Properties

Label 538.2.a.c.1.2
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 1.2
Root \(0.487928\) 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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +0.512072 q^{3} +1.00000 q^{4} -1.04948 q^{5} -0.512072 q^{6} -2.90570 q^{7} -1.00000 q^{8} -2.73778 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +0.512072 q^{3} +1.00000 q^{4} -1.04948 q^{5} -0.512072 q^{6} -2.90570 q^{7} -1.00000 q^{8} -2.73778 q^{9} +1.04948 q^{10} +2.34415 q^{11} +0.512072 q^{12} +4.24985 q^{13} +2.90570 q^{14} -0.537410 q^{15} +1.00000 q^{16} +7.66052 q^{17} +2.73778 q^{18} +5.81141 q^{19} -1.04948 q^{20} -1.48793 q^{21} -2.34415 q^{22} +6.53741 q^{23} -0.512072 q^{24} -3.89859 q^{25} -4.24985 q^{26} -2.93816 q^{27} -2.90570 q^{28} +0.119634 q^{29} +0.537410 q^{30} +3.75481 q^{31} -1.00000 q^{32} +1.20037 q^{33} -7.66052 q^{34} +3.04948 q^{35} -2.73778 q^{36} -3.27400 q^{37} -5.81141 q^{38} +2.17623 q^{39} +1.04948 q^{40} +4.46726 q^{41} +1.48793 q^{42} -3.34051 q^{43} +2.34415 q^{44} +2.87325 q^{45} -6.53741 q^{46} +2.61468 q^{47} +0.512072 q^{48} +1.44311 q^{49} +3.89859 q^{50} +3.92273 q^{51} +4.24985 q^{52} +2.87689 q^{53} +2.93816 q^{54} -2.46014 q^{55} +2.90570 q^{56} +2.97586 q^{57} -0.119634 q^{58} +0.549773 q^{59} -0.537410 q^{60} -8.24985 q^{61} -3.75481 q^{62} +7.95518 q^{63} +1.00000 q^{64} -4.46014 q^{65} -1.20037 q^{66} +1.09430 q^{67} +7.66052 q^{68} +3.34762 q^{69} -3.04948 q^{70} -8.10363 q^{71} +2.73778 q^{72} +9.15191 q^{73} +3.27400 q^{74} -1.99636 q^{75} +5.81141 q^{76} -6.81141 q^{77} -2.17623 q^{78} +0.426084 q^{79} -1.04948 q^{80} +6.70880 q^{81} -4.46726 q^{82} +7.84547 q^{83} -1.48793 q^{84} -8.03957 q^{85} +3.34051 q^{86} +0.0612614 q^{87} -2.34415 q^{88} +15.7424 q^{89} -2.87325 q^{90} -12.3488 q^{91} +6.53741 q^{92} +1.92273 q^{93} -2.61468 q^{94} -6.09896 q^{95} -0.512072 q^{96} -14.9051 q^{97} -1.44311 q^{98} -6.41778 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + 3 q^{3} + 4 q^{4} + 5 q^{5} - 3 q^{6} - q^{7} - 4 q^{8} + 3 q^{9}+O(q^{10}) \) 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
\(3\) 0.512072 0.295645 0.147822 0.989014i \(-0.452774\pi\)
0.147822 + 0.989014i \(0.452774\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.04948 −0.469342 −0.234671 0.972075i \(-0.575401\pi\)
−0.234671 + 0.972075i \(0.575401\pi\)
\(6\) −0.512072 −0.209052
\(7\) −2.90570 −1.09825 −0.549126 0.835739i \(-0.685039\pi\)
−0.549126 + 0.835739i \(0.685039\pi\)
\(8\) −1.00000 −0.353553
\(9\) −2.73778 −0.912594
\(10\) 1.04948 0.331875
\(11\) 2.34415 0.706788 0.353394 0.935475i \(-0.385028\pi\)
0.353394 + 0.935475i \(0.385028\pi\)
\(12\) 0.512072 0.147822
\(13\) 4.24985 1.17870 0.589349 0.807879i \(-0.299384\pi\)
0.589349 + 0.807879i \(0.299384\pi\)
\(14\) 2.90570 0.776582
\(15\) −0.537410 −0.138759
\(16\) 1.00000 0.250000
\(17\) 7.66052 1.85795 0.928974 0.370145i \(-0.120692\pi\)
0.928974 + 0.370145i \(0.120692\pi\)
\(18\) 2.73778 0.645302
\(19\) 5.81141 1.33323 0.666614 0.745403i \(-0.267743\pi\)
0.666614 + 0.745403i \(0.267743\pi\)
\(20\) −1.04948 −0.234671
\(21\) −1.48793 −0.324693
\(22\) −2.34415 −0.499775
\(23\) 6.53741 1.36314 0.681572 0.731751i \(-0.261297\pi\)
0.681572 + 0.731751i \(0.261297\pi\)
\(24\) −0.512072 −0.104526
\(25\) −3.89859 −0.779718
\(26\) −4.24985 −0.833465
\(27\) −2.93816 −0.565448
\(28\) −2.90570 −0.549126
\(29\) 0.119634 0.0222155 0.0111078 0.999938i \(-0.496464\pi\)
0.0111078 + 0.999938i \(0.496464\pi\)
\(30\) 0.537410 0.0981171
\(31\) 3.75481 0.674384 0.337192 0.941436i \(-0.390523\pi\)
0.337192 + 0.941436i \(0.390523\pi\)
\(32\) −1.00000 −0.176777
\(33\) 1.20037 0.208958
\(34\) −7.66052 −1.31377
\(35\) 3.04948 0.515456
\(36\) −2.73778 −0.456297
\(37\) −3.27400 −0.538242 −0.269121 0.963106i \(-0.586733\pi\)
−0.269121 + 0.963106i \(0.586733\pi\)
\(38\) −5.81141 −0.942735
\(39\) 2.17623 0.348476
\(40\) 1.04948 0.165938
\(41\) 4.46726 0.697668 0.348834 0.937184i \(-0.386578\pi\)
0.348834 + 0.937184i \(0.386578\pi\)
\(42\) 1.48793 0.229592
\(43\) −3.34051 −0.509423 −0.254711 0.967017i \(-0.581980\pi\)
−0.254711 + 0.967017i \(0.581980\pi\)
\(44\) 2.34415 0.353394
\(45\) 2.87325 0.428319
\(46\) −6.53741 −0.963888
\(47\) 2.61468 0.381390 0.190695 0.981649i \(-0.438926\pi\)
0.190695 + 0.981649i \(0.438926\pi\)
\(48\) 0.512072 0.0739112
\(49\) 1.44311 0.206159
\(50\) 3.89859 0.551344
\(51\) 3.92273 0.549292
\(52\) 4.24985 0.589349
\(53\) 2.87689 0.395172 0.197586 0.980286i \(-0.436690\pi\)
0.197586 + 0.980286i \(0.436690\pi\)
\(54\) 2.93816 0.399832
\(55\) −2.46014 −0.331725
\(56\) 2.90570 0.388291
\(57\) 2.97586 0.394162
\(58\) −0.119634 −0.0157088
\(59\) 0.549773 0.0715743 0.0357871 0.999359i \(-0.488606\pi\)
0.0357871 + 0.999359i \(0.488606\pi\)
\(60\) −0.537410 −0.0693793
\(61\) −8.24985 −1.05629 −0.528143 0.849156i \(-0.677111\pi\)
−0.528143 + 0.849156i \(0.677111\pi\)
\(62\) −3.75481 −0.476862
\(63\) 7.95518 1.00226
\(64\) 1.00000 0.125000
\(65\) −4.46014 −0.553213
\(66\) −1.20037 −0.147756
\(67\) 1.09430 0.133690 0.0668448 0.997763i \(-0.478707\pi\)
0.0668448 + 0.997763i \(0.478707\pi\)
\(68\) 7.66052 0.928974
\(69\) 3.34762 0.403006
\(70\) −3.04948 −0.364483
\(71\) −8.10363 −0.961724 −0.480862 0.876796i \(-0.659676\pi\)
−0.480862 + 0.876796i \(0.659676\pi\)
\(72\) 2.73778 0.322651
\(73\) 9.15191 1.07115 0.535575 0.844487i \(-0.320095\pi\)
0.535575 + 0.844487i \(0.320095\pi\)
\(74\) 3.27400 0.380594
\(75\) −1.99636 −0.230519
\(76\) 5.81141 0.666614
\(77\) −6.81141 −0.776232
\(78\) −2.17623 −0.246410
\(79\) 0.426084 0.0479382 0.0239691 0.999713i \(-0.492370\pi\)
0.0239691 + 0.999713i \(0.492370\pi\)
\(80\) −1.04948 −0.117336
\(81\) 6.70880 0.745422
\(82\) −4.46726 −0.493326
\(83\) 7.84547 0.861152 0.430576 0.902554i \(-0.358311\pi\)
0.430576 + 0.902554i \(0.358311\pi\)
\(84\) −1.48793 −0.162346
\(85\) −8.03957 −0.872013
\(86\) 3.34051 0.360216
\(87\) 0.0612614 0.00656791
\(88\) −2.34415 −0.249887
\(89\) 15.7424 1.66870 0.834348 0.551238i \(-0.185844\pi\)
0.834348 + 0.551238i \(0.185844\pi\)
\(90\) −2.87325 −0.302867
\(91\) −12.3488 −1.29451
\(92\) 6.53741 0.681572
\(93\) 1.92273 0.199378
\(94\) −2.61468 −0.269683
\(95\) −6.09896 −0.625740
\(96\) −0.512072 −0.0522631
\(97\) −14.9051 −1.51339 −0.756693 0.653771i \(-0.773186\pi\)
−0.756693 + 0.653771i \(0.773186\pi\)
\(98\) −1.44311 −0.145776
\(99\) −6.41778 −0.645011
\(100\) −3.89859 −0.389859
\(101\) −13.0300 −1.29653 −0.648267 0.761413i \(-0.724506\pi\)
−0.648267 + 0.761413i \(0.724506\pi\)
\(102\) −3.92273 −0.388408
\(103\) 8.52505 0.839998 0.419999 0.907525i \(-0.362030\pi\)
0.419999 + 0.907525i \(0.362030\pi\)
\(104\) −4.24985 −0.416732
\(105\) 1.56155 0.152392
\(106\) −2.87689 −0.279429
\(107\) 11.0371 1.06700 0.533499 0.845801i \(-0.320877\pi\)
0.533499 + 0.845801i \(0.320877\pi\)
\(108\) −2.93816 −0.282724
\(109\) −4.51919 −0.432860 −0.216430 0.976298i \(-0.569441\pi\)
−0.216430 + 0.976298i \(0.569441\pi\)
\(110\) 2.46014 0.234565
\(111\) −1.67652 −0.159128
\(112\) −2.90570 −0.274563
\(113\) −5.82616 −0.548079 −0.274039 0.961718i \(-0.588360\pi\)
−0.274039 + 0.961718i \(0.588360\pi\)
\(114\) −2.97586 −0.278715
\(115\) −6.86089 −0.639781
\(116\) 0.119634 0.0111078
\(117\) −11.6352 −1.07567
\(118\) −0.549773 −0.0506107
\(119\) −22.2592 −2.04050
\(120\) 0.537410 0.0490586
\(121\) −5.50496 −0.500451
\(122\) 8.24985 0.746907
\(123\) 2.28756 0.206262
\(124\) 3.75481 0.337192
\(125\) 9.33890 0.835297
\(126\) −7.95518 −0.708704
\(127\) −5.97102 −0.529842 −0.264921 0.964270i \(-0.585346\pi\)
−0.264921 + 0.964270i \(0.585346\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −1.71058 −0.150608
\(130\) 4.46014 0.391180
\(131\) 11.3535 0.991958 0.495979 0.868334i \(-0.334809\pi\)
0.495979 + 0.868334i \(0.334809\pi\)
\(132\) 1.20037 0.104479
\(133\) −16.8862 −1.46422
\(134\) −1.09430 −0.0945328
\(135\) 3.08354 0.265389
\(136\) −7.66052 −0.656884
\(137\) 10.6240 0.907670 0.453835 0.891086i \(-0.350055\pi\)
0.453835 + 0.891086i \(0.350055\pi\)
\(138\) −3.34762 −0.284969
\(139\) 8.41186 0.713484 0.356742 0.934203i \(-0.383887\pi\)
0.356742 + 0.934203i \(0.383887\pi\)
\(140\) 3.04948 0.257728
\(141\) 1.33890 0.112756
\(142\) 8.10363 0.680041
\(143\) 9.96230 0.833089
\(144\) −2.73778 −0.228149
\(145\) −0.125554 −0.0104267
\(146\) −9.15191 −0.757418
\(147\) 0.738977 0.0609498
\(148\) −3.27400 −0.269121
\(149\) −17.9321 −1.46905 −0.734526 0.678581i \(-0.762595\pi\)
−0.734526 + 0.678581i \(0.762595\pi\)
\(150\) 1.99636 0.163002
\(151\) −0.938156 −0.0763460 −0.0381730 0.999271i \(-0.512154\pi\)
−0.0381730 + 0.999271i \(0.512154\pi\)
\(152\) −5.81141 −0.471367
\(153\) −20.9728 −1.69555
\(154\) 6.81141 0.548879
\(155\) −3.94060 −0.316517
\(156\) 2.17623 0.174238
\(157\) −1.01948 −0.0813632 −0.0406816 0.999172i \(-0.512953\pi\)
−0.0406816 + 0.999172i \(0.512953\pi\)
\(158\) −0.426084 −0.0338974
\(159\) 1.47318 0.116830
\(160\) 1.04948 0.0829688
\(161\) −18.9958 −1.49708
\(162\) −6.70880 −0.527093
\(163\) 13.3440 1.04518 0.522591 0.852584i \(-0.324966\pi\)
0.522591 + 0.852584i \(0.324966\pi\)
\(164\) 4.46726 0.348834
\(165\) −1.25977 −0.0980729
\(166\) −7.84547 −0.608926
\(167\) 19.8414 1.53537 0.767687 0.640825i \(-0.221407\pi\)
0.767687 + 0.640825i \(0.221407\pi\)
\(168\) 1.48793 0.114796
\(169\) 5.06126 0.389328
\(170\) 8.03957 0.616607
\(171\) −15.9104 −1.21670
\(172\) −3.34051 −0.254711
\(173\) 14.7860 1.12416 0.562080 0.827083i \(-0.310001\pi\)
0.562080 + 0.827083i \(0.310001\pi\)
\(174\) −0.0612614 −0.00464421
\(175\) 11.3281 0.856327
\(176\) 2.34415 0.176697
\(177\) 0.281523 0.0211606
\(178\) −15.7424 −1.17995
\(179\) −13.1868 −0.985624 −0.492812 0.870136i \(-0.664031\pi\)
−0.492812 + 0.870136i \(0.664031\pi\)
\(180\) 2.87325 0.214160
\(181\) −21.7542 −1.61698 −0.808490 0.588511i \(-0.799715\pi\)
−0.808490 + 0.588511i \(0.799715\pi\)
\(182\) 12.3488 0.915355
\(183\) −4.22452 −0.312285
\(184\) −6.53741 −0.481944
\(185\) 3.43600 0.252620
\(186\) −1.92273 −0.140982
\(187\) 17.9574 1.31318
\(188\) 2.61468 0.190695
\(189\) 8.53741 0.621005
\(190\) 6.09896 0.442465
\(191\) −1.23630 −0.0894553 −0.0447276 0.998999i \(-0.514242\pi\)
−0.0447276 + 0.998999i \(0.514242\pi\)
\(192\) 0.512072 0.0369556
\(193\) 1.52749 0.109951 0.0549757 0.998488i \(-0.482492\pi\)
0.0549757 + 0.998488i \(0.482492\pi\)
\(194\) 14.9051 1.07013
\(195\) −2.28391 −0.163554
\(196\) 1.44311 0.103080
\(197\) −22.3689 −1.59372 −0.796859 0.604165i \(-0.793507\pi\)
−0.796859 + 0.604165i \(0.793507\pi\)
\(198\) 6.41778 0.456091
\(199\) −16.7266 −1.18572 −0.592859 0.805306i \(-0.702001\pi\)
−0.592859 + 0.805306i \(0.702001\pi\)
\(200\) 3.89859 0.275672
\(201\) 0.560358 0.0395246
\(202\) 13.0300 0.916788
\(203\) −0.347622 −0.0243983
\(204\) 3.92273 0.274646
\(205\) −4.68830 −0.327445
\(206\) −8.52505 −0.593968
\(207\) −17.8980 −1.24400
\(208\) 4.24985 0.294674
\(209\) 13.6228 0.942310
\(210\) −1.56155 −0.107757
\(211\) −2.00644 −0.138129 −0.0690646 0.997612i \(-0.522001\pi\)
−0.0690646 + 0.997612i \(0.522001\pi\)
\(212\) 2.87689 0.197586
\(213\) −4.14964 −0.284329
\(214\) −11.0371 −0.754482
\(215\) 3.50580 0.239094
\(216\) 2.93816 0.199916
\(217\) −10.9104 −0.740644
\(218\) 4.51919 0.306078
\(219\) 4.68644 0.316680
\(220\) −2.46014 −0.165863
\(221\) 32.5561 2.18996
\(222\) 1.67652 0.112521
\(223\) 17.0201 1.13975 0.569875 0.821732i \(-0.306992\pi\)
0.569875 + 0.821732i \(0.306992\pi\)
\(224\) 2.90570 0.194145
\(225\) 10.6735 0.711566
\(226\) 5.82616 0.387550
\(227\) 17.9605 1.19208 0.596041 0.802954i \(-0.296740\pi\)
0.596041 + 0.802954i \(0.296740\pi\)
\(228\) 2.97586 0.197081
\(229\) 3.85377 0.254665 0.127332 0.991860i \(-0.459359\pi\)
0.127332 + 0.991860i \(0.459359\pi\)
\(230\) 6.86089 0.452394
\(231\) −3.48793 −0.229489
\(232\) −0.119634 −0.00785438
\(233\) 19.8538 1.30066 0.650332 0.759650i \(-0.274630\pi\)
0.650332 + 0.759650i \(0.274630\pi\)
\(234\) 11.6352 0.760615
\(235\) −2.74405 −0.179002
\(236\) 0.549773 0.0357871
\(237\) 0.218186 0.0141727
\(238\) 22.2592 1.44285
\(239\) −11.1250 −0.719615 −0.359807 0.933027i \(-0.617158\pi\)
−0.359807 + 0.933027i \(0.617158\pi\)
\(240\) −0.537410 −0.0346896
\(241\) −17.2740 −1.11272 −0.556358 0.830943i \(-0.687802\pi\)
−0.556358 + 0.830943i \(0.687802\pi\)
\(242\) 5.50496 0.353872
\(243\) 12.2499 0.785829
\(244\) −8.24985 −0.528143
\(245\) −1.51452 −0.0967591
\(246\) −2.28756 −0.145849
\(247\) 24.6976 1.57147
\(248\) −3.75481 −0.238431
\(249\) 4.01744 0.254595
\(250\) −9.33890 −0.590644
\(251\) 2.07362 0.130886 0.0654430 0.997856i \(-0.479154\pi\)
0.0654430 + 0.997856i \(0.479154\pi\)
\(252\) 7.95518 0.501130
\(253\) 15.3247 0.963454
\(254\) 5.97102 0.374655
\(255\) −4.11683 −0.257806
\(256\) 1.00000 0.0625000
\(257\) −26.0799 −1.62682 −0.813411 0.581690i \(-0.802392\pi\)
−0.813411 + 0.581690i \(0.802392\pi\)
\(258\) 1.71058 0.106496
\(259\) 9.51327 0.591126
\(260\) −4.46014 −0.276606
\(261\) −0.327533 −0.0202738
\(262\) −11.3535 −0.701420
\(263\) −9.93935 −0.612887 −0.306443 0.951889i \(-0.599139\pi\)
−0.306443 + 0.951889i \(0.599139\pi\)
\(264\) −1.20037 −0.0738779
\(265\) −3.01925 −0.185471
\(266\) 16.8862 1.03536
\(267\) 8.06126 0.493341
\(268\) 1.09430 0.0668448
\(269\) 1.00000 0.0609711
\(270\) −3.08354 −0.187658
\(271\) 9.40702 0.571436 0.285718 0.958314i \(-0.407768\pi\)
0.285718 + 0.958314i \(0.407768\pi\)
\(272\) 7.66052 0.464487
\(273\) −6.32348 −0.382714
\(274\) −10.6240 −0.641820
\(275\) −9.13888 −0.551095
\(276\) 3.34762 0.201503
\(277\) 1.10380 0.0663209 0.0331605 0.999450i \(-0.489443\pi\)
0.0331605 + 0.999450i \(0.489443\pi\)
\(278\) −8.41186 −0.504510
\(279\) −10.2799 −0.615439
\(280\) −3.04948 −0.182241
\(281\) −16.2356 −0.968536 −0.484268 0.874920i \(-0.660914\pi\)
−0.484268 + 0.874920i \(0.660914\pi\)
\(282\) −1.33890 −0.0797304
\(283\) 22.0635 1.31154 0.655771 0.754960i \(-0.272344\pi\)
0.655771 + 0.754960i \(0.272344\pi\)
\(284\) −8.10363 −0.480862
\(285\) −3.12311 −0.184997
\(286\) −9.96230 −0.589083
\(287\) −12.9805 −0.766216
\(288\) 2.73778 0.161325
\(289\) 41.6835 2.45197
\(290\) 0.125554 0.00737279
\(291\) −7.63249 −0.447424
\(292\) 9.15191 0.535575
\(293\) 12.1496 0.709789 0.354895 0.934906i \(-0.384517\pi\)
0.354895 + 0.934906i \(0.384517\pi\)
\(294\) −0.738977 −0.0430980
\(295\) −0.576976 −0.0335928
\(296\) 3.27400 0.190297
\(297\) −6.88748 −0.399652
\(298\) 17.9321 1.03878
\(299\) 27.7830 1.60673
\(300\) −1.99636 −0.115260
\(301\) 9.70653 0.559475
\(302\) 0.938156 0.0539848
\(303\) −6.67230 −0.383313
\(304\) 5.81141 0.333307
\(305\) 8.65807 0.495759
\(306\) 20.9728 1.19894
\(307\) −22.7400 −1.29784 −0.648920 0.760856i \(-0.724779\pi\)
−0.648920 + 0.760856i \(0.724779\pi\)
\(308\) −6.81141 −0.388116
\(309\) 4.36543 0.248341
\(310\) 3.94060 0.223811
\(311\) −10.9941 −0.623417 −0.311709 0.950178i \(-0.600901\pi\)
−0.311709 + 0.950178i \(0.600901\pi\)
\(312\) −2.17623 −0.123205
\(313\) −25.0357 −1.41510 −0.707551 0.706663i \(-0.750200\pi\)
−0.707551 + 0.706663i \(0.750200\pi\)
\(314\) 1.01948 0.0575324
\(315\) −8.34882 −0.470403
\(316\) 0.426084 0.0239691
\(317\) 29.2168 1.64098 0.820490 0.571661i \(-0.193701\pi\)
0.820490 + 0.571661i \(0.193701\pi\)
\(318\) −1.47318 −0.0826116
\(319\) 0.280441 0.0157017
\(320\) −1.04948 −0.0586678
\(321\) 5.65180 0.315452
\(322\) 18.9958 1.05859
\(323\) 44.5184 2.47707
\(324\) 6.70880 0.372711
\(325\) −16.5684 −0.919051
\(326\) −13.3440 −0.739055
\(327\) −2.31415 −0.127973
\(328\) −4.46726 −0.246663
\(329\) −7.59748 −0.418862
\(330\) 1.25977 0.0693480
\(331\) 15.9156 0.874801 0.437401 0.899267i \(-0.355899\pi\)
0.437401 + 0.899267i \(0.355899\pi\)
\(332\) 7.84547 0.430576
\(333\) 8.96349 0.491196
\(334\) −19.8414 −1.08567
\(335\) −1.14844 −0.0627462
\(336\) −1.48793 −0.0811731
\(337\) −35.1956 −1.91722 −0.958612 0.284715i \(-0.908101\pi\)
−0.958612 + 0.284715i \(0.908101\pi\)
\(338\) −5.06126 −0.275296
\(339\) −2.98341 −0.162037
\(340\) −8.03957 −0.436007
\(341\) 8.80184 0.476647
\(342\) 15.9104 0.860334
\(343\) 16.1467 0.871838
\(344\) 3.34051 0.180108
\(345\) −3.51327 −0.189148
\(346\) −14.7860 −0.794901
\(347\) −24.8903 −1.33618 −0.668091 0.744080i \(-0.732888\pi\)
−0.668091 + 0.744080i \(0.732888\pi\)
\(348\) 0.0612614 0.00328395
\(349\) 1.73175 0.0926985 0.0463492 0.998925i \(-0.485241\pi\)
0.0463492 + 0.998925i \(0.485241\pi\)
\(350\) −11.3281 −0.605515
\(351\) −12.4867 −0.666493
\(352\) −2.34415 −0.124944
\(353\) −21.3682 −1.13732 −0.568658 0.822574i \(-0.692537\pi\)
−0.568658 + 0.822574i \(0.692537\pi\)
\(354\) −0.281523 −0.0149628
\(355\) 8.50461 0.451378
\(356\) 15.7424 0.834348
\(357\) −11.3983 −0.603262
\(358\) 13.1868 0.696942
\(359\) −19.8025 −1.04514 −0.522568 0.852597i \(-0.675026\pi\)
−0.522568 + 0.852597i \(0.675026\pi\)
\(360\) −2.87325 −0.151434
\(361\) 14.7725 0.777497
\(362\) 21.7542 1.14338
\(363\) −2.81893 −0.147956
\(364\) −12.3488 −0.647254
\(365\) −9.60476 −0.502736
\(366\) 4.22452 0.220819
\(367\) −1.72720 −0.0901590 −0.0450795 0.998983i \(-0.514354\pi\)
−0.0450795 + 0.998983i \(0.514354\pi\)
\(368\) 6.53741 0.340786
\(369\) −12.2304 −0.636688
\(370\) −3.43600 −0.178629
\(371\) −8.35940 −0.433999
\(372\) 1.92273 0.0996890
\(373\) −31.3423 −1.62284 −0.811422 0.584461i \(-0.801306\pi\)
−0.811422 + 0.584461i \(0.801306\pi\)
\(374\) −17.9574 −0.928555
\(375\) 4.78219 0.246951
\(376\) −2.61468 −0.134842
\(377\) 0.508429 0.0261854
\(378\) −8.53741 −0.439117
\(379\) 21.2263 1.09032 0.545161 0.838331i \(-0.316468\pi\)
0.545161 + 0.838331i \(0.316468\pi\)
\(380\) −6.09896 −0.312870
\(381\) −3.05759 −0.156645
\(382\) 1.23630 0.0632544
\(383\) −34.5798 −1.76695 −0.883473 0.468483i \(-0.844801\pi\)
−0.883473 + 0.468483i \(0.844801\pi\)
\(384\) −0.512072 −0.0261315
\(385\) 7.14844 0.364318
\(386\) −1.52749 −0.0777474
\(387\) 9.14558 0.464896
\(388\) −14.9051 −0.756693
\(389\) 2.21454 0.112282 0.0561409 0.998423i \(-0.482120\pi\)
0.0561409 + 0.998423i \(0.482120\pi\)
\(390\) 2.28391 0.115650
\(391\) 50.0799 2.53265
\(392\) −1.44311 −0.0728882
\(393\) 5.81380 0.293267
\(394\) 22.3689 1.12693
\(395\) −0.447167 −0.0224994
\(396\) −6.41778 −0.322505
\(397\) −28.7782 −1.44434 −0.722168 0.691717i \(-0.756854\pi\)
−0.722168 + 0.691717i \(0.756854\pi\)
\(398\) 16.7266 0.838429
\(399\) −8.64696 −0.432889
\(400\) −3.89859 −0.194929
\(401\) −9.00119 −0.449498 −0.224749 0.974417i \(-0.572156\pi\)
−0.224749 + 0.974417i \(0.572156\pi\)
\(402\) −0.560358 −0.0279481
\(403\) 15.9574 0.794895
\(404\) −13.0300 −0.648267
\(405\) −7.04076 −0.349858
\(406\) 0.347622 0.0172522
\(407\) −7.67474 −0.380423
\(408\) −3.92273 −0.194204
\(409\) 36.6526 1.81236 0.906178 0.422896i \(-0.138986\pi\)
0.906178 + 0.422896i \(0.138986\pi\)
\(410\) 4.68830 0.231539
\(411\) 5.44025 0.268348
\(412\) 8.52505 0.419999
\(413\) −1.59748 −0.0786067
\(414\) 17.8980 0.879639
\(415\) −8.23367 −0.404175
\(416\) −4.24985 −0.208366
\(417\) 4.30747 0.210938
\(418\) −13.6228 −0.666314
\(419\) −1.67822 −0.0819862 −0.0409931 0.999159i \(-0.513052\pi\)
−0.0409931 + 0.999159i \(0.513052\pi\)
\(420\) 1.56155 0.0761960
\(421\) −8.60926 −0.419589 −0.209795 0.977745i \(-0.567280\pi\)
−0.209795 + 0.977745i \(0.567280\pi\)
\(422\) 2.00644 0.0976721
\(423\) −7.15842 −0.348054
\(424\) −2.87689 −0.139714
\(425\) −29.8652 −1.44868
\(426\) 4.14964 0.201051
\(427\) 23.9716 1.16007
\(428\) 11.0371 0.533499
\(429\) 5.10141 0.246298
\(430\) −3.50580 −0.169065
\(431\) 20.5339 0.989081 0.494540 0.869155i \(-0.335336\pi\)
0.494540 + 0.869155i \(0.335336\pi\)
\(432\) −2.93816 −0.141362
\(433\) 0.678127 0.0325887 0.0162944 0.999867i \(-0.494813\pi\)
0.0162944 + 0.999867i \(0.494813\pi\)
\(434\) 10.9104 0.523714
\(435\) −0.0642926 −0.00308260
\(436\) −4.51919 −0.216430
\(437\) 37.9915 1.81738
\(438\) −4.68644 −0.223927
\(439\) −33.4206 −1.59508 −0.797539 0.603267i \(-0.793865\pi\)
−0.797539 + 0.603267i \(0.793865\pi\)
\(440\) 2.46014 0.117283
\(441\) −3.95093 −0.188140
\(442\) −32.5561 −1.54853
\(443\) 21.7843 1.03500 0.517501 0.855682i \(-0.326862\pi\)
0.517501 + 0.855682i \(0.326862\pi\)
\(444\) −1.67652 −0.0795642
\(445\) −16.5214 −0.783190
\(446\) −17.0201 −0.805925
\(447\) −9.18250 −0.434317
\(448\) −2.90570 −0.137282
\(449\) −15.6603 −0.739057 −0.369529 0.929219i \(-0.620481\pi\)
−0.369529 + 0.929219i \(0.620481\pi\)
\(450\) −10.6735 −0.503153
\(451\) 10.4719 0.493104
\(452\) −5.82616 −0.274039
\(453\) −0.480403 −0.0225713
\(454\) −17.9605 −0.842929
\(455\) 12.9599 0.607567
\(456\) −2.97586 −0.139357
\(457\) −4.59453 −0.214923 −0.107461 0.994209i \(-0.534272\pi\)
−0.107461 + 0.994209i \(0.534272\pi\)
\(458\) −3.85377 −0.180075
\(459\) −22.5078 −1.05057
\(460\) −6.86089 −0.319891
\(461\) −32.2651 −1.50274 −0.751368 0.659884i \(-0.770606\pi\)
−0.751368 + 0.659884i \(0.770606\pi\)
\(462\) 3.48793 0.162273
\(463\) 20.4853 0.952033 0.476017 0.879436i \(-0.342080\pi\)
0.476017 + 0.879436i \(0.342080\pi\)
\(464\) 0.119634 0.00555389
\(465\) −2.01787 −0.0935766
\(466\) −19.8538 −0.919708
\(467\) 36.5500 1.69133 0.845667 0.533711i \(-0.179203\pi\)
0.845667 + 0.533711i \(0.179203\pi\)
\(468\) −11.6352 −0.537836
\(469\) −3.17970 −0.146825
\(470\) 2.74405 0.126574
\(471\) −0.522045 −0.0240546
\(472\) −0.549773 −0.0253053
\(473\) −7.83065 −0.360054
\(474\) −0.218186 −0.0100216
\(475\) −22.6563 −1.03954
\(476\) −22.2592 −1.02025
\(477\) −7.87631 −0.360632
\(478\) 11.1250 0.508844
\(479\) −36.6751 −1.67573 −0.837864 0.545879i \(-0.816196\pi\)
−0.837864 + 0.545879i \(0.816196\pi\)
\(480\) 0.537410 0.0245293
\(481\) −13.9140 −0.634424
\(482\) 17.2740 0.786809
\(483\) −9.72720 −0.442603
\(484\) −5.50496 −0.250225
\(485\) 15.6426 0.710296
\(486\) −12.2499 −0.555665
\(487\) 34.8684 1.58004 0.790020 0.613081i \(-0.210070\pi\)
0.790020 + 0.613081i \(0.210070\pi\)
\(488\) 8.24985 0.373453
\(489\) 6.83307 0.309002
\(490\) 1.51452 0.0684190
\(491\) −2.32389 −0.104876 −0.0524378 0.998624i \(-0.516699\pi\)
−0.0524378 + 0.998624i \(0.516699\pi\)
\(492\) 2.28756 0.103131
\(493\) 0.916461 0.0412753
\(494\) −24.6976 −1.11120
\(495\) 6.73533 0.302731
\(496\) 3.75481 0.168596
\(497\) 23.5467 1.05622
\(498\) −4.01744 −0.180026
\(499\) 25.6747 1.14936 0.574680 0.818378i \(-0.305127\pi\)
0.574680 + 0.818378i \(0.305127\pi\)
\(500\) 9.33890 0.417648
\(501\) 10.1602 0.453925
\(502\) −2.07362 −0.0925504
\(503\) −35.8649 −1.59914 −0.799569 0.600575i \(-0.794939\pi\)
−0.799569 + 0.600575i \(0.794939\pi\)
\(504\) −7.95518 −0.354352
\(505\) 13.6747 0.608518
\(506\) −15.3247 −0.681265
\(507\) 2.59173 0.115103
\(508\) −5.97102 −0.264921
\(509\) 26.1779 1.16032 0.580158 0.814504i \(-0.302991\pi\)
0.580158 + 0.814504i \(0.302991\pi\)
\(510\) 4.11683 0.182296
\(511\) −26.5928 −1.17639
\(512\) −1.00000 −0.0441942
\(513\) −17.0748 −0.753872
\(514\) 26.0799 1.15034
\(515\) −8.94688 −0.394246
\(516\) −1.71058 −0.0753041
\(517\) 6.12920 0.269562
\(518\) −9.51327 −0.417989
\(519\) 7.57150 0.332352
\(520\) 4.46014 0.195590
\(521\) −30.9190 −1.35459 −0.677294 0.735713i \(-0.736847\pi\)
−0.677294 + 0.735713i \(0.736847\pi\)
\(522\) 0.327533 0.0143357
\(523\) −17.6068 −0.769892 −0.384946 0.922939i \(-0.625780\pi\)
−0.384946 + 0.922939i \(0.625780\pi\)
\(524\) 11.3535 0.495979
\(525\) 5.80082 0.253169
\(526\) 9.93935 0.433376
\(527\) 28.7638 1.25297
\(528\) 1.20037 0.0522395
\(529\) 19.7377 0.858162
\(530\) 3.01925 0.131148
\(531\) −1.50516 −0.0653183
\(532\) −16.8862 −0.732111
\(533\) 18.9852 0.822340
\(534\) −8.06126 −0.348845
\(535\) −11.5832 −0.500787
\(536\) −1.09430 −0.0472664
\(537\) −6.75256 −0.291395
\(538\) −1.00000 −0.0431131
\(539\) 3.38287 0.145711
\(540\) 3.08354 0.132694
\(541\) 4.03939 0.173667 0.0868336 0.996223i \(-0.472325\pi\)
0.0868336 + 0.996223i \(0.472325\pi\)
\(542\) −9.40702 −0.404066
\(543\) −11.1397 −0.478051
\(544\) −7.66052 −0.328442
\(545\) 4.74280 0.203159
\(546\) 6.32348 0.270620
\(547\) −40.7182 −1.74098 −0.870491 0.492184i \(-0.836199\pi\)
−0.870491 + 0.492184i \(0.836199\pi\)
\(548\) 10.6240 0.453835
\(549\) 22.5863 0.963960
\(550\) 9.13888 0.389683
\(551\) 0.695244 0.0296184
\(552\) −3.34762 −0.142484
\(553\) −1.23807 −0.0526483
\(554\) −1.10380 −0.0468960
\(555\) 1.75948 0.0746857
\(556\) 8.41186 0.356742
\(557\) 19.3259 0.818866 0.409433 0.912340i \(-0.365727\pi\)
0.409433 + 0.912340i \(0.365727\pi\)
\(558\) 10.2799 0.435181
\(559\) −14.1967 −0.600455
\(560\) 3.04948 0.128864
\(561\) 9.19548 0.388233
\(562\) 16.2356 0.684859
\(563\) −3.59240 −0.151402 −0.0757008 0.997131i \(-0.524119\pi\)
−0.0757008 + 0.997131i \(0.524119\pi\)
\(564\) 1.33890 0.0563779
\(565\) 6.11444 0.257237
\(566\) −22.0635 −0.927400
\(567\) −19.4938 −0.818662
\(568\) 8.10363 0.340021
\(569\) 15.0425 0.630616 0.315308 0.948989i \(-0.397892\pi\)
0.315308 + 0.948989i \(0.397892\pi\)
\(570\) 3.12311 0.130812
\(571\) −27.0643 −1.13261 −0.566303 0.824197i \(-0.691627\pi\)
−0.566303 + 0.824197i \(0.691627\pi\)
\(572\) 9.96230 0.416545
\(573\) −0.633072 −0.0264470
\(574\) 12.9805 0.541797
\(575\) −25.4867 −1.06287
\(576\) −2.73778 −0.114074
\(577\) 16.5926 0.690761 0.345380 0.938463i \(-0.387750\pi\)
0.345380 + 0.938463i \(0.387750\pi\)
\(578\) −41.6835 −1.73380
\(579\) 0.782187 0.0325066
\(580\) −0.125554 −0.00521335
\(581\) −22.7966 −0.945762
\(582\) 7.63249 0.316377
\(583\) 6.74387 0.279303
\(584\) −9.15191 −0.378709
\(585\) 12.2109 0.504859
\(586\) −12.1496 −0.501897
\(587\) −4.01362 −0.165660 −0.0828298 0.996564i \(-0.526396\pi\)
−0.0828298 + 0.996564i \(0.526396\pi\)
\(588\) 0.738977 0.0304749
\(589\) 21.8207 0.899108
\(590\) 0.576976 0.0237537
\(591\) −11.4545 −0.471174
\(592\) −3.27400 −0.134560
\(593\) 13.1761 0.541076 0.270538 0.962709i \(-0.412798\pi\)
0.270538 + 0.962709i \(0.412798\pi\)
\(594\) 6.88748 0.282597
\(595\) 23.3606 0.957691
\(596\) −17.9321 −0.734526
\(597\) −8.56522 −0.350551
\(598\) −27.7830 −1.13613
\(599\) −6.44903 −0.263500 −0.131750 0.991283i \(-0.542060\pi\)
−0.131750 + 0.991283i \(0.542060\pi\)
\(600\) 1.99636 0.0815009
\(601\) 25.6859 1.04775 0.523876 0.851795i \(-0.324486\pi\)
0.523876 + 0.851795i \(0.324486\pi\)
\(602\) −9.70653 −0.395608
\(603\) −2.99595 −0.122004
\(604\) −0.938156 −0.0381730
\(605\) 5.77735 0.234883
\(606\) 6.67230 0.271043
\(607\) −31.9037 −1.29493 −0.647466 0.762095i \(-0.724171\pi\)
−0.647466 + 0.762095i \(0.724171\pi\)
\(608\) −5.81141 −0.235684
\(609\) −0.178007 −0.00721322
\(610\) −8.65807 −0.350555
\(611\) 11.1120 0.449543
\(612\) −20.9728 −0.847776
\(613\) 20.5761 0.831062 0.415531 0.909579i \(-0.363596\pi\)
0.415531 + 0.909579i \(0.363596\pi\)
\(614\) 22.7400 0.917712
\(615\) −2.40075 −0.0968074
\(616\) 6.81141 0.274439
\(617\) 35.3614 1.42360 0.711798 0.702384i \(-0.247881\pi\)
0.711798 + 0.702384i \(0.247881\pi\)
\(618\) −4.36543 −0.175604
\(619\) 44.9921 1.80839 0.904193 0.427124i \(-0.140473\pi\)
0.904193 + 0.427124i \(0.140473\pi\)
\(620\) −3.94060 −0.158258
\(621\) −19.2079 −0.770788
\(622\) 10.9941 0.440822
\(623\) −45.7429 −1.83265
\(624\) 2.17623 0.0871189
\(625\) 9.69194 0.387678
\(626\) 25.0357 1.00063
\(627\) 6.97586 0.278589
\(628\) −1.01948 −0.0406816
\(629\) −25.0805 −1.00003
\(630\) 8.34882 0.332625
\(631\) −39.3156 −1.56513 −0.782565 0.622569i \(-0.786089\pi\)
−0.782565 + 0.622569i \(0.786089\pi\)
\(632\) −0.426084 −0.0169487
\(633\) −1.02744 −0.0408372
\(634\) −29.2168 −1.16035
\(635\) 6.26647 0.248677
\(636\) 1.47318 0.0584152
\(637\) 6.13302 0.242999
\(638\) −0.280441 −0.0111028
\(639\) 22.1860 0.877664
\(640\) 1.04948 0.0414844
\(641\) 28.3628 1.12026 0.560132 0.828403i \(-0.310750\pi\)
0.560132 + 0.828403i \(0.310750\pi\)
\(642\) −5.65180 −0.223059
\(643\) 46.4066 1.83010 0.915049 0.403343i \(-0.132152\pi\)
0.915049 + 0.403343i \(0.132152\pi\)
\(644\) −18.9958 −0.748538
\(645\) 1.79522 0.0706867
\(646\) −44.5184 −1.75155
\(647\) 9.47203 0.372384 0.186192 0.982513i \(-0.440385\pi\)
0.186192 + 0.982513i \(0.440385\pi\)
\(648\) −6.70880 −0.263547
\(649\) 1.28875 0.0505879
\(650\) 16.5684 0.649868
\(651\) −5.58689 −0.218968
\(652\) 13.3440 0.522591
\(653\) 25.5129 0.998398 0.499199 0.866487i \(-0.333628\pi\)
0.499199 + 0.866487i \(0.333628\pi\)
\(654\) 2.31415 0.0904903
\(655\) −11.9153 −0.465568
\(656\) 4.46726 0.174417
\(657\) −25.0560 −0.977526
\(658\) 7.59748 0.296180
\(659\) 11.7305 0.456955 0.228478 0.973549i \(-0.426625\pi\)
0.228478 + 0.973549i \(0.426625\pi\)
\(660\) −1.25977 −0.0490364
\(661\) −20.7855 −0.808462 −0.404231 0.914657i \(-0.632461\pi\)
−0.404231 + 0.914657i \(0.632461\pi\)
\(662\) −15.9156 −0.618578
\(663\) 16.6710 0.647450
\(664\) −7.84547 −0.304463
\(665\) 17.7218 0.687221
\(666\) −8.96349 −0.347328
\(667\) 0.782099 0.0302830
\(668\) 19.8414 0.767687
\(669\) 8.71550 0.336961
\(670\) 1.14844 0.0443682
\(671\) −19.3389 −0.746570
\(672\) 1.48793 0.0573981
\(673\) −5.05829 −0.194983 −0.0974914 0.995236i \(-0.531082\pi\)
−0.0974914 + 0.995236i \(0.531082\pi\)
\(674\) 35.1956 1.35568
\(675\) 11.4547 0.440890
\(676\) 5.06126 0.194664
\(677\) 14.5131 0.557784 0.278892 0.960323i \(-0.410033\pi\)
0.278892 + 0.960323i \(0.410033\pi\)
\(678\) 2.98341 0.114577
\(679\) 43.3099 1.66208
\(680\) 8.03957 0.308303
\(681\) 9.19707 0.352433
\(682\) −8.80184 −0.337040
\(683\) −48.9508 −1.87305 −0.936525 0.350601i \(-0.885977\pi\)
−0.936525 + 0.350601i \(0.885977\pi\)
\(684\) −15.9104 −0.608348
\(685\) −11.1497 −0.426008
\(686\) −16.1467 −0.616483
\(687\) 1.97341 0.0752902
\(688\) −3.34051 −0.127356
\(689\) 12.2264 0.465788
\(690\) 3.51327 0.133748
\(691\) −35.2573 −1.34125 −0.670626 0.741795i \(-0.733975\pi\)
−0.670626 + 0.741795i \(0.733975\pi\)
\(692\) 14.7860 0.562080
\(693\) 18.6482 0.708385
\(694\) 24.8903 0.944823
\(695\) −8.82808 −0.334868
\(696\) −0.0612614 −0.00232211
\(697\) 34.2215 1.29623
\(698\) −1.73175 −0.0655477
\(699\) 10.1666 0.384534
\(700\) 11.3281 0.428164
\(701\) 21.9096 0.827515 0.413757 0.910387i \(-0.364216\pi\)
0.413757 + 0.910387i \(0.364216\pi\)
\(702\) 12.4867 0.471281
\(703\) −19.0265 −0.717599
\(704\) 2.34415 0.0883485
\(705\) −1.40515 −0.0529211
\(706\) 21.3682 0.804204
\(707\) 37.8613 1.42392
\(708\) 0.281523 0.0105803
\(709\) 7.16159 0.268959 0.134480 0.990916i \(-0.457064\pi\)
0.134480 + 0.990916i \(0.457064\pi\)
\(710\) −8.50461 −0.319172
\(711\) −1.16653 −0.0437481
\(712\) −15.7424 −0.589973
\(713\) 24.5467 0.919283
\(714\) 11.3983 0.426571
\(715\) −10.4552 −0.391004
\(716\) −13.1868 −0.492812
\(717\) −5.69678 −0.212750
\(718\) 19.8025 0.739023
\(719\) 13.0574 0.486958 0.243479 0.969906i \(-0.421711\pi\)
0.243479 + 0.969906i \(0.421711\pi\)
\(720\) 2.87325 0.107080
\(721\) −24.7713 −0.922530
\(722\) −14.7725 −0.549774
\(723\) −8.84552 −0.328969
\(724\) −21.7542 −0.808490
\(725\) −0.466405 −0.0173219
\(726\) 2.81893 0.104620
\(727\) 39.8163 1.47670 0.738352 0.674416i \(-0.235604\pi\)
0.738352 + 0.674416i \(0.235604\pi\)
\(728\) 12.3488 0.457678
\(729\) −13.8536 −0.513096
\(730\) 9.60476 0.355488
\(731\) −25.5900 −0.946481
\(732\) −4.22452 −0.156143
\(733\) −10.9054 −0.402799 −0.201399 0.979509i \(-0.564549\pi\)
−0.201399 + 0.979509i \(0.564549\pi\)
\(734\) 1.72720 0.0637520
\(735\) −0.775543 −0.0286063
\(736\) −6.53741 −0.240972
\(737\) 2.56520 0.0944902
\(738\) 12.2304 0.450206
\(739\) −51.0337 −1.87731 −0.938653 0.344862i \(-0.887926\pi\)
−0.938653 + 0.344862i \(0.887926\pi\)
\(740\) 3.43600 0.126310
\(741\) 12.6470 0.464598
\(742\) 8.35940 0.306883
\(743\) −20.1783 −0.740269 −0.370135 0.928978i \(-0.620688\pi\)
−0.370135 + 0.928978i \(0.620688\pi\)
\(744\) −1.92273 −0.0704908
\(745\) 18.8194 0.689488
\(746\) 31.3423 1.14752
\(747\) −21.4792 −0.785882
\(748\) 17.9574 0.656588
\(749\) −32.0706 −1.17183
\(750\) −4.78219 −0.174621
\(751\) 5.83301 0.212850 0.106425 0.994321i \(-0.466060\pi\)
0.106425 + 0.994321i \(0.466060\pi\)
\(752\) 2.61468 0.0953475
\(753\) 1.06184 0.0386958
\(754\) −0.508429 −0.0185159
\(755\) 0.984577 0.0358324
\(756\) 8.53741 0.310503
\(757\) 37.8086 1.37418 0.687089 0.726573i \(-0.258888\pi\)
0.687089 + 0.726573i \(0.258888\pi\)
\(758\) −21.2263 −0.770975
\(759\) 7.84733 0.284840
\(760\) 6.09896 0.221233
\(761\) −22.2483 −0.806499 −0.403249 0.915090i \(-0.632119\pi\)
−0.403249 + 0.915090i \(0.632119\pi\)
\(762\) 3.05759 0.110765
\(763\) 13.1314 0.475389
\(764\) −1.23630 −0.0447276
\(765\) 22.0106 0.795794
\(766\) 34.5798 1.24942
\(767\) 2.33645 0.0843644
\(768\) 0.512072 0.0184778
\(769\) 7.40743 0.267119 0.133559 0.991041i \(-0.457359\pi\)
0.133559 + 0.991041i \(0.457359\pi\)
\(770\) −7.14844 −0.257612
\(771\) −13.3548 −0.480961
\(772\) 1.52749 0.0549757
\(773\) −28.3513 −1.01973 −0.509863 0.860256i \(-0.670304\pi\)
−0.509863 + 0.860256i \(0.670304\pi\)
\(774\) −9.14558 −0.328731
\(775\) −14.6385 −0.525829
\(776\) 14.9051 0.535063
\(777\) 4.87147 0.174763
\(778\) −2.21454 −0.0793952
\(779\) 25.9610 0.930151
\(780\) −2.28391 −0.0817772
\(781\) −18.9961 −0.679735
\(782\) −50.0799 −1.79085
\(783\) −0.351504 −0.0125617
\(784\) 1.44311 0.0515398
\(785\) 1.06992 0.0381872
\(786\) −5.81380 −0.207371
\(787\) −4.71084 −0.167923 −0.0839616 0.996469i \(-0.526757\pi\)
−0.0839616 + 0.996469i \(0.526757\pi\)
\(788\) −22.3689 −0.796859
\(789\) −5.08966 −0.181197
\(790\) 0.447167 0.0159095
\(791\) 16.9291 0.601929
\(792\) 6.41778 0.228046
\(793\) −35.0607 −1.24504
\(794\) 28.7782 1.02130
\(795\) −1.54607 −0.0548335
\(796\) −16.7266 −0.592859
\(797\) −17.8537 −0.632411 −0.316206 0.948691i \(-0.602409\pi\)
−0.316206 + 0.948691i \(0.602409\pi\)
\(798\) 8.64696 0.306099
\(799\) 20.0298 0.708602
\(800\) 3.89859 0.137836
\(801\) −43.0994 −1.52284
\(802\) 9.00119 0.317843
\(803\) 21.4535 0.757076
\(804\) 0.560358 0.0197623
\(805\) 19.9357 0.702641
\(806\) −15.9574 −0.562076
\(807\) 0.512072 0.0180258
\(808\) 13.0300 0.458394
\(809\) −34.3781 −1.20867 −0.604334 0.796731i \(-0.706561\pi\)
−0.604334 + 0.796731i \(0.706561\pi\)
\(810\) 7.04076 0.247387
\(811\) −29.4277 −1.03335 −0.516673 0.856183i \(-0.672830\pi\)
−0.516673 + 0.856183i \(0.672830\pi\)
\(812\) −0.347622 −0.0121991
\(813\) 4.81707 0.168942
\(814\) 7.67474 0.269000
\(815\) −14.0043 −0.490548
\(816\) 3.92273 0.137323
\(817\) −19.4131 −0.679177
\(818\) −36.6526 −1.28153
\(819\) 33.8084 1.18136
\(820\) −4.68830 −0.163723
\(821\) −29.8950 −1.04334 −0.521671 0.853147i \(-0.674691\pi\)
−0.521671 + 0.853147i \(0.674691\pi\)
\(822\) −5.44025 −0.189751
\(823\) 25.6212 0.893099 0.446550 0.894759i \(-0.352653\pi\)
0.446550 + 0.894759i \(0.352653\pi\)
\(824\) −8.52505 −0.296984
\(825\) −4.67976 −0.162928
\(826\) 1.59748 0.0555833
\(827\) 33.6271 1.16933 0.584664 0.811276i \(-0.301226\pi\)
0.584664 + 0.811276i \(0.301226\pi\)
\(828\) −17.8980 −0.621999
\(829\) −9.56047 −0.332049 −0.166024 0.986122i \(-0.553093\pi\)
−0.166024 + 0.986122i \(0.553093\pi\)
\(830\) 8.23367 0.285795
\(831\) 0.565225 0.0196074
\(832\) 4.24985 0.147337
\(833\) 11.0550 0.383033
\(834\) −4.30747 −0.149156
\(835\) −20.8232 −0.720616
\(836\) 13.6228 0.471155
\(837\) −11.0322 −0.381329
\(838\) 1.67822 0.0579730
\(839\) −3.75123 −0.129507 −0.0647534 0.997901i \(-0.520626\pi\)
−0.0647534 + 0.997901i \(0.520626\pi\)
\(840\) −1.56155 −0.0538787
\(841\) −28.9857 −0.999506
\(842\) 8.60926 0.296695
\(843\) −8.31380 −0.286343
\(844\) −2.00644 −0.0690646
\(845\) −5.31170 −0.182728
\(846\) 7.15842 0.246111
\(847\) 15.9958 0.549621
\(848\) 2.87689 0.0987930
\(849\) 11.2981 0.387750
\(850\) 29.8652 1.02437
\(851\) −21.4035 −0.733701
\(852\) −4.14964 −0.142164
\(853\) 23.1483 0.792583 0.396291 0.918125i \(-0.370297\pi\)
0.396291 + 0.918125i \(0.370297\pi\)
\(854\) −23.9716 −0.820292
\(855\) 16.6976 0.571047
\(856\) −11.0371 −0.377241
\(857\) −47.4439 −1.62065 −0.810326 0.585979i \(-0.800710\pi\)
−0.810326 + 0.585979i \(0.800710\pi\)
\(858\) −5.10141 −0.174159
\(859\) −10.3635 −0.353597 −0.176798 0.984247i \(-0.556574\pi\)
−0.176798 + 0.984247i \(0.556574\pi\)
\(860\) 3.50580 0.119547
\(861\) −6.64696 −0.226528
\(862\) −20.5339 −0.699386
\(863\) 19.6044 0.667342 0.333671 0.942690i \(-0.391713\pi\)
0.333671 + 0.942690i \(0.391713\pi\)
\(864\) 2.93816 0.0999581
\(865\) −15.5176 −0.527616
\(866\) −0.678127 −0.0230437
\(867\) 21.3449 0.724912
\(868\) −10.9104 −0.370322
\(869\) 0.998805 0.0338822
\(870\) 0.0642926 0.00217972
\(871\) 4.65060 0.157580
\(872\) 4.51919 0.153039
\(873\) 40.8070 1.38111
\(874\) −37.9915 −1.28508
\(875\) −27.1361 −0.917367
\(876\) 4.68644 0.158340
\(877\) 47.4632 1.60272 0.801359 0.598183i \(-0.204111\pi\)
0.801359 + 0.598183i \(0.204111\pi\)
\(878\) 33.4206 1.12789
\(879\) 6.22149 0.209845
\(880\) −2.46014 −0.0829314
\(881\) 56.7144 1.91076 0.955379 0.295383i \(-0.0954473\pi\)
0.955379 + 0.295383i \(0.0954473\pi\)
\(882\) 3.95093 0.133035
\(883\) −45.3481 −1.52609 −0.763043 0.646348i \(-0.776295\pi\)
−0.763043 + 0.646348i \(0.776295\pi\)
\(884\) 32.5561 1.09498
\(885\) −0.295453 −0.00993155
\(886\) −21.7843 −0.731858
\(887\) −4.41133 −0.148118 −0.0740590 0.997254i \(-0.523595\pi\)
−0.0740590 + 0.997254i \(0.523595\pi\)
\(888\) 1.67652 0.0562604
\(889\) 17.3500 0.581901
\(890\) 16.5214 0.553799
\(891\) 15.7264 0.526856
\(892\) 17.0201 0.569875
\(893\) 15.1950 0.508480
\(894\) 9.18250 0.307109
\(895\) 13.8393 0.462595
\(896\) 2.90570 0.0970727
\(897\) 14.2269 0.475023
\(898\) 15.6603 0.522592
\(899\) 0.449204 0.0149818
\(900\) 10.6735 0.355783
\(901\) 22.0385 0.734209
\(902\) −10.4719 −0.348677
\(903\) 4.97044 0.165406
\(904\) 5.82616 0.193775
\(905\) 22.8307 0.758917
\(906\) 0.480403 0.0159603
\(907\) 48.2246 1.60127 0.800635 0.599152i \(-0.204496\pi\)
0.800635 + 0.599152i \(0.204496\pi\)
\(908\) 17.9605 0.596041
\(909\) 35.6733 1.18321
\(910\) −12.9599 −0.429615
\(911\) −9.66308 −0.320152 −0.160076 0.987105i \(-0.551174\pi\)
−0.160076 + 0.987105i \(0.551174\pi\)
\(912\) 2.97586 0.0985405
\(913\) 18.3910 0.608652
\(914\) 4.59453 0.151974
\(915\) 4.43355 0.146569
\(916\) 3.85377 0.127332
\(917\) −32.9899 −1.08942
\(918\) 22.5078 0.742868
\(919\) −2.61634 −0.0863052 −0.0431526 0.999068i \(-0.513740\pi\)
−0.0431526 + 0.999068i \(0.513740\pi\)
\(920\) 6.86089 0.226197
\(921\) −11.6445 −0.383700
\(922\) 32.2651 1.06259
\(923\) −34.4392 −1.13358
\(924\) −3.48793 −0.114744
\(925\) 12.7640 0.419677
\(926\) −20.4853 −0.673189
\(927\) −23.3397 −0.766577
\(928\) −0.119634 −0.00392719
\(929\) 34.1534 1.12054 0.560268 0.828311i \(-0.310698\pi\)
0.560268 + 0.828311i \(0.310698\pi\)
\(930\) 2.01787 0.0661686
\(931\) 8.38652 0.274857
\(932\) 19.8538 0.650332
\(933\) −5.62976 −0.184310
\(934\) −36.5500 −1.19595
\(935\) −18.8460 −0.616329
\(936\) 11.6352 0.380308
\(937\) −0.840971 −0.0274733 −0.0137367 0.999906i \(-0.504373\pi\)
−0.0137367 + 0.999906i \(0.504373\pi\)
\(938\) 3.17970 0.103821
\(939\) −12.8201 −0.418367
\(940\) −2.74405 −0.0895012
\(941\) −16.3930 −0.534398 −0.267199 0.963641i \(-0.586098\pi\)
−0.267199 + 0.963641i \(0.586098\pi\)
\(942\) 0.522045 0.0170092
\(943\) 29.2043 0.951022
\(944\) 0.549773 0.0178936
\(945\) −8.95985 −0.291464
\(946\) 7.83065 0.254597
\(947\) −46.3550 −1.50634 −0.753168 0.657828i \(-0.771475\pi\)
−0.753168 + 0.657828i \(0.771475\pi\)
\(948\) 0.218186 0.00708634
\(949\) 38.8943 1.26256
\(950\) 22.6563 0.735067
\(951\) 14.9611 0.485147
\(952\) 22.2592 0.721424
\(953\) −56.3635 −1.82579 −0.912896 0.408191i \(-0.866160\pi\)
−0.912896 + 0.408191i \(0.866160\pi\)
\(954\) 7.87631 0.255005
\(955\) 1.29747 0.0419851
\(956\) −11.1250 −0.359807
\(957\) 0.143606 0.00464212
\(958\) 36.6751 1.18492
\(959\) −30.8702 −0.996851
\(960\) −0.537410 −0.0173448
\(961\) −16.9014 −0.545206
\(962\) 13.9140 0.448606
\(963\) −30.2172 −0.973737
\(964\) −17.2740 −0.556358
\(965\) −1.60308 −0.0516049
\(966\) 9.72720 0.312967
\(967\) −50.1806 −1.61370 −0.806849 0.590758i \(-0.798829\pi\)
−0.806849 + 0.590758i \(0.798829\pi\)
\(968\) 5.50496 0.176936
\(969\) 22.7966 0.732332
\(970\) −15.6426 −0.502255
\(971\) 20.7585 0.666173 0.333087 0.942896i \(-0.391910\pi\)
0.333087 + 0.942896i \(0.391910\pi\)
\(972\) 12.2499 0.392914
\(973\) −24.4424 −0.783586
\(974\) −34.8684 −1.11726
\(975\) −8.48423 −0.271713
\(976\) −8.24985 −0.264071
\(977\) 5.01009 0.160287 0.0801434 0.996783i \(-0.474462\pi\)
0.0801434 + 0.996783i \(0.474462\pi\)
\(978\) −6.83307 −0.218498
\(979\) 36.9027 1.17941
\(980\) −1.51452 −0.0483796
\(981\) 12.3725 0.395025
\(982\) 2.32389 0.0741583
\(983\) −20.6726 −0.659355 −0.329677 0.944094i \(-0.606940\pi\)
−0.329677 + 0.944094i \(0.606940\pi\)
\(984\) −2.28756 −0.0729246
\(985\) 23.4757 0.747999
\(986\) −0.916461 −0.0291861
\(987\) −3.89045 −0.123834
\(988\) 24.6976 0.785736
\(989\) −21.8383 −0.694416
\(990\) −6.73533 −0.214063
\(991\) 43.4299 1.37960 0.689799 0.724001i \(-0.257699\pi\)
0.689799 + 0.724001i \(0.257699\pi\)
\(992\) −3.75481 −0.119215
\(993\) 8.14994 0.258630
\(994\) −23.5467 −0.746857
\(995\) 17.5543 0.556508
\(996\) 4.01744 0.127297
\(997\) −4.39494 −0.139189 −0.0695946 0.997575i \(-0.522171\pi\)
−0.0695946 + 0.997575i \(0.522171\pi\)
\(998\) −25.6747 −0.812720
\(999\) 9.61951 0.304348
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.1.2 4
3.2 odd 2 4842.2.a.j.1.4 4
4.3 odd 2 4304.2.a.e.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
538.2.a.c.1.2 4 1.1 even 1 trivial
4304.2.a.e.1.3 4 4.3 odd 2
4842.2.a.j.1.4 4 3.2 odd 2