Properties

Label 8085.2.a.q.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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -2.00000 q^{4} +1.00000 q^{5} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -2.00000 q^{4} +1.00000 q^{5} +1.00000 q^{9} +1.00000 q^{11} -2.00000 q^{12} +1.00000 q^{15} +4.00000 q^{16} -3.00000 q^{17} +3.00000 q^{19} -2.00000 q^{20} -1.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} -7.00000 q^{29} -6.00000 q^{31} +1.00000 q^{33} -2.00000 q^{36} -8.00000 q^{37} +2.00000 q^{41} -5.00000 q^{43} -2.00000 q^{44} +1.00000 q^{45} +6.00000 q^{47} +4.00000 q^{48} -3.00000 q^{51} -3.00000 q^{53} +1.00000 q^{55} +3.00000 q^{57} +5.00000 q^{59} -2.00000 q^{60} +7.00000 q^{61} -8.00000 q^{64} +6.00000 q^{67} +6.00000 q^{68} -1.00000 q^{69} -10.0000 q^{71} -12.0000 q^{73} +1.00000 q^{75} -6.00000 q^{76} -16.0000 q^{79} +4.00000 q^{80} +1.00000 q^{81} +3.00000 q^{83} -3.00000 q^{85} -7.00000 q^{87} -3.00000 q^{89} +2.00000 q^{92} -6.00000 q^{93} +3.00000 q^{95} +1.00000 q^{97} +1.00000 q^{99} +O(q^{100})\)

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
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 4.00000 1.00000
\(17\) −3.00000 −0.727607 −0.363803 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 0 0
\(19\) 3.00000 0.688247 0.344124 0.938924i \(-0.388176\pi\)
0.344124 + 0.938924i \(0.388176\pi\)
\(20\) −2.00000 −0.447214
\(21\) 0 0
\(22\) 0 0
\(23\) −1.00000 −0.208514 −0.104257 0.994550i \(-0.533247\pi\)
−0.104257 + 0.994550i \(0.533247\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −7.00000 −1.29987 −0.649934 0.759991i \(-0.725203\pi\)
−0.649934 + 0.759991i \(0.725203\pi\)
\(30\) 0 0
\(31\) −6.00000 −1.07763 −0.538816 0.842424i \(-0.681128\pi\)
−0.538816 + 0.842424i \(0.681128\pi\)
\(32\) 0 0
\(33\) 1.00000 0.174078
\(34\) 0 0
\(35\) 0 0
\(36\) −2.00000 −0.333333
\(37\) −8.00000 −1.31519 −0.657596 0.753371i \(-0.728427\pi\)
−0.657596 + 0.753371i \(0.728427\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) −5.00000 −0.762493 −0.381246 0.924473i \(-0.624505\pi\)
−0.381246 + 0.924473i \(0.624505\pi\)
\(44\) −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
\(51\) −3.00000 −0.420084
\(52\) 0 0
\(53\) −3.00000 −0.412082 −0.206041 0.978543i \(-0.566058\pi\)
−0.206041 + 0.978543i \(0.566058\pi\)
\(54\) 0 0
\(55\) 1.00000 0.134840
\(56\) 0 0
\(57\) 3.00000 0.397360
\(58\) 0 0
\(59\) 5.00000 0.650945 0.325472 0.945552i \(-0.394477\pi\)
0.325472 + 0.945552i \(0.394477\pi\)
\(60\) −2.00000 −0.258199
\(61\) 7.00000 0.896258 0.448129 0.893969i \(-0.352090\pi\)
0.448129 + 0.893969i \(0.352090\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 6.00000 0.733017 0.366508 0.930415i \(-0.380553\pi\)
0.366508 + 0.930415i \(0.380553\pi\)
\(68\) 6.00000 0.727607
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −10.0000 −1.18678 −0.593391 0.804914i \(-0.702211\pi\)
−0.593391 + 0.804914i \(0.702211\pi\)
\(72\) 0 0
\(73\) −12.0000 −1.40449 −0.702247 0.711934i \(-0.747820\pi\)
−0.702247 + 0.711934i \(0.747820\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) −6.00000 −0.688247
\(77\) 0 0
\(78\) 0 0
\(79\) −16.0000 −1.80014 −0.900070 0.435745i \(-0.856485\pi\)
−0.900070 + 0.435745i \(0.856485\pi\)
\(80\) 4.00000 0.447214
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 3.00000 0.329293 0.164646 0.986353i \(-0.447352\pi\)
0.164646 + 0.986353i \(0.447352\pi\)
\(84\) 0 0
\(85\) −3.00000 −0.325396
\(86\) 0 0
\(87\) −7.00000 −0.750479
\(88\) 0 0
\(89\) −3.00000 −0.317999 −0.159000 0.987279i \(-0.550827\pi\)
−0.159000 + 0.987279i \(0.550827\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 2.00000 0.208514
\(93\) −6.00000 −0.622171
\(94\) 0 0
\(95\) 3.00000 0.307794
\(96\) 0 0
\(97\) 1.00000 0.101535 0.0507673 0.998711i \(-0.483833\pi\)
0.0507673 + 0.998711i \(0.483833\pi\)
\(98\) 0 0
\(99\) 1.00000 0.100504
\(100\) −2.00000 −0.200000
\(101\) 14.0000 1.39305 0.696526 0.717532i \(-0.254728\pi\)
0.696526 + 0.717532i \(0.254728\pi\)
\(102\) 0 0
\(103\) 5.00000 0.492665 0.246332 0.969185i \(-0.420775\pi\)
0.246332 + 0.969185i \(0.420775\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 6.00000 0.580042 0.290021 0.957020i \(-0.406338\pi\)
0.290021 + 0.957020i \(0.406338\pi\)
\(108\) −2.00000 −0.192450
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 0 0
\(111\) −8.00000 −0.759326
\(112\) 0 0
\(113\) −19.0000 −1.78737 −0.893685 0.448695i \(-0.851889\pi\)
−0.893685 + 0.448695i \(0.851889\pi\)
\(114\) 0 0
\(115\) −1.00000 −0.0932505
\(116\) 14.0000 1.29987
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 2.00000 0.180334
\(124\) 12.0000 1.07763
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 19.0000 1.68598 0.842989 0.537931i \(-0.180794\pi\)
0.842989 + 0.537931i \(0.180794\pi\)
\(128\) 0 0
\(129\) −5.00000 −0.440225
\(130\) 0 0
\(131\) −4.00000 −0.349482 −0.174741 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) −2.00000 −0.174078
\(133\) 0 0
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) −2.00000 −0.170872 −0.0854358 0.996344i \(-0.527228\pi\)
−0.0854358 + 0.996344i \(0.527228\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) 6.00000 0.505291
\(142\) 0 0
\(143\) 0 0
\(144\) 4.00000 0.333333
\(145\) −7.00000 −0.581318
\(146\) 0 0
\(147\) 0 0
\(148\) 16.0000 1.31519
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) 14.0000 1.13930 0.569652 0.821886i \(-0.307078\pi\)
0.569652 + 0.821886i \(0.307078\pi\)
\(152\) 0 0
\(153\) −3.00000 −0.242536
\(154\) 0 0
\(155\) −6.00000 −0.481932
\(156\) 0 0
\(157\) 1.00000 0.0798087 0.0399043 0.999204i \(-0.487295\pi\)
0.0399043 + 0.999204i \(0.487295\pi\)
\(158\) 0 0
\(159\) −3.00000 −0.237915
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 12.0000 0.939913 0.469956 0.882690i \(-0.344270\pi\)
0.469956 + 0.882690i \(0.344270\pi\)
\(164\) −4.00000 −0.312348
\(165\) 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
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) 3.00000 0.229416
\(172\) 10.0000 0.762493
\(173\) 14.0000 1.06440 0.532200 0.846619i \(-0.321365\pi\)
0.532200 + 0.846619i \(0.321365\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 4.00000 0.301511
\(177\) 5.00000 0.375823
\(178\) 0 0
\(179\) −18.0000 −1.34538 −0.672692 0.739923i \(-0.734862\pi\)
−0.672692 + 0.739923i \(0.734862\pi\)
\(180\) −2.00000 −0.149071
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 0 0
\(183\) 7.00000 0.517455
\(184\) 0 0
\(185\) −8.00000 −0.588172
\(186\) 0 0
\(187\) −3.00000 −0.219382
\(188\) −12.0000 −0.875190
\(189\) 0 0
\(190\) 0 0
\(191\) −16.0000 −1.15772 −0.578860 0.815427i \(-0.696502\pi\)
−0.578860 + 0.815427i \(0.696502\pi\)
\(192\) −8.00000 −0.577350
\(193\) 6.00000 0.431889 0.215945 0.976406i \(-0.430717\pi\)
0.215945 + 0.976406i \(0.430717\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) −24.0000 −1.70131 −0.850657 0.525720i \(-0.823796\pi\)
−0.850657 + 0.525720i \(0.823796\pi\)
\(200\) 0 0
\(201\) 6.00000 0.423207
\(202\) 0 0
\(203\) 0 0
\(204\) 6.00000 0.420084
\(205\) 2.00000 0.139686
\(206\) 0 0
\(207\) −1.00000 −0.0695048
\(208\) 0 0
\(209\) 3.00000 0.207514
\(210\) 0 0
\(211\) −18.0000 −1.23917 −0.619586 0.784929i \(-0.712699\pi\)
−0.619586 + 0.784929i \(0.712699\pi\)
\(212\) 6.00000 0.412082
\(213\) −10.0000 −0.685189
\(214\) 0 0
\(215\) −5.00000 −0.340997
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −12.0000 −0.810885
\(220\) −2.00000 −0.134840
\(221\) 0 0
\(222\) 0 0
\(223\) −1.00000 −0.0669650 −0.0334825 0.999439i \(-0.510660\pi\)
−0.0334825 + 0.999439i \(0.510660\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 3.00000 0.199117 0.0995585 0.995032i \(-0.468257\pi\)
0.0995585 + 0.995032i \(0.468257\pi\)
\(228\) −6.00000 −0.397360
\(229\) 14.0000 0.925146 0.462573 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −16.0000 −1.04819 −0.524097 0.851658i \(-0.675597\pi\)
−0.524097 + 0.851658i \(0.675597\pi\)
\(234\) 0 0
\(235\) 6.00000 0.391397
\(236\) −10.0000 −0.650945
\(237\) −16.0000 −1.03931
\(238\) 0 0
\(239\) −21.0000 −1.35838 −0.679189 0.733964i \(-0.737668\pi\)
−0.679189 + 0.733964i \(0.737668\pi\)
\(240\) 4.00000 0.258199
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) −14.0000 −0.896258
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 3.00000 0.190117
\(250\) 0 0
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 0 0
\(253\) −1.00000 −0.0628695
\(254\) 0 0
\(255\) −3.00000 −0.187867
\(256\) 16.0000 1.00000
\(257\) −20.0000 −1.24757 −0.623783 0.781598i \(-0.714405\pi\)
−0.623783 + 0.781598i \(0.714405\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −7.00000 −0.433289
\(262\) 0 0
\(263\) 16.0000 0.986602 0.493301 0.869859i \(-0.335790\pi\)
0.493301 + 0.869859i \(0.335790\pi\)
\(264\) 0 0
\(265\) −3.00000 −0.184289
\(266\) 0 0
\(267\) −3.00000 −0.183597
\(268\) −12.0000 −0.733017
\(269\) 1.00000 0.0609711 0.0304855 0.999535i \(-0.490295\pi\)
0.0304855 + 0.999535i \(0.490295\pi\)
\(270\) 0 0
\(271\) 1.00000 0.0607457 0.0303728 0.999539i \(-0.490331\pi\)
0.0303728 + 0.999539i \(0.490331\pi\)
\(272\) −12.0000 −0.727607
\(273\) 0 0
\(274\) 0 0
\(275\) 1.00000 0.0603023
\(276\) 2.00000 0.120386
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) 0 0
\(279\) −6.00000 −0.359211
\(280\) 0 0
\(281\) −22.0000 −1.31241 −0.656205 0.754583i \(-0.727839\pi\)
−0.656205 + 0.754583i \(0.727839\pi\)
\(282\) 0 0
\(283\) 8.00000 0.475551 0.237775 0.971320i \(-0.423582\pi\)
0.237775 + 0.971320i \(0.423582\pi\)
\(284\) 20.0000 1.18678
\(285\) 3.00000 0.177705
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) 1.00000 0.0586210
\(292\) 24.0000 1.40449
\(293\) −19.0000 −1.10999 −0.554996 0.831853i \(-0.687280\pi\)
−0.554996 + 0.831853i \(0.687280\pi\)
\(294\) 0 0
\(295\) 5.00000 0.291111
\(296\) 0 0
\(297\) 1.00000 0.0580259
\(298\) 0 0
\(299\) 0 0
\(300\) −2.00000 −0.115470
\(301\) 0 0
\(302\) 0 0
\(303\) 14.0000 0.804279
\(304\) 12.0000 0.688247
\(305\) 7.00000 0.400819
\(306\) 0 0
\(307\) −26.0000 −1.48390 −0.741949 0.670456i \(-0.766098\pi\)
−0.741949 + 0.670456i \(0.766098\pi\)
\(308\) 0 0
\(309\) 5.00000 0.284440
\(310\) 0 0
\(311\) 32.0000 1.81455 0.907277 0.420534i \(-0.138157\pi\)
0.907277 + 0.420534i \(0.138157\pi\)
\(312\) 0 0
\(313\) −21.0000 −1.18699 −0.593495 0.804838i \(-0.702252\pi\)
−0.593495 + 0.804838i \(0.702252\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 32.0000 1.80014
\(317\) 2.00000 0.112331 0.0561656 0.998421i \(-0.482113\pi\)
0.0561656 + 0.998421i \(0.482113\pi\)
\(318\) 0 0
\(319\) −7.00000 −0.391925
\(320\) −8.00000 −0.447214
\(321\) 6.00000 0.334887
\(322\) 0 0
\(323\) −9.00000 −0.500773
\(324\) −2.00000 −0.111111
\(325\) 0 0
\(326\) 0 0
\(327\) −2.00000 −0.110600
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 17.0000 0.934405 0.467202 0.884150i \(-0.345262\pi\)
0.467202 + 0.884150i \(0.345262\pi\)
\(332\) −6.00000 −0.329293
\(333\) −8.00000 −0.438397
\(334\) 0 0
\(335\) 6.00000 0.327815
\(336\) 0 0
\(337\) −1.00000 −0.0544735 −0.0272367 0.999629i \(-0.508671\pi\)
−0.0272367 + 0.999629i \(0.508671\pi\)
\(338\) 0 0
\(339\) −19.0000 −1.03194
\(340\) 6.00000 0.325396
\(341\) −6.00000 −0.324918
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −1.00000 −0.0538382
\(346\) 0 0
\(347\) −8.00000 −0.429463 −0.214731 0.976673i \(-0.568888\pi\)
−0.214731 + 0.976673i \(0.568888\pi\)
\(348\) 14.0000 0.750479
\(349\) 19.0000 1.01705 0.508523 0.861048i \(-0.330192\pi\)
0.508523 + 0.861048i \(0.330192\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 10.0000 0.532246 0.266123 0.963939i \(-0.414257\pi\)
0.266123 + 0.963939i \(0.414257\pi\)
\(354\) 0 0
\(355\) −10.0000 −0.530745
\(356\) 6.00000 0.317999
\(357\) 0 0
\(358\) 0 0
\(359\) −21.0000 −1.10834 −0.554169 0.832404i \(-0.686964\pi\)
−0.554169 + 0.832404i \(0.686964\pi\)
\(360\) 0 0
\(361\) −10.0000 −0.526316
\(362\) 0 0
\(363\) 1.00000 0.0524864
\(364\) 0 0
\(365\) −12.0000 −0.628109
\(366\) 0 0
\(367\) −7.00000 −0.365397 −0.182699 0.983169i \(-0.558483\pi\)
−0.182699 + 0.983169i \(0.558483\pi\)
\(368\) −4.00000 −0.208514
\(369\) 2.00000 0.104116
\(370\) 0 0
\(371\) 0 0
\(372\) 12.0000 0.622171
\(373\) 11.0000 0.569558 0.284779 0.958593i \(-0.408080\pi\)
0.284779 + 0.958593i \(0.408080\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 5.00000 0.256833 0.128416 0.991720i \(-0.459011\pi\)
0.128416 + 0.991720i \(0.459011\pi\)
\(380\) −6.00000 −0.307794
\(381\) 19.0000 0.973399
\(382\) 0 0
\(383\) −32.0000 −1.63512 −0.817562 0.575841i \(-0.804675\pi\)
−0.817562 + 0.575841i \(0.804675\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −5.00000 −0.254164
\(388\) −2.00000 −0.101535
\(389\) 12.0000 0.608424 0.304212 0.952604i \(-0.401607\pi\)
0.304212 + 0.952604i \(0.401607\pi\)
\(390\) 0 0
\(391\) 3.00000 0.151717
\(392\) 0 0
\(393\) −4.00000 −0.201773
\(394\) 0 0
\(395\) −16.0000 −0.805047
\(396\) −2.00000 −0.100504
\(397\) 10.0000 0.501886 0.250943 0.968002i \(-0.419259\pi\)
0.250943 + 0.968002i \(0.419259\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 4.00000 0.200000
\(401\) −10.0000 −0.499376 −0.249688 0.968326i \(-0.580328\pi\)
−0.249688 + 0.968326i \(0.580328\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −28.0000 −1.39305
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) −8.00000 −0.396545
\(408\) 0 0
\(409\) 6.00000 0.296681 0.148340 0.988936i \(-0.452607\pi\)
0.148340 + 0.988936i \(0.452607\pi\)
\(410\) 0 0
\(411\) −2.00000 −0.0986527
\(412\) −10.0000 −0.492665
\(413\) 0 0
\(414\) 0 0
\(415\) 3.00000 0.147264
\(416\) 0 0
\(417\) −4.00000 −0.195881
\(418\) 0 0
\(419\) −33.0000 −1.61216 −0.806078 0.591810i \(-0.798414\pi\)
−0.806078 + 0.591810i \(0.798414\pi\)
\(420\) 0 0
\(421\) −23.0000 −1.12095 −0.560476 0.828171i \(-0.689382\pi\)
−0.560476 + 0.828171i \(0.689382\pi\)
\(422\) 0 0
\(423\) 6.00000 0.291730
\(424\) 0 0
\(425\) −3.00000 −0.145521
\(426\) 0 0
\(427\) 0 0
\(428\) −12.0000 −0.580042
\(429\) 0 0
\(430\) 0 0
\(431\) 36.0000 1.73406 0.867029 0.498257i \(-0.166026\pi\)
0.867029 + 0.498257i \(0.166026\pi\)
\(432\) 4.00000 0.192450
\(433\) −22.0000 −1.05725 −0.528626 0.848855i \(-0.677293\pi\)
−0.528626 + 0.848855i \(0.677293\pi\)
\(434\) 0 0
\(435\) −7.00000 −0.335624
\(436\) 4.00000 0.191565
\(437\) −3.00000 −0.143509
\(438\) 0 0
\(439\) −1.00000 −0.0477274 −0.0238637 0.999715i \(-0.507597\pi\)
−0.0238637 + 0.999715i \(0.507597\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 32.0000 1.52037 0.760183 0.649709i \(-0.225109\pi\)
0.760183 + 0.649709i \(0.225109\pi\)
\(444\) 16.0000 0.759326
\(445\) −3.00000 −0.142214
\(446\) 0 0
\(447\) −6.00000 −0.283790
\(448\) 0 0
\(449\) 26.0000 1.22702 0.613508 0.789689i \(-0.289758\pi\)
0.613508 + 0.789689i \(0.289758\pi\)
\(450\) 0 0
\(451\) 2.00000 0.0941763
\(452\) 38.0000 1.78737
\(453\) 14.0000 0.657777
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −37.0000 −1.73079 −0.865393 0.501093i \(-0.832931\pi\)
−0.865393 + 0.501093i \(0.832931\pi\)
\(458\) 0 0
\(459\) −3.00000 −0.140028
\(460\) 2.00000 0.0932505
\(461\) −6.00000 −0.279448 −0.139724 0.990190i \(-0.544622\pi\)
−0.139724 + 0.990190i \(0.544622\pi\)
\(462\) 0 0
\(463\) −18.0000 −0.836531 −0.418265 0.908325i \(-0.637362\pi\)
−0.418265 + 0.908325i \(0.637362\pi\)
\(464\) −28.0000 −1.29987
\(465\) −6.00000 −0.278243
\(466\) 0 0
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 1.00000 0.0460776
\(472\) 0 0
\(473\) −5.00000 −0.229900
\(474\) 0 0
\(475\) 3.00000 0.137649
\(476\) 0 0
\(477\) −3.00000 −0.137361
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) −2.00000 −0.0909091
\(485\) 1.00000 0.0454077
\(486\) 0 0
\(487\) −22.0000 −0.996915 −0.498458 0.866914i \(-0.666100\pi\)
−0.498458 + 0.866914i \(0.666100\pi\)
\(488\) 0 0
\(489\) 12.0000 0.542659
\(490\) 0 0
\(491\) −15.0000 −0.676941 −0.338470 0.940977i \(-0.609909\pi\)
−0.338470 + 0.940977i \(0.609909\pi\)
\(492\) −4.00000 −0.180334
\(493\) 21.0000 0.945792
\(494\) 0 0
\(495\) 1.00000 0.0449467
\(496\) −24.0000 −1.07763
\(497\) 0 0
\(498\) 0 0
\(499\) −21.0000 −0.940089 −0.470045 0.882643i \(-0.655762\pi\)
−0.470045 + 0.882643i \(0.655762\pi\)
\(500\) −2.00000 −0.0894427
\(501\) −12.0000 −0.536120
\(502\) 0 0
\(503\) 13.0000 0.579641 0.289821 0.957081i \(-0.406404\pi\)
0.289821 + 0.957081i \(0.406404\pi\)
\(504\) 0 0
\(505\) 14.0000 0.622992
\(506\) 0 0
\(507\) −13.0000 −0.577350
\(508\) −38.0000 −1.68598
\(509\) 11.0000 0.487566 0.243783 0.969830i \(-0.421611\pi\)
0.243783 + 0.969830i \(0.421611\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 3.00000 0.132453
\(514\) 0 0
\(515\) 5.00000 0.220326
\(516\) 10.0000 0.440225
\(517\) 6.00000 0.263880
\(518\) 0 0
\(519\) 14.0000 0.614532
\(520\) 0 0
\(521\) 27.0000 1.18289 0.591446 0.806345i \(-0.298557\pi\)
0.591446 + 0.806345i \(0.298557\pi\)
\(522\) 0 0
\(523\) 20.0000 0.874539 0.437269 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) 8.00000 0.349482
\(525\) 0 0
\(526\) 0 0
\(527\) 18.0000 0.784092
\(528\) 4.00000 0.174078
\(529\) −22.0000 −0.956522
\(530\) 0 0
\(531\) 5.00000 0.216982
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 6.00000 0.259403
\(536\) 0 0
\(537\) −18.0000 −0.776757
\(538\) 0 0
\(539\) 0 0
\(540\) −2.00000 −0.0860663
\(541\) −14.0000 −0.601907 −0.300954 0.953639i \(-0.597305\pi\)
−0.300954 + 0.953639i \(0.597305\pi\)
\(542\) 0 0
\(543\) −10.0000 −0.429141
\(544\) 0 0
\(545\) −2.00000 −0.0856706
\(546\) 0 0
\(547\) 13.0000 0.555840 0.277920 0.960604i \(-0.410355\pi\)
0.277920 + 0.960604i \(0.410355\pi\)
\(548\) 4.00000 0.170872
\(549\) 7.00000 0.298753
\(550\) 0 0
\(551\) −21.0000 −0.894630
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −8.00000 −0.339581
\(556\) 8.00000 0.339276
\(557\) 6.00000 0.254228 0.127114 0.991888i \(-0.459429\pi\)
0.127114 + 0.991888i \(0.459429\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −3.00000 −0.126660
\(562\) 0 0
\(563\) 4.00000 0.168580 0.0842900 0.996441i \(-0.473138\pi\)
0.0842900 + 0.996441i \(0.473138\pi\)
\(564\) −12.0000 −0.505291
\(565\) −19.0000 −0.799336
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 13.0000 0.544988 0.272494 0.962157i \(-0.412151\pi\)
0.272494 + 0.962157i \(0.412151\pi\)
\(570\) 0 0
\(571\) 36.0000 1.50655 0.753277 0.657704i \(-0.228472\pi\)
0.753277 + 0.657704i \(0.228472\pi\)
\(572\) 0 0
\(573\) −16.0000 −0.668410
\(574\) 0 0
\(575\) −1.00000 −0.0417029
\(576\) −8.00000 −0.333333
\(577\) −18.0000 −0.749350 −0.374675 0.927156i \(-0.622246\pi\)
−0.374675 + 0.927156i \(0.622246\pi\)
\(578\) 0 0
\(579\) 6.00000 0.249351
\(580\) 14.0000 0.581318
\(581\) 0 0
\(582\) 0 0
\(583\) −3.00000 −0.124247
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −36.0000 −1.48588 −0.742940 0.669359i \(-0.766569\pi\)
−0.742940 + 0.669359i \(0.766569\pi\)
\(588\) 0 0
\(589\) −18.0000 −0.741677
\(590\) 0 0
\(591\) 0 0
\(592\) −32.0000 −1.31519
\(593\) 34.0000 1.39621 0.698106 0.715994i \(-0.254026\pi\)
0.698106 + 0.715994i \(0.254026\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 12.0000 0.491539
\(597\) −24.0000 −0.982255
\(598\) 0 0
\(599\) 30.0000 1.22577 0.612883 0.790173i \(-0.290010\pi\)
0.612883 + 0.790173i \(0.290010\pi\)
\(600\) 0 0
\(601\) −15.0000 −0.611863 −0.305931 0.952054i \(-0.598968\pi\)
−0.305931 + 0.952054i \(0.598968\pi\)
\(602\) 0 0
\(603\) 6.00000 0.244339
\(604\) −28.0000 −1.13930
\(605\) 1.00000 0.0406558
\(606\) 0 0
\(607\) 38.0000 1.54237 0.771186 0.636610i \(-0.219664\pi\)
0.771186 + 0.636610i \(0.219664\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 6.00000 0.242536
\(613\) −38.0000 −1.53481 −0.767403 0.641165i \(-0.778451\pi\)
−0.767403 + 0.641165i \(0.778451\pi\)
\(614\) 0 0
\(615\) 2.00000 0.0806478
\(616\) 0 0
\(617\) 2.00000 0.0805170 0.0402585 0.999189i \(-0.487182\pi\)
0.0402585 + 0.999189i \(0.487182\pi\)
\(618\) 0 0
\(619\) −46.0000 −1.84890 −0.924448 0.381308i \(-0.875474\pi\)
−0.924448 + 0.381308i \(0.875474\pi\)
\(620\) 12.0000 0.481932
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 3.00000 0.119808
\(628\) −2.00000 −0.0798087
\(629\) 24.0000 0.956943
\(630\) 0 0
\(631\) 29.0000 1.15447 0.577236 0.816577i \(-0.304131\pi\)
0.577236 + 0.816577i \(0.304131\pi\)
\(632\) 0 0
\(633\) −18.0000 −0.715436
\(634\) 0 0
\(635\) 19.0000 0.753992
\(636\) 6.00000 0.237915
\(637\) 0 0
\(638\) 0 0
\(639\) −10.0000 −0.395594
\(640\) 0 0
\(641\) −30.0000 −1.18493 −0.592464 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) 0 0
\(643\) −31.0000 −1.22252 −0.611260 0.791430i \(-0.709337\pi\)
−0.611260 + 0.791430i \(0.709337\pi\)
\(644\) 0 0
\(645\) −5.00000 −0.196875
\(646\) 0 0
\(647\) 4.00000 0.157256 0.0786281 0.996904i \(-0.474946\pi\)
0.0786281 + 0.996904i \(0.474946\pi\)
\(648\) 0 0
\(649\) 5.00000 0.196267
\(650\) 0 0
\(651\) 0 0
\(652\) −24.0000 −0.939913
\(653\) 39.0000 1.52619 0.763094 0.646288i \(-0.223679\pi\)
0.763094 + 0.646288i \(0.223679\pi\)
\(654\) 0 0
\(655\) −4.00000 −0.156293
\(656\) 8.00000 0.312348
\(657\) −12.0000 −0.468165
\(658\) 0 0
\(659\) −15.0000 −0.584317 −0.292159 0.956370i \(-0.594373\pi\)
−0.292159 + 0.956370i \(0.594373\pi\)
\(660\) −2.00000 −0.0778499
\(661\) 18.0000 0.700119 0.350059 0.936727i \(-0.386161\pi\)
0.350059 + 0.936727i \(0.386161\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 7.00000 0.271041
\(668\) 24.0000 0.928588
\(669\) −1.00000 −0.0386622
\(670\) 0 0
\(671\) 7.00000 0.270232
\(672\) 0 0
\(673\) −43.0000 −1.65753 −0.828764 0.559598i \(-0.810955\pi\)
−0.828764 + 0.559598i \(0.810955\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 26.0000 1.00000
\(677\) 25.0000 0.960828 0.480414 0.877042i \(-0.340486\pi\)
0.480414 + 0.877042i \(0.340486\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 3.00000 0.114960
\(682\) 0 0
\(683\) −4.00000 −0.153056 −0.0765279 0.997067i \(-0.524383\pi\)
−0.0765279 + 0.997067i \(0.524383\pi\)
\(684\) −6.00000 −0.229416
\(685\) −2.00000 −0.0764161
\(686\) 0 0
\(687\) 14.0000 0.534133
\(688\) −20.0000 −0.762493
\(689\) 0 0
\(690\) 0 0
\(691\) −2.00000 −0.0760836 −0.0380418 0.999276i \(-0.512112\pi\)
−0.0380418 + 0.999276i \(0.512112\pi\)
\(692\) −28.0000 −1.06440
\(693\) 0 0
\(694\) 0 0
\(695\) −4.00000 −0.151729
\(696\) 0 0
\(697\) −6.00000 −0.227266
\(698\) 0 0
\(699\) −16.0000 −0.605176
\(700\) 0 0
\(701\) 11.0000 0.415464 0.207732 0.978186i \(-0.433392\pi\)
0.207732 + 0.978186i \(0.433392\pi\)
\(702\) 0 0
\(703\) −24.0000 −0.905177
\(704\) −8.00000 −0.301511
\(705\) 6.00000 0.225973
\(706\) 0 0
\(707\) 0 0
\(708\) −10.0000 −0.375823
\(709\) −15.0000 −0.563337 −0.281668 0.959512i \(-0.590888\pi\)
−0.281668 + 0.959512i \(0.590888\pi\)
\(710\) 0 0
\(711\) −16.0000 −0.600047
\(712\) 0 0
\(713\) 6.00000 0.224702
\(714\) 0 0
\(715\) 0 0
\(716\) 36.0000 1.34538
\(717\) −21.0000 −0.784259
\(718\) 0 0
\(719\) −15.0000 −0.559406 −0.279703 0.960087i \(-0.590236\pi\)
−0.279703 + 0.960087i \(0.590236\pi\)
\(720\) 4.00000 0.149071
\(721\) 0 0
\(722\) 0 0
\(723\) −10.0000 −0.371904
\(724\) 20.0000 0.743294
\(725\) −7.00000 −0.259973
\(726\) 0 0
\(727\) 7.00000 0.259616 0.129808 0.991539i \(-0.458564\pi\)
0.129808 + 0.991539i \(0.458564\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 15.0000 0.554795
\(732\) −14.0000 −0.517455
\(733\) 6.00000 0.221615 0.110808 0.993842i \(-0.464656\pi\)
0.110808 + 0.993842i \(0.464656\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 6.00000 0.221013
\(738\) 0 0
\(739\) 22.0000 0.809283 0.404642 0.914475i \(-0.367396\pi\)
0.404642 + 0.914475i \(0.367396\pi\)
\(740\) 16.0000 0.588172
\(741\) 0 0
\(742\) 0 0
\(743\) 18.0000 0.660356 0.330178 0.943919i \(-0.392891\pi\)
0.330178 + 0.943919i \(0.392891\pi\)
\(744\) 0 0
\(745\) −6.00000 −0.219823
\(746\) 0 0
\(747\) 3.00000 0.109764
\(748\) 6.00000 0.219382
\(749\) 0 0
\(750\) 0 0
\(751\) 53.0000 1.93400 0.966999 0.254781i \(-0.0820034\pi\)
0.966999 + 0.254781i \(0.0820034\pi\)
\(752\) 24.0000 0.875190
\(753\) 12.0000 0.437304
\(754\) 0 0
\(755\) 14.0000 0.509512
\(756\) 0 0
\(757\) −6.00000 −0.218074 −0.109037 0.994038i \(-0.534777\pi\)
−0.109037 + 0.994038i \(0.534777\pi\)
\(758\) 0 0
\(759\) −1.00000 −0.0362977
\(760\) 0 0
\(761\) 30.0000 1.08750 0.543750 0.839248i \(-0.317004\pi\)
0.543750 + 0.839248i \(0.317004\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 32.0000 1.15772
\(765\) −3.00000 −0.108465
\(766\) 0 0
\(767\) 0 0
\(768\) 16.0000 0.577350
\(769\) −9.00000 −0.324548 −0.162274 0.986746i \(-0.551883\pi\)
−0.162274 + 0.986746i \(0.551883\pi\)
\(770\) 0 0
\(771\) −20.0000 −0.720282
\(772\) −12.0000 −0.431889
\(773\) −26.0000 −0.935155 −0.467578 0.883952i \(-0.654873\pi\)
−0.467578 + 0.883952i \(0.654873\pi\)
\(774\) 0 0
\(775\) −6.00000 −0.215526
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 6.00000 0.214972
\(780\) 0 0
\(781\) −10.0000 −0.357828
\(782\) 0 0
\(783\) −7.00000 −0.250160
\(784\) 0 0
\(785\) 1.00000 0.0356915
\(786\) 0 0
\(787\) 40.0000 1.42585 0.712923 0.701242i \(-0.247371\pi\)
0.712923 + 0.701242i \(0.247371\pi\)
\(788\) 0 0
\(789\) 16.0000 0.569615
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) −3.00000 −0.106399
\(796\) 48.0000 1.70131
\(797\) 30.0000 1.06265 0.531327 0.847167i \(-0.321693\pi\)
0.531327 + 0.847167i \(0.321693\pi\)
\(798\) 0 0
\(799\) −18.0000 −0.636794
\(800\) 0 0
\(801\) −3.00000 −0.106000
\(802\) 0 0
\(803\) −12.0000 −0.423471
\(804\) −12.0000 −0.423207
\(805\) 0 0
\(806\) 0 0
\(807\) 1.00000 0.0352017
\(808\) 0 0
\(809\) 10.0000 0.351581 0.175791 0.984428i \(-0.443752\pi\)
0.175791 + 0.984428i \(0.443752\pi\)
\(810\) 0 0
\(811\) −12.0000 −0.421377 −0.210688 0.977553i \(-0.567571\pi\)
−0.210688 + 0.977553i \(0.567571\pi\)
\(812\) 0 0
\(813\) 1.00000 0.0350715
\(814\) 0 0
\(815\) 12.0000 0.420342
\(816\) −12.0000 −0.420084
\(817\) −15.0000 −0.524784
\(818\) 0 0
\(819\) 0 0
\(820\) −4.00000 −0.139686
\(821\) 23.0000 0.802706 0.401353 0.915924i \(-0.368540\pi\)
0.401353 + 0.915924i \(0.368540\pi\)
\(822\) 0 0
\(823\) −24.0000 −0.836587 −0.418294 0.908312i \(-0.637372\pi\)
−0.418294 + 0.908312i \(0.637372\pi\)
\(824\) 0 0
\(825\) 1.00000 0.0348155
\(826\) 0 0
\(827\) −36.0000 −1.25184 −0.625921 0.779886i \(-0.715277\pi\)
−0.625921 + 0.779886i \(0.715277\pi\)
\(828\) 2.00000 0.0695048
\(829\) −38.0000 −1.31979 −0.659897 0.751356i \(-0.729400\pi\)
−0.659897 + 0.751356i \(0.729400\pi\)
\(830\) 0 0
\(831\) −10.0000 −0.346896
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −12.0000 −0.415277
\(836\) −6.00000 −0.207514
\(837\) −6.00000 −0.207390
\(838\) 0 0
\(839\) 17.0000 0.586905 0.293453 0.955974i \(-0.405196\pi\)
0.293453 + 0.955974i \(0.405196\pi\)
\(840\) 0 0
\(841\) 20.0000 0.689655
\(842\) 0 0
\(843\) −22.0000 −0.757720
\(844\) 36.0000 1.23917
\(845\) −13.0000 −0.447214
\(846\) 0 0
\(847\) 0 0
\(848\) −12.0000 −0.412082
\(849\) 8.00000 0.274559
\(850\) 0 0
\(851\) 8.00000 0.274236
\(852\) 20.0000 0.685189
\(853\) 10.0000 0.342393 0.171197 0.985237i \(-0.445237\pi\)
0.171197 + 0.985237i \(0.445237\pi\)
\(854\) 0 0
\(855\) 3.00000 0.102598
\(856\) 0 0
\(857\) −50.0000 −1.70797 −0.853984 0.520300i \(-0.825820\pi\)
−0.853984 + 0.520300i \(0.825820\pi\)
\(858\) 0 0
\(859\) −46.0000 −1.56950 −0.784750 0.619813i \(-0.787209\pi\)
−0.784750 + 0.619813i \(0.787209\pi\)
\(860\) 10.0000 0.340997
\(861\) 0 0
\(862\) 0 0
\(863\) 39.0000 1.32758 0.663788 0.747921i \(-0.268948\pi\)
0.663788 + 0.747921i \(0.268948\pi\)
\(864\) 0 0
\(865\) 14.0000 0.476014
\(866\) 0 0
\(867\) −8.00000 −0.271694
\(868\) 0 0
\(869\) −16.0000 −0.542763
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 1.00000 0.0338449
\(874\) 0 0
\(875\) 0 0
\(876\) 24.0000 0.810885
\(877\) 45.0000 1.51954 0.759771 0.650191i \(-0.225311\pi\)
0.759771 + 0.650191i \(0.225311\pi\)
\(878\) 0 0
\(879\) −19.0000 −0.640854
\(880\) 4.00000 0.134840
\(881\) −15.0000 −0.505363 −0.252681 0.967550i \(-0.581312\pi\)
−0.252681 + 0.967550i \(0.581312\pi\)
\(882\) 0 0
\(883\) 20.0000 0.673054 0.336527 0.941674i \(-0.390748\pi\)
0.336527 + 0.941674i \(0.390748\pi\)
\(884\) 0 0
\(885\) 5.00000 0.168073
\(886\) 0 0
\(887\) −41.0000 −1.37665 −0.688323 0.725405i \(-0.741653\pi\)
−0.688323 + 0.725405i \(0.741653\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 1.00000 0.0335013
\(892\) 2.00000 0.0669650
\(893\) 18.0000 0.602347
\(894\) 0 0
\(895\) −18.0000 −0.601674
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 42.0000 1.40078
\(900\) −2.00000 −0.0666667
\(901\) 9.00000 0.299833
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −10.0000 −0.332411
\(906\) 0 0
\(907\) −44.0000 −1.46100 −0.730498 0.682915i \(-0.760712\pi\)
−0.730498 + 0.682915i \(0.760712\pi\)
\(908\) −6.00000 −0.199117
\(909\) 14.0000 0.464351
\(910\) 0 0
\(911\) −4.00000 −0.132526 −0.0662630 0.997802i \(-0.521108\pi\)
−0.0662630 + 0.997802i \(0.521108\pi\)
\(912\) 12.0000 0.397360
\(913\) 3.00000 0.0992855
\(914\) 0 0
\(915\) 7.00000 0.231413
\(916\) −28.0000 −0.925146
\(917\) 0 0
\(918\) 0 0
\(919\) −14.0000 −0.461817 −0.230909 0.972975i \(-0.574170\pi\)
−0.230909 + 0.972975i \(0.574170\pi\)
\(920\) 0 0
\(921\) −26.0000 −0.856729
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −8.00000 −0.263038
\(926\) 0 0
\(927\) 5.00000 0.164222
\(928\) 0 0
\(929\) −6.00000 −0.196854 −0.0984268 0.995144i \(-0.531381\pi\)
−0.0984268 + 0.995144i \(0.531381\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 32.0000 1.04819
\(933\) 32.0000 1.04763
\(934\) 0 0
\(935\) −3.00000 −0.0981105
\(936\) 0 0
\(937\) 38.0000 1.24141 0.620703 0.784046i \(-0.286847\pi\)
0.620703 + 0.784046i \(0.286847\pi\)
\(938\) 0 0
\(939\) −21.0000 −0.685309
\(940\) −12.0000 −0.391397
\(941\) −40.0000 −1.30396 −0.651981 0.758235i \(-0.726062\pi\)
−0.651981 + 0.758235i \(0.726062\pi\)
\(942\) 0 0
\(943\) −2.00000 −0.0651290
\(944\) 20.0000 0.650945
\(945\) 0 0
\(946\) 0 0
\(947\) −45.0000 −1.46230 −0.731152 0.682215i \(-0.761017\pi\)
−0.731152 + 0.682215i \(0.761017\pi\)
\(948\) 32.0000 1.03931
\(949\) 0 0
\(950\) 0 0
\(951\) 2.00000 0.0648544
\(952\) 0 0
\(953\) 30.0000 0.971795 0.485898 0.874016i \(-0.338493\pi\)
0.485898 + 0.874016i \(0.338493\pi\)
\(954\) 0 0
\(955\) −16.0000 −0.517748
\(956\) 42.0000 1.35838
\(957\) −7.00000 −0.226278
\(958\) 0 0
\(959\) 0 0
\(960\) −8.00000 −0.258199
\(961\) 5.00000 0.161290
\(962\) 0 0
\(963\) 6.00000 0.193347
\(964\) 20.0000 0.644157
\(965\) 6.00000 0.193147
\(966\) 0 0
\(967\) −17.0000 −0.546683 −0.273342 0.961917i \(-0.588129\pi\)
−0.273342 + 0.961917i \(0.588129\pi\)
\(968\) 0 0
\(969\) −9.00000 −0.289122
\(970\) 0 0
\(971\) −47.0000 −1.50830 −0.754151 0.656701i \(-0.771951\pi\)
−0.754151 + 0.656701i \(0.771951\pi\)
\(972\) −2.00000 −0.0641500
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 28.0000 0.896258
\(977\) 49.0000 1.56765 0.783824 0.620982i \(-0.213266\pi\)
0.783824 + 0.620982i \(0.213266\pi\)
\(978\) 0 0
\(979\) −3.00000 −0.0958804
\(980\) 0 0
\(981\) −2.00000 −0.0638551
\(982\) 0 0
\(983\) 6.00000 0.191370 0.0956851 0.995412i \(-0.469496\pi\)
0.0956851 + 0.995412i \(0.469496\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 5.00000 0.158991
\(990\) 0 0
\(991\) 31.0000 0.984747 0.492374 0.870384i \(-0.336129\pi\)
0.492374 + 0.870384i \(0.336129\pi\)
\(992\) 0 0
\(993\) 17.0000 0.539479
\(994\) 0 0
\(995\) −24.0000 −0.760851
\(996\) −6.00000 −0.190117
\(997\) 42.0000 1.33015 0.665077 0.746775i \(-0.268399\pi\)
0.665077 + 0.746775i \(0.268399\pi\)
\(998\) 0 0
\(999\) −8.00000 −0.253109
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.q.1.1 1
7.6 odd 2 1155.2.a.g.1.1 1
21.20 even 2 3465.2.a.i.1.1 1
35.34 odd 2 5775.2.a.o.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1155.2.a.g.1.1 1 7.6 odd 2
3465.2.a.i.1.1 1 21.20 even 2
5775.2.a.o.1.1 1 35.34 odd 2
8085.2.a.q.1.1 1 1.1 even 1 trivial