Properties

Label 1840.2.m.g.1839.21
Level $1840$
Weight $2$
Character 1840.1839
Analytic conductor $14.692$
Analytic rank $0$
Dimension $40$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1840,2,Mod(1839,1840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1840, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1840.1839");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1840.m (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(14.6924739719\)
Analytic rank: \(0\)
Dimension: \(40\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1839.21
Character \(\chi\) \(=\) 1840.1839
Dual form 1840.2.m.g.1839.24

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.924187 q^{3} +(0.611037 + 2.15096i) q^{5} -1.51428i q^{7} -2.14588 q^{9} +O(q^{10})\) \(q+0.924187 q^{3} +(0.611037 + 2.15096i) q^{5} -1.51428i q^{7} -2.14588 q^{9} +0.770524 q^{11} +4.68356i q^{13} +(0.564712 + 1.98789i) q^{15} -4.00822 q^{17} -7.17499 q^{19} -1.39948i q^{21} +(4.38418 + 1.94395i) q^{23} +(-4.25327 + 2.62863i) q^{25} -4.75575 q^{27} -4.41697 q^{29} +3.07318i q^{31} +0.712108 q^{33} +(3.25717 - 0.925283i) q^{35} -5.52910 q^{37} +4.32849i q^{39} +7.75875 q^{41} +2.03824i q^{43} +(-1.31121 - 4.61570i) q^{45} +4.08211 q^{47} +4.70694 q^{49} -3.70434 q^{51} -10.2815 q^{53} +(0.470818 + 1.65737i) q^{55} -6.63104 q^{57} -0.831684i q^{59} -7.18803i q^{61} +3.24947i q^{63} +(-10.0742 + 2.86183i) q^{65} +12.6052i q^{67} +(4.05181 + 1.79657i) q^{69} -2.23430i q^{71} +11.1803i q^{73} +(-3.93082 + 2.42935i) q^{75} -1.16679i q^{77} +3.98050 q^{79} +2.04243 q^{81} +7.33338i q^{83} +(-2.44917 - 8.62152i) q^{85} -4.08211 q^{87} -1.08954i q^{89} +7.09225 q^{91} +2.84019i q^{93} +(-4.38418 - 15.4331i) q^{95} +2.74295 q^{97} -1.65345 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 40 q + 80 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 40 q + 80 q^{9} - 24 q^{25} + 24 q^{41} - 16 q^{49} + 80 q^{69} + 40 q^{81} - 8 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times\).

\(n\) \(737\) \(1151\) \(1201\) \(1381\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(1\)

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
\(3\) 0.924187 0.533580 0.266790 0.963755i \(-0.414037\pi\)
0.266790 + 0.963755i \(0.414037\pi\)
\(4\) 0 0
\(5\) 0.611037 + 2.15096i 0.273264 + 0.961939i
\(6\) 0 0
\(7\) 1.51428i 0.572346i −0.958178 0.286173i \(-0.907617\pi\)
0.958178 0.286173i \(-0.0923832\pi\)
\(8\) 0 0
\(9\) −2.14588 −0.715293
\(10\) 0 0
\(11\) 0.770524 0.232322 0.116161 0.993230i \(-0.462941\pi\)
0.116161 + 0.993230i \(0.462941\pi\)
\(12\) 0 0
\(13\) 4.68356i 1.29899i 0.760367 + 0.649493i \(0.225019\pi\)
−0.760367 + 0.649493i \(0.774981\pi\)
\(14\) 0 0
\(15\) 0.564712 + 1.98789i 0.145808 + 0.513271i
\(16\) 0 0
\(17\) −4.00822 −0.972136 −0.486068 0.873921i \(-0.661569\pi\)
−0.486068 + 0.873921i \(0.661569\pi\)
\(18\) 0 0
\(19\) −7.17499 −1.64606 −0.823028 0.568000i \(-0.807717\pi\)
−0.823028 + 0.568000i \(0.807717\pi\)
\(20\) 0 0
\(21\) 1.39948i 0.305392i
\(22\) 0 0
\(23\) 4.38418 + 1.94395i 0.914166 + 0.405341i
\(24\) 0 0
\(25\) −4.25327 + 2.62863i −0.850654 + 0.525726i
\(26\) 0 0
\(27\) −4.75575 −0.915245
\(28\) 0 0
\(29\) −4.41697 −0.820211 −0.410105 0.912038i \(-0.634508\pi\)
−0.410105 + 0.912038i \(0.634508\pi\)
\(30\) 0 0
\(31\) 3.07318i 0.551959i 0.961164 + 0.275979i \(0.0890021\pi\)
−0.961164 + 0.275979i \(0.910998\pi\)
\(32\) 0 0
\(33\) 0.712108 0.123962
\(34\) 0 0
\(35\) 3.25717 0.925283i 0.550562 0.156401i
\(36\) 0 0
\(37\) −5.52910 −0.908978 −0.454489 0.890752i \(-0.650178\pi\)
−0.454489 + 0.890752i \(0.650178\pi\)
\(38\) 0 0
\(39\) 4.32849i 0.693113i
\(40\) 0 0
\(41\) 7.75875 1.21171 0.605856 0.795574i \(-0.292831\pi\)
0.605856 + 0.795574i \(0.292831\pi\)
\(42\) 0 0
\(43\) 2.03824i 0.310829i 0.987849 + 0.155415i \(0.0496714\pi\)
−0.987849 + 0.155415i \(0.950329\pi\)
\(44\) 0 0
\(45\) −1.31121 4.61570i −0.195464 0.688068i
\(46\) 0 0
\(47\) 4.08211 0.595436 0.297718 0.954654i \(-0.403774\pi\)
0.297718 + 0.954654i \(0.403774\pi\)
\(48\) 0 0
\(49\) 4.70694 0.672420
\(50\) 0 0
\(51\) −3.70434 −0.518712
\(52\) 0 0
\(53\) −10.2815 −1.41227 −0.706137 0.708075i \(-0.749564\pi\)
−0.706137 + 0.708075i \(0.749564\pi\)
\(54\) 0 0
\(55\) 0.470818 + 1.65737i 0.0634851 + 0.223479i
\(56\) 0 0
\(57\) −6.63104 −0.878302
\(58\) 0 0
\(59\) 0.831684i 0.108276i −0.998533 0.0541380i \(-0.982759\pi\)
0.998533 0.0541380i \(-0.0172411\pi\)
\(60\) 0 0
\(61\) 7.18803i 0.920334i −0.887832 0.460167i \(-0.847790\pi\)
0.887832 0.460167i \(-0.152210\pi\)
\(62\) 0 0
\(63\) 3.24947i 0.409395i
\(64\) 0 0
\(65\) −10.0742 + 2.86183i −1.24955 + 0.354966i
\(66\) 0 0
\(67\) 12.6052i 1.53998i 0.638058 + 0.769988i \(0.279738\pi\)
−0.638058 + 0.769988i \(0.720262\pi\)
\(68\) 0 0
\(69\) 4.05181 + 1.79657i 0.487780 + 0.216282i
\(70\) 0 0
\(71\) 2.23430i 0.265162i −0.991172 0.132581i \(-0.957673\pi\)
0.991172 0.132581i \(-0.0423265\pi\)
\(72\) 0 0
\(73\) 11.1803i 1.30855i 0.756256 + 0.654276i \(0.227027\pi\)
−0.756256 + 0.654276i \(0.772973\pi\)
\(74\) 0 0
\(75\) −3.93082 + 2.42935i −0.453891 + 0.280517i
\(76\) 0 0
\(77\) 1.16679i 0.132968i
\(78\) 0 0
\(79\) 3.98050 0.447841 0.223920 0.974607i \(-0.428114\pi\)
0.223920 + 0.974607i \(0.428114\pi\)
\(80\) 0 0
\(81\) 2.04243 0.226937
\(82\) 0 0
\(83\) 7.33338i 0.804943i 0.915432 + 0.402472i \(0.131849\pi\)
−0.915432 + 0.402472i \(0.868151\pi\)
\(84\) 0 0
\(85\) −2.44917 8.62152i −0.265650 0.935135i
\(86\) 0 0
\(87\) −4.08211 −0.437648
\(88\) 0 0
\(89\) 1.08954i 0.115491i −0.998331 0.0577456i \(-0.981609\pi\)
0.998331 0.0577456i \(-0.0183912\pi\)
\(90\) 0 0
\(91\) 7.09225 0.743470
\(92\) 0 0
\(93\) 2.84019i 0.294514i
\(94\) 0 0
\(95\) −4.38418 15.4331i −0.449808 1.58341i
\(96\) 0 0
\(97\) 2.74295 0.278505 0.139252 0.990257i \(-0.455530\pi\)
0.139252 + 0.990257i \(0.455530\pi\)
\(98\) 0 0
\(99\) −1.65345 −0.166178
\(100\) 0 0
\(101\) −10.8717 −1.08177 −0.540887 0.841095i \(-0.681911\pi\)
−0.540887 + 0.841095i \(0.681911\pi\)
\(102\) 0 0
\(103\) 16.7824i 1.65362i −0.562484 0.826808i \(-0.690154\pi\)
0.562484 0.826808i \(-0.309846\pi\)
\(104\) 0 0
\(105\) 3.01023 0.855135i 0.293769 0.0834526i
\(106\) 0 0
\(107\) 11.3125i 1.09363i 0.837255 + 0.546813i \(0.184159\pi\)
−0.837255 + 0.546813i \(0.815841\pi\)
\(108\) 0 0
\(109\) 1.94861i 0.186643i −0.995636 0.0933217i \(-0.970251\pi\)
0.995636 0.0933217i \(-0.0297485\pi\)
\(110\) 0 0
\(111\) −5.10992 −0.485012
\(112\) 0 0
\(113\) −1.41749 −0.133346 −0.0666731 0.997775i \(-0.521238\pi\)
−0.0666731 + 0.997775i \(0.521238\pi\)
\(114\) 0 0
\(115\) −1.50246 + 10.6180i −0.140105 + 0.990137i
\(116\) 0 0
\(117\) 10.0504i 0.929156i
\(118\) 0 0
\(119\) 6.06958i 0.556398i
\(120\) 0 0
\(121\) −10.4063 −0.946027
\(122\) 0 0
\(123\) 7.17053 0.646545
\(124\) 0 0
\(125\) −8.25299 7.54243i −0.738170 0.674615i
\(126\) 0 0
\(127\) 11.9437 1.05984 0.529918 0.848049i \(-0.322223\pi\)
0.529918 + 0.848049i \(0.322223\pi\)
\(128\) 0 0
\(129\) 1.88372i 0.165852i
\(130\) 0 0
\(131\) 9.39358i 0.820721i 0.911923 + 0.410360i \(0.134597\pi\)
−0.911923 + 0.410360i \(0.865403\pi\)
\(132\) 0 0
\(133\) 10.8650i 0.942114i
\(134\) 0 0
\(135\) −2.90594 10.2294i −0.250103 0.880410i
\(136\) 0 0
\(137\) −13.8161 −1.18039 −0.590194 0.807261i \(-0.700949\pi\)
−0.590194 + 0.807261i \(0.700949\pi\)
\(138\) 0 0
\(139\) 20.0085i 1.69710i −0.529114 0.848551i \(-0.677476\pi\)
0.529114 0.848551i \(-0.322524\pi\)
\(140\) 0 0
\(141\) 3.77263 0.317713
\(142\) 0 0
\(143\) 3.60880i 0.301783i
\(144\) 0 0
\(145\) −2.69893 9.50073i −0.224134 0.788993i
\(146\) 0 0
\(147\) 4.35009 0.358790
\(148\) 0 0
\(149\) 5.70733i 0.467563i −0.972289 0.233781i \(-0.924890\pi\)
0.972289 0.233781i \(-0.0751100\pi\)
\(150\) 0 0
\(151\) 9.03536i 0.735287i 0.929967 + 0.367643i \(0.119835\pi\)
−0.929967 + 0.367643i \(0.880165\pi\)
\(152\) 0 0
\(153\) 8.60115 0.695362
\(154\) 0 0
\(155\) −6.61028 + 1.87782i −0.530951 + 0.150830i
\(156\) 0 0
\(157\) 3.88848 0.310335 0.155167 0.987888i \(-0.450408\pi\)
0.155167 + 0.987888i \(0.450408\pi\)
\(158\) 0 0
\(159\) −9.50204 −0.753561
\(160\) 0 0
\(161\) 2.94369 6.63890i 0.231995 0.523219i
\(162\) 0 0
\(163\) 24.1380 1.89064 0.945318 0.326151i \(-0.105752\pi\)
0.945318 + 0.326151i \(0.105752\pi\)
\(164\) 0 0
\(165\) 0.435124 + 1.53172i 0.0338744 + 0.119244i
\(166\) 0 0
\(167\) −6.51434 −0.504094 −0.252047 0.967715i \(-0.581104\pi\)
−0.252047 + 0.967715i \(0.581104\pi\)
\(168\) 0 0
\(169\) −8.93577 −0.687367
\(170\) 0 0
\(171\) 15.3967 1.17741
\(172\) 0 0
\(173\) 16.8909i 1.28419i 0.766625 + 0.642095i \(0.221934\pi\)
−0.766625 + 0.642095i \(0.778066\pi\)
\(174\) 0 0
\(175\) 3.98050 + 6.44066i 0.300897 + 0.486868i
\(176\) 0 0
\(177\) 0.768632i 0.0577739i
\(178\) 0 0
\(179\) 9.50098i 0.710137i −0.934840 0.355068i \(-0.884458\pi\)
0.934840 0.355068i \(-0.115542\pi\)
\(180\) 0 0
\(181\) 3.90483i 0.290244i 0.989414 + 0.145122i \(0.0463575\pi\)
−0.989414 + 0.145122i \(0.953643\pi\)
\(182\) 0 0
\(183\) 6.64309i 0.491071i
\(184\) 0 0
\(185\) −3.37848 11.8929i −0.248391 0.874382i
\(186\) 0 0
\(187\) −3.08843 −0.225848
\(188\) 0 0
\(189\) 7.20156i 0.523837i
\(190\) 0 0
\(191\) 13.2206 0.956607 0.478303 0.878195i \(-0.341252\pi\)
0.478303 + 0.878195i \(0.341252\pi\)
\(192\) 0 0
\(193\) 10.1726i 0.732242i 0.930567 + 0.366121i \(0.119314\pi\)
−0.930567 + 0.366121i \(0.880686\pi\)
\(194\) 0 0
\(195\) −9.31041 + 2.64486i −0.666732 + 0.189403i
\(196\) 0 0
\(197\) 18.2809i 1.30246i 0.758880 + 0.651231i \(0.225747\pi\)
−0.758880 + 0.651231i \(0.774253\pi\)
\(198\) 0 0
\(199\) 17.9189 1.27023 0.635117 0.772416i \(-0.280952\pi\)
0.635117 + 0.772416i \(0.280952\pi\)
\(200\) 0 0
\(201\) 11.6496i 0.821700i
\(202\) 0 0
\(203\) 6.68855i 0.469444i
\(204\) 0 0
\(205\) 4.74088 + 16.6888i 0.331117 + 1.16559i
\(206\) 0 0
\(207\) −9.40793 4.17147i −0.653896 0.289937i
\(208\) 0 0
\(209\) −5.52850 −0.382415
\(210\) 0 0
\(211\) 15.8222i 1.08925i 0.838681 + 0.544623i \(0.183327\pi\)
−0.838681 + 0.544623i \(0.816673\pi\)
\(212\) 0 0
\(213\) 2.06491i 0.141485i
\(214\) 0 0
\(215\) −4.38418 + 1.24544i −0.298999 + 0.0849384i
\(216\) 0 0
\(217\) 4.65366 0.315911
\(218\) 0 0
\(219\) 10.3327i 0.698217i
\(220\) 0 0
\(221\) 18.7727i 1.26279i
\(222\) 0 0
\(223\) −17.4686 −1.16978 −0.584891 0.811112i \(-0.698863\pi\)
−0.584891 + 0.811112i \(0.698863\pi\)
\(224\) 0 0
\(225\) 9.12700 5.64072i 0.608467 0.376048i
\(226\) 0 0
\(227\) 8.56046i 0.568178i 0.958798 + 0.284089i \(0.0916910\pi\)
−0.958798 + 0.284089i \(0.908309\pi\)
\(228\) 0 0
\(229\) 7.96756i 0.526511i −0.964726 0.263256i \(-0.915204\pi\)
0.964726 0.263256i \(-0.0847963\pi\)
\(230\) 0 0
\(231\) 1.07833i 0.0709492i
\(232\) 0 0
\(233\) 7.20846i 0.472242i 0.971724 + 0.236121i \(0.0758761\pi\)
−0.971724 + 0.236121i \(0.924124\pi\)
\(234\) 0 0
\(235\) 2.49432 + 8.78045i 0.162711 + 0.572774i
\(236\) 0 0
\(237\) 3.67872 0.238959
\(238\) 0 0
\(239\) 9.76559i 0.631683i −0.948812 0.315842i \(-0.897713\pi\)
0.948812 0.315842i \(-0.102287\pi\)
\(240\) 0 0
\(241\) 26.8111i 1.72706i 0.504300 + 0.863528i \(0.331750\pi\)
−0.504300 + 0.863528i \(0.668250\pi\)
\(242\) 0 0
\(243\) 16.1548 1.03633
\(244\) 0 0
\(245\) 2.87611 + 10.1245i 0.183748 + 0.646827i
\(246\) 0 0
\(247\) 33.6045i 2.13821i
\(248\) 0 0
\(249\) 6.77742i 0.429501i
\(250\) 0 0
\(251\) −26.6349 −1.68118 −0.840590 0.541672i \(-0.817792\pi\)
−0.840590 + 0.541672i \(0.817792\pi\)
\(252\) 0 0
\(253\) 3.37812 + 1.49786i 0.212380 + 0.0941695i
\(254\) 0 0
\(255\) −2.26349 7.96790i −0.141745 0.498969i
\(256\) 0 0
\(257\) 0.102882i 0.00641757i 0.999995 + 0.00320879i \(0.00102139\pi\)
−0.999995 + 0.00320879i \(0.998979\pi\)
\(258\) 0 0
\(259\) 8.37263i 0.520250i
\(260\) 0 0
\(261\) 9.47828 0.586691
\(262\) 0 0
\(263\) 0.710540i 0.0438138i −0.999760 0.0219069i \(-0.993026\pi\)
0.999760 0.0219069i \(-0.00697373\pi\)
\(264\) 0 0
\(265\) −6.28238 22.1151i −0.385924 1.35852i
\(266\) 0 0
\(267\) 1.00694i 0.0616238i
\(268\) 0 0
\(269\) 7.62225 0.464736 0.232368 0.972628i \(-0.425353\pi\)
0.232368 + 0.972628i \(0.425353\pi\)
\(270\) 0 0
\(271\) 19.3110i 1.17306i 0.809928 + 0.586530i \(0.199506\pi\)
−0.809928 + 0.586530i \(0.800494\pi\)
\(272\) 0 0
\(273\) 6.55456 0.396700
\(274\) 0 0
\(275\) −3.27724 + 2.02542i −0.197625 + 0.122138i
\(276\) 0 0
\(277\) 3.30436i 0.198540i −0.995061 0.0992698i \(-0.968349\pi\)
0.995061 0.0992698i \(-0.0316507\pi\)
\(278\) 0 0
\(279\) 6.59466i 0.394812i
\(280\) 0 0
\(281\) 21.8091i 1.30102i −0.759496 0.650512i \(-0.774554\pi\)
0.759496 0.650512i \(-0.225446\pi\)
\(282\) 0 0
\(283\) 17.5316i 1.04215i −0.853512 0.521073i \(-0.825532\pi\)
0.853512 0.521073i \(-0.174468\pi\)
\(284\) 0 0
\(285\) −4.05181 14.2631i −0.240008 0.844873i
\(286\) 0 0
\(287\) 11.7490i 0.693519i
\(288\) 0 0
\(289\) −0.934184 −0.0549520
\(290\) 0 0
\(291\) 2.53500 0.148604
\(292\) 0 0
\(293\) 23.9898 1.40150 0.700751 0.713406i \(-0.252848\pi\)
0.700751 + 0.713406i \(0.252848\pi\)
\(294\) 0 0
\(295\) 1.78892 0.508189i 0.104155 0.0295879i
\(296\) 0 0
\(297\) −3.66442 −0.212631
\(298\) 0 0
\(299\) −9.10460 + 20.5336i −0.526533 + 1.18749i
\(300\) 0 0
\(301\) 3.08648 0.177902
\(302\) 0 0
\(303\) −10.0475 −0.577213
\(304\) 0 0
\(305\) 15.4612 4.39215i 0.885305 0.251494i
\(306\) 0 0
\(307\) −6.35359 −0.362618 −0.181309 0.983426i \(-0.558033\pi\)
−0.181309 + 0.983426i \(0.558033\pi\)
\(308\) 0 0
\(309\) 15.5100i 0.882336i
\(310\) 0 0
\(311\) 21.3653i 1.21151i −0.795650 0.605757i \(-0.792870\pi\)
0.795650 0.605757i \(-0.207130\pi\)
\(312\) 0 0
\(313\) 13.9033 0.785860 0.392930 0.919568i \(-0.371462\pi\)
0.392930 + 0.919568i \(0.371462\pi\)
\(314\) 0 0
\(315\) −6.98949 + 1.98555i −0.393813 + 0.111873i
\(316\) 0 0
\(317\) 32.1985i 1.80845i −0.427058 0.904224i \(-0.640450\pi\)
0.427058 0.904224i \(-0.359550\pi\)
\(318\) 0 0
\(319\) −3.40338 −0.190553
\(320\) 0 0
\(321\) 10.4549i 0.583536i
\(322\) 0 0
\(323\) 28.7589 1.60019
\(324\) 0 0
\(325\) −12.3114 19.9205i −0.682912 1.10499i
\(326\) 0 0
\(327\) 1.80088i 0.0995891i
\(328\) 0 0
\(329\) 6.18147i 0.340795i
\(330\) 0 0
\(331\) 29.1630i 1.60294i −0.598032 0.801472i \(-0.704051\pi\)
0.598032 0.801472i \(-0.295949\pi\)
\(332\) 0 0
\(333\) 11.8648 0.650186
\(334\) 0 0
\(335\) −27.1134 + 7.70227i −1.48136 + 0.420820i
\(336\) 0 0
\(337\) 27.6280 1.50499 0.752495 0.658598i \(-0.228850\pi\)
0.752495 + 0.658598i \(0.228850\pi\)
\(338\) 0 0
\(339\) −1.31003 −0.0711508
\(340\) 0 0
\(341\) 2.36796i 0.128232i
\(342\) 0 0
\(343\) 17.7276i 0.957203i
\(344\) 0 0
\(345\) −1.38855 + 9.81305i −0.0747571 + 0.528317i
\(346\) 0 0
\(347\) −12.3319 −0.662014 −0.331007 0.943628i \(-0.607388\pi\)
−0.331007 + 0.943628i \(0.607388\pi\)
\(348\) 0 0
\(349\) 22.3284 1.19521 0.597605 0.801790i \(-0.296119\pi\)
0.597605 + 0.801790i \(0.296119\pi\)
\(350\) 0 0
\(351\) 22.2739i 1.18889i
\(352\) 0 0
\(353\) 23.1661i 1.23300i 0.787353 + 0.616502i \(0.211451\pi\)
−0.787353 + 0.616502i \(0.788549\pi\)
\(354\) 0 0
\(355\) 4.80589 1.36524i 0.255070 0.0724593i
\(356\) 0 0
\(357\) 5.60943i 0.296883i
\(358\) 0 0
\(359\) 23.1784 1.22331 0.611655 0.791124i \(-0.290504\pi\)
0.611655 + 0.791124i \(0.290504\pi\)
\(360\) 0 0
\(361\) 32.4805 1.70950
\(362\) 0 0
\(363\) −9.61736 −0.504781
\(364\) 0 0
\(365\) −24.0483 + 6.83156i −1.25875 + 0.357580i
\(366\) 0 0
\(367\) 13.5956i 0.709683i 0.934926 + 0.354842i \(0.115465\pi\)
−0.934926 + 0.354842i \(0.884535\pi\)
\(368\) 0 0
\(369\) −16.6493 −0.866730
\(370\) 0 0
\(371\) 15.5691i 0.808309i
\(372\) 0 0
\(373\) 28.2567 1.46307 0.731537 0.681802i \(-0.238803\pi\)
0.731537 + 0.681802i \(0.238803\pi\)
\(374\) 0 0
\(375\) −7.62730 6.97061i −0.393872 0.359961i
\(376\) 0 0
\(377\) 20.6872i 1.06544i
\(378\) 0 0
\(379\) −20.0649 −1.03067 −0.515333 0.856990i \(-0.672332\pi\)
−0.515333 + 0.856990i \(0.672332\pi\)
\(380\) 0 0
\(381\) 11.0382 0.565506
\(382\) 0 0
\(383\) 11.3479i 0.579850i −0.957049 0.289925i \(-0.906370\pi\)
0.957049 0.289925i \(-0.0936303\pi\)
\(384\) 0 0
\(385\) 2.50972 0.712953i 0.127907 0.0363354i
\(386\) 0 0
\(387\) 4.37382i 0.222334i
\(388\) 0 0
\(389\) 22.6863i 1.15024i 0.818069 + 0.575121i \(0.195045\pi\)
−0.818069 + 0.575121i \(0.804955\pi\)
\(390\) 0 0
\(391\) −17.5728 7.79176i −0.888693 0.394046i
\(392\) 0 0
\(393\) 8.68142i 0.437920i
\(394\) 0 0
\(395\) 2.43223 + 8.56189i 0.122379 + 0.430796i
\(396\) 0 0
\(397\) 9.68242i 0.485947i −0.970033 0.242973i \(-0.921877\pi\)
0.970033 0.242973i \(-0.0781228\pi\)
\(398\) 0 0
\(399\) 10.0413i 0.502693i
\(400\) 0 0
\(401\) 10.3976i 0.519232i 0.965712 + 0.259616i \(0.0835960\pi\)
−0.965712 + 0.259616i \(0.916404\pi\)
\(402\) 0 0
\(403\) −14.3934 −0.716987
\(404\) 0 0
\(405\) 1.24800 + 4.39319i 0.0620136 + 0.218299i
\(406\) 0 0
\(407\) −4.26030 −0.211175
\(408\) 0 0
\(409\) −15.1286 −0.748060 −0.374030 0.927417i \(-0.622024\pi\)
−0.374030 + 0.927417i \(0.622024\pi\)
\(410\) 0 0
\(411\) −12.7687 −0.629831
\(412\) 0 0
\(413\) −1.25941 −0.0619713
\(414\) 0 0
\(415\) −15.7738 + 4.48097i −0.774307 + 0.219962i
\(416\) 0 0
\(417\) 18.4916i 0.905538i
\(418\) 0 0
\(419\) 30.5455 1.49224 0.746122 0.665810i \(-0.231914\pi\)
0.746122 + 0.665810i \(0.231914\pi\)
\(420\) 0 0
\(421\) 32.6570i 1.59160i 0.605557 + 0.795802i \(0.292950\pi\)
−0.605557 + 0.795802i \(0.707050\pi\)
\(422\) 0 0
\(423\) −8.75970 −0.425911
\(424\) 0 0
\(425\) 17.0480 10.5361i 0.826951 0.511077i
\(426\) 0 0
\(427\) −10.8847 −0.526749
\(428\) 0 0
\(429\) 3.33520i 0.161025i
\(430\) 0 0
\(431\) −1.72165 −0.0829289 −0.0414645 0.999140i \(-0.513202\pi\)
−0.0414645 + 0.999140i \(0.513202\pi\)
\(432\) 0 0
\(433\) −11.1333 −0.535034 −0.267517 0.963553i \(-0.586203\pi\)
−0.267517 + 0.963553i \(0.586203\pi\)
\(434\) 0 0
\(435\) −2.49432 8.78045i −0.119593 0.420990i
\(436\) 0 0
\(437\) −31.4565 13.9478i −1.50477 0.667214i
\(438\) 0 0
\(439\) 32.9505i 1.57264i −0.617818 0.786321i \(-0.711983\pi\)
0.617818 0.786321i \(-0.288017\pi\)
\(440\) 0 0
\(441\) −10.1005 −0.480977
\(442\) 0 0
\(443\) 36.3281 1.72600 0.863001 0.505203i \(-0.168582\pi\)
0.863001 + 0.505203i \(0.168582\pi\)
\(444\) 0 0
\(445\) 2.34356 0.665750i 0.111096 0.0315596i
\(446\) 0 0
\(447\) 5.27464i 0.249482i
\(448\) 0 0
\(449\) −13.5895 −0.641327 −0.320664 0.947193i \(-0.603906\pi\)
−0.320664 + 0.947193i \(0.603906\pi\)
\(450\) 0 0
\(451\) 5.97830 0.281507
\(452\) 0 0
\(453\) 8.35036i 0.392334i
\(454\) 0 0
\(455\) 4.33362 + 15.2552i 0.203163 + 0.715172i
\(456\) 0 0
\(457\) 30.4328 1.42359 0.711794 0.702389i \(-0.247883\pi\)
0.711794 + 0.702389i \(0.247883\pi\)
\(458\) 0 0
\(459\) 19.0621 0.889743
\(460\) 0 0
\(461\) −21.1419 −0.984674 −0.492337 0.870405i \(-0.663857\pi\)
−0.492337 + 0.870405i \(0.663857\pi\)
\(462\) 0 0
\(463\) 29.4326 1.36785 0.683924 0.729553i \(-0.260272\pi\)
0.683924 + 0.729553i \(0.260272\pi\)
\(464\) 0 0
\(465\) −6.10914 + 1.73546i −0.283304 + 0.0804800i
\(466\) 0 0
\(467\) 41.4494i 1.91805i 0.283321 + 0.959025i \(0.408564\pi\)
−0.283321 + 0.959025i \(0.591436\pi\)
\(468\) 0 0
\(469\) 19.0879 0.881399
\(470\) 0 0
\(471\) 3.59368 0.165588
\(472\) 0 0
\(473\) 1.57052i 0.0722124i
\(474\) 0 0
\(475\) 30.5172 18.8604i 1.40022 0.865375i
\(476\) 0 0
\(477\) 22.0629 1.01019
\(478\) 0 0
\(479\) −26.2206 −1.19805 −0.599025 0.800730i \(-0.704445\pi\)
−0.599025 + 0.800730i \(0.704445\pi\)
\(480\) 0 0
\(481\) 25.8959i 1.18075i
\(482\) 0 0
\(483\) 2.72052 6.13559i 0.123788 0.279179i
\(484\) 0 0
\(485\) 1.67604 + 5.89999i 0.0761053 + 0.267905i
\(486\) 0 0
\(487\) 9.75788 0.442172 0.221086 0.975254i \(-0.429040\pi\)
0.221086 + 0.975254i \(0.429040\pi\)
\(488\) 0 0
\(489\) 22.3080 1.00880
\(490\) 0 0
\(491\) 7.43561i 0.335564i −0.985824 0.167782i \(-0.946339\pi\)
0.985824 0.167782i \(-0.0536605\pi\)
\(492\) 0 0
\(493\) 17.7042 0.797356
\(494\) 0 0
\(495\) −1.01032 3.55651i −0.0454104 0.159853i
\(496\) 0 0
\(497\) −3.38336 −0.151765
\(498\) 0 0
\(499\) 7.46322i 0.334100i −0.985948 0.167050i \(-0.946576\pi\)
0.985948 0.167050i \(-0.0534241\pi\)
\(500\) 0 0
\(501\) −6.02046 −0.268974
\(502\) 0 0
\(503\) 28.7105i 1.28014i −0.768318 0.640068i \(-0.778906\pi\)
0.768318 0.640068i \(-0.221094\pi\)
\(504\) 0 0
\(505\) −6.64301 23.3846i −0.295610 1.04060i
\(506\) 0 0
\(507\) −8.25832 −0.366765
\(508\) 0 0
\(509\) −22.9943 −1.01920 −0.509602 0.860410i \(-0.670208\pi\)
−0.509602 + 0.860410i \(0.670208\pi\)
\(510\) 0 0
\(511\) 16.9301 0.748945
\(512\) 0 0
\(513\) 34.1225 1.50655
\(514\) 0 0
\(515\) 36.0982 10.2546i 1.59068 0.451873i
\(516\) 0 0
\(517\) 3.14536 0.138333
\(518\) 0 0
\(519\) 15.6103i 0.685217i
\(520\) 0 0
\(521\) 34.5570i 1.51397i −0.653432 0.756985i \(-0.726671\pi\)
0.653432 0.756985i \(-0.273329\pi\)
\(522\) 0 0
\(523\) 9.88910i 0.432420i −0.976347 0.216210i \(-0.930630\pi\)
0.976347 0.216210i \(-0.0693696\pi\)
\(524\) 0 0
\(525\) 3.67872 + 5.95237i 0.160553 + 0.259783i
\(526\) 0 0
\(527\) 12.3180i 0.536579i
\(528\) 0 0
\(529\) 15.4421 + 17.0452i 0.671397 + 0.741097i
\(530\) 0 0
\(531\) 1.78469i 0.0774491i
\(532\) 0 0
\(533\) 36.3386i 1.57400i
\(534\) 0 0
\(535\) −24.3328 + 6.91238i −1.05200 + 0.298848i
\(536\) 0 0
\(537\) 8.78068i 0.378914i
\(538\) 0 0
\(539\) 3.62681 0.156218
\(540\) 0 0
\(541\) −15.9611 −0.686221 −0.343111 0.939295i \(-0.611481\pi\)
−0.343111 + 0.939295i \(0.611481\pi\)
\(542\) 0 0
\(543\) 3.60880i 0.154868i
\(544\) 0 0
\(545\) 4.19139 1.19067i 0.179540 0.0510029i
\(546\) 0 0
\(547\) −21.3204 −0.911596 −0.455798 0.890083i \(-0.650646\pi\)
−0.455798 + 0.890083i \(0.650646\pi\)
\(548\) 0 0
\(549\) 15.4246i 0.658308i
\(550\) 0 0
\(551\) 31.6917 1.35011
\(552\) 0 0
\(553\) 6.02761i 0.256320i
\(554\) 0 0
\(555\) −3.12235 10.9912i −0.132536 0.466552i
\(556\) 0 0
\(557\) 6.24545 0.264628 0.132314 0.991208i \(-0.457759\pi\)
0.132314 + 0.991208i \(0.457759\pi\)
\(558\) 0 0
\(559\) −9.54625 −0.403763
\(560\) 0 0
\(561\) −2.85428 −0.120508
\(562\) 0 0
\(563\) 17.4311i 0.734632i 0.930096 + 0.367316i \(0.119723\pi\)
−0.930096 + 0.367316i \(0.880277\pi\)
\(564\) 0 0
\(565\) −0.866138 3.04897i −0.0364387 0.128271i
\(566\) 0 0
\(567\) 3.09282i 0.129886i
\(568\) 0 0
\(569\) 23.7081i 0.993894i 0.867781 + 0.496947i \(0.165546\pi\)
−0.867781 + 0.496947i \(0.834454\pi\)
\(570\) 0 0
\(571\) −25.0536 −1.04846 −0.524230 0.851577i \(-0.675647\pi\)
−0.524230 + 0.851577i \(0.675647\pi\)
\(572\) 0 0
\(573\) 12.2183 0.510426
\(574\) 0 0
\(575\) −23.7570 + 3.25628i −0.990737 + 0.135796i
\(576\) 0 0
\(577\) 3.58387i 0.149198i −0.997214 0.0745992i \(-0.976232\pi\)
0.997214 0.0745992i \(-0.0237677\pi\)
\(578\) 0 0
\(579\) 9.40141i 0.390709i
\(580\) 0 0
\(581\) 11.1048 0.460706
\(582\) 0 0
\(583\) −7.92215 −0.328102
\(584\) 0 0
\(585\) 21.6179 6.14114i 0.893791 0.253905i
\(586\) 0 0
\(587\) −6.59996 −0.272410 −0.136205 0.990681i \(-0.543491\pi\)
−0.136205 + 0.990681i \(0.543491\pi\)
\(588\) 0 0
\(589\) 22.0500i 0.908555i
\(590\) 0 0
\(591\) 16.8950i 0.694967i
\(592\) 0 0
\(593\) 44.6733i 1.83451i 0.398295 + 0.917257i \(0.369602\pi\)
−0.398295 + 0.917257i \(0.630398\pi\)
\(594\) 0 0
\(595\) −13.0554 + 3.70874i −0.535221 + 0.152043i
\(596\) 0 0
\(597\) 16.5604 0.677771
\(598\) 0 0
\(599\) 5.33216i 0.217866i 0.994049 + 0.108933i \(0.0347434\pi\)
−0.994049 + 0.108933i \(0.965257\pi\)
\(600\) 0 0
\(601\) −15.8384 −0.646064 −0.323032 0.946388i \(-0.604702\pi\)
−0.323032 + 0.946388i \(0.604702\pi\)
\(602\) 0 0
\(603\) 27.0493i 1.10153i
\(604\) 0 0
\(605\) −6.35863 22.3835i −0.258515 0.910020i
\(606\) 0 0
\(607\) 20.3267 0.825037 0.412519 0.910949i \(-0.364649\pi\)
0.412519 + 0.910949i \(0.364649\pi\)
\(608\) 0 0
\(609\) 6.18147i 0.250486i
\(610\) 0 0
\(611\) 19.1188i 0.773464i
\(612\) 0 0
\(613\) 8.72981 0.352594 0.176297 0.984337i \(-0.443588\pi\)
0.176297 + 0.984337i \(0.443588\pi\)
\(614\) 0 0
\(615\) 4.38146 + 15.4235i 0.176677 + 0.621937i
\(616\) 0 0
\(617\) −23.7292 −0.955300 −0.477650 0.878550i \(-0.658511\pi\)
−0.477650 + 0.878550i \(0.658511\pi\)
\(618\) 0 0
\(619\) 22.6943 0.912162 0.456081 0.889938i \(-0.349253\pi\)
0.456081 + 0.889938i \(0.349253\pi\)
\(620\) 0 0
\(621\) −20.8501 9.24493i −0.836686 0.370986i
\(622\) 0 0
\(623\) −1.64988 −0.0661009
\(624\) 0 0
\(625\) 11.1806 22.3606i 0.447224 0.894422i
\(626\) 0 0
\(627\) −5.10937 −0.204049
\(628\) 0 0
\(629\) 22.1618 0.883650
\(630\) 0 0
\(631\) 23.7307 0.944706 0.472353 0.881409i \(-0.343405\pi\)
0.472353 + 0.881409i \(0.343405\pi\)
\(632\) 0 0
\(633\) 14.6227i 0.581199i
\(634\) 0 0
\(635\) 7.29806 + 25.6905i 0.289615 + 1.01950i
\(636\) 0 0
\(637\) 22.0453i 0.873465i
\(638\) 0 0
\(639\) 4.79453i 0.189669i
\(640\) 0 0
\(641\) 24.7220i 0.976460i 0.872715 + 0.488230i \(0.162357\pi\)
−0.872715 + 0.488230i \(0.837643\pi\)
\(642\) 0 0
\(643\) 37.7604i 1.48913i 0.667553 + 0.744563i \(0.267342\pi\)
−0.667553 + 0.744563i \(0.732658\pi\)
\(644\) 0 0
\(645\) −4.05181 + 1.15102i −0.159540 + 0.0453214i
\(646\) 0 0
\(647\) 13.8197 0.543308 0.271654 0.962395i \(-0.412429\pi\)
0.271654 + 0.962395i \(0.412429\pi\)
\(648\) 0 0
\(649\) 0.640832i 0.0251549i
\(650\) 0 0
\(651\) 4.30085 0.168564
\(652\) 0 0
\(653\) 10.1266i 0.396286i 0.980173 + 0.198143i \(0.0634910\pi\)
−0.980173 + 0.198143i \(0.936509\pi\)
\(654\) 0 0
\(655\) −20.2052 + 5.73982i −0.789483 + 0.224273i
\(656\) 0 0
\(657\) 23.9915i 0.935998i
\(658\) 0 0
\(659\) −4.08769 −0.159234 −0.0796168 0.996826i \(-0.525370\pi\)
−0.0796168 + 0.996826i \(0.525370\pi\)
\(660\) 0 0
\(661\) 4.20605i 0.163596i 0.996649 + 0.0817982i \(0.0260663\pi\)
−0.996649 + 0.0817982i \(0.973934\pi\)
\(662\) 0 0
\(663\) 17.3495i 0.673800i
\(664\) 0 0
\(665\) −23.3702 + 6.63890i −0.906256 + 0.257446i
\(666\) 0 0
\(667\) −19.3648 8.58635i −0.749808 0.332465i
\(668\) 0 0
\(669\) −16.1442 −0.624171
\(670\) 0 0
\(671\) 5.53855i 0.213813i
\(672\) 0 0
\(673\) 30.1518i 1.16227i −0.813808 0.581134i \(-0.802609\pi\)
0.813808 0.581134i \(-0.197391\pi\)
\(674\) 0 0
\(675\) 20.2275 12.5011i 0.778557 0.481169i
\(676\) 0 0
\(677\) −10.4970 −0.403434 −0.201717 0.979444i \(-0.564652\pi\)
−0.201717 + 0.979444i \(0.564652\pi\)
\(678\) 0 0
\(679\) 4.15361i 0.159401i
\(680\) 0 0
\(681\) 7.91146i 0.303168i
\(682\) 0 0
\(683\) 7.19811 0.275428 0.137714 0.990472i \(-0.456025\pi\)
0.137714 + 0.990472i \(0.456025\pi\)
\(684\) 0 0
\(685\) −8.44214 29.7179i −0.322558 1.13546i
\(686\) 0 0
\(687\) 7.36352i 0.280936i
\(688\) 0 0
\(689\) 48.1541i 1.83453i
\(690\) 0 0
\(691\) 44.4264i 1.69006i −0.534718 0.845030i \(-0.679582\pi\)
0.534718 0.845030i \(-0.320418\pi\)
\(692\) 0 0
\(693\) 2.50379i 0.0951113i
\(694\) 0 0
\(695\) 43.0376 12.2259i 1.63251 0.463756i
\(696\) 0 0
\(697\) −31.0988 −1.17795
\(698\) 0 0
\(699\) 6.66196i 0.251979i
\(700\) 0 0
\(701\) 51.4682i 1.94393i 0.235129 + 0.971964i \(0.424449\pi\)
−0.235129 + 0.971964i \(0.575551\pi\)
\(702\) 0 0
\(703\) 39.6713 1.49623
\(704\) 0 0
\(705\) 2.30521 + 8.11478i 0.0868194 + 0.305620i
\(706\) 0 0
\(707\) 16.4629i 0.619149i
\(708\) 0 0
\(709\) 0.106530i 0.00400082i −0.999998 0.00200041i \(-0.999363\pi\)
0.999998 0.00200041i \(-0.000636750\pi\)
\(710\) 0 0
\(711\) −8.54166 −0.320337
\(712\) 0 0
\(713\) −5.97409 + 13.4734i −0.223731 + 0.504582i
\(714\) 0 0
\(715\) −7.76238 + 2.20511i −0.290297 + 0.0824663i
\(716\) 0 0
\(717\) 9.02523i 0.337053i
\(718\) 0 0
\(719\) 12.5519i 0.468108i −0.972224 0.234054i \(-0.924801\pi\)
0.972224 0.234054i \(-0.0751993\pi\)
\(720\) 0 0
\(721\) −25.4133 −0.946440
\(722\) 0 0
\(723\) 24.7785i 0.921522i
\(724\) 0 0
\(725\) 18.7866 11.6106i 0.697715 0.431206i
\(726\) 0 0
\(727\) 41.4070i 1.53570i −0.640630 0.767850i \(-0.721327\pi\)
0.640630 0.767850i \(-0.278673\pi\)
\(728\) 0 0
\(729\) 8.80281 0.326030
\(730\) 0 0
\(731\) 8.16973i 0.302168i
\(732\) 0 0
\(733\) −8.31345 −0.307064 −0.153532 0.988144i \(-0.549065\pi\)
−0.153532 + 0.988144i \(0.549065\pi\)
\(734\) 0 0
\(735\) 2.65807 + 9.35688i 0.0980443 + 0.345134i
\(736\) 0 0
\(737\) 9.71264i 0.357770i
\(738\) 0 0
\(739\) 13.6173i 0.500921i 0.968127 + 0.250460i \(0.0805820\pi\)
−0.968127 + 0.250460i \(0.919418\pi\)
\(740\) 0 0
\(741\) 31.0569i 1.14090i
\(742\) 0 0
\(743\) 26.0822i 0.956864i 0.878125 + 0.478432i \(0.158795\pi\)
−0.878125 + 0.478432i \(0.841205\pi\)
\(744\) 0 0
\(745\) 12.2762 3.48739i 0.449767 0.127768i
\(746\) 0 0
\(747\) 15.7365i 0.575770i
\(748\) 0 0
\(749\) 17.1304 0.625932
\(750\) 0 0
\(751\) 7.61490 0.277872 0.138936 0.990301i \(-0.455632\pi\)
0.138936 + 0.990301i \(0.455632\pi\)
\(752\) 0 0
\(753\) −24.6156 −0.897043
\(754\) 0 0
\(755\) −19.4347 + 5.52093i −0.707301 + 0.200927i
\(756\) 0 0
\(757\) −28.9893 −1.05363 −0.526817 0.849979i \(-0.676615\pi\)
−0.526817 + 0.849979i \(0.676615\pi\)
\(758\) 0 0
\(759\) 3.12201 + 1.38430i 0.113322 + 0.0502469i
\(760\) 0 0
\(761\) 15.4769 0.561035 0.280518 0.959849i \(-0.409494\pi\)
0.280518 + 0.959849i \(0.409494\pi\)
\(762\) 0 0
\(763\) −2.95076 −0.106825
\(764\) 0 0
\(765\) 5.25562 + 18.5007i 0.190017 + 0.668896i
\(766\) 0 0
\(767\) 3.89524 0.140649
\(768\) 0 0
\(769\) 11.1918i 0.403586i −0.979428 0.201793i \(-0.935323\pi\)
0.979428 0.201793i \(-0.0646769\pi\)
\(770\) 0 0
\(771\) 0.0950817i 0.00342429i
\(772\) 0 0
\(773\) −55.2126 −1.98586 −0.992930 0.118698i \(-0.962128\pi\)
−0.992930 + 0.118698i \(0.962128\pi\)
\(774\) 0 0
\(775\) −8.07825 13.0710i −0.290179 0.469526i
\(776\) 0 0
\(777\) 7.73787i 0.277595i
\(778\) 0 0
\(779\) −55.6690 −1.99455
\(780\) 0 0
\(781\) 1.72158i 0.0616029i
\(782\) 0 0
\(783\) 21.0060 0.750694
\(784\) 0 0
\(785\) 2.37600 + 8.36397i 0.0848032 + 0.298523i
\(786\) 0 0
\(787\) 12.2547i 0.436833i 0.975856 + 0.218417i \(0.0700892\pi\)
−0.975856 + 0.218417i \(0.929911\pi\)
\(788\) 0 0
\(789\) 0.656672i 0.0233781i
\(790\) 0 0
\(791\) 2.14648i 0.0763201i
\(792\) 0 0
\(793\) 33.6656 1.19550
\(794\) 0 0
\(795\) −5.80609 20.4385i −0.205921 0.724880i
\(796\) 0 0
\(797\) −39.1677 −1.38739 −0.693695 0.720268i \(-0.744019\pi\)
−0.693695 + 0.720268i \(0.744019\pi\)
\(798\) 0 0
\(799\) −16.3620 −0.578845
\(800\) 0 0
\(801\) 2.33802i 0.0826100i
\(802\) 0 0
\(803\) 8.61467i 0.304005i
\(804\) 0 0
\(805\) 16.0787 + 2.27515i 0.566701 + 0.0801884i
\(806\) 0 0
\(807\) 7.04438 0.247974
\(808\) 0 0
\(809\) −31.3097 −1.10079 −0.550395 0.834904i \(-0.685523\pi\)
−0.550395 + 0.834904i \(0.685523\pi\)
\(810\) 0 0
\(811\) 8.77263i 0.308049i −0.988067 0.154024i \(-0.950777\pi\)
0.988067 0.154024i \(-0.0492234\pi\)
\(812\) 0 0
\(813\) 17.8470i 0.625921i
\(814\) 0 0
\(815\) 14.7492 + 51.9199i 0.516642 + 1.81868i
\(816\) 0 0
\(817\) 14.6244i 0.511643i
\(818\) 0 0
\(819\) −15.2191 −0.531798
\(820\) 0 0
\(821\) −33.0728 −1.15425 −0.577124 0.816657i \(-0.695825\pi\)
−0.577124 + 0.816657i \(0.695825\pi\)
\(822\) 0 0
\(823\) −26.5269 −0.924668 −0.462334 0.886706i \(-0.652988\pi\)
−0.462334 + 0.886706i \(0.652988\pi\)
\(824\) 0 0
\(825\) −3.02879 + 1.87187i −0.105449 + 0.0651701i
\(826\) 0 0
\(827\) 46.7494i 1.62564i 0.582517 + 0.812819i \(0.302068\pi\)
−0.582517 + 0.812819i \(0.697932\pi\)
\(828\) 0 0
\(829\) 22.9349 0.796564 0.398282 0.917263i \(-0.369607\pi\)
0.398282 + 0.917263i \(0.369607\pi\)
\(830\) 0 0
\(831\) 3.05385i 0.105937i
\(832\) 0 0
\(833\) −18.8665 −0.653684
\(834\) 0 0
\(835\) −3.98050 14.0121i −0.137751 0.484908i
\(836\) 0 0
\(837\) 14.6153i 0.505178i
\(838\) 0 0
\(839\) −46.2879 −1.59803 −0.799017 0.601308i \(-0.794646\pi\)
−0.799017 + 0.601308i \(0.794646\pi\)
\(840\) 0 0
\(841\) −9.49038 −0.327254
\(842\) 0 0
\(843\) 20.1557i 0.694200i
\(844\) 0 0
\(845\) −5.46008 19.2205i −0.187832 0.661205i
\(846\) 0 0
\(847\) 15.7581i 0.541454i
\(848\) 0 0
\(849\) 16.2025i 0.556068i
\(850\) 0 0
\(851\) −24.2406 10.7483i −0.830957 0.368446i
\(852\) 0 0
\(853\) 21.9817i 0.752640i −0.926490 0.376320i \(-0.877189\pi\)
0.926490 0.376320i \(-0.122811\pi\)
\(854\) 0 0
\(855\) 9.40793 + 33.1176i 0.321744 + 1.13260i
\(856\) 0 0
\(857\) 50.3790i 1.72091i 0.509523 + 0.860457i \(0.329822\pi\)
−0.509523 + 0.860457i \(0.670178\pi\)
\(858\) 0 0
\(859\) 38.4290i 1.31118i 0.755116 + 0.655591i \(0.227580\pi\)
−0.755116 + 0.655591i \(0.772420\pi\)
\(860\) 0 0
\(861\) 10.8582i 0.370047i
\(862\) 0 0
\(863\) −5.65548 −0.192515 −0.0962573 0.995356i \(-0.530687\pi\)
−0.0962573 + 0.995356i \(0.530687\pi\)
\(864\) 0 0
\(865\) −36.3316 + 10.3209i −1.23531 + 0.350923i
\(866\) 0 0
\(867\) −0.863361 −0.0293213
\(868\) 0 0
\(869\) 3.06707 0.104043
\(870\) 0 0
\(871\) −59.0375 −2.00041
\(872\) 0 0
\(873\) −5.88604 −0.199212
\(874\) 0 0
\(875\) −11.4214 + 12.4974i −0.386113 + 0.422488i
\(876\) 0 0
\(877\) 18.7077i 0.631712i 0.948807 + 0.315856i \(0.102292\pi\)
−0.948807 + 0.315856i \(0.897708\pi\)
\(878\) 0 0
\(879\) 22.1711 0.747813
\(880\) 0 0
\(881\) 16.1580i 0.544377i −0.962244 0.272188i \(-0.912253\pi\)
0.962244 0.272188i \(-0.0877474\pi\)
\(882\) 0 0
\(883\) −46.4724 −1.56392 −0.781961 0.623328i \(-0.785780\pi\)
−0.781961 + 0.623328i \(0.785780\pi\)
\(884\) 0 0
\(885\) 1.65330 0.469662i 0.0555750 0.0157875i
\(886\) 0 0
\(887\) −24.8088 −0.832998 −0.416499 0.909136i \(-0.636743\pi\)
−0.416499 + 0.909136i \(0.636743\pi\)
\(888\) 0 0
\(889\) 18.0862i 0.606592i
\(890\) 0 0
\(891\) 1.57374 0.0527223
\(892\) 0 0
\(893\) −29.2891 −0.980122
\(894\) 0 0
\(895\) 20.4362 5.80545i 0.683108 0.194055i
\(896\) 0 0
\(897\) −8.41435 + 18.9769i −0.280947 + 0.633620i
\(898\) 0 0
\(899\) 13.5741i 0.452722i
\(900\) 0 0
\(901\) 41.2106 1.37292
\(902\) 0 0
\(903\) 2.85249 0.0949248
\(904\) 0 0
\(905\) −8.39915 + 2.38600i −0.279197 + 0.0793132i
\(906\) 0 0
\(907\) 6.67649i 0.221689i −0.993838 0.110845i \(-0.964644\pi\)
0.993838 0.110845i \(-0.0353556\pi\)
\(908\) 0 0
\(909\) 23.3294 0.773786
\(910\) 0 0
\(911\) −26.7872 −0.887500 −0.443750 0.896151i \(-0.646352\pi\)
−0.443750 + 0.896151i \(0.646352\pi\)
\(912\) 0 0
\(913\) 5.65055i 0.187006i
\(914\) 0 0
\(915\) 14.2890 4.05917i 0.472381 0.134192i
\(916\) 0 0
\(917\) 14.2246 0.469736
\(918\) 0 0
\(919\) 40.3148 1.32986 0.664931 0.746905i \(-0.268461\pi\)
0.664931 + 0.746905i \(0.268461\pi\)
\(920\) 0 0
\(921\) −5.87190 −0.193486
\(922\) 0 0
\(923\) 10.4645 0.344442
\(924\) 0 0
\(925\) 23.5167 14.5340i 0.773226 0.477874i
\(926\) 0 0
\(927\) 36.0129i 1.18282i
\(928\) 0 0
\(929\) −26.8831 −0.882007 −0.441004 0.897505i \(-0.645377\pi\)
−0.441004 + 0.897505i \(0.645377\pi\)
\(930\) 0 0
\(931\) −33.7723 −1.10684
\(932\) 0 0
\(933\) 19.7455i 0.646439i
\(934\) 0 0
\(935\) −1.88714 6.64309i −0.0617161 0.217252i
\(936\) 0 0
\(937\) 23.0010 0.751410 0.375705 0.926739i \(-0.377401\pi\)
0.375705 + 0.926739i \(0.377401\pi\)
\(938\) 0 0
\(939\) 12.8492 0.419319
\(940\) 0 0
\(941\) 6.60800i 0.215415i −0.994183 0.107707i \(-0.965649\pi\)
0.994183 0.107707i \(-0.0343509\pi\)
\(942\) 0 0
\(943\) 34.0158 + 15.0826i 1.10771 + 0.491157i
\(944\) 0 0
\(945\) −15.4903 + 4.40042i −0.503899 + 0.143146i
\(946\) 0 0
\(947\) −11.7062 −0.380400 −0.190200 0.981745i \(-0.560914\pi\)
−0.190200 + 0.981745i \(0.560914\pi\)
\(948\) 0 0
\(949\) −52.3635 −1.69979
\(950\) 0 0
\(951\) 29.7574i 0.964951i
\(952\) 0 0
\(953\) 3.14979 0.102032 0.0510158 0.998698i \(-0.483754\pi\)
0.0510158 + 0.998698i \(0.483754\pi\)
\(954\) 0 0
\(955\) 8.07825 + 28.4369i 0.261406 + 0.920197i
\(956\) 0 0
\(957\) −3.14536 −0.101675
\(958\) 0 0
\(959\) 20.9215i 0.675591i
\(960\) 0 0
\(961\) 21.5556 0.695342
\(962\) 0 0
\(963\) 24.2753i 0.782262i
\(964\) 0 0
\(965\) −21.8809 + 6.21585i −0.704372 + 0.200095i
\(966\) 0 0
\(967\) 22.5256 0.724374 0.362187 0.932105i \(-0.382030\pi\)
0.362187 + 0.932105i \(0.382030\pi\)
\(968\) 0 0
\(969\) 26.5786 0.853829
\(970\) 0 0
\(971\) 48.9870 1.57207 0.786033 0.618184i \(-0.212131\pi\)
0.786033 + 0.618184i \(0.212131\pi\)
\(972\) 0 0
\(973\) −30.2986 −0.971329
\(974\) 0 0
\(975\) −11.3780 18.4102i −0.364388 0.589599i
\(976\) 0 0
\(977\) −40.8557 −1.30709 −0.653544 0.756888i \(-0.726719\pi\)
−0.653544 + 0.756888i \(0.726719\pi\)
\(978\) 0 0
\(979\) 0.839518i 0.0268311i
\(980\) 0 0
\(981\) 4.18149i 0.133505i
\(982\) 0 0
\(983\) 5.78502i 0.184513i −0.995735 0.0922567i \(-0.970592\pi\)
0.995735 0.0922567i \(-0.0294080\pi\)
\(984\) 0 0
\(985\) −39.3215 + 11.1703i −1.25289 + 0.355916i
\(986\) 0 0
\(987\) 5.71283i 0.181842i
\(988\) 0 0
\(989\) −3.96224 + 8.93604i −0.125992 + 0.284149i
\(990\) 0 0
\(991\) 47.0084i 1.49327i 0.665233 + 0.746636i \(0.268332\pi\)
−0.665233 + 0.746636i \(0.731668\pi\)
\(992\) 0 0
\(993\) 26.9521i 0.855298i
\(994\) 0 0
\(995\) 10.9491 + 38.5428i 0.347109 + 1.22189i
\(996\) 0 0
\(997\) 4.74045i 0.150132i −0.997179 0.0750658i \(-0.976083\pi\)
0.997179 0.0750658i \(-0.0239167\pi\)
\(998\) 0 0
\(999\) 26.2950 0.831938
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 1840.2.m.g.1839.21 yes 40
4.3 odd 2 inner 1840.2.m.g.1839.20 yes 40
5.4 even 2 inner 1840.2.m.g.1839.19 yes 40
20.19 odd 2 inner 1840.2.m.g.1839.22 yes 40
23.22 odd 2 inner 1840.2.m.g.1839.23 yes 40
92.91 even 2 inner 1840.2.m.g.1839.18 yes 40
115.114 odd 2 inner 1840.2.m.g.1839.17 40
460.459 even 2 inner 1840.2.m.g.1839.24 yes 40
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1840.2.m.g.1839.17 40 115.114 odd 2 inner
1840.2.m.g.1839.18 yes 40 92.91 even 2 inner
1840.2.m.g.1839.19 yes 40 5.4 even 2 inner
1840.2.m.g.1839.20 yes 40 4.3 odd 2 inner
1840.2.m.g.1839.21 yes 40 1.1 even 1 trivial
1840.2.m.g.1839.22 yes 40 20.19 odd 2 inner
1840.2.m.g.1839.23 yes 40 23.22 odd 2 inner
1840.2.m.g.1839.24 yes 40 460.459 even 2 inner