Properties

Label 538.2.a.a.1.2
Level $538$
Weight $2$
Character 538.1
Self dual yes
Analytic conductor $4.296$
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] = [538,2,Mod(1,538)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(538, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("538.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 538 = 2 \cdot 269 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 538.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: \(4.29595162874\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 538.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +0.618034 q^{3} +1.00000 q^{4} -3.61803 q^{5} +0.618034 q^{6} -4.23607 q^{7} +1.00000 q^{8} -2.61803 q^{9} -3.61803 q^{10} +0.236068 q^{11} +0.618034 q^{12} -1.00000 q^{13} -4.23607 q^{14} -2.23607 q^{15} +1.00000 q^{16} +0.236068 q^{17} -2.61803 q^{18} +2.00000 q^{19} -3.61803 q^{20} -2.61803 q^{21} +0.236068 q^{22} -8.23607 q^{23} +0.618034 q^{24} +8.09017 q^{25} -1.00000 q^{26} -3.47214 q^{27} -4.23607 q^{28} +0.854102 q^{29} -2.23607 q^{30} +7.47214 q^{31} +1.00000 q^{32} +0.145898 q^{33} +0.236068 q^{34} +15.3262 q^{35} -2.61803 q^{36} -8.70820 q^{37} +2.00000 q^{38} -0.618034 q^{39} -3.61803 q^{40} +7.70820 q^{41} -2.61803 q^{42} +2.23607 q^{43} +0.236068 q^{44} +9.47214 q^{45} -8.23607 q^{46} -6.85410 q^{47} +0.618034 q^{48} +10.9443 q^{49} +8.09017 q^{50} +0.145898 q^{51} -1.00000 q^{52} -3.47214 q^{54} -0.854102 q^{55} -4.23607 q^{56} +1.23607 q^{57} +0.854102 q^{58} -6.56231 q^{59} -2.23607 q^{60} -5.47214 q^{61} +7.47214 q^{62} +11.0902 q^{63} +1.00000 q^{64} +3.61803 q^{65} +0.145898 q^{66} +1.76393 q^{67} +0.236068 q^{68} -5.09017 q^{69} +15.3262 q^{70} +9.70820 q^{71} -2.61803 q^{72} -10.7082 q^{73} -8.70820 q^{74} +5.00000 q^{75} +2.00000 q^{76} -1.00000 q^{77} -0.618034 q^{78} -13.7984 q^{79} -3.61803 q^{80} +5.70820 q^{81} +7.70820 q^{82} -5.23607 q^{83} -2.61803 q^{84} -0.854102 q^{85} +2.23607 q^{86} +0.527864 q^{87} +0.236068 q^{88} -9.79837 q^{89} +9.47214 q^{90} +4.23607 q^{91} -8.23607 q^{92} +4.61803 q^{93} -6.85410 q^{94} -7.23607 q^{95} +0.618034 q^{96} +12.6525 q^{97} +10.9443 q^{98} -0.618034 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - q^{3} + 2 q^{4} - 5 q^{5} - q^{6} - 4 q^{7} + 2 q^{8} - 3 q^{9} - 5 q^{10} - 4 q^{11} - q^{12} - 2 q^{13} - 4 q^{14} + 2 q^{16} - 4 q^{17} - 3 q^{18} + 4 q^{19} - 5 q^{20} - 3 q^{21}+ \cdots + q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0.618034 0.356822 0.178411 0.983956i \(-0.442904\pi\)
0.178411 + 0.983956i \(0.442904\pi\)
\(4\) 1.00000 0.500000
\(5\) −3.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(6\) 0.618034 0.252311
\(7\) −4.23607 −1.60108 −0.800542 0.599277i \(-0.795455\pi\)
−0.800542 + 0.599277i \(0.795455\pi\)
\(8\) 1.00000 0.353553
\(9\) −2.61803 −0.872678
\(10\) −3.61803 −1.14412
\(11\) 0.236068 0.0711772 0.0355886 0.999367i \(-0.488669\pi\)
0.0355886 + 0.999367i \(0.488669\pi\)
\(12\) 0.618034 0.178411
\(13\) −1.00000 −0.277350 −0.138675 0.990338i \(-0.544284\pi\)
−0.138675 + 0.990338i \(0.544284\pi\)
\(14\) −4.23607 −1.13214
\(15\) −2.23607 −0.577350
\(16\) 1.00000 0.250000
\(17\) 0.236068 0.0572549 0.0286274 0.999590i \(-0.490886\pi\)
0.0286274 + 0.999590i \(0.490886\pi\)
\(18\) −2.61803 −0.617077
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) −3.61803 −0.809017
\(21\) −2.61803 −0.571302
\(22\) 0.236068 0.0503299
\(23\) −8.23607 −1.71734 −0.858669 0.512530i \(-0.828708\pi\)
−0.858669 + 0.512530i \(0.828708\pi\)
\(24\) 0.618034 0.126156
\(25\) 8.09017 1.61803
\(26\) −1.00000 −0.196116
\(27\) −3.47214 −0.668213
\(28\) −4.23607 −0.800542
\(29\) 0.854102 0.158603 0.0793014 0.996851i \(-0.474731\pi\)
0.0793014 + 0.996851i \(0.474731\pi\)
\(30\) −2.23607 −0.408248
\(31\) 7.47214 1.34204 0.671018 0.741441i \(-0.265857\pi\)
0.671018 + 0.741441i \(0.265857\pi\)
\(32\) 1.00000 0.176777
\(33\) 0.145898 0.0253976
\(34\) 0.236068 0.0404853
\(35\) 15.3262 2.59061
\(36\) −2.61803 −0.436339
\(37\) −8.70820 −1.43162 −0.715810 0.698295i \(-0.753942\pi\)
−0.715810 + 0.698295i \(0.753942\pi\)
\(38\) 2.00000 0.324443
\(39\) −0.618034 −0.0989646
\(40\) −3.61803 −0.572061
\(41\) 7.70820 1.20382 0.601910 0.798564i \(-0.294407\pi\)
0.601910 + 0.798564i \(0.294407\pi\)
\(42\) −2.61803 −0.403971
\(43\) 2.23607 0.340997 0.170499 0.985358i \(-0.445462\pi\)
0.170499 + 0.985358i \(0.445462\pi\)
\(44\) 0.236068 0.0355886
\(45\) 9.47214 1.41202
\(46\) −8.23607 −1.21434
\(47\) −6.85410 −0.999774 −0.499887 0.866091i \(-0.666625\pi\)
−0.499887 + 0.866091i \(0.666625\pi\)
\(48\) 0.618034 0.0892055
\(49\) 10.9443 1.56347
\(50\) 8.09017 1.14412
\(51\) 0.145898 0.0204298
\(52\) −1.00000 −0.138675
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) −3.47214 −0.472498
\(55\) −0.854102 −0.115167
\(56\) −4.23607 −0.566068
\(57\) 1.23607 0.163721
\(58\) 0.854102 0.112149
\(59\) −6.56231 −0.854339 −0.427170 0.904171i \(-0.640489\pi\)
−0.427170 + 0.904171i \(0.640489\pi\)
\(60\) −2.23607 −0.288675
\(61\) −5.47214 −0.700635 −0.350318 0.936631i \(-0.613926\pi\)
−0.350318 + 0.936631i \(0.613926\pi\)
\(62\) 7.47214 0.948962
\(63\) 11.0902 1.39723
\(64\) 1.00000 0.125000
\(65\) 3.61803 0.448762
\(66\) 0.145898 0.0179588
\(67\) 1.76393 0.215499 0.107749 0.994178i \(-0.465636\pi\)
0.107749 + 0.994178i \(0.465636\pi\)
\(68\) 0.236068 0.0286274
\(69\) −5.09017 −0.612784
\(70\) 15.3262 1.83184
\(71\) 9.70820 1.15215 0.576076 0.817396i \(-0.304583\pi\)
0.576076 + 0.817396i \(0.304583\pi\)
\(72\) −2.61803 −0.308538
\(73\) −10.7082 −1.25330 −0.626650 0.779301i \(-0.715575\pi\)
−0.626650 + 0.779301i \(0.715575\pi\)
\(74\) −8.70820 −1.01231
\(75\) 5.00000 0.577350
\(76\) 2.00000 0.229416
\(77\) −1.00000 −0.113961
\(78\) −0.618034 −0.0699786
\(79\) −13.7984 −1.55244 −0.776219 0.630463i \(-0.782865\pi\)
−0.776219 + 0.630463i \(0.782865\pi\)
\(80\) −3.61803 −0.404508
\(81\) 5.70820 0.634245
\(82\) 7.70820 0.851229
\(83\) −5.23607 −0.574733 −0.287367 0.957821i \(-0.592780\pi\)
−0.287367 + 0.957821i \(0.592780\pi\)
\(84\) −2.61803 −0.285651
\(85\) −0.854102 −0.0926404
\(86\) 2.23607 0.241121
\(87\) 0.527864 0.0565930
\(88\) 0.236068 0.0251649
\(89\) −9.79837 −1.03863 −0.519313 0.854584i \(-0.673812\pi\)
−0.519313 + 0.854584i \(0.673812\pi\)
\(90\) 9.47214 0.998451
\(91\) 4.23607 0.444061
\(92\) −8.23607 −0.858669
\(93\) 4.61803 0.478868
\(94\) −6.85410 −0.706947
\(95\) −7.23607 −0.742405
\(96\) 0.618034 0.0630778
\(97\) 12.6525 1.28466 0.642332 0.766426i \(-0.277967\pi\)
0.642332 + 0.766426i \(0.277967\pi\)
\(98\) 10.9443 1.10554
\(99\) −0.618034 −0.0621148
\(100\) 8.09017 0.809017
\(101\) −5.09017 −0.506491 −0.253245 0.967402i \(-0.581498\pi\)
−0.253245 + 0.967402i \(0.581498\pi\)
\(102\) 0.145898 0.0144461
\(103\) 10.8541 1.06949 0.534743 0.845015i \(-0.320408\pi\)
0.534743 + 0.845015i \(0.320408\pi\)
\(104\) −1.00000 −0.0980581
\(105\) 9.47214 0.924386
\(106\) 0 0
\(107\) −12.2361 −1.18291 −0.591453 0.806340i \(-0.701445\pi\)
−0.591453 + 0.806340i \(0.701445\pi\)
\(108\) −3.47214 −0.334106
\(109\) −4.70820 −0.450964 −0.225482 0.974247i \(-0.572396\pi\)
−0.225482 + 0.974247i \(0.572396\pi\)
\(110\) −0.854102 −0.0814354
\(111\) −5.38197 −0.510834
\(112\) −4.23607 −0.400271
\(113\) −15.3820 −1.44701 −0.723507 0.690317i \(-0.757471\pi\)
−0.723507 + 0.690317i \(0.757471\pi\)
\(114\) 1.23607 0.115768
\(115\) 29.7984 2.77871
\(116\) 0.854102 0.0793014
\(117\) 2.61803 0.242037
\(118\) −6.56231 −0.604109
\(119\) −1.00000 −0.0916698
\(120\) −2.23607 −0.204124
\(121\) −10.9443 −0.994934
\(122\) −5.47214 −0.495424
\(123\) 4.76393 0.429549
\(124\) 7.47214 0.671018
\(125\) −11.1803 −1.00000
\(126\) 11.0902 0.987991
\(127\) 20.7984 1.84556 0.922779 0.385331i \(-0.125913\pi\)
0.922779 + 0.385331i \(0.125913\pi\)
\(128\) 1.00000 0.0883883
\(129\) 1.38197 0.121675
\(130\) 3.61803 0.317323
\(131\) 2.76393 0.241486 0.120743 0.992684i \(-0.461472\pi\)
0.120743 + 0.992684i \(0.461472\pi\)
\(132\) 0.145898 0.0126988
\(133\) −8.47214 −0.734627
\(134\) 1.76393 0.152381
\(135\) 12.5623 1.08119
\(136\) 0.236068 0.0202427
\(137\) −15.6180 −1.33434 −0.667169 0.744906i \(-0.732494\pi\)
−0.667169 + 0.744906i \(0.732494\pi\)
\(138\) −5.09017 −0.433304
\(139\) 10.5623 0.895883 0.447942 0.894063i \(-0.352157\pi\)
0.447942 + 0.894063i \(0.352157\pi\)
\(140\) 15.3262 1.29530
\(141\) −4.23607 −0.356741
\(142\) 9.70820 0.814694
\(143\) −0.236068 −0.0197410
\(144\) −2.61803 −0.218169
\(145\) −3.09017 −0.256625
\(146\) −10.7082 −0.886217
\(147\) 6.76393 0.557880
\(148\) −8.70820 −0.715810
\(149\) 0.326238 0.0267265 0.0133632 0.999911i \(-0.495746\pi\)
0.0133632 + 0.999911i \(0.495746\pi\)
\(150\) 5.00000 0.408248
\(151\) −9.94427 −0.809253 −0.404627 0.914482i \(-0.632599\pi\)
−0.404627 + 0.914482i \(0.632599\pi\)
\(152\) 2.00000 0.162221
\(153\) −0.618034 −0.0499651
\(154\) −1.00000 −0.0805823
\(155\) −27.0344 −2.17146
\(156\) −0.618034 −0.0494823
\(157\) 22.6525 1.80786 0.903932 0.427676i \(-0.140668\pi\)
0.903932 + 0.427676i \(0.140668\pi\)
\(158\) −13.7984 −1.09774
\(159\) 0 0
\(160\) −3.61803 −0.286031
\(161\) 34.8885 2.74960
\(162\) 5.70820 0.448479
\(163\) 17.3262 1.35710 0.678548 0.734556i \(-0.262610\pi\)
0.678548 + 0.734556i \(0.262610\pi\)
\(164\) 7.70820 0.601910
\(165\) −0.527864 −0.0410942
\(166\) −5.23607 −0.406398
\(167\) 3.61803 0.279972 0.139986 0.990153i \(-0.455294\pi\)
0.139986 + 0.990153i \(0.455294\pi\)
\(168\) −2.61803 −0.201986
\(169\) −12.0000 −0.923077
\(170\) −0.854102 −0.0655066
\(171\) −5.23607 −0.400412
\(172\) 2.23607 0.170499
\(173\) −7.47214 −0.568096 −0.284048 0.958810i \(-0.591678\pi\)
−0.284048 + 0.958810i \(0.591678\pi\)
\(174\) 0.527864 0.0400173
\(175\) −34.2705 −2.59061
\(176\) 0.236068 0.0177943
\(177\) −4.05573 −0.304847
\(178\) −9.79837 −0.734419
\(179\) 18.7082 1.39832 0.699158 0.714967i \(-0.253558\pi\)
0.699158 + 0.714967i \(0.253558\pi\)
\(180\) 9.47214 0.706011
\(181\) 19.4164 1.44321 0.721605 0.692305i \(-0.243405\pi\)
0.721605 + 0.692305i \(0.243405\pi\)
\(182\) 4.23607 0.313998
\(183\) −3.38197 −0.250002
\(184\) −8.23607 −0.607171
\(185\) 31.5066 2.31641
\(186\) 4.61803 0.338611
\(187\) 0.0557281 0.00407524
\(188\) −6.85410 −0.499887
\(189\) 14.7082 1.06986
\(190\) −7.23607 −0.524960
\(191\) −5.05573 −0.365820 −0.182910 0.983130i \(-0.558552\pi\)
−0.182910 + 0.983130i \(0.558552\pi\)
\(192\) 0.618034 0.0446028
\(193\) −8.23607 −0.592845 −0.296423 0.955057i \(-0.595794\pi\)
−0.296423 + 0.955057i \(0.595794\pi\)
\(194\) 12.6525 0.908395
\(195\) 2.23607 0.160128
\(196\) 10.9443 0.781734
\(197\) −22.3820 −1.59465 −0.797325 0.603551i \(-0.793752\pi\)
−0.797325 + 0.603551i \(0.793752\pi\)
\(198\) −0.618034 −0.0439218
\(199\) −5.38197 −0.381517 −0.190759 0.981637i \(-0.561095\pi\)
−0.190759 + 0.981637i \(0.561095\pi\)
\(200\) 8.09017 0.572061
\(201\) 1.09017 0.0768947
\(202\) −5.09017 −0.358143
\(203\) −3.61803 −0.253936
\(204\) 0.145898 0.0102149
\(205\) −27.8885 −1.94782
\(206\) 10.8541 0.756241
\(207\) 21.5623 1.49868
\(208\) −1.00000 −0.0693375
\(209\) 0.472136 0.0326583
\(210\) 9.47214 0.653639
\(211\) 0.854102 0.0587988 0.0293994 0.999568i \(-0.490641\pi\)
0.0293994 + 0.999568i \(0.490641\pi\)
\(212\) 0 0
\(213\) 6.00000 0.411113
\(214\) −12.2361 −0.836440
\(215\) −8.09017 −0.551745
\(216\) −3.47214 −0.236249
\(217\) −31.6525 −2.14871
\(218\) −4.70820 −0.318880
\(219\) −6.61803 −0.447205
\(220\) −0.854102 −0.0575835
\(221\) −0.236068 −0.0158797
\(222\) −5.38197 −0.361214
\(223\) −5.32624 −0.356671 −0.178336 0.983970i \(-0.557071\pi\)
−0.178336 + 0.983970i \(0.557071\pi\)
\(224\) −4.23607 −0.283034
\(225\) −21.1803 −1.41202
\(226\) −15.3820 −1.02319
\(227\) −24.0344 −1.59522 −0.797611 0.603172i \(-0.793903\pi\)
−0.797611 + 0.603172i \(0.793903\pi\)
\(228\) 1.23607 0.0818606
\(229\) −10.7082 −0.707618 −0.353809 0.935318i \(-0.615114\pi\)
−0.353809 + 0.935318i \(0.615114\pi\)
\(230\) 29.7984 1.96485
\(231\) −0.618034 −0.0406637
\(232\) 0.854102 0.0560745
\(233\) 4.23607 0.277514 0.138757 0.990326i \(-0.455689\pi\)
0.138757 + 0.990326i \(0.455689\pi\)
\(234\) 2.61803 0.171146
\(235\) 24.7984 1.61767
\(236\) −6.56231 −0.427170
\(237\) −8.52786 −0.553944
\(238\) −1.00000 −0.0648204
\(239\) −19.2705 −1.24651 −0.623253 0.782020i \(-0.714189\pi\)
−0.623253 + 0.782020i \(0.714189\pi\)
\(240\) −2.23607 −0.144338
\(241\) −4.70820 −0.303282 −0.151641 0.988436i \(-0.548456\pi\)
−0.151641 + 0.988436i \(0.548456\pi\)
\(242\) −10.9443 −0.703524
\(243\) 13.9443 0.894525
\(244\) −5.47214 −0.350318
\(245\) −39.5967 −2.52974
\(246\) 4.76393 0.303737
\(247\) −2.00000 −0.127257
\(248\) 7.47214 0.474481
\(249\) −3.23607 −0.205077
\(250\) −11.1803 −0.707107
\(251\) 24.0902 1.52056 0.760279 0.649597i \(-0.225062\pi\)
0.760279 + 0.649597i \(0.225062\pi\)
\(252\) 11.0902 0.698615
\(253\) −1.94427 −0.122235
\(254\) 20.7984 1.30501
\(255\) −0.527864 −0.0330561
\(256\) 1.00000 0.0625000
\(257\) −5.47214 −0.341342 −0.170671 0.985328i \(-0.554594\pi\)
−0.170671 + 0.985328i \(0.554594\pi\)
\(258\) 1.38197 0.0860374
\(259\) 36.8885 2.29214
\(260\) 3.61803 0.224381
\(261\) −2.23607 −0.138409
\(262\) 2.76393 0.170756
\(263\) 23.2705 1.43492 0.717461 0.696599i \(-0.245304\pi\)
0.717461 + 0.696599i \(0.245304\pi\)
\(264\) 0.145898 0.00897940
\(265\) 0 0
\(266\) −8.47214 −0.519460
\(267\) −6.05573 −0.370605
\(268\) 1.76393 0.107749
\(269\) 1.00000 0.0609711
\(270\) 12.5623 0.764518
\(271\) 24.1246 1.46547 0.732733 0.680516i \(-0.238244\pi\)
0.732733 + 0.680516i \(0.238244\pi\)
\(272\) 0.236068 0.0143137
\(273\) 2.61803 0.158451
\(274\) −15.6180 −0.943520
\(275\) 1.90983 0.115167
\(276\) −5.09017 −0.306392
\(277\) −21.3820 −1.28472 −0.642359 0.766404i \(-0.722044\pi\)
−0.642359 + 0.766404i \(0.722044\pi\)
\(278\) 10.5623 0.633485
\(279\) −19.5623 −1.17116
\(280\) 15.3262 0.915918
\(281\) −29.1803 −1.74075 −0.870377 0.492387i \(-0.836125\pi\)
−0.870377 + 0.492387i \(0.836125\pi\)
\(282\) −4.23607 −0.252254
\(283\) 5.76393 0.342630 0.171315 0.985216i \(-0.445198\pi\)
0.171315 + 0.985216i \(0.445198\pi\)
\(284\) 9.70820 0.576076
\(285\) −4.47214 −0.264906
\(286\) −0.236068 −0.0139590
\(287\) −32.6525 −1.92741
\(288\) −2.61803 −0.154269
\(289\) −16.9443 −0.996722
\(290\) −3.09017 −0.181461
\(291\) 7.81966 0.458397
\(292\) −10.7082 −0.626650
\(293\) 7.52786 0.439783 0.219891 0.975524i \(-0.429430\pi\)
0.219891 + 0.975524i \(0.429430\pi\)
\(294\) 6.76393 0.394481
\(295\) 23.7426 1.38235
\(296\) −8.70820 −0.506154
\(297\) −0.819660 −0.0475615
\(298\) 0.326238 0.0188985
\(299\) 8.23607 0.476304
\(300\) 5.00000 0.288675
\(301\) −9.47214 −0.545965
\(302\) −9.94427 −0.572229
\(303\) −3.14590 −0.180727
\(304\) 2.00000 0.114708
\(305\) 19.7984 1.13365
\(306\) −0.618034 −0.0353307
\(307\) 32.1246 1.83345 0.916724 0.399521i \(-0.130823\pi\)
0.916724 + 0.399521i \(0.130823\pi\)
\(308\) −1.00000 −0.0569803
\(309\) 6.70820 0.381616
\(310\) −27.0344 −1.53545
\(311\) −17.2361 −0.977368 −0.488684 0.872461i \(-0.662523\pi\)
−0.488684 + 0.872461i \(0.662523\pi\)
\(312\) −0.618034 −0.0349893
\(313\) 13.1459 0.743050 0.371525 0.928423i \(-0.378835\pi\)
0.371525 + 0.928423i \(0.378835\pi\)
\(314\) 22.6525 1.27835
\(315\) −40.1246 −2.26077
\(316\) −13.7984 −0.776219
\(317\) −30.2361 −1.69823 −0.849113 0.528211i \(-0.822863\pi\)
−0.849113 + 0.528211i \(0.822863\pi\)
\(318\) 0 0
\(319\) 0.201626 0.0112889
\(320\) −3.61803 −0.202254
\(321\) −7.56231 −0.422087
\(322\) 34.8885 1.94426
\(323\) 0.472136 0.0262703
\(324\) 5.70820 0.317122
\(325\) −8.09017 −0.448762
\(326\) 17.3262 0.959612
\(327\) −2.90983 −0.160914
\(328\) 7.70820 0.425614
\(329\) 29.0344 1.60072
\(330\) −0.527864 −0.0290580
\(331\) 0.819660 0.0450526 0.0225263 0.999746i \(-0.492829\pi\)
0.0225263 + 0.999746i \(0.492829\pi\)
\(332\) −5.23607 −0.287367
\(333\) 22.7984 1.24934
\(334\) 3.61803 0.197970
\(335\) −6.38197 −0.348684
\(336\) −2.61803 −0.142825
\(337\) −27.9787 −1.52410 −0.762049 0.647520i \(-0.775806\pi\)
−0.762049 + 0.647520i \(0.775806\pi\)
\(338\) −12.0000 −0.652714
\(339\) −9.50658 −0.516326
\(340\) −0.854102 −0.0463202
\(341\) 1.76393 0.0955223
\(342\) −5.23607 −0.283134
\(343\) −16.7082 −0.902158
\(344\) 2.23607 0.120561
\(345\) 18.4164 0.991506
\(346\) −7.47214 −0.401705
\(347\) 13.3607 0.717239 0.358619 0.933484i \(-0.383248\pi\)
0.358619 + 0.933484i \(0.383248\pi\)
\(348\) 0.527864 0.0282965
\(349\) 21.2705 1.13858 0.569292 0.822135i \(-0.307217\pi\)
0.569292 + 0.822135i \(0.307217\pi\)
\(350\) −34.2705 −1.83184
\(351\) 3.47214 0.185329
\(352\) 0.236068 0.0125825
\(353\) −9.85410 −0.524481 −0.262240 0.965003i \(-0.584461\pi\)
−0.262240 + 0.965003i \(0.584461\pi\)
\(354\) −4.05573 −0.215560
\(355\) −35.1246 −1.86422
\(356\) −9.79837 −0.519313
\(357\) −0.618034 −0.0327098
\(358\) 18.7082 0.988759
\(359\) −16.9443 −0.894284 −0.447142 0.894463i \(-0.647558\pi\)
−0.447142 + 0.894463i \(0.647558\pi\)
\(360\) 9.47214 0.499225
\(361\) −15.0000 −0.789474
\(362\) 19.4164 1.02050
\(363\) −6.76393 −0.355014
\(364\) 4.23607 0.222030
\(365\) 38.7426 2.02788
\(366\) −3.38197 −0.176778
\(367\) −9.20163 −0.480321 −0.240160 0.970733i \(-0.577200\pi\)
−0.240160 + 0.970733i \(0.577200\pi\)
\(368\) −8.23607 −0.429335
\(369\) −20.1803 −1.05055
\(370\) 31.5066 1.63795
\(371\) 0 0
\(372\) 4.61803 0.239434
\(373\) 6.94427 0.359561 0.179780 0.983707i \(-0.442461\pi\)
0.179780 + 0.983707i \(0.442461\pi\)
\(374\) 0.0557281 0.00288163
\(375\) −6.90983 −0.356822
\(376\) −6.85410 −0.353473
\(377\) −0.854102 −0.0439885
\(378\) 14.7082 0.756508
\(379\) 6.14590 0.315694 0.157847 0.987464i \(-0.449545\pi\)
0.157847 + 0.987464i \(0.449545\pi\)
\(380\) −7.23607 −0.371202
\(381\) 12.8541 0.658536
\(382\) −5.05573 −0.258674
\(383\) −17.3820 −0.888177 −0.444088 0.895983i \(-0.646472\pi\)
−0.444088 + 0.895983i \(0.646472\pi\)
\(384\) 0.618034 0.0315389
\(385\) 3.61803 0.184392
\(386\) −8.23607 −0.419205
\(387\) −5.85410 −0.297581
\(388\) 12.6525 0.642332
\(389\) 28.9443 1.46753 0.733766 0.679402i \(-0.237761\pi\)
0.733766 + 0.679402i \(0.237761\pi\)
\(390\) 2.23607 0.113228
\(391\) −1.94427 −0.0983261
\(392\) 10.9443 0.552769
\(393\) 1.70820 0.0861675
\(394\) −22.3820 −1.12759
\(395\) 49.9230 2.51190
\(396\) −0.618034 −0.0310574
\(397\) −34.2705 −1.71999 −0.859994 0.510304i \(-0.829533\pi\)
−0.859994 + 0.510304i \(0.829533\pi\)
\(398\) −5.38197 −0.269774
\(399\) −5.23607 −0.262131
\(400\) 8.09017 0.404508
\(401\) −4.27051 −0.213259 −0.106630 0.994299i \(-0.534006\pi\)
−0.106630 + 0.994299i \(0.534006\pi\)
\(402\) 1.09017 0.0543727
\(403\) −7.47214 −0.372214
\(404\) −5.09017 −0.253245
\(405\) −20.6525 −1.02623
\(406\) −3.61803 −0.179560
\(407\) −2.05573 −0.101899
\(408\) 0.145898 0.00722303
\(409\) −11.6525 −0.576178 −0.288089 0.957604i \(-0.593020\pi\)
−0.288089 + 0.957604i \(0.593020\pi\)
\(410\) −27.8885 −1.37732
\(411\) −9.65248 −0.476122
\(412\) 10.8541 0.534743
\(413\) 27.7984 1.36787
\(414\) 21.5623 1.05973
\(415\) 18.9443 0.929938
\(416\) −1.00000 −0.0490290
\(417\) 6.52786 0.319671
\(418\) 0.472136 0.0230929
\(419\) −17.4377 −0.851887 −0.425944 0.904750i \(-0.640058\pi\)
−0.425944 + 0.904750i \(0.640058\pi\)
\(420\) 9.47214 0.462193
\(421\) 11.4721 0.559118 0.279559 0.960129i \(-0.409812\pi\)
0.279559 + 0.960129i \(0.409812\pi\)
\(422\) 0.854102 0.0415770
\(423\) 17.9443 0.872480
\(424\) 0 0
\(425\) 1.90983 0.0926404
\(426\) 6.00000 0.290701
\(427\) 23.1803 1.12178
\(428\) −12.2361 −0.591453
\(429\) −0.145898 −0.00704402
\(430\) −8.09017 −0.390143
\(431\) −15.4721 −0.745267 −0.372633 0.927979i \(-0.621545\pi\)
−0.372633 + 0.927979i \(0.621545\pi\)
\(432\) −3.47214 −0.167053
\(433\) 12.2016 0.586373 0.293186 0.956055i \(-0.405284\pi\)
0.293186 + 0.956055i \(0.405284\pi\)
\(434\) −31.6525 −1.51937
\(435\) −1.90983 −0.0915693
\(436\) −4.70820 −0.225482
\(437\) −16.4721 −0.787969
\(438\) −6.61803 −0.316222
\(439\) 19.1459 0.913784 0.456892 0.889522i \(-0.348963\pi\)
0.456892 + 0.889522i \(0.348963\pi\)
\(440\) −0.854102 −0.0407177
\(441\) −28.6525 −1.36440
\(442\) −0.236068 −0.0112286
\(443\) 33.4164 1.58766 0.793831 0.608139i \(-0.208084\pi\)
0.793831 + 0.608139i \(0.208084\pi\)
\(444\) −5.38197 −0.255417
\(445\) 35.4508 1.68053
\(446\) −5.32624 −0.252205
\(447\) 0.201626 0.00953659
\(448\) −4.23607 −0.200135
\(449\) 2.90983 0.137323 0.0686617 0.997640i \(-0.478127\pi\)
0.0686617 + 0.997640i \(0.478127\pi\)
\(450\) −21.1803 −0.998451
\(451\) 1.81966 0.0856844
\(452\) −15.3820 −0.723507
\(453\) −6.14590 −0.288759
\(454\) −24.0344 −1.12799
\(455\) −15.3262 −0.718505
\(456\) 1.23607 0.0578842
\(457\) 26.7984 1.25358 0.626788 0.779190i \(-0.284369\pi\)
0.626788 + 0.779190i \(0.284369\pi\)
\(458\) −10.7082 −0.500362
\(459\) −0.819660 −0.0382585
\(460\) 29.7984 1.38936
\(461\) 21.1246 0.983871 0.491936 0.870632i \(-0.336290\pi\)
0.491936 + 0.870632i \(0.336290\pi\)
\(462\) −0.618034 −0.0287535
\(463\) −20.0902 −0.933669 −0.466835 0.884345i \(-0.654606\pi\)
−0.466835 + 0.884345i \(0.654606\pi\)
\(464\) 0.854102 0.0396507
\(465\) −16.7082 −0.774824
\(466\) 4.23607 0.196232
\(467\) −6.58359 −0.304652 −0.152326 0.988330i \(-0.548676\pi\)
−0.152326 + 0.988330i \(0.548676\pi\)
\(468\) 2.61803 0.121019
\(469\) −7.47214 −0.345031
\(470\) 24.7984 1.14386
\(471\) 14.0000 0.645086
\(472\) −6.56231 −0.302055
\(473\) 0.527864 0.0242712
\(474\) −8.52786 −0.391698
\(475\) 16.1803 0.742405
\(476\) −1.00000 −0.0458349
\(477\) 0 0
\(478\) −19.2705 −0.881413
\(479\) −36.5967 −1.67215 −0.836074 0.548617i \(-0.815155\pi\)
−0.836074 + 0.548617i \(0.815155\pi\)
\(480\) −2.23607 −0.102062
\(481\) 8.70820 0.397060
\(482\) −4.70820 −0.214453
\(483\) 21.5623 0.981119
\(484\) −10.9443 −0.497467
\(485\) −45.7771 −2.07863
\(486\) 13.9443 0.632525
\(487\) −14.8328 −0.672139 −0.336070 0.941837i \(-0.609098\pi\)
−0.336070 + 0.941837i \(0.609098\pi\)
\(488\) −5.47214 −0.247712
\(489\) 10.7082 0.484242
\(490\) −39.5967 −1.78880
\(491\) 6.76393 0.305252 0.152626 0.988284i \(-0.451227\pi\)
0.152626 + 0.988284i \(0.451227\pi\)
\(492\) 4.76393 0.214775
\(493\) 0.201626 0.00908078
\(494\) −2.00000 −0.0899843
\(495\) 2.23607 0.100504
\(496\) 7.47214 0.335509
\(497\) −41.1246 −1.84469
\(498\) −3.23607 −0.145012
\(499\) 11.9443 0.534699 0.267350 0.963600i \(-0.413852\pi\)
0.267350 + 0.963600i \(0.413852\pi\)
\(500\) −11.1803 −0.500000
\(501\) 2.23607 0.0999001
\(502\) 24.0902 1.07520
\(503\) 34.3607 1.53207 0.766033 0.642801i \(-0.222228\pi\)
0.766033 + 0.642801i \(0.222228\pi\)
\(504\) 11.0902 0.493995
\(505\) 18.4164 0.819519
\(506\) −1.94427 −0.0864334
\(507\) −7.41641 −0.329374
\(508\) 20.7984 0.922779
\(509\) 26.6738 1.18229 0.591147 0.806564i \(-0.298675\pi\)
0.591147 + 0.806564i \(0.298675\pi\)
\(510\) −0.527864 −0.0233742
\(511\) 45.3607 2.00664
\(512\) 1.00000 0.0441942
\(513\) −6.94427 −0.306597
\(514\) −5.47214 −0.241366
\(515\) −39.2705 −1.73047
\(516\) 1.38197 0.0608377
\(517\) −1.61803 −0.0711611
\(518\) 36.8885 1.62079
\(519\) −4.61803 −0.202709
\(520\) 3.61803 0.158661
\(521\) 30.4164 1.33257 0.666284 0.745699i \(-0.267884\pi\)
0.666284 + 0.745699i \(0.267884\pi\)
\(522\) −2.23607 −0.0978700
\(523\) 27.6180 1.20765 0.603826 0.797116i \(-0.293642\pi\)
0.603826 + 0.797116i \(0.293642\pi\)
\(524\) 2.76393 0.120743
\(525\) −21.1803 −0.924386
\(526\) 23.2705 1.01464
\(527\) 1.76393 0.0768381
\(528\) 0.145898 0.00634940
\(529\) 44.8328 1.94925
\(530\) 0 0
\(531\) 17.1803 0.745563
\(532\) −8.47214 −0.367314
\(533\) −7.70820 −0.333879
\(534\) −6.05573 −0.262057
\(535\) 44.2705 1.91398
\(536\) 1.76393 0.0761903
\(537\) 11.5623 0.498950
\(538\) 1.00000 0.0431131
\(539\) 2.58359 0.111283
\(540\) 12.5623 0.540596
\(541\) 20.4721 0.880166 0.440083 0.897957i \(-0.354949\pi\)
0.440083 + 0.897957i \(0.354949\pi\)
\(542\) 24.1246 1.03624
\(543\) 12.0000 0.514969
\(544\) 0.236068 0.0101213
\(545\) 17.0344 0.729675
\(546\) 2.61803 0.112042
\(547\) 3.20163 0.136892 0.0684458 0.997655i \(-0.478196\pi\)
0.0684458 + 0.997655i \(0.478196\pi\)
\(548\) −15.6180 −0.667169
\(549\) 14.3262 0.611429
\(550\) 1.90983 0.0814354
\(551\) 1.70820 0.0727719
\(552\) −5.09017 −0.216652
\(553\) 58.4508 2.48558
\(554\) −21.3820 −0.908433
\(555\) 19.4721 0.826546
\(556\) 10.5623 0.447942
\(557\) 28.2918 1.19876 0.599381 0.800464i \(-0.295413\pi\)
0.599381 + 0.800464i \(0.295413\pi\)
\(558\) −19.5623 −0.828138
\(559\) −2.23607 −0.0945756
\(560\) 15.3262 0.647652
\(561\) 0.0344419 0.00145414
\(562\) −29.1803 −1.23090
\(563\) 7.29180 0.307313 0.153656 0.988124i \(-0.450895\pi\)
0.153656 + 0.988124i \(0.450895\pi\)
\(564\) −4.23607 −0.178371
\(565\) 55.6525 2.34132
\(566\) 5.76393 0.242276
\(567\) −24.1803 −1.01548
\(568\) 9.70820 0.407347
\(569\) −5.90983 −0.247753 −0.123876 0.992298i \(-0.539533\pi\)
−0.123876 + 0.992298i \(0.539533\pi\)
\(570\) −4.47214 −0.187317
\(571\) 10.0000 0.418487 0.209243 0.977864i \(-0.432900\pi\)
0.209243 + 0.977864i \(0.432900\pi\)
\(572\) −0.236068 −0.00987050
\(573\) −3.12461 −0.130533
\(574\) −32.6525 −1.36289
\(575\) −66.6312 −2.77871
\(576\) −2.61803 −0.109085
\(577\) 3.00000 0.124892 0.0624458 0.998048i \(-0.480110\pi\)
0.0624458 + 0.998048i \(0.480110\pi\)
\(578\) −16.9443 −0.704789
\(579\) −5.09017 −0.211540
\(580\) −3.09017 −0.128312
\(581\) 22.1803 0.920196
\(582\) 7.81966 0.324135
\(583\) 0 0
\(584\) −10.7082 −0.443109
\(585\) −9.47214 −0.391625
\(586\) 7.52786 0.310973
\(587\) −10.6738 −0.440553 −0.220277 0.975437i \(-0.570696\pi\)
−0.220277 + 0.975437i \(0.570696\pi\)
\(588\) 6.76393 0.278940
\(589\) 14.9443 0.615768
\(590\) 23.7426 0.977469
\(591\) −13.8328 −0.569006
\(592\) −8.70820 −0.357905
\(593\) −0.583592 −0.0239653 −0.0119826 0.999928i \(-0.503814\pi\)
−0.0119826 + 0.999928i \(0.503814\pi\)
\(594\) −0.819660 −0.0336311
\(595\) 3.61803 0.148325
\(596\) 0.326238 0.0133632
\(597\) −3.32624 −0.136134
\(598\) 8.23607 0.336798
\(599\) −15.5967 −0.637266 −0.318633 0.947878i \(-0.603224\pi\)
−0.318633 + 0.947878i \(0.603224\pi\)
\(600\) 5.00000 0.204124
\(601\) −25.6869 −1.04779 −0.523896 0.851782i \(-0.675522\pi\)
−0.523896 + 0.851782i \(0.675522\pi\)
\(602\) −9.47214 −0.386055
\(603\) −4.61803 −0.188061
\(604\) −9.94427 −0.404627
\(605\) 39.5967 1.60984
\(606\) −3.14590 −0.127793
\(607\) 47.5755 1.93103 0.965514 0.260350i \(-0.0838381\pi\)
0.965514 + 0.260350i \(0.0838381\pi\)
\(608\) 2.00000 0.0811107
\(609\) −2.23607 −0.0906100
\(610\) 19.7984 0.801613
\(611\) 6.85410 0.277287
\(612\) −0.618034 −0.0249825
\(613\) −26.5279 −1.07145 −0.535725 0.844392i \(-0.679962\pi\)
−0.535725 + 0.844392i \(0.679962\pi\)
\(614\) 32.1246 1.29644
\(615\) −17.2361 −0.695025
\(616\) −1.00000 −0.0402911
\(617\) 36.1591 1.45571 0.727854 0.685732i \(-0.240518\pi\)
0.727854 + 0.685732i \(0.240518\pi\)
\(618\) 6.70820 0.269844
\(619\) −40.8328 −1.64121 −0.820605 0.571496i \(-0.806363\pi\)
−0.820605 + 0.571496i \(0.806363\pi\)
\(620\) −27.0344 −1.08573
\(621\) 28.5967 1.14755
\(622\) −17.2361 −0.691103
\(623\) 41.5066 1.66293
\(624\) −0.618034 −0.0247412
\(625\) 0 0
\(626\) 13.1459 0.525416
\(627\) 0.291796 0.0116532
\(628\) 22.6525 0.903932
\(629\) −2.05573 −0.0819672
\(630\) −40.1246 −1.59860
\(631\) 5.20163 0.207073 0.103537 0.994626i \(-0.466984\pi\)
0.103537 + 0.994626i \(0.466984\pi\)
\(632\) −13.7984 −0.548870
\(633\) 0.527864 0.0209807
\(634\) −30.2361 −1.20083
\(635\) −75.2492 −2.98617
\(636\) 0 0
\(637\) −10.9443 −0.433628
\(638\) 0.201626 0.00798245
\(639\) −25.4164 −1.00546
\(640\) −3.61803 −0.143015
\(641\) 3.18034 0.125616 0.0628079 0.998026i \(-0.479994\pi\)
0.0628079 + 0.998026i \(0.479994\pi\)
\(642\) −7.56231 −0.298460
\(643\) −4.14590 −0.163498 −0.0817491 0.996653i \(-0.526051\pi\)
−0.0817491 + 0.996653i \(0.526051\pi\)
\(644\) 34.8885 1.37480
\(645\) −5.00000 −0.196875
\(646\) 0.472136 0.0185759
\(647\) −2.81966 −0.110852 −0.0554261 0.998463i \(-0.517652\pi\)
−0.0554261 + 0.998463i \(0.517652\pi\)
\(648\) 5.70820 0.224239
\(649\) −1.54915 −0.0608095
\(650\) −8.09017 −0.317323
\(651\) −19.5623 −0.766707
\(652\) 17.3262 0.678548
\(653\) −27.5410 −1.07776 −0.538882 0.842381i \(-0.681153\pi\)
−0.538882 + 0.842381i \(0.681153\pi\)
\(654\) −2.90983 −0.113783
\(655\) −10.0000 −0.390732
\(656\) 7.70820 0.300955
\(657\) 28.0344 1.09373
\(658\) 29.0344 1.13188
\(659\) −31.3262 −1.22030 −0.610148 0.792287i \(-0.708890\pi\)
−0.610148 + 0.792287i \(0.708890\pi\)
\(660\) −0.527864 −0.0205471
\(661\) −20.8541 −0.811131 −0.405565 0.914066i \(-0.632925\pi\)
−0.405565 + 0.914066i \(0.632925\pi\)
\(662\) 0.819660 0.0318570
\(663\) −0.145898 −0.00566621
\(664\) −5.23607 −0.203199
\(665\) 30.6525 1.18865
\(666\) 22.7984 0.883419
\(667\) −7.03444 −0.272375
\(668\) 3.61803 0.139986
\(669\) −3.29180 −0.127268
\(670\) −6.38197 −0.246557
\(671\) −1.29180 −0.0498692
\(672\) −2.61803 −0.100993
\(673\) 10.7082 0.412771 0.206385 0.978471i \(-0.433830\pi\)
0.206385 + 0.978471i \(0.433830\pi\)
\(674\) −27.9787 −1.07770
\(675\) −28.0902 −1.08119
\(676\) −12.0000 −0.461538
\(677\) −17.4377 −0.670185 −0.335093 0.942185i \(-0.608768\pi\)
−0.335093 + 0.942185i \(0.608768\pi\)
\(678\) −9.50658 −0.365098
\(679\) −53.5967 −2.05685
\(680\) −0.854102 −0.0327533
\(681\) −14.8541 −0.569210
\(682\) 1.76393 0.0675444
\(683\) −21.2361 −0.812576 −0.406288 0.913745i \(-0.633177\pi\)
−0.406288 + 0.913745i \(0.633177\pi\)
\(684\) −5.23607 −0.200206
\(685\) 56.5066 2.15901
\(686\) −16.7082 −0.637922
\(687\) −6.61803 −0.252494
\(688\) 2.23607 0.0852493
\(689\) 0 0
\(690\) 18.4164 0.701101
\(691\) −15.1246 −0.575367 −0.287684 0.957725i \(-0.592885\pi\)
−0.287684 + 0.957725i \(0.592885\pi\)
\(692\) −7.47214 −0.284048
\(693\) 2.61803 0.0994509
\(694\) 13.3607 0.507164
\(695\) −38.2148 −1.44957
\(696\) 0.527864 0.0200086
\(697\) 1.81966 0.0689245
\(698\) 21.2705 0.805101
\(699\) 2.61803 0.0990231
\(700\) −34.2705 −1.29530
\(701\) −10.9787 −0.414660 −0.207330 0.978271i \(-0.566477\pi\)
−0.207330 + 0.978271i \(0.566477\pi\)
\(702\) 3.47214 0.131047
\(703\) −17.4164 −0.656872
\(704\) 0.236068 0.00889715
\(705\) 15.3262 0.577220
\(706\) −9.85410 −0.370864
\(707\) 21.5623 0.810934
\(708\) −4.05573 −0.152424
\(709\) −37.9443 −1.42503 −0.712514 0.701658i \(-0.752443\pi\)
−0.712514 + 0.701658i \(0.752443\pi\)
\(710\) −35.1246 −1.31820
\(711\) 36.1246 1.35478
\(712\) −9.79837 −0.367210
\(713\) −61.5410 −2.30473
\(714\) −0.618034 −0.0231293
\(715\) 0.854102 0.0319416
\(716\) 18.7082 0.699158
\(717\) −11.9098 −0.444781
\(718\) −16.9443 −0.632355
\(719\) 42.1803 1.57306 0.786531 0.617551i \(-0.211875\pi\)
0.786531 + 0.617551i \(0.211875\pi\)
\(720\) 9.47214 0.353006
\(721\) −45.9787 −1.71234
\(722\) −15.0000 −0.558242
\(723\) −2.90983 −0.108218
\(724\) 19.4164 0.721605
\(725\) 6.90983 0.256625
\(726\) −6.76393 −0.251033
\(727\) −44.7082 −1.65814 −0.829068 0.559148i \(-0.811128\pi\)
−0.829068 + 0.559148i \(0.811128\pi\)
\(728\) 4.23607 0.156999
\(729\) −8.50658 −0.315058
\(730\) 38.7426 1.43393
\(731\) 0.527864 0.0195238
\(732\) −3.38197 −0.125001
\(733\) −33.3607 −1.23220 −0.616102 0.787666i \(-0.711289\pi\)
−0.616102 + 0.787666i \(0.711289\pi\)
\(734\) −9.20163 −0.339638
\(735\) −24.4721 −0.902668
\(736\) −8.23607 −0.303585
\(737\) 0.416408 0.0153386
\(738\) −20.1803 −0.742849
\(739\) 35.0344 1.28876 0.644381 0.764704i \(-0.277115\pi\)
0.644381 + 0.764704i \(0.277115\pi\)
\(740\) 31.5066 1.15820
\(741\) −1.23607 −0.0454081
\(742\) 0 0
\(743\) −9.20163 −0.337575 −0.168787 0.985652i \(-0.553985\pi\)
−0.168787 + 0.985652i \(0.553985\pi\)
\(744\) 4.61803 0.169305
\(745\) −1.18034 −0.0432443
\(746\) 6.94427 0.254248
\(747\) 13.7082 0.501557
\(748\) 0.0557281 0.00203762
\(749\) 51.8328 1.89393
\(750\) −6.90983 −0.252311
\(751\) 39.8673 1.45478 0.727388 0.686226i \(-0.240734\pi\)
0.727388 + 0.686226i \(0.240734\pi\)
\(752\) −6.85410 −0.249943
\(753\) 14.8885 0.542569
\(754\) −0.854102 −0.0311046
\(755\) 35.9787 1.30940
\(756\) 14.7082 0.534932
\(757\) −38.9443 −1.41545 −0.707727 0.706486i \(-0.750279\pi\)
−0.707727 + 0.706486i \(0.750279\pi\)
\(758\) 6.14590 0.223229
\(759\) −1.20163 −0.0436163
\(760\) −7.23607 −0.262480
\(761\) −1.41641 −0.0513447 −0.0256724 0.999670i \(-0.508173\pi\)
−0.0256724 + 0.999670i \(0.508173\pi\)
\(762\) 12.8541 0.465655
\(763\) 19.9443 0.722031
\(764\) −5.05573 −0.182910
\(765\) 2.23607 0.0808452
\(766\) −17.3820 −0.628036
\(767\) 6.56231 0.236951
\(768\) 0.618034 0.0223014
\(769\) −33.3262 −1.20177 −0.600887 0.799334i \(-0.705186\pi\)
−0.600887 + 0.799334i \(0.705186\pi\)
\(770\) 3.61803 0.130385
\(771\) −3.38197 −0.121799
\(772\) −8.23607 −0.296423
\(773\) 47.5967 1.71194 0.855968 0.517029i \(-0.172962\pi\)
0.855968 + 0.517029i \(0.172962\pi\)
\(774\) −5.85410 −0.210421
\(775\) 60.4508 2.17146
\(776\) 12.6525 0.454197
\(777\) 22.7984 0.817887
\(778\) 28.9443 1.03770
\(779\) 15.4164 0.552350
\(780\) 2.23607 0.0800641
\(781\) 2.29180 0.0820069
\(782\) −1.94427 −0.0695270
\(783\) −2.96556 −0.105980
\(784\) 10.9443 0.390867
\(785\) −81.9574 −2.92519
\(786\) 1.70820 0.0609296
\(787\) −2.34752 −0.0836802 −0.0418401 0.999124i \(-0.513322\pi\)
−0.0418401 + 0.999124i \(0.513322\pi\)
\(788\) −22.3820 −0.797325
\(789\) 14.3820 0.512012
\(790\) 49.9230 1.77618
\(791\) 65.1591 2.31679
\(792\) −0.618034 −0.0219609
\(793\) 5.47214 0.194321
\(794\) −34.2705 −1.21621
\(795\) 0 0
\(796\) −5.38197 −0.190759
\(797\) 13.2016 0.467626 0.233813 0.972282i \(-0.424880\pi\)
0.233813 + 0.972282i \(0.424880\pi\)
\(798\) −5.23607 −0.185355
\(799\) −1.61803 −0.0572419
\(800\) 8.09017 0.286031
\(801\) 25.6525 0.906386
\(802\) −4.27051 −0.150797
\(803\) −2.52786 −0.0892064
\(804\) 1.09017 0.0384473
\(805\) −126.228 −4.44895
\(806\) −7.47214 −0.263195
\(807\) 0.618034 0.0217558
\(808\) −5.09017 −0.179072
\(809\) 15.5279 0.545931 0.272965 0.962024i \(-0.411996\pi\)
0.272965 + 0.962024i \(0.411996\pi\)
\(810\) −20.6525 −0.725654
\(811\) −47.5967 −1.67135 −0.835674 0.549226i \(-0.814923\pi\)
−0.835674 + 0.549226i \(0.814923\pi\)
\(812\) −3.61803 −0.126968
\(813\) 14.9098 0.522911
\(814\) −2.05573 −0.0720532
\(815\) −62.6869 −2.19583
\(816\) 0.145898 0.00510745
\(817\) 4.47214 0.156460
\(818\) −11.6525 −0.407419
\(819\) −11.0902 −0.387522
\(820\) −27.8885 −0.973910
\(821\) 38.3607 1.33880 0.669398 0.742904i \(-0.266552\pi\)
0.669398 + 0.742904i \(0.266552\pi\)
\(822\) −9.65248 −0.336669
\(823\) −50.8885 −1.77386 −0.886932 0.461901i \(-0.847168\pi\)
−0.886932 + 0.461901i \(0.847168\pi\)
\(824\) 10.8541 0.378121
\(825\) 1.18034 0.0410942
\(826\) 27.7984 0.967229
\(827\) −37.7426 −1.31244 −0.656220 0.754569i \(-0.727846\pi\)
−0.656220 + 0.754569i \(0.727846\pi\)
\(828\) 21.5623 0.749342
\(829\) 43.7984 1.52118 0.760590 0.649232i \(-0.224910\pi\)
0.760590 + 0.649232i \(0.224910\pi\)
\(830\) 18.9443 0.657565
\(831\) −13.2148 −0.458416
\(832\) −1.00000 −0.0346688
\(833\) 2.58359 0.0895162
\(834\) 6.52786 0.226041
\(835\) −13.0902 −0.453004
\(836\) 0.472136 0.0163292
\(837\) −25.9443 −0.896765
\(838\) −17.4377 −0.602375
\(839\) 14.4508 0.498899 0.249449 0.968388i \(-0.419750\pi\)
0.249449 + 0.968388i \(0.419750\pi\)
\(840\) 9.47214 0.326820
\(841\) −28.2705 −0.974845
\(842\) 11.4721 0.395356
\(843\) −18.0344 −0.621139
\(844\) 0.854102 0.0293994
\(845\) 43.4164 1.49357
\(846\) 17.9443 0.616937
\(847\) 46.3607 1.59297
\(848\) 0 0
\(849\) 3.56231 0.122258
\(850\) 1.90983 0.0655066
\(851\) 71.7214 2.45858
\(852\) 6.00000 0.205557
\(853\) 51.8115 1.77399 0.886996 0.461776i \(-0.152788\pi\)
0.886996 + 0.461776i \(0.152788\pi\)
\(854\) 23.1803 0.793215
\(855\) 18.9443 0.647880
\(856\) −12.2361 −0.418220
\(857\) −42.0689 −1.43705 −0.718523 0.695503i \(-0.755181\pi\)
−0.718523 + 0.695503i \(0.755181\pi\)
\(858\) −0.145898 −0.00498088
\(859\) −12.7426 −0.434773 −0.217387 0.976086i \(-0.569753\pi\)
−0.217387 + 0.976086i \(0.569753\pi\)
\(860\) −8.09017 −0.275873
\(861\) −20.1803 −0.687744
\(862\) −15.4721 −0.526983
\(863\) 17.3262 0.589792 0.294896 0.955529i \(-0.404715\pi\)
0.294896 + 0.955529i \(0.404715\pi\)
\(864\) −3.47214 −0.118124
\(865\) 27.0344 0.919199
\(866\) 12.2016 0.414628
\(867\) −10.4721 −0.355652
\(868\) −31.6525 −1.07436
\(869\) −3.25735 −0.110498
\(870\) −1.90983 −0.0647493
\(871\) −1.76393 −0.0597686
\(872\) −4.70820 −0.159440
\(873\) −33.1246 −1.12110
\(874\) −16.4721 −0.557178
\(875\) 47.3607 1.60108
\(876\) −6.61803 −0.223603
\(877\) 42.5755 1.43767 0.718836 0.695180i \(-0.244675\pi\)
0.718836 + 0.695180i \(0.244675\pi\)
\(878\) 19.1459 0.646143
\(879\) 4.65248 0.156924
\(880\) −0.854102 −0.0287918
\(881\) −44.5066 −1.49946 −0.749732 0.661741i \(-0.769818\pi\)
−0.749732 + 0.661741i \(0.769818\pi\)
\(882\) −28.6525 −0.964779
\(883\) −50.1935 −1.68915 −0.844573 0.535441i \(-0.820146\pi\)
−0.844573 + 0.535441i \(0.820146\pi\)
\(884\) −0.236068 −0.00793983
\(885\) 14.6738 0.493253
\(886\) 33.4164 1.12265
\(887\) −13.8328 −0.464460 −0.232230 0.972661i \(-0.574602\pi\)
−0.232230 + 0.972661i \(0.574602\pi\)
\(888\) −5.38197 −0.180607
\(889\) −88.1033 −2.95489
\(890\) 35.4508 1.18832
\(891\) 1.34752 0.0451438
\(892\) −5.32624 −0.178336
\(893\) −13.7082 −0.458728
\(894\) 0.201626 0.00674339
\(895\) −67.6869 −2.26252
\(896\) −4.23607 −0.141517
\(897\) 5.09017 0.169956
\(898\) 2.90983 0.0971023
\(899\) 6.38197 0.212850
\(900\) −21.1803 −0.706011
\(901\) 0 0
\(902\) 1.81966 0.0605881
\(903\) −5.85410 −0.194812
\(904\) −15.3820 −0.511597
\(905\) −70.2492 −2.33516
\(906\) −6.14590 −0.204184
\(907\) −14.1246 −0.469000 −0.234500 0.972116i \(-0.575345\pi\)
−0.234500 + 0.972116i \(0.575345\pi\)
\(908\) −24.0344 −0.797611
\(909\) 13.3262 0.442003
\(910\) −15.3262 −0.508060
\(911\) 47.3394 1.56842 0.784212 0.620493i \(-0.213067\pi\)
0.784212 + 0.620493i \(0.213067\pi\)
\(912\) 1.23607 0.0409303
\(913\) −1.23607 −0.0409079
\(914\) 26.7984 0.886411
\(915\) 12.2361 0.404512
\(916\) −10.7082 −0.353809
\(917\) −11.7082 −0.386639
\(918\) −0.819660 −0.0270528
\(919\) −5.34752 −0.176399 −0.0881993 0.996103i \(-0.528111\pi\)
−0.0881993 + 0.996103i \(0.528111\pi\)
\(920\) 29.7984 0.982423
\(921\) 19.8541 0.654215
\(922\) 21.1246 0.695702
\(923\) −9.70820 −0.319549
\(924\) −0.618034 −0.0203318
\(925\) −70.4508 −2.31641
\(926\) −20.0902 −0.660204
\(927\) −28.4164 −0.933317
\(928\) 0.854102 0.0280373
\(929\) 17.8197 0.584644 0.292322 0.956320i \(-0.405572\pi\)
0.292322 + 0.956320i \(0.405572\pi\)
\(930\) −16.7082 −0.547884
\(931\) 21.8885 0.717368
\(932\) 4.23607 0.138757
\(933\) −10.6525 −0.348746
\(934\) −6.58359 −0.215422
\(935\) −0.201626 −0.00659388
\(936\) 2.61803 0.0855731
\(937\) −33.0902 −1.08101 −0.540504 0.841341i \(-0.681767\pi\)
−0.540504 + 0.841341i \(0.681767\pi\)
\(938\) −7.47214 −0.243974
\(939\) 8.12461 0.265137
\(940\) 24.7984 0.808834
\(941\) −46.9787 −1.53146 −0.765731 0.643161i \(-0.777623\pi\)
−0.765731 + 0.643161i \(0.777623\pi\)
\(942\) 14.0000 0.456145
\(943\) −63.4853 −2.06737
\(944\) −6.56231 −0.213585
\(945\) −53.2148 −1.73108
\(946\) 0.527864 0.0171623
\(947\) −50.2148 −1.63176 −0.815881 0.578220i \(-0.803747\pi\)
−0.815881 + 0.578220i \(0.803747\pi\)
\(948\) −8.52786 −0.276972
\(949\) 10.7082 0.347603
\(950\) 16.1803 0.524960
\(951\) −18.6869 −0.605965
\(952\) −1.00000 −0.0324102
\(953\) 25.3050 0.819708 0.409854 0.912151i \(-0.365580\pi\)
0.409854 + 0.912151i \(0.365580\pi\)
\(954\) 0 0
\(955\) 18.2918 0.591909
\(956\) −19.2705 −0.623253
\(957\) 0.124612 0.00402813
\(958\) −36.5967 −1.18239
\(959\) 66.1591 2.13639
\(960\) −2.23607 −0.0721688
\(961\) 24.8328 0.801059
\(962\) 8.70820 0.280764
\(963\) 32.0344 1.03230
\(964\) −4.70820 −0.151641
\(965\) 29.7984 0.959244
\(966\) 21.5623 0.693756
\(967\) 1.85410 0.0596239 0.0298119 0.999556i \(-0.490509\pi\)
0.0298119 + 0.999556i \(0.490509\pi\)
\(968\) −10.9443 −0.351762
\(969\) 0.291796 0.00937384
\(970\) −45.7771 −1.46981
\(971\) −0.652476 −0.0209389 −0.0104695 0.999945i \(-0.503333\pi\)
−0.0104695 + 0.999945i \(0.503333\pi\)
\(972\) 13.9443 0.447263
\(973\) −44.7426 −1.43438
\(974\) −14.8328 −0.475274
\(975\) −5.00000 −0.160128
\(976\) −5.47214 −0.175159
\(977\) 31.1459 0.996446 0.498223 0.867049i \(-0.333986\pi\)
0.498223 + 0.867049i \(0.333986\pi\)
\(978\) 10.7082 0.342411
\(979\) −2.31308 −0.0739264
\(980\) −39.5967 −1.26487
\(981\) 12.3262 0.393546
\(982\) 6.76393 0.215846
\(983\) −31.4377 −1.00271 −0.501353 0.865243i \(-0.667164\pi\)
−0.501353 + 0.865243i \(0.667164\pi\)
\(984\) 4.76393 0.151869
\(985\) 80.9787 2.58020
\(986\) 0.201626 0.00642108
\(987\) 17.9443 0.571172
\(988\) −2.00000 −0.0636285
\(989\) −18.4164 −0.585608
\(990\) 2.23607 0.0710669
\(991\) −16.3820 −0.520390 −0.260195 0.965556i \(-0.583787\pi\)
−0.260195 + 0.965556i \(0.583787\pi\)
\(992\) 7.47214 0.237241
\(993\) 0.506578 0.0160758
\(994\) −41.1246 −1.30439
\(995\) 19.4721 0.617308
\(996\) −3.23607 −0.102539
\(997\) −17.5836 −0.556878 −0.278439 0.960454i \(-0.589817\pi\)
−0.278439 + 0.960454i \(0.589817\pi\)
\(998\) 11.9443 0.378089
\(999\) 30.2361 0.956627
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 538.2.a.a.1.2 2
3.2 odd 2 4842.2.a.f.1.2 2
4.3 odd 2 4304.2.a.d.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
538.2.a.a.1.2 2 1.1 even 1 trivial
4304.2.a.d.1.1 2 4.3 odd 2
4842.2.a.f.1.2 2 3.2 odd 2