Properties

Label 5390.2.a.ca.1.2
Level $5390$
Weight $2$
Character 5390.1
Self dual yes
Analytic conductor $43.039$
Analytic rank $0$
Dimension $3$
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] = [5390,2,Mod(1,5390)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5390, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("5390.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5390 = 2 \cdot 5 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5390.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: \(43.0393666895\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 770)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.34292\) of defining polynomial
Character \(\chi\) \(=\) 5390.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.14637 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.14637 q^{6} +1.00000 q^{8} -1.68585 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.14637 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.14637 q^{6} +1.00000 q^{8} -1.68585 q^{9} -1.00000 q^{10} +1.00000 q^{11} -1.14637 q^{12} -4.68585 q^{13} +1.14637 q^{15} +1.00000 q^{16} +0.292731 q^{17} -1.68585 q^{18} -6.51806 q^{19} -1.00000 q^{20} +1.00000 q^{22} +2.85363 q^{23} -1.14637 q^{24} +1.00000 q^{25} -4.68585 q^{26} +5.37169 q^{27} -1.43910 q^{29} +1.14637 q^{30} -0.978577 q^{31} +1.00000 q^{32} -1.14637 q^{33} +0.292731 q^{34} -1.68585 q^{36} +0.853635 q^{37} -6.51806 q^{38} +5.37169 q^{39} -1.00000 q^{40} +6.22533 q^{41} -10.3503 q^{43} +1.00000 q^{44} +1.68585 q^{45} +2.85363 q^{46} +9.95715 q^{47} -1.14637 q^{48} +1.00000 q^{50} -0.335577 q^{51} -4.68585 q^{52} +5.43910 q^{53} +5.37169 q^{54} -1.00000 q^{55} +7.47208 q^{57} -1.43910 q^{58} +9.37169 q^{59} +1.14637 q^{60} +11.9572 q^{61} -0.978577 q^{62} +1.00000 q^{64} +4.68585 q^{65} -1.14637 q^{66} -0.585462 q^{67} +0.292731 q^{68} -3.27131 q^{69} -0.335577 q^{71} -1.68585 q^{72} +3.70727 q^{73} +0.853635 q^{74} -1.14637 q^{75} -6.51806 q^{76} +5.37169 q^{78} -2.51806 q^{79} -1.00000 q^{80} -1.10038 q^{81} +6.22533 q^{82} +1.70727 q^{83} -0.292731 q^{85} -10.3503 q^{86} +1.64973 q^{87} +1.00000 q^{88} -13.0790 q^{89} +1.68585 q^{90} +2.85363 q^{92} +1.12181 q^{93} +9.95715 q^{94} +6.51806 q^{95} -1.14637 q^{96} -9.10352 q^{97} -1.68585 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} - 2 q^{3} + 3 q^{4} - 3 q^{5} - 2 q^{6} + 3 q^{8} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{2} - 2 q^{3} + 3 q^{4} - 3 q^{5} - 2 q^{6} + 3 q^{8} + 7 q^{9} - 3 q^{10} + 3 q^{11} - 2 q^{12} - 2 q^{13} + 2 q^{15} + 3 q^{16} - 2 q^{17} + 7 q^{18} + 6 q^{19} - 3 q^{20} + 3 q^{22} + 10 q^{23} - 2 q^{24} + 3 q^{25} - 2 q^{26} - 8 q^{27} + 2 q^{30} + 12 q^{31} + 3 q^{32} - 2 q^{33} - 2 q^{34} + 7 q^{36} + 4 q^{37} + 6 q^{38} - 8 q^{39} - 3 q^{40} - 4 q^{41} + 8 q^{43} + 3 q^{44} - 7 q^{45} + 10 q^{46} - 2 q^{48} + 3 q^{50} - 28 q^{51} - 2 q^{52} + 12 q^{53} - 8 q^{54} - 3 q^{55} - 8 q^{57} + 4 q^{59} + 2 q^{60} + 6 q^{61} + 12 q^{62} + 3 q^{64} + 2 q^{65} - 2 q^{66} + 4 q^{67} - 2 q^{68} + 8 q^{69} - 28 q^{71} + 7 q^{72} + 14 q^{73} + 4 q^{74} - 2 q^{75} + 6 q^{76} - 8 q^{78} + 18 q^{79} - 3 q^{80} + 3 q^{81} - 4 q^{82} + 8 q^{83} + 2 q^{85} + 8 q^{86} + 44 q^{87} + 3 q^{88} - 18 q^{89} - 7 q^{90} + 10 q^{92} + 12 q^{93} - 6 q^{95} - 2 q^{96} + 4 q^{97} + 7 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.14637 −0.661854 −0.330927 0.943656i \(-0.607361\pi\)
−0.330927 + 0.943656i \(0.607361\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) −1.14637 −0.468002
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) −1.68585 −0.561949
\(10\) −1.00000 −0.316228
\(11\) 1.00000 0.301511
\(12\) −1.14637 −0.330927
\(13\) −4.68585 −1.29962 −0.649810 0.760097i \(-0.725152\pi\)
−0.649810 + 0.760097i \(0.725152\pi\)
\(14\) 0 0
\(15\) 1.14637 0.295990
\(16\) 1.00000 0.250000
\(17\) 0.292731 0.0709977 0.0354988 0.999370i \(-0.488698\pi\)
0.0354988 + 0.999370i \(0.488698\pi\)
\(18\) −1.68585 −0.397358
\(19\) −6.51806 −1.49535 −0.747673 0.664068i \(-0.768829\pi\)
−0.747673 + 0.664068i \(0.768829\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) 2.85363 0.595024 0.297512 0.954718i \(-0.403843\pi\)
0.297512 + 0.954718i \(0.403843\pi\)
\(24\) −1.14637 −0.234001
\(25\) 1.00000 0.200000
\(26\) −4.68585 −0.918970
\(27\) 5.37169 1.03378
\(28\) 0 0
\(29\) −1.43910 −0.267234 −0.133617 0.991033i \(-0.542659\pi\)
−0.133617 + 0.991033i \(0.542659\pi\)
\(30\) 1.14637 0.209297
\(31\) −0.978577 −0.175758 −0.0878788 0.996131i \(-0.528009\pi\)
−0.0878788 + 0.996131i \(0.528009\pi\)
\(32\) 1.00000 0.176777
\(33\) −1.14637 −0.199557
\(34\) 0.292731 0.0502029
\(35\) 0 0
\(36\) −1.68585 −0.280974
\(37\) 0.853635 0.140337 0.0701683 0.997535i \(-0.477646\pi\)
0.0701683 + 0.997535i \(0.477646\pi\)
\(38\) −6.51806 −1.05737
\(39\) 5.37169 0.860159
\(40\) −1.00000 −0.158114
\(41\) 6.22533 0.972233 0.486116 0.873894i \(-0.338413\pi\)
0.486116 + 0.873894i \(0.338413\pi\)
\(42\) 0 0
\(43\) −10.3503 −1.57840 −0.789201 0.614135i \(-0.789505\pi\)
−0.789201 + 0.614135i \(0.789505\pi\)
\(44\) 1.00000 0.150756
\(45\) 1.68585 0.251311
\(46\) 2.85363 0.420745
\(47\) 9.95715 1.45240 0.726200 0.687483i \(-0.241285\pi\)
0.726200 + 0.687483i \(0.241285\pi\)
\(48\) −1.14637 −0.165464
\(49\) 0 0
\(50\) 1.00000 0.141421
\(51\) −0.335577 −0.0469901
\(52\) −4.68585 −0.649810
\(53\) 5.43910 0.747117 0.373559 0.927607i \(-0.378137\pi\)
0.373559 + 0.927607i \(0.378137\pi\)
\(54\) 5.37169 0.730995
\(55\) −1.00000 −0.134840
\(56\) 0 0
\(57\) 7.47208 0.989701
\(58\) −1.43910 −0.188963
\(59\) 9.37169 1.22009 0.610045 0.792367i \(-0.291151\pi\)
0.610045 + 0.792367i \(0.291151\pi\)
\(60\) 1.14637 0.147995
\(61\) 11.9572 1.53096 0.765478 0.643462i \(-0.222502\pi\)
0.765478 + 0.643462i \(0.222502\pi\)
\(62\) −0.978577 −0.124279
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 4.68585 0.581208
\(66\) −1.14637 −0.141108
\(67\) −0.585462 −0.0715256 −0.0357628 0.999360i \(-0.511386\pi\)
−0.0357628 + 0.999360i \(0.511386\pi\)
\(68\) 0.292731 0.0354988
\(69\) −3.27131 −0.393819
\(70\) 0 0
\(71\) −0.335577 −0.0398256 −0.0199128 0.999802i \(-0.506339\pi\)
−0.0199128 + 0.999802i \(0.506339\pi\)
\(72\) −1.68585 −0.198679
\(73\) 3.70727 0.433903 0.216952 0.976182i \(-0.430389\pi\)
0.216952 + 0.976182i \(0.430389\pi\)
\(74\) 0.853635 0.0992330
\(75\) −1.14637 −0.132371
\(76\) −6.51806 −0.747673
\(77\) 0 0
\(78\) 5.37169 0.608224
\(79\) −2.51806 −0.283304 −0.141652 0.989917i \(-0.545241\pi\)
−0.141652 + 0.989917i \(0.545241\pi\)
\(80\) −1.00000 −0.111803
\(81\) −1.10038 −0.122265
\(82\) 6.22533 0.687472
\(83\) 1.70727 0.187397 0.0936986 0.995601i \(-0.470131\pi\)
0.0936986 + 0.995601i \(0.470131\pi\)
\(84\) 0 0
\(85\) −0.292731 −0.0317511
\(86\) −10.3503 −1.11610
\(87\) 1.64973 0.176870
\(88\) 1.00000 0.106600
\(89\) −13.0790 −1.38637 −0.693184 0.720761i \(-0.743792\pi\)
−0.693184 + 0.720761i \(0.743792\pi\)
\(90\) 1.68585 0.177704
\(91\) 0 0
\(92\) 2.85363 0.297512
\(93\) 1.12181 0.116326
\(94\) 9.95715 1.02700
\(95\) 6.51806 0.668739
\(96\) −1.14637 −0.117000
\(97\) −9.10352 −0.924322 −0.462161 0.886796i \(-0.652926\pi\)
−0.462161 + 0.886796i \(0.652926\pi\)
\(98\) 0 0
\(99\) −1.68585 −0.169434
\(100\) 1.00000 0.100000
\(101\) 10.7862 1.07327 0.536635 0.843814i \(-0.319695\pi\)
0.536635 + 0.843814i \(0.319695\pi\)
\(102\) −0.335577 −0.0332270
\(103\) −7.66442 −0.755198 −0.377599 0.925969i \(-0.623250\pi\)
−0.377599 + 0.925969i \(0.623250\pi\)
\(104\) −4.68585 −0.459485
\(105\) 0 0
\(106\) 5.43910 0.528292
\(107\) 9.56404 0.924591 0.462295 0.886726i \(-0.347026\pi\)
0.462295 + 0.886726i \(0.347026\pi\)
\(108\) 5.37169 0.516891
\(109\) 3.14637 0.301367 0.150684 0.988582i \(-0.451853\pi\)
0.150684 + 0.988582i \(0.451853\pi\)
\(110\) −1.00000 −0.0953463
\(111\) −0.978577 −0.0928824
\(112\) 0 0
\(113\) −2.58546 −0.243220 −0.121610 0.992578i \(-0.538806\pi\)
−0.121610 + 0.992578i \(0.538806\pi\)
\(114\) 7.47208 0.699824
\(115\) −2.85363 −0.266103
\(116\) −1.43910 −0.133617
\(117\) 7.89962 0.730320
\(118\) 9.37169 0.862734
\(119\) 0 0
\(120\) 1.14637 0.104648
\(121\) 1.00000 0.0909091
\(122\) 11.9572 1.08255
\(123\) −7.13650 −0.643477
\(124\) −0.978577 −0.0878788
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −13.3717 −1.18655 −0.593273 0.805001i \(-0.702164\pi\)
−0.593273 + 0.805001i \(0.702164\pi\)
\(128\) 1.00000 0.0883883
\(129\) 11.8652 1.04467
\(130\) 4.68585 0.410976
\(131\) 11.4391 0.999438 0.499719 0.866187i \(-0.333437\pi\)
0.499719 + 0.866187i \(0.333437\pi\)
\(132\) −1.14637 −0.0997783
\(133\) 0 0
\(134\) −0.585462 −0.0505762
\(135\) −5.37169 −0.462322
\(136\) 0.292731 0.0251015
\(137\) 10.0000 0.854358 0.427179 0.904167i \(-0.359507\pi\)
0.427179 + 0.904167i \(0.359507\pi\)
\(138\) −3.27131 −0.278472
\(139\) 14.1825 1.20294 0.601471 0.798895i \(-0.294581\pi\)
0.601471 + 0.798895i \(0.294581\pi\)
\(140\) 0 0
\(141\) −11.4145 −0.961278
\(142\) −0.335577 −0.0281610
\(143\) −4.68585 −0.391850
\(144\) −1.68585 −0.140487
\(145\) 1.43910 0.119510
\(146\) 3.70727 0.306816
\(147\) 0 0
\(148\) 0.853635 0.0701683
\(149\) 11.5970 0.950065 0.475032 0.879968i \(-0.342436\pi\)
0.475032 + 0.879968i \(0.342436\pi\)
\(150\) −1.14637 −0.0936004
\(151\) 1.73183 0.140934 0.0704671 0.997514i \(-0.477551\pi\)
0.0704671 + 0.997514i \(0.477551\pi\)
\(152\) −6.51806 −0.528684
\(153\) −0.493499 −0.0398971
\(154\) 0 0
\(155\) 0.978577 0.0786012
\(156\) 5.37169 0.430080
\(157\) 12.2927 0.981067 0.490533 0.871422i \(-0.336802\pi\)
0.490533 + 0.871422i \(0.336802\pi\)
\(158\) −2.51806 −0.200326
\(159\) −6.23519 −0.494483
\(160\) −1.00000 −0.0790569
\(161\) 0 0
\(162\) −1.10038 −0.0864543
\(163\) 23.9143 1.87311 0.936557 0.350516i \(-0.113994\pi\)
0.936557 + 0.350516i \(0.113994\pi\)
\(164\) 6.22533 0.486116
\(165\) 1.14637 0.0892444
\(166\) 1.70727 0.132510
\(167\) 8.00000 0.619059 0.309529 0.950890i \(-0.399829\pi\)
0.309529 + 0.950890i \(0.399829\pi\)
\(168\) 0 0
\(169\) 8.95715 0.689012
\(170\) −0.292731 −0.0224514
\(171\) 10.9884 0.840307
\(172\) −10.3503 −0.789201
\(173\) 7.37169 0.560459 0.280230 0.959933i \(-0.409589\pi\)
0.280230 + 0.959933i \(0.409589\pi\)
\(174\) 1.64973 0.125066
\(175\) 0 0
\(176\) 1.00000 0.0753778
\(177\) −10.7434 −0.807522
\(178\) −13.0790 −0.980310
\(179\) 1.56404 0.116902 0.0584509 0.998290i \(-0.481384\pi\)
0.0584509 + 0.998290i \(0.481384\pi\)
\(180\) 1.68585 0.125656
\(181\) −14.7862 −1.09905 −0.549526 0.835477i \(-0.685192\pi\)
−0.549526 + 0.835477i \(0.685192\pi\)
\(182\) 0 0
\(183\) −13.7073 −1.01327
\(184\) 2.85363 0.210373
\(185\) −0.853635 −0.0627605
\(186\) 1.12181 0.0822549
\(187\) 0.292731 0.0214066
\(188\) 9.95715 0.726200
\(189\) 0 0
\(190\) 6.51806 0.472870
\(191\) −17.9572 −1.29933 −0.649667 0.760219i \(-0.725092\pi\)
−0.649667 + 0.760219i \(0.725092\pi\)
\(192\) −1.14637 −0.0827318
\(193\) 18.6430 1.34195 0.670976 0.741479i \(-0.265875\pi\)
0.670976 + 0.741479i \(0.265875\pi\)
\(194\) −9.10352 −0.653595
\(195\) −5.37169 −0.384675
\(196\) 0 0
\(197\) −15.0361 −1.07128 −0.535639 0.844447i \(-0.679929\pi\)
−0.535639 + 0.844447i \(0.679929\pi\)
\(198\) −1.68585 −0.119808
\(199\) −15.3288 −1.08663 −0.543317 0.839528i \(-0.682832\pi\)
−0.543317 + 0.839528i \(0.682832\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0.671153 0.0473395
\(202\) 10.7862 0.758917
\(203\) 0 0
\(204\) −0.335577 −0.0234951
\(205\) −6.22533 −0.434796
\(206\) −7.66442 −0.534006
\(207\) −4.81079 −0.334373
\(208\) −4.68585 −0.324905
\(209\) −6.51806 −0.450863
\(210\) 0 0
\(211\) 2.04285 0.140635 0.0703176 0.997525i \(-0.477599\pi\)
0.0703176 + 0.997525i \(0.477599\pi\)
\(212\) 5.43910 0.373559
\(213\) 0.384694 0.0263588
\(214\) 9.56404 0.653784
\(215\) 10.3503 0.705883
\(216\) 5.37169 0.365497
\(217\) 0 0
\(218\) 3.14637 0.213099
\(219\) −4.24989 −0.287181
\(220\) −1.00000 −0.0674200
\(221\) −1.37169 −0.0922700
\(222\) −0.978577 −0.0656778
\(223\) −11.0790 −0.741902 −0.370951 0.928652i \(-0.620968\pi\)
−0.370951 + 0.928652i \(0.620968\pi\)
\(224\) 0 0
\(225\) −1.68585 −0.112390
\(226\) −2.58546 −0.171982
\(227\) 18.5426 1.23072 0.615358 0.788248i \(-0.289011\pi\)
0.615358 + 0.788248i \(0.289011\pi\)
\(228\) 7.47208 0.494850
\(229\) 8.01469 0.529626 0.264813 0.964300i \(-0.414690\pi\)
0.264813 + 0.964300i \(0.414690\pi\)
\(230\) −2.85363 −0.188163
\(231\) 0 0
\(232\) −1.43910 −0.0944813
\(233\) 27.9572 1.83153 0.915767 0.401710i \(-0.131584\pi\)
0.915767 + 0.401710i \(0.131584\pi\)
\(234\) 7.89962 0.516414
\(235\) −9.95715 −0.649533
\(236\) 9.37169 0.610045
\(237\) 2.88661 0.187506
\(238\) 0 0
\(239\) −28.8108 −1.86362 −0.931808 0.362953i \(-0.881769\pi\)
−0.931808 + 0.362953i \(0.881769\pi\)
\(240\) 1.14637 0.0739976
\(241\) 8.51806 0.548696 0.274348 0.961630i \(-0.411538\pi\)
0.274348 + 0.961630i \(0.411538\pi\)
\(242\) 1.00000 0.0642824
\(243\) −14.8536 −0.952861
\(244\) 11.9572 0.765478
\(245\) 0 0
\(246\) −7.13650 −0.455007
\(247\) 30.5426 1.94338
\(248\) −0.978577 −0.0621397
\(249\) −1.95715 −0.124030
\(250\) −1.00000 −0.0632456
\(251\) 23.1281 1.45983 0.729916 0.683537i \(-0.239559\pi\)
0.729916 + 0.683537i \(0.239559\pi\)
\(252\) 0 0
\(253\) 2.85363 0.179406
\(254\) −13.3717 −0.839015
\(255\) 0.335577 0.0210146
\(256\) 1.00000 0.0625000
\(257\) −14.8108 −0.923872 −0.461936 0.886913i \(-0.652845\pi\)
−0.461936 + 0.886913i \(0.652845\pi\)
\(258\) 11.8652 0.738695
\(259\) 0 0
\(260\) 4.68585 0.290604
\(261\) 2.42610 0.150172
\(262\) 11.4391 0.706710
\(263\) 18.7434 1.15577 0.577883 0.816119i \(-0.303879\pi\)
0.577883 + 0.816119i \(0.303879\pi\)
\(264\) −1.14637 −0.0705539
\(265\) −5.43910 −0.334121
\(266\) 0 0
\(267\) 14.9933 0.917573
\(268\) −0.585462 −0.0357628
\(269\) −12.6858 −0.773470 −0.386735 0.922191i \(-0.626397\pi\)
−0.386735 + 0.922191i \(0.626397\pi\)
\(270\) −5.37169 −0.326911
\(271\) −1.12181 −0.0681449 −0.0340725 0.999419i \(-0.510848\pi\)
−0.0340725 + 0.999419i \(0.510848\pi\)
\(272\) 0.292731 0.0177494
\(273\) 0 0
\(274\) 10.0000 0.604122
\(275\) 1.00000 0.0603023
\(276\) −3.27131 −0.196910
\(277\) 30.1151 1.80944 0.904720 0.426007i \(-0.140080\pi\)
0.904720 + 0.426007i \(0.140080\pi\)
\(278\) 14.1825 0.850609
\(279\) 1.64973 0.0987668
\(280\) 0 0
\(281\) −14.3356 −0.855189 −0.427594 0.903971i \(-0.640639\pi\)
−0.427594 + 0.903971i \(0.640639\pi\)
\(282\) −11.4145 −0.679726
\(283\) −3.21377 −0.191039 −0.0955194 0.995428i \(-0.530451\pi\)
−0.0955194 + 0.995428i \(0.530451\pi\)
\(284\) −0.335577 −0.0199128
\(285\) −7.47208 −0.442608
\(286\) −4.68585 −0.277080
\(287\) 0 0
\(288\) −1.68585 −0.0993394
\(289\) −16.9143 −0.994959
\(290\) 1.43910 0.0845067
\(291\) 10.4360 0.611767
\(292\) 3.70727 0.216952
\(293\) 16.6858 0.974798 0.487399 0.873179i \(-0.337946\pi\)
0.487399 + 0.873179i \(0.337946\pi\)
\(294\) 0 0
\(295\) −9.37169 −0.545641
\(296\) 0.853635 0.0496165
\(297\) 5.37169 0.311697
\(298\) 11.5970 0.671797
\(299\) −13.3717 −0.773305
\(300\) −1.14637 −0.0661854
\(301\) 0 0
\(302\) 1.73183 0.0996555
\(303\) −12.3650 −0.710349
\(304\) −6.51806 −0.373836
\(305\) −11.9572 −0.684665
\(306\) −0.493499 −0.0282115
\(307\) 6.29273 0.359145 0.179573 0.983745i \(-0.442529\pi\)
0.179573 + 0.983745i \(0.442529\pi\)
\(308\) 0 0
\(309\) 8.78623 0.499831
\(310\) 0.978577 0.0555794
\(311\) 20.3931 1.15639 0.578194 0.815900i \(-0.303758\pi\)
0.578194 + 0.815900i \(0.303758\pi\)
\(312\) 5.37169 0.304112
\(313\) −12.0674 −0.682090 −0.341045 0.940047i \(-0.610781\pi\)
−0.341045 + 0.940047i \(0.610781\pi\)
\(314\) 12.2927 0.693719
\(315\) 0 0
\(316\) −2.51806 −0.141652
\(317\) 28.7679 1.61577 0.807884 0.589341i \(-0.200613\pi\)
0.807884 + 0.589341i \(0.200613\pi\)
\(318\) −6.23519 −0.349652
\(319\) −1.43910 −0.0805739
\(320\) −1.00000 −0.0559017
\(321\) −10.9639 −0.611944
\(322\) 0 0
\(323\) −1.90804 −0.106166
\(324\) −1.10038 −0.0611325
\(325\) −4.68585 −0.259924
\(326\) 23.9143 1.32449
\(327\) −3.60688 −0.199461
\(328\) 6.22533 0.343736
\(329\) 0 0
\(330\) 1.14637 0.0631053
\(331\) −3.80765 −0.209288 −0.104644 0.994510i \(-0.533370\pi\)
−0.104644 + 0.994510i \(0.533370\pi\)
\(332\) 1.70727 0.0936986
\(333\) −1.43910 −0.0788620
\(334\) 8.00000 0.437741
\(335\) 0.585462 0.0319872
\(336\) 0 0
\(337\) 15.3717 0.837349 0.418675 0.908136i \(-0.362495\pi\)
0.418675 + 0.908136i \(0.362495\pi\)
\(338\) 8.95715 0.487205
\(339\) 2.96388 0.160976
\(340\) −0.292731 −0.0158756
\(341\) −0.978577 −0.0529929
\(342\) 10.9884 0.594187
\(343\) 0 0
\(344\) −10.3503 −0.558049
\(345\) 3.27131 0.176121
\(346\) 7.37169 0.396305
\(347\) 3.02142 0.162198 0.0810992 0.996706i \(-0.474157\pi\)
0.0810992 + 0.996706i \(0.474157\pi\)
\(348\) 1.64973 0.0884348
\(349\) −11.0361 −0.590750 −0.295375 0.955381i \(-0.595445\pi\)
−0.295375 + 0.955381i \(0.595445\pi\)
\(350\) 0 0
\(351\) −25.1709 −1.34352
\(352\) 1.00000 0.0533002
\(353\) −23.9817 −1.27642 −0.638209 0.769863i \(-0.720324\pi\)
−0.638209 + 0.769863i \(0.720324\pi\)
\(354\) −10.7434 −0.571004
\(355\) 0.335577 0.0178106
\(356\) −13.0790 −0.693184
\(357\) 0 0
\(358\) 1.56404 0.0826620
\(359\) −10.5181 −0.555122 −0.277561 0.960708i \(-0.589526\pi\)
−0.277561 + 0.960708i \(0.589526\pi\)
\(360\) 1.68585 0.0888519
\(361\) 23.4851 1.23606
\(362\) −14.7862 −0.777147
\(363\) −1.14637 −0.0601686
\(364\) 0 0
\(365\) −3.70727 −0.194047
\(366\) −13.7073 −0.716490
\(367\) 35.5787 1.85719 0.928597 0.371089i \(-0.121015\pi\)
0.928597 + 0.371089i \(0.121015\pi\)
\(368\) 2.85363 0.148756
\(369\) −10.4949 −0.546345
\(370\) −0.853635 −0.0443783
\(371\) 0 0
\(372\) 1.12181 0.0581630
\(373\) 7.50650 0.388672 0.194336 0.980935i \(-0.437745\pi\)
0.194336 + 0.980935i \(0.437745\pi\)
\(374\) 0.292731 0.0151368
\(375\) 1.14637 0.0591981
\(376\) 9.95715 0.513501
\(377\) 6.74338 0.347302
\(378\) 0 0
\(379\) 33.4868 1.72010 0.860050 0.510210i \(-0.170432\pi\)
0.860050 + 0.510210i \(0.170432\pi\)
\(380\) 6.51806 0.334369
\(381\) 15.3288 0.785321
\(382\) −17.9572 −0.918768
\(383\) −15.6644 −0.800415 −0.400207 0.916425i \(-0.631062\pi\)
−0.400207 + 0.916425i \(0.631062\pi\)
\(384\) −1.14637 −0.0585002
\(385\) 0 0
\(386\) 18.6430 0.948904
\(387\) 17.4490 0.886981
\(388\) −9.10352 −0.462161
\(389\) −19.6216 −0.994853 −0.497427 0.867506i \(-0.665722\pi\)
−0.497427 + 0.867506i \(0.665722\pi\)
\(390\) −5.37169 −0.272006
\(391\) 0.835347 0.0422453
\(392\) 0 0
\(393\) −13.1134 −0.661483
\(394\) −15.0361 −0.757509
\(395\) 2.51806 0.126697
\(396\) −1.68585 −0.0847170
\(397\) −10.2499 −0.514427 −0.257213 0.966355i \(-0.582804\pi\)
−0.257213 + 0.966355i \(0.582804\pi\)
\(398\) −15.3288 −0.768366
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −16.8866 −0.843277 −0.421639 0.906764i \(-0.638545\pi\)
−0.421639 + 0.906764i \(0.638545\pi\)
\(402\) 0.671153 0.0334741
\(403\) 4.58546 0.228418
\(404\) 10.7862 0.536635
\(405\) 1.10038 0.0546785
\(406\) 0 0
\(407\) 0.853635 0.0423131
\(408\) −0.335577 −0.0166135
\(409\) −18.2744 −0.903613 −0.451807 0.892116i \(-0.649220\pi\)
−0.451807 + 0.892116i \(0.649220\pi\)
\(410\) −6.22533 −0.307447
\(411\) −11.4637 −0.565460
\(412\) −7.66442 −0.377599
\(413\) 0 0
\(414\) −4.81079 −0.236437
\(415\) −1.70727 −0.0838065
\(416\) −4.68585 −0.229743
\(417\) −16.2583 −0.796173
\(418\) −6.51806 −0.318809
\(419\) 37.2860 1.82154 0.910770 0.412914i \(-0.135489\pi\)
0.910770 + 0.412914i \(0.135489\pi\)
\(420\) 0 0
\(421\) −2.78623 −0.135793 −0.0678963 0.997692i \(-0.521629\pi\)
−0.0678963 + 0.997692i \(0.521629\pi\)
\(422\) 2.04285 0.0994442
\(423\) −16.7862 −0.816174
\(424\) 5.43910 0.264146
\(425\) 0.292731 0.0141995
\(426\) 0.384694 0.0186385
\(427\) 0 0
\(428\) 9.56404 0.462295
\(429\) 5.37169 0.259348
\(430\) 10.3503 0.499134
\(431\) 38.0477 1.83269 0.916346 0.400387i \(-0.131124\pi\)
0.916346 + 0.400387i \(0.131124\pi\)
\(432\) 5.37169 0.258446
\(433\) 10.8108 0.519533 0.259767 0.965671i \(-0.416354\pi\)
0.259767 + 0.965671i \(0.416354\pi\)
\(434\) 0 0
\(435\) −1.64973 −0.0790985
\(436\) 3.14637 0.150684
\(437\) −18.6002 −0.889766
\(438\) −4.24989 −0.203067
\(439\) −30.9933 −1.47923 −0.739614 0.673031i \(-0.764992\pi\)
−0.739614 + 0.673031i \(0.764992\pi\)
\(440\) −1.00000 −0.0476731
\(441\) 0 0
\(442\) −1.37169 −0.0652448
\(443\) 25.3717 1.20545 0.602723 0.797951i \(-0.294083\pi\)
0.602723 + 0.797951i \(0.294083\pi\)
\(444\) −0.978577 −0.0464412
\(445\) 13.0790 0.620002
\(446\) −11.0790 −0.524604
\(447\) −13.2944 −0.628805
\(448\) 0 0
\(449\) −0.886615 −0.0418419 −0.0209210 0.999781i \(-0.506660\pi\)
−0.0209210 + 0.999781i \(0.506660\pi\)
\(450\) −1.68585 −0.0794716
\(451\) 6.22533 0.293139
\(452\) −2.58546 −0.121610
\(453\) −1.98531 −0.0932779
\(454\) 18.5426 0.870248
\(455\) 0 0
\(456\) 7.47208 0.349912
\(457\) 29.9143 1.39933 0.699666 0.714470i \(-0.253332\pi\)
0.699666 + 0.714470i \(0.253332\pi\)
\(458\) 8.01469 0.374502
\(459\) 1.57246 0.0733962
\(460\) −2.85363 −0.133051
\(461\) 2.33558 0.108779 0.0543893 0.998520i \(-0.482679\pi\)
0.0543893 + 0.998520i \(0.482679\pi\)
\(462\) 0 0
\(463\) 0.110250 0.00512374 0.00256187 0.999997i \(-0.499185\pi\)
0.00256187 + 0.999997i \(0.499185\pi\)
\(464\) −1.43910 −0.0668084
\(465\) −1.12181 −0.0520226
\(466\) 27.9572 1.29509
\(467\) −36.6760 −1.69716 −0.848581 0.529066i \(-0.822543\pi\)
−0.848581 + 0.529066i \(0.822543\pi\)
\(468\) 7.89962 0.365160
\(469\) 0 0
\(470\) −9.95715 −0.459289
\(471\) −14.0920 −0.649323
\(472\) 9.37169 0.431367
\(473\) −10.3503 −0.475906
\(474\) 2.88661 0.132587
\(475\) −6.51806 −0.299069
\(476\) 0 0
\(477\) −9.16948 −0.419842
\(478\) −28.8108 −1.31777
\(479\) 42.7434 1.95300 0.976498 0.215529i \(-0.0691474\pi\)
0.976498 + 0.215529i \(0.0691474\pi\)
\(480\) 1.14637 0.0523242
\(481\) −4.00000 −0.182384
\(482\) 8.51806 0.387987
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 9.10352 0.413370
\(486\) −14.8536 −0.673775
\(487\) −36.1396 −1.63764 −0.818822 0.574048i \(-0.805372\pi\)
−0.818822 + 0.574048i \(0.805372\pi\)
\(488\) 11.9572 0.541275
\(489\) −27.4145 −1.23973
\(490\) 0 0
\(491\) 2.54262 0.114747 0.0573733 0.998353i \(-0.481727\pi\)
0.0573733 + 0.998353i \(0.481727\pi\)
\(492\) −7.13650 −0.321738
\(493\) −0.421268 −0.0189730
\(494\) 30.5426 1.37418
\(495\) 1.68585 0.0757732
\(496\) −0.978577 −0.0439394
\(497\) 0 0
\(498\) −1.95715 −0.0877022
\(499\) 3.13650 0.140409 0.0702045 0.997533i \(-0.477635\pi\)
0.0702045 + 0.997533i \(0.477635\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −9.17092 −0.409727
\(502\) 23.1281 1.03226
\(503\) 28.5855 1.27456 0.637281 0.770631i \(-0.280059\pi\)
0.637281 + 0.770631i \(0.280059\pi\)
\(504\) 0 0
\(505\) −10.7862 −0.479981
\(506\) 2.85363 0.126860
\(507\) −10.2682 −0.456026
\(508\) −13.3717 −0.593273
\(509\) −2.20077 −0.0975473 −0.0487737 0.998810i \(-0.515531\pi\)
−0.0487737 + 0.998810i \(0.515531\pi\)
\(510\) 0.335577 0.0148596
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) −35.0130 −1.54586
\(514\) −14.8108 −0.653276
\(515\) 7.66442 0.337735
\(516\) 11.8652 0.522336
\(517\) 9.95715 0.437915
\(518\) 0 0
\(519\) −8.45065 −0.370943
\(520\) 4.68585 0.205488
\(521\) −37.1940 −1.62950 −0.814750 0.579812i \(-0.803126\pi\)
−0.814750 + 0.579812i \(0.803126\pi\)
\(522\) 2.42610 0.106187
\(523\) −30.4078 −1.32964 −0.664820 0.747003i \(-0.731492\pi\)
−0.664820 + 0.747003i \(0.731492\pi\)
\(524\) 11.4391 0.499719
\(525\) 0 0
\(526\) 18.7434 0.817250
\(527\) −0.286460 −0.0124784
\(528\) −1.14637 −0.0498892
\(529\) −14.8568 −0.645947
\(530\) −5.43910 −0.236259
\(531\) −15.7992 −0.685628
\(532\) 0 0
\(533\) −29.1709 −1.26353
\(534\) 14.9933 0.648822
\(535\) −9.56404 −0.413489
\(536\) −0.585462 −0.0252881
\(537\) −1.79296 −0.0773720
\(538\) −12.6858 −0.546926
\(539\) 0 0
\(540\) −5.37169 −0.231161
\(541\) 39.0607 1.67935 0.839675 0.543090i \(-0.182746\pi\)
0.839675 + 0.543090i \(0.182746\pi\)
\(542\) −1.12181 −0.0481857
\(543\) 16.9504 0.727412
\(544\) 0.292731 0.0125507
\(545\) −3.14637 −0.134775
\(546\) 0 0
\(547\) −0.585462 −0.0250325 −0.0125163 0.999922i \(-0.503984\pi\)
−0.0125163 + 0.999922i \(0.503984\pi\)
\(548\) 10.0000 0.427179
\(549\) −20.1579 −0.860319
\(550\) 1.00000 0.0426401
\(551\) 9.38011 0.399606
\(552\) −3.27131 −0.139236
\(553\) 0 0
\(554\) 30.1151 1.27947
\(555\) 0.978577 0.0415383
\(556\) 14.1825 0.601471
\(557\) 17.4637 0.739959 0.369979 0.929040i \(-0.379365\pi\)
0.369979 + 0.929040i \(0.379365\pi\)
\(558\) 1.64973 0.0698387
\(559\) 48.4998 2.05132
\(560\) 0 0
\(561\) −0.335577 −0.0141681
\(562\) −14.3356 −0.604710
\(563\) 18.1579 0.765265 0.382633 0.923901i \(-0.375018\pi\)
0.382633 + 0.923901i \(0.375018\pi\)
\(564\) −11.4145 −0.480639
\(565\) 2.58546 0.108771
\(566\) −3.21377 −0.135085
\(567\) 0 0
\(568\) −0.335577 −0.0140805
\(569\) −22.0000 −0.922288 −0.461144 0.887325i \(-0.652561\pi\)
−0.461144 + 0.887325i \(0.652561\pi\)
\(570\) −7.47208 −0.312971
\(571\) −9.03612 −0.378150 −0.189075 0.981963i \(-0.560549\pi\)
−0.189075 + 0.981963i \(0.560549\pi\)
\(572\) −4.68585 −0.195925
\(573\) 20.5855 0.859970
\(574\) 0 0
\(575\) 2.85363 0.119005
\(576\) −1.68585 −0.0702436
\(577\) −19.3963 −0.807476 −0.403738 0.914875i \(-0.632289\pi\)
−0.403738 + 0.914875i \(0.632289\pi\)
\(578\) −16.9143 −0.703542
\(579\) −21.3717 −0.888177
\(580\) 1.43910 0.0597552
\(581\) 0 0
\(582\) 10.4360 0.432584
\(583\) 5.43910 0.225264
\(584\) 3.70727 0.153408
\(585\) −7.89962 −0.326609
\(586\) 16.6858 0.689286
\(587\) −17.2614 −0.712456 −0.356228 0.934399i \(-0.615937\pi\)
−0.356228 + 0.934399i \(0.615937\pi\)
\(588\) 0 0
\(589\) 6.37842 0.262818
\(590\) −9.37169 −0.385826
\(591\) 17.2369 0.709031
\(592\) 0.853635 0.0350842
\(593\) 11.0361 0.453199 0.226599 0.973988i \(-0.427239\pi\)
0.226599 + 0.973988i \(0.427239\pi\)
\(594\) 5.37169 0.220403
\(595\) 0 0
\(596\) 11.5970 0.475032
\(597\) 17.5725 0.719193
\(598\) −13.3717 −0.546809
\(599\) −41.7367 −1.70531 −0.852657 0.522472i \(-0.825010\pi\)
−0.852657 + 0.522472i \(0.825010\pi\)
\(600\) −1.14637 −0.0468002
\(601\) −39.3106 −1.60351 −0.801756 0.597652i \(-0.796100\pi\)
−0.801756 + 0.597652i \(0.796100\pi\)
\(602\) 0 0
\(603\) 0.986999 0.0401937
\(604\) 1.73183 0.0704671
\(605\) −1.00000 −0.0406558
\(606\) −12.3650 −0.502292
\(607\) 6.35027 0.257749 0.128875 0.991661i \(-0.458863\pi\)
0.128875 + 0.991661i \(0.458863\pi\)
\(608\) −6.51806 −0.264342
\(609\) 0 0
\(610\) −11.9572 −0.484131
\(611\) −46.6577 −1.88757
\(612\) −0.493499 −0.0199485
\(613\) −42.4507 −1.71457 −0.857283 0.514846i \(-0.827849\pi\)
−0.857283 + 0.514846i \(0.827849\pi\)
\(614\) 6.29273 0.253954
\(615\) 7.13650 0.287771
\(616\) 0 0
\(617\) −0.677425 −0.0272721 −0.0136360 0.999907i \(-0.504341\pi\)
−0.0136360 + 0.999907i \(0.504341\pi\)
\(618\) 8.78623 0.353434
\(619\) 18.5426 0.745291 0.372645 0.927974i \(-0.378451\pi\)
0.372645 + 0.927974i \(0.378451\pi\)
\(620\) 0.978577 0.0393006
\(621\) 15.3288 0.615125
\(622\) 20.3931 0.817689
\(623\) 0 0
\(624\) 5.37169 0.215040
\(625\) 1.00000 0.0400000
\(626\) −12.0674 −0.482310
\(627\) 7.47208 0.298406
\(628\) 12.2927 0.490533
\(629\) 0.249885 0.00996358
\(630\) 0 0
\(631\) 21.3717 0.850794 0.425397 0.905007i \(-0.360135\pi\)
0.425397 + 0.905007i \(0.360135\pi\)
\(632\) −2.51806 −0.100163
\(633\) −2.34185 −0.0930801
\(634\) 28.7679 1.14252
\(635\) 13.3717 0.530639
\(636\) −6.23519 −0.247241
\(637\) 0 0
\(638\) −1.43910 −0.0569744
\(639\) 0.565731 0.0223800
\(640\) −1.00000 −0.0395285
\(641\) 23.8715 0.942866 0.471433 0.881902i \(-0.343737\pi\)
0.471433 + 0.881902i \(0.343737\pi\)
\(642\) −10.9639 −0.432710
\(643\) −0.311018 −0.0122654 −0.00613268 0.999981i \(-0.501952\pi\)
−0.00613268 + 0.999981i \(0.501952\pi\)
\(644\) 0 0
\(645\) −11.8652 −0.467191
\(646\) −1.90804 −0.0750707
\(647\) 9.62158 0.378263 0.189132 0.981952i \(-0.439433\pi\)
0.189132 + 0.981952i \(0.439433\pi\)
\(648\) −1.10038 −0.0432272
\(649\) 9.37169 0.367871
\(650\) −4.68585 −0.183794
\(651\) 0 0
\(652\) 23.9143 0.936557
\(653\) −27.0607 −1.05897 −0.529483 0.848321i \(-0.677614\pi\)
−0.529483 + 0.848321i \(0.677614\pi\)
\(654\) −3.60688 −0.141040
\(655\) −11.4391 −0.446962
\(656\) 6.22533 0.243058
\(657\) −6.24989 −0.243831
\(658\) 0 0
\(659\) −26.4935 −1.03204 −0.516020 0.856576i \(-0.672587\pi\)
−0.516020 + 0.856576i \(0.672587\pi\)
\(660\) 1.14637 0.0446222
\(661\) −35.0852 −1.36466 −0.682329 0.731046i \(-0.739033\pi\)
−0.682329 + 0.731046i \(0.739033\pi\)
\(662\) −3.80765 −0.147989
\(663\) 1.57246 0.0610693
\(664\) 1.70727 0.0662549
\(665\) 0 0
\(666\) −1.43910 −0.0557639
\(667\) −4.10666 −0.159010
\(668\) 8.00000 0.309529
\(669\) 12.7005 0.491031
\(670\) 0.585462 0.0226184
\(671\) 11.9572 0.461601
\(672\) 0 0
\(673\) 32.6577 1.25886 0.629431 0.777057i \(-0.283288\pi\)
0.629431 + 0.777057i \(0.283288\pi\)
\(674\) 15.3717 0.592095
\(675\) 5.37169 0.206757
\(676\) 8.95715 0.344506
\(677\) −9.12808 −0.350821 −0.175410 0.984495i \(-0.556125\pi\)
−0.175410 + 0.984495i \(0.556125\pi\)
\(678\) 2.96388 0.113827
\(679\) 0 0
\(680\) −0.292731 −0.0112257
\(681\) −21.2566 −0.814555
\(682\) −0.978577 −0.0374717
\(683\) 10.4935 0.401523 0.200761 0.979640i \(-0.435658\pi\)
0.200761 + 0.979640i \(0.435658\pi\)
\(684\) 10.9884 0.420154
\(685\) −10.0000 −0.382080
\(686\) 0 0
\(687\) −9.18777 −0.350535
\(688\) −10.3503 −0.394600
\(689\) −25.4868 −0.970969
\(690\) 3.27131 0.124537
\(691\) −1.70727 −0.0649476 −0.0324738 0.999473i \(-0.510339\pi\)
−0.0324738 + 0.999473i \(0.510339\pi\)
\(692\) 7.37169 0.280230
\(693\) 0 0
\(694\) 3.02142 0.114692
\(695\) −14.1825 −0.537972
\(696\) 1.64973 0.0625329
\(697\) 1.82235 0.0690263
\(698\) −11.0361 −0.417723
\(699\) −32.0491 −1.21221
\(700\) 0 0
\(701\) −37.3534 −1.41082 −0.705409 0.708800i \(-0.749237\pi\)
−0.705409 + 0.708800i \(0.749237\pi\)
\(702\) −25.1709 −0.950015
\(703\) −5.56404 −0.209852
\(704\) 1.00000 0.0376889
\(705\) 11.4145 0.429896
\(706\) −23.9817 −0.902564
\(707\) 0 0
\(708\) −10.7434 −0.403761
\(709\) −8.20704 −0.308222 −0.154111 0.988054i \(-0.549251\pi\)
−0.154111 + 0.988054i \(0.549251\pi\)
\(710\) 0.335577 0.0125940
\(711\) 4.24506 0.159202
\(712\) −13.0790 −0.490155
\(713\) −2.79250 −0.104580
\(714\) 0 0
\(715\) 4.68585 0.175241
\(716\) 1.56404 0.0584509
\(717\) 33.0277 1.23344
\(718\) −10.5181 −0.392530
\(719\) 8.00000 0.298350 0.149175 0.988811i \(-0.452338\pi\)
0.149175 + 0.988811i \(0.452338\pi\)
\(720\) 1.68585 0.0628278
\(721\) 0 0
\(722\) 23.4851 0.874024
\(723\) −9.76481 −0.363157
\(724\) −14.7862 −0.549526
\(725\) −1.43910 −0.0534467
\(726\) −1.14637 −0.0425456
\(727\) 46.9442 1.74106 0.870531 0.492113i \(-0.163775\pi\)
0.870531 + 0.492113i \(0.163775\pi\)
\(728\) 0 0
\(729\) 20.3288 0.752920
\(730\) −3.70727 −0.137212
\(731\) −3.02984 −0.112063
\(732\) −13.7073 −0.506635
\(733\) 42.0000 1.55131 0.775653 0.631160i \(-0.217421\pi\)
0.775653 + 0.631160i \(0.217421\pi\)
\(734\) 35.5787 1.31323
\(735\) 0 0
\(736\) 2.85363 0.105186
\(737\) −0.585462 −0.0215658
\(738\) −10.4949 −0.386324
\(739\) −0.871922 −0.0320742 −0.0160371 0.999871i \(-0.505105\pi\)
−0.0160371 + 0.999871i \(0.505105\pi\)
\(740\) −0.853635 −0.0313802
\(741\) −35.0130 −1.28623
\(742\) 0 0
\(743\) −24.6712 −0.905097 −0.452548 0.891740i \(-0.649485\pi\)
−0.452548 + 0.891740i \(0.649485\pi\)
\(744\) 1.12181 0.0411274
\(745\) −11.5970 −0.424882
\(746\) 7.50650 0.274833
\(747\) −2.87819 −0.105308
\(748\) 0.292731 0.0107033
\(749\) 0 0
\(750\) 1.14637 0.0418593
\(751\) −3.75011 −0.136844 −0.0684218 0.997656i \(-0.521796\pi\)
−0.0684218 + 0.997656i \(0.521796\pi\)
\(752\) 9.95715 0.363100
\(753\) −26.5132 −0.966196
\(754\) 6.74338 0.245580
\(755\) −1.73183 −0.0630277
\(756\) 0 0
\(757\) 15.7318 0.571783 0.285891 0.958262i \(-0.407710\pi\)
0.285891 + 0.958262i \(0.407710\pi\)
\(758\) 33.4868 1.21629
\(759\) −3.27131 −0.118741
\(760\) 6.51806 0.236435
\(761\) −15.2614 −0.553227 −0.276613 0.960981i \(-0.589212\pi\)
−0.276613 + 0.960981i \(0.589212\pi\)
\(762\) 15.3288 0.555306
\(763\) 0 0
\(764\) −17.9572 −0.649667
\(765\) 0.493499 0.0178425
\(766\) −15.6644 −0.565979
\(767\) −43.9143 −1.58565
\(768\) −1.14637 −0.0413659
\(769\) 9.63986 0.347622 0.173811 0.984779i \(-0.444392\pi\)
0.173811 + 0.984779i \(0.444392\pi\)
\(770\) 0 0
\(771\) 16.9786 0.611469
\(772\) 18.6430 0.670976
\(773\) −12.8291 −0.461430 −0.230715 0.973021i \(-0.574106\pi\)
−0.230715 + 0.973021i \(0.574106\pi\)
\(774\) 17.4490 0.627190
\(775\) −0.978577 −0.0351515
\(776\) −9.10352 −0.326797
\(777\) 0 0
\(778\) −19.6216 −0.703468
\(779\) −40.5770 −1.45382
\(780\) −5.37169 −0.192337
\(781\) −0.335577 −0.0120079
\(782\) 0.835347 0.0298720
\(783\) −7.73038 −0.276261
\(784\) 0 0
\(785\) −12.2927 −0.438746
\(786\) −13.1134 −0.467739
\(787\) −10.8291 −0.386015 −0.193007 0.981197i \(-0.561824\pi\)
−0.193007 + 0.981197i \(0.561824\pi\)
\(788\) −15.0361 −0.535639
\(789\) −21.4868 −0.764949
\(790\) 2.51806 0.0895885
\(791\) 0 0
\(792\) −1.68585 −0.0599039
\(793\) −56.0294 −1.98966
\(794\) −10.2499 −0.363755
\(795\) 6.23519 0.221139
\(796\) −15.3288 −0.543317
\(797\) 21.0790 0.746655 0.373328 0.927700i \(-0.378217\pi\)
0.373328 + 0.927700i \(0.378217\pi\)
\(798\) 0 0
\(799\) 2.91477 0.103117
\(800\) 1.00000 0.0353553
\(801\) 22.0491 0.779067
\(802\) −16.8866 −0.596287
\(803\) 3.70727 0.130827
\(804\) 0.671153 0.0236698
\(805\) 0 0
\(806\) 4.58546 0.161516
\(807\) 14.5426 0.511924
\(808\) 10.7862 0.379458
\(809\) 3.62158 0.127328 0.0636639 0.997971i \(-0.479721\pi\)
0.0636639 + 0.997971i \(0.479721\pi\)
\(810\) 1.10038 0.0386636
\(811\) −34.7188 −1.21914 −0.609571 0.792731i \(-0.708658\pi\)
−0.609571 + 0.792731i \(0.708658\pi\)
\(812\) 0 0
\(813\) 1.28600 0.0451020
\(814\) 0.853635 0.0299199
\(815\) −23.9143 −0.837682
\(816\) −0.335577 −0.0117475
\(817\) 67.4637 2.36025
\(818\) −18.2744 −0.638951
\(819\) 0 0
\(820\) −6.22533 −0.217398
\(821\) −37.7549 −1.31766 −0.658828 0.752293i \(-0.728948\pi\)
−0.658828 + 0.752293i \(0.728948\pi\)
\(822\) −11.4637 −0.399841
\(823\) 11.1892 0.390031 0.195016 0.980800i \(-0.437524\pi\)
0.195016 + 0.980800i \(0.437524\pi\)
\(824\) −7.66442 −0.267003
\(825\) −1.14637 −0.0399113
\(826\) 0 0
\(827\) −7.91431 −0.275207 −0.137604 0.990487i \(-0.543940\pi\)
−0.137604 + 0.990487i \(0.543940\pi\)
\(828\) −4.81079 −0.167186
\(829\) 44.7152 1.55302 0.776512 0.630102i \(-0.216987\pi\)
0.776512 + 0.630102i \(0.216987\pi\)
\(830\) −1.70727 −0.0592602
\(831\) −34.5229 −1.19759
\(832\) −4.68585 −0.162452
\(833\) 0 0
\(834\) −16.2583 −0.562979
\(835\) −8.00000 −0.276851
\(836\) −6.51806 −0.225432
\(837\) −5.25662 −0.181695
\(838\) 37.2860 1.28802
\(839\) −21.6791 −0.748446 −0.374223 0.927339i \(-0.622091\pi\)
−0.374223 + 0.927339i \(0.622091\pi\)
\(840\) 0 0
\(841\) −26.9290 −0.928586
\(842\) −2.78623 −0.0960198
\(843\) 16.4338 0.566010
\(844\) 2.04285 0.0703176
\(845\) −8.95715 −0.308135
\(846\) −16.7862 −0.577122
\(847\) 0 0
\(848\) 5.43910 0.186779
\(849\) 3.68415 0.126440
\(850\) 0.292731 0.0100406
\(851\) 2.43596 0.0835037
\(852\) 0.384694 0.0131794
\(853\) −9.41454 −0.322348 −0.161174 0.986926i \(-0.551528\pi\)
−0.161174 + 0.986926i \(0.551528\pi\)
\(854\) 0 0
\(855\) −10.9884 −0.375797
\(856\) 9.56404 0.326892
\(857\) 25.8652 0.883538 0.441769 0.897129i \(-0.354351\pi\)
0.441769 + 0.897129i \(0.354351\pi\)
\(858\) 5.37169 0.183387
\(859\) 29.2860 0.999225 0.499613 0.866249i \(-0.333476\pi\)
0.499613 + 0.866249i \(0.333476\pi\)
\(860\) 10.3503 0.352941
\(861\) 0 0
\(862\) 38.0477 1.29591
\(863\) −20.3110 −0.691395 −0.345698 0.938346i \(-0.612358\pi\)
−0.345698 + 0.938346i \(0.612358\pi\)
\(864\) 5.37169 0.182749
\(865\) −7.37169 −0.250645
\(866\) 10.8108 0.367366
\(867\) 19.3900 0.658518
\(868\) 0 0
\(869\) −2.51806 −0.0854193
\(870\) −1.64973 −0.0559311
\(871\) 2.74338 0.0929560
\(872\) 3.14637 0.106549
\(873\) 15.3471 0.519422
\(874\) −18.6002 −0.629160
\(875\) 0 0
\(876\) −4.24989 −0.143590
\(877\) −38.3650 −1.29549 −0.647746 0.761856i \(-0.724288\pi\)
−0.647746 + 0.761856i \(0.724288\pi\)
\(878\) −30.9933 −1.04597
\(879\) −19.1281 −0.645174
\(880\) −1.00000 −0.0337100
\(881\) −40.5426 −1.36592 −0.682958 0.730458i \(-0.739307\pi\)
−0.682958 + 0.730458i \(0.739307\pi\)
\(882\) 0 0
\(883\) 28.0491 0.943928 0.471964 0.881618i \(-0.343545\pi\)
0.471964 + 0.881618i \(0.343545\pi\)
\(884\) −1.37169 −0.0461350
\(885\) 10.7434 0.361135
\(886\) 25.3717 0.852379
\(887\) 6.65769 0.223543 0.111772 0.993734i \(-0.464347\pi\)
0.111772 + 0.993734i \(0.464347\pi\)
\(888\) −0.978577 −0.0328389
\(889\) 0 0
\(890\) 13.0790 0.438408
\(891\) −1.10038 −0.0368643
\(892\) −11.0790 −0.370951
\(893\) −64.9013 −2.17184
\(894\) −13.2944 −0.444632
\(895\) −1.56404 −0.0522801
\(896\) 0 0
\(897\) 15.3288 0.511815
\(898\) −0.886615 −0.0295867
\(899\) 1.40827 0.0469683
\(900\) −1.68585 −0.0561949
\(901\) 1.59219 0.0530436
\(902\) 6.22533 0.207281
\(903\) 0 0
\(904\) −2.58546 −0.0859912
\(905\) 14.7862 0.491511
\(906\) −1.98531 −0.0659574
\(907\) 39.5787 1.31419 0.657095 0.753808i \(-0.271785\pi\)
0.657095 + 0.753808i \(0.271785\pi\)
\(908\) 18.5426 0.615358
\(909\) −18.1839 −0.603123
\(910\) 0 0
\(911\) −24.3356 −0.806274 −0.403137 0.915140i \(-0.632080\pi\)
−0.403137 + 0.915140i \(0.632080\pi\)
\(912\) 7.47208 0.247425
\(913\) 1.70727 0.0565024
\(914\) 29.9143 0.989477
\(915\) 13.7073 0.453148
\(916\) 8.01469 0.264813
\(917\) 0 0
\(918\) 1.57246 0.0518989
\(919\) 50.6331 1.67023 0.835117 0.550073i \(-0.185400\pi\)
0.835117 + 0.550073i \(0.185400\pi\)
\(920\) −2.85363 −0.0940815
\(921\) −7.21377 −0.237702
\(922\) 2.33558 0.0769181
\(923\) 1.57246 0.0517582
\(924\) 0 0
\(925\) 0.853635 0.0280673
\(926\) 0.110250 0.00362303
\(927\) 12.9210 0.424383
\(928\) −1.43910 −0.0472407
\(929\) −47.1512 −1.54698 −0.773490 0.633808i \(-0.781491\pi\)
−0.773490 + 0.633808i \(0.781491\pi\)
\(930\) −1.12181 −0.0367855
\(931\) 0 0
\(932\) 27.9572 0.915767
\(933\) −23.3780 −0.765360
\(934\) −36.6760 −1.20007
\(935\) −0.292731 −0.00957333
\(936\) 7.89962 0.258207
\(937\) 42.0294 1.37304 0.686520 0.727111i \(-0.259137\pi\)
0.686520 + 0.727111i \(0.259137\pi\)
\(938\) 0 0
\(939\) 13.8337 0.451444
\(940\) −9.95715 −0.324767
\(941\) 36.7434 1.19780 0.598900 0.800824i \(-0.295605\pi\)
0.598900 + 0.800824i \(0.295605\pi\)
\(942\) −14.0920 −0.459141
\(943\) 17.7648 0.578502
\(944\) 9.37169 0.305023
\(945\) 0 0
\(946\) −10.3503 −0.336516
\(947\) 18.8782 0.613459 0.306729 0.951797i \(-0.400765\pi\)
0.306729 + 0.951797i \(0.400765\pi\)
\(948\) 2.88661 0.0937529
\(949\) −17.3717 −0.563909
\(950\) −6.51806 −0.211474
\(951\) −32.9786 −1.06940
\(952\) 0 0
\(953\) −43.2285 −1.40031 −0.700154 0.713992i \(-0.746885\pi\)
−0.700154 + 0.713992i \(0.746885\pi\)
\(954\) −9.16948 −0.296873
\(955\) 17.9572 0.581080
\(956\) −28.8108 −0.931808
\(957\) 1.64973 0.0533282
\(958\) 42.7434 1.38098
\(959\) 0 0
\(960\) 1.14637 0.0369988
\(961\) −30.0424 −0.969109
\(962\) −4.00000 −0.128965
\(963\) −16.1235 −0.519572
\(964\) 8.51806 0.274348
\(965\) −18.6430 −0.600139
\(966\) 0 0
\(967\) 48.7299 1.56705 0.783524 0.621361i \(-0.213420\pi\)
0.783524 + 0.621361i \(0.213420\pi\)
\(968\) 1.00000 0.0321412
\(969\) 2.18731 0.0702665
\(970\) 9.10352 0.292296
\(971\) 25.9865 0.833948 0.416974 0.908918i \(-0.363091\pi\)
0.416974 + 0.908918i \(0.363091\pi\)
\(972\) −14.8536 −0.476431
\(973\) 0 0
\(974\) −36.1396 −1.15799
\(975\) 5.37169 0.172032
\(976\) 11.9572 0.382739
\(977\) 2.67115 0.0854578 0.0427289 0.999087i \(-0.486395\pi\)
0.0427289 + 0.999087i \(0.486395\pi\)
\(978\) −27.4145 −0.876620
\(979\) −13.0790 −0.418005
\(980\) 0 0
\(981\) −5.30429 −0.169353
\(982\) 2.54262 0.0811381
\(983\) 8.33558 0.265864 0.132932 0.991125i \(-0.457561\pi\)
0.132932 + 0.991125i \(0.457561\pi\)
\(984\) −7.13650 −0.227503
\(985\) 15.0361 0.479090
\(986\) −0.421268 −0.0134159
\(987\) 0 0
\(988\) 30.5426 0.971690
\(989\) −29.5359 −0.939187
\(990\) 1.68585 0.0535797
\(991\) 46.2730 1.46991 0.734955 0.678116i \(-0.237203\pi\)
0.734955 + 0.678116i \(0.237203\pi\)
\(992\) −0.978577 −0.0310699
\(993\) 4.36496 0.138518
\(994\) 0 0
\(995\) 15.3288 0.485957
\(996\) −1.95715 −0.0620148
\(997\) −35.8715 −1.13606 −0.568030 0.823008i \(-0.692294\pi\)
−0.568030 + 0.823008i \(0.692294\pi\)
\(998\) 3.13650 0.0992842
\(999\) 4.58546 0.145078
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 5390.2.a.ca.1.2 3
7.6 odd 2 770.2.a.m.1.2 3
21.20 even 2 6930.2.a.ce.1.3 3
28.27 even 2 6160.2.a.bf.1.2 3
35.13 even 4 3850.2.c.ba.1849.2 6
35.27 even 4 3850.2.c.ba.1849.5 6
35.34 odd 2 3850.2.a.bt.1.2 3
77.76 even 2 8470.2.a.ci.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.a.m.1.2 3 7.6 odd 2
3850.2.a.bt.1.2 3 35.34 odd 2
3850.2.c.ba.1849.2 6 35.13 even 4
3850.2.c.ba.1849.5 6 35.27 even 4
5390.2.a.ca.1.2 3 1.1 even 1 trivial
6160.2.a.bf.1.2 3 28.27 even 2
6930.2.a.ce.1.3 3 21.20 even 2
8470.2.a.ci.1.2 3 77.76 even 2