Properties

Label 2880.2.d.g.289.5
Level $2880$
Weight $2$
Character 2880.289
Analytic conductor $22.997$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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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: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: no (minimal twist has level 320)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 289.5
Root \(0.965926 + 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 2880.289
Dual form 2880.2.d.g.289.8

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.41421 - 1.73205i) q^{5} -1.41421i q^{7} +O(q^{10})\) \(q+(1.41421 - 1.73205i) q^{5} -1.41421i q^{7} +2.00000i q^{11} +5.65685 q^{13} +4.89898i q^{17} -6.00000i q^{19} -7.07107i q^{23} +(-1.00000 - 4.89898i) q^{25} +6.92820i q^{29} +6.92820 q^{31} +(-2.44949 - 2.00000i) q^{35} -2.82843 q^{37} +4.00000 q^{41} -2.44949 q^{43} +4.24264i q^{47} +5.00000 q^{49} +(3.46410 + 2.82843i) q^{55} -2.00000i q^{59} +3.46410i q^{61} +(8.00000 - 9.79796i) q^{65} +2.44949 q^{67} +6.92820 q^{71} -4.89898i q^{73} +2.82843 q^{77} -6.92820 q^{79} -12.2474 q^{83} +(8.48528 + 6.92820i) q^{85} +2.00000 q^{89} -8.00000i q^{91} +(-10.3923 - 8.48528i) q^{95} -14.6969i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{25} + 32 q^{41} + 40 q^{49} + 64 q^{65} + 16 q^{89}+O(q^{100}) \) 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
\(5\) 1.41421 1.73205i 0.632456 0.774597i
\(6\) 0 0
\(7\) 1.41421i 0.534522i −0.963624 0.267261i \(-0.913881\pi\)
0.963624 0.267261i \(-0.0861187\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.00000i 0.603023i 0.953463 + 0.301511i \(0.0974911\pi\)
−0.953463 + 0.301511i \(0.902509\pi\)
\(12\) 0 0
\(13\) 5.65685 1.56893 0.784465 0.620174i \(-0.212938\pi\)
0.784465 + 0.620174i \(0.212938\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.89898i 1.18818i 0.804400 + 0.594089i \(0.202487\pi\)
−0.804400 + 0.594089i \(0.797513\pi\)
\(18\) 0 0
\(19\) 6.00000i 1.37649i −0.725476 0.688247i \(-0.758380\pi\)
0.725476 0.688247i \(-0.241620\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 7.07107i 1.47442i −0.675664 0.737210i \(-0.736143\pi\)
0.675664 0.737210i \(-0.263857\pi\)
\(24\) 0 0
\(25\) −1.00000 4.89898i −0.200000 0.979796i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 6.92820i 1.28654i 0.765641 + 0.643268i \(0.222422\pi\)
−0.765641 + 0.643268i \(0.777578\pi\)
\(30\) 0 0
\(31\) 6.92820 1.24434 0.622171 0.782881i \(-0.286251\pi\)
0.622171 + 0.782881i \(0.286251\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2.44949 2.00000i −0.414039 0.338062i
\(36\) 0 0
\(37\) −2.82843 −0.464991 −0.232495 0.972598i \(-0.574689\pi\)
−0.232495 + 0.972598i \(0.574689\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 4.00000 0.624695 0.312348 0.949968i \(-0.398885\pi\)
0.312348 + 0.949968i \(0.398885\pi\)
\(42\) 0 0
\(43\) −2.44949 −0.373544 −0.186772 0.982403i \(-0.559803\pi\)
−0.186772 + 0.982403i \(0.559803\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.24264i 0.618853i 0.950923 + 0.309426i \(0.100137\pi\)
−0.950923 + 0.309426i \(0.899863\pi\)
\(48\) 0 0
\(49\) 5.00000 0.714286
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 3.46410 + 2.82843i 0.467099 + 0.381385i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 2.00000i 0.260378i −0.991489 0.130189i \(-0.958442\pi\)
0.991489 0.130189i \(-0.0415584\pi\)
\(60\) 0 0
\(61\) 3.46410i 0.443533i 0.975100 + 0.221766i \(0.0711822\pi\)
−0.975100 + 0.221766i \(0.928818\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 8.00000 9.79796i 0.992278 1.21529i
\(66\) 0 0
\(67\) 2.44949 0.299253 0.149626 0.988743i \(-0.452193\pi\)
0.149626 + 0.988743i \(0.452193\pi\)
\(68\) 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
\(73\) 4.89898i 0.573382i −0.958023 0.286691i \(-0.907445\pi\)
0.958023 0.286691i \(-0.0925553\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.82843 0.322329
\(78\) 0 0
\(79\) −6.92820 −0.779484 −0.389742 0.920924i \(-0.627436\pi\)
−0.389742 + 0.920924i \(0.627436\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −12.2474 −1.34433 −0.672166 0.740400i \(-0.734636\pi\)
−0.672166 + 0.740400i \(0.734636\pi\)
\(84\) 0 0
\(85\) 8.48528 + 6.92820i 0.920358 + 0.751469i
\(86\) 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
\(91\) 8.00000i 0.838628i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −10.3923 8.48528i −1.06623 0.870572i
\(96\) 0 0
\(97\) 14.6969i 1.49225i −0.665807 0.746124i \(-0.731913\pi\)
0.665807 0.746124i \(-0.268087\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 6.92820i 0.689382i 0.938716 + 0.344691i \(0.112016\pi\)
−0.938716 + 0.344691i \(0.887984\pi\)
\(102\) 0 0
\(103\) 9.89949i 0.975426i −0.873004 0.487713i \(-0.837831\pi\)
0.873004 0.487713i \(-0.162169\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −2.44949 −0.236801 −0.118401 0.992966i \(-0.537777\pi\)
−0.118401 + 0.992966i \(0.537777\pi\)
\(108\) 0 0
\(109\) 3.46410i 0.331801i −0.986143 0.165900i \(-0.946947\pi\)
0.986143 0.165900i \(-0.0530530\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) −12.2474 10.0000i −1.14208 0.932505i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 6.92820 0.635107
\(120\) 0 0
\(121\) 7.00000 0.636364
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −9.89949 5.19615i −0.885438 0.464758i
\(126\) 0 0
\(127\) 7.07107i 0.627456i −0.949513 0.313728i \(-0.898422\pi\)
0.949513 0.313728i \(-0.101578\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 10.0000i 0.873704i −0.899533 0.436852i \(-0.856093\pi\)
0.899533 0.436852i \(-0.143907\pi\)
\(132\) 0 0
\(133\) −8.48528 −0.735767
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 9.79796i 0.837096i −0.908195 0.418548i \(-0.862539\pi\)
0.908195 0.418548i \(-0.137461\pi\)
\(138\) 0 0
\(139\) 10.0000i 0.848189i −0.905618 0.424094i \(-0.860592\pi\)
0.905618 0.424094i \(-0.139408\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 11.3137i 0.946100i
\(144\) 0 0
\(145\) 12.0000 + 9.79796i 0.996546 + 0.813676i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 10.3923i 0.851371i −0.904871 0.425685i \(-0.860033\pi\)
0.904871 0.425685i \(-0.139967\pi\)
\(150\) 0 0
\(151\) 13.8564 1.12762 0.563809 0.825905i \(-0.309335\pi\)
0.563809 + 0.825905i \(0.309335\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 9.79796 12.0000i 0.786991 0.963863i
\(156\) 0 0
\(157\) 2.82843 0.225733 0.112867 0.993610i \(-0.463997\pi\)
0.112867 + 0.993610i \(0.463997\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −10.0000 −0.788110
\(162\) 0 0
\(163\) −22.0454 −1.72673 −0.863365 0.504580i \(-0.831647\pi\)
−0.863365 + 0.504580i \(0.831647\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 9.89949i 0.766046i 0.923739 + 0.383023i \(0.125117\pi\)
−0.923739 + 0.383023i \(0.874883\pi\)
\(168\) 0 0
\(169\) 19.0000 1.46154
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −8.48528 −0.645124 −0.322562 0.946548i \(-0.604544\pi\)
−0.322562 + 0.946548i \(0.604544\pi\)
\(174\) 0 0
\(175\) −6.92820 + 1.41421i −0.523723 + 0.106904i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 2.00000i 0.149487i 0.997203 + 0.0747435i \(0.0238138\pi\)
−0.997203 + 0.0747435i \(0.976186\pi\)
\(180\) 0 0
\(181\) 20.7846i 1.54491i 0.635071 + 0.772454i \(0.280971\pi\)
−0.635071 + 0.772454i \(0.719029\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −4.00000 + 4.89898i −0.294086 + 0.360180i
\(186\) 0 0
\(187\) −9.79796 −0.716498
\(188\) 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
\(193\) 4.89898i 0.352636i −0.984333 0.176318i \(-0.943581\pi\)
0.984333 0.176318i \(-0.0564187\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 22.6274 1.61214 0.806068 0.591822i \(-0.201591\pi\)
0.806068 + 0.591822i \(0.201591\pi\)
\(198\) 0 0
\(199\) 20.7846 1.47338 0.736691 0.676230i \(-0.236387\pi\)
0.736691 + 0.676230i \(0.236387\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 9.79796 0.687682
\(204\) 0 0
\(205\) 5.65685 6.92820i 0.395092 0.483887i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 12.0000 0.830057
\(210\) 0 0
\(211\) 10.0000i 0.688428i 0.938891 + 0.344214i \(0.111855\pi\)
−0.938891 + 0.344214i \(0.888145\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −3.46410 + 4.24264i −0.236250 + 0.289346i
\(216\) 0 0
\(217\) 9.79796i 0.665129i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 27.7128i 1.86417i
\(222\) 0 0
\(223\) 26.8701i 1.79935i 0.436558 + 0.899676i \(0.356197\pi\)
−0.436558 + 0.899676i \(0.643803\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −17.1464 −1.13805 −0.569024 0.822321i \(-0.692679\pi\)
−0.569024 + 0.822321i \(0.692679\pi\)
\(228\) 0 0
\(229\) 6.92820i 0.457829i 0.973447 + 0.228914i \(0.0735176\pi\)
−0.973447 + 0.228914i \(0.926482\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 14.6969i 0.962828i −0.876493 0.481414i \(-0.840123\pi\)
0.876493 0.481414i \(-0.159877\pi\)
\(234\) 0 0
\(235\) 7.34847 + 6.00000i 0.479361 + 0.391397i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −6.92820 −0.448148 −0.224074 0.974572i \(-0.571936\pi\)
−0.224074 + 0.974572i \(0.571936\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 7.07107 8.66025i 0.451754 0.553283i
\(246\) 0 0
\(247\) 33.9411i 2.15962i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 26.0000i 1.64111i 0.571571 + 0.820553i \(0.306334\pi\)
−0.571571 + 0.820553i \(0.693666\pi\)
\(252\) 0 0
\(253\) 14.1421 0.889108
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 9.79796i 0.611180i −0.952163 0.305590i \(-0.901146\pi\)
0.952163 0.305590i \(-0.0988537\pi\)
\(258\) 0 0
\(259\) 4.00000i 0.248548i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 26.8701i 1.65688i −0.560079 0.828439i \(-0.689229\pi\)
0.560079 0.828439i \(-0.310771\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 17.3205i 1.05605i −0.849229 0.528025i \(-0.822933\pi\)
0.849229 0.528025i \(-0.177067\pi\)
\(270\) 0 0
\(271\) −20.7846 −1.26258 −0.631288 0.775549i \(-0.717473\pi\)
−0.631288 + 0.775549i \(0.717473\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 9.79796 2.00000i 0.590839 0.120605i
\(276\) 0 0
\(277\) −14.1421 −0.849719 −0.424859 0.905259i \(-0.639676\pi\)
−0.424859 + 0.905259i \(0.639676\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −8.00000 −0.477240 −0.238620 0.971113i \(-0.576695\pi\)
−0.238620 + 0.971113i \(0.576695\pi\)
\(282\) 0 0
\(283\) 2.44949 0.145607 0.0728035 0.997346i \(-0.476805\pi\)
0.0728035 + 0.997346i \(0.476805\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 5.65685i 0.333914i
\(288\) 0 0
\(289\) −7.00000 −0.411765
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 2.82843 0.165238 0.0826192 0.996581i \(-0.473671\pi\)
0.0826192 + 0.996581i \(0.473671\pi\)
\(294\) 0 0
\(295\) −3.46410 2.82843i −0.201688 0.164677i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 40.0000i 2.31326i
\(300\) 0 0
\(301\) 3.46410i 0.199667i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 6.00000 + 4.89898i 0.343559 + 0.280515i
\(306\) 0 0
\(307\) 2.44949 0.139800 0.0698999 0.997554i \(-0.477732\pi\)
0.0698999 + 0.997554i \(0.477732\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 27.7128 1.57145 0.785725 0.618576i \(-0.212290\pi\)
0.785725 + 0.618576i \(0.212290\pi\)
\(312\) 0 0
\(313\) 29.3939i 1.66144i 0.556690 + 0.830720i \(0.312071\pi\)
−0.556690 + 0.830720i \(0.687929\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 22.6274 1.27088 0.635441 0.772149i \(-0.280818\pi\)
0.635441 + 0.772149i \(0.280818\pi\)
\(318\) 0 0
\(319\) −13.8564 −0.775810
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 29.3939 1.63552
\(324\) 0 0
\(325\) −5.65685 27.7128i −0.313786 1.53723i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 6.00000 0.330791
\(330\) 0 0
\(331\) 6.00000i 0.329790i −0.986311 0.164895i \(-0.947272\pi\)
0.986311 0.164895i \(-0.0527285\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 3.46410 4.24264i 0.189264 0.231800i
\(336\) 0 0
\(337\) 19.5959i 1.06746i 0.845656 + 0.533729i \(0.179210\pi\)
−0.845656 + 0.533729i \(0.820790\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 13.8564i 0.750366i
\(342\) 0 0
\(343\) 16.9706i 0.916324i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −17.1464 −0.920468 −0.460234 0.887798i \(-0.652235\pi\)
−0.460234 + 0.887798i \(0.652235\pi\)
\(348\) 0 0
\(349\) 6.92820i 0.370858i 0.982658 + 0.185429i \(0.0593675\pi\)
−0.982658 + 0.185429i \(0.940632\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 9.79796i 0.521493i 0.965407 + 0.260746i \(0.0839686\pi\)
−0.965407 + 0.260746i \(0.916031\pi\)
\(354\) 0 0
\(355\) 9.79796 12.0000i 0.520022 0.636894i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −27.7128 −1.46263 −0.731313 0.682042i \(-0.761092\pi\)
−0.731313 + 0.682042i \(0.761092\pi\)
\(360\) 0 0
\(361\) −17.0000 −0.894737
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −8.48528 6.92820i −0.444140 0.362639i
\(366\) 0 0
\(367\) 9.89949i 0.516749i 0.966045 + 0.258375i \(0.0831869\pi\)
−0.966045 + 0.258375i \(0.916813\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 19.7990 1.02515 0.512576 0.858642i \(-0.328691\pi\)
0.512576 + 0.858642i \(0.328691\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 39.1918i 2.01848i
\(378\) 0 0
\(379\) 14.0000i 0.719132i 0.933120 + 0.359566i \(0.117075\pi\)
−0.933120 + 0.359566i \(0.882925\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 12.7279i 0.650366i 0.945651 + 0.325183i \(0.105426\pi\)
−0.945651 + 0.325183i \(0.894574\pi\)
\(384\) 0 0
\(385\) 4.00000 4.89898i 0.203859 0.249675i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 3.46410i 0.175637i 0.996136 + 0.0878185i \(0.0279895\pi\)
−0.996136 + 0.0878185i \(0.972010\pi\)
\(390\) 0 0
\(391\) 34.6410 1.75187
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −9.79796 + 12.0000i −0.492989 + 0.603786i
\(396\) 0 0
\(397\) −11.3137 −0.567819 −0.283909 0.958851i \(-0.591631\pi\)
−0.283909 + 0.958851i \(0.591631\pi\)
\(398\) 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
\(403\) 39.1918 1.95228
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 5.65685i 0.280400i
\(408\) 0 0
\(409\) −20.0000 −0.988936 −0.494468 0.869196i \(-0.664637\pi\)
−0.494468 + 0.869196i \(0.664637\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −2.82843 −0.139178
\(414\) 0 0
\(415\) −17.3205 + 21.2132i −0.850230 + 1.04132i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 26.0000i 1.27018i 0.772437 + 0.635092i \(0.219038\pi\)
−0.772437 + 0.635092i \(0.780962\pi\)
\(420\) 0 0
\(421\) 3.46410i 0.168830i −0.996431 0.0844150i \(-0.973098\pi\)
0.996431 0.0844150i \(-0.0269021\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 24.0000 4.89898i 1.16417 0.237635i
\(426\) 0 0
\(427\) 4.89898 0.237078
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 6.92820 0.333720 0.166860 0.985981i \(-0.446637\pi\)
0.166860 + 0.985981i \(0.446637\pi\)
\(432\) 0 0
\(433\) 34.2929i 1.64801i −0.566583 0.824005i \(-0.691735\pi\)
0.566583 0.824005i \(-0.308265\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −42.4264 −2.02953
\(438\) 0 0
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −22.0454 −1.04741 −0.523704 0.851900i \(-0.675450\pi\)
−0.523704 + 0.851900i \(0.675450\pi\)
\(444\) 0 0
\(445\) 2.82843 3.46410i 0.134080 0.164214i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −16.0000 −0.755087 −0.377543 0.925992i \(-0.623231\pi\)
−0.377543 + 0.925992i \(0.623231\pi\)
\(450\) 0 0
\(451\) 8.00000i 0.376705i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −13.8564 11.3137i −0.649598 0.530395i
\(456\) 0 0
\(457\) 19.5959i 0.916658i −0.888783 0.458329i \(-0.848448\pi\)
0.888783 0.458329i \(-0.151552\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 20.7846i 0.968036i 0.875058 + 0.484018i \(0.160823\pi\)
−0.875058 + 0.484018i \(0.839177\pi\)
\(462\) 0 0
\(463\) 1.41421i 0.0657241i 0.999460 + 0.0328620i \(0.0104622\pi\)
−0.999460 + 0.0328620i \(0.989538\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 31.8434 1.47354 0.736768 0.676146i \(-0.236351\pi\)
0.736768 + 0.676146i \(0.236351\pi\)
\(468\) 0 0
\(469\) 3.46410i 0.159957i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 4.89898i 0.225255i
\(474\) 0 0
\(475\) −29.3939 + 6.00000i −1.34868 + 0.275299i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −41.5692 −1.89935 −0.949673 0.313243i \(-0.898585\pi\)
−0.949673 + 0.313243i \(0.898585\pi\)
\(480\) 0 0
\(481\) −16.0000 −0.729537
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −25.4558 20.7846i −1.15589 0.943781i
\(486\) 0 0
\(487\) 26.8701i 1.21760i 0.793324 + 0.608799i \(0.208349\pi\)
−0.793324 + 0.608799i \(0.791651\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 38.0000i 1.71492i 0.514554 + 0.857458i \(0.327958\pi\)
−0.514554 + 0.857458i \(0.672042\pi\)
\(492\) 0 0
\(493\) −33.9411 −1.52863
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 9.79796i 0.439499i
\(498\) 0 0
\(499\) 6.00000i 0.268597i 0.990941 + 0.134298i \(0.0428781\pi\)
−0.990941 + 0.134298i \(0.957122\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 21.2132i 0.945850i 0.881103 + 0.472925i \(0.156802\pi\)
−0.881103 + 0.472925i \(0.843198\pi\)
\(504\) 0 0
\(505\) 12.0000 + 9.79796i 0.533993 + 0.436003i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 20.7846i 0.921262i −0.887592 0.460631i \(-0.847623\pi\)
0.887592 0.460631i \(-0.152377\pi\)
\(510\) 0 0
\(511\) −6.92820 −0.306486
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −17.1464 14.0000i −0.755562 0.616914i
\(516\) 0 0
\(517\) −8.48528 −0.373182
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −34.0000 −1.48957 −0.744784 0.667306i \(-0.767447\pi\)
−0.744784 + 0.667306i \(0.767447\pi\)
\(522\) 0 0
\(523\) 2.44949 0.107109 0.0535544 0.998565i \(-0.482945\pi\)
0.0535544 + 0.998565i \(0.482945\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 33.9411i 1.47850i
\(528\) 0 0
\(529\) −27.0000 −1.17391
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 22.6274 0.980102
\(534\) 0 0
\(535\) −3.46410 + 4.24264i −0.149766 + 0.183425i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 10.0000i 0.430730i
\(540\) 0 0
\(541\) 6.92820i 0.297867i 0.988847 + 0.148933i \(0.0475840\pi\)
−0.988847 + 0.148933i \(0.952416\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −6.00000 4.89898i −0.257012 0.209849i
\(546\) 0 0
\(547\) 26.9444 1.15206 0.576029 0.817429i \(-0.304601\pi\)
0.576029 + 0.817429i \(0.304601\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 41.5692 1.77091
\(552\) 0 0
\(553\) 9.79796i 0.416652i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −25.4558 −1.07860 −0.539299 0.842114i \(-0.681311\pi\)
−0.539299 + 0.842114i \(0.681311\pi\)
\(558\) 0 0
\(559\) −13.8564 −0.586064
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −7.34847 −0.309701 −0.154851 0.987938i \(-0.549490\pi\)
−0.154851 + 0.987938i \(0.549490\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 28.0000 1.17382 0.586911 0.809652i \(-0.300344\pi\)
0.586911 + 0.809652i \(0.300344\pi\)
\(570\) 0 0
\(571\) 30.0000i 1.25546i 0.778431 + 0.627730i \(0.216016\pi\)
−0.778431 + 0.627730i \(0.783984\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −34.6410 + 7.07107i −1.44463 + 0.294884i
\(576\) 0 0
\(577\) 29.3939i 1.22368i 0.790980 + 0.611842i \(0.209571\pi\)
−0.790980 + 0.611842i \(0.790429\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 17.3205i 0.718576i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 12.2474 0.505506 0.252753 0.967531i \(-0.418664\pi\)
0.252753 + 0.967531i \(0.418664\pi\)
\(588\) 0 0
\(589\) 41.5692i 1.71283i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 29.3939i 1.20706i 0.797340 + 0.603531i \(0.206240\pi\)
−0.797340 + 0.603531i \(0.793760\pi\)
\(594\) 0 0
\(595\) 9.79796 12.0000i 0.401677 0.491952i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 20.7846 0.849236 0.424618 0.905373i \(-0.360408\pi\)
0.424618 + 0.905373i \(0.360408\pi\)
\(600\) 0 0
\(601\) −24.0000 −0.978980 −0.489490 0.872009i \(-0.662817\pi\)
−0.489490 + 0.872009i \(0.662817\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 9.89949 12.1244i 0.402472 0.492925i
\(606\) 0 0
\(607\) 41.0122i 1.66463i 0.554300 + 0.832317i \(0.312986\pi\)
−0.554300 + 0.832317i \(0.687014\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 24.0000i 0.970936i
\(612\) 0 0
\(613\) 5.65685 0.228478 0.114239 0.993453i \(-0.463557\pi\)
0.114239 + 0.993453i \(0.463557\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 24.4949i 0.986127i 0.869993 + 0.493064i \(0.164123\pi\)
−0.869993 + 0.493064i \(0.835877\pi\)
\(618\) 0 0
\(619\) 14.0000i 0.562708i −0.959604 0.281354i \(-0.909217\pi\)
0.959604 0.281354i \(-0.0907834\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 2.82843i 0.113319i
\(624\) 0 0
\(625\) −23.0000 + 9.79796i −0.920000 + 0.391918i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 13.8564i 0.552491i
\(630\) 0 0
\(631\) 13.8564 0.551615 0.275807 0.961213i \(-0.411055\pi\)
0.275807 + 0.961213i \(0.411055\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −12.2474 10.0000i −0.486025 0.396838i
\(636\) 0 0
\(637\) 28.2843 1.12066
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 40.0000 1.57991 0.789953 0.613168i \(-0.210105\pi\)
0.789953 + 0.613168i \(0.210105\pi\)
\(642\) 0 0
\(643\) 2.44949 0.0965984 0.0482992 0.998833i \(-0.484620\pi\)
0.0482992 + 0.998833i \(0.484620\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 4.24264i 0.166795i −0.996516 0.0833977i \(-0.973423\pi\)
0.996516 0.0833977i \(-0.0265772\pi\)
\(648\) 0 0
\(649\) 4.00000 0.157014
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −5.65685 −0.221370 −0.110685 0.993856i \(-0.535304\pi\)
−0.110685 + 0.993856i \(0.535304\pi\)
\(654\) 0 0
\(655\) −17.3205 14.1421i −0.676768 0.552579i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 2.00000i 0.0779089i 0.999241 + 0.0389545i \(0.0124027\pi\)
−0.999241 + 0.0389545i \(0.987597\pi\)
\(660\) 0 0
\(661\) 10.3923i 0.404214i −0.979363 0.202107i \(-0.935221\pi\)
0.979363 0.202107i \(-0.0647788\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −12.0000 + 14.6969i −0.465340 + 0.569923i
\(666\) 0 0
\(667\) 48.9898 1.89689
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −6.92820 −0.267460
\(672\) 0 0
\(673\) 24.4949i 0.944209i 0.881543 + 0.472104i \(0.156505\pi\)
−0.881543 + 0.472104i \(0.843495\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −5.65685 −0.217411 −0.108705 0.994074i \(-0.534670\pi\)
−0.108705 + 0.994074i \(0.534670\pi\)
\(678\) 0 0
\(679\) −20.7846 −0.797640
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 2.44949 0.0937271 0.0468636 0.998901i \(-0.485077\pi\)
0.0468636 + 0.998901i \(0.485077\pi\)
\(684\) 0 0
\(685\) −16.9706 13.8564i −0.648412 0.529426i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 42.0000i 1.59776i 0.601494 + 0.798878i \(0.294573\pi\)
−0.601494 + 0.798878i \(0.705427\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −17.3205 14.1421i −0.657004 0.536442i
\(696\) 0 0
\(697\) 19.5959i 0.742248i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 3.46410i 0.130837i 0.997858 + 0.0654187i \(0.0208383\pi\)
−0.997858 + 0.0654187i \(0.979162\pi\)
\(702\) 0 0
\(703\) 16.9706i 0.640057i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 9.79796 0.368490
\(708\) 0 0
\(709\) 34.6410i 1.30097i −0.759519 0.650485i \(-0.774566\pi\)
0.759519 0.650485i \(-0.225434\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 48.9898i 1.83468i
\(714\) 0 0
\(715\) 19.5959 + 16.0000i 0.732846 + 0.598366i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 6.92820 0.258378 0.129189 0.991620i \(-0.458763\pi\)
0.129189 + 0.991620i \(0.458763\pi\)
\(720\) 0 0
\(721\) −14.0000 −0.521387
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 33.9411 6.92820i 1.26054 0.257307i
\(726\) 0 0
\(727\) 18.3848i 0.681854i −0.940090 0.340927i \(-0.889259\pi\)
0.940090 0.340927i \(-0.110741\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 12.0000i 0.443836i
\(732\) 0 0
\(733\) −19.7990 −0.731292 −0.365646 0.930754i \(-0.619152\pi\)
−0.365646 + 0.930754i \(0.619152\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 4.89898i 0.180456i
\(738\) 0 0
\(739\) 30.0000i 1.10357i 0.833987 + 0.551784i \(0.186053\pi\)
−0.833987 + 0.551784i \(0.813947\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 4.24264i 0.155647i −0.996967 0.0778237i \(-0.975203\pi\)
0.996967 0.0778237i \(-0.0247971\pi\)
\(744\) 0 0
\(745\) −18.0000 14.6969i −0.659469 0.538454i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 3.46410i 0.126576i
\(750\) 0 0
\(751\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 19.5959 24.0000i 0.713168 0.873449i
\(756\) 0 0
\(757\) 2.82843 0.102801 0.0514005 0.998678i \(-0.483632\pi\)
0.0514005 + 0.998678i \(0.483632\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −38.0000 −1.37750 −0.688749 0.724999i \(-0.741840\pi\)
−0.688749 + 0.724999i \(0.741840\pi\)
\(762\) 0 0
\(763\) −4.89898 −0.177355
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 11.3137i 0.408514i
\(768\) 0 0
\(769\) 42.0000 1.51456 0.757279 0.653091i \(-0.226528\pi\)
0.757279 + 0.653091i \(0.226528\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −45.2548 −1.62770 −0.813852 0.581073i \(-0.802633\pi\)
−0.813852 + 0.581073i \(0.802633\pi\)
\(774\) 0 0
\(775\) −6.92820 33.9411i −0.248868 1.21920i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 24.0000i 0.859889i
\(780\) 0 0
\(781\) 13.8564i 0.495821i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 4.00000 4.89898i 0.142766 0.174852i
\(786\) 0 0
\(787\) −46.5403 −1.65898 −0.829491 0.558520i \(-0.811370\pi\)
−0.829491 + 0.558520i \(0.811370\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 19.5959i 0.695871i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 33.9411 1.20226 0.601128 0.799153i \(-0.294718\pi\)
0.601128 + 0.799153i \(0.294718\pi\)
\(798\) 0 0
\(799\) −20.7846 −0.735307
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 9.79796 0.345762
\(804\) 0 0
\(805\) −14.1421 + 17.3205i −0.498445 + 0.610468i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −50.0000 −1.75791 −0.878953 0.476908i \(-0.841757\pi\)
−0.878953 + 0.476908i \(0.841757\pi\)
\(810\) 0 0
\(811\) 22.0000i 0.772524i 0.922389 + 0.386262i \(0.126234\pi\)
−0.922389 + 0.386262i \(0.873766\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −31.1769 + 38.1838i −1.09208 + 1.33752i
\(816\) 0 0
\(817\) 14.6969i 0.514181i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 51.9615i 1.81347i 0.421701 + 0.906735i \(0.361433\pi\)
−0.421701 + 0.906735i \(0.638567\pi\)
\(822\) 0 0
\(823\) 35.3553i 1.23241i 0.787586 + 0.616205i \(0.211331\pi\)
−0.787586 + 0.616205i \(0.788669\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 31.8434 1.10730 0.553651 0.832749i \(-0.313234\pi\)
0.553651 + 0.832749i \(0.313234\pi\)
\(828\) 0 0
\(829\) 45.0333i 1.56407i 0.623233 + 0.782036i \(0.285819\pi\)
−0.623233 + 0.782036i \(0.714181\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 24.4949i 0.848698i
\(834\) 0 0
\(835\) 17.1464 + 14.0000i 0.593377 + 0.484490i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 41.5692 1.43513 0.717564 0.696492i \(-0.245257\pi\)
0.717564 + 0.696492i \(0.245257\pi\)
\(840\) 0 0
\(841\) −19.0000 −0.655172
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 26.8701 32.9090i 0.924358 1.13210i
\(846\) 0 0
\(847\) 9.89949i 0.340151i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 20.0000i 0.685591i
\(852\) 0 0
\(853\) 5.65685 0.193687 0.0968435 0.995300i \(-0.469125\pi\)
0.0968435 + 0.995300i \(0.469125\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 39.1918i 1.33877i −0.742917 0.669384i \(-0.766558\pi\)
0.742917 0.669384i \(-0.233442\pi\)
\(858\) 0 0
\(859\) 22.0000i 0.750630i −0.926897 0.375315i \(-0.877534\pi\)
0.926897 0.375315i \(-0.122466\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 24.0416i 0.818387i −0.912448 0.409193i \(-0.865810\pi\)
0.912448 0.409193i \(-0.134190\pi\)
\(864\) 0 0
\(865\) −12.0000 + 14.6969i −0.408012 + 0.499711i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 13.8564i 0.470046i
\(870\) 0 0
\(871\) 13.8564 0.469506
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −7.34847 + 14.0000i −0.248424 + 0.473286i
\(876\) 0 0
\(877\) −36.7696 −1.24162 −0.620810 0.783961i \(-0.713196\pi\)
−0.620810 + 0.783961i \(0.713196\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −52.0000 −1.75192 −0.875962 0.482380i \(-0.839773\pi\)
−0.875962 + 0.482380i \(0.839773\pi\)
\(882\) 0 0
\(883\) −12.2474 −0.412159 −0.206080 0.978535i \(-0.566071\pi\)
−0.206080 + 0.978535i \(0.566071\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 49.4975i 1.66196i −0.556300 0.830981i \(-0.687780\pi\)
0.556300 0.830981i \(-0.312220\pi\)
\(888\) 0 0
\(889\) −10.0000 −0.335389
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 25.4558 0.851847
\(894\) 0 0
\(895\) 3.46410 + 2.82843i 0.115792 + 0.0945439i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 48.0000i 1.60089i
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 36.0000 + 29.3939i 1.19668 + 0.977086i
\(906\) 0 0
\(907\) −26.9444 −0.894674 −0.447337 0.894366i \(-0.647627\pi\)
−0.447337 + 0.894366i \(0.647627\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 48.4974 1.60679 0.803396 0.595446i \(-0.203024\pi\)
0.803396 + 0.595446i \(0.203024\pi\)
\(912\) 0 0
\(913\) 24.4949i 0.810663i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −14.1421 −0.467014
\(918\) 0 0
\(919\) 6.92820 0.228540 0.114270 0.993450i \(-0.463547\pi\)
0.114270 + 0.993450i \(0.463547\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 39.1918 1.29001
\(924\) 0 0
\(925\) 2.82843 + 13.8564i 0.0929981 + 0.455596i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 8.00000 0.262471 0.131236 0.991351i \(-0.458106\pi\)
0.131236 + 0.991351i \(0.458106\pi\)
\(930\) 0 0
\(931\) 30.0000i 0.983210i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −13.8564 + 16.9706i −0.453153 + 0.554997i
\(936\) 0 0
\(937\) 24.4949i 0.800213i 0.916469 + 0.400107i \(0.131027\pi\)
−0.916469 + 0.400107i \(0.868973\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 34.6410i 1.12926i −0.825342 0.564632i \(-0.809018\pi\)
0.825342 0.564632i \(-0.190982\pi\)
\(942\) 0 0
\(943\) 28.2843i 0.921063i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −41.6413 −1.35316 −0.676581 0.736369i \(-0.736539\pi\)
−0.676581 + 0.736369i \(0.736539\pi\)
\(948\) 0 0
\(949\) 27.7128i 0.899596i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 39.1918i 1.26955i 0.772698 + 0.634774i \(0.218907\pi\)
−0.772698 + 0.634774i \(0.781093\pi\)
\(954\) 0 0
\(955\) 19.5959 24.0000i 0.634109 0.776622i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −13.8564 −0.447447
\(960\) 0 0
\(961\) 17.0000 0.548387
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −8.48528 6.92820i −0.273151 0.223027i
\(966\) 0 0
\(967\) 15.5563i 0.500258i −0.968212 0.250129i \(-0.919527\pi\)
0.968212 0.250129i \(-0.0804731\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 22.0000i 0.706014i 0.935621 + 0.353007i \(0.114841\pi\)
−0.935621 + 0.353007i \(0.885159\pi\)
\(972\) 0 0
\(973\) −14.1421 −0.453376
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 4.89898i 0.156732i −0.996925 0.0783661i \(-0.975030\pi\)
0.996925 0.0783661i \(-0.0249703\pi\)
\(978\) 0 0
\(979\) 4.00000i 0.127841i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 46.6690i 1.48851i 0.667895 + 0.744256i \(0.267196\pi\)
−0.667895 + 0.744256i \(0.732804\pi\)
\(984\) 0 0
\(985\) 32.0000 39.1918i 1.01960 1.24876i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 17.3205i 0.550760i
\(990\) 0 0
\(991\) 27.7128 0.880327 0.440163 0.897918i \(-0.354921\pi\)
0.440163 + 0.897918i \(0.354921\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 29.3939 36.0000i 0.931849 1.14128i
\(996\) 0 0
\(997\) 56.5685 1.79154 0.895772 0.444514i \(-0.146624\pi\)
0.895772 + 0.444514i \(0.146624\pi\)
\(998\) 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.g.289.5 8
3.2 odd 2 320.2.f.b.289.2 yes 8
4.3 odd 2 inner 2880.2.d.g.289.6 8
5.4 even 2 inner 2880.2.d.g.289.2 8
8.3 odd 2 inner 2880.2.d.g.289.4 8
8.5 even 2 inner 2880.2.d.g.289.3 8
12.11 even 2 320.2.f.b.289.6 yes 8
15.2 even 4 1600.2.d.i.801.7 8
15.8 even 4 1600.2.d.i.801.1 8
15.14 odd 2 320.2.f.b.289.8 yes 8
20.19 odd 2 inner 2880.2.d.g.289.1 8
24.5 odd 2 320.2.f.b.289.7 yes 8
24.11 even 2 320.2.f.b.289.3 yes 8
40.19 odd 2 inner 2880.2.d.g.289.7 8
40.29 even 2 inner 2880.2.d.g.289.8 8
48.5 odd 4 1280.2.c.h.769.3 4
48.11 even 4 1280.2.c.g.769.1 4
48.29 odd 4 1280.2.c.g.769.2 4
48.35 even 4 1280.2.c.h.769.4 4
60.23 odd 4 1600.2.d.i.801.8 8
60.47 odd 4 1600.2.d.i.801.2 8
60.59 even 2 320.2.f.b.289.4 yes 8
120.29 odd 2 320.2.f.b.289.1 8
120.53 even 4 1600.2.d.i.801.6 8
120.59 even 2 320.2.f.b.289.5 yes 8
120.77 even 4 1600.2.d.i.801.4 8
120.83 odd 4 1600.2.d.i.801.3 8
120.107 odd 4 1600.2.d.i.801.5 8
240.29 odd 4 1280.2.c.g.769.4 4
240.53 even 4 6400.2.a.cu.1.2 4
240.59 even 4 1280.2.c.g.769.3 4
240.77 even 4 6400.2.a.ct.1.1 4
240.83 odd 4 6400.2.a.cu.1.1 4
240.107 odd 4 6400.2.a.ct.1.2 4
240.149 odd 4 1280.2.c.h.769.1 4
240.173 even 4 6400.2.a.ct.1.4 4
240.179 even 4 1280.2.c.h.769.2 4
240.197 even 4 6400.2.a.cu.1.3 4
240.203 odd 4 6400.2.a.ct.1.3 4
240.227 odd 4 6400.2.a.cu.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
320.2.f.b.289.1 8 120.29 odd 2
320.2.f.b.289.2 yes 8 3.2 odd 2
320.2.f.b.289.3 yes 8 24.11 even 2
320.2.f.b.289.4 yes 8 60.59 even 2
320.2.f.b.289.5 yes 8 120.59 even 2
320.2.f.b.289.6 yes 8 12.11 even 2
320.2.f.b.289.7 yes 8 24.5 odd 2
320.2.f.b.289.8 yes 8 15.14 odd 2
1280.2.c.g.769.1 4 48.11 even 4
1280.2.c.g.769.2 4 48.29 odd 4
1280.2.c.g.769.3 4 240.59 even 4
1280.2.c.g.769.4 4 240.29 odd 4
1280.2.c.h.769.1 4 240.149 odd 4
1280.2.c.h.769.2 4 240.179 even 4
1280.2.c.h.769.3 4 48.5 odd 4
1280.2.c.h.769.4 4 48.35 even 4
1600.2.d.i.801.1 8 15.8 even 4
1600.2.d.i.801.2 8 60.47 odd 4
1600.2.d.i.801.3 8 120.83 odd 4
1600.2.d.i.801.4 8 120.77 even 4
1600.2.d.i.801.5 8 120.107 odd 4
1600.2.d.i.801.6 8 120.53 even 4
1600.2.d.i.801.7 8 15.2 even 4
1600.2.d.i.801.8 8 60.23 odd 4
2880.2.d.g.289.1 8 20.19 odd 2 inner
2880.2.d.g.289.2 8 5.4 even 2 inner
2880.2.d.g.289.3 8 8.5 even 2 inner
2880.2.d.g.289.4 8 8.3 odd 2 inner
2880.2.d.g.289.5 8 1.1 even 1 trivial
2880.2.d.g.289.6 8 4.3 odd 2 inner
2880.2.d.g.289.7 8 40.19 odd 2 inner
2880.2.d.g.289.8 8 40.29 even 2 inner
6400.2.a.ct.1.1 4 240.77 even 4
6400.2.a.ct.1.2 4 240.107 odd 4
6400.2.a.ct.1.3 4 240.203 odd 4
6400.2.a.ct.1.4 4 240.173 even 4
6400.2.a.cu.1.1 4 240.83 odd 4
6400.2.a.cu.1.2 4 240.53 even 4
6400.2.a.cu.1.3 4 240.197 even 4
6400.2.a.cu.1.4 4 240.227 odd 4