Properties

Label 9680.2.a.cg.1.2
Level $9680$
Weight $2$
Character 9680.1
Self dual yes
Analytic conductor $77.295$
Analytic rank $0$
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] = [9680,2,Mod(1,9680)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9680, 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("9680.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9680 = 2^{4} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9680.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: \(77.2951891566\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.568.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 6x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 4840)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(3.12489\) of defining polynomial
Character \(\chi\) \(=\) 9680.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.484862 q^{3} +1.00000 q^{5} -0.484862 q^{7} -2.76491 q^{9} +O(q^{10})\) \(q+0.484862 q^{3} +1.00000 q^{5} -0.484862 q^{7} -2.76491 q^{9} -5.28005 q^{13} +0.484862 q^{15} -2.48486 q^{17} -4.73463 q^{19} -0.235091 q^{21} -4.24977 q^{23} +1.00000 q^{25} -2.79518 q^{27} +3.76491 q^{29} +0.235091 q^{31} -0.484862 q^{35} +5.76491 q^{37} -2.56009 q^{39} +0.969724 q^{41} -0.249771 q^{43} -2.76491 q^{45} -3.28005 q^{47} -6.76491 q^{49} -1.20482 q^{51} +5.76491 q^{53} -2.29564 q^{57} +12.4995 q^{59} -9.70436 q^{61} +1.34060 q^{63} -5.28005 q^{65} +10.3103 q^{67} -2.06055 q^{69} +5.70436 q^{71} +1.75023 q^{73} +0.484862 q^{75} +12.0000 q^{79} +6.93945 q^{81} +16.7493 q^{83} -2.48486 q^{85} +1.82546 q^{87} +3.82546 q^{89} +2.56009 q^{91} +0.113987 q^{93} -4.73463 q^{95} +6.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{3} + 3 q^{5} - q^{7} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + q^{3} + 3 q^{5} - q^{7} + 8 q^{9} + q^{15} - 7 q^{17} + 3 q^{19} - 17 q^{21} + 4 q^{23} + 3 q^{25} + 7 q^{27} - 5 q^{29} + 17 q^{31} - q^{35} + q^{37} + 24 q^{39} + 2 q^{41} + 16 q^{43} + 8 q^{45} + 6 q^{47} - 4 q^{49} - 19 q^{51} + q^{53} - 25 q^{57} + 4 q^{59} - 11 q^{61} - 10 q^{63} + 16 q^{67} - 8 q^{69} - q^{71} + 22 q^{73} + q^{75} + 36 q^{79} + 19 q^{81} - 7 q^{85} - 9 q^{87} - 3 q^{89} - 24 q^{91} + 13 q^{93} + 3 q^{95} + 18 q^{97}+O(q^{100}) \) 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
\(3\) 0.484862 0.279935 0.139968 0.990156i \(-0.455300\pi\)
0.139968 + 0.990156i \(0.455300\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −0.484862 −0.183261 −0.0916303 0.995793i \(-0.529208\pi\)
−0.0916303 + 0.995793i \(0.529208\pi\)
\(8\) 0 0
\(9\) −2.76491 −0.921636
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) −5.28005 −1.46442 −0.732211 0.681078i \(-0.761511\pi\)
−0.732211 + 0.681078i \(0.761511\pi\)
\(14\) 0 0
\(15\) 0.484862 0.125191
\(16\) 0 0
\(17\) −2.48486 −0.602668 −0.301334 0.953519i \(-0.597432\pi\)
−0.301334 + 0.953519i \(0.597432\pi\)
\(18\) 0 0
\(19\) −4.73463 −1.08620 −0.543100 0.839668i \(-0.682749\pi\)
−0.543100 + 0.839668i \(0.682749\pi\)
\(20\) 0 0
\(21\) −0.235091 −0.0513011
\(22\) 0 0
\(23\) −4.24977 −0.886138 −0.443069 0.896487i \(-0.646110\pi\)
−0.443069 + 0.896487i \(0.646110\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −2.79518 −0.537934
\(28\) 0 0
\(29\) 3.76491 0.699126 0.349563 0.936913i \(-0.386330\pi\)
0.349563 + 0.936913i \(0.386330\pi\)
\(30\) 0 0
\(31\) 0.235091 0.0422236 0.0211118 0.999777i \(-0.493279\pi\)
0.0211118 + 0.999777i \(0.493279\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −0.484862 −0.0819566
\(36\) 0 0
\(37\) 5.76491 0.947745 0.473873 0.880593i \(-0.342856\pi\)
0.473873 + 0.880593i \(0.342856\pi\)
\(38\) 0 0
\(39\) −2.56009 −0.409943
\(40\) 0 0
\(41\) 0.969724 0.151445 0.0757227 0.997129i \(-0.475874\pi\)
0.0757227 + 0.997129i \(0.475874\pi\)
\(42\) 0 0
\(43\) −0.249771 −0.0380897 −0.0190448 0.999819i \(-0.506063\pi\)
−0.0190448 + 0.999819i \(0.506063\pi\)
\(44\) 0 0
\(45\) −2.76491 −0.412168
\(46\) 0 0
\(47\) −3.28005 −0.478444 −0.239222 0.970965i \(-0.576892\pi\)
−0.239222 + 0.970965i \(0.576892\pi\)
\(48\) 0 0
\(49\) −6.76491 −0.966416
\(50\) 0 0
\(51\) −1.20482 −0.168708
\(52\) 0 0
\(53\) 5.76491 0.791871 0.395936 0.918278i \(-0.370420\pi\)
0.395936 + 0.918278i \(0.370420\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −2.29564 −0.304065
\(58\) 0 0
\(59\) 12.4995 1.62730 0.813651 0.581354i \(-0.197477\pi\)
0.813651 + 0.581354i \(0.197477\pi\)
\(60\) 0 0
\(61\) −9.70436 −1.24252 −0.621258 0.783606i \(-0.713378\pi\)
−0.621258 + 0.783606i \(0.713378\pi\)
\(62\) 0 0
\(63\) 1.34060 0.168900
\(64\) 0 0
\(65\) −5.28005 −0.654909
\(66\) 0 0
\(67\) 10.3103 1.25961 0.629803 0.776755i \(-0.283136\pi\)
0.629803 + 0.776755i \(0.283136\pi\)
\(68\) 0 0
\(69\) −2.06055 −0.248061
\(70\) 0 0
\(71\) 5.70436 0.676983 0.338491 0.940970i \(-0.390083\pi\)
0.338491 + 0.940970i \(0.390083\pi\)
\(72\) 0 0
\(73\) 1.75023 0.204849 0.102424 0.994741i \(-0.467340\pi\)
0.102424 + 0.994741i \(0.467340\pi\)
\(74\) 0 0
\(75\) 0.484862 0.0559870
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 12.0000 1.35011 0.675053 0.737769i \(-0.264121\pi\)
0.675053 + 0.737769i \(0.264121\pi\)
\(80\) 0 0
\(81\) 6.93945 0.771050
\(82\) 0 0
\(83\) 16.7493 1.83848 0.919238 0.393702i \(-0.128806\pi\)
0.919238 + 0.393702i \(0.128806\pi\)
\(84\) 0 0
\(85\) −2.48486 −0.269521
\(86\) 0 0
\(87\) 1.82546 0.195710
\(88\) 0 0
\(89\) 3.82546 0.405498 0.202749 0.979231i \(-0.435012\pi\)
0.202749 + 0.979231i \(0.435012\pi\)
\(90\) 0 0
\(91\) 2.56009 0.268371
\(92\) 0 0
\(93\) 0.113987 0.0118199
\(94\) 0 0
\(95\) −4.73463 −0.485763
\(96\) 0 0
\(97\) 6.00000 0.609208 0.304604 0.952479i \(-0.401476\pi\)
0.304604 + 0.952479i \(0.401476\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −3.03028 −0.301524 −0.150762 0.988570i \(-0.548173\pi\)
−0.150762 + 0.988570i \(0.548173\pi\)
\(102\) 0 0
\(103\) 4.71995 0.465071 0.232535 0.972588i \(-0.425298\pi\)
0.232535 + 0.972588i \(0.425298\pi\)
\(104\) 0 0
\(105\) −0.235091 −0.0229425
\(106\) 0 0
\(107\) 1.21949 0.117893 0.0589465 0.998261i \(-0.481226\pi\)
0.0589465 + 0.998261i \(0.481226\pi\)
\(108\) 0 0
\(109\) −16.0294 −1.53533 −0.767667 0.640849i \(-0.778583\pi\)
−0.767667 + 0.640849i \(0.778583\pi\)
\(110\) 0 0
\(111\) 2.79518 0.265307
\(112\) 0 0
\(113\) −11.4693 −1.07894 −0.539469 0.842006i \(-0.681375\pi\)
−0.539469 + 0.842006i \(0.681375\pi\)
\(114\) 0 0
\(115\) −4.24977 −0.396293
\(116\) 0 0
\(117\) 14.5988 1.34966
\(118\) 0 0
\(119\) 1.20482 0.110445
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) 0.470182 0.0423949
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 4.24977 0.377106 0.188553 0.982063i \(-0.439620\pi\)
0.188553 + 0.982063i \(0.439620\pi\)
\(128\) 0 0
\(129\) −0.121104 −0.0106626
\(130\) 0 0
\(131\) 15.7649 1.37739 0.688693 0.725053i \(-0.258185\pi\)
0.688693 + 0.725053i \(0.258185\pi\)
\(132\) 0 0
\(133\) 2.29564 0.199058
\(134\) 0 0
\(135\) −2.79518 −0.240571
\(136\) 0 0
\(137\) 13.5298 1.15593 0.577965 0.816061i \(-0.303847\pi\)
0.577965 + 0.816061i \(0.303847\pi\)
\(138\) 0 0
\(139\) −16.6206 −1.40974 −0.704872 0.709334i \(-0.748996\pi\)
−0.704872 + 0.709334i \(0.748996\pi\)
\(140\) 0 0
\(141\) −1.59037 −0.133933
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 3.76491 0.312659
\(146\) 0 0
\(147\) −3.28005 −0.270534
\(148\) 0 0
\(149\) −5.20482 −0.426395 −0.213198 0.977009i \(-0.568388\pi\)
−0.213198 + 0.977009i \(0.568388\pi\)
\(150\) 0 0
\(151\) 22.5601 1.83591 0.917957 0.396679i \(-0.129838\pi\)
0.917957 + 0.396679i \(0.129838\pi\)
\(152\) 0 0
\(153\) 6.87042 0.555440
\(154\) 0 0
\(155\) 0.235091 0.0188830
\(156\) 0 0
\(157\) 3.70436 0.295640 0.147820 0.989014i \(-0.452774\pi\)
0.147820 + 0.989014i \(0.452774\pi\)
\(158\) 0 0
\(159\) 2.79518 0.221673
\(160\) 0 0
\(161\) 2.06055 0.162394
\(162\) 0 0
\(163\) 15.5445 1.21754 0.608770 0.793347i \(-0.291663\pi\)
0.608770 + 0.793347i \(0.291663\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 22.0752 1.70823 0.854116 0.520082i \(-0.174099\pi\)
0.854116 + 0.520082i \(0.174099\pi\)
\(168\) 0 0
\(169\) 14.8789 1.14453
\(170\) 0 0
\(171\) 13.0908 1.00108
\(172\) 0 0
\(173\) −2.24977 −0.171047 −0.0855235 0.996336i \(-0.527256\pi\)
−0.0855235 + 0.996336i \(0.527256\pi\)
\(174\) 0 0
\(175\) −0.484862 −0.0366521
\(176\) 0 0
\(177\) 6.06055 0.455539
\(178\) 0 0
\(179\) −5.09083 −0.380506 −0.190253 0.981735i \(-0.560931\pi\)
−0.190253 + 0.981735i \(0.560931\pi\)
\(180\) 0 0
\(181\) 7.46927 0.555186 0.277593 0.960699i \(-0.410463\pi\)
0.277593 + 0.960699i \(0.410463\pi\)
\(182\) 0 0
\(183\) −4.70527 −0.347824
\(184\) 0 0
\(185\) 5.76491 0.423845
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 1.35528 0.0985820
\(190\) 0 0
\(191\) 12.0000 0.868290 0.434145 0.900843i \(-0.357051\pi\)
0.434145 + 0.900843i \(0.357051\pi\)
\(192\) 0 0
\(193\) −2.48486 −0.178864 −0.0894321 0.995993i \(-0.528505\pi\)
−0.0894321 + 0.995993i \(0.528505\pi\)
\(194\) 0 0
\(195\) −2.56009 −0.183332
\(196\) 0 0
\(197\) 16.6888 1.18902 0.594512 0.804086i \(-0.297345\pi\)
0.594512 + 0.804086i \(0.297345\pi\)
\(198\) 0 0
\(199\) −4.73463 −0.335629 −0.167815 0.985819i \(-0.553671\pi\)
−0.167815 + 0.985819i \(0.553671\pi\)
\(200\) 0 0
\(201\) 4.99908 0.352608
\(202\) 0 0
\(203\) −1.82546 −0.128122
\(204\) 0 0
\(205\) 0.969724 0.0677285
\(206\) 0 0
\(207\) 11.7502 0.816697
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −0.235091 −0.0161843 −0.00809217 0.999967i \(-0.502576\pi\)
−0.00809217 + 0.999967i \(0.502576\pi\)
\(212\) 0 0
\(213\) 2.76583 0.189511
\(214\) 0 0
\(215\) −0.249771 −0.0170342
\(216\) 0 0
\(217\) −0.113987 −0.00773792
\(218\) 0 0
\(219\) 0.848620 0.0573444
\(220\) 0 0
\(221\) 13.1202 0.882559
\(222\) 0 0
\(223\) −19.7796 −1.32454 −0.662270 0.749266i \(-0.730407\pi\)
−0.662270 + 0.749266i \(0.730407\pi\)
\(224\) 0 0
\(225\) −2.76491 −0.184327
\(226\) 0 0
\(227\) −17.2195 −1.14290 −0.571449 0.820638i \(-0.693619\pi\)
−0.571449 + 0.820638i \(0.693619\pi\)
\(228\) 0 0
\(229\) 9.40871 0.621745 0.310873 0.950452i \(-0.399379\pi\)
0.310873 + 0.950452i \(0.399379\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −24.9239 −1.63282 −0.816408 0.577476i \(-0.804038\pi\)
−0.816408 + 0.577476i \(0.804038\pi\)
\(234\) 0 0
\(235\) −3.28005 −0.213967
\(236\) 0 0
\(237\) 5.81834 0.377942
\(238\) 0 0
\(239\) −11.5298 −0.745802 −0.372901 0.927871i \(-0.621637\pi\)
−0.372901 + 0.927871i \(0.621637\pi\)
\(240\) 0 0
\(241\) −15.0303 −0.968185 −0.484093 0.875017i \(-0.660850\pi\)
−0.484093 + 0.875017i \(0.660850\pi\)
\(242\) 0 0
\(243\) 11.7502 0.753778
\(244\) 0 0
\(245\) −6.76491 −0.432194
\(246\) 0 0
\(247\) 24.9991 1.59065
\(248\) 0 0
\(249\) 8.12110 0.514654
\(250\) 0 0
\(251\) −5.93945 −0.374895 −0.187447 0.982275i \(-0.560021\pi\)
−0.187447 + 0.982275i \(0.560021\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) −1.20482 −0.0754484
\(256\) 0 0
\(257\) −3.59037 −0.223961 −0.111981 0.993710i \(-0.535719\pi\)
−0.111981 + 0.993710i \(0.535719\pi\)
\(258\) 0 0
\(259\) −2.79518 −0.173684
\(260\) 0 0
\(261\) −10.4096 −0.644340
\(262\) 0 0
\(263\) 16.9844 1.04730 0.523652 0.851933i \(-0.324569\pi\)
0.523652 + 0.851933i \(0.324569\pi\)
\(264\) 0 0
\(265\) 5.76491 0.354136
\(266\) 0 0
\(267\) 1.85482 0.113513
\(268\) 0 0
\(269\) 7.59037 0.462793 0.231397 0.972860i \(-0.425671\pi\)
0.231397 + 0.972860i \(0.425671\pi\)
\(270\) 0 0
\(271\) −14.5601 −0.884463 −0.442231 0.896901i \(-0.645813\pi\)
−0.442231 + 0.896901i \(0.645813\pi\)
\(272\) 0 0
\(273\) 1.24129 0.0751264
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −28.8099 −1.73102 −0.865509 0.500894i \(-0.833005\pi\)
−0.865509 + 0.500894i \(0.833005\pi\)
\(278\) 0 0
\(279\) −0.650006 −0.0389148
\(280\) 0 0
\(281\) −16.9697 −1.01233 −0.506164 0.862437i \(-0.668937\pi\)
−0.506164 + 0.862437i \(0.668937\pi\)
\(282\) 0 0
\(283\) 9.68968 0.575992 0.287996 0.957632i \(-0.407011\pi\)
0.287996 + 0.957632i \(0.407011\pi\)
\(284\) 0 0
\(285\) −2.29564 −0.135982
\(286\) 0 0
\(287\) −0.470182 −0.0277540
\(288\) 0 0
\(289\) −10.8255 −0.636792
\(290\) 0 0
\(291\) 2.90917 0.170539
\(292\) 0 0
\(293\) 6.37088 0.372191 0.186095 0.982532i \(-0.440417\pi\)
0.186095 + 0.982532i \(0.440417\pi\)
\(294\) 0 0
\(295\) 12.4995 0.727751
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 22.4390 1.29768
\(300\) 0 0
\(301\) 0.121104 0.00698034
\(302\) 0 0
\(303\) −1.46927 −0.0844071
\(304\) 0 0
\(305\) −9.70436 −0.555670
\(306\) 0 0
\(307\) −27.6585 −1.57855 −0.789277 0.614038i \(-0.789544\pi\)
−0.789277 + 0.614038i \(0.789544\pi\)
\(308\) 0 0
\(309\) 2.28853 0.130190
\(310\) 0 0
\(311\) −13.7044 −0.777103 −0.388551 0.921427i \(-0.627024\pi\)
−0.388551 + 0.921427i \(0.627024\pi\)
\(312\) 0 0
\(313\) 2.00000 0.113047 0.0565233 0.998401i \(-0.481998\pi\)
0.0565233 + 0.998401i \(0.481998\pi\)
\(314\) 0 0
\(315\) 1.34060 0.0755342
\(316\) 0 0
\(317\) 12.1745 0.683790 0.341895 0.939738i \(-0.388931\pi\)
0.341895 + 0.939738i \(0.388931\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0.591287 0.0330024
\(322\) 0 0
\(323\) 11.7649 0.654617
\(324\) 0 0
\(325\) −5.28005 −0.292884
\(326\) 0 0
\(327\) −7.77203 −0.429794
\(328\) 0 0
\(329\) 1.59037 0.0876799
\(330\) 0 0
\(331\) 29.1202 1.60059 0.800295 0.599606i \(-0.204676\pi\)
0.800295 + 0.599606i \(0.204676\pi\)
\(332\) 0 0
\(333\) −15.9394 −0.873476
\(334\) 0 0
\(335\) 10.3103 0.563313
\(336\) 0 0
\(337\) 16.9239 0.921901 0.460950 0.887426i \(-0.347509\pi\)
0.460950 + 0.887426i \(0.347509\pi\)
\(338\) 0 0
\(339\) −5.56101 −0.302033
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 6.67408 0.360366
\(344\) 0 0
\(345\) −2.06055 −0.110936
\(346\) 0 0
\(347\) −7.77959 −0.417630 −0.208815 0.977955i \(-0.566961\pi\)
−0.208815 + 0.977955i \(0.566961\pi\)
\(348\) 0 0
\(349\) −7.52982 −0.403062 −0.201531 0.979482i \(-0.564592\pi\)
−0.201531 + 0.979482i \(0.564592\pi\)
\(350\) 0 0
\(351\) 14.7587 0.787762
\(352\) 0 0
\(353\) −22.5289 −1.19909 −0.599546 0.800340i \(-0.704652\pi\)
−0.599546 + 0.800340i \(0.704652\pi\)
\(354\) 0 0
\(355\) 5.70436 0.302756
\(356\) 0 0
\(357\) 0.584169 0.0309175
\(358\) 0 0
\(359\) 17.3482 0.915601 0.457800 0.889055i \(-0.348637\pi\)
0.457800 + 0.889055i \(0.348637\pi\)
\(360\) 0 0
\(361\) 3.41675 0.179829
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 1.75023 0.0916112
\(366\) 0 0
\(367\) 27.3094 1.42554 0.712770 0.701398i \(-0.247440\pi\)
0.712770 + 0.701398i \(0.247440\pi\)
\(368\) 0 0
\(369\) −2.68120 −0.139578
\(370\) 0 0
\(371\) −2.79518 −0.145119
\(372\) 0 0
\(373\) −2.24977 −0.116489 −0.0582444 0.998302i \(-0.518550\pi\)
−0.0582444 + 0.998302i \(0.518550\pi\)
\(374\) 0 0
\(375\) 0.484862 0.0250382
\(376\) 0 0
\(377\) −19.8789 −1.02382
\(378\) 0 0
\(379\) −22.4390 −1.15261 −0.576307 0.817233i \(-0.695507\pi\)
−0.576307 + 0.817233i \(0.695507\pi\)
\(380\) 0 0
\(381\) 2.06055 0.105565
\(382\) 0 0
\(383\) −5.21949 −0.266704 −0.133352 0.991069i \(-0.542574\pi\)
−0.133352 + 0.991069i \(0.542574\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0.690594 0.0351048
\(388\) 0 0
\(389\) 27.5904 1.39889 0.699444 0.714688i \(-0.253431\pi\)
0.699444 + 0.714688i \(0.253431\pi\)
\(390\) 0 0
\(391\) 10.5601 0.534047
\(392\) 0 0
\(393\) 7.64380 0.385579
\(394\) 0 0
\(395\) 12.0000 0.603786
\(396\) 0 0
\(397\) 32.9385 1.65314 0.826569 0.562836i \(-0.190290\pi\)
0.826569 + 0.562836i \(0.190290\pi\)
\(398\) 0 0
\(399\) 1.11307 0.0557232
\(400\) 0 0
\(401\) 0.295643 0.0147637 0.00738186 0.999973i \(-0.497650\pi\)
0.00738186 + 0.999973i \(0.497650\pi\)
\(402\) 0 0
\(403\) −1.24129 −0.0618332
\(404\) 0 0
\(405\) 6.93945 0.344824
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −31.4087 −1.55306 −0.776530 0.630080i \(-0.783022\pi\)
−0.776530 + 0.630080i \(0.783022\pi\)
\(410\) 0 0
\(411\) 6.56009 0.323586
\(412\) 0 0
\(413\) −6.06055 −0.298220
\(414\) 0 0
\(415\) 16.7493 0.822191
\(416\) 0 0
\(417\) −8.05872 −0.394637
\(418\) 0 0
\(419\) 32.0294 1.56474 0.782368 0.622816i \(-0.214012\pi\)
0.782368 + 0.622816i \(0.214012\pi\)
\(420\) 0 0
\(421\) −28.9385 −1.41038 −0.705189 0.709020i \(-0.749138\pi\)
−0.705189 + 0.709020i \(0.749138\pi\)
\(422\) 0 0
\(423\) 9.06903 0.440951
\(424\) 0 0
\(425\) −2.48486 −0.120534
\(426\) 0 0
\(427\) 4.70527 0.227704
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −33.6197 −1.61941 −0.809703 0.586840i \(-0.800372\pi\)
−0.809703 + 0.586840i \(0.800372\pi\)
\(432\) 0 0
\(433\) 29.0596 1.39652 0.698258 0.715846i \(-0.253959\pi\)
0.698258 + 0.715846i \(0.253959\pi\)
\(434\) 0 0
\(435\) 1.82546 0.0875242
\(436\) 0 0
\(437\) 20.1211 0.962523
\(438\) 0 0
\(439\) −15.5298 −0.741198 −0.370599 0.928793i \(-0.620848\pi\)
−0.370599 + 0.928793i \(0.620848\pi\)
\(440\) 0 0
\(441\) 18.7044 0.890684
\(442\) 0 0
\(443\) −24.6282 −1.17012 −0.585061 0.810989i \(-0.698929\pi\)
−0.585061 + 0.810989i \(0.698929\pi\)
\(444\) 0 0
\(445\) 3.82546 0.181344
\(446\) 0 0
\(447\) −2.52362 −0.119363
\(448\) 0 0
\(449\) −8.06055 −0.380401 −0.190200 0.981745i \(-0.560914\pi\)
−0.190200 + 0.981745i \(0.560914\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 10.9385 0.513937
\(454\) 0 0
\(455\) 2.56009 0.120019
\(456\) 0 0
\(457\) 10.4849 0.490461 0.245231 0.969465i \(-0.421136\pi\)
0.245231 + 0.969465i \(0.421136\pi\)
\(458\) 0 0
\(459\) 6.94565 0.324195
\(460\) 0 0
\(461\) −36.3856 −1.69464 −0.847322 0.531079i \(-0.821787\pi\)
−0.847322 + 0.531079i \(0.821787\pi\)
\(462\) 0 0
\(463\) −27.3094 −1.26918 −0.634588 0.772851i \(-0.718830\pi\)
−0.634588 + 0.772851i \(0.718830\pi\)
\(464\) 0 0
\(465\) 0.113987 0.00528601
\(466\) 0 0
\(467\) 12.6060 0.583335 0.291667 0.956520i \(-0.405790\pi\)
0.291667 + 0.956520i \(0.405790\pi\)
\(468\) 0 0
\(469\) −4.99908 −0.230836
\(470\) 0 0
\(471\) 1.79610 0.0827600
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −4.73463 −0.217240
\(476\) 0 0
\(477\) −15.9394 −0.729817
\(478\) 0 0
\(479\) 20.5289 0.937989 0.468995 0.883201i \(-0.344616\pi\)
0.468995 + 0.883201i \(0.344616\pi\)
\(480\) 0 0
\(481\) −30.4390 −1.38790
\(482\) 0 0
\(483\) 0.999083 0.0454599
\(484\) 0 0
\(485\) 6.00000 0.272446
\(486\) 0 0
\(487\) −6.68876 −0.303097 −0.151548 0.988450i \(-0.548426\pi\)
−0.151548 + 0.988450i \(0.548426\pi\)
\(488\) 0 0
\(489\) 7.53694 0.340832
\(490\) 0 0
\(491\) 40.8851 1.84512 0.922559 0.385855i \(-0.126094\pi\)
0.922559 + 0.385855i \(0.126094\pi\)
\(492\) 0 0
\(493\) −9.35528 −0.421341
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −2.76583 −0.124064
\(498\) 0 0
\(499\) 20.0294 0.896637 0.448319 0.893874i \(-0.352023\pi\)
0.448319 + 0.893874i \(0.352023\pi\)
\(500\) 0 0
\(501\) 10.7034 0.478194
\(502\) 0 0
\(503\) −17.8108 −0.794143 −0.397072 0.917788i \(-0.629974\pi\)
−0.397072 + 0.917788i \(0.629974\pi\)
\(504\) 0 0
\(505\) −3.03028 −0.134846
\(506\) 0 0
\(507\) 7.21421 0.320394
\(508\) 0 0
\(509\) 8.93853 0.396193 0.198097 0.980182i \(-0.436524\pi\)
0.198097 + 0.980182i \(0.436524\pi\)
\(510\) 0 0
\(511\) −0.848620 −0.0375407
\(512\) 0 0
\(513\) 13.2342 0.584303
\(514\) 0 0
\(515\) 4.71995 0.207986
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −1.09083 −0.0478820
\(520\) 0 0
\(521\) 29.0596 1.27313 0.636563 0.771225i \(-0.280356\pi\)
0.636563 + 0.771225i \(0.280356\pi\)
\(522\) 0 0
\(523\) −32.2791 −1.41147 −0.705734 0.708477i \(-0.749383\pi\)
−0.705734 + 0.708477i \(0.749383\pi\)
\(524\) 0 0
\(525\) −0.235091 −0.0102602
\(526\) 0 0
\(527\) −0.584169 −0.0254468
\(528\) 0 0
\(529\) −4.93945 −0.214759
\(530\) 0 0
\(531\) −34.5601 −1.49978
\(532\) 0 0
\(533\) −5.12019 −0.221780
\(534\) 0 0
\(535\) 1.21949 0.0527234
\(536\) 0 0
\(537\) −2.46835 −0.106517
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 38.3250 1.64772 0.823860 0.566793i \(-0.191816\pi\)
0.823860 + 0.566793i \(0.191816\pi\)
\(542\) 0 0
\(543\) 3.62156 0.155416
\(544\) 0 0
\(545\) −16.0294 −0.686622
\(546\) 0 0
\(547\) 9.71904 0.415556 0.207778 0.978176i \(-0.433377\pi\)
0.207778 + 0.978176i \(0.433377\pi\)
\(548\) 0 0
\(549\) 26.8317 1.14515
\(550\) 0 0
\(551\) −17.8255 −0.759390
\(552\) 0 0
\(553\) −5.81834 −0.247421
\(554\) 0 0
\(555\) 2.79518 0.118649
\(556\) 0 0
\(557\) −16.7805 −0.711013 −0.355506 0.934674i \(-0.615692\pi\)
−0.355506 + 0.934674i \(0.615692\pi\)
\(558\) 0 0
\(559\) 1.31880 0.0557794
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 25.7190 1.08393 0.541964 0.840402i \(-0.317681\pi\)
0.541964 + 0.840402i \(0.317681\pi\)
\(564\) 0 0
\(565\) −11.4693 −0.482516
\(566\) 0 0
\(567\) −3.36467 −0.141303
\(568\) 0 0
\(569\) 39.4087 1.65210 0.826050 0.563597i \(-0.190583\pi\)
0.826050 + 0.563597i \(0.190583\pi\)
\(570\) 0 0
\(571\) 40.4149 1.69131 0.845656 0.533729i \(-0.179210\pi\)
0.845656 + 0.533729i \(0.179210\pi\)
\(572\) 0 0
\(573\) 5.81834 0.243065
\(574\) 0 0
\(575\) −4.24977 −0.177228
\(576\) 0 0
\(577\) −7.34816 −0.305908 −0.152954 0.988233i \(-0.548879\pi\)
−0.152954 + 0.988233i \(0.548879\pi\)
\(578\) 0 0
\(579\) −1.20482 −0.0500704
\(580\) 0 0
\(581\) −8.12110 −0.336920
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 14.5988 0.603588
\(586\) 0 0
\(587\) −41.9835 −1.73284 −0.866422 0.499312i \(-0.833586\pi\)
−0.866422 + 0.499312i \(0.833586\pi\)
\(588\) 0 0
\(589\) −1.11307 −0.0458633
\(590\) 0 0
\(591\) 8.09174 0.332850
\(592\) 0 0
\(593\) −19.3700 −0.795429 −0.397714 0.917509i \(-0.630197\pi\)
−0.397714 + 0.917509i \(0.630197\pi\)
\(594\) 0 0
\(595\) 1.20482 0.0493926
\(596\) 0 0
\(597\) −2.29564 −0.0939544
\(598\) 0 0
\(599\) −4.14335 −0.169293 −0.0846463 0.996411i \(-0.526976\pi\)
−0.0846463 + 0.996411i \(0.526976\pi\)
\(600\) 0 0
\(601\) −15.0303 −0.613098 −0.306549 0.951855i \(-0.599174\pi\)
−0.306549 + 0.951855i \(0.599174\pi\)
\(602\) 0 0
\(603\) −28.5071 −1.16090
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 12.6060 0.511660 0.255830 0.966722i \(-0.417651\pi\)
0.255830 + 0.966722i \(0.417651\pi\)
\(608\) 0 0
\(609\) −0.885097 −0.0358659
\(610\) 0 0
\(611\) 17.3188 0.700644
\(612\) 0 0
\(613\) 14.2791 0.576729 0.288364 0.957521i \(-0.406889\pi\)
0.288364 + 0.957521i \(0.406889\pi\)
\(614\) 0 0
\(615\) 0.470182 0.0189596
\(616\) 0 0
\(617\) 8.18166 0.329381 0.164691 0.986345i \(-0.447337\pi\)
0.164691 + 0.986345i \(0.447337\pi\)
\(618\) 0 0
\(619\) −39.0890 −1.57112 −0.785560 0.618786i \(-0.787625\pi\)
−0.785560 + 0.618786i \(0.787625\pi\)
\(620\) 0 0
\(621\) 11.8789 0.476684
\(622\) 0 0
\(623\) −1.85482 −0.0743118
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −14.3250 −0.571175
\(630\) 0 0
\(631\) 23.8860 0.950887 0.475444 0.879746i \(-0.342288\pi\)
0.475444 + 0.879746i \(0.342288\pi\)
\(632\) 0 0
\(633\) −0.113987 −0.00453057
\(634\) 0 0
\(635\) 4.24977 0.168647
\(636\) 0 0
\(637\) 35.7190 1.41524
\(638\) 0 0
\(639\) −15.7720 −0.623932
\(640\) 0 0
\(641\) −9.64380 −0.380907 −0.190454 0.981696i \(-0.560996\pi\)
−0.190454 + 0.981696i \(0.560996\pi\)
\(642\) 0 0
\(643\) −39.5151 −1.55832 −0.779162 0.626822i \(-0.784355\pi\)
−0.779162 + 0.626822i \(0.784355\pi\)
\(644\) 0 0
\(645\) −0.121104 −0.00476848
\(646\) 0 0
\(647\) 41.8695 1.64606 0.823030 0.567998i \(-0.192282\pi\)
0.823030 + 0.567998i \(0.192282\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −0.0552678 −0.00216612
\(652\) 0 0
\(653\) 9.41583 0.368470 0.184235 0.982882i \(-0.441019\pi\)
0.184235 + 0.982882i \(0.441019\pi\)
\(654\) 0 0
\(655\) 15.7649 0.615986
\(656\) 0 0
\(657\) −4.83922 −0.188796
\(658\) 0 0
\(659\) −12.1433 −0.473038 −0.236519 0.971627i \(-0.576006\pi\)
−0.236519 + 0.971627i \(0.576006\pi\)
\(660\) 0 0
\(661\) −26.0587 −1.01357 −0.506783 0.862073i \(-0.669166\pi\)
−0.506783 + 0.862073i \(0.669166\pi\)
\(662\) 0 0
\(663\) 6.36148 0.247059
\(664\) 0 0
\(665\) 2.29564 0.0890212
\(666\) 0 0
\(667\) −16.0000 −0.619522
\(668\) 0 0
\(669\) −9.59037 −0.370785
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 34.6353 1.33509 0.667547 0.744568i \(-0.267344\pi\)
0.667547 + 0.744568i \(0.267344\pi\)
\(674\) 0 0
\(675\) −2.79518 −0.107587
\(676\) 0 0
\(677\) 22.3709 0.859783 0.429891 0.902881i \(-0.358552\pi\)
0.429891 + 0.902881i \(0.358552\pi\)
\(678\) 0 0
\(679\) −2.90917 −0.111644
\(680\) 0 0
\(681\) −8.34908 −0.319937
\(682\) 0 0
\(683\) 28.6353 1.09570 0.547850 0.836576i \(-0.315446\pi\)
0.547850 + 0.836576i \(0.315446\pi\)
\(684\) 0 0
\(685\) 13.5298 0.516948
\(686\) 0 0
\(687\) 4.56193 0.174048
\(688\) 0 0
\(689\) −30.4390 −1.15963
\(690\) 0 0
\(691\) −7.65092 −0.291055 −0.145527 0.989354i \(-0.546488\pi\)
−0.145527 + 0.989354i \(0.546488\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −16.6206 −0.630457
\(696\) 0 0
\(697\) −2.40963 −0.0912712
\(698\) 0 0
\(699\) −12.0846 −0.457083
\(700\) 0 0
\(701\) −28.8851 −1.09098 −0.545488 0.838119i \(-0.683655\pi\)
−0.545488 + 0.838119i \(0.683655\pi\)
\(702\) 0 0
\(703\) −27.2947 −1.02944
\(704\) 0 0
\(705\) −1.59037 −0.0598968
\(706\) 0 0
\(707\) 1.46927 0.0552574
\(708\) 0 0
\(709\) −42.9991 −1.61486 −0.807432 0.589960i \(-0.799143\pi\)
−0.807432 + 0.589960i \(0.799143\pi\)
\(710\) 0 0
\(711\) −33.1789 −1.24431
\(712\) 0 0
\(713\) −0.999083 −0.0374160
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −5.59037 −0.208776
\(718\) 0 0
\(719\) 42.6741 1.59147 0.795737 0.605642i \(-0.207084\pi\)
0.795737 + 0.605642i \(0.207084\pi\)
\(720\) 0 0
\(721\) −2.28853 −0.0852291
\(722\) 0 0
\(723\) −7.28761 −0.271029
\(724\) 0 0
\(725\) 3.76491 0.139825
\(726\) 0 0
\(727\) 39.9301 1.48092 0.740462 0.672098i \(-0.234607\pi\)
0.740462 + 0.672098i \(0.234607\pi\)
\(728\) 0 0
\(729\) −15.1211 −0.560041
\(730\) 0 0
\(731\) 0.620646 0.0229554
\(732\) 0 0
\(733\) −17.7796 −0.656704 −0.328352 0.944555i \(-0.606493\pi\)
−0.328352 + 0.944555i \(0.606493\pi\)
\(734\) 0 0
\(735\) −3.28005 −0.120986
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −22.4390 −0.825432 −0.412716 0.910860i \(-0.635420\pi\)
−0.412716 + 0.910860i \(0.635420\pi\)
\(740\) 0 0
\(741\) 12.1211 0.445280
\(742\) 0 0
\(743\) −11.6656 −0.427969 −0.213985 0.976837i \(-0.568644\pi\)
−0.213985 + 0.976837i \(0.568644\pi\)
\(744\) 0 0
\(745\) −5.20482 −0.190690
\(746\) 0 0
\(747\) −46.3103 −1.69441
\(748\) 0 0
\(749\) −0.591287 −0.0216051
\(750\) 0 0
\(751\) 50.3544 1.83746 0.918728 0.394890i \(-0.129217\pi\)
0.918728 + 0.394890i \(0.129217\pi\)
\(752\) 0 0
\(753\) −2.87981 −0.104946
\(754\) 0 0
\(755\) 22.5601 0.821046
\(756\) 0 0
\(757\) 50.1798 1.82382 0.911908 0.410394i \(-0.134609\pi\)
0.911908 + 0.410394i \(0.134609\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 35.4087 1.28356 0.641782 0.766887i \(-0.278195\pi\)
0.641782 + 0.766887i \(0.278195\pi\)
\(762\) 0 0
\(763\) 7.77203 0.281366
\(764\) 0 0
\(765\) 6.87042 0.248400
\(766\) 0 0
\(767\) −65.9982 −2.38306
\(768\) 0 0
\(769\) 54.9679 1.98219 0.991096 0.133146i \(-0.0425080\pi\)
0.991096 + 0.133146i \(0.0425080\pi\)
\(770\) 0 0
\(771\) −1.74083 −0.0626946
\(772\) 0 0
\(773\) −25.2947 −0.909788 −0.454894 0.890546i \(-0.650323\pi\)
−0.454894 + 0.890546i \(0.650323\pi\)
\(774\) 0 0
\(775\) 0.235091 0.00844472
\(776\) 0 0
\(777\) −1.35528 −0.0486204
\(778\) 0 0
\(779\) −4.59129 −0.164500
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −10.5236 −0.376083
\(784\) 0 0
\(785\) 3.70436 0.132214
\(786\) 0 0
\(787\) −33.9612 −1.21059 −0.605294 0.796002i \(-0.706944\pi\)
−0.605294 + 0.796002i \(0.706944\pi\)
\(788\) 0 0
\(789\) 8.23509 0.293177
\(790\) 0 0
\(791\) 5.56101 0.197727
\(792\) 0 0
\(793\) 51.2395 1.81957
\(794\) 0 0
\(795\) 2.79518 0.0991350
\(796\) 0 0
\(797\) 48.9385 1.73349 0.866746 0.498750i \(-0.166207\pi\)
0.866746 + 0.498750i \(0.166207\pi\)
\(798\) 0 0
\(799\) 8.15046 0.288343
\(800\) 0 0
\(801\) −10.5771 −0.373722
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 2.06055 0.0726249
\(806\) 0 0
\(807\) 3.68028 0.129552
\(808\) 0 0
\(809\) 39.6803 1.39508 0.697542 0.716544i \(-0.254277\pi\)
0.697542 + 0.716544i \(0.254277\pi\)
\(810\) 0 0
\(811\) −6.70344 −0.235390 −0.117695 0.993050i \(-0.537550\pi\)
−0.117695 + 0.993050i \(0.537550\pi\)
\(812\) 0 0
\(813\) −7.05964 −0.247592
\(814\) 0 0
\(815\) 15.5445 0.544500
\(816\) 0 0
\(817\) 1.18257 0.0413730
\(818\) 0 0
\(819\) −7.07843 −0.247340
\(820\) 0 0
\(821\) −22.4096 −0.782101 −0.391051 0.920369i \(-0.627888\pi\)
−0.391051 + 0.920369i \(0.627888\pi\)
\(822\) 0 0
\(823\) −11.0378 −0.384755 −0.192377 0.981321i \(-0.561620\pi\)
−0.192377 + 0.981321i \(0.561620\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 20.4002 0.709386 0.354693 0.934983i \(-0.384585\pi\)
0.354693 + 0.934983i \(0.384585\pi\)
\(828\) 0 0
\(829\) 21.5298 0.747761 0.373881 0.927477i \(-0.378027\pi\)
0.373881 + 0.927477i \(0.378027\pi\)
\(830\) 0 0
\(831\) −13.9688 −0.484573
\(832\) 0 0
\(833\) 16.8099 0.582427
\(834\) 0 0
\(835\) 22.0752 0.763945
\(836\) 0 0
\(837\) −0.657123 −0.0227135
\(838\) 0 0
\(839\) 23.5592 0.813353 0.406677 0.913572i \(-0.366688\pi\)
0.406677 + 0.913572i \(0.366688\pi\)
\(840\) 0 0
\(841\) −14.8255 −0.511223
\(842\) 0 0
\(843\) −8.22797 −0.283386
\(844\) 0 0
\(845\) 14.8789 0.511850
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 4.69816 0.161240
\(850\) 0 0
\(851\) −24.4995 −0.839833
\(852\) 0 0
\(853\) 19.3406 0.662210 0.331105 0.943594i \(-0.392579\pi\)
0.331105 + 0.943594i \(0.392579\pi\)
\(854\) 0 0
\(855\) 13.0908 0.447697
\(856\) 0 0
\(857\) 37.5739 1.28350 0.641749 0.766915i \(-0.278209\pi\)
0.641749 + 0.766915i \(0.278209\pi\)
\(858\) 0 0
\(859\) 10.4390 0.356174 0.178087 0.984015i \(-0.443009\pi\)
0.178087 + 0.984015i \(0.443009\pi\)
\(860\) 0 0
\(861\) −0.227973 −0.00776932
\(862\) 0 0
\(863\) 43.7796 1.49027 0.745137 0.666911i \(-0.232384\pi\)
0.745137 + 0.666911i \(0.232384\pi\)
\(864\) 0 0
\(865\) −2.24977 −0.0764945
\(866\) 0 0
\(867\) −5.24885 −0.178260
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −54.4390 −1.84459
\(872\) 0 0
\(873\) −16.5895 −0.561468
\(874\) 0 0
\(875\) −0.484862 −0.0163913
\(876\) 0 0
\(877\) 35.6273 1.20305 0.601524 0.798855i \(-0.294560\pi\)
0.601524 + 0.798855i \(0.294560\pi\)
\(878\) 0 0
\(879\) 3.08899 0.104189
\(880\) 0 0
\(881\) −8.18166 −0.275647 −0.137824 0.990457i \(-0.544011\pi\)
−0.137824 + 0.990457i \(0.544011\pi\)
\(882\) 0 0
\(883\) −4.86330 −0.163663 −0.0818315 0.996646i \(-0.526077\pi\)
−0.0818315 + 0.996646i \(0.526077\pi\)
\(884\) 0 0
\(885\) 6.06055 0.203723
\(886\) 0 0
\(887\) 11.2507 0.377761 0.188881 0.982000i \(-0.439514\pi\)
0.188881 + 0.982000i \(0.439514\pi\)
\(888\) 0 0
\(889\) −2.06055 −0.0691087
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 15.5298 0.519686
\(894\) 0 0
\(895\) −5.09083 −0.170168
\(896\) 0 0
\(897\) 10.8798 0.363266
\(898\) 0 0
\(899\) 0.885097 0.0295196
\(900\) 0 0
\(901\) −14.3250 −0.477235
\(902\) 0 0
\(903\) 0.0587189 0.00195404
\(904\) 0 0
\(905\) 7.46927 0.248287
\(906\) 0 0
\(907\) −9.69679 −0.321977 −0.160988 0.986956i \(-0.551468\pi\)
−0.160988 + 0.986956i \(0.551468\pi\)
\(908\) 0 0
\(909\) 8.37844 0.277895
\(910\) 0 0
\(911\) 39.8236 1.31942 0.659708 0.751522i \(-0.270680\pi\)
0.659708 + 0.751522i \(0.270680\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) −4.70527 −0.155552
\(916\) 0 0
\(917\) −7.64380 −0.252421
\(918\) 0 0
\(919\) 1.93945 0.0639765 0.0319882 0.999488i \(-0.489816\pi\)
0.0319882 + 0.999488i \(0.489816\pi\)
\(920\) 0 0
\(921\) −13.4105 −0.441893
\(922\) 0 0
\(923\) −30.1193 −0.991388
\(924\) 0 0
\(925\) 5.76491 0.189549
\(926\) 0 0
\(927\) −13.0502 −0.428626
\(928\) 0 0
\(929\) −48.5823 −1.59393 −0.796967 0.604022i \(-0.793564\pi\)
−0.796967 + 0.604022i \(0.793564\pi\)
\(930\) 0 0
\(931\) 32.0294 1.04972
\(932\) 0 0
\(933\) −6.64472 −0.217538
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −8.80986 −0.287806 −0.143903 0.989592i \(-0.545965\pi\)
−0.143903 + 0.989592i \(0.545965\pi\)
\(938\) 0 0
\(939\) 0.969724 0.0316457
\(940\) 0 0
\(941\) −44.9145 −1.46417 −0.732085 0.681214i \(-0.761452\pi\)
−0.732085 + 0.681214i \(0.761452\pi\)
\(942\) 0 0
\(943\) −4.12110 −0.134202
\(944\) 0 0
\(945\) 1.35528 0.0440872
\(946\) 0 0
\(947\) 13.0761 0.424918 0.212459 0.977170i \(-0.431853\pi\)
0.212459 + 0.977170i \(0.431853\pi\)
\(948\) 0 0
\(949\) −9.24129 −0.299985
\(950\) 0 0
\(951\) 5.90297 0.191417
\(952\) 0 0
\(953\) −25.6362 −0.830439 −0.415220 0.909721i \(-0.636295\pi\)
−0.415220 + 0.909721i \(0.636295\pi\)
\(954\) 0 0
\(955\) 12.0000 0.388311
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −6.56009 −0.211836
\(960\) 0 0
\(961\) −30.9447 −0.998217
\(962\) 0 0
\(963\) −3.37179 −0.108654
\(964\) 0 0
\(965\) −2.48486 −0.0799905
\(966\) 0 0
\(967\) −41.4546 −1.33309 −0.666545 0.745465i \(-0.732227\pi\)
−0.666545 + 0.745465i \(0.732227\pi\)
\(968\) 0 0
\(969\) 5.70436 0.183250
\(970\) 0 0
\(971\) −36.4078 −1.16838 −0.584191 0.811616i \(-0.698588\pi\)
−0.584191 + 0.811616i \(0.698588\pi\)
\(972\) 0 0
\(973\) 8.05872 0.258351
\(974\) 0 0
\(975\) −2.56009 −0.0819886
\(976\) 0 0
\(977\) 30.0587 0.961664 0.480832 0.876813i \(-0.340335\pi\)
0.480832 + 0.876813i \(0.340335\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 44.3197 1.41502
\(982\) 0 0
\(983\) 26.3103 0.839169 0.419584 0.907716i \(-0.362176\pi\)
0.419584 + 0.907716i \(0.362176\pi\)
\(984\) 0 0
\(985\) 16.6888 0.531748
\(986\) 0 0
\(987\) 0.771110 0.0245447
\(988\) 0 0
\(989\) 1.06147 0.0337527
\(990\) 0 0
\(991\) −11.0596 −0.351321 −0.175660 0.984451i \(-0.556206\pi\)
−0.175660 + 0.984451i \(0.556206\pi\)
\(992\) 0 0
\(993\) 14.1193 0.448062
\(994\) 0 0
\(995\) −4.73463 −0.150098
\(996\) 0 0
\(997\) 24.4608 0.774681 0.387340 0.921937i \(-0.373394\pi\)
0.387340 + 0.921937i \(0.373394\pi\)
\(998\) 0 0
\(999\) −16.1140 −0.509824
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 9680.2.a.cg.1.2 3
4.3 odd 2 4840.2.a.r.1.2 yes 3
11.10 odd 2 9680.2.a.ch.1.2 3
44.43 even 2 4840.2.a.q.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4840.2.a.q.1.2 3 44.43 even 2
4840.2.a.r.1.2 yes 3 4.3 odd 2
9680.2.a.cg.1.2 3 1.1 even 1 trivial
9680.2.a.ch.1.2 3 11.10 odd 2