Properties

Label 8568.2.a.bj.1.2
Level $8568$
Weight $2$
Character 8568.1
Self dual yes
Analytic conductor $68.416$
Analytic rank $1$
Dimension $4$
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] = [8568,2,Mod(1,8568)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8568, 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("8568.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8568 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8568.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: \(68.4158244518\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.5225.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 8x^{2} + x + 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 952)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.26498\) of defining polynomial
Character \(\chi\) \(=\) 8568.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.163765 q^{5} -1.00000 q^{7} +O(q^{10})\) \(q-0.163765 q^{5} -1.00000 q^{7} -3.23607 q^{11} +2.47214 q^{13} -1.00000 q^{17} +1.47005 q^{19} -4.85748 q^{23} -4.97318 q^{25} -0.378584 q^{29} +8.78727 q^{31} +0.163765 q^{35} +8.56569 q^{37} -5.97318 q^{41} +10.0310 q^{43} +8.18710 q^{47} +1.00000 q^{49} +8.72945 q^{53} +0.529954 q^{55} -2.85748 q^{59} -13.2671 q^{61} -0.404849 q^{65} +14.9319 q^{67} -14.3317 q^{71} -4.91327 q^{73} +3.23607 q^{77} -16.1871 q^{79} +5.65715 q^{83} +0.163765 q^{85} -13.5167 q^{89} -2.47214 q^{91} -0.240742 q^{95} -12.0744 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{5} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{5} - 4 q^{7} - 4 q^{11} - 8 q^{13} - 4 q^{17} + 14 q^{19} - 8 q^{23} + 11 q^{25} - 4 q^{29} + 5 q^{31} - q^{35} - 4 q^{37} + 7 q^{41} + 19 q^{43} - 8 q^{47} + 4 q^{49} - 5 q^{53} - 6 q^{55} - 23 q^{61} + 8 q^{65} + 15 q^{67} - 2 q^{71} - 5 q^{73} + 4 q^{77} - 24 q^{79} - 10 q^{83} - q^{85} + 16 q^{89} + 8 q^{91} - 22 q^{95} - 15 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −0.163765 −0.0732379 −0.0366189 0.999329i \(-0.511659\pi\)
−0.0366189 + 0.999329i \(0.511659\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −3.23607 −0.975711 −0.487856 0.872924i \(-0.662221\pi\)
−0.487856 + 0.872924i \(0.662221\pi\)
\(12\) 0 0
\(13\) 2.47214 0.685647 0.342824 0.939400i \(-0.388617\pi\)
0.342824 + 0.939400i \(0.388617\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.00000 −0.242536
\(18\) 0 0
\(19\) 1.47005 0.337252 0.168626 0.985680i \(-0.446067\pi\)
0.168626 + 0.985680i \(0.446067\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4.85748 −1.01286 −0.506428 0.862282i \(-0.669034\pi\)
−0.506428 + 0.862282i \(0.669034\pi\)
\(24\) 0 0
\(25\) −4.97318 −0.994636
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −0.378584 −0.0703013 −0.0351506 0.999382i \(-0.511191\pi\)
−0.0351506 + 0.999382i \(0.511191\pi\)
\(30\) 0 0
\(31\) 8.78727 1.57824 0.789120 0.614239i \(-0.210537\pi\)
0.789120 + 0.614239i \(0.210537\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0.163765 0.0276813
\(36\) 0 0
\(37\) 8.56569 1.40819 0.704095 0.710106i \(-0.251353\pi\)
0.704095 + 0.710106i \(0.251353\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −5.97318 −0.932854 −0.466427 0.884560i \(-0.654459\pi\)
−0.466427 + 0.884560i \(0.654459\pi\)
\(42\) 0 0
\(43\) 10.0310 1.52971 0.764857 0.644201i \(-0.222810\pi\)
0.764857 + 0.644201i \(0.222810\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 8.18710 1.19421 0.597106 0.802162i \(-0.296317\pi\)
0.597106 + 0.802162i \(0.296317\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 8.72945 1.19908 0.599541 0.800344i \(-0.295350\pi\)
0.599541 + 0.800344i \(0.295350\pi\)
\(54\) 0 0
\(55\) 0.529954 0.0714590
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −2.85748 −0.372013 −0.186006 0.982549i \(-0.559555\pi\)
−0.186006 + 0.982549i \(0.559555\pi\)
\(60\) 0 0
\(61\) −13.2671 −1.69867 −0.849337 0.527851i \(-0.822998\pi\)
−0.849337 + 0.527851i \(0.822998\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −0.404849 −0.0502153
\(66\) 0 0
\(67\) 14.9319 1.82422 0.912110 0.409946i \(-0.134453\pi\)
0.912110 + 0.409946i \(0.134453\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −14.3317 −1.70086 −0.850431 0.526087i \(-0.823658\pi\)
−0.850431 + 0.526087i \(0.823658\pi\)
\(72\) 0 0
\(73\) −4.91327 −0.575055 −0.287528 0.957772i \(-0.592833\pi\)
−0.287528 + 0.957772i \(0.592833\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.23607 0.368784
\(78\) 0 0
\(79\) −16.1871 −1.82119 −0.910596 0.413298i \(-0.864377\pi\)
−0.910596 + 0.413298i \(0.864377\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 5.65715 0.620953 0.310476 0.950581i \(-0.399511\pi\)
0.310476 + 0.950581i \(0.399511\pi\)
\(84\) 0 0
\(85\) 0.163765 0.0177628
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −13.5167 −1.43277 −0.716385 0.697705i \(-0.754205\pi\)
−0.716385 + 0.697705i \(0.754205\pi\)
\(90\) 0 0
\(91\) −2.47214 −0.259150
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −0.240742 −0.0246996
\(96\) 0 0
\(97\) −12.0744 −1.22597 −0.612984 0.790095i \(-0.710031\pi\)
−0.612984 + 0.790095i \(0.710031\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −15.3916 −1.53152 −0.765762 0.643124i \(-0.777638\pi\)
−0.765762 + 0.643124i \(0.777638\pi\)
\(102\) 0 0
\(103\) 10.7997 1.06412 0.532061 0.846706i \(-0.321418\pi\)
0.532061 + 0.846706i \(0.321418\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −9.27857 −0.896993 −0.448496 0.893785i \(-0.648040\pi\)
−0.448496 + 0.893785i \(0.648040\pi\)
\(108\) 0 0
\(109\) −2.90436 −0.278187 −0.139094 0.990279i \(-0.544419\pi\)
−0.139094 + 0.990279i \(0.544419\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −8.67456 −0.816034 −0.408017 0.912974i \(-0.633780\pi\)
−0.408017 + 0.912974i \(0.633780\pi\)
\(114\) 0 0
\(115\) 0.795485 0.0741794
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 1.00000 0.0916698
\(120\) 0 0
\(121\) −0.527864 −0.0479876
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.63326 0.146083
\(126\) 0 0
\(127\) 6.19950 0.550117 0.275058 0.961428i \(-0.411303\pi\)
0.275058 + 0.961428i \(0.411303\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 8.23816 0.719771 0.359886 0.932996i \(-0.382816\pi\)
0.359886 + 0.932996i \(0.382816\pi\)
\(132\) 0 0
\(133\) −1.47005 −0.127469
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −2.02682 −0.173163 −0.0865814 0.996245i \(-0.527594\pi\)
−0.0865814 + 0.996245i \(0.527594\pi\)
\(138\) 0 0
\(139\) −16.6979 −1.41630 −0.708149 0.706063i \(-0.750469\pi\)
−0.708149 + 0.706063i \(0.750469\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −8.00000 −0.668994
\(144\) 0 0
\(145\) 0.0619988 0.00514872
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 13.1004 1.07322 0.536612 0.843829i \(-0.319704\pi\)
0.536612 + 0.843829i \(0.319704\pi\)
\(150\) 0 0
\(151\) 12.5031 1.01749 0.508745 0.860917i \(-0.330109\pi\)
0.508745 + 0.860917i \(0.330109\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −1.43905 −0.115587
\(156\) 0 0
\(157\) −16.7171 −1.33417 −0.667083 0.744983i \(-0.732457\pi\)
−0.667083 + 0.744983i \(0.732457\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 4.85748 0.382823
\(162\) 0 0
\(163\) −11.0089 −0.862280 −0.431140 0.902285i \(-0.641888\pi\)
−0.431140 + 0.902285i \(0.641888\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 4.97109 0.384675 0.192337 0.981329i \(-0.438393\pi\)
0.192337 + 0.981329i \(0.438393\pi\)
\(168\) 0 0
\(169\) −6.88854 −0.529888
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 17.4542 1.32702 0.663508 0.748169i \(-0.269067\pi\)
0.663508 + 0.748169i \(0.269067\pi\)
\(174\) 0 0
\(175\) 4.97318 0.375937
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 2.38332 0.178138 0.0890688 0.996025i \(-0.471611\pi\)
0.0890688 + 0.996025i \(0.471611\pi\)
\(180\) 0 0
\(181\) −24.7103 −1.83670 −0.918351 0.395767i \(-0.870479\pi\)
−0.918351 + 0.395767i \(0.870479\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −1.40276 −0.103133
\(186\) 0 0
\(187\) 3.23607 0.236645
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 20.3193 1.47025 0.735127 0.677929i \(-0.237122\pi\)
0.735127 + 0.677929i \(0.237122\pi\)
\(192\) 0 0
\(193\) 16.4568 1.18459 0.592294 0.805722i \(-0.298223\pi\)
0.592294 + 0.805722i \(0.298223\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 3.38067 0.240863 0.120432 0.992722i \(-0.461572\pi\)
0.120432 + 0.992722i \(0.461572\pi\)
\(198\) 0 0
\(199\) 5.14873 0.364984 0.182492 0.983207i \(-0.441584\pi\)
0.182492 + 0.983207i \(0.441584\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0.378584 0.0265714
\(204\) 0 0
\(205\) 0.978197 0.0683203
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −4.75717 −0.329060
\(210\) 0 0
\(211\) −11.6256 −0.800339 −0.400170 0.916441i \(-0.631049\pi\)
−0.400170 + 0.916441i \(0.631049\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −1.64273 −0.112033
\(216\) 0 0
\(217\) −8.78727 −0.596519
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −2.47214 −0.166294
\(222\) 0 0
\(223\) −11.6725 −0.781646 −0.390823 0.920466i \(-0.627810\pi\)
−0.390823 + 0.920466i \(0.627810\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −14.9056 −0.989320 −0.494660 0.869087i \(-0.664707\pi\)
−0.494660 + 0.869087i \(0.664707\pi\)
\(228\) 0 0
\(229\) −18.7997 −1.24232 −0.621158 0.783685i \(-0.713338\pi\)
−0.621158 + 0.783685i \(0.713338\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −16.7997 −1.10058 −0.550291 0.834973i \(-0.685483\pi\)
−0.550291 + 0.834973i \(0.685483\pi\)
\(234\) 0 0
\(235\) −1.34076 −0.0874615
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −16.5031 −1.06750 −0.533750 0.845643i \(-0.679217\pi\)
−0.533750 + 0.845643i \(0.679217\pi\)
\(240\) 0 0
\(241\) 19.1768 1.23529 0.617643 0.786458i \(-0.288088\pi\)
0.617643 + 0.786458i \(0.288088\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −0.163765 −0.0104626
\(246\) 0 0
\(247\) 3.63415 0.231236
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −21.8307 −1.37794 −0.688972 0.724788i \(-0.741938\pi\)
−0.688972 + 0.724788i \(0.741938\pi\)
\(252\) 0 0
\(253\) 15.7191 0.988254
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 21.5167 1.34218 0.671088 0.741378i \(-0.265827\pi\)
0.671088 + 0.741378i \(0.265827\pi\)
\(258\) 0 0
\(259\) −8.56569 −0.532246
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 5.71497 0.352400 0.176200 0.984354i \(-0.443619\pi\)
0.176200 + 0.984354i \(0.443619\pi\)
\(264\) 0 0
\(265\) −1.42958 −0.0878183
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 23.8375 1.45340 0.726699 0.686956i \(-0.241054\pi\)
0.726699 + 0.686956i \(0.241054\pi\)
\(270\) 0 0
\(271\) −27.9930 −1.70046 −0.850228 0.526415i \(-0.823536\pi\)
−0.850228 + 0.526415i \(0.823536\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 16.0936 0.970478
\(276\) 0 0
\(277\) 2.03155 0.122064 0.0610321 0.998136i \(-0.480561\pi\)
0.0610321 + 0.998136i \(0.480561\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −30.4444 −1.81616 −0.908081 0.418795i \(-0.862453\pi\)
−0.908081 + 0.418795i \(0.862453\pi\)
\(282\) 0 0
\(283\) −7.19482 −0.427688 −0.213844 0.976868i \(-0.568598\pi\)
−0.213844 + 0.976868i \(0.568598\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 5.97318 0.352586
\(288\) 0 0
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −22.6592 −1.32377 −0.661883 0.749607i \(-0.730243\pi\)
−0.661883 + 0.749607i \(0.730243\pi\)
\(294\) 0 0
\(295\) 0.467956 0.0272454
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −12.0084 −0.694461
\(300\) 0 0
\(301\) −10.0310 −0.578177
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 2.17268 0.124407
\(306\) 0 0
\(307\) 8.44735 0.482116 0.241058 0.970511i \(-0.422506\pi\)
0.241058 + 0.970511i \(0.422506\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0.203679 0.0115496 0.00577479 0.999983i \(-0.498162\pi\)
0.00577479 + 0.999983i \(0.498162\pi\)
\(312\) 0 0
\(313\) −2.60964 −0.147505 −0.0737527 0.997277i \(-0.523498\pi\)
−0.0737527 + 0.997277i \(0.523498\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −28.4253 −1.59652 −0.798261 0.602312i \(-0.794246\pi\)
−0.798261 + 0.602312i \(0.794246\pi\)
\(318\) 0 0
\(319\) 1.22512 0.0685937
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −1.47005 −0.0817955
\(324\) 0 0
\(325\) −12.2944 −0.681969
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −8.18710 −0.451370
\(330\) 0 0
\(331\) −28.8204 −1.58411 −0.792057 0.610447i \(-0.790990\pi\)
−0.792057 + 0.610447i \(0.790990\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −2.44532 −0.133602
\(336\) 0 0
\(337\) 6.95959 0.379113 0.189557 0.981870i \(-0.439295\pi\)
0.189557 + 0.981870i \(0.439295\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −28.4362 −1.53991
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 4.06876 0.218423 0.109211 0.994019i \(-0.465167\pi\)
0.109211 + 0.994019i \(0.465167\pi\)
\(348\) 0 0
\(349\) −3.58568 −0.191937 −0.0959686 0.995384i \(-0.530595\pi\)
−0.0959686 + 0.995384i \(0.530595\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −19.1219 −1.01776 −0.508878 0.860839i \(-0.669940\pi\)
−0.508878 + 0.860839i \(0.669940\pi\)
\(354\) 0 0
\(355\) 2.34703 0.124567
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 11.8047 0.623027 0.311514 0.950242i \(-0.399164\pi\)
0.311514 + 0.950242i \(0.399164\pi\)
\(360\) 0 0
\(361\) −16.8390 −0.886261
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0.804621 0.0421158
\(366\) 0 0
\(367\) 8.73035 0.455721 0.227860 0.973694i \(-0.426827\pi\)
0.227860 + 0.973694i \(0.426827\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −8.72945 −0.453211
\(372\) 0 0
\(373\) 20.0930 1.04038 0.520188 0.854052i \(-0.325862\pi\)
0.520188 + 0.854052i \(0.325862\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −0.935911 −0.0482019
\(378\) 0 0
\(379\) 16.5852 0.851924 0.425962 0.904741i \(-0.359936\pi\)
0.425962 + 0.904741i \(0.359936\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −13.5745 −0.693627 −0.346813 0.937934i \(-0.612736\pi\)
−0.346813 + 0.937934i \(0.612736\pi\)
\(384\) 0 0
\(385\) −0.529954 −0.0270090
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −9.46183 −0.479734 −0.239867 0.970806i \(-0.577104\pi\)
−0.239867 + 0.970806i \(0.577104\pi\)
\(390\) 0 0
\(391\) 4.85748 0.245654
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 2.65088 0.133380
\(396\) 0 0
\(397\) −13.7383 −0.689506 −0.344753 0.938693i \(-0.612037\pi\)
−0.344753 + 0.938693i \(0.612037\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 1.20451 0.0601506 0.0300753 0.999548i \(-0.490425\pi\)
0.0300753 + 0.999548i \(0.490425\pi\)
\(402\) 0 0
\(403\) 21.7233 1.08212
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −27.7191 −1.37399
\(408\) 0 0
\(409\) 33.1739 1.64034 0.820171 0.572118i \(-0.193878\pi\)
0.820171 + 0.572118i \(0.193878\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 2.85748 0.140608
\(414\) 0 0
\(415\) −0.926442 −0.0454773
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 29.0969 1.42148 0.710738 0.703457i \(-0.248361\pi\)
0.710738 + 0.703457i \(0.248361\pi\)
\(420\) 0 0
\(421\) 9.31812 0.454137 0.227069 0.973879i \(-0.427086\pi\)
0.227069 + 0.973879i \(0.427086\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 4.97318 0.241235
\(426\) 0 0
\(427\) 13.2671 0.642038
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −8.88227 −0.427844 −0.213922 0.976851i \(-0.568624\pi\)
−0.213922 + 0.976851i \(0.568624\pi\)
\(432\) 0 0
\(433\) 16.7749 0.806149 0.403075 0.915167i \(-0.367942\pi\)
0.403075 + 0.915167i \(0.367942\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −7.14072 −0.341587
\(438\) 0 0
\(439\) −38.0488 −1.81597 −0.907986 0.419001i \(-0.862380\pi\)
−0.907986 + 0.419001i \(0.862380\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −25.8171 −1.22661 −0.613303 0.789848i \(-0.710160\pi\)
−0.613303 + 0.789848i \(0.710160\pi\)
\(444\) 0 0
\(445\) 2.21356 0.104933
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −19.2470 −0.908323 −0.454161 0.890919i \(-0.650061\pi\)
−0.454161 + 0.890919i \(0.650061\pi\)
\(450\) 0 0
\(451\) 19.3296 0.910196
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0.404849 0.0189796
\(456\) 0 0
\(457\) −3.86250 −0.180680 −0.0903401 0.995911i \(-0.528795\pi\)
−0.0903401 + 0.995911i \(0.528795\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 4.55474 0.212136 0.106068 0.994359i \(-0.466174\pi\)
0.106068 + 0.994359i \(0.466174\pi\)
\(462\) 0 0
\(463\) −16.1603 −0.751032 −0.375516 0.926816i \(-0.622535\pi\)
−0.375516 + 0.926816i \(0.622535\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 3.29659 0.152548 0.0762740 0.997087i \(-0.475698\pi\)
0.0762740 + 0.997087i \(0.475698\pi\)
\(468\) 0 0
\(469\) −14.9319 −0.689490
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −32.4610 −1.49256
\(474\) 0 0
\(475\) −7.31080 −0.335443
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 15.7996 0.721902 0.360951 0.932585i \(-0.382452\pi\)
0.360951 + 0.932585i \(0.382452\pi\)
\(480\) 0 0
\(481\) 21.1755 0.965522
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 1.97736 0.0897874
\(486\) 0 0
\(487\) −22.8369 −1.03484 −0.517419 0.855732i \(-0.673107\pi\)
−0.517419 + 0.855732i \(0.673107\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 4.20368 0.189709 0.0948547 0.995491i \(-0.469761\pi\)
0.0948547 + 0.995491i \(0.469761\pi\)
\(492\) 0 0
\(493\) 0.378584 0.0170506
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 14.3317 0.642865
\(498\) 0 0
\(499\) 8.86425 0.396818 0.198409 0.980119i \(-0.436423\pi\)
0.198409 + 0.980119i \(0.436423\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 20.6333 0.919994 0.459997 0.887920i \(-0.347850\pi\)
0.459997 + 0.887920i \(0.347850\pi\)
\(504\) 0 0
\(505\) 2.52061 0.112166
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −1.96697 −0.0871844 −0.0435922 0.999049i \(-0.513880\pi\)
−0.0435922 + 0.999049i \(0.513880\pi\)
\(510\) 0 0
\(511\) 4.91327 0.217350
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −1.76861 −0.0779341
\(516\) 0 0
\(517\) −26.4940 −1.16521
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 40.9435 1.79377 0.896884 0.442266i \(-0.145825\pi\)
0.896884 + 0.442266i \(0.145825\pi\)
\(522\) 0 0
\(523\) 37.4879 1.63923 0.819615 0.572914i \(-0.194187\pi\)
0.819615 + 0.572914i \(0.194187\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −8.78727 −0.382780
\(528\) 0 0
\(529\) 0.595151 0.0258761
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −14.7665 −0.639609
\(534\) 0 0
\(535\) 1.51950 0.0656939
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −3.23607 −0.139387
\(540\) 0 0
\(541\) −8.60580 −0.369992 −0.184996 0.982739i \(-0.559227\pi\)
−0.184996 + 0.982739i \(0.559227\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0.475632 0.0203738
\(546\) 0 0
\(547\) −15.6887 −0.670800 −0.335400 0.942076i \(-0.608872\pi\)
−0.335400 + 0.942076i \(0.608872\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −0.556536 −0.0237092
\(552\) 0 0
\(553\) 16.1871 0.688346
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −12.0673 −0.511307 −0.255654 0.966768i \(-0.582291\pi\)
−0.255654 + 0.966768i \(0.582291\pi\)
\(558\) 0 0
\(559\) 24.7980 1.04884
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 5.97700 0.251901 0.125950 0.992037i \(-0.459802\pi\)
0.125950 + 0.992037i \(0.459802\pi\)
\(564\) 0 0
\(565\) 1.42059 0.0597646
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −26.1373 −1.09573 −0.547866 0.836566i \(-0.684560\pi\)
−0.547866 + 0.836566i \(0.684560\pi\)
\(570\) 0 0
\(571\) 7.33818 0.307093 0.153547 0.988141i \(-0.450930\pi\)
0.153547 + 0.988141i \(0.450930\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 24.1571 1.00742
\(576\) 0 0
\(577\) 0.169395 0.00705202 0.00352601 0.999994i \(-0.498878\pi\)
0.00352601 + 0.999994i \(0.498878\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −5.65715 −0.234698
\(582\) 0 0
\(583\) −28.2491 −1.16996
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −10.4184 −0.430012 −0.215006 0.976613i \(-0.568977\pi\)
−0.215006 + 0.976613i \(0.568977\pi\)
\(588\) 0 0
\(589\) 12.9177 0.532264
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −21.7811 −0.894445 −0.447222 0.894423i \(-0.647587\pi\)
−0.447222 + 0.894423i \(0.647587\pi\)
\(594\) 0 0
\(595\) −0.163765 −0.00671371
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −18.9847 −0.775695 −0.387848 0.921723i \(-0.626781\pi\)
−0.387848 + 0.921723i \(0.626781\pi\)
\(600\) 0 0
\(601\) 17.1797 0.700776 0.350388 0.936605i \(-0.386050\pi\)
0.350388 + 0.936605i \(0.386050\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0.0864456 0.00351451
\(606\) 0 0
\(607\) 27.7740 1.12731 0.563657 0.826009i \(-0.309394\pi\)
0.563657 + 0.826009i \(0.309394\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 20.2396 0.818808
\(612\) 0 0
\(613\) −37.0901 −1.49806 −0.749028 0.662538i \(-0.769479\pi\)
−0.749028 + 0.662538i \(0.769479\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −39.8484 −1.60424 −0.802119 0.597165i \(-0.796294\pi\)
−0.802119 + 0.597165i \(0.796294\pi\)
\(618\) 0 0
\(619\) 21.4852 0.863562 0.431781 0.901978i \(-0.357885\pi\)
0.431781 + 0.901978i \(0.357885\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 13.5167 0.541536
\(624\) 0 0
\(625\) 24.5984 0.983937
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −8.56569 −0.341536
\(630\) 0 0
\(631\) 37.2884 1.48443 0.742213 0.670164i \(-0.233776\pi\)
0.742213 + 0.670164i \(0.233776\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −1.01526 −0.0402894
\(636\) 0 0
\(637\) 2.47214 0.0979496
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 40.5188 1.60040 0.800198 0.599735i \(-0.204727\pi\)
0.800198 + 0.599735i \(0.204727\pi\)
\(642\) 0 0
\(643\) −28.2963 −1.11590 −0.557949 0.829875i \(-0.688412\pi\)
−0.557949 + 0.829875i \(0.688412\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −37.6767 −1.48122 −0.740611 0.671934i \(-0.765464\pi\)
−0.740611 + 0.671934i \(0.765464\pi\)
\(648\) 0 0
\(649\) 9.24701 0.362977
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −23.0915 −0.903639 −0.451819 0.892109i \(-0.649225\pi\)
−0.451819 + 0.892109i \(0.649225\pi\)
\(654\) 0 0
\(655\) −1.34912 −0.0527145
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −9.40401 −0.366328 −0.183164 0.983082i \(-0.558634\pi\)
−0.183164 + 0.983082i \(0.558634\pi\)
\(660\) 0 0
\(661\) −48.9621 −1.90441 −0.952203 0.305467i \(-0.901187\pi\)
−0.952203 + 0.305467i \(0.901187\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0.240742 0.00933557
\(666\) 0 0
\(667\) 1.83897 0.0712050
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 42.9331 1.65742
\(672\) 0 0
\(673\) −28.8959 −1.11386 −0.556928 0.830561i \(-0.688020\pi\)
−0.556928 + 0.830561i \(0.688020\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 39.7744 1.52865 0.764327 0.644829i \(-0.223071\pi\)
0.764327 + 0.644829i \(0.223071\pi\)
\(678\) 0 0
\(679\) 12.0744 0.463373
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −16.6165 −0.635814 −0.317907 0.948122i \(-0.602980\pi\)
−0.317907 + 0.948122i \(0.602980\pi\)
\(684\) 0 0
\(685\) 0.331922 0.0126821
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 21.5804 0.822148
\(690\) 0 0
\(691\) 46.7768 1.77947 0.889737 0.456473i \(-0.150887\pi\)
0.889737 + 0.456473i \(0.150887\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 2.73453 0.103727
\(696\) 0 0
\(697\) 5.97318 0.226250
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 6.59195 0.248975 0.124487 0.992221i \(-0.460271\pi\)
0.124487 + 0.992221i \(0.460271\pi\)
\(702\) 0 0
\(703\) 12.5920 0.474915
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 15.3916 0.578861
\(708\) 0 0
\(709\) −43.7396 −1.64267 −0.821337 0.570443i \(-0.806771\pi\)
−0.821337 + 0.570443i \(0.806771\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −42.6840 −1.59853
\(714\) 0 0
\(715\) 1.31012 0.0489957
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 9.08673 0.338878 0.169439 0.985541i \(-0.445804\pi\)
0.169439 + 0.985541i \(0.445804\pi\)
\(720\) 0 0
\(721\) −10.7997 −0.402201
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 1.88277 0.0699242
\(726\) 0 0
\(727\) 18.7843 0.696673 0.348336 0.937370i \(-0.386747\pi\)
0.348336 + 0.937370i \(0.386747\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −10.0310 −0.371010
\(732\) 0 0
\(733\) −25.1602 −0.929314 −0.464657 0.885491i \(-0.653822\pi\)
−0.464657 + 0.885491i \(0.653822\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −48.3206 −1.77991
\(738\) 0 0
\(739\) −30.5073 −1.12223 −0.561115 0.827738i \(-0.689627\pi\)
−0.561115 + 0.827738i \(0.689627\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −24.0773 −0.883311 −0.441656 0.897185i \(-0.645609\pi\)
−0.441656 + 0.897185i \(0.645609\pi\)
\(744\) 0 0
\(745\) −2.14538 −0.0786007
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 9.27857 0.339031
\(750\) 0 0
\(751\) −10.6840 −0.389866 −0.194933 0.980817i \(-0.562449\pi\)
−0.194933 + 0.980817i \(0.562449\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −2.04757 −0.0745189
\(756\) 0 0
\(757\) −39.0013 −1.41753 −0.708764 0.705446i \(-0.750747\pi\)
−0.708764 + 0.705446i \(0.750747\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 52.6056 1.90695 0.953476 0.301470i \(-0.0974772\pi\)
0.953476 + 0.301470i \(0.0974772\pi\)
\(762\) 0 0
\(763\) 2.90436 0.105145
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −7.06409 −0.255069
\(768\) 0 0
\(769\) −21.8596 −0.788276 −0.394138 0.919051i \(-0.628957\pi\)
−0.394138 + 0.919051i \(0.628957\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −16.5478 −0.595182 −0.297591 0.954693i \(-0.596183\pi\)
−0.297591 + 0.954693i \(0.596183\pi\)
\(774\) 0 0
\(775\) −43.7007 −1.56978
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −8.78085 −0.314607
\(780\) 0 0
\(781\) 46.3784 1.65955
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 2.73767 0.0977115
\(786\) 0 0
\(787\) −31.1476 −1.11029 −0.555146 0.831753i \(-0.687338\pi\)
−0.555146 + 0.831753i \(0.687338\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 8.67456 0.308432
\(792\) 0 0
\(793\) −32.7980 −1.16469
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −7.15198 −0.253336 −0.126668 0.991945i \(-0.540428\pi\)
−0.126668 + 0.991945i \(0.540428\pi\)
\(798\) 0 0
\(799\) −8.18710 −0.289639
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 15.8997 0.561088
\(804\) 0 0
\(805\) −0.795485 −0.0280372
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 34.4048 1.20961 0.604805 0.796374i \(-0.293251\pi\)
0.604805 + 0.796374i \(0.293251\pi\)
\(810\) 0 0
\(811\) −29.0670 −1.02068 −0.510341 0.859972i \(-0.670481\pi\)
−0.510341 + 0.859972i \(0.670481\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 1.80286 0.0631516
\(816\) 0 0
\(817\) 14.7460 0.515898
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −19.7064 −0.687759 −0.343879 0.939014i \(-0.611741\pi\)
−0.343879 + 0.939014i \(0.611741\pi\)
\(822\) 0 0
\(823\) 3.35262 0.116865 0.0584324 0.998291i \(-0.481390\pi\)
0.0584324 + 0.998291i \(0.481390\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 32.8201 1.14127 0.570633 0.821205i \(-0.306698\pi\)
0.570633 + 0.821205i \(0.306698\pi\)
\(828\) 0 0
\(829\) −24.4048 −0.847615 −0.423808 0.905752i \(-0.639307\pi\)
−0.423808 + 0.905752i \(0.639307\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −1.00000 −0.0346479
\(834\) 0 0
\(835\) −0.814090 −0.0281727
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 22.7307 0.784751 0.392376 0.919805i \(-0.371653\pi\)
0.392376 + 0.919805i \(0.371653\pi\)
\(840\) 0 0
\(841\) −28.8567 −0.995058
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 1.12810 0.0388079
\(846\) 0 0
\(847\) 0.527864 0.0181376
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −41.6077 −1.42629
\(852\) 0 0
\(853\) 14.8218 0.507487 0.253744 0.967272i \(-0.418338\pi\)
0.253744 + 0.967272i \(0.418338\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −0.355245 −0.0121349 −0.00606747 0.999982i \(-0.501931\pi\)
−0.00606747 + 0.999982i \(0.501931\pi\)
\(858\) 0 0
\(859\) 37.2665 1.27152 0.635759 0.771888i \(-0.280687\pi\)
0.635759 + 0.771888i \(0.280687\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0.0548924 0.00186856 0.000934279 1.00000i \(-0.499703\pi\)
0.000934279 1.00000i \(0.499703\pi\)
\(864\) 0 0
\(865\) −2.85838 −0.0971878
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 52.3826 1.77696
\(870\) 0 0
\(871\) 36.9136 1.25077
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −1.63326 −0.0552142
\(876\) 0 0
\(877\) −11.7330 −0.396195 −0.198098 0.980182i \(-0.563476\pi\)
−0.198098 + 0.980182i \(0.563476\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −42.6705 −1.43761 −0.718803 0.695214i \(-0.755310\pi\)
−0.718803 + 0.695214i \(0.755310\pi\)
\(882\) 0 0
\(883\) 0.555998 0.0187108 0.00935541 0.999956i \(-0.497022\pi\)
0.00935541 + 0.999956i \(0.497022\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 16.2420 0.545353 0.272676 0.962106i \(-0.412091\pi\)
0.272676 + 0.962106i \(0.412091\pi\)
\(888\) 0 0
\(889\) −6.19950 −0.207925
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 12.0354 0.402750
\(894\) 0 0
\(895\) −0.390304 −0.0130464
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −3.32672 −0.110952
\(900\) 0 0
\(901\) −8.72945 −0.290820
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 4.04668 0.134516
\(906\) 0 0
\(907\) −13.6050 −0.451746 −0.225873 0.974157i \(-0.572523\pi\)
−0.225873 + 0.974157i \(0.572523\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 13.4163 0.444501 0.222251 0.974990i \(-0.428660\pi\)
0.222251 + 0.974990i \(0.428660\pi\)
\(912\) 0 0
\(913\) −18.3069 −0.605871
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −8.23816 −0.272048
\(918\) 0 0
\(919\) −17.9373 −0.591695 −0.295848 0.955235i \(-0.595602\pi\)
−0.295848 + 0.955235i \(0.595602\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −35.4299 −1.16619
\(924\) 0 0
\(925\) −42.5987 −1.40064
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 19.5156 0.640286 0.320143 0.947369i \(-0.396269\pi\)
0.320143 + 0.947369i \(0.396269\pi\)
\(930\) 0 0
\(931\) 1.47005 0.0481788
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −0.529954 −0.0173314
\(936\) 0 0
\(937\) −48.4958 −1.58429 −0.792145 0.610333i \(-0.791035\pi\)
−0.792145 + 0.610333i \(0.791035\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −12.3406 −0.402293 −0.201146 0.979561i \(-0.564467\pi\)
−0.201146 + 0.979561i \(0.564467\pi\)
\(942\) 0 0
\(943\) 29.0146 0.944846
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −26.3633 −0.856691 −0.428345 0.903615i \(-0.640903\pi\)
−0.428345 + 0.903615i \(0.640903\pi\)
\(948\) 0 0
\(949\) −12.1463 −0.394285
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 3.60644 0.116824 0.0584120 0.998293i \(-0.481396\pi\)
0.0584120 + 0.998293i \(0.481396\pi\)
\(954\) 0 0
\(955\) −3.32759 −0.107678
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 2.02682 0.0654494
\(960\) 0 0
\(961\) 46.2161 1.49084
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −2.69505 −0.0867567
\(966\) 0 0
\(967\) 28.2139 0.907299 0.453649 0.891180i \(-0.350122\pi\)
0.453649 + 0.891180i \(0.350122\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 20.0355 0.642971 0.321486 0.946914i \(-0.395818\pi\)
0.321486 + 0.946914i \(0.395818\pi\)
\(972\) 0 0
\(973\) 16.6979 0.535310
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 44.5259 1.42451 0.712255 0.701921i \(-0.247674\pi\)
0.712255 + 0.701921i \(0.247674\pi\)
\(978\) 0 0
\(979\) 43.7410 1.39797
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 2.54931 0.0813103 0.0406552 0.999173i \(-0.487055\pi\)
0.0406552 + 0.999173i \(0.487055\pi\)
\(984\) 0 0
\(985\) −0.553636 −0.0176403
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −48.7254 −1.54938
\(990\) 0 0
\(991\) 24.8794 0.790319 0.395160 0.918612i \(-0.370689\pi\)
0.395160 + 0.918612i \(0.370689\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −0.843181 −0.0267306
\(996\) 0 0
\(997\) 3.68557 0.116723 0.0583615 0.998296i \(-0.481412\pi\)
0.0583615 + 0.998296i \(0.481412\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 8568.2.a.bj.1.2 4
3.2 odd 2 952.2.a.g.1.2 4
12.11 even 2 1904.2.a.q.1.3 4
21.20 even 2 6664.2.a.o.1.3 4
24.5 odd 2 7616.2.a.bj.1.3 4
24.11 even 2 7616.2.a.bp.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
952.2.a.g.1.2 4 3.2 odd 2
1904.2.a.q.1.3 4 12.11 even 2
6664.2.a.o.1.3 4 21.20 even 2
7616.2.a.bj.1.3 4 24.5 odd 2
7616.2.a.bp.1.2 4 24.11 even 2
8568.2.a.bj.1.2 4 1.1 even 1 trivial