Properties

Label 4006.2.a.h.1.8
Level $4006$
Weight $2$
Character 4006.1
Self dual yes
Analytic conductor $31.988$
Analytic rank $0$
Dimension $42$
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] = [4006,2,Mod(1,4006)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4006, 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("4006.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4006 = 2 \cdot 2003 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4006.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: \(31.9880710497\)
Analytic rank: \(0\)
Dimension: \(42\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Character \(\chi\) \(=\) 4006.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -2.67759 q^{3} +1.00000 q^{4} +2.43345 q^{5} +2.67759 q^{6} -4.11067 q^{7} -1.00000 q^{8} +4.16951 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -2.67759 q^{3} +1.00000 q^{4} +2.43345 q^{5} +2.67759 q^{6} -4.11067 q^{7} -1.00000 q^{8} +4.16951 q^{9} -2.43345 q^{10} +1.05949 q^{11} -2.67759 q^{12} -4.86114 q^{13} +4.11067 q^{14} -6.51579 q^{15} +1.00000 q^{16} -5.22105 q^{17} -4.16951 q^{18} -4.63882 q^{19} +2.43345 q^{20} +11.0067 q^{21} -1.05949 q^{22} -7.57779 q^{23} +2.67759 q^{24} +0.921680 q^{25} +4.86114 q^{26} -3.13148 q^{27} -4.11067 q^{28} +0.794274 q^{29} +6.51579 q^{30} -5.03209 q^{31} -1.00000 q^{32} -2.83689 q^{33} +5.22105 q^{34} -10.0031 q^{35} +4.16951 q^{36} +2.84483 q^{37} +4.63882 q^{38} +13.0162 q^{39} -2.43345 q^{40} -5.07384 q^{41} -11.0067 q^{42} -0.955623 q^{43} +1.05949 q^{44} +10.1463 q^{45} +7.57779 q^{46} +11.0124 q^{47} -2.67759 q^{48} +9.89761 q^{49} -0.921680 q^{50} +13.9798 q^{51} -4.86114 q^{52} -6.98643 q^{53} +3.13148 q^{54} +2.57822 q^{55} +4.11067 q^{56} +12.4209 q^{57} -0.794274 q^{58} -4.06419 q^{59} -6.51579 q^{60} +5.19597 q^{61} +5.03209 q^{62} -17.1395 q^{63} +1.00000 q^{64} -11.8293 q^{65} +2.83689 q^{66} -8.04508 q^{67} -5.22105 q^{68} +20.2902 q^{69} +10.0031 q^{70} -7.05132 q^{71} -4.16951 q^{72} +2.51912 q^{73} -2.84483 q^{74} -2.46789 q^{75} -4.63882 q^{76} -4.35523 q^{77} -13.0162 q^{78} +2.83622 q^{79} +2.43345 q^{80} -4.12370 q^{81} +5.07384 q^{82} -9.91499 q^{83} +11.0067 q^{84} -12.7052 q^{85} +0.955623 q^{86} -2.12674 q^{87} -1.05949 q^{88} +6.57163 q^{89} -10.1463 q^{90} +19.9826 q^{91} -7.57779 q^{92} +13.4739 q^{93} -11.0124 q^{94} -11.2883 q^{95} +2.67759 q^{96} +13.2264 q^{97} -9.89761 q^{98} +4.41757 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 42 q - 42 q^{2} + 42 q^{4} + 27 q^{5} - 10 q^{7} - 42 q^{8} + 56 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 42 q - 42 q^{2} + 42 q^{4} + 27 q^{5} - 10 q^{7} - 42 q^{8} + 56 q^{9} - 27 q^{10} + 23 q^{11} + 15 q^{13} + 10 q^{14} + 6 q^{15} + 42 q^{16} + 14 q^{17} - 56 q^{18} - 4 q^{19} + 27 q^{20} + 26 q^{21} - 23 q^{22} + 12 q^{23} + 45 q^{25} - 15 q^{26} - 3 q^{27} - 10 q^{28} + 41 q^{29} - 6 q^{30} + 18 q^{31} - 42 q^{32} + 25 q^{33} - 14 q^{34} + 8 q^{35} + 56 q^{36} + 33 q^{37} + 4 q^{38} + 10 q^{39} - 27 q^{40} + 84 q^{41} - 26 q^{42} - 36 q^{43} + 23 q^{44} + 66 q^{45} - 12 q^{46} + 28 q^{47} + 58 q^{49} - 45 q^{50} + 17 q^{51} + 15 q^{52} + 68 q^{53} + 3 q^{54} - 28 q^{55} + 10 q^{56} - 9 q^{57} - 41 q^{58} + 59 q^{59} + 6 q^{60} + 41 q^{61} - 18 q^{62} - 28 q^{63} + 42 q^{64} + 44 q^{65} - 25 q^{66} + 14 q^{68} + 67 q^{69} - 8 q^{70} + 69 q^{71} - 56 q^{72} - 27 q^{73} - 33 q^{74} + 14 q^{75} - 4 q^{76} + 43 q^{77} - 10 q^{78} - 19 q^{79} + 27 q^{80} + 74 q^{81} - 84 q^{82} + 20 q^{83} + 26 q^{84} + 16 q^{85} + 36 q^{86} - 28 q^{87} - 23 q^{88} + 123 q^{89} - 66 q^{90} - 9 q^{91} + 12 q^{92} + 48 q^{93} - 28 q^{94} + 28 q^{95} + 10 q^{97} - 58 q^{98} + 75 q^{99}+O(q^{100}) \) 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
\(3\) −2.67759 −1.54591 −0.772955 0.634461i \(-0.781222\pi\)
−0.772955 + 0.634461i \(0.781222\pi\)
\(4\) 1.00000 0.500000
\(5\) 2.43345 1.08827 0.544136 0.838997i \(-0.316858\pi\)
0.544136 + 0.838997i \(0.316858\pi\)
\(6\) 2.67759 1.09312
\(7\) −4.11067 −1.55369 −0.776844 0.629694i \(-0.783180\pi\)
−0.776844 + 0.629694i \(0.783180\pi\)
\(8\) −1.00000 −0.353553
\(9\) 4.16951 1.38984
\(10\) −2.43345 −0.769525
\(11\) 1.05949 0.319449 0.159725 0.987162i \(-0.448939\pi\)
0.159725 + 0.987162i \(0.448939\pi\)
\(12\) −2.67759 −0.772955
\(13\) −4.86114 −1.34824 −0.674119 0.738623i \(-0.735477\pi\)
−0.674119 + 0.738623i \(0.735477\pi\)
\(14\) 4.11067 1.09862
\(15\) −6.51579 −1.68237
\(16\) 1.00000 0.250000
\(17\) −5.22105 −1.26629 −0.633145 0.774033i \(-0.718236\pi\)
−0.633145 + 0.774033i \(0.718236\pi\)
\(18\) −4.16951 −0.982764
\(19\) −4.63882 −1.06422 −0.532109 0.846676i \(-0.678600\pi\)
−0.532109 + 0.846676i \(0.678600\pi\)
\(20\) 2.43345 0.544136
\(21\) 11.0067 2.40186
\(22\) −1.05949 −0.225885
\(23\) −7.57779 −1.58008 −0.790039 0.613057i \(-0.789940\pi\)
−0.790039 + 0.613057i \(0.789940\pi\)
\(24\) 2.67759 0.546562
\(25\) 0.921680 0.184336
\(26\) 4.86114 0.953348
\(27\) −3.13148 −0.602654
\(28\) −4.11067 −0.776844
\(29\) 0.794274 0.147493 0.0737465 0.997277i \(-0.476504\pi\)
0.0737465 + 0.997277i \(0.476504\pi\)
\(30\) 6.51579 1.18962
\(31\) −5.03209 −0.903791 −0.451895 0.892071i \(-0.649252\pi\)
−0.451895 + 0.892071i \(0.649252\pi\)
\(32\) −1.00000 −0.176777
\(33\) −2.83689 −0.493840
\(34\) 5.22105 0.895402
\(35\) −10.0031 −1.69083
\(36\) 4.16951 0.694919
\(37\) 2.84483 0.467687 0.233843 0.972274i \(-0.424870\pi\)
0.233843 + 0.972274i \(0.424870\pi\)
\(38\) 4.63882 0.752516
\(39\) 13.0162 2.08426
\(40\) −2.43345 −0.384762
\(41\) −5.07384 −0.792401 −0.396201 0.918164i \(-0.629672\pi\)
−0.396201 + 0.918164i \(0.629672\pi\)
\(42\) −11.0067 −1.69837
\(43\) −0.955623 −0.145731 −0.0728656 0.997342i \(-0.523214\pi\)
−0.0728656 + 0.997342i \(0.523214\pi\)
\(44\) 1.05949 0.159725
\(45\) 10.1463 1.51252
\(46\) 7.57779 1.11728
\(47\) 11.0124 1.60633 0.803164 0.595758i \(-0.203148\pi\)
0.803164 + 0.595758i \(0.203148\pi\)
\(48\) −2.67759 −0.386477
\(49\) 9.89761 1.41394
\(50\) −0.921680 −0.130345
\(51\) 13.9798 1.95757
\(52\) −4.86114 −0.674119
\(53\) −6.98643 −0.959660 −0.479830 0.877361i \(-0.659302\pi\)
−0.479830 + 0.877361i \(0.659302\pi\)
\(54\) 3.13148 0.426140
\(55\) 2.57822 0.347648
\(56\) 4.11067 0.549311
\(57\) 12.4209 1.64518
\(58\) −0.794274 −0.104293
\(59\) −4.06419 −0.529113 −0.264556 0.964370i \(-0.585225\pi\)
−0.264556 + 0.964370i \(0.585225\pi\)
\(60\) −6.51579 −0.841185
\(61\) 5.19597 0.665276 0.332638 0.943055i \(-0.392061\pi\)
0.332638 + 0.943055i \(0.392061\pi\)
\(62\) 5.03209 0.639076
\(63\) −17.1395 −2.15937
\(64\) 1.00000 0.125000
\(65\) −11.8293 −1.46725
\(66\) 2.83689 0.349197
\(67\) −8.04508 −0.982863 −0.491432 0.870916i \(-0.663526\pi\)
−0.491432 + 0.870916i \(0.663526\pi\)
\(68\) −5.22105 −0.633145
\(69\) 20.2902 2.44266
\(70\) 10.0031 1.19560
\(71\) −7.05132 −0.836837 −0.418419 0.908254i \(-0.637415\pi\)
−0.418419 + 0.908254i \(0.637415\pi\)
\(72\) −4.16951 −0.491382
\(73\) 2.51912 0.294841 0.147420 0.989074i \(-0.452903\pi\)
0.147420 + 0.989074i \(0.452903\pi\)
\(74\) −2.84483 −0.330705
\(75\) −2.46789 −0.284967
\(76\) −4.63882 −0.532109
\(77\) −4.35523 −0.496324
\(78\) −13.0162 −1.47379
\(79\) 2.83622 0.319100 0.159550 0.987190i \(-0.448996\pi\)
0.159550 + 0.987190i \(0.448996\pi\)
\(80\) 2.43345 0.272068
\(81\) −4.12370 −0.458189
\(82\) 5.07384 0.560312
\(83\) −9.91499 −1.08831 −0.544156 0.838984i \(-0.683150\pi\)
−0.544156 + 0.838984i \(0.683150\pi\)
\(84\) 11.0067 1.20093
\(85\) −12.7052 −1.37807
\(86\) 0.955623 0.103047
\(87\) −2.12674 −0.228011
\(88\) −1.05949 −0.112942
\(89\) 6.57163 0.696591 0.348296 0.937385i \(-0.386761\pi\)
0.348296 + 0.937385i \(0.386761\pi\)
\(90\) −10.1463 −1.06951
\(91\) 19.9826 2.09474
\(92\) −7.57779 −0.790039
\(93\) 13.4739 1.39718
\(94\) −11.0124 −1.13585
\(95\) −11.2883 −1.15816
\(96\) 2.67759 0.273281
\(97\) 13.2264 1.34293 0.671467 0.741035i \(-0.265665\pi\)
0.671467 + 0.741035i \(0.265665\pi\)
\(98\) −9.89761 −0.999809
\(99\) 4.41757 0.443983
\(100\) 0.921680 0.0921680
\(101\) −1.97607 −0.196626 −0.0983132 0.995156i \(-0.531345\pi\)
−0.0983132 + 0.995156i \(0.531345\pi\)
\(102\) −13.9798 −1.38421
\(103\) −6.94001 −0.683820 −0.341910 0.939733i \(-0.611074\pi\)
−0.341910 + 0.939733i \(0.611074\pi\)
\(104\) 4.86114 0.476674
\(105\) 26.7843 2.61388
\(106\) 6.98643 0.678582
\(107\) −12.9585 −1.25274 −0.626371 0.779525i \(-0.715461\pi\)
−0.626371 + 0.779525i \(0.715461\pi\)
\(108\) −3.13148 −0.301327
\(109\) 14.4654 1.38554 0.692768 0.721160i \(-0.256391\pi\)
0.692768 + 0.721160i \(0.256391\pi\)
\(110\) −2.57822 −0.245824
\(111\) −7.61730 −0.723002
\(112\) −4.11067 −0.388422
\(113\) 14.2872 1.34403 0.672015 0.740538i \(-0.265429\pi\)
0.672015 + 0.740538i \(0.265429\pi\)
\(114\) −12.4209 −1.16332
\(115\) −18.4402 −1.71955
\(116\) 0.794274 0.0737465
\(117\) −20.2686 −1.87383
\(118\) 4.06419 0.374139
\(119\) 21.4620 1.96742
\(120\) 6.51579 0.594808
\(121\) −9.87747 −0.897952
\(122\) −5.19597 −0.470421
\(123\) 13.5857 1.22498
\(124\) −5.03209 −0.451895
\(125\) −9.92439 −0.887664
\(126\) 17.1395 1.52691
\(127\) 12.7024 1.12715 0.563576 0.826064i \(-0.309425\pi\)
0.563576 + 0.826064i \(0.309425\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 2.55877 0.225287
\(130\) 11.8293 1.03750
\(131\) 2.68811 0.234862 0.117431 0.993081i \(-0.462534\pi\)
0.117431 + 0.993081i \(0.462534\pi\)
\(132\) −2.83689 −0.246920
\(133\) 19.0687 1.65346
\(134\) 8.04508 0.694989
\(135\) −7.62030 −0.655851
\(136\) 5.22105 0.447701
\(137\) 1.54509 0.132006 0.0660031 0.997819i \(-0.478975\pi\)
0.0660031 + 0.997819i \(0.478975\pi\)
\(138\) −20.2902 −1.72722
\(139\) −4.95650 −0.420404 −0.210202 0.977658i \(-0.567412\pi\)
−0.210202 + 0.977658i \(0.567412\pi\)
\(140\) −10.0031 −0.845417
\(141\) −29.4868 −2.48324
\(142\) 7.05132 0.591733
\(143\) −5.15035 −0.430694
\(144\) 4.16951 0.347459
\(145\) 1.93283 0.160512
\(146\) −2.51912 −0.208484
\(147\) −26.5018 −2.18583
\(148\) 2.84483 0.233843
\(149\) −4.48625 −0.367528 −0.183764 0.982970i \(-0.558828\pi\)
−0.183764 + 0.982970i \(0.558828\pi\)
\(150\) 2.46789 0.201502
\(151\) −18.6905 −1.52101 −0.760507 0.649330i \(-0.775050\pi\)
−0.760507 + 0.649330i \(0.775050\pi\)
\(152\) 4.63882 0.376258
\(153\) −21.7692 −1.75994
\(154\) 4.35523 0.350954
\(155\) −12.2453 −0.983570
\(156\) 13.0162 1.04213
\(157\) −4.68508 −0.373910 −0.186955 0.982368i \(-0.559862\pi\)
−0.186955 + 0.982368i \(0.559862\pi\)
\(158\) −2.83622 −0.225638
\(159\) 18.7068 1.48355
\(160\) −2.43345 −0.192381
\(161\) 31.1498 2.45495
\(162\) 4.12370 0.323989
\(163\) −2.98365 −0.233698 −0.116849 0.993150i \(-0.537279\pi\)
−0.116849 + 0.993150i \(0.537279\pi\)
\(164\) −5.07384 −0.396201
\(165\) −6.90344 −0.537432
\(166\) 9.91499 0.769552
\(167\) −14.9757 −1.15885 −0.579427 0.815024i \(-0.696724\pi\)
−0.579427 + 0.815024i \(0.696724\pi\)
\(168\) −11.0067 −0.849186
\(169\) 10.6307 0.817747
\(170\) 12.7052 0.974441
\(171\) −19.3416 −1.47909
\(172\) −0.955623 −0.0728656
\(173\) −3.82867 −0.291089 −0.145544 0.989352i \(-0.546493\pi\)
−0.145544 + 0.989352i \(0.546493\pi\)
\(174\) 2.12674 0.161228
\(175\) −3.78872 −0.286401
\(176\) 1.05949 0.0798623
\(177\) 10.8823 0.817960
\(178\) −6.57163 −0.492564
\(179\) 6.70714 0.501315 0.250657 0.968076i \(-0.419353\pi\)
0.250657 + 0.968076i \(0.419353\pi\)
\(180\) 10.1463 0.756261
\(181\) 15.0132 1.11592 0.557962 0.829866i \(-0.311583\pi\)
0.557962 + 0.829866i \(0.311583\pi\)
\(182\) −19.9826 −1.48121
\(183\) −13.9127 −1.02846
\(184\) 7.57779 0.558642
\(185\) 6.92275 0.508971
\(186\) −13.4739 −0.987955
\(187\) −5.53167 −0.404515
\(188\) 11.0124 0.803164
\(189\) 12.8725 0.936335
\(190\) 11.2883 0.818942
\(191\) −17.7619 −1.28520 −0.642602 0.766200i \(-0.722145\pi\)
−0.642602 + 0.766200i \(0.722145\pi\)
\(192\) −2.67759 −0.193239
\(193\) −14.1089 −1.01558 −0.507790 0.861481i \(-0.669537\pi\)
−0.507790 + 0.861481i \(0.669537\pi\)
\(194\) −13.2264 −0.949597
\(195\) 31.6742 2.26824
\(196\) 9.89761 0.706972
\(197\) −20.3268 −1.44823 −0.724113 0.689681i \(-0.757751\pi\)
−0.724113 + 0.689681i \(0.757751\pi\)
\(198\) −4.41757 −0.313943
\(199\) −6.28032 −0.445200 −0.222600 0.974910i \(-0.571454\pi\)
−0.222600 + 0.974910i \(0.571454\pi\)
\(200\) −0.921680 −0.0651726
\(201\) 21.5415 1.51942
\(202\) 1.97607 0.139036
\(203\) −3.26500 −0.229158
\(204\) 13.9798 0.978785
\(205\) −12.3469 −0.862348
\(206\) 6.94001 0.483534
\(207\) −31.5957 −2.19605
\(208\) −4.86114 −0.337060
\(209\) −4.91480 −0.339964
\(210\) −26.7843 −1.84829
\(211\) −7.77030 −0.534929 −0.267465 0.963568i \(-0.586186\pi\)
−0.267465 + 0.963568i \(0.586186\pi\)
\(212\) −6.98643 −0.479830
\(213\) 18.8806 1.29367
\(214\) 12.9585 0.885823
\(215\) −2.32546 −0.158595
\(216\) 3.13148 0.213070
\(217\) 20.6853 1.40421
\(218\) −14.4654 −0.979722
\(219\) −6.74518 −0.455797
\(220\) 2.57822 0.173824
\(221\) 25.3803 1.70726
\(222\) 7.61730 0.511239
\(223\) 22.9131 1.53437 0.767187 0.641424i \(-0.221656\pi\)
0.767187 + 0.641424i \(0.221656\pi\)
\(224\) 4.11067 0.274656
\(225\) 3.84296 0.256197
\(226\) −14.2872 −0.950373
\(227\) 0.0390277 0.00259036 0.00129518 0.999999i \(-0.499588\pi\)
0.00129518 + 0.999999i \(0.499588\pi\)
\(228\) 12.4209 0.822592
\(229\) 9.09822 0.601227 0.300614 0.953746i \(-0.402809\pi\)
0.300614 + 0.953746i \(0.402809\pi\)
\(230\) 18.4402 1.21591
\(231\) 11.6615 0.767273
\(232\) −0.794274 −0.0521466
\(233\) 8.16189 0.534703 0.267352 0.963599i \(-0.413851\pi\)
0.267352 + 0.963599i \(0.413851\pi\)
\(234\) 20.2686 1.32500
\(235\) 26.7982 1.74812
\(236\) −4.06419 −0.264556
\(237\) −7.59426 −0.493300
\(238\) −21.4620 −1.39118
\(239\) 22.4745 1.45375 0.726876 0.686768i \(-0.240971\pi\)
0.726876 + 0.686768i \(0.240971\pi\)
\(240\) −6.51579 −0.420593
\(241\) 8.44935 0.544271 0.272135 0.962259i \(-0.412270\pi\)
0.272135 + 0.962259i \(0.412270\pi\)
\(242\) 9.87747 0.634948
\(243\) 20.4360 1.31097
\(244\) 5.19597 0.332638
\(245\) 24.0853 1.53876
\(246\) −13.5857 −0.866192
\(247\) 22.5500 1.43482
\(248\) 5.03209 0.319538
\(249\) 26.5483 1.68243
\(250\) 9.92439 0.627673
\(251\) 16.8167 1.06146 0.530729 0.847541i \(-0.321918\pi\)
0.530729 + 0.847541i \(0.321918\pi\)
\(252\) −17.1395 −1.07969
\(253\) −8.02862 −0.504755
\(254\) −12.7024 −0.797016
\(255\) 34.0193 2.13037
\(256\) 1.00000 0.0625000
\(257\) 22.3936 1.39688 0.698438 0.715671i \(-0.253879\pi\)
0.698438 + 0.715671i \(0.253879\pi\)
\(258\) −2.55877 −0.159302
\(259\) −11.6941 −0.726639
\(260\) −11.8293 −0.733625
\(261\) 3.31173 0.204991
\(262\) −2.68811 −0.166072
\(263\) −12.8444 −0.792018 −0.396009 0.918247i \(-0.629605\pi\)
−0.396009 + 0.918247i \(0.629605\pi\)
\(264\) 2.83689 0.174599
\(265\) −17.0011 −1.04437
\(266\) −19.0687 −1.16917
\(267\) −17.5962 −1.07687
\(268\) −8.04508 −0.491432
\(269\) 0.237234 0.0144644 0.00723221 0.999974i \(-0.497698\pi\)
0.00723221 + 0.999974i \(0.497698\pi\)
\(270\) 7.62030 0.463757
\(271\) 2.48344 0.150858 0.0754291 0.997151i \(-0.475967\pi\)
0.0754291 + 0.997151i \(0.475967\pi\)
\(272\) −5.22105 −0.316572
\(273\) −53.5052 −3.23828
\(274\) −1.54509 −0.0933425
\(275\) 0.976514 0.0588860
\(276\) 20.2902 1.22133
\(277\) −18.6309 −1.11942 −0.559711 0.828688i \(-0.689088\pi\)
−0.559711 + 0.828688i \(0.689088\pi\)
\(278\) 4.95650 0.297271
\(279\) −20.9814 −1.25612
\(280\) 10.0031 0.597800
\(281\) 13.3646 0.797263 0.398632 0.917111i \(-0.369485\pi\)
0.398632 + 0.917111i \(0.369485\pi\)
\(282\) 29.4868 1.75591
\(283\) −24.6956 −1.46800 −0.734001 0.679148i \(-0.762349\pi\)
−0.734001 + 0.679148i \(0.762349\pi\)
\(284\) −7.05132 −0.418419
\(285\) 30.2256 1.79041
\(286\) 5.15035 0.304546
\(287\) 20.8569 1.23114
\(288\) −4.16951 −0.245691
\(289\) 10.2593 0.603490
\(290\) −1.93283 −0.113499
\(291\) −35.4148 −2.07605
\(292\) 2.51912 0.147420
\(293\) 28.5865 1.67004 0.835021 0.550217i \(-0.185455\pi\)
0.835021 + 0.550217i \(0.185455\pi\)
\(294\) 26.5018 1.54562
\(295\) −9.89001 −0.575818
\(296\) −2.84483 −0.165352
\(297\) −3.31778 −0.192517
\(298\) 4.48625 0.259881
\(299\) 36.8367 2.13032
\(300\) −2.46789 −0.142483
\(301\) 3.92825 0.226421
\(302\) 18.6905 1.07552
\(303\) 5.29112 0.303967
\(304\) −4.63882 −0.266054
\(305\) 12.6441 0.724001
\(306\) 21.7692 1.24446
\(307\) −4.44773 −0.253845 −0.126923 0.991913i \(-0.540510\pi\)
−0.126923 + 0.991913i \(0.540510\pi\)
\(308\) −4.35523 −0.248162
\(309\) 18.5825 1.05712
\(310\) 12.2453 0.695489
\(311\) 31.1713 1.76756 0.883781 0.467901i \(-0.154990\pi\)
0.883781 + 0.467901i \(0.154990\pi\)
\(312\) −13.0162 −0.736895
\(313\) 16.8653 0.953282 0.476641 0.879098i \(-0.341854\pi\)
0.476641 + 0.879098i \(0.341854\pi\)
\(314\) 4.68508 0.264395
\(315\) −41.7081 −2.34999
\(316\) 2.83622 0.159550
\(317\) −27.4996 −1.54453 −0.772267 0.635298i \(-0.780877\pi\)
−0.772267 + 0.635298i \(0.780877\pi\)
\(318\) −18.7068 −1.04903
\(319\) 0.841528 0.0471165
\(320\) 2.43345 0.136034
\(321\) 34.6975 1.93663
\(322\) −31.1498 −1.73591
\(323\) 24.2195 1.34761
\(324\) −4.12370 −0.229095
\(325\) −4.48042 −0.248529
\(326\) 2.98365 0.165249
\(327\) −38.7325 −2.14191
\(328\) 5.07384 0.280156
\(329\) −45.2685 −2.49573
\(330\) 6.90344 0.380022
\(331\) 14.0627 0.772955 0.386478 0.922299i \(-0.373692\pi\)
0.386478 + 0.922299i \(0.373692\pi\)
\(332\) −9.91499 −0.544156
\(333\) 11.8615 0.650009
\(334\) 14.9757 0.819434
\(335\) −19.5773 −1.06962
\(336\) 11.0067 0.600465
\(337\) 31.8017 1.73235 0.866174 0.499742i \(-0.166572\pi\)
0.866174 + 0.499742i \(0.166572\pi\)
\(338\) −10.6307 −0.578234
\(339\) −38.2554 −2.07775
\(340\) −12.7052 −0.689034
\(341\) −5.33147 −0.288715
\(342\) 19.3416 1.04587
\(343\) −11.9111 −0.643140
\(344\) 0.955623 0.0515237
\(345\) 49.3753 2.65828
\(346\) 3.82867 0.205831
\(347\) 32.8021 1.76091 0.880454 0.474131i \(-0.157238\pi\)
0.880454 + 0.474131i \(0.157238\pi\)
\(348\) −2.12674 −0.114005
\(349\) 10.3535 0.554213 0.277106 0.960839i \(-0.410625\pi\)
0.277106 + 0.960839i \(0.410625\pi\)
\(350\) 3.78872 0.202516
\(351\) 15.2226 0.812521
\(352\) −1.05949 −0.0564712
\(353\) −8.48982 −0.451868 −0.225934 0.974143i \(-0.572543\pi\)
−0.225934 + 0.974143i \(0.572543\pi\)
\(354\) −10.8823 −0.578385
\(355\) −17.1590 −0.910706
\(356\) 6.57163 0.348296
\(357\) −57.4665 −3.04145
\(358\) −6.70714 −0.354483
\(359\) 3.52096 0.185829 0.0929146 0.995674i \(-0.470382\pi\)
0.0929146 + 0.995674i \(0.470382\pi\)
\(360\) −10.1463 −0.534757
\(361\) 2.51863 0.132560
\(362\) −15.0132 −0.789078
\(363\) 26.4479 1.38815
\(364\) 19.9826 1.04737
\(365\) 6.13015 0.320867
\(366\) 13.9127 0.727229
\(367\) −2.87976 −0.150322 −0.0751612 0.997171i \(-0.523947\pi\)
−0.0751612 + 0.997171i \(0.523947\pi\)
\(368\) −7.57779 −0.395019
\(369\) −21.1555 −1.10131
\(370\) −6.92275 −0.359897
\(371\) 28.7189 1.49101
\(372\) 13.4739 0.698589
\(373\) 4.69567 0.243133 0.121566 0.992583i \(-0.461208\pi\)
0.121566 + 0.992583i \(0.461208\pi\)
\(374\) 5.53167 0.286036
\(375\) 26.5735 1.37225
\(376\) −11.0124 −0.567923
\(377\) −3.86108 −0.198856
\(378\) −12.8725 −0.662089
\(379\) 17.5661 0.902310 0.451155 0.892446i \(-0.351012\pi\)
0.451155 + 0.892446i \(0.351012\pi\)
\(380\) −11.2883 −0.579079
\(381\) −34.0117 −1.74247
\(382\) 17.7619 0.908776
\(383\) 29.3181 1.49809 0.749043 0.662521i \(-0.230514\pi\)
0.749043 + 0.662521i \(0.230514\pi\)
\(384\) 2.67759 0.136640
\(385\) −10.5982 −0.540136
\(386\) 14.1089 0.718123
\(387\) −3.98448 −0.202543
\(388\) 13.2264 0.671467
\(389\) −24.4342 −1.23886 −0.619431 0.785051i \(-0.712636\pi\)
−0.619431 + 0.785051i \(0.712636\pi\)
\(390\) −31.6742 −1.60389
\(391\) 39.5640 2.00084
\(392\) −9.89761 −0.499905
\(393\) −7.19768 −0.363075
\(394\) 20.3268 1.02405
\(395\) 6.90181 0.347268
\(396\) 4.41757 0.221991
\(397\) −17.8022 −0.893467 −0.446734 0.894667i \(-0.647413\pi\)
−0.446734 + 0.894667i \(0.647413\pi\)
\(398\) 6.28032 0.314804
\(399\) −51.0581 −2.55610
\(400\) 0.921680 0.0460840
\(401\) −27.1614 −1.35638 −0.678189 0.734888i \(-0.737235\pi\)
−0.678189 + 0.734888i \(0.737235\pi\)
\(402\) −21.5415 −1.07439
\(403\) 24.4617 1.21853
\(404\) −1.97607 −0.0983132
\(405\) −10.0348 −0.498635
\(406\) 3.26500 0.162039
\(407\) 3.01408 0.149402
\(408\) −13.9798 −0.692106
\(409\) 0.0400064 0.00197819 0.000989094 1.00000i \(-0.499685\pi\)
0.000989094 1.00000i \(0.499685\pi\)
\(410\) 12.3469 0.609772
\(411\) −4.13713 −0.204070
\(412\) −6.94001 −0.341910
\(413\) 16.7065 0.822075
\(414\) 31.5957 1.55284
\(415\) −24.1276 −1.18438
\(416\) 4.86114 0.238337
\(417\) 13.2715 0.649907
\(418\) 4.91480 0.240391
\(419\) 3.34725 0.163524 0.0817619 0.996652i \(-0.473945\pi\)
0.0817619 + 0.996652i \(0.473945\pi\)
\(420\) 26.7843 1.30694
\(421\) 39.2445 1.91266 0.956329 0.292292i \(-0.0944179\pi\)
0.956329 + 0.292292i \(0.0944179\pi\)
\(422\) 7.77030 0.378252
\(423\) 45.9165 2.23253
\(424\) 6.98643 0.339291
\(425\) −4.81214 −0.233423
\(426\) −18.8806 −0.914766
\(427\) −21.3589 −1.03363
\(428\) −12.9585 −0.626371
\(429\) 13.7905 0.665814
\(430\) 2.32546 0.112144
\(431\) −8.10983 −0.390637 −0.195318 0.980740i \(-0.562574\pi\)
−0.195318 + 0.980740i \(0.562574\pi\)
\(432\) −3.13148 −0.150663
\(433\) −3.52249 −0.169280 −0.0846400 0.996412i \(-0.526974\pi\)
−0.0846400 + 0.996412i \(0.526974\pi\)
\(434\) −20.6853 −0.992925
\(435\) −5.17532 −0.248138
\(436\) 14.4654 0.692768
\(437\) 35.1520 1.68155
\(438\) 6.74518 0.322297
\(439\) −20.7205 −0.988936 −0.494468 0.869196i \(-0.664637\pi\)
−0.494468 + 0.869196i \(0.664637\pi\)
\(440\) −2.57822 −0.122912
\(441\) 41.2682 1.96515
\(442\) −25.3803 −1.20722
\(443\) 32.4349 1.54103 0.770514 0.637424i \(-0.220000\pi\)
0.770514 + 0.637424i \(0.220000\pi\)
\(444\) −7.61730 −0.361501
\(445\) 15.9917 0.758081
\(446\) −22.9131 −1.08497
\(447\) 12.0123 0.568164
\(448\) −4.11067 −0.194211
\(449\) −7.36674 −0.347658 −0.173829 0.984776i \(-0.555614\pi\)
−0.173829 + 0.984776i \(0.555614\pi\)
\(450\) −3.84296 −0.181159
\(451\) −5.37570 −0.253132
\(452\) 14.2872 0.672015
\(453\) 50.0457 2.35135
\(454\) −0.0390277 −0.00183166
\(455\) 48.6265 2.27965
\(456\) −12.4209 −0.581661
\(457\) −27.9691 −1.30834 −0.654171 0.756347i \(-0.726982\pi\)
−0.654171 + 0.756347i \(0.726982\pi\)
\(458\) −9.09822 −0.425132
\(459\) 16.3496 0.763134
\(460\) −18.4402 −0.859777
\(461\) 10.2155 0.475785 0.237893 0.971291i \(-0.423543\pi\)
0.237893 + 0.971291i \(0.423543\pi\)
\(462\) −11.6615 −0.542544
\(463\) −8.00352 −0.371955 −0.185978 0.982554i \(-0.559545\pi\)
−0.185978 + 0.982554i \(0.559545\pi\)
\(464\) 0.794274 0.0368732
\(465\) 32.7881 1.52051
\(466\) −8.16189 −0.378092
\(467\) 7.17708 0.332116 0.166058 0.986116i \(-0.446896\pi\)
0.166058 + 0.986116i \(0.446896\pi\)
\(468\) −20.2686 −0.936916
\(469\) 33.0707 1.52706
\(470\) −26.7982 −1.23611
\(471\) 12.5448 0.578032
\(472\) 4.06419 0.187070
\(473\) −1.01248 −0.0465537
\(474\) 7.59426 0.348816
\(475\) −4.27551 −0.196174
\(476\) 21.4620 0.983709
\(477\) −29.1300 −1.33377
\(478\) −22.4745 −1.02796
\(479\) 13.6050 0.621630 0.310815 0.950470i \(-0.399398\pi\)
0.310815 + 0.950470i \(0.399398\pi\)
\(480\) 6.51579 0.297404
\(481\) −13.8291 −0.630553
\(482\) −8.44935 −0.384857
\(483\) −83.4065 −3.79513
\(484\) −9.87747 −0.448976
\(485\) 32.1857 1.46148
\(486\) −20.4360 −0.926998
\(487\) −28.4639 −1.28982 −0.644911 0.764257i \(-0.723106\pi\)
−0.644911 + 0.764257i \(0.723106\pi\)
\(488\) −5.19597 −0.235211
\(489\) 7.98902 0.361276
\(490\) −24.0853 −1.08806
\(491\) −42.5299 −1.91935 −0.959673 0.281117i \(-0.909295\pi\)
−0.959673 + 0.281117i \(0.909295\pi\)
\(492\) 13.5857 0.612490
\(493\) −4.14694 −0.186769
\(494\) −22.5500 −1.01457
\(495\) 10.7499 0.483174
\(496\) −5.03209 −0.225948
\(497\) 28.9856 1.30018
\(498\) −26.5483 −1.18966
\(499\) 28.8151 1.28994 0.644970 0.764208i \(-0.276870\pi\)
0.644970 + 0.764208i \(0.276870\pi\)
\(500\) −9.92439 −0.443832
\(501\) 40.0989 1.79148
\(502\) −16.8167 −0.750565
\(503\) −1.20412 −0.0536888 −0.0268444 0.999640i \(-0.508546\pi\)
−0.0268444 + 0.999640i \(0.508546\pi\)
\(504\) 17.1395 0.763454
\(505\) −4.80867 −0.213983
\(506\) 8.02862 0.356915
\(507\) −28.4647 −1.26416
\(508\) 12.7024 0.563576
\(509\) 8.30873 0.368278 0.184139 0.982900i \(-0.441050\pi\)
0.184139 + 0.982900i \(0.441050\pi\)
\(510\) −34.0193 −1.50640
\(511\) −10.3553 −0.458090
\(512\) −1.00000 −0.0441942
\(513\) 14.5264 0.641355
\(514\) −22.3936 −0.987740
\(515\) −16.8882 −0.744182
\(516\) 2.55877 0.112644
\(517\) 11.6676 0.513140
\(518\) 11.6941 0.513811
\(519\) 10.2516 0.449997
\(520\) 11.8293 0.518751
\(521\) −5.14716 −0.225501 −0.112751 0.993623i \(-0.535966\pi\)
−0.112751 + 0.993623i \(0.535966\pi\)
\(522\) −3.31173 −0.144951
\(523\) 38.8548 1.69900 0.849500 0.527588i \(-0.176904\pi\)
0.849500 + 0.527588i \(0.176904\pi\)
\(524\) 2.68811 0.117431
\(525\) 10.1447 0.442750
\(526\) 12.8444 0.560041
\(527\) 26.2728 1.14446
\(528\) −2.83689 −0.123460
\(529\) 34.4229 1.49665
\(530\) 17.0011 0.738482
\(531\) −16.9457 −0.735380
\(532\) 19.0687 0.826731
\(533\) 24.6647 1.06835
\(534\) 17.5962 0.761460
\(535\) −31.5338 −1.36332
\(536\) 8.04508 0.347495
\(537\) −17.9590 −0.774988
\(538\) −0.237234 −0.0102279
\(539\) 10.4865 0.451683
\(540\) −7.62030 −0.327926
\(541\) −20.1648 −0.866951 −0.433475 0.901165i \(-0.642713\pi\)
−0.433475 + 0.901165i \(0.642713\pi\)
\(542\) −2.48344 −0.106673
\(543\) −40.1993 −1.72512
\(544\) 5.22105 0.223851
\(545\) 35.2009 1.50784
\(546\) 53.5052 2.28981
\(547\) −30.3506 −1.29770 −0.648850 0.760917i \(-0.724750\pi\)
−0.648850 + 0.760917i \(0.724750\pi\)
\(548\) 1.54509 0.0660031
\(549\) 21.6647 0.924626
\(550\) −0.976514 −0.0416387
\(551\) −3.68449 −0.156965
\(552\) −20.2902 −0.863610
\(553\) −11.6588 −0.495782
\(554\) 18.6309 0.791551
\(555\) −18.5363 −0.786823
\(556\) −4.95650 −0.210202
\(557\) 33.6954 1.42772 0.713860 0.700288i \(-0.246945\pi\)
0.713860 + 0.700288i \(0.246945\pi\)
\(558\) 20.9814 0.888212
\(559\) 4.64542 0.196480
\(560\) −10.0031 −0.422709
\(561\) 14.8116 0.625344
\(562\) −13.3646 −0.563750
\(563\) −23.5299 −0.991669 −0.495834 0.868417i \(-0.665138\pi\)
−0.495834 + 0.868417i \(0.665138\pi\)
\(564\) −29.4868 −1.24162
\(565\) 34.7673 1.46267
\(566\) 24.6956 1.03803
\(567\) 16.9512 0.711883
\(568\) 7.05132 0.295867
\(569\) −42.2987 −1.77325 −0.886627 0.462486i \(-0.846958\pi\)
−0.886627 + 0.462486i \(0.846958\pi\)
\(570\) −30.2256 −1.26601
\(571\) 3.36129 0.140666 0.0703329 0.997524i \(-0.477594\pi\)
0.0703329 + 0.997524i \(0.477594\pi\)
\(572\) −5.15035 −0.215347
\(573\) 47.5591 1.98681
\(574\) −20.8569 −0.870550
\(575\) −6.98430 −0.291265
\(576\) 4.16951 0.173730
\(577\) −18.3257 −0.762908 −0.381454 0.924388i \(-0.624576\pi\)
−0.381454 + 0.924388i \(0.624576\pi\)
\(578\) −10.2593 −0.426732
\(579\) 37.7779 1.56999
\(580\) 1.93283 0.0802562
\(581\) 40.7572 1.69090
\(582\) 35.4148 1.46799
\(583\) −7.40208 −0.306563
\(584\) −2.51912 −0.104242
\(585\) −49.3226 −2.03924
\(586\) −28.5865 −1.18090
\(587\) 34.2116 1.41206 0.706032 0.708180i \(-0.250484\pi\)
0.706032 + 0.708180i \(0.250484\pi\)
\(588\) −26.5018 −1.09292
\(589\) 23.3430 0.961830
\(590\) 9.89001 0.407165
\(591\) 54.4270 2.23883
\(592\) 2.84483 0.116922
\(593\) −22.1432 −0.909310 −0.454655 0.890668i \(-0.650237\pi\)
−0.454655 + 0.890668i \(0.650237\pi\)
\(594\) 3.31778 0.136130
\(595\) 52.2267 2.14109
\(596\) −4.48625 −0.183764
\(597\) 16.8162 0.688240
\(598\) −36.8367 −1.50636
\(599\) −19.8175 −0.809722 −0.404861 0.914378i \(-0.632680\pi\)
−0.404861 + 0.914378i \(0.632680\pi\)
\(600\) 2.46789 0.100751
\(601\) −1.11887 −0.0456396 −0.0228198 0.999740i \(-0.507264\pi\)
−0.0228198 + 0.999740i \(0.507264\pi\)
\(602\) −3.92825 −0.160104
\(603\) −33.5441 −1.36602
\(604\) −18.6905 −0.760507
\(605\) −24.0363 −0.977216
\(606\) −5.29112 −0.214937
\(607\) 23.9587 0.972453 0.486226 0.873833i \(-0.338373\pi\)
0.486226 + 0.873833i \(0.338373\pi\)
\(608\) 4.63882 0.188129
\(609\) 8.74234 0.354257
\(610\) −12.6441 −0.511946
\(611\) −53.5330 −2.16571
\(612\) −21.7692 −0.879969
\(613\) −27.2649 −1.10122 −0.550609 0.834763i \(-0.685605\pi\)
−0.550609 + 0.834763i \(0.685605\pi\)
\(614\) 4.44773 0.179496
\(615\) 33.0601 1.33311
\(616\) 4.35523 0.175477
\(617\) −6.90660 −0.278049 −0.139025 0.990289i \(-0.544397\pi\)
−0.139025 + 0.990289i \(0.544397\pi\)
\(618\) −18.5825 −0.747500
\(619\) −18.2143 −0.732096 −0.366048 0.930596i \(-0.619289\pi\)
−0.366048 + 0.930596i \(0.619289\pi\)
\(620\) −12.2453 −0.491785
\(621\) 23.7297 0.952240
\(622\) −31.1713 −1.24985
\(623\) −27.0138 −1.08229
\(624\) 13.0162 0.521064
\(625\) −28.7589 −1.15036
\(626\) −16.8653 −0.674072
\(627\) 13.1598 0.525553
\(628\) −4.68508 −0.186955
\(629\) −14.8530 −0.592227
\(630\) 41.7081 1.66169
\(631\) −28.9573 −1.15277 −0.576386 0.817178i \(-0.695537\pi\)
−0.576386 + 0.817178i \(0.695537\pi\)
\(632\) −2.83622 −0.112819
\(633\) 20.8057 0.826953
\(634\) 27.4996 1.09215
\(635\) 30.9105 1.22665
\(636\) 18.7068 0.741774
\(637\) −48.1137 −1.90633
\(638\) −0.841528 −0.0333164
\(639\) −29.4005 −1.16307
\(640\) −2.43345 −0.0961906
\(641\) −20.2384 −0.799368 −0.399684 0.916653i \(-0.630880\pi\)
−0.399684 + 0.916653i \(0.630880\pi\)
\(642\) −34.6975 −1.36940
\(643\) −29.6131 −1.16783 −0.583914 0.811816i \(-0.698479\pi\)
−0.583914 + 0.811816i \(0.698479\pi\)
\(644\) 31.1498 1.22747
\(645\) 6.22664 0.245174
\(646\) −24.2195 −0.952903
\(647\) 32.4517 1.27581 0.637905 0.770115i \(-0.279801\pi\)
0.637905 + 0.770115i \(0.279801\pi\)
\(648\) 4.12370 0.161994
\(649\) −4.30598 −0.169025
\(650\) 4.48042 0.175737
\(651\) −55.3868 −2.17078
\(652\) −2.98365 −0.116849
\(653\) −8.07390 −0.315956 −0.157978 0.987443i \(-0.550498\pi\)
−0.157978 + 0.987443i \(0.550498\pi\)
\(654\) 38.7325 1.51456
\(655\) 6.54139 0.255593
\(656\) −5.07384 −0.198100
\(657\) 10.5035 0.409780
\(658\) 45.2685 1.76475
\(659\) −4.36527 −0.170047 −0.0850234 0.996379i \(-0.527097\pi\)
−0.0850234 + 0.996379i \(0.527097\pi\)
\(660\) −6.90344 −0.268716
\(661\) −42.6328 −1.65822 −0.829112 0.559082i \(-0.811154\pi\)
−0.829112 + 0.559082i \(0.811154\pi\)
\(662\) −14.0627 −0.546562
\(663\) −67.9580 −2.63927
\(664\) 9.91499 0.384776
\(665\) 46.4026 1.79942
\(666\) −11.8615 −0.459626
\(667\) −6.01884 −0.233050
\(668\) −14.9757 −0.579427
\(669\) −61.3519 −2.37200
\(670\) 19.5773 0.756338
\(671\) 5.50510 0.212522
\(672\) −11.0067 −0.424593
\(673\) −5.20170 −0.200511 −0.100255 0.994962i \(-0.531966\pi\)
−0.100255 + 0.994962i \(0.531966\pi\)
\(674\) −31.8017 −1.22496
\(675\) −2.88622 −0.111091
\(676\) 10.6307 0.408873
\(677\) −5.41537 −0.208130 −0.104065 0.994571i \(-0.533185\pi\)
−0.104065 + 0.994571i \(0.533185\pi\)
\(678\) 38.2554 1.46919
\(679\) −54.3692 −2.08650
\(680\) 12.7052 0.487221
\(681\) −0.104500 −0.00400446
\(682\) 5.33147 0.204153
\(683\) −0.760953 −0.0291171 −0.0145585 0.999894i \(-0.504634\pi\)
−0.0145585 + 0.999894i \(0.504634\pi\)
\(684\) −19.3416 −0.739545
\(685\) 3.75991 0.143659
\(686\) 11.9111 0.454768
\(687\) −24.3613 −0.929443
\(688\) −0.955623 −0.0364328
\(689\) 33.9620 1.29385
\(690\) −49.3753 −1.87969
\(691\) 43.8137 1.66675 0.833375 0.552708i \(-0.186405\pi\)
0.833375 + 0.552708i \(0.186405\pi\)
\(692\) −3.82867 −0.145544
\(693\) −18.1592 −0.689810
\(694\) −32.8021 −1.24515
\(695\) −12.0614 −0.457514
\(696\) 2.12674 0.0806140
\(697\) 26.4908 1.00341
\(698\) −10.3535 −0.391888
\(699\) −21.8542 −0.826603
\(700\) −3.78872 −0.143200
\(701\) 42.7844 1.61595 0.807973 0.589220i \(-0.200565\pi\)
0.807973 + 0.589220i \(0.200565\pi\)
\(702\) −15.2226 −0.574539
\(703\) −13.1966 −0.497721
\(704\) 1.05949 0.0399312
\(705\) −71.7547 −2.70244
\(706\) 8.48982 0.319519
\(707\) 8.12298 0.305496
\(708\) 10.8823 0.408980
\(709\) −3.38059 −0.126961 −0.0634803 0.997983i \(-0.520220\pi\)
−0.0634803 + 0.997983i \(0.520220\pi\)
\(710\) 17.1590 0.643967
\(711\) 11.8257 0.443497
\(712\) −6.57163 −0.246282
\(713\) 38.1321 1.42806
\(714\) 57.4665 2.15063
\(715\) −12.5331 −0.468712
\(716\) 6.70714 0.250657
\(717\) −60.1775 −2.24737
\(718\) −3.52096 −0.131401
\(719\) −21.5712 −0.804471 −0.402236 0.915536i \(-0.631767\pi\)
−0.402236 + 0.915536i \(0.631767\pi\)
\(720\) 10.1463 0.378130
\(721\) 28.5281 1.06244
\(722\) −2.51863 −0.0937338
\(723\) −22.6239 −0.841393
\(724\) 15.0132 0.557962
\(725\) 0.732066 0.0271883
\(726\) −26.4479 −0.981573
\(727\) 33.7621 1.25217 0.626083 0.779756i \(-0.284657\pi\)
0.626083 + 0.779756i \(0.284657\pi\)
\(728\) −19.9826 −0.740603
\(729\) −42.3483 −1.56846
\(730\) −6.13015 −0.226887
\(731\) 4.98935 0.184538
\(732\) −13.9127 −0.514228
\(733\) 37.5615 1.38736 0.693682 0.720281i \(-0.255987\pi\)
0.693682 + 0.720281i \(0.255987\pi\)
\(734\) 2.87976 0.106294
\(735\) −64.4908 −2.37878
\(736\) 7.57779 0.279321
\(737\) −8.52371 −0.313975
\(738\) 21.1555 0.778743
\(739\) −44.8819 −1.65101 −0.825504 0.564397i \(-0.809109\pi\)
−0.825504 + 0.564397i \(0.809109\pi\)
\(740\) 6.92275 0.254485
\(741\) −60.3796 −2.21810
\(742\) −28.7189 −1.05430
\(743\) −11.8117 −0.433331 −0.216665 0.976246i \(-0.569518\pi\)
−0.216665 + 0.976246i \(0.569518\pi\)
\(744\) −13.4739 −0.493977
\(745\) −10.9171 −0.399970
\(746\) −4.69567 −0.171921
\(747\) −41.3407 −1.51258
\(748\) −5.53167 −0.202258
\(749\) 53.2680 1.94637
\(750\) −26.5735 −0.970327
\(751\) 9.97303 0.363921 0.181960 0.983306i \(-0.441756\pi\)
0.181960 + 0.983306i \(0.441756\pi\)
\(752\) 11.0124 0.401582
\(753\) −45.0282 −1.64092
\(754\) 3.86108 0.140612
\(755\) −45.4825 −1.65528
\(756\) 12.8725 0.468168
\(757\) −52.9459 −1.92435 −0.962175 0.272433i \(-0.912172\pi\)
−0.962175 + 0.272433i \(0.912172\pi\)
\(758\) −17.5661 −0.638029
\(759\) 21.4974 0.780305
\(760\) 11.2883 0.409471
\(761\) 26.3388 0.954780 0.477390 0.878692i \(-0.341583\pi\)
0.477390 + 0.878692i \(0.341583\pi\)
\(762\) 34.0117 1.23212
\(763\) −59.4626 −2.15269
\(764\) −17.7619 −0.642602
\(765\) −52.9743 −1.91529
\(766\) −29.3181 −1.05931
\(767\) 19.7566 0.713370
\(768\) −2.67759 −0.0966194
\(769\) −38.7934 −1.39892 −0.699462 0.714669i \(-0.746577\pi\)
−0.699462 + 0.714669i \(0.746577\pi\)
\(770\) 10.5982 0.381934
\(771\) −59.9610 −2.15944
\(772\) −14.1089 −0.507790
\(773\) 47.7909 1.71892 0.859459 0.511204i \(-0.170800\pi\)
0.859459 + 0.511204i \(0.170800\pi\)
\(774\) 3.98448 0.143219
\(775\) −4.63798 −0.166601
\(776\) −13.2264 −0.474799
\(777\) 31.3122 1.12332
\(778\) 24.4342 0.876007
\(779\) 23.5366 0.843288
\(780\) 31.6742 1.13412
\(781\) −7.47082 −0.267327
\(782\) −39.5640 −1.41481
\(783\) −2.48725 −0.0888871
\(784\) 9.89761 0.353486
\(785\) −11.4009 −0.406916
\(786\) 7.19768 0.256733
\(787\) −51.3390 −1.83004 −0.915018 0.403412i \(-0.867824\pi\)
−0.915018 + 0.403412i \(0.867824\pi\)
\(788\) −20.3268 −0.724113
\(789\) 34.3920 1.22439
\(790\) −6.90181 −0.245555
\(791\) −58.7301 −2.08820
\(792\) −4.41757 −0.156972
\(793\) −25.2584 −0.896951
\(794\) 17.8022 0.631777
\(795\) 45.5221 1.61450
\(796\) −6.28032 −0.222600
\(797\) −31.6422 −1.12082 −0.560412 0.828214i \(-0.689357\pi\)
−0.560412 + 0.828214i \(0.689357\pi\)
\(798\) 51.0581 1.80744
\(799\) −57.4964 −2.03408
\(800\) −0.921680 −0.0325863
\(801\) 27.4005 0.968149
\(802\) 27.1614 0.959104
\(803\) 2.66899 0.0941866
\(804\) 21.5415 0.759709
\(805\) 75.8014 2.67165
\(806\) −24.4617 −0.861627
\(807\) −0.635217 −0.0223607
\(808\) 1.97607 0.0695179
\(809\) 43.4501 1.52763 0.763813 0.645438i \(-0.223325\pi\)
0.763813 + 0.645438i \(0.223325\pi\)
\(810\) 10.0348 0.352588
\(811\) −3.77107 −0.132420 −0.0662101 0.997806i \(-0.521091\pi\)
−0.0662101 + 0.997806i \(0.521091\pi\)
\(812\) −3.26500 −0.114579
\(813\) −6.64965 −0.233213
\(814\) −3.01408 −0.105643
\(815\) −7.26057 −0.254327
\(816\) 13.9798 0.489393
\(817\) 4.43296 0.155090
\(818\) −0.0400064 −0.00139879
\(819\) 83.3175 2.91135
\(820\) −12.3469 −0.431174
\(821\) 20.6666 0.721270 0.360635 0.932707i \(-0.382560\pi\)
0.360635 + 0.932707i \(0.382560\pi\)
\(822\) 4.13713 0.144299
\(823\) 8.43910 0.294168 0.147084 0.989124i \(-0.453011\pi\)
0.147084 + 0.989124i \(0.453011\pi\)
\(824\) 6.94001 0.241767
\(825\) −2.61471 −0.0910325
\(826\) −16.7065 −0.581295
\(827\) 15.1689 0.527475 0.263737 0.964595i \(-0.415045\pi\)
0.263737 + 0.964595i \(0.415045\pi\)
\(828\) −31.5957 −1.09803
\(829\) −9.10167 −0.316114 −0.158057 0.987430i \(-0.550523\pi\)
−0.158057 + 0.987430i \(0.550523\pi\)
\(830\) 24.1276 0.837482
\(831\) 49.8860 1.73053
\(832\) −4.86114 −0.168530
\(833\) −51.6759 −1.79046
\(834\) −13.2715 −0.459554
\(835\) −36.4426 −1.26115
\(836\) −4.91480 −0.169982
\(837\) 15.7579 0.544673
\(838\) −3.34725 −0.115629
\(839\) 16.8815 0.582816 0.291408 0.956599i \(-0.405876\pi\)
0.291408 + 0.956599i \(0.405876\pi\)
\(840\) −26.7843 −0.924145
\(841\) −28.3691 −0.978246
\(842\) −39.2445 −1.35245
\(843\) −35.7849 −1.23250
\(844\) −7.77030 −0.267465
\(845\) 25.8693 0.889931
\(846\) −45.9165 −1.57864
\(847\) 40.6030 1.39514
\(848\) −6.98643 −0.239915
\(849\) 66.1249 2.26940
\(850\) 4.81214 0.165055
\(851\) −21.5575 −0.738982
\(852\) 18.8806 0.646837
\(853\) −22.4956 −0.770236 −0.385118 0.922867i \(-0.625839\pi\)
−0.385118 + 0.922867i \(0.625839\pi\)
\(854\) 21.3589 0.730887
\(855\) −47.0668 −1.60965
\(856\) 12.9585 0.442911
\(857\) 3.62386 0.123789 0.0618943 0.998083i \(-0.480286\pi\)
0.0618943 + 0.998083i \(0.480286\pi\)
\(858\) −13.7905 −0.470801
\(859\) 13.7846 0.470324 0.235162 0.971956i \(-0.424438\pi\)
0.235162 + 0.971956i \(0.424438\pi\)
\(860\) −2.32546 −0.0792976
\(861\) −55.8463 −1.90324
\(862\) 8.10983 0.276222
\(863\) 13.0348 0.443711 0.221855 0.975080i \(-0.428789\pi\)
0.221855 + 0.975080i \(0.428789\pi\)
\(864\) 3.13148 0.106535
\(865\) −9.31688 −0.316784
\(866\) 3.52249 0.119699
\(867\) −27.4703 −0.932942
\(868\) 20.6853 0.702104
\(869\) 3.00496 0.101936
\(870\) 5.17532 0.175460
\(871\) 39.1083 1.32513
\(872\) −14.4654 −0.489861
\(873\) 55.1475 1.86646
\(874\) −35.1520 −1.18903
\(875\) 40.7959 1.37915
\(876\) −6.74518 −0.227898
\(877\) 19.8386 0.669902 0.334951 0.942236i \(-0.391280\pi\)
0.334951 + 0.942236i \(0.391280\pi\)
\(878\) 20.7205 0.699283
\(879\) −76.5431 −2.58174
\(880\) 2.57822 0.0869119
\(881\) −23.3312 −0.786047 −0.393024 0.919528i \(-0.628571\pi\)
−0.393024 + 0.919528i \(0.628571\pi\)
\(882\) −41.2682 −1.38957
\(883\) −27.3927 −0.921837 −0.460919 0.887442i \(-0.652480\pi\)
−0.460919 + 0.887442i \(0.652480\pi\)
\(884\) 25.3803 0.853630
\(885\) 26.4814 0.890163
\(886\) −32.4349 −1.08967
\(887\) −25.3344 −0.850647 −0.425323 0.905041i \(-0.639840\pi\)
−0.425323 + 0.905041i \(0.639840\pi\)
\(888\) 7.61730 0.255620
\(889\) −52.2152 −1.75124
\(890\) −15.9917 −0.536044
\(891\) −4.36904 −0.146368
\(892\) 22.9131 0.767187
\(893\) −51.0847 −1.70948
\(894\) −12.0123 −0.401753
\(895\) 16.3215 0.545567
\(896\) 4.11067 0.137328
\(897\) −98.6338 −3.29329
\(898\) 7.36674 0.245831
\(899\) −3.99686 −0.133303
\(900\) 3.84296 0.128099
\(901\) 36.4765 1.21521
\(902\) 5.37570 0.178991
\(903\) −10.5183 −0.350026
\(904\) −14.2872 −0.475186
\(905\) 36.5339 1.21443
\(906\) −50.0457 −1.66266
\(907\) 37.4225 1.24260 0.621298 0.783575i \(-0.286606\pi\)
0.621298 + 0.783575i \(0.286606\pi\)
\(908\) 0.0390277 0.00129518
\(909\) −8.23925 −0.273279
\(910\) −48.6265 −1.61195
\(911\) 37.6270 1.24664 0.623320 0.781967i \(-0.285784\pi\)
0.623320 + 0.781967i \(0.285784\pi\)
\(912\) 12.4209 0.411296
\(913\) −10.5049 −0.347660
\(914\) 27.9691 0.925137
\(915\) −33.8559 −1.11924
\(916\) 9.09822 0.300614
\(917\) −11.0500 −0.364902
\(918\) −16.3496 −0.539617
\(919\) 40.9577 1.35107 0.675535 0.737328i \(-0.263913\pi\)
0.675535 + 0.737328i \(0.263913\pi\)
\(920\) 18.4402 0.607954
\(921\) 11.9092 0.392422
\(922\) −10.2155 −0.336431
\(923\) 34.2774 1.12826
\(924\) 11.6615 0.383636
\(925\) 2.62202 0.0862115
\(926\) 8.00352 0.263012
\(927\) −28.9365 −0.950399
\(928\) −0.794274 −0.0260733
\(929\) 30.6539 1.00572 0.502860 0.864368i \(-0.332281\pi\)
0.502860 + 0.864368i \(0.332281\pi\)
\(930\) −32.7881 −1.07516
\(931\) −45.9132 −1.50474
\(932\) 8.16189 0.267352
\(933\) −83.4641 −2.73249
\(934\) −7.17708 −0.234841
\(935\) −13.4610 −0.440223
\(936\) 20.2686 0.662500
\(937\) −21.8677 −0.714385 −0.357193 0.934031i \(-0.616266\pi\)
−0.357193 + 0.934031i \(0.616266\pi\)
\(938\) −33.0707 −1.07980
\(939\) −45.1584 −1.47369
\(940\) 26.7982 0.874061
\(941\) 16.5698 0.540162 0.270081 0.962838i \(-0.412950\pi\)
0.270081 + 0.962838i \(0.412950\pi\)
\(942\) −12.5448 −0.408730
\(943\) 38.4485 1.25206
\(944\) −4.06419 −0.132278
\(945\) 31.3245 1.01899
\(946\) 1.01248 0.0329184
\(947\) −31.6363 −1.02804 −0.514021 0.857778i \(-0.671845\pi\)
−0.514021 + 0.857778i \(0.671845\pi\)
\(948\) −7.59426 −0.246650
\(949\) −12.2458 −0.397515
\(950\) 4.27551 0.138716
\(951\) 73.6329 2.38771
\(952\) −21.4620 −0.695588
\(953\) −54.9702 −1.78066 −0.890330 0.455316i \(-0.849526\pi\)
−0.890330 + 0.455316i \(0.849526\pi\)
\(954\) 29.1300 0.943119
\(955\) −43.2226 −1.39865
\(956\) 22.4745 0.726876
\(957\) −2.25327 −0.0728379
\(958\) −13.6050 −0.439559
\(959\) −6.35137 −0.205096
\(960\) −6.51579 −0.210296
\(961\) −5.67804 −0.183163
\(962\) 13.8291 0.445869
\(963\) −54.0305 −1.74111
\(964\) 8.44935 0.272135
\(965\) −34.3333 −1.10523
\(966\) 83.4065 2.68356
\(967\) −12.7845 −0.411123 −0.205562 0.978644i \(-0.565902\pi\)
−0.205562 + 0.978644i \(0.565902\pi\)
\(968\) 9.87747 0.317474
\(969\) −64.8500 −2.08328
\(970\) −32.1857 −1.03342
\(971\) 0.136359 0.00437597 0.00218799 0.999998i \(-0.499304\pi\)
0.00218799 + 0.999998i \(0.499304\pi\)
\(972\) 20.4360 0.655486
\(973\) 20.3745 0.653177
\(974\) 28.4639 0.912042
\(975\) 11.9967 0.384203
\(976\) 5.19597 0.166319
\(977\) −9.80076 −0.313554 −0.156777 0.987634i \(-0.550110\pi\)
−0.156777 + 0.987634i \(0.550110\pi\)
\(978\) −7.98902 −0.255461
\(979\) 6.96260 0.222526
\(980\) 24.0853 0.769378
\(981\) 60.3138 1.92567
\(982\) 42.5299 1.35718
\(983\) −7.83488 −0.249894 −0.124947 0.992163i \(-0.539876\pi\)
−0.124947 + 0.992163i \(0.539876\pi\)
\(984\) −13.5857 −0.433096
\(985\) −49.4643 −1.57606
\(986\) 4.14694 0.132065
\(987\) 121.211 3.85818
\(988\) 22.5500 0.717410
\(989\) 7.24151 0.230267
\(990\) −10.7499 −0.341656
\(991\) −24.7734 −0.786952 −0.393476 0.919335i \(-0.628728\pi\)
−0.393476 + 0.919335i \(0.628728\pi\)
\(992\) 5.03209 0.159769
\(993\) −37.6542 −1.19492
\(994\) −28.9856 −0.919368
\(995\) −15.2829 −0.484499
\(996\) 26.5483 0.841216
\(997\) 9.83713 0.311545 0.155772 0.987793i \(-0.450213\pi\)
0.155772 + 0.987793i \(0.450213\pi\)
\(998\) −28.8151 −0.912125
\(999\) −8.90852 −0.281853
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 4006.2.a.h.1.8 42
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4006.2.a.h.1.8 42 1.1 even 1 trivial