Properties

Label 2880.2.d.i.289.4
Level $2880$
Weight $2$
Character 2880.289
Analytic conductor $22.997$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2880,2,Mod(289,2880)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2880, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2880.289");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2880.d (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(22.9969157821\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.49787136.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: no (minimal twist has level 960)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 289.4
Root \(-1.09445 + 0.895644i\) of defining polynomial
Character \(\chi\) \(=\) 2880.289
Dual form 2880.2.d.i.289.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.456850 + 2.18890i) q^{5} +4.37780i q^{7} +O(q^{10})\) \(q+(-0.456850 + 2.18890i) q^{5} +4.37780i q^{7} +5.58258i q^{11} +4.37780 q^{13} +5.58258i q^{17} +4.00000i q^{19} +(-4.58258 - 2.00000i) q^{25} -2.55040i q^{29} +5.29150 q^{31} +(-9.58258 - 2.00000i) q^{35} +2.55040 q^{37} -6.00000 q^{41} +11.1652 q^{43} -6.92820i q^{47} -12.1652 q^{49} +7.84190 q^{53} +(-12.2197 - 2.55040i) q^{55} +1.58258i q^{59} -10.5830i q^{61} +(-2.00000 + 9.58258i) q^{65} +3.16515 q^{67} +6.92820 q^{71} -12.0000i q^{73} -24.4394 q^{77} -5.29150 q^{79} +7.16515 q^{83} +(-12.2197 - 2.55040i) q^{85} +2.00000 q^{89} +19.1652i q^{91} +(-8.75560 - 1.82740i) q^{95} +11.1652i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 40 q^{35} - 48 q^{41} + 16 q^{43} - 24 q^{49} - 16 q^{65} - 48 q^{67} - 16 q^{83} + 16 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2880\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(641\) \(901\) \(2431\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −0.456850 + 2.18890i −0.204310 + 0.978906i
\(6\) 0 0
\(7\) 4.37780i 1.65465i 0.561721 + 0.827327i \(0.310140\pi\)
−0.561721 + 0.827327i \(0.689860\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 5.58258i 1.68321i 0.540094 + 0.841605i \(0.318389\pi\)
−0.540094 + 0.841605i \(0.681611\pi\)
\(12\) 0 0
\(13\) 4.37780 1.21418 0.607092 0.794632i \(-0.292336\pi\)
0.607092 + 0.794632i \(0.292336\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.58258i 1.35397i 0.735995 + 0.676987i \(0.236715\pi\)
−0.735995 + 0.676987i \(0.763285\pi\)
\(18\) 0 0
\(19\) 4.00000i 0.917663i 0.888523 + 0.458831i \(0.151732\pi\)
−0.888523 + 0.458831i \(0.848268\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) −4.58258 2.00000i −0.916515 0.400000i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 2.55040i 0.473598i −0.971559 0.236799i \(-0.923902\pi\)
0.971559 0.236799i \(-0.0760982\pi\)
\(30\) 0 0
\(31\) 5.29150 0.950382 0.475191 0.879883i \(-0.342379\pi\)
0.475191 + 0.879883i \(0.342379\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −9.58258 2.00000i −1.61975 0.338062i
\(36\) 0 0
\(37\) 2.55040 0.419283 0.209642 0.977778i \(-0.432770\pi\)
0.209642 + 0.977778i \(0.432770\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 11.1652 1.70267 0.851335 0.524623i \(-0.175794\pi\)
0.851335 + 0.524623i \(0.175794\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.92820i 1.01058i −0.862949 0.505291i \(-0.831385\pi\)
0.862949 0.505291i \(-0.168615\pi\)
\(48\) 0 0
\(49\) −12.1652 −1.73788
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 7.84190 1.07717 0.538584 0.842572i \(-0.318959\pi\)
0.538584 + 0.842572i \(0.318959\pi\)
\(54\) 0 0
\(55\) −12.2197 2.55040i −1.64770 0.343896i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 1.58258i 0.206034i 0.994680 + 0.103017i \(0.0328496\pi\)
−0.994680 + 0.103017i \(0.967150\pi\)
\(60\) 0 0
\(61\) 10.5830i 1.35501i −0.735516 0.677507i \(-0.763060\pi\)
0.735516 0.677507i \(-0.236940\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −2.00000 + 9.58258i −0.248069 + 1.18857i
\(66\) 0 0
\(67\) 3.16515 0.386685 0.193342 0.981131i \(-0.438067\pi\)
0.193342 + 0.981131i \(0.438067\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 6.92820 0.822226 0.411113 0.911584i \(-0.365140\pi\)
0.411113 + 0.911584i \(0.365140\pi\)
\(72\) 0 0
\(73\) 12.0000i 1.40449i −0.711934 0.702247i \(-0.752180\pi\)
0.711934 0.702247i \(-0.247820\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −24.4394 −2.78513
\(78\) 0 0
\(79\) −5.29150 −0.595341 −0.297670 0.954669i \(-0.596210\pi\)
−0.297670 + 0.954669i \(0.596210\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 7.16515 0.786478 0.393239 0.919436i \(-0.371355\pi\)
0.393239 + 0.919436i \(0.371355\pi\)
\(84\) 0 0
\(85\) −12.2197 2.55040i −1.32541 0.276630i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 2.00000 0.212000 0.106000 0.994366i \(-0.466196\pi\)
0.106000 + 0.994366i \(0.466196\pi\)
\(90\) 0 0
\(91\) 19.1652i 2.00905i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −8.75560 1.82740i −0.898306 0.187487i
\(96\) 0 0
\(97\) 11.1652i 1.13365i 0.823838 + 0.566825i \(0.191828\pi\)
−0.823838 + 0.566825i \(0.808172\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0.723000i 0.0719412i 0.999353 + 0.0359706i \(0.0114523\pi\)
−0.999353 + 0.0359706i \(0.988548\pi\)
\(102\) 0 0
\(103\) 0.723000i 0.0712393i −0.999365 0.0356197i \(-0.988660\pi\)
0.999365 0.0356197i \(-0.0113405\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 4.00000 0.386695 0.193347 0.981130i \(-0.438066\pi\)
0.193347 + 0.981130i \(0.438066\pi\)
\(108\) 0 0
\(109\) 1.82740i 0.175033i −0.996163 0.0875166i \(-0.972107\pi\)
0.996163 0.0875166i \(-0.0278931\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 13.5826i 1.27774i −0.769314 0.638871i \(-0.779402\pi\)
0.769314 0.638871i \(-0.220598\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −24.4394 −2.24036
\(120\) 0 0
\(121\) −20.1652 −1.83320
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 6.47135 9.11710i 0.578815 0.815459i
\(126\) 0 0
\(127\) 9.47860i 0.841090i 0.907272 + 0.420545i \(0.138161\pi\)
−0.907272 + 0.420545i \(0.861839\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 6.41742i 0.560693i 0.959899 + 0.280346i \(0.0904494\pi\)
−0.959899 + 0.280346i \(0.909551\pi\)
\(132\) 0 0
\(133\) −17.5112 −1.51841
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 10.4174i 0.890021i −0.895525 0.445010i \(-0.853200\pi\)
0.895525 0.445010i \(-0.146800\pi\)
\(138\) 0 0
\(139\) 7.16515i 0.607740i 0.952713 + 0.303870i \(0.0982789\pi\)
−0.952713 + 0.303870i \(0.901721\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 24.4394i 2.04373i
\(144\) 0 0
\(145\) 5.58258 + 1.16515i 0.463608 + 0.0967606i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 14.9608i 1.22564i −0.790224 0.612819i \(-0.790036\pi\)
0.790224 0.612819i \(-0.209964\pi\)
\(150\) 0 0
\(151\) −15.8745 −1.29185 −0.645925 0.763401i \(-0.723528\pi\)
−0.645925 + 0.763401i \(0.723528\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −2.41742 + 11.5826i −0.194172 + 0.930335i
\(156\) 0 0
\(157\) 16.4068 1.30941 0.654703 0.755886i \(-0.272794\pi\)
0.654703 + 0.755886i \(0.272794\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 12.0290i 0.930832i −0.885092 0.465416i \(-0.845905\pi\)
0.885092 0.465416i \(-0.154095\pi\)
\(168\) 0 0
\(169\) 6.16515 0.474242
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 6.01450 0.457274 0.228637 0.973512i \(-0.426573\pi\)
0.228637 + 0.973512i \(0.426573\pi\)
\(174\) 0 0
\(175\) 8.75560 20.0616i 0.661861 1.51652i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 1.58258i 0.118287i −0.998249 0.0591436i \(-0.981163\pi\)
0.998249 0.0591436i \(-0.0188370\pi\)
\(180\) 0 0
\(181\) 1.82740i 0.135830i 0.997691 + 0.0679148i \(0.0216346\pi\)
−0.997691 + 0.0679148i \(0.978365\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −1.16515 + 5.58258i −0.0856636 + 0.410439i
\(186\) 0 0
\(187\) −31.1652 −2.27902
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 13.8564 1.00261 0.501307 0.865269i \(-0.332853\pi\)
0.501307 + 0.865269i \(0.332853\pi\)
\(192\) 0 0
\(193\) 15.1652i 1.09161i 0.837912 + 0.545806i \(0.183776\pi\)
−0.837912 + 0.545806i \(0.816224\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 22.0797 1.57311 0.786557 0.617518i \(-0.211862\pi\)
0.786557 + 0.617518i \(0.211862\pi\)
\(198\) 0 0
\(199\) −12.2197 −0.866232 −0.433116 0.901338i \(-0.642586\pi\)
−0.433116 + 0.901338i \(0.642586\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 11.1652 0.783640
\(204\) 0 0
\(205\) 2.74110 13.1334i 0.191447 0.917277i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −22.3303 −1.54462
\(210\) 0 0
\(211\) 7.16515i 0.493269i 0.969109 + 0.246635i \(0.0793248\pi\)
−0.969109 + 0.246635i \(0.920675\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −5.10080 + 24.4394i −0.347872 + 1.66675i
\(216\) 0 0
\(217\) 23.1652i 1.57255i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 24.4394i 1.64397i
\(222\) 0 0
\(223\) 4.37780i 0.293159i 0.989199 + 0.146580i \(0.0468265\pi\)
−0.989199 + 0.146580i \(0.953174\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 15.1652 1.00655 0.503273 0.864127i \(-0.332129\pi\)
0.503273 + 0.864127i \(0.332129\pi\)
\(228\) 0 0
\(229\) 12.0290i 0.794899i −0.917624 0.397450i \(-0.869895\pi\)
0.917624 0.397450i \(-0.130105\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 10.4174i 0.682468i 0.939978 + 0.341234i \(0.110845\pi\)
−0.939978 + 0.341234i \(0.889155\pi\)
\(234\) 0 0
\(235\) 15.1652 + 3.16515i 0.989265 + 0.206472i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 21.1660 1.36912 0.684558 0.728959i \(-0.259995\pi\)
0.684558 + 0.728959i \(0.259995\pi\)
\(240\) 0 0
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 5.55765 26.6283i 0.355065 1.70122i
\(246\) 0 0
\(247\) 17.5112i 1.11421i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 2.41742i 0.152586i 0.997085 + 0.0762932i \(0.0243085\pi\)
−0.997085 + 0.0762932i \(0.975691\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 24.7477i 1.54372i −0.635792 0.771860i \(-0.719326\pi\)
0.635792 0.771860i \(-0.280674\pi\)
\(258\) 0 0
\(259\) 11.1652i 0.693769i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 1.82740i 0.112682i 0.998412 + 0.0563412i \(0.0179435\pi\)
−0.998412 + 0.0563412i \(0.982057\pi\)
\(264\) 0 0
\(265\) −3.58258 + 17.1652i −0.220076 + 1.05445i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 6.20520i 0.378338i 0.981945 + 0.189169i \(0.0605794\pi\)
−0.981945 + 0.189169i \(0.939421\pi\)
\(270\) 0 0
\(271\) 15.4931 0.941139 0.470570 0.882363i \(-0.344048\pi\)
0.470570 + 0.882363i \(0.344048\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 11.1652 25.5826i 0.673284 1.54269i
\(276\) 0 0
\(277\) 9.47860 0.569514 0.284757 0.958600i \(-0.408087\pi\)
0.284757 + 0.958600i \(0.408087\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −32.3303 −1.92866 −0.964332 0.264695i \(-0.914729\pi\)
−0.964332 + 0.264695i \(0.914729\pi\)
\(282\) 0 0
\(283\) −11.1652 −0.663699 −0.331850 0.943332i \(-0.607673\pi\)
−0.331850 + 0.943332i \(0.607673\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 26.2668i 1.55048i
\(288\) 0 0
\(289\) −14.1652 −0.833244
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −0.913701 −0.0533790 −0.0266895 0.999644i \(-0.508497\pi\)
−0.0266895 + 0.999644i \(0.508497\pi\)
\(294\) 0 0
\(295\) −3.46410 0.723000i −0.201688 0.0420947i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 48.8788i 2.81733i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 23.1652 + 4.83485i 1.32643 + 0.276843i
\(306\) 0 0
\(307\) −4.00000 −0.228292 −0.114146 0.993464i \(-0.536413\pi\)
−0.114146 + 0.993464i \(0.536413\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −3.65480 −0.207245 −0.103622 0.994617i \(-0.533043\pi\)
−0.103622 + 0.994617i \(0.533043\pi\)
\(312\) 0 0
\(313\) 0.834849i 0.0471884i −0.999722 0.0235942i \(-0.992489\pi\)
0.999722 0.0235942i \(-0.00751097\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −34.1087 −1.91574 −0.957868 0.287208i \(-0.907273\pi\)
−0.957868 + 0.287208i \(0.907273\pi\)
\(318\) 0 0
\(319\) 14.2378 0.797164
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −22.3303 −1.24249
\(324\) 0 0
\(325\) −20.0616 8.75560i −1.11282 0.485674i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 30.3303 1.67216
\(330\) 0 0
\(331\) 10.3303i 0.567805i −0.958853 0.283902i \(-0.908371\pi\)
0.958853 0.283902i \(-0.0916292\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −1.44600 + 6.92820i −0.0790034 + 0.378528i
\(336\) 0 0
\(337\) 26.3303i 1.43430i −0.696917 0.717151i \(-0.745446\pi\)
0.696917 0.717151i \(-0.254554\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 29.5402i 1.59969i
\(342\) 0 0
\(343\) 22.6120i 1.22093i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0.834849 0.0448170 0.0224085 0.999749i \(-0.492867\pi\)
0.0224085 + 0.999749i \(0.492867\pi\)
\(348\) 0 0
\(349\) 24.4394i 1.30821i −0.756403 0.654106i \(-0.773045\pi\)
0.756403 0.654106i \(-0.226955\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 11.9129i 0.634059i 0.948416 + 0.317029i \(0.102685\pi\)
−0.948416 + 0.317029i \(0.897315\pi\)
\(354\) 0 0
\(355\) −3.16515 + 15.1652i −0.167989 + 0.804883i
\(356\) 0 0
\(357\) 0 0
\(358\) 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
\(363\) 0 0
\(364\) 0 0
\(365\) 26.2668 + 5.48220i 1.37487 + 0.286952i
\(366\) 0 0
\(367\) 11.6874i 0.610078i −0.952340 0.305039i \(-0.901331\pi\)
0.952340 0.305039i \(-0.0986694\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 34.3303i 1.78234i
\(372\) 0 0
\(373\) −11.3060 −0.585403 −0.292701 0.956204i \(-0.594554\pi\)
−0.292701 + 0.956204i \(0.594554\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 11.1652i 0.575035i
\(378\) 0 0
\(379\) 23.1652i 1.18991i −0.803758 0.594957i \(-0.797169\pi\)
0.803758 0.594957i \(-0.202831\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 29.9216i 1.52892i −0.644669 0.764462i \(-0.723005\pi\)
0.644669 0.764462i \(-0.276995\pi\)
\(384\) 0 0
\(385\) 11.1652 53.4955i 0.569029 2.72638i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 8.03260i 0.407269i 0.979047 + 0.203635i \(0.0652755\pi\)
−0.979047 + 0.203635i \(0.934725\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 2.41742 11.5826i 0.121634 0.582783i
\(396\) 0 0
\(397\) −2.55040 −0.128001 −0.0640005 0.997950i \(-0.520386\pi\)
−0.0640005 + 0.997950i \(0.520386\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −22.0000 −1.09863 −0.549314 0.835616i \(-0.685111\pi\)
−0.549314 + 0.835616i \(0.685111\pi\)
\(402\) 0 0
\(403\) 23.1652 1.15394
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 14.2378i 0.705742i
\(408\) 0 0
\(409\) 37.1652 1.83770 0.918849 0.394609i \(-0.129120\pi\)
0.918849 + 0.394609i \(0.129120\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −6.92820 −0.340915
\(414\) 0 0
\(415\) −3.27340 + 15.6838i −0.160685 + 0.769888i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 16.7477i 0.818180i 0.912494 + 0.409090i \(0.134154\pi\)
−0.912494 + 0.409090i \(0.865846\pi\)
\(420\) 0 0
\(421\) 29.5402i 1.43970i 0.694129 + 0.719851i \(0.255790\pi\)
−0.694129 + 0.719851i \(0.744210\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 11.1652 25.5826i 0.541589 1.24094i
\(426\) 0 0
\(427\) 46.3303 2.24208
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 38.2958 1.84464 0.922322 0.386421i \(-0.126289\pi\)
0.922322 + 0.386421i \(0.126289\pi\)
\(432\) 0 0
\(433\) 1.66970i 0.0802405i 0.999195 + 0.0401203i \(0.0127741\pi\)
−0.999195 + 0.0401203i \(0.987226\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −29.7309 −1.41898 −0.709490 0.704716i \(-0.751074\pi\)
−0.709490 + 0.704716i \(0.751074\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −31.1652 −1.48070 −0.740351 0.672221i \(-0.765340\pi\)
−0.740351 + 0.672221i \(0.765340\pi\)
\(444\) 0 0
\(445\) −0.913701 + 4.37780i −0.0433136 + 0.207528i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 28.3303 1.33699 0.668495 0.743717i \(-0.266939\pi\)
0.668495 + 0.743717i \(0.266939\pi\)
\(450\) 0 0
\(451\) 33.4955i 1.57724i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −41.9506 8.75560i −1.96668 0.410469i
\(456\) 0 0
\(457\) 4.83485i 0.226165i 0.993586 + 0.113082i \(0.0360724\pi\)
−0.993586 + 0.113082i \(0.963928\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 1.10440i 0.0514371i −0.999669 0.0257185i \(-0.991813\pi\)
0.999669 0.0257185i \(-0.00818737\pi\)
\(462\) 0 0
\(463\) 29.1986i 1.35697i −0.734612 0.678487i \(-0.762636\pi\)
0.734612 0.678487i \(-0.237364\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0.834849 0.0386322 0.0193161 0.999813i \(-0.493851\pi\)
0.0193161 + 0.999813i \(0.493851\pi\)
\(468\) 0 0
\(469\) 13.8564i 0.639829i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 62.3303i 2.86595i
\(474\) 0 0
\(475\) 8.00000 18.3303i 0.367065 0.841052i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −31.7490 −1.45065 −0.725325 0.688407i \(-0.758310\pi\)
−0.725325 + 0.688407i \(0.758310\pi\)
\(480\) 0 0
\(481\) 11.1652 0.509087
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −24.4394 5.10080i −1.10974 0.231615i
\(486\) 0 0
\(487\) 16.7882i 0.760746i 0.924833 + 0.380373i \(0.124204\pi\)
−0.924833 + 0.380373i \(0.875796\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 12.7477i 0.575297i −0.957736 0.287648i \(-0.907127\pi\)
0.957736 0.287648i \(-0.0928735\pi\)
\(492\) 0 0
\(493\) 14.2378 0.641239
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 30.3303i 1.36050i
\(498\) 0 0
\(499\) 18.3303i 0.820577i −0.911956 0.410289i \(-0.865428\pi\)
0.911956 0.410289i \(-0.134572\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 28.0942i 1.25266i −0.779558 0.626330i \(-0.784556\pi\)
0.779558 0.626330i \(-0.215444\pi\)
\(504\) 0 0
\(505\) −1.58258 0.330303i −0.0704237 0.0146983i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 39.4002i 1.74638i −0.487376 0.873192i \(-0.662046\pi\)
0.487376 0.873192i \(-0.337954\pi\)
\(510\) 0 0
\(511\) 52.5336 2.32395
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 1.58258 + 0.330303i 0.0697366 + 0.0145549i
\(516\) 0 0
\(517\) 38.6772 1.70102
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 34.6606 1.51851 0.759254 0.650794i \(-0.225564\pi\)
0.759254 + 0.650794i \(0.225564\pi\)
\(522\) 0 0
\(523\) −18.3303 −0.801528 −0.400764 0.916181i \(-0.631255\pi\)
−0.400764 + 0.916181i \(0.631255\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 29.5402i 1.28679i
\(528\) 0 0
\(529\) 23.0000 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −26.2668 −1.13774
\(534\) 0 0
\(535\) −1.82740 + 8.75560i −0.0790054 + 0.378538i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 67.9129i 2.92521i
\(540\) 0 0
\(541\) 36.8498i 1.58430i −0.610328 0.792149i \(-0.708962\pi\)
0.610328 0.792149i \(-0.291038\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 4.00000 + 0.834849i 0.171341 + 0.0357610i
\(546\) 0 0
\(547\) −25.4955 −1.09011 −0.545053 0.838401i \(-0.683491\pi\)
−0.545053 + 0.838401i \(0.683491\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 10.2016 0.434603
\(552\) 0 0
\(553\) 23.1652i 0.985082i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −4.18710 −0.177413 −0.0887066 0.996058i \(-0.528273\pi\)
−0.0887066 + 0.996058i \(0.528273\pi\)
\(558\) 0 0
\(559\) 48.8788 2.06735
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −29.4955 −1.24309 −0.621543 0.783380i \(-0.713494\pi\)
−0.621543 + 0.783380i \(0.713494\pi\)
\(564\) 0 0
\(565\) 29.7309 + 6.20520i 1.25079 + 0.261055i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −36.3303 −1.52305 −0.761523 0.648138i \(-0.775548\pi\)
−0.761523 + 0.648138i \(0.775548\pi\)
\(570\) 0 0
\(571\) 20.0000i 0.836974i 0.908223 + 0.418487i \(0.137439\pi\)
−0.908223 + 0.418487i \(0.862561\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 15.1652i 0.631334i −0.948870 0.315667i \(-0.897772\pi\)
0.948870 0.315667i \(-0.102228\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 31.3676i 1.30135i
\(582\) 0 0
\(583\) 43.7780i 1.81310i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −5.66970 −0.234013 −0.117007 0.993131i \(-0.537330\pi\)
−0.117007 + 0.993131i \(0.537330\pi\)
\(588\) 0 0
\(589\) 21.1660i 0.872130i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 35.9129i 1.47477i −0.675475 0.737383i \(-0.736062\pi\)
0.675475 0.737383i \(-0.263938\pi\)
\(594\) 0 0
\(595\) 11.1652 53.4955i 0.457727 2.19310i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 14.2378 0.581741 0.290871 0.956762i \(-0.406055\pi\)
0.290871 + 0.956762i \(0.406055\pi\)
\(600\) 0 0
\(601\) 13.1652 0.537018 0.268509 0.963277i \(-0.413469\pi\)
0.268509 + 0.963277i \(0.413469\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 9.21245 44.1395i 0.374540 1.79453i
\(606\) 0 0
\(607\) 35.7454i 1.45086i 0.688295 + 0.725431i \(0.258359\pi\)
−0.688295 + 0.725431i \(0.741641\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 30.3303i 1.22703i
\(612\) 0 0
\(613\) 32.4720 1.31153 0.655766 0.754964i \(-0.272346\pi\)
0.655766 + 0.754964i \(0.272346\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 13.5826i 0.546814i 0.961898 + 0.273407i \(0.0881506\pi\)
−0.961898 + 0.273407i \(0.911849\pi\)
\(618\) 0 0
\(619\) 45.4955i 1.82862i −0.405019 0.914308i \(-0.632735\pi\)
0.405019 0.914308i \(-0.367265\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 8.75560i 0.350786i
\(624\) 0 0
\(625\) 17.0000 + 18.3303i 0.680000 + 0.733212i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 14.2378i 0.567699i
\(630\) 0 0
\(631\) 12.2197 0.486459 0.243229 0.969969i \(-0.421793\pi\)
0.243229 + 0.969969i \(0.421793\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −20.7477 4.33030i −0.823348 0.171843i
\(636\) 0 0
\(637\) −53.2566 −2.11010
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 4.33030 0.171037 0.0855183 0.996337i \(-0.472745\pi\)
0.0855183 + 0.996337i \(0.472745\pi\)
\(642\) 0 0
\(643\) −18.3303 −0.722877 −0.361438 0.932396i \(-0.617714\pi\)
−0.361438 + 0.932396i \(0.617714\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 17.8926i 0.703431i −0.936107 0.351716i \(-0.885598\pi\)
0.936107 0.351716i \(-0.114402\pi\)
\(648\) 0 0
\(649\) −8.83485 −0.346798
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 22.0797 0.864046 0.432023 0.901863i \(-0.357800\pi\)
0.432023 + 0.901863i \(0.357800\pi\)
\(654\) 0 0
\(655\) −14.0471 2.93180i −0.548866 0.114555i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 25.5826i 0.996556i 0.867017 + 0.498278i \(0.166034\pi\)
−0.867017 + 0.498278i \(0.833966\pi\)
\(660\) 0 0
\(661\) 17.8926i 0.695942i −0.937505 0.347971i \(-0.886871\pi\)
0.937505 0.347971i \(-0.113129\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 8.00000 38.3303i 0.310227 1.48639i
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 59.0804 2.28077
\(672\) 0 0
\(673\) 40.0000i 1.54189i 0.636904 + 0.770943i \(0.280215\pi\)
−0.636904 + 0.770943i \(0.719785\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 28.6265 1.10021 0.550103 0.835097i \(-0.314588\pi\)
0.550103 + 0.835097i \(0.314588\pi\)
\(678\) 0 0
\(679\) −48.8788 −1.87580
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −5.49545 −0.210278 −0.105139 0.994458i \(-0.533529\pi\)
−0.105139 + 0.994458i \(0.533529\pi\)
\(684\) 0 0
\(685\) 22.8027 + 4.75920i 0.871247 + 0.181840i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 34.3303 1.30788
\(690\) 0 0
\(691\) 40.6606i 1.54680i 0.633917 + 0.773401i \(0.281446\pi\)
−0.633917 + 0.773401i \(0.718554\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −15.6838 3.27340i −0.594921 0.124167i
\(696\) 0 0
\(697\) 33.4955i 1.26873i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 45.9470i 1.73540i 0.497093 + 0.867698i \(0.334401\pi\)
−0.497093 + 0.867698i \(0.665599\pi\)
\(702\) 0 0
\(703\) 10.2016i 0.384761i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −3.16515 −0.119038
\(708\) 0 0
\(709\) 15.6838i 0.589018i 0.955649 + 0.294509i \(0.0951561\pi\)
−0.955649 + 0.294509i \(0.904844\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) −53.4955 11.1652i −2.00062 0.417553i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −30.9862 −1.15559 −0.577795 0.816182i \(-0.696087\pi\)
−0.577795 + 0.816182i \(0.696087\pi\)
\(720\) 0 0
\(721\) 3.16515 0.117876
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −5.10080 + 11.6874i −0.189439 + 0.434059i
\(726\) 0 0
\(727\) 49.6018i 1.83963i −0.392353 0.919815i \(-0.628339\pi\)
0.392353 0.919815i \(-0.371661\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 62.3303i 2.30537i
\(732\) 0 0
\(733\) −29.1986 −1.07848 −0.539238 0.842154i \(-0.681288\pi\)
−0.539238 + 0.842154i \(0.681288\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 17.6697i 0.650872i
\(738\) 0 0
\(739\) 20.0000i 0.735712i 0.929883 + 0.367856i \(0.119908\pi\)
−0.929883 + 0.367856i \(0.880092\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 47.4328i 1.74014i 0.492927 + 0.870071i \(0.335927\pi\)
−0.492927 + 0.870071i \(0.664073\pi\)
\(744\) 0 0
\(745\) 32.7477 + 6.83485i 1.19978 + 0.250409i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 17.5112i 0.639846i
\(750\) 0 0
\(751\) 33.0043 1.20434 0.602172 0.798366i \(-0.294302\pi\)
0.602172 + 0.798366i \(0.294302\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 7.25227 34.7477i 0.263937 1.26460i
\(756\) 0 0
\(757\) −18.2342 −0.662734 −0.331367 0.943502i \(-0.607510\pi\)
−0.331367 + 0.943502i \(0.607510\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 22.0000 0.797499 0.398750 0.917060i \(-0.369444\pi\)
0.398750 + 0.917060i \(0.369444\pi\)
\(762\) 0 0
\(763\) 8.00000 0.289619
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 6.92820i 0.250163i
\(768\) 0 0
\(769\) 23.4955 0.847268 0.423634 0.905834i \(-0.360754\pi\)
0.423634 + 0.905834i \(0.360754\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −30.4539 −1.09535 −0.547676 0.836691i \(-0.684487\pi\)
−0.547676 + 0.836691i \(0.684487\pi\)
\(774\) 0 0
\(775\) −24.2487 10.5830i −0.871039 0.380153i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 24.0000i 0.859889i
\(780\) 0 0
\(781\) 38.6772i 1.38398i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −7.49545 + 35.9129i −0.267524 + 1.28179i
\(786\) 0 0
\(787\) 36.0000 1.28326 0.641631 0.767014i \(-0.278258\pi\)
0.641631 + 0.767014i \(0.278258\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 59.4618 2.11422
\(792\) 0 0
\(793\) 46.3303i 1.64524i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 53.0659 1.87969 0.939846 0.341599i \(-0.110969\pi\)
0.939846 + 0.341599i \(0.110969\pi\)
\(798\) 0 0
\(799\) 38.6772 1.36830
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 66.9909 2.36406
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 30.0000 1.05474 0.527372 0.849635i \(-0.323177\pi\)
0.527372 + 0.849635i \(0.323177\pi\)
\(810\) 0 0
\(811\) 0.834849i 0.0293155i −0.999893 0.0146577i \(-0.995334\pi\)
0.999893 0.0146577i \(-0.00466587\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 1.82740 8.75560i 0.0640111 0.306695i
\(816\) 0 0
\(817\) 44.6606i 1.56248i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 6.20520i 0.216563i −0.994120 0.108282i \(-0.965465\pi\)
0.994120 0.108282i \(-0.0345348\pi\)
\(822\) 0 0
\(823\) 9.47860i 0.330403i −0.986260 0.165202i \(-0.947172\pi\)
0.986260 0.165202i \(-0.0528275\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 28.0000 0.973655 0.486828 0.873498i \(-0.338154\pi\)
0.486828 + 0.873498i \(0.338154\pi\)
\(828\) 0 0
\(829\) 36.8498i 1.27985i 0.768439 + 0.639924i \(0.221034\pi\)
−0.768439 + 0.639924i \(0.778966\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 67.9129i 2.35304i
\(834\) 0 0
\(835\) 26.3303 + 5.49545i 0.911198 + 0.190178i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −55.8070 −1.92667 −0.963336 0.268297i \(-0.913539\pi\)
−0.963336 + 0.268297i \(0.913539\pi\)
\(840\) 0 0
\(841\) 22.4955 0.775705
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −2.81655 + 13.4949i −0.0968923 + 0.464239i
\(846\) 0 0
\(847\) 88.2790i 3.03330i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 7.65120 0.261972 0.130986 0.991384i \(-0.458186\pi\)
0.130986 + 0.991384i \(0.458186\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 29.5826i 1.01052i −0.862967 0.505261i \(-0.831396\pi\)
0.862967 0.505261i \(-0.168604\pi\)
\(858\) 0 0
\(859\) 24.8348i 0.847354i −0.905813 0.423677i \(-0.860739\pi\)
0.905813 0.423677i \(-0.139261\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 50.7062i 1.72606i −0.505153 0.863030i \(-0.668564\pi\)
0.505153 0.863030i \(-0.331436\pi\)
\(864\) 0 0
\(865\) −2.74773 + 13.1652i −0.0934255 + 0.447629i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 29.5402i 1.00208i
\(870\) 0 0
\(871\) 13.8564 0.469506
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 39.9129 + 28.3303i 1.34930 + 0.957739i
\(876\) 0 0
\(877\) −17.8528 −0.602846 −0.301423 0.953490i \(-0.597462\pi\)
−0.301423 + 0.953490i \(0.597462\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −36.3303 −1.22400 −0.612000 0.790858i \(-0.709635\pi\)
−0.612000 + 0.790858i \(0.709635\pi\)
\(882\) 0 0
\(883\) −34.3303 −1.15531 −0.577653 0.816282i \(-0.696031\pi\)
−0.577653 + 0.816282i \(0.696031\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 47.0514i 1.57983i 0.613215 + 0.789916i \(0.289876\pi\)
−0.613215 + 0.789916i \(0.710124\pi\)
\(888\) 0 0
\(889\) −41.4955 −1.39171
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 27.7128 0.927374
\(894\) 0 0
\(895\) 3.46410 + 0.723000i 0.115792 + 0.0241672i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 13.4955i 0.450099i
\(900\) 0 0
\(901\) 43.7780i 1.45846i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −4.00000 0.834849i −0.132964 0.0277513i
\(906\) 0 0
\(907\) −17.4955 −0.580927 −0.290464 0.956886i \(-0.593809\pi\)
−0.290464 + 0.956886i \(0.593809\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 7.30960 0.242178 0.121089 0.992642i \(-0.461361\pi\)
0.121089 + 0.992642i \(0.461361\pi\)
\(912\) 0 0
\(913\) 40.0000i 1.32381i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −28.0942 −0.927753
\(918\) 0 0
\(919\) 11.8383 0.390510 0.195255 0.980753i \(-0.437447\pi\)
0.195255 + 0.980753i \(0.437447\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 30.3303 0.998334
\(924\) 0 0
\(925\) −11.6874 5.10080i −0.384280 0.167713i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −30.6606 −1.00594 −0.502971 0.864303i \(-0.667760\pi\)
−0.502971 + 0.864303i \(0.667760\pi\)
\(930\) 0 0
\(931\) 48.6606i 1.59479i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 14.2378 68.2174i 0.465626 2.23095i
\(936\) 0 0
\(937\) 14.3303i 0.468151i 0.972218 + 0.234075i \(0.0752062\pi\)
−0.972218 + 0.234075i \(0.924794\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 28.4358i 0.926981i 0.886102 + 0.463491i \(0.153403\pi\)
−0.886102 + 0.463491i \(0.846597\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −52.0000 −1.68977 −0.844886 0.534946i \(-0.820332\pi\)
−0.844886 + 0.534946i \(0.820332\pi\)
\(948\) 0 0
\(949\) 52.5336i 1.70531i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 26.2432i 0.850100i −0.905170 0.425050i \(-0.860257\pi\)
0.905170 0.425050i \(-0.139743\pi\)
\(954\) 0 0
\(955\) −6.33030 + 30.3303i −0.204844 + 0.981466i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 45.6054 1.47268
\(960\) 0 0
\(961\) −3.00000 −0.0967742
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −33.1950 6.92820i −1.06859 0.223027i
\(966\) 0 0
\(967\) 26.9898i 0.867934i −0.900929 0.433967i \(-0.857113\pi\)
0.900929 0.433967i \(-0.142887\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 4.08712i 0.131162i 0.997847 + 0.0655810i \(0.0208901\pi\)
−0.997847 + 0.0655810i \(0.979110\pi\)
\(972\) 0 0
\(973\) −31.3676 −1.00560
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 7.25227i 0.232021i 0.993248 + 0.116010i \(0.0370106\pi\)
−0.993248 + 0.116010i \(0.962989\pi\)
\(978\) 0 0
\(979\) 11.1652i 0.356840i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 2.20880i 0.0704498i 0.999379 + 0.0352249i \(0.0112148\pi\)
−0.999379 + 0.0352249i \(0.988785\pi\)
\(984\) 0 0
\(985\) −10.0871 + 48.3303i −0.321402 + 1.53993i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −11.8383 −0.376056 −0.188028 0.982164i \(-0.560210\pi\)
−0.188028 + 0.982164i \(0.560210\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 5.58258 26.7477i 0.176980 0.847960i
\(996\) 0 0
\(997\) −12.7520 −0.403860 −0.201930 0.979400i \(-0.564721\pi\)
−0.201930 + 0.979400i \(0.564721\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2880.2.d.i.289.4 8
3.2 odd 2 960.2.d.f.289.5 yes 8
4.3 odd 2 2880.2.d.j.289.4 8
5.4 even 2 2880.2.d.j.289.6 8
8.3 odd 2 inner 2880.2.d.i.289.5 8
8.5 even 2 2880.2.d.j.289.5 8
12.11 even 2 960.2.d.e.289.5 yes 8
15.2 even 4 4800.2.k.r.2401.1 8
15.8 even 4 4800.2.k.q.2401.8 8
15.14 odd 2 960.2.d.e.289.3 8
20.19 odd 2 inner 2880.2.d.i.289.6 8
24.5 odd 2 960.2.d.e.289.4 yes 8
24.11 even 2 960.2.d.f.289.4 yes 8
40.19 odd 2 2880.2.d.j.289.3 8
40.29 even 2 inner 2880.2.d.i.289.3 8
48.5 odd 4 3840.2.f.k.769.4 8
48.11 even 4 3840.2.f.i.769.8 8
48.29 odd 4 3840.2.f.i.769.5 8
48.35 even 4 3840.2.f.k.769.1 8
60.23 odd 4 4800.2.k.q.2401.1 8
60.47 odd 4 4800.2.k.r.2401.8 8
60.59 even 2 960.2.d.f.289.3 yes 8
120.29 odd 2 960.2.d.f.289.6 yes 8
120.53 even 4 4800.2.k.q.2401.4 8
120.59 even 2 960.2.d.e.289.6 yes 8
120.77 even 4 4800.2.k.r.2401.5 8
120.83 odd 4 4800.2.k.q.2401.5 8
120.107 odd 4 4800.2.k.r.2401.4 8
240.29 odd 4 3840.2.f.i.769.1 8
240.59 even 4 3840.2.f.i.769.4 8
240.149 odd 4 3840.2.f.k.769.8 8
240.179 even 4 3840.2.f.k.769.5 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
960.2.d.e.289.3 8 15.14 odd 2
960.2.d.e.289.4 yes 8 24.5 odd 2
960.2.d.e.289.5 yes 8 12.11 even 2
960.2.d.e.289.6 yes 8 120.59 even 2
960.2.d.f.289.3 yes 8 60.59 even 2
960.2.d.f.289.4 yes 8 24.11 even 2
960.2.d.f.289.5 yes 8 3.2 odd 2
960.2.d.f.289.6 yes 8 120.29 odd 2
2880.2.d.i.289.3 8 40.29 even 2 inner
2880.2.d.i.289.4 8 1.1 even 1 trivial
2880.2.d.i.289.5 8 8.3 odd 2 inner
2880.2.d.i.289.6 8 20.19 odd 2 inner
2880.2.d.j.289.3 8 40.19 odd 2
2880.2.d.j.289.4 8 4.3 odd 2
2880.2.d.j.289.5 8 8.5 even 2
2880.2.d.j.289.6 8 5.4 even 2
3840.2.f.i.769.1 8 240.29 odd 4
3840.2.f.i.769.4 8 240.59 even 4
3840.2.f.i.769.5 8 48.29 odd 4
3840.2.f.i.769.8 8 48.11 even 4
3840.2.f.k.769.1 8 48.35 even 4
3840.2.f.k.769.4 8 48.5 odd 4
3840.2.f.k.769.5 8 240.179 even 4
3840.2.f.k.769.8 8 240.149 odd 4
4800.2.k.q.2401.1 8 60.23 odd 4
4800.2.k.q.2401.4 8 120.53 even 4
4800.2.k.q.2401.5 8 120.83 odd 4
4800.2.k.q.2401.8 8 15.8 even 4
4800.2.k.r.2401.1 8 15.2 even 4
4800.2.k.r.2401.4 8 120.107 odd 4
4800.2.k.r.2401.5 8 120.77 even 4
4800.2.k.r.2401.8 8 60.47 odd 4