Properties

Label 539.2.a.l.1.8
Level $539$
Weight $2$
Character 539.1
Self dual yes
Analytic conductor $4.304$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [539,2,Mod(1,539)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(539, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("539.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 539 = 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 539.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: \(4.30393666895\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 26x^{8} + 245x^{6} - 1038x^{4} + 1884x^{2} - 968 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(3.27614\) of defining polynomial
Character \(\chi\) \(=\) 539.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.70296 q^{2} +3.27614 q^{3} +0.900071 q^{4} -0.246676 q^{5} +5.57913 q^{6} -1.87313 q^{8} +7.73309 q^{9} +O(q^{10})\) \(q+1.70296 q^{2} +3.27614 q^{3} +0.900071 q^{4} -0.246676 q^{5} +5.57913 q^{6} -1.87313 q^{8} +7.73309 q^{9} -0.420079 q^{10} +1.00000 q^{11} +2.94876 q^{12} -3.17078 q^{13} -0.808144 q^{15} -4.99001 q^{16} -6.49256 q^{17} +13.1691 q^{18} -4.32335 q^{19} -0.222026 q^{20} +1.70296 q^{22} +3.15700 q^{23} -6.13665 q^{24} -4.93915 q^{25} -5.39972 q^{26} +15.5063 q^{27} +6.48417 q^{29} -1.37624 q^{30} +1.78122 q^{31} -4.75152 q^{32} +3.27614 q^{33} -11.0566 q^{34} +6.96033 q^{36} +8.38424 q^{37} -7.36248 q^{38} -10.3879 q^{39} +0.462057 q^{40} -0.553067 q^{41} +5.69023 q^{43} +0.900071 q^{44} -1.90756 q^{45} +5.37624 q^{46} +10.2971 q^{47} -16.3480 q^{48} -8.41117 q^{50} -21.2705 q^{51} -2.85393 q^{52} -10.1744 q^{53} +26.4065 q^{54} -0.246676 q^{55} -14.1639 q^{57} +11.0423 q^{58} -6.45660 q^{59} -0.727387 q^{60} -3.38149 q^{61} +3.03335 q^{62} +1.88838 q^{64} +0.782155 q^{65} +5.57913 q^{66} -3.65484 q^{67} -5.84377 q^{68} +10.3428 q^{69} +0.345158 q^{71} -14.4851 q^{72} +2.97942 q^{73} +14.2780 q^{74} -16.1813 q^{75} -3.89132 q^{76} -17.6902 q^{78} -3.77595 q^{79} +1.23091 q^{80} +27.6014 q^{81} -0.941851 q^{82} -6.34157 q^{83} +1.60156 q^{85} +9.69023 q^{86} +21.2430 q^{87} -1.87313 q^{88} -0.246676 q^{89} -3.24851 q^{90} +2.84152 q^{92} +5.83553 q^{93} +17.5356 q^{94} +1.06646 q^{95} -15.5667 q^{96} -4.81068 q^{97} +7.73309 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 2 q^{2} + 18 q^{4} - 6 q^{8} + 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 2 q^{2} + 18 q^{4} - 6 q^{8} + 22 q^{9} + 10 q^{11} + 8 q^{15} + 42 q^{16} + 6 q^{18} + 2 q^{22} + 4 q^{23} + 18 q^{25} + 12 q^{29} - 4 q^{30} - 30 q^{32} - 2 q^{36} + 40 q^{37} - 16 q^{39} - 8 q^{43} + 18 q^{44} + 44 q^{46} - 62 q^{50} + 16 q^{53} - 8 q^{57} - 28 q^{58} + 36 q^{60} + 106 q^{64} - 32 q^{65} - 4 q^{67} + 36 q^{71} - 90 q^{72} - 28 q^{74} - 112 q^{78} + 8 q^{79} - 6 q^{81} + 88 q^{85} + 32 q^{86} - 6 q^{88} - 52 q^{92} + 44 q^{93} - 64 q^{95} + 22 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.70296 1.20417 0.602087 0.798430i \(-0.294336\pi\)
0.602087 + 0.798430i \(0.294336\pi\)
\(3\) 3.27614 1.89148 0.945740 0.324924i \(-0.105339\pi\)
0.945740 + 0.324924i \(0.105339\pi\)
\(4\) 0.900071 0.450036
\(5\) −0.246676 −0.110317 −0.0551584 0.998478i \(-0.517566\pi\)
−0.0551584 + 0.998478i \(0.517566\pi\)
\(6\) 5.57913 2.27767
\(7\) 0 0
\(8\) −1.87313 −0.662253
\(9\) 7.73309 2.57770
\(10\) −0.420079 −0.132841
\(11\) 1.00000 0.301511
\(12\) 2.94876 0.851233
\(13\) −3.17078 −0.879417 −0.439709 0.898140i \(-0.644918\pi\)
−0.439709 + 0.898140i \(0.644918\pi\)
\(14\) 0 0
\(15\) −0.808144 −0.208662
\(16\) −4.99001 −1.24750
\(17\) −6.49256 −1.57468 −0.787339 0.616520i \(-0.788542\pi\)
−0.787339 + 0.616520i \(0.788542\pi\)
\(18\) 13.1691 3.10400
\(19\) −4.32335 −0.991844 −0.495922 0.868367i \(-0.665170\pi\)
−0.495922 + 0.868367i \(0.665170\pi\)
\(20\) −0.222026 −0.0496464
\(21\) 0 0
\(22\) 1.70296 0.363072
\(23\) 3.15700 0.658279 0.329140 0.944281i \(-0.393241\pi\)
0.329140 + 0.944281i \(0.393241\pi\)
\(24\) −6.13665 −1.25264
\(25\) −4.93915 −0.987830
\(26\) −5.39972 −1.05897
\(27\) 15.5063 2.98418
\(28\) 0 0
\(29\) 6.48417 1.20408 0.602040 0.798466i \(-0.294355\pi\)
0.602040 + 0.798466i \(0.294355\pi\)
\(30\) −1.37624 −0.251265
\(31\) 1.78122 0.319917 0.159958 0.987124i \(-0.448864\pi\)
0.159958 + 0.987124i \(0.448864\pi\)
\(32\) −4.75152 −0.839959
\(33\) 3.27614 0.570303
\(34\) −11.0566 −1.89619
\(35\) 0 0
\(36\) 6.96033 1.16006
\(37\) 8.38424 1.37836 0.689180 0.724590i \(-0.257971\pi\)
0.689180 + 0.724590i \(0.257971\pi\)
\(38\) −7.36248 −1.19435
\(39\) −10.3879 −1.66340
\(40\) 0.462057 0.0730576
\(41\) −0.553067 −0.0863746 −0.0431873 0.999067i \(-0.513751\pi\)
−0.0431873 + 0.999067i \(0.513751\pi\)
\(42\) 0 0
\(43\) 5.69023 0.867752 0.433876 0.900973i \(-0.357146\pi\)
0.433876 + 0.900973i \(0.357146\pi\)
\(44\) 0.900071 0.135691
\(45\) −1.90756 −0.284363
\(46\) 5.37624 0.792683
\(47\) 10.2971 1.50199 0.750995 0.660308i \(-0.229574\pi\)
0.750995 + 0.660308i \(0.229574\pi\)
\(48\) −16.3480 −2.35963
\(49\) 0 0
\(50\) −8.41117 −1.18952
\(51\) −21.2705 −2.97847
\(52\) −2.85393 −0.395769
\(53\) −10.1744 −1.39756 −0.698780 0.715336i \(-0.746273\pi\)
−0.698780 + 0.715336i \(0.746273\pi\)
\(54\) 26.4065 3.59347
\(55\) −0.246676 −0.0332617
\(56\) 0 0
\(57\) −14.1639 −1.87605
\(58\) 11.0423 1.44992
\(59\) −6.45660 −0.840577 −0.420289 0.907390i \(-0.638071\pi\)
−0.420289 + 0.907390i \(0.638071\pi\)
\(60\) −0.727387 −0.0939053
\(61\) −3.38149 −0.432956 −0.216478 0.976287i \(-0.569457\pi\)
−0.216478 + 0.976287i \(0.569457\pi\)
\(62\) 3.03335 0.385235
\(63\) 0 0
\(64\) 1.88838 0.236047
\(65\) 0.782155 0.0970144
\(66\) 5.57913 0.686744
\(67\) −3.65484 −0.446510 −0.223255 0.974760i \(-0.571668\pi\)
−0.223255 + 0.974760i \(0.571668\pi\)
\(68\) −5.84377 −0.708661
\(69\) 10.3428 1.24512
\(70\) 0 0
\(71\) 0.345158 0.0409627 0.0204813 0.999790i \(-0.493480\pi\)
0.0204813 + 0.999790i \(0.493480\pi\)
\(72\) −14.4851 −1.70709
\(73\) 2.97942 0.348715 0.174357 0.984682i \(-0.444215\pi\)
0.174357 + 0.984682i \(0.444215\pi\)
\(74\) 14.2780 1.65979
\(75\) −16.1813 −1.86846
\(76\) −3.89132 −0.446365
\(77\) 0 0
\(78\) −17.6902 −2.00302
\(79\) −3.77595 −0.424828 −0.212414 0.977180i \(-0.568132\pi\)
−0.212414 + 0.977180i \(0.568132\pi\)
\(80\) 1.23091 0.137620
\(81\) 27.6014 3.06682
\(82\) −0.941851 −0.104010
\(83\) −6.34157 −0.696078 −0.348039 0.937480i \(-0.613152\pi\)
−0.348039 + 0.937480i \(0.613152\pi\)
\(84\) 0 0
\(85\) 1.60156 0.173713
\(86\) 9.69023 1.04492
\(87\) 21.2430 2.27749
\(88\) −1.87313 −0.199677
\(89\) −0.246676 −0.0261476 −0.0130738 0.999915i \(-0.504162\pi\)
−0.0130738 + 0.999915i \(0.504162\pi\)
\(90\) −3.24851 −0.342423
\(91\) 0 0
\(92\) 2.84152 0.296249
\(93\) 5.83553 0.605116
\(94\) 17.5356 1.80866
\(95\) 1.06646 0.109417
\(96\) −15.5667 −1.58877
\(97\) −4.81068 −0.488451 −0.244226 0.969718i \(-0.578534\pi\)
−0.244226 + 0.969718i \(0.578534\pi\)
\(98\) 0 0
\(99\) 7.73309 0.777205
\(100\) −4.44559 −0.444559
\(101\) −6.86468 −0.683061 −0.341531 0.939871i \(-0.610945\pi\)
−0.341531 + 0.939871i \(0.610945\pi\)
\(102\) −36.2229 −3.58660
\(103\) 5.07835 0.500385 0.250192 0.968196i \(-0.419506\pi\)
0.250192 + 0.968196i \(0.419506\pi\)
\(104\) 5.93930 0.582397
\(105\) 0 0
\(106\) −17.3266 −1.68291
\(107\) 0.897562 0.0867706 0.0433853 0.999058i \(-0.486186\pi\)
0.0433853 + 0.999058i \(0.486186\pi\)
\(108\) 13.9567 1.34299
\(109\) 5.71569 0.547464 0.273732 0.961806i \(-0.411742\pi\)
0.273732 + 0.961806i \(0.411742\pi\)
\(110\) −0.420079 −0.0400529
\(111\) 27.4679 2.60714
\(112\) 0 0
\(113\) −5.99001 −0.563493 −0.281747 0.959489i \(-0.590914\pi\)
−0.281747 + 0.959489i \(0.590914\pi\)
\(114\) −24.1205 −2.25909
\(115\) −0.778754 −0.0726192
\(116\) 5.83621 0.541879
\(117\) −24.5200 −2.26687
\(118\) −10.9953 −1.01220
\(119\) 0 0
\(120\) 1.51376 0.138187
\(121\) 1.00000 0.0909091
\(122\) −5.75855 −0.521354
\(123\) −1.81193 −0.163376
\(124\) 1.60323 0.143974
\(125\) 2.45175 0.219291
\(126\) 0 0
\(127\) 2.31399 0.205334 0.102667 0.994716i \(-0.467262\pi\)
0.102667 + 0.994716i \(0.467262\pi\)
\(128\) 12.7189 1.12420
\(129\) 18.6420 1.64133
\(130\) 1.33198 0.116822
\(131\) 14.4800 1.26513 0.632564 0.774508i \(-0.282003\pi\)
0.632564 + 0.774508i \(0.282003\pi\)
\(132\) 2.94876 0.256657
\(133\) 0 0
\(134\) −6.22405 −0.537676
\(135\) −3.82502 −0.329205
\(136\) 12.1614 1.04284
\(137\) −10.9962 −0.939470 −0.469735 0.882807i \(-0.655651\pi\)
−0.469735 + 0.882807i \(0.655651\pi\)
\(138\) 17.6133 1.49934
\(139\) 1.05087 0.0891334 0.0445667 0.999006i \(-0.485809\pi\)
0.0445667 + 0.999006i \(0.485809\pi\)
\(140\) 0 0
\(141\) 33.7348 2.84098
\(142\) 0.587790 0.0493262
\(143\) −3.17078 −0.265154
\(144\) −38.5882 −3.21569
\(145\) −1.59949 −0.132830
\(146\) 5.07383 0.419913
\(147\) 0 0
\(148\) 7.54641 0.620311
\(149\) 18.4185 1.50890 0.754452 0.656356i \(-0.227903\pi\)
0.754452 + 0.656356i \(0.227903\pi\)
\(150\) −27.5562 −2.24995
\(151\) 19.1564 1.55893 0.779463 0.626448i \(-0.215492\pi\)
0.779463 + 0.626448i \(0.215492\pi\)
\(152\) 8.09821 0.656851
\(153\) −50.2076 −4.05904
\(154\) 0 0
\(155\) −0.439384 −0.0352922
\(156\) −9.34988 −0.748589
\(157\) 23.0214 1.83731 0.918653 0.395065i \(-0.129278\pi\)
0.918653 + 0.395065i \(0.129278\pi\)
\(158\) −6.43029 −0.511567
\(159\) −33.3327 −2.64346
\(160\) 1.17209 0.0926615
\(161\) 0 0
\(162\) 47.0041 3.69299
\(163\) −4.93865 −0.386825 −0.193412 0.981118i \(-0.561956\pi\)
−0.193412 + 0.981118i \(0.561956\pi\)
\(164\) −0.497800 −0.0388716
\(165\) −0.808144 −0.0629139
\(166\) −10.7994 −0.838199
\(167\) 7.39244 0.572044 0.286022 0.958223i \(-0.407667\pi\)
0.286022 + 0.958223i \(0.407667\pi\)
\(168\) 0 0
\(169\) −2.94613 −0.226625
\(170\) 2.72739 0.209181
\(171\) −33.4328 −2.55667
\(172\) 5.12161 0.390519
\(173\) −18.2383 −1.38663 −0.693316 0.720634i \(-0.743851\pi\)
−0.693316 + 0.720634i \(0.743851\pi\)
\(174\) 36.1760 2.74250
\(175\) 0 0
\(176\) −4.99001 −0.376136
\(177\) −21.1527 −1.58994
\(178\) −0.420079 −0.0314862
\(179\) −2.37373 −0.177421 −0.0887103 0.996057i \(-0.528275\pi\)
−0.0887103 + 0.996057i \(0.528275\pi\)
\(180\) −1.71694 −0.127973
\(181\) −21.6459 −1.60893 −0.804463 0.594002i \(-0.797547\pi\)
−0.804463 + 0.594002i \(0.797547\pi\)
\(182\) 0 0
\(183\) −11.0782 −0.818928
\(184\) −5.91348 −0.435947
\(185\) −2.06819 −0.152056
\(186\) 9.93767 0.728665
\(187\) −6.49256 −0.474783
\(188\) 9.26814 0.675949
\(189\) 0 0
\(190\) 1.81615 0.131757
\(191\) 7.22154 0.522532 0.261266 0.965267i \(-0.415860\pi\)
0.261266 + 0.965267i \(0.415860\pi\)
\(192\) 6.18658 0.446478
\(193\) −3.66604 −0.263887 −0.131944 0.991257i \(-0.542122\pi\)
−0.131944 + 0.991257i \(0.542122\pi\)
\(194\) −8.19240 −0.588180
\(195\) 2.56245 0.183501
\(196\) 0 0
\(197\) −3.81615 −0.271889 −0.135945 0.990716i \(-0.543407\pi\)
−0.135945 + 0.990716i \(0.543407\pi\)
\(198\) 13.1691 0.935890
\(199\) −19.3819 −1.37395 −0.686973 0.726683i \(-0.741061\pi\)
−0.686973 + 0.726683i \(0.741061\pi\)
\(200\) 9.25169 0.654193
\(201\) −11.9738 −0.844565
\(202\) −11.6903 −0.822525
\(203\) 0 0
\(204\) −19.1450 −1.34042
\(205\) 0.136428 0.00952856
\(206\) 8.64822 0.602550
\(207\) 24.4133 1.69684
\(208\) 15.8223 1.09708
\(209\) −4.32335 −0.299052
\(210\) 0 0
\(211\) 24.1279 1.66103 0.830517 0.556993i \(-0.188045\pi\)
0.830517 + 0.556993i \(0.188045\pi\)
\(212\) −9.15768 −0.628952
\(213\) 1.13078 0.0774801
\(214\) 1.52851 0.104487
\(215\) −1.40364 −0.0957275
\(216\) −29.0453 −1.97628
\(217\) 0 0
\(218\) 9.73359 0.659242
\(219\) 9.76100 0.659587
\(220\) −0.222026 −0.0149690
\(221\) 20.5865 1.38480
\(222\) 46.7768 3.13945
\(223\) 14.6952 0.984061 0.492030 0.870578i \(-0.336255\pi\)
0.492030 + 0.870578i \(0.336255\pi\)
\(224\) 0 0
\(225\) −38.1949 −2.54633
\(226\) −10.2008 −0.678544
\(227\) 9.19531 0.610314 0.305157 0.952302i \(-0.401291\pi\)
0.305157 + 0.952302i \(0.401291\pi\)
\(228\) −12.7485 −0.844290
\(229\) 17.4339 1.15207 0.576033 0.817427i \(-0.304600\pi\)
0.576033 + 0.817427i \(0.304600\pi\)
\(230\) −1.32619 −0.0874461
\(231\) 0 0
\(232\) −12.1457 −0.797405
\(233\) −27.7505 −1.81799 −0.908997 0.416802i \(-0.863151\pi\)
−0.908997 + 0.416802i \(0.863151\pi\)
\(234\) −41.7565 −2.72971
\(235\) −2.54005 −0.165695
\(236\) −5.81140 −0.378290
\(237\) −12.3705 −0.803553
\(238\) 0 0
\(239\) −22.5326 −1.45752 −0.728758 0.684772i \(-0.759902\pi\)
−0.728758 + 0.684772i \(0.759902\pi\)
\(240\) 4.03265 0.260306
\(241\) −7.22658 −0.465505 −0.232752 0.972536i \(-0.574773\pi\)
−0.232752 + 0.972536i \(0.574773\pi\)
\(242\) 1.70296 0.109470
\(243\) 43.9073 2.81665
\(244\) −3.04359 −0.194846
\(245\) 0 0
\(246\) −3.08564 −0.196733
\(247\) 13.7084 0.872244
\(248\) −3.33647 −0.211866
\(249\) −20.7759 −1.31662
\(250\) 4.17523 0.264064
\(251\) 12.8842 0.813245 0.406622 0.913596i \(-0.366706\pi\)
0.406622 + 0.913596i \(0.366706\pi\)
\(252\) 0 0
\(253\) 3.15700 0.198479
\(254\) 3.94063 0.247257
\(255\) 5.24692 0.328575
\(256\) 17.8830 1.11769
\(257\) 31.0932 1.93954 0.969771 0.244016i \(-0.0784650\pi\)
0.969771 + 0.244016i \(0.0784650\pi\)
\(258\) 31.7465 1.97645
\(259\) 0 0
\(260\) 0.703995 0.0436599
\(261\) 50.1426 3.10375
\(262\) 24.6589 1.52343
\(263\) −7.50215 −0.462603 −0.231301 0.972882i \(-0.574298\pi\)
−0.231301 + 0.972882i \(0.574298\pi\)
\(264\) −6.13665 −0.377685
\(265\) 2.50978 0.154174
\(266\) 0 0
\(267\) −0.808144 −0.0494576
\(268\) −3.28962 −0.200945
\(269\) −1.93752 −0.118133 −0.0590663 0.998254i \(-0.518812\pi\)
−0.0590663 + 0.998254i \(0.518812\pi\)
\(270\) −6.51385 −0.396420
\(271\) 22.9786 1.39585 0.697926 0.716170i \(-0.254107\pi\)
0.697926 + 0.716170i \(0.254107\pi\)
\(272\) 32.3980 1.96442
\(273\) 0 0
\(274\) −18.7261 −1.13129
\(275\) −4.93915 −0.297842
\(276\) 9.30922 0.560349
\(277\) −5.52095 −0.331722 −0.165861 0.986149i \(-0.553040\pi\)
−0.165861 + 0.986149i \(0.553040\pi\)
\(278\) 1.78958 0.107332
\(279\) 13.7743 0.824648
\(280\) 0 0
\(281\) −7.28000 −0.434288 −0.217144 0.976140i \(-0.569674\pi\)
−0.217144 + 0.976140i \(0.569674\pi\)
\(282\) 57.4490 3.42104
\(283\) −13.2600 −0.788225 −0.394113 0.919062i \(-0.628948\pi\)
−0.394113 + 0.919062i \(0.628948\pi\)
\(284\) 0.310666 0.0184347
\(285\) 3.49389 0.206960
\(286\) −5.39972 −0.319292
\(287\) 0 0
\(288\) −36.7440 −2.16516
\(289\) 25.1534 1.47961
\(290\) −2.72386 −0.159951
\(291\) −15.7605 −0.923895
\(292\) 2.68169 0.156934
\(293\) −5.56718 −0.325238 −0.162619 0.986689i \(-0.551994\pi\)
−0.162619 + 0.986689i \(0.551994\pi\)
\(294\) 0 0
\(295\) 1.59269 0.0927297
\(296\) −15.7048 −0.912823
\(297\) 15.5063 0.899765
\(298\) 31.3660 1.81698
\(299\) −10.0102 −0.578902
\(300\) −14.5644 −0.840874
\(301\) 0 0
\(302\) 32.6226 1.87722
\(303\) −22.4897 −1.29200
\(304\) 21.5736 1.23733
\(305\) 0.834132 0.0477623
\(306\) −85.5015 −4.88779
\(307\) −14.2467 −0.813102 −0.406551 0.913628i \(-0.633269\pi\)
−0.406551 + 0.913628i \(0.633269\pi\)
\(308\) 0 0
\(309\) 16.6374 0.946468
\(310\) −0.748253 −0.0424979
\(311\) −3.56834 −0.202342 −0.101171 0.994869i \(-0.532259\pi\)
−0.101171 + 0.994869i \(0.532259\pi\)
\(312\) 19.4580 1.10159
\(313\) 9.11394 0.515150 0.257575 0.966258i \(-0.417076\pi\)
0.257575 + 0.966258i \(0.417076\pi\)
\(314\) 39.2045 2.21244
\(315\) 0 0
\(316\) −3.39863 −0.191188
\(317\) −11.7542 −0.660181 −0.330090 0.943949i \(-0.607079\pi\)
−0.330090 + 0.943949i \(0.607079\pi\)
\(318\) −56.7643 −3.18318
\(319\) 6.48417 0.363044
\(320\) −0.465816 −0.0260399
\(321\) 2.94054 0.164125
\(322\) 0 0
\(323\) 28.0696 1.56183
\(324\) 24.8432 1.38018
\(325\) 15.6610 0.868715
\(326\) −8.41032 −0.465805
\(327\) 18.7254 1.03552
\(328\) 1.03597 0.0572018
\(329\) 0 0
\(330\) −1.37624 −0.0757593
\(331\) −25.6748 −1.41122 −0.705609 0.708602i \(-0.749326\pi\)
−0.705609 + 0.708602i \(0.749326\pi\)
\(332\) −5.70786 −0.313260
\(333\) 64.8361 3.55299
\(334\) 12.5890 0.688841
\(335\) 0.901561 0.0492575
\(336\) 0 0
\(337\) 33.5644 1.82837 0.914185 0.405298i \(-0.132832\pi\)
0.914185 + 0.405298i \(0.132832\pi\)
\(338\) −5.01713 −0.272896
\(339\) −19.6241 −1.06584
\(340\) 1.44152 0.0781772
\(341\) 1.78122 0.0964585
\(342\) −56.9347 −3.07868
\(343\) 0 0
\(344\) −10.6586 −0.574671
\(345\) −2.55131 −0.137358
\(346\) −31.0591 −1.66975
\(347\) −26.7567 −1.43637 −0.718187 0.695850i \(-0.755028\pi\)
−0.718187 + 0.695850i \(0.755028\pi\)
\(348\) 19.1202 1.02495
\(349\) 19.9411 1.06742 0.533711 0.845667i \(-0.320797\pi\)
0.533711 + 0.845667i \(0.320797\pi\)
\(350\) 0 0
\(351\) −49.1670 −2.62434
\(352\) −4.75152 −0.253257
\(353\) 10.0436 0.534566 0.267283 0.963618i \(-0.413874\pi\)
0.267283 + 0.963618i \(0.413874\pi\)
\(354\) −36.0222 −1.91456
\(355\) −0.0851420 −0.00451887
\(356\) −0.222026 −0.0117673
\(357\) 0 0
\(358\) −4.04236 −0.213645
\(359\) −17.0519 −0.899964 −0.449982 0.893038i \(-0.648570\pi\)
−0.449982 + 0.893038i \(0.648570\pi\)
\(360\) 3.57313 0.188320
\(361\) −0.308682 −0.0162464
\(362\) −36.8621 −1.93743
\(363\) 3.27614 0.171953
\(364\) 0 0
\(365\) −0.734951 −0.0384691
\(366\) −18.8658 −0.986132
\(367\) −0.115028 −0.00600443 −0.00300221 0.999995i \(-0.500956\pi\)
−0.00300221 + 0.999995i \(0.500956\pi\)
\(368\) −15.7535 −0.821206
\(369\) −4.27692 −0.222648
\(370\) −3.52204 −0.183102
\(371\) 0 0
\(372\) 5.25239 0.272324
\(373\) −17.5106 −0.906664 −0.453332 0.891342i \(-0.649765\pi\)
−0.453332 + 0.891342i \(0.649765\pi\)
\(374\) −11.0566 −0.571722
\(375\) 8.03226 0.414784
\(376\) −19.2879 −0.994697
\(377\) −20.5599 −1.05889
\(378\) 0 0
\(379\) 7.72561 0.396838 0.198419 0.980117i \(-0.436419\pi\)
0.198419 + 0.980117i \(0.436419\pi\)
\(380\) 0.959894 0.0492415
\(381\) 7.58096 0.388384
\(382\) 12.2980 0.629220
\(383\) −10.6044 −0.541860 −0.270930 0.962599i \(-0.587331\pi\)
−0.270930 + 0.962599i \(0.587331\pi\)
\(384\) 41.6688 2.12640
\(385\) 0 0
\(386\) −6.24311 −0.317766
\(387\) 44.0030 2.23680
\(388\) −4.32996 −0.219820
\(389\) −22.2172 −1.12646 −0.563228 0.826302i \(-0.690441\pi\)
−0.563228 + 0.826302i \(0.690441\pi\)
\(390\) 4.36375 0.220967
\(391\) −20.4970 −1.03658
\(392\) 0 0
\(393\) 47.4386 2.39296
\(394\) −6.49874 −0.327402
\(395\) 0.931435 0.0468656
\(396\) 6.96033 0.349770
\(397\) 19.8781 0.997654 0.498827 0.866702i \(-0.333764\pi\)
0.498827 + 0.866702i \(0.333764\pi\)
\(398\) −33.0066 −1.65447
\(399\) 0 0
\(400\) 24.6464 1.23232
\(401\) −24.7759 −1.23725 −0.618624 0.785687i \(-0.712309\pi\)
−0.618624 + 0.785687i \(0.712309\pi\)
\(402\) −20.3909 −1.01700
\(403\) −5.64787 −0.281340
\(404\) −6.17870 −0.307402
\(405\) −6.80859 −0.338322
\(406\) 0 0
\(407\) 8.38424 0.415591
\(408\) 39.8426 1.97250
\(409\) −21.7215 −1.07406 −0.537029 0.843564i \(-0.680454\pi\)
−0.537029 + 0.843564i \(0.680454\pi\)
\(410\) 0.232332 0.0114740
\(411\) −36.0251 −1.77699
\(412\) 4.57088 0.225191
\(413\) 0 0
\(414\) 41.5749 2.04330
\(415\) 1.56431 0.0767890
\(416\) 15.0661 0.738674
\(417\) 3.44279 0.168594
\(418\) −7.36248 −0.360111
\(419\) −25.6815 −1.25462 −0.627311 0.778769i \(-0.715845\pi\)
−0.627311 + 0.778769i \(0.715845\pi\)
\(420\) 0 0
\(421\) 28.7580 1.40158 0.700789 0.713369i \(-0.252832\pi\)
0.700789 + 0.713369i \(0.252832\pi\)
\(422\) 41.0889 2.00017
\(423\) 79.6285 3.87167
\(424\) 19.0580 0.925539
\(425\) 32.0677 1.55551
\(426\) 1.92568 0.0932995
\(427\) 0 0
\(428\) 0.807869 0.0390498
\(429\) −10.3879 −0.501534
\(430\) −2.39034 −0.115273
\(431\) −9.84034 −0.473992 −0.236996 0.971511i \(-0.576163\pi\)
−0.236996 + 0.971511i \(0.576163\pi\)
\(432\) −77.3765 −3.72278
\(433\) −0.424750 −0.0204122 −0.0102061 0.999948i \(-0.503249\pi\)
−0.0102061 + 0.999948i \(0.503249\pi\)
\(434\) 0 0
\(435\) −5.24014 −0.251245
\(436\) 5.14453 0.246378
\(437\) −13.6488 −0.652910
\(438\) 16.6226 0.794258
\(439\) −3.98983 −0.190424 −0.0952121 0.995457i \(-0.530353\pi\)
−0.0952121 + 0.995457i \(0.530353\pi\)
\(440\) 0.462057 0.0220277
\(441\) 0 0
\(442\) 35.0580 1.66754
\(443\) −19.9380 −0.947285 −0.473642 0.880717i \(-0.657061\pi\)
−0.473642 + 0.880717i \(0.657061\pi\)
\(444\) 24.7231 1.17331
\(445\) 0.0608489 0.00288451
\(446\) 25.0253 1.18498
\(447\) 60.3416 2.85406
\(448\) 0 0
\(449\) 9.97100 0.470561 0.235280 0.971928i \(-0.424399\pi\)
0.235280 + 0.971928i \(0.424399\pi\)
\(450\) −65.0444 −3.06622
\(451\) −0.553067 −0.0260429
\(452\) −5.39144 −0.253592
\(453\) 62.7591 2.94868
\(454\) 15.6592 0.734925
\(455\) 0 0
\(456\) 26.5309 1.24242
\(457\) 34.9248 1.63371 0.816856 0.576841i \(-0.195715\pi\)
0.816856 + 0.576841i \(0.195715\pi\)
\(458\) 29.6893 1.38729
\(459\) −100.675 −4.69912
\(460\) −0.700934 −0.0326812
\(461\) 26.8042 1.24839 0.624197 0.781267i \(-0.285426\pi\)
0.624197 + 0.781267i \(0.285426\pi\)
\(462\) 0 0
\(463\) −28.1253 −1.30709 −0.653547 0.756886i \(-0.726720\pi\)
−0.653547 + 0.756886i \(0.726720\pi\)
\(464\) −32.3561 −1.50209
\(465\) −1.43948 −0.0667544
\(466\) −47.2580 −2.18918
\(467\) −27.7857 −1.28577 −0.642884 0.765964i \(-0.722262\pi\)
−0.642884 + 0.765964i \(0.722262\pi\)
\(468\) −22.0697 −1.02017
\(469\) 0 0
\(470\) −4.32560 −0.199525
\(471\) 75.4213 3.47523
\(472\) 12.0941 0.556675
\(473\) 5.69023 0.261637
\(474\) −21.0665 −0.967618
\(475\) 21.3537 0.979773
\(476\) 0 0
\(477\) −78.6795 −3.60249
\(478\) −38.3722 −1.75510
\(479\) −9.27456 −0.423766 −0.211883 0.977295i \(-0.567960\pi\)
−0.211883 + 0.977295i \(0.567960\pi\)
\(480\) 3.83991 0.175267
\(481\) −26.5846 −1.21215
\(482\) −12.3066 −0.560549
\(483\) 0 0
\(484\) 0.900071 0.0409123
\(485\) 1.18668 0.0538843
\(486\) 74.7723 3.39174
\(487\) 10.6356 0.481944 0.240972 0.970532i \(-0.422534\pi\)
0.240972 + 0.970532i \(0.422534\pi\)
\(488\) 6.33399 0.286726
\(489\) −16.1797 −0.731672
\(490\) 0 0
\(491\) −19.9832 −0.901829 −0.450915 0.892567i \(-0.648902\pi\)
−0.450915 + 0.892567i \(0.648902\pi\)
\(492\) −1.63086 −0.0735249
\(493\) −42.0989 −1.89604
\(494\) 23.3448 1.05033
\(495\) −1.90756 −0.0857387
\(496\) −8.88832 −0.399097
\(497\) 0 0
\(498\) −35.3805 −1.58544
\(499\) −10.9536 −0.490351 −0.245175 0.969479i \(-0.578846\pi\)
−0.245175 + 0.969479i \(0.578846\pi\)
\(500\) 2.20675 0.0986887
\(501\) 24.2187 1.08201
\(502\) 21.9413 0.979288
\(503\) −25.5069 −1.13729 −0.568647 0.822582i \(-0.692533\pi\)
−0.568647 + 0.822582i \(0.692533\pi\)
\(504\) 0 0
\(505\) 1.69335 0.0753531
\(506\) 5.37624 0.239003
\(507\) −9.65192 −0.428657
\(508\) 2.08276 0.0924074
\(509\) 5.16877 0.229102 0.114551 0.993417i \(-0.463457\pi\)
0.114551 + 0.993417i \(0.463457\pi\)
\(510\) 8.93530 0.395662
\(511\) 0 0
\(512\) 5.01624 0.221689
\(513\) −67.0389 −2.95984
\(514\) 52.9505 2.33555
\(515\) −1.25271 −0.0552008
\(516\) 16.7791 0.738659
\(517\) 10.2971 0.452867
\(518\) 0 0
\(519\) −59.7512 −2.62279
\(520\) −1.46508 −0.0642481
\(521\) 15.4837 0.678352 0.339176 0.940723i \(-0.389852\pi\)
0.339176 + 0.940723i \(0.389852\pi\)
\(522\) 85.3909 3.73746
\(523\) −10.2761 −0.449344 −0.224672 0.974434i \(-0.572131\pi\)
−0.224672 + 0.974434i \(0.572131\pi\)
\(524\) 13.0331 0.569352
\(525\) 0 0
\(526\) −12.7759 −0.557054
\(527\) −11.5647 −0.503766
\(528\) −16.3480 −0.711455
\(529\) −13.0334 −0.566669
\(530\) 4.27405 0.185653
\(531\) −49.9294 −2.16675
\(532\) 0 0
\(533\) 1.75366 0.0759593
\(534\) −1.37624 −0.0595556
\(535\) −0.221407 −0.00957224
\(536\) 6.84601 0.295703
\(537\) −7.77666 −0.335588
\(538\) −3.29952 −0.142252
\(539\) 0 0
\(540\) −3.44279 −0.148154
\(541\) 32.5212 1.39820 0.699099 0.715025i \(-0.253585\pi\)
0.699099 + 0.715025i \(0.253585\pi\)
\(542\) 39.1317 1.68085
\(543\) −70.9150 −3.04325
\(544\) 30.8496 1.32266
\(545\) −1.40992 −0.0603944
\(546\) 0 0
\(547\) −32.7759 −1.40139 −0.700697 0.713459i \(-0.747128\pi\)
−0.700697 + 0.713459i \(0.747128\pi\)
\(548\) −9.89738 −0.422795
\(549\) −26.1494 −1.11603
\(550\) −8.41117 −0.358654
\(551\) −28.0333 −1.19426
\(552\) −19.3734 −0.824586
\(553\) 0 0
\(554\) −9.40195 −0.399451
\(555\) −6.77567 −0.287611
\(556\) 0.945855 0.0401132
\(557\) −14.3574 −0.608341 −0.304170 0.952618i \(-0.598379\pi\)
−0.304170 + 0.952618i \(0.598379\pi\)
\(558\) 23.4571 0.993020
\(559\) −18.0425 −0.763116
\(560\) 0 0
\(561\) −21.2705 −0.898043
\(562\) −12.3975 −0.522959
\(563\) −13.3422 −0.562305 −0.281152 0.959663i \(-0.590717\pi\)
−0.281152 + 0.959663i \(0.590717\pi\)
\(564\) 30.3637 1.27854
\(565\) 1.47759 0.0621627
\(566\) −22.5812 −0.949160
\(567\) 0 0
\(568\) −0.646527 −0.0271277
\(569\) −18.6491 −0.781809 −0.390904 0.920431i \(-0.627838\pi\)
−0.390904 + 0.920431i \(0.627838\pi\)
\(570\) 5.94994 0.249216
\(571\) 32.0508 1.34128 0.670642 0.741781i \(-0.266019\pi\)
0.670642 + 0.741781i \(0.266019\pi\)
\(572\) −2.85393 −0.119329
\(573\) 23.6588 0.988359
\(574\) 0 0
\(575\) −15.5929 −0.650268
\(576\) 14.6030 0.608457
\(577\) −16.4352 −0.684206 −0.342103 0.939662i \(-0.611139\pi\)
−0.342103 + 0.939662i \(0.611139\pi\)
\(578\) 42.8352 1.78171
\(579\) −12.0104 −0.499137
\(580\) −1.43965 −0.0597783
\(581\) 0 0
\(582\) −26.8394 −1.11253
\(583\) −10.1744 −0.421380
\(584\) −5.58086 −0.230937
\(585\) 6.04848 0.250074
\(586\) −9.48069 −0.391644
\(587\) 11.8148 0.487647 0.243824 0.969820i \(-0.421598\pi\)
0.243824 + 0.969820i \(0.421598\pi\)
\(588\) 0 0
\(589\) −7.70083 −0.317307
\(590\) 2.71228 0.111663
\(591\) −12.5022 −0.514273
\(592\) −41.8375 −1.71951
\(593\) −20.7452 −0.851905 −0.425952 0.904746i \(-0.640061\pi\)
−0.425952 + 0.904746i \(0.640061\pi\)
\(594\) 26.4065 1.08347
\(595\) 0 0
\(596\) 16.5780 0.679060
\(597\) −63.4978 −2.59879
\(598\) −17.0469 −0.697099
\(599\) 8.71038 0.355897 0.177948 0.984040i \(-0.443054\pi\)
0.177948 + 0.984040i \(0.443054\pi\)
\(600\) 30.3098 1.23739
\(601\) 46.2310 1.88580 0.942902 0.333072i \(-0.108085\pi\)
0.942902 + 0.333072i \(0.108085\pi\)
\(602\) 0 0
\(603\) −28.2632 −1.15097
\(604\) 17.2421 0.701572
\(605\) −0.246676 −0.0100288
\(606\) −38.2990 −1.55579
\(607\) −30.1304 −1.22295 −0.611477 0.791262i \(-0.709424\pi\)
−0.611477 + 0.791262i \(0.709424\pi\)
\(608\) 20.5425 0.833108
\(609\) 0 0
\(610\) 1.42049 0.0575141
\(611\) −32.6499 −1.32088
\(612\) −45.1904 −1.82671
\(613\) 41.7643 1.68684 0.843421 0.537253i \(-0.180538\pi\)
0.843421 + 0.537253i \(0.180538\pi\)
\(614\) −24.2616 −0.979117
\(615\) 0.446958 0.0180231
\(616\) 0 0
\(617\) 8.94182 0.359984 0.179992 0.983668i \(-0.442393\pi\)
0.179992 + 0.983668i \(0.442393\pi\)
\(618\) 28.3328 1.13971
\(619\) −30.9126 −1.24248 −0.621241 0.783620i \(-0.713371\pi\)
−0.621241 + 0.783620i \(0.713371\pi\)
\(620\) −0.395477 −0.0158827
\(621\) 48.9532 1.96442
\(622\) −6.07674 −0.243655
\(623\) 0 0
\(624\) 51.8359 2.07510
\(625\) 24.0910 0.963639
\(626\) 15.5207 0.620331
\(627\) −14.1639 −0.565651
\(628\) 20.7209 0.826853
\(629\) −54.4352 −2.17047
\(630\) 0 0
\(631\) 30.5300 1.21538 0.607691 0.794174i \(-0.292096\pi\)
0.607691 + 0.794174i \(0.292096\pi\)
\(632\) 7.07286 0.281343
\(633\) 79.0464 3.14181
\(634\) −20.0169 −0.794973
\(635\) −0.570805 −0.0226517
\(636\) −30.0018 −1.18965
\(637\) 0 0
\(638\) 11.0423 0.437168
\(639\) 2.66914 0.105589
\(640\) −3.13744 −0.124018
\(641\) −10.4287 −0.411907 −0.205953 0.978562i \(-0.566030\pi\)
−0.205953 + 0.978562i \(0.566030\pi\)
\(642\) 5.00762 0.197635
\(643\) −16.4973 −0.650590 −0.325295 0.945613i \(-0.605464\pi\)
−0.325295 + 0.945613i \(0.605464\pi\)
\(644\) 0 0
\(645\) −4.59852 −0.181067
\(646\) 47.8014 1.88072
\(647\) 41.0615 1.61429 0.807147 0.590350i \(-0.201010\pi\)
0.807147 + 0.590350i \(0.201010\pi\)
\(648\) −51.7011 −2.03101
\(649\) −6.45660 −0.253444
\(650\) 26.6700 1.04608
\(651\) 0 0
\(652\) −4.44514 −0.174085
\(653\) −20.1454 −0.788350 −0.394175 0.919035i \(-0.628970\pi\)
−0.394175 + 0.919035i \(0.628970\pi\)
\(654\) 31.8886 1.24694
\(655\) −3.57187 −0.139565
\(656\) 2.75981 0.107753
\(657\) 23.0401 0.898881
\(658\) 0 0
\(659\) 6.46125 0.251694 0.125847 0.992050i \(-0.459835\pi\)
0.125847 + 0.992050i \(0.459835\pi\)
\(660\) −0.727387 −0.0283135
\(661\) 14.3775 0.559221 0.279611 0.960114i \(-0.409795\pi\)
0.279611 + 0.960114i \(0.409795\pi\)
\(662\) −43.7232 −1.69935
\(663\) 67.4443 2.61932
\(664\) 11.8786 0.460979
\(665\) 0 0
\(666\) 110.413 4.27842
\(667\) 20.4705 0.792620
\(668\) 6.65372 0.257440
\(669\) 48.1434 1.86133
\(670\) 1.53532 0.0593146
\(671\) −3.38149 −0.130541
\(672\) 0 0
\(673\) 27.0790 1.04382 0.521909 0.853001i \(-0.325220\pi\)
0.521909 + 0.853001i \(0.325220\pi\)
\(674\) 57.1588 2.20168
\(675\) −76.5878 −2.94786
\(676\) −2.65172 −0.101989
\(677\) 15.8329 0.608506 0.304253 0.952591i \(-0.401593\pi\)
0.304253 + 0.952591i \(0.401593\pi\)
\(678\) −33.4191 −1.28345
\(679\) 0 0
\(680\) −2.99993 −0.115042
\(681\) 30.1251 1.15440
\(682\) 3.03335 0.116153
\(683\) 37.0041 1.41592 0.707962 0.706251i \(-0.249615\pi\)
0.707962 + 0.706251i \(0.249615\pi\)
\(684\) −30.0919 −1.15059
\(685\) 2.71250 0.103639
\(686\) 0 0
\(687\) 57.1159 2.17911
\(688\) −28.3943 −1.08252
\(689\) 32.2608 1.22904
\(690\) −4.34477 −0.165403
\(691\) 14.5966 0.555283 0.277641 0.960685i \(-0.410447\pi\)
0.277641 + 0.960685i \(0.410447\pi\)
\(692\) −16.4158 −0.624034
\(693\) 0 0
\(694\) −45.5656 −1.72965
\(695\) −0.259223 −0.00983290
\(696\) −39.7911 −1.50828
\(697\) 3.59082 0.136012
\(698\) 33.9589 1.28536
\(699\) −90.9145 −3.43870
\(700\) 0 0
\(701\) −40.3475 −1.52390 −0.761952 0.647633i \(-0.775759\pi\)
−0.761952 + 0.647633i \(0.775759\pi\)
\(702\) −83.7294 −3.16016
\(703\) −36.2480 −1.36712
\(704\) 1.88838 0.0711708
\(705\) −8.32155 −0.313408
\(706\) 17.1038 0.643710
\(707\) 0 0
\(708\) −19.0389 −0.715527
\(709\) 0.763572 0.0286766 0.0143383 0.999897i \(-0.495436\pi\)
0.0143383 + 0.999897i \(0.495436\pi\)
\(710\) −0.144993 −0.00544150
\(711\) −29.1998 −1.09508
\(712\) 0.462057 0.0173163
\(713\) 5.62331 0.210594
\(714\) 0 0
\(715\) 0.782155 0.0292509
\(716\) −2.13652 −0.0798456
\(717\) −73.8201 −2.75686
\(718\) −29.0387 −1.08371
\(719\) −7.59172 −0.283123 −0.141562 0.989929i \(-0.545212\pi\)
−0.141562 + 0.989929i \(0.545212\pi\)
\(720\) 9.51878 0.354744
\(721\) 0 0
\(722\) −0.525673 −0.0195635
\(723\) −23.6753 −0.880493
\(724\) −19.4828 −0.724074
\(725\) −32.0263 −1.18943
\(726\) 5.57913 0.207061
\(727\) 2.08478 0.0773204 0.0386602 0.999252i \(-0.487691\pi\)
0.0386602 + 0.999252i \(0.487691\pi\)
\(728\) 0 0
\(729\) 61.0421 2.26082
\(730\) −1.25159 −0.0463235
\(731\) −36.9442 −1.36643
\(732\) −9.97121 −0.368547
\(733\) −37.7859 −1.39566 −0.697828 0.716266i \(-0.745850\pi\)
−0.697828 + 0.716266i \(0.745850\pi\)
\(734\) −0.195889 −0.00723038
\(735\) 0 0
\(736\) −15.0005 −0.552927
\(737\) −3.65484 −0.134628
\(738\) −7.28342 −0.268106
\(739\) 36.8466 1.35543 0.677713 0.735327i \(-0.262971\pi\)
0.677713 + 0.735327i \(0.262971\pi\)
\(740\) −1.86152 −0.0684307
\(741\) 44.9106 1.64983
\(742\) 0 0
\(743\) −44.4863 −1.63204 −0.816022 0.578020i \(-0.803825\pi\)
−0.816022 + 0.578020i \(0.803825\pi\)
\(744\) −10.9307 −0.400740
\(745\) −4.54340 −0.166457
\(746\) −29.8198 −1.09178
\(747\) −49.0399 −1.79428
\(748\) −5.84377 −0.213669
\(749\) 0 0
\(750\) 13.6786 0.499473
\(751\) 26.6146 0.971179 0.485590 0.874187i \(-0.338605\pi\)
0.485590 + 0.874187i \(0.338605\pi\)
\(752\) −51.3828 −1.87374
\(753\) 42.2105 1.53824
\(754\) −35.0127 −1.27509
\(755\) −4.72542 −0.171976
\(756\) 0 0
\(757\) −6.38731 −0.232151 −0.116075 0.993240i \(-0.537031\pi\)
−0.116075 + 0.993240i \(0.537031\pi\)
\(758\) 13.1564 0.477862
\(759\) 10.3428 0.375418
\(760\) −1.99763 −0.0724617
\(761\) 40.1614 1.45585 0.727925 0.685657i \(-0.240485\pi\)
0.727925 + 0.685657i \(0.240485\pi\)
\(762\) 12.9101 0.467682
\(763\) 0 0
\(764\) 6.49990 0.235158
\(765\) 12.3850 0.447780
\(766\) −18.0589 −0.652494
\(767\) 20.4725 0.739218
\(768\) 58.5871 2.11408
\(769\) −0.481136 −0.0173502 −0.00867510 0.999962i \(-0.502761\pi\)
−0.00867510 + 0.999962i \(0.502761\pi\)
\(770\) 0 0
\(771\) 101.866 3.66861
\(772\) −3.29969 −0.118759
\(773\) 24.1864 0.869924 0.434962 0.900449i \(-0.356762\pi\)
0.434962 + 0.900449i \(0.356762\pi\)
\(774\) 74.9354 2.69350
\(775\) −8.79772 −0.316023
\(776\) 9.01106 0.323478
\(777\) 0 0
\(778\) −37.8349 −1.35645
\(779\) 2.39110 0.0856701
\(780\) 2.30639 0.0825819
\(781\) 0.345158 0.0123507
\(782\) −34.9056 −1.24822
\(783\) 100.545 3.59319
\(784\) 0 0
\(785\) −5.67881 −0.202686
\(786\) 80.7861 2.88154
\(787\) 37.8595 1.34954 0.674772 0.738026i \(-0.264242\pi\)
0.674772 + 0.738026i \(0.264242\pi\)
\(788\) −3.43480 −0.122360
\(789\) −24.5781 −0.875004
\(790\) 1.58620 0.0564343
\(791\) 0 0
\(792\) −14.4851 −0.514706
\(793\) 10.7220 0.380749
\(794\) 33.8516 1.20135
\(795\) 8.22237 0.291618
\(796\) −17.4451 −0.618325
\(797\) −27.0930 −0.959683 −0.479842 0.877355i \(-0.659306\pi\)
−0.479842 + 0.877355i \(0.659306\pi\)
\(798\) 0 0
\(799\) −66.8547 −2.36515
\(800\) 23.4685 0.829737
\(801\) −1.90756 −0.0674005
\(802\) −42.1923 −1.48986
\(803\) 2.97942 0.105141
\(804\) −10.7772 −0.380084
\(805\) 0 0
\(806\) −9.61809 −0.338783
\(807\) −6.34758 −0.223445
\(808\) 12.8585 0.452359
\(809\) 15.7254 0.552876 0.276438 0.961032i \(-0.410846\pi\)
0.276438 + 0.961032i \(0.410846\pi\)
\(810\) −11.5948 −0.407398
\(811\) 32.3327 1.13536 0.567678 0.823251i \(-0.307842\pi\)
0.567678 + 0.823251i \(0.307842\pi\)
\(812\) 0 0
\(813\) 75.2811 2.64023
\(814\) 14.2780 0.500444
\(815\) 1.21824 0.0426732
\(816\) 106.140 3.71565
\(817\) −24.6008 −0.860674
\(818\) −36.9908 −1.29335
\(819\) 0 0
\(820\) 0.122795 0.00428819
\(821\) 9.55190 0.333364 0.166682 0.986011i \(-0.446695\pi\)
0.166682 + 0.986011i \(0.446695\pi\)
\(822\) −61.3494 −2.13980
\(823\) 28.6991 1.00039 0.500194 0.865913i \(-0.333262\pi\)
0.500194 + 0.865913i \(0.333262\pi\)
\(824\) −9.51243 −0.331381
\(825\) −16.1813 −0.563362
\(826\) 0 0
\(827\) −9.66064 −0.335933 −0.167967 0.985793i \(-0.553720\pi\)
−0.167967 + 0.985793i \(0.553720\pi\)
\(828\) 21.9737 0.763640
\(829\) 16.4010 0.569629 0.284815 0.958583i \(-0.408068\pi\)
0.284815 + 0.958583i \(0.408068\pi\)
\(830\) 2.66396 0.0924673
\(831\) −18.0874 −0.627445
\(832\) −5.98763 −0.207584
\(833\) 0 0
\(834\) 5.86293 0.203017
\(835\) −1.82353 −0.0631060
\(836\) −3.89132 −0.134584
\(837\) 27.6201 0.954690
\(838\) −43.7345 −1.51078
\(839\) −15.0128 −0.518301 −0.259150 0.965837i \(-0.583443\pi\)
−0.259150 + 0.965837i \(0.583443\pi\)
\(840\) 0 0
\(841\) 13.0444 0.449807
\(842\) 48.9736 1.68774
\(843\) −23.8503 −0.821448
\(844\) 21.7168 0.747524
\(845\) 0.726738 0.0250005
\(846\) 135.604 4.66217
\(847\) 0 0
\(848\) 50.7704 1.74346
\(849\) −43.4416 −1.49091
\(850\) 54.6101 1.87311
\(851\) 26.4690 0.907346
\(852\) 1.01779 0.0348688
\(853\) 57.4026 1.96543 0.982714 0.185133i \(-0.0592715\pi\)
0.982714 + 0.185133i \(0.0592715\pi\)
\(854\) 0 0
\(855\) 8.24706 0.282044
\(856\) −1.68125 −0.0574641
\(857\) 32.1817 1.09931 0.549654 0.835393i \(-0.314760\pi\)
0.549654 + 0.835393i \(0.314760\pi\)
\(858\) −17.6902 −0.603934
\(859\) 23.9856 0.818380 0.409190 0.912449i \(-0.365811\pi\)
0.409190 + 0.912449i \(0.365811\pi\)
\(860\) −1.26338 −0.0430808
\(861\) 0 0
\(862\) −16.7577 −0.570769
\(863\) 36.7485 1.25093 0.625467 0.780251i \(-0.284909\pi\)
0.625467 + 0.780251i \(0.284909\pi\)
\(864\) −73.6784 −2.50659
\(865\) 4.49894 0.152969
\(866\) −0.723331 −0.0245798
\(867\) 82.4060 2.79865
\(868\) 0 0
\(869\) −3.77595 −0.128090
\(870\) −8.92374 −0.302543
\(871\) 11.5887 0.392669
\(872\) −10.7063 −0.362560
\(873\) −37.2015 −1.25908
\(874\) −23.2433 −0.786217
\(875\) 0 0
\(876\) 8.78559 0.296838
\(877\) −25.3446 −0.855826 −0.427913 0.903820i \(-0.640751\pi\)
−0.427913 + 0.903820i \(0.640751\pi\)
\(878\) −6.79452 −0.229304
\(879\) −18.2389 −0.615182
\(880\) 1.23091 0.0414941
\(881\) −12.7302 −0.428890 −0.214445 0.976736i \(-0.568794\pi\)
−0.214445 + 0.976736i \(0.568794\pi\)
\(882\) 0 0
\(883\) 10.0508 0.338235 0.169117 0.985596i \(-0.445908\pi\)
0.169117 + 0.985596i \(0.445908\pi\)
\(884\) 18.5293 0.623209
\(885\) 5.21786 0.175396
\(886\) −33.9537 −1.14070
\(887\) −18.9974 −0.637871 −0.318936 0.947776i \(-0.603325\pi\)
−0.318936 + 0.947776i \(0.603325\pi\)
\(888\) −51.4511 −1.72659
\(889\) 0 0
\(890\) 0.103623 0.00347346
\(891\) 27.6014 0.924682
\(892\) 13.2267 0.442862
\(893\) −44.5180 −1.48974
\(894\) 102.759 3.43679
\(895\) 0.585541 0.0195725
\(896\) 0 0
\(897\) −32.7947 −1.09498
\(898\) 16.9802 0.566637
\(899\) 11.5497 0.385205
\(900\) −34.3781 −1.14594
\(901\) 66.0579 2.20071
\(902\) −0.941851 −0.0313602
\(903\) 0 0
\(904\) 11.2201 0.373175
\(905\) 5.33951 0.177491
\(906\) 106.876 3.55072
\(907\) −59.1419 −1.96378 −0.981888 0.189464i \(-0.939325\pi\)
−0.981888 + 0.189464i \(0.939325\pi\)
\(908\) 8.27643 0.274663
\(909\) −53.0852 −1.76073
\(910\) 0 0
\(911\) −11.5307 −0.382028 −0.191014 0.981587i \(-0.561178\pi\)
−0.191014 + 0.981587i \(0.561178\pi\)
\(912\) 70.6780 2.34038
\(913\) −6.34157 −0.209875
\(914\) 59.4755 1.96728
\(915\) 2.73273 0.0903414
\(916\) 15.6918 0.518471
\(917\) 0 0
\(918\) −171.446 −5.65856
\(919\) 27.8583 0.918962 0.459481 0.888188i \(-0.348036\pi\)
0.459481 + 0.888188i \(0.348036\pi\)
\(920\) 1.45871 0.0480923
\(921\) −46.6742 −1.53797
\(922\) 45.6464 1.50328
\(923\) −1.09442 −0.0360233
\(924\) 0 0
\(925\) −41.4110 −1.36159
\(926\) −47.8963 −1.57397
\(927\) 39.2713 1.28984
\(928\) −30.8097 −1.01138
\(929\) −51.3944 −1.68619 −0.843097 0.537761i \(-0.819270\pi\)
−0.843097 + 0.537761i \(0.819270\pi\)
\(930\) −2.45138 −0.0803839
\(931\) 0 0
\(932\) −24.9774 −0.818162
\(933\) −11.6904 −0.382726
\(934\) −47.3179 −1.54829
\(935\) 1.60156 0.0523765
\(936\) 45.9292 1.50124
\(937\) 8.07437 0.263778 0.131889 0.991264i \(-0.457896\pi\)
0.131889 + 0.991264i \(0.457896\pi\)
\(938\) 0 0
\(939\) 29.8585 0.974397
\(940\) −2.28622 −0.0745684
\(941\) −24.3020 −0.792223 −0.396112 0.918202i \(-0.629641\pi\)
−0.396112 + 0.918202i \(0.629641\pi\)
\(942\) 128.439 4.18478
\(943\) −1.74603 −0.0568586
\(944\) 32.2185 1.04862
\(945\) 0 0
\(946\) 9.69023 0.315056
\(947\) 40.5660 1.31822 0.659109 0.752047i \(-0.270933\pi\)
0.659109 + 0.752047i \(0.270933\pi\)
\(948\) −11.1344 −0.361628
\(949\) −9.44710 −0.306666
\(950\) 36.3644 1.17982
\(951\) −38.5083 −1.24872
\(952\) 0 0
\(953\) −4.52347 −0.146529 −0.0732647 0.997313i \(-0.523342\pi\)
−0.0732647 + 0.997313i \(0.523342\pi\)
\(954\) −133.988 −4.33802
\(955\) −1.78138 −0.0576440
\(956\) −20.2810 −0.655934
\(957\) 21.2430 0.686690
\(958\) −15.7942 −0.510288
\(959\) 0 0
\(960\) −1.52608 −0.0492540
\(961\) −27.8273 −0.897653
\(962\) −45.2725 −1.45964
\(963\) 6.94092 0.223668
\(964\) −6.50443 −0.209494
\(965\) 0.904322 0.0291112
\(966\) 0 0
\(967\) −48.3508 −1.55486 −0.777428 0.628972i \(-0.783476\pi\)
−0.777428 + 0.628972i \(0.783476\pi\)
\(968\) −1.87313 −0.0602048
\(969\) 91.9599 2.95418
\(970\) 2.02087 0.0648861
\(971\) 12.7515 0.409217 0.204608 0.978844i \(-0.434408\pi\)
0.204608 + 0.978844i \(0.434408\pi\)
\(972\) 39.5197 1.26759
\(973\) 0 0
\(974\) 18.1120 0.580345
\(975\) 51.3076 1.64316
\(976\) 16.8737 0.540114
\(977\) 21.9973 0.703757 0.351879 0.936046i \(-0.385543\pi\)
0.351879 + 0.936046i \(0.385543\pi\)
\(978\) −27.5534 −0.881060
\(979\) −0.246676 −0.00788379
\(980\) 0 0
\(981\) 44.2000 1.41120
\(982\) −34.0306 −1.08596
\(983\) −51.4813 −1.64200 −0.820999 0.570930i \(-0.806583\pi\)
−0.820999 + 0.570930i \(0.806583\pi\)
\(984\) 3.39398 0.108196
\(985\) 0.941350 0.0299939
\(986\) −71.6927 −2.28316
\(987\) 0 0
\(988\) 12.3385 0.392541
\(989\) 17.9640 0.571223
\(990\) −3.24851 −0.103244
\(991\) −12.8667 −0.408724 −0.204362 0.978895i \(-0.565512\pi\)
−0.204362 + 0.978895i \(0.565512\pi\)
\(992\) −8.46351 −0.268717
\(993\) −84.1144 −2.66929
\(994\) 0 0
\(995\) 4.78104 0.151569
\(996\) −18.6998 −0.592524
\(997\) −30.0438 −0.951496 −0.475748 0.879582i \(-0.657823\pi\)
−0.475748 + 0.879582i \(0.657823\pi\)
\(998\) −18.6535 −0.590468
\(999\) 130.008 4.11328
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 539.2.a.l.1.8 yes 10
3.2 odd 2 4851.2.a.cg.1.4 10
4.3 odd 2 8624.2.a.df.1.1 10
7.2 even 3 539.2.e.o.67.3 20
7.3 odd 6 539.2.e.o.177.4 20
7.4 even 3 539.2.e.o.177.3 20
7.5 odd 6 539.2.e.o.67.4 20
7.6 odd 2 inner 539.2.a.l.1.7 10
11.10 odd 2 5929.2.a.bv.1.4 10
21.20 even 2 4851.2.a.cg.1.3 10
28.27 even 2 8624.2.a.df.1.10 10
77.76 even 2 5929.2.a.bv.1.3 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
539.2.a.l.1.7 10 7.6 odd 2 inner
539.2.a.l.1.8 yes 10 1.1 even 1 trivial
539.2.e.o.67.3 20 7.2 even 3
539.2.e.o.67.4 20 7.5 odd 6
539.2.e.o.177.3 20 7.4 even 3
539.2.e.o.177.4 20 7.3 odd 6
4851.2.a.cg.1.3 10 21.20 even 2
4851.2.a.cg.1.4 10 3.2 odd 2
5929.2.a.bv.1.3 10 77.76 even 2
5929.2.a.bv.1.4 10 11.10 odd 2
8624.2.a.df.1.1 10 4.3 odd 2
8624.2.a.df.1.10 10 28.27 even 2