Properties

Label 1520.2.q.h.961.1
Level $1520$
Weight $2$
Character 1520.961
Analytic conductor $12.137$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1520,2,Mod(881,1520)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1520, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1520.881");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1520.q (of order \(3\), 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: \(12.1372611072\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{17})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 5x^{2} + 4x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 190)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 961.1
Root \(1.28078 - 2.21837i\) of defining polynomial
Character \(\chi\) \(=\) 1520.961
Dual form 1520.2.q.h.881.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.28078 + 2.21837i) q^{3} +(-0.500000 + 0.866025i) q^{5} -0.438447 q^{7} +(-1.78078 - 3.08440i) q^{9} +O(q^{10})\) \(q+(-1.28078 + 2.21837i) q^{3} +(-0.500000 + 0.866025i) q^{5} -0.438447 q^{7} +(-1.78078 - 3.08440i) q^{9} -1.00000 q^{11} +(1.00000 + 1.73205i) q^{13} +(-1.28078 - 2.21837i) q^{15} +(-2.56155 + 4.43674i) q^{17} +(2.50000 + 3.57071i) q^{19} +(0.561553 - 0.972638i) q^{21} +(-2.34233 - 4.05703i) q^{23} +(-0.500000 - 0.866025i) q^{25} +1.43845 q^{27} +(-1.00000 - 1.73205i) q^{29} -10.2462 q^{31} +(1.28078 - 2.21837i) q^{33} +(0.219224 - 0.379706i) q^{35} -4.68466 q^{37} -5.12311 q^{39} +(3.06155 - 5.30277i) q^{41} +(1.56155 - 2.70469i) q^{43} +3.56155 q^{45} +(1.43845 + 2.49146i) q^{47} -6.80776 q^{49} +(-6.56155 - 11.3649i) q^{51} +(3.78078 + 6.54850i) q^{53} +(0.500000 - 0.866025i) q^{55} +(-11.1231 + 0.972638i) q^{57} +(7.28078 - 12.6107i) q^{59} +(-2.56155 - 4.43674i) q^{61} +(0.780776 + 1.35234i) q^{63} -2.00000 q^{65} +(4.71922 + 8.17394i) q^{67} +12.0000 q^{69} +(8.12311 - 14.0696i) q^{71} +(0.842329 - 1.45896i) q^{73} +2.56155 q^{75} +0.438447 q^{77} +(-5.56155 + 9.63289i) q^{79} +(3.50000 - 6.06218i) q^{81} +10.8078 q^{83} +(-2.56155 - 4.43674i) q^{85} +5.12311 q^{87} +(1.34233 + 2.32498i) q^{89} +(-0.438447 - 0.759413i) q^{91} +(13.1231 - 22.7299i) q^{93} +(-4.34233 + 0.379706i) q^{95} +(-0.842329 + 1.45896i) q^{97} +(1.78078 + 3.08440i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{3} - 2 q^{5} - 10 q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{3} - 2 q^{5} - 10 q^{7} - 3 q^{9} - 4 q^{11} + 4 q^{13} - q^{15} - 2 q^{17} + 10 q^{19} - 6 q^{21} + 3 q^{23} - 2 q^{25} + 14 q^{27} - 4 q^{29} - 8 q^{31} + q^{33} + 5 q^{35} + 6 q^{37} - 4 q^{39} + 4 q^{41} - 2 q^{43} + 6 q^{45} + 14 q^{47} + 14 q^{49} - 18 q^{51} + 11 q^{53} + 2 q^{55} - 28 q^{57} + 25 q^{59} - 2 q^{61} - q^{63} - 8 q^{65} + 23 q^{67} + 48 q^{69} + 16 q^{71} - 9 q^{73} + 2 q^{75} + 10 q^{77} - 14 q^{79} + 14 q^{81} + 2 q^{83} - 2 q^{85} + 4 q^{87} - 7 q^{89} - 10 q^{91} + 36 q^{93} - 5 q^{95} + 9 q^{97} + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1520\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(401\) \(1141\) \(1217\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.28078 + 2.21837i −0.739457 + 1.28078i 0.213284 + 0.976990i \(0.431584\pi\)
−0.952740 + 0.303786i \(0.901749\pi\)
\(4\) 0 0
\(5\) −0.500000 + 0.866025i −0.223607 + 0.387298i
\(6\) 0 0
\(7\) −0.438447 −0.165717 −0.0828587 0.996561i \(-0.526405\pi\)
−0.0828587 + 0.996561i \(0.526405\pi\)
\(8\) 0 0
\(9\) −1.78078 3.08440i −0.593592 1.02813i
\(10\) 0 0
\(11\) −1.00000 −0.301511 −0.150756 0.988571i \(-0.548171\pi\)
−0.150756 + 0.988571i \(0.548171\pi\)
\(12\) 0 0
\(13\) 1.00000 + 1.73205i 0.277350 + 0.480384i 0.970725 0.240192i \(-0.0772105\pi\)
−0.693375 + 0.720577i \(0.743877\pi\)
\(14\) 0 0
\(15\) −1.28078 2.21837i −0.330695 0.572781i
\(16\) 0 0
\(17\) −2.56155 + 4.43674i −0.621268 + 1.07607i 0.367982 + 0.929833i \(0.380049\pi\)
−0.989250 + 0.146235i \(0.953285\pi\)
\(18\) 0 0
\(19\) 2.50000 + 3.57071i 0.573539 + 0.819178i
\(20\) 0 0
\(21\) 0.561553 0.972638i 0.122541 0.212247i
\(22\) 0 0
\(23\) −2.34233 4.05703i −0.488409 0.845950i 0.511502 0.859282i \(-0.329089\pi\)
−0.999911 + 0.0133324i \(0.995756\pi\)
\(24\) 0 0
\(25\) −0.500000 0.866025i −0.100000 0.173205i
\(26\) 0 0
\(27\) 1.43845 0.276829
\(28\) 0 0
\(29\) −1.00000 1.73205i −0.185695 0.321634i 0.758115 0.652121i \(-0.226120\pi\)
−0.943811 + 0.330487i \(0.892787\pi\)
\(30\) 0 0
\(31\) −10.2462 −1.84027 −0.920137 0.391597i \(-0.871923\pi\)
−0.920137 + 0.391597i \(0.871923\pi\)
\(32\) 0 0
\(33\) 1.28078 2.21837i 0.222955 0.386169i
\(34\) 0 0
\(35\) 0.219224 0.379706i 0.0370556 0.0641821i
\(36\) 0 0
\(37\) −4.68466 −0.770153 −0.385077 0.922885i \(-0.625825\pi\)
−0.385077 + 0.922885i \(0.625825\pi\)
\(38\) 0 0
\(39\) −5.12311 −0.820353
\(40\) 0 0
\(41\) 3.06155 5.30277i 0.478134 0.828153i −0.521552 0.853220i \(-0.674647\pi\)
0.999686 + 0.0250670i \(0.00797991\pi\)
\(42\) 0 0
\(43\) 1.56155 2.70469i 0.238135 0.412461i −0.722044 0.691847i \(-0.756797\pi\)
0.960179 + 0.279385i \(0.0901307\pi\)
\(44\) 0 0
\(45\) 3.56155 0.530925
\(46\) 0 0
\(47\) 1.43845 + 2.49146i 0.209819 + 0.363417i 0.951657 0.307161i \(-0.0993792\pi\)
−0.741838 + 0.670579i \(0.766046\pi\)
\(48\) 0 0
\(49\) −6.80776 −0.972538
\(50\) 0 0
\(51\) −6.56155 11.3649i −0.918801 1.59141i
\(52\) 0 0
\(53\) 3.78078 + 6.54850i 0.519330 + 0.899505i 0.999748 + 0.0224656i \(0.00715162\pi\)
−0.480418 + 0.877040i \(0.659515\pi\)
\(54\) 0 0
\(55\) 0.500000 0.866025i 0.0674200 0.116775i
\(56\) 0 0
\(57\) −11.1231 + 0.972638i −1.47329 + 0.128829i
\(58\) 0 0
\(59\) 7.28078 12.6107i 0.947876 1.64177i 0.197989 0.980204i \(-0.436559\pi\)
0.749887 0.661566i \(-0.230108\pi\)
\(60\) 0 0
\(61\) −2.56155 4.43674i −0.327973 0.568066i 0.654136 0.756377i \(-0.273032\pi\)
−0.982110 + 0.188310i \(0.939699\pi\)
\(62\) 0 0
\(63\) 0.780776 + 1.35234i 0.0983686 + 0.170379i
\(64\) 0 0
\(65\) −2.00000 −0.248069
\(66\) 0 0
\(67\) 4.71922 + 8.17394i 0.576545 + 0.998605i 0.995872 + 0.0907698i \(0.0289328\pi\)
−0.419327 + 0.907835i \(0.637734\pi\)
\(68\) 0 0
\(69\) 12.0000 1.44463
\(70\) 0 0
\(71\) 8.12311 14.0696i 0.964035 1.66976i 0.251850 0.967766i \(-0.418961\pi\)
0.712185 0.701992i \(-0.247706\pi\)
\(72\) 0 0
\(73\) 0.842329 1.45896i 0.0985872 0.170758i −0.812513 0.582943i \(-0.801901\pi\)
0.911100 + 0.412185i \(0.135234\pi\)
\(74\) 0 0
\(75\) 2.56155 0.295783
\(76\) 0 0
\(77\) 0.438447 0.0499657
\(78\) 0 0
\(79\) −5.56155 + 9.63289i −0.625724 + 1.08379i 0.362677 + 0.931915i \(0.381863\pi\)
−0.988400 + 0.151870i \(0.951470\pi\)
\(80\) 0 0
\(81\) 3.50000 6.06218i 0.388889 0.673575i
\(82\) 0 0
\(83\) 10.8078 1.18631 0.593153 0.805090i \(-0.297883\pi\)
0.593153 + 0.805090i \(0.297883\pi\)
\(84\) 0 0
\(85\) −2.56155 4.43674i −0.277839 0.481232i
\(86\) 0 0
\(87\) 5.12311 0.549255
\(88\) 0 0
\(89\) 1.34233 + 2.32498i 0.142287 + 0.246448i 0.928357 0.371689i \(-0.121221\pi\)
−0.786071 + 0.618137i \(0.787888\pi\)
\(90\) 0 0
\(91\) −0.438447 0.759413i −0.0459618 0.0796081i
\(92\) 0 0
\(93\) 13.1231 22.7299i 1.36080 2.35698i
\(94\) 0 0
\(95\) −4.34233 + 0.379706i −0.445514 + 0.0389571i
\(96\) 0 0
\(97\) −0.842329 + 1.45896i −0.0855256 + 0.148135i −0.905615 0.424101i \(-0.860590\pi\)
0.820089 + 0.572235i \(0.193924\pi\)
\(98\) 0 0
\(99\) 1.78078 + 3.08440i 0.178975 + 0.309993i
\(100\) 0 0
\(101\) −5.00000 8.66025i −0.497519 0.861727i 0.502477 0.864590i \(-0.332422\pi\)
−0.999996 + 0.00286291i \(0.999089\pi\)
\(102\) 0 0
\(103\) 5.80776 0.572256 0.286128 0.958191i \(-0.407632\pi\)
0.286128 + 0.958191i \(0.407632\pi\)
\(104\) 0 0
\(105\) 0.561553 + 0.972638i 0.0548019 + 0.0949197i
\(106\) 0 0
\(107\) −2.24621 −0.217149 −0.108575 0.994088i \(-0.534629\pi\)
−0.108575 + 0.994088i \(0.534629\pi\)
\(108\) 0 0
\(109\) 7.12311 12.3376i 0.682270 1.18173i −0.292017 0.956413i \(-0.594326\pi\)
0.974286 0.225313i \(-0.0723404\pi\)
\(110\) 0 0
\(111\) 6.00000 10.3923i 0.569495 0.986394i
\(112\) 0 0
\(113\) −16.8078 −1.58114 −0.790571 0.612371i \(-0.790216\pi\)
−0.790571 + 0.612371i \(0.790216\pi\)
\(114\) 0 0
\(115\) 4.68466 0.436847
\(116\) 0 0
\(117\) 3.56155 6.16879i 0.329266 0.570305i
\(118\) 0 0
\(119\) 1.12311 1.94528i 0.102955 0.178323i
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) 0 0
\(123\) 7.84233 + 13.5833i 0.707119 + 1.22477i
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −7.78078 13.4767i −0.690432 1.19586i −0.971696 0.236233i \(-0.924087\pi\)
0.281264 0.959630i \(-0.409246\pi\)
\(128\) 0 0
\(129\) 4.00000 + 6.92820i 0.352180 + 0.609994i
\(130\) 0 0
\(131\) −8.06155 + 13.9630i −0.704341 + 1.21995i 0.262588 + 0.964908i \(0.415424\pi\)
−0.966929 + 0.255046i \(0.917909\pi\)
\(132\) 0 0
\(133\) −1.09612 1.56557i −0.0950455 0.135752i
\(134\) 0 0
\(135\) −0.719224 + 1.24573i −0.0619009 + 0.107216i
\(136\) 0 0
\(137\) 2.71922 + 4.70983i 0.232319 + 0.402388i 0.958490 0.285126i \(-0.0920354\pi\)
−0.726171 + 0.687514i \(0.758702\pi\)
\(138\) 0 0
\(139\) 4.40388 + 7.62775i 0.373532 + 0.646977i 0.990106 0.140320i \(-0.0448131\pi\)
−0.616574 + 0.787297i \(0.711480\pi\)
\(140\) 0 0
\(141\) −7.36932 −0.620608
\(142\) 0 0
\(143\) −1.00000 1.73205i −0.0836242 0.144841i
\(144\) 0 0
\(145\) 2.00000 0.166091
\(146\) 0 0
\(147\) 8.71922 15.1021i 0.719149 1.24560i
\(148\) 0 0
\(149\) −8.00000 + 13.8564i −0.655386 + 1.13516i 0.326411 + 0.945228i \(0.394160\pi\)
−0.981797 + 0.189933i \(0.939173\pi\)
\(150\) 0 0
\(151\) −20.4924 −1.66765 −0.833825 0.552029i \(-0.813854\pi\)
−0.833825 + 0.552029i \(0.813854\pi\)
\(152\) 0 0
\(153\) 18.2462 1.47512
\(154\) 0 0
\(155\) 5.12311 8.87348i 0.411498 0.712735i
\(156\) 0 0
\(157\) −1.21922 + 2.11176i −0.0973046 + 0.168537i −0.910568 0.413359i \(-0.864355\pi\)
0.813263 + 0.581896i \(0.197689\pi\)
\(158\) 0 0
\(159\) −19.3693 −1.53609
\(160\) 0 0
\(161\) 1.02699 + 1.77879i 0.0809380 + 0.140189i
\(162\) 0 0
\(163\) −17.0540 −1.33577 −0.667885 0.744264i \(-0.732800\pi\)
−0.667885 + 0.744264i \(0.732800\pi\)
\(164\) 0 0
\(165\) 1.28078 + 2.21837i 0.0997083 + 0.172700i
\(166\) 0 0
\(167\) 0.342329 + 0.592932i 0.0264902 + 0.0458824i 0.878967 0.476883i \(-0.158234\pi\)
−0.852476 + 0.522766i \(0.824900\pi\)
\(168\) 0 0
\(169\) 4.50000 7.79423i 0.346154 0.599556i
\(170\) 0 0
\(171\) 6.56155 14.0696i 0.501774 1.07593i
\(172\) 0 0
\(173\) 2.90388 5.02967i 0.220778 0.382399i −0.734266 0.678861i \(-0.762474\pi\)
0.955044 + 0.296463i \(0.0958070\pi\)
\(174\) 0 0
\(175\) 0.219224 + 0.379706i 0.0165717 + 0.0287031i
\(176\) 0 0
\(177\) 18.6501 + 32.3029i 1.40183 + 2.42804i
\(178\) 0 0
\(179\) −11.4924 −0.858984 −0.429492 0.903071i \(-0.641307\pi\)
−0.429492 + 0.903071i \(0.641307\pi\)
\(180\) 0 0
\(181\) −10.6847 18.5064i −0.794184 1.37557i −0.923356 0.383945i \(-0.874565\pi\)
0.129171 0.991622i \(-0.458768\pi\)
\(182\) 0 0
\(183\) 13.1231 0.970088
\(184\) 0 0
\(185\) 2.34233 4.05703i 0.172211 0.298279i
\(186\) 0 0
\(187\) 2.56155 4.43674i 0.187319 0.324447i
\(188\) 0 0
\(189\) −0.630683 −0.0458754
\(190\) 0 0
\(191\) 5.36932 0.388510 0.194255 0.980951i \(-0.437771\pi\)
0.194255 + 0.980951i \(0.437771\pi\)
\(192\) 0 0
\(193\) −12.8078 + 22.1837i −0.921923 + 1.59682i −0.125487 + 0.992095i \(0.540049\pi\)
−0.796436 + 0.604722i \(0.793284\pi\)
\(194\) 0 0
\(195\) 2.56155 4.43674i 0.183437 0.317722i
\(196\) 0 0
\(197\) 14.4384 1.02870 0.514348 0.857581i \(-0.328034\pi\)
0.514348 + 0.857581i \(0.328034\pi\)
\(198\) 0 0
\(199\) −1.43845 2.49146i −0.101969 0.176615i 0.810527 0.585701i \(-0.199181\pi\)
−0.912496 + 0.409086i \(0.865848\pi\)
\(200\) 0 0
\(201\) −24.1771 −1.70532
\(202\) 0 0
\(203\) 0.438447 + 0.759413i 0.0307730 + 0.0533003i
\(204\) 0 0
\(205\) 3.06155 + 5.30277i 0.213828 + 0.370361i
\(206\) 0 0
\(207\) −8.34233 + 14.4493i −0.579832 + 1.00430i
\(208\) 0 0
\(209\) −2.50000 3.57071i −0.172929 0.246991i
\(210\) 0 0
\(211\) −1.65767 + 2.87117i −0.114119 + 0.197659i −0.917427 0.397904i \(-0.869738\pi\)
0.803308 + 0.595563i \(0.203071\pi\)
\(212\) 0 0
\(213\) 20.8078 + 36.0401i 1.42572 + 2.46943i
\(214\) 0 0
\(215\) 1.56155 + 2.70469i 0.106497 + 0.184458i
\(216\) 0 0
\(217\) 4.49242 0.304966
\(218\) 0 0
\(219\) 2.15767 + 3.73720i 0.145802 + 0.252536i
\(220\) 0 0
\(221\) −10.2462 −0.689235
\(222\) 0 0
\(223\) 5.65767 9.79937i 0.378866 0.656215i −0.612032 0.790833i \(-0.709648\pi\)
0.990897 + 0.134619i \(0.0429809\pi\)
\(224\) 0 0
\(225\) −1.78078 + 3.08440i −0.118718 + 0.205626i
\(226\) 0 0
\(227\) −19.9309 −1.32286 −0.661429 0.750008i \(-0.730050\pi\)
−0.661429 + 0.750008i \(0.730050\pi\)
\(228\) 0 0
\(229\) 12.8769 0.850929 0.425465 0.904975i \(-0.360111\pi\)
0.425465 + 0.904975i \(0.360111\pi\)
\(230\) 0 0
\(231\) −0.561553 + 0.972638i −0.0369475 + 0.0639949i
\(232\) 0 0
\(233\) −1.15767 + 2.00514i −0.0758415 + 0.131361i −0.901452 0.432879i \(-0.857498\pi\)
0.825610 + 0.564241i \(0.190831\pi\)
\(234\) 0 0
\(235\) −2.87689 −0.187668
\(236\) 0 0
\(237\) −14.2462 24.6752i −0.925391 1.60282i
\(238\) 0 0
\(239\) −18.2462 −1.18025 −0.590125 0.807312i \(-0.700921\pi\)
−0.590125 + 0.807312i \(0.700921\pi\)
\(240\) 0 0
\(241\) 4.28078 + 7.41452i 0.275749 + 0.477611i 0.970324 0.241809i \(-0.0777408\pi\)
−0.694575 + 0.719421i \(0.744407\pi\)
\(242\) 0 0
\(243\) 11.1231 + 19.2658i 0.713548 + 1.23590i
\(244\) 0 0
\(245\) 3.40388 5.89570i 0.217466 0.376662i
\(246\) 0 0
\(247\) −3.68466 + 7.90084i −0.234449 + 0.502718i
\(248\) 0 0
\(249\) −13.8423 + 23.9756i −0.877222 + 1.51939i
\(250\) 0 0
\(251\) 12.9654 + 22.4568i 0.818371 + 1.41746i 0.906882 + 0.421385i \(0.138456\pi\)
−0.0885109 + 0.996075i \(0.528211\pi\)
\(252\) 0 0
\(253\) 2.34233 + 4.05703i 0.147261 + 0.255063i
\(254\) 0 0
\(255\) 13.1231 0.821801
\(256\) 0 0
\(257\) −6.52699 11.3051i −0.407142 0.705191i 0.587426 0.809278i \(-0.300141\pi\)
−0.994568 + 0.104087i \(0.966808\pi\)
\(258\) 0 0
\(259\) 2.05398 0.127628
\(260\) 0 0
\(261\) −3.56155 + 6.16879i −0.220455 + 0.381839i
\(262\) 0 0
\(263\) −1.09612 + 1.89853i −0.0675895 + 0.117068i −0.897840 0.440322i \(-0.854864\pi\)
0.830250 + 0.557391i \(0.188197\pi\)
\(264\) 0 0
\(265\) −7.56155 −0.464502
\(266\) 0 0
\(267\) −6.87689 −0.420859
\(268\) 0 0
\(269\) −2.00000 + 3.46410i −0.121942 + 0.211210i −0.920534 0.390664i \(-0.872246\pi\)
0.798591 + 0.601874i \(0.205579\pi\)
\(270\) 0 0
\(271\) −4.12311 + 7.14143i −0.250461 + 0.433811i −0.963653 0.267158i \(-0.913915\pi\)
0.713192 + 0.700969i \(0.247249\pi\)
\(272\) 0 0
\(273\) 2.24621 0.135947
\(274\) 0 0
\(275\) 0.500000 + 0.866025i 0.0301511 + 0.0522233i
\(276\) 0 0
\(277\) 4.24621 0.255130 0.127565 0.991830i \(-0.459284\pi\)
0.127565 + 0.991830i \(0.459284\pi\)
\(278\) 0 0
\(279\) 18.2462 + 31.6034i 1.09237 + 1.89204i
\(280\) 0 0
\(281\) −4.18466 7.24804i −0.249636 0.432382i 0.713789 0.700361i \(-0.246978\pi\)
−0.963425 + 0.267979i \(0.913644\pi\)
\(282\) 0 0
\(283\) −9.71922 + 16.8342i −0.577748 + 1.00069i 0.417989 + 0.908452i \(0.362735\pi\)
−0.995737 + 0.0922367i \(0.970598\pi\)
\(284\) 0 0
\(285\) 4.71922 10.1192i 0.279543 0.599410i
\(286\) 0 0
\(287\) −1.34233 + 2.32498i −0.0792352 + 0.137239i
\(288\) 0 0
\(289\) −4.62311 8.00745i −0.271947 0.471027i
\(290\) 0 0
\(291\) −2.15767 3.73720i −0.126485 0.219078i
\(292\) 0 0
\(293\) 23.5616 1.37648 0.688240 0.725483i \(-0.258383\pi\)
0.688240 + 0.725483i \(0.258383\pi\)
\(294\) 0 0
\(295\) 7.28078 + 12.6107i 0.423903 + 0.734222i
\(296\) 0 0
\(297\) −1.43845 −0.0834672
\(298\) 0 0
\(299\) 4.68466 8.11407i 0.270921 0.469249i
\(300\) 0 0
\(301\) −0.684658 + 1.18586i −0.0394631 + 0.0683520i
\(302\) 0 0
\(303\) 25.6155 1.47157
\(304\) 0 0
\(305\) 5.12311 0.293348
\(306\) 0 0
\(307\) −11.9654 + 20.7247i −0.682903 + 1.18282i 0.291187 + 0.956666i \(0.405950\pi\)
−0.974091 + 0.226157i \(0.927384\pi\)
\(308\) 0 0
\(309\) −7.43845 + 12.8838i −0.423158 + 0.732932i
\(310\) 0 0
\(311\) −4.00000 −0.226819 −0.113410 0.993548i \(-0.536177\pi\)
−0.113410 + 0.993548i \(0.536177\pi\)
\(312\) 0 0
\(313\) −2.15767 3.73720i −0.121959 0.211239i 0.798581 0.601887i \(-0.205584\pi\)
−0.920540 + 0.390648i \(0.872251\pi\)
\(314\) 0 0
\(315\) −1.56155 −0.0879835
\(316\) 0 0
\(317\) −9.34233 16.1814i −0.524717 0.908837i −0.999586 0.0287805i \(-0.990838\pi\)
0.474868 0.880057i \(-0.342496\pi\)
\(318\) 0 0
\(319\) 1.00000 + 1.73205i 0.0559893 + 0.0969762i
\(320\) 0 0
\(321\) 2.87689 4.98293i 0.160573 0.278120i
\(322\) 0 0
\(323\) −22.2462 + 1.94528i −1.23781 + 0.108238i
\(324\) 0 0
\(325\) 1.00000 1.73205i 0.0554700 0.0960769i
\(326\) 0 0
\(327\) 18.2462 + 31.6034i 1.00902 + 1.74767i
\(328\) 0 0
\(329\) −0.630683 1.09238i −0.0347707 0.0602246i
\(330\) 0 0
\(331\) −23.4924 −1.29126 −0.645630 0.763650i \(-0.723405\pi\)
−0.645630 + 0.763650i \(0.723405\pi\)
\(332\) 0 0
\(333\) 8.34233 + 14.4493i 0.457157 + 0.791819i
\(334\) 0 0
\(335\) −9.43845 −0.515677
\(336\) 0 0
\(337\) −10.5270 + 18.2333i −0.573442 + 0.993230i 0.422767 + 0.906238i \(0.361059\pi\)
−0.996209 + 0.0869917i \(0.972275\pi\)
\(338\) 0 0
\(339\) 21.5270 37.2858i 1.16919 2.02509i
\(340\) 0 0
\(341\) 10.2462 0.554863
\(342\) 0 0
\(343\) 6.05398 0.326884
\(344\) 0 0
\(345\) −6.00000 + 10.3923i −0.323029 + 0.559503i
\(346\) 0 0
\(347\) 0.842329 1.45896i 0.0452186 0.0783209i −0.842530 0.538649i \(-0.818935\pi\)
0.887749 + 0.460328i \(0.152268\pi\)
\(348\) 0 0
\(349\) −14.2462 −0.762582 −0.381291 0.924455i \(-0.624520\pi\)
−0.381291 + 0.924455i \(0.624520\pi\)
\(350\) 0 0
\(351\) 1.43845 + 2.49146i 0.0767786 + 0.132984i
\(352\) 0 0
\(353\) 24.1771 1.28682 0.643408 0.765523i \(-0.277520\pi\)
0.643408 + 0.765523i \(0.277520\pi\)
\(354\) 0 0
\(355\) 8.12311 + 14.0696i 0.431130 + 0.746739i
\(356\) 0 0
\(357\) 2.87689 + 4.98293i 0.152261 + 0.263724i
\(358\) 0 0
\(359\) 7.56155 13.0970i 0.399083 0.691233i −0.594530 0.804074i \(-0.702662\pi\)
0.993613 + 0.112841i \(0.0359950\pi\)
\(360\) 0 0
\(361\) −6.50000 + 17.8536i −0.342105 + 0.939662i
\(362\) 0 0
\(363\) 12.8078 22.1837i 0.672233 1.16434i
\(364\) 0 0
\(365\) 0.842329 + 1.45896i 0.0440895 + 0.0763653i
\(366\) 0 0
\(367\) −8.87689 15.3752i −0.463370 0.802581i 0.535756 0.844373i \(-0.320027\pi\)
−0.999126 + 0.0417921i \(0.986693\pi\)
\(368\) 0 0
\(369\) −21.8078 −1.13527
\(370\) 0 0
\(371\) −1.65767 2.87117i −0.0860620 0.149064i
\(372\) 0 0
\(373\) −35.8078 −1.85406 −0.927028 0.374992i \(-0.877645\pi\)
−0.927028 + 0.374992i \(0.877645\pi\)
\(374\) 0 0
\(375\) −1.28078 + 2.21837i −0.0661390 + 0.114556i
\(376\) 0 0
\(377\) 2.00000 3.46410i 0.103005 0.178410i
\(378\) 0 0
\(379\) 0.492423 0.0252940 0.0126470 0.999920i \(-0.495974\pi\)
0.0126470 + 0.999920i \(0.495974\pi\)
\(380\) 0 0
\(381\) 39.8617 2.04218
\(382\) 0 0
\(383\) 2.56155 4.43674i 0.130889 0.226707i −0.793130 0.609052i \(-0.791550\pi\)
0.924020 + 0.382345i \(0.124883\pi\)
\(384\) 0 0
\(385\) −0.219224 + 0.379706i −0.0111727 + 0.0193516i
\(386\) 0 0
\(387\) −11.1231 −0.565419
\(388\) 0 0
\(389\) 9.56155 + 16.5611i 0.484790 + 0.839681i 0.999847 0.0174749i \(-0.00556271\pi\)
−0.515057 + 0.857156i \(0.672229\pi\)
\(390\) 0 0
\(391\) 24.0000 1.21373
\(392\) 0 0
\(393\) −20.6501 35.7670i −1.04166 1.80421i
\(394\) 0 0
\(395\) −5.56155 9.63289i −0.279832 0.484683i
\(396\) 0 0
\(397\) −14.1501 + 24.5087i −0.710173 + 1.23006i 0.254619 + 0.967041i \(0.418050\pi\)
−0.964792 + 0.263014i \(0.915283\pi\)
\(398\) 0 0
\(399\) 4.87689 0.426450i 0.244150 0.0213492i
\(400\) 0 0
\(401\) −13.9654 + 24.1888i −0.697401 + 1.20793i 0.271964 + 0.962307i \(0.412327\pi\)
−0.969365 + 0.245626i \(0.921007\pi\)
\(402\) 0 0
\(403\) −10.2462 17.7470i −0.510400 0.884039i
\(404\) 0 0
\(405\) 3.50000 + 6.06218i 0.173916 + 0.301232i
\(406\) 0 0
\(407\) 4.68466 0.232210
\(408\) 0 0
\(409\) −10.5000 18.1865i −0.519192 0.899266i −0.999751 0.0223042i \(-0.992900\pi\)
0.480560 0.876962i \(-0.340434\pi\)
\(410\) 0 0
\(411\) −13.9309 −0.687159
\(412\) 0 0
\(413\) −3.19224 + 5.52911i −0.157080 + 0.272070i
\(414\) 0 0
\(415\) −5.40388 + 9.35980i −0.265266 + 0.459454i
\(416\) 0 0
\(417\) −22.5616 −1.10484
\(418\) 0 0
\(419\) 17.5616 0.857938 0.428969 0.903319i \(-0.358877\pi\)
0.428969 + 0.903319i \(0.358877\pi\)
\(420\) 0 0
\(421\) −16.2462 + 28.1393i −0.791792 + 1.37142i 0.133065 + 0.991107i \(0.457518\pi\)
−0.924856 + 0.380316i \(0.875815\pi\)
\(422\) 0 0
\(423\) 5.12311 8.87348i 0.249094 0.431443i
\(424\) 0 0
\(425\) 5.12311 0.248507
\(426\) 0 0
\(427\) 1.12311 + 1.94528i 0.0543509 + 0.0941385i
\(428\) 0 0
\(429\) 5.12311 0.247346
\(430\) 0 0
\(431\) −3.68466 6.38202i −0.177484 0.307411i 0.763534 0.645767i \(-0.223462\pi\)
−0.941018 + 0.338356i \(0.890129\pi\)
\(432\) 0 0
\(433\) 18.3693 + 31.8166i 0.882773 + 1.52901i 0.848245 + 0.529604i \(0.177659\pi\)
0.0345280 + 0.999404i \(0.489007\pi\)
\(434\) 0 0
\(435\) −2.56155 + 4.43674i −0.122817 + 0.212725i
\(436\) 0 0
\(437\) 8.63068 18.5064i 0.412862 0.885280i
\(438\) 0 0
\(439\) −8.24621 + 14.2829i −0.393570 + 0.681684i −0.992918 0.118806i \(-0.962094\pi\)
0.599347 + 0.800489i \(0.295427\pi\)
\(440\) 0 0
\(441\) 12.1231 + 20.9978i 0.577291 + 0.999897i
\(442\) 0 0
\(443\) 16.9654 + 29.3850i 0.806052 + 1.39612i 0.915578 + 0.402139i \(0.131733\pi\)
−0.109526 + 0.993984i \(0.534933\pi\)
\(444\) 0 0
\(445\) −2.68466 −0.127265
\(446\) 0 0
\(447\) −20.4924 35.4939i −0.969258 1.67880i
\(448\) 0 0
\(449\) 29.0000 1.36859 0.684297 0.729203i \(-0.260109\pi\)
0.684297 + 0.729203i \(0.260109\pi\)
\(450\) 0 0
\(451\) −3.06155 + 5.30277i −0.144163 + 0.249697i
\(452\) 0 0
\(453\) 26.2462 45.4598i 1.23315 2.13589i
\(454\) 0 0
\(455\) 0.876894 0.0411094
\(456\) 0 0
\(457\) 7.05398 0.329971 0.164986 0.986296i \(-0.447242\pi\)
0.164986 + 0.986296i \(0.447242\pi\)
\(458\) 0 0
\(459\) −3.68466 + 6.38202i −0.171985 + 0.297887i
\(460\) 0 0
\(461\) 19.4924 33.7619i 0.907853 1.57245i 0.0908110 0.995868i \(-0.471054\pi\)
0.817042 0.576579i \(-0.195613\pi\)
\(462\) 0 0
\(463\) 19.5616 0.909102 0.454551 0.890721i \(-0.349800\pi\)
0.454551 + 0.890721i \(0.349800\pi\)
\(464\) 0 0
\(465\) 13.1231 + 22.7299i 0.608569 + 1.05407i
\(466\) 0 0
\(467\) −16.5616 −0.766377 −0.383189 0.923670i \(-0.625174\pi\)
−0.383189 + 0.923670i \(0.625174\pi\)
\(468\) 0 0
\(469\) −2.06913 3.58384i −0.0955436 0.165486i
\(470\) 0 0
\(471\) −3.12311 5.40938i −0.143905 0.249251i
\(472\) 0 0
\(473\) −1.56155 + 2.70469i −0.0718003 + 0.124362i
\(474\) 0 0
\(475\) 1.84233 3.95042i 0.0845319 0.181258i
\(476\) 0 0
\(477\) 13.4654 23.3228i 0.616540 1.06788i
\(478\) 0 0
\(479\) 14.6847 + 25.4346i 0.670959 + 1.16214i 0.977632 + 0.210321i \(0.0674508\pi\)
−0.306673 + 0.951815i \(0.599216\pi\)
\(480\) 0 0
\(481\) −4.68466 8.11407i −0.213602 0.369970i
\(482\) 0 0
\(483\) −5.26137 −0.239400
\(484\) 0 0
\(485\) −0.842329 1.45896i −0.0382482 0.0662478i
\(486\) 0 0
\(487\) 6.93087 0.314068 0.157034 0.987593i \(-0.449807\pi\)
0.157034 + 0.987593i \(0.449807\pi\)
\(488\) 0 0
\(489\) 21.8423 37.8320i 0.987744 1.71082i
\(490\) 0 0
\(491\) −4.58854 + 7.94759i −0.207078 + 0.358670i −0.950793 0.309827i \(-0.899729\pi\)
0.743715 + 0.668497i \(0.233062\pi\)
\(492\) 0 0
\(493\) 10.2462 0.461466
\(494\) 0 0
\(495\) −3.56155 −0.160080
\(496\) 0 0
\(497\) −3.56155 + 6.16879i −0.159757 + 0.276708i
\(498\) 0 0
\(499\) 14.5000 25.1147i 0.649109 1.12429i −0.334227 0.942493i \(-0.608475\pi\)
0.983336 0.181797i \(-0.0581915\pi\)
\(500\) 0 0
\(501\) −1.75379 −0.0783535
\(502\) 0 0
\(503\) 13.7116 + 23.7493i 0.611372 + 1.05893i 0.991009 + 0.133792i \(0.0427154\pi\)
−0.379637 + 0.925135i \(0.623951\pi\)
\(504\) 0 0
\(505\) 10.0000 0.444994
\(506\) 0 0
\(507\) 11.5270 + 19.9653i 0.511931 + 0.886691i
\(508\) 0 0
\(509\) 0.438447 + 0.759413i 0.0194338 + 0.0336604i 0.875579 0.483075i \(-0.160480\pi\)
−0.856145 + 0.516736i \(0.827147\pi\)
\(510\) 0 0
\(511\) −0.369317 + 0.639676i −0.0163376 + 0.0282976i
\(512\) 0 0
\(513\) 3.59612 + 5.13628i 0.158772 + 0.226772i
\(514\) 0 0
\(515\) −2.90388 + 5.02967i −0.127960 + 0.221634i
\(516\) 0 0
\(517\) −1.43845 2.49146i −0.0632628 0.109574i
\(518\) 0 0
\(519\) 7.43845 + 12.8838i 0.326512 + 0.565535i
\(520\) 0 0
\(521\) −22.8078 −0.999226 −0.499613 0.866249i \(-0.666524\pi\)
−0.499613 + 0.866249i \(0.666524\pi\)
\(522\) 0 0
\(523\) 7.31534 + 12.6705i 0.319878 + 0.554044i 0.980462 0.196707i \(-0.0630249\pi\)
−0.660585 + 0.750752i \(0.729692\pi\)
\(524\) 0 0
\(525\) −1.12311 −0.0490163
\(526\) 0 0
\(527\) 26.2462 45.4598i 1.14330 1.98026i
\(528\) 0 0
\(529\) 0.526988 0.912769i 0.0229125 0.0396856i
\(530\) 0 0
\(531\) −51.8617 −2.25061
\(532\) 0 0
\(533\) 12.2462 0.530442
\(534\) 0 0
\(535\) 1.12311 1.94528i 0.0485561 0.0841016i
\(536\) 0 0
\(537\) 14.7192 25.4944i 0.635181 1.10017i
\(538\) 0 0
\(539\) 6.80776 0.293231
\(540\) 0 0
\(541\) 14.2462 + 24.6752i 0.612492 + 1.06087i 0.990819 + 0.135196i \(0.0431664\pi\)
−0.378326 + 0.925672i \(0.623500\pi\)
\(542\) 0 0
\(543\) 54.7386 2.34906
\(544\) 0 0
\(545\) 7.12311 + 12.3376i 0.305120 + 0.528484i
\(546\) 0 0
\(547\) −2.43845 4.22351i −0.104260 0.180584i 0.809175 0.587567i \(-0.199914\pi\)
−0.913436 + 0.406983i \(0.866581\pi\)
\(548\) 0 0
\(549\) −9.12311 + 15.8017i −0.389365 + 0.674399i
\(550\) 0 0
\(551\) 3.68466 7.90084i 0.156972 0.336587i
\(552\) 0 0
\(553\) 2.43845 4.22351i 0.103693 0.179602i
\(554\) 0 0
\(555\) 6.00000 + 10.3923i 0.254686 + 0.441129i
\(556\) 0 0
\(557\) −20.8348 36.0868i −0.882797 1.52905i −0.848218 0.529647i \(-0.822324\pi\)
−0.0345785 0.999402i \(-0.511009\pi\)
\(558\) 0 0
\(559\) 6.24621 0.264187
\(560\) 0 0
\(561\) 6.56155 + 11.3649i 0.277029 + 0.479828i
\(562\) 0 0
\(563\) −21.3002 −0.897696 −0.448848 0.893608i \(-0.648166\pi\)
−0.448848 + 0.893608i \(0.648166\pi\)
\(564\) 0 0
\(565\) 8.40388 14.5560i 0.353554 0.612373i
\(566\) 0 0
\(567\) −1.53457 + 2.65794i −0.0644457 + 0.111623i
\(568\) 0 0
\(569\) −11.1771 −0.468568 −0.234284 0.972168i \(-0.575274\pi\)
−0.234284 + 0.972168i \(0.575274\pi\)
\(570\) 0 0
\(571\) 4.80776 0.201199 0.100599 0.994927i \(-0.467924\pi\)
0.100599 + 0.994927i \(0.467924\pi\)
\(572\) 0 0
\(573\) −6.87689 + 11.9111i −0.287286 + 0.497595i
\(574\) 0 0
\(575\) −2.34233 + 4.05703i −0.0976819 + 0.169190i
\(576\) 0 0
\(577\) −16.3153 −0.679217 −0.339608 0.940567i \(-0.610295\pi\)
−0.339608 + 0.940567i \(0.610295\pi\)
\(578\) 0 0
\(579\) −32.8078 56.8247i −1.36344 2.36155i
\(580\) 0 0
\(581\) −4.73863 −0.196592
\(582\) 0 0
\(583\) −3.78078 6.54850i −0.156584 0.271211i
\(584\) 0 0
\(585\) 3.56155 + 6.16879i 0.147252 + 0.255048i
\(586\) 0 0
\(587\) 13.3693 23.1563i 0.551811 0.955764i −0.446333 0.894867i \(-0.647270\pi\)
0.998144 0.0608975i \(-0.0193963\pi\)
\(588\) 0 0
\(589\) −25.6155 36.5863i −1.05547 1.50751i
\(590\) 0 0
\(591\) −18.4924 + 32.0298i −0.760677 + 1.31753i
\(592\) 0 0
\(593\) −12.7732 22.1238i −0.524532 0.908517i −0.999592 0.0285632i \(-0.990907\pi\)
0.475060 0.879954i \(-0.342427\pi\)
\(594\) 0 0
\(595\) 1.12311 + 1.94528i 0.0460428 + 0.0797485i
\(596\) 0 0
\(597\) 7.36932 0.301606
\(598\) 0 0
\(599\) −16.2462 28.1393i −0.663802 1.14974i −0.979609 0.200915i \(-0.935608\pi\)
0.315806 0.948824i \(-0.397725\pi\)
\(600\) 0 0
\(601\) −11.6307 −0.474425 −0.237213 0.971458i \(-0.576234\pi\)
−0.237213 + 0.971458i \(0.576234\pi\)
\(602\) 0 0
\(603\) 16.8078 29.1119i 0.684465 1.18553i
\(604\) 0 0
\(605\) 5.00000 8.66025i 0.203279 0.352089i
\(606\) 0 0
\(607\) −10.1922 −0.413690 −0.206845 0.978374i \(-0.566320\pi\)
−0.206845 + 0.978374i \(0.566320\pi\)
\(608\) 0 0
\(609\) −2.24621 −0.0910211
\(610\) 0 0
\(611\) −2.87689 + 4.98293i −0.116387 + 0.201588i
\(612\) 0 0
\(613\) −15.3423 + 26.5737i −0.619671 + 1.07330i 0.369875 + 0.929082i \(0.379401\pi\)
−0.989546 + 0.144220i \(0.953933\pi\)
\(614\) 0 0
\(615\) −15.6847 −0.632466
\(616\) 0 0
\(617\) −3.59612 6.22866i −0.144774 0.250756i 0.784514 0.620111i \(-0.212912\pi\)
−0.929289 + 0.369354i \(0.879579\pi\)
\(618\) 0 0
\(619\) −26.0540 −1.04720 −0.523599 0.851965i \(-0.675411\pi\)
−0.523599 + 0.851965i \(0.675411\pi\)
\(620\) 0 0
\(621\) −3.36932 5.83583i −0.135206 0.234184i
\(622\) 0 0
\(623\) −0.588540 1.01938i −0.0235794 0.0408407i
\(624\) 0 0
\(625\) −0.500000 + 0.866025i −0.0200000 + 0.0346410i
\(626\) 0 0
\(627\) 11.1231 0.972638i 0.444214 0.0388434i
\(628\) 0 0
\(629\) 12.0000 20.7846i 0.478471 0.828737i
\(630\) 0 0
\(631\) 2.12311 + 3.67733i 0.0845195 + 0.146392i 0.905186 0.425015i \(-0.139731\pi\)
−0.820667 + 0.571407i \(0.806398\pi\)
\(632\) 0 0
\(633\) −4.24621 7.35465i −0.168772 0.292321i
\(634\) 0 0
\(635\) 15.5616 0.617541
\(636\) 0 0
\(637\) −6.80776 11.7914i −0.269733 0.467192i
\(638\) 0 0
\(639\) −57.8617 −2.28898
\(640\) 0 0
\(641\) −3.71922 + 6.44188i −0.146900 + 0.254439i −0.930080 0.367356i \(-0.880263\pi\)
0.783180 + 0.621795i \(0.213596\pi\)
\(642\) 0 0
\(643\) −11.2808 + 19.5389i −0.444870 + 0.770538i −0.998043 0.0625284i \(-0.980084\pi\)
0.553173 + 0.833067i \(0.313417\pi\)
\(644\) 0 0
\(645\) −8.00000 −0.315000
\(646\) 0 0
\(647\) −3.17708 −0.124904 −0.0624520 0.998048i \(-0.519892\pi\)
−0.0624520 + 0.998048i \(0.519892\pi\)
\(648\) 0 0
\(649\) −7.28078 + 12.6107i −0.285795 + 0.495012i
\(650\) 0 0
\(651\) −5.75379 + 9.96585i −0.225509 + 0.390593i
\(652\) 0 0
\(653\) −5.06913 −0.198370 −0.0991852 0.995069i \(-0.531624\pi\)
−0.0991852 + 0.995069i \(0.531624\pi\)
\(654\) 0 0
\(655\) −8.06155 13.9630i −0.314991 0.545580i
\(656\) 0 0
\(657\) −6.00000 −0.234082
\(658\) 0 0
\(659\) 9.46543 + 16.3946i 0.368721 + 0.638643i 0.989366 0.145448i \(-0.0464624\pi\)
−0.620645 + 0.784092i \(0.713129\pi\)
\(660\) 0 0
\(661\) 19.9309 + 34.5213i 0.775221 + 1.34272i 0.934670 + 0.355516i \(0.115695\pi\)
−0.159449 + 0.987206i \(0.550972\pi\)
\(662\) 0 0
\(663\) 13.1231 22.7299i 0.509659 0.882756i
\(664\) 0 0
\(665\) 1.90388 0.166481i 0.0738294 0.00645586i
\(666\) 0 0
\(667\) −4.68466 + 8.11407i −0.181391 + 0.314178i
\(668\) 0 0
\(669\) 14.4924 + 25.1016i 0.560309 + 0.970484i
\(670\) 0 0
\(671\) 2.56155 + 4.43674i 0.0988876 + 0.171278i
\(672\) 0 0
\(673\) 46.1080 1.77733 0.888665 0.458556i \(-0.151633\pi\)
0.888665 + 0.458556i \(0.151633\pi\)
\(674\) 0 0
\(675\) −0.719224 1.24573i −0.0276829 0.0479482i
\(676\) 0 0
\(677\) −23.1771 −0.890768 −0.445384 0.895340i \(-0.646933\pi\)
−0.445384 + 0.895340i \(0.646933\pi\)
\(678\) 0 0
\(679\) 0.369317 0.639676i 0.0141731 0.0245485i
\(680\) 0 0
\(681\) 25.5270 44.2140i 0.978196 1.69429i
\(682\) 0 0
\(683\) −28.0000 −1.07139 −0.535695 0.844411i \(-0.679950\pi\)
−0.535695 + 0.844411i \(0.679950\pi\)
\(684\) 0 0
\(685\) −5.43845 −0.207792
\(686\) 0 0
\(687\) −16.4924 + 28.5657i −0.629225 + 1.08985i
\(688\) 0 0
\(689\) −7.56155 + 13.0970i −0.288072 + 0.498956i
\(690\) 0 0
\(691\) 21.0691 0.801507 0.400754 0.916186i \(-0.368748\pi\)
0.400754 + 0.916186i \(0.368748\pi\)
\(692\) 0 0
\(693\) −0.780776 1.35234i −0.0296592 0.0513713i
\(694\) 0 0
\(695\) −8.80776 −0.334098
\(696\) 0 0
\(697\) 15.6847 + 27.1666i 0.594099 + 1.02901i
\(698\) 0 0
\(699\) −2.96543 5.13628i −0.112163 0.194272i
\(700\) 0 0
\(701\) −3.24621 + 5.62260i −0.122608 + 0.212363i −0.920795 0.390046i \(-0.872459\pi\)
0.798188 + 0.602409i \(0.205792\pi\)
\(702\) 0 0
\(703\) −11.7116 16.7276i −0.441713 0.630892i
\(704\) 0 0
\(705\) 3.68466 6.38202i 0.138772 0.240361i
\(706\) 0 0
\(707\) 2.19224 + 3.79706i 0.0824475 + 0.142803i
\(708\) 0 0
\(709\) 20.0000 + 34.6410i 0.751116 + 1.30097i 0.947282 + 0.320400i \(0.103817\pi\)
−0.196167 + 0.980571i \(0.562849\pi\)
\(710\) 0 0
\(711\) 39.6155 1.48570
\(712\) 0 0
\(713\) 24.0000 + 41.5692i 0.898807 + 1.55678i
\(714\) 0 0
\(715\) 2.00000 0.0747958
\(716\) 0 0
\(717\) 23.3693 40.4768i 0.872743 1.51164i
\(718\) 0 0
\(719\) 7.43845 12.8838i 0.277407 0.480483i −0.693332 0.720618i \(-0.743858\pi\)
0.970740 + 0.240134i \(0.0771915\pi\)
\(720\) 0 0
\(721\) −2.54640 −0.0948328
\(722\) 0 0
\(723\) −21.9309 −0.815618
\(724\) 0 0
\(725\) −1.00000 + 1.73205i −0.0371391 + 0.0643268i
\(726\) 0 0
\(727\) −4.00000 + 6.92820i −0.148352 + 0.256953i −0.930618 0.365991i \(-0.880730\pi\)
0.782267 + 0.622944i \(0.214063\pi\)
\(728\) 0 0
\(729\) −35.9848 −1.33277
\(730\) 0 0
\(731\) 8.00000 + 13.8564i 0.295891 + 0.512498i
\(732\) 0 0
\(733\) −26.9309 −0.994714 −0.497357 0.867546i \(-0.665696\pi\)
−0.497357 + 0.867546i \(0.665696\pi\)
\(734\) 0 0
\(735\) 8.71922 + 15.1021i 0.321613 + 0.557051i
\(736\) 0 0
\(737\) −4.71922 8.17394i −0.173835 0.301091i
\(738\) 0 0
\(739\) 9.37689 16.2413i 0.344935 0.597444i −0.640407 0.768036i \(-0.721234\pi\)
0.985342 + 0.170591i \(0.0545677\pi\)
\(740\) 0 0
\(741\) −12.8078 18.2931i −0.470505 0.672016i
\(742\) 0 0
\(743\) −9.21922 + 15.9682i −0.338221 + 0.585815i −0.984098 0.177626i \(-0.943158\pi\)
0.645878 + 0.763441i \(0.276492\pi\)
\(744\) 0 0
\(745\) −8.00000 13.8564i −0.293097 0.507659i
\(746\) 0 0
\(747\) −19.2462 33.3354i −0.704182 1.21968i
\(748\) 0 0
\(749\) 0.984845 0.0359855
\(750\) 0 0
\(751\) −17.4384 30.2043i −0.636338 1.10217i −0.986230 0.165380i \(-0.947115\pi\)
0.349892 0.936790i \(-0.386218\pi\)
\(752\) 0 0
\(753\) −66.4233 −2.42060
\(754\) 0 0
\(755\) 10.2462 17.7470i 0.372898 0.645878i
\(756\) 0 0
\(757\) 9.78078 16.9408i 0.355488 0.615724i −0.631713 0.775202i \(-0.717648\pi\)
0.987201 + 0.159478i \(0.0509812\pi\)
\(758\) 0 0
\(759\) −12.0000 −0.435572
\(760\) 0 0
\(761\) −51.9848 −1.88445 −0.942225 0.334982i \(-0.891270\pi\)
−0.942225 + 0.334982i \(0.891270\pi\)
\(762\) 0 0
\(763\) −3.12311 + 5.40938i −0.113064 + 0.195833i
\(764\) 0 0
\(765\) −9.12311 + 15.8017i −0.329847 + 0.571311i
\(766\) 0 0
\(767\) 29.1231 1.05157
\(768\) 0 0
\(769\) −13.2462 22.9431i −0.477671 0.827350i 0.522002 0.852944i \(-0.325185\pi\)
−0.999672 + 0.0255946i \(0.991852\pi\)
\(770\) 0 0
\(771\) 33.4384 1.20426
\(772\) 0 0
\(773\) 7.15009 + 12.3843i 0.257171 + 0.445433i 0.965483 0.260466i \(-0.0838763\pi\)
−0.708312 + 0.705900i \(0.750543\pi\)
\(774\) 0 0
\(775\) 5.12311 + 8.87348i 0.184027 + 0.318745i
\(776\) 0 0
\(777\) −2.63068 + 4.55648i −0.0943752 + 0.163463i
\(778\) 0 0
\(779\) 26.5885 2.32498i 0.952633 0.0833011i
\(780\) 0 0
\(781\) −8.12311 + 14.0696i −0.290668 + 0.503451i
\(782\) 0 0
\(783\) −1.43845 2.49146i −0.0514059 0.0890376i
\(784\) 0 0
\(785\) −1.21922 2.11176i −0.0435160 0.0753718i
\(786\) 0 0
\(787\) −8.56155 −0.305186 −0.152593 0.988289i \(-0.548762\pi\)
−0.152593 + 0.988289i \(0.548762\pi\)
\(788\) 0 0
\(789\) −2.80776 4.86319i −0.0999590 0.173134i
\(790\) 0 0
\(791\) 7.36932 0.262023
\(792\) 0 0
\(793\) 5.12311 8.87348i 0.181927 0.315106i
\(794\) 0 0
\(795\) 9.68466 16.7743i 0.343479 0.594924i
\(796\) 0 0
\(797\) −4.19224 −0.148497 −0.0742483 0.997240i \(-0.523656\pi\)
−0.0742483 + 0.997240i \(0.523656\pi\)
\(798\) 0 0
\(799\) −14.7386 −0.521415
\(800\) 0 0
\(801\) 4.78078 8.28055i 0.168920 0.292579i
\(802\) 0 0
\(803\) −0.842329 + 1.45896i −0.0297252 + 0.0514855i
\(804\) 0 0
\(805\) −2.05398 −0.0723931
\(806\) 0 0
\(807\) −5.12311 8.87348i −0.180342 0.312361i
\(808\) 0 0
\(809\) 23.4384 0.824052 0.412026 0.911172i \(-0.364821\pi\)
0.412026 + 0.911172i \(0.364821\pi\)
\(810\) 0 0
\(811\) −4.58854 7.94759i −0.161125 0.279077i 0.774147 0.633006i \(-0.218179\pi\)
−0.935273 + 0.353928i \(0.884846\pi\)
\(812\) 0 0
\(813\) −10.5616 18.2931i −0.370410 0.641569i
\(814\) 0 0
\(815\) 8.52699 14.7692i 0.298687 0.517342i
\(816\) 0 0
\(817\) 13.5616 1.18586i 0.474459 0.0414881i
\(818\) 0 0
\(819\) −1.56155 + 2.70469i −0.0545651 + 0.0945095i
\(820\) 0 0
\(821\) 7.00000 + 12.1244i 0.244302 + 0.423143i 0.961935 0.273278i \(-0.0881079\pi\)
−0.717633 + 0.696421i \(0.754775\pi\)
\(822\) 0 0
\(823\) −13.5885 23.5360i −0.473667 0.820415i 0.525879 0.850560i \(-0.323737\pi\)
−0.999546 + 0.0301446i \(0.990403\pi\)
\(824\) 0 0
\(825\) −2.56155 −0.0891818
\(826\) 0 0
\(827\) −15.9654 27.6529i −0.555173 0.961587i −0.997890 0.0649260i \(-0.979319\pi\)
0.442718 0.896661i \(-0.354014\pi\)
\(828\) 0 0
\(829\) 26.7386 0.928671 0.464336 0.885659i \(-0.346293\pi\)
0.464336 + 0.885659i \(0.346293\pi\)
\(830\) 0 0
\(831\) −5.43845 + 9.41967i −0.188658 + 0.326765i
\(832\) 0 0
\(833\) 17.4384 30.2043i 0.604206 1.04652i
\(834\) 0 0
\(835\) −0.684658 −0.0236936
\(836\) 0 0
\(837\) −14.7386 −0.509442
\(838\) 0 0
\(839\) 1.24621 2.15850i 0.0430240 0.0745197i −0.843712 0.536797i \(-0.819634\pi\)
0.886735 + 0.462277i \(0.152967\pi\)
\(840\) 0 0
\(841\) 12.5000 21.6506i 0.431034 0.746574i
\(842\) 0 0
\(843\) 21.4384 0.738379
\(844\) 0 0
\(845\) 4.50000 + 7.79423i 0.154805 + 0.268130i
\(846\) 0 0
\(847\) 4.38447 0.150652
\(848\) 0 0
\(849\) −24.8963 43.1217i −0.854439 1.47993i
\(850\) 0 0
\(851\) 10.9730 + 19.0058i 0.376150 + 0.651511i
\(852\) 0 0
\(853\) −0.369317 + 0.639676i −0.0126452 + 0.0219021i −0.872279 0.489009i \(-0.837359\pi\)
0.859634 + 0.510911i \(0.170692\pi\)
\(854\) 0 0
\(855\) 8.90388 + 12.7173i 0.304506 + 0.434922i
\(856\) 0 0
\(857\) −14.8963 + 25.8012i −0.508848 + 0.881351i 0.491099 + 0.871104i \(0.336595\pi\)
−0.999947 + 0.0102472i \(0.996738\pi\)
\(858\) 0 0
\(859\) 23.7462 + 41.1296i 0.810210 + 1.40333i 0.912717 + 0.408593i \(0.133980\pi\)
−0.102506 + 0.994732i \(0.532686\pi\)
\(860\) 0 0
\(861\) −3.43845 5.95557i −0.117182 0.202965i
\(862\) 0 0
\(863\) −52.3002 −1.78032 −0.890160 0.455649i \(-0.849407\pi\)
−0.890160 + 0.455649i \(0.849407\pi\)
\(864\) 0 0
\(865\) 2.90388 + 5.02967i 0.0987350 + 0.171014i
\(866\) 0 0
\(867\) 23.6847 0.804373
\(868\) 0 0
\(869\) 5.56155 9.63289i 0.188663 0.326773i
\(870\) 0 0
\(871\) −9.43845 + 16.3479i −0.319810 + 0.553926i
\(872\) 0 0
\(873\) 6.00000 0.203069
\(874\) 0 0
\(875\) −0.438447 −0.0148222
\(876\) 0 0
\(877\) −8.90388 + 15.4220i −0.300663 + 0.520763i −0.976286 0.216484i \(-0.930541\pi\)
0.675623 + 0.737247i \(0.263875\pi\)
\(878\) 0 0
\(879\) −30.1771 + 52.2682i −1.01785 + 1.76296i
\(880\) 0 0
\(881\) −24.1231 −0.812728 −0.406364 0.913711i \(-0.633204\pi\)
−0.406364 + 0.913711i \(0.633204\pi\)
\(882\) 0 0
\(883\) −4.40388 7.62775i −0.148202 0.256694i 0.782361 0.622826i \(-0.214015\pi\)
−0.930563 + 0.366131i \(0.880682\pi\)
\(884\) 0 0
\(885\) −37.3002 −1.25383
\(886\) 0 0
\(887\) −13.1231 22.7299i −0.440631 0.763195i 0.557106 0.830442i \(-0.311912\pi\)
−0.997736 + 0.0672468i \(0.978579\pi\)
\(888\) 0 0
\(889\) 3.41146 + 5.90882i 0.114417 + 0.198176i
\(890\) 0 0
\(891\) −3.50000 + 6.06218i −0.117254 + 0.203091i
\(892\) 0 0
\(893\) −5.30019 + 11.3649i −0.177364 + 0.380313i
\(894\) 0 0
\(895\) 5.74621 9.95273i 0.192075 0.332683i
\(896\) 0 0
\(897\) 12.0000 + 20.7846i 0.400668 + 0.693978i
\(898\) 0 0
\(899\) 10.2462 + 17.7470i 0.341730 + 0.591894i
\(900\) 0 0
\(901\) −38.7386 −1.29057
\(902\) 0 0
\(903\) −1.75379 3.03765i −0.0583624 0.101087i
\(904\) 0 0
\(905\) 21.3693 0.710340
\(906\) 0 0
\(907\) −25.0885 + 43.4546i −0.833051 + 1.44289i 0.0625559 + 0.998041i \(0.480075\pi\)
−0.895607 + 0.444846i \(0.853259\pi\)
\(908\) 0 0
\(909\) −17.8078 + 30.8440i −0.590646 + 1.02303i
\(910\) 0 0
\(911\) −12.3845 −0.410316 −0.205158 0.978729i \(-0.565771\pi\)
−0.205158 + 0.978729i \(0.565771\pi\)
\(912\) 0 0
\(913\) −10.8078 −0.357685
\(914\) 0 0
\(915\) −6.56155 + 11.3649i −0.216918 + 0.375713i
\(916\) 0 0
\(917\) 3.53457 6.12205i 0.116722 0.202168i
\(918\) 0 0
\(919\) −20.2462 −0.667861 −0.333930 0.942598i \(-0.608375\pi\)
−0.333930 + 0.942598i \(0.608375\pi\)
\(920\) 0 0
\(921\) −30.6501 53.0875i −1.00995 1.74929i
\(922\) 0 0
\(923\) 32.4924 1.06950
\(924\) 0 0
\(925\) 2.34233 + 4.05703i 0.0770153 + 0.133394i
\(926\) 0 0
\(927\) −10.3423 17.9134i −0.339687 0.588355i
\(928\) 0 0
\(929\) 17.9924 31.1638i 0.590312 1.02245i −0.403878 0.914813i \(-0.632338\pi\)
0.994190 0.107638i \(-0.0343287\pi\)
\(930\) 0 0
\(931\) −17.0194 24.3086i −0.557789 0.796682i
\(932\) 0 0
\(933\) 5.12311 8.87348i 0.167723 0.290505i
\(934\) 0 0
\(935\) 2.56155 + 4.43674i 0.0837717 + 0.145097i
\(936\) 0 0
\(937\) 25.0885 + 43.4546i 0.819607 + 1.41960i 0.905972 + 0.423337i \(0.139141\pi\)
−0.0863653 + 0.996264i \(0.527525\pi\)
\(938\) 0 0
\(939\) 11.0540 0.360733
\(940\) 0 0
\(941\) 8.12311 + 14.0696i 0.264806 + 0.458657i 0.967513 0.252822i \(-0.0813589\pi\)
−0.702707 + 0.711479i \(0.748026\pi\)
\(942\) 0 0
\(943\) −28.6847 −0.934101
\(944\) 0 0
\(945\) 0.315342 0.546188i 0.0102581 0.0177675i
\(946\) 0 0
\(947\) 28.2462 48.9239i 0.917879 1.58981i 0.115247 0.993337i \(-0.463234\pi\)
0.802631 0.596475i \(-0.203433\pi\)
\(948\) 0 0
\(949\) 3.36932 0.109373
\(950\) 0 0
\(951\) 47.8617 1.55202
\(952\) 0 0
\(953\) 7.84233 13.5833i 0.254038 0.440007i −0.710596 0.703600i \(-0.751575\pi\)
0.964634 + 0.263594i \(0.0849078\pi\)
\(954\) 0 0
\(955\) −2.68466 + 4.64996i −0.0868735 + 0.150469i
\(956\) 0 0
\(957\) −5.12311 −0.165606
\(958\) 0 0
\(959\) −1.19224 2.06501i −0.0384993 0.0666828i
\(960\) 0 0
\(961\) 73.9848 2.38661
\(962\) 0 0
\(963\) 4.00000 + 6.92820i 0.128898 + 0.223258i
\(964\) 0 0
\(965\) −12.8078 22.1837i −0.412297 0.714119i
\(966\) 0 0
\(967\) 14.5616 25.2213i 0.468268 0.811064i −0.531074 0.847325i \(-0.678212\pi\)
0.999342 + 0.0362613i \(0.0115449\pi\)
\(968\) 0 0
\(969\) 24.1771 51.8418i 0.776680 1.66540i
\(970\) 0 0
\(971\) 8.96543 15.5286i 0.287714 0.498336i −0.685549 0.728026i \(-0.740438\pi\)
0.973264 + 0.229690i \(0.0737712\pi\)
\(972\) 0 0
\(973\) −1.93087 3.34436i −0.0619008 0.107215i
\(974\) 0 0
\(975\) 2.56155 + 4.43674i 0.0820353 + 0.142089i
\(976\) 0 0
\(977\) −30.3153 −0.969874 −0.484937 0.874549i \(-0.661157\pi\)
−0.484937 + 0.874549i \(0.661157\pi\)
\(978\) 0 0
\(979\) −1.34233 2.32498i −0.0429010 0.0743068i
\(980\) 0 0
\(981\) −50.7386 −1.61996
\(982\) 0 0
\(983\) 20.7808 35.9934i 0.662804 1.14801i −0.317072 0.948401i \(-0.602700\pi\)
0.979876 0.199608i \(-0.0639670\pi\)
\(984\) 0 0
\(985\) −7.21922 + 12.5041i −0.230024 + 0.398413i
\(986\) 0 0
\(987\) 3.23106 0.102846
\(988\) 0 0
\(989\) −14.6307 −0.465229
\(990\) 0 0
\(991\) −4.19224 + 7.26117i −0.133171 + 0.230659i −0.924897 0.380217i \(-0.875849\pi\)
0.791726 + 0.610876i \(0.209183\pi\)
\(992\) 0 0
\(993\) 30.0885 52.1149i 0.954831 1.65382i
\(994\) 0 0
\(995\) 2.87689 0.0912037
\(996\) 0 0
\(997\) 1.34233 + 2.32498i 0.0425120 + 0.0736329i 0.886498 0.462732i \(-0.153131\pi\)
−0.843986 + 0.536364i \(0.819797\pi\)
\(998\) 0 0
\(999\) −6.73863 −0.213201
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 1520.2.q.h.961.1 4
4.3 odd 2 190.2.e.c.11.2 4
12.11 even 2 1710.2.l.m.1531.1 4
19.7 even 3 inner 1520.2.q.h.881.1 4
20.3 even 4 950.2.j.f.49.2 8
20.7 even 4 950.2.j.f.49.3 8
20.19 odd 2 950.2.e.h.201.1 4
76.7 odd 6 190.2.e.c.121.2 yes 4
76.11 odd 6 3610.2.a.k.1.1 2
76.27 even 6 3610.2.a.u.1.2 2
228.83 even 6 1710.2.l.m.1261.1 4
380.7 even 12 950.2.j.f.349.2 8
380.83 even 12 950.2.j.f.349.3 8
380.159 odd 6 950.2.e.h.501.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
190.2.e.c.11.2 4 4.3 odd 2
190.2.e.c.121.2 yes 4 76.7 odd 6
950.2.e.h.201.1 4 20.19 odd 2
950.2.e.h.501.1 4 380.159 odd 6
950.2.j.f.49.2 8 20.3 even 4
950.2.j.f.49.3 8 20.7 even 4
950.2.j.f.349.2 8 380.7 even 12
950.2.j.f.349.3 8 380.83 even 12
1520.2.q.h.881.1 4 19.7 even 3 inner
1520.2.q.h.961.1 4 1.1 even 1 trivial
1710.2.l.m.1261.1 4 228.83 even 6
1710.2.l.m.1531.1 4 12.11 even 2
3610.2.a.k.1.1 2 76.11 odd 6
3610.2.a.u.1.2 2 76.27 even 6