Properties

Label 8016.2.a.l.1.2
Level $8016$
Weight $2$
Character 8016.1
Self dual yes
Analytic conductor $64.008$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8016,2,Mod(1,8016)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8016, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("8016.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8016 = 2^{4} \cdot 3 \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8016.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: \(64.0080822603\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1002)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.17009\) of defining polynomial
Character \(\chi\) \(=\) 8016.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -0.460811 q^{5} +0.369102 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -0.460811 q^{5} +0.369102 q^{7} +1.00000 q^{9} +6.04945 q^{11} -1.36910 q^{13} -0.460811 q^{15} -7.70928 q^{17} -2.78765 q^{19} +0.369102 q^{21} +2.63090 q^{23} -4.78765 q^{25} +1.00000 q^{27} +6.04945 q^{29} -9.09890 q^{31} +6.04945 q^{33} -0.170086 q^{35} -6.95774 q^{37} -1.36910 q^{39} -7.26180 q^{41} +0.447480 q^{43} -0.460811 q^{45} +4.89269 q^{47} -6.86376 q^{49} -7.70928 q^{51} +2.66701 q^{53} -2.78765 q^{55} -2.78765 q^{57} -5.14116 q^{59} -10.8371 q^{61} +0.369102 q^{63} +0.630898 q^{65} +11.6381 q^{67} +2.63090 q^{69} +1.55252 q^{71} +4.81432 q^{73} -4.78765 q^{75} +2.23287 q^{77} -15.4186 q^{79} +1.00000 q^{81} +4.51026 q^{83} +3.55252 q^{85} +6.04945 q^{87} -17.2062 q^{89} -0.505339 q^{91} -9.09890 q^{93} +1.28458 q^{95} +1.44748 q^{97} +6.04945 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} - 3 q^{5} + 5 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{3} - 3 q^{5} + 5 q^{7} + 3 q^{9} - 8 q^{13} - 3 q^{15} - 16 q^{17} + 2 q^{19} + 5 q^{21} + 4 q^{23} - 4 q^{25} + 3 q^{27} + 9 q^{31} + 5 q^{35} - 5 q^{37} - 8 q^{39} - 14 q^{41} + 2 q^{43} - 3 q^{45} + 3 q^{47} + 6 q^{49} - 16 q^{51} - 15 q^{53} + 2 q^{55} + 2 q^{57} + 5 q^{59} - 4 q^{61} + 5 q^{63} - 2 q^{65} - 3 q^{67} + 4 q^{69} + 4 q^{71} + 6 q^{73} - 4 q^{75} - 16 q^{77} - 32 q^{79} + 3 q^{81} - 3 q^{83} + 10 q^{85} - 27 q^{89} - 32 q^{91} + 9 q^{93} - 24 q^{95} + 5 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) −0.460811 −0.206081 −0.103041 0.994677i \(-0.532857\pi\)
−0.103041 + 0.994677i \(0.532857\pi\)
\(6\) 0 0
\(7\) 0.369102 0.139508 0.0697538 0.997564i \(-0.477779\pi\)
0.0697538 + 0.997564i \(0.477779\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 6.04945 1.82398 0.911989 0.410215i \(-0.134546\pi\)
0.911989 + 0.410215i \(0.134546\pi\)
\(12\) 0 0
\(13\) −1.36910 −0.379721 −0.189860 0.981811i \(-0.560804\pi\)
−0.189860 + 0.981811i \(0.560804\pi\)
\(14\) 0 0
\(15\) −0.460811 −0.118981
\(16\) 0 0
\(17\) −7.70928 −1.86977 −0.934887 0.354946i \(-0.884499\pi\)
−0.934887 + 0.354946i \(0.884499\pi\)
\(18\) 0 0
\(19\) −2.78765 −0.639531 −0.319766 0.947497i \(-0.603604\pi\)
−0.319766 + 0.947497i \(0.603604\pi\)
\(20\) 0 0
\(21\) 0.369102 0.0805447
\(22\) 0 0
\(23\) 2.63090 0.548580 0.274290 0.961647i \(-0.411557\pi\)
0.274290 + 0.961647i \(0.411557\pi\)
\(24\) 0 0
\(25\) −4.78765 −0.957531
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 6.04945 1.12335 0.561677 0.827356i \(-0.310156\pi\)
0.561677 + 0.827356i \(0.310156\pi\)
\(30\) 0 0
\(31\) −9.09890 −1.63421 −0.817105 0.576489i \(-0.804422\pi\)
−0.817105 + 0.576489i \(0.804422\pi\)
\(32\) 0 0
\(33\) 6.04945 1.05307
\(34\) 0 0
\(35\) −0.170086 −0.0287499
\(36\) 0 0
\(37\) −6.95774 −1.14385 −0.571923 0.820308i \(-0.693802\pi\)
−0.571923 + 0.820308i \(0.693802\pi\)
\(38\) 0 0
\(39\) −1.36910 −0.219232
\(40\) 0 0
\(41\) −7.26180 −1.13410 −0.567051 0.823683i \(-0.691916\pi\)
−0.567051 + 0.823683i \(0.691916\pi\)
\(42\) 0 0
\(43\) 0.447480 0.0682401 0.0341200 0.999418i \(-0.489137\pi\)
0.0341200 + 0.999418i \(0.489137\pi\)
\(44\) 0 0
\(45\) −0.460811 −0.0686937
\(46\) 0 0
\(47\) 4.89269 0.713673 0.356836 0.934167i \(-0.383855\pi\)
0.356836 + 0.934167i \(0.383855\pi\)
\(48\) 0 0
\(49\) −6.86376 −0.980538
\(50\) 0 0
\(51\) −7.70928 −1.07951
\(52\) 0 0
\(53\) 2.66701 0.366343 0.183171 0.983081i \(-0.441364\pi\)
0.183171 + 0.983081i \(0.441364\pi\)
\(54\) 0 0
\(55\) −2.78765 −0.375887
\(56\) 0 0
\(57\) −2.78765 −0.369234
\(58\) 0 0
\(59\) −5.14116 −0.669322 −0.334661 0.942339i \(-0.608622\pi\)
−0.334661 + 0.942339i \(0.608622\pi\)
\(60\) 0 0
\(61\) −10.8371 −1.38755 −0.693774 0.720192i \(-0.744053\pi\)
−0.693774 + 0.720192i \(0.744053\pi\)
\(62\) 0 0
\(63\) 0.369102 0.0465025
\(64\) 0 0
\(65\) 0.630898 0.0782532
\(66\) 0 0
\(67\) 11.6381 1.42182 0.710909 0.703284i \(-0.248284\pi\)
0.710909 + 0.703284i \(0.248284\pi\)
\(68\) 0 0
\(69\) 2.63090 0.316723
\(70\) 0 0
\(71\) 1.55252 0.184250 0.0921251 0.995747i \(-0.470634\pi\)
0.0921251 + 0.995747i \(0.470634\pi\)
\(72\) 0 0
\(73\) 4.81432 0.563473 0.281736 0.959492i \(-0.409090\pi\)
0.281736 + 0.959492i \(0.409090\pi\)
\(74\) 0 0
\(75\) −4.78765 −0.552831
\(76\) 0 0
\(77\) 2.23287 0.254459
\(78\) 0 0
\(79\) −15.4186 −1.73472 −0.867361 0.497679i \(-0.834186\pi\)
−0.867361 + 0.497679i \(0.834186\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 4.51026 0.495065 0.247533 0.968880i \(-0.420380\pi\)
0.247533 + 0.968880i \(0.420380\pi\)
\(84\) 0 0
\(85\) 3.55252 0.385325
\(86\) 0 0
\(87\) 6.04945 0.648569
\(88\) 0 0
\(89\) −17.2062 −1.82385 −0.911927 0.410353i \(-0.865406\pi\)
−0.911927 + 0.410353i \(0.865406\pi\)
\(90\) 0 0
\(91\) −0.505339 −0.0529739
\(92\) 0 0
\(93\) −9.09890 −0.943512
\(94\) 0 0
\(95\) 1.28458 0.131795
\(96\) 0 0
\(97\) 1.44748 0.146969 0.0734847 0.997296i \(-0.476588\pi\)
0.0734847 + 0.997296i \(0.476588\pi\)
\(98\) 0 0
\(99\) 6.04945 0.607992
\(100\) 0 0
\(101\) 1.27513 0.126880 0.0634399 0.997986i \(-0.479793\pi\)
0.0634399 + 0.997986i \(0.479793\pi\)
\(102\) 0 0
\(103\) −13.9155 −1.37113 −0.685566 0.728010i \(-0.740445\pi\)
−0.685566 + 0.728010i \(0.740445\pi\)
\(104\) 0 0
\(105\) −0.170086 −0.0165987
\(106\) 0 0
\(107\) −20.4391 −1.97592 −0.987960 0.154711i \(-0.950555\pi\)
−0.987960 + 0.154711i \(0.950555\pi\)
\(108\) 0 0
\(109\) 19.3112 1.84968 0.924841 0.380354i \(-0.124198\pi\)
0.924841 + 0.380354i \(0.124198\pi\)
\(110\) 0 0
\(111\) −6.95774 −0.660399
\(112\) 0 0
\(113\) −11.7854 −1.10868 −0.554338 0.832292i \(-0.687028\pi\)
−0.554338 + 0.832292i \(0.687028\pi\)
\(114\) 0 0
\(115\) −1.21235 −0.113052
\(116\) 0 0
\(117\) −1.36910 −0.126574
\(118\) 0 0
\(119\) −2.84551 −0.260848
\(120\) 0 0
\(121\) 25.5958 2.32689
\(122\) 0 0
\(123\) −7.26180 −0.654774
\(124\) 0 0
\(125\) 4.51026 0.403410
\(126\) 0 0
\(127\) 12.7587 1.13215 0.566077 0.824352i \(-0.308461\pi\)
0.566077 + 0.824352i \(0.308461\pi\)
\(128\) 0 0
\(129\) 0.447480 0.0393984
\(130\) 0 0
\(131\) −11.5525 −1.00935 −0.504674 0.863310i \(-0.668387\pi\)
−0.504674 + 0.863310i \(0.668387\pi\)
\(132\) 0 0
\(133\) −1.02893 −0.0892195
\(134\) 0 0
\(135\) −0.460811 −0.0396603
\(136\) 0 0
\(137\) 0.552520 0.0472050 0.0236025 0.999721i \(-0.492486\pi\)
0.0236025 + 0.999721i \(0.492486\pi\)
\(138\) 0 0
\(139\) 6.55971 0.556387 0.278194 0.960525i \(-0.410264\pi\)
0.278194 + 0.960525i \(0.410264\pi\)
\(140\) 0 0
\(141\) 4.89269 0.412039
\(142\) 0 0
\(143\) −8.28231 −0.692602
\(144\) 0 0
\(145\) −2.78765 −0.231502
\(146\) 0 0
\(147\) −6.86376 −0.566114
\(148\) 0 0
\(149\) 10.4836 0.858850 0.429425 0.903103i \(-0.358716\pi\)
0.429425 + 0.903103i \(0.358716\pi\)
\(150\) 0 0
\(151\) 1.97334 0.160588 0.0802940 0.996771i \(-0.474414\pi\)
0.0802940 + 0.996771i \(0.474414\pi\)
\(152\) 0 0
\(153\) −7.70928 −0.623258
\(154\) 0 0
\(155\) 4.19287 0.336780
\(156\) 0 0
\(157\) −2.76487 −0.220660 −0.110330 0.993895i \(-0.535191\pi\)
−0.110330 + 0.993895i \(0.535191\pi\)
\(158\) 0 0
\(159\) 2.66701 0.211508
\(160\) 0 0
\(161\) 0.971071 0.0765311
\(162\) 0 0
\(163\) −9.40522 −0.736674 −0.368337 0.929692i \(-0.620073\pi\)
−0.368337 + 0.929692i \(0.620073\pi\)
\(164\) 0 0
\(165\) −2.78765 −0.217018
\(166\) 0 0
\(167\) −1.00000 −0.0773823
\(168\) 0 0
\(169\) −11.1256 −0.855812
\(170\) 0 0
\(171\) −2.78765 −0.213177
\(172\) 0 0
\(173\) −23.3340 −1.77405 −0.887027 0.461718i \(-0.847233\pi\)
−0.887027 + 0.461718i \(0.847233\pi\)
\(174\) 0 0
\(175\) −1.76713 −0.133583
\(176\) 0 0
\(177\) −5.14116 −0.386433
\(178\) 0 0
\(179\) 18.1483 1.35647 0.678235 0.734845i \(-0.262745\pi\)
0.678235 + 0.734845i \(0.262745\pi\)
\(180\) 0 0
\(181\) 8.54638 0.635247 0.317624 0.948217i \(-0.397115\pi\)
0.317624 + 0.948217i \(0.397115\pi\)
\(182\) 0 0
\(183\) −10.8371 −0.801102
\(184\) 0 0
\(185\) 3.20620 0.235725
\(186\) 0 0
\(187\) −46.6369 −3.41043
\(188\) 0 0
\(189\) 0.369102 0.0268482
\(190\) 0 0
\(191\) −12.5753 −0.909917 −0.454959 0.890513i \(-0.650346\pi\)
−0.454959 + 0.890513i \(0.650346\pi\)
\(192\) 0 0
\(193\) 16.7031 1.20232 0.601159 0.799130i \(-0.294706\pi\)
0.601159 + 0.799130i \(0.294706\pi\)
\(194\) 0 0
\(195\) 0.630898 0.0451795
\(196\) 0 0
\(197\) 20.9854 1.49515 0.747576 0.664177i \(-0.231218\pi\)
0.747576 + 0.664177i \(0.231218\pi\)
\(198\) 0 0
\(199\) 9.17727 0.650560 0.325280 0.945618i \(-0.394541\pi\)
0.325280 + 0.945618i \(0.394541\pi\)
\(200\) 0 0
\(201\) 11.6381 0.820887
\(202\) 0 0
\(203\) 2.23287 0.156716
\(204\) 0 0
\(205\) 3.34632 0.233717
\(206\) 0 0
\(207\) 2.63090 0.182860
\(208\) 0 0
\(209\) −16.8638 −1.16649
\(210\) 0 0
\(211\) 2.99386 0.206106 0.103053 0.994676i \(-0.467139\pi\)
0.103053 + 0.994676i \(0.467139\pi\)
\(212\) 0 0
\(213\) 1.55252 0.106377
\(214\) 0 0
\(215\) −0.206204 −0.0140630
\(216\) 0 0
\(217\) −3.35842 −0.227985
\(218\) 0 0
\(219\) 4.81432 0.325321
\(220\) 0 0
\(221\) 10.5548 0.709992
\(222\) 0 0
\(223\) 18.8660 1.26336 0.631681 0.775228i \(-0.282365\pi\)
0.631681 + 0.775228i \(0.282365\pi\)
\(224\) 0 0
\(225\) −4.78765 −0.319177
\(226\) 0 0
\(227\) 11.7948 0.782851 0.391426 0.920210i \(-0.371982\pi\)
0.391426 + 0.920210i \(0.371982\pi\)
\(228\) 0 0
\(229\) 26.0410 1.72084 0.860420 0.509585i \(-0.170201\pi\)
0.860420 + 0.509585i \(0.170201\pi\)
\(230\) 0 0
\(231\) 2.23287 0.146912
\(232\) 0 0
\(233\) −3.97948 −0.260704 −0.130352 0.991468i \(-0.541611\pi\)
−0.130352 + 0.991468i \(0.541611\pi\)
\(234\) 0 0
\(235\) −2.25461 −0.147074
\(236\) 0 0
\(237\) −15.4186 −1.00154
\(238\) 0 0
\(239\) 14.6537 0.947868 0.473934 0.880560i \(-0.342834\pi\)
0.473934 + 0.880560i \(0.342834\pi\)
\(240\) 0 0
\(241\) −10.1483 −0.653712 −0.326856 0.945074i \(-0.605989\pi\)
−0.326856 + 0.945074i \(0.605989\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 3.16290 0.202070
\(246\) 0 0
\(247\) 3.81658 0.242843
\(248\) 0 0
\(249\) 4.51026 0.285826
\(250\) 0 0
\(251\) 19.7503 1.24663 0.623314 0.781971i \(-0.285786\pi\)
0.623314 + 0.781971i \(0.285786\pi\)
\(252\) 0 0
\(253\) 15.9155 1.00060
\(254\) 0 0
\(255\) 3.55252 0.222467
\(256\) 0 0
\(257\) −5.62475 −0.350863 −0.175431 0.984492i \(-0.556132\pi\)
−0.175431 + 0.984492i \(0.556132\pi\)
\(258\) 0 0
\(259\) −2.56812 −0.159575
\(260\) 0 0
\(261\) 6.04945 0.374451
\(262\) 0 0
\(263\) −8.56690 −0.528257 −0.264129 0.964487i \(-0.585084\pi\)
−0.264129 + 0.964487i \(0.585084\pi\)
\(264\) 0 0
\(265\) −1.22899 −0.0754963
\(266\) 0 0
\(267\) −17.2062 −1.05300
\(268\) 0 0
\(269\) −14.2195 −0.866980 −0.433490 0.901158i \(-0.642718\pi\)
−0.433490 + 0.901158i \(0.642718\pi\)
\(270\) 0 0
\(271\) 13.2267 0.803466 0.401733 0.915757i \(-0.368408\pi\)
0.401733 + 0.915757i \(0.368408\pi\)
\(272\) 0 0
\(273\) −0.505339 −0.0305845
\(274\) 0 0
\(275\) −28.9627 −1.74651
\(276\) 0 0
\(277\) −16.6670 −1.00142 −0.500712 0.865614i \(-0.666928\pi\)
−0.500712 + 0.865614i \(0.666928\pi\)
\(278\) 0 0
\(279\) −9.09890 −0.544737
\(280\) 0 0
\(281\) −17.4391 −1.04033 −0.520164 0.854066i \(-0.674129\pi\)
−0.520164 + 0.854066i \(0.674129\pi\)
\(282\) 0 0
\(283\) −25.9565 −1.54295 −0.771477 0.636257i \(-0.780482\pi\)
−0.771477 + 0.636257i \(0.780482\pi\)
\(284\) 0 0
\(285\) 1.28458 0.0760920
\(286\) 0 0
\(287\) −2.68035 −0.158216
\(288\) 0 0
\(289\) 42.4329 2.49605
\(290\) 0 0
\(291\) 1.44748 0.0848528
\(292\) 0 0
\(293\) −14.6225 −0.854255 −0.427127 0.904191i \(-0.640474\pi\)
−0.427127 + 0.904191i \(0.640474\pi\)
\(294\) 0 0
\(295\) 2.36910 0.137934
\(296\) 0 0
\(297\) 6.04945 0.351025
\(298\) 0 0
\(299\) −3.60197 −0.208307
\(300\) 0 0
\(301\) 0.165166 0.00952001
\(302\) 0 0
\(303\) 1.27513 0.0732541
\(304\) 0 0
\(305\) 4.99386 0.285947
\(306\) 0 0
\(307\) 0.170086 0.00970735 0.00485367 0.999988i \(-0.498455\pi\)
0.00485367 + 0.999988i \(0.498455\pi\)
\(308\) 0 0
\(309\) −13.9155 −0.791624
\(310\) 0 0
\(311\) −4.60197 −0.260954 −0.130477 0.991451i \(-0.541651\pi\)
−0.130477 + 0.991451i \(0.541651\pi\)
\(312\) 0 0
\(313\) −15.5525 −0.879080 −0.439540 0.898223i \(-0.644859\pi\)
−0.439540 + 0.898223i \(0.644859\pi\)
\(314\) 0 0
\(315\) −0.170086 −0.00958329
\(316\) 0 0
\(317\) 9.68261 0.543830 0.271915 0.962321i \(-0.412343\pi\)
0.271915 + 0.962321i \(0.412343\pi\)
\(318\) 0 0
\(319\) 36.5958 2.04897
\(320\) 0 0
\(321\) −20.4391 −1.14080
\(322\) 0 0
\(323\) 21.4908 1.19578
\(324\) 0 0
\(325\) 6.55479 0.363594
\(326\) 0 0
\(327\) 19.3112 1.06791
\(328\) 0 0
\(329\) 1.80590 0.0995627
\(330\) 0 0
\(331\) −32.4307 −1.78255 −0.891275 0.453463i \(-0.850188\pi\)
−0.891275 + 0.453463i \(0.850188\pi\)
\(332\) 0 0
\(333\) −6.95774 −0.381282
\(334\) 0 0
\(335\) −5.36296 −0.293010
\(336\) 0 0
\(337\) −3.09890 −0.168808 −0.0844038 0.996432i \(-0.526899\pi\)
−0.0844038 + 0.996432i \(0.526899\pi\)
\(338\) 0 0
\(339\) −11.7854 −0.640094
\(340\) 0 0
\(341\) −55.0433 −2.98076
\(342\) 0 0
\(343\) −5.11715 −0.276300
\(344\) 0 0
\(345\) −1.21235 −0.0652706
\(346\) 0 0
\(347\) −31.2979 −1.68016 −0.840080 0.542463i \(-0.817492\pi\)
−0.840080 + 0.542463i \(0.817492\pi\)
\(348\) 0 0
\(349\) −17.7226 −0.948669 −0.474335 0.880345i \(-0.657311\pi\)
−0.474335 + 0.880345i \(0.657311\pi\)
\(350\) 0 0
\(351\) −1.36910 −0.0730773
\(352\) 0 0
\(353\) −17.1568 −0.913162 −0.456581 0.889682i \(-0.650926\pi\)
−0.456581 + 0.889682i \(0.650926\pi\)
\(354\) 0 0
\(355\) −0.715418 −0.0379705
\(356\) 0 0
\(357\) −2.84551 −0.150600
\(358\) 0 0
\(359\) 29.5441 1.55928 0.779639 0.626229i \(-0.215402\pi\)
0.779639 + 0.626229i \(0.215402\pi\)
\(360\) 0 0
\(361\) −11.2290 −0.590999
\(362\) 0 0
\(363\) 25.5958 1.34343
\(364\) 0 0
\(365\) −2.21849 −0.116121
\(366\) 0 0
\(367\) 14.6537 0.764916 0.382458 0.923973i \(-0.375078\pi\)
0.382458 + 0.923973i \(0.375078\pi\)
\(368\) 0 0
\(369\) −7.26180 −0.378034
\(370\) 0 0
\(371\) 0.984402 0.0511076
\(372\) 0 0
\(373\) −33.6030 −1.73990 −0.869949 0.493142i \(-0.835848\pi\)
−0.869949 + 0.493142i \(0.835848\pi\)
\(374\) 0 0
\(375\) 4.51026 0.232909
\(376\) 0 0
\(377\) −8.28231 −0.426561
\(378\) 0 0
\(379\) −13.2218 −0.679158 −0.339579 0.940577i \(-0.610285\pi\)
−0.339579 + 0.940577i \(0.610285\pi\)
\(380\) 0 0
\(381\) 12.7587 0.653649
\(382\) 0 0
\(383\) −2.92162 −0.149288 −0.0746440 0.997210i \(-0.523782\pi\)
−0.0746440 + 0.997210i \(0.523782\pi\)
\(384\) 0 0
\(385\) −1.02893 −0.0524391
\(386\) 0 0
\(387\) 0.447480 0.0227467
\(388\) 0 0
\(389\) 24.0060 1.21715 0.608575 0.793496i \(-0.291741\pi\)
0.608575 + 0.793496i \(0.291741\pi\)
\(390\) 0 0
\(391\) −20.2823 −1.02572
\(392\) 0 0
\(393\) −11.5525 −0.582748
\(394\) 0 0
\(395\) 7.10504 0.357493
\(396\) 0 0
\(397\) 16.7031 0.838306 0.419153 0.907916i \(-0.362327\pi\)
0.419153 + 0.907916i \(0.362327\pi\)
\(398\) 0 0
\(399\) −1.02893 −0.0515109
\(400\) 0 0
\(401\) −37.5897 −1.87714 −0.938570 0.345090i \(-0.887848\pi\)
−0.938570 + 0.345090i \(0.887848\pi\)
\(402\) 0 0
\(403\) 12.4573 0.620543
\(404\) 0 0
\(405\) −0.460811 −0.0228979
\(406\) 0 0
\(407\) −42.0905 −2.08635
\(408\) 0 0
\(409\) −4.47027 −0.221040 −0.110520 0.993874i \(-0.535252\pi\)
−0.110520 + 0.993874i \(0.535252\pi\)
\(410\) 0 0
\(411\) 0.552520 0.0272538
\(412\) 0 0
\(413\) −1.89761 −0.0933754
\(414\) 0 0
\(415\) −2.07838 −0.102024
\(416\) 0 0
\(417\) 6.55971 0.321230
\(418\) 0 0
\(419\) 20.9939 1.02562 0.512809 0.858503i \(-0.328605\pi\)
0.512809 + 0.858503i \(0.328605\pi\)
\(420\) 0 0
\(421\) −20.2907 −0.988909 −0.494455 0.869203i \(-0.664632\pi\)
−0.494455 + 0.869203i \(0.664632\pi\)
\(422\) 0 0
\(423\) 4.89269 0.237891
\(424\) 0 0
\(425\) 36.9093 1.79037
\(426\) 0 0
\(427\) −4.00000 −0.193574
\(428\) 0 0
\(429\) −8.28231 −0.399874
\(430\) 0 0
\(431\) 0.789921 0.0380491 0.0190246 0.999819i \(-0.493944\pi\)
0.0190246 + 0.999819i \(0.493944\pi\)
\(432\) 0 0
\(433\) −10.1362 −0.487116 −0.243558 0.969886i \(-0.578315\pi\)
−0.243558 + 0.969886i \(0.578315\pi\)
\(434\) 0 0
\(435\) −2.78765 −0.133658
\(436\) 0 0
\(437\) −7.33403 −0.350834
\(438\) 0 0
\(439\) −9.17727 −0.438007 −0.219004 0.975724i \(-0.570281\pi\)
−0.219004 + 0.975724i \(0.570281\pi\)
\(440\) 0 0
\(441\) −6.86376 −0.326846
\(442\) 0 0
\(443\) −18.7431 −0.890513 −0.445256 0.895403i \(-0.646887\pi\)
−0.445256 + 0.895403i \(0.646887\pi\)
\(444\) 0 0
\(445\) 7.92881 0.375862
\(446\) 0 0
\(447\) 10.4836 0.495857
\(448\) 0 0
\(449\) −11.2641 −0.531584 −0.265792 0.964030i \(-0.585633\pi\)
−0.265792 + 0.964030i \(0.585633\pi\)
\(450\) 0 0
\(451\) −43.9299 −2.06858
\(452\) 0 0
\(453\) 1.97334 0.0927155
\(454\) 0 0
\(455\) 0.232866 0.0109169
\(456\) 0 0
\(457\) 18.0989 0.846631 0.423315 0.905982i \(-0.360866\pi\)
0.423315 + 0.905982i \(0.360866\pi\)
\(458\) 0 0
\(459\) −7.70928 −0.359838
\(460\) 0 0
\(461\) −9.76713 −0.454901 −0.227450 0.973790i \(-0.573039\pi\)
−0.227450 + 0.973790i \(0.573039\pi\)
\(462\) 0 0
\(463\) −24.3545 −1.13185 −0.565926 0.824456i \(-0.691481\pi\)
−0.565926 + 0.824456i \(0.691481\pi\)
\(464\) 0 0
\(465\) 4.19287 0.194440
\(466\) 0 0
\(467\) 6.00000 0.277647 0.138823 0.990317i \(-0.455668\pi\)
0.138823 + 0.990317i \(0.455668\pi\)
\(468\) 0 0
\(469\) 4.29565 0.198354
\(470\) 0 0
\(471\) −2.76487 −0.127398
\(472\) 0 0
\(473\) 2.70701 0.124468
\(474\) 0 0
\(475\) 13.3463 0.612371
\(476\) 0 0
\(477\) 2.66701 0.122114
\(478\) 0 0
\(479\) −32.7480 −1.49630 −0.748148 0.663532i \(-0.769057\pi\)
−0.748148 + 0.663532i \(0.769057\pi\)
\(480\) 0 0
\(481\) 9.52586 0.434342
\(482\) 0 0
\(483\) 0.971071 0.0441852
\(484\) 0 0
\(485\) −0.667015 −0.0302876
\(486\) 0 0
\(487\) 34.4619 1.56162 0.780808 0.624771i \(-0.214808\pi\)
0.780808 + 0.624771i \(0.214808\pi\)
\(488\) 0 0
\(489\) −9.40522 −0.425319
\(490\) 0 0
\(491\) −29.7731 −1.34364 −0.671821 0.740714i \(-0.734487\pi\)
−0.671821 + 0.740714i \(0.734487\pi\)
\(492\) 0 0
\(493\) −46.6369 −2.10042
\(494\) 0 0
\(495\) −2.78765 −0.125296
\(496\) 0 0
\(497\) 0.573039 0.0257043
\(498\) 0 0
\(499\) −21.4413 −0.959846 −0.479923 0.877311i \(-0.659335\pi\)
−0.479923 + 0.877311i \(0.659335\pi\)
\(500\) 0 0
\(501\) −1.00000 −0.0446767
\(502\) 0 0
\(503\) 32.2823 1.43940 0.719699 0.694287i \(-0.244280\pi\)
0.719699 + 0.694287i \(0.244280\pi\)
\(504\) 0 0
\(505\) −0.587592 −0.0261475
\(506\) 0 0
\(507\) −11.1256 −0.494103
\(508\) 0 0
\(509\) 4.08452 0.181043 0.0905216 0.995894i \(-0.471147\pi\)
0.0905216 + 0.995894i \(0.471147\pi\)
\(510\) 0 0
\(511\) 1.77698 0.0786088
\(512\) 0 0
\(513\) −2.78765 −0.123078
\(514\) 0 0
\(515\) 6.41241 0.282564
\(516\) 0 0
\(517\) 29.5981 1.30172
\(518\) 0 0
\(519\) −23.3340 −1.02425
\(520\) 0 0
\(521\) −22.6719 −0.993276 −0.496638 0.867958i \(-0.665432\pi\)
−0.496638 + 0.867958i \(0.665432\pi\)
\(522\) 0 0
\(523\) 21.8660 0.956135 0.478067 0.878323i \(-0.341337\pi\)
0.478067 + 0.878323i \(0.341337\pi\)
\(524\) 0 0
\(525\) −1.76713 −0.0771241
\(526\) 0 0
\(527\) 70.1459 3.05560
\(528\) 0 0
\(529\) −16.0784 −0.699060
\(530\) 0 0
\(531\) −5.14116 −0.223107
\(532\) 0 0
\(533\) 9.94214 0.430642
\(534\) 0 0
\(535\) 9.41855 0.407199
\(536\) 0 0
\(537\) 18.1483 0.783159
\(538\) 0 0
\(539\) −41.5220 −1.78848
\(540\) 0 0
\(541\) 21.3474 0.917795 0.458897 0.888489i \(-0.348245\pi\)
0.458897 + 0.888489i \(0.348245\pi\)
\(542\) 0 0
\(543\) 8.54638 0.366760
\(544\) 0 0
\(545\) −8.89884 −0.381184
\(546\) 0 0
\(547\) 27.9783 1.19626 0.598132 0.801398i \(-0.295910\pi\)
0.598132 + 0.801398i \(0.295910\pi\)
\(548\) 0 0
\(549\) −10.8371 −0.462516
\(550\) 0 0
\(551\) −16.8638 −0.718420
\(552\) 0 0
\(553\) −5.69102 −0.242007
\(554\) 0 0
\(555\) 3.20620 0.136096
\(556\) 0 0
\(557\) −20.8554 −0.883670 −0.441835 0.897096i \(-0.645672\pi\)
−0.441835 + 0.897096i \(0.645672\pi\)
\(558\) 0 0
\(559\) −0.612646 −0.0259122
\(560\) 0 0
\(561\) −46.6369 −1.96901
\(562\) 0 0
\(563\) 20.3630 0.858196 0.429098 0.903258i \(-0.358831\pi\)
0.429098 + 0.903258i \(0.358831\pi\)
\(564\) 0 0
\(565\) 5.43084 0.228477
\(566\) 0 0
\(567\) 0.369102 0.0155008
\(568\) 0 0
\(569\) −15.7093 −0.658567 −0.329284 0.944231i \(-0.606807\pi\)
−0.329284 + 0.944231i \(0.606807\pi\)
\(570\) 0 0
\(571\) 24.3991 1.02107 0.510535 0.859857i \(-0.329447\pi\)
0.510535 + 0.859857i \(0.329447\pi\)
\(572\) 0 0
\(573\) −12.5753 −0.525341
\(574\) 0 0
\(575\) −12.5958 −0.525282
\(576\) 0 0
\(577\) −39.9276 −1.66221 −0.831104 0.556118i \(-0.812290\pi\)
−0.831104 + 0.556118i \(0.812290\pi\)
\(578\) 0 0
\(579\) 16.7031 0.694158
\(580\) 0 0
\(581\) 1.66475 0.0690654
\(582\) 0 0
\(583\) 16.1340 0.668201
\(584\) 0 0
\(585\) 0.630898 0.0260844
\(586\) 0 0
\(587\) −32.0133 −1.32133 −0.660666 0.750680i \(-0.729726\pi\)
−0.660666 + 0.750680i \(0.729726\pi\)
\(588\) 0 0
\(589\) 25.3646 1.04513
\(590\) 0 0
\(591\) 20.9854 0.863226
\(592\) 0 0
\(593\) −16.4163 −0.674136 −0.337068 0.941480i \(-0.609435\pi\)
−0.337068 + 0.941480i \(0.609435\pi\)
\(594\) 0 0
\(595\) 1.31124 0.0537557
\(596\) 0 0
\(597\) 9.17727 0.375601
\(598\) 0 0
\(599\) −46.9526 −1.91843 −0.959216 0.282673i \(-0.908779\pi\)
−0.959216 + 0.282673i \(0.908779\pi\)
\(600\) 0 0
\(601\) 36.5136 1.48942 0.744710 0.667388i \(-0.232588\pi\)
0.744710 + 0.667388i \(0.232588\pi\)
\(602\) 0 0
\(603\) 11.6381 0.473939
\(604\) 0 0
\(605\) −11.7948 −0.479528
\(606\) 0 0
\(607\) 37.1727 1.50879 0.754397 0.656418i \(-0.227929\pi\)
0.754397 + 0.656418i \(0.227929\pi\)
\(608\) 0 0
\(609\) 2.23287 0.0904803
\(610\) 0 0
\(611\) −6.69860 −0.270996
\(612\) 0 0
\(613\) −42.3896 −1.71210 −0.856050 0.516892i \(-0.827089\pi\)
−0.856050 + 0.516892i \(0.827089\pi\)
\(614\) 0 0
\(615\) 3.34632 0.134936
\(616\) 0 0
\(617\) 28.8059 1.15968 0.579841 0.814730i \(-0.303115\pi\)
0.579841 + 0.814730i \(0.303115\pi\)
\(618\) 0 0
\(619\) 42.0372 1.68962 0.844808 0.535069i \(-0.179715\pi\)
0.844808 + 0.535069i \(0.179715\pi\)
\(620\) 0 0
\(621\) 2.63090 0.105574
\(622\) 0 0
\(623\) −6.35085 −0.254441
\(624\) 0 0
\(625\) 21.8599 0.874396
\(626\) 0 0
\(627\) −16.8638 −0.673474
\(628\) 0 0
\(629\) 53.6391 2.13873
\(630\) 0 0
\(631\) 39.1711 1.55938 0.779689 0.626167i \(-0.215377\pi\)
0.779689 + 0.626167i \(0.215377\pi\)
\(632\) 0 0
\(633\) 2.99386 0.118995
\(634\) 0 0
\(635\) −5.87936 −0.233315
\(636\) 0 0
\(637\) 9.39719 0.372330
\(638\) 0 0
\(639\) 1.55252 0.0614167
\(640\) 0 0
\(641\) 45.8310 1.81021 0.905107 0.425184i \(-0.139791\pi\)
0.905107 + 0.425184i \(0.139791\pi\)
\(642\) 0 0
\(643\) 7.55252 0.297842 0.148921 0.988849i \(-0.452420\pi\)
0.148921 + 0.988849i \(0.452420\pi\)
\(644\) 0 0
\(645\) −0.206204 −0.00811927
\(646\) 0 0
\(647\) 0.764867 0.0300700 0.0150350 0.999887i \(-0.495214\pi\)
0.0150350 + 0.999887i \(0.495214\pi\)
\(648\) 0 0
\(649\) −31.1012 −1.22083
\(650\) 0 0
\(651\) −3.35842 −0.131627
\(652\) 0 0
\(653\) 38.0866 1.49044 0.745222 0.666816i \(-0.232343\pi\)
0.745222 + 0.666816i \(0.232343\pi\)
\(654\) 0 0
\(655\) 5.32353 0.208008
\(656\) 0 0
\(657\) 4.81432 0.187824
\(658\) 0 0
\(659\) −4.38470 −0.170804 −0.0854018 0.996347i \(-0.527217\pi\)
−0.0854018 + 0.996347i \(0.527217\pi\)
\(660\) 0 0
\(661\) 5.99612 0.233222 0.116611 0.993178i \(-0.462797\pi\)
0.116611 + 0.993178i \(0.462797\pi\)
\(662\) 0 0
\(663\) 10.5548 0.409914
\(664\) 0 0
\(665\) 0.474142 0.0183864
\(666\) 0 0
\(667\) 15.9155 0.616250
\(668\) 0 0
\(669\) 18.8660 0.729403
\(670\) 0 0
\(671\) −65.5585 −2.53086
\(672\) 0 0
\(673\) −12.6719 −0.488467 −0.244234 0.969716i \(-0.578536\pi\)
−0.244234 + 0.969716i \(0.578536\pi\)
\(674\) 0 0
\(675\) −4.78765 −0.184277
\(676\) 0 0
\(677\) 3.39576 0.130510 0.0652549 0.997869i \(-0.479214\pi\)
0.0652549 + 0.997869i \(0.479214\pi\)
\(678\) 0 0
\(679\) 0.534268 0.0205033
\(680\) 0 0
\(681\) 11.7948 0.451979
\(682\) 0 0
\(683\) 45.7237 1.74957 0.874783 0.484514i \(-0.161004\pi\)
0.874783 + 0.484514i \(0.161004\pi\)
\(684\) 0 0
\(685\) −0.254607 −0.00972805
\(686\) 0 0
\(687\) 26.0410 0.993528
\(688\) 0 0
\(689\) −3.65142 −0.139108
\(690\) 0 0
\(691\) −9.80817 −0.373120 −0.186560 0.982444i \(-0.559734\pi\)
−0.186560 + 0.982444i \(0.559734\pi\)
\(692\) 0 0
\(693\) 2.23287 0.0848196
\(694\) 0 0
\(695\) −3.02279 −0.114661
\(696\) 0 0
\(697\) 55.9832 2.12051
\(698\) 0 0
\(699\) −3.97948 −0.150518
\(700\) 0 0
\(701\) −35.0349 −1.32325 −0.661625 0.749835i \(-0.730133\pi\)
−0.661625 + 0.749835i \(0.730133\pi\)
\(702\) 0 0
\(703\) 19.3958 0.731525
\(704\) 0 0
\(705\) −2.25461 −0.0849134
\(706\) 0 0
\(707\) 0.470652 0.0177007
\(708\) 0 0
\(709\) −34.7392 −1.30466 −0.652330 0.757935i \(-0.726208\pi\)
−0.652330 + 0.757935i \(0.726208\pi\)
\(710\) 0 0
\(711\) −15.4186 −0.578241
\(712\) 0 0
\(713\) −23.9383 −0.896495
\(714\) 0 0
\(715\) 3.81658 0.142732
\(716\) 0 0
\(717\) 14.6537 0.547252
\(718\) 0 0
\(719\) −4.48255 −0.167171 −0.0835855 0.996501i \(-0.526637\pi\)
−0.0835855 + 0.996501i \(0.526637\pi\)
\(720\) 0 0
\(721\) −5.13624 −0.191283
\(722\) 0 0
\(723\) −10.1483 −0.377421
\(724\) 0 0
\(725\) −28.9627 −1.07565
\(726\) 0 0
\(727\) 18.7154 0.694116 0.347058 0.937844i \(-0.387181\pi\)
0.347058 + 0.937844i \(0.387181\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −3.44975 −0.127594
\(732\) 0 0
\(733\) 0.971071 0.0358673 0.0179337 0.999839i \(-0.494291\pi\)
0.0179337 + 0.999839i \(0.494291\pi\)
\(734\) 0 0
\(735\) 3.16290 0.116665
\(736\) 0 0
\(737\) 70.4040 2.59336
\(738\) 0 0
\(739\) 5.12291 0.188449 0.0942246 0.995551i \(-0.469963\pi\)
0.0942246 + 0.995551i \(0.469963\pi\)
\(740\) 0 0
\(741\) 3.81658 0.140206
\(742\) 0 0
\(743\) 28.3523 1.04014 0.520072 0.854122i \(-0.325905\pi\)
0.520072 + 0.854122i \(0.325905\pi\)
\(744\) 0 0
\(745\) −4.83096 −0.176993
\(746\) 0 0
\(747\) 4.51026 0.165022
\(748\) 0 0
\(749\) −7.54411 −0.275656
\(750\) 0 0
\(751\) 7.93827 0.289671 0.144836 0.989456i \(-0.453735\pi\)
0.144836 + 0.989456i \(0.453735\pi\)
\(752\) 0 0
\(753\) 19.7503 0.719741
\(754\) 0 0
\(755\) −0.909336 −0.0330941
\(756\) 0 0
\(757\) 6.93003 0.251876 0.125938 0.992038i \(-0.459806\pi\)
0.125938 + 0.992038i \(0.459806\pi\)
\(758\) 0 0
\(759\) 15.9155 0.577695
\(760\) 0 0
\(761\) −17.7382 −0.643009 −0.321505 0.946908i \(-0.604189\pi\)
−0.321505 + 0.946908i \(0.604189\pi\)
\(762\) 0 0
\(763\) 7.12783 0.258045
\(764\) 0 0
\(765\) 3.55252 0.128442
\(766\) 0 0
\(767\) 7.03877 0.254155
\(768\) 0 0
\(769\) 29.5090 1.06412 0.532062 0.846706i \(-0.321417\pi\)
0.532062 + 0.846706i \(0.321417\pi\)
\(770\) 0 0
\(771\) −5.62475 −0.202571
\(772\) 0 0
\(773\) 11.3207 0.407177 0.203589 0.979057i \(-0.434739\pi\)
0.203589 + 0.979057i \(0.434739\pi\)
\(774\) 0 0
\(775\) 43.5624 1.56481
\(776\) 0 0
\(777\) −2.56812 −0.0921307
\(778\) 0 0
\(779\) 20.2434 0.725294
\(780\) 0 0
\(781\) 9.39189 0.336068
\(782\) 0 0
\(783\) 6.04945 0.216190
\(784\) 0 0
\(785\) 1.27408 0.0454739
\(786\) 0 0
\(787\) −36.2784 −1.29319 −0.646593 0.762835i \(-0.723807\pi\)
−0.646593 + 0.762835i \(0.723807\pi\)
\(788\) 0 0
\(789\) −8.56690 −0.304990
\(790\) 0 0
\(791\) −4.35001 −0.154669
\(792\) 0 0
\(793\) 14.8371 0.526881
\(794\) 0 0
\(795\) −1.22899 −0.0435878
\(796\) 0 0
\(797\) −16.6264 −0.588936 −0.294468 0.955661i \(-0.595142\pi\)
−0.294468 + 0.955661i \(0.595142\pi\)
\(798\) 0 0
\(799\) −37.7191 −1.33441
\(800\) 0 0
\(801\) −17.2062 −0.607951
\(802\) 0 0
\(803\) 29.1240 1.02776
\(804\) 0 0
\(805\) −0.447480 −0.0157716
\(806\) 0 0
\(807\) −14.2195 −0.500551
\(808\) 0 0
\(809\) 4.35473 0.153104 0.0765520 0.997066i \(-0.475609\pi\)
0.0765520 + 0.997066i \(0.475609\pi\)
\(810\) 0 0
\(811\) 32.5597 1.14333 0.571663 0.820489i \(-0.306298\pi\)
0.571663 + 0.820489i \(0.306298\pi\)
\(812\) 0 0
\(813\) 13.2267 0.463881
\(814\) 0 0
\(815\) 4.33403 0.151814
\(816\) 0 0
\(817\) −1.24742 −0.0436417
\(818\) 0 0
\(819\) −0.505339 −0.0176580
\(820\) 0 0
\(821\) 31.7770 1.10902 0.554512 0.832176i \(-0.312905\pi\)
0.554512 + 0.832176i \(0.312905\pi\)
\(822\) 0 0
\(823\) 2.40400 0.0837981 0.0418990 0.999122i \(-0.486659\pi\)
0.0418990 + 0.999122i \(0.486659\pi\)
\(824\) 0 0
\(825\) −28.9627 −1.00835
\(826\) 0 0
\(827\) −39.8036 −1.38411 −0.692054 0.721846i \(-0.743294\pi\)
−0.692054 + 0.721846i \(0.743294\pi\)
\(828\) 0 0
\(829\) 13.1184 0.455620 0.227810 0.973706i \(-0.426844\pi\)
0.227810 + 0.973706i \(0.426844\pi\)
\(830\) 0 0
\(831\) −16.6670 −0.578172
\(832\) 0 0
\(833\) 52.9146 1.83338
\(834\) 0 0
\(835\) 0.460811 0.0159470
\(836\) 0 0
\(837\) −9.09890 −0.314504
\(838\) 0 0
\(839\) −48.6574 −1.67984 −0.839920 0.542711i \(-0.817398\pi\)
−0.839920 + 0.542711i \(0.817398\pi\)
\(840\) 0 0
\(841\) 7.59583 0.261925
\(842\) 0 0
\(843\) −17.4391 −0.600633
\(844\) 0 0
\(845\) 5.12678 0.176367
\(846\) 0 0
\(847\) 9.44748 0.324619
\(848\) 0 0
\(849\) −25.9565 −0.890825
\(850\) 0 0
\(851\) −18.3051 −0.627491
\(852\) 0 0
\(853\) −29.8720 −1.02280 −0.511399 0.859343i \(-0.670873\pi\)
−0.511399 + 0.859343i \(0.670873\pi\)
\(854\) 0 0
\(855\) 1.28458 0.0439318
\(856\) 0 0
\(857\) 0.633165 0.0216285 0.0108143 0.999942i \(-0.496558\pi\)
0.0108143 + 0.999942i \(0.496558\pi\)
\(858\) 0 0
\(859\) −17.5837 −0.599949 −0.299974 0.953947i \(-0.596978\pi\)
−0.299974 + 0.953947i \(0.596978\pi\)
\(860\) 0 0
\(861\) −2.68035 −0.0913459
\(862\) 0 0
\(863\) 23.3197 0.793810 0.396905 0.917860i \(-0.370084\pi\)
0.396905 + 0.917860i \(0.370084\pi\)
\(864\) 0 0
\(865\) 10.7526 0.365599
\(866\) 0 0
\(867\) 42.4329 1.44110
\(868\) 0 0
\(869\) −93.2737 −3.16409
\(870\) 0 0
\(871\) −15.9337 −0.539894
\(872\) 0 0
\(873\) 1.44748 0.0489898
\(874\) 0 0
\(875\) 1.66475 0.0562787
\(876\) 0 0
\(877\) 14.2641 0.481663 0.240832 0.970567i \(-0.422580\pi\)
0.240832 + 0.970567i \(0.422580\pi\)
\(878\) 0 0
\(879\) −14.6225 −0.493204
\(880\) 0 0
\(881\) −10.8227 −0.364627 −0.182313 0.983240i \(-0.558359\pi\)
−0.182313 + 0.983240i \(0.558359\pi\)
\(882\) 0 0
\(883\) 11.8783 0.399737 0.199869 0.979823i \(-0.435948\pi\)
0.199869 + 0.979823i \(0.435948\pi\)
\(884\) 0 0
\(885\) 2.36910 0.0796365
\(886\) 0 0
\(887\) −4.07223 −0.136732 −0.0683661 0.997660i \(-0.521779\pi\)
−0.0683661 + 0.997660i \(0.521779\pi\)
\(888\) 0 0
\(889\) 4.70928 0.157944
\(890\) 0 0
\(891\) 6.04945 0.202664
\(892\) 0 0
\(893\) −13.6391 −0.456416
\(894\) 0 0
\(895\) −8.36296 −0.279543
\(896\) 0 0
\(897\) −3.60197 −0.120266
\(898\) 0 0
\(899\) −55.0433 −1.83580
\(900\) 0 0
\(901\) −20.5608 −0.684978
\(902\) 0 0
\(903\) 0.165166 0.00549638
\(904\) 0 0
\(905\) −3.93827 −0.130912
\(906\) 0 0
\(907\) −11.1422 −0.369971 −0.184985 0.982741i \(-0.559224\pi\)
−0.184985 + 0.982741i \(0.559224\pi\)
\(908\) 0 0
\(909\) 1.27513 0.0422933
\(910\) 0 0
\(911\) −25.7670 −0.853697 −0.426849 0.904323i \(-0.640376\pi\)
−0.426849 + 0.904323i \(0.640376\pi\)
\(912\) 0 0
\(913\) 27.2846 0.902988
\(914\) 0 0
\(915\) 4.99386 0.165092
\(916\) 0 0
\(917\) −4.26406 −0.140812
\(918\) 0 0
\(919\) −4.01664 −0.132497 −0.0662484 0.997803i \(-0.521103\pi\)
−0.0662484 + 0.997803i \(0.521103\pi\)
\(920\) 0 0
\(921\) 0.170086 0.00560454
\(922\) 0 0
\(923\) −2.12556 −0.0699636
\(924\) 0 0
\(925\) 33.3112 1.09527
\(926\) 0 0
\(927\) −13.9155 −0.457044
\(928\) 0 0
\(929\) 39.4924 1.29570 0.647852 0.761766i \(-0.275668\pi\)
0.647852 + 0.761766i \(0.275668\pi\)
\(930\) 0 0
\(931\) 19.1338 0.627085
\(932\) 0 0
\(933\) −4.60197 −0.150662
\(934\) 0 0
\(935\) 21.4908 0.702824
\(936\) 0 0
\(937\) −3.23126 −0.105561 −0.0527803 0.998606i \(-0.516808\pi\)
−0.0527803 + 0.998606i \(0.516808\pi\)
\(938\) 0 0
\(939\) −15.5525 −0.507537
\(940\) 0 0
\(941\) −31.0966 −1.01372 −0.506861 0.862028i \(-0.669194\pi\)
−0.506861 + 0.862028i \(0.669194\pi\)
\(942\) 0 0
\(943\) −19.1050 −0.622146
\(944\) 0 0
\(945\) −0.170086 −0.00553291
\(946\) 0 0
\(947\) 36.8203 1.19650 0.598249 0.801310i \(-0.295863\pi\)
0.598249 + 0.801310i \(0.295863\pi\)
\(948\) 0 0
\(949\) −6.59129 −0.213962
\(950\) 0 0
\(951\) 9.68261 0.313980
\(952\) 0 0
\(953\) −6.13850 −0.198846 −0.0994228 0.995045i \(-0.531700\pi\)
−0.0994228 + 0.995045i \(0.531700\pi\)
\(954\) 0 0
\(955\) 5.79484 0.187517
\(956\) 0 0
\(957\) 36.5958 1.18298
\(958\) 0 0
\(959\) 0.203936 0.00658545
\(960\) 0 0
\(961\) 51.7899 1.67064
\(962\) 0 0
\(963\) −20.4391 −0.658640
\(964\) 0 0
\(965\) −7.69699 −0.247775
\(966\) 0 0
\(967\) 21.6742 0.696995 0.348498 0.937310i \(-0.386692\pi\)
0.348498 + 0.937310i \(0.386692\pi\)
\(968\) 0 0
\(969\) 21.4908 0.690383
\(970\) 0 0
\(971\) −21.8299 −0.700555 −0.350278 0.936646i \(-0.613913\pi\)
−0.350278 + 0.936646i \(0.613913\pi\)
\(972\) 0 0
\(973\) 2.42120 0.0776202
\(974\) 0 0
\(975\) 6.55479 0.209921
\(976\) 0 0
\(977\) 24.4741 0.782997 0.391499 0.920179i \(-0.371957\pi\)
0.391499 + 0.920179i \(0.371957\pi\)
\(978\) 0 0
\(979\) −104.088 −3.32667
\(980\) 0 0
\(981\) 19.3112 0.616561
\(982\) 0 0
\(983\) −34.3402 −1.09528 −0.547641 0.836714i \(-0.684474\pi\)
−0.547641 + 0.836714i \(0.684474\pi\)
\(984\) 0 0
\(985\) −9.67033 −0.308122
\(986\) 0 0
\(987\) 1.80590 0.0574826
\(988\) 0 0
\(989\) 1.17727 0.0374351
\(990\) 0 0
\(991\) −26.1340 −0.830173 −0.415086 0.909782i \(-0.636249\pi\)
−0.415086 + 0.909782i \(0.636249\pi\)
\(992\) 0 0
\(993\) −32.4307 −1.02916
\(994\) 0 0
\(995\) −4.22899 −0.134068
\(996\) 0 0
\(997\) −2.78765 −0.0882859 −0.0441429 0.999025i \(-0.514056\pi\)
−0.0441429 + 0.999025i \(0.514056\pi\)
\(998\) 0 0
\(999\) −6.95774 −0.220133
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 8016.2.a.l.1.2 3
4.3 odd 2 1002.2.a.h.1.2 3
12.11 even 2 3006.2.a.o.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1002.2.a.h.1.2 3 4.3 odd 2
3006.2.a.o.1.2 3 12.11 even 2
8016.2.a.l.1.2 3 1.1 even 1 trivial