Properties

Label 124.3.o.a.17.2
Level $124$
Weight $3$
Character 124.17
Analytic conductor $3.379$
Analytic rank $0$
Dimension $40$
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] = [124,3,Mod(13,124)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(124, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([0, 11]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("124.13");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 124 = 2^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 124.o (of order \(30\), degree \(8\), 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: \(3.37875527807\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(5\) over \(\Q(\zeta_{30})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label 17.2
Character \(\chi\) \(=\) 124.17
Dual form 124.3.o.a.73.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.31136 - 1.18075i) q^{3} +(2.72362 - 4.71744i) q^{5} +(-0.568373 + 5.40771i) q^{7} +(-0.615273 - 5.85393i) q^{9} +O(q^{10})\) \(q+(-1.31136 - 1.18075i) q^{3} +(2.72362 - 4.71744i) q^{5} +(-0.568373 + 5.40771i) q^{7} +(-0.615273 - 5.85393i) q^{9} +(-8.24988 - 18.5295i) q^{11} +(0.300183 - 1.41225i) q^{13} +(-9.14175 + 2.97034i) q^{15} +(8.32365 - 18.6952i) q^{17} +(9.86752 - 2.09741i) q^{19} +(7.13049 - 6.42032i) q^{21} +(21.5145 + 29.6121i) q^{23} +(-2.33618 - 4.04638i) q^{25} +(-15.4401 + 21.2514i) q^{27} +(-33.4954 - 10.8833i) q^{29} +(-16.7164 + 26.1067i) q^{31} +(-11.0602 + 34.0399i) q^{33} +(23.9625 + 17.4098i) q^{35} +(34.1947 - 19.7423i) q^{37} +(-2.06116 + 1.49752i) q^{39} +(37.9235 + 42.1183i) q^{41} +(-1.99301 - 9.37635i) q^{43} +(-29.2914 - 13.0413i) q^{45} +(1.15874 + 3.56625i) q^{47} +(19.0090 + 4.04048i) q^{49} +(-32.9897 + 14.6879i) q^{51} +(65.8818 - 6.92446i) q^{53} +(-109.882 - 11.5490i) q^{55} +(-15.4163 - 8.90063i) q^{57} +(15.1814 - 16.8606i) q^{59} +60.4313i q^{61} +32.0061 q^{63} +(-5.84462 - 5.26252i) q^{65} +(-31.6110 + 54.7518i) q^{67} +(6.75140 - 64.2353i) q^{69} +(1.20157 + 11.4322i) q^{71} +(-31.1377 - 69.9365i) q^{73} +(-1.71420 + 8.06469i) q^{75} +(104.891 - 34.0813i) q^{77} +(31.2003 - 70.0771i) q^{79} +(-6.47792 + 1.37692i) q^{81} +(90.5466 - 81.5285i) q^{83} +(-65.5232 - 90.1850i) q^{85} +(31.0739 + 53.8215i) q^{87} +(-42.3350 + 58.2691i) q^{89} +(7.46642 + 2.42599i) q^{91} +(52.7467 - 14.4973i) q^{93} +(16.9809 - 52.2620i) q^{95} +(-103.021 - 74.8494i) q^{97} +(-103.395 + 59.6950i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 40 q - 3 q^{3} - 3 q^{5} + 19 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 40 q - 3 q^{3} - 3 q^{5} + 19 q^{7} - 2 q^{9} + 2 q^{11} - 18 q^{13} + 35 q^{15} + 25 q^{17} - 11 q^{19} + 54 q^{21} + 25 q^{23} - 75 q^{25} + 225 q^{27} + 20 q^{29} + 59 q^{31} - 303 q^{33} - 66 q^{35} - 222 q^{37} - 169 q^{39} + q^{41} + 122 q^{43} + 54 q^{45} - 120 q^{47} - 118 q^{49} - 515 q^{51} + 61 q^{53} - 121 q^{55} - 201 q^{57} - 257 q^{59} - 158 q^{63} + 182 q^{65} - q^{67} + 510 q^{69} + 459 q^{71} + 253 q^{73} + 651 q^{75} + 670 q^{77} + 385 q^{79} + 974 q^{81} + 375 q^{83} - 370 q^{85} - 344 q^{87} + 245 q^{89} + 960 q^{91} - 212 q^{93} - 851 q^{95} - 797 q^{97} - 21 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(63\) \(65\)
\(\chi(n)\) \(1\) \(e\left(\frac{7}{30}\right)\)

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
\(3\) −1.31136 1.18075i −0.437119 0.393583i 0.420979 0.907071i \(-0.361687\pi\)
−0.858097 + 0.513487i \(0.828353\pi\)
\(4\) 0 0
\(5\) 2.72362 4.71744i 0.544723 0.943489i −0.453901 0.891052i \(-0.649968\pi\)
0.998624 0.0524364i \(-0.0166987\pi\)
\(6\) 0 0
\(7\) −0.568373 + 5.40771i −0.0811962 + 0.772530i 0.875848 + 0.482587i \(0.160303\pi\)
−0.957044 + 0.289943i \(0.906364\pi\)
\(8\) 0 0
\(9\) −0.615273 5.85393i −0.0683637 0.650437i
\(10\) 0 0
\(11\) −8.24988 18.5295i −0.749989 1.68450i −0.728798 0.684729i \(-0.759921\pi\)
−0.0211913 0.999775i \(-0.506746\pi\)
\(12\) 0 0
\(13\) 0.300183 1.41225i 0.0230910 0.108635i −0.965093 0.261908i \(-0.915648\pi\)
0.988184 + 0.153274i \(0.0489816\pi\)
\(14\) 0 0
\(15\) −9.14175 + 2.97034i −0.609450 + 0.198022i
\(16\) 0 0
\(17\) 8.32365 18.6952i 0.489627 1.09972i −0.484721 0.874669i \(-0.661079\pi\)
0.974347 0.225050i \(-0.0722546\pi\)
\(18\) 0 0
\(19\) 9.86752 2.09741i 0.519343 0.110390i 0.0592200 0.998245i \(-0.481139\pi\)
0.460123 + 0.887855i \(0.347805\pi\)
\(20\) 0 0
\(21\) 7.13049 6.42032i 0.339547 0.305730i
\(22\) 0 0
\(23\) 21.5145 + 29.6121i 0.935412 + 1.28748i 0.957711 + 0.287733i \(0.0929014\pi\)
−0.0222990 + 0.999751i \(0.507099\pi\)
\(24\) 0 0
\(25\) −2.33618 4.04638i −0.0934472 0.161855i
\(26\) 0 0
\(27\) −15.4401 + 21.2514i −0.571854 + 0.787089i
\(28\) 0 0
\(29\) −33.4954 10.8833i −1.15501 0.375286i −0.331983 0.943285i \(-0.607718\pi\)
−0.823029 + 0.567999i \(0.807718\pi\)
\(30\) 0 0
\(31\) −16.7164 + 26.1067i −0.539240 + 0.842152i
\(32\) 0 0
\(33\) −11.0602 + 34.0399i −0.335158 + 1.03151i
\(34\) 0 0
\(35\) 23.9625 + 17.4098i 0.684644 + 0.497423i
\(36\) 0 0
\(37\) 34.1947 19.7423i 0.924182 0.533577i 0.0392151 0.999231i \(-0.487514\pi\)
0.884967 + 0.465654i \(0.154181\pi\)
\(38\) 0 0
\(39\) −2.06116 + 1.49752i −0.0528503 + 0.0383980i
\(40\) 0 0
\(41\) 37.9235 + 42.1183i 0.924964 + 1.02728i 0.999548 + 0.0300615i \(0.00957030\pi\)
−0.0745845 + 0.997215i \(0.523763\pi\)
\(42\) 0 0
\(43\) −1.99301 9.37635i −0.0463490 0.218055i 0.948878 0.315643i \(-0.102220\pi\)
−0.995227 + 0.0975887i \(0.968887\pi\)
\(44\) 0 0
\(45\) −29.2914 13.0413i −0.650919 0.289808i
\(46\) 0 0
\(47\) 1.15874 + 3.56625i 0.0246541 + 0.0758776i 0.962627 0.270832i \(-0.0872989\pi\)
−0.937972 + 0.346710i \(0.887299\pi\)
\(48\) 0 0
\(49\) 19.0090 + 4.04048i 0.387938 + 0.0824587i
\(50\) 0 0
\(51\) −32.9897 + 14.6879i −0.646856 + 0.287999i
\(52\) 0 0
\(53\) 65.8818 6.92446i 1.24305 0.130650i 0.539925 0.841713i \(-0.318453\pi\)
0.703128 + 0.711063i \(0.251786\pi\)
\(54\) 0 0
\(55\) −109.882 11.5490i −1.99785 0.209982i
\(56\) 0 0
\(57\) −15.4163 8.90063i −0.270462 0.156151i
\(58\) 0 0
\(59\) 15.1814 16.8606i 0.257312 0.285773i −0.600623 0.799532i \(-0.705081\pi\)
0.857935 + 0.513759i \(0.171747\pi\)
\(60\) 0 0
\(61\) 60.4313i 0.990677i 0.868700 + 0.495339i \(0.164956\pi\)
−0.868700 + 0.495339i \(0.835044\pi\)
\(62\) 0 0
\(63\) 32.0061 0.508033
\(64\) 0 0
\(65\) −5.84462 5.26252i −0.0899173 0.0809619i
\(66\) 0 0
\(67\) −31.6110 + 54.7518i −0.471806 + 0.817191i −0.999480 0.0322558i \(-0.989731\pi\)
0.527674 + 0.849447i \(0.323064\pi\)
\(68\) 0 0
\(69\) 6.75140 64.2353i 0.0978463 0.930946i
\(70\) 0 0
\(71\) 1.20157 + 11.4322i 0.0169235 + 0.161016i 0.999720 0.0236772i \(-0.00753739\pi\)
−0.982796 + 0.184694i \(0.940871\pi\)
\(72\) 0 0
\(73\) −31.1377 69.9365i −0.426544 0.958035i −0.991159 0.132680i \(-0.957642\pi\)
0.564614 0.825355i \(-0.309025\pi\)
\(74\) 0 0
\(75\) −1.71420 + 8.06469i −0.0228560 + 0.107529i
\(76\) 0 0
\(77\) 104.891 34.0813i 1.36223 0.442614i
\(78\) 0 0
\(79\) 31.2003 70.0771i 0.394941 0.887052i −0.601185 0.799110i \(-0.705304\pi\)
0.996126 0.0879416i \(-0.0280289\pi\)
\(80\) 0 0
\(81\) −6.47792 + 1.37692i −0.0799743 + 0.0169991i
\(82\) 0 0
\(83\) 90.5466 81.5285i 1.09092 0.982271i 0.0910151 0.995850i \(-0.470989\pi\)
0.999908 + 0.0135781i \(0.00432219\pi\)
\(84\) 0 0
\(85\) −65.5232 90.1850i −0.770861 1.06100i
\(86\) 0 0
\(87\) 31.0739 + 53.8215i 0.357171 + 0.618638i
\(88\) 0 0
\(89\) −42.3350 + 58.2691i −0.475674 + 0.654709i −0.977666 0.210163i \(-0.932600\pi\)
0.501993 + 0.864872i \(0.332600\pi\)
\(90\) 0 0
\(91\) 7.46642 + 2.42599i 0.0820486 + 0.0266592i
\(92\) 0 0
\(93\) 52.7467 14.4973i 0.567169 0.155885i
\(94\) 0 0
\(95\) 16.9809 52.2620i 0.178747 0.550126i
\(96\) 0 0
\(97\) −103.021 74.8494i −1.06208 0.771643i −0.0876041 0.996155i \(-0.527921\pi\)
−0.974471 + 0.224512i \(0.927921\pi\)
\(98\) 0 0
\(99\) −103.395 + 59.6950i −1.04439 + 0.602980i
\(100\) 0 0
\(101\) 18.1564 13.1914i 0.179766 0.130608i −0.494263 0.869312i \(-0.664562\pi\)
0.674029 + 0.738705i \(0.264562\pi\)
\(102\) 0 0
\(103\) −55.5245 61.6662i −0.539073 0.598701i 0.410650 0.911793i \(-0.365302\pi\)
−0.949723 + 0.313092i \(0.898635\pi\)
\(104\) 0 0
\(105\) −10.8668 51.1242i −0.103493 0.486897i
\(106\) 0 0
\(107\) −104.129 46.3614i −0.973173 0.433284i −0.142347 0.989817i \(-0.545465\pi\)
−0.830826 + 0.556532i \(0.812132\pi\)
\(108\) 0 0
\(109\) 43.2016 + 132.961i 0.396345 + 1.21982i 0.927909 + 0.372806i \(0.121604\pi\)
−0.531564 + 0.847018i \(0.678396\pi\)
\(110\) 0 0
\(111\) −68.1522 14.4862i −0.613984 0.130506i
\(112\) 0 0
\(113\) 17.2923 7.69903i 0.153029 0.0681330i −0.328793 0.944402i \(-0.606642\pi\)
0.481822 + 0.876269i \(0.339975\pi\)
\(114\) 0 0
\(115\) 198.291 20.8412i 1.72427 0.181228i
\(116\) 0 0
\(117\) −8.45190 0.888331i −0.0722385 0.00759257i
\(118\) 0 0
\(119\) 96.3674 + 55.6378i 0.809810 + 0.467544i
\(120\) 0 0
\(121\) −194.319 + 215.813i −1.60594 + 1.78358i
\(122\) 0 0
\(123\) 100.010i 0.813092i
\(124\) 0 0
\(125\) 110.729 0.885835
\(126\) 0 0
\(127\) 48.6007 + 43.7602i 0.382682 + 0.344569i 0.837912 0.545805i \(-0.183776\pi\)
−0.455230 + 0.890374i \(0.650443\pi\)
\(128\) 0 0
\(129\) −8.45759 + 14.6490i −0.0655627 + 0.113558i
\(130\) 0 0
\(131\) −7.34935 + 69.9244i −0.0561019 + 0.533774i 0.929991 + 0.367581i \(0.119814\pi\)
−0.986093 + 0.166193i \(0.946853\pi\)
\(132\) 0 0
\(133\) 5.73373 + 54.5528i 0.0431107 + 0.410171i
\(134\) 0 0
\(135\) 58.1995 + 130.718i 0.431108 + 0.968284i
\(136\) 0 0
\(137\) 21.9062 103.060i 0.159899 0.752266i −0.822988 0.568058i \(-0.807695\pi\)
0.982887 0.184207i \(-0.0589718\pi\)
\(138\) 0 0
\(139\) 127.623 41.4671i 0.918149 0.298325i 0.188441 0.982084i \(-0.439656\pi\)
0.729707 + 0.683760i \(0.239656\pi\)
\(140\) 0 0
\(141\) 2.69132 6.04481i 0.0190874 0.0428710i
\(142\) 0 0
\(143\) −28.6448 + 6.08864i −0.200313 + 0.0425779i
\(144\) 0 0
\(145\) −142.570 + 128.370i −0.983240 + 0.885314i
\(146\) 0 0
\(147\) −20.1567 27.7433i −0.137120 0.188730i
\(148\) 0 0
\(149\) 73.3036 + 126.966i 0.491970 + 0.852118i 0.999957 0.00924706i \(-0.00294347\pi\)
−0.507987 + 0.861365i \(0.669610\pi\)
\(150\) 0 0
\(151\) −68.9723 + 94.9322i −0.456770 + 0.628690i −0.973835 0.227256i \(-0.927025\pi\)
0.517065 + 0.855946i \(0.327025\pi\)
\(152\) 0 0
\(153\) −114.562 37.2234i −0.748770 0.243290i
\(154\) 0 0
\(155\) 77.6278 + 149.964i 0.500825 + 0.967507i
\(156\) 0 0
\(157\) 1.09721 3.37686i 0.00698859 0.0215087i −0.947501 0.319752i \(-0.896401\pi\)
0.954490 + 0.298243i \(0.0964005\pi\)
\(158\) 0 0
\(159\) −94.5706 68.7095i −0.594784 0.432136i
\(160\) 0 0
\(161\) −172.362 + 99.5133i −1.07057 + 0.618095i
\(162\) 0 0
\(163\) 169.366 123.051i 1.03905 0.754917i 0.0689531 0.997620i \(-0.478034\pi\)
0.970101 + 0.242703i \(0.0780341\pi\)
\(164\) 0 0
\(165\) 130.457 + 144.888i 0.790651 + 0.878106i
\(166\) 0 0
\(167\) 7.67326 + 36.0999i 0.0459477 + 0.216167i 0.995132 0.0985551i \(-0.0314221\pi\)
−0.949184 + 0.314722i \(0.898089\pi\)
\(168\) 0 0
\(169\) 152.485 + 67.8906i 0.902277 + 0.401720i
\(170\) 0 0
\(171\) −18.3493 56.4733i −0.107306 0.330253i
\(172\) 0 0
\(173\) 114.821 + 24.4061i 0.663708 + 0.141075i 0.527434 0.849596i \(-0.323154\pi\)
0.136274 + 0.990671i \(0.456487\pi\)
\(174\) 0 0
\(175\) 23.2095 10.3335i 0.132626 0.0590487i
\(176\) 0 0
\(177\) −39.8164 + 4.18487i −0.224951 + 0.0236433i
\(178\) 0 0
\(179\) −215.899 22.6919i −1.20614 0.126770i −0.519952 0.854196i \(-0.674050\pi\)
−0.686187 + 0.727425i \(0.740717\pi\)
\(180\) 0 0
\(181\) −192.495 111.137i −1.06351 0.614018i −0.137109 0.990556i \(-0.543781\pi\)
−0.926401 + 0.376538i \(0.877115\pi\)
\(182\) 0 0
\(183\) 71.3543 79.2470i 0.389914 0.433043i
\(184\) 0 0
\(185\) 215.082i 1.16261i
\(186\) 0 0
\(187\) −415.083 −2.21970
\(188\) 0 0
\(189\) −106.146 95.5741i −0.561618 0.505683i
\(190\) 0 0
\(191\) 184.857 320.181i 0.967836 1.67634i 0.266042 0.963962i \(-0.414284\pi\)
0.701794 0.712380i \(-0.252383\pi\)
\(192\) 0 0
\(193\) −38.6748 + 367.966i −0.200388 + 1.90656i 0.183232 + 0.983070i \(0.441344\pi\)
−0.383619 + 0.923491i \(0.625322\pi\)
\(194\) 0 0
\(195\) 1.45066 + 13.8021i 0.00743926 + 0.0707799i
\(196\) 0 0
\(197\) 98.5340 + 221.311i 0.500172 + 1.12341i 0.970541 + 0.240936i \(0.0774543\pi\)
−0.470369 + 0.882470i \(0.655879\pi\)
\(198\) 0 0
\(199\) 39.8170 187.324i 0.200086 0.941328i −0.757423 0.652924i \(-0.773542\pi\)
0.957509 0.288404i \(-0.0931247\pi\)
\(200\) 0 0
\(201\) 106.101 34.4744i 0.527868 0.171515i
\(202\) 0 0
\(203\) 77.8916 174.947i 0.383702 0.861810i
\(204\) 0 0
\(205\) 301.980 64.1878i 1.47307 0.313111i
\(206\) 0 0
\(207\) 160.110 144.164i 0.773479 0.696443i
\(208\) 0 0
\(209\) −120.270 165.537i −0.575454 0.792044i
\(210\) 0 0
\(211\) 76.4808 + 132.469i 0.362468 + 0.627813i 0.988366 0.152091i \(-0.0486008\pi\)
−0.625898 + 0.779905i \(0.715267\pi\)
\(212\) 0 0
\(213\) 11.9228 16.4104i 0.0559758 0.0770441i
\(214\) 0 0
\(215\) −49.6606 16.1357i −0.230980 0.0750498i
\(216\) 0 0
\(217\) −131.676 105.236i −0.606804 0.484958i
\(218\) 0 0
\(219\) −41.7449 + 128.478i −0.190616 + 0.586656i
\(220\) 0 0
\(221\) −23.9037 17.3671i −0.108162 0.0785840i
\(222\) 0 0
\(223\) 210.185 121.350i 0.942532 0.544171i 0.0517787 0.998659i \(-0.483511\pi\)
0.890753 + 0.454488i \(0.150178\pi\)
\(224\) 0 0
\(225\) −22.2498 + 16.1655i −0.0988882 + 0.0718465i
\(226\) 0 0
\(227\) 37.7395 + 41.9139i 0.166253 + 0.184643i 0.820515 0.571626i \(-0.193687\pi\)
−0.654261 + 0.756269i \(0.727020\pi\)
\(228\) 0 0
\(229\) −55.6536 261.830i −0.243029 1.14336i −0.915211 0.402975i \(-0.867976\pi\)
0.672182 0.740386i \(-0.265357\pi\)
\(230\) 0 0
\(231\) −177.791 79.1578i −0.769660 0.342675i
\(232\) 0 0
\(233\) 82.0335 + 252.473i 0.352075 + 1.08358i 0.957686 + 0.287815i \(0.0929289\pi\)
−0.605611 + 0.795761i \(0.707071\pi\)
\(234\) 0 0
\(235\) 19.9795 + 4.24678i 0.0850193 + 0.0180714i
\(236\) 0 0
\(237\) −123.658 + 55.0562i −0.521765 + 0.232305i
\(238\) 0 0
\(239\) −163.946 + 17.2314i −0.685966 + 0.0720979i −0.441098 0.897459i \(-0.645411\pi\)
−0.244867 + 0.969557i \(0.578744\pi\)
\(240\) 0 0
\(241\) −270.189 28.3980i −1.12112 0.117834i −0.474215 0.880409i \(-0.657268\pi\)
−0.646900 + 0.762575i \(0.723935\pi\)
\(242\) 0 0
\(243\) 214.861 + 124.050i 0.884202 + 0.510494i
\(244\) 0 0
\(245\) 70.8338 78.6689i 0.289118 0.321098i
\(246\) 0 0
\(247\) 14.5650i 0.0589676i
\(248\) 0 0
\(249\) −215.004 −0.863468
\(250\) 0 0
\(251\) −266.154 239.646i −1.06038 0.954766i −0.0613153 0.998118i \(-0.519530\pi\)
−0.999060 + 0.0433524i \(0.986196\pi\)
\(252\) 0 0
\(253\) 371.207 642.950i 1.46722 2.54130i
\(254\) 0 0
\(255\) −20.5617 + 195.631i −0.0806340 + 0.767181i
\(256\) 0 0
\(257\) 38.2716 + 364.130i 0.148917 + 1.41685i 0.772462 + 0.635061i \(0.219025\pi\)
−0.623545 + 0.781788i \(0.714308\pi\)
\(258\) 0 0
\(259\) 87.3255 + 196.136i 0.337164 + 0.757283i
\(260\) 0 0
\(261\) −43.1013 + 202.776i −0.165139 + 0.776918i
\(262\) 0 0
\(263\) 39.0657 12.6932i 0.148539 0.0482631i −0.233804 0.972284i \(-0.575117\pi\)
0.382343 + 0.924021i \(0.375117\pi\)
\(264\) 0 0
\(265\) 146.771 329.653i 0.553853 1.24397i
\(266\) 0 0
\(267\) 124.317 26.4245i 0.465608 0.0989681i
\(268\) 0 0
\(269\) 48.9120 44.0406i 0.181829 0.163720i −0.573190 0.819422i \(-0.694294\pi\)
0.755019 + 0.655703i \(0.227628\pi\)
\(270\) 0 0
\(271\) 142.795 + 196.541i 0.526919 + 0.725242i 0.986657 0.162813i \(-0.0520567\pi\)
−0.459738 + 0.888055i \(0.652057\pi\)
\(272\) 0 0
\(273\) −6.92665 11.9973i −0.0253723 0.0439462i
\(274\) 0 0
\(275\) −55.7044 + 76.6705i −0.202561 + 0.278802i
\(276\) 0 0
\(277\) −238.373 77.4522i −0.860554 0.279611i −0.154694 0.987962i \(-0.549439\pi\)
−0.705860 + 0.708351i \(0.749439\pi\)
\(278\) 0 0
\(279\) 163.112 + 81.7940i 0.584631 + 0.293169i
\(280\) 0 0
\(281\) 15.5682 47.9141i 0.0554029 0.170513i −0.919526 0.393029i \(-0.871427\pi\)
0.974929 + 0.222517i \(0.0714271\pi\)
\(282\) 0 0
\(283\) −42.7805 31.0818i −0.151168 0.109830i 0.509630 0.860393i \(-0.329782\pi\)
−0.660798 + 0.750564i \(0.729782\pi\)
\(284\) 0 0
\(285\) −83.9764 + 48.4838i −0.294654 + 0.170119i
\(286\) 0 0
\(287\) −249.318 + 181.140i −0.868705 + 0.631151i
\(288\) 0 0
\(289\) −86.8496 96.4562i −0.300518 0.333759i
\(290\) 0 0
\(291\) 46.7192 + 219.797i 0.160547 + 0.755315i
\(292\) 0 0
\(293\) −382.239 170.184i −1.30457 0.580832i −0.367516 0.930017i \(-0.619792\pi\)
−0.937054 + 0.349185i \(0.886458\pi\)
\(294\) 0 0
\(295\) −38.1908 117.539i −0.129460 0.398438i
\(296\) 0 0
\(297\) 521.158 + 110.775i 1.75474 + 0.372981i
\(298\) 0 0
\(299\) 48.2780 21.4947i 0.161465 0.0718888i
\(300\) 0 0
\(301\) 51.8374 5.44833i 0.172217 0.0181008i
\(302\) 0 0
\(303\) −39.3852 4.13955i −0.129984 0.0136619i
\(304\) 0 0
\(305\) 285.081 + 164.592i 0.934693 + 0.539645i
\(306\) 0 0
\(307\) −144.980 + 161.017i −0.472249 + 0.524486i −0.931461 0.363841i \(-0.881465\pi\)
0.459212 + 0.888327i \(0.348132\pi\)
\(308\) 0 0
\(309\) 146.427i 0.473873i
\(310\) 0 0
\(311\) −557.952 −1.79406 −0.897029 0.441971i \(-0.854279\pi\)
−0.897029 + 0.441971i \(0.854279\pi\)
\(312\) 0 0
\(313\) 333.403 + 300.198i 1.06519 + 0.959098i 0.999250 0.0387263i \(-0.0123301\pi\)
0.0659359 + 0.997824i \(0.478997\pi\)
\(314\) 0 0
\(315\) 87.1723 150.987i 0.276737 0.479323i
\(316\) 0 0
\(317\) −26.3308 + 250.521i −0.0830625 + 0.790287i 0.871122 + 0.491067i \(0.163393\pi\)
−0.954184 + 0.299220i \(0.903274\pi\)
\(318\) 0 0
\(319\) 74.6702 + 710.440i 0.234076 + 2.22708i
\(320\) 0 0
\(321\) 81.8095 + 183.747i 0.254858 + 0.572421i
\(322\) 0 0
\(323\) 42.9223 201.934i 0.132886 0.625181i
\(324\) 0 0
\(325\) −6.41578 + 2.08461i −0.0197409 + 0.00641419i
\(326\) 0 0
\(327\) 100.341 225.369i 0.306853 0.689203i
\(328\) 0 0
\(329\) −19.9438 + 4.23919i −0.0606195 + 0.0128851i
\(330\) 0 0
\(331\) −344.829 + 310.486i −1.04178 + 0.938023i −0.998136 0.0610239i \(-0.980563\pi\)
−0.0436440 + 0.999047i \(0.513897\pi\)
\(332\) 0 0
\(333\) −136.609 188.027i −0.410238 0.564645i
\(334\) 0 0
\(335\) 172.192 + 298.246i 0.514007 + 0.890286i
\(336\) 0 0
\(337\) −155.928 + 214.617i −0.462695 + 0.636845i −0.975065 0.221920i \(-0.928768\pi\)
0.512370 + 0.858765i \(0.328768\pi\)
\(338\) 0 0
\(339\) −31.7670 10.3217i −0.0937080 0.0304476i
\(340\) 0 0
\(341\) 621.654 + 94.3703i 1.82303 + 0.276746i
\(342\) 0 0
\(343\) −114.988 + 353.895i −0.335241 + 1.03176i
\(344\) 0 0
\(345\) −284.638 206.802i −0.825037 0.599425i
\(346\) 0 0
\(347\) −207.914 + 120.039i −0.599177 + 0.345935i −0.768718 0.639588i \(-0.779105\pi\)
0.169541 + 0.985523i \(0.445772\pi\)
\(348\) 0 0
\(349\) 444.157 322.699i 1.27266 0.924638i 0.273351 0.961914i \(-0.411868\pi\)
0.999305 + 0.0372760i \(0.0118681\pi\)
\(350\) 0 0
\(351\) 25.3775 + 28.1845i 0.0723004 + 0.0802978i
\(352\) 0 0
\(353\) −139.657 657.032i −0.395628 1.86128i −0.498946 0.866633i \(-0.666279\pi\)
0.103318 0.994648i \(-0.467054\pi\)
\(354\) 0 0
\(355\) 57.2032 + 25.4685i 0.161136 + 0.0717423i
\(356\) 0 0
\(357\) −60.6777 186.747i −0.169966 0.523100i
\(358\) 0 0
\(359\) −495.721 105.369i −1.38084 0.293506i −0.543142 0.839641i \(-0.682765\pi\)
−0.837698 + 0.546134i \(0.816099\pi\)
\(360\) 0 0
\(361\) −236.821 + 105.440i −0.656014 + 0.292076i
\(362\) 0 0
\(363\) 509.642 53.5655i 1.40397 0.147563i
\(364\) 0 0
\(365\) −414.729 43.5898i −1.13624 0.119424i
\(366\) 0 0
\(367\) 268.992 + 155.302i 0.732948 + 0.423167i 0.819500 0.573080i \(-0.194252\pi\)
−0.0865519 + 0.996247i \(0.527585\pi\)
\(368\) 0 0
\(369\) 223.224 247.916i 0.604944 0.671859i
\(370\) 0 0
\(371\) 360.206i 0.970904i
\(372\) 0 0
\(373\) 672.226 1.80221 0.901107 0.433597i \(-0.142756\pi\)
0.901107 + 0.433597i \(0.142756\pi\)
\(374\) 0 0
\(375\) −145.206 130.744i −0.387215 0.348650i
\(376\) 0 0
\(377\) −25.4247 + 44.0368i −0.0674394 + 0.116809i
\(378\) 0 0
\(379\) 20.0763 191.013i 0.0529718 0.503993i −0.935580 0.353114i \(-0.885123\pi\)
0.988552 0.150879i \(-0.0482104\pi\)
\(380\) 0 0
\(381\) −12.0629 114.770i −0.0316611 0.301235i
\(382\) 0 0
\(383\) −70.8050 159.031i −0.184869 0.415223i 0.797207 0.603706i \(-0.206310\pi\)
−0.982077 + 0.188482i \(0.939643\pi\)
\(384\) 0 0
\(385\) 124.908 587.644i 0.324435 1.52635i
\(386\) 0 0
\(387\) −53.6623 + 17.4359i −0.138662 + 0.0450541i
\(388\) 0 0
\(389\) −213.012 + 478.432i −0.547587 + 1.22990i 0.401781 + 0.915736i \(0.368391\pi\)
−0.949368 + 0.314166i \(0.898275\pi\)
\(390\) 0 0
\(391\) 732.684 155.737i 1.87387 0.398304i
\(392\) 0 0
\(393\) 92.2009 83.0180i 0.234608 0.211242i
\(394\) 0 0
\(395\) −245.607 338.049i −0.621790 0.855820i
\(396\) 0 0
\(397\) 102.045 + 176.746i 0.257039 + 0.445205i 0.965447 0.260598i \(-0.0839197\pi\)
−0.708408 + 0.705803i \(0.750586\pi\)
\(398\) 0 0
\(399\) 56.8942 78.3082i 0.142592 0.196261i
\(400\) 0 0
\(401\) −103.355 33.5822i −0.257744 0.0837461i 0.177295 0.984158i \(-0.443265\pi\)
−0.435039 + 0.900412i \(0.643265\pi\)
\(402\) 0 0
\(403\) 31.8512 + 31.4446i 0.0790353 + 0.0780262i
\(404\) 0 0
\(405\) −11.1478 + 34.3094i −0.0275255 + 0.0847147i
\(406\) 0 0
\(407\) −647.919 470.741i −1.59194 1.15661i
\(408\) 0 0
\(409\) −333.237 + 192.395i −0.814761 + 0.470403i −0.848607 0.529024i \(-0.822558\pi\)
0.0338454 + 0.999427i \(0.489225\pi\)
\(410\) 0 0
\(411\) −150.415 + 109.283i −0.365974 + 0.265896i
\(412\) 0 0
\(413\) 82.5487 + 91.6797i 0.199876 + 0.221985i
\(414\) 0 0
\(415\) −137.992 649.201i −0.332511 1.56434i
\(416\) 0 0
\(417\) −216.321 96.3123i −0.518755 0.230965i
\(418\) 0 0
\(419\) −102.457 315.330i −0.244527 0.752578i −0.995714 0.0924880i \(-0.970518\pi\)
0.751186 0.660090i \(-0.229482\pi\)
\(420\) 0 0
\(421\) −673.209 143.095i −1.59907 0.339893i −0.679763 0.733432i \(-0.737917\pi\)
−0.919308 + 0.393539i \(0.871251\pi\)
\(422\) 0 0
\(423\) 20.1636 8.97742i 0.0476681 0.0212232i
\(424\) 0 0
\(425\) −95.0935 + 9.99473i −0.223749 + 0.0235170i
\(426\) 0 0
\(427\) −326.795 34.3475i −0.765328 0.0804392i
\(428\) 0 0
\(429\) 44.7527 + 25.8380i 0.104319 + 0.0602284i
\(430\) 0 0
\(431\) −149.138 + 165.635i −0.346028 + 0.384303i −0.890887 0.454224i \(-0.849916\pi\)
0.544859 + 0.838528i \(0.316583\pi\)
\(432\) 0 0
\(433\) 443.738i 1.02480i −0.858747 0.512399i \(-0.828757\pi\)
0.858747 0.512399i \(-0.171243\pi\)
\(434\) 0 0
\(435\) 338.533 0.778237
\(436\) 0 0
\(437\) 274.403 + 247.074i 0.627925 + 0.565386i
\(438\) 0 0
\(439\) −154.468 + 267.547i −0.351864 + 0.609447i −0.986576 0.163302i \(-0.947785\pi\)
0.634712 + 0.772749i \(0.281119\pi\)
\(440\) 0 0
\(441\) 11.9570 113.763i 0.0271133 0.257966i
\(442\) 0 0
\(443\) 53.0759 + 504.983i 0.119810 + 1.13992i 0.874903 + 0.484299i \(0.160925\pi\)
−0.755092 + 0.655618i \(0.772408\pi\)
\(444\) 0 0
\(445\) 159.577 + 358.415i 0.358600 + 0.805428i
\(446\) 0 0
\(447\) 53.7875 253.050i 0.120330 0.566108i
\(448\) 0 0
\(449\) 753.741 244.905i 1.67871 0.545446i 0.694050 0.719926i \(-0.255824\pi\)
0.984661 + 0.174480i \(0.0558244\pi\)
\(450\) 0 0
\(451\) 467.569 1050.18i 1.03674 2.32855i
\(452\) 0 0
\(453\) 202.538 43.0509i 0.447105 0.0950350i
\(454\) 0 0
\(455\) 31.7801 28.6149i 0.0698464 0.0628900i
\(456\) 0 0
\(457\) 367.652 + 506.030i 0.804491 + 1.10729i 0.992150 + 0.125053i \(0.0399100\pi\)
−0.187659 + 0.982234i \(0.560090\pi\)
\(458\) 0 0
\(459\) 268.782 + 465.545i 0.585582 + 1.01426i
\(460\) 0 0
\(461\) −29.2786 + 40.2985i −0.0635110 + 0.0874154i −0.839591 0.543219i \(-0.817205\pi\)
0.776080 + 0.630635i \(0.217205\pi\)
\(462\) 0 0
\(463\) 340.483 + 110.630i 0.735385 + 0.238941i 0.652681 0.757633i \(-0.273644\pi\)
0.0827045 + 0.996574i \(0.473644\pi\)
\(464\) 0 0
\(465\) 75.2717 288.315i 0.161875 0.620031i
\(466\) 0 0
\(467\) 59.8831 184.301i 0.128229 0.394649i −0.866246 0.499617i \(-0.833474\pi\)
0.994476 + 0.104968i \(0.0334740\pi\)
\(468\) 0 0
\(469\) −278.115 202.062i −0.592996 0.430837i
\(470\) 0 0
\(471\) −5.42606 + 3.13274i −0.0115203 + 0.00665124i
\(472\) 0 0
\(473\) −157.298 + 114.283i −0.332553 + 0.241614i
\(474\) 0 0
\(475\) −31.5392 35.0278i −0.0663983 0.0737428i
\(476\) 0 0
\(477\) −81.0706 381.407i −0.169959 0.799596i
\(478\) 0 0
\(479\) 366.581 + 163.212i 0.765305 + 0.340736i 0.751989 0.659176i \(-0.229095\pi\)
0.0133158 + 0.999911i \(0.495761\pi\)
\(480\) 0 0
\(481\) −17.6164 54.2178i −0.0366246 0.112719i
\(482\) 0 0
\(483\) 343.528 + 73.0192i 0.711239 + 0.151178i
\(484\) 0 0
\(485\) −633.688 + 282.136i −1.30657 + 0.581724i
\(486\) 0 0
\(487\) 131.620 13.8339i 0.270268 0.0284063i 0.0315749 0.999501i \(-0.489948\pi\)
0.238693 + 0.971095i \(0.423281\pi\)
\(488\) 0 0
\(489\) −367.392 38.6144i −0.751312 0.0789661i
\(490\) 0 0
\(491\) −434.939 251.112i −0.885823 0.511430i −0.0132491 0.999912i \(-0.504217\pi\)
−0.872574 + 0.488482i \(0.837551\pi\)
\(492\) 0 0
\(493\) −482.269 + 535.614i −0.978234 + 1.08644i
\(494\) 0 0
\(495\) 650.345i 1.31383i
\(496\) 0 0
\(497\) −62.5048 −0.125764
\(498\) 0 0
\(499\) 244.205 + 219.883i 0.489389 + 0.440648i 0.876512 0.481380i \(-0.159864\pi\)
−0.387123 + 0.922028i \(0.626531\pi\)
\(500\) 0 0
\(501\) 32.5625 56.4000i 0.0649951 0.112575i
\(502\) 0 0
\(503\) −12.9927 + 123.617i −0.0258304 + 0.245759i 0.973986 + 0.226607i \(0.0727634\pi\)
−0.999817 + 0.0191522i \(0.993903\pi\)
\(504\) 0 0
\(505\) −12.7786 121.580i −0.0253041 0.240752i
\(506\) 0 0
\(507\) −119.800 269.075i −0.236292 0.530720i
\(508\) 0 0
\(509\) −39.6424 + 186.503i −0.0778828 + 0.366410i −0.999779 0.0210273i \(-0.993306\pi\)
0.921896 + 0.387437i \(0.126640\pi\)
\(510\) 0 0
\(511\) 395.894 128.634i 0.774744 0.251730i
\(512\) 0 0
\(513\) −107.782 + 242.083i −0.210102 + 0.471896i
\(514\) 0 0
\(515\) −442.134 + 93.9786i −0.858513 + 0.182483i
\(516\) 0 0
\(517\) 56.5214 50.8921i 0.109326 0.0984374i
\(518\) 0 0
\(519\) −121.754 167.580i −0.234594 0.322891i
\(520\) 0 0
\(521\) −77.5760 134.365i −0.148898 0.257899i 0.781922 0.623376i \(-0.214239\pi\)
−0.930821 + 0.365477i \(0.880906\pi\)
\(522\) 0 0
\(523\) −246.486 + 339.259i −0.471293 + 0.648680i −0.976803 0.214142i \(-0.931305\pi\)
0.505509 + 0.862821i \(0.331305\pi\)
\(524\) 0 0
\(525\) −42.6372 13.8537i −0.0812137 0.0263879i
\(526\) 0 0
\(527\) 348.929 + 529.821i 0.662105 + 1.00535i
\(528\) 0 0
\(529\) −250.536 + 771.070i −0.473603 + 1.45760i
\(530\) 0 0
\(531\) −108.042 78.4969i −0.203468 0.147828i
\(532\) 0 0
\(533\) 70.8656 40.9142i 0.132956 0.0767622i
\(534\) 0 0
\(535\) −502.316 + 364.954i −0.938909 + 0.682157i
\(536\) 0 0
\(537\) 256.327 + 284.680i 0.477331 + 0.530130i
\(538\) 0 0
\(539\) −81.9535 385.561i −0.152047 0.715326i
\(540\) 0 0
\(541\) −424.878 189.168i −0.785357 0.349663i −0.0254337 0.999677i \(-0.508097\pi\)
−0.759923 + 0.650013i \(0.774763\pi\)
\(542\) 0 0
\(543\) 121.205 + 373.029i 0.223213 + 0.686979i
\(544\) 0 0
\(545\) 744.900 + 158.333i 1.36679 + 0.290520i
\(546\) 0 0
\(547\) 119.820 53.3474i 0.219050 0.0975272i −0.294278 0.955720i \(-0.595079\pi\)
0.513328 + 0.858193i \(0.328413\pi\)
\(548\) 0 0
\(549\) 353.761 37.1818i 0.644373 0.0677263i
\(550\) 0 0
\(551\) −353.343 37.1378i −0.641275 0.0674007i
\(552\) 0 0
\(553\) 361.223 + 208.552i 0.653206 + 0.377129i
\(554\) 0 0
\(555\) −253.958 + 282.049i −0.457583 + 0.508197i
\(556\) 0 0
\(557\) 253.453i 0.455033i −0.973774 0.227516i \(-0.926939\pi\)
0.973774 0.227516i \(-0.0730605\pi\)
\(558\) 0 0
\(559\) −13.8400 −0.0247585
\(560\) 0 0
\(561\) 544.322 + 490.109i 0.970270 + 0.873635i
\(562\) 0 0
\(563\) 422.134 731.157i 0.749793 1.29868i −0.198128 0.980176i \(-0.563486\pi\)
0.947921 0.318504i \(-0.103180\pi\)
\(564\) 0 0
\(565\) 10.7779 102.545i 0.0190759 0.181495i
\(566\) 0 0
\(567\) −3.76413 35.8133i −0.00663868 0.0631628i
\(568\) 0 0
\(569\) −161.017 361.650i −0.282983 0.635589i 0.714997 0.699128i \(-0.246428\pi\)
−0.997979 + 0.0635385i \(0.979761\pi\)
\(570\) 0 0
\(571\) −190.116 + 894.423i −0.332952 + 1.56642i 0.419492 + 0.907759i \(0.362208\pi\)
−0.752444 + 0.658657i \(0.771125\pi\)
\(572\) 0 0
\(573\) −620.467 + 201.602i −1.08284 + 0.351836i
\(574\) 0 0
\(575\) 69.5603 156.235i 0.120974 0.271713i
\(576\) 0 0
\(577\) −86.4764 + 18.3811i −0.149872 + 0.0318564i −0.282237 0.959345i \(-0.591076\pi\)
0.132364 + 0.991201i \(0.457743\pi\)
\(578\) 0 0
\(579\) 485.193 436.869i 0.837984 0.754524i
\(580\) 0 0
\(581\) 389.418 + 535.988i 0.670255 + 0.922527i
\(582\) 0 0
\(583\) −671.825 1163.63i −1.15236 1.99594i
\(584\) 0 0
\(585\) −27.2104 + 37.4519i −0.0465135 + 0.0640203i
\(586\) 0 0
\(587\) −111.610 36.2643i −0.190136 0.0617790i 0.212401 0.977183i \(-0.431872\pi\)
−0.402537 + 0.915404i \(0.631872\pi\)
\(588\) 0 0
\(589\) −110.193 + 292.670i −0.187085 + 0.496892i
\(590\) 0 0
\(591\) 132.100 406.561i 0.223519 0.687921i
\(592\) 0 0
\(593\) −137.176 99.6640i −0.231325 0.168067i 0.466085 0.884740i \(-0.345664\pi\)
−0.697410 + 0.716673i \(0.745664\pi\)
\(594\) 0 0
\(595\) 524.936 303.072i 0.882245 0.509364i
\(596\) 0 0
\(597\) −273.398 + 198.635i −0.457952 + 0.332722i
\(598\) 0 0
\(599\) 679.245 + 754.378i 1.13396 + 1.25940i 0.961628 + 0.274357i \(0.0884651\pi\)
0.172337 + 0.985038i \(0.444868\pi\)
\(600\) 0 0
\(601\) 118.607 + 558.003i 0.197350 + 0.928457i 0.959642 + 0.281224i \(0.0907405\pi\)
−0.762292 + 0.647233i \(0.775926\pi\)
\(602\) 0 0
\(603\) 339.963 + 151.361i 0.563785 + 0.251013i
\(604\) 0 0
\(605\) 488.835 + 1504.48i 0.807991 + 2.48674i
\(606\) 0 0
\(607\) 16.2476 + 3.45353i 0.0267670 + 0.00568951i 0.221276 0.975211i \(-0.428978\pi\)
−0.194509 + 0.980901i \(0.562311\pi\)
\(608\) 0 0
\(609\) −308.713 + 137.448i −0.506917 + 0.225694i
\(610\) 0 0
\(611\) 5.38426 0.565909i 0.00881222 0.000926201i
\(612\) 0 0
\(613\) −395.523 41.5712i −0.645225 0.0678159i −0.223737 0.974650i \(-0.571826\pi\)
−0.421489 + 0.906834i \(0.638492\pi\)
\(614\) 0 0
\(615\) −471.793 272.390i −0.767143 0.442910i
\(616\) 0 0
\(617\) 601.066 667.552i 0.974175 1.08193i −0.0224420 0.999748i \(-0.507144\pi\)
0.996617 0.0821830i \(-0.0261892\pi\)
\(618\) 0 0
\(619\) 496.975i 0.802868i −0.915888 0.401434i \(-0.868512\pi\)
0.915888 0.401434i \(-0.131488\pi\)
\(620\) 0 0
\(621\) −961.484 −1.54828
\(622\) 0 0
\(623\) −291.040 262.054i −0.467159 0.420632i
\(624\) 0 0
\(625\) 359.989 623.519i 0.575982 0.997631i
\(626\) 0 0
\(627\) −37.7415 + 359.087i −0.0601938 + 0.572706i
\(628\) 0 0
\(629\) −84.4625 803.607i −0.134281 1.27759i
\(630\) 0 0
\(631\) 197.075 + 442.637i 0.312321 + 0.701485i 0.999692 0.0248218i \(-0.00790183\pi\)
−0.687371 + 0.726307i \(0.741235\pi\)
\(632\) 0 0
\(633\) 56.1188 264.018i 0.0886553 0.417090i
\(634\) 0 0
\(635\) 338.806 110.085i 0.533553 0.173362i
\(636\) 0 0
\(637\) 11.4123 25.6325i 0.0179157 0.0402394i
\(638\) 0 0
\(639\) 66.1838 14.0678i 0.103574 0.0220153i
\(640\) 0 0
\(641\) 210.134 189.206i 0.327823 0.295173i −0.488740 0.872429i \(-0.662543\pi\)
0.816563 + 0.577257i \(0.195877\pi\)
\(642\) 0 0
\(643\) −218.286 300.445i −0.339481 0.467255i 0.604809 0.796371i \(-0.293250\pi\)
−0.944290 + 0.329115i \(0.893250\pi\)
\(644\) 0 0
\(645\) 46.0705 + 79.7964i 0.0714271 + 0.123715i
\(646\) 0 0
\(647\) −118.897 + 163.648i −0.183767 + 0.252933i −0.890954 0.454093i \(-0.849964\pi\)
0.707188 + 0.707026i \(0.249964\pi\)
\(648\) 0 0
\(649\) −437.665 142.206i −0.674368 0.219115i
\(650\) 0 0
\(651\) 48.4173 + 293.479i 0.0743737 + 0.450812i
\(652\) 0 0
\(653\) −67.0601 + 206.390i −0.102695 + 0.316064i −0.989183 0.146689i \(-0.953138\pi\)
0.886487 + 0.462753i \(0.153138\pi\)
\(654\) 0 0
\(655\) 309.848 + 225.117i 0.473050 + 0.343691i
\(656\) 0 0
\(657\) −390.245 + 225.308i −0.593981 + 0.342935i
\(658\) 0 0
\(659\) −286.417 + 208.094i −0.434624 + 0.315773i −0.783495 0.621398i \(-0.786565\pi\)
0.348871 + 0.937171i \(0.386565\pi\)
\(660\) 0 0
\(661\) −544.518 604.748i −0.823779 0.914899i 0.173776 0.984785i \(-0.444403\pi\)
−0.997555 + 0.0698860i \(0.977736\pi\)
\(662\) 0 0
\(663\) 10.8401 + 50.9987i 0.0163501 + 0.0769211i
\(664\) 0 0
\(665\) 272.966 + 121.532i 0.410475 + 0.182755i
\(666\) 0 0
\(667\) −398.357 1226.02i −0.597237 1.83811i
\(668\) 0 0
\(669\) −418.911 89.0423i −0.626175 0.133098i
\(670\) 0 0
\(671\) 1119.76 498.551i 1.66880 0.742998i
\(672\) 0 0
\(673\) −191.630 + 20.1412i −0.284741 + 0.0299275i −0.245822 0.969315i \(-0.579058\pi\)
−0.0389185 + 0.999242i \(0.512391\pi\)
\(674\) 0 0
\(675\) 122.062 + 12.8292i 0.180833 + 0.0190063i
\(676\) 0 0
\(677\) −401.649 231.892i −0.593277 0.342529i 0.173115 0.984902i \(-0.444617\pi\)
−0.766392 + 0.642373i \(0.777950\pi\)
\(678\) 0 0
\(679\) 463.318 514.567i 0.682354 0.757831i
\(680\) 0 0
\(681\) 99.5249i 0.146145i
\(682\) 0 0
\(683\) 303.747 0.444725 0.222363 0.974964i \(-0.428623\pi\)
0.222363 + 0.974964i \(0.428623\pi\)
\(684\) 0 0
\(685\) −426.517 384.038i −0.622653 0.560639i
\(686\) 0 0
\(687\) −236.174 + 409.065i −0.343775 + 0.595436i
\(688\) 0 0
\(689\) 9.99753 95.1202i 0.0145102 0.138055i
\(690\) 0 0
\(691\) 34.2665 + 326.024i 0.0495897 + 0.471815i 0.990932 + 0.134362i \(0.0428986\pi\)
−0.941343 + 0.337453i \(0.890435\pi\)
\(692\) 0 0
\(693\) −264.046 593.058i −0.381019 0.855783i
\(694\) 0 0
\(695\) 151.976 714.993i 0.218671 1.02877i
\(696\) 0 0
\(697\) 1103.07 358.410i 1.58260 0.514219i
\(698\) 0 0
\(699\) 190.533 427.943i 0.272579 0.612222i
\(700\) 0 0
\(701\) 187.693 39.8953i 0.267750 0.0569120i −0.0720802 0.997399i \(-0.522964\pi\)
0.339830 + 0.940487i \(0.389630\pi\)
\(702\) 0 0
\(703\) 296.009 266.528i 0.421066 0.379130i
\(704\) 0 0
\(705\) −21.1859 29.1599i −0.0300509 0.0413615i
\(706\) 0 0
\(707\) 61.0155 + 105.682i 0.0863020 + 0.149480i
\(708\) 0 0
\(709\) −106.548 + 146.651i −0.150279 + 0.206842i −0.877519 0.479542i \(-0.840803\pi\)
0.727240 + 0.686384i \(0.240803\pi\)
\(710\) 0 0
\(711\) −429.423 139.528i −0.603971 0.196242i
\(712\) 0 0
\(713\) −1132.72 + 66.6633i −1.58867 + 0.0934970i
\(714\) 0 0
\(715\) −49.2947 + 151.713i −0.0689436 + 0.212187i
\(716\) 0 0
\(717\) 235.337 + 170.982i 0.328225 + 0.238469i
\(718\) 0 0
\(719\) 12.4104 7.16515i 0.0172606 0.00996544i −0.491345 0.870965i \(-0.663495\pi\)
0.508605 + 0.861000i \(0.330161\pi\)
\(720\) 0 0
\(721\) 365.032 265.211i 0.506285 0.367838i
\(722\) 0 0
\(723\) 320.783 + 356.265i 0.443683 + 0.492760i
\(724\) 0 0
\(725\) 34.2132 + 160.960i 0.0471906 + 0.222014i
\(726\) 0 0
\(727\) −358.176 159.470i −0.492677 0.219354i 0.145332 0.989383i \(-0.453575\pi\)
−0.638009 + 0.770029i \(0.720242\pi\)
\(728\) 0 0
\(729\) −116.868 359.684i −0.160313 0.493394i
\(730\) 0 0
\(731\) −191.882 40.7858i −0.262493 0.0557945i
\(732\) 0 0
\(733\) −1043.68 + 464.677i −1.42385 + 0.633938i −0.966807 0.255508i \(-0.917757\pi\)
−0.457041 + 0.889446i \(0.651091\pi\)
\(734\) 0 0
\(735\) −185.777 + 19.5259i −0.252757 + 0.0265659i
\(736\) 0 0
\(737\) 1275.31 + 134.041i 1.73041 + 0.181874i
\(738\) 0 0
\(739\) −172.810 99.7716i −0.233842 0.135009i 0.378501 0.925601i \(-0.376440\pi\)
−0.612343 + 0.790592i \(0.709773\pi\)
\(740\) 0 0
\(741\) −17.1976 + 19.0999i −0.0232087 + 0.0257758i
\(742\) 0 0
\(743\) 962.274i 1.29512i −0.762015 0.647560i \(-0.775790\pi\)
0.762015 0.647560i \(-0.224210\pi\)
\(744\) 0 0
\(745\) 798.604 1.07195
\(746\) 0 0
\(747\) −532.973 479.891i −0.713485 0.642425i
\(748\) 0 0
\(749\) 309.894 536.751i 0.413743 0.716624i
\(750\) 0 0
\(751\) −24.2010 + 230.257i −0.0322250 + 0.306601i 0.966523 + 0.256580i \(0.0825956\pi\)
−0.998748 + 0.0500212i \(0.984071\pi\)
\(752\) 0 0
\(753\) 66.0604 + 628.523i 0.0877297 + 0.834692i
\(754\) 0 0
\(755\) 259.983 + 583.932i 0.344348 + 0.773419i
\(756\) 0 0
\(757\) −35.5524 + 167.261i −0.0469648 + 0.220952i −0.995371 0.0961053i \(-0.969361\pi\)
0.948406 + 0.317057i \(0.102695\pi\)
\(758\) 0 0
\(759\) −1245.95 + 404.833i −1.64157 + 0.533377i
\(760\) 0 0
\(761\) −107.789 + 242.097i −0.141641 + 0.318131i −0.970411 0.241458i \(-0.922374\pi\)
0.828770 + 0.559589i \(0.189041\pi\)
\(762\) 0 0
\(763\) −743.568 + 158.050i −0.974533 + 0.207143i
\(764\) 0 0
\(765\) −487.622 + 439.057i −0.637414 + 0.573930i
\(766\) 0 0
\(767\) −19.2542 26.5012i −0.0251033 0.0345517i
\(768\) 0 0
\(769\) 666.556 + 1154.51i 0.866783 + 1.50131i 0.865265 + 0.501314i \(0.167150\pi\)
0.00151784 + 0.999999i \(0.499517\pi\)
\(770\) 0 0
\(771\) 379.759 522.693i 0.492554 0.677942i
\(772\) 0 0
\(773\) 960.282 + 312.014i 1.24228 + 0.403641i 0.855148 0.518384i \(-0.173466\pi\)
0.387131 + 0.922025i \(0.373466\pi\)
\(774\) 0 0
\(775\) 144.690 + 6.65105i 0.186697 + 0.00858200i
\(776\) 0 0
\(777\) 117.073 360.314i 0.150673 0.463724i
\(778\) 0 0
\(779\) 462.550 + 336.062i 0.593774 + 0.431402i
\(780\) 0 0
\(781\) 201.920 116.579i 0.258540 0.149268i
\(782\) 0 0
\(783\) 748.456 543.785i 0.955882 0.694489i
\(784\) 0 0
\(785\) −12.9418 14.3733i −0.0164863 0.0183099i
\(786\) 0 0
\(787\) 301.179 + 1416.94i 0.382693 + 1.80043i 0.573952 + 0.818889i \(0.305410\pi\)
−0.191259 + 0.981540i \(0.561257\pi\)
\(788\) 0 0
\(789\) −66.2165 29.4815i −0.0839246 0.0373656i
\(790\) 0 0
\(791\) 31.8056 + 97.8877i 0.0402094 + 0.123752i
\(792\) 0 0
\(793\) 85.3441 + 18.1404i 0.107622 + 0.0228757i
\(794\) 0 0
\(795\) −581.707 + 258.993i −0.731707 + 0.325777i
\(796\) 0 0
\(797\) 500.302 52.5839i 0.627731 0.0659772i 0.214679 0.976685i \(-0.431129\pi\)
0.413052 + 0.910707i \(0.364463\pi\)
\(798\) 0 0
\(799\) 76.3168 + 8.02122i 0.0955154 + 0.0100391i
\(800\) 0 0
\(801\) 367.151 + 211.975i 0.458365 + 0.264637i
\(802\) 0 0
\(803\) −1039.01 + 1153.94i −1.29391 + 1.43703i
\(804\) 0 0
\(805\) 1084.14i 1.34676i
\(806\) 0 0
\(807\) −116.142 −0.143918
\(808\) 0 0
\(809\) −336.433 302.925i −0.415862 0.374444i 0.434473 0.900685i \(-0.356934\pi\)
−0.850336 + 0.526241i \(0.823601\pi\)
\(810\) 0 0
\(811\) −723.353 + 1252.88i −0.891928 + 1.54486i −0.0543658 + 0.998521i \(0.517314\pi\)
−0.837562 + 0.546343i \(0.816020\pi\)
\(812\) 0 0
\(813\) 44.8101 426.340i 0.0551170 0.524403i
\(814\) 0 0
\(815\) −119.201 1134.12i −0.146258 1.39156i
\(816\) 0 0
\(817\) −39.3320 88.3412i −0.0481420 0.108129i
\(818\) 0 0
\(819\) 9.60767 45.2005i 0.0117310 0.0551899i
\(820\) 0 0
\(821\) −279.062 + 90.6727i −0.339905 + 0.110442i −0.473996 0.880527i \(-0.657189\pi\)
0.134091 + 0.990969i \(0.457189\pi\)
\(822\) 0 0
\(823\) −542.691 + 1218.90i −0.659405 + 1.48105i 0.205264 + 0.978707i \(0.434195\pi\)
−0.864669 + 0.502342i \(0.832472\pi\)
\(824\) 0 0
\(825\) 163.577 34.7694i 0.198275 0.0421447i
\(826\) 0 0
\(827\) −778.763 + 701.202i −0.941673 + 0.847886i −0.988531 0.151016i \(-0.951746\pi\)
0.0468587 + 0.998902i \(0.485079\pi\)
\(828\) 0 0
\(829\) −412.388 567.604i −0.497453 0.684685i 0.484288 0.874909i \(-0.339079\pi\)
−0.981741 + 0.190224i \(0.939079\pi\)
\(830\) 0 0
\(831\) 221.141 + 383.027i 0.266114 + 0.460923i
\(832\) 0 0
\(833\) 233.762 321.745i 0.280626 0.386249i
\(834\) 0 0
\(835\) 191.198 + 62.1240i 0.228980 + 0.0744000i
\(836\) 0 0
\(837\) −296.702 758.337i −0.354483 0.906018i
\(838\) 0 0
\(839\) 221.778 682.564i 0.264337 0.813545i −0.727509 0.686098i \(-0.759322\pi\)
0.991845 0.127446i \(-0.0406780\pi\)
\(840\) 0 0
\(841\) 323.109 + 234.753i 0.384197 + 0.279135i
\(842\) 0 0
\(843\) −76.9900 + 44.4502i −0.0913286 + 0.0527286i
\(844\) 0 0
\(845\) 735.580 534.430i 0.870509 0.632462i
\(846\) 0 0
\(847\) −1056.61 1173.48i −1.24747 1.38546i
\(848\) 0 0
\(849\) 19.4005 + 91.2724i 0.0228511 + 0.107506i
\(850\) 0 0
\(851\) 1320.29 + 587.833i 1.55146 + 0.690755i
\(852\) 0 0
\(853\) 229.416 + 706.071i 0.268952 + 0.827751i 0.990756 + 0.135653i \(0.0433132\pi\)
−0.721804 + 0.692098i \(0.756687\pi\)
\(854\) 0 0
\(855\) −316.386 67.2499i −0.370042 0.0786548i
\(856\) 0 0
\(857\) 1341.35 597.209i 1.56517 0.696860i 0.572750 0.819730i \(-0.305877\pi\)
0.992423 + 0.122871i \(0.0392101\pi\)
\(858\) 0 0
\(859\) −440.624 + 46.3114i −0.512950 + 0.0539132i −0.357468 0.933925i \(-0.616360\pi\)
−0.155482 + 0.987839i \(0.549693\pi\)
\(860\) 0 0
\(861\) 540.827 + 56.8432i 0.628138 + 0.0660199i
\(862\) 0 0
\(863\) 1449.29 + 836.749i 1.67936 + 0.969582i 0.962066 + 0.272815i \(0.0879548\pi\)
0.717298 + 0.696766i \(0.245379\pi\)
\(864\) 0 0
\(865\) 427.864 475.191i 0.494640 0.549354i
\(866\) 0 0
\(867\) 229.036i 0.264171i
\(868\) 0 0
\(869\) −1555.90 −1.79044
\(870\) 0 0
\(871\) 67.8341 + 61.0781i 0.0778807 + 0.0701241i
\(872\) 0 0
\(873\) −374.777 + 649.132i −0.429298 + 0.743565i
\(874\) 0 0
\(875\) −62.9356 + 598.793i −0.0719264 + 0.684334i
\(876\) 0 0
\(877\) 64.5896 + 614.529i 0.0736483 + 0.700717i 0.967589 + 0.252531i \(0.0812630\pi\)
−0.893940 + 0.448186i \(0.852070\pi\)
\(878\) 0 0
\(879\) 300.307 + 674.500i 0.341646 + 0.767350i
\(880\) 0 0
\(881\) 242.220 1139.56i 0.274938 1.29348i −0.596356 0.802720i \(-0.703385\pi\)
0.871294 0.490761i \(-0.163281\pi\)
\(882\) 0 0
\(883\) −619.778 + 201.378i −0.701900 + 0.228061i −0.638158 0.769905i \(-0.720303\pi\)
−0.0637420 + 0.997966i \(0.520303\pi\)
\(884\) 0 0
\(885\) −88.7027 + 199.230i −0.100229 + 0.225118i
\(886\) 0 0
\(887\) −356.160 + 75.7041i −0.401533 + 0.0853485i −0.404252 0.914648i \(-0.632468\pi\)
0.00271850 + 0.999996i \(0.499135\pi\)
\(888\) 0 0
\(889\) −264.266 + 237.946i −0.297262 + 0.267656i
\(890\) 0 0
\(891\) 78.9559 + 108.673i 0.0886149 + 0.121968i
\(892\) 0 0
\(893\) 18.9138 + 32.7596i 0.0211801 + 0.0366849i
\(894\) 0 0
\(895\) −695.073 + 956.687i −0.776618 + 1.06892i
\(896\) 0 0
\(897\) −88.6895 28.8170i −0.0988735 0.0321259i
\(898\) 0 0
\(899\) 844.050 692.524i 0.938877 0.770327i
\(900\) 0 0
\(901\) 418.923 1289.31i 0.464953 1.43098i
\(902\) 0 0
\(903\) −74.4104 54.0623i −0.0824035 0.0598696i
\(904\) 0 0
\(905\) −1048.57 + 605.391i −1.15864 + 0.668940i
\(906\) 0 0
\(907\) 851.493 618.646i 0.938801 0.682079i −0.00933072 0.999956i \(-0.502970\pi\)
0.948132 + 0.317878i \(0.102970\pi\)
\(908\) 0 0
\(909\) −88.3925 98.1699i −0.0972415 0.107998i
\(910\) 0 0
\(911\) 122.199 + 574.902i 0.134137 + 0.631067i 0.992926 + 0.118737i \(0.0378844\pi\)
−0.858788 + 0.512331i \(0.828782\pi\)
\(912\) 0 0
\(913\) −2257.69 1005.19i −2.47282 1.10097i
\(914\) 0 0
\(915\) −179.501 552.448i −0.196176 0.603768i
\(916\) 0 0
\(917\) −373.954 79.4863i −0.407801 0.0866808i
\(918\) 0 0
\(919\) 1461.88 650.871i 1.59073 0.708238i 0.595279 0.803519i \(-0.297042\pi\)
0.995450 + 0.0952811i \(0.0303750\pi\)
\(920\) 0 0
\(921\) 380.242 39.9650i 0.412858 0.0433931i
\(922\) 0 0
\(923\) 16.5058 + 1.73483i 0.0178827 + 0.00187955i
\(924\) 0 0
\(925\) −159.770 92.2433i −0.172724 0.0997224i
\(926\) 0 0
\(927\) −326.827 + 362.978i −0.352564 + 0.391562i
\(928\) 0 0
\(929\) 634.016i 0.682472i 0.939978 + 0.341236i \(0.110845\pi\)
−0.939978 + 0.341236i \(0.889155\pi\)
\(930\) 0 0
\(931\) 196.046 0.210575
\(932\) 0 0
\(933\) 731.674 + 658.802i 0.784216 + 0.706112i
\(934\) 0 0
\(935\) −1130.53 + 1958.13i −1.20912 + 2.09426i
\(936\) 0 0
\(937\) 165.438 1574.04i 0.176562 1.67987i −0.444243 0.895906i \(-0.646527\pi\)
0.620805 0.783965i \(-0.286806\pi\)
\(938\) 0 0
\(939\) −82.7519 787.332i −0.0881277 0.838479i
\(940\) 0 0
\(941\) −322.581 724.528i −0.342806 0.769955i −0.999870 0.0161203i \(-0.994869\pi\)
0.657064 0.753835i \(-0.271798\pi\)
\(942\) 0 0
\(943\) −431.309 + 2029.15i −0.457380 + 2.15180i
\(944\) 0 0
\(945\) −739.966 + 240.429i −0.783032 + 0.254423i
\(946\) 0 0
\(947\) 94.2280 211.639i 0.0995015 0.223484i −0.856935 0.515425i \(-0.827634\pi\)
0.956437 + 0.291940i \(0.0943009\pi\)
\(948\) 0 0
\(949\) −108.115 + 22.9805i −0.113925 + 0.0242155i
\(950\) 0 0
\(951\) 330.332 297.432i 0.347352 0.312757i
\(952\) 0 0
\(953\) −444.716 612.099i −0.466648 0.642286i 0.509223 0.860635i \(-0.329933\pi\)
−0.975871 + 0.218349i \(0.929933\pi\)
\(954\) 0 0
\(955\) −1006.96 1744.10i −1.05441 1.82628i
\(956\) 0 0
\(957\) 740.932 1019.81i 0.774224 1.06563i
\(958\) 0 0
\(959\) 544.870 + 177.039i 0.568164 + 0.184608i
\(960\) 0 0
\(961\) −402.122 872.822i −0.418441 0.908244i
\(962\) 0 0
\(963\) −207.329 + 638.092i −0.215294 + 0.662608i
\(964\) 0 0
\(965\) 1630.52 + 1184.65i 1.68966 + 1.22761i
\(966\) 0 0
\(967\) 588.730 339.903i 0.608821 0.351503i −0.163683 0.986513i \(-0.552337\pi\)
0.772504 + 0.635010i \(0.219004\pi\)
\(968\) 0 0
\(969\) −294.719 + 214.126i −0.304148 + 0.220976i
\(970\) 0 0
\(971\) −906.331 1006.58i −0.933400 1.03665i −0.999246 0.0388159i \(-0.987641\pi\)
0.0658464 0.997830i \(-0.479025\pi\)
\(972\) 0 0
\(973\) 151.705 + 713.715i 0.155915 + 0.733520i
\(974\) 0 0
\(975\) 10.8748 + 4.84176i 0.0111536 + 0.00496591i
\(976\) 0 0
\(977\) 14.1767 + 43.6314i 0.0145104 + 0.0446585i 0.958050 0.286603i \(-0.0925259\pi\)
−0.943539 + 0.331261i \(0.892526\pi\)
\(978\) 0 0
\(979\) 1428.96 + 303.734i 1.45961 + 0.310250i
\(980\) 0 0
\(981\) 751.763 334.706i 0.766323 0.341189i
\(982\) 0 0
\(983\) −291.460 + 30.6337i −0.296501 + 0.0311635i −0.251611 0.967828i \(-0.580960\pi\)
−0.0448897 + 0.998992i \(0.514294\pi\)
\(984\) 0 0
\(985\) 1312.39 + 137.938i 1.33238 + 0.140038i
\(986\) 0 0
\(987\) 31.1589 + 17.9896i 0.0315693 + 0.0182265i
\(988\) 0 0
\(989\) 234.775 260.744i 0.237387 0.263645i
\(990\) 0 0
\(991\) 490.075i 0.494525i −0.968948 0.247263i \(-0.920469\pi\)
0.968948 0.247263i \(-0.0795311\pi\)
\(992\) 0 0
\(993\) 818.800 0.824572
\(994\) 0 0
\(995\) −775.246 698.034i −0.779141 0.701542i
\(996\) 0 0
\(997\) 514.371 890.916i 0.515919 0.893597i −0.483911 0.875117i \(-0.660784\pi\)
0.999829 0.0184798i \(-0.00588262\pi\)
\(998\) 0 0
\(999\) −108.416 + 1031.51i −0.108525 + 1.03254i
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 124.3.o.a.17.2 40
31.11 odd 30 inner 124.3.o.a.73.2 yes 40
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
124.3.o.a.17.2 40 1.1 even 1 trivial
124.3.o.a.73.2 yes 40 31.11 odd 30 inner