Properties

Label 2736.2.dc.c.1889.2
Level $2736$
Weight $2$
Character 2736.1889
Analytic conductor $21.847$
Analytic rank $0$
Dimension $16$
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] = [2736,2,Mod(449,2736)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2736, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 3, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2736.449");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2736.dc (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: \(21.8470699930\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 16x^{14} + 174x^{12} + 1012x^{10} + 4243x^{8} + 9708x^{6} + 15858x^{4} + 12150x^{2} + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 171)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 1889.2
Root \(-1.13921 + 1.97317i\) of defining polynomial
Character \(\chi\) \(=\) 2736.1889
Dual form 2736.2.dc.c.449.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-2.90154 - 1.67520i) q^{5} -3.54697 q^{7} +O(q^{10})\) \(q+(-2.90154 - 1.67520i) q^{5} -3.54697 q^{7} -0.251548i q^{11} +(-4.29049 + 2.47712i) q^{13} +(3.11939 + 1.80098i) q^{17} +(-4.15959 + 1.30301i) q^{19} +(-3.89208 + 2.24709i) q^{23} +(3.11262 + 5.39122i) q^{25} +(2.27843 + 3.94635i) q^{29} -5.28544i q^{31} +(10.2917 + 5.94190i) q^{35} -6.47474i q^{37} +(-4.96212 + 8.59464i) q^{41} +(3.38610 - 5.86490i) q^{43} +(8.96412 - 5.17544i) q^{47} +5.58098 q^{49} +(-0.217847 - 0.377322i) q^{53} +(-0.421394 + 0.729876i) q^{55} +(-2.06058 + 3.56903i) q^{59} +(-6.46836 - 11.2035i) q^{61} +16.5987 q^{65} +(-0.273084 + 0.157665i) q^{67} +(4.10993 - 7.11860i) q^{71} +(0.356142 - 0.616857i) q^{73} +0.892232i q^{77} +(7.57733 + 4.37477i) q^{79} +1.31440i q^{83} +(-6.03401 - 10.4512i) q^{85} +(-1.20839 - 2.09299i) q^{89} +(15.2182 - 8.78625i) q^{91} +(14.2520 + 3.18743i) q^{95} +(9.36417 + 5.40640i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 8 q^{7} - 24 q^{13} + 12 q^{19} + 20 q^{25} + 4 q^{55} - 44 q^{61} + 24 q^{67} - 20 q^{73} + 48 q^{79} - 56 q^{85} + 24 q^{91} + 48 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times\).

\(n\) \(1009\) \(1217\) \(1711\) \(2053\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(-1\) \(1\) \(1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) −2.90154 1.67520i −1.29761 0.749174i −0.317617 0.948219i \(-0.602883\pi\)
−0.979990 + 0.199045i \(0.936216\pi\)
\(6\) 0 0
\(7\) −3.54697 −1.34063 −0.670314 0.742078i \(-0.733841\pi\)
−0.670314 + 0.742078i \(0.733841\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0.251548i 0.0758445i −0.999281 0.0379223i \(-0.987926\pi\)
0.999281 0.0379223i \(-0.0120739\pi\)
\(12\) 0 0
\(13\) −4.29049 + 2.47712i −1.18997 + 0.687028i −0.958299 0.285766i \(-0.907752\pi\)
−0.231669 + 0.972795i \(0.574419\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.11939 + 1.80098i 0.756562 + 0.436801i 0.828060 0.560639i \(-0.189445\pi\)
−0.0714979 + 0.997441i \(0.522778\pi\)
\(18\) 0 0
\(19\) −4.15959 + 1.30301i −0.954275 + 0.298931i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −3.89208 + 2.24709i −0.811555 + 0.468552i −0.847496 0.530802i \(-0.821891\pi\)
0.0359405 + 0.999354i \(0.488557\pi\)
\(24\) 0 0
\(25\) 3.11262 + 5.39122i 0.622524 + 1.07824i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 2.27843 + 3.94635i 0.423093 + 0.732819i 0.996240 0.0866337i \(-0.0276110\pi\)
−0.573147 + 0.819452i \(0.694278\pi\)
\(30\) 0 0
\(31\) 5.28544i 0.949294i −0.880176 0.474647i \(-0.842576\pi\)
0.880176 0.474647i \(-0.157424\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 10.2917 + 5.94190i 1.73961 + 1.00436i
\(36\) 0 0
\(37\) 6.47474i 1.06444i −0.846606 0.532220i \(-0.821358\pi\)
0.846606 0.532220i \(-0.178642\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −4.96212 + 8.59464i −0.774953 + 1.34226i 0.159868 + 0.987138i \(0.448893\pi\)
−0.934821 + 0.355119i \(0.884440\pi\)
\(42\) 0 0
\(43\) 3.38610 5.86490i 0.516376 0.894389i −0.483443 0.875376i \(-0.660614\pi\)
0.999819 0.0190137i \(-0.00605261\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 8.96412 5.17544i 1.30755 0.754915i 0.325865 0.945416i \(-0.394345\pi\)
0.981687 + 0.190501i \(0.0610113\pi\)
\(48\) 0 0
\(49\) 5.58098 0.797283
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −0.217847 0.377322i −0.0299236 0.0518291i 0.850676 0.525691i \(-0.176193\pi\)
−0.880599 + 0.473862i \(0.842860\pi\)
\(54\) 0 0
\(55\) −0.421394 + 0.729876i −0.0568207 + 0.0984164i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −2.06058 + 3.56903i −0.268265 + 0.464648i −0.968414 0.249349i \(-0.919783\pi\)
0.700149 + 0.713997i \(0.253117\pi\)
\(60\) 0 0
\(61\) −6.46836 11.2035i −0.828189 1.43447i −0.899458 0.437008i \(-0.856038\pi\)
0.0712687 0.997457i \(-0.477295\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 16.5987 2.05882
\(66\) 0 0
\(67\) −0.273084 + 0.157665i −0.0333625 + 0.0192618i −0.516588 0.856234i \(-0.672798\pi\)
0.483226 + 0.875496i \(0.339465\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 4.10993 7.11860i 0.487759 0.844823i −0.512142 0.858901i \(-0.671148\pi\)
0.999901 + 0.0140778i \(0.00448125\pi\)
\(72\) 0 0
\(73\) 0.356142 0.616857i 0.0416833 0.0721976i −0.844431 0.535664i \(-0.820061\pi\)
0.886114 + 0.463467i \(0.153395\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0.892232i 0.101679i
\(78\) 0 0
\(79\) 7.57733 + 4.37477i 0.852516 + 0.492200i 0.861499 0.507759i \(-0.169526\pi\)
−0.00898288 + 0.999960i \(0.502859\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 1.31440i 0.144274i 0.997395 + 0.0721369i \(0.0229819\pi\)
−0.997395 + 0.0721369i \(0.977018\pi\)
\(84\) 0 0
\(85\) −6.03401 10.4512i −0.654481 1.13359i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −1.20839 2.09299i −0.128089 0.221857i 0.794847 0.606810i \(-0.207551\pi\)
−0.922936 + 0.384953i \(0.874218\pi\)
\(90\) 0 0
\(91\) 15.2182 8.78625i 1.59530 0.921049i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 14.2520 + 3.18743i 1.46223 + 0.327023i
\(96\) 0 0
\(97\) 9.36417 + 5.40640i 0.950787 + 0.548937i 0.893325 0.449410i \(-0.148366\pi\)
0.0574618 + 0.998348i \(0.481699\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 5.40904 3.12291i 0.538220 0.310741i −0.206137 0.978523i \(-0.566089\pi\)
0.744357 + 0.667782i \(0.232756\pi\)
\(102\) 0 0
\(103\) 4.38110i 0.431683i −0.976428 0.215841i \(-0.930751\pi\)
0.976428 0.215841i \(-0.0692494\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −1.98108 −0.191519 −0.0957593 0.995405i \(-0.530528\pi\)
−0.0957593 + 0.995405i \(0.530528\pi\)
\(108\) 0 0
\(109\) −0.897781 0.518334i −0.0859918 0.0496474i 0.456387 0.889781i \(-0.349143\pi\)
−0.542379 + 0.840134i \(0.682476\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1.32954 0.125073 0.0625364 0.998043i \(-0.480081\pi\)
0.0625364 + 0.998043i \(0.480081\pi\)
\(114\) 0 0
\(115\) 15.0574 1.40411
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −11.0644 6.38801i −1.01427 0.585588i
\(120\) 0 0
\(121\) 10.9367 0.994248
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 4.10505i 0.367167i
\(126\) 0 0
\(127\) −7.10729 + 4.10340i −0.630670 + 0.364118i −0.781012 0.624517i \(-0.785296\pi\)
0.150341 + 0.988634i \(0.451963\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 4.70643 + 2.71726i 0.411203 + 0.237408i 0.691307 0.722562i \(-0.257035\pi\)
−0.280103 + 0.959970i \(0.590369\pi\)
\(132\) 0 0
\(133\) 14.7539 4.62173i 1.27933 0.400755i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 11.8941 6.86706i 1.01618 0.586692i 0.103185 0.994662i \(-0.467097\pi\)
0.912995 + 0.407970i \(0.133763\pi\)
\(138\) 0 0
\(139\) 1.17422 + 2.03381i 0.0995959 + 0.172505i 0.911517 0.411261i \(-0.134912\pi\)
−0.811922 + 0.583767i \(0.801578\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0.623113 + 1.07926i 0.0521073 + 0.0902525i
\(144\) 0 0
\(145\) 15.2673i 1.26788i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 8.31058 + 4.79812i 0.680829 + 0.393077i 0.800167 0.599777i \(-0.204744\pi\)
−0.119338 + 0.992854i \(0.538077\pi\)
\(150\) 0 0
\(151\) 7.06053i 0.574577i −0.957844 0.287289i \(-0.907246\pi\)
0.957844 0.287289i \(-0.0927539\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −8.85420 + 15.3359i −0.711186 + 1.23181i
\(156\) 0 0
\(157\) −8.17492 + 14.1594i −0.652429 + 1.13004i 0.330102 + 0.943945i \(0.392917\pi\)
−0.982532 + 0.186096i \(0.940417\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 13.8051 7.97037i 1.08799 0.628153i
\(162\) 0 0
\(163\) −12.3119 −0.964340 −0.482170 0.876078i \(-0.660151\pi\)
−0.482170 + 0.876078i \(0.660151\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −0.623113 1.07926i −0.0482179 0.0835159i 0.840909 0.541176i \(-0.182021\pi\)
−0.889127 + 0.457660i \(0.848688\pi\)
\(168\) 0 0
\(169\) 5.77221 9.99776i 0.444016 0.769058i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 1.24623 2.15853i 0.0947488 0.164110i −0.814755 0.579805i \(-0.803129\pi\)
0.909504 + 0.415696i \(0.136462\pi\)
\(174\) 0 0
\(175\) −11.0404 19.1225i −0.834573 1.44552i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 21.0722 1.57501 0.787507 0.616306i \(-0.211372\pi\)
0.787507 + 0.616306i \(0.211372\pi\)
\(180\) 0 0
\(181\) −18.4387 + 10.6456i −1.37054 + 0.791280i −0.990996 0.133894i \(-0.957252\pi\)
−0.379542 + 0.925174i \(0.623918\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −10.8465 + 18.7867i −0.797452 + 1.38123i
\(186\) 0 0
\(187\) 0.453032 0.784675i 0.0331290 0.0573811i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 10.1888i 0.737237i 0.929581 + 0.368618i \(0.120169\pi\)
−0.929581 + 0.368618i \(0.879831\pi\)
\(192\) 0 0
\(193\) −8.93139 5.15654i −0.642896 0.371176i 0.142833 0.989747i \(-0.454379\pi\)
−0.785729 + 0.618571i \(0.787712\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 14.3901i 1.02525i −0.858613 0.512625i \(-0.828673\pi\)
0.858613 0.512625i \(-0.171327\pi\)
\(198\) 0 0
\(199\) 11.3022 + 19.5759i 0.801189 + 1.38770i 0.918834 + 0.394645i \(0.129132\pi\)
−0.117644 + 0.993056i \(0.537534\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −8.08150 13.9976i −0.567210 0.982437i
\(204\) 0 0
\(205\) 28.7956 16.6251i 2.01117 1.16115i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0.327769 + 1.04633i 0.0226723 + 0.0723765i
\(210\) 0 0
\(211\) 8.09839 + 4.67561i 0.557516 + 0.321882i 0.752148 0.658994i \(-0.229018\pi\)
−0.194632 + 0.980876i \(0.562351\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −19.6498 + 11.3448i −1.34011 + 0.773711i
\(216\) 0 0
\(217\) 18.7473i 1.27265i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −17.8449 −1.20038
\(222\) 0 0
\(223\) 2.63200 + 1.51959i 0.176252 + 0.101759i 0.585530 0.810650i \(-0.300886\pi\)
−0.409279 + 0.912409i \(0.634220\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −2.35592 −0.156368 −0.0781841 0.996939i \(-0.524912\pi\)
−0.0781841 + 0.996939i \(0.524912\pi\)
\(228\) 0 0
\(229\) −3.15801 −0.208687 −0.104344 0.994541i \(-0.533274\pi\)
−0.104344 + 0.994541i \(0.533274\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 20.4642 + 11.8150i 1.34065 + 0.774026i 0.986903 0.161314i \(-0.0515731\pi\)
0.353750 + 0.935340i \(0.384906\pi\)
\(234\) 0 0
\(235\) −34.6797 −2.26225
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 20.5878i 1.33171i 0.746080 + 0.665857i \(0.231934\pi\)
−0.746080 + 0.665857i \(0.768066\pi\)
\(240\) 0 0
\(241\) 3.12835 1.80615i 0.201515 0.116345i −0.395847 0.918316i \(-0.629549\pi\)
0.597362 + 0.801972i \(0.296216\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −16.1934 9.34928i −1.03456 0.597304i
\(246\) 0 0
\(247\) 14.6190 15.8943i 0.930183 1.01133i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 6.28043 3.62601i 0.396417 0.228872i −0.288520 0.957474i \(-0.593163\pi\)
0.684937 + 0.728602i \(0.259830\pi\)
\(252\) 0 0
\(253\) 0.565252 + 0.979044i 0.0355371 + 0.0615520i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −9.40905 16.2970i −0.586920 1.01658i −0.994633 0.103466i \(-0.967007\pi\)
0.407713 0.913110i \(-0.366326\pi\)
\(258\) 0 0
\(259\) 22.9657i 1.42702i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 5.21963 + 3.01355i 0.321856 + 0.185824i 0.652220 0.758030i \(-0.273838\pi\)
−0.330364 + 0.943854i \(0.607171\pi\)
\(264\) 0 0
\(265\) 1.45975i 0.0896718i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −3.29557 + 5.70810i −0.200935 + 0.348029i −0.948830 0.315788i \(-0.897731\pi\)
0.747895 + 0.663817i \(0.231065\pi\)
\(270\) 0 0
\(271\) −10.4497 + 18.0994i −0.634773 + 1.09946i 0.351790 + 0.936079i \(0.385573\pi\)
−0.986563 + 0.163380i \(0.947760\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 1.35615 0.782973i 0.0817788 0.0472150i
\(276\) 0 0
\(277\) −25.8223 −1.55151 −0.775755 0.631034i \(-0.782631\pi\)
−0.775755 + 0.631034i \(0.782631\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 9.43948 + 16.3497i 0.563112 + 0.975338i 0.997223 + 0.0744789i \(0.0237293\pi\)
−0.434111 + 0.900860i \(0.642937\pi\)
\(282\) 0 0
\(283\) 8.38245 14.5188i 0.498285 0.863055i −0.501713 0.865034i \(-0.667297\pi\)
0.999998 + 0.00197917i \(0.000629990\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 17.6005 30.4849i 1.03892 1.79947i
\(288\) 0 0
\(289\) −2.01295 3.48654i −0.118409 0.205091i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 28.0683 1.63977 0.819885 0.572529i \(-0.194037\pi\)
0.819885 + 0.572529i \(0.194037\pi\)
\(294\) 0 0
\(295\) 11.9577 6.90378i 0.696204 0.401954i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 11.1326 19.2823i 0.643816 1.11512i
\(300\) 0 0
\(301\) −12.0104 + 20.8026i −0.692268 + 1.19904i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 43.3433i 2.48183i
\(306\) 0 0
\(307\) −19.6261 11.3311i −1.12012 0.646703i −0.178690 0.983905i \(-0.557186\pi\)
−0.941432 + 0.337203i \(0.890519\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 17.8544i 1.01243i 0.862407 + 0.506216i \(0.168956\pi\)
−0.862407 + 0.506216i \(0.831044\pi\)
\(312\) 0 0
\(313\) 7.18630 + 12.4470i 0.406193 + 0.703548i 0.994460 0.105120i \(-0.0335226\pi\)
−0.588266 + 0.808667i \(0.700189\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −8.89586 15.4081i −0.499641 0.865404i 0.500359 0.865818i \(-0.333201\pi\)
−1.00000 0.000414314i \(0.999868\pi\)
\(318\) 0 0
\(319\) 0.992696 0.573133i 0.0555803 0.0320893i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −15.3220 3.42674i −0.852542 0.190669i
\(324\) 0 0
\(325\) −26.7093 15.4206i −1.48157 0.855383i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −31.7955 + 18.3571i −1.75294 + 1.01206i
\(330\) 0 0
\(331\) 27.4202i 1.50715i −0.657361 0.753575i \(-0.728327\pi\)
0.657361 0.753575i \(-0.271673\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 1.05648 0.0577219
\(336\) 0 0
\(337\) −14.9188 8.61340i −0.812681 0.469202i 0.0352050 0.999380i \(-0.488792\pi\)
−0.847886 + 0.530179i \(0.822125\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −1.32954 −0.0719987
\(342\) 0 0
\(343\) 5.03321 0.271768
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −7.28635 4.20677i −0.391152 0.225832i 0.291507 0.956569i \(-0.405843\pi\)
−0.682659 + 0.730737i \(0.739177\pi\)
\(348\) 0 0
\(349\) −11.5389 −0.617661 −0.308831 0.951117i \(-0.599938\pi\)
−0.308831 + 0.951117i \(0.599938\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 12.0544i 0.641592i 0.947148 + 0.320796i \(0.103950\pi\)
−0.947148 + 0.320796i \(0.896050\pi\)
\(354\) 0 0
\(355\) −23.8502 + 13.7699i −1.26584 + 0.730833i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 27.7710 + 16.0336i 1.46570 + 0.846220i 0.999265 0.0383367i \(-0.0122060\pi\)
0.466432 + 0.884557i \(0.345539\pi\)
\(360\) 0 0
\(361\) 15.6043 10.8400i 0.821281 0.570524i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −2.06672 + 1.19322i −0.108177 + 0.0624561i
\(366\) 0 0
\(367\) 1.47004 + 2.54618i 0.0767354 + 0.132910i 0.901840 0.432071i \(-0.142217\pi\)
−0.825104 + 0.564981i \(0.808884\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0.772695 + 1.33835i 0.0401164 + 0.0694836i
\(372\) 0 0
\(373\) 22.4374i 1.16177i 0.813987 + 0.580883i \(0.197293\pi\)
−0.813987 + 0.580883i \(0.802707\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −19.5511 11.2879i −1.00693 0.581354i
\(378\) 0 0
\(379\) 9.54429i 0.490257i −0.969491 0.245129i \(-0.921170\pi\)
0.969491 0.245129i \(-0.0788302\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 7.61920 13.1968i 0.389323 0.674327i −0.603035 0.797714i \(-0.706042\pi\)
0.992359 + 0.123387i \(0.0393756\pi\)
\(384\) 0 0
\(385\) 1.49467 2.58885i 0.0761755 0.131940i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 15.6174 9.01671i 0.791833 0.457165i −0.0487743 0.998810i \(-0.515531\pi\)
0.840608 + 0.541645i \(0.182198\pi\)
\(390\) 0 0
\(391\) −16.1879 −0.818656
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −14.6573 25.3872i −0.737488 1.27737i
\(396\) 0 0
\(397\) 7.09927 12.2963i 0.356302 0.617133i −0.631038 0.775752i \(-0.717371\pi\)
0.987340 + 0.158619i \(0.0507041\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 5.24623 9.08674i 0.261984 0.453770i −0.704785 0.709421i \(-0.748956\pi\)
0.966769 + 0.255651i \(0.0822898\pi\)
\(402\) 0 0
\(403\) 13.0927 + 22.6772i 0.652192 + 1.12963i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −1.62871 −0.0807320
\(408\) 0 0
\(409\) 10.9650 6.33065i 0.542185 0.313031i −0.203779 0.979017i \(-0.565322\pi\)
0.745964 + 0.665986i \(0.231989\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 7.30881 12.6592i 0.359643 0.622920i
\(414\) 0 0
\(415\) 2.20188 3.81378i 0.108086 0.187211i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 27.3392i 1.33561i 0.744338 + 0.667804i \(0.232765\pi\)
−0.744338 + 0.667804i \(0.767235\pi\)
\(420\) 0 0
\(421\) 22.3913 + 12.9276i 1.09129 + 0.630055i 0.933919 0.357485i \(-0.116366\pi\)
0.157368 + 0.987540i \(0.449699\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 22.4230i 1.08768i
\(426\) 0 0
\(427\) 22.9431 + 39.7386i 1.11029 + 1.92308i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 8.09274 + 14.0170i 0.389813 + 0.675176i 0.992424 0.122859i \(-0.0392062\pi\)
−0.602611 + 0.798035i \(0.705873\pi\)
\(432\) 0 0
\(433\) 14.4504 8.34294i 0.694442 0.400936i −0.110832 0.993839i \(-0.535352\pi\)
0.805274 + 0.592903i \(0.202018\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 13.2615 14.4184i 0.634382 0.689726i
\(438\) 0 0
\(439\) 29.2233 + 16.8721i 1.39475 + 0.805261i 0.993837 0.110854i \(-0.0353587\pi\)
0.400916 + 0.916115i \(0.368692\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 16.3770 9.45529i 0.778096 0.449234i −0.0576588 0.998336i \(-0.518364\pi\)
0.835755 + 0.549102i \(0.185030\pi\)
\(444\) 0 0
\(445\) 8.09719i 0.383844i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −32.0530 −1.51267 −0.756336 0.654183i \(-0.773013\pi\)
−0.756336 + 0.654183i \(0.773013\pi\)
\(450\) 0 0
\(451\) 2.16196 + 1.24821i 0.101803 + 0.0587759i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −58.8751 −2.76011
\(456\) 0 0
\(457\) 27.5605 1.28922 0.644612 0.764510i \(-0.277019\pi\)
0.644612 + 0.764510i \(0.277019\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −15.4207 8.90317i −0.718215 0.414662i 0.0958800 0.995393i \(-0.469433\pi\)
−0.814096 + 0.580731i \(0.802767\pi\)
\(462\) 0 0
\(463\) −31.5005 −1.46395 −0.731977 0.681329i \(-0.761402\pi\)
−0.731977 + 0.681329i \(0.761402\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 34.7591i 1.60846i −0.594317 0.804231i \(-0.702578\pi\)
0.594317 0.804231i \(-0.297422\pi\)
\(468\) 0 0
\(469\) 0.968620 0.559233i 0.0447267 0.0258230i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −1.47530 0.851767i −0.0678345 0.0391643i
\(474\) 0 0
\(475\) −19.9720 18.3695i −0.916379 0.842849i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 4.62312 2.66916i 0.211236 0.121957i −0.390650 0.920539i \(-0.627750\pi\)
0.601886 + 0.798582i \(0.294416\pi\)
\(480\) 0 0
\(481\) 16.0387 + 27.7798i 0.731301 + 1.26665i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −18.1137 31.3738i −0.822499 1.42461i
\(486\) 0 0
\(487\) 37.8144i 1.71353i −0.515705 0.856766i \(-0.672470\pi\)
0.515705 0.856766i \(-0.327530\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 8.95830 + 5.17208i 0.404283 + 0.233413i 0.688330 0.725398i \(-0.258344\pi\)
−0.284048 + 0.958810i \(0.591677\pi\)
\(492\) 0 0
\(493\) 16.4136i 0.739231i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −14.5778 + 25.2495i −0.653903 + 1.13259i
\(498\) 0 0
\(499\) 15.6603 27.1244i 0.701051 1.21426i −0.267047 0.963683i \(-0.586048\pi\)
0.968098 0.250572i \(-0.0806187\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −19.1154 + 11.0363i −0.852315 + 0.492084i −0.861431 0.507874i \(-0.830432\pi\)
0.00911608 + 0.999958i \(0.497098\pi\)
\(504\) 0 0
\(505\) −20.9261 −0.931198
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −5.73681 9.93645i −0.254280 0.440425i 0.710420 0.703778i \(-0.248505\pi\)
−0.964700 + 0.263353i \(0.915172\pi\)
\(510\) 0 0
\(511\) −1.26323 + 2.18797i −0.0558818 + 0.0967901i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −7.33924 + 12.7119i −0.323405 + 0.560155i
\(516\) 0 0
\(517\) −1.30187 2.25490i −0.0572562 0.0991706i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 16.0265 0.702133 0.351066 0.936351i \(-0.385819\pi\)
0.351066 + 0.936351i \(0.385819\pi\)
\(522\) 0 0
\(523\) 10.1731 5.87344i 0.444839 0.256828i −0.260809 0.965390i \(-0.583989\pi\)
0.705648 + 0.708563i \(0.250656\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 9.51897 16.4873i 0.414653 0.718200i
\(528\) 0 0
\(529\) −1.40114 + 2.42684i −0.0609189 + 0.105515i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 49.1670i 2.12966i
\(534\) 0 0
\(535\) 5.74819 + 3.31872i 0.248516 + 0.143481i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 1.40388i 0.0604695i
\(540\) 0 0
\(541\) 19.5906 + 33.9319i 0.842265 + 1.45885i 0.887975 + 0.459891i \(0.152111\pi\)
−0.0457105 + 0.998955i \(0.514555\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 1.73663 + 3.00793i 0.0743891 + 0.128846i
\(546\) 0 0
\(547\) −8.36275 + 4.82824i −0.357565 + 0.206440i −0.668012 0.744150i \(-0.732855\pi\)
0.310447 + 0.950591i \(0.399521\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −14.6194 13.4464i −0.622809 0.572835i
\(552\) 0 0
\(553\) −26.8765 15.5172i −1.14291 0.659858i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 26.1760 15.1127i 1.10911 0.640346i 0.170512 0.985356i \(-0.445458\pi\)
0.938599 + 0.345010i \(0.112124\pi\)
\(558\) 0 0
\(559\) 33.5511i 1.41906i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −39.7803 −1.67654 −0.838269 0.545257i \(-0.816432\pi\)
−0.838269 + 0.545257i \(0.816432\pi\)
\(564\) 0 0
\(565\) −3.85772 2.22725i −0.162295 0.0937013i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −12.7383 −0.534018 −0.267009 0.963694i \(-0.586035\pi\)
−0.267009 + 0.963694i \(0.586035\pi\)
\(570\) 0 0
\(571\) 4.21174 0.176256 0.0881280 0.996109i \(-0.471912\pi\)
0.0881280 + 0.996109i \(0.471912\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −24.2291 13.9887i −1.01042 0.583369i
\(576\) 0 0
\(577\) −4.82818 −0.201000 −0.100500 0.994937i \(-0.532044\pi\)
−0.100500 + 0.994937i \(0.532044\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 4.66212i 0.193418i
\(582\) 0 0
\(583\) −0.0949144 + 0.0547989i −0.00393095 + 0.00226954i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −15.4624 8.92722i −0.638201 0.368466i 0.145720 0.989326i \(-0.453450\pi\)
−0.783921 + 0.620860i \(0.786784\pi\)
\(588\) 0 0
\(589\) 6.88698 + 21.9853i 0.283773 + 0.905887i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 20.4379 11.7998i 0.839284 0.484561i −0.0177369 0.999843i \(-0.505646\pi\)
0.857021 + 0.515282i \(0.172313\pi\)
\(594\) 0 0
\(595\) 21.4025 + 37.0701i 0.877415 + 1.51973i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −23.1857 40.1588i −0.947342 1.64084i −0.750992 0.660311i \(-0.770424\pi\)
−0.196350 0.980534i \(-0.562909\pi\)
\(600\) 0 0
\(601\) 20.8985i 0.852466i −0.904613 0.426233i \(-0.859840\pi\)
0.904613 0.426233i \(-0.140160\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −31.7333 18.3212i −1.29014 0.744865i
\(606\) 0 0
\(607\) 47.0730i 1.91064i 0.295580 + 0.955318i \(0.404487\pi\)
−0.295580 + 0.955318i \(0.595513\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −25.6403 + 44.4103i −1.03730 + 1.79665i
\(612\) 0 0
\(613\) 0.887781 1.53768i 0.0358571 0.0621063i −0.847540 0.530732i \(-0.821917\pi\)
0.883397 + 0.468625i \(0.155251\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −6.39218 + 3.69052i −0.257339 + 0.148575i −0.623120 0.782126i \(-0.714135\pi\)
0.365781 + 0.930701i \(0.380802\pi\)
\(618\) 0 0
\(619\) −25.0914 −1.00851 −0.504254 0.863555i \(-0.668232\pi\)
−0.504254 + 0.863555i \(0.668232\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 4.28612 + 7.42377i 0.171720 + 0.297427i
\(624\) 0 0
\(625\) 8.68630 15.0451i 0.347452 0.601804i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 11.6609 20.1972i 0.464949 0.805316i
\(630\) 0 0
\(631\) 2.73692 + 4.74048i 0.108955 + 0.188715i 0.915347 0.402666i \(-0.131916\pi\)
−0.806392 + 0.591381i \(0.798583\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 27.4961 1.09115
\(636\) 0 0
\(637\) −23.9451 + 13.8247i −0.948741 + 0.547756i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −20.6782 + 35.8157i −0.816740 + 1.41464i 0.0913319 + 0.995821i \(0.470888\pi\)
−0.908072 + 0.418815i \(0.862446\pi\)
\(642\) 0 0
\(643\) −21.1035 + 36.5523i −0.832239 + 1.44148i 0.0640194 + 0.997949i \(0.479608\pi\)
−0.896259 + 0.443532i \(0.853725\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 12.7852i 0.502639i 0.967904 + 0.251319i \(0.0808645\pi\)
−0.967904 + 0.251319i \(0.919136\pi\)
\(648\) 0 0
\(649\) 0.897781 + 0.518334i 0.0352410 + 0.0203464i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 12.4197i 0.486021i −0.970024 0.243011i \(-0.921865\pi\)
0.970024 0.243011i \(-0.0781350\pi\)
\(654\) 0 0
\(655\) −9.10394 15.7685i −0.355720 0.616126i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 1.38658 + 2.40162i 0.0540133 + 0.0935538i 0.891768 0.452493i \(-0.149465\pi\)
−0.837755 + 0.546047i \(0.816132\pi\)
\(660\) 0 0
\(661\) 7.90526 4.56411i 0.307479 0.177523i −0.338319 0.941032i \(-0.609858\pi\)
0.645798 + 0.763508i \(0.276525\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −50.5514 11.3057i −1.96030 0.438416i
\(666\) 0 0
\(667\) −17.7356 10.2397i −0.686727 0.396482i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −2.81822 + 1.62710i −0.108796 + 0.0628136i
\(672\) 0 0
\(673\) 3.33862i 0.128694i −0.997928 0.0643472i \(-0.979503\pi\)
0.997928 0.0643472i \(-0.0204965\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −45.3794 −1.74407 −0.872037 0.489441i \(-0.837201\pi\)
−0.872037 + 0.489441i \(0.837201\pi\)
\(678\) 0 0
\(679\) −33.2144 19.1763i −1.27465 0.735921i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −33.7656 −1.29201 −0.646003 0.763335i \(-0.723561\pi\)
−0.646003 + 0.763335i \(0.723561\pi\)
\(684\) 0 0
\(685\) −46.0149 −1.75814
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 1.86934 + 1.07926i 0.0712161 + 0.0411167i
\(690\) 0 0
\(691\) 47.6057 1.81101 0.905504 0.424338i \(-0.139493\pi\)
0.905504 + 0.424338i \(0.139493\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 7.86823i 0.298459i
\(696\) 0 0
\(697\) −30.9575 + 17.8733i −1.17260 + 0.677001i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 9.81431 + 5.66630i 0.370682 + 0.214013i 0.673756 0.738954i \(-0.264680\pi\)
−0.303075 + 0.952967i \(0.598013\pi\)
\(702\) 0 0
\(703\) 8.43664 + 26.9323i 0.318194 + 1.01577i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −19.1857 + 11.0769i −0.721553 + 0.416589i
\(708\) 0 0
\(709\) 8.10276 + 14.0344i 0.304306 + 0.527073i 0.977106 0.212751i \(-0.0682423\pi\)
−0.672801 + 0.739824i \(0.734909\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 11.8769 + 20.5714i 0.444793 + 0.770404i
\(714\) 0 0
\(715\) 4.17537i 0.156150i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 6.73659 + 3.88937i 0.251232 + 0.145049i 0.620328 0.784342i \(-0.286999\pi\)
−0.369096 + 0.929391i \(0.620333\pi\)
\(720\) 0 0
\(721\) 15.5396i 0.578726i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −14.1837 + 24.5670i −0.526771 + 0.912394i
\(726\) 0 0
\(727\) −15.7320 + 27.2486i −0.583467 + 1.01059i 0.411598 + 0.911366i \(0.364971\pi\)
−0.995065 + 0.0992290i \(0.968362\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 21.1251 12.1966i 0.781341 0.451107i
\(732\) 0 0
\(733\) −8.80147 −0.325090 −0.162545 0.986701i \(-0.551970\pi\)
−0.162545 + 0.986701i \(0.551970\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0.0396603 + 0.0686936i 0.00146091 + 0.00253036i
\(738\) 0 0
\(739\) −3.08996 + 5.35197i −0.113666 + 0.196876i −0.917246 0.398322i \(-0.869593\pi\)
0.803580 + 0.595197i \(0.202926\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 20.3126 35.1825i 0.745197 1.29072i −0.204906 0.978782i \(-0.565689\pi\)
0.950103 0.311937i \(-0.100978\pi\)
\(744\) 0 0
\(745\) −16.0757 27.8438i −0.588966 1.02012i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 7.02684 0.256755
\(750\) 0 0
\(751\) 10.4204 6.01624i 0.380247 0.219536i −0.297679 0.954666i \(-0.596212\pi\)
0.677926 + 0.735130i \(0.262879\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −11.8278 + 20.4864i −0.430459 + 0.745576i
\(756\) 0 0
\(757\) 4.43515 7.68190i 0.161198 0.279204i −0.774100 0.633063i \(-0.781797\pi\)
0.935299 + 0.353859i \(0.115131\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 10.3745i 0.376076i −0.982162 0.188038i \(-0.939787\pi\)
0.982162 0.188038i \(-0.0602127\pi\)
\(762\) 0 0
\(763\) 3.18440 + 1.83851i 0.115283 + 0.0665587i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 20.4172i 0.737222i
\(768\) 0 0
\(769\) −17.1749 29.7478i −0.619343 1.07273i −0.989606 0.143807i \(-0.954066\pi\)
0.370262 0.928927i \(-0.379268\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 26.4831 + 45.8701i 0.952531 + 1.64983i 0.739920 + 0.672695i \(0.234864\pi\)
0.212612 + 0.977137i \(0.431803\pi\)
\(774\) 0 0
\(775\) 28.4950 16.4516i 1.02357 0.590958i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 9.44147 42.2158i 0.338276 1.51254i
\(780\) 0 0
\(781\) −1.79067 1.03384i −0.0640752 0.0369938i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 47.4397 27.3893i 1.69319 0.977567i
\(786\) 0 0
\(787\) 1.75236i 0.0624648i −0.999512 0.0312324i \(-0.990057\pi\)
0.999512 0.0312324i \(-0.00994319\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −4.71584 −0.167676
\(792\) 0 0
\(793\) 55.5049 + 32.0458i 1.97104 + 1.13798i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 43.7424 1.54944 0.774718 0.632307i \(-0.217892\pi\)
0.774718 + 0.632307i \(0.217892\pi\)
\(798\) 0 0
\(799\) 37.2834 1.31899
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −0.155169 0.0895868i −0.00547579 0.00316145i
\(804\) 0 0
\(805\) −53.4080 −1.88238
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 1.97955i 0.0695974i 0.999394 + 0.0347987i \(0.0110790\pi\)
−0.999394 + 0.0347987i \(0.988921\pi\)
\(810\) 0 0
\(811\) 17.9304 10.3521i 0.629620 0.363511i −0.150985 0.988536i \(-0.548244\pi\)
0.780605 + 0.625025i \(0.214911\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 35.7234 + 20.6249i 1.25134 + 0.722459i
\(816\) 0 0
\(817\) −6.44277 + 28.8077i −0.225404 + 1.00785i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −22.7618 + 13.1415i −0.794392 + 0.458643i −0.841507 0.540247i \(-0.818331\pi\)
0.0471143 + 0.998890i \(0.484998\pi\)
\(822\) 0 0
\(823\) −9.16801 15.8795i −0.319577 0.553523i 0.660823 0.750542i \(-0.270207\pi\)
−0.980400 + 0.197019i \(0.936874\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 25.7899 + 44.6694i 0.896803 + 1.55331i 0.831557 + 0.555439i \(0.187450\pi\)
0.0652454 + 0.997869i \(0.479217\pi\)
\(828\) 0 0
\(829\) 30.7526i 1.06808i −0.845458 0.534042i \(-0.820673\pi\)
0.845458 0.534042i \(-0.179327\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 17.4092 + 10.0512i 0.603194 + 0.348254i
\(834\) 0 0
\(835\) 4.17537i 0.144495i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −21.2481 + 36.8028i −0.733566 + 1.27057i 0.221784 + 0.975096i \(0.428812\pi\)
−0.955350 + 0.295478i \(0.904521\pi\)
\(840\) 0 0
\(841\) 4.11755 7.13180i 0.141984 0.245924i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −33.4966 + 19.3393i −1.15232 + 0.665291i
\(846\) 0 0
\(847\) −38.7922 −1.33292
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 14.5494 + 25.2002i 0.498745 + 0.863852i
\(852\) 0 0
\(853\) −0.296220 + 0.513068i −0.0101424 + 0.0175671i −0.871052 0.491191i \(-0.836562\pi\)
0.860910 + 0.508758i \(0.169895\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 13.8976 24.0714i 0.474734 0.822264i −0.524847 0.851196i \(-0.675878\pi\)
0.999581 + 0.0289327i \(0.00921085\pi\)
\(858\) 0 0
\(859\) −11.8549 20.5332i −0.404483 0.700585i 0.589778 0.807565i \(-0.299215\pi\)
−0.994261 + 0.106980i \(0.965882\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −14.2464 −0.484954 −0.242477 0.970157i \(-0.577960\pi\)
−0.242477 + 0.970157i \(0.577960\pi\)
\(864\) 0 0
\(865\) −7.23195 + 4.17537i −0.245894 + 0.141967i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 1.10046 1.90606i 0.0373307 0.0646587i
\(870\) 0 0
\(871\) 0.781109 1.35292i 0.0264669 0.0458420i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 14.5605i 0.492234i
\(876\) 0 0
\(877\) −16.5049 9.52910i −0.557331 0.321775i 0.194743 0.980854i \(-0.437613\pi\)
−0.752073 + 0.659079i \(0.770946\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 16.0700i 0.541411i 0.962662 + 0.270706i \(0.0872570\pi\)
−0.962662 + 0.270706i \(0.912743\pi\)
\(882\) 0 0
\(883\) 13.6274 + 23.6033i 0.458598 + 0.794316i 0.998887 0.0471641i \(-0.0150184\pi\)
−0.540289 + 0.841480i \(0.681685\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −8.46016 14.6534i −0.284065 0.492014i 0.688317 0.725410i \(-0.258350\pi\)
−0.972382 + 0.233395i \(0.925016\pi\)
\(888\) 0 0
\(889\) 25.2093 14.5546i 0.845494 0.488146i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −30.5434 + 33.2080i −1.02210 + 1.11126i
\(894\) 0 0
\(895\) −61.1419 35.3003i −2.04375 1.17996i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 20.8582 12.0425i 0.695660 0.401640i
\(900\) 0 0
\(901\) 1.56935i 0.0522826i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 71.3341 2.37123
\(906\) 0 0
\(907\) −2.16196 1.24821i −0.0717868 0.0414461i 0.463677 0.886004i \(-0.346530\pi\)
−0.535464 + 0.844558i \(0.679863\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 31.0838 1.02985 0.514926 0.857235i \(-0.327819\pi\)
0.514926 + 0.857235i \(0.327819\pi\)
\(912\) 0 0
\(913\) 0.330634 0.0109424
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −16.6936 9.63804i −0.551270 0.318276i
\(918\) 0 0
\(919\) −13.1750 −0.434604 −0.217302 0.976104i \(-0.569726\pi\)
−0.217302 + 0.976104i \(0.569726\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 40.7231i 1.34042i
\(924\) 0 0
\(925\) 34.9067 20.1534i 1.14773 0.662640i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −1.08127 0.624270i −0.0354752 0.0204816i 0.482157 0.876085i \(-0.339853\pi\)
−0.517633 + 0.855603i \(0.673187\pi\)
\(930\) 0 0
\(931\) −23.2146 + 7.27207i −0.760827 + 0.238332i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −2.62898 + 1.51784i −0.0859769 + 0.0496388i
\(936\) 0 0
\(937\) 2.58551 + 4.47824i 0.0844649 + 0.146298i 0.905163 0.425064i \(-0.139749\pi\)
−0.820698 + 0.571362i \(0.806415\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −18.7406 32.4597i −0.610926 1.05816i −0.991085 0.133234i \(-0.957464\pi\)
0.380159 0.924921i \(-0.375869\pi\)
\(942\) 0 0
\(943\) 44.6014i 1.45242i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 4.42574 + 2.55520i 0.143817 + 0.0830328i 0.570182 0.821518i \(-0.306873\pi\)
−0.426365 + 0.904551i \(0.640206\pi\)
\(948\) 0 0
\(949\) 3.52882i 0.114550i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −6.30089 + 10.9135i −0.204106 + 0.353522i −0.949848 0.312713i \(-0.898762\pi\)
0.745742 + 0.666235i \(0.232095\pi\)
\(954\) 0 0
\(955\) 17.0683 29.5632i 0.552319 0.956644i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −42.1880 + 24.3572i −1.36232 + 0.786536i
\(960\) 0 0
\(961\) 3.06408 0.0988412
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 17.2765 + 29.9238i 0.556151 + 0.963282i
\(966\) 0 0
\(967\) −3.11699 + 5.39879i −0.100236 + 0.173613i −0.911782 0.410675i \(-0.865293\pi\)
0.811546 + 0.584289i \(0.198626\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 4.10993 7.11860i 0.131894 0.228447i −0.792513 0.609855i \(-0.791228\pi\)
0.924407 + 0.381408i \(0.124561\pi\)
\(972\) 0 0
\(973\) −4.16492 7.21385i −0.133521 0.231265i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −46.0799 −1.47423 −0.737113 0.675769i \(-0.763812\pi\)
−0.737113 + 0.675769i \(0.763812\pi\)
\(978\) 0 0
\(979\) −0.526487 + 0.303968i −0.0168266 + 0.00971485i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 3.60597 6.24572i 0.115013 0.199208i −0.802772 0.596286i \(-0.796642\pi\)
0.917785 + 0.397078i \(0.129976\pi\)
\(984\) 0 0
\(985\) −24.1063 + 41.7534i −0.768091 + 1.33037i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 30.4356i 0.967795i
\(990\) 0 0
\(991\) 17.3753 + 10.0316i 0.551945 + 0.318665i 0.749906 0.661544i \(-0.230099\pi\)
−0.197961 + 0.980210i \(0.563432\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 75.7338i 2.40092i
\(996\) 0 0
\(997\) −0.0129544 0.0224377i −0.000410270 0.000710608i 0.865820 0.500355i \(-0.166797\pi\)
−0.866230 + 0.499645i \(0.833464\pi\)
\(998\) 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 2736.2.dc.c.1889.2 16
3.2 odd 2 inner 2736.2.dc.c.1889.7 16
4.3 odd 2 171.2.m.a.8.7 yes 16
12.11 even 2 171.2.m.a.8.2 16
19.12 odd 6 inner 2736.2.dc.c.449.7 16
57.50 even 6 inner 2736.2.dc.c.449.2 16
76.31 even 6 171.2.m.a.107.2 yes 16
228.107 odd 6 171.2.m.a.107.7 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
171.2.m.a.8.2 16 12.11 even 2
171.2.m.a.8.7 yes 16 4.3 odd 2
171.2.m.a.107.2 yes 16 76.31 even 6
171.2.m.a.107.7 yes 16 228.107 odd 6
2736.2.dc.c.449.2 16 57.50 even 6 inner
2736.2.dc.c.449.7 16 19.12 odd 6 inner
2736.2.dc.c.1889.2 16 1.1 even 1 trivial
2736.2.dc.c.1889.7 16 3.2 odd 2 inner