Properties

Label 1120.2.bq.b.719.15
Level $1120$
Weight $2$
Character 1120.719
Analytic conductor $8.943$
Analytic rank $0$
Dimension $80$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1120,2,Mod(719,1120)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1120, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 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("1120.719");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1120 = 2^{5} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1120.bq (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: \(8.94324502638\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(40\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 280)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 719.15
Character \(\chi\) \(=\) 1120.719
Dual form 1120.2.bq.b.1039.15

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.653375 - 1.13168i) q^{3} +(1.67181 + 1.48495i) q^{5} +(-1.03631 - 2.43435i) q^{7} +(0.646203 - 1.11926i) q^{9} +O(q^{10})\) \(q+(-0.653375 - 1.13168i) q^{3} +(1.67181 + 1.48495i) q^{5} +(-1.03631 - 2.43435i) q^{7} +(0.646203 - 1.11926i) q^{9} +(-0.168671 - 0.292146i) q^{11} +1.29865i q^{13} +(0.588166 - 2.86217i) q^{15} +(-3.41832 - 5.92071i) q^{17} +(0.551990 + 0.318691i) q^{19} +(-2.07781 + 2.76331i) q^{21} +(1.51602 - 2.62582i) q^{23} +(0.589869 + 4.96508i) q^{25} -5.60910 q^{27} +5.90687i q^{29} +(-1.08115 - 1.87261i) q^{31} +(-0.220410 + 0.381762i) q^{33} +(1.88238 - 5.60862i) q^{35} +(4.30421 - 7.45510i) q^{37} +(1.46966 - 0.848507i) q^{39} -5.28231i q^{41} -10.6010i q^{43} +(2.74236 - 0.911602i) q^{45} +(7.78384 + 4.49400i) q^{47} +(-4.85214 + 5.04547i) q^{49} +(-4.46689 + 7.73688i) q^{51} +(-3.63168 - 6.29026i) q^{53} +(0.151837 - 0.738878i) q^{55} -0.832900i q^{57} +(-7.50354 + 4.33217i) q^{59} +(3.97207 - 6.87982i) q^{61} +(-3.39433 - 0.413194i) q^{63} +(-1.92843 + 2.17110i) q^{65} +(-10.9447 + 6.31895i) q^{67} -3.96211 q^{69} -3.53507i q^{71} +(-5.93618 - 10.2818i) q^{73} +(5.23347 - 3.91160i) q^{75} +(-0.536392 + 0.713356i) q^{77} +(8.96466 + 5.17575i) q^{79} +(1.72624 + 2.98993i) q^{81} +3.76821 q^{83} +(3.07716 - 14.9743i) q^{85} +(6.68467 - 3.85940i) q^{87} +(-6.43915 - 3.71765i) q^{89} +(3.16138 - 1.34580i) q^{91} +(-1.41279 + 2.44703i) q^{93} +(0.449580 + 1.35247i) q^{95} +6.03558 q^{97} -0.435982 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 80 q - 52 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 80 q - 52 q^{9} + 48 q^{19} - 22 q^{25} + 22 q^{35} + 16 q^{49} - 20 q^{51} + 60 q^{59} - 12 q^{65} + 6 q^{75} - 36 q^{89} - 72 q^{91} - 104 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1120\mathbb{Z}\right)^\times\).

\(n\) \(351\) \(421\) \(801\) \(897\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{5}{6}\right)\) \(-1\)

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
\(3\) −0.653375 1.13168i −0.377226 0.653375i 0.613431 0.789748i \(-0.289789\pi\)
−0.990658 + 0.136373i \(0.956455\pi\)
\(4\) 0 0
\(5\) 1.67181 + 1.48495i 0.747654 + 0.664088i
\(6\) 0 0
\(7\) −1.03631 2.43435i −0.391687 0.920099i
\(8\) 0 0
\(9\) 0.646203 1.11926i 0.215401 0.373085i
\(10\) 0 0
\(11\) −0.168671 0.292146i −0.0508561 0.0880853i 0.839477 0.543396i \(-0.182862\pi\)
−0.890333 + 0.455310i \(0.849528\pi\)
\(12\) 0 0
\(13\) 1.29865i 0.360182i 0.983650 + 0.180091i \(0.0576391\pi\)
−0.983650 + 0.180091i \(0.942361\pi\)
\(14\) 0 0
\(15\) 0.588166 2.86217i 0.151864 0.739010i
\(16\) 0 0
\(17\) −3.41832 5.92071i −0.829065 1.43598i −0.898773 0.438415i \(-0.855540\pi\)
0.0697076 0.997567i \(-0.477793\pi\)
\(18\) 0 0
\(19\) 0.551990 + 0.318691i 0.126635 + 0.0731128i 0.561979 0.827151i \(-0.310040\pi\)
−0.435344 + 0.900264i \(0.643373\pi\)
\(20\) 0 0
\(21\) −2.07781 + 2.76331i −0.453415 + 0.603003i
\(22\) 0 0
\(23\) 1.51602 2.62582i 0.316111 0.547521i −0.663562 0.748121i \(-0.730956\pi\)
0.979673 + 0.200601i \(0.0642894\pi\)
\(24\) 0 0
\(25\) 0.589869 + 4.96508i 0.117974 + 0.993017i
\(26\) 0 0
\(27\) −5.60910 −1.07947
\(28\) 0 0
\(29\) 5.90687i 1.09688i 0.836191 + 0.548439i \(0.184778\pi\)
−0.836191 + 0.548439i \(0.815222\pi\)
\(30\) 0 0
\(31\) −1.08115 1.87261i −0.194181 0.336331i 0.752451 0.658648i \(-0.228871\pi\)
−0.946632 + 0.322318i \(0.895538\pi\)
\(32\) 0 0
\(33\) −0.220410 + 0.381762i −0.0383685 + 0.0664562i
\(34\) 0 0
\(35\) 1.88238 5.60862i 0.318180 0.948030i
\(36\) 0 0
\(37\) 4.30421 7.45510i 0.707607 1.22561i −0.258135 0.966109i \(-0.583108\pi\)
0.965742 0.259503i \(-0.0835587\pi\)
\(38\) 0 0
\(39\) 1.46966 0.848507i 0.235334 0.135870i
\(40\) 0 0
\(41\) 5.28231i 0.824959i −0.910967 0.412479i \(-0.864663\pi\)
0.910967 0.412479i \(-0.135337\pi\)
\(42\) 0 0
\(43\) 10.6010i 1.61664i −0.588744 0.808320i \(-0.700377\pi\)
0.588744 0.808320i \(-0.299623\pi\)
\(44\) 0 0
\(45\) 2.74236 0.911602i 0.408807 0.135894i
\(46\) 0 0
\(47\) 7.78384 + 4.49400i 1.13539 + 0.655517i 0.945285 0.326247i \(-0.105784\pi\)
0.190105 + 0.981764i \(0.439117\pi\)
\(48\) 0 0
\(49\) −4.85214 + 5.04547i −0.693163 + 0.720781i
\(50\) 0 0
\(51\) −4.46689 + 7.73688i −0.625490 + 1.08338i
\(52\) 0 0
\(53\) −3.63168 6.29026i −0.498850 0.864034i 0.501149 0.865361i \(-0.332911\pi\)
−0.999999 + 0.00132737i \(0.999577\pi\)
\(54\) 0 0
\(55\) 0.151837 0.738878i 0.0204737 0.0996303i
\(56\) 0 0
\(57\) 0.832900i 0.110320i
\(58\) 0 0
\(59\) −7.50354 + 4.33217i −0.976877 + 0.564000i −0.901326 0.433142i \(-0.857405\pi\)
−0.0755513 + 0.997142i \(0.524072\pi\)
\(60\) 0 0
\(61\) 3.97207 6.87982i 0.508571 0.880871i −0.491380 0.870945i \(-0.663507\pi\)
0.999951 0.00992531i \(-0.00315937\pi\)
\(62\) 0 0
\(63\) −3.39433 0.413194i −0.427645 0.0520575i
\(64\) 0 0
\(65\) −1.92843 + 2.17110i −0.239192 + 0.269291i
\(66\) 0 0
\(67\) −10.9447 + 6.31895i −1.33711 + 0.771983i −0.986378 0.164492i \(-0.947401\pi\)
−0.350735 + 0.936475i \(0.614068\pi\)
\(68\) 0 0
\(69\) −3.96211 −0.476981
\(70\) 0 0
\(71\) 3.53507i 0.419536i −0.977751 0.209768i \(-0.932729\pi\)
0.977751 0.209768i \(-0.0672708\pi\)
\(72\) 0 0
\(73\) −5.93618 10.2818i −0.694777 1.20339i −0.970256 0.242082i \(-0.922170\pi\)
0.275479 0.961307i \(-0.411164\pi\)
\(74\) 0 0
\(75\) 5.23347 3.91160i 0.604309 0.451673i
\(76\) 0 0
\(77\) −0.536392 + 0.713356i −0.0611275 + 0.0812945i
\(78\) 0 0
\(79\) 8.96466 + 5.17575i 1.00860 + 0.582317i 0.910782 0.412887i \(-0.135479\pi\)
0.0978208 + 0.995204i \(0.468813\pi\)
\(80\) 0 0
\(81\) 1.72624 + 2.98993i 0.191804 + 0.332214i
\(82\) 0 0
\(83\) 3.76821 0.413615 0.206807 0.978382i \(-0.433693\pi\)
0.206807 + 0.978382i \(0.433693\pi\)
\(84\) 0 0
\(85\) 3.07716 14.9743i 0.333765 1.62419i
\(86\) 0 0
\(87\) 6.68467 3.85940i 0.716672 0.413771i
\(88\) 0 0
\(89\) −6.43915 3.71765i −0.682549 0.394070i 0.118266 0.992982i \(-0.462266\pi\)
−0.800815 + 0.598912i \(0.795600\pi\)
\(90\) 0 0
\(91\) 3.16138 1.34580i 0.331403 0.141078i
\(92\) 0 0
\(93\) −1.41279 + 2.44703i −0.146500 + 0.253745i
\(94\) 0 0
\(95\) 0.449580 + 1.35247i 0.0461259 + 0.138760i
\(96\) 0 0
\(97\) 6.03558 0.612820 0.306410 0.951900i \(-0.400872\pi\)
0.306410 + 0.951900i \(0.400872\pi\)
\(98\) 0 0
\(99\) −0.435982 −0.0438178
\(100\) 0 0
\(101\) 0.757059 + 1.31127i 0.0753302 + 0.130476i 0.901230 0.433341i \(-0.142666\pi\)
−0.825900 + 0.563817i \(0.809332\pi\)
\(102\) 0 0
\(103\) −6.60049 3.81079i −0.650366 0.375489i 0.138231 0.990400i \(-0.455858\pi\)
−0.788596 + 0.614911i \(0.789192\pi\)
\(104\) 0 0
\(105\) −7.57706 + 1.53428i −0.739445 + 0.149731i
\(106\) 0 0
\(107\) −1.68270 0.971509i −0.162673 0.0939193i 0.416453 0.909157i \(-0.363273\pi\)
−0.579126 + 0.815238i \(0.696606\pi\)
\(108\) 0 0
\(109\) 1.32933 0.767488i 0.127326 0.0735120i −0.434984 0.900438i \(-0.643246\pi\)
0.562310 + 0.826926i \(0.309913\pi\)
\(110\) 0 0
\(111\) −11.2490 −1.06771
\(112\) 0 0
\(113\) 6.74556i 0.634569i 0.948330 + 0.317285i \(0.102771\pi\)
−0.948330 + 0.317285i \(0.897229\pi\)
\(114\) 0 0
\(115\) 6.43368 2.13865i 0.599944 0.199430i
\(116\) 0 0
\(117\) 1.45353 + 0.839193i 0.134378 + 0.0775834i
\(118\) 0 0
\(119\) −10.8707 + 14.4571i −0.996512 + 1.32528i
\(120\) 0 0
\(121\) 5.44310 9.42773i 0.494827 0.857066i
\(122\) 0 0
\(123\) −5.97788 + 3.45133i −0.539007 + 0.311196i
\(124\) 0 0
\(125\) −6.38674 + 9.17658i −0.571247 + 0.820778i
\(126\) 0 0
\(127\) −4.49296 −0.398686 −0.199343 0.979930i \(-0.563881\pi\)
−0.199343 + 0.979930i \(0.563881\pi\)
\(128\) 0 0
\(129\) −11.9969 + 6.92644i −1.05627 + 0.609839i
\(130\) 0 0
\(131\) 4.26159 + 2.46043i 0.372336 + 0.214969i 0.674479 0.738294i \(-0.264368\pi\)
−0.302142 + 0.953263i \(0.597702\pi\)
\(132\) 0 0
\(133\) 0.203777 1.67400i 0.0176697 0.145154i
\(134\) 0 0
\(135\) −9.37732 8.32921i −0.807072 0.716864i
\(136\) 0 0
\(137\) 14.2291 8.21516i 1.21567 0.701868i 0.251682 0.967810i \(-0.419016\pi\)
0.963989 + 0.265942i \(0.0856829\pi\)
\(138\) 0 0
\(139\) 9.85042i 0.835502i −0.908562 0.417751i \(-0.862818\pi\)
0.908562 0.417751i \(-0.137182\pi\)
\(140\) 0 0
\(141\) 11.7451i 0.989113i
\(142\) 0 0
\(143\) 0.379396 0.219044i 0.0317267 0.0183174i
\(144\) 0 0
\(145\) −8.77138 + 9.87513i −0.728423 + 0.820085i
\(146\) 0 0
\(147\) 8.88011 + 2.19448i 0.732419 + 0.180998i
\(148\) 0 0
\(149\) 12.5917 + 7.26981i 1.03155 + 0.595566i 0.917428 0.397901i \(-0.130261\pi\)
0.114122 + 0.993467i \(0.463595\pi\)
\(150\) 0 0
\(151\) 6.58039 3.79919i 0.535504 0.309174i −0.207751 0.978182i \(-0.566614\pi\)
0.743255 + 0.669008i \(0.233281\pi\)
\(152\) 0 0
\(153\) −8.83572 −0.714326
\(154\) 0 0
\(155\) 0.973249 4.73609i 0.0781732 0.380412i
\(156\) 0 0
\(157\) −15.6636 + 9.04339i −1.25009 + 0.721741i −0.971128 0.238561i \(-0.923324\pi\)
−0.278964 + 0.960302i \(0.589991\pi\)
\(158\) 0 0
\(159\) −4.74570 + 8.21979i −0.376358 + 0.651872i
\(160\) 0 0
\(161\) −7.96322 0.969367i −0.627589 0.0763968i
\(162\) 0 0
\(163\) 6.40416 + 3.69745i 0.501613 + 0.289606i 0.729379 0.684109i \(-0.239809\pi\)
−0.227766 + 0.973716i \(0.573142\pi\)
\(164\) 0 0
\(165\) −0.935378 + 0.310934i −0.0728191 + 0.0242062i
\(166\) 0 0
\(167\) 18.7215i 1.44872i 0.689424 + 0.724358i \(0.257864\pi\)
−0.689424 + 0.724358i \(0.742136\pi\)
\(168\) 0 0
\(169\) 11.3135 0.870269
\(170\) 0 0
\(171\) 0.713395 0.411879i 0.0545547 0.0314971i
\(172\) 0 0
\(173\) 8.41281 + 4.85714i 0.639614 + 0.369281i 0.784466 0.620172i \(-0.212937\pi\)
−0.144852 + 0.989453i \(0.546271\pi\)
\(174\) 0 0
\(175\) 11.4755 6.58129i 0.867465 0.497499i
\(176\) 0 0
\(177\) 9.80524 + 5.66106i 0.737007 + 0.425511i
\(178\) 0 0
\(179\) −1.19445 2.06885i −0.0892775 0.154633i 0.817928 0.575320i \(-0.195122\pi\)
−0.907206 + 0.420687i \(0.861789\pi\)
\(180\) 0 0
\(181\) −9.87158 −0.733749 −0.366874 0.930270i \(-0.619572\pi\)
−0.366874 + 0.930270i \(0.619572\pi\)
\(182\) 0 0
\(183\) −10.3810 −0.767385
\(184\) 0 0
\(185\) 18.2662 6.07197i 1.34296 0.446420i
\(186\) 0 0
\(187\) −1.15314 + 1.99730i −0.0843260 + 0.146057i
\(188\) 0 0
\(189\) 5.81274 + 13.6545i 0.422815 + 0.993220i
\(190\) 0 0
\(191\) 11.1524 + 6.43881i 0.806956 + 0.465896i 0.845898 0.533345i \(-0.179065\pi\)
−0.0389418 + 0.999241i \(0.512399\pi\)
\(192\) 0 0
\(193\) 4.28850 2.47597i 0.308693 0.178224i −0.337649 0.941272i \(-0.609632\pi\)
0.646341 + 0.763048i \(0.276298\pi\)
\(194\) 0 0
\(195\) 3.71697 + 0.763823i 0.266178 + 0.0546985i
\(196\) 0 0
\(197\) −6.56633 −0.467831 −0.233916 0.972257i \(-0.575154\pi\)
−0.233916 + 0.972257i \(0.575154\pi\)
\(198\) 0 0
\(199\) 13.8184 + 23.9342i 0.979560 + 1.69665i 0.663983 + 0.747748i \(0.268865\pi\)
0.315577 + 0.948900i \(0.397802\pi\)
\(200\) 0 0
\(201\) 14.3020 + 8.25729i 1.00879 + 0.582424i
\(202\) 0 0
\(203\) 14.3794 6.12132i 1.00924 0.429632i
\(204\) 0 0
\(205\) 7.84395 8.83100i 0.547845 0.616784i
\(206\) 0 0
\(207\) −1.95931 3.39362i −0.136181 0.235873i
\(208\) 0 0
\(209\) 0.215015i 0.0148729i
\(210\) 0 0
\(211\) 16.2713 1.12016 0.560080 0.828439i \(-0.310770\pi\)
0.560080 + 0.828439i \(0.310770\pi\)
\(212\) 0 0
\(213\) −4.00056 + 2.30973i −0.274114 + 0.158260i
\(214\) 0 0
\(215\) 15.7419 17.7228i 1.07359 1.20869i
\(216\) 0 0
\(217\) −3.43819 + 4.57250i −0.233399 + 0.310401i
\(218\) 0 0
\(219\) −7.75709 + 13.4357i −0.524176 + 0.907899i
\(220\) 0 0
\(221\) 7.68895 4.43921i 0.517214 0.298614i
\(222\) 0 0
\(223\) 1.78444i 0.119495i 0.998214 + 0.0597475i \(0.0190296\pi\)
−0.998214 + 0.0597475i \(0.980970\pi\)
\(224\) 0 0
\(225\) 5.93838 + 2.54824i 0.395892 + 0.169882i
\(226\) 0 0
\(227\) 10.5973 + 18.3551i 0.703368 + 1.21827i 0.967277 + 0.253722i \(0.0816547\pi\)
−0.263909 + 0.964548i \(0.585012\pi\)
\(228\) 0 0
\(229\) −4.07746 + 7.06236i −0.269446 + 0.466694i −0.968719 0.248161i \(-0.920174\pi\)
0.699273 + 0.714855i \(0.253507\pi\)
\(230\) 0 0
\(231\) 1.15775 + 0.140934i 0.0761746 + 0.00927279i
\(232\) 0 0
\(233\) 23.4709 + 13.5509i 1.53763 + 0.887750i 0.998977 + 0.0452221i \(0.0143995\pi\)
0.538652 + 0.842528i \(0.318934\pi\)
\(234\) 0 0
\(235\) 6.33971 + 19.0717i 0.413557 + 1.24410i
\(236\) 0 0
\(237\) 13.5268i 0.878661i
\(238\) 0 0
\(239\) 18.6789i 1.20824i −0.796894 0.604119i \(-0.793525\pi\)
0.796894 0.604119i \(-0.206475\pi\)
\(240\) 0 0
\(241\) −17.4676 + 10.0849i −1.12518 + 0.649625i −0.942719 0.333587i \(-0.891741\pi\)
−0.182465 + 0.983212i \(0.558408\pi\)
\(242\) 0 0
\(243\) −6.15789 + 10.6658i −0.395029 + 0.684210i
\(244\) 0 0
\(245\) −15.6041 + 1.22987i −0.996908 + 0.0785736i
\(246\) 0 0
\(247\) −0.413870 + 0.716843i −0.0263339 + 0.0456116i
\(248\) 0 0
\(249\) −2.46205 4.26440i −0.156026 0.270245i
\(250\) 0 0
\(251\) 21.5319i 1.35908i −0.733637 0.679542i \(-0.762179\pi\)
0.733637 0.679542i \(-0.237821\pi\)
\(252\) 0 0
\(253\) −1.02283 −0.0643047
\(254\) 0 0
\(255\) −18.9566 + 6.30147i −1.18711 + 0.394613i
\(256\) 0 0
\(257\) 4.89663 8.48121i 0.305443 0.529043i −0.671917 0.740627i \(-0.734529\pi\)
0.977360 + 0.211583i \(0.0678620\pi\)
\(258\) 0 0
\(259\) −22.6088 2.75219i −1.40484 0.171013i
\(260\) 0 0
\(261\) 6.61130 + 3.81703i 0.409229 + 0.236268i
\(262\) 0 0
\(263\) 8.74263 + 15.1427i 0.539094 + 0.933737i 0.998953 + 0.0457458i \(0.0145664\pi\)
−0.459860 + 0.887992i \(0.652100\pi\)
\(264\) 0 0
\(265\) 3.26923 15.9089i 0.200827 0.977279i
\(266\) 0 0
\(267\) 9.71606i 0.594613i
\(268\) 0 0
\(269\) 9.86300 + 17.0832i 0.601358 + 1.04158i 0.992616 + 0.121301i \(0.0387067\pi\)
−0.391258 + 0.920281i \(0.627960\pi\)
\(270\) 0 0
\(271\) 2.58340 4.47458i 0.156930 0.271811i −0.776830 0.629710i \(-0.783174\pi\)
0.933760 + 0.357899i \(0.116507\pi\)
\(272\) 0 0
\(273\) −3.58858 2.69835i −0.217191 0.163312i
\(274\) 0 0
\(275\) 1.35104 1.00979i 0.0814705 0.0608927i
\(276\) 0 0
\(277\) 7.49971 + 12.9899i 0.450614 + 0.780486i 0.998424 0.0561162i \(-0.0178717\pi\)
−0.547810 + 0.836603i \(0.684538\pi\)
\(278\) 0 0
\(279\) −2.79457 −0.167307
\(280\) 0 0
\(281\) 23.7107 1.41446 0.707232 0.706981i \(-0.249944\pi\)
0.707232 + 0.706981i \(0.249944\pi\)
\(282\) 0 0
\(283\) 5.91985 + 10.2535i 0.351899 + 0.609506i 0.986582 0.163266i \(-0.0522028\pi\)
−0.634683 + 0.772772i \(0.718869\pi\)
\(284\) 0 0
\(285\) 1.23681 1.39245i 0.0732624 0.0824814i
\(286\) 0 0
\(287\) −12.8590 + 5.47409i −0.759043 + 0.323125i
\(288\) 0 0
\(289\) −14.8699 + 25.7553i −0.874697 + 1.51502i
\(290\) 0 0
\(291\) −3.94349 6.83033i −0.231172 0.400401i
\(292\) 0 0
\(293\) 5.93168i 0.346533i 0.984875 + 0.173266i \(0.0554321\pi\)
−0.984875 + 0.173266i \(0.944568\pi\)
\(294\) 0 0
\(295\) −18.9775 3.89980i −1.10491 0.227055i
\(296\) 0 0
\(297\) 0.946090 + 1.63868i 0.0548977 + 0.0950856i
\(298\) 0 0
\(299\) 3.41002 + 1.96878i 0.197207 + 0.113857i
\(300\) 0 0
\(301\) −25.8066 + 10.9859i −1.48747 + 0.633216i
\(302\) 0 0
\(303\) 0.989287 1.71350i 0.0568331 0.0984377i
\(304\) 0 0
\(305\) 16.8567 5.60342i 0.965211 0.320851i
\(306\) 0 0
\(307\) −8.83855 −0.504443 −0.252221 0.967670i \(-0.581161\pi\)
−0.252221 + 0.967670i \(0.581161\pi\)
\(308\) 0 0
\(309\) 9.95951i 0.566577i
\(310\) 0 0
\(311\) −11.3386 19.6391i −0.642955 1.11363i −0.984770 0.173864i \(-0.944375\pi\)
0.341814 0.939768i \(-0.388959\pi\)
\(312\) 0 0
\(313\) −10.2270 + 17.7137i −0.578064 + 1.00124i 0.417637 + 0.908614i \(0.362858\pi\)
−0.995701 + 0.0926229i \(0.970475\pi\)
\(314\) 0 0
\(315\) −5.06109 5.73117i −0.285160 0.322915i
\(316\) 0 0
\(317\) −1.04378 + 1.80788i −0.0586245 + 0.101541i −0.893848 0.448370i \(-0.852005\pi\)
0.835224 + 0.549910i \(0.185338\pi\)
\(318\) 0 0
\(319\) 1.72567 0.996314i 0.0966188 0.0557829i
\(320\) 0 0
\(321\) 2.53904i 0.141715i
\(322\) 0 0
\(323\) 4.35756i 0.242461i
\(324\) 0 0
\(325\) −6.44792 + 0.766035i −0.357666 + 0.0424920i
\(326\) 0 0
\(327\) −1.73710 1.00291i −0.0960617 0.0554613i
\(328\) 0 0
\(329\) 2.87354 23.6058i 0.158424 1.30143i
\(330\) 0 0
\(331\) 6.62025 11.4666i 0.363882 0.630262i −0.624714 0.780854i \(-0.714784\pi\)
0.988596 + 0.150591i \(0.0481178\pi\)
\(332\) 0 0
\(333\) −5.56278 9.63502i −0.304839 0.527996i
\(334\) 0 0
\(335\) −27.6808 5.68830i −1.51236 0.310785i
\(336\) 0 0
\(337\) 13.0464i 0.710681i 0.934737 + 0.355341i \(0.115635\pi\)
−0.934737 + 0.355341i \(0.884365\pi\)
\(338\) 0 0
\(339\) 7.63381 4.40738i 0.414612 0.239376i
\(340\) 0 0
\(341\) −0.364717 + 0.631708i −0.0197505 + 0.0342089i
\(342\) 0 0
\(343\) 17.3107 + 6.58317i 0.934692 + 0.355458i
\(344\) 0 0
\(345\) −6.62387 5.88351i −0.356617 0.316758i
\(346\) 0 0
\(347\) 22.2014 12.8180i 1.19184 0.688107i 0.233113 0.972450i \(-0.425109\pi\)
0.958723 + 0.284343i \(0.0917754\pi\)
\(348\) 0 0
\(349\) −24.9774 −1.33701 −0.668505 0.743708i \(-0.733065\pi\)
−0.668505 + 0.743708i \(0.733065\pi\)
\(350\) 0 0
\(351\) 7.28427i 0.388806i
\(352\) 0 0
\(353\) 4.78385 + 8.28587i 0.254619 + 0.441012i 0.964792 0.263015i \(-0.0847168\pi\)
−0.710173 + 0.704027i \(0.751383\pi\)
\(354\) 0 0
\(355\) 5.24939 5.90995i 0.278609 0.313668i
\(356\) 0 0
\(357\) 23.4634 + 2.85621i 1.24181 + 0.151167i
\(358\) 0 0
\(359\) 17.0899 + 9.86683i 0.901968 + 0.520752i 0.877838 0.478957i \(-0.158985\pi\)
0.0241299 + 0.999709i \(0.492318\pi\)
\(360\) 0 0
\(361\) −9.29687 16.1027i −0.489309 0.847508i
\(362\) 0 0
\(363\) −14.2255 −0.746647
\(364\) 0 0
\(365\) 5.34373 26.0040i 0.279703 1.36111i
\(366\) 0 0
\(367\) 8.78783 5.07365i 0.458721 0.264843i −0.252785 0.967522i \(-0.581347\pi\)
0.711506 + 0.702680i \(0.248013\pi\)
\(368\) 0 0
\(369\) −5.91226 3.41345i −0.307780 0.177697i
\(370\) 0 0
\(371\) −11.5492 + 15.3594i −0.599603 + 0.797422i
\(372\) 0 0
\(373\) −9.28187 + 16.0767i −0.480597 + 0.832419i −0.999752 0.0222614i \(-0.992913\pi\)
0.519155 + 0.854680i \(0.326247\pi\)
\(374\) 0 0
\(375\) 14.5579 + 1.23199i 0.751765 + 0.0636195i
\(376\) 0 0
\(377\) −7.67097 −0.395075
\(378\) 0 0
\(379\) −23.9371 −1.22956 −0.614782 0.788697i \(-0.710756\pi\)
−0.614782 + 0.788697i \(0.710756\pi\)
\(380\) 0 0
\(381\) 2.93559 + 5.08459i 0.150395 + 0.260491i
\(382\) 0 0
\(383\) −19.1286 11.0439i −0.977428 0.564318i −0.0759355 0.997113i \(-0.524194\pi\)
−0.901493 + 0.432794i \(0.857528\pi\)
\(384\) 0 0
\(385\) −1.95604 + 0.396080i −0.0996889 + 0.0201861i
\(386\) 0 0
\(387\) −11.8653 6.85041i −0.603145 0.348226i
\(388\) 0 0
\(389\) 2.69778 1.55757i 0.136783 0.0789717i −0.430047 0.902807i \(-0.641503\pi\)
0.566830 + 0.823835i \(0.308170\pi\)
\(390\) 0 0
\(391\) −20.7289 −1.04831
\(392\) 0 0
\(393\) 6.43032i 0.324367i
\(394\) 0 0
\(395\) 7.30146 + 21.9649i 0.367376 + 1.10517i
\(396\) 0 0
\(397\) −21.1570 12.2150i −1.06184 0.613054i −0.135900 0.990723i \(-0.543393\pi\)
−0.925941 + 0.377668i \(0.876726\pi\)
\(398\) 0 0
\(399\) −2.02757 + 0.863139i −0.101506 + 0.0432110i
\(400\) 0 0
\(401\) −12.5658 + 21.7647i −0.627508 + 1.08688i 0.360542 + 0.932743i \(0.382592\pi\)
−0.988050 + 0.154133i \(0.950742\pi\)
\(402\) 0 0
\(403\) 2.43187 1.40404i 0.121140 0.0699402i
\(404\) 0 0
\(405\) −1.55395 + 7.56194i −0.0772165 + 0.375756i
\(406\) 0 0
\(407\) −2.90397 −0.143945
\(408\) 0 0
\(409\) 9.63575 5.56320i 0.476457 0.275083i −0.242482 0.970156i \(-0.577961\pi\)
0.718939 + 0.695073i \(0.244628\pi\)
\(410\) 0 0
\(411\) −18.5938 10.7352i −0.917166 0.529526i
\(412\) 0 0
\(413\) 18.3220 + 13.7768i 0.901566 + 0.677912i
\(414\) 0 0
\(415\) 6.29971 + 5.59559i 0.309241 + 0.274677i
\(416\) 0 0
\(417\) −11.1475 + 6.43602i −0.545896 + 0.315173i
\(418\) 0 0
\(419\) 4.05377i 0.198040i 0.995085 + 0.0990198i \(0.0315707\pi\)
−0.995085 + 0.0990198i \(0.968429\pi\)
\(420\) 0 0
\(421\) 22.4223i 1.09279i −0.837526 0.546397i \(-0.815999\pi\)
0.837526 0.546397i \(-0.184001\pi\)
\(422\) 0 0
\(423\) 10.0599 5.80807i 0.489128 0.282398i
\(424\) 0 0
\(425\) 27.3805 20.4647i 1.32815 0.992684i
\(426\) 0 0
\(427\) −20.8642 2.53981i −1.00969 0.122910i
\(428\) 0 0
\(429\) −0.495776 0.286236i −0.0239363 0.0138196i
\(430\) 0 0
\(431\) 14.0817 8.13005i 0.678289 0.391611i −0.120921 0.992662i \(-0.538585\pi\)
0.799210 + 0.601052i \(0.205251\pi\)
\(432\) 0 0
\(433\) 2.53783 0.121960 0.0609801 0.998139i \(-0.480577\pi\)
0.0609801 + 0.998139i \(0.480577\pi\)
\(434\) 0 0
\(435\) 16.9065 + 3.47422i 0.810603 + 0.166576i
\(436\) 0 0
\(437\) 1.67365 0.966283i 0.0800616 0.0462236i
\(438\) 0 0
\(439\) −11.3888 + 19.7261i −0.543560 + 0.941473i 0.455136 + 0.890422i \(0.349591\pi\)
−0.998696 + 0.0510514i \(0.983743\pi\)
\(440\) 0 0
\(441\) 2.51170 + 8.69118i 0.119605 + 0.413866i
\(442\) 0 0
\(443\) −1.64896 0.952029i −0.0783446 0.0452323i 0.460316 0.887755i \(-0.347736\pi\)
−0.538661 + 0.842523i \(0.681069\pi\)
\(444\) 0 0
\(445\) −5.24451 15.7770i −0.248613 0.747900i
\(446\) 0 0
\(447\) 18.9996i 0.898652i
\(448\) 0 0
\(449\) −22.8903 −1.08026 −0.540130 0.841582i \(-0.681625\pi\)
−0.540130 + 0.841582i \(0.681625\pi\)
\(450\) 0 0
\(451\) −1.54321 + 0.890970i −0.0726667 + 0.0419542i
\(452\) 0 0
\(453\) −8.59892 4.96459i −0.404012 0.233257i
\(454\) 0 0
\(455\) 7.28365 + 2.44456i 0.341463 + 0.114603i
\(456\) 0 0
\(457\) −32.0413 18.4990i −1.49883 0.865349i −0.498829 0.866701i \(-0.666236\pi\)
−0.999999 + 0.00135206i \(0.999570\pi\)
\(458\) 0 0
\(459\) 19.1737 + 33.2098i 0.894952 + 1.55010i
\(460\) 0 0
\(461\) 7.04683 0.328204 0.164102 0.986443i \(-0.447527\pi\)
0.164102 + 0.986443i \(0.447527\pi\)
\(462\) 0 0
\(463\) 36.8182 1.71109 0.855543 0.517731i \(-0.173223\pi\)
0.855543 + 0.517731i \(0.173223\pi\)
\(464\) 0 0
\(465\) −5.99563 + 1.99304i −0.278041 + 0.0924249i
\(466\) 0 0
\(467\) 13.6516 23.6452i 0.631719 1.09417i −0.355481 0.934683i \(-0.615683\pi\)
0.987200 0.159486i \(-0.0509837\pi\)
\(468\) 0 0
\(469\) 26.7247 + 20.0950i 1.23403 + 0.927901i
\(470\) 0 0
\(471\) 20.4684 + 11.8174i 0.943134 + 0.544519i
\(472\) 0 0
\(473\) −3.09704 + 1.78808i −0.142402 + 0.0822160i
\(474\) 0 0
\(475\) −1.25673 + 2.92866i −0.0576626 + 0.134376i
\(476\) 0 0
\(477\) −9.38722 −0.429811
\(478\) 0 0
\(479\) 8.50010 + 14.7226i 0.388379 + 0.672693i 0.992232 0.124403i \(-0.0397017\pi\)
−0.603852 + 0.797096i \(0.706368\pi\)
\(480\) 0 0
\(481\) 9.68159 + 5.58967i 0.441443 + 0.254867i
\(482\) 0 0
\(483\) 4.10595 + 9.64516i 0.186827 + 0.438870i
\(484\) 0 0
\(485\) 10.0903 + 8.96251i 0.458177 + 0.406966i
\(486\) 0 0
\(487\) −19.8165 34.3232i −0.897973 1.55533i −0.830082 0.557642i \(-0.811706\pi\)
−0.0678912 0.997693i \(-0.521627\pi\)
\(488\) 0 0
\(489\) 9.66327i 0.436988i
\(490\) 0 0
\(491\) 42.8252 1.93267 0.966337 0.257280i \(-0.0828263\pi\)
0.966337 + 0.257280i \(0.0828263\pi\)
\(492\) 0 0
\(493\) 34.9728 20.1916i 1.57510 0.909383i
\(494\) 0 0
\(495\) −0.728876 0.647409i −0.0327606 0.0290989i
\(496\) 0 0
\(497\) −8.60561 + 3.66342i −0.386014 + 0.164327i
\(498\) 0 0
\(499\) 0.363871 0.630243i 0.0162891 0.0282136i −0.857766 0.514040i \(-0.828148\pi\)
0.874055 + 0.485827i \(0.161481\pi\)
\(500\) 0 0
\(501\) 21.1868 12.2322i 0.946555 0.546494i
\(502\) 0 0
\(503\) 4.71017i 0.210016i −0.994471 0.105008i \(-0.966513\pi\)
0.994471 0.105008i \(-0.0334868\pi\)
\(504\) 0 0
\(505\) −0.681503 + 3.31637i −0.0303265 + 0.147577i
\(506\) 0 0
\(507\) −7.39196 12.8032i −0.328288 0.568612i
\(508\) 0 0
\(509\) 16.9005 29.2726i 0.749103 1.29749i −0.199150 0.979969i \(-0.563818\pi\)
0.948253 0.317516i \(-0.102849\pi\)
\(510\) 0 0
\(511\) −18.8777 + 25.1058i −0.835101 + 1.11061i
\(512\) 0 0
\(513\) −3.09617 1.78757i −0.136699 0.0789232i
\(514\) 0 0
\(515\) −5.37591 16.1723i −0.236891 0.712636i
\(516\) 0 0
\(517\) 3.03202i 0.133348i
\(518\) 0 0
\(519\) 12.6941i 0.557210i
\(520\) 0 0
\(521\) −1.19800 + 0.691667i −0.0524855 + 0.0303025i −0.526013 0.850476i \(-0.676314\pi\)
0.473528 + 0.880779i \(0.342980\pi\)
\(522\) 0 0
\(523\) −10.5809 + 18.3267i −0.462671 + 0.801370i −0.999093 0.0425801i \(-0.986442\pi\)
0.536422 + 0.843950i \(0.319776\pi\)
\(524\) 0 0
\(525\) −14.9457 8.68650i −0.652284 0.379110i
\(526\) 0 0
\(527\) −7.39145 + 12.8024i −0.321977 + 0.557680i
\(528\) 0 0
\(529\) 6.90339 + 11.9570i 0.300147 + 0.519871i
\(530\) 0 0
\(531\) 11.1978i 0.485945i
\(532\) 0 0
\(533\) 6.85989 0.297135
\(534\) 0 0
\(535\) −1.37051 4.12290i −0.0592525 0.178248i
\(536\) 0 0
\(537\) −1.56085 + 2.70347i −0.0673556 + 0.116663i
\(538\) 0 0
\(539\) 2.29243 + 0.566512i 0.0987418 + 0.0244014i
\(540\) 0 0
\(541\) −2.03605 1.17551i −0.0875366 0.0505393i 0.455593 0.890188i \(-0.349427\pi\)
−0.543129 + 0.839649i \(0.682761\pi\)
\(542\) 0 0
\(543\) 6.44984 + 11.1715i 0.276789 + 0.479413i
\(544\) 0 0
\(545\) 3.36206 + 0.690890i 0.144015 + 0.0295945i
\(546\) 0 0
\(547\) 37.9822i 1.62400i −0.583656 0.812001i \(-0.698378\pi\)
0.583656 0.812001i \(-0.301622\pi\)
\(548\) 0 0
\(549\) −5.13352 8.89152i −0.219093 0.379481i
\(550\) 0 0
\(551\) −1.88247 + 3.26053i −0.0801958 + 0.138903i
\(552\) 0 0
\(553\) 3.30947 27.1868i 0.140733 1.15610i
\(554\) 0 0
\(555\) −18.8062 16.7042i −0.798279 0.709055i
\(556\) 0 0
\(557\) 12.5827 + 21.7938i 0.533144 + 0.923433i 0.999251 + 0.0387043i \(0.0123230\pi\)
−0.466106 + 0.884729i \(0.654344\pi\)
\(558\) 0 0
\(559\) 13.7670 0.582284
\(560\) 0 0
\(561\) 3.01373 0.127240
\(562\) 0 0
\(563\) −22.7306 39.3705i −0.957980 1.65927i −0.727396 0.686218i \(-0.759269\pi\)
−0.230585 0.973052i \(-0.574064\pi\)
\(564\) 0 0
\(565\) −10.0168 + 11.2773i −0.421410 + 0.474438i
\(566\) 0 0
\(567\) 5.48963 7.30074i 0.230543 0.306602i
\(568\) 0 0
\(569\) −5.41670 + 9.38201i −0.227080 + 0.393314i −0.956941 0.290281i \(-0.906251\pi\)
0.729861 + 0.683595i \(0.239585\pi\)
\(570\) 0 0
\(571\) −0.714547 1.23763i −0.0299029 0.0517933i 0.850687 0.525673i \(-0.176186\pi\)
−0.880590 + 0.473880i \(0.842853\pi\)
\(572\) 0 0
\(573\) 16.8278i 0.702993i
\(574\) 0 0
\(575\) 13.9316 + 5.97826i 0.580990 + 0.249311i
\(576\) 0 0
\(577\) 7.32076 + 12.6799i 0.304767 + 0.527872i 0.977209 0.212277i \(-0.0680880\pi\)
−0.672442 + 0.740149i \(0.734755\pi\)
\(578\) 0 0
\(579\) −5.60399 3.23547i −0.232894 0.134461i
\(580\) 0 0
\(581\) −3.90502 9.17314i −0.162007 0.380566i
\(582\) 0 0
\(583\) −1.22512 + 2.12196i −0.0507391 + 0.0878827i
\(584\) 0 0
\(585\) 1.18385 + 3.56138i 0.0489464 + 0.147245i
\(586\) 0 0
\(587\) 20.7299 0.855616 0.427808 0.903870i \(-0.359286\pi\)
0.427808 + 0.903870i \(0.359286\pi\)
\(588\) 0 0
\(589\) 1.37821i 0.0567884i
\(590\) 0 0
\(591\) 4.29027 + 7.43097i 0.176478 + 0.305669i
\(592\) 0 0
\(593\) −0.830997 + 1.43933i −0.0341249 + 0.0591061i −0.882583 0.470156i \(-0.844198\pi\)
0.848459 + 0.529262i \(0.177531\pi\)
\(594\) 0 0
\(595\) −39.6416 + 8.02706i −1.62515 + 0.329077i
\(596\) 0 0
\(597\) 18.0572 31.2760i 0.739031 1.28004i
\(598\) 0 0
\(599\) −31.9015 + 18.4183i −1.30346 + 0.752553i −0.980996 0.194029i \(-0.937844\pi\)
−0.322464 + 0.946582i \(0.604511\pi\)
\(600\) 0 0
\(601\) 36.8443i 1.50291i −0.659785 0.751455i \(-0.729353\pi\)
0.659785 0.751455i \(-0.270647\pi\)
\(602\) 0 0
\(603\) 16.3333i 0.665143i
\(604\) 0 0
\(605\) 23.0995 7.67861i 0.939127 0.312180i
\(606\) 0 0
\(607\) 16.7562 + 9.67418i 0.680112 + 0.392663i 0.799897 0.600137i \(-0.204887\pi\)
−0.119785 + 0.992800i \(0.538221\pi\)
\(608\) 0 0
\(609\) −16.3225 12.2733i −0.661421 0.497340i
\(610\) 0 0
\(611\) −5.83615 + 10.1085i −0.236105 + 0.408946i
\(612\) 0 0
\(613\) −6.61363 11.4551i −0.267122 0.462669i 0.700995 0.713166i \(-0.252739\pi\)
−0.968117 + 0.250497i \(0.919406\pi\)
\(614\) 0 0
\(615\) −15.1189 3.10688i −0.609652 0.125281i
\(616\) 0 0
\(617\) 12.0831i 0.486448i −0.969970 0.243224i \(-0.921795\pi\)
0.969970 0.243224i \(-0.0782049\pi\)
\(618\) 0 0
\(619\) 10.4861 6.05415i 0.421472 0.243337i −0.274235 0.961663i \(-0.588425\pi\)
0.695707 + 0.718326i \(0.255091\pi\)
\(620\) 0 0
\(621\) −8.50348 + 14.7285i −0.341233 + 0.591033i
\(622\) 0 0
\(623\) −2.37713 + 19.5278i −0.0952376 + 0.782364i
\(624\) 0 0
\(625\) −24.3041 + 5.85749i −0.972164 + 0.234300i
\(626\) 0 0
\(627\) −0.243328 + 0.140486i −0.00971760 + 0.00561046i
\(628\) 0 0
\(629\) −58.8527 −2.34661
\(630\) 0 0
\(631\) 3.98549i 0.158660i −0.996848 0.0793299i \(-0.974722\pi\)
0.996848 0.0793299i \(-0.0252780\pi\)
\(632\) 0 0
\(633\) −10.6312 18.4138i −0.422554 0.731884i
\(634\) 0 0
\(635\) −7.51136 6.67181i −0.298079 0.264763i
\(636\) 0 0
\(637\) −6.55231 6.30125i −0.259612 0.249664i
\(638\) 0 0
\(639\) −3.95665 2.28437i −0.156523 0.0903684i
\(640\) 0 0
\(641\) −3.32962 5.76707i −0.131512 0.227786i 0.792748 0.609550i \(-0.208650\pi\)
−0.924260 + 0.381764i \(0.875317\pi\)
\(642\) 0 0
\(643\) −22.3506 −0.881421 −0.440711 0.897649i \(-0.645273\pi\)
−0.440711 + 0.897649i \(0.645273\pi\)
\(644\) 0 0
\(645\) −30.3419 6.23516i −1.19471 0.245509i
\(646\) 0 0
\(647\) 9.62380 5.55631i 0.378351 0.218441i −0.298750 0.954331i \(-0.596570\pi\)
0.677100 + 0.735891i \(0.263236\pi\)
\(648\) 0 0
\(649\) 2.53125 + 1.46142i 0.0993603 + 0.0573657i
\(650\) 0 0
\(651\) 7.42102 + 0.903366i 0.290853 + 0.0354057i
\(652\) 0 0
\(653\) −7.41053 + 12.8354i −0.289997 + 0.502289i −0.973809 0.227369i \(-0.926988\pi\)
0.683812 + 0.729658i \(0.260321\pi\)
\(654\) 0 0
\(655\) 3.47094 + 10.4416i 0.135621 + 0.407986i
\(656\) 0 0
\(657\) −15.3439 −0.598622
\(658\) 0 0
\(659\) 0.263237 0.0102542 0.00512712 0.999987i \(-0.498368\pi\)
0.00512712 + 0.999987i \(0.498368\pi\)
\(660\) 0 0
\(661\) −13.3229 23.0759i −0.518199 0.897547i −0.999776 0.0211434i \(-0.993269\pi\)
0.481577 0.876404i \(-0.340064\pi\)
\(662\) 0 0
\(663\) −10.0475 5.80094i −0.390214 0.225290i
\(664\) 0 0
\(665\) 2.82647 2.49600i 0.109606 0.0967909i
\(666\) 0 0
\(667\) 15.5103 + 8.95490i 0.600563 + 0.346735i
\(668\) 0 0
\(669\) 2.01941 1.16591i 0.0780750 0.0450766i
\(670\) 0 0
\(671\) −2.67988 −0.103456
\(672\) 0 0
\(673\) 36.5349i 1.40832i −0.710043 0.704158i \(-0.751324\pi\)
0.710043 0.704158i \(-0.248676\pi\)
\(674\) 0 0
\(675\) −3.30863 27.8496i −0.127349 1.07193i
\(676\) 0 0
\(677\) 17.0196 + 9.82627i 0.654116 + 0.377654i 0.790031 0.613066i \(-0.210064\pi\)
−0.135915 + 0.990720i \(0.543398\pi\)
\(678\) 0 0
\(679\) −6.25470 14.6927i −0.240033 0.563855i
\(680\) 0 0
\(681\) 13.8480 23.9855i 0.530658 0.919126i
\(682\) 0 0
\(683\) 5.62027 3.24486i 0.215054 0.124161i −0.388604 0.921405i \(-0.627043\pi\)
0.603658 + 0.797244i \(0.293709\pi\)
\(684\) 0 0
\(685\) 35.9873 + 7.39526i 1.37500 + 0.282558i
\(686\) 0 0
\(687\) 10.6564 0.406568
\(688\) 0 0
\(689\) 8.16886 4.71630i 0.311209 0.179677i
\(690\) 0 0
\(691\) 14.9183 + 8.61308i 0.567518 + 0.327657i 0.756158 0.654390i \(-0.227074\pi\)
−0.188639 + 0.982046i \(0.560408\pi\)
\(692\) 0 0
\(693\) 0.451810 + 1.06133i 0.0171629 + 0.0403167i
\(694\) 0 0
\(695\) 14.6274 16.4680i 0.554847 0.624667i
\(696\) 0 0
\(697\) −31.2750 + 18.0566i −1.18463 + 0.683944i
\(698\) 0 0
\(699\) 35.4153i 1.33953i
\(700\) 0 0
\(701\) 20.0480i 0.757201i 0.925560 + 0.378600i \(0.123595\pi\)
−0.925560 + 0.378600i \(0.876405\pi\)
\(702\) 0 0
\(703\) 4.75176 2.74343i 0.179216 0.103470i
\(704\) 0 0
\(705\) 17.4408 19.6355i 0.656858 0.739514i
\(706\) 0 0
\(707\) 2.40754 3.20182i 0.0905447 0.120417i
\(708\) 0 0
\(709\) 13.6342 + 7.87172i 0.512044 + 0.295629i 0.733673 0.679502i \(-0.237804\pi\)
−0.221630 + 0.975131i \(0.571138\pi\)
\(710\) 0 0
\(711\) 11.5860 6.68917i 0.434508 0.250863i
\(712\) 0 0
\(713\) −6.55617 −0.245531
\(714\) 0 0
\(715\) 0.959546 + 0.197183i 0.0358850 + 0.00737423i
\(716\) 0 0
\(717\) −21.1385 + 12.2043i −0.789432 + 0.455779i
\(718\) 0 0
\(719\) 11.4051 19.7541i 0.425337 0.736705i −0.571115 0.820870i \(-0.693489\pi\)
0.996452 + 0.0841652i \(0.0268223\pi\)
\(720\) 0 0
\(721\) −2.43669 + 20.0171i −0.0907471 + 0.745474i
\(722\) 0 0
\(723\) 22.8257 + 13.1784i 0.848898 + 0.490111i
\(724\) 0 0
\(725\) −29.3281 + 3.48427i −1.08922 + 0.129403i
\(726\) 0 0
\(727\) 43.1926i 1.60193i 0.598714 + 0.800963i \(0.295679\pi\)
−0.598714 + 0.800963i \(0.704321\pi\)
\(728\) 0 0
\(729\) 26.4511 0.979669
\(730\) 0 0
\(731\) −62.7655 + 36.2377i −2.32147 + 1.34030i
\(732\) 0 0
\(733\) −12.9526 7.47818i −0.478415 0.276213i 0.241341 0.970440i \(-0.422413\pi\)
−0.719756 + 0.694227i \(0.755746\pi\)
\(734\) 0 0
\(735\) 11.5871 + 16.8552i 0.427398 + 0.621715i
\(736\) 0 0
\(737\) 3.69211 + 2.13164i 0.136001 + 0.0785200i
\(738\) 0 0
\(739\) 1.37228 + 2.37687i 0.0504803 + 0.0874344i 0.890161 0.455645i \(-0.150591\pi\)
−0.839681 + 0.543080i \(0.817258\pi\)
\(740\) 0 0
\(741\) 1.08165 0.0397353
\(742\) 0 0
\(743\) −20.5678 −0.754559 −0.377279 0.926099i \(-0.623140\pi\)
−0.377279 + 0.926099i \(0.623140\pi\)
\(744\) 0 0
\(745\) 10.2556 + 30.8517i 0.375735 + 1.13032i
\(746\) 0 0
\(747\) 2.43503 4.21759i 0.0890930 0.154314i
\(748\) 0 0
\(749\) −0.621200 + 5.10307i −0.0226982 + 0.186462i
\(750\) 0 0
\(751\) −31.6827 18.2920i −1.15612 0.667486i −0.205748 0.978605i \(-0.565963\pi\)
−0.950371 + 0.311119i \(0.899296\pi\)
\(752\) 0 0
\(753\) −24.3672 + 14.0684i −0.887991 + 0.512682i
\(754\) 0 0
\(755\) 16.6427 + 3.42002i 0.605691 + 0.124467i
\(756\) 0 0
\(757\) 34.7609 1.26341 0.631704 0.775210i \(-0.282356\pi\)
0.631704 + 0.775210i \(0.282356\pi\)
\(758\) 0 0
\(759\) 0.668291 + 1.15751i 0.0242574 + 0.0420151i
\(760\) 0 0
\(761\) −7.09181 4.09446i −0.257078 0.148424i 0.365923 0.930645i \(-0.380753\pi\)
−0.623001 + 0.782221i \(0.714087\pi\)
\(762\) 0 0
\(763\) −3.24592 2.44070i −0.117510 0.0883593i
\(764\) 0 0
\(765\) −14.7716 13.1206i −0.534069 0.474375i
\(766\) 0 0
\(767\) −5.62598 9.74449i −0.203142 0.351853i
\(768\) 0 0
\(769\) 29.2850i 1.05604i 0.849231 + 0.528021i \(0.177066\pi\)
−0.849231 + 0.528021i \(0.822934\pi\)
\(770\) 0 0
\(771\) −12.7973 −0.460885
\(772\) 0 0
\(773\) 14.5234 8.38511i 0.522372 0.301591i −0.215533 0.976497i \(-0.569149\pi\)
0.737904 + 0.674905i \(0.235815\pi\)
\(774\) 0 0
\(775\) 8.65992 6.47260i 0.311074 0.232503i
\(776\) 0 0
\(777\) 11.6574 + 27.3841i 0.418209 + 0.982400i
\(778\) 0 0
\(779\) 1.68343 2.91578i 0.0603151 0.104469i
\(780\) 0 0
\(781\) −1.03276 + 0.596262i −0.0369549 + 0.0213359i
\(782\) 0 0
\(783\) 33.1322i 1.18405i
\(784\) 0 0
\(785\) −39.6154 8.14083i −1.41394 0.290559i
\(786\) 0 0
\(787\) −0.115323 0.199746i −0.00411083 0.00712017i 0.863963 0.503556i \(-0.167975\pi\)
−0.868074 + 0.496436i \(0.834642\pi\)
\(788\) 0 0
\(789\) 11.4244 19.7877i 0.406720 0.704460i
\(790\) 0 0
\(791\) 16.4211 6.99047i 0.583866 0.248552i
\(792\) 0 0
\(793\) 8.93450 + 5.15834i 0.317273 + 0.183178i
\(794\) 0 0
\(795\) −20.1398 + 6.69479i −0.714287 + 0.237440i
\(796\) 0 0
\(797\) 25.6496i 0.908554i 0.890860 + 0.454277i \(0.150102\pi\)
−0.890860 + 0.454277i \(0.849898\pi\)
\(798\) 0 0
\(799\) 61.4478i 2.17387i
\(800\) 0 0
\(801\) −8.32200 + 4.80471i −0.294043 + 0.169766i
\(802\) 0 0
\(803\) −2.00252 + 3.46846i −0.0706673 + 0.122399i
\(804\) 0 0
\(805\) −11.8735 13.4455i −0.418486 0.473893i
\(806\) 0 0
\(807\) 12.8885 22.3235i 0.453696 0.785824i
\(808\) 0 0
\(809\) 13.0378 + 22.5821i 0.458385 + 0.793946i 0.998876 0.0474041i \(-0.0150948\pi\)
−0.540491 + 0.841350i \(0.681762\pi\)
\(810\) 0 0
\(811\) 16.3763i 0.575051i 0.957773 + 0.287525i \(0.0928325\pi\)
−0.957773 + 0.287525i \(0.907167\pi\)
\(812\) 0 0
\(813\) −6.75171 −0.236793
\(814\) 0 0
\(815\) 5.21601 + 15.6913i 0.182709 + 0.549641i
\(816\) 0 0
\(817\) 3.37845 5.85165i 0.118197 0.204723i
\(818\) 0 0
\(819\) 0.536595 4.40805i 0.0187501 0.154030i
\(820\) 0 0
\(821\) 34.7768 + 20.0784i 1.21372 + 0.700740i 0.963567 0.267467i \(-0.0861866\pi\)
0.250150 + 0.968207i \(0.419520\pi\)
\(822\) 0 0
\(823\) −27.4665 47.5734i −0.957422 1.65830i −0.728725 0.684806i \(-0.759887\pi\)
−0.228697 0.973498i \(-0.573447\pi\)
\(824\) 0 0
\(825\) −2.02549 0.869166i −0.0705186 0.0302605i
\(826\) 0 0
\(827\) 36.3142i 1.26277i 0.775471 + 0.631384i \(0.217513\pi\)
−0.775471 + 0.631384i \(0.782487\pi\)
\(828\) 0 0
\(829\) −2.57148 4.45393i −0.0893111 0.154691i 0.817909 0.575348i \(-0.195133\pi\)
−0.907220 + 0.420656i \(0.861800\pi\)
\(830\) 0 0
\(831\) 9.80025 16.9745i 0.339967 0.588840i
\(832\) 0 0
\(833\) 46.4589 + 11.4811i 1.60971 + 0.397796i
\(834\) 0 0
\(835\) −27.8005 + 31.2988i −0.962075 + 1.08314i
\(836\) 0 0
\(837\) 6.06429 + 10.5037i 0.209612 + 0.363059i
\(838\) 0 0
\(839\) 34.7304 1.19903 0.599514 0.800365i \(-0.295361\pi\)
0.599514 + 0.800365i \(0.295361\pi\)
\(840\) 0 0
\(841\) −5.89106 −0.203140
\(842\) 0 0
\(843\) −15.4920 26.8329i −0.533573 0.924175i
\(844\) 0 0
\(845\) 18.9140 + 16.7999i 0.650661 + 0.577936i
\(846\) 0 0
\(847\) −28.5911 3.48041i −0.982403 0.119588i
\(848\) 0 0
\(849\) 7.73576 13.3987i 0.265491 0.459843i
\(850\) 0 0
\(851\) −13.0505 22.6041i −0.447365 0.774859i
\(852\) 0 0
\(853\) 1.93358i 0.0662045i −0.999452 0.0331023i \(-0.989461\pi\)
0.999452 0.0331023i \(-0.0105387\pi\)
\(854\) 0 0
\(855\) 1.80428 + 0.370772i 0.0617049 + 0.0126801i
\(856\) 0 0
\(857\) 6.25417 + 10.8325i 0.213638 + 0.370032i 0.952850 0.303440i \(-0.0981353\pi\)
−0.739212 + 0.673473i \(0.764802\pi\)
\(858\) 0 0
\(859\) −10.9495 6.32172i −0.373594 0.215694i 0.301434 0.953487i \(-0.402535\pi\)
−0.675027 + 0.737793i \(0.735868\pi\)
\(860\) 0 0
\(861\) 14.5967 + 10.9756i 0.497453 + 0.374048i
\(862\) 0 0
\(863\) −22.6283 + 39.1934i −0.770276 + 1.33416i 0.167135 + 0.985934i \(0.446548\pi\)
−0.937411 + 0.348224i \(0.886785\pi\)
\(864\) 0 0
\(865\) 6.85199 + 20.6128i 0.232975 + 0.700855i
\(866\) 0 0
\(867\) 38.8624 1.31983
\(868\) 0 0
\(869\) 3.49199i 0.118458i
\(870\) 0 0
\(871\) −8.20612 14.2134i −0.278054 0.481603i
\(872\) 0 0
\(873\) 3.90021 6.75536i 0.132002 0.228634i
\(874\) 0 0
\(875\) 28.9576 + 6.03783i 0.978947 + 0.204116i
\(876\) 0 0
\(877\) −21.2278 + 36.7676i −0.716810 + 1.24155i 0.245447 + 0.969410i \(0.421065\pi\)
−0.962257 + 0.272142i \(0.912268\pi\)
\(878\) 0 0
\(879\) 6.71276 3.87561i 0.226416 0.130721i
\(880\) 0 0
\(881\) 12.5917i 0.424225i −0.977245 0.212113i \(-0.931966\pi\)
0.977245 0.212113i \(-0.0680344\pi\)
\(882\) 0 0
\(883\) 9.44218i 0.317755i 0.987298 + 0.158877i \(0.0507874\pi\)
−0.987298 + 0.158877i \(0.949213\pi\)
\(884\) 0 0
\(885\) 7.98609 + 24.0245i 0.268450 + 0.807573i
\(886\) 0 0
\(887\) −6.38100 3.68407i −0.214253 0.123699i 0.389033 0.921224i \(-0.372809\pi\)
−0.603286 + 0.797525i \(0.706142\pi\)
\(888\) 0 0
\(889\) 4.65609 + 10.9375i 0.156160 + 0.366831i
\(890\) 0 0
\(891\) 0.582330 1.00863i 0.0195088 0.0337902i
\(892\) 0 0
\(893\) 2.86440 + 4.96128i 0.0958535 + 0.166023i
\(894\) 0 0
\(895\) 1.07524 5.23241i 0.0359414 0.174900i
\(896\) 0 0
\(897\) 5.14540i 0.171800i
\(898\) 0 0
\(899\) 11.0612 6.38622i 0.368913 0.212992i
\(900\) 0 0
\(901\) −24.8285 + 43.0043i −0.827158 + 1.43268i
\(902\) 0 0
\(903\) 29.2939 + 22.0269i 0.974839 + 0.733008i
\(904\) 0 0
\(905\) −16.5034 14.6588i −0.548590 0.487274i
\(906\) 0 0
\(907\) 29.0520 16.7732i 0.964657 0.556945i 0.0670537 0.997749i \(-0.478640\pi\)
0.897603 + 0.440804i \(0.145307\pi\)
\(908\) 0 0
\(909\) 1.95686 0.0649048
\(910\) 0 0
\(911\) 49.7262i 1.64750i 0.566951 + 0.823751i \(0.308123\pi\)
−0.566951 + 0.823751i \(0.691877\pi\)
\(912\) 0 0
\(913\) −0.635586 1.10087i −0.0210348 0.0364334i
\(914\) 0 0
\(915\) −17.3550 15.4152i −0.573739 0.509611i
\(916\) 0 0
\(917\) 1.57324 12.9240i 0.0519530 0.426787i
\(918\) 0 0
\(919\) −15.1509 8.74736i −0.499781 0.288549i 0.228842 0.973464i \(-0.426506\pi\)
−0.728623 + 0.684915i \(0.759839\pi\)
\(920\) 0 0
\(921\) 5.77489 + 10.0024i 0.190289 + 0.329590i
\(922\) 0 0
\(923\) 4.59083 0.151109
\(924\) 0 0
\(925\) 39.5541 + 16.9732i 1.30053 + 0.558076i
\(926\) 0 0
\(927\) −8.53051 + 4.92509i −0.280179 + 0.161761i
\(928\) 0 0
\(929\) −14.5419 8.39576i −0.477104 0.275456i 0.242105 0.970250i \(-0.422162\pi\)
−0.719209 + 0.694794i \(0.755496\pi\)
\(930\) 0 0
\(931\) −4.28628 + 1.23871i −0.140477 + 0.0405971i
\(932\) 0 0
\(933\) −14.8168 + 25.6634i −0.485079 + 0.840182i
\(934\) 0 0
\(935\) −4.89371 + 1.62674i −0.160041 + 0.0532002i
\(936\) 0 0
\(937\) 43.8764 1.43338 0.716690 0.697392i \(-0.245656\pi\)
0.716690 + 0.697392i \(0.245656\pi\)
\(938\) 0 0
\(939\) 26.7283 0.872244
\(940\) 0 0
\(941\) 1.66745 + 2.88811i 0.0543573 + 0.0941495i 0.891924 0.452186i \(-0.149356\pi\)
−0.837566 + 0.546336i \(0.816022\pi\)
\(942\) 0 0
\(943\) −13.8704 8.00807i −0.451682 0.260779i
\(944\) 0 0
\(945\) −10.5585 + 31.4593i −0.343467 + 1.02337i
\(946\) 0 0
\(947\) −13.9434 8.05021i −0.453099 0.261597i 0.256039 0.966666i \(-0.417582\pi\)
−0.709138 + 0.705070i \(0.750916\pi\)
\(948\) 0 0
\(949\) 13.3524 7.70903i 0.433438 0.250246i
\(950\) 0 0
\(951\) 2.72791 0.0884587
\(952\) 0 0
\(953\) 3.07172i 0.0995026i 0.998762 + 0.0497513i \(0.0158429\pi\)
−0.998762 + 0.0497513i \(0.984157\pi\)
\(954\) 0 0
\(955\) 9.08327 + 27.3251i 0.293928 + 0.884219i
\(956\) 0 0
\(957\) −2.25501 1.30193i −0.0728943 0.0420855i
\(958\) 0 0
\(959\) −34.7442 26.1251i −1.12195 0.843625i
\(960\) 0 0
\(961\) 13.1622 22.7976i 0.424588 0.735408i
\(962\) 0 0
\(963\) −2.17473 + 1.25558i −0.0700798 + 0.0404606i
\(964\) 0 0
\(965\) 10.8462 + 2.22886i 0.349152 + 0.0717494i
\(966\) 0 0
\(967\) −19.1674 −0.616383 −0.308192 0.951324i \(-0.599724\pi\)
−0.308192 + 0.951324i \(0.599724\pi\)
\(968\) 0 0
\(969\) −4.93136 + 2.84712i −0.158418 + 0.0914627i
\(970\) 0 0
\(971\) 36.7871 + 21.2390i 1.18055 + 0.681593i 0.956143 0.292901i \(-0.0946206\pi\)
0.224412 + 0.974494i \(0.427954\pi\)
\(972\) 0 0
\(973\) −23.9794 + 10.2081i −0.768744 + 0.327255i
\(974\) 0 0
\(975\) 5.07981 + 6.79646i 0.162684 + 0.217661i
\(976\) 0 0
\(977\) 4.83162 2.78954i 0.154577 0.0892452i −0.420716 0.907192i \(-0.638221\pi\)
0.575293 + 0.817947i \(0.304888\pi\)
\(978\) 0 0
\(979\) 2.50823i 0.0801634i
\(980\) 0 0
\(981\) 1.98381i 0.0633382i
\(982\) 0 0
\(983\) 2.32060 1.33980i 0.0740156 0.0427329i −0.462535 0.886601i \(-0.653060\pi\)
0.536551 + 0.843868i \(0.319727\pi\)
\(984\) 0 0
\(985\) −10.9776 9.75064i −0.349776 0.310681i
\(986\) 0 0
\(987\) −28.5916 + 12.1715i −0.910081 + 0.387422i
\(988\) 0 0
\(989\) −27.8363 16.0713i −0.885143 0.511038i
\(990\) 0 0
\(991\) −0.289819 + 0.167327i −0.00920640 + 0.00531532i −0.504596 0.863356i \(-0.668359\pi\)
0.495390 + 0.868671i \(0.335025\pi\)
\(992\) 0 0
\(993\) −17.3020 −0.549063
\(994\) 0 0
\(995\) −12.4393 + 60.5328i −0.394352 + 1.91902i
\(996\) 0 0
\(997\) 26.3710 15.2253i 0.835179 0.482191i −0.0204437 0.999791i \(-0.506508\pi\)
0.855623 + 0.517600i \(0.173175\pi\)
\(998\) 0 0
\(999\) −24.1427 + 41.8164i −0.763842 + 1.32301i
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 1120.2.bq.b.719.15 80
4.3 odd 2 280.2.ba.b.19.1 80
5.4 even 2 inner 1120.2.bq.b.719.25 80
7.3 odd 6 inner 1120.2.bq.b.1039.26 80
8.3 odd 2 inner 1120.2.bq.b.719.16 80
8.5 even 2 280.2.ba.b.19.14 yes 80
20.19 odd 2 280.2.ba.b.19.40 yes 80
28.3 even 6 280.2.ba.b.59.27 yes 80
35.24 odd 6 inner 1120.2.bq.b.1039.16 80
40.19 odd 2 inner 1120.2.bq.b.719.26 80
40.29 even 2 280.2.ba.b.19.27 yes 80
56.3 even 6 inner 1120.2.bq.b.1039.25 80
56.45 odd 6 280.2.ba.b.59.40 yes 80
140.59 even 6 280.2.ba.b.59.14 yes 80
280.59 even 6 inner 1120.2.bq.b.1039.15 80
280.269 odd 6 280.2.ba.b.59.1 yes 80
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
280.2.ba.b.19.1 80 4.3 odd 2
280.2.ba.b.19.14 yes 80 8.5 even 2
280.2.ba.b.19.27 yes 80 40.29 even 2
280.2.ba.b.19.40 yes 80 20.19 odd 2
280.2.ba.b.59.1 yes 80 280.269 odd 6
280.2.ba.b.59.14 yes 80 140.59 even 6
280.2.ba.b.59.27 yes 80 28.3 even 6
280.2.ba.b.59.40 yes 80 56.45 odd 6
1120.2.bq.b.719.15 80 1.1 even 1 trivial
1120.2.bq.b.719.16 80 8.3 odd 2 inner
1120.2.bq.b.719.25 80 5.4 even 2 inner
1120.2.bq.b.719.26 80 40.19 odd 2 inner
1120.2.bq.b.1039.15 80 280.59 even 6 inner
1120.2.bq.b.1039.16 80 35.24 odd 6 inner
1120.2.bq.b.1039.25 80 56.3 even 6 inner
1120.2.bq.b.1039.26 80 7.3 odd 6 inner