Properties

Label 504.2.t.d.193.3
Level $504$
Weight $2$
Character 504.193
Analytic conductor $4.024$
Analytic rank $0$
Dimension $22$
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] = [504,2,Mod(193,504)]
 
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, 2, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("504.193");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 504.t (of order \(3\), 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: \(22\)
Relative dimension: \(11\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 193.3
Character \(\chi\) \(=\) 504.193
Dual form 504.2.t.d.457.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.21966 + 1.22980i) q^{3} -0.481387 q^{5} +(2.53326 - 0.763277i) q^{7} +(-0.0248369 - 2.99990i) q^{9} +O(q^{10})\) \(q+(-1.21966 + 1.22980i) q^{3} -0.481387 q^{5} +(2.53326 - 0.763277i) q^{7} +(-0.0248369 - 2.99990i) q^{9} +3.38159 q^{11} +(-2.86067 + 4.95482i) q^{13} +(0.587131 - 0.592012i) q^{15} +(2.75605 - 4.77362i) q^{17} +(2.18023 + 3.77626i) q^{19} +(-2.15105 + 4.04636i) q^{21} +3.62585 q^{23} -4.76827 q^{25} +(3.71958 + 3.62832i) q^{27} +(1.53131 + 2.65231i) q^{29} +(4.67459 + 8.09663i) q^{31} +(-4.12441 + 4.15870i) q^{33} +(-1.21948 + 0.367431i) q^{35} +(1.48552 + 2.57299i) q^{37} +(-2.60441 - 9.56128i) q^{39} +(-6.29558 + 10.9043i) q^{41} +(1.90827 + 3.30522i) q^{43} +(0.0119562 + 1.44411i) q^{45} +(1.88282 - 3.26114i) q^{47} +(5.83482 - 3.86716i) q^{49} +(2.50916 + 9.21161i) q^{51} +(5.57860 - 9.66242i) q^{53} -1.62786 q^{55} +(-7.30321 - 1.92452i) q^{57} +(-4.21141 - 7.29438i) q^{59} +(3.64312 - 6.31007i) q^{61} +(-2.35267 - 7.58056i) q^{63} +(1.37709 - 2.38519i) q^{65} +(-1.28571 - 2.22692i) q^{67} +(-4.42232 + 4.45909i) q^{69} -3.94304 q^{71} +(-0.862216 + 1.49340i) q^{73} +(5.81569 - 5.86403i) q^{75} +(8.56646 - 2.58109i) q^{77} +(2.79980 - 4.84940i) q^{79} +(-8.99877 + 0.149016i) q^{81} +(-0.119494 - 0.206970i) q^{83} +(-1.32673 + 2.29796i) q^{85} +(-5.12952 - 1.35172i) q^{87} +(0.648116 + 1.12257i) q^{89} +(-3.46492 + 14.7353i) q^{91} +(-15.6587 - 4.12634i) q^{93} +(-1.04953 - 1.81784i) q^{95} +(-7.02669 - 12.1706i) q^{97} +(-0.0839884 - 10.1444i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 22 q + 2 q^{3} - 6 q^{5} + 7 q^{7} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 22 q + 2 q^{3} - 6 q^{5} + 7 q^{7} - 8 q^{9} + 6 q^{11} - 3 q^{13} - q^{15} + 7 q^{17} - q^{19} - 15 q^{21} - 4 q^{23} + 20 q^{25} - 4 q^{27} + 9 q^{29} - 4 q^{31} - 31 q^{33} + 14 q^{35} + 2 q^{37} + 8 q^{39} + 16 q^{41} + 22 q^{45} + 5 q^{47} - 15 q^{49} + 7 q^{51} + 11 q^{53} + 22 q^{55} + 7 q^{57} - 19 q^{59} - 13 q^{61} + 21 q^{63} + 13 q^{65} + 26 q^{67} - 4 q^{69} - 48 q^{71} - 35 q^{73} - 8 q^{75} - 4 q^{77} + 10 q^{79} - 8 q^{81} - 28 q^{83} - 20 q^{85} + 9 q^{87} + 6 q^{89} - 37 q^{91} - 32 q^{93} + 12 q^{95} - 29 q^{97} - 56 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(73\) \(127\) \(253\) \(281\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\)

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
\(3\) −1.21966 + 1.22980i −0.704174 + 0.710028i
\(4\) 0 0
\(5\) −0.481387 −0.215283 −0.107641 0.994190i \(-0.534330\pi\)
−0.107641 + 0.994190i \(0.534330\pi\)
\(6\) 0 0
\(7\) 2.53326 0.763277i 0.957482 0.288491i
\(8\) 0 0
\(9\) −0.0248369 2.99990i −0.00827898 0.999966i
\(10\) 0 0
\(11\) 3.38159 1.01959 0.509794 0.860296i \(-0.329721\pi\)
0.509794 + 0.860296i \(0.329721\pi\)
\(12\) 0 0
\(13\) −2.86067 + 4.95482i −0.793406 + 1.37422i 0.130440 + 0.991456i \(0.458361\pi\)
−0.923846 + 0.382764i \(0.874972\pi\)
\(14\) 0 0
\(15\) 0.587131 0.592012i 0.151596 0.152857i
\(16\) 0 0
\(17\) 2.75605 4.77362i 0.668440 1.15777i −0.309900 0.950769i \(-0.600296\pi\)
0.978340 0.207003i \(-0.0663711\pi\)
\(18\) 0 0
\(19\) 2.18023 + 3.77626i 0.500178 + 0.866334i 1.00000 0.000205746i \(6.54909e-5\pi\)
−0.499822 + 0.866128i \(0.666601\pi\)
\(20\) 0 0
\(21\) −2.15105 + 4.04636i −0.469397 + 0.882987i
\(22\) 0 0
\(23\) 3.62585 0.756042 0.378021 0.925797i \(-0.376605\pi\)
0.378021 + 0.925797i \(0.376605\pi\)
\(24\) 0 0
\(25\) −4.76827 −0.953653
\(26\) 0 0
\(27\) 3.71958 + 3.62832i 0.715833 + 0.698271i
\(28\) 0 0
\(29\) 1.53131 + 2.65231i 0.284358 + 0.492522i 0.972453 0.233098i \(-0.0748863\pi\)
−0.688095 + 0.725620i \(0.741553\pi\)
\(30\) 0 0
\(31\) 4.67459 + 8.09663i 0.839581 + 1.45420i 0.890245 + 0.455481i \(0.150533\pi\)
−0.0506646 + 0.998716i \(0.516134\pi\)
\(32\) 0 0
\(33\) −4.12441 + 4.15870i −0.717968 + 0.723936i
\(34\) 0 0
\(35\) −1.21948 + 0.367431i −0.206130 + 0.0621072i
\(36\) 0 0
\(37\) 1.48552 + 2.57299i 0.244218 + 0.422997i 0.961911 0.273361i \(-0.0881355\pi\)
−0.717694 + 0.696359i \(0.754802\pi\)
\(38\) 0 0
\(39\) −2.60441 9.56128i −0.417039 1.53103i
\(40\) 0 0
\(41\) −6.29558 + 10.9043i −0.983204 + 1.70296i −0.333545 + 0.942734i \(0.608245\pi\)
−0.649659 + 0.760226i \(0.725088\pi\)
\(42\) 0 0
\(43\) 1.90827 + 3.30522i 0.291009 + 0.504042i 0.974049 0.226339i \(-0.0726758\pi\)
−0.683040 + 0.730381i \(0.739342\pi\)
\(44\) 0 0
\(45\) 0.0119562 + 1.44411i 0.00178232 + 0.215275i
\(46\) 0 0
\(47\) 1.88282 3.26114i 0.274638 0.475687i −0.695406 0.718617i \(-0.744775\pi\)
0.970044 + 0.242930i \(0.0781087\pi\)
\(48\) 0 0
\(49\) 5.83482 3.86716i 0.833545 0.552451i
\(50\) 0 0
\(51\) 2.50916 + 9.21161i 0.351353 + 1.28988i
\(52\) 0 0
\(53\) 5.57860 9.66242i 0.766280 1.32724i −0.173287 0.984871i \(-0.555439\pi\)
0.939567 0.342364i \(-0.111228\pi\)
\(54\) 0 0
\(55\) −1.62786 −0.219500
\(56\) 0 0
\(57\) −7.30321 1.92452i −0.967334 0.254909i
\(58\) 0 0
\(59\) −4.21141 7.29438i −0.548279 0.949647i −0.998393 0.0566756i \(-0.981950\pi\)
0.450114 0.892971i \(-0.351383\pi\)
\(60\) 0 0
\(61\) 3.64312 6.31007i 0.466454 0.807922i −0.532812 0.846234i \(-0.678865\pi\)
0.999266 + 0.0383116i \(0.0121979\pi\)
\(62\) 0 0
\(63\) −2.35267 7.58056i −0.296409 0.955061i
\(64\) 0 0
\(65\) 1.37709 2.38519i 0.170807 0.295846i
\(66\) 0 0
\(67\) −1.28571 2.22692i −0.157075 0.272062i 0.776738 0.629824i \(-0.216873\pi\)
−0.933813 + 0.357763i \(0.883540\pi\)
\(68\) 0 0
\(69\) −4.42232 + 4.45909i −0.532385 + 0.536811i
\(70\) 0 0
\(71\) −3.94304 −0.467953 −0.233977 0.972242i \(-0.575174\pi\)
−0.233977 + 0.972242i \(0.575174\pi\)
\(72\) 0 0
\(73\) −0.862216 + 1.49340i −0.100915 + 0.174790i −0.912062 0.410053i \(-0.865510\pi\)
0.811147 + 0.584842i \(0.198844\pi\)
\(74\) 0 0
\(75\) 5.81569 5.86403i 0.671538 0.677120i
\(76\) 0 0
\(77\) 8.56646 2.58109i 0.976238 0.294143i
\(78\) 0 0
\(79\) 2.79980 4.84940i 0.315002 0.545600i −0.664436 0.747345i \(-0.731328\pi\)
0.979438 + 0.201746i \(0.0646614\pi\)
\(80\) 0 0
\(81\) −8.99877 + 0.149016i −0.999863 + 0.0165574i
\(82\) 0 0
\(83\) −0.119494 0.206970i −0.0131162 0.0227179i 0.859393 0.511316i \(-0.170842\pi\)
−0.872509 + 0.488598i \(0.837508\pi\)
\(84\) 0 0
\(85\) −1.32673 + 2.29796i −0.143904 + 0.249249i
\(86\) 0 0
\(87\) −5.12952 1.35172i −0.549942 0.144919i
\(88\) 0 0
\(89\) 0.648116 + 1.12257i 0.0687002 + 0.118992i 0.898329 0.439323i \(-0.144781\pi\)
−0.829629 + 0.558315i \(0.811448\pi\)
\(90\) 0 0
\(91\) −3.46492 + 14.7353i −0.363222 + 1.54468i
\(92\) 0 0
\(93\) −15.6587 4.12634i −1.62373 0.427881i
\(94\) 0 0
\(95\) −1.04953 1.81784i −0.107680 0.186507i
\(96\) 0 0
\(97\) −7.02669 12.1706i −0.713452 1.23574i −0.963553 0.267516i \(-0.913797\pi\)
0.250101 0.968220i \(-0.419536\pi\)
\(98\) 0 0
\(99\) −0.0839884 10.1444i −0.00844115 1.01955i
\(100\) 0 0
\(101\) 10.6064 1.05538 0.527690 0.849437i \(-0.323058\pi\)
0.527690 + 0.849437i \(0.323058\pi\)
\(102\) 0 0
\(103\) −0.159416 −0.0157077 −0.00785385 0.999969i \(-0.502500\pi\)
−0.00785385 + 0.999969i \(0.502500\pi\)
\(104\) 0 0
\(105\) 1.03549 1.94786i 0.101053 0.190092i
\(106\) 0 0
\(107\) −3.99030 6.91140i −0.385757 0.668150i 0.606117 0.795375i \(-0.292726\pi\)
−0.991874 + 0.127225i \(0.959393\pi\)
\(108\) 0 0
\(109\) −6.85612 + 11.8751i −0.656697 + 1.13743i 0.324769 + 0.945793i \(0.394714\pi\)
−0.981466 + 0.191639i \(0.938620\pi\)
\(110\) 0 0
\(111\) −4.97611 1.31129i −0.472312 0.124462i
\(112\) 0 0
\(113\) 8.98656 15.5652i 0.845384 1.46425i −0.0399031 0.999204i \(-0.512705\pi\)
0.885287 0.465045i \(-0.153962\pi\)
\(114\) 0 0
\(115\) −1.74544 −0.162763
\(116\) 0 0
\(117\) 14.9350 + 8.45865i 1.38074 + 0.782002i
\(118\) 0 0
\(119\) 3.33820 14.1964i 0.306012 1.30139i
\(120\) 0 0
\(121\) 0.435176 0.0395615
\(122\) 0 0
\(123\) −5.73161 21.0419i −0.516802 1.89728i
\(124\) 0 0
\(125\) 4.70232 0.420588
\(126\) 0 0
\(127\) 18.9684 1.68317 0.841587 0.540121i \(-0.181622\pi\)
0.841587 + 0.540121i \(0.181622\pi\)
\(128\) 0 0
\(129\) −6.39223 1.68446i −0.562804 0.148309i
\(130\) 0 0
\(131\) −4.88232 −0.426570 −0.213285 0.976990i \(-0.568416\pi\)
−0.213285 + 0.976990i \(0.568416\pi\)
\(132\) 0 0
\(133\) 8.40541 + 7.90214i 0.728842 + 0.685203i
\(134\) 0 0
\(135\) −1.79056 1.74663i −0.154107 0.150326i
\(136\) 0 0
\(137\) −6.47482 −0.553181 −0.276591 0.960988i \(-0.589205\pi\)
−0.276591 + 0.960988i \(0.589205\pi\)
\(138\) 0 0
\(139\) −11.3740 + 19.7003i −0.964727 + 1.67096i −0.254381 + 0.967104i \(0.581872\pi\)
−0.710346 + 0.703852i \(0.751462\pi\)
\(140\) 0 0
\(141\) 1.71416 + 6.29300i 0.144358 + 0.529967i
\(142\) 0 0
\(143\) −9.67362 + 16.7552i −0.808948 + 1.40114i
\(144\) 0 0
\(145\) −0.737155 1.27679i −0.0612174 0.106032i
\(146\) 0 0
\(147\) −2.36067 + 11.8923i −0.194705 + 0.980862i
\(148\) 0 0
\(149\) −14.2046 −1.16369 −0.581843 0.813301i \(-0.697668\pi\)
−0.581843 + 0.813301i \(0.697668\pi\)
\(150\) 0 0
\(151\) 2.52259 0.205285 0.102643 0.994718i \(-0.467270\pi\)
0.102643 + 0.994718i \(0.467270\pi\)
\(152\) 0 0
\(153\) −14.3888 8.14930i −1.16327 0.658832i
\(154\) 0 0
\(155\) −2.25029 3.89761i −0.180747 0.313064i
\(156\) 0 0
\(157\) −8.74064 15.1392i −0.697579 1.20824i −0.969303 0.245867i \(-0.920927\pi\)
0.271724 0.962375i \(-0.412406\pi\)
\(158\) 0 0
\(159\) 5.07887 + 18.6455i 0.402780 + 1.47868i
\(160\) 0 0
\(161\) 9.18523 2.76753i 0.723897 0.218112i
\(162\) 0 0
\(163\) 0.881184 + 1.52625i 0.0690196 + 0.119546i 0.898470 0.439035i \(-0.144680\pi\)
−0.829450 + 0.558580i \(0.811346\pi\)
\(164\) 0 0
\(165\) 1.98544 2.00194i 0.154566 0.155851i
\(166\) 0 0
\(167\) −3.57220 + 6.18723i −0.276425 + 0.478782i −0.970494 0.241127i \(-0.922483\pi\)
0.694069 + 0.719909i \(0.255816\pi\)
\(168\) 0 0
\(169\) −9.86684 17.0899i −0.758988 1.31461i
\(170\) 0 0
\(171\) 11.2742 6.63424i 0.862163 0.507333i
\(172\) 0 0
\(173\) −4.94691 + 8.56830i −0.376107 + 0.651436i −0.990492 0.137570i \(-0.956071\pi\)
0.614385 + 0.789006i \(0.289404\pi\)
\(174\) 0 0
\(175\) −12.0793 + 3.63951i −0.913106 + 0.275121i
\(176\) 0 0
\(177\) 14.1072 + 3.71748i 1.06036 + 0.279423i
\(178\) 0 0
\(179\) 2.02967 3.51550i 0.151705 0.262761i −0.780149 0.625593i \(-0.784857\pi\)
0.931854 + 0.362833i \(0.118190\pi\)
\(180\) 0 0
\(181\) 4.58084 0.340491 0.170246 0.985402i \(-0.445544\pi\)
0.170246 + 0.985402i \(0.445544\pi\)
\(182\) 0 0
\(183\) 3.31677 + 12.1765i 0.245183 + 0.900113i
\(184\) 0 0
\(185\) −0.715109 1.23861i −0.0525759 0.0910641i
\(186\) 0 0
\(187\) 9.31984 16.1424i 0.681534 1.18045i
\(188\) 0 0
\(189\) 12.1921 + 6.35242i 0.886843 + 0.462071i
\(190\) 0 0
\(191\) −5.59624 + 9.69298i −0.404930 + 0.701359i −0.994313 0.106494i \(-0.966037\pi\)
0.589383 + 0.807853i \(0.299371\pi\)
\(192\) 0 0
\(193\) −8.14679 14.1106i −0.586419 1.01571i −0.994697 0.102849i \(-0.967204\pi\)
0.408278 0.912857i \(-0.366129\pi\)
\(194\) 0 0
\(195\) 1.25373 + 4.60268i 0.0897813 + 0.329605i
\(196\) 0 0
\(197\) −3.17438 −0.226165 −0.113082 0.993586i \(-0.536072\pi\)
−0.113082 + 0.993586i \(0.536072\pi\)
\(198\) 0 0
\(199\) 1.44140 2.49658i 0.102178 0.176978i −0.810404 0.585872i \(-0.800752\pi\)
0.912582 + 0.408894i \(0.134085\pi\)
\(200\) 0 0
\(201\) 4.30681 + 1.13492i 0.303779 + 0.0800511i
\(202\) 0 0
\(203\) 5.90367 + 5.55019i 0.414356 + 0.389547i
\(204\) 0 0
\(205\) 3.03061 5.24917i 0.211667 0.366618i
\(206\) 0 0
\(207\) −0.0900550 10.8772i −0.00625925 0.756016i
\(208\) 0 0
\(209\) 7.37264 + 12.7698i 0.509976 + 0.883305i
\(210\) 0 0
\(211\) −0.242718 + 0.420400i −0.0167094 + 0.0289415i −0.874259 0.485459i \(-0.838652\pi\)
0.857550 + 0.514401i \(0.171986\pi\)
\(212\) 0 0
\(213\) 4.80919 4.84917i 0.329520 0.332260i
\(214\) 0 0
\(215\) −0.918617 1.59109i −0.0626492 0.108512i
\(216\) 0 0
\(217\) 18.0219 + 16.9429i 1.22341 + 1.15016i
\(218\) 0 0
\(219\) −0.784978 2.88181i −0.0530439 0.194734i
\(220\) 0 0
\(221\) 15.7683 + 27.3115i 1.06069 + 1.83717i
\(222\) 0 0
\(223\) 2.14795 + 3.72037i 0.143838 + 0.249134i 0.928939 0.370234i \(-0.120722\pi\)
−0.785101 + 0.619368i \(0.787389\pi\)
\(224\) 0 0
\(225\) 0.118429 + 14.3043i 0.00789527 + 0.953621i
\(226\) 0 0
\(227\) −17.3827 −1.15373 −0.576866 0.816839i \(-0.695725\pi\)
−0.576866 + 0.816839i \(0.695725\pi\)
\(228\) 0 0
\(229\) 7.33125 0.484463 0.242231 0.970218i \(-0.422121\pi\)
0.242231 + 0.970218i \(0.422121\pi\)
\(230\) 0 0
\(231\) −7.27397 + 13.6831i −0.478592 + 0.900284i
\(232\) 0 0
\(233\) −2.16624 3.75205i −0.141915 0.245805i 0.786302 0.617842i \(-0.211993\pi\)
−0.928218 + 0.372037i \(0.878659\pi\)
\(234\) 0 0
\(235\) −0.906366 + 1.56987i −0.0591248 + 0.102407i
\(236\) 0 0
\(237\) 2.54899 + 9.35784i 0.165575 + 0.607857i
\(238\) 0 0
\(239\) 1.77960 3.08236i 0.115113 0.199381i −0.802712 0.596367i \(-0.796610\pi\)
0.917825 + 0.396986i \(0.129944\pi\)
\(240\) 0 0
\(241\) 16.0185 1.03184 0.515921 0.856636i \(-0.327450\pi\)
0.515921 + 0.856636i \(0.327450\pi\)
\(242\) 0 0
\(243\) 10.7922 11.2485i 0.692321 0.721590i
\(244\) 0 0
\(245\) −2.80881 + 1.86160i −0.179448 + 0.118933i
\(246\) 0 0
\(247\) −24.9476 −1.58738
\(248\) 0 0
\(249\) 0.400276 + 0.105480i 0.0253665 + 0.00668450i
\(250\) 0 0
\(251\) −12.8007 −0.807972 −0.403986 0.914765i \(-0.632375\pi\)
−0.403986 + 0.914765i \(0.632375\pi\)
\(252\) 0 0
\(253\) 12.2612 0.770852
\(254\) 0 0
\(255\) −1.20788 4.43435i −0.0756402 0.277690i
\(256\) 0 0
\(257\) 16.4154 1.02396 0.511981 0.858997i \(-0.328912\pi\)
0.511981 + 0.858997i \(0.328912\pi\)
\(258\) 0 0
\(259\) 5.72711 + 5.38420i 0.355865 + 0.334558i
\(260\) 0 0
\(261\) 7.91864 4.65966i 0.490151 0.288426i
\(262\) 0 0
\(263\) 25.6528 1.58182 0.790910 0.611933i \(-0.209608\pi\)
0.790910 + 0.611933i \(0.209608\pi\)
\(264\) 0 0
\(265\) −2.68547 + 4.65136i −0.164967 + 0.285731i
\(266\) 0 0
\(267\) −2.17103 0.572103i −0.132865 0.0350121i
\(268\) 0 0
\(269\) 5.35397 9.27335i 0.326437 0.565406i −0.655365 0.755313i \(-0.727485\pi\)
0.981802 + 0.189906i \(0.0608184\pi\)
\(270\) 0 0
\(271\) −12.7513 22.0859i −0.774587 1.34162i −0.935026 0.354578i \(-0.884624\pi\)
0.160439 0.987046i \(-0.448709\pi\)
\(272\) 0 0
\(273\) −13.8955 22.2333i −0.840996 1.34562i
\(274\) 0 0
\(275\) −16.1243 −0.972334
\(276\) 0 0
\(277\) −12.7825 −0.768023 −0.384012 0.923328i \(-0.625458\pi\)
−0.384012 + 0.923328i \(0.625458\pi\)
\(278\) 0 0
\(279\) 24.1729 14.2244i 1.44720 0.851591i
\(280\) 0 0
\(281\) −10.4763 18.1454i −0.624961 1.08246i −0.988548 0.150904i \(-0.951781\pi\)
0.363587 0.931560i \(-0.381552\pi\)
\(282\) 0 0
\(283\) −7.53085 13.0438i −0.447663 0.775374i 0.550571 0.834788i \(-0.314410\pi\)
−0.998233 + 0.0594141i \(0.981077\pi\)
\(284\) 0 0
\(285\) 3.51567 + 0.926440i 0.208250 + 0.0548776i
\(286\) 0 0
\(287\) −7.62537 + 32.4286i −0.450112 + 1.91420i
\(288\) 0 0
\(289\) −6.69162 11.5902i −0.393625 0.681778i
\(290\) 0 0
\(291\) 23.5376 + 6.20258i 1.37980 + 0.363602i
\(292\) 0 0
\(293\) 0.134459 0.232890i 0.00785519 0.0136056i −0.862071 0.506787i \(-0.830833\pi\)
0.869926 + 0.493182i \(0.164166\pi\)
\(294\) 0 0
\(295\) 2.02732 + 3.51142i 0.118035 + 0.204443i
\(296\) 0 0
\(297\) 12.5781 + 12.2695i 0.729856 + 0.711950i
\(298\) 0 0
\(299\) −10.3724 + 17.9654i −0.599849 + 1.03897i
\(300\) 0 0
\(301\) 7.35695 + 6.91645i 0.424048 + 0.398658i
\(302\) 0 0
\(303\) −12.9363 + 13.0439i −0.743171 + 0.749350i
\(304\) 0 0
\(305\) −1.75375 + 3.03759i −0.100420 + 0.173932i
\(306\) 0 0
\(307\) −5.03514 −0.287371 −0.143685 0.989623i \(-0.545895\pi\)
−0.143685 + 0.989623i \(0.545895\pi\)
\(308\) 0 0
\(309\) 0.194434 0.196050i 0.0110609 0.0111529i
\(310\) 0 0
\(311\) −2.23815 3.87659i −0.126914 0.219821i 0.795566 0.605868i \(-0.207174\pi\)
−0.922479 + 0.386046i \(0.873841\pi\)
\(312\) 0 0
\(313\) −5.48895 + 9.50715i −0.310254 + 0.537376i −0.978417 0.206639i \(-0.933747\pi\)
0.668163 + 0.744015i \(0.267081\pi\)
\(314\) 0 0
\(315\) 1.13254 + 3.64918i 0.0638117 + 0.205608i
\(316\) 0 0
\(317\) −12.4826 + 21.6205i −0.701092 + 1.21433i 0.266992 + 0.963699i \(0.413970\pi\)
−0.968084 + 0.250628i \(0.919363\pi\)
\(318\) 0 0
\(319\) 5.17828 + 8.96905i 0.289928 + 0.502170i
\(320\) 0 0
\(321\) 13.3665 + 3.52230i 0.746045 + 0.196596i
\(322\) 0 0
\(323\) 24.0352 1.33736
\(324\) 0 0
\(325\) 13.6404 23.6259i 0.756635 1.31053i
\(326\) 0 0
\(327\) −6.24194 22.9154i −0.345180 1.26722i
\(328\) 0 0
\(329\) 2.28052 9.69844i 0.125729 0.534692i
\(330\) 0 0
\(331\) −6.01206 + 10.4132i −0.330453 + 0.572361i −0.982601 0.185731i \(-0.940535\pi\)
0.652148 + 0.758092i \(0.273868\pi\)
\(332\) 0 0
\(333\) 7.68182 4.52031i 0.420961 0.247711i
\(334\) 0 0
\(335\) 0.618925 + 1.07201i 0.0338155 + 0.0585702i
\(336\) 0 0
\(337\) −14.1286 + 24.4715i −0.769636 + 1.33305i 0.168125 + 0.985766i \(0.446229\pi\)
−0.937761 + 0.347282i \(0.887105\pi\)
\(338\) 0 0
\(339\) 8.18153 + 30.0360i 0.444360 + 1.63133i
\(340\) 0 0
\(341\) 15.8076 + 27.3795i 0.856027 + 1.48268i
\(342\) 0 0
\(343\) 11.8294 14.2501i 0.638728 0.769433i
\(344\) 0 0
\(345\) 2.12885 2.14655i 0.114613 0.115566i
\(346\) 0 0
\(347\) −9.80293 16.9792i −0.526249 0.911490i −0.999532 0.0305797i \(-0.990265\pi\)
0.473283 0.880910i \(-0.343069\pi\)
\(348\) 0 0
\(349\) −8.22904 14.2531i −0.440490 0.762952i 0.557236 0.830354i \(-0.311862\pi\)
−0.997726 + 0.0674029i \(0.978529\pi\)
\(350\) 0 0
\(351\) −28.6182 + 8.05042i −1.52753 + 0.429700i
\(352\) 0 0
\(353\) −27.3709 −1.45680 −0.728402 0.685150i \(-0.759737\pi\)
−0.728402 + 0.685150i \(0.759737\pi\)
\(354\) 0 0
\(355\) 1.89813 0.100742
\(356\) 0 0
\(357\) 13.3874 + 21.4202i 0.708535 + 1.13368i
\(358\) 0 0
\(359\) 7.88714 + 13.6609i 0.416267 + 0.720996i 0.995561 0.0941231i \(-0.0300047\pi\)
−0.579293 + 0.815119i \(0.696671\pi\)
\(360\) 0 0
\(361\) −0.00677168 + 0.0117289i −0.000356404 + 0.000617310i
\(362\) 0 0
\(363\) −0.530769 + 0.535182i −0.0278582 + 0.0280898i
\(364\) 0 0
\(365\) 0.415060 0.718905i 0.0217252 0.0376292i
\(366\) 0 0
\(367\) 18.8137 0.982066 0.491033 0.871141i \(-0.336619\pi\)
0.491033 + 0.871141i \(0.336619\pi\)
\(368\) 0 0
\(369\) 32.8680 + 18.6153i 1.71104 + 0.969072i
\(370\) 0 0
\(371\) 6.75695 28.7355i 0.350803 1.49187i
\(372\) 0 0
\(373\) −17.5737 −0.909934 −0.454967 0.890508i \(-0.650349\pi\)
−0.454967 + 0.890508i \(0.650349\pi\)
\(374\) 0 0
\(375\) −5.73525 + 5.78293i −0.296167 + 0.298629i
\(376\) 0 0
\(377\) −17.5223 −0.902446
\(378\) 0 0
\(379\) 34.4618 1.77018 0.885091 0.465419i \(-0.154096\pi\)
0.885091 + 0.465419i \(0.154096\pi\)
\(380\) 0 0
\(381\) −23.1351 + 23.3274i −1.18525 + 1.19510i
\(382\) 0 0
\(383\) −22.7410 −1.16201 −0.581005 0.813900i \(-0.697340\pi\)
−0.581005 + 0.813900i \(0.697340\pi\)
\(384\) 0 0
\(385\) −4.12378 + 1.24250i −0.210167 + 0.0633239i
\(386\) 0 0
\(387\) 9.86794 5.80671i 0.501615 0.295172i
\(388\) 0 0
\(389\) 15.7638 0.799255 0.399628 0.916678i \(-0.369139\pi\)
0.399628 + 0.916678i \(0.369139\pi\)
\(390\) 0 0
\(391\) 9.99303 17.3084i 0.505369 0.875325i
\(392\) 0 0
\(393\) 5.95479 6.00429i 0.300379 0.302877i
\(394\) 0 0
\(395\) −1.34779 + 2.33444i −0.0678145 + 0.117458i
\(396\) 0 0
\(397\) −5.39875 9.35091i −0.270955 0.469308i 0.698151 0.715950i \(-0.254006\pi\)
−0.969107 + 0.246642i \(0.920673\pi\)
\(398\) 0 0
\(399\) −19.9699 + 0.699052i −0.999744 + 0.0349964i
\(400\) 0 0
\(401\) 12.3295 0.615704 0.307852 0.951434i \(-0.400390\pi\)
0.307852 + 0.951434i \(0.400390\pi\)
\(402\) 0 0
\(403\) −53.4898 −2.66452
\(404\) 0 0
\(405\) 4.33189 0.0717346i 0.215253 0.00356452i
\(406\) 0 0
\(407\) 5.02342 + 8.70082i 0.249002 + 0.431284i
\(408\) 0 0
\(409\) −9.31771 16.1387i −0.460731 0.798010i 0.538266 0.842775i \(-0.319079\pi\)
−0.998998 + 0.0447650i \(0.985746\pi\)
\(410\) 0 0
\(411\) 7.89711 7.96276i 0.389536 0.392774i
\(412\) 0 0
\(413\) −16.2362 15.2641i −0.798932 0.751096i
\(414\) 0 0
\(415\) 0.0575230 + 0.0996328i 0.00282369 + 0.00489078i
\(416\) 0 0
\(417\) −10.3551 38.0155i −0.507090 1.86163i
\(418\) 0 0
\(419\) −5.90976 + 10.2360i −0.288711 + 0.500062i −0.973502 0.228677i \(-0.926560\pi\)
0.684791 + 0.728739i \(0.259893\pi\)
\(420\) 0 0
\(421\) 4.81800 + 8.34503i 0.234815 + 0.406712i 0.959219 0.282664i \(-0.0912182\pi\)
−0.724404 + 0.689376i \(0.757885\pi\)
\(422\) 0 0
\(423\) −9.82986 5.56728i −0.477944 0.270690i
\(424\) 0 0
\(425\) −13.1416 + 22.7619i −0.637460 + 1.10411i
\(426\) 0 0
\(427\) 4.41265 18.7658i 0.213543 0.908139i
\(428\) 0 0
\(429\) −8.80704 32.3324i −0.425208 1.56102i
\(430\) 0 0
\(431\) −18.2913 + 31.6815i −0.881062 + 1.52604i −0.0309004 + 0.999522i \(0.509837\pi\)
−0.850162 + 0.526522i \(0.823496\pi\)
\(432\) 0 0
\(433\) −7.69388 −0.369744 −0.184872 0.982763i \(-0.559187\pi\)
−0.184872 + 0.982763i \(0.559187\pi\)
\(434\) 0 0
\(435\) 2.46928 + 0.650699i 0.118393 + 0.0311986i
\(436\) 0 0
\(437\) 7.90518 + 13.6922i 0.378156 + 0.654985i
\(438\) 0 0
\(439\) 10.2717 17.7911i 0.490241 0.849122i −0.509696 0.860355i \(-0.670242\pi\)
0.999937 + 0.0112324i \(0.00357545\pi\)
\(440\) 0 0
\(441\) −11.7460 17.4078i −0.559333 0.828943i
\(442\) 0 0
\(443\) 16.8401 29.1679i 0.800098 1.38581i −0.119454 0.992840i \(-0.538114\pi\)
0.919551 0.392970i \(-0.128552\pi\)
\(444\) 0 0
\(445\) −0.311995 0.540391i −0.0147900 0.0256170i
\(446\) 0 0
\(447\) 17.3249 17.4689i 0.819438 0.826250i
\(448\) 0 0
\(449\) −22.5141 −1.06250 −0.531252 0.847214i \(-0.678278\pi\)
−0.531252 + 0.847214i \(0.678278\pi\)
\(450\) 0 0
\(451\) −21.2891 + 36.8738i −1.00246 + 1.73632i
\(452\) 0 0
\(453\) −3.07671 + 3.10229i −0.144556 + 0.145758i
\(454\) 0 0
\(455\) 1.66797 7.09340i 0.0781955 0.332544i
\(456\) 0 0
\(457\) 11.8559 20.5349i 0.554594 0.960584i −0.443342 0.896353i \(-0.646207\pi\)
0.997935 0.0642314i \(-0.0204596\pi\)
\(458\) 0 0
\(459\) 27.5716 7.75601i 1.28693 0.362020i
\(460\) 0 0
\(461\) −5.57340 9.65342i −0.259579 0.449605i 0.706550 0.707663i \(-0.250251\pi\)
−0.966129 + 0.258059i \(0.916917\pi\)
\(462\) 0 0
\(463\) 10.3208 17.8761i 0.479647 0.830773i −0.520080 0.854117i \(-0.674098\pi\)
0.999727 + 0.0233441i \(0.00743133\pi\)
\(464\) 0 0
\(465\) 7.53789 + 1.98637i 0.349561 + 0.0921155i
\(466\) 0 0
\(467\) −8.68477 15.0425i −0.401883 0.696082i 0.592070 0.805887i \(-0.298311\pi\)
−0.993953 + 0.109804i \(0.964978\pi\)
\(468\) 0 0
\(469\) −4.95680 4.66001i −0.228884 0.215179i
\(470\) 0 0
\(471\) 29.2789 + 7.71551i 1.34910 + 0.355512i
\(472\) 0 0
\(473\) 6.45300 + 11.1769i 0.296709 + 0.513916i
\(474\) 0 0
\(475\) −10.3959 18.0062i −0.476997 0.826182i
\(476\) 0 0
\(477\) −29.1248 16.4952i −1.33353 0.755266i
\(478\) 0 0
\(479\) −4.09033 −0.186892 −0.0934461 0.995624i \(-0.529788\pi\)
−0.0934461 + 0.995624i \(0.529788\pi\)
\(480\) 0 0
\(481\) −16.9983 −0.775056
\(482\) 0 0
\(483\) −7.79938 + 14.6715i −0.354884 + 0.667576i
\(484\) 0 0
\(485\) 3.38256 + 5.85876i 0.153594 + 0.266033i
\(486\) 0 0
\(487\) −0.843065 + 1.46023i −0.0382029 + 0.0661694i −0.884495 0.466550i \(-0.845497\pi\)
0.846292 + 0.532720i \(0.178830\pi\)
\(488\) 0 0
\(489\) −2.95174 0.777835i −0.133482 0.0351749i
\(490\) 0 0
\(491\) 6.85070 11.8658i 0.309168 0.535494i −0.669013 0.743251i \(-0.733283\pi\)
0.978181 + 0.207757i \(0.0666162\pi\)
\(492\) 0 0
\(493\) 16.8815 0.760305
\(494\) 0 0
\(495\) 0.0404309 + 4.88340i 0.00181723 + 0.219492i
\(496\) 0 0
\(497\) −9.98875 + 3.00963i −0.448057 + 0.135000i
\(498\) 0 0
\(499\) −6.55655 −0.293511 −0.146756 0.989173i \(-0.546883\pi\)
−0.146756 + 0.989173i \(0.546883\pi\)
\(500\) 0 0
\(501\) −3.25220 11.9394i −0.145297 0.533415i
\(502\) 0 0
\(503\) 26.8584 1.19756 0.598779 0.800914i \(-0.295653\pi\)
0.598779 + 0.800914i \(0.295653\pi\)
\(504\) 0 0
\(505\) −5.10580 −0.227205
\(506\) 0 0
\(507\) 33.0514 + 8.70962i 1.46787 + 0.386808i
\(508\) 0 0
\(509\) −39.7337 −1.76117 −0.880584 0.473891i \(-0.842849\pi\)
−0.880584 + 0.473891i \(0.842849\pi\)
\(510\) 0 0
\(511\) −1.04434 + 4.44129i −0.0461989 + 0.196471i
\(512\) 0 0
\(513\) −5.59198 + 21.9567i −0.246892 + 0.969411i
\(514\) 0 0
\(515\) 0.0767406 0.00338160
\(516\) 0 0
\(517\) 6.36694 11.0279i 0.280018 0.485005i
\(518\) 0 0
\(519\) −4.50376 16.5342i −0.197693 0.725770i
\(520\) 0 0
\(521\) 11.7585 20.3663i 0.515148 0.892262i −0.484698 0.874682i \(-0.661070\pi\)
0.999845 0.0175802i \(-0.00559623\pi\)
\(522\) 0 0
\(523\) −10.9289 18.9294i −0.477887 0.827725i 0.521791 0.853073i \(-0.325264\pi\)
−0.999679 + 0.0253481i \(0.991931\pi\)
\(524\) 0 0
\(525\) 10.2568 19.2941i 0.447642 0.842064i
\(526\) 0 0
\(527\) 51.5336 2.24484
\(528\) 0 0
\(529\) −9.85320 −0.428400
\(530\) 0 0
\(531\) −21.7778 + 12.8150i −0.945075 + 0.556122i
\(532\) 0 0
\(533\) −36.0191 62.3869i −1.56016 2.70228i
\(534\) 0 0
\(535\) 1.92088 + 3.32706i 0.0830468 + 0.143841i
\(536\) 0 0
\(537\) 1.84785 + 6.78383i 0.0797407 + 0.292744i
\(538\) 0 0
\(539\) 19.7310 13.0772i 0.849874 0.563273i
\(540\) 0 0
\(541\) 14.0063 + 24.2596i 0.602178 + 1.04300i 0.992491 + 0.122320i \(0.0390334\pi\)
−0.390313 + 0.920682i \(0.627633\pi\)
\(542\) 0 0
\(543\) −5.58709 + 5.63354i −0.239765 + 0.241758i
\(544\) 0 0
\(545\) 3.30045 5.71654i 0.141376 0.244870i
\(546\) 0 0
\(547\) −2.02714 3.51112i −0.0866744 0.150124i 0.819429 0.573181i \(-0.194291\pi\)
−0.906103 + 0.423056i \(0.860957\pi\)
\(548\) 0 0
\(549\) −19.0201 10.7723i −0.811756 0.459749i
\(550\) 0 0
\(551\) −6.67722 + 11.5653i −0.284459 + 0.492698i
\(552\) 0 0
\(553\) 3.39119 14.4218i 0.144208 0.613278i
\(554\) 0 0
\(555\) 2.39544 + 0.631239i 0.101681 + 0.0267946i
\(556\) 0 0
\(557\) −0.926620 + 1.60495i −0.0392621 + 0.0680040i −0.884989 0.465612i \(-0.845834\pi\)
0.845727 + 0.533616i \(0.179167\pi\)
\(558\) 0 0
\(559\) −21.8357 −0.923553
\(560\) 0 0
\(561\) 8.48496 + 31.1499i 0.358235 + 1.31515i
\(562\) 0 0
\(563\) −22.2331 38.5088i −0.937013 1.62295i −0.771005 0.636830i \(-0.780245\pi\)
−0.166008 0.986124i \(-0.553088\pi\)
\(564\) 0 0
\(565\) −4.32601 + 7.49287i −0.181997 + 0.315227i
\(566\) 0 0
\(567\) −22.6825 + 7.24605i −0.952575 + 0.304305i
\(568\) 0 0
\(569\) −7.63116 + 13.2176i −0.319915 + 0.554109i −0.980470 0.196669i \(-0.936988\pi\)
0.660555 + 0.750778i \(0.270321\pi\)
\(570\) 0 0
\(571\) 12.7634 + 22.1068i 0.534130 + 0.925140i 0.999205 + 0.0398690i \(0.0126940\pi\)
−0.465075 + 0.885271i \(0.653973\pi\)
\(572\) 0 0
\(573\) −5.09493 18.7045i −0.212844 0.781390i
\(574\) 0 0
\(575\) −17.2890 −0.721002
\(576\) 0 0
\(577\) −3.26981 + 5.66348i −0.136124 + 0.235774i −0.926026 0.377459i \(-0.876798\pi\)
0.789902 + 0.613233i \(0.210131\pi\)
\(578\) 0 0
\(579\) 27.2897 + 7.19130i 1.13412 + 0.298860i
\(580\) 0 0
\(581\) −0.460686 0.433102i −0.0191125 0.0179681i
\(582\) 0 0
\(583\) 18.8646 32.6744i 0.781291 1.35323i
\(584\) 0 0
\(585\) −7.18952 4.07188i −0.297250 0.168352i
\(586\) 0 0
\(587\) 15.2055 + 26.3366i 0.627597 + 1.08703i 0.988033 + 0.154245i \(0.0492947\pi\)
−0.360436 + 0.932784i \(0.617372\pi\)
\(588\) 0 0
\(589\) −20.3833 + 35.3049i −0.839880 + 1.45471i
\(590\) 0 0
\(591\) 3.87167 3.90386i 0.159259 0.160583i
\(592\) 0 0
\(593\) −21.3291 36.9432i −0.875883 1.51707i −0.855819 0.517275i \(-0.826947\pi\)
−0.0200633 0.999799i \(-0.506387\pi\)
\(594\) 0 0
\(595\) −1.60697 + 6.83399i −0.0658792 + 0.280166i
\(596\) 0 0
\(597\) 1.31228 + 4.81763i 0.0537079 + 0.197172i
\(598\) 0 0
\(599\) 22.2787 + 38.5879i 0.910284 + 1.57666i 0.813663 + 0.581337i \(0.197470\pi\)
0.0966209 + 0.995321i \(0.469197\pi\)
\(600\) 0 0
\(601\) 14.1961 + 24.5884i 0.579071 + 1.00298i 0.995586 + 0.0938518i \(0.0299180\pi\)
−0.416515 + 0.909129i \(0.636749\pi\)
\(602\) 0 0
\(603\) −6.64860 + 3.91232i −0.270752 + 0.159322i
\(604\) 0 0
\(605\) −0.209488 −0.00851691
\(606\) 0 0
\(607\) 14.0278 0.569372 0.284686 0.958621i \(-0.408111\pi\)
0.284686 + 0.958621i \(0.408111\pi\)
\(608\) 0 0
\(609\) −14.0261 + 0.490990i −0.568368 + 0.0198959i
\(610\) 0 0
\(611\) 10.7723 + 18.6581i 0.435799 + 0.754826i
\(612\) 0 0
\(613\) −9.97062 + 17.2696i −0.402709 + 0.697513i −0.994052 0.108907i \(-0.965265\pi\)
0.591342 + 0.806421i \(0.298598\pi\)
\(614\) 0 0
\(615\) 2.75912 + 10.1293i 0.111259 + 0.408452i
\(616\) 0 0
\(617\) −1.51584 + 2.62551i −0.0610254 + 0.105699i −0.894924 0.446219i \(-0.852770\pi\)
0.833899 + 0.551918i \(0.186104\pi\)
\(618\) 0 0
\(619\) 2.55431 0.102666 0.0513331 0.998682i \(-0.483653\pi\)
0.0513331 + 0.998682i \(0.483653\pi\)
\(620\) 0 0
\(621\) 13.4866 + 13.1558i 0.541200 + 0.527923i
\(622\) 0 0
\(623\) 2.49868 + 2.34907i 0.100107 + 0.0941136i
\(624\) 0 0
\(625\) 21.5777 0.863108
\(626\) 0 0
\(627\) −24.6965 6.50795i −0.986282 0.259903i
\(628\) 0 0
\(629\) 16.3766 0.652980
\(630\) 0 0
\(631\) 37.1162 1.47757 0.738786 0.673941i \(-0.235400\pi\)
0.738786 + 0.673941i \(0.235400\pi\)
\(632\) 0 0
\(633\) −0.220975 0.811243i −0.00878297 0.0322440i
\(634\) 0 0
\(635\) −9.13115 −0.362358
\(636\) 0 0
\(637\) 2.46960 + 39.9731i 0.0978491 + 1.58379i
\(638\) 0 0
\(639\) 0.0979331 + 11.8287i 0.00387417 + 0.467937i
\(640\) 0 0
\(641\) 21.0968 0.833275 0.416638 0.909073i \(-0.363208\pi\)
0.416638 + 0.909073i \(0.363208\pi\)
\(642\) 0 0
\(643\) 9.31948 16.1418i 0.367524 0.636571i −0.621654 0.783292i \(-0.713539\pi\)
0.989178 + 0.146722i \(0.0468722\pi\)
\(644\) 0 0
\(645\) 3.07714 + 0.810879i 0.121162 + 0.0319283i
\(646\) 0 0
\(647\) 4.78509 8.28801i 0.188121 0.325835i −0.756503 0.653991i \(-0.773094\pi\)
0.944624 + 0.328155i \(0.106427\pi\)
\(648\) 0 0
\(649\) −14.2413 24.6666i −0.559019 0.968249i
\(650\) 0 0
\(651\) −42.8171 + 1.49883i −1.67813 + 0.0587437i
\(652\) 0 0
\(653\) −32.8560 −1.28575 −0.642877 0.765970i \(-0.722259\pi\)
−0.642877 + 0.765970i \(0.722259\pi\)
\(654\) 0 0
\(655\) 2.35028 0.0918332
\(656\) 0 0
\(657\) 4.50147 + 2.54947i 0.175619 + 0.0994642i
\(658\) 0 0
\(659\) 24.8011 + 42.9568i 0.966114 + 1.67336i 0.706593 + 0.707620i \(0.250231\pi\)
0.259521 + 0.965738i \(0.416435\pi\)
\(660\) 0 0
\(661\) 1.65895 + 2.87338i 0.0645255 + 0.111761i 0.896483 0.443077i \(-0.146113\pi\)
−0.831958 + 0.554839i \(0.812780\pi\)
\(662\) 0 0
\(663\) −52.8198 13.9189i −2.05135 0.540566i
\(664\) 0 0
\(665\) −4.04626 3.80399i −0.156907 0.147512i
\(666\) 0 0
\(667\) 5.55232 + 9.61690i 0.214987 + 0.372368i
\(668\) 0 0
\(669\) −7.19511 1.89604i −0.278179 0.0733050i
\(670\) 0 0
\(671\) 12.3196 21.3381i 0.475591 0.823748i
\(672\) 0 0
\(673\) 21.8005 + 37.7597i 0.840349 + 1.45553i 0.889600 + 0.456741i \(0.150983\pi\)
−0.0492503 + 0.998786i \(0.515683\pi\)
\(674\) 0 0
\(675\) −17.7359 17.3008i −0.682657 0.665909i
\(676\) 0 0
\(677\) 9.14039 15.8316i 0.351294 0.608459i −0.635183 0.772362i \(-0.719075\pi\)
0.986476 + 0.163903i \(0.0524085\pi\)
\(678\) 0 0
\(679\) −27.0900 25.4680i −1.03962 0.977371i
\(680\) 0 0
\(681\) 21.2011 21.3774i 0.812428 0.819182i
\(682\) 0 0
\(683\) 22.5380 39.0369i 0.862391 1.49371i −0.00722317 0.999974i \(-0.502299\pi\)
0.869614 0.493732i \(-0.164367\pi\)
\(684\) 0 0
\(685\) 3.11689 0.119090
\(686\) 0 0
\(687\) −8.94167 + 9.01601i −0.341146 + 0.343982i
\(688\) 0 0
\(689\) 31.9171 + 55.2820i 1.21594 + 2.10608i
\(690\) 0 0
\(691\) −20.8977 + 36.1960i −0.794988 + 1.37696i 0.127859 + 0.991792i \(0.459189\pi\)
−0.922847 + 0.385167i \(0.874144\pi\)
\(692\) 0 0
\(693\) −7.95577 25.6344i −0.302215 0.973770i
\(694\) 0 0
\(695\) 5.47528 9.48346i 0.207689 0.359728i
\(696\) 0 0
\(697\) 34.7019 + 60.1054i 1.31443 + 2.27665i
\(698\) 0 0
\(699\) 7.25637 + 1.91218i 0.274461 + 0.0723253i
\(700\) 0 0
\(701\) 19.3967 0.732604 0.366302 0.930496i \(-0.380624\pi\)
0.366302 + 0.930496i \(0.380624\pi\)
\(702\) 0 0
\(703\) −6.47753 + 11.2194i −0.244305 + 0.423148i
\(704\) 0 0
\(705\) −0.825173 3.02937i −0.0310778 0.114093i
\(706\) 0 0
\(707\) 26.8689 8.09565i 1.01051 0.304468i
\(708\) 0 0
\(709\) 8.61542 14.9223i 0.323559 0.560420i −0.657661 0.753314i \(-0.728454\pi\)
0.981220 + 0.192894i \(0.0617873\pi\)
\(710\) 0 0
\(711\) −14.6172 8.27867i −0.548189 0.310474i
\(712\) 0 0
\(713\) 16.9494 + 29.3572i 0.634759 + 1.09943i
\(714\) 0 0
\(715\) 4.65675 8.06573i 0.174153 0.301641i
\(716\) 0 0
\(717\) 1.62018 + 5.94800i 0.0605068 + 0.222132i
\(718\) 0 0
\(719\) 5.08444 + 8.80650i 0.189617 + 0.328427i 0.945123 0.326715i \(-0.105942\pi\)
−0.755505 + 0.655143i \(0.772609\pi\)
\(720\) 0 0
\(721\) −0.403841 + 0.121678i −0.0150398 + 0.00453153i
\(722\) 0 0
\(723\) −19.5372 + 19.6996i −0.726596 + 0.732637i
\(724\) 0 0
\(725\) −7.30172 12.6469i −0.271179 0.469696i
\(726\) 0 0
\(727\) −0.0914356 0.158371i −0.00339116 0.00587366i 0.864325 0.502934i \(-0.167746\pi\)
−0.867716 + 0.497060i \(0.834413\pi\)
\(728\) 0 0
\(729\) 0.670536 + 26.9917i 0.0248347 + 0.999692i
\(730\) 0 0
\(731\) 21.0372 0.778088
\(732\) 0 0
\(733\) 41.9343 1.54888 0.774440 0.632647i \(-0.218032\pi\)
0.774440 + 0.632647i \(0.218032\pi\)
\(734\) 0 0
\(735\) 1.13640 5.72481i 0.0419167 0.211163i
\(736\) 0 0
\(737\) −4.34776 7.53054i −0.160152 0.277391i
\(738\) 0 0
\(739\) −11.8013 + 20.4404i −0.434116 + 0.751911i −0.997223 0.0744729i \(-0.976273\pi\)
0.563107 + 0.826384i \(0.309606\pi\)
\(740\) 0 0
\(741\) 30.4277 30.6807i 1.11779 1.12708i
\(742\) 0 0
\(743\) −11.1821 + 19.3680i −0.410233 + 0.710544i −0.994915 0.100718i \(-0.967886\pi\)
0.584682 + 0.811263i \(0.301219\pi\)
\(744\) 0 0
\(745\) 6.83791 0.250522
\(746\) 0 0
\(747\) −0.617922 + 0.363611i −0.0226086 + 0.0133038i
\(748\) 0 0
\(749\) −15.3838 14.4627i −0.562111 0.528454i
\(750\) 0 0
\(751\) 31.0462 1.13289 0.566445 0.824099i \(-0.308318\pi\)
0.566445 + 0.824099i \(0.308318\pi\)
\(752\) 0 0
\(753\) 15.6125 15.7423i 0.568952 0.573682i
\(754\) 0 0
\(755\) −1.21434 −0.0441943
\(756\) 0 0
\(757\) −44.0639 −1.60153 −0.800764 0.598980i \(-0.795573\pi\)
−0.800764 + 0.598980i \(0.795573\pi\)
\(758\) 0 0
\(759\) −14.9545 + 15.0788i −0.542814 + 0.547327i
\(760\) 0 0
\(761\) 5.74941 0.208416 0.104208 0.994556i \(-0.466769\pi\)
0.104208 + 0.994556i \(0.466769\pi\)
\(762\) 0 0
\(763\) −8.30431 + 35.3159i −0.300636 + 1.27852i
\(764\) 0 0
\(765\) 6.92659 + 3.92297i 0.250431 + 0.141835i
\(766\) 0 0
\(767\) 48.1898 1.74003
\(768\) 0 0
\(769\) −7.48401 + 12.9627i −0.269880 + 0.467446i −0.968831 0.247724i \(-0.920317\pi\)
0.698950 + 0.715170i \(0.253651\pi\)
\(770\) 0 0
\(771\) −20.0212 + 20.1877i −0.721048 + 0.727042i
\(772\) 0 0
\(773\) −10.0605 + 17.4253i −0.361850 + 0.626743i −0.988265 0.152747i \(-0.951188\pi\)
0.626415 + 0.779490i \(0.284521\pi\)
\(774\) 0 0
\(775\) −22.2897 38.6069i −0.800669 1.38680i
\(776\) 0 0
\(777\) −13.6067 + 0.476306i −0.488136 + 0.0170874i
\(778\) 0 0
\(779\) −54.9031 −1.96711
\(780\) 0 0
\(781\) −13.3338 −0.477120
\(782\) 0 0
\(783\) −3.92761 + 15.4216i −0.140361 + 0.551123i
\(784\) 0 0
\(785\) 4.20763 + 7.28783i 0.150177 + 0.260114i
\(786\) 0 0
\(787\) 12.0572 + 20.8837i 0.429794 + 0.744425i 0.996855 0.0792508i \(-0.0252528\pi\)
−0.567061 + 0.823676i \(0.691919\pi\)
\(788\) 0 0
\(789\) −31.2878 + 31.5479i −1.11388 + 1.12314i
\(790\) 0 0
\(791\) 10.8848 46.2899i 0.387017 1.64588i
\(792\) 0 0
\(793\) 20.8435 + 36.1021i 0.740175 + 1.28202i
\(794\) 0 0
\(795\) −2.44490 8.97570i −0.0867117 0.318335i
\(796\) 0 0
\(797\) −1.54611 + 2.67794i −0.0547661 + 0.0948576i −0.892109 0.451821i \(-0.850775\pi\)
0.837343 + 0.546678i \(0.184108\pi\)
\(798\) 0 0
\(799\) −10.3783 17.9758i −0.367158 0.635936i
\(800\) 0 0
\(801\) 3.35150 1.97216i 0.118419 0.0696830i
\(802\) 0 0
\(803\) −2.91567 + 5.05008i −0.102892 + 0.178213i
\(804\) 0 0
\(805\) −4.42165 + 1.33225i −0.155843 + 0.0469557i
\(806\) 0 0
\(807\) 4.87436 + 17.8947i 0.171586 + 0.629924i
\(808\) 0 0
\(809\) −3.32067 + 5.75157i −0.116749 + 0.202215i −0.918477 0.395473i \(-0.870581\pi\)
0.801729 + 0.597688i \(0.203914\pi\)
\(810\) 0 0
\(811\) −23.7806 −0.835049 −0.417525 0.908666i \(-0.637102\pi\)
−0.417525 + 0.908666i \(0.637102\pi\)
\(812\) 0 0
\(813\) 42.7137 + 11.2558i 1.49803 + 0.394758i
\(814\) 0 0
\(815\) −0.424190 0.734719i −0.0148587 0.0257361i
\(816\) 0 0
\(817\) −8.32093 + 14.4123i −0.291112 + 0.504222i
\(818\) 0 0
\(819\) 44.2905 + 10.0284i 1.54764 + 0.350421i
\(820\) 0 0
\(821\) 24.9151 43.1543i 0.869544 1.50609i 0.00708033 0.999975i \(-0.497746\pi\)
0.862464 0.506119i \(-0.168920\pi\)
\(822\) 0 0
\(823\) 4.72691 + 8.18726i 0.164770 + 0.285390i 0.936574 0.350471i \(-0.113979\pi\)
−0.771804 + 0.635861i \(0.780645\pi\)
\(824\) 0 0
\(825\) 19.6663 19.8298i 0.684692 0.690384i
\(826\) 0 0
\(827\) −7.95385 −0.276582 −0.138291 0.990392i \(-0.544161\pi\)
−0.138291 + 0.990392i \(0.544161\pi\)
\(828\) 0 0
\(829\) 6.22333 10.7791i 0.216145 0.374374i −0.737481 0.675368i \(-0.763985\pi\)
0.953626 + 0.300993i \(0.0973182\pi\)
\(830\) 0 0
\(831\) 15.5903 15.7199i 0.540822 0.545318i
\(832\) 0 0
\(833\) −2.37928 38.5113i −0.0824373 1.33434i
\(834\) 0 0
\(835\) 1.71961 2.97845i 0.0595096 0.103074i
\(836\) 0 0
\(837\) −11.9897 + 47.0770i −0.414424 + 1.62722i
\(838\) 0 0
\(839\) −22.5163 38.9994i −0.777350 1.34641i −0.933464 0.358671i \(-0.883230\pi\)
0.156114 0.987739i \(-0.450103\pi\)
\(840\) 0 0
\(841\) 9.81015 16.9917i 0.338281 0.585920i
\(842\) 0 0
\(843\) 35.0928 + 9.24757i 1.20866 + 0.318503i
\(844\) 0 0
\(845\) 4.74977 + 8.22684i 0.163397 + 0.283012i
\(846\) 0 0
\(847\) 1.10242 0.332160i 0.0378794 0.0114131i
\(848\) 0 0
\(849\) 25.2264 + 6.64761i 0.865770 + 0.228145i
\(850\) 0 0
\(851\) 5.38627 + 9.32929i 0.184639 + 0.319804i
\(852\) 0 0
\(853\) −1.15007 1.99198i −0.0393777 0.0682042i 0.845665 0.533714i \(-0.179204\pi\)
−0.885043 + 0.465510i \(0.845871\pi\)
\(854\) 0 0
\(855\) −5.42728 + 3.19364i −0.185609 + 0.109220i
\(856\) 0 0
\(857\) 12.6695 0.432780 0.216390 0.976307i \(-0.430572\pi\)
0.216390 + 0.976307i \(0.430572\pi\)
\(858\) 0 0
\(859\) 47.4748 1.61982 0.809910 0.586554i \(-0.199516\pi\)
0.809910 + 0.586554i \(0.199516\pi\)
\(860\) 0 0
\(861\) −30.5804 48.9297i −1.04218 1.66752i
\(862\) 0 0
\(863\) −16.1445 27.9630i −0.549564 0.951873i −0.998304 0.0582109i \(-0.981460\pi\)
0.448740 0.893662i \(-0.351873\pi\)
\(864\) 0 0
\(865\) 2.38138 4.12467i 0.0809693 0.140243i
\(866\) 0 0
\(867\) 22.4152 + 5.90681i 0.761262 + 0.200606i
\(868\) 0 0
\(869\) 9.46779 16.3987i 0.321173 0.556287i
\(870\) 0 0
\(871\) 14.7120 0.498497
\(872\) 0 0
\(873\) −36.3360 + 21.3816i −1.22979 + 0.723659i
\(874\) 0 0
\(875\) 11.9122 3.58917i 0.402706 0.121336i
\(876\) 0 0
\(877\) 27.7589 0.937352 0.468676 0.883370i \(-0.344731\pi\)
0.468676 + 0.883370i \(0.344731\pi\)
\(878\) 0 0
\(879\) 0.122414 + 0.449406i 0.00412893 + 0.0151581i
\(880\) 0 0
\(881\) −22.7696 −0.767126 −0.383563 0.923515i \(-0.625303\pi\)
−0.383563 + 0.923515i \(0.625303\pi\)
\(882\) 0 0
\(883\) 4.65312 0.156590 0.0782950 0.996930i \(-0.475052\pi\)
0.0782950 + 0.996930i \(0.475052\pi\)
\(884\) 0 0
\(885\) −6.79100 1.78955i −0.228277 0.0601550i
\(886\) 0 0
\(887\) −47.2867 −1.58773 −0.793866 0.608093i \(-0.791935\pi\)
−0.793866 + 0.608093i \(0.791935\pi\)
\(888\) 0 0
\(889\) 48.0519 14.4781i 1.61161 0.485581i
\(890\) 0 0
\(891\) −30.4302 + 0.503913i −1.01945 + 0.0168817i
\(892\) 0 0
\(893\) 16.4199 0.549471
\(894\) 0 0
\(895\) −0.977058 + 1.69231i −0.0326595 + 0.0565678i
\(896\) 0 0
\(897\) −9.44319 34.6678i −0.315299 1.15752i
\(898\) 0 0
\(899\) −14.3165 + 24.7970i −0.477483 + 0.827025i
\(900\) 0 0
\(901\) −30.7498 53.2602i −1.02442 1.77436i
\(902\) 0 0
\(903\) −17.4789 + 0.611855i −0.581661 + 0.0203613i
\(904\) 0 0
\(905\) −2.20516 −0.0733019
\(906\) 0 0
\(907\) −6.06612 −0.201422 −0.100711 0.994916i \(-0.532112\pi\)
−0.100711 + 0.994916i \(0.532112\pi\)
\(908\) 0 0
\(909\) −0.263431 31.8182i −0.00873747 1.05534i
\(910\) 0 0
\(911\) 11.3223 + 19.6109i 0.375126 + 0.649737i 0.990346 0.138618i \(-0.0442661\pi\)
−0.615220 + 0.788356i \(0.710933\pi\)
\(912\) 0 0
\(913\) −0.404081 0.699889i −0.0133731 0.0231630i
\(914\) 0 0
\(915\) −1.59665 5.86161i −0.0527836 0.193779i
\(916\) 0 0
\(917\) −12.3682 + 3.72656i −0.408433 + 0.123062i
\(918\) 0 0
\(919\) −22.3902 38.7810i −0.738585 1.27927i −0.953133 0.302553i \(-0.902161\pi\)
0.214548 0.976713i \(-0.431172\pi\)
\(920\) 0 0
\(921\) 6.14118 6.19224i 0.202359 0.204041i
\(922\) 0 0
\(923\) 11.2797 19.5371i 0.371277 0.643071i
\(924\) 0 0
\(925\) −7.08335 12.2687i −0.232899 0.403393i
\(926\) 0 0
\(927\) 0.00395939 + 0.478231i 0.000130044 + 0.0157072i
\(928\) 0 0
\(929\) 0.552620 0.957166i 0.0181309 0.0314036i −0.856818 0.515620i \(-0.827562\pi\)
0.874948 + 0.484216i \(0.160895\pi\)
\(930\) 0 0
\(931\) 27.3246 + 13.6025i 0.895528 + 0.445805i
\(932\) 0 0
\(933\) 7.49724 + 1.97565i 0.245449 + 0.0646800i
\(934\) 0 0
\(935\) −4.48645 + 7.77076i −0.146723 + 0.254131i
\(936\) 0 0
\(937\) 44.7012 1.46033 0.730163 0.683273i \(-0.239444\pi\)
0.730163 + 0.683273i \(0.239444\pi\)
\(938\) 0 0
\(939\) −4.99725 18.3459i −0.163079 0.598695i
\(940\) 0 0
\(941\) 0.749661 + 1.29845i 0.0244383 + 0.0423283i 0.877986 0.478687i \(-0.158887\pi\)
−0.853548 + 0.521015i \(0.825554\pi\)
\(942\) 0 0
\(943\) −22.8268 + 39.5372i −0.743344 + 1.28751i
\(944\) 0 0
\(945\) −5.86911 3.05797i −0.190922 0.0994759i
\(946\) 0 0
\(947\) −4.75447 + 8.23498i −0.154499 + 0.267601i −0.932877 0.360196i \(-0.882710\pi\)
0.778377 + 0.627797i \(0.216043\pi\)
\(948\) 0 0
\(949\) −4.93303 8.54426i −0.160133 0.277358i
\(950\) 0 0
\(951\) −11.3644 41.7209i −0.368515 1.35289i
\(952\) 0 0
\(953\) 15.2064 0.492585 0.246292 0.969196i \(-0.420788\pi\)
0.246292 + 0.969196i \(0.420788\pi\)
\(954\) 0 0
\(955\) 2.69396 4.66607i 0.0871745 0.150991i
\(956\) 0 0
\(957\) −17.3459 4.57096i −0.560715 0.147758i
\(958\) 0 0
\(959\) −16.4024 + 4.94208i −0.529661 + 0.159588i
\(960\) 0 0
\(961\) −28.2036 + 48.8500i −0.909792 + 1.57581i
\(962\) 0 0
\(963\) −20.6344 + 12.1421i −0.664933 + 0.391275i
\(964\) 0 0
\(965\) 3.92176 + 6.79268i 0.126246 + 0.218664i
\(966\) 0 0
\(967\) −17.9319 + 31.0589i −0.576649 + 0.998786i 0.419211 + 0.907889i \(0.362307\pi\)
−0.995860 + 0.0908973i \(0.971027\pi\)
\(968\) 0 0
\(969\) −29.3149 + 29.5586i −0.941731 + 0.949561i
\(970\) 0 0
\(971\) −3.72746 6.45615i −0.119620 0.207188i 0.799997 0.600004i \(-0.204834\pi\)
−0.919617 + 0.392816i \(0.871501\pi\)
\(972\) 0 0
\(973\) −13.7765 + 58.5874i −0.441653 + 1.87823i
\(974\) 0 0
\(975\) 12.4185 + 45.5907i 0.397710 + 1.46007i
\(976\) 0 0
\(977\) 19.9756 + 34.5988i 0.639076 + 1.10691i 0.985636 + 0.168885i \(0.0540166\pi\)
−0.346559 + 0.938028i \(0.612650\pi\)
\(978\) 0 0
\(979\) 2.19167 + 3.79608i 0.0700460 + 0.121323i
\(980\) 0 0
\(981\) 35.7945 + 20.2727i 1.14283 + 0.647258i
\(982\) 0 0
\(983\) 8.38416 0.267413 0.133707 0.991021i \(-0.457312\pi\)
0.133707 + 0.991021i \(0.457312\pi\)
\(984\) 0 0
\(985\) 1.52810 0.0486894
\(986\) 0 0
\(987\) 9.14571 + 14.6334i 0.291111 + 0.465788i
\(988\) 0 0
\(989\) 6.91911 + 11.9843i 0.220015 + 0.381077i
\(990\) 0 0
\(991\) −21.2345 + 36.7792i −0.674536 + 1.16833i 0.302068 + 0.953286i \(0.402323\pi\)
−0.976604 + 0.215044i \(0.931010\pi\)
\(992\) 0 0
\(993\) −5.47349 20.0942i −0.173696 0.637672i
\(994\) 0 0
\(995\) −0.693871 + 1.20182i −0.0219972 + 0.0381003i
\(996\) 0 0
\(997\) −30.9632 −0.980613 −0.490306 0.871550i \(-0.663115\pi\)
−0.490306 + 0.871550i \(0.663115\pi\)
\(998\) 0 0
\(999\) −3.81015 + 14.9604i −0.120548 + 0.473326i
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.t.d.193.3 yes 22
3.2 odd 2 1512.2.t.d.361.6 22
4.3 odd 2 1008.2.t.k.193.9 22
7.2 even 3 504.2.q.d.121.5 yes 22
9.2 odd 6 1512.2.q.c.1369.6 22
9.7 even 3 504.2.q.d.25.5 22
12.11 even 2 3024.2.t.l.1873.6 22
21.2 odd 6 1512.2.q.c.793.6 22
28.23 odd 6 1008.2.q.k.625.7 22
36.7 odd 6 1008.2.q.k.529.7 22
36.11 even 6 3024.2.q.k.2881.6 22
63.2 odd 6 1512.2.t.d.289.6 22
63.16 even 3 inner 504.2.t.d.457.3 yes 22
84.23 even 6 3024.2.q.k.2305.6 22
252.79 odd 6 1008.2.t.k.961.9 22
252.191 even 6 3024.2.t.l.289.6 22
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
504.2.q.d.25.5 22 9.7 even 3
504.2.q.d.121.5 yes 22 7.2 even 3
504.2.t.d.193.3 yes 22 1.1 even 1 trivial
504.2.t.d.457.3 yes 22 63.16 even 3 inner
1008.2.q.k.529.7 22 36.7 odd 6
1008.2.q.k.625.7 22 28.23 odd 6
1008.2.t.k.193.9 22 4.3 odd 2
1008.2.t.k.961.9 22 252.79 odd 6
1512.2.q.c.793.6 22 21.2 odd 6
1512.2.q.c.1369.6 22 9.2 odd 6
1512.2.t.d.289.6 22 63.2 odd 6
1512.2.t.d.361.6 22 3.2 odd 2
3024.2.q.k.2305.6 22 84.23 even 6
3024.2.q.k.2881.6 22 36.11 even 6
3024.2.t.l.289.6 22 252.191 even 6
3024.2.t.l.1873.6 22 12.11 even 2