 1.41812 0.709060 0.705148i \(-0.249120\pi\)
0.709060 + 0.705148i \(0.249120\pi\)
\(240\) 0 0
\(241\) 17.7964 1.14636 0.573182 0.819428i \(-0.305709\pi\)
0.573182 + 0.819428i \(0.305709\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 5.61301 0.358602
\(246\) 0 0
\(247\) 23.4320 1.49094
\(248\) 0 0
\(249\) −5.83097 −0.369523
\(250\) 0 0
\(251\) −23.2785 −1.46932 −0.734661 0.678434i \(-0.762659\pi\)
−0.734661 + 0.678434i \(0.762659\pi\)
\(252\) 0 0
\(253\) 2.79304 0.175597
\(254\) 0 0
\(255\) 7.83097 0.490394
\(256\) 0 0
\(257\) 25.5884 1.59616 0.798079 0.602552i \(-0.205850\pi\)
0.798079 + 0.602552i \(0.205850\pi\)
\(258\) 0 0
\(259\) 5.51005 0.342378
\(260\) 0 0
\(261\) 2.75844 0.170743
\(262\) 0 0
\(263\) −8.57051 −0.528481 −0.264240 0.964457i \(-0.585121\pi\)
−0.264240 + 0.964457i \(0.585121\pi\)
\(264\) 0 0
\(265\) 8.89250 0.546262
\(266\) 0 0
\(267\) −0.390487 −0.0238974
\(268\) 0 0
\(269\) −13.4405 −0.819481 −0.409740 0.912202i \(-0.634381\pi\)
−0.409740 + 0.912202i \(0.634381\pi\)
\(270\) 0 0
\(271\) 3.38699 0.205745 0.102872 0.994695i \(-0.467197\pi\)
0.102872 + 0.994695i \(0.467197\pi\)
\(272\) 0 0
\(273\) 13.2524 0.802070
\(274\) 0 0
\(275\) −2.79304 −0.168427
\(276\) 0 0
\(277\) −18.7469 −1.12639 −0.563196 0.826323i \(-0.690428\pi\)
−0.563196 + 0.826323i \(0.690428\pi\)
\(278\) 0 0
\(279\) −2.48995 −0.149069
\(280\) 0 0
\(281\) −5.21396 −0.311039 −0.155519 0.987833i \(-0.549705\pi\)
−0.155519 + 0.987833i \(0.549705\pi\)
\(282\) 0 0
\(283\) −18.9771 −1.12807 −0.564035 0.825751i \(-0.690752\pi\)
−0.564035 + 0.825751i \(0.690752\pi\)
\(284\) 0 0
\(285\) −6.27949 −0.371965
\(286\) 0 0
\(287\) −20.5614 −1.21370
\(288\) 0 0
\(289\) 44.3240 2.60730
\(290\) 0 0
\(291\) −0.961896 −0.0563873
\(292\) 0 0
\(293\) −27.3976 −1.60059 −0.800293 0.599609i \(-0.795323\pi\)
−0.800293 + 0.599609i \(0.795323\pi\)
\(294\) 0 0
\(295\) 3.31342 0.192915
\(296\) 0 0
\(297\) 2.79304 0.162069
\(298\) 0 0
\(299\) −3.73151 −0.215799
\(300\) 0 0
\(301\) 16.1521 0.930989
\(302\) 0 0
\(303\) 16.9035 0.971080
\(304\) 0 0
\(305\) 11.2180 0.642338
\(306\) 0 0
\(307\) 1.18353 0.0675475 0.0337738 0.999430i \(-0.489247\pi\)
0.0337738 + 0.999430i \(0.489247\pi\)
\(308\) 0 0
\(309\) 6.04143 0.343685
\(310\) 0 0
\(311\) 18.5770 1.05340 0.526702 0.850050i \(-0.323428\pi\)
0.526702 + 0.850050i \(0.323428\pi\)
\(312\) 0 0
\(313\) −8.50094 −0.480502 −0.240251 0.970711i \(-0.577230\pi\)
−0.240251 + 0.970711i \(0.577230\pi\)
\(314\) 0 0
\(315\) −3.55148 −0.200103
\(316\) 0 0
\(317\) 21.5884 1.21252 0.606262 0.795265i \(-0.292668\pi\)
0.606262 + 0.795265i \(0.292668\pi\)
\(318\) 0 0
\(319\) −7.70443 −0.431366
\(320\) 0 0
\(321\) −12.3141 −0.687305
\(322\) 0 0
\(323\) −49.1745 −2.73614
\(324\) 0 0
\(325\) 3.73151 0.206987
\(326\) 0 0
\(327\) −14.8579 −0.821644
\(328\) 0 0
\(329\) 22.3015 1.22952
\(330\) 0 0
\(331\) 24.0288 1.32074 0.660372 0.750939i \(-0.270399\pi\)
0.660372 + 0.750939i \(0.270399\pi\)
\(332\) 0 0
\(333\) −1.55148 −0.0850206
\(334\) 0 0
\(335\) 5.52105 0.301647
\(336\) 0 0
\(337\) −19.7960 −1.07836 −0.539178 0.842192i \(-0.681265\pi\)
−0.539178 + 0.842192i \(0.681265\pi\)
\(338\) 0 0
\(339\) −3.21113 −0.174405
\(340\) 0 0
\(341\) 6.95452 0.376609
\(342\) 0 0
\(343\) 4.92585 0.265971
\(344\) 0 0
\(345\) 1.00000 0.0538382
\(346\) 0 0
\(347\) 3.52388 0.189172 0.0945859 0.995517i \(-0.469847\pi\)
0.0945859 + 0.995517i \(0.469847\pi\)
\(348\) 0 0
\(349\) 9.10979 0.487636 0.243818 0.969821i \(-0.421600\pi\)
0.243818 + 0.969821i \(0.421600\pi\)
\(350\) 0 0
\(351\) −3.73151 −0.199173
\(352\) 0 0
\(353\) −24.7224 −1.31584 −0.657920 0.753088i \(-0.728564\pi\)
−0.657920 + 0.753088i \(0.728564\pi\)
\(354\) 0 0
\(355\) 6.34452 0.336732
\(356\) 0 0
\(357\) −27.8115 −1.47194
\(358\) 0 0
\(359\) 11.3895 0.601112 0.300556 0.953764i \(-0.402828\pi\)
0.300556 + 0.953764i \(0.402828\pi\)
\(360\) 0 0
\(361\) 20.4320 1.07537
\(362\) 0 0
\(363\) 3.19892 0.167900
\(364\) 0 0
\(365\) 15.3824 0.805154
\(366\) 0 0
\(367\) 11.1306 0.581011 0.290505 0.956873i \(-0.406177\pi\)
0.290505 + 0.956873i \(0.406177\pi\)
\(368\) 0 0
\(369\) 5.78954 0.301391
\(370\) 0 0
\(371\) −31.5815 −1.63963
\(372\) 0 0
\(373\) 8.51755 0.441022 0.220511 0.975385i \(-0.429228\pi\)
0.220511 + 0.975385i \(0.429228\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) 10.2931 0.530123
\(378\) 0 0
\(379\) −21.1320 −1.08548 −0.542738 0.839902i \(-0.682612\pi\)
−0.542738 + 0.839902i \(0.682612\pi\)
\(380\) 0 0
\(381\) −0.145425 −0.00745033
\(382\) 0 0
\(383\) 9.37562 0.479072 0.239536 0.970887i \(-0.423005\pi\)
0.239536 + 0.970887i \(0.423005\pi\)
\(384\) 0 0
\(385\) 9.91943 0.505541
\(386\) 0 0
\(387\) −4.54798 −0.231187
\(388\) 0 0
\(389\) −16.6352 −0.843437 −0.421719 0.906727i \(-0.638573\pi\)
−0.421719 + 0.906727i \(0.638573\pi\)
\(390\) 0 0
\(391\) 7.83097 0.396029
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −9.62051 −0.484060
\(396\) 0 0
\(397\) 23.9261 1.20081 0.600407 0.799694i \(-0.295005\pi\)
0.600407 + 0.799694i \(0.295005\pi\)
\(398\) 0 0
\(399\) 22.3015 1.11647
\(400\) 0 0
\(401\) −12.4457 −0.621508 −0.310754 0.950490i \(-0.600582\pi\)
−0.310754 + 0.950490i \(0.600582\pi\)
\(402\) 0 0
\(403\) −9.29125 −0.462830
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) 4.33335 0.214796
\(408\) 0 0
\(409\) 27.0419 1.33714 0.668568 0.743651i \(-0.266907\pi\)
0.668568 + 0.743651i \(0.266907\pi\)
\(410\) 0 0
\(411\) 6.16450 0.304072
\(412\) 0 0
\(413\) −11.7676 −0.579043
\(414\) 0 0
\(415\) 5.83097 0.286231
\(416\) 0 0
\(417\) −14.0690 −0.688963
\(418\) 0 0
\(419\) 10.0949 0.493169 0.246585 0.969121i \(-0.420692\pi\)
0.246585 + 0.969121i \(0.420692\pi\)
\(420\) 0 0
\(421\) −28.5268 −1.39031 −0.695156 0.718858i \(-0.744665\pi\)
−0.695156 + 0.718858i \(0.744665\pi\)
\(422\) 0 0
\(423\) −6.27949 −0.305319
\(424\) 0 0
\(425\) −7.83097 −0.379858
\(426\) 0 0
\(427\) −39.8403 −1.92801
\(428\) 0 0
\(429\) 10.4223 0.503191
\(430\) 0 0
\(431\) 21.6081 1.04082 0.520412 0.853915i \(-0.325778\pi\)
0.520412 + 0.853915i \(0.325778\pi\)
\(432\) 0 0
\(433\) −8.21009 −0.394552 −0.197276 0.980348i \(-0.563209\pi\)
−0.197276 + 0.980348i \(0.563209\pi\)
\(434\) 0 0
\(435\) −2.75844 −0.132257
\(436\) 0 0
\(437\) −6.27949 −0.300389
\(438\) 0 0
\(439\) 24.4429 1.16660 0.583298 0.812258i \(-0.301762\pi\)
0.583298 + 0.812258i \(0.301762\pi\)
\(440\) 0 0
\(441\) 5.61301 0.267286
\(442\) 0 0
\(443\) 32.5074 1.54447 0.772237 0.635335i \(-0.219138\pi\)
0.772237 + 0.635335i \(0.219138\pi\)
\(444\) 0 0
\(445\) 0.390487 0.0185109
\(446\) 0 0
\(447\) −13.7964 −0.652546
\(448\) 0 0
\(449\) −24.5635 −1.15922 −0.579612 0.814893i \(-0.696796\pi\)
−0.579612 + 0.814893i \(0.696796\pi\)
\(450\) 0 0
\(451\) −16.1704 −0.761436
\(452\) 0 0
\(453\) −16.1989 −0.761092
\(454\) 0 0
\(455\) −13.2524 −0.621281
\(456\) 0 0
\(457\) −29.6798 −1.38836 −0.694180 0.719801i \(-0.744233\pi\)
−0.694180 + 0.719801i \(0.744233\pi\)
\(458\) 0 0
\(459\) 7.83097 0.365518
\(460\) 0 0
\(461\) −34.2239 −1.59397 −0.796983 0.604001i \(-0.793572\pi\)
−0.796983 + 0.604001i \(0.793572\pi\)
\(462\) 0 0
\(463\) −27.8074 −1.29232 −0.646159 0.763203i \(-0.723626\pi\)
−0.646159 + 0.763203i \(0.723626\pi\)
\(464\) 0 0
\(465\) 2.48995 0.115469
\(466\) 0 0
\(467\) 22.3207 1.03288 0.516440 0.856323i \(-0.327257\pi\)
0.516440 + 0.856323i \(0.327257\pi\)
\(468\) 0 0
\(469\) −19.6079 −0.905408
\(470\) 0 0
\(471\) 14.1105 0.650175
\(472\) 0 0
\(473\) 12.7027 0.584070
\(474\) 0 0
\(475\) 6.27949 0.288123
\(476\) 0 0
\(477\) 8.89250 0.407160
\(478\) 0 0
\(479\) −32.7535 −1.49655 −0.748274 0.663390i \(-0.769117\pi\)
−0.748274 + 0.663390i \(0.769117\pi\)
\(480\) 0 0
\(481\) −5.78936 −0.263972
\(482\) 0 0
\(483\) −3.55148 −0.161598
\(484\) 0 0
\(485\) 0.961896 0.0436774
\(486\) 0 0
\(487\) −25.9996 −1.17816 −0.589078 0.808076i \(-0.700509\pi\)
−0.589078 + 0.808076i \(0.700509\pi\)
\(488\) 0 0
\(489\) −21.2480 −0.960869
\(490\) 0 0
\(491\) −7.67348 −0.346299 −0.173150 0.984896i \(-0.555394\pi\)
−0.173150 + 0.984896i \(0.555394\pi\)
\(492\) 0 0
\(493\) −21.6012 −0.972871
\(494\) 0 0
\(495\) −2.79304 −0.125538
\(496\) 0 0
\(497\) −22.5324 −1.01072
\(498\) 0 0
\(499\) 15.2951 0.684701 0.342350 0.939572i \(-0.388777\pi\)
0.342350 + 0.939572i \(0.388777\pi\)
\(500\) 0 0
\(501\) −19.3895 −0.866257
\(502\) 0 0
\(503\) −9.79654 −0.436806 −0.218403 0.975859i \(-0.570085\pi\)
−0.218403 + 0.975859i \(0.570085\pi\)
\(504\) 0 0
\(505\) −16.9035 −0.752196
\(506\) 0 0
\(507\) −0.924149 −0.0410429
\(508\) 0 0
\(509\) 6.25713 0.277342 0.138671 0.990338i \(-0.455717\pi\)
0.138671 + 0.990338i \(0.455717\pi\)
\(510\) 0 0
\(511\) −54.6305 −2.41671
\(512\) 0 0
\(513\) −6.27949 −0.277246
\(514\) 0 0
\(515\) −6.04143 −0.266217
\(516\) 0 0
\(517\) 17.5389 0.771358
\(518\) 0 0
\(519\) 13.1030 0.575156
\(520\) 0 0
\(521\) −8.72384 −0.382198 −0.191099 0.981571i \(-0.561205\pi\)
−0.191099 + 0.981571i \(0.561205\pi\)
\(522\) 0 0
\(523\) −8.66404 −0.378852 −0.189426 0.981895i \(-0.560663\pi\)
−0.189426 + 0.981895i \(0.560663\pi\)
\(524\) 0 0
\(525\) 3.55148 0.154999
\(526\) 0 0
\(527\) 19.4987 0.849376
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 3.31342 0.143790
\(532\) 0 0
\(533\) 21.6037 0.935761
\(534\) 0 0
\(535\) 12.3141 0.532384
\(536\) 0 0
\(537\) 14.0759 0.607418
\(538\) 0 0
\(539\) −15.6774 −0.675273
\(540\) 0 0
\(541\) −36.5660 −1.57209 −0.786047 0.618167i \(-0.787876\pi\)
−0.786047 + 0.618167i \(0.787876\pi\)
\(542\) 0 0
\(543\) −21.1374 −0.907092
\(544\) 0 0
\(545\) 14.8579 0.636442
\(546\) 0 0
\(547\) −0.827834 −0.0353956 −0.0176978 0.999843i \(-0.505634\pi\)
−0.0176978 + 0.999843i \(0.505634\pi\)
\(548\) 0 0
\(549\) 11.2180 0.478771
\(550\) 0 0
\(551\) 17.3216 0.737924
\(552\) 0 0
\(553\) 34.1670 1.45293
\(554\) 0 0
\(555\) 1.55148 0.0658567
\(556\) 0 0
\(557\) 0.125533 0.00531902 0.00265951 0.999996i \(-0.499153\pi\)
0.00265951 + 0.999996i \(0.499153\pi\)
\(558\) 0 0
\(559\) −16.9708 −0.717789
\(560\) 0 0
\(561\) −21.8722 −0.923446
\(562\) 0 0
\(563\) −38.3207 −1.61503 −0.807513 0.589849i \(-0.799187\pi\)
−0.807513 + 0.589849i \(0.799187\pi\)
\(564\) 0 0
\(565\) 3.21113 0.135093
\(566\) 0 0
\(567\) −3.55148 −0.149148
\(568\) 0 0
\(569\) −28.8715 −1.21036 −0.605179 0.796090i \(-0.706898\pi\)
−0.605179 + 0.796090i \(0.706898\pi\)
\(570\) 0 0
\(571\) 15.0514 0.629881 0.314940 0.949111i \(-0.398015\pi\)
0.314940 + 0.949111i \(0.398015\pi\)
\(572\) 0 0
\(573\) −2.81647 −0.117660
\(574\) 0 0
\(575\) −1.00000 −0.0417029
\(576\) 0 0
\(577\) −21.0561 −0.876577 −0.438288 0.898834i \(-0.644415\pi\)
−0.438288 + 0.898834i \(0.644415\pi\)
\(578\) 0 0
\(579\) −15.4560 −0.642330
\(580\) 0 0
\(581\) −20.7086 −0.859136
\(582\) 0 0
\(583\) −24.8371 −1.02865
\(584\) 0 0
\(585\) 3.73151 0.154279
\(586\) 0 0
\(587\) 22.2751 0.919393 0.459696 0.888076i \(-0.347958\pi\)
0.459696 + 0.888076i \(0.347958\pi\)
\(588\) 0 0
\(589\) −15.6356 −0.644253
\(590\) 0 0
\(591\) −25.7311 −1.05844
\(592\) 0 0
\(593\) −8.45642 −0.347263 −0.173632 0.984811i \(-0.555550\pi\)
−0.173632 + 0.984811i \(0.555550\pi\)
\(594\) 0 0
\(595\) 27.8115 1.14016
\(596\) 0 0
\(597\) −5.95857 −0.243868
\(598\) 0 0
\(599\) −10.0040 −0.408752 −0.204376 0.978892i \(-0.565517\pi\)
−0.204376 + 0.978892i \(0.565517\pi\)
\(600\) 0 0
\(601\) 14.8119 0.604191 0.302096 0.953278i \(-0.402314\pi\)
0.302096 + 0.953278i \(0.402314\pi\)
\(602\) 0 0
\(603\) 5.52105 0.224834
\(604\) 0 0
\(605\) −3.19892 −0.130055
\(606\) 0 0
\(607\) 10.8883 0.441944 0.220972 0.975280i \(-0.429077\pi\)
0.220972 + 0.975280i \(0.429077\pi\)
\(608\) 0 0
\(609\) 9.79654 0.396976
\(610\) 0 0
\(611\) −23.4320 −0.947955
\(612\) 0 0
\(613\) 3.35238 0.135401 0.0677007 0.997706i \(-0.478434\pi\)
0.0677007 + 0.997706i \(0.478434\pi\)
\(614\) 0 0
\(615\) −5.78954 −0.233457
\(616\) 0 0
\(617\) −21.8848 −0.881050 −0.440525 0.897740i \(-0.645208\pi\)
−0.440525 + 0.897740i \(0.645208\pi\)
\(618\) 0 0
\(619\) −10.0762 −0.404997 −0.202498 0.979283i \(-0.564906\pi\)
−0.202498 + 0.979283i \(0.564906\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 0 0
\(623\) −1.38681 −0.0555613
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 17.5389 0.700435
\(628\) 0 0
\(629\) 12.1496 0.484436
\(630\) 0 0
\(631\) 6.50485 0.258954 0.129477 0.991582i \(-0.458670\pi\)
0.129477 + 0.991582i \(0.458670\pi\)
\(632\) 0 0
\(633\) 6.06903 0.241222
\(634\) 0 0
\(635\) 0.145425 0.00577100
\(636\) 0 0
\(637\) 20.9450 0.829871
\(638\) 0 0
\(639\) 6.34452 0.250985
\(640\) 0 0
\(641\) −1.60723 −0.0634816 −0.0317408 0.999496i \(-0.510105\pi\)
−0.0317408 + 0.999496i \(0.510105\pi\)
\(642\) 0 0
\(643\) −7.73605 −0.305080 −0.152540 0.988297i \(-0.548745\pi\)
−0.152540 + 0.988297i \(0.548745\pi\)
\(644\) 0 0
\(645\) 4.54798 0.179076
\(646\) 0 0
\(647\) −8.89232 −0.349593 −0.174797 0.984605i \(-0.555927\pi\)
−0.174797 + 0.984605i \(0.555927\pi\)
\(648\) 0 0
\(649\) −9.25452 −0.363272
\(650\) 0 0
\(651\) −8.84300 −0.346584
\(652\) 0 0
\(653\) 31.2055 1.22117 0.610583 0.791952i \(-0.290935\pi\)
0.610583 + 0.791952i \(0.290935\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 15.3824 0.600126
\(658\) 0 0
\(659\) −21.7051 −0.845509 −0.422755 0.906244i \(-0.638937\pi\)
−0.422755 + 0.906244i \(0.638937\pi\)
\(660\) 0 0
\(661\) 30.0564 1.16906 0.584529 0.811372i \(-0.301279\pi\)
0.584529 + 0.811372i \(0.301279\pi\)
\(662\) 0 0
\(663\) 29.2213 1.13486
\(664\) 0 0
\(665\) −22.3015 −0.864814
\(666\) 0 0
\(667\) −2.75844 −0.106807
\(668\) 0 0
\(669\) 1.57509 0.0608965
\(670\) 0 0
\(671\) −31.3322 −1.20957
\(672\) 0 0
\(673\) 16.2795 0.627528 0.313764 0.949501i \(-0.398410\pi\)
0.313764 + 0.949501i \(0.398410\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) −4.70723 −0.180914 −0.0904568 0.995900i \(-0.528833\pi\)
−0.0904568 + 0.995900i \(0.528833\pi\)
\(678\) 0 0
\(679\) −3.41616 −0.131100
\(680\) 0 0
\(681\) 21.7034 0.831675
\(682\) 0 0
\(683\) 9.82536 0.375957 0.187978 0.982173i \(-0.439807\pi\)
0.187978 + 0.982173i \(0.439807\pi\)
\(684\) 0 0
\(685\) −6.16450 −0.235533
\(686\) 0 0
\(687\) −23.2136 −0.885654
\(688\) 0 0
\(689\) 33.1824 1.26415
\(690\) 0 0
\(691\) −43.1718 −1.64233 −0.821167 0.570689i \(-0.806676\pi\)
−0.821167 + 0.570689i \(0.806676\pi\)
\(692\) 0 0
\(693\) 9.91943 0.376808
\(694\) 0 0
\(695\) 14.0690 0.533669
\(696\) 0 0
\(697\) −45.3377 −1.71729
\(698\) 0 0
\(699\) 18.1989 0.688346
\(700\) 0 0
\(701\) −1.44539 −0.0545915 −0.0272957 0.999627i \(-0.508690\pi\)
−0.0272957 + 0.999627i \(0.508690\pi\)
\(702\) 0 0
\(703\) −9.74250 −0.367445
\(704\) 0 0
\(705\) 6.27949 0.236499
\(706\) 0 0
\(707\) 60.0324 2.25775
\(708\) 0 0
\(709\) −28.9697 −1.08798 −0.543991 0.839091i \(-0.683087\pi\)
−0.543991 + 0.839091i \(0.683087\pi\)
\(710\) 0 0
\(711\) −9.62051 −0.360797
\(712\) 0 0
\(713\) 2.48995 0.0932492
\(714\) 0 0
\(715\) −10.4223 −0.389770
\(716\) 0 0
\(717\) −21.9236 −0.818752
\(718\) 0 0
\(719\) 42.4742 1.58402 0.792011 0.610507i \(-0.209034\pi\)
0.792011 + 0.610507i \(0.209034\pi\)
\(720\) 0 0
\(721\) 21.4560 0.799064
\(722\) 0 0
\(723\) −17.7964 −0.661854
\(724\) 0 0
\(725\) 2.75844 0.102446
\(726\) 0 0
\(727\) −3.10857 −0.115291 −0.0576453 0.998337i \(-0.518359\pi\)
−0.0576453 + 0.998337i \(0.518359\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 35.6151 1.31727
\(732\) 0 0
\(733\) 6.20453 0.229170 0.114585 0.993413i \(-0.463446\pi\)
0.114585 + 0.993413i \(0.463446\pi\)
\(734\) 0 0
\(735\) −5.61301 −0.207039
\(736\) 0 0
\(737\) −15.4205 −0.568022
\(738\) 0 0
\(739\) −21.5010 −0.790926 −0.395463 0.918482i \(-0.629416\pi\)
−0.395463 + 0.918482i \(0.629416\pi\)
\(740\) 0 0
\(741\) −23.4320 −0.860794
\(742\) 0 0
\(743\) −25.4469 −0.933558 −0.466779 0.884374i \(-0.654586\pi\)
−0.466779 + 0.884374i \(0.654586\pi\)
\(744\) 0 0
\(745\) 13.7964 0.505460
\(746\) 0 0
\(747\) 5.83097 0.213344
\(748\) 0 0
\(749\) −43.7332 −1.59798
\(750\) 0 0
\(751\) 49.9066 1.82112 0.910560 0.413378i \(-0.135651\pi\)
0.910560 + 0.413378i \(0.135651\pi\)
\(752\) 0 0
\(753\) 23.2785 0.848314
\(754\) 0 0
\(755\) 16.1989 0.589539
\(756\) 0 0
\(757\) 31.8486 1.15756 0.578778 0.815485i \(-0.303530\pi\)
0.578778 + 0.815485i \(0.303530\pi\)
\(758\) 0 0
\(759\) −2.79304 −0.101381
\(760\) 0 0
\(761\) 7.63541 0.276783 0.138392 0.990378i \(-0.455807\pi\)
0.138392 + 0.990378i \(0.455807\pi\)
\(762\) 0 0
\(763\) −52.7675 −1.91031
\(764\) 0 0
\(765\) −7.83097 −0.283129
\(766\) 0 0
\(767\) 12.3641 0.446440
\(768\) 0 0
\(769\) −3.36880 −0.121482 −0.0607410 0.998154i \(-0.519346\pi\)
−0.0607410 + 0.998154i \(0.519346\pi\)
\(770\) 0 0
\(771\) −25.5884 −0.921543
\(772\) 0 0
\(773\) 5.77800 0.207820 0.103910 0.994587i \(-0.466865\pi\)
0.103910 + 0.994587i \(0.466865\pi\)
\(774\) 0 0
\(775\) −2.48995 −0.0894415
\(776\) 0 0
\(777\) −5.51005 −0.197672
\(778\) 0 0
\(779\) 36.3553 1.30257
\(780\) 0 0
\(781\) −17.7205 −0.634090
\(782\) 0 0
\(783\) −2.75844 −0.0985786
\(784\) 0 0
\(785\) −14.1105 −0.503624
\(786\) 0 0
\(787\) 6.46988 0.230626 0.115313 0.993329i \(-0.463213\pi\)
0.115313 + 0.993329i \(0.463213\pi\)
\(788\) 0 0
\(789\) 8.57051 0.305118
\(790\) 0 0
\(791\) −11.4043 −0.405489
\(792\) 0 0
\(793\) 41.8599 1.48649
\(794\) 0 0
\(795\) −8.89250 −0.315385
\(796\) 0 0
\(797\) −21.9636 −0.777990 −0.388995 0.921240i \(-0.627178\pi\)
−0.388995 + 0.921240i \(0.627178\pi\)
\(798\) 0 0
\(799\) 49.1745 1.73967
\(800\) 0 0
\(801\) 0.390487 0.0137972
\(802\) 0 0
\(803\) −42.9638 −1.51616
\(804\) 0 0
\(805\) 3.55148 0.125173
\(806\) 0 0
\(807\) 13.4405 0.473127
\(808\) 0 0
\(809\) −42.7222 −1.50203 −0.751017 0.660283i \(-0.770436\pi\)
−0.751017 + 0.660283i \(0.770436\pi\)
\(810\) 0 0
\(811\) −30.2347 −1.06169 −0.530843 0.847470i \(-0.678124\pi\)
−0.530843 + 0.847470i \(0.678124\pi\)
\(812\) 0 0
\(813\) −3.38699 −0.118787
\(814\) 0 0
\(815\) 21.2480 0.744286
\(816\) 0 0
\(817\) −28.5590 −0.999152
\(818\) 0 0
\(819\) −13.2524 −0.463076
\(820\) 0 0
\(821\) −18.3260 −0.639581 −0.319791 0.947488i \(-0.603613\pi\)
−0.319791 + 0.947488i \(0.603613\pi\)
\(822\) 0 0
\(823\) −14.5081 −0.505721 −0.252861 0.967503i \(-0.581371\pi\)
−0.252861 + 0.967503i \(0.581371\pi\)
\(824\) 0 0
\(825\) 2.79304 0.0972412
\(826\) 0 0
\(827\) −28.0452 −0.975228 −0.487614 0.873059i \(-0.662133\pi\)
−0.487614 + 0.873059i \(0.662133\pi\)
\(828\) 0 0
\(829\) −26.2281 −0.910939 −0.455470 0.890251i \(-0.650529\pi\)
−0.455470 + 0.890251i \(0.650529\pi\)
\(830\) 0 0
\(831\) 18.7469 0.650323
\(832\) 0 0
\(833\) −43.9553 −1.52296
\(834\) 0 0
\(835\) 19.3895 0.671000
\(836\) 0 0
\(837\) 2.48995 0.0860651
\(838\) 0 0
\(839\) 12.2129 0.421637 0.210819 0.977525i \(-0.432387\pi\)
0.210819 + 0.977525i \(0.432387\pi\)
\(840\) 0 0
\(841\) −21.3910 −0.737621
\(842\) 0 0
\(843\) 5.21396 0.179578
\(844\) 0 0
\(845\) 0.924149 0.0317917
\(846\) 0 0
\(847\) 11.3609 0.390365
\(848\) 0 0
\(849\) 18.9771 0.651291
\(850\) 0 0
\(851\) 1.55148 0.0531841
\(852\) 0 0
\(853\) 54.2928 1.85895 0.929474 0.368886i \(-0.120261\pi\)
0.929474 + 0.368886i \(0.120261\pi\)
\(854\) 0 0
\(855\) 6.27949 0.214754
\(856\) 0 0
\(857\) 15.0491 0.514067 0.257034 0.966402i \(-0.417255\pi\)
0.257034 + 0.966402i \(0.417255\pi\)
\(858\) 0 0
\(859\) 34.2351 1.16809 0.584043 0.811723i \(-0.301470\pi\)
0.584043 + 0.811723i \(0.301470\pi\)
\(860\) 0 0
\(861\) 20.5614 0.700732
\(862\) 0 0
\(863\) 14.6488 0.498652 0.249326 0.968420i \(-0.419791\pi\)
0.249326 + 0.968420i \(0.419791\pi\)
\(864\) 0 0
\(865\) −13.1030 −0.445514
\(866\) 0 0
\(867\) −44.3240 −1.50532
\(868\) 0 0
\(869\) 26.8705 0.911518
\(870\) 0 0
\(871\) 20.6018 0.698066
\(872\) 0 0
\(873\) 0.961896 0.0325552
\(874\) 0 0
\(875\) −3.55148 −0.120062
\(876\) 0 0
\(877\) −54.2319 −1.83128 −0.915641 0.401998i \(-0.868316\pi\)
−0.915641 + 0.401998i \(0.868316\pi\)
\(878\) 0 0
\(879\) 27.3976 0.924098
\(880\) 0 0
\(881\) 44.3162 1.49305 0.746525 0.665357i \(-0.231721\pi\)
0.746525 + 0.665357i \(0.231721\pi\)
\(882\) 0 0
\(883\) 38.2704 1.28790 0.643951 0.765067i \(-0.277294\pi\)
0.643951 + 0.765067i \(0.277294\pi\)
\(884\) 0 0
\(885\) −3.31342 −0.111379
\(886\) 0 0
\(887\) −5.80699 −0.194980 −0.0974898 0.995237i \(-0.531081\pi\)
−0.0974898 + 0.995237i \(0.531081\pi\)
\(888\) 0 0
\(889\) −0.516473 −0.0173219
\(890\) 0 0
\(891\) −2.79304 −0.0935704
\(892\) 0 0
\(893\) −39.4320 −1.31954
\(894\) 0 0
\(895\) −14.0759 −0.470504
\(896\) 0 0
\(897\) 3.73151 0.124591
\(898\) 0 0
\(899\) −6.86837 −0.229073
\(900\) 0 0
\(901\) −69.6369 −2.31994
\(902\) 0 0
\(903\) −16.1521 −0.537507
\(904\) 0 0
\(905\) 21.1374 0.702630
\(906\) 0 0
\(907\) −12.3419 −0.409805 −0.204903 0.978782i \(-0.565688\pi\)
−0.204903 + 0.978782i \(0.565688\pi\)
\(908\) 0 0
\(909\) −16.9035 −0.560654
\(910\) 0 0
\(911\) 27.2190 0.901807 0.450903 0.892573i \(-0.351102\pi\)
0.450903 + 0.892573i \(0.351102\pi\)
\(912\) 0 0
\(913\) −16.2861 −0.538992
\(914\) 0 0
\(915\) −11.2180 −0.370854
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 38.6836 1.27605 0.638027 0.770014i \(-0.279751\pi\)
0.638027 + 0.770014i \(0.279751\pi\)
\(920\) 0 0
\(921\) −1.18353 −0.0389986
\(922\) 0 0
\(923\) 23.6746 0.779260
\(924\) 0 0
\(925\) −1.55148 −0.0510124
\(926\) 0 0
\(927\) −6.04143 −0.198427
\(928\) 0 0
\(929\) −17.1365 −0.562230 −0.281115 0.959674i \(-0.590704\pi\)
−0.281115 + 0.959674i \(0.590704\pi\)
\(930\) 0 0
\(931\) 35.2468 1.15517
\(932\) 0 0
\(933\) −18.5770 −0.608183
\(934\) 0 0
\(935\) 21.8722 0.715298
\(936\) 0 0
\(937\) 44.4382 1.45173 0.725866 0.687836i \(-0.241439\pi\)
0.725866 + 0.687836i \(0.241439\pi\)
\(938\) 0 0
\(939\) 8.50094 0.277418
\(940\) 0 0
\(941\) −19.1745 −0.625069 −0.312535 0.949906i \(-0.601178\pi\)
−0.312535 + 0.949906i \(0.601178\pi\)
\(942\) 0 0
\(943\) −5.78954 −0.188533
\(944\) 0 0
\(945\) 3.55148 0.115530
\(946\) 0 0
\(947\) 56.7662 1.84465 0.922327 0.386410i \(-0.126285\pi\)
0.922327 + 0.386410i \(0.126285\pi\)
\(948\) 0 0
\(949\) 57.3997 1.86327
\(950\) 0 0
\(951\) −21.5884 −0.700051
\(952\) 0 0
\(953\) 37.8953 1.22755 0.613774 0.789482i \(-0.289651\pi\)
0.613774 + 0.789482i \(0.289651\pi\)
\(954\) 0 0
\(955\) 2.81647 0.0911389
\(956\) 0 0
\(957\) 7.70443 0.249049
\(958\) 0 0
\(959\) 21.8931 0.706965
\(960\) 0 0
\(961\) −24.8002 −0.800005
\(962\) 0 0
\(963\) 12.3141 0.396816
\(964\) 0 0
\(965\) 15.4560 0.497547
\(966\) 0 0
\(967\) 58.9924 1.89707 0.948534 0.316674i \(-0.102566\pi\)
0.948534 + 0.316674i \(0.102566\pi\)
\(968\) 0 0
\(969\) 49.1745 1.57971
\(970\) 0 0
\(971\) 19.9675 0.640787 0.320394 0.947284i \(-0.396185\pi\)
0.320394 + 0.947284i \(0.396185\pi\)
\(972\) 0 0
\(973\) −49.9659 −1.60183
\(974\) 0 0
\(975\) −3.73151 −0.119504
\(976\) 0 0
\(977\) −5.54712 −0.177468 −0.0887341 0.996055i \(-0.528282\pi\)
−0.0887341 + 0.996055i \(0.528282\pi\)
\(978\) 0 0
\(979\) −1.09065 −0.0348572
\(980\) 0 0
\(981\) 14.8579 0.474376
\(982\) 0 0
\(983\) 12.1636 0.387959 0.193979 0.981006i \(-0.437861\pi\)
0.193979 + 0.981006i \(0.437861\pi\)
\(984\) 0 0
\(985\) 25.7311 0.819862
\(986\) 0 0
\(987\) −22.3015 −0.709864
\(988\) 0 0
\(989\) 4.54798 0.144617
\(990\) 0 0
\(991\) 18.1991 0.578113 0.289057 0.957312i \(-0.406658\pi\)
0.289057 + 0.957312i \(0.406658\pi\)
\(992\) 0 0
\(993\) −24.0288 −0.762531
\(994\) 0 0
\(995\) 5.95857 0.188899
\(996\) 0 0
\(997\) 49.4097 1.56482 0.782411 0.622762i \(-0.213990\pi\)
0.782411 + 0.622762i \(0.213990\pi\)
\(998\) 0 0
\(999\) 1.55148 0.0490867
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 2760.2.a.w.1.2 5
3.2 odd 2 8280.2.a.br.1.2 5
4.3 odd 2 5520.2.a.cc.1.4 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2760.2.a.w.1.2 5 1.1 even 1 trivial
5520.2.a.cc.1.4 5 4.3 odd 2
8280.2.a.br.1.2 5 3.2 odd 2