Properties

Label 4560.2.a.bj.1.2
Level $4560$
Weight $2$
Character 4560.1
Self dual yes
Analytic conductor $36.412$
Analytic rank $1$
Dimension $2$
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] = [4560,2,Mod(1,4560)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4560, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4560.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4560.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: \(36.4117833217\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 285)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 4560.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -1.00000 q^{5} -0.585786 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -1.00000 q^{5} -0.585786 q^{7} +1.00000 q^{9} -1.41421 q^{11} +5.41421 q^{13} -1.00000 q^{15} -1.17157 q^{17} -1.00000 q^{19} -0.585786 q^{21} -7.65685 q^{23} +1.00000 q^{25} +1.00000 q^{27} -9.07107 q^{29} -6.48528 q^{31} -1.41421 q^{33} +0.585786 q^{35} +11.0711 q^{37} +5.41421 q^{39} -7.41421 q^{41} -0.585786 q^{43} -1.00000 q^{45} -0.343146 q^{47} -6.65685 q^{49} -1.17157 q^{51} +4.00000 q^{53} +1.41421 q^{55} -1.00000 q^{57} -8.48528 q^{59} +5.65685 q^{61} -0.585786 q^{63} -5.41421 q^{65} -12.0000 q^{67} -7.65685 q^{69} +4.48528 q^{71} -2.00000 q^{73} +1.00000 q^{75} +0.828427 q^{77} +11.3137 q^{79} +1.00000 q^{81} +10.4853 q^{83} +1.17157 q^{85} -9.07107 q^{87} +10.7279 q^{89} -3.17157 q^{91} -6.48528 q^{93} +1.00000 q^{95} -4.24264 q^{97} -1.41421 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 2 q^{5} - 4 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} - 2 q^{5} - 4 q^{7} + 2 q^{9} + 8 q^{13} - 2 q^{15} - 8 q^{17} - 2 q^{19} - 4 q^{21} - 4 q^{23} + 2 q^{25} + 2 q^{27} - 4 q^{29} + 4 q^{31} + 4 q^{35} + 8 q^{37} + 8 q^{39} - 12 q^{41} - 4 q^{43} - 2 q^{45} - 12 q^{47} - 2 q^{49} - 8 q^{51} + 8 q^{53} - 2 q^{57} - 4 q^{63} - 8 q^{65} - 24 q^{67} - 4 q^{69} - 8 q^{71} - 4 q^{73} + 2 q^{75} - 4 q^{77} + 2 q^{81} + 4 q^{83} + 8 q^{85} - 4 q^{87} - 4 q^{89} - 12 q^{91} + 4 q^{93} + 2 q^{95}+O(q^{100}) \) 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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −0.585786 −0.221406 −0.110703 0.993854i \(-0.535310\pi\)
−0.110703 + 0.993854i \(0.535310\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.41421 −0.426401 −0.213201 0.977008i \(-0.568389\pi\)
−0.213201 + 0.977008i \(0.568389\pi\)
\(12\) 0 0
\(13\) 5.41421 1.50163 0.750816 0.660511i \(-0.229660\pi\)
0.750816 + 0.660511i \(0.229660\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) −1.17157 −0.284148 −0.142074 0.989856i \(-0.545377\pi\)
−0.142074 + 0.989856i \(0.545377\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −0.585786 −0.127829
\(22\) 0 0
\(23\) −7.65685 −1.59656 −0.798282 0.602284i \(-0.794258\pi\)
−0.798282 + 0.602284i \(0.794258\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −9.07107 −1.68446 −0.842228 0.539122i \(-0.818756\pi\)
−0.842228 + 0.539122i \(0.818756\pi\)
\(30\) 0 0
\(31\) −6.48528 −1.16479 −0.582395 0.812906i \(-0.697884\pi\)
−0.582395 + 0.812906i \(0.697884\pi\)
\(32\) 0 0
\(33\) −1.41421 −0.246183
\(34\) 0 0
\(35\) 0.585786 0.0990160
\(36\) 0 0
\(37\) 11.0711 1.82007 0.910036 0.414529i \(-0.136054\pi\)
0.910036 + 0.414529i \(0.136054\pi\)
\(38\) 0 0
\(39\) 5.41421 0.866968
\(40\) 0 0
\(41\) −7.41421 −1.15791 −0.578953 0.815361i \(-0.696538\pi\)
−0.578953 + 0.815361i \(0.696538\pi\)
\(42\) 0 0
\(43\) −0.585786 −0.0893316 −0.0446658 0.999002i \(-0.514222\pi\)
−0.0446658 + 0.999002i \(0.514222\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) −0.343146 −0.0500530 −0.0250265 0.999687i \(-0.507967\pi\)
−0.0250265 + 0.999687i \(0.507967\pi\)
\(48\) 0 0
\(49\) −6.65685 −0.950979
\(50\) 0 0
\(51\) −1.17157 −0.164053
\(52\) 0 0
\(53\) 4.00000 0.549442 0.274721 0.961524i \(-0.411414\pi\)
0.274721 + 0.961524i \(0.411414\pi\)
\(54\) 0 0
\(55\) 1.41421 0.190693
\(56\) 0 0
\(57\) −1.00000 −0.132453
\(58\) 0 0
\(59\) −8.48528 −1.10469 −0.552345 0.833616i \(-0.686267\pi\)
−0.552345 + 0.833616i \(0.686267\pi\)
\(60\) 0 0
\(61\) 5.65685 0.724286 0.362143 0.932123i \(-0.382045\pi\)
0.362143 + 0.932123i \(0.382045\pi\)
\(62\) 0 0
\(63\) −0.585786 −0.0738022
\(64\) 0 0
\(65\) −5.41421 −0.671551
\(66\) 0 0
\(67\) −12.0000 −1.46603 −0.733017 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) 0 0
\(69\) −7.65685 −0.921777
\(70\) 0 0
\(71\) 4.48528 0.532305 0.266152 0.963931i \(-0.414248\pi\)
0.266152 + 0.963931i \(0.414248\pi\)
\(72\) 0 0
\(73\) −2.00000 −0.234082 −0.117041 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) 0.828427 0.0944080
\(78\) 0 0
\(79\) 11.3137 1.27289 0.636446 0.771321i \(-0.280404\pi\)
0.636446 + 0.771321i \(0.280404\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 10.4853 1.15091 0.575455 0.817834i \(-0.304825\pi\)
0.575455 + 0.817834i \(0.304825\pi\)
\(84\) 0 0
\(85\) 1.17157 0.127075
\(86\) 0 0
\(87\) −9.07107 −0.972521
\(88\) 0 0
\(89\) 10.7279 1.13716 0.568579 0.822629i \(-0.307493\pi\)
0.568579 + 0.822629i \(0.307493\pi\)
\(90\) 0 0
\(91\) −3.17157 −0.332471
\(92\) 0 0
\(93\) −6.48528 −0.672492
\(94\) 0 0
\(95\) 1.00000 0.102598
\(96\) 0 0
\(97\) −4.24264 −0.430775 −0.215387 0.976529i \(-0.569101\pi\)
−0.215387 + 0.976529i \(0.569101\pi\)
\(98\) 0 0
\(99\) −1.41421 −0.142134
\(100\) 0 0
\(101\) −4.82843 −0.480446 −0.240223 0.970718i \(-0.577221\pi\)
−0.240223 + 0.970718i \(0.577221\pi\)
\(102\) 0 0
\(103\) 1.65685 0.163255 0.0816274 0.996663i \(-0.473988\pi\)
0.0816274 + 0.996663i \(0.473988\pi\)
\(104\) 0 0
\(105\) 0.585786 0.0571669
\(106\) 0 0
\(107\) −8.00000 −0.773389 −0.386695 0.922208i \(-0.626383\pi\)
−0.386695 + 0.922208i \(0.626383\pi\)
\(108\) 0 0
\(109\) 3.17157 0.303782 0.151891 0.988397i \(-0.451464\pi\)
0.151891 + 0.988397i \(0.451464\pi\)
\(110\) 0 0
\(111\) 11.0711 1.05082
\(112\) 0 0
\(113\) 12.4853 1.17452 0.587258 0.809400i \(-0.300207\pi\)
0.587258 + 0.809400i \(0.300207\pi\)
\(114\) 0 0
\(115\) 7.65685 0.714005
\(116\) 0 0
\(117\) 5.41421 0.500544
\(118\) 0 0
\(119\) 0.686292 0.0629122
\(120\) 0 0
\(121\) −9.00000 −0.818182
\(122\) 0 0
\(123\) −7.41421 −0.668517
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 0 0
\(129\) −0.585786 −0.0515756
\(130\) 0 0
\(131\) −16.7279 −1.46153 −0.730763 0.682632i \(-0.760835\pi\)
−0.730763 + 0.682632i \(0.760835\pi\)
\(132\) 0 0
\(133\) 0.585786 0.0507941
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) −14.0000 −1.19610 −0.598050 0.801459i \(-0.704058\pi\)
−0.598050 + 0.801459i \(0.704058\pi\)
\(138\) 0 0
\(139\) −1.17157 −0.0993715 −0.0496858 0.998765i \(-0.515822\pi\)
−0.0496858 + 0.998765i \(0.515822\pi\)
\(140\) 0 0
\(141\) −0.343146 −0.0288981
\(142\) 0 0
\(143\) −7.65685 −0.640298
\(144\) 0 0
\(145\) 9.07107 0.753311
\(146\) 0 0
\(147\) −6.65685 −0.549048
\(148\) 0 0
\(149\) −3.65685 −0.299581 −0.149791 0.988718i \(-0.547860\pi\)
−0.149791 + 0.988718i \(0.547860\pi\)
\(150\) 0 0
\(151\) −17.7990 −1.44846 −0.724231 0.689558i \(-0.757805\pi\)
−0.724231 + 0.689558i \(0.757805\pi\)
\(152\) 0 0
\(153\) −1.17157 −0.0947161
\(154\) 0 0
\(155\) 6.48528 0.520910
\(156\) 0 0
\(157\) −21.7990 −1.73975 −0.869874 0.493273i \(-0.835800\pi\)
−0.869874 + 0.493273i \(0.835800\pi\)
\(158\) 0 0
\(159\) 4.00000 0.317221
\(160\) 0 0
\(161\) 4.48528 0.353490
\(162\) 0 0
\(163\) −7.89949 −0.618736 −0.309368 0.950942i \(-0.600118\pi\)
−0.309368 + 0.950942i \(0.600118\pi\)
\(164\) 0 0
\(165\) 1.41421 0.110096
\(166\) 0 0
\(167\) −10.0000 −0.773823 −0.386912 0.922117i \(-0.626458\pi\)
−0.386912 + 0.922117i \(0.626458\pi\)
\(168\) 0 0
\(169\) 16.3137 1.25490
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) 0 0
\(173\) −6.14214 −0.466978 −0.233489 0.972359i \(-0.575014\pi\)
−0.233489 + 0.972359i \(0.575014\pi\)
\(174\) 0 0
\(175\) −0.585786 −0.0442813
\(176\) 0 0
\(177\) −8.48528 −0.637793
\(178\) 0 0
\(179\) −17.1716 −1.28346 −0.641732 0.766929i \(-0.721784\pi\)
−0.641732 + 0.766929i \(0.721784\pi\)
\(180\) 0 0
\(181\) −19.1716 −1.42501 −0.712506 0.701666i \(-0.752440\pi\)
−0.712506 + 0.701666i \(0.752440\pi\)
\(182\) 0 0
\(183\) 5.65685 0.418167
\(184\) 0 0
\(185\) −11.0711 −0.813961
\(186\) 0 0
\(187\) 1.65685 0.121161
\(188\) 0 0
\(189\) −0.585786 −0.0426097
\(190\) 0 0
\(191\) −1.89949 −0.137443 −0.0687213 0.997636i \(-0.521892\pi\)
−0.0687213 + 0.997636i \(0.521892\pi\)
\(192\) 0 0
\(193\) −15.0711 −1.08484 −0.542420 0.840108i \(-0.682492\pi\)
−0.542420 + 0.840108i \(0.682492\pi\)
\(194\) 0 0
\(195\) −5.41421 −0.387720
\(196\) 0 0
\(197\) −14.8284 −1.05648 −0.528241 0.849095i \(-0.677148\pi\)
−0.528241 + 0.849095i \(0.677148\pi\)
\(198\) 0 0
\(199\) 16.4853 1.16861 0.584305 0.811534i \(-0.301367\pi\)
0.584305 + 0.811534i \(0.301367\pi\)
\(200\) 0 0
\(201\) −12.0000 −0.846415
\(202\) 0 0
\(203\) 5.31371 0.372949
\(204\) 0 0
\(205\) 7.41421 0.517831
\(206\) 0 0
\(207\) −7.65685 −0.532188
\(208\) 0 0
\(209\) 1.41421 0.0978232
\(210\) 0 0
\(211\) 15.3137 1.05424 0.527120 0.849791i \(-0.323272\pi\)
0.527120 + 0.849791i \(0.323272\pi\)
\(212\) 0 0
\(213\) 4.48528 0.307326
\(214\) 0 0
\(215\) 0.585786 0.0399503
\(216\) 0 0
\(217\) 3.79899 0.257892
\(218\) 0 0
\(219\) −2.00000 −0.135147
\(220\) 0 0
\(221\) −6.34315 −0.426686
\(222\) 0 0
\(223\) −6.34315 −0.424768 −0.212384 0.977186i \(-0.568123\pi\)
−0.212384 + 0.977186i \(0.568123\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 18.9706 1.25912 0.629560 0.776952i \(-0.283235\pi\)
0.629560 + 0.776952i \(0.283235\pi\)
\(228\) 0 0
\(229\) −1.65685 −0.109488 −0.0547440 0.998500i \(-0.517434\pi\)
−0.0547440 + 0.998500i \(0.517434\pi\)
\(230\) 0 0
\(231\) 0.828427 0.0545065
\(232\) 0 0
\(233\) −8.34315 −0.546578 −0.273289 0.961932i \(-0.588111\pi\)
−0.273289 + 0.961932i \(0.588111\pi\)
\(234\) 0 0
\(235\) 0.343146 0.0223844
\(236\) 0 0
\(237\) 11.3137 0.734904
\(238\) 0 0
\(239\) −2.58579 −0.167261 −0.0836303 0.996497i \(-0.526651\pi\)
−0.0836303 + 0.996497i \(0.526651\pi\)
\(240\) 0 0
\(241\) −14.9706 −0.964339 −0.482169 0.876078i \(-0.660151\pi\)
−0.482169 + 0.876078i \(0.660151\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 6.65685 0.425291
\(246\) 0 0
\(247\) −5.41421 −0.344498
\(248\) 0 0
\(249\) 10.4853 0.664478
\(250\) 0 0
\(251\) −12.9289 −0.816067 −0.408033 0.912967i \(-0.633785\pi\)
−0.408033 + 0.912967i \(0.633785\pi\)
\(252\) 0 0
\(253\) 10.8284 0.680777
\(254\) 0 0
\(255\) 1.17157 0.0733667
\(256\) 0 0
\(257\) 1.17157 0.0730807 0.0365404 0.999332i \(-0.488366\pi\)
0.0365404 + 0.999332i \(0.488366\pi\)
\(258\) 0 0
\(259\) −6.48528 −0.402976
\(260\) 0 0
\(261\) −9.07107 −0.561485
\(262\) 0 0
\(263\) 32.1421 1.98197 0.990984 0.133977i \(-0.0427747\pi\)
0.990984 + 0.133977i \(0.0427747\pi\)
\(264\) 0 0
\(265\) −4.00000 −0.245718
\(266\) 0 0
\(267\) 10.7279 0.656538
\(268\) 0 0
\(269\) 8.38478 0.511229 0.255614 0.966779i \(-0.417722\pi\)
0.255614 + 0.966779i \(0.417722\pi\)
\(270\) 0 0
\(271\) 30.8284 1.87269 0.936347 0.351076i \(-0.114184\pi\)
0.936347 + 0.351076i \(0.114184\pi\)
\(272\) 0 0
\(273\) −3.17157 −0.191952
\(274\) 0 0
\(275\) −1.41421 −0.0852803
\(276\) 0 0
\(277\) 10.9706 0.659157 0.329579 0.944128i \(-0.393093\pi\)
0.329579 + 0.944128i \(0.393093\pi\)
\(278\) 0 0
\(279\) −6.48528 −0.388264
\(280\) 0 0
\(281\) −14.7279 −0.878594 −0.439297 0.898342i \(-0.644772\pi\)
−0.439297 + 0.898342i \(0.644772\pi\)
\(282\) 0 0
\(283\) −6.24264 −0.371086 −0.185543 0.982636i \(-0.559404\pi\)
−0.185543 + 0.982636i \(0.559404\pi\)
\(284\) 0 0
\(285\) 1.00000 0.0592349
\(286\) 0 0
\(287\) 4.34315 0.256368
\(288\) 0 0
\(289\) −15.6274 −0.919260
\(290\) 0 0
\(291\) −4.24264 −0.248708
\(292\) 0 0
\(293\) −31.7990 −1.85772 −0.928858 0.370435i \(-0.879209\pi\)
−0.928858 + 0.370435i \(0.879209\pi\)
\(294\) 0 0
\(295\) 8.48528 0.494032
\(296\) 0 0
\(297\) −1.41421 −0.0820610
\(298\) 0 0
\(299\) −41.4558 −2.39745
\(300\) 0 0
\(301\) 0.343146 0.0197786
\(302\) 0 0
\(303\) −4.82843 −0.277386
\(304\) 0 0
\(305\) −5.65685 −0.323911
\(306\) 0 0
\(307\) −7.79899 −0.445112 −0.222556 0.974920i \(-0.571440\pi\)
−0.222556 + 0.974920i \(0.571440\pi\)
\(308\) 0 0
\(309\) 1.65685 0.0942551
\(310\) 0 0
\(311\) 32.2426 1.82831 0.914156 0.405362i \(-0.132855\pi\)
0.914156 + 0.405362i \(0.132855\pi\)
\(312\) 0 0
\(313\) −9.51472 −0.537804 −0.268902 0.963168i \(-0.586661\pi\)
−0.268902 + 0.963168i \(0.586661\pi\)
\(314\) 0 0
\(315\) 0.585786 0.0330053
\(316\) 0 0
\(317\) 11.3137 0.635441 0.317721 0.948184i \(-0.397083\pi\)
0.317721 + 0.948184i \(0.397083\pi\)
\(318\) 0 0
\(319\) 12.8284 0.718254
\(320\) 0 0
\(321\) −8.00000 −0.446516
\(322\) 0 0
\(323\) 1.17157 0.0651881
\(324\) 0 0
\(325\) 5.41421 0.300327
\(326\) 0 0
\(327\) 3.17157 0.175388
\(328\) 0 0
\(329\) 0.201010 0.0110820
\(330\) 0 0
\(331\) −7.17157 −0.394185 −0.197093 0.980385i \(-0.563150\pi\)
−0.197093 + 0.980385i \(0.563150\pi\)
\(332\) 0 0
\(333\) 11.0711 0.606691
\(334\) 0 0
\(335\) 12.0000 0.655630
\(336\) 0 0
\(337\) 28.2426 1.53847 0.769237 0.638963i \(-0.220636\pi\)
0.769237 + 0.638963i \(0.220636\pi\)
\(338\) 0 0
\(339\) 12.4853 0.678107
\(340\) 0 0
\(341\) 9.17157 0.496669
\(342\) 0 0
\(343\) 8.00000 0.431959
\(344\) 0 0
\(345\) 7.65685 0.412231
\(346\) 0 0
\(347\) −10.4853 −0.562879 −0.281440 0.959579i \(-0.590812\pi\)
−0.281440 + 0.959579i \(0.590812\pi\)
\(348\) 0 0
\(349\) 29.3137 1.56913 0.784563 0.620049i \(-0.212887\pi\)
0.784563 + 0.620049i \(0.212887\pi\)
\(350\) 0 0
\(351\) 5.41421 0.288989
\(352\) 0 0
\(353\) 3.65685 0.194635 0.0973174 0.995253i \(-0.468974\pi\)
0.0973174 + 0.995253i \(0.468974\pi\)
\(354\) 0 0
\(355\) −4.48528 −0.238054
\(356\) 0 0
\(357\) 0.686292 0.0363224
\(358\) 0 0
\(359\) −9.89949 −0.522475 −0.261238 0.965275i \(-0.584131\pi\)
−0.261238 + 0.965275i \(0.584131\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −9.00000 −0.472377
\(364\) 0 0
\(365\) 2.00000 0.104685
\(366\) 0 0
\(367\) 19.4142 1.01341 0.506707 0.862118i \(-0.330863\pi\)
0.506707 + 0.862118i \(0.330863\pi\)
\(368\) 0 0
\(369\) −7.41421 −0.385969
\(370\) 0 0
\(371\) −2.34315 −0.121650
\(372\) 0 0
\(373\) −9.89949 −0.512576 −0.256288 0.966600i \(-0.582500\pi\)
−0.256288 + 0.966600i \(0.582500\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) −49.1127 −2.52943
\(378\) 0 0
\(379\) −24.1421 −1.24010 −0.620049 0.784563i \(-0.712887\pi\)
−0.620049 + 0.784563i \(0.712887\pi\)
\(380\) 0 0
\(381\) −8.00000 −0.409852
\(382\) 0 0
\(383\) −12.0000 −0.613171 −0.306586 0.951843i \(-0.599187\pi\)
−0.306586 + 0.951843i \(0.599187\pi\)
\(384\) 0 0
\(385\) −0.828427 −0.0422206
\(386\) 0 0
\(387\) −0.585786 −0.0297772
\(388\) 0 0
\(389\) 14.9706 0.759038 0.379519 0.925184i \(-0.376090\pi\)
0.379519 + 0.925184i \(0.376090\pi\)
\(390\) 0 0
\(391\) 8.97056 0.453661
\(392\) 0 0
\(393\) −16.7279 −0.843812
\(394\) 0 0
\(395\) −11.3137 −0.569254
\(396\) 0 0
\(397\) 35.6569 1.78957 0.894783 0.446501i \(-0.147330\pi\)
0.894783 + 0.446501i \(0.147330\pi\)
\(398\) 0 0
\(399\) 0.585786 0.0293260
\(400\) 0 0
\(401\) 30.0416 1.50021 0.750104 0.661320i \(-0.230004\pi\)
0.750104 + 0.661320i \(0.230004\pi\)
\(402\) 0 0
\(403\) −35.1127 −1.74909
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) −15.6569 −0.776081
\(408\) 0 0
\(409\) 9.51472 0.470473 0.235236 0.971938i \(-0.424414\pi\)
0.235236 + 0.971938i \(0.424414\pi\)
\(410\) 0 0
\(411\) −14.0000 −0.690569
\(412\) 0 0
\(413\) 4.97056 0.244585
\(414\) 0 0
\(415\) −10.4853 −0.514702
\(416\) 0 0
\(417\) −1.17157 −0.0573722
\(418\) 0 0
\(419\) 34.8701 1.70351 0.851757 0.523937i \(-0.175537\pi\)
0.851757 + 0.523937i \(0.175537\pi\)
\(420\) 0 0
\(421\) −14.6863 −0.715766 −0.357883 0.933766i \(-0.616501\pi\)
−0.357883 + 0.933766i \(0.616501\pi\)
\(422\) 0 0
\(423\) −0.343146 −0.0166843
\(424\) 0 0
\(425\) −1.17157 −0.0568296
\(426\) 0 0
\(427\) −3.31371 −0.160362
\(428\) 0 0
\(429\) −7.65685 −0.369676
\(430\) 0 0
\(431\) −3.51472 −0.169298 −0.0846490 0.996411i \(-0.526977\pi\)
−0.0846490 + 0.996411i \(0.526977\pi\)
\(432\) 0 0
\(433\) 0.928932 0.0446416 0.0223208 0.999751i \(-0.492894\pi\)
0.0223208 + 0.999751i \(0.492894\pi\)
\(434\) 0 0
\(435\) 9.07107 0.434924
\(436\) 0 0
\(437\) 7.65685 0.366277
\(438\) 0 0
\(439\) −0.970563 −0.0463224 −0.0231612 0.999732i \(-0.507373\pi\)
−0.0231612 + 0.999732i \(0.507373\pi\)
\(440\) 0 0
\(441\) −6.65685 −0.316993
\(442\) 0 0
\(443\) 1.31371 0.0624162 0.0312081 0.999513i \(-0.490065\pi\)
0.0312081 + 0.999513i \(0.490065\pi\)
\(444\) 0 0
\(445\) −10.7279 −0.508552
\(446\) 0 0
\(447\) −3.65685 −0.172963
\(448\) 0 0
\(449\) 7.89949 0.372800 0.186400 0.982474i \(-0.440318\pi\)
0.186400 + 0.982474i \(0.440318\pi\)
\(450\) 0 0
\(451\) 10.4853 0.493733
\(452\) 0 0
\(453\) −17.7990 −0.836269
\(454\) 0 0
\(455\) 3.17157 0.148686
\(456\) 0 0
\(457\) 28.8284 1.34854 0.674268 0.738486i \(-0.264459\pi\)
0.674268 + 0.738486i \(0.264459\pi\)
\(458\) 0 0
\(459\) −1.17157 −0.0546843
\(460\) 0 0
\(461\) 10.6863 0.497710 0.248855 0.968541i \(-0.419946\pi\)
0.248855 + 0.968541i \(0.419946\pi\)
\(462\) 0 0
\(463\) 8.10051 0.376462 0.188231 0.982125i \(-0.439725\pi\)
0.188231 + 0.982125i \(0.439725\pi\)
\(464\) 0 0
\(465\) 6.48528 0.300748
\(466\) 0 0
\(467\) −16.3431 −0.756271 −0.378135 0.925750i \(-0.623435\pi\)
−0.378135 + 0.925750i \(0.623435\pi\)
\(468\) 0 0
\(469\) 7.02944 0.324589
\(470\) 0 0
\(471\) −21.7990 −1.00444
\(472\) 0 0
\(473\) 0.828427 0.0380911
\(474\) 0 0
\(475\) −1.00000 −0.0458831
\(476\) 0 0
\(477\) 4.00000 0.183147
\(478\) 0 0
\(479\) 38.1838 1.74466 0.872330 0.488917i \(-0.162608\pi\)
0.872330 + 0.488917i \(0.162608\pi\)
\(480\) 0 0
\(481\) 59.9411 2.73308
\(482\) 0 0
\(483\) 4.48528 0.204087
\(484\) 0 0
\(485\) 4.24264 0.192648
\(486\) 0 0
\(487\) 2.82843 0.128168 0.0640841 0.997944i \(-0.479587\pi\)
0.0640841 + 0.997944i \(0.479587\pi\)
\(488\) 0 0
\(489\) −7.89949 −0.357228
\(490\) 0 0
\(491\) −21.8995 −0.988310 −0.494155 0.869374i \(-0.664523\pi\)
−0.494155 + 0.869374i \(0.664523\pi\)
\(492\) 0 0
\(493\) 10.6274 0.478635
\(494\) 0 0
\(495\) 1.41421 0.0635642
\(496\) 0 0
\(497\) −2.62742 −0.117856
\(498\) 0 0
\(499\) 23.7990 1.06539 0.532695 0.846308i \(-0.321179\pi\)
0.532695 + 0.846308i \(0.321179\pi\)
\(500\) 0 0
\(501\) −10.0000 −0.446767
\(502\) 0 0
\(503\) −0.828427 −0.0369377 −0.0184689 0.999829i \(-0.505879\pi\)
−0.0184689 + 0.999829i \(0.505879\pi\)
\(504\) 0 0
\(505\) 4.82843 0.214862
\(506\) 0 0
\(507\) 16.3137 0.724517
\(508\) 0 0
\(509\) 32.3848 1.43543 0.717715 0.696337i \(-0.245188\pi\)
0.717715 + 0.696337i \(0.245188\pi\)
\(510\) 0 0
\(511\) 1.17157 0.0518273
\(512\) 0 0
\(513\) −1.00000 −0.0441511
\(514\) 0 0
\(515\) −1.65685 −0.0730097
\(516\) 0 0
\(517\) 0.485281 0.0213427
\(518\) 0 0
\(519\) −6.14214 −0.269610
\(520\) 0 0
\(521\) −24.3848 −1.06832 −0.534158 0.845385i \(-0.679371\pi\)
−0.534158 + 0.845385i \(0.679371\pi\)
\(522\) 0 0
\(523\) 15.7990 0.690842 0.345421 0.938448i \(-0.387736\pi\)
0.345421 + 0.938448i \(0.387736\pi\)
\(524\) 0 0
\(525\) −0.585786 −0.0255658
\(526\) 0 0
\(527\) 7.59798 0.330973
\(528\) 0 0
\(529\) 35.6274 1.54902
\(530\) 0 0
\(531\) −8.48528 −0.368230
\(532\) 0 0
\(533\) −40.1421 −1.73875
\(534\) 0 0
\(535\) 8.00000 0.345870
\(536\) 0 0
\(537\) −17.1716 −0.741008
\(538\) 0 0
\(539\) 9.41421 0.405499
\(540\) 0 0
\(541\) −32.6274 −1.40276 −0.701381 0.712786i \(-0.747433\pi\)
−0.701381 + 0.712786i \(0.747433\pi\)
\(542\) 0 0
\(543\) −19.1716 −0.822731
\(544\) 0 0
\(545\) −3.17157 −0.135855
\(546\) 0 0
\(547\) 34.1421 1.45981 0.729906 0.683547i \(-0.239564\pi\)
0.729906 + 0.683547i \(0.239564\pi\)
\(548\) 0 0
\(549\) 5.65685 0.241429
\(550\) 0 0
\(551\) 9.07107 0.386440
\(552\) 0 0
\(553\) −6.62742 −0.281826
\(554\) 0 0
\(555\) −11.0711 −0.469941
\(556\) 0 0
\(557\) 10.0000 0.423714 0.211857 0.977301i \(-0.432049\pi\)
0.211857 + 0.977301i \(0.432049\pi\)
\(558\) 0 0
\(559\) −3.17157 −0.134143
\(560\) 0 0
\(561\) 1.65685 0.0699524
\(562\) 0 0
\(563\) −42.2843 −1.78207 −0.891035 0.453935i \(-0.850020\pi\)
−0.891035 + 0.453935i \(0.850020\pi\)
\(564\) 0 0
\(565\) −12.4853 −0.525260
\(566\) 0 0
\(567\) −0.585786 −0.0246007
\(568\) 0 0
\(569\) −6.72792 −0.282049 −0.141025 0.990006i \(-0.545040\pi\)
−0.141025 + 0.990006i \(0.545040\pi\)
\(570\) 0 0
\(571\) −19.7990 −0.828562 −0.414281 0.910149i \(-0.635967\pi\)
−0.414281 + 0.910149i \(0.635967\pi\)
\(572\) 0 0
\(573\) −1.89949 −0.0793525
\(574\) 0 0
\(575\) −7.65685 −0.319313
\(576\) 0 0
\(577\) −37.7990 −1.57359 −0.786796 0.617213i \(-0.788262\pi\)
−0.786796 + 0.617213i \(0.788262\pi\)
\(578\) 0 0
\(579\) −15.0711 −0.626332
\(580\) 0 0
\(581\) −6.14214 −0.254819
\(582\) 0 0
\(583\) −5.65685 −0.234283
\(584\) 0 0
\(585\) −5.41421 −0.223850
\(586\) 0 0
\(587\) −11.6569 −0.481130 −0.240565 0.970633i \(-0.577333\pi\)
−0.240565 + 0.970633i \(0.577333\pi\)
\(588\) 0 0
\(589\) 6.48528 0.267221
\(590\) 0 0
\(591\) −14.8284 −0.609960
\(592\) 0 0
\(593\) −29.3137 −1.20377 −0.601885 0.798583i \(-0.705583\pi\)
−0.601885 + 0.798583i \(0.705583\pi\)
\(594\) 0 0
\(595\) −0.686292 −0.0281352
\(596\) 0 0
\(597\) 16.4853 0.674698
\(598\) 0 0
\(599\) 21.9411 0.896490 0.448245 0.893911i \(-0.352049\pi\)
0.448245 + 0.893911i \(0.352049\pi\)
\(600\) 0 0
\(601\) −28.8284 −1.17594 −0.587968 0.808884i \(-0.700072\pi\)
−0.587968 + 0.808884i \(0.700072\pi\)
\(602\) 0 0
\(603\) −12.0000 −0.488678
\(604\) 0 0
\(605\) 9.00000 0.365902
\(606\) 0 0
\(607\) 2.14214 0.0869466 0.0434733 0.999055i \(-0.486158\pi\)
0.0434733 + 0.999055i \(0.486158\pi\)
\(608\) 0 0
\(609\) 5.31371 0.215322
\(610\) 0 0
\(611\) −1.85786 −0.0751611
\(612\) 0 0
\(613\) 25.1127 1.01429 0.507146 0.861860i \(-0.330700\pi\)
0.507146 + 0.861860i \(0.330700\pi\)
\(614\) 0 0
\(615\) 7.41421 0.298970
\(616\) 0 0
\(617\) 10.1421 0.408307 0.204154 0.978939i \(-0.434556\pi\)
0.204154 + 0.978939i \(0.434556\pi\)
\(618\) 0 0
\(619\) 43.7990 1.76043 0.880215 0.474575i \(-0.157398\pi\)
0.880215 + 0.474575i \(0.157398\pi\)
\(620\) 0 0
\(621\) −7.65685 −0.307259
\(622\) 0 0
\(623\) −6.28427 −0.251774
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 1.41421 0.0564782
\(628\) 0 0
\(629\) −12.9706 −0.517170
\(630\) 0 0
\(631\) −22.6274 −0.900783 −0.450392 0.892831i \(-0.648716\pi\)
−0.450392 + 0.892831i \(0.648716\pi\)
\(632\) 0 0
\(633\) 15.3137 0.608665
\(634\) 0 0
\(635\) 8.00000 0.317470
\(636\) 0 0
\(637\) −36.0416 −1.42802
\(638\) 0 0
\(639\) 4.48528 0.177435
\(640\) 0 0
\(641\) −8.58579 −0.339118 −0.169559 0.985520i \(-0.554234\pi\)
−0.169559 + 0.985520i \(0.554234\pi\)
\(642\) 0 0
\(643\) −6.04163 −0.238259 −0.119129 0.992879i \(-0.538010\pi\)
−0.119129 + 0.992879i \(0.538010\pi\)
\(644\) 0 0
\(645\) 0.585786 0.0230653
\(646\) 0 0
\(647\) −45.1127 −1.77356 −0.886782 0.462189i \(-0.847064\pi\)
−0.886782 + 0.462189i \(0.847064\pi\)
\(648\) 0 0
\(649\) 12.0000 0.471041
\(650\) 0 0
\(651\) 3.79899 0.148894
\(652\) 0 0
\(653\) 42.4264 1.66027 0.830137 0.557560i \(-0.188262\pi\)
0.830137 + 0.557560i \(0.188262\pi\)
\(654\) 0 0
\(655\) 16.7279 0.653614
\(656\) 0 0
\(657\) −2.00000 −0.0780274
\(658\) 0 0
\(659\) 20.0000 0.779089 0.389545 0.921008i \(-0.372632\pi\)
0.389545 + 0.921008i \(0.372632\pi\)
\(660\) 0 0
\(661\) 18.4853 0.718994 0.359497 0.933146i \(-0.382948\pi\)
0.359497 + 0.933146i \(0.382948\pi\)
\(662\) 0 0
\(663\) −6.34315 −0.246347
\(664\) 0 0
\(665\) −0.585786 −0.0227158
\(666\) 0 0
\(667\) 69.4558 2.68934
\(668\) 0 0
\(669\) −6.34315 −0.245240
\(670\) 0 0
\(671\) −8.00000 −0.308837
\(672\) 0 0
\(673\) 21.8995 0.844163 0.422082 0.906558i \(-0.361300\pi\)
0.422082 + 0.906558i \(0.361300\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) −44.9706 −1.72836 −0.864180 0.503184i \(-0.832162\pi\)
−0.864180 + 0.503184i \(0.832162\pi\)
\(678\) 0 0
\(679\) 2.48528 0.0953763
\(680\) 0 0
\(681\) 18.9706 0.726954
\(682\) 0 0
\(683\) −5.65685 −0.216454 −0.108227 0.994126i \(-0.534517\pi\)
−0.108227 + 0.994126i \(0.534517\pi\)
\(684\) 0 0
\(685\) 14.0000 0.534913
\(686\) 0 0
\(687\) −1.65685 −0.0632129
\(688\) 0 0
\(689\) 21.6569 0.825060
\(690\) 0 0
\(691\) 23.1127 0.879248 0.439624 0.898182i \(-0.355112\pi\)
0.439624 + 0.898182i \(0.355112\pi\)
\(692\) 0 0
\(693\) 0.828427 0.0314693
\(694\) 0 0
\(695\) 1.17157 0.0444403
\(696\) 0 0
\(697\) 8.68629 0.329017
\(698\) 0 0
\(699\) −8.34315 −0.315567
\(700\) 0 0
\(701\) −0.343146 −0.0129604 −0.00648022 0.999979i \(-0.502063\pi\)
−0.00648022 + 0.999979i \(0.502063\pi\)
\(702\) 0 0
\(703\) −11.0711 −0.417553
\(704\) 0 0
\(705\) 0.343146 0.0129236
\(706\) 0 0
\(707\) 2.82843 0.106374
\(708\) 0 0
\(709\) −35.3137 −1.32623 −0.663117 0.748516i \(-0.730767\pi\)
−0.663117 + 0.748516i \(0.730767\pi\)
\(710\) 0 0
\(711\) 11.3137 0.424297
\(712\) 0 0
\(713\) 49.6569 1.85966
\(714\) 0 0
\(715\) 7.65685 0.286350
\(716\) 0 0
\(717\) −2.58579 −0.0965680
\(718\) 0 0
\(719\) 16.4437 0.613245 0.306622 0.951831i \(-0.400801\pi\)
0.306622 + 0.951831i \(0.400801\pi\)
\(720\) 0 0
\(721\) −0.970563 −0.0361456
\(722\) 0 0
\(723\) −14.9706 −0.556761
\(724\) 0 0
\(725\) −9.07107 −0.336891
\(726\) 0 0
\(727\) 4.58579 0.170077 0.0850387 0.996378i \(-0.472899\pi\)
0.0850387 + 0.996378i \(0.472899\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 0.686292 0.0253834
\(732\) 0 0
\(733\) −1.31371 −0.0485229 −0.0242615 0.999706i \(-0.507723\pi\)
−0.0242615 + 0.999706i \(0.507723\pi\)
\(734\) 0 0
\(735\) 6.65685 0.245542
\(736\) 0 0
\(737\) 16.9706 0.625119
\(738\) 0 0
\(739\) 9.65685 0.355233 0.177617 0.984100i \(-0.443161\pi\)
0.177617 + 0.984100i \(0.443161\pi\)
\(740\) 0 0
\(741\) −5.41421 −0.198896
\(742\) 0 0
\(743\) 15.3137 0.561805 0.280903 0.959736i \(-0.409366\pi\)
0.280903 + 0.959736i \(0.409366\pi\)
\(744\) 0 0
\(745\) 3.65685 0.133977
\(746\) 0 0
\(747\) 10.4853 0.383636
\(748\) 0 0
\(749\) 4.68629 0.171233
\(750\) 0 0
\(751\) 37.1127 1.35426 0.677131 0.735863i \(-0.263223\pi\)
0.677131 + 0.735863i \(0.263223\pi\)
\(752\) 0 0
\(753\) −12.9289 −0.471156
\(754\) 0 0
\(755\) 17.7990 0.647772
\(756\) 0 0
\(757\) 16.8284 0.611640 0.305820 0.952089i \(-0.401069\pi\)
0.305820 + 0.952089i \(0.401069\pi\)
\(758\) 0 0
\(759\) 10.8284 0.393047
\(760\) 0 0
\(761\) −10.2843 −0.372805 −0.186402 0.982474i \(-0.559683\pi\)
−0.186402 + 0.982474i \(0.559683\pi\)
\(762\) 0 0
\(763\) −1.85786 −0.0672592
\(764\) 0 0
\(765\) 1.17157 0.0423583
\(766\) 0 0
\(767\) −45.9411 −1.65884
\(768\) 0 0
\(769\) −35.6569 −1.28582 −0.642910 0.765942i \(-0.722273\pi\)
−0.642910 + 0.765942i \(0.722273\pi\)
\(770\) 0 0
\(771\) 1.17157 0.0421932
\(772\) 0 0
\(773\) −32.9706 −1.18587 −0.592934 0.805251i \(-0.702031\pi\)
−0.592934 + 0.805251i \(0.702031\pi\)
\(774\) 0 0
\(775\) −6.48528 −0.232958
\(776\) 0 0
\(777\) −6.48528 −0.232658
\(778\) 0 0
\(779\) 7.41421 0.265642
\(780\) 0 0
\(781\) −6.34315 −0.226976
\(782\) 0 0
\(783\) −9.07107 −0.324174
\(784\) 0 0
\(785\) 21.7990 0.778039
\(786\) 0 0
\(787\) −13.4558 −0.479649 −0.239825 0.970816i \(-0.577090\pi\)
−0.239825 + 0.970816i \(0.577090\pi\)
\(788\) 0 0
\(789\) 32.1421 1.14429
\(790\) 0 0
\(791\) −7.31371 −0.260046
\(792\) 0 0
\(793\) 30.6274 1.08761
\(794\) 0 0
\(795\) −4.00000 −0.141865
\(796\) 0 0
\(797\) 10.8284 0.383563 0.191781 0.981438i \(-0.438574\pi\)
0.191781 + 0.981438i \(0.438574\pi\)
\(798\) 0 0
\(799\) 0.402020 0.0142225
\(800\) 0 0
\(801\) 10.7279 0.379052
\(802\) 0 0
\(803\) 2.82843 0.0998130
\(804\) 0 0
\(805\) −4.48528 −0.158085
\(806\) 0 0
\(807\) 8.38478 0.295158
\(808\) 0 0
\(809\) 49.3137 1.73378 0.866889 0.498502i \(-0.166116\pi\)
0.866889 + 0.498502i \(0.166116\pi\)
\(810\) 0 0
\(811\) 15.3137 0.537737 0.268869 0.963177i \(-0.413350\pi\)
0.268869 + 0.963177i \(0.413350\pi\)
\(812\) 0 0
\(813\) 30.8284 1.08120
\(814\) 0 0
\(815\) 7.89949 0.276707
\(816\) 0 0
\(817\) 0.585786 0.0204941
\(818\) 0 0
\(819\) −3.17157 −0.110824
\(820\) 0 0
\(821\) 51.4558 1.79582 0.897911 0.440178i \(-0.145085\pi\)
0.897911 + 0.440178i \(0.145085\pi\)
\(822\) 0 0
\(823\) 2.72792 0.0950894 0.0475447 0.998869i \(-0.484860\pi\)
0.0475447 + 0.998869i \(0.484860\pi\)
\(824\) 0 0
\(825\) −1.41421 −0.0492366
\(826\) 0 0
\(827\) −48.6274 −1.69094 −0.845470 0.534022i \(-0.820680\pi\)
−0.845470 + 0.534022i \(0.820680\pi\)
\(828\) 0 0
\(829\) −38.4853 −1.33665 −0.668325 0.743870i \(-0.732988\pi\)
−0.668325 + 0.743870i \(0.732988\pi\)
\(830\) 0 0
\(831\) 10.9706 0.380565
\(832\) 0 0
\(833\) 7.79899 0.270219
\(834\) 0 0
\(835\) 10.0000 0.346064
\(836\) 0 0
\(837\) −6.48528 −0.224164
\(838\) 0 0
\(839\) −27.1127 −0.936034 −0.468017 0.883719i \(-0.655031\pi\)
−0.468017 + 0.883719i \(0.655031\pi\)
\(840\) 0 0
\(841\) 53.2843 1.83739
\(842\) 0 0
\(843\) −14.7279 −0.507257
\(844\) 0 0
\(845\) −16.3137 −0.561209
\(846\) 0 0
\(847\) 5.27208 0.181151
\(848\) 0 0
\(849\) −6.24264 −0.214247
\(850\) 0 0
\(851\) −84.7696 −2.90586
\(852\) 0 0
\(853\) −9.51472 −0.325778 −0.162889 0.986644i \(-0.552081\pi\)
−0.162889 + 0.986644i \(0.552081\pi\)
\(854\) 0 0
\(855\) 1.00000 0.0341993
\(856\) 0 0
\(857\) 37.9411 1.29604 0.648022 0.761622i \(-0.275596\pi\)
0.648022 + 0.761622i \(0.275596\pi\)
\(858\) 0 0
\(859\) −25.9411 −0.885100 −0.442550 0.896744i \(-0.645926\pi\)
−0.442550 + 0.896744i \(0.645926\pi\)
\(860\) 0 0
\(861\) 4.34315 0.148014
\(862\) 0 0
\(863\) 31.3137 1.06593 0.532966 0.846137i \(-0.321078\pi\)
0.532966 + 0.846137i \(0.321078\pi\)
\(864\) 0 0
\(865\) 6.14214 0.208839
\(866\) 0 0
\(867\) −15.6274 −0.530735
\(868\) 0 0
\(869\) −16.0000 −0.542763
\(870\) 0 0
\(871\) −64.9706 −2.20144
\(872\) 0 0
\(873\) −4.24264 −0.143592
\(874\) 0 0
\(875\) 0.585786 0.0198032
\(876\) 0 0
\(877\) 17.8995 0.604423 0.302211 0.953241i \(-0.402275\pi\)
0.302211 + 0.953241i \(0.402275\pi\)
\(878\) 0 0
\(879\) −31.7990 −1.07255
\(880\) 0 0
\(881\) −47.4558 −1.59883 −0.799414 0.600781i \(-0.794857\pi\)
−0.799414 + 0.600781i \(0.794857\pi\)
\(882\) 0 0
\(883\) −46.5269 −1.56576 −0.782878 0.622176i \(-0.786249\pi\)
−0.782878 + 0.622176i \(0.786249\pi\)
\(884\) 0 0
\(885\) 8.48528 0.285230
\(886\) 0 0
\(887\) 8.00000 0.268614 0.134307 0.990940i \(-0.457119\pi\)
0.134307 + 0.990940i \(0.457119\pi\)
\(888\) 0 0
\(889\) 4.68629 0.157173
\(890\) 0 0
\(891\) −1.41421 −0.0473779
\(892\) 0 0
\(893\) 0.343146 0.0114829
\(894\) 0 0
\(895\) 17.1716 0.573982
\(896\) 0 0
\(897\) −41.4558 −1.38417
\(898\) 0 0
\(899\) 58.8284 1.96204
\(900\) 0 0
\(901\) −4.68629 −0.156123
\(902\) 0 0
\(903\) 0.343146 0.0114192
\(904\) 0 0
\(905\) 19.1716 0.637285
\(906\) 0 0
\(907\) −38.1421 −1.26649 −0.633244 0.773952i \(-0.718277\pi\)
−0.633244 + 0.773952i \(0.718277\pi\)
\(908\) 0 0
\(909\) −4.82843 −0.160149
\(910\) 0 0
\(911\) 23.3137 0.772418 0.386209 0.922411i \(-0.373784\pi\)
0.386209 + 0.922411i \(0.373784\pi\)
\(912\) 0 0
\(913\) −14.8284 −0.490749
\(914\) 0 0
\(915\) −5.65685 −0.187010
\(916\) 0 0
\(917\) 9.79899 0.323591
\(918\) 0 0
\(919\) 12.0000 0.395843 0.197922 0.980218i \(-0.436581\pi\)
0.197922 + 0.980218i \(0.436581\pi\)
\(920\) 0 0
\(921\) −7.79899 −0.256985
\(922\) 0 0
\(923\) 24.2843 0.799327
\(924\) 0 0
\(925\) 11.0711 0.364014
\(926\) 0 0
\(927\) 1.65685 0.0544182
\(928\) 0 0
\(929\) −51.4558 −1.68821 −0.844106 0.536177i \(-0.819868\pi\)
−0.844106 + 0.536177i \(0.819868\pi\)
\(930\) 0 0
\(931\) 6.65685 0.218170
\(932\) 0 0
\(933\) 32.2426 1.05558
\(934\) 0 0
\(935\) −1.65685 −0.0541849
\(936\) 0 0
\(937\) −18.7696 −0.613175 −0.306587 0.951843i \(-0.599187\pi\)
−0.306587 + 0.951843i \(0.599187\pi\)
\(938\) 0 0
\(939\) −9.51472 −0.310501
\(940\) 0 0
\(941\) −17.5563 −0.572321 −0.286160 0.958182i \(-0.592379\pi\)
−0.286160 + 0.958182i \(0.592379\pi\)
\(942\) 0 0
\(943\) 56.7696 1.84867
\(944\) 0 0
\(945\) 0.585786 0.0190556
\(946\) 0 0
\(947\) −12.8284 −0.416868 −0.208434 0.978036i \(-0.566837\pi\)
−0.208434 + 0.978036i \(0.566837\pi\)
\(948\) 0 0
\(949\) −10.8284 −0.351506
\(950\) 0 0
\(951\) 11.3137 0.366872
\(952\) 0 0
\(953\) −5.85786 −0.189755 −0.0948774 0.995489i \(-0.530246\pi\)
−0.0948774 + 0.995489i \(0.530246\pi\)
\(954\) 0 0
\(955\) 1.89949 0.0614662
\(956\) 0 0
\(957\) 12.8284 0.414684
\(958\) 0 0
\(959\) 8.20101 0.264824
\(960\) 0 0
\(961\) 11.0589 0.356738
\(962\) 0 0
\(963\) −8.00000 −0.257796
\(964\) 0 0
\(965\) 15.0711 0.485155
\(966\) 0 0
\(967\) 27.8995 0.897187 0.448594 0.893736i \(-0.351925\pi\)
0.448594 + 0.893736i \(0.351925\pi\)
\(968\) 0 0
\(969\) 1.17157 0.0376363
\(970\) 0 0
\(971\) −17.6569 −0.566635 −0.283318 0.959026i \(-0.591435\pi\)
−0.283318 + 0.959026i \(0.591435\pi\)
\(972\) 0 0
\(973\) 0.686292 0.0220015
\(974\) 0 0
\(975\) 5.41421 0.173394
\(976\) 0 0
\(977\) 39.5980 1.26685 0.633426 0.773803i \(-0.281648\pi\)
0.633426 + 0.773803i \(0.281648\pi\)
\(978\) 0 0
\(979\) −15.1716 −0.484886
\(980\) 0 0
\(981\) 3.17157 0.101261
\(982\) 0 0
\(983\) −31.9411 −1.01876 −0.509382 0.860541i \(-0.670126\pi\)
−0.509382 + 0.860541i \(0.670126\pi\)
\(984\) 0 0
\(985\) 14.8284 0.472473
\(986\) 0 0
\(987\) 0.201010 0.00639822
\(988\) 0 0
\(989\) 4.48528 0.142624
\(990\) 0 0
\(991\) 61.6569 1.95859 0.979297 0.202427i \(-0.0648830\pi\)
0.979297 + 0.202427i \(0.0648830\pi\)
\(992\) 0 0
\(993\) −7.17157 −0.227583
\(994\) 0 0
\(995\) −16.4853 −0.522619
\(996\) 0 0
\(997\) −50.4853 −1.59888 −0.799442 0.600743i \(-0.794872\pi\)
−0.799442 + 0.600743i \(0.794872\pi\)
\(998\) 0 0
\(999\) 11.0711 0.350273
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 4560.2.a.bj.1.2 2
4.3 odd 2 285.2.a.f.1.1 2
12.11 even 2 855.2.a.e.1.2 2
20.3 even 4 1425.2.c.j.799.3 4
20.7 even 4 1425.2.c.j.799.2 4
20.19 odd 2 1425.2.a.l.1.2 2
60.59 even 2 4275.2.a.x.1.1 2
76.75 even 2 5415.2.a.p.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
285.2.a.f.1.1 2 4.3 odd 2
855.2.a.e.1.2 2 12.11 even 2
1425.2.a.l.1.2 2 20.19 odd 2
1425.2.c.j.799.2 4 20.7 even 4
1425.2.c.j.799.3 4 20.3 even 4
4275.2.a.x.1.1 2 60.59 even 2
4560.2.a.bj.1.2 2 1.1 even 1 trivial
5415.2.a.p.1.2 2 76.75 even 2