Properties

Label 1176.2.u.b.1097.2
Level $1176$
Weight $2$
Character 1176.1097
Analytic conductor $9.390$
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] = [1176,2,Mod(521,1176)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1176, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1176.521");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1176 = 2^{3} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1176.u (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: \(9.39040727770\)
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: \( x^{16} - 6 x^{15} + 19 x^{14} - 42 x^{13} + 65 x^{12} - 48 x^{11} - 94 x^{10} + 444 x^{9} - 962 x^{8} + 1332 x^{7} - 846 x^{6} - 1296 x^{5} + 5265 x^{4} - 10206 x^{3} + 13851 x^{2} + \cdots + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 168)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 1097.2
Root \(-1.70742 - 0.291063i\) of defining polynomial
Character \(\chi\) \(=\) 1176.1097
Dual form 1176.2.u.b.521.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.62420 - 0.601642i) q^{3} +(-0.0726693 + 0.125867i) q^{5} +(2.27605 + 1.95437i) q^{9} +O(q^{10})\) \(q+(-1.62420 - 0.601642i) q^{3} +(-0.0726693 + 0.125867i) q^{5} +(2.27605 + 1.95437i) q^{9} +(-2.13889 + 1.23489i) q^{11} -2.04143i q^{13} +(0.193756 - 0.160712i) q^{15} +(-0.878419 - 1.52147i) q^{17} +(3.68319 + 2.12649i) q^{19} +(-7.46351 - 4.30906i) q^{23} +(2.48944 + 4.31183i) q^{25} +(-2.52094 - 4.54366i) q^{27} -7.08790i q^{29} +(-3.11812 + 1.80025i) q^{31} +(4.21694 - 0.718860i) q^{33} +(-2.93493 + 5.08345i) q^{37} +(-1.22821 + 3.31569i) q^{39} +5.33255 q^{41} -9.19692 q^{43} +(-0.411390 + 0.144457i) q^{45} +(-4.65190 + 8.05733i) q^{47} +(0.511351 + 2.99966i) q^{51} +(-4.49578 + 2.59564i) q^{53} -0.358953i q^{55} +(-4.70286 - 5.66982i) q^{57} +(-5.60299 - 9.70466i) q^{59} +(-4.66353 - 2.69249i) q^{61} +(0.256949 + 0.148349i) q^{65} +(2.57417 + 4.45860i) q^{67} +(9.52973 + 11.4891i) q^{69} +7.79323i q^{71} +(-11.3013 + 6.52482i) q^{73} +(-1.44917 - 8.50103i) q^{75} +(2.86075 - 4.95497i) q^{79} +(1.36085 + 8.89652i) q^{81} -15.9818 q^{83} +0.255336 q^{85} +(-4.26437 + 11.5122i) q^{87} +(-4.34252 + 7.52147i) q^{89} +(6.14756 - 1.04797i) q^{93} +(-0.535310 + 0.309061i) q^{95} +6.65337i q^{97} +(-7.28165 - 1.36951i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 2 q^{9} + 8 q^{15} + 6 q^{19} - 18 q^{25} + 48 q^{31} + 12 q^{33} - 2 q^{37} - 22 q^{39} + 20 q^{43} + 42 q^{45} + 6 q^{51} - 8 q^{57} - 36 q^{61} + 14 q^{67} - 30 q^{73} - 54 q^{75} + 28 q^{79} + 30 q^{81} + 16 q^{85} - 78 q^{87} + 16 q^{93} + 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1176\mathbb{Z}\right)^\times\).

\(n\) \(295\) \(589\) \(785\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(e\left(\frac{5}{6}\right)\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.62420 0.601642i −0.937733 0.347358i
\(4\) 0 0
\(5\) −0.0726693 + 0.125867i −0.0324987 + 0.0562894i −0.881817 0.471591i \(-0.843680\pi\)
0.849319 + 0.527881i \(0.177013\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 2.27605 + 1.95437i 0.758685 + 0.651458i
\(10\) 0 0
\(11\) −2.13889 + 1.23489i −0.644899 + 0.372332i −0.786499 0.617592i \(-0.788109\pi\)
0.141600 + 0.989924i \(0.454775\pi\)
\(12\) 0 0
\(13\) 2.04143i 0.566191i −0.959092 0.283096i \(-0.908639\pi\)
0.959092 0.283096i \(-0.0913613\pi\)
\(14\) 0 0
\(15\) 0.193756 0.160712i 0.0500276 0.0414957i
\(16\) 0 0
\(17\) −0.878419 1.52147i −0.213048 0.369010i 0.739619 0.673026i \(-0.235006\pi\)
−0.952667 + 0.304016i \(0.901672\pi\)
\(18\) 0 0
\(19\) 3.68319 + 2.12649i 0.844983 + 0.487851i 0.858955 0.512051i \(-0.171114\pi\)
−0.0139720 + 0.999902i \(0.504448\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −7.46351 4.30906i −1.55625 0.898501i −0.997610 0.0690910i \(-0.977990\pi\)
−0.558640 0.829410i \(-0.688677\pi\)
\(24\) 0 0
\(25\) 2.48944 + 4.31183i 0.497888 + 0.862367i
\(26\) 0 0
\(27\) −2.52094 4.54366i −0.485154 0.874429i
\(28\) 0 0
\(29\) 7.08790i 1.31619i −0.752935 0.658095i \(-0.771363\pi\)
0.752935 0.658095i \(-0.228637\pi\)
\(30\) 0 0
\(31\) −3.11812 + 1.80025i −0.560031 + 0.323334i −0.753158 0.657840i \(-0.771470\pi\)
0.193127 + 0.981174i \(0.438137\pi\)
\(32\) 0 0
\(33\) 4.21694 0.718860i 0.734075 0.125138i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −2.93493 + 5.08345i −0.482499 + 0.835713i −0.999798 0.0200916i \(-0.993604\pi\)
0.517299 + 0.855805i \(0.326938\pi\)
\(38\) 0 0
\(39\) −1.22821 + 3.31569i −0.196671 + 0.530936i
\(40\) 0 0
\(41\) 5.33255 0.832804 0.416402 0.909181i \(-0.363291\pi\)
0.416402 + 0.909181i \(0.363291\pi\)
\(42\) 0 0
\(43\) −9.19692 −1.40252 −0.701258 0.712907i \(-0.747378\pi\)
−0.701258 + 0.712907i \(0.747378\pi\)
\(44\) 0 0
\(45\) −0.411390 + 0.144457i −0.0613264 + 0.0215344i
\(46\) 0 0
\(47\) −4.65190 + 8.05733i −0.678549 + 1.17528i 0.296868 + 0.954918i \(0.404058\pi\)
−0.975418 + 0.220364i \(0.929276\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0.511351 + 2.99966i 0.0716034 + 0.420036i
\(52\) 0 0
\(53\) −4.49578 + 2.59564i −0.617543 + 0.356539i −0.775912 0.630841i \(-0.782710\pi\)
0.158369 + 0.987380i \(0.449377\pi\)
\(54\) 0 0
\(55\) 0.358953i 0.0484013i
\(56\) 0 0
\(57\) −4.70286 5.66982i −0.622909 0.750985i
\(58\) 0 0
\(59\) −5.60299 9.70466i −0.729447 1.26344i −0.957117 0.289701i \(-0.906444\pi\)
0.227670 0.973738i \(-0.426889\pi\)
\(60\) 0 0
\(61\) −4.66353 2.69249i −0.597104 0.344738i 0.170798 0.985306i \(-0.445366\pi\)
−0.767901 + 0.640568i \(0.778699\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0.256949 + 0.148349i 0.0318705 + 0.0184005i
\(66\) 0 0
\(67\) 2.57417 + 4.45860i 0.314485 + 0.544705i 0.979328 0.202279i \(-0.0648347\pi\)
−0.664843 + 0.746984i \(0.731501\pi\)
\(68\) 0 0
\(69\) 9.52973 + 11.4891i 1.14724 + 1.38313i
\(70\) 0 0
\(71\) 7.79323i 0.924886i 0.886649 + 0.462443i \(0.153027\pi\)
−0.886649 + 0.462443i \(0.846973\pi\)
\(72\) 0 0
\(73\) −11.3013 + 6.52482i −1.32272 + 0.763672i −0.984162 0.177273i \(-0.943272\pi\)
−0.338558 + 0.940946i \(0.609939\pi\)
\(74\) 0 0
\(75\) −1.44917 8.50103i −0.167335 0.981615i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 2.86075 4.95497i 0.321860 0.557478i −0.659012 0.752133i \(-0.729025\pi\)
0.980872 + 0.194655i \(0.0623586\pi\)
\(80\) 0 0
\(81\) 1.36085 + 8.89652i 0.151205 + 0.988502i
\(82\) 0 0
\(83\) −15.9818 −1.75423 −0.877115 0.480280i \(-0.840535\pi\)
−0.877115 + 0.480280i \(0.840535\pi\)
\(84\) 0 0
\(85\) 0.255336 0.0276951
\(86\) 0 0
\(87\) −4.26437 + 11.5122i −0.457189 + 1.23423i
\(88\) 0 0
\(89\) −4.34252 + 7.52147i −0.460306 + 0.797274i −0.998976 0.0452432i \(-0.985594\pi\)
0.538670 + 0.842517i \(0.318927\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 6.14756 1.04797i 0.637472 0.108670i
\(94\) 0 0
\(95\) −0.535310 + 0.309061i −0.0549217 + 0.0317090i
\(96\) 0 0
\(97\) 6.65337i 0.675547i 0.941227 + 0.337774i \(0.109674\pi\)
−0.941227 + 0.337774i \(0.890326\pi\)
\(98\) 0 0
\(99\) −7.28165 1.36951i −0.731834 0.137641i
\(100\) 0 0
\(101\) −8.06357 13.9665i −0.802355 1.38972i −0.918062 0.396437i \(-0.870247\pi\)
0.115707 0.993283i \(-0.463087\pi\)
\(102\) 0 0
\(103\) 0.147333 + 0.0850626i 0.0145171 + 0.00838147i 0.507241 0.861804i \(-0.330665\pi\)
−0.492724 + 0.870186i \(0.663999\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −6.03900 3.48662i −0.583813 0.337064i 0.178835 0.983879i \(-0.442767\pi\)
−0.762647 + 0.646815i \(0.776101\pi\)
\(108\) 0 0
\(109\) −0.677559 1.17357i −0.0648984 0.112407i 0.831751 0.555150i \(-0.187339\pi\)
−0.896649 + 0.442742i \(0.854006\pi\)
\(110\) 0 0
\(111\) 7.82532 6.49076i 0.742747 0.616076i
\(112\) 0 0
\(113\) 4.00000i 0.376288i 0.982141 + 0.188144i \(0.0602472\pi\)
−0.982141 + 0.188144i \(0.939753\pi\)
\(114\) 0 0
\(115\) 1.08474 0.626273i 0.101152 0.0584002i
\(116\) 0 0
\(117\) 3.98972 4.64641i 0.368850 0.429561i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −2.45011 + 4.24371i −0.222737 + 0.385792i
\(122\) 0 0
\(123\) −8.66113 3.20828i −0.780948 0.289281i
\(124\) 0 0
\(125\) −1.45032 −0.129720
\(126\) 0 0
\(127\) −7.33399 −0.650787 −0.325393 0.945579i \(-0.605497\pi\)
−0.325393 + 0.945579i \(0.605497\pi\)
\(128\) 0 0
\(129\) 14.9376 + 5.53325i 1.31519 + 0.487175i
\(130\) 0 0
\(131\) 3.04832 5.27985i 0.266333 0.461303i −0.701579 0.712592i \(-0.747521\pi\)
0.967912 + 0.251289i \(0.0808545\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0.755091 + 0.0128824i 0.0649879 + 0.00110874i
\(136\) 0 0
\(137\) 17.3832 10.0362i 1.48515 0.857451i 0.485291 0.874353i \(-0.338714\pi\)
0.999857 + 0.0169018i \(0.00538026\pi\)
\(138\) 0 0
\(139\) 0.117694i 0.00998266i −0.999988 0.00499133i \(-0.998411\pi\)
0.999988 0.00499133i \(-0.00158880\pi\)
\(140\) 0 0
\(141\) 12.4032 10.2879i 1.04454 0.866401i
\(142\) 0 0
\(143\) 2.52094 + 4.36639i 0.210811 + 0.365136i
\(144\) 0 0
\(145\) 0.892131 + 0.515072i 0.0740875 + 0.0427744i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −7.75705 4.47853i −0.635482 0.366896i 0.147390 0.989078i \(-0.452913\pi\)
−0.782872 + 0.622183i \(0.786246\pi\)
\(150\) 0 0
\(151\) 1.37132 + 2.37519i 0.111596 + 0.193290i 0.916414 0.400232i \(-0.131070\pi\)
−0.804818 + 0.593522i \(0.797737\pi\)
\(152\) 0 0
\(153\) 0.974184 5.17970i 0.0787581 0.418754i
\(154\) 0 0
\(155\) 0.523291i 0.0420318i
\(156\) 0 0
\(157\) 11.7303 6.77249i 0.936180 0.540504i 0.0474193 0.998875i \(-0.484900\pi\)
0.888761 + 0.458371i \(0.151567\pi\)
\(158\) 0 0
\(159\) 8.86370 1.51099i 0.702937 0.119829i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 2.02428 3.50616i 0.158554 0.274624i −0.775793 0.630987i \(-0.782650\pi\)
0.934347 + 0.356363i \(0.115983\pi\)
\(164\) 0 0
\(165\) −0.215961 + 0.583012i −0.0168126 + 0.0453874i
\(166\) 0 0
\(167\) 3.70521 0.286717 0.143359 0.989671i \(-0.454210\pi\)
0.143359 + 0.989671i \(0.454210\pi\)
\(168\) 0 0
\(169\) 8.83256 0.679428
\(170\) 0 0
\(171\) 4.22719 + 12.0384i 0.323261 + 0.920596i
\(172\) 0 0
\(173\) −11.2370 + 19.4630i −0.854333 + 1.47975i 0.0229296 + 0.999737i \(0.492701\pi\)
−0.877263 + 0.480011i \(0.840633\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 3.26165 + 19.1333i 0.245160 + 1.43815i
\(178\) 0 0
\(179\) −3.18574 + 1.83929i −0.238113 + 0.137475i −0.614309 0.789065i \(-0.710565\pi\)
0.376196 + 0.926540i \(0.377232\pi\)
\(180\) 0 0
\(181\) 8.01062i 0.595425i −0.954656 0.297712i \(-0.903776\pi\)
0.954656 0.297712i \(-0.0962237\pi\)
\(182\) 0 0
\(183\) 5.95459 + 7.17892i 0.440176 + 0.530681i
\(184\) 0 0
\(185\) −0.426558 0.738821i −0.0313612 0.0543192i
\(186\) 0 0
\(187\) 3.75768 + 2.16950i 0.274789 + 0.158649i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0.971326 + 0.560795i 0.0702827 + 0.0405777i 0.534730 0.845023i \(-0.320413\pi\)
−0.464447 + 0.885601i \(0.653747\pi\)
\(192\) 0 0
\(193\) −9.18421 15.9075i −0.661094 1.14505i −0.980329 0.197373i \(-0.936759\pi\)
0.319235 0.947676i \(-0.396574\pi\)
\(194\) 0 0
\(195\) −0.328083 0.395540i −0.0234945 0.0283252i
\(196\) 0 0
\(197\) 0.296699i 0.0211389i −0.999944 0.0105695i \(-0.996636\pi\)
0.999944 0.0105695i \(-0.00336442\pi\)
\(198\) 0 0
\(199\) 23.6874 13.6759i 1.67915 0.969460i 0.716951 0.697124i \(-0.245537\pi\)
0.962202 0.272336i \(-0.0877962\pi\)
\(200\) 0 0
\(201\) −1.49850 8.79039i −0.105696 0.620027i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −0.387513 + 0.671191i −0.0270650 + 0.0468780i
\(206\) 0 0
\(207\) −8.56585 24.3942i −0.595368 1.69551i
\(208\) 0 0
\(209\) −10.5039 −0.726571
\(210\) 0 0
\(211\) 21.0295 1.44773 0.723864 0.689942i \(-0.242364\pi\)
0.723864 + 0.689942i \(0.242364\pi\)
\(212\) 0 0
\(213\) 4.68873 12.6578i 0.321267 0.867296i
\(214\) 0 0
\(215\) 0.668333 1.15759i 0.0455800 0.0789468i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 22.2812 3.79827i 1.50562 0.256663i
\(220\) 0 0
\(221\) −3.10597 + 1.79323i −0.208930 + 0.120626i
\(222\) 0 0
\(223\) 6.89447i 0.461688i −0.972991 0.230844i \(-0.925851\pi\)
0.972991 0.230844i \(-0.0741487\pi\)
\(224\) 0 0
\(225\) −2.76084 + 14.6793i −0.184056 + 0.978617i
\(226\) 0 0
\(227\) 6.70734 + 11.6174i 0.445182 + 0.771077i 0.998065 0.0621816i \(-0.0198058\pi\)
−0.552883 + 0.833259i \(0.686472\pi\)
\(228\) 0 0
\(229\) 5.51012 + 3.18127i 0.364119 + 0.210224i 0.670886 0.741560i \(-0.265914\pi\)
−0.306767 + 0.951785i \(0.599247\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −4.29295 2.47853i −0.281240 0.162374i 0.352744 0.935720i \(-0.385248\pi\)
−0.633985 + 0.773346i \(0.718582\pi\)
\(234\) 0 0
\(235\) −0.676100 1.17104i −0.0441039 0.0763903i
\(236\) 0 0
\(237\) −7.62755 + 6.32672i −0.495463 + 0.410964i
\(238\) 0 0
\(239\) 17.3756i 1.12394i 0.827159 + 0.561968i \(0.189956\pi\)
−0.827159 + 0.561968i \(0.810044\pi\)
\(240\) 0 0
\(241\) −12.5626 + 7.25302i −0.809228 + 0.467208i −0.846688 0.532090i \(-0.821407\pi\)
0.0374597 + 0.999298i \(0.488073\pi\)
\(242\) 0 0
\(243\) 3.14223 15.2685i 0.201574 0.979473i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 4.34109 7.51899i 0.276217 0.478422i
\(248\) 0 0
\(249\) 25.9577 + 9.61532i 1.64500 + 0.609346i
\(250\) 0 0
\(251\) −3.49783 −0.220781 −0.110391 0.993888i \(-0.535210\pi\)
−0.110391 + 0.993888i \(0.535210\pi\)
\(252\) 0 0
\(253\) 21.2848 1.33816
\(254\) 0 0
\(255\) −0.414717 0.153621i −0.0259706 0.00962012i
\(256\) 0 0
\(257\) 7.96781 13.8006i 0.497018 0.860861i −0.502976 0.864300i \(-0.667762\pi\)
0.999994 + 0.00343985i \(0.00109494\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 13.8524 16.1324i 0.857442 0.998573i
\(262\) 0 0
\(263\) 12.4343 7.17892i 0.766729 0.442671i −0.0649777 0.997887i \(-0.520698\pi\)
0.831706 + 0.555216i \(0.187364\pi\)
\(264\) 0 0
\(265\) 0.754493i 0.0463482i
\(266\) 0 0
\(267\) 11.5784 9.60373i 0.708584 0.587739i
\(268\) 0 0
\(269\) 3.68211 + 6.37760i 0.224502 + 0.388849i 0.956170 0.292812i \(-0.0945911\pi\)
−0.731668 + 0.681661i \(0.761258\pi\)
\(270\) 0 0
\(271\) −10.8537 6.26636i −0.659313 0.380654i 0.132702 0.991156i \(-0.457635\pi\)
−0.792015 + 0.610501i \(0.790968\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −10.6493 6.14835i −0.642174 0.370759i
\(276\) 0 0
\(277\) −16.2409 28.1300i −0.975819 1.69017i −0.677205 0.735794i \(-0.736809\pi\)
−0.298614 0.954374i \(-0.596524\pi\)
\(278\) 0 0
\(279\) −10.6154 1.99651i −0.635526 0.119528i
\(280\) 0 0
\(281\) 10.1758i 0.607037i 0.952826 + 0.303518i \(0.0981614\pi\)
−0.952826 + 0.303518i \(0.901839\pi\)
\(282\) 0 0
\(283\) 1.18666 0.685120i 0.0705397 0.0407261i −0.464315 0.885670i \(-0.653700\pi\)
0.534855 + 0.844944i \(0.320366\pi\)
\(284\) 0 0
\(285\) 1.05540 0.179913i 0.0625162 0.0106571i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 6.95676 12.0495i 0.409221 0.708792i
\(290\) 0 0
\(291\) 4.00295 10.8064i 0.234657 0.633483i
\(292\) 0 0
\(293\) 16.9961 0.992923 0.496461 0.868059i \(-0.334632\pi\)
0.496461 + 0.868059i \(0.334632\pi\)
\(294\) 0 0
\(295\) 1.62866 0.0948243
\(296\) 0 0
\(297\) 11.0029 + 6.60531i 0.638453 + 0.383279i
\(298\) 0 0
\(299\) −8.79665 + 15.2362i −0.508723 + 0.881135i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 4.69402 + 27.5358i 0.269664 + 1.58189i
\(304\) 0 0
\(305\) 0.677791 0.391323i 0.0388102 0.0224071i
\(306\) 0 0
\(307\) 20.9023i 1.19296i 0.802629 + 0.596479i \(0.203434\pi\)
−0.802629 + 0.596479i \(0.796566\pi\)
\(308\) 0 0
\(309\) −0.188121 0.226800i −0.0107018 0.0129022i
\(310\) 0 0
\(311\) −5.74040 9.94267i −0.325508 0.563797i 0.656107 0.754668i \(-0.272202\pi\)
−0.981615 + 0.190871i \(0.938869\pi\)
\(312\) 0 0
\(313\) −8.57172 4.94889i −0.484502 0.279728i 0.237788 0.971317i \(-0.423577\pi\)
−0.722291 + 0.691589i \(0.756911\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −5.74547 3.31715i −0.322698 0.186310i 0.329897 0.944017i \(-0.392986\pi\)
−0.652594 + 0.757707i \(0.726319\pi\)
\(318\) 0 0
\(319\) 8.75275 + 15.1602i 0.490060 + 0.848809i
\(320\) 0 0
\(321\) 7.71086 + 9.29629i 0.430378 + 0.518868i
\(322\) 0 0
\(323\) 7.47181i 0.415742i
\(324\) 0 0
\(325\) 8.80231 5.08202i 0.488264 0.281900i
\(326\) 0 0
\(327\) 0.394425 + 2.31376i 0.0218118 + 0.127951i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −7.36537 + 12.7572i −0.404837 + 0.701199i −0.994303 0.106595i \(-0.966005\pi\)
0.589465 + 0.807794i \(0.299339\pi\)
\(332\) 0 0
\(333\) −16.6150 + 5.83425i −0.910497 + 0.319715i
\(334\) 0 0
\(335\) −0.748254 −0.0408815
\(336\) 0 0
\(337\) −30.7209 −1.67347 −0.836737 0.547605i \(-0.815540\pi\)
−0.836737 + 0.547605i \(0.815540\pi\)
\(338\) 0 0
\(339\) 2.40657 6.49680i 0.130707 0.352858i
\(340\) 0 0
\(341\) 4.44621 7.70106i 0.240776 0.417036i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −2.13862 + 0.364570i −0.115139 + 0.0196278i
\(346\) 0 0
\(347\) 14.5124 8.37875i 0.779068 0.449795i −0.0570320 0.998372i \(-0.518164\pi\)
0.836100 + 0.548577i \(0.184830\pi\)
\(348\) 0 0
\(349\) 3.12385i 0.167216i 0.996499 + 0.0836080i \(0.0266443\pi\)
−0.996499 + 0.0836080i \(0.973356\pi\)
\(350\) 0 0
\(351\) −9.27558 + 5.14632i −0.495094 + 0.274690i
\(352\) 0 0
\(353\) 17.7450 + 30.7353i 0.944473 + 1.63587i 0.756804 + 0.653642i \(0.226760\pi\)
0.187669 + 0.982232i \(0.439907\pi\)
\(354\) 0 0
\(355\) −0.980910 0.566328i −0.0520613 0.0300576i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 5.42817 + 3.13395i 0.286488 + 0.165404i 0.636357 0.771395i \(-0.280441\pi\)
−0.349869 + 0.936799i \(0.613774\pi\)
\(360\) 0 0
\(361\) −0.456052 0.789905i −0.0240027 0.0415739i
\(362\) 0 0
\(363\) 6.53266 5.41855i 0.342876 0.284400i
\(364\) 0 0
\(365\) 1.89662i 0.0992734i
\(366\) 0 0
\(367\) 14.5823 8.41907i 0.761188 0.439472i −0.0685342 0.997649i \(-0.521832\pi\)
0.829722 + 0.558177i \(0.188499\pi\)
\(368\) 0 0
\(369\) 12.1372 + 10.4218i 0.631836 + 0.542537i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0.617106 1.06886i 0.0319526 0.0553435i −0.849607 0.527416i \(-0.823161\pi\)
0.881559 + 0.472073i \(0.156494\pi\)
\(374\) 0 0
\(375\) 2.35560 + 0.872570i 0.121643 + 0.0450593i
\(376\) 0 0
\(377\) −14.4695 −0.745215
\(378\) 0 0
\(379\) −14.3895 −0.739141 −0.369571 0.929203i \(-0.620495\pi\)
−0.369571 + 0.929203i \(0.620495\pi\)
\(380\) 0 0
\(381\) 11.9119 + 4.41244i 0.610264 + 0.226056i
\(382\) 0 0
\(383\) −4.95842 + 8.58824i −0.253364 + 0.438839i −0.964450 0.264266i \(-0.914870\pi\)
0.711086 + 0.703105i \(0.248204\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −20.9327 17.9742i −1.06407 0.913680i
\(388\) 0 0
\(389\) −11.5061 + 6.64306i −0.583383 + 0.336816i −0.762477 0.647016i \(-0.776017\pi\)
0.179094 + 0.983832i \(0.442683\pi\)
\(390\) 0 0
\(391\) 15.1406i 0.765695i
\(392\) 0 0
\(393\) −8.12767 + 6.74154i −0.409987 + 0.340066i
\(394\) 0 0
\(395\) 0.415778 + 0.720148i 0.0209201 + 0.0362346i
\(396\) 0 0
\(397\) −21.0410 12.1480i −1.05602 0.609693i −0.131691 0.991291i \(-0.542041\pi\)
−0.924328 + 0.381598i \(0.875374\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 12.4125 + 7.16635i 0.619850 + 0.357870i 0.776810 0.629735i \(-0.216836\pi\)
−0.156961 + 0.987605i \(0.550170\pi\)
\(402\) 0 0
\(403\) 3.67508 + 6.36543i 0.183069 + 0.317085i
\(404\) 0 0
\(405\) −1.21867 0.475218i −0.0605562 0.0236138i
\(406\) 0 0
\(407\) 14.4972i 0.718600i
\(408\) 0 0
\(409\) −17.3256 + 10.0029i −0.856695 + 0.494613i −0.862904 0.505368i \(-0.831357\pi\)
0.00620937 + 0.999981i \(0.498023\pi\)
\(410\) 0 0
\(411\) −34.2720 + 5.84234i −1.69051 + 0.288181i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 1.16139 2.01158i 0.0570102 0.0987446i
\(416\) 0 0
\(417\) −0.0708095 + 0.191158i −0.00346756 + 0.00936107i
\(418\) 0 0
\(419\) 27.7445 1.35541 0.677704 0.735335i \(-0.262975\pi\)
0.677704 + 0.735335i \(0.262975\pi\)
\(420\) 0 0
\(421\) −1.53586 −0.0748533 −0.0374267 0.999299i \(-0.511916\pi\)
−0.0374267 + 0.999299i \(0.511916\pi\)
\(422\) 0 0
\(423\) −26.3350 + 9.24737i −1.28045 + 0.449622i
\(424\) 0 0
\(425\) 4.37354 7.57519i 0.212148 0.367451i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −1.46750 8.60859i −0.0708517 0.415627i
\(430\) 0 0
\(431\) −14.8277 + 8.56080i −0.714227 + 0.412359i −0.812624 0.582788i \(-0.801962\pi\)
0.0983974 + 0.995147i \(0.468628\pi\)
\(432\) 0 0
\(433\) 27.5219i 1.32262i −0.750113 0.661310i \(-0.770001\pi\)
0.750113 0.661310i \(-0.229999\pi\)
\(434\) 0 0
\(435\) −1.13911 1.37332i −0.0546162 0.0658459i
\(436\) 0 0
\(437\) −18.3264 31.7422i −0.876670 1.51844i
\(438\) 0 0
\(439\) 18.9922 + 10.9651i 0.906446 + 0.523337i 0.879286 0.476294i \(-0.158020\pi\)
0.0271602 + 0.999631i \(0.491354\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 17.7589 + 10.2531i 0.843750 + 0.487139i 0.858537 0.512752i \(-0.171374\pi\)
−0.0147873 + 0.999891i \(0.504707\pi\)
\(444\) 0 0
\(445\) −0.631136 1.09316i −0.0299187 0.0518207i
\(446\) 0 0
\(447\) 9.90453 + 11.9410i 0.468468 + 0.564790i
\(448\) 0 0
\(449\) 18.7692i 0.885773i 0.896578 + 0.442886i \(0.146046\pi\)
−0.896578 + 0.442886i \(0.853954\pi\)
\(450\) 0 0
\(451\) −11.4057 + 6.58509i −0.537074 + 0.310080i
\(452\) 0 0
\(453\) −0.798279 4.68282i −0.0375064 0.220018i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −3.79670 + 6.57607i −0.177602 + 0.307616i −0.941059 0.338243i \(-0.890167\pi\)
0.763457 + 0.645859i \(0.223501\pi\)
\(458\) 0 0
\(459\) −4.69859 + 7.82676i −0.219312 + 0.365322i
\(460\) 0 0
\(461\) −29.2727 −1.36337 −0.681683 0.731648i \(-0.738752\pi\)
−0.681683 + 0.731648i \(0.738752\pi\)
\(462\) 0 0
\(463\) 11.8326 0.549906 0.274953 0.961458i \(-0.411338\pi\)
0.274953 + 0.961458i \(0.411338\pi\)
\(464\) 0 0
\(465\) −0.314834 + 0.849930i −0.0146001 + 0.0394146i
\(466\) 0 0
\(467\) 2.58282 4.47358i 0.119519 0.207013i −0.800058 0.599922i \(-0.795198\pi\)
0.919577 + 0.392910i \(0.128531\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −23.1270 + 3.94245i −1.06563 + 0.181658i
\(472\) 0 0
\(473\) 19.6712 11.3572i 0.904481 0.522202i
\(474\) 0 0
\(475\) 21.1751i 0.971580i
\(476\) 0 0
\(477\) −15.3055 2.87862i −0.700790 0.131803i
\(478\) 0 0
\(479\) −9.85496 17.0693i −0.450284 0.779915i 0.548119 0.836400i \(-0.315344\pi\)
−0.998403 + 0.0564848i \(0.982011\pi\)
\(480\) 0 0
\(481\) 10.3775 + 5.99145i 0.473173 + 0.273187i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −0.837439 0.483496i −0.0380261 0.0219544i
\(486\) 0 0
\(487\) 2.50360 + 4.33637i 0.113449 + 0.196500i 0.917159 0.398522i \(-0.130477\pi\)
−0.803710 + 0.595022i \(0.797143\pi\)
\(488\) 0 0
\(489\) −5.39730 + 4.47682i −0.244074 + 0.202449i
\(490\) 0 0
\(491\) 3.55902i 0.160616i 0.996770 + 0.0803081i \(0.0255904\pi\)
−0.996770 + 0.0803081i \(0.974410\pi\)
\(492\) 0 0
\(493\) −10.7840 + 6.22614i −0.485687 + 0.280411i
\(494\) 0 0
\(495\) 0.701529 0.816997i 0.0315314 0.0367213i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0.404702 0.700965i 0.0181170 0.0313795i −0.856825 0.515608i \(-0.827566\pi\)
0.874942 + 0.484228i \(0.160900\pi\)
\(500\) 0 0
\(501\) −6.01800 2.22921i −0.268864 0.0995936i
\(502\) 0 0
\(503\) −9.47070 −0.422278 −0.211139 0.977456i \(-0.567717\pi\)
−0.211139 + 0.977456i \(0.567717\pi\)
\(504\) 0 0
\(505\) 2.34390 0.104302
\(506\) 0 0
\(507\) −14.3458 5.31404i −0.637122 0.236005i
\(508\) 0 0
\(509\) 5.24404 9.08294i 0.232438 0.402594i −0.726087 0.687603i \(-0.758663\pi\)
0.958525 + 0.285009i \(0.0919965\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0.376973 22.0960i 0.0166438 0.975560i
\(514\) 0 0
\(515\) −0.0214131 + 0.0123629i −0.000943576 + 0.000544774i
\(516\) 0 0
\(517\) 22.9783i 1.01058i
\(518\) 0 0
\(519\) 29.9609 24.8512i 1.31514 1.09085i
\(520\) 0 0
\(521\) 4.77854 + 8.27667i 0.209351 + 0.362607i 0.951510 0.307617i \(-0.0995314\pi\)
−0.742159 + 0.670224i \(0.766198\pi\)
\(522\) 0 0
\(523\) 24.0305 + 13.8740i 1.05078 + 0.606668i 0.922868 0.385117i \(-0.125839\pi\)
0.127912 + 0.991785i \(0.459172\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 5.47804 + 3.16275i 0.238627 + 0.137771i
\(528\) 0 0
\(529\) 25.6360 + 44.4029i 1.11461 + 1.93056i
\(530\) 0 0
\(531\) 6.21383 33.0387i 0.269657 1.43376i
\(532\) 0 0
\(533\) 10.8860i 0.471526i
\(534\) 0 0
\(535\) 0.877700 0.506740i 0.0379463 0.0219083i
\(536\) 0 0
\(537\) 6.28087 1.07070i 0.271040 0.0462040i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 0.577777 1.00074i 0.0248406 0.0430251i −0.853338 0.521358i \(-0.825426\pi\)
0.878178 + 0.478333i \(0.158759\pi\)
\(542\) 0 0
\(543\) −4.81953 + 13.0109i −0.206826 + 0.558349i
\(544\) 0 0
\(545\) 0.196951 0.00843645
\(546\) 0 0
\(547\) 16.1394 0.690070 0.345035 0.938590i \(-0.387867\pi\)
0.345035 + 0.938590i \(0.387867\pi\)
\(548\) 0 0
\(549\) −5.35232 15.2425i −0.228431 0.650536i
\(550\) 0 0
\(551\) 15.0724 26.1061i 0.642104 1.11216i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0.248311 + 1.45663i 0.0105402 + 0.0618304i
\(556\) 0 0
\(557\) −32.1074 + 18.5372i −1.36043 + 0.785447i −0.989682 0.143284i \(-0.954234\pi\)
−0.370753 + 0.928732i \(0.620900\pi\)
\(558\) 0 0
\(559\) 18.7749i 0.794092i
\(560\) 0 0
\(561\) −4.79796 5.78447i −0.202570 0.244221i
\(562\) 0 0
\(563\) −7.79584 13.5028i −0.328556 0.569075i 0.653670 0.756780i \(-0.273228\pi\)
−0.982225 + 0.187705i \(0.939895\pi\)
\(564\) 0 0
\(565\) −0.503468 0.290677i −0.0211810 0.0122289i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 13.0276 + 7.52147i 0.546144 + 0.315316i 0.747565 0.664188i \(-0.231223\pi\)
−0.201421 + 0.979505i \(0.564556\pi\)
\(570\) 0 0
\(571\) −2.81334 4.87284i −0.117735 0.203922i 0.801135 0.598484i \(-0.204230\pi\)
−0.918870 + 0.394561i \(0.870897\pi\)
\(572\) 0 0
\(573\) −1.24023 1.49523i −0.0518114 0.0624643i
\(574\) 0 0
\(575\) 42.9086i 1.78941i
\(576\) 0 0
\(577\) −19.2278 + 11.1012i −0.800465 + 0.462149i −0.843634 0.536919i \(-0.819588\pi\)
0.0431688 + 0.999068i \(0.486255\pi\)
\(578\) 0 0
\(579\) 5.34637 + 31.3626i 0.222188 + 1.30339i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 6.41064 11.1036i 0.265502 0.459863i
\(584\) 0 0
\(585\) 0.294899 + 0.839825i 0.0121926 + 0.0347225i
\(586\) 0 0
\(587\) 20.9245 0.863648 0.431824 0.901958i \(-0.357870\pi\)
0.431824 + 0.901958i \(0.357870\pi\)
\(588\) 0 0
\(589\) −15.3129 −0.630956
\(590\) 0 0
\(591\) −0.178506 + 0.481898i −0.00734277 + 0.0198226i
\(592\) 0 0
\(593\) −10.5845 + 18.3329i −0.434654 + 0.752842i −0.997267 0.0738778i \(-0.976463\pi\)
0.562614 + 0.826720i \(0.309796\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −46.7010 + 7.96111i −1.91135 + 0.325826i
\(598\) 0 0
\(599\) 4.58648 2.64801i 0.187399 0.108195i −0.403366 0.915039i \(-0.632160\pi\)
0.590764 + 0.806844i \(0.298826\pi\)
\(600\) 0 0
\(601\) 37.5346i 1.53107i −0.643396 0.765533i \(-0.722475\pi\)
0.643396 0.765533i \(-0.277525\pi\)
\(602\) 0 0
\(603\) −2.85481 + 15.1789i −0.116257 + 0.618133i
\(604\) 0 0
\(605\) −0.356095 0.616775i −0.0144773 0.0250755i
\(606\) 0 0
\(607\) −34.2123 19.7525i −1.38864 0.801729i −0.395474 0.918477i \(-0.629420\pi\)
−0.993162 + 0.116748i \(0.962753\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 16.4485 + 9.49653i 0.665434 + 0.384189i
\(612\) 0 0
\(613\) −7.19736 12.4662i −0.290699 0.503505i 0.683277 0.730160i \(-0.260554\pi\)
−0.973975 + 0.226655i \(0.927221\pi\)
\(614\) 0 0
\(615\) 1.03321 0.857006i 0.0416632 0.0345578i
\(616\) 0 0
\(617\) 12.1573i 0.489435i 0.969594 + 0.244718i \(0.0786952\pi\)
−0.969594 + 0.244718i \(0.921305\pi\)
\(618\) 0 0
\(619\) 16.8732 9.74173i 0.678190 0.391553i −0.120983 0.992655i \(-0.538605\pi\)
0.799173 + 0.601101i \(0.205271\pi\)
\(620\) 0 0
\(621\) −0.763888 + 44.7746i −0.0306538 + 1.79674i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −12.3418 + 21.3766i −0.493672 + 0.855065i
\(626\) 0 0
\(627\) 17.0605 + 6.31959i 0.681329 + 0.252380i
\(628\) 0 0
\(629\) 10.3124 0.411182
\(630\) 0 0
\(631\) −31.3846 −1.24940 −0.624701 0.780864i \(-0.714779\pi\)
−0.624701 + 0.780864i \(0.714779\pi\)
\(632\) 0 0
\(633\) −34.1561 12.6522i −1.35758 0.502880i
\(634\) 0 0
\(635\) 0.532956 0.923107i 0.0211497 0.0366324i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −15.2309 + 17.7378i −0.602525 + 0.701697i
\(640\) 0 0
\(641\) −36.0118 + 20.7914i −1.42238 + 0.821211i −0.996502 0.0835697i \(-0.973368\pi\)
−0.425878 + 0.904781i \(0.640035\pi\)
\(642\) 0 0
\(643\) 13.5290i 0.533531i −0.963761 0.266766i \(-0.914045\pi\)
0.963761 0.266766i \(-0.0859549\pi\)
\(644\) 0 0
\(645\) −1.78196 + 1.47806i −0.0701646 + 0.0581984i
\(646\) 0 0
\(647\) 15.0442 + 26.0573i 0.591449 + 1.02442i 0.994038 + 0.109038i \(0.0347771\pi\)
−0.402589 + 0.915381i \(0.631890\pi\)
\(648\) 0 0
\(649\) 23.9683 + 13.8381i 0.940839 + 0.543193i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −18.3717 10.6069i −0.718941 0.415081i 0.0954221 0.995437i \(-0.469580\pi\)
−0.814363 + 0.580356i \(0.802913\pi\)
\(654\) 0 0
\(655\) 0.443039 + 0.767366i 0.0173110 + 0.0299835i
\(656\) 0 0
\(657\) −38.4743 7.23615i −1.50103 0.282309i
\(658\) 0 0
\(659\) 2.67926i 0.104369i −0.998637 0.0521846i \(-0.983382\pi\)
0.998637 0.0521846i \(-0.0166184\pi\)
\(660\) 0 0
\(661\) −4.79785 + 2.77004i −0.186615 + 0.107742i −0.590397 0.807113i \(-0.701029\pi\)
0.403782 + 0.914855i \(0.367695\pi\)
\(662\) 0 0
\(663\) 6.12360 1.04389i 0.237821 0.0405412i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −30.5422 + 52.9006i −1.18260 + 2.04832i
\(668\) 0 0
\(669\) −4.14800 + 11.1980i −0.160371 + 0.432940i
\(670\) 0 0
\(671\) 13.2997 0.513429
\(672\) 0 0
\(673\) −17.1946 −0.662804 −0.331402 0.943490i \(-0.607522\pi\)
−0.331402 + 0.943490i \(0.607522\pi\)
\(674\) 0 0
\(675\) 13.3158 22.1810i 0.512526 0.853748i
\(676\) 0 0
\(677\) 7.23319 12.5283i 0.277994 0.481500i −0.692892 0.721041i \(-0.743664\pi\)
0.970886 + 0.239541i \(0.0769971\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −3.90452 22.9045i −0.149621 0.877702i
\(682\) 0 0
\(683\) −28.1356 + 16.2441i −1.07658 + 0.621564i −0.929972 0.367630i \(-0.880169\pi\)
−0.146609 + 0.989195i \(0.546836\pi\)
\(684\) 0 0
\(685\) 2.91730i 0.111464i
\(686\) 0 0
\(687\) −7.03555 8.48214i −0.268423 0.323614i
\(688\) 0 0
\(689\) 5.29882 + 9.17783i 0.201869 + 0.349647i
\(690\) 0 0
\(691\) 28.0961 + 16.2213i 1.06883 + 0.617087i 0.927861 0.372927i \(-0.121646\pi\)
0.140966 + 0.990014i \(0.454979\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0.0148138 + 0.00855273i 0.000561918 + 0.000324423i
\(696\) 0 0
\(697\) −4.68421 8.11329i −0.177427 0.307313i
\(698\) 0 0
\(699\) 5.48142 + 6.60845i 0.207326 + 0.249955i
\(700\) 0 0
\(701\) 30.3777i 1.14735i −0.819084 0.573674i \(-0.805518\pi\)
0.819084 0.573674i \(-0.194482\pi\)
\(702\) 0 0
\(703\) −21.6198 + 12.4822i −0.815407 + 0.470776i
\(704\) 0 0
\(705\) 0.393576 + 2.30877i 0.0148229 + 0.0869535i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 16.2569 28.1578i 0.610542 1.05749i −0.380607 0.924737i \(-0.624285\pi\)
0.991149 0.132753i \(-0.0423816\pi\)
\(710\) 0 0
\(711\) 16.1951 5.68680i 0.607364 0.213272i
\(712\) 0 0
\(713\) 31.0295 1.16207
\(714\) 0 0
\(715\) −0.732778 −0.0274044
\(716\) 0 0
\(717\) 10.4539 28.2215i 0.390408 1.05395i
\(718\) 0 0
\(719\) −5.29867 + 9.17757i −0.197607 + 0.342266i −0.947752 0.319008i \(-0.896650\pi\)
0.750145 + 0.661273i \(0.229984\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 24.7679 4.22217i 0.921128 0.157024i
\(724\) 0 0
\(725\) 30.5618 17.6449i 1.13504 0.655314i
\(726\) 0 0
\(727\) 31.3600i 1.16308i −0.813518 0.581540i \(-0.802451\pi\)
0.813518 0.581540i \(-0.197549\pi\)
\(728\) 0 0
\(729\) −14.2898 + 22.9086i −0.529251 + 0.848466i
\(730\) 0 0
\(731\) 8.07874 + 13.9928i 0.298803 + 0.517542i
\(732\) 0 0
\(733\) 3.13184 + 1.80817i 0.115677 + 0.0667863i 0.556722 0.830699i \(-0.312059\pi\)
−0.441045 + 0.897485i \(0.645392\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −11.0117 6.35763i −0.405623 0.234186i
\(738\) 0 0
\(739\) 19.3463 + 33.5087i 0.711665 + 1.23264i 0.964232 + 0.265060i \(0.0853917\pi\)
−0.252567 + 0.967579i \(0.581275\pi\)
\(740\) 0 0
\(741\) −11.5745 + 9.60056i −0.425201 + 0.352685i
\(742\) 0 0
\(743\) 45.1194i 1.65527i −0.561266 0.827635i \(-0.689686\pi\)
0.561266 0.827635i \(-0.310314\pi\)
\(744\) 0 0
\(745\) 1.12740 0.650904i 0.0413047 0.0238473i
\(746\) 0 0
\(747\) −36.3755 31.2344i −1.33091 1.14281i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 11.7841 20.4107i 0.430009 0.744797i −0.566865 0.823811i \(-0.691844\pi\)
0.996874 + 0.0790136i \(0.0251771\pi\)
\(752\) 0 0
\(753\) 5.68118 + 2.10444i 0.207034 + 0.0766901i
\(754\) 0 0
\(755\) −0.398610 −0.0145069
\(756\) 0 0
\(757\) 26.2967 0.955770 0.477885 0.878422i \(-0.341404\pi\)
0.477885 + 0.878422i \(0.341404\pi\)
\(758\) 0 0
\(759\) −34.5708 12.8058i −1.25484 0.464822i
\(760\) 0 0
\(761\) 12.9780 22.4785i 0.470452 0.814846i −0.528977 0.848636i \(-0.677424\pi\)
0.999429 + 0.0337898i \(0.0107577\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0.581159 + 0.499022i 0.0210119 + 0.0180422i
\(766\) 0 0
\(767\) −19.8114 + 11.4381i −0.715348 + 0.413006i
\(768\) 0 0
\(769\) 36.9215i 1.33142i 0.746209 + 0.665712i \(0.231872\pi\)
−0.746209 + 0.665712i \(0.768128\pi\)
\(770\) 0 0
\(771\) −21.2444 + 17.6212i −0.765097 + 0.634614i
\(772\) 0 0
\(773\) 5.79284 + 10.0335i 0.208354 + 0.360879i 0.951196 0.308587i \(-0.0998561\pi\)
−0.742842 + 0.669466i \(0.766523\pi\)
\(774\) 0 0
\(775\) −15.5247 8.96322i −0.557665 0.321968i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 19.6408 + 11.3396i 0.703705 + 0.406284i
\(780\) 0 0
\(781\) −9.62376 16.6688i −0.344365 0.596458i
\(782\) 0 0
\(783\) −32.2050 + 17.8681i −1.15091 + 0.638555i
\(784\) 0 0
\(785\) 1.96861i 0.0702627i
\(786\) 0 0
\(787\) −25.9153 + 14.9622i −0.923779 + 0.533344i −0.884839 0.465897i \(-0.845732\pi\)
−0.0389406 + 0.999242i \(0.512398\pi\)
\(788\) 0 0
\(789\) −24.5149 + 4.17904i −0.872752 + 0.148778i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −5.49653 + 9.52027i −0.195188 + 0.338075i
\(794\) 0 0
\(795\) −0.453935 + 1.22545i −0.0160994 + 0.0434622i
\(796\) 0 0
\(797\) 20.2866 0.718587 0.359293 0.933225i \(-0.383018\pi\)
0.359293 + 0.933225i \(0.383018\pi\)
\(798\) 0 0
\(799\) 16.3453 0.578254
\(800\) 0 0
\(801\) −24.5836 + 8.63236i −0.868618 + 0.305009i
\(802\) 0 0
\(803\) 16.1148 27.9117i 0.568680 0.984982i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −2.14345 12.5738i −0.0754531 0.442619i
\(808\) 0 0
\(809\) 2.15641 1.24501i 0.0758155 0.0437721i −0.461613 0.887081i \(-0.652729\pi\)
0.537429 + 0.843309i \(0.319396\pi\)
\(810\) 0 0
\(811\) 15.9838i 0.561269i 0.959815 + 0.280634i \(0.0905448\pi\)
−0.959815 + 0.280634i \(0.909455\pi\)
\(812\) 0 0
\(813\) 13.8584 + 16.7078i 0.486036 + 0.585970i
\(814\) 0 0
\(815\) 0.294206 + 0.509581i 0.0103056 + 0.0178498i
\(816\) 0 0
\(817\) −33.8740 19.5572i −1.18510 0.684219i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −2.38960 1.37963i −0.0833975 0.0481495i 0.457721 0.889096i \(-0.348666\pi\)
−0.541119 + 0.840946i \(0.681999\pi\)
\(822\) 0 0
\(823\) 14.2212 + 24.6318i 0.495720 + 0.858612i 0.999988 0.00493523i \(-0.00157094\pi\)
−0.504268 + 0.863547i \(0.668238\pi\)
\(824\) 0 0
\(825\) 13.5974 + 16.3932i 0.473401 + 0.570738i
\(826\) 0 0
\(827\) 30.6070i 1.06431i 0.846647 + 0.532154i \(0.178617\pi\)
−0.846647 + 0.532154i \(0.821383\pi\)
\(828\) 0 0
\(829\) −8.71397 + 5.03101i −0.302649 + 0.174734i −0.643632 0.765335i \(-0.722573\pi\)
0.340983 + 0.940069i \(0.389240\pi\)
\(830\) 0 0
\(831\) 9.45424 + 55.4599i 0.327964 + 1.92388i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −0.269255 + 0.466363i −0.00931794 + 0.0161391i
\(836\) 0 0
\(837\) 16.0403 + 9.62939i 0.554434 + 0.332840i
\(838\) 0 0
\(839\) 13.3067 0.459400 0.229700 0.973262i \(-0.426226\pi\)
0.229700 + 0.973262i \(0.426226\pi\)
\(840\) 0 0
\(841\) −21.2383 −0.732354
\(842\) 0 0
\(843\) 6.12218 16.5275i 0.210859 0.569238i
\(844\) 0 0
\(845\) −0.641856 + 1.11173i −0.0220805 + 0.0382446i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −2.33957 + 0.398826i −0.0802940 + 0.0136877i
\(850\) 0 0
\(851\) 43.8098 25.2936i 1.50178 0.867053i
\(852\) 0 0
\(853\) 42.4736i 1.45427i 0.686495 + 0.727134i \(0.259148\pi\)
−0.686495 + 0.727134i \(0.740852\pi\)
\(854\) 0 0
\(855\) −1.82242 0.342755i −0.0623253 0.0117220i
\(856\) 0 0
\(857\) 4.77854 + 8.27667i 0.163232 + 0.282726i 0.936026 0.351931i \(-0.114475\pi\)
−0.772794 + 0.634657i \(0.781142\pi\)
\(858\) 0 0
\(859\) 1.82940 + 1.05620i 0.0624183 + 0.0360372i 0.530884 0.847444i \(-0.321860\pi\)
−0.468466 + 0.883481i \(0.655193\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 30.5684 + 17.6487i 1.04056 + 0.600768i 0.919992 0.391936i \(-0.128195\pi\)
0.120569 + 0.992705i \(0.461528\pi\)
\(864\) 0 0
\(865\) −1.63317 2.82873i −0.0555294 0.0961797i
\(866\) 0 0
\(867\) −18.5486 + 15.3853i −0.629945 + 0.522511i
\(868\) 0 0
\(869\) 14.1308i 0.479356i
\(870\) 0 0
\(871\) 9.10193 5.25500i 0.308407 0.178059i
\(872\) 0 0
\(873\) −13.0032 + 15.1434i −0.440091 + 0.512528i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 8.36886 14.4953i 0.282596 0.489471i −0.689427 0.724355i \(-0.742138\pi\)
0.972023 + 0.234884i \(0.0754709\pi\)
\(878\) 0 0
\(879\) −27.6051 10.2256i −0.931096 0.344900i
\(880\) 0 0
\(881\) −42.6152 −1.43574 −0.717871 0.696176i \(-0.754883\pi\)
−0.717871 + 0.696176i \(0.754883\pi\)
\(882\) 0 0
\(883\) −15.2392 −0.512839 −0.256419 0.966566i \(-0.582543\pi\)
−0.256419 + 0.966566i \(0.582543\pi\)
\(884\) 0 0
\(885\) −2.64527 0.979870i −0.0889198 0.0329380i
\(886\) 0 0
\(887\) 28.0633 48.6071i 0.942275 1.63207i 0.181157 0.983454i \(-0.442016\pi\)
0.761118 0.648613i \(-0.224651\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −13.8969 17.3482i −0.465564 0.581185i
\(892\) 0 0
\(893\) −34.2677 + 19.7845i −1.14673 + 0.662062i
\(894\) 0 0
\(895\) 0.534639i 0.0178710i
\(896\) 0 0
\(897\) 23.4543 19.4543i 0.783116 0.649560i
\(898\) 0 0
\(899\) 12.7600 + 22.1009i 0.425569 + 0.737107i
\(900\) 0 0
\(901\) 7.89836 + 4.56012i 0.263132 + 0.151920i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 1.00827 + 0.582126i 0.0335161 + 0.0193505i
\(906\) 0 0
\(907\) −20.3625 35.2688i −0.676124 1.17108i −0.976139 0.217147i \(-0.930325\pi\)
0.300015 0.953935i \(-0.403008\pi\)
\(908\) 0 0
\(909\) 8.94266 47.5478i 0.296609 1.57706i
\(910\) 0 0
\(911\) 31.0308i 1.02810i 0.857761 + 0.514049i \(0.171855\pi\)
−0.857761 + 0.514049i \(0.828145\pi\)
\(912\) 0 0
\(913\) 34.1833 19.7357i 1.13130 0.653157i
\(914\) 0 0
\(915\) −1.33630 + 0.227799i −0.0441769 + 0.00753081i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 25.6178 44.3713i 0.845053 1.46367i −0.0405222 0.999179i \(-0.512902\pi\)
0.885575 0.464496i \(-0.153765\pi\)
\(920\) 0 0
\(921\) 12.5757 33.9495i 0.414383 1.11868i
\(922\) 0 0
\(923\) 15.9093 0.523662
\(924\) 0 0
\(925\) −29.2253 −0.960922
\(926\) 0 0
\(927\) 0.169093 + 0.481550i 0.00555375 + 0.0158162i
\(928\) 0 0
\(929\) −1.87116 + 3.24094i −0.0613907 + 0.106332i −0.895087 0.445891i \(-0.852887\pi\)
0.833697 + 0.552223i \(0.186220\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 3.34164 + 19.6026i 0.109400 + 0.641759i
\(934\) 0 0
\(935\) −0.546135 + 0.315311i −0.0178605 + 0.0103118i
\(936\) 0 0
\(937\) 3.23951i 0.105830i 0.998599 + 0.0529150i \(0.0168512\pi\)
−0.998599 + 0.0529150i \(0.983149\pi\)
\(938\) 0 0
\(939\) 10.9447 + 13.1951i 0.357168 + 0.430606i
\(940\) 0 0
\(941\) −11.0342 19.1117i −0.359703 0.623024i 0.628208 0.778046i \(-0.283789\pi\)
−0.987911 + 0.155021i \(0.950455\pi\)
\(942\) 0 0
\(943\) −39.7996 22.9783i −1.29605 0.748276i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −31.4016 18.1297i −1.02042 0.589137i −0.106191 0.994346i \(-0.533866\pi\)
−0.914224 + 0.405208i \(0.867199\pi\)
\(948\) 0 0
\(949\) 13.3200 + 23.0709i 0.432384 + 0.748912i
\(950\) 0 0
\(951\) 7.33606 + 8.84443i 0.237888 + 0.286800i
\(952\) 0 0
\(953\) 13.4656i 0.436192i −0.975927 0.218096i \(-0.930015\pi\)
0.975927 0.218096i \(-0.0699846\pi\)
\(954\) 0 0
\(955\) −0.141171 + 0.0815052i −0.00456819 + 0.00263745i
\(956\) 0 0
\(957\) −5.09521 29.8892i −0.164705 0.966182i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −9.01821 + 15.6200i −0.290910 + 0.503871i
\(962\) 0 0
\(963\) −6.93094 19.7382i −0.223347 0.636055i
\(964\) 0 0
\(965\) 2.66964 0.0859388
\(966\) 0 0
\(967\) −9.25940 −0.297762 −0.148881 0.988855i \(-0.547567\pi\)
−0.148881 + 0.988855i \(0.547567\pi\)
\(968\) 0 0
\(969\) −4.49535 + 12.1357i −0.144411 + 0.389855i
\(970\) 0 0
\(971\) −19.9645 + 34.5795i −0.640691 + 1.10971i 0.344587 + 0.938754i \(0.388019\pi\)
−0.985279 + 0.170956i \(0.945314\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −17.3543 + 2.95838i −0.555781 + 0.0947439i
\(976\) 0 0
\(977\) 52.4299 30.2704i 1.67738 0.968436i 0.714060 0.700084i \(-0.246854\pi\)
0.963321 0.268352i \(-0.0864791\pi\)
\(978\) 0 0
\(979\) 21.4501i 0.685548i
\(980\) 0 0
\(981\) 0.751427 3.99531i 0.0239912 0.127560i
\(982\) 0 0
\(983\) 19.7027 + 34.1261i 0.628419 + 1.08845i 0.987869 + 0.155289i \(0.0496308\pi\)
−0.359450 + 0.933164i \(0.617036\pi\)
\(984\) 0 0
\(985\) 0.0373445 + 0.0215609i 0.00118990 + 0.000686987i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 68.6413 + 39.6301i 2.18267 + 1.26016i
\(990\) 0 0
\(991\) −15.0507 26.0686i −0.478102 0.828098i 0.521583 0.853201i \(-0.325342\pi\)
−0.999685 + 0.0251033i \(0.992009\pi\)
\(992\) 0 0
\(993\) 19.6381 16.2889i 0.623196 0.516914i
\(994\) 0 0
\(995\) 3.97527i 0.126025i
\(996\) 0 0
\(997\) 6.10467 3.52453i 0.193337 0.111623i −0.400207 0.916425i \(-0.631062\pi\)
0.593544 + 0.804802i \(0.297728\pi\)
\(998\) 0 0
\(999\) 30.4962 + 0.520289i 0.964858 + 0.0164612i
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 1176.2.u.b.1097.2 16
3.2 odd 2 inner 1176.2.u.b.1097.4 16
7.2 even 3 1176.2.k.a.881.13 16
7.3 odd 6 inner 1176.2.u.b.521.4 16
7.4 even 3 168.2.u.a.17.5 16
7.5 odd 6 1176.2.k.a.881.4 16
7.6 odd 2 168.2.u.a.89.7 yes 16
21.2 odd 6 1176.2.k.a.881.3 16
21.5 even 6 1176.2.k.a.881.14 16
21.11 odd 6 168.2.u.a.17.7 yes 16
21.17 even 6 inner 1176.2.u.b.521.2 16
21.20 even 2 168.2.u.a.89.5 yes 16
28.11 odd 6 336.2.bc.f.17.4 16
28.19 even 6 2352.2.k.i.881.13 16
28.23 odd 6 2352.2.k.i.881.4 16
28.27 even 2 336.2.bc.f.257.2 16
84.11 even 6 336.2.bc.f.17.2 16
84.23 even 6 2352.2.k.i.881.14 16
84.47 odd 6 2352.2.k.i.881.3 16
84.83 odd 2 336.2.bc.f.257.4 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.2.u.a.17.5 16 7.4 even 3
168.2.u.a.17.7 yes 16 21.11 odd 6
168.2.u.a.89.5 yes 16 21.20 even 2
168.2.u.a.89.7 yes 16 7.6 odd 2
336.2.bc.f.17.2 16 84.11 even 6
336.2.bc.f.17.4 16 28.11 odd 6
336.2.bc.f.257.2 16 28.27 even 2
336.2.bc.f.257.4 16 84.83 odd 2
1176.2.k.a.881.3 16 21.2 odd 6
1176.2.k.a.881.4 16 7.5 odd 6
1176.2.k.a.881.13 16 7.2 even 3
1176.2.k.a.881.14 16 21.5 even 6
1176.2.u.b.521.2 16 21.17 even 6 inner
1176.2.u.b.521.4 16 7.3 odd 6 inner
1176.2.u.b.1097.2 16 1.1 even 1 trivial
1176.2.u.b.1097.4 16 3.2 odd 2 inner
2352.2.k.i.881.3 16 84.47 odd 6
2352.2.k.i.881.4 16 28.23 odd 6
2352.2.k.i.881.13 16 28.19 even 6
2352.2.k.i.881.14 16 84.23 even 6