Properties

Label 504.2.bu.a.41.13
Level $504$
Weight $2$
Character 504.41
Analytic conductor $4.024$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [504,2,Mod(41,504)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(504, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 5, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("504.41");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 504.bu (of order \(6\), degree \(2\), 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: \(4.02446026187\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 41.13
Character \(\chi\) \(=\) 504.41
Dual form 504.2.bu.a.209.13

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.0936255 + 1.72952i) q^{3} +(1.26858 - 2.19724i) q^{5} +(0.521158 + 2.59391i) q^{7} +(-2.98247 + 0.323854i) q^{9} +O(q^{10})\) \(q+(0.0936255 + 1.72952i) q^{3} +(1.26858 - 2.19724i) q^{5} +(0.521158 + 2.59391i) q^{7} +(-2.98247 + 0.323854i) q^{9} +(-5.68321 + 3.28120i) q^{11} +(5.13890 + 2.96694i) q^{13} +(3.91894 + 1.98831i) q^{15} +3.52698 q^{17} +0.261355i q^{19} +(-4.43743 + 1.14421i) q^{21} +(2.89967 + 1.67412i) q^{23} +(-0.718584 - 1.24462i) q^{25} +(-0.839346 - 5.12791i) q^{27} +(-1.00813 + 0.582041i) q^{29} +(1.69360 + 0.977800i) q^{31} +(-6.20700 - 9.52202i) q^{33} +(6.36059 + 2.14547i) q^{35} -1.16528 q^{37} +(-4.65025 + 9.16560i) q^{39} +(-2.85769 + 4.94966i) q^{41} +(2.67364 + 4.63088i) q^{43} +(-3.07191 + 6.96404i) q^{45} +(-3.79767 - 6.57775i) q^{47} +(-6.45679 + 2.70368i) q^{49} +(0.330215 + 6.09998i) q^{51} -3.54792i q^{53} +16.6499i q^{55} +(-0.452019 + 0.0244695i) q^{57} +(5.47899 - 9.48989i) q^{59} +(-5.53699 + 3.19678i) q^{61} +(-2.39439 - 7.56749i) q^{63} +(13.0382 - 7.52760i) q^{65} +(4.54164 - 7.86635i) q^{67} +(-2.62394 + 5.17177i) q^{69} -10.5889i q^{71} -2.72923i q^{73} +(2.08532 - 1.35933i) q^{75} +(-11.4730 - 13.0317i) q^{77} +(-0.652687 - 1.13049i) q^{79} +(8.79024 - 1.93177i) q^{81} +(-4.53241 - 7.85037i) q^{83} +(4.47426 - 7.74964i) q^{85} +(-1.10104 - 1.68908i) q^{87} +14.2599 q^{89} +(-5.01782 + 14.8761i) q^{91} +(-1.53256 + 3.02066i) q^{93} +(0.574261 + 0.331550i) q^{95} +(9.95686 - 5.74859i) q^{97} +(15.8874 - 11.6266i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 48 q - 4 q^{9} + 8 q^{15} - 4 q^{21} + 12 q^{23} - 24 q^{25} - 36 q^{29} + 32 q^{39} + 12 q^{43} + 6 q^{49} + 24 q^{51} + 28 q^{57} - 14 q^{63} + 36 q^{65} - 60 q^{77} - 12 q^{79} - 36 q^{81} - 12 q^{91} + 16 q^{93} - 108 q^{95} + 44 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/504\mathbb{Z}\right)^\times\).

\(n\) \(73\) \(127\) \(253\) \(281\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(e\left(\frac{5}{6}\right)\)

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
\(3\) 0.0936255 + 1.72952i 0.0540547 + 0.998538i
\(4\) 0 0
\(5\) 1.26858 2.19724i 0.567326 0.982637i −0.429503 0.903065i \(-0.641311\pi\)
0.996829 0.0795717i \(-0.0253553\pi\)
\(6\) 0 0
\(7\) 0.521158 + 2.59391i 0.196979 + 0.980408i
\(8\) 0 0
\(9\) −2.98247 + 0.323854i −0.994156 + 0.107951i
\(10\) 0 0
\(11\) −5.68321 + 3.28120i −1.71355 + 0.989320i −0.783898 + 0.620889i \(0.786772\pi\)
−0.929655 + 0.368431i \(0.879895\pi\)
\(12\) 0 0
\(13\) 5.13890 + 2.96694i 1.42527 + 0.822882i 0.996743 0.0806450i \(-0.0256980\pi\)
0.428531 + 0.903527i \(0.359031\pi\)
\(14\) 0 0
\(15\) 3.91894 + 1.98831i 1.01187 + 0.513380i
\(16\) 0 0
\(17\) 3.52698 0.855419 0.427709 0.903916i \(-0.359321\pi\)
0.427709 + 0.903916i \(0.359321\pi\)
\(18\) 0 0
\(19\) 0.261355i 0.0599590i 0.999551 + 0.0299795i \(0.00954420\pi\)
−0.999551 + 0.0299795i \(0.990456\pi\)
\(20\) 0 0
\(21\) −4.43743 + 1.14421i −0.968327 + 0.249687i
\(22\) 0 0
\(23\) 2.89967 + 1.67412i 0.604622 + 0.349079i 0.770858 0.637007i \(-0.219828\pi\)
−0.166236 + 0.986086i \(0.553161\pi\)
\(24\) 0 0
\(25\) −0.718584 1.24462i −0.143717 0.248925i
\(26\) 0 0
\(27\) −0.839346 5.12791i −0.161532 0.986867i
\(28\) 0 0
\(29\) −1.00813 + 0.582041i −0.187204 + 0.108082i −0.590673 0.806911i \(-0.701138\pi\)
0.403469 + 0.914993i \(0.367804\pi\)
\(30\) 0 0
\(31\) 1.69360 + 0.977800i 0.304179 + 0.175618i 0.644319 0.764757i \(-0.277141\pi\)
−0.340139 + 0.940375i \(0.610474\pi\)
\(32\) 0 0
\(33\) −6.20700 9.52202i −1.08050 1.65757i
\(34\) 0 0
\(35\) 6.36059 + 2.14547i 1.07514 + 0.362651i
\(36\) 0 0
\(37\) −1.16528 −0.191570 −0.0957851 0.995402i \(-0.530536\pi\)
−0.0957851 + 0.995402i \(0.530536\pi\)
\(38\) 0 0
\(39\) −4.65025 + 9.16560i −0.744636 + 1.46767i
\(40\) 0 0
\(41\) −2.85769 + 4.94966i −0.446296 + 0.773007i −0.998141 0.0609391i \(-0.980590\pi\)
0.551846 + 0.833946i \(0.313924\pi\)
\(42\) 0 0
\(43\) 2.67364 + 4.63088i 0.407726 + 0.706203i 0.994635 0.103451i \(-0.0329884\pi\)
−0.586908 + 0.809654i \(0.699655\pi\)
\(44\) 0 0
\(45\) −3.07191 + 6.96404i −0.457933 + 1.03814i
\(46\) 0 0
\(47\) −3.79767 6.57775i −0.553947 0.959464i −0.997985 0.0634560i \(-0.979788\pi\)
0.444038 0.896008i \(-0.353546\pi\)
\(48\) 0 0
\(49\) −6.45679 + 2.70368i −0.922398 + 0.386240i
\(50\) 0 0
\(51\) 0.330215 + 6.09998i 0.0462394 + 0.854168i
\(52\) 0 0
\(53\) 3.54792i 0.487344i −0.969858 0.243672i \(-0.921648\pi\)
0.969858 0.243672i \(-0.0783521\pi\)
\(54\) 0 0
\(55\) 16.6499i 2.24507i
\(56\) 0 0
\(57\) −0.452019 + 0.0244695i −0.0598714 + 0.00324107i
\(58\) 0 0
\(59\) 5.47899 9.48989i 0.713304 1.23548i −0.250306 0.968167i \(-0.580531\pi\)
0.963610 0.267312i \(-0.0861353\pi\)
\(60\) 0 0
\(61\) −5.53699 + 3.19678i −0.708939 + 0.409306i −0.810668 0.585506i \(-0.800896\pi\)
0.101729 + 0.994812i \(0.467562\pi\)
\(62\) 0 0
\(63\) −2.39439 7.56749i −0.301664 0.953414i
\(64\) 0 0
\(65\) 13.0382 7.52760i 1.61719 0.933684i
\(66\) 0 0
\(67\) 4.54164 7.86635i 0.554850 0.961028i −0.443066 0.896489i \(-0.646109\pi\)
0.997915 0.0645384i \(-0.0205575\pi\)
\(68\) 0 0
\(69\) −2.62394 + 5.17177i −0.315886 + 0.622608i
\(70\) 0 0
\(71\) 10.5889i 1.25667i −0.777944 0.628334i \(-0.783737\pi\)
0.777944 0.628334i \(-0.216263\pi\)
\(72\) 0 0
\(73\) 2.72923i 0.319433i −0.987163 0.159716i \(-0.948942\pi\)
0.987163 0.159716i \(-0.0510579\pi\)
\(74\) 0 0
\(75\) 2.08532 1.35933i 0.240792 0.156962i
\(76\) 0 0
\(77\) −11.4730 13.0317i −1.30747 1.48511i
\(78\) 0 0
\(79\) −0.652687 1.13049i −0.0734330 0.127190i 0.826971 0.562245i \(-0.190062\pi\)
−0.900404 + 0.435055i \(0.856729\pi\)
\(80\) 0 0
\(81\) 8.79024 1.93177i 0.976693 0.214641i
\(82\) 0 0
\(83\) −4.53241 7.85037i −0.497497 0.861690i 0.502499 0.864578i \(-0.332414\pi\)
−0.999996 + 0.00288802i \(0.999081\pi\)
\(84\) 0 0
\(85\) 4.47426 7.74964i 0.485301 0.840566i
\(86\) 0 0
\(87\) −1.10104 1.68908i −0.118044 0.181088i
\(88\) 0 0
\(89\) 14.2599 1.51155 0.755776 0.654830i \(-0.227260\pi\)
0.755776 + 0.654830i \(0.227260\pi\)
\(90\) 0 0
\(91\) −5.01782 + 14.8761i −0.526011 + 1.55944i
\(92\) 0 0
\(93\) −1.53256 + 3.02066i −0.158919 + 0.313228i
\(94\) 0 0
\(95\) 0.574261 + 0.331550i 0.0589180 + 0.0340163i
\(96\) 0 0
\(97\) 9.95686 5.74859i 1.01097 0.583681i 0.0994914 0.995038i \(-0.468278\pi\)
0.911474 + 0.411357i \(0.134945\pi\)
\(98\) 0 0
\(99\) 15.8874 11.6266i 1.59674 1.16852i
\(100\) 0 0
\(101\) −1.09191 1.89124i −0.108649 0.188186i 0.806574 0.591133i \(-0.201319\pi\)
−0.915223 + 0.402947i \(0.867986\pi\)
\(102\) 0 0
\(103\) 9.75576 + 5.63249i 0.961264 + 0.554986i 0.896562 0.442919i \(-0.146057\pi\)
0.0647021 + 0.997905i \(0.479390\pi\)
\(104\) 0 0
\(105\) −3.11512 + 11.2016i −0.304005 + 1.09317i
\(106\) 0 0
\(107\) 7.82576i 0.756545i 0.925694 + 0.378272i \(0.123482\pi\)
−0.925694 + 0.378272i \(0.876518\pi\)
\(108\) 0 0
\(109\) −17.5453 −1.68053 −0.840266 0.542174i \(-0.817601\pi\)
−0.840266 + 0.542174i \(0.817601\pi\)
\(110\) 0 0
\(111\) −0.109100 2.01537i −0.0103553 0.191290i
\(112\) 0 0
\(113\) 3.81112 + 2.20035i 0.358520 + 0.206992i 0.668431 0.743774i \(-0.266966\pi\)
−0.309911 + 0.950765i \(0.600299\pi\)
\(114\) 0 0
\(115\) 7.35691 4.24751i 0.686035 0.396083i
\(116\) 0 0
\(117\) −16.2875 7.18456i −1.50578 0.664213i
\(118\) 0 0
\(119\) 1.83812 + 9.14869i 0.168500 + 0.838659i
\(120\) 0 0
\(121\) 16.0326 27.7693i 1.45751 2.52448i
\(122\) 0 0
\(123\) −8.82808 4.47901i −0.796001 0.403859i
\(124\) 0 0
\(125\) 9.03946 0.808514
\(126\) 0 0
\(127\) 19.5689 1.73646 0.868228 0.496165i \(-0.165259\pi\)
0.868228 + 0.496165i \(0.165259\pi\)
\(128\) 0 0
\(129\) −7.75888 + 5.05768i −0.683131 + 0.445304i
\(130\) 0 0
\(131\) 0.00967786 0.0167625i 0.000845559 0.00146455i −0.865602 0.500732i \(-0.833064\pi\)
0.866448 + 0.499268i \(0.166398\pi\)
\(132\) 0 0
\(133\) −0.677933 + 0.136207i −0.0587843 + 0.0118107i
\(134\) 0 0
\(135\) −12.3320 4.66091i −1.06137 0.401148i
\(136\) 0 0
\(137\) −3.67032 + 2.11906i −0.313577 + 0.181044i −0.648526 0.761193i \(-0.724614\pi\)
0.334949 + 0.942236i \(0.391281\pi\)
\(138\) 0 0
\(139\) −17.6403 10.1847i −1.49623 0.863851i −0.496244 0.868183i \(-0.665288\pi\)
−0.999991 + 0.00433182i \(0.998621\pi\)
\(140\) 0 0
\(141\) 11.0208 7.18398i 0.928118 0.605000i
\(142\) 0 0
\(143\) −38.9406 −3.25638
\(144\) 0 0
\(145\) 2.95346i 0.245272i
\(146\) 0 0
\(147\) −5.28058 10.9140i −0.435535 0.900172i
\(148\) 0 0
\(149\) −6.40370 3.69718i −0.524611 0.302885i 0.214208 0.976788i \(-0.431283\pi\)
−0.738819 + 0.673904i \(0.764616\pi\)
\(150\) 0 0
\(151\) 6.94189 + 12.0237i 0.564923 + 0.978475i 0.997057 + 0.0766658i \(0.0244275\pi\)
−0.432134 + 0.901809i \(0.642239\pi\)
\(152\) 0 0
\(153\) −10.5191 + 1.14223i −0.850420 + 0.0923436i
\(154\) 0 0
\(155\) 4.29693 2.48083i 0.345138 0.199265i
\(156\) 0 0
\(157\) 1.86713 + 1.07799i 0.149013 + 0.0860329i 0.572653 0.819798i \(-0.305914\pi\)
−0.423639 + 0.905831i \(0.639248\pi\)
\(158\) 0 0
\(159\) 6.13619 0.332176i 0.486632 0.0263432i
\(160\) 0 0
\(161\) −2.83135 + 8.39397i −0.223142 + 0.661538i
\(162\) 0 0
\(163\) −17.1866 −1.34616 −0.673079 0.739571i \(-0.735029\pi\)
−0.673079 + 0.739571i \(0.735029\pi\)
\(164\) 0 0
\(165\) −28.7963 + 1.55885i −2.24179 + 0.121356i
\(166\) 0 0
\(167\) 5.40101 9.35482i 0.417943 0.723898i −0.577790 0.816186i \(-0.696085\pi\)
0.995732 + 0.0922879i \(0.0294180\pi\)
\(168\) 0 0
\(169\) 11.1055 + 19.2353i 0.854270 + 1.47964i
\(170\) 0 0
\(171\) −0.0846409 0.779484i −0.00647265 0.0596086i
\(172\) 0 0
\(173\) 9.55489 + 16.5495i 0.726444 + 1.25824i 0.958377 + 0.285507i \(0.0921619\pi\)
−0.231932 + 0.972732i \(0.574505\pi\)
\(174\) 0 0
\(175\) 2.85395 2.51259i 0.215739 0.189934i
\(176\) 0 0
\(177\) 16.9259 + 8.58752i 1.27223 + 0.645477i
\(178\) 0 0
\(179\) 14.9224i 1.11536i −0.830057 0.557678i \(-0.811692\pi\)
0.830057 0.557678i \(-0.188308\pi\)
\(180\) 0 0
\(181\) 18.5179i 1.37642i −0.725511 0.688211i \(-0.758396\pi\)
0.725511 0.688211i \(-0.241604\pi\)
\(182\) 0 0
\(183\) −6.04730 9.27703i −0.447029 0.685777i
\(184\) 0 0
\(185\) −1.47824 + 2.56039i −0.108683 + 0.188244i
\(186\) 0 0
\(187\) −20.0446 + 11.5728i −1.46581 + 0.846283i
\(188\) 0 0
\(189\) 12.8639 4.84965i 0.935714 0.352760i
\(190\) 0 0
\(191\) −5.08192 + 2.93405i −0.367715 + 0.212300i −0.672460 0.740134i \(-0.734762\pi\)
0.304745 + 0.952434i \(0.401429\pi\)
\(192\) 0 0
\(193\) −8.19577 + 14.1955i −0.589945 + 1.02181i 0.404294 + 0.914629i \(0.367517\pi\)
−0.994239 + 0.107185i \(0.965816\pi\)
\(194\) 0 0
\(195\) 14.2398 + 21.8450i 1.01974 + 1.56435i
\(196\) 0 0
\(197\) 6.90103i 0.491678i 0.969311 + 0.245839i \(0.0790635\pi\)
−0.969311 + 0.245839i \(0.920936\pi\)
\(198\) 0 0
\(199\) 10.3555i 0.734078i −0.930205 0.367039i \(-0.880371\pi\)
0.930205 0.367039i \(-0.119629\pi\)
\(200\) 0 0
\(201\) 14.0302 + 7.11836i 0.989615 + 0.502090i
\(202\) 0 0
\(203\) −2.03516 2.31166i −0.142840 0.162246i
\(204\) 0 0
\(205\) 7.25041 + 12.5581i 0.506390 + 0.877094i
\(206\) 0 0
\(207\) −9.19034 4.05395i −0.638772 0.281769i
\(208\) 0 0
\(209\) −0.857560 1.48534i −0.0593187 0.102743i
\(210\) 0 0
\(211\) 3.43331 5.94667i 0.236359 0.409385i −0.723308 0.690526i \(-0.757379\pi\)
0.959667 + 0.281140i \(0.0907126\pi\)
\(212\) 0 0
\(213\) 18.3136 0.991387i 1.25483 0.0679287i
\(214\) 0 0
\(215\) 13.5669 0.925255
\(216\) 0 0
\(217\) −1.65370 + 4.90264i −0.112260 + 0.332813i
\(218\) 0 0
\(219\) 4.72026 0.255526i 0.318966 0.0172668i
\(220\) 0 0
\(221\) 18.1248 + 10.4644i 1.21921 + 0.703909i
\(222\) 0 0
\(223\) 0.467052 0.269653i 0.0312761 0.0180573i −0.484280 0.874913i \(-0.660919\pi\)
0.515557 + 0.856856i \(0.327585\pi\)
\(224\) 0 0
\(225\) 2.54623 + 3.47934i 0.169749 + 0.231956i
\(226\) 0 0
\(227\) −0.610518 1.05745i −0.0405215 0.0701852i 0.845053 0.534682i \(-0.179569\pi\)
−0.885575 + 0.464497i \(0.846235\pi\)
\(228\) 0 0
\(229\) −10.4049 6.00725i −0.687573 0.396970i 0.115129 0.993351i \(-0.463272\pi\)
−0.802702 + 0.596380i \(0.796605\pi\)
\(230\) 0 0
\(231\) 21.4645 21.0629i 1.41226 1.38584i
\(232\) 0 0
\(233\) 5.13142i 0.336171i 0.985772 + 0.168085i \(0.0537584\pi\)
−0.985772 + 0.168085i \(0.946242\pi\)
\(234\) 0 0
\(235\) −19.2706 −1.25707
\(236\) 0 0
\(237\) 1.89409 1.23468i 0.123034 0.0802008i
\(238\) 0 0
\(239\) 10.5981 + 6.11881i 0.685533 + 0.395793i 0.801937 0.597409i \(-0.203803\pi\)
−0.116403 + 0.993202i \(0.537137\pi\)
\(240\) 0 0
\(241\) 24.9320 14.3945i 1.60601 0.927231i 0.615761 0.787933i \(-0.288849\pi\)
0.990251 0.139298i \(-0.0444846\pi\)
\(242\) 0 0
\(243\) 4.16402 + 15.0220i 0.267122 + 0.963663i
\(244\) 0 0
\(245\) −2.25030 + 17.6170i −0.143767 + 1.12551i
\(246\) 0 0
\(247\) −0.775427 + 1.34308i −0.0493392 + 0.0854580i
\(248\) 0 0
\(249\) 13.1530 8.57388i 0.833538 0.543348i
\(250\) 0 0
\(251\) 4.82335 0.304447 0.152223 0.988346i \(-0.451357\pi\)
0.152223 + 0.988346i \(0.451357\pi\)
\(252\) 0 0
\(253\) −21.9726 −1.38140
\(254\) 0 0
\(255\) 13.8220 + 7.01274i 0.865570 + 0.439155i
\(256\) 0 0
\(257\) −2.15022 + 3.72430i −0.134127 + 0.232315i −0.925264 0.379324i \(-0.876156\pi\)
0.791136 + 0.611640i \(0.209490\pi\)
\(258\) 0 0
\(259\) −0.607293 3.02263i −0.0377354 0.187817i
\(260\) 0 0
\(261\) 2.81821 2.06241i 0.174443 0.127660i
\(262\) 0 0
\(263\) −19.2444 + 11.1107i −1.18666 + 0.685118i −0.957546 0.288282i \(-0.906916\pi\)
−0.229113 + 0.973400i \(0.573583\pi\)
\(264\) 0 0
\(265\) −7.79564 4.50082i −0.478883 0.276483i
\(266\) 0 0
\(267\) 1.33509 + 24.6628i 0.0817064 + 1.50934i
\(268\) 0 0
\(269\) −21.6564 −1.32042 −0.660208 0.751083i \(-0.729532\pi\)
−0.660208 + 0.751083i \(0.729532\pi\)
\(270\) 0 0
\(271\) 17.0808i 1.03758i 0.854901 + 0.518792i \(0.173618\pi\)
−0.854901 + 0.518792i \(0.826382\pi\)
\(272\) 0 0
\(273\) −26.1983 7.28563i −1.58559 0.440947i
\(274\) 0 0
\(275\) 8.16774 + 4.71565i 0.492533 + 0.284364i
\(276\) 0 0
\(277\) −1.60480 2.77959i −0.0964230 0.167009i 0.813779 0.581175i \(-0.197407\pi\)
−0.910202 + 0.414165i \(0.864073\pi\)
\(278\) 0 0
\(279\) −5.36777 2.36778i −0.321360 0.141755i
\(280\) 0 0
\(281\) 18.0599 10.4269i 1.07736 0.622016i 0.147179 0.989110i \(-0.452981\pi\)
0.930184 + 0.367094i \(0.119647\pi\)
\(282\) 0 0
\(283\) 9.99941 + 5.77316i 0.594403 + 0.343179i 0.766837 0.641842i \(-0.221830\pi\)
−0.172433 + 0.985021i \(0.555163\pi\)
\(284\) 0 0
\(285\) −0.519656 + 1.02424i −0.0307818 + 0.0606706i
\(286\) 0 0
\(287\) −14.3283 4.83304i −0.845773 0.285286i
\(288\) 0 0
\(289\) −4.56040 −0.268259
\(290\) 0 0
\(291\) 10.8745 + 16.6824i 0.637475 + 0.977937i
\(292\) 0 0
\(293\) −14.1600 + 24.5259i −0.827238 + 1.43282i 0.0729596 + 0.997335i \(0.476756\pi\)
−0.900197 + 0.435483i \(0.856578\pi\)
\(294\) 0 0
\(295\) −13.9011 24.0773i −0.809351 1.40184i
\(296\) 0 0
\(297\) 21.5959 + 26.3890i 1.25312 + 1.53124i
\(298\) 0 0
\(299\) 9.93406 + 17.2063i 0.574501 + 0.995066i
\(300\) 0 0
\(301\) −10.6187 + 9.34862i −0.612053 + 0.538845i
\(302\) 0 0
\(303\) 3.16871 2.06554i 0.182037 0.118662i
\(304\) 0 0
\(305\) 16.2215i 0.928839i
\(306\) 0 0
\(307\) 24.2818i 1.38584i −0.721016 0.692918i \(-0.756325\pi\)
0.721016 0.692918i \(-0.243675\pi\)
\(308\) 0 0
\(309\) −8.82811 + 17.4001i −0.502214 + 0.989858i
\(310\) 0 0
\(311\) −16.3119 + 28.2531i −0.924964 + 1.60209i −0.133346 + 0.991070i \(0.542572\pi\)
−0.791619 + 0.611016i \(0.790761\pi\)
\(312\) 0 0
\(313\) 12.5650 7.25443i 0.710218 0.410045i −0.100924 0.994894i \(-0.532180\pi\)
0.811142 + 0.584850i \(0.198846\pi\)
\(314\) 0 0
\(315\) −19.6651 4.33891i −1.10800 0.244470i
\(316\) 0 0
\(317\) 14.0182 8.09338i 0.787338 0.454570i −0.0516867 0.998663i \(-0.516460\pi\)
0.839024 + 0.544094i \(0.183126\pi\)
\(318\) 0 0
\(319\) 3.81959 6.61573i 0.213856 0.370410i
\(320\) 0 0
\(321\) −13.5348 + 0.732691i −0.755439 + 0.0408948i
\(322\) 0 0
\(323\) 0.921796i 0.0512901i
\(324\) 0 0
\(325\) 8.52800i 0.473048i
\(326\) 0 0
\(327\) −1.64268 30.3449i −0.0908407 1.67808i
\(328\) 0 0
\(329\) 15.0829 13.2789i 0.831550 0.732088i
\(330\) 0 0
\(331\) 14.6054 + 25.2973i 0.802787 + 1.39047i 0.917775 + 0.397100i \(0.129984\pi\)
−0.114989 + 0.993367i \(0.536683\pi\)
\(332\) 0 0
\(333\) 3.47540 0.377379i 0.190451 0.0206803i
\(334\) 0 0
\(335\) −11.5229 19.9582i −0.629561 1.09043i
\(336\) 0 0
\(337\) −8.46812 + 14.6672i −0.461288 + 0.798974i −0.999025 0.0441385i \(-0.985946\pi\)
0.537738 + 0.843112i \(0.319279\pi\)
\(338\) 0 0
\(339\) −3.44873 + 6.79741i −0.187309 + 0.369185i
\(340\) 0 0
\(341\) −12.8334 −0.694970
\(342\) 0 0
\(343\) −10.3781 15.3393i −0.560366 0.828245i
\(344\) 0 0
\(345\) 8.03495 + 12.3262i 0.432587 + 0.663622i
\(346\) 0 0
\(347\) −17.7384 10.2413i −0.952247 0.549780i −0.0584689 0.998289i \(-0.518622\pi\)
−0.893778 + 0.448509i \(0.851955\pi\)
\(348\) 0 0
\(349\) 5.98087 3.45306i 0.320149 0.184838i −0.331310 0.943522i \(-0.607491\pi\)
0.651459 + 0.758684i \(0.274157\pi\)
\(350\) 0 0
\(351\) 10.9009 28.8421i 0.581848 1.53948i
\(352\) 0 0
\(353\) −10.5025 18.1908i −0.558990 0.968198i −0.997581 0.0695118i \(-0.977856\pi\)
0.438592 0.898686i \(-0.355477\pi\)
\(354\) 0 0
\(355\) −23.2663 13.4328i −1.23485 0.712939i
\(356\) 0 0
\(357\) −15.6507 + 4.03560i −0.828325 + 0.213587i
\(358\) 0 0
\(359\) 2.61864i 0.138206i 0.997610 + 0.0691032i \(0.0220138\pi\)
−0.997610 + 0.0691032i \(0.977986\pi\)
\(360\) 0 0
\(361\) 18.9317 0.996405
\(362\) 0 0
\(363\) 49.5286 + 25.1288i 2.59958 + 1.31892i
\(364\) 0 0
\(365\) −5.99679 3.46225i −0.313886 0.181222i
\(366\) 0 0
\(367\) 23.8455 13.7672i 1.24472 0.718641i 0.274671 0.961538i \(-0.411431\pi\)
0.970052 + 0.242897i \(0.0780977\pi\)
\(368\) 0 0
\(369\) 6.92000 15.6877i 0.360241 0.816668i
\(370\) 0 0
\(371\) 9.20300 1.84903i 0.477796 0.0959967i
\(372\) 0 0
\(373\) −2.67768 + 4.63788i −0.138645 + 0.240140i −0.926984 0.375101i \(-0.877608\pi\)
0.788339 + 0.615241i \(0.210941\pi\)
\(374\) 0 0
\(375\) 0.846324 + 15.6339i 0.0437040 + 0.807332i
\(376\) 0 0
\(377\) −6.90754 −0.355756
\(378\) 0 0
\(379\) 21.5330 1.10608 0.553038 0.833156i \(-0.313468\pi\)
0.553038 + 0.833156i \(0.313468\pi\)
\(380\) 0 0
\(381\) 1.83214 + 33.8447i 0.0938636 + 1.73392i
\(382\) 0 0
\(383\) −1.85948 + 3.22071i −0.0950149 + 0.164571i −0.909615 0.415453i \(-0.863623\pi\)
0.814600 + 0.580023i \(0.196957\pi\)
\(384\) 0 0
\(385\) −43.1883 + 8.67721i −2.20108 + 0.442232i
\(386\) 0 0
\(387\) −9.47378 12.9456i −0.481579 0.658061i
\(388\) 0 0
\(389\) 7.72516 4.46012i 0.391681 0.226137i −0.291207 0.956660i \(-0.594057\pi\)
0.682888 + 0.730523i \(0.260724\pi\)
\(390\) 0 0
\(391\) 10.2271 + 5.90460i 0.517205 + 0.298609i
\(392\) 0 0
\(393\) 0.0298972 + 0.0151686i 0.00150812 + 0.000765157i
\(394\) 0 0
\(395\) −3.31194 −0.166642
\(396\) 0 0
\(397\) 13.3591i 0.670476i 0.942134 + 0.335238i \(0.108817\pi\)
−0.942134 + 0.335238i \(0.891183\pi\)
\(398\) 0 0
\(399\) −0.299045 1.15975i −0.0149710 0.0580599i
\(400\) 0 0
\(401\) −18.6966 10.7945i −0.933663 0.539051i −0.0456950 0.998955i \(-0.514550\pi\)
−0.887968 + 0.459905i \(0.847884\pi\)
\(402\) 0 0
\(403\) 5.80216 + 10.0496i 0.289026 + 0.500608i
\(404\) 0 0
\(405\) 6.90654 21.7649i 0.343189 1.08151i
\(406\) 0 0
\(407\) 6.62251 3.82351i 0.328266 0.189524i
\(408\) 0 0
\(409\) −25.8784 14.9409i −1.27960 0.738779i −0.302827 0.953045i \(-0.597931\pi\)
−0.976775 + 0.214266i \(0.931264\pi\)
\(410\) 0 0
\(411\) −4.00859 6.14950i −0.197729 0.303332i
\(412\) 0 0
\(413\) 27.4714 + 9.26630i 1.35178 + 0.455965i
\(414\) 0 0
\(415\) −22.9989 −1.12897
\(416\) 0 0
\(417\) 15.9630 31.4628i 0.781710 1.54074i
\(418\) 0 0
\(419\) −9.30438 + 16.1157i −0.454549 + 0.787302i −0.998662 0.0517100i \(-0.983533\pi\)
0.544113 + 0.839012i \(0.316866\pi\)
\(420\) 0 0
\(421\) −9.79010 16.9570i −0.477140 0.826431i 0.522516 0.852629i \(-0.324993\pi\)
−0.999657 + 0.0261979i \(0.991660\pi\)
\(422\) 0 0
\(423\) 13.4567 + 18.3881i 0.654285 + 0.894058i
\(424\) 0 0
\(425\) −2.53443 4.38977i −0.122938 0.212935i
\(426\) 0 0
\(427\) −11.1778 12.6965i −0.540933 0.614424i
\(428\) 0 0
\(429\) −3.64583 67.3485i −0.176022 3.25162i
\(430\) 0 0
\(431\) 13.3632i 0.643683i 0.946794 + 0.321842i \(0.104302\pi\)
−0.946794 + 0.321842i \(0.895698\pi\)
\(432\) 0 0
\(433\) 3.05246i 0.146692i −0.997307 0.0733458i \(-0.976632\pi\)
0.997307 0.0733458i \(-0.0233677\pi\)
\(434\) 0 0
\(435\) −5.10807 + 0.276519i −0.244913 + 0.0132581i
\(436\) 0 0
\(437\) −0.437541 + 0.757843i −0.0209304 + 0.0362526i
\(438\) 0 0
\(439\) −16.8139 + 9.70754i −0.802486 + 0.463316i −0.844340 0.535808i \(-0.820007\pi\)
0.0418537 + 0.999124i \(0.486674\pi\)
\(440\) 0 0
\(441\) 18.3816 10.1547i 0.875313 0.483557i
\(442\) 0 0
\(443\) −28.5753 + 16.4979i −1.35765 + 0.783841i −0.989307 0.145848i \(-0.953409\pi\)
−0.368345 + 0.929689i \(0.620076\pi\)
\(444\) 0 0
\(445\) 18.0899 31.3326i 0.857542 1.48531i
\(446\) 0 0
\(447\) 5.79479 11.4215i 0.274084 0.540217i
\(448\) 0 0
\(449\) 9.72985i 0.459180i 0.973287 + 0.229590i \(0.0737385\pi\)
−0.973287 + 0.229590i \(0.926262\pi\)
\(450\) 0 0
\(451\) 37.5066i 1.76612i
\(452\) 0 0
\(453\) −20.1453 + 13.1319i −0.946508 + 0.616988i
\(454\) 0 0
\(455\) 26.3209 + 29.8969i 1.23394 + 1.40159i
\(456\) 0 0
\(457\) 7.14815 + 12.3810i 0.334376 + 0.579157i 0.983365 0.181641i \(-0.0581410\pi\)
−0.648989 + 0.760798i \(0.724808\pi\)
\(458\) 0 0
\(459\) −2.96036 18.0861i −0.138178 0.844185i
\(460\) 0 0
\(461\) −3.95893 6.85706i −0.184386 0.319365i 0.758984 0.651110i \(-0.225696\pi\)
−0.943369 + 0.331744i \(0.892363\pi\)
\(462\) 0 0
\(463\) 8.81566 15.2692i 0.409698 0.709618i −0.585157 0.810920i \(-0.698967\pi\)
0.994856 + 0.101301i \(0.0323006\pi\)
\(464\) 0 0
\(465\) 4.69295 + 7.19935i 0.217630 + 0.333862i
\(466\) 0 0
\(467\) 25.1745 1.16494 0.582469 0.812853i \(-0.302087\pi\)
0.582469 + 0.812853i \(0.302087\pi\)
\(468\) 0 0
\(469\) 22.7716 + 7.68101i 1.05149 + 0.354676i
\(470\) 0 0
\(471\) −1.68959 + 3.33017i −0.0778523 + 0.153446i
\(472\) 0 0
\(473\) −30.3897 17.5455i −1.39732 0.806744i
\(474\) 0 0
\(475\) 0.325289 0.187806i 0.0149253 0.00861712i
\(476\) 0 0
\(477\) 1.14901 + 10.5816i 0.0526095 + 0.484496i
\(478\) 0 0
\(479\) 11.3258 + 19.6168i 0.517488 + 0.896316i 0.999794 + 0.0203125i \(0.00646613\pi\)
−0.482306 + 0.876003i \(0.660201\pi\)
\(480\) 0 0
\(481\) −5.98823 3.45731i −0.273040 0.157640i
\(482\) 0 0
\(483\) −14.7826 4.11098i −0.672632 0.187056i
\(484\) 0 0
\(485\) 29.1702i 1.32455i
\(486\) 0 0
\(487\) −6.17391 −0.279767 −0.139883 0.990168i \(-0.544673\pi\)
−0.139883 + 0.990168i \(0.544673\pi\)
\(488\) 0 0
\(489\) −1.60910 29.7245i −0.0727661 1.34419i
\(490\) 0 0
\(491\) 20.8632 + 12.0454i 0.941544 + 0.543601i 0.890444 0.455093i \(-0.150394\pi\)
0.0511002 + 0.998694i \(0.483727\pi\)
\(492\) 0 0
\(493\) −3.55564 + 2.05285i −0.160138 + 0.0924557i
\(494\) 0 0
\(495\) −5.39212 49.6577i −0.242358 2.23195i
\(496\) 0 0
\(497\) 27.4666 5.51847i 1.23205 0.247537i
\(498\) 0 0
\(499\) −7.18584 + 12.4462i −0.321682 + 0.557170i −0.980835 0.194839i \(-0.937582\pi\)
0.659153 + 0.752009i \(0.270915\pi\)
\(500\) 0 0
\(501\) 16.6850 + 8.46529i 0.745431 + 0.378201i
\(502\) 0 0
\(503\) −18.4937 −0.824593 −0.412296 0.911050i \(-0.635273\pi\)
−0.412296 + 0.911050i \(0.635273\pi\)
\(504\) 0 0
\(505\) −5.54069 −0.246557
\(506\) 0 0
\(507\) −32.2281 + 21.0081i −1.43130 + 0.933003i
\(508\) 0 0
\(509\) −7.48887 + 12.9711i −0.331938 + 0.574934i −0.982892 0.184184i \(-0.941036\pi\)
0.650954 + 0.759118i \(0.274369\pi\)
\(510\) 0 0
\(511\) 7.07940 1.42236i 0.313174 0.0629216i
\(512\) 0 0
\(513\) 1.34021 0.219368i 0.0591716 0.00968532i
\(514\) 0 0
\(515\) 24.7519 14.2905i 1.09070 0.629716i
\(516\) 0 0
\(517\) 43.1659 + 24.9219i 1.89843 + 1.09606i
\(518\) 0 0
\(519\) −27.7282 + 18.0748i −1.21713 + 0.793396i
\(520\) 0 0
\(521\) −15.4043 −0.674873 −0.337436 0.941348i \(-0.609560\pi\)
−0.337436 + 0.941348i \(0.609560\pi\)
\(522\) 0 0
\(523\) 16.9319i 0.740381i −0.928956 0.370191i \(-0.879292\pi\)
0.928956 0.370191i \(-0.120708\pi\)
\(524\) 0 0
\(525\) 4.61278 + 4.70072i 0.201318 + 0.205156i
\(526\) 0 0
\(527\) 5.97329 + 3.44868i 0.260201 + 0.150227i
\(528\) 0 0
\(529\) −5.89462 10.2098i −0.256288 0.443904i
\(530\) 0 0
\(531\) −13.2676 + 30.0777i −0.575764 + 1.30526i
\(532\) 0 0
\(533\) −29.3707 + 16.9572i −1.27219 + 0.734498i
\(534\) 0 0
\(535\) 17.1951 + 9.92760i 0.743409 + 0.429207i
\(536\) 0 0
\(537\) 25.8087 1.39712i 1.11373 0.0602902i
\(538\) 0 0
\(539\) 27.8240 36.5516i 1.19846 1.57439i
\(540\) 0 0
\(541\) 6.44265 0.276991 0.138496 0.990363i \(-0.455773\pi\)
0.138496 + 0.990363i \(0.455773\pi\)
\(542\) 0 0
\(543\) 32.0270 1.73374i 1.37441 0.0744021i
\(544\) 0 0
\(545\) −22.2576 + 38.5512i −0.953409 + 1.65135i
\(546\) 0 0
\(547\) −2.89191 5.00893i −0.123649 0.214166i 0.797555 0.603246i \(-0.206126\pi\)
−0.921204 + 0.389080i \(0.872793\pi\)
\(548\) 0 0
\(549\) 15.4786 11.3275i 0.660611 0.483445i
\(550\) 0 0
\(551\) −0.152120 0.263479i −0.00648051 0.0112246i
\(552\) 0 0
\(553\) 2.59223 2.28218i 0.110233 0.0970480i
\(554\) 0 0
\(555\) −4.56665 2.31693i −0.193844 0.0983483i
\(556\) 0 0
\(557\) 3.86933i 0.163949i −0.996634 0.0819745i \(-0.973877\pi\)
0.996634 0.0819745i \(-0.0261226\pi\)
\(558\) 0 0
\(559\) 31.7302i 1.34204i
\(560\) 0 0
\(561\) −21.8920 33.5840i −0.924280 1.41792i
\(562\) 0 0
\(563\) 12.2712 21.2544i 0.517171 0.895766i −0.482630 0.875824i \(-0.660319\pi\)
0.999801 0.0199420i \(-0.00634815\pi\)
\(564\) 0 0
\(565\) 9.66941 5.58264i 0.406795 0.234863i
\(566\) 0 0
\(567\) 9.59195 + 21.7944i 0.402824 + 0.915278i
\(568\) 0 0
\(569\) −12.0119 + 6.93510i −0.503567 + 0.290735i −0.730185 0.683249i \(-0.760566\pi\)
0.226618 + 0.973984i \(0.427233\pi\)
\(570\) 0 0
\(571\) −3.32890 + 5.76582i −0.139310 + 0.241292i −0.927236 0.374478i \(-0.877822\pi\)
0.787926 + 0.615770i \(0.211155\pi\)
\(572\) 0 0
\(573\) −5.55028 8.51457i −0.231866 0.355701i
\(574\) 0 0
\(575\) 4.81200i 0.200674i
\(576\) 0 0
\(577\) 20.4063i 0.849525i −0.905305 0.424763i \(-0.860358\pi\)
0.905305 0.424763i \(-0.139642\pi\)
\(578\) 0 0
\(579\) −25.3187 12.8457i −1.05221 0.533848i
\(580\) 0 0
\(581\) 18.0011 15.8480i 0.746811 0.657485i
\(582\) 0 0
\(583\) 11.6415 + 20.1636i 0.482140 + 0.835090i
\(584\) 0 0
\(585\) −36.4482 + 26.6733i −1.50695 + 1.10281i
\(586\) 0 0
\(587\) −3.43838 5.95544i −0.141917 0.245807i 0.786302 0.617843i \(-0.211993\pi\)
−0.928218 + 0.372036i \(0.878660\pi\)
\(588\) 0 0
\(589\) −0.255553 + 0.442631i −0.0105299 + 0.0182383i
\(590\) 0 0
\(591\) −11.9355 + 0.646113i −0.490960 + 0.0265775i
\(592\) 0 0
\(593\) 13.7068 0.562871 0.281435 0.959580i \(-0.409189\pi\)
0.281435 + 0.959580i \(0.409189\pi\)
\(594\) 0 0
\(595\) 22.4337 + 7.56705i 0.919692 + 0.310219i
\(596\) 0 0
\(597\) 17.9099 0.969534i 0.733005 0.0396804i
\(598\) 0 0
\(599\) 15.3640 + 8.87044i 0.627758 + 0.362436i 0.779883 0.625925i \(-0.215278\pi\)
−0.152125 + 0.988361i \(0.548612\pi\)
\(600\) 0 0
\(601\) −6.31369 + 3.64521i −0.257541 + 0.148691i −0.623212 0.782053i \(-0.714173\pi\)
0.365671 + 0.930744i \(0.380839\pi\)
\(602\) 0 0
\(603\) −10.9977 + 24.9320i −0.447863 + 1.01531i
\(604\) 0 0
\(605\) −40.6773 70.4551i −1.65377 2.86441i
\(606\) 0 0
\(607\) 6.00042 + 3.46434i 0.243550 + 0.140613i 0.616807 0.787114i \(-0.288426\pi\)
−0.373258 + 0.927728i \(0.621759\pi\)
\(608\) 0 0
\(609\) 3.80751 3.73627i 0.154288 0.151401i
\(610\) 0 0
\(611\) 45.0699i 1.82333i
\(612\) 0 0
\(613\) −4.60656 −0.186057 −0.0930287 0.995663i \(-0.529655\pi\)
−0.0930287 + 0.995663i \(0.529655\pi\)
\(614\) 0 0
\(615\) −21.0406 + 13.7155i −0.848439 + 0.553061i
\(616\) 0 0
\(617\) −25.4618 14.7004i −1.02505 0.591816i −0.109491 0.993988i \(-0.534922\pi\)
−0.915564 + 0.402172i \(0.868255\pi\)
\(618\) 0 0
\(619\) 28.9537 16.7165i 1.16375 0.671891i 0.211550 0.977367i \(-0.432149\pi\)
0.952200 + 0.305476i \(0.0988157\pi\)
\(620\) 0 0
\(621\) 6.15093 16.2744i 0.246829 0.653070i
\(622\) 0 0
\(623\) 7.43169 + 36.9891i 0.297744 + 1.48194i
\(624\) 0 0
\(625\) 15.0602 26.0850i 0.602408 1.04340i
\(626\) 0 0
\(627\) 2.48863 1.62223i 0.0993863 0.0647857i
\(628\) 0 0
\(629\) −4.10991 −0.163873
\(630\) 0 0
\(631\) −26.3381 −1.04850 −0.524251 0.851564i \(-0.675655\pi\)
−0.524251 + 0.851564i \(0.675655\pi\)
\(632\) 0 0
\(633\) 10.6063 + 5.38121i 0.421563 + 0.213884i
\(634\) 0 0
\(635\) 24.8247 42.9976i 0.985136 1.70631i
\(636\) 0 0
\(637\) −41.2024 5.26300i −1.63250 0.208528i
\(638\) 0 0
\(639\) 3.42925 + 31.5810i 0.135659 + 1.24932i
\(640\) 0 0
\(641\) 11.6878 6.74797i 0.461641 0.266529i −0.251093 0.967963i \(-0.580790\pi\)
0.712734 + 0.701434i \(0.247457\pi\)
\(642\) 0 0
\(643\) −19.2270 11.1007i −0.758239 0.437769i 0.0704244 0.997517i \(-0.477565\pi\)
−0.828663 + 0.559748i \(0.810898\pi\)
\(644\) 0 0
\(645\) 1.27021 + 23.4642i 0.0500144 + 0.923902i
\(646\) 0 0
\(647\) −11.6716 −0.458856 −0.229428 0.973326i \(-0.573686\pi\)
−0.229428 + 0.973326i \(0.573686\pi\)
\(648\) 0 0
\(649\) 71.9107i 2.82274i
\(650\) 0 0
\(651\) −8.63404 2.40109i −0.338395 0.0941060i
\(652\) 0 0
\(653\) 20.3964 + 11.7759i 0.798174 + 0.460826i 0.842832 0.538177i \(-0.180887\pi\)
−0.0446586 + 0.999002i \(0.514220\pi\)
\(654\) 0 0
\(655\) −0.0245543 0.0425292i −0.000959414 0.00166175i
\(656\) 0 0
\(657\) 0.883873 + 8.13985i 0.0344832 + 0.317566i
\(658\) 0 0
\(659\) −9.47020 + 5.46763i −0.368907 + 0.212988i −0.672981 0.739660i \(-0.734986\pi\)
0.304074 + 0.952648i \(0.401653\pi\)
\(660\) 0 0
\(661\) −9.96651 5.75417i −0.387652 0.223811i 0.293490 0.955962i \(-0.405183\pi\)
−0.681142 + 0.732151i \(0.738517\pi\)
\(662\) 0 0
\(663\) −16.4014 + 32.3269i −0.636976 + 1.25547i
\(664\) 0 0
\(665\) −0.560731 + 1.66237i −0.0217442 + 0.0644641i
\(666\) 0 0
\(667\) −3.89764 −0.150917
\(668\) 0 0
\(669\) 0.510098 + 0.782529i 0.0197215 + 0.0302543i
\(670\) 0 0
\(671\) 20.9786 36.3360i 0.809870 1.40274i
\(672\) 0 0
\(673\) −14.5621 25.2224i −0.561329 0.972251i −0.997381 0.0723293i \(-0.976957\pi\)
0.436051 0.899922i \(-0.356377\pi\)
\(674\) 0 0
\(675\) −5.77919 + 4.72951i −0.222441 + 0.182039i
\(676\) 0 0
\(677\) −4.60972 7.98426i −0.177166 0.306860i 0.763743 0.645521i \(-0.223360\pi\)
−0.940909 + 0.338660i \(0.890026\pi\)
\(678\) 0 0
\(679\) 20.1005 + 22.8313i 0.771385 + 0.876185i
\(680\) 0 0
\(681\) 1.77171 1.15491i 0.0678923 0.0442561i
\(682\) 0 0
\(683\) 17.2370i 0.659554i 0.944059 + 0.329777i \(0.106974\pi\)
−0.944059 + 0.329777i \(0.893026\pi\)
\(684\) 0 0
\(685\) 10.7528i 0.410843i
\(686\) 0 0
\(687\) 9.41549 18.5578i 0.359223 0.708026i
\(688\) 0 0
\(689\) 10.5265 18.2324i 0.401027 0.694599i
\(690\) 0 0
\(691\) −13.2875 + 7.67157i −0.505482 + 0.291840i −0.730975 0.682405i \(-0.760934\pi\)
0.225492 + 0.974245i \(0.427601\pi\)
\(692\) 0 0
\(693\) 38.4383 + 35.1512i 1.46015 + 1.33528i
\(694\) 0 0
\(695\) −44.7563 + 25.8401i −1.69770 + 0.980170i
\(696\) 0 0
\(697\) −10.0790 + 17.4574i −0.381770 + 0.661245i
\(698\) 0 0
\(699\) −8.87489 + 0.480432i −0.335679 + 0.0181716i
\(700\) 0 0
\(701\) 13.1791i 0.497767i 0.968533 + 0.248883i \(0.0800636\pi\)
−0.968533 + 0.248883i \(0.919936\pi\)
\(702\) 0 0
\(703\) 0.304551i 0.0114864i
\(704\) 0 0
\(705\) −1.80422 33.3288i −0.0679507 1.25524i
\(706\) 0 0
\(707\) 4.33666 3.81795i 0.163097 0.143589i
\(708\) 0 0
\(709\) 8.23396 + 14.2616i 0.309233 + 0.535607i 0.978195 0.207690i \(-0.0665944\pi\)
−0.668962 + 0.743297i \(0.733261\pi\)
\(710\) 0 0
\(711\) 2.31273 + 3.16027i 0.0867341 + 0.118519i
\(712\) 0 0
\(713\) 3.27392 + 5.67059i 0.122609 + 0.212365i
\(714\) 0 0
\(715\) −49.3992 + 85.5620i −1.84743 + 3.19984i
\(716\) 0 0
\(717\) −9.59034 + 18.9025i −0.358158 + 0.705925i
\(718\) 0 0
\(719\) 13.3816 0.499048 0.249524 0.968369i \(-0.419726\pi\)
0.249524 + 0.968369i \(0.419726\pi\)
\(720\) 0 0
\(721\) −9.52591 + 28.2410i −0.354763 + 1.05175i
\(722\) 0 0
\(723\) 27.2298 + 41.7727i 1.01269 + 1.55354i
\(724\) 0 0
\(725\) 1.44885 + 0.836492i 0.0538088 + 0.0310665i
\(726\) 0 0
\(727\) −7.16036 + 4.13403i −0.265563 + 0.153323i −0.626870 0.779124i \(-0.715664\pi\)
0.361307 + 0.932447i \(0.382331\pi\)
\(728\) 0 0
\(729\) −25.5910 + 8.60819i −0.947815 + 0.318822i
\(730\) 0 0
\(731\) 9.42988 + 16.3330i 0.348777 + 0.604099i
\(732\) 0 0
\(733\) −12.1773 7.03056i −0.449778 0.259679i 0.257958 0.966156i \(-0.416950\pi\)
−0.707736 + 0.706477i \(0.750284\pi\)
\(734\) 0 0
\(735\) −30.6796 2.24255i −1.13163 0.0827176i
\(736\) 0 0
\(737\) 59.6082i 2.19570i
\(738\) 0 0
\(739\) 25.8860 0.952231 0.476115 0.879383i \(-0.342044\pi\)
0.476115 + 0.879383i \(0.342044\pi\)
\(740\) 0 0
\(741\) −2.39548 1.21537i −0.0880001 0.0446477i
\(742\) 0 0
\(743\) −36.5554 21.1053i −1.34109 0.774277i −0.354120 0.935200i \(-0.615220\pi\)
−0.986967 + 0.160923i \(0.948553\pi\)
\(744\) 0 0
\(745\) −16.2472 + 9.38032i −0.595251 + 0.343668i
\(746\) 0 0
\(747\) 16.0601 + 21.9456i 0.587610 + 0.802949i
\(748\) 0 0
\(749\) −20.2994 + 4.07846i −0.741723 + 0.149024i
\(750\) 0 0
\(751\) 21.3157 36.9198i 0.777820 1.34722i −0.155375 0.987856i \(-0.549659\pi\)
0.933196 0.359369i \(-0.117008\pi\)
\(752\) 0 0
\(753\) 0.451588 + 8.34207i 0.0164568 + 0.304002i
\(754\) 0 0
\(755\) 35.2253 1.28198
\(756\) 0 0
\(757\) 51.5092 1.87213 0.936067 0.351823i \(-0.114438\pi\)
0.936067 + 0.351823i \(0.114438\pi\)
\(758\) 0 0
\(759\) −2.05719 38.0020i −0.0746713 1.37938i
\(760\) 0 0
\(761\) 15.4845 26.8199i 0.561313 0.972222i −0.436070 0.899913i \(-0.643630\pi\)
0.997382 0.0723091i \(-0.0230368\pi\)
\(762\) 0 0
\(763\) −9.14386 45.5110i −0.331030 1.64761i
\(764\) 0 0
\(765\) −10.8346 + 24.5621i −0.391725 + 0.888043i
\(766\) 0 0
\(767\) 56.3119 32.5117i 2.03331 1.17393i
\(768\) 0 0
\(769\) 41.2004 + 23.7871i 1.48572 + 0.857783i 0.999868 0.0162553i \(-0.00517444\pi\)
0.485856 + 0.874039i \(0.338508\pi\)
\(770\) 0 0
\(771\) −6.64256 3.37016i −0.239226 0.121374i
\(772\) 0 0
\(773\) −27.9971 −1.00699 −0.503493 0.864000i \(-0.667952\pi\)
−0.503493 + 0.864000i \(0.667952\pi\)
\(774\) 0 0
\(775\) 2.81053i 0.100957i
\(776\) 0 0
\(777\) 5.17083 1.33332i 0.185503 0.0478326i
\(778\) 0 0
\(779\) −1.29362 0.746872i −0.0463488 0.0267595i
\(780\) 0 0
\(781\) 34.7442 + 60.1788i 1.24325 + 2.15337i
\(782\) 0 0
\(783\) 3.83082 + 4.68105i 0.136902 + 0.167287i
\(784\) 0 0
\(785\) 4.73721 2.73503i 0.169078 0.0976174i
\(786\) 0 0
\(787\) 8.86189 + 5.11642i 0.315892 + 0.182381i 0.649560 0.760310i \(-0.274953\pi\)
−0.333668 + 0.942691i \(0.608286\pi\)
\(788\) 0 0
\(789\) −21.0180 32.2432i −0.748260 1.14789i
\(790\) 0 0
\(791\) −3.72133 + 11.0324i −0.132315 + 0.392269i
\(792\) 0 0
\(793\) −37.9387 −1.34724
\(794\) 0 0
\(795\) 7.05437 13.9041i 0.250193 0.493128i
\(796\) 0 0
\(797\) 17.7557 30.7538i 0.628940 1.08936i −0.358825 0.933405i \(-0.616823\pi\)
0.987765 0.155951i \(-0.0498441\pi\)
\(798\) 0 0
\(799\) −13.3943 23.1996i −0.473857 0.820744i
\(800\) 0 0
\(801\) −42.5299 + 4.61814i −1.50272 + 0.163174i
\(802\) 0 0
\(803\) 8.95517 + 15.5108i 0.316021 + 0.547365i
\(804\) 0 0
\(805\) 14.8518 + 16.8696i 0.523457 + 0.594574i
\(806\) 0 0
\(807\) −2.02759 37.4552i −0.0713747 1.31849i
\(808\) 0 0
\(809\) 18.4932i 0.650188i −0.945682 0.325094i \(-0.894604\pi\)
0.945682 0.325094i \(-0.105396\pi\)
\(810\) 0 0
\(811\) 35.6320i 1.25121i −0.780140 0.625605i \(-0.784852\pi\)
0.780140 0.625605i \(-0.215148\pi\)
\(812\) 0 0
\(813\) −29.5415 + 1.59920i −1.03607 + 0.0560862i
\(814\) 0 0
\(815\) −21.8026 + 37.7631i −0.763710 + 1.32278i
\(816\) 0 0
\(817\) −1.21031 + 0.698770i −0.0423432 + 0.0244469i
\(818\) 0 0
\(819\) 10.1478 45.9926i 0.354593 1.60711i
\(820\) 0 0
\(821\) 17.1389 9.89514i 0.598151 0.345343i −0.170163 0.985416i \(-0.554429\pi\)
0.768314 + 0.640073i \(0.221096\pi\)
\(822\) 0 0
\(823\) −3.47939 + 6.02647i −0.121284 + 0.210070i −0.920274 0.391274i \(-0.872034\pi\)
0.798990 + 0.601344i \(0.205368\pi\)
\(824\) 0 0
\(825\) −7.39109 + 14.5678i −0.257325 + 0.507184i
\(826\) 0 0
\(827\) 4.92505i 0.171261i 0.996327 + 0.0856304i \(0.0272904\pi\)
−0.996327 + 0.0856304i \(0.972710\pi\)
\(828\) 0 0
\(829\) 42.9274i 1.49093i 0.666544 + 0.745466i \(0.267773\pi\)
−0.666544 + 0.745466i \(0.732227\pi\)
\(830\) 0 0
\(831\) 4.65710 3.03577i 0.161553 0.105310i
\(832\) 0 0
\(833\) −22.7730 + 9.53583i −0.789037 + 0.330397i
\(834\) 0 0
\(835\) −13.7032 23.7347i −0.474219 0.821372i
\(836\) 0 0
\(837\) 3.59256 9.50534i 0.124177 0.328553i
\(838\) 0 0
\(839\) 8.84639 + 15.3224i 0.305411 + 0.528988i 0.977353 0.211616i \(-0.0678727\pi\)
−0.671941 + 0.740604i \(0.734539\pi\)
\(840\) 0 0
\(841\) −13.8225 + 23.9412i −0.476636 + 0.825558i
\(842\) 0 0
\(843\) 19.7244 + 30.2587i 0.679343 + 1.04217i
\(844\) 0 0
\(845\) 56.3529 1.93860
\(846\) 0 0
\(847\) 80.3867 + 27.1150i 2.76212 + 0.931683i
\(848\) 0 0
\(849\) −9.04859 + 17.8347i −0.310547 + 0.612085i
\(850\) 0 0
\(851\) −3.37891 1.95082i −0.115828 0.0668731i
\(852\) 0 0
\(853\) 23.5730 13.6099i 0.807123 0.465993i −0.0388328 0.999246i \(-0.512364\pi\)
0.845956 + 0.533253i \(0.179031\pi\)
\(854\) 0 0
\(855\) −1.82009 0.802860i −0.0622457 0.0274572i
\(856\) 0 0
\(857\) 12.8303 + 22.2227i 0.438274 + 0.759113i 0.997557 0.0698641i \(-0.0222566\pi\)
−0.559282 + 0.828977i \(0.688923\pi\)
\(858\) 0 0
\(859\) 16.8087 + 9.70452i 0.573506 + 0.331114i 0.758549 0.651617i \(-0.225909\pi\)
−0.185042 + 0.982731i \(0.559242\pi\)
\(860\) 0 0
\(861\) 7.01734 25.2336i 0.239150 0.859958i
\(862\) 0 0
\(863\) 22.6418i 0.770737i −0.922763 0.385368i \(-0.874074\pi\)
0.922763 0.385368i \(-0.125926\pi\)
\(864\) 0 0
\(865\) 48.4845 1.64852
\(866\) 0 0
\(867\) −0.426969 7.88729i −0.0145006 0.267866i
\(868\) 0 0
\(869\) 7.41871 + 4.28320i 0.251663 + 0.145298i
\(870\) 0 0
\(871\) 46.6780 26.9496i 1.58162 0.913152i
\(872\) 0 0
\(873\) −27.8343 + 20.3696i −0.942049 + 0.689405i
\(874\) 0 0
\(875\) 4.71099 + 23.4476i 0.159261 + 0.792674i
\(876\) 0 0
\(877\) −17.6395 + 30.5525i −0.595644 + 1.03169i 0.397812 + 0.917467i \(0.369770\pi\)
−0.993456 + 0.114219i \(0.963564\pi\)
\(878\) 0 0
\(879\) −43.7437 22.1938i −1.47544 0.748578i
\(880\) 0 0
\(881\) −0.975907 −0.0328791 −0.0164396 0.999865i \(-0.505233\pi\)
−0.0164396 + 0.999865i \(0.505233\pi\)
\(882\) 0 0
\(883\) 3.26917 0.110016 0.0550082 0.998486i \(-0.482481\pi\)
0.0550082 + 0.998486i \(0.482481\pi\)
\(884\) 0 0
\(885\) 40.3407 26.2964i 1.35604 0.883944i
\(886\) 0 0
\(887\) −12.1653 + 21.0709i −0.408470 + 0.707490i −0.994718 0.102641i \(-0.967271\pi\)
0.586249 + 0.810131i \(0.300604\pi\)
\(888\) 0 0
\(889\) 10.1985 + 50.7600i 0.342046 + 1.70244i
\(890\) 0 0
\(891\) −43.6183 + 39.8212i −1.46127 + 1.33406i
\(892\) 0 0
\(893\) 1.71913 0.992541i 0.0575285 0.0332141i
\(894\) 0 0
\(895\) −32.7882 18.9303i −1.09599 0.632770i
\(896\) 0 0
\(897\) −28.8285 + 18.7921i −0.962556 + 0.627449i
\(898\) 0 0
\(899\) −2.27648 −0.0759249
\(900\) 0 0
\(901\) 12.5135i 0.416884i
\(902\) 0 0
\(903\) −17.1628 17.4900i −0.571142 0.582031i
\(904\) 0 0
\(905\) −40.6882 23.4914i −1.35252 0.780880i
\(906\) 0 0
\(907\) −22.0508 38.1932i −0.732186 1.26818i −0.955947 0.293539i \(-0.905167\pi\)
0.223761 0.974644i \(-0.428166\pi\)
\(908\) 0 0
\(909\) 3.86907 + 5.28695i 0.128329 + 0.175357i
\(910\) 0 0
\(911\) −20.1215 + 11.6171i −0.666654 + 0.384893i −0.794807 0.606862i \(-0.792428\pi\)
0.128154 + 0.991754i \(0.459095\pi\)
\(912\) 0 0
\(913\) 51.5173 + 29.7435i 1.70497 + 0.984368i
\(914\) 0 0
\(915\) −28.0554 + 1.51874i −0.927481 + 0.0502081i
\(916\) 0 0
\(917\) 0.0485243 + 0.0163676i 0.00160241 + 0.000540506i
\(918\) 0 0
\(919\) 2.91416 0.0961291 0.0480646 0.998844i \(-0.484695\pi\)
0.0480646 + 0.998844i \(0.484695\pi\)
\(920\) 0 0
\(921\) 41.9959 2.27340i 1.38381 0.0749110i
\(922\) 0 0
\(923\) 31.4166 54.4151i 1.03409 1.79109i
\(924\) 0 0
\(925\) 0.837349 + 1.45033i 0.0275319 + 0.0476866i
\(926\) 0 0
\(927\) −30.9204 13.6393i −1.01556 0.447973i
\(928\) 0 0
\(929\) 8.34767 + 14.4586i 0.273878 + 0.474371i 0.969852 0.243697i \(-0.0783601\pi\)
−0.695973 + 0.718068i \(0.745027\pi\)
\(930\) 0 0
\(931\) −0.706621 1.68752i −0.0231586 0.0553061i
\(932\) 0 0
\(933\) −50.3914 25.5666i −1.64974 0.837012i
\(934\) 0 0
\(935\) 58.7238i 1.92047i
\(936\) 0 0
\(937\) 40.4272i 1.32070i −0.750959 0.660349i \(-0.770408\pi\)
0.750959 0.660349i \(-0.229592\pi\)
\(938\) 0 0
\(939\) 13.7231 + 21.0523i 0.447836 + 0.687015i
\(940\) 0 0
\(941\) 27.9889 48.4782i 0.912411 1.58034i 0.101764 0.994809i \(-0.467551\pi\)
0.810648 0.585534i \(-0.199115\pi\)
\(942\) 0 0
\(943\) −16.5727 + 9.56824i −0.539681 + 0.311585i
\(944\) 0 0
\(945\) 5.66307 34.4174i 0.184220 1.11960i
\(946\) 0 0
\(947\) 17.8498 10.3056i 0.580041 0.334887i −0.181109 0.983463i \(-0.557969\pi\)
0.761150 + 0.648576i \(0.224635\pi\)
\(948\) 0 0
\(949\) 8.09748 14.0253i 0.262855 0.455279i
\(950\) 0 0
\(951\) 15.3101 + 23.4869i 0.496464 + 0.761615i
\(952\) 0 0
\(953\) 17.3803i 0.563005i 0.959561 + 0.281502i \(0.0908327\pi\)
−0.959561 + 0.281502i \(0.909167\pi\)
\(954\) 0 0
\(955\) 14.8883i 0.481773i
\(956\) 0 0
\(957\) 11.7996 + 5.98666i 0.381428 + 0.193521i
\(958\) 0 0
\(959\) −7.40949 8.41614i −0.239265 0.271771i
\(960\) 0 0
\(961\) −13.5878 23.5348i −0.438317 0.759187i
\(962\) 0 0
\(963\) −2.53440 23.3401i −0.0816700 0.752124i
\(964\) 0 0
\(965\) 20.7940 + 36.0162i 0.669381 + 1.15940i
\(966\) 0 0
\(967\) 1.32992 2.30349i 0.0427674 0.0740753i −0.843849 0.536580i \(-0.819716\pi\)
0.886617 + 0.462505i \(0.153049\pi\)
\(968\) 0 0
\(969\) −1.59426 + 0.0863035i −0.0512151 + 0.00277247i
\(970\) 0 0
\(971\) −34.9474 −1.12151 −0.560757 0.827980i \(-0.689490\pi\)
−0.560757 + 0.827980i \(0.689490\pi\)
\(972\) 0 0
\(973\) 17.2247 51.0654i 0.552199 1.63708i
\(974\) 0 0
\(975\) 14.7493 0.798438i 0.472357 0.0255705i
\(976\) 0 0
\(977\) 13.6938 + 7.90614i 0.438105 + 0.252940i 0.702793 0.711394i \(-0.251936\pi\)
−0.264689 + 0.964334i \(0.585269\pi\)
\(978\) 0 0
\(979\) −81.0423 + 46.7898i −2.59012 + 1.49541i
\(980\) 0 0
\(981\) 52.3282 5.68211i 1.67071 0.181416i
\(982\) 0 0
\(983\) −30.2832 52.4521i −0.965885 1.67296i −0.707219 0.706994i \(-0.750051\pi\)
−0.258666 0.965967i \(-0.583283\pi\)
\(984\) 0 0
\(985\) 15.1632 + 8.75451i 0.483141 + 0.278942i
\(986\) 0 0
\(987\) 24.3782 + 24.8430i 0.775967 + 0.790761i
\(988\) 0 0
\(989\) 17.9040i 0.569315i
\(990\) 0 0
\(991\) 8.00131 0.254170 0.127085 0.991892i \(-0.459438\pi\)
0.127085 + 0.991892i \(0.459438\pi\)
\(992\) 0 0
\(993\) −42.3848 + 27.6288i −1.34504 + 0.876774i
\(994\) 0 0
\(995\) −22.7534 13.1367i −0.721333 0.416462i
\(996\) 0 0
\(997\) −37.8349 + 21.8440i −1.19824 + 0.691805i −0.960163 0.279440i \(-0.909851\pi\)
−0.238079 + 0.971246i \(0.576518\pi\)
\(998\) 0 0
\(999\) 0.978070 + 5.97544i 0.0309448 + 0.189054i
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 504.2.bu.a.41.13 yes 48
3.2 odd 2 1512.2.bu.a.881.5 48
4.3 odd 2 1008.2.cc.d.545.12 48
7.6 odd 2 inner 504.2.bu.a.41.12 48
9.2 odd 6 inner 504.2.bu.a.209.12 yes 48
9.4 even 3 4536.2.k.a.3401.10 48
9.5 odd 6 4536.2.k.a.3401.39 48
9.7 even 3 1512.2.bu.a.1385.20 48
12.11 even 2 3024.2.cc.d.881.5 48
21.20 even 2 1512.2.bu.a.881.20 48
28.27 even 2 1008.2.cc.d.545.13 48
36.7 odd 6 3024.2.cc.d.2897.20 48
36.11 even 6 1008.2.cc.d.209.13 48
63.13 odd 6 4536.2.k.a.3401.40 48
63.20 even 6 inner 504.2.bu.a.209.13 yes 48
63.34 odd 6 1512.2.bu.a.1385.5 48
63.41 even 6 4536.2.k.a.3401.9 48
84.83 odd 2 3024.2.cc.d.881.20 48
252.83 odd 6 1008.2.cc.d.209.12 48
252.223 even 6 3024.2.cc.d.2897.5 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
504.2.bu.a.41.12 48 7.6 odd 2 inner
504.2.bu.a.41.13 yes 48 1.1 even 1 trivial
504.2.bu.a.209.12 yes 48 9.2 odd 6 inner
504.2.bu.a.209.13 yes 48 63.20 even 6 inner
1008.2.cc.d.209.12 48 252.83 odd 6
1008.2.cc.d.209.13 48 36.11 even 6
1008.2.cc.d.545.12 48 4.3 odd 2
1008.2.cc.d.545.13 48 28.27 even 2
1512.2.bu.a.881.5 48 3.2 odd 2
1512.2.bu.a.881.20 48 21.20 even 2
1512.2.bu.a.1385.5 48 63.34 odd 6
1512.2.bu.a.1385.20 48 9.7 even 3
3024.2.cc.d.881.5 48 12.11 even 2
3024.2.cc.d.881.20 48 84.83 odd 2
3024.2.cc.d.2897.5 48 252.223 even 6
3024.2.cc.d.2897.20 48 36.7 odd 6
4536.2.k.a.3401.9 48 63.41 even 6
4536.2.k.a.3401.10 48 9.4 even 3
4536.2.k.a.3401.39 48 9.5 odd 6
4536.2.k.a.3401.40 48 63.13 odd 6