Properties

Label 1280.2.d.h.641.1
Level $1280$
Weight $2$
Character 1280.641
Analytic conductor $10.221$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1280,2,Mod(641,1280)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1280, 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("1280.641");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1280 = 2^{8} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1280.d (of order \(2\), degree \(1\), not 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: \(10.2208514587\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 160)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 641.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1280.641
Dual form 1280.2.d.h.641.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000i q^{3} +1.00000i q^{5} +2.00000 q^{7} -1.00000 q^{9} +O(q^{10})\) \(q-2.00000i q^{3} +1.00000i q^{5} +2.00000 q^{7} -1.00000 q^{9} +4.00000i q^{11} -6.00000i q^{13} +2.00000 q^{15} +2.00000 q^{17} +8.00000i q^{19} -4.00000i q^{21} +6.00000 q^{23} -1.00000 q^{25} -4.00000i q^{27} -2.00000i q^{29} +4.00000 q^{31} +8.00000 q^{33} +2.00000i q^{35} -2.00000i q^{37} -12.0000 q^{39} +10.0000 q^{41} +2.00000i q^{43} -1.00000i q^{45} -2.00000 q^{47} -3.00000 q^{49} -4.00000i q^{51} -2.00000i q^{53} -4.00000 q^{55} +16.0000 q^{57} +2.00000i q^{61} -2.00000 q^{63} +6.00000 q^{65} -6.00000i q^{67} -12.0000i q^{69} +12.0000 q^{71} -10.0000 q^{73} +2.00000i q^{75} +8.00000i q^{77} -8.00000 q^{79} -11.0000 q^{81} -10.0000i q^{83} +2.00000i q^{85} -4.00000 q^{87} +6.00000 q^{89} -12.0000i q^{91} -8.00000i q^{93} -8.00000 q^{95} +10.0000 q^{97} -4.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{7} - 2 q^{9} + 4 q^{15} + 4 q^{17} + 12 q^{23} - 2 q^{25} + 8 q^{31} + 16 q^{33} - 24 q^{39} + 20 q^{41} - 4 q^{47} - 6 q^{49} - 8 q^{55} + 32 q^{57} - 4 q^{63} + 12 q^{65} + 24 q^{71} - 20 q^{73} - 16 q^{79} - 22 q^{81} - 8 q^{87} + 12 q^{89} - 16 q^{95} + 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1280\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(261\) \(511\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) − 2.00000i − 1.15470i −0.816497 0.577350i \(-0.804087\pi\)
0.816497 0.577350i \(-0.195913\pi\)
\(4\) 0 0
\(5\) 1.00000i 0.447214i
\(6\) 0 0
\(7\) 2.00000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 4.00000i 1.20605i 0.797724 + 0.603023i \(0.206037\pi\)
−0.797724 + 0.603023i \(0.793963\pi\)
\(12\) 0 0
\(13\) − 6.00000i − 1.66410i −0.554700 0.832050i \(-0.687167\pi\)
0.554700 0.832050i \(-0.312833\pi\)
\(14\) 0 0
\(15\) 2.00000 0.516398
\(16\) 0 0
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) 8.00000i 1.83533i 0.397360 + 0.917663i \(0.369927\pi\)
−0.397360 + 0.917663i \(0.630073\pi\)
\(20\) 0 0
\(21\) − 4.00000i − 0.872872i
\(22\) 0 0
\(23\) 6.00000 1.25109 0.625543 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) − 4.00000i − 0.769800i
\(28\) 0 0
\(29\) − 2.00000i − 0.371391i −0.982607 0.185695i \(-0.940546\pi\)
0.982607 0.185695i \(-0.0594537\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 0 0
\(33\) 8.00000 1.39262
\(34\) 0 0
\(35\) 2.00000i 0.338062i
\(36\) 0 0
\(37\) − 2.00000i − 0.328798i −0.986394 0.164399i \(-0.947432\pi\)
0.986394 0.164399i \(-0.0525685\pi\)
\(38\) 0 0
\(39\) −12.0000 −1.92154
\(40\) 0 0
\(41\) 10.0000 1.56174 0.780869 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) 0 0
\(43\) 2.00000i 0.304997i 0.988304 + 0.152499i \(0.0487319\pi\)
−0.988304 + 0.152499i \(0.951268\pi\)
\(44\) 0 0
\(45\) − 1.00000i − 0.149071i
\(46\) 0 0
\(47\) −2.00000 −0.291730 −0.145865 0.989305i \(-0.546597\pi\)
−0.145865 + 0.989305i \(0.546597\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) − 4.00000i − 0.560112i
\(52\) 0 0
\(53\) − 2.00000i − 0.274721i −0.990521 0.137361i \(-0.956138\pi\)
0.990521 0.137361i \(-0.0438619\pi\)
\(54\) 0 0
\(55\) −4.00000 −0.539360
\(56\) 0 0
\(57\) 16.0000 2.11925
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 2.00000i 0.256074i 0.991769 + 0.128037i \(0.0408676\pi\)
−0.991769 + 0.128037i \(0.959132\pi\)
\(62\) 0 0
\(63\) −2.00000 −0.251976
\(64\) 0 0
\(65\) 6.00000 0.744208
\(66\) 0 0
\(67\) − 6.00000i − 0.733017i −0.930415 0.366508i \(-0.880553\pi\)
0.930415 0.366508i \(-0.119447\pi\)
\(68\) 0 0
\(69\) − 12.0000i − 1.44463i
\(70\) 0 0
\(71\) 12.0000 1.42414 0.712069 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 0 0
\(73\) −10.0000 −1.17041 −0.585206 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) 0 0
\(75\) 2.00000i 0.230940i
\(76\) 0 0
\(77\) 8.00000i 0.911685i
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) − 10.0000i − 1.09764i −0.835940 0.548821i \(-0.815077\pi\)
0.835940 0.548821i \(-0.184923\pi\)
\(84\) 0 0
\(85\) 2.00000i 0.216930i
\(86\) 0 0
\(87\) −4.00000 −0.428845
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) − 12.0000i − 1.25794i
\(92\) 0 0
\(93\) − 8.00000i − 0.829561i
\(94\) 0 0
\(95\) −8.00000 −0.820783
\(96\) 0 0
\(97\) 10.0000 1.01535 0.507673 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) 0 0
\(99\) − 4.00000i − 0.402015i
\(100\) 0 0
\(101\) − 14.0000i − 1.39305i −0.717532 0.696526i \(-0.754728\pi\)
0.717532 0.696526i \(-0.245272\pi\)
\(102\) 0 0
\(103\) −2.00000 −0.197066 −0.0985329 0.995134i \(-0.531415\pi\)
−0.0985329 + 0.995134i \(0.531415\pi\)
\(104\) 0 0
\(105\) 4.00000 0.390360
\(106\) 0 0
\(107\) 6.00000i 0.580042i 0.957020 + 0.290021i \(0.0936623\pi\)
−0.957020 + 0.290021i \(0.906338\pi\)
\(108\) 0 0
\(109\) − 14.0000i − 1.34096i −0.741929 0.670478i \(-0.766089\pi\)
0.741929 0.670478i \(-0.233911\pi\)
\(110\) 0 0
\(111\) −4.00000 −0.379663
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 6.00000i 0.559503i
\(116\) 0 0
\(117\) 6.00000i 0.554700i
\(118\) 0 0
\(119\) 4.00000 0.366679
\(120\) 0 0
\(121\) −5.00000 −0.454545
\(122\) 0 0
\(123\) − 20.0000i − 1.80334i
\(124\) 0 0
\(125\) − 1.00000i − 0.0894427i
\(126\) 0 0
\(127\) 6.00000 0.532414 0.266207 0.963916i \(-0.414230\pi\)
0.266207 + 0.963916i \(0.414230\pi\)
\(128\) 0 0
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) 4.00000i 0.349482i 0.984614 + 0.174741i \(0.0559088\pi\)
−0.984614 + 0.174741i \(0.944091\pi\)
\(132\) 0 0
\(133\) 16.0000i 1.38738i
\(134\) 0 0
\(135\) 4.00000 0.344265
\(136\) 0 0
\(137\) −2.00000 −0.170872 −0.0854358 0.996344i \(-0.527228\pi\)
−0.0854358 + 0.996344i \(0.527228\pi\)
\(138\) 0 0
\(139\) 16.0000i 1.35710i 0.734553 + 0.678551i \(0.237392\pi\)
−0.734553 + 0.678551i \(0.762608\pi\)
\(140\) 0 0
\(141\) 4.00000i 0.336861i
\(142\) 0 0
\(143\) 24.0000 2.00698
\(144\) 0 0
\(145\) 2.00000 0.166091
\(146\) 0 0
\(147\) 6.00000i 0.494872i
\(148\) 0 0
\(149\) − 10.0000i − 0.819232i −0.912258 0.409616i \(-0.865663\pi\)
0.912258 0.409616i \(-0.134337\pi\)
\(150\) 0 0
\(151\) −20.0000 −1.62758 −0.813788 0.581161i \(-0.802599\pi\)
−0.813788 + 0.581161i \(0.802599\pi\)
\(152\) 0 0
\(153\) −2.00000 −0.161690
\(154\) 0 0
\(155\) 4.00000i 0.321288i
\(156\) 0 0
\(157\) 10.0000i 0.798087i 0.916932 + 0.399043i \(0.130658\pi\)
−0.916932 + 0.399043i \(0.869342\pi\)
\(158\) 0 0
\(159\) −4.00000 −0.317221
\(160\) 0 0
\(161\) 12.0000 0.945732
\(162\) 0 0
\(163\) 6.00000i 0.469956i 0.972001 + 0.234978i \(0.0755019\pi\)
−0.972001 + 0.234978i \(0.924498\pi\)
\(164\) 0 0
\(165\) 8.00000i 0.622799i
\(166\) 0 0
\(167\) −14.0000 −1.08335 −0.541676 0.840587i \(-0.682210\pi\)
−0.541676 + 0.840587i \(0.682210\pi\)
\(168\) 0 0
\(169\) −23.0000 −1.76923
\(170\) 0 0
\(171\) − 8.00000i − 0.611775i
\(172\) 0 0
\(173\) − 6.00000i − 0.456172i −0.973641 0.228086i \(-0.926753\pi\)
0.973641 0.228086i \(-0.0732467\pi\)
\(174\) 0 0
\(175\) −2.00000 −0.151186
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) 18.0000i 1.33793i 0.743294 + 0.668965i \(0.233262\pi\)
−0.743294 + 0.668965i \(0.766738\pi\)
\(182\) 0 0
\(183\) 4.00000 0.295689
\(184\) 0 0
\(185\) 2.00000 0.147043
\(186\) 0 0
\(187\) 8.00000i 0.585018i
\(188\) 0 0
\(189\) − 8.00000i − 0.581914i
\(190\) 0 0
\(191\) −4.00000 −0.289430 −0.144715 0.989473i \(-0.546227\pi\)
−0.144715 + 0.989473i \(0.546227\pi\)
\(192\) 0 0
\(193\) 2.00000 0.143963 0.0719816 0.997406i \(-0.477068\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) 0 0
\(195\) − 12.0000i − 0.859338i
\(196\) 0 0
\(197\) 22.0000i 1.56744i 0.621117 + 0.783718i \(0.286679\pi\)
−0.621117 + 0.783718i \(0.713321\pi\)
\(198\) 0 0
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 0 0
\(201\) −12.0000 −0.846415
\(202\) 0 0
\(203\) − 4.00000i − 0.280745i
\(204\) 0 0
\(205\) 10.0000i 0.698430i
\(206\) 0 0
\(207\) −6.00000 −0.417029
\(208\) 0 0
\(209\) −32.0000 −2.21349
\(210\) 0 0
\(211\) 4.00000i 0.275371i 0.990476 + 0.137686i \(0.0439664\pi\)
−0.990476 + 0.137686i \(0.956034\pi\)
\(212\) 0 0
\(213\) − 24.0000i − 1.64445i
\(214\) 0 0
\(215\) −2.00000 −0.136399
\(216\) 0 0
\(217\) 8.00000 0.543075
\(218\) 0 0
\(219\) 20.0000i 1.35147i
\(220\) 0 0
\(221\) − 12.0000i − 0.807207i
\(222\) 0 0
\(223\) 2.00000 0.133930 0.0669650 0.997755i \(-0.478668\pi\)
0.0669650 + 0.997755i \(0.478668\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 18.0000i 1.19470i 0.801980 + 0.597351i \(0.203780\pi\)
−0.801980 + 0.597351i \(0.796220\pi\)
\(228\) 0 0
\(229\) 10.0000i 0.660819i 0.943838 + 0.330409i \(0.107187\pi\)
−0.943838 + 0.330409i \(0.892813\pi\)
\(230\) 0 0
\(231\) 16.0000 1.05272
\(232\) 0 0
\(233\) −2.00000 −0.131024 −0.0655122 0.997852i \(-0.520868\pi\)
−0.0655122 + 0.997852i \(0.520868\pi\)
\(234\) 0 0
\(235\) − 2.00000i − 0.130466i
\(236\) 0 0
\(237\) 16.0000i 1.03931i
\(238\) 0 0
\(239\) −8.00000 −0.517477 −0.258738 0.965947i \(-0.583307\pi\)
−0.258738 + 0.965947i \(0.583307\pi\)
\(240\) 0 0
\(241\) −18.0000 −1.15948 −0.579741 0.814801i \(-0.696846\pi\)
−0.579741 + 0.814801i \(0.696846\pi\)
\(242\) 0 0
\(243\) 10.0000i 0.641500i
\(244\) 0 0
\(245\) − 3.00000i − 0.191663i
\(246\) 0 0
\(247\) 48.0000 3.05417
\(248\) 0 0
\(249\) −20.0000 −1.26745
\(250\) 0 0
\(251\) 20.0000i 1.26239i 0.775625 + 0.631194i \(0.217435\pi\)
−0.775625 + 0.631194i \(0.782565\pi\)
\(252\) 0 0
\(253\) 24.0000i 1.50887i
\(254\) 0 0
\(255\) 4.00000 0.250490
\(256\) 0 0
\(257\) 26.0000 1.62184 0.810918 0.585160i \(-0.198968\pi\)
0.810918 + 0.585160i \(0.198968\pi\)
\(258\) 0 0
\(259\) − 4.00000i − 0.248548i
\(260\) 0 0
\(261\) 2.00000i 0.123797i
\(262\) 0 0
\(263\) −2.00000 −0.123325 −0.0616626 0.998097i \(-0.519640\pi\)
−0.0616626 + 0.998097i \(0.519640\pi\)
\(264\) 0 0
\(265\) 2.00000 0.122859
\(266\) 0 0
\(267\) − 12.0000i − 0.734388i
\(268\) 0 0
\(269\) 18.0000i 1.09748i 0.835993 + 0.548740i \(0.184892\pi\)
−0.835993 + 0.548740i \(0.815108\pi\)
\(270\) 0 0
\(271\) 28.0000 1.70088 0.850439 0.526073i \(-0.176336\pi\)
0.850439 + 0.526073i \(0.176336\pi\)
\(272\) 0 0
\(273\) −24.0000 −1.45255
\(274\) 0 0
\(275\) − 4.00000i − 0.241209i
\(276\) 0 0
\(277\) 22.0000i 1.32185i 0.750451 + 0.660926i \(0.229836\pi\)
−0.750451 + 0.660926i \(0.770164\pi\)
\(278\) 0 0
\(279\) −4.00000 −0.239474
\(280\) 0 0
\(281\) 10.0000 0.596550 0.298275 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 0 0
\(283\) 26.0000i 1.54554i 0.634686 + 0.772770i \(0.281129\pi\)
−0.634686 + 0.772770i \(0.718871\pi\)
\(284\) 0 0
\(285\) 16.0000i 0.947758i
\(286\) 0 0
\(287\) 20.0000 1.18056
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) − 20.0000i − 1.17242i
\(292\) 0 0
\(293\) − 18.0000i − 1.05157i −0.850617 0.525786i \(-0.823771\pi\)
0.850617 0.525786i \(-0.176229\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 16.0000 0.928414
\(298\) 0 0
\(299\) − 36.0000i − 2.08193i
\(300\) 0 0
\(301\) 4.00000i 0.230556i
\(302\) 0 0
\(303\) −28.0000 −1.60856
\(304\) 0 0
\(305\) −2.00000 −0.114520
\(306\) 0 0
\(307\) − 22.0000i − 1.25561i −0.778372 0.627803i \(-0.783954\pi\)
0.778372 0.627803i \(-0.216046\pi\)
\(308\) 0 0
\(309\) 4.00000i 0.227552i
\(310\) 0 0
\(311\) −28.0000 −1.58773 −0.793867 0.608091i \(-0.791935\pi\)
−0.793867 + 0.608091i \(0.791935\pi\)
\(312\) 0 0
\(313\) 6.00000 0.339140 0.169570 0.985518i \(-0.445762\pi\)
0.169570 + 0.985518i \(0.445762\pi\)
\(314\) 0 0
\(315\) − 2.00000i − 0.112687i
\(316\) 0 0
\(317\) 2.00000i 0.112331i 0.998421 + 0.0561656i \(0.0178875\pi\)
−0.998421 + 0.0561656i \(0.982113\pi\)
\(318\) 0 0
\(319\) 8.00000 0.447914
\(320\) 0 0
\(321\) 12.0000 0.669775
\(322\) 0 0
\(323\) 16.0000i 0.890264i
\(324\) 0 0
\(325\) 6.00000i 0.332820i
\(326\) 0 0
\(327\) −28.0000 −1.54840
\(328\) 0 0
\(329\) −4.00000 −0.220527
\(330\) 0 0
\(331\) − 4.00000i − 0.219860i −0.993939 0.109930i \(-0.964937\pi\)
0.993939 0.109930i \(-0.0350627\pi\)
\(332\) 0 0
\(333\) 2.00000i 0.109599i
\(334\) 0 0
\(335\) 6.00000 0.327815
\(336\) 0 0
\(337\) −14.0000 −0.762629 −0.381314 0.924445i \(-0.624528\pi\)
−0.381314 + 0.924445i \(0.624528\pi\)
\(338\) 0 0
\(339\) 12.0000i 0.651751i
\(340\) 0 0
\(341\) 16.0000i 0.866449i
\(342\) 0 0
\(343\) −20.0000 −1.07990
\(344\) 0 0
\(345\) 12.0000 0.646058
\(346\) 0 0
\(347\) − 18.0000i − 0.966291i −0.875540 0.483145i \(-0.839494\pi\)
0.875540 0.483145i \(-0.160506\pi\)
\(348\) 0 0
\(349\) − 2.00000i − 0.107058i −0.998566 0.0535288i \(-0.982953\pi\)
0.998566 0.0535288i \(-0.0170469\pi\)
\(350\) 0 0
\(351\) −24.0000 −1.28103
\(352\) 0 0
\(353\) −30.0000 −1.59674 −0.798369 0.602168i \(-0.794304\pi\)
−0.798369 + 0.602168i \(0.794304\pi\)
\(354\) 0 0
\(355\) 12.0000i 0.636894i
\(356\) 0 0
\(357\) − 8.00000i − 0.423405i
\(358\) 0 0
\(359\) −16.0000 −0.844448 −0.422224 0.906492i \(-0.638750\pi\)
−0.422224 + 0.906492i \(0.638750\pi\)
\(360\) 0 0
\(361\) −45.0000 −2.36842
\(362\) 0 0
\(363\) 10.0000i 0.524864i
\(364\) 0 0
\(365\) − 10.0000i − 0.523424i
\(366\) 0 0
\(367\) −18.0000 −0.939592 −0.469796 0.882775i \(-0.655673\pi\)
−0.469796 + 0.882775i \(0.655673\pi\)
\(368\) 0 0
\(369\) −10.0000 −0.520579
\(370\) 0 0
\(371\) − 4.00000i − 0.207670i
\(372\) 0 0
\(373\) − 10.0000i − 0.517780i −0.965907 0.258890i \(-0.916643\pi\)
0.965907 0.258890i \(-0.0833568\pi\)
\(374\) 0 0
\(375\) −2.00000 −0.103280
\(376\) 0 0
\(377\) −12.0000 −0.618031
\(378\) 0 0
\(379\) − 24.0000i − 1.23280i −0.787434 0.616399i \(-0.788591\pi\)
0.787434 0.616399i \(-0.211409\pi\)
\(380\) 0 0
\(381\) − 12.0000i − 0.614779i
\(382\) 0 0
\(383\) −14.0000 −0.715367 −0.357683 0.933843i \(-0.616433\pi\)
−0.357683 + 0.933843i \(0.616433\pi\)
\(384\) 0 0
\(385\) −8.00000 −0.407718
\(386\) 0 0
\(387\) − 2.00000i − 0.101666i
\(388\) 0 0
\(389\) − 26.0000i − 1.31825i −0.752032 0.659126i \(-0.770926\pi\)
0.752032 0.659126i \(-0.229074\pi\)
\(390\) 0 0
\(391\) 12.0000 0.606866
\(392\) 0 0
\(393\) 8.00000 0.403547
\(394\) 0 0
\(395\) − 8.00000i − 0.402524i
\(396\) 0 0
\(397\) 26.0000i 1.30490i 0.757831 + 0.652451i \(0.226259\pi\)
−0.757831 + 0.652451i \(0.773741\pi\)
\(398\) 0 0
\(399\) 32.0000 1.60200
\(400\) 0 0
\(401\) 18.0000 0.898877 0.449439 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) 0 0
\(403\) − 24.0000i − 1.19553i
\(404\) 0 0
\(405\) − 11.0000i − 0.546594i
\(406\) 0 0
\(407\) 8.00000 0.396545
\(408\) 0 0
\(409\) −14.0000 −0.692255 −0.346128 0.938187i \(-0.612504\pi\)
−0.346128 + 0.938187i \(0.612504\pi\)
\(410\) 0 0
\(411\) 4.00000i 0.197305i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 10.0000 0.490881
\(416\) 0 0
\(417\) 32.0000 1.56705
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) − 10.0000i − 0.487370i −0.969854 0.243685i \(-0.921644\pi\)
0.969854 0.243685i \(-0.0783563\pi\)
\(422\) 0 0
\(423\) 2.00000 0.0972433
\(424\) 0 0
\(425\) −2.00000 −0.0970143
\(426\) 0 0
\(427\) 4.00000i 0.193574i
\(428\) 0 0
\(429\) − 48.0000i − 2.31746i
\(430\) 0 0
\(431\) −20.0000 −0.963366 −0.481683 0.876346i \(-0.659974\pi\)
−0.481683 + 0.876346i \(0.659974\pi\)
\(432\) 0 0
\(433\) −22.0000 −1.05725 −0.528626 0.848855i \(-0.677293\pi\)
−0.528626 + 0.848855i \(0.677293\pi\)
\(434\) 0 0
\(435\) − 4.00000i − 0.191785i
\(436\) 0 0
\(437\) 48.0000i 2.29615i
\(438\) 0 0
\(439\) 16.0000 0.763638 0.381819 0.924237i \(-0.375298\pi\)
0.381819 + 0.924237i \(0.375298\pi\)
\(440\) 0 0
\(441\) 3.00000 0.142857
\(442\) 0 0
\(443\) 18.0000i 0.855206i 0.903967 + 0.427603i \(0.140642\pi\)
−0.903967 + 0.427603i \(0.859358\pi\)
\(444\) 0 0
\(445\) 6.00000i 0.284427i
\(446\) 0 0
\(447\) −20.0000 −0.945968
\(448\) 0 0
\(449\) 22.0000 1.03824 0.519122 0.854700i \(-0.326259\pi\)
0.519122 + 0.854700i \(0.326259\pi\)
\(450\) 0 0
\(451\) 40.0000i 1.88353i
\(452\) 0 0
\(453\) 40.0000i 1.87936i
\(454\) 0 0
\(455\) 12.0000 0.562569
\(456\) 0 0
\(457\) 6.00000 0.280668 0.140334 0.990104i \(-0.455182\pi\)
0.140334 + 0.990104i \(0.455182\pi\)
\(458\) 0 0
\(459\) − 8.00000i − 0.373408i
\(460\) 0 0
\(461\) 6.00000i 0.279448i 0.990190 + 0.139724i \(0.0446215\pi\)
−0.990190 + 0.139724i \(0.955378\pi\)
\(462\) 0 0
\(463\) −38.0000 −1.76601 −0.883005 0.469364i \(-0.844483\pi\)
−0.883005 + 0.469364i \(0.844483\pi\)
\(464\) 0 0
\(465\) 8.00000 0.370991
\(466\) 0 0
\(467\) − 6.00000i − 0.277647i −0.990317 0.138823i \(-0.955668\pi\)
0.990317 0.138823i \(-0.0443321\pi\)
\(468\) 0 0
\(469\) − 12.0000i − 0.554109i
\(470\) 0 0
\(471\) 20.0000 0.921551
\(472\) 0 0
\(473\) −8.00000 −0.367840
\(474\) 0 0
\(475\) − 8.00000i − 0.367065i
\(476\) 0 0
\(477\) 2.00000i 0.0915737i
\(478\) 0 0
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 0 0
\(481\) −12.0000 −0.547153
\(482\) 0 0
\(483\) − 24.0000i − 1.09204i
\(484\) 0 0
\(485\) 10.0000i 0.454077i
\(486\) 0 0
\(487\) −38.0000 −1.72194 −0.860972 0.508652i \(-0.830144\pi\)
−0.860972 + 0.508652i \(0.830144\pi\)
\(488\) 0 0
\(489\) 12.0000 0.542659
\(490\) 0 0
\(491\) 20.0000i 0.902587i 0.892375 + 0.451294i \(0.149037\pi\)
−0.892375 + 0.451294i \(0.850963\pi\)
\(492\) 0 0
\(493\) − 4.00000i − 0.180151i
\(494\) 0 0
\(495\) 4.00000 0.179787
\(496\) 0 0
\(497\) 24.0000 1.07655
\(498\) 0 0
\(499\) − 40.0000i − 1.79065i −0.445418 0.895323i \(-0.646945\pi\)
0.445418 0.895323i \(-0.353055\pi\)
\(500\) 0 0
\(501\) 28.0000i 1.25095i
\(502\) 0 0
\(503\) 14.0000 0.624229 0.312115 0.950044i \(-0.398963\pi\)
0.312115 + 0.950044i \(0.398963\pi\)
\(504\) 0 0
\(505\) 14.0000 0.622992
\(506\) 0 0
\(507\) 46.0000i 2.04293i
\(508\) 0 0
\(509\) − 18.0000i − 0.797836i −0.916987 0.398918i \(-0.869386\pi\)
0.916987 0.398918i \(-0.130614\pi\)
\(510\) 0 0
\(511\) −20.0000 −0.884748
\(512\) 0 0
\(513\) 32.0000 1.41283
\(514\) 0 0
\(515\) − 2.00000i − 0.0881305i
\(516\) 0 0
\(517\) − 8.00000i − 0.351840i
\(518\) 0 0
\(519\) −12.0000 −0.526742
\(520\) 0 0
\(521\) 6.00000 0.262865 0.131432 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) 0 0
\(523\) − 22.0000i − 0.961993i −0.876723 0.480996i \(-0.840275\pi\)
0.876723 0.480996i \(-0.159725\pi\)
\(524\) 0 0
\(525\) 4.00000i 0.174574i
\(526\) 0 0
\(527\) 8.00000 0.348485
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 60.0000i − 2.59889i
\(534\) 0 0
\(535\) −6.00000 −0.259403
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) − 12.0000i − 0.516877i
\(540\) 0 0
\(541\) − 10.0000i − 0.429934i −0.976621 0.214967i \(-0.931036\pi\)
0.976621 0.214967i \(-0.0689643\pi\)
\(542\) 0 0
\(543\) 36.0000 1.54491
\(544\) 0 0
\(545\) 14.0000 0.599694
\(546\) 0 0
\(547\) − 14.0000i − 0.598597i −0.954160 0.299298i \(-0.903247\pi\)
0.954160 0.299298i \(-0.0967526\pi\)
\(548\) 0 0
\(549\) − 2.00000i − 0.0853579i
\(550\) 0 0
\(551\) 16.0000 0.681623
\(552\) 0 0
\(553\) −16.0000 −0.680389
\(554\) 0 0
\(555\) − 4.00000i − 0.169791i
\(556\) 0 0
\(557\) − 46.0000i − 1.94908i −0.224208 0.974541i \(-0.571980\pi\)
0.224208 0.974541i \(-0.428020\pi\)
\(558\) 0 0
\(559\) 12.0000 0.507546
\(560\) 0 0
\(561\) 16.0000 0.675521
\(562\) 0 0
\(563\) 30.0000i 1.26435i 0.774826 + 0.632175i \(0.217837\pi\)
−0.774826 + 0.632175i \(0.782163\pi\)
\(564\) 0 0
\(565\) − 6.00000i − 0.252422i
\(566\) 0 0
\(567\) −22.0000 −0.923913
\(568\) 0 0
\(569\) 18.0000 0.754599 0.377300 0.926091i \(-0.376853\pi\)
0.377300 + 0.926091i \(0.376853\pi\)
\(570\) 0 0
\(571\) − 4.00000i − 0.167395i −0.996491 0.0836974i \(-0.973327\pi\)
0.996491 0.0836974i \(-0.0266729\pi\)
\(572\) 0 0
\(573\) 8.00000i 0.334205i
\(574\) 0 0
\(575\) −6.00000 −0.250217
\(576\) 0 0
\(577\) −38.0000 −1.58196 −0.790980 0.611842i \(-0.790429\pi\)
−0.790980 + 0.611842i \(0.790429\pi\)
\(578\) 0 0
\(579\) − 4.00000i − 0.166234i
\(580\) 0 0
\(581\) − 20.0000i − 0.829740i
\(582\) 0 0
\(583\) 8.00000 0.331326
\(584\) 0 0
\(585\) −6.00000 −0.248069
\(586\) 0 0
\(587\) − 10.0000i − 0.412744i −0.978474 0.206372i \(-0.933834\pi\)
0.978474 0.206372i \(-0.0661657\pi\)
\(588\) 0 0
\(589\) 32.0000i 1.31854i
\(590\) 0 0
\(591\) 44.0000 1.80992
\(592\) 0 0
\(593\) 18.0000 0.739171 0.369586 0.929197i \(-0.379500\pi\)
0.369586 + 0.929197i \(0.379500\pi\)
\(594\) 0 0
\(595\) 4.00000i 0.163984i
\(596\) 0 0
\(597\) − 32.0000i − 1.30967i
\(598\) 0 0
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 0 0
\(601\) 26.0000 1.06056 0.530281 0.847822i \(-0.322086\pi\)
0.530281 + 0.847822i \(0.322086\pi\)
\(602\) 0 0
\(603\) 6.00000i 0.244339i
\(604\) 0 0
\(605\) − 5.00000i − 0.203279i
\(606\) 0 0
\(607\) −2.00000 −0.0811775 −0.0405887 0.999176i \(-0.512923\pi\)
−0.0405887 + 0.999176i \(0.512923\pi\)
\(608\) 0 0
\(609\) −8.00000 −0.324176
\(610\) 0 0
\(611\) 12.0000i 0.485468i
\(612\) 0 0
\(613\) 22.0000i 0.888572i 0.895885 + 0.444286i \(0.146543\pi\)
−0.895885 + 0.444286i \(0.853457\pi\)
\(614\) 0 0
\(615\) 20.0000 0.806478
\(616\) 0 0
\(617\) 30.0000 1.20775 0.603877 0.797077i \(-0.293622\pi\)
0.603877 + 0.797077i \(0.293622\pi\)
\(618\) 0 0
\(619\) − 8.00000i − 0.321547i −0.986991 0.160774i \(-0.948601\pi\)
0.986991 0.160774i \(-0.0513989\pi\)
\(620\) 0 0
\(621\) − 24.0000i − 0.963087i
\(622\) 0 0
\(623\) 12.0000 0.480770
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 64.0000i 2.55591i
\(628\) 0 0
\(629\) − 4.00000i − 0.159490i
\(630\) 0 0
\(631\) −36.0000 −1.43314 −0.716569 0.697517i \(-0.754288\pi\)
−0.716569 + 0.697517i \(0.754288\pi\)
\(632\) 0 0
\(633\) 8.00000 0.317971
\(634\) 0 0
\(635\) 6.00000i 0.238103i
\(636\) 0 0
\(637\) 18.0000i 0.713186i
\(638\) 0 0
\(639\) −12.0000 −0.474713
\(640\) 0 0
\(641\) −18.0000 −0.710957 −0.355479 0.934684i \(-0.615682\pi\)
−0.355479 + 0.934684i \(0.615682\pi\)
\(642\) 0 0
\(643\) 30.0000i 1.18308i 0.806274 + 0.591542i \(0.201481\pi\)
−0.806274 + 0.591542i \(0.798519\pi\)
\(644\) 0 0
\(645\) 4.00000i 0.157500i
\(646\) 0 0
\(647\) 42.0000 1.65119 0.825595 0.564263i \(-0.190840\pi\)
0.825595 + 0.564263i \(0.190840\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) − 16.0000i − 0.627089i
\(652\) 0 0
\(653\) − 30.0000i − 1.17399i −0.809590 0.586995i \(-0.800311\pi\)
0.809590 0.586995i \(-0.199689\pi\)
\(654\) 0 0
\(655\) −4.00000 −0.156293
\(656\) 0 0
\(657\) 10.0000 0.390137
\(658\) 0 0
\(659\) − 16.0000i − 0.623272i −0.950202 0.311636i \(-0.899123\pi\)
0.950202 0.311636i \(-0.100877\pi\)
\(660\) 0 0
\(661\) − 10.0000i − 0.388955i −0.980907 0.194477i \(-0.937699\pi\)
0.980907 0.194477i \(-0.0623011\pi\)
\(662\) 0 0
\(663\) −24.0000 −0.932083
\(664\) 0 0
\(665\) −16.0000 −0.620453
\(666\) 0 0
\(667\) − 12.0000i − 0.464642i
\(668\) 0 0
\(669\) − 4.00000i − 0.154649i
\(670\) 0 0
\(671\) −8.00000 −0.308837
\(672\) 0 0
\(673\) 10.0000 0.385472 0.192736 0.981251i \(-0.438264\pi\)
0.192736 + 0.981251i \(0.438264\pi\)
\(674\) 0 0
\(675\) 4.00000i 0.153960i
\(676\) 0 0
\(677\) − 26.0000i − 0.999261i −0.866239 0.499631i \(-0.833469\pi\)
0.866239 0.499631i \(-0.166531\pi\)
\(678\) 0 0
\(679\) 20.0000 0.767530
\(680\) 0 0
\(681\) 36.0000 1.37952
\(682\) 0 0
\(683\) 34.0000i 1.30097i 0.759517 + 0.650487i \(0.225435\pi\)
−0.759517 + 0.650487i \(0.774565\pi\)
\(684\) 0 0
\(685\) − 2.00000i − 0.0764161i
\(686\) 0 0
\(687\) 20.0000 0.763048
\(688\) 0 0
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) 12.0000i 0.456502i 0.973602 + 0.228251i \(0.0733006\pi\)
−0.973602 + 0.228251i \(0.926699\pi\)
\(692\) 0 0
\(693\) − 8.00000i − 0.303895i
\(694\) 0 0
\(695\) −16.0000 −0.606915
\(696\) 0 0
\(697\) 20.0000 0.757554
\(698\) 0 0
\(699\) 4.00000i 0.151294i
\(700\) 0 0
\(701\) 18.0000i 0.679851i 0.940452 + 0.339925i \(0.110402\pi\)
−0.940452 + 0.339925i \(0.889598\pi\)
\(702\) 0 0
\(703\) 16.0000 0.603451
\(704\) 0 0
\(705\) −4.00000 −0.150649
\(706\) 0 0
\(707\) − 28.0000i − 1.05305i
\(708\) 0 0
\(709\) 10.0000i 0.375558i 0.982211 + 0.187779i \(0.0601289\pi\)
−0.982211 + 0.187779i \(0.939871\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) 0 0
\(713\) 24.0000 0.898807
\(714\) 0 0
\(715\) 24.0000i 0.897549i
\(716\) 0 0
\(717\) 16.0000i 0.597531i
\(718\) 0 0
\(719\) −40.0000 −1.49175 −0.745874 0.666087i \(-0.767968\pi\)
−0.745874 + 0.666087i \(0.767968\pi\)
\(720\) 0 0
\(721\) −4.00000 −0.148968
\(722\) 0 0
\(723\) 36.0000i 1.33885i
\(724\) 0 0
\(725\) 2.00000i 0.0742781i
\(726\) 0 0
\(727\) 2.00000 0.0741759 0.0370879 0.999312i \(-0.488192\pi\)
0.0370879 + 0.999312i \(0.488192\pi\)
\(728\) 0 0
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) 4.00000i 0.147945i
\(732\) 0 0
\(733\) − 6.00000i − 0.221615i −0.993842 0.110808i \(-0.964656\pi\)
0.993842 0.110808i \(-0.0353437\pi\)
\(734\) 0 0
\(735\) −6.00000 −0.221313
\(736\) 0 0
\(737\) 24.0000 0.884051
\(738\) 0 0
\(739\) 16.0000i 0.588570i 0.955718 + 0.294285i \(0.0950814\pi\)
−0.955718 + 0.294285i \(0.904919\pi\)
\(740\) 0 0
\(741\) − 96.0000i − 3.52665i
\(742\) 0 0
\(743\) −42.0000 −1.54083 −0.770415 0.637542i \(-0.779951\pi\)
−0.770415 + 0.637542i \(0.779951\pi\)
\(744\) 0 0
\(745\) 10.0000 0.366372
\(746\) 0 0
\(747\) 10.0000i 0.365881i
\(748\) 0 0
\(749\) 12.0000i 0.438470i
\(750\) 0 0
\(751\) 4.00000 0.145962 0.0729810 0.997333i \(-0.476749\pi\)
0.0729810 + 0.997333i \(0.476749\pi\)
\(752\) 0 0
\(753\) 40.0000 1.45768
\(754\) 0 0
\(755\) − 20.0000i − 0.727875i
\(756\) 0 0
\(757\) − 10.0000i − 0.363456i −0.983349 0.181728i \(-0.941831\pi\)
0.983349 0.181728i \(-0.0581691\pi\)
\(758\) 0 0
\(759\) 48.0000 1.74229
\(760\) 0 0
\(761\) −10.0000 −0.362500 −0.181250 0.983437i \(-0.558014\pi\)
−0.181250 + 0.983437i \(0.558014\pi\)
\(762\) 0 0
\(763\) − 28.0000i − 1.01367i
\(764\) 0 0
\(765\) − 2.00000i − 0.0723102i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −30.0000 −1.08183 −0.540914 0.841078i \(-0.681921\pi\)
−0.540914 + 0.841078i \(0.681921\pi\)
\(770\) 0 0
\(771\) − 52.0000i − 1.87273i
\(772\) 0 0
\(773\) 38.0000i 1.36677i 0.730061 + 0.683383i \(0.239492\pi\)
−0.730061 + 0.683383i \(0.760508\pi\)
\(774\) 0 0
\(775\) −4.00000 −0.143684
\(776\) 0 0
\(777\) −8.00000 −0.286998
\(778\) 0 0
\(779\) 80.0000i 2.86630i
\(780\) 0 0
\(781\) 48.0000i 1.71758i
\(782\) 0 0
\(783\) −8.00000 −0.285897
\(784\) 0 0
\(785\) −10.0000 −0.356915
\(786\) 0 0
\(787\) − 14.0000i − 0.499046i −0.968369 0.249523i \(-0.919726\pi\)
0.968369 0.249523i \(-0.0802738\pi\)
\(788\) 0 0
\(789\) 4.00000i 0.142404i
\(790\) 0 0
\(791\) −12.0000 −0.426671
\(792\) 0 0
\(793\) 12.0000 0.426132
\(794\) 0 0
\(795\) − 4.00000i − 0.141865i
\(796\) 0 0
\(797\) − 30.0000i − 1.06265i −0.847167 0.531327i \(-0.821693\pi\)
0.847167 0.531327i \(-0.178307\pi\)
\(798\) 0 0
\(799\) −4.00000 −0.141510
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) 0 0
\(803\) − 40.0000i − 1.41157i
\(804\) 0 0
\(805\) 12.0000i 0.422944i
\(806\) 0 0
\(807\) 36.0000 1.26726
\(808\) 0 0
\(809\) 6.00000 0.210949 0.105474 0.994422i \(-0.466364\pi\)
0.105474 + 0.994422i \(0.466364\pi\)
\(810\) 0 0
\(811\) 20.0000i 0.702295i 0.936320 + 0.351147i \(0.114208\pi\)
−0.936320 + 0.351147i \(0.885792\pi\)
\(812\) 0 0
\(813\) − 56.0000i − 1.96401i
\(814\) 0 0
\(815\) −6.00000 −0.210171
\(816\) 0 0
\(817\) −16.0000 −0.559769
\(818\) 0 0
\(819\) 12.0000i 0.419314i
\(820\) 0 0
\(821\) 38.0000i 1.32621i 0.748527 + 0.663105i \(0.230762\pi\)
−0.748527 + 0.663105i \(0.769238\pi\)
\(822\) 0 0
\(823\) 6.00000 0.209147 0.104573 0.994517i \(-0.466652\pi\)
0.104573 + 0.994517i \(0.466652\pi\)
\(824\) 0 0
\(825\) −8.00000 −0.278524
\(826\) 0 0
\(827\) − 50.0000i − 1.73867i −0.494223 0.869335i \(-0.664547\pi\)
0.494223 0.869335i \(-0.335453\pi\)
\(828\) 0 0
\(829\) 2.00000i 0.0694629i 0.999397 + 0.0347314i \(0.0110576\pi\)
−0.999397 + 0.0347314i \(0.988942\pi\)
\(830\) 0 0
\(831\) 44.0000 1.52634
\(832\) 0 0
\(833\) −6.00000 −0.207888
\(834\) 0 0
\(835\) − 14.0000i − 0.484490i
\(836\) 0 0
\(837\) − 16.0000i − 0.553041i
\(838\) 0 0
\(839\) −24.0000 −0.828572 −0.414286 0.910147i \(-0.635969\pi\)
−0.414286 + 0.910147i \(0.635969\pi\)
\(840\) 0 0
\(841\) 25.0000 0.862069
\(842\) 0 0
\(843\) − 20.0000i − 0.688837i
\(844\) 0 0
\(845\) − 23.0000i − 0.791224i
\(846\) 0 0
\(847\) −10.0000 −0.343604
\(848\) 0 0
\(849\) 52.0000 1.78464
\(850\) 0 0
\(851\) − 12.0000i − 0.411355i
\(852\) 0 0
\(853\) − 10.0000i − 0.342393i −0.985237 0.171197i \(-0.945237\pi\)
0.985237 0.171197i \(-0.0547634\pi\)
\(854\) 0 0
\(855\) 8.00000 0.273594
\(856\) 0 0
\(857\) −18.0000 −0.614868 −0.307434 0.951569i \(-0.599470\pi\)
−0.307434 + 0.951569i \(0.599470\pi\)
\(858\) 0 0
\(859\) 16.0000i 0.545913i 0.962026 + 0.272956i \(0.0880015\pi\)
−0.962026 + 0.272956i \(0.911998\pi\)
\(860\) 0 0
\(861\) − 40.0000i − 1.36320i
\(862\) 0 0
\(863\) −46.0000 −1.56586 −0.782929 0.622111i \(-0.786275\pi\)
−0.782929 + 0.622111i \(0.786275\pi\)
\(864\) 0 0
\(865\) 6.00000 0.204006
\(866\) 0 0
\(867\) 26.0000i 0.883006i
\(868\) 0 0
\(869\) − 32.0000i − 1.08553i
\(870\) 0 0
\(871\) −36.0000 −1.21981
\(872\) 0 0
\(873\) −10.0000 −0.338449
\(874\) 0 0
\(875\) − 2.00000i − 0.0676123i
\(876\) 0 0
\(877\) 10.0000i 0.337676i 0.985644 + 0.168838i \(0.0540015\pi\)
−0.985644 + 0.168838i \(0.945999\pi\)
\(878\) 0 0
\(879\) −36.0000 −1.21425
\(880\) 0 0
\(881\) 14.0000 0.471672 0.235836 0.971793i \(-0.424217\pi\)
0.235836 + 0.971793i \(0.424217\pi\)
\(882\) 0 0
\(883\) − 2.00000i − 0.0673054i −0.999434 0.0336527i \(-0.989286\pi\)
0.999434 0.0336527i \(-0.0107140\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 2.00000 0.0671534 0.0335767 0.999436i \(-0.489310\pi\)
0.0335767 + 0.999436i \(0.489310\pi\)
\(888\) 0 0
\(889\) 12.0000 0.402467
\(890\) 0 0
\(891\) − 44.0000i − 1.47406i
\(892\) 0 0
\(893\) − 16.0000i − 0.535420i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −72.0000 −2.40401
\(898\) 0 0
\(899\) − 8.00000i − 0.266815i
\(900\) 0 0
\(901\) − 4.00000i − 0.133259i
\(902\) 0 0
\(903\) 8.00000 0.266223
\(904\) 0 0
\(905\) −18.0000 −0.598340
\(906\) 0 0
\(907\) − 2.00000i − 0.0664089i −0.999449 0.0332045i \(-0.989429\pi\)
0.999449 0.0332045i \(-0.0105712\pi\)
\(908\) 0 0
\(909\) 14.0000i 0.464351i
\(910\) 0 0
\(911\) −28.0000 −0.927681 −0.463841 0.885919i \(-0.653529\pi\)
−0.463841 + 0.885919i \(0.653529\pi\)
\(912\) 0 0
\(913\) 40.0000 1.32381
\(914\) 0 0
\(915\) 4.00000i 0.132236i
\(916\) 0 0
\(917\) 8.00000i 0.264183i
\(918\) 0 0
\(919\) −56.0000 −1.84727 −0.923635 0.383274i \(-0.874797\pi\)
−0.923635 + 0.383274i \(0.874797\pi\)
\(920\) 0 0
\(921\) −44.0000 −1.44985
\(922\) 0 0
\(923\) − 72.0000i − 2.36991i
\(924\) 0 0
\(925\) 2.00000i 0.0657596i
\(926\) 0 0
\(927\) 2.00000 0.0656886
\(928\) 0 0
\(929\) 6.00000 0.196854 0.0984268 0.995144i \(-0.468619\pi\)
0.0984268 + 0.995144i \(0.468619\pi\)
\(930\) 0 0
\(931\) − 24.0000i − 0.786568i
\(932\) 0 0
\(933\) 56.0000i 1.83336i
\(934\) 0 0
\(935\) −8.00000 −0.261628
\(936\) 0 0
\(937\) −34.0000 −1.11073 −0.555366 0.831606i \(-0.687422\pi\)
−0.555366 + 0.831606i \(0.687422\pi\)
\(938\) 0 0
\(939\) − 12.0000i − 0.391605i
\(940\) 0 0
\(941\) 38.0000i 1.23876i 0.785090 + 0.619382i \(0.212617\pi\)
−0.785090 + 0.619382i \(0.787383\pi\)
\(942\) 0 0
\(943\) 60.0000 1.95387
\(944\) 0 0
\(945\) 8.00000 0.260240
\(946\) 0 0
\(947\) − 38.0000i − 1.23483i −0.786636 0.617417i \(-0.788179\pi\)
0.786636 0.617417i \(-0.211821\pi\)
\(948\) 0 0
\(949\) 60.0000i 1.94768i
\(950\) 0 0
\(951\) 4.00000 0.129709
\(952\) 0 0
\(953\) −58.0000 −1.87880 −0.939402 0.342817i \(-0.888619\pi\)
−0.939402 + 0.342817i \(0.888619\pi\)
\(954\) 0 0
\(955\) − 4.00000i − 0.129437i
\(956\) 0 0
\(957\) − 16.0000i − 0.517207i
\(958\) 0 0
\(959\) −4.00000 −0.129167
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) − 6.00000i − 0.193347i
\(964\) 0 0
\(965\) 2.00000i 0.0643823i
\(966\) 0 0
\(967\) −22.0000 −0.707472 −0.353736 0.935345i \(-0.615089\pi\)
−0.353736 + 0.935345i \(0.615089\pi\)
\(968\) 0 0
\(969\) 32.0000 1.02799
\(970\) 0 0
\(971\) − 36.0000i − 1.15529i −0.816286 0.577647i \(-0.803971\pi\)
0.816286 0.577647i \(-0.196029\pi\)
\(972\) 0 0
\(973\) 32.0000i 1.02587i
\(974\) 0 0
\(975\) 12.0000 0.384308
\(976\) 0 0
\(977\) 42.0000 1.34370 0.671850 0.740688i \(-0.265500\pi\)
0.671850 + 0.740688i \(0.265500\pi\)
\(978\) 0 0
\(979\) 24.0000i 0.767043i
\(980\) 0 0
\(981\) 14.0000i 0.446986i
\(982\) 0 0
\(983\) 46.0000 1.46717 0.733586 0.679597i \(-0.237845\pi\)
0.733586 + 0.679597i \(0.237845\pi\)
\(984\) 0 0
\(985\) −22.0000 −0.700978
\(986\) 0 0
\(987\) 8.00000i 0.254643i
\(988\) 0 0
\(989\) 12.0000i 0.381578i
\(990\) 0 0
\(991\) −44.0000 −1.39771 −0.698853 0.715265i \(-0.746306\pi\)
−0.698853 + 0.715265i \(0.746306\pi\)
\(992\) 0 0
\(993\) −8.00000 −0.253872
\(994\) 0 0
\(995\) 16.0000i 0.507234i
\(996\) 0 0
\(997\) − 18.0000i − 0.570066i −0.958518 0.285033i \(-0.907995\pi\)
0.958518 0.285033i \(-0.0920045\pi\)
\(998\) 0 0
\(999\) −8.00000 −0.253109
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 1280.2.d.h.641.1 2
4.3 odd 2 1280.2.d.b.641.2 2
8.3 odd 2 1280.2.d.b.641.1 2
8.5 even 2 inner 1280.2.d.h.641.2 2
16.3 odd 4 320.2.a.b.1.1 1
16.5 even 4 160.2.a.a.1.1 1
16.11 odd 4 160.2.a.b.1.1 yes 1
16.13 even 4 320.2.a.e.1.1 1
48.5 odd 4 1440.2.a.i.1.1 1
48.11 even 4 1440.2.a.l.1.1 1
48.29 odd 4 2880.2.a.d.1.1 1
48.35 even 4 2880.2.a.o.1.1 1
80.3 even 4 1600.2.c.c.449.1 2
80.13 odd 4 1600.2.c.f.449.2 2
80.19 odd 4 1600.2.a.t.1.1 1
80.27 even 4 800.2.c.b.449.1 2
80.29 even 4 1600.2.a.e.1.1 1
80.37 odd 4 800.2.c.a.449.2 2
80.43 even 4 800.2.c.b.449.2 2
80.53 odd 4 800.2.c.a.449.1 2
80.59 odd 4 800.2.a.a.1.1 1
80.67 even 4 1600.2.c.c.449.2 2
80.69 even 4 800.2.a.i.1.1 1
80.77 odd 4 1600.2.c.f.449.1 2
112.27 even 4 7840.2.a.e.1.1 1
112.69 odd 4 7840.2.a.w.1.1 1
240.53 even 4 7200.2.f.w.6049.2 2
240.59 even 4 7200.2.a.l.1.1 1
240.107 odd 4 7200.2.f.g.6049.2 2
240.149 odd 4 7200.2.a.bp.1.1 1
240.197 even 4 7200.2.f.w.6049.1 2
240.203 odd 4 7200.2.f.g.6049.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
160.2.a.a.1.1 1 16.5 even 4
160.2.a.b.1.1 yes 1 16.11 odd 4
320.2.a.b.1.1 1 16.3 odd 4
320.2.a.e.1.1 1 16.13 even 4
800.2.a.a.1.1 1 80.59 odd 4
800.2.a.i.1.1 1 80.69 even 4
800.2.c.a.449.1 2 80.53 odd 4
800.2.c.a.449.2 2 80.37 odd 4
800.2.c.b.449.1 2 80.27 even 4
800.2.c.b.449.2 2 80.43 even 4
1280.2.d.b.641.1 2 8.3 odd 2
1280.2.d.b.641.2 2 4.3 odd 2
1280.2.d.h.641.1 2 1.1 even 1 trivial
1280.2.d.h.641.2 2 8.5 even 2 inner
1440.2.a.i.1.1 1 48.5 odd 4
1440.2.a.l.1.1 1 48.11 even 4
1600.2.a.e.1.1 1 80.29 even 4
1600.2.a.t.1.1 1 80.19 odd 4
1600.2.c.c.449.1 2 80.3 even 4
1600.2.c.c.449.2 2 80.67 even 4
1600.2.c.f.449.1 2 80.77 odd 4
1600.2.c.f.449.2 2 80.13 odd 4
2880.2.a.d.1.1 1 48.29 odd 4
2880.2.a.o.1.1 1 48.35 even 4
7200.2.a.l.1.1 1 240.59 even 4
7200.2.a.bp.1.1 1 240.149 odd 4
7200.2.f.g.6049.1 2 240.203 odd 4
7200.2.f.g.6049.2 2 240.107 odd 4
7200.2.f.w.6049.1 2 240.197 even 4
7200.2.f.w.6049.2 2 240.53 even 4
7840.2.a.e.1.1 1 112.27 even 4
7840.2.a.w.1.1 1 112.69 odd 4