Properties

Label 8008.2.a.w.1.9
Level $8008$
Weight $2$
Character 8008.1
Self dual yes
Analytic conductor $63.944$
Analytic rank $0$
Dimension $11$
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: \(0\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{11} - 3 x^{10} - 19 x^{9} + 55 x^{8} + 128 x^{7} - 361 x^{6} - 343 x^{5} + 1012 x^{4} + 215 x^{3} - 1090 x^{2} + 240 x + 160 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.9
Root \(2.48144\) 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)
 
\(f(q)\) \(=\) \(q+2.48144 q^{3} +1.56116 q^{5} -1.00000 q^{7} +3.15756 q^{9} +O(q^{10})\) \(q+2.48144 q^{3} +1.56116 q^{5} -1.00000 q^{7} +3.15756 q^{9} +1.00000 q^{11} +1.00000 q^{13} +3.87394 q^{15} +1.17712 q^{17} +6.31074 q^{19} -2.48144 q^{21} +4.27004 q^{23} -2.56277 q^{25} +0.390980 q^{27} -6.71487 q^{29} -3.78667 q^{31} +2.48144 q^{33} -1.56116 q^{35} +2.14323 q^{37} +2.48144 q^{39} +11.7416 q^{41} -4.31235 q^{43} +4.92947 q^{45} -3.97107 q^{47} +1.00000 q^{49} +2.92097 q^{51} +4.15591 q^{53} +1.56116 q^{55} +15.6597 q^{57} +5.96907 q^{59} +6.28945 q^{61} -3.15756 q^{63} +1.56116 q^{65} +14.3339 q^{67} +10.5959 q^{69} -0.0156182 q^{71} +16.3588 q^{73} -6.35937 q^{75} -1.00000 q^{77} -2.88084 q^{79} -8.50249 q^{81} -4.75310 q^{83} +1.83768 q^{85} -16.6626 q^{87} +5.87297 q^{89} -1.00000 q^{91} -9.39640 q^{93} +9.85210 q^{95} -13.2741 q^{97} +3.15756 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11 q + 3 q^{3} - 2 q^{5} - 11 q^{7} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 11 q + 3 q^{3} - 2 q^{5} - 11 q^{7} + 14 q^{9} + 11 q^{11} + 11 q^{13} + 7 q^{15} + 9 q^{17} + 20 q^{19} - 3 q^{21} + 12 q^{23} + 13 q^{25} + 15 q^{27} + 8 q^{29} + 7 q^{31} + 3 q^{33} + 2 q^{35} - 10 q^{37} + 3 q^{39} - 2 q^{41} + 24 q^{43} - 6 q^{45} + 2 q^{47} + 11 q^{49} + 17 q^{51} + 3 q^{53} - 2 q^{55} - 16 q^{57} + q^{59} - 22 q^{61} - 14 q^{63} - 2 q^{65} + 14 q^{67} - 22 q^{69} + 6 q^{71} + 3 q^{73} - 11 q^{77} + 8 q^{79} - 9 q^{81} + 29 q^{83} - 9 q^{85} + 19 q^{87} + 20 q^{89} - 11 q^{91} - q^{93} + 18 q^{95} - 25 q^{97} + 14 q^{99}+O(q^{100}) \) 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
\(3\) 2.48144 1.43266 0.716331 0.697761i \(-0.245820\pi\)
0.716331 + 0.697761i \(0.245820\pi\)
\(4\) 0 0
\(5\) 1.56116 0.698173 0.349087 0.937090i \(-0.386492\pi\)
0.349087 + 0.937090i \(0.386492\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 3.15756 1.05252
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) 3.87394 1.00025
\(16\) 0 0
\(17\) 1.17712 0.285494 0.142747 0.989759i \(-0.454406\pi\)
0.142747 + 0.989759i \(0.454406\pi\)
\(18\) 0 0
\(19\) 6.31074 1.44778 0.723892 0.689914i \(-0.242351\pi\)
0.723892 + 0.689914i \(0.242351\pi\)
\(20\) 0 0
\(21\) −2.48144 −0.541495
\(22\) 0 0
\(23\) 4.27004 0.890366 0.445183 0.895440i \(-0.353139\pi\)
0.445183 + 0.895440i \(0.353139\pi\)
\(24\) 0 0
\(25\) −2.56277 −0.512554
\(26\) 0 0
\(27\) 0.390980 0.0752440
\(28\) 0 0
\(29\) −6.71487 −1.24692 −0.623460 0.781855i \(-0.714274\pi\)
−0.623460 + 0.781855i \(0.714274\pi\)
\(30\) 0 0
\(31\) −3.78667 −0.680106 −0.340053 0.940406i \(-0.610445\pi\)
−0.340053 + 0.940406i \(0.610445\pi\)
\(32\) 0 0
\(33\) 2.48144 0.431964
\(34\) 0 0
\(35\) −1.56116 −0.263885
\(36\) 0 0
\(37\) 2.14323 0.352345 0.176172 0.984359i \(-0.443628\pi\)
0.176172 + 0.984359i \(0.443628\pi\)
\(38\) 0 0
\(39\) 2.48144 0.397349
\(40\) 0 0
\(41\) 11.7416 1.83372 0.916861 0.399206i \(-0.130714\pi\)
0.916861 + 0.399206i \(0.130714\pi\)
\(42\) 0 0
\(43\) −4.31235 −0.657627 −0.328814 0.944395i \(-0.606649\pi\)
−0.328814 + 0.944395i \(0.606649\pi\)
\(44\) 0 0
\(45\) 4.92947 0.734842
\(46\) 0 0
\(47\) −3.97107 −0.579240 −0.289620 0.957142i \(-0.593529\pi\)
−0.289620 + 0.957142i \(0.593529\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 2.92097 0.409017
\(52\) 0 0
\(53\) 4.15591 0.570858 0.285429 0.958400i \(-0.407864\pi\)
0.285429 + 0.958400i \(0.407864\pi\)
\(54\) 0 0
\(55\) 1.56116 0.210507
\(56\) 0 0
\(57\) 15.6597 2.07418
\(58\) 0 0
\(59\) 5.96907 0.777106 0.388553 0.921426i \(-0.372975\pi\)
0.388553 + 0.921426i \(0.372975\pi\)
\(60\) 0 0
\(61\) 6.28945 0.805282 0.402641 0.915358i \(-0.368092\pi\)
0.402641 + 0.915358i \(0.368092\pi\)
\(62\) 0 0
\(63\) −3.15756 −0.397815
\(64\) 0 0
\(65\) 1.56116 0.193638
\(66\) 0 0
\(67\) 14.3339 1.75116 0.875582 0.483070i \(-0.160478\pi\)
0.875582 + 0.483070i \(0.160478\pi\)
\(68\) 0 0
\(69\) 10.5959 1.27559
\(70\) 0 0
\(71\) −0.0156182 −0.00185353 −0.000926767 1.00000i \(-0.500295\pi\)
−0.000926767 1.00000i \(0.500295\pi\)
\(72\) 0 0
\(73\) 16.3588 1.91466 0.957328 0.289002i \(-0.0933235\pi\)
0.957328 + 0.289002i \(0.0933235\pi\)
\(74\) 0 0
\(75\) −6.35937 −0.734317
\(76\) 0 0
\(77\) −1.00000 −0.113961
\(78\) 0 0
\(79\) −2.88084 −0.324119 −0.162060 0.986781i \(-0.551814\pi\)
−0.162060 + 0.986781i \(0.551814\pi\)
\(80\) 0 0
\(81\) −8.50249 −0.944721
\(82\) 0 0
\(83\) −4.75310 −0.521720 −0.260860 0.965377i \(-0.584006\pi\)
−0.260860 + 0.965377i \(0.584006\pi\)
\(84\) 0 0
\(85\) 1.83768 0.199325
\(86\) 0 0
\(87\) −16.6626 −1.78641
\(88\) 0 0
\(89\) 5.87297 0.622533 0.311267 0.950323i \(-0.399247\pi\)
0.311267 + 0.950323i \(0.399247\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) 0 0
\(93\) −9.39640 −0.974362
\(94\) 0 0
\(95\) 9.85210 1.01080
\(96\) 0 0
\(97\) −13.2741 −1.34778 −0.673892 0.738830i \(-0.735379\pi\)
−0.673892 + 0.738830i \(0.735379\pi\)
\(98\) 0 0
\(99\) 3.15756 0.317347
\(100\) 0 0
\(101\) 10.8013 1.07477 0.537384 0.843338i \(-0.319413\pi\)
0.537384 + 0.843338i \(0.319413\pi\)
\(102\) 0 0
\(103\) 16.9337 1.66852 0.834262 0.551368i \(-0.185894\pi\)
0.834262 + 0.551368i \(0.185894\pi\)
\(104\) 0 0
\(105\) −3.87394 −0.378058
\(106\) 0 0
\(107\) 7.72746 0.747041 0.373521 0.927622i \(-0.378150\pi\)
0.373521 + 0.927622i \(0.378150\pi\)
\(108\) 0 0
\(109\) 0.625583 0.0599200 0.0299600 0.999551i \(-0.490462\pi\)
0.0299600 + 0.999551i \(0.490462\pi\)
\(110\) 0 0
\(111\) 5.31830 0.504791
\(112\) 0 0
\(113\) −13.6749 −1.28643 −0.643213 0.765688i \(-0.722399\pi\)
−0.643213 + 0.765688i \(0.722399\pi\)
\(114\) 0 0
\(115\) 6.66623 0.621630
\(116\) 0 0
\(117\) 3.15756 0.291917
\(118\) 0 0
\(119\) −1.17712 −0.107907
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 29.1360 2.62710
\(124\) 0 0
\(125\) −11.8067 −1.05602
\(126\) 0 0
\(127\) −2.30079 −0.204162 −0.102081 0.994776i \(-0.532550\pi\)
−0.102081 + 0.994776i \(0.532550\pi\)
\(128\) 0 0
\(129\) −10.7009 −0.942158
\(130\) 0 0
\(131\) 6.90098 0.602941 0.301471 0.953475i \(-0.402522\pi\)
0.301471 + 0.953475i \(0.402522\pi\)
\(132\) 0 0
\(133\) −6.31074 −0.547211
\(134\) 0 0
\(135\) 0.610383 0.0525334
\(136\) 0 0
\(137\) −6.25875 −0.534721 −0.267361 0.963597i \(-0.586151\pi\)
−0.267361 + 0.963597i \(0.586151\pi\)
\(138\) 0 0
\(139\) −2.04281 −0.173269 −0.0866346 0.996240i \(-0.527611\pi\)
−0.0866346 + 0.996240i \(0.527611\pi\)
\(140\) 0 0
\(141\) −9.85398 −0.829855
\(142\) 0 0
\(143\) 1.00000 0.0836242
\(144\) 0 0
\(145\) −10.4830 −0.870566
\(146\) 0 0
\(147\) 2.48144 0.204666
\(148\) 0 0
\(149\) −15.3696 −1.25913 −0.629564 0.776949i \(-0.716766\pi\)
−0.629564 + 0.776949i \(0.716766\pi\)
\(150\) 0 0
\(151\) −19.7034 −1.60344 −0.801722 0.597697i \(-0.796082\pi\)
−0.801722 + 0.597697i \(0.796082\pi\)
\(152\) 0 0
\(153\) 3.71684 0.300489
\(154\) 0 0
\(155\) −5.91161 −0.474832
\(156\) 0 0
\(157\) −10.8416 −0.865256 −0.432628 0.901572i \(-0.642414\pi\)
−0.432628 + 0.901572i \(0.642414\pi\)
\(158\) 0 0
\(159\) 10.3126 0.817846
\(160\) 0 0
\(161\) −4.27004 −0.336527
\(162\) 0 0
\(163\) −12.3512 −0.967423 −0.483711 0.875228i \(-0.660712\pi\)
−0.483711 + 0.875228i \(0.660712\pi\)
\(164\) 0 0
\(165\) 3.87394 0.301586
\(166\) 0 0
\(167\) −3.57583 −0.276706 −0.138353 0.990383i \(-0.544181\pi\)
−0.138353 + 0.990383i \(0.544181\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 19.9266 1.52382
\(172\) 0 0
\(173\) 5.10474 0.388106 0.194053 0.980991i \(-0.437837\pi\)
0.194053 + 0.980991i \(0.437837\pi\)
\(174\) 0 0
\(175\) 2.56277 0.193727
\(176\) 0 0
\(177\) 14.8119 1.11333
\(178\) 0 0
\(179\) −2.46690 −0.184385 −0.0921924 0.995741i \(-0.529387\pi\)
−0.0921924 + 0.995741i \(0.529387\pi\)
\(180\) 0 0
\(181\) −10.5309 −0.782755 −0.391378 0.920230i \(-0.628001\pi\)
−0.391378 + 0.920230i \(0.628001\pi\)
\(182\) 0 0
\(183\) 15.6069 1.15370
\(184\) 0 0
\(185\) 3.34593 0.245998
\(186\) 0 0
\(187\) 1.17712 0.0860798
\(188\) 0 0
\(189\) −0.390980 −0.0284396
\(190\) 0 0
\(191\) 5.58755 0.404301 0.202150 0.979354i \(-0.435207\pi\)
0.202150 + 0.979354i \(0.435207\pi\)
\(192\) 0 0
\(193\) 10.5914 0.762389 0.381194 0.924495i \(-0.375513\pi\)
0.381194 + 0.924495i \(0.375513\pi\)
\(194\) 0 0
\(195\) 3.87394 0.277418
\(196\) 0 0
\(197\) 18.2836 1.30265 0.651326 0.758798i \(-0.274213\pi\)
0.651326 + 0.758798i \(0.274213\pi\)
\(198\) 0 0
\(199\) 13.2091 0.936368 0.468184 0.883631i \(-0.344908\pi\)
0.468184 + 0.883631i \(0.344908\pi\)
\(200\) 0 0
\(201\) 35.5687 2.50883
\(202\) 0 0
\(203\) 6.71487 0.471291
\(204\) 0 0
\(205\) 18.3305 1.28026
\(206\) 0 0
\(207\) 13.4829 0.937128
\(208\) 0 0
\(209\) 6.31074 0.436523
\(210\) 0 0
\(211\) 3.59626 0.247577 0.123788 0.992309i \(-0.460496\pi\)
0.123788 + 0.992309i \(0.460496\pi\)
\(212\) 0 0
\(213\) −0.0387556 −0.00265549
\(214\) 0 0
\(215\) −6.73228 −0.459138
\(216\) 0 0
\(217\) 3.78667 0.257056
\(218\) 0 0
\(219\) 40.5935 2.74306
\(220\) 0 0
\(221\) 1.17712 0.0791819
\(222\) 0 0
\(223\) −1.67011 −0.111839 −0.0559194 0.998435i \(-0.517809\pi\)
−0.0559194 + 0.998435i \(0.517809\pi\)
\(224\) 0 0
\(225\) −8.09211 −0.539474
\(226\) 0 0
\(227\) 5.32857 0.353670 0.176835 0.984241i \(-0.443414\pi\)
0.176835 + 0.984241i \(0.443414\pi\)
\(228\) 0 0
\(229\) 1.02612 0.0678078 0.0339039 0.999425i \(-0.489206\pi\)
0.0339039 + 0.999425i \(0.489206\pi\)
\(230\) 0 0
\(231\) −2.48144 −0.163267
\(232\) 0 0
\(233\) 4.41024 0.288924 0.144462 0.989510i \(-0.453855\pi\)
0.144462 + 0.989510i \(0.453855\pi\)
\(234\) 0 0
\(235\) −6.19948 −0.404410
\(236\) 0 0
\(237\) −7.14863 −0.464354
\(238\) 0 0
\(239\) −20.3307 −1.31508 −0.657542 0.753417i \(-0.728404\pi\)
−0.657542 + 0.753417i \(0.728404\pi\)
\(240\) 0 0
\(241\) 11.7834 0.759036 0.379518 0.925184i \(-0.376090\pi\)
0.379518 + 0.925184i \(0.376090\pi\)
\(242\) 0 0
\(243\) −22.2714 −1.42871
\(244\) 0 0
\(245\) 1.56116 0.0997390
\(246\) 0 0
\(247\) 6.31074 0.401543
\(248\) 0 0
\(249\) −11.7945 −0.747448
\(250\) 0 0
\(251\) −0.399012 −0.0251854 −0.0125927 0.999921i \(-0.504008\pi\)
−0.0125927 + 0.999921i \(0.504008\pi\)
\(252\) 0 0
\(253\) 4.27004 0.268455
\(254\) 0 0
\(255\) 4.56010 0.285565
\(256\) 0 0
\(257\) −18.1347 −1.13121 −0.565605 0.824676i \(-0.691357\pi\)
−0.565605 + 0.824676i \(0.691357\pi\)
\(258\) 0 0
\(259\) −2.14323 −0.133174
\(260\) 0 0
\(261\) −21.2026 −1.31241
\(262\) 0 0
\(263\) −15.5704 −0.960110 −0.480055 0.877238i \(-0.659383\pi\)
−0.480055 + 0.877238i \(0.659383\pi\)
\(264\) 0 0
\(265\) 6.48805 0.398558
\(266\) 0 0
\(267\) 14.5734 0.891880
\(268\) 0 0
\(269\) 0.238391 0.0145350 0.00726748 0.999974i \(-0.497687\pi\)
0.00726748 + 0.999974i \(0.497687\pi\)
\(270\) 0 0
\(271\) 28.7051 1.74371 0.871857 0.489761i \(-0.162916\pi\)
0.871857 + 0.489761i \(0.162916\pi\)
\(272\) 0 0
\(273\) −2.48144 −0.150184
\(274\) 0 0
\(275\) −2.56277 −0.154541
\(276\) 0 0
\(277\) 8.48559 0.509850 0.254925 0.966961i \(-0.417949\pi\)
0.254925 + 0.966961i \(0.417949\pi\)
\(278\) 0 0
\(279\) −11.9566 −0.715825
\(280\) 0 0
\(281\) −0.618209 −0.0368792 −0.0184396 0.999830i \(-0.505870\pi\)
−0.0184396 + 0.999830i \(0.505870\pi\)
\(282\) 0 0
\(283\) 28.1327 1.67231 0.836157 0.548490i \(-0.184797\pi\)
0.836157 + 0.548490i \(0.184797\pi\)
\(284\) 0 0
\(285\) 24.4474 1.44814
\(286\) 0 0
\(287\) −11.7416 −0.693082
\(288\) 0 0
\(289\) −15.6144 −0.918493
\(290\) 0 0
\(291\) −32.9390 −1.93092
\(292\) 0 0
\(293\) −12.3144 −0.719413 −0.359707 0.933065i \(-0.617123\pi\)
−0.359707 + 0.933065i \(0.617123\pi\)
\(294\) 0 0
\(295\) 9.31868 0.542555
\(296\) 0 0
\(297\) 0.390980 0.0226869
\(298\) 0 0
\(299\) 4.27004 0.246943
\(300\) 0 0
\(301\) 4.31235 0.248560
\(302\) 0 0
\(303\) 26.8028 1.53978
\(304\) 0 0
\(305\) 9.81886 0.562226
\(306\) 0 0
\(307\) −17.5236 −1.00013 −0.500064 0.865989i \(-0.666690\pi\)
−0.500064 + 0.865989i \(0.666690\pi\)
\(308\) 0 0
\(309\) 42.0199 2.39043
\(310\) 0 0
\(311\) 0.0147336 0.000835468 0 0.000417734 1.00000i \(-0.499867\pi\)
0.000417734 1.00000i \(0.499867\pi\)
\(312\) 0 0
\(313\) −1.73651 −0.0981531 −0.0490766 0.998795i \(-0.515628\pi\)
−0.0490766 + 0.998795i \(0.515628\pi\)
\(314\) 0 0
\(315\) −4.92947 −0.277744
\(316\) 0 0
\(317\) −15.6520 −0.879104 −0.439552 0.898217i \(-0.644863\pi\)
−0.439552 + 0.898217i \(0.644863\pi\)
\(318\) 0 0
\(319\) −6.71487 −0.375960
\(320\) 0 0
\(321\) 19.1752 1.07026
\(322\) 0 0
\(323\) 7.42852 0.413334
\(324\) 0 0
\(325\) −2.56277 −0.142157
\(326\) 0 0
\(327\) 1.55235 0.0858451
\(328\) 0 0
\(329\) 3.97107 0.218932
\(330\) 0 0
\(331\) −23.0136 −1.26494 −0.632471 0.774584i \(-0.717959\pi\)
−0.632471 + 0.774584i \(0.717959\pi\)
\(332\) 0 0
\(333\) 6.76738 0.370850
\(334\) 0 0
\(335\) 22.3775 1.22262
\(336\) 0 0
\(337\) 16.7006 0.909738 0.454869 0.890558i \(-0.349686\pi\)
0.454869 + 0.890558i \(0.349686\pi\)
\(338\) 0 0
\(339\) −33.9335 −1.84301
\(340\) 0 0
\(341\) −3.78667 −0.205060
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 16.5419 0.890585
\(346\) 0 0
\(347\) 17.4530 0.936927 0.468463 0.883483i \(-0.344808\pi\)
0.468463 + 0.883483i \(0.344808\pi\)
\(348\) 0 0
\(349\) −34.0812 −1.82432 −0.912161 0.409831i \(-0.865588\pi\)
−0.912161 + 0.409831i \(0.865588\pi\)
\(350\) 0 0
\(351\) 0.390980 0.0208689
\(352\) 0 0
\(353\) −2.72405 −0.144986 −0.0724932 0.997369i \(-0.523096\pi\)
−0.0724932 + 0.997369i \(0.523096\pi\)
\(354\) 0 0
\(355\) −0.0243825 −0.00129409
\(356\) 0 0
\(357\) −2.92097 −0.154594
\(358\) 0 0
\(359\) 12.3466 0.651630 0.325815 0.945433i \(-0.394361\pi\)
0.325815 + 0.945433i \(0.394361\pi\)
\(360\) 0 0
\(361\) 20.8255 1.09608
\(362\) 0 0
\(363\) 2.48144 0.130242
\(364\) 0 0
\(365\) 25.5388 1.33676
\(366\) 0 0
\(367\) 3.46233 0.180732 0.0903662 0.995909i \(-0.471196\pi\)
0.0903662 + 0.995909i \(0.471196\pi\)
\(368\) 0 0
\(369\) 37.0747 1.93003
\(370\) 0 0
\(371\) −4.15591 −0.215764
\(372\) 0 0
\(373\) −10.6744 −0.552701 −0.276350 0.961057i \(-0.589125\pi\)
−0.276350 + 0.961057i \(0.589125\pi\)
\(374\) 0 0
\(375\) −29.2977 −1.51293
\(376\) 0 0
\(377\) −6.71487 −0.345833
\(378\) 0 0
\(379\) −15.6381 −0.803276 −0.401638 0.915799i \(-0.631559\pi\)
−0.401638 + 0.915799i \(0.631559\pi\)
\(380\) 0 0
\(381\) −5.70928 −0.292495
\(382\) 0 0
\(383\) 18.9362 0.967594 0.483797 0.875180i \(-0.339257\pi\)
0.483797 + 0.875180i \(0.339257\pi\)
\(384\) 0 0
\(385\) −1.56116 −0.0795642
\(386\) 0 0
\(387\) −13.6165 −0.692166
\(388\) 0 0
\(389\) 2.50949 0.127236 0.0636182 0.997974i \(-0.479736\pi\)
0.0636182 + 0.997974i \(0.479736\pi\)
\(390\) 0 0
\(391\) 5.02637 0.254194
\(392\) 0 0
\(393\) 17.1244 0.863811
\(394\) 0 0
\(395\) −4.49745 −0.226292
\(396\) 0 0
\(397\) −12.6202 −0.633390 −0.316695 0.948528i \(-0.602573\pi\)
−0.316695 + 0.948528i \(0.602573\pi\)
\(398\) 0 0
\(399\) −15.6597 −0.783968
\(400\) 0 0
\(401\) 15.6824 0.783139 0.391570 0.920148i \(-0.371932\pi\)
0.391570 + 0.920148i \(0.371932\pi\)
\(402\) 0 0
\(403\) −3.78667 −0.188627
\(404\) 0 0
\(405\) −13.2738 −0.659579
\(406\) 0 0
\(407\) 2.14323 0.106236
\(408\) 0 0
\(409\) −13.5252 −0.668780 −0.334390 0.942435i \(-0.608530\pi\)
−0.334390 + 0.942435i \(0.608530\pi\)
\(410\) 0 0
\(411\) −15.5307 −0.766075
\(412\) 0 0
\(413\) −5.96907 −0.293719
\(414\) 0 0
\(415\) −7.42036 −0.364251
\(416\) 0 0
\(417\) −5.06913 −0.248236
\(418\) 0 0
\(419\) 28.3526 1.38512 0.692558 0.721362i \(-0.256484\pi\)
0.692558 + 0.721362i \(0.256484\pi\)
\(420\) 0 0
\(421\) 9.94894 0.484882 0.242441 0.970166i \(-0.422052\pi\)
0.242441 + 0.970166i \(0.422052\pi\)
\(422\) 0 0
\(423\) −12.5389 −0.609662
\(424\) 0 0
\(425\) −3.01670 −0.146331
\(426\) 0 0
\(427\) −6.28945 −0.304368
\(428\) 0 0
\(429\) 2.48144 0.119805
\(430\) 0 0
\(431\) −9.81615 −0.472827 −0.236414 0.971652i \(-0.575972\pi\)
−0.236414 + 0.971652i \(0.575972\pi\)
\(432\) 0 0
\(433\) 22.8631 1.09873 0.549366 0.835582i \(-0.314869\pi\)
0.549366 + 0.835582i \(0.314869\pi\)
\(434\) 0 0
\(435\) −26.0130 −1.24723
\(436\) 0 0
\(437\) 26.9471 1.28906
\(438\) 0 0
\(439\) 9.74098 0.464912 0.232456 0.972607i \(-0.425324\pi\)
0.232456 + 0.972607i \(0.425324\pi\)
\(440\) 0 0
\(441\) 3.15756 0.150360
\(442\) 0 0
\(443\) −41.0526 −1.95047 −0.975233 0.221178i \(-0.929010\pi\)
−0.975233 + 0.221178i \(0.929010\pi\)
\(444\) 0 0
\(445\) 9.16866 0.434636
\(446\) 0 0
\(447\) −38.1388 −1.80390
\(448\) 0 0
\(449\) 9.43267 0.445155 0.222578 0.974915i \(-0.428553\pi\)
0.222578 + 0.974915i \(0.428553\pi\)
\(450\) 0 0
\(451\) 11.7416 0.552888
\(452\) 0 0
\(453\) −48.8930 −2.29719
\(454\) 0 0
\(455\) −1.56116 −0.0731884
\(456\) 0 0
\(457\) −13.5005 −0.631528 −0.315764 0.948838i \(-0.602261\pi\)
−0.315764 + 0.948838i \(0.602261\pi\)
\(458\) 0 0
\(459\) 0.460231 0.0214818
\(460\) 0 0
\(461\) −1.78455 −0.0831148 −0.0415574 0.999136i \(-0.513232\pi\)
−0.0415574 + 0.999136i \(0.513232\pi\)
\(462\) 0 0
\(463\) −3.84604 −0.178741 −0.0893704 0.995998i \(-0.528485\pi\)
−0.0893704 + 0.995998i \(0.528485\pi\)
\(464\) 0 0
\(465\) −14.6693 −0.680273
\(466\) 0 0
\(467\) −5.33881 −0.247050 −0.123525 0.992341i \(-0.539420\pi\)
−0.123525 + 0.992341i \(0.539420\pi\)
\(468\) 0 0
\(469\) −14.3339 −0.661878
\(470\) 0 0
\(471\) −26.9029 −1.23962
\(472\) 0 0
\(473\) −4.31235 −0.198282
\(474\) 0 0
\(475\) −16.1730 −0.742067
\(476\) 0 0
\(477\) 13.1225 0.600839
\(478\) 0 0
\(479\) −14.7598 −0.674395 −0.337197 0.941434i \(-0.609479\pi\)
−0.337197 + 0.941434i \(0.609479\pi\)
\(480\) 0 0
\(481\) 2.14323 0.0977229
\(482\) 0 0
\(483\) −10.5959 −0.482129
\(484\) 0 0
\(485\) −20.7231 −0.940987
\(486\) 0 0
\(487\) −16.6715 −0.755456 −0.377728 0.925917i \(-0.623295\pi\)
−0.377728 + 0.925917i \(0.623295\pi\)
\(488\) 0 0
\(489\) −30.6489 −1.38599
\(490\) 0 0
\(491\) 34.3758 1.55136 0.775680 0.631127i \(-0.217407\pi\)
0.775680 + 0.631127i \(0.217407\pi\)
\(492\) 0 0
\(493\) −7.90423 −0.355989
\(494\) 0 0
\(495\) 4.92947 0.221563
\(496\) 0 0
\(497\) 0.0156182 0.000700570 0
\(498\) 0 0
\(499\) 26.2659 1.17582 0.587912 0.808925i \(-0.299950\pi\)
0.587912 + 0.808925i \(0.299950\pi\)
\(500\) 0 0
\(501\) −8.87322 −0.396426
\(502\) 0 0
\(503\) 41.2508 1.83928 0.919640 0.392762i \(-0.128480\pi\)
0.919640 + 0.392762i \(0.128480\pi\)
\(504\) 0 0
\(505\) 16.8626 0.750374
\(506\) 0 0
\(507\) 2.48144 0.110205
\(508\) 0 0
\(509\) 11.2542 0.498833 0.249417 0.968396i \(-0.419761\pi\)
0.249417 + 0.968396i \(0.419761\pi\)
\(510\) 0 0
\(511\) −16.3588 −0.723672
\(512\) 0 0
\(513\) 2.46737 0.108937
\(514\) 0 0
\(515\) 26.4362 1.16492
\(516\) 0 0
\(517\) −3.97107 −0.174647
\(518\) 0 0
\(519\) 12.6671 0.556025
\(520\) 0 0
\(521\) 1.82010 0.0797399 0.0398700 0.999205i \(-0.487306\pi\)
0.0398700 + 0.999205i \(0.487306\pi\)
\(522\) 0 0
\(523\) 15.6531 0.684463 0.342231 0.939616i \(-0.388817\pi\)
0.342231 + 0.939616i \(0.388817\pi\)
\(524\) 0 0
\(525\) 6.35937 0.277546
\(526\) 0 0
\(527\) −4.45738 −0.194166
\(528\) 0 0
\(529\) −4.76672 −0.207249
\(530\) 0 0
\(531\) 18.8477 0.817920
\(532\) 0 0
\(533\) 11.7416 0.508583
\(534\) 0 0
\(535\) 12.0638 0.521564
\(536\) 0 0
\(537\) −6.12147 −0.264161
\(538\) 0 0
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) −34.1889 −1.46990 −0.734948 0.678123i \(-0.762794\pi\)
−0.734948 + 0.678123i \(0.762794\pi\)
\(542\) 0 0
\(543\) −26.1318 −1.12142
\(544\) 0 0
\(545\) 0.976637 0.0418345
\(546\) 0 0
\(547\) 36.5322 1.56200 0.781001 0.624529i \(-0.214709\pi\)
0.781001 + 0.624529i \(0.214709\pi\)
\(548\) 0 0
\(549\) 19.8593 0.847575
\(550\) 0 0
\(551\) −42.3758 −1.80527
\(552\) 0 0
\(553\) 2.88084 0.122506
\(554\) 0 0
\(555\) 8.30274 0.352432
\(556\) 0 0
\(557\) −3.84658 −0.162985 −0.0814925 0.996674i \(-0.525969\pi\)
−0.0814925 + 0.996674i \(0.525969\pi\)
\(558\) 0 0
\(559\) −4.31235 −0.182393
\(560\) 0 0
\(561\) 2.92097 0.123323
\(562\) 0 0
\(563\) −18.2475 −0.769038 −0.384519 0.923117i \(-0.625633\pi\)
−0.384519 + 0.923117i \(0.625633\pi\)
\(564\) 0 0
\(565\) −21.3487 −0.898148
\(566\) 0 0
\(567\) 8.50249 0.357071
\(568\) 0 0
\(569\) −29.5637 −1.23937 −0.619687 0.784849i \(-0.712740\pi\)
−0.619687 + 0.784849i \(0.712740\pi\)
\(570\) 0 0
\(571\) 11.2496 0.470781 0.235390 0.971901i \(-0.424363\pi\)
0.235390 + 0.971901i \(0.424363\pi\)
\(572\) 0 0
\(573\) 13.8652 0.579226
\(574\) 0 0
\(575\) −10.9431 −0.456361
\(576\) 0 0
\(577\) −19.2782 −0.802562 −0.401281 0.915955i \(-0.631435\pi\)
−0.401281 + 0.915955i \(0.631435\pi\)
\(578\) 0 0
\(579\) 26.2821 1.09225
\(580\) 0 0
\(581\) 4.75310 0.197192
\(582\) 0 0
\(583\) 4.15591 0.172120
\(584\) 0 0
\(585\) 4.92947 0.203808
\(586\) 0 0
\(587\) −24.8675 −1.02639 −0.513196 0.858271i \(-0.671539\pi\)
−0.513196 + 0.858271i \(0.671539\pi\)
\(588\) 0 0
\(589\) −23.8967 −0.984646
\(590\) 0 0
\(591\) 45.3697 1.86626
\(592\) 0 0
\(593\) 36.3691 1.49350 0.746749 0.665106i \(-0.231614\pi\)
0.746749 + 0.665106i \(0.231614\pi\)
\(594\) 0 0
\(595\) −1.83768 −0.0753376
\(596\) 0 0
\(597\) 32.7776 1.34150
\(598\) 0 0
\(599\) 0.250843 0.0102492 0.00512458 0.999987i \(-0.498369\pi\)
0.00512458 + 0.999987i \(0.498369\pi\)
\(600\) 0 0
\(601\) 26.7065 1.08938 0.544691 0.838637i \(-0.316647\pi\)
0.544691 + 0.838637i \(0.316647\pi\)
\(602\) 0 0
\(603\) 45.2601 1.84314
\(604\) 0 0
\(605\) 1.56116 0.0634703
\(606\) 0 0
\(607\) 37.2503 1.51194 0.755972 0.654604i \(-0.227165\pi\)
0.755972 + 0.654604i \(0.227165\pi\)
\(608\) 0 0
\(609\) 16.6626 0.675201
\(610\) 0 0
\(611\) −3.97107 −0.160652
\(612\) 0 0
\(613\) −32.9308 −1.33006 −0.665032 0.746815i \(-0.731582\pi\)
−0.665032 + 0.746815i \(0.731582\pi\)
\(614\) 0 0
\(615\) 45.4860 1.83417
\(616\) 0 0
\(617\) −27.9539 −1.12538 −0.562691 0.826667i \(-0.690234\pi\)
−0.562691 + 0.826667i \(0.690234\pi\)
\(618\) 0 0
\(619\) −31.3143 −1.25863 −0.629313 0.777152i \(-0.716664\pi\)
−0.629313 + 0.777152i \(0.716664\pi\)
\(620\) 0 0
\(621\) 1.66950 0.0669947
\(622\) 0 0
\(623\) −5.87297 −0.235295
\(624\) 0 0
\(625\) −5.61835 −0.224734
\(626\) 0 0
\(627\) 15.6597 0.625390
\(628\) 0 0
\(629\) 2.52285 0.100592
\(630\) 0 0
\(631\) 2.55579 0.101744 0.0508722 0.998705i \(-0.483800\pi\)
0.0508722 + 0.998705i \(0.483800\pi\)
\(632\) 0 0
\(633\) 8.92392 0.354694
\(634\) 0 0
\(635\) −3.59191 −0.142541
\(636\) 0 0
\(637\) 1.00000 0.0396214
\(638\) 0 0
\(639\) −0.0493153 −0.00195088
\(640\) 0 0
\(641\) −40.5791 −1.60278 −0.801389 0.598144i \(-0.795905\pi\)
−0.801389 + 0.598144i \(0.795905\pi\)
\(642\) 0 0
\(643\) 11.4614 0.451992 0.225996 0.974128i \(-0.427436\pi\)
0.225996 + 0.974128i \(0.427436\pi\)
\(644\) 0 0
\(645\) −16.7058 −0.657789
\(646\) 0 0
\(647\) −27.1042 −1.06557 −0.532787 0.846249i \(-0.678855\pi\)
−0.532787 + 0.846249i \(0.678855\pi\)
\(648\) 0 0
\(649\) 5.96907 0.234306
\(650\) 0 0
\(651\) 9.39640 0.368274
\(652\) 0 0
\(653\) −37.9089 −1.48349 −0.741745 0.670682i \(-0.766002\pi\)
−0.741745 + 0.670682i \(0.766002\pi\)
\(654\) 0 0
\(655\) 10.7736 0.420957
\(656\) 0 0
\(657\) 51.6540 2.01522
\(658\) 0 0
\(659\) −23.2350 −0.905105 −0.452553 0.891738i \(-0.649487\pi\)
−0.452553 + 0.891738i \(0.649487\pi\)
\(660\) 0 0
\(661\) −18.7507 −0.729318 −0.364659 0.931141i \(-0.618814\pi\)
−0.364659 + 0.931141i \(0.618814\pi\)
\(662\) 0 0
\(663\) 2.92097 0.113441
\(664\) 0 0
\(665\) −9.85210 −0.382048
\(666\) 0 0
\(667\) −28.6728 −1.11021
\(668\) 0 0
\(669\) −4.14428 −0.160227
\(670\) 0 0
\(671\) 6.28945 0.242802
\(672\) 0 0
\(673\) 29.2275 1.12664 0.563318 0.826240i \(-0.309525\pi\)
0.563318 + 0.826240i \(0.309525\pi\)
\(674\) 0 0
\(675\) −1.00199 −0.0385666
\(676\) 0 0
\(677\) −24.2094 −0.930441 −0.465221 0.885195i \(-0.654025\pi\)
−0.465221 + 0.885195i \(0.654025\pi\)
\(678\) 0 0
\(679\) 13.2741 0.509415
\(680\) 0 0
\(681\) 13.2226 0.506689
\(682\) 0 0
\(683\) 41.4345 1.58545 0.792724 0.609581i \(-0.208662\pi\)
0.792724 + 0.609581i \(0.208662\pi\)
\(684\) 0 0
\(685\) −9.77093 −0.373328
\(686\) 0 0
\(687\) 2.54625 0.0971456
\(688\) 0 0
\(689\) 4.15591 0.158327
\(690\) 0 0
\(691\) 19.8913 0.756701 0.378351 0.925662i \(-0.376491\pi\)
0.378351 + 0.925662i \(0.376491\pi\)
\(692\) 0 0
\(693\) −3.15756 −0.119946
\(694\) 0 0
\(695\) −3.18917 −0.120972
\(696\) 0 0
\(697\) 13.8213 0.523518
\(698\) 0 0
\(699\) 10.9438 0.413931
\(700\) 0 0
\(701\) 3.82418 0.144437 0.0722187 0.997389i \(-0.476992\pi\)
0.0722187 + 0.997389i \(0.476992\pi\)
\(702\) 0 0
\(703\) 13.5254 0.510119
\(704\) 0 0
\(705\) −15.3837 −0.579383
\(706\) 0 0
\(707\) −10.8013 −0.406224
\(708\) 0 0
\(709\) 5.53919 0.208029 0.104014 0.994576i \(-0.466831\pi\)
0.104014 + 0.994576i \(0.466831\pi\)
\(710\) 0 0
\(711\) −9.09642 −0.341142
\(712\) 0 0
\(713\) −16.1692 −0.605543
\(714\) 0 0
\(715\) 1.56116 0.0583842
\(716\) 0 0
\(717\) −50.4495 −1.88407
\(718\) 0 0
\(719\) 45.1864 1.68517 0.842585 0.538564i \(-0.181033\pi\)
0.842585 + 0.538564i \(0.181033\pi\)
\(720\) 0 0
\(721\) −16.9337 −0.630643
\(722\) 0 0
\(723\) 29.2399 1.08744
\(724\) 0 0
\(725\) 17.2087 0.639114
\(726\) 0 0
\(727\) −47.6293 −1.76647 −0.883236 0.468929i \(-0.844640\pi\)
−0.883236 + 0.468929i \(0.844640\pi\)
\(728\) 0 0
\(729\) −29.7577 −1.10214
\(730\) 0 0
\(731\) −5.07617 −0.187749
\(732\) 0 0
\(733\) 23.7008 0.875408 0.437704 0.899119i \(-0.355792\pi\)
0.437704 + 0.899119i \(0.355792\pi\)
\(734\) 0 0
\(735\) 3.87394 0.142892
\(736\) 0 0
\(737\) 14.3339 0.527996
\(738\) 0 0
\(739\) −51.6597 −1.90033 −0.950167 0.311741i \(-0.899088\pi\)
−0.950167 + 0.311741i \(0.899088\pi\)
\(740\) 0 0
\(741\) 15.6597 0.575275
\(742\) 0 0
\(743\) −36.3330 −1.33293 −0.666464 0.745537i \(-0.732193\pi\)
−0.666464 + 0.745537i \(0.732193\pi\)
\(744\) 0 0
\(745\) −23.9945 −0.879089
\(746\) 0 0
\(747\) −15.0082 −0.549121
\(748\) 0 0
\(749\) −7.72746 −0.282355
\(750\) 0 0
\(751\) −18.3407 −0.669263 −0.334631 0.942349i \(-0.608612\pi\)
−0.334631 + 0.942349i \(0.608612\pi\)
\(752\) 0 0
\(753\) −0.990125 −0.0360821
\(754\) 0 0
\(755\) −30.7603 −1.11948
\(756\) 0 0
\(757\) −36.9776 −1.34397 −0.671987 0.740563i \(-0.734559\pi\)
−0.671987 + 0.740563i \(0.734559\pi\)
\(758\) 0 0
\(759\) 10.5959 0.384606
\(760\) 0 0
\(761\) −2.89156 −0.104819 −0.0524094 0.998626i \(-0.516690\pi\)
−0.0524094 + 0.998626i \(0.516690\pi\)
\(762\) 0 0
\(763\) −0.625583 −0.0226476
\(764\) 0 0
\(765\) 5.80259 0.209793
\(766\) 0 0
\(767\) 5.96907 0.215530
\(768\) 0 0
\(769\) −4.38294 −0.158053 −0.0790265 0.996873i \(-0.525181\pi\)
−0.0790265 + 0.996873i \(0.525181\pi\)
\(770\) 0 0
\(771\) −45.0002 −1.62064
\(772\) 0 0
\(773\) −28.1000 −1.01069 −0.505344 0.862918i \(-0.668634\pi\)
−0.505344 + 0.862918i \(0.668634\pi\)
\(774\) 0 0
\(775\) 9.70436 0.348591
\(776\) 0 0
\(777\) −5.31830 −0.190793
\(778\) 0 0
\(779\) 74.0979 2.65483
\(780\) 0 0
\(781\) −0.0156182 −0.000558862 0
\(782\) 0 0
\(783\) −2.62538 −0.0938233
\(784\) 0 0
\(785\) −16.9255 −0.604099
\(786\) 0 0
\(787\) 7.36322 0.262471 0.131235 0.991351i \(-0.458106\pi\)
0.131235 + 0.991351i \(0.458106\pi\)
\(788\) 0 0
\(789\) −38.6370 −1.37551
\(790\) 0 0
\(791\) 13.6749 0.486223
\(792\) 0 0
\(793\) 6.28945 0.223345
\(794\) 0 0
\(795\) 16.0997 0.570998
\(796\) 0 0
\(797\) 4.93083 0.174659 0.0873295 0.996179i \(-0.472167\pi\)
0.0873295 + 0.996179i \(0.472167\pi\)
\(798\) 0 0
\(799\) −4.67444 −0.165370
\(800\) 0 0
\(801\) 18.5442 0.655229
\(802\) 0 0
\(803\) 16.3588 0.577291
\(804\) 0 0
\(805\) −6.66623 −0.234954
\(806\) 0 0
\(807\) 0.591554 0.0208237
\(808\) 0 0
\(809\) 2.58531 0.0908945 0.0454473 0.998967i \(-0.485529\pi\)
0.0454473 + 0.998967i \(0.485529\pi\)
\(810\) 0 0
\(811\) −0.721237 −0.0253260 −0.0126630 0.999920i \(-0.504031\pi\)
−0.0126630 + 0.999920i \(0.504031\pi\)
\(812\) 0 0
\(813\) 71.2302 2.49815
\(814\) 0 0
\(815\) −19.2823 −0.675429
\(816\) 0 0
\(817\) −27.2141 −0.952102
\(818\) 0 0
\(819\) −3.15756 −0.110334
\(820\) 0 0
\(821\) −49.9930 −1.74477 −0.872384 0.488821i \(-0.837427\pi\)
−0.872384 + 0.488821i \(0.837427\pi\)
\(822\) 0 0
\(823\) 0.705645 0.0245972 0.0122986 0.999924i \(-0.496085\pi\)
0.0122986 + 0.999924i \(0.496085\pi\)
\(824\) 0 0
\(825\) −6.35937 −0.221405
\(826\) 0 0
\(827\) 12.4921 0.434393 0.217197 0.976128i \(-0.430309\pi\)
0.217197 + 0.976128i \(0.430309\pi\)
\(828\) 0 0
\(829\) −2.70790 −0.0940493 −0.0470246 0.998894i \(-0.514974\pi\)
−0.0470246 + 0.998894i \(0.514974\pi\)
\(830\) 0 0
\(831\) 21.0565 0.730443
\(832\) 0 0
\(833\) 1.17712 0.0407849
\(834\) 0 0
\(835\) −5.58245 −0.193189
\(836\) 0 0
\(837\) −1.48051 −0.0511739
\(838\) 0 0
\(839\) −15.5682 −0.537475 −0.268738 0.963213i \(-0.586606\pi\)
−0.268738 + 0.963213i \(0.586606\pi\)
\(840\) 0 0
\(841\) 16.0894 0.554809
\(842\) 0 0
\(843\) −1.53405 −0.0528355
\(844\) 0 0
\(845\) 1.56116 0.0537056
\(846\) 0 0
\(847\) −1.00000 −0.0343604
\(848\) 0 0
\(849\) 69.8096 2.39586
\(850\) 0 0
\(851\) 9.15169 0.313716
\(852\) 0 0
\(853\) −9.41907 −0.322503 −0.161251 0.986913i \(-0.551553\pi\)
−0.161251 + 0.986913i \(0.551553\pi\)
\(854\) 0 0
\(855\) 31.1086 1.06389
\(856\) 0 0
\(857\) 15.3585 0.524638 0.262319 0.964981i \(-0.415513\pi\)
0.262319 + 0.964981i \(0.415513\pi\)
\(858\) 0 0
\(859\) 8.74147 0.298255 0.149128 0.988818i \(-0.452354\pi\)
0.149128 + 0.988818i \(0.452354\pi\)
\(860\) 0 0
\(861\) −29.1360 −0.992952
\(862\) 0 0
\(863\) −4.63880 −0.157907 −0.0789533 0.996878i \(-0.525158\pi\)
−0.0789533 + 0.996878i \(0.525158\pi\)
\(864\) 0 0
\(865\) 7.96933 0.270965
\(866\) 0 0
\(867\) −38.7462 −1.31589
\(868\) 0 0
\(869\) −2.88084 −0.0977257
\(870\) 0 0
\(871\) 14.3339 0.485685
\(872\) 0 0
\(873\) −41.9139 −1.41857
\(874\) 0 0
\(875\) 11.8067 0.399140
\(876\) 0 0
\(877\) −47.5610 −1.60602 −0.803010 0.595966i \(-0.796769\pi\)
−0.803010 + 0.595966i \(0.796769\pi\)
\(878\) 0 0
\(879\) −30.5574 −1.03068
\(880\) 0 0
\(881\) 37.7015 1.27020 0.635098 0.772432i \(-0.280960\pi\)
0.635098 + 0.772432i \(0.280960\pi\)
\(882\) 0 0
\(883\) 26.9623 0.907353 0.453676 0.891167i \(-0.350112\pi\)
0.453676 + 0.891167i \(0.350112\pi\)
\(884\) 0 0
\(885\) 23.1238 0.777298
\(886\) 0 0
\(887\) 27.7931 0.933200 0.466600 0.884468i \(-0.345479\pi\)
0.466600 + 0.884468i \(0.345479\pi\)
\(888\) 0 0
\(889\) 2.30079 0.0771661
\(890\) 0 0
\(891\) −8.50249 −0.284844
\(892\) 0 0
\(893\) −25.0604 −0.838614
\(894\) 0 0
\(895\) −3.85123 −0.128733
\(896\) 0 0
\(897\) 10.5959 0.353786
\(898\) 0 0
\(899\) 25.4270 0.848037
\(900\) 0 0
\(901\) 4.89202 0.162977
\(902\) 0 0
\(903\) 10.7009 0.356102
\(904\) 0 0
\(905\) −16.4404 −0.546499
\(906\) 0 0
\(907\) −40.5750 −1.34727 −0.673636 0.739063i \(-0.735268\pi\)
−0.673636 + 0.739063i \(0.735268\pi\)
\(908\) 0 0
\(909\) 34.1057 1.13121
\(910\) 0 0
\(911\) −59.0732 −1.95718 −0.978591 0.205813i \(-0.934016\pi\)
−0.978591 + 0.205813i \(0.934016\pi\)
\(912\) 0 0
\(913\) −4.75310 −0.157304
\(914\) 0 0
\(915\) 24.3649 0.805480
\(916\) 0 0
\(917\) −6.90098 −0.227890
\(918\) 0 0
\(919\) −10.4062 −0.343269 −0.171634 0.985161i \(-0.554905\pi\)
−0.171634 + 0.985161i \(0.554905\pi\)
\(920\) 0 0
\(921\) −43.4839 −1.43284
\(922\) 0 0
\(923\) −0.0156182 −0.000514078 0
\(924\) 0 0
\(925\) −5.49261 −0.180596
\(926\) 0 0
\(927\) 53.4691 1.75616
\(928\) 0 0
\(929\) −40.4171 −1.32604 −0.663021 0.748601i \(-0.730726\pi\)
−0.663021 + 0.748601i \(0.730726\pi\)
\(930\) 0 0
\(931\) 6.31074 0.206826
\(932\) 0 0
\(933\) 0.0365607 0.00119694
\(934\) 0 0
\(935\) 1.83768 0.0600986
\(936\) 0 0
\(937\) −31.3396 −1.02382 −0.511910 0.859039i \(-0.671062\pi\)
−0.511910 + 0.859039i \(0.671062\pi\)
\(938\) 0 0
\(939\) −4.30904 −0.140620
\(940\) 0 0
\(941\) −18.9408 −0.617451 −0.308726 0.951151i \(-0.599902\pi\)
−0.308726 + 0.951151i \(0.599902\pi\)
\(942\) 0 0
\(943\) 50.1369 1.63268
\(944\) 0 0
\(945\) −0.610383 −0.0198557
\(946\) 0 0
\(947\) −7.62876 −0.247901 −0.123951 0.992288i \(-0.539556\pi\)
−0.123951 + 0.992288i \(0.539556\pi\)
\(948\) 0 0
\(949\) 16.3588 0.531030
\(950\) 0 0
\(951\) −38.8396 −1.25946
\(952\) 0 0
\(953\) 17.2639 0.559232 0.279616 0.960112i \(-0.409793\pi\)
0.279616 + 0.960112i \(0.409793\pi\)
\(954\) 0 0
\(955\) 8.72307 0.282272
\(956\) 0 0
\(957\) −16.6626 −0.538624
\(958\) 0 0
\(959\) 6.25875 0.202106
\(960\) 0 0
\(961\) −16.6611 −0.537456
\(962\) 0 0
\(963\) 24.3999 0.786276
\(964\) 0 0
\(965\) 16.5350 0.532279
\(966\) 0 0
\(967\) −3.96313 −0.127446 −0.0637228 0.997968i \(-0.520297\pi\)
−0.0637228 + 0.997968i \(0.520297\pi\)
\(968\) 0 0
\(969\) 18.4335 0.592168
\(970\) 0 0
\(971\) −11.4619 −0.367830 −0.183915 0.982942i \(-0.558877\pi\)
−0.183915 + 0.982942i \(0.558877\pi\)
\(972\) 0 0
\(973\) 2.04281 0.0654896
\(974\) 0 0
\(975\) −6.35937 −0.203663
\(976\) 0 0
\(977\) −38.3767 −1.22778 −0.613890 0.789392i \(-0.710396\pi\)
−0.613890 + 0.789392i \(0.710396\pi\)
\(978\) 0 0
\(979\) 5.87297 0.187701
\(980\) 0 0
\(981\) 1.97532 0.0630670
\(982\) 0 0
\(983\) 25.8034 0.823000 0.411500 0.911410i \(-0.365005\pi\)
0.411500 + 0.911410i \(0.365005\pi\)
\(984\) 0 0
\(985\) 28.5437 0.909476
\(986\) 0 0
\(987\) 9.85398 0.313656
\(988\) 0 0
\(989\) −18.4139 −0.585529
\(990\) 0 0
\(991\) 7.47383 0.237414 0.118707 0.992929i \(-0.462125\pi\)
0.118707 + 0.992929i \(0.462125\pi\)
\(992\) 0 0
\(993\) −57.1069 −1.81223
\(994\) 0 0
\(995\) 20.6215 0.653747
\(996\) 0 0
\(997\) −34.9687 −1.10747 −0.553734 0.832693i \(-0.686798\pi\)
−0.553734 + 0.832693i \(0.686798\pi\)
\(998\) 0 0
\(999\) 0.837959 0.0265118
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.w.1.9 11
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8008.2.a.w.1.9 11 1.1 even 1 trivial