Properties

Label 4140.3.c.a.4049.17
Level $4140$
Weight $3$
Character 4140.4049
Analytic conductor $112.807$
Analytic rank $0$
Dimension $88$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4140,3,Mod(4049,4140)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4140, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4140.4049");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 4140.c (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: \(112.806829445\)
Analytic rank: \(0\)
Dimension: \(88\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 4049.17
Character \(\chi\) \(=\) 4140.4049
Dual form 4140.3.c.a.4049.18

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-3.90943 - 3.11711i) q^{5} +7.62373i q^{7} +O(q^{10})\) \(q+(-3.90943 - 3.11711i) q^{5} +7.62373i q^{7} +3.21524i q^{11} -21.1842i q^{13} +2.40382 q^{17} +5.95402 q^{19} -4.79583 q^{23} +(5.56729 + 24.3722i) q^{25} -22.0845i q^{29} -10.8439 q^{31} +(23.7640 - 29.8044i) q^{35} -7.39190i q^{37} +18.9081i q^{41} +58.6048i q^{43} +33.2863 q^{47} -9.12130 q^{49} -34.2495 q^{53} +(10.0222 - 12.5698i) q^{55} -63.6692i q^{59} +78.7741 q^{61} +(-66.0334 + 82.8181i) q^{65} -79.0164i q^{67} -46.2778i q^{71} +125.253i q^{73} -24.5121 q^{77} +6.15717 q^{79} -70.5276 q^{83} +(-9.39755 - 7.49295i) q^{85} +139.383i q^{89} +161.503 q^{91} +(-23.2768 - 18.5593i) q^{95} +145.917i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 88 q+O(q^{10}) \) Copy content Toggle raw display \( 88 q - 16 q^{19} - 48 q^{25} + 272 q^{31} - 600 q^{49} + 112 q^{55} + 448 q^{61} - 32 q^{79} - 264 q^{85} - 16 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4140\mathbb{Z}\right)^\times\).

\(n\) \(461\) \(1657\) \(2071\) \(3961\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(1\)

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
\(5\) −3.90943 3.11711i −0.781886 0.623421i
\(6\) 0 0
\(7\) 7.62373i 1.08910i 0.838727 + 0.544552i \(0.183300\pi\)
−0.838727 + 0.544552i \(0.816700\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.21524i 0.292294i 0.989263 + 0.146147i \(0.0466873\pi\)
−0.989263 + 0.146147i \(0.953313\pi\)
\(12\) 0 0
\(13\) 21.1842i 1.62955i −0.579775 0.814776i \(-0.696860\pi\)
0.579775 0.814776i \(-0.303140\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.40382 0.141401 0.0707005 0.997498i \(-0.477477\pi\)
0.0707005 + 0.997498i \(0.477477\pi\)
\(18\) 0 0
\(19\) 5.95402 0.313370 0.156685 0.987649i \(-0.449919\pi\)
0.156685 + 0.987649i \(0.449919\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4.79583 −0.208514
\(24\) 0 0
\(25\) 5.56729 + 24.3722i 0.222691 + 0.974889i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 22.0845i 0.761535i −0.924671 0.380767i \(-0.875660\pi\)
0.924671 0.380767i \(-0.124340\pi\)
\(30\) 0 0
\(31\) −10.8439 −0.349802 −0.174901 0.984586i \(-0.555961\pi\)
−0.174901 + 0.984586i \(0.555961\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 23.7640 29.8044i 0.678971 0.851556i
\(36\) 0 0
\(37\) 7.39190i 0.199781i −0.994998 0.0998905i \(-0.968151\pi\)
0.994998 0.0998905i \(-0.0318493\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 18.9081i 0.461173i 0.973052 + 0.230587i \(0.0740645\pi\)
−0.973052 + 0.230587i \(0.925936\pi\)
\(42\) 0 0
\(43\) 58.6048i 1.36290i 0.731864 + 0.681451i \(0.238651\pi\)
−0.731864 + 0.681451i \(0.761349\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 33.2863 0.708219 0.354110 0.935204i \(-0.384784\pi\)
0.354110 + 0.935204i \(0.384784\pi\)
\(48\) 0 0
\(49\) −9.12130 −0.186149
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −34.2495 −0.646216 −0.323108 0.946362i \(-0.604728\pi\)
−0.323108 + 0.946362i \(0.604728\pi\)
\(54\) 0 0
\(55\) 10.0222 12.5698i 0.182223 0.228541i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 63.6692i 1.07914i −0.841941 0.539569i \(-0.818587\pi\)
0.841941 0.539569i \(-0.181413\pi\)
\(60\) 0 0
\(61\) 78.7741 1.29138 0.645690 0.763600i \(-0.276570\pi\)
0.645690 + 0.763600i \(0.276570\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −66.0334 + 82.8181i −1.01590 + 1.27412i
\(66\) 0 0
\(67\) 79.0164i 1.17935i −0.807641 0.589675i \(-0.799256\pi\)
0.807641 0.589675i \(-0.200744\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 46.2778i 0.651800i −0.945404 0.325900i \(-0.894333\pi\)
0.945404 0.325900i \(-0.105667\pi\)
\(72\) 0 0
\(73\) 125.253i 1.71580i 0.513819 + 0.857899i \(0.328230\pi\)
−0.513819 + 0.857899i \(0.671770\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −24.5121 −0.318339
\(78\) 0 0
\(79\) 6.15717 0.0779389 0.0389694 0.999240i \(-0.487593\pi\)
0.0389694 + 0.999240i \(0.487593\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −70.5276 −0.849731 −0.424865 0.905257i \(-0.639679\pi\)
−0.424865 + 0.905257i \(0.639679\pi\)
\(84\) 0 0
\(85\) −9.39755 7.49295i −0.110559 0.0881524i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 139.383i 1.56610i 0.621960 + 0.783049i \(0.286337\pi\)
−0.621960 + 0.783049i \(0.713663\pi\)
\(90\) 0 0
\(91\) 161.503 1.77475
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −23.2768 18.5593i −0.245019 0.195361i
\(96\) 0 0
\(97\) 145.917i 1.50429i 0.658995 + 0.752147i \(0.270982\pi\)
−0.658995 + 0.752147i \(0.729018\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 175.345i 1.73609i −0.496486 0.868045i \(-0.665377\pi\)
0.496486 0.868045i \(-0.334623\pi\)
\(102\) 0 0
\(103\) 58.8536i 0.571394i −0.958320 0.285697i \(-0.907775\pi\)
0.958320 0.285697i \(-0.0922251\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −88.2174 −0.824462 −0.412231 0.911079i \(-0.635250\pi\)
−0.412231 + 0.911079i \(0.635250\pi\)
\(108\) 0 0
\(109\) −29.8711 −0.274047 −0.137023 0.990568i \(-0.543754\pi\)
−0.137023 + 0.990568i \(0.543754\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −29.5966 −0.261917 −0.130959 0.991388i \(-0.541805\pi\)
−0.130959 + 0.991388i \(0.541805\pi\)
\(114\) 0 0
\(115\) 18.7490 + 14.9491i 0.163035 + 0.129992i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 18.3260i 0.154000i
\(120\) 0 0
\(121\) 110.662 0.914564
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 54.2059 112.635i 0.433647 0.901083i
\(126\) 0 0
\(127\) 36.9306i 0.290793i 0.989374 + 0.145396i \(0.0464457\pi\)
−0.989374 + 0.145396i \(0.953554\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 203.365i 1.55241i −0.630482 0.776204i \(-0.717143\pi\)
0.630482 0.776204i \(-0.282857\pi\)
\(132\) 0 0
\(133\) 45.3919i 0.341292i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −177.216 −1.29354 −0.646772 0.762683i \(-0.723881\pi\)
−0.646772 + 0.762683i \(0.723881\pi\)
\(138\) 0 0
\(139\) −228.337 −1.64271 −0.821356 0.570415i \(-0.806782\pi\)
−0.821356 + 0.570415i \(0.806782\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 68.1122 0.476309
\(144\) 0 0
\(145\) −68.8398 + 86.3378i −0.474757 + 0.595433i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 98.0909i 0.658328i 0.944273 + 0.329164i \(0.106767\pi\)
−0.944273 + 0.329164i \(0.893233\pi\)
\(150\) 0 0
\(151\) 78.0414 0.516831 0.258415 0.966034i \(-0.416800\pi\)
0.258415 + 0.966034i \(0.416800\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 42.3933 + 33.8015i 0.273505 + 0.218074i
\(156\) 0 0
\(157\) 15.4850i 0.0986304i −0.998783 0.0493152i \(-0.984296\pi\)
0.998783 0.0493152i \(-0.0157039\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 36.5621i 0.227094i
\(162\) 0 0
\(163\) 29.5903i 0.181536i −0.995872 0.0907678i \(-0.971068\pi\)
0.995872 0.0907678i \(-0.0289321\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 58.6076 0.350944 0.175472 0.984484i \(-0.443855\pi\)
0.175472 + 0.984484i \(0.443855\pi\)
\(168\) 0 0
\(169\) −279.770 −1.65544
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −80.8229 −0.467184 −0.233592 0.972335i \(-0.575048\pi\)
−0.233592 + 0.972335i \(0.575048\pi\)
\(174\) 0 0
\(175\) −185.807 + 42.4435i −1.06176 + 0.242534i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 155.177i 0.866910i −0.901175 0.433455i \(-0.857294\pi\)
0.901175 0.433455i \(-0.142706\pi\)
\(180\) 0 0
\(181\) −22.1896 −0.122594 −0.0612972 0.998120i \(-0.519524\pi\)
−0.0612972 + 0.998120i \(0.519524\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −23.0413 + 28.8981i −0.124548 + 0.156206i
\(186\) 0 0
\(187\) 7.72884i 0.0413307i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 198.804i 1.04086i −0.853905 0.520429i \(-0.825772\pi\)
0.853905 0.520429i \(-0.174228\pi\)
\(192\) 0 0
\(193\) 303.541i 1.57275i −0.617748 0.786376i \(-0.711955\pi\)
0.617748 0.786376i \(-0.288045\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 98.2896 0.498932 0.249466 0.968384i \(-0.419745\pi\)
0.249466 + 0.968384i \(0.419745\pi\)
\(198\) 0 0
\(199\) −39.3046 −0.197511 −0.0987553 0.995112i \(-0.531486\pi\)
−0.0987553 + 0.995112i \(0.531486\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 168.366 0.829391
\(204\) 0 0
\(205\) 58.9386 73.9199i 0.287505 0.360585i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 19.1436i 0.0915962i
\(210\) 0 0
\(211\) −324.356 −1.53723 −0.768615 0.639712i \(-0.779054\pi\)
−0.768615 + 0.639712i \(0.779054\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 182.677 229.111i 0.849662 1.06563i
\(216\) 0 0
\(217\) 82.6707i 0.380971i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 50.9229i 0.230420i
\(222\) 0 0
\(223\) 64.1839i 0.287820i 0.989591 + 0.143910i \(0.0459676\pi\)
−0.989591 + 0.143910i \(0.954032\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 66.2929 0.292039 0.146020 0.989282i \(-0.453354\pi\)
0.146020 + 0.989282i \(0.453354\pi\)
\(228\) 0 0
\(229\) −445.764 −1.94657 −0.973284 0.229602i \(-0.926257\pi\)
−0.973284 + 0.229602i \(0.926257\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −361.910 −1.55326 −0.776632 0.629955i \(-0.783073\pi\)
−0.776632 + 0.629955i \(0.783073\pi\)
\(234\) 0 0
\(235\) −130.130 103.757i −0.553747 0.441519i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 442.262i 1.85047i −0.379397 0.925234i \(-0.623868\pi\)
0.379397 0.925234i \(-0.376132\pi\)
\(240\) 0 0
\(241\) −4.24860 −0.0176290 −0.00881451 0.999961i \(-0.502806\pi\)
−0.00881451 + 0.999961i \(0.502806\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 35.6591 + 28.4321i 0.145547 + 0.116049i
\(246\) 0 0
\(247\) 126.131i 0.510653i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 346.867i 1.38194i −0.722883 0.690970i \(-0.757184\pi\)
0.722883 0.690970i \(-0.242816\pi\)
\(252\) 0 0
\(253\) 15.4197i 0.0609476i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 126.259 0.491280 0.245640 0.969361i \(-0.421002\pi\)
0.245640 + 0.969361i \(0.421002\pi\)
\(258\) 0 0
\(259\) 56.3539 0.217582
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −7.29073 −0.0277214 −0.0138607 0.999904i \(-0.504412\pi\)
−0.0138607 + 0.999904i \(0.504412\pi\)
\(264\) 0 0
\(265\) 133.896 + 106.759i 0.505267 + 0.402865i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 257.321i 0.956584i 0.878201 + 0.478292i \(0.158744\pi\)
−0.878201 + 0.478292i \(0.841256\pi\)
\(270\) 0 0
\(271\) −108.965 −0.402084 −0.201042 0.979583i \(-0.564433\pi\)
−0.201042 + 0.979583i \(0.564433\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −78.3625 + 17.9002i −0.284955 + 0.0650915i
\(276\) 0 0
\(277\) 397.844i 1.43626i 0.695909 + 0.718130i \(0.255002\pi\)
−0.695909 + 0.718130i \(0.744998\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 325.194i 1.15727i 0.815585 + 0.578637i \(0.196415\pi\)
−0.815585 + 0.578637i \(0.803585\pi\)
\(282\) 0 0
\(283\) 157.751i 0.557423i −0.960375 0.278712i \(-0.910093\pi\)
0.960375 0.278712i \(-0.0899073\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −144.150 −0.502266
\(288\) 0 0
\(289\) −283.222 −0.980006
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −389.693 −1.33001 −0.665005 0.746839i \(-0.731571\pi\)
−0.665005 + 0.746839i \(0.731571\pi\)
\(294\) 0 0
\(295\) −198.464 + 248.910i −0.672758 + 0.843764i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 101.596i 0.339785i
\(300\) 0 0
\(301\) −446.787 −1.48434
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −307.962 245.547i −1.00971 0.805074i
\(306\) 0 0
\(307\) 81.0767i 0.264094i 0.991243 + 0.132047i \(0.0421549\pi\)
−0.991243 + 0.132047i \(0.957845\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 92.4751i 0.297348i 0.988886 + 0.148674i \(0.0475004\pi\)
−0.988886 + 0.148674i \(0.952500\pi\)
\(312\) 0 0
\(313\) 363.385i 1.16097i 0.814270 + 0.580487i \(0.197138\pi\)
−0.814270 + 0.580487i \(0.802862\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 51.1777 0.161444 0.0807219 0.996737i \(-0.474277\pi\)
0.0807219 + 0.996737i \(0.474277\pi\)
\(318\) 0 0
\(319\) 71.0070 0.222592
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 14.3124 0.0443108
\(324\) 0 0
\(325\) 516.306 117.938i 1.58863 0.362887i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 253.766i 0.771325i
\(330\) 0 0
\(331\) 441.929 1.33513 0.667567 0.744550i \(-0.267336\pi\)
0.667567 + 0.744550i \(0.267336\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −246.303 + 308.909i −0.735231 + 0.922116i
\(336\) 0 0
\(337\) 288.958i 0.857441i 0.903437 + 0.428720i \(0.141035\pi\)
−0.903437 + 0.428720i \(0.858965\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 34.8656i 0.102245i
\(342\) 0 0
\(343\) 304.025i 0.886369i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −639.619 −1.84328 −0.921641 0.388044i \(-0.873151\pi\)
−0.921641 + 0.388044i \(0.873151\pi\)
\(348\) 0 0
\(349\) −452.401 −1.29628 −0.648139 0.761522i \(-0.724452\pi\)
−0.648139 + 0.761522i \(0.724452\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −297.051 −0.841503 −0.420751 0.907176i \(-0.638234\pi\)
−0.420751 + 0.907176i \(0.638234\pi\)
\(354\) 0 0
\(355\) −144.253 + 180.920i −0.406346 + 0.509633i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 40.7532i 0.113519i 0.998388 + 0.0567593i \(0.0180768\pi\)
−0.998388 + 0.0567593i \(0.981923\pi\)
\(360\) 0 0
\(361\) −325.550 −0.901799
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 390.428 489.669i 1.06967 1.34156i
\(366\) 0 0
\(367\) 450.788i 1.22830i 0.789188 + 0.614152i \(0.210502\pi\)
−0.789188 + 0.614152i \(0.789498\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 261.109i 0.703797i
\(372\) 0 0
\(373\) 75.9119i 0.203517i −0.994809 0.101759i \(-0.967553\pi\)
0.994809 0.101759i \(-0.0324469\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −467.842 −1.24096
\(378\) 0 0
\(379\) −550.199 −1.45171 −0.725856 0.687847i \(-0.758556\pi\)
−0.725856 + 0.687847i \(0.758556\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −751.680 −1.96261 −0.981306 0.192456i \(-0.938355\pi\)
−0.981306 + 0.192456i \(0.938355\pi\)
\(384\) 0 0
\(385\) 95.8284 + 76.4069i 0.248905 + 0.198460i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 27.8168i 0.0715084i −0.999361 0.0357542i \(-0.988617\pi\)
0.999361 0.0357542i \(-0.0113833\pi\)
\(390\) 0 0
\(391\) −11.5283 −0.0294841
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −24.0710 19.1926i −0.0609393 0.0485888i
\(396\) 0 0
\(397\) 507.708i 1.27886i −0.768849 0.639431i \(-0.779170\pi\)
0.768849 0.639431i \(-0.220830\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 270.184i 0.673774i −0.941545 0.336887i \(-0.890626\pi\)
0.941545 0.336887i \(-0.109374\pi\)
\(402\) 0 0
\(403\) 229.718i 0.570021i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 23.7667 0.0583949
\(408\) 0 0
\(409\) −95.9356 −0.234561 −0.117281 0.993099i \(-0.537418\pi\)
−0.117281 + 0.993099i \(0.537418\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 485.397 1.17530
\(414\) 0 0
\(415\) 275.723 + 219.842i 0.664392 + 0.529740i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 485.607i 1.15897i −0.814984 0.579483i \(-0.803254\pi\)
0.814984 0.579483i \(-0.196746\pi\)
\(420\) 0 0
\(421\) −37.0169 −0.0879261 −0.0439631 0.999033i \(-0.513998\pi\)
−0.0439631 + 0.999033i \(0.513998\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 13.3827 + 58.5863i 0.0314888 + 0.137850i
\(426\) 0 0
\(427\) 600.553i 1.40645i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 264.228i 0.613057i 0.951861 + 0.306529i \(0.0991675\pi\)
−0.951861 + 0.306529i \(0.900832\pi\)
\(432\) 0 0
\(433\) 223.032i 0.515087i −0.966267 0.257543i \(-0.917087\pi\)
0.966267 0.257543i \(-0.0829130\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −28.5545 −0.0653421
\(438\) 0 0
\(439\) 856.080 1.95007 0.975034 0.222056i \(-0.0712766\pi\)
0.975034 + 0.222056i \(0.0712766\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −327.558 −0.739408 −0.369704 0.929150i \(-0.620541\pi\)
−0.369704 + 0.929150i \(0.620541\pi\)
\(444\) 0 0
\(445\) 434.471 544.907i 0.976339 1.22451i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 220.079i 0.490154i 0.969504 + 0.245077i \(0.0788133\pi\)
−0.969504 + 0.245077i \(0.921187\pi\)
\(450\) 0 0
\(451\) −60.7940 −0.134798
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −631.383 503.421i −1.38766 1.10642i
\(456\) 0 0
\(457\) 184.792i 0.404358i −0.979349 0.202179i \(-0.935198\pi\)
0.979349 0.202179i \(-0.0648023\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 526.656i 1.14242i 0.820804 + 0.571210i \(0.193526\pi\)
−0.820804 + 0.571210i \(0.806474\pi\)
\(462\) 0 0
\(463\) 622.938i 1.34544i −0.739898 0.672719i \(-0.765126\pi\)
0.739898 0.672719i \(-0.234874\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −282.557 −0.605048 −0.302524 0.953142i \(-0.597829\pi\)
−0.302524 + 0.953142i \(0.597829\pi\)
\(468\) 0 0
\(469\) 602.400 1.28443
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −188.428 −0.398369
\(474\) 0 0
\(475\) 33.1478 + 145.113i 0.0697848 + 0.305501i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 24.5133i 0.0511761i −0.999673 0.0255880i \(-0.991854\pi\)
0.999673 0.0255880i \(-0.00814581\pi\)
\(480\) 0 0
\(481\) −156.591 −0.325554
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 454.837 570.450i 0.937809 1.17619i
\(486\) 0 0
\(487\) 502.670i 1.03218i 0.856536 + 0.516088i \(0.172612\pi\)
−0.856536 + 0.516088i \(0.827388\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 358.223i 0.729578i −0.931090 0.364789i \(-0.881141\pi\)
0.931090 0.364789i \(-0.118859\pi\)
\(492\) 0 0
\(493\) 53.0871i 0.107682i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 352.809 0.709878
\(498\) 0 0
\(499\) −352.130 −0.705672 −0.352836 0.935685i \(-0.614783\pi\)
−0.352836 + 0.935685i \(0.614783\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −668.684 −1.32939 −0.664695 0.747114i \(-0.731439\pi\)
−0.664695 + 0.747114i \(0.731439\pi\)
\(504\) 0 0
\(505\) −546.569 + 685.499i −1.08232 + 1.35742i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 743.009i 1.45974i −0.683585 0.729871i \(-0.739580\pi\)
0.683585 0.729871i \(-0.260420\pi\)
\(510\) 0 0
\(511\) −954.897 −1.86868
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −183.453 + 230.084i −0.356219 + 0.446765i
\(516\) 0 0
\(517\) 107.023i 0.207009i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 26.9328i 0.0516944i −0.999666 0.0258472i \(-0.991772\pi\)
0.999666 0.0258472i \(-0.00822833\pi\)
\(522\) 0 0
\(523\) 645.889i 1.23497i 0.786583 + 0.617484i \(0.211848\pi\)
−0.786583 + 0.617484i \(0.788152\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −26.0666 −0.0494623
\(528\) 0 0
\(529\) 23.0000 0.0434783
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 400.553 0.751506
\(534\) 0 0
\(535\) 344.880 + 274.983i 0.644635 + 0.513987i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 29.3272i 0.0544103i
\(540\) 0 0
\(541\) −668.559 −1.23578 −0.617892 0.786263i \(-0.712013\pi\)
−0.617892 + 0.786263i \(0.712013\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 116.779 + 93.1113i 0.214273 + 0.170846i
\(546\) 0 0
\(547\) 432.280i 0.790274i −0.918622 0.395137i \(-0.870697\pi\)
0.918622 0.395137i \(-0.129303\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 131.492i 0.238642i
\(552\) 0 0
\(553\) 46.9406i 0.0848836i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 289.965 0.520583 0.260291 0.965530i \(-0.416181\pi\)
0.260291 + 0.965530i \(0.416181\pi\)
\(558\) 0 0
\(559\) 1241.49 2.22092
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 993.271 1.76425 0.882123 0.471018i \(-0.156113\pi\)
0.882123 + 0.471018i \(0.156113\pi\)
\(564\) 0 0
\(565\) 115.706 + 92.2559i 0.204789 + 0.163285i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 591.750i 1.03998i −0.854171 0.519991i \(-0.825935\pi\)
0.854171 0.519991i \(-0.174065\pi\)
\(570\) 0 0
\(571\) −8.49753 −0.0148818 −0.00744092 0.999972i \(-0.502369\pi\)
−0.00744092 + 0.999972i \(0.502369\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −26.6998 116.885i −0.0464344 0.203278i
\(576\) 0 0
\(577\) 98.2139i 0.170215i 0.996372 + 0.0851074i \(0.0271233\pi\)
−0.996372 + 0.0851074i \(0.972877\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 537.684i 0.925446i
\(582\) 0 0
\(583\) 110.120i 0.188885i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 462.810 0.788432 0.394216 0.919018i \(-0.371016\pi\)
0.394216 + 0.919018i \(0.371016\pi\)
\(588\) 0 0
\(589\) −64.5646 −0.109617
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −385.900 −0.650759 −0.325380 0.945583i \(-0.605492\pi\)
−0.325380 + 0.945583i \(0.605492\pi\)
\(594\) 0 0
\(595\) 57.1243 71.6444i 0.0960072 0.120411i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 300.516i 0.501695i −0.968027 0.250848i \(-0.919291\pi\)
0.968027 0.250848i \(-0.0807093\pi\)
\(600\) 0 0
\(601\) 735.625 1.22400 0.612000 0.790857i \(-0.290365\pi\)
0.612000 + 0.790857i \(0.290365\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −432.626 344.946i −0.715085 0.570159i
\(606\) 0 0
\(607\) 520.875i 0.858113i −0.903278 0.429057i \(-0.858846\pi\)
0.903278 0.429057i \(-0.141154\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 705.143i 1.15408i
\(612\) 0 0
\(613\) 720.853i 1.17594i −0.808882 0.587971i \(-0.799927\pi\)
0.808882 0.587971i \(-0.200073\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 159.490 0.258493 0.129246 0.991613i \(-0.458744\pi\)
0.129246 + 0.991613i \(0.458744\pi\)
\(618\) 0 0
\(619\) −681.600 −1.10113 −0.550565 0.834792i \(-0.685588\pi\)
−0.550565 + 0.834792i \(0.685588\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −1062.62 −1.70565
\(624\) 0 0
\(625\) −563.011 + 271.374i −0.900817 + 0.434199i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 17.7688i 0.0282492i
\(630\) 0 0
\(631\) 849.204 1.34581 0.672903 0.739730i \(-0.265047\pi\)
0.672903 + 0.739730i \(0.265047\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 115.117 144.378i 0.181286 0.227367i
\(636\) 0 0
\(637\) 193.227i 0.303340i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 395.174i 0.616496i 0.951306 + 0.308248i \(0.0997427\pi\)
−0.951306 + 0.308248i \(0.900257\pi\)
\(642\) 0 0
\(643\) 545.806i 0.848843i −0.905465 0.424422i \(-0.860477\pi\)
0.905465 0.424422i \(-0.139523\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 445.714 0.688893 0.344446 0.938806i \(-0.388067\pi\)
0.344446 + 0.938806i \(0.388067\pi\)
\(648\) 0 0
\(649\) 204.712 0.315426
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 857.780 1.31360 0.656800 0.754065i \(-0.271910\pi\)
0.656800 + 0.754065i \(0.271910\pi\)
\(654\) 0 0
\(655\) −633.912 + 795.043i −0.967804 + 1.21381i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 850.528i 1.29063i −0.763915 0.645317i \(-0.776725\pi\)
0.763915 0.645317i \(-0.223275\pi\)
\(660\) 0 0
\(661\) −331.769 −0.501919 −0.250960 0.967998i \(-0.580746\pi\)
−0.250960 + 0.967998i \(0.580746\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 141.491 177.456i 0.212769 0.266852i
\(666\) 0 0
\(667\) 105.914i 0.158791i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 253.278i 0.377463i
\(672\) 0 0
\(673\) 987.624i 1.46750i −0.679422 0.733748i \(-0.737770\pi\)
0.679422 0.733748i \(-0.262230\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −396.595 −0.585812 −0.292906 0.956141i \(-0.594622\pi\)
−0.292906 + 0.956141i \(0.594622\pi\)
\(678\) 0 0
\(679\) −1112.43 −1.63833
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 33.9937 0.0497711 0.0248855 0.999690i \(-0.492078\pi\)
0.0248855 + 0.999690i \(0.492078\pi\)
\(684\) 0 0
\(685\) 692.812 + 552.400i 1.01140 + 0.806423i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 725.547i 1.05304i
\(690\) 0 0
\(691\) −995.845 −1.44117 −0.720583 0.693369i \(-0.756126\pi\)
−0.720583 + 0.693369i \(0.756126\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 892.668 + 711.751i 1.28441 + 1.02410i
\(696\) 0 0
\(697\) 45.4516i 0.0652103i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 400.145i 0.570820i −0.958406 0.285410i \(-0.907870\pi\)
0.958406 0.285410i \(-0.0921297\pi\)
\(702\) 0 0
\(703\) 44.0116i 0.0626053i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 1336.78 1.89078
\(708\) 0 0
\(709\) 376.581 0.531144 0.265572 0.964091i \(-0.414439\pi\)
0.265572 + 0.964091i \(0.414439\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 52.0053 0.0729387
\(714\) 0 0
\(715\) −266.280 212.313i −0.372420 0.296941i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 560.265i 0.779229i −0.920978 0.389614i \(-0.872608\pi\)
0.920978 0.389614i \(-0.127392\pi\)
\(720\) 0 0
\(721\) 448.684 0.622308
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 538.249 122.951i 0.742412 0.169587i
\(726\) 0 0
\(727\) 407.313i 0.560266i −0.959961 0.280133i \(-0.909621\pi\)
0.959961 0.280133i \(-0.0903785\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 140.875i 0.192716i
\(732\) 0 0
\(733\) 66.8245i 0.0911658i 0.998961 + 0.0455829i \(0.0145145\pi\)
−0.998961 + 0.0455829i \(0.985485\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 254.057 0.344717
\(738\) 0 0
\(739\) −36.2775 −0.0490900 −0.0245450 0.999699i \(-0.507814\pi\)
−0.0245450 + 0.999699i \(0.507814\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −390.878 −0.526081 −0.263040 0.964785i \(-0.584725\pi\)
−0.263040 + 0.964785i \(0.584725\pi\)
\(744\) 0 0
\(745\) 305.760 383.479i 0.410416 0.514738i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 672.546i 0.897925i
\(750\) 0 0
\(751\) −13.2884 −0.0176943 −0.00884717 0.999961i \(-0.502816\pi\)
−0.00884717 + 0.999961i \(0.502816\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −305.097 243.263i −0.404103 0.322203i
\(756\) 0 0
\(757\) 975.953i 1.28924i 0.764504 + 0.644619i \(0.222984\pi\)
−0.764504 + 0.644619i \(0.777016\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 616.620i 0.810277i 0.914255 + 0.405138i \(0.132777\pi\)
−0.914255 + 0.405138i \(0.867223\pi\)
\(762\) 0 0
\(763\) 227.729i 0.298465i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −1348.78 −1.75851
\(768\) 0 0
\(769\) 288.659 0.375369 0.187685 0.982229i \(-0.439902\pi\)
0.187685 + 0.982229i \(0.439902\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −826.552 −1.06928 −0.534639 0.845081i \(-0.679552\pi\)
−0.534639 + 0.845081i \(0.679552\pi\)
\(774\) 0 0
\(775\) −60.3709 264.289i −0.0778979 0.341018i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 112.579i 0.144518i
\(780\) 0 0
\(781\) 148.794 0.190517
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −48.2683 + 60.5374i −0.0614883 + 0.0771177i
\(786\) 0 0
\(787\) 739.060i 0.939085i 0.882910 + 0.469543i \(0.155581\pi\)
−0.882910 + 0.469543i \(0.844419\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 225.637i 0.285255i
\(792\) 0 0
\(793\) 1668.77i 2.10437i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −1030.90 −1.29348 −0.646739 0.762711i \(-0.723868\pi\)
−0.646739 + 0.762711i \(0.723868\pi\)
\(798\) 0 0
\(799\) 80.0141 0.100143
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −402.719 −0.501518
\(804\) 0 0
\(805\) −113.968 + 142.937i −0.141575 + 0.177562i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 741.519i 0.916587i 0.888801 + 0.458294i \(0.151539\pi\)
−0.888801 + 0.458294i \(0.848461\pi\)
\(810\) 0 0
\(811\) 723.965 0.892682 0.446341 0.894863i \(-0.352727\pi\)
0.446341 + 0.894863i \(0.352727\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −92.2362 + 115.681i −0.113173 + 0.141940i
\(816\) 0 0
\(817\) 348.934i 0.427092i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 1522.59i 1.85456i 0.374370 + 0.927279i \(0.377859\pi\)
−0.374370 + 0.927279i \(0.622141\pi\)
\(822\) 0 0
\(823\) 1134.31i 1.37826i 0.724637 + 0.689131i \(0.242007\pi\)
−0.724637 + 0.689131i \(0.757993\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −545.247 −0.659308 −0.329654 0.944102i \(-0.606932\pi\)
−0.329654 + 0.944102i \(0.606932\pi\)
\(828\) 0 0
\(829\) −117.564 −0.141814 −0.0709071 0.997483i \(-0.522589\pi\)
−0.0709071 + 0.997483i \(0.522589\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −21.9259 −0.0263216
\(834\) 0 0
\(835\) −229.122 182.686i −0.274398 0.218786i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 719.479i 0.857543i 0.903413 + 0.428772i \(0.141054\pi\)
−0.903413 + 0.428772i \(0.858946\pi\)
\(840\) 0 0
\(841\) 353.274 0.420065
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 1093.74 + 872.072i 1.29437 + 1.03204i
\(846\) 0 0
\(847\) 843.659i 0.996056i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 35.4503i 0.0416572i
\(852\) 0 0
\(853\) 780.772i 0.915325i 0.889126 + 0.457663i \(0.151313\pi\)
−0.889126 + 0.457663i \(0.848687\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −183.816 −0.214488 −0.107244 0.994233i \(-0.534203\pi\)
−0.107244 + 0.994233i \(0.534203\pi\)
\(858\) 0 0
\(859\) 1051.01 1.22352 0.611761 0.791042i \(-0.290461\pi\)
0.611761 + 0.791042i \(0.290461\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 357.434 0.414177 0.207088 0.978322i \(-0.433601\pi\)
0.207088 + 0.978322i \(0.433601\pi\)
\(864\) 0 0
\(865\) 315.971 + 251.933i 0.365285 + 0.291253i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 19.7968i 0.0227811i
\(870\) 0 0
\(871\) −1673.90 −1.92181
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 858.702 + 413.251i 0.981373 + 0.472287i
\(876\) 0 0
\(877\) 968.591i 1.10444i −0.833700 0.552218i \(-0.813782\pi\)
0.833700 0.552218i \(-0.186218\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 984.881i 1.11791i −0.829197 0.558956i \(-0.811202\pi\)
0.829197 0.558956i \(-0.188798\pi\)
\(882\) 0 0
\(883\) 1399.06i 1.58444i −0.610238 0.792218i \(-0.708926\pi\)
0.610238 0.792218i \(-0.291074\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −1071.56 −1.20807 −0.604035 0.796958i \(-0.706441\pi\)
−0.604035 + 0.796958i \(0.706441\pi\)
\(888\) 0 0
\(889\) −281.549 −0.316703
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 198.187 0.221934
\(894\) 0 0
\(895\) −483.703 + 606.654i −0.540451 + 0.677825i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 239.481i 0.266386i
\(900\) 0 0
\(901\) −82.3294 −0.0913756
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 86.7486 + 69.1673i 0.0958548 + 0.0764280i
\(906\) 0 0
\(907\) 62.2104i 0.0685892i −0.999412 0.0342946i \(-0.989082\pi\)
0.999412 0.0342946i \(-0.0109185\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 38.4024i 0.0421541i −0.999778 0.0210771i \(-0.993290\pi\)
0.999778 0.0210771i \(-0.00670954\pi\)
\(912\) 0 0
\(913\) 226.763i 0.248372i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 1550.40 1.69073
\(918\) 0 0
\(919\) −612.247 −0.666210 −0.333105 0.942890i \(-0.608096\pi\)
−0.333105 + 0.942890i \(0.608096\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −980.357 −1.06214
\(924\) 0 0
\(925\) 180.157 41.1528i 0.194764 0.0444895i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 110.243i 0.118668i −0.998238 0.0593340i \(-0.981102\pi\)
0.998238 0.0593340i \(-0.0188977\pi\)
\(930\) 0 0
\(931\) −54.3084 −0.0583335
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 24.0916 30.2154i 0.0257664 0.0323159i
\(936\) 0 0
\(937\) 327.959i 0.350010i −0.984568 0.175005i \(-0.944006\pi\)
0.984568 0.175005i \(-0.0559942\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 1671.55i 1.77636i 0.459497 + 0.888179i \(0.348030\pi\)
−0.459497 + 0.888179i \(0.651970\pi\)
\(942\) 0 0
\(943\) 90.6800i 0.0961612i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 982.416 1.03740 0.518699 0.854957i \(-0.326417\pi\)
0.518699 + 0.854957i \(0.326417\pi\)
\(948\) 0 0
\(949\) 2653.39 2.79598
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 957.587 1.00481 0.502406 0.864632i \(-0.332448\pi\)
0.502406 + 0.864632i \(0.332448\pi\)
\(954\) 0 0
\(955\) −619.693 + 777.210i −0.648893 + 0.813832i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 1351.04i 1.40880i
\(960\) 0 0
\(961\) −843.411 −0.877639
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −946.170 + 1186.67i −0.980487 + 1.22971i
\(966\) 0 0
\(967\) 298.269i 0.308448i −0.988036 0.154224i \(-0.950712\pi\)
0.988036 0.154224i \(-0.0492877\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 1568.44i 1.61529i 0.589670 + 0.807644i \(0.299258\pi\)
−0.589670 + 0.807644i \(0.700742\pi\)
\(972\) 0 0
\(973\) 1740.78i 1.78909i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 671.177 0.686977 0.343489 0.939157i \(-0.388391\pi\)
0.343489 + 0.939157i \(0.388391\pi\)
\(978\) 0 0
\(979\) −448.149 −0.457762
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −1102.45 −1.12152 −0.560759 0.827979i \(-0.689491\pi\)
−0.560759 + 0.827979i \(0.689491\pi\)
\(984\) 0 0
\(985\) −384.256 306.379i −0.390108 0.311045i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 281.059i 0.284185i
\(990\) 0 0
\(991\) 126.154 0.127300 0.0636498 0.997972i \(-0.479726\pi\)
0.0636498 + 0.997972i \(0.479726\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 153.659 + 122.517i 0.154431 + 0.123132i
\(996\) 0 0
\(997\) 133.802i 0.134204i −0.997746 0.0671021i \(-0.978625\pi\)
0.997746 0.0671021i \(-0.0213753\pi\)
\(998\) 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 4140.3.c.a.4049.17 88
3.2 odd 2 inner 4140.3.c.a.4049.72 yes 88
5.4 even 2 inner 4140.3.c.a.4049.71 yes 88
15.14 odd 2 inner 4140.3.c.a.4049.18 yes 88
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4140.3.c.a.4049.17 88 1.1 even 1 trivial
4140.3.c.a.4049.18 yes 88 15.14 odd 2 inner
4140.3.c.a.4049.71 yes 88 5.4 even 2 inner
4140.3.c.a.4049.72 yes 88 3.2 odd 2 inner