Properties

Label 124.2.f.a.33.1
Level $124$
Weight $2$
Character 124.33
Analytic conductor $0.990$
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] = [124,2,Mod(33,124)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(124, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("124.33");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 124 = 2^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 124.f (of order \(5\), 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: \(0.990144985064\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label 33.1
Root \(0.809017 + 0.587785i\) of defining polynomial
Character \(\chi\) \(=\) 124.33
Dual form 124.2.f.a.109.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.30902 + 0.951057i) q^{3} +2.00000 q^{5} +(0.809017 + 2.48990i) q^{7} +(-0.118034 + 0.363271i) q^{9} +O(q^{10})\) \(q+(-1.30902 + 0.951057i) q^{3} +2.00000 q^{5} +(0.809017 + 2.48990i) q^{7} +(-0.118034 + 0.363271i) q^{9} +(0.809017 + 2.48990i) q^{11} +(1.30902 - 0.951057i) q^{13} +(-2.61803 + 1.90211i) q^{15} +(1.42705 - 4.39201i) q^{17} +(-3.92705 - 2.85317i) q^{19} +(-3.42705 - 2.48990i) q^{21} +(1.66312 - 5.11855i) q^{23} -1.00000 q^{25} +(-1.69098 - 5.20431i) q^{27} +(-0.690983 - 0.502029i) q^{29} +(3.23607 + 4.53077i) q^{31} +(-3.42705 - 2.48990i) q^{33} +(1.61803 + 4.97980i) q^{35} -4.47214 q^{37} +(-0.809017 + 2.48990i) q^{39} +(9.78115 + 7.10642i) q^{41} +(-7.16312 - 5.20431i) q^{43} +(-0.236068 + 0.726543i) q^{45} +(0.690983 - 0.502029i) q^{47} +(0.118034 - 0.0857567i) q^{49} +(2.30902 + 7.10642i) q^{51} +(2.66312 - 8.19624i) q^{53} +(1.61803 + 4.97980i) q^{55} +7.85410 q^{57} +(7.16312 - 5.20431i) q^{59} -13.4164 q^{61} -1.00000 q^{63} +(2.61803 - 1.90211i) q^{65} +1.52786 q^{67} +(2.69098 + 8.28199i) q^{69} +(-1.28115 + 3.94298i) q^{71} +(3.89919 + 12.0005i) q^{73} +(1.30902 - 0.951057i) q^{75} +(-5.54508 + 4.02874i) q^{77} +(1.19098 - 3.66547i) q^{79} +(6.23607 + 4.53077i) q^{81} +(2.54508 + 1.84911i) q^{83} +(2.85410 - 8.78402i) q^{85} +1.38197 q^{87} +(-2.57295 - 7.91872i) q^{89} +(3.42705 + 2.48990i) q^{91} +(-8.54508 - 2.85317i) q^{93} +(-7.85410 - 5.70634i) q^{95} +(5.13525 + 15.8047i) q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 3 q^{3} + 8 q^{5} + q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 3 q^{3} + 8 q^{5} + q^{7} + 4 q^{9} + q^{11} + 3 q^{13} - 6 q^{15} - q^{17} - 9 q^{19} - 7 q^{21} - 9 q^{23} - 4 q^{25} - 9 q^{27} - 5 q^{29} + 4 q^{31} - 7 q^{33} + 2 q^{35} - q^{39} + 19 q^{41} - 13 q^{43} + 8 q^{45} + 5 q^{47} - 4 q^{49} + 7 q^{51} - 5 q^{53} + 2 q^{55} + 18 q^{57} + 13 q^{59} - 4 q^{63} + 6 q^{65} + 24 q^{67} + 13 q^{69} + 15 q^{71} - 9 q^{73} + 3 q^{75} - 11 q^{77} + 7 q^{79} + 16 q^{81} - q^{83} - 2 q^{85} + 10 q^{87} - 17 q^{89} + 7 q^{91} - 23 q^{93} - 18 q^{95} - 13 q^{97} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/124\mathbb{Z}\right)^\times\).

\(n\) \(63\) \(65\)
\(\chi(n)\) \(1\) \(e\left(\frac{4}{5}\right)\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.30902 + 0.951057i −0.755761 + 0.549093i −0.897607 0.440796i \(-0.854696\pi\)
0.141846 + 0.989889i \(0.454696\pi\)
\(4\) 0 0
\(5\) 2.00000 0.894427 0.447214 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 0 0
\(7\) 0.809017 + 2.48990i 0.305780 + 0.941093i 0.979385 + 0.202002i \(0.0647447\pi\)
−0.673605 + 0.739091i \(0.735255\pi\)
\(8\) 0 0
\(9\) −0.118034 + 0.363271i −0.0393447 + 0.121090i
\(10\) 0 0
\(11\) 0.809017 + 2.48990i 0.243928 + 0.750733i 0.995811 + 0.0914362i \(0.0291457\pi\)
−0.751883 + 0.659296i \(0.770854\pi\)
\(12\) 0 0
\(13\) 1.30902 0.951057i 0.363056 0.263776i −0.391270 0.920276i \(-0.627964\pi\)
0.754326 + 0.656500i \(0.227964\pi\)
\(14\) 0 0
\(15\) −2.61803 + 1.90211i −0.675973 + 0.491123i
\(16\) 0 0
\(17\) 1.42705 4.39201i 0.346111 1.06522i −0.614876 0.788624i \(-0.710794\pi\)
0.960987 0.276595i \(-0.0892061\pi\)
\(18\) 0 0
\(19\) −3.92705 2.85317i −0.900927 0.654562i 0.0377767 0.999286i \(-0.487972\pi\)
−0.938704 + 0.344724i \(0.887972\pi\)
\(20\) 0 0
\(21\) −3.42705 2.48990i −0.747844 0.543340i
\(22\) 0 0
\(23\) 1.66312 5.11855i 0.346784 1.06729i −0.613837 0.789433i \(-0.710375\pi\)
0.960622 0.277860i \(-0.0896251\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) −1.69098 5.20431i −0.325430 1.00157i
\(28\) 0 0
\(29\) −0.690983 0.502029i −0.128312 0.0932244i 0.521778 0.853081i \(-0.325269\pi\)
−0.650090 + 0.759857i \(0.725269\pi\)
\(30\) 0 0
\(31\) 3.23607 + 4.53077i 0.581215 + 0.813750i
\(32\) 0 0
\(33\) −3.42705 2.48990i −0.596573 0.433436i
\(34\) 0 0
\(35\) 1.61803 + 4.97980i 0.273498 + 0.841739i
\(36\) 0 0
\(37\) −4.47214 −0.735215 −0.367607 0.929981i \(-0.619823\pi\)
−0.367607 + 0.929981i \(0.619823\pi\)
\(38\) 0 0
\(39\) −0.809017 + 2.48990i −0.129546 + 0.398703i
\(40\) 0 0
\(41\) 9.78115 + 7.10642i 1.52756 + 1.10984i 0.957576 + 0.288181i \(0.0930504\pi\)
0.569983 + 0.821656i \(0.306950\pi\)
\(42\) 0 0
\(43\) −7.16312 5.20431i −1.09237 0.793650i −0.112568 0.993644i \(-0.535908\pi\)
−0.979797 + 0.199994i \(0.935908\pi\)
\(44\) 0 0
\(45\) −0.236068 + 0.726543i −0.0351909 + 0.108307i
\(46\) 0 0
\(47\) 0.690983 0.502029i 0.100790 0.0732284i −0.536249 0.844060i \(-0.680159\pi\)
0.637039 + 0.770832i \(0.280159\pi\)
\(48\) 0 0
\(49\) 0.118034 0.0857567i 0.0168620 0.0122510i
\(50\) 0 0
\(51\) 2.30902 + 7.10642i 0.323327 + 0.995098i
\(52\) 0 0
\(53\) 2.66312 8.19624i 0.365808 1.12584i −0.583666 0.811994i \(-0.698382\pi\)
0.949474 0.313846i \(-0.101618\pi\)
\(54\) 0 0
\(55\) 1.61803 + 4.97980i 0.218176 + 0.671476i
\(56\) 0 0
\(57\) 7.85410 1.04030
\(58\) 0 0
\(59\) 7.16312 5.20431i 0.932559 0.677544i −0.0140593 0.999901i \(-0.504475\pi\)
0.946618 + 0.322358i \(0.104475\pi\)
\(60\) 0 0
\(61\) −13.4164 −1.71780 −0.858898 0.512148i \(-0.828850\pi\)
−0.858898 + 0.512148i \(0.828850\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) 2.61803 1.90211i 0.324727 0.235928i
\(66\) 0 0
\(67\) 1.52786 0.186658 0.0933292 0.995635i \(-0.470249\pi\)
0.0933292 + 0.995635i \(0.470249\pi\)
\(68\) 0 0
\(69\) 2.69098 + 8.28199i 0.323956 + 0.997035i
\(70\) 0 0
\(71\) −1.28115 + 3.94298i −0.152045 + 0.467946i −0.997849 0.0655468i \(-0.979121\pi\)
0.845805 + 0.533493i \(0.179121\pi\)
\(72\) 0 0
\(73\) 3.89919 + 12.0005i 0.456365 + 1.40455i 0.869525 + 0.493889i \(0.164425\pi\)
−0.413159 + 0.910659i \(0.635575\pi\)
\(74\) 0 0
\(75\) 1.30902 0.951057i 0.151152 0.109819i
\(76\) 0 0
\(77\) −5.54508 + 4.02874i −0.631921 + 0.459118i
\(78\) 0 0
\(79\) 1.19098 3.66547i 0.133996 0.412397i −0.861436 0.507865i \(-0.830435\pi\)
0.995433 + 0.0954679i \(0.0304347\pi\)
\(80\) 0 0
\(81\) 6.23607 + 4.53077i 0.692896 + 0.503419i
\(82\) 0 0
\(83\) 2.54508 + 1.84911i 0.279359 + 0.202966i 0.718638 0.695384i \(-0.244766\pi\)
−0.439279 + 0.898351i \(0.644766\pi\)
\(84\) 0 0
\(85\) 2.85410 8.78402i 0.309571 0.952761i
\(86\) 0 0
\(87\) 1.38197 0.148162
\(88\) 0 0
\(89\) −2.57295 7.91872i −0.272732 0.839383i −0.989811 0.142391i \(-0.954521\pi\)
0.717079 0.696992i \(-0.245479\pi\)
\(90\) 0 0
\(91\) 3.42705 + 2.48990i 0.359253 + 0.261012i
\(92\) 0 0
\(93\) −8.54508 2.85317i −0.886084 0.295860i
\(94\) 0 0
\(95\) −7.85410 5.70634i −0.805814 0.585458i
\(96\) 0 0
\(97\) 5.13525 + 15.8047i 0.521406 + 1.60472i 0.771315 + 0.636454i \(0.219599\pi\)
−0.249909 + 0.968269i \(0.580401\pi\)
\(98\) 0 0
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) −2.57295 + 7.91872i −0.256018 + 0.787942i 0.737609 + 0.675228i \(0.235955\pi\)
−0.993627 + 0.112715i \(0.964045\pi\)
\(102\) 0 0
\(103\) −14.3992 10.4616i −1.41879 1.03081i −0.991969 0.126480i \(-0.959632\pi\)
−0.426825 0.904334i \(-0.640368\pi\)
\(104\) 0 0
\(105\) −6.85410 4.97980i −0.668892 0.485978i
\(106\) 0 0
\(107\) 3.95492 12.1720i 0.382336 1.17671i −0.556058 0.831143i \(-0.687687\pi\)
0.938394 0.345566i \(-0.112313\pi\)
\(108\) 0 0
\(109\) −6.39919 + 4.64928i −0.612931 + 0.445320i −0.850445 0.526064i \(-0.823667\pi\)
0.237514 + 0.971384i \(0.423667\pi\)
\(110\) 0 0
\(111\) 5.85410 4.25325i 0.555647 0.403701i
\(112\) 0 0
\(113\) −3.33688 10.2699i −0.313907 0.966108i −0.976202 0.216864i \(-0.930417\pi\)
0.662294 0.749244i \(-0.269583\pi\)
\(114\) 0 0
\(115\) 3.32624 10.2371i 0.310173 0.954615i
\(116\) 0 0
\(117\) 0.190983 + 0.587785i 0.0176564 + 0.0543408i
\(118\) 0 0
\(119\) 12.0902 1.10830
\(120\) 0 0
\(121\) 3.35410 2.43690i 0.304918 0.221536i
\(122\) 0 0
\(123\) −19.5623 −1.76387
\(124\) 0 0
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) 2.69098 1.95511i 0.238786 0.173488i −0.461956 0.886903i \(-0.652852\pi\)
0.700742 + 0.713414i \(0.252852\pi\)
\(128\) 0 0
\(129\) 14.3262 1.26135
\(130\) 0 0
\(131\) 4.33688 + 13.3475i 0.378915 + 1.16618i 0.940799 + 0.338965i \(0.110077\pi\)
−0.561884 + 0.827216i \(0.689923\pi\)
\(132\) 0 0
\(133\) 3.92705 12.0862i 0.340519 1.04801i
\(134\) 0 0
\(135\) −3.38197 10.4086i −0.291073 0.895831i
\(136\) 0 0
\(137\) −10.3992 + 7.55545i −0.888462 + 0.645506i −0.935477 0.353388i \(-0.885029\pi\)
0.0470142 + 0.998894i \(0.485029\pi\)
\(138\) 0 0
\(139\) 6.39919 4.64928i 0.542772 0.394347i −0.282342 0.959314i \(-0.591111\pi\)
0.825113 + 0.564967i \(0.191111\pi\)
\(140\) 0 0
\(141\) −0.427051 + 1.31433i −0.0359642 + 0.110686i
\(142\) 0 0
\(143\) 3.42705 + 2.48990i 0.286584 + 0.208216i
\(144\) 0 0
\(145\) −1.38197 1.00406i −0.114766 0.0833824i
\(146\) 0 0
\(147\) −0.0729490 + 0.224514i −0.00601673 + 0.0185176i
\(148\) 0 0
\(149\) −21.4164 −1.75450 −0.877250 0.480033i \(-0.840625\pi\)
−0.877250 + 0.480033i \(0.840625\pi\)
\(150\) 0 0
\(151\) −3.66312 11.2739i −0.298100 0.917459i −0.982162 0.188035i \(-0.939788\pi\)
0.684062 0.729424i \(-0.260212\pi\)
\(152\) 0 0
\(153\) 1.42705 + 1.03681i 0.115370 + 0.0838214i
\(154\) 0 0
\(155\) 6.47214 + 9.06154i 0.519854 + 0.727840i
\(156\) 0 0
\(157\) 8.54508 + 6.20837i 0.681972 + 0.495482i 0.874011 0.485906i \(-0.161510\pi\)
−0.192039 + 0.981387i \(0.561510\pi\)
\(158\) 0 0
\(159\) 4.30902 + 13.2618i 0.341727 + 1.05173i
\(160\) 0 0
\(161\) 14.0902 1.11046
\(162\) 0 0
\(163\) −5.57295 + 17.1518i −0.436507 + 1.34343i 0.455027 + 0.890477i \(0.349629\pi\)
−0.891534 + 0.452953i \(0.850371\pi\)
\(164\) 0 0
\(165\) −6.85410 4.97980i −0.533591 0.387677i
\(166\) 0 0
\(167\) 4.07295 + 2.95917i 0.315174 + 0.228988i 0.734114 0.679027i \(-0.237598\pi\)
−0.418939 + 0.908014i \(0.637598\pi\)
\(168\) 0 0
\(169\) −3.20820 + 9.87384i −0.246785 + 0.759526i
\(170\) 0 0
\(171\) 1.50000 1.08981i 0.114708 0.0833401i
\(172\) 0 0
\(173\) −3.92705 + 2.85317i −0.298568 + 0.216922i −0.726976 0.686663i \(-0.759075\pi\)
0.428408 + 0.903586i \(0.359075\pi\)
\(174\) 0 0
\(175\) −0.809017 2.48990i −0.0611559 0.188219i
\(176\) 0 0
\(177\) −4.42705 + 13.6251i −0.332758 + 1.02412i
\(178\) 0 0
\(179\) 5.57295 + 17.1518i 0.416542 + 1.28198i 0.910865 + 0.412705i \(0.135416\pi\)
−0.494323 + 0.869278i \(0.664584\pi\)
\(180\) 0 0
\(181\) 24.4721 1.81900 0.909500 0.415704i \(-0.136465\pi\)
0.909500 + 0.415704i \(0.136465\pi\)
\(182\) 0 0
\(183\) 17.5623 12.7598i 1.29824 0.943229i
\(184\) 0 0
\(185\) −8.94427 −0.657596
\(186\) 0 0
\(187\) 12.0902 0.884121
\(188\) 0 0
\(189\) 11.5902 8.42075i 0.843061 0.612520i
\(190\) 0 0
\(191\) −7.41641 −0.536632 −0.268316 0.963331i \(-0.586467\pi\)
−0.268316 + 0.963331i \(0.586467\pi\)
\(192\) 0 0
\(193\) −4.75329 14.6291i −0.342149 1.05303i −0.963092 0.269171i \(-0.913250\pi\)
0.620943 0.783856i \(-0.286750\pi\)
\(194\) 0 0
\(195\) −1.61803 + 4.97980i −0.115870 + 0.356611i
\(196\) 0 0
\(197\) 1.89919 + 5.84510i 0.135311 + 0.416446i 0.995638 0.0932969i \(-0.0297406\pi\)
−0.860327 + 0.509743i \(0.829741\pi\)
\(198\) 0 0
\(199\) −11.7812 + 8.55951i −0.835144 + 0.606767i −0.921010 0.389539i \(-0.872634\pi\)
0.0858661 + 0.996307i \(0.472634\pi\)
\(200\) 0 0
\(201\) −2.00000 + 1.45309i −0.141069 + 0.102493i
\(202\) 0 0
\(203\) 0.690983 2.12663i 0.0484975 0.149260i
\(204\) 0 0
\(205\) 19.5623 + 14.2128i 1.36629 + 0.992668i
\(206\) 0 0
\(207\) 1.66312 + 1.20833i 0.115595 + 0.0839845i
\(208\) 0 0
\(209\) 3.92705 12.0862i 0.271640 0.836021i
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) 0 0
\(213\) −2.07295 6.37988i −0.142036 0.437142i
\(214\) 0 0
\(215\) −14.3262 10.4086i −0.977041 0.709862i
\(216\) 0 0
\(217\) −8.66312 + 11.7229i −0.588091 + 0.795806i
\(218\) 0 0
\(219\) −16.5172 12.0005i −1.11613 0.810916i
\(220\) 0 0
\(221\) −2.30902 7.10642i −0.155321 0.478030i
\(222\) 0 0
\(223\) 0.944272 0.0632331 0.0316166 0.999500i \(-0.489934\pi\)
0.0316166 + 0.999500i \(0.489934\pi\)
\(224\) 0 0
\(225\) 0.118034 0.363271i 0.00786893 0.0242181i
\(226\) 0 0
\(227\) −3.16312 2.29814i −0.209944 0.152533i 0.477845 0.878444i \(-0.341418\pi\)
−0.687788 + 0.725911i \(0.741418\pi\)
\(228\) 0 0
\(229\) 0.0729490 + 0.0530006i 0.00482061 + 0.00350238i 0.590193 0.807262i \(-0.299052\pi\)
−0.585372 + 0.810765i \(0.699052\pi\)
\(230\) 0 0
\(231\) 3.42705 10.5474i 0.225483 0.693967i
\(232\) 0 0
\(233\) 12.5451 9.11454i 0.821856 0.597113i −0.0953876 0.995440i \(-0.530409\pi\)
0.917243 + 0.398327i \(0.130409\pi\)
\(234\) 0 0
\(235\) 1.38197 1.00406i 0.0901495 0.0654975i
\(236\) 0 0
\(237\) 1.92705 + 5.93085i 0.125175 + 0.385250i
\(238\) 0 0
\(239\) −8.33688 + 25.6583i −0.539268 + 1.65970i 0.194974 + 0.980808i \(0.437538\pi\)
−0.734242 + 0.678888i \(0.762462\pi\)
\(240\) 0 0
\(241\) 0.190983 + 0.587785i 0.0123023 + 0.0378626i 0.957019 0.290024i \(-0.0936635\pi\)
−0.944717 + 0.327887i \(0.893663\pi\)
\(242\) 0 0
\(243\) 3.94427 0.253025
\(244\) 0 0
\(245\) 0.236068 0.171513i 0.0150818 0.0109576i
\(246\) 0 0
\(247\) −7.85410 −0.499745
\(248\) 0 0
\(249\) −5.09017 −0.322576
\(250\) 0 0
\(251\) −2.07295 + 1.50609i −0.130843 + 0.0950633i −0.651282 0.758836i \(-0.725768\pi\)
0.520439 + 0.853899i \(0.325768\pi\)
\(252\) 0 0
\(253\) 14.0902 0.885841
\(254\) 0 0
\(255\) 4.61803 + 14.2128i 0.289193 + 0.890043i
\(256\) 0 0
\(257\) 8.19098 25.2093i 0.510939 1.57251i −0.279611 0.960113i \(-0.590206\pi\)
0.790551 0.612397i \(-0.209794\pi\)
\(258\) 0 0
\(259\) −3.61803 11.1352i −0.224814 0.691905i
\(260\) 0 0
\(261\) 0.263932 0.191758i 0.0163370 0.0118695i
\(262\) 0 0
\(263\) −17.0172 + 12.3637i −1.04933 + 0.762381i −0.972084 0.234632i \(-0.924612\pi\)
−0.0772425 + 0.997012i \(0.524612\pi\)
\(264\) 0 0
\(265\) 5.32624 16.3925i 0.327188 1.00698i
\(266\) 0 0
\(267\) 10.8992 + 7.91872i 0.667019 + 0.484618i
\(268\) 0 0
\(269\) −12.3992 9.00854i −0.755992 0.549260i 0.141687 0.989912i \(-0.454747\pi\)
−0.897678 + 0.440651i \(0.854747\pi\)
\(270\) 0 0
\(271\) −7.10081 + 21.8541i −0.431344 + 1.32754i 0.465444 + 0.885078i \(0.345895\pi\)
−0.896787 + 0.442462i \(0.854105\pi\)
\(272\) 0 0
\(273\) −6.85410 −0.414829
\(274\) 0 0
\(275\) −0.809017 2.48990i −0.0487856 0.150147i
\(276\) 0 0
\(277\) −14.3992 10.4616i −0.865163 0.628578i 0.0641214 0.997942i \(-0.479576\pi\)
−0.929285 + 0.369364i \(0.879576\pi\)
\(278\) 0 0
\(279\) −2.02786 + 0.640786i −0.121405 + 0.0383628i
\(280\) 0 0
\(281\) −15.6353 11.3597i −0.932721 0.677662i 0.0139364 0.999903i \(-0.495564\pi\)
−0.946658 + 0.322241i \(0.895564\pi\)
\(282\) 0 0
\(283\) −5.48278 16.8743i −0.325917 1.00307i −0.971025 0.238979i \(-0.923187\pi\)
0.645107 0.764092i \(-0.276813\pi\)
\(284\) 0 0
\(285\) 15.7082 0.930474
\(286\) 0 0
\(287\) −9.78115 + 30.1033i −0.577363 + 1.77694i
\(288\) 0 0
\(289\) −3.50000 2.54290i −0.205882 0.149582i
\(290\) 0 0
\(291\) −21.7533 15.8047i −1.27520 0.926487i
\(292\) 0 0
\(293\) −1.33688 + 4.11450i −0.0781014 + 0.240371i −0.982483 0.186354i \(-0.940333\pi\)
0.904381 + 0.426725i \(0.140333\pi\)
\(294\) 0 0
\(295\) 14.3262 10.4086i 0.834106 0.606013i
\(296\) 0 0
\(297\) 11.5902 8.42075i 0.672530 0.488622i
\(298\) 0 0
\(299\) −2.69098 8.28199i −0.155624 0.478960i
\(300\) 0 0
\(301\) 7.16312 22.0458i 0.412875 1.27070i
\(302\) 0 0
\(303\) −4.16312 12.8128i −0.239165 0.736074i
\(304\) 0 0
\(305\) −26.8328 −1.53644
\(306\) 0 0
\(307\) 6.69098 4.86128i 0.381875 0.277448i −0.380243 0.924887i \(-0.624160\pi\)
0.762118 + 0.647438i \(0.224160\pi\)
\(308\) 0 0
\(309\) 28.7984 1.63828
\(310\) 0 0
\(311\) 21.8885 1.24119 0.620593 0.784133i \(-0.286892\pi\)
0.620593 + 0.784133i \(0.286892\pi\)
\(312\) 0 0
\(313\) −1.92705 + 1.40008i −0.108923 + 0.0791375i −0.640914 0.767613i \(-0.721444\pi\)
0.531990 + 0.846751i \(0.321444\pi\)
\(314\) 0 0
\(315\) −2.00000 −0.112687
\(316\) 0 0
\(317\) −0.572949 1.76336i −0.0321800 0.0990399i 0.933676 0.358118i \(-0.116581\pi\)
−0.965856 + 0.259078i \(0.916581\pi\)
\(318\) 0 0
\(319\) 0.690983 2.12663i 0.0386876 0.119068i
\(320\) 0 0
\(321\) 6.39919 + 19.6947i 0.357168 + 1.09925i
\(322\) 0 0
\(323\) −18.1353 + 13.1760i −1.00907 + 0.733134i
\(324\) 0 0
\(325\) −1.30902 + 0.951057i −0.0726112 + 0.0527551i
\(326\) 0 0
\(327\) 3.95492 12.1720i 0.218707 0.673112i
\(328\) 0 0
\(329\) 1.80902 + 1.31433i 0.0997343 + 0.0724612i
\(330\) 0 0
\(331\) 8.54508 + 6.20837i 0.469680 + 0.341243i 0.797317 0.603561i \(-0.206252\pi\)
−0.327636 + 0.944804i \(0.606252\pi\)
\(332\) 0 0
\(333\) 0.527864 1.62460i 0.0289268 0.0890274i
\(334\) 0 0
\(335\) 3.05573 0.166952
\(336\) 0 0
\(337\) 3.13525 + 9.64932i 0.170788 + 0.525632i 0.999416 0.0341677i \(-0.0108780\pi\)
−0.828628 + 0.559800i \(0.810878\pi\)
\(338\) 0 0
\(339\) 14.1353 + 10.2699i 0.767722 + 0.557782i
\(340\) 0 0
\(341\) −8.66312 + 11.7229i −0.469134 + 0.634833i
\(342\) 0 0
\(343\) 15.1353 + 10.9964i 0.817227 + 0.593750i
\(344\) 0 0
\(345\) 5.38197 + 16.5640i 0.289755 + 0.891775i
\(346\) 0 0
\(347\) −22.8328 −1.22573 −0.612865 0.790188i \(-0.709983\pi\)
−0.612865 + 0.790188i \(0.709983\pi\)
\(348\) 0 0
\(349\) −2.28115 + 7.02067i −0.122107 + 0.375808i −0.993363 0.115022i \(-0.963306\pi\)
0.871256 + 0.490829i \(0.163306\pi\)
\(350\) 0 0
\(351\) −7.16312 5.20431i −0.382339 0.277786i
\(352\) 0 0
\(353\) 13.7812 + 10.0126i 0.733497 + 0.532917i 0.890668 0.454655i \(-0.150237\pi\)
−0.157171 + 0.987571i \(0.550237\pi\)
\(354\) 0 0
\(355\) −2.56231 + 7.88597i −0.135993 + 0.418544i
\(356\) 0 0
\(357\) −15.8262 + 11.4984i −0.837613 + 0.608562i
\(358\) 0 0
\(359\) 9.63525 7.00042i 0.508529 0.369468i −0.303736 0.952756i \(-0.598234\pi\)
0.812265 + 0.583288i \(0.198234\pi\)
\(360\) 0 0
\(361\) 1.40983 + 4.33901i 0.0742016 + 0.228369i
\(362\) 0 0
\(363\) −2.07295 + 6.37988i −0.108802 + 0.334857i
\(364\) 0 0
\(365\) 7.79837 + 24.0009i 0.408186 + 1.25627i
\(366\) 0 0
\(367\) 20.9443 1.09328 0.546641 0.837367i \(-0.315906\pi\)
0.546641 + 0.837367i \(0.315906\pi\)
\(368\) 0 0
\(369\) −3.73607 + 2.71441i −0.194492 + 0.141307i
\(370\) 0 0
\(371\) 22.5623 1.17138
\(372\) 0 0
\(373\) −2.94427 −0.152449 −0.0762243 0.997091i \(-0.524287\pi\)
−0.0762243 + 0.997091i \(0.524287\pi\)
\(374\) 0 0
\(375\) 15.7082 11.4127i 0.811168 0.589348i
\(376\) 0 0
\(377\) −1.38197 −0.0711749
\(378\) 0 0
\(379\) 5.57295 + 17.1518i 0.286263 + 0.881027i 0.986017 + 0.166644i \(0.0532929\pi\)
−0.699754 + 0.714384i \(0.746707\pi\)
\(380\) 0 0
\(381\) −1.66312 + 5.11855i −0.0852042 + 0.262231i
\(382\) 0 0
\(383\) 1.86475 + 5.73910i 0.0952840 + 0.293254i 0.987328 0.158695i \(-0.0507288\pi\)
−0.892044 + 0.451949i \(0.850729\pi\)
\(384\) 0 0
\(385\) −11.0902 + 8.05748i −0.565207 + 0.410647i
\(386\) 0 0
\(387\) 2.73607 1.98787i 0.139082 0.101049i
\(388\) 0 0
\(389\) 4.37132 13.4535i 0.221635 0.682122i −0.776981 0.629524i \(-0.783250\pi\)
0.998616 0.0525978i \(-0.0167501\pi\)
\(390\) 0 0
\(391\) −20.1074 14.6089i −1.01687 0.738803i
\(392\) 0 0
\(393\) −18.3713 13.3475i −0.926711 0.673295i
\(394\) 0 0
\(395\) 2.38197 7.33094i 0.119850 0.368860i
\(396\) 0 0
\(397\) 1.41641 0.0710875 0.0355437 0.999368i \(-0.488684\pi\)
0.0355437 + 0.999368i \(0.488684\pi\)
\(398\) 0 0
\(399\) 6.35410 + 19.5559i 0.318103 + 0.979020i
\(400\) 0 0
\(401\) −13.1631 9.56357i −0.657335 0.477582i 0.208427 0.978038i \(-0.433166\pi\)
−0.865762 + 0.500456i \(0.833166\pi\)
\(402\) 0 0
\(403\) 8.54508 + 2.85317i 0.425661 + 0.142126i
\(404\) 0 0
\(405\) 12.4721 + 9.06154i 0.619745 + 0.450271i
\(406\) 0 0
\(407\) −3.61803 11.1352i −0.179339 0.551950i
\(408\) 0 0
\(409\) −12.4721 −0.616707 −0.308354 0.951272i \(-0.599778\pi\)
−0.308354 + 0.951272i \(0.599778\pi\)
\(410\) 0 0
\(411\) 6.42705 19.7804i 0.317023 0.975697i
\(412\) 0 0
\(413\) 18.7533 + 13.6251i 0.922789 + 0.670445i
\(414\) 0 0
\(415\) 5.09017 + 3.69822i 0.249867 + 0.181539i
\(416\) 0 0
\(417\) −3.95492 + 12.1720i −0.193673 + 0.596064i
\(418\) 0 0
\(419\) 16.8713 12.2577i 0.824218 0.598829i −0.0936995 0.995601i \(-0.529869\pi\)
0.917918 + 0.396771i \(0.129869\pi\)
\(420\) 0 0
\(421\) 13.0172 9.45756i 0.634421 0.460933i −0.223508 0.974702i \(-0.571751\pi\)
0.857929 + 0.513769i \(0.171751\pi\)
\(422\) 0 0
\(423\) 0.100813 + 0.310271i 0.00490170 + 0.0150859i
\(424\) 0 0
\(425\) −1.42705 + 4.39201i −0.0692221 + 0.213044i
\(426\) 0 0
\(427\) −10.8541 33.4055i −0.525267 1.61661i
\(428\) 0 0
\(429\) −6.85410 −0.330919
\(430\) 0 0
\(431\) −5.01722 + 3.64522i −0.241671 + 0.175584i −0.702027 0.712150i \(-0.747722\pi\)
0.460356 + 0.887734i \(0.347722\pi\)
\(432\) 0 0
\(433\) −23.8885 −1.14801 −0.574005 0.818852i \(-0.694611\pi\)
−0.574005 + 0.818852i \(0.694611\pi\)
\(434\) 0 0
\(435\) 2.76393 0.132520
\(436\) 0 0
\(437\) −21.1353 + 15.3557i −1.01104 + 0.734561i
\(438\) 0 0
\(439\) −31.4164 −1.49942 −0.749712 0.661765i \(-0.769808\pi\)
−0.749712 + 0.661765i \(0.769808\pi\)
\(440\) 0 0
\(441\) 0.0172209 + 0.0530006i 0.000820044 + 0.00252384i
\(442\) 0 0
\(443\) 2.42705 7.46969i 0.115313 0.354896i −0.876699 0.481039i \(-0.840260\pi\)
0.992012 + 0.126143i \(0.0402597\pi\)
\(444\) 0 0
\(445\) −5.14590 15.8374i −0.243939 0.750767i
\(446\) 0 0
\(447\) 28.0344 20.3682i 1.32598 0.963383i
\(448\) 0 0
\(449\) 25.0172 18.1761i 1.18064 0.857782i 0.188392 0.982094i \(-0.439672\pi\)
0.992243 + 0.124312i \(0.0396723\pi\)
\(450\) 0 0
\(451\) −9.78115 + 30.1033i −0.460577 + 1.41751i
\(452\) 0 0
\(453\) 15.5172 + 11.2739i 0.729063 + 0.529695i
\(454\) 0 0
\(455\) 6.85410 + 4.97980i 0.321325 + 0.233456i
\(456\) 0 0
\(457\) 0.663119 2.04087i 0.0310194 0.0954679i −0.934348 0.356361i \(-0.884017\pi\)
0.965368 + 0.260894i \(0.0840172\pi\)
\(458\) 0 0
\(459\) −25.2705 −1.17953
\(460\) 0 0
\(461\) −8.57295 26.3848i −0.399282 1.22886i −0.925576 0.378561i \(-0.876419\pi\)
0.526294 0.850303i \(-0.323581\pi\)
\(462\) 0 0
\(463\) 9.30902 + 6.76340i 0.432627 + 0.314322i 0.782698 0.622401i \(-0.213843\pi\)
−0.350072 + 0.936723i \(0.613843\pi\)
\(464\) 0 0
\(465\) −17.0902 5.70634i −0.792538 0.264625i
\(466\) 0 0
\(467\) −5.45492 3.96323i −0.252423 0.183396i 0.454377 0.890810i \(-0.349862\pi\)
−0.706800 + 0.707413i \(0.749862\pi\)
\(468\) 0 0
\(469\) 1.23607 + 3.80423i 0.0570763 + 0.175663i
\(470\) 0 0
\(471\) −17.0902 −0.787473
\(472\) 0 0
\(473\) 7.16312 22.0458i 0.329361 1.01367i
\(474\) 0 0
\(475\) 3.92705 + 2.85317i 0.180185 + 0.130912i
\(476\) 0 0
\(477\) 2.66312 + 1.93487i 0.121936 + 0.0885916i
\(478\) 0 0
\(479\) 5.84346 17.9843i 0.266994 0.821724i −0.724233 0.689556i \(-0.757806\pi\)
0.991227 0.132169i \(-0.0421941\pi\)
\(480\) 0 0
\(481\) −5.85410 + 4.25325i −0.266924 + 0.193932i
\(482\) 0 0
\(483\) −18.4443 + 13.4005i −0.839243 + 0.609746i
\(484\) 0 0
\(485\) 10.2705 + 31.6094i 0.466360 + 1.43531i
\(486\) 0 0
\(487\) 7.19098 22.1316i 0.325855 1.00288i −0.645199 0.764015i \(-0.723225\pi\)
0.971053 0.238862i \(-0.0767745\pi\)
\(488\) 0 0
\(489\) −9.01722 27.7522i −0.407773 1.25500i
\(490\) 0 0
\(491\) 8.36068 0.377312 0.188656 0.982043i \(-0.439587\pi\)
0.188656 + 0.982043i \(0.439587\pi\)
\(492\) 0 0
\(493\) −3.19098 + 2.31838i −0.143715 + 0.104415i
\(494\) 0 0
\(495\) −2.00000 −0.0898933
\(496\) 0 0
\(497\) −10.8541 −0.486873
\(498\) 0 0
\(499\) −1.30902 + 0.951057i −0.0585996 + 0.0425751i −0.616699 0.787199i \(-0.711531\pi\)
0.558100 + 0.829774i \(0.311531\pi\)
\(500\) 0 0
\(501\) −8.14590 −0.363932
\(502\) 0 0
\(503\) −5.19098 15.9762i −0.231454 0.712344i −0.997572 0.0696431i \(-0.977814\pi\)
0.766117 0.642701i \(-0.222186\pi\)
\(504\) 0 0
\(505\) −5.14590 + 15.8374i −0.228989 + 0.704757i
\(506\) 0 0
\(507\) −5.19098 15.9762i −0.230540 0.709528i
\(508\) 0 0
\(509\) 17.0172 12.3637i 0.754275 0.548013i −0.142874 0.989741i \(-0.545634\pi\)
0.897149 + 0.441728i \(0.145634\pi\)
\(510\) 0 0
\(511\) −26.7254 + 19.4172i −1.18226 + 0.858964i
\(512\) 0 0
\(513\) −8.20820 + 25.2623i −0.362401 + 1.11536i
\(514\) 0 0
\(515\) −28.7984 20.9232i −1.26901 0.921988i
\(516\) 0 0
\(517\) 1.80902 + 1.31433i 0.0795605 + 0.0578041i
\(518\) 0 0
\(519\) 2.42705 7.46969i 0.106536 0.327883i
\(520\) 0 0
\(521\) 34.3607 1.50537 0.752684 0.658382i \(-0.228759\pi\)
0.752684 + 0.658382i \(0.228759\pi\)
\(522\) 0 0
\(523\) 9.86475 + 30.3606i 0.431355 + 1.32757i 0.896776 + 0.442485i \(0.145903\pi\)
−0.465421 + 0.885090i \(0.654097\pi\)
\(524\) 0 0
\(525\) 3.42705 + 2.48990i 0.149569 + 0.108668i
\(526\) 0 0
\(527\) 24.5172 7.74721i 1.06799 0.337474i
\(528\) 0 0
\(529\) −4.82624 3.50647i −0.209836 0.152455i
\(530\) 0 0
\(531\) 1.04508 + 3.21644i 0.0453528 + 0.139582i
\(532\) 0 0
\(533\) 19.5623 0.847338
\(534\) 0 0
\(535\) 7.90983 24.3440i 0.341972 1.05248i
\(536\) 0 0
\(537\) −23.6074 17.1518i −1.01873 0.740154i
\(538\) 0 0
\(539\) 0.309017 + 0.224514i 0.0133103 + 0.00967050i
\(540\) 0 0
\(541\) −11.0451 + 33.9933i −0.474865 + 1.46149i 0.371274 + 0.928523i \(0.378921\pi\)
−0.846139 + 0.532962i \(0.821079\pi\)
\(542\) 0 0
\(543\) −32.0344 + 23.2744i −1.37473 + 0.998799i
\(544\) 0 0
\(545\) −12.7984 + 9.29856i −0.548222 + 0.398307i
\(546\) 0 0
\(547\) −5.84346 17.9843i −0.249848 0.768954i −0.994801 0.101837i \(-0.967528\pi\)
0.744953 0.667117i \(-0.232472\pi\)
\(548\) 0 0
\(549\) 1.58359 4.87380i 0.0675861 0.208009i
\(550\) 0 0
\(551\) 1.28115 + 3.94298i 0.0545790 + 0.167977i
\(552\) 0 0
\(553\) 10.0902 0.429078
\(554\) 0 0
\(555\) 11.7082 8.50651i 0.496986 0.361081i
\(556\) 0 0
\(557\) 26.3607 1.11694 0.558469 0.829525i \(-0.311389\pi\)
0.558469 + 0.829525i \(0.311389\pi\)
\(558\) 0 0
\(559\) −14.3262 −0.605935
\(560\) 0 0
\(561\) −15.8262 + 11.4984i −0.668184 + 0.485464i
\(562\) 0 0
\(563\) 28.9443 1.21986 0.609928 0.792457i \(-0.291198\pi\)
0.609928 + 0.792457i \(0.291198\pi\)
\(564\) 0 0
\(565\) −6.67376 20.5397i −0.280767 0.864113i
\(566\) 0 0
\(567\) −6.23607 + 19.1926i −0.261890 + 0.806015i
\(568\) 0 0
\(569\) 6.01064 + 18.4989i 0.251979 + 0.775512i 0.994410 + 0.105589i \(0.0336729\pi\)
−0.742431 + 0.669923i \(0.766327\pi\)
\(570\) 0 0
\(571\) 20.1074 14.6089i 0.841468 0.611362i −0.0813123 0.996689i \(-0.525911\pi\)
0.922780 + 0.385326i \(0.125911\pi\)
\(572\) 0 0
\(573\) 9.70820 7.05342i 0.405566 0.294661i
\(574\) 0 0
\(575\) −1.66312 + 5.11855i −0.0693569 + 0.213458i
\(576\) 0 0
\(577\) −9.92705 7.21242i −0.413269 0.300257i 0.361655 0.932312i \(-0.382212\pi\)
−0.774924 + 0.632055i \(0.782212\pi\)
\(578\) 0 0
\(579\) 20.1353 + 14.6291i 0.836793 + 0.607965i
\(580\) 0 0
\(581\) −2.54508 + 7.83297i −0.105588 + 0.324966i
\(582\) 0 0
\(583\) 22.5623 0.934435
\(584\) 0 0
\(585\) 0.381966 + 1.17557i 0.0157924 + 0.0486039i
\(586\) 0 0
\(587\) 17.0172 + 12.3637i 0.702376 + 0.510306i 0.880705 0.473665i \(-0.157069\pi\)
−0.178329 + 0.983971i \(0.557069\pi\)
\(588\) 0 0
\(589\) 0.218847 27.0256i 0.00901744 1.11357i
\(590\) 0 0
\(591\) −8.04508 5.84510i −0.330931 0.240435i
\(592\) 0 0
\(593\) −1.69756 5.22455i −0.0697105 0.214547i 0.910132 0.414318i \(-0.135980\pi\)
−0.979842 + 0.199772i \(0.935980\pi\)
\(594\) 0 0
\(595\) 24.1803 0.991297
\(596\) 0 0
\(597\) 7.28115 22.4091i 0.297998 0.917143i
\(598\) 0 0
\(599\) 21.4894 + 15.6129i 0.878031 + 0.637927i 0.932730 0.360576i \(-0.117420\pi\)
−0.0546986 + 0.998503i \(0.517420\pi\)
\(600\) 0 0
\(601\) 4.54508 + 3.30220i 0.185398 + 0.134699i 0.676613 0.736339i \(-0.263447\pi\)
−0.491215 + 0.871038i \(0.663447\pi\)
\(602\) 0 0
\(603\) −0.180340 + 0.555029i −0.00734401 + 0.0226025i
\(604\) 0 0
\(605\) 6.70820 4.87380i 0.272727 0.198148i
\(606\) 0 0
\(607\) −5.30902 + 3.85723i −0.215486 + 0.156560i −0.690293 0.723530i \(-0.742518\pi\)
0.474806 + 0.880090i \(0.342518\pi\)
\(608\) 0 0
\(609\) 1.11803 + 3.44095i 0.0453050 + 0.139435i
\(610\) 0 0
\(611\) 0.427051 1.31433i 0.0172766 0.0531720i
\(612\) 0 0
\(613\) 10.9549 + 33.7158i 0.442465 + 1.36177i 0.885240 + 0.465134i \(0.153994\pi\)
−0.442775 + 0.896633i \(0.646006\pi\)
\(614\) 0 0
\(615\) −39.1246 −1.57766
\(616\) 0 0
\(617\) −37.3435 + 27.1316i −1.50339 + 1.09228i −0.534385 + 0.845241i \(0.679457\pi\)
−0.969006 + 0.247036i \(0.920543\pi\)
\(618\) 0 0
\(619\) −11.0557 −0.444367 −0.222184 0.975005i \(-0.571318\pi\)
−0.222184 + 0.975005i \(0.571318\pi\)
\(620\) 0 0
\(621\) −29.4508 −1.18182
\(622\) 0 0
\(623\) 17.6353 12.8128i 0.706542 0.513332i
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 0 0
\(627\) 6.35410 + 19.5559i 0.253758 + 0.780988i
\(628\) 0 0
\(629\) −6.38197 + 19.6417i −0.254466 + 0.783165i
\(630\) 0 0
\(631\) −3.37132 10.3759i −0.134210 0.413057i 0.861256 0.508171i \(-0.169678\pi\)
−0.995466 + 0.0951145i \(0.969678\pi\)
\(632\) 0 0
\(633\) 5.23607 3.80423i 0.208115 0.151204i
\(634\) 0 0
\(635\) 5.38197 3.91023i 0.213577 0.155173i
\(636\) 0 0
\(637\) 0.0729490 0.224514i 0.00289035 0.00889557i
\(638\) 0 0
\(639\) −1.28115 0.930812i −0.0506816 0.0368224i
\(640\) 0 0
\(641\) 2.83688 + 2.06111i 0.112050 + 0.0814091i 0.642399 0.766370i \(-0.277939\pi\)
−0.530349 + 0.847779i \(0.677939\pi\)
\(642\) 0 0
\(643\) −7.93363 + 24.4172i −0.312872 + 0.962920i 0.663750 + 0.747955i \(0.268964\pi\)
−0.976622 + 0.214966i \(0.931036\pi\)
\(644\) 0 0
\(645\) 28.6525 1.12819
\(646\) 0 0
\(647\) 3.10081 + 9.54332i 0.121906 + 0.375187i 0.993325 0.115353i \(-0.0367998\pi\)
−0.871419 + 0.490539i \(0.836800\pi\)
\(648\) 0 0
\(649\) 18.7533 + 13.6251i 0.736131 + 0.534830i
\(650\) 0 0
\(651\) 0.190983 23.5847i 0.00748521 0.924356i
\(652\) 0 0
\(653\) 35.9615 + 26.1276i 1.40728 + 1.02245i 0.993711 + 0.111978i \(0.0357188\pi\)
0.413571 + 0.910472i \(0.364281\pi\)
\(654\) 0 0
\(655\) 8.67376 + 26.6951i 0.338912 + 1.04306i
\(656\) 0 0
\(657\) −4.81966 −0.188033
\(658\) 0 0
\(659\) −4.62868 + 14.2456i −0.180308 + 0.554930i −0.999836 0.0181087i \(-0.994236\pi\)
0.819528 + 0.573039i \(0.194236\pi\)
\(660\) 0 0
\(661\) 24.7254 + 17.9641i 0.961708 + 0.698721i 0.953547 0.301245i \(-0.0974023\pi\)
0.00816087 + 0.999967i \(0.497402\pi\)
\(662\) 0 0
\(663\) 9.78115 + 7.10642i 0.379869 + 0.275991i
\(664\) 0 0
\(665\) 7.85410 24.1724i 0.304569 0.937367i
\(666\) 0 0
\(667\) −3.71885 + 2.70190i −0.143994 + 0.104618i
\(668\) 0 0
\(669\) −1.23607 + 0.898056i −0.0477891 + 0.0347208i
\(670\) 0 0
\(671\) −10.8541 33.4055i −0.419018 1.28960i
\(672\) 0 0
\(673\) 2.30244 7.08618i 0.0887525 0.273152i −0.896823 0.442390i \(-0.854131\pi\)
0.985575 + 0.169238i \(0.0541307\pi\)
\(674\) 0 0
\(675\) 1.69098 + 5.20431i 0.0650860 + 0.200314i
\(676\) 0 0
\(677\) −47.3050 −1.81808 −0.909038 0.416712i \(-0.863182\pi\)
−0.909038 + 0.416712i \(0.863182\pi\)
\(678\) 0 0
\(679\) −35.1976 + 25.5725i −1.35076 + 0.981383i
\(680\) 0 0
\(681\) 6.32624 0.242422
\(682\) 0 0
\(683\) 48.7214 1.86427 0.932136 0.362110i \(-0.117943\pi\)
0.932136 + 0.362110i \(0.117943\pi\)
\(684\) 0 0
\(685\) −20.7984 + 15.1109i −0.794665 + 0.577358i
\(686\) 0 0
\(687\) −0.145898 −0.00556636
\(688\) 0 0
\(689\) −4.30902 13.2618i −0.164160 0.505234i
\(690\) 0 0
\(691\) −1.39261 + 4.28601i −0.0529773 + 0.163048i −0.974045 0.226356i \(-0.927319\pi\)
0.921067 + 0.389403i \(0.127319\pi\)
\(692\) 0 0
\(693\) −0.809017 2.48990i −0.0307320 0.0945834i
\(694\) 0 0
\(695\) 12.7984 9.29856i 0.485470 0.352715i
\(696\) 0 0
\(697\) 45.1697 32.8177i 1.71092 1.24306i
\(698\) 0 0
\(699\) −7.75329 + 23.8622i −0.293256 + 0.902550i
\(700\) 0 0
\(701\) −23.1631 16.8290i −0.874859 0.635622i 0.0570274 0.998373i \(-0.481838\pi\)
−0.931886 + 0.362750i \(0.881838\pi\)
\(702\) 0 0
\(703\) 17.5623 + 12.7598i 0.662375 + 0.481244i
\(704\) 0 0
\(705\) −0.854102 + 2.62866i −0.0321673 + 0.0990009i
\(706\) 0 0
\(707\) −21.7984 −0.819812
\(708\) 0 0
\(709\) −12.4615 38.3525i −0.468001 1.44036i −0.855169 0.518350i \(-0.826546\pi\)
0.387167 0.922009i \(-0.373454\pi\)
\(710\) 0 0
\(711\) 1.19098 + 0.865300i 0.0446654 + 0.0324513i
\(712\) 0 0
\(713\) 28.5729 9.02878i 1.07007 0.338131i
\(714\) 0 0
\(715\) 6.85410 + 4.97980i 0.256329 + 0.186234i
\(716\) 0 0
\(717\) −13.4894 41.5160i −0.503769 1.55044i
\(718\) 0 0
\(719\) −32.9443 −1.22861 −0.614307 0.789067i \(-0.710564\pi\)
−0.614307 + 0.789067i \(0.710564\pi\)
\(720\) 0 0
\(721\) 14.3992 44.3161i 0.536254 1.65042i
\(722\) 0 0
\(723\) −0.809017 0.587785i −0.0300877 0.0218600i
\(724\) 0 0
\(725\) 0.690983 + 0.502029i 0.0256625 + 0.0186449i
\(726\) 0 0
\(727\) 10.3156 31.7481i 0.382584 1.17747i −0.555633 0.831427i \(-0.687524\pi\)
0.938218 0.346046i \(-0.112476\pi\)
\(728\) 0 0
\(729\) −23.8713 + 17.3435i −0.884123 + 0.642353i
\(730\) 0 0
\(731\) −33.0795 + 24.0337i −1.22349 + 0.888918i
\(732\) 0 0
\(733\) −2.86475 8.81678i −0.105812 0.325655i 0.884108 0.467282i \(-0.154767\pi\)
−0.989920 + 0.141627i \(0.954767\pi\)
\(734\) 0 0
\(735\) −0.145898 + 0.449028i −0.00538153 + 0.0165626i
\(736\) 0 0
\(737\) 1.23607 + 3.80423i 0.0455311 + 0.140130i
\(738\) 0 0
\(739\) −41.5279 −1.52763 −0.763814 0.645437i \(-0.776675\pi\)
−0.763814 + 0.645437i \(0.776675\pi\)
\(740\) 0 0
\(741\) 10.2812 7.46969i 0.377688 0.274406i
\(742\) 0 0
\(743\) −23.0557 −0.845833 −0.422916 0.906169i \(-0.638994\pi\)
−0.422916 + 0.906169i \(0.638994\pi\)
\(744\) 0 0
\(745\) −42.8328 −1.56927
\(746\) 0 0
\(747\) −0.972136 + 0.706298i −0.0355686 + 0.0258421i
\(748\) 0 0
\(749\) 33.5066 1.22430
\(750\) 0 0
\(751\) −15.0106 46.1980i −0.547746 1.68579i −0.714370 0.699769i \(-0.753286\pi\)
0.166623 0.986021i \(-0.446714\pi\)
\(752\) 0 0
\(753\) 1.28115 3.94298i 0.0466878 0.143690i
\(754\) 0 0
\(755\) −7.32624 22.5478i −0.266629 0.820600i
\(756\) 0 0
\(757\) −14.1074 + 10.2496i −0.512742 + 0.372529i −0.813863 0.581057i \(-0.802639\pi\)
0.301121 + 0.953586i \(0.402639\pi\)
\(758\) 0 0
\(759\) −18.4443 + 13.4005i −0.669485 + 0.486409i
\(760\) 0 0
\(761\) 14.0795 43.3323i 0.510382 1.57080i −0.281147 0.959665i \(-0.590715\pi\)
0.791529 0.611131i \(-0.209285\pi\)
\(762\) 0 0
\(763\) −16.7533 12.1720i −0.606510 0.440655i
\(764\) 0 0
\(765\) 2.85410 + 2.07363i 0.103190 + 0.0749721i
\(766\) 0 0
\(767\) 4.42705 13.6251i 0.159852 0.491972i
\(768\) 0 0
\(769\) 36.8328 1.32823 0.664113 0.747633i \(-0.268810\pi\)
0.664113 + 0.747633i \(0.268810\pi\)
\(770\) 0 0
\(771\) 13.2533 + 40.7894i 0.477306 + 1.46900i
\(772\) 0 0
\(773\) −4.39919 3.19620i −0.158228 0.114959i 0.505854 0.862619i \(-0.331177\pi\)
−0.664082 + 0.747660i \(0.731177\pi\)
\(774\) 0 0
\(775\) −3.23607 4.53077i −0.116243 0.162750i
\(776\) 0 0
\(777\) 15.3262 + 11.1352i 0.549826 + 0.399472i
\(778\) 0 0
\(779\) −18.1353 55.8146i −0.649763 1.99976i
\(780\) 0 0
\(781\) −10.8541 −0.388390
\(782\) 0 0
\(783\) −1.44427 + 4.44501i −0.0516141 + 0.158852i
\(784\) 0 0
\(785\) 17.0902 + 12.4167i 0.609974 + 0.443172i
\(786\) 0 0
\(787\) 9.48936 + 6.89442i 0.338259 + 0.245760i 0.743927 0.668261i \(-0.232961\pi\)
−0.405668 + 0.914021i \(0.632961\pi\)
\(788\) 0 0
\(789\) 10.5172 32.3687i 0.374423 1.15236i
\(790\) 0 0
\(791\) 22.8713 16.6170i 0.813211 0.590832i
\(792\) 0 0
\(793\) −17.5623 + 12.7598i −0.623656 + 0.453112i
\(794\) 0 0
\(795\) 8.61803 + 26.5236i 0.305650 + 0.940694i
\(796\) 0 0
\(797\) −8.46149 + 26.0418i −0.299721 + 0.922448i 0.681873 + 0.731471i \(0.261166\pi\)
−0.981594 + 0.190977i \(0.938834\pi\)
\(798\) 0 0
\(799\) −1.21885 3.75123i −0.0431197 0.132709i
\(800\) 0 0
\(801\) 3.18034 0.112372
\(802\) 0 0
\(803\) −26.7254 + 19.4172i −0.943120 + 0.685217i
\(804\) 0 0
\(805\) 28.1803 0.993226
\(806\) 0 0
\(807\) 24.7984 0.872944
\(808\) 0 0
\(809\) 6.25329 4.54328i 0.219854 0.159733i −0.472407 0.881381i \(-0.656615\pi\)
0.692261 + 0.721647i \(0.256615\pi\)
\(810\) 0 0
\(811\) −26.4721 −0.929562 −0.464781 0.885426i \(-0.653867\pi\)
−0.464781 + 0.885426i \(0.653867\pi\)
\(812\) 0 0
\(813\) −11.4894 35.3606i −0.402949 1.24015i
\(814\) 0 0
\(815\) −11.1459 + 34.3035i −0.390424 + 1.20160i
\(816\) 0 0
\(817\) 13.2812 + 40.8752i 0.464649 + 1.43004i
\(818\) 0 0
\(819\) −1.30902 + 0.951057i −0.0457408 + 0.0332326i
\(820\) 0 0
\(821\) −25.8156 + 18.7561i −0.900970 + 0.654593i −0.938715 0.344694i \(-0.887983\pi\)
0.0377446 + 0.999287i \(0.487983\pi\)
\(822\) 0 0
\(823\) −15.1697 + 46.6875i −0.528782 + 1.62742i 0.227930 + 0.973677i \(0.426804\pi\)
−0.756713 + 0.653747i \(0.773196\pi\)
\(824\) 0 0
\(825\) 3.42705 + 2.48990i 0.119315 + 0.0866871i
\(826\) 0 0
\(827\) −32.2877 23.4584i −1.12275 0.815729i −0.138130 0.990414i \(-0.544109\pi\)
−0.984624 + 0.174685i \(0.944109\pi\)
\(828\) 0 0
\(829\) 3.53851 10.8904i 0.122897 0.378239i −0.870615 0.491965i \(-0.836279\pi\)
0.993512 + 0.113726i \(0.0362786\pi\)
\(830\) 0 0
\(831\) 28.7984 0.999005
\(832\) 0 0
\(833\) −0.208204 0.640786i −0.00721384 0.0222019i
\(834\) 0 0
\(835\) 8.14590 + 5.91834i 0.281900 + 0.204813i
\(836\) 0 0
\(837\) 18.1074 24.5030i 0.625883 0.846946i
\(838\) 0 0
\(839\) 31.6697 + 23.0094i 1.09336 + 0.794372i 0.979963 0.199178i \(-0.0638271\pi\)
0.113396 + 0.993550i \(0.463827\pi\)
\(840\) 0 0
\(841\) −8.73607 26.8869i −0.301244 0.927133i
\(842\) 0 0
\(843\) 31.2705 1.07701
\(844\) 0 0
\(845\) −6.41641 + 19.7477i −0.220731 + 0.679341i
\(846\) 0 0
\(847\) 8.78115 + 6.37988i 0.301724 + 0.219215i
\(848\) 0 0
\(849\) 23.2254 + 16.8743i 0.797095 + 0.579123i
\(850\) 0 0
\(851\) −7.43769 + 22.8909i −0.254961 + 0.784689i
\(852\) 0 0
\(853\) −28.1074 + 20.4212i −0.962379 + 0.699209i −0.953702 0.300753i \(-0.902762\pi\)
−0.00867675 + 0.999962i \(0.502762\pi\)
\(854\) 0 0
\(855\) 3.00000 2.17963i 0.102598 0.0745417i
\(856\) 0 0
\(857\) 5.60739 + 17.2578i 0.191545 + 0.589514i 1.00000 0.000955970i \(0.000304295\pi\)
−0.808455 + 0.588558i \(0.799696\pi\)
\(858\) 0 0
\(859\) −3.75329 + 11.5514i −0.128061 + 0.394130i −0.994446 0.105246i \(-0.966437\pi\)
0.866386 + 0.499375i \(0.166437\pi\)
\(860\) 0 0
\(861\) −15.8262 48.7082i −0.539357 1.65997i
\(862\) 0 0
\(863\) −28.3607 −0.965409 −0.482704 0.875783i \(-0.660345\pi\)
−0.482704 + 0.875783i \(0.660345\pi\)
\(864\) 0 0
\(865\) −7.85410 + 5.70634i −0.267047 + 0.194021i
\(866\) 0 0
\(867\) 7.00000 0.237732
\(868\) 0 0
\(869\) 10.0902 0.342286
\(870\) 0 0
\(871\) 2.00000 1.45309i 0.0677674 0.0492359i
\(872\) 0 0
\(873\) −6.34752 −0.214831
\(874\) 0 0
\(875\) −9.70820 29.8788i −0.328197 1.01009i
\(876\) 0 0
\(877\) 1.42705 4.39201i 0.0481881 0.148308i −0.924067 0.382230i \(-0.875156\pi\)
0.972255 + 0.233922i \(0.0751562\pi\)
\(878\) 0 0
\(879\) −2.16312 6.65740i −0.0729602 0.224548i
\(880\) 0 0
\(881\) 8.25329 5.99637i 0.278060 0.202023i −0.440011 0.897993i \(-0.645025\pi\)
0.718071 + 0.695970i \(0.245025\pi\)
\(882\) 0 0
\(883\) 36.1074 26.2336i 1.21511 0.882829i 0.219426 0.975629i \(-0.429582\pi\)
0.995685 + 0.0927998i \(0.0295817\pi\)
\(884\) 0 0
\(885\) −8.85410 + 27.2501i −0.297627 + 0.916003i
\(886\) 0 0
\(887\) −19.3435 14.0538i −0.649490 0.471882i 0.213608 0.976920i \(-0.431479\pi\)
−0.863097 + 0.505038i \(0.831479\pi\)
\(888\) 0 0
\(889\) 7.04508 + 5.11855i 0.236285 + 0.171671i
\(890\) 0 0
\(891\) −6.23607 + 19.1926i −0.208916 + 0.642978i
\(892\) 0 0
\(893\) −4.14590 −0.138737
\(894\) 0 0
\(895\) 11.1459 + 34.3035i 0.372566 + 1.14664i
\(896\) 0 0
\(897\) 11.3992 + 8.28199i 0.380608 + 0.276528i
\(898\) 0 0
\(899\) 0.0385072 4.75528i 0.00128429 0.158598i
\(900\) 0 0
\(901\) −32.1976 23.3929i −1.07266 0.779330i
\(902\) 0 0
\(903\) 11.5902 + 35.6709i 0.385697 + 1.18705i
\(904\) 0 0
\(905\) 48.9443 1.62696
\(906\) 0 0
\(907\) 3.66312 11.2739i 0.121632 0.374344i −0.871641 0.490146i \(-0.836944\pi\)
0.993272 + 0.115801i \(0.0369436\pi\)
\(908\) 0 0
\(909\) −2.57295 1.86936i −0.0853393 0.0620027i
\(910\) 0 0
\(911\) −20.1074 14.6089i −0.666188 0.484014i 0.202559 0.979270i \(-0.435074\pi\)
−0.868747 + 0.495256i \(0.835074\pi\)
\(912\) 0 0
\(913\) −2.54508 + 7.83297i −0.0842300 + 0.259233i
\(914\) 0 0
\(915\) 35.1246 25.5195i 1.16118 0.843649i
\(916\) 0 0
\(917\) −29.7254 + 21.5968i −0.981620 + 0.713189i
\(918\) 0 0
\(919\) 0.628677 + 1.93487i 0.0207381 + 0.0638254i 0.960890 0.276930i \(-0.0893171\pi\)
−0.940152 + 0.340756i \(0.889317\pi\)
\(920\) 0 0
\(921\) −4.13525 + 12.7270i −0.136261 + 0.419369i
\(922\) 0 0
\(923\) 2.07295 + 6.37988i 0.0682319 + 0.209996i
\(924\) 0 0
\(925\) 4.47214 0.147043
\(926\) 0 0
\(927\) 5.50000 3.99598i 0.180644 0.131245i
\(928\) 0 0
\(929\) 50.3607 1.65228 0.826140 0.563465i \(-0.190532\pi\)
0.826140 + 0.563465i \(0.190532\pi\)
\(930\) 0 0
\(931\) −0.708204 −0.0232104
\(932\) 0 0
\(933\) −28.6525 + 20.8172i −0.938040 + 0.681526i
\(934\) 0 0
\(935\) 24.1803 0.790782
\(936\) 0 0
\(937\) 9.60739 + 29.5685i 0.313860 + 0.965961i 0.976221 + 0.216777i \(0.0695543\pi\)
−0.662361 + 0.749184i \(0.730446\pi\)
\(938\) 0 0
\(939\) 1.19098 3.66547i 0.0388663 0.119618i
\(940\) 0 0
\(941\) −4.46149 13.7311i −0.145441 0.447620i 0.851627 0.524149i \(-0.175616\pi\)
−0.997067 + 0.0765287i \(0.975616\pi\)
\(942\) 0 0
\(943\) 52.6418 38.2465i 1.71425 1.24548i
\(944\) 0 0
\(945\) 23.1803 16.8415i 0.754057 0.547854i
\(946\) 0 0
\(947\) 6.89919 21.2335i 0.224193 0.689996i −0.774179 0.632967i \(-0.781837\pi\)
0.998372 0.0570298i \(-0.0181630\pi\)
\(948\) 0 0
\(949\) 16.5172 + 12.0005i 0.536172 + 0.389552i
\(950\) 0 0
\(951\) 2.42705 + 1.76336i 0.0787025 + 0.0571807i
\(952\) 0 0
\(953\) −16.1697 + 49.7652i −0.523788 + 1.61205i 0.242913 + 0.970048i \(0.421897\pi\)
−0.766700 + 0.642005i \(0.778103\pi\)
\(954\) 0 0
\(955\) −14.8328 −0.479979
\(956\) 0 0
\(957\) 1.11803 + 3.44095i 0.0361409 + 0.111230i
\(958\) 0 0
\(959\) −27.2254 19.7804i −0.879155 0.638743i
\(960\) 0 0
\(961\) −10.0557 + 29.3238i −0.324378 + 0.945927i
\(962\) 0 0
\(963\) 3.95492 + 2.87341i 0.127445 + 0.0925945i
\(964\) 0 0
\(965\) −9.50658 29.2582i −0.306028 0.941856i
\(966\) 0 0
\(967\) −23.4164 −0.753021 −0.376510 0.926412i \(-0.622876\pi\)
−0.376510 + 0.926412i \(0.622876\pi\)
\(968\) 0 0
\(969\) 11.2082 34.4953i 0.360059 1.10815i
\(970\) 0 0
\(971\) 33.4894 + 24.3314i 1.07472 + 0.780833i 0.976756 0.214356i \(-0.0687654\pi\)
0.0979691 + 0.995189i \(0.468765\pi\)
\(972\) 0 0
\(973\) 16.7533 + 12.1720i 0.537086 + 0.390216i
\(974\) 0 0
\(975\) 0.809017 2.48990i 0.0259093 0.0797406i
\(976\) 0 0
\(977\) −7.45492 + 5.41631i −0.238504 + 0.173283i −0.700616 0.713538i \(-0.747092\pi\)
0.462113 + 0.886821i \(0.347092\pi\)
\(978\) 0 0
\(979\) 17.6353 12.8128i 0.563625 0.409498i
\(980\) 0 0
\(981\) −0.933629 2.87341i −0.0298085 0.0917410i
\(982\) 0 0
\(983\) −14.8090 + 45.5775i −0.472334 + 1.45370i 0.377185 + 0.926138i \(0.376892\pi\)
−0.849519 + 0.527558i \(0.823108\pi\)
\(984\) 0 0
\(985\) 3.79837 + 11.6902i 0.121026 + 0.372481i
\(986\) 0 0
\(987\) −3.61803 −0.115163
\(988\) 0 0
\(989\) −38.5517 + 28.0094i −1.22587 + 0.890648i
\(990\) 0 0
\(991\) 55.7771 1.77182 0.885909 0.463859i \(-0.153536\pi\)
0.885909 + 0.463859i \(0.153536\pi\)
\(992\) 0 0
\(993\) −17.0902 −0.542340
\(994\) 0 0
\(995\) −23.5623 + 17.1190i −0.746975 + 0.542709i
\(996\) 0 0
\(997\) −26.0000 −0.823428 −0.411714 0.911313i \(-0.635070\pi\)
−0.411714 + 0.911313i \(0.635070\pi\)
\(998\) 0 0
\(999\) 7.56231 + 23.2744i 0.239261 + 0.736369i
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 124.2.f.a.33.1 4
3.2 odd 2 1116.2.m.a.901.1 4
4.3 odd 2 496.2.n.c.33.1 4
31.4 even 5 3844.2.a.h.1.2 2
31.16 even 5 inner 124.2.f.a.109.1 yes 4
31.27 odd 10 3844.2.a.e.1.1 2
93.47 odd 10 1116.2.m.a.109.1 4
124.47 odd 10 496.2.n.c.481.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
124.2.f.a.33.1 4 1.1 even 1 trivial
124.2.f.a.109.1 yes 4 31.16 even 5 inner
496.2.n.c.33.1 4 4.3 odd 2
496.2.n.c.481.1 4 124.47 odd 10
1116.2.m.a.109.1 4 93.47 odd 10
1116.2.m.a.901.1 4 3.2 odd 2
3844.2.a.e.1.1 2 31.27 odd 10
3844.2.a.h.1.2 2 31.4 even 5