Properties

Label 8008.2.a.s.1.2
Level $8008$
Weight $2$
Character 8008.1
Self dual yes
Analytic conductor $63.944$
Analytic rank $1$
Dimension $10$
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] = [8008,2,Mod(1,8008)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8008, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("8008.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8008.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: \(63.9442019386\)
Analytic rank: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 3x^{9} - 15x^{8} + 43x^{7} + 66x^{6} - 173x^{5} - 127x^{4} + 246x^{3} + 99x^{2} - 82x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.46329\) of defining polynomial
Character \(\chi\) \(=\) 8008.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.46329 q^{3} -1.35093 q^{5} +1.00000 q^{7} +3.06778 q^{9} +O(q^{10})\) \(q-2.46329 q^{3} -1.35093 q^{5} +1.00000 q^{7} +3.06778 q^{9} +1.00000 q^{11} -1.00000 q^{13} +3.32773 q^{15} +3.18596 q^{17} +4.39961 q^{19} -2.46329 q^{21} -2.64117 q^{23} -3.17498 q^{25} -0.166958 q^{27} -5.83924 q^{29} -8.29622 q^{31} -2.46329 q^{33} -1.35093 q^{35} +5.93067 q^{37} +2.46329 q^{39} +0.838410 q^{41} -4.98346 q^{43} -4.14436 q^{45} -3.71058 q^{47} +1.00000 q^{49} -7.84793 q^{51} +11.1825 q^{53} -1.35093 q^{55} -10.8375 q^{57} +11.1625 q^{59} -12.9239 q^{61} +3.06778 q^{63} +1.35093 q^{65} +2.65052 q^{67} +6.50595 q^{69} +9.04103 q^{71} -7.48913 q^{73} +7.82088 q^{75} +1.00000 q^{77} -5.13527 q^{79} -8.79207 q^{81} -6.21047 q^{83} -4.30402 q^{85} +14.3837 q^{87} +16.5794 q^{89} -1.00000 q^{91} +20.4360 q^{93} -5.94358 q^{95} +4.42426 q^{97} +3.06778 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 3 q^{3} - 4 q^{5} + 10 q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 3 q^{3} - 4 q^{5} + 10 q^{7} + 9 q^{9} + 10 q^{11} - 10 q^{13} - 5 q^{15} - 11 q^{17} + 2 q^{19} - 3 q^{21} - 8 q^{23} + 2 q^{25} - 15 q^{27} - 8 q^{29} - 23 q^{31} - 3 q^{33} - 4 q^{35} + 10 q^{37} + 3 q^{39} - 18 q^{41} + 12 q^{43} - 10 q^{45} - 36 q^{47} + 10 q^{49} + 9 q^{51} - 21 q^{53} - 4 q^{55} - 30 q^{57} - 13 q^{59} - 2 q^{61} + 9 q^{63} + 4 q^{65} - 4 q^{67} - 26 q^{69} - 24 q^{71} - 23 q^{73} - 28 q^{75} + 10 q^{77} + 14 q^{79} + 30 q^{81} - 9 q^{83} - 17 q^{85} + 7 q^{87} - 18 q^{89} - 10 q^{91} + q^{93} - 4 q^{95} - 9 q^{97} + 9 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −2.46329 −1.42218 −0.711089 0.703102i \(-0.751798\pi\)
−0.711089 + 0.703102i \(0.751798\pi\)
\(4\) 0 0
\(5\) −1.35093 −0.604156 −0.302078 0.953283i \(-0.597680\pi\)
−0.302078 + 0.953283i \(0.597680\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 3.06778 1.02259
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) 3.32773 0.859217
\(16\) 0 0
\(17\) 3.18596 0.772709 0.386355 0.922350i \(-0.373734\pi\)
0.386355 + 0.922350i \(0.373734\pi\)
\(18\) 0 0
\(19\) 4.39961 1.00934 0.504670 0.863312i \(-0.331614\pi\)
0.504670 + 0.863312i \(0.331614\pi\)
\(20\) 0 0
\(21\) −2.46329 −0.537533
\(22\) 0 0
\(23\) −2.64117 −0.550721 −0.275361 0.961341i \(-0.588797\pi\)
−0.275361 + 0.961341i \(0.588797\pi\)
\(24\) 0 0
\(25\) −3.17498 −0.634996
\(26\) 0 0
\(27\) −0.166958 −0.0321311
\(28\) 0 0
\(29\) −5.83924 −1.08432 −0.542160 0.840276i \(-0.682393\pi\)
−0.542160 + 0.840276i \(0.682393\pi\)
\(30\) 0 0
\(31\) −8.29622 −1.49005 −0.745023 0.667039i \(-0.767561\pi\)
−0.745023 + 0.667039i \(0.767561\pi\)
\(32\) 0 0
\(33\) −2.46329 −0.428803
\(34\) 0 0
\(35\) −1.35093 −0.228349
\(36\) 0 0
\(37\) 5.93067 0.974996 0.487498 0.873124i \(-0.337910\pi\)
0.487498 + 0.873124i \(0.337910\pi\)
\(38\) 0 0
\(39\) 2.46329 0.394441
\(40\) 0 0
\(41\) 0.838410 0.130938 0.0654688 0.997855i \(-0.479146\pi\)
0.0654688 + 0.997855i \(0.479146\pi\)
\(42\) 0 0
\(43\) −4.98346 −0.759970 −0.379985 0.924993i \(-0.624071\pi\)
−0.379985 + 0.924993i \(0.624071\pi\)
\(44\) 0 0
\(45\) −4.14436 −0.617805
\(46\) 0 0
\(47\) −3.71058 −0.541244 −0.270622 0.962686i \(-0.587229\pi\)
−0.270622 + 0.962686i \(0.587229\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −7.84793 −1.09893
\(52\) 0 0
\(53\) 11.1825 1.53603 0.768017 0.640430i \(-0.221244\pi\)
0.768017 + 0.640430i \(0.221244\pi\)
\(54\) 0 0
\(55\) −1.35093 −0.182160
\(56\) 0 0
\(57\) −10.8375 −1.43546
\(58\) 0 0
\(59\) 11.1625 1.45323 0.726616 0.687044i \(-0.241092\pi\)
0.726616 + 0.687044i \(0.241092\pi\)
\(60\) 0 0
\(61\) −12.9239 −1.65474 −0.827369 0.561659i \(-0.810163\pi\)
−0.827369 + 0.561659i \(0.810163\pi\)
\(62\) 0 0
\(63\) 3.06778 0.386504
\(64\) 0 0
\(65\) 1.35093 0.167563
\(66\) 0 0
\(67\) 2.65052 0.323812 0.161906 0.986806i \(-0.448236\pi\)
0.161906 + 0.986806i \(0.448236\pi\)
\(68\) 0 0
\(69\) 6.50595 0.783224
\(70\) 0 0
\(71\) 9.04103 1.07297 0.536486 0.843909i \(-0.319751\pi\)
0.536486 + 0.843909i \(0.319751\pi\)
\(72\) 0 0
\(73\) −7.48913 −0.876536 −0.438268 0.898844i \(-0.644408\pi\)
−0.438268 + 0.898844i \(0.644408\pi\)
\(74\) 0 0
\(75\) 7.82088 0.903078
\(76\) 0 0
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) −5.13527 −0.577763 −0.288882 0.957365i \(-0.593283\pi\)
−0.288882 + 0.957365i \(0.593283\pi\)
\(80\) 0 0
\(81\) −8.79207 −0.976897
\(82\) 0 0
\(83\) −6.21047 −0.681687 −0.340844 0.940120i \(-0.610713\pi\)
−0.340844 + 0.940120i \(0.610713\pi\)
\(84\) 0 0
\(85\) −4.30402 −0.466837
\(86\) 0 0
\(87\) 14.3837 1.54210
\(88\) 0 0
\(89\) 16.5794 1.75742 0.878709 0.477358i \(-0.158405\pi\)
0.878709 + 0.477358i \(0.158405\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) 0 0
\(93\) 20.4360 2.11911
\(94\) 0 0
\(95\) −5.94358 −0.609798
\(96\) 0 0
\(97\) 4.42426 0.449215 0.224608 0.974449i \(-0.427890\pi\)
0.224608 + 0.974449i \(0.427890\pi\)
\(98\) 0 0
\(99\) 3.06778 0.308323
\(100\) 0 0
\(101\) −15.9983 −1.59190 −0.795948 0.605366i \(-0.793027\pi\)
−0.795948 + 0.605366i \(0.793027\pi\)
\(102\) 0 0
\(103\) 0.599693 0.0590895 0.0295447 0.999563i \(-0.490594\pi\)
0.0295447 + 0.999563i \(0.490594\pi\)
\(104\) 0 0
\(105\) 3.32773 0.324754
\(106\) 0 0
\(107\) 18.3701 1.77590 0.887952 0.459936i \(-0.152128\pi\)
0.887952 + 0.459936i \(0.152128\pi\)
\(108\) 0 0
\(109\) 9.37221 0.897695 0.448847 0.893608i \(-0.351835\pi\)
0.448847 + 0.893608i \(0.351835\pi\)
\(110\) 0 0
\(111\) −14.6089 −1.38662
\(112\) 0 0
\(113\) −8.62416 −0.811293 −0.405647 0.914030i \(-0.632954\pi\)
−0.405647 + 0.914030i \(0.632954\pi\)
\(114\) 0 0
\(115\) 3.56804 0.332721
\(116\) 0 0
\(117\) −3.06778 −0.283616
\(118\) 0 0
\(119\) 3.18596 0.292057
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) −2.06524 −0.186217
\(124\) 0 0
\(125\) 11.0438 0.987792
\(126\) 0 0
\(127\) 8.08723 0.717626 0.358813 0.933410i \(-0.383182\pi\)
0.358813 + 0.933410i \(0.383182\pi\)
\(128\) 0 0
\(129\) 12.2757 1.08081
\(130\) 0 0
\(131\) −3.46938 −0.303121 −0.151560 0.988448i \(-0.548430\pi\)
−0.151560 + 0.988448i \(0.548430\pi\)
\(132\) 0 0
\(133\) 4.39961 0.381495
\(134\) 0 0
\(135\) 0.225549 0.0194122
\(136\) 0 0
\(137\) −4.59224 −0.392342 −0.196171 0.980570i \(-0.562851\pi\)
−0.196171 + 0.980570i \(0.562851\pi\)
\(138\) 0 0
\(139\) 9.60157 0.814395 0.407197 0.913340i \(-0.366506\pi\)
0.407197 + 0.913340i \(0.366506\pi\)
\(140\) 0 0
\(141\) 9.14023 0.769746
\(142\) 0 0
\(143\) −1.00000 −0.0836242
\(144\) 0 0
\(145\) 7.88842 0.655097
\(146\) 0 0
\(147\) −2.46329 −0.203168
\(148\) 0 0
\(149\) −12.0202 −0.984733 −0.492366 0.870388i \(-0.663868\pi\)
−0.492366 + 0.870388i \(0.663868\pi\)
\(150\) 0 0
\(151\) 17.2257 1.40181 0.700906 0.713254i \(-0.252779\pi\)
0.700906 + 0.713254i \(0.252779\pi\)
\(152\) 0 0
\(153\) 9.77382 0.790167
\(154\) 0 0
\(155\) 11.2076 0.900219
\(156\) 0 0
\(157\) 1.54531 0.123329 0.0616647 0.998097i \(-0.480359\pi\)
0.0616647 + 0.998097i \(0.480359\pi\)
\(158\) 0 0
\(159\) −27.5457 −2.18451
\(160\) 0 0
\(161\) −2.64117 −0.208153
\(162\) 0 0
\(163\) 12.7567 0.999183 0.499592 0.866261i \(-0.333483\pi\)
0.499592 + 0.866261i \(0.333483\pi\)
\(164\) 0 0
\(165\) 3.32773 0.259064
\(166\) 0 0
\(167\) −10.5517 −0.816513 −0.408256 0.912867i \(-0.633863\pi\)
−0.408256 + 0.912867i \(0.633863\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 13.4970 1.03214
\(172\) 0 0
\(173\) −23.4206 −1.78064 −0.890318 0.455340i \(-0.849518\pi\)
−0.890318 + 0.455340i \(0.849518\pi\)
\(174\) 0 0
\(175\) −3.17498 −0.240006
\(176\) 0 0
\(177\) −27.4964 −2.06676
\(178\) 0 0
\(179\) 24.8944 1.86069 0.930347 0.366682i \(-0.119506\pi\)
0.930347 + 0.366682i \(0.119506\pi\)
\(180\) 0 0
\(181\) −20.4521 −1.52019 −0.760095 0.649812i \(-0.774847\pi\)
−0.760095 + 0.649812i \(0.774847\pi\)
\(182\) 0 0
\(183\) 31.8353 2.35333
\(184\) 0 0
\(185\) −8.01193 −0.589049
\(186\) 0 0
\(187\) 3.18596 0.232981
\(188\) 0 0
\(189\) −0.166958 −0.0121444
\(190\) 0 0
\(191\) −5.62671 −0.407134 −0.203567 0.979061i \(-0.565253\pi\)
−0.203567 + 0.979061i \(0.565253\pi\)
\(192\) 0 0
\(193\) −2.56450 −0.184597 −0.0922984 0.995731i \(-0.529421\pi\)
−0.0922984 + 0.995731i \(0.529421\pi\)
\(194\) 0 0
\(195\) −3.32773 −0.238304
\(196\) 0 0
\(197\) 10.1710 0.724656 0.362328 0.932051i \(-0.381982\pi\)
0.362328 + 0.932051i \(0.381982\pi\)
\(198\) 0 0
\(199\) 12.3576 0.876010 0.438005 0.898973i \(-0.355685\pi\)
0.438005 + 0.898973i \(0.355685\pi\)
\(200\) 0 0
\(201\) −6.52898 −0.460519
\(202\) 0 0
\(203\) −5.83924 −0.409834
\(204\) 0 0
\(205\) −1.13264 −0.0791067
\(206\) 0 0
\(207\) −8.10251 −0.563164
\(208\) 0 0
\(209\) 4.39961 0.304327
\(210\) 0 0
\(211\) 12.7165 0.875437 0.437718 0.899112i \(-0.355787\pi\)
0.437718 + 0.899112i \(0.355787\pi\)
\(212\) 0 0
\(213\) −22.2706 −1.52596
\(214\) 0 0
\(215\) 6.73232 0.459140
\(216\) 0 0
\(217\) −8.29622 −0.563184
\(218\) 0 0
\(219\) 18.4479 1.24659
\(220\) 0 0
\(221\) −3.18596 −0.214311
\(222\) 0 0
\(223\) −17.6861 −1.18435 −0.592175 0.805809i \(-0.701731\pi\)
−0.592175 + 0.805809i \(0.701731\pi\)
\(224\) 0 0
\(225\) −9.74014 −0.649342
\(226\) 0 0
\(227\) 12.5324 0.831804 0.415902 0.909409i \(-0.363466\pi\)
0.415902 + 0.909409i \(0.363466\pi\)
\(228\) 0 0
\(229\) 3.68995 0.243839 0.121919 0.992540i \(-0.461095\pi\)
0.121919 + 0.992540i \(0.461095\pi\)
\(230\) 0 0
\(231\) −2.46329 −0.162072
\(232\) 0 0
\(233\) −15.6168 −1.02309 −0.511546 0.859256i \(-0.670927\pi\)
−0.511546 + 0.859256i \(0.670927\pi\)
\(234\) 0 0
\(235\) 5.01275 0.326996
\(236\) 0 0
\(237\) 12.6496 0.821683
\(238\) 0 0
\(239\) −22.0562 −1.42669 −0.713347 0.700811i \(-0.752822\pi\)
−0.713347 + 0.700811i \(0.752822\pi\)
\(240\) 0 0
\(241\) 8.92740 0.575065 0.287532 0.957771i \(-0.407165\pi\)
0.287532 + 0.957771i \(0.407165\pi\)
\(242\) 0 0
\(243\) 22.1583 1.42145
\(244\) 0 0
\(245\) −1.35093 −0.0863079
\(246\) 0 0
\(247\) −4.39961 −0.279941
\(248\) 0 0
\(249\) 15.2982 0.969481
\(250\) 0 0
\(251\) −17.9062 −1.13023 −0.565116 0.825011i \(-0.691169\pi\)
−0.565116 + 0.825011i \(0.691169\pi\)
\(252\) 0 0
\(253\) −2.64117 −0.166049
\(254\) 0 0
\(255\) 10.6020 0.663925
\(256\) 0 0
\(257\) −26.9545 −1.68138 −0.840688 0.541519i \(-0.817849\pi\)
−0.840688 + 0.541519i \(0.817849\pi\)
\(258\) 0 0
\(259\) 5.93067 0.368514
\(260\) 0 0
\(261\) −17.9135 −1.10882
\(262\) 0 0
\(263\) −5.99043 −0.369386 −0.184693 0.982796i \(-0.559129\pi\)
−0.184693 + 0.982796i \(0.559129\pi\)
\(264\) 0 0
\(265\) −15.1068 −0.928003
\(266\) 0 0
\(267\) −40.8399 −2.49936
\(268\) 0 0
\(269\) −29.4761 −1.79719 −0.898594 0.438782i \(-0.855410\pi\)
−0.898594 + 0.438782i \(0.855410\pi\)
\(270\) 0 0
\(271\) −2.05759 −0.124989 −0.0624947 0.998045i \(-0.519906\pi\)
−0.0624947 + 0.998045i \(0.519906\pi\)
\(272\) 0 0
\(273\) 2.46329 0.149085
\(274\) 0 0
\(275\) −3.17498 −0.191459
\(276\) 0 0
\(277\) 10.1057 0.607195 0.303598 0.952800i \(-0.401812\pi\)
0.303598 + 0.952800i \(0.401812\pi\)
\(278\) 0 0
\(279\) −25.4510 −1.52371
\(280\) 0 0
\(281\) −11.0777 −0.660841 −0.330420 0.943834i \(-0.607190\pi\)
−0.330420 + 0.943834i \(0.607190\pi\)
\(282\) 0 0
\(283\) 4.59180 0.272954 0.136477 0.990643i \(-0.456422\pi\)
0.136477 + 0.990643i \(0.456422\pi\)
\(284\) 0 0
\(285\) 14.6407 0.867242
\(286\) 0 0
\(287\) 0.838410 0.0494898
\(288\) 0 0
\(289\) −6.84965 −0.402921
\(290\) 0 0
\(291\) −10.8982 −0.638865
\(292\) 0 0
\(293\) −32.6070 −1.90492 −0.952461 0.304659i \(-0.901457\pi\)
−0.952461 + 0.304659i \(0.901457\pi\)
\(294\) 0 0
\(295\) −15.0798 −0.877978
\(296\) 0 0
\(297\) −0.166958 −0.00968789
\(298\) 0 0
\(299\) 2.64117 0.152743
\(300\) 0 0
\(301\) −4.98346 −0.287242
\(302\) 0 0
\(303\) 39.4085 2.26396
\(304\) 0 0
\(305\) 17.4593 0.999719
\(306\) 0 0
\(307\) 10.7352 0.612692 0.306346 0.951920i \(-0.400894\pi\)
0.306346 + 0.951920i \(0.400894\pi\)
\(308\) 0 0
\(309\) −1.47721 −0.0840358
\(310\) 0 0
\(311\) −14.1123 −0.800233 −0.400117 0.916464i \(-0.631030\pi\)
−0.400117 + 0.916464i \(0.631030\pi\)
\(312\) 0 0
\(313\) 4.90962 0.277508 0.138754 0.990327i \(-0.455690\pi\)
0.138754 + 0.990327i \(0.455690\pi\)
\(314\) 0 0
\(315\) −4.14436 −0.233508
\(316\) 0 0
\(317\) 14.8092 0.831768 0.415884 0.909418i \(-0.363472\pi\)
0.415884 + 0.909418i \(0.363472\pi\)
\(318\) 0 0
\(319\) −5.83924 −0.326935
\(320\) 0 0
\(321\) −45.2508 −2.52565
\(322\) 0 0
\(323\) 14.0170 0.779926
\(324\) 0 0
\(325\) 3.17498 0.176116
\(326\) 0 0
\(327\) −23.0864 −1.27668
\(328\) 0 0
\(329\) −3.71058 −0.204571
\(330\) 0 0
\(331\) 9.20520 0.505963 0.252982 0.967471i \(-0.418589\pi\)
0.252982 + 0.967471i \(0.418589\pi\)
\(332\) 0 0
\(333\) 18.1940 0.997024
\(334\) 0 0
\(335\) −3.58067 −0.195633
\(336\) 0 0
\(337\) −9.43834 −0.514139 −0.257070 0.966393i \(-0.582757\pi\)
−0.257070 + 0.966393i \(0.582757\pi\)
\(338\) 0 0
\(339\) 21.2438 1.15380
\(340\) 0 0
\(341\) −8.29622 −0.449266
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) −8.78910 −0.473189
\(346\) 0 0
\(347\) −26.8945 −1.44377 −0.721886 0.692012i \(-0.756725\pi\)
−0.721886 + 0.692012i \(0.756725\pi\)
\(348\) 0 0
\(349\) 15.5615 0.832990 0.416495 0.909138i \(-0.363258\pi\)
0.416495 + 0.909138i \(0.363258\pi\)
\(350\) 0 0
\(351\) 0.166958 0.00891156
\(352\) 0 0
\(353\) −29.0537 −1.54637 −0.773186 0.634180i \(-0.781338\pi\)
−0.773186 + 0.634180i \(0.781338\pi\)
\(354\) 0 0
\(355\) −12.2138 −0.648242
\(356\) 0 0
\(357\) −7.84793 −0.415357
\(358\) 0 0
\(359\) −35.5627 −1.87693 −0.938463 0.345380i \(-0.887750\pi\)
−0.938463 + 0.345380i \(0.887750\pi\)
\(360\) 0 0
\(361\) 0.356572 0.0187669
\(362\) 0 0
\(363\) −2.46329 −0.129289
\(364\) 0 0
\(365\) 10.1173 0.529564
\(366\) 0 0
\(367\) 28.0017 1.46168 0.730839 0.682550i \(-0.239129\pi\)
0.730839 + 0.682550i \(0.239129\pi\)
\(368\) 0 0
\(369\) 2.57206 0.133896
\(370\) 0 0
\(371\) 11.1825 0.580566
\(372\) 0 0
\(373\) −8.59794 −0.445185 −0.222592 0.974912i \(-0.571452\pi\)
−0.222592 + 0.974912i \(0.571452\pi\)
\(374\) 0 0
\(375\) −27.2042 −1.40482
\(376\) 0 0
\(377\) 5.83924 0.300736
\(378\) 0 0
\(379\) 9.29924 0.477670 0.238835 0.971060i \(-0.423235\pi\)
0.238835 + 0.971060i \(0.423235\pi\)
\(380\) 0 0
\(381\) −19.9212 −1.02059
\(382\) 0 0
\(383\) −11.2423 −0.574455 −0.287227 0.957862i \(-0.592734\pi\)
−0.287227 + 0.957862i \(0.592734\pi\)
\(384\) 0 0
\(385\) −1.35093 −0.0688499
\(386\) 0 0
\(387\) −15.2881 −0.777140
\(388\) 0 0
\(389\) 31.1249 1.57809 0.789046 0.614334i \(-0.210575\pi\)
0.789046 + 0.614334i \(0.210575\pi\)
\(390\) 0 0
\(391\) −8.41465 −0.425547
\(392\) 0 0
\(393\) 8.54606 0.431092
\(394\) 0 0
\(395\) 6.93741 0.349059
\(396\) 0 0
\(397\) 3.66752 0.184067 0.0920337 0.995756i \(-0.470663\pi\)
0.0920337 + 0.995756i \(0.470663\pi\)
\(398\) 0 0
\(399\) −10.8375 −0.542554
\(400\) 0 0
\(401\) −7.01974 −0.350549 −0.175275 0.984520i \(-0.556081\pi\)
−0.175275 + 0.984520i \(0.556081\pi\)
\(402\) 0 0
\(403\) 8.29622 0.413264
\(404\) 0 0
\(405\) 11.8775 0.590198
\(406\) 0 0
\(407\) 5.93067 0.293972
\(408\) 0 0
\(409\) 17.8214 0.881212 0.440606 0.897701i \(-0.354764\pi\)
0.440606 + 0.897701i \(0.354764\pi\)
\(410\) 0 0
\(411\) 11.3120 0.557980
\(412\) 0 0
\(413\) 11.1625 0.549270
\(414\) 0 0
\(415\) 8.38993 0.411845
\(416\) 0 0
\(417\) −23.6514 −1.15821
\(418\) 0 0
\(419\) −14.1126 −0.689444 −0.344722 0.938705i \(-0.612027\pi\)
−0.344722 + 0.938705i \(0.612027\pi\)
\(420\) 0 0
\(421\) 1.33945 0.0652808 0.0326404 0.999467i \(-0.489608\pi\)
0.0326404 + 0.999467i \(0.489608\pi\)
\(422\) 0 0
\(423\) −11.3832 −0.553473
\(424\) 0 0
\(425\) −10.1154 −0.490667
\(426\) 0 0
\(427\) −12.9239 −0.625432
\(428\) 0 0
\(429\) 2.46329 0.118929
\(430\) 0 0
\(431\) 12.8072 0.616902 0.308451 0.951240i \(-0.400189\pi\)
0.308451 + 0.951240i \(0.400189\pi\)
\(432\) 0 0
\(433\) −14.3722 −0.690683 −0.345341 0.938477i \(-0.612237\pi\)
−0.345341 + 0.938477i \(0.612237\pi\)
\(434\) 0 0
\(435\) −19.4314 −0.931666
\(436\) 0 0
\(437\) −11.6201 −0.555865
\(438\) 0 0
\(439\) −22.4871 −1.07325 −0.536625 0.843821i \(-0.680301\pi\)
−0.536625 + 0.843821i \(0.680301\pi\)
\(440\) 0 0
\(441\) 3.06778 0.146085
\(442\) 0 0
\(443\) 10.5139 0.499528 0.249764 0.968307i \(-0.419647\pi\)
0.249764 + 0.968307i \(0.419647\pi\)
\(444\) 0 0
\(445\) −22.3977 −1.06175
\(446\) 0 0
\(447\) 29.6092 1.40047
\(448\) 0 0
\(449\) 21.5042 1.01484 0.507422 0.861698i \(-0.330598\pi\)
0.507422 + 0.861698i \(0.330598\pi\)
\(450\) 0 0
\(451\) 0.838410 0.0394792
\(452\) 0 0
\(453\) −42.4320 −1.99363
\(454\) 0 0
\(455\) 1.35093 0.0633327
\(456\) 0 0
\(457\) −22.2943 −1.04288 −0.521442 0.853287i \(-0.674606\pi\)
−0.521442 + 0.853287i \(0.674606\pi\)
\(458\) 0 0
\(459\) −0.531922 −0.0248280
\(460\) 0 0
\(461\) 32.8453 1.52976 0.764880 0.644173i \(-0.222798\pi\)
0.764880 + 0.644173i \(0.222798\pi\)
\(462\) 0 0
\(463\) −27.0927 −1.25910 −0.629552 0.776958i \(-0.716762\pi\)
−0.629552 + 0.776958i \(0.716762\pi\)
\(464\) 0 0
\(465\) −27.6076 −1.28027
\(466\) 0 0
\(467\) 0.0125807 0.000582167 0 0.000291083 1.00000i \(-0.499907\pi\)
0.000291083 1.00000i \(0.499907\pi\)
\(468\) 0 0
\(469\) 2.65052 0.122389
\(470\) 0 0
\(471\) −3.80655 −0.175396
\(472\) 0 0
\(473\) −4.98346 −0.229140
\(474\) 0 0
\(475\) −13.9687 −0.640927
\(476\) 0 0
\(477\) 34.3054 1.57074
\(478\) 0 0
\(479\) 6.92578 0.316447 0.158223 0.987403i \(-0.449423\pi\)
0.158223 + 0.987403i \(0.449423\pi\)
\(480\) 0 0
\(481\) −5.93067 −0.270415
\(482\) 0 0
\(483\) 6.50595 0.296031
\(484\) 0 0
\(485\) −5.97688 −0.271396
\(486\) 0 0
\(487\) 25.7243 1.16568 0.582840 0.812587i \(-0.301941\pi\)
0.582840 + 0.812587i \(0.301941\pi\)
\(488\) 0 0
\(489\) −31.4234 −1.42102
\(490\) 0 0
\(491\) 35.1520 1.58639 0.793193 0.608970i \(-0.208417\pi\)
0.793193 + 0.608970i \(0.208417\pi\)
\(492\) 0 0
\(493\) −18.6036 −0.837863
\(494\) 0 0
\(495\) −4.14436 −0.186275
\(496\) 0 0
\(497\) 9.04103 0.405545
\(498\) 0 0
\(499\) 10.0030 0.447797 0.223898 0.974612i \(-0.428122\pi\)
0.223898 + 0.974612i \(0.428122\pi\)
\(500\) 0 0
\(501\) 25.9918 1.16123
\(502\) 0 0
\(503\) 1.72411 0.0768742 0.0384371 0.999261i \(-0.487762\pi\)
0.0384371 + 0.999261i \(0.487762\pi\)
\(504\) 0 0
\(505\) 21.6127 0.961752
\(506\) 0 0
\(507\) −2.46329 −0.109398
\(508\) 0 0
\(509\) −31.7114 −1.40558 −0.702792 0.711395i \(-0.748064\pi\)
−0.702792 + 0.711395i \(0.748064\pi\)
\(510\) 0 0
\(511\) −7.48913 −0.331300
\(512\) 0 0
\(513\) −0.734551 −0.0324312
\(514\) 0 0
\(515\) −0.810145 −0.0356992
\(516\) 0 0
\(517\) −3.71058 −0.163191
\(518\) 0 0
\(519\) 57.6916 2.53238
\(520\) 0 0
\(521\) 13.5878 0.595294 0.297647 0.954676i \(-0.403798\pi\)
0.297647 + 0.954676i \(0.403798\pi\)
\(522\) 0 0
\(523\) −22.2291 −0.972010 −0.486005 0.873956i \(-0.661546\pi\)
−0.486005 + 0.873956i \(0.661546\pi\)
\(524\) 0 0
\(525\) 7.82088 0.341331
\(526\) 0 0
\(527\) −26.4314 −1.15137
\(528\) 0 0
\(529\) −16.0242 −0.696706
\(530\) 0 0
\(531\) 34.2440 1.48607
\(532\) 0 0
\(533\) −0.838410 −0.0363156
\(534\) 0 0
\(535\) −24.8168 −1.07292
\(536\) 0 0
\(537\) −61.3220 −2.64624
\(538\) 0 0
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) 19.7593 0.849517 0.424759 0.905307i \(-0.360359\pi\)
0.424759 + 0.905307i \(0.360359\pi\)
\(542\) 0 0
\(543\) 50.3793 2.16198
\(544\) 0 0
\(545\) −12.6612 −0.542347
\(546\) 0 0
\(547\) −16.6384 −0.711408 −0.355704 0.934599i \(-0.615759\pi\)
−0.355704 + 0.934599i \(0.615759\pi\)
\(548\) 0 0
\(549\) −39.6477 −1.69212
\(550\) 0 0
\(551\) −25.6904 −1.09445
\(552\) 0 0
\(553\) −5.13527 −0.218374
\(554\) 0 0
\(555\) 19.7357 0.837733
\(556\) 0 0
\(557\) −29.5860 −1.25360 −0.626799 0.779181i \(-0.715635\pi\)
−0.626799 + 0.779181i \(0.715635\pi\)
\(558\) 0 0
\(559\) 4.98346 0.210778
\(560\) 0 0
\(561\) −7.84793 −0.331340
\(562\) 0 0
\(563\) −8.06054 −0.339711 −0.169856 0.985469i \(-0.554330\pi\)
−0.169856 + 0.985469i \(0.554330\pi\)
\(564\) 0 0
\(565\) 11.6507 0.490147
\(566\) 0 0
\(567\) −8.79207 −0.369232
\(568\) 0 0
\(569\) −28.1597 −1.18052 −0.590258 0.807215i \(-0.700974\pi\)
−0.590258 + 0.807215i \(0.700974\pi\)
\(570\) 0 0
\(571\) −8.77328 −0.367150 −0.183575 0.983006i \(-0.558767\pi\)
−0.183575 + 0.983006i \(0.558767\pi\)
\(572\) 0 0
\(573\) 13.8602 0.579018
\(574\) 0 0
\(575\) 8.38565 0.349706
\(576\) 0 0
\(577\) −46.3842 −1.93100 −0.965499 0.260405i \(-0.916144\pi\)
−0.965499 + 0.260405i \(0.916144\pi\)
\(578\) 0 0
\(579\) 6.31710 0.262530
\(580\) 0 0
\(581\) −6.21047 −0.257654
\(582\) 0 0
\(583\) 11.1825 0.463131
\(584\) 0 0
\(585\) 4.14436 0.171348
\(586\) 0 0
\(587\) 4.55730 0.188100 0.0940499 0.995567i \(-0.470019\pi\)
0.0940499 + 0.995567i \(0.470019\pi\)
\(588\) 0 0
\(589\) −36.5001 −1.50396
\(590\) 0 0
\(591\) −25.0542 −1.03059
\(592\) 0 0
\(593\) 0.399117 0.0163898 0.00819489 0.999966i \(-0.497391\pi\)
0.00819489 + 0.999966i \(0.497391\pi\)
\(594\) 0 0
\(595\) −4.30402 −0.176448
\(596\) 0 0
\(597\) −30.4404 −1.24584
\(598\) 0 0
\(599\) −15.5508 −0.635388 −0.317694 0.948193i \(-0.602908\pi\)
−0.317694 + 0.948193i \(0.602908\pi\)
\(600\) 0 0
\(601\) 7.98613 0.325761 0.162881 0.986646i \(-0.447921\pi\)
0.162881 + 0.986646i \(0.447921\pi\)
\(602\) 0 0
\(603\) 8.13120 0.331128
\(604\) 0 0
\(605\) −1.35093 −0.0549232
\(606\) 0 0
\(607\) −40.6607 −1.65037 −0.825183 0.564865i \(-0.808928\pi\)
−0.825183 + 0.564865i \(0.808928\pi\)
\(608\) 0 0
\(609\) 14.3837 0.582857
\(610\) 0 0
\(611\) 3.71058 0.150114
\(612\) 0 0
\(613\) −31.1689 −1.25890 −0.629451 0.777040i \(-0.716720\pi\)
−0.629451 + 0.777040i \(0.716720\pi\)
\(614\) 0 0
\(615\) 2.79001 0.112504
\(616\) 0 0
\(617\) −28.3109 −1.13975 −0.569877 0.821730i \(-0.693009\pi\)
−0.569877 + 0.821730i \(0.693009\pi\)
\(618\) 0 0
\(619\) −22.2838 −0.895661 −0.447831 0.894118i \(-0.647803\pi\)
−0.447831 + 0.894118i \(0.647803\pi\)
\(620\) 0 0
\(621\) 0.440964 0.0176953
\(622\) 0 0
\(623\) 16.5794 0.664242
\(624\) 0 0
\(625\) 0.955401 0.0382160
\(626\) 0 0
\(627\) −10.8375 −0.432808
\(628\) 0 0
\(629\) 18.8949 0.753388
\(630\) 0 0
\(631\) −34.9786 −1.39248 −0.696238 0.717811i \(-0.745144\pi\)
−0.696238 + 0.717811i \(0.745144\pi\)
\(632\) 0 0
\(633\) −31.3243 −1.24503
\(634\) 0 0
\(635\) −10.9253 −0.433558
\(636\) 0 0
\(637\) −1.00000 −0.0396214
\(638\) 0 0
\(639\) 27.7359 1.09721
\(640\) 0 0
\(641\) 18.4872 0.730199 0.365099 0.930969i \(-0.381035\pi\)
0.365099 + 0.930969i \(0.381035\pi\)
\(642\) 0 0
\(643\) −36.8122 −1.45173 −0.725866 0.687836i \(-0.758561\pi\)
−0.725866 + 0.687836i \(0.758561\pi\)
\(644\) 0 0
\(645\) −16.5836 −0.652980
\(646\) 0 0
\(647\) −28.5174 −1.12114 −0.560568 0.828108i \(-0.689417\pi\)
−0.560568 + 0.828108i \(0.689417\pi\)
\(648\) 0 0
\(649\) 11.1625 0.438166
\(650\) 0 0
\(651\) 20.4360 0.800949
\(652\) 0 0
\(653\) 39.8728 1.56034 0.780171 0.625567i \(-0.215132\pi\)
0.780171 + 0.625567i \(0.215132\pi\)
\(654\) 0 0
\(655\) 4.68689 0.183132
\(656\) 0 0
\(657\) −22.9750 −0.896340
\(658\) 0 0
\(659\) −38.0278 −1.48135 −0.740677 0.671862i \(-0.765495\pi\)
−0.740677 + 0.671862i \(0.765495\pi\)
\(660\) 0 0
\(661\) 47.7002 1.85532 0.927661 0.373424i \(-0.121816\pi\)
0.927661 + 0.373424i \(0.121816\pi\)
\(662\) 0 0
\(663\) 7.84793 0.304789
\(664\) 0 0
\(665\) −5.94358 −0.230482
\(666\) 0 0
\(667\) 15.4224 0.597158
\(668\) 0 0
\(669\) 43.5660 1.68436
\(670\) 0 0
\(671\) −12.9239 −0.498922
\(672\) 0 0
\(673\) −19.9971 −0.770831 −0.385415 0.922743i \(-0.625942\pi\)
−0.385415 + 0.922743i \(0.625942\pi\)
\(674\) 0 0
\(675\) 0.530089 0.0204031
\(676\) 0 0
\(677\) −16.5933 −0.637731 −0.318865 0.947800i \(-0.603302\pi\)
−0.318865 + 0.947800i \(0.603302\pi\)
\(678\) 0 0
\(679\) 4.42426 0.169787
\(680\) 0 0
\(681\) −30.8709 −1.18297
\(682\) 0 0
\(683\) −3.36101 −0.128605 −0.0643027 0.997930i \(-0.520482\pi\)
−0.0643027 + 0.997930i \(0.520482\pi\)
\(684\) 0 0
\(685\) 6.20381 0.237035
\(686\) 0 0
\(687\) −9.08940 −0.346782
\(688\) 0 0
\(689\) −11.1825 −0.426019
\(690\) 0 0
\(691\) 11.9872 0.456015 0.228008 0.973659i \(-0.426779\pi\)
0.228008 + 0.973659i \(0.426779\pi\)
\(692\) 0 0
\(693\) 3.06778 0.116535
\(694\) 0 0
\(695\) −12.9711 −0.492021
\(696\) 0 0
\(697\) 2.67114 0.101177
\(698\) 0 0
\(699\) 38.4687 1.45502
\(700\) 0 0
\(701\) −42.4807 −1.60448 −0.802238 0.597005i \(-0.796357\pi\)
−0.802238 + 0.597005i \(0.796357\pi\)
\(702\) 0 0
\(703\) 26.0926 0.984102
\(704\) 0 0
\(705\) −12.3478 −0.465046
\(706\) 0 0
\(707\) −15.9983 −0.601680
\(708\) 0 0
\(709\) 11.6705 0.438293 0.219146 0.975692i \(-0.429673\pi\)
0.219146 + 0.975692i \(0.429673\pi\)
\(710\) 0 0
\(711\) −15.7539 −0.590817
\(712\) 0 0
\(713\) 21.9117 0.820600
\(714\) 0 0
\(715\) 1.35093 0.0505220
\(716\) 0 0
\(717\) 54.3306 2.02901
\(718\) 0 0
\(719\) −17.1474 −0.639491 −0.319746 0.947503i \(-0.603597\pi\)
−0.319746 + 0.947503i \(0.603597\pi\)
\(720\) 0 0
\(721\) 0.599693 0.0223337
\(722\) 0 0
\(723\) −21.9908 −0.817845
\(724\) 0 0
\(725\) 18.5395 0.688538
\(726\) 0 0
\(727\) −20.1774 −0.748339 −0.374170 0.927360i \(-0.622072\pi\)
−0.374170 + 0.927360i \(0.622072\pi\)
\(728\) 0 0
\(729\) −28.2059 −1.04466
\(730\) 0 0
\(731\) −15.8771 −0.587236
\(732\) 0 0
\(733\) −28.2515 −1.04349 −0.521746 0.853101i \(-0.674719\pi\)
−0.521746 + 0.853101i \(0.674719\pi\)
\(734\) 0 0
\(735\) 3.32773 0.122745
\(736\) 0 0
\(737\) 2.65052 0.0976330
\(738\) 0 0
\(739\) 22.9454 0.844061 0.422031 0.906582i \(-0.361317\pi\)
0.422031 + 0.906582i \(0.361317\pi\)
\(740\) 0 0
\(741\) 10.8375 0.398125
\(742\) 0 0
\(743\) 28.2336 1.03579 0.517895 0.855444i \(-0.326716\pi\)
0.517895 + 0.855444i \(0.326716\pi\)
\(744\) 0 0
\(745\) 16.2385 0.594932
\(746\) 0 0
\(747\) −19.0523 −0.697089
\(748\) 0 0
\(749\) 18.3701 0.671229
\(750\) 0 0
\(751\) −3.83280 −0.139861 −0.0699304 0.997552i \(-0.522278\pi\)
−0.0699304 + 0.997552i \(0.522278\pi\)
\(752\) 0 0
\(753\) 44.1082 1.60739
\(754\) 0 0
\(755\) −23.2708 −0.846912
\(756\) 0 0
\(757\) −5.27925 −0.191878 −0.0959389 0.995387i \(-0.530585\pi\)
−0.0959389 + 0.995387i \(0.530585\pi\)
\(758\) 0 0
\(759\) 6.50595 0.236151
\(760\) 0 0
\(761\) −36.3162 −1.31646 −0.658231 0.752816i \(-0.728695\pi\)
−0.658231 + 0.752816i \(0.728695\pi\)
\(762\) 0 0
\(763\) 9.37221 0.339297
\(764\) 0 0
\(765\) −13.2038 −0.477384
\(766\) 0 0
\(767\) −11.1625 −0.403054
\(768\) 0 0
\(769\) −11.4002 −0.411101 −0.205550 0.978647i \(-0.565898\pi\)
−0.205550 + 0.978647i \(0.565898\pi\)
\(770\) 0 0
\(771\) 66.3967 2.39122
\(772\) 0 0
\(773\) −10.1555 −0.365268 −0.182634 0.983181i \(-0.558462\pi\)
−0.182634 + 0.983181i \(0.558462\pi\)
\(774\) 0 0
\(775\) 26.3403 0.946173
\(776\) 0 0
\(777\) −14.6089 −0.524092
\(778\) 0 0
\(779\) 3.68868 0.132161
\(780\) 0 0
\(781\) 9.04103 0.323513
\(782\) 0 0
\(783\) 0.974908 0.0348404
\(784\) 0 0
\(785\) −2.08761 −0.0745102
\(786\) 0 0
\(787\) −9.57018 −0.341140 −0.170570 0.985346i \(-0.554561\pi\)
−0.170570 + 0.985346i \(0.554561\pi\)
\(788\) 0 0
\(789\) 14.7561 0.525333
\(790\) 0 0
\(791\) −8.62416 −0.306640
\(792\) 0 0
\(793\) 12.9239 0.458942
\(794\) 0 0
\(795\) 37.2124 1.31979
\(796\) 0 0
\(797\) −12.7480 −0.451557 −0.225779 0.974179i \(-0.572493\pi\)
−0.225779 + 0.974179i \(0.572493\pi\)
\(798\) 0 0
\(799\) −11.8218 −0.418224
\(800\) 0 0
\(801\) 50.8621 1.79712
\(802\) 0 0
\(803\) −7.48913 −0.264286
\(804\) 0 0
\(805\) 3.56804 0.125757
\(806\) 0 0
\(807\) 72.6080 2.55592
\(808\) 0 0
\(809\) 12.8669 0.452377 0.226189 0.974084i \(-0.427373\pi\)
0.226189 + 0.974084i \(0.427373\pi\)
\(810\) 0 0
\(811\) −15.4923 −0.544010 −0.272005 0.962296i \(-0.587687\pi\)
−0.272005 + 0.962296i \(0.587687\pi\)
\(812\) 0 0
\(813\) 5.06842 0.177757
\(814\) 0 0
\(815\) −17.2335 −0.603662
\(816\) 0 0
\(817\) −21.9253 −0.767068
\(818\) 0 0
\(819\) −3.06778 −0.107197
\(820\) 0 0
\(821\) 23.7513 0.828927 0.414463 0.910066i \(-0.363969\pi\)
0.414463 + 0.910066i \(0.363969\pi\)
\(822\) 0 0
\(823\) 52.3347 1.82427 0.912137 0.409886i \(-0.134432\pi\)
0.912137 + 0.409886i \(0.134432\pi\)
\(824\) 0 0
\(825\) 7.82088 0.272288
\(826\) 0 0
\(827\) 34.2952 1.19256 0.596280 0.802777i \(-0.296645\pi\)
0.596280 + 0.802777i \(0.296645\pi\)
\(828\) 0 0
\(829\) −12.3995 −0.430652 −0.215326 0.976542i \(-0.569081\pi\)
−0.215326 + 0.976542i \(0.569081\pi\)
\(830\) 0 0
\(831\) −24.8933 −0.863540
\(832\) 0 0
\(833\) 3.18596 0.110387
\(834\) 0 0
\(835\) 14.2546 0.493301
\(836\) 0 0
\(837\) 1.38512 0.0478768
\(838\) 0 0
\(839\) −19.9994 −0.690456 −0.345228 0.938519i \(-0.612198\pi\)
−0.345228 + 0.938519i \(0.612198\pi\)
\(840\) 0 0
\(841\) 5.09669 0.175748
\(842\) 0 0
\(843\) 27.2876 0.939834
\(844\) 0 0
\(845\) −1.35093 −0.0464735
\(846\) 0 0
\(847\) 1.00000 0.0343604
\(848\) 0 0
\(849\) −11.3109 −0.388190
\(850\) 0 0
\(851\) −15.6639 −0.536951
\(852\) 0 0
\(853\) −27.2841 −0.934190 −0.467095 0.884207i \(-0.654699\pi\)
−0.467095 + 0.884207i \(0.654699\pi\)
\(854\) 0 0
\(855\) −18.2336 −0.623575
\(856\) 0 0
\(857\) −25.9605 −0.886795 −0.443397 0.896325i \(-0.646227\pi\)
−0.443397 + 0.896325i \(0.646227\pi\)
\(858\) 0 0
\(859\) 16.5988 0.566345 0.283173 0.959069i \(-0.408613\pi\)
0.283173 + 0.959069i \(0.408613\pi\)
\(860\) 0 0
\(861\) −2.06524 −0.0703833
\(862\) 0 0
\(863\) −44.6487 −1.51986 −0.759929 0.650006i \(-0.774767\pi\)
−0.759929 + 0.650006i \(0.774767\pi\)
\(864\) 0 0
\(865\) 31.6397 1.07578
\(866\) 0 0
\(867\) 16.8726 0.573025
\(868\) 0 0
\(869\) −5.13527 −0.174202
\(870\) 0 0
\(871\) −2.65052 −0.0898093
\(872\) 0 0
\(873\) 13.5726 0.459364
\(874\) 0 0
\(875\) 11.0438 0.373350
\(876\) 0 0
\(877\) −15.7613 −0.532221 −0.266110 0.963943i \(-0.585739\pi\)
−0.266110 + 0.963943i \(0.585739\pi\)
\(878\) 0 0
\(879\) 80.3204 2.70914
\(880\) 0 0
\(881\) −42.9162 −1.44588 −0.722942 0.690909i \(-0.757211\pi\)
−0.722942 + 0.690909i \(0.757211\pi\)
\(882\) 0 0
\(883\) −10.2080 −0.343527 −0.171764 0.985138i \(-0.554946\pi\)
−0.171764 + 0.985138i \(0.554946\pi\)
\(884\) 0 0
\(885\) 37.1458 1.24864
\(886\) 0 0
\(887\) 32.4015 1.08794 0.543968 0.839106i \(-0.316921\pi\)
0.543968 + 0.839106i \(0.316921\pi\)
\(888\) 0 0
\(889\) 8.08723 0.271237
\(890\) 0 0
\(891\) −8.79207 −0.294545
\(892\) 0 0
\(893\) −16.3251 −0.546299
\(894\) 0 0
\(895\) −33.6306 −1.12415
\(896\) 0 0
\(897\) −6.50595 −0.217227
\(898\) 0 0
\(899\) 48.4436 1.61568
\(900\) 0 0
\(901\) 35.6270 1.18691
\(902\) 0 0
\(903\) 12.2757 0.408509
\(904\) 0 0
\(905\) 27.6294 0.918431
\(906\) 0 0
\(907\) −37.2661 −1.23740 −0.618700 0.785628i \(-0.712340\pi\)
−0.618700 + 0.785628i \(0.712340\pi\)
\(908\) 0 0
\(909\) −49.0794 −1.62786
\(910\) 0 0
\(911\) −7.99443 −0.264867 −0.132434 0.991192i \(-0.542279\pi\)
−0.132434 + 0.991192i \(0.542279\pi\)
\(912\) 0 0
\(913\) −6.21047 −0.205536
\(914\) 0 0
\(915\) −43.0073 −1.42178
\(916\) 0 0
\(917\) −3.46938 −0.114569
\(918\) 0 0
\(919\) 42.4336 1.39976 0.699878 0.714262i \(-0.253238\pi\)
0.699878 + 0.714262i \(0.253238\pi\)
\(920\) 0 0
\(921\) −26.4439 −0.871357
\(922\) 0 0
\(923\) −9.04103 −0.297589
\(924\) 0 0
\(925\) −18.8298 −0.619118
\(926\) 0 0
\(927\) 1.83972 0.0604245
\(928\) 0 0
\(929\) −37.1310 −1.21823 −0.609114 0.793082i \(-0.708475\pi\)
−0.609114 + 0.793082i \(0.708475\pi\)
\(930\) 0 0
\(931\) 4.39961 0.144191
\(932\) 0 0
\(933\) 34.7626 1.13807
\(934\) 0 0
\(935\) −4.30402 −0.140757
\(936\) 0 0
\(937\) 44.7878 1.46315 0.731576 0.681760i \(-0.238785\pi\)
0.731576 + 0.681760i \(0.238785\pi\)
\(938\) 0 0
\(939\) −12.0938 −0.394666
\(940\) 0 0
\(941\) 24.4694 0.797679 0.398840 0.917021i \(-0.369413\pi\)
0.398840 + 0.917021i \(0.369413\pi\)
\(942\) 0 0
\(943\) −2.21438 −0.0721101
\(944\) 0 0
\(945\) 0.225549 0.00733712
\(946\) 0 0
\(947\) −6.01950 −0.195607 −0.0978037 0.995206i \(-0.531182\pi\)
−0.0978037 + 0.995206i \(0.531182\pi\)
\(948\) 0 0
\(949\) 7.48913 0.243107
\(950\) 0 0
\(951\) −36.4793 −1.18292
\(952\) 0 0
\(953\) −11.4091 −0.369577 −0.184788 0.982778i \(-0.559160\pi\)
−0.184788 + 0.982778i \(0.559160\pi\)
\(954\) 0 0
\(955\) 7.60130 0.245972
\(956\) 0 0
\(957\) 14.3837 0.464959
\(958\) 0 0
\(959\) −4.59224 −0.148291
\(960\) 0 0
\(961\) 37.8273 1.22024
\(962\) 0 0
\(963\) 56.3554 1.81603
\(964\) 0 0
\(965\) 3.46447 0.111525
\(966\) 0 0
\(967\) 30.3666 0.976524 0.488262 0.872697i \(-0.337631\pi\)
0.488262 + 0.872697i \(0.337631\pi\)
\(968\) 0 0
\(969\) −34.5279 −1.10919
\(970\) 0 0
\(971\) −37.7717 −1.21215 −0.606075 0.795407i \(-0.707257\pi\)
−0.606075 + 0.795407i \(0.707257\pi\)
\(972\) 0 0
\(973\) 9.60157 0.307812
\(974\) 0 0
\(975\) −7.82088 −0.250469
\(976\) 0 0
\(977\) 31.3212 1.00205 0.501027 0.865432i \(-0.332956\pi\)
0.501027 + 0.865432i \(0.332956\pi\)
\(978\) 0 0
\(979\) 16.5794 0.529881
\(980\) 0 0
\(981\) 28.7519 0.917976
\(982\) 0 0
\(983\) 46.9288 1.49679 0.748397 0.663250i \(-0.230824\pi\)
0.748397 + 0.663250i \(0.230824\pi\)
\(984\) 0 0
\(985\) −13.7404 −0.437805
\(986\) 0 0
\(987\) 9.14023 0.290937
\(988\) 0 0
\(989\) 13.1621 0.418532
\(990\) 0 0
\(991\) 40.3861 1.28291 0.641453 0.767163i \(-0.278332\pi\)
0.641453 + 0.767163i \(0.278332\pi\)
\(992\) 0 0
\(993\) −22.6750 −0.719570
\(994\) 0 0
\(995\) −16.6943 −0.529246
\(996\) 0 0
\(997\) 53.2863 1.68759 0.843797 0.536662i \(-0.180315\pi\)
0.843797 + 0.536662i \(0.180315\pi\)
\(998\) 0 0
\(999\) −0.990173 −0.0313277
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 8008.2.a.s.1.2 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8008.2.a.s.1.2 10 1.1 even 1 trivial