Properties

Label 768.2.a.j.1.2
Level $768$
Weight $2$
Character 768.1
Self dual yes
Analytic conductor $6.133$
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] = [768,2,Mod(1,768)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(768, 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("768.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 768.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: \(6.13251087523\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 192)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 768.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} +3.46410 q^{5} +3.46410 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +3.46410 q^{5} +3.46410 q^{7} +1.00000 q^{9} -3.46410 q^{15} +6.00000 q^{17} -4.00000 q^{19} -3.46410 q^{21} -6.92820 q^{23} +7.00000 q^{25} -1.00000 q^{27} -3.46410 q^{29} +3.46410 q^{31} +12.0000 q^{35} -6.92820 q^{37} +6.00000 q^{41} -4.00000 q^{43} +3.46410 q^{45} +6.92820 q^{47} +5.00000 q^{49} -6.00000 q^{51} -3.46410 q^{53} +4.00000 q^{57} +12.0000 q^{59} +6.92820 q^{61} +3.46410 q^{63} +4.00000 q^{67} +6.92820 q^{69} +6.92820 q^{71} -2.00000 q^{73} -7.00000 q^{75} -10.3923 q^{79} +1.00000 q^{81} +20.7846 q^{85} +3.46410 q^{87} -6.00000 q^{89} -3.46410 q^{93} -13.8564 q^{95} -2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} + 2 q^{9} + 12 q^{17} - 8 q^{19} + 14 q^{25} - 2 q^{27} + 24 q^{35} + 12 q^{41} - 8 q^{43} + 10 q^{49} - 12 q^{51} + 8 q^{57} + 24 q^{59} + 8 q^{67} - 4 q^{73} - 14 q^{75} + 2 q^{81} - 12 q^{89} - 4 q^{97}+O(q^{100}) \) 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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 3.46410 1.54919 0.774597 0.632456i \(-0.217953\pi\)
0.774597 + 0.632456i \(0.217953\pi\)
\(6\) 0 0
\(7\) 3.46410 1.30931 0.654654 0.755929i \(-0.272814\pi\)
0.654654 + 0.755929i \(0.272814\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) −3.46410 −0.894427
\(16\) 0 0
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0 0
\(21\) −3.46410 −0.755929
\(22\) 0 0
\(23\) −6.92820 −1.44463 −0.722315 0.691564i \(-0.756922\pi\)
−0.722315 + 0.691564i \(0.756922\pi\)
\(24\) 0 0
\(25\) 7.00000 1.40000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −3.46410 −0.643268 −0.321634 0.946864i \(-0.604232\pi\)
−0.321634 + 0.946864i \(0.604232\pi\)
\(30\) 0 0
\(31\) 3.46410 0.622171 0.311086 0.950382i \(-0.399307\pi\)
0.311086 + 0.950382i \(0.399307\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 12.0000 2.02837
\(36\) 0 0
\(37\) −6.92820 −1.13899 −0.569495 0.821995i \(-0.692861\pi\)
−0.569495 + 0.821995i \(0.692861\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) 3.46410 0.516398
\(46\) 0 0
\(47\) 6.92820 1.01058 0.505291 0.862949i \(-0.331385\pi\)
0.505291 + 0.862949i \(0.331385\pi\)
\(48\) 0 0
\(49\) 5.00000 0.714286
\(50\) 0 0
\(51\) −6.00000 −0.840168
\(52\) 0 0
\(53\) −3.46410 −0.475831 −0.237915 0.971286i \(-0.576464\pi\)
−0.237915 + 0.971286i \(0.576464\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 4.00000 0.529813
\(58\) 0 0
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) 6.92820 0.887066 0.443533 0.896258i \(-0.353725\pi\)
0.443533 + 0.896258i \(0.353725\pi\)
\(62\) 0 0
\(63\) 3.46410 0.436436
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 0 0
\(69\) 6.92820 0.834058
\(70\) 0 0
\(71\) 6.92820 0.822226 0.411113 0.911584i \(-0.365140\pi\)
0.411113 + 0.911584i \(0.365140\pi\)
\(72\) 0 0
\(73\) −2.00000 −0.234082 −0.117041 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) 0 0
\(75\) −7.00000 −0.808290
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −10.3923 −1.16923 −0.584613 0.811312i \(-0.698754\pi\)
−0.584613 + 0.811312i \(0.698754\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 20.7846 2.25441
\(86\) 0 0
\(87\) 3.46410 0.371391
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −3.46410 −0.359211
\(94\) 0 0
\(95\) −13.8564 −1.42164
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −3.46410 −0.344691 −0.172345 0.985037i \(-0.555135\pi\)
−0.172345 + 0.985037i \(0.555135\pi\)
\(102\) 0 0
\(103\) −17.3205 −1.70664 −0.853320 0.521387i \(-0.825415\pi\)
−0.853320 + 0.521387i \(0.825415\pi\)
\(104\) 0 0
\(105\) −12.0000 −1.17108
\(106\) 0 0
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 0 0
\(109\) −13.8564 −1.32720 −0.663602 0.748086i \(-0.730973\pi\)
−0.663602 + 0.748086i \(0.730973\pi\)
\(110\) 0 0
\(111\) 6.92820 0.657596
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) −24.0000 −2.23801
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 20.7846 1.90532
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) −6.00000 −0.541002
\(124\) 0 0
\(125\) 6.92820 0.619677
\(126\) 0 0
\(127\) −3.46410 −0.307389 −0.153695 0.988118i \(-0.549117\pi\)
−0.153695 + 0.988118i \(0.549117\pi\)
\(128\) 0 0
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 0 0
\(133\) −13.8564 −1.20150
\(134\) 0 0
\(135\) −3.46410 −0.298142
\(136\) 0 0
\(137\) 6.00000 0.512615 0.256307 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) 0 0
\(139\) −20.0000 −1.69638 −0.848189 0.529694i \(-0.822307\pi\)
−0.848189 + 0.529694i \(0.822307\pi\)
\(140\) 0 0
\(141\) −6.92820 −0.583460
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −12.0000 −0.996546
\(146\) 0 0
\(147\) −5.00000 −0.412393
\(148\) 0 0
\(149\) −10.3923 −0.851371 −0.425685 0.904871i \(-0.639967\pi\)
−0.425685 + 0.904871i \(0.639967\pi\)
\(150\) 0 0
\(151\) 3.46410 0.281905 0.140952 0.990016i \(-0.454984\pi\)
0.140952 + 0.990016i \(0.454984\pi\)
\(152\) 0 0
\(153\) 6.00000 0.485071
\(154\) 0 0
\(155\) 12.0000 0.963863
\(156\) 0 0
\(157\) −6.92820 −0.552931 −0.276465 0.961024i \(-0.589163\pi\)
−0.276465 + 0.961024i \(0.589163\pi\)
\(158\) 0 0
\(159\) 3.46410 0.274721
\(160\) 0 0
\(161\) −24.0000 −1.89146
\(162\) 0 0
\(163\) 20.0000 1.56652 0.783260 0.621694i \(-0.213555\pi\)
0.783260 + 0.621694i \(0.213555\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −13.8564 −1.07224 −0.536120 0.844141i \(-0.680111\pi\)
−0.536120 + 0.844141i \(0.680111\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) −4.00000 −0.305888
\(172\) 0 0
\(173\) 17.3205 1.31685 0.658427 0.752645i \(-0.271222\pi\)
0.658427 + 0.752645i \(0.271222\pi\)
\(174\) 0 0
\(175\) 24.2487 1.83303
\(176\) 0 0
\(177\) −12.0000 −0.901975
\(178\) 0 0
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) −13.8564 −1.02994 −0.514969 0.857209i \(-0.672197\pi\)
−0.514969 + 0.857209i \(0.672197\pi\)
\(182\) 0 0
\(183\) −6.92820 −0.512148
\(184\) 0 0
\(185\) −24.0000 −1.76452
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −3.46410 −0.251976
\(190\) 0 0
\(191\) 13.8564 1.00261 0.501307 0.865269i \(-0.332853\pi\)
0.501307 + 0.865269i \(0.332853\pi\)
\(192\) 0 0
\(193\) 2.00000 0.143963 0.0719816 0.997406i \(-0.477068\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 10.3923 0.740421 0.370211 0.928948i \(-0.379286\pi\)
0.370211 + 0.928948i \(0.379286\pi\)
\(198\) 0 0
\(199\) 10.3923 0.736691 0.368345 0.929689i \(-0.379924\pi\)
0.368345 + 0.929689i \(0.379924\pi\)
\(200\) 0 0
\(201\) −4.00000 −0.282138
\(202\) 0 0
\(203\) −12.0000 −0.842235
\(204\) 0 0
\(205\) 20.7846 1.45166
\(206\) 0 0
\(207\) −6.92820 −0.481543
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) 0 0
\(213\) −6.92820 −0.474713
\(214\) 0 0
\(215\) −13.8564 −0.944999
\(216\) 0 0
\(217\) 12.0000 0.814613
\(218\) 0 0
\(219\) 2.00000 0.135147
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −3.46410 −0.231973 −0.115987 0.993251i \(-0.537003\pi\)
−0.115987 + 0.993251i \(0.537003\pi\)
\(224\) 0 0
\(225\) 7.00000 0.466667
\(226\) 0 0
\(227\) 24.0000 1.59294 0.796468 0.604681i \(-0.206699\pi\)
0.796468 + 0.604681i \(0.206699\pi\)
\(228\) 0 0
\(229\) 27.7128 1.83131 0.915657 0.401960i \(-0.131671\pi\)
0.915657 + 0.401960i \(0.131671\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 18.0000 1.17922 0.589610 0.807688i \(-0.299282\pi\)
0.589610 + 0.807688i \(0.299282\pi\)
\(234\) 0 0
\(235\) 24.0000 1.56559
\(236\) 0 0
\(237\) 10.3923 0.675053
\(238\) 0 0
\(239\) −27.7128 −1.79259 −0.896296 0.443455i \(-0.853752\pi\)
−0.896296 + 0.443455i \(0.853752\pi\)
\(240\) 0 0
\(241\) −22.0000 −1.41714 −0.708572 0.705638i \(-0.750660\pi\)
−0.708572 + 0.705638i \(0.750660\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 17.3205 1.10657
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) −20.7846 −1.30158
\(256\) 0 0
\(257\) 18.0000 1.12281 0.561405 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) 0 0
\(259\) −24.0000 −1.49129
\(260\) 0 0
\(261\) −3.46410 −0.214423
\(262\) 0 0
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) −12.0000 −0.737154
\(266\) 0 0
\(267\) 6.00000 0.367194
\(268\) 0 0
\(269\) −31.1769 −1.90089 −0.950445 0.310893i \(-0.899372\pi\)
−0.950445 + 0.310893i \(0.899372\pi\)
\(270\) 0 0
\(271\) −3.46410 −0.210429 −0.105215 0.994450i \(-0.533553\pi\)
−0.105215 + 0.994450i \(0.533553\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −13.8564 −0.832551 −0.416275 0.909239i \(-0.636665\pi\)
−0.416275 + 0.909239i \(0.636665\pi\)
\(278\) 0 0
\(279\) 3.46410 0.207390
\(280\) 0 0
\(281\) −30.0000 −1.78965 −0.894825 0.446417i \(-0.852700\pi\)
−0.894825 + 0.446417i \(0.852700\pi\)
\(282\) 0 0
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) 0 0
\(285\) 13.8564 0.820783
\(286\) 0 0
\(287\) 20.7846 1.22688
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 2.00000 0.117242
\(292\) 0 0
\(293\) −3.46410 −0.202375 −0.101187 0.994867i \(-0.532264\pi\)
−0.101187 + 0.994867i \(0.532264\pi\)
\(294\) 0 0
\(295\) 41.5692 2.42025
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −13.8564 −0.798670
\(302\) 0 0
\(303\) 3.46410 0.199007
\(304\) 0 0
\(305\) 24.0000 1.37424
\(306\) 0 0
\(307\) 4.00000 0.228292 0.114146 0.993464i \(-0.463587\pi\)
0.114146 + 0.993464i \(0.463587\pi\)
\(308\) 0 0
\(309\) 17.3205 0.985329
\(310\) 0 0
\(311\) −13.8564 −0.785725 −0.392862 0.919597i \(-0.628515\pi\)
−0.392862 + 0.919597i \(0.628515\pi\)
\(312\) 0 0
\(313\) 26.0000 1.46961 0.734803 0.678280i \(-0.237274\pi\)
0.734803 + 0.678280i \(0.237274\pi\)
\(314\) 0 0
\(315\) 12.0000 0.676123
\(316\) 0 0
\(317\) −31.1769 −1.75107 −0.875535 0.483155i \(-0.839491\pi\)
−0.875535 + 0.483155i \(0.839491\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −12.0000 −0.669775
\(322\) 0 0
\(323\) −24.0000 −1.33540
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 13.8564 0.766261
\(328\) 0 0
\(329\) 24.0000 1.32316
\(330\) 0 0
\(331\) −20.0000 −1.09930 −0.549650 0.835395i \(-0.685239\pi\)
−0.549650 + 0.835395i \(0.685239\pi\)
\(332\) 0 0
\(333\) −6.92820 −0.379663
\(334\) 0 0
\(335\) 13.8564 0.757056
\(336\) 0 0
\(337\) 22.0000 1.19842 0.599208 0.800593i \(-0.295482\pi\)
0.599208 + 0.800593i \(0.295482\pi\)
\(338\) 0 0
\(339\) 6.00000 0.325875
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −6.92820 −0.374088
\(344\) 0 0
\(345\) 24.0000 1.29212
\(346\) 0 0
\(347\) −24.0000 −1.28839 −0.644194 0.764862i \(-0.722807\pi\)
−0.644194 + 0.764862i \(0.722807\pi\)
\(348\) 0 0
\(349\) 6.92820 0.370858 0.185429 0.982658i \(-0.440632\pi\)
0.185429 + 0.982658i \(0.440632\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −6.00000 −0.319348 −0.159674 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 0 0
\(355\) 24.0000 1.27379
\(356\) 0 0
\(357\) −20.7846 −1.10004
\(358\) 0 0
\(359\) 6.92820 0.365657 0.182828 0.983145i \(-0.441475\pi\)
0.182828 + 0.983145i \(0.441475\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 0 0
\(363\) 11.0000 0.577350
\(364\) 0 0
\(365\) −6.92820 −0.362639
\(366\) 0 0
\(367\) −24.2487 −1.26577 −0.632886 0.774245i \(-0.718130\pi\)
−0.632886 + 0.774245i \(0.718130\pi\)
\(368\) 0 0
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) −12.0000 −0.623009
\(372\) 0 0
\(373\) −20.7846 −1.07619 −0.538093 0.842885i \(-0.680855\pi\)
−0.538093 + 0.842885i \(0.680855\pi\)
\(374\) 0 0
\(375\) −6.92820 −0.357771
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 20.0000 1.02733 0.513665 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 0 0
\(381\) 3.46410 0.177471
\(382\) 0 0
\(383\) 13.8564 0.708029 0.354015 0.935240i \(-0.384816\pi\)
0.354015 + 0.935240i \(0.384816\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −4.00000 −0.203331
\(388\) 0 0
\(389\) −24.2487 −1.22946 −0.614729 0.788738i \(-0.710735\pi\)
−0.614729 + 0.788738i \(0.710735\pi\)
\(390\) 0 0
\(391\) −41.5692 −2.10225
\(392\) 0 0
\(393\) −12.0000 −0.605320
\(394\) 0 0
\(395\) −36.0000 −1.81136
\(396\) 0 0
\(397\) 34.6410 1.73858 0.869291 0.494300i \(-0.164576\pi\)
0.869291 + 0.494300i \(0.164576\pi\)
\(398\) 0 0
\(399\) 13.8564 0.693688
\(400\) 0 0
\(401\) 6.00000 0.299626 0.149813 0.988714i \(-0.452133\pi\)
0.149813 + 0.988714i \(0.452133\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 3.46410 0.172133
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 22.0000 1.08783 0.543915 0.839140i \(-0.316941\pi\)
0.543915 + 0.839140i \(0.316941\pi\)
\(410\) 0 0
\(411\) −6.00000 −0.295958
\(412\) 0 0
\(413\) 41.5692 2.04549
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 20.0000 0.979404
\(418\) 0 0
\(419\) −24.0000 −1.17248 −0.586238 0.810139i \(-0.699392\pi\)
−0.586238 + 0.810139i \(0.699392\pi\)
\(420\) 0 0
\(421\) 13.8564 0.675320 0.337660 0.941268i \(-0.390365\pi\)
0.337660 + 0.941268i \(0.390365\pi\)
\(422\) 0 0
\(423\) 6.92820 0.336861
\(424\) 0 0
\(425\) 42.0000 2.03730
\(426\) 0 0
\(427\) 24.0000 1.16144
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −6.92820 −0.333720 −0.166860 0.985981i \(-0.553363\pi\)
−0.166860 + 0.985981i \(0.553363\pi\)
\(432\) 0 0
\(433\) −2.00000 −0.0961139 −0.0480569 0.998845i \(-0.515303\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) 0 0
\(435\) 12.0000 0.575356
\(436\) 0 0
\(437\) 27.7128 1.32568
\(438\) 0 0
\(439\) −31.1769 −1.48799 −0.743996 0.668184i \(-0.767072\pi\)
−0.743996 + 0.668184i \(0.767072\pi\)
\(440\) 0 0
\(441\) 5.00000 0.238095
\(442\) 0 0
\(443\) −24.0000 −1.14027 −0.570137 0.821549i \(-0.693110\pi\)
−0.570137 + 0.821549i \(0.693110\pi\)
\(444\) 0 0
\(445\) −20.7846 −0.985285
\(446\) 0 0
\(447\) 10.3923 0.491539
\(448\) 0 0
\(449\) 6.00000 0.283158 0.141579 0.989927i \(-0.454782\pi\)
0.141579 + 0.989927i \(0.454782\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −3.46410 −0.162758
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 22.0000 1.02912 0.514558 0.857455i \(-0.327956\pi\)
0.514558 + 0.857455i \(0.327956\pi\)
\(458\) 0 0
\(459\) −6.00000 −0.280056
\(460\) 0 0
\(461\) 31.1769 1.45205 0.726027 0.687666i \(-0.241365\pi\)
0.726027 + 0.687666i \(0.241365\pi\)
\(462\) 0 0
\(463\) 10.3923 0.482971 0.241486 0.970404i \(-0.422365\pi\)
0.241486 + 0.970404i \(0.422365\pi\)
\(464\) 0 0
\(465\) −12.0000 −0.556487
\(466\) 0 0
\(467\) 24.0000 1.11059 0.555294 0.831654i \(-0.312606\pi\)
0.555294 + 0.831654i \(0.312606\pi\)
\(468\) 0 0
\(469\) 13.8564 0.639829
\(470\) 0 0
\(471\) 6.92820 0.319235
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −28.0000 −1.28473
\(476\) 0 0
\(477\) −3.46410 −0.158610
\(478\) 0 0
\(479\) −20.7846 −0.949673 −0.474837 0.880074i \(-0.657493\pi\)
−0.474837 + 0.880074i \(0.657493\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 24.0000 1.09204
\(484\) 0 0
\(485\) −6.92820 −0.314594
\(486\) 0 0
\(487\) −10.3923 −0.470920 −0.235460 0.971884i \(-0.575660\pi\)
−0.235460 + 0.971884i \(0.575660\pi\)
\(488\) 0 0
\(489\) −20.0000 −0.904431
\(490\) 0 0
\(491\) 36.0000 1.62466 0.812329 0.583200i \(-0.198200\pi\)
0.812329 + 0.583200i \(0.198200\pi\)
\(492\) 0 0
\(493\) −20.7846 −0.936092
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 24.0000 1.07655
\(498\) 0 0
\(499\) 28.0000 1.25345 0.626726 0.779240i \(-0.284395\pi\)
0.626726 + 0.779240i \(0.284395\pi\)
\(500\) 0 0
\(501\) 13.8564 0.619059
\(502\) 0 0
\(503\) 34.6410 1.54457 0.772283 0.635278i \(-0.219115\pi\)
0.772283 + 0.635278i \(0.219115\pi\)
\(504\) 0 0
\(505\) −12.0000 −0.533993
\(506\) 0 0
\(507\) 13.0000 0.577350
\(508\) 0 0
\(509\) 24.2487 1.07481 0.537403 0.843326i \(-0.319406\pi\)
0.537403 + 0.843326i \(0.319406\pi\)
\(510\) 0 0
\(511\) −6.92820 −0.306486
\(512\) 0 0
\(513\) 4.00000 0.176604
\(514\) 0 0
\(515\) −60.0000 −2.64392
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −17.3205 −0.760286
\(520\) 0 0
\(521\) −18.0000 −0.788594 −0.394297 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) 0 0
\(523\) 20.0000 0.874539 0.437269 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) 0 0
\(525\) −24.2487 −1.05830
\(526\) 0 0
\(527\) 20.7846 0.905392
\(528\) 0 0
\(529\) 25.0000 1.08696
\(530\) 0 0
\(531\) 12.0000 0.520756
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 41.5692 1.79719
\(536\) 0 0
\(537\) 12.0000 0.517838
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −13.8564 −0.595733 −0.297867 0.954607i \(-0.596275\pi\)
−0.297867 + 0.954607i \(0.596275\pi\)
\(542\) 0 0
\(543\) 13.8564 0.594635
\(544\) 0 0
\(545\) −48.0000 −2.05609
\(546\) 0 0
\(547\) −28.0000 −1.19719 −0.598597 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) 0 0
\(549\) 6.92820 0.295689
\(550\) 0 0
\(551\) 13.8564 0.590303
\(552\) 0 0
\(553\) −36.0000 −1.53088
\(554\) 0 0
\(555\) 24.0000 1.01874
\(556\) 0 0
\(557\) −10.3923 −0.440336 −0.220168 0.975462i \(-0.570661\pi\)
−0.220168 + 0.975462i \(0.570661\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −24.0000 −1.01148 −0.505740 0.862686i \(-0.668780\pi\)
−0.505740 + 0.862686i \(0.668780\pi\)
\(564\) 0 0
\(565\) −20.7846 −0.874415
\(566\) 0 0
\(567\) 3.46410 0.145479
\(568\) 0 0
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 0 0
\(571\) 4.00000 0.167395 0.0836974 0.996491i \(-0.473327\pi\)
0.0836974 + 0.996491i \(0.473327\pi\)
\(572\) 0 0
\(573\) −13.8564 −0.578860
\(574\) 0 0
\(575\) −48.4974 −2.02248
\(576\) 0 0
\(577\) −2.00000 −0.0832611 −0.0416305 0.999133i \(-0.513255\pi\)
−0.0416305 + 0.999133i \(0.513255\pi\)
\(578\) 0 0
\(579\) −2.00000 −0.0831172
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −12.0000 −0.495293 −0.247647 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) 0 0
\(589\) −13.8564 −0.570943
\(590\) 0 0
\(591\) −10.3923 −0.427482
\(592\) 0 0
\(593\) −6.00000 −0.246390 −0.123195 0.992382i \(-0.539314\pi\)
−0.123195 + 0.992382i \(0.539314\pi\)
\(594\) 0 0
\(595\) 72.0000 2.95171
\(596\) 0 0
\(597\) −10.3923 −0.425329
\(598\) 0 0
\(599\) 34.6410 1.41539 0.707697 0.706516i \(-0.249734\pi\)
0.707697 + 0.706516i \(0.249734\pi\)
\(600\) 0 0
\(601\) −26.0000 −1.06056 −0.530281 0.847822i \(-0.677914\pi\)
−0.530281 + 0.847822i \(0.677914\pi\)
\(602\) 0 0
\(603\) 4.00000 0.162893
\(604\) 0 0
\(605\) −38.1051 −1.54919
\(606\) 0 0
\(607\) −3.46410 −0.140604 −0.0703018 0.997526i \(-0.522396\pi\)
−0.0703018 + 0.997526i \(0.522396\pi\)
\(608\) 0 0
\(609\) 12.0000 0.486265
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 34.6410 1.39914 0.699569 0.714565i \(-0.253375\pi\)
0.699569 + 0.714565i \(0.253375\pi\)
\(614\) 0 0
\(615\) −20.7846 −0.838116
\(616\) 0 0
\(617\) 18.0000 0.724653 0.362326 0.932051i \(-0.381983\pi\)
0.362326 + 0.932051i \(0.381983\pi\)
\(618\) 0 0
\(619\) 4.00000 0.160774 0.0803868 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) 0 0
\(621\) 6.92820 0.278019
\(622\) 0 0
\(623\) −20.7846 −0.832718
\(624\) 0 0
\(625\) −11.0000 −0.440000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −41.5692 −1.65747
\(630\) 0 0
\(631\) 17.3205 0.689519 0.344759 0.938691i \(-0.387961\pi\)
0.344759 + 0.938691i \(0.387961\pi\)
\(632\) 0 0
\(633\) −4.00000 −0.158986
\(634\) 0 0
\(635\) −12.0000 −0.476205
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 6.92820 0.274075
\(640\) 0 0
\(641\) −18.0000 −0.710957 −0.355479 0.934684i \(-0.615682\pi\)
−0.355479 + 0.934684i \(0.615682\pi\)
\(642\) 0 0
\(643\) −28.0000 −1.10421 −0.552106 0.833774i \(-0.686176\pi\)
−0.552106 + 0.833774i \(0.686176\pi\)
\(644\) 0 0
\(645\) 13.8564 0.545595
\(646\) 0 0
\(647\) −34.6410 −1.36188 −0.680939 0.732340i \(-0.738428\pi\)
−0.680939 + 0.732340i \(0.738428\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −12.0000 −0.470317
\(652\) 0 0
\(653\) −10.3923 −0.406682 −0.203341 0.979108i \(-0.565180\pi\)
−0.203341 + 0.979108i \(0.565180\pi\)
\(654\) 0 0
\(655\) 41.5692 1.62424
\(656\) 0 0
\(657\) −2.00000 −0.0780274
\(658\) 0 0
\(659\) 12.0000 0.467454 0.233727 0.972302i \(-0.424908\pi\)
0.233727 + 0.972302i \(0.424908\pi\)
\(660\) 0 0
\(661\) −6.92820 −0.269476 −0.134738 0.990881i \(-0.543019\pi\)
−0.134738 + 0.990881i \(0.543019\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −48.0000 −1.86136
\(666\) 0 0
\(667\) 24.0000 0.929284
\(668\) 0 0
\(669\) 3.46410 0.133930
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −2.00000 −0.0770943 −0.0385472 0.999257i \(-0.512273\pi\)
−0.0385472 + 0.999257i \(0.512273\pi\)
\(674\) 0 0
\(675\) −7.00000 −0.269430
\(676\) 0 0
\(677\) 3.46410 0.133136 0.0665681 0.997782i \(-0.478795\pi\)
0.0665681 + 0.997782i \(0.478795\pi\)
\(678\) 0 0
\(679\) −6.92820 −0.265880
\(680\) 0 0
\(681\) −24.0000 −0.919682
\(682\) 0 0
\(683\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 0 0
\(685\) 20.7846 0.794139
\(686\) 0 0
\(687\) −27.7128 −1.05731
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −28.0000 −1.06517 −0.532585 0.846376i \(-0.678779\pi\)
−0.532585 + 0.846376i \(0.678779\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −69.2820 −2.62802
\(696\) 0 0
\(697\) 36.0000 1.36360
\(698\) 0 0
\(699\) −18.0000 −0.680823
\(700\) 0 0
\(701\) 10.3923 0.392512 0.196256 0.980553i \(-0.437122\pi\)
0.196256 + 0.980553i \(0.437122\pi\)
\(702\) 0 0
\(703\) 27.7128 1.04521
\(704\) 0 0
\(705\) −24.0000 −0.903892
\(706\) 0 0
\(707\) −12.0000 −0.451306
\(708\) 0 0
\(709\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(710\) 0 0
\(711\) −10.3923 −0.389742
\(712\) 0 0
\(713\) −24.0000 −0.898807
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 27.7128 1.03495
\(718\) 0 0
\(719\) 20.7846 0.775135 0.387568 0.921841i \(-0.373315\pi\)
0.387568 + 0.921841i \(0.373315\pi\)
\(720\) 0 0
\(721\) −60.0000 −2.23452
\(722\) 0 0
\(723\) 22.0000 0.818189
\(724\) 0 0
\(725\) −24.2487 −0.900575
\(726\) 0 0
\(727\) −10.3923 −0.385429 −0.192715 0.981255i \(-0.561729\pi\)
−0.192715 + 0.981255i \(0.561729\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −24.0000 −0.887672
\(732\) 0 0
\(733\) 27.7128 1.02360 0.511798 0.859106i \(-0.328980\pi\)
0.511798 + 0.859106i \(0.328980\pi\)
\(734\) 0 0
\(735\) −17.3205 −0.638877
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 28.0000 1.03000 0.514998 0.857191i \(-0.327793\pi\)
0.514998 + 0.857191i \(0.327793\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 27.7128 1.01668 0.508342 0.861155i \(-0.330258\pi\)
0.508342 + 0.861155i \(0.330258\pi\)
\(744\) 0 0
\(745\) −36.0000 −1.31894
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 41.5692 1.51891
\(750\) 0 0
\(751\) −24.2487 −0.884848 −0.442424 0.896806i \(-0.645881\pi\)
−0.442424 + 0.896806i \(0.645881\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 12.0000 0.436725
\(756\) 0 0
\(757\) 27.7128 1.00724 0.503620 0.863925i \(-0.332001\pi\)
0.503620 + 0.863925i \(0.332001\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −18.0000 −0.652499 −0.326250 0.945284i \(-0.605785\pi\)
−0.326250 + 0.945284i \(0.605785\pi\)
\(762\) 0 0
\(763\) −48.0000 −1.73772
\(764\) 0 0
\(765\) 20.7846 0.751469
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −46.0000 −1.65880 −0.829401 0.558653i \(-0.811318\pi\)
−0.829401 + 0.558653i \(0.811318\pi\)
\(770\) 0 0
\(771\) −18.0000 −0.648254
\(772\) 0 0
\(773\) 38.1051 1.37055 0.685273 0.728286i \(-0.259683\pi\)
0.685273 + 0.728286i \(0.259683\pi\)
\(774\) 0 0
\(775\) 24.2487 0.871039
\(776\) 0 0
\(777\) 24.0000 0.860995
\(778\) 0 0
\(779\) −24.0000 −0.859889
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 3.46410 0.123797
\(784\) 0 0
\(785\) −24.0000 −0.856597
\(786\) 0 0
\(787\) −4.00000 −0.142585 −0.0712923 0.997455i \(-0.522712\pi\)
−0.0712923 + 0.997455i \(0.522712\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −20.7846 −0.739016
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 12.0000 0.425596
\(796\) 0 0
\(797\) −24.2487 −0.858933 −0.429467 0.903083i \(-0.641298\pi\)
−0.429467 + 0.903083i \(0.641298\pi\)
\(798\) 0 0
\(799\) 41.5692 1.47061
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) −83.1384 −2.93024
\(806\) 0 0
\(807\) 31.1769 1.09748
\(808\) 0 0
\(809\) 54.0000 1.89854 0.949269 0.314464i \(-0.101825\pi\)
0.949269 + 0.314464i \(0.101825\pi\)
\(810\) 0 0
\(811\) 20.0000 0.702295 0.351147 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) 0 0
\(813\) 3.46410 0.121491
\(814\) 0 0
\(815\) 69.2820 2.42684
\(816\) 0 0
\(817\) 16.0000 0.559769
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 24.2487 0.846286 0.423143 0.906063i \(-0.360927\pi\)
0.423143 + 0.906063i \(0.360927\pi\)
\(822\) 0 0
\(823\) 31.1769 1.08676 0.543379 0.839487i \(-0.317144\pi\)
0.543379 + 0.839487i \(0.317144\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 12.0000 0.417281 0.208640 0.977992i \(-0.433096\pi\)
0.208640 + 0.977992i \(0.433096\pi\)
\(828\) 0 0
\(829\) 41.5692 1.44376 0.721879 0.692019i \(-0.243279\pi\)
0.721879 + 0.692019i \(0.243279\pi\)
\(830\) 0 0
\(831\) 13.8564 0.480673
\(832\) 0 0
\(833\) 30.0000 1.03944
\(834\) 0 0
\(835\) −48.0000 −1.66111
\(836\) 0 0
\(837\) −3.46410 −0.119737
\(838\) 0 0
\(839\) −6.92820 −0.239188 −0.119594 0.992823i \(-0.538159\pi\)
−0.119594 + 0.992823i \(0.538159\pi\)
\(840\) 0 0
\(841\) −17.0000 −0.586207
\(842\) 0 0
\(843\) 30.0000 1.03325
\(844\) 0 0
\(845\) −45.0333 −1.54919
\(846\) 0 0
\(847\) −38.1051 −1.30931
\(848\) 0 0
\(849\) −4.00000 −0.137280
\(850\) 0 0
\(851\) 48.0000 1.64542
\(852\) 0 0
\(853\) 48.4974 1.66052 0.830260 0.557376i \(-0.188192\pi\)
0.830260 + 0.557376i \(0.188192\pi\)
\(854\) 0 0
\(855\) −13.8564 −0.473879
\(856\) 0 0
\(857\) −42.0000 −1.43469 −0.717346 0.696717i \(-0.754643\pi\)
−0.717346 + 0.696717i \(0.754643\pi\)
\(858\) 0 0
\(859\) −4.00000 −0.136478 −0.0682391 0.997669i \(-0.521738\pi\)
−0.0682391 + 0.997669i \(0.521738\pi\)
\(860\) 0 0
\(861\) −20.7846 −0.708338
\(862\) 0 0
\(863\) 13.8564 0.471678 0.235839 0.971792i \(-0.424216\pi\)
0.235839 + 0.971792i \(0.424216\pi\)
\(864\) 0 0
\(865\) 60.0000 2.04006
\(866\) 0 0
\(867\) −19.0000 −0.645274
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −2.00000 −0.0676897
\(874\) 0 0
\(875\) 24.0000 0.811348
\(876\) 0 0
\(877\) 6.92820 0.233949 0.116974 0.993135i \(-0.462680\pi\)
0.116974 + 0.993135i \(0.462680\pi\)
\(878\) 0 0
\(879\) 3.46410 0.116841
\(880\) 0 0
\(881\) −30.0000 −1.01073 −0.505363 0.862907i \(-0.668641\pi\)
−0.505363 + 0.862907i \(0.668641\pi\)
\(882\) 0 0
\(883\) 20.0000 0.673054 0.336527 0.941674i \(-0.390748\pi\)
0.336527 + 0.941674i \(0.390748\pi\)
\(884\) 0 0
\(885\) −41.5692 −1.39733
\(886\) 0 0
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) −12.0000 −0.402467
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −27.7128 −0.927374
\(894\) 0 0
\(895\) −41.5692 −1.38951
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −12.0000 −0.400222
\(900\) 0 0
\(901\) −20.7846 −0.692436
\(902\) 0 0
\(903\) 13.8564 0.461112
\(904\) 0 0
\(905\) −48.0000 −1.59557
\(906\) 0 0
\(907\) −52.0000 −1.72663 −0.863316 0.504664i \(-0.831616\pi\)
−0.863316 + 0.504664i \(0.831616\pi\)
\(908\) 0 0
\(909\) −3.46410 −0.114897
\(910\) 0 0
\(911\) −55.4256 −1.83633 −0.918166 0.396195i \(-0.870330\pi\)
−0.918166 + 0.396195i \(0.870330\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) −24.0000 −0.793416
\(916\) 0 0
\(917\) 41.5692 1.37274
\(918\) 0 0
\(919\) −51.9615 −1.71405 −0.857026 0.515273i \(-0.827691\pi\)
−0.857026 + 0.515273i \(0.827691\pi\)
\(920\) 0 0
\(921\) −4.00000 −0.131804
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −48.4974 −1.59459
\(926\) 0 0
\(927\) −17.3205 −0.568880
\(928\) 0 0
\(929\) 6.00000 0.196854 0.0984268 0.995144i \(-0.468619\pi\)
0.0984268 + 0.995144i \(0.468619\pi\)
\(930\) 0 0
\(931\) −20.0000 −0.655474
\(932\) 0 0
\(933\) 13.8564 0.453638
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 26.0000 0.849383 0.424691 0.905338i \(-0.360383\pi\)
0.424691 + 0.905338i \(0.360383\pi\)
\(938\) 0 0
\(939\) −26.0000 −0.848478
\(940\) 0 0
\(941\) −24.2487 −0.790485 −0.395243 0.918577i \(-0.629340\pi\)
−0.395243 + 0.918577i \(0.629340\pi\)
\(942\) 0 0
\(943\) −41.5692 −1.35368
\(944\) 0 0
\(945\) −12.0000 −0.390360
\(946\) 0 0
\(947\) 36.0000 1.16984 0.584921 0.811090i \(-0.301125\pi\)
0.584921 + 0.811090i \(0.301125\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 31.1769 1.01098
\(952\) 0 0
\(953\) 30.0000 0.971795 0.485898 0.874016i \(-0.338493\pi\)
0.485898 + 0.874016i \(0.338493\pi\)
\(954\) 0 0
\(955\) 48.0000 1.55324
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 20.7846 0.671170
\(960\) 0 0
\(961\) −19.0000 −0.612903
\(962\) 0 0
\(963\) 12.0000 0.386695
\(964\) 0 0
\(965\) 6.92820 0.223027
\(966\) 0 0
\(967\) 38.1051 1.22538 0.612689 0.790324i \(-0.290088\pi\)
0.612689 + 0.790324i \(0.290088\pi\)
\(968\) 0 0
\(969\) 24.0000 0.770991
\(970\) 0 0
\(971\) 48.0000 1.54039 0.770197 0.637806i \(-0.220158\pi\)
0.770197 + 0.637806i \(0.220158\pi\)
\(972\) 0 0
\(973\) −69.2820 −2.22108
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 54.0000 1.72761 0.863807 0.503824i \(-0.168074\pi\)
0.863807 + 0.503824i \(0.168074\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −13.8564 −0.442401
\(982\) 0 0
\(983\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 0 0
\(985\) 36.0000 1.14706
\(986\) 0 0
\(987\) −24.0000 −0.763928
\(988\) 0 0
\(989\) 27.7128 0.881216
\(990\) 0 0
\(991\) −24.2487 −0.770286 −0.385143 0.922857i \(-0.625848\pi\)
−0.385143 + 0.922857i \(0.625848\pi\)
\(992\) 0 0
\(993\) 20.0000 0.634681
\(994\) 0 0
\(995\) 36.0000 1.14128
\(996\) 0 0
\(997\) −20.7846 −0.658255 −0.329128 0.944285i \(-0.606755\pi\)
−0.329128 + 0.944285i \(0.606755\pi\)
\(998\) 0 0
\(999\) 6.92820 0.219199
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 768.2.a.j.1.2 2
3.2 odd 2 2304.2.a.s.1.1 2
4.3 odd 2 768.2.a.k.1.2 2
8.3 odd 2 inner 768.2.a.j.1.1 2
8.5 even 2 768.2.a.k.1.1 2
12.11 even 2 2304.2.a.u.1.1 2
16.3 odd 4 192.2.d.a.97.3 yes 4
16.5 even 4 192.2.d.a.97.4 yes 4
16.11 odd 4 192.2.d.a.97.2 yes 4
16.13 even 4 192.2.d.a.97.1 4
24.5 odd 2 2304.2.a.u.1.2 2
24.11 even 2 2304.2.a.s.1.2 2
48.5 odd 4 576.2.d.b.289.1 4
48.11 even 4 576.2.d.b.289.2 4
48.29 odd 4 576.2.d.b.289.3 4
48.35 even 4 576.2.d.b.289.4 4
80.3 even 4 4800.2.d.j.1249.1 4
80.13 odd 4 4800.2.d.o.1249.3 4
80.19 odd 4 4800.2.k.j.2401.1 4
80.27 even 4 4800.2.d.j.1249.4 4
80.29 even 4 4800.2.k.j.2401.4 4
80.37 odd 4 4800.2.d.o.1249.2 4
80.43 even 4 4800.2.d.o.1249.1 4
80.53 odd 4 4800.2.d.j.1249.3 4
80.59 odd 4 4800.2.k.j.2401.3 4
80.67 even 4 4800.2.d.o.1249.4 4
80.69 even 4 4800.2.k.j.2401.2 4
80.77 odd 4 4800.2.d.j.1249.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
192.2.d.a.97.1 4 16.13 even 4
192.2.d.a.97.2 yes 4 16.11 odd 4
192.2.d.a.97.3 yes 4 16.3 odd 4
192.2.d.a.97.4 yes 4 16.5 even 4
576.2.d.b.289.1 4 48.5 odd 4
576.2.d.b.289.2 4 48.11 even 4
576.2.d.b.289.3 4 48.29 odd 4
576.2.d.b.289.4 4 48.35 even 4
768.2.a.j.1.1 2 8.3 odd 2 inner
768.2.a.j.1.2 2 1.1 even 1 trivial
768.2.a.k.1.1 2 8.5 even 2
768.2.a.k.1.2 2 4.3 odd 2
2304.2.a.s.1.1 2 3.2 odd 2
2304.2.a.s.1.2 2 24.11 even 2
2304.2.a.u.1.1 2 12.11 even 2
2304.2.a.u.1.2 2 24.5 odd 2
4800.2.d.j.1249.1 4 80.3 even 4
4800.2.d.j.1249.2 4 80.77 odd 4
4800.2.d.j.1249.3 4 80.53 odd 4
4800.2.d.j.1249.4 4 80.27 even 4
4800.2.d.o.1249.1 4 80.43 even 4
4800.2.d.o.1249.2 4 80.37 odd 4
4800.2.d.o.1249.3 4 80.13 odd 4
4800.2.d.o.1249.4 4 80.67 even 4
4800.2.k.j.2401.1 4 80.19 odd 4
4800.2.k.j.2401.2 4 80.69 even 4
4800.2.k.j.2401.3 4 80.59 odd 4
4800.2.k.j.2401.4 4 80.29 even 4