Properties

Label 1280.2.a.f.1.2
Level $1280$
Weight $2$
Character 1280.1
Self dual yes
Analytic conductor $10.221$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1280,2,Mod(1,1280)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1280, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1280.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1280 = 2^{8} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1280.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: \(10.2208514587\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 640)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 1280.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.41421 q^{3} -1.00000 q^{5} -4.24264 q^{7} -1.00000 q^{9} +O(q^{10})\) \(q+1.41421 q^{3} -1.00000 q^{5} -4.24264 q^{7} -1.00000 q^{9} +5.65685 q^{11} +2.00000 q^{13} -1.41421 q^{15} +6.00000 q^{17} -2.82843 q^{19} -6.00000 q^{21} +7.07107 q^{23} +1.00000 q^{25} -5.65685 q^{27} +4.00000 q^{29} +2.82843 q^{31} +8.00000 q^{33} +4.24264 q^{35} -2.00000 q^{37} +2.82843 q^{39} +8.00000 q^{41} -1.41421 q^{43} +1.00000 q^{45} +1.41421 q^{47} +11.0000 q^{49} +8.48528 q^{51} +2.00000 q^{53} -5.65685 q^{55} -4.00000 q^{57} +2.82843 q^{59} +14.0000 q^{61} +4.24264 q^{63} -2.00000 q^{65} -4.24264 q^{67} +10.0000 q^{69} +2.82843 q^{71} +6.00000 q^{73} +1.41421 q^{75} -24.0000 q^{77} -16.9706 q^{79} -5.00000 q^{81} -12.7279 q^{83} -6.00000 q^{85} +5.65685 q^{87} +6.00000 q^{89} -8.48528 q^{91} +4.00000 q^{93} +2.82843 q^{95} +10.0000 q^{97} -5.65685 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{5} - 2 q^{9} + 4 q^{13} + 12 q^{17} - 12 q^{21} + 2 q^{25} + 8 q^{29} + 16 q^{33} - 4 q^{37} + 16 q^{41} + 2 q^{45} + 22 q^{49} + 4 q^{53} - 8 q^{57} + 28 q^{61} - 4 q^{65} + 20 q^{69} + 12 q^{73} - 48 q^{77} - 10 q^{81} - 12 q^{85} + 12 q^{89} + 8 q^{93} + 20 q^{97}+O(q^{100}) \) 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
\(3\) 1.41421 0.816497 0.408248 0.912871i \(-0.366140\pi\)
0.408248 + 0.912871i \(0.366140\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −4.24264 −1.60357 −0.801784 0.597614i \(-0.796115\pi\)
−0.801784 + 0.597614i \(0.796115\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 5.65685 1.70561 0.852803 0.522233i \(-0.174901\pi\)
0.852803 + 0.522233i \(0.174901\pi\)
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) −1.41421 −0.365148
\(16\) 0 0
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 0 0
\(19\) −2.82843 −0.648886 −0.324443 0.945905i \(-0.605177\pi\)
−0.324443 + 0.945905i \(0.605177\pi\)
\(20\) 0 0
\(21\) −6.00000 −1.30931
\(22\) 0 0
\(23\) 7.07107 1.47442 0.737210 0.675664i \(-0.236143\pi\)
0.737210 + 0.675664i \(0.236143\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −5.65685 −1.08866
\(28\) 0 0
\(29\) 4.00000 0.742781 0.371391 0.928477i \(-0.378881\pi\)
0.371391 + 0.928477i \(0.378881\pi\)
\(30\) 0 0
\(31\) 2.82843 0.508001 0.254000 0.967204i \(-0.418254\pi\)
0.254000 + 0.967204i \(0.418254\pi\)
\(32\) 0 0
\(33\) 8.00000 1.39262
\(34\) 0 0
\(35\) 4.24264 0.717137
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 0 0
\(39\) 2.82843 0.452911
\(40\) 0 0
\(41\) 8.00000 1.24939 0.624695 0.780869i \(-0.285223\pi\)
0.624695 + 0.780869i \(0.285223\pi\)
\(42\) 0 0
\(43\) −1.41421 −0.215666 −0.107833 0.994169i \(-0.534391\pi\)
−0.107833 + 0.994169i \(0.534391\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) 1.41421 0.206284 0.103142 0.994667i \(-0.467110\pi\)
0.103142 + 0.994667i \(0.467110\pi\)
\(48\) 0 0
\(49\) 11.0000 1.57143
\(50\) 0 0
\(51\) 8.48528 1.18818
\(52\) 0 0
\(53\) 2.00000 0.274721 0.137361 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) 0 0
\(55\) −5.65685 −0.762770
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) 0 0
\(59\) 2.82843 0.368230 0.184115 0.982905i \(-0.441058\pi\)
0.184115 + 0.982905i \(0.441058\pi\)
\(60\) 0 0
\(61\) 14.0000 1.79252 0.896258 0.443533i \(-0.146275\pi\)
0.896258 + 0.443533i \(0.146275\pi\)
\(62\) 0 0
\(63\) 4.24264 0.534522
\(64\) 0 0
\(65\) −2.00000 −0.248069
\(66\) 0 0
\(67\) −4.24264 −0.518321 −0.259161 0.965834i \(-0.583446\pi\)
−0.259161 + 0.965834i \(0.583446\pi\)
\(68\) 0 0
\(69\) 10.0000 1.20386
\(70\) 0 0
\(71\) 2.82843 0.335673 0.167836 0.985815i \(-0.446322\pi\)
0.167836 + 0.985815i \(0.446322\pi\)
\(72\) 0 0
\(73\) 6.00000 0.702247 0.351123 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) 0 0
\(75\) 1.41421 0.163299
\(76\) 0 0
\(77\) −24.0000 −2.73505
\(78\) 0 0
\(79\) −16.9706 −1.90934 −0.954669 0.297670i \(-0.903790\pi\)
−0.954669 + 0.297670i \(0.903790\pi\)
\(80\) 0 0
\(81\) −5.00000 −0.555556
\(82\) 0 0
\(83\) −12.7279 −1.39707 −0.698535 0.715575i \(-0.746165\pi\)
−0.698535 + 0.715575i \(0.746165\pi\)
\(84\) 0 0
\(85\) −6.00000 −0.650791
\(86\) 0 0
\(87\) 5.65685 0.606478
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) −8.48528 −0.889499
\(92\) 0 0
\(93\) 4.00000 0.414781
\(94\) 0 0
\(95\) 2.82843 0.290191
\(96\) 0 0
\(97\) 10.0000 1.01535 0.507673 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) 0 0
\(99\) −5.65685 −0.568535
\(100\) 0 0
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) −9.89949 −0.975426 −0.487713 0.873004i \(-0.662169\pi\)
−0.487713 + 0.873004i \(0.662169\pi\)
\(104\) 0 0
\(105\) 6.00000 0.585540
\(106\) 0 0
\(107\) −7.07107 −0.683586 −0.341793 0.939775i \(-0.611034\pi\)
−0.341793 + 0.939775i \(0.611034\pi\)
\(108\) 0 0
\(109\) −6.00000 −0.574696 −0.287348 0.957826i \(-0.592774\pi\)
−0.287348 + 0.957826i \(0.592774\pi\)
\(110\) 0 0
\(111\) −2.82843 −0.268462
\(112\) 0 0
\(113\) −10.0000 −0.940721 −0.470360 0.882474i \(-0.655876\pi\)
−0.470360 + 0.882474i \(0.655876\pi\)
\(114\) 0 0
\(115\) −7.07107 −0.659380
\(116\) 0 0
\(117\) −2.00000 −0.184900
\(118\) 0 0
\(119\) −25.4558 −2.33353
\(120\) 0 0
\(121\) 21.0000 1.90909
\(122\) 0 0
\(123\) 11.3137 1.02012
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 9.89949 0.878438 0.439219 0.898380i \(-0.355255\pi\)
0.439219 + 0.898380i \(0.355255\pi\)
\(128\) 0 0
\(129\) −2.00000 −0.176090
\(130\) 0 0
\(131\) −11.3137 −0.988483 −0.494242 0.869325i \(-0.664554\pi\)
−0.494242 + 0.869325i \(0.664554\pi\)
\(132\) 0 0
\(133\) 12.0000 1.04053
\(134\) 0 0
\(135\) 5.65685 0.486864
\(136\) 0 0
\(137\) −2.00000 −0.170872 −0.0854358 0.996344i \(-0.527228\pi\)
−0.0854358 + 0.996344i \(0.527228\pi\)
\(138\) 0 0
\(139\) −8.48528 −0.719712 −0.359856 0.933008i \(-0.617174\pi\)
−0.359856 + 0.933008i \(0.617174\pi\)
\(140\) 0 0
\(141\) 2.00000 0.168430
\(142\) 0 0
\(143\) 11.3137 0.946100
\(144\) 0 0
\(145\) −4.00000 −0.332182
\(146\) 0 0
\(147\) 15.5563 1.28307
\(148\) 0 0
\(149\) 22.0000 1.80231 0.901155 0.433497i \(-0.142720\pi\)
0.901155 + 0.433497i \(0.142720\pi\)
\(150\) 0 0
\(151\) 14.1421 1.15087 0.575435 0.817847i \(-0.304833\pi\)
0.575435 + 0.817847i \(0.304833\pi\)
\(152\) 0 0
\(153\) −6.00000 −0.485071
\(154\) 0 0
\(155\) −2.82843 −0.227185
\(156\) 0 0
\(157\) −18.0000 −1.43656 −0.718278 0.695756i \(-0.755069\pi\)
−0.718278 + 0.695756i \(0.755069\pi\)
\(158\) 0 0
\(159\) 2.82843 0.224309
\(160\) 0 0
\(161\) −30.0000 −2.36433
\(162\) 0 0
\(163\) 18.3848 1.44001 0.720003 0.693971i \(-0.244140\pi\)
0.720003 + 0.693971i \(0.244140\pi\)
\(164\) 0 0
\(165\) −8.00000 −0.622799
\(166\) 0 0
\(167\) 9.89949 0.766046 0.383023 0.923739i \(-0.374883\pi\)
0.383023 + 0.923739i \(0.374883\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 2.82843 0.216295
\(172\) 0 0
\(173\) −18.0000 −1.36851 −0.684257 0.729241i \(-0.739873\pi\)
−0.684257 + 0.729241i \(0.739873\pi\)
\(174\) 0 0
\(175\) −4.24264 −0.320713
\(176\) 0 0
\(177\) 4.00000 0.300658
\(178\) 0 0
\(179\) 19.7990 1.47985 0.739923 0.672692i \(-0.234862\pi\)
0.739923 + 0.672692i \(0.234862\pi\)
\(180\) 0 0
\(181\) −16.0000 −1.18927 −0.594635 0.803996i \(-0.702704\pi\)
−0.594635 + 0.803996i \(0.702704\pi\)
\(182\) 0 0
\(183\) 19.7990 1.46358
\(184\) 0 0
\(185\) 2.00000 0.147043
\(186\) 0 0
\(187\) 33.9411 2.48202
\(188\) 0 0
\(189\) 24.0000 1.74574
\(190\) 0 0
\(191\) −2.82843 −0.204658 −0.102329 0.994751i \(-0.532629\pi\)
−0.102329 + 0.994751i \(0.532629\pi\)
\(192\) 0 0
\(193\) 6.00000 0.431889 0.215945 0.976406i \(-0.430717\pi\)
0.215945 + 0.976406i \(0.430717\pi\)
\(194\) 0 0
\(195\) −2.82843 −0.202548
\(196\) 0 0
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 0 0
\(199\) 11.3137 0.802008 0.401004 0.916076i \(-0.368661\pi\)
0.401004 + 0.916076i \(0.368661\pi\)
\(200\) 0 0
\(201\) −6.00000 −0.423207
\(202\) 0 0
\(203\) −16.9706 −1.19110
\(204\) 0 0
\(205\) −8.00000 −0.558744
\(206\) 0 0
\(207\) −7.07107 −0.491473
\(208\) 0 0
\(209\) −16.0000 −1.10674
\(210\) 0 0
\(211\) −16.9706 −1.16830 −0.584151 0.811645i \(-0.698572\pi\)
−0.584151 + 0.811645i \(0.698572\pi\)
\(212\) 0 0
\(213\) 4.00000 0.274075
\(214\) 0 0
\(215\) 1.41421 0.0964486
\(216\) 0 0
\(217\) −12.0000 −0.814613
\(218\) 0 0
\(219\) 8.48528 0.573382
\(220\) 0 0
\(221\) 12.0000 0.807207
\(222\) 0 0
\(223\) −15.5563 −1.04173 −0.520865 0.853639i \(-0.674391\pi\)
−0.520865 + 0.853639i \(0.674391\pi\)
\(224\) 0 0
\(225\) −1.00000 −0.0666667
\(226\) 0 0
\(227\) −7.07107 −0.469323 −0.234662 0.972077i \(-0.575398\pi\)
−0.234662 + 0.972077i \(0.575398\pi\)
\(228\) 0 0
\(229\) 20.0000 1.32164 0.660819 0.750546i \(-0.270209\pi\)
0.660819 + 0.750546i \(0.270209\pi\)
\(230\) 0 0
\(231\) −33.9411 −2.23316
\(232\) 0 0
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 0 0
\(235\) −1.41421 −0.0922531
\(236\) 0 0
\(237\) −24.0000 −1.55897
\(238\) 0 0
\(239\) 11.3137 0.731823 0.365911 0.930650i \(-0.380757\pi\)
0.365911 + 0.930650i \(0.380757\pi\)
\(240\) 0 0
\(241\) 8.00000 0.515325 0.257663 0.966235i \(-0.417048\pi\)
0.257663 + 0.966235i \(0.417048\pi\)
\(242\) 0 0
\(243\) 9.89949 0.635053
\(244\) 0 0
\(245\) −11.0000 −0.702764
\(246\) 0 0
\(247\) −5.65685 −0.359937
\(248\) 0 0
\(249\) −18.0000 −1.14070
\(250\) 0 0
\(251\) −22.6274 −1.42823 −0.714115 0.700028i \(-0.753171\pi\)
−0.714115 + 0.700028i \(0.753171\pi\)
\(252\) 0 0
\(253\) 40.0000 2.51478
\(254\) 0 0
\(255\) −8.48528 −0.531369
\(256\) 0 0
\(257\) −18.0000 −1.12281 −0.561405 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) 0 0
\(259\) 8.48528 0.527250
\(260\) 0 0
\(261\) −4.00000 −0.247594
\(262\) 0 0
\(263\) −7.07107 −0.436021 −0.218010 0.975946i \(-0.569957\pi\)
−0.218010 + 0.975946i \(0.569957\pi\)
\(264\) 0 0
\(265\) −2.00000 −0.122859
\(266\) 0 0
\(267\) 8.48528 0.519291
\(268\) 0 0
\(269\) 6.00000 0.365826 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) 0 0
\(271\) 25.4558 1.54633 0.773166 0.634203i \(-0.218672\pi\)
0.773166 + 0.634203i \(0.218672\pi\)
\(272\) 0 0
\(273\) −12.0000 −0.726273
\(274\) 0 0
\(275\) 5.65685 0.341121
\(276\) 0 0
\(277\) −30.0000 −1.80253 −0.901263 0.433273i \(-0.857359\pi\)
−0.901263 + 0.433273i \(0.857359\pi\)
\(278\) 0 0
\(279\) −2.82843 −0.169334
\(280\) 0 0
\(281\) −8.00000 −0.477240 −0.238620 0.971113i \(-0.576695\pi\)
−0.238620 + 0.971113i \(0.576695\pi\)
\(282\) 0 0
\(283\) −9.89949 −0.588464 −0.294232 0.955734i \(-0.595064\pi\)
−0.294232 + 0.955734i \(0.595064\pi\)
\(284\) 0 0
\(285\) 4.00000 0.236940
\(286\) 0 0
\(287\) −33.9411 −2.00348
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 14.1421 0.829027
\(292\) 0 0
\(293\) 6.00000 0.350524 0.175262 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 0 0
\(295\) −2.82843 −0.164677
\(296\) 0 0
\(297\) −32.0000 −1.85683
\(298\) 0 0
\(299\) 14.1421 0.817861
\(300\) 0 0
\(301\) 6.00000 0.345834
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −14.0000 −0.801638
\(306\) 0 0
\(307\) −15.5563 −0.887848 −0.443924 0.896065i \(-0.646414\pi\)
−0.443924 + 0.896065i \(0.646414\pi\)
\(308\) 0 0
\(309\) −14.0000 −0.796432
\(310\) 0 0
\(311\) 2.82843 0.160385 0.0801927 0.996779i \(-0.474446\pi\)
0.0801927 + 0.996779i \(0.474446\pi\)
\(312\) 0 0
\(313\) −6.00000 −0.339140 −0.169570 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) 0 0
\(315\) −4.24264 −0.239046
\(316\) 0 0
\(317\) −2.00000 −0.112331 −0.0561656 0.998421i \(-0.517887\pi\)
−0.0561656 + 0.998421i \(0.517887\pi\)
\(318\) 0 0
\(319\) 22.6274 1.26689
\(320\) 0 0
\(321\) −10.0000 −0.558146
\(322\) 0 0
\(323\) −16.9706 −0.944267
\(324\) 0 0
\(325\) 2.00000 0.110940
\(326\) 0 0
\(327\) −8.48528 −0.469237
\(328\) 0 0
\(329\) −6.00000 −0.330791
\(330\) 0 0
\(331\) 22.6274 1.24372 0.621858 0.783130i \(-0.286378\pi\)
0.621858 + 0.783130i \(0.286378\pi\)
\(332\) 0 0
\(333\) 2.00000 0.109599
\(334\) 0 0
\(335\) 4.24264 0.231800
\(336\) 0 0
\(337\) 18.0000 0.980522 0.490261 0.871576i \(-0.336901\pi\)
0.490261 + 0.871576i \(0.336901\pi\)
\(338\) 0 0
\(339\) −14.1421 −0.768095
\(340\) 0 0
\(341\) 16.0000 0.866449
\(342\) 0 0
\(343\) −16.9706 −0.916324
\(344\) 0 0
\(345\) −10.0000 −0.538382
\(346\) 0 0
\(347\) −4.24264 −0.227757 −0.113878 0.993495i \(-0.536327\pi\)
−0.113878 + 0.993495i \(0.536327\pi\)
\(348\) 0 0
\(349\) 12.0000 0.642345 0.321173 0.947021i \(-0.395923\pi\)
0.321173 + 0.947021i \(0.395923\pi\)
\(350\) 0 0
\(351\) −11.3137 −0.603881
\(352\) 0 0
\(353\) 10.0000 0.532246 0.266123 0.963939i \(-0.414257\pi\)
0.266123 + 0.963939i \(0.414257\pi\)
\(354\) 0 0
\(355\) −2.82843 −0.150117
\(356\) 0 0
\(357\) −36.0000 −1.90532
\(358\) 0 0
\(359\) 28.2843 1.49279 0.746393 0.665505i \(-0.231784\pi\)
0.746393 + 0.665505i \(0.231784\pi\)
\(360\) 0 0
\(361\) −11.0000 −0.578947
\(362\) 0 0
\(363\) 29.6985 1.55877
\(364\) 0 0
\(365\) −6.00000 −0.314054
\(366\) 0 0
\(367\) 4.24264 0.221464 0.110732 0.993850i \(-0.464680\pi\)
0.110732 + 0.993850i \(0.464680\pi\)
\(368\) 0 0
\(369\) −8.00000 −0.416463
\(370\) 0 0
\(371\) −8.48528 −0.440534
\(372\) 0 0
\(373\) −14.0000 −0.724893 −0.362446 0.932005i \(-0.618058\pi\)
−0.362446 + 0.932005i \(0.618058\pi\)
\(374\) 0 0
\(375\) −1.41421 −0.0730297
\(376\) 0 0
\(377\) 8.00000 0.412021
\(378\) 0 0
\(379\) −31.1127 −1.59815 −0.799076 0.601230i \(-0.794678\pi\)
−0.799076 + 0.601230i \(0.794678\pi\)
\(380\) 0 0
\(381\) 14.0000 0.717242
\(382\) 0 0
\(383\) 7.07107 0.361315 0.180657 0.983546i \(-0.442177\pi\)
0.180657 + 0.983546i \(0.442177\pi\)
\(384\) 0 0
\(385\) 24.0000 1.22315
\(386\) 0 0
\(387\) 1.41421 0.0718885
\(388\) 0 0
\(389\) 14.0000 0.709828 0.354914 0.934899i \(-0.384510\pi\)
0.354914 + 0.934899i \(0.384510\pi\)
\(390\) 0 0
\(391\) 42.4264 2.14560
\(392\) 0 0
\(393\) −16.0000 −0.807093
\(394\) 0 0
\(395\) 16.9706 0.853882
\(396\) 0 0
\(397\) −14.0000 −0.702640 −0.351320 0.936255i \(-0.614267\pi\)
−0.351320 + 0.936255i \(0.614267\pi\)
\(398\) 0 0
\(399\) 16.9706 0.849591
\(400\) 0 0
\(401\) 18.0000 0.898877 0.449439 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) 0 0
\(403\) 5.65685 0.281788
\(404\) 0 0
\(405\) 5.00000 0.248452
\(406\) 0 0
\(407\) −11.3137 −0.560800
\(408\) 0 0
\(409\) −4.00000 −0.197787 −0.0988936 0.995098i \(-0.531530\pi\)
−0.0988936 + 0.995098i \(0.531530\pi\)
\(410\) 0 0
\(411\) −2.82843 −0.139516
\(412\) 0 0
\(413\) −12.0000 −0.590481
\(414\) 0 0
\(415\) 12.7279 0.624789
\(416\) 0 0
\(417\) −12.0000 −0.587643
\(418\) 0 0
\(419\) −8.48528 −0.414533 −0.207267 0.978285i \(-0.566457\pi\)
−0.207267 + 0.978285i \(0.566457\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) 0 0
\(423\) −1.41421 −0.0687614
\(424\) 0 0
\(425\) 6.00000 0.291043
\(426\) 0 0
\(427\) −59.3970 −2.87442
\(428\) 0 0
\(429\) 16.0000 0.772487
\(430\) 0 0
\(431\) −31.1127 −1.49865 −0.749323 0.662205i \(-0.769621\pi\)
−0.749323 + 0.662205i \(0.769621\pi\)
\(432\) 0 0
\(433\) −14.0000 −0.672797 −0.336399 0.941720i \(-0.609209\pi\)
−0.336399 + 0.941720i \(0.609209\pi\)
\(434\) 0 0
\(435\) −5.65685 −0.271225
\(436\) 0 0
\(437\) −20.0000 −0.956730
\(438\) 0 0
\(439\) 16.9706 0.809961 0.404980 0.914325i \(-0.367278\pi\)
0.404980 + 0.914325i \(0.367278\pi\)
\(440\) 0 0
\(441\) −11.0000 −0.523810
\(442\) 0 0
\(443\) −15.5563 −0.739104 −0.369552 0.929210i \(-0.620489\pi\)
−0.369552 + 0.929210i \(0.620489\pi\)
\(444\) 0 0
\(445\) −6.00000 −0.284427
\(446\) 0 0
\(447\) 31.1127 1.47158
\(448\) 0 0
\(449\) −20.0000 −0.943858 −0.471929 0.881636i \(-0.656442\pi\)
−0.471929 + 0.881636i \(0.656442\pi\)
\(450\) 0 0
\(451\) 45.2548 2.13097
\(452\) 0 0
\(453\) 20.0000 0.939682
\(454\) 0 0
\(455\) 8.48528 0.397796
\(456\) 0 0
\(457\) 34.0000 1.59045 0.795226 0.606313i \(-0.207352\pi\)
0.795226 + 0.606313i \(0.207352\pi\)
\(458\) 0 0
\(459\) −33.9411 −1.58424
\(460\) 0 0
\(461\) −24.0000 −1.11779 −0.558896 0.829238i \(-0.688775\pi\)
−0.558896 + 0.829238i \(0.688775\pi\)
\(462\) 0 0
\(463\) −7.07107 −0.328620 −0.164310 0.986409i \(-0.552540\pi\)
−0.164310 + 0.986409i \(0.552540\pi\)
\(464\) 0 0
\(465\) −4.00000 −0.185496
\(466\) 0 0
\(467\) 9.89949 0.458094 0.229047 0.973415i \(-0.426439\pi\)
0.229047 + 0.973415i \(0.426439\pi\)
\(468\) 0 0
\(469\) 18.0000 0.831163
\(470\) 0 0
\(471\) −25.4558 −1.17294
\(472\) 0 0
\(473\) −8.00000 −0.367840
\(474\) 0 0
\(475\) −2.82843 −0.129777
\(476\) 0 0
\(477\) −2.00000 −0.0915737
\(478\) 0 0
\(479\) −33.9411 −1.55081 −0.775405 0.631464i \(-0.782454\pi\)
−0.775405 + 0.631464i \(0.782454\pi\)
\(480\) 0 0
\(481\) −4.00000 −0.182384
\(482\) 0 0
\(483\) −42.4264 −1.93047
\(484\) 0 0
\(485\) −10.0000 −0.454077
\(486\) 0 0
\(487\) 12.7279 0.576757 0.288379 0.957516i \(-0.406884\pi\)
0.288379 + 0.957516i \(0.406884\pi\)
\(488\) 0 0
\(489\) 26.0000 1.17576
\(490\) 0 0
\(491\) 16.9706 0.765871 0.382935 0.923775i \(-0.374913\pi\)
0.382935 + 0.923775i \(0.374913\pi\)
\(492\) 0 0
\(493\) 24.0000 1.08091
\(494\) 0 0
\(495\) 5.65685 0.254257
\(496\) 0 0
\(497\) −12.0000 −0.538274
\(498\) 0 0
\(499\) −25.4558 −1.13956 −0.569780 0.821797i \(-0.692972\pi\)
−0.569780 + 0.821797i \(0.692972\pi\)
\(500\) 0 0
\(501\) 14.0000 0.625474
\(502\) 0 0
\(503\) −32.5269 −1.45030 −0.725152 0.688589i \(-0.758230\pi\)
−0.725152 + 0.688589i \(0.758230\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −12.7279 −0.565267
\(508\) 0 0
\(509\) 20.0000 0.886484 0.443242 0.896402i \(-0.353828\pi\)
0.443242 + 0.896402i \(0.353828\pi\)
\(510\) 0 0
\(511\) −25.4558 −1.12610
\(512\) 0 0
\(513\) 16.0000 0.706417
\(514\) 0 0
\(515\) 9.89949 0.436224
\(516\) 0 0
\(517\) 8.00000 0.351840
\(518\) 0 0
\(519\) −25.4558 −1.11739
\(520\) 0 0
\(521\) −18.0000 −0.788594 −0.394297 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) 0 0
\(523\) 12.7279 0.556553 0.278277 0.960501i \(-0.410237\pi\)
0.278277 + 0.960501i \(0.410237\pi\)
\(524\) 0 0
\(525\) −6.00000 −0.261861
\(526\) 0 0
\(527\) 16.9706 0.739249
\(528\) 0 0
\(529\) 27.0000 1.17391
\(530\) 0 0
\(531\) −2.82843 −0.122743
\(532\) 0 0
\(533\) 16.0000 0.693037
\(534\) 0 0
\(535\) 7.07107 0.305709
\(536\) 0 0
\(537\) 28.0000 1.20829
\(538\) 0 0
\(539\) 62.2254 2.68024
\(540\) 0 0
\(541\) −40.0000 −1.71973 −0.859867 0.510518i \(-0.829454\pi\)
−0.859867 + 0.510518i \(0.829454\pi\)
\(542\) 0 0
\(543\) −22.6274 −0.971035
\(544\) 0 0
\(545\) 6.00000 0.257012
\(546\) 0 0
\(547\) 4.24264 0.181402 0.0907011 0.995878i \(-0.471089\pi\)
0.0907011 + 0.995878i \(0.471089\pi\)
\(548\) 0 0
\(549\) −14.0000 −0.597505
\(550\) 0 0
\(551\) −11.3137 −0.481980
\(552\) 0 0
\(553\) 72.0000 3.06175
\(554\) 0 0
\(555\) 2.82843 0.120060
\(556\) 0 0
\(557\) 18.0000 0.762684 0.381342 0.924434i \(-0.375462\pi\)
0.381342 + 0.924434i \(0.375462\pi\)
\(558\) 0 0
\(559\) −2.82843 −0.119630
\(560\) 0 0
\(561\) 48.0000 2.02656
\(562\) 0 0
\(563\) 35.3553 1.49005 0.745025 0.667037i \(-0.232438\pi\)
0.745025 + 0.667037i \(0.232438\pi\)
\(564\) 0 0
\(565\) 10.0000 0.420703
\(566\) 0 0
\(567\) 21.2132 0.890871
\(568\) 0 0
\(569\) 28.0000 1.17382 0.586911 0.809652i \(-0.300344\pi\)
0.586911 + 0.809652i \(0.300344\pi\)
\(570\) 0 0
\(571\) 22.6274 0.946928 0.473464 0.880813i \(-0.343003\pi\)
0.473464 + 0.880813i \(0.343003\pi\)
\(572\) 0 0
\(573\) −4.00000 −0.167102
\(574\) 0 0
\(575\) 7.07107 0.294884
\(576\) 0 0
\(577\) −10.0000 −0.416305 −0.208153 0.978096i \(-0.566745\pi\)
−0.208153 + 0.978096i \(0.566745\pi\)
\(578\) 0 0
\(579\) 8.48528 0.352636
\(580\) 0 0
\(581\) 54.0000 2.24030
\(582\) 0 0
\(583\) 11.3137 0.468566
\(584\) 0 0
\(585\) 2.00000 0.0826898
\(586\) 0 0
\(587\) 12.7279 0.525338 0.262669 0.964886i \(-0.415397\pi\)
0.262669 + 0.964886i \(0.415397\pi\)
\(588\) 0 0
\(589\) −8.00000 −0.329634
\(590\) 0 0
\(591\) 8.48528 0.349038
\(592\) 0 0
\(593\) −30.0000 −1.23195 −0.615976 0.787765i \(-0.711238\pi\)
−0.615976 + 0.787765i \(0.711238\pi\)
\(594\) 0 0
\(595\) 25.4558 1.04359
\(596\) 0 0
\(597\) 16.0000 0.654836
\(598\) 0 0
\(599\) 39.5980 1.61793 0.808965 0.587857i \(-0.200028\pi\)
0.808965 + 0.587857i \(0.200028\pi\)
\(600\) 0 0
\(601\) 32.0000 1.30531 0.652654 0.757656i \(-0.273656\pi\)
0.652654 + 0.757656i \(0.273656\pi\)
\(602\) 0 0
\(603\) 4.24264 0.172774
\(604\) 0 0
\(605\) −21.0000 −0.853771
\(606\) 0 0
\(607\) −9.89949 −0.401808 −0.200904 0.979611i \(-0.564388\pi\)
−0.200904 + 0.979611i \(0.564388\pi\)
\(608\) 0 0
\(609\) −24.0000 −0.972529
\(610\) 0 0
\(611\) 2.82843 0.114426
\(612\) 0 0
\(613\) −26.0000 −1.05013 −0.525065 0.851062i \(-0.675959\pi\)
−0.525065 + 0.851062i \(0.675959\pi\)
\(614\) 0 0
\(615\) −11.3137 −0.456213
\(616\) 0 0
\(617\) 2.00000 0.0805170 0.0402585 0.999189i \(-0.487182\pi\)
0.0402585 + 0.999189i \(0.487182\pi\)
\(618\) 0 0
\(619\) 19.7990 0.795789 0.397894 0.917431i \(-0.369741\pi\)
0.397894 + 0.917431i \(0.369741\pi\)
\(620\) 0 0
\(621\) −40.0000 −1.60514
\(622\) 0 0
\(623\) −25.4558 −1.01987
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −22.6274 −0.903652
\(628\) 0 0
\(629\) −12.0000 −0.478471
\(630\) 0 0
\(631\) −36.7696 −1.46377 −0.731886 0.681427i \(-0.761360\pi\)
−0.731886 + 0.681427i \(0.761360\pi\)
\(632\) 0 0
\(633\) −24.0000 −0.953914
\(634\) 0 0
\(635\) −9.89949 −0.392849
\(636\) 0 0
\(637\) 22.0000 0.871672
\(638\) 0 0
\(639\) −2.82843 −0.111891
\(640\) 0 0
\(641\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) −7.07107 −0.278856 −0.139428 0.990232i \(-0.544526\pi\)
−0.139428 + 0.990232i \(0.544526\pi\)
\(644\) 0 0
\(645\) 2.00000 0.0787499
\(646\) 0 0
\(647\) −32.5269 −1.27876 −0.639382 0.768889i \(-0.720810\pi\)
−0.639382 + 0.768889i \(0.720810\pi\)
\(648\) 0 0
\(649\) 16.0000 0.628055
\(650\) 0 0
\(651\) −16.9706 −0.665129
\(652\) 0 0
\(653\) −18.0000 −0.704394 −0.352197 0.935926i \(-0.614565\pi\)
−0.352197 + 0.935926i \(0.614565\pi\)
\(654\) 0 0
\(655\) 11.3137 0.442063
\(656\) 0 0
\(657\) −6.00000 −0.234082
\(658\) 0 0
\(659\) 25.4558 0.991619 0.495809 0.868431i \(-0.334871\pi\)
0.495809 + 0.868431i \(0.334871\pi\)
\(660\) 0 0
\(661\) 10.0000 0.388955 0.194477 0.980907i \(-0.437699\pi\)
0.194477 + 0.980907i \(0.437699\pi\)
\(662\) 0 0
\(663\) 16.9706 0.659082
\(664\) 0 0
\(665\) −12.0000 −0.465340
\(666\) 0 0
\(667\) 28.2843 1.09517
\(668\) 0 0
\(669\) −22.0000 −0.850569
\(670\) 0 0
\(671\) 79.1960 3.05733
\(672\) 0 0
\(673\) 50.0000 1.92736 0.963679 0.267063i \(-0.0860531\pi\)
0.963679 + 0.267063i \(0.0860531\pi\)
\(674\) 0 0
\(675\) −5.65685 −0.217732
\(676\) 0 0
\(677\) −10.0000 −0.384331 −0.192166 0.981363i \(-0.561551\pi\)
−0.192166 + 0.981363i \(0.561551\pi\)
\(678\) 0 0
\(679\) −42.4264 −1.62818
\(680\) 0 0
\(681\) −10.0000 −0.383201
\(682\) 0 0
\(683\) −35.3553 −1.35283 −0.676417 0.736519i \(-0.736468\pi\)
−0.676417 + 0.736519i \(0.736468\pi\)
\(684\) 0 0
\(685\) 2.00000 0.0764161
\(686\) 0 0
\(687\) 28.2843 1.07911
\(688\) 0 0
\(689\) 4.00000 0.152388
\(690\) 0 0
\(691\) −22.6274 −0.860788 −0.430394 0.902641i \(-0.641625\pi\)
−0.430394 + 0.902641i \(0.641625\pi\)
\(692\) 0 0
\(693\) 24.0000 0.911685
\(694\) 0 0
\(695\) 8.48528 0.321865
\(696\) 0 0
\(697\) 48.0000 1.81813
\(698\) 0 0
\(699\) −8.48528 −0.320943
\(700\) 0 0
\(701\) 18.0000 0.679851 0.339925 0.940452i \(-0.389598\pi\)
0.339925 + 0.940452i \(0.389598\pi\)
\(702\) 0 0
\(703\) 5.65685 0.213352
\(704\) 0 0
\(705\) −2.00000 −0.0753244
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 28.0000 1.05156 0.525781 0.850620i \(-0.323773\pi\)
0.525781 + 0.850620i \(0.323773\pi\)
\(710\) 0 0
\(711\) 16.9706 0.636446
\(712\) 0 0
\(713\) 20.0000 0.749006
\(714\) 0 0
\(715\) −11.3137 −0.423109
\(716\) 0 0
\(717\) 16.0000 0.597531
\(718\) 0 0
\(719\) 11.3137 0.421930 0.210965 0.977494i \(-0.432339\pi\)
0.210965 + 0.977494i \(0.432339\pi\)
\(720\) 0 0
\(721\) 42.0000 1.56416
\(722\) 0 0
\(723\) 11.3137 0.420761
\(724\) 0 0
\(725\) 4.00000 0.148556
\(726\) 0 0
\(727\) −4.24264 −0.157351 −0.0786754 0.996900i \(-0.525069\pi\)
−0.0786754 + 0.996900i \(0.525069\pi\)
\(728\) 0 0
\(729\) 29.0000 1.07407
\(730\) 0 0
\(731\) −8.48528 −0.313839
\(732\) 0 0
\(733\) −34.0000 −1.25582 −0.627909 0.778287i \(-0.716089\pi\)
−0.627909 + 0.778287i \(0.716089\pi\)
\(734\) 0 0
\(735\) −15.5563 −0.573805
\(736\) 0 0
\(737\) −24.0000 −0.884051
\(738\) 0 0
\(739\) −25.4558 −0.936408 −0.468204 0.883620i \(-0.655099\pi\)
−0.468204 + 0.883620i \(0.655099\pi\)
\(740\) 0 0
\(741\) −8.00000 −0.293887
\(742\) 0 0
\(743\) 32.5269 1.19330 0.596648 0.802503i \(-0.296499\pi\)
0.596648 + 0.802503i \(0.296499\pi\)
\(744\) 0 0
\(745\) −22.0000 −0.806018
\(746\) 0 0
\(747\) 12.7279 0.465690
\(748\) 0 0
\(749\) 30.0000 1.09618
\(750\) 0 0
\(751\) −8.48528 −0.309632 −0.154816 0.987943i \(-0.549479\pi\)
−0.154816 + 0.987943i \(0.549479\pi\)
\(752\) 0 0
\(753\) −32.0000 −1.16614
\(754\) 0 0
\(755\) −14.1421 −0.514685
\(756\) 0 0
\(757\) −46.0000 −1.67190 −0.835949 0.548807i \(-0.815082\pi\)
−0.835949 + 0.548807i \(0.815082\pi\)
\(758\) 0 0
\(759\) 56.5685 2.05331
\(760\) 0 0
\(761\) −14.0000 −0.507500 −0.253750 0.967270i \(-0.581664\pi\)
−0.253750 + 0.967270i \(0.581664\pi\)
\(762\) 0 0
\(763\) 25.4558 0.921563
\(764\) 0 0
\(765\) 6.00000 0.216930
\(766\) 0 0
\(767\) 5.65685 0.204257
\(768\) 0 0
\(769\) 14.0000 0.504853 0.252426 0.967616i \(-0.418771\pi\)
0.252426 + 0.967616i \(0.418771\pi\)
\(770\) 0 0
\(771\) −25.4558 −0.916770
\(772\) 0 0
\(773\) −26.0000 −0.935155 −0.467578 0.883952i \(-0.654873\pi\)
−0.467578 + 0.883952i \(0.654873\pi\)
\(774\) 0 0
\(775\) 2.82843 0.101600
\(776\) 0 0
\(777\) 12.0000 0.430498
\(778\) 0 0
\(779\) −22.6274 −0.810711
\(780\) 0 0
\(781\) 16.0000 0.572525
\(782\) 0 0
\(783\) −22.6274 −0.808638
\(784\) 0 0
\(785\) 18.0000 0.642448
\(786\) 0 0
\(787\) 46.6690 1.66357 0.831786 0.555097i \(-0.187319\pi\)
0.831786 + 0.555097i \(0.187319\pi\)
\(788\) 0 0
\(789\) −10.0000 −0.356009
\(790\) 0 0
\(791\) 42.4264 1.50851
\(792\) 0 0
\(793\) 28.0000 0.994309
\(794\) 0 0
\(795\) −2.82843 −0.100314
\(796\) 0 0
\(797\) −10.0000 −0.354218 −0.177109 0.984191i \(-0.556675\pi\)
−0.177109 + 0.984191i \(0.556675\pi\)
\(798\) 0 0
\(799\) 8.48528 0.300188
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) 0 0
\(803\) 33.9411 1.19776
\(804\) 0 0
\(805\) 30.0000 1.05736
\(806\) 0 0
\(807\) 8.48528 0.298696
\(808\) 0 0
\(809\) −10.0000 −0.351581 −0.175791 0.984428i \(-0.556248\pi\)
−0.175791 + 0.984428i \(0.556248\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(812\) 0 0
\(813\) 36.0000 1.26258
\(814\) 0 0
\(815\) −18.3848 −0.643991
\(816\) 0 0
\(817\) 4.00000 0.139942
\(818\) 0 0
\(819\) 8.48528 0.296500
\(820\) 0 0
\(821\) 26.0000 0.907406 0.453703 0.891153i \(-0.350103\pi\)
0.453703 + 0.891153i \(0.350103\pi\)
\(822\) 0 0
\(823\) −26.8701 −0.936631 −0.468316 0.883561i \(-0.655139\pi\)
−0.468316 + 0.883561i \(0.655139\pi\)
\(824\) 0 0
\(825\) 8.00000 0.278524
\(826\) 0 0
\(827\) 7.07107 0.245885 0.122943 0.992414i \(-0.460767\pi\)
0.122943 + 0.992414i \(0.460767\pi\)
\(828\) 0 0
\(829\) −26.0000 −0.903017 −0.451509 0.892267i \(-0.649114\pi\)
−0.451509 + 0.892267i \(0.649114\pi\)
\(830\) 0 0
\(831\) −42.4264 −1.47176
\(832\) 0 0
\(833\) 66.0000 2.28676
\(834\) 0 0
\(835\) −9.89949 −0.342586
\(836\) 0 0
\(837\) −16.0000 −0.553041
\(838\) 0 0
\(839\) −28.2843 −0.976481 −0.488241 0.872709i \(-0.662361\pi\)
−0.488241 + 0.872709i \(0.662361\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) 0 0
\(843\) −11.3137 −0.389665
\(844\) 0 0
\(845\) 9.00000 0.309609
\(846\) 0 0
\(847\) −89.0955 −3.06136
\(848\) 0 0
\(849\) −14.0000 −0.480479
\(850\) 0 0
\(851\) −14.1421 −0.484786
\(852\) 0 0
\(853\) −26.0000 −0.890223 −0.445112 0.895475i \(-0.646836\pi\)
−0.445112 + 0.895475i \(0.646836\pi\)
\(854\) 0 0
\(855\) −2.82843 −0.0967302
\(856\) 0 0
\(857\) −18.0000 −0.614868 −0.307434 0.951569i \(-0.599470\pi\)
−0.307434 + 0.951569i \(0.599470\pi\)
\(858\) 0 0
\(859\) 2.82843 0.0965047 0.0482523 0.998835i \(-0.484635\pi\)
0.0482523 + 0.998835i \(0.484635\pi\)
\(860\) 0 0
\(861\) −48.0000 −1.63584
\(862\) 0 0
\(863\) 55.1543 1.87748 0.938738 0.344633i \(-0.111997\pi\)
0.938738 + 0.344633i \(0.111997\pi\)
\(864\) 0 0
\(865\) 18.0000 0.612018
\(866\) 0 0
\(867\) 26.8701 0.912555
\(868\) 0 0
\(869\) −96.0000 −3.25658
\(870\) 0 0
\(871\) −8.48528 −0.287513
\(872\) 0 0
\(873\) −10.0000 −0.338449
\(874\) 0 0
\(875\) 4.24264 0.143427
\(876\) 0 0
\(877\) 38.0000 1.28317 0.641584 0.767052i \(-0.278277\pi\)
0.641584 + 0.767052i \(0.278277\pi\)
\(878\) 0 0
\(879\) 8.48528 0.286201
\(880\) 0 0
\(881\) 48.0000 1.61716 0.808581 0.588386i \(-0.200236\pi\)
0.808581 + 0.588386i \(0.200236\pi\)
\(882\) 0 0
\(883\) −21.2132 −0.713881 −0.356941 0.934127i \(-0.616180\pi\)
−0.356941 + 0.934127i \(0.616180\pi\)
\(884\) 0 0
\(885\) −4.00000 −0.134459
\(886\) 0 0
\(887\) −26.8701 −0.902208 −0.451104 0.892471i \(-0.648970\pi\)
−0.451104 + 0.892471i \(0.648970\pi\)
\(888\) 0 0
\(889\) −42.0000 −1.40863
\(890\) 0 0
\(891\) −28.2843 −0.947559
\(892\) 0 0
\(893\) −4.00000 −0.133855
\(894\) 0 0
\(895\) −19.7990 −0.661807
\(896\) 0 0
\(897\) 20.0000 0.667781
\(898\) 0 0
\(899\) 11.3137 0.377333
\(900\) 0 0
\(901\) 12.0000 0.399778
\(902\) 0 0
\(903\) 8.48528 0.282372
\(904\) 0 0
\(905\) 16.0000 0.531858
\(906\) 0 0
\(907\) −1.41421 −0.0469582 −0.0234791 0.999724i \(-0.507474\pi\)
−0.0234791 + 0.999724i \(0.507474\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −31.1127 −1.03081 −0.515405 0.856947i \(-0.672358\pi\)
−0.515405 + 0.856947i \(0.672358\pi\)
\(912\) 0 0
\(913\) −72.0000 −2.38285
\(914\) 0 0
\(915\) −19.7990 −0.654534
\(916\) 0 0
\(917\) 48.0000 1.58510
\(918\) 0 0
\(919\) −39.5980 −1.30622 −0.653108 0.757264i \(-0.726535\pi\)
−0.653108 + 0.757264i \(0.726535\pi\)
\(920\) 0 0
\(921\) −22.0000 −0.724925
\(922\) 0 0
\(923\) 5.65685 0.186198
\(924\) 0 0
\(925\) −2.00000 −0.0657596
\(926\) 0 0
\(927\) 9.89949 0.325142
\(928\) 0 0
\(929\) −28.0000 −0.918650 −0.459325 0.888268i \(-0.651909\pi\)
−0.459325 + 0.888268i \(0.651909\pi\)
\(930\) 0 0
\(931\) −31.1127 −1.01968
\(932\) 0 0
\(933\) 4.00000 0.130954
\(934\) 0 0
\(935\) −33.9411 −1.10999
\(936\) 0 0
\(937\) 2.00000 0.0653372 0.0326686 0.999466i \(-0.489599\pi\)
0.0326686 + 0.999466i \(0.489599\pi\)
\(938\) 0 0
\(939\) −8.48528 −0.276907
\(940\) 0 0
\(941\) 48.0000 1.56476 0.782378 0.622804i \(-0.214007\pi\)
0.782378 + 0.622804i \(0.214007\pi\)
\(942\) 0 0
\(943\) 56.5685 1.84213
\(944\) 0 0
\(945\) −24.0000 −0.780720
\(946\) 0 0
\(947\) −15.5563 −0.505513 −0.252757 0.967530i \(-0.581337\pi\)
−0.252757 + 0.967530i \(0.581337\pi\)
\(948\) 0 0
\(949\) 12.0000 0.389536
\(950\) 0 0
\(951\) −2.82843 −0.0917180
\(952\) 0 0
\(953\) −54.0000 −1.74923 −0.874616 0.484817i \(-0.838886\pi\)
−0.874616 + 0.484817i \(0.838886\pi\)
\(954\) 0 0
\(955\) 2.82843 0.0915258
\(956\) 0 0
\(957\) 32.0000 1.03441
\(958\) 0 0
\(959\) 8.48528 0.274004
\(960\) 0 0
\(961\) −23.0000 −0.741935
\(962\) 0 0
\(963\) 7.07107 0.227862
\(964\) 0 0
\(965\) −6.00000 −0.193147
\(966\) 0 0
\(967\) 4.24264 0.136434 0.0682171 0.997671i \(-0.478269\pi\)
0.0682171 + 0.997671i \(0.478269\pi\)
\(968\) 0 0
\(969\) −24.0000 −0.770991
\(970\) 0 0
\(971\) −28.2843 −0.907685 −0.453843 0.891082i \(-0.649947\pi\)
−0.453843 + 0.891082i \(0.649947\pi\)
\(972\) 0 0
\(973\) 36.0000 1.15411
\(974\) 0 0
\(975\) 2.82843 0.0905822
\(976\) 0 0
\(977\) 42.0000 1.34370 0.671850 0.740688i \(-0.265500\pi\)
0.671850 + 0.740688i \(0.265500\pi\)
\(978\) 0 0
\(979\) 33.9411 1.08476
\(980\) 0 0
\(981\) 6.00000 0.191565
\(982\) 0 0
\(983\) −18.3848 −0.586383 −0.293192 0.956054i \(-0.594717\pi\)
−0.293192 + 0.956054i \(0.594717\pi\)
\(984\) 0 0
\(985\) −6.00000 −0.191176
\(986\) 0 0
\(987\) −8.48528 −0.270089
\(988\) 0 0
\(989\) −10.0000 −0.317982
\(990\) 0 0
\(991\) 8.48528 0.269544 0.134772 0.990877i \(-0.456970\pi\)
0.134772 + 0.990877i \(0.456970\pi\)
\(992\) 0 0
\(993\) 32.0000 1.01549
\(994\) 0 0
\(995\) −11.3137 −0.358669
\(996\) 0 0
\(997\) 10.0000 0.316703 0.158352 0.987383i \(-0.449382\pi\)
0.158352 + 0.987383i \(0.449382\pi\)
\(998\) 0 0
\(999\) 11.3137 0.357950
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 1280.2.a.f.1.2 2
4.3 odd 2 inner 1280.2.a.f.1.1 2
5.4 even 2 6400.2.a.bl.1.1 2
8.3 odd 2 1280.2.a.j.1.2 2
8.5 even 2 1280.2.a.j.1.1 2
16.3 odd 4 640.2.d.d.321.2 yes 4
16.5 even 4 640.2.d.d.321.1 4
16.11 odd 4 640.2.d.d.321.3 yes 4
16.13 even 4 640.2.d.d.321.4 yes 4
20.19 odd 2 6400.2.a.bl.1.2 2
40.19 odd 2 6400.2.a.bn.1.1 2
40.29 even 2 6400.2.a.bn.1.2 2
48.5 odd 4 5760.2.k.o.2881.4 4
48.11 even 4 5760.2.k.o.2881.3 4
48.29 odd 4 5760.2.k.o.2881.2 4
48.35 even 4 5760.2.k.o.2881.1 4
80.3 even 4 3200.2.f.i.449.4 4
80.13 odd 4 3200.2.f.i.449.1 4
80.19 odd 4 3200.2.d.s.1601.4 4
80.27 even 4 3200.2.f.i.449.3 4
80.29 even 4 3200.2.d.s.1601.1 4
80.37 odd 4 3200.2.f.i.449.2 4
80.43 even 4 3200.2.f.j.449.2 4
80.53 odd 4 3200.2.f.j.449.3 4
80.59 odd 4 3200.2.d.s.1601.2 4
80.67 even 4 3200.2.f.j.449.1 4
80.69 even 4 3200.2.d.s.1601.3 4
80.77 odd 4 3200.2.f.j.449.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
640.2.d.d.321.1 4 16.5 even 4
640.2.d.d.321.2 yes 4 16.3 odd 4
640.2.d.d.321.3 yes 4 16.11 odd 4
640.2.d.d.321.4 yes 4 16.13 even 4
1280.2.a.f.1.1 2 4.3 odd 2 inner
1280.2.a.f.1.2 2 1.1 even 1 trivial
1280.2.a.j.1.1 2 8.5 even 2
1280.2.a.j.1.2 2 8.3 odd 2
3200.2.d.s.1601.1 4 80.29 even 4
3200.2.d.s.1601.2 4 80.59 odd 4
3200.2.d.s.1601.3 4 80.69 even 4
3200.2.d.s.1601.4 4 80.19 odd 4
3200.2.f.i.449.1 4 80.13 odd 4
3200.2.f.i.449.2 4 80.37 odd 4
3200.2.f.i.449.3 4 80.27 even 4
3200.2.f.i.449.4 4 80.3 even 4
3200.2.f.j.449.1 4 80.67 even 4
3200.2.f.j.449.2 4 80.43 even 4
3200.2.f.j.449.3 4 80.53 odd 4
3200.2.f.j.449.4 4 80.77 odd 4
5760.2.k.o.2881.1 4 48.35 even 4
5760.2.k.o.2881.2 4 48.29 odd 4
5760.2.k.o.2881.3 4 48.11 even 4
5760.2.k.o.2881.4 4 48.5 odd 4
6400.2.a.bl.1.1 2 5.4 even 2
6400.2.a.bl.1.2 2 20.19 odd 2
6400.2.a.bn.1.1 2 40.19 odd 2
6400.2.a.bn.1.2 2 40.29 even 2