Properties

Label 538.2.a.c.1.3
Level $538$
Weight $2$
Character 538.1
Self dual yes
Analytic conductor $4.296$
Analytic rank $0$
Dimension $4$
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: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.4913.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 6x^{2} + x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-0.344151\) 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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.34415 q^{3} +1.00000 q^{4} +3.90570 q^{5} -1.34415 q^{6} -0.487928 q^{7} -1.00000 q^{8} -1.19326 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.34415 q^{3} +1.00000 q^{4} +3.90570 q^{5} -1.34415 q^{6} -0.487928 q^{7} -1.00000 q^{8} -1.19326 q^{9} -3.90570 q^{10} +4.04948 q^{11} +1.34415 q^{12} +3.53741 q^{13} +0.487928 q^{14} +5.24985 q^{15} +1.00000 q^{16} -6.37296 q^{17} +1.19326 q^{18} +0.975857 q^{19} +3.90570 q^{20} -0.655849 q^{21} -4.04948 q^{22} +0.750146 q^{23} -1.34415 q^{24} +10.2545 q^{25} -3.53741 q^{26} -5.63637 q^{27} -0.487928 q^{28} -4.08193 q^{29} -5.24985 q^{30} -7.86089 q^{31} -1.00000 q^{32} +5.44311 q^{33} +6.37296 q^{34} -1.90570 q^{35} -1.19326 q^{36} -4.22571 q^{37} -0.975857 q^{38} +4.75481 q^{39} -3.90570 q^{40} -2.07362 q^{41} +0.655849 q^{42} +10.7341 q^{43} +4.04948 q^{44} -4.66052 q^{45} -0.750146 q^{46} +9.31636 q^{47} +1.34415 q^{48} -6.76193 q^{49} -10.2545 q^{50} -8.56622 q^{51} +3.53741 q^{52} +11.1231 q^{53} +5.63637 q^{54} +15.8161 q^{55} +0.487928 q^{56} +1.31170 q^{57} +4.08193 q^{58} -2.98052 q^{59} +5.24985 q^{60} -7.53741 q^{61} +7.86089 q^{62} +0.582225 q^{63} +1.00000 q^{64} +13.8161 q^{65} -5.44311 q^{66} +3.51207 q^{67} -6.37296 q^{68} +1.00831 q^{69} +1.90570 q^{70} +14.1349 q^{71} +1.19326 q^{72} -9.75828 q^{73} +4.22571 q^{74} +13.7836 q^{75} +0.975857 q^{76} -1.97586 q^{77} -4.75481 q^{78} +2.29222 q^{79} +3.90570 q^{80} -3.99636 q^{81} +2.07362 q^{82} -17.1324 q^{83} -0.655849 q^{84} -24.8909 q^{85} -10.7341 q^{86} -5.48673 q^{87} -4.04948 q^{88} +1.86978 q^{89} +4.66052 q^{90} -1.72600 q^{91} +0.750146 q^{92} -10.5662 q^{93} -9.31636 q^{94} +3.81141 q^{95} -1.34415 q^{96} -9.63757 q^{97} +6.76193 q^{98} -4.83208 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + 3 q^{3} + 4 q^{4} + 5 q^{5} - 3 q^{6} - q^{7} - 4 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} + 3 q^{3} + 4 q^{4} + 5 q^{5} - 3 q^{6} - q^{7} - 4 q^{8} + 3 q^{9} - 5 q^{10} + 7 q^{11} + 3 q^{12} + 4 q^{13} + q^{14} + 8 q^{15} + 4 q^{16} + 4 q^{17} - 3 q^{18} + 2 q^{19} + 5 q^{20} - 5 q^{21} - 7 q^{22} + 16 q^{23} - 3 q^{24} - q^{25} - 4 q^{26} + 6 q^{27} - q^{28} - 8 q^{30} - q^{31} - 4 q^{32} + q^{33} - 4 q^{34} + 3 q^{35} + 3 q^{36} - 2 q^{37} - 2 q^{38} + 3 q^{39} - 5 q^{40} - q^{41} + 5 q^{42} + 9 q^{43} + 7 q^{44} + 8 q^{45} - 16 q^{46} + 13 q^{47} + 3 q^{48} - 15 q^{49} + q^{50} + 3 q^{51} + 4 q^{52} + 28 q^{53} - 6 q^{54} + 13 q^{55} + q^{56} + 10 q^{57} + 19 q^{59} + 8 q^{60} - 20 q^{61} + q^{62} + 12 q^{63} + 4 q^{64} + 5 q^{65} - q^{66} + 15 q^{67} + 4 q^{68} - 5 q^{69} - 3 q^{70} + 15 q^{71} - 3 q^{72} - 7 q^{73} + 2 q^{74} + 12 q^{75} + 2 q^{76} - 6 q^{77} - 3 q^{78} - 17 q^{79} + 5 q^{80} + 4 q^{81} + q^{82} + 6 q^{83} - 5 q^{84} - 29 q^{85} - 9 q^{86} - 34 q^{87} - 7 q^{88} + 20 q^{89} - 8 q^{90} - 18 q^{91} + 16 q^{92} - 5 q^{93} - 13 q^{94} - 6 q^{95} - 3 q^{96} + 3 q^{97} + 15 q^{98} - 16 q^{99}+O(q^{100}) \) 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
\(3\) 1.34415 0.776046 0.388023 0.921650i \(-0.373158\pi\)
0.388023 + 0.921650i \(0.373158\pi\)
\(4\) 1.00000 0.500000
\(5\) 3.90570 1.74668 0.873342 0.487108i \(-0.161948\pi\)
0.873342 + 0.487108i \(0.161948\pi\)
\(6\) −1.34415 −0.548747
\(7\) −0.487928 −0.184420 −0.0922098 0.995740i \(-0.529393\pi\)
−0.0922098 + 0.995740i \(0.529393\pi\)
\(8\) −1.00000 −0.353553
\(9\) −1.19326 −0.397753
\(10\) −3.90570 −1.23509
\(11\) 4.04948 1.22096 0.610482 0.792030i \(-0.290976\pi\)
0.610482 + 0.792030i \(0.290976\pi\)
\(12\) 1.34415 0.388023
\(13\) 3.53741 0.981101 0.490550 0.871413i \(-0.336796\pi\)
0.490550 + 0.871413i \(0.336796\pi\)
\(14\) 0.487928 0.130404
\(15\) 5.24985 1.35551
\(16\) 1.00000 0.250000
\(17\) −6.37296 −1.54567 −0.772835 0.634607i \(-0.781162\pi\)
−0.772835 + 0.634607i \(0.781162\pi\)
\(18\) 1.19326 0.281254
\(19\) 0.975857 0.223877 0.111938 0.993715i \(-0.464294\pi\)
0.111938 + 0.993715i \(0.464294\pi\)
\(20\) 3.90570 0.873342
\(21\) −0.655849 −0.143118
\(22\) −4.04948 −0.863352
\(23\) 0.750146 0.156416 0.0782081 0.996937i \(-0.475080\pi\)
0.0782081 + 0.996937i \(0.475080\pi\)
\(24\) −1.34415 −0.274374
\(25\) 10.2545 2.05090
\(26\) −3.53741 −0.693743
\(27\) −5.63637 −1.08472
\(28\) −0.487928 −0.0922098
\(29\) −4.08193 −0.757996 −0.378998 0.925397i \(-0.623731\pi\)
−0.378998 + 0.925397i \(0.623731\pi\)
\(30\) −5.24985 −0.958488
\(31\) −7.86089 −1.41186 −0.705929 0.708283i \(-0.749470\pi\)
−0.705929 + 0.708283i \(0.749470\pi\)
\(32\) −1.00000 −0.176777
\(33\) 5.44311 0.947524
\(34\) 6.37296 1.09295
\(35\) −1.90570 −0.322123
\(36\) −1.19326 −0.198876
\(37\) −4.22571 −0.694703 −0.347351 0.937735i \(-0.612919\pi\)
−0.347351 + 0.937735i \(0.612919\pi\)
\(38\) −0.975857 −0.158305
\(39\) 4.75481 0.761379
\(40\) −3.90570 −0.617546
\(41\) −2.07362 −0.323846 −0.161923 0.986803i \(-0.551770\pi\)
−0.161923 + 0.986803i \(0.551770\pi\)
\(42\) 0.655849 0.101200
\(43\) 10.7341 1.63694 0.818470 0.574549i \(-0.194822\pi\)
0.818470 + 0.574549i \(0.194822\pi\)
\(44\) 4.04948 0.610482
\(45\) −4.66052 −0.694749
\(46\) −0.750146 −0.110603
\(47\) 9.31636 1.35893 0.679466 0.733707i \(-0.262212\pi\)
0.679466 + 0.733707i \(0.262212\pi\)
\(48\) 1.34415 0.194011
\(49\) −6.76193 −0.965989
\(50\) −10.2545 −1.45021
\(51\) −8.56622 −1.19951
\(52\) 3.53741 0.490550
\(53\) 11.1231 1.52788 0.763938 0.645290i \(-0.223263\pi\)
0.763938 + 0.645290i \(0.223263\pi\)
\(54\) 5.63637 0.767013
\(55\) 15.8161 2.13264
\(56\) 0.487928 0.0652022
\(57\) 1.31170 0.173739
\(58\) 4.08193 0.535984
\(59\) −2.98052 −0.388031 −0.194015 0.980998i \(-0.562151\pi\)
−0.194015 + 0.980998i \(0.562151\pi\)
\(60\) 5.24985 0.677753
\(61\) −7.53741 −0.965066 −0.482533 0.875878i \(-0.660283\pi\)
−0.482533 + 0.875878i \(0.660283\pi\)
\(62\) 7.86089 0.998334
\(63\) 0.582225 0.0733534
\(64\) 1.00000 0.125000
\(65\) 13.8161 1.71367
\(66\) −5.44311 −0.670001
\(67\) 3.51207 0.429068 0.214534 0.976717i \(-0.431177\pi\)
0.214534 + 0.976717i \(0.431177\pi\)
\(68\) −6.37296 −0.772835
\(69\) 1.00831 0.121386
\(70\) 1.90570 0.227775
\(71\) 14.1349 1.67750 0.838751 0.544515i \(-0.183286\pi\)
0.838751 + 0.544515i \(0.183286\pi\)
\(72\) 1.19326 0.140627
\(73\) −9.75828 −1.14212 −0.571060 0.820908i \(-0.693468\pi\)
−0.571060 + 0.820908i \(0.693468\pi\)
\(74\) 4.22571 0.491229
\(75\) 13.7836 1.59160
\(76\) 0.975857 0.111938
\(77\) −1.97586 −0.225170
\(78\) −4.75481 −0.538376
\(79\) 2.29222 0.257895 0.128948 0.991651i \(-0.458840\pi\)
0.128948 + 0.991651i \(0.458840\pi\)
\(80\) 3.90570 0.436671
\(81\) −3.99636 −0.444040
\(82\) 2.07362 0.228994
\(83\) −17.1324 −1.88053 −0.940265 0.340444i \(-0.889422\pi\)
−0.940265 + 0.340444i \(0.889422\pi\)
\(84\) −0.655849 −0.0715590
\(85\) −24.8909 −2.69980
\(86\) −10.7341 −1.15749
\(87\) −5.48673 −0.588240
\(88\) −4.04948 −0.431676
\(89\) 1.86978 0.198196 0.0990981 0.995078i \(-0.468404\pi\)
0.0990981 + 0.995078i \(0.468404\pi\)
\(90\) 4.66052 0.491261
\(91\) −1.72600 −0.180934
\(92\) 0.750146 0.0782081
\(93\) −10.5662 −1.09567
\(94\) −9.31636 −0.960910
\(95\) 3.81141 0.391042
\(96\) −1.34415 −0.137187
\(97\) −9.63757 −0.978547 −0.489273 0.872131i \(-0.662738\pi\)
−0.489273 + 0.872131i \(0.662738\pi\)
\(98\) 6.76193 0.683058
\(99\) −4.83208 −0.485642
\(100\) 10.2545 1.02545
\(101\) 5.91748 0.588812 0.294406 0.955680i \(-0.404878\pi\)
0.294406 + 0.955680i \(0.404878\pi\)
\(102\) 8.56622 0.848182
\(103\) 0.480814 0.0473760 0.0236880 0.999719i \(-0.492459\pi\)
0.0236880 + 0.999719i \(0.492459\pi\)
\(104\) −3.53741 −0.346872
\(105\) −2.56155 −0.249982
\(106\) −11.1231 −1.08037
\(107\) 3.82496 0.369773 0.184887 0.982760i \(-0.440808\pi\)
0.184887 + 0.982760i \(0.440808\pi\)
\(108\) −5.63637 −0.542360
\(109\) −17.0866 −1.63660 −0.818300 0.574792i \(-0.805083\pi\)
−0.818300 + 0.574792i \(0.805083\pi\)
\(110\) −15.8161 −1.50800
\(111\) −5.67999 −0.539121
\(112\) −0.487928 −0.0461049
\(113\) 13.3194 1.25299 0.626493 0.779427i \(-0.284490\pi\)
0.626493 + 0.779427i \(0.284490\pi\)
\(114\) −1.31170 −0.122852
\(115\) 2.92985 0.273210
\(116\) −4.08193 −0.378998
\(117\) −4.22105 −0.390236
\(118\) 2.98052 0.274379
\(119\) 3.10955 0.285052
\(120\) −5.24985 −0.479244
\(121\) 5.39830 0.490754
\(122\) 7.53741 0.682405
\(123\) −2.78726 −0.251319
\(124\) −7.86089 −0.705929
\(125\) 20.5226 1.83560
\(126\) −0.582225 −0.0518687
\(127\) 3.18962 0.283033 0.141516 0.989936i \(-0.454802\pi\)
0.141516 + 0.989936i \(0.454802\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 14.4283 1.27034
\(130\) −13.8161 −1.21175
\(131\) −11.5975 −1.01328 −0.506638 0.862159i \(-0.669112\pi\)
−0.506638 + 0.862159i \(0.669112\pi\)
\(132\) 5.44311 0.473762
\(133\) −0.476148 −0.0412873
\(134\) −3.51207 −0.303397
\(135\) −22.0140 −1.89466
\(136\) 6.37296 0.546477
\(137\) −7.33059 −0.626295 −0.313147 0.949705i \(-0.601383\pi\)
−0.313147 + 0.949705i \(0.601383\pi\)
\(138\) −1.00831 −0.0858330
\(139\) −13.1927 −1.11899 −0.559494 0.828834i \(-0.689005\pi\)
−0.559494 + 0.828834i \(0.689005\pi\)
\(140\) −1.90570 −0.161061
\(141\) 12.5226 1.05459
\(142\) −14.1349 −1.18617
\(143\) 14.3247 1.19789
\(144\) −1.19326 −0.0994382
\(145\) −15.9428 −1.32398
\(146\) 9.75828 0.807601
\(147\) −9.08905 −0.749652
\(148\) −4.22571 −0.347351
\(149\) 19.2132 1.57400 0.787002 0.616950i \(-0.211632\pi\)
0.787002 + 0.616950i \(0.211632\pi\)
\(150\) −13.7836 −1.12543
\(151\) −3.63637 −0.295924 −0.147962 0.988993i \(-0.547271\pi\)
−0.147962 + 0.988993i \(0.547271\pi\)
\(152\) −0.975857 −0.0791524
\(153\) 7.60459 0.614795
\(154\) 1.97586 0.159219
\(155\) −30.7023 −2.46607
\(156\) 4.75481 0.380690
\(157\) −15.0118 −1.19807 −0.599035 0.800723i \(-0.704449\pi\)
−0.599035 + 0.800723i \(0.704449\pi\)
\(158\) −2.29222 −0.182359
\(159\) 14.9511 1.18570
\(160\) −3.90570 −0.308773
\(161\) −0.366017 −0.0288462
\(162\) 3.99636 0.313983
\(163\) −4.77531 −0.374031 −0.187016 0.982357i \(-0.559882\pi\)
−0.187016 + 0.982357i \(0.559882\pi\)
\(164\) −2.07362 −0.161923
\(165\) 21.2592 1.65503
\(166\) 17.1324 1.32974
\(167\) −3.94163 −0.305012 −0.152506 0.988303i \(-0.548734\pi\)
−0.152506 + 0.988303i \(0.548734\pi\)
\(168\) 0.655849 0.0505999
\(169\) −0.486734 −0.0374411
\(170\) 24.8909 1.90904
\(171\) −1.16445 −0.0890477
\(172\) 10.7341 0.818470
\(173\) −24.0607 −1.82930 −0.914650 0.404247i \(-0.867534\pi\)
−0.914650 + 0.404247i \(0.867534\pi\)
\(174\) 5.48673 0.415948
\(175\) −5.00347 −0.378227
\(176\) 4.04948 0.305241
\(177\) −4.00627 −0.301130
\(178\) −1.86978 −0.140146
\(179\) 17.1745 1.28368 0.641839 0.766839i \(-0.278172\pi\)
0.641839 + 0.766839i \(0.278172\pi\)
\(180\) −4.66052 −0.347374
\(181\) −7.28875 −0.541769 −0.270884 0.962612i \(-0.587316\pi\)
−0.270884 + 0.962612i \(0.587316\pi\)
\(182\) 1.72600 0.127940
\(183\) −10.1314 −0.748936
\(184\) −0.750146 −0.0553015
\(185\) −16.5044 −1.21343
\(186\) 10.5662 0.774753
\(187\) −25.8072 −1.88721
\(188\) 9.31636 0.679466
\(189\) 2.75015 0.200044
\(190\) −3.81141 −0.276509
\(191\) −6.55038 −0.473969 −0.236985 0.971513i \(-0.576159\pi\)
−0.236985 + 0.971513i \(0.576159\pi\)
\(192\) 1.34415 0.0970057
\(193\) 17.5467 1.26304 0.631521 0.775359i \(-0.282431\pi\)
0.631521 + 0.775359i \(0.282431\pi\)
\(194\) 9.63757 0.691937
\(195\) 18.5709 1.32989
\(196\) −6.76193 −0.482995
\(197\) −14.6051 −1.04057 −0.520286 0.853992i \(-0.674175\pi\)
−0.520286 + 0.853992i \(0.674175\pi\)
\(198\) 4.83208 0.343401
\(199\) −4.64162 −0.329036 −0.164518 0.986374i \(-0.552607\pi\)
−0.164518 + 0.986374i \(0.552607\pi\)
\(200\) −10.2545 −0.725104
\(201\) 4.72075 0.332976
\(202\) −5.91748 −0.416353
\(203\) 1.99169 0.139789
\(204\) −8.56622 −0.599755
\(205\) −8.09896 −0.565656
\(206\) −0.480814 −0.0334999
\(207\) −0.895118 −0.0622150
\(208\) 3.53741 0.245275
\(209\) 3.95171 0.273346
\(210\) 2.56155 0.176764
\(211\) 15.7554 1.08465 0.542324 0.840169i \(-0.317545\pi\)
0.542324 + 0.840169i \(0.317545\pi\)
\(212\) 11.1231 0.763938
\(213\) 18.9994 1.30182
\(214\) −3.82496 −0.261469
\(215\) 41.9244 2.85922
\(216\) 5.63637 0.383507
\(217\) 3.83555 0.260374
\(218\) 17.0866 1.15725
\(219\) −13.1166 −0.886338
\(220\) 15.8161 1.06632
\(221\) −22.5438 −1.51646
\(222\) 5.67999 0.381216
\(223\) 19.8791 1.33120 0.665602 0.746307i \(-0.268175\pi\)
0.665602 + 0.746307i \(0.268175\pi\)
\(224\) 0.487928 0.0326011
\(225\) −12.2363 −0.815753
\(226\) −13.3194 −0.885995
\(227\) 27.7566 1.84227 0.921136 0.389242i \(-0.127263\pi\)
0.921136 + 0.389242i \(0.127263\pi\)
\(228\) 1.31170 0.0868694
\(229\) −17.6723 −1.16782 −0.583909 0.811819i \(-0.698478\pi\)
−0.583909 + 0.811819i \(0.698478\pi\)
\(230\) −2.92985 −0.193188
\(231\) −2.65585 −0.174742
\(232\) 4.08193 0.267992
\(233\) −1.67230 −0.109556 −0.0547779 0.998499i \(-0.517445\pi\)
−0.0547779 + 0.998499i \(0.517445\pi\)
\(234\) 4.22105 0.275938
\(235\) 36.3870 2.37362
\(236\) −2.98052 −0.194015
\(237\) 3.08109 0.200138
\(238\) −3.10955 −0.201562
\(239\) −24.0925 −1.55841 −0.779206 0.626768i \(-0.784377\pi\)
−0.779206 + 0.626768i \(0.784377\pi\)
\(240\) 5.24985 0.338877
\(241\) −18.2257 −1.17402 −0.587011 0.809579i \(-0.699695\pi\)
−0.587011 + 0.809579i \(0.699695\pi\)
\(242\) −5.39830 −0.347016
\(243\) 11.5374 0.740125
\(244\) −7.53741 −0.482533
\(245\) −26.4101 −1.68728
\(246\) 2.78726 0.177709
\(247\) 3.45200 0.219646
\(248\) 7.86089 0.499167
\(249\) −23.0286 −1.45938
\(250\) −20.5226 −1.29796
\(251\) −1.21740 −0.0768417 −0.0384209 0.999262i \(-0.512233\pi\)
−0.0384209 + 0.999262i \(0.512233\pi\)
\(252\) 0.582225 0.0366767
\(253\) 3.03770 0.190979
\(254\) −3.18962 −0.200134
\(255\) −33.4571 −2.09517
\(256\) 1.00000 0.0625000
\(257\) 28.7806 1.79529 0.897644 0.440722i \(-0.145277\pi\)
0.897644 + 0.440722i \(0.145277\pi\)
\(258\) −14.4283 −0.898267
\(259\) 2.06184 0.128117
\(260\) 13.8161 0.856836
\(261\) 4.87080 0.301495
\(262\) 11.5975 0.716494
\(263\) −4.35407 −0.268483 −0.134242 0.990949i \(-0.542860\pi\)
−0.134242 + 0.990949i \(0.542860\pi\)
\(264\) −5.44311 −0.335000
\(265\) 43.4436 2.66872
\(266\) 0.476148 0.0291945
\(267\) 2.51327 0.153809
\(268\) 3.51207 0.214534
\(269\) 1.00000 0.0609711
\(270\) 22.0140 1.33973
\(271\) −19.6940 −1.19632 −0.598162 0.801375i \(-0.704102\pi\)
−0.598162 + 0.801375i \(0.704102\pi\)
\(272\) −6.37296 −0.386417
\(273\) −2.32001 −0.140413
\(274\) 7.33059 0.442857
\(275\) 41.5255 2.50408
\(276\) 1.00831 0.0606931
\(277\) −1.31009 −0.0787158 −0.0393579 0.999225i \(-0.512531\pi\)
−0.0393579 + 0.999225i \(0.512531\pi\)
\(278\) 13.1927 0.791244
\(279\) 9.38007 0.561570
\(280\) 1.90570 0.113888
\(281\) 7.94749 0.474107 0.237054 0.971497i \(-0.423818\pi\)
0.237054 + 0.971497i \(0.423818\pi\)
\(282\) −12.5226 −0.745710
\(283\) 20.7544 1.23372 0.616861 0.787072i \(-0.288404\pi\)
0.616861 + 0.787072i \(0.288404\pi\)
\(284\) 14.1349 0.838751
\(285\) 5.12311 0.303467
\(286\) −14.3247 −0.847036
\(287\) 1.01178 0.0597235
\(288\) 1.19326 0.0703134
\(289\) 23.6146 1.38910
\(290\) 15.9428 0.936195
\(291\) −12.9543 −0.759397
\(292\) −9.75828 −0.571060
\(293\) −10.9994 −0.642593 −0.321296 0.946979i \(-0.604119\pi\)
−0.321296 + 0.946979i \(0.604119\pi\)
\(294\) 9.08905 0.530084
\(295\) −11.6410 −0.677767
\(296\) 4.22571 0.245614
\(297\) −22.8244 −1.32440
\(298\) −19.2132 −1.11299
\(299\) 2.65357 0.153460
\(300\) 13.7836 0.795798
\(301\) −5.23749 −0.301884
\(302\) 3.63637 0.209250
\(303\) 7.95399 0.456945
\(304\) 0.975857 0.0559692
\(305\) −29.4389 −1.68567
\(306\) −7.60459 −0.434726
\(307\) 15.1961 0.867290 0.433645 0.901084i \(-0.357227\pi\)
0.433645 + 0.901084i \(0.357227\pi\)
\(308\) −1.97586 −0.112585
\(309\) 0.646287 0.0367660
\(310\) 30.7023 1.74377
\(311\) 9.02476 0.511747 0.255873 0.966710i \(-0.417637\pi\)
0.255873 + 0.966710i \(0.417637\pi\)
\(312\) −4.75481 −0.269188
\(313\) 34.3481 1.94147 0.970734 0.240159i \(-0.0771995\pi\)
0.970734 + 0.240159i \(0.0771995\pi\)
\(314\) 15.0118 0.847164
\(315\) 2.27400 0.128125
\(316\) 2.29222 0.128948
\(317\) 20.5386 1.15356 0.576781 0.816898i \(-0.304308\pi\)
0.576781 + 0.816898i \(0.304308\pi\)
\(318\) −14.9511 −0.838418
\(319\) −16.5297 −0.925486
\(320\) 3.90570 0.218335
\(321\) 5.14133 0.286961
\(322\) 0.366017 0.0203974
\(323\) −6.21910 −0.346040
\(324\) −3.99636 −0.222020
\(325\) 36.2744 2.01214
\(326\) 4.77531 0.264480
\(327\) −22.9670 −1.27008
\(328\) 2.07362 0.114497
\(329\) −4.54572 −0.250614
\(330\) −21.2592 −1.17028
\(331\) −8.30867 −0.456686 −0.228343 0.973581i \(-0.573331\pi\)
−0.228343 + 0.973581i \(0.573331\pi\)
\(332\) −17.1324 −0.940265
\(333\) 5.04237 0.276320
\(334\) 3.94163 0.215676
\(335\) 13.7171 0.749446
\(336\) −0.655849 −0.0357795
\(337\) −17.9588 −0.978276 −0.489138 0.872206i \(-0.662689\pi\)
−0.489138 + 0.872206i \(0.662689\pi\)
\(338\) 0.486734 0.0264748
\(339\) 17.9033 0.972375
\(340\) −24.8909 −1.34990
\(341\) −31.8325 −1.72383
\(342\) 1.16445 0.0629662
\(343\) 6.71483 0.362567
\(344\) −10.7341 −0.578746
\(345\) 3.93816 0.212023
\(346\) 24.0607 1.29351
\(347\) 6.69769 0.359551 0.179775 0.983708i \(-0.442463\pi\)
0.179775 + 0.983708i \(0.442463\pi\)
\(348\) −5.48673 −0.294120
\(349\) 0.974251 0.0521505 0.0260752 0.999660i \(-0.491699\pi\)
0.0260752 + 0.999660i \(0.491699\pi\)
\(350\) 5.00347 0.267447
\(351\) −19.9382 −1.06422
\(352\) −4.04948 −0.215838
\(353\) 15.8928 0.845886 0.422943 0.906156i \(-0.360997\pi\)
0.422943 + 0.906156i \(0.360997\pi\)
\(354\) 4.00627 0.212931
\(355\) 55.2067 2.93007
\(356\) 1.86978 0.0990981
\(357\) 4.17970 0.221213
\(358\) −17.1745 −0.907698
\(359\) −8.66535 −0.457340 −0.228670 0.973504i \(-0.573438\pi\)
−0.228670 + 0.973504i \(0.573438\pi\)
\(360\) 4.66052 0.245631
\(361\) −18.0477 −0.949879
\(362\) 7.28875 0.383088
\(363\) 7.25613 0.380848
\(364\) −1.72600 −0.0904671
\(365\) −38.1130 −1.99492
\(366\) 10.1314 0.529578
\(367\) 7.50802 0.391915 0.195958 0.980612i \(-0.437218\pi\)
0.195958 + 0.980612i \(0.437218\pi\)
\(368\) 0.750146 0.0391040
\(369\) 2.47437 0.128811
\(370\) 16.5044 0.858022
\(371\) −5.42728 −0.281770
\(372\) −10.5662 −0.547833
\(373\) 2.14911 0.111277 0.0556385 0.998451i \(-0.482281\pi\)
0.0556385 + 0.998451i \(0.482281\pi\)
\(374\) 25.8072 1.33446
\(375\) 27.5855 1.42451
\(376\) −9.31636 −0.480455
\(377\) −14.4395 −0.743671
\(378\) −2.75015 −0.141452
\(379\) 7.71644 0.396367 0.198183 0.980165i \(-0.436496\pi\)
0.198183 + 0.980165i \(0.436496\pi\)
\(380\) 3.81141 0.195521
\(381\) 4.28732 0.219646
\(382\) 6.55038 0.335147
\(383\) 1.88104 0.0961165 0.0480582 0.998845i \(-0.484697\pi\)
0.0480582 + 0.998845i \(0.484697\pi\)
\(384\) −1.34415 −0.0685934
\(385\) −7.71711 −0.393300
\(386\) −17.5467 −0.893106
\(387\) −12.8086 −0.651098
\(388\) −9.63757 −0.489273
\(389\) −10.7025 −0.542640 −0.271320 0.962489i \(-0.587460\pi\)
−0.271320 + 0.962489i \(0.587460\pi\)
\(390\) −18.5709 −0.940373
\(391\) −4.78065 −0.241768
\(392\) 6.76193 0.341529
\(393\) −15.5888 −0.786349
\(394\) 14.6051 0.735795
\(395\) 8.95274 0.450461
\(396\) −4.83208 −0.242821
\(397\) 3.84774 0.193113 0.0965563 0.995328i \(-0.469217\pi\)
0.0965563 + 0.995328i \(0.469217\pi\)
\(398\) 4.64162 0.232663
\(399\) −0.640015 −0.0320408
\(400\) 10.2545 0.512726
\(401\) −0.717694 −0.0358399 −0.0179200 0.999839i \(-0.505704\pi\)
−0.0179200 + 0.999839i \(0.505704\pi\)
\(402\) −4.72075 −0.235450
\(403\) −27.8072 −1.38517
\(404\) 5.91748 0.294406
\(405\) −15.6086 −0.775597
\(406\) −1.99169 −0.0988460
\(407\) −17.1119 −0.848207
\(408\) 8.56622 0.424091
\(409\) −11.7906 −0.583006 −0.291503 0.956570i \(-0.594155\pi\)
−0.291503 + 0.956570i \(0.594155\pi\)
\(410\) 8.09896 0.399979
\(411\) −9.85342 −0.486033
\(412\) 0.480814 0.0236880
\(413\) 1.45428 0.0715605
\(414\) 0.895118 0.0439926
\(415\) −66.9142 −3.28469
\(416\) −3.53741 −0.173436
\(417\) −17.7329 −0.868386
\(418\) −3.95171 −0.193285
\(419\) −7.07076 −0.345429 −0.172715 0.984972i \(-0.555254\pi\)
−0.172715 + 0.984972i \(0.555254\pi\)
\(420\) −2.56155 −0.124991
\(421\) −4.96469 −0.241964 −0.120982 0.992655i \(-0.538604\pi\)
−0.120982 + 0.992655i \(0.538604\pi\)
\(422\) −15.7554 −0.766962
\(423\) −11.1168 −0.540519
\(424\) −11.1231 −0.540186
\(425\) −65.3516 −3.17002
\(426\) −18.9994 −0.920525
\(427\) 3.67772 0.177977
\(428\) 3.82496 0.184887
\(429\) 19.2545 0.929617
\(430\) −41.9244 −2.02177
\(431\) 25.6140 1.23378 0.616892 0.787048i \(-0.288391\pi\)
0.616892 + 0.787048i \(0.288391\pi\)
\(432\) −5.63637 −0.271180
\(433\) −20.5767 −0.988855 −0.494428 0.869219i \(-0.664622\pi\)
−0.494428 + 0.869219i \(0.664622\pi\)
\(434\) −3.83555 −0.184112
\(435\) −21.4296 −1.02747
\(436\) −17.0866 −0.818300
\(437\) 0.732035 0.0350180
\(438\) 13.1166 0.626736
\(439\) 15.6900 0.748843 0.374421 0.927259i \(-0.377841\pi\)
0.374421 + 0.927259i \(0.377841\pi\)
\(440\) −15.8161 −0.754002
\(441\) 8.06873 0.384225
\(442\) 22.5438 1.07230
\(443\) 29.0018 1.37792 0.688959 0.724801i \(-0.258068\pi\)
0.688959 + 0.724801i \(0.258068\pi\)
\(444\) −5.67999 −0.269561
\(445\) 7.30281 0.346186
\(446\) −19.8791 −0.941303
\(447\) 25.8254 1.22150
\(448\) −0.487928 −0.0230524
\(449\) 18.1978 0.858805 0.429403 0.903113i \(-0.358724\pi\)
0.429403 + 0.903113i \(0.358724\pi\)
\(450\) 12.2363 0.576825
\(451\) −8.39710 −0.395404
\(452\) 13.3194 0.626493
\(453\) −4.88783 −0.229650
\(454\) −27.7566 −1.30268
\(455\) −6.74125 −0.316035
\(456\) −1.31170 −0.0614259
\(457\) 32.1933 1.50594 0.752969 0.658056i \(-0.228621\pi\)
0.752969 + 0.658056i \(0.228621\pi\)
\(458\) 17.6723 0.825772
\(459\) 35.9204 1.67662
\(460\) 2.92985 0.136605
\(461\) −26.9152 −1.25357 −0.626783 0.779194i \(-0.715629\pi\)
−0.626783 + 0.779194i \(0.715629\pi\)
\(462\) 2.65585 0.123561
\(463\) −24.2349 −1.12629 −0.563145 0.826358i \(-0.690409\pi\)
−0.563145 + 0.826358i \(0.690409\pi\)
\(464\) −4.08193 −0.189499
\(465\) −41.2685 −1.91378
\(466\) 1.67230 0.0774676
\(467\) −17.7628 −0.821963 −0.410982 0.911644i \(-0.634814\pi\)
−0.410982 + 0.911644i \(0.634814\pi\)
\(468\) −4.22105 −0.195118
\(469\) −1.71364 −0.0791285
\(470\) −36.3870 −1.67841
\(471\) −20.1781 −0.929758
\(472\) 2.98052 0.137190
\(473\) 43.4677 1.99865
\(474\) −3.08109 −0.141519
\(475\) 10.0069 0.459150
\(476\) 3.10955 0.142526
\(477\) −13.2727 −0.607717
\(478\) 24.0925 1.10196
\(479\) 11.4930 0.525130 0.262565 0.964914i \(-0.415432\pi\)
0.262565 + 0.964914i \(0.415432\pi\)
\(480\) −5.24985 −0.239622
\(481\) −14.9481 −0.681573
\(482\) 18.2257 0.830158
\(483\) −0.491983 −0.0223860
\(484\) 5.39830 0.245377
\(485\) −37.6415 −1.70921
\(486\) −11.5374 −0.523348
\(487\) 5.85514 0.265322 0.132661 0.991161i \(-0.457648\pi\)
0.132661 + 0.991161i \(0.457648\pi\)
\(488\) 7.53741 0.341202
\(489\) −6.41874 −0.290265
\(490\) 26.4101 1.19309
\(491\) 18.6544 0.841862 0.420931 0.907093i \(-0.361703\pi\)
0.420931 + 0.907093i \(0.361703\pi\)
\(492\) −2.78726 −0.125660
\(493\) 26.0140 1.17161
\(494\) −3.45200 −0.155313
\(495\) −18.8727 −0.848263
\(496\) −7.86089 −0.352964
\(497\) −6.89681 −0.309364
\(498\) 23.0286 1.03194
\(499\) 35.1119 1.57183 0.785913 0.618337i \(-0.212193\pi\)
0.785913 + 0.618337i \(0.212193\pi\)
\(500\) 20.5226 0.917798
\(501\) −5.29814 −0.236703
\(502\) 1.21740 0.0543353
\(503\) 29.7342 1.32578 0.662892 0.748715i \(-0.269329\pi\)
0.662892 + 0.748715i \(0.269329\pi\)
\(504\) −0.582225 −0.0259344
\(505\) 23.1119 1.02847
\(506\) −3.03770 −0.135042
\(507\) −0.654243 −0.0290560
\(508\) 3.18962 0.141516
\(509\) 30.1456 1.33618 0.668090 0.744081i \(-0.267112\pi\)
0.668090 + 0.744081i \(0.267112\pi\)
\(510\) 33.4571 1.48151
\(511\) 4.76134 0.210629
\(512\) −1.00000 −0.0441942
\(513\) −5.50029 −0.242844
\(514\) −28.7806 −1.26946
\(515\) 1.87792 0.0827509
\(516\) 14.4283 0.635171
\(517\) 37.7264 1.65921
\(518\) −2.06184 −0.0905922
\(519\) −32.3412 −1.41962
\(520\) −13.8161 −0.605875
\(521\) 37.9804 1.66395 0.831975 0.554813i \(-0.187210\pi\)
0.831975 + 0.554813i \(0.187210\pi\)
\(522\) −4.87080 −0.213189
\(523\) 10.1012 0.441696 0.220848 0.975308i \(-0.429117\pi\)
0.220848 + 0.975308i \(0.429117\pi\)
\(524\) −11.5975 −0.506638
\(525\) −6.72542 −0.293521
\(526\) 4.35407 0.189846
\(527\) 50.0971 2.18227
\(528\) 5.44311 0.236881
\(529\) −22.4373 −0.975534
\(530\) −43.4436 −1.88707
\(531\) 3.55653 0.154340
\(532\) −0.476148 −0.0206436
\(533\) −7.33526 −0.317725
\(534\) −2.51327 −0.108760
\(535\) 14.9392 0.645877
\(536\) −3.51207 −0.151698
\(537\) 23.0851 0.996194
\(538\) −1.00000 −0.0431131
\(539\) −27.3823 −1.17944
\(540\) −22.0140 −0.947332
\(541\) 1.06610 0.0458352 0.0229176 0.999737i \(-0.492704\pi\)
0.0229176 + 0.999737i \(0.492704\pi\)
\(542\) 19.6940 0.845929
\(543\) −9.79718 −0.420437
\(544\) 6.37296 0.273238
\(545\) −66.7352 −2.85862
\(546\) 2.32001 0.0992872
\(547\) 8.62634 0.368836 0.184418 0.982848i \(-0.440960\pi\)
0.184418 + 0.982848i \(0.440960\pi\)
\(548\) −7.33059 −0.313147
\(549\) 8.99408 0.383858
\(550\) −41.5255 −1.77065
\(551\) −3.98338 −0.169698
\(552\) −1.00831 −0.0429165
\(553\) −1.11844 −0.0475609
\(554\) 1.31009 0.0556605
\(555\) −22.1844 −0.941674
\(556\) −13.1927 −0.559494
\(557\) 39.3859 1.66884 0.834418 0.551132i \(-0.185804\pi\)
0.834418 + 0.551132i \(0.185804\pi\)
\(558\) −9.38007 −0.397090
\(559\) 37.9710 1.60600
\(560\) −1.90570 −0.0805307
\(561\) −34.6887 −1.46456
\(562\) −7.94749 −0.335245
\(563\) −29.8436 −1.25776 −0.628879 0.777503i \(-0.716486\pi\)
−0.628879 + 0.777503i \(0.716486\pi\)
\(564\) 12.5226 0.527297
\(565\) 52.0217 2.18857
\(566\) −20.7544 −0.872373
\(567\) 1.94994 0.0818896
\(568\) −14.1349 −0.593087
\(569\) 18.1766 0.762004 0.381002 0.924574i \(-0.375579\pi\)
0.381002 + 0.924574i \(0.375579\pi\)
\(570\) −5.12311 −0.214583
\(571\) −34.4465 −1.44154 −0.720771 0.693173i \(-0.756212\pi\)
−0.720771 + 0.693173i \(0.756212\pi\)
\(572\) 14.3247 0.598945
\(573\) −8.80470 −0.367822
\(574\) −1.01178 −0.0422309
\(575\) 7.69238 0.320795
\(576\) −1.19326 −0.0497191
\(577\) 6.04440 0.251632 0.125816 0.992054i \(-0.459845\pi\)
0.125816 + 0.992054i \(0.459845\pi\)
\(578\) −23.6146 −0.982239
\(579\) 23.5855 0.980178
\(580\) −15.9428 −0.661990
\(581\) 8.35940 0.346806
\(582\) 12.9543 0.536975
\(583\) 45.0428 1.86548
\(584\) 9.75828 0.403801
\(585\) −16.4862 −0.681618
\(586\) 10.9994 0.454382
\(587\) −38.6176 −1.59392 −0.796959 0.604034i \(-0.793559\pi\)
−0.796959 + 0.604034i \(0.793559\pi\)
\(588\) −9.08905 −0.374826
\(589\) −7.67110 −0.316082
\(590\) 11.6410 0.479254
\(591\) −19.6315 −0.807531
\(592\) −4.22571 −0.173676
\(593\) −4.06998 −0.167134 −0.0835671 0.996502i \(-0.526631\pi\)
−0.0835671 + 0.996502i \(0.526631\pi\)
\(594\) 22.8244 0.936496
\(595\) 12.1450 0.497895
\(596\) 19.2132 0.787002
\(597\) −6.23904 −0.255347
\(598\) −2.65357 −0.108513
\(599\) −18.2628 −0.746199 −0.373099 0.927791i \(-0.621705\pi\)
−0.373099 + 0.927791i \(0.621705\pi\)
\(600\) −13.7836 −0.562714
\(601\) 31.6805 1.29228 0.646138 0.763220i \(-0.276383\pi\)
0.646138 + 0.763220i \(0.276383\pi\)
\(602\) 5.23749 0.213464
\(603\) −4.19081 −0.170663
\(604\) −3.63637 −0.147962
\(605\) 21.0842 0.857193
\(606\) −7.95399 −0.323109
\(607\) 25.5355 1.03645 0.518226 0.855244i \(-0.326593\pi\)
0.518226 + 0.855244i \(0.326593\pi\)
\(608\) −0.975857 −0.0395762
\(609\) 2.67713 0.108483
\(610\) 29.4389 1.19195
\(611\) 32.9558 1.33325
\(612\) 7.60459 0.307397
\(613\) −17.6816 −0.714154 −0.357077 0.934075i \(-0.616227\pi\)
−0.357077 + 0.934075i \(0.616227\pi\)
\(614\) −15.1961 −0.613267
\(615\) −10.8862 −0.438975
\(616\) 1.97586 0.0796095
\(617\) 32.8371 1.32197 0.660986 0.750398i \(-0.270138\pi\)
0.660986 + 0.750398i \(0.270138\pi\)
\(618\) −0.646287 −0.0259975
\(619\) 10.5824 0.425342 0.212671 0.977124i \(-0.431784\pi\)
0.212671 + 0.977124i \(0.431784\pi\)
\(620\) −30.7023 −1.23303
\(621\) −4.22810 −0.169668
\(622\) −9.02476 −0.361860
\(623\) −0.912319 −0.0365513
\(624\) 4.75481 0.190345
\(625\) 28.8826 1.15530
\(626\) −34.3481 −1.37282
\(627\) 5.31170 0.212129
\(628\) −15.0118 −0.599035
\(629\) 26.9303 1.07378
\(630\) −2.27400 −0.0905982
\(631\) −0.902404 −0.0359241 −0.0179621 0.999839i \(-0.505718\pi\)
−0.0179621 + 0.999839i \(0.505718\pi\)
\(632\) −2.29222 −0.0911797
\(633\) 21.1777 0.841737
\(634\) −20.5386 −0.815692
\(635\) 12.4577 0.494368
\(636\) 14.9511 0.592851
\(637\) −23.9197 −0.947733
\(638\) 16.5297 0.654418
\(639\) −16.8666 −0.667231
\(640\) −3.90570 −0.154386
\(641\) −19.2444 −0.760109 −0.380055 0.924964i \(-0.624095\pi\)
−0.380055 + 0.924964i \(0.624095\pi\)
\(642\) −5.14133 −0.202912
\(643\) 34.2804 1.35189 0.675944 0.736953i \(-0.263736\pi\)
0.675944 + 0.736953i \(0.263736\pi\)
\(644\) −0.366017 −0.0144231
\(645\) 56.3527 2.21888
\(646\) 6.21910 0.244687
\(647\) −30.2029 −1.18740 −0.593698 0.804688i \(-0.702333\pi\)
−0.593698 + 0.804688i \(0.702333\pi\)
\(648\) 3.99636 0.156992
\(649\) −12.0696 −0.473772
\(650\) −36.2744 −1.42280
\(651\) 5.15556 0.202062
\(652\) −4.77531 −0.187016
\(653\) −21.5877 −0.844794 −0.422397 0.906411i \(-0.638811\pi\)
−0.422397 + 0.906411i \(0.638811\pi\)
\(654\) 22.9670 0.898079
\(655\) −45.2963 −1.76987
\(656\) −2.07362 −0.0809614
\(657\) 11.6442 0.454282
\(658\) 4.54572 0.177211
\(659\) −21.3740 −0.832612 −0.416306 0.909225i \(-0.636676\pi\)
−0.416306 + 0.909225i \(0.636676\pi\)
\(660\) 21.2592 0.827513
\(661\) −19.7195 −0.767000 −0.383500 0.923541i \(-0.625281\pi\)
−0.383500 + 0.923541i \(0.625281\pi\)
\(662\) 8.30867 0.322926
\(663\) −30.3022 −1.17684
\(664\) 17.1324 0.664868
\(665\) −1.85969 −0.0721158
\(666\) −5.04237 −0.195388
\(667\) −3.06204 −0.118563
\(668\) −3.94163 −0.152506
\(669\) 26.7205 1.03308
\(670\) −13.7171 −0.529938
\(671\) −30.5226 −1.17831
\(672\) 0.655849 0.0252999
\(673\) −13.2275 −0.509883 −0.254942 0.966956i \(-0.582056\pi\)
−0.254942 + 0.966956i \(0.582056\pi\)
\(674\) 17.9588 0.691746
\(675\) −57.7983 −2.22466
\(676\) −0.486734 −0.0187205
\(677\) −12.7629 −0.490520 −0.245260 0.969457i \(-0.578873\pi\)
−0.245260 + 0.969457i \(0.578873\pi\)
\(678\) −17.9033 −0.687573
\(679\) 4.70244 0.180463
\(680\) 24.8909 0.954522
\(681\) 37.3091 1.42969
\(682\) 31.8325 1.21893
\(683\) −3.12345 −0.119515 −0.0597577 0.998213i \(-0.519033\pi\)
−0.0597577 + 0.998213i \(0.519033\pi\)
\(684\) −1.16445 −0.0445238
\(685\) −28.6311 −1.09394
\(686\) −6.71483 −0.256374
\(687\) −23.7542 −0.906280
\(688\) 10.7341 0.409235
\(689\) 39.3470 1.49900
\(690\) −3.93816 −0.149923
\(691\) 25.3081 0.962767 0.481384 0.876510i \(-0.340134\pi\)
0.481384 + 0.876510i \(0.340134\pi\)
\(692\) −24.0607 −0.914650
\(693\) 2.35771 0.0895619
\(694\) −6.69769 −0.254241
\(695\) −51.5267 −1.95452
\(696\) 5.48673 0.207974
\(697\) 13.2151 0.500559
\(698\) −0.974251 −0.0368759
\(699\) −2.24782 −0.0850202
\(700\) −5.00347 −0.189113
\(701\) −15.5107 −0.585831 −0.292916 0.956138i \(-0.594626\pi\)
−0.292916 + 0.956138i \(0.594626\pi\)
\(702\) 19.9382 0.752517
\(703\) −4.12369 −0.155528
\(704\) 4.04948 0.152621
\(705\) 48.9096 1.84204
\(706\) −15.8928 −0.598132
\(707\) −2.88731 −0.108588
\(708\) −4.00627 −0.150565
\(709\) 3.24435 0.121844 0.0609220 0.998143i \(-0.480596\pi\)
0.0609220 + 0.998143i \(0.480596\pi\)
\(710\) −55.2067 −2.07187
\(711\) −2.73521 −0.102579
\(712\) −1.86978 −0.0700730
\(713\) −5.89681 −0.220837
\(714\) −4.17970 −0.156421
\(715\) 55.9479 2.09233
\(716\) 17.1745 0.641839
\(717\) −32.3839 −1.20940
\(718\) 8.66535 0.323388
\(719\) 28.5289 1.06395 0.531974 0.846761i \(-0.321451\pi\)
0.531974 + 0.846761i \(0.321451\pi\)
\(720\) −4.66052 −0.173687
\(721\) −0.234603 −0.00873707
\(722\) 18.0477 0.671666
\(723\) −24.4981 −0.911094
\(724\) −7.28875 −0.270884
\(725\) −41.8583 −1.55458
\(726\) −7.25613 −0.269300
\(727\) 28.4941 1.05679 0.528395 0.848999i \(-0.322794\pi\)
0.528395 + 0.848999i \(0.322794\pi\)
\(728\) 1.72600 0.0639699
\(729\) 27.4971 1.01841
\(730\) 38.1130 1.41062
\(731\) −68.4082 −2.53017
\(732\) −10.1314 −0.374468
\(733\) 31.1617 1.15098 0.575491 0.817808i \(-0.304811\pi\)
0.575491 + 0.817808i \(0.304811\pi\)
\(734\) −7.50802 −0.277126
\(735\) −35.4991 −1.30940
\(736\) −0.750146 −0.0276507
\(737\) 14.2221 0.523877
\(738\) −2.47437 −0.0910828
\(739\) 22.7409 0.836538 0.418269 0.908323i \(-0.362637\pi\)
0.418269 + 0.908323i \(0.362637\pi\)
\(740\) −16.5044 −0.606713
\(741\) 4.64001 0.170455
\(742\) 5.42728 0.199242
\(743\) 33.4594 1.22751 0.613753 0.789498i \(-0.289659\pi\)
0.613753 + 0.789498i \(0.289659\pi\)
\(744\) 10.5662 0.387376
\(745\) 75.0410 2.74929
\(746\) −2.14911 −0.0786847
\(747\) 20.4434 0.747986
\(748\) −25.8072 −0.943604
\(749\) −1.86631 −0.0681934
\(750\) −27.5855 −1.00728
\(751\) −48.0493 −1.75334 −0.876672 0.481089i \(-0.840241\pi\)
−0.876672 + 0.481089i \(0.840241\pi\)
\(752\) 9.31636 0.339733
\(753\) −1.63637 −0.0596327
\(754\) 14.4395 0.525854
\(755\) −14.2026 −0.516885
\(756\) 2.75015 0.100022
\(757\) 52.5319 1.90930 0.954652 0.297725i \(-0.0962279\pi\)
0.954652 + 0.297725i \(0.0962279\pi\)
\(758\) −7.71644 −0.280274
\(759\) 4.08313 0.148208
\(760\) −3.81141 −0.138254
\(761\) 50.4604 1.82919 0.914594 0.404373i \(-0.132510\pi\)
0.914594 + 0.404373i \(0.132510\pi\)
\(762\) −4.28732 −0.155313
\(763\) 8.33704 0.301821
\(764\) −6.55038 −0.236985
\(765\) 29.7013 1.07385
\(766\) −1.88104 −0.0679646
\(767\) −10.5433 −0.380698
\(768\) 1.34415 0.0485029
\(769\) −38.6684 −1.39442 −0.697209 0.716867i \(-0.745575\pi\)
−0.697209 + 0.716867i \(0.745575\pi\)
\(770\) 7.71711 0.278105
\(771\) 38.6855 1.39323
\(772\) 17.5467 0.631521
\(773\) −27.8089 −1.00022 −0.500108 0.865963i \(-0.666706\pi\)
−0.500108 + 0.865963i \(0.666706\pi\)
\(774\) 12.8086 0.460396
\(775\) −80.6096 −2.89558
\(776\) 9.63757 0.345968
\(777\) 2.77143 0.0994245
\(778\) 10.7025 0.383704
\(779\) −2.02356 −0.0725016
\(780\) 18.5709 0.664944
\(781\) 57.2390 2.04817
\(782\) 4.78065 0.170956
\(783\) 23.0073 0.822214
\(784\) −6.76193 −0.241497
\(785\) −58.6316 −2.09265
\(786\) 15.5888 0.556033
\(787\) −35.0440 −1.24918 −0.624592 0.780951i \(-0.714735\pi\)
−0.624592 + 0.780951i \(0.714735\pi\)
\(788\) −14.6051 −0.520286
\(789\) −5.85252 −0.208355
\(790\) −8.95274 −0.318524
\(791\) −6.49893 −0.231075
\(792\) 4.83208 0.171700
\(793\) −26.6629 −0.946827
\(794\) −3.84774 −0.136551
\(795\) 58.3947 2.07105
\(796\) −4.64162 −0.164518
\(797\) 44.3028 1.56929 0.784643 0.619947i \(-0.212846\pi\)
0.784643 + 0.619947i \(0.212846\pi\)
\(798\) 0.640015 0.0226563
\(799\) −59.3728 −2.10046
\(800\) −10.2545 −0.362552
\(801\) −2.23113 −0.0788332
\(802\) 0.717694 0.0253426
\(803\) −39.5160 −1.39449
\(804\) 4.72075 0.166488
\(805\) −1.42956 −0.0503852
\(806\) 27.8072 0.979466
\(807\) 1.34415 0.0473163
\(808\) −5.91748 −0.208176
\(809\) 17.8666 0.628157 0.314079 0.949397i \(-0.398304\pi\)
0.314079 + 0.949397i \(0.398304\pi\)
\(810\) 15.6086 0.548430
\(811\) 7.94755 0.279076 0.139538 0.990217i \(-0.455438\pi\)
0.139538 + 0.990217i \(0.455438\pi\)
\(812\) 1.99169 0.0698947
\(813\) −26.4717 −0.928403
\(814\) 17.1119 0.599773
\(815\) −18.6510 −0.653314
\(816\) −8.56622 −0.299878
\(817\) 10.4750 0.366473
\(818\) 11.7906 0.412247
\(819\) 2.05957 0.0719671
\(820\) −8.09896 −0.282828
\(821\) 14.0212 0.489342 0.244671 0.969606i \(-0.421320\pi\)
0.244671 + 0.969606i \(0.421320\pi\)
\(822\) 9.85342 0.343678
\(823\) −56.0461 −1.95364 −0.976822 0.214052i \(-0.931334\pi\)
−0.976822 + 0.214052i \(0.931334\pi\)
\(824\) −0.480814 −0.0167500
\(825\) 55.8165 1.94328
\(826\) −1.45428 −0.0506009
\(827\) 28.6027 0.994612 0.497306 0.867575i \(-0.334323\pi\)
0.497306 + 0.867575i \(0.334323\pi\)
\(828\) −0.895118 −0.0311075
\(829\) 7.08499 0.246072 0.123036 0.992402i \(-0.460737\pi\)
0.123036 + 0.992402i \(0.460737\pi\)
\(830\) 66.9142 2.32263
\(831\) −1.76096 −0.0610871
\(832\) 3.53741 0.122638
\(833\) 43.0935 1.49310
\(834\) 17.7329 0.614042
\(835\) −15.3948 −0.532760
\(836\) 3.95171 0.136673
\(837\) 44.3069 1.53147
\(838\) 7.07076 0.244256
\(839\) −16.9860 −0.586423 −0.293211 0.956048i \(-0.594724\pi\)
−0.293211 + 0.956048i \(0.594724\pi\)
\(840\) 2.56155 0.0883820
\(841\) −12.3378 −0.425442
\(842\) 4.96469 0.171094
\(843\) 10.6826 0.367929
\(844\) 15.7554 0.542324
\(845\) −1.90104 −0.0653977
\(846\) 11.1168 0.382205
\(847\) −2.63398 −0.0905047
\(848\) 11.1231 0.381969
\(849\) 27.8970 0.957424
\(850\) 65.3516 2.24154
\(851\) −3.16990 −0.108663
\(852\) 18.9994 0.650909
\(853\) −25.5249 −0.873957 −0.436979 0.899472i \(-0.643951\pi\)
−0.436979 + 0.899472i \(0.643951\pi\)
\(854\) −3.67772 −0.125849
\(855\) −4.54800 −0.155538
\(856\) −3.82496 −0.130735
\(857\) −47.9302 −1.63726 −0.818632 0.574318i \(-0.805267\pi\)
−0.818632 + 0.574318i \(0.805267\pi\)
\(858\) −19.2545 −0.657338
\(859\) −6.23647 −0.212786 −0.106393 0.994324i \(-0.533930\pi\)
−0.106393 + 0.994324i \(0.533930\pi\)
\(860\) 41.9244 1.42961
\(861\) 1.35999 0.0463482
\(862\) −25.6140 −0.872417
\(863\) 8.46337 0.288097 0.144048 0.989571i \(-0.453988\pi\)
0.144048 + 0.989571i \(0.453988\pi\)
\(864\) 5.63637 0.191753
\(865\) −93.9739 −3.19521
\(866\) 20.5767 0.699226
\(867\) 31.7416 1.07800
\(868\) 3.83555 0.130187
\(869\) 9.28231 0.314881
\(870\) 21.4296 0.726530
\(871\) 12.4236 0.420959
\(872\) 17.0866 0.578625
\(873\) 11.5001 0.389220
\(874\) −0.732035 −0.0247614
\(875\) −10.0136 −0.338520
\(876\) −13.1166 −0.443169
\(877\) 42.1172 1.42220 0.711098 0.703093i \(-0.248198\pi\)
0.711098 + 0.703093i \(0.248198\pi\)
\(878\) −15.6900 −0.529512
\(879\) −14.7849 −0.498681
\(880\) 15.8161 0.533160
\(881\) −35.0575 −1.18112 −0.590558 0.806995i \(-0.701092\pi\)
−0.590558 + 0.806995i \(0.701092\pi\)
\(882\) −8.06873 −0.271688
\(883\) 49.3854 1.66195 0.830976 0.556308i \(-0.187783\pi\)
0.830976 + 0.556308i \(0.187783\pi\)
\(884\) −22.5438 −0.758229
\(885\) −15.6473 −0.525979
\(886\) −29.0018 −0.974335
\(887\) −20.5875 −0.691261 −0.345630 0.938371i \(-0.612335\pi\)
−0.345630 + 0.938371i \(0.612335\pi\)
\(888\) 5.67999 0.190608
\(889\) −1.55630 −0.0521968
\(890\) −7.30281 −0.244791
\(891\) −16.1832 −0.542157
\(892\) 19.8791 0.665602
\(893\) 9.09144 0.304233
\(894\) −25.8254 −0.863731
\(895\) 67.0783 2.24218
\(896\) 0.487928 0.0163005
\(897\) 3.56680 0.119092
\(898\) −18.1978 −0.607267
\(899\) 32.0876 1.07018
\(900\) −12.2363 −0.407877
\(901\) −70.8871 −2.36159
\(902\) 8.39710 0.279593
\(903\) −7.03998 −0.234276
\(904\) −13.3194 −0.442997
\(905\) −28.4677 −0.946298
\(906\) 4.88783 0.162387
\(907\) −57.1062 −1.89618 −0.948090 0.318003i \(-0.896988\pi\)
−0.948090 + 0.318003i \(0.896988\pi\)
\(908\) 27.7566 0.921136
\(909\) −7.06109 −0.234202
\(910\) 6.74125 0.223470
\(911\) 1.11278 0.0368680 0.0184340 0.999830i \(-0.494132\pi\)
0.0184340 + 0.999830i \(0.494132\pi\)
\(912\) 1.31170 0.0434347
\(913\) −69.3775 −2.29606
\(914\) −32.1933 −1.06486
\(915\) −39.5703 −1.30815
\(916\) −17.6723 −0.583909
\(917\) 5.65874 0.186868
\(918\) −35.9204 −1.18555
\(919\) −24.6902 −0.814453 −0.407227 0.913327i \(-0.633504\pi\)
−0.407227 + 0.913327i \(0.633504\pi\)
\(920\) −2.92985 −0.0965942
\(921\) 20.4259 0.673057
\(922\) 26.9152 0.886405
\(923\) 50.0009 1.64580
\(924\) −2.65585 −0.0873710
\(925\) −43.3326 −1.42477
\(926\) 24.2349 0.796407
\(927\) −0.573736 −0.0188440
\(928\) 4.08193 0.133996
\(929\) 53.4317 1.75304 0.876519 0.481368i \(-0.159860\pi\)
0.876519 + 0.481368i \(0.159860\pi\)
\(930\) 41.2685 1.35325
\(931\) −6.59867 −0.216263
\(932\) −1.67230 −0.0547779
\(933\) 12.1306 0.397139
\(934\) 17.7628 0.581216
\(935\) −100.795 −3.29636
\(936\) 4.22105 0.137969
\(937\) −8.01583 −0.261866 −0.130933 0.991391i \(-0.541797\pi\)
−0.130933 + 0.991391i \(0.541797\pi\)
\(938\) 1.71364 0.0559523
\(939\) 46.1690 1.50667
\(940\) 36.3870 1.18681
\(941\) −10.2934 −0.335556 −0.167778 0.985825i \(-0.553659\pi\)
−0.167778 + 0.985825i \(0.553659\pi\)
\(942\) 20.1781 0.657438
\(943\) −1.55552 −0.0506547
\(944\) −2.98052 −0.0970077
\(945\) 10.7413 0.349413
\(946\) −43.4677 −1.41326
\(947\) 42.4848 1.38057 0.690285 0.723538i \(-0.257485\pi\)
0.690285 + 0.723538i \(0.257485\pi\)
\(948\) 3.08109 0.100069
\(949\) −34.5190 −1.12054
\(950\) −10.0069 −0.324668
\(951\) 27.6070 0.895218
\(952\) −3.10955 −0.100781
\(953\) 59.0011 1.91123 0.955617 0.294612i \(-0.0951905\pi\)
0.955617 + 0.294612i \(0.0951905\pi\)
\(954\) 13.2727 0.429721
\(955\) −25.5839 −0.827874
\(956\) −24.0925 −0.779206
\(957\) −22.2184 −0.718220
\(958\) −11.4930 −0.371323
\(959\) 3.57680 0.115501
\(960\) 5.24985 0.169438
\(961\) 30.7936 0.993341
\(962\) 14.9481 0.481945
\(963\) −4.56417 −0.147078
\(964\) −18.2257 −0.587011
\(965\) 68.5324 2.20614
\(966\) 0.491983 0.0158293
\(967\) −0.781747 −0.0251393 −0.0125696 0.999921i \(-0.504001\pi\)
−0.0125696 + 0.999921i \(0.504001\pi\)
\(968\) −5.39830 −0.173508
\(969\) −8.35940 −0.268543
\(970\) 37.6415 1.20860
\(971\) 51.5702 1.65497 0.827484 0.561489i \(-0.189771\pi\)
0.827484 + 0.561489i \(0.189771\pi\)
\(972\) 11.5374 0.370063
\(973\) 6.43708 0.206363
\(974\) −5.85514 −0.187611
\(975\) 48.7583 1.56152
\(976\) −7.53741 −0.241267
\(977\) 3.02820 0.0968806 0.0484403 0.998826i \(-0.484575\pi\)
0.0484403 + 0.998826i \(0.484575\pi\)
\(978\) 6.41874 0.205249
\(979\) 7.57164 0.241991
\(980\) −26.4101 −0.843639
\(981\) 20.3887 0.650962
\(982\) −18.6544 −0.595287
\(983\) −45.6956 −1.45746 −0.728732 0.684799i \(-0.759890\pi\)
−0.728732 + 0.684799i \(0.759890\pi\)
\(984\) 2.78726 0.0888547
\(985\) −57.0432 −1.81755
\(986\) −26.0140 −0.828454
\(987\) −6.11013 −0.194488
\(988\) 3.45200 0.109823
\(989\) 8.05217 0.256044
\(990\) 18.8727 0.599813
\(991\) −30.3370 −0.963685 −0.481843 0.876258i \(-0.660032\pi\)
−0.481843 + 0.876258i \(0.660032\pi\)
\(992\) 7.86089 0.249583
\(993\) −11.1681 −0.354409
\(994\) 6.89681 0.218754
\(995\) −18.1288 −0.574721
\(996\) −23.0286 −0.729688
\(997\) 27.9443 0.885004 0.442502 0.896767i \(-0.354091\pi\)
0.442502 + 0.896767i \(0.354091\pi\)
\(998\) −35.1119 −1.11145
\(999\) 23.8177 0.753558
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.c.1.3 4
3.2 odd 2 4842.2.a.j.1.1 4
4.3 odd 2 4304.2.a.e.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
538.2.a.c.1.3 4 1.1 even 1 trivial
4304.2.a.e.1.2 4 4.3 odd 2
4842.2.a.j.1.1 4 3.2 odd 2