Properties

Label 7728.2.a.bw.1.2
Level $7728$
Weight $2$
Character 7728.1
Self dual yes
Analytic conductor $61.708$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7728.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(61.7083906820\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.229.1
Defining polynomial: \(x^{3} - 4 x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3864)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.86081\) of defining polynomial
Character \(\chi\) \(=\) 7728.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +0.462598 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +0.462598 q^{5} +1.00000 q^{7} +1.00000 q^{9} +0.398207 q^{11} -0.860806 q^{13} +0.462598 q^{15} -1.60179 q^{17} -6.11982 q^{19} +1.00000 q^{21} -1.00000 q^{23} -4.78600 q^{25} +1.00000 q^{27} -8.24860 q^{29} +1.32340 q^{31} +0.398207 q^{33} +0.462598 q^{35} +10.7666 q^{37} -0.860806 q^{39} +1.04502 q^{41} +2.86081 q^{43} +0.462598 q^{45} -2.27839 q^{47} +1.00000 q^{49} -1.60179 q^{51} -4.46260 q^{53} +0.184210 q^{55} -6.11982 q^{57} +1.90582 q^{59} -13.3580 q^{61} +1.00000 q^{63} -0.398207 q^{65} -2.33382 q^{67} -1.00000 q^{69} -14.8012 q^{71} -5.84143 q^{73} -4.78600 q^{75} +0.398207 q^{77} +1.75140 q^{79} +1.00000 q^{81} +8.76663 q^{83} -0.740987 q^{85} -8.24860 q^{87} +1.91478 q^{89} -0.860806 q^{91} +1.32340 q^{93} -2.83102 q^{95} -13.8414 q^{97} +0.398207 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} - q^{5} + 3 q^{7} + 3 q^{9} + O(q^{10}) \) \( 3 q + 3 q^{3} - q^{5} + 3 q^{7} + 3 q^{9} - 2 q^{11} + 3 q^{13} - q^{15} - 8 q^{17} - 4 q^{19} + 3 q^{21} - 3 q^{23} - 4 q^{25} + 3 q^{27} - 12 q^{29} - 4 q^{31} - 2 q^{33} - q^{35} + 2 q^{37} + 3 q^{39} - 16 q^{41} + 3 q^{43} - q^{45} - 18 q^{47} + 3 q^{49} - 8 q^{51} - 11 q^{53} - 13 q^{55} - 4 q^{57} - 19 q^{59} - 9 q^{61} + 3 q^{63} + 2 q^{65} - 3 q^{67} - 3 q^{69} + 9 q^{71} + 8 q^{73} - 4 q^{75} - 2 q^{77} + 18 q^{79} + 3 q^{81} - 4 q^{83} - 11 q^{85} - 12 q^{87} - 3 q^{89} + 3 q^{91} - 4 q^{93} + 21 q^{95} - 16 q^{97} - 2 q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 0.462598 0.206880 0.103440 0.994636i \(-0.467015\pi\)
0.103440 + 0.994636i \(0.467015\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0.398207 0.120064 0.0600320 0.998196i \(-0.480880\pi\)
0.0600320 + 0.998196i \(0.480880\pi\)
\(12\) 0 0
\(13\) −0.860806 −0.238745 −0.119372 0.992850i \(-0.538088\pi\)
−0.119372 + 0.992850i \(0.538088\pi\)
\(14\) 0 0
\(15\) 0.462598 0.119442
\(16\) 0 0
\(17\) −1.60179 −0.388492 −0.194246 0.980953i \(-0.562226\pi\)
−0.194246 + 0.980953i \(0.562226\pi\)
\(18\) 0 0
\(19\) −6.11982 −1.40398 −0.701991 0.712185i \(-0.747706\pi\)
−0.701991 + 0.712185i \(0.747706\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) −4.78600 −0.957201
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −8.24860 −1.53173 −0.765863 0.643003i \(-0.777688\pi\)
−0.765863 + 0.643003i \(0.777688\pi\)
\(30\) 0 0
\(31\) 1.32340 0.237690 0.118845 0.992913i \(-0.462081\pi\)
0.118845 + 0.992913i \(0.462081\pi\)
\(32\) 0 0
\(33\) 0.398207 0.0693190
\(34\) 0 0
\(35\) 0.462598 0.0781934
\(36\) 0 0
\(37\) 10.7666 1.77002 0.885011 0.465569i \(-0.154150\pi\)
0.885011 + 0.465569i \(0.154150\pi\)
\(38\) 0 0
\(39\) −0.860806 −0.137839
\(40\) 0 0
\(41\) 1.04502 0.163204 0.0816020 0.996665i \(-0.473996\pi\)
0.0816020 + 0.996665i \(0.473996\pi\)
\(42\) 0 0
\(43\) 2.86081 0.436269 0.218134 0.975919i \(-0.430003\pi\)
0.218134 + 0.975919i \(0.430003\pi\)
\(44\) 0 0
\(45\) 0.462598 0.0689601
\(46\) 0 0
\(47\) −2.27839 −0.332337 −0.166169 0.986097i \(-0.553140\pi\)
−0.166169 + 0.986097i \(0.553140\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −1.60179 −0.224296
\(52\) 0 0
\(53\) −4.46260 −0.612985 −0.306493 0.951873i \(-0.599155\pi\)
−0.306493 + 0.951873i \(0.599155\pi\)
\(54\) 0 0
\(55\) 0.184210 0.0248389
\(56\) 0 0
\(57\) −6.11982 −0.810590
\(58\) 0 0
\(59\) 1.90582 0.248117 0.124058 0.992275i \(-0.460409\pi\)
0.124058 + 0.992275i \(0.460409\pi\)
\(60\) 0 0
\(61\) −13.3580 −1.71032 −0.855159 0.518366i \(-0.826540\pi\)
−0.855159 + 0.518366i \(0.826540\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) −0.398207 −0.0493916
\(66\) 0 0
\(67\) −2.33382 −0.285121 −0.142561 0.989786i \(-0.545534\pi\)
−0.142561 + 0.989786i \(0.545534\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −14.8012 −1.75658 −0.878292 0.478126i \(-0.841316\pi\)
−0.878292 + 0.478126i \(0.841316\pi\)
\(72\) 0 0
\(73\) −5.84143 −0.683688 −0.341844 0.939757i \(-0.611051\pi\)
−0.341844 + 0.939757i \(0.611051\pi\)
\(74\) 0 0
\(75\) −4.78600 −0.552640
\(76\) 0 0
\(77\) 0.398207 0.0453799
\(78\) 0 0
\(79\) 1.75140 0.197048 0.0985239 0.995135i \(-0.468588\pi\)
0.0985239 + 0.995135i \(0.468588\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 8.76663 0.962262 0.481131 0.876649i \(-0.340226\pi\)
0.481131 + 0.876649i \(0.340226\pi\)
\(84\) 0 0
\(85\) −0.740987 −0.0803713
\(86\) 0 0
\(87\) −8.24860 −0.884343
\(88\) 0 0
\(89\) 1.91478 0.202967 0.101483 0.994837i \(-0.467641\pi\)
0.101483 + 0.994837i \(0.467641\pi\)
\(90\) 0 0
\(91\) −0.860806 −0.0902370
\(92\) 0 0
\(93\) 1.32340 0.137231
\(94\) 0 0
\(95\) −2.83102 −0.290456
\(96\) 0 0
\(97\) −13.8414 −1.40538 −0.702692 0.711494i \(-0.748019\pi\)
−0.702692 + 0.711494i \(0.748019\pi\)
\(98\) 0 0
\(99\) 0.398207 0.0400214
\(100\) 0 0
\(101\) −12.8310 −1.27673 −0.638367 0.769732i \(-0.720390\pi\)
−0.638367 + 0.769732i \(0.720390\pi\)
\(102\) 0 0
\(103\) −14.8864 −1.46681 −0.733403 0.679795i \(-0.762069\pi\)
−0.733403 + 0.679795i \(0.762069\pi\)
\(104\) 0 0
\(105\) 0.462598 0.0451450
\(106\) 0 0
\(107\) −5.50761 −0.532441 −0.266221 0.963912i \(-0.585775\pi\)
−0.266221 + 0.963912i \(0.585775\pi\)
\(108\) 0 0
\(109\) −3.66618 −0.351157 −0.175578 0.984465i \(-0.556180\pi\)
−0.175578 + 0.984465i \(0.556180\pi\)
\(110\) 0 0
\(111\) 10.7666 1.02192
\(112\) 0 0
\(113\) 2.55263 0.240131 0.120066 0.992766i \(-0.461689\pi\)
0.120066 + 0.992766i \(0.461689\pi\)
\(114\) 0 0
\(115\) −0.462598 −0.0431375
\(116\) 0 0
\(117\) −0.860806 −0.0795815
\(118\) 0 0
\(119\) −1.60179 −0.146836
\(120\) 0 0
\(121\) −10.8414 −0.985585
\(122\) 0 0
\(123\) 1.04502 0.0942259
\(124\) 0 0
\(125\) −4.52699 −0.404906
\(126\) 0 0
\(127\) 12.1544 1.07853 0.539265 0.842136i \(-0.318702\pi\)
0.539265 + 0.842136i \(0.318702\pi\)
\(128\) 0 0
\(129\) 2.86081 0.251880
\(130\) 0 0
\(131\) 12.0990 1.05709 0.528547 0.848904i \(-0.322737\pi\)
0.528547 + 0.848904i \(0.322737\pi\)
\(132\) 0 0
\(133\) −6.11982 −0.530656
\(134\) 0 0
\(135\) 0.462598 0.0398141
\(136\) 0 0
\(137\) −14.8954 −1.27260 −0.636300 0.771441i \(-0.719536\pi\)
−0.636300 + 0.771441i \(0.719536\pi\)
\(138\) 0 0
\(139\) −15.1994 −1.28920 −0.644600 0.764520i \(-0.722976\pi\)
−0.644600 + 0.764520i \(0.722976\pi\)
\(140\) 0 0
\(141\) −2.27839 −0.191875
\(142\) 0 0
\(143\) −0.342779 −0.0286646
\(144\) 0 0
\(145\) −3.81579 −0.316884
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 0 0
\(149\) −16.5180 −1.35321 −0.676605 0.736346i \(-0.736549\pi\)
−0.676605 + 0.736346i \(0.736549\pi\)
\(150\) 0 0
\(151\) 9.16484 0.745824 0.372912 0.927867i \(-0.378359\pi\)
0.372912 + 0.927867i \(0.378359\pi\)
\(152\) 0 0
\(153\) −1.60179 −0.129497
\(154\) 0 0
\(155\) 0.612205 0.0491735
\(156\) 0 0
\(157\) 23.1648 1.84876 0.924378 0.381479i \(-0.124585\pi\)
0.924378 + 0.381479i \(0.124585\pi\)
\(158\) 0 0
\(159\) −4.46260 −0.353907
\(160\) 0 0
\(161\) −1.00000 −0.0788110
\(162\) 0 0
\(163\) 19.0707 1.49373 0.746865 0.664976i \(-0.231558\pi\)
0.746865 + 0.664976i \(0.231558\pi\)
\(164\) 0 0
\(165\) 0.184210 0.0143407
\(166\) 0 0
\(167\) 18.7458 1.45059 0.725297 0.688436i \(-0.241702\pi\)
0.725297 + 0.688436i \(0.241702\pi\)
\(168\) 0 0
\(169\) −12.2590 −0.943001
\(170\) 0 0
\(171\) −6.11982 −0.467994
\(172\) 0 0
\(173\) 19.3926 1.47439 0.737196 0.675678i \(-0.236149\pi\)
0.737196 + 0.675678i \(0.236149\pi\)
\(174\) 0 0
\(175\) −4.78600 −0.361788
\(176\) 0 0
\(177\) 1.90582 0.143250
\(178\) 0 0
\(179\) 14.8310 1.10852 0.554261 0.832343i \(-0.313001\pi\)
0.554261 + 0.832343i \(0.313001\pi\)
\(180\) 0 0
\(181\) 1.60179 0.119060 0.0595302 0.998227i \(-0.481040\pi\)
0.0595302 + 0.998227i \(0.481040\pi\)
\(182\) 0 0
\(183\) −13.3580 −0.987452
\(184\) 0 0
\(185\) 4.98062 0.366183
\(186\) 0 0
\(187\) −0.637846 −0.0466439
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) 13.7216 0.992861 0.496430 0.868076i \(-0.334644\pi\)
0.496430 + 0.868076i \(0.334644\pi\)
\(192\) 0 0
\(193\) 22.6081 1.62736 0.813682 0.581310i \(-0.197460\pi\)
0.813682 + 0.581310i \(0.197460\pi\)
\(194\) 0 0
\(195\) −0.398207 −0.0285162
\(196\) 0 0
\(197\) 7.07962 0.504402 0.252201 0.967675i \(-0.418846\pi\)
0.252201 + 0.967675i \(0.418846\pi\)
\(198\) 0 0
\(199\) −12.6814 −0.898961 −0.449481 0.893290i \(-0.648391\pi\)
−0.449481 + 0.893290i \(0.648391\pi\)
\(200\) 0 0
\(201\) −2.33382 −0.164615
\(202\) 0 0
\(203\) −8.24860 −0.578938
\(204\) 0 0
\(205\) 0.483423 0.0337637
\(206\) 0 0
\(207\) −1.00000 −0.0695048
\(208\) 0 0
\(209\) −2.43696 −0.168568
\(210\) 0 0
\(211\) 3.47301 0.239092 0.119546 0.992829i \(-0.461856\pi\)
0.119546 + 0.992829i \(0.461856\pi\)
\(212\) 0 0
\(213\) −14.8012 −1.01416
\(214\) 0 0
\(215\) 1.32340 0.0902554
\(216\) 0 0
\(217\) 1.32340 0.0898385
\(218\) 0 0
\(219\) −5.84143 −0.394727
\(220\) 0 0
\(221\) 1.37883 0.0927503
\(222\) 0 0
\(223\) 7.38780 0.494723 0.247362 0.968923i \(-0.420436\pi\)
0.247362 + 0.968923i \(0.420436\pi\)
\(224\) 0 0
\(225\) −4.78600 −0.319067
\(226\) 0 0
\(227\) −18.6427 −1.23736 −0.618678 0.785644i \(-0.712332\pi\)
−0.618678 + 0.785644i \(0.712332\pi\)
\(228\) 0 0
\(229\) 8.03460 0.530942 0.265471 0.964119i \(-0.414473\pi\)
0.265471 + 0.964119i \(0.414473\pi\)
\(230\) 0 0
\(231\) 0.398207 0.0262001
\(232\) 0 0
\(233\) −28.2742 −1.85231 −0.926154 0.377147i \(-0.876905\pi\)
−0.926154 + 0.377147i \(0.876905\pi\)
\(234\) 0 0
\(235\) −1.05398 −0.0687540
\(236\) 0 0
\(237\) 1.75140 0.113766
\(238\) 0 0
\(239\) 16.8102 1.08736 0.543681 0.839292i \(-0.317030\pi\)
0.543681 + 0.839292i \(0.317030\pi\)
\(240\) 0 0
\(241\) 1.84143 0.118617 0.0593085 0.998240i \(-0.481110\pi\)
0.0593085 + 0.998240i \(0.481110\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 0.462598 0.0295543
\(246\) 0 0
\(247\) 5.26798 0.335193
\(248\) 0 0
\(249\) 8.76663 0.555562
\(250\) 0 0
\(251\) −15.8116 −0.998022 −0.499011 0.866596i \(-0.666303\pi\)
−0.499011 + 0.866596i \(0.666303\pi\)
\(252\) 0 0
\(253\) −0.398207 −0.0250351
\(254\) 0 0
\(255\) −0.740987 −0.0464024
\(256\) 0 0
\(257\) −27.4045 −1.70944 −0.854722 0.519086i \(-0.826272\pi\)
−0.854722 + 0.519086i \(0.826272\pi\)
\(258\) 0 0
\(259\) 10.7666 0.669006
\(260\) 0 0
\(261\) −8.24860 −0.510576
\(262\) 0 0
\(263\) 5.56304 0.343032 0.171516 0.985181i \(-0.445133\pi\)
0.171516 + 0.985181i \(0.445133\pi\)
\(264\) 0 0
\(265\) −2.06439 −0.126815
\(266\) 0 0
\(267\) 1.91478 0.117183
\(268\) 0 0
\(269\) −11.5076 −0.701632 −0.350816 0.936444i \(-0.614096\pi\)
−0.350816 + 0.936444i \(0.614096\pi\)
\(270\) 0 0
\(271\) 27.2251 1.65381 0.826903 0.562345i \(-0.190101\pi\)
0.826903 + 0.562345i \(0.190101\pi\)
\(272\) 0 0
\(273\) −0.860806 −0.0520983
\(274\) 0 0
\(275\) −1.90582 −0.114925
\(276\) 0 0
\(277\) −3.07962 −0.185036 −0.0925182 0.995711i \(-0.529492\pi\)
−0.0925182 + 0.995711i \(0.529492\pi\)
\(278\) 0 0
\(279\) 1.32340 0.0792301
\(280\) 0 0
\(281\) −23.1648 −1.38190 −0.690949 0.722903i \(-0.742807\pi\)
−0.690949 + 0.722903i \(0.742807\pi\)
\(282\) 0 0
\(283\) 19.4391 1.15553 0.577767 0.816202i \(-0.303924\pi\)
0.577767 + 0.816202i \(0.303924\pi\)
\(284\) 0 0
\(285\) −2.83102 −0.167695
\(286\) 0 0
\(287\) 1.04502 0.0616853
\(288\) 0 0
\(289\) −14.4343 −0.849074
\(290\) 0 0
\(291\) −13.8414 −0.811399
\(292\) 0 0
\(293\) −25.6829 −1.50041 −0.750204 0.661206i \(-0.770045\pi\)
−0.750204 + 0.661206i \(0.770045\pi\)
\(294\) 0 0
\(295\) 0.881630 0.0513305
\(296\) 0 0
\(297\) 0.398207 0.0231063
\(298\) 0 0
\(299\) 0.860806 0.0497817
\(300\) 0 0
\(301\) 2.86081 0.164894
\(302\) 0 0
\(303\) −12.8310 −0.737123
\(304\) 0 0
\(305\) −6.17939 −0.353831
\(306\) 0 0
\(307\) −28.2099 −1.61002 −0.805011 0.593260i \(-0.797840\pi\)
−0.805011 + 0.593260i \(0.797840\pi\)
\(308\) 0 0
\(309\) −14.8864 −0.846860
\(310\) 0 0
\(311\) −22.5437 −1.27833 −0.639167 0.769068i \(-0.720721\pi\)
−0.639167 + 0.769068i \(0.720721\pi\)
\(312\) 0 0
\(313\) 1.20359 0.0680307 0.0340153 0.999421i \(-0.489170\pi\)
0.0340153 + 0.999421i \(0.489170\pi\)
\(314\) 0 0
\(315\) 0.462598 0.0260645
\(316\) 0 0
\(317\) 0.831019 0.0466747 0.0233373 0.999728i \(-0.492571\pi\)
0.0233373 + 0.999728i \(0.492571\pi\)
\(318\) 0 0
\(319\) −3.28465 −0.183905
\(320\) 0 0
\(321\) −5.50761 −0.307405
\(322\) 0 0
\(323\) 9.80268 0.545436
\(324\) 0 0
\(325\) 4.11982 0.228526
\(326\) 0 0
\(327\) −3.66618 −0.202740
\(328\) 0 0
\(329\) −2.27839 −0.125612
\(330\) 0 0
\(331\) −5.05398 −0.277792 −0.138896 0.990307i \(-0.544355\pi\)
−0.138896 + 0.990307i \(0.544355\pi\)
\(332\) 0 0
\(333\) 10.7666 0.590008
\(334\) 0 0
\(335\) −1.07962 −0.0589859
\(336\) 0 0
\(337\) −15.1392 −0.824684 −0.412342 0.911029i \(-0.635289\pi\)
−0.412342 + 0.911029i \(0.635289\pi\)
\(338\) 0 0
\(339\) 2.55263 0.138640
\(340\) 0 0
\(341\) 0.526989 0.0285381
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) −0.462598 −0.0249055
\(346\) 0 0
\(347\) 29.2638 1.57096 0.785482 0.618884i \(-0.212415\pi\)
0.785482 + 0.618884i \(0.212415\pi\)
\(348\) 0 0
\(349\) −10.1842 −0.545148 −0.272574 0.962135i \(-0.587875\pi\)
−0.272574 + 0.962135i \(0.587875\pi\)
\(350\) 0 0
\(351\) −0.860806 −0.0459464
\(352\) 0 0
\(353\) −2.39821 −0.127644 −0.0638219 0.997961i \(-0.520329\pi\)
−0.0638219 + 0.997961i \(0.520329\pi\)
\(354\) 0 0
\(355\) −6.84703 −0.363402
\(356\) 0 0
\(357\) −1.60179 −0.0847759
\(358\) 0 0
\(359\) 14.5914 0.770104 0.385052 0.922895i \(-0.374184\pi\)
0.385052 + 0.922895i \(0.374184\pi\)
\(360\) 0 0
\(361\) 18.4522 0.971168
\(362\) 0 0
\(363\) −10.8414 −0.569028
\(364\) 0 0
\(365\) −2.70224 −0.141442
\(366\) 0 0
\(367\) 0.453636 0.0236796 0.0118398 0.999930i \(-0.496231\pi\)
0.0118398 + 0.999930i \(0.496231\pi\)
\(368\) 0 0
\(369\) 1.04502 0.0544014
\(370\) 0 0
\(371\) −4.46260 −0.231687
\(372\) 0 0
\(373\) 32.0602 1.66002 0.830008 0.557751i \(-0.188336\pi\)
0.830008 + 0.557751i \(0.188336\pi\)
\(374\) 0 0
\(375\) −4.52699 −0.233773
\(376\) 0 0
\(377\) 7.10044 0.365691
\(378\) 0 0
\(379\) 4.00000 0.205466 0.102733 0.994709i \(-0.467241\pi\)
0.102733 + 0.994709i \(0.467241\pi\)
\(380\) 0 0
\(381\) 12.1544 0.622690
\(382\) 0 0
\(383\) −5.62262 −0.287302 −0.143651 0.989628i \(-0.545884\pi\)
−0.143651 + 0.989628i \(0.545884\pi\)
\(384\) 0 0
\(385\) 0.184210 0.00938822
\(386\) 0 0
\(387\) 2.86081 0.145423
\(388\) 0 0
\(389\) −33.0963 −1.67805 −0.839024 0.544094i \(-0.816874\pi\)
−0.839024 + 0.544094i \(0.816874\pi\)
\(390\) 0 0
\(391\) 1.60179 0.0810061
\(392\) 0 0
\(393\) 12.0990 0.610314
\(394\) 0 0
\(395\) 0.810194 0.0407653
\(396\) 0 0
\(397\) 18.9252 0.949828 0.474914 0.880032i \(-0.342479\pi\)
0.474914 + 0.880032i \(0.342479\pi\)
\(398\) 0 0
\(399\) −6.11982 −0.306374
\(400\) 0 0
\(401\) −1.23337 −0.0615917 −0.0307958 0.999526i \(-0.509804\pi\)
−0.0307958 + 0.999526i \(0.509804\pi\)
\(402\) 0 0
\(403\) −1.13919 −0.0567473
\(404\) 0 0
\(405\) 0.462598 0.0229867
\(406\) 0 0
\(407\) 4.28735 0.212516
\(408\) 0 0
\(409\) −5.47301 −0.270623 −0.135311 0.990803i \(-0.543204\pi\)
−0.135311 + 0.990803i \(0.543204\pi\)
\(410\) 0 0
\(411\) −14.8954 −0.734736
\(412\) 0 0
\(413\) 1.90582 0.0937794
\(414\) 0 0
\(415\) 4.05543 0.199073
\(416\) 0 0
\(417\) −15.1994 −0.744320
\(418\) 0 0
\(419\) 12.0138 0.586912 0.293456 0.955973i \(-0.405195\pi\)
0.293456 + 0.955973i \(0.405195\pi\)
\(420\) 0 0
\(421\) 33.4093 1.62827 0.814135 0.580676i \(-0.197212\pi\)
0.814135 + 0.580676i \(0.197212\pi\)
\(422\) 0 0
\(423\) −2.27839 −0.110779
\(424\) 0 0
\(425\) 7.66618 0.371865
\(426\) 0 0
\(427\) −13.3580 −0.646439
\(428\) 0 0
\(429\) −0.342779 −0.0165495
\(430\) 0 0
\(431\) 10.6212 0.511604 0.255802 0.966729i \(-0.417661\pi\)
0.255802 + 0.966729i \(0.417661\pi\)
\(432\) 0 0
\(433\) 7.46405 0.358699 0.179350 0.983785i \(-0.442601\pi\)
0.179350 + 0.983785i \(0.442601\pi\)
\(434\) 0 0
\(435\) −3.81579 −0.182953
\(436\) 0 0
\(437\) 6.11982 0.292751
\(438\) 0 0
\(439\) −21.8206 −1.04144 −0.520720 0.853727i \(-0.674337\pi\)
−0.520720 + 0.853727i \(0.674337\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −20.7368 −0.985237 −0.492619 0.870245i \(-0.663960\pi\)
−0.492619 + 0.870245i \(0.663960\pi\)
\(444\) 0 0
\(445\) 0.885776 0.0419898
\(446\) 0 0
\(447\) −16.5180 −0.781276
\(448\) 0 0
\(449\) −8.46260 −0.399375 −0.199687 0.979860i \(-0.563993\pi\)
−0.199687 + 0.979860i \(0.563993\pi\)
\(450\) 0 0
\(451\) 0.416133 0.0195949
\(452\) 0 0
\(453\) 9.16484 0.430602
\(454\) 0 0
\(455\) −0.398207 −0.0186683
\(456\) 0 0
\(457\) −10.4834 −0.490394 −0.245197 0.969473i \(-0.578853\pi\)
−0.245197 + 0.969473i \(0.578853\pi\)
\(458\) 0 0
\(459\) −1.60179 −0.0747653
\(460\) 0 0
\(461\) −19.2382 −0.896012 −0.448006 0.894031i \(-0.647866\pi\)
−0.448006 + 0.894031i \(0.647866\pi\)
\(462\) 0 0
\(463\) 9.63225 0.447649 0.223824 0.974630i \(-0.428146\pi\)
0.223824 + 0.974630i \(0.428146\pi\)
\(464\) 0 0
\(465\) 0.612205 0.0283903
\(466\) 0 0
\(467\) −32.6475 −1.51075 −0.755373 0.655296i \(-0.772544\pi\)
−0.755373 + 0.655296i \(0.772544\pi\)
\(468\) 0 0
\(469\) −2.33382 −0.107766
\(470\) 0 0
\(471\) 23.1648 1.06738
\(472\) 0 0
\(473\) 1.13919 0.0523802
\(474\) 0 0
\(475\) 29.2895 1.34389
\(476\) 0 0
\(477\) −4.46260 −0.204328
\(478\) 0 0
\(479\) −7.90101 −0.361006 −0.180503 0.983574i \(-0.557773\pi\)
−0.180503 + 0.983574i \(0.557773\pi\)
\(480\) 0 0
\(481\) −9.26798 −0.422583
\(482\) 0 0
\(483\) −1.00000 −0.0455016
\(484\) 0 0
\(485\) −6.40302 −0.290746
\(486\) 0 0
\(487\) −20.0811 −0.909960 −0.454980 0.890502i \(-0.650354\pi\)
−0.454980 + 0.890502i \(0.650354\pi\)
\(488\) 0 0
\(489\) 19.0707 0.862405
\(490\) 0 0
\(491\) −28.7112 −1.29572 −0.647859 0.761760i \(-0.724335\pi\)
−0.647859 + 0.761760i \(0.724335\pi\)
\(492\) 0 0
\(493\) 13.2125 0.595063
\(494\) 0 0
\(495\) 0.184210 0.00827963
\(496\) 0 0
\(497\) −14.8012 −0.663926
\(498\) 0 0
\(499\) −23.1994 −1.03855 −0.519275 0.854607i \(-0.673798\pi\)
−0.519275 + 0.854607i \(0.673798\pi\)
\(500\) 0 0
\(501\) 18.7458 0.837501
\(502\) 0 0
\(503\) 36.4238 1.62406 0.812030 0.583616i \(-0.198363\pi\)
0.812030 + 0.583616i \(0.198363\pi\)
\(504\) 0 0
\(505\) −5.93561 −0.264131
\(506\) 0 0
\(507\) −12.2590 −0.544442
\(508\) 0 0
\(509\) −20.2188 −0.896183 −0.448092 0.893988i \(-0.647896\pi\)
−0.448092 + 0.893988i \(0.647896\pi\)
\(510\) 0 0
\(511\) −5.84143 −0.258410
\(512\) 0 0
\(513\) −6.11982 −0.270197
\(514\) 0 0
\(515\) −6.88645 −0.303453
\(516\) 0 0
\(517\) −0.907271 −0.0399017
\(518\) 0 0
\(519\) 19.3926 0.851241
\(520\) 0 0
\(521\) 4.32967 0.189686 0.0948431 0.995492i \(-0.469765\pi\)
0.0948431 + 0.995492i \(0.469765\pi\)
\(522\) 0 0
\(523\) 6.64681 0.290645 0.145322 0.989384i \(-0.453578\pi\)
0.145322 + 0.989384i \(0.453578\pi\)
\(524\) 0 0
\(525\) −4.78600 −0.208878
\(526\) 0 0
\(527\) −2.11982 −0.0923408
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 1.90582 0.0827056
\(532\) 0 0
\(533\) −0.899556 −0.0389641
\(534\) 0 0
\(535\) −2.54781 −0.110152
\(536\) 0 0
\(537\) 14.8310 0.640006
\(538\) 0 0
\(539\) 0.398207 0.0171520
\(540\) 0 0
\(541\) −39.6620 −1.70520 −0.852602 0.522561i \(-0.824977\pi\)
−0.852602 + 0.522561i \(0.824977\pi\)
\(542\) 0 0
\(543\) 1.60179 0.0687395
\(544\) 0 0
\(545\) −1.69597 −0.0726474
\(546\) 0 0
\(547\) −15.4570 −0.660894 −0.330447 0.943825i \(-0.607199\pi\)
−0.330447 + 0.943825i \(0.607199\pi\)
\(548\) 0 0
\(549\) −13.3580 −0.570106
\(550\) 0 0
\(551\) 50.4799 2.15052
\(552\) 0 0
\(553\) 1.75140 0.0744771
\(554\) 0 0
\(555\) 4.98062 0.211416
\(556\) 0 0
\(557\) 43.6025 1.84750 0.923748 0.383001i \(-0.125110\pi\)
0.923748 + 0.383001i \(0.125110\pi\)
\(558\) 0 0
\(559\) −2.46260 −0.104157
\(560\) 0 0
\(561\) −0.637846 −0.0269299
\(562\) 0 0
\(563\) −14.5616 −0.613698 −0.306849 0.951758i \(-0.599275\pi\)
−0.306849 + 0.951758i \(0.599275\pi\)
\(564\) 0 0
\(565\) 1.18084 0.0496784
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) −15.9100 −0.666981 −0.333490 0.942754i \(-0.608226\pi\)
−0.333490 + 0.942754i \(0.608226\pi\)
\(570\) 0 0
\(571\) 30.4287 1.27340 0.636700 0.771112i \(-0.280299\pi\)
0.636700 + 0.771112i \(0.280299\pi\)
\(572\) 0 0
\(573\) 13.7216 0.573229
\(574\) 0 0
\(575\) 4.78600 0.199590
\(576\) 0 0
\(577\) 9.74244 0.405583 0.202791 0.979222i \(-0.434999\pi\)
0.202791 + 0.979222i \(0.434999\pi\)
\(578\) 0 0
\(579\) 22.6081 0.939559
\(580\) 0 0
\(581\) 8.76663 0.363701
\(582\) 0 0
\(583\) −1.77704 −0.0735975
\(584\) 0 0
\(585\) −0.398207 −0.0164639
\(586\) 0 0
\(587\) 20.7625 0.856959 0.428480 0.903551i \(-0.359049\pi\)
0.428480 + 0.903551i \(0.359049\pi\)
\(588\) 0 0
\(589\) −8.09899 −0.333713
\(590\) 0 0
\(591\) 7.07962 0.291217
\(592\) 0 0
\(593\) 11.1648 0.458485 0.229242 0.973369i \(-0.426375\pi\)
0.229242 + 0.973369i \(0.426375\pi\)
\(594\) 0 0
\(595\) −0.740987 −0.0303775
\(596\) 0 0
\(597\) −12.6814 −0.519016
\(598\) 0 0
\(599\) −14.2624 −0.582745 −0.291373 0.956610i \(-0.594112\pi\)
−0.291373 + 0.956610i \(0.594112\pi\)
\(600\) 0 0
\(601\) 4.83102 0.197061 0.0985307 0.995134i \(-0.468586\pi\)
0.0985307 + 0.995134i \(0.468586\pi\)
\(602\) 0 0
\(603\) −2.33382 −0.0950404
\(604\) 0 0
\(605\) −5.01523 −0.203898
\(606\) 0 0
\(607\) −43.2984 −1.75743 −0.878715 0.477348i \(-0.841598\pi\)
−0.878715 + 0.477348i \(0.841598\pi\)
\(608\) 0 0
\(609\) −8.24860 −0.334250
\(610\) 0 0
\(611\) 1.96125 0.0793437
\(612\) 0 0
\(613\) 3.51176 0.141839 0.0709193 0.997482i \(-0.477407\pi\)
0.0709193 + 0.997482i \(0.477407\pi\)
\(614\) 0 0
\(615\) 0.483423 0.0194935
\(616\) 0 0
\(617\) −41.5560 −1.67298 −0.836491 0.547981i \(-0.815397\pi\)
−0.836491 + 0.547981i \(0.815397\pi\)
\(618\) 0 0
\(619\) −38.8704 −1.56233 −0.781167 0.624322i \(-0.785375\pi\)
−0.781167 + 0.624322i \(0.785375\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) 1.91478 0.0767142
\(624\) 0 0
\(625\) 21.8358 0.873433
\(626\) 0 0
\(627\) −2.43696 −0.0973227
\(628\) 0 0
\(629\) −17.2459 −0.687639
\(630\) 0 0
\(631\) −29.5035 −1.17451 −0.587257 0.809400i \(-0.699792\pi\)
−0.587257 + 0.809400i \(0.699792\pi\)
\(632\) 0 0
\(633\) 3.47301 0.138040
\(634\) 0 0
\(635\) 5.62262 0.223127
\(636\) 0 0
\(637\) −0.860806 −0.0341064
\(638\) 0 0
\(639\) −14.8012 −0.585528
\(640\) 0 0
\(641\) 33.4868 1.32265 0.661324 0.750100i \(-0.269995\pi\)
0.661324 + 0.750100i \(0.269995\pi\)
\(642\) 0 0
\(643\) 5.56719 0.219548 0.109774 0.993957i \(-0.464987\pi\)
0.109774 + 0.993957i \(0.464987\pi\)
\(644\) 0 0
\(645\) 1.32340 0.0521090
\(646\) 0 0
\(647\) −24.4418 −0.960905 −0.480453 0.877021i \(-0.659528\pi\)
−0.480453 + 0.877021i \(0.659528\pi\)
\(648\) 0 0
\(649\) 0.758912 0.0297899
\(650\) 0 0
\(651\) 1.32340 0.0518683
\(652\) 0 0
\(653\) 22.6122 0.884884 0.442442 0.896797i \(-0.354112\pi\)
0.442442 + 0.896797i \(0.354112\pi\)
\(654\) 0 0
\(655\) 5.59698 0.218692
\(656\) 0 0
\(657\) −5.84143 −0.227896
\(658\) 0 0
\(659\) −34.1890 −1.33182 −0.665908 0.746034i \(-0.731956\pi\)
−0.665908 + 0.746034i \(0.731956\pi\)
\(660\) 0 0
\(661\) 16.0090 0.622676 0.311338 0.950299i \(-0.399223\pi\)
0.311338 + 0.950299i \(0.399223\pi\)
\(662\) 0 0
\(663\) 1.37883 0.0535494
\(664\) 0 0
\(665\) −2.83102 −0.109782
\(666\) 0 0
\(667\) 8.24860 0.319387
\(668\) 0 0
\(669\) 7.38780 0.285629
\(670\) 0 0
\(671\) −5.31926 −0.205348
\(672\) 0 0
\(673\) 18.5783 0.716140 0.358070 0.933695i \(-0.383435\pi\)
0.358070 + 0.933695i \(0.383435\pi\)
\(674\) 0 0
\(675\) −4.78600 −0.184213
\(676\) 0 0
\(677\) 43.5768 1.67479 0.837397 0.546596i \(-0.184077\pi\)
0.837397 + 0.546596i \(0.184077\pi\)
\(678\) 0 0
\(679\) −13.8414 −0.531185
\(680\) 0 0
\(681\) −18.6427 −0.714388
\(682\) 0 0
\(683\) −24.9646 −0.955245 −0.477622 0.878565i \(-0.658501\pi\)
−0.477622 + 0.878565i \(0.658501\pi\)
\(684\) 0 0
\(685\) −6.89059 −0.263276
\(686\) 0 0
\(687\) 8.03460 0.306539
\(688\) 0 0
\(689\) 3.84143 0.146347
\(690\) 0 0
\(691\) 8.15442 0.310209 0.155104 0.987898i \(-0.450429\pi\)
0.155104 + 0.987898i \(0.450429\pi\)
\(692\) 0 0
\(693\) 0.398207 0.0151266
\(694\) 0 0
\(695\) −7.03124 −0.266710
\(696\) 0 0
\(697\) −1.67390 −0.0634034
\(698\) 0 0
\(699\) −28.2742 −1.06943
\(700\) 0 0
\(701\) −16.5318 −0.624398 −0.312199 0.950017i \(-0.601066\pi\)
−0.312199 + 0.950017i \(0.601066\pi\)
\(702\) 0 0
\(703\) −65.8898 −2.48508
\(704\) 0 0
\(705\) −1.05398 −0.0396951
\(706\) 0 0
\(707\) −12.8310 −0.482560
\(708\) 0 0
\(709\) −4.13360 −0.155241 −0.0776203 0.996983i \(-0.524732\pi\)
−0.0776203 + 0.996983i \(0.524732\pi\)
\(710\) 0 0
\(711\) 1.75140 0.0656826
\(712\) 0 0
\(713\) −1.32340 −0.0495619
\(714\) 0 0
\(715\) −0.158569 −0.00593015
\(716\) 0 0
\(717\) 16.8102 0.627788
\(718\) 0 0
\(719\) −16.0119 −0.597142 −0.298571 0.954387i \(-0.596510\pi\)
−0.298571 + 0.954387i \(0.596510\pi\)
\(720\) 0 0
\(721\) −14.8864 −0.554400
\(722\) 0 0
\(723\) 1.84143 0.0684835
\(724\) 0 0
\(725\) 39.4778 1.46617
\(726\) 0 0
\(727\) 0.269425 0.00999244 0.00499622 0.999988i \(-0.498410\pi\)
0.00499622 + 0.999988i \(0.498410\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −4.58242 −0.169487
\(732\) 0 0
\(733\) 48.1592 1.77880 0.889401 0.457128i \(-0.151122\pi\)
0.889401 + 0.457128i \(0.151122\pi\)
\(734\) 0 0
\(735\) 0.462598 0.0170632
\(736\) 0 0
\(737\) −0.929343 −0.0342328
\(738\) 0 0
\(739\) 43.6114 1.60427 0.802136 0.597141i \(-0.203697\pi\)
0.802136 + 0.597141i \(0.203697\pi\)
\(740\) 0 0
\(741\) 5.26798 0.193524
\(742\) 0 0
\(743\) 41.1786 1.51070 0.755348 0.655323i \(-0.227468\pi\)
0.755348 + 0.655323i \(0.227468\pi\)
\(744\) 0 0
\(745\) −7.64121 −0.279952
\(746\) 0 0
\(747\) 8.76663 0.320754
\(748\) 0 0
\(749\) −5.50761 −0.201244
\(750\) 0 0
\(751\) 45.1607 1.64794 0.823968 0.566636i \(-0.191755\pi\)
0.823968 + 0.566636i \(0.191755\pi\)
\(752\) 0 0
\(753\) −15.8116 −0.576208
\(754\) 0 0
\(755\) 4.23964 0.154296
\(756\) 0 0
\(757\) 32.3774 1.17678 0.588388 0.808579i \(-0.299763\pi\)
0.588388 + 0.808579i \(0.299763\pi\)
\(758\) 0 0
\(759\) −0.398207 −0.0144540
\(760\) 0 0
\(761\) 35.5333 1.28808 0.644040 0.764992i \(-0.277257\pi\)
0.644040 + 0.764992i \(0.277257\pi\)
\(762\) 0 0
\(763\) −3.66618 −0.132725
\(764\) 0 0
\(765\) −0.740987 −0.0267904
\(766\) 0 0
\(767\) −1.64054 −0.0592366
\(768\) 0 0
\(769\) −12.4585 −0.449263 −0.224632 0.974444i \(-0.572118\pi\)
−0.224632 + 0.974444i \(0.572118\pi\)
\(770\) 0 0
\(771\) −27.4045 −0.986948
\(772\) 0 0
\(773\) −0.359457 −0.0129288 −0.00646439 0.999979i \(-0.502058\pi\)
−0.00646439 + 0.999979i \(0.502058\pi\)
\(774\) 0 0
\(775\) −6.33382 −0.227517
\(776\) 0 0
\(777\) 10.7666 0.386251
\(778\) 0 0
\(779\) −6.39531 −0.229136
\(780\) 0 0
\(781\) −5.89396 −0.210902
\(782\) 0 0
\(783\) −8.24860 −0.294781
\(784\) 0 0
\(785\) 10.7160 0.382471
\(786\) 0 0
\(787\) 40.6731 1.44984 0.724920 0.688833i \(-0.241877\pi\)
0.724920 + 0.688833i \(0.241877\pi\)
\(788\) 0 0
\(789\) 5.56304 0.198050
\(790\) 0 0
\(791\) 2.55263 0.0907611
\(792\) 0 0
\(793\) 11.4987 0.408329
\(794\) 0 0
\(795\) −2.06439 −0.0732164
\(796\) 0 0
\(797\) −42.1109 −1.49164 −0.745822 0.666146i \(-0.767943\pi\)
−0.745822 + 0.666146i \(0.767943\pi\)
\(798\) 0 0
\(799\) 3.64951 0.129110
\(800\) 0 0
\(801\) 1.91478 0.0676556
\(802\) 0 0
\(803\) −2.32610 −0.0820863
\(804\) 0 0
\(805\) −0.462598 −0.0163045
\(806\) 0 0
\(807\) −11.5076 −0.405087
\(808\) 0 0
\(809\) −14.5408 −0.511226 −0.255613 0.966779i \(-0.582277\pi\)
−0.255613 + 0.966779i \(0.582277\pi\)
\(810\) 0 0
\(811\) 52.0990 1.82944 0.914722 0.404085i \(-0.132410\pi\)
0.914722 + 0.404085i \(0.132410\pi\)
\(812\) 0 0
\(813\) 27.2251 0.954825
\(814\) 0 0
\(815\) 8.82206 0.309023
\(816\) 0 0
\(817\) −17.5076 −0.612514
\(818\) 0 0
\(819\) −0.860806 −0.0300790
\(820\) 0 0
\(821\) 24.9557 0.870958 0.435479 0.900199i \(-0.356579\pi\)
0.435479 + 0.900199i \(0.356579\pi\)
\(822\) 0 0
\(823\) −17.4958 −0.609864 −0.304932 0.952374i \(-0.598634\pi\)
−0.304932 + 0.952374i \(0.598634\pi\)
\(824\) 0 0
\(825\) −1.90582 −0.0663522
\(826\) 0 0
\(827\) −0.570556 −0.0198402 −0.00992009 0.999951i \(-0.503158\pi\)
−0.00992009 + 0.999951i \(0.503158\pi\)
\(828\) 0 0
\(829\) 52.0186 1.80668 0.903340 0.428925i \(-0.141107\pi\)
0.903340 + 0.428925i \(0.141107\pi\)
\(830\) 0 0
\(831\) −3.07962 −0.106831
\(832\) 0 0
\(833\) −1.60179 −0.0554988
\(834\) 0 0
\(835\) 8.67178 0.300099
\(836\) 0 0
\(837\) 1.32340 0.0457435
\(838\) 0 0
\(839\) 5.75555 0.198703 0.0993517 0.995052i \(-0.468323\pi\)
0.0993517 + 0.995052i \(0.468323\pi\)
\(840\) 0 0
\(841\) 39.0394 1.34619
\(842\) 0 0
\(843\) −23.1648 −0.797839
\(844\) 0 0
\(845\) −5.67100 −0.195088
\(846\) 0 0
\(847\) −10.8414 −0.372516
\(848\) 0 0
\(849\) 19.4391 0.667147
\(850\) 0 0
\(851\) −10.7666 −0.369075
\(852\) 0 0
\(853\) 50.2313 1.71989 0.859944 0.510388i \(-0.170498\pi\)
0.859944 + 0.510388i \(0.170498\pi\)
\(854\) 0 0
\(855\) −2.83102 −0.0968188
\(856\) 0 0
\(857\) −43.3449 −1.48063 −0.740317 0.672258i \(-0.765324\pi\)
−0.740317 + 0.672258i \(0.765324\pi\)
\(858\) 0 0
\(859\) −10.4667 −0.357121 −0.178560 0.983929i \(-0.557144\pi\)
−0.178560 + 0.983929i \(0.557144\pi\)
\(860\) 0 0
\(861\) 1.04502 0.0356140
\(862\) 0 0
\(863\) 18.6981 0.636490 0.318245 0.948008i \(-0.396906\pi\)
0.318245 + 0.948008i \(0.396906\pi\)
\(864\) 0 0
\(865\) 8.97099 0.305023
\(866\) 0 0
\(867\) −14.4343 −0.490213
\(868\) 0 0
\(869\) 0.697420 0.0236584
\(870\) 0 0
\(871\) 2.00896 0.0680711
\(872\) 0 0
\(873\) −13.8414 −0.468461
\(874\) 0 0
\(875\) −4.52699 −0.153040
\(876\) 0 0
\(877\) −11.3115 −0.381964 −0.190982 0.981594i \(-0.561167\pi\)
−0.190982 + 0.981594i \(0.561167\pi\)
\(878\) 0 0
\(879\) −25.6829 −0.866261
\(880\) 0 0
\(881\) −32.7577 −1.10363 −0.551817 0.833965i \(-0.686065\pi\)
−0.551817 + 0.833965i \(0.686065\pi\)
\(882\) 0 0
\(883\) −29.7293 −1.00047 −0.500236 0.865889i \(-0.666753\pi\)
−0.500236 + 0.865889i \(0.666753\pi\)
\(884\) 0 0
\(885\) 0.881630 0.0296357
\(886\) 0 0
\(887\) −35.5797 −1.19465 −0.597325 0.801999i \(-0.703770\pi\)
−0.597325 + 0.801999i \(0.703770\pi\)
\(888\) 0 0
\(889\) 12.1544 0.407646
\(890\) 0 0
\(891\) 0.398207 0.0133405
\(892\) 0 0
\(893\) 13.9433 0.466596
\(894\) 0 0
\(895\) 6.86081 0.229331
\(896\) 0 0
\(897\) 0.860806 0.0287415
\(898\) 0 0
\(899\) −10.9162 −0.364077
\(900\) 0 0
\(901\) 7.14816 0.238140
\(902\) 0 0
\(903\) 2.86081 0.0952017
\(904\) 0 0
\(905\) 0.740987 0.0246312
\(906\) 0 0
\(907\) −13.3490 −0.443248 −0.221624 0.975132i \(-0.571136\pi\)
−0.221624 + 0.975132i \(0.571136\pi\)
\(908\) 0 0
\(909\) −12.8310 −0.425578
\(910\) 0 0
\(911\) −2.79352 −0.0925533 −0.0462767 0.998929i \(-0.514736\pi\)
−0.0462767 + 0.998929i \(0.514736\pi\)
\(912\) 0 0
\(913\) 3.49094 0.115533
\(914\) 0 0
\(915\) −6.17939 −0.204284
\(916\) 0 0
\(917\) 12.0990 0.399544
\(918\) 0 0
\(919\) −1.32340 −0.0436551 −0.0218275 0.999762i \(-0.506948\pi\)
−0.0218275 + 0.999762i \(0.506948\pi\)
\(920\) 0 0
\(921\) −28.2099 −0.929546
\(922\) 0 0
\(923\) 12.7410 0.419375
\(924\) 0 0
\(925\) −51.5291 −1.69427
\(926\) 0 0
\(927\) −14.8864 −0.488935
\(928\) 0 0
\(929\) 45.3699 1.48854 0.744269 0.667880i \(-0.232798\pi\)
0.744269 + 0.667880i \(0.232798\pi\)
\(930\) 0 0
\(931\) −6.11982 −0.200569
\(932\) 0 0
\(933\) −22.5437 −0.738047
\(934\) 0 0
\(935\) −0.295066 −0.00964970
\(936\) 0 0
\(937\) 55.4439 1.81127 0.905637 0.424055i \(-0.139394\pi\)
0.905637 + 0.424055i \(0.139394\pi\)
\(938\) 0 0
\(939\) 1.20359 0.0392775
\(940\) 0 0
\(941\) −7.83247 −0.255331 −0.127666 0.991817i \(-0.540748\pi\)
−0.127666 + 0.991817i \(0.540748\pi\)
\(942\) 0 0
\(943\) −1.04502 −0.0340304
\(944\) 0 0
\(945\) 0.462598 0.0150483
\(946\) 0 0
\(947\) 10.3476 0.336252 0.168126 0.985766i \(-0.446229\pi\)
0.168126 + 0.985766i \(0.446229\pi\)
\(948\) 0 0
\(949\) 5.02834 0.163227
\(950\) 0 0
\(951\) 0.831019 0.0269476
\(952\) 0 0
\(953\) 9.28880 0.300894 0.150447 0.988618i \(-0.451929\pi\)
0.150447 + 0.988618i \(0.451929\pi\)
\(954\) 0 0
\(955\) 6.34760 0.205403
\(956\) 0 0
\(957\) −3.28465 −0.106178
\(958\) 0 0
\(959\) −14.8954 −0.480998
\(960\) 0 0
\(961\) −29.2486 −0.943503
\(962\) 0 0
\(963\) −5.50761 −0.177480
\(964\) 0 0
\(965\) 10.4585 0.336669
\(966\) 0 0
\(967\) 14.0900 0.453105 0.226552 0.973999i \(-0.427255\pi\)
0.226552 + 0.973999i \(0.427255\pi\)
\(968\) 0 0
\(969\) 9.80268 0.314907
\(970\) 0 0
\(971\) 3.97436 0.127543 0.0637716 0.997965i \(-0.479687\pi\)
0.0637716 + 0.997965i \(0.479687\pi\)
\(972\) 0 0
\(973\) −15.1994 −0.487272
\(974\) 0 0
\(975\) 4.11982 0.131940
\(976\) 0 0
\(977\) −9.83728 −0.314723 −0.157361 0.987541i \(-0.550299\pi\)
−0.157361 + 0.987541i \(0.550299\pi\)
\(978\) 0 0
\(979\) 0.762481 0.0243690
\(980\) 0 0
\(981\) −3.66618 −0.117052
\(982\) 0 0
\(983\) 0.269425 0.00859334 0.00429667 0.999991i \(-0.498632\pi\)
0.00429667 + 0.999991i \(0.498632\pi\)
\(984\) 0 0
\(985\) 3.27502 0.104351
\(986\) 0 0
\(987\) −2.27839 −0.0725219
\(988\) 0 0
\(989\) −2.86081 −0.0909683
\(990\) 0 0
\(991\) 9.99518 0.317507 0.158754 0.987318i \(-0.449252\pi\)
0.158754 + 0.987318i \(0.449252\pi\)
\(992\) 0 0
\(993\) −5.05398 −0.160383
\(994\) 0 0
\(995\) −5.86640 −0.185977
\(996\) 0 0
\(997\) 48.9162 1.54919 0.774596 0.632456i \(-0.217953\pi\)
0.774596 + 0.632456i \(0.217953\pi\)
\(998\) 0 0
\(999\) 10.7666 0.340641
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 7728.2.a.bw.1.2 3
4.3 odd 2 3864.2.a.l.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3864.2.a.l.1.2 3 4.3 odd 2
7728.2.a.bw.1.2 3 1.1 even 1 trivial