Properties

Label 483.2.a.d.1.2
Level $483$
Weight $2$
Character 483.1
Self dual yes
Analytic conductor $3.857$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [483,2,Mod(1,483)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(483, 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("483.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 483 = 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 483.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: \(3.85677441763\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 483.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.618034 q^{2} -1.00000 q^{3} -1.61803 q^{4} +1.61803 q^{5} -0.618034 q^{6} -1.00000 q^{7} -2.23607 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+0.618034 q^{2} -1.00000 q^{3} -1.61803 q^{4} +1.61803 q^{5} -0.618034 q^{6} -1.00000 q^{7} -2.23607 q^{8} +1.00000 q^{9} +1.00000 q^{10} -2.23607 q^{11} +1.61803 q^{12} -4.61803 q^{13} -0.618034 q^{14} -1.61803 q^{15} +1.85410 q^{16} -6.70820 q^{17} +0.618034 q^{18} +5.47214 q^{19} -2.61803 q^{20} +1.00000 q^{21} -1.38197 q^{22} -1.00000 q^{23} +2.23607 q^{24} -2.38197 q^{25} -2.85410 q^{26} -1.00000 q^{27} +1.61803 q^{28} -3.76393 q^{29} -1.00000 q^{30} -6.70820 q^{31} +5.61803 q^{32} +2.23607 q^{33} -4.14590 q^{34} -1.61803 q^{35} -1.61803 q^{36} -11.0000 q^{37} +3.38197 q^{38} +4.61803 q^{39} -3.61803 q^{40} +7.47214 q^{41} +0.618034 q^{42} +0.618034 q^{43} +3.61803 q^{44} +1.61803 q^{45} -0.618034 q^{46} -2.76393 q^{47} -1.85410 q^{48} +1.00000 q^{49} -1.47214 q^{50} +6.70820 q^{51} +7.47214 q^{52} -1.90983 q^{53} -0.618034 q^{54} -3.61803 q^{55} +2.23607 q^{56} -5.47214 q^{57} -2.32624 q^{58} +11.6180 q^{59} +2.61803 q^{60} +1.85410 q^{61} -4.14590 q^{62} -1.00000 q^{63} -0.236068 q^{64} -7.47214 q^{65} +1.38197 q^{66} +6.09017 q^{67} +10.8541 q^{68} +1.00000 q^{69} -1.00000 q^{70} +6.61803 q^{71} -2.23607 q^{72} +0.708204 q^{73} -6.79837 q^{74} +2.38197 q^{75} -8.85410 q^{76} +2.23607 q^{77} +2.85410 q^{78} -0.527864 q^{79} +3.00000 q^{80} +1.00000 q^{81} +4.61803 q^{82} +13.1803 q^{83} -1.61803 q^{84} -10.8541 q^{85} +0.381966 q^{86} +3.76393 q^{87} +5.00000 q^{88} +9.38197 q^{89} +1.00000 q^{90} +4.61803 q^{91} +1.61803 q^{92} +6.70820 q^{93} -1.70820 q^{94} +8.85410 q^{95} -5.61803 q^{96} -16.4164 q^{97} +0.618034 q^{98} -2.23607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - 2 q^{3} - q^{4} + q^{5} + q^{6} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} - 2 q^{3} - q^{4} + q^{5} + q^{6} - 2 q^{7} + 2 q^{9} + 2 q^{10} + q^{12} - 7 q^{13} + q^{14} - q^{15} - 3 q^{16} - q^{18} + 2 q^{19} - 3 q^{20} + 2 q^{21} - 5 q^{22} - 2 q^{23} - 7 q^{25} + q^{26} - 2 q^{27} + q^{28} - 12 q^{29} - 2 q^{30} + 9 q^{32} - 15 q^{34} - q^{35} - q^{36} - 22 q^{37} + 9 q^{38} + 7 q^{39} - 5 q^{40} + 6 q^{41} - q^{42} - q^{43} + 5 q^{44} + q^{45} + q^{46} - 10 q^{47} + 3 q^{48} + 2 q^{49} + 6 q^{50} + 6 q^{52} - 15 q^{53} + q^{54} - 5 q^{55} - 2 q^{57} + 11 q^{58} + 21 q^{59} + 3 q^{60} - 3 q^{61} - 15 q^{62} - 2 q^{63} + 4 q^{64} - 6 q^{65} + 5 q^{66} + q^{67} + 15 q^{68} + 2 q^{69} - 2 q^{70} + 11 q^{71} - 12 q^{73} + 11 q^{74} + 7 q^{75} - 11 q^{76} - q^{78} - 10 q^{79} + 6 q^{80} + 2 q^{81} + 7 q^{82} + 4 q^{83} - q^{84} - 15 q^{85} + 3 q^{86} + 12 q^{87} + 10 q^{88} + 21 q^{89} + 2 q^{90} + 7 q^{91} + q^{92} + 10 q^{94} + 11 q^{95} - 9 q^{96} - 6 q^{97} - q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.618034 0.437016 0.218508 0.975835i \(-0.429881\pi\)
0.218508 + 0.975835i \(0.429881\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.61803 −0.809017
\(5\) 1.61803 0.723607 0.361803 0.932254i \(-0.382161\pi\)
0.361803 + 0.932254i \(0.382161\pi\)
\(6\) −0.618034 −0.252311
\(7\) −1.00000 −0.377964
\(8\) −2.23607 −0.790569
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) −2.23607 −0.674200 −0.337100 0.941469i \(-0.609446\pi\)
−0.337100 + 0.941469i \(0.609446\pi\)
\(12\) 1.61803 0.467086
\(13\) −4.61803 −1.28081 −0.640406 0.768036i \(-0.721234\pi\)
−0.640406 + 0.768036i \(0.721234\pi\)
\(14\) −0.618034 −0.165177
\(15\) −1.61803 −0.417775
\(16\) 1.85410 0.463525
\(17\) −6.70820 −1.62698 −0.813489 0.581580i \(-0.802435\pi\)
−0.813489 + 0.581580i \(0.802435\pi\)
\(18\) 0.618034 0.145672
\(19\) 5.47214 1.25539 0.627697 0.778458i \(-0.283998\pi\)
0.627697 + 0.778458i \(0.283998\pi\)
\(20\) −2.61803 −0.585410
\(21\) 1.00000 0.218218
\(22\) −1.38197 −0.294636
\(23\) −1.00000 −0.208514
\(24\) 2.23607 0.456435
\(25\) −2.38197 −0.476393
\(26\) −2.85410 −0.559735
\(27\) −1.00000 −0.192450
\(28\) 1.61803 0.305780
\(29\) −3.76393 −0.698945 −0.349472 0.936947i \(-0.613639\pi\)
−0.349472 + 0.936947i \(0.613639\pi\)
\(30\) −1.00000 −0.182574
\(31\) −6.70820 −1.20483 −0.602414 0.798183i \(-0.705795\pi\)
−0.602414 + 0.798183i \(0.705795\pi\)
\(32\) 5.61803 0.993137
\(33\) 2.23607 0.389249
\(34\) −4.14590 −0.711016
\(35\) −1.61803 −0.273498
\(36\) −1.61803 −0.269672
\(37\) −11.0000 −1.80839 −0.904194 0.427121i \(-0.859528\pi\)
−0.904194 + 0.427121i \(0.859528\pi\)
\(38\) 3.38197 0.548627
\(39\) 4.61803 0.739477
\(40\) −3.61803 −0.572061
\(41\) 7.47214 1.16695 0.583476 0.812131i \(-0.301692\pi\)
0.583476 + 0.812131i \(0.301692\pi\)
\(42\) 0.618034 0.0953647
\(43\) 0.618034 0.0942493 0.0471246 0.998889i \(-0.484994\pi\)
0.0471246 + 0.998889i \(0.484994\pi\)
\(44\) 3.61803 0.545439
\(45\) 1.61803 0.241202
\(46\) −0.618034 −0.0911241
\(47\) −2.76393 −0.403161 −0.201580 0.979472i \(-0.564608\pi\)
−0.201580 + 0.979472i \(0.564608\pi\)
\(48\) −1.85410 −0.267617
\(49\) 1.00000 0.142857
\(50\) −1.47214 −0.208191
\(51\) 6.70820 0.939336
\(52\) 7.47214 1.03620
\(53\) −1.90983 −0.262335 −0.131168 0.991360i \(-0.541873\pi\)
−0.131168 + 0.991360i \(0.541873\pi\)
\(54\) −0.618034 −0.0841038
\(55\) −3.61803 −0.487856
\(56\) 2.23607 0.298807
\(57\) −5.47214 −0.724802
\(58\) −2.32624 −0.305450
\(59\) 11.6180 1.51254 0.756270 0.654260i \(-0.227020\pi\)
0.756270 + 0.654260i \(0.227020\pi\)
\(60\) 2.61803 0.337987
\(61\) 1.85410 0.237393 0.118697 0.992931i \(-0.462128\pi\)
0.118697 + 0.992931i \(0.462128\pi\)
\(62\) −4.14590 −0.526530
\(63\) −1.00000 −0.125988
\(64\) −0.236068 −0.0295085
\(65\) −7.47214 −0.926804
\(66\) 1.38197 0.170108
\(67\) 6.09017 0.744033 0.372016 0.928226i \(-0.378667\pi\)
0.372016 + 0.928226i \(0.378667\pi\)
\(68\) 10.8541 1.31625
\(69\) 1.00000 0.120386
\(70\) −1.00000 −0.119523
\(71\) 6.61803 0.785416 0.392708 0.919663i \(-0.371538\pi\)
0.392708 + 0.919663i \(0.371538\pi\)
\(72\) −2.23607 −0.263523
\(73\) 0.708204 0.0828890 0.0414445 0.999141i \(-0.486804\pi\)
0.0414445 + 0.999141i \(0.486804\pi\)
\(74\) −6.79837 −0.790295
\(75\) 2.38197 0.275046
\(76\) −8.85410 −1.01564
\(77\) 2.23607 0.254824
\(78\) 2.85410 0.323163
\(79\) −0.527864 −0.0593893 −0.0296947 0.999559i \(-0.509453\pi\)
−0.0296947 + 0.999559i \(0.509453\pi\)
\(80\) 3.00000 0.335410
\(81\) 1.00000 0.111111
\(82\) 4.61803 0.509977
\(83\) 13.1803 1.44673 0.723365 0.690466i \(-0.242594\pi\)
0.723365 + 0.690466i \(0.242594\pi\)
\(84\) −1.61803 −0.176542
\(85\) −10.8541 −1.17729
\(86\) 0.381966 0.0411885
\(87\) 3.76393 0.403536
\(88\) 5.00000 0.533002
\(89\) 9.38197 0.994486 0.497243 0.867611i \(-0.334346\pi\)
0.497243 + 0.867611i \(0.334346\pi\)
\(90\) 1.00000 0.105409
\(91\) 4.61803 0.484102
\(92\) 1.61803 0.168692
\(93\) 6.70820 0.695608
\(94\) −1.70820 −0.176188
\(95\) 8.85410 0.908412
\(96\) −5.61803 −0.573388
\(97\) −16.4164 −1.66683 −0.833417 0.552645i \(-0.813619\pi\)
−0.833417 + 0.552645i \(0.813619\pi\)
\(98\) 0.618034 0.0624309
\(99\) −2.23607 −0.224733
\(100\) 3.85410 0.385410
\(101\) 10.8541 1.08002 0.540012 0.841657i \(-0.318420\pi\)
0.540012 + 0.841657i \(0.318420\pi\)
\(102\) 4.14590 0.410505
\(103\) 19.4164 1.91316 0.956578 0.291477i \(-0.0941468\pi\)
0.956578 + 0.291477i \(0.0941468\pi\)
\(104\) 10.3262 1.01257
\(105\) 1.61803 0.157904
\(106\) −1.18034 −0.114645
\(107\) −12.3262 −1.19162 −0.595811 0.803125i \(-0.703169\pi\)
−0.595811 + 0.803125i \(0.703169\pi\)
\(108\) 1.61803 0.155695
\(109\) −14.2705 −1.36687 −0.683433 0.730013i \(-0.739514\pi\)
−0.683433 + 0.730013i \(0.739514\pi\)
\(110\) −2.23607 −0.213201
\(111\) 11.0000 1.04407
\(112\) −1.85410 −0.175196
\(113\) −12.0902 −1.13735 −0.568674 0.822563i \(-0.692543\pi\)
−0.568674 + 0.822563i \(0.692543\pi\)
\(114\) −3.38197 −0.316750
\(115\) −1.61803 −0.150882
\(116\) 6.09017 0.565458
\(117\) −4.61803 −0.426937
\(118\) 7.18034 0.661004
\(119\) 6.70820 0.614940
\(120\) 3.61803 0.330280
\(121\) −6.00000 −0.545455
\(122\) 1.14590 0.103745
\(123\) −7.47214 −0.673740
\(124\) 10.8541 0.974727
\(125\) −11.9443 −1.06833
\(126\) −0.618034 −0.0550588
\(127\) −11.2705 −1.00010 −0.500048 0.865998i \(-0.666684\pi\)
−0.500048 + 0.865998i \(0.666684\pi\)
\(128\) −11.3820 −1.00603
\(129\) −0.618034 −0.0544149
\(130\) −4.61803 −0.405028
\(131\) −7.18034 −0.627349 −0.313675 0.949531i \(-0.601560\pi\)
−0.313675 + 0.949531i \(0.601560\pi\)
\(132\) −3.61803 −0.314909
\(133\) −5.47214 −0.474494
\(134\) 3.76393 0.325154
\(135\) −1.61803 −0.139258
\(136\) 15.0000 1.28624
\(137\) 10.4164 0.889934 0.444967 0.895547i \(-0.353215\pi\)
0.444967 + 0.895547i \(0.353215\pi\)
\(138\) 0.618034 0.0526105
\(139\) −13.6180 −1.15507 −0.577533 0.816367i \(-0.695985\pi\)
−0.577533 + 0.816367i \(0.695985\pi\)
\(140\) 2.61803 0.221264
\(141\) 2.76393 0.232765
\(142\) 4.09017 0.343239
\(143\) 10.3262 0.863523
\(144\) 1.85410 0.154508
\(145\) −6.09017 −0.505761
\(146\) 0.437694 0.0362238
\(147\) −1.00000 −0.0824786
\(148\) 17.7984 1.46302
\(149\) −14.7639 −1.20951 −0.604754 0.796412i \(-0.706729\pi\)
−0.604754 + 0.796412i \(0.706729\pi\)
\(150\) 1.47214 0.120199
\(151\) −10.7639 −0.875956 −0.437978 0.898986i \(-0.644305\pi\)
−0.437978 + 0.898986i \(0.644305\pi\)
\(152\) −12.2361 −0.992476
\(153\) −6.70820 −0.542326
\(154\) 1.38197 0.111362
\(155\) −10.8541 −0.871822
\(156\) −7.47214 −0.598250
\(157\) −13.7082 −1.09403 −0.547017 0.837122i \(-0.684237\pi\)
−0.547017 + 0.837122i \(0.684237\pi\)
\(158\) −0.326238 −0.0259541
\(159\) 1.90983 0.151459
\(160\) 9.09017 0.718641
\(161\) 1.00000 0.0788110
\(162\) 0.618034 0.0485573
\(163\) 1.61803 0.126734 0.0633671 0.997990i \(-0.479816\pi\)
0.0633671 + 0.997990i \(0.479816\pi\)
\(164\) −12.0902 −0.944084
\(165\) 3.61803 0.281664
\(166\) 8.14590 0.632244
\(167\) 15.1803 1.17469 0.587345 0.809337i \(-0.300173\pi\)
0.587345 + 0.809337i \(0.300173\pi\)
\(168\) −2.23607 −0.172516
\(169\) 8.32624 0.640480
\(170\) −6.70820 −0.514496
\(171\) 5.47214 0.418465
\(172\) −1.00000 −0.0762493
\(173\) 5.47214 0.416039 0.208019 0.978125i \(-0.433298\pi\)
0.208019 + 0.978125i \(0.433298\pi\)
\(174\) 2.32624 0.176352
\(175\) 2.38197 0.180060
\(176\) −4.14590 −0.312509
\(177\) −11.6180 −0.873265
\(178\) 5.79837 0.434606
\(179\) −4.85410 −0.362813 −0.181406 0.983408i \(-0.558065\pi\)
−0.181406 + 0.983408i \(0.558065\pi\)
\(180\) −2.61803 −0.195137
\(181\) 15.9443 1.18513 0.592564 0.805523i \(-0.298116\pi\)
0.592564 + 0.805523i \(0.298116\pi\)
\(182\) 2.85410 0.211560
\(183\) −1.85410 −0.137059
\(184\) 2.23607 0.164845
\(185\) −17.7984 −1.30856
\(186\) 4.14590 0.303992
\(187\) 15.0000 1.09691
\(188\) 4.47214 0.326164
\(189\) 1.00000 0.0727393
\(190\) 5.47214 0.396990
\(191\) −6.18034 −0.447194 −0.223597 0.974682i \(-0.571780\pi\)
−0.223597 + 0.974682i \(0.571780\pi\)
\(192\) 0.236068 0.0170367
\(193\) 21.7082 1.56259 0.781295 0.624161i \(-0.214559\pi\)
0.781295 + 0.624161i \(0.214559\pi\)
\(194\) −10.1459 −0.728433
\(195\) 7.47214 0.535091
\(196\) −1.61803 −0.115574
\(197\) 12.5066 0.891057 0.445528 0.895268i \(-0.353016\pi\)
0.445528 + 0.895268i \(0.353016\pi\)
\(198\) −1.38197 −0.0982120
\(199\) −18.0902 −1.28238 −0.641189 0.767383i \(-0.721559\pi\)
−0.641189 + 0.767383i \(0.721559\pi\)
\(200\) 5.32624 0.376622
\(201\) −6.09017 −0.429567
\(202\) 6.70820 0.471988
\(203\) 3.76393 0.264176
\(204\) −10.8541 −0.759939
\(205\) 12.0902 0.844414
\(206\) 12.0000 0.836080
\(207\) −1.00000 −0.0695048
\(208\) −8.56231 −0.593689
\(209\) −12.2361 −0.846387
\(210\) 1.00000 0.0690066
\(211\) −16.4164 −1.13015 −0.565076 0.825039i \(-0.691153\pi\)
−0.565076 + 0.825039i \(0.691153\pi\)
\(212\) 3.09017 0.212234
\(213\) −6.61803 −0.453460
\(214\) −7.61803 −0.520758
\(215\) 1.00000 0.0681994
\(216\) 2.23607 0.152145
\(217\) 6.70820 0.455383
\(218\) −8.81966 −0.597343
\(219\) −0.708204 −0.0478560
\(220\) 5.85410 0.394683
\(221\) 30.9787 2.08385
\(222\) 6.79837 0.456277
\(223\) −11.1459 −0.746385 −0.373192 0.927754i \(-0.621737\pi\)
−0.373192 + 0.927754i \(0.621737\pi\)
\(224\) −5.61803 −0.375371
\(225\) −2.38197 −0.158798
\(226\) −7.47214 −0.497039
\(227\) 8.67376 0.575698 0.287849 0.957676i \(-0.407060\pi\)
0.287849 + 0.957676i \(0.407060\pi\)
\(228\) 8.85410 0.586377
\(229\) −0.673762 −0.0445235 −0.0222617 0.999752i \(-0.507087\pi\)
−0.0222617 + 0.999752i \(0.507087\pi\)
\(230\) −1.00000 −0.0659380
\(231\) −2.23607 −0.147122
\(232\) 8.41641 0.552564
\(233\) 0.0901699 0.00590723 0.00295361 0.999996i \(-0.499060\pi\)
0.00295361 + 0.999996i \(0.499060\pi\)
\(234\) −2.85410 −0.186578
\(235\) −4.47214 −0.291730
\(236\) −18.7984 −1.22367
\(237\) 0.527864 0.0342885
\(238\) 4.14590 0.268739
\(239\) 19.7984 1.28065 0.640325 0.768104i \(-0.278800\pi\)
0.640325 + 0.768104i \(0.278800\pi\)
\(240\) −3.00000 −0.193649
\(241\) −11.0000 −0.708572 −0.354286 0.935137i \(-0.615276\pi\)
−0.354286 + 0.935137i \(0.615276\pi\)
\(242\) −3.70820 −0.238372
\(243\) −1.00000 −0.0641500
\(244\) −3.00000 −0.192055
\(245\) 1.61803 0.103372
\(246\) −4.61803 −0.294435
\(247\) −25.2705 −1.60792
\(248\) 15.0000 0.952501
\(249\) −13.1803 −0.835270
\(250\) −7.38197 −0.466877
\(251\) −17.1246 −1.08090 −0.540448 0.841377i \(-0.681745\pi\)
−0.540448 + 0.841377i \(0.681745\pi\)
\(252\) 1.61803 0.101927
\(253\) 2.23607 0.140580
\(254\) −6.96556 −0.437058
\(255\) 10.8541 0.679710
\(256\) −6.56231 −0.410144
\(257\) 1.23607 0.0771038 0.0385519 0.999257i \(-0.487726\pi\)
0.0385519 + 0.999257i \(0.487726\pi\)
\(258\) −0.381966 −0.0237802
\(259\) 11.0000 0.683507
\(260\) 12.0902 0.749801
\(261\) −3.76393 −0.232982
\(262\) −4.43769 −0.274162
\(263\) −17.9443 −1.10649 −0.553246 0.833018i \(-0.686611\pi\)
−0.553246 + 0.833018i \(0.686611\pi\)
\(264\) −5.00000 −0.307729
\(265\) −3.09017 −0.189828
\(266\) −3.38197 −0.207362
\(267\) −9.38197 −0.574167
\(268\) −9.85410 −0.601935
\(269\) −20.9098 −1.27489 −0.637447 0.770494i \(-0.720010\pi\)
−0.637447 + 0.770494i \(0.720010\pi\)
\(270\) −1.00000 −0.0608581
\(271\) 4.05573 0.246368 0.123184 0.992384i \(-0.460689\pi\)
0.123184 + 0.992384i \(0.460689\pi\)
\(272\) −12.4377 −0.754146
\(273\) −4.61803 −0.279496
\(274\) 6.43769 0.388915
\(275\) 5.32624 0.321184
\(276\) −1.61803 −0.0973942
\(277\) −29.2705 −1.75869 −0.879347 0.476181i \(-0.842021\pi\)
−0.879347 + 0.476181i \(0.842021\pi\)
\(278\) −8.41641 −0.504783
\(279\) −6.70820 −0.401610
\(280\) 3.61803 0.216219
\(281\) 9.70820 0.579143 0.289571 0.957156i \(-0.406487\pi\)
0.289571 + 0.957156i \(0.406487\pi\)
\(282\) 1.70820 0.101722
\(283\) 25.2148 1.49886 0.749432 0.662082i \(-0.230327\pi\)
0.749432 + 0.662082i \(0.230327\pi\)
\(284\) −10.7082 −0.635415
\(285\) −8.85410 −0.524472
\(286\) 6.38197 0.377374
\(287\) −7.47214 −0.441066
\(288\) 5.61803 0.331046
\(289\) 28.0000 1.64706
\(290\) −3.76393 −0.221026
\(291\) 16.4164 0.962347
\(292\) −1.14590 −0.0670586
\(293\) 12.0000 0.701047 0.350524 0.936554i \(-0.386004\pi\)
0.350524 + 0.936554i \(0.386004\pi\)
\(294\) −0.618034 −0.0360445
\(295\) 18.7984 1.09448
\(296\) 24.5967 1.42966
\(297\) 2.23607 0.129750
\(298\) −9.12461 −0.528575
\(299\) 4.61803 0.267068
\(300\) −3.85410 −0.222517
\(301\) −0.618034 −0.0356229
\(302\) −6.65248 −0.382807
\(303\) −10.8541 −0.623552
\(304\) 10.1459 0.581907
\(305\) 3.00000 0.171780
\(306\) −4.14590 −0.237005
\(307\) −24.1246 −1.37686 −0.688432 0.725301i \(-0.741701\pi\)
−0.688432 + 0.725301i \(0.741701\pi\)
\(308\) −3.61803 −0.206157
\(309\) −19.4164 −1.10456
\(310\) −6.70820 −0.381000
\(311\) −0.326238 −0.0184993 −0.00924963 0.999957i \(-0.502944\pi\)
−0.00924963 + 0.999957i \(0.502944\pi\)
\(312\) −10.3262 −0.584608
\(313\) 6.47214 0.365827 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(314\) −8.47214 −0.478110
\(315\) −1.61803 −0.0911659
\(316\) 0.854102 0.0480470
\(317\) −23.5066 −1.32026 −0.660130 0.751151i \(-0.729499\pi\)
−0.660130 + 0.751151i \(0.729499\pi\)
\(318\) 1.18034 0.0661902
\(319\) 8.41641 0.471228
\(320\) −0.381966 −0.0213525
\(321\) 12.3262 0.687984
\(322\) 0.618034 0.0344417
\(323\) −36.7082 −2.04250
\(324\) −1.61803 −0.0898908
\(325\) 11.0000 0.610170
\(326\) 1.00000 0.0553849
\(327\) 14.2705 0.789161
\(328\) −16.7082 −0.922556
\(329\) 2.76393 0.152381
\(330\) 2.23607 0.123091
\(331\) 13.4164 0.737432 0.368716 0.929542i \(-0.379797\pi\)
0.368716 + 0.929542i \(0.379797\pi\)
\(332\) −21.3262 −1.17043
\(333\) −11.0000 −0.602796
\(334\) 9.38197 0.513358
\(335\) 9.85410 0.538387
\(336\) 1.85410 0.101150
\(337\) 30.5066 1.66180 0.830900 0.556422i \(-0.187826\pi\)
0.830900 + 0.556422i \(0.187826\pi\)
\(338\) 5.14590 0.279900
\(339\) 12.0902 0.656648
\(340\) 17.5623 0.952450
\(341\) 15.0000 0.812296
\(342\) 3.38197 0.182876
\(343\) −1.00000 −0.0539949
\(344\) −1.38197 −0.0745106
\(345\) 1.61803 0.0871120
\(346\) 3.38197 0.181816
\(347\) −17.1803 −0.922289 −0.461144 0.887325i \(-0.652561\pi\)
−0.461144 + 0.887325i \(0.652561\pi\)
\(348\) −6.09017 −0.326467
\(349\) −5.14590 −0.275454 −0.137727 0.990470i \(-0.543980\pi\)
−0.137727 + 0.990470i \(0.543980\pi\)
\(350\) 1.47214 0.0786890
\(351\) 4.61803 0.246492
\(352\) −12.5623 −0.669573
\(353\) −32.3050 −1.71942 −0.859710 0.510783i \(-0.829355\pi\)
−0.859710 + 0.510783i \(0.829355\pi\)
\(354\) −7.18034 −0.381631
\(355\) 10.7082 0.568332
\(356\) −15.1803 −0.804556
\(357\) −6.70820 −0.355036
\(358\) −3.00000 −0.158555
\(359\) 11.9098 0.628577 0.314288 0.949328i \(-0.398234\pi\)
0.314288 + 0.949328i \(0.398234\pi\)
\(360\) −3.61803 −0.190687
\(361\) 10.9443 0.576014
\(362\) 9.85410 0.517920
\(363\) 6.00000 0.314918
\(364\) −7.47214 −0.391646
\(365\) 1.14590 0.0599790
\(366\) −1.14590 −0.0598970
\(367\) 4.14590 0.216414 0.108207 0.994128i \(-0.465489\pi\)
0.108207 + 0.994128i \(0.465489\pi\)
\(368\) −1.85410 −0.0966517
\(369\) 7.47214 0.388984
\(370\) −11.0000 −0.571863
\(371\) 1.90983 0.0991534
\(372\) −10.8541 −0.562759
\(373\) −2.41641 −0.125117 −0.0625584 0.998041i \(-0.519926\pi\)
−0.0625584 + 0.998041i \(0.519926\pi\)
\(374\) 9.27051 0.479367
\(375\) 11.9443 0.616800
\(376\) 6.18034 0.318727
\(377\) 17.3820 0.895217
\(378\) 0.618034 0.0317882
\(379\) −19.4164 −0.997354 −0.498677 0.866788i \(-0.666181\pi\)
−0.498677 + 0.866788i \(0.666181\pi\)
\(380\) −14.3262 −0.734920
\(381\) 11.2705 0.577406
\(382\) −3.81966 −0.195431
\(383\) 12.7082 0.649359 0.324679 0.945824i \(-0.394744\pi\)
0.324679 + 0.945824i \(0.394744\pi\)
\(384\) 11.3820 0.580834
\(385\) 3.61803 0.184392
\(386\) 13.4164 0.682877
\(387\) 0.618034 0.0314164
\(388\) 26.5623 1.34850
\(389\) 18.7082 0.948544 0.474272 0.880378i \(-0.342711\pi\)
0.474272 + 0.880378i \(0.342711\pi\)
\(390\) 4.61803 0.233843
\(391\) 6.70820 0.339248
\(392\) −2.23607 −0.112938
\(393\) 7.18034 0.362200
\(394\) 7.72949 0.389406
\(395\) −0.854102 −0.0429745
\(396\) 3.61803 0.181813
\(397\) −36.0689 −1.81025 −0.905123 0.425150i \(-0.860221\pi\)
−0.905123 + 0.425150i \(0.860221\pi\)
\(398\) −11.1803 −0.560420
\(399\) 5.47214 0.273949
\(400\) −4.41641 −0.220820
\(401\) 10.8197 0.540308 0.270154 0.962817i \(-0.412925\pi\)
0.270154 + 0.962817i \(0.412925\pi\)
\(402\) −3.76393 −0.187728
\(403\) 30.9787 1.54316
\(404\) −17.5623 −0.873757
\(405\) 1.61803 0.0804008
\(406\) 2.32624 0.115449
\(407\) 24.5967 1.21922
\(408\) −15.0000 −0.742611
\(409\) −20.4164 −1.00953 −0.504763 0.863258i \(-0.668420\pi\)
−0.504763 + 0.863258i \(0.668420\pi\)
\(410\) 7.47214 0.369022
\(411\) −10.4164 −0.513804
\(412\) −31.4164 −1.54778
\(413\) −11.6180 −0.571686
\(414\) −0.618034 −0.0303747
\(415\) 21.3262 1.04686
\(416\) −25.9443 −1.27202
\(417\) 13.6180 0.666878
\(418\) −7.56231 −0.369884
\(419\) 31.3262 1.53039 0.765193 0.643800i \(-0.222643\pi\)
0.765193 + 0.643800i \(0.222643\pi\)
\(420\) −2.61803 −0.127747
\(421\) 23.2705 1.13414 0.567068 0.823671i \(-0.308078\pi\)
0.567068 + 0.823671i \(0.308078\pi\)
\(422\) −10.1459 −0.493895
\(423\) −2.76393 −0.134387
\(424\) 4.27051 0.207394
\(425\) 15.9787 0.775081
\(426\) −4.09017 −0.198169
\(427\) −1.85410 −0.0897263
\(428\) 19.9443 0.964043
\(429\) −10.3262 −0.498555
\(430\) 0.618034 0.0298042
\(431\) 16.3262 0.786407 0.393204 0.919451i \(-0.371367\pi\)
0.393204 + 0.919451i \(0.371367\pi\)
\(432\) −1.85410 −0.0892055
\(433\) −21.1246 −1.01518 −0.507592 0.861598i \(-0.669464\pi\)
−0.507592 + 0.861598i \(0.669464\pi\)
\(434\) 4.14590 0.199009
\(435\) 6.09017 0.292001
\(436\) 23.0902 1.10582
\(437\) −5.47214 −0.261768
\(438\) −0.437694 −0.0209138
\(439\) −7.65248 −0.365233 −0.182616 0.983184i \(-0.558457\pi\)
−0.182616 + 0.983184i \(0.558457\pi\)
\(440\) 8.09017 0.385684
\(441\) 1.00000 0.0476190
\(442\) 19.1459 0.910678
\(443\) −18.4721 −0.877638 −0.438819 0.898576i \(-0.644603\pi\)
−0.438819 + 0.898576i \(0.644603\pi\)
\(444\) −17.7984 −0.844673
\(445\) 15.1803 0.719617
\(446\) −6.88854 −0.326182
\(447\) 14.7639 0.698310
\(448\) 0.236068 0.0111532
\(449\) −38.5623 −1.81987 −0.909934 0.414753i \(-0.863868\pi\)
−0.909934 + 0.414753i \(0.863868\pi\)
\(450\) −1.47214 −0.0693972
\(451\) −16.7082 −0.786759
\(452\) 19.5623 0.920133
\(453\) 10.7639 0.505734
\(454\) 5.36068 0.251589
\(455\) 7.47214 0.350299
\(456\) 12.2361 0.573006
\(457\) 1.32624 0.0620388 0.0310194 0.999519i \(-0.490125\pi\)
0.0310194 + 0.999519i \(0.490125\pi\)
\(458\) −0.416408 −0.0194575
\(459\) 6.70820 0.313112
\(460\) 2.61803 0.122066
\(461\) −30.4508 −1.41824 −0.709119 0.705089i \(-0.750907\pi\)
−0.709119 + 0.705089i \(0.750907\pi\)
\(462\) −1.38197 −0.0642949
\(463\) 30.7082 1.42713 0.713566 0.700588i \(-0.247079\pi\)
0.713566 + 0.700588i \(0.247079\pi\)
\(464\) −6.97871 −0.323979
\(465\) 10.8541 0.503347
\(466\) 0.0557281 0.00258155
\(467\) 7.18034 0.332267 0.166133 0.986103i \(-0.446872\pi\)
0.166133 + 0.986103i \(0.446872\pi\)
\(468\) 7.47214 0.345400
\(469\) −6.09017 −0.281218
\(470\) −2.76393 −0.127491
\(471\) 13.7082 0.631641
\(472\) −25.9787 −1.19577
\(473\) −1.38197 −0.0635429
\(474\) 0.326238 0.0149846
\(475\) −13.0344 −0.598061
\(476\) −10.8541 −0.497497
\(477\) −1.90983 −0.0874451
\(478\) 12.2361 0.559665
\(479\) 21.0000 0.959514 0.479757 0.877401i \(-0.340725\pi\)
0.479757 + 0.877401i \(0.340725\pi\)
\(480\) −9.09017 −0.414908
\(481\) 50.7984 2.31621
\(482\) −6.79837 −0.309657
\(483\) −1.00000 −0.0455016
\(484\) 9.70820 0.441282
\(485\) −26.5623 −1.20613
\(486\) −0.618034 −0.0280346
\(487\) −42.7082 −1.93529 −0.967647 0.252309i \(-0.918810\pi\)
−0.967647 + 0.252309i \(0.918810\pi\)
\(488\) −4.14590 −0.187676
\(489\) −1.61803 −0.0731700
\(490\) 1.00000 0.0451754
\(491\) −14.7984 −0.667841 −0.333921 0.942601i \(-0.608372\pi\)
−0.333921 + 0.942601i \(0.608372\pi\)
\(492\) 12.0902 0.545067
\(493\) 25.2492 1.13717
\(494\) −15.6180 −0.702689
\(495\) −3.61803 −0.162619
\(496\) −12.4377 −0.558469
\(497\) −6.61803 −0.296859
\(498\) −8.14590 −0.365026
\(499\) 35.3951 1.58450 0.792252 0.610195i \(-0.208909\pi\)
0.792252 + 0.610195i \(0.208909\pi\)
\(500\) 19.3262 0.864296
\(501\) −15.1803 −0.678208
\(502\) −10.5836 −0.472369
\(503\) 20.6738 0.921797 0.460899 0.887453i \(-0.347527\pi\)
0.460899 + 0.887453i \(0.347527\pi\)
\(504\) 2.23607 0.0996024
\(505\) 17.5623 0.781512
\(506\) 1.38197 0.0614359
\(507\) −8.32624 −0.369781
\(508\) 18.2361 0.809095
\(509\) 33.5967 1.48915 0.744575 0.667539i \(-0.232652\pi\)
0.744575 + 0.667539i \(0.232652\pi\)
\(510\) 6.70820 0.297044
\(511\) −0.708204 −0.0313291
\(512\) 18.7082 0.826794
\(513\) −5.47214 −0.241601
\(514\) 0.763932 0.0336956
\(515\) 31.4164 1.38437
\(516\) 1.00000 0.0440225
\(517\) 6.18034 0.271811
\(518\) 6.79837 0.298703
\(519\) −5.47214 −0.240200
\(520\) 16.7082 0.732703
\(521\) 12.4721 0.546414 0.273207 0.961955i \(-0.411916\pi\)
0.273207 + 0.961955i \(0.411916\pi\)
\(522\) −2.32624 −0.101817
\(523\) −5.41641 −0.236843 −0.118421 0.992963i \(-0.537783\pi\)
−0.118421 + 0.992963i \(0.537783\pi\)
\(524\) 11.6180 0.507536
\(525\) −2.38197 −0.103958
\(526\) −11.0902 −0.483554
\(527\) 45.0000 1.96023
\(528\) 4.14590 0.180427
\(529\) 1.00000 0.0434783
\(530\) −1.90983 −0.0829577
\(531\) 11.6180 0.504180
\(532\) 8.85410 0.383874
\(533\) −34.5066 −1.49465
\(534\) −5.79837 −0.250920
\(535\) −19.9443 −0.862266
\(536\) −13.6180 −0.588209
\(537\) 4.85410 0.209470
\(538\) −12.9230 −0.557149
\(539\) −2.23607 −0.0963143
\(540\) 2.61803 0.112662
\(541\) 35.1246 1.51013 0.755063 0.655653i \(-0.227606\pi\)
0.755063 + 0.655653i \(0.227606\pi\)
\(542\) 2.50658 0.107667
\(543\) −15.9443 −0.684234
\(544\) −37.6869 −1.61581
\(545\) −23.0902 −0.989074
\(546\) −2.85410 −0.122144
\(547\) −11.7984 −0.504462 −0.252231 0.967667i \(-0.581164\pi\)
−0.252231 + 0.967667i \(0.581164\pi\)
\(548\) −16.8541 −0.719972
\(549\) 1.85410 0.0791311
\(550\) 3.29180 0.140363
\(551\) −20.5967 −0.877451
\(552\) −2.23607 −0.0951734
\(553\) 0.527864 0.0224471
\(554\) −18.0902 −0.768578
\(555\) 17.7984 0.755499
\(556\) 22.0344 0.934468
\(557\) −28.1803 −1.19404 −0.597020 0.802227i \(-0.703649\pi\)
−0.597020 + 0.802227i \(0.703649\pi\)
\(558\) −4.14590 −0.175510
\(559\) −2.85410 −0.120716
\(560\) −3.00000 −0.126773
\(561\) −15.0000 −0.633300
\(562\) 6.00000 0.253095
\(563\) 14.5066 0.611379 0.305690 0.952131i \(-0.401113\pi\)
0.305690 + 0.952131i \(0.401113\pi\)
\(564\) −4.47214 −0.188311
\(565\) −19.5623 −0.822992
\(566\) 15.5836 0.655027
\(567\) −1.00000 −0.0419961
\(568\) −14.7984 −0.620926
\(569\) 28.3607 1.18894 0.594471 0.804117i \(-0.297362\pi\)
0.594471 + 0.804117i \(0.297362\pi\)
\(570\) −5.47214 −0.229203
\(571\) −17.5836 −0.735850 −0.367925 0.929855i \(-0.619932\pi\)
−0.367925 + 0.929855i \(0.619932\pi\)
\(572\) −16.7082 −0.698605
\(573\) 6.18034 0.258187
\(574\) −4.61803 −0.192753
\(575\) 2.38197 0.0993348
\(576\) −0.236068 −0.00983617
\(577\) 16.8328 0.700759 0.350380 0.936608i \(-0.386053\pi\)
0.350380 + 0.936608i \(0.386053\pi\)
\(578\) 17.3050 0.719791
\(579\) −21.7082 −0.902162
\(580\) 9.85410 0.409169
\(581\) −13.1803 −0.546813
\(582\) 10.1459 0.420561
\(583\) 4.27051 0.176866
\(584\) −1.58359 −0.0655295
\(585\) −7.47214 −0.308935
\(586\) 7.41641 0.306369
\(587\) −1.96556 −0.0811273 −0.0405636 0.999177i \(-0.512915\pi\)
−0.0405636 + 0.999177i \(0.512915\pi\)
\(588\) 1.61803 0.0667266
\(589\) −36.7082 −1.51254
\(590\) 11.6180 0.478307
\(591\) −12.5066 −0.514452
\(592\) −20.3951 −0.838234
\(593\) 26.0689 1.07052 0.535260 0.844687i \(-0.320213\pi\)
0.535260 + 0.844687i \(0.320213\pi\)
\(594\) 1.38197 0.0567028
\(595\) 10.8541 0.444975
\(596\) 23.8885 0.978513
\(597\) 18.0902 0.740381
\(598\) 2.85410 0.116713
\(599\) 22.5066 0.919594 0.459797 0.888024i \(-0.347922\pi\)
0.459797 + 0.888024i \(0.347922\pi\)
\(600\) −5.32624 −0.217443
\(601\) 12.0902 0.493168 0.246584 0.969121i \(-0.420692\pi\)
0.246584 + 0.969121i \(0.420692\pi\)
\(602\) −0.381966 −0.0155678
\(603\) 6.09017 0.248011
\(604\) 17.4164 0.708664
\(605\) −9.70820 −0.394695
\(606\) −6.70820 −0.272502
\(607\) −9.20163 −0.373482 −0.186741 0.982409i \(-0.559793\pi\)
−0.186741 + 0.982409i \(0.559793\pi\)
\(608\) 30.7426 1.24678
\(609\) −3.76393 −0.152522
\(610\) 1.85410 0.0750704
\(611\) 12.7639 0.516373
\(612\) 10.8541 0.438751
\(613\) 19.6525 0.793756 0.396878 0.917871i \(-0.370094\pi\)
0.396878 + 0.917871i \(0.370094\pi\)
\(614\) −14.9098 −0.601712
\(615\) −12.0902 −0.487523
\(616\) −5.00000 −0.201456
\(617\) −18.3262 −0.737787 −0.368893 0.929472i \(-0.620263\pi\)
−0.368893 + 0.929472i \(0.620263\pi\)
\(618\) −12.0000 −0.482711
\(619\) −17.3820 −0.698640 −0.349320 0.937003i \(-0.613587\pi\)
−0.349320 + 0.937003i \(0.613587\pi\)
\(620\) 17.5623 0.705319
\(621\) 1.00000 0.0401286
\(622\) −0.201626 −0.00808447
\(623\) −9.38197 −0.375881
\(624\) 8.56231 0.342767
\(625\) −7.41641 −0.296656
\(626\) 4.00000 0.159872
\(627\) 12.2361 0.488661
\(628\) 22.1803 0.885092
\(629\) 73.7902 2.94221
\(630\) −1.00000 −0.0398410
\(631\) −24.1246 −0.960386 −0.480193 0.877163i \(-0.659433\pi\)
−0.480193 + 0.877163i \(0.659433\pi\)
\(632\) 1.18034 0.0469514
\(633\) 16.4164 0.652494
\(634\) −14.5279 −0.576975
\(635\) −18.2361 −0.723676
\(636\) −3.09017 −0.122533
\(637\) −4.61803 −0.182973
\(638\) 5.20163 0.205934
\(639\) 6.61803 0.261805
\(640\) −18.4164 −0.727972
\(641\) −6.79837 −0.268520 −0.134260 0.990946i \(-0.542866\pi\)
−0.134260 + 0.990946i \(0.542866\pi\)
\(642\) 7.61803 0.300660
\(643\) 34.9787 1.37943 0.689713 0.724083i \(-0.257737\pi\)
0.689713 + 0.724083i \(0.257737\pi\)
\(644\) −1.61803 −0.0637595
\(645\) −1.00000 −0.0393750
\(646\) −22.6869 −0.892605
\(647\) 5.03444 0.197924 0.0989622 0.995091i \(-0.468448\pi\)
0.0989622 + 0.995091i \(0.468448\pi\)
\(648\) −2.23607 −0.0878410
\(649\) −25.9787 −1.01975
\(650\) 6.79837 0.266654
\(651\) −6.70820 −0.262915
\(652\) −2.61803 −0.102530
\(653\) −14.7426 −0.576924 −0.288462 0.957491i \(-0.593144\pi\)
−0.288462 + 0.957491i \(0.593144\pi\)
\(654\) 8.81966 0.344876
\(655\) −11.6180 −0.453954
\(656\) 13.8541 0.540912
\(657\) 0.708204 0.0276297
\(658\) 1.70820 0.0665927
\(659\) −21.6525 −0.843461 −0.421730 0.906721i \(-0.638577\pi\)
−0.421730 + 0.906721i \(0.638577\pi\)
\(660\) −5.85410 −0.227871
\(661\) −12.4164 −0.482942 −0.241471 0.970408i \(-0.577630\pi\)
−0.241471 + 0.970408i \(0.577630\pi\)
\(662\) 8.29180 0.322270
\(663\) −30.9787 −1.20311
\(664\) −29.4721 −1.14374
\(665\) −8.85410 −0.343347
\(666\) −6.79837 −0.263432
\(667\) 3.76393 0.145740
\(668\) −24.5623 −0.950344
\(669\) 11.1459 0.430925
\(670\) 6.09017 0.235284
\(671\) −4.14590 −0.160051
\(672\) 5.61803 0.216720
\(673\) −14.4164 −0.555712 −0.277856 0.960623i \(-0.589624\pi\)
−0.277856 + 0.960623i \(0.589624\pi\)
\(674\) 18.8541 0.726233
\(675\) 2.38197 0.0916819
\(676\) −13.4721 −0.518159
\(677\) −43.7426 −1.68117 −0.840583 0.541682i \(-0.817788\pi\)
−0.840583 + 0.541682i \(0.817788\pi\)
\(678\) 7.47214 0.286966
\(679\) 16.4164 0.630004
\(680\) 24.2705 0.930732
\(681\) −8.67376 −0.332379
\(682\) 9.27051 0.354986
\(683\) −50.0132 −1.91370 −0.956850 0.290582i \(-0.906151\pi\)
−0.956850 + 0.290582i \(0.906151\pi\)
\(684\) −8.85410 −0.338545
\(685\) 16.8541 0.643962
\(686\) −0.618034 −0.0235966
\(687\) 0.673762 0.0257056
\(688\) 1.14590 0.0436870
\(689\) 8.81966 0.336002
\(690\) 1.00000 0.0380693
\(691\) −41.2705 −1.57000 −0.785002 0.619493i \(-0.787338\pi\)
−0.785002 + 0.619493i \(0.787338\pi\)
\(692\) −8.85410 −0.336582
\(693\) 2.23607 0.0849412
\(694\) −10.6180 −0.403055
\(695\) −22.0344 −0.835814
\(696\) −8.41641 −0.319023
\(697\) −50.1246 −1.89861
\(698\) −3.18034 −0.120378
\(699\) −0.0901699 −0.00341054
\(700\) −3.85410 −0.145671
\(701\) −18.4508 −0.696879 −0.348439 0.937331i \(-0.613288\pi\)
−0.348439 + 0.937331i \(0.613288\pi\)
\(702\) 2.85410 0.107721
\(703\) −60.1935 −2.27024
\(704\) 0.527864 0.0198946
\(705\) 4.47214 0.168430
\(706\) −19.9656 −0.751414
\(707\) −10.8541 −0.408211
\(708\) 18.7984 0.706486
\(709\) 17.5623 0.659566 0.329783 0.944057i \(-0.393024\pi\)
0.329783 + 0.944057i \(0.393024\pi\)
\(710\) 6.61803 0.248370
\(711\) −0.527864 −0.0197964
\(712\) −20.9787 −0.786211
\(713\) 6.70820 0.251224
\(714\) −4.14590 −0.155156
\(715\) 16.7082 0.624851
\(716\) 7.85410 0.293522
\(717\) −19.7984 −0.739384
\(718\) 7.36068 0.274698
\(719\) −3.00000 −0.111881 −0.0559406 0.998434i \(-0.517816\pi\)
−0.0559406 + 0.998434i \(0.517816\pi\)
\(720\) 3.00000 0.111803
\(721\) −19.4164 −0.723105
\(722\) 6.76393 0.251727
\(723\) 11.0000 0.409094
\(724\) −25.7984 −0.958789
\(725\) 8.96556 0.332972
\(726\) 3.70820 0.137624
\(727\) −16.8885 −0.626361 −0.313181 0.949694i \(-0.601395\pi\)
−0.313181 + 0.949694i \(0.601395\pi\)
\(728\) −10.3262 −0.382716
\(729\) 1.00000 0.0370370
\(730\) 0.708204 0.0262118
\(731\) −4.14590 −0.153342
\(732\) 3.00000 0.110883
\(733\) −23.5967 −0.871566 −0.435783 0.900052i \(-0.643528\pi\)
−0.435783 + 0.900052i \(0.643528\pi\)
\(734\) 2.56231 0.0945764
\(735\) −1.61803 −0.0596821
\(736\) −5.61803 −0.207083
\(737\) −13.6180 −0.501627
\(738\) 4.61803 0.169992
\(739\) −35.2492 −1.29666 −0.648332 0.761358i \(-0.724533\pi\)
−0.648332 + 0.761358i \(0.724533\pi\)
\(740\) 28.7984 1.05865
\(741\) 25.2705 0.928335
\(742\) 1.18034 0.0433316
\(743\) 27.3262 1.00250 0.501251 0.865302i \(-0.332873\pi\)
0.501251 + 0.865302i \(0.332873\pi\)
\(744\) −15.0000 −0.549927
\(745\) −23.8885 −0.875209
\(746\) −1.49342 −0.0546781
\(747\) 13.1803 0.482243
\(748\) −24.2705 −0.887418
\(749\) 12.3262 0.450391
\(750\) 7.38197 0.269551
\(751\) −13.9787 −0.510091 −0.255045 0.966929i \(-0.582090\pi\)
−0.255045 + 0.966929i \(0.582090\pi\)
\(752\) −5.12461 −0.186875
\(753\) 17.1246 0.624056
\(754\) 10.7426 0.391224
\(755\) −17.4164 −0.633848
\(756\) −1.61803 −0.0588473
\(757\) −17.0000 −0.617876 −0.308938 0.951082i \(-0.599973\pi\)
−0.308938 + 0.951082i \(0.599973\pi\)
\(758\) −12.0000 −0.435860
\(759\) −2.23607 −0.0811641
\(760\) −19.7984 −0.718163
\(761\) −5.52786 −0.200385 −0.100192 0.994968i \(-0.531946\pi\)
−0.100192 + 0.994968i \(0.531946\pi\)
\(762\) 6.96556 0.252336
\(763\) 14.2705 0.516627
\(764\) 10.0000 0.361787
\(765\) −10.8541 −0.392431
\(766\) 7.85410 0.283780
\(767\) −53.6525 −1.93728
\(768\) 6.56231 0.236797
\(769\) 28.7639 1.03725 0.518627 0.855001i \(-0.326443\pi\)
0.518627 + 0.855001i \(0.326443\pi\)
\(770\) 2.23607 0.0805823
\(771\) −1.23607 −0.0445159
\(772\) −35.1246 −1.26416
\(773\) −2.05573 −0.0739394 −0.0369697 0.999316i \(-0.511771\pi\)
−0.0369697 + 0.999316i \(0.511771\pi\)
\(774\) 0.381966 0.0137295
\(775\) 15.9787 0.573972
\(776\) 36.7082 1.31775
\(777\) −11.0000 −0.394623
\(778\) 11.5623 0.414529
\(779\) 40.8885 1.46498
\(780\) −12.0902 −0.432898
\(781\) −14.7984 −0.529527
\(782\) 4.14590 0.148257
\(783\) 3.76393 0.134512
\(784\) 1.85410 0.0662179
\(785\) −22.1803 −0.791650
\(786\) 4.43769 0.158287
\(787\) 47.5755 1.69588 0.847941 0.530091i \(-0.177842\pi\)
0.847941 + 0.530091i \(0.177842\pi\)
\(788\) −20.2361 −0.720880
\(789\) 17.9443 0.638833
\(790\) −0.527864 −0.0187806
\(791\) 12.0902 0.429877
\(792\) 5.00000 0.177667
\(793\) −8.56231 −0.304056
\(794\) −22.2918 −0.791106
\(795\) 3.09017 0.109597
\(796\) 29.2705 1.03747
\(797\) 21.8197 0.772892 0.386446 0.922312i \(-0.373703\pi\)
0.386446 + 0.922312i \(0.373703\pi\)
\(798\) 3.38197 0.119720
\(799\) 18.5410 0.655934
\(800\) −13.3820 −0.473124
\(801\) 9.38197 0.331495
\(802\) 6.68692 0.236123
\(803\) −1.58359 −0.0558838
\(804\) 9.85410 0.347527
\(805\) 1.61803 0.0570282
\(806\) 19.1459 0.674385
\(807\) 20.9098 0.736061
\(808\) −24.2705 −0.853834
\(809\) −29.5623 −1.03936 −0.519678 0.854362i \(-0.673948\pi\)
−0.519678 + 0.854362i \(0.673948\pi\)
\(810\) 1.00000 0.0351364
\(811\) 17.4721 0.613530 0.306765 0.951785i \(-0.400753\pi\)
0.306765 + 0.951785i \(0.400753\pi\)
\(812\) −6.09017 −0.213723
\(813\) −4.05573 −0.142241
\(814\) 15.2016 0.532817
\(815\) 2.61803 0.0917057
\(816\) 12.4377 0.435406
\(817\) 3.38197 0.118320
\(818\) −12.6180 −0.441179
\(819\) 4.61803 0.161367
\(820\) −19.5623 −0.683145
\(821\) 11.1246 0.388252 0.194126 0.980977i \(-0.437813\pi\)
0.194126 + 0.980977i \(0.437813\pi\)
\(822\) −6.43769 −0.224540
\(823\) −33.1459 −1.15539 −0.577697 0.816252i \(-0.696048\pi\)
−0.577697 + 0.816252i \(0.696048\pi\)
\(824\) −43.4164 −1.51248
\(825\) −5.32624 −0.185436
\(826\) −7.18034 −0.249836
\(827\) 23.2148 0.807257 0.403629 0.914923i \(-0.367749\pi\)
0.403629 + 0.914923i \(0.367749\pi\)
\(828\) 1.61803 0.0562306
\(829\) −3.58359 −0.124463 −0.0622316 0.998062i \(-0.519822\pi\)
−0.0622316 + 0.998062i \(0.519822\pi\)
\(830\) 13.1803 0.457496
\(831\) 29.2705 1.01538
\(832\) 1.09017 0.0377948
\(833\) −6.70820 −0.232425
\(834\) 8.41641 0.291436
\(835\) 24.5623 0.850014
\(836\) 19.7984 0.684741
\(837\) 6.70820 0.231869
\(838\) 19.3607 0.668804
\(839\) −25.7426 −0.888735 −0.444367 0.895845i \(-0.646572\pi\)
−0.444367 + 0.895845i \(0.646572\pi\)
\(840\) −3.61803 −0.124834
\(841\) −14.8328 −0.511476
\(842\) 14.3820 0.495635
\(843\) −9.70820 −0.334368
\(844\) 26.5623 0.914312
\(845\) 13.4721 0.463456
\(846\) −1.70820 −0.0587293
\(847\) 6.00000 0.206162
\(848\) −3.54102 −0.121599
\(849\) −25.2148 −0.865369
\(850\) 9.87539 0.338723
\(851\) 11.0000 0.377075
\(852\) 10.7082 0.366857
\(853\) −6.87539 −0.235409 −0.117704 0.993049i \(-0.537554\pi\)
−0.117704 + 0.993049i \(0.537554\pi\)
\(854\) −1.14590 −0.0392118
\(855\) 8.85410 0.302804
\(856\) 27.5623 0.942060
\(857\) 26.2918 0.898111 0.449055 0.893504i \(-0.351761\pi\)
0.449055 + 0.893504i \(0.351761\pi\)
\(858\) −6.38197 −0.217877
\(859\) −14.0000 −0.477674 −0.238837 0.971060i \(-0.576766\pi\)
−0.238837 + 0.971060i \(0.576766\pi\)
\(860\) −1.61803 −0.0551745
\(861\) 7.47214 0.254650
\(862\) 10.0902 0.343673
\(863\) 50.0689 1.70436 0.852182 0.523245i \(-0.175279\pi\)
0.852182 + 0.523245i \(0.175279\pi\)
\(864\) −5.61803 −0.191129
\(865\) 8.85410 0.301048
\(866\) −13.0557 −0.443652
\(867\) −28.0000 −0.950930
\(868\) −10.8541 −0.368412
\(869\) 1.18034 0.0400403
\(870\) 3.76393 0.127609
\(871\) −28.1246 −0.952966
\(872\) 31.9098 1.08060
\(873\) −16.4164 −0.555611
\(874\) −3.38197 −0.114397
\(875\) 11.9443 0.403790
\(876\) 1.14590 0.0387163
\(877\) 43.4164 1.46607 0.733034 0.680192i \(-0.238104\pi\)
0.733034 + 0.680192i \(0.238104\pi\)
\(878\) −4.72949 −0.159613
\(879\) −12.0000 −0.404750
\(880\) −6.70820 −0.226134
\(881\) 24.1803 0.814656 0.407328 0.913282i \(-0.366461\pi\)
0.407328 + 0.913282i \(0.366461\pi\)
\(882\) 0.618034 0.0208103
\(883\) −15.0902 −0.507825 −0.253912 0.967227i \(-0.581717\pi\)
−0.253912 + 0.967227i \(0.581717\pi\)
\(884\) −50.1246 −1.68587
\(885\) −18.7984 −0.631900
\(886\) −11.4164 −0.383542
\(887\) −14.7984 −0.496881 −0.248440 0.968647i \(-0.579918\pi\)
−0.248440 + 0.968647i \(0.579918\pi\)
\(888\) −24.5967 −0.825413
\(889\) 11.2705 0.378001
\(890\) 9.38197 0.314484
\(891\) −2.23607 −0.0749111
\(892\) 18.0344 0.603838
\(893\) −15.1246 −0.506126
\(894\) 9.12461 0.305173
\(895\) −7.85410 −0.262534
\(896\) 11.3820 0.380245
\(897\) −4.61803 −0.154192
\(898\) −23.8328 −0.795311
\(899\) 25.2492 0.842109
\(900\) 3.85410 0.128470
\(901\) 12.8115 0.426814
\(902\) −10.3262 −0.343826
\(903\) 0.618034 0.0205669
\(904\) 27.0344 0.899152
\(905\) 25.7984 0.857567
\(906\) 6.65248 0.221014
\(907\) 2.02129 0.0671157 0.0335579 0.999437i \(-0.489316\pi\)
0.0335579 + 0.999437i \(0.489316\pi\)
\(908\) −14.0344 −0.465749
\(909\) 10.8541 0.360008
\(910\) 4.61803 0.153086
\(911\) 41.5967 1.37816 0.689081 0.724684i \(-0.258014\pi\)
0.689081 + 0.724684i \(0.258014\pi\)
\(912\) −10.1459 −0.335964
\(913\) −29.4721 −0.975385
\(914\) 0.819660 0.0271119
\(915\) −3.00000 −0.0991769
\(916\) 1.09017 0.0360202
\(917\) 7.18034 0.237116
\(918\) 4.14590 0.136835
\(919\) −9.87539 −0.325759 −0.162879 0.986646i \(-0.552078\pi\)
−0.162879 + 0.986646i \(0.552078\pi\)
\(920\) 3.61803 0.119283
\(921\) 24.1246 0.794933
\(922\) −18.8197 −0.619792
\(923\) −30.5623 −1.00597
\(924\) 3.61803 0.119025
\(925\) 26.2016 0.861504
\(926\) 18.9787 0.623679
\(927\) 19.4164 0.637719
\(928\) −21.1459 −0.694148
\(929\) 40.0902 1.31532 0.657658 0.753317i \(-0.271547\pi\)
0.657658 + 0.753317i \(0.271547\pi\)
\(930\) 6.70820 0.219971
\(931\) 5.47214 0.179342
\(932\) −0.145898 −0.00477905
\(933\) 0.326238 0.0106806
\(934\) 4.43769 0.145206
\(935\) 24.2705 0.793731
\(936\) 10.3262 0.337524
\(937\) −10.7082 −0.349822 −0.174911 0.984584i \(-0.555964\pi\)
−0.174911 + 0.984584i \(0.555964\pi\)
\(938\) −3.76393 −0.122897
\(939\) −6.47214 −0.211210
\(940\) 7.23607 0.236015
\(941\) −32.9443 −1.07395 −0.536976 0.843597i \(-0.680434\pi\)
−0.536976 + 0.843597i \(0.680434\pi\)
\(942\) 8.47214 0.276037
\(943\) −7.47214 −0.243326
\(944\) 21.5410 0.701100
\(945\) 1.61803 0.0526346
\(946\) −0.854102 −0.0277693
\(947\) −1.05573 −0.0343066 −0.0171533 0.999853i \(-0.505460\pi\)
−0.0171533 + 0.999853i \(0.505460\pi\)
\(948\) −0.854102 −0.0277399
\(949\) −3.27051 −0.106165
\(950\) −8.05573 −0.261362
\(951\) 23.5066 0.762253
\(952\) −15.0000 −0.486153
\(953\) −57.1591 −1.85156 −0.925782 0.378059i \(-0.876592\pi\)
−0.925782 + 0.378059i \(0.876592\pi\)
\(954\) −1.18034 −0.0382149
\(955\) −10.0000 −0.323592
\(956\) −32.0344 −1.03607
\(957\) −8.41641 −0.272064
\(958\) 12.9787 0.419323
\(959\) −10.4164 −0.336363
\(960\) 0.381966 0.0123279
\(961\) 14.0000 0.451613
\(962\) 31.3951 1.01222
\(963\) −12.3262 −0.397207
\(964\) 17.7984 0.573247
\(965\) 35.1246 1.13070
\(966\) −0.618034 −0.0198849
\(967\) 5.63932 0.181348 0.0906742 0.995881i \(-0.471098\pi\)
0.0906742 + 0.995881i \(0.471098\pi\)
\(968\) 13.4164 0.431220
\(969\) 36.7082 1.17924
\(970\) −16.4164 −0.527099
\(971\) 40.6869 1.30571 0.652853 0.757485i \(-0.273572\pi\)
0.652853 + 0.757485i \(0.273572\pi\)
\(972\) 1.61803 0.0518985
\(973\) 13.6180 0.436574
\(974\) −26.3951 −0.845754
\(975\) −11.0000 −0.352282
\(976\) 3.43769 0.110038
\(977\) 25.2016 0.806271 0.403136 0.915140i \(-0.367920\pi\)
0.403136 + 0.915140i \(0.367920\pi\)
\(978\) −1.00000 −0.0319765
\(979\) −20.9787 −0.670483
\(980\) −2.61803 −0.0836300
\(981\) −14.2705 −0.455622
\(982\) −9.14590 −0.291857
\(983\) −56.5967 −1.80516 −0.902578 0.430526i \(-0.858328\pi\)
−0.902578 + 0.430526i \(0.858328\pi\)
\(984\) 16.7082 0.532638
\(985\) 20.2361 0.644775
\(986\) 15.6049 0.496961
\(987\) −2.76393 −0.0879769
\(988\) 40.8885 1.30084
\(989\) −0.618034 −0.0196523
\(990\) −2.23607 −0.0710669
\(991\) 60.5755 1.92424 0.962121 0.272621i \(-0.0878905\pi\)
0.962121 + 0.272621i \(0.0878905\pi\)
\(992\) −37.6869 −1.19656
\(993\) −13.4164 −0.425757
\(994\) −4.09017 −0.129732
\(995\) −29.2705 −0.927938
\(996\) 21.3262 0.675748
\(997\) −31.7214 −1.00463 −0.502313 0.864686i \(-0.667517\pi\)
−0.502313 + 0.864686i \(0.667517\pi\)
\(998\) 21.8754 0.692453
\(999\) 11.0000 0.348025
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 483.2.a.d.1.2 2
3.2 odd 2 1449.2.a.h.1.1 2
4.3 odd 2 7728.2.a.bn.1.2 2
7.6 odd 2 3381.2.a.r.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
483.2.a.d.1.2 2 1.1 even 1 trivial
1449.2.a.h.1.1 2 3.2 odd 2
3381.2.a.r.1.2 2 7.6 odd 2
7728.2.a.bn.1.2 2 4.3 odd 2