Properties

Label 4840.2.a.bf.1.2
Level $4840$
Weight $2$
Character 4840.1
Self dual yes
Analytic conductor $38.648$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4840,2,Mod(1,4840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4840, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4840.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4840 = 2^{3} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4840.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: \(38.6475945783\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.45753625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 13x^{4} + 11x^{3} + 41x^{2} - 30x - 20 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 440)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-2.80540\) of defining polynomial
Character \(\chi\) \(=\) 4840.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.73383 q^{3} -1.00000 q^{5} -1.25225 q^{7} +0.00617996 q^{9} +O(q^{10})\) \(q-1.73383 q^{3} -1.00000 q^{5} -1.25225 q^{7} +0.00617996 q^{9} -2.50777 q^{13} +1.73383 q^{15} -3.60412 q^{17} -7.05765 q^{19} +2.17119 q^{21} -2.20737 q^{23} +1.00000 q^{25} +5.19079 q^{27} +2.69960 q^{29} -2.83345 q^{31} +1.25225 q^{35} -8.07294 q^{37} +4.34805 q^{39} -10.3266 q^{41} +6.88581 q^{43} -0.00617996 q^{45} +1.59656 q^{47} -5.43188 q^{49} +6.24894 q^{51} +1.77393 q^{53} +12.2368 q^{57} -4.36887 q^{59} -8.33324 q^{61} -0.00773884 q^{63} +2.50777 q^{65} +5.43293 q^{67} +3.82721 q^{69} -4.63664 q^{71} -2.03474 q^{73} -1.73383 q^{75} +16.7625 q^{79} -9.01850 q^{81} -5.62561 q^{83} +3.60412 q^{85} -4.68066 q^{87} -4.72832 q^{89} +3.14034 q^{91} +4.91274 q^{93} +7.05765 q^{95} +19.0943 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{3} - 6 q^{5} + 6 q^{7} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 2 q^{3} - 6 q^{5} + 6 q^{7} + 10 q^{9} - 6 q^{13} - 2 q^{15} + 11 q^{17} - 11 q^{19} + 2 q^{21} + 18 q^{23} + 6 q^{25} - q^{27} - 6 q^{29} + q^{31} - 6 q^{35} + 4 q^{37} + 27 q^{39} - 4 q^{41} + 3 q^{43} - 10 q^{45} + 14 q^{47} + 8 q^{49} + 31 q^{51} + 14 q^{53} - 5 q^{57} + 2 q^{59} - 4 q^{61} - 16 q^{63} + 6 q^{65} + 11 q^{67} + 8 q^{69} + 7 q^{71} - 9 q^{73} + 2 q^{75} + 36 q^{79} + 30 q^{81} - 45 q^{83} - 11 q^{85} + 25 q^{87} + q^{89} - 8 q^{91} + 55 q^{93} + 11 q^{95} + 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) −1.73383 −1.00103 −0.500515 0.865728i \(-0.666856\pi\)
−0.500515 + 0.865728i \(0.666856\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −1.25225 −0.473305 −0.236652 0.971594i \(-0.576050\pi\)
−0.236652 + 0.971594i \(0.576050\pi\)
\(8\) 0 0
\(9\) 0.00617996 0.00205999
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) −2.50777 −0.695529 −0.347764 0.937582i \(-0.613059\pi\)
−0.347764 + 0.937582i \(0.613059\pi\)
\(14\) 0 0
\(15\) 1.73383 0.447674
\(16\) 0 0
\(17\) −3.60412 −0.874126 −0.437063 0.899431i \(-0.643981\pi\)
−0.437063 + 0.899431i \(0.643981\pi\)
\(18\) 0 0
\(19\) −7.05765 −1.61914 −0.809568 0.587026i \(-0.800299\pi\)
−0.809568 + 0.587026i \(0.800299\pi\)
\(20\) 0 0
\(21\) 2.17119 0.473792
\(22\) 0 0
\(23\) −2.20737 −0.460268 −0.230134 0.973159i \(-0.573916\pi\)
−0.230134 + 0.973159i \(0.573916\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 5.19079 0.998967
\(28\) 0 0
\(29\) 2.69960 0.501303 0.250652 0.968077i \(-0.419355\pi\)
0.250652 + 0.968077i \(0.419355\pi\)
\(30\) 0 0
\(31\) −2.83345 −0.508903 −0.254452 0.967085i \(-0.581895\pi\)
−0.254452 + 0.967085i \(0.581895\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.25225 0.211668
\(36\) 0 0
\(37\) −8.07294 −1.32718 −0.663592 0.748095i \(-0.730969\pi\)
−0.663592 + 0.748095i \(0.730969\pi\)
\(38\) 0 0
\(39\) 4.34805 0.696245
\(40\) 0 0
\(41\) −10.3266 −1.61274 −0.806371 0.591410i \(-0.798571\pi\)
−0.806371 + 0.591410i \(0.798571\pi\)
\(42\) 0 0
\(43\) 6.88581 1.05008 0.525038 0.851079i \(-0.324051\pi\)
0.525038 + 0.851079i \(0.324051\pi\)
\(44\) 0 0
\(45\) −0.00617996 −0.000921254 0
\(46\) 0 0
\(47\) 1.59656 0.232883 0.116441 0.993198i \(-0.462851\pi\)
0.116441 + 0.993198i \(0.462851\pi\)
\(48\) 0 0
\(49\) −5.43188 −0.775982
\(50\) 0 0
\(51\) 6.24894 0.875026
\(52\) 0 0
\(53\) 1.77393 0.243668 0.121834 0.992550i \(-0.461122\pi\)
0.121834 + 0.992550i \(0.461122\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 12.2368 1.62080
\(58\) 0 0
\(59\) −4.36887 −0.568779 −0.284389 0.958709i \(-0.591791\pi\)
−0.284389 + 0.958709i \(0.591791\pi\)
\(60\) 0 0
\(61\) −8.33324 −1.06696 −0.533481 0.845812i \(-0.679116\pi\)
−0.533481 + 0.845812i \(0.679116\pi\)
\(62\) 0 0
\(63\) −0.00773884 −0.000975002 0
\(64\) 0 0
\(65\) 2.50777 0.311050
\(66\) 0 0
\(67\) 5.43293 0.663738 0.331869 0.943326i \(-0.392321\pi\)
0.331869 + 0.943326i \(0.392321\pi\)
\(68\) 0 0
\(69\) 3.82721 0.460742
\(70\) 0 0
\(71\) −4.63664 −0.550267 −0.275134 0.961406i \(-0.588722\pi\)
−0.275134 + 0.961406i \(0.588722\pi\)
\(72\) 0 0
\(73\) −2.03474 −0.238148 −0.119074 0.992885i \(-0.537993\pi\)
−0.119074 + 0.992885i \(0.537993\pi\)
\(74\) 0 0
\(75\) −1.73383 −0.200206
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 16.7625 1.88593 0.942967 0.332887i \(-0.108023\pi\)
0.942967 + 0.332887i \(0.108023\pi\)
\(80\) 0 0
\(81\) −9.01850 −1.00206
\(82\) 0 0
\(83\) −5.62561 −0.617491 −0.308746 0.951145i \(-0.599909\pi\)
−0.308746 + 0.951145i \(0.599909\pi\)
\(84\) 0 0
\(85\) 3.60412 0.390921
\(86\) 0 0
\(87\) −4.68066 −0.501820
\(88\) 0 0
\(89\) −4.72832 −0.501200 −0.250600 0.968091i \(-0.580628\pi\)
−0.250600 + 0.968091i \(0.580628\pi\)
\(90\) 0 0
\(91\) 3.14034 0.329197
\(92\) 0 0
\(93\) 4.91274 0.509427
\(94\) 0 0
\(95\) 7.05765 0.724100
\(96\) 0 0
\(97\) 19.0943 1.93873 0.969365 0.245625i \(-0.0789933\pi\)
0.969365 + 0.245625i \(0.0789933\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −14.2252 −1.41546 −0.707732 0.706481i \(-0.750281\pi\)
−0.707732 + 0.706481i \(0.750281\pi\)
\(102\) 0 0
\(103\) −15.8815 −1.56485 −0.782426 0.622744i \(-0.786018\pi\)
−0.782426 + 0.622744i \(0.786018\pi\)
\(104\) 0 0
\(105\) −2.17119 −0.211886
\(106\) 0 0
\(107\) −4.04935 −0.391465 −0.195733 0.980657i \(-0.562708\pi\)
−0.195733 + 0.980657i \(0.562708\pi\)
\(108\) 0 0
\(109\) 5.89844 0.564968 0.282484 0.959272i \(-0.408842\pi\)
0.282484 + 0.959272i \(0.408842\pi\)
\(110\) 0 0
\(111\) 13.9971 1.32855
\(112\) 0 0
\(113\) 3.41401 0.321163 0.160582 0.987023i \(-0.448663\pi\)
0.160582 + 0.987023i \(0.448663\pi\)
\(114\) 0 0
\(115\) 2.20737 0.205838
\(116\) 0 0
\(117\) −0.0154979 −0.00143278
\(118\) 0 0
\(119\) 4.51324 0.413728
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) 17.9046 1.61440
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 6.19208 0.549458 0.274729 0.961522i \(-0.411412\pi\)
0.274729 + 0.961522i \(0.411412\pi\)
\(128\) 0 0
\(129\) −11.9389 −1.05116
\(130\) 0 0
\(131\) −7.39924 −0.646475 −0.323237 0.946318i \(-0.604771\pi\)
−0.323237 + 0.946318i \(0.604771\pi\)
\(132\) 0 0
\(133\) 8.83792 0.766345
\(134\) 0 0
\(135\) −5.19079 −0.446752
\(136\) 0 0
\(137\) 18.4830 1.57911 0.789553 0.613683i \(-0.210313\pi\)
0.789553 + 0.613683i \(0.210313\pi\)
\(138\) 0 0
\(139\) −17.8424 −1.51337 −0.756687 0.653778i \(-0.773183\pi\)
−0.756687 + 0.653778i \(0.773183\pi\)
\(140\) 0 0
\(141\) −2.76817 −0.233122
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −2.69960 −0.224190
\(146\) 0 0
\(147\) 9.41797 0.776781
\(148\) 0 0
\(149\) −7.39910 −0.606157 −0.303079 0.952966i \(-0.598015\pi\)
−0.303079 + 0.952966i \(0.598015\pi\)
\(150\) 0 0
\(151\) 4.41515 0.359300 0.179650 0.983731i \(-0.442504\pi\)
0.179650 + 0.983731i \(0.442504\pi\)
\(152\) 0 0
\(153\) −0.0222733 −0.00180069
\(154\) 0 0
\(155\) 2.83345 0.227589
\(156\) 0 0
\(157\) −2.33057 −0.186000 −0.0929998 0.995666i \(-0.529646\pi\)
−0.0929998 + 0.995666i \(0.529646\pi\)
\(158\) 0 0
\(159\) −3.07570 −0.243919
\(160\) 0 0
\(161\) 2.76417 0.217847
\(162\) 0 0
\(163\) 1.50615 0.117971 0.0589855 0.998259i \(-0.481213\pi\)
0.0589855 + 0.998259i \(0.481213\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 9.79732 0.758139 0.379070 0.925368i \(-0.376244\pi\)
0.379070 + 0.925368i \(0.376244\pi\)
\(168\) 0 0
\(169\) −6.71111 −0.516239
\(170\) 0 0
\(171\) −0.0436160 −0.00333540
\(172\) 0 0
\(173\) −20.0936 −1.52769 −0.763845 0.645400i \(-0.776691\pi\)
−0.763845 + 0.645400i \(0.776691\pi\)
\(174\) 0 0
\(175\) −1.25225 −0.0946610
\(176\) 0 0
\(177\) 7.57490 0.569364
\(178\) 0 0
\(179\) 16.2049 1.21121 0.605606 0.795764i \(-0.292931\pi\)
0.605606 + 0.795764i \(0.292931\pi\)
\(180\) 0 0
\(181\) −9.42873 −0.700832 −0.350416 0.936594i \(-0.613960\pi\)
−0.350416 + 0.936594i \(0.613960\pi\)
\(182\) 0 0
\(183\) 14.4485 1.06806
\(184\) 0 0
\(185\) 8.07294 0.593534
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −6.50015 −0.472816
\(190\) 0 0
\(191\) 8.17556 0.591563 0.295781 0.955256i \(-0.404420\pi\)
0.295781 + 0.955256i \(0.404420\pi\)
\(192\) 0 0
\(193\) −4.99518 −0.359561 −0.179781 0.983707i \(-0.557539\pi\)
−0.179781 + 0.983707i \(0.557539\pi\)
\(194\) 0 0
\(195\) −4.34805 −0.311370
\(196\) 0 0
\(197\) −7.00790 −0.499292 −0.249646 0.968337i \(-0.580314\pi\)
−0.249646 + 0.968337i \(0.580314\pi\)
\(198\) 0 0
\(199\) 18.8762 1.33810 0.669050 0.743217i \(-0.266701\pi\)
0.669050 + 0.743217i \(0.266701\pi\)
\(200\) 0 0
\(201\) −9.41979 −0.664421
\(202\) 0 0
\(203\) −3.38057 −0.237269
\(204\) 0 0
\(205\) 10.3266 0.721240
\(206\) 0 0
\(207\) −0.0136414 −0.000948145 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 13.7688 0.947881 0.473940 0.880557i \(-0.342831\pi\)
0.473940 + 0.880557i \(0.342831\pi\)
\(212\) 0 0
\(213\) 8.03916 0.550834
\(214\) 0 0
\(215\) −6.88581 −0.469608
\(216\) 0 0
\(217\) 3.54819 0.240867
\(218\) 0 0
\(219\) 3.52790 0.238394
\(220\) 0 0
\(221\) 9.03828 0.607980
\(222\) 0 0
\(223\) 28.7976 1.92843 0.964214 0.265125i \(-0.0854133\pi\)
0.964214 + 0.265125i \(0.0854133\pi\)
\(224\) 0 0
\(225\) 0.00617996 0.000411997 0
\(226\) 0 0
\(227\) 24.2769 1.61131 0.805656 0.592383i \(-0.201813\pi\)
0.805656 + 0.592383i \(0.201813\pi\)
\(228\) 0 0
\(229\) −16.3933 −1.08330 −0.541650 0.840604i \(-0.682200\pi\)
−0.541650 + 0.840604i \(0.682200\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −2.51844 −0.164988 −0.0824941 0.996592i \(-0.526289\pi\)
−0.0824941 + 0.996592i \(0.526289\pi\)
\(234\) 0 0
\(235\) −1.59656 −0.104148
\(236\) 0 0
\(237\) −29.0635 −1.88787
\(238\) 0 0
\(239\) 6.67824 0.431979 0.215990 0.976396i \(-0.430702\pi\)
0.215990 + 0.976396i \(0.430702\pi\)
\(240\) 0 0
\(241\) −2.14310 −0.138049 −0.0690247 0.997615i \(-0.521989\pi\)
−0.0690247 + 0.997615i \(0.521989\pi\)
\(242\) 0 0
\(243\) 0.0642239 0.00411997
\(244\) 0 0
\(245\) 5.43188 0.347030
\(246\) 0 0
\(247\) 17.6989 1.12616
\(248\) 0 0
\(249\) 9.75388 0.618127
\(250\) 0 0
\(251\) 14.3906 0.908325 0.454162 0.890919i \(-0.349939\pi\)
0.454162 + 0.890919i \(0.349939\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) −6.24894 −0.391324
\(256\) 0 0
\(257\) 24.4034 1.52224 0.761120 0.648612i \(-0.224650\pi\)
0.761120 + 0.648612i \(0.224650\pi\)
\(258\) 0 0
\(259\) 10.1093 0.628162
\(260\) 0 0
\(261\) 0.0166834 0.00103268
\(262\) 0 0
\(263\) 0.992957 0.0612283 0.0306142 0.999531i \(-0.490254\pi\)
0.0306142 + 0.999531i \(0.490254\pi\)
\(264\) 0 0
\(265\) −1.77393 −0.108972
\(266\) 0 0
\(267\) 8.19811 0.501716
\(268\) 0 0
\(269\) 30.4230 1.85493 0.927463 0.373915i \(-0.121985\pi\)
0.927463 + 0.373915i \(0.121985\pi\)
\(270\) 0 0
\(271\) 25.9425 1.57589 0.787946 0.615744i \(-0.211144\pi\)
0.787946 + 0.615744i \(0.211144\pi\)
\(272\) 0 0
\(273\) −5.44483 −0.329536
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −5.03851 −0.302735 −0.151367 0.988478i \(-0.548368\pi\)
−0.151367 + 0.988478i \(0.548368\pi\)
\(278\) 0 0
\(279\) −0.0175106 −0.00104833
\(280\) 0 0
\(281\) 20.7739 1.23927 0.619634 0.784891i \(-0.287281\pi\)
0.619634 + 0.784891i \(0.287281\pi\)
\(282\) 0 0
\(283\) −4.55629 −0.270843 −0.135422 0.990788i \(-0.543239\pi\)
−0.135422 + 0.990788i \(0.543239\pi\)
\(284\) 0 0
\(285\) −12.2368 −0.724845
\(286\) 0 0
\(287\) 12.9314 0.763319
\(288\) 0 0
\(289\) −4.01035 −0.235903
\(290\) 0 0
\(291\) −33.1063 −1.94073
\(292\) 0 0
\(293\) −2.39092 −0.139679 −0.0698394 0.997558i \(-0.522249\pi\)
−0.0698394 + 0.997558i \(0.522249\pi\)
\(294\) 0 0
\(295\) 4.36887 0.254366
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 5.53556 0.320130
\(300\) 0 0
\(301\) −8.62274 −0.497006
\(302\) 0 0
\(303\) 24.6642 1.41692
\(304\) 0 0
\(305\) 8.33324 0.477160
\(306\) 0 0
\(307\) 3.26648 0.186428 0.0932138 0.995646i \(-0.470286\pi\)
0.0932138 + 0.995646i \(0.470286\pi\)
\(308\) 0 0
\(309\) 27.5359 1.56646
\(310\) 0 0
\(311\) 2.73567 0.155125 0.0775627 0.996987i \(-0.475286\pi\)
0.0775627 + 0.996987i \(0.475286\pi\)
\(312\) 0 0
\(313\) −10.3586 −0.585501 −0.292750 0.956189i \(-0.594570\pi\)
−0.292750 + 0.956189i \(0.594570\pi\)
\(314\) 0 0
\(315\) 0.00773884 0.000436034 0
\(316\) 0 0
\(317\) 31.2560 1.75551 0.877755 0.479110i \(-0.159041\pi\)
0.877755 + 0.479110i \(0.159041\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 7.02090 0.391868
\(322\) 0 0
\(323\) 25.4366 1.41533
\(324\) 0 0
\(325\) −2.50777 −0.139106
\(326\) 0 0
\(327\) −10.2269 −0.565550
\(328\) 0 0
\(329\) −1.99929 −0.110224
\(330\) 0 0
\(331\) −29.7878 −1.63729 −0.818644 0.574302i \(-0.805274\pi\)
−0.818644 + 0.574302i \(0.805274\pi\)
\(332\) 0 0
\(333\) −0.0498904 −0.00273398
\(334\) 0 0
\(335\) −5.43293 −0.296832
\(336\) 0 0
\(337\) −29.8351 −1.62522 −0.812610 0.582808i \(-0.801954\pi\)
−0.812610 + 0.582808i \(0.801954\pi\)
\(338\) 0 0
\(339\) −5.91933 −0.321494
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 15.5678 0.840581
\(344\) 0 0
\(345\) −3.82721 −0.206050
\(346\) 0 0
\(347\) −22.2170 −1.19267 −0.596336 0.802735i \(-0.703377\pi\)
−0.596336 + 0.802735i \(0.703377\pi\)
\(348\) 0 0
\(349\) −19.2134 −1.02847 −0.514234 0.857650i \(-0.671924\pi\)
−0.514234 + 0.857650i \(0.671924\pi\)
\(350\) 0 0
\(351\) −13.0173 −0.694811
\(352\) 0 0
\(353\) 31.0691 1.65364 0.826821 0.562466i \(-0.190147\pi\)
0.826821 + 0.562466i \(0.190147\pi\)
\(354\) 0 0
\(355\) 4.63664 0.246087
\(356\) 0 0
\(357\) −7.82521 −0.414154
\(358\) 0 0
\(359\) −27.5365 −1.45332 −0.726660 0.686998i \(-0.758928\pi\)
−0.726660 + 0.686998i \(0.758928\pi\)
\(360\) 0 0
\(361\) 30.8104 1.62160
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 2.03474 0.106503
\(366\) 0 0
\(367\) 23.7891 1.24178 0.620890 0.783898i \(-0.286771\pi\)
0.620890 + 0.783898i \(0.286771\pi\)
\(368\) 0 0
\(369\) −0.0638179 −0.00332223
\(370\) 0 0
\(371\) −2.22140 −0.115329
\(372\) 0 0
\(373\) −15.5085 −0.803000 −0.401500 0.915859i \(-0.631511\pi\)
−0.401500 + 0.915859i \(0.631511\pi\)
\(374\) 0 0
\(375\) 1.73383 0.0895348
\(376\) 0 0
\(377\) −6.76997 −0.348671
\(378\) 0 0
\(379\) −19.0461 −0.978335 −0.489167 0.872190i \(-0.662699\pi\)
−0.489167 + 0.872190i \(0.662699\pi\)
\(380\) 0 0
\(381\) −10.7360 −0.550023
\(382\) 0 0
\(383\) −15.8224 −0.808488 −0.404244 0.914651i \(-0.632465\pi\)
−0.404244 + 0.914651i \(0.632465\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0.0425540 0.00216314
\(388\) 0 0
\(389\) −16.2272 −0.822752 −0.411376 0.911466i \(-0.634952\pi\)
−0.411376 + 0.911466i \(0.634952\pi\)
\(390\) 0 0
\(391\) 7.95560 0.402332
\(392\) 0 0
\(393\) 12.8291 0.647140
\(394\) 0 0
\(395\) −16.7625 −0.843415
\(396\) 0 0
\(397\) 31.3002 1.57091 0.785455 0.618919i \(-0.212429\pi\)
0.785455 + 0.618919i \(0.212429\pi\)
\(398\) 0 0
\(399\) −15.3235 −0.767134
\(400\) 0 0
\(401\) −6.86096 −0.342620 −0.171310 0.985217i \(-0.554800\pi\)
−0.171310 + 0.985217i \(0.554800\pi\)
\(402\) 0 0
\(403\) 7.10564 0.353957
\(404\) 0 0
\(405\) 9.01850 0.448133
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −22.8291 −1.12882 −0.564412 0.825493i \(-0.690897\pi\)
−0.564412 + 0.825493i \(0.690897\pi\)
\(410\) 0 0
\(411\) −32.0464 −1.58073
\(412\) 0 0
\(413\) 5.47091 0.269206
\(414\) 0 0
\(415\) 5.62561 0.276150
\(416\) 0 0
\(417\) 30.9358 1.51493
\(418\) 0 0
\(419\) 5.98106 0.292194 0.146097 0.989270i \(-0.453329\pi\)
0.146097 + 0.989270i \(0.453329\pi\)
\(420\) 0 0
\(421\) 37.8255 1.84350 0.921751 0.387781i \(-0.126758\pi\)
0.921751 + 0.387781i \(0.126758\pi\)
\(422\) 0 0
\(423\) 0.00986669 0.000479735 0
\(424\) 0 0
\(425\) −3.60412 −0.174825
\(426\) 0 0
\(427\) 10.4353 0.504999
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 6.51913 0.314015 0.157008 0.987597i \(-0.449815\pi\)
0.157008 + 0.987597i \(0.449815\pi\)
\(432\) 0 0
\(433\) −29.0198 −1.39460 −0.697301 0.716778i \(-0.745616\pi\)
−0.697301 + 0.716778i \(0.745616\pi\)
\(434\) 0 0
\(435\) 4.68066 0.224421
\(436\) 0 0
\(437\) 15.5788 0.745236
\(438\) 0 0
\(439\) 12.2777 0.585984 0.292992 0.956115i \(-0.405349\pi\)
0.292992 + 0.956115i \(0.405349\pi\)
\(440\) 0 0
\(441\) −0.0335688 −0.00159851
\(442\) 0 0
\(443\) 8.75445 0.415936 0.207968 0.978136i \(-0.433315\pi\)
0.207968 + 0.978136i \(0.433315\pi\)
\(444\) 0 0
\(445\) 4.72832 0.224144
\(446\) 0 0
\(447\) 12.8288 0.606782
\(448\) 0 0
\(449\) −21.0104 −0.991543 −0.495771 0.868453i \(-0.665115\pi\)
−0.495771 + 0.868453i \(0.665115\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −7.65513 −0.359670
\(454\) 0 0
\(455\) −3.14034 −0.147222
\(456\) 0 0
\(457\) −13.8185 −0.646402 −0.323201 0.946330i \(-0.604759\pi\)
−0.323201 + 0.946330i \(0.604759\pi\)
\(458\) 0 0
\(459\) −18.7082 −0.873224
\(460\) 0 0
\(461\) −38.6522 −1.80021 −0.900106 0.435672i \(-0.856511\pi\)
−0.900106 + 0.435672i \(0.856511\pi\)
\(462\) 0 0
\(463\) −15.0206 −0.698066 −0.349033 0.937110i \(-0.613490\pi\)
−0.349033 + 0.937110i \(0.613490\pi\)
\(464\) 0 0
\(465\) −4.91274 −0.227823
\(466\) 0 0
\(467\) 11.7903 0.545591 0.272796 0.962072i \(-0.412052\pi\)
0.272796 + 0.962072i \(0.412052\pi\)
\(468\) 0 0
\(469\) −6.80337 −0.314150
\(470\) 0 0
\(471\) 4.04082 0.186191
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −7.05765 −0.323827
\(476\) 0 0
\(477\) 0.0109628 0.000501953 0
\(478\) 0 0
\(479\) 18.3651 0.839125 0.419562 0.907726i \(-0.362184\pi\)
0.419562 + 0.907726i \(0.362184\pi\)
\(480\) 0 0
\(481\) 20.2450 0.923094
\(482\) 0 0
\(483\) −4.79261 −0.218071
\(484\) 0 0
\(485\) −19.0943 −0.867026
\(486\) 0 0
\(487\) 22.3361 1.01214 0.506072 0.862492i \(-0.331097\pi\)
0.506072 + 0.862492i \(0.331097\pi\)
\(488\) 0 0
\(489\) −2.61142 −0.118092
\(490\) 0 0
\(491\) 8.71330 0.393226 0.196613 0.980481i \(-0.437006\pi\)
0.196613 + 0.980481i \(0.437006\pi\)
\(492\) 0 0
\(493\) −9.72968 −0.438203
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 5.80621 0.260444
\(498\) 0 0
\(499\) 38.9860 1.74525 0.872626 0.488390i \(-0.162415\pi\)
0.872626 + 0.488390i \(0.162415\pi\)
\(500\) 0 0
\(501\) −16.9869 −0.758920
\(502\) 0 0
\(503\) 12.7798 0.569821 0.284910 0.958554i \(-0.408036\pi\)
0.284910 + 0.958554i \(0.408036\pi\)
\(504\) 0 0
\(505\) 14.2252 0.633015
\(506\) 0 0
\(507\) 11.6360 0.516771
\(508\) 0 0
\(509\) 36.9761 1.63894 0.819469 0.573124i \(-0.194269\pi\)
0.819469 + 0.573124i \(0.194269\pi\)
\(510\) 0 0
\(511\) 2.54800 0.112717
\(512\) 0 0
\(513\) −36.6348 −1.61746
\(514\) 0 0
\(515\) 15.8815 0.699823
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 34.8390 1.52926
\(520\) 0 0
\(521\) −8.97119 −0.393035 −0.196518 0.980500i \(-0.562963\pi\)
−0.196518 + 0.980500i \(0.562963\pi\)
\(522\) 0 0
\(523\) 3.09464 0.135319 0.0676596 0.997708i \(-0.478447\pi\)
0.0676596 + 0.997708i \(0.478447\pi\)
\(524\) 0 0
\(525\) 2.17119 0.0947584
\(526\) 0 0
\(527\) 10.2121 0.444846
\(528\) 0 0
\(529\) −18.1275 −0.788154
\(530\) 0 0
\(531\) −0.0269994 −0.00117168
\(532\) 0 0
\(533\) 25.8967 1.12171
\(534\) 0 0
\(535\) 4.04935 0.175069
\(536\) 0 0
\(537\) −28.0966 −1.21246
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −18.8882 −0.812065 −0.406033 0.913859i \(-0.633088\pi\)
−0.406033 + 0.913859i \(0.633088\pi\)
\(542\) 0 0
\(543\) 16.3478 0.701553
\(544\) 0 0
\(545\) −5.89844 −0.252661
\(546\) 0 0
\(547\) −34.0440 −1.45562 −0.727808 0.685781i \(-0.759461\pi\)
−0.727808 + 0.685781i \(0.759461\pi\)
\(548\) 0 0
\(549\) −0.0514991 −0.00219793
\(550\) 0 0
\(551\) −19.0528 −0.811678
\(552\) 0 0
\(553\) −20.9908 −0.892622
\(554\) 0 0
\(555\) −13.9971 −0.594145
\(556\) 0 0
\(557\) 19.8622 0.841589 0.420795 0.907156i \(-0.361751\pi\)
0.420795 + 0.907156i \(0.361751\pi\)
\(558\) 0 0
\(559\) −17.2680 −0.730359
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −23.7516 −1.00101 −0.500505 0.865734i \(-0.666852\pi\)
−0.500505 + 0.865734i \(0.666852\pi\)
\(564\) 0 0
\(565\) −3.41401 −0.143628
\(566\) 0 0
\(567\) 11.2934 0.474278
\(568\) 0 0
\(569\) 36.4083 1.52632 0.763158 0.646211i \(-0.223648\pi\)
0.763158 + 0.646211i \(0.223648\pi\)
\(570\) 0 0
\(571\) −1.49673 −0.0626362 −0.0313181 0.999509i \(-0.509970\pi\)
−0.0313181 + 0.999509i \(0.509970\pi\)
\(572\) 0 0
\(573\) −14.1751 −0.592172
\(574\) 0 0
\(575\) −2.20737 −0.0920536
\(576\) 0 0
\(577\) 41.2565 1.71753 0.858766 0.512368i \(-0.171232\pi\)
0.858766 + 0.512368i \(0.171232\pi\)
\(578\) 0 0
\(579\) 8.66082 0.359931
\(580\) 0 0
\(581\) 7.04466 0.292262
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0.0154979 0.000640759 0
\(586\) 0 0
\(587\) 41.6959 1.72097 0.860486 0.509474i \(-0.170160\pi\)
0.860486 + 0.509474i \(0.170160\pi\)
\(588\) 0 0
\(589\) 19.9975 0.823984
\(590\) 0 0
\(591\) 12.1505 0.499806
\(592\) 0 0
\(593\) 16.0656 0.659736 0.329868 0.944027i \(-0.392996\pi\)
0.329868 + 0.944027i \(0.392996\pi\)
\(594\) 0 0
\(595\) −4.51324 −0.185025
\(596\) 0 0
\(597\) −32.7282 −1.33948
\(598\) 0 0
\(599\) −25.4316 −1.03911 −0.519553 0.854438i \(-0.673902\pi\)
−0.519553 + 0.854438i \(0.673902\pi\)
\(600\) 0 0
\(601\) 13.4615 0.549104 0.274552 0.961572i \(-0.411470\pi\)
0.274552 + 0.961572i \(0.411470\pi\)
\(602\) 0 0
\(603\) 0.0335753 0.00136729
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −2.05778 −0.0835226 −0.0417613 0.999128i \(-0.513297\pi\)
−0.0417613 + 0.999128i \(0.513297\pi\)
\(608\) 0 0
\(609\) 5.86134 0.237514
\(610\) 0 0
\(611\) −4.00380 −0.161977
\(612\) 0 0
\(613\) −38.3437 −1.54869 −0.774344 0.632765i \(-0.781920\pi\)
−0.774344 + 0.632765i \(0.781920\pi\)
\(614\) 0 0
\(615\) −17.9046 −0.721982
\(616\) 0 0
\(617\) 7.25724 0.292166 0.146083 0.989272i \(-0.453333\pi\)
0.146083 + 0.989272i \(0.453333\pi\)
\(618\) 0 0
\(619\) −17.6113 −0.707860 −0.353930 0.935272i \(-0.615155\pi\)
−0.353930 + 0.935272i \(0.615155\pi\)
\(620\) 0 0
\(621\) −11.4580 −0.459793
\(622\) 0 0
\(623\) 5.92102 0.237221
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 29.0958 1.16013
\(630\) 0 0
\(631\) −28.4353 −1.13199 −0.565996 0.824408i \(-0.691508\pi\)
−0.565996 + 0.824408i \(0.691508\pi\)
\(632\) 0 0
\(633\) −23.8727 −0.948857
\(634\) 0 0
\(635\) −6.19208 −0.245725
\(636\) 0 0
\(637\) 13.6219 0.539718
\(638\) 0 0
\(639\) −0.0286542 −0.00113354
\(640\) 0 0
\(641\) −42.5129 −1.67916 −0.839579 0.543237i \(-0.817199\pi\)
−0.839579 + 0.543237i \(0.817199\pi\)
\(642\) 0 0
\(643\) −22.9742 −0.906015 −0.453007 0.891507i \(-0.649649\pi\)
−0.453007 + 0.891507i \(0.649649\pi\)
\(644\) 0 0
\(645\) 11.9389 0.470092
\(646\) 0 0
\(647\) −14.2632 −0.560746 −0.280373 0.959891i \(-0.590458\pi\)
−0.280373 + 0.959891i \(0.590458\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −6.15196 −0.241115
\(652\) 0 0
\(653\) −26.8378 −1.05024 −0.525122 0.851027i \(-0.675980\pi\)
−0.525122 + 0.851027i \(0.675980\pi\)
\(654\) 0 0
\(655\) 7.39924 0.289112
\(656\) 0 0
\(657\) −0.0125746 −0.000490582 0
\(658\) 0 0
\(659\) 28.8747 1.12480 0.562400 0.826865i \(-0.309878\pi\)
0.562400 + 0.826865i \(0.309878\pi\)
\(660\) 0 0
\(661\) −23.6938 −0.921582 −0.460791 0.887509i \(-0.652434\pi\)
−0.460791 + 0.887509i \(0.652434\pi\)
\(662\) 0 0
\(663\) −15.6709 −0.608606
\(664\) 0 0
\(665\) −8.83792 −0.342720
\(666\) 0 0
\(667\) −5.95901 −0.230734
\(668\) 0 0
\(669\) −49.9302 −1.93041
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −5.90541 −0.227637 −0.113818 0.993502i \(-0.536308\pi\)
−0.113818 + 0.993502i \(0.536308\pi\)
\(674\) 0 0
\(675\) 5.19079 0.199793
\(676\) 0 0
\(677\) 30.4595 1.17065 0.585327 0.810797i \(-0.300966\pi\)
0.585327 + 0.810797i \(0.300966\pi\)
\(678\) 0 0
\(679\) −23.9108 −0.917610
\(680\) 0 0
\(681\) −42.0921 −1.61297
\(682\) 0 0
\(683\) −2.95068 −0.112905 −0.0564524 0.998405i \(-0.517979\pi\)
−0.0564524 + 0.998405i \(0.517979\pi\)
\(684\) 0 0
\(685\) −18.4830 −0.706198
\(686\) 0 0
\(687\) 28.4233 1.08442
\(688\) 0 0
\(689\) −4.44860 −0.169478
\(690\) 0 0
\(691\) −38.6147 −1.46897 −0.734487 0.678623i \(-0.762577\pi\)
−0.734487 + 0.678623i \(0.762577\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 17.8424 0.676801
\(696\) 0 0
\(697\) 37.2182 1.40974
\(698\) 0 0
\(699\) 4.36655 0.165158
\(700\) 0 0
\(701\) 33.1129 1.25066 0.625328 0.780362i \(-0.284965\pi\)
0.625328 + 0.780362i \(0.284965\pi\)
\(702\) 0 0
\(703\) 56.9760 2.14889
\(704\) 0 0
\(705\) 2.76817 0.104255
\(706\) 0 0
\(707\) 17.8135 0.669946
\(708\) 0 0
\(709\) 21.9947 0.826030 0.413015 0.910724i \(-0.364476\pi\)
0.413015 + 0.910724i \(0.364476\pi\)
\(710\) 0 0
\(711\) 0.103592 0.00388500
\(712\) 0 0
\(713\) 6.25447 0.234232
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −11.5790 −0.432424
\(718\) 0 0
\(719\) −5.99121 −0.223434 −0.111717 0.993740i \(-0.535635\pi\)
−0.111717 + 0.993740i \(0.535635\pi\)
\(720\) 0 0
\(721\) 19.8876 0.740652
\(722\) 0 0
\(723\) 3.71579 0.138192
\(724\) 0 0
\(725\) 2.69960 0.100261
\(726\) 0 0
\(727\) −4.06512 −0.150767 −0.0753834 0.997155i \(-0.524018\pi\)
−0.0753834 + 0.997155i \(0.524018\pi\)
\(728\) 0 0
\(729\) 26.9442 0.997932
\(730\) 0 0
\(731\) −24.8173 −0.917899
\(732\) 0 0
\(733\) −29.9274 −1.10539 −0.552697 0.833382i \(-0.686401\pi\)
−0.552697 + 0.833382i \(0.686401\pi\)
\(734\) 0 0
\(735\) −9.41797 −0.347387
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −19.6749 −0.723755 −0.361877 0.932226i \(-0.617864\pi\)
−0.361877 + 0.932226i \(0.617864\pi\)
\(740\) 0 0
\(741\) −30.6870 −1.12732
\(742\) 0 0
\(743\) −26.6733 −0.978548 −0.489274 0.872130i \(-0.662738\pi\)
−0.489274 + 0.872130i \(0.662738\pi\)
\(744\) 0 0
\(745\) 7.39910 0.271082
\(746\) 0 0
\(747\) −0.0347660 −0.00127202
\(748\) 0 0
\(749\) 5.07079 0.185282
\(750\) 0 0
\(751\) 7.86537 0.287011 0.143506 0.989649i \(-0.454162\pi\)
0.143506 + 0.989649i \(0.454162\pi\)
\(752\) 0 0
\(753\) −24.9509 −0.909260
\(754\) 0 0
\(755\) −4.41515 −0.160684
\(756\) 0 0
\(757\) −24.4559 −0.888866 −0.444433 0.895812i \(-0.646595\pi\)
−0.444433 + 0.895812i \(0.646595\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −52.2028 −1.89235 −0.946174 0.323658i \(-0.895087\pi\)
−0.946174 + 0.323658i \(0.895087\pi\)
\(762\) 0 0
\(763\) −7.38631 −0.267402
\(764\) 0 0
\(765\) 0.0222733 0.000805292 0
\(766\) 0 0
\(767\) 10.9561 0.395602
\(768\) 0 0
\(769\) 26.7480 0.964558 0.482279 0.876018i \(-0.339809\pi\)
0.482279 + 0.876018i \(0.339809\pi\)
\(770\) 0 0
\(771\) −42.3114 −1.52381
\(772\) 0 0
\(773\) −50.2424 −1.80709 −0.903546 0.428490i \(-0.859046\pi\)
−0.903546 + 0.428490i \(0.859046\pi\)
\(774\) 0 0
\(775\) −2.83345 −0.101781
\(776\) 0 0
\(777\) −17.5279 −0.628809
\(778\) 0 0
\(779\) 72.8814 2.61125
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 14.0131 0.500786
\(784\) 0 0
\(785\) 2.33057 0.0831816
\(786\) 0 0
\(787\) −32.3414 −1.15285 −0.576423 0.817152i \(-0.695552\pi\)
−0.576423 + 0.817152i \(0.695552\pi\)
\(788\) 0 0
\(789\) −1.72162 −0.0612914
\(790\) 0 0
\(791\) −4.27518 −0.152008
\(792\) 0 0
\(793\) 20.8978 0.742103
\(794\) 0 0
\(795\) 3.07570 0.109084
\(796\) 0 0
\(797\) 38.6252 1.36817 0.684087 0.729400i \(-0.260201\pi\)
0.684087 + 0.729400i \(0.260201\pi\)
\(798\) 0 0
\(799\) −5.75420 −0.203569
\(800\) 0 0
\(801\) −0.0292208 −0.00103247
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) −2.76417 −0.0974242
\(806\) 0 0
\(807\) −52.7485 −1.85684
\(808\) 0 0
\(809\) −29.6195 −1.04137 −0.520684 0.853750i \(-0.674323\pi\)
−0.520684 + 0.853750i \(0.674323\pi\)
\(810\) 0 0
\(811\) −14.6182 −0.513315 −0.256657 0.966502i \(-0.582621\pi\)
−0.256657 + 0.966502i \(0.582621\pi\)
\(812\) 0 0
\(813\) −44.9799 −1.57751
\(814\) 0 0
\(815\) −1.50615 −0.0527583
\(816\) 0 0
\(817\) −48.5976 −1.70022
\(818\) 0 0
\(819\) 0.0194072 0.000678142 0
\(820\) 0 0
\(821\) 27.9890 0.976822 0.488411 0.872614i \(-0.337577\pi\)
0.488411 + 0.872614i \(0.337577\pi\)
\(822\) 0 0
\(823\) 17.6590 0.615556 0.307778 0.951458i \(-0.400415\pi\)
0.307778 + 0.951458i \(0.400415\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −3.87712 −0.134821 −0.0674103 0.997725i \(-0.521474\pi\)
−0.0674103 + 0.997725i \(0.521474\pi\)
\(828\) 0 0
\(829\) −20.8274 −0.723366 −0.361683 0.932301i \(-0.617798\pi\)
−0.361683 + 0.932301i \(0.617798\pi\)
\(830\) 0 0
\(831\) 8.73593 0.303046
\(832\) 0 0
\(833\) 19.5771 0.678307
\(834\) 0 0
\(835\) −9.79732 −0.339050
\(836\) 0 0
\(837\) −14.7079 −0.508378
\(838\) 0 0
\(839\) −29.2063 −1.00831 −0.504157 0.863612i \(-0.668197\pi\)
−0.504157 + 0.863612i \(0.668197\pi\)
\(840\) 0 0
\(841\) −21.7122 −0.748695
\(842\) 0 0
\(843\) −36.0185 −1.24054
\(844\) 0 0
\(845\) 6.71111 0.230869
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 7.89985 0.271122
\(850\) 0 0
\(851\) 17.8199 0.610860
\(852\) 0 0
\(853\) 57.3270 1.96284 0.981420 0.191870i \(-0.0614553\pi\)
0.981420 + 0.191870i \(0.0614553\pi\)
\(854\) 0 0
\(855\) 0.0436160 0.00149163
\(856\) 0 0
\(857\) 43.7938 1.49597 0.747984 0.663717i \(-0.231022\pi\)
0.747984 + 0.663717i \(0.231022\pi\)
\(858\) 0 0
\(859\) −32.0382 −1.09313 −0.546564 0.837417i \(-0.684065\pi\)
−0.546564 + 0.837417i \(0.684065\pi\)
\(860\) 0 0
\(861\) −22.4210 −0.764105
\(862\) 0 0
\(863\) 50.5619 1.72115 0.860574 0.509325i \(-0.170105\pi\)
0.860574 + 0.509325i \(0.170105\pi\)
\(864\) 0 0
\(865\) 20.0936 0.683204
\(866\) 0 0
\(867\) 6.95329 0.236146
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −13.6245 −0.461649
\(872\) 0 0
\(873\) 0.118002 0.00399376
\(874\) 0 0
\(875\) 1.25225 0.0423337
\(876\) 0 0
\(877\) 19.9976 0.675270 0.337635 0.941277i \(-0.390373\pi\)
0.337635 + 0.941277i \(0.390373\pi\)
\(878\) 0 0
\(879\) 4.14545 0.139823
\(880\) 0 0
\(881\) −19.7712 −0.666108 −0.333054 0.942908i \(-0.608079\pi\)
−0.333054 + 0.942908i \(0.608079\pi\)
\(882\) 0 0
\(883\) −8.72958 −0.293774 −0.146887 0.989153i \(-0.546925\pi\)
−0.146887 + 0.989153i \(0.546925\pi\)
\(884\) 0 0
\(885\) −7.57490 −0.254627
\(886\) 0 0
\(887\) −50.1444 −1.68368 −0.841842 0.539724i \(-0.818529\pi\)
−0.841842 + 0.539724i \(0.818529\pi\)
\(888\) 0 0
\(889\) −7.75401 −0.260061
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −11.2680 −0.377069
\(894\) 0 0
\(895\) −16.2049 −0.541671
\(896\) 0 0
\(897\) −9.59774 −0.320459
\(898\) 0 0
\(899\) −7.64920 −0.255115
\(900\) 0 0
\(901\) −6.39345 −0.212997
\(902\) 0 0
\(903\) 14.9504 0.497518
\(904\) 0 0
\(905\) 9.42873 0.313422
\(906\) 0 0
\(907\) 21.5643 0.716031 0.358015 0.933716i \(-0.383453\pi\)
0.358015 + 0.933716i \(0.383453\pi\)
\(908\) 0 0
\(909\) −0.0879113 −0.00291584
\(910\) 0 0
\(911\) 25.9189 0.858731 0.429365 0.903131i \(-0.358737\pi\)
0.429365 + 0.903131i \(0.358737\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) −14.4485 −0.477651
\(916\) 0 0
\(917\) 9.26568 0.305980
\(918\) 0 0
\(919\) 13.1654 0.434286 0.217143 0.976140i \(-0.430326\pi\)
0.217143 + 0.976140i \(0.430326\pi\)
\(920\) 0 0
\(921\) −5.66353 −0.186620
\(922\) 0 0
\(923\) 11.6276 0.382727
\(924\) 0 0
\(925\) −8.07294 −0.265437
\(926\) 0 0
\(927\) −0.0981471 −0.00322357
\(928\) 0 0
\(929\) 40.2771 1.32145 0.660725 0.750628i \(-0.270249\pi\)
0.660725 + 0.750628i \(0.270249\pi\)
\(930\) 0 0
\(931\) 38.3363 1.25642
\(932\) 0 0
\(933\) −4.74319 −0.155285
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 37.3771 1.22106 0.610528 0.791995i \(-0.290957\pi\)
0.610528 + 0.791995i \(0.290957\pi\)
\(938\) 0 0
\(939\) 17.9600 0.586103
\(940\) 0 0
\(941\) −38.2224 −1.24601 −0.623007 0.782216i \(-0.714089\pi\)
−0.623007 + 0.782216i \(0.714089\pi\)
\(942\) 0 0
\(943\) 22.7946 0.742293
\(944\) 0 0
\(945\) 6.50015 0.211450
\(946\) 0 0
\(947\) −58.9505 −1.91564 −0.957818 0.287377i \(-0.907217\pi\)
−0.957818 + 0.287377i \(0.907217\pi\)
\(948\) 0 0
\(949\) 5.10265 0.165639
\(950\) 0 0
\(951\) −54.1926 −1.75732
\(952\) 0 0
\(953\) 20.6255 0.668126 0.334063 0.942551i \(-0.391580\pi\)
0.334063 + 0.942551i \(0.391580\pi\)
\(954\) 0 0
\(955\) −8.17556 −0.264555
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −23.1452 −0.747399
\(960\) 0 0
\(961\) −22.9715 −0.741017
\(962\) 0 0
\(963\) −0.0250248 −0.000806413 0
\(964\) 0 0
\(965\) 4.99518 0.160801
\(966\) 0 0
\(967\) −35.8174 −1.15181 −0.575904 0.817517i \(-0.695350\pi\)
−0.575904 + 0.817517i \(0.695350\pi\)
\(968\) 0 0
\(969\) −44.1028 −1.41679
\(970\) 0 0
\(971\) 11.3094 0.362937 0.181469 0.983397i \(-0.441915\pi\)
0.181469 + 0.983397i \(0.441915\pi\)
\(972\) 0 0
\(973\) 22.3431 0.716287
\(974\) 0 0
\(975\) 4.34805 0.139249
\(976\) 0 0
\(977\) 23.2137 0.742671 0.371336 0.928499i \(-0.378900\pi\)
0.371336 + 0.928499i \(0.378900\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0.0364521 0.00116383
\(982\) 0 0
\(983\) 34.1621 1.08960 0.544801 0.838565i \(-0.316605\pi\)
0.544801 + 0.838565i \(0.316605\pi\)
\(984\) 0 0
\(985\) 7.00790 0.223290
\(986\) 0 0
\(987\) 3.46644 0.110338
\(988\) 0 0
\(989\) −15.1995 −0.483316
\(990\) 0 0
\(991\) −27.7516 −0.881560 −0.440780 0.897615i \(-0.645298\pi\)
−0.440780 + 0.897615i \(0.645298\pi\)
\(992\) 0 0
\(993\) 51.6472 1.63897
\(994\) 0 0
\(995\) −18.8762 −0.598416
\(996\) 0 0
\(997\) −43.1978 −1.36809 −0.684044 0.729440i \(-0.739781\pi\)
−0.684044 + 0.729440i \(0.739781\pi\)
\(998\) 0 0
\(999\) −41.9049 −1.32581
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 4840.2.a.bf.1.2 6
4.3 odd 2 9680.2.a.cx.1.5 6
11.5 even 5 440.2.y.b.201.1 yes 12
11.9 even 5 440.2.y.b.81.1 12
11.10 odd 2 4840.2.a.be.1.2 6
44.27 odd 10 880.2.bo.j.641.3 12
44.31 odd 10 880.2.bo.j.81.3 12
44.43 even 2 9680.2.a.cy.1.5 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.y.b.81.1 12 11.9 even 5
440.2.y.b.201.1 yes 12 11.5 even 5
880.2.bo.j.81.3 12 44.31 odd 10
880.2.bo.j.641.3 12 44.27 odd 10
4840.2.a.be.1.2 6 11.10 odd 2
4840.2.a.bf.1.2 6 1.1 even 1 trivial
9680.2.a.cx.1.5 6 4.3 odd 2
9680.2.a.cy.1.5 6 44.43 even 2