Properties

Label 1024.2.g.e.129.1
Level $1024$
Weight $2$
Character 1024.129
Analytic conductor $8.177$
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] = [1024,2,Mod(129,1024)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1024, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([0, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1024.129");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1024 = 2^{10} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1024.g (of order \(8\), degree \(4\), 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: \(8.17668116698\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{8})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8x^{14} + 32x^{12} - 64x^{10} + 127x^{8} - 576x^{6} + 2592x^{4} - 5832x^{2} + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label 129.1
Root \(1.66798 - 0.466730i\) of defining polynomial
Character \(\chi\) \(=\) 1024.129
Dual form 1024.2.g.e.897.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.20125 + 2.90008i) q^{3} +(2.92673 - 1.21229i) q^{5} +(0.933460 - 0.933460i) q^{7} +(-4.84613 - 4.84613i) q^{9} +O(q^{10})\) \(q+(-1.20125 + 2.90008i) q^{3} +(2.92673 - 1.21229i) q^{5} +(0.933460 - 0.933460i) q^{7} +(-4.84613 - 4.84613i) q^{9} +(-0.435885 - 1.05232i) q^{11} +(2.65154 + 1.09830i) q^{13} +9.94402i q^{15} -1.61082i q^{17} +(5.66214 + 2.34533i) q^{19} +(1.58579 + 3.82843i) q^{21} +(1.67967 + 1.67967i) q^{23} +(3.56057 - 3.56057i) q^{25} +(11.1753 - 4.62898i) q^{27} +(2.92673 - 7.06575i) q^{29} -1.53073 q^{31} +3.57542 q^{33} +(1.60036 - 3.86361i) q^{35} +(5.04072 - 2.08793i) q^{37} +(-6.37033 + 6.37033i) q^{39} +(1.03540 + 1.03540i) q^{41} +(1.64959 + 3.98247i) q^{43} +(-20.0582 - 8.30839i) q^{45} -2.97641i q^{47} +5.25731i q^{49} +(4.67151 + 1.93500i) q^{51} +(4.21229 + 10.1694i) q^{53} +(-2.55144 - 2.55144i) q^{55} +(-13.6033 + 13.6033i) q^{57} +(-8.08380 + 3.34842i) q^{59} +(-5.23733 + 12.6440i) q^{61} -9.04733 q^{63} +9.09181 q^{65} +(4.91593 - 11.8681i) q^{67} +(-6.88887 + 2.85346i) q^{69} +(-5.88894 + 5.88894i) q^{71} +(-3.71444 - 3.71444i) q^{73} +(6.04879 + 14.6031i) q^{75} +(-1.38918 - 0.575417i) q^{77} -10.4058i q^{79} +17.4096i q^{81} +(-2.73199 - 1.13163i) q^{83} +(-1.95278 - 4.71444i) q^{85} +(16.9755 + 16.9755i) q^{87} +(6.56361 - 6.56361i) q^{89} +(3.50033 - 1.44988i) q^{91} +(1.83880 - 4.43925i) q^{93} +19.4148 q^{95} +1.01037 q^{97} +(-2.98732 + 7.21203i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{5} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{5} - 16 q^{9} - 24 q^{13} + 48 q^{21} - 32 q^{25} + 8 q^{29} + 80 q^{33} + 24 q^{37} - 16 q^{41} - 104 q^{45} + 56 q^{53} - 80 q^{57} - 40 q^{61} + 32 q^{65} - 32 q^{73} - 32 q^{77} + 32 q^{85} + 32 q^{89} - 16 q^{93} + 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1024\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(1023\)
\(\chi(n)\) \(e\left(\frac{7}{8}\right)\) \(1\)

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
\(3\) −1.20125 + 2.90008i −0.693543 + 1.67436i 0.0439740 + 0.999033i \(0.485998\pi\)
−0.737517 + 0.675328i \(0.764002\pi\)
\(4\) 0 0
\(5\) 2.92673 1.21229i 1.30887 0.542153i 0.384318 0.923201i \(-0.374437\pi\)
0.924556 + 0.381047i \(0.124437\pi\)
\(6\) 0 0
\(7\) 0.933460 0.933460i 0.352815 0.352815i −0.508341 0.861156i \(-0.669741\pi\)
0.861156 + 0.508341i \(0.169741\pi\)
\(8\) 0 0
\(9\) −4.84613 4.84613i −1.61538 1.61538i
\(10\) 0 0
\(11\) −0.435885 1.05232i −0.131424 0.317286i 0.844445 0.535643i \(-0.179931\pi\)
−0.975869 + 0.218356i \(0.929931\pi\)
\(12\) 0 0
\(13\) 2.65154 + 1.09830i 0.735405 + 0.304615i 0.718771 0.695247i \(-0.244705\pi\)
0.0166339 + 0.999862i \(0.494705\pi\)
\(14\) 0 0
\(15\) 9.94402i 2.56753i
\(16\) 0 0
\(17\) 1.61082i 0.390681i −0.980735 0.195341i \(-0.937419\pi\)
0.980735 0.195341i \(-0.0625813\pi\)
\(18\) 0 0
\(19\) 5.66214 + 2.34533i 1.29898 + 0.538056i 0.921650 0.388021i \(-0.126841\pi\)
0.377333 + 0.926078i \(0.376841\pi\)
\(20\) 0 0
\(21\) 1.58579 + 3.82843i 0.346047 + 0.835431i
\(22\) 0 0
\(23\) 1.67967 + 1.67967i 0.350235 + 0.350235i 0.860197 0.509962i \(-0.170341\pi\)
−0.509962 + 0.860197i \(0.670341\pi\)
\(24\) 0 0
\(25\) 3.56057 3.56057i 0.712114 0.712114i
\(26\) 0 0
\(27\) 11.1753 4.62898i 2.15070 0.890847i
\(28\) 0 0
\(29\) 2.92673 7.06575i 0.543480 1.31208i −0.378772 0.925490i \(-0.623654\pi\)
0.922253 0.386588i \(-0.126346\pi\)
\(30\) 0 0
\(31\) −1.53073 −0.274928 −0.137464 0.990507i \(-0.543895\pi\)
−0.137464 + 0.990507i \(0.543895\pi\)
\(32\) 0 0
\(33\) 3.57542 0.622400
\(34\) 0 0
\(35\) 1.60036 3.86361i 0.270510 0.653069i
\(36\) 0 0
\(37\) 5.04072 2.08793i 0.828689 0.343254i 0.0723054 0.997383i \(-0.476964\pi\)
0.756384 + 0.654128i \(0.226964\pi\)
\(38\) 0 0
\(39\) −6.37033 + 6.37033i −1.02007 + 1.02007i
\(40\) 0 0
\(41\) 1.03540 + 1.03540i 0.161703 + 0.161703i 0.783321 0.621618i \(-0.213524\pi\)
−0.621618 + 0.783321i \(0.713524\pi\)
\(42\) 0 0
\(43\) 1.64959 + 3.98247i 0.251561 + 0.607321i 0.998330 0.0577615i \(-0.0183963\pi\)
−0.746770 + 0.665083i \(0.768396\pi\)
\(44\) 0 0
\(45\) −20.0582 8.30839i −2.99011 1.23854i
\(46\) 0 0
\(47\) 2.97641i 0.434154i −0.976154 0.217077i \(-0.930348\pi\)
0.976154 0.217077i \(-0.0696523\pi\)
\(48\) 0 0
\(49\) 5.25731i 0.751044i
\(50\) 0 0
\(51\) 4.67151 + 1.93500i 0.654142 + 0.270954i
\(52\) 0 0
\(53\) 4.21229 + 10.1694i 0.578603 + 1.39687i 0.894067 + 0.447933i \(0.147840\pi\)
−0.315464 + 0.948937i \(0.602160\pi\)
\(54\) 0 0
\(55\) −2.55144 2.55144i −0.344036 0.344036i
\(56\) 0 0
\(57\) −13.6033 + 13.6033i −1.80180 + 1.80180i
\(58\) 0 0
\(59\) −8.08380 + 3.34842i −1.05242 + 0.435927i −0.840756 0.541415i \(-0.817889\pi\)
−0.211666 + 0.977342i \(0.567889\pi\)
\(60\) 0 0
\(61\) −5.23733 + 12.6440i −0.670571 + 1.61890i 0.110072 + 0.993924i \(0.464892\pi\)
−0.780643 + 0.624978i \(0.785108\pi\)
\(62\) 0 0
\(63\) −9.04733 −1.13986
\(64\) 0 0
\(65\) 9.09181 1.12770
\(66\) 0 0
\(67\) 4.91593 11.8681i 0.600577 1.44992i −0.272413 0.962180i \(-0.587822\pi\)
0.872989 0.487739i \(-0.162178\pi\)
\(68\) 0 0
\(69\) −6.88887 + 2.85346i −0.829322 + 0.343516i
\(70\) 0 0
\(71\) −5.88894 + 5.88894i −0.698889 + 0.698889i −0.964171 0.265282i \(-0.914535\pi\)
0.265282 + 0.964171i \(0.414535\pi\)
\(72\) 0 0
\(73\) −3.71444 3.71444i −0.434742 0.434742i 0.455496 0.890238i \(-0.349462\pi\)
−0.890238 + 0.455496i \(0.849462\pi\)
\(74\) 0 0
\(75\) 6.04879 + 14.6031i 0.698454 + 1.68622i
\(76\) 0 0
\(77\) −1.38918 0.575417i −0.158312 0.0655748i
\(78\) 0 0
\(79\) 10.4058i 1.17074i −0.810766 0.585370i \(-0.800949\pi\)
0.810766 0.585370i \(-0.199051\pi\)
\(80\) 0 0
\(81\) 17.4096i 1.93439i
\(82\) 0 0
\(83\) −2.73199 1.13163i −0.299874 0.124212i 0.227673 0.973738i \(-0.426888\pi\)
−0.527547 + 0.849526i \(0.676888\pi\)
\(84\) 0 0
\(85\) −1.95278 4.71444i −0.211809 0.511353i
\(86\) 0 0
\(87\) 16.9755 + 16.9755i 1.81996 + 1.81996i
\(88\) 0 0
\(89\) 6.56361 6.56361i 0.695741 0.695741i −0.267748 0.963489i \(-0.586280\pi\)
0.963489 + 0.267748i \(0.0862795\pi\)
\(90\) 0 0
\(91\) 3.50033 1.44988i 0.366934 0.151989i
\(92\) 0 0
\(93\) 1.83880 4.43925i 0.190674 0.460329i
\(94\) 0 0
\(95\) 19.4148 1.99191
\(96\) 0 0
\(97\) 1.01037 0.102587 0.0512937 0.998684i \(-0.483666\pi\)
0.0512937 + 0.998684i \(0.483666\pi\)
\(98\) 0 0
\(99\) −2.98732 + 7.21203i −0.300237 + 0.724836i
\(100\) 0 0
\(101\) −10.0658 + 4.16937i −1.00158 + 0.414868i −0.822376 0.568944i \(-0.807352\pi\)
−0.179204 + 0.983812i \(0.557352\pi\)
\(102\) 0 0
\(103\) −10.1802 + 10.1802i −1.00308 + 1.00308i −0.00308887 + 0.999995i \(0.500983\pi\)
−0.999995 + 0.00308887i \(0.999017\pi\)
\(104\) 0 0
\(105\) 9.28234 + 9.28234i 0.905864 + 0.905864i
\(106\) 0 0
\(107\) −3.79522 9.16246i −0.366897 0.885769i −0.994255 0.107038i \(-0.965864\pi\)
0.627357 0.778731i \(-0.284136\pi\)
\(108\) 0 0
\(109\) 9.75516 + 4.04072i 0.934375 + 0.387031i 0.797336 0.603535i \(-0.206242\pi\)
0.137038 + 0.990566i \(0.456242\pi\)
\(110\) 0 0
\(111\) 17.1266i 1.62559i
\(112\) 0 0
\(113\) 2.57112i 0.241871i 0.992660 + 0.120935i \(0.0385894\pi\)
−0.992660 + 0.120935i \(0.961411\pi\)
\(114\) 0 0
\(115\) 6.95218 + 2.87969i 0.648294 + 0.268532i
\(116\) 0 0
\(117\) −7.52718 18.1722i −0.695888 1.68002i
\(118\) 0 0
\(119\) −1.50364 1.50364i −0.137838 0.137838i
\(120\) 0 0
\(121\) 6.86079 6.86079i 0.623709 0.623709i
\(122\) 0 0
\(123\) −4.24653 + 1.75897i −0.382897 + 0.158601i
\(124\) 0 0
\(125\) 0.0429201 0.103618i 0.00383889 0.00926790i
\(126\) 0 0
\(127\) 6.16126 0.546723 0.273362 0.961911i \(-0.411864\pi\)
0.273362 + 0.961911i \(0.411864\pi\)
\(128\) 0 0
\(129\) −13.5311 −1.19134
\(130\) 0 0
\(131\) −0.796984 + 1.92409i −0.0696328 + 0.168108i −0.954864 0.297042i \(-0.904000\pi\)
0.885232 + 0.465151i \(0.154000\pi\)
\(132\) 0 0
\(133\) 7.47465 3.09610i 0.648135 0.268466i
\(134\) 0 0
\(135\) 27.0955 27.0955i 2.33201 2.33201i
\(136\) 0 0
\(137\) −11.1315 11.1315i −0.951029 0.951029i 0.0478269 0.998856i \(-0.484770\pi\)
−0.998856 + 0.0478269i \(0.984770\pi\)
\(138\) 0 0
\(139\) 4.12347 + 9.95493i 0.349748 + 0.844366i 0.996649 + 0.0817925i \(0.0260645\pi\)
−0.646902 + 0.762574i \(0.723936\pi\)
\(140\) 0 0
\(141\) 8.63182 + 3.57542i 0.726930 + 0.301104i
\(142\) 0 0
\(143\) 3.26900i 0.273368i
\(144\) 0 0
\(145\) 24.2276i 2.01199i
\(146\) 0 0
\(147\) −15.2466 6.31535i −1.25752 0.520881i
\(148\) 0 0
\(149\) 5.41201 + 13.0658i 0.443369 + 1.07039i 0.974759 + 0.223261i \(0.0716701\pi\)
−0.531389 + 0.847128i \(0.678330\pi\)
\(150\) 0 0
\(151\) −14.0916 14.0916i −1.14676 1.14676i −0.987186 0.159572i \(-0.948988\pi\)
−0.159572 0.987186i \(-0.551012\pi\)
\(152\) 0 0
\(153\) −7.80625 + 7.80625i −0.631098 + 0.631098i
\(154\) 0 0
\(155\) −4.48005 + 1.85570i −0.359846 + 0.149053i
\(156\) 0 0
\(157\) −1.18726 + 2.86629i −0.0947534 + 0.228755i −0.964149 0.265362i \(-0.914509\pi\)
0.869395 + 0.494117i \(0.164509\pi\)
\(158\) 0 0
\(159\) −34.5520 −2.74015
\(160\) 0 0
\(161\) 3.13580 0.247136
\(162\) 0 0
\(163\) −3.81438 + 9.20872i −0.298765 + 0.721283i 0.701200 + 0.712964i \(0.252648\pi\)
−0.999965 + 0.00831846i \(0.997352\pi\)
\(164\) 0 0
\(165\) 10.4643 4.33445i 0.814643 0.337436i
\(166\) 0 0
\(167\) 6.15971 6.15971i 0.476653 0.476653i −0.427407 0.904059i \(-0.640573\pi\)
0.904059 + 0.427407i \(0.140573\pi\)
\(168\) 0 0
\(169\) −3.36800 3.36800i −0.259077 0.259077i
\(170\) 0 0
\(171\) −16.0737 38.8052i −1.22918 2.96751i
\(172\) 0 0
\(173\) 9.71198 + 4.02283i 0.738388 + 0.305850i 0.719994 0.693980i \(-0.244145\pi\)
0.0183942 + 0.999831i \(0.494145\pi\)
\(174\) 0 0
\(175\) 6.64730i 0.502488i
\(176\) 0 0
\(177\) 27.4660i 2.06447i
\(178\) 0 0
\(179\) −15.3222 6.34665i −1.14523 0.474371i −0.272301 0.962212i \(-0.587785\pi\)
−0.872932 + 0.487841i \(0.837785\pi\)
\(180\) 0 0
\(181\) 1.70912 + 4.12619i 0.127038 + 0.306697i 0.974583 0.224026i \(-0.0719202\pi\)
−0.847545 + 0.530724i \(0.821920\pi\)
\(182\) 0 0
\(183\) −30.3773 30.3773i −2.24556 2.24556i
\(184\) 0 0
\(185\) 12.2216 12.2216i 0.898553 0.898553i
\(186\) 0 0
\(187\) −1.69510 + 0.702133i −0.123958 + 0.0513450i
\(188\) 0 0
\(189\) 6.11077 14.7527i 0.444493 1.07310i
\(190\) 0 0
\(191\) 14.7979 1.07074 0.535371 0.844617i \(-0.320172\pi\)
0.535371 + 0.844617i \(0.320172\pi\)
\(192\) 0 0
\(193\) −6.66722 −0.479917 −0.239959 0.970783i \(-0.577134\pi\)
−0.239959 + 0.970783i \(0.577134\pi\)
\(194\) 0 0
\(195\) −10.9216 + 26.3670i −0.782108 + 1.88818i
\(196\) 0 0
\(197\) 15.0936 6.25199i 1.07538 0.445436i 0.226492 0.974013i \(-0.427274\pi\)
0.848885 + 0.528577i \(0.177274\pi\)
\(198\) 0 0
\(199\) 0.682332 0.682332i 0.0483693 0.0483693i −0.682508 0.730878i \(-0.739111\pi\)
0.730878 + 0.682508i \(0.239111\pi\)
\(200\) 0 0
\(201\) 28.5132 + 28.5132i 2.01116 + 2.01116i
\(202\) 0 0
\(203\) −3.86361 9.32758i −0.271172 0.654668i
\(204\) 0 0
\(205\) 4.28556 + 1.77514i 0.299317 + 0.123981i
\(206\) 0 0
\(207\) 16.2798i 1.13152i
\(208\) 0 0
\(209\) 6.98067i 0.482863i
\(210\) 0 0
\(211\) 15.5403 + 6.43698i 1.06984 + 0.443140i 0.846933 0.531700i \(-0.178447\pi\)
0.222903 + 0.974841i \(0.428447\pi\)
\(212\) 0 0
\(213\) −10.0043 24.1525i −0.685483 1.65490i
\(214\) 0 0
\(215\) 9.65583 + 9.65583i 0.658522 + 0.658522i
\(216\) 0 0
\(217\) −1.42888 + 1.42888i −0.0969986 + 0.0969986i
\(218\) 0 0
\(219\) 15.2341 6.31019i 1.02943 0.426403i
\(220\) 0 0
\(221\) 1.76917 4.27116i 0.119007 0.287309i
\(222\) 0 0
\(223\) −1.47654 −0.0988764 −0.0494382 0.998777i \(-0.515743\pi\)
−0.0494382 + 0.998777i \(0.515743\pi\)
\(224\) 0 0
\(225\) −34.5099 −2.30066
\(226\) 0 0
\(227\) 10.8203 26.1225i 0.718169 1.73381i 0.0396690 0.999213i \(-0.487370\pi\)
0.678500 0.734600i \(-0.262630\pi\)
\(228\) 0 0
\(229\) −8.73012 + 3.61614i −0.576903 + 0.238961i −0.652005 0.758215i \(-0.726072\pi\)
0.0751019 + 0.997176i \(0.476072\pi\)
\(230\) 0 0
\(231\) 3.33751 3.33751i 0.219592 0.219592i
\(232\) 0 0
\(233\) −9.99248 9.99248i −0.654629 0.654629i 0.299475 0.954104i \(-0.403189\pi\)
−0.954104 + 0.299475i \(0.903189\pi\)
\(234\) 0 0
\(235\) −3.60828 8.71115i −0.235378 0.568253i
\(236\) 0 0
\(237\) 30.1775 + 12.4999i 1.96024 + 0.811959i
\(238\) 0 0
\(239\) 10.5759i 0.684097i −0.939682 0.342049i \(-0.888879\pi\)
0.939682 0.342049i \(-0.111121\pi\)
\(240\) 0 0
\(241\) 0.690846i 0.0445013i −0.999752 0.0222507i \(-0.992917\pi\)
0.999752 0.0222507i \(-0.00708319\pi\)
\(242\) 0 0
\(243\) −16.9630 7.02632i −1.08818 0.450739i
\(244\) 0 0
\(245\) 6.37339 + 15.3867i 0.407181 + 0.983021i
\(246\) 0 0
\(247\) 12.4375 + 12.4375i 0.791379 + 0.791379i
\(248\) 0 0
\(249\) 6.56361 6.56361i 0.415952 0.415952i
\(250\) 0 0
\(251\) −14.6903 + 6.08493i −0.927244 + 0.384077i −0.794632 0.607091i \(-0.792336\pi\)
−0.132612 + 0.991168i \(0.542336\pi\)
\(252\) 0 0
\(253\) 1.03540 2.49969i 0.0650953 0.157154i
\(254\) 0 0
\(255\) 16.0180 1.00309
\(256\) 0 0
\(257\) −11.7928 −0.735612 −0.367806 0.929902i \(-0.619891\pi\)
−0.367806 + 0.929902i \(0.619891\pi\)
\(258\) 0 0
\(259\) 2.75631 6.65431i 0.171269 0.413479i
\(260\) 0 0
\(261\) −48.4249 + 20.0582i −2.99742 + 1.24157i
\(262\) 0 0
\(263\) 20.4934 20.4934i 1.26368 1.26368i 0.314384 0.949296i \(-0.398202\pi\)
0.949296 0.314384i \(-0.101798\pi\)
\(264\) 0 0
\(265\) 24.6565 + 24.6565i 1.51464 + 1.51464i
\(266\) 0 0
\(267\) 11.1504 + 26.9195i 0.682395 + 1.64745i
\(268\) 0 0
\(269\) −9.24018 3.82741i −0.563384 0.233361i 0.0827697 0.996569i \(-0.473623\pi\)
−0.646153 + 0.763208i \(0.723623\pi\)
\(270\) 0 0
\(271\) 25.2037i 1.53102i 0.643426 + 0.765508i \(0.277512\pi\)
−0.643426 + 0.765508i \(0.722488\pi\)
\(272\) 0 0
\(273\) 11.8929i 0.719791i
\(274\) 0 0
\(275\) −5.29885 2.19486i −0.319533 0.132355i
\(276\) 0 0
\(277\) −5.65583 13.6544i −0.339826 0.820413i −0.997732 0.0673127i \(-0.978557\pi\)
0.657906 0.753100i \(-0.271443\pi\)
\(278\) 0 0
\(279\) 7.41813 + 7.41813i 0.444112 + 0.444112i
\(280\) 0 0
\(281\) −4.45247 + 4.45247i −0.265612 + 0.265612i −0.827329 0.561717i \(-0.810141\pi\)
0.561717 + 0.827329i \(0.310141\pi\)
\(282\) 0 0
\(283\) −22.5158 + 9.32635i −1.33843 + 0.554394i −0.933047 0.359755i \(-0.882860\pi\)
−0.405379 + 0.914149i \(0.632860\pi\)
\(284\) 0 0
\(285\) −23.3220 + 56.3044i −1.38148 + 3.33518i
\(286\) 0 0
\(287\) 1.93302 0.114102
\(288\) 0 0
\(289\) 14.4053 0.847368
\(290\) 0 0
\(291\) −1.21371 + 2.93015i −0.0711488 + 0.171769i
\(292\) 0 0
\(293\) −6.66906 + 2.76241i −0.389611 + 0.161382i −0.568885 0.822417i \(-0.692625\pi\)
0.179274 + 0.983799i \(0.442625\pi\)
\(294\) 0 0
\(295\) −19.5998 + 19.5998i −1.14115 + 1.14115i
\(296\) 0 0
\(297\) −9.74233 9.74233i −0.565307 0.565307i
\(298\) 0 0
\(299\) 2.60892 + 6.29848i 0.150878 + 0.364251i
\(300\) 0 0
\(301\) 5.25731 + 2.17765i 0.303026 + 0.125518i
\(302\) 0 0
\(303\) 34.1999i 1.96474i
\(304\) 0 0
\(305\) 43.3548i 2.48249i
\(306\) 0 0
\(307\) −16.3327 6.76523i −0.932158 0.386112i −0.135661 0.990755i \(-0.543316\pi\)
−0.796497 + 0.604643i \(0.793316\pi\)
\(308\) 0 0
\(309\) −17.2944 41.7523i −0.983843 2.37521i
\(310\) 0 0
\(311\) −9.48818 9.48818i −0.538025 0.538025i 0.384923 0.922949i \(-0.374228\pi\)
−0.922949 + 0.384923i \(0.874228\pi\)
\(312\) 0 0
\(313\) −8.39347 + 8.39347i −0.474427 + 0.474427i −0.903344 0.428917i \(-0.858895\pi\)
0.428917 + 0.903344i \(0.358895\pi\)
\(314\) 0 0
\(315\) −26.4791 + 10.9680i −1.49193 + 0.617977i
\(316\) 0 0
\(317\) 11.0142 26.5907i 0.618621 1.49348i −0.234683 0.972072i \(-0.575405\pi\)
0.853305 0.521413i \(-0.174595\pi\)
\(318\) 0 0
\(319\) −8.71115 −0.487731
\(320\) 0 0
\(321\) 31.1309 1.73756
\(322\) 0 0
\(323\) 3.77791 9.12069i 0.210209 0.507489i
\(324\) 0 0
\(325\) 13.3516 5.53040i 0.740612 0.306772i
\(326\) 0 0
\(327\) −23.4368 + 23.4368i −1.29606 + 1.29606i
\(328\) 0 0
\(329\) −2.77836 2.77836i −0.153176 0.153176i
\(330\) 0 0
\(331\) 3.90491 + 9.42728i 0.214633 + 0.518170i 0.994124 0.108243i \(-0.0345224\pi\)
−0.779491 + 0.626413i \(0.784522\pi\)
\(332\) 0 0
\(333\) −34.5464 14.3096i −1.89313 0.784160i
\(334\) 0 0
\(335\) 40.6943i 2.22337i
\(336\) 0 0
\(337\) 2.82843i 0.154074i −0.997028 0.0770371i \(-0.975454\pi\)
0.997028 0.0770371i \(-0.0245460\pi\)
\(338\) 0 0
\(339\) −7.45645 3.08856i −0.404979 0.167748i
\(340\) 0 0
\(341\) 0.667224 + 1.61082i 0.0361322 + 0.0872308i
\(342\) 0 0
\(343\) 11.4417 + 11.4417i 0.617794 + 0.617794i
\(344\) 0 0
\(345\) −16.7026 + 16.7026i −0.899239 + 0.899239i
\(346\) 0 0
\(347\) −0.244423 + 0.101243i −0.0131213 + 0.00543503i −0.389234 0.921139i \(-0.627260\pi\)
0.376113 + 0.926574i \(0.377260\pi\)
\(348\) 0 0
\(349\) 0.715459 1.72727i 0.0382976 0.0924586i −0.903573 0.428433i \(-0.859066\pi\)
0.941871 + 0.335975i \(0.109066\pi\)
\(350\) 0 0
\(351\) 34.7159 1.85300
\(352\) 0 0
\(353\) 22.0270 1.17238 0.586189 0.810175i \(-0.300628\pi\)
0.586189 + 0.810175i \(0.300628\pi\)
\(354\) 0 0
\(355\) −10.0962 + 24.3745i −0.535853 + 1.29366i
\(356\) 0 0
\(357\) 6.16691 2.55442i 0.326388 0.135194i
\(358\) 0 0
\(359\) 9.40689 9.40689i 0.496477 0.496477i −0.413863 0.910339i \(-0.635821\pi\)
0.910339 + 0.413863i \(0.135821\pi\)
\(360\) 0 0
\(361\) 13.1242 + 13.1242i 0.690746 + 0.690746i
\(362\) 0 0
\(363\) 11.6553 + 28.1384i 0.611745 + 1.47688i
\(364\) 0 0
\(365\) −15.3741 6.36818i −0.804720 0.333326i
\(366\) 0 0
\(367\) 6.57941i 0.343442i 0.985146 + 0.171721i \(0.0549328\pi\)
−0.985146 + 0.171721i \(0.945067\pi\)
\(368\) 0 0
\(369\) 10.0354i 0.522422i
\(370\) 0 0
\(371\) 13.4247 + 5.56069i 0.696976 + 0.288697i
\(372\) 0 0
\(373\) 5.47997 + 13.2298i 0.283742 + 0.685014i 0.999917 0.0129074i \(-0.00410866\pi\)
−0.716175 + 0.697921i \(0.754109\pi\)
\(374\) 0 0
\(375\) 0.248943 + 0.248943i 0.0128554 + 0.0128554i
\(376\) 0 0
\(377\) 15.5207 15.5207i 0.799356 0.799356i
\(378\) 0 0
\(379\) −8.57046 + 3.55000i −0.440235 + 0.182351i −0.591781 0.806099i \(-0.701575\pi\)
0.151546 + 0.988450i \(0.451575\pi\)
\(380\) 0 0
\(381\) −7.40122 + 17.8681i −0.379176 + 0.915412i
\(382\) 0 0
\(383\) −21.6316 −1.10532 −0.552661 0.833406i \(-0.686388\pi\)
−0.552661 + 0.833406i \(0.686388\pi\)
\(384\) 0 0
\(385\) −4.76333 −0.242762
\(386\) 0 0
\(387\) 11.3054 27.2937i 0.574687 1.38742i
\(388\) 0 0
\(389\) −7.99494 + 3.31161i −0.405360 + 0.167906i −0.576041 0.817421i \(-0.695403\pi\)
0.170681 + 0.985326i \(0.445403\pi\)
\(390\) 0 0
\(391\) 2.70564 2.70564i 0.136830 0.136830i
\(392\) 0 0
\(393\) −4.62263 4.62263i −0.233181 0.233181i
\(394\) 0 0
\(395\) −12.6148 30.4549i −0.634721 1.53235i
\(396\) 0 0
\(397\) 2.97276 + 1.23136i 0.149199 + 0.0618001i 0.456033 0.889963i \(-0.349270\pi\)
−0.306834 + 0.951763i \(0.599270\pi\)
\(398\) 0 0
\(399\) 25.3963i 1.27140i
\(400\) 0 0
\(401\) 22.2175i 1.10949i 0.832021 + 0.554744i \(0.187184\pi\)
−0.832021 + 0.554744i \(0.812816\pi\)
\(402\) 0 0
\(403\) −4.05880 1.68121i −0.202183 0.0837471i
\(404\) 0 0
\(405\) 21.1055 + 50.9531i 1.04874 + 2.53188i
\(406\) 0 0
\(407\) −4.39435 4.39435i −0.217820 0.217820i
\(408\) 0 0
\(409\) −17.8932 + 17.8932i −0.884760 + 0.884760i −0.994014 0.109254i \(-0.965154\pi\)
0.109254 + 0.994014i \(0.465154\pi\)
\(410\) 0 0
\(411\) 45.6540 18.9105i 2.25195 0.932786i
\(412\) 0 0
\(413\) −4.42029 + 10.6715i −0.217508 + 0.525111i
\(414\) 0 0
\(415\) −9.36765 −0.459840
\(416\) 0 0
\(417\) −33.8234 −1.65634
\(418\) 0 0
\(419\) 9.62029 23.2254i 0.469982 1.13464i −0.494189 0.869354i \(-0.664535\pi\)
0.964171 0.265282i \(-0.0854650\pi\)
\(420\) 0 0
\(421\) −20.3467 + 8.42786i −0.991635 + 0.410749i −0.818723 0.574189i \(-0.805318\pi\)
−0.172912 + 0.984937i \(0.555318\pi\)
\(422\) 0 0
\(423\) −14.4241 + 14.4241i −0.701322 + 0.701322i
\(424\) 0 0
\(425\) −5.73544 5.73544i −0.278210 0.278210i
\(426\) 0 0
\(427\) 6.91385 + 16.6915i 0.334585 + 0.807759i
\(428\) 0 0
\(429\) 9.48036 + 3.92689i 0.457716 + 0.189592i
\(430\) 0 0
\(431\) 32.2075i 1.55138i 0.631115 + 0.775689i \(0.282598\pi\)
−0.631115 + 0.775689i \(0.717402\pi\)
\(432\) 0 0
\(433\) 29.4806i 1.41675i −0.705837 0.708374i \(-0.749429\pi\)
0.705837 0.708374i \(-0.250571\pi\)
\(434\) 0 0
\(435\) 70.2620 + 29.1035i 3.36880 + 1.39540i
\(436\) 0 0
\(437\) 5.57112 + 13.4499i 0.266503 + 0.643395i
\(438\) 0 0
\(439\) 13.2529 + 13.2529i 0.632526 + 0.632526i 0.948701 0.316175i \(-0.102399\pi\)
−0.316175 + 0.948701i \(0.602399\pi\)
\(440\) 0 0
\(441\) 25.4776 25.4776i 1.21322 1.21322i
\(442\) 0 0
\(443\) 31.7407 13.1474i 1.50805 0.624653i 0.532893 0.846183i \(-0.321105\pi\)
0.975154 + 0.221529i \(0.0711049\pi\)
\(444\) 0 0
\(445\) 11.2529 27.1669i 0.533439 1.28784i
\(446\) 0 0
\(447\) −44.3929 −2.09971
\(448\) 0 0
\(449\) −34.2227 −1.61507 −0.807534 0.589821i \(-0.799198\pi\)
−0.807534 + 0.589821i \(0.799198\pi\)
\(450\) 0 0
\(451\) 0.638259 1.54089i 0.0300544 0.0725578i
\(452\) 0 0
\(453\) 57.7943 23.9392i 2.71541 1.12476i
\(454\) 0 0
\(455\) 8.48684 8.48684i 0.397869 0.397869i
\(456\) 0 0
\(457\) −9.79873 9.79873i −0.458365 0.458365i 0.439753 0.898119i \(-0.355066\pi\)
−0.898119 + 0.439753i \(0.855066\pi\)
\(458\) 0 0
\(459\) −7.45645 18.0015i −0.348037 0.840237i
\(460\) 0 0
\(461\) −4.43134 1.83552i −0.206388 0.0854887i 0.277095 0.960843i \(-0.410628\pi\)
−0.483483 + 0.875354i \(0.660628\pi\)
\(462\) 0 0
\(463\) 24.8164i 1.15332i 0.816985 + 0.576658i \(0.195644\pi\)
−0.816985 + 0.576658i \(0.804356\pi\)
\(464\) 0 0
\(465\) 15.2216i 0.705887i
\(466\) 0 0
\(467\) −15.7099 6.50725i −0.726967 0.301120i −0.0116621 0.999932i \(-0.503712\pi\)
−0.715305 + 0.698812i \(0.753712\pi\)
\(468\) 0 0
\(469\) −6.48958 15.6672i −0.299661 0.723445i
\(470\) 0 0
\(471\) −6.88628 6.88628i −0.317303 0.317303i
\(472\) 0 0
\(473\) 3.47180 3.47180i 0.159633 0.159633i
\(474\) 0 0
\(475\) 28.5111 11.8097i 1.30818 0.541866i
\(476\) 0 0
\(477\) 28.8688 69.6954i 1.32181 3.19113i
\(478\) 0 0
\(479\) 35.7254 1.63234 0.816168 0.577815i \(-0.196095\pi\)
0.816168 + 0.577815i \(0.196095\pi\)
\(480\) 0 0
\(481\) 15.6589 0.713982
\(482\) 0 0
\(483\) −3.76689 + 9.09407i −0.171399 + 0.413795i
\(484\) 0 0
\(485\) 2.95708 1.22486i 0.134274 0.0556181i
\(486\) 0 0
\(487\) −5.34809 + 5.34809i −0.242345 + 0.242345i −0.817820 0.575475i \(-0.804817\pi\)
0.575475 + 0.817820i \(0.304817\pi\)
\(488\) 0 0
\(489\) −22.1240 22.1240i −1.00048 1.00048i
\(490\) 0 0
\(491\) −16.8853 40.7647i −0.762022 1.83968i −0.467291 0.884104i \(-0.654770\pi\)
−0.294731 0.955580i \(-0.595230\pi\)
\(492\) 0 0
\(493\) −11.3817 4.71444i −0.512604 0.212328i
\(494\) 0 0
\(495\) 24.7292i 1.11149i
\(496\) 0 0
\(497\) 10.9942i 0.493157i
\(498\) 0 0
\(499\) −4.71852 1.95447i −0.211230 0.0874943i 0.274560 0.961570i \(-0.411468\pi\)
−0.485790 + 0.874076i \(0.661468\pi\)
\(500\) 0 0
\(501\) 10.4643 + 25.2630i 0.467510 + 1.12867i
\(502\) 0 0
\(503\) −4.81469 4.81469i −0.214676 0.214676i 0.591574 0.806251i \(-0.298507\pi\)
−0.806251 + 0.591574i \(0.798507\pi\)
\(504\) 0 0
\(505\) −24.4053 + 24.4053i −1.08602 + 1.08602i
\(506\) 0 0
\(507\) 13.8133 5.72164i 0.613469 0.254107i
\(508\) 0 0
\(509\) −2.36909 + 5.71949i −0.105008 + 0.253512i −0.967647 0.252308i \(-0.918810\pi\)
0.862639 + 0.505821i \(0.168810\pi\)
\(510\) 0 0
\(511\) −6.93456 −0.306767
\(512\) 0 0
\(513\) 74.1328 3.27304
\(514\) 0 0
\(515\) −17.4533 + 42.1360i −0.769085 + 1.85674i
\(516\) 0 0
\(517\) −3.13213 + 1.29737i −0.137751 + 0.0570584i
\(518\) 0 0
\(519\) −23.3331 + 23.3331i −1.02421 + 1.02421i
\(520\) 0 0
\(521\) −13.8240 13.8240i −0.605642 0.605642i 0.336162 0.941804i \(-0.390871\pi\)
−0.941804 + 0.336162i \(0.890871\pi\)
\(522\) 0 0
\(523\) −3.11134 7.51144i −0.136049 0.328452i 0.841141 0.540815i \(-0.181884\pi\)
−0.977191 + 0.212363i \(0.931884\pi\)
\(524\) 0 0
\(525\) 19.2777 + 7.98508i 0.841347 + 0.348497i
\(526\) 0 0
\(527\) 2.46574i 0.107409i
\(528\) 0 0
\(529\) 17.3574i 0.754671i
\(530\) 0 0
\(531\) 55.4020 + 22.9483i 2.40424 + 0.995870i
\(532\) 0 0
\(533\) 1.60823 + 3.88260i 0.0696600 + 0.168174i
\(534\) 0 0
\(535\) −22.2152 22.2152i −0.960445 0.960445i
\(536\) 0 0
\(537\) 36.8116 36.8116i 1.58854 1.58854i
\(538\) 0 0
\(539\) 5.53237 2.29158i 0.238296 0.0987054i
\(540\) 0 0
\(541\) −9.10142 + 21.9728i −0.391300 + 0.944683i 0.598357 + 0.801230i \(0.295821\pi\)
−0.989657 + 0.143453i \(0.954179\pi\)
\(542\) 0 0
\(543\) −14.0194 −0.601629
\(544\) 0 0
\(545\) 33.4493 1.43281
\(546\) 0 0
\(547\) −16.2561 + 39.2456i −0.695060 + 1.67802i 0.0392692 + 0.999229i \(0.487497\pi\)
−0.734329 + 0.678794i \(0.762503\pi\)
\(548\) 0 0
\(549\) 86.6553 35.8938i 3.69836 1.53191i
\(550\) 0 0
\(551\) 33.1431 33.1431i 1.41194 1.41194i
\(552\) 0 0
\(553\) −9.71336 9.71336i −0.413054 0.413054i
\(554\) 0 0
\(555\) 20.7625 + 50.1250i 0.881317 + 2.12769i
\(556\) 0 0
\(557\) −38.8291 16.0835i −1.64524 0.681481i −0.648429 0.761275i \(-0.724574\pi\)
−0.996811 + 0.0797936i \(0.974574\pi\)
\(558\) 0 0
\(559\) 12.3714i 0.523256i
\(560\) 0 0
\(561\) 5.75936i 0.243160i
\(562\) 0 0
\(563\) −37.0331 15.3396i −1.56076 0.646487i −0.575536 0.817776i \(-0.695207\pi\)
−0.985221 + 0.171289i \(0.945207\pi\)
\(564\) 0 0
\(565\) 3.11695 + 7.52498i 0.131131 + 0.316578i
\(566\) 0 0
\(567\) 16.2511 + 16.2511i 0.682483 + 0.682483i
\(568\) 0 0
\(569\) −18.5518 + 18.5518i −0.777732 + 0.777732i −0.979445 0.201713i \(-0.935349\pi\)
0.201713 + 0.979445i \(0.435349\pi\)
\(570\) 0 0
\(571\) 4.69785 1.94591i 0.196599 0.0814340i −0.282212 0.959352i \(-0.591068\pi\)
0.478811 + 0.877918i \(0.341068\pi\)
\(572\) 0 0
\(573\) −17.7761 + 42.9152i −0.742606 + 1.79281i
\(574\) 0 0
\(575\) 11.9611 0.498814
\(576\) 0 0
\(577\) −36.3839 −1.51468 −0.757341 0.653020i \(-0.773502\pi\)
−0.757341 + 0.653020i \(0.773502\pi\)
\(578\) 0 0
\(579\) 8.00902 19.3355i 0.332843 0.803555i
\(580\) 0 0
\(581\) −3.60653 + 1.49387i −0.149624 + 0.0619762i
\(582\) 0 0
\(583\) 8.86535 8.86535i 0.367165 0.367165i
\(584\) 0 0
\(585\) −44.0601 44.0601i −1.82166 1.82166i
\(586\) 0 0
\(587\) 10.8778 + 26.2613i 0.448974 + 1.08392i 0.972707 + 0.232037i \(0.0745391\pi\)
−0.523733 + 0.851883i \(0.675461\pi\)
\(588\) 0 0
\(589\) −8.66722 3.59008i −0.357127 0.147927i
\(590\) 0 0
\(591\) 51.2830i 2.10950i
\(592\) 0 0
\(593\) 8.36388i 0.343464i 0.985144 + 0.171732i \(0.0549362\pi\)
−0.985144 + 0.171732i \(0.945064\pi\)
\(594\) 0 0
\(595\) −6.22359 2.57789i −0.255142 0.105683i
\(596\) 0 0
\(597\) 1.15916 + 2.79847i 0.0474414 + 0.114534i
\(598\) 0 0
\(599\) 10.8797 + 10.8797i 0.444531 + 0.444531i 0.893531 0.449001i \(-0.148220\pi\)
−0.449001 + 0.893531i \(0.648220\pi\)
\(600\) 0 0
\(601\) 13.9066 13.9066i 0.567264 0.567264i −0.364097 0.931361i \(-0.618622\pi\)
0.931361 + 0.364097i \(0.118622\pi\)
\(602\) 0 0
\(603\) −81.3376 + 33.6911i −3.31232 + 1.37201i
\(604\) 0 0
\(605\) 11.7624 28.3970i 0.478210 1.15450i
\(606\) 0 0
\(607\) −42.3665 −1.71960 −0.859801 0.510629i \(-0.829412\pi\)
−0.859801 + 0.510629i \(0.829412\pi\)
\(608\) 0 0
\(609\) 31.6919 1.28422
\(610\) 0 0
\(611\) 3.26900 7.89207i 0.132250 0.319279i
\(612\) 0 0
\(613\) −13.2902 + 5.50500i −0.536788 + 0.222345i −0.634573 0.772863i \(-0.718824\pi\)
0.0977853 + 0.995208i \(0.468824\pi\)
\(614\) 0 0
\(615\) −10.2961 + 10.2961i −0.415178 + 0.415178i
\(616\) 0 0
\(617\) −17.8105 17.8105i −0.717025 0.717025i 0.250970 0.967995i \(-0.419251\pi\)
−0.967995 + 0.250970i \(0.919251\pi\)
\(618\) 0 0
\(619\) 4.80624 + 11.6033i 0.193179 + 0.466376i 0.990556 0.137105i \(-0.0437799\pi\)
−0.797377 + 0.603481i \(0.793780\pi\)
\(620\) 0 0
\(621\) 26.5460 + 10.9957i 1.06525 + 0.441242i
\(622\) 0 0
\(623\) 12.2537i 0.490935i
\(624\) 0 0
\(625\) 24.8217i 0.992869i
\(626\) 0 0
\(627\) 20.2445 + 8.38555i 0.808487 + 0.334886i
\(628\) 0 0
\(629\) −3.36329 8.11970i −0.134103 0.323753i
\(630\) 0 0
\(631\) 12.8865 + 12.8865i 0.513004 + 0.513004i 0.915446 0.402442i \(-0.131838\pi\)
−0.402442 + 0.915446i \(0.631838\pi\)
\(632\) 0 0
\(633\) −37.3355 + 37.3355i −1.48395 + 1.48395i
\(634\) 0 0
\(635\) 18.0323 7.46924i 0.715592 0.296408i
\(636\) 0 0
\(637\) −5.77412 + 13.9400i −0.228779 + 0.552321i
\(638\) 0 0
\(639\) 57.0772 2.25794
\(640\) 0 0
\(641\) −12.6050 −0.497868 −0.248934 0.968520i \(-0.580080\pi\)
−0.248934 + 0.968520i \(0.580080\pi\)
\(642\) 0 0
\(643\) −5.90799 + 14.2632i −0.232988 + 0.562484i −0.996526 0.0832793i \(-0.973461\pi\)
0.763538 + 0.645763i \(0.223461\pi\)
\(644\) 0 0
\(645\) −39.6018 + 16.4036i −1.55932 + 0.645891i
\(646\) 0 0
\(647\) −19.4412 + 19.4412i −0.764314 + 0.764314i −0.977099 0.212785i \(-0.931747\pi\)
0.212785 + 0.977099i \(0.431747\pi\)
\(648\) 0 0
\(649\) 7.04722 + 7.04722i 0.276627 + 0.276627i
\(650\) 0 0
\(651\) −2.42742 5.86030i −0.0951380 0.229683i
\(652\) 0 0
\(653\) 3.38672 + 1.40282i 0.132533 + 0.0548968i 0.447964 0.894052i \(-0.352149\pi\)
−0.315432 + 0.948948i \(0.602149\pi\)
\(654\) 0 0
\(655\) 6.59747i 0.257784i
\(656\) 0 0
\(657\) 36.0013i 1.40454i
\(658\) 0 0
\(659\) 15.5711 + 6.44977i 0.606565 + 0.251247i 0.664759 0.747058i \(-0.268534\pi\)
−0.0581942 + 0.998305i \(0.518534\pi\)
\(660\) 0 0
\(661\) −6.03928 14.5801i −0.234901 0.567100i 0.761841 0.647764i \(-0.224296\pi\)
−0.996741 + 0.0806641i \(0.974296\pi\)
\(662\) 0 0
\(663\) 10.2615 + 10.2615i 0.398522 + 0.398522i
\(664\) 0 0
\(665\) 18.1229 18.1229i 0.702777 0.702777i
\(666\) 0 0
\(667\) 16.7840 6.95218i 0.649880 0.269189i
\(668\) 0 0
\(669\) 1.77370 4.28208i 0.0685750 0.165555i
\(670\) 0 0
\(671\) 15.5884 0.601784
\(672\) 0 0
\(673\) −18.1812 −0.700834 −0.350417 0.936594i \(-0.613960\pi\)
−0.350417 + 0.936594i \(0.613960\pi\)
\(674\) 0 0
\(675\) 23.3088 56.2724i 0.897155 2.16592i
\(676\) 0 0
\(677\) −35.5504 + 14.7255i −1.36631 + 0.565945i −0.940786 0.339000i \(-0.889911\pi\)
−0.425527 + 0.904946i \(0.639911\pi\)
\(678\) 0 0
\(679\) 0.943139 0.943139i 0.0361944 0.0361944i
\(680\) 0 0
\(681\) 62.7595 + 62.7595i 2.40495 + 2.40495i
\(682\) 0 0
\(683\) −2.35810 5.69296i −0.0902303 0.217835i 0.872322 0.488932i \(-0.162614\pi\)
−0.962552 + 0.271097i \(0.912614\pi\)
\(684\) 0 0
\(685\) −46.0736 19.0843i −1.76038 0.729173i
\(686\) 0 0
\(687\) 29.6619i 1.13167i
\(688\) 0 0
\(689\) 31.5909i 1.20352i
\(690\) 0 0
\(691\) 6.48765 + 2.68727i 0.246802 + 0.102229i 0.502655 0.864487i \(-0.332356\pi\)
−0.255854 + 0.966716i \(0.582356\pi\)
\(692\) 0 0
\(693\) 3.94360 + 9.52069i 0.149805 + 0.361661i
\(694\) 0 0
\(695\) 24.1366 + 24.1366i 0.915552 + 0.915552i
\(696\) 0 0
\(697\) 1.66785 1.66785i 0.0631743 0.0631743i
\(698\) 0 0
\(699\) 40.9825 16.9755i 1.55010 0.642072i
\(700\) 0 0
\(701\) 4.81730 11.6300i 0.181947 0.439258i −0.806421 0.591342i \(-0.798598\pi\)
0.988368 + 0.152084i \(0.0485983\pi\)
\(702\) 0 0
\(703\) 33.4381 1.26114
\(704\) 0 0
\(705\) 29.5975 1.11470
\(706\) 0 0
\(707\) −5.50404 + 13.2879i −0.207001 + 0.499744i
\(708\) 0 0
\(709\) −2.06979 + 0.857335i −0.0777326 + 0.0321979i −0.421211 0.906963i \(-0.638395\pi\)
0.343478 + 0.939161i \(0.388395\pi\)
\(710\) 0 0
\(711\) −50.4277 + 50.4277i −1.89119 + 1.89119i
\(712\) 0 0
\(713\) −2.57112 2.57112i −0.0962893 0.0962893i
\(714\) 0 0
\(715\) −3.96298 9.56749i −0.148207 0.357804i
\(716\) 0 0
\(717\) 30.6709 + 12.7043i 1.14543 + 0.474451i
\(718\) 0 0
\(719\) 29.4245i 1.09735i −0.836036 0.548674i \(-0.815133\pi\)
0.836036 0.548674i \(-0.184867\pi\)
\(720\) 0 0
\(721\) 19.0056i 0.707806i
\(722\) 0 0
\(723\) 2.00351 + 0.829880i 0.0745113 + 0.0308636i
\(724\) 0 0
\(725\) −14.7373 35.5789i −0.547329 1.32137i
\(726\) 0 0
\(727\) −1.75350 1.75350i −0.0650337 0.0650337i 0.673842 0.738876i \(-0.264643\pi\)
−0.738876 + 0.673842i \(0.764643\pi\)
\(728\) 0 0
\(729\) 3.82254 3.82254i 0.141575 0.141575i
\(730\) 0 0
\(731\) 6.41505 2.65720i 0.237269 0.0982801i
\(732\) 0 0
\(733\) −0.447524 + 1.08042i −0.0165297 + 0.0399062i −0.931929 0.362641i \(-0.881875\pi\)
0.915399 + 0.402547i \(0.131875\pi\)
\(734\) 0 0
\(735\) −52.2787 −1.92833
\(736\) 0 0
\(737\) −14.6318 −0.538970
\(738\) 0 0
\(739\) 15.0760 36.3968i 0.554581 1.33888i −0.359424 0.933174i \(-0.617027\pi\)
0.914005 0.405703i \(-0.132973\pi\)
\(740\) 0 0
\(741\) −51.0103 + 21.1291i −1.87391 + 0.776198i
\(742\) 0 0
\(743\) 1.45255 1.45255i 0.0532888 0.0532888i −0.679960 0.733249i \(-0.738003\pi\)
0.733249 + 0.679960i \(0.238003\pi\)
\(744\) 0 0
\(745\) 31.6790 + 31.6790i 1.16063 + 1.16063i
\(746\) 0 0
\(747\) 7.75555 + 18.7236i 0.283761 + 0.685059i
\(748\) 0 0
\(749\) −12.0955 5.01011i −0.441959 0.183065i
\(750\) 0 0
\(751\) 17.4254i 0.635861i −0.948114 0.317931i \(-0.897012\pi\)
0.948114 0.317931i \(-0.102988\pi\)
\(752\) 0 0
\(753\) 49.9126i 1.81892i
\(754\) 0 0
\(755\) −58.3255 24.1592i −2.12268 0.879243i
\(756\) 0 0
\(757\) −10.8868 26.2830i −0.395686 0.955272i −0.988677 0.150062i \(-0.952053\pi\)
0.592990 0.805210i \(-0.297947\pi\)
\(758\) 0 0
\(759\) 6.00551 + 6.00551i 0.217986 + 0.217986i
\(760\) 0 0
\(761\) −13.6001 + 13.6001i −0.493003 + 0.493003i −0.909251 0.416248i \(-0.863345\pi\)
0.416248 + 0.909251i \(0.363345\pi\)
\(762\) 0 0
\(763\) 12.8779 5.33420i 0.466211 0.193111i
\(764\) 0 0
\(765\) −13.3833 + 32.3102i −0.483875 + 1.16818i
\(766\) 0 0
\(767\) −25.1121 −0.906745
\(768\) 0 0
\(769\) −8.19053 −0.295358 −0.147679 0.989035i \(-0.547180\pi\)
−0.147679 + 0.989035i \(0.547180\pi\)
\(770\) 0 0
\(771\) 14.1661 34.1999i 0.510179 1.23168i
\(772\) 0 0
\(773\) 29.7373 12.3176i 1.06958 0.443033i 0.222735 0.974879i \(-0.428502\pi\)
0.846841 + 0.531847i \(0.178502\pi\)
\(774\) 0 0
\(775\) −5.45028 + 5.45028i −0.195780 + 0.195780i
\(776\) 0 0
\(777\) 15.9870 + 15.9870i 0.573531 + 0.573531i
\(778\) 0 0
\(779\) 3.43423 + 8.29097i 0.123044 + 0.297055i
\(780\) 0 0
\(781\) 8.76395 + 3.63015i 0.313599 + 0.129897i
\(782\) 0 0
\(783\) 92.5100i 3.30604i
\(784\) 0 0
\(785\) 9.82817i 0.350782i
\(786\) 0 0
\(787\) 9.40919 + 3.89741i 0.335401 + 0.138928i 0.544027 0.839068i \(-0.316899\pi\)
−0.208626 + 0.977996i \(0.566899\pi\)
\(788\) 0 0
\(789\) 34.8148 + 84.0504i 1.23944 + 2.99227i
\(790\) 0 0
\(791\) 2.40004 + 2.40004i 0.0853356 + 0.0853356i
\(792\) 0 0
\(793\) −27.7740 + 27.7740i −0.986282 + 0.986282i
\(794\) 0 0
\(795\) −101.124 + 41.8871i −3.58651 + 1.48558i
\(796\) 0 0
\(797\) 7.15756 17.2799i 0.253534 0.612085i −0.744951 0.667120i \(-0.767527\pi\)
0.998484 + 0.0550349i \(0.0175270\pi\)
\(798\) 0 0
\(799\) −4.79446 −0.169616
\(800\) 0 0
\(801\) −63.6162 −2.24777
\(802\) 0 0
\(803\) −2.28971 + 5.52785i −0.0808021 + 0.195073i
\(804\) 0 0
\(805\) 9.17765 3.80151i 0.323470 0.133985i
\(806\) 0 0
\(807\) 22.1996 22.1996i 0.781462 0.781462i
\(808\) 0 0
\(809\) 28.3548 + 28.3548i 0.996902 + 0.996902i 0.999995 0.00309284i \(-0.000984483\pi\)
−0.00309284 + 0.999995i \(0.500984\pi\)
\(810\) 0 0
\(811\) 1.61674 + 3.90316i 0.0567716 + 0.137059i 0.949720 0.313100i \(-0.101367\pi\)
−0.892949 + 0.450159i \(0.851367\pi\)
\(812\) 0 0
\(813\) −73.0927 30.2760i −2.56347 1.06183i
\(814\) 0 0
\(815\) 31.5756i 1.10604i
\(816\) 0 0
\(817\) 26.4181i 0.924254i
\(818\) 0 0
\(819\) −23.9894 9.93672i −0.838256 0.347217i
\(820\) 0 0
\(821\) 11.5300 + 27.8360i 0.402401 + 0.971482i 0.987082 + 0.160218i \(0.0512199\pi\)
−0.584681 + 0.811264i \(0.698780\pi\)
\(822\) 0 0
\(823\) −30.7602 30.7602i −1.07223 1.07223i −0.997179 0.0750551i \(-0.976087\pi\)
−0.0750551 0.997179i \(-0.523913\pi\)
\(824\) 0 0
\(825\) 12.7305 12.7305i 0.443220 0.443220i
\(826\) 0 0
\(827\) −23.1882 + 9.60486i −0.806332 + 0.333994i −0.747489 0.664274i \(-0.768741\pi\)
−0.0588423 + 0.998267i \(0.518741\pi\)
\(828\) 0 0
\(829\) −17.1975 + 41.5185i −0.597295 + 1.44200i 0.279033 + 0.960281i \(0.409986\pi\)
−0.876328 + 0.481715i \(0.840014\pi\)
\(830\) 0 0
\(831\) 46.3929 1.60935
\(832\) 0 0
\(833\) 8.46858 0.293419
\(834\) 0 0
\(835\) 10.5605 25.4952i 0.365459 0.882297i
\(836\) 0 0
\(837\) −17.1065 + 7.08573i −0.591286 + 0.244919i
\(838\) 0 0
\(839\) 31.1198 31.1198i 1.07437 1.07437i 0.0773708 0.997002i \(-0.475347\pi\)
0.997002 0.0773708i \(-0.0246525\pi\)
\(840\) 0 0
\(841\) −20.8530 20.8530i −0.719070 0.719070i
\(842\) 0 0
\(843\) −7.56398 18.2611i −0.260517 0.628944i
\(844\) 0 0
\(845\) −13.9402 5.77423i −0.479558 0.198639i
\(846\) 0 0
\(847\) 12.8086i 0.440107i
\(848\) 0 0
\(849\) 76.5009i 2.62550i
\(850\) 0 0
\(851\) 11.9738 + 4.95969i 0.410455 + 0.170016i
\(852\) 0 0
\(853\) −9.80845 23.6797i −0.335835 0.810777i −0.998106 0.0615130i \(-0.980407\pi\)
0.662271 0.749264i \(-0.269593\pi\)
\(854\) 0 0
\(855\) −94.0865 94.0865i −3.21769 3.21769i
\(856\) 0 0
\(857\) −24.7590 + 24.7590i −0.845752 + 0.845752i −0.989600 0.143847i \(-0.954053\pi\)
0.143847 + 0.989600i \(0.454053\pi\)
\(858\) 0 0
\(859\) −7.98006 + 3.30545i −0.272276 + 0.112780i −0.514644 0.857404i \(-0.672076\pi\)
0.242368 + 0.970184i \(0.422076\pi\)
\(860\) 0 0
\(861\) −2.32204 + 5.60590i −0.0791349 + 0.191049i
\(862\) 0 0
\(863\) −35.4394 −1.20637 −0.603185 0.797601i \(-0.706102\pi\)
−0.603185 + 0.797601i \(0.706102\pi\)
\(864\) 0 0
\(865\) 33.3012 1.13227
\(866\) 0 0
\(867\) −17.3043 + 41.7764i −0.587686 + 1.41880i
\(868\) 0 0
\(869\) −10.9502 + 4.53572i −0.371460 + 0.153864i
\(870\) 0 0
\(871\) 26.0696 26.0696i 0.883334 0.883334i
\(872\) 0 0
\(873\) −4.89638 4.89638i −0.165717 0.165717i
\(874\) 0 0
\(875\) −0.0566593 0.136788i −0.00191543 0.00462427i
\(876\) 0 0
\(877\) 43.2968 + 17.9341i 1.46203 + 0.605593i 0.965026 0.262155i \(-0.0844332\pi\)
0.497005 + 0.867748i \(0.334433\pi\)
\(878\) 0 0
\(879\) 22.6591i 0.764274i
\(880\) 0 0
\(881\) 4.84917i 0.163373i −0.996658 0.0816863i \(-0.973969\pi\)
0.996658 0.0816863i \(-0.0260306\pi\)
\(882\) 0 0
\(883\) 40.9077 + 16.9445i 1.37665 + 0.570228i 0.943584 0.331133i \(-0.107431\pi\)
0.433068 + 0.901361i \(0.357431\pi\)
\(884\) 0 0
\(885\) −33.2967 80.3855i −1.11926 2.70213i
\(886\) 0 0
\(887\) 7.77550 + 7.77550i 0.261076 + 0.261076i 0.825491 0.564415i \(-0.190898\pi\)
−0.564415 + 0.825491i \(0.690898\pi\)
\(888\) 0 0
\(889\) 5.75128 5.75128i 0.192892 0.192892i
\(890\) 0 0
\(891\) 18.3204 7.58856i 0.613757 0.254226i
\(892\) 0 0
\(893\) 6.98067 16.8528i 0.233599 0.563959i
\(894\) 0 0
\(895\) −52.5379 −1.75615
\(896\) 0 0
\(897\) −21.4001 −0.714527
\(898\) 0 0
\(899\) −4.48005 + 10.8158i −0.149418 + 0.360727i
\(900\) 0 0
\(901\) 16.3810 6.78525i 0.545731 0.226049i
\(902\) 0 0
\(903\) −12.6307 + 12.6307i −0.420323 + 0.420323i
\(904\) 0 0
\(905\) 10.0043 + 10.0043i 0.332554 + 0.332554i
\(906\) 0 0
\(907\) −2.38548 5.75906i −0.0792086 0.191226i 0.879314 0.476242i \(-0.158001\pi\)
−0.958523 + 0.285015i \(0.908001\pi\)
\(908\) 0 0
\(909\) 68.9853 + 28.5746i 2.28810 + 0.947761i
\(910\) 0 0
\(911\) 15.3924i 0.509973i −0.966945 0.254986i \(-0.917929\pi\)
0.966945 0.254986i \(-0.0820710\pi\)
\(912\) 0 0
\(913\) 3.36818i 0.111470i
\(914\) 0 0
\(915\) −125.732 52.0801i −4.15658 1.72171i
\(916\) 0 0
\(917\) 1.05211 + 2.54001i 0.0347437 + 0.0838786i
\(918\) 0 0
\(919\) 13.0672 + 13.0672i 0.431046 + 0.431046i 0.888984 0.457938i \(-0.151412\pi\)
−0.457938 + 0.888984i \(0.651412\pi\)
\(920\) 0 0
\(921\) 39.2394 39.2394i 1.29298 1.29298i
\(922\) 0 0
\(923\) −22.0826 + 9.14692i −0.726858 + 0.301075i
\(924\) 0 0
\(925\) 10.5136 25.3821i 0.345685 0.834557i
\(926\) 0 0
\(927\) 98.6690 3.24072
\(928\) 0 0
\(929\) 14.4450 0.473924 0.236962 0.971519i \(-0.423848\pi\)
0.236962 + 0.971519i \(0.423848\pi\)
\(930\) 0 0
\(931\) −12.3301 + 29.7676i −0.404104 + 0.975593i
\(932\) 0 0
\(933\) 38.9142 16.1188i 1.27399 0.527705i
\(934\) 0 0
\(935\) −4.10991 + 4.10991i −0.134408 + 0.134408i
\(936\) 0 0
\(937\) −4.56931 4.56931i −0.149273 0.149273i 0.628520 0.777793i \(-0.283661\pi\)
−0.777793 + 0.628520i \(0.783661\pi\)
\(938\) 0 0
\(939\) −14.2591 34.4244i −0.465327 1.12340i
\(940\) 0 0
\(941\) 17.6192 + 7.29813i 0.574371 + 0.237912i 0.650911 0.759154i \(-0.274387\pi\)
−0.0765397 + 0.997067i \(0.524387\pi\)
\(942\) 0 0
\(943\) 3.47827i 0.113268i
\(944\) 0 0
\(945\) 50.5852i 1.64554i
\(946\) 0 0
\(947\) 10.8687 + 4.50198i 0.353187 + 0.146295i 0.552221 0.833698i \(-0.313780\pi\)
−0.199034 + 0.979993i \(0.563780\pi\)
\(948\) 0 0
\(949\) −5.76940 13.9286i −0.187283 0.452140i
\(950\) 0 0
\(951\) 63.8843 + 63.8843i 2.07159 + 2.07159i
\(952\) 0 0
\(953\) 9.88290 9.88290i 0.320138 0.320138i −0.528682 0.848820i \(-0.677313\pi\)
0.848820 + 0.528682i \(0.177313\pi\)
\(954\) 0 0
\(955\) 43.3096 17.9394i 1.40147 0.580506i
\(956\) 0 0
\(957\) 10.4643 25.2630i 0.338262 0.816637i
\(958\) 0 0
\(959\) −20.7816 −0.671074
\(960\) 0 0
\(961\) −28.6569 −0.924415
\(962\) 0 0
\(963\) −26.0104 + 62.7946i −0.838173 + 2.02353i
\(964\) 0 0
\(965\) −19.5132 + 8.08262i −0.628151 + 0.260189i
\(966\) 0 0
\(967\) 27.4467 27.4467i 0.882626 0.882626i −0.111175 0.993801i \(-0.535461\pi\)
0.993801 + 0.111175i \(0.0354614\pi\)
\(968\) 0 0
\(969\) 21.9125 + 21.9125i 0.703931 + 0.703931i
\(970\) 0 0
\(971\) −21.6361 52.2342i −0.694335 1.67627i −0.735858 0.677136i \(-0.763221\pi\)
0.0415225 0.999138i \(-0.486779\pi\)
\(972\) 0 0
\(973\) 13.1416 + 5.44344i 0.421301 + 0.174509i
\(974\) 0 0
\(975\) 45.3640i 1.45281i
\(976\) 0 0
\(977\) 23.1041i 0.739165i −0.929198 0.369582i \(-0.879501\pi\)
0.929198 0.369582i \(-0.120499\pi\)
\(978\) 0 0
\(979\) −9.76799 4.04603i −0.312186 0.129312i
\(980\) 0 0
\(981\) −27.6929 66.8566i −0.884167 2.13457i
\(982\) 0 0
\(983\) −17.7442 17.7442i −0.565951 0.565951i 0.365041 0.930991i \(-0.381055\pi\)
−0.930991 + 0.365041i \(0.881055\pi\)
\(984\) 0 0
\(985\) 36.5958 36.5958i 1.16604 1.16604i
\(986\) 0 0
\(987\) 11.3950 4.71995i 0.362706 0.150238i
\(988\) 0 0
\(989\) −3.91846 + 9.45999i −0.124600 + 0.300810i
\(990\) 0 0
\(991\) 6.68692 0.212417 0.106208 0.994344i \(-0.466129\pi\)
0.106208 + 0.994344i \(0.466129\pi\)
\(992\) 0 0
\(993\) −32.0306 −1.01646
\(994\) 0 0
\(995\) 1.16982 2.82419i 0.0370857 0.0895328i
\(996\) 0 0
\(997\) −3.36920 + 1.39557i −0.106704 + 0.0441981i −0.435397 0.900239i \(-0.643392\pi\)
0.328693 + 0.944437i \(0.393392\pi\)
\(998\) 0 0
\(999\) 46.6668 46.6668i 1.47647 1.47647i
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 1024.2.g.e.129.1 yes 16
4.3 odd 2 inner 1024.2.g.e.129.4 yes 16
8.3 odd 2 1024.2.g.b.129.1 16
8.5 even 2 1024.2.g.b.129.4 yes 16
16.3 odd 4 1024.2.g.c.641.4 yes 16
16.5 even 4 1024.2.g.h.641.4 yes 16
16.11 odd 4 1024.2.g.h.641.1 yes 16
16.13 even 4 1024.2.g.c.641.1 yes 16
32.3 odd 8 1024.2.g.c.385.4 yes 16
32.5 even 8 inner 1024.2.g.e.897.1 yes 16
32.11 odd 8 1024.2.g.b.897.1 yes 16
32.13 even 8 1024.2.g.h.385.4 yes 16
32.19 odd 8 1024.2.g.h.385.1 yes 16
32.21 even 8 1024.2.g.b.897.4 yes 16
32.27 odd 8 inner 1024.2.g.e.897.4 yes 16
32.29 even 8 1024.2.g.c.385.1 yes 16
64.5 even 16 4096.2.a.n.1.2 8
64.27 odd 16 4096.2.a.n.1.1 8
64.37 even 16 4096.2.a.o.1.7 8
64.59 odd 16 4096.2.a.o.1.8 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1024.2.g.b.129.1 16 8.3 odd 2
1024.2.g.b.129.4 yes 16 8.5 even 2
1024.2.g.b.897.1 yes 16 32.11 odd 8
1024.2.g.b.897.4 yes 16 32.21 even 8
1024.2.g.c.385.1 yes 16 32.29 even 8
1024.2.g.c.385.4 yes 16 32.3 odd 8
1024.2.g.c.641.1 yes 16 16.13 even 4
1024.2.g.c.641.4 yes 16 16.3 odd 4
1024.2.g.e.129.1 yes 16 1.1 even 1 trivial
1024.2.g.e.129.4 yes 16 4.3 odd 2 inner
1024.2.g.e.897.1 yes 16 32.5 even 8 inner
1024.2.g.e.897.4 yes 16 32.27 odd 8 inner
1024.2.g.h.385.1 yes 16 32.19 odd 8
1024.2.g.h.385.4 yes 16 32.13 even 8
1024.2.g.h.641.1 yes 16 16.11 odd 4
1024.2.g.h.641.4 yes 16 16.5 even 4
4096.2.a.n.1.1 8 64.27 odd 16
4096.2.a.n.1.2 8 64.5 even 16
4096.2.a.o.1.7 8 64.37 even 16
4096.2.a.o.1.8 8 64.59 odd 16