Properties

Label 8036.2.a.n.1.5
Level $8036$
Weight $2$
Character 8036.1
Self dual yes
Analytic conductor $64.168$
Analytic rank $1$
Dimension $8$
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] = [8036,2,Mod(1,8036)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8036, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8036.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8036 = 2^{2} \cdot 7^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8036.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: \(64.1677830643\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 14x^{6} - 4x^{5} + 60x^{4} + 31x^{3} - 75x^{2} - 60x - 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1148)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-0.308117\) of defining polynomial
Character \(\chi\) \(=\) 8036.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.308117 q^{3} +3.30124 q^{5} -2.90506 q^{9} +O(q^{10})\) \(q-0.308117 q^{3} +3.30124 q^{5} -2.90506 q^{9} +1.51731 q^{11} -1.10922 q^{13} -1.01717 q^{15} -1.60677 q^{17} -0.880518 q^{19} +0.682060 q^{23} +5.89819 q^{25} +1.81945 q^{27} -1.88250 q^{29} -4.71409 q^{31} -0.467508 q^{33} -3.22136 q^{37} +0.341769 q^{39} -1.00000 q^{41} -9.57315 q^{43} -9.59031 q^{45} -4.78592 q^{47} +0.495073 q^{51} -1.63217 q^{53} +5.00899 q^{55} +0.271303 q^{57} -1.01005 q^{59} +0.380672 q^{61} -3.66180 q^{65} +8.48699 q^{67} -0.210154 q^{69} +7.90937 q^{71} -1.56461 q^{73} -1.81733 q^{75} -8.41469 q^{79} +8.15459 q^{81} -1.55396 q^{83} -5.30433 q^{85} +0.580030 q^{87} -13.9216 q^{89} +1.45249 q^{93} -2.90680 q^{95} -0.601041 q^{97} -4.40787 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{9} - 8 q^{11} + 7 q^{13} - q^{15} + q^{17} - 4 q^{19} - 3 q^{23} - 4 q^{25} + 12 q^{27} - 4 q^{29} - 4 q^{31} - 23 q^{33} - 31 q^{37} + 5 q^{39} - 8 q^{41} - 8 q^{43} - q^{45} - 24 q^{47} - 23 q^{51} - q^{53} - 2 q^{55} - 15 q^{57} - 4 q^{59} + 4 q^{61} - 24 q^{65} + 21 q^{69} + 8 q^{71} - 11 q^{73} + 15 q^{75} + 14 q^{79} - 28 q^{81} + 42 q^{83} - 20 q^{85} - 25 q^{87} + 11 q^{89} - 27 q^{93} - 15 q^{95} + 16 q^{97} - 8 q^{99}+O(q^{100}) \) 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
\(3\) −0.308117 −0.177891 −0.0889457 0.996036i \(-0.528350\pi\)
−0.0889457 + 0.996036i \(0.528350\pi\)
\(4\) 0 0
\(5\) 3.30124 1.47636 0.738180 0.674604i \(-0.235686\pi\)
0.738180 + 0.674604i \(0.235686\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −2.90506 −0.968355
\(10\) 0 0
\(11\) 1.51731 0.457485 0.228743 0.973487i \(-0.426539\pi\)
0.228743 + 0.973487i \(0.426539\pi\)
\(12\) 0 0
\(13\) −1.10922 −0.307642 −0.153821 0.988099i \(-0.549158\pi\)
−0.153821 + 0.988099i \(0.549158\pi\)
\(14\) 0 0
\(15\) −1.01717 −0.262632
\(16\) 0 0
\(17\) −1.60677 −0.389699 −0.194850 0.980833i \(-0.562422\pi\)
−0.194850 + 0.980833i \(0.562422\pi\)
\(18\) 0 0
\(19\) −0.880518 −0.202005 −0.101002 0.994886i \(-0.532205\pi\)
−0.101002 + 0.994886i \(0.532205\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0.682060 0.142219 0.0711097 0.997469i \(-0.477346\pi\)
0.0711097 + 0.997469i \(0.477346\pi\)
\(24\) 0 0
\(25\) 5.89819 1.17964
\(26\) 0 0
\(27\) 1.81945 0.350153
\(28\) 0 0
\(29\) −1.88250 −0.349572 −0.174786 0.984606i \(-0.555923\pi\)
−0.174786 + 0.984606i \(0.555923\pi\)
\(30\) 0 0
\(31\) −4.71409 −0.846675 −0.423337 0.905972i \(-0.639142\pi\)
−0.423337 + 0.905972i \(0.639142\pi\)
\(32\) 0 0
\(33\) −0.467508 −0.0813826
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −3.22136 −0.529588 −0.264794 0.964305i \(-0.585304\pi\)
−0.264794 + 0.964305i \(0.585304\pi\)
\(38\) 0 0
\(39\) 0.341769 0.0547268
\(40\) 0 0
\(41\) −1.00000 −0.156174
\(42\) 0 0
\(43\) −9.57315 −1.45989 −0.729946 0.683505i \(-0.760455\pi\)
−0.729946 + 0.683505i \(0.760455\pi\)
\(44\) 0 0
\(45\) −9.59031 −1.42964
\(46\) 0 0
\(47\) −4.78592 −0.698098 −0.349049 0.937105i \(-0.613495\pi\)
−0.349049 + 0.937105i \(0.613495\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0.495073 0.0693241
\(52\) 0 0
\(53\) −1.63217 −0.224196 −0.112098 0.993697i \(-0.535757\pi\)
−0.112098 + 0.993697i \(0.535757\pi\)
\(54\) 0 0
\(55\) 5.00899 0.675412
\(56\) 0 0
\(57\) 0.271303 0.0359349
\(58\) 0 0
\(59\) −1.01005 −0.131497 −0.0657485 0.997836i \(-0.520944\pi\)
−0.0657485 + 0.997836i \(0.520944\pi\)
\(60\) 0 0
\(61\) 0.380672 0.0487400 0.0243700 0.999703i \(-0.492242\pi\)
0.0243700 + 0.999703i \(0.492242\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −3.66180 −0.454190
\(66\) 0 0
\(67\) 8.48699 1.03685 0.518425 0.855123i \(-0.326518\pi\)
0.518425 + 0.855123i \(0.326518\pi\)
\(68\) 0 0
\(69\) −0.210154 −0.0252996
\(70\) 0 0
\(71\) 7.90937 0.938669 0.469335 0.883020i \(-0.344494\pi\)
0.469335 + 0.883020i \(0.344494\pi\)
\(72\) 0 0
\(73\) −1.56461 −0.183124 −0.0915618 0.995799i \(-0.529186\pi\)
−0.0915618 + 0.995799i \(0.529186\pi\)
\(74\) 0 0
\(75\) −1.81733 −0.209847
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −8.41469 −0.946727 −0.473364 0.880867i \(-0.656960\pi\)
−0.473364 + 0.880867i \(0.656960\pi\)
\(80\) 0 0
\(81\) 8.15459 0.906065
\(82\) 0 0
\(83\) −1.55396 −0.170570 −0.0852848 0.996357i \(-0.527180\pi\)
−0.0852848 + 0.996357i \(0.527180\pi\)
\(84\) 0 0
\(85\) −5.30433 −0.575336
\(86\) 0 0
\(87\) 0.580030 0.0621858
\(88\) 0 0
\(89\) −13.9216 −1.47568 −0.737841 0.674974i \(-0.764155\pi\)
−0.737841 + 0.674974i \(0.764155\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 1.45249 0.150616
\(94\) 0 0
\(95\) −2.90680 −0.298232
\(96\) 0 0
\(97\) −0.601041 −0.0610264 −0.0305132 0.999534i \(-0.509714\pi\)
−0.0305132 + 0.999534i \(0.509714\pi\)
\(98\) 0 0
\(99\) −4.40787 −0.443008
\(100\) 0 0
\(101\) −3.79423 −0.377540 −0.188770 0.982021i \(-0.560450\pi\)
−0.188770 + 0.982021i \(0.560450\pi\)
\(102\) 0 0
\(103\) −0.520009 −0.0512380 −0.0256190 0.999672i \(-0.508156\pi\)
−0.0256190 + 0.999672i \(0.508156\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −9.35377 −0.904263 −0.452132 0.891951i \(-0.649336\pi\)
−0.452132 + 0.891951i \(0.649336\pi\)
\(108\) 0 0
\(109\) 5.25348 0.503192 0.251596 0.967832i \(-0.419045\pi\)
0.251596 + 0.967832i \(0.419045\pi\)
\(110\) 0 0
\(111\) 0.992554 0.0942091
\(112\) 0 0
\(113\) −0.945705 −0.0889644 −0.0444822 0.999010i \(-0.514164\pi\)
−0.0444822 + 0.999010i \(0.514164\pi\)
\(114\) 0 0
\(115\) 2.25164 0.209967
\(116\) 0 0
\(117\) 3.22235 0.297906
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −8.69778 −0.790707
\(122\) 0 0
\(123\) 0.308117 0.0277820
\(124\) 0 0
\(125\) 2.96513 0.265210
\(126\) 0 0
\(127\) 11.5966 1.02903 0.514517 0.857480i \(-0.327971\pi\)
0.514517 + 0.857480i \(0.327971\pi\)
\(128\) 0 0
\(129\) 2.94965 0.259702
\(130\) 0 0
\(131\) 8.01947 0.700664 0.350332 0.936626i \(-0.386069\pi\)
0.350332 + 0.936626i \(0.386069\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 6.00644 0.516952
\(136\) 0 0
\(137\) −0.817226 −0.0698203 −0.0349102 0.999390i \(-0.511115\pi\)
−0.0349102 + 0.999390i \(0.511115\pi\)
\(138\) 0 0
\(139\) −2.61528 −0.221825 −0.110912 0.993830i \(-0.535377\pi\)
−0.110912 + 0.993830i \(0.535377\pi\)
\(140\) 0 0
\(141\) 1.47462 0.124186
\(142\) 0 0
\(143\) −1.68302 −0.140742
\(144\) 0 0
\(145\) −6.21459 −0.516093
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −16.0183 −1.31227 −0.656134 0.754644i \(-0.727809\pi\)
−0.656134 + 0.754644i \(0.727809\pi\)
\(150\) 0 0
\(151\) −9.67825 −0.787605 −0.393802 0.919195i \(-0.628841\pi\)
−0.393802 + 0.919195i \(0.628841\pi\)
\(152\) 0 0
\(153\) 4.66777 0.377367
\(154\) 0 0
\(155\) −15.5623 −1.25000
\(156\) 0 0
\(157\) −8.78060 −0.700768 −0.350384 0.936606i \(-0.613949\pi\)
−0.350384 + 0.936606i \(0.613949\pi\)
\(158\) 0 0
\(159\) 0.502900 0.0398825
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −11.5969 −0.908335 −0.454168 0.890916i \(-0.650063\pi\)
−0.454168 + 0.890916i \(0.650063\pi\)
\(164\) 0 0
\(165\) −1.54336 −0.120150
\(166\) 0 0
\(167\) 3.23013 0.249955 0.124977 0.992160i \(-0.460114\pi\)
0.124977 + 0.992160i \(0.460114\pi\)
\(168\) 0 0
\(169\) −11.7696 −0.905356
\(170\) 0 0
\(171\) 2.55796 0.195612
\(172\) 0 0
\(173\) −9.85729 −0.749436 −0.374718 0.927139i \(-0.622260\pi\)
−0.374718 + 0.927139i \(0.622260\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0.311213 0.0233922
\(178\) 0 0
\(179\) −20.0945 −1.50193 −0.750967 0.660339i \(-0.770412\pi\)
−0.750967 + 0.660339i \(0.770412\pi\)
\(180\) 0 0
\(181\) −3.60880 −0.268240 −0.134120 0.990965i \(-0.542821\pi\)
−0.134120 + 0.990965i \(0.542821\pi\)
\(182\) 0 0
\(183\) −0.117291 −0.00867042
\(184\) 0 0
\(185\) −10.6345 −0.781862
\(186\) 0 0
\(187\) −2.43796 −0.178282
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −19.5419 −1.41400 −0.707000 0.707214i \(-0.749952\pi\)
−0.707000 + 0.707214i \(0.749952\pi\)
\(192\) 0 0
\(193\) −26.9641 −1.94092 −0.970459 0.241266i \(-0.922437\pi\)
−0.970459 + 0.241266i \(0.922437\pi\)
\(194\) 0 0
\(195\) 1.12826 0.0807965
\(196\) 0 0
\(197\) 10.5882 0.754381 0.377191 0.926136i \(-0.376890\pi\)
0.377191 + 0.926136i \(0.376890\pi\)
\(198\) 0 0
\(199\) 19.8558 1.40754 0.703770 0.710428i \(-0.251499\pi\)
0.703770 + 0.710428i \(0.251499\pi\)
\(200\) 0 0
\(201\) −2.61498 −0.184447
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −3.30124 −0.230569
\(206\) 0 0
\(207\) −1.98143 −0.137719
\(208\) 0 0
\(209\) −1.33602 −0.0924142
\(210\) 0 0
\(211\) 9.05365 0.623279 0.311639 0.950200i \(-0.399122\pi\)
0.311639 + 0.950200i \(0.399122\pi\)
\(212\) 0 0
\(213\) −2.43701 −0.166981
\(214\) 0 0
\(215\) −31.6033 −2.15532
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0.482082 0.0325761
\(220\) 0 0
\(221\) 1.78226 0.119888
\(222\) 0 0
\(223\) 11.4325 0.765580 0.382790 0.923835i \(-0.374963\pi\)
0.382790 + 0.923835i \(0.374963\pi\)
\(224\) 0 0
\(225\) −17.1346 −1.14231
\(226\) 0 0
\(227\) 21.2978 1.41358 0.706792 0.707421i \(-0.250142\pi\)
0.706792 + 0.707421i \(0.250142\pi\)
\(228\) 0 0
\(229\) 21.1127 1.39517 0.697583 0.716504i \(-0.254259\pi\)
0.697583 + 0.716504i \(0.254259\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 22.6782 1.48570 0.742849 0.669459i \(-0.233474\pi\)
0.742849 + 0.669459i \(0.233474\pi\)
\(234\) 0 0
\(235\) −15.7995 −1.03064
\(236\) 0 0
\(237\) 2.59271 0.168415
\(238\) 0 0
\(239\) 17.7226 1.14638 0.573191 0.819422i \(-0.305705\pi\)
0.573191 + 0.819422i \(0.305705\pi\)
\(240\) 0 0
\(241\) 20.1713 1.29935 0.649675 0.760212i \(-0.274905\pi\)
0.649675 + 0.760212i \(0.274905\pi\)
\(242\) 0 0
\(243\) −7.97092 −0.511335
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0.976687 0.0621451
\(248\) 0 0
\(249\) 0.478802 0.0303429
\(250\) 0 0
\(251\) 16.5113 1.04219 0.521093 0.853500i \(-0.325524\pi\)
0.521093 + 0.853500i \(0.325524\pi\)
\(252\) 0 0
\(253\) 1.03489 0.0650632
\(254\) 0 0
\(255\) 1.63436 0.102347
\(256\) 0 0
\(257\) −17.1369 −1.06897 −0.534485 0.845178i \(-0.679494\pi\)
−0.534485 + 0.845178i \(0.679494\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 5.46878 0.338509
\(262\) 0 0
\(263\) −17.2443 −1.06333 −0.531665 0.846955i \(-0.678433\pi\)
−0.531665 + 0.846955i \(0.678433\pi\)
\(264\) 0 0
\(265\) −5.38819 −0.330994
\(266\) 0 0
\(267\) 4.28947 0.262511
\(268\) 0 0
\(269\) 14.5073 0.884524 0.442262 0.896886i \(-0.354176\pi\)
0.442262 + 0.896886i \(0.354176\pi\)
\(270\) 0 0
\(271\) 23.8779 1.45048 0.725240 0.688496i \(-0.241729\pi\)
0.725240 + 0.688496i \(0.241729\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 8.94936 0.539667
\(276\) 0 0
\(277\) −15.6351 −0.939422 −0.469711 0.882820i \(-0.655642\pi\)
−0.469711 + 0.882820i \(0.655642\pi\)
\(278\) 0 0
\(279\) 13.6947 0.819881
\(280\) 0 0
\(281\) 5.17005 0.308419 0.154210 0.988038i \(-0.450717\pi\)
0.154210 + 0.988038i \(0.450717\pi\)
\(282\) 0 0
\(283\) 10.0611 0.598071 0.299035 0.954242i \(-0.403335\pi\)
0.299035 + 0.954242i \(0.403335\pi\)
\(284\) 0 0
\(285\) 0.895635 0.0530528
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −14.4183 −0.848135
\(290\) 0 0
\(291\) 0.185191 0.0108561
\(292\) 0 0
\(293\) 9.32341 0.544680 0.272340 0.962201i \(-0.412203\pi\)
0.272340 + 0.962201i \(0.412203\pi\)
\(294\) 0 0
\(295\) −3.33441 −0.194137
\(296\) 0 0
\(297\) 2.76066 0.160190
\(298\) 0 0
\(299\) −0.756554 −0.0437526
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 1.16906 0.0671610
\(304\) 0 0
\(305\) 1.25669 0.0719578
\(306\) 0 0
\(307\) 10.3822 0.592544 0.296272 0.955104i \(-0.404257\pi\)
0.296272 + 0.955104i \(0.404257\pi\)
\(308\) 0 0
\(309\) 0.160224 0.00911480
\(310\) 0 0
\(311\) −31.7534 −1.80057 −0.900284 0.435302i \(-0.856642\pi\)
−0.900284 + 0.435302i \(0.856642\pi\)
\(312\) 0 0
\(313\) 24.5198 1.38594 0.692971 0.720966i \(-0.256301\pi\)
0.692971 + 0.720966i \(0.256301\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −7.89454 −0.443402 −0.221701 0.975115i \(-0.571161\pi\)
−0.221701 + 0.975115i \(0.571161\pi\)
\(318\) 0 0
\(319\) −2.85633 −0.159924
\(320\) 0 0
\(321\) 2.88206 0.160861
\(322\) 0 0
\(323\) 1.41479 0.0787211
\(324\) 0 0
\(325\) −6.54238 −0.362906
\(326\) 0 0
\(327\) −1.61868 −0.0895135
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −14.1270 −0.776489 −0.388244 0.921557i \(-0.626918\pi\)
−0.388244 + 0.921557i \(0.626918\pi\)
\(332\) 0 0
\(333\) 9.35824 0.512829
\(334\) 0 0
\(335\) 28.0176 1.53076
\(336\) 0 0
\(337\) 16.5644 0.902319 0.451160 0.892443i \(-0.351011\pi\)
0.451160 + 0.892443i \(0.351011\pi\)
\(338\) 0 0
\(339\) 0.291388 0.0158260
\(340\) 0 0
\(341\) −7.15271 −0.387341
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −0.693769 −0.0373513
\(346\) 0 0
\(347\) −13.1015 −0.703328 −0.351664 0.936126i \(-0.614384\pi\)
−0.351664 + 0.936126i \(0.614384\pi\)
\(348\) 0 0
\(349\) −22.0563 −1.18065 −0.590324 0.807167i \(-0.701000\pi\)
−0.590324 + 0.807167i \(0.701000\pi\)
\(350\) 0 0
\(351\) −2.01817 −0.107722
\(352\) 0 0
\(353\) −20.6032 −1.09660 −0.548298 0.836283i \(-0.684724\pi\)
−0.548298 + 0.836283i \(0.684724\pi\)
\(354\) 0 0
\(355\) 26.1107 1.38581
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −15.1830 −0.801328 −0.400664 0.916225i \(-0.631220\pi\)
−0.400664 + 0.916225i \(0.631220\pi\)
\(360\) 0 0
\(361\) −18.2247 −0.959194
\(362\) 0 0
\(363\) 2.67993 0.140660
\(364\) 0 0
\(365\) −5.16515 −0.270356
\(366\) 0 0
\(367\) −1.02831 −0.0536773 −0.0268386 0.999640i \(-0.508544\pi\)
−0.0268386 + 0.999640i \(0.508544\pi\)
\(368\) 0 0
\(369\) 2.90506 0.151232
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −25.0723 −1.29820 −0.649098 0.760704i \(-0.724854\pi\)
−0.649098 + 0.760704i \(0.724854\pi\)
\(374\) 0 0
\(375\) −0.913607 −0.0471785
\(376\) 0 0
\(377\) 2.08810 0.107543
\(378\) 0 0
\(379\) 7.20297 0.369992 0.184996 0.982739i \(-0.440773\pi\)
0.184996 + 0.982739i \(0.440773\pi\)
\(380\) 0 0
\(381\) −3.57311 −0.183056
\(382\) 0 0
\(383\) 10.9882 0.561469 0.280734 0.959786i \(-0.409422\pi\)
0.280734 + 0.959786i \(0.409422\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 27.8106 1.41369
\(388\) 0 0
\(389\) 6.82819 0.346203 0.173102 0.984904i \(-0.444621\pi\)
0.173102 + 0.984904i \(0.444621\pi\)
\(390\) 0 0
\(391\) −1.09591 −0.0554227
\(392\) 0 0
\(393\) −2.47093 −0.124642
\(394\) 0 0
\(395\) −27.7789 −1.39771
\(396\) 0 0
\(397\) −1.82458 −0.0915733 −0.0457866 0.998951i \(-0.514579\pi\)
−0.0457866 + 0.998951i \(0.514579\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0.510011 0.0254687 0.0127344 0.999919i \(-0.495946\pi\)
0.0127344 + 0.999919i \(0.495946\pi\)
\(402\) 0 0
\(403\) 5.22895 0.260473
\(404\) 0 0
\(405\) 26.9203 1.33768
\(406\) 0 0
\(407\) −4.88778 −0.242278
\(408\) 0 0
\(409\) −23.4467 −1.15936 −0.579681 0.814843i \(-0.696823\pi\)
−0.579681 + 0.814843i \(0.696823\pi\)
\(410\) 0 0
\(411\) 0.251801 0.0124204
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −5.13001 −0.251822
\(416\) 0 0
\(417\) 0.805811 0.0394607
\(418\) 0 0
\(419\) 38.8743 1.89913 0.949566 0.313567i \(-0.101524\pi\)
0.949566 + 0.313567i \(0.101524\pi\)
\(420\) 0 0
\(421\) 3.62394 0.176620 0.0883101 0.996093i \(-0.471853\pi\)
0.0883101 + 0.996093i \(0.471853\pi\)
\(422\) 0 0
\(423\) 13.9034 0.676006
\(424\) 0 0
\(425\) −9.47703 −0.459704
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0.518568 0.0250367
\(430\) 0 0
\(431\) −8.37794 −0.403551 −0.201776 0.979432i \(-0.564671\pi\)
−0.201776 + 0.979432i \(0.564671\pi\)
\(432\) 0 0
\(433\) 5.27478 0.253490 0.126745 0.991935i \(-0.459547\pi\)
0.126745 + 0.991935i \(0.459547\pi\)
\(434\) 0 0
\(435\) 1.91482 0.0918085
\(436\) 0 0
\(437\) −0.600566 −0.0287290
\(438\) 0 0
\(439\) −28.5702 −1.36358 −0.681790 0.731548i \(-0.738798\pi\)
−0.681790 + 0.731548i \(0.738798\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −12.3310 −0.585863 −0.292932 0.956133i \(-0.594631\pi\)
−0.292932 + 0.956133i \(0.594631\pi\)
\(444\) 0 0
\(445\) −45.9584 −2.17864
\(446\) 0 0
\(447\) 4.93550 0.233441
\(448\) 0 0
\(449\) −36.7971 −1.73656 −0.868280 0.496074i \(-0.834775\pi\)
−0.868280 + 0.496074i \(0.834775\pi\)
\(450\) 0 0
\(451\) −1.51731 −0.0714472
\(452\) 0 0
\(453\) 2.98203 0.140108
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 4.46379 0.208807 0.104404 0.994535i \(-0.466707\pi\)
0.104404 + 0.994535i \(0.466707\pi\)
\(458\) 0 0
\(459\) −2.92344 −0.136454
\(460\) 0 0
\(461\) −10.8591 −0.505757 −0.252878 0.967498i \(-0.581377\pi\)
−0.252878 + 0.967498i \(0.581377\pi\)
\(462\) 0 0
\(463\) 17.7680 0.825747 0.412874 0.910788i \(-0.364525\pi\)
0.412874 + 0.910788i \(0.364525\pi\)
\(464\) 0 0
\(465\) 4.79502 0.222364
\(466\) 0 0
\(467\) 3.65317 0.169048 0.0845242 0.996421i \(-0.473063\pi\)
0.0845242 + 0.996421i \(0.473063\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 2.70545 0.124661
\(472\) 0 0
\(473\) −14.5254 −0.667878
\(474\) 0 0
\(475\) −5.19346 −0.238292
\(476\) 0 0
\(477\) 4.74156 0.217101
\(478\) 0 0
\(479\) −1.84559 −0.0843270 −0.0421635 0.999111i \(-0.513425\pi\)
−0.0421635 + 0.999111i \(0.513425\pi\)
\(480\) 0 0
\(481\) 3.57319 0.162923
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −1.98418 −0.0900970
\(486\) 0 0
\(487\) 6.98021 0.316304 0.158152 0.987415i \(-0.449446\pi\)
0.158152 + 0.987415i \(0.449446\pi\)
\(488\) 0 0
\(489\) 3.57319 0.161585
\(490\) 0 0
\(491\) 4.53099 0.204481 0.102240 0.994760i \(-0.467399\pi\)
0.102240 + 0.994760i \(0.467399\pi\)
\(492\) 0 0
\(493\) 3.02475 0.136228
\(494\) 0 0
\(495\) −14.5514 −0.654039
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 28.8012 1.28932 0.644659 0.764471i \(-0.277000\pi\)
0.644659 + 0.764471i \(0.277000\pi\)
\(500\) 0 0
\(501\) −0.995257 −0.0444648
\(502\) 0 0
\(503\) 12.0511 0.537330 0.268665 0.963234i \(-0.413417\pi\)
0.268665 + 0.963234i \(0.413417\pi\)
\(504\) 0 0
\(505\) −12.5256 −0.557384
\(506\) 0 0
\(507\) 3.62642 0.161055
\(508\) 0 0
\(509\) −15.2708 −0.676868 −0.338434 0.940990i \(-0.609897\pi\)
−0.338434 + 0.940990i \(0.609897\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −1.60206 −0.0707326
\(514\) 0 0
\(515\) −1.71668 −0.0756458
\(516\) 0 0
\(517\) −7.26170 −0.319369
\(518\) 0 0
\(519\) 3.03720 0.133318
\(520\) 0 0
\(521\) −22.1754 −0.971522 −0.485761 0.874092i \(-0.661458\pi\)
−0.485761 + 0.874092i \(0.661458\pi\)
\(522\) 0 0
\(523\) 34.0779 1.49012 0.745062 0.666995i \(-0.232420\pi\)
0.745062 + 0.666995i \(0.232420\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 7.57445 0.329948
\(528\) 0 0
\(529\) −22.5348 −0.979774
\(530\) 0 0
\(531\) 2.93425 0.127336
\(532\) 0 0
\(533\) 1.10922 0.0480456
\(534\) 0 0
\(535\) −30.8791 −1.33502
\(536\) 0 0
\(537\) 6.19146 0.267181
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −29.3752 −1.26294 −0.631470 0.775400i \(-0.717548\pi\)
−0.631470 + 0.775400i \(0.717548\pi\)
\(542\) 0 0
\(543\) 1.11193 0.0477176
\(544\) 0 0
\(545\) 17.3430 0.742892
\(546\) 0 0
\(547\) −4.60982 −0.197102 −0.0985509 0.995132i \(-0.531421\pi\)
−0.0985509 + 0.995132i \(0.531421\pi\)
\(548\) 0 0
\(549\) −1.10588 −0.0471976
\(550\) 0 0
\(551\) 1.65758 0.0706151
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 3.27666 0.139086
\(556\) 0 0
\(557\) −1.15369 −0.0488836 −0.0244418 0.999701i \(-0.507781\pi\)
−0.0244418 + 0.999701i \(0.507781\pi\)
\(558\) 0 0
\(559\) 10.6187 0.449124
\(560\) 0 0
\(561\) 0.751178 0.0317147
\(562\) 0 0
\(563\) 0.652565 0.0275024 0.0137512 0.999905i \(-0.495623\pi\)
0.0137512 + 0.999905i \(0.495623\pi\)
\(564\) 0 0
\(565\) −3.12200 −0.131343
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −4.56467 −0.191361 −0.0956804 0.995412i \(-0.530503\pi\)
−0.0956804 + 0.995412i \(0.530503\pi\)
\(570\) 0 0
\(571\) 32.6522 1.36645 0.683226 0.730207i \(-0.260577\pi\)
0.683226 + 0.730207i \(0.260577\pi\)
\(572\) 0 0
\(573\) 6.02118 0.251538
\(574\) 0 0
\(575\) 4.02292 0.167767
\(576\) 0 0
\(577\) 15.9542 0.664183 0.332091 0.943247i \(-0.392246\pi\)
0.332091 + 0.943247i \(0.392246\pi\)
\(578\) 0 0
\(579\) 8.30809 0.345273
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −2.47650 −0.102566
\(584\) 0 0
\(585\) 10.6378 0.439817
\(586\) 0 0
\(587\) −7.90734 −0.326371 −0.163185 0.986595i \(-0.552177\pi\)
−0.163185 + 0.986595i \(0.552177\pi\)
\(588\) 0 0
\(589\) 4.15084 0.171032
\(590\) 0 0
\(591\) −3.26242 −0.134198
\(592\) 0 0
\(593\) −27.1920 −1.11664 −0.558320 0.829625i \(-0.688554\pi\)
−0.558320 + 0.829625i \(0.688554\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −6.11790 −0.250389
\(598\) 0 0
\(599\) 24.0880 0.984211 0.492105 0.870536i \(-0.336228\pi\)
0.492105 + 0.870536i \(0.336228\pi\)
\(600\) 0 0
\(601\) 13.6546 0.556984 0.278492 0.960439i \(-0.410165\pi\)
0.278492 + 0.960439i \(0.410165\pi\)
\(602\) 0 0
\(603\) −24.6552 −1.00404
\(604\) 0 0
\(605\) −28.7135 −1.16737
\(606\) 0 0
\(607\) −15.3900 −0.624662 −0.312331 0.949973i \(-0.601110\pi\)
−0.312331 + 0.949973i \(0.601110\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 5.30863 0.214764
\(612\) 0 0
\(613\) 10.7887 0.435753 0.217876 0.975976i \(-0.430087\pi\)
0.217876 + 0.975976i \(0.430087\pi\)
\(614\) 0 0
\(615\) 1.01717 0.0410162
\(616\) 0 0
\(617\) −26.2621 −1.05727 −0.528637 0.848848i \(-0.677297\pi\)
−0.528637 + 0.848848i \(0.677297\pi\)
\(618\) 0 0
\(619\) −16.0257 −0.644127 −0.322063 0.946718i \(-0.604376\pi\)
−0.322063 + 0.946718i \(0.604376\pi\)
\(620\) 0 0
\(621\) 1.24097 0.0497986
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −19.7023 −0.788093
\(626\) 0 0
\(627\) 0.411649 0.0164397
\(628\) 0 0
\(629\) 5.17598 0.206380
\(630\) 0 0
\(631\) −3.37257 −0.134260 −0.0671299 0.997744i \(-0.521384\pi\)
−0.0671299 + 0.997744i \(0.521384\pi\)
\(632\) 0 0
\(633\) −2.78958 −0.110876
\(634\) 0 0
\(635\) 38.2832 1.51922
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −22.9772 −0.908965
\(640\) 0 0
\(641\) 35.4637 1.40073 0.700365 0.713785i \(-0.253021\pi\)
0.700365 + 0.713785i \(0.253021\pi\)
\(642\) 0 0
\(643\) −30.4094 −1.19923 −0.599615 0.800288i \(-0.704680\pi\)
−0.599615 + 0.800288i \(0.704680\pi\)
\(644\) 0 0
\(645\) 9.73750 0.383414
\(646\) 0 0
\(647\) −30.5807 −1.20225 −0.601126 0.799154i \(-0.705281\pi\)
−0.601126 + 0.799154i \(0.705281\pi\)
\(648\) 0 0
\(649\) −1.53255 −0.0601579
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −47.1664 −1.84576 −0.922882 0.385083i \(-0.874173\pi\)
−0.922882 + 0.385083i \(0.874173\pi\)
\(654\) 0 0
\(655\) 26.4742 1.03443
\(656\) 0 0
\(657\) 4.54529 0.177329
\(658\) 0 0
\(659\) 2.62840 0.102388 0.0511940 0.998689i \(-0.483697\pi\)
0.0511940 + 0.998689i \(0.483697\pi\)
\(660\) 0 0
\(661\) −23.7133 −0.922341 −0.461171 0.887311i \(-0.652570\pi\)
−0.461171 + 0.887311i \(0.652570\pi\)
\(662\) 0 0
\(663\) −0.549144 −0.0213270
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −1.28398 −0.0497158
\(668\) 0 0
\(669\) −3.52256 −0.136190
\(670\) 0 0
\(671\) 0.577595 0.0222978
\(672\) 0 0
\(673\) −22.4850 −0.866734 −0.433367 0.901218i \(-0.642675\pi\)
−0.433367 + 0.901218i \(0.642675\pi\)
\(674\) 0 0
\(675\) 10.7315 0.413054
\(676\) 0 0
\(677\) −25.4832 −0.979397 −0.489699 0.871892i \(-0.662893\pi\)
−0.489699 + 0.871892i \(0.662893\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −6.56221 −0.251464
\(682\) 0 0
\(683\) 10.0754 0.385525 0.192762 0.981245i \(-0.438255\pi\)
0.192762 + 0.981245i \(0.438255\pi\)
\(684\) 0 0
\(685\) −2.69786 −0.103080
\(686\) 0 0
\(687\) −6.50518 −0.248188
\(688\) 0 0
\(689\) 1.81043 0.0689721
\(690\) 0 0
\(691\) 21.0068 0.799138 0.399569 0.916703i \(-0.369160\pi\)
0.399569 + 0.916703i \(0.369160\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −8.63366 −0.327493
\(696\) 0 0
\(697\) 1.60677 0.0608608
\(698\) 0 0
\(699\) −6.98754 −0.264293
\(700\) 0 0
\(701\) 48.8524 1.84513 0.922564 0.385844i \(-0.126090\pi\)
0.922564 + 0.385844i \(0.126090\pi\)
\(702\) 0 0
\(703\) 2.83646 0.106979
\(704\) 0 0
\(705\) 4.86808 0.183342
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 41.0714 1.54247 0.771234 0.636551i \(-0.219640\pi\)
0.771234 + 0.636551i \(0.219640\pi\)
\(710\) 0 0
\(711\) 24.4452 0.916768
\(712\) 0 0
\(713\) −3.21529 −0.120414
\(714\) 0 0
\(715\) −5.55607 −0.207785
\(716\) 0 0
\(717\) −5.46065 −0.203932
\(718\) 0 0
\(719\) −35.1717 −1.31168 −0.655841 0.754899i \(-0.727686\pi\)
−0.655841 + 0.754899i \(0.727686\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −6.21513 −0.231143
\(724\) 0 0
\(725\) −11.1033 −0.412368
\(726\) 0 0
\(727\) 31.0287 1.15079 0.575396 0.817875i \(-0.304848\pi\)
0.575396 + 0.817875i \(0.304848\pi\)
\(728\) 0 0
\(729\) −22.0078 −0.815103
\(730\) 0 0
\(731\) 15.3818 0.568918
\(732\) 0 0
\(733\) −2.72105 −0.100504 −0.0502522 0.998737i \(-0.516003\pi\)
−0.0502522 + 0.998737i \(0.516003\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 12.8774 0.474344
\(738\) 0 0
\(739\) 21.0421 0.774045 0.387022 0.922070i \(-0.373504\pi\)
0.387022 + 0.922070i \(0.373504\pi\)
\(740\) 0 0
\(741\) −0.300934 −0.0110551
\(742\) 0 0
\(743\) 30.2470 1.10965 0.554827 0.831966i \(-0.312785\pi\)
0.554827 + 0.831966i \(0.312785\pi\)
\(744\) 0 0
\(745\) −52.8802 −1.93738
\(746\) 0 0
\(747\) 4.51436 0.165172
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −47.0853 −1.71817 −0.859083 0.511836i \(-0.828965\pi\)
−0.859083 + 0.511836i \(0.828965\pi\)
\(752\) 0 0
\(753\) −5.08742 −0.185396
\(754\) 0 0
\(755\) −31.9502 −1.16279
\(756\) 0 0
\(757\) −20.1878 −0.733738 −0.366869 0.930273i \(-0.619570\pi\)
−0.366869 + 0.930273i \(0.619570\pi\)
\(758\) 0 0
\(759\) −0.318868 −0.0115742
\(760\) 0 0
\(761\) −51.2177 −1.85664 −0.928320 0.371782i \(-0.878747\pi\)
−0.928320 + 0.371782i \(0.878747\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 15.4094 0.557129
\(766\) 0 0
\(767\) 1.12036 0.0404540
\(768\) 0 0
\(769\) −0.804472 −0.0290100 −0.0145050 0.999895i \(-0.504617\pi\)
−0.0145050 + 0.999895i \(0.504617\pi\)
\(770\) 0 0
\(771\) 5.28017 0.190160
\(772\) 0 0
\(773\) 40.0107 1.43909 0.719543 0.694448i \(-0.244351\pi\)
0.719543 + 0.694448i \(0.244351\pi\)
\(774\) 0 0
\(775\) −27.8046 −0.998769
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0.880518 0.0315478
\(780\) 0 0
\(781\) 12.0009 0.429427
\(782\) 0 0
\(783\) −3.42511 −0.122404
\(784\) 0 0
\(785\) −28.9869 −1.03459
\(786\) 0 0
\(787\) 45.1844 1.61065 0.805326 0.592833i \(-0.201990\pi\)
0.805326 + 0.592833i \(0.201990\pi\)
\(788\) 0 0
\(789\) 5.31326 0.189157
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −0.422248 −0.0149945
\(794\) 0 0
\(795\) 1.66019 0.0588809
\(796\) 0 0
\(797\) −42.7382 −1.51386 −0.756932 0.653494i \(-0.773303\pi\)
−0.756932 + 0.653494i \(0.773303\pi\)
\(798\) 0 0
\(799\) 7.68987 0.272048
\(800\) 0 0
\(801\) 40.4430 1.42898
\(802\) 0 0
\(803\) −2.37399 −0.0837763
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −4.46993 −0.157349
\(808\) 0 0
\(809\) 13.0164 0.457632 0.228816 0.973470i \(-0.426515\pi\)
0.228816 + 0.973470i \(0.426515\pi\)
\(810\) 0 0
\(811\) 21.9400 0.770418 0.385209 0.922829i \(-0.374129\pi\)
0.385209 + 0.922829i \(0.374129\pi\)
\(812\) 0 0
\(813\) −7.35719 −0.258028
\(814\) 0 0
\(815\) −38.2840 −1.34103
\(816\) 0 0
\(817\) 8.42933 0.294905
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −27.5081 −0.960040 −0.480020 0.877257i \(-0.659371\pi\)
−0.480020 + 0.877257i \(0.659371\pi\)
\(822\) 0 0
\(823\) −29.8968 −1.04214 −0.521068 0.853515i \(-0.674466\pi\)
−0.521068 + 0.853515i \(0.674466\pi\)
\(824\) 0 0
\(825\) −2.75745 −0.0960020
\(826\) 0 0
\(827\) 8.18193 0.284513 0.142257 0.989830i \(-0.454564\pi\)
0.142257 + 0.989830i \(0.454564\pi\)
\(828\) 0 0
\(829\) 56.3549 1.95729 0.978643 0.205565i \(-0.0659033\pi\)
0.978643 + 0.205565i \(0.0659033\pi\)
\(830\) 0 0
\(831\) 4.81744 0.167115
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 10.6634 0.369023
\(836\) 0 0
\(837\) −8.57704 −0.296466
\(838\) 0 0
\(839\) −26.0609 −0.899724 −0.449862 0.893098i \(-0.648527\pi\)
−0.449862 + 0.893098i \(0.648527\pi\)
\(840\) 0 0
\(841\) −25.4562 −0.877800
\(842\) 0 0
\(843\) −1.59298 −0.0548651
\(844\) 0 0
\(845\) −38.8544 −1.33663
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −3.10000 −0.106392
\(850\) 0 0
\(851\) −2.19716 −0.0753176
\(852\) 0 0
\(853\) −34.0375 −1.16542 −0.582712 0.812679i \(-0.698008\pi\)
−0.582712 + 0.812679i \(0.698008\pi\)
\(854\) 0 0
\(855\) 8.44445 0.288794
\(856\) 0 0
\(857\) 40.0483 1.36802 0.684011 0.729471i \(-0.260234\pi\)
0.684011 + 0.729471i \(0.260234\pi\)
\(858\) 0 0
\(859\) 25.4840 0.869504 0.434752 0.900550i \(-0.356836\pi\)
0.434752 + 0.900550i \(0.356836\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −0.747306 −0.0254386 −0.0127193 0.999919i \(-0.504049\pi\)
−0.0127193 + 0.999919i \(0.504049\pi\)
\(864\) 0 0
\(865\) −32.5413 −1.10644
\(866\) 0 0
\(867\) 4.44252 0.150876
\(868\) 0 0
\(869\) −12.7677 −0.433114
\(870\) 0 0
\(871\) −9.41393 −0.318979
\(872\) 0 0
\(873\) 1.74606 0.0590952
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 16.5402 0.558523 0.279262 0.960215i \(-0.409910\pi\)
0.279262 + 0.960215i \(0.409910\pi\)
\(878\) 0 0
\(879\) −2.87270 −0.0968938
\(880\) 0 0
\(881\) 50.7950 1.71133 0.855663 0.517533i \(-0.173150\pi\)
0.855663 + 0.517533i \(0.173150\pi\)
\(882\) 0 0
\(883\) −26.1612 −0.880394 −0.440197 0.897901i \(-0.645091\pi\)
−0.440197 + 0.897901i \(0.645091\pi\)
\(884\) 0 0
\(885\) 1.02739 0.0345353
\(886\) 0 0
\(887\) −35.5282 −1.19292 −0.596461 0.802642i \(-0.703427\pi\)
−0.596461 + 0.802642i \(0.703427\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 12.3730 0.414511
\(892\) 0 0
\(893\) 4.21409 0.141019
\(894\) 0 0
\(895\) −66.3368 −2.21740
\(896\) 0 0
\(897\) 0.233107 0.00778321
\(898\) 0 0
\(899\) 8.87427 0.295973
\(900\) 0 0
\(901\) 2.62252 0.0873690
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −11.9135 −0.396019
\(906\) 0 0
\(907\) 29.0487 0.964545 0.482273 0.876021i \(-0.339811\pi\)
0.482273 + 0.876021i \(0.339811\pi\)
\(908\) 0 0
\(909\) 11.0225 0.365592
\(910\) 0 0
\(911\) 10.5596 0.349855 0.174927 0.984581i \(-0.444031\pi\)
0.174927 + 0.984581i \(0.444031\pi\)
\(912\) 0 0
\(913\) −2.35784 −0.0780331
\(914\) 0 0
\(915\) −0.387207 −0.0128007
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 15.5083 0.511572 0.255786 0.966733i \(-0.417666\pi\)
0.255786 + 0.966733i \(0.417666\pi\)
\(920\) 0 0
\(921\) −3.19893 −0.105408
\(922\) 0 0
\(923\) −8.77322 −0.288774
\(924\) 0 0
\(925\) −19.0002 −0.624721
\(926\) 0 0
\(927\) 1.51066 0.0496166
\(928\) 0 0
\(929\) −5.10650 −0.167539 −0.0837694 0.996485i \(-0.526696\pi\)
−0.0837694 + 0.996485i \(0.526696\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 9.78375 0.320306
\(934\) 0 0
\(935\) −8.04830 −0.263208
\(936\) 0 0
\(937\) 6.24701 0.204081 0.102040 0.994780i \(-0.467463\pi\)
0.102040 + 0.994780i \(0.467463\pi\)
\(938\) 0 0
\(939\) −7.55497 −0.246547
\(940\) 0 0
\(941\) 30.3734 0.990145 0.495073 0.868852i \(-0.335141\pi\)
0.495073 + 0.868852i \(0.335141\pi\)
\(942\) 0 0
\(943\) −0.682060 −0.0222109
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 42.1003 1.36808 0.684038 0.729446i \(-0.260222\pi\)
0.684038 + 0.729446i \(0.260222\pi\)
\(948\) 0 0
\(949\) 1.73549 0.0563365
\(950\) 0 0
\(951\) 2.43244 0.0788773
\(952\) 0 0
\(953\) −48.9720 −1.58636 −0.793180 0.608988i \(-0.791576\pi\)
−0.793180 + 0.608988i \(0.791576\pi\)
\(954\) 0 0
\(955\) −64.5124 −2.08757
\(956\) 0 0
\(957\) 0.880083 0.0284491
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −8.77739 −0.283142
\(962\) 0 0
\(963\) 27.1733 0.875648
\(964\) 0 0
\(965\) −89.0150 −2.86549
\(966\) 0 0
\(967\) 33.8769 1.08941 0.544704 0.838628i \(-0.316642\pi\)
0.544704 + 0.838628i \(0.316642\pi\)
\(968\) 0 0
\(969\) −0.435921 −0.0140038
\(970\) 0 0
\(971\) 0.943295 0.0302718 0.0151359 0.999885i \(-0.495182\pi\)
0.0151359 + 0.999885i \(0.495182\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 2.01582 0.0645578
\(976\) 0 0
\(977\) −17.1014 −0.547124 −0.273562 0.961854i \(-0.588202\pi\)
−0.273562 + 0.961854i \(0.588202\pi\)
\(978\) 0 0
\(979\) −21.1233 −0.675103
\(980\) 0 0
\(981\) −15.2617 −0.487268
\(982\) 0 0
\(983\) −1.88552 −0.0601388 −0.0300694 0.999548i \(-0.509573\pi\)
−0.0300694 + 0.999548i \(0.509573\pi\)
\(984\) 0 0
\(985\) 34.9544 1.11374
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −6.52946 −0.207625
\(990\) 0 0
\(991\) 22.9252 0.728244 0.364122 0.931351i \(-0.381369\pi\)
0.364122 + 0.931351i \(0.381369\pi\)
\(992\) 0 0
\(993\) 4.35276 0.138131
\(994\) 0 0
\(995\) 65.5487 2.07803
\(996\) 0 0
\(997\) 48.2258 1.52733 0.763663 0.645615i \(-0.223399\pi\)
0.763663 + 0.645615i \(0.223399\pi\)
\(998\) 0 0
\(999\) −5.86110 −0.185437
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 8036.2.a.n.1.5 8
7.3 odd 6 1148.2.i.d.821.5 yes 16
7.5 odd 6 1148.2.i.d.165.5 16
7.6 odd 2 8036.2.a.m.1.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1148.2.i.d.165.5 16 7.5 odd 6
1148.2.i.d.821.5 yes 16 7.3 odd 6
8036.2.a.m.1.4 8 7.6 odd 2
8036.2.a.n.1.5 8 1.1 even 1 trivial