Properties

Label 1600.2.d.f.801.3
Level $1600$
Weight $2$
Character 1600.801
Analytic conductor $12.776$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1600,2,Mod(801,1600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1600, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1600.801");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1600.d (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(12.7760643234\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 801.3
Root \(-0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1600.801
Dual form 1600.2.d.f.801.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{3} -3.46410 q^{7} +2.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{3} -3.46410 q^{7} +2.00000 q^{9} -3.00000i q^{11} +3.46410i q^{13} +3.00000 q^{17} -1.00000i q^{19} -3.46410i q^{21} +5.00000i q^{27} +10.3923i q^{29} +6.92820 q^{31} +3.00000 q^{33} +10.3923i q^{37} -3.46410 q^{39} -9.00000 q^{41} -4.00000i q^{43} -10.3923 q^{47} +5.00000 q^{49} +3.00000i q^{51} +1.00000 q^{57} +12.0000i q^{59} -3.46410i q^{61} -6.92820 q^{63} +11.0000i q^{67} +10.3923 q^{71} +7.00000 q^{73} +10.3923i q^{77} -10.3923 q^{79} +1.00000 q^{81} +15.0000i q^{83} -10.3923 q^{87} -3.00000 q^{89} -12.0000i q^{91} +6.92820i q^{93} +14.0000 q^{97} -6.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{9} + 12 q^{17} + 12 q^{33} - 36 q^{41} + 20 q^{49} + 4 q^{57} + 28 q^{73} + 4 q^{81} - 12 q^{89} + 56 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1151\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000i 0.577350i 0.957427 + 0.288675i \(0.0932147\pi\)
−0.957427 + 0.288675i \(0.906785\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −3.46410 −1.30931 −0.654654 0.755929i \(-0.727186\pi\)
−0.654654 + 0.755929i \(0.727186\pi\)
\(8\) 0 0
\(9\) 2.00000 0.666667
\(10\) 0 0
\(11\) − 3.00000i − 0.904534i −0.891883 0.452267i \(-0.850615\pi\)
0.891883 0.452267i \(-0.149385\pi\)
\(12\) 0 0
\(13\) 3.46410i 0.960769i 0.877058 + 0.480384i \(0.159503\pi\)
−0.877058 + 0.480384i \(0.840497\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.00000 0.727607 0.363803 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) 0 0
\(19\) − 1.00000i − 0.229416i −0.993399 0.114708i \(-0.963407\pi\)
0.993399 0.114708i \(-0.0365932\pi\)
\(20\) 0 0
\(21\) − 3.46410i − 0.755929i
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 5.00000i 0.962250i
\(28\) 0 0
\(29\) 10.3923i 1.92980i 0.262613 + 0.964901i \(0.415416\pi\)
−0.262613 + 0.964901i \(0.584584\pi\)
\(30\) 0 0
\(31\) 6.92820 1.24434 0.622171 0.782881i \(-0.286251\pi\)
0.622171 + 0.782881i \(0.286251\pi\)
\(32\) 0 0
\(33\) 3.00000 0.522233
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 10.3923i 1.70848i 0.519875 + 0.854242i \(0.325978\pi\)
−0.519875 + 0.854242i \(0.674022\pi\)
\(38\) 0 0
\(39\) −3.46410 −0.554700
\(40\) 0 0
\(41\) −9.00000 −1.40556 −0.702782 0.711405i \(-0.748059\pi\)
−0.702782 + 0.711405i \(0.748059\pi\)
\(42\) 0 0
\(43\) − 4.00000i − 0.609994i −0.952353 0.304997i \(-0.901344\pi\)
0.952353 0.304997i \(-0.0986555\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −10.3923 −1.51587 −0.757937 0.652328i \(-0.773792\pi\)
−0.757937 + 0.652328i \(0.773792\pi\)
\(48\) 0 0
\(49\) 5.00000 0.714286
\(50\) 0 0
\(51\) 3.00000i 0.420084i
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 1.00000 0.132453
\(58\) 0 0
\(59\) 12.0000i 1.56227i 0.624364 + 0.781133i \(0.285358\pi\)
−0.624364 + 0.781133i \(0.714642\pi\)
\(60\) 0 0
\(61\) − 3.46410i − 0.443533i −0.975100 0.221766i \(-0.928818\pi\)
0.975100 0.221766i \(-0.0711822\pi\)
\(62\) 0 0
\(63\) −6.92820 −0.872872
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 11.0000i 1.34386i 0.740613 + 0.671932i \(0.234535\pi\)
−0.740613 + 0.671932i \(0.765465\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 10.3923 1.23334 0.616670 0.787222i \(-0.288481\pi\)
0.616670 + 0.787222i \(0.288481\pi\)
\(72\) 0 0
\(73\) 7.00000 0.819288 0.409644 0.912245i \(-0.365653\pi\)
0.409644 + 0.912245i \(0.365653\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 10.3923i 1.18431i
\(78\) 0 0
\(79\) −10.3923 −1.16923 −0.584613 0.811312i \(-0.698754\pi\)
−0.584613 + 0.811312i \(0.698754\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 15.0000i 1.64646i 0.567705 + 0.823232i \(0.307831\pi\)
−0.567705 + 0.823232i \(0.692169\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −10.3923 −1.11417
\(88\) 0 0
\(89\) −3.00000 −0.317999 −0.159000 0.987279i \(-0.550827\pi\)
−0.159000 + 0.987279i \(0.550827\pi\)
\(90\) 0 0
\(91\) − 12.0000i − 1.25794i
\(92\) 0 0
\(93\) 6.92820i 0.718421i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 14.0000 1.42148 0.710742 0.703452i \(-0.248359\pi\)
0.710742 + 0.703452i \(0.248359\pi\)
\(98\) 0 0
\(99\) − 6.00000i − 0.603023i
\(100\) 0 0
\(101\) − 10.3923i − 1.03407i −0.855963 0.517036i \(-0.827035\pi\)
0.855963 0.517036i \(-0.172965\pi\)
\(102\) 0 0
\(103\) −10.3923 −1.02398 −0.511992 0.858990i \(-0.671092\pi\)
−0.511992 + 0.858990i \(0.671092\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 9.00000i − 0.870063i −0.900415 0.435031i \(-0.856737\pi\)
0.900415 0.435031i \(-0.143263\pi\)
\(108\) 0 0
\(109\) − 3.46410i − 0.331801i −0.986143 0.165900i \(-0.946947\pi\)
0.986143 0.165900i \(-0.0530530\pi\)
\(110\) 0 0
\(111\) −10.3923 −0.986394
\(112\) 0 0
\(113\) −15.0000 −1.41108 −0.705541 0.708669i \(-0.749296\pi\)
−0.705541 + 0.708669i \(0.749296\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 6.92820i 0.640513i
\(118\) 0 0
\(119\) −10.3923 −0.952661
\(120\) 0 0
\(121\) 2.00000 0.181818
\(122\) 0 0
\(123\) − 9.00000i − 0.811503i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 0 0
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 3.46410i 0.300376i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 9.00000 0.768922 0.384461 0.923141i \(-0.374387\pi\)
0.384461 + 0.923141i \(0.374387\pi\)
\(138\) 0 0
\(139\) 5.00000i 0.424094i 0.977259 + 0.212047i \(0.0680131\pi\)
−0.977259 + 0.212047i \(0.931987\pi\)
\(140\) 0 0
\(141\) − 10.3923i − 0.875190i
\(142\) 0 0
\(143\) 10.3923 0.869048
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 5.00000i 0.412393i
\(148\) 0 0
\(149\) 10.3923i 0.851371i 0.904871 + 0.425685i \(0.139967\pi\)
−0.904871 + 0.425685i \(0.860033\pi\)
\(150\) 0 0
\(151\) −3.46410 −0.281905 −0.140952 0.990016i \(-0.545016\pi\)
−0.140952 + 0.990016i \(0.545016\pi\)
\(152\) 0 0
\(153\) 6.00000 0.485071
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 6.92820i 0.552931i 0.961024 + 0.276465i \(0.0891631\pi\)
−0.961024 + 0.276465i \(0.910837\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 19.0000i 1.48819i 0.668071 + 0.744097i \(0.267120\pi\)
−0.668071 + 0.744097i \(0.732880\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 10.3923 0.804181 0.402090 0.915600i \(-0.368284\pi\)
0.402090 + 0.915600i \(0.368284\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) − 2.00000i − 0.152944i
\(172\) 0 0
\(173\) − 20.7846i − 1.58022i −0.612962 0.790112i \(-0.710022\pi\)
0.612962 0.790112i \(-0.289978\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −12.0000 −0.901975
\(178\) 0 0
\(179\) 3.00000i 0.224231i 0.993695 + 0.112115i \(0.0357626\pi\)
−0.993695 + 0.112115i \(0.964237\pi\)
\(180\) 0 0
\(181\) 13.8564i 1.02994i 0.857209 + 0.514969i \(0.172197\pi\)
−0.857209 + 0.514969i \(0.827803\pi\)
\(182\) 0 0
\(183\) 3.46410 0.256074
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 9.00000i − 0.658145i
\(188\) 0 0
\(189\) − 17.3205i − 1.25988i
\(190\) 0 0
\(191\) 20.7846 1.50392 0.751961 0.659208i \(-0.229108\pi\)
0.751961 + 0.659208i \(0.229108\pi\)
\(192\) 0 0
\(193\) 7.00000 0.503871 0.251936 0.967744i \(-0.418933\pi\)
0.251936 + 0.967744i \(0.418933\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 20.7846i − 1.48084i −0.672143 0.740421i \(-0.734626\pi\)
0.672143 0.740421i \(-0.265374\pi\)
\(198\) 0 0
\(199\) 13.8564 0.982255 0.491127 0.871088i \(-0.336585\pi\)
0.491127 + 0.871088i \(0.336585\pi\)
\(200\) 0 0
\(201\) −11.0000 −0.775880
\(202\) 0 0
\(203\) − 36.0000i − 2.52670i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −3.00000 −0.207514
\(210\) 0 0
\(211\) 25.0000i 1.72107i 0.509390 + 0.860535i \(0.329871\pi\)
−0.509390 + 0.860535i \(0.670129\pi\)
\(212\) 0 0
\(213\) 10.3923i 0.712069i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −24.0000 −1.62923
\(218\) 0 0
\(219\) 7.00000i 0.473016i
\(220\) 0 0
\(221\) 10.3923i 0.699062i
\(222\) 0 0
\(223\) −13.8564 −0.927894 −0.463947 0.885863i \(-0.653567\pi\)
−0.463947 + 0.885863i \(0.653567\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 12.0000i 0.796468i 0.917284 + 0.398234i \(0.130377\pi\)
−0.917284 + 0.398234i \(0.869623\pi\)
\(228\) 0 0
\(229\) 6.92820i 0.457829i 0.973447 + 0.228914i \(0.0735176\pi\)
−0.973447 + 0.228914i \(0.926482\pi\)
\(230\) 0 0
\(231\) −10.3923 −0.683763
\(232\) 0 0
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 10.3923i − 0.675053i
\(238\) 0 0
\(239\) 10.3923 0.672222 0.336111 0.941822i \(-0.390888\pi\)
0.336111 + 0.941822i \(0.390888\pi\)
\(240\) 0 0
\(241\) 5.00000 0.322078 0.161039 0.986948i \(-0.448515\pi\)
0.161039 + 0.986948i \(0.448515\pi\)
\(242\) 0 0
\(243\) 16.0000i 1.02640i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 3.46410 0.220416
\(248\) 0 0
\(249\) −15.0000 −0.950586
\(250\) 0 0
\(251\) − 9.00000i − 0.568075i −0.958813 0.284037i \(-0.908326\pi\)
0.958813 0.284037i \(-0.0916740\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −6.00000 −0.374270 −0.187135 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) 0 0
\(259\) − 36.0000i − 2.23693i
\(260\) 0 0
\(261\) 20.7846i 1.28654i
\(262\) 0 0
\(263\) 20.7846 1.28163 0.640817 0.767694i \(-0.278596\pi\)
0.640817 + 0.767694i \(0.278596\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 3.00000i − 0.183597i
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) −20.7846 −1.26258 −0.631288 0.775549i \(-0.717473\pi\)
−0.631288 + 0.775549i \(0.717473\pi\)
\(272\) 0 0
\(273\) 12.0000 0.726273
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 17.3205i − 1.04069i −0.853957 0.520344i \(-0.825804\pi\)
0.853957 0.520344i \(-0.174196\pi\)
\(278\) 0 0
\(279\) 13.8564 0.829561
\(280\) 0 0
\(281\) 18.0000 1.07379 0.536895 0.843649i \(-0.319597\pi\)
0.536895 + 0.843649i \(0.319597\pi\)
\(282\) 0 0
\(283\) − 11.0000i − 0.653882i −0.945045 0.326941i \(-0.893982\pi\)
0.945045 0.326941i \(-0.106018\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 31.1769 1.84032
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) 14.0000i 0.820695i
\(292\) 0 0
\(293\) − 10.3923i − 0.607125i −0.952812 0.303562i \(-0.901824\pi\)
0.952812 0.303562i \(-0.0981761\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 15.0000 0.870388
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 13.8564i 0.798670i
\(302\) 0 0
\(303\) 10.3923 0.597022
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 19.0000i − 1.08439i −0.840254 0.542194i \(-0.817594\pi\)
0.840254 0.542194i \(-0.182406\pi\)
\(308\) 0 0
\(309\) − 10.3923i − 0.591198i
\(310\) 0 0
\(311\) −10.3923 −0.589294 −0.294647 0.955606i \(-0.595202\pi\)
−0.294647 + 0.955606i \(0.595202\pi\)
\(312\) 0 0
\(313\) −22.0000 −1.24351 −0.621757 0.783210i \(-0.713581\pi\)
−0.621757 + 0.783210i \(0.713581\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 20.7846i − 1.16738i −0.811977 0.583690i \(-0.801608\pi\)
0.811977 0.583690i \(-0.198392\pi\)
\(318\) 0 0
\(319\) 31.1769 1.74557
\(320\) 0 0
\(321\) 9.00000 0.502331
\(322\) 0 0
\(323\) − 3.00000i − 0.166924i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 3.46410 0.191565
\(328\) 0 0
\(329\) 36.0000 1.98474
\(330\) 0 0
\(331\) − 19.0000i − 1.04433i −0.852843 0.522167i \(-0.825124\pi\)
0.852843 0.522167i \(-0.174876\pi\)
\(332\) 0 0
\(333\) 20.7846i 1.13899i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −7.00000 −0.381314 −0.190657 0.981657i \(-0.561062\pi\)
−0.190657 + 0.981657i \(0.561062\pi\)
\(338\) 0 0
\(339\) − 15.0000i − 0.814688i
\(340\) 0 0
\(341\) − 20.7846i − 1.12555i
\(342\) 0 0
\(343\) 6.92820 0.374088
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 33.0000i − 1.77153i −0.464131 0.885766i \(-0.653633\pi\)
0.464131 0.885766i \(-0.346367\pi\)
\(348\) 0 0
\(349\) − 10.3923i − 0.556287i −0.960539 0.278144i \(-0.910281\pi\)
0.960539 0.278144i \(-0.0897191\pi\)
\(350\) 0 0
\(351\) −17.3205 −0.924500
\(352\) 0 0
\(353\) −6.00000 −0.319348 −0.159674 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 10.3923i − 0.550019i
\(358\) 0 0
\(359\) −10.3923 −0.548485 −0.274242 0.961661i \(-0.588427\pi\)
−0.274242 + 0.961661i \(0.588427\pi\)
\(360\) 0 0
\(361\) 18.0000 0.947368
\(362\) 0 0
\(363\) 2.00000i 0.104973i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 6.92820 0.361649 0.180825 0.983515i \(-0.442123\pi\)
0.180825 + 0.983515i \(0.442123\pi\)
\(368\) 0 0
\(369\) −18.0000 −0.937043
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) − 13.8564i − 0.717458i −0.933442 0.358729i \(-0.883210\pi\)
0.933442 0.358729i \(-0.116790\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −36.0000 −1.85409
\(378\) 0 0
\(379\) 19.0000i 0.975964i 0.872854 + 0.487982i \(0.162267\pi\)
−0.872854 + 0.487982i \(0.837733\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 31.1769 1.59307 0.796533 0.604595i \(-0.206665\pi\)
0.796533 + 0.604595i \(0.206665\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 8.00000i − 0.406663i
\(388\) 0 0
\(389\) − 20.7846i − 1.05382i −0.849921 0.526911i \(-0.823350\pi\)
0.849921 0.526911i \(-0.176650\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 27.7128i − 1.39087i −0.718591 0.695433i \(-0.755213\pi\)
0.718591 0.695433i \(-0.244787\pi\)
\(398\) 0 0
\(399\) −3.46410 −0.173422
\(400\) 0 0
\(401\) −3.00000 −0.149813 −0.0749064 0.997191i \(-0.523866\pi\)
−0.0749064 + 0.997191i \(0.523866\pi\)
\(402\) 0 0
\(403\) 24.0000i 1.19553i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 31.1769 1.54538
\(408\) 0 0
\(409\) −25.0000 −1.23617 −0.618085 0.786111i \(-0.712091\pi\)
−0.618085 + 0.786111i \(0.712091\pi\)
\(410\) 0 0
\(411\) 9.00000i 0.443937i
\(412\) 0 0
\(413\) − 41.5692i − 2.04549i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −5.00000 −0.244851
\(418\) 0 0
\(419\) − 21.0000i − 1.02592i −0.858413 0.512959i \(-0.828549\pi\)
0.858413 0.512959i \(-0.171451\pi\)
\(420\) 0 0
\(421\) 20.7846i 1.01298i 0.862246 + 0.506490i \(0.169057\pi\)
−0.862246 + 0.506490i \(0.830943\pi\)
\(422\) 0 0
\(423\) −20.7846 −1.01058
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 12.0000i 0.580721i
\(428\) 0 0
\(429\) 10.3923i 0.501745i
\(430\) 0 0
\(431\) 10.3923 0.500580 0.250290 0.968171i \(-0.419474\pi\)
0.250290 + 0.968171i \(0.419474\pi\)
\(432\) 0 0
\(433\) −1.00000 −0.0480569 −0.0240285 0.999711i \(-0.507649\pi\)
−0.0240285 + 0.999711i \(0.507649\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 17.3205 0.826663 0.413331 0.910581i \(-0.364365\pi\)
0.413331 + 0.910581i \(0.364365\pi\)
\(440\) 0 0
\(441\) 10.0000 0.476190
\(442\) 0 0
\(443\) − 39.0000i − 1.85295i −0.376361 0.926473i \(-0.622825\pi\)
0.376361 0.926473i \(-0.377175\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −10.3923 −0.491539
\(448\) 0 0
\(449\) 9.00000 0.424736 0.212368 0.977190i \(-0.431882\pi\)
0.212368 + 0.977190i \(0.431882\pi\)
\(450\) 0 0
\(451\) 27.0000i 1.27138i
\(452\) 0 0
\(453\) − 3.46410i − 0.162758i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 37.0000 1.73079 0.865393 0.501093i \(-0.167069\pi\)
0.865393 + 0.501093i \(0.167069\pi\)
\(458\) 0 0
\(459\) 15.0000i 0.700140i
\(460\) 0 0
\(461\) 20.7846i 0.968036i 0.875058 + 0.484018i \(0.160823\pi\)
−0.875058 + 0.484018i \(0.839177\pi\)
\(462\) 0 0
\(463\) −20.7846 −0.965943 −0.482971 0.875636i \(-0.660442\pi\)
−0.482971 + 0.875636i \(0.660442\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 12.0000i 0.555294i 0.960683 + 0.277647i \(0.0895545\pi\)
−0.960683 + 0.277647i \(0.910445\pi\)
\(468\) 0 0
\(469\) − 38.1051i − 1.75953i
\(470\) 0 0
\(471\) −6.92820 −0.319235
\(472\) 0 0
\(473\) −12.0000 −0.551761
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −41.5692 −1.89935 −0.949673 0.313243i \(-0.898585\pi\)
−0.949673 + 0.313243i \(0.898585\pi\)
\(480\) 0 0
\(481\) −36.0000 −1.64146
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 13.8564 0.627894 0.313947 0.949441i \(-0.398349\pi\)
0.313947 + 0.949441i \(0.398349\pi\)
\(488\) 0 0
\(489\) −19.0000 −0.859210
\(490\) 0 0
\(491\) − 12.0000i − 0.541552i −0.962642 0.270776i \(-0.912720\pi\)
0.962642 0.270776i \(-0.0872803\pi\)
\(492\) 0 0
\(493\) 31.1769i 1.40414i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −36.0000 −1.61482
\(498\) 0 0
\(499\) 16.0000i 0.716258i 0.933672 + 0.358129i \(0.116585\pi\)
−0.933672 + 0.358129i \(0.883415\pi\)
\(500\) 0 0
\(501\) 10.3923i 0.464294i
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 1.00000i 0.0444116i
\(508\) 0 0
\(509\) 20.7846i 0.921262i 0.887592 + 0.460631i \(0.152377\pi\)
−0.887592 + 0.460631i \(0.847623\pi\)
\(510\) 0 0
\(511\) −24.2487 −1.07270
\(512\) 0 0
\(513\) 5.00000 0.220755
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 31.1769i 1.37116i
\(518\) 0 0
\(519\) 20.7846 0.912343
\(520\) 0 0
\(521\) −21.0000 −0.920027 −0.460013 0.887912i \(-0.652155\pi\)
−0.460013 + 0.887912i \(0.652155\pi\)
\(522\) 0 0
\(523\) − 1.00000i − 0.0437269i −0.999761 0.0218635i \(-0.993040\pi\)
0.999761 0.0218635i \(-0.00695991\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 20.7846 0.905392
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) 24.0000i 1.04151i
\(532\) 0 0
\(533\) − 31.1769i − 1.35042i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −3.00000 −0.129460
\(538\) 0 0
\(539\) − 15.0000i − 0.646096i
\(540\) 0 0
\(541\) − 6.92820i − 0.297867i −0.988847 0.148933i \(-0.952416\pi\)
0.988847 0.148933i \(-0.0475840\pi\)
\(542\) 0 0
\(543\) −13.8564 −0.594635
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 1.00000i 0.0427569i 0.999771 + 0.0213785i \(0.00680549\pi\)
−0.999771 + 0.0213785i \(0.993195\pi\)
\(548\) 0 0
\(549\) − 6.92820i − 0.295689i
\(550\) 0 0
\(551\) 10.3923 0.442727
\(552\) 0 0
\(553\) 36.0000 1.53088
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 10.3923i 0.440336i 0.975462 + 0.220168i \(0.0706606\pi\)
−0.975462 + 0.220168i \(0.929339\pi\)
\(558\) 0 0
\(559\) 13.8564 0.586064
\(560\) 0 0
\(561\) 9.00000 0.379980
\(562\) 0 0
\(563\) 12.0000i 0.505740i 0.967500 + 0.252870i \(0.0813744\pi\)
−0.967500 + 0.252870i \(0.918626\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −3.46410 −0.145479
\(568\) 0 0
\(569\) 3.00000 0.125767 0.0628833 0.998021i \(-0.479970\pi\)
0.0628833 + 0.998021i \(0.479970\pi\)
\(570\) 0 0
\(571\) 20.0000i 0.836974i 0.908223 + 0.418487i \(0.137439\pi\)
−0.908223 + 0.418487i \(0.862561\pi\)
\(572\) 0 0
\(573\) 20.7846i 0.868290i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 17.0000 0.707719 0.353860 0.935299i \(-0.384869\pi\)
0.353860 + 0.935299i \(0.384869\pi\)
\(578\) 0 0
\(579\) 7.00000i 0.290910i
\(580\) 0 0
\(581\) − 51.9615i − 2.15573i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 15.0000i − 0.619116i −0.950881 0.309558i \(-0.899819\pi\)
0.950881 0.309558i \(-0.100181\pi\)
\(588\) 0 0
\(589\) − 6.92820i − 0.285472i
\(590\) 0 0
\(591\) 20.7846 0.854965
\(592\) 0 0
\(593\) −3.00000 −0.123195 −0.0615976 0.998101i \(-0.519620\pi\)
−0.0615976 + 0.998101i \(0.519620\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 13.8564i 0.567105i
\(598\) 0 0
\(599\) −31.1769 −1.27385 −0.636927 0.770924i \(-0.719795\pi\)
−0.636927 + 0.770924i \(0.719795\pi\)
\(600\) 0 0
\(601\) 41.0000 1.67242 0.836212 0.548406i \(-0.184765\pi\)
0.836212 + 0.548406i \(0.184765\pi\)
\(602\) 0 0
\(603\) 22.0000i 0.895909i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −20.7846 −0.843621 −0.421811 0.906684i \(-0.638605\pi\)
−0.421811 + 0.906684i \(0.638605\pi\)
\(608\) 0 0
\(609\) 36.0000 1.45879
\(610\) 0 0
\(611\) − 36.0000i − 1.45640i
\(612\) 0 0
\(613\) 20.7846i 0.839482i 0.907644 + 0.419741i \(0.137879\pi\)
−0.907644 + 0.419741i \(0.862121\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 30.0000 1.20775 0.603877 0.797077i \(-0.293622\pi\)
0.603877 + 0.797077i \(0.293622\pi\)
\(618\) 0 0
\(619\) − 4.00000i − 0.160774i −0.996764 0.0803868i \(-0.974384\pi\)
0.996764 0.0803868i \(-0.0256155\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 10.3923 0.416359
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 3.00000i − 0.119808i
\(628\) 0 0
\(629\) 31.1769i 1.24310i
\(630\) 0 0
\(631\) 13.8564 0.551615 0.275807 0.961213i \(-0.411055\pi\)
0.275807 + 0.961213i \(0.411055\pi\)
\(632\) 0 0
\(633\) −25.0000 −0.993661
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 17.3205i 0.686264i
\(638\) 0 0
\(639\) 20.7846 0.822226
\(640\) 0 0
\(641\) −30.0000 −1.18493 −0.592464 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) 0 0
\(643\) 4.00000i 0.157745i 0.996885 + 0.0788723i \(0.0251319\pi\)
−0.996885 + 0.0788723i \(0.974868\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −41.5692 −1.63425 −0.817127 0.576457i \(-0.804435\pi\)
−0.817127 + 0.576457i \(0.804435\pi\)
\(648\) 0 0
\(649\) 36.0000 1.41312
\(650\) 0 0
\(651\) − 24.0000i − 0.940634i
\(652\) 0 0
\(653\) − 31.1769i − 1.22005i −0.792383 0.610023i \(-0.791160\pi\)
0.792383 0.610023i \(-0.208840\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 14.0000 0.546192
\(658\) 0 0
\(659\) 33.0000i 1.28550i 0.766077 + 0.642749i \(0.222206\pi\)
−0.766077 + 0.642749i \(0.777794\pi\)
\(660\) 0 0
\(661\) 10.3923i 0.404214i 0.979363 + 0.202107i \(0.0647788\pi\)
−0.979363 + 0.202107i \(0.935221\pi\)
\(662\) 0 0
\(663\) −10.3923 −0.403604
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) − 13.8564i − 0.535720i
\(670\) 0 0
\(671\) −10.3923 −0.401190
\(672\) 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
\(677\) − 20.7846i − 0.798817i −0.916773 0.399409i \(-0.869215\pi\)
0.916773 0.399409i \(-0.130785\pi\)
\(678\) 0 0
\(679\) −48.4974 −1.86116
\(680\) 0 0
\(681\) −12.0000 −0.459841
\(682\) 0 0
\(683\) − 9.00000i − 0.344375i −0.985064 0.172188i \(-0.944916\pi\)
0.985064 0.172188i \(-0.0550836\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −6.92820 −0.264327
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) − 37.0000i − 1.40755i −0.710425 0.703773i \(-0.751497\pi\)
0.710425 0.703773i \(-0.248503\pi\)
\(692\) 0 0
\(693\) 20.7846i 0.789542i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −27.0000 −1.02270
\(698\) 0 0
\(699\) − 6.00000i − 0.226941i
\(700\) 0 0
\(701\) − 31.1769i − 1.17754i −0.808302 0.588768i \(-0.799613\pi\)
0.808302 0.588768i \(-0.200387\pi\)
\(702\) 0 0
\(703\) 10.3923 0.391953
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 36.0000i 1.35392i
\(708\) 0 0
\(709\) 34.6410i 1.30097i 0.759519 + 0.650485i \(0.225434\pi\)
−0.759519 + 0.650485i \(0.774566\pi\)
\(710\) 0 0
\(711\) −20.7846 −0.779484
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 10.3923i 0.388108i
\(718\) 0 0
\(719\) −10.3923 −0.387568 −0.193784 0.981044i \(-0.562076\pi\)
−0.193784 + 0.981044i \(0.562076\pi\)
\(720\) 0 0
\(721\) 36.0000 1.34071
\(722\) 0 0
\(723\) 5.00000i 0.185952i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −45.0333 −1.67019 −0.835097 0.550103i \(-0.814588\pi\)
−0.835097 + 0.550103i \(0.814588\pi\)
\(728\) 0 0
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) − 12.0000i − 0.443836i
\(732\) 0 0
\(733\) 31.1769i 1.15155i 0.817610 + 0.575773i \(0.195299\pi\)
−0.817610 + 0.575773i \(0.804701\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 33.0000 1.21557
\(738\) 0 0
\(739\) − 20.0000i − 0.735712i −0.929883 0.367856i \(-0.880092\pi\)
0.929883 0.367856i \(-0.119908\pi\)
\(740\) 0 0
\(741\) 3.46410i 0.127257i
\(742\) 0 0
\(743\) 41.5692 1.52503 0.762513 0.646972i \(-0.223965\pi\)
0.762513 + 0.646972i \(0.223965\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 30.0000i 1.09764i
\(748\) 0 0
\(749\) 31.1769i 1.13918i
\(750\) 0 0
\(751\) 51.9615 1.89610 0.948051 0.318117i \(-0.103050\pi\)
0.948051 + 0.318117i \(0.103050\pi\)
\(752\) 0 0
\(753\) 9.00000 0.327978
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 6.92820i 0.251810i 0.992042 + 0.125905i \(0.0401834\pi\)
−0.992042 + 0.125905i \(0.959817\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 3.00000 0.108750 0.0543750 0.998521i \(-0.482683\pi\)
0.0543750 + 0.998521i \(0.482683\pi\)
\(762\) 0 0
\(763\) 12.0000i 0.434429i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −41.5692 −1.50098
\(768\) 0 0
\(769\) −7.00000 −0.252426 −0.126213 0.992003i \(-0.540282\pi\)
−0.126213 + 0.992003i \(0.540282\pi\)
\(770\) 0 0
\(771\) − 6.00000i − 0.216085i
\(772\) 0 0
\(773\) 10.3923i 0.373785i 0.982380 + 0.186893i \(0.0598416\pi\)
−0.982380 + 0.186893i \(0.940158\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 36.0000 1.29149
\(778\) 0 0
\(779\) 9.00000i 0.322458i
\(780\) 0 0
\(781\) − 31.1769i − 1.11560i
\(782\) 0 0
\(783\) −51.9615 −1.85695
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 16.0000i 0.570338i 0.958477 + 0.285169i \(0.0920498\pi\)
−0.958477 + 0.285169i \(0.907950\pi\)
\(788\) 0 0
\(789\) 20.7846i 0.739952i
\(790\) 0 0
\(791\) 51.9615 1.84754
\(792\) 0 0
\(793\) 12.0000 0.426132
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 20.7846i 0.736229i 0.929781 + 0.368114i \(0.119996\pi\)
−0.929781 + 0.368114i \(0.880004\pi\)
\(798\) 0 0
\(799\) −31.1769 −1.10296
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) 0 0
\(803\) − 21.0000i − 0.741074i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 30.0000 1.05474 0.527372 0.849635i \(-0.323177\pi\)
0.527372 + 0.849635i \(0.323177\pi\)
\(810\) 0 0
\(811\) 28.0000i 0.983213i 0.870817 + 0.491606i \(0.163590\pi\)
−0.870817 + 0.491606i \(0.836410\pi\)
\(812\) 0 0
\(813\) − 20.7846i − 0.728948i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −4.00000 −0.139942
\(818\) 0 0
\(819\) − 24.0000i − 0.838628i
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) −34.6410 −1.20751 −0.603755 0.797170i \(-0.706329\pi\)
−0.603755 + 0.797170i \(0.706329\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 3.00000i − 0.104320i −0.998639 0.0521601i \(-0.983389\pi\)
0.998639 0.0521601i \(-0.0166106\pi\)
\(828\) 0 0
\(829\) 10.3923i 0.360940i 0.983581 + 0.180470i \(0.0577618\pi\)
−0.983581 + 0.180470i \(0.942238\pi\)
\(830\) 0 0
\(831\) 17.3205 0.600842
\(832\) 0 0
\(833\) 15.0000 0.519719
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 34.6410i 1.19737i
\(838\) 0 0
\(839\) 41.5692 1.43513 0.717564 0.696492i \(-0.245257\pi\)
0.717564 + 0.696492i \(0.245257\pi\)
\(840\) 0 0
\(841\) −79.0000 −2.72414
\(842\) 0 0
\(843\) 18.0000i 0.619953i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −6.92820 −0.238056
\(848\) 0 0
\(849\) 11.0000 0.377519
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 55.4256i 1.89774i 0.315671 + 0.948869i \(0.397770\pi\)
−0.315671 + 0.948869i \(0.602230\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −39.0000 −1.33221 −0.666107 0.745856i \(-0.732041\pi\)
−0.666107 + 0.745856i \(0.732041\pi\)
\(858\) 0 0
\(859\) 13.0000i 0.443554i 0.975097 + 0.221777i \(0.0711857\pi\)
−0.975097 + 0.221777i \(0.928814\pi\)
\(860\) 0 0
\(861\) 31.1769i 1.06251i
\(862\) 0 0
\(863\) 10.3923 0.353758 0.176879 0.984233i \(-0.443400\pi\)
0.176879 + 0.984233i \(0.443400\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 8.00000i − 0.271694i
\(868\) 0 0
\(869\) 31.1769i 1.05760i
\(870\) 0 0
\(871\) −38.1051 −1.29114
\(872\) 0 0
\(873\) 28.0000 0.947656
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 48.4974i 1.63764i 0.574049 + 0.818821i \(0.305372\pi\)
−0.574049 + 0.818821i \(0.694628\pi\)
\(878\) 0 0
\(879\) 10.3923 0.350524
\(880\) 0 0
\(881\) −18.0000 −0.606435 −0.303218 0.952921i \(-0.598061\pi\)
−0.303218 + 0.952921i \(0.598061\pi\)
\(882\) 0 0
\(883\) − 35.0000i − 1.17784i −0.808190 0.588922i \(-0.799553\pi\)
0.808190 0.588922i \(-0.200447\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) − 3.00000i − 0.100504i
\(892\) 0 0
\(893\) 10.3923i 0.347765i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 72.0000i 2.40133i
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) −13.8564 −0.461112
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 44.0000i 1.46100i 0.682915 + 0.730498i \(0.260712\pi\)
−0.682915 + 0.730498i \(0.739288\pi\)
\(908\) 0 0
\(909\) − 20.7846i − 0.689382i
\(910\) 0 0
\(911\) 20.7846 0.688625 0.344312 0.938855i \(-0.388112\pi\)
0.344312 + 0.938855i \(0.388112\pi\)
\(912\) 0 0
\(913\) 45.0000 1.48928
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 27.7128 0.914161 0.457081 0.889425i \(-0.348895\pi\)
0.457081 + 0.889425i \(0.348895\pi\)
\(920\) 0 0
\(921\) 19.0000 0.626071
\(922\) 0 0
\(923\) 36.0000i 1.18495i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −20.7846 −0.682656
\(928\) 0 0
\(929\) 18.0000 0.590561 0.295280 0.955411i \(-0.404587\pi\)
0.295280 + 0.955411i \(0.404587\pi\)
\(930\) 0 0
\(931\) − 5.00000i − 0.163868i
\(932\) 0 0
\(933\) − 10.3923i − 0.340229i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 35.0000 1.14340 0.571700 0.820463i \(-0.306284\pi\)
0.571700 + 0.820463i \(0.306284\pi\)
\(938\) 0 0
\(939\) − 22.0000i − 0.717943i
\(940\) 0 0
\(941\) − 51.9615i − 1.69390i −0.531675 0.846949i \(-0.678437\pi\)
0.531675 0.846949i \(-0.321563\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 12.0000i 0.389948i 0.980808 + 0.194974i \(0.0624622\pi\)
−0.980808 + 0.194974i \(0.937538\pi\)
\(948\) 0 0
\(949\) 24.2487i 0.787146i
\(950\) 0 0
\(951\) 20.7846 0.673987
\(952\) 0 0
\(953\) −39.0000 −1.26333 −0.631667 0.775240i \(-0.717629\pi\)
−0.631667 + 0.775240i \(0.717629\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 31.1769i 1.00781i
\(958\) 0 0
\(959\) −31.1769 −1.00676
\(960\) 0 0
\(961\) 17.0000 0.548387
\(962\) 0 0
\(963\) − 18.0000i − 0.580042i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −20.7846 −0.668388 −0.334194 0.942504i \(-0.608464\pi\)
−0.334194 + 0.942504i \(0.608464\pi\)
\(968\) 0 0
\(969\) 3.00000 0.0963739
\(970\) 0 0
\(971\) 57.0000i 1.82922i 0.404341 + 0.914609i \(0.367501\pi\)
−0.404341 + 0.914609i \(0.632499\pi\)
\(972\) 0 0
\(973\) − 17.3205i − 0.555270i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 27.0000 0.863807 0.431903 0.901920i \(-0.357842\pi\)
0.431903 + 0.901920i \(0.357842\pi\)
\(978\) 0 0
\(979\) 9.00000i 0.287641i
\(980\) 0 0
\(981\) − 6.92820i − 0.221201i
\(982\) 0 0
\(983\) 10.3923 0.331463 0.165732 0.986171i \(-0.447001\pi\)
0.165732 + 0.986171i \(0.447001\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 36.0000i 1.14589i
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −24.2487 −0.770286 −0.385143 0.922857i \(-0.625848\pi\)
−0.385143 + 0.922857i \(0.625848\pi\)
\(992\) 0 0
\(993\) 19.0000 0.602947
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) 0 0
\(999\) −51.9615 −1.64399
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 1600.2.d.f.801.3 yes 4
4.3 odd 2 inner 1600.2.d.f.801.2 yes 4
5.2 odd 4 1600.2.f.g.1249.1 4
5.3 odd 4 1600.2.f.c.1249.3 4
5.4 even 2 1600.2.d.e.801.2 yes 4
8.3 odd 2 inner 1600.2.d.f.801.4 yes 4
8.5 even 2 inner 1600.2.d.f.801.1 yes 4
16.3 odd 4 6400.2.a.cg.1.1 2
16.5 even 4 6400.2.a.cg.1.2 2
16.11 odd 4 6400.2.a.ba.1.1 2
16.13 even 4 6400.2.a.ba.1.2 2
20.3 even 4 1600.2.f.g.1249.2 4
20.7 even 4 1600.2.f.c.1249.4 4
20.19 odd 2 1600.2.d.e.801.3 yes 4
40.3 even 4 1600.2.f.c.1249.1 4
40.13 odd 4 1600.2.f.g.1249.4 4
40.19 odd 2 1600.2.d.e.801.1 4
40.27 even 4 1600.2.f.g.1249.3 4
40.29 even 2 1600.2.d.e.801.4 yes 4
40.37 odd 4 1600.2.f.c.1249.2 4
80.19 odd 4 6400.2.a.bb.1.2 2
80.29 even 4 6400.2.a.cf.1.1 2
80.59 odd 4 6400.2.a.cf.1.2 2
80.69 even 4 6400.2.a.bb.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1600.2.d.e.801.1 4 40.19 odd 2
1600.2.d.e.801.2 yes 4 5.4 even 2
1600.2.d.e.801.3 yes 4 20.19 odd 2
1600.2.d.e.801.4 yes 4 40.29 even 2
1600.2.d.f.801.1 yes 4 8.5 even 2 inner
1600.2.d.f.801.2 yes 4 4.3 odd 2 inner
1600.2.d.f.801.3 yes 4 1.1 even 1 trivial
1600.2.d.f.801.4 yes 4 8.3 odd 2 inner
1600.2.f.c.1249.1 4 40.3 even 4
1600.2.f.c.1249.2 4 40.37 odd 4
1600.2.f.c.1249.3 4 5.3 odd 4
1600.2.f.c.1249.4 4 20.7 even 4
1600.2.f.g.1249.1 4 5.2 odd 4
1600.2.f.g.1249.2 4 20.3 even 4
1600.2.f.g.1249.3 4 40.27 even 4
1600.2.f.g.1249.4 4 40.13 odd 4
6400.2.a.ba.1.1 2 16.11 odd 4
6400.2.a.ba.1.2 2 16.13 even 4
6400.2.a.bb.1.1 2 80.69 even 4
6400.2.a.bb.1.2 2 80.19 odd 4
6400.2.a.cf.1.1 2 80.29 even 4
6400.2.a.cf.1.2 2 80.59 odd 4
6400.2.a.cg.1.1 2 16.3 odd 4
6400.2.a.cg.1.2 2 16.5 even 4