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Properties

Label 4200.2.a.a.1.1
Level $4200$
Weight $2$
Character 4200.1
Self dual yes
Analytic conductor $33.537$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4200,2,Mod(1,4200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4200, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("4200.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4200 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4200.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: \(33.5371688489\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 4200.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -1.00000 q^{7} +1.00000 q^{9} -3.00000 q^{11} -2.00000 q^{13} +4.00000 q^{17} -2.00000 q^{19} +1.00000 q^{21} +9.00000 q^{23} -1.00000 q^{27} -7.00000 q^{29} +10.0000 q^{31} +3.00000 q^{33} -5.00000 q^{37} +2.00000 q^{39} -8.00000 q^{41} +7.00000 q^{43} +10.0000 q^{47} +1.00000 q^{49} -4.00000 q^{51} -10.0000 q^{53} +2.00000 q^{57} -6.00000 q^{59} -8.00000 q^{61} -1.00000 q^{63} +13.0000 q^{67} -9.00000 q^{69} +1.00000 q^{71} -4.00000 q^{73} +3.00000 q^{77} -13.0000 q^{79} +1.00000 q^{81} -14.0000 q^{83} +7.00000 q^{87} +6.00000 q^{89} +2.00000 q^{91} -10.0000 q^{93} +2.00000 q^{97} -3.00000 q^{99} +O(q^{100})\)

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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −3.00000 −0.904534 −0.452267 − 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.00000 0.970143 0.485071 − 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 0 0
\(19\) −2.00000 −0.458831 −0.229416 − 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) 9.00000 1.87663 0.938315 − 0.345782i \(-0.112386\pi\)
0.938315 + 0.345782i \(0.112386\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 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
\(31\) 10.0000 1.79605 0.898027 − 0.439941i \(-0.145001\pi\)
0.898027 + 0.439941i \(0.145001\pi\)
\(32\) 0 0
\(33\) 3.00000 0.522233
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −5.00000 −0.821995 −0.410997 − 0.911636i \(-0.634819\pi\)
−0.410997 + 0.911636i \(0.634819\pi\)
\(38\) 0 0
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) −8.00000 −1.24939 −0.624695 − 0.780869i \(-0.714777\pi\)
−0.624695 + 0.780869i \(0.714777\pi\)
\(42\) 0 0
\(43\) 7.00000 1.06749 0.533745 − 0.845645i \(-0.320784\pi\)
0.533745 + 0.845645i \(0.320784\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 10.0000 1.45865 0.729325 − 0.684167i \(-0.239834\pi\)
0.729325 + 0.684167i \(0.239834\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −4.00000 −0.560112
\(52\) 0 0
\(53\) −10.0000 −1.37361 −0.686803 − 0.726844i \(-0.740986\pi\)
−0.686803 + 0.726844i \(0.740986\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 2.00000 0.264906
\(58\) 0 0
\(59\) −6.00000 −0.781133 −0.390567 − 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 0 0
\(61\) −8.00000 −1.02430 −0.512148 − 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 13.0000 1.58820 0.794101 − 0.607785i \(-0.207942\pi\)
0.794101 + 0.607785i \(0.207942\pi\)
\(68\) 0 0
\(69\) −9.00000 −1.08347
\(70\) 0 0
\(71\) 1.00000 0.118678 0.0593391 − 0.998238i \(-0.481101\pi\)
0.0593391 + 0.998238i \(0.481101\pi\)
\(72\) 0 0
\(73\) −4.00000 −0.468165 −0.234082 − 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.00000 0.341882
\(78\) 0 0
\(79\) −13.0000 −1.46261 −0.731307 − 0.682048i \(-0.761089\pi\)
−0.731307 + 0.682048i \(0.761089\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −14.0000 −1.53670 −0.768350 − 0.640030i \(-0.778922\pi\)
−0.768350 + 0.640030i \(0.778922\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 7.00000 0.750479
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 − 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) 0 0
\(93\) −10.0000 −1.03695
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 2.00000 0.203069 0.101535 − 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 0 0
\(99\) −3.00000 −0.301511
\(100\) 0 0
\(101\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) −18.0000 −1.77359 −0.886796 − 0.462160i \(-0.847074\pi\)
−0.886796 + 0.462160i \(0.847074\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −4.00000 −0.386695 −0.193347 − 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) 0 0
\(109\) −11.0000 −1.05361 −0.526804 − 0.849987i \(-0.676610\pi\)
−0.526804 + 0.849987i \(0.676610\pi\)
\(110\) 0 0
\(111\) 5.00000 0.474579
\(112\) 0 0
\(113\) 13.0000 1.22294 0.611469 − 0.791269i \(-0.290579\pi\)
0.611469 + 0.791269i \(0.290579\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −2.00000 −0.184900
\(118\) 0 0
\(119\) −4.00000 −0.366679
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 0 0
\(123\) 8.00000 0.721336
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −7.00000 −0.621150 −0.310575 − 0.950549i \(-0.600522\pi\)
−0.310575 + 0.950549i \(0.600522\pi\)
\(128\) 0 0
\(129\) −7.00000 −0.616316
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 − 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 0 0
\(133\) 2.00000 0.173422
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −10.0000 −0.854358 −0.427179 − 0.904167i \(-0.640493\pi\)
−0.427179 + 0.904167i \(0.640493\pi\)
\(138\) 0 0
\(139\) −16.0000 −1.35710 −0.678551 − 0.734553i \(-0.737392\pi\)
−0.678551 + 0.734553i \(0.737392\pi\)
\(140\) 0 0
\(141\) −10.0000 −0.842152
\(142\) 0 0
\(143\) 6.00000 0.501745
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −1.00000 −0.0824786
\(148\) 0 0
\(149\) 3.00000 0.245770 0.122885 − 0.992421i \(-0.460785\pi\)
0.122885 + 0.992421i \(0.460785\pi\)
\(150\) 0 0
\(151\) 1.00000 0.0813788 0.0406894 − 0.999172i \(-0.487045\pi\)
0.0406894 + 0.999172i \(0.487045\pi\)
\(152\) 0 0
\(153\) 4.00000 0.323381
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 4.00000 0.319235 0.159617 − 0.987179i \(-0.448974\pi\)
0.159617 + 0.987179i \(0.448974\pi\)
\(158\) 0 0
\(159\) 10.0000 0.793052
\(160\) 0 0
\(161\) −9.00000 −0.709299
\(162\) 0 0
\(163\) −16.0000 −1.25322 −0.626608 − 0.779334i \(-0.715557\pi\)
−0.626608 + 0.779334i \(0.715557\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 18.0000 1.39288 0.696441 − 0.717614i \(-0.254766\pi\)
0.696441 + 0.717614i \(0.254766\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) −2.00000 −0.152944
\(172\) 0 0
\(173\) −2.00000 −0.152057 −0.0760286 − 0.997106i \(-0.524224\pi\)
−0.0760286 + 0.997106i \(0.524224\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 6.00000 0.450988
\(178\) 0 0
\(179\) −12.0000 −0.896922 −0.448461 − 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) 16.0000 1.18927 0.594635 − 0.803996i \(-0.297296\pi\)
0.594635 + 0.803996i \(0.297296\pi\)
\(182\) 0 0
\(183\) 8.00000 0.591377
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −12.0000 −0.877527
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) −8.00000 −0.578860 −0.289430 − 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) 0 0
\(193\) −19.0000 −1.36765 −0.683825 − 0.729646i \(-0.739685\pi\)
−0.683825 + 0.729646i \(0.739685\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 21.0000 1.49619 0.748094 − 0.663593i \(-0.230969\pi\)
0.748094 + 0.663593i \(0.230969\pi\)
\(198\) 0 0
\(199\) −8.00000 −0.567105 −0.283552 − 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) 0 0
\(201\) −13.0000 −0.916949
\(202\) 0 0
\(203\) 7.00000 0.491304
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 9.00000 0.625543
\(208\) 0 0
\(209\) 6.00000 0.415029
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 − 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) 0 0
\(213\) −1.00000 −0.0685189
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −10.0000 −0.678844
\(218\) 0 0
\(219\) 4.00000 0.270295
\(220\) 0 0
\(221\) −8.00000 −0.538138
\(222\) 0 0
\(223\) 24.0000 1.60716 0.803579 − 0.595198i \(-0.202926\pi\)
0.803579 + 0.595198i \(0.202926\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 14.0000 0.929213 0.464606 − 0.885517i \(-0.346196\pi\)
0.464606 + 0.885517i \(0.346196\pi\)
\(228\) 0 0
\(229\) −14.0000 −0.925146 −0.462573 − 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) 0 0
\(231\) −3.00000 −0.197386
\(232\) 0 0
\(233\) −21.0000 −1.37576 −0.687878 − 0.725826i \(-0.741458\pi\)
−0.687878 + 0.725826i \(0.741458\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 13.0000 0.844441
\(238\) 0 0
\(239\) −20.0000 −1.29369 −0.646846 − 0.762620i \(-0.723912\pi\)
−0.646846 + 0.762620i \(0.723912\pi\)
\(240\) 0 0
\(241\) −10.0000 −0.644157 −0.322078 − 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 4.00000 0.254514
\(248\) 0 0
\(249\) 14.0000 0.887214
\(250\) 0 0
\(251\) 8.00000 0.504956 0.252478 − 0.967603i \(-0.418755\pi\)
0.252478 + 0.967603i \(0.418755\pi\)
\(252\) 0 0
\(253\) −27.0000 −1.69748
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −14.0000 −0.873296 −0.436648 − 0.899632i \(-0.643834\pi\)
−0.436648 + 0.899632i \(0.643834\pi\)
\(258\) 0 0
\(259\) 5.00000 0.310685
\(260\) 0 0
\(261\) −7.00000 −0.433289
\(262\) 0 0
\(263\) −17.0000 −1.04826 −0.524132 − 0.851637i \(-0.675610\pi\)
−0.524132 + 0.851637i \(0.675610\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −6.00000 −0.367194
\(268\) 0 0
\(269\) 14.0000 0.853595 0.426798 − 0.904347i \(-0.359642\pi\)
0.426798 + 0.904347i \(0.359642\pi\)
\(270\) 0 0
\(271\) −26.0000 −1.57939 −0.789694 − 0.613501i \(-0.789761\pi\)
−0.789694 + 0.613501i \(0.789761\pi\)
\(272\) 0 0
\(273\) −2.00000 −0.121046
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −26.0000 −1.56219 −0.781094 − 0.624413i \(-0.785338\pi\)
−0.781094 + 0.624413i \(0.785338\pi\)
\(278\) 0 0
\(279\) 10.0000 0.598684
\(280\) 0 0
\(281\) −3.00000 −0.178965 −0.0894825 − 0.995988i \(-0.528521\pi\)
−0.0894825 + 0.995988i \(0.528521\pi\)
\(282\) 0 0
\(283\) 10.0000 0.594438 0.297219 − 0.954809i \(-0.403941\pi\)
0.297219 + 0.954809i \(0.403941\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 8.00000 0.472225
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) −2.00000 −0.117242
\(292\) 0 0
\(293\) −18.0000 −1.05157 −0.525786 − 0.850617i \(-0.676229\pi\)
−0.525786 + 0.850617i \(0.676229\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 3.00000 0.174078
\(298\) 0 0
\(299\) −18.0000 −1.04097
\(300\) 0 0
\(301\) −7.00000 −0.403473
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 0 0
\(309\) 18.0000 1.02398
\(310\) 0 0
\(311\) −8.00000 −0.453638 −0.226819 − 0.973937i \(-0.572833\pi\)
−0.226819 + 0.973937i \(0.572833\pi\)
\(312\) 0 0
\(313\) −14.0000 −0.791327 −0.395663 − 0.918396i \(-0.629485\pi\)
−0.395663 + 0.918396i \(0.629485\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −7.00000 −0.393159 −0.196580 − 0.980488i \(-0.562983\pi\)
−0.196580 + 0.980488i \(0.562983\pi\)
\(318\) 0 0
\(319\) 21.0000 1.17577
\(320\) 0 0
\(321\) 4.00000 0.223258
\(322\) 0 0
\(323\) −8.00000 −0.445132
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 11.0000 0.608301
\(328\) 0 0
\(329\) −10.0000 −0.551318
\(330\) 0 0
\(331\) 11.0000 0.604615 0.302307 − 0.953211i \(-0.402243\pi\)
0.302307 + 0.953211i \(0.402243\pi\)
\(332\) 0 0
\(333\) −5.00000 −0.273998
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 2.00000 0.108947 0.0544735 − 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) 0 0
\(339\) −13.0000 −0.706063
\(340\) 0 0
\(341\) −30.0000 −1.62459
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 3.00000 0.161048 0.0805242 − 0.996753i \(-0.474341\pi\)
0.0805242 + 0.996753i \(0.474341\pi\)
\(348\) 0 0
\(349\) −2.00000 −0.107058 −0.0535288 − 0.998566i \(-0.517047\pi\)
−0.0535288 + 0.998566i \(0.517047\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) 0 0
\(353\) 14.0000 0.745145 0.372572 − 0.928003i \(-0.378476\pi\)
0.372572 + 0.928003i \(0.378476\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 4.00000 0.211702
\(358\) 0 0
\(359\) −19.0000 −1.00278 −0.501391 − 0.865221i \(-0.667178\pi\)
−0.501391 + 0.865221i \(0.667178\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) 2.00000 0.104973
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −4.00000 −0.208798 −0.104399 − 0.994535i \(-0.533292\pi\)
−0.104399 + 0.994535i \(0.533292\pi\)
\(368\) 0 0
\(369\) −8.00000 −0.416463
\(370\) 0 0
\(371\) 10.0000 0.519174
\(372\) 0 0
\(373\) 13.0000 0.673114 0.336557 − 0.941663i \(-0.390737\pi\)
0.336557 + 0.941663i \(0.390737\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 14.0000 0.721037
\(378\) 0 0
\(379\) −29.0000 −1.48963 −0.744815 − 0.667271i \(-0.767462\pi\)
−0.744815 + 0.667271i \(0.767462\pi\)
\(380\) 0 0
\(381\) 7.00000 0.358621
\(382\) 0 0
\(383\) −6.00000 −0.306586 −0.153293 − 0.988181i \(-0.548988\pi\)
−0.153293 + 0.988181i \(0.548988\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 7.00000 0.355830
\(388\) 0 0
\(389\) 17.0000 0.861934 0.430967 − 0.902368i \(-0.358172\pi\)
0.430967 + 0.902368i \(0.358172\pi\)
\(390\) 0 0
\(391\) 36.0000 1.82060
\(392\) 0 0
\(393\) −12.0000 −0.605320
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 14.0000 0.702640 0.351320 − 0.936255i \(-0.385733\pi\)
0.351320 + 0.936255i \(0.385733\pi\)
\(398\) 0 0
\(399\) −2.00000 −0.100125
\(400\) 0 0
\(401\) −3.00000 −0.149813 −0.0749064 − 0.997191i \(-0.523866\pi\)
−0.0749064 + 0.997191i \(0.523866\pi\)
\(402\) 0 0
\(403\) −20.0000 −0.996271
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 15.0000 0.743522
\(408\) 0 0
\(409\) −30.0000 −1.48340 −0.741702 − 0.670729i \(-0.765981\pi\)
−0.741702 + 0.670729i \(0.765981\pi\)
\(410\) 0 0
\(411\) 10.0000 0.493264
\(412\) 0 0
\(413\) 6.00000 0.295241
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 16.0000 0.783523
\(418\) 0 0
\(419\) 6.00000 0.293119 0.146560 − 0.989202i \(-0.453180\pi\)
0.146560 + 0.989202i \(0.453180\pi\)
\(420\) 0 0
\(421\) 19.0000 0.926003 0.463002 − 0.886357i \(-0.346772\pi\)
0.463002 + 0.886357i \(0.346772\pi\)
\(422\) 0 0
\(423\) 10.0000 0.486217
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 8.00000 0.387147
\(428\) 0 0
\(429\) −6.00000 −0.289683
\(430\) 0 0
\(431\) −24.0000 −1.15604 −0.578020 − 0.816023i \(-0.696174\pi\)
−0.578020 + 0.816023i \(0.696174\pi\)
\(432\) 0 0
\(433\) −28.0000 −1.34559 −0.672797 − 0.739827i \(-0.734907\pi\)
−0.672797 + 0.739827i \(0.734907\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −18.0000 −0.861057
\(438\) 0 0
\(439\) −26.0000 −1.24091 −0.620456 − 0.784241i \(-0.713053\pi\)
−0.620456 + 0.784241i \(0.713053\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 16.0000 0.760183 0.380091 − 0.924949i \(-0.375893\pi\)
0.380091 + 0.924949i \(0.375893\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −3.00000 −0.141895
\(448\) 0 0
\(449\) 27.0000 1.27421 0.637104 − 0.770778i \(-0.280132\pi\)
0.637104 + 0.770778i \(0.280132\pi\)
\(450\) 0 0
\(451\) 24.0000 1.13012
\(452\) 0 0
\(453\) −1.00000 −0.0469841
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −15.0000 −0.701670 −0.350835 − 0.936437i \(-0.614102\pi\)
−0.350835 + 0.936437i \(0.614102\pi\)
\(458\) 0 0
\(459\) −4.00000 −0.186704
\(460\) 0 0
\(461\) 18.0000 0.838344 0.419172 − 0.907907i \(-0.362320\pi\)
0.419172 + 0.907907i \(0.362320\pi\)
\(462\) 0 0
\(463\) 32.0000 1.48717 0.743583 − 0.668644i \(-0.233125\pi\)
0.743583 + 0.668644i \(0.233125\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −26.0000 −1.20314 −0.601568 − 0.798821i \(-0.705457\pi\)
−0.601568 + 0.798821i \(0.705457\pi\)
\(468\) 0 0
\(469\) −13.0000 −0.600284
\(470\) 0 0
\(471\) −4.00000 −0.184310
\(472\) 0 0
\(473\) −21.0000 −0.965581
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −10.0000 −0.457869
\(478\) 0 0
\(479\) 22.0000 1.00521 0.502603 − 0.864517i \(-0.332376\pi\)
0.502603 + 0.864517i \(0.332376\pi\)
\(480\) 0 0
\(481\) 10.0000 0.455961
\(482\) 0 0
\(483\) 9.00000 0.409514
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −29.0000 −1.31412 −0.657058 − 0.753840i \(-0.728199\pi\)
−0.657058 + 0.753840i \(0.728199\pi\)
\(488\) 0 0
\(489\) 16.0000 0.723545
\(490\) 0 0
\(491\) −39.0000 −1.76005 −0.880023 − 0.474932i \(-0.842473\pi\)
−0.880023 + 0.474932i \(0.842473\pi\)
\(492\) 0 0
\(493\) −28.0000 −1.26106
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −1.00000 −0.0448561
\(498\) 0 0
\(499\) 20.0000 0.895323 0.447661 − 0.894203i \(-0.352257\pi\)
0.447661 + 0.894203i \(0.352257\pi\)
\(500\) 0 0
\(501\) −18.0000 −0.804181
\(502\) 0 0
\(503\) 34.0000 1.51599 0.757993 − 0.652263i \(-0.226180\pi\)
0.757993 + 0.652263i \(0.226180\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 9.00000 0.399704
\(508\) 0 0
\(509\) 42.0000 1.86162 0.930809 − 0.365507i \(-0.119104\pi\)
0.930809 + 0.365507i \(0.119104\pi\)
\(510\) 0 0
\(511\) 4.00000 0.176950
\(512\) 0 0
\(513\) 2.00000 0.0883022
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −30.0000 −1.31940
\(518\) 0 0
\(519\) 2.00000 0.0877903
\(520\) 0 0
\(521\) 14.0000 0.613351 0.306676 − 0.951814i \(-0.400783\pi\)
0.306676 + 0.951814i \(0.400783\pi\)
\(522\) 0 0
\(523\) 16.0000 0.699631 0.349816 − 0.936819i \(-0.386244\pi\)
0.349816 + 0.936819i \(0.386244\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 40.0000 1.74243
\(528\) 0 0
\(529\) 58.0000 2.52174
\(530\) 0 0
\(531\) −6.00000 −0.260378
\(532\) 0 0
\(533\) 16.0000 0.693037
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 12.0000 0.517838
\(538\) 0 0
\(539\) −3.00000 −0.129219
\(540\) 0 0
\(541\) −25.0000 −1.07483 −0.537417 − 0.843317i \(-0.680600\pi\)
−0.537417 + 0.843317i \(0.680600\pi\)
\(542\) 0 0
\(543\) −16.0000 −0.686626
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −35.0000 −1.49649 −0.748246 − 0.663421i \(-0.769104\pi\)
−0.748246 + 0.663421i \(0.769104\pi\)
\(548\) 0 0
\(549\) −8.00000 −0.341432
\(550\) 0 0
\(551\) 14.0000 0.596420
\(552\) 0 0
\(553\) 13.0000 0.552816
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 37.0000 1.56774 0.783870 − 0.620925i \(-0.213243\pi\)
0.783870 + 0.620925i \(0.213243\pi\)
\(558\) 0 0
\(559\) −14.0000 −0.592137
\(560\) 0 0
\(561\) 12.0000 0.506640
\(562\) 0 0
\(563\) 26.0000 1.09577 0.547885 − 0.836554i \(-0.315433\pi\)
0.547885 + 0.836554i \(0.315433\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) 13.0000 0.544988 0.272494 − 0.962157i \(-0.412151\pi\)
0.272494 + 0.962157i \(0.412151\pi\)
\(570\) 0 0
\(571\) 31.0000 1.29731 0.648655 − 0.761083i \(-0.275332\pi\)
0.648655 + 0.761083i \(0.275332\pi\)
\(572\) 0 0
\(573\) 8.00000 0.334205
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −26.0000 −1.08239 −0.541197 − 0.840896i \(-0.682029\pi\)
−0.541197 + 0.840896i \(0.682029\pi\)
\(578\) 0 0
\(579\) 19.0000 0.789613
\(580\) 0 0
\(581\) 14.0000 0.580818
\(582\) 0 0
\(583\) 30.0000 1.24247
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 16.0000 0.660391 0.330195 − 0.943913i \(-0.392885\pi\)
0.330195 + 0.943913i \(0.392885\pi\)
\(588\) 0 0
\(589\) −20.0000 −0.824086
\(590\) 0 0
\(591\) −21.0000 −0.863825
\(592\) 0 0
\(593\) 16.0000 0.657041 0.328521 − 0.944497i \(-0.393450\pi\)
0.328521 + 0.944497i \(0.393450\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 8.00000 0.327418
\(598\) 0 0
\(599\) 11.0000 0.449448 0.224724 − 0.974422i \(-0.427852\pi\)
0.224724 + 0.974422i \(0.427852\pi\)
\(600\) 0 0
\(601\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(602\) 0 0
\(603\) 13.0000 0.529401
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −18.0000 −0.730597 −0.365299 − 0.930890i \(-0.619033\pi\)
−0.365299 + 0.930890i \(0.619033\pi\)
\(608\) 0 0
\(609\) −7.00000 −0.283654
\(610\) 0 0
\(611\) −20.0000 −0.809113
\(612\) 0 0
\(613\) −29.0000 −1.17130 −0.585649 − 0.810564i \(-0.699160\pi\)
−0.585649 + 0.810564i \(0.699160\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −25.0000 −1.00646 −0.503231 − 0.864152i \(-0.667856\pi\)
−0.503231 + 0.864152i \(0.667856\pi\)
\(618\) 0 0
\(619\) −10.0000 −0.401934 −0.200967 − 0.979598i \(-0.564408\pi\)
−0.200967 + 0.979598i \(0.564408\pi\)
\(620\) 0 0
\(621\) −9.00000 −0.361158
\(622\) 0 0
\(623\) −6.00000 −0.240385
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −6.00000 −0.239617
\(628\) 0 0
\(629\) −20.0000 −0.797452
\(630\) 0 0
\(631\) −25.0000 −0.995234 −0.497617 − 0.867397i \(-0.665792\pi\)
−0.497617 + 0.867397i \(0.665792\pi\)
\(632\) 0 0
\(633\) −4.00000 −0.158986
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −2.00000 −0.0792429
\(638\) 0 0
\(639\) 1.00000 0.0395594
\(640\) 0 0
\(641\) 33.0000 1.30342 0.651711 − 0.758468i \(-0.274052\pi\)
0.651711 + 0.758468i \(0.274052\pi\)
\(642\) 0 0
\(643\) −4.00000 −0.157745 −0.0788723 − 0.996885i \(-0.525132\pi\)
−0.0788723 + 0.996885i \(0.525132\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 18.0000 0.707653 0.353827 − 0.935311i \(-0.384880\pi\)
0.353827 + 0.935311i \(0.384880\pi\)
\(648\) 0 0
\(649\) 18.0000 0.706562
\(650\) 0 0
\(651\) 10.0000 0.391931
\(652\) 0 0
\(653\) −6.00000 −0.234798 −0.117399 − 0.993085i \(-0.537456\pi\)
−0.117399 + 0.993085i \(0.537456\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −4.00000 −0.156055
\(658\) 0 0
\(659\) −20.0000 −0.779089 −0.389545 − 0.921008i \(-0.627368\pi\)
−0.389545 + 0.921008i \(0.627368\pi\)
\(660\) 0 0
\(661\) 22.0000 0.855701 0.427850 − 0.903850i \(-0.359271\pi\)
0.427850 + 0.903850i \(0.359271\pi\)
\(662\) 0 0
\(663\) 8.00000 0.310694
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −63.0000 −2.43937
\(668\) 0 0
\(669\) −24.0000 −0.927894
\(670\) 0 0
\(671\) 24.0000 0.926510
\(672\) 0 0
\(673\) −10.0000 −0.385472 −0.192736 − 0.981251i \(-0.561736\pi\)
−0.192736 + 0.981251i \(0.561736\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 10.0000 0.384331 0.192166 − 0.981363i \(-0.438449\pi\)
0.192166 + 0.981363i \(0.438449\pi\)
\(678\) 0 0
\(679\) −2.00000 −0.0767530
\(680\) 0 0
\(681\) −14.0000 −0.536481
\(682\) 0 0
\(683\) −45.0000 −1.72188 −0.860939 − 0.508709i \(-0.830123\pi\)
−0.860939 + 0.508709i \(0.830123\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 14.0000 0.534133
\(688\) 0 0
\(689\) 20.0000 0.761939
\(690\) 0 0
\(691\) 48.0000 1.82601 0.913003 − 0.407953i \(-0.133757\pi\)
0.913003 + 0.407953i \(0.133757\pi\)
\(692\) 0 0
\(693\) 3.00000 0.113961
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −32.0000 −1.21209
\(698\) 0 0
\(699\) 21.0000 0.794293
\(700\) 0 0
\(701\) 22.0000 0.830929 0.415464 − 0.909610i \(-0.363619\pi\)
0.415464 + 0.909610i \(0.363619\pi\)
\(702\) 0 0
\(703\) 10.0000 0.377157
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −34.0000 −1.27690 −0.638448 − 0.769665i \(-0.720423\pi\)
−0.638448 + 0.769665i \(0.720423\pi\)
\(710\) 0 0
\(711\) −13.0000 −0.487538
\(712\) 0 0
\(713\) 90.0000 3.37053
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 20.0000 0.746914
\(718\) 0 0
\(719\) 20.0000 0.745874 0.372937 − 0.927857i \(-0.378351\pi\)
0.372937 + 0.927857i \(0.378351\pi\)
\(720\) 0 0
\(721\) 18.0000 0.670355
\(722\) 0 0
\(723\) 10.0000 0.371904
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 40.0000 1.48352 0.741759 − 0.670667i \(-0.233992\pi\)
0.741759 + 0.670667i \(0.233992\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 28.0000 1.03562
\(732\) 0 0
\(733\) −28.0000 −1.03420 −0.517102 − 0.855924i \(-0.672989\pi\)
−0.517102 + 0.855924i \(0.672989\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −39.0000 −1.43658
\(738\) 0 0
\(739\) 25.0000 0.919640 0.459820 − 0.888012i \(-0.347914\pi\)
0.459820 + 0.888012i \(0.347914\pi\)
\(740\) 0 0
\(741\) −4.00000 −0.146944
\(742\) 0 0
\(743\) 20.0000 0.733729 0.366864 − 0.930274i \(-0.380431\pi\)
0.366864 + 0.930274i \(0.380431\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −14.0000 −0.512233
\(748\) 0 0
\(749\) 4.00000 0.146157
\(750\) 0 0
\(751\) −12.0000 −0.437886 −0.218943 − 0.975738i \(-0.570261\pi\)
−0.218943 + 0.975738i \(0.570261\pi\)
\(752\) 0 0
\(753\) −8.00000 −0.291536
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 47.0000 1.70824 0.854122 − 0.520073i \(-0.174095\pi\)
0.854122 + 0.520073i \(0.174095\pi\)
\(758\) 0 0
\(759\) 27.0000 0.980038
\(760\) 0 0
\(761\) −32.0000 −1.16000 −0.580000 − 0.814617i \(-0.696947\pi\)
−0.580000 + 0.814617i \(0.696947\pi\)
\(762\) 0 0
\(763\) 11.0000 0.398227
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 12.0000 0.433295
\(768\) 0 0
\(769\) −36.0000 −1.29819 −0.649097 − 0.760706i \(-0.724853\pi\)
−0.649097 + 0.760706i \(0.724853\pi\)
\(770\) 0 0
\(771\) 14.0000 0.504198
\(772\) 0 0
\(773\) 2.00000 0.0719350 0.0359675 − 0.999353i \(-0.488549\pi\)
0.0359675 + 0.999353i \(0.488549\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −5.00000 −0.179374
\(778\) 0 0
\(779\) 16.0000 0.573259
\(780\) 0 0
\(781\) −3.00000 −0.107348
\(782\) 0 0
\(783\) 7.00000 0.250160
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −32.0000 −1.14068 −0.570338 − 0.821410i \(-0.693188\pi\)
−0.570338 + 0.821410i \(0.693188\pi\)
\(788\) 0 0
\(789\) 17.0000 0.605216
\(790\) 0 0
\(791\) −13.0000 −0.462227
\(792\) 0 0
\(793\) 16.0000 0.568177
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) 0 0
\(799\) 40.0000 1.41510
\(800\) 0 0
\(801\) 6.00000 0.212000
\(802\) 0 0
\(803\) 12.0000 0.423471
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −14.0000 −0.492823
\(808\) 0 0
\(809\) 23.0000 0.808637 0.404318 − 0.914618i \(-0.367509\pi\)
0.404318 + 0.914618i \(0.367509\pi\)
\(810\) 0 0
\(811\) 42.0000 1.47482 0.737410 − 0.675446i \(-0.236049\pi\)
0.737410 + 0.675446i \(0.236049\pi\)
\(812\) 0 0
\(813\) 26.0000 0.911860
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −14.0000 −0.489798
\(818\) 0 0
\(819\) 2.00000 0.0698857
\(820\) 0 0
\(821\) −30.0000 −1.04701 −0.523504 − 0.852023i \(-0.675375\pi\)
−0.523504 + 0.852023i \(0.675375\pi\)
\(822\) 0 0
\(823\) 29.0000 1.01088 0.505438 − 0.862863i \(-0.331331\pi\)
0.505438 + 0.862863i \(0.331331\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −9.00000 −0.312961 −0.156480 − 0.987681i \(-0.550015\pi\)
−0.156480 + 0.987681i \(0.550015\pi\)
\(828\) 0 0
\(829\) 16.0000 0.555703 0.277851 − 0.960624i \(-0.410378\pi\)
0.277851 + 0.960624i \(0.410378\pi\)
\(830\) 0 0
\(831\) 26.0000 0.901930
\(832\) 0 0
\(833\) 4.00000 0.138592
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −10.0000 −0.345651
\(838\) 0 0
\(839\) −36.0000 −1.24286 −0.621429 − 0.783470i \(-0.713448\pi\)
−0.621429 + 0.783470i \(0.713448\pi\)
\(840\) 0 0
\(841\) 20.0000 0.689655
\(842\) 0 0
\(843\) 3.00000 0.103325
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 2.00000 0.0687208
\(848\) 0 0
\(849\) −10.0000 −0.343199
\(850\) 0 0
\(851\) −45.0000 −1.54258
\(852\) 0 0
\(853\) 40.0000 1.36957 0.684787 − 0.728743i \(-0.259895\pi\)
0.684787 + 0.728743i \(0.259895\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 24.0000 0.819824 0.409912 − 0.912125i \(-0.365559\pi\)
0.409912 + 0.912125i \(0.365559\pi\)
\(858\) 0 0
\(859\) −22.0000 −0.750630 −0.375315 − 0.926897i \(-0.622466\pi\)
−0.375315 + 0.926897i \(0.622466\pi\)
\(860\) 0 0
\(861\) −8.00000 −0.272639
\(862\) 0 0
\(863\) −33.0000 −1.12333 −0.561667 − 0.827364i \(-0.689840\pi\)
−0.561667 + 0.827364i \(0.689840\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 1.00000 0.0339618
\(868\) 0 0
\(869\) 39.0000 1.32298
\(870\) 0 0
\(871\) −26.0000 −0.880976
\(872\) 0 0
\(873\) 2.00000 0.0676897
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −6.00000 −0.202606 −0.101303 − 0.994856i \(-0.532301\pi\)
−0.101303 + 0.994856i \(0.532301\pi\)
\(878\) 0 0
\(879\) 18.0000 0.607125
\(880\) 0 0
\(881\) −2.00000 −0.0673817 −0.0336909 − 0.999432i \(-0.510726\pi\)
−0.0336909 + 0.999432i \(0.510726\pi\)
\(882\) 0 0
\(883\) 13.0000 0.437485 0.218742 − 0.975783i \(-0.429805\pi\)
0.218742 + 0.975783i \(0.429805\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 8.00000 0.268614 0.134307 − 0.990940i \(-0.457119\pi\)
0.134307 + 0.990940i \(0.457119\pi\)
\(888\) 0 0
\(889\) 7.00000 0.234772
\(890\) 0 0
\(891\) −3.00000 −0.100504
\(892\) 0 0
\(893\) −20.0000 −0.669274
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 18.0000 0.601003
\(898\) 0 0
\(899\) −70.0000 −2.33463
\(900\) 0 0
\(901\) −40.0000 −1.33259
\(902\) 0 0
\(903\) 7.00000 0.232945
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −52.0000 −1.72663 −0.863316 − 0.504664i \(-0.831616\pi\)
−0.863316 + 0.504664i \(0.831616\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 3.00000 0.0993944 0.0496972 − 0.998764i \(-0.484174\pi\)
0.0496972 + 0.998764i \(0.484174\pi\)
\(912\) 0 0
\(913\) 42.0000 1.39000
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −12.0000 −0.396275
\(918\) 0 0
\(919\) 51.0000 1.68233 0.841167 − 0.540775i \(-0.181869\pi\)
0.841167 + 0.540775i \(0.181869\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −2.00000 −0.0658308
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −18.0000 −0.591198
\(928\) 0 0
\(929\) 24.0000 0.787414 0.393707 − 0.919236i \(-0.371192\pi\)
0.393707 + 0.919236i \(0.371192\pi\)
\(930\) 0 0
\(931\) −2.00000 −0.0655474
\(932\) 0 0
\(933\) 8.00000 0.261908
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 56.0000 1.82944 0.914720 − 0.404088i \(-0.132411\pi\)
0.914720 + 0.404088i \(0.132411\pi\)
\(938\) 0 0
\(939\) 14.0000 0.456873
\(940\) 0 0
\(941\) 46.0000 1.49956 0.749779 − 0.661689i \(-0.230160\pi\)
0.749779 + 0.661689i \(0.230160\pi\)
\(942\) 0 0
\(943\) −72.0000 −2.34464
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 4.00000 0.129983 0.0649913 − 0.997886i \(-0.479298\pi\)
0.0649913 + 0.997886i \(0.479298\pi\)
\(948\) 0 0
\(949\) 8.00000 0.259691
\(950\) 0 0
\(951\) 7.00000 0.226991
\(952\) 0 0
\(953\) −47.0000 −1.52248 −0.761240 − 0.648471i \(-0.775409\pi\)
−0.761240 + 0.648471i \(0.775409\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −21.0000 −0.678834
\(958\) 0 0
\(959\) 10.0000 0.322917
\(960\) 0 0
\(961\) 69.0000 2.22581
\(962\) 0 0
\(963\) −4.00000 −0.128898
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −24.0000 −0.771788 −0.385894 − 0.922543i \(-0.626107\pi\)
−0.385894 + 0.922543i \(0.626107\pi\)
\(968\) 0 0
\(969\) 8.00000 0.256997
\(970\) 0 0
\(971\) −30.0000 −0.962746 −0.481373 − 0.876516i \(-0.659862\pi\)
−0.481373 + 0.876516i \(0.659862\pi\)
\(972\) 0 0
\(973\) 16.0000 0.512936
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −33.0000 −1.05576 −0.527882 − 0.849318i \(-0.677014\pi\)
−0.527882 + 0.849318i \(0.677014\pi\)
\(978\) 0 0
\(979\) −18.0000 −0.575282
\(980\) 0 0
\(981\) −11.0000 −0.351203
\(982\) 0 0
\(983\) 54.0000 1.72233 0.861166 − 0.508323i \(-0.169735\pi\)
0.861166 + 0.508323i \(0.169735\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 10.0000 0.318304
\(988\) 0 0
\(989\) 63.0000 2.00328
\(990\) 0 0
\(991\) −17.0000 −0.540023 −0.270011 − 0.962857i \(-0.587027\pi\)
−0.270011 + 0.962857i \(0.587027\pi\)
\(992\) 0 0
\(993\) −11.0000 −0.349074
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 38.0000 1.20347 0.601736 − 0.798695i \(-0.294476\pi\)
0.601736 + 0.798695i \(0.294476\pi\)
\(998\) 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 4200.2.a.a.1.1 ✓ 1
4.3 odd 2 8400.2.a.cr.1.1 1
5.2 odd 4 4200.2.t.f.1849.2 2
5.3 odd 4 4200.2.t.f.1849.1 2
5.4 even 2 4200.2.a.bc.1.1 yes 1
20.19 odd 2 8400.2.a.j.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4200.2.a.a.1.1 ✓ 1 1.1 even 1 trivial
4200.2.a.bc.1.1 yes 1 5.4 even 2
4200.2.t.f.1849.1 2 5.3 odd 4
4200.2.t.f.1849.2 2 5.2 odd 4
8400.2.a.j.1.1 1 20.19 odd 2
8400.2.a.cr.1.1 1 4.3 odd 2