Properties

Label 2052.3.m.a.881.6
Level $2052$
Weight $3$
Character 2052.881
Analytic conductor $55.913$
Analytic rank $0$
Dimension $80$
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] = [2052,3,Mod(881,2052)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2052, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 5, 2]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2052.881");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2052 = 2^{2} \cdot 3^{3} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2052.m (of order \(6\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(55.9129502467\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(40\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 684)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 881.6
Character \(\chi\) \(=\) 2052.881
Dual form 2052.3.m.a.1493.35

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-6.34957i q^{5} +(-3.73312 - 6.46596i) q^{7} +O(q^{10})\) \(q-6.34957i q^{5} +(-3.73312 - 6.46596i) q^{7} +(4.85076 - 2.80059i) q^{11} +(-2.59582 - 4.49610i) q^{13} +(10.9238 - 6.30688i) q^{17} +(18.2190 + 5.39160i) q^{19} +(22.3510 - 12.9044i) q^{23} -15.3170 q^{25} +9.16161i q^{29} +(23.3648 - 40.4690i) q^{31} +(-41.0560 + 23.7037i) q^{35} +46.3137 q^{37} -26.3532i q^{41} +(27.9247 - 48.3669i) q^{43} +20.4907i q^{47} +(-3.37239 + 5.84114i) q^{49} +(45.9725 + 26.5422i) q^{53} +(-17.7825 - 30.8003i) q^{55} +20.6874i q^{59} -35.4492 q^{61} +(-28.5483 + 16.4823i) q^{65} +(-53.7493 - 93.0966i) q^{67} +(-74.9285 + 43.2600i) q^{71} +(-25.3385 - 43.8876i) q^{73} +(-36.2170 - 20.9099i) q^{77} +(20.8830 - 36.1704i) q^{79} +(-90.2997 + 52.1346i) q^{83} +(-40.0459 - 69.3616i) q^{85} +(-32.7765 - 18.9235i) q^{89} +(-19.3810 + 33.5689i) q^{91} +(34.2343 - 115.683i) q^{95} +(81.1072 - 140.482i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 80 q + q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 80 q + q^{7} - 18 q^{11} - 5 q^{13} + 9 q^{17} + 20 q^{19} - 72 q^{23} - 400 q^{25} - 8 q^{31} + 22 q^{37} - 44 q^{43} - 267 q^{49} + 36 q^{53} - 14 q^{61} + 144 q^{65} - 77 q^{67} + 135 q^{71} + 43 q^{73} - 216 q^{77} - 17 q^{79} + 171 q^{83} - 216 q^{89} + 122 q^{91} + 288 q^{95} - 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2052\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1027\) \(1217\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(e\left(\frac{5}{6}\right)\)

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 0
\(4\) 0 0
\(5\) 6.34957i 1.26991i −0.772548 0.634957i \(-0.781018\pi\)
0.772548 0.634957i \(-0.218982\pi\)
\(6\) 0 0
\(7\) −3.73312 6.46596i −0.533303 0.923708i −0.999243 0.0388918i \(-0.987617\pi\)
0.465940 0.884816i \(-0.345716\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 4.85076 2.80059i 0.440978 0.254599i −0.263034 0.964787i \(-0.584723\pi\)
0.704013 + 0.710187i \(0.251390\pi\)
\(12\) 0 0
\(13\) −2.59582 4.49610i −0.199679 0.345853i 0.748746 0.662857i \(-0.230656\pi\)
−0.948424 + 0.317004i \(0.897323\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 10.9238 6.30688i 0.642578 0.370993i −0.143029 0.989719i \(-0.545684\pi\)
0.785607 + 0.618726i \(0.212351\pi\)
\(18\) 0 0
\(19\) 18.2190 + 5.39160i 0.958893 + 0.283768i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 22.3510 12.9044i 0.971784 0.561060i 0.0720043 0.997404i \(-0.477060\pi\)
0.899780 + 0.436345i \(0.143727\pi\)
\(24\) 0 0
\(25\) −15.3170 −0.612681
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 9.16161i 0.315918i 0.987446 + 0.157959i \(0.0504913\pi\)
−0.987446 + 0.157959i \(0.949509\pi\)
\(30\) 0 0
\(31\) 23.3648 40.4690i 0.753703 1.30545i −0.192313 0.981334i \(-0.561599\pi\)
0.946016 0.324119i \(-0.105068\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −41.0560 + 23.7037i −1.17303 + 0.677249i
\(36\) 0 0
\(37\) 46.3137 1.25172 0.625860 0.779935i \(-0.284748\pi\)
0.625860 + 0.779935i \(0.284748\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 26.3532i 0.642761i −0.946950 0.321380i \(-0.895853\pi\)
0.946950 0.321380i \(-0.104147\pi\)
\(42\) 0 0
\(43\) 27.9247 48.3669i 0.649411 1.12481i −0.333853 0.942625i \(-0.608349\pi\)
0.983264 0.182187i \(-0.0583177\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 20.4907i 0.435972i 0.975952 + 0.217986i \(0.0699487\pi\)
−0.975952 + 0.217986i \(0.930051\pi\)
\(48\) 0 0
\(49\) −3.37239 + 5.84114i −0.0688242 + 0.119207i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 45.9725 + 26.5422i 0.867406 + 0.500797i 0.866485 0.499203i \(-0.166374\pi\)
0.000920543 1.00000i \(0.499707\pi\)
\(54\) 0 0
\(55\) −17.7825 30.8003i −0.323319 0.560005i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 20.6874i 0.350634i 0.984512 + 0.175317i \(0.0560951\pi\)
−0.984512 + 0.175317i \(0.943905\pi\)
\(60\) 0 0
\(61\) −35.4492 −0.581134 −0.290567 0.956855i \(-0.593844\pi\)
−0.290567 + 0.956855i \(0.593844\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −28.5483 + 16.4823i −0.439204 + 0.253575i
\(66\) 0 0
\(67\) −53.7493 93.0966i −0.802229 1.38950i −0.918146 0.396242i \(-0.870314\pi\)
0.115917 0.993259i \(-0.463019\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −74.9285 + 43.2600i −1.05533 + 0.609296i −0.924137 0.382061i \(-0.875214\pi\)
−0.131194 + 0.991357i \(0.541881\pi\)
\(72\) 0 0
\(73\) −25.3385 43.8876i −0.347103 0.601200i 0.638630 0.769514i \(-0.279501\pi\)
−0.985734 + 0.168313i \(0.946168\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −36.2170 20.9099i −0.470350 0.271557i
\(78\) 0 0
\(79\) 20.8830 36.1704i 0.264342 0.457853i −0.703049 0.711141i \(-0.748179\pi\)
0.967391 + 0.253288i \(0.0815120\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −90.2997 + 52.1346i −1.08795 + 0.628127i −0.933029 0.359800i \(-0.882845\pi\)
−0.154919 + 0.987927i \(0.549512\pi\)
\(84\) 0 0
\(85\) −40.0459 69.3616i −0.471129 0.816019i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −32.7765 18.9235i −0.368276 0.212624i 0.304429 0.952535i \(-0.401534\pi\)
−0.672705 + 0.739911i \(0.734868\pi\)
\(90\) 0 0
\(91\) −19.3810 + 33.5689i −0.212978 + 0.368889i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 34.2343 115.683i 0.360361 1.21771i
\(96\) 0 0
\(97\) 81.1072 140.482i 0.836157 1.44827i −0.0569277 0.998378i \(-0.518130\pi\)
0.893085 0.449888i \(-0.148536\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 185.798i 1.83958i 0.392406 + 0.919792i \(0.371643\pi\)
−0.392406 + 0.919792i \(0.628357\pi\)
\(102\) 0 0
\(103\) −89.3215 + 154.709i −0.867199 + 1.50203i −0.00235139 + 0.999997i \(0.500748\pi\)
−0.864847 + 0.502035i \(0.832585\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 143.064i 1.33705i 0.743690 + 0.668524i \(0.233074\pi\)
−0.743690 + 0.668524i \(0.766926\pi\)
\(108\) 0 0
\(109\) −13.3471 23.1179i −0.122451 0.212091i 0.798283 0.602283i \(-0.205742\pi\)
−0.920734 + 0.390192i \(0.872409\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −113.859 65.7363i −1.00760 0.581737i −0.0971109 0.995274i \(-0.530960\pi\)
−0.910488 + 0.413536i \(0.864293\pi\)
\(114\) 0 0
\(115\) −81.9372 141.919i −0.712497 1.23408i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −81.5599 47.0887i −0.685378 0.395703i
\(120\) 0 0
\(121\) −44.8134 + 77.6191i −0.370359 + 0.641480i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 61.4828i 0.491862i
\(126\) 0 0
\(127\) −45.9185 + 79.5332i −0.361563 + 0.626245i −0.988218 0.153051i \(-0.951090\pi\)
0.626655 + 0.779297i \(0.284423\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 6.77581i 0.0517237i 0.999666 + 0.0258619i \(0.00823300\pi\)
−0.999666 + 0.0258619i \(0.991767\pi\)
\(132\) 0 0
\(133\) −33.1518 137.931i −0.249261 1.03707i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 18.8796i 0.137807i −0.997623 0.0689037i \(-0.978050\pi\)
0.997623 0.0689037i \(-0.0219501\pi\)
\(138\) 0 0
\(139\) 111.807 + 193.656i 0.804369 + 1.39321i 0.916716 + 0.399539i \(0.130830\pi\)
−0.112347 + 0.993669i \(0.535837\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −25.1834 14.5397i −0.176108 0.101676i
\(144\) 0 0
\(145\) 58.1723 0.401188
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 32.8124i 0.220218i −0.993920 0.110109i \(-0.964880\pi\)
0.993920 0.110109i \(-0.0351200\pi\)
\(150\) 0 0
\(151\) 57.6278 + 99.8143i 0.381641 + 0.661022i 0.991297 0.131644i \(-0.0420256\pi\)
−0.609656 + 0.792666i \(0.708692\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −256.961 148.356i −1.65781 0.957138i
\(156\) 0 0
\(157\) −17.2911 −0.110134 −0.0550672 0.998483i \(-0.517537\pi\)
−0.0550672 + 0.998483i \(0.517537\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −166.878 96.3472i −1.03651 0.598430i
\(162\) 0 0
\(163\) −149.764 −0.918798 −0.459399 0.888230i \(-0.651935\pi\)
−0.459399 + 0.888230i \(0.651935\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −34.6487 + 20.0044i −0.207477 + 0.119787i −0.600138 0.799896i \(-0.704888\pi\)
0.392661 + 0.919683i \(0.371554\pi\)
\(168\) 0 0
\(169\) 71.0234 123.016i 0.420257 0.727906i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 44.8672 + 25.9041i 0.259348 + 0.149735i 0.624037 0.781395i \(-0.285491\pi\)
−0.364689 + 0.931129i \(0.618825\pi\)
\(174\) 0 0
\(175\) 57.1803 + 99.0392i 0.326744 + 0.565938i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 246.793i 1.37873i −0.724413 0.689366i \(-0.757889\pi\)
0.724413 0.689366i \(-0.242111\pi\)
\(180\) 0 0
\(181\) −7.45828 + 12.9181i −0.0412060 + 0.0713709i −0.885893 0.463890i \(-0.846453\pi\)
0.844687 + 0.535261i \(0.179787\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 294.072i 1.58958i
\(186\) 0 0
\(187\) 35.3259 61.1863i 0.188909 0.327200i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −221.125 + 127.667i −1.15772 + 0.668412i −0.950758 0.309936i \(-0.899693\pi\)
−0.206967 + 0.978348i \(0.566359\pi\)
\(192\) 0 0
\(193\) 110.136 0.570653 0.285326 0.958430i \(-0.407898\pi\)
0.285326 + 0.958430i \(0.407898\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 237.875i 1.20749i 0.797178 + 0.603744i \(0.206325\pi\)
−0.797178 + 0.603744i \(0.793675\pi\)
\(198\) 0 0
\(199\) 41.7395 72.2950i 0.209746 0.363291i −0.741888 0.670524i \(-0.766070\pi\)
0.951635 + 0.307232i \(0.0994029\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 59.2386 34.2014i 0.291816 0.168480i
\(204\) 0 0
\(205\) −167.331 −0.816251
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 103.476 24.8705i 0.495098 0.118997i
\(210\) 0 0
\(211\) 96.3856 0.456804 0.228402 0.973567i \(-0.426650\pi\)
0.228402 + 0.973567i \(0.426650\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −307.109 177.310i −1.42841 0.824696i
\(216\) 0 0
\(217\) −348.895 −1.60781
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −56.7126 32.7430i −0.256618 0.148159i
\(222\) 0 0
\(223\) −20.9762 + 36.3319i −0.0940637 + 0.162923i −0.909217 0.416322i \(-0.863319\pi\)
0.815154 + 0.579245i \(0.196652\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 249.167 143.857i 1.09765 0.633731i 0.162050 0.986783i \(-0.448189\pi\)
0.935604 + 0.353052i \(0.114856\pi\)
\(228\) 0 0
\(229\) 46.6191 80.7466i 0.203577 0.352605i −0.746102 0.665832i \(-0.768077\pi\)
0.949678 + 0.313227i \(0.101410\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 55.4506 32.0144i 0.237986 0.137401i −0.376265 0.926512i \(-0.622792\pi\)
0.614251 + 0.789111i \(0.289458\pi\)
\(234\) 0 0
\(235\) 130.107 0.553647
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 28.1997 + 16.2811i 0.117990 + 0.0681217i 0.557834 0.829953i \(-0.311633\pi\)
−0.439843 + 0.898075i \(0.644966\pi\)
\(240\) 0 0
\(241\) −27.7563 −0.115171 −0.0575857 0.998341i \(-0.518340\pi\)
−0.0575857 + 0.998341i \(0.518340\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 37.0887 + 21.4132i 0.151383 + 0.0874008i
\(246\) 0 0
\(247\) −23.0520 95.9098i −0.0933281 0.388299i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 234.569 + 135.428i 0.934537 + 0.539555i 0.888244 0.459373i \(-0.151926\pi\)
0.0462932 + 0.998928i \(0.485259\pi\)
\(252\) 0 0
\(253\) 72.2797 125.192i 0.285691 0.494831i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 402.613 232.449i 1.56659 0.904469i 0.570024 0.821628i \(-0.306934\pi\)
0.996563 0.0828408i \(-0.0263993\pi\)
\(258\) 0 0
\(259\) −172.894 299.462i −0.667546 1.15622i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 318.571 + 183.927i 1.21130 + 0.699342i 0.963042 0.269353i \(-0.0868098\pi\)
0.248254 + 0.968695i \(0.420143\pi\)
\(264\) 0 0
\(265\) 168.532 291.906i 0.635969 1.10153i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −154.326 + 89.1001i −0.573702 + 0.331227i −0.758627 0.651526i \(-0.774129\pi\)
0.184925 + 0.982753i \(0.440796\pi\)
\(270\) 0 0
\(271\) −127.792 221.342i −0.471558 0.816762i 0.527913 0.849299i \(-0.322975\pi\)
−0.999471 + 0.0325367i \(0.989641\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −74.2992 + 42.8967i −0.270179 + 0.155988i
\(276\) 0 0
\(277\) 148.439 + 257.103i 0.535880 + 0.928171i 0.999120 + 0.0419380i \(0.0133532\pi\)
−0.463241 + 0.886233i \(0.653313\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 95.4738i 0.339764i −0.985464 0.169882i \(-0.945661\pi\)
0.985464 0.169882i \(-0.0543387\pi\)
\(282\) 0 0
\(283\) −169.157 −0.597727 −0.298864 0.954296i \(-0.596608\pi\)
−0.298864 + 0.954296i \(0.596608\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −170.399 + 98.3797i −0.593723 + 0.342786i
\(288\) 0 0
\(289\) −64.9467 + 112.491i −0.224729 + 0.389242i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −25.7450 14.8639i −0.0878669 0.0507300i 0.455423 0.890275i \(-0.349488\pi\)
−0.543290 + 0.839545i \(0.682821\pi\)
\(294\) 0 0
\(295\) 131.356 0.445275
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −116.039 66.9949i −0.388089 0.224063i
\(300\) 0 0
\(301\) −416.985 −1.38533
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 225.087i 0.737990i
\(306\) 0 0
\(307\) −175.632 304.203i −0.572090 0.990889i −0.996351 0.0853488i \(-0.972800\pi\)
0.424261 0.905540i \(-0.360534\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −62.2618 35.9469i −0.200199 0.115585i 0.396549 0.918013i \(-0.370208\pi\)
−0.596748 + 0.802429i \(0.703541\pi\)
\(312\) 0 0
\(313\) 394.525 1.26046 0.630231 0.776408i \(-0.282960\pi\)
0.630231 + 0.776408i \(0.282960\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 400.928i 1.26476i −0.774660 0.632378i \(-0.782079\pi\)
0.774660 0.632378i \(-0.217921\pi\)
\(318\) 0 0
\(319\) 25.6579 + 44.4408i 0.0804323 + 0.139313i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 233.025 56.0078i 0.721440 0.173399i
\(324\) 0 0
\(325\) 39.7603 + 68.8668i 0.122339 + 0.211898i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 132.492 76.4942i 0.402711 0.232505i
\(330\) 0 0
\(331\) −67.5617 117.020i −0.204114 0.353536i 0.745736 0.666241i \(-0.232098\pi\)
−0.949850 + 0.312706i \(0.898765\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −591.123 + 341.285i −1.76455 + 1.01876i
\(336\) 0 0
\(337\) 128.296 0.380701 0.190350 0.981716i \(-0.439038\pi\)
0.190350 + 0.981716i \(0.439038\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 261.741i 0.767569i
\(342\) 0 0
\(343\) −315.488 −0.919789
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 251.966i 0.726127i −0.931764 0.363063i \(-0.881731\pi\)
0.931764 0.363063i \(-0.118269\pi\)
\(348\) 0 0
\(349\) −69.2104 119.876i −0.198310 0.343484i 0.749670 0.661812i \(-0.230212\pi\)
−0.947981 + 0.318328i \(0.896879\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −105.800 + 61.0837i −0.299717 + 0.173042i −0.642316 0.766440i \(-0.722026\pi\)
0.342599 + 0.939482i \(0.388693\pi\)
\(354\) 0 0
\(355\) 274.682 + 475.764i 0.773753 + 1.34018i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −469.734 + 271.201i −1.30845 + 0.755434i −0.981838 0.189724i \(-0.939241\pi\)
−0.326613 + 0.945158i \(0.605907\pi\)
\(360\) 0 0
\(361\) 302.861 + 196.459i 0.838951 + 0.544207i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −278.667 + 160.889i −0.763472 + 0.440791i
\(366\) 0 0
\(367\) 24.7886 0.0675440 0.0337720 0.999430i \(-0.489248\pi\)
0.0337720 + 0.999430i \(0.489248\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 396.342i 1.06831i
\(372\) 0 0
\(373\) 328.888 569.650i 0.881736 1.52721i 0.0323264 0.999477i \(-0.489708\pi\)
0.849410 0.527734i \(-0.176958\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 41.1915 23.7819i 0.109261 0.0630820i
\(378\) 0 0
\(379\) −352.306 −0.929567 −0.464784 0.885424i \(-0.653868\pi\)
−0.464784 + 0.885424i \(0.653868\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 665.936i 1.73874i 0.494166 + 0.869368i \(0.335474\pi\)
−0.494166 + 0.869368i \(0.664526\pi\)
\(384\) 0 0
\(385\) −132.769 + 229.962i −0.344854 + 0.597304i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 348.817i 0.896703i 0.893857 + 0.448352i \(0.147989\pi\)
−0.893857 + 0.448352i \(0.852011\pi\)
\(390\) 0 0
\(391\) 162.773 281.930i 0.416298 0.721049i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −229.667 132.598i −0.581434 0.335691i
\(396\) 0 0
\(397\) −318.706 552.015i −0.802787 1.39047i −0.917775 0.397100i \(-0.870016\pi\)
0.114989 0.993367i \(-0.463317\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 256.162i 0.638807i −0.947619 0.319404i \(-0.896517\pi\)
0.947619 0.319404i \(-0.103483\pi\)
\(402\) 0 0
\(403\) −242.603 −0.601994
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 224.657 129.706i 0.551982 0.318687i
\(408\) 0 0
\(409\) −261.426 452.803i −0.639183 1.10710i −0.985612 0.169022i \(-0.945939\pi\)
0.346429 0.938076i \(-0.387394\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 133.764 77.2287i 0.323884 0.186994i
\(414\) 0 0
\(415\) 331.032 + 573.364i 0.797667 + 1.38160i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −69.0044 39.8397i −0.164688 0.0950828i 0.415391 0.909643i \(-0.363645\pi\)
−0.580079 + 0.814560i \(0.696978\pi\)
\(420\) 0 0
\(421\) −124.880 + 216.298i −0.296627 + 0.513773i −0.975362 0.220610i \(-0.929195\pi\)
0.678735 + 0.734383i \(0.262528\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −167.320 + 96.6025i −0.393695 + 0.227300i
\(426\) 0 0
\(427\) 132.336 + 229.213i 0.309921 + 0.536798i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 362.196 + 209.114i 0.840362 + 0.485183i 0.857387 0.514672i \(-0.172086\pi\)
−0.0170255 + 0.999855i \(0.505420\pi\)
\(432\) 0 0
\(433\) 43.8115 75.8837i 0.101181 0.175251i −0.810990 0.585060i \(-0.801071\pi\)
0.912172 + 0.409808i \(0.134404\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 476.788 114.597i 1.09105 0.262235i
\(438\) 0 0
\(439\) −82.6557 + 143.164i −0.188282 + 0.326114i −0.944677 0.328001i \(-0.893625\pi\)
0.756396 + 0.654114i \(0.226958\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 772.675i 1.74419i 0.489339 + 0.872094i \(0.337238\pi\)
−0.489339 + 0.872094i \(0.662762\pi\)
\(444\) 0 0
\(445\) −120.156 + 208.117i −0.270014 + 0.467678i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 14.1079i 0.0314208i −0.999877 0.0157104i \(-0.994999\pi\)
0.999877 0.0157104i \(-0.00500097\pi\)
\(450\) 0 0
\(451\) −73.8045 127.833i −0.163646 0.283444i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 213.148 + 123.061i 0.468458 + 0.270464i
\(456\) 0 0
\(457\) 337.400 + 584.395i 0.738294 + 1.27876i 0.953263 + 0.302142i \(0.0977016\pi\)
−0.214969 + 0.976621i \(0.568965\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 235.868 + 136.178i 0.511644 + 0.295398i 0.733509 0.679680i \(-0.237881\pi\)
−0.221865 + 0.975077i \(0.571215\pi\)
\(462\) 0 0
\(463\) −205.309 + 355.606i −0.443432 + 0.768047i −0.997942 0.0641303i \(-0.979573\pi\)
0.554509 + 0.832178i \(0.312906\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 577.112i 1.23579i 0.786262 + 0.617893i \(0.212014\pi\)
−0.786262 + 0.617893i \(0.787986\pi\)
\(468\) 0 0
\(469\) −401.306 + 695.082i −0.855662 + 1.48205i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 312.822i 0.661357i
\(474\) 0 0
\(475\) −279.060 82.5832i −0.587495 0.173859i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 874.756i 1.82621i 0.407722 + 0.913106i \(0.366323\pi\)
−0.407722 + 0.913106i \(0.633677\pi\)
\(480\) 0 0
\(481\) −120.222 208.231i −0.249942 0.432912i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −891.999 514.996i −1.83917 1.06185i
\(486\) 0 0
\(487\) 779.378 1.60037 0.800183 0.599756i \(-0.204736\pi\)
0.800183 + 0.599756i \(0.204736\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 886.844i 1.80620i −0.429430 0.903100i \(-0.641286\pi\)
0.429430 0.903100i \(-0.358714\pi\)
\(492\) 0 0
\(493\) 57.7811 + 100.080i 0.117203 + 0.203002i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 559.434 + 322.990i 1.12562 + 0.649878i
\(498\) 0 0
\(499\) −85.7107 −0.171765 −0.0858824 0.996305i \(-0.527371\pi\)
−0.0858824 + 0.996305i \(0.527371\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 84.7516 + 48.9314i 0.168492 + 0.0972791i 0.581875 0.813278i \(-0.302319\pi\)
−0.413382 + 0.910558i \(0.635653\pi\)
\(504\) 0 0
\(505\) 1179.74 2.33611
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −319.490 + 184.458i −0.627682 + 0.362392i −0.779854 0.625962i \(-0.784707\pi\)
0.152172 + 0.988354i \(0.451373\pi\)
\(510\) 0 0
\(511\) −189.184 + 327.676i −0.370222 + 0.641244i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 982.337 + 567.153i 1.90745 + 1.10127i
\(516\) 0 0
\(517\) 57.3860 + 99.3954i 0.110998 + 0.192254i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 340.779i 0.654087i −0.945009 0.327044i \(-0.893948\pi\)
0.945009 0.327044i \(-0.106052\pi\)
\(522\) 0 0
\(523\) 363.130 628.959i 0.694321 1.20260i −0.276088 0.961132i \(-0.589038\pi\)
0.970409 0.241467i \(-0.0776284\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 589.436i 1.11847i
\(528\) 0 0
\(529\) 68.5457 118.725i 0.129576 0.224432i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −118.486 + 68.4082i −0.222301 + 0.128346i
\(534\) 0 0
\(535\) 908.396 1.69794
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 37.7787i 0.0700903i
\(540\) 0 0
\(541\) −6.83691 + 11.8419i −0.0126375 + 0.0218889i −0.872275 0.489016i \(-0.837356\pi\)
0.859637 + 0.510905i \(0.170689\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −146.789 + 84.7486i −0.269337 + 0.155502i
\(546\) 0 0
\(547\) 566.818 1.03623 0.518115 0.855311i \(-0.326634\pi\)
0.518115 + 0.855311i \(0.326634\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −49.3957 + 166.915i −0.0896474 + 0.302931i
\(552\) 0 0
\(553\) −311.835 −0.563897
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −550.401 317.774i −0.988153 0.570511i −0.0834316 0.996514i \(-0.526588\pi\)
−0.904722 + 0.426003i \(0.859921\pi\)
\(558\) 0 0
\(559\) −289.950 −0.518694
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −230.616 133.146i −0.409619 0.236494i 0.281007 0.959706i \(-0.409332\pi\)
−0.690626 + 0.723212i \(0.742665\pi\)
\(564\) 0 0
\(565\) −417.397 + 722.953i −0.738756 + 1.27956i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 119.581 69.0401i 0.210160 0.121336i −0.391226 0.920295i \(-0.627949\pi\)
0.601386 + 0.798959i \(0.294616\pi\)
\(570\) 0 0
\(571\) −316.361 + 547.954i −0.554048 + 0.959639i 0.443929 + 0.896062i \(0.353584\pi\)
−0.997977 + 0.0635774i \(0.979749\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −342.351 + 197.657i −0.595393 + 0.343750i
\(576\) 0 0
\(577\) −99.7397 −0.172859 −0.0864296 0.996258i \(-0.527546\pi\)
−0.0864296 + 0.996258i \(0.527546\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 674.199 + 389.249i 1.16041 + 0.669964i
\(582\) 0 0
\(583\) 297.336 0.510010
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 794.396 + 458.645i 1.35332 + 0.781337i 0.988712 0.149826i \(-0.0478714\pi\)
0.364603 + 0.931163i \(0.381205\pi\)
\(588\) 0 0
\(589\) 643.875 611.330i 1.09317 1.03791i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −675.475 389.986i −1.13908 0.657649i −0.192878 0.981223i \(-0.561782\pi\)
−0.946203 + 0.323574i \(0.895116\pi\)
\(594\) 0 0
\(595\) −298.993 + 517.870i −0.502509 + 0.870371i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 719.650 415.490i 1.20142 0.693640i 0.240548 0.970637i \(-0.422673\pi\)
0.960871 + 0.276997i \(0.0893394\pi\)
\(600\) 0 0
\(601\) 160.866 + 278.628i 0.267664 + 0.463607i 0.968258 0.249953i \(-0.0804151\pi\)
−0.700594 + 0.713560i \(0.747082\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 492.848 + 284.546i 0.814624 + 0.470324i
\(606\) 0 0
\(607\) 353.985 613.121i 0.583172 1.01008i −0.411929 0.911216i \(-0.635145\pi\)
0.995101 0.0988672i \(-0.0315219\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 92.1281 53.1902i 0.150782 0.0870543i
\(612\) 0 0
\(613\) −239.920 415.553i −0.391386 0.677901i 0.601246 0.799064i \(-0.294671\pi\)
−0.992633 + 0.121163i \(0.961338\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −616.930 + 356.185i −0.999887 + 0.577285i −0.908215 0.418504i \(-0.862554\pi\)
−0.0916720 + 0.995789i \(0.529221\pi\)
\(618\) 0 0
\(619\) −238.695 413.431i −0.385613 0.667902i 0.606241 0.795281i \(-0.292677\pi\)
−0.991854 + 0.127379i \(0.959343\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 282.575i 0.453572i
\(624\) 0 0
\(625\) −773.314 −1.23730
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 505.922 292.094i 0.804328 0.464379i
\(630\) 0 0
\(631\) −138.225 + 239.412i −0.219057 + 0.379417i −0.954520 0.298147i \(-0.903631\pi\)
0.735463 + 0.677565i \(0.236965\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 505.001 + 291.563i 0.795278 + 0.459154i
\(636\) 0 0
\(637\) 35.0165 0.0549709
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −578.579 334.043i −0.902619 0.521127i −0.0245700 0.999698i \(-0.507822\pi\)
−0.878049 + 0.478571i \(0.841155\pi\)
\(642\) 0 0
\(643\) −155.953 −0.242540 −0.121270 0.992620i \(-0.538697\pi\)
−0.121270 + 0.992620i \(0.538697\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 325.510i 0.503107i 0.967843 + 0.251554i \(0.0809415\pi\)
−0.967843 + 0.251554i \(0.919058\pi\)
\(648\) 0 0
\(649\) 57.9370 + 100.350i 0.0892712 + 0.154622i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 970.469 + 560.300i 1.48617 + 0.858040i 0.999876 0.0157554i \(-0.00501530\pi\)
0.486293 + 0.873796i \(0.338349\pi\)
\(654\) 0 0
\(655\) 43.0235 0.0656847
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 1156.40i 1.75479i 0.479771 + 0.877394i \(0.340720\pi\)
−0.479771 + 0.877394i \(0.659280\pi\)
\(660\) 0 0
\(661\) 409.948 + 710.051i 0.620194 + 1.07421i 0.989449 + 0.144880i \(0.0462795\pi\)
−0.369255 + 0.929328i \(0.620387\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −875.799 + 210.499i −1.31699 + 0.316540i
\(666\) 0 0
\(667\) 118.225 + 204.771i 0.177249 + 0.307004i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −171.956 + 99.2786i −0.256268 + 0.147956i
\(672\) 0 0
\(673\) 307.896 + 533.292i 0.457498 + 0.792410i 0.998828 0.0484006i \(-0.0154124\pi\)
−0.541330 + 0.840810i \(0.682079\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 721.894 416.786i 1.06631 0.615636i 0.139141 0.990273i \(-0.455566\pi\)
0.927172 + 0.374636i \(0.122232\pi\)
\(678\) 0 0
\(679\) −1211.13 −1.78370
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 162.807i 0.238370i 0.992872 + 0.119185i \(0.0380282\pi\)
−0.992872 + 0.119185i \(0.961972\pi\)
\(684\) 0 0
\(685\) −119.877 −0.175003
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 275.596i 0.399994i
\(690\) 0 0
\(691\) −113.291 196.226i −0.163952 0.283974i 0.772330 0.635221i \(-0.219091\pi\)
−0.936283 + 0.351247i \(0.885758\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 1229.63 709.928i 1.76925 1.02148i
\(696\) 0 0
\(697\) −166.206 287.878i −0.238460 0.413024i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 603.483 348.421i 0.860888 0.497034i −0.00342124 0.999994i \(-0.501089\pi\)
0.864310 + 0.502960i \(0.167756\pi\)
\(702\) 0 0
\(703\) 843.787 + 249.705i 1.20027 + 0.355199i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 1201.36 693.606i 1.69924 0.981056i
\(708\) 0 0
\(709\) −476.252 −0.671724 −0.335862 0.941911i \(-0.609028\pi\)
−0.335862 + 0.941911i \(0.609028\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 1206.03i 1.69149i
\(714\) 0 0
\(715\) −92.3206 + 159.904i −0.129120 + 0.223642i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −134.505 + 77.6567i −0.187073 + 0.108007i −0.590611 0.806956i \(-0.701113\pi\)
0.403539 + 0.914963i \(0.367780\pi\)
\(720\) 0 0
\(721\) 1333.79 1.84992
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 140.329i 0.193557i
\(726\) 0 0
\(727\) −583.550 + 1010.74i −0.802683 + 1.39029i 0.115162 + 0.993347i \(0.463261\pi\)
−0.917844 + 0.396940i \(0.870072\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 704.469i 0.963706i
\(732\) 0 0
\(733\) 435.020 753.476i 0.593478 1.02793i −0.400281 0.916392i \(-0.631088\pi\)
0.993760 0.111542i \(-0.0355790\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −521.451 301.060i −0.707531 0.408493i
\(738\) 0 0
\(739\) −264.833 458.704i −0.358367 0.620709i 0.629322 0.777145i \(-0.283333\pi\)
−0.987688 + 0.156436i \(0.950000\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 478.400i 0.643876i 0.946761 + 0.321938i \(0.104334\pi\)
−0.946761 + 0.321938i \(0.895666\pi\)
\(744\) 0 0
\(745\) −208.345 −0.279658
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 925.047 534.076i 1.23504 0.713052i
\(750\) 0 0
\(751\) 380.129 + 658.403i 0.506164 + 0.876701i 0.999975 + 0.00713180i \(0.00227014\pi\)
−0.493811 + 0.869569i \(0.664397\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 633.778 365.912i 0.839441 0.484652i
\(756\) 0 0
\(757\) −189.020 327.392i −0.249696 0.432486i 0.713746 0.700405i \(-0.246997\pi\)
−0.963441 + 0.267919i \(0.913664\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −362.168 209.098i −0.475911 0.274767i 0.242800 0.970076i \(-0.421934\pi\)
−0.718711 + 0.695309i \(0.755268\pi\)
\(762\) 0 0
\(763\) −99.6530 + 172.604i −0.130607 + 0.226218i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 93.0126 53.7009i 0.121268 0.0700142i
\(768\) 0 0
\(769\) −650.948 1127.47i −0.846486 1.46616i −0.884325 0.466873i \(-0.845381\pi\)
0.0378388 0.999284i \(-0.487953\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −0.961798 0.555294i −0.00124424 0.000718362i 0.499378 0.866384i \(-0.333562\pi\)
−0.500622 + 0.865666i \(0.666895\pi\)
\(774\) 0 0
\(775\) −357.879 + 619.865i −0.461779 + 0.799826i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 142.086 480.128i 0.182395 0.616339i
\(780\) 0 0
\(781\) −242.307 + 419.688i −0.310252 + 0.537373i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 109.791i 0.139861i
\(786\) 0 0
\(787\) −726.228 + 1257.86i −0.922780 + 1.59830i −0.127687 + 0.991814i \(0.540755\pi\)
−0.795093 + 0.606488i \(0.792578\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 981.606i 1.24097i
\(792\) 0 0
\(793\) 92.0198 + 159.383i 0.116040 + 0.200987i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 565.944 + 326.748i 0.710093 + 0.409972i 0.811095 0.584914i \(-0.198872\pi\)
−0.101003 + 0.994886i \(0.532205\pi\)
\(798\) 0 0
\(799\) 129.232 + 223.837i 0.161742 + 0.280146i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −245.822 141.926i −0.306130 0.176744i
\(804\) 0 0
\(805\) −611.763 + 1059.60i −0.759954 + 1.31628i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 307.348i 0.379911i 0.981793 + 0.189956i \(0.0608344\pi\)
−0.981793 + 0.189956i \(0.939166\pi\)
\(810\) 0 0
\(811\) −483.368 + 837.219i −0.596015 + 1.03233i 0.397388 + 0.917651i \(0.369917\pi\)
−0.993403 + 0.114678i \(0.963416\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 950.937i 1.16679i
\(816\) 0 0
\(817\) 769.534 730.637i 0.941901 0.894292i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 1061.23i 1.29261i 0.763080 + 0.646304i \(0.223686\pi\)
−0.763080 + 0.646304i \(0.776314\pi\)
\(822\) 0 0
\(823\) 568.833 + 985.248i 0.691170 + 1.19714i 0.971455 + 0.237225i \(0.0762380\pi\)
−0.280284 + 0.959917i \(0.590429\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −1240.79 716.370i −1.50035 0.866227i −1.00000 0.000403585i \(-0.999872\pi\)
−0.500349 0.865824i \(-0.666795\pi\)
\(828\) 0 0
\(829\) 476.786 0.575133 0.287567 0.957761i \(-0.407154\pi\)
0.287567 + 0.957761i \(0.407154\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 85.0769i 0.102133i
\(834\) 0 0
\(835\) 127.019 + 220.004i 0.152119 + 0.263478i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 100.555 + 58.0553i 0.119851 + 0.0691959i 0.558727 0.829352i \(-0.311290\pi\)
−0.438876 + 0.898548i \(0.644623\pi\)
\(840\) 0 0
\(841\) 757.065 0.900196
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −781.100 450.968i −0.924378 0.533690i
\(846\) 0 0
\(847\) 669.175 0.790054
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 1035.16 597.649i 1.21640 0.702290i
\(852\) 0 0
\(853\) 660.180 1143.47i 0.773951 1.34052i −0.161432 0.986884i \(-0.551611\pi\)
0.935382 0.353638i \(-0.115055\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 59.5634 + 34.3889i 0.0695022 + 0.0401271i 0.534348 0.845264i \(-0.320557\pi\)
−0.464846 + 0.885391i \(0.653890\pi\)
\(858\) 0 0
\(859\) −33.4964 58.0175i −0.0389947 0.0675407i 0.845869 0.533390i \(-0.179082\pi\)
−0.884864 + 0.465849i \(0.845749\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 983.878i 1.14007i 0.821621 + 0.570034i \(0.193070\pi\)
−0.821621 + 0.570034i \(0.806930\pi\)
\(864\) 0 0
\(865\) 164.480 284.887i 0.190150 0.329349i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 233.939i 0.269205i
\(870\) 0 0
\(871\) −279.047 + 483.324i −0.320376 + 0.554907i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −397.545 + 229.523i −0.454337 + 0.262312i
\(876\) 0 0
\(877\) 795.482 0.907049 0.453524 0.891244i \(-0.350166\pi\)
0.453524 + 0.891244i \(0.350166\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 1278.78i 1.45151i −0.687955 0.725753i \(-0.741491\pi\)
0.687955 0.725753i \(-0.258509\pi\)
\(882\) 0 0
\(883\) −423.514 + 733.548i −0.479631 + 0.830745i −0.999727 0.0233629i \(-0.992563\pi\)
0.520096 + 0.854108i \(0.325896\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −712.045 + 411.099i −0.802756 + 0.463471i −0.844434 0.535660i \(-0.820063\pi\)
0.0416779 + 0.999131i \(0.486730\pi\)
\(888\) 0 0
\(889\) 685.677 0.771291
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −110.478 + 373.319i −0.123715 + 0.418050i
\(894\) 0 0
\(895\) −1567.03 −1.75087
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 370.761 + 214.059i 0.412415 + 0.238108i
\(900\) 0 0
\(901\) 669.594 0.743168
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 82.0245 + 47.3569i 0.0906348 + 0.0523280i
\(906\) 0 0
\(907\) 375.839 650.973i 0.414376 0.717721i −0.580986 0.813913i \(-0.697333\pi\)
0.995363 + 0.0961924i \(0.0306664\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 1194.64 689.727i 1.31135 0.757110i 0.329033 0.944318i \(-0.393277\pi\)
0.982320 + 0.187208i \(0.0599439\pi\)
\(912\) 0 0
\(913\) −292.015 + 505.785i −0.319841 + 0.553981i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 43.8121 25.2949i 0.0477776 0.0275844i
\(918\) 0 0
\(919\) 149.957 0.163174 0.0815868 0.996666i \(-0.474001\pi\)
0.0815868 + 0.996666i \(0.474001\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 389.002 + 224.590i 0.421454 + 0.243327i
\(924\) 0 0
\(925\) −709.387 −0.766905
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 140.260 + 80.9793i 0.150980 + 0.0871683i 0.573587 0.819145i \(-0.305552\pi\)
−0.422607 + 0.906313i \(0.638885\pi\)
\(930\) 0 0
\(931\) −92.9345 + 88.2370i −0.0998222 + 0.0947766i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −388.507 224.304i −0.415515 0.239898i
\(936\) 0 0
\(937\) −705.449 + 1221.87i −0.752881 + 1.30403i 0.193540 + 0.981092i \(0.438003\pi\)
−0.946421 + 0.322935i \(0.895330\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 296.238 171.033i 0.314812 0.181757i −0.334266 0.942479i \(-0.608488\pi\)
0.649078 + 0.760722i \(0.275155\pi\)
\(942\) 0 0
\(943\) −340.072 589.021i −0.360627 0.624625i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −310.543 179.292i −0.327922 0.189326i 0.326996 0.945026i \(-0.393964\pi\)
−0.654918 + 0.755700i \(0.727297\pi\)
\(948\) 0 0
\(949\) −131.549 + 227.849i −0.138618 + 0.240094i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 228.456 131.899i 0.239723 0.138404i −0.375326 0.926893i \(-0.622469\pi\)
0.615050 + 0.788488i \(0.289136\pi\)
\(954\) 0 0
\(955\) 810.629 + 1404.05i 0.848826 + 1.47021i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −122.075 + 70.4799i −0.127294 + 0.0734931i
\(960\) 0 0
\(961\) −611.328 1058.85i −0.636137 1.10182i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 699.316i 0.724680i
\(966\) 0 0
\(967\) −746.026 −0.771485 −0.385743 0.922606i \(-0.626055\pi\)
−0.385743 + 0.922606i \(0.626055\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 1462.99 844.659i 1.50669 0.869886i 0.506717 0.862112i \(-0.330859\pi\)
0.999970 0.00777358i \(-0.00247443\pi\)
\(972\) 0 0
\(973\) 834.781 1445.88i 0.857945 1.48600i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −1140.33 658.373i −1.16718 0.673872i −0.214166 0.976797i \(-0.568703\pi\)
−0.953014 + 0.302926i \(0.902037\pi\)
\(978\) 0 0
\(979\) −211.988 −0.216536
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 650.198 + 375.392i 0.661442 + 0.381884i 0.792826 0.609448i \(-0.208609\pi\)
−0.131384 + 0.991332i \(0.541942\pi\)
\(984\) 0 0
\(985\) 1510.41 1.53341
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 1441.40i 1.45743i
\(990\) 0 0
\(991\) −140.771 243.823i −0.142050 0.246038i 0.786219 0.617949i \(-0.212036\pi\)
−0.928268 + 0.371911i \(0.878703\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −459.042 265.028i −0.461349 0.266360i
\(996\) 0 0
\(997\) 516.351 0.517905 0.258952 0.965890i \(-0.416623\pi\)
0.258952 + 0.965890i \(0.416623\pi\)
\(998\) 0 0
\(999\) 0 0
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 2052.3.m.a.881.6 80
3.2 odd 2 684.3.m.a.653.7 yes 80
9.2 odd 6 2052.3.be.a.197.6 80
9.7 even 3 684.3.be.a.425.20 yes 80
19.11 even 3 2052.3.be.a.125.6 80
57.11 odd 6 684.3.be.a.581.20 yes 80
171.11 odd 6 inner 2052.3.m.a.1493.35 80
171.106 even 3 684.3.m.a.353.7 80
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
684.3.m.a.353.7 80 171.106 even 3
684.3.m.a.653.7 yes 80 3.2 odd 2
684.3.be.a.425.20 yes 80 9.7 even 3
684.3.be.a.581.20 yes 80 57.11 odd 6
2052.3.m.a.881.6 80 1.1 even 1 trivial
2052.3.m.a.1493.35 80 171.11 odd 6 inner
2052.3.be.a.125.6 80 19.11 even 3
2052.3.be.a.197.6 80 9.2 odd 6