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Properties

Label 6384.2.a.r.1.1
Level $6384$
Weight $2$
Character 6384.1
Self dual yes
Analytic conductor $50.976$
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] = [6384,2,Mod(1,6384)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6384, 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("6384.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6384 = 2^{4} \cdot 3 \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6384.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: \(50.9764966504\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3192)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 6384.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -4.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -4.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} -2.00000 q^{13} -4.00000 q^{15} +4.00000 q^{17} -1.00000 q^{19} +1.00000 q^{21} -4.00000 q^{23} +11.0000 q^{25} +1.00000 q^{27} +4.00000 q^{29} -4.00000 q^{35} -10.0000 q^{37} -2.00000 q^{39} +10.0000 q^{41} -4.00000 q^{43} -4.00000 q^{45} +6.00000 q^{47} +1.00000 q^{49} +4.00000 q^{51} -1.00000 q^{57} -4.00000 q^{59} -10.0000 q^{61} +1.00000 q^{63} +8.00000 q^{65} +4.00000 q^{67} -4.00000 q^{69} +2.00000 q^{71} -2.00000 q^{73} +11.0000 q^{75} -4.00000 q^{79} +1.00000 q^{81} +14.0000 q^{83} -16.0000 q^{85} +4.00000 q^{87} -6.00000 q^{89} -2.00000 q^{91} +4.00000 q^{95} -18.0000 q^{97} +O(q^{100})\)

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
\(5\) −4.00000 −1.78885 −0.894427 − 0.447214i \(-0.852416\pi\)
−0.894427 + 0.447214i \(0.852416\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) −4.00000 −1.03280
\(16\) 0 0
\(17\) 4.00000 0.970143 0.485071 − 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) −4.00000 −0.834058 −0.417029 − 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) 11.0000 2.20000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 4.00000 0.742781 0.371391 − 0.928477i \(-0.378881\pi\)
0.371391 + 0.928477i \(0.378881\pi\)
\(30\) 0 0
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −4.00000 −0.676123
\(36\) 0 0
\(37\) −10.0000 −1.64399 −0.821995 − 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) 0 0
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) 10.0000 1.56174 0.780869 − 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) −4.00000 −0.596285
\(46\) 0 0
\(47\) 6.00000 0.875190 0.437595 − 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 4.00000 0.560112
\(52\) 0 0
\(53\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −1.00000 −0.132453
\(58\) 0 0
\(59\) −4.00000 −0.520756 −0.260378 − 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 − 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) 8.00000 0.992278
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 − 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 0 0
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) 2.00000 0.237356 0.118678 − 0.992933i \(-0.462134\pi\)
0.118678 + 0.992933i \(0.462134\pi\)
\(72\) 0 0
\(73\) −2.00000 −0.234082 −0.117041 − 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) 0 0
\(75\) 11.0000 1.27017
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 − 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 14.0000 1.53670 0.768350 − 0.640030i \(-0.221078\pi\)
0.768350 + 0.640030i \(0.221078\pi\)
\(84\) 0 0
\(85\) −16.0000 −1.73544
\(86\) 0 0
\(87\) 4.00000 0.428845
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 − 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) −2.00000 −0.209657
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 4.00000 0.410391
\(96\) 0 0
\(97\) −18.0000 −1.82762 −0.913812 − 0.406138i \(-0.866875\pi\)
−0.913812 + 0.406138i \(0.866875\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 4.00000 0.398015 0.199007 − 0.979998i \(-0.436228\pi\)
0.199007 + 0.979998i \(0.436228\pi\)
\(102\) 0 0
\(103\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) −4.00000 −0.390360
\(106\) 0 0
\(107\) −18.0000 −1.74013 −0.870063 − 0.492941i \(-0.835922\pi\)
−0.870063 + 0.492941i \(0.835922\pi\)
\(108\) 0 0
\(109\) 2.00000 0.191565 0.0957826 − 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) −10.0000 −0.949158
\(112\) 0 0
\(113\) 16.0000 1.50515 0.752577 − 0.658505i \(-0.228811\pi\)
0.752577 + 0.658505i \(0.228811\pi\)
\(114\) 0 0
\(115\) 16.0000 1.49201
\(116\) 0 0
\(117\) −2.00000 −0.184900
\(118\) 0 0
\(119\) 4.00000 0.366679
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) 10.0000 0.901670
\(124\) 0 0
\(125\) −24.0000 −2.14663
\(126\) 0 0
\(127\) −8.00000 −0.709885 −0.354943 − 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 0 0
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) −6.00000 −0.524222 −0.262111 − 0.965038i \(-0.584419\pi\)
−0.262111 + 0.965038i \(0.584419\pi\)
\(132\) 0 0
\(133\) −1.00000 −0.0867110
\(134\) 0 0
\(135\) −4.00000 −0.344265
\(136\) 0 0
\(137\) −14.0000 −1.19610 −0.598050 − 0.801459i \(-0.704058\pi\)
−0.598050 + 0.801459i \(0.704058\pi\)
\(138\) 0 0
\(139\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 6.00000 0.505291
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −16.0000 −1.32873
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 0 0
\(149\) −6.00000 −0.491539 −0.245770 − 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) 4.00000 0.323381
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −2.00000 −0.159617 −0.0798087 − 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −4.00000 −0.315244
\(162\) 0 0
\(163\) 4.00000 0.313304 0.156652 − 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 12.0000 0.928588 0.464294 − 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) 0 0
\(173\) −18.0000 −1.36851 −0.684257 − 0.729241i \(-0.739873\pi\)
−0.684257 + 0.729241i \(0.739873\pi\)
\(174\) 0 0
\(175\) 11.0000 0.831522
\(176\) 0 0
\(177\) −4.00000 −0.300658
\(178\) 0 0
\(179\) −6.00000 −0.448461 −0.224231 − 0.974536i \(-0.571987\pi\)
−0.224231 + 0.974536i \(0.571987\pi\)
\(180\) 0 0
\(181\) −22.0000 −1.63525 −0.817624 − 0.575753i \(-0.804709\pi\)
−0.817624 + 0.575753i \(0.804709\pi\)
\(182\) 0 0
\(183\) −10.0000 −0.739221
\(184\) 0 0
\(185\) 40.0000 2.94086
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) −20.0000 −1.44715 −0.723575 − 0.690246i \(-0.757502\pi\)
−0.723575 + 0.690246i \(0.757502\pi\)
\(192\) 0 0
\(193\) −6.00000 −0.431889 −0.215945 − 0.976406i \(-0.569283\pi\)
−0.215945 + 0.976406i \(0.569283\pi\)
\(194\) 0 0
\(195\) 8.00000 0.572892
\(196\) 0 0
\(197\) 6.00000 0.427482 0.213741 − 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 0 0
\(199\) −4.00000 −0.283552 −0.141776 − 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) 0 0
\(201\) 4.00000 0.282138
\(202\) 0 0
\(203\) 4.00000 0.280745
\(204\) 0 0
\(205\) −40.0000 −2.79372
\(206\) 0 0
\(207\) −4.00000 −0.278019
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 8.00000 0.550743 0.275371 − 0.961338i \(-0.411199\pi\)
0.275371 + 0.961338i \(0.411199\pi\)
\(212\) 0 0
\(213\) 2.00000 0.137038
\(214\) 0 0
\(215\) 16.0000 1.09119
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −2.00000 −0.135147
\(220\) 0 0
\(221\) −8.00000 −0.538138
\(222\) 0 0
\(223\) 16.0000 1.07144 0.535720 − 0.844396i \(-0.320040\pi\)
0.535720 + 0.844396i \(0.320040\pi\)
\(224\) 0 0
\(225\) 11.0000 0.733333
\(226\) 0 0
\(227\) 16.0000 1.06196 0.530979 − 0.847385i \(-0.321824\pi\)
0.530979 + 0.847385i \(0.321824\pi\)
\(228\) 0 0
\(229\) −6.00000 −0.396491 −0.198246 − 0.980152i \(-0.563524\pi\)
−0.198246 + 0.980152i \(0.563524\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −18.0000 −1.17922 −0.589610 − 0.807688i \(-0.700718\pi\)
−0.589610 + 0.807688i \(0.700718\pi\)
\(234\) 0 0
\(235\) −24.0000 −1.56559
\(236\) 0 0
\(237\) −4.00000 −0.259828
\(238\) 0 0
\(239\) 24.0000 1.55243 0.776215 − 0.630468i \(-0.217137\pi\)
0.776215 + 0.630468i \(0.217137\pi\)
\(240\) 0 0
\(241\) −2.00000 −0.128831 −0.0644157 − 0.997923i \(-0.520518\pi\)
−0.0644157 + 0.997923i \(0.520518\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −4.00000 −0.255551
\(246\) 0 0
\(247\) 2.00000 0.127257
\(248\) 0 0
\(249\) 14.0000 0.887214
\(250\) 0 0
\(251\) −6.00000 −0.378717 −0.189358 − 0.981908i \(-0.560641\pi\)
−0.189358 + 0.981908i \(0.560641\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) −16.0000 −1.00196
\(256\) 0 0
\(257\) 14.0000 0.873296 0.436648 − 0.899632i \(-0.356166\pi\)
0.436648 + 0.899632i \(0.356166\pi\)
\(258\) 0 0
\(259\) −10.0000 −0.621370
\(260\) 0 0
\(261\) 4.00000 0.247594
\(262\) 0 0
\(263\) −32.0000 −1.97320 −0.986602 − 0.163144i \(-0.947836\pi\)
−0.986602 + 0.163144i \(0.947836\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −6.00000 −0.367194
\(268\) 0 0
\(269\) 18.0000 1.09748 0.548740 − 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) 0 0
\(271\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 0 0
\(273\) −2.00000 −0.121046
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 10.0000 0.600842 0.300421 − 0.953807i \(-0.402873\pi\)
0.300421 + 0.953807i \(0.402873\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −32.0000 −1.90896 −0.954480 − 0.298275i \(-0.903589\pi\)
−0.954480 + 0.298275i \(0.903589\pi\)
\(282\) 0 0
\(283\) −8.00000 −0.475551 −0.237775 − 0.971320i \(-0.576418\pi\)
−0.237775 + 0.971320i \(0.576418\pi\)
\(284\) 0 0
\(285\) 4.00000 0.236940
\(286\) 0 0
\(287\) 10.0000 0.590281
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) −18.0000 −1.05518
\(292\) 0 0
\(293\) −18.0000 −1.05157 −0.525786 − 0.850617i \(-0.676229\pi\)
−0.525786 + 0.850617i \(0.676229\pi\)
\(294\) 0 0
\(295\) 16.0000 0.931556
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 8.00000 0.462652
\(300\) 0 0
\(301\) −4.00000 −0.230556
\(302\) 0 0
\(303\) 4.00000 0.229794
\(304\) 0 0
\(305\) 40.0000 2.29039
\(306\) 0 0
\(307\) −20.0000 −1.14146 −0.570730 − 0.821138i \(-0.693340\pi\)
−0.570730 + 0.821138i \(0.693340\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −18.0000 −1.02069 −0.510343 − 0.859971i \(-0.670482\pi\)
−0.510343 + 0.859971i \(0.670482\pi\)
\(312\) 0 0
\(313\) −14.0000 −0.791327 −0.395663 − 0.918396i \(-0.629485\pi\)
−0.395663 + 0.918396i \(0.629485\pi\)
\(314\) 0 0
\(315\) −4.00000 −0.225374
\(316\) 0 0
\(317\) 4.00000 0.224662 0.112331 − 0.993671i \(-0.464168\pi\)
0.112331 + 0.993671i \(0.464168\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −18.0000 −1.00466
\(322\) 0 0
\(323\) −4.00000 −0.222566
\(324\) 0 0
\(325\) −22.0000 −1.22034
\(326\) 0 0
\(327\) 2.00000 0.110600
\(328\) 0 0
\(329\) 6.00000 0.330791
\(330\) 0 0
\(331\) −12.0000 −0.659580 −0.329790 − 0.944054i \(-0.606978\pi\)
−0.329790 + 0.944054i \(0.606978\pi\)
\(332\) 0 0
\(333\) −10.0000 −0.547997
\(334\) 0 0
\(335\) −16.0000 −0.874173
\(336\) 0 0
\(337\) 22.0000 1.19842 0.599208 − 0.800593i \(-0.295482\pi\)
0.599208 + 0.800593i \(0.295482\pi\)
\(338\) 0 0
\(339\) 16.0000 0.869001
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 16.0000 0.861411
\(346\) 0 0
\(347\) 4.00000 0.214731 0.107366 − 0.994220i \(-0.465758\pi\)
0.107366 + 0.994220i \(0.465758\pi\)
\(348\) 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
\(353\) 4.00000 0.212899 0.106449 − 0.994318i \(-0.466052\pi\)
0.106449 + 0.994318i \(0.466052\pi\)
\(354\) 0 0
\(355\) −8.00000 −0.424596
\(356\) 0 0
\(357\) 4.00000 0.211702
\(358\) 0 0
\(359\) −4.00000 −0.211112 −0.105556 − 0.994413i \(-0.533662\pi\)
−0.105556 + 0.994413i \(0.533662\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −11.0000 −0.577350
\(364\) 0 0
\(365\) 8.00000 0.418739
\(366\) 0 0
\(367\) −12.0000 −0.626395 −0.313197 − 0.949688i \(-0.601400\pi\)
−0.313197 + 0.949688i \(0.601400\pi\)
\(368\) 0 0
\(369\) 10.0000 0.520579
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −26.0000 −1.34623 −0.673114 − 0.739538i \(-0.735044\pi\)
−0.673114 + 0.739538i \(0.735044\pi\)
\(374\) 0 0
\(375\) −24.0000 −1.23935
\(376\) 0 0
\(377\) −8.00000 −0.412021
\(378\) 0 0
\(379\) −36.0000 −1.84920 −0.924598 − 0.380945i \(-0.875599\pi\)
−0.924598 + 0.380945i \(0.875599\pi\)
\(380\) 0 0
\(381\) −8.00000 −0.409852
\(382\) 0 0
\(383\) 36.0000 1.83951 0.919757 − 0.392488i \(-0.128386\pi\)
0.919757 + 0.392488i \(0.128386\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −4.00000 −0.203331
\(388\) 0 0
\(389\) −30.0000 −1.52106 −0.760530 − 0.649303i \(-0.775061\pi\)
−0.760530 + 0.649303i \(0.775061\pi\)
\(390\) 0 0
\(391\) −16.0000 −0.809155
\(392\) 0 0
\(393\) −6.00000 −0.302660
\(394\) 0 0
\(395\) 16.0000 0.805047
\(396\) 0 0
\(397\) 6.00000 0.301131 0.150566 − 0.988600i \(-0.451890\pi\)
0.150566 + 0.988600i \(0.451890\pi\)
\(398\) 0 0
\(399\) −1.00000 −0.0500626
\(400\) 0 0
\(401\) −36.0000 −1.79775 −0.898877 − 0.438201i \(-0.855616\pi\)
−0.898877 + 0.438201i \(0.855616\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −4.00000 −0.198762
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −22.0000 −1.08783 −0.543915 − 0.839140i \(-0.683059\pi\)
−0.543915 + 0.839140i \(0.683059\pi\)
\(410\) 0 0
\(411\) −14.0000 −0.690569
\(412\) 0 0
\(413\) −4.00000 −0.196827
\(414\) 0 0
\(415\) −56.0000 −2.74893
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −2.00000 −0.0977064 −0.0488532 − 0.998806i \(-0.515557\pi\)
−0.0488532 + 0.998806i \(0.515557\pi\)
\(420\) 0 0
\(421\) −14.0000 −0.682318 −0.341159 − 0.940006i \(-0.610819\pi\)
−0.341159 + 0.940006i \(0.610819\pi\)
\(422\) 0 0
\(423\) 6.00000 0.291730
\(424\) 0 0
\(425\) 44.0000 2.13431
\(426\) 0 0
\(427\) −10.0000 −0.483934
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −30.0000 −1.44505 −0.722525 − 0.691345i \(-0.757018\pi\)
−0.722525 + 0.691345i \(0.757018\pi\)
\(432\) 0 0
\(433\) 14.0000 0.672797 0.336399 − 0.941720i \(-0.390791\pi\)
0.336399 + 0.941720i \(0.390791\pi\)
\(434\) 0 0
\(435\) −16.0000 −0.767141
\(436\) 0 0
\(437\) 4.00000 0.191346
\(438\) 0 0
\(439\) 40.0000 1.90910 0.954548 − 0.298057i \(-0.0963387\pi\)
0.954548 + 0.298057i \(0.0963387\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 12.0000 0.570137 0.285069 − 0.958507i \(-0.407984\pi\)
0.285069 + 0.958507i \(0.407984\pi\)
\(444\) 0 0
\(445\) 24.0000 1.13771
\(446\) 0 0
\(447\) −6.00000 −0.283790
\(448\) 0 0
\(449\) 20.0000 0.943858 0.471929 − 0.881636i \(-0.343558\pi\)
0.471929 + 0.881636i \(0.343558\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 8.00000 0.375046
\(456\) 0 0
\(457\) −10.0000 −0.467780 −0.233890 − 0.972263i \(-0.575146\pi\)
−0.233890 + 0.972263i \(0.575146\pi\)
\(458\) 0 0
\(459\) 4.00000 0.186704
\(460\) 0 0
\(461\) −28.0000 −1.30409 −0.652045 − 0.758180i \(-0.726089\pi\)
−0.652045 + 0.758180i \(0.726089\pi\)
\(462\) 0 0
\(463\) −24.0000 −1.11537 −0.557687 − 0.830051i \(-0.688311\pi\)
−0.557687 + 0.830051i \(0.688311\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 10.0000 0.462745 0.231372 − 0.972865i \(-0.425678\pi\)
0.231372 + 0.972865i \(0.425678\pi\)
\(468\) 0 0
\(469\) 4.00000 0.184703
\(470\) 0 0
\(471\) −2.00000 −0.0921551
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −11.0000 −0.504715
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −14.0000 −0.639676 −0.319838 − 0.947472i \(-0.603629\pi\)
−0.319838 + 0.947472i \(0.603629\pi\)
\(480\) 0 0
\(481\) 20.0000 0.911922
\(482\) 0 0
\(483\) −4.00000 −0.182006
\(484\) 0 0
\(485\) 72.0000 3.26935
\(486\) 0 0
\(487\) −4.00000 −0.181257 −0.0906287 − 0.995885i \(-0.528888\pi\)
−0.0906287 + 0.995885i \(0.528888\pi\)
\(488\) 0 0
\(489\) 4.00000 0.180886
\(490\) 0 0
\(491\) 36.0000 1.62466 0.812329 − 0.583200i \(-0.198200\pi\)
0.812329 + 0.583200i \(0.198200\pi\)
\(492\) 0 0
\(493\) 16.0000 0.720604
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 2.00000 0.0897123
\(498\) 0 0
\(499\) −4.00000 −0.179065 −0.0895323 − 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) 0 0
\(501\) 12.0000 0.536120
\(502\) 0 0
\(503\) 30.0000 1.33763 0.668817 − 0.743427i \(-0.266801\pi\)
0.668817 + 0.743427i \(0.266801\pi\)
\(504\) 0 0
\(505\) −16.0000 −0.711991
\(506\) 0 0
\(507\) −9.00000 −0.399704
\(508\) 0 0
\(509\) −6.00000 −0.265945 −0.132973 − 0.991120i \(-0.542452\pi\)
−0.132973 + 0.991120i \(0.542452\pi\)
\(510\) 0 0
\(511\) −2.00000 −0.0884748
\(512\) 0 0
\(513\) −1.00000 −0.0441511
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −18.0000 −0.790112
\(520\) 0 0
\(521\) 38.0000 1.66481 0.832405 − 0.554168i \(-0.186963\pi\)
0.832405 + 0.554168i \(0.186963\pi\)
\(522\) 0 0
\(523\) 28.0000 1.22435 0.612177 − 0.790721i \(-0.290294\pi\)
0.612177 + 0.790721i \(0.290294\pi\)
\(524\) 0 0
\(525\) 11.0000 0.480079
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) −4.00000 −0.173585
\(532\) 0 0
\(533\) −20.0000 −0.866296
\(534\) 0 0
\(535\) 72.0000 3.11283
\(536\) 0 0
\(537\) −6.00000 −0.258919
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 30.0000 1.28980 0.644900 − 0.764267i \(-0.276899\pi\)
0.644900 + 0.764267i \(0.276899\pi\)
\(542\) 0 0
\(543\) −22.0000 −0.944110
\(544\) 0 0
\(545\) −8.00000 −0.342682
\(546\) 0 0
\(547\) 40.0000 1.71028 0.855138 − 0.518400i \(-0.173472\pi\)
0.855138 + 0.518400i \(0.173472\pi\)
\(548\) 0 0
\(549\) −10.0000 −0.426790
\(550\) 0 0
\(551\) −4.00000 −0.170406
\(552\) 0 0
\(553\) −4.00000 −0.170097
\(554\) 0 0
\(555\) 40.0000 1.69791
\(556\) 0 0
\(557\) 10.0000 0.423714 0.211857 − 0.977301i \(-0.432049\pi\)
0.211857 + 0.977301i \(0.432049\pi\)
\(558\) 0 0
\(559\) 8.00000 0.338364
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 0 0
\(565\) −64.0000 −2.69250
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) 24.0000 1.00613 0.503066 − 0.864248i \(-0.332205\pi\)
0.503066 + 0.864248i \(0.332205\pi\)
\(570\) 0 0
\(571\) 36.0000 1.50655 0.753277 − 0.657704i \(-0.228472\pi\)
0.753277 + 0.657704i \(0.228472\pi\)
\(572\) 0 0
\(573\) −20.0000 −0.835512
\(574\) 0 0
\(575\) −44.0000 −1.83493
\(576\) 0 0
\(577\) 10.0000 0.416305 0.208153 − 0.978096i \(-0.433255\pi\)
0.208153 + 0.978096i \(0.433255\pi\)
\(578\) 0 0
\(579\) −6.00000 −0.249351
\(580\) 0 0
\(581\) 14.0000 0.580818
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 8.00000 0.330759
\(586\) 0 0
\(587\) −30.0000 −1.23823 −0.619116 − 0.785299i \(-0.712509\pi\)
−0.619116 + 0.785299i \(0.712509\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 6.00000 0.246807
\(592\) 0 0
\(593\) 16.0000 0.657041 0.328521 − 0.944497i \(-0.393450\pi\)
0.328521 + 0.944497i \(0.393450\pi\)
\(594\) 0 0
\(595\) −16.0000 −0.655936
\(596\) 0 0
\(597\) −4.00000 −0.163709
\(598\) 0 0
\(599\) 14.0000 0.572024 0.286012 − 0.958226i \(-0.407670\pi\)
0.286012 + 0.958226i \(0.407670\pi\)
\(600\) 0 0
\(601\) 6.00000 0.244745 0.122373 − 0.992484i \(-0.460950\pi\)
0.122373 + 0.992484i \(0.460950\pi\)
\(602\) 0 0
\(603\) 4.00000 0.162893
\(604\) 0 0
\(605\) 44.0000 1.78885
\(606\) 0 0
\(607\) −8.00000 −0.324710 −0.162355 − 0.986732i \(-0.551909\pi\)
−0.162355 + 0.986732i \(0.551909\pi\)
\(608\) 0 0
\(609\) 4.00000 0.162088
\(610\) 0 0
\(611\) −12.0000 −0.485468
\(612\) 0 0
\(613\) −30.0000 −1.21169 −0.605844 − 0.795583i \(-0.707165\pi\)
−0.605844 + 0.795583i \(0.707165\pi\)
\(614\) 0 0
\(615\) −40.0000 −1.61296
\(616\) 0 0
\(617\) −38.0000 −1.52982 −0.764911 − 0.644136i \(-0.777217\pi\)
−0.764911 + 0.644136i \(0.777217\pi\)
\(618\) 0 0
\(619\) 40.0000 1.60774 0.803868 − 0.594808i \(-0.202772\pi\)
0.803868 + 0.594808i \(0.202772\pi\)
\(620\) 0 0
\(621\) −4.00000 −0.160514
\(622\) 0 0
\(623\) −6.00000 −0.240385
\(624\) 0 0
\(625\) 41.0000 1.64000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −40.0000 −1.59490
\(630\) 0 0
\(631\) 32.0000 1.27390 0.636950 − 0.770905i \(-0.280196\pi\)
0.636950 + 0.770905i \(0.280196\pi\)
\(632\) 0 0
\(633\) 8.00000 0.317971
\(634\) 0 0
\(635\) 32.0000 1.26988
\(636\) 0 0
\(637\) −2.00000 −0.0792429
\(638\) 0 0
\(639\) 2.00000 0.0791188
\(640\) 0 0
\(641\) 4.00000 0.157991 0.0789953 − 0.996875i \(-0.474829\pi\)
0.0789953 + 0.996875i \(0.474829\pi\)
\(642\) 0 0
\(643\) 24.0000 0.946468 0.473234 − 0.880937i \(-0.343087\pi\)
0.473234 + 0.880937i \(0.343087\pi\)
\(644\) 0 0
\(645\) 16.0000 0.629999
\(646\) 0 0
\(647\) −10.0000 −0.393141 −0.196570 − 0.980490i \(-0.562980\pi\)
−0.196570 + 0.980490i \(0.562980\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −34.0000 −1.33052 −0.665261 − 0.746611i \(-0.731680\pi\)
−0.665261 + 0.746611i \(0.731680\pi\)
\(654\) 0 0
\(655\) 24.0000 0.937758
\(656\) 0 0
\(657\) −2.00000 −0.0780274
\(658\) 0 0
\(659\) −6.00000 −0.233727 −0.116863 − 0.993148i \(-0.537284\pi\)
−0.116863 + 0.993148i \(0.537284\pi\)
\(660\) 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
\(665\) 4.00000 0.155113
\(666\) 0 0
\(667\) −16.0000 −0.619522
\(668\) 0 0
\(669\) 16.0000 0.618596
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −14.0000 −0.539660 −0.269830 − 0.962908i \(-0.586968\pi\)
−0.269830 + 0.962908i \(0.586968\pi\)
\(674\) 0 0
\(675\) 11.0000 0.423390
\(676\) 0 0
\(677\) −22.0000 −0.845529 −0.422764 − 0.906240i \(-0.638940\pi\)
−0.422764 + 0.906240i \(0.638940\pi\)
\(678\) 0 0
\(679\) −18.0000 −0.690777
\(680\) 0 0
\(681\) 16.0000 0.613121
\(682\) 0 0
\(683\) −14.0000 −0.535695 −0.267848 − 0.963461i \(-0.586312\pi\)
−0.267848 + 0.963461i \(0.586312\pi\)
\(684\) 0 0
\(685\) 56.0000 2.13965
\(686\) 0 0
\(687\) −6.00000 −0.228914
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −20.0000 −0.760836 −0.380418 − 0.924815i \(-0.624220\pi\)
−0.380418 + 0.924815i \(0.624220\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 40.0000 1.51511
\(698\) 0 0
\(699\) −18.0000 −0.680823
\(700\) 0 0
\(701\) −18.0000 −0.679851 −0.339925 − 0.940452i \(-0.610402\pi\)
−0.339925 + 0.940452i \(0.610402\pi\)
\(702\) 0 0
\(703\) 10.0000 0.377157
\(704\) 0 0
\(705\) −24.0000 −0.903892
\(706\) 0 0
\(707\) 4.00000 0.150435
\(708\) 0 0
\(709\) 26.0000 0.976450 0.488225 − 0.872718i \(-0.337644\pi\)
0.488225 + 0.872718i \(0.337644\pi\)
\(710\) 0 0
\(711\) −4.00000 −0.150012
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 24.0000 0.896296
\(718\) 0 0
\(719\) −38.0000 −1.41716 −0.708580 − 0.705630i \(-0.750664\pi\)
−0.708580 + 0.705630i \(0.750664\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −2.00000 −0.0743808
\(724\) 0 0
\(725\) 44.0000 1.63412
\(726\) 0 0
\(727\) 16.0000 0.593407 0.296704 − 0.954970i \(-0.404113\pi\)
0.296704 + 0.954970i \(0.404113\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −16.0000 −0.591781
\(732\) 0 0
\(733\) 34.0000 1.25582 0.627909 − 0.778287i \(-0.283911\pi\)
0.627909 + 0.778287i \(0.283911\pi\)
\(734\) 0 0
\(735\) −4.00000 −0.147542
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −28.0000 −1.03000 −0.514998 − 0.857191i \(-0.672207\pi\)
−0.514998 + 0.857191i \(0.672207\pi\)
\(740\) 0 0
\(741\) 2.00000 0.0734718
\(742\) 0 0
\(743\) −6.00000 −0.220119 −0.110059 − 0.993925i \(-0.535104\pi\)
−0.110059 + 0.993925i \(0.535104\pi\)
\(744\) 0 0
\(745\) 24.0000 0.879292
\(746\) 0 0
\(747\) 14.0000 0.512233
\(748\) 0 0
\(749\) −18.0000 −0.657706
\(750\) 0 0
\(751\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) 0 0
\(753\) −6.00000 −0.218652
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −50.0000 −1.81728 −0.908640 − 0.417579i \(-0.862879\pi\)
−0.908640 + 0.417579i \(0.862879\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 24.0000 0.869999 0.435000 − 0.900431i \(-0.356748\pi\)
0.435000 + 0.900431i \(0.356748\pi\)
\(762\) 0 0
\(763\) 2.00000 0.0724049
\(764\) 0 0
\(765\) −16.0000 −0.578481
\(766\) 0 0
\(767\) 8.00000 0.288863
\(768\) 0 0
\(769\) 18.0000 0.649097 0.324548 − 0.945869i \(-0.394788\pi\)
0.324548 + 0.945869i \(0.394788\pi\)
\(770\) 0 0
\(771\) 14.0000 0.504198
\(772\) 0 0
\(773\) 14.0000 0.503545 0.251773 − 0.967786i \(-0.418987\pi\)
0.251773 + 0.967786i \(0.418987\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −10.0000 −0.358748
\(778\) 0 0
\(779\) −10.0000 −0.358287
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 4.00000 0.142948
\(784\) 0 0
\(785\) 8.00000 0.285532
\(786\) 0 0
\(787\) 4.00000 0.142585 0.0712923 − 0.997455i \(-0.477288\pi\)
0.0712923 + 0.997455i \(0.477288\pi\)
\(788\) 0 0
\(789\) −32.0000 −1.13923
\(790\) 0 0
\(791\) 16.0000 0.568895
\(792\) 0 0
\(793\) 20.0000 0.710221
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 30.0000 1.06265 0.531327 − 0.847167i \(-0.321693\pi\)
0.531327 + 0.847167i \(0.321693\pi\)
\(798\) 0 0
\(799\) 24.0000 0.849059
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 16.0000 0.563926
\(806\) 0 0
\(807\) 18.0000 0.633630
\(808\) 0 0
\(809\) 2.00000 0.0703163 0.0351581 − 0.999382i \(-0.488807\pi\)
0.0351581 + 0.999382i \(0.488807\pi\)
\(810\) 0 0
\(811\) −36.0000 −1.26413 −0.632065 − 0.774915i \(-0.717793\pi\)
−0.632065 + 0.774915i \(0.717793\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −16.0000 −0.560456
\(816\) 0 0
\(817\) 4.00000 0.139942
\(818\) 0 0
\(819\) −2.00000 −0.0698857
\(820\) 0 0
\(821\) 26.0000 0.907406 0.453703 − 0.891153i \(-0.350103\pi\)
0.453703 + 0.891153i \(0.350103\pi\)
\(822\) 0 0
\(823\) 40.0000 1.39431 0.697156 − 0.716919i \(-0.254448\pi\)
0.697156 + 0.716919i \(0.254448\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 38.0000 1.32139 0.660695 − 0.750655i \(-0.270262\pi\)
0.660695 + 0.750655i \(0.270262\pi\)
\(828\) 0 0
\(829\) 34.0000 1.18087 0.590434 − 0.807086i \(-0.298956\pi\)
0.590434 + 0.807086i \(0.298956\pi\)
\(830\) 0 0
\(831\) 10.0000 0.346896
\(832\) 0 0
\(833\) 4.00000 0.138592
\(834\) 0 0
\(835\) −48.0000 −1.66111
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −48.0000 −1.65714 −0.828572 − 0.559883i \(-0.810846\pi\)
−0.828572 + 0.559883i \(0.810846\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) 0 0
\(843\) −32.0000 −1.10214
\(844\) 0 0
\(845\) 36.0000 1.23844
\(846\) 0 0
\(847\) −11.0000 −0.377964
\(848\) 0 0
\(849\) −8.00000 −0.274559
\(850\) 0 0
\(851\) 40.0000 1.37118
\(852\) 0 0
\(853\) 46.0000 1.57501 0.787505 − 0.616308i \(-0.211372\pi\)
0.787505 + 0.616308i \(0.211372\pi\)
\(854\) 0 0
\(855\) 4.00000 0.136797
\(856\) 0 0
\(857\) 2.00000 0.0683187 0.0341593 − 0.999416i \(-0.489125\pi\)
0.0341593 + 0.999416i \(0.489125\pi\)
\(858\) 0 0
\(859\) 20.0000 0.682391 0.341196 − 0.939992i \(-0.389168\pi\)
0.341196 + 0.939992i \(0.389168\pi\)
\(860\) 0 0
\(861\) 10.0000 0.340799
\(862\) 0 0
\(863\) −18.0000 −0.612727 −0.306364 − 0.951915i \(-0.599112\pi\)
−0.306364 + 0.951915i \(0.599112\pi\)
\(864\) 0 0
\(865\) 72.0000 2.44807
\(866\) 0 0
\(867\) −1.00000 −0.0339618
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −8.00000 −0.271070
\(872\) 0 0
\(873\) −18.0000 −0.609208
\(874\) 0 0
\(875\) −24.0000 −0.811348
\(876\) 0 0
\(877\) −2.00000 −0.0675352 −0.0337676 − 0.999430i \(-0.510751\pi\)
−0.0337676 + 0.999430i \(0.510751\pi\)
\(878\) 0 0
\(879\) −18.0000 −0.607125
\(880\) 0 0
\(881\) −52.0000 −1.75192 −0.875962 − 0.482380i \(-0.839773\pi\)
−0.875962 + 0.482380i \(0.839773\pi\)
\(882\) 0 0
\(883\) 36.0000 1.21150 0.605748 − 0.795656i \(-0.292874\pi\)
0.605748 + 0.795656i \(0.292874\pi\)
\(884\) 0 0
\(885\) 16.0000 0.537834
\(886\) 0 0
\(887\) 44.0000 1.47738 0.738688 − 0.674048i \(-0.235446\pi\)
0.738688 + 0.674048i \(0.235446\pi\)
\(888\) 0 0
\(889\) −8.00000 −0.268311
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −6.00000 −0.200782
\(894\) 0 0
\(895\) 24.0000 0.802232
\(896\) 0 0
\(897\) 8.00000 0.267112
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) −4.00000 −0.133112
\(904\) 0 0
\(905\) 88.0000 2.92522
\(906\) 0 0
\(907\) −40.0000 −1.32818 −0.664089 − 0.747653i \(-0.731180\pi\)
−0.664089 + 0.747653i \(0.731180\pi\)
\(908\) 0 0
\(909\) 4.00000 0.132672
\(910\) 0 0
\(911\) −10.0000 −0.331315 −0.165657 − 0.986183i \(-0.552975\pi\)
−0.165657 + 0.986183i \(0.552975\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 40.0000 1.32236
\(916\) 0 0
\(917\) −6.00000 −0.198137
\(918\) 0 0
\(919\) 24.0000 0.791687 0.395843 − 0.918318i \(-0.370452\pi\)
0.395843 + 0.918318i \(0.370452\pi\)
\(920\) 0 0
\(921\) −20.0000 −0.659022
\(922\) 0 0
\(923\) −4.00000 −0.131662
\(924\) 0 0
\(925\) −110.000 −3.61678
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 36.0000 1.18112 0.590561 − 0.806993i \(-0.298907\pi\)
0.590561 + 0.806993i \(0.298907\pi\)
\(930\) 0 0
\(931\) −1.00000 −0.0327737
\(932\) 0 0
\(933\) −18.0000 −0.589294
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 2.00000 0.0653372 0.0326686 − 0.999466i \(-0.489599\pi\)
0.0326686 + 0.999466i \(0.489599\pi\)
\(938\) 0 0
\(939\) −14.0000 −0.456873
\(940\) 0 0
\(941\) 42.0000 1.36916 0.684580 − 0.728937i \(-0.259985\pi\)
0.684580 + 0.728937i \(0.259985\pi\)
\(942\) 0 0
\(943\) −40.0000 −1.30258
\(944\) 0 0
\(945\) −4.00000 −0.130120
\(946\) 0 0
\(947\) −24.0000 −0.779895 −0.389948 − 0.920837i \(-0.627507\pi\)
−0.389948 + 0.920837i \(0.627507\pi\)
\(948\) 0 0
\(949\) 4.00000 0.129845
\(950\) 0 0
\(951\) 4.00000 0.129709
\(952\) 0 0
\(953\) −28.0000 −0.907009 −0.453504 − 0.891254i \(-0.649826\pi\)
−0.453504 + 0.891254i \(0.649826\pi\)
\(954\) 0 0
\(955\) 80.0000 2.58874
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −14.0000 −0.452084
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) −18.0000 −0.580042
\(964\) 0 0
\(965\) 24.0000 0.772587
\(966\) 0 0
\(967\) −48.0000 −1.54358 −0.771788 − 0.635880i \(-0.780637\pi\)
−0.771788 + 0.635880i \(0.780637\pi\)
\(968\) 0 0
\(969\) −4.00000 −0.128499
\(970\) 0 0
\(971\) 12.0000 0.385098 0.192549 − 0.981287i \(-0.438325\pi\)
0.192549 + 0.981287i \(0.438325\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −22.0000 −0.704564
\(976\) 0 0
\(977\) −60.0000 −1.91957 −0.959785 − 0.280736i \(-0.909421\pi\)
−0.959785 + 0.280736i \(0.909421\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 2.00000 0.0638551
\(982\) 0 0
\(983\) 8.00000 0.255160 0.127580 − 0.991828i \(-0.459279\pi\)
0.127580 + 0.991828i \(0.459279\pi\)
\(984\) 0 0
\(985\) −24.0000 −0.764704
\(986\) 0 0
\(987\) 6.00000 0.190982
\(988\) 0 0
\(989\) 16.0000 0.508770
\(990\) 0 0
\(991\) −28.0000 −0.889449 −0.444725 − 0.895667i \(-0.646698\pi\)
−0.444725 + 0.895667i \(0.646698\pi\)
\(992\) 0 0
\(993\) −12.0000 −0.380808
\(994\) 0 0
\(995\) 16.0000 0.507234
\(996\) 0 0
\(997\) −50.0000 −1.58352 −0.791758 − 0.610835i \(-0.790834\pi\)
−0.791758 + 0.610835i \(0.790834\pi\)
\(998\) 0 0
\(999\) −10.0000 −0.316386
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 6384.2.a.r.1.1 1
4.3 odd 2 3192.2.a.a.1.1 ✓ 1
12.11 even 2 9576.2.a.ba.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3192.2.a.a.1.1 ✓ 1 4.3 odd 2
6384.2.a.r.1.1 1 1.1 even 1 trivial
9576.2.a.ba.1.1 1 12.11 even 2