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Properties

Label 8085.2.a.n.1.1
Level $8085$
Weight $2$
Character 8085.1
Self dual yes
Analytic conductor $64.559$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8085,2,Mod(1,8085)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8085, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8085.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8085 = 3 \cdot 5 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8085.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: \(64.5590500342\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1155)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 8085.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -2.00000 q^{4} +1.00000 q^{5} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -2.00000 q^{4} +1.00000 q^{5} +1.00000 q^{9} -1.00000 q^{11} -2.00000 q^{12} -1.00000 q^{13} +1.00000 q^{15} +4.00000 q^{16} -6.00000 q^{17} -7.00000 q^{19} -2.00000 q^{20} +6.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} +6.00000 q^{29} +5.00000 q^{31} -1.00000 q^{33} -2.00000 q^{36} +5.00000 q^{37} -1.00000 q^{39} -1.00000 q^{43} +2.00000 q^{44} +1.00000 q^{45} +6.00000 q^{47} +4.00000 q^{48} -6.00000 q^{51} +2.00000 q^{52} -12.0000 q^{53} -1.00000 q^{55} -7.00000 q^{57} -6.00000 q^{59} -2.00000 q^{60} +2.00000 q^{61} -8.00000 q^{64} -1.00000 q^{65} +5.00000 q^{67} +12.0000 q^{68} +6.00000 q^{69} -13.0000 q^{73} +1.00000 q^{75} +14.0000 q^{76} -1.00000 q^{79} +4.00000 q^{80} +1.00000 q^{81} -6.00000 q^{85} +6.00000 q^{87} -6.00000 q^{89} -12.0000 q^{92} +5.00000 q^{93} -7.00000 q^{95} +14.0000 q^{97} -1.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(3\) 1.00000 0.577350
\(4\) −2.00000 −1.00000
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) −2.00000 −0.577350
\(13\) −1.00000 −0.277350 −0.138675 − 0.990338i \(-0.544284\pi\)
−0.138675 + 0.990338i \(0.544284\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 4.00000 1.00000
\(17\) −6.00000 −1.45521 −0.727607 − 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 0 0
\(19\) −7.00000 −1.60591 −0.802955 − 0.596040i \(-0.796740\pi\)
−0.802955 + 0.596040i \(0.796740\pi\)
\(20\) −2.00000 −0.447214
\(21\) 0 0
\(22\) 0 0
\(23\) 6.00000 1.25109 0.625543 − 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 − 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) 5.00000 0.898027 0.449013 − 0.893525i \(-0.351776\pi\)
0.449013 + 0.893525i \(0.351776\pi\)
\(32\) 0 0
\(33\) −1.00000 −0.174078
\(34\) 0 0
\(35\) 0 0
\(36\) −2.00000 −0.333333
\(37\) 5.00000 0.821995 0.410997 − 0.911636i \(-0.365181\pi\)
0.410997 + 0.911636i \(0.365181\pi\)
\(38\) 0 0
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) −1.00000 −0.152499 −0.0762493 − 0.997089i \(-0.524294\pi\)
−0.0762493 + 0.997089i \(0.524294\pi\)
\(44\) 2.00000 0.301511
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) 6.00000 0.875190 0.437595 − 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 4.00000 0.577350
\(49\) 0 0
\(50\) 0 0
\(51\) −6.00000 −0.840168
\(52\) 2.00000 0.277350
\(53\) −12.0000 −1.64833 −0.824163 − 0.566352i \(-0.808354\pi\)
−0.824163 + 0.566352i \(0.808354\pi\)
\(54\) 0 0
\(55\) −1.00000 −0.134840
\(56\) 0 0
\(57\) −7.00000 −0.927173
\(58\) 0 0
\(59\) −6.00000 −0.781133 −0.390567 − 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) −2.00000 −0.258199
\(61\) 2.00000 0.256074 0.128037 − 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) −1.00000 −0.124035
\(66\) 0 0
\(67\) 5.00000 0.610847 0.305424 − 0.952217i \(-0.401202\pi\)
0.305424 + 0.952217i \(0.401202\pi\)
\(68\) 12.0000 1.45521
\(69\) 6.00000 0.722315
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −13.0000 −1.52153 −0.760767 − 0.649025i \(-0.775177\pi\)
−0.760767 + 0.649025i \(0.775177\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 14.0000 1.60591
\(77\) 0 0
\(78\) 0 0
\(79\) −1.00000 −0.112509 −0.0562544 − 0.998416i \(-0.517916\pi\)
−0.0562544 + 0.998416i \(0.517916\pi\)
\(80\) 4.00000 0.447214
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) −6.00000 −0.650791
\(86\) 0 0
\(87\) 6.00000 0.643268
\(88\) 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
\(92\) −12.0000 −1.25109
\(93\) 5.00000 0.518476
\(94\) 0 0
\(95\) −7.00000 −0.718185
\(96\) 0 0
\(97\) 14.0000 1.42148 0.710742 − 0.703452i \(-0.248359\pi\)
0.710742 + 0.703452i \(0.248359\pi\)
\(98\) 0 0
\(99\) −1.00000 −0.100504
\(100\) −2.00000 −0.200000
\(101\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) −13.0000 −1.28093 −0.640464 − 0.767988i \(-0.721258\pi\)
−0.640464 + 0.767988i \(0.721258\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −6.00000 −0.580042 −0.290021 − 0.957020i \(-0.593662\pi\)
−0.290021 + 0.957020i \(0.593662\pi\)
\(108\) −2.00000 −0.192450
\(109\) −19.0000 −1.81987 −0.909935 − 0.414751i \(-0.863869\pi\)
−0.909935 + 0.414751i \(0.863869\pi\)
\(110\) 0 0
\(111\) 5.00000 0.474579
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 − 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 6.00000 0.559503
\(116\) −12.0000 −1.11417
\(117\) −1.00000 −0.0924500
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 0 0
\(124\) −10.0000 −0.898027
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −13.0000 −1.15356 −0.576782 − 0.816898i \(-0.695692\pi\)
−0.576782 + 0.816898i \(0.695692\pi\)
\(128\) 0 0
\(129\) −1.00000 −0.0880451
\(130\) 0 0
\(131\) −18.0000 −1.57267 −0.786334 − 0.617802i \(-0.788023\pi\)
−0.786334 + 0.617802i \(0.788023\pi\)
\(132\) 2.00000 0.174078
\(133\) 0 0
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) 18.0000 1.53784 0.768922 − 0.639343i \(-0.220793\pi\)
0.768922 + 0.639343i \(0.220793\pi\)
\(138\) 0 0
\(139\) −7.00000 −0.593732 −0.296866 − 0.954919i \(-0.595942\pi\)
−0.296866 + 0.954919i \(0.595942\pi\)
\(140\) 0 0
\(141\) 6.00000 0.505291
\(142\) 0 0
\(143\) 1.00000 0.0836242
\(144\) 4.00000 0.333333
\(145\) 6.00000 0.498273
\(146\) 0 0
\(147\) 0 0
\(148\) −10.0000 −0.821995
\(149\) −12.0000 −0.983078 −0.491539 − 0.870855i \(-0.663566\pi\)
−0.491539 + 0.870855i \(0.663566\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 − 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 0 0
\(153\) −6.00000 −0.485071
\(154\) 0 0
\(155\) 5.00000 0.401610
\(156\) 2.00000 0.160128
\(157\) −10.0000 −0.798087 −0.399043 − 0.916932i \(-0.630658\pi\)
−0.399043 + 0.916932i \(0.630658\pi\)
\(158\) 0 0
\(159\) −12.0000 −0.951662
\(160\) 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
\(165\) −1.00000 −0.0778499
\(166\) 0 0
\(167\) −12.0000 −0.928588 −0.464294 − 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) −7.00000 −0.535303
\(172\) 2.00000 0.152499
\(173\) −12.0000 −0.912343 −0.456172 − 0.889892i \(-0.650780\pi\)
−0.456172 + 0.889892i \(0.650780\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −4.00000 −0.301511
\(177\) −6.00000 −0.450988
\(178\) 0 0
\(179\) 6.00000 0.448461 0.224231 − 0.974536i \(-0.428013\pi\)
0.224231 + 0.974536i \(0.428013\pi\)
\(180\) −2.00000 −0.149071
\(181\) −7.00000 −0.520306 −0.260153 − 0.965567i \(-0.583773\pi\)
−0.260153 + 0.965567i \(0.583773\pi\)
\(182\) 0 0
\(183\) 2.00000 0.147844
\(184\) 0 0
\(185\) 5.00000 0.367607
\(186\) 0 0
\(187\) 6.00000 0.438763
\(188\) −12.0000 −0.875190
\(189\) 0 0
\(190\) 0 0
\(191\) 18.0000 1.30243 0.651217 − 0.758891i \(-0.274259\pi\)
0.651217 + 0.758891i \(0.274259\pi\)
\(192\) −8.00000 −0.577350
\(193\) 5.00000 0.359908 0.179954 − 0.983675i \(-0.442405\pi\)
0.179954 + 0.983675i \(0.442405\pi\)
\(194\) 0 0
\(195\) −1.00000 −0.0716115
\(196\) 0 0
\(197\) −18.0000 −1.28245 −0.641223 − 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 0 0
\(199\) 8.00000 0.567105 0.283552 − 0.958957i \(-0.408487\pi\)
0.283552 + 0.958957i \(0.408487\pi\)
\(200\) 0 0
\(201\) 5.00000 0.352673
\(202\) 0 0
\(203\) 0 0
\(204\) 12.0000 0.840168
\(205\) 0 0
\(206\) 0 0
\(207\) 6.00000 0.417029
\(208\) −4.00000 −0.277350
\(209\) 7.00000 0.484200
\(210\) 0 0
\(211\) 20.0000 1.37686 0.688428 − 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) 24.0000 1.64833
\(213\) 0 0
\(214\) 0 0
\(215\) −1.00000 −0.0681994
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −13.0000 −0.878459
\(220\) 2.00000 0.134840
\(221\) 6.00000 0.403604
\(222\) 0 0
\(223\) 20.0000 1.33930 0.669650 − 0.742677i \(-0.266444\pi\)
0.669650 + 0.742677i \(0.266444\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 12.0000 0.796468 0.398234 − 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) 14.0000 0.927173
\(229\) −19.0000 −1.25556 −0.627778 − 0.778393i \(-0.716035\pi\)
−0.627778 + 0.778393i \(0.716035\pi\)
\(230\) 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
\(235\) 6.00000 0.391397
\(236\) 12.0000 0.781133
\(237\) −1.00000 −0.0649570
\(238\) 0 0
\(239\) −18.0000 −1.16432 −0.582162 − 0.813073i \(-0.697793\pi\)
−0.582162 + 0.813073i \(0.697793\pi\)
\(240\) 4.00000 0.258199
\(241\) 26.0000 1.67481 0.837404 − 0.546585i \(-0.184072\pi\)
0.837404 + 0.546585i \(0.184072\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) −4.00000 −0.256074
\(245\) 0 0
\(246\) 0 0
\(247\) 7.00000 0.445399
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −30.0000 −1.89358 −0.946792 − 0.321847i \(-0.895696\pi\)
−0.946792 + 0.321847i \(0.895696\pi\)
\(252\) 0 0
\(253\) −6.00000 −0.377217
\(254\) 0 0
\(255\) −6.00000 −0.375735
\(256\) 16.0000 1.00000
\(257\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 2.00000 0.124035
\(261\) 6.00000 0.371391
\(262\) 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
\(268\) −10.0000 −0.610847
\(269\) 6.00000 0.365826 0.182913 − 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) 0 0
\(271\) −16.0000 −0.971931 −0.485965 − 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) −24.0000 −1.45521
\(273\) 0 0
\(274\) 0 0
\(275\) −1.00000 −0.0603023
\(276\) −12.0000 −0.722315
\(277\) −1.00000 −0.0600842 −0.0300421 − 0.999549i \(-0.509564\pi\)
−0.0300421 + 0.999549i \(0.509564\pi\)
\(278\) 0 0
\(279\) 5.00000 0.299342
\(280\) 0 0
\(281\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) −13.0000 −0.772770 −0.386385 − 0.922338i \(-0.626276\pi\)
−0.386385 + 0.922338i \(0.626276\pi\)
\(284\) 0 0
\(285\) −7.00000 −0.414644
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 14.0000 0.820695
\(292\) 26.0000 1.52153
\(293\) −12.0000 −0.701047 −0.350524 − 0.936554i \(-0.613996\pi\)
−0.350524 + 0.936554i \(0.613996\pi\)
\(294\) 0 0
\(295\) −6.00000 −0.349334
\(296\) 0 0
\(297\) −1.00000 −0.0580259
\(298\) 0 0
\(299\) −6.00000 −0.346989
\(300\) −2.00000 −0.115470
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) −28.0000 −1.60591
\(305\) 2.00000 0.114520
\(306\) 0 0
\(307\) −7.00000 −0.399511 −0.199756 − 0.979846i \(-0.564015\pi\)
−0.199756 + 0.979846i \(0.564015\pi\)
\(308\) 0 0
\(309\) −13.0000 −0.739544
\(310\) 0 0
\(311\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) −19.0000 −1.07394 −0.536972 − 0.843600i \(-0.680432\pi\)
−0.536972 + 0.843600i \(0.680432\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 2.00000 0.112509
\(317\) −6.00000 −0.336994 −0.168497 − 0.985702i \(-0.553891\pi\)
−0.168497 + 0.985702i \(0.553891\pi\)
\(318\) 0 0
\(319\) −6.00000 −0.335936
\(320\) −8.00000 −0.447214
\(321\) −6.00000 −0.334887
\(322\) 0 0
\(323\) 42.0000 2.33694
\(324\) −2.00000 −0.111111
\(325\) −1.00000 −0.0554700
\(326\) 0 0
\(327\) −19.0000 −1.05070
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −25.0000 −1.37412 −0.687062 − 0.726599i \(-0.741100\pi\)
−0.687062 + 0.726599i \(0.741100\pi\)
\(332\) 0 0
\(333\) 5.00000 0.273998
\(334\) 0 0
\(335\) 5.00000 0.273179
\(336\) 0 0
\(337\) −13.0000 −0.708155 −0.354078 − 0.935216i \(-0.615205\pi\)
−0.354078 + 0.935216i \(0.615205\pi\)
\(338\) 0 0
\(339\) −6.00000 −0.325875
\(340\) 12.0000 0.650791
\(341\) −5.00000 −0.270765
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 6.00000 0.323029
\(346\) 0 0
\(347\) −12.0000 −0.644194 −0.322097 − 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) −12.0000 −0.643268
\(349\) 26.0000 1.39175 0.695874 − 0.718164i \(-0.255017\pi\)
0.695874 + 0.718164i \(0.255017\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) 0 0
\(353\) 24.0000 1.27739 0.638696 − 0.769460i \(-0.279474\pi\)
0.638696 + 0.769460i \(0.279474\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 12.0000 0.635999
\(357\) 0 0
\(358\) 0 0
\(359\) −30.0000 −1.58334 −0.791670 − 0.610949i \(-0.790788\pi\)
−0.791670 + 0.610949i \(0.790788\pi\)
\(360\) 0 0
\(361\) 30.0000 1.57895
\(362\) 0 0
\(363\) 1.00000 0.0524864
\(364\) 0 0
\(365\) −13.0000 −0.680451
\(366\) 0 0
\(367\) 17.0000 0.887393 0.443696 − 0.896177i \(-0.353667\pi\)
0.443696 + 0.896177i \(0.353667\pi\)
\(368\) 24.0000 1.25109
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) −10.0000 −0.518476
\(373\) −7.00000 −0.362446 −0.181223 − 0.983442i \(-0.558006\pi\)
−0.181223 + 0.983442i \(0.558006\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) −6.00000 −0.309016
\(378\) 0 0
\(379\) 23.0000 1.18143 0.590715 − 0.806880i \(-0.298846\pi\)
0.590715 + 0.806880i \(0.298846\pi\)
\(380\) 14.0000 0.718185
\(381\) −13.0000 −0.666010
\(382\) 0 0
\(383\) −36.0000 −1.83951 −0.919757 − 0.392488i \(-0.871614\pi\)
−0.919757 + 0.392488i \(0.871614\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −1.00000 −0.0508329
\(388\) −28.0000 −1.42148
\(389\) −12.0000 −0.608424 −0.304212 − 0.952604i \(-0.598393\pi\)
−0.304212 + 0.952604i \(0.598393\pi\)
\(390\) 0 0
\(391\) −36.0000 −1.82060
\(392\) 0 0
\(393\) −18.0000 −0.907980
\(394\) 0 0
\(395\) −1.00000 −0.0503155
\(396\) 2.00000 0.100504
\(397\) −7.00000 −0.351320 −0.175660 − 0.984451i \(-0.556206\pi\)
−0.175660 + 0.984451i \(0.556206\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 4.00000 0.200000
\(401\) −24.0000 −1.19850 −0.599251 − 0.800561i \(-0.704535\pi\)
−0.599251 + 0.800561i \(0.704535\pi\)
\(402\) 0 0
\(403\) −5.00000 −0.249068
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) −5.00000 −0.247841
\(408\) 0 0
\(409\) −7.00000 −0.346128 −0.173064 − 0.984911i \(-0.555367\pi\)
−0.173064 + 0.984911i \(0.555367\pi\)
\(410\) 0 0
\(411\) 18.0000 0.887875
\(412\) 26.0000 1.28093
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −7.00000 −0.342791
\(418\) 0 0
\(419\) 30.0000 1.46560 0.732798 − 0.680446i \(-0.238214\pi\)
0.732798 + 0.680446i \(0.238214\pi\)
\(420\) 0 0
\(421\) −31.0000 −1.51085 −0.755424 − 0.655237i \(-0.772569\pi\)
−0.755424 + 0.655237i \(0.772569\pi\)
\(422\) 0 0
\(423\) 6.00000 0.291730
\(424\) 0 0
\(425\) −6.00000 −0.291043
\(426\) 0 0
\(427\) 0 0
\(428\) 12.0000 0.580042
\(429\) 1.00000 0.0482805
\(430\) 0 0
\(431\) −18.0000 −0.867029 −0.433515 − 0.901146i \(-0.642727\pi\)
−0.433515 + 0.901146i \(0.642727\pi\)
\(432\) 4.00000 0.192450
\(433\) −7.00000 −0.336399 −0.168199 − 0.985753i \(-0.553795\pi\)
−0.168199 + 0.985753i \(0.553795\pi\)
\(434\) 0 0
\(435\) 6.00000 0.287678
\(436\) 38.0000 1.81987
\(437\) −42.0000 −2.00913
\(438\) 0 0
\(439\) −4.00000 −0.190910 −0.0954548 − 0.995434i \(-0.530431\pi\)
−0.0954548 + 0.995434i \(0.530431\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) −10.0000 −0.474579
\(445\) −6.00000 −0.284427
\(446\) 0 0
\(447\) −12.0000 −0.567581
\(448\) 0 0
\(449\) −6.00000 −0.283158 −0.141579 − 0.989927i \(-0.545218\pi\)
−0.141579 + 0.989927i \(0.545218\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 12.0000 0.564433
\(453\) 8.00000 0.375873
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −13.0000 −0.608114 −0.304057 − 0.952654i \(-0.598341\pi\)
−0.304057 + 0.952654i \(0.598341\pi\)
\(458\) 0 0
\(459\) −6.00000 −0.280056
\(460\) −12.0000 −0.559503
\(461\) 18.0000 0.838344 0.419172 − 0.907907i \(-0.362320\pi\)
0.419172 + 0.907907i \(0.362320\pi\)
\(462\) 0 0
\(463\) 29.0000 1.34774 0.673872 − 0.738848i \(-0.264630\pi\)
0.673872 + 0.738848i \(0.264630\pi\)
\(464\) 24.0000 1.11417
\(465\) 5.00000 0.231869
\(466\) 0 0
\(467\) −6.00000 −0.277647 −0.138823 − 0.990317i \(-0.544332\pi\)
−0.138823 + 0.990317i \(0.544332\pi\)
\(468\) 2.00000 0.0924500
\(469\) 0 0
\(470\) 0 0
\(471\) −10.0000 −0.460776
\(472\) 0 0
\(473\) 1.00000 0.0459800
\(474\) 0 0
\(475\) −7.00000 −0.321182
\(476\) 0 0
\(477\) −12.0000 −0.549442
\(478\) 0 0
\(479\) 12.0000 0.548294 0.274147 − 0.961688i \(-0.411605\pi\)
0.274147 + 0.961688i \(0.411605\pi\)
\(480\) 0 0
\(481\) −5.00000 −0.227980
\(482\) 0 0
\(483\) 0 0
\(484\) −2.00000 −0.0909091
\(485\) 14.0000 0.635707
\(486\) 0 0
\(487\) −43.0000 −1.94852 −0.974258 − 0.225436i \(-0.927619\pi\)
−0.974258 + 0.225436i \(0.927619\pi\)
\(488\) 0 0
\(489\) −4.00000 −0.180886
\(490\) 0 0
\(491\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) −36.0000 −1.62136
\(494\) 0 0
\(495\) −1.00000 −0.0449467
\(496\) 20.0000 0.898027
\(497\) 0 0
\(498\) 0 0
\(499\) −7.00000 −0.313363 −0.156682 − 0.987649i \(-0.550080\pi\)
−0.156682 + 0.987649i \(0.550080\pi\)
\(500\) −2.00000 −0.0894427
\(501\) −12.0000 −0.536120
\(502\) 0 0
\(503\) −42.0000 −1.87269 −0.936344 − 0.351085i \(-0.885813\pi\)
−0.936344 + 0.351085i \(0.885813\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −12.0000 −0.532939
\(508\) 26.0000 1.15356
\(509\) 6.00000 0.265945 0.132973 − 0.991120i \(-0.457548\pi\)
0.132973 + 0.991120i \(0.457548\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −7.00000 −0.309058
\(514\) 0 0
\(515\) −13.0000 −0.572848
\(516\) 2.00000 0.0880451
\(517\) −6.00000 −0.263880
\(518\) 0 0
\(519\) −12.0000 −0.526742
\(520\) 0 0
\(521\) −12.0000 −0.525730 −0.262865 − 0.964833i \(-0.584667\pi\)
−0.262865 + 0.964833i \(0.584667\pi\)
\(522\) 0 0
\(523\) 29.0000 1.26808 0.634041 − 0.773300i \(-0.281395\pi\)
0.634041 + 0.773300i \(0.281395\pi\)
\(524\) 36.0000 1.57267
\(525\) 0 0
\(526\) 0 0
\(527\) −30.0000 −1.30682
\(528\) −4.00000 −0.174078
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) −6.00000 −0.260378
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −6.00000 −0.259403
\(536\) 0 0
\(537\) 6.00000 0.258919
\(538\) 0 0
\(539\) 0 0
\(540\) −2.00000 −0.0860663
\(541\) 41.0000 1.76273 0.881364 − 0.472438i \(-0.156626\pi\)
0.881364 + 0.472438i \(0.156626\pi\)
\(542\) 0 0
\(543\) −7.00000 −0.300399
\(544\) 0 0
\(545\) −19.0000 −0.813871
\(546\) 0 0
\(547\) 8.00000 0.342055 0.171028 − 0.985266i \(-0.445291\pi\)
0.171028 + 0.985266i \(0.445291\pi\)
\(548\) −36.0000 −1.53784
\(549\) 2.00000 0.0853579
\(550\) 0 0
\(551\) −42.0000 −1.78926
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 5.00000 0.212238
\(556\) 14.0000 0.593732
\(557\) 30.0000 1.27114 0.635570 − 0.772043i \(-0.280765\pi\)
0.635570 + 0.772043i \(0.280765\pi\)
\(558\) 0 0
\(559\) 1.00000 0.0422955
\(560\) 0 0
\(561\) 6.00000 0.253320
\(562\) 0 0
\(563\) −18.0000 −0.758610 −0.379305 − 0.925272i \(-0.623837\pi\)
−0.379305 + 0.925272i \(0.623837\pi\)
\(564\) −12.0000 −0.505291
\(565\) −6.00000 −0.252422
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 24.0000 1.00613 0.503066 − 0.864248i \(-0.332205\pi\)
0.503066 + 0.864248i \(0.332205\pi\)
\(570\) 0 0
\(571\) 29.0000 1.21361 0.606806 − 0.794850i \(-0.292450\pi\)
0.606806 + 0.794850i \(0.292450\pi\)
\(572\) −2.00000 −0.0836242
\(573\) 18.0000 0.751961
\(574\) 0 0
\(575\) 6.00000 0.250217
\(576\) −8.00000 −0.333333
\(577\) −43.0000 −1.79011 −0.895057 − 0.445952i \(-0.852865\pi\)
−0.895057 + 0.445952i \(0.852865\pi\)
\(578\) 0 0
\(579\) 5.00000 0.207793
\(580\) −12.0000 −0.498273
\(581\) 0 0
\(582\) 0 0
\(583\) 12.0000 0.496989
\(584\) 0 0
\(585\) −1.00000 −0.0413449
\(586\) 0 0
\(587\) −42.0000 −1.73353 −0.866763 − 0.498721i \(-0.833803\pi\)
−0.866763 + 0.498721i \(0.833803\pi\)
\(588\) 0 0
\(589\) −35.0000 −1.44215
\(590\) 0 0
\(591\) −18.0000 −0.740421
\(592\) 20.0000 0.821995
\(593\) 36.0000 1.47834 0.739171 − 0.673517i \(-0.235217\pi\)
0.739171 + 0.673517i \(0.235217\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 24.0000 0.983078
\(597\) 8.00000 0.327418
\(598\) 0 0
\(599\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) −37.0000 −1.50926 −0.754631 − 0.656150i \(-0.772184\pi\)
−0.754631 + 0.656150i \(0.772184\pi\)
\(602\) 0 0
\(603\) 5.00000 0.203616
\(604\) −16.0000 −0.651031
\(605\) 1.00000 0.0406558
\(606\) 0 0
\(607\) 29.0000 1.17707 0.588537 − 0.808470i \(-0.299704\pi\)
0.588537 + 0.808470i \(0.299704\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −6.00000 −0.242734
\(612\) 12.0000 0.485071
\(613\) −34.0000 −1.37325 −0.686624 − 0.727013i \(-0.740908\pi\)
−0.686624 + 0.727013i \(0.740908\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 42.0000 1.69086 0.845428 − 0.534089i \(-0.179345\pi\)
0.845428 + 0.534089i \(0.179345\pi\)
\(618\) 0 0
\(619\) −37.0000 −1.48716 −0.743578 − 0.668649i \(-0.766873\pi\)
−0.743578 + 0.668649i \(0.766873\pi\)
\(620\) −10.0000 −0.401610
\(621\) 6.00000 0.240772
\(622\) 0 0
\(623\) 0 0
\(624\) −4.00000 −0.160128
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 7.00000 0.279553
\(628\) 20.0000 0.798087
\(629\) −30.0000 −1.19618
\(630\) 0 0
\(631\) −16.0000 −0.636950 −0.318475 − 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) 0 0
\(633\) 20.0000 0.794929
\(634\) 0 0
\(635\) −13.0000 −0.515889
\(636\) 24.0000 0.951662
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 18.0000 0.710957 0.355479 − 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) 0 0
\(643\) −37.0000 −1.45914 −0.729569 − 0.683907i \(-0.760279\pi\)
−0.729569 + 0.683907i \(0.760279\pi\)
\(644\) 0 0
\(645\) −1.00000 −0.0393750
\(646\) 0 0
\(647\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) 6.00000 0.235521
\(650\) 0 0
\(651\) 0 0
\(652\) 8.00000 0.313304
\(653\) 36.0000 1.40879 0.704394 − 0.709809i \(-0.251219\pi\)
0.704394 + 0.709809i \(0.251219\pi\)
\(654\) 0 0
\(655\) −18.0000 −0.703318
\(656\) 0 0
\(657\) −13.0000 −0.507178
\(658\) 0 0
\(659\) 48.0000 1.86981 0.934907 − 0.354892i \(-0.115482\pi\)
0.934907 + 0.354892i \(0.115482\pi\)
\(660\) 2.00000 0.0778499
\(661\) 35.0000 1.36134 0.680671 − 0.732589i \(-0.261688\pi\)
0.680671 + 0.732589i \(0.261688\pi\)
\(662\) 0 0
\(663\) 6.00000 0.233021
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 36.0000 1.39393
\(668\) 24.0000 0.928588
\(669\) 20.0000 0.773245
\(670\) 0 0
\(671\) −2.00000 −0.0772091
\(672\) 0 0
\(673\) −7.00000 −0.269830 −0.134915 − 0.990857i \(-0.543076\pi\)
−0.134915 + 0.990857i \(0.543076\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 24.0000 0.923077
\(677\) −18.0000 −0.691796 −0.345898 − 0.938272i \(-0.612426\pi\)
−0.345898 + 0.938272i \(0.612426\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 12.0000 0.459841
\(682\) 0 0
\(683\) 18.0000 0.688751 0.344375 − 0.938832i \(-0.388091\pi\)
0.344375 + 0.938832i \(0.388091\pi\)
\(684\) 14.0000 0.535303
\(685\) 18.0000 0.687745
\(686\) 0 0
\(687\) −19.0000 −0.724895
\(688\) −4.00000 −0.152499
\(689\) 12.0000 0.457164
\(690\) 0 0
\(691\) −37.0000 −1.40755 −0.703773 − 0.710425i \(-0.748503\pi\)
−0.703773 + 0.710425i \(0.748503\pi\)
\(692\) 24.0000 0.912343
\(693\) 0 0
\(694\) 0 0
\(695\) −7.00000 −0.265525
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 18.0000 0.680823
\(700\) 0 0
\(701\) 42.0000 1.58632 0.793159 − 0.609015i \(-0.208435\pi\)
0.793159 + 0.609015i \(0.208435\pi\)
\(702\) 0 0
\(703\) −35.0000 −1.32005
\(704\) 8.00000 0.301511
\(705\) 6.00000 0.225973
\(706\) 0 0
\(707\) 0 0
\(708\) 12.0000 0.450988
\(709\) −10.0000 −0.375558 −0.187779 − 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) 0 0
\(711\) −1.00000 −0.0375029
\(712\) 0 0
\(713\) 30.0000 1.12351
\(714\) 0 0
\(715\) 1.00000 0.0373979
\(716\) −12.0000 −0.448461
\(717\) −18.0000 −0.672222
\(718\) 0 0
\(719\) −18.0000 −0.671287 −0.335643 − 0.941989i \(-0.608954\pi\)
−0.335643 + 0.941989i \(0.608954\pi\)
\(720\) 4.00000 0.149071
\(721\) 0 0
\(722\) 0 0
\(723\) 26.0000 0.966950
\(724\) 14.0000 0.520306
\(725\) 6.00000 0.222834
\(726\) 0 0
\(727\) 29.0000 1.07555 0.537775 − 0.843088i \(-0.319265\pi\)
0.537775 + 0.843088i \(0.319265\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 6.00000 0.221918
\(732\) −4.00000 −0.147844
\(733\) 11.0000 0.406294 0.203147 − 0.979148i \(-0.434883\pi\)
0.203147 + 0.979148i \(0.434883\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −5.00000 −0.184177
\(738\) 0 0
\(739\) −43.0000 −1.58178 −0.790890 − 0.611958i \(-0.790382\pi\)
−0.790890 + 0.611958i \(0.790382\pi\)
\(740\) −10.0000 −0.367607
\(741\) 7.00000 0.257151
\(742\) 0 0
\(743\) 12.0000 0.440237 0.220119 − 0.975473i \(-0.429356\pi\)
0.220119 + 0.975473i \(0.429356\pi\)
\(744\) 0 0
\(745\) −12.0000 −0.439646
\(746\) 0 0
\(747\) 0 0
\(748\) −12.0000 −0.438763
\(749\) 0 0
\(750\) 0 0
\(751\) −19.0000 −0.693320 −0.346660 − 0.937991i \(-0.612684\pi\)
−0.346660 + 0.937991i \(0.612684\pi\)
\(752\) 24.0000 0.875190
\(753\) −30.0000 −1.09326
\(754\) 0 0
\(755\) 8.00000 0.291150
\(756\) 0 0
\(757\) −46.0000 −1.67190 −0.835949 − 0.548807i \(-0.815082\pi\)
−0.835949 + 0.548807i \(0.815082\pi\)
\(758\) 0 0
\(759\) −6.00000 −0.217786
\(760\) 0 0
\(761\) 18.0000 0.652499 0.326250 − 0.945284i \(-0.394215\pi\)
0.326250 + 0.945284i \(0.394215\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −36.0000 −1.30243
\(765\) −6.00000 −0.216930
\(766\) 0 0
\(767\) 6.00000 0.216647
\(768\) 16.0000 0.577350
\(769\) 23.0000 0.829401 0.414701 − 0.909958i \(-0.363886\pi\)
0.414701 + 0.909958i \(0.363886\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −10.0000 −0.359908
\(773\) −24.0000 −0.863220 −0.431610 − 0.902060i \(-0.642054\pi\)
−0.431610 + 0.902060i \(0.642054\pi\)
\(774\) 0 0
\(775\) 5.00000 0.179605
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 2.00000 0.0716115
\(781\) 0 0
\(782\) 0 0
\(783\) 6.00000 0.214423
\(784\) 0 0
\(785\) −10.0000 −0.356915
\(786\) 0 0
\(787\) −4.00000 −0.142585 −0.0712923 − 0.997455i \(-0.522712\pi\)
−0.0712923 + 0.997455i \(0.522712\pi\)
\(788\) 36.0000 1.28245
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −2.00000 −0.0710221
\(794\) 0 0
\(795\) −12.0000 −0.425596
\(796\) −16.0000 −0.567105
\(797\) −30.0000 −1.06265 −0.531327 − 0.847167i \(-0.678307\pi\)
−0.531327 + 0.847167i \(0.678307\pi\)
\(798\) 0 0
\(799\) −36.0000 −1.27359
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) 0 0
\(803\) 13.0000 0.458760
\(804\) −10.0000 −0.352673
\(805\) 0 0
\(806\) 0 0
\(807\) 6.00000 0.211210
\(808\) 0 0
\(809\) 18.0000 0.632846 0.316423 − 0.948618i \(-0.397518\pi\)
0.316423 + 0.948618i \(0.397518\pi\)
\(810\) 0 0
\(811\) 44.0000 1.54505 0.772524 − 0.634985i \(-0.218994\pi\)
0.772524 + 0.634985i \(0.218994\pi\)
\(812\) 0 0
\(813\) −16.0000 −0.561144
\(814\) 0 0
\(815\) −4.00000 −0.140114
\(816\) −24.0000 −0.840168
\(817\) 7.00000 0.244899
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 36.0000 1.25641 0.628204 − 0.778048i \(-0.283790\pi\)
0.628204 + 0.778048i \(0.283790\pi\)
\(822\) 0 0
\(823\) 56.0000 1.95204 0.976019 − 0.217687i \(-0.0698512\pi\)
0.976019 + 0.217687i \(0.0698512\pi\)
\(824\) 0 0
\(825\) −1.00000 −0.0348155
\(826\) 0 0
\(827\) 42.0000 1.46048 0.730242 − 0.683189i \(-0.239408\pi\)
0.730242 + 0.683189i \(0.239408\pi\)
\(828\) −12.0000 −0.417029
\(829\) −43.0000 −1.49345 −0.746726 − 0.665132i \(-0.768375\pi\)
−0.746726 + 0.665132i \(0.768375\pi\)
\(830\) 0 0
\(831\) −1.00000 −0.0346896
\(832\) 8.00000 0.277350
\(833\) 0 0
\(834\) 0 0
\(835\) −12.0000 −0.415277
\(836\) −14.0000 −0.484200
\(837\) 5.00000 0.172825
\(838\) 0 0
\(839\) −18.0000 −0.621429 −0.310715 − 0.950503i \(-0.600568\pi\)
−0.310715 + 0.950503i \(0.600568\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) 0 0
\(844\) −40.0000 −1.37686
\(845\) −12.0000 −0.412813
\(846\) 0 0
\(847\) 0 0
\(848\) −48.0000 −1.64833
\(849\) −13.0000 −0.446159
\(850\) 0 0
\(851\) 30.0000 1.02839
\(852\) 0 0
\(853\) −55.0000 −1.88316 −0.941582 − 0.336784i \(-0.890661\pi\)
−0.941582 + 0.336784i \(0.890661\pi\)
\(854\) 0 0
\(855\) −7.00000 −0.239395
\(856\) 0 0
\(857\) 42.0000 1.43469 0.717346 − 0.696717i \(-0.245357\pi\)
0.717346 + 0.696717i \(0.245357\pi\)
\(858\) 0 0
\(859\) −4.00000 −0.136478 −0.0682391 − 0.997669i \(-0.521738\pi\)
−0.0682391 + 0.997669i \(0.521738\pi\)
\(860\) 2.00000 0.0681994
\(861\) 0 0
\(862\) 0 0
\(863\) 6.00000 0.204242 0.102121 − 0.994772i \(-0.467437\pi\)
0.102121 + 0.994772i \(0.467437\pi\)
\(864\) 0 0
\(865\) −12.0000 −0.408012
\(866\) 0 0
\(867\) 19.0000 0.645274
\(868\) 0 0
\(869\) 1.00000 0.0339227
\(870\) 0 0
\(871\) −5.00000 −0.169419
\(872\) 0 0
\(873\) 14.0000 0.473828
\(874\) 0 0
\(875\) 0 0
\(876\) 26.0000 0.878459
\(877\) −46.0000 −1.55331 −0.776655 − 0.629926i \(-0.783085\pi\)
−0.776655 + 0.629926i \(0.783085\pi\)
\(878\) 0 0
\(879\) −12.0000 −0.404750
\(880\) −4.00000 −0.134840
\(881\) 24.0000 0.808581 0.404290 − 0.914631i \(-0.367519\pi\)
0.404290 + 0.914631i \(0.367519\pi\)
\(882\) 0 0
\(883\) −19.0000 −0.639401 −0.319700 − 0.947519i \(-0.603582\pi\)
−0.319700 + 0.947519i \(0.603582\pi\)
\(884\) −12.0000 −0.403604
\(885\) −6.00000 −0.201688
\(886\) 0 0
\(887\) −12.0000 −0.402921 −0.201460 − 0.979497i \(-0.564569\pi\)
−0.201460 + 0.979497i \(0.564569\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) −40.0000 −1.33930
\(893\) −42.0000 −1.40548
\(894\) 0 0
\(895\) 6.00000 0.200558
\(896\) 0 0
\(897\) −6.00000 −0.200334
\(898\) 0 0
\(899\) 30.0000 1.00056
\(900\) −2.00000 −0.0666667
\(901\) 72.0000 2.39867
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −7.00000 −0.232688
\(906\) 0 0
\(907\) −43.0000 −1.42779 −0.713896 − 0.700252i \(-0.753071\pi\)
−0.713896 + 0.700252i \(0.753071\pi\)
\(908\) −24.0000 −0.796468
\(909\) 0 0
\(910\) 0 0
\(911\) 30.0000 0.993944 0.496972 − 0.867766i \(-0.334445\pi\)
0.496972 + 0.867766i \(0.334445\pi\)
\(912\) −28.0000 −0.927173
\(913\) 0 0
\(914\) 0 0
\(915\) 2.00000 0.0661180
\(916\) 38.0000 1.25556
\(917\) 0 0
\(918\) 0 0
\(919\) 11.0000 0.362857 0.181428 − 0.983404i \(-0.441928\pi\)
0.181428 + 0.983404i \(0.441928\pi\)
\(920\) 0 0
\(921\) −7.00000 −0.230658
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 5.00000 0.164399
\(926\) 0 0
\(927\) −13.0000 −0.426976
\(928\) 0 0
\(929\) −48.0000 −1.57483 −0.787414 − 0.616424i \(-0.788581\pi\)
−0.787414 + 0.616424i \(0.788581\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −36.0000 −1.17922
\(933\) 0 0
\(934\) 0 0
\(935\) 6.00000 0.196221
\(936\) 0 0
\(937\) 29.0000 0.947389 0.473694 − 0.880689i \(-0.342920\pi\)
0.473694 + 0.880689i \(0.342920\pi\)
\(938\) 0 0
\(939\) −19.0000 −0.620042
\(940\) −12.0000 −0.391397
\(941\) 18.0000 0.586783 0.293392 − 0.955992i \(-0.405216\pi\)
0.293392 + 0.955992i \(0.405216\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) −24.0000 −0.781133
\(945\) 0 0
\(946\) 0 0
\(947\) 24.0000 0.779895 0.389948 − 0.920837i \(-0.372493\pi\)
0.389948 + 0.920837i \(0.372493\pi\)
\(948\) 2.00000 0.0649570
\(949\) 13.0000 0.421998
\(950\) 0 0
\(951\) −6.00000 −0.194563
\(952\) 0 0
\(953\) 36.0000 1.16615 0.583077 − 0.812417i \(-0.301849\pi\)
0.583077 + 0.812417i \(0.301849\pi\)
\(954\) 0 0
\(955\) 18.0000 0.582466
\(956\) 36.0000 1.16432
\(957\) −6.00000 −0.193952
\(958\) 0 0
\(959\) 0 0
\(960\) −8.00000 −0.258199
\(961\) −6.00000 −0.193548
\(962\) 0 0
\(963\) −6.00000 −0.193347
\(964\) −52.0000 −1.67481
\(965\) 5.00000 0.160956
\(966\) 0 0
\(967\) 17.0000 0.546683 0.273342 − 0.961917i \(-0.411871\pi\)
0.273342 + 0.961917i \(0.411871\pi\)
\(968\) 0 0
\(969\) 42.0000 1.34923
\(970\) 0 0
\(971\) 48.0000 1.54039 0.770197 − 0.637806i \(-0.220158\pi\)
0.770197 + 0.637806i \(0.220158\pi\)
\(972\) −2.00000 −0.0641500
\(973\) 0 0
\(974\) 0 0
\(975\) −1.00000 −0.0320256
\(976\) 8.00000 0.256074
\(977\) 48.0000 1.53566 0.767828 − 0.640656i \(-0.221338\pi\)
0.767828 + 0.640656i \(0.221338\pi\)
\(978\) 0 0
\(979\) 6.00000 0.191761
\(980\) 0 0
\(981\) −19.0000 −0.606623
\(982\) 0 0
\(983\) 6.00000 0.191370 0.0956851 − 0.995412i \(-0.469496\pi\)
0.0956851 + 0.995412i \(0.469496\pi\)
\(984\) 0 0
\(985\) −18.0000 −0.573528
\(986\) 0 0
\(987\) 0 0
\(988\) −14.0000 −0.445399
\(989\) −6.00000 −0.190789
\(990\) 0 0
\(991\) −43.0000 −1.36594 −0.682970 − 0.730446i \(-0.739312\pi\)
−0.682970 + 0.730446i \(0.739312\pi\)
\(992\) 0 0
\(993\) −25.0000 −0.793351
\(994\) 0 0
\(995\) 8.00000 0.253617
\(996\) 0 0
\(997\) 17.0000 0.538395 0.269198 − 0.963085i \(-0.413241\pi\)
0.269198 + 0.963085i \(0.413241\pi\)
\(998\) 0 0
\(999\) 5.00000 0.158193
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 8085.2.a.n.1.1 1
7.2 even 3 1155.2.q.c.991.1 yes 2
7.4 even 3 1155.2.q.c.331.1 ✓ 2
7.6 odd 2 8085.2.a.j.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1155.2.q.c.331.1 ✓ 2 7.4 even 3
1155.2.q.c.991.1 yes 2 7.2 even 3
8085.2.a.j.1.1 1 7.6 odd 2
8085.2.a.n.1.1 1 1.1 even 1 trivial