Properties

Label 8001.2.a.ba.1.14
Level $8001$
Weight $2$
Character 8001.1
Self dual yes
Analytic conductor $63.888$
Analytic rank $0$
Dimension $40$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8001,2,Mod(1,8001)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8001, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8001.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8001.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(63.8883066572\)
Analytic rank: \(0\)
Dimension: \(40\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.14
Character \(\chi\) \(=\) 8001.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.13347 q^{2} -0.715244 q^{4} -1.11166 q^{5} +1.00000 q^{7} +3.07765 q^{8} +O(q^{10})\) \(q-1.13347 q^{2} -0.715244 q^{4} -1.11166 q^{5} +1.00000 q^{7} +3.07765 q^{8} +1.26004 q^{10} -3.25471 q^{11} +6.02587 q^{13} -1.13347 q^{14} -2.05794 q^{16} -2.98181 q^{17} -6.12147 q^{19} +0.795111 q^{20} +3.68912 q^{22} -5.74306 q^{23} -3.76420 q^{25} -6.83015 q^{26} -0.715244 q^{28} -7.65283 q^{29} +7.30587 q^{31} -3.82268 q^{32} +3.37980 q^{34} -1.11166 q^{35} +5.93067 q^{37} +6.93851 q^{38} -3.42131 q^{40} +4.23604 q^{41} -10.6860 q^{43} +2.32791 q^{44} +6.50959 q^{46} +10.9140 q^{47} +1.00000 q^{49} +4.26661 q^{50} -4.30996 q^{52} +4.13316 q^{53} +3.61815 q^{55} +3.07765 q^{56} +8.67426 q^{58} -7.82101 q^{59} -4.32320 q^{61} -8.28100 q^{62} +8.44878 q^{64} -6.69874 q^{65} -1.91712 q^{67} +2.13272 q^{68} +1.26004 q^{70} +1.98654 q^{71} +0.455884 q^{73} -6.72225 q^{74} +4.37834 q^{76} -3.25471 q^{77} +8.00445 q^{79} +2.28774 q^{80} -4.80143 q^{82} +4.44407 q^{83} +3.31478 q^{85} +12.1123 q^{86} -10.0169 q^{88} +6.16675 q^{89} +6.02587 q^{91} +4.10769 q^{92} -12.3706 q^{94} +6.80502 q^{95} +3.77960 q^{97} -1.13347 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 40 q + 54 q^{4} + 40 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 40 q + 54 q^{4} + 40 q^{7} + 20 q^{10} + 10 q^{13} + 90 q^{16} + 38 q^{19} + 14 q^{22} + 84 q^{25} + 54 q^{28} + 66 q^{31} + 22 q^{34} + 40 q^{37} + 26 q^{40} + 38 q^{43} + 28 q^{46} + 40 q^{49} + 28 q^{52} + 60 q^{55} + 42 q^{58} + 54 q^{61} + 124 q^{64} + 48 q^{67} + 20 q^{70} + 16 q^{76} + 102 q^{79} + 48 q^{82} + 104 q^{85} + 48 q^{88} + 10 q^{91} - 10 q^{94} + 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.13347 −0.801485 −0.400743 0.916191i \(-0.631248\pi\)
−0.400743 + 0.916191i \(0.631248\pi\)
\(3\) 0 0
\(4\) −0.715244 −0.357622
\(5\) −1.11166 −0.497151 −0.248576 0.968612i \(-0.579962\pi\)
−0.248576 + 0.968612i \(0.579962\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 3.07765 1.08811
\(9\) 0 0
\(10\) 1.26004 0.398459
\(11\) −3.25471 −0.981333 −0.490667 0.871347i \(-0.663247\pi\)
−0.490667 + 0.871347i \(0.663247\pi\)
\(12\) 0 0
\(13\) 6.02587 1.67128 0.835638 0.549281i \(-0.185098\pi\)
0.835638 + 0.549281i \(0.185098\pi\)
\(14\) −1.13347 −0.302933
\(15\) 0 0
\(16\) −2.05794 −0.514485
\(17\) −2.98181 −0.723196 −0.361598 0.932334i \(-0.617769\pi\)
−0.361598 + 0.932334i \(0.617769\pi\)
\(18\) 0 0
\(19\) −6.12147 −1.40436 −0.702181 0.711998i \(-0.747790\pi\)
−0.702181 + 0.711998i \(0.747790\pi\)
\(20\) 0.795111 0.177792
\(21\) 0 0
\(22\) 3.68912 0.786524
\(23\) −5.74306 −1.19751 −0.598755 0.800932i \(-0.704338\pi\)
−0.598755 + 0.800932i \(0.704338\pi\)
\(24\) 0 0
\(25\) −3.76420 −0.752840
\(26\) −6.83015 −1.33950
\(27\) 0 0
\(28\) −0.715244 −0.135168
\(29\) −7.65283 −1.42109 −0.710547 0.703650i \(-0.751552\pi\)
−0.710547 + 0.703650i \(0.751552\pi\)
\(30\) 0 0
\(31\) 7.30587 1.31217 0.656087 0.754685i \(-0.272211\pi\)
0.656087 + 0.754685i \(0.272211\pi\)
\(32\) −3.82268 −0.675762
\(33\) 0 0
\(34\) 3.37980 0.579631
\(35\) −1.11166 −0.187906
\(36\) 0 0
\(37\) 5.93067 0.974997 0.487498 0.873124i \(-0.337910\pi\)
0.487498 + 0.873124i \(0.337910\pi\)
\(38\) 6.93851 1.12558
\(39\) 0 0
\(40\) −3.42131 −0.540957
\(41\) 4.23604 0.661558 0.330779 0.943708i \(-0.392688\pi\)
0.330779 + 0.943708i \(0.392688\pi\)
\(42\) 0 0
\(43\) −10.6860 −1.62961 −0.814803 0.579737i \(-0.803155\pi\)
−0.814803 + 0.579737i \(0.803155\pi\)
\(44\) 2.32791 0.350946
\(45\) 0 0
\(46\) 6.50959 0.959787
\(47\) 10.9140 1.59196 0.795982 0.605320i \(-0.206955\pi\)
0.795982 + 0.605320i \(0.206955\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 4.26661 0.603390
\(51\) 0 0
\(52\) −4.30996 −0.597684
\(53\) 4.13316 0.567733 0.283866 0.958864i \(-0.408383\pi\)
0.283866 + 0.958864i \(0.408383\pi\)
\(54\) 0 0
\(55\) 3.61815 0.487871
\(56\) 3.07765 0.411268
\(57\) 0 0
\(58\) 8.67426 1.13899
\(59\) −7.82101 −1.01821 −0.509104 0.860705i \(-0.670023\pi\)
−0.509104 + 0.860705i \(0.670023\pi\)
\(60\) 0 0
\(61\) −4.32320 −0.553529 −0.276764 0.960938i \(-0.589262\pi\)
−0.276764 + 0.960938i \(0.589262\pi\)
\(62\) −8.28100 −1.05169
\(63\) 0 0
\(64\) 8.44878 1.05610
\(65\) −6.69874 −0.830877
\(66\) 0 0
\(67\) −1.91712 −0.234214 −0.117107 0.993119i \(-0.537362\pi\)
−0.117107 + 0.993119i \(0.537362\pi\)
\(68\) 2.13272 0.258631
\(69\) 0 0
\(70\) 1.26004 0.150604
\(71\) 1.98654 0.235759 0.117880 0.993028i \(-0.462390\pi\)
0.117880 + 0.993028i \(0.462390\pi\)
\(72\) 0 0
\(73\) 0.455884 0.0533572 0.0266786 0.999644i \(-0.491507\pi\)
0.0266786 + 0.999644i \(0.491507\pi\)
\(74\) −6.72225 −0.781445
\(75\) 0 0
\(76\) 4.37834 0.502230
\(77\) −3.25471 −0.370909
\(78\) 0 0
\(79\) 8.00445 0.900571 0.450286 0.892885i \(-0.351322\pi\)
0.450286 + 0.892885i \(0.351322\pi\)
\(80\) 2.28774 0.255777
\(81\) 0 0
\(82\) −4.80143 −0.530229
\(83\) 4.44407 0.487800 0.243900 0.969800i \(-0.421573\pi\)
0.243900 + 0.969800i \(0.421573\pi\)
\(84\) 0 0
\(85\) 3.31478 0.359538
\(86\) 12.1123 1.30611
\(87\) 0 0
\(88\) −10.0169 −1.06780
\(89\) 6.16675 0.653674 0.326837 0.945081i \(-0.394017\pi\)
0.326837 + 0.945081i \(0.394017\pi\)
\(90\) 0 0
\(91\) 6.02587 0.631683
\(92\) 4.10769 0.428256
\(93\) 0 0
\(94\) −12.3706 −1.27594
\(95\) 6.80502 0.698181
\(96\) 0 0
\(97\) 3.77960 0.383760 0.191880 0.981418i \(-0.438542\pi\)
0.191880 + 0.981418i \(0.438542\pi\)
\(98\) −1.13347 −0.114498
\(99\) 0 0
\(100\) 2.69232 0.269232
\(101\) −4.23530 −0.421428 −0.210714 0.977548i \(-0.567579\pi\)
−0.210714 + 0.977548i \(0.567579\pi\)
\(102\) 0 0
\(103\) 18.4322 1.81618 0.908091 0.418773i \(-0.137540\pi\)
0.908091 + 0.418773i \(0.137540\pi\)
\(104\) 18.5455 1.81854
\(105\) 0 0
\(106\) −4.68482 −0.455029
\(107\) −5.51966 −0.533606 −0.266803 0.963751i \(-0.585967\pi\)
−0.266803 + 0.963751i \(0.585967\pi\)
\(108\) 0 0
\(109\) −13.3691 −1.28052 −0.640262 0.768156i \(-0.721174\pi\)
−0.640262 + 0.768156i \(0.721174\pi\)
\(110\) −4.10107 −0.391022
\(111\) 0 0
\(112\) −2.05794 −0.194457
\(113\) −9.04421 −0.850807 −0.425404 0.905004i \(-0.639868\pi\)
−0.425404 + 0.905004i \(0.639868\pi\)
\(114\) 0 0
\(115\) 6.38435 0.595344
\(116\) 5.47363 0.508214
\(117\) 0 0
\(118\) 8.86489 0.816079
\(119\) −2.98181 −0.273342
\(120\) 0 0
\(121\) −0.406833 −0.0369848
\(122\) 4.90022 0.443645
\(123\) 0 0
\(124\) −5.22548 −0.469262
\(125\) 9.74285 0.871427
\(126\) 0 0
\(127\) −1.00000 −0.0887357
\(128\) −1.93108 −0.170685
\(129\) 0 0
\(130\) 7.59283 0.665935
\(131\) −14.6257 −1.27785 −0.638927 0.769268i \(-0.720621\pi\)
−0.638927 + 0.769268i \(0.720621\pi\)
\(132\) 0 0
\(133\) −6.12147 −0.530799
\(134\) 2.17301 0.187719
\(135\) 0 0
\(136\) −9.17698 −0.786920
\(137\) 3.21791 0.274925 0.137462 0.990507i \(-0.456105\pi\)
0.137462 + 0.990507i \(0.456105\pi\)
\(138\) 0 0
\(139\) 20.2380 1.71657 0.858283 0.513177i \(-0.171532\pi\)
0.858283 + 0.513177i \(0.171532\pi\)
\(140\) 0.795111 0.0671991
\(141\) 0 0
\(142\) −2.25169 −0.188957
\(143\) −19.6125 −1.64008
\(144\) 0 0
\(145\) 8.50737 0.706499
\(146\) −0.516731 −0.0427650
\(147\) 0 0
\(148\) −4.24188 −0.348680
\(149\) −21.4254 −1.75524 −0.877619 0.479359i \(-0.840869\pi\)
−0.877619 + 0.479359i \(0.840869\pi\)
\(150\) 0 0
\(151\) −21.1145 −1.71827 −0.859137 0.511746i \(-0.828999\pi\)
−0.859137 + 0.511746i \(0.828999\pi\)
\(152\) −18.8397 −1.52811
\(153\) 0 0
\(154\) 3.68912 0.297278
\(155\) −8.12168 −0.652349
\(156\) 0 0
\(157\) −7.49426 −0.598107 −0.299053 0.954236i \(-0.596671\pi\)
−0.299053 + 0.954236i \(0.596671\pi\)
\(158\) −9.07282 −0.721794
\(159\) 0 0
\(160\) 4.24954 0.335956
\(161\) −5.74306 −0.452616
\(162\) 0 0
\(163\) 2.72881 0.213737 0.106868 0.994273i \(-0.465918\pi\)
0.106868 + 0.994273i \(0.465918\pi\)
\(164\) −3.02980 −0.236588
\(165\) 0 0
\(166\) −5.03722 −0.390964
\(167\) −9.06194 −0.701234 −0.350617 0.936519i \(-0.614028\pi\)
−0.350617 + 0.936519i \(0.614028\pi\)
\(168\) 0 0
\(169\) 23.3111 1.79316
\(170\) −3.75720 −0.288164
\(171\) 0 0
\(172\) 7.64313 0.582783
\(173\) 8.33046 0.633353 0.316677 0.948534i \(-0.397433\pi\)
0.316677 + 0.948534i \(0.397433\pi\)
\(174\) 0 0
\(175\) −3.76420 −0.284547
\(176\) 6.69801 0.504881
\(177\) 0 0
\(178\) −6.98984 −0.523910
\(179\) −15.9945 −1.19549 −0.597743 0.801688i \(-0.703936\pi\)
−0.597743 + 0.801688i \(0.703936\pi\)
\(180\) 0 0
\(181\) −6.75440 −0.502051 −0.251025 0.967981i \(-0.580768\pi\)
−0.251025 + 0.967981i \(0.580768\pi\)
\(182\) −6.83015 −0.506284
\(183\) 0 0
\(184\) −17.6751 −1.30303
\(185\) −6.59292 −0.484721
\(186\) 0 0
\(187\) 9.70495 0.709697
\(188\) −7.80613 −0.569321
\(189\) 0 0
\(190\) −7.71330 −0.559581
\(191\) −4.83394 −0.349772 −0.174886 0.984589i \(-0.555956\pi\)
−0.174886 + 0.984589i \(0.555956\pi\)
\(192\) 0 0
\(193\) 5.83823 0.420245 0.210122 0.977675i \(-0.432614\pi\)
0.210122 + 0.977675i \(0.432614\pi\)
\(194\) −4.28407 −0.307578
\(195\) 0 0
\(196\) −0.715244 −0.0510888
\(197\) −6.38000 −0.454556 −0.227278 0.973830i \(-0.572983\pi\)
−0.227278 + 0.973830i \(0.572983\pi\)
\(198\) 0 0
\(199\) 12.5731 0.891284 0.445642 0.895211i \(-0.352975\pi\)
0.445642 + 0.895211i \(0.352975\pi\)
\(200\) −11.5849 −0.819176
\(201\) 0 0
\(202\) 4.80059 0.337769
\(203\) −7.65283 −0.537123
\(204\) 0 0
\(205\) −4.70906 −0.328895
\(206\) −20.8924 −1.45564
\(207\) 0 0
\(208\) −12.4009 −0.859846
\(209\) 19.9236 1.37815
\(210\) 0 0
\(211\) 19.1458 1.31805 0.659027 0.752120i \(-0.270968\pi\)
0.659027 + 0.752120i \(0.270968\pi\)
\(212\) −2.95621 −0.203034
\(213\) 0 0
\(214\) 6.25638 0.427677
\(215\) 11.8793 0.810161
\(216\) 0 0
\(217\) 7.30587 0.495955
\(218\) 15.1535 1.02632
\(219\) 0 0
\(220\) −2.58786 −0.174473
\(221\) −17.9680 −1.20866
\(222\) 0 0
\(223\) 16.9013 1.13180 0.565898 0.824475i \(-0.308530\pi\)
0.565898 + 0.824475i \(0.308530\pi\)
\(224\) −3.82268 −0.255414
\(225\) 0 0
\(226\) 10.2513 0.681909
\(227\) 2.26636 0.150423 0.0752116 0.997168i \(-0.476037\pi\)
0.0752116 + 0.997168i \(0.476037\pi\)
\(228\) 0 0
\(229\) −5.42113 −0.358238 −0.179119 0.983827i \(-0.557325\pi\)
−0.179119 + 0.983827i \(0.557325\pi\)
\(230\) −7.23648 −0.477159
\(231\) 0 0
\(232\) −23.5527 −1.54631
\(233\) 28.1801 1.84614 0.923069 0.384635i \(-0.125673\pi\)
0.923069 + 0.384635i \(0.125673\pi\)
\(234\) 0 0
\(235\) −12.1327 −0.791447
\(236\) 5.59393 0.364134
\(237\) 0 0
\(238\) 3.37980 0.219080
\(239\) 11.6171 0.751449 0.375725 0.926731i \(-0.377394\pi\)
0.375725 + 0.926731i \(0.377394\pi\)
\(240\) 0 0
\(241\) −17.9005 −1.15308 −0.576538 0.817071i \(-0.695597\pi\)
−0.576538 + 0.817071i \(0.695597\pi\)
\(242\) 0.461134 0.0296428
\(243\) 0 0
\(244\) 3.09214 0.197954
\(245\) −1.11166 −0.0710216
\(246\) 0 0
\(247\) −36.8872 −2.34708
\(248\) 22.4849 1.42779
\(249\) 0 0
\(250\) −11.0432 −0.698436
\(251\) 8.13941 0.513755 0.256877 0.966444i \(-0.417306\pi\)
0.256877 + 0.966444i \(0.417306\pi\)
\(252\) 0 0
\(253\) 18.6920 1.17516
\(254\) 1.13347 0.0711203
\(255\) 0 0
\(256\) −14.7087 −0.919296
\(257\) 16.1852 1.00961 0.504804 0.863234i \(-0.331565\pi\)
0.504804 + 0.863234i \(0.331565\pi\)
\(258\) 0 0
\(259\) 5.93067 0.368514
\(260\) 4.79123 0.297140
\(261\) 0 0
\(262\) 16.5778 1.02418
\(263\) 7.37626 0.454840 0.227420 0.973797i \(-0.426971\pi\)
0.227420 + 0.973797i \(0.426971\pi\)
\(264\) 0 0
\(265\) −4.59468 −0.282249
\(266\) 6.93851 0.425427
\(267\) 0 0
\(268\) 1.37121 0.0837601
\(269\) −28.8536 −1.75923 −0.879617 0.475683i \(-0.842201\pi\)
−0.879617 + 0.475683i \(0.842201\pi\)
\(270\) 0 0
\(271\) 30.8082 1.87146 0.935732 0.352712i \(-0.114740\pi\)
0.935732 + 0.352712i \(0.114740\pi\)
\(272\) 6.13639 0.372074
\(273\) 0 0
\(274\) −3.64741 −0.220348
\(275\) 12.2514 0.738787
\(276\) 0 0
\(277\) −5.75477 −0.345770 −0.172885 0.984942i \(-0.555309\pi\)
−0.172885 + 0.984942i \(0.555309\pi\)
\(278\) −22.9392 −1.37580
\(279\) 0 0
\(280\) −3.42131 −0.204463
\(281\) 11.4370 0.682273 0.341136 0.940014i \(-0.389188\pi\)
0.341136 + 0.940014i \(0.389188\pi\)
\(282\) 0 0
\(283\) 21.9866 1.30697 0.653484 0.756940i \(-0.273307\pi\)
0.653484 + 0.756940i \(0.273307\pi\)
\(284\) −1.42086 −0.0843125
\(285\) 0 0
\(286\) 22.2302 1.31450
\(287\) 4.23604 0.250046
\(288\) 0 0
\(289\) −8.10878 −0.476987
\(290\) −9.64286 −0.566248
\(291\) 0 0
\(292\) −0.326068 −0.0190817
\(293\) 24.5513 1.43430 0.717152 0.696917i \(-0.245445\pi\)
0.717152 + 0.696917i \(0.245445\pi\)
\(294\) 0 0
\(295\) 8.69434 0.506204
\(296\) 18.2525 1.06091
\(297\) 0 0
\(298\) 24.2851 1.40680
\(299\) −34.6069 −2.00137
\(300\) 0 0
\(301\) −10.6860 −0.615933
\(302\) 23.9327 1.37717
\(303\) 0 0
\(304\) 12.5976 0.722523
\(305\) 4.80595 0.275188
\(306\) 0 0
\(307\) 20.0864 1.14639 0.573197 0.819418i \(-0.305703\pi\)
0.573197 + 0.819418i \(0.305703\pi\)
\(308\) 2.32791 0.132645
\(309\) 0 0
\(310\) 9.20569 0.522848
\(311\) 6.92673 0.392779 0.196389 0.980526i \(-0.437078\pi\)
0.196389 + 0.980526i \(0.437078\pi\)
\(312\) 0 0
\(313\) −6.04097 −0.341456 −0.170728 0.985318i \(-0.554612\pi\)
−0.170728 + 0.985318i \(0.554612\pi\)
\(314\) 8.49452 0.479374
\(315\) 0 0
\(316\) −5.72513 −0.322064
\(317\) −25.3538 −1.42401 −0.712007 0.702173i \(-0.752213\pi\)
−0.712007 + 0.702173i \(0.752213\pi\)
\(318\) 0 0
\(319\) 24.9078 1.39457
\(320\) −9.39221 −0.525040
\(321\) 0 0
\(322\) 6.50959 0.362765
\(323\) 18.2531 1.01563
\(324\) 0 0
\(325\) −22.6826 −1.25820
\(326\) −3.09303 −0.171307
\(327\) 0 0
\(328\) 13.0371 0.719851
\(329\) 10.9140 0.601706
\(330\) 0 0
\(331\) −27.2950 −1.50027 −0.750135 0.661285i \(-0.770011\pi\)
−0.750135 + 0.661285i \(0.770011\pi\)
\(332\) −3.17859 −0.174448
\(333\) 0 0
\(334\) 10.2715 0.562029
\(335\) 2.13120 0.116440
\(336\) 0 0
\(337\) 16.5755 0.902924 0.451462 0.892290i \(-0.350903\pi\)
0.451462 + 0.892290i \(0.350903\pi\)
\(338\) −26.4225 −1.43719
\(339\) 0 0
\(340\) −2.37087 −0.128579
\(341\) −23.7785 −1.28768
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) −32.8879 −1.77320
\(345\) 0 0
\(346\) −9.44234 −0.507623
\(347\) 33.5428 1.80067 0.900336 0.435195i \(-0.143320\pi\)
0.900336 + 0.435195i \(0.143320\pi\)
\(348\) 0 0
\(349\) 1.49714 0.0801399 0.0400699 0.999197i \(-0.487242\pi\)
0.0400699 + 0.999197i \(0.487242\pi\)
\(350\) 4.26661 0.228060
\(351\) 0 0
\(352\) 12.4417 0.663147
\(353\) −6.55775 −0.349034 −0.174517 0.984654i \(-0.555836\pi\)
−0.174517 + 0.984654i \(0.555836\pi\)
\(354\) 0 0
\(355\) −2.20837 −0.117208
\(356\) −4.41073 −0.233768
\(357\) 0 0
\(358\) 18.1293 0.958164
\(359\) −25.8101 −1.36220 −0.681102 0.732188i \(-0.738499\pi\)
−0.681102 + 0.732188i \(0.738499\pi\)
\(360\) 0 0
\(361\) 18.4724 0.972233
\(362\) 7.65592 0.402386
\(363\) 0 0
\(364\) −4.30996 −0.225903
\(365\) −0.506790 −0.0265266
\(366\) 0 0
\(367\) 28.6181 1.49385 0.746927 0.664906i \(-0.231528\pi\)
0.746927 + 0.664906i \(0.231528\pi\)
\(368\) 11.8189 0.616101
\(369\) 0 0
\(370\) 7.47288 0.388497
\(371\) 4.13316 0.214583
\(372\) 0 0
\(373\) 25.7901 1.33536 0.667680 0.744448i \(-0.267287\pi\)
0.667680 + 0.744448i \(0.267287\pi\)
\(374\) −11.0003 −0.568811
\(375\) 0 0
\(376\) 33.5893 1.73224
\(377\) −46.1149 −2.37504
\(378\) 0 0
\(379\) 21.0276 1.08011 0.540056 0.841629i \(-0.318403\pi\)
0.540056 + 0.841629i \(0.318403\pi\)
\(380\) −4.86725 −0.249685
\(381\) 0 0
\(382\) 5.47913 0.280337
\(383\) 35.9979 1.83941 0.919704 0.392612i \(-0.128429\pi\)
0.919704 + 0.392612i \(0.128429\pi\)
\(384\) 0 0
\(385\) 3.61815 0.184398
\(386\) −6.61746 −0.336820
\(387\) 0 0
\(388\) −2.70333 −0.137241
\(389\) −4.67570 −0.237067 −0.118534 0.992950i \(-0.537819\pi\)
−0.118534 + 0.992950i \(0.537819\pi\)
\(390\) 0 0
\(391\) 17.1247 0.866035
\(392\) 3.07765 0.155445
\(393\) 0 0
\(394\) 7.23155 0.364320
\(395\) −8.89827 −0.447720
\(396\) 0 0
\(397\) −17.8933 −0.898041 −0.449021 0.893521i \(-0.648227\pi\)
−0.449021 + 0.893521i \(0.648227\pi\)
\(398\) −14.2512 −0.714351
\(399\) 0 0
\(400\) 7.74650 0.387325
\(401\) −25.2484 −1.26084 −0.630422 0.776253i \(-0.717118\pi\)
−0.630422 + 0.776253i \(0.717118\pi\)
\(402\) 0 0
\(403\) 44.0242 2.19300
\(404\) 3.02927 0.150712
\(405\) 0 0
\(406\) 8.67426 0.430496
\(407\) −19.3026 −0.956797
\(408\) 0 0
\(409\) 15.1682 0.750017 0.375009 0.927021i \(-0.377640\pi\)
0.375009 + 0.927021i \(0.377640\pi\)
\(410\) 5.33758 0.263604
\(411\) 0 0
\(412\) −13.1835 −0.649506
\(413\) −7.82101 −0.384847
\(414\) 0 0
\(415\) −4.94031 −0.242510
\(416\) −23.0350 −1.12938
\(417\) 0 0
\(418\) −22.5829 −1.10456
\(419\) −2.22005 −0.108457 −0.0542283 0.998529i \(-0.517270\pi\)
−0.0542283 + 0.998529i \(0.517270\pi\)
\(420\) 0 0
\(421\) −3.48135 −0.169671 −0.0848353 0.996395i \(-0.527036\pi\)
−0.0848353 + 0.996395i \(0.527036\pi\)
\(422\) −21.7012 −1.05640
\(423\) 0 0
\(424\) 12.7204 0.617758
\(425\) 11.2242 0.544451
\(426\) 0 0
\(427\) −4.32320 −0.209214
\(428\) 3.94790 0.190829
\(429\) 0 0
\(430\) −13.4648 −0.649332
\(431\) 19.6652 0.947240 0.473620 0.880729i \(-0.342947\pi\)
0.473620 + 0.880729i \(0.342947\pi\)
\(432\) 0 0
\(433\) −23.8927 −1.14821 −0.574106 0.818781i \(-0.694650\pi\)
−0.574106 + 0.818781i \(0.694650\pi\)
\(434\) −8.28100 −0.397500
\(435\) 0 0
\(436\) 9.56214 0.457944
\(437\) 35.1560 1.68174
\(438\) 0 0
\(439\) −3.49404 −0.166761 −0.0833806 0.996518i \(-0.526572\pi\)
−0.0833806 + 0.996518i \(0.526572\pi\)
\(440\) 11.1354 0.530859
\(441\) 0 0
\(442\) 20.3662 0.968723
\(443\) 18.6063 0.884011 0.442006 0.897012i \(-0.354267\pi\)
0.442006 + 0.897012i \(0.354267\pi\)
\(444\) 0 0
\(445\) −6.85536 −0.324975
\(446\) −19.1571 −0.907117
\(447\) 0 0
\(448\) 8.44878 0.399167
\(449\) −20.4734 −0.966201 −0.483101 0.875565i \(-0.660490\pi\)
−0.483101 + 0.875565i \(0.660490\pi\)
\(450\) 0 0
\(451\) −13.7871 −0.649209
\(452\) 6.46881 0.304267
\(453\) 0 0
\(454\) −2.56885 −0.120562
\(455\) −6.69874 −0.314042
\(456\) 0 0
\(457\) 30.8314 1.44223 0.721115 0.692815i \(-0.243630\pi\)
0.721115 + 0.692815i \(0.243630\pi\)
\(458\) 6.14469 0.287123
\(459\) 0 0
\(460\) −4.56637 −0.212908
\(461\) 33.3908 1.55517 0.777583 0.628781i \(-0.216446\pi\)
0.777583 + 0.628781i \(0.216446\pi\)
\(462\) 0 0
\(463\) 4.46991 0.207734 0.103867 0.994591i \(-0.466878\pi\)
0.103867 + 0.994591i \(0.466878\pi\)
\(464\) 15.7491 0.731131
\(465\) 0 0
\(466\) −31.9413 −1.47965
\(467\) 35.4354 1.63976 0.819878 0.572538i \(-0.194041\pi\)
0.819878 + 0.572538i \(0.194041\pi\)
\(468\) 0 0
\(469\) −1.91712 −0.0885246
\(470\) 13.7520 0.634333
\(471\) 0 0
\(472\) −24.0703 −1.10793
\(473\) 34.7800 1.59919
\(474\) 0 0
\(475\) 23.0425 1.05726
\(476\) 2.13272 0.0977532
\(477\) 0 0
\(478\) −13.1677 −0.602275
\(479\) −16.7718 −0.766324 −0.383162 0.923681i \(-0.625165\pi\)
−0.383162 + 0.923681i \(0.625165\pi\)
\(480\) 0 0
\(481\) 35.7375 1.62949
\(482\) 20.2897 0.924173
\(483\) 0 0
\(484\) 0.290985 0.0132266
\(485\) −4.20165 −0.190787
\(486\) 0 0
\(487\) 26.8363 1.21607 0.608034 0.793911i \(-0.291958\pi\)
0.608034 + 0.793911i \(0.291958\pi\)
\(488\) −13.3053 −0.602302
\(489\) 0 0
\(490\) 1.26004 0.0569228
\(491\) 40.7790 1.84033 0.920165 0.391530i \(-0.128054\pi\)
0.920165 + 0.391530i \(0.128054\pi\)
\(492\) 0 0
\(493\) 22.8193 1.02773
\(494\) 41.8106 1.88115
\(495\) 0 0
\(496\) −15.0350 −0.675094
\(497\) 1.98654 0.0891085
\(498\) 0 0
\(499\) 4.03766 0.180751 0.0903753 0.995908i \(-0.471193\pi\)
0.0903753 + 0.995908i \(0.471193\pi\)
\(500\) −6.96851 −0.311641
\(501\) 0 0
\(502\) −9.22578 −0.411767
\(503\) −40.3253 −1.79802 −0.899008 0.437932i \(-0.855711\pi\)
−0.899008 + 0.437932i \(0.855711\pi\)
\(504\) 0 0
\(505\) 4.70824 0.209514
\(506\) −21.1869 −0.941871
\(507\) 0 0
\(508\) 0.715244 0.0317338
\(509\) 17.0735 0.756771 0.378386 0.925648i \(-0.376479\pi\)
0.378386 + 0.925648i \(0.376479\pi\)
\(510\) 0 0
\(511\) 0.455884 0.0201671
\(512\) 20.5341 0.907487
\(513\) 0 0
\(514\) −18.3455 −0.809185
\(515\) −20.4905 −0.902918
\(516\) 0 0
\(517\) −35.5218 −1.56225
\(518\) −6.72225 −0.295359
\(519\) 0 0
\(520\) −20.6164 −0.904088
\(521\) −10.5698 −0.463073 −0.231536 0.972826i \(-0.574375\pi\)
−0.231536 + 0.972826i \(0.574375\pi\)
\(522\) 0 0
\(523\) −11.8649 −0.518815 −0.259407 0.965768i \(-0.583527\pi\)
−0.259407 + 0.965768i \(0.583527\pi\)
\(524\) 10.4609 0.456988
\(525\) 0 0
\(526\) −8.36078 −0.364547
\(527\) −21.7848 −0.948959
\(528\) 0 0
\(529\) 9.98272 0.434031
\(530\) 5.20794 0.226219
\(531\) 0 0
\(532\) 4.37834 0.189825
\(533\) 25.5258 1.10565
\(534\) 0 0
\(535\) 6.13601 0.265283
\(536\) −5.90024 −0.254852
\(537\) 0 0
\(538\) 32.7047 1.41000
\(539\) −3.25471 −0.140190
\(540\) 0 0
\(541\) −8.62403 −0.370776 −0.185388 0.982665i \(-0.559354\pi\)
−0.185388 + 0.982665i \(0.559354\pi\)
\(542\) −34.9202 −1.49995
\(543\) 0 0
\(544\) 11.3985 0.488708
\(545\) 14.8619 0.636615
\(546\) 0 0
\(547\) 33.5125 1.43289 0.716445 0.697644i \(-0.245768\pi\)
0.716445 + 0.697644i \(0.245768\pi\)
\(548\) −2.30159 −0.0983190
\(549\) 0 0
\(550\) −13.8866 −0.592127
\(551\) 46.8466 1.99573
\(552\) 0 0
\(553\) 8.00445 0.340384
\(554\) 6.52286 0.277130
\(555\) 0 0
\(556\) −14.4751 −0.613881
\(557\) −15.0410 −0.637306 −0.318653 0.947871i \(-0.603230\pi\)
−0.318653 + 0.947871i \(0.603230\pi\)
\(558\) 0 0
\(559\) −64.3927 −2.72352
\(560\) 2.28774 0.0966746
\(561\) 0 0
\(562\) −12.9635 −0.546831
\(563\) −15.7124 −0.662197 −0.331099 0.943596i \(-0.607419\pi\)
−0.331099 + 0.943596i \(0.607419\pi\)
\(564\) 0 0
\(565\) 10.0541 0.422980
\(566\) −24.9212 −1.04751
\(567\) 0 0
\(568\) 6.11388 0.256533
\(569\) 34.6546 1.45280 0.726399 0.687273i \(-0.241193\pi\)
0.726399 + 0.687273i \(0.241193\pi\)
\(570\) 0 0
\(571\) 9.56214 0.400163 0.200082 0.979779i \(-0.435879\pi\)
0.200082 + 0.979779i \(0.435879\pi\)
\(572\) 14.0277 0.586528
\(573\) 0 0
\(574\) −4.80143 −0.200408
\(575\) 21.6180 0.901534
\(576\) 0 0
\(577\) 18.8185 0.783424 0.391712 0.920088i \(-0.371883\pi\)
0.391712 + 0.920088i \(0.371883\pi\)
\(578\) 9.19107 0.382298
\(579\) 0 0
\(580\) −6.08484 −0.252659
\(581\) 4.44407 0.184371
\(582\) 0 0
\(583\) −13.4523 −0.557135
\(584\) 1.40305 0.0580586
\(585\) 0 0
\(586\) −27.8282 −1.14957
\(587\) 29.9799 1.23740 0.618701 0.785627i \(-0.287659\pi\)
0.618701 + 0.785627i \(0.287659\pi\)
\(588\) 0 0
\(589\) −44.7227 −1.84277
\(590\) −9.85478 −0.405715
\(591\) 0 0
\(592\) −12.2050 −0.501621
\(593\) −20.0227 −0.822232 −0.411116 0.911583i \(-0.634861\pi\)
−0.411116 + 0.911583i \(0.634861\pi\)
\(594\) 0 0
\(595\) 3.31478 0.135893
\(596\) 15.3244 0.627711
\(597\) 0 0
\(598\) 39.2259 1.60407
\(599\) −42.4949 −1.73630 −0.868148 0.496305i \(-0.834690\pi\)
−0.868148 + 0.496305i \(0.834690\pi\)
\(600\) 0 0
\(601\) 20.1112 0.820352 0.410176 0.912006i \(-0.365467\pi\)
0.410176 + 0.912006i \(0.365467\pi\)
\(602\) 12.1123 0.493661
\(603\) 0 0
\(604\) 15.1020 0.614492
\(605\) 0.452262 0.0183871
\(606\) 0 0
\(607\) −18.3581 −0.745133 −0.372567 0.928005i \(-0.621522\pi\)
−0.372567 + 0.928005i \(0.621522\pi\)
\(608\) 23.4005 0.949014
\(609\) 0 0
\(610\) −5.44740 −0.220559
\(611\) 65.7661 2.66061
\(612\) 0 0
\(613\) −19.0981 −0.771364 −0.385682 0.922632i \(-0.626034\pi\)
−0.385682 + 0.922632i \(0.626034\pi\)
\(614\) −22.7674 −0.918817
\(615\) 0 0
\(616\) −10.0169 −0.403591
\(617\) −18.3387 −0.738290 −0.369145 0.929372i \(-0.620349\pi\)
−0.369145 + 0.929372i \(0.620349\pi\)
\(618\) 0 0
\(619\) −13.7695 −0.553445 −0.276722 0.960950i \(-0.589248\pi\)
−0.276722 + 0.960950i \(0.589248\pi\)
\(620\) 5.80898 0.233294
\(621\) 0 0
\(622\) −7.85124 −0.314806
\(623\) 6.16675 0.247066
\(624\) 0 0
\(625\) 7.99023 0.319609
\(626\) 6.84726 0.273672
\(627\) 0 0
\(628\) 5.36022 0.213896
\(629\) −17.6842 −0.705114
\(630\) 0 0
\(631\) −30.3936 −1.20995 −0.604976 0.796244i \(-0.706817\pi\)
−0.604976 + 0.796244i \(0.706817\pi\)
\(632\) 24.6349 0.979924
\(633\) 0 0
\(634\) 28.7378 1.14133
\(635\) 1.11166 0.0441151
\(636\) 0 0
\(637\) 6.02587 0.238754
\(638\) −28.2322 −1.11772
\(639\) 0 0
\(640\) 2.14671 0.0848563
\(641\) 15.3445 0.606071 0.303036 0.952979i \(-0.402000\pi\)
0.303036 + 0.952979i \(0.402000\pi\)
\(642\) 0 0
\(643\) 6.64052 0.261876 0.130938 0.991391i \(-0.458201\pi\)
0.130938 + 0.991391i \(0.458201\pi\)
\(644\) 4.10769 0.161865
\(645\) 0 0
\(646\) −20.6894 −0.814012
\(647\) 31.1079 1.22298 0.611488 0.791253i \(-0.290571\pi\)
0.611488 + 0.791253i \(0.290571\pi\)
\(648\) 0 0
\(649\) 25.4552 0.999202
\(650\) 25.7101 1.00843
\(651\) 0 0
\(652\) −1.95176 −0.0764369
\(653\) 13.8619 0.542460 0.271230 0.962515i \(-0.412570\pi\)
0.271230 + 0.962515i \(0.412570\pi\)
\(654\) 0 0
\(655\) 16.2589 0.635287
\(656\) −8.71752 −0.340362
\(657\) 0 0
\(658\) −12.3706 −0.482258
\(659\) −32.7089 −1.27416 −0.637079 0.770798i \(-0.719858\pi\)
−0.637079 + 0.770798i \(0.719858\pi\)
\(660\) 0 0
\(661\) 9.66172 0.375797 0.187899 0.982188i \(-0.439832\pi\)
0.187899 + 0.982188i \(0.439832\pi\)
\(662\) 30.9381 1.20244
\(663\) 0 0
\(664\) 13.6773 0.530781
\(665\) 6.80502 0.263887
\(666\) 0 0
\(667\) 43.9506 1.70177
\(668\) 6.48150 0.250777
\(669\) 0 0
\(670\) −2.41565 −0.0933248
\(671\) 14.0708 0.543196
\(672\) 0 0
\(673\) 10.2693 0.395852 0.197926 0.980217i \(-0.436579\pi\)
0.197926 + 0.980217i \(0.436579\pi\)
\(674\) −18.7878 −0.723680
\(675\) 0 0
\(676\) −16.6731 −0.641273
\(677\) 47.3679 1.82049 0.910247 0.414065i \(-0.135891\pi\)
0.910247 + 0.414065i \(0.135891\pi\)
\(678\) 0 0
\(679\) 3.77960 0.145048
\(680\) 10.2017 0.391218
\(681\) 0 0
\(682\) 26.9523 1.03206
\(683\) −44.8136 −1.71475 −0.857373 0.514696i \(-0.827905\pi\)
−0.857373 + 0.514696i \(0.827905\pi\)
\(684\) 0 0
\(685\) −3.57724 −0.136679
\(686\) −1.13347 −0.0432761
\(687\) 0 0
\(688\) 21.9912 0.838408
\(689\) 24.9059 0.948838
\(690\) 0 0
\(691\) −4.80863 −0.182929 −0.0914645 0.995808i \(-0.529155\pi\)
−0.0914645 + 0.995808i \(0.529155\pi\)
\(692\) −5.95831 −0.226501
\(693\) 0 0
\(694\) −38.0198 −1.44321
\(695\) −22.4979 −0.853393
\(696\) 0 0
\(697\) −12.6311 −0.478437
\(698\) −1.69696 −0.0642309
\(699\) 0 0
\(700\) 2.69232 0.101760
\(701\) 0.948236 0.0358144 0.0179072 0.999840i \(-0.494300\pi\)
0.0179072 + 0.999840i \(0.494300\pi\)
\(702\) 0 0
\(703\) −36.3045 −1.36925
\(704\) −27.4984 −1.03638
\(705\) 0 0
\(706\) 7.43302 0.279745
\(707\) −4.23530 −0.159285
\(708\) 0 0
\(709\) −36.4973 −1.37068 −0.685342 0.728221i \(-0.740347\pi\)
−0.685342 + 0.728221i \(0.740347\pi\)
\(710\) 2.50312 0.0939404
\(711\) 0 0
\(712\) 18.9791 0.711272
\(713\) −41.9581 −1.57134
\(714\) 0 0
\(715\) 21.8025 0.815367
\(716\) 11.4400 0.427532
\(717\) 0 0
\(718\) 29.2550 1.09179
\(719\) 37.5806 1.40152 0.700760 0.713397i \(-0.252844\pi\)
0.700760 + 0.713397i \(0.252844\pi\)
\(720\) 0 0
\(721\) 18.4322 0.686452
\(722\) −20.9380 −0.779230
\(723\) 0 0
\(724\) 4.83104 0.179544
\(725\) 28.8068 1.06986
\(726\) 0 0
\(727\) −45.1600 −1.67489 −0.837446 0.546521i \(-0.815952\pi\)
−0.837446 + 0.546521i \(0.815952\pi\)
\(728\) 18.5455 0.687342
\(729\) 0 0
\(730\) 0.574431 0.0212607
\(731\) 31.8638 1.17853
\(732\) 0 0
\(733\) −23.7090 −0.875712 −0.437856 0.899045i \(-0.644262\pi\)
−0.437856 + 0.899045i \(0.644262\pi\)
\(734\) −32.4378 −1.19730
\(735\) 0 0
\(736\) 21.9539 0.809231
\(737\) 6.23969 0.229842
\(738\) 0 0
\(739\) −6.95413 −0.255812 −0.127906 0.991786i \(-0.540826\pi\)
−0.127906 + 0.991786i \(0.540826\pi\)
\(740\) 4.71554 0.173347
\(741\) 0 0
\(742\) −4.68482 −0.171985
\(743\) 31.6115 1.15971 0.579856 0.814719i \(-0.303109\pi\)
0.579856 + 0.814719i \(0.303109\pi\)
\(744\) 0 0
\(745\) 23.8179 0.872619
\(746\) −29.2323 −1.07027
\(747\) 0 0
\(748\) −6.94141 −0.253803
\(749\) −5.51966 −0.201684
\(750\) 0 0
\(751\) 8.32654 0.303840 0.151920 0.988393i \(-0.451454\pi\)
0.151920 + 0.988393i \(0.451454\pi\)
\(752\) −22.4603 −0.819041
\(753\) 0 0
\(754\) 52.2699 1.90356
\(755\) 23.4722 0.854242
\(756\) 0 0
\(757\) −40.4829 −1.47138 −0.735688 0.677320i \(-0.763141\pi\)
−0.735688 + 0.677320i \(0.763141\pi\)
\(758\) −23.8341 −0.865694
\(759\) 0 0
\(760\) 20.9435 0.759700
\(761\) 0.945201 0.0342635 0.0171318 0.999853i \(-0.494547\pi\)
0.0171318 + 0.999853i \(0.494547\pi\)
\(762\) 0 0
\(763\) −13.3691 −0.483993
\(764\) 3.45745 0.125086
\(765\) 0 0
\(766\) −40.8026 −1.47426
\(767\) −47.1284 −1.70171
\(768\) 0 0
\(769\) 14.2859 0.515161 0.257581 0.966257i \(-0.417075\pi\)
0.257581 + 0.966257i \(0.417075\pi\)
\(770\) −4.10107 −0.147792
\(771\) 0 0
\(772\) −4.17575 −0.150289
\(773\) −47.1453 −1.69570 −0.847850 0.530237i \(-0.822103\pi\)
−0.847850 + 0.530237i \(0.822103\pi\)
\(774\) 0 0
\(775\) −27.5008 −0.987857
\(776\) 11.6323 0.417575
\(777\) 0 0
\(778\) 5.29977 0.190006
\(779\) −25.9308 −0.929068
\(780\) 0 0
\(781\) −6.46562 −0.231358
\(782\) −19.4104 −0.694114
\(783\) 0 0
\(784\) −2.05794 −0.0734978
\(785\) 8.33110 0.297350
\(786\) 0 0
\(787\) 49.9185 1.77940 0.889702 0.456542i \(-0.150912\pi\)
0.889702 + 0.456542i \(0.150912\pi\)
\(788\) 4.56326 0.162559
\(789\) 0 0
\(790\) 10.0859 0.358841
\(791\) −9.04421 −0.321575
\(792\) 0 0
\(793\) −26.0510 −0.925099
\(794\) 20.2816 0.719767
\(795\) 0 0
\(796\) −8.99283 −0.318742
\(797\) 28.1911 0.998580 0.499290 0.866435i \(-0.333594\pi\)
0.499290 + 0.866435i \(0.333594\pi\)
\(798\) 0 0
\(799\) −32.5434 −1.15130
\(800\) 14.3894 0.508741
\(801\) 0 0
\(802\) 28.6183 1.01055
\(803\) −1.48377 −0.0523612
\(804\) 0 0
\(805\) 6.38435 0.225019
\(806\) −49.9002 −1.75766
\(807\) 0 0
\(808\) −13.0348 −0.458562
\(809\) 32.9492 1.15843 0.579216 0.815174i \(-0.303359\pi\)
0.579216 + 0.815174i \(0.303359\pi\)
\(810\) 0 0
\(811\) −45.6791 −1.60401 −0.802004 0.597318i \(-0.796233\pi\)
−0.802004 + 0.597318i \(0.796233\pi\)
\(812\) 5.47363 0.192087
\(813\) 0 0
\(814\) 21.8790 0.766858
\(815\) −3.03352 −0.106260
\(816\) 0 0
\(817\) 65.4143 2.28856
\(818\) −17.1927 −0.601127
\(819\) 0 0
\(820\) 3.36812 0.117620
\(821\) −12.0427 −0.420294 −0.210147 0.977670i \(-0.567394\pi\)
−0.210147 + 0.977670i \(0.567394\pi\)
\(822\) 0 0
\(823\) 54.4766 1.89894 0.949468 0.313865i \(-0.101624\pi\)
0.949468 + 0.313865i \(0.101624\pi\)
\(824\) 56.7280 1.97621
\(825\) 0 0
\(826\) 8.86489 0.308449
\(827\) 5.46702 0.190107 0.0950534 0.995472i \(-0.469698\pi\)
0.0950534 + 0.995472i \(0.469698\pi\)
\(828\) 0 0
\(829\) −6.91752 −0.240255 −0.120128 0.992758i \(-0.538330\pi\)
−0.120128 + 0.992758i \(0.538330\pi\)
\(830\) 5.59970 0.194368
\(831\) 0 0
\(832\) 50.9112 1.76503
\(833\) −2.98181 −0.103314
\(834\) 0 0
\(835\) 10.0738 0.348620
\(836\) −14.2503 −0.492856
\(837\) 0 0
\(838\) 2.51637 0.0869264
\(839\) 20.4883 0.707335 0.353668 0.935371i \(-0.384934\pi\)
0.353668 + 0.935371i \(0.384934\pi\)
\(840\) 0 0
\(841\) 29.5657 1.01951
\(842\) 3.94601 0.135988
\(843\) 0 0
\(844\) −13.6939 −0.471364
\(845\) −25.9141 −0.891473
\(846\) 0 0
\(847\) −0.406833 −0.0139790
\(848\) −8.50579 −0.292090
\(849\) 0 0
\(850\) −12.7223 −0.436370
\(851\) −34.0602 −1.16757
\(852\) 0 0
\(853\) 0.0273175 0.000935333 0 0.000467666 1.00000i \(-0.499851\pi\)
0.000467666 1.00000i \(0.499851\pi\)
\(854\) 4.90022 0.167682
\(855\) 0 0
\(856\) −16.9876 −0.580624
\(857\) 46.5967 1.59171 0.795856 0.605486i \(-0.207021\pi\)
0.795856 + 0.605486i \(0.207021\pi\)
\(858\) 0 0
\(859\) −14.7542 −0.503408 −0.251704 0.967804i \(-0.580991\pi\)
−0.251704 + 0.967804i \(0.580991\pi\)
\(860\) −8.49659 −0.289731
\(861\) 0 0
\(862\) −22.2900 −0.759199
\(863\) 52.6741 1.79305 0.896523 0.442997i \(-0.146085\pi\)
0.896523 + 0.442997i \(0.146085\pi\)
\(864\) 0 0
\(865\) −9.26068 −0.314873
\(866\) 27.0817 0.920275
\(867\) 0 0
\(868\) −5.22548 −0.177364
\(869\) −26.0522 −0.883761
\(870\) 0 0
\(871\) −11.5523 −0.391436
\(872\) −41.1453 −1.39336
\(873\) 0 0
\(874\) −39.8483 −1.34789
\(875\) 9.74285 0.329368
\(876\) 0 0
\(877\) −37.3389 −1.26085 −0.630423 0.776252i \(-0.717118\pi\)
−0.630423 + 0.776252i \(0.717118\pi\)
\(878\) 3.96039 0.133657
\(879\) 0 0
\(880\) −7.44593 −0.251002
\(881\) 19.6295 0.661334 0.330667 0.943748i \(-0.392726\pi\)
0.330667 + 0.943748i \(0.392726\pi\)
\(882\) 0 0
\(883\) 54.8052 1.84434 0.922172 0.386781i \(-0.126413\pi\)
0.922172 + 0.386781i \(0.126413\pi\)
\(884\) 12.8515 0.432243
\(885\) 0 0
\(886\) −21.0897 −0.708522
\(887\) −11.0653 −0.371536 −0.185768 0.982594i \(-0.559477\pi\)
−0.185768 + 0.982594i \(0.559477\pi\)
\(888\) 0 0
\(889\) −1.00000 −0.0335389
\(890\) 7.77035 0.260463
\(891\) 0 0
\(892\) −12.0886 −0.404755
\(893\) −66.8095 −2.23569
\(894\) 0 0
\(895\) 17.7805 0.594337
\(896\) −1.93108 −0.0645128
\(897\) 0 0
\(898\) 23.2061 0.774396
\(899\) −55.9106 −1.86472
\(900\) 0 0
\(901\) −12.3243 −0.410582
\(902\) 15.6273 0.520332
\(903\) 0 0
\(904\) −27.8349 −0.925775
\(905\) 7.50863 0.249595
\(906\) 0 0
\(907\) 48.1569 1.59902 0.799511 0.600651i \(-0.205092\pi\)
0.799511 + 0.600651i \(0.205092\pi\)
\(908\) −1.62100 −0.0537946
\(909\) 0 0
\(910\) 7.59283 0.251700
\(911\) 19.6703 0.651705 0.325852 0.945421i \(-0.394349\pi\)
0.325852 + 0.945421i \(0.394349\pi\)
\(912\) 0 0
\(913\) −14.4642 −0.478694
\(914\) −34.9464 −1.15593
\(915\) 0 0
\(916\) 3.87743 0.128114
\(917\) −14.6257 −0.482983
\(918\) 0 0
\(919\) −50.5830 −1.66858 −0.834290 0.551327i \(-0.814122\pi\)
−0.834290 + 0.551327i \(0.814122\pi\)
\(920\) 19.6488 0.647802
\(921\) 0 0
\(922\) −37.8475 −1.24644
\(923\) 11.9706 0.394018
\(924\) 0 0
\(925\) −22.3243 −0.734017
\(926\) −5.06651 −0.166496
\(927\) 0 0
\(928\) 29.2543 0.960321
\(929\) −27.0185 −0.886449 −0.443225 0.896411i \(-0.646166\pi\)
−0.443225 + 0.896411i \(0.646166\pi\)
\(930\) 0 0
\(931\) −6.12147 −0.200623
\(932\) −20.1556 −0.660219
\(933\) 0 0
\(934\) −40.1650 −1.31424
\(935\) −10.7887 −0.352827
\(936\) 0 0
\(937\) −23.2124 −0.758317 −0.379159 0.925332i \(-0.623787\pi\)
−0.379159 + 0.925332i \(0.623787\pi\)
\(938\) 2.17301 0.0709511
\(939\) 0 0
\(940\) 8.67780 0.283039
\(941\) 14.3851 0.468941 0.234471 0.972123i \(-0.424664\pi\)
0.234471 + 0.972123i \(0.424664\pi\)
\(942\) 0 0
\(943\) −24.3278 −0.792223
\(944\) 16.0952 0.523853
\(945\) 0 0
\(946\) −39.4222 −1.28172
\(947\) −4.40420 −0.143117 −0.0715586 0.997436i \(-0.522797\pi\)
−0.0715586 + 0.997436i \(0.522797\pi\)
\(948\) 0 0
\(949\) 2.74710 0.0891745
\(950\) −26.1180 −0.847379
\(951\) 0 0
\(952\) −9.17698 −0.297428
\(953\) 4.33111 0.140298 0.0701492 0.997537i \(-0.477652\pi\)
0.0701492 + 0.997537i \(0.477652\pi\)
\(954\) 0 0
\(955\) 5.37372 0.173890
\(956\) −8.30907 −0.268735
\(957\) 0 0
\(958\) 19.0104 0.614197
\(959\) 3.21791 0.103912
\(960\) 0 0
\(961\) 22.3758 0.721799
\(962\) −40.5074 −1.30601
\(963\) 0 0
\(964\) 12.8032 0.412365
\(965\) −6.49015 −0.208925
\(966\) 0 0
\(967\) −15.9148 −0.511784 −0.255892 0.966705i \(-0.582369\pi\)
−0.255892 + 0.966705i \(0.582369\pi\)
\(968\) −1.25209 −0.0402437
\(969\) 0 0
\(970\) 4.76244 0.152913
\(971\) −25.9388 −0.832415 −0.416208 0.909270i \(-0.636641\pi\)
−0.416208 + 0.909270i \(0.636641\pi\)
\(972\) 0 0
\(973\) 20.2380 0.648801
\(974\) −30.4182 −0.974660
\(975\) 0 0
\(976\) 8.89688 0.284782
\(977\) 15.3064 0.489694 0.244847 0.969562i \(-0.421262\pi\)
0.244847 + 0.969562i \(0.421262\pi\)
\(978\) 0 0
\(979\) −20.0710 −0.641473
\(980\) 0.795111 0.0253989
\(981\) 0 0
\(982\) −46.2218 −1.47500
\(983\) 58.9802 1.88118 0.940588 0.339551i \(-0.110275\pi\)
0.940588 + 0.339551i \(0.110275\pi\)
\(984\) 0 0
\(985\) 7.09242 0.225983
\(986\) −25.8650 −0.823710
\(987\) 0 0
\(988\) 26.3833 0.839365
\(989\) 61.3706 1.95147
\(990\) 0 0
\(991\) 27.5746 0.875936 0.437968 0.898991i \(-0.355698\pi\)
0.437968 + 0.898991i \(0.355698\pi\)
\(992\) −27.9280 −0.886716
\(993\) 0 0
\(994\) −2.25169 −0.0714191
\(995\) −13.9771 −0.443103
\(996\) 0 0
\(997\) −40.1943 −1.27297 −0.636483 0.771291i \(-0.719611\pi\)
−0.636483 + 0.771291i \(0.719611\pi\)
\(998\) −4.57658 −0.144869
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 8001.2.a.ba.1.14 40
3.2 odd 2 inner 8001.2.a.ba.1.27 yes 40
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8001.2.a.ba.1.14 40 1.1 even 1 trivial
8001.2.a.ba.1.27 yes 40 3.2 odd 2 inner