Properties

Label 8624.2.a.de.1.2
Level $8624$
Weight $2$
Character 8624.1
Self dual yes
Analytic conductor $68.863$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8624,2,Mod(1,8624)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8624, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8624.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8624 = 2^{4} \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8624.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: \(68.8629867032\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.8.98988261376.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 12x^{6} + 37x^{4} - 28x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 4312)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.917853\) of defining polynomial
Character \(\chi\) \(=\) 8624.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.61511 q^{3} -0.436116 q^{5} +3.83882 q^{9} +O(q^{10})\) \(q-2.61511 q^{3} -0.436116 q^{5} +3.83882 q^{9} +1.00000 q^{11} -5.35072 q^{13} +1.14049 q^{15} -7.06594 q^{17} -8.05866 q^{19} +6.66843 q^{23} -4.80980 q^{25} -2.19362 q^{27} -4.59608 q^{29} -3.04174 q^{31} -2.61511 q^{33} -9.94108 q^{37} +13.9927 q^{39} -6.37935 q^{41} -11.2263 q^{43} -1.67417 q^{45} +8.72113 q^{47} +18.4782 q^{51} +4.97098 q^{53} -0.436116 q^{55} +21.0743 q^{57} +2.27182 q^{59} +3.55092 q^{61} +2.33354 q^{65} -9.31705 q^{67} -17.4387 q^{69} +0.846083 q^{71} -4.23751 q^{73} +12.5782 q^{75} +8.05255 q^{79} -5.77990 q^{81} -9.08729 q^{83} +3.08157 q^{85} +12.0193 q^{87} -14.5680 q^{89} +7.95450 q^{93} +3.51451 q^{95} +7.77894 q^{97} +3.83882 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{9} + 8 q^{11} + 20 q^{15} + 16 q^{23} + 24 q^{25} - 24 q^{29} - 36 q^{37} + 32 q^{39} - 8 q^{43} + 48 q^{51} + 56 q^{57} - 32 q^{65} + 48 q^{67} + 64 q^{71} - 8 q^{79} + 20 q^{81} - 8 q^{85} - 48 q^{93} + 48 q^{95} + 8 q^{99}+O(q^{100}) \) 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
\(3\) −2.61511 −1.50984 −0.754919 0.655819i \(-0.772324\pi\)
−0.754919 + 0.655819i \(0.772324\pi\)
\(4\) 0 0
\(5\) −0.436116 −0.195037 −0.0975186 0.995234i \(-0.531091\pi\)
−0.0975186 + 0.995234i \(0.531091\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 3.83882 1.27961
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) −5.35072 −1.48402 −0.742011 0.670388i \(-0.766128\pi\)
−0.742011 + 0.670388i \(0.766128\pi\)
\(14\) 0 0
\(15\) 1.14049 0.294474
\(16\) 0 0
\(17\) −7.06594 −1.71374 −0.856870 0.515532i \(-0.827594\pi\)
−0.856870 + 0.515532i \(0.827594\pi\)
\(18\) 0 0
\(19\) −8.05866 −1.84878 −0.924391 0.381446i \(-0.875426\pi\)
−0.924391 + 0.381446i \(0.875426\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 6.66843 1.39046 0.695232 0.718786i \(-0.255302\pi\)
0.695232 + 0.718786i \(0.255302\pi\)
\(24\) 0 0
\(25\) −4.80980 −0.961961
\(26\) 0 0
\(27\) −2.19362 −0.422163
\(28\) 0 0
\(29\) −4.59608 −0.853471 −0.426735 0.904377i \(-0.640336\pi\)
−0.426735 + 0.904377i \(0.640336\pi\)
\(30\) 0 0
\(31\) −3.04174 −0.546313 −0.273156 0.961970i \(-0.588068\pi\)
−0.273156 + 0.961970i \(0.588068\pi\)
\(32\) 0 0
\(33\) −2.61511 −0.455233
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −9.94108 −1.63430 −0.817151 0.576423i \(-0.804448\pi\)
−0.817151 + 0.576423i \(0.804448\pi\)
\(38\) 0 0
\(39\) 13.9927 2.24063
\(40\) 0 0
\(41\) −6.37935 −0.996287 −0.498144 0.867095i \(-0.665985\pi\)
−0.498144 + 0.867095i \(0.665985\pi\)
\(42\) 0 0
\(43\) −11.2263 −1.71199 −0.855995 0.516985i \(-0.827054\pi\)
−0.855995 + 0.516985i \(0.827054\pi\)
\(44\) 0 0
\(45\) −1.67417 −0.249571
\(46\) 0 0
\(47\) 8.72113 1.27211 0.636054 0.771645i \(-0.280566\pi\)
0.636054 + 0.771645i \(0.280566\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 18.4782 2.58747
\(52\) 0 0
\(53\) 4.97098 0.682816 0.341408 0.939915i \(-0.389096\pi\)
0.341408 + 0.939915i \(0.389096\pi\)
\(54\) 0 0
\(55\) −0.436116 −0.0588059
\(56\) 0 0
\(57\) 21.0743 2.79136
\(58\) 0 0
\(59\) 2.27182 0.295766 0.147883 0.989005i \(-0.452754\pi\)
0.147883 + 0.989005i \(0.452754\pi\)
\(60\) 0 0
\(61\) 3.55092 0.454649 0.227325 0.973819i \(-0.427002\pi\)
0.227325 + 0.973819i \(0.427002\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 2.33354 0.289439
\(66\) 0 0
\(67\) −9.31705 −1.13826 −0.569130 0.822248i \(-0.692720\pi\)
−0.569130 + 0.822248i \(0.692720\pi\)
\(68\) 0 0
\(69\) −17.4387 −2.09937
\(70\) 0 0
\(71\) 0.846083 0.100412 0.0502058 0.998739i \(-0.484012\pi\)
0.0502058 + 0.998739i \(0.484012\pi\)
\(72\) 0 0
\(73\) −4.23751 −0.495963 −0.247981 0.968765i \(-0.579767\pi\)
−0.247981 + 0.968765i \(0.579767\pi\)
\(74\) 0 0
\(75\) 12.5782 1.45240
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 8.05255 0.905982 0.452991 0.891515i \(-0.350357\pi\)
0.452991 + 0.891515i \(0.350357\pi\)
\(80\) 0 0
\(81\) −5.77990 −0.642211
\(82\) 0 0
\(83\) −9.08729 −0.997459 −0.498730 0.866758i \(-0.666200\pi\)
−0.498730 + 0.866758i \(0.666200\pi\)
\(84\) 0 0
\(85\) 3.08157 0.334243
\(86\) 0 0
\(87\) 12.0193 1.28860
\(88\) 0 0
\(89\) −14.5680 −1.54420 −0.772102 0.635499i \(-0.780794\pi\)
−0.772102 + 0.635499i \(0.780794\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 7.95450 0.824843
\(94\) 0 0
\(95\) 3.51451 0.360581
\(96\) 0 0
\(97\) 7.77894 0.789832 0.394916 0.918717i \(-0.370774\pi\)
0.394916 + 0.918717i \(0.370774\pi\)
\(98\) 0 0
\(99\) 3.83882 0.385816
\(100\) 0 0
\(101\) 15.3656 1.52893 0.764466 0.644664i \(-0.223003\pi\)
0.764466 + 0.644664i \(0.223003\pi\)
\(102\) 0 0
\(103\) 0.848117 0.0835675 0.0417837 0.999127i \(-0.486696\pi\)
0.0417837 + 0.999127i \(0.486696\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −0.706671 −0.0683165 −0.0341582 0.999416i \(-0.510875\pi\)
−0.0341582 + 0.999416i \(0.510875\pi\)
\(108\) 0 0
\(109\) −10.3151 −0.988007 −0.494003 0.869460i \(-0.664467\pi\)
−0.494003 + 0.869460i \(0.664467\pi\)
\(110\) 0 0
\(111\) 25.9971 2.46753
\(112\) 0 0
\(113\) −7.02990 −0.661318 −0.330659 0.943750i \(-0.607271\pi\)
−0.330659 + 0.943750i \(0.607271\pi\)
\(114\) 0 0
\(115\) −2.90821 −0.271192
\(116\) 0 0
\(117\) −20.5405 −1.89897
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 16.6827 1.50423
\(124\) 0 0
\(125\) 4.27821 0.382655
\(126\) 0 0
\(127\) −10.6486 −0.944913 −0.472456 0.881354i \(-0.656633\pi\)
−0.472456 + 0.881354i \(0.656633\pi\)
\(128\) 0 0
\(129\) 29.3580 2.58482
\(130\) 0 0
\(131\) 7.21567 0.630436 0.315218 0.949019i \(-0.397922\pi\)
0.315218 + 0.949019i \(0.397922\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0.956674 0.0823374
\(136\) 0 0
\(137\) −1.31089 −0.111997 −0.0559985 0.998431i \(-0.517834\pi\)
−0.0559985 + 0.998431i \(0.517834\pi\)
\(138\) 0 0
\(139\) −23.1181 −1.96085 −0.980425 0.196891i \(-0.936915\pi\)
−0.980425 + 0.196891i \(0.936915\pi\)
\(140\) 0 0
\(141\) −22.8067 −1.92067
\(142\) 0 0
\(143\) −5.35072 −0.447450
\(144\) 0 0
\(145\) 2.00443 0.166458
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −4.76648 −0.390485 −0.195243 0.980755i \(-0.562549\pi\)
−0.195243 + 0.980755i \(0.562549\pi\)
\(150\) 0 0
\(151\) −14.4369 −1.17486 −0.587428 0.809277i \(-0.699859\pi\)
−0.587428 + 0.809277i \(0.699859\pi\)
\(152\) 0 0
\(153\) −27.1249 −2.19292
\(154\) 0 0
\(155\) 1.32655 0.106551
\(156\) 0 0
\(157\) 0.808657 0.0645378 0.0322689 0.999479i \(-0.489727\pi\)
0.0322689 + 0.999479i \(0.489727\pi\)
\(158\) 0 0
\(159\) −12.9997 −1.03094
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −3.14137 −0.246051 −0.123026 0.992403i \(-0.539260\pi\)
−0.123026 + 0.992403i \(0.539260\pi\)
\(164\) 0 0
\(165\) 1.14049 0.0887873
\(166\) 0 0
\(167\) −12.6576 −0.979477 −0.489738 0.871869i \(-0.662908\pi\)
−0.489738 + 0.871869i \(0.662908\pi\)
\(168\) 0 0
\(169\) 15.6302 1.20232
\(170\) 0 0
\(171\) −30.9358 −2.36572
\(172\) 0 0
\(173\) 17.0515 1.29640 0.648202 0.761468i \(-0.275521\pi\)
0.648202 + 0.761468i \(0.275521\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −5.94108 −0.446559
\(178\) 0 0
\(179\) 14.2221 1.06301 0.531503 0.847056i \(-0.321627\pi\)
0.531503 + 0.847056i \(0.321627\pi\)
\(180\) 0 0
\(181\) −14.2350 −1.05808 −0.529038 0.848598i \(-0.677447\pi\)
−0.529038 + 0.848598i \(0.677447\pi\)
\(182\) 0 0
\(183\) −9.28607 −0.686446
\(184\) 0 0
\(185\) 4.33546 0.318750
\(186\) 0 0
\(187\) −7.06594 −0.516712
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 11.9411 0.864026 0.432013 0.901867i \(-0.357803\pi\)
0.432013 + 0.901867i \(0.357803\pi\)
\(192\) 0 0
\(193\) −21.3821 −1.53912 −0.769560 0.638574i \(-0.779525\pi\)
−0.769560 + 0.638574i \(0.779525\pi\)
\(194\) 0 0
\(195\) −6.10246 −0.437006
\(196\) 0 0
\(197\) −19.5184 −1.39063 −0.695316 0.718705i \(-0.744735\pi\)
−0.695316 + 0.718705i \(0.744735\pi\)
\(198\) 0 0
\(199\) 16.9552 1.20192 0.600960 0.799279i \(-0.294785\pi\)
0.600960 + 0.799279i \(0.294785\pi\)
\(200\) 0 0
\(201\) 24.3652 1.71859
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 2.78214 0.194313
\(206\) 0 0
\(207\) 25.5989 1.77925
\(208\) 0 0
\(209\) −8.05866 −0.557429
\(210\) 0 0
\(211\) 14.9112 1.02653 0.513264 0.858231i \(-0.328436\pi\)
0.513264 + 0.858231i \(0.328436\pi\)
\(212\) 0 0
\(213\) −2.21260 −0.151605
\(214\) 0 0
\(215\) 4.89596 0.333901
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 11.0816 0.748823
\(220\) 0 0
\(221\) 37.8078 2.54323
\(222\) 0 0
\(223\) −11.6191 −0.778071 −0.389035 0.921223i \(-0.627192\pi\)
−0.389035 + 0.921223i \(0.627192\pi\)
\(224\) 0 0
\(225\) −18.4640 −1.23093
\(226\) 0 0
\(227\) 4.51440 0.299631 0.149815 0.988714i \(-0.452132\pi\)
0.149815 + 0.988714i \(0.452132\pi\)
\(228\) 0 0
\(229\) −22.4513 −1.48362 −0.741811 0.670610i \(-0.766033\pi\)
−0.741811 + 0.670610i \(0.766033\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −0.00725878 −0.000475538 0 −0.000237769 1.00000i \(-0.500076\pi\)
−0.000237769 1.00000i \(0.500076\pi\)
\(234\) 0 0
\(235\) −3.80342 −0.248108
\(236\) 0 0
\(237\) −21.0583 −1.36789
\(238\) 0 0
\(239\) −15.0145 −0.971208 −0.485604 0.874179i \(-0.661400\pi\)
−0.485604 + 0.874179i \(0.661400\pi\)
\(240\) 0 0
\(241\) 9.70871 0.625393 0.312697 0.949853i \(-0.398768\pi\)
0.312697 + 0.949853i \(0.398768\pi\)
\(242\) 0 0
\(243\) 21.6960 1.39180
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 43.1196 2.74363
\(248\) 0 0
\(249\) 23.7643 1.50600
\(250\) 0 0
\(251\) −13.9288 −0.879180 −0.439590 0.898199i \(-0.644876\pi\)
−0.439590 + 0.898199i \(0.644876\pi\)
\(252\) 0 0
\(253\) 6.66843 0.419240
\(254\) 0 0
\(255\) −8.05866 −0.504653
\(256\) 0 0
\(257\) −0.271755 −0.0169516 −0.00847582 0.999964i \(-0.502698\pi\)
−0.00847582 + 0.999964i \(0.502698\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −17.6435 −1.09211
\(262\) 0 0
\(263\) 11.5145 0.710015 0.355008 0.934863i \(-0.384478\pi\)
0.355008 + 0.934863i \(0.384478\pi\)
\(264\) 0 0
\(265\) −2.16792 −0.133175
\(266\) 0 0
\(267\) 38.0970 2.33150
\(268\) 0 0
\(269\) 23.1884 1.41382 0.706910 0.707303i \(-0.250088\pi\)
0.706910 + 0.707303i \(0.250088\pi\)
\(270\) 0 0
\(271\) −7.18642 −0.436544 −0.218272 0.975888i \(-0.570042\pi\)
−0.218272 + 0.975888i \(0.570042\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −4.80980 −0.290042
\(276\) 0 0
\(277\) −8.57038 −0.514944 −0.257472 0.966286i \(-0.582890\pi\)
−0.257472 + 0.966286i \(0.582890\pi\)
\(278\) 0 0
\(279\) −11.6767 −0.699066
\(280\) 0 0
\(281\) 10.3151 0.615347 0.307673 0.951492i \(-0.400450\pi\)
0.307673 + 0.951492i \(0.400450\pi\)
\(282\) 0 0
\(283\) −20.4650 −1.21652 −0.608260 0.793738i \(-0.708132\pi\)
−0.608260 + 0.793738i \(0.708132\pi\)
\(284\) 0 0
\(285\) −9.19085 −0.544419
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 32.9274 1.93691
\(290\) 0 0
\(291\) −20.3428 −1.19252
\(292\) 0 0
\(293\) −2.10593 −0.123030 −0.0615149 0.998106i \(-0.519593\pi\)
−0.0615149 + 0.998106i \(0.519593\pi\)
\(294\) 0 0
\(295\) −0.990779 −0.0576854
\(296\) 0 0
\(297\) −2.19362 −0.127287
\(298\) 0 0
\(299\) −35.6809 −2.06348
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −40.1827 −2.30844
\(304\) 0 0
\(305\) −1.54862 −0.0886734
\(306\) 0 0
\(307\) 14.3468 0.818813 0.409407 0.912352i \(-0.365736\pi\)
0.409407 + 0.912352i \(0.365736\pi\)
\(308\) 0 0
\(309\) −2.21792 −0.126173
\(310\) 0 0
\(311\) −12.5229 −0.710106 −0.355053 0.934846i \(-0.615537\pi\)
−0.355053 + 0.934846i \(0.615537\pi\)
\(312\) 0 0
\(313\) −14.6964 −0.830689 −0.415344 0.909664i \(-0.636339\pi\)
−0.415344 + 0.909664i \(0.636339\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −26.2674 −1.47532 −0.737661 0.675172i \(-0.764070\pi\)
−0.737661 + 0.675172i \(0.764070\pi\)
\(318\) 0 0
\(319\) −4.59608 −0.257331
\(320\) 0 0
\(321\) 1.84803 0.103147
\(322\) 0 0
\(323\) 56.9419 3.16833
\(324\) 0 0
\(325\) 25.7359 1.42757
\(326\) 0 0
\(327\) 26.9751 1.49173
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 9.25285 0.508582 0.254291 0.967128i \(-0.418158\pi\)
0.254291 + 0.967128i \(0.418158\pi\)
\(332\) 0 0
\(333\) −38.1620 −2.09127
\(334\) 0 0
\(335\) 4.06332 0.222003
\(336\) 0 0
\(337\) −8.92236 −0.486032 −0.243016 0.970022i \(-0.578137\pi\)
−0.243016 + 0.970022i \(0.578137\pi\)
\(338\) 0 0
\(339\) 18.3840 0.998482
\(340\) 0 0
\(341\) −3.04174 −0.164719
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 7.60530 0.409456
\(346\) 0 0
\(347\) −16.5631 −0.889155 −0.444577 0.895740i \(-0.646646\pi\)
−0.444577 + 0.895740i \(0.646646\pi\)
\(348\) 0 0
\(349\) −32.5218 −1.74085 −0.870425 0.492300i \(-0.836156\pi\)
−0.870425 + 0.492300i \(0.836156\pi\)
\(350\) 0 0
\(351\) 11.7375 0.626499
\(352\) 0 0
\(353\) 12.5285 0.666823 0.333411 0.942781i \(-0.391800\pi\)
0.333411 + 0.942781i \(0.391800\pi\)
\(354\) 0 0
\(355\) −0.368991 −0.0195840
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 7.26763 0.383571 0.191785 0.981437i \(-0.438572\pi\)
0.191785 + 0.981437i \(0.438572\pi\)
\(360\) 0 0
\(361\) 45.9419 2.41800
\(362\) 0 0
\(363\) −2.61511 −0.137258
\(364\) 0 0
\(365\) 1.84805 0.0967312
\(366\) 0 0
\(367\) −11.2478 −0.587129 −0.293565 0.955939i \(-0.594842\pi\)
−0.293565 + 0.955939i \(0.594842\pi\)
\(368\) 0 0
\(369\) −24.4892 −1.27486
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −0.952535 −0.0493204 −0.0246602 0.999696i \(-0.507850\pi\)
−0.0246602 + 0.999696i \(0.507850\pi\)
\(374\) 0 0
\(375\) −11.1880 −0.577747
\(376\) 0 0
\(377\) 24.5923 1.26657
\(378\) 0 0
\(379\) 25.4617 1.30788 0.653941 0.756545i \(-0.273114\pi\)
0.653941 + 0.756545i \(0.273114\pi\)
\(380\) 0 0
\(381\) 27.8474 1.42666
\(382\) 0 0
\(383\) 9.64514 0.492844 0.246422 0.969163i \(-0.420745\pi\)
0.246422 + 0.969163i \(0.420745\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −43.0957 −2.19068
\(388\) 0 0
\(389\) −37.0886 −1.88047 −0.940234 0.340530i \(-0.889394\pi\)
−0.940234 + 0.340530i \(0.889394\pi\)
\(390\) 0 0
\(391\) −47.1187 −2.38289
\(392\) 0 0
\(393\) −18.8698 −0.951856
\(394\) 0 0
\(395\) −3.51185 −0.176700
\(396\) 0 0
\(397\) 12.4869 0.626702 0.313351 0.949637i \(-0.398548\pi\)
0.313351 + 0.949637i \(0.398548\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −5.60116 −0.279709 −0.139854 0.990172i \(-0.544663\pi\)
−0.139854 + 0.990172i \(0.544663\pi\)
\(402\) 0 0
\(403\) 16.2755 0.810740
\(404\) 0 0
\(405\) 2.52071 0.125255
\(406\) 0 0
\(407\) −9.94108 −0.492761
\(408\) 0 0
\(409\) 21.1717 1.04687 0.523437 0.852064i \(-0.324649\pi\)
0.523437 + 0.852064i \(0.324649\pi\)
\(410\) 0 0
\(411\) 3.42813 0.169097
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 3.96311 0.194542
\(416\) 0 0
\(417\) 60.4565 2.96057
\(418\) 0 0
\(419\) −32.2398 −1.57502 −0.787508 0.616305i \(-0.788629\pi\)
−0.787508 + 0.616305i \(0.788629\pi\)
\(420\) 0 0
\(421\) 30.2682 1.47518 0.737592 0.675247i \(-0.235963\pi\)
0.737592 + 0.675247i \(0.235963\pi\)
\(422\) 0 0
\(423\) 33.4789 1.62780
\(424\) 0 0
\(425\) 33.9858 1.64855
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 13.9927 0.675576
\(430\) 0 0
\(431\) 22.6229 1.08971 0.544854 0.838531i \(-0.316585\pi\)
0.544854 + 0.838531i \(0.316585\pi\)
\(432\) 0 0
\(433\) 22.5421 1.08330 0.541651 0.840603i \(-0.317799\pi\)
0.541651 + 0.840603i \(0.317799\pi\)
\(434\) 0 0
\(435\) −5.24180 −0.251325
\(436\) 0 0
\(437\) −53.7386 −2.57066
\(438\) 0 0
\(439\) 39.5964 1.88983 0.944917 0.327309i \(-0.106142\pi\)
0.944917 + 0.327309i \(0.106142\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 2.23658 0.106263 0.0531316 0.998588i \(-0.483080\pi\)
0.0531316 + 0.998588i \(0.483080\pi\)
\(444\) 0 0
\(445\) 6.35334 0.301177
\(446\) 0 0
\(447\) 12.4649 0.589569
\(448\) 0 0
\(449\) 36.9835 1.74536 0.872680 0.488292i \(-0.162380\pi\)
0.872680 + 0.488292i \(0.162380\pi\)
\(450\) 0 0
\(451\) −6.37935 −0.300392
\(452\) 0 0
\(453\) 37.7541 1.77384
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −24.3347 −1.13833 −0.569164 0.822224i \(-0.692733\pi\)
−0.569164 + 0.822224i \(0.692733\pi\)
\(458\) 0 0
\(459\) 15.5000 0.723478
\(460\) 0 0
\(461\) 30.1287 1.40323 0.701617 0.712555i \(-0.252462\pi\)
0.701617 + 0.712555i \(0.252462\pi\)
\(462\) 0 0
\(463\) 31.7841 1.47713 0.738566 0.674181i \(-0.235503\pi\)
0.738566 + 0.674181i \(0.235503\pi\)
\(464\) 0 0
\(465\) −3.46909 −0.160875
\(466\) 0 0
\(467\) 18.6186 0.861566 0.430783 0.902456i \(-0.358237\pi\)
0.430783 + 0.902456i \(0.358237\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −2.11473 −0.0974416
\(472\) 0 0
\(473\) −11.2263 −0.516184
\(474\) 0 0
\(475\) 38.7605 1.77846
\(476\) 0 0
\(477\) 19.0827 0.873737
\(478\) 0 0
\(479\) −3.03306 −0.138584 −0.0692920 0.997596i \(-0.522074\pi\)
−0.0692920 + 0.997596i \(0.522074\pi\)
\(480\) 0 0
\(481\) 53.1919 2.42534
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −3.39252 −0.154046
\(486\) 0 0
\(487\) 26.1684 1.18580 0.592902 0.805274i \(-0.297982\pi\)
0.592902 + 0.805274i \(0.297982\pi\)
\(488\) 0 0
\(489\) 8.21506 0.371498
\(490\) 0 0
\(491\) −8.27373 −0.373388 −0.186694 0.982418i \(-0.559777\pi\)
−0.186694 + 0.982418i \(0.559777\pi\)
\(492\) 0 0
\(493\) 32.4756 1.46263
\(494\) 0 0
\(495\) −1.67417 −0.0752485
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 5.19942 0.232758 0.116379 0.993205i \(-0.462871\pi\)
0.116379 + 0.993205i \(0.462871\pi\)
\(500\) 0 0
\(501\) 33.1012 1.47885
\(502\) 0 0
\(503\) 5.64659 0.251769 0.125884 0.992045i \(-0.459823\pi\)
0.125884 + 0.992045i \(0.459823\pi\)
\(504\) 0 0
\(505\) −6.70117 −0.298198
\(506\) 0 0
\(507\) −40.8747 −1.81531
\(508\) 0 0
\(509\) 36.1578 1.60266 0.801332 0.598220i \(-0.204125\pi\)
0.801332 + 0.598220i \(0.204125\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 17.6776 0.780487
\(514\) 0 0
\(515\) −0.369878 −0.0162988
\(516\) 0 0
\(517\) 8.72113 0.383555
\(518\) 0 0
\(519\) −44.5917 −1.95736
\(520\) 0 0
\(521\) −35.1005 −1.53778 −0.768891 0.639379i \(-0.779191\pi\)
−0.768891 + 0.639379i \(0.779191\pi\)
\(522\) 0 0
\(523\) −18.7048 −0.817902 −0.408951 0.912556i \(-0.634105\pi\)
−0.408951 + 0.912556i \(0.634105\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 21.4927 0.936238
\(528\) 0 0
\(529\) 21.4679 0.933388
\(530\) 0 0
\(531\) 8.72113 0.378465
\(532\) 0 0
\(533\) 34.1341 1.47851
\(534\) 0 0
\(535\) 0.308191 0.0133242
\(536\) 0 0
\(537\) −37.1923 −1.60497
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 28.4129 1.22157 0.610784 0.791798i \(-0.290855\pi\)
0.610784 + 0.791798i \(0.290855\pi\)
\(542\) 0 0
\(543\) 37.2261 1.59752
\(544\) 0 0
\(545\) 4.49858 0.192698
\(546\) 0 0
\(547\) 9.19216 0.393028 0.196514 0.980501i \(-0.437038\pi\)
0.196514 + 0.980501i \(0.437038\pi\)
\(548\) 0 0
\(549\) 13.6314 0.581773
\(550\) 0 0
\(551\) 37.0382 1.57788
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −11.3377 −0.481260
\(556\) 0 0
\(557\) −13.6178 −0.577007 −0.288503 0.957479i \(-0.593158\pi\)
−0.288503 + 0.957479i \(0.593158\pi\)
\(558\) 0 0
\(559\) 60.0686 2.54063
\(560\) 0 0
\(561\) 18.4782 0.780151
\(562\) 0 0
\(563\) −15.3944 −0.648796 −0.324398 0.945921i \(-0.605162\pi\)
−0.324398 + 0.945921i \(0.605162\pi\)
\(564\) 0 0
\(565\) 3.06585 0.128981
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −3.99274 −0.167384 −0.0836922 0.996492i \(-0.526671\pi\)
−0.0836922 + 0.996492i \(0.526671\pi\)
\(570\) 0 0
\(571\) −34.4481 −1.44161 −0.720803 0.693140i \(-0.756227\pi\)
−0.720803 + 0.693140i \(0.756227\pi\)
\(572\) 0 0
\(573\) −31.2273 −1.30454
\(574\) 0 0
\(575\) −32.0738 −1.33757
\(576\) 0 0
\(577\) −20.9857 −0.873648 −0.436824 0.899547i \(-0.643897\pi\)
−0.436824 + 0.899547i \(0.643897\pi\)
\(578\) 0 0
\(579\) 55.9168 2.32382
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 4.97098 0.205877
\(584\) 0 0
\(585\) 8.95803 0.370369
\(586\) 0 0
\(587\) −0.525053 −0.0216713 −0.0108356 0.999941i \(-0.503449\pi\)
−0.0108356 + 0.999941i \(0.503449\pi\)
\(588\) 0 0
\(589\) 24.5123 1.01001
\(590\) 0 0
\(591\) 51.0429 2.09963
\(592\) 0 0
\(593\) 11.3978 0.468054 0.234027 0.972230i \(-0.424810\pi\)
0.234027 + 0.972230i \(0.424810\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −44.3397 −1.81470
\(598\) 0 0
\(599\) 3.48217 0.142278 0.0711388 0.997466i \(-0.477337\pi\)
0.0711388 + 0.997466i \(0.477337\pi\)
\(600\) 0 0
\(601\) −11.0794 −0.451938 −0.225969 0.974135i \(-0.572555\pi\)
−0.225969 + 0.974135i \(0.572555\pi\)
\(602\) 0 0
\(603\) −35.7665 −1.45653
\(604\) 0 0
\(605\) −0.436116 −0.0177306
\(606\) 0 0
\(607\) −11.2845 −0.458022 −0.229011 0.973424i \(-0.573549\pi\)
−0.229011 + 0.973424i \(0.573549\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −46.6643 −1.88784
\(612\) 0 0
\(613\) −14.0939 −0.569248 −0.284624 0.958639i \(-0.591869\pi\)
−0.284624 + 0.958639i \(0.591869\pi\)
\(614\) 0 0
\(615\) −7.27561 −0.293381
\(616\) 0 0
\(617\) −12.7251 −0.512294 −0.256147 0.966638i \(-0.582453\pi\)
−0.256147 + 0.966638i \(0.582453\pi\)
\(618\) 0 0
\(619\) 9.41318 0.378348 0.189174 0.981944i \(-0.439419\pi\)
0.189174 + 0.981944i \(0.439419\pi\)
\(620\) 0 0
\(621\) −14.6280 −0.587002
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 22.1832 0.887329
\(626\) 0 0
\(627\) 21.0743 0.841627
\(628\) 0 0
\(629\) 70.2430 2.80077
\(630\) 0 0
\(631\) −6.18518 −0.246228 −0.123114 0.992393i \(-0.539288\pi\)
−0.123114 + 0.992393i \(0.539288\pi\)
\(632\) 0 0
\(633\) −38.9944 −1.54989
\(634\) 0 0
\(635\) 4.64404 0.184293
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 3.24797 0.128488
\(640\) 0 0
\(641\) −29.4679 −1.16391 −0.581956 0.813220i \(-0.697713\pi\)
−0.581956 + 0.813220i \(0.697713\pi\)
\(642\) 0 0
\(643\) −12.7043 −0.501008 −0.250504 0.968116i \(-0.580596\pi\)
−0.250504 + 0.968116i \(0.580596\pi\)
\(644\) 0 0
\(645\) −12.8035 −0.504137
\(646\) 0 0
\(647\) 44.9669 1.76783 0.883916 0.467646i \(-0.154898\pi\)
0.883916 + 0.467646i \(0.154898\pi\)
\(648\) 0 0
\(649\) 2.27182 0.0891768
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −15.0416 −0.588622 −0.294311 0.955710i \(-0.595090\pi\)
−0.294311 + 0.955710i \(0.595090\pi\)
\(654\) 0 0
\(655\) −3.14687 −0.122958
\(656\) 0 0
\(657\) −16.2670 −0.634638
\(658\) 0 0
\(659\) 28.5218 1.11105 0.555525 0.831500i \(-0.312517\pi\)
0.555525 + 0.831500i \(0.312517\pi\)
\(660\) 0 0
\(661\) −20.2979 −0.789497 −0.394749 0.918789i \(-0.629168\pi\)
−0.394749 + 0.918789i \(0.629168\pi\)
\(662\) 0 0
\(663\) −98.8718 −3.83986
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −30.6486 −1.18672
\(668\) 0 0
\(669\) 30.3852 1.17476
\(670\) 0 0
\(671\) 3.55092 0.137082
\(672\) 0 0
\(673\) −2.39334 −0.0922565 −0.0461283 0.998936i \(-0.514688\pi\)
−0.0461283 + 0.998936i \(0.514688\pi\)
\(674\) 0 0
\(675\) 10.5509 0.406104
\(676\) 0 0
\(677\) −18.4381 −0.708635 −0.354318 0.935125i \(-0.615287\pi\)
−0.354318 + 0.935125i \(0.615287\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −11.8057 −0.452394
\(682\) 0 0
\(683\) 25.1922 0.963951 0.481976 0.876185i \(-0.339919\pi\)
0.481976 + 0.876185i \(0.339919\pi\)
\(684\) 0 0
\(685\) 0.571701 0.0218436
\(686\) 0 0
\(687\) 58.7126 2.24003
\(688\) 0 0
\(689\) −26.5983 −1.01331
\(690\) 0 0
\(691\) 24.6772 0.938767 0.469383 0.882995i \(-0.344476\pi\)
0.469383 + 0.882995i \(0.344476\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 10.0822 0.382439
\(696\) 0 0
\(697\) 45.0761 1.70738
\(698\) 0 0
\(699\) 0.0189825 0.000717986 0
\(700\) 0 0
\(701\) 6.68273 0.252403 0.126202 0.992005i \(-0.459721\pi\)
0.126202 + 0.992005i \(0.459721\pi\)
\(702\) 0 0
\(703\) 80.1117 3.02147
\(704\) 0 0
\(705\) 9.94639 0.374603
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −24.4584 −0.918556 −0.459278 0.888293i \(-0.651892\pi\)
−0.459278 + 0.888293i \(0.651892\pi\)
\(710\) 0 0
\(711\) 30.9123 1.15930
\(712\) 0 0
\(713\) −20.2836 −0.759628
\(714\) 0 0
\(715\) 2.33354 0.0872693
\(716\) 0 0
\(717\) 39.2647 1.46637
\(718\) 0 0
\(719\) −27.7612 −1.03532 −0.517659 0.855587i \(-0.673197\pi\)
−0.517659 + 0.855587i \(0.673197\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −25.3894 −0.944242
\(724\) 0 0
\(725\) 22.1062 0.821005
\(726\) 0 0
\(727\) 4.47341 0.165909 0.0829547 0.996553i \(-0.473564\pi\)
0.0829547 + 0.996553i \(0.473564\pi\)
\(728\) 0 0
\(729\) −39.3977 −1.45918
\(730\) 0 0
\(731\) 79.3241 2.93391
\(732\) 0 0
\(733\) −17.3028 −0.639093 −0.319546 0.947571i \(-0.603531\pi\)
−0.319546 + 0.947571i \(0.603531\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −9.31705 −0.343198
\(738\) 0 0
\(739\) 38.9235 1.43182 0.715912 0.698190i \(-0.246011\pi\)
0.715912 + 0.698190i \(0.246011\pi\)
\(740\) 0 0
\(741\) −112.763 −4.14244
\(742\) 0 0
\(743\) 18.0704 0.662938 0.331469 0.943466i \(-0.392456\pi\)
0.331469 + 0.943466i \(0.392456\pi\)
\(744\) 0 0
\(745\) 2.07874 0.0761591
\(746\) 0 0
\(747\) −34.8845 −1.27636
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 10.6433 0.388381 0.194191 0.980964i \(-0.437792\pi\)
0.194191 + 0.980964i \(0.437792\pi\)
\(752\) 0 0
\(753\) 36.4255 1.32742
\(754\) 0 0
\(755\) 6.29615 0.229140
\(756\) 0 0
\(757\) −12.5106 −0.454704 −0.227352 0.973813i \(-0.573007\pi\)
−0.227352 + 0.973813i \(0.573007\pi\)
\(758\) 0 0
\(759\) −17.4387 −0.632985
\(760\) 0 0
\(761\) −3.89546 −0.141210 −0.0706052 0.997504i \(-0.522493\pi\)
−0.0706052 + 0.997504i \(0.522493\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 11.8296 0.427700
\(766\) 0 0
\(767\) −12.1559 −0.438923
\(768\) 0 0
\(769\) 30.2686 1.09151 0.545757 0.837944i \(-0.316242\pi\)
0.545757 + 0.837944i \(0.316242\pi\)
\(770\) 0 0
\(771\) 0.710672 0.0255942
\(772\) 0 0
\(773\) 7.84529 0.282175 0.141088 0.989997i \(-0.454940\pi\)
0.141088 + 0.989997i \(0.454940\pi\)
\(774\) 0 0
\(775\) 14.6302 0.525531
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 51.4090 1.84192
\(780\) 0 0
\(781\) 0.846083 0.0302752
\(782\) 0 0
\(783\) 10.0821 0.360304
\(784\) 0 0
\(785\) −0.352668 −0.0125873
\(786\) 0 0
\(787\) −20.1333 −0.717673 −0.358837 0.933400i \(-0.616826\pi\)
−0.358837 + 0.933400i \(0.616826\pi\)
\(788\) 0 0
\(789\) −30.1118 −1.07201
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −19.0000 −0.674709
\(794\) 0 0
\(795\) 5.66937 0.201072
\(796\) 0 0
\(797\) −13.1420 −0.465513 −0.232756 0.972535i \(-0.574774\pi\)
−0.232756 + 0.972535i \(0.574774\pi\)
\(798\) 0 0
\(799\) −61.6229 −2.18006
\(800\) 0 0
\(801\) −55.9239 −1.97598
\(802\) 0 0
\(803\) −4.23751 −0.149538
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −60.6403 −2.13464
\(808\) 0 0
\(809\) −13.6861 −0.481176 −0.240588 0.970627i \(-0.577340\pi\)
−0.240588 + 0.970627i \(0.577340\pi\)
\(810\) 0 0
\(811\) −4.85644 −0.170533 −0.0852664 0.996358i \(-0.527174\pi\)
−0.0852664 + 0.996358i \(0.527174\pi\)
\(812\) 0 0
\(813\) 18.7933 0.659111
\(814\) 0 0
\(815\) 1.37000 0.0479892
\(816\) 0 0
\(817\) 90.4686 3.16510
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 2.05140 0.0715945 0.0357972 0.999359i \(-0.488603\pi\)
0.0357972 + 0.999359i \(0.488603\pi\)
\(822\) 0 0
\(823\) −48.4903 −1.69027 −0.845133 0.534557i \(-0.820479\pi\)
−0.845133 + 0.534557i \(0.820479\pi\)
\(824\) 0 0
\(825\) 12.5782 0.437916
\(826\) 0 0
\(827\) −33.9782 −1.18154 −0.590769 0.806841i \(-0.701176\pi\)
−0.590769 + 0.806841i \(0.701176\pi\)
\(828\) 0 0
\(829\) 7.72044 0.268142 0.134071 0.990972i \(-0.457195\pi\)
0.134071 + 0.990972i \(0.457195\pi\)
\(830\) 0 0
\(831\) 22.4125 0.777482
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 5.52020 0.191034
\(836\) 0 0
\(837\) 6.67243 0.230633
\(838\) 0 0
\(839\) 0.333364 0.0115090 0.00575450 0.999983i \(-0.498168\pi\)
0.00575450 + 0.999983i \(0.498168\pi\)
\(840\) 0 0
\(841\) −7.87605 −0.271588
\(842\) 0 0
\(843\) −26.9751 −0.929073
\(844\) 0 0
\(845\) −6.81658 −0.234497
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 53.5184 1.83675
\(850\) 0 0
\(851\) −66.2913 −2.27244
\(852\) 0 0
\(853\) 1.39326 0.0477044 0.0238522 0.999715i \(-0.492407\pi\)
0.0238522 + 0.999715i \(0.492407\pi\)
\(854\) 0 0
\(855\) 13.4916 0.461403
\(856\) 0 0
\(857\) −13.5232 −0.461944 −0.230972 0.972960i \(-0.574191\pi\)
−0.230972 + 0.972960i \(0.574191\pi\)
\(858\) 0 0
\(859\) −33.5872 −1.14598 −0.572991 0.819562i \(-0.694217\pi\)
−0.572991 + 0.819562i \(0.694217\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −30.5039 −1.03837 −0.519183 0.854663i \(-0.673764\pi\)
−0.519183 + 0.854663i \(0.673764\pi\)
\(864\) 0 0
\(865\) −7.43645 −0.252847
\(866\) 0 0
\(867\) −86.1090 −2.92442
\(868\) 0 0
\(869\) 8.05255 0.273164
\(870\) 0 0
\(871\) 49.8529 1.68920
\(872\) 0 0
\(873\) 29.8620 1.01068
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 8.25529 0.278761 0.139381 0.990239i \(-0.455489\pi\)
0.139381 + 0.990239i \(0.455489\pi\)
\(878\) 0 0
\(879\) 5.50725 0.185755
\(880\) 0 0
\(881\) 25.1220 0.846383 0.423192 0.906040i \(-0.360910\pi\)
0.423192 + 0.906040i \(0.360910\pi\)
\(882\) 0 0
\(883\) −23.3039 −0.784238 −0.392119 0.919914i \(-0.628258\pi\)
−0.392119 + 0.919914i \(0.628258\pi\)
\(884\) 0 0
\(885\) 2.59100 0.0870955
\(886\) 0 0
\(887\) 51.9301 1.74364 0.871820 0.489826i \(-0.162940\pi\)
0.871820 + 0.489826i \(0.162940\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −5.77990 −0.193634
\(892\) 0 0
\(893\) −70.2806 −2.35185
\(894\) 0 0
\(895\) −6.20247 −0.207326
\(896\) 0 0
\(897\) 93.3096 3.11552
\(898\) 0 0
\(899\) 13.9801 0.466262
\(900\) 0 0
\(901\) −35.1246 −1.17017
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 6.20810 0.206364
\(906\) 0 0
\(907\) −28.0978 −0.932973 −0.466487 0.884528i \(-0.654480\pi\)
−0.466487 + 0.884528i \(0.654480\pi\)
\(908\) 0 0
\(909\) 58.9857 1.95643
\(910\) 0 0
\(911\) −51.1123 −1.69343 −0.846714 0.532049i \(-0.821422\pi\)
−0.846714 + 0.532049i \(0.821422\pi\)
\(912\) 0 0
\(913\) −9.08729 −0.300745
\(914\) 0 0
\(915\) 4.04981 0.133882
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −4.05588 −0.133791 −0.0668956 0.997760i \(-0.521309\pi\)
−0.0668956 + 0.997760i \(0.521309\pi\)
\(920\) 0 0
\(921\) −37.5184 −1.23627
\(922\) 0 0
\(923\) −4.52715 −0.149013
\(924\) 0 0
\(925\) 47.8146 1.57213
\(926\) 0 0
\(927\) 3.25577 0.106934
\(928\) 0 0
\(929\) −1.71675 −0.0563246 −0.0281623 0.999603i \(-0.508966\pi\)
−0.0281623 + 0.999603i \(0.508966\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 32.7487 1.07214
\(934\) 0 0
\(935\) 3.08157 0.100778
\(936\) 0 0
\(937\) −3.89546 −0.127259 −0.0636296 0.997974i \(-0.520268\pi\)
−0.0636296 + 0.997974i \(0.520268\pi\)
\(938\) 0 0
\(939\) 38.4327 1.25420
\(940\) 0 0
\(941\) 37.6407 1.22705 0.613525 0.789675i \(-0.289751\pi\)
0.613525 + 0.789675i \(0.289751\pi\)
\(942\) 0 0
\(943\) −42.5402 −1.38530
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 11.0897 0.360367 0.180184 0.983633i \(-0.442331\pi\)
0.180184 + 0.983633i \(0.442331\pi\)
\(948\) 0 0
\(949\) 22.6737 0.736020
\(950\) 0 0
\(951\) 68.6921 2.22750
\(952\) 0 0
\(953\) 23.4833 0.760699 0.380350 0.924843i \(-0.375804\pi\)
0.380350 + 0.924843i \(0.375804\pi\)
\(954\) 0 0
\(955\) −5.20770 −0.168517
\(956\) 0 0
\(957\) 12.0193 0.388528
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −21.7478 −0.701543
\(962\) 0 0
\(963\) −2.71279 −0.0874183
\(964\) 0 0
\(965\) 9.32510 0.300186
\(966\) 0 0
\(967\) −22.1743 −0.713078 −0.356539 0.934280i \(-0.616043\pi\)
−0.356539 + 0.934280i \(0.616043\pi\)
\(968\) 0 0
\(969\) −148.910 −4.78367
\(970\) 0 0
\(971\) 41.5159 1.33231 0.666155 0.745813i \(-0.267939\pi\)
0.666155 + 0.745813i \(0.267939\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −67.3023 −2.15540
\(976\) 0 0
\(977\) −27.9729 −0.894934 −0.447467 0.894301i \(-0.647674\pi\)
−0.447467 + 0.894301i \(0.647674\pi\)
\(978\) 0 0
\(979\) −14.5680 −0.465595
\(980\) 0 0
\(981\) −39.5978 −1.26426
\(982\) 0 0
\(983\) −27.3066 −0.870945 −0.435473 0.900202i \(-0.643419\pi\)
−0.435473 + 0.900202i \(0.643419\pi\)
\(984\) 0 0
\(985\) 8.51231 0.271225
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −74.8615 −2.38046
\(990\) 0 0
\(991\) −45.5649 −1.44742 −0.723708 0.690106i \(-0.757564\pi\)
−0.723708 + 0.690106i \(0.757564\pi\)
\(992\) 0 0
\(993\) −24.1973 −0.767877
\(994\) 0 0
\(995\) −7.39442 −0.234419
\(996\) 0 0
\(997\) −20.6148 −0.652877 −0.326438 0.945219i \(-0.605849\pi\)
−0.326438 + 0.945219i \(0.605849\pi\)
\(998\) 0 0
\(999\) 21.8070 0.689942
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 8624.2.a.de.1.2 8
4.3 odd 2 4312.2.a.bi.1.7 yes 8
7.6 odd 2 inner 8624.2.a.de.1.7 8
28.27 even 2 4312.2.a.bi.1.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4312.2.a.bi.1.2 8 28.27 even 2
4312.2.a.bi.1.7 yes 8 4.3 odd 2
8624.2.a.de.1.2 8 1.1 even 1 trivial
8624.2.a.de.1.7 8 7.6 odd 2 inner