Properties

Label 1368.2.a.o.1.1
Level $1368$
Weight $2$
Character 1368.1
Self dual yes
Analytic conductor $10.924$
Analytic rank $1$
Dimension $3$
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] = [1368,2,Mod(1,1368)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1368, 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("1368.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1368 = 2^{3} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1368.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: \(10.9235349965\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.892.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 8x + 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.59774\) of defining polynomial
Character \(\chi\) \(=\) 1368.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.59774 q^{5} +4.94370 q^{7} +O(q^{10})\) \(q-1.59774 q^{5} +4.94370 q^{7} -5.59774 q^{11} -3.49646 q^{13} -7.09419 q^{17} -1.00000 q^{19} +3.19547 q^{23} -2.44724 q^{25} +4.69193 q^{29} -6.00000 q^{31} -7.89872 q^{35} +4.00000 q^{37} -2.50354 q^{41} -8.94370 q^{43} +6.28966 q^{47} +17.4402 q^{49} -5.69901 q^{53} +8.94370 q^{55} +6.99291 q^{59} -3.44724 q^{61} +5.58641 q^{65} -13.3839 q^{67} -13.3839 q^{71} -2.44015 q^{73} -27.6735 q^{77} -9.49646 q^{79} +0.691927 q^{83} +11.3346 q^{85} -8.69193 q^{89} -17.2854 q^{91} +1.59774 q^{95} +16.3909 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{5} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 2 q^{5} - 2 q^{7} - 10 q^{11} - 4 q^{13} - 8 q^{17} - 3 q^{19} - 4 q^{23} + 3 q^{25} - 6 q^{29} - 18 q^{31} - 24 q^{35} + 12 q^{37} - 14 q^{41} - 10 q^{43} - 8 q^{47} + 29 q^{49} - 10 q^{53} + 10 q^{55} + 8 q^{59} - 24 q^{65} + 16 q^{73} - 16 q^{77} - 22 q^{79} - 18 q^{83} - 10 q^{85} - 6 q^{89} - 4 q^{91} - 2 q^{95} + 22 q^{97}+O(q^{100}) \) 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
\(5\) −1.59774 −0.714529 −0.357264 0.934003i \(-0.616291\pi\)
−0.357264 + 0.934003i \(0.616291\pi\)
\(6\) 0 0
\(7\) 4.94370 1.86854 0.934271 0.356563i \(-0.116052\pi\)
0.934271 + 0.356563i \(0.116052\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −5.59774 −1.68778 −0.843890 0.536516i \(-0.819740\pi\)
−0.843890 + 0.536516i \(0.819740\pi\)
\(12\) 0 0
\(13\) −3.49646 −0.969742 −0.484871 0.874586i \(-0.661134\pi\)
−0.484871 + 0.874586i \(0.661134\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −7.09419 −1.72059 −0.860297 0.509793i \(-0.829722\pi\)
−0.860297 + 0.509793i \(0.829722\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.19547 0.666302 0.333151 0.942874i \(-0.391888\pi\)
0.333151 + 0.942874i \(0.391888\pi\)
\(24\) 0 0
\(25\) −2.44724 −0.489448
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 4.69193 0.871269 0.435634 0.900124i \(-0.356524\pi\)
0.435634 + 0.900124i \(0.356524\pi\)
\(30\) 0 0
\(31\) −6.00000 −1.07763 −0.538816 0.842424i \(-0.681128\pi\)
−0.538816 + 0.842424i \(0.681128\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −7.89872 −1.33513
\(36\) 0 0
\(37\) 4.00000 0.657596 0.328798 0.944400i \(-0.393356\pi\)
0.328798 + 0.944400i \(0.393356\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −2.50354 −0.390988 −0.195494 0.980705i \(-0.562631\pi\)
−0.195494 + 0.980705i \(0.562631\pi\)
\(42\) 0 0
\(43\) −8.94370 −1.36390 −0.681951 0.731398i \(-0.738868\pi\)
−0.681951 + 0.731398i \(0.738868\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.28966 0.917441 0.458721 0.888580i \(-0.348308\pi\)
0.458721 + 0.888580i \(0.348308\pi\)
\(48\) 0 0
\(49\) 17.4402 2.49145
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −5.69901 −0.782820 −0.391410 0.920216i \(-0.628013\pi\)
−0.391410 + 0.920216i \(0.628013\pi\)
\(54\) 0 0
\(55\) 8.94370 1.20597
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 6.99291 0.910400 0.455200 0.890389i \(-0.349568\pi\)
0.455200 + 0.890389i \(0.349568\pi\)
\(60\) 0 0
\(61\) −3.44724 −0.441374 −0.220687 0.975345i \(-0.570830\pi\)
−0.220687 + 0.975345i \(0.570830\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 5.58641 0.692909
\(66\) 0 0
\(67\) −13.3839 −1.63510 −0.817549 0.575859i \(-0.804668\pi\)
−0.817549 + 0.575859i \(0.804668\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −13.3839 −1.58837 −0.794186 0.607675i \(-0.792102\pi\)
−0.794186 + 0.607675i \(0.792102\pi\)
\(72\) 0 0
\(73\) −2.44015 −0.285599 −0.142799 0.989752i \(-0.545610\pi\)
−0.142799 + 0.989752i \(0.545610\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −27.6735 −3.15369
\(78\) 0 0
\(79\) −9.49646 −1.06843 −0.534217 0.845347i \(-0.679394\pi\)
−0.534217 + 0.845347i \(0.679394\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0.691927 0.0759488 0.0379744 0.999279i \(-0.487909\pi\)
0.0379744 + 0.999279i \(0.487909\pi\)
\(84\) 0 0
\(85\) 11.3346 1.22941
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −8.69193 −0.921342 −0.460671 0.887571i \(-0.652391\pi\)
−0.460671 + 0.887571i \(0.652391\pi\)
\(90\) 0 0
\(91\) −17.2854 −1.81200
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1.59774 0.163924
\(96\) 0 0
\(97\) 16.3909 1.66425 0.832124 0.554590i \(-0.187125\pi\)
0.832124 + 0.554590i \(0.187125\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −5.49646 −0.546918 −0.273459 0.961884i \(-0.588168\pi\)
−0.273459 + 0.961884i \(0.588168\pi\)
\(102\) 0 0
\(103\) 0.992912 0.0978346 0.0489173 0.998803i \(-0.484423\pi\)
0.0489173 + 0.998803i \(0.484423\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −20.5793 −1.98948 −0.994739 0.102440i \(-0.967335\pi\)
−0.994739 + 0.102440i \(0.967335\pi\)
\(108\) 0 0
\(109\) −10.3909 −0.995272 −0.497636 0.867386i \(-0.665798\pi\)
−0.497636 + 0.867386i \(0.665798\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 19.0829 1.79517 0.897583 0.440846i \(-0.145322\pi\)
0.897583 + 0.440846i \(0.145322\pi\)
\(114\) 0 0
\(115\) −5.10552 −0.476092
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −35.0715 −3.21500
\(120\) 0 0
\(121\) 20.3346 1.84860
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 11.8987 1.06425
\(126\) 0 0
\(127\) 15.3839 1.36510 0.682548 0.730841i \(-0.260872\pi\)
0.682548 + 0.730841i \(0.260872\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −6.98159 −0.609984 −0.304992 0.952355i \(-0.598654\pi\)
−0.304992 + 0.952355i \(0.598654\pi\)
\(132\) 0 0
\(133\) −4.94370 −0.428673
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 5.48513 0.468626 0.234313 0.972161i \(-0.424716\pi\)
0.234313 + 0.972161i \(0.424716\pi\)
\(138\) 0 0
\(139\) 0.943698 0.0800435 0.0400217 0.999199i \(-0.487257\pi\)
0.0400217 + 0.999199i \(0.487257\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 19.5722 1.63671
\(144\) 0 0
\(145\) −7.49646 −0.622547
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 4.79321 0.392675 0.196337 0.980536i \(-0.437095\pi\)
0.196337 + 0.980536i \(0.437095\pi\)
\(150\) 0 0
\(151\) −9.49646 −0.772811 −0.386405 0.922329i \(-0.626283\pi\)
−0.386405 + 0.922329i \(0.626283\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 9.58641 0.769999
\(156\) 0 0
\(157\) 2.00000 0.159617 0.0798087 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 15.7974 1.24501
\(162\) 0 0
\(163\) −2.99291 −0.234423 −0.117211 0.993107i \(-0.537396\pi\)
−0.117211 + 0.993107i \(0.537396\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 15.1955 1.17586 0.587930 0.808912i \(-0.299943\pi\)
0.587930 + 0.808912i \(0.299943\pi\)
\(168\) 0 0
\(169\) −0.774794 −0.0595995
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 15.0829 1.14673 0.573365 0.819300i \(-0.305638\pi\)
0.573365 + 0.819300i \(0.305638\pi\)
\(174\) 0 0
\(175\) −12.0984 −0.914555
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 8.57932 0.641249 0.320624 0.947206i \(-0.396107\pi\)
0.320624 + 0.947206i \(0.396107\pi\)
\(180\) 0 0
\(181\) 12.8803 0.957386 0.478693 0.877982i \(-0.341111\pi\)
0.478693 + 0.877982i \(0.341111\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −6.39094 −0.469871
\(186\) 0 0
\(187\) 39.7114 2.90399
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −6.28966 −0.455104 −0.227552 0.973766i \(-0.573072\pi\)
−0.227552 + 0.973766i \(0.573072\pi\)
\(192\) 0 0
\(193\) 9.49646 0.683570 0.341785 0.939778i \(-0.388969\pi\)
0.341785 + 0.939778i \(0.388969\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −21.4965 −1.53156 −0.765780 0.643103i \(-0.777647\pi\)
−0.765780 + 0.643103i \(0.777647\pi\)
\(198\) 0 0
\(199\) −8.94370 −0.634002 −0.317001 0.948425i \(-0.602676\pi\)
−0.317001 + 0.948425i \(0.602676\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 23.1955 1.62800
\(204\) 0 0
\(205\) 4.00000 0.279372
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 5.59774 0.387203
\(210\) 0 0
\(211\) −2.39094 −0.164599 −0.0822996 0.996608i \(-0.526226\pi\)
−0.0822996 + 0.996608i \(0.526226\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 14.2897 0.974547
\(216\) 0 0
\(217\) −29.6622 −2.01360
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 24.8045 1.66853
\(222\) 0 0
\(223\) −4.89448 −0.327759 −0.163879 0.986480i \(-0.552401\pi\)
−0.163879 + 0.986480i \(0.552401\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 15.1955 1.00856 0.504279 0.863541i \(-0.331758\pi\)
0.504279 + 0.863541i \(0.331758\pi\)
\(228\) 0 0
\(229\) 6.44015 0.425577 0.212789 0.977098i \(-0.431745\pi\)
0.212789 + 0.977098i \(0.431745\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −6.89163 −0.451486 −0.225743 0.974187i \(-0.572481\pi\)
−0.225743 + 0.974187i \(0.572481\pi\)
\(234\) 0 0
\(235\) −10.0492 −0.655538
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −2.89163 −0.187044 −0.0935221 0.995617i \(-0.529813\pi\)
−0.0935221 + 0.995617i \(0.529813\pi\)
\(240\) 0 0
\(241\) 4.89448 0.315281 0.157641 0.987497i \(-0.449611\pi\)
0.157641 + 0.987497i \(0.449611\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −27.8647 −1.78021
\(246\) 0 0
\(247\) 3.49646 0.222474
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 24.5906 1.55215 0.776074 0.630642i \(-0.217208\pi\)
0.776074 + 0.630642i \(0.217208\pi\)
\(252\) 0 0
\(253\) −17.8874 −1.12457
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 2.50354 0.156167 0.0780834 0.996947i \(-0.475120\pi\)
0.0780834 + 0.996947i \(0.475120\pi\)
\(258\) 0 0
\(259\) 19.7748 1.22875
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −12.9058 −0.795806 −0.397903 0.917427i \(-0.630262\pi\)
−0.397903 + 0.917427i \(0.630262\pi\)
\(264\) 0 0
\(265\) 9.10552 0.559347
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0.691927 0.0421875 0.0210938 0.999778i \(-0.493285\pi\)
0.0210938 + 0.999778i \(0.493285\pi\)
\(270\) 0 0
\(271\) −16.7819 −1.01943 −0.509713 0.860344i \(-0.670249\pi\)
−0.509713 + 0.860344i \(0.670249\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 13.6990 0.826082
\(276\) 0 0
\(277\) −29.3346 −1.76255 −0.881274 0.472606i \(-0.843313\pi\)
−0.881274 + 0.472606i \(0.843313\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −32.4667 −1.93680 −0.968401 0.249398i \(-0.919767\pi\)
−0.968401 + 0.249398i \(0.919767\pi\)
\(282\) 0 0
\(283\) −17.9508 −1.06706 −0.533532 0.845780i \(-0.679136\pi\)
−0.533532 + 0.845780i \(0.679136\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −12.3768 −0.730577
\(288\) 0 0
\(289\) 33.3276 1.96044
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 22.8803 1.33668 0.668341 0.743855i \(-0.267005\pi\)
0.668341 + 0.743855i \(0.267005\pi\)
\(294\) 0 0
\(295\) −11.1728 −0.650507
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −11.1728 −0.646141
\(300\) 0 0
\(301\) −44.2149 −2.54851
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 5.50778 0.315375
\(306\) 0 0
\(307\) −2.39094 −0.136458 −0.0682291 0.997670i \(-0.521735\pi\)
−0.0682291 + 0.997670i \(0.521735\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 7.89872 0.447895 0.223948 0.974601i \(-0.428106\pi\)
0.223948 + 0.974601i \(0.428106\pi\)
\(312\) 0 0
\(313\) 22.7819 1.28771 0.643854 0.765148i \(-0.277334\pi\)
0.643854 + 0.765148i \(0.277334\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 9.49646 0.533374 0.266687 0.963783i \(-0.414071\pi\)
0.266687 + 0.963783i \(0.414071\pi\)
\(318\) 0 0
\(319\) −26.2642 −1.47051
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 7.09419 0.394731
\(324\) 0 0
\(325\) 8.55668 0.474639
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 31.0942 1.71428
\(330\) 0 0
\(331\) 3.39803 0.186773 0.0933863 0.995630i \(-0.470231\pi\)
0.0933863 + 0.995630i \(0.470231\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 21.3839 1.16832
\(336\) 0 0
\(337\) −8.39094 −0.457084 −0.228542 0.973534i \(-0.573396\pi\)
−0.228542 + 0.973534i \(0.573396\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 33.5864 1.81881
\(342\) 0 0
\(343\) 51.6130 2.78684
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −19.9887 −1.07305 −0.536524 0.843885i \(-0.680263\pi\)
−0.536524 + 0.843885i \(0.680263\pi\)
\(348\) 0 0
\(349\) −21.4331 −1.14729 −0.573643 0.819106i \(-0.694470\pi\)
−0.573643 + 0.819106i \(0.694470\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −23.7748 −1.26540 −0.632702 0.774395i \(-0.718054\pi\)
−0.632702 + 0.774395i \(0.718054\pi\)
\(354\) 0 0
\(355\) 21.3839 1.13494
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 34.8690 1.84031 0.920157 0.391549i \(-0.128061\pi\)
0.920157 + 0.391549i \(0.128061\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 3.89872 0.204068
\(366\) 0 0
\(367\) 5.00709 0.261368 0.130684 0.991424i \(-0.458283\pi\)
0.130684 + 0.991424i \(0.458283\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −28.1742 −1.46273
\(372\) 0 0
\(373\) 24.7819 1.28316 0.641579 0.767057i \(-0.278280\pi\)
0.641579 + 0.767057i \(0.278280\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −16.4051 −0.844906
\(378\) 0 0
\(379\) 13.8874 0.713348 0.356674 0.934229i \(-0.383911\pi\)
0.356674 + 0.934229i \(0.383911\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −15.7974 −0.807212 −0.403606 0.914933i \(-0.632243\pi\)
−0.403606 + 0.914933i \(0.632243\pi\)
\(384\) 0 0
\(385\) 44.2149 2.25340
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 12.9958 0.658911 0.329456 0.944171i \(-0.393135\pi\)
0.329456 + 0.944171i \(0.393135\pi\)
\(390\) 0 0
\(391\) −22.6693 −1.14643
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 15.1728 0.763428
\(396\) 0 0
\(397\) −15.4472 −0.775275 −0.387637 0.921812i \(-0.626709\pi\)
−0.387637 + 0.921812i \(0.626709\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 28.8945 1.44292 0.721461 0.692455i \(-0.243471\pi\)
0.721461 + 0.692455i \(0.243471\pi\)
\(402\) 0 0
\(403\) 20.9787 1.04503
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −22.3909 −1.10988
\(408\) 0 0
\(409\) 28.7677 1.42247 0.711236 0.702954i \(-0.248136\pi\)
0.711236 + 0.702954i \(0.248136\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 34.5709 1.70112
\(414\) 0 0
\(415\) −1.10552 −0.0542676
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 7.08287 0.346021 0.173010 0.984920i \(-0.444651\pi\)
0.173010 + 0.984920i \(0.444651\pi\)
\(420\) 0 0
\(421\) 21.6622 1.05575 0.527875 0.849322i \(-0.322989\pi\)
0.527875 + 0.849322i \(0.322989\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 17.3612 0.842142
\(426\) 0 0
\(427\) −17.0421 −0.824726
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 16.8045 0.809446 0.404723 0.914439i \(-0.367368\pi\)
0.404723 + 0.914439i \(0.367368\pi\)
\(432\) 0 0
\(433\) 11.3839 0.547073 0.273537 0.961862i \(-0.411807\pi\)
0.273537 + 0.961862i \(0.411807\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −3.19547 −0.152860
\(438\) 0 0
\(439\) −35.1586 −1.67803 −0.839015 0.544108i \(-0.816868\pi\)
−0.839015 + 0.544108i \(0.816868\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −18.3796 −0.873242 −0.436621 0.899646i \(-0.643825\pi\)
−0.436621 + 0.899646i \(0.643825\pi\)
\(444\) 0 0
\(445\) 13.8874 0.658326
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −19.8874 −0.938544 −0.469272 0.883054i \(-0.655484\pi\)
−0.469272 + 0.883054i \(0.655484\pi\)
\(450\) 0 0
\(451\) 14.0142 0.659902
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 27.6175 1.29473
\(456\) 0 0
\(457\) −37.1094 −1.73591 −0.867953 0.496646i \(-0.834565\pi\)
−0.867953 + 0.496646i \(0.834565\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −11.2068 −0.521952 −0.260976 0.965345i \(-0.584044\pi\)
−0.260976 + 0.965345i \(0.584044\pi\)
\(462\) 0 0
\(463\) −0.0633891 −0.00294594 −0.00147297 0.999999i \(-0.500469\pi\)
−0.00147297 + 0.999999i \(0.500469\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 41.7493 1.93193 0.965963 0.258678i \(-0.0832870\pi\)
0.965963 + 0.258678i \(0.0832870\pi\)
\(468\) 0 0
\(469\) −66.1657 −3.05525
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 50.0645 2.30197
\(474\) 0 0
\(475\) 2.44724 0.112287
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −7.79744 −0.356274 −0.178137 0.984006i \(-0.557007\pi\)
−0.178137 + 0.984006i \(0.557007\pi\)
\(480\) 0 0
\(481\) −13.9858 −0.637699
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −26.1884 −1.18915
\(486\) 0 0
\(487\) 10.5035 0.475961 0.237981 0.971270i \(-0.423515\pi\)
0.237981 + 0.971270i \(0.423515\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −4.31516 −0.194740 −0.0973702 0.995248i \(-0.531043\pi\)
−0.0973702 + 0.995248i \(0.531043\pi\)
\(492\) 0 0
\(493\) −33.2854 −1.49910
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −66.1657 −2.96794
\(498\) 0 0
\(499\) 11.8382 0.529950 0.264975 0.964255i \(-0.414636\pi\)
0.264975 + 0.964255i \(0.414636\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −15.5722 −0.694332 −0.347166 0.937804i \(-0.612856\pi\)
−0.347166 + 0.937804i \(0.612856\pi\)
\(504\) 0 0
\(505\) 8.78188 0.390789
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −15.6848 −0.695218 −0.347609 0.937640i \(-0.613006\pi\)
−0.347609 + 0.937640i \(0.613006\pi\)
\(510\) 0 0
\(511\) −12.0634 −0.533653
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −1.58641 −0.0699056
\(516\) 0 0
\(517\) −35.2079 −1.54844
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −30.2783 −1.32652 −0.663259 0.748390i \(-0.730827\pi\)
−0.663259 + 0.748390i \(0.730827\pi\)
\(522\) 0 0
\(523\) −0.880309 −0.0384932 −0.0192466 0.999815i \(-0.506127\pi\)
−0.0192466 + 0.999815i \(0.506127\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 42.5651 1.85417
\(528\) 0 0
\(529\) −12.7890 −0.556042
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 8.75353 0.379158
\(534\) 0 0
\(535\) 32.8803 1.42154
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −97.6254 −4.20502
\(540\) 0 0
\(541\) 9.43307 0.405559 0.202780 0.979224i \(-0.435002\pi\)
0.202780 + 0.979224i \(0.435002\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 16.6020 0.711150
\(546\) 0 0
\(547\) 22.4894 0.961576 0.480788 0.876837i \(-0.340351\pi\)
0.480788 + 0.876837i \(0.340351\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −4.69193 −0.199883
\(552\) 0 0
\(553\) −46.9476 −1.99642
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 2.98159 0.126334 0.0631670 0.998003i \(-0.479880\pi\)
0.0631670 + 0.998003i \(0.479880\pi\)
\(558\) 0 0
\(559\) 31.2713 1.32263
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 19.3980 0.817529 0.408765 0.912640i \(-0.365960\pi\)
0.408765 + 0.912640i \(0.365960\pi\)
\(564\) 0 0
\(565\) −30.4894 −1.28270
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 11.3081 0.474059 0.237030 0.971502i \(-0.423826\pi\)
0.237030 + 0.971502i \(0.423826\pi\)
\(570\) 0 0
\(571\) 10.9929 0.460039 0.230020 0.973186i \(-0.426121\pi\)
0.230020 + 0.973186i \(0.426121\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −7.82009 −0.326120
\(576\) 0 0
\(577\) 44.1023 1.83600 0.918002 0.396575i \(-0.129801\pi\)
0.918002 + 0.396575i \(0.129801\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 3.42068 0.141914
\(582\) 0 0
\(583\) 31.9016 1.32123
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −2.40226 −0.0991521 −0.0495760 0.998770i \(-0.515787\pi\)
−0.0495760 + 0.998770i \(0.515787\pi\)
\(588\) 0 0
\(589\) 6.00000 0.247226
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −26.7677 −1.09922 −0.549609 0.835422i \(-0.685223\pi\)
−0.549609 + 0.835422i \(0.685223\pi\)
\(594\) 0 0
\(595\) 56.0350 2.29721
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −8.60197 −0.351467 −0.175734 0.984438i \(-0.556230\pi\)
−0.175734 + 0.984438i \(0.556230\pi\)
\(600\) 0 0
\(601\) 29.4965 1.20319 0.601593 0.798803i \(-0.294533\pi\)
0.601593 + 0.798803i \(0.294533\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −32.4894 −1.32088
\(606\) 0 0
\(607\) −12.6161 −0.512074 −0.256037 0.966667i \(-0.582417\pi\)
−0.256037 + 0.966667i \(0.582417\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −21.9915 −0.889682
\(612\) 0 0
\(613\) −17.3346 −0.700139 −0.350070 0.936724i \(-0.613842\pi\)
−0.350070 + 0.936724i \(0.613842\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −22.6664 −0.912516 −0.456258 0.889848i \(-0.650811\pi\)
−0.456258 + 0.889848i \(0.650811\pi\)
\(618\) 0 0
\(619\) −28.0000 −1.12542 −0.562708 0.826656i \(-0.690240\pi\)
−0.562708 + 0.826656i \(0.690240\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −42.9703 −1.72157
\(624\) 0 0
\(625\) −6.77479 −0.270992
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −28.3768 −1.13146
\(630\) 0 0
\(631\) 2.95787 0.117751 0.0588755 0.998265i \(-0.481248\pi\)
0.0588755 + 0.998265i \(0.481248\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −24.5793 −0.975401
\(636\) 0 0
\(637\) −60.9787 −2.41607
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −11.3081 −0.446642 −0.223321 0.974745i \(-0.571690\pi\)
−0.223321 + 0.974745i \(0.571690\pi\)
\(642\) 0 0
\(643\) −21.7256 −0.856773 −0.428387 0.903596i \(-0.640918\pi\)
−0.428387 + 0.903596i \(0.640918\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −27.0942 −1.06518 −0.532591 0.846373i \(-0.678782\pi\)
−0.532591 + 0.846373i \(0.678782\pi\)
\(648\) 0 0
\(649\) −39.1445 −1.53655
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −32.1912 −1.25974 −0.629870 0.776700i \(-0.716892\pi\)
−0.629870 + 0.776700i \(0.716892\pi\)
\(654\) 0 0
\(655\) 11.1547 0.435851
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 10.9703 0.427341 0.213670 0.976906i \(-0.431458\pi\)
0.213670 + 0.976906i \(0.431458\pi\)
\(660\) 0 0
\(661\) 10.9929 0.427575 0.213787 0.976880i \(-0.431420\pi\)
0.213787 + 0.976880i \(0.431420\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 7.89872 0.306299
\(666\) 0 0
\(667\) 14.9929 0.580528
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 19.2967 0.744943
\(672\) 0 0
\(673\) −10.2783 −0.396201 −0.198100 0.980182i \(-0.563477\pi\)
−0.198100 + 0.980182i \(0.563477\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 36.4667 1.40153 0.700765 0.713392i \(-0.252842\pi\)
0.700765 + 0.713392i \(0.252842\pi\)
\(678\) 0 0
\(679\) 81.0319 3.10972
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −30.5651 −1.16954 −0.584771 0.811198i \(-0.698816\pi\)
−0.584771 + 0.811198i \(0.698816\pi\)
\(684\) 0 0
\(685\) −8.76379 −0.334847
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 19.9264 0.759134
\(690\) 0 0
\(691\) −9.82401 −0.373723 −0.186861 0.982386i \(-0.559832\pi\)
−0.186861 + 0.982386i \(0.559832\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −1.50778 −0.0571934
\(696\) 0 0
\(697\) 17.7606 0.672731
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 34.0531 1.28617 0.643085 0.765795i \(-0.277654\pi\)
0.643085 + 0.765795i \(0.277654\pi\)
\(702\) 0 0
\(703\) −4.00000 −0.150863
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −27.1728 −1.02194
\(708\) 0 0
\(709\) 8.76771 0.329278 0.164639 0.986354i \(-0.447354\pi\)
0.164639 + 0.986354i \(0.447354\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −19.1728 −0.718028
\(714\) 0 0
\(715\) −31.2713 −1.16948
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −24.1013 −0.898826 −0.449413 0.893324i \(-0.648367\pi\)
−0.449413 + 0.893324i \(0.648367\pi\)
\(720\) 0 0
\(721\) 4.90866 0.182808
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −11.4823 −0.426441
\(726\) 0 0
\(727\) −37.8240 −1.40281 −0.701407 0.712761i \(-0.747445\pi\)
−0.701407 + 0.712761i \(0.747445\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 63.4483 2.34672
\(732\) 0 0
\(733\) −6.00000 −0.221615 −0.110808 0.993842i \(-0.535344\pi\)
−0.110808 + 0.993842i \(0.535344\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 74.9193 2.75969
\(738\) 0 0
\(739\) −13.0421 −0.479762 −0.239881 0.970802i \(-0.577108\pi\)
−0.239881 + 0.970802i \(0.577108\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −13.7890 −0.505868 −0.252934 0.967484i \(-0.581396\pi\)
−0.252934 + 0.967484i \(0.581396\pi\)
\(744\) 0 0
\(745\) −7.65827 −0.280577
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −101.738 −3.71742
\(750\) 0 0
\(751\) 19.8874 0.725701 0.362851 0.931847i \(-0.381803\pi\)
0.362851 + 0.931847i \(0.381803\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 15.1728 0.552196
\(756\) 0 0
\(757\) −24.4543 −0.888808 −0.444404 0.895827i \(-0.646584\pi\)
−0.444404 + 0.895827i \(0.646584\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 42.4696 1.53952 0.769760 0.638333i \(-0.220376\pi\)
0.769760 + 0.638333i \(0.220376\pi\)
\(762\) 0 0
\(763\) −51.3697 −1.85971
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −24.4504 −0.882853
\(768\) 0 0
\(769\) 7.54567 0.272104 0.136052 0.990702i \(-0.456559\pi\)
0.136052 + 0.990702i \(0.456559\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 22.5035 0.809396 0.404698 0.914450i \(-0.367377\pi\)
0.404698 + 0.914450i \(0.367377\pi\)
\(774\) 0 0
\(775\) 14.6835 0.527445
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 2.50354 0.0896988
\(780\) 0 0
\(781\) 74.9193 2.68082
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −3.19547 −0.114051
\(786\) 0 0
\(787\) 24.7819 0.883379 0.441689 0.897168i \(-0.354379\pi\)
0.441689 + 0.897168i \(0.354379\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 94.3399 3.35434
\(792\) 0 0
\(793\) 12.0531 0.428019
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 8.11260 0.287363 0.143682 0.989624i \(-0.454106\pi\)
0.143682 + 0.989624i \(0.454106\pi\)
\(798\) 0 0
\(799\) −44.6201 −1.57854
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 13.6593 0.482028
\(804\) 0 0
\(805\) −25.2401 −0.889598
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −18.6948 −0.657273 −0.328637 0.944456i \(-0.606589\pi\)
−0.328637 + 0.944456i \(0.606589\pi\)
\(810\) 0 0
\(811\) −31.0460 −1.09017 −0.545087 0.838379i \(-0.683503\pi\)
−0.545087 + 0.838379i \(0.683503\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 4.78188 0.167502
\(816\) 0 0
\(817\) 8.94370 0.312900
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 17.5977 0.614165 0.307083 0.951683i \(-0.400647\pi\)
0.307083 + 0.951683i \(0.400647\pi\)
\(822\) 0 0
\(823\) 14.7327 0.513549 0.256774 0.966471i \(-0.417340\pi\)
0.256774 + 0.966471i \(0.417340\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −45.5638 −1.58441 −0.792204 0.610257i \(-0.791066\pi\)
−0.792204 + 0.610257i \(0.791066\pi\)
\(828\) 0 0
\(829\) −25.7890 −0.895688 −0.447844 0.894112i \(-0.647808\pi\)
−0.447844 + 0.894112i \(0.647808\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −123.724 −4.28678
\(834\) 0 0
\(835\) −24.2783 −0.840187
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 2.97026 0.102545 0.0512725 0.998685i \(-0.483672\pi\)
0.0512725 + 0.998685i \(0.483672\pi\)
\(840\) 0 0
\(841\) −6.98582 −0.240891
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 1.23792 0.0425856
\(846\) 0 0
\(847\) 100.528 3.45419
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 12.7819 0.438157
\(852\) 0 0
\(853\) 54.5567 1.86798 0.933992 0.357293i \(-0.116300\pi\)
0.933992 + 0.357293i \(0.116300\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 9.27125 0.316700 0.158350 0.987383i \(-0.449383\pi\)
0.158350 + 0.987383i \(0.449383\pi\)
\(858\) 0 0
\(859\) −5.85236 −0.199680 −0.0998399 0.995004i \(-0.531833\pi\)
−0.0998399 + 0.995004i \(0.531833\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −47.5496 −1.61861 −0.809303 0.587391i \(-0.800155\pi\)
−0.809303 + 0.587391i \(0.800155\pi\)
\(864\) 0 0
\(865\) −24.0984 −0.819371
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 53.1586 1.80328
\(870\) 0 0
\(871\) 46.7961 1.58562
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 58.8237 1.98860
\(876\) 0 0
\(877\) 33.6622 1.13669 0.568346 0.822790i \(-0.307584\pi\)
0.568346 + 0.822790i \(0.307584\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −30.6664 −1.03318 −0.516589 0.856233i \(-0.672799\pi\)
−0.516589 + 0.856233i \(0.672799\pi\)
\(882\) 0 0
\(883\) −30.6059 −1.02997 −0.514985 0.857199i \(-0.672203\pi\)
−0.514985 + 0.857199i \(0.672203\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −33.7606 −1.13357 −0.566785 0.823866i \(-0.691813\pi\)
−0.566785 + 0.823866i \(0.691813\pi\)
\(888\) 0 0
\(889\) 76.0531 2.55074
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −6.28966 −0.210476
\(894\) 0 0
\(895\) −13.7075 −0.458191
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −28.1516 −0.938907
\(900\) 0 0
\(901\) 40.4299 1.34692
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −20.5793 −0.684080
\(906\) 0 0
\(907\) −7.87322 −0.261426 −0.130713 0.991420i \(-0.541727\pi\)
−0.130713 + 0.991420i \(0.541727\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 5.02974 0.166643 0.0833213 0.996523i \(-0.473447\pi\)
0.0833213 + 0.996523i \(0.473447\pi\)
\(912\) 0 0
\(913\) −3.87322 −0.128185
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −34.5149 −1.13978
\(918\) 0 0
\(919\) −33.7890 −1.11460 −0.557298 0.830313i \(-0.688162\pi\)
−0.557298 + 0.830313i \(0.688162\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 46.7961 1.54031
\(924\) 0 0
\(925\) −9.78897 −0.321859
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −1.38385 −0.0454028 −0.0227014 0.999742i \(-0.507227\pi\)
−0.0227014 + 0.999742i \(0.507227\pi\)
\(930\) 0 0
\(931\) −17.4402 −0.571578
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −63.4483 −2.07498
\(936\) 0 0
\(937\) −33.2362 −1.08578 −0.542890 0.839804i \(-0.682670\pi\)
−0.542890 + 0.839804i \(0.682670\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −31.4880 −1.02648 −0.513239 0.858245i \(-0.671555\pi\)
−0.513239 + 0.858245i \(0.671555\pi\)
\(942\) 0 0
\(943\) −8.00000 −0.260516
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −46.6778 −1.51682 −0.758412 0.651776i \(-0.774024\pi\)
−0.758412 + 0.651776i \(0.774024\pi\)
\(948\) 0 0
\(949\) 8.53189 0.276957
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −44.4894 −1.44115 −0.720576 0.693376i \(-0.756123\pi\)
−0.720576 + 0.693376i \(0.756123\pi\)
\(954\) 0 0
\(955\) 10.0492 0.325185
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 27.1168 0.875648
\(960\) 0 0
\(961\) 5.00000 0.161290
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −15.1728 −0.488430
\(966\) 0 0
\(967\) −21.9858 −0.707016 −0.353508 0.935431i \(-0.615011\pi\)
−0.353508 + 0.935431i \(0.615011\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 40.0000 1.28366 0.641831 0.766846i \(-0.278175\pi\)
0.641831 + 0.766846i \(0.278175\pi\)
\(972\) 0 0
\(973\) 4.66536 0.149565
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 5.69901 0.182328 0.0911638 0.995836i \(-0.470941\pi\)
0.0911638 + 0.995836i \(0.470941\pi\)
\(978\) 0 0
\(979\) 48.6551 1.55502
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −49.1586 −1.56792 −0.783959 0.620813i \(-0.786803\pi\)
−0.783959 + 0.620813i \(0.786803\pi\)
\(984\) 0 0
\(985\) 34.3456 1.09434
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −28.5793 −0.908770
\(990\) 0 0
\(991\) −2.90866 −0.0923966 −0.0461983 0.998932i \(-0.514711\pi\)
−0.0461983 + 0.998932i \(0.514711\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 14.2897 0.453013
\(996\) 0 0
\(997\) −58.0882 −1.83967 −0.919835 0.392305i \(-0.871678\pi\)
−0.919835 + 0.392305i \(0.871678\pi\)
\(998\) 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 1368.2.a.o.1.1 yes 3
3.2 odd 2 1368.2.a.m.1.3 3
4.3 odd 2 2736.2.a.be.1.1 3
12.11 even 2 2736.2.a.bc.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1368.2.a.m.1.3 3 3.2 odd 2
1368.2.a.o.1.1 yes 3 1.1 even 1 trivial
2736.2.a.bc.1.3 3 12.11 even 2
2736.2.a.be.1.1 3 4.3 odd 2