Properties

Label 483.2.a.g.1.2
Level $483$
Weight $2$
Character 483.1
Self dual yes
Analytic conductor $3.857$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [483,2,Mod(1,483)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(483, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("483.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 483 = 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 483.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: \(3.85677441763\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 483.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.61803 q^{2} -1.00000 q^{3} +0.618034 q^{4} -2.61803 q^{5} -1.61803 q^{6} +1.00000 q^{7} -2.23607 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.61803 q^{2} -1.00000 q^{3} +0.618034 q^{4} -2.61803 q^{5} -1.61803 q^{6} +1.00000 q^{7} -2.23607 q^{8} +1.00000 q^{9} -4.23607 q^{10} -2.23607 q^{11} -0.618034 q^{12} -6.85410 q^{13} +1.61803 q^{14} +2.61803 q^{15} -4.85410 q^{16} +3.47214 q^{17} +1.61803 q^{18} -3.76393 q^{19} -1.61803 q^{20} -1.00000 q^{21} -3.61803 q^{22} +1.00000 q^{23} +2.23607 q^{24} +1.85410 q^{25} -11.0902 q^{26} -1.00000 q^{27} +0.618034 q^{28} +10.7082 q^{29} +4.23607 q^{30} -5.00000 q^{31} -3.38197 q^{32} +2.23607 q^{33} +5.61803 q^{34} -2.61803 q^{35} +0.618034 q^{36} -9.47214 q^{37} -6.09017 q^{38} +6.85410 q^{39} +5.85410 q^{40} -4.70820 q^{41} -1.61803 q^{42} +7.32624 q^{43} -1.38197 q^{44} -2.61803 q^{45} +1.61803 q^{46} +0.763932 q^{47} +4.85410 q^{48} +1.00000 q^{49} +3.00000 q^{50} -3.47214 q^{51} -4.23607 q^{52} -1.38197 q^{53} -1.61803 q^{54} +5.85410 q^{55} -2.23607 q^{56} +3.76393 q^{57} +17.3262 q^{58} +1.85410 q^{59} +1.61803 q^{60} +2.09017 q^{61} -8.09017 q^{62} +1.00000 q^{63} +4.23607 q^{64} +17.9443 q^{65} +3.61803 q^{66} -0.145898 q^{67} +2.14590 q^{68} -1.00000 q^{69} -4.23607 q^{70} +11.6180 q^{71} -2.23607 q^{72} +5.47214 q^{73} -15.3262 q^{74} -1.85410 q^{75} -2.32624 q^{76} -2.23607 q^{77} +11.0902 q^{78} -11.9443 q^{79} +12.7082 q^{80} +1.00000 q^{81} -7.61803 q^{82} -9.94427 q^{83} -0.618034 q^{84} -9.09017 q^{85} +11.8541 q^{86} -10.7082 q^{87} +5.00000 q^{88} -0.673762 q^{89} -4.23607 q^{90} -6.85410 q^{91} +0.618034 q^{92} +5.00000 q^{93} +1.23607 q^{94} +9.85410 q^{95} +3.38197 q^{96} +5.18034 q^{97} +1.61803 q^{98} -2.23607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - 2 q^{3} - q^{4} - 3 q^{5} - q^{6} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - 2 q^{3} - q^{4} - 3 q^{5} - q^{6} + 2 q^{7} + 2 q^{9} - 4 q^{10} + q^{12} - 7 q^{13} + q^{14} + 3 q^{15} - 3 q^{16} - 2 q^{17} + q^{18} - 12 q^{19} - q^{20} - 2 q^{21} - 5 q^{22} + 2 q^{23} - 3 q^{25} - 11 q^{26} - 2 q^{27} - q^{28} + 8 q^{29} + 4 q^{30} - 10 q^{31} - 9 q^{32} + 9 q^{34} - 3 q^{35} - q^{36} - 10 q^{37} - q^{38} + 7 q^{39} + 5 q^{40} + 4 q^{41} - q^{42} - q^{43} - 5 q^{44} - 3 q^{45} + q^{46} + 6 q^{47} + 3 q^{48} + 2 q^{49} + 6 q^{50} + 2 q^{51} - 4 q^{52} - 5 q^{53} - q^{54} + 5 q^{55} + 12 q^{57} + 19 q^{58} - 3 q^{59} + q^{60} - 7 q^{61} - 5 q^{62} + 2 q^{63} + 4 q^{64} + 18 q^{65} + 5 q^{66} - 7 q^{67} + 11 q^{68} - 2 q^{69} - 4 q^{70} + 21 q^{71} + 2 q^{73} - 15 q^{74} + 3 q^{75} + 11 q^{76} + 11 q^{78} - 6 q^{79} + 12 q^{80} + 2 q^{81} - 13 q^{82} - 2 q^{83} + q^{84} - 7 q^{85} + 17 q^{86} - 8 q^{87} + 10 q^{88} - 17 q^{89} - 4 q^{90} - 7 q^{91} - q^{92} + 10 q^{93} - 2 q^{94} + 13 q^{95} + 9 q^{96} - 12 q^{97} + q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.61803 1.14412 0.572061 0.820211i \(-0.306144\pi\)
0.572061 + 0.820211i \(0.306144\pi\)
\(3\) −1.00000 −0.577350
\(4\) 0.618034 0.309017
\(5\) −2.61803 −1.17082 −0.585410 0.810737i \(-0.699067\pi\)
−0.585410 + 0.810737i \(0.699067\pi\)
\(6\) −1.61803 −0.660560
\(7\) 1.00000 0.377964
\(8\) −2.23607 −0.790569
\(9\) 1.00000 0.333333
\(10\) −4.23607 −1.33956
\(11\) −2.23607 −0.674200 −0.337100 0.941469i \(-0.609446\pi\)
−0.337100 + 0.941469i \(0.609446\pi\)
\(12\) −0.618034 −0.178411
\(13\) −6.85410 −1.90099 −0.950493 0.310746i \(-0.899421\pi\)
−0.950493 + 0.310746i \(0.899421\pi\)
\(14\) 1.61803 0.432438
\(15\) 2.61803 0.675973
\(16\) −4.85410 −1.21353
\(17\) 3.47214 0.842117 0.421058 0.907034i \(-0.361659\pi\)
0.421058 + 0.907034i \(0.361659\pi\)
\(18\) 1.61803 0.381374
\(19\) −3.76393 −0.863505 −0.431753 0.901992i \(-0.642105\pi\)
−0.431753 + 0.901992i \(0.642105\pi\)
\(20\) −1.61803 −0.361803
\(21\) −1.00000 −0.218218
\(22\) −3.61803 −0.771367
\(23\) 1.00000 0.208514
\(24\) 2.23607 0.456435
\(25\) 1.85410 0.370820
\(26\) −11.0902 −2.17496
\(27\) −1.00000 −0.192450
\(28\) 0.618034 0.116797
\(29\) 10.7082 1.98846 0.994232 0.107253i \(-0.0342054\pi\)
0.994232 + 0.107253i \(0.0342054\pi\)
\(30\) 4.23607 0.773397
\(31\) −5.00000 −0.898027 −0.449013 0.893525i \(-0.648224\pi\)
−0.449013 + 0.893525i \(0.648224\pi\)
\(32\) −3.38197 −0.597853
\(33\) 2.23607 0.389249
\(34\) 5.61803 0.963485
\(35\) −2.61803 −0.442529
\(36\) 0.618034 0.103006
\(37\) −9.47214 −1.55721 −0.778605 0.627515i \(-0.784072\pi\)
−0.778605 + 0.627515i \(0.784072\pi\)
\(38\) −6.09017 −0.987956
\(39\) 6.85410 1.09753
\(40\) 5.85410 0.925615
\(41\) −4.70820 −0.735298 −0.367649 0.929965i \(-0.619837\pi\)
−0.367649 + 0.929965i \(0.619837\pi\)
\(42\) −1.61803 −0.249668
\(43\) 7.32624 1.11724 0.558620 0.829423i \(-0.311331\pi\)
0.558620 + 0.829423i \(0.311331\pi\)
\(44\) −1.38197 −0.208339
\(45\) −2.61803 −0.390273
\(46\) 1.61803 0.238566
\(47\) 0.763932 0.111431 0.0557155 0.998447i \(-0.482256\pi\)
0.0557155 + 0.998447i \(0.482256\pi\)
\(48\) 4.85410 0.700629
\(49\) 1.00000 0.142857
\(50\) 3.00000 0.424264
\(51\) −3.47214 −0.486196
\(52\) −4.23607 −0.587437
\(53\) −1.38197 −0.189828 −0.0949138 0.995485i \(-0.530258\pi\)
−0.0949138 + 0.995485i \(0.530258\pi\)
\(54\) −1.61803 −0.220187
\(55\) 5.85410 0.789367
\(56\) −2.23607 −0.298807
\(57\) 3.76393 0.498545
\(58\) 17.3262 2.27505
\(59\) 1.85410 0.241384 0.120692 0.992690i \(-0.461489\pi\)
0.120692 + 0.992690i \(0.461489\pi\)
\(60\) 1.61803 0.208887
\(61\) 2.09017 0.267619 0.133809 0.991007i \(-0.457279\pi\)
0.133809 + 0.991007i \(0.457279\pi\)
\(62\) −8.09017 −1.02745
\(63\) 1.00000 0.125988
\(64\) 4.23607 0.529508
\(65\) 17.9443 2.22571
\(66\) 3.61803 0.445349
\(67\) −0.145898 −0.0178243 −0.00891214 0.999960i \(-0.502837\pi\)
−0.00891214 + 0.999960i \(0.502837\pi\)
\(68\) 2.14590 0.260228
\(69\) −1.00000 −0.120386
\(70\) −4.23607 −0.506307
\(71\) 11.6180 1.37881 0.689403 0.724378i \(-0.257873\pi\)
0.689403 + 0.724378i \(0.257873\pi\)
\(72\) −2.23607 −0.263523
\(73\) 5.47214 0.640465 0.320233 0.947339i \(-0.396239\pi\)
0.320233 + 0.947339i \(0.396239\pi\)
\(74\) −15.3262 −1.78164
\(75\) −1.85410 −0.214093
\(76\) −2.32624 −0.266838
\(77\) −2.23607 −0.254824
\(78\) 11.0902 1.25571
\(79\) −11.9443 −1.34384 −0.671918 0.740626i \(-0.734529\pi\)
−0.671918 + 0.740626i \(0.734529\pi\)
\(80\) 12.7082 1.42082
\(81\) 1.00000 0.111111
\(82\) −7.61803 −0.841271
\(83\) −9.94427 −1.09153 −0.545763 0.837940i \(-0.683760\pi\)
−0.545763 + 0.837940i \(0.683760\pi\)
\(84\) −0.618034 −0.0674330
\(85\) −9.09017 −0.985967
\(86\) 11.8541 1.27826
\(87\) −10.7082 −1.14804
\(88\) 5.00000 0.533002
\(89\) −0.673762 −0.0714186 −0.0357093 0.999362i \(-0.511369\pi\)
−0.0357093 + 0.999362i \(0.511369\pi\)
\(90\) −4.23607 −0.446521
\(91\) −6.85410 −0.718505
\(92\) 0.618034 0.0644345
\(93\) 5.00000 0.518476
\(94\) 1.23607 0.127491
\(95\) 9.85410 1.01101
\(96\) 3.38197 0.345170
\(97\) 5.18034 0.525984 0.262992 0.964798i \(-0.415291\pi\)
0.262992 + 0.964798i \(0.415291\pi\)
\(98\) 1.61803 0.163446
\(99\) −2.23607 −0.224733
\(100\) 1.14590 0.114590
\(101\) −4.90983 −0.488546 −0.244273 0.969706i \(-0.578549\pi\)
−0.244273 + 0.969706i \(0.578549\pi\)
\(102\) −5.61803 −0.556268
\(103\) −6.47214 −0.637719 −0.318859 0.947802i \(-0.603300\pi\)
−0.318859 + 0.947802i \(0.603300\pi\)
\(104\) 15.3262 1.50286
\(105\) 2.61803 0.255494
\(106\) −2.23607 −0.217186
\(107\) −0.854102 −0.0825692 −0.0412846 0.999147i \(-0.513145\pi\)
−0.0412846 + 0.999147i \(0.513145\pi\)
\(108\) −0.618034 −0.0594703
\(109\) −14.0344 −1.34426 −0.672128 0.740435i \(-0.734619\pi\)
−0.672128 + 0.740435i \(0.734619\pi\)
\(110\) 9.47214 0.903133
\(111\) 9.47214 0.899055
\(112\) −4.85410 −0.458670
\(113\) −7.09017 −0.666987 −0.333494 0.942752i \(-0.608228\pi\)
−0.333494 + 0.942752i \(0.608228\pi\)
\(114\) 6.09017 0.570397
\(115\) −2.61803 −0.244133
\(116\) 6.61803 0.614469
\(117\) −6.85410 −0.633662
\(118\) 3.00000 0.276172
\(119\) 3.47214 0.318290
\(120\) −5.85410 −0.534404
\(121\) −6.00000 −0.545455
\(122\) 3.38197 0.306189
\(123\) 4.70820 0.424524
\(124\) −3.09017 −0.277505
\(125\) 8.23607 0.736656
\(126\) 1.61803 0.144146
\(127\) −2.67376 −0.237258 −0.118629 0.992939i \(-0.537850\pi\)
−0.118629 + 0.992939i \(0.537850\pi\)
\(128\) 13.6180 1.20368
\(129\) −7.32624 −0.645039
\(130\) 29.0344 2.54649
\(131\) −12.4164 −1.08483 −0.542413 0.840112i \(-0.682489\pi\)
−0.542413 + 0.840112i \(0.682489\pi\)
\(132\) 1.38197 0.120285
\(133\) −3.76393 −0.326374
\(134\) −0.236068 −0.0203932
\(135\) 2.61803 0.225324
\(136\) −7.76393 −0.665752
\(137\) −19.9443 −1.70395 −0.851977 0.523579i \(-0.824596\pi\)
−0.851977 + 0.523579i \(0.824596\pi\)
\(138\) −1.61803 −0.137736
\(139\) 19.8541 1.68400 0.842001 0.539475i \(-0.181377\pi\)
0.842001 + 0.539475i \(0.181377\pi\)
\(140\) −1.61803 −0.136749
\(141\) −0.763932 −0.0643347
\(142\) 18.7984 1.57752
\(143\) 15.3262 1.28164
\(144\) −4.85410 −0.404508
\(145\) −28.0344 −2.32813
\(146\) 8.85410 0.732771
\(147\) −1.00000 −0.0824786
\(148\) −5.85410 −0.481204
\(149\) −14.6525 −1.20038 −0.600189 0.799858i \(-0.704908\pi\)
−0.600189 + 0.799858i \(0.704908\pi\)
\(150\) −3.00000 −0.244949
\(151\) −1.70820 −0.139012 −0.0695058 0.997582i \(-0.522142\pi\)
−0.0695058 + 0.997582i \(0.522142\pi\)
\(152\) 8.41641 0.682661
\(153\) 3.47214 0.280706
\(154\) −3.61803 −0.291549
\(155\) 13.0902 1.05143
\(156\) 4.23607 0.339157
\(157\) −14.1803 −1.13171 −0.565857 0.824503i \(-0.691455\pi\)
−0.565857 + 0.824503i \(0.691455\pi\)
\(158\) −19.3262 −1.53751
\(159\) 1.38197 0.109597
\(160\) 8.85410 0.699978
\(161\) 1.00000 0.0788110
\(162\) 1.61803 0.127125
\(163\) −22.5066 −1.76285 −0.881426 0.472323i \(-0.843416\pi\)
−0.881426 + 0.472323i \(0.843416\pi\)
\(164\) −2.90983 −0.227220
\(165\) −5.85410 −0.455741
\(166\) −16.0902 −1.24884
\(167\) 11.4721 0.887741 0.443870 0.896091i \(-0.353605\pi\)
0.443870 + 0.896091i \(0.353605\pi\)
\(168\) 2.23607 0.172516
\(169\) 33.9787 2.61375
\(170\) −14.7082 −1.12807
\(171\) −3.76393 −0.287835
\(172\) 4.52786 0.345246
\(173\) −11.2918 −0.858499 −0.429250 0.903186i \(-0.641222\pi\)
−0.429250 + 0.903186i \(0.641222\pi\)
\(174\) −17.3262 −1.31350
\(175\) 1.85410 0.140157
\(176\) 10.8541 0.818159
\(177\) −1.85410 −0.139363
\(178\) −1.09017 −0.0817117
\(179\) 20.0344 1.49744 0.748722 0.662884i \(-0.230668\pi\)
0.748722 + 0.662884i \(0.230668\pi\)
\(180\) −1.61803 −0.120601
\(181\) −15.1803 −1.12835 −0.564173 0.825657i \(-0.690805\pi\)
−0.564173 + 0.825657i \(0.690805\pi\)
\(182\) −11.0902 −0.822058
\(183\) −2.09017 −0.154510
\(184\) −2.23607 −0.164845
\(185\) 24.7984 1.82321
\(186\) 8.09017 0.593200
\(187\) −7.76393 −0.567755
\(188\) 0.472136 0.0344341
\(189\) −1.00000 −0.0727393
\(190\) 15.9443 1.15672
\(191\) 21.7082 1.57075 0.785375 0.619020i \(-0.212470\pi\)
0.785375 + 0.619020i \(0.212470\pi\)
\(192\) −4.23607 −0.305712
\(193\) −18.1803 −1.30865 −0.654325 0.756214i \(-0.727047\pi\)
−0.654325 + 0.756214i \(0.727047\pi\)
\(194\) 8.38197 0.601790
\(195\) −17.9443 −1.28502
\(196\) 0.618034 0.0441453
\(197\) −25.3262 −1.80442 −0.902210 0.431297i \(-0.858056\pi\)
−0.902210 + 0.431297i \(0.858056\pi\)
\(198\) −3.61803 −0.257122
\(199\) −9.56231 −0.677854 −0.338927 0.940813i \(-0.610064\pi\)
−0.338927 + 0.940813i \(0.610064\pi\)
\(200\) −4.14590 −0.293159
\(201\) 0.145898 0.0102909
\(202\) −7.94427 −0.558957
\(203\) 10.7082 0.751569
\(204\) −2.14590 −0.150243
\(205\) 12.3262 0.860902
\(206\) −10.4721 −0.729628
\(207\) 1.00000 0.0695048
\(208\) 33.2705 2.30689
\(209\) 8.41641 0.582175
\(210\) 4.23607 0.292316
\(211\) 15.0000 1.03264 0.516321 0.856395i \(-0.327301\pi\)
0.516321 + 0.856395i \(0.327301\pi\)
\(212\) −0.854102 −0.0586600
\(213\) −11.6180 −0.796055
\(214\) −1.38197 −0.0944693
\(215\) −19.1803 −1.30809
\(216\) 2.23607 0.152145
\(217\) −5.00000 −0.339422
\(218\) −22.7082 −1.53799
\(219\) −5.47214 −0.369773
\(220\) 3.61803 0.243928
\(221\) −23.7984 −1.60085
\(222\) 15.3262 1.02863
\(223\) 18.3262 1.22722 0.613608 0.789611i \(-0.289718\pi\)
0.613608 + 0.789611i \(0.289718\pi\)
\(224\) −3.38197 −0.225967
\(225\) 1.85410 0.123607
\(226\) −11.4721 −0.763115
\(227\) 15.2705 1.01354 0.506770 0.862081i \(-0.330839\pi\)
0.506770 + 0.862081i \(0.330839\pi\)
\(228\) 2.32624 0.154059
\(229\) −5.67376 −0.374933 −0.187466 0.982271i \(-0.560028\pi\)
−0.187466 + 0.982271i \(0.560028\pi\)
\(230\) −4.23607 −0.279318
\(231\) 2.23607 0.147122
\(232\) −23.9443 −1.57202
\(233\) 7.09017 0.464492 0.232246 0.972657i \(-0.425392\pi\)
0.232246 + 0.972657i \(0.425392\pi\)
\(234\) −11.0902 −0.724987
\(235\) −2.00000 −0.130466
\(236\) 1.14590 0.0745916
\(237\) 11.9443 0.775864
\(238\) 5.61803 0.364163
\(239\) −6.61803 −0.428085 −0.214043 0.976824i \(-0.568663\pi\)
−0.214043 + 0.976824i \(0.568663\pi\)
\(240\) −12.7082 −0.820311
\(241\) 28.7082 1.84926 0.924629 0.380869i \(-0.124375\pi\)
0.924629 + 0.380869i \(0.124375\pi\)
\(242\) −9.70820 −0.624067
\(243\) −1.00000 −0.0641500
\(244\) 1.29180 0.0826988
\(245\) −2.61803 −0.167260
\(246\) 7.61803 0.485708
\(247\) 25.7984 1.64151
\(248\) 11.1803 0.709952
\(249\) 9.94427 0.630193
\(250\) 13.3262 0.842825
\(251\) −12.2918 −0.775851 −0.387926 0.921691i \(-0.626808\pi\)
−0.387926 + 0.921691i \(0.626808\pi\)
\(252\) 0.618034 0.0389325
\(253\) −2.23607 −0.140580
\(254\) −4.32624 −0.271452
\(255\) 9.09017 0.569249
\(256\) 13.5623 0.847644
\(257\) 29.7082 1.85315 0.926573 0.376114i \(-0.122740\pi\)
0.926573 + 0.376114i \(0.122740\pi\)
\(258\) −11.8541 −0.738004
\(259\) −9.47214 −0.588570
\(260\) 11.0902 0.687783
\(261\) 10.7082 0.662821
\(262\) −20.0902 −1.24117
\(263\) 5.00000 0.308313 0.154157 0.988046i \(-0.450734\pi\)
0.154157 + 0.988046i \(0.450734\pi\)
\(264\) −5.00000 −0.307729
\(265\) 3.61803 0.222254
\(266\) −6.09017 −0.373412
\(267\) 0.673762 0.0412336
\(268\) −0.0901699 −0.00550801
\(269\) −2.85410 −0.174018 −0.0870088 0.996208i \(-0.527731\pi\)
−0.0870088 + 0.996208i \(0.527731\pi\)
\(270\) 4.23607 0.257799
\(271\) −17.1803 −1.04363 −0.521816 0.853058i \(-0.674745\pi\)
−0.521816 + 0.853058i \(0.674745\pi\)
\(272\) −16.8541 −1.02193
\(273\) 6.85410 0.414829
\(274\) −32.2705 −1.94953
\(275\) −4.14590 −0.250007
\(276\) −0.618034 −0.0372013
\(277\) 21.2148 1.27467 0.637336 0.770586i \(-0.280036\pi\)
0.637336 + 0.770586i \(0.280036\pi\)
\(278\) 32.1246 1.92671
\(279\) −5.00000 −0.299342
\(280\) 5.85410 0.349850
\(281\) −21.5967 −1.28835 −0.644177 0.764876i \(-0.722800\pi\)
−0.644177 + 0.764876i \(0.722800\pi\)
\(282\) −1.23607 −0.0736068
\(283\) 14.3262 0.851606 0.425803 0.904816i \(-0.359992\pi\)
0.425803 + 0.904816i \(0.359992\pi\)
\(284\) 7.18034 0.426075
\(285\) −9.85410 −0.583707
\(286\) 24.7984 1.46636
\(287\) −4.70820 −0.277916
\(288\) −3.38197 −0.199284
\(289\) −4.94427 −0.290840
\(290\) −45.3607 −2.66367
\(291\) −5.18034 −0.303677
\(292\) 3.38197 0.197915
\(293\) −9.52786 −0.556624 −0.278312 0.960491i \(-0.589775\pi\)
−0.278312 + 0.960491i \(0.589775\pi\)
\(294\) −1.61803 −0.0943657
\(295\) −4.85410 −0.282617
\(296\) 21.1803 1.23108
\(297\) 2.23607 0.129750
\(298\) −23.7082 −1.37338
\(299\) −6.85410 −0.396383
\(300\) −1.14590 −0.0661585
\(301\) 7.32624 0.422277
\(302\) −2.76393 −0.159046
\(303\) 4.90983 0.282062
\(304\) 18.2705 1.04789
\(305\) −5.47214 −0.313334
\(306\) 5.61803 0.321162
\(307\) 25.9443 1.48072 0.740359 0.672212i \(-0.234656\pi\)
0.740359 + 0.672212i \(0.234656\pi\)
\(308\) −1.38197 −0.0787448
\(309\) 6.47214 0.368187
\(310\) 21.1803 1.20296
\(311\) 7.61803 0.431979 0.215990 0.976396i \(-0.430702\pi\)
0.215990 + 0.976396i \(0.430702\pi\)
\(312\) −15.3262 −0.867677
\(313\) −1.52786 −0.0863600 −0.0431800 0.999067i \(-0.513749\pi\)
−0.0431800 + 0.999067i \(0.513749\pi\)
\(314\) −22.9443 −1.29482
\(315\) −2.61803 −0.147510
\(316\) −7.38197 −0.415268
\(317\) −22.1459 −1.24384 −0.621919 0.783082i \(-0.713647\pi\)
−0.621919 + 0.783082i \(0.713647\pi\)
\(318\) 2.23607 0.125392
\(319\) −23.9443 −1.34062
\(320\) −11.0902 −0.619959
\(321\) 0.854102 0.0476713
\(322\) 1.61803 0.0901695
\(323\) −13.0689 −0.727172
\(324\) 0.618034 0.0343352
\(325\) −12.7082 −0.704924
\(326\) −36.4164 −2.01692
\(327\) 14.0344 0.776106
\(328\) 10.5279 0.581304
\(329\) 0.763932 0.0421169
\(330\) −9.47214 −0.521424
\(331\) 10.9443 0.601552 0.300776 0.953695i \(-0.402754\pi\)
0.300776 + 0.953695i \(0.402754\pi\)
\(332\) −6.14590 −0.337300
\(333\) −9.47214 −0.519070
\(334\) 18.5623 1.01568
\(335\) 0.381966 0.0208690
\(336\) 4.85410 0.264813
\(337\) −18.5623 −1.01115 −0.505577 0.862782i \(-0.668720\pi\)
−0.505577 + 0.862782i \(0.668720\pi\)
\(338\) 54.9787 2.99045
\(339\) 7.09017 0.385085
\(340\) −5.61803 −0.304681
\(341\) 11.1803 0.605449
\(342\) −6.09017 −0.329319
\(343\) 1.00000 0.0539949
\(344\) −16.3820 −0.883256
\(345\) 2.61803 0.140950
\(346\) −18.2705 −0.982229
\(347\) −3.65248 −0.196075 −0.0980376 0.995183i \(-0.531257\pi\)
−0.0980376 + 0.995183i \(0.531257\pi\)
\(348\) −6.61803 −0.354764
\(349\) −6.50658 −0.348289 −0.174145 0.984720i \(-0.555716\pi\)
−0.174145 + 0.984720i \(0.555716\pi\)
\(350\) 3.00000 0.160357
\(351\) 6.85410 0.365845
\(352\) 7.56231 0.403072
\(353\) −26.5967 −1.41560 −0.707801 0.706412i \(-0.750313\pi\)
−0.707801 + 0.706412i \(0.750313\pi\)
\(354\) −3.00000 −0.159448
\(355\) −30.4164 −1.61434
\(356\) −0.416408 −0.0220696
\(357\) −3.47214 −0.183765
\(358\) 32.4164 1.71326
\(359\) 22.7984 1.20325 0.601626 0.798778i \(-0.294520\pi\)
0.601626 + 0.798778i \(0.294520\pi\)
\(360\) 5.85410 0.308538
\(361\) −4.83282 −0.254359
\(362\) −24.5623 −1.29097
\(363\) 6.00000 0.314918
\(364\) −4.23607 −0.222030
\(365\) −14.3262 −0.749870
\(366\) −3.38197 −0.176778
\(367\) −32.4508 −1.69392 −0.846960 0.531656i \(-0.821570\pi\)
−0.846960 + 0.531656i \(0.821570\pi\)
\(368\) −4.85410 −0.253038
\(369\) −4.70820 −0.245099
\(370\) 40.1246 2.08598
\(371\) −1.38197 −0.0717481
\(372\) 3.09017 0.160218
\(373\) 2.88854 0.149563 0.0747816 0.997200i \(-0.476174\pi\)
0.0747816 + 0.997200i \(0.476174\pi\)
\(374\) −12.5623 −0.649581
\(375\) −8.23607 −0.425309
\(376\) −1.70820 −0.0880939
\(377\) −73.3951 −3.78004
\(378\) −1.61803 −0.0832227
\(379\) −20.9443 −1.07583 −0.537917 0.842997i \(-0.680789\pi\)
−0.537917 + 0.842997i \(0.680789\pi\)
\(380\) 6.09017 0.312419
\(381\) 2.67376 0.136981
\(382\) 35.1246 1.79713
\(383\) −4.52786 −0.231363 −0.115682 0.993286i \(-0.536905\pi\)
−0.115682 + 0.993286i \(0.536905\pi\)
\(384\) −13.6180 −0.694942
\(385\) 5.85410 0.298353
\(386\) −29.4164 −1.49726
\(387\) 7.32624 0.372414
\(388\) 3.20163 0.162538
\(389\) 22.1246 1.12176 0.560881 0.827896i \(-0.310462\pi\)
0.560881 + 0.827896i \(0.310462\pi\)
\(390\) −29.0344 −1.47022
\(391\) 3.47214 0.175593
\(392\) −2.23607 −0.112938
\(393\) 12.4164 0.626325
\(394\) −40.9787 −2.06448
\(395\) 31.2705 1.57339
\(396\) −1.38197 −0.0694464
\(397\) 15.2361 0.764676 0.382338 0.924022i \(-0.375119\pi\)
0.382338 + 0.924022i \(0.375119\pi\)
\(398\) −15.4721 −0.775548
\(399\) 3.76393 0.188432
\(400\) −9.00000 −0.450000
\(401\) 6.81966 0.340558 0.170279 0.985396i \(-0.445533\pi\)
0.170279 + 0.985396i \(0.445533\pi\)
\(402\) 0.236068 0.0117740
\(403\) 34.2705 1.70714
\(404\) −3.03444 −0.150969
\(405\) −2.61803 −0.130091
\(406\) 17.3262 0.859887
\(407\) 21.1803 1.04987
\(408\) 7.76393 0.384372
\(409\) 23.6525 1.16954 0.584770 0.811199i \(-0.301185\pi\)
0.584770 + 0.811199i \(0.301185\pi\)
\(410\) 19.9443 0.984977
\(411\) 19.9443 0.983778
\(412\) −4.00000 −0.197066
\(413\) 1.85410 0.0912344
\(414\) 1.61803 0.0795220
\(415\) 26.0344 1.27798
\(416\) 23.1803 1.13651
\(417\) −19.8541 −0.972260
\(418\) 13.6180 0.666080
\(419\) 19.5623 0.955681 0.477841 0.878447i \(-0.341420\pi\)
0.477841 + 0.878447i \(0.341420\pi\)
\(420\) 1.61803 0.0789520
\(421\) 10.6738 0.520207 0.260104 0.965581i \(-0.416243\pi\)
0.260104 + 0.965581i \(0.416243\pi\)
\(422\) 24.2705 1.18147
\(423\) 0.763932 0.0371436
\(424\) 3.09017 0.150072
\(425\) 6.43769 0.312274
\(426\) −18.7984 −0.910784
\(427\) 2.09017 0.101150
\(428\) −0.527864 −0.0255153
\(429\) −15.3262 −0.739958
\(430\) −31.0344 −1.49661
\(431\) 19.9098 0.959023 0.479511 0.877536i \(-0.340814\pi\)
0.479511 + 0.877536i \(0.340814\pi\)
\(432\) 4.85410 0.233543
\(433\) 23.1246 1.11130 0.555649 0.831417i \(-0.312470\pi\)
0.555649 + 0.831417i \(0.312470\pi\)
\(434\) −8.09017 −0.388341
\(435\) 28.0344 1.34415
\(436\) −8.67376 −0.415398
\(437\) −3.76393 −0.180053
\(438\) −8.85410 −0.423065
\(439\) −17.9443 −0.856433 −0.428217 0.903676i \(-0.640858\pi\)
−0.428217 + 0.903676i \(0.640858\pi\)
\(440\) −13.0902 −0.624049
\(441\) 1.00000 0.0476190
\(442\) −38.5066 −1.83157
\(443\) −4.00000 −0.190046 −0.0950229 0.995475i \(-0.530292\pi\)
−0.0950229 + 0.995475i \(0.530292\pi\)
\(444\) 5.85410 0.277823
\(445\) 1.76393 0.0836184
\(446\) 29.6525 1.40409
\(447\) 14.6525 0.693038
\(448\) 4.23607 0.200135
\(449\) 12.9098 0.609253 0.304626 0.952472i \(-0.401468\pi\)
0.304626 + 0.952472i \(0.401468\pi\)
\(450\) 3.00000 0.141421
\(451\) 10.5279 0.495738
\(452\) −4.38197 −0.206110
\(453\) 1.70820 0.0802584
\(454\) 24.7082 1.15961
\(455\) 17.9443 0.841240
\(456\) −8.41641 −0.394134
\(457\) −32.7984 −1.53424 −0.767122 0.641502i \(-0.778312\pi\)
−0.767122 + 0.641502i \(0.778312\pi\)
\(458\) −9.18034 −0.428969
\(459\) −3.47214 −0.162065
\(460\) −1.61803 −0.0754412
\(461\) 6.14590 0.286243 0.143122 0.989705i \(-0.454286\pi\)
0.143122 + 0.989705i \(0.454286\pi\)
\(462\) 3.61803 0.168326
\(463\) 14.5967 0.678368 0.339184 0.940720i \(-0.389849\pi\)
0.339184 + 0.940720i \(0.389849\pi\)
\(464\) −51.9787 −2.41305
\(465\) −13.0902 −0.607042
\(466\) 11.4721 0.531436
\(467\) 18.8885 0.874058 0.437029 0.899448i \(-0.356031\pi\)
0.437029 + 0.899448i \(0.356031\pi\)
\(468\) −4.23607 −0.195812
\(469\) −0.145898 −0.00673695
\(470\) −3.23607 −0.149269
\(471\) 14.1803 0.653396
\(472\) −4.14590 −0.190830
\(473\) −16.3820 −0.753244
\(474\) 19.3262 0.887684
\(475\) −6.97871 −0.320205
\(476\) 2.14590 0.0983571
\(477\) −1.38197 −0.0632759
\(478\) −10.7082 −0.489782
\(479\) −5.65248 −0.258268 −0.129134 0.991627i \(-0.541220\pi\)
−0.129134 + 0.991627i \(0.541220\pi\)
\(480\) −8.85410 −0.404133
\(481\) 64.9230 2.96023
\(482\) 46.4508 2.11578
\(483\) −1.00000 −0.0455016
\(484\) −3.70820 −0.168555
\(485\) −13.5623 −0.615833
\(486\) −1.61803 −0.0733955
\(487\) 17.7639 0.804961 0.402480 0.915429i \(-0.368148\pi\)
0.402480 + 0.915429i \(0.368148\pi\)
\(488\) −4.67376 −0.211571
\(489\) 22.5066 1.01778
\(490\) −4.23607 −0.191366
\(491\) 28.0902 1.26769 0.633846 0.773459i \(-0.281475\pi\)
0.633846 + 0.773459i \(0.281475\pi\)
\(492\) 2.90983 0.131185
\(493\) 37.1803 1.67452
\(494\) 41.7426 1.87809
\(495\) 5.85410 0.263122
\(496\) 24.2705 1.08978
\(497\) 11.6180 0.521140
\(498\) 16.0902 0.721018
\(499\) 12.7984 0.572934 0.286467 0.958090i \(-0.407519\pi\)
0.286467 + 0.958090i \(0.407519\pi\)
\(500\) 5.09017 0.227639
\(501\) −11.4721 −0.512537
\(502\) −19.8885 −0.887669
\(503\) −21.0902 −0.940364 −0.470182 0.882569i \(-0.655812\pi\)
−0.470182 + 0.882569i \(0.655812\pi\)
\(504\) −2.23607 −0.0996024
\(505\) 12.8541 0.572000
\(506\) −3.61803 −0.160841
\(507\) −33.9787 −1.50905
\(508\) −1.65248 −0.0733168
\(509\) 25.7082 1.13950 0.569748 0.821819i \(-0.307041\pi\)
0.569748 + 0.821819i \(0.307041\pi\)
\(510\) 14.7082 0.651290
\(511\) 5.47214 0.242073
\(512\) −5.29180 −0.233867
\(513\) 3.76393 0.166182
\(514\) 48.0689 2.12023
\(515\) 16.9443 0.746654
\(516\) −4.52786 −0.199328
\(517\) −1.70820 −0.0751267
\(518\) −15.3262 −0.673396
\(519\) 11.2918 0.495655
\(520\) −40.1246 −1.75958
\(521\) −3.52786 −0.154559 −0.0772793 0.997009i \(-0.524623\pi\)
−0.0772793 + 0.997009i \(0.524623\pi\)
\(522\) 17.3262 0.758349
\(523\) 32.8328 1.43568 0.717839 0.696209i \(-0.245131\pi\)
0.717839 + 0.696209i \(0.245131\pi\)
\(524\) −7.67376 −0.335230
\(525\) −1.85410 −0.0809196
\(526\) 8.09017 0.352748
\(527\) −17.3607 −0.756243
\(528\) −10.8541 −0.472364
\(529\) 1.00000 0.0434783
\(530\) 5.85410 0.254286
\(531\) 1.85410 0.0804612
\(532\) −2.32624 −0.100855
\(533\) 32.2705 1.39779
\(534\) 1.09017 0.0471763
\(535\) 2.23607 0.0966736
\(536\) 0.326238 0.0140913
\(537\) −20.0344 −0.864550
\(538\) −4.61803 −0.199098
\(539\) −2.23607 −0.0963143
\(540\) 1.61803 0.0696291
\(541\) −23.2361 −0.998997 −0.499498 0.866315i \(-0.666482\pi\)
−0.499498 + 0.866315i \(0.666482\pi\)
\(542\) −27.7984 −1.19404
\(543\) 15.1803 0.651451
\(544\) −11.7426 −0.503462
\(545\) 36.7426 1.57388
\(546\) 11.0902 0.474615
\(547\) −26.6180 −1.13810 −0.569052 0.822301i \(-0.692690\pi\)
−0.569052 + 0.822301i \(0.692690\pi\)
\(548\) −12.3262 −0.526551
\(549\) 2.09017 0.0892063
\(550\) −6.70820 −0.286039
\(551\) −40.3050 −1.71705
\(552\) 2.23607 0.0951734
\(553\) −11.9443 −0.507922
\(554\) 34.3262 1.45838
\(555\) −24.7984 −1.05263
\(556\) 12.2705 0.520386
\(557\) −17.2361 −0.730316 −0.365158 0.930946i \(-0.618985\pi\)
−0.365158 + 0.930946i \(0.618985\pi\)
\(558\) −8.09017 −0.342484
\(559\) −50.2148 −2.12386
\(560\) 12.7082 0.537020
\(561\) 7.76393 0.327793
\(562\) −34.9443 −1.47404
\(563\) 8.56231 0.360858 0.180429 0.983588i \(-0.442251\pi\)
0.180429 + 0.983588i \(0.442251\pi\)
\(564\) −0.472136 −0.0198805
\(565\) 18.5623 0.780922
\(566\) 23.1803 0.974342
\(567\) 1.00000 0.0419961
\(568\) −25.9787 −1.09004
\(569\) 27.7771 1.16448 0.582238 0.813018i \(-0.302177\pi\)
0.582238 + 0.813018i \(0.302177\pi\)
\(570\) −15.9443 −0.667832
\(571\) −44.7771 −1.87386 −0.936931 0.349513i \(-0.886347\pi\)
−0.936931 + 0.349513i \(0.886347\pi\)
\(572\) 9.47214 0.396050
\(573\) −21.7082 −0.906873
\(574\) −7.61803 −0.317971
\(575\) 1.85410 0.0773214
\(576\) 4.23607 0.176503
\(577\) −17.4164 −0.725055 −0.362527 0.931973i \(-0.618086\pi\)
−0.362527 + 0.931973i \(0.618086\pi\)
\(578\) −8.00000 −0.332756
\(579\) 18.1803 0.755549
\(580\) −17.3262 −0.719433
\(581\) −9.94427 −0.412558
\(582\) −8.38197 −0.347444
\(583\) 3.09017 0.127982
\(584\) −12.2361 −0.506332
\(585\) 17.9443 0.741904
\(586\) −15.4164 −0.636846
\(587\) 2.56231 0.105758 0.0528788 0.998601i \(-0.483160\pi\)
0.0528788 + 0.998601i \(0.483160\pi\)
\(588\) −0.618034 −0.0254873
\(589\) 18.8197 0.775451
\(590\) −7.85410 −0.323348
\(591\) 25.3262 1.04178
\(592\) 45.9787 1.88971
\(593\) −24.2918 −0.997545 −0.498772 0.866733i \(-0.666216\pi\)
−0.498772 + 0.866733i \(0.666216\pi\)
\(594\) 3.61803 0.148450
\(595\) −9.09017 −0.372661
\(596\) −9.05573 −0.370937
\(597\) 9.56231 0.391359
\(598\) −11.0902 −0.453511
\(599\) 7.72949 0.315818 0.157909 0.987454i \(-0.449525\pi\)
0.157909 + 0.987454i \(0.449525\pi\)
\(600\) 4.14590 0.169256
\(601\) 12.5066 0.510154 0.255077 0.966921i \(-0.417899\pi\)
0.255077 + 0.966921i \(0.417899\pi\)
\(602\) 11.8541 0.483137
\(603\) −0.145898 −0.00594143
\(604\) −1.05573 −0.0429570
\(605\) 15.7082 0.638629
\(606\) 7.94427 0.322714
\(607\) −8.38197 −0.340214 −0.170107 0.985426i \(-0.554411\pi\)
−0.170107 + 0.985426i \(0.554411\pi\)
\(608\) 12.7295 0.516249
\(609\) −10.7082 −0.433918
\(610\) −8.85410 −0.358492
\(611\) −5.23607 −0.211829
\(612\) 2.14590 0.0867428
\(613\) −6.34752 −0.256374 −0.128187 0.991750i \(-0.540916\pi\)
−0.128187 + 0.991750i \(0.540916\pi\)
\(614\) 41.9787 1.69412
\(615\) −12.3262 −0.497042
\(616\) 5.00000 0.201456
\(617\) −4.74265 −0.190932 −0.0954659 0.995433i \(-0.530434\pi\)
−0.0954659 + 0.995433i \(0.530434\pi\)
\(618\) 10.4721 0.421251
\(619\) 0.270510 0.0108727 0.00543635 0.999985i \(-0.498270\pi\)
0.00543635 + 0.999985i \(0.498270\pi\)
\(620\) 8.09017 0.324909
\(621\) −1.00000 −0.0401286
\(622\) 12.3262 0.494237
\(623\) −0.673762 −0.0269937
\(624\) −33.2705 −1.33189
\(625\) −30.8328 −1.23331
\(626\) −2.47214 −0.0988064
\(627\) −8.41641 −0.336119
\(628\) −8.76393 −0.349719
\(629\) −32.8885 −1.31135
\(630\) −4.23607 −0.168769
\(631\) −33.7639 −1.34412 −0.672060 0.740496i \(-0.734591\pi\)
−0.672060 + 0.740496i \(0.734591\pi\)
\(632\) 26.7082 1.06240
\(633\) −15.0000 −0.596196
\(634\) −35.8328 −1.42310
\(635\) 7.00000 0.277787
\(636\) 0.854102 0.0338673
\(637\) −6.85410 −0.271569
\(638\) −38.7426 −1.53384
\(639\) 11.6180 0.459602
\(640\) −35.6525 −1.40929
\(641\) −26.3820 −1.04203 −0.521013 0.853549i \(-0.674446\pi\)
−0.521013 + 0.853549i \(0.674446\pi\)
\(642\) 1.38197 0.0545418
\(643\) −35.5066 −1.40024 −0.700121 0.714024i \(-0.746871\pi\)
−0.700121 + 0.714024i \(0.746871\pi\)
\(644\) 0.618034 0.0243540
\(645\) 19.1803 0.755225
\(646\) −21.1459 −0.831974
\(647\) −9.09017 −0.357371 −0.178686 0.983906i \(-0.557185\pi\)
−0.178686 + 0.983906i \(0.557185\pi\)
\(648\) −2.23607 −0.0878410
\(649\) −4.14590 −0.162741
\(650\) −20.5623 −0.806520
\(651\) 5.00000 0.195965
\(652\) −13.9098 −0.544751
\(653\) −20.9098 −0.818265 −0.409132 0.912475i \(-0.634169\pi\)
−0.409132 + 0.912475i \(0.634169\pi\)
\(654\) 22.7082 0.887961
\(655\) 32.5066 1.27014
\(656\) 22.8541 0.892303
\(657\) 5.47214 0.213488
\(658\) 1.23607 0.0481869
\(659\) 20.5967 0.802335 0.401168 0.916005i \(-0.368604\pi\)
0.401168 + 0.916005i \(0.368604\pi\)
\(660\) −3.61803 −0.140832
\(661\) 29.1803 1.13498 0.567492 0.823379i \(-0.307914\pi\)
0.567492 + 0.823379i \(0.307914\pi\)
\(662\) 17.7082 0.688249
\(663\) 23.7984 0.924252
\(664\) 22.2361 0.862927
\(665\) 9.85410 0.382126
\(666\) −15.3262 −0.593880
\(667\) 10.7082 0.414623
\(668\) 7.09017 0.274327
\(669\) −18.3262 −0.708533
\(670\) 0.618034 0.0238767
\(671\) −4.67376 −0.180429
\(672\) 3.38197 0.130462
\(673\) 4.05573 0.156337 0.0781684 0.996940i \(-0.475093\pi\)
0.0781684 + 0.996940i \(0.475093\pi\)
\(674\) −30.0344 −1.15688
\(675\) −1.85410 −0.0713644
\(676\) 21.0000 0.807692
\(677\) −23.7984 −0.914646 −0.457323 0.889301i \(-0.651192\pi\)
−0.457323 + 0.889301i \(0.651192\pi\)
\(678\) 11.4721 0.440585
\(679\) 5.18034 0.198803
\(680\) 20.3262 0.779476
\(681\) −15.2705 −0.585167
\(682\) 18.0902 0.692708
\(683\) 27.2918 1.04429 0.522146 0.852856i \(-0.325132\pi\)
0.522146 + 0.852856i \(0.325132\pi\)
\(684\) −2.32624 −0.0889459
\(685\) 52.2148 1.99502
\(686\) 1.61803 0.0617768
\(687\) 5.67376 0.216468
\(688\) −35.5623 −1.35580
\(689\) 9.47214 0.360860
\(690\) 4.23607 0.161264
\(691\) 3.79837 0.144497 0.0722485 0.997387i \(-0.476983\pi\)
0.0722485 + 0.997387i \(0.476983\pi\)
\(692\) −6.97871 −0.265291
\(693\) −2.23607 −0.0849412
\(694\) −5.90983 −0.224334
\(695\) −51.9787 −1.97166
\(696\) 23.9443 0.907605
\(697\) −16.3475 −0.619207
\(698\) −10.5279 −0.398486
\(699\) −7.09017 −0.268175
\(700\) 1.14590 0.0433109
\(701\) −7.20163 −0.272002 −0.136001 0.990709i \(-0.543425\pi\)
−0.136001 + 0.990709i \(0.543425\pi\)
\(702\) 11.0902 0.418571
\(703\) 35.6525 1.34466
\(704\) −9.47214 −0.356995
\(705\) 2.00000 0.0753244
\(706\) −43.0344 −1.61962
\(707\) −4.90983 −0.184653
\(708\) −1.14590 −0.0430655
\(709\) 3.32624 0.124919 0.0624597 0.998047i \(-0.480105\pi\)
0.0624597 + 0.998047i \(0.480105\pi\)
\(710\) −49.2148 −1.84700
\(711\) −11.9443 −0.447945
\(712\) 1.50658 0.0564614
\(713\) −5.00000 −0.187251
\(714\) −5.61803 −0.210250
\(715\) −40.1246 −1.50058
\(716\) 12.3820 0.462736
\(717\) 6.61803 0.247155
\(718\) 36.8885 1.37667
\(719\) 22.7082 0.846873 0.423437 0.905926i \(-0.360824\pi\)
0.423437 + 0.905926i \(0.360824\pi\)
\(720\) 12.7082 0.473607
\(721\) −6.47214 −0.241035
\(722\) −7.81966 −0.291018
\(723\) −28.7082 −1.06767
\(724\) −9.38197 −0.348678
\(725\) 19.8541 0.737363
\(726\) 9.70820 0.360305
\(727\) −25.7639 −0.955531 −0.477766 0.878487i \(-0.658553\pi\)
−0.477766 + 0.878487i \(0.658553\pi\)
\(728\) 15.3262 0.568028
\(729\) 1.00000 0.0370370
\(730\) −23.1803 −0.857943
\(731\) 25.4377 0.940847
\(732\) −1.29180 −0.0477462
\(733\) −38.5410 −1.42355 −0.711773 0.702410i \(-0.752107\pi\)
−0.711773 + 0.702410i \(0.752107\pi\)
\(734\) −52.5066 −1.93805
\(735\) 2.61803 0.0965676
\(736\) −3.38197 −0.124661
\(737\) 0.326238 0.0120171
\(738\) −7.61803 −0.280424
\(739\) −13.5836 −0.499681 −0.249840 0.968287i \(-0.580378\pi\)
−0.249840 + 0.968287i \(0.580378\pi\)
\(740\) 15.3262 0.563404
\(741\) −25.7984 −0.947727
\(742\) −2.23607 −0.0820886
\(743\) −48.3951 −1.77544 −0.887722 0.460379i \(-0.847713\pi\)
−0.887722 + 0.460379i \(0.847713\pi\)
\(744\) −11.1803 −0.409891
\(745\) 38.3607 1.40543
\(746\) 4.67376 0.171119
\(747\) −9.94427 −0.363842
\(748\) −4.79837 −0.175446
\(749\) −0.854102 −0.0312082
\(750\) −13.3262 −0.486605
\(751\) −20.9098 −0.763011 −0.381505 0.924367i \(-0.624594\pi\)
−0.381505 + 0.924367i \(0.624594\pi\)
\(752\) −3.70820 −0.135224
\(753\) 12.2918 0.447938
\(754\) −118.756 −4.32483
\(755\) 4.47214 0.162758
\(756\) −0.618034 −0.0224777
\(757\) −5.58359 −0.202939 −0.101470 0.994839i \(-0.532354\pi\)
−0.101470 + 0.994839i \(0.532354\pi\)
\(758\) −33.8885 −1.23089
\(759\) 2.23607 0.0811641
\(760\) −22.0344 −0.799273
\(761\) 18.1115 0.656540 0.328270 0.944584i \(-0.393534\pi\)
0.328270 + 0.944584i \(0.393534\pi\)
\(762\) 4.32624 0.156723
\(763\) −14.0344 −0.508081
\(764\) 13.4164 0.485389
\(765\) −9.09017 −0.328656
\(766\) −7.32624 −0.264708
\(767\) −12.7082 −0.458867
\(768\) −13.5623 −0.489388
\(769\) −45.9574 −1.65727 −0.828634 0.559791i \(-0.810881\pi\)
−0.828634 + 0.559791i \(0.810881\pi\)
\(770\) 9.47214 0.341352
\(771\) −29.7082 −1.06991
\(772\) −11.2361 −0.404395
\(773\) 40.0132 1.43917 0.719587 0.694403i \(-0.244331\pi\)
0.719587 + 0.694403i \(0.244331\pi\)
\(774\) 11.8541 0.426087
\(775\) −9.27051 −0.333007
\(776\) −11.5836 −0.415827
\(777\) 9.47214 0.339811
\(778\) 35.7984 1.28343
\(779\) 17.7214 0.634934
\(780\) −11.0902 −0.397092
\(781\) −25.9787 −0.929591
\(782\) 5.61803 0.200900
\(783\) −10.7082 −0.382680
\(784\) −4.85410 −0.173361
\(785\) 37.1246 1.32503
\(786\) 20.0902 0.716593
\(787\) 22.7984 0.812674 0.406337 0.913723i \(-0.366806\pi\)
0.406337 + 0.913723i \(0.366806\pi\)
\(788\) −15.6525 −0.557596
\(789\) −5.00000 −0.178005
\(790\) 50.5967 1.80015
\(791\) −7.09017 −0.252097
\(792\) 5.00000 0.177667
\(793\) −14.3262 −0.508740
\(794\) 24.6525 0.874884
\(795\) −3.61803 −0.128318
\(796\) −5.90983 −0.209468
\(797\) 9.70820 0.343882 0.171941 0.985107i \(-0.444996\pi\)
0.171941 + 0.985107i \(0.444996\pi\)
\(798\) 6.09017 0.215590
\(799\) 2.65248 0.0938378
\(800\) −6.27051 −0.221696
\(801\) −0.673762 −0.0238062
\(802\) 11.0344 0.389640
\(803\) −12.2361 −0.431801
\(804\) 0.0901699 0.00318005
\(805\) −2.61803 −0.0922736
\(806\) 55.4508 1.95317
\(807\) 2.85410 0.100469
\(808\) 10.9787 0.386230
\(809\) −22.5623 −0.793248 −0.396624 0.917981i \(-0.629818\pi\)
−0.396624 + 0.917981i \(0.629818\pi\)
\(810\) −4.23607 −0.148840
\(811\) 41.5410 1.45870 0.729351 0.684139i \(-0.239822\pi\)
0.729351 + 0.684139i \(0.239822\pi\)
\(812\) 6.61803 0.232247
\(813\) 17.1803 0.602541
\(814\) 34.2705 1.20118
\(815\) 58.9230 2.06398
\(816\) 16.8541 0.590012
\(817\) −27.5755 −0.964743
\(818\) 38.2705 1.33810
\(819\) −6.85410 −0.239502
\(820\) 7.61803 0.266033
\(821\) 25.7082 0.897223 0.448611 0.893727i \(-0.351919\pi\)
0.448611 + 0.893727i \(0.351919\pi\)
\(822\) 32.2705 1.12556
\(823\) 54.9787 1.91644 0.958219 0.286036i \(-0.0923378\pi\)
0.958219 + 0.286036i \(0.0923378\pi\)
\(824\) 14.4721 0.504161
\(825\) 4.14590 0.144342
\(826\) 3.00000 0.104383
\(827\) −16.1459 −0.561448 −0.280724 0.959789i \(-0.590575\pi\)
−0.280724 + 0.959789i \(0.590575\pi\)
\(828\) 0.618034 0.0214782
\(829\) −23.0689 −0.801215 −0.400608 0.916250i \(-0.631201\pi\)
−0.400608 + 0.916250i \(0.631201\pi\)
\(830\) 42.1246 1.46217
\(831\) −21.2148 −0.735933
\(832\) −29.0344 −1.00659
\(833\) 3.47214 0.120302
\(834\) −32.1246 −1.11238
\(835\) −30.0344 −1.03938
\(836\) 5.20163 0.179902
\(837\) 5.00000 0.172825
\(838\) 31.6525 1.09342
\(839\) −36.2705 −1.25220 −0.626099 0.779744i \(-0.715349\pi\)
−0.626099 + 0.779744i \(0.715349\pi\)
\(840\) −5.85410 −0.201986
\(841\) 85.6656 2.95399
\(842\) 17.2705 0.595181
\(843\) 21.5967 0.743832
\(844\) 9.27051 0.319104
\(845\) −88.9574 −3.06023
\(846\) 1.23607 0.0424969
\(847\) −6.00000 −0.206162
\(848\) 6.70820 0.230361
\(849\) −14.3262 −0.491675
\(850\) 10.4164 0.357280
\(851\) −9.47214 −0.324701
\(852\) −7.18034 −0.245994
\(853\) −11.3475 −0.388532 −0.194266 0.980949i \(-0.562232\pi\)
−0.194266 + 0.980949i \(0.562232\pi\)
\(854\) 3.38197 0.115728
\(855\) 9.85410 0.337003
\(856\) 1.90983 0.0652766
\(857\) −49.9574 −1.70651 −0.853257 0.521491i \(-0.825376\pi\)
−0.853257 + 0.521491i \(0.825376\pi\)
\(858\) −24.7984 −0.846603
\(859\) −25.4164 −0.867197 −0.433598 0.901106i \(-0.642756\pi\)
−0.433598 + 0.901106i \(0.642756\pi\)
\(860\) −11.8541 −0.404222
\(861\) 4.70820 0.160455
\(862\) 32.2148 1.09724
\(863\) −17.5967 −0.599000 −0.299500 0.954096i \(-0.596820\pi\)
−0.299500 + 0.954096i \(0.596820\pi\)
\(864\) 3.38197 0.115057
\(865\) 29.5623 1.00515
\(866\) 37.4164 1.27146
\(867\) 4.94427 0.167916
\(868\) −3.09017 −0.104887
\(869\) 26.7082 0.906014
\(870\) 45.3607 1.53787
\(871\) 1.00000 0.0338837
\(872\) 31.3820 1.06273
\(873\) 5.18034 0.175328
\(874\) −6.09017 −0.206003
\(875\) 8.23607 0.278430
\(876\) −3.38197 −0.114266
\(877\) −17.5279 −0.591874 −0.295937 0.955207i \(-0.595632\pi\)
−0.295937 + 0.955207i \(0.595632\pi\)
\(878\) −29.0344 −0.979865
\(879\) 9.52786 0.321367
\(880\) −28.4164 −0.957917
\(881\) 26.5410 0.894190 0.447095 0.894487i \(-0.352459\pi\)
0.447095 + 0.894487i \(0.352459\pi\)
\(882\) 1.61803 0.0544820
\(883\) −34.7426 −1.16918 −0.584592 0.811328i \(-0.698745\pi\)
−0.584592 + 0.811328i \(0.698745\pi\)
\(884\) −14.7082 −0.494690
\(885\) 4.85410 0.163169
\(886\) −6.47214 −0.217436
\(887\) 6.67376 0.224083 0.112041 0.993704i \(-0.464261\pi\)
0.112041 + 0.993704i \(0.464261\pi\)
\(888\) −21.1803 −0.710766
\(889\) −2.67376 −0.0896751
\(890\) 2.85410 0.0956697
\(891\) −2.23607 −0.0749111
\(892\) 11.3262 0.379230
\(893\) −2.87539 −0.0962212
\(894\) 23.7082 0.792921
\(895\) −52.4508 −1.75324
\(896\) 13.6180 0.454947
\(897\) 6.85410 0.228852
\(898\) 20.8885 0.697060
\(899\) −53.5410 −1.78569
\(900\) 1.14590 0.0381966
\(901\) −4.79837 −0.159857
\(902\) 17.0344 0.567185
\(903\) −7.32624 −0.243802
\(904\) 15.8541 0.527300
\(905\) 39.7426 1.32109
\(906\) 2.76393 0.0918255
\(907\) −16.3262 −0.542104 −0.271052 0.962565i \(-0.587372\pi\)
−0.271052 + 0.962565i \(0.587372\pi\)
\(908\) 9.43769 0.313201
\(909\) −4.90983 −0.162849
\(910\) 29.0344 0.962482
\(911\) −18.2918 −0.606034 −0.303017 0.952985i \(-0.597994\pi\)
−0.303017 + 0.952985i \(0.597994\pi\)
\(912\) −18.2705 −0.604997
\(913\) 22.2361 0.735906
\(914\) −53.0689 −1.75536
\(915\) 5.47214 0.180903
\(916\) −3.50658 −0.115861
\(917\) −12.4164 −0.410026
\(918\) −5.61803 −0.185423
\(919\) 36.4853 1.20354 0.601769 0.798670i \(-0.294463\pi\)
0.601769 + 0.798670i \(0.294463\pi\)
\(920\) 5.85410 0.193004
\(921\) −25.9443 −0.854893
\(922\) 9.94427 0.327497
\(923\) −79.6312 −2.62109
\(924\) 1.38197 0.0454633
\(925\) −17.5623 −0.577445
\(926\) 23.6180 0.776137
\(927\) −6.47214 −0.212573
\(928\) −36.2148 −1.18881
\(929\) 33.7426 1.10706 0.553530 0.832829i \(-0.313280\pi\)
0.553530 + 0.832829i \(0.313280\pi\)
\(930\) −21.1803 −0.694531
\(931\) −3.76393 −0.123358
\(932\) 4.38197 0.143536
\(933\) −7.61803 −0.249403
\(934\) 30.5623 1.00003
\(935\) 20.3262 0.664739
\(936\) 15.3262 0.500954
\(937\) −33.3607 −1.08985 −0.544923 0.838486i \(-0.683441\pi\)
−0.544923 + 0.838486i \(0.683441\pi\)
\(938\) −0.236068 −0.00770789
\(939\) 1.52786 0.0498600
\(940\) −1.23607 −0.0403161
\(941\) −34.8328 −1.13552 −0.567759 0.823195i \(-0.692189\pi\)
−0.567759 + 0.823195i \(0.692189\pi\)
\(942\) 22.9443 0.747565
\(943\) −4.70820 −0.153320
\(944\) −9.00000 −0.292925
\(945\) 2.61803 0.0851647
\(946\) −26.5066 −0.861803
\(947\) −46.7214 −1.51824 −0.759120 0.650951i \(-0.774370\pi\)
−0.759120 + 0.650951i \(0.774370\pi\)
\(948\) 7.38197 0.239755
\(949\) −37.5066 −1.21752
\(950\) −11.2918 −0.366354
\(951\) 22.1459 0.718130
\(952\) −7.76393 −0.251630
\(953\) 48.9230 1.58477 0.792385 0.610021i \(-0.208839\pi\)
0.792385 + 0.610021i \(0.208839\pi\)
\(954\) −2.23607 −0.0723954
\(955\) −56.8328 −1.83907
\(956\) −4.09017 −0.132286
\(957\) 23.9443 0.774008
\(958\) −9.14590 −0.295491
\(959\) −19.9443 −0.644034
\(960\) 11.0902 0.357934
\(961\) −6.00000 −0.193548
\(962\) 105.048 3.38687
\(963\) −0.854102 −0.0275231
\(964\) 17.7426 0.571452
\(965\) 47.5967 1.53219
\(966\) −1.61803 −0.0520594
\(967\) −16.4721 −0.529708 −0.264854 0.964288i \(-0.585324\pi\)
−0.264854 + 0.964288i \(0.585324\pi\)
\(968\) 13.4164 0.431220
\(969\) 13.0689 0.419833
\(970\) −21.9443 −0.704588
\(971\) −14.0902 −0.452175 −0.226088 0.974107i \(-0.572594\pi\)
−0.226088 + 0.974107i \(0.572594\pi\)
\(972\) −0.618034 −0.0198234
\(973\) 19.8541 0.636493
\(974\) 28.7426 0.920974
\(975\) 12.7082 0.406988
\(976\) −10.1459 −0.324762
\(977\) 53.9787 1.72693 0.863466 0.504407i \(-0.168289\pi\)
0.863466 + 0.504407i \(0.168289\pi\)
\(978\) 36.4164 1.16447
\(979\) 1.50658 0.0481504
\(980\) −1.61803 −0.0516862
\(981\) −14.0344 −0.448085
\(982\) 45.4508 1.45039
\(983\) −59.3607 −1.89331 −0.946656 0.322246i \(-0.895562\pi\)
−0.946656 + 0.322246i \(0.895562\pi\)
\(984\) −10.5279 −0.335616
\(985\) 66.3050 2.11265
\(986\) 60.1591 1.91585
\(987\) −0.763932 −0.0243162
\(988\) 15.9443 0.507255
\(989\) 7.32624 0.232961
\(990\) 9.47214 0.301044
\(991\) 2.67376 0.0849349 0.0424674 0.999098i \(-0.486478\pi\)
0.0424674 + 0.999098i \(0.486478\pi\)
\(992\) 16.9098 0.536888
\(993\) −10.9443 −0.347306
\(994\) 18.7984 0.596248
\(995\) 25.0344 0.793645
\(996\) 6.14590 0.194740
\(997\) −25.5410 −0.808892 −0.404446 0.914562i \(-0.632536\pi\)
−0.404446 + 0.914562i \(0.632536\pi\)
\(998\) 20.7082 0.655507
\(999\) 9.47214 0.299685
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 483.2.a.g.1.2 2
3.2 odd 2 1449.2.a.f.1.1 2
4.3 odd 2 7728.2.a.bg.1.1 2
7.6 odd 2 3381.2.a.u.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
483.2.a.g.1.2 2 1.1 even 1 trivial
1449.2.a.f.1.1 2 3.2 odd 2
3381.2.a.u.1.2 2 7.6 odd 2
7728.2.a.bg.1.1 2 4.3 odd 2