Properties

Label 6080.2.a.bq.1.2
Level $6080$
Weight $2$
Character 6080.1
Self dual yes
Analytic conductor $48.549$
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] = [6080,2,Mod(1,6080)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6080, 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("6080.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6080 = 2^{6} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6080.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: \(48.5490444289\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.229.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 4x - 1 \) 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 760)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.11491\) of defining polynomial
Character \(\chi\) \(=\) 6080.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.642074 q^{3} -1.00000 q^{5} -3.58774 q^{7} -2.58774 q^{9} +O(q^{10})\) \(q-0.642074 q^{3} -1.00000 q^{5} -3.58774 q^{7} -2.58774 q^{9} -1.35793 q^{13} +0.642074 q^{15} +5.58774 q^{17} +1.00000 q^{19} +2.30359 q^{21} +4.87189 q^{23} +1.00000 q^{25} +3.58774 q^{27} -9.58774 q^{29} +7.17548 q^{31} +3.58774 q^{35} +0.945668 q^{37} +0.871889 q^{39} +10.4596 q^{41} -2.71585 q^{43} +2.58774 q^{45} +5.89134 q^{47} +5.87189 q^{49} -3.58774 q^{51} -9.81756 q^{53} -0.642074 q^{57} +10.1560 q^{59} -3.28415 q^{61} +9.28415 q^{63} +1.35793 q^{65} +10.3859 q^{67} -3.12811 q^{69} -14.3510 q^{71} -4.15604 q^{73} -0.642074 q^{75} -1.28415 q^{79} +5.45963 q^{81} -11.1755 q^{83} -5.58774 q^{85} +6.15604 q^{87} -6.45963 q^{89} +4.87189 q^{91} -4.60719 q^{93} -1.00000 q^{95} -13.4053 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{3} - 3 q^{5} + q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{3} - 3 q^{5} + q^{7} + 4 q^{9} - 5 q^{13} + q^{15} + 5 q^{17} + 3 q^{19} - 3 q^{21} + q^{23} + 3 q^{25} - q^{27} - 17 q^{29} - 2 q^{31} - q^{35} - 8 q^{37} - 11 q^{39} + 6 q^{41} - 10 q^{43} - 4 q^{45} - 4 q^{47} + 4 q^{49} + q^{51} - 5 q^{53} - q^{57} + 15 q^{59} - 8 q^{61} + 26 q^{63} + 5 q^{65} + 3 q^{67} - 23 q^{69} + 4 q^{71} + 3 q^{73} - q^{75} - 2 q^{79} - 9 q^{81} - 10 q^{83} - 5 q^{85} + 3 q^{87} + 6 q^{89} + q^{91} + 6 q^{93} - 3 q^{95} - 4 q^{97}+O(q^{100}) \) 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
\(3\) −0.642074 −0.370701 −0.185351 0.982672i \(-0.559342\pi\)
−0.185351 + 0.982672i \(0.559342\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −3.58774 −1.35604 −0.678019 0.735044i \(-0.737161\pi\)
−0.678019 + 0.735044i \(0.737161\pi\)
\(8\) 0 0
\(9\) −2.58774 −0.862580
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) −1.35793 −0.376621 −0.188311 0.982110i \(-0.560301\pi\)
−0.188311 + 0.982110i \(0.560301\pi\)
\(14\) 0 0
\(15\) 0.642074 0.165783
\(16\) 0 0
\(17\) 5.58774 1.35523 0.677613 0.735419i \(-0.263014\pi\)
0.677613 + 0.735419i \(0.263014\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 2.30359 0.502685
\(22\) 0 0
\(23\) 4.87189 1.01586 0.507930 0.861399i \(-0.330411\pi\)
0.507930 + 0.861399i \(0.330411\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 3.58774 0.690461
\(28\) 0 0
\(29\) −9.58774 −1.78040 −0.890199 0.455571i \(-0.849435\pi\)
−0.890199 + 0.455571i \(0.849435\pi\)
\(30\) 0 0
\(31\) 7.17548 1.28875 0.644377 0.764708i \(-0.277117\pi\)
0.644377 + 0.764708i \(0.277117\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 3.58774 0.606439
\(36\) 0 0
\(37\) 0.945668 0.155467 0.0777334 0.996974i \(-0.475232\pi\)
0.0777334 + 0.996974i \(0.475232\pi\)
\(38\) 0 0
\(39\) 0.871889 0.139614
\(40\) 0 0
\(41\) 10.4596 1.63352 0.816760 0.576978i \(-0.195768\pi\)
0.816760 + 0.576978i \(0.195768\pi\)
\(42\) 0 0
\(43\) −2.71585 −0.414164 −0.207082 0.978324i \(-0.566397\pi\)
−0.207082 + 0.978324i \(0.566397\pi\)
\(44\) 0 0
\(45\) 2.58774 0.385758
\(46\) 0 0
\(47\) 5.89134 0.859340 0.429670 0.902986i \(-0.358630\pi\)
0.429670 + 0.902986i \(0.358630\pi\)
\(48\) 0 0
\(49\) 5.87189 0.838841
\(50\) 0 0
\(51\) −3.58774 −0.502384
\(52\) 0 0
\(53\) −9.81756 −1.34855 −0.674273 0.738483i \(-0.735543\pi\)
−0.674273 + 0.738483i \(0.735543\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −0.642074 −0.0850447
\(58\) 0 0
\(59\) 10.1560 1.32220 0.661102 0.750296i \(-0.270089\pi\)
0.661102 + 0.750296i \(0.270089\pi\)
\(60\) 0 0
\(61\) −3.28415 −0.420492 −0.210246 0.977649i \(-0.567426\pi\)
−0.210246 + 0.977649i \(0.567426\pi\)
\(62\) 0 0
\(63\) 9.28415 1.16969
\(64\) 0 0
\(65\) 1.35793 0.168430
\(66\) 0 0
\(67\) 10.3859 1.26883 0.634417 0.772991i \(-0.281240\pi\)
0.634417 + 0.772991i \(0.281240\pi\)
\(68\) 0 0
\(69\) −3.12811 −0.376580
\(70\) 0 0
\(71\) −14.3510 −1.70315 −0.851573 0.524236i \(-0.824351\pi\)
−0.851573 + 0.524236i \(0.824351\pi\)
\(72\) 0 0
\(73\) −4.15604 −0.486427 −0.243214 0.969973i \(-0.578202\pi\)
−0.243214 + 0.969973i \(0.578202\pi\)
\(74\) 0 0
\(75\) −0.642074 −0.0741403
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −1.28415 −0.144478 −0.0722389 0.997387i \(-0.523014\pi\)
−0.0722389 + 0.997387i \(0.523014\pi\)
\(80\) 0 0
\(81\) 5.45963 0.606626
\(82\) 0 0
\(83\) −11.1755 −1.22667 −0.613334 0.789823i \(-0.710172\pi\)
−0.613334 + 0.789823i \(0.710172\pi\)
\(84\) 0 0
\(85\) −5.58774 −0.606076
\(86\) 0 0
\(87\) 6.15604 0.659996
\(88\) 0 0
\(89\) −6.45963 −0.684719 −0.342360 0.939569i \(-0.611226\pi\)
−0.342360 + 0.939569i \(0.611226\pi\)
\(90\) 0 0
\(91\) 4.87189 0.510713
\(92\) 0 0
\(93\) −4.60719 −0.477743
\(94\) 0 0
\(95\) −1.00000 −0.102598
\(96\) 0 0
\(97\) −13.4053 −1.36110 −0.680551 0.732701i \(-0.738259\pi\)
−0.680551 + 0.732701i \(0.738259\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 19.0668 1.89722 0.948610 0.316449i \(-0.102490\pi\)
0.948610 + 0.316449i \(0.102490\pi\)
\(102\) 0 0
\(103\) 9.05433 0.892150 0.446075 0.894996i \(-0.352821\pi\)
0.446075 + 0.894996i \(0.352821\pi\)
\(104\) 0 0
\(105\) −2.30359 −0.224808
\(106\) 0 0
\(107\) −5.24926 −0.507465 −0.253733 0.967274i \(-0.581658\pi\)
−0.253733 + 0.967274i \(0.581658\pi\)
\(108\) 0 0
\(109\) 6.04737 0.579233 0.289617 0.957143i \(-0.406472\pi\)
0.289617 + 0.957143i \(0.406472\pi\)
\(110\) 0 0
\(111\) −0.607188 −0.0576318
\(112\) 0 0
\(113\) −6.22982 −0.586052 −0.293026 0.956105i \(-0.594662\pi\)
−0.293026 + 0.956105i \(0.594662\pi\)
\(114\) 0 0
\(115\) −4.87189 −0.454306
\(116\) 0 0
\(117\) 3.51396 0.324866
\(118\) 0 0
\(119\) −20.0474 −1.83774
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) −6.71585 −0.605548
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 14.1212 1.25305 0.626525 0.779402i \(-0.284477\pi\)
0.626525 + 0.779402i \(0.284477\pi\)
\(128\) 0 0
\(129\) 1.74378 0.153531
\(130\) 0 0
\(131\) −18.3510 −1.60333 −0.801666 0.597773i \(-0.796053\pi\)
−0.801666 + 0.597773i \(0.796053\pi\)
\(132\) 0 0
\(133\) −3.58774 −0.311097
\(134\) 0 0
\(135\) −3.58774 −0.308784
\(136\) 0 0
\(137\) 3.01945 0.257969 0.128984 0.991647i \(-0.458828\pi\)
0.128984 + 0.991647i \(0.458828\pi\)
\(138\) 0 0
\(139\) 11.7827 0.999393 0.499697 0.866201i \(-0.333445\pi\)
0.499697 + 0.866201i \(0.333445\pi\)
\(140\) 0 0
\(141\) −3.78267 −0.318558
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 9.58774 0.796219
\(146\) 0 0
\(147\) −3.77018 −0.310960
\(148\) 0 0
\(149\) −20.4985 −1.67930 −0.839652 0.543124i \(-0.817241\pi\)
−0.839652 + 0.543124i \(0.817241\pi\)
\(150\) 0 0
\(151\) 3.85244 0.313507 0.156754 0.987638i \(-0.449897\pi\)
0.156754 + 0.987638i \(0.449897\pi\)
\(152\) 0 0
\(153\) −14.4596 −1.16899
\(154\) 0 0
\(155\) −7.17548 −0.576349
\(156\) 0 0
\(157\) −3.89134 −0.310562 −0.155281 0.987870i \(-0.549628\pi\)
−0.155281 + 0.987870i \(0.549628\pi\)
\(158\) 0 0
\(159\) 6.30359 0.499908
\(160\) 0 0
\(161\) −17.4791 −1.37754
\(162\) 0 0
\(163\) −18.7159 −1.46594 −0.732969 0.680262i \(-0.761866\pi\)
−0.732969 + 0.680262i \(0.761866\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 3.09323 0.239361 0.119681 0.992812i \(-0.461813\pi\)
0.119681 + 0.992812i \(0.461813\pi\)
\(168\) 0 0
\(169\) −11.1560 −0.858157
\(170\) 0 0
\(171\) −2.58774 −0.197890
\(172\) 0 0
\(173\) −7.51396 −0.571276 −0.285638 0.958338i \(-0.592205\pi\)
−0.285638 + 0.958338i \(0.592205\pi\)
\(174\) 0 0
\(175\) −3.58774 −0.271208
\(176\) 0 0
\(177\) −6.52092 −0.490143
\(178\) 0 0
\(179\) −4.45963 −0.333328 −0.166664 0.986014i \(-0.553300\pi\)
−0.166664 + 0.986014i \(0.553300\pi\)
\(180\) 0 0
\(181\) −8.56829 −0.636876 −0.318438 0.947944i \(-0.603158\pi\)
−0.318438 + 0.947944i \(0.603158\pi\)
\(182\) 0 0
\(183\) 2.10866 0.155877
\(184\) 0 0
\(185\) −0.945668 −0.0695269
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −12.8719 −0.936292
\(190\) 0 0
\(191\) −7.73530 −0.559707 −0.279853 0.960043i \(-0.590286\pi\)
−0.279853 + 0.960043i \(0.590286\pi\)
\(192\) 0 0
\(193\) 6.08226 0.437810 0.218905 0.975746i \(-0.429751\pi\)
0.218905 + 0.975746i \(0.429751\pi\)
\(194\) 0 0
\(195\) −0.871889 −0.0624372
\(196\) 0 0
\(197\) 2.00000 0.142494 0.0712470 0.997459i \(-0.477302\pi\)
0.0712470 + 0.997459i \(0.477302\pi\)
\(198\) 0 0
\(199\) 6.91078 0.489892 0.244946 0.969537i \(-0.421230\pi\)
0.244946 + 0.969537i \(0.421230\pi\)
\(200\) 0 0
\(201\) −6.66848 −0.470358
\(202\) 0 0
\(203\) 34.3983 2.41429
\(204\) 0 0
\(205\) −10.4596 −0.730532
\(206\) 0 0
\(207\) −12.6072 −0.876260
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −12.1949 −0.839534 −0.419767 0.907632i \(-0.637888\pi\)
−0.419767 + 0.907632i \(0.637888\pi\)
\(212\) 0 0
\(213\) 9.21438 0.631359
\(214\) 0 0
\(215\) 2.71585 0.185220
\(216\) 0 0
\(217\) −25.7438 −1.74760
\(218\) 0 0
\(219\) 2.66848 0.180319
\(220\) 0 0
\(221\) −7.58774 −0.510407
\(222\) 0 0
\(223\) 1.87885 0.125817 0.0629085 0.998019i \(-0.479962\pi\)
0.0629085 + 0.998019i \(0.479962\pi\)
\(224\) 0 0
\(225\) −2.58774 −0.172516
\(226\) 0 0
\(227\) 7.35793 0.488363 0.244181 0.969730i \(-0.421481\pi\)
0.244181 + 0.969730i \(0.421481\pi\)
\(228\) 0 0
\(229\) −13.0279 −0.860909 −0.430455 0.902612i \(-0.641647\pi\)
−0.430455 + 0.902612i \(0.641647\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −26.7019 −1.74930 −0.874651 0.484753i \(-0.838909\pi\)
−0.874651 + 0.484753i \(0.838909\pi\)
\(234\) 0 0
\(235\) −5.89134 −0.384308
\(236\) 0 0
\(237\) 0.824517 0.0535581
\(238\) 0 0
\(239\) 9.47908 0.613151 0.306575 0.951846i \(-0.400817\pi\)
0.306575 + 0.951846i \(0.400817\pi\)
\(240\) 0 0
\(241\) −24.2034 −1.55908 −0.779539 0.626353i \(-0.784547\pi\)
−0.779539 + 0.626353i \(0.784547\pi\)
\(242\) 0 0
\(243\) −14.2687 −0.915338
\(244\) 0 0
\(245\) −5.87189 −0.375141
\(246\) 0 0
\(247\) −1.35793 −0.0864028
\(248\) 0 0
\(249\) 7.17548 0.454728
\(250\) 0 0
\(251\) 9.43171 0.595324 0.297662 0.954671i \(-0.403793\pi\)
0.297662 + 0.954671i \(0.403793\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 3.58774 0.224673
\(256\) 0 0
\(257\) 25.1102 1.56633 0.783165 0.621814i \(-0.213604\pi\)
0.783165 + 0.621814i \(0.213604\pi\)
\(258\) 0 0
\(259\) −3.39281 −0.210819
\(260\) 0 0
\(261\) 24.8106 1.53574
\(262\) 0 0
\(263\) 28.7019 1.76984 0.884918 0.465746i \(-0.154214\pi\)
0.884918 + 0.465746i \(0.154214\pi\)
\(264\) 0 0
\(265\) 9.81756 0.603088
\(266\) 0 0
\(267\) 4.14756 0.253826
\(268\) 0 0
\(269\) 0.350966 0.0213988 0.0106994 0.999943i \(-0.496594\pi\)
0.0106994 + 0.999943i \(0.496594\pi\)
\(270\) 0 0
\(271\) −16.8719 −1.02489 −0.512447 0.858719i \(-0.671261\pi\)
−0.512447 + 0.858719i \(0.671261\pi\)
\(272\) 0 0
\(273\) −3.12811 −0.189322
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 6.14756 0.369371 0.184685 0.982798i \(-0.440873\pi\)
0.184685 + 0.982798i \(0.440873\pi\)
\(278\) 0 0
\(279\) −18.5683 −1.11165
\(280\) 0 0
\(281\) −17.0279 −1.01580 −0.507900 0.861416i \(-0.669578\pi\)
−0.507900 + 0.861416i \(0.669578\pi\)
\(282\) 0 0
\(283\) 5.74378 0.341432 0.170716 0.985320i \(-0.445392\pi\)
0.170716 + 0.985320i \(0.445392\pi\)
\(284\) 0 0
\(285\) 0.642074 0.0380332
\(286\) 0 0
\(287\) −37.5264 −2.21512
\(288\) 0 0
\(289\) 14.2229 0.836639
\(290\) 0 0
\(291\) 8.60719 0.504562
\(292\) 0 0
\(293\) 14.2772 0.834082 0.417041 0.908888i \(-0.363067\pi\)
0.417041 + 0.908888i \(0.363067\pi\)
\(294\) 0 0
\(295\) −10.1560 −0.591307
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −6.61567 −0.382594
\(300\) 0 0
\(301\) 9.74378 0.561622
\(302\) 0 0
\(303\) −12.2423 −0.703302
\(304\) 0 0
\(305\) 3.28415 0.188050
\(306\) 0 0
\(307\) 8.83700 0.504354 0.252177 0.967681i \(-0.418853\pi\)
0.252177 + 0.967681i \(0.418853\pi\)
\(308\) 0 0
\(309\) −5.81355 −0.330721
\(310\) 0 0
\(311\) −23.2229 −1.31685 −0.658424 0.752648i \(-0.728776\pi\)
−0.658424 + 0.752648i \(0.728776\pi\)
\(312\) 0 0
\(313\) 30.1421 1.70373 0.851867 0.523759i \(-0.175471\pi\)
0.851867 + 0.523759i \(0.175471\pi\)
\(314\) 0 0
\(315\) −9.28415 −0.523102
\(316\) 0 0
\(317\) −5.96511 −0.335034 −0.167517 0.985869i \(-0.553575\pi\)
−0.167517 + 0.985869i \(0.553575\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 3.37041 0.188118
\(322\) 0 0
\(323\) 5.58774 0.310910
\(324\) 0 0
\(325\) −1.35793 −0.0753242
\(326\) 0 0
\(327\) −3.88286 −0.214723
\(328\) 0 0
\(329\) −21.1366 −1.16530
\(330\) 0 0
\(331\) −2.15604 −0.118506 −0.0592532 0.998243i \(-0.518872\pi\)
−0.0592532 + 0.998243i \(0.518872\pi\)
\(332\) 0 0
\(333\) −2.44714 −0.134103
\(334\) 0 0
\(335\) −10.3859 −0.567440
\(336\) 0 0
\(337\) 0.945668 0.0515138 0.0257569 0.999668i \(-0.491800\pi\)
0.0257569 + 0.999668i \(0.491800\pi\)
\(338\) 0 0
\(339\) 4.00000 0.217250
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 4.04737 0.218538
\(344\) 0 0
\(345\) 3.12811 0.168412
\(346\) 0 0
\(347\) 10.2562 0.550583 0.275291 0.961361i \(-0.411226\pi\)
0.275291 + 0.961361i \(0.411226\pi\)
\(348\) 0 0
\(349\) 24.0558 1.28768 0.643840 0.765160i \(-0.277340\pi\)
0.643840 + 0.765160i \(0.277340\pi\)
\(350\) 0 0
\(351\) −4.87189 −0.260042
\(352\) 0 0
\(353\) 15.3315 0.816014 0.408007 0.912979i \(-0.366224\pi\)
0.408007 + 0.912979i \(0.366224\pi\)
\(354\) 0 0
\(355\) 14.3510 0.761670
\(356\) 0 0
\(357\) 12.8719 0.681253
\(358\) 0 0
\(359\) −15.1281 −0.798431 −0.399216 0.916857i \(-0.630718\pi\)
−0.399216 + 0.916857i \(0.630718\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 7.06281 0.370701
\(364\) 0 0
\(365\) 4.15604 0.217537
\(366\) 0 0
\(367\) 33.6740 1.75777 0.878884 0.477035i \(-0.158288\pi\)
0.878884 + 0.477035i \(0.158288\pi\)
\(368\) 0 0
\(369\) −27.0668 −1.40904
\(370\) 0 0
\(371\) 35.2229 1.82868
\(372\) 0 0
\(373\) −16.0738 −0.832269 −0.416134 0.909303i \(-0.636615\pi\)
−0.416134 + 0.909303i \(0.636615\pi\)
\(374\) 0 0
\(375\) 0.642074 0.0331565
\(376\) 0 0
\(377\) 13.0194 0.670536
\(378\) 0 0
\(379\) −6.00848 −0.308635 −0.154317 0.988021i \(-0.549318\pi\)
−0.154317 + 0.988021i \(0.549318\pi\)
\(380\) 0 0
\(381\) −9.06682 −0.464507
\(382\) 0 0
\(383\) 15.8649 0.810660 0.405330 0.914170i \(-0.367157\pi\)
0.405330 + 0.914170i \(0.367157\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 7.02792 0.357249
\(388\) 0 0
\(389\) −15.7438 −0.798241 −0.399121 0.916898i \(-0.630685\pi\)
−0.399121 + 0.916898i \(0.630685\pi\)
\(390\) 0 0
\(391\) 27.2229 1.37672
\(392\) 0 0
\(393\) 11.7827 0.594357
\(394\) 0 0
\(395\) 1.28415 0.0646125
\(396\) 0 0
\(397\) 23.9861 1.20383 0.601913 0.798561i \(-0.294405\pi\)
0.601913 + 0.798561i \(0.294405\pi\)
\(398\) 0 0
\(399\) 2.30359 0.115324
\(400\) 0 0
\(401\) −24.5683 −1.22688 −0.613441 0.789741i \(-0.710215\pi\)
−0.613441 + 0.789741i \(0.710215\pi\)
\(402\) 0 0
\(403\) −9.74378 −0.485372
\(404\) 0 0
\(405\) −5.45963 −0.271291
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −14.9193 −0.737710 −0.368855 0.929487i \(-0.620250\pi\)
−0.368855 + 0.929487i \(0.620250\pi\)
\(410\) 0 0
\(411\) −1.93871 −0.0956294
\(412\) 0 0
\(413\) −36.4372 −1.79296
\(414\) 0 0
\(415\) 11.1755 0.548583
\(416\) 0 0
\(417\) −7.56534 −0.370476
\(418\) 0 0
\(419\) 14.0389 0.685845 0.342922 0.939364i \(-0.388583\pi\)
0.342922 + 0.939364i \(0.388583\pi\)
\(420\) 0 0
\(421\) −39.9387 −1.94649 −0.973247 0.229762i \(-0.926205\pi\)
−0.973247 + 0.229762i \(0.926205\pi\)
\(422\) 0 0
\(423\) −15.2453 −0.741250
\(424\) 0 0
\(425\) 5.58774 0.271045
\(426\) 0 0
\(427\) 11.7827 0.570203
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −27.0279 −1.30189 −0.650945 0.759125i \(-0.725627\pi\)
−0.650945 + 0.759125i \(0.725627\pi\)
\(432\) 0 0
\(433\) −40.0683 −1.92556 −0.962781 0.270284i \(-0.912882\pi\)
−0.962781 + 0.270284i \(0.912882\pi\)
\(434\) 0 0
\(435\) −6.15604 −0.295159
\(436\) 0 0
\(437\) 4.87189 0.233054
\(438\) 0 0
\(439\) −6.42074 −0.306445 −0.153223 0.988192i \(-0.548965\pi\)
−0.153223 + 0.988192i \(0.548965\pi\)
\(440\) 0 0
\(441\) −15.1949 −0.723568
\(442\) 0 0
\(443\) −35.2702 −1.67574 −0.837870 0.545871i \(-0.816199\pi\)
−0.837870 + 0.545871i \(0.816199\pi\)
\(444\) 0 0
\(445\) 6.45963 0.306216
\(446\) 0 0
\(447\) 13.1616 0.622521
\(448\) 0 0
\(449\) 5.78267 0.272901 0.136451 0.990647i \(-0.456431\pi\)
0.136451 + 0.990647i \(0.456431\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −2.47355 −0.116218
\(454\) 0 0
\(455\) −4.87189 −0.228398
\(456\) 0 0
\(457\) −9.12811 −0.426995 −0.213498 0.976944i \(-0.568486\pi\)
−0.213498 + 0.976944i \(0.568486\pi\)
\(458\) 0 0
\(459\) 20.0474 0.935731
\(460\) 0 0
\(461\) −8.86341 −0.412810 −0.206405 0.978467i \(-0.566176\pi\)
−0.206405 + 0.978467i \(0.566176\pi\)
\(462\) 0 0
\(463\) 21.1366 0.982301 0.491150 0.871075i \(-0.336577\pi\)
0.491150 + 0.871075i \(0.336577\pi\)
\(464\) 0 0
\(465\) 4.60719 0.213653
\(466\) 0 0
\(467\) 7.78267 0.360139 0.180070 0.983654i \(-0.442368\pi\)
0.180070 + 0.983654i \(0.442368\pi\)
\(468\) 0 0
\(469\) −37.2617 −1.72059
\(470\) 0 0
\(471\) 2.49852 0.115126
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 1.00000 0.0458831
\(476\) 0 0
\(477\) 25.4053 1.16323
\(478\) 0 0
\(479\) −40.4068 −1.84623 −0.923117 0.384519i \(-0.874367\pi\)
−0.923117 + 0.384519i \(0.874367\pi\)
\(480\) 0 0
\(481\) −1.28415 −0.0585521
\(482\) 0 0
\(483\) 11.2229 0.510658
\(484\) 0 0
\(485\) 13.4053 0.608703
\(486\) 0 0
\(487\) −5.82603 −0.264003 −0.132001 0.991250i \(-0.542140\pi\)
−0.132001 + 0.991250i \(0.542140\pi\)
\(488\) 0 0
\(489\) 12.0170 0.543426
\(490\) 0 0
\(491\) 22.6630 1.02277 0.511384 0.859352i \(-0.329133\pi\)
0.511384 + 0.859352i \(0.329133\pi\)
\(492\) 0 0
\(493\) −53.5738 −2.41284
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 51.4876 2.30953
\(498\) 0 0
\(499\) −25.7438 −1.15245 −0.576225 0.817291i \(-0.695475\pi\)
−0.576225 + 0.817291i \(0.695475\pi\)
\(500\) 0 0
\(501\) −1.98608 −0.0887315
\(502\) 0 0
\(503\) −25.4791 −1.13606 −0.568028 0.823009i \(-0.692293\pi\)
−0.568028 + 0.823009i \(0.692293\pi\)
\(504\) 0 0
\(505\) −19.0668 −0.848462
\(506\) 0 0
\(507\) 7.16300 0.318120
\(508\) 0 0
\(509\) 9.54037 0.422869 0.211435 0.977392i \(-0.432186\pi\)
0.211435 + 0.977392i \(0.432186\pi\)
\(510\) 0 0
\(511\) 14.9108 0.659614
\(512\) 0 0
\(513\) 3.58774 0.158403
\(514\) 0 0
\(515\) −9.05433 −0.398982
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 4.82452 0.211773
\(520\) 0 0
\(521\) 6.31207 0.276537 0.138268 0.990395i \(-0.455846\pi\)
0.138268 + 0.990395i \(0.455846\pi\)
\(522\) 0 0
\(523\) 8.93719 0.390796 0.195398 0.980724i \(-0.437400\pi\)
0.195398 + 0.980724i \(0.437400\pi\)
\(524\) 0 0
\(525\) 2.30359 0.100537
\(526\) 0 0
\(527\) 40.0947 1.74655
\(528\) 0 0
\(529\) 0.735300 0.0319696
\(530\) 0 0
\(531\) −26.2812 −1.14051
\(532\) 0 0
\(533\) −14.2034 −0.615218
\(534\) 0 0
\(535\) 5.24926 0.226945
\(536\) 0 0
\(537\) 2.86341 0.123565
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −8.71585 −0.374724 −0.187362 0.982291i \(-0.559994\pi\)
−0.187362 + 0.982291i \(0.559994\pi\)
\(542\) 0 0
\(543\) 5.50148 0.236091
\(544\) 0 0
\(545\) −6.04737 −0.259041
\(546\) 0 0
\(547\) −26.9457 −1.15211 −0.576057 0.817410i \(-0.695409\pi\)
−0.576057 + 0.817410i \(0.695409\pi\)
\(548\) 0 0
\(549\) 8.49852 0.362708
\(550\) 0 0
\(551\) −9.58774 −0.408452
\(552\) 0 0
\(553\) 4.60719 0.195918
\(554\) 0 0
\(555\) 0.607188 0.0257737
\(556\) 0 0
\(557\) 23.3619 0.989877 0.494938 0.868928i \(-0.335191\pi\)
0.494938 + 0.868928i \(0.335191\pi\)
\(558\) 0 0
\(559\) 3.68793 0.155983
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 31.6476 1.33379 0.666894 0.745153i \(-0.267624\pi\)
0.666894 + 0.745153i \(0.267624\pi\)
\(564\) 0 0
\(565\) 6.22982 0.262090
\(566\) 0 0
\(567\) −19.5877 −0.822608
\(568\) 0 0
\(569\) −20.8804 −0.875350 −0.437675 0.899133i \(-0.644198\pi\)
−0.437675 + 0.899133i \(0.644198\pi\)
\(570\) 0 0
\(571\) 42.6630 1.78539 0.892696 0.450659i \(-0.148811\pi\)
0.892696 + 0.450659i \(0.148811\pi\)
\(572\) 0 0
\(573\) 4.96663 0.207484
\(574\) 0 0
\(575\) 4.87189 0.203172
\(576\) 0 0
\(577\) −8.59871 −0.357969 −0.178985 0.983852i \(-0.557281\pi\)
−0.178985 + 0.983852i \(0.557281\pi\)
\(578\) 0 0
\(579\) −3.90526 −0.162297
\(580\) 0 0
\(581\) 40.0947 1.66341
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −3.51396 −0.145284
\(586\) 0 0
\(587\) 29.7438 1.22766 0.613829 0.789439i \(-0.289629\pi\)
0.613829 + 0.789439i \(0.289629\pi\)
\(588\) 0 0
\(589\) 7.17548 0.295661
\(590\) 0 0
\(591\) −1.28415 −0.0528228
\(592\) 0 0
\(593\) 4.56829 0.187597 0.0937987 0.995591i \(-0.470099\pi\)
0.0937987 + 0.995591i \(0.470099\pi\)
\(594\) 0 0
\(595\) 20.0474 0.821862
\(596\) 0 0
\(597\) −4.43723 −0.181604
\(598\) 0 0
\(599\) −14.7159 −0.601273 −0.300637 0.953739i \(-0.597199\pi\)
−0.300637 + 0.953739i \(0.597199\pi\)
\(600\) 0 0
\(601\) 37.3230 1.52244 0.761219 0.648495i \(-0.224601\pi\)
0.761219 + 0.648495i \(0.224601\pi\)
\(602\) 0 0
\(603\) −26.8759 −1.09447
\(604\) 0 0
\(605\) 11.0000 0.447214
\(606\) 0 0
\(607\) −23.0404 −0.935181 −0.467591 0.883945i \(-0.654878\pi\)
−0.467591 + 0.883945i \(0.654878\pi\)
\(608\) 0 0
\(609\) −22.0863 −0.894981
\(610\) 0 0
\(611\) −8.00000 −0.323645
\(612\) 0 0
\(613\) −32.4985 −1.31260 −0.656302 0.754499i \(-0.727880\pi\)
−0.656302 + 0.754499i \(0.727880\pi\)
\(614\) 0 0
\(615\) 6.71585 0.270809
\(616\) 0 0
\(617\) −32.2982 −1.30027 −0.650137 0.759817i \(-0.725289\pi\)
−0.650137 + 0.759817i \(0.725289\pi\)
\(618\) 0 0
\(619\) −34.5683 −1.38942 −0.694709 0.719291i \(-0.744467\pi\)
−0.694709 + 0.719291i \(0.744467\pi\)
\(620\) 0 0
\(621\) 17.4791 0.701411
\(622\) 0 0
\(623\) 23.1755 0.928506
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 5.28415 0.210693
\(630\) 0 0
\(631\) 19.2702 0.767136 0.383568 0.923513i \(-0.374695\pi\)
0.383568 + 0.923513i \(0.374695\pi\)
\(632\) 0 0
\(633\) 7.83004 0.311216
\(634\) 0 0
\(635\) −14.1212 −0.560381
\(636\) 0 0
\(637\) −7.97359 −0.315925
\(638\) 0 0
\(639\) 37.1366 1.46910
\(640\) 0 0
\(641\) −3.06682 −0.121132 −0.0605660 0.998164i \(-0.519291\pi\)
−0.0605660 + 0.998164i \(0.519291\pi\)
\(642\) 0 0
\(643\) −25.5264 −1.00666 −0.503332 0.864093i \(-0.667893\pi\)
−0.503332 + 0.864093i \(0.667893\pi\)
\(644\) 0 0
\(645\) −1.74378 −0.0686612
\(646\) 0 0
\(647\) −25.4791 −1.00169 −0.500843 0.865538i \(-0.666977\pi\)
−0.500843 + 0.865538i \(0.666977\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 16.5294 0.647838
\(652\) 0 0
\(653\) 36.2034 1.41675 0.708374 0.705837i \(-0.249429\pi\)
0.708374 + 0.705837i \(0.249429\pi\)
\(654\) 0 0
\(655\) 18.3510 0.717032
\(656\) 0 0
\(657\) 10.7547 0.419583
\(658\) 0 0
\(659\) −42.6406 −1.66104 −0.830522 0.556986i \(-0.811958\pi\)
−0.830522 + 0.556986i \(0.811958\pi\)
\(660\) 0 0
\(661\) −24.4681 −0.951699 −0.475850 0.879527i \(-0.657859\pi\)
−0.475850 + 0.879527i \(0.657859\pi\)
\(662\) 0 0
\(663\) 4.87189 0.189208
\(664\) 0 0
\(665\) 3.58774 0.139127
\(666\) 0 0
\(667\) −46.7104 −1.80863
\(668\) 0 0
\(669\) −1.20636 −0.0466406
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −26.7672 −1.03180 −0.515901 0.856649i \(-0.672543\pi\)
−0.515901 + 0.856649i \(0.672543\pi\)
\(674\) 0 0
\(675\) 3.58774 0.138092
\(676\) 0 0
\(677\) 5.65304 0.217264 0.108632 0.994082i \(-0.465353\pi\)
0.108632 + 0.994082i \(0.465353\pi\)
\(678\) 0 0
\(679\) 48.0947 1.84571
\(680\) 0 0
\(681\) −4.72433 −0.181037
\(682\) 0 0
\(683\) −11.1102 −0.425119 −0.212560 0.977148i \(-0.568180\pi\)
−0.212560 + 0.977148i \(0.568180\pi\)
\(684\) 0 0
\(685\) −3.01945 −0.115367
\(686\) 0 0
\(687\) 8.36489 0.319140
\(688\) 0 0
\(689\) 13.3315 0.507890
\(690\) 0 0
\(691\) −5.96111 −0.226771 −0.113386 0.993551i \(-0.536170\pi\)
−0.113386 + 0.993551i \(0.536170\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −11.7827 −0.446942
\(696\) 0 0
\(697\) 58.4457 2.21379
\(698\) 0 0
\(699\) 17.1446 0.648469
\(700\) 0 0
\(701\) 25.8524 0.976433 0.488217 0.872722i \(-0.337648\pi\)
0.488217 + 0.872722i \(0.337648\pi\)
\(702\) 0 0
\(703\) 0.945668 0.0356665
\(704\) 0 0
\(705\) 3.78267 0.142464
\(706\) 0 0
\(707\) −68.4068 −2.57270
\(708\) 0 0
\(709\) 23.4317 0.879996 0.439998 0.897999i \(-0.354979\pi\)
0.439998 + 0.897999i \(0.354979\pi\)
\(710\) 0 0
\(711\) 3.32304 0.124624
\(712\) 0 0
\(713\) 34.9582 1.30919
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −6.08627 −0.227296
\(718\) 0 0
\(719\) 6.20885 0.231551 0.115776 0.993275i \(-0.463065\pi\)
0.115776 + 0.993275i \(0.463065\pi\)
\(720\) 0 0
\(721\) −32.4846 −1.20979
\(722\) 0 0
\(723\) 15.5404 0.577953
\(724\) 0 0
\(725\) −9.58774 −0.356080
\(726\) 0 0
\(727\) −29.7214 −1.10230 −0.551152 0.834405i \(-0.685812\pi\)
−0.551152 + 0.834405i \(0.685812\pi\)
\(728\) 0 0
\(729\) −7.21733 −0.267308
\(730\) 0 0
\(731\) −15.1755 −0.561286
\(732\) 0 0
\(733\) 7.52645 0.277996 0.138998 0.990293i \(-0.455612\pi\)
0.138998 + 0.990293i \(0.455612\pi\)
\(734\) 0 0
\(735\) 3.77018 0.139065
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 22.9582 0.844529 0.422265 0.906473i \(-0.361235\pi\)
0.422265 + 0.906473i \(0.361235\pi\)
\(740\) 0 0
\(741\) 0.871889 0.0320296
\(742\) 0 0
\(743\) −0.524931 −0.0192579 −0.00962893 0.999954i \(-0.503065\pi\)
−0.00962893 + 0.999954i \(0.503065\pi\)
\(744\) 0 0
\(745\) 20.4985 0.751008
\(746\) 0 0
\(747\) 28.9193 1.05810
\(748\) 0 0
\(749\) 18.8330 0.688143
\(750\) 0 0
\(751\) −21.3619 −0.779508 −0.389754 0.920919i \(-0.627440\pi\)
−0.389754 + 0.920919i \(0.627440\pi\)
\(752\) 0 0
\(753\) −6.05585 −0.220687
\(754\) 0 0
\(755\) −3.85244 −0.140205
\(756\) 0 0
\(757\) −50.4068 −1.83207 −0.916033 0.401102i \(-0.868627\pi\)
−0.916033 + 0.401102i \(0.868627\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −20.2338 −0.733476 −0.366738 0.930324i \(-0.619525\pi\)
−0.366738 + 0.930324i \(0.619525\pi\)
\(762\) 0 0
\(763\) −21.6964 −0.785463
\(764\) 0 0
\(765\) 14.4596 0.522789
\(766\) 0 0
\(767\) −13.7911 −0.497970
\(768\) 0 0
\(769\) −27.1142 −0.977763 −0.488881 0.872350i \(-0.662595\pi\)
−0.488881 + 0.872350i \(0.662595\pi\)
\(770\) 0 0
\(771\) −16.1226 −0.580641
\(772\) 0 0
\(773\) 12.1685 0.437671 0.218836 0.975762i \(-0.429774\pi\)
0.218836 + 0.975762i \(0.429774\pi\)
\(774\) 0 0
\(775\) 7.17548 0.257751
\(776\) 0 0
\(777\) 2.17843 0.0781509
\(778\) 0 0
\(779\) 10.4596 0.374755
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −34.3983 −1.22930
\(784\) 0 0
\(785\) 3.89134 0.138888
\(786\) 0 0
\(787\) −36.1296 −1.28788 −0.643941 0.765075i \(-0.722702\pi\)
−0.643941 + 0.765075i \(0.722702\pi\)
\(788\) 0 0
\(789\) −18.4288 −0.656081
\(790\) 0 0
\(791\) 22.3510 0.794709
\(792\) 0 0
\(793\) 4.45963 0.158366
\(794\) 0 0
\(795\) −6.30359 −0.223565
\(796\) 0 0
\(797\) −42.4417 −1.50336 −0.751681 0.659527i \(-0.770757\pi\)
−0.751681 + 0.659527i \(0.770757\pi\)
\(798\) 0 0
\(799\) 32.9193 1.16460
\(800\) 0 0
\(801\) 16.7159 0.590626
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 17.4791 0.616057
\(806\) 0 0
\(807\) −0.225346 −0.00793255
\(808\) 0 0
\(809\) −37.3006 −1.31142 −0.655710 0.755013i \(-0.727631\pi\)
−0.655710 + 0.755013i \(0.727631\pi\)
\(810\) 0 0
\(811\) 27.4402 0.963555 0.481778 0.876294i \(-0.339991\pi\)
0.481778 + 0.876294i \(0.339991\pi\)
\(812\) 0 0
\(813\) 10.8330 0.379930
\(814\) 0 0
\(815\) 18.7159 0.655588
\(816\) 0 0
\(817\) −2.71585 −0.0950157
\(818\) 0 0
\(819\) −12.6072 −0.440531
\(820\) 0 0
\(821\) −47.0140 −1.64080 −0.820400 0.571790i \(-0.806249\pi\)
−0.820400 + 0.571790i \(0.806249\pi\)
\(822\) 0 0
\(823\) −35.9776 −1.25410 −0.627050 0.778979i \(-0.715738\pi\)
−0.627050 + 0.778979i \(0.715738\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 14.6032 0.507802 0.253901 0.967230i \(-0.418286\pi\)
0.253901 + 0.967230i \(0.418286\pi\)
\(828\) 0 0
\(829\) 6.96663 0.241961 0.120981 0.992655i \(-0.461396\pi\)
0.120981 + 0.992655i \(0.461396\pi\)
\(830\) 0 0
\(831\) −3.94719 −0.136926
\(832\) 0 0
\(833\) 32.8106 1.13682
\(834\) 0 0
\(835\) −3.09323 −0.107046
\(836\) 0 0
\(837\) 25.7438 0.889835
\(838\) 0 0
\(839\) 0.459630 0.0158682 0.00793410 0.999969i \(-0.497474\pi\)
0.00793410 + 0.999969i \(0.497474\pi\)
\(840\) 0 0
\(841\) 62.9248 2.16982
\(842\) 0 0
\(843\) 10.9332 0.376559
\(844\) 0 0
\(845\) 11.1560 0.383779
\(846\) 0 0
\(847\) 39.4652 1.35604
\(848\) 0 0
\(849\) −3.68793 −0.126569
\(850\) 0 0
\(851\) 4.60719 0.157932
\(852\) 0 0
\(853\) 5.48755 0.187890 0.0939451 0.995577i \(-0.470052\pi\)
0.0939451 + 0.995577i \(0.470052\pi\)
\(854\) 0 0
\(855\) 2.58774 0.0884989
\(856\) 0 0
\(857\) 52.6586 1.79878 0.899391 0.437145i \(-0.144010\pi\)
0.899391 + 0.437145i \(0.144010\pi\)
\(858\) 0 0
\(859\) −4.29512 −0.146547 −0.0732737 0.997312i \(-0.523345\pi\)
−0.0732737 + 0.997312i \(0.523345\pi\)
\(860\) 0 0
\(861\) 24.0947 0.821147
\(862\) 0 0
\(863\) 14.6506 0.498711 0.249355 0.968412i \(-0.419781\pi\)
0.249355 + 0.968412i \(0.419781\pi\)
\(864\) 0 0
\(865\) 7.51396 0.255482
\(866\) 0 0
\(867\) −9.13212 −0.310143
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −14.1032 −0.477869
\(872\) 0 0
\(873\) 34.6894 1.17406
\(874\) 0 0
\(875\) 3.58774 0.121288
\(876\) 0 0
\(877\) 19.7089 0.665522 0.332761 0.943011i \(-0.392020\pi\)
0.332761 + 0.943011i \(0.392020\pi\)
\(878\) 0 0
\(879\) −9.16701 −0.309195
\(880\) 0 0
\(881\) 1.32304 0.0445744 0.0222872 0.999752i \(-0.492905\pi\)
0.0222872 + 0.999752i \(0.492905\pi\)
\(882\) 0 0
\(883\) −25.0668 −0.843566 −0.421783 0.906697i \(-0.638596\pi\)
−0.421783 + 0.906697i \(0.638596\pi\)
\(884\) 0 0
\(885\) 6.52092 0.219198
\(886\) 0 0
\(887\) 36.5669 1.22779 0.613897 0.789386i \(-0.289601\pi\)
0.613897 + 0.789386i \(0.289601\pi\)
\(888\) 0 0
\(889\) −50.6630 −1.69918
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 5.89134 0.197146
\(894\) 0 0
\(895\) 4.45963 0.149069
\(896\) 0 0
\(897\) 4.24774 0.141828
\(898\) 0 0
\(899\) −68.7967 −2.29450
\(900\) 0 0
\(901\) −54.8580 −1.82758
\(902\) 0 0
\(903\) −6.25622 −0.208194
\(904\) 0 0
\(905\) 8.56829 0.284820
\(906\) 0 0
\(907\) −40.7368 −1.35264 −0.676322 0.736606i \(-0.736427\pi\)
−0.676322 + 0.736606i \(0.736427\pi\)
\(908\) 0 0
\(909\) −49.3400 −1.63650
\(910\) 0 0
\(911\) 58.2982 1.93150 0.965752 0.259467i \(-0.0835469\pi\)
0.965752 + 0.259467i \(0.0835469\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) −2.10866 −0.0697103
\(916\) 0 0
\(917\) 65.8385 2.17418
\(918\) 0 0
\(919\) 14.5209 0.479001 0.239501 0.970896i \(-0.423016\pi\)
0.239501 + 0.970896i \(0.423016\pi\)
\(920\) 0 0
\(921\) −5.67401 −0.186965
\(922\) 0 0
\(923\) 19.4876 0.641441
\(924\) 0 0
\(925\) 0.945668 0.0310934
\(926\) 0 0
\(927\) −23.4303 −0.769551
\(928\) 0 0
\(929\) 13.7523 0.451197 0.225598 0.974220i \(-0.427566\pi\)
0.225598 + 0.974220i \(0.427566\pi\)
\(930\) 0 0
\(931\) 5.87189 0.192443
\(932\) 0 0
\(933\) 14.9108 0.488157
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −12.4512 −0.406761 −0.203381 0.979100i \(-0.565193\pi\)
−0.203381 + 0.979100i \(0.565193\pi\)
\(938\) 0 0
\(939\) −19.3535 −0.631576
\(940\) 0 0
\(941\) 34.3594 1.12009 0.560043 0.828464i \(-0.310785\pi\)
0.560043 + 0.828464i \(0.310785\pi\)
\(942\) 0 0
\(943\) 50.9582 1.65943
\(944\) 0 0
\(945\) 12.8719 0.418723
\(946\) 0 0
\(947\) 23.4876 0.763243 0.381621 0.924319i \(-0.375366\pi\)
0.381621 + 0.924319i \(0.375366\pi\)
\(948\) 0 0
\(949\) 5.64359 0.183199
\(950\) 0 0
\(951\) 3.83004 0.124198
\(952\) 0 0
\(953\) 48.5808 1.57369 0.786843 0.617153i \(-0.211714\pi\)
0.786843 + 0.617153i \(0.211714\pi\)
\(954\) 0 0
\(955\) 7.73530 0.250308
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −10.8330 −0.349816
\(960\) 0 0
\(961\) 20.4876 0.660889
\(962\) 0 0
\(963\) 13.5837 0.437730
\(964\) 0 0
\(965\) −6.08226 −0.195795
\(966\) 0 0
\(967\) −53.3261 −1.71485 −0.857425 0.514608i \(-0.827937\pi\)
−0.857425 + 0.514608i \(0.827937\pi\)
\(968\) 0 0
\(969\) −3.58774 −0.115255
\(970\) 0 0
\(971\) −33.1366 −1.06340 −0.531702 0.846932i \(-0.678447\pi\)
−0.531702 + 0.846932i \(0.678447\pi\)
\(972\) 0 0
\(973\) −42.2732 −1.35522
\(974\) 0 0
\(975\) 0.871889 0.0279228
\(976\) 0 0
\(977\) 5.62263 0.179884 0.0899419 0.995947i \(-0.471332\pi\)
0.0899419 + 0.995947i \(0.471332\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −15.6490 −0.499635
\(982\) 0 0
\(983\) −21.3664 −0.681482 −0.340741 0.940157i \(-0.610678\pi\)
−0.340741 + 0.940157i \(0.610678\pi\)
\(984\) 0 0
\(985\) −2.00000 −0.0637253
\(986\) 0 0
\(987\) 13.5712 0.431978
\(988\) 0 0
\(989\) −13.2313 −0.420732
\(990\) 0 0
\(991\) −41.1446 −1.30700 −0.653501 0.756926i \(-0.726700\pi\)
−0.653501 + 0.756926i \(0.726700\pi\)
\(992\) 0 0
\(993\) 1.38433 0.0439305
\(994\) 0 0
\(995\) −6.91078 −0.219087
\(996\) 0 0
\(997\) 48.5155 1.53650 0.768250 0.640150i \(-0.221128\pi\)
0.768250 + 0.640150i \(0.221128\pi\)
\(998\) 0 0
\(999\) 3.39281 0.107344
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 6080.2.a.bq.1.2 3
4.3 odd 2 6080.2.a.bv.1.2 3
8.3 odd 2 760.2.a.j.1.2 3
8.5 even 2 1520.2.a.s.1.2 3
24.11 even 2 6840.2.a.bg.1.3 3
40.3 even 4 3800.2.d.l.3649.3 6
40.19 odd 2 3800.2.a.x.1.2 3
40.27 even 4 3800.2.d.l.3649.4 6
40.29 even 2 7600.2.a.bq.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
760.2.a.j.1.2 3 8.3 odd 2
1520.2.a.s.1.2 3 8.5 even 2
3800.2.a.x.1.2 3 40.19 odd 2
3800.2.d.l.3649.3 6 40.3 even 4
3800.2.d.l.3649.4 6 40.27 even 4
6080.2.a.bq.1.2 3 1.1 even 1 trivial
6080.2.a.bv.1.2 3 4.3 odd 2
6840.2.a.bg.1.3 3 24.11 even 2
7600.2.a.bq.1.2 3 40.29 even 2