Properties

Label 252.4.x.a.209.16
Level $252$
Weight $4$
Character 252.209
Analytic conductor $14.868$
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] = [252,4,Mod(41,252)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(252, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 5, 3]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("252.41");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 252.x (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: \(14.8684813214\)
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 209.16
Character \(\chi\) \(=\) 252.209
Dual form 252.4.x.a.41.16

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(2.29905 + 4.65987i) q^{3} +(6.03570 + 10.4541i) q^{5} +(2.10370 + 18.4004i) q^{7} +(-16.4287 + 21.4265i) q^{9} +O(q^{10})\) \(q+(2.29905 + 4.65987i) q^{3} +(6.03570 + 10.4541i) q^{5} +(2.10370 + 18.4004i) q^{7} +(-16.4287 + 21.4265i) q^{9} +(0.00221312 + 0.00127775i) q^{11} +(6.06841 - 3.50360i) q^{13} +(-34.8386 + 52.1602i) q^{15} -28.3143 q^{17} -49.1973i q^{19} +(-80.9069 + 52.1064i) q^{21} +(44.4689 - 25.6741i) q^{23} +(-10.3594 + 17.9431i) q^{25} +(-137.615 - 27.2951i) q^{27} +(97.9469 + 56.5497i) q^{29} +(-28.4271 + 16.4124i) q^{31} +(-0.000866054 + 0.0132505i) q^{33} +(-179.663 + 133.052i) q^{35} -101.961 q^{37} +(30.2779 + 20.2230i) q^{39} +(-11.2449 - 19.4767i) q^{41} +(-227.203 + 393.527i) q^{43} +(-323.155 - 42.4243i) q^{45} +(231.260 - 400.554i) q^{47} +(-334.149 + 77.4179i) q^{49} +(-65.0960 - 131.941i) q^{51} +567.586i q^{53} +0.0308484i q^{55} +(229.253 - 113.107i) q^{57} +(-145.654 - 252.280i) q^{59} +(592.003 + 341.793i) q^{61} +(-428.818 - 257.220i) q^{63} +(73.2542 + 42.2933i) q^{65} +(269.780 + 467.273i) q^{67} +(221.874 + 148.193i) q^{69} -307.517i q^{71} +495.192i q^{73} +(-107.429 - 7.02162i) q^{75} +(-0.0188553 + 0.0434103i) q^{77} +(-324.865 + 562.683i) q^{79} +(-189.193 - 704.022i) q^{81} +(565.581 - 979.615i) q^{83} +(-170.897 - 296.002i) q^{85} +(-38.3293 + 586.430i) q^{87} +130.217 q^{89} +(77.2337 + 104.291i) q^{91} +(-141.835 - 94.7337i) q^{93} +(514.315 - 296.940i) q^{95} +(1260.41 + 727.696i) q^{97} +(-0.0637365 + 0.0264277i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q + 6 q^{7} + 60 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 48 q + 6 q^{7} + 60 q^{9} - 12 q^{11} + 192 q^{15} - 72 q^{21} - 408 q^{23} - 600 q^{25} - 84 q^{29} + 336 q^{37} + 36 q^{39} + 84 q^{43} + 318 q^{49} - 1812 q^{51} - 852 q^{57} - 564 q^{63} + 2964 q^{65} - 588 q^{67} + 2400 q^{77} + 204 q^{79} + 1980 q^{81} - 360 q^{85} - 1080 q^{91} + 2496 q^{93} + 300 q^{95} - 4968 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/252\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(73\) \(127\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(-1\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2.29905 + 4.65987i 0.442452 + 0.896792i
\(4\) 0 0
\(5\) 6.03570 + 10.4541i 0.539850 + 0.935047i 0.998912 + 0.0466430i \(0.0148523\pi\)
−0.459062 + 0.888404i \(0.651814\pi\)
\(6\) 0 0
\(7\) 2.10370 + 18.4004i 0.113589 + 0.993528i
\(8\) 0 0
\(9\) −16.4287 + 21.4265i −0.608472 + 0.793575i
\(10\) 0 0
\(11\) 0.00221312 + 0.00127775i 6.06619e−5 + 3.50232e-5i 0.500030 0.866008i \(-0.333322\pi\)
−0.499970 + 0.866043i \(0.666656\pi\)
\(12\) 0 0
\(13\) 6.06841 3.50360i 0.129467 0.0747479i −0.433868 0.900977i \(-0.642851\pi\)
0.563335 + 0.826229i \(0.309518\pi\)
\(14\) 0 0
\(15\) −34.8386 + 52.1602i −0.599685 + 0.897847i
\(16\) 0 0
\(17\) −28.3143 −0.403955 −0.201977 0.979390i \(-0.564737\pi\)
−0.201977 + 0.979390i \(0.564737\pi\)
\(18\) 0 0
\(19\) 49.1973i 0.594033i −0.954872 0.297016i \(-0.904008\pi\)
0.954872 0.297016i \(-0.0959916\pi\)
\(20\) 0 0
\(21\) −80.9069 + 52.1064i −0.840730 + 0.541455i
\(22\) 0 0
\(23\) 44.4689 25.6741i 0.403148 0.232758i −0.284693 0.958619i \(-0.591892\pi\)
0.687841 + 0.725861i \(0.258558\pi\)
\(24\) 0 0
\(25\) −10.3594 + 17.9431i −0.0828756 + 0.143545i
\(26\) 0 0
\(27\) −137.615 27.2951i −0.980892 0.194554i
\(28\) 0 0
\(29\) 97.9469 + 56.5497i 0.627182 + 0.362104i 0.779660 0.626203i \(-0.215392\pi\)
−0.152478 + 0.988307i \(0.548725\pi\)
\(30\) 0 0
\(31\) −28.4271 + 16.4124i −0.164699 + 0.0950889i −0.580084 0.814557i \(-0.696980\pi\)
0.415385 + 0.909646i \(0.363647\pi\)
\(32\) 0 0
\(33\) −0.000866054 0.0132505i −4.56850e−6 6.98972e-5i
\(34\) 0 0
\(35\) −179.663 + 133.052i −0.867674 + 0.642567i
\(36\) 0 0
\(37\) −101.961 −0.453034 −0.226517 0.974007i \(-0.572734\pi\)
−0.226517 + 0.974007i \(0.572734\pi\)
\(38\) 0 0
\(39\) 30.2779 + 20.2230i 0.124316 + 0.0830328i
\(40\) 0 0
\(41\) −11.2449 19.4767i −0.0428331 0.0741891i 0.843814 0.536636i \(-0.180305\pi\)
−0.886647 + 0.462447i \(0.846972\pi\)
\(42\) 0 0
\(43\) −227.203 + 393.527i −0.805771 + 1.39564i 0.109998 + 0.993932i \(0.464916\pi\)
−0.915769 + 0.401705i \(0.868418\pi\)
\(44\) 0 0
\(45\) −323.155 42.4243i −1.07051 0.140539i
\(46\) 0 0
\(47\) 231.260 400.554i 0.717717 1.24312i −0.244185 0.969729i \(-0.578520\pi\)
0.961902 0.273394i \(-0.0881464\pi\)
\(48\) 0 0
\(49\) −334.149 + 77.4179i −0.974195 + 0.225708i
\(50\) 0 0
\(51\) −65.0960 131.941i −0.178731 0.362263i
\(52\) 0 0
\(53\) 567.586i 1.47102i 0.677515 + 0.735509i \(0.263057\pi\)
−0.677515 + 0.735509i \(0.736943\pi\)
\(54\) 0 0
\(55\) 0.0308484i 7.56290e-5i
\(56\) 0 0
\(57\) 229.253 113.107i 0.532724 0.262831i
\(58\) 0 0
\(59\) −145.654 252.280i −0.321399 0.556680i 0.659378 0.751812i \(-0.270820\pi\)
−0.980777 + 0.195132i \(0.937486\pi\)
\(60\) 0 0
\(61\) 592.003 + 341.793i 1.24259 + 0.717412i 0.969621 0.244610i \(-0.0786601\pi\)
0.272972 + 0.962022i \(0.411993\pi\)
\(62\) 0 0
\(63\) −428.818 257.220i −0.857555 0.514392i
\(64\) 0 0
\(65\) 73.2542 + 42.2933i 0.139786 + 0.0807053i
\(66\) 0 0
\(67\) 269.780 + 467.273i 0.491924 + 0.852037i 0.999957 0.00930051i \(-0.00296049\pi\)
−0.508033 + 0.861338i \(0.669627\pi\)
\(68\) 0 0
\(69\) 221.874 + 148.193i 0.387109 + 0.258556i
\(70\) 0 0
\(71\) 307.517i 0.514022i −0.966408 0.257011i \(-0.917262\pi\)
0.966408 0.257011i \(-0.0827377\pi\)
\(72\) 0 0
\(73\) 495.192i 0.793943i 0.917831 + 0.396971i \(0.129939\pi\)
−0.917831 + 0.396971i \(0.870061\pi\)
\(74\) 0 0
\(75\) −107.429 7.02162i −0.165398 0.0108105i
\(76\) 0 0
\(77\) −0.0188553 + 0.0434103i −2.79060e−5 + 6.42476e-5i
\(78\) 0 0
\(79\) −324.865 + 562.683i −0.462661 + 0.801352i −0.999093 0.0425920i \(-0.986438\pi\)
0.536432 + 0.843944i \(0.319772\pi\)
\(80\) 0 0
\(81\) −189.193 704.022i −0.259524 0.965737i
\(82\) 0 0
\(83\) 565.581 979.615i 0.747959 1.29550i −0.200841 0.979624i \(-0.564367\pi\)
0.948800 0.315879i \(-0.102299\pi\)
\(84\) 0 0
\(85\) −170.897 296.002i −0.218075 0.377717i
\(86\) 0 0
\(87\) −38.3293 + 586.430i −0.0472336 + 0.722665i
\(88\) 0 0
\(89\) 130.217 0.155089 0.0775447 0.996989i \(-0.475292\pi\)
0.0775447 + 0.996989i \(0.475292\pi\)
\(90\) 0 0
\(91\) 77.2337 + 104.291i 0.0889702 + 0.120139i
\(92\) 0 0
\(93\) −141.835 94.7337i −0.158146 0.105628i
\(94\) 0 0
\(95\) 514.315 296.940i 0.555449 0.320688i
\(96\) 0 0
\(97\) 1260.41 + 727.696i 1.31933 + 0.761715i 0.983621 0.180251i \(-0.0576909\pi\)
0.335709 + 0.941966i \(0.391024\pi\)
\(98\) 0 0
\(99\) −0.0637365 + 0.0264277i −6.47046e−5 + 2.68292e-5i
\(100\) 0 0
\(101\) 197.600 342.254i 0.194673 0.337183i −0.752120 0.659026i \(-0.770969\pi\)
0.946793 + 0.321842i \(0.104302\pi\)
\(102\) 0 0
\(103\) 545.260 314.806i 0.521612 0.301153i −0.215982 0.976397i \(-0.569295\pi\)
0.737594 + 0.675244i \(0.235962\pi\)
\(104\) 0 0
\(105\) −1033.06 531.314i −0.960154 0.493818i
\(106\) 0 0
\(107\) 515.616i 0.465855i −0.972494 0.232927i \(-0.925170\pi\)
0.972494 0.232927i \(-0.0748305\pi\)
\(108\) 0 0
\(109\) 794.011 0.697729 0.348864 0.937173i \(-0.386567\pi\)
0.348864 + 0.937173i \(0.386567\pi\)
\(110\) 0 0
\(111\) −234.413 475.124i −0.200446 0.406277i
\(112\) 0 0
\(113\) 1782.31 1029.02i 1.48377 0.856654i 0.483939 0.875102i \(-0.339206\pi\)
0.999830 + 0.0184476i \(0.00587238\pi\)
\(114\) 0 0
\(115\) 536.802 + 309.923i 0.435279 + 0.251308i
\(116\) 0 0
\(117\) −24.6264 + 187.585i −0.0194591 + 0.148224i
\(118\) 0 0
\(119\) −59.5649 520.994i −0.0458849 0.401340i
\(120\) 0 0
\(121\) −665.500 1152.68i −0.500000 0.866025i
\(122\) 0 0
\(123\) 64.9064 97.1776i 0.0475806 0.0712375i
\(124\) 0 0
\(125\) 1258.82 0.900738
\(126\) 0 0
\(127\) −2071.63 −1.44746 −0.723729 0.690084i \(-0.757573\pi\)
−0.723729 + 0.690084i \(0.757573\pi\)
\(128\) 0 0
\(129\) −2356.14 153.998i −1.60811 0.105107i
\(130\) 0 0
\(131\) −467.210 809.232i −0.311606 0.539717i 0.667104 0.744964i \(-0.267533\pi\)
−0.978710 + 0.205247i \(0.934200\pi\)
\(132\) 0 0
\(133\) 905.249 103.496i 0.590188 0.0674758i
\(134\) 0 0
\(135\) −545.258 1603.40i −0.347617 1.02221i
\(136\) 0 0
\(137\) 2320.59 + 1339.79i 1.44716 + 0.835521i 0.998312 0.0580849i \(-0.0184994\pi\)
0.448853 + 0.893606i \(0.351833\pi\)
\(138\) 0 0
\(139\) 794.727 458.836i 0.484949 0.279985i −0.237528 0.971381i \(-0.576337\pi\)
0.722476 + 0.691395i \(0.243004\pi\)
\(140\) 0 0
\(141\) 2398.21 + 156.748i 1.43238 + 0.0936207i
\(142\) 0 0
\(143\) 0.0179068 1.04716e−5
\(144\) 0 0
\(145\) 1365.27i 0.781926i
\(146\) 0 0
\(147\) −1128.98 1379.10i −0.633448 0.773785i
\(148\) 0 0
\(149\) 1187.44 685.572i 0.652881 0.376941i −0.136678 0.990616i \(-0.543643\pi\)
0.789559 + 0.613674i \(0.210309\pi\)
\(150\) 0 0
\(151\) 127.186 220.292i 0.0685445 0.118723i −0.829716 0.558185i \(-0.811498\pi\)
0.898261 + 0.439463i \(0.144831\pi\)
\(152\) 0 0
\(153\) 465.169 606.678i 0.245795 0.320568i
\(154\) 0 0
\(155\) −343.155 198.121i −0.177825 0.102667i
\(156\) 0 0
\(157\) 511.747 295.457i 0.260139 0.150191i −0.364259 0.931298i \(-0.618678\pi\)
0.624398 + 0.781106i \(0.285344\pi\)
\(158\) 0 0
\(159\) −2644.87 + 1304.91i −1.31920 + 0.650855i
\(160\) 0 0
\(161\) 565.963 + 764.234i 0.277044 + 0.374100i
\(162\) 0 0
\(163\) −2197.17 −1.05580 −0.527901 0.849306i \(-0.677021\pi\)
−0.527901 + 0.849306i \(0.677021\pi\)
\(164\) 0 0
\(165\) −0.143749 + 0.0709220i −6.78235e−5 + 3.34622e-5i
\(166\) 0 0
\(167\) 210.903 + 365.295i 0.0977256 + 0.169266i 0.910743 0.412974i \(-0.135510\pi\)
−0.813017 + 0.582240i \(0.802177\pi\)
\(168\) 0 0
\(169\) −1073.95 + 1860.14i −0.488826 + 0.846671i
\(170\) 0 0
\(171\) 1054.13 + 808.249i 0.471410 + 0.361452i
\(172\) 0 0
\(173\) 872.428 1511.09i 0.383407 0.664081i −0.608139 0.793830i \(-0.708084\pi\)
0.991547 + 0.129749i \(0.0414172\pi\)
\(174\) 0 0
\(175\) −351.953 152.871i −0.152029 0.0660341i
\(176\) 0 0
\(177\) 840.727 1258.73i 0.357022 0.534532i
\(178\) 0 0
\(179\) 4311.61i 1.80036i 0.435518 + 0.900180i \(0.356565\pi\)
−0.435518 + 0.900180i \(0.643435\pi\)
\(180\) 0 0
\(181\) 1307.99i 0.537141i 0.963260 + 0.268570i \(0.0865512\pi\)
−0.963260 + 0.268570i \(0.913449\pi\)
\(182\) 0 0
\(183\) −231.667 + 3544.45i −0.0935808 + 1.43177i
\(184\) 0 0
\(185\) −615.405 1065.91i −0.244570 0.423608i
\(186\) 0 0
\(187\) −0.0626630 0.0361785i −2.45047e−5 1.41478e-5i
\(188\) 0 0
\(189\) 212.739 2589.60i 0.0818757 0.996643i
\(190\) 0 0
\(191\) −3495.99 2018.41i −1.32440 0.764644i −0.339975 0.940435i \(-0.610419\pi\)
−0.984428 + 0.175791i \(0.943752\pi\)
\(192\) 0 0
\(193\) 2209.64 + 3827.21i 0.824112 + 1.42740i 0.902596 + 0.430488i \(0.141658\pi\)
−0.0784849 + 0.996915i \(0.525008\pi\)
\(194\) 0 0
\(195\) −28.6663 + 438.589i −0.0105274 + 0.161067i
\(196\) 0 0
\(197\) 151.673i 0.0548540i 0.999624 + 0.0274270i \(0.00873138\pi\)
−0.999624 + 0.0274270i \(0.991269\pi\)
\(198\) 0 0
\(199\) 1838.65i 0.654966i 0.944857 + 0.327483i \(0.106200\pi\)
−0.944857 + 0.327483i \(0.893800\pi\)
\(200\) 0 0
\(201\) −1557.19 + 2331.42i −0.546447 + 0.818139i
\(202\) 0 0
\(203\) −834.485 + 1921.22i −0.288519 + 0.664254i
\(204\) 0 0
\(205\) 135.742 235.111i 0.0462468 0.0801019i
\(206\) 0 0
\(207\) −180.461 + 1374.61i −0.0605936 + 0.461555i
\(208\) 0 0
\(209\) 0.0628616 0.108879i 2.08049e−5 3.60352e-5i
\(210\) 0 0
\(211\) 1229.27 + 2129.15i 0.401072 + 0.694678i 0.993856 0.110684i \(-0.0353042\pi\)
−0.592783 + 0.805362i \(0.701971\pi\)
\(212\) 0 0
\(213\) 1432.99 706.997i 0.460971 0.227430i
\(214\) 0 0
\(215\) −5485.32 −1.73998
\(216\) 0 0
\(217\) −361.797 488.543i −0.113181 0.152832i
\(218\) 0 0
\(219\) −2307.53 + 1138.47i −0.712002 + 0.351282i
\(220\) 0 0
\(221\) −171.823 + 99.2019i −0.0522989 + 0.0301948i
\(222\) 0 0
\(223\) −4491.19 2592.99i −1.34867 0.778653i −0.360606 0.932718i \(-0.617430\pi\)
−0.988061 + 0.154066i \(0.950763\pi\)
\(224\) 0 0
\(225\) −214.266 516.750i −0.0634861 0.153111i
\(226\) 0 0
\(227\) 497.531 861.749i 0.145473 0.251966i −0.784077 0.620664i \(-0.786863\pi\)
0.929549 + 0.368698i \(0.120196\pi\)
\(228\) 0 0
\(229\) 101.864 58.8111i 0.0293946 0.0169710i −0.485231 0.874386i \(-0.661264\pi\)
0.514625 + 0.857415i \(0.327931\pi\)
\(230\) 0 0
\(231\) −0.245635 + 0.0119393i −6.99637e−5 + 3.40064e-6i
\(232\) 0 0
\(233\) 1484.08i 0.417276i 0.977993 + 0.208638i \(0.0669030\pi\)
−0.977993 + 0.208638i \(0.933097\pi\)
\(234\) 0 0
\(235\) 5583.26 1.54984
\(236\) 0 0
\(237\) −3368.91 220.193i −0.923351 0.0603505i
\(238\) 0 0
\(239\) 2392.08 1381.07i 0.647409 0.373782i −0.140054 0.990144i \(-0.544728\pi\)
0.787463 + 0.616362i \(0.211394\pi\)
\(240\) 0 0
\(241\) −3362.99 1941.62i −0.898877 0.518967i −0.0220413 0.999757i \(-0.507017\pi\)
−0.876836 + 0.480790i \(0.840350\pi\)
\(242\) 0 0
\(243\) 2845.69 2500.19i 0.751238 0.660031i
\(244\) 0 0
\(245\) −2826.16 3025.97i −0.736967 0.789070i
\(246\) 0 0
\(247\) −172.367 298.549i −0.0444027 0.0769078i
\(248\) 0 0
\(249\) 5865.18 + 383.350i 1.49273 + 0.0975655i
\(250\) 0 0
\(251\) −5549.57 −1.39556 −0.697780 0.716312i \(-0.745829\pi\)
−0.697780 + 0.716312i \(0.745829\pi\)
\(252\) 0 0
\(253\) 0.131220 3.26076e−5
\(254\) 0 0
\(255\) 986.430 1476.88i 0.242246 0.362689i
\(256\) 0 0
\(257\) 809.585 + 1402.24i 0.196500 + 0.340348i 0.947391 0.320078i \(-0.103709\pi\)
−0.750891 + 0.660426i \(0.770376\pi\)
\(258\) 0 0
\(259\) −214.495 1876.12i −0.0514598 0.450102i
\(260\) 0 0
\(261\) −2820.81 + 1169.62i −0.668979 + 0.277386i
\(262\) 0 0
\(263\) −594.813 343.415i −0.139459 0.0805167i 0.428647 0.903472i \(-0.358990\pi\)
−0.568106 + 0.822955i \(0.692324\pi\)
\(264\) 0 0
\(265\) −5933.62 + 3425.78i −1.37547 + 0.794128i
\(266\) 0 0
\(267\) 299.375 + 606.793i 0.0686196 + 0.139083i
\(268\) 0 0
\(269\) −1805.03 −0.409124 −0.204562 0.978854i \(-0.565577\pi\)
−0.204562 + 0.978854i \(0.565577\pi\)
\(270\) 0 0
\(271\) 6726.19i 1.50770i −0.657046 0.753851i \(-0.728194\pi\)
0.657046 0.753851i \(-0.271806\pi\)
\(272\) 0 0
\(273\) −308.416 + 599.668i −0.0683743 + 0.132943i
\(274\) 0 0
\(275\) −0.0458534 + 0.0264735i −1.00548e−5 + 5.80513e-6i
\(276\) 0 0
\(277\) −336.124 + 582.184i −0.0729088 + 0.126282i −0.900175 0.435528i \(-0.856562\pi\)
0.827266 + 0.561810i \(0.189895\pi\)
\(278\) 0 0
\(279\) 115.361 878.730i 0.0247544 0.188560i
\(280\) 0 0
\(281\) −2007.12 1158.81i −0.426102 0.246010i 0.271583 0.962415i \(-0.412453\pi\)
−0.697685 + 0.716405i \(0.745786\pi\)
\(282\) 0 0
\(283\) 8014.87 4627.39i 1.68352 0.971978i 0.724222 0.689567i \(-0.242199\pi\)
0.959294 0.282411i \(-0.0911342\pi\)
\(284\) 0 0
\(285\) 2566.14 + 1713.96i 0.533350 + 0.356233i
\(286\) 0 0
\(287\) 334.723 247.883i 0.0688435 0.0509829i
\(288\) 0 0
\(289\) −4111.30 −0.836821
\(290\) 0 0
\(291\) −493.231 + 7546.34i −0.0993599 + 1.52019i
\(292\) 0 0
\(293\) 4392.72 + 7608.42i 0.875855 + 1.51703i 0.855849 + 0.517225i \(0.173035\pi\)
0.0200056 + 0.999800i \(0.493632\pi\)
\(294\) 0 0
\(295\) 1758.25 3045.38i 0.347015 0.601047i
\(296\) 0 0
\(297\) −0.269683 0.236245i −5.26889e−5 4.61559e-5i
\(298\) 0 0
\(299\) 179.904 311.602i 0.0347963 0.0602689i
\(300\) 0 0
\(301\) −7719.03 3352.76i −1.47813 0.642027i
\(302\) 0 0
\(303\) 2049.15 + 133.933i 0.388517 + 0.0253936i
\(304\) 0 0
\(305\) 8251.85i 1.54918i
\(306\) 0 0
\(307\) 599.516i 0.111453i 0.998446 + 0.0557267i \(0.0177476\pi\)
−0.998446 + 0.0557267i \(0.982252\pi\)
\(308\) 0 0
\(309\) 2720.53 + 1817.08i 0.500860 + 0.334532i
\(310\) 0 0
\(311\) 4494.29 + 7784.34i 0.819447 + 1.41932i 0.906090 + 0.423084i \(0.139052\pi\)
−0.0866437 + 0.996239i \(0.527614\pi\)
\(312\) 0 0
\(313\) 7102.69 + 4100.74i 1.28264 + 0.740535i 0.977331 0.211717i \(-0.0679054\pi\)
0.305314 + 0.952252i \(0.401239\pi\)
\(314\) 0 0
\(315\) 100.801 6035.43i 0.0180301 1.07955i
\(316\) 0 0
\(317\) −5894.42 3403.15i −1.04436 0.602964i −0.123298 0.992370i \(-0.539347\pi\)
−0.921066 + 0.389405i \(0.872681\pi\)
\(318\) 0 0
\(319\) 0.144512 + 0.250302i 2.53640e−5 + 4.39318e-5i
\(320\) 0 0
\(321\) 2402.70 1185.43i 0.417775 0.206118i
\(322\) 0 0
\(323\) 1392.99i 0.239962i
\(324\) 0 0
\(325\) 145.181i 0.0247791i
\(326\) 0 0
\(327\) 1825.47 + 3699.99i 0.308712 + 0.625718i
\(328\) 0 0
\(329\) 7856.85 + 3412.63i 1.31660 + 0.571867i
\(330\) 0 0
\(331\) 3875.41 6712.41i 0.643540 1.11464i −0.341097 0.940028i \(-0.610798\pi\)
0.984637 0.174616i \(-0.0558684\pi\)
\(332\) 0 0
\(333\) 1675.09 2184.67i 0.275658 0.359517i
\(334\) 0 0
\(335\) −3256.63 + 5640.64i −0.531130 + 0.919944i
\(336\) 0 0
\(337\) −1544.24 2674.70i −0.249614 0.432344i 0.713805 0.700345i \(-0.246970\pi\)
−0.963419 + 0.268001i \(0.913637\pi\)
\(338\) 0 0
\(339\) 8892.72 + 5939.58i 1.42474 + 0.951603i
\(340\) 0 0
\(341\) −0.0838835 −1.33213e−5
\(342\) 0 0
\(343\) −2127.47 5985.61i −0.334906 0.942252i
\(344\) 0 0
\(345\) −210.065 + 3213.95i −0.0327812 + 0.501546i
\(346\) 0 0
\(347\) 10112.9 5838.71i 1.56453 0.903281i 0.567739 0.823209i \(-0.307818\pi\)
0.996789 0.0800721i \(-0.0255151\pi\)
\(348\) 0 0
\(349\) −6683.19 3858.54i −1.02505 0.591814i −0.109489 0.993988i \(-0.534921\pi\)
−0.915563 + 0.402174i \(0.868255\pi\)
\(350\) 0 0
\(351\) −930.737 + 316.511i −0.141536 + 0.0481313i
\(352\) 0 0
\(353\) 2760.12 4780.66i 0.416165 0.720818i −0.579385 0.815054i \(-0.696707\pi\)
0.995550 + 0.0942355i \(0.0300407\pi\)
\(354\) 0 0
\(355\) 3214.83 1856.08i 0.480635 0.277495i
\(356\) 0 0
\(357\) 2290.82 1475.36i 0.339617 0.218723i
\(358\) 0 0
\(359\) 6290.95i 0.924857i −0.886657 0.462429i \(-0.846978\pi\)
0.886657 0.462429i \(-0.153022\pi\)
\(360\) 0 0
\(361\) 4438.63 0.647125
\(362\) 0 0
\(363\) 3841.32 5751.21i 0.555419 0.831571i
\(364\) 0 0
\(365\) −5176.81 + 2988.83i −0.742374 + 0.428610i
\(366\) 0 0
\(367\) −3342.42 1929.75i −0.475404 0.274474i 0.243095 0.970002i \(-0.421837\pi\)
−0.718499 + 0.695528i \(0.755171\pi\)
\(368\) 0 0
\(369\) 602.058 + 79.0390i 0.0849373 + 0.0111507i
\(370\) 0 0
\(371\) −10443.8 + 1194.03i −1.46150 + 0.167092i
\(372\) 0 0
\(373\) −4308.29 7462.17i −0.598055 1.03586i −0.993108 0.117204i \(-0.962607\pi\)
0.395052 0.918659i \(-0.370726\pi\)
\(374\) 0 0
\(375\) 2894.09 + 5865.93i 0.398534 + 0.807775i
\(376\) 0 0
\(377\) 792.509 0.108266
\(378\) 0 0
\(379\) 11100.5 1.50447 0.752237 0.658893i \(-0.228975\pi\)
0.752237 + 0.658893i \(0.228975\pi\)
\(380\) 0 0
\(381\) −4762.77 9653.51i −0.640431 1.29807i
\(382\) 0 0
\(383\) −3294.29 5705.87i −0.439505 0.761244i 0.558147 0.829742i \(-0.311513\pi\)
−0.997651 + 0.0684980i \(0.978179\pi\)
\(384\) 0 0
\(385\) −0.567622 + 0.0648959i −7.51395e−5 + 8.59065e-6i
\(386\) 0 0
\(387\) −4699.27 11333.3i −0.617253 1.48865i
\(388\) 0 0
\(389\) −6913.20 3991.34i −0.901062 0.520228i −0.0235174 0.999723i \(-0.507487\pi\)
−0.877545 + 0.479495i \(0.840820\pi\)
\(390\) 0 0
\(391\) −1259.11 + 726.945i −0.162854 + 0.0940235i
\(392\) 0 0
\(393\) 2696.77 4037.60i 0.346143 0.518245i
\(394\) 0 0
\(395\) −7843.16 −0.999069
\(396\) 0 0
\(397\) 9040.06i 1.14284i −0.820658 0.571420i \(-0.806393\pi\)
0.820658 0.571420i \(-0.193607\pi\)
\(398\) 0 0
\(399\) 2563.49 + 3980.40i 0.321642 + 0.499421i
\(400\) 0 0
\(401\) −836.173 + 482.765i −0.104131 + 0.0601200i −0.551161 0.834399i \(-0.685815\pi\)
0.447030 + 0.894519i \(0.352482\pi\)
\(402\) 0 0
\(403\) −115.005 + 199.194i −0.0142154 + 0.0246218i
\(404\) 0 0
\(405\) 6218.04 6227.12i 0.762906 0.764020i
\(406\) 0 0
\(407\) −0.225652 0.130280i −2.74819e−5 1.58667e-5i
\(408\) 0 0
\(409\) −2868.77 + 1656.28i −0.346825 + 0.200240i −0.663286 0.748366i \(-0.730839\pi\)
0.316461 + 0.948606i \(0.397505\pi\)
\(410\) 0 0
\(411\) −908.110 + 13893.9i −0.108987 + 1.66748i
\(412\) 0 0
\(413\) 4335.64 3210.82i 0.516569 0.382552i
\(414\) 0 0
\(415\) 13654.7 1.61514
\(416\) 0 0
\(417\) 3965.23 + 2648.44i 0.465655 + 0.311018i
\(418\) 0 0
\(419\) 4172.32 + 7226.67i 0.486470 + 0.842591i 0.999879 0.0155530i \(-0.00495087\pi\)
−0.513409 + 0.858144i \(0.671618\pi\)
\(420\) 0 0
\(421\) 3350.59 5803.38i 0.387880 0.671828i −0.604284 0.796769i \(-0.706541\pi\)
0.992164 + 0.124941i \(0.0398741\pi\)
\(422\) 0 0
\(423\) 4783.17 + 11535.7i 0.549801 + 1.32597i
\(424\) 0 0
\(425\) 293.321 508.046i 0.0334780 0.0579856i
\(426\) 0 0
\(427\) −5043.73 + 11612.1i −0.571623 + 1.31604i
\(428\) 0 0
\(429\) 0.0411687 + 0.0834434i 4.63320e−6 + 9.39088e-6i
\(430\) 0 0
\(431\) 9168.55i 1.02467i −0.858785 0.512336i \(-0.828780\pi\)
0.858785 0.512336i \(-0.171220\pi\)
\(432\) 0 0
\(433\) 1346.90i 0.149487i −0.997203 0.0747436i \(-0.976186\pi\)
0.997203 0.0747436i \(-0.0238138\pi\)
\(434\) 0 0
\(435\) −6361.97 + 3138.82i −0.701225 + 0.345965i
\(436\) 0 0
\(437\) −1263.10 2187.75i −0.138266 0.239483i
\(438\) 0 0
\(439\) 4892.07 + 2824.44i 0.531858 + 0.307069i 0.741773 0.670651i \(-0.233985\pi\)
−0.209914 + 0.977720i \(0.567319\pi\)
\(440\) 0 0
\(441\) 3830.85 8431.53i 0.413654 0.910434i
\(442\) 0 0
\(443\) −7525.68 4344.95i −0.807124 0.465993i 0.0388322 0.999246i \(-0.487636\pi\)
−0.845956 + 0.533253i \(0.820970\pi\)
\(444\) 0 0
\(445\) 785.950 + 1361.31i 0.0837249 + 0.145016i
\(446\) 0 0
\(447\) 5924.67 + 3957.17i 0.626906 + 0.418720i
\(448\) 0 0
\(449\) 14492.3i 1.52324i 0.648026 + 0.761618i \(0.275595\pi\)
−0.648026 + 0.761618i \(0.724405\pi\)
\(450\) 0 0
\(451\) 0.0574724i 6.00060e-6i
\(452\) 0 0
\(453\) 1318.94 + 86.2062i 0.136797 + 0.00894111i
\(454\) 0 0
\(455\) −624.109 + 1436.88i −0.0643048 + 0.148048i
\(456\) 0 0
\(457\) −4131.68 + 7156.27i −0.422914 + 0.732508i −0.996223 0.0868308i \(-0.972326\pi\)
0.573309 + 0.819339i \(0.305659\pi\)
\(458\) 0 0
\(459\) 3896.48 + 772.843i 0.396236 + 0.0785909i
\(460\) 0 0
\(461\) 4948.33 8570.75i 0.499928 0.865900i −0.500072 0.865984i \(-0.666693\pi\)
1.00000 8.36677e-5i \(2.66323e-5\pi\)
\(462\) 0 0
\(463\) 1987.79 + 3442.95i 0.199525 + 0.345588i 0.948375 0.317152i \(-0.102727\pi\)
−0.748849 + 0.662740i \(0.769393\pi\)
\(464\) 0 0
\(465\) 134.286 2054.55i 0.0133922 0.204898i
\(466\) 0 0
\(467\) −17044.5 −1.68892 −0.844460 0.535619i \(-0.820079\pi\)
−0.844460 + 0.535619i \(0.820079\pi\)
\(468\) 0 0
\(469\) −8030.47 + 5947.07i −0.790645 + 0.585522i
\(470\) 0 0
\(471\) 2553.32 + 1705.40i 0.249790 + 0.166838i
\(472\) 0 0
\(473\) −1.00566 + 0.580616i −9.77592e−5 + 5.64413e-5i
\(474\) 0 0
\(475\) 882.751 + 509.656i 0.0852703 + 0.0492308i
\(476\) 0 0
\(477\) −12161.4 9324.72i −1.16736 0.895073i
\(478\) 0 0
\(479\) −5364.93 + 9292.33i −0.511753 + 0.886383i 0.488154 + 0.872758i \(0.337670\pi\)
−0.999907 + 0.0136250i \(0.995663\pi\)
\(480\) 0 0
\(481\) −618.740 + 357.230i −0.0586530 + 0.0338633i
\(482\) 0 0
\(483\) −2260.05 + 4394.33i −0.212911 + 0.413973i
\(484\) 0 0
\(485\) 17568.6i 1.64485i
\(486\) 0 0
\(487\) −12461.1 −1.15948 −0.579741 0.814801i \(-0.696846\pi\)
−0.579741 + 0.814801i \(0.696846\pi\)
\(488\) 0 0
\(489\) −5051.40 10238.5i −0.467142 0.946834i
\(490\) 0 0
\(491\) −4052.21 + 2339.55i −0.372452 + 0.215035i −0.674529 0.738248i \(-0.735653\pi\)
0.302077 + 0.953283i \(0.402320\pi\)
\(492\) 0 0
\(493\) −2773.30 1601.16i −0.253353 0.146273i
\(494\) 0 0
\(495\) −0.660974 0.506800i −6.00173e−5 4.60181e-5i
\(496\) 0 0
\(497\) 5658.43 646.925i 0.510695 0.0583874i
\(498\) 0 0
\(499\) 2685.54 + 4651.49i 0.240924 + 0.417293i 0.960978 0.276625i \(-0.0892161\pi\)
−0.720054 + 0.693918i \(0.755883\pi\)
\(500\) 0 0
\(501\) −1217.35 + 1822.61i −0.108557 + 0.162532i
\(502\) 0 0
\(503\) 22263.9 1.97355 0.986777 0.162086i \(-0.0518222\pi\)
0.986777 + 0.162086i \(0.0518222\pi\)
\(504\) 0 0
\(505\) 4770.63 0.420376
\(506\) 0 0
\(507\) −11137.0 727.921i −0.975569 0.0637635i
\(508\) 0 0
\(509\) −7249.76 12556.9i −0.631316 1.09347i −0.987283 0.158973i \(-0.949182\pi\)
0.355967 0.934498i \(-0.384152\pi\)
\(510\) 0 0
\(511\) −9111.72 + 1041.74i −0.788804 + 0.0901834i
\(512\) 0 0
\(513\) −1342.85 + 6770.30i −0.115571 + 0.582682i
\(514\) 0 0
\(515\) 6582.05 + 3800.15i 0.563184 + 0.325155i
\(516\) 0 0
\(517\) 1.02361 0.590983i 8.70762e−5 5.02735e-5i
\(518\) 0 0
\(519\) 9047.24 + 591.330i 0.765182 + 0.0500126i
\(520\) 0 0
\(521\) −7009.79 −0.589452 −0.294726 0.955582i \(-0.595228\pi\)
−0.294726 + 0.955582i \(0.595228\pi\)
\(522\) 0 0
\(523\) 3631.76i 0.303644i −0.988408 0.151822i \(-0.951486\pi\)
0.988408 0.151822i \(-0.0485141\pi\)
\(524\) 0 0
\(525\) −96.7990 1991.51i −0.00804696 0.165556i
\(526\) 0 0
\(527\) 804.894 464.706i 0.0665308 0.0384116i
\(528\) 0 0
\(529\) −4765.18 + 8253.53i −0.391648 + 0.678354i
\(530\) 0 0
\(531\) 7798.41 + 1023.79i 0.637330 + 0.0836696i
\(532\) 0 0
\(533\) −136.477 78.7951i −0.0110910 0.00640336i
\(534\) 0 0
\(535\) 5390.32 3112.10i 0.435596 0.251492i
\(536\) 0 0
\(537\) −20091.5 + 9912.59i −1.61455 + 0.796573i
\(538\) 0 0
\(539\) −0.838432 0.255622i −6.70015e−5 2.04275e-5i
\(540\) 0 0
\(541\) −12760.4 −1.01407 −0.507037 0.861925i \(-0.669259\pi\)
−0.507037 + 0.861925i \(0.669259\pi\)
\(542\) 0 0
\(543\) −6095.08 + 3007.14i −0.481704 + 0.237659i
\(544\) 0 0
\(545\) 4792.42 + 8300.71i 0.376669 + 0.652410i
\(546\) 0 0
\(547\) −3919.62 + 6788.98i −0.306382 + 0.530669i −0.977568 0.210620i \(-0.932452\pi\)
0.671186 + 0.741289i \(0.265785\pi\)
\(548\) 0 0
\(549\) −17049.3 + 7069.34i −1.32540 + 0.549566i
\(550\) 0 0
\(551\) 2782.09 4818.72i 0.215101 0.372567i
\(552\) 0 0
\(553\) −11037.0 4793.93i −0.848718 0.368641i
\(554\) 0 0
\(555\) 3552.17 5318.29i 0.271678 0.406755i
\(556\) 0 0
\(557\) 9443.19i 0.718350i 0.933270 + 0.359175i \(0.116942\pi\)
−0.933270 + 0.359175i \(0.883058\pi\)
\(558\) 0 0
\(559\) 3184.11i 0.240919i
\(560\) 0 0
\(561\) 0.0245217 0.375177i 1.84547e−6 2.82353e-5i
\(562\) 0 0
\(563\) −8022.13 13894.7i −0.600519 1.04013i −0.992742 0.120260i \(-0.961627\pi\)
0.392223 0.919870i \(-0.371706\pi\)
\(564\) 0 0
\(565\) 21515.0 + 12421.7i 1.60202 + 0.924929i
\(566\) 0 0
\(567\) 12556.3 4962.27i 0.930007 0.367541i
\(568\) 0 0
\(569\) 19213.9 + 11093.2i 1.41562 + 0.817310i 0.995910 0.0903469i \(-0.0287976\pi\)
0.419712 + 0.907657i \(0.362131\pi\)
\(570\) 0 0
\(571\) −3211.11 5561.80i −0.235343 0.407625i 0.724030 0.689769i \(-0.242288\pi\)
−0.959372 + 0.282144i \(0.908955\pi\)
\(572\) 0 0
\(573\) 1368.07 20931.3i 0.0997419 1.52603i
\(574\) 0 0
\(575\) 1063.88i 0.0771597i
\(576\) 0 0
\(577\) 21441.1i 1.54697i 0.633813 + 0.773486i \(0.281489\pi\)
−0.633813 + 0.773486i \(0.718511\pi\)
\(578\) 0 0
\(579\) −12754.2 + 19095.6i −0.915454 + 1.37061i
\(580\) 0 0
\(581\) 19215.1 + 8346.09i 1.37208 + 0.595963i
\(582\) 0 0
\(583\) −0.725230 + 1.25614i −5.15197e−5 + 8.92347e-5i
\(584\) 0 0
\(585\) −2109.67 + 874.757i −0.149101 + 0.0618235i
\(586\) 0 0
\(587\) 342.762 593.681i 0.0241010 0.0417442i −0.853723 0.520727i \(-0.825661\pi\)
0.877824 + 0.478983i \(0.158994\pi\)
\(588\) 0 0
\(589\) 807.445 + 1398.54i 0.0564859 + 0.0978365i
\(590\) 0 0
\(591\) −706.775 + 348.703i −0.0491926 + 0.0242703i
\(592\) 0 0
\(593\) −11503.8 −0.796635 −0.398317 0.917248i \(-0.630406\pi\)
−0.398317 + 0.917248i \(0.630406\pi\)
\(594\) 0 0
\(595\) 5087.04 3767.27i 0.350501 0.259568i
\(596\) 0 0
\(597\) −8567.85 + 4227.14i −0.587368 + 0.289791i
\(598\) 0 0
\(599\) 5124.45 2958.60i 0.349548 0.201812i −0.314938 0.949112i \(-0.601984\pi\)
0.664486 + 0.747300i \(0.268650\pi\)
\(600\) 0 0
\(601\) −10168.3 5870.67i −0.690139 0.398452i 0.113525 0.993535i \(-0.463786\pi\)
−0.803664 + 0.595083i \(0.797119\pi\)
\(602\) 0 0
\(603\) −14444.2 1896.25i −0.975478 0.128062i
\(604\) 0 0
\(605\) 8033.52 13914.5i 0.539850 0.935047i
\(606\) 0 0
\(607\) 24018.0 13866.8i 1.60603 0.927241i 0.615782 0.787916i \(-0.288840\pi\)
0.990247 0.139325i \(-0.0444933\pi\)
\(608\) 0 0
\(609\) −10871.2 + 528.402i −0.723353 + 0.0351591i
\(610\) 0 0
\(611\) 3240.96i 0.214592i
\(612\) 0 0
\(613\) −19338.2 −1.27416 −0.637081 0.770797i \(-0.719858\pi\)
−0.637081 + 0.770797i \(0.719858\pi\)
\(614\) 0 0
\(615\) 1407.66 + 92.0054i 0.0922968 + 0.00603255i
\(616\) 0 0
\(617\) 9927.33 5731.55i 0.647746 0.373976i −0.139846 0.990173i \(-0.544661\pi\)
0.787592 + 0.616197i \(0.211327\pi\)
\(618\) 0 0
\(619\) −3429.59 1980.07i −0.222693 0.128572i 0.384504 0.923123i \(-0.374373\pi\)
−0.607196 + 0.794552i \(0.707706\pi\)
\(620\) 0 0
\(621\) −6820.38 + 2319.37i −0.440728 + 0.149876i
\(622\) 0 0
\(623\) 273.937 + 2396.04i 0.0176165 + 0.154086i
\(624\) 0 0
\(625\) 8892.79 + 15402.8i 0.569139 + 0.985777i
\(626\) 0 0
\(627\) 0.651886 + 0.0426075i 4.15212e−5 + 2.71384e-6i
\(628\) 0 0
\(629\) 2886.95 0.183005
\(630\) 0 0
\(631\) −5409.20 −0.341263 −0.170631 0.985335i \(-0.554581\pi\)
−0.170631 + 0.985335i \(0.554581\pi\)
\(632\) 0 0
\(633\) −7095.43 + 10623.3i −0.445526 + 0.667040i
\(634\) 0 0
\(635\) −12503.7 21657.1i −0.781410 1.35344i
\(636\) 0 0
\(637\) −1756.51 + 1640.53i −0.109255 + 0.102041i
\(638\) 0 0
\(639\) 6589.02 + 5052.12i 0.407915 + 0.312768i
\(640\) 0 0
\(641\) −20809.6 12014.4i −1.28226 0.740313i −0.304999 0.952353i \(-0.598656\pi\)
−0.977261 + 0.212040i \(0.931989\pi\)
\(642\) 0 0
\(643\) −3581.87 + 2068.00i −0.219682 + 0.126833i −0.605803 0.795615i \(-0.707148\pi\)
0.386121 + 0.922448i \(0.373815\pi\)
\(644\) 0 0
\(645\) −12611.0 25560.9i −0.769859 1.56040i
\(646\) 0 0
\(647\) 1393.75 0.0846895 0.0423447 0.999103i \(-0.486517\pi\)
0.0423447 + 0.999103i \(0.486517\pi\)
\(648\) 0 0
\(649\) 0.744436i 4.50257e-5i
\(650\) 0 0
\(651\) 1444.76 2809.11i 0.0869809 0.169121i
\(652\) 0 0
\(653\) 7846.10 4529.95i 0.470202 0.271471i −0.246122 0.969239i \(-0.579157\pi\)
0.716324 + 0.697768i \(0.245823\pi\)
\(654\) 0 0
\(655\) 5639.89 9768.57i 0.336441 0.582732i
\(656\) 0 0
\(657\) −10610.2 8135.38i −0.630053 0.483092i
\(658\) 0 0
\(659\) 14018.1 + 8093.34i 0.828629 + 0.478409i 0.853383 0.521284i \(-0.174547\pi\)
−0.0247539 + 0.999694i \(0.507880\pi\)
\(660\) 0 0
\(661\) 24821.5 14330.7i 1.46058 0.843269i 0.461546 0.887116i \(-0.347295\pi\)
0.999038 + 0.0438473i \(0.0139615\pi\)
\(662\) 0 0
\(663\) −857.297 572.601i −0.0502182 0.0335415i
\(664\) 0 0
\(665\) 6545.78 + 8838.93i 0.381706 + 0.515427i
\(666\) 0 0
\(667\) 5807.45 0.337130
\(668\) 0 0
\(669\) 1757.53 26889.8i 0.101569 1.55399i
\(670\) 0 0
\(671\) 0.873449 + 1.51286i 5.02521e−5 + 8.70391e-5i
\(672\) 0 0
\(673\) 3670.88 6358.15i 0.210255 0.364173i −0.741539 0.670910i \(-0.765904\pi\)
0.951794 + 0.306737i \(0.0992371\pi\)
\(674\) 0 0
\(675\) 1915.38 2186.48i 0.109219 0.124678i
\(676\) 0 0
\(677\) 16478.4 28541.4i 0.935474 1.62029i 0.161687 0.986842i \(-0.448307\pi\)
0.773787 0.633446i \(-0.218360\pi\)
\(678\) 0 0
\(679\) −10738.4 + 24722.8i −0.606924 + 1.39731i
\(680\) 0 0
\(681\) 5159.48 + 337.226i 0.290326 + 0.0189758i
\(682\) 0 0
\(683\) 22213.0i 1.24444i −0.782841 0.622222i \(-0.786230\pi\)
0.782841 0.622222i \(-0.213770\pi\)
\(684\) 0 0
\(685\) 32346.4i 1.80422i
\(686\) 0 0
\(687\) 508.242 + 339.463i 0.0282251 + 0.0188520i
\(688\) 0 0
\(689\) 1988.59 + 3444.34i 0.109955 + 0.190448i
\(690\) 0 0
\(691\) 22072.1 + 12743.3i 1.21514 + 0.701563i 0.963875 0.266355i \(-0.0858194\pi\)
0.251267 + 0.967918i \(0.419153\pi\)
\(692\) 0 0
\(693\) −0.620364 1.11718i −3.40053e−5 6.12383e-5i
\(694\) 0 0
\(695\) 9593.47 + 5538.79i 0.523599 + 0.302300i
\(696\) 0 0
\(697\) 318.391 + 551.470i 0.0173026 + 0.0299690i
\(698\) 0 0
\(699\) −6915.61 + 3411.97i −0.374209 + 0.184625i
\(700\) 0 0
\(701\) 22316.4i 1.20239i −0.799101 0.601197i \(-0.794691\pi\)
0.799101 0.601197i \(-0.205309\pi\)
\(702\) 0 0
\(703\) 5016.19i 0.269117i
\(704\) 0 0
\(705\) 12836.2 + 26017.3i 0.685729 + 1.38988i
\(706\) 0 0
\(707\) 6713.29 + 2915.92i 0.357114 + 0.155113i
\(708\) 0 0
\(709\) −620.251 + 1074.31i −0.0328547 + 0.0569061i −0.881985 0.471277i \(-0.843793\pi\)
0.849130 + 0.528183i \(0.177127\pi\)
\(710\) 0 0
\(711\) −6719.22 16204.9i −0.354417 0.854756i
\(712\) 0 0
\(713\) −842.748 + 1459.68i −0.0442653 + 0.0766698i
\(714\) 0 0
\(715\) 0.108080 + 0.187201i 5.65311e−6 + 9.79147e-6i
\(716\) 0 0
\(717\) 11935.1 + 7971.63i 0.621652 + 0.415211i
\(718\) 0 0
\(719\) 1718.52 0.0891374 0.0445687 0.999006i \(-0.485809\pi\)
0.0445687 + 0.999006i \(0.485809\pi\)
\(720\) 0 0
\(721\) 6939.62 + 9370.74i 0.358453 + 0.484028i
\(722\) 0 0
\(723\) 1316.03 20135.0i 0.0676952 1.03572i
\(724\) 0 0
\(725\) −2029.35 + 1171.65i −0.103956 + 0.0600191i
\(726\) 0 0
\(727\) 2204.74 + 1272.91i 0.112475 + 0.0649376i 0.555182 0.831729i \(-0.312649\pi\)
−0.442707 + 0.896666i \(0.645982\pi\)
\(728\) 0 0
\(729\) 18193.0 + 7512.46i 0.924298 + 0.381672i
\(730\) 0 0
\(731\) 6433.10 11142.5i 0.325495 0.563774i
\(732\) 0 0
\(733\) 19126.0 11042.4i 0.963761 0.556427i 0.0664323 0.997791i \(-0.478838\pi\)
0.897328 + 0.441363i \(0.145505\pi\)
\(734\) 0 0
\(735\) 7603.13 20126.4i 0.381559 1.01003i
\(736\) 0 0
\(737\) 1.37884i 6.89149e-5i
\(738\) 0 0
\(739\) −5401.50 −0.268873 −0.134437 0.990922i \(-0.542922\pi\)
−0.134437 + 0.990922i \(0.542922\pi\)
\(740\) 0 0
\(741\) 994.918 1489.59i 0.0493242 0.0738480i
\(742\) 0 0
\(743\) −26971.1 + 15571.7i −1.33173 + 0.768872i −0.985564 0.169303i \(-0.945848\pi\)
−0.346161 + 0.938175i \(0.612515\pi\)
\(744\) 0 0
\(745\) 14334.1 + 8275.82i 0.704915 + 0.406983i
\(746\) 0 0
\(747\) 11698.0 + 28212.3i 0.572967 + 1.38184i
\(748\) 0 0
\(749\) 9487.53 1084.70i 0.462840 0.0529161i
\(750\) 0 0
\(751\) −2644.61 4580.61i −0.128500 0.222568i 0.794596 0.607139i \(-0.207683\pi\)
−0.923096 + 0.384571i \(0.874350\pi\)
\(752\) 0 0
\(753\) −12758.7 25860.2i −0.617468 1.25153i
\(754\) 0 0
\(755\) 3070.62 0.148015
\(756\) 0 0
\(757\) −20794.2 −0.998384 −0.499192 0.866491i \(-0.666370\pi\)
−0.499192 + 0.866491i \(0.666370\pi\)
\(758\) 0 0
\(759\) 0.301681 + 0.611468i 1.44273e−5 + 2.92423e-5i
\(760\) 0 0
\(761\) −9126.94 15808.3i −0.434759 0.753024i 0.562517 0.826785i \(-0.309833\pi\)
−0.997276 + 0.0737616i \(0.976500\pi\)
\(762\) 0 0
\(763\) 1670.36 + 14610.1i 0.0792546 + 0.693213i
\(764\) 0 0
\(765\) 9149.92 + 1201.21i 0.432439 + 0.0567712i
\(766\) 0 0
\(767\) −1767.78 1020.63i −0.0832213 0.0480478i
\(768\) 0 0
\(769\) −28098.3 + 16222.6i −1.31762 + 0.760730i −0.983346 0.181745i \(-0.941826\pi\)
−0.334278 + 0.942475i \(0.608492\pi\)
\(770\) 0 0
\(771\) −4672.99 + 6996.39i −0.218280 + 0.326808i
\(772\) 0 0
\(773\) 20600.1 0.958519 0.479259 0.877673i \(-0.340905\pi\)
0.479259 + 0.877673i \(0.340905\pi\)
\(774\) 0 0
\(775\) 680.094i 0.0315222i
\(776\) 0 0
\(777\) 8249.33 5312.81i 0.380879 0.245297i
\(778\) 0 0
\(779\) −958.201 + 553.217i −0.0440707 + 0.0254442i
\(780\) 0 0
\(781\) 0.392929 0.680572i 1.80027e−5 3.11816e-5i
\(782\) 0 0
\(783\) −11935.5 10455.6i −0.544749 0.477205i
\(784\) 0 0
\(785\) 6177.51 + 3566.59i 0.280872 + 0.162162i
\(786\) 0 0
\(787\) 12253.1 7074.35i 0.554990 0.320424i −0.196142 0.980575i \(-0.562841\pi\)
0.751132 + 0.660152i \(0.229508\pi\)
\(788\) 0 0
\(789\) 232.766 3561.28i 0.0105028 0.160691i
\(790\) 0 0
\(791\) 22683.8 + 30630.5i 1.01965 + 1.37686i
\(792\) 0 0
\(793\) 4790.02 0.214500
\(794\) 0 0
\(795\) −29605.4 19773.9i −1.32075 0.882147i
\(796\) 0 0
\(797\) 9980.73 + 17287.1i 0.443583 + 0.768308i 0.997952 0.0639626i \(-0.0203739\pi\)
−0.554369 + 0.832271i \(0.687041\pi\)
\(798\) 0 0
\(799\) −6547.96 + 11341.4i −0.289925 + 0.502165i
\(800\) 0 0
\(801\) −2139.30 + 2790.09i −0.0943675 + 0.123075i
\(802\) 0 0
\(803\) −0.632729 + 1.09592i −2.78064e−5 + 4.81621e-5i
\(804\) 0 0
\(805\) −4573.43 + 10529.4i −0.200239 + 0.461007i
\(806\) 0 0
\(807\) −4149.84 8411.18i −0.181018 0.366899i
\(808\) 0 0
\(809\) 13027.7i 0.566169i 0.959095 + 0.283085i \(0.0913578\pi\)
−0.959095 + 0.283085i \(0.908642\pi\)
\(810\) 0 0
\(811\) 2467.58i 0.106841i −0.998572 0.0534207i \(-0.982988\pi\)
0.998572 0.0534207i \(-0.0170124\pi\)
\(812\) 0 0
\(813\) 31343.2 15463.8i 1.35209 0.667086i
\(814\) 0 0
\(815\) −13261.5 22969.5i −0.569974 0.987224i
\(816\) 0 0
\(817\) 19360.5 + 11177.8i 0.829054 + 0.478654i
\(818\) 0 0
\(819\) −3503.44 58.5127i −0.149475 0.00249646i
\(820\) 0 0
\(821\) 17302.8 + 9989.80i 0.735534 + 0.424661i 0.820443 0.571728i \(-0.193727\pi\)
−0.0849092 + 0.996389i \(0.527060\pi\)
\(822\) 0 0
\(823\) −20481.8 35475.5i −0.867497 1.50255i −0.864546 0.502553i \(-0.832394\pi\)
−0.00295098 0.999996i \(-0.500939\pi\)
\(824\) 0 0
\(825\) −0.228782 0.152807i −9.65476e−6 6.44856e-6i
\(826\) 0 0
\(827\) 38066.2i 1.60059i −0.599604 0.800297i \(-0.704675\pi\)
0.599604 0.800297i \(-0.295325\pi\)
\(828\) 0 0
\(829\) 5157.09i 0.216059i 0.994148 + 0.108030i \(0.0344542\pi\)
−0.994148 + 0.108030i \(0.965546\pi\)
\(830\) 0 0
\(831\) −3485.67 227.824i −0.145507 0.00951039i
\(832\) 0 0
\(833\) 9461.19 2192.04i 0.393531 0.0911759i
\(834\) 0 0
\(835\) −2545.90 + 4409.62i −0.105514 + 0.182756i
\(836\) 0 0
\(837\) 4359.99 1482.68i 0.180052 0.0612291i
\(838\) 0 0
\(839\) −12192.2 + 21117.5i −0.501695 + 0.868960i 0.498304 + 0.867003i \(0.333957\pi\)
−0.999998 + 0.00195781i \(0.999377\pi\)
\(840\) 0 0
\(841\) −5798.77 10043.8i −0.237762 0.411816i
\(842\) 0 0
\(843\) 785.439 12017.1i 0.0320901 0.490972i
\(844\) 0 0
\(845\) −25928.2 −1.05557
\(846\) 0 0
\(847\) 19809.7 14670.4i 0.803626 0.595135i
\(848\) 0 0
\(849\) 39989.6 + 26709.7i 1.61654 + 1.07971i
\(850\) 0 0
\(851\) −4534.08 + 2617.75i −0.182640 + 0.105447i
\(852\) 0 0
\(853\) −35213.6 20330.6i −1.41347 0.816068i −0.417757 0.908559i \(-0.637184\pi\)
−0.995714 + 0.0924912i \(0.970517\pi\)
\(854\) 0 0
\(855\) −2087.16 + 15898.3i −0.0834846 + 0.635920i
\(856\) 0 0
\(857\) −5424.95 + 9396.29i −0.216234 + 0.374529i −0.953654 0.300906i \(-0.902711\pi\)
0.737419 + 0.675435i \(0.236044\pi\)
\(858\) 0 0
\(859\) 29442.4 16998.6i 1.16945 0.675184i 0.215902 0.976415i \(-0.430731\pi\)
0.953551 + 0.301230i \(0.0973974\pi\)
\(860\) 0 0
\(861\) 1924.65 + 989.870i 0.0761810 + 0.0391808i
\(862\) 0 0
\(863\) 21617.2i 0.852675i −0.904564 0.426337i \(-0.859804\pi\)
0.904564 0.426337i \(-0.140196\pi\)
\(864\) 0 0
\(865\) 21062.9 0.827930
\(866\) 0 0
\(867\) −9452.08 19158.1i −0.370253 0.750454i
\(868\) 0 0
\(869\) −1.43793 + 0.830190i −5.61317e−5 + 3.24077e-5i
\(870\) 0 0
\(871\) 3274.27 + 1890.40i 0.127376 + 0.0735406i
\(872\) 0 0
\(873\) −36298.9 + 15051.0i −1.40725 + 0.583505i
\(874\) 0 0
\(875\) 2648.18 + 23162.8i 0.102314 + 0.894908i
\(876\) 0 0
\(877\) 7452.42 + 12908.0i 0.286945 + 0.497003i 0.973079 0.230472i \(-0.0740271\pi\)
−0.686134 + 0.727475i \(0.740694\pi\)
\(878\) 0 0
\(879\) −25355.1 + 37961.6i −0.972932 + 1.45667i
\(880\) 0 0
\(881\) 18956.5 0.724928 0.362464 0.931998i \(-0.381936\pi\)
0.362464 + 0.931998i \(0.381936\pi\)
\(882\) 0 0
\(883\) −35931.1 −1.36940 −0.684700 0.728825i \(-0.740067\pi\)
−0.684700 + 0.728825i \(0.740067\pi\)
\(884\) 0 0
\(885\) 18233.4 + 1191.74i 0.692551 + 0.0452654i
\(886\) 0 0
\(887\) −19322.4 33467.3i −0.731434 1.26688i −0.956270 0.292484i \(-0.905518\pi\)
0.224837 0.974396i \(-0.427815\pi\)
\(888\) 0 0
\(889\) −4358.09 38118.7i −0.164416 1.43809i
\(890\) 0 0
\(891\) 0.480855 1.79983i 1.80800e−5 6.76728e-5i
\(892\) 0 0
\(893\) −19706.1 11377.3i −0.738456 0.426348i
\(894\) 0 0
\(895\) −45074.2 + 26023.6i −1.68342 + 0.971924i
\(896\) 0 0
\(897\) 1865.63 + 121.938i 0.0694444 + 0.00453891i
\(898\) 0 0
\(899\) −3712.46 −0.137728
\(900\) 0 0
\(901\) 16070.8i 0.594224i
\(902\) 0 0
\(903\) −2122.99 43677.8i −0.0782378 1.60964i
\(904\) 0 0
\(905\) −13674.0 + 7894.67i −0.502252 + 0.289975i
\(906\) 0 0
\(907\) −286.156 + 495.636i −0.0104759 + 0.0181448i −0.871216 0.490900i \(-0.836668\pi\)
0.860740 + 0.509045i \(0.170001\pi\)
\(908\) 0 0
\(909\) 4086.99 + 9856.69i 0.149127 + 0.359654i
\(910\) 0 0
\(911\) 24634.1 + 14222.5i 0.895901 + 0.517248i 0.875868 0.482551i \(-0.160290\pi\)
0.0200327 + 0.999799i \(0.493623\pi\)
\(912\) 0 0
\(913\) 2.50340 1.44534i 9.07452e−5 5.23918e-5i
\(914\) 0 0
\(915\) −38452.5 + 18971.4i −1.38929 + 0.685437i
\(916\) 0 0
\(917\) 13907.3 10299.2i 0.500829 0.370895i
\(918\) 0 0
\(919\) 20019.4 0.718585 0.359292 0.933225i \(-0.383018\pi\)
0.359292 + 0.933225i \(0.383018\pi\)
\(920\) 0 0
\(921\) −2793.67 + 1378.32i −0.0999506 + 0.0493128i
\(922\) 0 0
\(923\) −1077.42 1866.14i −0.0384221 0.0665490i
\(924\) 0 0
\(925\) 1056.26 1829.49i 0.0375455 0.0650306i
\(926\) 0 0
\(927\) −2212.74 + 16854.9i −0.0783989 + 0.597182i
\(928\) 0 0
\(929\) −15791.3 + 27351.3i −0.557690 + 0.965948i 0.439999 + 0.897999i \(0.354979\pi\)
−0.997689 + 0.0679494i \(0.978354\pi\)
\(930\) 0 0
\(931\) 3808.75 + 16439.2i 0.134078 + 0.578704i
\(932\) 0 0
\(933\) −25941.4 + 38839.4i −0.910272 + 1.36286i
\(934\) 0 0
\(935\) 0.873451i 3.05507e-5i
\(936\) 0 0
\(937\) 39841.1i 1.38906i −0.719463 0.694531i \(-0.755612\pi\)
0.719463 0.694531i \(-0.244388\pi\)
\(938\) 0 0
\(939\) −2779.47 + 42525.4i −0.0965971 + 1.47792i
\(940\) 0 0
\(941\) −5443.60 9428.59i −0.188583 0.326635i 0.756195 0.654346i \(-0.227056\pi\)
−0.944778 + 0.327711i \(0.893723\pi\)
\(942\) 0 0
\(943\) −1000.09 577.405i −0.0345361 0.0199394i
\(944\) 0 0
\(945\) 28356.1 13406.0i 0.976108 0.461480i
\(946\) 0 0
\(947\) 19780.5 + 11420.3i 0.678755 + 0.391879i 0.799386 0.600818i \(-0.205158\pi\)
−0.120631 + 0.992697i \(0.538492\pi\)
\(948\) 0 0
\(949\) 1734.95 + 3005.03i 0.0593456 + 0.102790i
\(950\) 0 0
\(951\) 2306.65 35291.2i 0.0786521 1.20336i
\(952\) 0 0
\(953\) 51243.3i 1.74180i −0.491462 0.870899i \(-0.663537\pi\)
0.491462 0.870899i \(-0.336463\pi\)
\(954\) 0 0
\(955\) 48730.1i 1.65117i
\(956\) 0 0
\(957\) −0.834136 + 1.24887i −2.81753e−5 + 4.21840e-5i
\(958\) 0 0
\(959\) −19770.9 + 45518.3i −0.665731 + 1.53270i
\(960\) 0 0
\(961\) −14356.8 + 24866.6i −0.481916 + 0.834703i
\(962\) 0 0
\(963\) 11047.9 + 8470.92i 0.369691 + 0.283460i
\(964\) 0 0
\(965\) −26673.5 + 46199.8i −0.889793 + 1.54117i
\(966\) 0 0
\(967\) −7489.56 12972.3i −0.249067 0.431397i 0.714200 0.699942i \(-0.246791\pi\)
−0.963267 + 0.268544i \(0.913457\pi\)
\(968\) 0 0
\(969\) −6491.13 + 3202.54i −0.215196 + 0.106172i
\(970\) 0 0
\(971\) −12705.2 −0.419908 −0.209954 0.977711i \(-0.567331\pi\)
−0.209954 + 0.977711i \(0.567331\pi\)
\(972\) 0 0
\(973\) 10114.6 + 13658.0i 0.333258 + 0.450007i
\(974\) 0 0
\(975\) −676.526 + 333.779i −0.0222217 + 0.0109636i
\(976\) 0 0
\(977\) −17602.3 + 10162.7i −0.576405 + 0.332788i −0.759703 0.650270i \(-0.774656\pi\)
0.183298 + 0.983057i \(0.441323\pi\)
\(978\) 0 0
\(979\) 0.288185 + 0.166384i 9.40802e−6 + 5.43172e-6i
\(980\) 0 0
\(981\) −13044.6 + 17012.9i −0.424549 + 0.553700i
\(982\) 0 0
\(983\) −20257.1 + 35086.3i −0.657274 + 1.13843i 0.324044 + 0.946042i \(0.394957\pi\)
−0.981318 + 0.192390i \(0.938376\pi\)
\(984\) 0 0
\(985\) −1585.61 + 915.452i −0.0512911 + 0.0296129i
\(986\) 0 0
\(987\) 2160.90 + 44457.7i 0.0696881 + 1.43374i
\(988\) 0 0
\(989\) 23333.0i 0.750197i
\(990\) 0 0
\(991\) −5083.59 −0.162952 −0.0814761 0.996675i \(-0.525963\pi\)
−0.0814761 + 0.996675i \(0.525963\pi\)
\(992\) 0 0
\(993\) 40188.7 + 2626.75i 1.28434 + 0.0839449i
\(994\) 0 0
\(995\) −19221.5 + 11097.5i −0.612424 + 0.353583i
\(996\) 0 0
\(997\) 15996.3 + 9235.49i 0.508133 + 0.293371i 0.732066 0.681234i \(-0.238556\pi\)
−0.223933 + 0.974605i \(0.571890\pi\)
\(998\) 0 0
\(999\) 14031.4 + 2783.03i 0.444377 + 0.0881394i
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 252.4.x.a.209.16 yes 48
3.2 odd 2 756.4.x.a.629.5 48
7.6 odd 2 inner 252.4.x.a.209.9 yes 48
9.2 odd 6 2268.4.f.a.1133.39 48
9.4 even 3 756.4.x.a.125.20 48
9.5 odd 6 inner 252.4.x.a.41.9 48
9.7 even 3 2268.4.f.a.1133.10 48
21.20 even 2 756.4.x.a.629.20 48
63.13 odd 6 756.4.x.a.125.5 48
63.20 even 6 2268.4.f.a.1133.9 48
63.34 odd 6 2268.4.f.a.1133.40 48
63.41 even 6 inner 252.4.x.a.41.16 yes 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
252.4.x.a.41.9 48 9.5 odd 6 inner
252.4.x.a.41.16 yes 48 63.41 even 6 inner
252.4.x.a.209.9 yes 48 7.6 odd 2 inner
252.4.x.a.209.16 yes 48 1.1 even 1 trivial
756.4.x.a.125.5 48 63.13 odd 6
756.4.x.a.125.20 48 9.4 even 3
756.4.x.a.629.5 48 3.2 odd 2
756.4.x.a.629.20 48 21.20 even 2
2268.4.f.a.1133.9 48 63.20 even 6
2268.4.f.a.1133.10 48 9.7 even 3
2268.4.f.a.1133.39 48 9.2 odd 6
2268.4.f.a.1133.40 48 63.34 odd 6