Properties

Label 2640.2.a.bb.1.2
Level $2640$
Weight $2$
Character 2640.1
Self dual yes
Analytic conductor $21.081$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2640 = 2^{4} \cdot 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2640.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(21.0805061336\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 2640.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} -1.00000 q^{5} +4.82843 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -1.00000 q^{5} +4.82843 q^{7} +1.00000 q^{9} +1.00000 q^{11} +5.65685 q^{13} -1.00000 q^{15} -6.82843 q^{17} +1.17157 q^{19} +4.82843 q^{21} +4.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} +0.828427 q^{29} +1.00000 q^{33} -4.82843 q^{35} +0.343146 q^{37} +5.65685 q^{39} -0.828427 q^{41} +3.17157 q^{43} -1.00000 q^{45} +4.00000 q^{47} +16.3137 q^{49} -6.82843 q^{51} -13.3137 q^{53} -1.00000 q^{55} +1.17157 q^{57} +4.00000 q^{59} -0.343146 q^{61} +4.82843 q^{63} -5.65685 q^{65} -5.65685 q^{67} +4.00000 q^{69} -13.6569 q^{71} -11.3137 q^{73} +1.00000 q^{75} +4.82843 q^{77} +8.48528 q^{79} +1.00000 q^{81} +10.0000 q^{83} +6.82843 q^{85} +0.828427 q^{87} -7.65685 q^{89} +27.3137 q^{91} -1.17157 q^{95} +0.343146 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} - 2q^{5} + 4q^{7} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{3} - 2q^{5} + 4q^{7} + 2q^{9} + 2q^{11} - 2q^{15} - 8q^{17} + 8q^{19} + 4q^{21} + 8q^{23} + 2q^{25} + 2q^{27} - 4q^{29} + 2q^{33} - 4q^{35} + 12q^{37} + 4q^{41} + 12q^{43} - 2q^{45} + 8q^{47} + 10q^{49} - 8q^{51} - 4q^{53} - 2q^{55} + 8q^{57} + 8q^{59} - 12q^{61} + 4q^{63} + 8q^{69} - 16q^{71} + 2q^{75} + 4q^{77} + 2q^{81} + 20q^{83} + 8q^{85} - 4q^{87} - 4q^{89} + 32q^{91} - 8q^{95} + 12q^{97} + 2q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 4.82843 1.82497 0.912487 0.409106i \(-0.134159\pi\)
0.912487 + 0.409106i \(0.134159\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 5.65685 1.56893 0.784465 0.620174i \(-0.212938\pi\)
0.784465 + 0.620174i \(0.212938\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) −6.82843 −1.65614 −0.828068 0.560627i \(-0.810560\pi\)
−0.828068 + 0.560627i \(0.810560\pi\)
\(18\) 0 0
\(19\) 1.17157 0.268777 0.134389 0.990929i \(-0.457093\pi\)
0.134389 + 0.990929i \(0.457093\pi\)
\(20\) 0 0
\(21\) 4.82843 1.05365
\(22\) 0 0
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 0.828427 0.153835 0.0769175 0.997037i \(-0.475492\pi\)
0.0769175 + 0.997037i \(0.475492\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 1.00000 0.174078
\(34\) 0 0
\(35\) −4.82843 −0.816153
\(36\) 0 0
\(37\) 0.343146 0.0564128 0.0282064 0.999602i \(-0.491020\pi\)
0.0282064 + 0.999602i \(0.491020\pi\)
\(38\) 0 0
\(39\) 5.65685 0.905822
\(40\) 0 0
\(41\) −0.828427 −0.129379 −0.0646893 0.997905i \(-0.520606\pi\)
−0.0646893 + 0.997905i \(0.520606\pi\)
\(42\) 0 0
\(43\) 3.17157 0.483660 0.241830 0.970319i \(-0.422252\pi\)
0.241830 + 0.970319i \(0.422252\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 4.00000 0.583460 0.291730 0.956501i \(-0.405769\pi\)
0.291730 + 0.956501i \(0.405769\pi\)
\(48\) 0 0
\(49\) 16.3137 2.33053
\(50\) 0 0
\(51\) −6.82843 −0.956171
\(52\) 0 0
\(53\) −13.3137 −1.82878 −0.914389 0.404836i \(-0.867329\pi\)
−0.914389 + 0.404836i \(0.867329\pi\)
\(54\) 0 0
\(55\) −1.00000 −0.134840
\(56\) 0 0
\(57\) 1.17157 0.155179
\(58\) 0 0
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) −0.343146 −0.0439353 −0.0219677 0.999759i \(-0.506993\pi\)
−0.0219677 + 0.999759i \(0.506993\pi\)
\(62\) 0 0
\(63\) 4.82843 0.608325
\(64\) 0 0
\(65\) −5.65685 −0.701646
\(66\) 0 0
\(67\) −5.65685 −0.691095 −0.345547 0.938401i \(-0.612307\pi\)
−0.345547 + 0.938401i \(0.612307\pi\)
\(68\) 0 0
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) −13.6569 −1.62077 −0.810385 0.585897i \(-0.800742\pi\)
−0.810385 + 0.585897i \(0.800742\pi\)
\(72\) 0 0
\(73\) −11.3137 −1.32417 −0.662085 0.749429i \(-0.730328\pi\)
−0.662085 + 0.749429i \(0.730328\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) 4.82843 0.550250
\(78\) 0 0
\(79\) 8.48528 0.954669 0.477334 0.878722i \(-0.341603\pi\)
0.477334 + 0.878722i \(0.341603\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 10.0000 1.09764 0.548821 0.835940i \(-0.315077\pi\)
0.548821 + 0.835940i \(0.315077\pi\)
\(84\) 0 0
\(85\) 6.82843 0.740647
\(86\) 0 0
\(87\) 0.828427 0.0888167
\(88\) 0 0
\(89\) −7.65685 −0.811625 −0.405812 0.913956i \(-0.633011\pi\)
−0.405812 + 0.913956i \(0.633011\pi\)
\(90\) 0 0
\(91\) 27.3137 2.86325
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1.17157 −0.120201
\(96\) 0 0
\(97\) 0.343146 0.0348412 0.0174206 0.999848i \(-0.494455\pi\)
0.0174206 + 0.999848i \(0.494455\pi\)
\(98\) 0 0
\(99\) 1.00000 0.100504
\(100\) 0 0
\(101\) 4.82843 0.480446 0.240223 0.970718i \(-0.422779\pi\)
0.240223 + 0.970718i \(0.422779\pi\)
\(102\) 0 0
\(103\) −19.3137 −1.90304 −0.951518 0.307593i \(-0.900477\pi\)
−0.951518 + 0.307593i \(0.900477\pi\)
\(104\) 0 0
\(105\) −4.82843 −0.471206
\(106\) 0 0
\(107\) 5.31371 0.513696 0.256848 0.966452i \(-0.417316\pi\)
0.256848 + 0.966452i \(0.417316\pi\)
\(108\) 0 0
\(109\) −5.31371 −0.508961 −0.254480 0.967078i \(-0.581904\pi\)
−0.254480 + 0.967078i \(0.581904\pi\)
\(110\) 0 0
\(111\) 0.343146 0.0325700
\(112\) 0 0
\(113\) 14.9706 1.40831 0.704156 0.710045i \(-0.251326\pi\)
0.704156 + 0.710045i \(0.251326\pi\)
\(114\) 0 0
\(115\) −4.00000 −0.373002
\(116\) 0 0
\(117\) 5.65685 0.522976
\(118\) 0 0
\(119\) −32.9706 −3.02241
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) −0.828427 −0.0746968
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −2.48528 −0.220533 −0.110267 0.993902i \(-0.535170\pi\)
−0.110267 + 0.993902i \(0.535170\pi\)
\(128\) 0 0
\(129\) 3.17157 0.279241
\(130\) 0 0
\(131\) 19.3137 1.68745 0.843723 0.536778i \(-0.180359\pi\)
0.843723 + 0.536778i \(0.180359\pi\)
\(132\) 0 0
\(133\) 5.65685 0.490511
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) 9.31371 0.795724 0.397862 0.917445i \(-0.369752\pi\)
0.397862 + 0.917445i \(0.369752\pi\)
\(138\) 0 0
\(139\) 16.4853 1.39826 0.699132 0.714993i \(-0.253570\pi\)
0.699132 + 0.714993i \(0.253570\pi\)
\(140\) 0 0
\(141\) 4.00000 0.336861
\(142\) 0 0
\(143\) 5.65685 0.473050
\(144\) 0 0
\(145\) −0.828427 −0.0687971
\(146\) 0 0
\(147\) 16.3137 1.34553
\(148\) 0 0
\(149\) 18.4853 1.51437 0.757187 0.653199i \(-0.226573\pi\)
0.757187 + 0.653199i \(0.226573\pi\)
\(150\) 0 0
\(151\) 0.485281 0.0394916 0.0197458 0.999805i \(-0.493714\pi\)
0.0197458 + 0.999805i \(0.493714\pi\)
\(152\) 0 0
\(153\) −6.82843 −0.552046
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 18.0000 1.43656 0.718278 0.695756i \(-0.244931\pi\)
0.718278 + 0.695756i \(0.244931\pi\)
\(158\) 0 0
\(159\) −13.3137 −1.05585
\(160\) 0 0
\(161\) 19.3137 1.52213
\(162\) 0 0
\(163\) −15.3137 −1.19946 −0.599731 0.800202i \(-0.704726\pi\)
−0.599731 + 0.800202i \(0.704726\pi\)
\(164\) 0 0
\(165\) −1.00000 −0.0778499
\(166\) 0 0
\(167\) −9.31371 −0.720716 −0.360358 0.932814i \(-0.617346\pi\)
−0.360358 + 0.932814i \(0.617346\pi\)
\(168\) 0 0
\(169\) 19.0000 1.46154
\(170\) 0 0
\(171\) 1.17157 0.0895924
\(172\) 0 0
\(173\) −2.82843 −0.215041 −0.107521 0.994203i \(-0.534291\pi\)
−0.107521 + 0.994203i \(0.534291\pi\)
\(174\) 0 0
\(175\) 4.82843 0.364995
\(176\) 0 0
\(177\) 4.00000 0.300658
\(178\) 0 0
\(179\) 6.34315 0.474109 0.237054 0.971496i \(-0.423818\pi\)
0.237054 + 0.971496i \(0.423818\pi\)
\(180\) 0 0
\(181\) −14.0000 −1.04061 −0.520306 0.853980i \(-0.674182\pi\)
−0.520306 + 0.853980i \(0.674182\pi\)
\(182\) 0 0
\(183\) −0.343146 −0.0253661
\(184\) 0 0
\(185\) −0.343146 −0.0252286
\(186\) 0 0
\(187\) −6.82843 −0.499344
\(188\) 0 0
\(189\) 4.82843 0.351216
\(190\) 0 0
\(191\) 5.65685 0.409316 0.204658 0.978834i \(-0.434392\pi\)
0.204658 + 0.978834i \(0.434392\pi\)
\(192\) 0 0
\(193\) 2.34315 0.168663 0.0843317 0.996438i \(-0.473124\pi\)
0.0843317 + 0.996438i \(0.473124\pi\)
\(194\) 0 0
\(195\) −5.65685 −0.405096
\(196\) 0 0
\(197\) 8.48528 0.604551 0.302276 0.953221i \(-0.402254\pi\)
0.302276 + 0.953221i \(0.402254\pi\)
\(198\) 0 0
\(199\) 10.3431 0.733206 0.366603 0.930377i \(-0.380521\pi\)
0.366603 + 0.930377i \(0.380521\pi\)
\(200\) 0 0
\(201\) −5.65685 −0.399004
\(202\) 0 0
\(203\) 4.00000 0.280745
\(204\) 0 0
\(205\) 0.828427 0.0578599
\(206\) 0 0
\(207\) 4.00000 0.278019
\(208\) 0 0
\(209\) 1.17157 0.0810394
\(210\) 0 0
\(211\) −6.82843 −0.470088 −0.235044 0.971985i \(-0.575523\pi\)
−0.235044 + 0.971985i \(0.575523\pi\)
\(212\) 0 0
\(213\) −13.6569 −0.935752
\(214\) 0 0
\(215\) −3.17157 −0.216299
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −11.3137 −0.764510
\(220\) 0 0
\(221\) −38.6274 −2.59836
\(222\) 0 0
\(223\) 17.6569 1.18239 0.591195 0.806529i \(-0.298656\pi\)
0.591195 + 0.806529i \(0.298656\pi\)
\(224\) 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
\(229\) −2.00000 −0.132164 −0.0660819 0.997814i \(-0.521050\pi\)
−0.0660819 + 0.997814i \(0.521050\pi\)
\(230\) 0 0
\(231\) 4.82843 0.317687
\(232\) 0 0
\(233\) −13.1716 −0.862898 −0.431449 0.902137i \(-0.641998\pi\)
−0.431449 + 0.902137i \(0.641998\pi\)
\(234\) 0 0
\(235\) −4.00000 −0.260931
\(236\) 0 0
\(237\) 8.48528 0.551178
\(238\) 0 0
\(239\) −6.34315 −0.410304 −0.205152 0.978730i \(-0.565769\pi\)
−0.205152 + 0.978730i \(0.565769\pi\)
\(240\) 0 0
\(241\) −23.6569 −1.52387 −0.761936 0.647652i \(-0.775751\pi\)
−0.761936 + 0.647652i \(0.775751\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −16.3137 −1.04224
\(246\) 0 0
\(247\) 6.62742 0.421692
\(248\) 0 0
\(249\) 10.0000 0.633724
\(250\) 0 0
\(251\) 12.9706 0.818695 0.409347 0.912379i \(-0.365756\pi\)
0.409347 + 0.912379i \(0.365756\pi\)
\(252\) 0 0
\(253\) 4.00000 0.251478
\(254\) 0 0
\(255\) 6.82843 0.427613
\(256\) 0 0
\(257\) −27.6569 −1.72519 −0.862594 0.505898i \(-0.831161\pi\)
−0.862594 + 0.505898i \(0.831161\pi\)
\(258\) 0 0
\(259\) 1.65685 0.102952
\(260\) 0 0
\(261\) 0.828427 0.0512784
\(262\) 0 0
\(263\) −18.0000 −1.10993 −0.554964 0.831875i \(-0.687268\pi\)
−0.554964 + 0.831875i \(0.687268\pi\)
\(264\) 0 0
\(265\) 13.3137 0.817855
\(266\) 0 0
\(267\) −7.65685 −0.468592
\(268\) 0 0
\(269\) −24.6274 −1.50156 −0.750780 0.660552i \(-0.770322\pi\)
−0.750780 + 0.660552i \(0.770322\pi\)
\(270\) 0 0
\(271\) −27.7990 −1.68867 −0.844334 0.535817i \(-0.820004\pi\)
−0.844334 + 0.535817i \(0.820004\pi\)
\(272\) 0 0
\(273\) 27.3137 1.65310
\(274\) 0 0
\(275\) 1.00000 0.0603023
\(276\) 0 0
\(277\) −13.6569 −0.820561 −0.410280 0.911959i \(-0.634569\pi\)
−0.410280 + 0.911959i \(0.634569\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −16.8284 −1.00390 −0.501950 0.864897i \(-0.667384\pi\)
−0.501950 + 0.864897i \(0.667384\pi\)
\(282\) 0 0
\(283\) 3.17157 0.188530 0.0942652 0.995547i \(-0.469950\pi\)
0.0942652 + 0.995547i \(0.469950\pi\)
\(284\) 0 0
\(285\) −1.17157 −0.0693980
\(286\) 0 0
\(287\) −4.00000 −0.236113
\(288\) 0 0
\(289\) 29.6274 1.74279
\(290\) 0 0
\(291\) 0.343146 0.0201156
\(292\) 0 0
\(293\) −1.17157 −0.0684440 −0.0342220 0.999414i \(-0.510895\pi\)
−0.0342220 + 0.999414i \(0.510895\pi\)
\(294\) 0 0
\(295\) −4.00000 −0.232889
\(296\) 0 0
\(297\) 1.00000 0.0580259
\(298\) 0 0
\(299\) 22.6274 1.30858
\(300\) 0 0
\(301\) 15.3137 0.882667
\(302\) 0 0
\(303\) 4.82843 0.277386
\(304\) 0 0
\(305\) 0.343146 0.0196485
\(306\) 0 0
\(307\) 8.82843 0.503865 0.251932 0.967745i \(-0.418934\pi\)
0.251932 + 0.967745i \(0.418934\pi\)
\(308\) 0 0
\(309\) −19.3137 −1.09872
\(310\) 0 0
\(311\) −19.3137 −1.09518 −0.547590 0.836747i \(-0.684455\pi\)
−0.547590 + 0.836747i \(0.684455\pi\)
\(312\) 0 0
\(313\) 4.34315 0.245489 0.122745 0.992438i \(-0.460830\pi\)
0.122745 + 0.992438i \(0.460830\pi\)
\(314\) 0 0
\(315\) −4.82843 −0.272051
\(316\) 0 0
\(317\) 30.2843 1.70093 0.850467 0.526028i \(-0.176319\pi\)
0.850467 + 0.526028i \(0.176319\pi\)
\(318\) 0 0
\(319\) 0.828427 0.0463830
\(320\) 0 0
\(321\) 5.31371 0.296582
\(322\) 0 0
\(323\) −8.00000 −0.445132
\(324\) 0 0
\(325\) 5.65685 0.313786
\(326\) 0 0
\(327\) −5.31371 −0.293849
\(328\) 0 0
\(329\) 19.3137 1.06480
\(330\) 0 0
\(331\) −17.6569 −0.970508 −0.485254 0.874373i \(-0.661273\pi\)
−0.485254 + 0.874373i \(0.661273\pi\)
\(332\) 0 0
\(333\) 0.343146 0.0188043
\(334\) 0 0
\(335\) 5.65685 0.309067
\(336\) 0 0
\(337\) −19.3137 −1.05208 −0.526042 0.850458i \(-0.676325\pi\)
−0.526042 + 0.850458i \(0.676325\pi\)
\(338\) 0 0
\(339\) 14.9706 0.813089
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 44.9706 2.42818
\(344\) 0 0
\(345\) −4.00000 −0.215353
\(346\) 0 0
\(347\) 6.68629 0.358939 0.179469 0.983764i \(-0.442562\pi\)
0.179469 + 0.983764i \(0.442562\pi\)
\(348\) 0 0
\(349\) −22.9706 −1.22959 −0.614793 0.788688i \(-0.710760\pi\)
−0.614793 + 0.788688i \(0.710760\pi\)
\(350\) 0 0
\(351\) 5.65685 0.301941
\(352\) 0 0
\(353\) 26.0000 1.38384 0.691920 0.721974i \(-0.256765\pi\)
0.691920 + 0.721974i \(0.256765\pi\)
\(354\) 0 0
\(355\) 13.6569 0.724831
\(356\) 0 0
\(357\) −32.9706 −1.74499
\(358\) 0 0
\(359\) −12.0000 −0.633336 −0.316668 0.948536i \(-0.602564\pi\)
−0.316668 + 0.948536i \(0.602564\pi\)
\(360\) 0 0
\(361\) −17.6274 −0.927759
\(362\) 0 0
\(363\) 1.00000 0.0524864
\(364\) 0 0
\(365\) 11.3137 0.592187
\(366\) 0 0
\(367\) 1.65685 0.0864871 0.0432435 0.999065i \(-0.486231\pi\)
0.0432435 + 0.999065i \(0.486231\pi\)
\(368\) 0 0
\(369\) −0.828427 −0.0431262
\(370\) 0 0
\(371\) −64.2843 −3.33747
\(372\) 0 0
\(373\) −34.6274 −1.79294 −0.896470 0.443105i \(-0.853877\pi\)
−0.896470 + 0.443105i \(0.853877\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) 4.68629 0.241356
\(378\) 0 0
\(379\) 0.686292 0.0352524 0.0176262 0.999845i \(-0.494389\pi\)
0.0176262 + 0.999845i \(0.494389\pi\)
\(380\) 0 0
\(381\) −2.48528 −0.127325
\(382\) 0 0
\(383\) 8.00000 0.408781 0.204390 0.978889i \(-0.434479\pi\)
0.204390 + 0.978889i \(0.434479\pi\)
\(384\) 0 0
\(385\) −4.82843 −0.246079
\(386\) 0 0
\(387\) 3.17157 0.161220
\(388\) 0 0
\(389\) −12.3431 −0.625822 −0.312911 0.949782i \(-0.601304\pi\)
−0.312911 + 0.949782i \(0.601304\pi\)
\(390\) 0 0
\(391\) −27.3137 −1.38131
\(392\) 0 0
\(393\) 19.3137 0.974248
\(394\) 0 0
\(395\) −8.48528 −0.426941
\(396\) 0 0
\(397\) 18.9706 0.952105 0.476053 0.879417i \(-0.342067\pi\)
0.476053 + 0.879417i \(0.342067\pi\)
\(398\) 0 0
\(399\) 5.65685 0.283197
\(400\) 0 0
\(401\) −29.3137 −1.46386 −0.731928 0.681382i \(-0.761379\pi\)
−0.731928 + 0.681382i \(0.761379\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) 0.343146 0.0170091
\(408\) 0 0
\(409\) −8.34315 −0.412542 −0.206271 0.978495i \(-0.566133\pi\)
−0.206271 + 0.978495i \(0.566133\pi\)
\(410\) 0 0
\(411\) 9.31371 0.459411
\(412\) 0 0
\(413\) 19.3137 0.950365
\(414\) 0 0
\(415\) −10.0000 −0.490881
\(416\) 0 0
\(417\) 16.4853 0.807288
\(418\) 0 0
\(419\) 3.02944 0.147998 0.0739988 0.997258i \(-0.476424\pi\)
0.0739988 + 0.997258i \(0.476424\pi\)
\(420\) 0 0
\(421\) −6.00000 −0.292422 −0.146211 0.989253i \(-0.546708\pi\)
−0.146211 + 0.989253i \(0.546708\pi\)
\(422\) 0 0
\(423\) 4.00000 0.194487
\(424\) 0 0
\(425\) −6.82843 −0.331227
\(426\) 0 0
\(427\) −1.65685 −0.0801808
\(428\) 0 0
\(429\) 5.65685 0.273115
\(430\) 0 0
\(431\) −10.3431 −0.498212 −0.249106 0.968476i \(-0.580137\pi\)
−0.249106 + 0.968476i \(0.580137\pi\)
\(432\) 0 0
\(433\) −4.34315 −0.208718 −0.104359 0.994540i \(-0.533279\pi\)
−0.104359 + 0.994540i \(0.533279\pi\)
\(434\) 0 0
\(435\) −0.828427 −0.0397200
\(436\) 0 0
\(437\) 4.68629 0.224176
\(438\) 0 0
\(439\) 3.51472 0.167748 0.0838742 0.996476i \(-0.473271\pi\)
0.0838742 + 0.996476i \(0.473271\pi\)
\(440\) 0 0
\(441\) 16.3137 0.776843
\(442\) 0 0
\(443\) −12.0000 −0.570137 −0.285069 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) 0 0
\(445\) 7.65685 0.362970
\(446\) 0 0
\(447\) 18.4853 0.874324
\(448\) 0 0
\(449\) −2.97056 −0.140190 −0.0700948 0.997540i \(-0.522330\pi\)
−0.0700948 + 0.997540i \(0.522330\pi\)
\(450\) 0 0
\(451\) −0.828427 −0.0390091
\(452\) 0 0
\(453\) 0.485281 0.0228005
\(454\) 0 0
\(455\) −27.3137 −1.28049
\(456\) 0 0
\(457\) −0.686292 −0.0321034 −0.0160517 0.999871i \(-0.505110\pi\)
−0.0160517 + 0.999871i \(0.505110\pi\)
\(458\) 0 0
\(459\) −6.82843 −0.318724
\(460\) 0 0
\(461\) −28.1421 −1.31071 −0.655355 0.755321i \(-0.727481\pi\)
−0.655355 + 0.755321i \(0.727481\pi\)
\(462\) 0 0
\(463\) 28.9706 1.34638 0.673188 0.739471i \(-0.264924\pi\)
0.673188 + 0.739471i \(0.264924\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −22.6274 −1.04707 −0.523536 0.852004i \(-0.675387\pi\)
−0.523536 + 0.852004i \(0.675387\pi\)
\(468\) 0 0
\(469\) −27.3137 −1.26123
\(470\) 0 0
\(471\) 18.0000 0.829396
\(472\) 0 0
\(473\) 3.17157 0.145829
\(474\) 0 0
\(475\) 1.17157 0.0537555
\(476\) 0 0
\(477\) −13.3137 −0.609593
\(478\) 0 0
\(479\) −3.02944 −0.138419 −0.0692093 0.997602i \(-0.522048\pi\)
−0.0692093 + 0.997602i \(0.522048\pi\)
\(480\) 0 0
\(481\) 1.94113 0.0885077
\(482\) 0 0
\(483\) 19.3137 0.878804
\(484\) 0 0
\(485\) −0.343146 −0.0155814
\(486\) 0 0
\(487\) −20.9706 −0.950267 −0.475133 0.879914i \(-0.657600\pi\)
−0.475133 + 0.879914i \(0.657600\pi\)
\(488\) 0 0
\(489\) −15.3137 −0.692510
\(490\) 0 0
\(491\) −25.6569 −1.15788 −0.578939 0.815371i \(-0.696533\pi\)
−0.578939 + 0.815371i \(0.696533\pi\)
\(492\) 0 0
\(493\) −5.65685 −0.254772
\(494\) 0 0
\(495\) −1.00000 −0.0449467
\(496\) 0 0
\(497\) −65.9411 −2.95786
\(498\) 0 0
\(499\) 33.6569 1.50669 0.753344 0.657627i \(-0.228440\pi\)
0.753344 + 0.657627i \(0.228440\pi\)
\(500\) 0 0
\(501\) −9.31371 −0.416106
\(502\) 0 0
\(503\) 5.31371 0.236927 0.118463 0.992958i \(-0.462203\pi\)
0.118463 + 0.992958i \(0.462203\pi\)
\(504\) 0 0
\(505\) −4.82843 −0.214862
\(506\) 0 0
\(507\) 19.0000 0.843820
\(508\) 0 0
\(509\) 41.3137 1.83120 0.915599 0.402093i \(-0.131717\pi\)
0.915599 + 0.402093i \(0.131717\pi\)
\(510\) 0 0
\(511\) −54.6274 −2.41657
\(512\) 0 0
\(513\) 1.17157 0.0517262
\(514\) 0 0
\(515\) 19.3137 0.851064
\(516\) 0 0
\(517\) 4.00000 0.175920
\(518\) 0 0
\(519\) −2.82843 −0.124154
\(520\) 0 0
\(521\) 12.6274 0.553217 0.276609 0.960983i \(-0.410789\pi\)
0.276609 + 0.960983i \(0.410789\pi\)
\(522\) 0 0
\(523\) 26.4853 1.15812 0.579060 0.815285i \(-0.303420\pi\)
0.579060 + 0.815285i \(0.303420\pi\)
\(524\) 0 0
\(525\) 4.82843 0.210730
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 4.00000 0.173585
\(532\) 0 0
\(533\) −4.68629 −0.202986
\(534\) 0 0
\(535\) −5.31371 −0.229732
\(536\) 0 0
\(537\) 6.34315 0.273727
\(538\) 0 0
\(539\) 16.3137 0.702681
\(540\) 0 0
\(541\) −5.31371 −0.228454 −0.114227 0.993455i \(-0.536439\pi\)
−0.114227 + 0.993455i \(0.536439\pi\)
\(542\) 0 0
\(543\) −14.0000 −0.600798
\(544\) 0 0
\(545\) 5.31371 0.227614
\(546\) 0 0
\(547\) −20.1421 −0.861216 −0.430608 0.902539i \(-0.641701\pi\)
−0.430608 + 0.902539i \(0.641701\pi\)
\(548\) 0 0
\(549\) −0.343146 −0.0146451
\(550\) 0 0
\(551\) 0.970563 0.0413474
\(552\) 0 0
\(553\) 40.9706 1.74225
\(554\) 0 0
\(555\) −0.343146 −0.0145657
\(556\) 0 0
\(557\) 10.8284 0.458815 0.229408 0.973330i \(-0.426321\pi\)
0.229408 + 0.973330i \(0.426321\pi\)
\(558\) 0 0
\(559\) 17.9411 0.758829
\(560\) 0 0
\(561\) −6.82843 −0.288296
\(562\) 0 0
\(563\) −20.3431 −0.857361 −0.428681 0.903456i \(-0.641021\pi\)
−0.428681 + 0.903456i \(0.641021\pi\)
\(564\) 0 0
\(565\) −14.9706 −0.629816
\(566\) 0 0
\(567\) 4.82843 0.202775
\(568\) 0 0
\(569\) 15.4558 0.647943 0.323971 0.946067i \(-0.394982\pi\)
0.323971 + 0.946067i \(0.394982\pi\)
\(570\) 0 0
\(571\) −0.485281 −0.0203084 −0.0101542 0.999948i \(-0.503232\pi\)
−0.0101542 + 0.999948i \(0.503232\pi\)
\(572\) 0 0
\(573\) 5.65685 0.236318
\(574\) 0 0
\(575\) 4.00000 0.166812
\(576\) 0 0
\(577\) 14.0000 0.582828 0.291414 0.956597i \(-0.405874\pi\)
0.291414 + 0.956597i \(0.405874\pi\)
\(578\) 0 0
\(579\) 2.34315 0.0973778
\(580\) 0 0
\(581\) 48.2843 2.00317
\(582\) 0 0
\(583\) −13.3137 −0.551397
\(584\) 0 0
\(585\) −5.65685 −0.233882
\(586\) 0 0
\(587\) 30.6274 1.26413 0.632064 0.774916i \(-0.282208\pi\)
0.632064 + 0.774916i \(0.282208\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 8.48528 0.349038
\(592\) 0 0
\(593\) −17.1716 −0.705152 −0.352576 0.935783i \(-0.614694\pi\)
−0.352576 + 0.935783i \(0.614694\pi\)
\(594\) 0 0
\(595\) 32.9706 1.35166
\(596\) 0 0
\(597\) 10.3431 0.423317
\(598\) 0 0
\(599\) −4.68629 −0.191477 −0.0957383 0.995407i \(-0.530521\pi\)
−0.0957383 + 0.995407i \(0.530521\pi\)
\(600\) 0 0
\(601\) 17.3137 0.706241 0.353120 0.935578i \(-0.385121\pi\)
0.353120 + 0.935578i \(0.385121\pi\)
\(602\) 0 0
\(603\) −5.65685 −0.230365
\(604\) 0 0
\(605\) −1.00000 −0.0406558
\(606\) 0 0
\(607\) −18.4853 −0.750294 −0.375147 0.926965i \(-0.622408\pi\)
−0.375147 + 0.926965i \(0.622408\pi\)
\(608\) 0 0
\(609\) 4.00000 0.162088
\(610\) 0 0
\(611\) 22.6274 0.915407
\(612\) 0 0
\(613\) 21.9411 0.886194 0.443097 0.896474i \(-0.353880\pi\)
0.443097 + 0.896474i \(0.353880\pi\)
\(614\) 0 0
\(615\) 0.828427 0.0334054
\(616\) 0 0
\(617\) 11.6569 0.469287 0.234644 0.972081i \(-0.424608\pi\)
0.234644 + 0.972081i \(0.424608\pi\)
\(618\) 0 0
\(619\) 25.6569 1.03124 0.515618 0.856819i \(-0.327562\pi\)
0.515618 + 0.856819i \(0.327562\pi\)
\(620\) 0 0
\(621\) 4.00000 0.160514
\(622\) 0 0
\(623\) −36.9706 −1.48119
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 1.17157 0.0467881
\(628\) 0 0
\(629\) −2.34315 −0.0934273
\(630\) 0 0
\(631\) −34.3431 −1.36718 −0.683590 0.729867i \(-0.739582\pi\)
−0.683590 + 0.729867i \(0.739582\pi\)
\(632\) 0 0
\(633\) −6.82843 −0.271406
\(634\) 0 0
\(635\) 2.48528 0.0986254
\(636\) 0 0
\(637\) 92.2843 3.65644
\(638\) 0 0
\(639\) −13.6569 −0.540257
\(640\) 0 0
\(641\) −26.9706 −1.06527 −0.532637 0.846344i \(-0.678799\pi\)
−0.532637 + 0.846344i \(0.678799\pi\)
\(642\) 0 0
\(643\) 29.9411 1.18076 0.590381 0.807124i \(-0.298977\pi\)
0.590381 + 0.807124i \(0.298977\pi\)
\(644\) 0 0
\(645\) −3.17157 −0.124881
\(646\) 0 0
\(647\) −27.3137 −1.07381 −0.536906 0.843642i \(-0.680407\pi\)
−0.536906 + 0.843642i \(0.680407\pi\)
\(648\) 0 0
\(649\) 4.00000 0.157014
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −26.9706 −1.05544 −0.527720 0.849418i \(-0.676953\pi\)
−0.527720 + 0.849418i \(0.676953\pi\)
\(654\) 0 0
\(655\) −19.3137 −0.754649
\(656\) 0 0
\(657\) −11.3137 −0.441390
\(658\) 0 0
\(659\) −7.31371 −0.284902 −0.142451 0.989802i \(-0.545498\pi\)
−0.142451 + 0.989802i \(0.545498\pi\)
\(660\) 0 0
\(661\) −13.3137 −0.517843 −0.258922 0.965898i \(-0.583367\pi\)
−0.258922 + 0.965898i \(0.583367\pi\)
\(662\) 0 0
\(663\) −38.6274 −1.50016
\(664\) 0 0
\(665\) −5.65685 −0.219363
\(666\) 0 0
\(667\) 3.31371 0.128307
\(668\) 0 0
\(669\) 17.6569 0.682653
\(670\) 0 0
\(671\) −0.343146 −0.0132470
\(672\) 0 0
\(673\) 29.6569 1.14319 0.571594 0.820537i \(-0.306325\pi\)
0.571594 + 0.820537i \(0.306325\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) −21.4558 −0.824615 −0.412308 0.911045i \(-0.635277\pi\)
−0.412308 + 0.911045i \(0.635277\pi\)
\(678\) 0 0
\(679\) 1.65685 0.0635842
\(680\) 0 0
\(681\) 14.0000 0.536481
\(682\) 0 0
\(683\) −24.0000 −0.918334 −0.459167 0.888350i \(-0.651852\pi\)
−0.459167 + 0.888350i \(0.651852\pi\)
\(684\) 0 0
\(685\) −9.31371 −0.355859
\(686\) 0 0
\(687\) −2.00000 −0.0763048
\(688\) 0 0
\(689\) −75.3137 −2.86922
\(690\) 0 0
\(691\) −20.0000 −0.760836 −0.380418 0.924815i \(-0.624220\pi\)
−0.380418 + 0.924815i \(0.624220\pi\)
\(692\) 0 0
\(693\) 4.82843 0.183417
\(694\) 0 0
\(695\) −16.4853 −0.625322
\(696\) 0 0
\(697\) 5.65685 0.214269
\(698\) 0 0
\(699\) −13.1716 −0.498195
\(700\) 0 0
\(701\) 7.85786 0.296787 0.148394 0.988928i \(-0.452590\pi\)
0.148394 + 0.988928i \(0.452590\pi\)
\(702\) 0 0
\(703\) 0.402020 0.0151625
\(704\) 0 0
\(705\) −4.00000 −0.150649
\(706\) 0 0
\(707\) 23.3137 0.876802
\(708\) 0 0
\(709\) 29.3137 1.10090 0.550450 0.834868i \(-0.314456\pi\)
0.550450 + 0.834868i \(0.314456\pi\)
\(710\) 0 0
\(711\) 8.48528 0.318223
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) −5.65685 −0.211554
\(716\) 0 0
\(717\) −6.34315 −0.236889
\(718\) 0 0
\(719\) −31.5980 −1.17841 −0.589203 0.807985i \(-0.700558\pi\)
−0.589203 + 0.807985i \(0.700558\pi\)
\(720\) 0 0
\(721\) −93.2548 −3.47299
\(722\) 0 0
\(723\) −23.6569 −0.879808
\(724\) 0 0
\(725\) 0.828427 0.0307670
\(726\) 0 0
\(727\) 33.9411 1.25881 0.629403 0.777079i \(-0.283299\pi\)
0.629403 + 0.777079i \(0.283299\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −21.6569 −0.801008
\(732\) 0 0
\(733\) −17.6569 −0.652171 −0.326085 0.945340i \(-0.605730\pi\)
−0.326085 + 0.945340i \(0.605730\pi\)
\(734\) 0 0
\(735\) −16.3137 −0.601740
\(736\) 0 0
\(737\) −5.65685 −0.208373
\(738\) 0 0
\(739\) −47.1127 −1.73307 −0.866534 0.499118i \(-0.833658\pi\)
−0.866534 + 0.499118i \(0.833658\pi\)
\(740\) 0 0
\(741\) 6.62742 0.243464
\(742\) 0 0
\(743\) −47.6569 −1.74836 −0.874180 0.485602i \(-0.838601\pi\)
−0.874180 + 0.485602i \(0.838601\pi\)
\(744\) 0 0
\(745\) −18.4853 −0.677248
\(746\) 0 0
\(747\) 10.0000 0.365881
\(748\) 0 0
\(749\) 25.6569 0.937481
\(750\) 0 0
\(751\) 36.2843 1.32403 0.662016 0.749490i \(-0.269701\pi\)
0.662016 + 0.749490i \(0.269701\pi\)
\(752\) 0 0
\(753\) 12.9706 0.472674
\(754\) 0 0
\(755\) −0.485281 −0.0176612
\(756\) 0 0
\(757\) 8.62742 0.313569 0.156784 0.987633i \(-0.449887\pi\)
0.156784 + 0.987633i \(0.449887\pi\)
\(758\) 0 0
\(759\) 4.00000 0.145191
\(760\) 0 0
\(761\) −23.1716 −0.839969 −0.419984 0.907531i \(-0.637964\pi\)
−0.419984 + 0.907531i \(0.637964\pi\)
\(762\) 0 0
\(763\) −25.6569 −0.928840
\(764\) 0 0
\(765\) 6.82843 0.246882
\(766\) 0 0
\(767\) 22.6274 0.817029
\(768\) 0 0
\(769\) 33.3137 1.20132 0.600662 0.799503i \(-0.294904\pi\)
0.600662 + 0.799503i \(0.294904\pi\)
\(770\) 0 0
\(771\) −27.6569 −0.996037
\(772\) 0 0
\(773\) −7.65685 −0.275398 −0.137699 0.990474i \(-0.543971\pi\)
−0.137699 + 0.990474i \(0.543971\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 1.65685 0.0594393
\(778\) 0 0
\(779\) −0.970563 −0.0347740
\(780\) 0 0
\(781\) −13.6569 −0.488681
\(782\) 0 0
\(783\) 0.828427 0.0296056
\(784\) 0 0
\(785\) −18.0000 −0.642448
\(786\) 0 0
\(787\) −8.14214 −0.290236 −0.145118 0.989414i \(-0.546356\pi\)
−0.145118 + 0.989414i \(0.546356\pi\)
\(788\) 0 0
\(789\) −18.0000 −0.640817
\(790\) 0 0
\(791\) 72.2843 2.57013
\(792\) 0 0
\(793\) −1.94113 −0.0689314
\(794\) 0 0
\(795\) 13.3137 0.472189
\(796\) 0 0
\(797\) 1.02944 0.0364645 0.0182323 0.999834i \(-0.494196\pi\)
0.0182323 + 0.999834i \(0.494196\pi\)
\(798\) 0 0
\(799\) −27.3137 −0.966290
\(800\) 0 0
\(801\) −7.65685 −0.270542
\(802\) 0 0
\(803\) −11.3137 −0.399252
\(804\) 0 0
\(805\) −19.3137 −0.680719
\(806\) 0 0
\(807\) −24.6274 −0.866926
\(808\) 0 0
\(809\) −56.4264 −1.98385 −0.991923 0.126838i \(-0.959517\pi\)
−0.991923 + 0.126838i \(0.959517\pi\)
\(810\) 0 0
\(811\) 16.4853 0.578877 0.289438 0.957197i \(-0.406532\pi\)
0.289438 + 0.957197i \(0.406532\pi\)
\(812\) 0 0
\(813\) −27.7990 −0.974953
\(814\) 0 0
\(815\) 15.3137 0.536416
\(816\) 0 0
\(817\) 3.71573 0.129997
\(818\) 0 0
\(819\) 27.3137 0.954418
\(820\) 0 0
\(821\) −7.17157 −0.250290 −0.125145 0.992138i \(-0.539940\pi\)
−0.125145 + 0.992138i \(0.539940\pi\)
\(822\) 0 0
\(823\) −16.0000 −0.557725 −0.278862 0.960331i \(-0.589957\pi\)
−0.278862 + 0.960331i \(0.589957\pi\)
\(824\) 0 0
\(825\) 1.00000 0.0348155
\(826\) 0 0
\(827\) −18.6863 −0.649786 −0.324893 0.945751i \(-0.605328\pi\)
−0.324893 + 0.945751i \(0.605328\pi\)
\(828\) 0 0
\(829\) −38.0000 −1.31979 −0.659897 0.751356i \(-0.729400\pi\)
−0.659897 + 0.751356i \(0.729400\pi\)
\(830\) 0 0
\(831\) −13.6569 −0.473751
\(832\) 0 0
\(833\) −111.397 −3.85968
\(834\) 0 0
\(835\) 9.31371 0.322314
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −22.6274 −0.781185 −0.390593 0.920564i \(-0.627730\pi\)
−0.390593 + 0.920564i \(0.627730\pi\)
\(840\) 0 0
\(841\) −28.3137 −0.976335
\(842\) 0 0
\(843\) −16.8284 −0.579602
\(844\) 0 0
\(845\) −19.0000 −0.653620
\(846\) 0 0
\(847\) 4.82843 0.165907
\(848\) 0 0
\(849\) 3.17157 0.108848
\(850\) 0 0
\(851\) 1.37258 0.0470515
\(852\) 0 0
\(853\) −31.3137 −1.07216 −0.536080 0.844167i \(-0.680096\pi\)
−0.536080 + 0.844167i \(0.680096\pi\)
\(854\) 0 0
\(855\) −1.17157 −0.0400669
\(856\) 0 0
\(857\) −11.5147 −0.393335 −0.196668 0.980470i \(-0.563012\pi\)
−0.196668 + 0.980470i \(0.563012\pi\)
\(858\) 0 0
\(859\) 19.0294 0.649276 0.324638 0.945838i \(-0.394758\pi\)
0.324638 + 0.945838i \(0.394758\pi\)
\(860\) 0 0
\(861\) −4.00000 −0.136320
\(862\) 0 0
\(863\) 43.3137 1.47442 0.737208 0.675666i \(-0.236144\pi\)
0.737208 + 0.675666i \(0.236144\pi\)
\(864\) 0 0
\(865\) 2.82843 0.0961694
\(866\) 0 0
\(867\) 29.6274 1.00620
\(868\) 0 0
\(869\) 8.48528 0.287843
\(870\) 0 0
\(871\) −32.0000 −1.08428
\(872\) 0 0
\(873\) 0.343146 0.0116137
\(874\) 0 0
\(875\) −4.82843 −0.163231
\(876\) 0 0
\(877\) −42.6274 −1.43943 −0.719713 0.694272i \(-0.755727\pi\)
−0.719713 + 0.694272i \(0.755727\pi\)
\(878\) 0 0
\(879\) −1.17157 −0.0395162
\(880\) 0 0
\(881\) −13.0294 −0.438973 −0.219486 0.975616i \(-0.570438\pi\)
−0.219486 + 0.975616i \(0.570438\pi\)
\(882\) 0 0
\(883\) 50.6274 1.70375 0.851874 0.523747i \(-0.175466\pi\)
0.851874 + 0.523747i \(0.175466\pi\)
\(884\) 0 0
\(885\) −4.00000 −0.134459
\(886\) 0 0
\(887\) 4.34315 0.145829 0.0729143 0.997338i \(-0.476770\pi\)
0.0729143 + 0.997338i \(0.476770\pi\)
\(888\) 0 0
\(889\) −12.0000 −0.402467
\(890\) 0 0
\(891\) 1.00000 0.0335013
\(892\) 0 0
\(893\) 4.68629 0.156821
\(894\) 0 0
\(895\) −6.34315 −0.212028
\(896\) 0 0
\(897\) 22.6274 0.755507
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 90.9117 3.02871
\(902\) 0 0
\(903\) 15.3137 0.509608
\(904\) 0 0
\(905\) 14.0000 0.465376
\(906\) 0 0
\(907\) −7.02944 −0.233409 −0.116704 0.993167i \(-0.537233\pi\)
−0.116704 + 0.993167i \(0.537233\pi\)
\(908\) 0 0
\(909\) 4.82843 0.160149
\(910\) 0 0
\(911\) 15.0294 0.497947 0.248974 0.968510i \(-0.419907\pi\)
0.248974 + 0.968510i \(0.419907\pi\)
\(912\) 0 0
\(913\) 10.0000 0.330952
\(914\) 0 0
\(915\) 0.343146 0.0113440
\(916\) 0 0
\(917\) 93.2548 3.07955
\(918\) 0 0
\(919\) −28.4853 −0.939643 −0.469821 0.882762i \(-0.655682\pi\)
−0.469821 + 0.882762i \(0.655682\pi\)
\(920\) 0 0
\(921\) 8.82843 0.290907
\(922\) 0 0
\(923\) −77.2548 −2.54287
\(924\) 0 0
\(925\) 0.343146 0.0112826
\(926\) 0 0
\(927\) −19.3137 −0.634345
\(928\) 0 0
\(929\) 33.5980 1.10231 0.551157 0.834402i \(-0.314187\pi\)
0.551157 + 0.834402i \(0.314187\pi\)
\(930\) 0 0
\(931\) 19.1127 0.626393
\(932\) 0 0
\(933\) −19.3137 −0.632302
\(934\) 0 0
\(935\) 6.82843 0.223313
\(936\) 0 0
\(937\) 44.9706 1.46912 0.734562 0.678541i \(-0.237388\pi\)
0.734562 + 0.678541i \(0.237388\pi\)
\(938\) 0 0
\(939\) 4.34315 0.141733
\(940\) 0 0
\(941\) 38.7696 1.26385 0.631926 0.775029i \(-0.282265\pi\)
0.631926 + 0.775029i \(0.282265\pi\)
\(942\) 0 0
\(943\) −3.31371 −0.107909
\(944\) 0 0
\(945\) −4.82843 −0.157069
\(946\) 0 0
\(947\) 38.6274 1.25522 0.627611 0.778527i \(-0.284033\pi\)
0.627611 + 0.778527i \(0.284033\pi\)
\(948\) 0 0
\(949\) −64.0000 −2.07753
\(950\) 0 0
\(951\) 30.2843 0.982035
\(952\) 0 0
\(953\) 27.7990 0.900498 0.450249 0.892903i \(-0.351335\pi\)
0.450249 + 0.892903i \(0.351335\pi\)
\(954\) 0 0
\(955\) −5.65685 −0.183052
\(956\) 0 0
\(957\) 0.828427 0.0267792
\(958\) 0 0
\(959\) 44.9706 1.45218
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) 5.31371 0.171232
\(964\) 0 0
\(965\) −2.34315 −0.0754285
\(966\) 0 0
\(967\) 39.4558 1.26881 0.634407 0.772999i \(-0.281244\pi\)
0.634407 + 0.772999i \(0.281244\pi\)
\(968\) 0 0
\(969\) −8.00000 −0.256997
\(970\) 0 0
\(971\) −10.6274 −0.341050 −0.170525 0.985353i \(-0.554546\pi\)
−0.170525 + 0.985353i \(0.554546\pi\)
\(972\) 0 0
\(973\) 79.5980 2.55179
\(974\) 0 0
\(975\) 5.65685 0.181164
\(976\) 0 0
\(977\) 25.3137 0.809857 0.404929 0.914348i \(-0.367296\pi\)
0.404929 + 0.914348i \(0.367296\pi\)
\(978\) 0 0
\(979\) −7.65685 −0.244714
\(980\) 0 0
\(981\) −5.31371 −0.169654
\(982\) 0 0
\(983\) −14.6274 −0.466542 −0.233271 0.972412i \(-0.574943\pi\)
−0.233271 + 0.972412i \(0.574943\pi\)
\(984\) 0 0
\(985\) −8.48528 −0.270364
\(986\) 0 0
\(987\) 19.3137 0.614762
\(988\) 0 0
\(989\) 12.6863 0.403401
\(990\) 0 0
\(991\) −14.6274 −0.464655 −0.232328 0.972638i \(-0.574634\pi\)
−0.232328 + 0.972638i \(0.574634\pi\)
\(992\) 0 0
\(993\) −17.6569 −0.560323
\(994\) 0 0
\(995\) −10.3431 −0.327900
\(996\) 0 0
\(997\) −16.6863 −0.528460 −0.264230 0.964460i \(-0.585118\pi\)
−0.264230 + 0.964460i \(0.585118\pi\)
\(998\) 0 0
\(999\) 0.343146 0.0108567
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 2640.2.a.bb.1.2 2
3.2 odd 2 7920.2.a.cg.1.2 2
4.3 odd 2 165.2.a.a.1.2 2
12.11 even 2 495.2.a.d.1.1 2
20.3 even 4 825.2.c.e.199.2 4
20.7 even 4 825.2.c.e.199.3 4
20.19 odd 2 825.2.a.g.1.1 2
28.27 even 2 8085.2.a.ba.1.2 2
44.43 even 2 1815.2.a.k.1.1 2
60.23 odd 4 2475.2.c.m.199.3 4
60.47 odd 4 2475.2.c.m.199.2 4
60.59 even 2 2475.2.a.m.1.2 2
132.131 odd 2 5445.2.a.m.1.2 2
220.219 even 2 9075.2.a.v.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.a.a.1.2 2 4.3 odd 2
495.2.a.d.1.1 2 12.11 even 2
825.2.a.g.1.1 2 20.19 odd 2
825.2.c.e.199.2 4 20.3 even 4
825.2.c.e.199.3 4 20.7 even 4
1815.2.a.k.1.1 2 44.43 even 2
2475.2.a.m.1.2 2 60.59 even 2
2475.2.c.m.199.2 4 60.47 odd 4
2475.2.c.m.199.3 4 60.23 odd 4
2640.2.a.bb.1.2 2 1.1 even 1 trivial
5445.2.a.m.1.2 2 132.131 odd 2
7920.2.a.cg.1.2 2 3.2 odd 2
8085.2.a.ba.1.2 2 28.27 even 2
9075.2.a.v.1.2 2 220.219 even 2