Properties

Label 1075.2.a.c.1.1
Level $1075$
Weight $2$
Character 1075.1
Self dual yes
Analytic conductor $8.584$
Analytic rank $0$
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] = [1075,2,Mod(1,1075)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1075, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1075.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1075 = 5^{2} \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1075.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: \(8.58391821729\)
Analytic rank: \(0\)
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 \(\chi\) \(=\) 1075.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{3} -2.00000 q^{4} +4.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-2.00000 q^{3} -2.00000 q^{4} +4.00000 q^{7} +1.00000 q^{9} +1.00000 q^{11} +4.00000 q^{12} -5.00000 q^{13} +4.00000 q^{16} -7.00000 q^{17} -8.00000 q^{21} +4.00000 q^{27} -8.00000 q^{28} +8.00000 q^{29} +4.00000 q^{31} -2.00000 q^{33} -2.00000 q^{36} -10.0000 q^{37} +10.0000 q^{39} +2.00000 q^{41} -1.00000 q^{43} -2.00000 q^{44} +7.00000 q^{47} -8.00000 q^{48} +9.00000 q^{49} +14.0000 q^{51} +10.0000 q^{52} +10.0000 q^{53} +5.00000 q^{59} +12.0000 q^{61} +4.00000 q^{63} -8.00000 q^{64} +9.00000 q^{67} +14.0000 q^{68} -8.00000 q^{71} +4.00000 q^{73} +4.00000 q^{77} +1.00000 q^{79} -11.0000 q^{81} +9.00000 q^{83} +16.0000 q^{84} -16.0000 q^{87} +6.00000 q^{89} -20.0000 q^{91} -8.00000 q^{93} +6.00000 q^{97} +1.00000 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(3\) −2.00000 −1.15470 −0.577350 0.816497i \(-0.695913\pi\)
−0.577350 + 0.816497i \(0.695913\pi\)
\(4\) −2.00000 −1.00000
\(5\) 0 0
\(6\) 0 0
\(7\) 4.00000 1.51186 0.755929 0.654654i \(-0.227186\pi\)
0.755929 + 0.654654i \(0.227186\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.00000 0.301511 0.150756 0.988571i \(-0.451829\pi\)
0.150756 + 0.988571i \(0.451829\pi\)
\(12\) 4.00000 1.15470
\(13\) −5.00000 −1.38675 −0.693375 0.720577i \(-0.743877\pi\)
−0.693375 + 0.720577i \(0.743877\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 4.00000 1.00000
\(17\) −7.00000 −1.69775 −0.848875 0.528594i \(-0.822719\pi\)
−0.848875 + 0.528594i \(0.822719\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) −8.00000 −1.74574
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 4.00000 0.769800
\(28\) −8.00000 −1.51186
\(29\) 8.00000 1.48556 0.742781 0.669534i \(-0.233506\pi\)
0.742781 + 0.669534i \(0.233506\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 0 0
\(33\) −2.00000 −0.348155
\(34\) 0 0
\(35\) 0 0
\(36\) −2.00000 −0.333333
\(37\) −10.0000 −1.64399 −0.821995 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) 0 0
\(39\) 10.0000 1.60128
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) −1.00000 −0.152499
\(44\) −2.00000 −0.301511
\(45\) 0 0
\(46\) 0 0
\(47\) 7.00000 1.02105 0.510527 0.859861i \(-0.329450\pi\)
0.510527 + 0.859861i \(0.329450\pi\)
\(48\) −8.00000 −1.15470
\(49\) 9.00000 1.28571
\(50\) 0 0
\(51\) 14.0000 1.96039
\(52\) 10.0000 1.38675
\(53\) 10.0000 1.37361 0.686803 0.726844i \(-0.259014\pi\)
0.686803 + 0.726844i \(0.259014\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 5.00000 0.650945 0.325472 0.945552i \(-0.394477\pi\)
0.325472 + 0.945552i \(0.394477\pi\)
\(60\) 0 0
\(61\) 12.0000 1.53644 0.768221 0.640184i \(-0.221142\pi\)
0.768221 + 0.640184i \(0.221142\pi\)
\(62\) 0 0
\(63\) 4.00000 0.503953
\(64\) −8.00000 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 9.00000 1.09952 0.549762 0.835321i \(-0.314718\pi\)
0.549762 + 0.835321i \(0.314718\pi\)
\(68\) 14.0000 1.69775
\(69\) 0 0
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) 4.00000 0.468165 0.234082 0.972217i \(-0.424791\pi\)
0.234082 + 0.972217i \(0.424791\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 4.00000 0.455842
\(78\) 0 0
\(79\) 1.00000 0.112509 0.0562544 0.998416i \(-0.482084\pi\)
0.0562544 + 0.998416i \(0.482084\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) 9.00000 0.987878 0.493939 0.869496i \(-0.335557\pi\)
0.493939 + 0.869496i \(0.335557\pi\)
\(84\) 16.0000 1.74574
\(85\) 0 0
\(86\) 0 0
\(87\) −16.0000 −1.71538
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) −20.0000 −2.09657
\(92\) 0 0
\(93\) −8.00000 −0.829561
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 6.00000 0.609208 0.304604 0.952479i \(-0.401476\pi\)
0.304604 + 0.952479i \(0.401476\pi\)
\(98\) 0 0
\(99\) 1.00000 0.100504
\(100\) 0 0
\(101\) 19.0000 1.89057 0.945285 0.326245i \(-0.105783\pi\)
0.945285 + 0.326245i \(0.105783\pi\)
\(102\) 0 0
\(103\) 1.00000 0.0985329 0.0492665 0.998786i \(-0.484312\pi\)
0.0492665 + 0.998786i \(0.484312\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 11.0000 1.06341 0.531705 0.846930i \(-0.321551\pi\)
0.531705 + 0.846930i \(0.321551\pi\)
\(108\) −8.00000 −0.769800
\(109\) 10.0000 0.957826 0.478913 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) 0 0
\(111\) 20.0000 1.89832
\(112\) 16.0000 1.51186
\(113\) −12.0000 −1.12887 −0.564433 0.825479i \(-0.690905\pi\)
−0.564433 + 0.825479i \(0.690905\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −16.0000 −1.48556
\(117\) −5.00000 −0.462250
\(118\) 0 0
\(119\) −28.0000 −2.56676
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) 0 0
\(123\) −4.00000 −0.360668
\(124\) −8.00000 −0.718421
\(125\) 0 0
\(126\) 0 0
\(127\) −17.0000 −1.50851 −0.754253 0.656584i \(-0.772001\pi\)
−0.754253 + 0.656584i \(0.772001\pi\)
\(128\) 0 0
\(129\) 2.00000 0.176090
\(130\) 0 0
\(131\) 6.00000 0.524222 0.262111 0.965038i \(-0.415581\pi\)
0.262111 + 0.965038i \(0.415581\pi\)
\(132\) 4.00000 0.348155
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 2.00000 0.170872 0.0854358 0.996344i \(-0.472772\pi\)
0.0854358 + 0.996344i \(0.472772\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) −14.0000 −1.17901
\(142\) 0 0
\(143\) −5.00000 −0.418121
\(144\) 4.00000 0.333333
\(145\) 0 0
\(146\) 0 0
\(147\) −18.0000 −1.48461
\(148\) 20.0000 1.64399
\(149\) −14.0000 −1.14692 −0.573462 0.819232i \(-0.694400\pi\)
−0.573462 + 0.819232i \(0.694400\pi\)
\(150\) 0 0
\(151\) −2.00000 −0.162758 −0.0813788 0.996683i \(-0.525932\pi\)
−0.0813788 + 0.996683i \(0.525932\pi\)
\(152\) 0 0
\(153\) −7.00000 −0.565916
\(154\) 0 0
\(155\) 0 0
\(156\) −20.0000 −1.60128
\(157\) 8.00000 0.638470 0.319235 0.947676i \(-0.396574\pi\)
0.319235 + 0.947676i \(0.396574\pi\)
\(158\) 0 0
\(159\) −20.0000 −1.58610
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −14.0000 −1.09656 −0.548282 0.836293i \(-0.684718\pi\)
−0.548282 + 0.836293i \(0.684718\pi\)
\(164\) −4.00000 −0.312348
\(165\) 0 0
\(166\) 0 0
\(167\) −21.0000 −1.62503 −0.812514 0.582941i \(-0.801902\pi\)
−0.812514 + 0.582941i \(0.801902\pi\)
\(168\) 0 0
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) 0 0
\(172\) 2.00000 0.152499
\(173\) 21.0000 1.59660 0.798300 0.602260i \(-0.205733\pi\)
0.798300 + 0.602260i \(0.205733\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 4.00000 0.301511
\(177\) −10.0000 −0.751646
\(178\) 0 0
\(179\) −6.00000 −0.448461 −0.224231 0.974536i \(-0.571987\pi\)
−0.224231 + 0.974536i \(0.571987\pi\)
\(180\) 0 0
\(181\) 13.0000 0.966282 0.483141 0.875542i \(-0.339496\pi\)
0.483141 + 0.875542i \(0.339496\pi\)
\(182\) 0 0
\(183\) −24.0000 −1.77413
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −7.00000 −0.511891
\(188\) −14.0000 −1.02105
\(189\) 16.0000 1.16383
\(190\) 0 0
\(191\) 22.0000 1.59186 0.795932 0.605386i \(-0.206981\pi\)
0.795932 + 0.605386i \(0.206981\pi\)
\(192\) 16.0000 1.15470
\(193\) 2.00000 0.143963 0.0719816 0.997406i \(-0.477068\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −18.0000 −1.28571
\(197\) −26.0000 −1.85242 −0.926212 0.377004i \(-0.876954\pi\)
−0.926212 + 0.377004i \(0.876954\pi\)
\(198\) 0 0
\(199\) 4.00000 0.283552 0.141776 0.989899i \(-0.454719\pi\)
0.141776 + 0.989899i \(0.454719\pi\)
\(200\) 0 0
\(201\) −18.0000 −1.26962
\(202\) 0 0
\(203\) 32.0000 2.24596
\(204\) −28.0000 −1.96039
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) −20.0000 −1.38675
\(209\) 0 0
\(210\) 0 0
\(211\) 14.0000 0.963800 0.481900 0.876226i \(-0.339947\pi\)
0.481900 + 0.876226i \(0.339947\pi\)
\(212\) −20.0000 −1.37361
\(213\) 16.0000 1.09630
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 16.0000 1.08615
\(218\) 0 0
\(219\) −8.00000 −0.540590
\(220\) 0 0
\(221\) 35.0000 2.35435
\(222\) 0 0
\(223\) 4.00000 0.267860 0.133930 0.990991i \(-0.457240\pi\)
0.133930 + 0.990991i \(0.457240\pi\)
\(224\) 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
\(229\) −11.0000 −0.726900 −0.363450 0.931614i \(-0.618401\pi\)
−0.363450 + 0.931614i \(0.618401\pi\)
\(230\) 0 0
\(231\) −8.00000 −0.526361
\(232\) 0 0
\(233\) 4.00000 0.262049 0.131024 0.991379i \(-0.458173\pi\)
0.131024 + 0.991379i \(0.458173\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −10.0000 −0.650945
\(237\) −2.00000 −0.129914
\(238\) 0 0
\(239\) −13.0000 −0.840900 −0.420450 0.907316i \(-0.638128\pi\)
−0.420450 + 0.907316i \(0.638128\pi\)
\(240\) 0 0
\(241\) −12.0000 −0.772988 −0.386494 0.922292i \(-0.626314\pi\)
−0.386494 + 0.922292i \(0.626314\pi\)
\(242\) 0 0
\(243\) 10.0000 0.641500
\(244\) −24.0000 −1.53644
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −18.0000 −1.14070
\(250\) 0 0
\(251\) 20.0000 1.26239 0.631194 0.775625i \(-0.282565\pi\)
0.631194 + 0.775625i \(0.282565\pi\)
\(252\) −8.00000 −0.503953
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) 22.0000 1.37232 0.686161 0.727450i \(-0.259294\pi\)
0.686161 + 0.727450i \(0.259294\pi\)
\(258\) 0 0
\(259\) −40.0000 −2.48548
\(260\) 0 0
\(261\) 8.00000 0.495188
\(262\) 0 0
\(263\) 18.0000 1.10993 0.554964 0.831875i \(-0.312732\pi\)
0.554964 + 0.831875i \(0.312732\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −12.0000 −0.734388
\(268\) −18.0000 −1.09952
\(269\) 15.0000 0.914566 0.457283 0.889321i \(-0.348823\pi\)
0.457283 + 0.889321i \(0.348823\pi\)
\(270\) 0 0
\(271\) −9.00000 −0.546711 −0.273356 0.961913i \(-0.588134\pi\)
−0.273356 + 0.961913i \(0.588134\pi\)
\(272\) −28.0000 −1.69775
\(273\) 40.0000 2.42091
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 8.00000 0.480673 0.240337 0.970690i \(-0.422742\pi\)
0.240337 + 0.970690i \(0.422742\pi\)
\(278\) 0 0
\(279\) 4.00000 0.239474
\(280\) 0 0
\(281\) 3.00000 0.178965 0.0894825 0.995988i \(-0.471479\pi\)
0.0894825 + 0.995988i \(0.471479\pi\)
\(282\) 0 0
\(283\) 28.0000 1.66443 0.832214 0.554455i \(-0.187073\pi\)
0.832214 + 0.554455i \(0.187073\pi\)
\(284\) 16.0000 0.949425
\(285\) 0 0
\(286\) 0 0
\(287\) 8.00000 0.472225
\(288\) 0 0
\(289\) 32.0000 1.88235
\(290\) 0 0
\(291\) −12.0000 −0.703452
\(292\) −8.00000 −0.468165
\(293\) 3.00000 0.175262 0.0876309 0.996153i \(-0.472070\pi\)
0.0876309 + 0.996153i \(0.472070\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 4.00000 0.232104
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −4.00000 −0.230556
\(302\) 0 0
\(303\) −38.0000 −2.18304
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −21.0000 −1.19853 −0.599267 0.800549i \(-0.704541\pi\)
−0.599267 + 0.800549i \(0.704541\pi\)
\(308\) −8.00000 −0.455842
\(309\) −2.00000 −0.113776
\(310\) 0 0
\(311\) −5.00000 −0.283524 −0.141762 0.989901i \(-0.545277\pi\)
−0.141762 + 0.989901i \(0.545277\pi\)
\(312\) 0 0
\(313\) 22.0000 1.24351 0.621757 0.783210i \(-0.286419\pi\)
0.621757 + 0.783210i \(0.286419\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −2.00000 −0.112509
\(317\) 17.0000 0.954815 0.477408 0.878682i \(-0.341577\pi\)
0.477408 + 0.878682i \(0.341577\pi\)
\(318\) 0 0
\(319\) 8.00000 0.447914
\(320\) 0 0
\(321\) −22.0000 −1.22792
\(322\) 0 0
\(323\) 0 0
\(324\) 22.0000 1.22222
\(325\) 0 0
\(326\) 0 0
\(327\) −20.0000 −1.10600
\(328\) 0 0
\(329\) 28.0000 1.54369
\(330\) 0 0
\(331\) −28.0000 −1.53902 −0.769510 0.638635i \(-0.779499\pi\)
−0.769510 + 0.638635i \(0.779499\pi\)
\(332\) −18.0000 −0.987878
\(333\) −10.0000 −0.547997
\(334\) 0 0
\(335\) 0 0
\(336\) −32.0000 −1.74574
\(337\) −3.00000 −0.163420 −0.0817102 0.996656i \(-0.526038\pi\)
−0.0817102 + 0.996656i \(0.526038\pi\)
\(338\) 0 0
\(339\) 24.0000 1.30350
\(340\) 0 0
\(341\) 4.00000 0.216612
\(342\) 0 0
\(343\) 8.00000 0.431959
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −6.00000 −0.322097 −0.161048 0.986947i \(-0.551488\pi\)
−0.161048 + 0.986947i \(0.551488\pi\)
\(348\) 32.0000 1.71538
\(349\) −30.0000 −1.60586 −0.802932 0.596071i \(-0.796728\pi\)
−0.802932 + 0.596071i \(0.796728\pi\)
\(350\) 0 0
\(351\) −20.0000 −1.06752
\(352\) 0 0
\(353\) −19.0000 −1.01127 −0.505634 0.862748i \(-0.668741\pi\)
−0.505634 + 0.862748i \(0.668741\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −12.0000 −0.635999
\(357\) 56.0000 2.96383
\(358\) 0 0
\(359\) −9.00000 −0.475002 −0.237501 0.971387i \(-0.576328\pi\)
−0.237501 + 0.971387i \(0.576328\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 0 0
\(363\) 20.0000 1.04973
\(364\) 40.0000 2.09657
\(365\) 0 0
\(366\) 0 0
\(367\) 1.00000 0.0521996 0.0260998 0.999659i \(-0.491691\pi\)
0.0260998 + 0.999659i \(0.491691\pi\)
\(368\) 0 0
\(369\) 2.00000 0.104116
\(370\) 0 0
\(371\) 40.0000 2.07670
\(372\) 16.0000 0.829561
\(373\) 14.0000 0.724893 0.362446 0.932005i \(-0.381942\pi\)
0.362446 + 0.932005i \(0.381942\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −40.0000 −2.06010
\(378\) 0 0
\(379\) −29.0000 −1.48963 −0.744815 0.667271i \(-0.767462\pi\)
−0.744815 + 0.667271i \(0.767462\pi\)
\(380\) 0 0
\(381\) 34.0000 1.74187
\(382\) 0 0
\(383\) −18.0000 −0.919757 −0.459879 0.887982i \(-0.652107\pi\)
−0.459879 + 0.887982i \(0.652107\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −1.00000 −0.0508329
\(388\) −12.0000 −0.609208
\(389\) 2.00000 0.101404 0.0507020 0.998714i \(-0.483854\pi\)
0.0507020 + 0.998714i \(0.483854\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) −12.0000 −0.605320
\(394\) 0 0
\(395\) 0 0
\(396\) −2.00000 −0.100504
\(397\) 2.00000 0.100377 0.0501886 0.998740i \(-0.484018\pi\)
0.0501886 + 0.998740i \(0.484018\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 9.00000 0.449439 0.224719 0.974424i \(-0.427853\pi\)
0.224719 + 0.974424i \(0.427853\pi\)
\(402\) 0 0
\(403\) −20.0000 −0.996271
\(404\) −38.0000 −1.89057
\(405\) 0 0
\(406\) 0 0
\(407\) −10.0000 −0.495682
\(408\) 0 0
\(409\) −32.0000 −1.58230 −0.791149 0.611623i \(-0.790517\pi\)
−0.791149 + 0.611623i \(0.790517\pi\)
\(410\) 0 0
\(411\) −4.00000 −0.197305
\(412\) −2.00000 −0.0985329
\(413\) 20.0000 0.984136
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 8.00000 0.391762
\(418\) 0 0
\(419\) 10.0000 0.488532 0.244266 0.969708i \(-0.421453\pi\)
0.244266 + 0.969708i \(0.421453\pi\)
\(420\) 0 0
\(421\) −22.0000 −1.07221 −0.536107 0.844150i \(-0.680106\pi\)
−0.536107 + 0.844150i \(0.680106\pi\)
\(422\) 0 0
\(423\) 7.00000 0.340352
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 48.0000 2.32288
\(428\) −22.0000 −1.06341
\(429\) 10.0000 0.482805
\(430\) 0 0
\(431\) 33.0000 1.58955 0.794777 0.606902i \(-0.207588\pi\)
0.794777 + 0.606902i \(0.207588\pi\)
\(432\) 16.0000 0.769800
\(433\) 14.0000 0.672797 0.336399 0.941720i \(-0.390791\pi\)
0.336399 + 0.941720i \(0.390791\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −20.0000 −0.957826
\(437\) 0 0
\(438\) 0 0
\(439\) −5.00000 −0.238637 −0.119318 0.992856i \(-0.538071\pi\)
−0.119318 + 0.992856i \(0.538071\pi\)
\(440\) 0 0
\(441\) 9.00000 0.428571
\(442\) 0 0
\(443\) −20.0000 −0.950229 −0.475114 0.879924i \(-0.657593\pi\)
−0.475114 + 0.879924i \(0.657593\pi\)
\(444\) −40.0000 −1.89832
\(445\) 0 0
\(446\) 0 0
\(447\) 28.0000 1.32435
\(448\) −32.0000 −1.51186
\(449\) 34.0000 1.60456 0.802280 0.596948i \(-0.203620\pi\)
0.802280 + 0.596948i \(0.203620\pi\)
\(450\) 0 0
\(451\) 2.00000 0.0941763
\(452\) 24.0000 1.12887
\(453\) 4.00000 0.187936
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 30.0000 1.40334 0.701670 0.712502i \(-0.252438\pi\)
0.701670 + 0.712502i \(0.252438\pi\)
\(458\) 0 0
\(459\) −28.0000 −1.30693
\(460\) 0 0
\(461\) −39.0000 −1.81641 −0.908206 0.418524i \(-0.862547\pi\)
−0.908206 + 0.418524i \(0.862547\pi\)
\(462\) 0 0
\(463\) −24.0000 −1.11537 −0.557687 0.830051i \(-0.688311\pi\)
−0.557687 + 0.830051i \(0.688311\pi\)
\(464\) 32.0000 1.48556
\(465\) 0 0
\(466\) 0 0
\(467\) 8.00000 0.370196 0.185098 0.982720i \(-0.440740\pi\)
0.185098 + 0.982720i \(0.440740\pi\)
\(468\) 10.0000 0.462250
\(469\) 36.0000 1.66233
\(470\) 0 0
\(471\) −16.0000 −0.737241
\(472\) 0 0
\(473\) −1.00000 −0.0459800
\(474\) 0 0
\(475\) 0 0
\(476\) 56.0000 2.56676
\(477\) 10.0000 0.457869
\(478\) 0 0
\(479\) 33.0000 1.50781 0.753904 0.656984i \(-0.228168\pi\)
0.753904 + 0.656984i \(0.228168\pi\)
\(480\) 0 0
\(481\) 50.0000 2.27980
\(482\) 0 0
\(483\) 0 0
\(484\) 20.0000 0.909091
\(485\) 0 0
\(486\) 0 0
\(487\) −24.0000 −1.08754 −0.543772 0.839233i \(-0.683004\pi\)
−0.543772 + 0.839233i \(0.683004\pi\)
\(488\) 0 0
\(489\) 28.0000 1.26620
\(490\) 0 0
\(491\) −30.0000 −1.35388 −0.676941 0.736038i \(-0.736695\pi\)
−0.676941 + 0.736038i \(0.736695\pi\)
\(492\) 8.00000 0.360668
\(493\) −56.0000 −2.52211
\(494\) 0 0
\(495\) 0 0
\(496\) 16.0000 0.718421
\(497\) −32.0000 −1.43540
\(498\) 0 0
\(499\) 32.0000 1.43252 0.716258 0.697835i \(-0.245853\pi\)
0.716258 + 0.697835i \(0.245853\pi\)
\(500\) 0 0
\(501\) 42.0000 1.87642
\(502\) 0 0
\(503\) 6.00000 0.267527 0.133763 0.991013i \(-0.457294\pi\)
0.133763 + 0.991013i \(0.457294\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −24.0000 −1.06588
\(508\) 34.0000 1.50851
\(509\) −19.0000 −0.842160 −0.421080 0.907023i \(-0.638349\pi\)
−0.421080 + 0.907023i \(0.638349\pi\)
\(510\) 0 0
\(511\) 16.0000 0.707798
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) −4.00000 −0.176090
\(517\) 7.00000 0.307860
\(518\) 0 0
\(519\) −42.0000 −1.84360
\(520\) 0 0
\(521\) −10.0000 −0.438108 −0.219054 0.975713i \(-0.570297\pi\)
−0.219054 + 0.975713i \(0.570297\pi\)
\(522\) 0 0
\(523\) 16.0000 0.699631 0.349816 0.936819i \(-0.386244\pi\)
0.349816 + 0.936819i \(0.386244\pi\)
\(524\) −12.0000 −0.524222
\(525\) 0 0
\(526\) 0 0
\(527\) −28.0000 −1.21970
\(528\) −8.00000 −0.348155
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) 5.00000 0.216982
\(532\) 0 0
\(533\) −10.0000 −0.433148
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 12.0000 0.517838
\(538\) 0 0
\(539\) 9.00000 0.387657
\(540\) 0 0
\(541\) −39.0000 −1.67674 −0.838370 0.545101i \(-0.816491\pi\)
−0.838370 + 0.545101i \(0.816491\pi\)
\(542\) 0 0
\(543\) −26.0000 −1.11577
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −4.00000 −0.171028 −0.0855138 0.996337i \(-0.527253\pi\)
−0.0855138 + 0.996337i \(0.527253\pi\)
\(548\) −4.00000 −0.170872
\(549\) 12.0000 0.512148
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 4.00000 0.170097
\(554\) 0 0
\(555\) 0 0
\(556\) 8.00000 0.339276
\(557\) 22.0000 0.932170 0.466085 0.884740i \(-0.345664\pi\)
0.466085 + 0.884740i \(0.345664\pi\)
\(558\) 0 0
\(559\) 5.00000 0.211477
\(560\) 0 0
\(561\) 14.0000 0.591080
\(562\) 0 0
\(563\) −24.0000 −1.01148 −0.505740 0.862686i \(-0.668780\pi\)
−0.505740 + 0.862686i \(0.668780\pi\)
\(564\) 28.0000 1.17901
\(565\) 0 0
\(566\) 0 0
\(567\) −44.0000 −1.84783
\(568\) 0 0
\(569\) 23.0000 0.964210 0.482105 0.876113i \(-0.339872\pi\)
0.482105 + 0.876113i \(0.339872\pi\)
\(570\) 0 0
\(571\) 10.0000 0.418487 0.209243 0.977864i \(-0.432900\pi\)
0.209243 + 0.977864i \(0.432900\pi\)
\(572\) 10.0000 0.418121
\(573\) −44.0000 −1.83813
\(574\) 0 0
\(575\) 0 0
\(576\) −8.00000 −0.333333
\(577\) 32.0000 1.33218 0.666089 0.745873i \(-0.267967\pi\)
0.666089 + 0.745873i \(0.267967\pi\)
\(578\) 0 0
\(579\) −4.00000 −0.166234
\(580\) 0 0
\(581\) 36.0000 1.49353
\(582\) 0 0
\(583\) 10.0000 0.414158
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 18.0000 0.742940 0.371470 0.928445i \(-0.378854\pi\)
0.371470 + 0.928445i \(0.378854\pi\)
\(588\) 36.0000 1.48461
\(589\) 0 0
\(590\) 0 0
\(591\) 52.0000 2.13899
\(592\) −40.0000 −1.64399
\(593\) −18.0000 −0.739171 −0.369586 0.929197i \(-0.620500\pi\)
−0.369586 + 0.929197i \(0.620500\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 28.0000 1.14692
\(597\) −8.00000 −0.327418
\(598\) 0 0
\(599\) 48.0000 1.96123 0.980613 0.195952i \(-0.0627798\pi\)
0.980613 + 0.195952i \(0.0627798\pi\)
\(600\) 0 0
\(601\) −10.0000 −0.407909 −0.203954 0.978980i \(-0.565379\pi\)
−0.203954 + 0.978980i \(0.565379\pi\)
\(602\) 0 0
\(603\) 9.00000 0.366508
\(604\) 4.00000 0.162758
\(605\) 0 0
\(606\) 0 0
\(607\) −38.0000 −1.54237 −0.771186 0.636610i \(-0.780336\pi\)
−0.771186 + 0.636610i \(0.780336\pi\)
\(608\) 0 0
\(609\) −64.0000 −2.59341
\(610\) 0 0
\(611\) −35.0000 −1.41595
\(612\) 14.0000 0.565916
\(613\) −43.0000 −1.73675 −0.868377 0.495905i \(-0.834836\pi\)
−0.868377 + 0.495905i \(0.834836\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −13.0000 −0.523360 −0.261680 0.965155i \(-0.584277\pi\)
−0.261680 + 0.965155i \(0.584277\pi\)
\(618\) 0 0
\(619\) −23.0000 −0.924448 −0.462224 0.886763i \(-0.652948\pi\)
−0.462224 + 0.886763i \(0.652948\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 24.0000 0.961540
\(624\) 40.0000 1.60128
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) −16.0000 −0.638470
\(629\) 70.0000 2.79108
\(630\) 0 0
\(631\) −20.0000 −0.796187 −0.398094 0.917345i \(-0.630328\pi\)
−0.398094 + 0.917345i \(0.630328\pi\)
\(632\) 0 0
\(633\) −28.0000 −1.11290
\(634\) 0 0
\(635\) 0 0
\(636\) 40.0000 1.58610
\(637\) −45.0000 −1.78296
\(638\) 0 0
\(639\) −8.00000 −0.316475
\(640\) 0 0
\(641\) −2.00000 −0.0789953 −0.0394976 0.999220i \(-0.512576\pi\)
−0.0394976 + 0.999220i \(0.512576\pi\)
\(642\) 0 0
\(643\) 33.0000 1.30139 0.650696 0.759338i \(-0.274477\pi\)
0.650696 + 0.759338i \(0.274477\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) 5.00000 0.196267
\(650\) 0 0
\(651\) −32.0000 −1.25418
\(652\) 28.0000 1.09656
\(653\) −6.00000 −0.234798 −0.117399 0.993085i \(-0.537456\pi\)
−0.117399 + 0.993085i \(0.537456\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 8.00000 0.312348
\(657\) 4.00000 0.156055
\(658\) 0 0
\(659\) 32.0000 1.24654 0.623272 0.782006i \(-0.285803\pi\)
0.623272 + 0.782006i \(0.285803\pi\)
\(660\) 0 0
\(661\) 18.0000 0.700119 0.350059 0.936727i \(-0.386161\pi\)
0.350059 + 0.936727i \(0.386161\pi\)
\(662\) 0 0
\(663\) −70.0000 −2.71857
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 42.0000 1.62503
\(669\) −8.00000 −0.309298
\(670\) 0 0
\(671\) 12.0000 0.463255
\(672\) 0 0
\(673\) 12.0000 0.462566 0.231283 0.972887i \(-0.425708\pi\)
0.231283 + 0.972887i \(0.425708\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) −24.0000 −0.923077
\(677\) 50.0000 1.92166 0.960828 0.277145i \(-0.0893883\pi\)
0.960828 + 0.277145i \(0.0893883\pi\)
\(678\) 0 0
\(679\) 24.0000 0.921035
\(680\) 0 0
\(681\) −28.0000 −1.07296
\(682\) 0 0
\(683\) 27.0000 1.03313 0.516563 0.856249i \(-0.327211\pi\)
0.516563 + 0.856249i \(0.327211\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 22.0000 0.839352
\(688\) −4.00000 −0.152499
\(689\) −50.0000 −1.90485
\(690\) 0 0
\(691\) −14.0000 −0.532585 −0.266293 0.963892i \(-0.585799\pi\)
−0.266293 + 0.963892i \(0.585799\pi\)
\(692\) −42.0000 −1.59660
\(693\) 4.00000 0.151947
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −14.0000 −0.530288
\(698\) 0 0
\(699\) −8.00000 −0.302588
\(700\) 0 0
\(701\) 15.0000 0.566542 0.283271 0.959040i \(-0.408580\pi\)
0.283271 + 0.959040i \(0.408580\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −8.00000 −0.301511
\(705\) 0 0
\(706\) 0 0
\(707\) 76.0000 2.85827
\(708\) 20.0000 0.751646
\(709\) 35.0000 1.31445 0.657226 0.753693i \(-0.271730\pi\)
0.657226 + 0.753693i \(0.271730\pi\)
\(710\) 0 0
\(711\) 1.00000 0.0375029
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 12.0000 0.448461
\(717\) 26.0000 0.970988
\(718\) 0 0
\(719\) −33.0000 −1.23069 −0.615346 0.788257i \(-0.710984\pi\)
−0.615346 + 0.788257i \(0.710984\pi\)
\(720\) 0 0
\(721\) 4.00000 0.148968
\(722\) 0 0
\(723\) 24.0000 0.892570
\(724\) −26.0000 −0.966282
\(725\) 0 0
\(726\) 0 0
\(727\) −22.0000 −0.815935 −0.407967 0.912996i \(-0.633762\pi\)
−0.407967 + 0.912996i \(0.633762\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) 7.00000 0.258904
\(732\) 48.0000 1.77413
\(733\) 2.00000 0.0738717 0.0369358 0.999318i \(-0.488240\pi\)
0.0369358 + 0.999318i \(0.488240\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 9.00000 0.331519
\(738\) 0 0
\(739\) −16.0000 −0.588570 −0.294285 0.955718i \(-0.595081\pi\)
−0.294285 + 0.955718i \(0.595081\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 34.0000 1.24734 0.623670 0.781688i \(-0.285641\pi\)
0.623670 + 0.781688i \(0.285641\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 9.00000 0.329293
\(748\) 14.0000 0.511891
\(749\) 44.0000 1.60773
\(750\) 0 0
\(751\) −10.0000 −0.364905 −0.182453 0.983215i \(-0.558404\pi\)
−0.182453 + 0.983215i \(0.558404\pi\)
\(752\) 28.0000 1.02105
\(753\) −40.0000 −1.45768
\(754\) 0 0
\(755\) 0 0
\(756\) −32.0000 −1.16383
\(757\) −2.00000 −0.0726912 −0.0363456 0.999339i \(-0.511572\pi\)
−0.0363456 + 0.999339i \(0.511572\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −18.0000 −0.652499 −0.326250 0.945284i \(-0.605785\pi\)
−0.326250 + 0.945284i \(0.605785\pi\)
\(762\) 0 0
\(763\) 40.0000 1.44810
\(764\) −44.0000 −1.59186
\(765\) 0 0
\(766\) 0 0
\(767\) −25.0000 −0.902698
\(768\) −32.0000 −1.15470
\(769\) 15.0000 0.540914 0.270457 0.962732i \(-0.412825\pi\)
0.270457 + 0.962732i \(0.412825\pi\)
\(770\) 0 0
\(771\) −44.0000 −1.58462
\(772\) −4.00000 −0.143963
\(773\) 46.0000 1.65451 0.827253 0.561830i \(-0.189903\pi\)
0.827253 + 0.561830i \(0.189903\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 80.0000 2.86998
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) −8.00000 −0.286263
\(782\) 0 0
\(783\) 32.0000 1.14359
\(784\) 36.0000 1.28571
\(785\) 0 0
\(786\) 0 0
\(787\) 28.0000 0.998092 0.499046 0.866575i \(-0.333684\pi\)
0.499046 + 0.866575i \(0.333684\pi\)
\(788\) 52.0000 1.85242
\(789\) −36.0000 −1.28163
\(790\) 0 0
\(791\) −48.0000 −1.70668
\(792\) 0 0
\(793\) −60.0000 −2.13066
\(794\) 0 0
\(795\) 0 0
\(796\) −8.00000 −0.283552
\(797\) −42.0000 −1.48772 −0.743858 0.668338i \(-0.767006\pi\)
−0.743858 + 0.668338i \(0.767006\pi\)
\(798\) 0 0
\(799\) −49.0000 −1.73350
\(800\) 0 0
\(801\) 6.00000 0.212000
\(802\) 0 0
\(803\) 4.00000 0.141157
\(804\) 36.0000 1.26962
\(805\) 0 0
\(806\) 0 0
\(807\) −30.0000 −1.05605
\(808\) 0 0
\(809\) 53.0000 1.86338 0.931690 0.363253i \(-0.118334\pi\)
0.931690 + 0.363253i \(0.118334\pi\)
\(810\) 0 0
\(811\) −14.0000 −0.491606 −0.245803 0.969320i \(-0.579052\pi\)
−0.245803 + 0.969320i \(0.579052\pi\)
\(812\) −64.0000 −2.24596
\(813\) 18.0000 0.631288
\(814\) 0 0
\(815\) 0 0
\(816\) 56.0000 1.96039
\(817\) 0 0
\(818\) 0 0
\(819\) −20.0000 −0.698857
\(820\) 0 0
\(821\) −19.0000 −0.663105 −0.331552 0.943437i \(-0.607572\pi\)
−0.331552 + 0.943437i \(0.607572\pi\)
\(822\) 0 0
\(823\) −31.0000 −1.08059 −0.540296 0.841475i \(-0.681688\pi\)
−0.540296 + 0.841475i \(0.681688\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 43.0000 1.49526 0.747628 0.664117i \(-0.231193\pi\)
0.747628 + 0.664117i \(0.231193\pi\)
\(828\) 0 0
\(829\) 34.0000 1.18087 0.590434 0.807086i \(-0.298956\pi\)
0.590434 + 0.807086i \(0.298956\pi\)
\(830\) 0 0
\(831\) −16.0000 −0.555034
\(832\) 40.0000 1.38675
\(833\) −63.0000 −2.18282
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 16.0000 0.553041
\(838\) 0 0
\(839\) 12.0000 0.414286 0.207143 0.978311i \(-0.433583\pi\)
0.207143 + 0.978311i \(0.433583\pi\)
\(840\) 0 0
\(841\) 35.0000 1.20690
\(842\) 0 0
\(843\) −6.00000 −0.206651
\(844\) −28.0000 −0.963800
\(845\) 0 0
\(846\) 0 0
\(847\) −40.0000 −1.37442
\(848\) 40.0000 1.37361
\(849\) −56.0000 −1.92192
\(850\) 0 0
\(851\) 0 0
\(852\) −32.0000 −1.09630
\(853\) −1.00000 −0.0342393 −0.0171197 0.999853i \(-0.505450\pi\)
−0.0171197 + 0.999853i \(0.505450\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −5.00000 −0.170797 −0.0853984 0.996347i \(-0.527216\pi\)
−0.0853984 + 0.996347i \(0.527216\pi\)
\(858\) 0 0
\(859\) −32.0000 −1.09183 −0.545913 0.837842i \(-0.683817\pi\)
−0.545913 + 0.837842i \(0.683817\pi\)
\(860\) 0 0
\(861\) −16.0000 −0.545279
\(862\) 0 0
\(863\) 36.0000 1.22545 0.612727 0.790295i \(-0.290072\pi\)
0.612727 + 0.790295i \(0.290072\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −64.0000 −2.17355
\(868\) −32.0000 −1.08615
\(869\) 1.00000 0.0339227
\(870\) 0 0
\(871\) −45.0000 −1.52477
\(872\) 0 0
\(873\) 6.00000 0.203069
\(874\) 0 0
\(875\) 0 0
\(876\) 16.0000 0.540590
\(877\) −55.0000 −1.85722 −0.928609 0.371060i \(-0.878995\pi\)
−0.928609 + 0.371060i \(0.878995\pi\)
\(878\) 0 0
\(879\) −6.00000 −0.202375
\(880\) 0 0
\(881\) 57.0000 1.92038 0.960189 0.279350i \(-0.0901189\pi\)
0.960189 + 0.279350i \(0.0901189\pi\)
\(882\) 0 0
\(883\) 1.00000 0.0336527 0.0168263 0.999858i \(-0.494644\pi\)
0.0168263 + 0.999858i \(0.494644\pi\)
\(884\) −70.0000 −2.35435
\(885\) 0 0
\(886\) 0 0
\(887\) −14.0000 −0.470074 −0.235037 0.971986i \(-0.575521\pi\)
−0.235037 + 0.971986i \(0.575521\pi\)
\(888\) 0 0
\(889\) −68.0000 −2.28065
\(890\) 0 0
\(891\) −11.0000 −0.368514
\(892\) −8.00000 −0.267860
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 32.0000 1.06726
\(900\) 0 0
\(901\) −70.0000 −2.33204
\(902\) 0 0
\(903\) 8.00000 0.266223
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 33.0000 1.09575 0.547874 0.836561i \(-0.315438\pi\)
0.547874 + 0.836561i \(0.315438\pi\)
\(908\) −28.0000 −0.929213
\(909\) 19.0000 0.630190
\(910\) 0 0
\(911\) −40.0000 −1.32526 −0.662630 0.748947i \(-0.730560\pi\)
−0.662630 + 0.748947i \(0.730560\pi\)
\(912\) 0 0
\(913\) 9.00000 0.297857
\(914\) 0 0
\(915\) 0 0
\(916\) 22.0000 0.726900
\(917\) 24.0000 0.792550
\(918\) 0 0
\(919\) 47.0000 1.55039 0.775193 0.631724i \(-0.217652\pi\)
0.775193 + 0.631724i \(0.217652\pi\)
\(920\) 0 0
\(921\) 42.0000 1.38395
\(922\) 0 0
\(923\) 40.0000 1.31662
\(924\) 16.0000 0.526361
\(925\) 0 0
\(926\) 0 0
\(927\) 1.00000 0.0328443
\(928\) 0 0
\(929\) 10.0000 0.328089 0.164045 0.986453i \(-0.447546\pi\)
0.164045 + 0.986453i \(0.447546\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −8.00000 −0.262049
\(933\) 10.0000 0.327385
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(938\) 0 0
\(939\) −44.0000 −1.43589
\(940\) 0 0
\(941\) −27.0000 −0.880175 −0.440087 0.897955i \(-0.645053\pi\)
−0.440087 + 0.897955i \(0.645053\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 20.0000 0.650945
\(945\) 0 0
\(946\) 0 0
\(947\) −27.0000 −0.877382 −0.438691 0.898638i \(-0.644558\pi\)
−0.438691 + 0.898638i \(0.644558\pi\)
\(948\) 4.00000 0.129914
\(949\) −20.0000 −0.649227
\(950\) 0 0
\(951\) −34.0000 −1.10253
\(952\) 0 0
\(953\) 20.0000 0.647864 0.323932 0.946080i \(-0.394995\pi\)
0.323932 + 0.946080i \(0.394995\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 26.0000 0.840900
\(957\) −16.0000 −0.517207
\(958\) 0 0
\(959\) 8.00000 0.258333
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) 11.0000 0.354470
\(964\) 24.0000 0.772988
\(965\) 0 0
\(966\) 0 0
\(967\) 16.0000 0.514525 0.257263 0.966342i \(-0.417179\pi\)
0.257263 + 0.966342i \(0.417179\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −13.0000 −0.417190 −0.208595 0.978002i \(-0.566889\pi\)
−0.208595 + 0.978002i \(0.566889\pi\)
\(972\) −20.0000 −0.641500
\(973\) −16.0000 −0.512936
\(974\) 0 0
\(975\) 0 0
\(976\) 48.0000 1.53644
\(977\) −21.0000 −0.671850 −0.335925 0.941889i \(-0.609049\pi\)
−0.335925 + 0.941889i \(0.609049\pi\)
\(978\) 0 0
\(979\) 6.00000 0.191761
\(980\) 0 0
\(981\) 10.0000 0.319275
\(982\) 0 0
\(983\) −44.0000 −1.40338 −0.701691 0.712481i \(-0.747571\pi\)
−0.701691 + 0.712481i \(0.747571\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −56.0000 −1.78250
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −8.00000 −0.254128 −0.127064 0.991894i \(-0.540555\pi\)
−0.127064 + 0.991894i \(0.540555\pi\)
\(992\) 0 0
\(993\) 56.0000 1.77711
\(994\) 0 0
\(995\) 0 0
\(996\) 36.0000 1.14070
\(997\) 44.0000 1.39349 0.696747 0.717317i \(-0.254630\pi\)
0.696747 + 0.717317i \(0.254630\pi\)
\(998\) 0 0
\(999\) −40.0000 −1.26554
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 1075.2.a.c.1.1 1
3.2 odd 2 9675.2.a.o.1.1 1
5.2 odd 4 1075.2.b.d.474.2 2
5.3 odd 4 1075.2.b.d.474.1 2
5.4 even 2 1075.2.a.e.1.1 yes 1
15.14 odd 2 9675.2.a.i.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1075.2.a.c.1.1 1 1.1 even 1 trivial
1075.2.a.e.1.1 yes 1 5.4 even 2
1075.2.b.d.474.1 2 5.3 odd 4
1075.2.b.d.474.2 2 5.2 odd 4
9675.2.a.i.1.1 1 15.14 odd 2
9675.2.a.o.1.1 1 3.2 odd 2