Properties

Label 640.2.s.d.287.3
Level $640$
Weight $2$
Character 640.287
Analytic conductor $5.110$
Analytic rank $0$
Dimension $18$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [640,2,Mod(223,640)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(640, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("640.223");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 640 = 2^{7} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 640.s (of order \(4\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.11042572936\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(9\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} + 2 x^{16} - 4 x^{15} - 5 x^{14} - 14 x^{13} - 10 x^{12} + 6 x^{11} + 37 x^{10} + 70 x^{9} + 74 x^{8} + 24 x^{7} - 80 x^{6} - 224 x^{5} - 160 x^{4} - 256 x^{3} + 256 x^{2} + 512 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{13} \)
Twist minimal: no (minimal twist has level 80)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 287.3
Root \(1.41323 - 0.0526497i\) of defining polynomial
Character \(\chi\) \(=\) 640.287
Dual form 640.2.s.d.223.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.28110 q^{3} +(-2.07160 + 0.841703i) q^{5} +(-1.13975 - 1.13975i) q^{7} -1.35879 q^{9} +O(q^{10})\) \(q-1.28110 q^{3} +(-2.07160 + 0.841703i) q^{5} +(-1.13975 - 1.13975i) q^{7} -1.35879 q^{9} +(2.32204 - 2.32204i) q^{11} +1.36502i q^{13} +(2.65392 - 1.07830i) q^{15} +(5.25380 + 5.25380i) q^{17} +(3.69752 - 3.69752i) q^{19} +(1.46013 + 1.46013i) q^{21} +(-0.911118 + 0.911118i) q^{23} +(3.58307 - 3.48735i) q^{25} +5.58403 q^{27} +(2.37343 + 2.37343i) q^{29} +0.242577i q^{31} +(-2.97475 + 2.97475i) q^{33} +(3.32044 + 1.40178i) q^{35} +3.34494i q^{37} -1.74872i q^{39} +2.66956i q^{41} +9.04874i q^{43} +(2.81488 - 1.14370i) q^{45} +(7.87820 - 7.87820i) q^{47} -4.40194i q^{49} +(-6.73063 - 6.73063i) q^{51} +5.80113 q^{53} +(-2.85587 + 6.76480i) q^{55} +(-4.73688 + 4.73688i) q^{57} +(-5.91474 - 5.91474i) q^{59} +(6.67404 - 6.67404i) q^{61} +(1.54868 + 1.54868i) q^{63} +(-1.14894 - 2.82778i) q^{65} +4.54673i q^{67} +(1.16723 - 1.16723i) q^{69} +15.4389 q^{71} +(-1.49307 - 1.49307i) q^{73} +(-4.59026 + 4.46763i) q^{75} -5.29308 q^{77} -10.3024 q^{79} -3.07731 q^{81} -3.26589 q^{83} +(-15.3059 - 6.46165i) q^{85} +(-3.04060 - 3.04060i) q^{87} +9.77206 q^{89} +(1.55578 - 1.55578i) q^{91} -0.310765i q^{93} +(-4.54758 + 10.7720i) q^{95} +(-1.63587 - 1.63587i) q^{97} +(-3.15516 + 3.15516i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 2 q^{5} + 2 q^{7} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 2 q^{5} + 2 q^{7} + 10 q^{9} + 2 q^{11} - 20 q^{15} - 6 q^{17} + 2 q^{19} + 16 q^{21} - 2 q^{23} - 6 q^{25} + 24 q^{27} - 14 q^{29} - 8 q^{33} - 2 q^{35} + 14 q^{45} + 38 q^{47} - 8 q^{51} - 12 q^{53} - 6 q^{55} - 24 q^{57} - 10 q^{59} - 14 q^{61} - 6 q^{63} + 32 q^{69} + 24 q^{71} - 14 q^{73} - 16 q^{75} + 44 q^{77} - 16 q^{79} + 2 q^{81} - 40 q^{83} - 14 q^{85} + 24 q^{87} + 12 q^{89} + 34 q^{95} + 18 q^{97} - 22 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/640\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(261\) \(511\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(e\left(\frac{1}{4}\right)\) \(-1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.28110 −0.739642 −0.369821 0.929103i \(-0.620581\pi\)
−0.369821 + 0.929103i \(0.620581\pi\)
\(4\) 0 0
\(5\) −2.07160 + 0.841703i −0.926449 + 0.376421i
\(6\) 0 0
\(7\) −1.13975 1.13975i −0.430785 0.430785i 0.458111 0.888895i \(-0.348526\pi\)
−0.888895 + 0.458111i \(0.848526\pi\)
\(8\) 0 0
\(9\) −1.35879 −0.452930
\(10\) 0 0
\(11\) 2.32204 2.32204i 0.700120 0.700120i −0.264316 0.964436i \(-0.585146\pi\)
0.964436 + 0.264316i \(0.0851462\pi\)
\(12\) 0 0
\(13\) 1.36502i 0.378589i 0.981920 + 0.189294i \(0.0606201\pi\)
−0.981920 + 0.189294i \(0.939380\pi\)
\(14\) 0 0
\(15\) 2.65392 1.07830i 0.685240 0.278416i
\(16\) 0 0
\(17\) 5.25380 + 5.25380i 1.27423 + 1.27423i 0.943845 + 0.330389i \(0.107180\pi\)
0.330389 + 0.943845i \(0.392820\pi\)
\(18\) 0 0
\(19\) 3.69752 3.69752i 0.848269 0.848269i −0.141648 0.989917i \(-0.545240\pi\)
0.989917 + 0.141648i \(0.0452403\pi\)
\(20\) 0 0
\(21\) 1.46013 + 1.46013i 0.318626 + 0.318626i
\(22\) 0 0
\(23\) −0.911118 + 0.911118i −0.189981 + 0.189981i −0.795688 0.605707i \(-0.792890\pi\)
0.605707 + 0.795688i \(0.292890\pi\)
\(24\) 0 0
\(25\) 3.58307 3.48735i 0.716615 0.697469i
\(26\) 0 0
\(27\) 5.58403 1.07465
\(28\) 0 0
\(29\) 2.37343 + 2.37343i 0.440736 + 0.440736i 0.892259 0.451524i \(-0.149119\pi\)
−0.451524 + 0.892259i \(0.649119\pi\)
\(30\) 0 0
\(31\) 0.242577i 0.0435681i 0.999763 + 0.0217841i \(0.00693463\pi\)
−0.999763 + 0.0217841i \(0.993065\pi\)
\(32\) 0 0
\(33\) −2.97475 + 2.97475i −0.517838 + 0.517838i
\(34\) 0 0
\(35\) 3.32044 + 1.40178i 0.561256 + 0.236944i
\(36\) 0 0
\(37\) 3.34494i 0.549905i 0.961458 + 0.274953i \(0.0886621\pi\)
−0.961458 + 0.274953i \(0.911338\pi\)
\(38\) 0 0
\(39\) 1.74872i 0.280020i
\(40\) 0 0
\(41\) 2.66956i 0.416915i 0.978031 + 0.208457i \(0.0668442\pi\)
−0.978031 + 0.208457i \(0.933156\pi\)
\(42\) 0 0
\(43\) 9.04874i 1.37992i 0.723847 + 0.689960i \(0.242372\pi\)
−0.723847 + 0.689960i \(0.757628\pi\)
\(44\) 0 0
\(45\) 2.81488 1.14370i 0.419617 0.170492i
\(46\) 0 0
\(47\) 7.87820 7.87820i 1.14915 1.14915i 0.162435 0.986719i \(-0.448065\pi\)
0.986719 0.162435i \(-0.0519348\pi\)
\(48\) 0 0
\(49\) 4.40194i 0.628849i
\(50\) 0 0
\(51\) −6.73063 6.73063i −0.942476 0.942476i
\(52\) 0 0
\(53\) 5.80113 0.796846 0.398423 0.917202i \(-0.369558\pi\)
0.398423 + 0.917202i \(0.369558\pi\)
\(54\) 0 0
\(55\) −2.85587 + 6.76480i −0.385086 + 0.912165i
\(56\) 0 0
\(57\) −4.73688 + 4.73688i −0.627415 + 0.627415i
\(58\) 0 0
\(59\) −5.91474 5.91474i −0.770033 0.770033i 0.208079 0.978112i \(-0.433279\pi\)
−0.978112 + 0.208079i \(0.933279\pi\)
\(60\) 0 0
\(61\) 6.67404 6.67404i 0.854523 0.854523i −0.136163 0.990686i \(-0.543477\pi\)
0.990686 + 0.136163i \(0.0434772\pi\)
\(62\) 0 0
\(63\) 1.54868 + 1.54868i 0.195116 + 0.195116i
\(64\) 0 0
\(65\) −1.14894 2.82778i −0.142509 0.350743i
\(66\) 0 0
\(67\) 4.54673i 0.555471i 0.960658 + 0.277736i \(0.0895839\pi\)
−0.960658 + 0.277736i \(0.910416\pi\)
\(68\) 0 0
\(69\) 1.16723 1.16723i 0.140518 0.140518i
\(70\) 0 0
\(71\) 15.4389 1.83226 0.916128 0.400885i \(-0.131297\pi\)
0.916128 + 0.400885i \(0.131297\pi\)
\(72\) 0 0
\(73\) −1.49307 1.49307i −0.174750 0.174750i 0.614313 0.789063i \(-0.289433\pi\)
−0.789063 + 0.614313i \(0.789433\pi\)
\(74\) 0 0
\(75\) −4.59026 + 4.46763i −0.530038 + 0.515877i
\(76\) 0 0
\(77\) −5.29308 −0.603202
\(78\) 0 0
\(79\) −10.3024 −1.15911 −0.579556 0.814932i \(-0.696774\pi\)
−0.579556 + 0.814932i \(0.696774\pi\)
\(80\) 0 0
\(81\) −3.07731 −0.341924
\(82\) 0 0
\(83\) −3.26589 −0.358478 −0.179239 0.983806i \(-0.557364\pi\)
−0.179239 + 0.983806i \(0.557364\pi\)
\(84\) 0 0
\(85\) −15.3059 6.46165i −1.66016 0.700864i
\(86\) 0 0
\(87\) −3.04060 3.04060i −0.325986 0.325986i
\(88\) 0 0
\(89\) 9.77206 1.03584 0.517918 0.855430i \(-0.326707\pi\)
0.517918 + 0.855430i \(0.326707\pi\)
\(90\) 0 0
\(91\) 1.55578 1.55578i 0.163090 0.163090i
\(92\) 0 0
\(93\) 0.310765i 0.0322248i
\(94\) 0 0
\(95\) −4.54758 + 10.7720i −0.466571 + 1.10518i
\(96\) 0 0
\(97\) −1.63587 1.63587i −0.166097 0.166097i 0.619164 0.785262i \(-0.287472\pi\)
−0.785262 + 0.619164i \(0.787472\pi\)
\(98\) 0 0
\(99\) −3.15516 + 3.15516i −0.317106 + 0.317106i
\(100\) 0 0
\(101\) 6.63953 + 6.63953i 0.660658 + 0.660658i 0.955535 0.294877i \(-0.0952787\pi\)
−0.294877 + 0.955535i \(0.595279\pi\)
\(102\) 0 0
\(103\) 1.62219 1.62219i 0.159839 0.159839i −0.622656 0.782496i \(-0.713946\pi\)
0.782496 + 0.622656i \(0.213946\pi\)
\(104\) 0 0
\(105\) −4.25380 1.79581i −0.415129 0.175253i
\(106\) 0 0
\(107\) −3.65206 −0.353058 −0.176529 0.984295i \(-0.556487\pi\)
−0.176529 + 0.984295i \(0.556487\pi\)
\(108\) 0 0
\(109\) −5.20757 5.20757i −0.498795 0.498795i 0.412268 0.911063i \(-0.364737\pi\)
−0.911063 + 0.412268i \(0.864737\pi\)
\(110\) 0 0
\(111\) 4.28519i 0.406733i
\(112\) 0 0
\(113\) −4.27905 + 4.27905i −0.402539 + 0.402539i −0.879127 0.476588i \(-0.841873\pi\)
0.476588 + 0.879127i \(0.341873\pi\)
\(114\) 0 0
\(115\) 1.12058 2.65437i 0.104495 0.247521i
\(116\) 0 0
\(117\) 1.85478i 0.171474i
\(118\) 0 0
\(119\) 11.9760i 1.09784i
\(120\) 0 0
\(121\) 0.216302i 0.0196639i
\(122\) 0 0
\(123\) 3.41996i 0.308367i
\(124\) 0 0
\(125\) −4.48739 + 10.2403i −0.401365 + 0.915918i
\(126\) 0 0
\(127\) −7.29257 + 7.29257i −0.647111 + 0.647111i −0.952294 0.305183i \(-0.901282\pi\)
0.305183 + 0.952294i \(0.401282\pi\)
\(128\) 0 0
\(129\) 11.5923i 1.02065i
\(130\) 0 0
\(131\) 11.9793 + 11.9793i 1.04664 + 1.04664i 0.998858 + 0.0477778i \(0.0152139\pi\)
0.0477778 + 0.998858i \(0.484786\pi\)
\(132\) 0 0
\(133\) −8.42848 −0.730842
\(134\) 0 0
\(135\) −11.5679 + 4.70010i −0.995606 + 0.404520i
\(136\) 0 0
\(137\) −4.92762 + 4.92762i −0.420995 + 0.420995i −0.885546 0.464551i \(-0.846216\pi\)
0.464551 + 0.885546i \(0.346216\pi\)
\(138\) 0 0
\(139\) −10.3015 10.3015i −0.873761 0.873761i 0.119119 0.992880i \(-0.461993\pi\)
−0.992880 + 0.119119i \(0.961993\pi\)
\(140\) 0 0
\(141\) −10.0927 + 10.0927i −0.849962 + 0.849962i
\(142\) 0 0
\(143\) 3.16963 + 3.16963i 0.265058 + 0.265058i
\(144\) 0 0
\(145\) −6.91454 2.91909i −0.574221 0.242417i
\(146\) 0 0
\(147\) 5.63931i 0.465123i
\(148\) 0 0
\(149\) 15.2040 15.2040i 1.24556 1.24556i 0.287896 0.957662i \(-0.407044\pi\)
0.957662 0.287896i \(-0.0929557\pi\)
\(150\) 0 0
\(151\) −10.7055 −0.871204 −0.435602 0.900139i \(-0.643464\pi\)
−0.435602 + 0.900139i \(0.643464\pi\)
\(152\) 0 0
\(153\) −7.13882 7.13882i −0.577139 0.577139i
\(154\) 0 0
\(155\) −0.204178 0.502523i −0.0164000 0.0403636i
\(156\) 0 0
\(157\) −2.34588 −0.187222 −0.0936108 0.995609i \(-0.529841\pi\)
−0.0936108 + 0.995609i \(0.529841\pi\)
\(158\) 0 0
\(159\) −7.43180 −0.589380
\(160\) 0 0
\(161\) 2.07689 0.163682
\(162\) 0 0
\(163\) 2.73625 0.214319 0.107160 0.994242i \(-0.465824\pi\)
0.107160 + 0.994242i \(0.465824\pi\)
\(164\) 0 0
\(165\) 3.65865 8.66636i 0.284825 0.674675i
\(166\) 0 0
\(167\) 10.1328 + 10.1328i 0.784097 + 0.784097i 0.980519 0.196423i \(-0.0629325\pi\)
−0.196423 + 0.980519i \(0.562932\pi\)
\(168\) 0 0
\(169\) 11.1367 0.856670
\(170\) 0 0
\(171\) −5.02415 + 5.02415i −0.384207 + 0.384207i
\(172\) 0 0
\(173\) 8.79590i 0.668740i −0.942442 0.334370i \(-0.891477\pi\)
0.942442 0.334370i \(-0.108523\pi\)
\(174\) 0 0
\(175\) −8.05851 0.109105i −0.609166 0.00824753i
\(176\) 0 0
\(177\) 7.57735 + 7.57735i 0.569549 + 0.569549i
\(178\) 0 0
\(179\) −6.62071 + 6.62071i −0.494855 + 0.494855i −0.909832 0.414977i \(-0.863790\pi\)
0.414977 + 0.909832i \(0.363790\pi\)
\(180\) 0 0
\(181\) 5.84339 + 5.84339i 0.434336 + 0.434336i 0.890100 0.455765i \(-0.150634\pi\)
−0.455765 + 0.890100i \(0.650634\pi\)
\(182\) 0 0
\(183\) −8.55009 + 8.55009i −0.632041 + 0.632041i
\(184\) 0 0
\(185\) −2.81545 6.92939i −0.206996 0.509459i
\(186\) 0 0
\(187\) 24.3990 1.78423
\(188\) 0 0
\(189\) −6.36440 6.36440i −0.462942 0.462942i
\(190\) 0 0
\(191\) 1.83906i 0.133070i 0.997784 + 0.0665349i \(0.0211944\pi\)
−0.997784 + 0.0665349i \(0.978806\pi\)
\(192\) 0 0
\(193\) 6.18343 6.18343i 0.445093 0.445093i −0.448626 0.893719i \(-0.648087\pi\)
0.893719 + 0.448626i \(0.148087\pi\)
\(194\) 0 0
\(195\) 1.47191 + 3.62266i 0.105405 + 0.259424i
\(196\) 0 0
\(197\) 5.55669i 0.395898i −0.980212 0.197949i \(-0.936572\pi\)
0.980212 0.197949i \(-0.0634280\pi\)
\(198\) 0 0
\(199\) 6.96413i 0.493674i 0.969057 + 0.246837i \(0.0793912\pi\)
−0.969057 + 0.246837i \(0.920609\pi\)
\(200\) 0 0
\(201\) 5.82480i 0.410850i
\(202\) 0 0
\(203\) 5.41024i 0.379724i
\(204\) 0 0
\(205\) −2.24697 5.53026i −0.156935 0.386250i
\(206\) 0 0
\(207\) 1.23802 1.23802i 0.0860483 0.0860483i
\(208\) 0 0
\(209\) 17.1715i 1.18778i
\(210\) 0 0
\(211\) −5.43389 5.43389i −0.374084 0.374084i 0.494878 0.868962i \(-0.335213\pi\)
−0.868962 + 0.494878i \(0.835213\pi\)
\(212\) 0 0
\(213\) −19.7787 −1.35521
\(214\) 0 0
\(215\) −7.61635 18.7454i −0.519431 1.27843i
\(216\) 0 0
\(217\) 0.276477 0.276477i 0.0187685 0.0187685i
\(218\) 0 0
\(219\) 1.91276 + 1.91276i 0.129253 + 0.129253i
\(220\) 0 0
\(221\) −7.17155 + 7.17155i −0.482411 + 0.482411i
\(222\) 0 0
\(223\) 8.61776 + 8.61776i 0.577088 + 0.577088i 0.934100 0.357012i \(-0.116204\pi\)
−0.357012 + 0.934100i \(0.616204\pi\)
\(224\) 0 0
\(225\) −4.86865 + 4.73858i −0.324577 + 0.315905i
\(226\) 0 0
\(227\) 6.01977i 0.399546i −0.979842 0.199773i \(-0.935980\pi\)
0.979842 0.199773i \(-0.0640205\pi\)
\(228\) 0 0
\(229\) 0.568504 0.568504i 0.0375678 0.0375678i −0.688073 0.725641i \(-0.741543\pi\)
0.725641 + 0.688073i \(0.241543\pi\)
\(230\) 0 0
\(231\) 6.78094 0.446153
\(232\) 0 0
\(233\) 12.6979 + 12.6979i 0.831869 + 0.831869i 0.987772 0.155904i \(-0.0498289\pi\)
−0.155904 + 0.987772i \(0.549829\pi\)
\(234\) 0 0
\(235\) −9.68940 + 22.9516i −0.632067 + 1.49720i
\(236\) 0 0
\(237\) 13.1984 0.857327
\(238\) 0 0
\(239\) 1.78306 0.115336 0.0576682 0.998336i \(-0.481633\pi\)
0.0576682 + 0.998336i \(0.481633\pi\)
\(240\) 0 0
\(241\) 10.4440 0.672754 0.336377 0.941727i \(-0.390798\pi\)
0.336377 + 0.941727i \(0.390798\pi\)
\(242\) 0 0
\(243\) −12.8098 −0.821747
\(244\) 0 0
\(245\) 3.70513 + 9.11908i 0.236712 + 0.582596i
\(246\) 0 0
\(247\) 5.04719 + 5.04719i 0.321145 + 0.321145i
\(248\) 0 0
\(249\) 4.18392 0.265145
\(250\) 0 0
\(251\) 12.6497 12.6497i 0.798445 0.798445i −0.184406 0.982850i \(-0.559036\pi\)
0.982850 + 0.184406i \(0.0590360\pi\)
\(252\) 0 0
\(253\) 4.23130i 0.266019i
\(254\) 0 0
\(255\) 19.6084 + 8.27800i 1.22792 + 0.518388i
\(256\) 0 0
\(257\) −4.13062 4.13062i −0.257661 0.257661i 0.566441 0.824102i \(-0.308320\pi\)
−0.824102 + 0.566441i \(0.808320\pi\)
\(258\) 0 0
\(259\) 3.81240 3.81240i 0.236891 0.236891i
\(260\) 0 0
\(261\) −3.22500 3.22500i −0.199623 0.199623i
\(262\) 0 0
\(263\) 17.1303 17.1303i 1.05630 1.05630i 0.0579798 0.998318i \(-0.481534\pi\)
0.998318 0.0579798i \(-0.0184659\pi\)
\(264\) 0 0
\(265\) −12.0176 + 4.88282i −0.738237 + 0.299949i
\(266\) 0 0
\(267\) −12.5190 −0.766147
\(268\) 0 0
\(269\) 19.8075 + 19.8075i 1.20768 + 1.20768i 0.971775 + 0.235910i \(0.0758070\pi\)
0.235910 + 0.971775i \(0.424193\pi\)
\(270\) 0 0
\(271\) 27.9542i 1.69810i −0.528316 0.849048i \(-0.677176\pi\)
0.528316 0.849048i \(-0.322824\pi\)
\(272\) 0 0
\(273\) −1.99311 + 1.99311i −0.120628 + 0.120628i
\(274\) 0 0
\(275\) 0.222281 16.4178i 0.0134041 0.990029i
\(276\) 0 0
\(277\) 26.0257i 1.56373i 0.623447 + 0.781866i \(0.285732\pi\)
−0.623447 + 0.781866i \(0.714268\pi\)
\(278\) 0 0
\(279\) 0.329612i 0.0197333i
\(280\) 0 0
\(281\) 24.1001i 1.43769i 0.695170 + 0.718846i \(0.255329\pi\)
−0.695170 + 0.718846i \(0.744671\pi\)
\(282\) 0 0
\(283\) 4.73708i 0.281590i 0.990039 + 0.140795i \(0.0449658\pi\)
−0.990039 + 0.140795i \(0.955034\pi\)
\(284\) 0 0
\(285\) 5.82588 13.8000i 0.345096 0.817439i
\(286\) 0 0
\(287\) 3.04262 3.04262i 0.179600 0.179600i
\(288\) 0 0
\(289\) 38.2049i 2.24734i
\(290\) 0 0
\(291\) 2.09571 + 2.09571i 0.122852 + 0.122852i
\(292\) 0 0
\(293\) 3.11001 0.181689 0.0908445 0.995865i \(-0.471043\pi\)
0.0908445 + 0.995865i \(0.471043\pi\)
\(294\) 0 0
\(295\) 17.2314 + 7.27454i 1.00325 + 0.423540i
\(296\) 0 0
\(297\) 12.9663 12.9663i 0.752382 0.752382i
\(298\) 0 0
\(299\) −1.24370 1.24370i −0.0719248 0.0719248i
\(300\) 0 0
\(301\) 10.3133 10.3133i 0.594449 0.594449i
\(302\) 0 0
\(303\) −8.50588 8.50588i −0.488650 0.488650i
\(304\) 0 0
\(305\) −8.20840 + 19.4435i −0.470012 + 1.11333i
\(306\) 0 0
\(307\) 14.5670i 0.831382i −0.909506 0.415691i \(-0.863540\pi\)
0.909506 0.415691i \(-0.136460\pi\)
\(308\) 0 0
\(309\) −2.07819 + 2.07819i −0.118224 + 0.118224i
\(310\) 0 0
\(311\) −14.4572 −0.819791 −0.409896 0.912132i \(-0.634435\pi\)
−0.409896 + 0.912132i \(0.634435\pi\)
\(312\) 0 0
\(313\) −10.1273 10.1273i −0.572429 0.572429i 0.360377 0.932807i \(-0.382648\pi\)
−0.932807 + 0.360377i \(0.882648\pi\)
\(314\) 0 0
\(315\) −4.51178 1.90472i −0.254210 0.107319i
\(316\) 0 0
\(317\) 13.8750 0.779295 0.389648 0.920964i \(-0.372597\pi\)
0.389648 + 0.920964i \(0.372597\pi\)
\(318\) 0 0
\(319\) 11.0224 0.617136
\(320\) 0 0
\(321\) 4.67864 0.261136
\(322\) 0 0
\(323\) 38.8520 2.16179
\(324\) 0 0
\(325\) 4.76030 + 4.89097i 0.264054 + 0.271302i
\(326\) 0 0
\(327\) 6.67140 + 6.67140i 0.368930 + 0.368930i
\(328\) 0 0
\(329\) −17.9584 −0.990076
\(330\) 0 0
\(331\) −1.69458 + 1.69458i −0.0931425 + 0.0931425i −0.752143 0.659000i \(-0.770980\pi\)
0.659000 + 0.752143i \(0.270980\pi\)
\(332\) 0 0
\(333\) 4.54508i 0.249069i
\(334\) 0 0
\(335\) −3.82699 9.41902i −0.209091 0.514616i
\(336\) 0 0
\(337\) −9.53338 9.53338i −0.519316 0.519316i 0.398048 0.917364i \(-0.369688\pi\)
−0.917364 + 0.398048i \(0.869688\pi\)
\(338\) 0 0
\(339\) 5.48188 5.48188i 0.297735 0.297735i
\(340\) 0 0
\(341\) 0.563273 + 0.563273i 0.0305029 + 0.0305029i
\(342\) 0 0
\(343\) −12.9954 + 12.9954i −0.701683 + 0.701683i
\(344\) 0 0
\(345\) −1.43558 + 3.40050i −0.0772888 + 0.183077i
\(346\) 0 0
\(347\) 6.67273 0.358211 0.179105 0.983830i \(-0.442680\pi\)
0.179105 + 0.983830i \(0.442680\pi\)
\(348\) 0 0
\(349\) −2.02618 2.02618i −0.108459 0.108459i 0.650795 0.759254i \(-0.274436\pi\)
−0.759254 + 0.650795i \(0.774436\pi\)
\(350\) 0 0
\(351\) 7.62233i 0.406850i
\(352\) 0 0
\(353\) −5.36542 + 5.36542i −0.285572 + 0.285572i −0.835327 0.549754i \(-0.814721\pi\)
0.549754 + 0.835327i \(0.314721\pi\)
\(354\) 0 0
\(355\) −31.9832 + 12.9949i −1.69749 + 0.689700i
\(356\) 0 0
\(357\) 15.3425i 0.812009i
\(358\) 0 0
\(359\) 7.76117i 0.409619i −0.978802 0.204809i \(-0.934343\pi\)
0.978802 0.204809i \(-0.0656574\pi\)
\(360\) 0 0
\(361\) 8.34326i 0.439119i
\(362\) 0 0
\(363\) 0.277104i 0.0145442i
\(364\) 0 0
\(365\) 4.34976 + 1.83632i 0.227677 + 0.0961175i
\(366\) 0 0
\(367\) 18.0536 18.0536i 0.942389 0.942389i −0.0560392 0.998429i \(-0.517847\pi\)
0.998429 + 0.0560392i \(0.0178472\pi\)
\(368\) 0 0
\(369\) 3.62737i 0.188833i
\(370\) 0 0
\(371\) −6.61183 6.61183i −0.343269 0.343269i
\(372\) 0 0
\(373\) 4.36197 0.225854 0.112927 0.993603i \(-0.463977\pi\)
0.112927 + 0.993603i \(0.463977\pi\)
\(374\) 0 0
\(375\) 5.74879 13.1188i 0.296866 0.677451i
\(376\) 0 0
\(377\) −3.23979 + 3.23979i −0.166858 + 0.166858i
\(378\) 0 0
\(379\) 5.93072 + 5.93072i 0.304641 + 0.304641i 0.842826 0.538186i \(-0.180890\pi\)
−0.538186 + 0.842826i \(0.680890\pi\)
\(380\) 0 0
\(381\) 9.34249 9.34249i 0.478630 0.478630i
\(382\) 0 0
\(383\) −19.3340 19.3340i −0.987922 0.987922i 0.0120057 0.999928i \(-0.496178\pi\)
−0.999928 + 0.0120057i \(0.996178\pi\)
\(384\) 0 0
\(385\) 10.9652 4.45520i 0.558836 0.227058i
\(386\) 0 0
\(387\) 12.2954i 0.625008i
\(388\) 0 0
\(389\) −6.28607 + 6.28607i −0.318716 + 0.318716i −0.848274 0.529558i \(-0.822358\pi\)
0.529558 + 0.848274i \(0.322358\pi\)
\(390\) 0 0
\(391\) −9.57367 −0.484161
\(392\) 0 0
\(393\) −15.3466 15.3466i −0.774135 0.774135i
\(394\) 0 0
\(395\) 21.3425 8.67157i 1.07386 0.436314i
\(396\) 0 0
\(397\) −6.58413 −0.330448 −0.165224 0.986256i \(-0.552835\pi\)
−0.165224 + 0.986256i \(0.552835\pi\)
\(398\) 0 0
\(399\) 10.7977 0.540561
\(400\) 0 0
\(401\) 19.7951 0.988522 0.494261 0.869313i \(-0.335439\pi\)
0.494261 + 0.869313i \(0.335439\pi\)
\(402\) 0 0
\(403\) −0.331123 −0.0164944
\(404\) 0 0
\(405\) 6.37497 2.59018i 0.316775 0.128707i
\(406\) 0 0
\(407\) 7.76707 + 7.76707i 0.385000 + 0.385000i
\(408\) 0 0
\(409\) 5.76937 0.285277 0.142638 0.989775i \(-0.454441\pi\)
0.142638 + 0.989775i \(0.454441\pi\)
\(410\) 0 0
\(411\) 6.31276 6.31276i 0.311385 0.311385i
\(412\) 0 0
\(413\) 13.4826i 0.663437i
\(414\) 0 0
\(415\) 6.76563 2.74891i 0.332112 0.134939i
\(416\) 0 0
\(417\) 13.1972 + 13.1972i 0.646270 + 0.646270i
\(418\) 0 0
\(419\) −8.68932 + 8.68932i −0.424501 + 0.424501i −0.886750 0.462249i \(-0.847043\pi\)
0.462249 + 0.886750i \(0.347043\pi\)
\(420\) 0 0
\(421\) −20.1193 20.1193i −0.980555 0.980555i 0.0192594 0.999815i \(-0.493869\pi\)
−0.999815 + 0.0192594i \(0.993869\pi\)
\(422\) 0 0
\(423\) −10.7048 + 10.7048i −0.520487 + 0.520487i
\(424\) 0 0
\(425\) 37.1466 + 0.502930i 1.80187 + 0.0243957i
\(426\) 0 0
\(427\) −15.2135 −0.736231
\(428\) 0 0
\(429\) −4.06060 4.06060i −0.196048 0.196048i
\(430\) 0 0
\(431\) 33.6247i 1.61965i 0.586675 + 0.809823i \(0.300437\pi\)
−0.586675 + 0.809823i \(0.699563\pi\)
\(432\) 0 0
\(433\) 7.46558 7.46558i 0.358773 0.358773i −0.504588 0.863361i \(-0.668355\pi\)
0.863361 + 0.504588i \(0.168355\pi\)
\(434\) 0 0
\(435\) 8.85819 + 3.73963i 0.424718 + 0.179302i
\(436\) 0 0
\(437\) 6.73775i 0.322310i
\(438\) 0 0
\(439\) 7.91929i 0.377967i −0.981980 0.188984i \(-0.939481\pi\)
0.981980 0.188984i \(-0.0605193\pi\)
\(440\) 0 0
\(441\) 5.98132i 0.284825i
\(442\) 0 0
\(443\) 10.6463i 0.505823i 0.967489 + 0.252911i \(0.0813881\pi\)
−0.967489 + 0.252911i \(0.918612\pi\)
\(444\) 0 0
\(445\) −20.2438 + 8.22517i −0.959649 + 0.389910i
\(446\) 0 0
\(447\) −19.4778 + 19.4778i −0.921266 + 0.921266i
\(448\) 0 0
\(449\) 6.08115i 0.286987i −0.989651 0.143494i \(-0.954166\pi\)
0.989651 0.143494i \(-0.0458336\pi\)
\(450\) 0 0
\(451\) 6.19880 + 6.19880i 0.291890 + 0.291890i
\(452\) 0 0
\(453\) 13.7148 0.644379
\(454\) 0 0
\(455\) −1.91346 + 4.53247i −0.0897042 + 0.212485i
\(456\) 0 0
\(457\) 0.313815 0.313815i 0.0146796 0.0146796i −0.699729 0.714409i \(-0.746696\pi\)
0.714409 + 0.699729i \(0.246696\pi\)
\(458\) 0 0
\(459\) 29.3374 + 29.3374i 1.36935 + 1.36935i
\(460\) 0 0
\(461\) −9.90949 + 9.90949i −0.461531 + 0.461531i −0.899157 0.437626i \(-0.855819\pi\)
0.437626 + 0.899157i \(0.355819\pi\)
\(462\) 0 0
\(463\) 17.3430 + 17.3430i 0.805999 + 0.805999i 0.984026 0.178027i \(-0.0569714\pi\)
−0.178027 + 0.984026i \(0.556971\pi\)
\(464\) 0 0
\(465\) 0.261571 + 0.643781i 0.0121301 + 0.0298546i
\(466\) 0 0
\(467\) 1.52267i 0.0704606i −0.999379 0.0352303i \(-0.988784\pi\)
0.999379 0.0352303i \(-0.0112165\pi\)
\(468\) 0 0
\(469\) 5.18213 5.18213i 0.239289 0.239289i
\(470\) 0 0
\(471\) 3.00530 0.138477
\(472\) 0 0
\(473\) 21.0115 + 21.0115i 0.966110 + 0.966110i
\(474\) 0 0
\(475\) 0.353952 26.1430i 0.0162404 1.19952i
\(476\) 0 0
\(477\) −7.88252 −0.360916
\(478\) 0 0
\(479\) 0.507657 0.0231955 0.0115977 0.999933i \(-0.496308\pi\)
0.0115977 + 0.999933i \(0.496308\pi\)
\(480\) 0 0
\(481\) −4.56592 −0.208188
\(482\) 0 0
\(483\) −2.66070 −0.121066
\(484\) 0 0
\(485\) 4.76578 + 2.01195i 0.216403 + 0.0913581i
\(486\) 0 0
\(487\) −25.9809 25.9809i −1.17730 1.17730i −0.980428 0.196876i \(-0.936920\pi\)
−0.196876 0.980428i \(-0.563080\pi\)
\(488\) 0 0
\(489\) −3.50539 −0.158519
\(490\) 0 0
\(491\) 3.28208 3.28208i 0.148118 0.148118i −0.629159 0.777277i \(-0.716600\pi\)
0.777277 + 0.629159i \(0.216600\pi\)
\(492\) 0 0
\(493\) 24.9391i 1.12320i
\(494\) 0 0
\(495\) 3.88053 9.19195i 0.174417 0.413147i
\(496\) 0 0
\(497\) −17.5964 17.5964i −0.789308 0.789308i
\(498\) 0 0
\(499\) 6.73907 6.73907i 0.301682 0.301682i −0.539990 0.841672i \(-0.681572\pi\)
0.841672 + 0.539990i \(0.181572\pi\)
\(500\) 0 0
\(501\) −12.9810 12.9810i −0.579950 0.579950i
\(502\) 0 0
\(503\) −6.12090 + 6.12090i −0.272918 + 0.272918i −0.830274 0.557356i \(-0.811816\pi\)
0.557356 + 0.830274i \(0.311816\pi\)
\(504\) 0 0
\(505\) −19.3430 8.16596i −0.860752 0.363380i
\(506\) 0 0
\(507\) −14.2672 −0.633629
\(508\) 0 0
\(509\) −13.8727 13.8727i −0.614894 0.614894i 0.329323 0.944217i \(-0.393180\pi\)
−0.944217 + 0.329323i \(0.893180\pi\)
\(510\) 0 0
\(511\) 3.40344i 0.150559i
\(512\) 0 0
\(513\) 20.6471 20.6471i 0.911590 0.911590i
\(514\) 0 0
\(515\) −1.99514 + 4.72594i −0.0879162 + 0.208250i
\(516\) 0 0
\(517\) 36.5869i 1.60909i
\(518\) 0 0
\(519\) 11.2684i 0.494628i
\(520\) 0 0
\(521\) 5.87686i 0.257470i −0.991679 0.128735i \(-0.958908\pi\)
0.991679 0.128735i \(-0.0410917\pi\)
\(522\) 0 0
\(523\) 26.0176i 1.13767i −0.822452 0.568834i \(-0.807395\pi\)
0.822452 0.568834i \(-0.192605\pi\)
\(524\) 0 0
\(525\) 10.3237 + 0.139774i 0.450564 + 0.00610022i
\(526\) 0 0
\(527\) −1.27445 + 1.27445i −0.0555160 + 0.0555160i
\(528\) 0 0
\(529\) 21.3397i 0.927814i
\(530\) 0 0
\(531\) 8.03690 + 8.03690i 0.348772 + 0.348772i
\(532\) 0 0
\(533\) −3.64400 −0.157839
\(534\) 0 0
\(535\) 7.56561 3.07394i 0.327090 0.132898i
\(536\) 0 0
\(537\) 8.48177 8.48177i 0.366016 0.366016i
\(538\) 0 0
\(539\) −10.2215 10.2215i −0.440270 0.440270i
\(540\) 0 0
\(541\) 6.57691 6.57691i 0.282764 0.282764i −0.551447 0.834210i \(-0.685924\pi\)
0.834210 + 0.551447i \(0.185924\pi\)
\(542\) 0 0
\(543\) −7.48594 7.48594i −0.321253 0.321253i
\(544\) 0 0
\(545\) 15.1712 + 6.40479i 0.649865 + 0.274351i
\(546\) 0 0
\(547\) 10.6170i 0.453951i 0.973900 + 0.226976i \(0.0728838\pi\)
−0.973900 + 0.226976i \(0.927116\pi\)
\(548\) 0 0
\(549\) −9.06863 + 9.06863i −0.387040 + 0.387040i
\(550\) 0 0
\(551\) 17.5516 0.747724
\(552\) 0 0
\(553\) 11.7422 + 11.7422i 0.499328 + 0.499328i
\(554\) 0 0
\(555\) 3.60686 + 8.87722i 0.153103 + 0.376817i
\(556\) 0 0
\(557\) −20.9610 −0.888146 −0.444073 0.895991i \(-0.646467\pi\)
−0.444073 + 0.895991i \(0.646467\pi\)
\(558\) 0 0
\(559\) −12.3517 −0.522422
\(560\) 0 0
\(561\) −31.2575 −1.31969
\(562\) 0 0
\(563\) −16.5598 −0.697911 −0.348955 0.937139i \(-0.613464\pi\)
−0.348955 + 0.937139i \(0.613464\pi\)
\(564\) 0 0
\(565\) 5.26280 12.4662i 0.221408 0.524456i
\(566\) 0 0
\(567\) 3.50736 + 3.50736i 0.147295 + 0.147295i
\(568\) 0 0
\(569\) −39.6751 −1.66327 −0.831634 0.555325i \(-0.812594\pi\)
−0.831634 + 0.555325i \(0.812594\pi\)
\(570\) 0 0
\(571\) −24.0292 + 24.0292i −1.00559 + 1.00559i −0.00560819 + 0.999984i \(0.501785\pi\)
−0.999984 + 0.00560819i \(0.998215\pi\)
\(572\) 0 0
\(573\) 2.35602i 0.0984240i
\(574\) 0 0
\(575\) −0.0872185 + 6.44199i −0.00363726 + 0.268649i
\(576\) 0 0
\(577\) −28.7705 28.7705i −1.19773 1.19773i −0.974844 0.222888i \(-0.928451\pi\)
−0.222888 0.974844i \(-0.571549\pi\)
\(578\) 0 0
\(579\) −7.92157 + 7.92157i −0.329209 + 0.329209i
\(580\) 0 0
\(581\) 3.72230 + 3.72230i 0.154427 + 0.154427i
\(582\) 0 0
\(583\) 13.4704 13.4704i 0.557888 0.557888i
\(584\) 0 0
\(585\) 1.56117 + 3.84237i 0.0645466 + 0.158862i
\(586\) 0 0
\(587\) −33.4854 −1.38209 −0.691046 0.722811i \(-0.742850\pi\)
−0.691046 + 0.722811i \(0.742850\pi\)
\(588\) 0 0
\(589\) 0.896933 + 0.896933i 0.0369575 + 0.0369575i
\(590\) 0 0
\(591\) 7.11866i 0.292822i
\(592\) 0 0
\(593\) 11.5298 11.5298i 0.473472 0.473472i −0.429564 0.903036i \(-0.641333\pi\)
0.903036 + 0.429564i \(0.141333\pi\)
\(594\) 0 0
\(595\) 10.0803 + 24.8096i 0.413250 + 1.01709i
\(596\) 0 0
\(597\) 8.92172i 0.365142i
\(598\) 0 0
\(599\) 20.0148i 0.817781i 0.912583 + 0.408891i \(0.134084\pi\)
−0.912583 + 0.408891i \(0.865916\pi\)
\(600\) 0 0
\(601\) 27.5924i 1.12552i −0.826621 0.562759i \(-0.809740\pi\)
0.826621 0.562759i \(-0.190260\pi\)
\(602\) 0 0
\(603\) 6.17806i 0.251590i
\(604\) 0 0
\(605\) −0.182062 0.448093i −0.00740188 0.0182176i
\(606\) 0 0
\(607\) −30.4850 + 30.4850i −1.23735 + 1.23735i −0.276265 + 0.961081i \(0.589097\pi\)
−0.961081 + 0.276265i \(0.910903\pi\)
\(608\) 0 0
\(609\) 6.93104i 0.280860i
\(610\) 0 0
\(611\) 10.7539 + 10.7539i 0.435057 + 0.435057i
\(612\) 0 0
\(613\) 20.2657 0.818523 0.409261 0.912417i \(-0.365786\pi\)
0.409261 + 0.912417i \(0.365786\pi\)
\(614\) 0 0
\(615\) 2.87859 + 7.08480i 0.116076 + 0.285687i
\(616\) 0 0
\(617\) 1.61302 1.61302i 0.0649378 0.0649378i −0.673892 0.738830i \(-0.735379\pi\)
0.738830 + 0.673892i \(0.235379\pi\)
\(618\) 0 0
\(619\) 2.46756 + 2.46756i 0.0991797 + 0.0991797i 0.754956 0.655776i \(-0.227658\pi\)
−0.655776 + 0.754956i \(0.727658\pi\)
\(620\) 0 0
\(621\) −5.08771 + 5.08771i −0.204163 + 0.204163i
\(622\) 0 0
\(623\) −11.1377 11.1377i −0.446222 0.446222i
\(624\) 0 0
\(625\) 0.676829 24.9908i 0.0270732 0.999633i
\(626\) 0 0
\(627\) 21.9984i 0.878531i
\(628\) 0 0
\(629\) −17.5737 + 17.5737i −0.700708 + 0.700708i
\(630\) 0 0
\(631\) −29.9602 −1.19270 −0.596348 0.802726i \(-0.703382\pi\)
−0.596348 + 0.802726i \(0.703382\pi\)
\(632\) 0 0
\(633\) 6.96133 + 6.96133i 0.276688 + 0.276688i
\(634\) 0 0
\(635\) 8.96913 21.2455i 0.355929 0.843101i
\(636\) 0 0
\(637\) 6.00875 0.238075
\(638\) 0 0
\(639\) −20.9782 −0.829885
\(640\) 0 0
\(641\) −37.3386 −1.47478 −0.737392 0.675465i \(-0.763943\pi\)
−0.737392 + 0.675465i \(0.763943\pi\)
\(642\) 0 0
\(643\) −24.5635 −0.968691 −0.484345 0.874877i \(-0.660942\pi\)
−0.484345 + 0.874877i \(0.660942\pi\)
\(644\) 0 0
\(645\) 9.75728 + 24.0147i 0.384193 + 0.945577i
\(646\) 0 0
\(647\) −23.1347 23.1347i −0.909519 0.909519i 0.0867142 0.996233i \(-0.472363\pi\)
−0.996233 + 0.0867142i \(0.972363\pi\)
\(648\) 0 0
\(649\) −27.4685 −1.07823
\(650\) 0 0
\(651\) −0.354194 + 0.354194i −0.0138820 + 0.0138820i
\(652\) 0 0
\(653\) 50.8060i 1.98819i −0.108496 0.994097i \(-0.534603\pi\)
0.108496 0.994097i \(-0.465397\pi\)
\(654\) 0 0
\(655\) −34.8993 14.7333i −1.36363 0.575679i
\(656\) 0 0
\(657\) 2.02877 + 2.02877i 0.0791497 + 0.0791497i
\(658\) 0 0
\(659\) 9.97780 9.97780i 0.388680 0.388680i −0.485537 0.874216i \(-0.661376\pi\)
0.874216 + 0.485537i \(0.161376\pi\)
\(660\) 0 0
\(661\) 5.09643 + 5.09643i 0.198228 + 0.198228i 0.799240 0.601012i \(-0.205236\pi\)
−0.601012 + 0.799240i \(0.705236\pi\)
\(662\) 0 0
\(663\) 9.18745 9.18745i 0.356811 0.356811i
\(664\) 0 0
\(665\) 17.4605 7.09428i 0.677088 0.275104i
\(666\) 0 0
\(667\) −4.32496 −0.167463
\(668\) 0 0
\(669\) −11.0402 11.0402i −0.426838 0.426838i
\(670\) 0 0
\(671\) 30.9947i 1.19654i
\(672\) 0 0
\(673\) −31.6322 + 31.6322i −1.21933 + 1.21933i −0.251464 + 0.967867i \(0.580912\pi\)
−0.967867 + 0.251464i \(0.919088\pi\)
\(674\) 0 0
\(675\) 20.0080 19.4735i 0.770108 0.749534i
\(676\) 0 0
\(677\) 25.6600i 0.986196i −0.869974 0.493098i \(-0.835864\pi\)
0.869974 0.493098i \(-0.164136\pi\)
\(678\) 0 0
\(679\) 3.72896i 0.143104i
\(680\) 0 0
\(681\) 7.71190i 0.295521i
\(682\) 0 0
\(683\) 12.3536i 0.472698i 0.971668 + 0.236349i \(0.0759509\pi\)
−0.971668 + 0.236349i \(0.924049\pi\)
\(684\) 0 0
\(685\) 6.06048 14.3557i 0.231559 0.548501i
\(686\) 0 0
\(687\) −0.728309 + 0.728309i −0.0277867 + 0.0277867i
\(688\) 0 0
\(689\) 7.91866i 0.301677i
\(690\) 0 0
\(691\) −22.5426 22.5426i −0.857561 0.857561i 0.133489 0.991050i \(-0.457382\pi\)
−0.991050 + 0.133489i \(0.957382\pi\)
\(692\) 0 0
\(693\) 7.19219 0.273209
\(694\) 0 0
\(695\) 30.0114 + 12.6698i 1.13840 + 0.480593i
\(696\) 0 0
\(697\) −14.0253 + 14.0253i −0.531247 + 0.531247i
\(698\) 0 0
\(699\) −16.2673 16.2673i −0.615285 0.615285i
\(700\) 0 0
\(701\) −26.9530 + 26.9530i −1.01800 + 1.01800i −0.0181663 + 0.999835i \(0.505783\pi\)
−0.999835 + 0.0181663i \(0.994217\pi\)
\(702\) 0 0
\(703\) 12.3680 + 12.3680i 0.466467 + 0.466467i
\(704\) 0 0
\(705\) 12.4131 29.4032i 0.467503 1.10739i
\(706\) 0 0
\(707\) 15.1348i 0.569203i
\(708\) 0 0
\(709\) −7.78615 + 7.78615i −0.292415 + 0.292415i −0.838034 0.545619i \(-0.816295\pi\)
0.545619 + 0.838034i \(0.316295\pi\)
\(710\) 0 0
\(711\) 13.9988 0.524997
\(712\) 0 0
\(713\) −0.221016 0.221016i −0.00827713 0.00827713i
\(714\) 0 0
\(715\) −9.23410 3.89833i −0.345336 0.145789i
\(716\) 0 0
\(717\) −2.28427 −0.0853076
\(718\) 0 0
\(719\) 20.6777 0.771150 0.385575 0.922677i \(-0.374003\pi\)
0.385575 + 0.922677i \(0.374003\pi\)
\(720\) 0 0
\(721\) −3.69779 −0.137713
\(722\) 0 0
\(723\) −13.3797 −0.497597
\(724\) 0 0
\(725\) 16.7812 + 0.227201i 0.623237 + 0.00843805i
\(726\) 0 0
\(727\) 20.4994 + 20.4994i 0.760280 + 0.760280i 0.976373 0.216093i \(-0.0693315\pi\)
−0.216093 + 0.976373i \(0.569331\pi\)
\(728\) 0 0
\(729\) 25.6425 0.949722
\(730\) 0 0
\(731\) −47.5403 + 47.5403i −1.75834 + 1.75834i
\(732\) 0 0
\(733\) 10.7306i 0.396344i 0.980167 + 0.198172i \(0.0635005\pi\)
−0.980167 + 0.198172i \(0.936500\pi\)
\(734\) 0 0
\(735\) −4.74663 11.6824i −0.175082 0.430912i
\(736\) 0 0
\(737\) 10.5577 + 10.5577i 0.388897 + 0.388897i
\(738\) 0 0
\(739\) −2.93837 + 2.93837i −0.108090 + 0.108090i −0.759083 0.650994i \(-0.774352\pi\)
0.650994 + 0.759083i \(0.274352\pi\)
\(740\) 0 0
\(741\) −6.46594 6.46594i −0.237532 0.237532i
\(742\) 0 0
\(743\) −0.223404 + 0.223404i −0.00819590 + 0.00819590i −0.711193 0.702997i \(-0.751845\pi\)
0.702997 + 0.711193i \(0.251845\pi\)
\(744\) 0 0
\(745\) −18.6994 + 44.2938i −0.685091 + 1.62280i
\(746\) 0 0
\(747\) 4.43766 0.162366
\(748\) 0 0
\(749\) 4.16243 + 4.16243i 0.152092 + 0.152092i
\(750\) 0 0
\(751\) 39.9939i 1.45940i 0.683769 + 0.729699i \(0.260340\pi\)
−0.683769 + 0.729699i \(0.739660\pi\)
\(752\) 0 0
\(753\) −16.2055 + 16.2055i −0.590563 + 0.590563i
\(754\) 0 0
\(755\) 22.1776 9.01088i 0.807126 0.327939i
\(756\) 0 0
\(757\) 32.9120i 1.19621i −0.801419 0.598103i \(-0.795921\pi\)
0.801419 0.598103i \(-0.204079\pi\)
\(758\) 0 0
\(759\) 5.42070i 0.196759i
\(760\) 0 0
\(761\) 33.9591i 1.23102i −0.788130 0.615509i \(-0.788951\pi\)
0.788130 0.615509i \(-0.211049\pi\)
\(762\) 0 0
\(763\) 11.8707i 0.429747i
\(764\) 0 0
\(765\) 20.7976 + 8.78003i 0.751937 + 0.317443i
\(766\) 0 0
\(767\) 8.07375 8.07375i 0.291526 0.291526i
\(768\) 0 0
\(769\) 40.2535i 1.45158i −0.687917 0.725789i \(-0.741475\pi\)
0.687917 0.725789i \(-0.258525\pi\)
\(770\) 0 0
\(771\) 5.29172 + 5.29172i 0.190577 + 0.190577i
\(772\) 0 0
\(773\) −9.47175 −0.340675 −0.170338 0.985386i \(-0.554486\pi\)
−0.170338 + 0.985386i \(0.554486\pi\)
\(774\) 0 0
\(775\) 0.845950 + 0.869172i 0.0303874 + 0.0312216i
\(776\) 0 0
\(777\) −4.88405 + 4.88405i −0.175214 + 0.175214i
\(778\) 0 0
\(779\) 9.87073 + 9.87073i 0.353656 + 0.353656i
\(780\) 0 0
\(781\) 35.8496 35.8496i 1.28280 1.28280i
\(782\) 0 0
\(783\) 13.2533 + 13.2533i 0.473636 + 0.473636i
\(784\) 0 0
\(785\) 4.85973 1.97453i 0.173451 0.0704741i
\(786\) 0 0
\(787\) 48.1367i 1.71589i 0.513742 + 0.857945i \(0.328259\pi\)
−0.513742 + 0.857945i \(0.671741\pi\)
\(788\) 0 0
\(789\) −21.9455 + 21.9455i −0.781282 + 0.781282i
\(790\) 0 0
\(791\) 9.75409 0.346815
\(792\) 0 0
\(793\) 9.11021 + 9.11021i 0.323513 + 0.323513i
\(794\) 0 0
\(795\) 15.3957 6.25537i 0.546031 0.221855i
\(796\) 0 0
\(797\) 33.8962 1.20066 0.600332 0.799751i \(-0.295035\pi\)
0.600332 + 0.799751i \(0.295035\pi\)
\(798\) 0 0
\(799\) 82.7810 2.92858
\(800\) 0 0
\(801\) −13.2782 −0.469162
\(802\) 0 0
\(803\) −6.93391 −0.244692
\(804\) 0 0
\(805\) −4.30250 + 1.74813i −0.151643 + 0.0616133i
\(806\) 0 0
\(807\) −25.3753 25.3753i −0.893254 0.893254i
\(808\) 0 0
\(809\) 27.5625 0.969047 0.484523 0.874778i \(-0.338993\pi\)
0.484523 + 0.874778i \(0.338993\pi\)
\(810\) 0 0
\(811\) 24.1817 24.1817i 0.849133 0.849133i −0.140892 0.990025i \(-0.544997\pi\)
0.990025 + 0.140892i \(0.0449970\pi\)
\(812\) 0 0
\(813\) 35.8120i 1.25598i
\(814\) 0 0
\(815\) −5.66841 + 2.30310i −0.198556 + 0.0806742i
\(816\) 0 0
\(817\) 33.4579 + 33.4579i 1.17054 + 1.17054i
\(818\) 0 0
\(819\) −2.11398 + 2.11398i −0.0738686 + 0.0738686i
\(820\) 0 0
\(821\) −0.0575735 0.0575735i −0.00200933 0.00200933i 0.706101 0.708111i \(-0.250452\pi\)
−0.708111 + 0.706101i \(0.750452\pi\)
\(822\) 0 0
\(823\) 28.5594 28.5594i 0.995518 0.995518i −0.00447159 0.999990i \(-0.501423\pi\)
0.999990 + 0.00447159i \(0.00142335\pi\)
\(824\) 0 0
\(825\) −0.284764 + 21.0327i −0.00991420 + 0.732266i
\(826\) 0 0
\(827\) 23.0863 0.802788 0.401394 0.915905i \(-0.368526\pi\)
0.401394 + 0.915905i \(0.368526\pi\)
\(828\) 0 0
\(829\) 33.3543 + 33.3543i 1.15844 + 1.15844i 0.984811 + 0.173631i \(0.0555499\pi\)
0.173631 + 0.984811i \(0.444450\pi\)
\(830\) 0 0
\(831\) 33.3414i 1.15660i
\(832\) 0 0
\(833\) 23.1269 23.1269i 0.801301 0.801301i
\(834\) 0 0
\(835\) −29.5198 12.4623i −1.02158 0.431275i
\(836\) 0 0
\(837\) 1.35456i 0.0468204i
\(838\) 0 0
\(839\) 49.4524i 1.70729i 0.520859 + 0.853643i \(0.325612\pi\)
−0.520859 + 0.853643i \(0.674388\pi\)
\(840\) 0 0
\(841\) 17.7336i 0.611504i
\(842\) 0 0
\(843\) 30.8746i 1.06338i
\(844\) 0 0
\(845\) −23.0708 + 9.37380i −0.793661 + 0.322469i
\(846\) 0 0
\(847\) 0.246530 0.246530i 0.00847089 0.00847089i
\(848\) 0 0
\(849\) 6.06865i 0.208276i
\(850\) 0 0
\(851\) −3.04764 3.04764i −0.104472 0.104472i
\(852\) 0 0
\(853\) 31.3639 1.07388 0.536939 0.843621i \(-0.319581\pi\)
0.536939 + 0.843621i \(0.319581\pi\)
\(854\) 0 0
\(855\) 6.17921 14.6369i 0.211324 0.500571i
\(856\) 0 0
\(857\) 16.1594 16.1594i 0.551996 0.551996i −0.375021 0.927016i \(-0.622364\pi\)
0.927016 + 0.375021i \(0.122364\pi\)
\(858\) 0 0
\(859\) 30.7369 + 30.7369i 1.04873 + 1.04873i 0.998750 + 0.0499792i \(0.0159155\pi\)
0.0499792 + 0.998750i \(0.484085\pi\)
\(860\) 0 0
\(861\) −3.89790 + 3.89790i −0.132840 + 0.132840i
\(862\) 0 0
\(863\) −18.9353 18.9353i −0.644565 0.644565i 0.307109 0.951674i \(-0.400638\pi\)
−0.951674 + 0.307109i \(0.900638\pi\)
\(864\) 0 0
\(865\) 7.40353 + 18.2216i 0.251728 + 0.619553i
\(866\) 0 0
\(867\) 48.9441i 1.66223i
\(868\) 0 0
\(869\) −23.9226 + 23.9226i −0.811517 + 0.811517i
\(870\) 0 0
\(871\) −6.20638 −0.210295
\(872\) 0 0
\(873\) 2.22280 + 2.22280i 0.0752305 + 0.0752305i
\(874\) 0 0
\(875\) 16.7859 6.55684i 0.567465 0.221662i
\(876\) 0 0
\(877\) −49.7461 −1.67981 −0.839903 0.542737i \(-0.817388\pi\)
−0.839903 + 0.542737i \(0.817388\pi\)
\(878\) 0 0
\(879\) −3.98423 −0.134385
\(880\) 0 0
\(881\) 27.7694 0.935574 0.467787 0.883841i \(-0.345051\pi\)
0.467787 + 0.883841i \(0.345051\pi\)
\(882\) 0 0
\(883\) 42.4602 1.42890 0.714450 0.699686i \(-0.246677\pi\)
0.714450 + 0.699686i \(0.246677\pi\)
\(884\) 0 0
\(885\) −22.0751 9.31938i −0.742048 0.313268i
\(886\) 0 0
\(887\) −16.1076 16.1076i −0.540842 0.540842i 0.382934 0.923776i \(-0.374914\pi\)
−0.923776 + 0.382934i \(0.874914\pi\)
\(888\) 0 0
\(889\) 16.6234 0.557531
\(890\) 0 0
\(891\) −7.14563 + 7.14563i −0.239388 + 0.239388i
\(892\) 0 0
\(893\) 58.2596i 1.94958i
\(894\) 0 0
\(895\) 8.14281 19.2882i 0.272184 0.644732i
\(896\) 0 0
\(897\) 1.59329 + 1.59329i 0.0531986 + 0.0531986i
\(898\) 0 0
\(899\) −0.575741 + 0.575741i −0.0192020 + 0.0192020i
\(900\) 0 0
\(901\) 30.4780 + 30.4780i 1.01537 + 1.01537i
\(902\) 0 0
\(903\) −13.2123 + 13.2123i −0.439679 + 0.439679i
\(904\) 0 0
\(905\) −17.0236 7.18678i −0.565883 0.238897i
\(906\) 0 0
\(907\) 9.20991 0.305810 0.152905 0.988241i \(-0.451137\pi\)
0.152905 + 0.988241i \(0.451137\pi\)
\(908\) 0 0
\(909\) −9.02174 9.02174i −0.299232 0.299232i
\(910\) 0 0
\(911\) 45.8065i 1.51764i 0.651302 + 0.758819i \(0.274223\pi\)
−0.651302 + 0.758819i \(0.725777\pi\)
\(912\) 0 0
\(913\) −7.58351 + 7.58351i −0.250978 + 0.250978i
\(914\) 0 0
\(915\) 10.5158 24.9090i 0.347640 0.823467i
\(916\) 0 0
\(917\) 27.3068i 0.901749i
\(918\) 0 0
\(919\) 5.52468i 0.182242i 0.995840 + 0.0911211i \(0.0290450\pi\)
−0.995840 + 0.0911211i \(0.970955\pi\)
\(920\) 0 0
\(921\) 18.6617i 0.614924i
\(922\) 0 0
\(923\) 21.0744i 0.693672i
\(924\) 0 0
\(925\) 11.6650 + 11.9852i 0.383542 + 0.394070i
\(926\) 0 0
\(927\) −2.20422 + 2.20422i −0.0723961 + 0.0723961i
\(928\) 0 0
\(929\) 43.4288i 1.42485i 0.701746 + 0.712427i \(0.252404\pi\)
−0.701746 + 0.712427i \(0.747596\pi\)
\(930\) 0 0
\(931\) −16.2763 16.2763i −0.533433 0.533433i
\(932\) 0 0
\(933\) 18.5210 0.606352
\(934\) 0 0
\(935\) −50.5451 + 20.5367i −1.65300 + 0.671623i
\(936\) 0 0
\(937\) 20.7275 20.7275i 0.677138 0.677138i −0.282213 0.959352i \(-0.591069\pi\)
0.959352 + 0.282213i \(0.0910686\pi\)
\(938\) 0 0
\(939\) 12.9741 + 12.9741i 0.423392 + 0.423392i
\(940\) 0 0
\(941\) −12.3393 + 12.3393i −0.402251 + 0.402251i −0.879026 0.476775i \(-0.841806\pi\)
0.476775 + 0.879026i \(0.341806\pi\)
\(942\) 0 0
\(943\) −2.43228 2.43228i −0.0792060 0.0792060i
\(944\) 0 0
\(945\) 18.5414 + 7.82757i 0.603153 + 0.254631i
\(946\) 0 0
\(947\) 48.3611i 1.57152i 0.618529 + 0.785762i \(0.287729\pi\)
−0.618529 + 0.785762i \(0.712271\pi\)
\(948\) 0 0
\(949\) 2.03807 2.03807i 0.0661585 0.0661585i
\(950\) 0 0
\(951\) −17.7752 −0.576399
\(952\) 0 0
\(953\) −34.0371 34.0371i −1.10257 1.10257i −0.994100 0.108471i \(-0.965405\pi\)
−0.108471 0.994100i \(-0.534595\pi\)
\(954\) 0 0
\(955\) −1.54794 3.80980i −0.0500903 0.123282i
\(956\) 0 0
\(957\) −14.1208 −0.456459
\(958\) 0 0
\(959\) 11.2325 0.362716
\(960\) 0 0
\(961\) 30.9412 0.998102
\(962\) 0 0
\(963\) 4.96238 0.159911
\(964\) 0 0
\(965\) −7.60500 + 18.0142i −0.244814 + 0.579898i
\(966\) 0 0
\(967\) −18.9307 18.9307i −0.608770 0.608770i 0.333855 0.942625i \(-0.391651\pi\)
−0.942625 + 0.333855i \(0.891651\pi\)
\(968\) 0 0
\(969\) −49.7732 −1.59895
\(970\) 0 0
\(971\) 21.2698 21.2698i 0.682580 0.682580i −0.278001 0.960581i \(-0.589672\pi\)
0.960581 + 0.278001i \(0.0896719\pi\)
\(972\) 0 0
\(973\) 23.4822i 0.752805i
\(974\) 0 0
\(975\) −6.09841 6.26581i −0.195305 0.200666i
\(976\) 0 0
\(977\) 2.13884 + 2.13884i 0.0684275 + 0.0684275i 0.740492 0.672065i \(-0.234592\pi\)
−0.672065 + 0.740492i \(0.734592\pi\)
\(978\) 0 0
\(979\) 22.6911 22.6911i 0.725210 0.725210i
\(980\) 0 0
\(981\) 7.07601 + 7.07601i 0.225919 + 0.225919i
\(982\) 0 0
\(983\) −6.18193 + 6.18193i −0.197173 + 0.197173i −0.798787 0.601614i \(-0.794525\pi\)
0.601614 + 0.798787i \(0.294525\pi\)
\(984\) 0 0
\(985\) 4.67708 + 11.5113i 0.149024 + 0.366779i
\(986\) 0 0
\(987\) 23.0064 0.732302
\(988\) 0 0
\(989\) −8.24447 8.24447i −0.262159 0.262159i
\(990\) 0 0
\(991\) 43.4847i 1.38134i −0.723172 0.690668i \(-0.757317\pi\)
0.723172 0.690668i \(-0.242683\pi\)
\(992\) 0 0
\(993\) 2.17092 2.17092i 0.0688921 0.0688921i
\(994\) 0 0
\(995\) −5.86173 14.4269i −0.185829 0.457364i
\(996\) 0 0
\(997\) 33.4043i 1.05793i 0.848645 + 0.528963i \(0.177419\pi\)
−0.848645 + 0.528963i \(0.822581\pi\)
\(998\) 0 0
\(999\) 18.6783i 0.590954i
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 640.2.s.d.287.3 18
4.3 odd 2 640.2.s.c.287.7 18
5.3 odd 4 640.2.j.d.543.3 18
8.3 odd 2 320.2.s.b.207.3 18
8.5 even 2 80.2.s.b.27.4 yes 18
16.3 odd 4 640.2.j.d.607.7 18
16.5 even 4 320.2.j.b.47.7 18
16.11 odd 4 80.2.j.b.67.2 yes 18
16.13 even 4 640.2.j.c.607.3 18
20.3 even 4 640.2.j.c.543.7 18
24.5 odd 2 720.2.z.g.667.6 18
40.3 even 4 320.2.j.b.143.3 18
40.13 odd 4 80.2.j.b.43.2 18
40.19 odd 2 1600.2.s.d.207.7 18
40.27 even 4 1600.2.j.d.143.7 18
40.29 even 2 400.2.s.d.107.6 18
40.37 odd 4 400.2.j.d.43.8 18
48.11 even 4 720.2.bd.g.307.8 18
80.3 even 4 inner 640.2.s.d.223.3 18
80.13 odd 4 640.2.s.c.223.7 18
80.27 even 4 400.2.s.d.243.6 18
80.37 odd 4 1600.2.s.d.943.7 18
80.43 even 4 80.2.s.b.3.4 yes 18
80.53 odd 4 320.2.s.b.303.3 18
80.59 odd 4 400.2.j.d.307.8 18
80.69 even 4 1600.2.j.d.1007.3 18
120.53 even 4 720.2.bd.g.523.8 18
240.203 odd 4 720.2.z.g.163.6 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.2.j.b.43.2 18 40.13 odd 4
80.2.j.b.67.2 yes 18 16.11 odd 4
80.2.s.b.3.4 yes 18 80.43 even 4
80.2.s.b.27.4 yes 18 8.5 even 2
320.2.j.b.47.7 18 16.5 even 4
320.2.j.b.143.3 18 40.3 even 4
320.2.s.b.207.3 18 8.3 odd 2
320.2.s.b.303.3 18 80.53 odd 4
400.2.j.d.43.8 18 40.37 odd 4
400.2.j.d.307.8 18 80.59 odd 4
400.2.s.d.107.6 18 40.29 even 2
400.2.s.d.243.6 18 80.27 even 4
640.2.j.c.543.7 18 20.3 even 4
640.2.j.c.607.3 18 16.13 even 4
640.2.j.d.543.3 18 5.3 odd 4
640.2.j.d.607.7 18 16.3 odd 4
640.2.s.c.223.7 18 80.13 odd 4
640.2.s.c.287.7 18 4.3 odd 2
640.2.s.d.223.3 18 80.3 even 4 inner
640.2.s.d.287.3 18 1.1 even 1 trivial
720.2.z.g.163.6 18 240.203 odd 4
720.2.z.g.667.6 18 24.5 odd 2
720.2.bd.g.307.8 18 48.11 even 4
720.2.bd.g.523.8 18 120.53 even 4
1600.2.j.d.143.7 18 40.27 even 4
1600.2.j.d.1007.3 18 80.69 even 4
1600.2.s.d.207.7 18 40.19 odd 2
1600.2.s.d.943.7 18 80.37 odd 4