Properties

Label 531.2.a.d.1.3
Level $531$
Weight $2$
Character 531.1
Self dual yes
Analytic conductor $4.240$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [531,2,Mod(1,531)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(531, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("531.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 531 = 3^{2} \cdot 59 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 531.a (trivial)

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: yes
Analytic conductor: \(4.24005634733\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.229.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 4x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 177)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.86081\) of defining polynomial
Character \(\chi\) \(=\) 531.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.86081 q^{2} +1.46260 q^{4} +3.32340 q^{5} +1.13919 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q+1.86081 q^{2} +1.46260 q^{4} +3.32340 q^{5} +1.13919 q^{7} -1.00000 q^{8} +6.18421 q^{10} -0.398207 q^{11} +2.39821 q^{13} +2.11982 q^{14} -4.78600 q^{16} -7.10941 q^{17} +1.53740 q^{19} +4.86081 q^{20} -0.740987 q^{22} -3.25901 q^{23} +6.04502 q^{25} +4.46260 q^{26} +1.66618 q^{28} +3.93561 q^{29} +7.78600 q^{31} -6.90582 q^{32} -13.2292 q^{34} +3.78600 q^{35} +1.25901 q^{37} +2.86081 q^{38} -3.32340 q^{40} +7.50761 q^{41} -9.69182 q^{43} -0.582418 q^{44} -6.06439 q^{46} -8.71120 q^{47} -5.70224 q^{49} +11.2486 q^{50} +3.50761 q^{52} -10.7666 q^{53} -1.32340 q^{55} -1.13919 q^{56} +7.32340 q^{58} -1.00000 q^{59} +0.989588 q^{61} +14.4882 q^{62} -3.27839 q^{64} +7.97021 q^{65} +5.45219 q^{67} -10.3982 q^{68} +7.04502 q^{70} -15.0450 q^{71} +4.73202 q^{73} +2.34278 q^{74} +2.24860 q^{76} -0.453636 q^{77} +7.04502 q^{79} -15.9058 q^{80} +13.9702 q^{82} -13.0796 q^{83} -23.6274 q^{85} -18.0346 q^{86} +0.398207 q^{88} +16.4328 q^{89} +2.73202 q^{91} -4.76663 q^{92} -16.2099 q^{94} +5.10941 q^{95} +14.7666 q^{97} -10.6108 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{4} + 2 q^{5} + 9 q^{7} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 2 q^{4} + 2 q^{5} + 9 q^{7} - 3 q^{8} + 5 q^{10} + 2 q^{11} + 4 q^{13} - 8 q^{14} - 4 q^{16} - 3 q^{17} + 7 q^{19} + 9 q^{20} - 11 q^{22} - q^{23} - q^{25} + 11 q^{26} + 9 q^{28} + 11 q^{29} + 13 q^{31} + 4 q^{32} - 7 q^{34} + q^{35} - 5 q^{37} + 3 q^{38} - 2 q^{40} + q^{41} + 6 q^{43} + 15 q^{44} - 19 q^{46} - 11 q^{47} + 14 q^{49} + 21 q^{50} - 11 q^{52} - 2 q^{53} + 4 q^{55} - 9 q^{56} + 14 q^{58} - 3 q^{59} - q^{61} + 2 q^{62} - 21 q^{64} + 10 q^{67} - 28 q^{68} + 2 q^{70} - 26 q^{71} + 7 q^{73} + 19 q^{74} - 6 q^{76} + 17 q^{77} + 2 q^{79} - 23 q^{80} + 18 q^{82} + 3 q^{83} - 35 q^{85} - 31 q^{86} - 2 q^{88} + 23 q^{89} + q^{91} + 16 q^{92} + 4 q^{94} - 3 q^{95} + 14 q^{97} - 51 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.86081 1.31579 0.657894 0.753110i \(-0.271447\pi\)
0.657894 + 0.753110i \(0.271447\pi\)
\(3\) 0 0
\(4\) 1.46260 0.731299
\(5\) 3.32340 1.48627 0.743136 0.669141i \(-0.233338\pi\)
0.743136 + 0.669141i \(0.233338\pi\)
\(6\) 0 0
\(7\) 1.13919 0.430575 0.215287 0.976551i \(-0.430931\pi\)
0.215287 + 0.976551i \(0.430931\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 6.18421 1.95562
\(11\) −0.398207 −0.120064 −0.0600320 0.998196i \(-0.519120\pi\)
−0.0600320 + 0.998196i \(0.519120\pi\)
\(12\) 0 0
\(13\) 2.39821 0.665143 0.332572 0.943078i \(-0.392084\pi\)
0.332572 + 0.943078i \(0.392084\pi\)
\(14\) 2.11982 0.566545
\(15\) 0 0
\(16\) −4.78600 −1.19650
\(17\) −7.10941 −1.72428 −0.862142 0.506666i \(-0.830878\pi\)
−0.862142 + 0.506666i \(0.830878\pi\)
\(18\) 0 0
\(19\) 1.53740 0.352704 0.176352 0.984327i \(-0.443570\pi\)
0.176352 + 0.984327i \(0.443570\pi\)
\(20\) 4.86081 1.08691
\(21\) 0 0
\(22\) −0.740987 −0.157979
\(23\) −3.25901 −0.679551 −0.339776 0.940507i \(-0.610351\pi\)
−0.339776 + 0.940507i \(0.610351\pi\)
\(24\) 0 0
\(25\) 6.04502 1.20900
\(26\) 4.46260 0.875188
\(27\) 0 0
\(28\) 1.66618 0.314879
\(29\) 3.93561 0.730824 0.365412 0.930846i \(-0.380928\pi\)
0.365412 + 0.930846i \(0.380928\pi\)
\(30\) 0 0
\(31\) 7.78600 1.39841 0.699204 0.714923i \(-0.253538\pi\)
0.699204 + 0.714923i \(0.253538\pi\)
\(32\) −6.90582 −1.22079
\(33\) 0 0
\(34\) −13.2292 −2.26879
\(35\) 3.78600 0.639951
\(36\) 0 0
\(37\) 1.25901 0.206981 0.103490 0.994630i \(-0.466999\pi\)
0.103490 + 0.994630i \(0.466999\pi\)
\(38\) 2.86081 0.464084
\(39\) 0 0
\(40\) −3.32340 −0.525476
\(41\) 7.50761 1.17249 0.586246 0.810133i \(-0.300605\pi\)
0.586246 + 0.810133i \(0.300605\pi\)
\(42\) 0 0
\(43\) −9.69182 −1.47799 −0.738995 0.673711i \(-0.764699\pi\)
−0.738995 + 0.673711i \(0.764699\pi\)
\(44\) −0.582418 −0.0878028
\(45\) 0 0
\(46\) −6.06439 −0.894146
\(47\) −8.71120 −1.27066 −0.635330 0.772241i \(-0.719136\pi\)
−0.635330 + 0.772241i \(0.719136\pi\)
\(48\) 0 0
\(49\) −5.70224 −0.814605
\(50\) 11.2486 1.59079
\(51\) 0 0
\(52\) 3.50761 0.486419
\(53\) −10.7666 −1.47891 −0.739455 0.673206i \(-0.764917\pi\)
−0.739455 + 0.673206i \(0.764917\pi\)
\(54\) 0 0
\(55\) −1.32340 −0.178448
\(56\) −1.13919 −0.152231
\(57\) 0 0
\(58\) 7.32340 0.961610
\(59\) −1.00000 −0.130189
\(60\) 0 0
\(61\) 0.989588 0.126704 0.0633519 0.997991i \(-0.479821\pi\)
0.0633519 + 0.997991i \(0.479821\pi\)
\(62\) 14.4882 1.84001
\(63\) 0 0
\(64\) −3.27839 −0.409799
\(65\) 7.97021 0.988583
\(66\) 0 0
\(67\) 5.45219 0.666091 0.333045 0.942911i \(-0.391924\pi\)
0.333045 + 0.942911i \(0.391924\pi\)
\(68\) −10.3982 −1.26097
\(69\) 0 0
\(70\) 7.04502 0.842040
\(71\) −15.0450 −1.78551 −0.892757 0.450538i \(-0.851232\pi\)
−0.892757 + 0.450538i \(0.851232\pi\)
\(72\) 0 0
\(73\) 4.73202 0.553842 0.276921 0.960893i \(-0.410686\pi\)
0.276921 + 0.960893i \(0.410686\pi\)
\(74\) 2.34278 0.272343
\(75\) 0 0
\(76\) 2.24860 0.257932
\(77\) −0.453636 −0.0516966
\(78\) 0 0
\(79\) 7.04502 0.792626 0.396313 0.918115i \(-0.370289\pi\)
0.396313 + 0.918115i \(0.370289\pi\)
\(80\) −15.9058 −1.77832
\(81\) 0 0
\(82\) 13.9702 1.54275
\(83\) −13.0796 −1.43567 −0.717837 0.696211i \(-0.754868\pi\)
−0.717837 + 0.696211i \(0.754868\pi\)
\(84\) 0 0
\(85\) −23.6274 −2.56275
\(86\) −18.0346 −1.94472
\(87\) 0 0
\(88\) 0.398207 0.0424491
\(89\) 16.4328 1.74187 0.870937 0.491394i \(-0.163513\pi\)
0.870937 + 0.491394i \(0.163513\pi\)
\(90\) 0 0
\(91\) 2.73202 0.286394
\(92\) −4.76663 −0.496955
\(93\) 0 0
\(94\) −16.2099 −1.67192
\(95\) 5.10941 0.524214
\(96\) 0 0
\(97\) 14.7666 1.49932 0.749662 0.661821i \(-0.230216\pi\)
0.749662 + 0.661821i \(0.230216\pi\)
\(98\) −10.6108 −1.07185
\(99\) 0 0
\(100\) 8.84143 0.884143
\(101\) 0.278388 0.0277007 0.0138503 0.999904i \(-0.495591\pi\)
0.0138503 + 0.999904i \(0.495591\pi\)
\(102\) 0 0
\(103\) −15.0152 −1.47949 −0.739747 0.672885i \(-0.765055\pi\)
−0.739747 + 0.672885i \(0.765055\pi\)
\(104\) −2.39821 −0.235164
\(105\) 0 0
\(106\) −20.0346 −1.94593
\(107\) −1.16898 −0.113010 −0.0565048 0.998402i \(-0.517996\pi\)
−0.0565048 + 0.998402i \(0.517996\pi\)
\(108\) 0 0
\(109\) −4.89059 −0.468434 −0.234217 0.972184i \(-0.575253\pi\)
−0.234217 + 0.972184i \(0.575253\pi\)
\(110\) −2.46260 −0.234800
\(111\) 0 0
\(112\) −5.45219 −0.515183
\(113\) 15.5630 1.46405 0.732024 0.681279i \(-0.238576\pi\)
0.732024 + 0.681279i \(0.238576\pi\)
\(114\) 0 0
\(115\) −10.8310 −1.01000
\(116\) 5.75622 0.534451
\(117\) 0 0
\(118\) −1.86081 −0.171301
\(119\) −8.09899 −0.742434
\(120\) 0 0
\(121\) −10.8414 −0.985585
\(122\) 1.84143 0.166715
\(123\) 0 0
\(124\) 11.3878 1.02265
\(125\) 3.47301 0.310636
\(126\) 0 0
\(127\) 4.39821 0.390278 0.195139 0.980776i \(-0.437484\pi\)
0.195139 + 0.980776i \(0.437484\pi\)
\(128\) 7.71120 0.681580
\(129\) 0 0
\(130\) 14.8310 1.30077
\(131\) −17.6620 −1.54314 −0.771570 0.636145i \(-0.780528\pi\)
−0.771570 + 0.636145i \(0.780528\pi\)
\(132\) 0 0
\(133\) 1.75140 0.151866
\(134\) 10.1455 0.876434
\(135\) 0 0
\(136\) 7.10941 0.609627
\(137\) 10.7666 0.919855 0.459928 0.887956i \(-0.347875\pi\)
0.459928 + 0.887956i \(0.347875\pi\)
\(138\) 0 0
\(139\) 9.82061 0.832973 0.416486 0.909142i \(-0.363261\pi\)
0.416486 + 0.909142i \(0.363261\pi\)
\(140\) 5.53740 0.467996
\(141\) 0 0
\(142\) −27.9959 −2.34936
\(143\) −0.954984 −0.0798598
\(144\) 0 0
\(145\) 13.0796 1.08620
\(146\) 8.80538 0.728738
\(147\) 0 0
\(148\) 1.84143 0.151365
\(149\) 5.01938 0.411203 0.205602 0.978636i \(-0.434085\pi\)
0.205602 + 0.978636i \(0.434085\pi\)
\(150\) 0 0
\(151\) 7.35801 0.598786 0.299393 0.954130i \(-0.403216\pi\)
0.299393 + 0.954130i \(0.403216\pi\)
\(152\) −1.53740 −0.124700
\(153\) 0 0
\(154\) −0.844128 −0.0680218
\(155\) 25.8760 2.07841
\(156\) 0 0
\(157\) −1.73057 −0.138115 −0.0690574 0.997613i \(-0.521999\pi\)
−0.0690574 + 0.997613i \(0.521999\pi\)
\(158\) 13.1094 1.04293
\(159\) 0 0
\(160\) −22.9508 −1.81442
\(161\) −3.71265 −0.292598
\(162\) 0 0
\(163\) −8.58242 −0.672227 −0.336113 0.941822i \(-0.609113\pi\)
−0.336113 + 0.941822i \(0.609113\pi\)
\(164\) 10.9806 0.857443
\(165\) 0 0
\(166\) −24.3386 −1.88904
\(167\) −6.49239 −0.502396 −0.251198 0.967936i \(-0.580825\pi\)
−0.251198 + 0.967936i \(0.580825\pi\)
\(168\) 0 0
\(169\) −7.24860 −0.557585
\(170\) −43.9661 −3.37204
\(171\) 0 0
\(172\) −14.1752 −1.08085
\(173\) 20.5437 1.56191 0.780953 0.624590i \(-0.214734\pi\)
0.780953 + 0.624590i \(0.214734\pi\)
\(174\) 0 0
\(175\) 6.88645 0.520566
\(176\) 1.90582 0.143657
\(177\) 0 0
\(178\) 30.5783 2.29194
\(179\) 23.9702 1.79162 0.895809 0.444439i \(-0.146597\pi\)
0.895809 + 0.444439i \(0.146597\pi\)
\(180\) 0 0
\(181\) 14.7112 1.09347 0.546737 0.837304i \(-0.315870\pi\)
0.546737 + 0.837304i \(0.315870\pi\)
\(182\) 5.08377 0.376834
\(183\) 0 0
\(184\) 3.25901 0.240258
\(185\) 4.18421 0.307629
\(186\) 0 0
\(187\) 2.83102 0.207025
\(188\) −12.7410 −0.929232
\(189\) 0 0
\(190\) 9.50761 0.689755
\(191\) −2.91623 −0.211011 −0.105506 0.994419i \(-0.533646\pi\)
−0.105506 + 0.994419i \(0.533646\pi\)
\(192\) 0 0
\(193\) −0.646809 −0.0465583 −0.0232791 0.999729i \(-0.507411\pi\)
−0.0232791 + 0.999729i \(0.507411\pi\)
\(194\) 27.4778 1.97279
\(195\) 0 0
\(196\) −8.34008 −0.595720
\(197\) 5.87122 0.418307 0.209153 0.977883i \(-0.432929\pi\)
0.209153 + 0.977883i \(0.432929\pi\)
\(198\) 0 0
\(199\) 11.1004 0.786890 0.393445 0.919348i \(-0.371283\pi\)
0.393445 + 0.919348i \(0.371283\pi\)
\(200\) −6.04502 −0.427447
\(201\) 0 0
\(202\) 0.518027 0.0364482
\(203\) 4.48342 0.314675
\(204\) 0 0
\(205\) 24.9508 1.74264
\(206\) −27.9404 −1.94670
\(207\) 0 0
\(208\) −11.4778 −0.795844
\(209\) −0.612205 −0.0423471
\(210\) 0 0
\(211\) −13.4778 −0.927852 −0.463926 0.885874i \(-0.653560\pi\)
−0.463926 + 0.885874i \(0.653560\pi\)
\(212\) −15.7473 −1.08153
\(213\) 0 0
\(214\) −2.17525 −0.148697
\(215\) −32.2099 −2.19669
\(216\) 0 0
\(217\) 8.86977 0.602119
\(218\) −9.10044 −0.616360
\(219\) 0 0
\(220\) −1.93561 −0.130499
\(221\) −17.0498 −1.14690
\(222\) 0 0
\(223\) 16.7368 1.12078 0.560391 0.828228i \(-0.310651\pi\)
0.560391 + 0.828228i \(0.310651\pi\)
\(224\) −7.86707 −0.525641
\(225\) 0 0
\(226\) 28.9598 1.92638
\(227\) 14.0346 0.931509 0.465755 0.884914i \(-0.345783\pi\)
0.465755 + 0.884914i \(0.345783\pi\)
\(228\) 0 0
\(229\) −12.2445 −0.809136 −0.404568 0.914508i \(-0.632578\pi\)
−0.404568 + 0.914508i \(0.632578\pi\)
\(230\) −20.1544 −1.32894
\(231\) 0 0
\(232\) −3.93561 −0.258385
\(233\) −1.87122 −0.122588 −0.0612938 0.998120i \(-0.519523\pi\)
−0.0612938 + 0.998120i \(0.519523\pi\)
\(234\) 0 0
\(235\) −28.9508 −1.88854
\(236\) −1.46260 −0.0952070
\(237\) 0 0
\(238\) −15.0707 −0.976886
\(239\) 13.9910 0.905005 0.452502 0.891763i \(-0.350531\pi\)
0.452502 + 0.891763i \(0.350531\pi\)
\(240\) 0 0
\(241\) 27.5333 1.77357 0.886786 0.462179i \(-0.152932\pi\)
0.886786 + 0.462179i \(0.152932\pi\)
\(242\) −20.1738 −1.29682
\(243\) 0 0
\(244\) 1.44737 0.0926583
\(245\) −18.9508 −1.21072
\(246\) 0 0
\(247\) 3.68701 0.234599
\(248\) −7.78600 −0.494412
\(249\) 0 0
\(250\) 6.46260 0.408731
\(251\) 2.61702 0.165185 0.0825925 0.996583i \(-0.473680\pi\)
0.0825925 + 0.996583i \(0.473680\pi\)
\(252\) 0 0
\(253\) 1.29776 0.0815897
\(254\) 8.18421 0.513523
\(255\) 0 0
\(256\) 20.9058 1.30661
\(257\) −20.3892 −1.27185 −0.635923 0.771752i \(-0.719380\pi\)
−0.635923 + 0.771752i \(0.719380\pi\)
\(258\) 0 0
\(259\) 1.43426 0.0891206
\(260\) 11.6572 0.722950
\(261\) 0 0
\(262\) −32.8656 −2.03044
\(263\) 23.8552 1.47098 0.735488 0.677538i \(-0.236953\pi\)
0.735488 + 0.677538i \(0.236953\pi\)
\(264\) 0 0
\(265\) −35.7819 −2.19806
\(266\) 3.25901 0.199823
\(267\) 0 0
\(268\) 7.97436 0.487112
\(269\) 28.2999 1.72547 0.862737 0.505653i \(-0.168748\pi\)
0.862737 + 0.505653i \(0.168748\pi\)
\(270\) 0 0
\(271\) −23.4134 −1.42226 −0.711132 0.703058i \(-0.751817\pi\)
−0.711132 + 0.703058i \(0.751817\pi\)
\(272\) 34.0256 2.06311
\(273\) 0 0
\(274\) 20.0346 1.21033
\(275\) −2.40717 −0.145158
\(276\) 0 0
\(277\) 15.6918 0.942830 0.471415 0.881911i \(-0.343743\pi\)
0.471415 + 0.881911i \(0.343743\pi\)
\(278\) 18.2742 1.09602
\(279\) 0 0
\(280\) −3.78600 −0.226257
\(281\) 5.63158 0.335952 0.167976 0.985791i \(-0.446277\pi\)
0.167976 + 0.985791i \(0.446277\pi\)
\(282\) 0 0
\(283\) 10.8310 0.643837 0.321919 0.946767i \(-0.395672\pi\)
0.321919 + 0.946767i \(0.395672\pi\)
\(284\) −22.0048 −1.30575
\(285\) 0 0
\(286\) −1.77704 −0.105079
\(287\) 8.55263 0.504846
\(288\) 0 0
\(289\) 33.5437 1.97316
\(290\) 24.3386 1.42921
\(291\) 0 0
\(292\) 6.92105 0.405024
\(293\) −33.9959 −1.98606 −0.993029 0.117866i \(-0.962395\pi\)
−0.993029 + 0.117866i \(0.962395\pi\)
\(294\) 0 0
\(295\) −3.32340 −0.193496
\(296\) −1.25901 −0.0731787
\(297\) 0 0
\(298\) 9.34008 0.541056
\(299\) −7.81579 −0.451999
\(300\) 0 0
\(301\) −11.0409 −0.636385
\(302\) 13.6918 0.787876
\(303\) 0 0
\(304\) −7.35801 −0.422011
\(305\) 3.28880 0.188316
\(306\) 0 0
\(307\) −4.25756 −0.242992 −0.121496 0.992592i \(-0.538769\pi\)
−0.121496 + 0.992592i \(0.538769\pi\)
\(308\) −0.663487 −0.0378057
\(309\) 0 0
\(310\) 48.1503 2.73475
\(311\) −21.5374 −1.22127 −0.610637 0.791911i \(-0.709087\pi\)
−0.610637 + 0.791911i \(0.709087\pi\)
\(312\) 0 0
\(313\) −3.96540 −0.224137 −0.112069 0.993700i \(-0.535748\pi\)
−0.112069 + 0.993700i \(0.535748\pi\)
\(314\) −3.22026 −0.181730
\(315\) 0 0
\(316\) 10.3040 0.579647
\(317\) −7.58097 −0.425790 −0.212895 0.977075i \(-0.568289\pi\)
−0.212895 + 0.977075i \(0.568289\pi\)
\(318\) 0 0
\(319\) −1.56719 −0.0877457
\(320\) −10.8954 −0.609072
\(321\) 0 0
\(322\) −6.90852 −0.384997
\(323\) −10.9300 −0.608162
\(324\) 0 0
\(325\) 14.4972 0.804160
\(326\) −15.9702 −0.884508
\(327\) 0 0
\(328\) −7.50761 −0.414539
\(329\) −9.92375 −0.547114
\(330\) 0 0
\(331\) −0.667633 −0.0366964 −0.0183482 0.999832i \(-0.505841\pi\)
−0.0183482 + 0.999832i \(0.505841\pi\)
\(332\) −19.1302 −1.04991
\(333\) 0 0
\(334\) −12.0811 −0.661047
\(335\) 18.1198 0.989991
\(336\) 0 0
\(337\) 7.47783 0.407343 0.203672 0.979039i \(-0.434713\pi\)
0.203672 + 0.979039i \(0.434713\pi\)
\(338\) −13.4882 −0.733664
\(339\) 0 0
\(340\) −34.5574 −1.87414
\(341\) −3.10044 −0.167898
\(342\) 0 0
\(343\) −14.4703 −0.781323
\(344\) 9.69182 0.522548
\(345\) 0 0
\(346\) 38.2278 2.05514
\(347\) 29.0200 1.55788 0.778939 0.627100i \(-0.215758\pi\)
0.778939 + 0.627100i \(0.215758\pi\)
\(348\) 0 0
\(349\) −16.8746 −0.903276 −0.451638 0.892201i \(-0.649160\pi\)
−0.451638 + 0.892201i \(0.649160\pi\)
\(350\) 12.8143 0.684955
\(351\) 0 0
\(352\) 2.74995 0.146573
\(353\) 2.98062 0.158643 0.0793213 0.996849i \(-0.474725\pi\)
0.0793213 + 0.996849i \(0.474725\pi\)
\(354\) 0 0
\(355\) −50.0007 −2.65376
\(356\) 24.0346 1.27383
\(357\) 0 0
\(358\) 44.6039 2.35739
\(359\) 1.85039 0.0976600 0.0488300 0.998807i \(-0.484451\pi\)
0.0488300 + 0.998807i \(0.484451\pi\)
\(360\) 0 0
\(361\) −16.6364 −0.875600
\(362\) 27.3747 1.43878
\(363\) 0 0
\(364\) 3.99585 0.209440
\(365\) 15.7264 0.823159
\(366\) 0 0
\(367\) 7.97021 0.416042 0.208021 0.978124i \(-0.433298\pi\)
0.208021 + 0.978124i \(0.433298\pi\)
\(368\) 15.5976 0.813084
\(369\) 0 0
\(370\) 7.78600 0.404775
\(371\) −12.2653 −0.636782
\(372\) 0 0
\(373\) 1.31859 0.0682739 0.0341369 0.999417i \(-0.489132\pi\)
0.0341369 + 0.999417i \(0.489132\pi\)
\(374\) 5.26798 0.272401
\(375\) 0 0
\(376\) 8.71120 0.449246
\(377\) 9.43841 0.486103
\(378\) 0 0
\(379\) 20.5783 1.05703 0.528517 0.848922i \(-0.322748\pi\)
0.528517 + 0.848922i \(0.322748\pi\)
\(380\) 7.47301 0.383357
\(381\) 0 0
\(382\) −5.42655 −0.277646
\(383\) −21.8802 −1.11803 −0.559013 0.829159i \(-0.688820\pi\)
−0.559013 + 0.829159i \(0.688820\pi\)
\(384\) 0 0
\(385\) −1.50761 −0.0768351
\(386\) −1.20359 −0.0612609
\(387\) 0 0
\(388\) 21.5976 1.09645
\(389\) 6.29921 0.319383 0.159691 0.987167i \(-0.448950\pi\)
0.159691 + 0.987167i \(0.448950\pi\)
\(390\) 0 0
\(391\) 23.1697 1.17174
\(392\) 5.70224 0.288006
\(393\) 0 0
\(394\) 10.9252 0.550403
\(395\) 23.4134 1.17806
\(396\) 0 0
\(397\) −12.1198 −0.608276 −0.304138 0.952628i \(-0.598368\pi\)
−0.304138 + 0.952628i \(0.598368\pi\)
\(398\) 20.6558 1.03538
\(399\) 0 0
\(400\) −28.9315 −1.44657
\(401\) 16.3338 0.815672 0.407836 0.913055i \(-0.366284\pi\)
0.407836 + 0.913055i \(0.366284\pi\)
\(402\) 0 0
\(403\) 18.6724 0.930141
\(404\) 0.407170 0.0202575
\(405\) 0 0
\(406\) 8.34278 0.414045
\(407\) −0.501348 −0.0248509
\(408\) 0 0
\(409\) 37.0665 1.83282 0.916410 0.400240i \(-0.131073\pi\)
0.916410 + 0.400240i \(0.131073\pi\)
\(410\) 46.4287 2.29295
\(411\) 0 0
\(412\) −21.9612 −1.08195
\(413\) −1.13919 −0.0560561
\(414\) 0 0
\(415\) −43.4689 −2.13380
\(416\) −16.5616 −0.811999
\(417\) 0 0
\(418\) −1.13919 −0.0557198
\(419\) −14.6468 −0.715543 −0.357772 0.933809i \(-0.616463\pi\)
−0.357772 + 0.933809i \(0.616463\pi\)
\(420\) 0 0
\(421\) −3.32340 −0.161973 −0.0809864 0.996715i \(-0.525807\pi\)
−0.0809864 + 0.996715i \(0.525807\pi\)
\(422\) −25.0796 −1.22086
\(423\) 0 0
\(424\) 10.7666 0.522874
\(425\) −42.9765 −2.08467
\(426\) 0 0
\(427\) 1.12733 0.0545555
\(428\) −1.70975 −0.0826439
\(429\) 0 0
\(430\) −59.9363 −2.89038
\(431\) −2.73202 −0.131597 −0.0657985 0.997833i \(-0.520959\pi\)
−0.0657985 + 0.997833i \(0.520959\pi\)
\(432\) 0 0
\(433\) −36.1038 −1.73504 −0.867519 0.497404i \(-0.834287\pi\)
−0.867519 + 0.497404i \(0.834287\pi\)
\(434\) 16.5049 0.792261
\(435\) 0 0
\(436\) −7.15297 −0.342565
\(437\) −5.01041 −0.239681
\(438\) 0 0
\(439\) −37.9571 −1.81159 −0.905797 0.423712i \(-0.860727\pi\)
−0.905797 + 0.423712i \(0.860727\pi\)
\(440\) 1.32340 0.0630908
\(441\) 0 0
\(442\) −31.7264 −1.50907
\(443\) −1.84625 −0.0877179 −0.0438589 0.999038i \(-0.513965\pi\)
−0.0438589 + 0.999038i \(0.513965\pi\)
\(444\) 0 0
\(445\) 54.6129 2.58890
\(446\) 31.1440 1.47471
\(447\) 0 0
\(448\) −3.73472 −0.176449
\(449\) −27.9494 −1.31901 −0.659507 0.751699i \(-0.729235\pi\)
−0.659507 + 0.751699i \(0.729235\pi\)
\(450\) 0 0
\(451\) −2.98959 −0.140774
\(452\) 22.7625 1.07066
\(453\) 0 0
\(454\) 26.1157 1.22567
\(455\) 9.07962 0.425659
\(456\) 0 0
\(457\) −23.2501 −1.08759 −0.543796 0.839218i \(-0.683013\pi\)
−0.543796 + 0.839218i \(0.683013\pi\)
\(458\) −22.7846 −1.06465
\(459\) 0 0
\(460\) −15.8414 −0.738611
\(461\) −20.4447 −0.952203 −0.476102 0.879390i \(-0.657951\pi\)
−0.476102 + 0.879390i \(0.657951\pi\)
\(462\) 0 0
\(463\) −12.6122 −0.586139 −0.293069 0.956091i \(-0.594677\pi\)
−0.293069 + 0.956091i \(0.594677\pi\)
\(464\) −18.8358 −0.874432
\(465\) 0 0
\(466\) −3.48197 −0.161299
\(467\) 1.89204 0.0875533 0.0437766 0.999041i \(-0.486061\pi\)
0.0437766 + 0.999041i \(0.486061\pi\)
\(468\) 0 0
\(469\) 6.21110 0.286802
\(470\) −53.8719 −2.48492
\(471\) 0 0
\(472\) 1.00000 0.0460287
\(473\) 3.85936 0.177453
\(474\) 0 0
\(475\) 9.29362 0.426420
\(476\) −11.8456 −0.542941
\(477\) 0 0
\(478\) 26.0346 1.19080
\(479\) 3.51658 0.160677 0.0803383 0.996768i \(-0.474400\pi\)
0.0803383 + 0.996768i \(0.474400\pi\)
\(480\) 0 0
\(481\) 3.01938 0.137672
\(482\) 51.2340 2.33365
\(483\) 0 0
\(484\) −15.8567 −0.720757
\(485\) 49.0755 2.22840
\(486\) 0 0
\(487\) 21.2203 0.961582 0.480791 0.876835i \(-0.340350\pi\)
0.480791 + 0.876835i \(0.340350\pi\)
\(488\) −0.989588 −0.0447965
\(489\) 0 0
\(490\) −35.2638 −1.59306
\(491\) −12.4536 −0.562025 −0.281012 0.959704i \(-0.590670\pi\)
−0.281012 + 0.959704i \(0.590670\pi\)
\(492\) 0 0
\(493\) −27.9798 −1.26015
\(494\) 6.86081 0.308682
\(495\) 0 0
\(496\) −37.2638 −1.67320
\(497\) −17.1392 −0.768798
\(498\) 0 0
\(499\) 33.0755 1.48066 0.740331 0.672243i \(-0.234669\pi\)
0.740331 + 0.672243i \(0.234669\pi\)
\(500\) 5.07962 0.227168
\(501\) 0 0
\(502\) 4.86977 0.217348
\(503\) −12.5824 −0.561022 −0.280511 0.959851i \(-0.590504\pi\)
−0.280511 + 0.959851i \(0.590504\pi\)
\(504\) 0 0
\(505\) 0.925197 0.0411707
\(506\) 2.41489 0.107355
\(507\) 0 0
\(508\) 6.43281 0.285410
\(509\) −28.0465 −1.24314 −0.621569 0.783360i \(-0.713504\pi\)
−0.621569 + 0.783360i \(0.713504\pi\)
\(510\) 0 0
\(511\) 5.39069 0.238470
\(512\) 23.4793 1.03765
\(513\) 0 0
\(514\) −37.9404 −1.67348
\(515\) −49.9017 −2.19893
\(516\) 0 0
\(517\) 3.46886 0.152560
\(518\) 2.66888 0.117264
\(519\) 0 0
\(520\) −7.97021 −0.349517
\(521\) 37.7956 1.65586 0.827928 0.560834i \(-0.189519\pi\)
0.827928 + 0.560834i \(0.189519\pi\)
\(522\) 0 0
\(523\) −17.1946 −0.751868 −0.375934 0.926646i \(-0.622678\pi\)
−0.375934 + 0.926646i \(0.622678\pi\)
\(524\) −25.8325 −1.12850
\(525\) 0 0
\(526\) 44.3899 1.93549
\(527\) −55.3539 −2.41125
\(528\) 0 0
\(529\) −12.3788 −0.538210
\(530\) −66.5831 −2.89218
\(531\) 0 0
\(532\) 2.56159 0.111059
\(533\) 18.0048 0.779875
\(534\) 0 0
\(535\) −3.88500 −0.167963
\(536\) −5.45219 −0.235499
\(537\) 0 0
\(538\) 52.6606 2.27036
\(539\) 2.27067 0.0978048
\(540\) 0 0
\(541\) −3.94939 −0.169797 −0.0848987 0.996390i \(-0.527057\pi\)
−0.0848987 + 0.996390i \(0.527057\pi\)
\(542\) −43.5679 −1.87140
\(543\) 0 0
\(544\) 49.0963 2.10499
\(545\) −16.2534 −0.696220
\(546\) 0 0
\(547\) 17.1094 0.731545 0.365773 0.930704i \(-0.380805\pi\)
0.365773 + 0.930704i \(0.380805\pi\)
\(548\) 15.7473 0.672689
\(549\) 0 0
\(550\) −4.47928 −0.190997
\(551\) 6.05061 0.257765
\(552\) 0 0
\(553\) 8.02564 0.341285
\(554\) 29.1994 1.24057
\(555\) 0 0
\(556\) 14.3636 0.609152
\(557\) −3.29362 −0.139555 −0.0697775 0.997563i \(-0.522229\pi\)
−0.0697775 + 0.997563i \(0.522229\pi\)
\(558\) 0 0
\(559\) −23.2430 −0.983074
\(560\) −18.1198 −0.765702
\(561\) 0 0
\(562\) 10.4793 0.442042
\(563\) −3.10459 −0.130843 −0.0654214 0.997858i \(-0.520839\pi\)
−0.0654214 + 0.997858i \(0.520839\pi\)
\(564\) 0 0
\(565\) 51.7223 2.17597
\(566\) 20.1544 0.847154
\(567\) 0 0
\(568\) 15.0450 0.631275
\(569\) −14.4024 −0.603778 −0.301889 0.953343i \(-0.597617\pi\)
−0.301889 + 0.953343i \(0.597617\pi\)
\(570\) 0 0
\(571\) −34.1198 −1.42787 −0.713935 0.700212i \(-0.753089\pi\)
−0.713935 + 0.700212i \(0.753089\pi\)
\(572\) −1.39676 −0.0584014
\(573\) 0 0
\(574\) 15.9148 0.664270
\(575\) −19.7008 −0.821580
\(576\) 0 0
\(577\) −43.6143 −1.81569 −0.907844 0.419308i \(-0.862273\pi\)
−0.907844 + 0.419308i \(0.862273\pi\)
\(578\) 62.4183 2.59626
\(579\) 0 0
\(580\) 19.1302 0.794340
\(581\) −14.9002 −0.618166
\(582\) 0 0
\(583\) 4.28735 0.177564
\(584\) −4.73202 −0.195813
\(585\) 0 0
\(586\) −63.2597 −2.61323
\(587\) −10.6170 −0.438211 −0.219106 0.975701i \(-0.570314\pi\)
−0.219106 + 0.975701i \(0.570314\pi\)
\(588\) 0 0
\(589\) 11.9702 0.493224
\(590\) −6.18421 −0.254600
\(591\) 0 0
\(592\) −6.02564 −0.247652
\(593\) 0.621168 0.0255083 0.0127541 0.999919i \(-0.495940\pi\)
0.0127541 + 0.999919i \(0.495940\pi\)
\(594\) 0 0
\(595\) −26.9162 −1.10346
\(596\) 7.34133 0.300713
\(597\) 0 0
\(598\) −14.5437 −0.594735
\(599\) −3.01523 −0.123199 −0.0615995 0.998101i \(-0.519620\pi\)
−0.0615995 + 0.998101i \(0.519620\pi\)
\(600\) 0 0
\(601\) −6.24378 −0.254689 −0.127345 0.991859i \(-0.540645\pi\)
−0.127345 + 0.991859i \(0.540645\pi\)
\(602\) −20.5449 −0.837348
\(603\) 0 0
\(604\) 10.7618 0.437892
\(605\) −36.0305 −1.46485
\(606\) 0 0
\(607\) −0.526989 −0.0213898 −0.0106949 0.999943i \(-0.503404\pi\)
−0.0106949 + 0.999943i \(0.503404\pi\)
\(608\) −10.6170 −0.430577
\(609\) 0 0
\(610\) 6.11982 0.247784
\(611\) −20.8913 −0.845170
\(612\) 0 0
\(613\) 44.6564 1.80366 0.901828 0.432094i \(-0.142225\pi\)
0.901828 + 0.432094i \(0.142225\pi\)
\(614\) −7.92250 −0.319726
\(615\) 0 0
\(616\) 0.453636 0.0182775
\(617\) 44.5831 1.79485 0.897424 0.441170i \(-0.145436\pi\)
0.897424 + 0.441170i \(0.145436\pi\)
\(618\) 0 0
\(619\) −39.6531 −1.59379 −0.796896 0.604117i \(-0.793526\pi\)
−0.796896 + 0.604117i \(0.793526\pi\)
\(620\) 37.8462 1.51994
\(621\) 0 0
\(622\) −40.0769 −1.60694
\(623\) 18.7202 0.750007
\(624\) 0 0
\(625\) −18.6829 −0.747314
\(626\) −7.37883 −0.294917
\(627\) 0 0
\(628\) −2.53114 −0.101003
\(629\) −8.95084 −0.356893
\(630\) 0 0
\(631\) −6.30818 −0.251125 −0.125562 0.992086i \(-0.540073\pi\)
−0.125562 + 0.992086i \(0.540073\pi\)
\(632\) −7.04502 −0.280236
\(633\) 0 0
\(634\) −14.1067 −0.560249
\(635\) 14.6170 0.580059
\(636\) 0 0
\(637\) −13.6751 −0.541829
\(638\) −2.91623 −0.115455
\(639\) 0 0
\(640\) 25.6274 1.01301
\(641\) −16.5062 −0.651954 −0.325977 0.945378i \(-0.605693\pi\)
−0.325977 + 0.945378i \(0.605693\pi\)
\(642\) 0 0
\(643\) −22.1455 −0.873332 −0.436666 0.899624i \(-0.643841\pi\)
−0.436666 + 0.899624i \(0.643841\pi\)
\(644\) −5.43011 −0.213976
\(645\) 0 0
\(646\) −20.3386 −0.800213
\(647\) 12.4376 0.488974 0.244487 0.969653i \(-0.421381\pi\)
0.244487 + 0.969653i \(0.421381\pi\)
\(648\) 0 0
\(649\) 0.398207 0.0156310
\(650\) 26.9765 1.05810
\(651\) 0 0
\(652\) −12.5526 −0.491599
\(653\) 10.3941 0.406751 0.203376 0.979101i \(-0.434809\pi\)
0.203376 + 0.979101i \(0.434809\pi\)
\(654\) 0 0
\(655\) −58.6981 −2.29352
\(656\) −35.9315 −1.40289
\(657\) 0 0
\(658\) −18.4662 −0.719886
\(659\) 46.5139 1.81192 0.905962 0.423360i \(-0.139149\pi\)
0.905962 + 0.423360i \(0.139149\pi\)
\(660\) 0 0
\(661\) −31.3788 −1.22050 −0.610248 0.792211i \(-0.708930\pi\)
−0.610248 + 0.792211i \(0.708930\pi\)
\(662\) −1.24234 −0.0482847
\(663\) 0 0
\(664\) 13.0796 0.507588
\(665\) 5.82061 0.225713
\(666\) 0 0
\(667\) −12.8262 −0.496633
\(668\) −9.49575 −0.367402
\(669\) 0 0
\(670\) 33.7175 1.30262
\(671\) −0.394061 −0.0152126
\(672\) 0 0
\(673\) 23.2936 0.897903 0.448951 0.893556i \(-0.351798\pi\)
0.448951 + 0.893556i \(0.351798\pi\)
\(674\) 13.9148 0.535977
\(675\) 0 0
\(676\) −10.6018 −0.407761
\(677\) 0.775591 0.0298084 0.0149042 0.999889i \(-0.495256\pi\)
0.0149042 + 0.999889i \(0.495256\pi\)
\(678\) 0 0
\(679\) 16.8221 0.645571
\(680\) 23.6274 0.906071
\(681\) 0 0
\(682\) −5.76932 −0.220919
\(683\) 13.5810 0.519661 0.259831 0.965654i \(-0.416333\pi\)
0.259831 + 0.965654i \(0.416333\pi\)
\(684\) 0 0
\(685\) 35.7819 1.36715
\(686\) −26.9264 −1.02806
\(687\) 0 0
\(688\) 46.3851 1.76842
\(689\) −25.8206 −0.983687
\(690\) 0 0
\(691\) 18.6981 0.711309 0.355654 0.934618i \(-0.384258\pi\)
0.355654 + 0.934618i \(0.384258\pi\)
\(692\) 30.0471 1.14222
\(693\) 0 0
\(694\) 54.0007 2.04984
\(695\) 32.6378 1.23802
\(696\) 0 0
\(697\) −53.3747 −2.02171
\(698\) −31.4003 −1.18852
\(699\) 0 0
\(700\) 10.0721 0.380690
\(701\) 46.7625 1.76619 0.883097 0.469190i \(-0.155454\pi\)
0.883097 + 0.469190i \(0.155454\pi\)
\(702\) 0 0
\(703\) 1.93561 0.0730029
\(704\) 1.30548 0.0492021
\(705\) 0 0
\(706\) 5.54636 0.208740
\(707\) 0.317138 0.0119272
\(708\) 0 0
\(709\) 2.98477 0.112095 0.0560477 0.998428i \(-0.482150\pi\)
0.0560477 + 0.998428i \(0.482150\pi\)
\(710\) −93.0415 −3.49179
\(711\) 0 0
\(712\) −16.4328 −0.615846
\(713\) −25.3747 −0.950289
\(714\) 0 0
\(715\) −3.17380 −0.118693
\(716\) 35.0588 1.31021
\(717\) 0 0
\(718\) 3.44322 0.128500
\(719\) 33.0915 1.23410 0.617052 0.786922i \(-0.288327\pi\)
0.617052 + 0.786922i \(0.288327\pi\)
\(720\) 0 0
\(721\) −17.1053 −0.637033
\(722\) −30.9571 −1.15210
\(723\) 0 0
\(724\) 21.5166 0.799657
\(725\) 23.7908 0.883569
\(726\) 0 0
\(727\) 25.9100 0.960948 0.480474 0.877009i \(-0.340465\pi\)
0.480474 + 0.877009i \(0.340465\pi\)
\(728\) −2.73202 −0.101256
\(729\) 0 0
\(730\) 29.2638 1.08310
\(731\) 68.9031 2.54847
\(732\) 0 0
\(733\) −3.43426 −0.126847 −0.0634237 0.997987i \(-0.520202\pi\)
−0.0634237 + 0.997987i \(0.520202\pi\)
\(734\) 14.8310 0.547423
\(735\) 0 0
\(736\) 22.5062 0.829588
\(737\) −2.17110 −0.0799735
\(738\) 0 0
\(739\) 19.0014 0.698980 0.349490 0.936940i \(-0.386355\pi\)
0.349490 + 0.936940i \(0.386355\pi\)
\(740\) 6.11982 0.224969
\(741\) 0 0
\(742\) −22.8233 −0.837870
\(743\) 13.1607 0.482819 0.241409 0.970423i \(-0.422390\pi\)
0.241409 + 0.970423i \(0.422390\pi\)
\(744\) 0 0
\(745\) 16.6814 0.611160
\(746\) 2.45364 0.0898340
\(747\) 0 0
\(748\) 4.14064 0.151397
\(749\) −1.33170 −0.0486591
\(750\) 0 0
\(751\) −14.6981 −0.536341 −0.268170 0.963371i \(-0.586419\pi\)
−0.268170 + 0.963371i \(0.586419\pi\)
\(752\) 41.6918 1.52034
\(753\) 0 0
\(754\) 17.5630 0.639608
\(755\) 24.4536 0.889959
\(756\) 0 0
\(757\) 3.27984 0.119208 0.0596039 0.998222i \(-0.481016\pi\)
0.0596039 + 0.998222i \(0.481016\pi\)
\(758\) 38.2922 1.39083
\(759\) 0 0
\(760\) −5.10941 −0.185338
\(761\) 15.5810 0.564810 0.282405 0.959295i \(-0.408868\pi\)
0.282405 + 0.959295i \(0.408868\pi\)
\(762\) 0 0
\(763\) −5.57133 −0.201696
\(764\) −4.26528 −0.154312
\(765\) 0 0
\(766\) −40.7148 −1.47108
\(767\) −2.39821 −0.0865943
\(768\) 0 0
\(769\) 11.0402 0.398120 0.199060 0.979987i \(-0.436211\pi\)
0.199060 + 0.979987i \(0.436211\pi\)
\(770\) −2.80538 −0.101099
\(771\) 0 0
\(772\) −0.946021 −0.0340480
\(773\) −39.8600 −1.43367 −0.716833 0.697245i \(-0.754409\pi\)
−0.716833 + 0.697245i \(0.754409\pi\)
\(774\) 0 0
\(775\) 47.0665 1.69068
\(776\) −14.7666 −0.530091
\(777\) 0 0
\(778\) 11.7216 0.420240
\(779\) 11.5422 0.413543
\(780\) 0 0
\(781\) 5.99104 0.214376
\(782\) 43.1142 1.54176
\(783\) 0 0
\(784\) 27.2909 0.974676
\(785\) −5.75140 −0.205276
\(786\) 0 0
\(787\) 24.0692 0.857975 0.428987 0.903311i \(-0.358871\pi\)
0.428987 + 0.903311i \(0.358871\pi\)
\(788\) 8.58723 0.305908
\(789\) 0 0
\(790\) 43.5679 1.55007
\(791\) 17.7293 0.630382
\(792\) 0 0
\(793\) 2.37324 0.0842761
\(794\) −22.5526 −0.800363
\(795\) 0 0
\(796\) 16.2355 0.575452
\(797\) −10.3088 −0.365158 −0.182579 0.983191i \(-0.558445\pi\)
−0.182579 + 0.983191i \(0.558445\pi\)
\(798\) 0 0
\(799\) 61.9315 2.19098
\(800\) −41.7458 −1.47594
\(801\) 0 0
\(802\) 30.3941 1.07325
\(803\) −1.88433 −0.0664965
\(804\) 0 0
\(805\) −12.3386 −0.434880
\(806\) 34.7458 1.22387
\(807\) 0 0
\(808\) −0.278388 −0.00979367
\(809\) −24.7119 −0.868823 −0.434412 0.900715i \(-0.643044\pi\)
−0.434412 + 0.900715i \(0.643044\pi\)
\(810\) 0 0
\(811\) −16.3088 −0.572681 −0.286341 0.958128i \(-0.592439\pi\)
−0.286341 + 0.958128i \(0.592439\pi\)
\(812\) 6.55745 0.230121
\(813\) 0 0
\(814\) −0.932912 −0.0326986
\(815\) −28.5228 −0.999112
\(816\) 0 0
\(817\) −14.9002 −0.521293
\(818\) 68.9736 2.41160
\(819\) 0 0
\(820\) 36.4931 1.27439
\(821\) 39.5076 1.37883 0.689413 0.724369i \(-0.257869\pi\)
0.689413 + 0.724369i \(0.257869\pi\)
\(822\) 0 0
\(823\) −0.655771 −0.0228588 −0.0114294 0.999935i \(-0.503638\pi\)
−0.0114294 + 0.999935i \(0.503638\pi\)
\(824\) 15.0152 0.523080
\(825\) 0 0
\(826\) −2.11982 −0.0737579
\(827\) −33.5035 −1.16503 −0.582515 0.812820i \(-0.697931\pi\)
−0.582515 + 0.812820i \(0.697931\pi\)
\(828\) 0 0
\(829\) 28.1752 0.978567 0.489283 0.872125i \(-0.337258\pi\)
0.489283 + 0.872125i \(0.337258\pi\)
\(830\) −80.8871 −2.80763
\(831\) 0 0
\(832\) −7.86226 −0.272575
\(833\) 40.5395 1.40461
\(834\) 0 0
\(835\) −21.5768 −0.746697
\(836\) −0.895410 −0.0309684
\(837\) 0 0
\(838\) −27.2549 −0.941504
\(839\) 21.9315 0.757158 0.378579 0.925569i \(-0.376413\pi\)
0.378579 + 0.925569i \(0.376413\pi\)
\(840\) 0 0
\(841\) −13.5110 −0.465896
\(842\) −6.18421 −0.213122
\(843\) 0 0
\(844\) −19.7126 −0.678537
\(845\) −24.0900 −0.828722
\(846\) 0 0
\(847\) −12.3505 −0.424368
\(848\) 51.5291 1.76952
\(849\) 0 0
\(850\) −79.9709 −2.74298
\(851\) −4.10314 −0.140654
\(852\) 0 0
\(853\) −22.9903 −0.787171 −0.393586 0.919288i \(-0.628766\pi\)
−0.393586 + 0.919288i \(0.628766\pi\)
\(854\) 2.09775 0.0717834
\(855\) 0 0
\(856\) 1.16898 0.0399550
\(857\) 16.5574 0.565592 0.282796 0.959180i \(-0.408738\pi\)
0.282796 + 0.959180i \(0.408738\pi\)
\(858\) 0 0
\(859\) −3.50280 −0.119514 −0.0597570 0.998213i \(-0.519033\pi\)
−0.0597570 + 0.998213i \(0.519033\pi\)
\(860\) −47.1101 −1.60644
\(861\) 0 0
\(862\) −5.08377 −0.173154
\(863\) 19.1994 0.653557 0.326778 0.945101i \(-0.394037\pi\)
0.326778 + 0.945101i \(0.394037\pi\)
\(864\) 0 0
\(865\) 68.2749 2.32142
\(866\) −67.1822 −2.28294
\(867\) 0 0
\(868\) 12.9729 0.440329
\(869\) −2.80538 −0.0951659
\(870\) 0 0
\(871\) 13.0755 0.443046
\(872\) 4.89059 0.165616
\(873\) 0 0
\(874\) −9.32340 −0.315369
\(875\) 3.95643 0.133752
\(876\) 0 0
\(877\) 52.8864 1.78585 0.892924 0.450207i \(-0.148650\pi\)
0.892924 + 0.450207i \(0.148650\pi\)
\(878\) −70.6308 −2.38367
\(879\) 0 0
\(880\) 6.33382 0.213513
\(881\) 6.86562 0.231309 0.115654 0.993290i \(-0.463104\pi\)
0.115654 + 0.993290i \(0.463104\pi\)
\(882\) 0 0
\(883\) −51.6233 −1.73726 −0.868631 0.495460i \(-0.835000\pi\)
−0.868631 + 0.495460i \(0.835000\pi\)
\(884\) −24.9371 −0.838724
\(885\) 0 0
\(886\) −3.43551 −0.115418
\(887\) −19.7126 −0.661886 −0.330943 0.943651i \(-0.607367\pi\)
−0.330943 + 0.943651i \(0.607367\pi\)
\(888\) 0 0
\(889\) 5.01041 0.168044
\(890\) 101.624 3.40644
\(891\) 0 0
\(892\) 24.4793 0.819627
\(893\) −13.3926 −0.448167
\(894\) 0 0
\(895\) 79.6627 2.66283
\(896\) 8.78455 0.293471
\(897\) 0 0
\(898\) −52.0084 −1.73554
\(899\) 30.6427 1.02199
\(900\) 0 0
\(901\) 76.5443 2.55006
\(902\) −5.56304 −0.185229
\(903\) 0 0
\(904\) −15.5630 −0.517619
\(905\) 48.8913 1.62520
\(906\) 0 0
\(907\) −40.5366 −1.34600 −0.672998 0.739644i \(-0.734994\pi\)
−0.672998 + 0.739644i \(0.734994\pi\)
\(908\) 20.5270 0.681212
\(909\) 0 0
\(910\) 16.8954 0.560077
\(911\) −54.5949 −1.80881 −0.904406 0.426674i \(-0.859685\pi\)
−0.904406 + 0.426674i \(0.859685\pi\)
\(912\) 0 0
\(913\) 5.20840 0.172373
\(914\) −43.2638 −1.43104
\(915\) 0 0
\(916\) −17.9087 −0.591721
\(917\) −20.1205 −0.664437
\(918\) 0 0
\(919\) −6.95565 −0.229446 −0.114723 0.993398i \(-0.536598\pi\)
−0.114723 + 0.993398i \(0.536598\pi\)
\(920\) 10.8310 0.357088
\(921\) 0 0
\(922\) −38.0436 −1.25290
\(923\) −36.0811 −1.18762
\(924\) 0 0
\(925\) 7.61076 0.250240
\(926\) −23.4689 −0.771235
\(927\) 0 0
\(928\) −27.1786 −0.892182
\(929\) −16.5478 −0.542916 −0.271458 0.962450i \(-0.587506\pi\)
−0.271458 + 0.962450i \(0.587506\pi\)
\(930\) 0 0
\(931\) −8.76663 −0.287315
\(932\) −2.73684 −0.0896482
\(933\) 0 0
\(934\) 3.52072 0.115202
\(935\) 9.40862 0.307695
\(936\) 0 0
\(937\) −28.6766 −0.936824 −0.468412 0.883510i \(-0.655174\pi\)
−0.468412 + 0.883510i \(0.655174\pi\)
\(938\) 11.5576 0.377371
\(939\) 0 0
\(940\) −42.3434 −1.38109
\(941\) 24.1198 0.786284 0.393142 0.919478i \(-0.371388\pi\)
0.393142 + 0.919478i \(0.371388\pi\)
\(942\) 0 0
\(943\) −24.4674 −0.796769
\(944\) 4.78600 0.155771
\(945\) 0 0
\(946\) 7.18151 0.233491
\(947\) −33.0617 −1.07436 −0.537180 0.843467i \(-0.680511\pi\)
−0.537180 + 0.843467i \(0.680511\pi\)
\(948\) 0 0
\(949\) 11.3484 0.368384
\(950\) 17.2936 0.561079
\(951\) 0 0
\(952\) 8.09899 0.262490
\(953\) 6.79641 0.220157 0.110079 0.993923i \(-0.464890\pi\)
0.110079 + 0.993923i \(0.464890\pi\)
\(954\) 0 0
\(955\) −9.69182 −0.313620
\(956\) 20.4633 0.661829
\(957\) 0 0
\(958\) 6.54367 0.211416
\(959\) 12.2653 0.396067
\(960\) 0 0
\(961\) 29.6218 0.955543
\(962\) 5.61847 0.181147
\(963\) 0 0
\(964\) 40.2701 1.29701
\(965\) −2.14961 −0.0691983
\(966\) 0 0
\(967\) −43.7479 −1.40684 −0.703419 0.710775i \(-0.748344\pi\)
−0.703419 + 0.710775i \(0.748344\pi\)
\(968\) 10.8414 0.348457
\(969\) 0 0
\(970\) 91.3199 2.93211
\(971\) −30.3434 −0.973768 −0.486884 0.873467i \(-0.661866\pi\)
−0.486884 + 0.873467i \(0.661866\pi\)
\(972\) 0 0
\(973\) 11.1876 0.358657
\(974\) 39.4868 1.26524
\(975\) 0 0
\(976\) −4.73617 −0.151601
\(977\) −47.2638 −1.51210 −0.756052 0.654512i \(-0.772874\pi\)
−0.756052 + 0.654512i \(0.772874\pi\)
\(978\) 0 0
\(979\) −6.54367 −0.209137
\(980\) −27.7175 −0.885402
\(981\) 0 0
\(982\) −23.1738 −0.739506
\(983\) −30.9854 −0.988282 −0.494141 0.869382i \(-0.664517\pi\)
−0.494141 + 0.869382i \(0.664517\pi\)
\(984\) 0 0
\(985\) 19.5124 0.621718
\(986\) −52.0651 −1.65809
\(987\) 0 0
\(988\) 5.39261 0.171562
\(989\) 31.5858 1.00437
\(990\) 0 0
\(991\) −18.8131 −0.597618 −0.298809 0.954313i \(-0.596589\pi\)
−0.298809 + 0.954313i \(0.596589\pi\)
\(992\) −53.7687 −1.70716
\(993\) 0 0
\(994\) −31.8927 −1.01158
\(995\) 36.8913 1.16953
\(996\) 0 0
\(997\) 16.5020 0.522624 0.261312 0.965254i \(-0.415845\pi\)
0.261312 + 0.965254i \(0.415845\pi\)
\(998\) 61.5470 1.94824
\(999\) 0 0
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 531.2.a.d.1.3 3
3.2 odd 2 177.2.a.d.1.1 3
4.3 odd 2 8496.2.a.bl.1.3 3
12.11 even 2 2832.2.a.t.1.1 3
15.14 odd 2 4425.2.a.w.1.3 3
21.20 even 2 8673.2.a.s.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.2.a.d.1.1 3 3.2 odd 2
531.2.a.d.1.3 3 1.1 even 1 trivial
2832.2.a.t.1.1 3 12.11 even 2
4425.2.a.w.1.3 3 15.14 odd 2
8496.2.a.bl.1.3 3 4.3 odd 2
8673.2.a.s.1.1 3 21.20 even 2