Properties

Label 285.2.a.f.1.2
Level $285$
Weight $2$
Character 285.1
Self dual yes
Analytic conductor $2.276$
Analytic rank $0$
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] = [285,2,Mod(1,285)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(285, 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("285.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 285 = 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 285.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: \(2.27573645761\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) 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 \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 285.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.41421 q^{2} -1.00000 q^{3} +3.82843 q^{4} -1.00000 q^{5} -2.41421 q^{6} +3.41421 q^{7} +4.41421 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+2.41421 q^{2} -1.00000 q^{3} +3.82843 q^{4} -1.00000 q^{5} -2.41421 q^{6} +3.41421 q^{7} +4.41421 q^{8} +1.00000 q^{9} -2.41421 q^{10} -1.41421 q^{11} -3.82843 q^{12} +2.58579 q^{13} +8.24264 q^{14} +1.00000 q^{15} +3.00000 q^{16} -6.82843 q^{17} +2.41421 q^{18} +1.00000 q^{19} -3.82843 q^{20} -3.41421 q^{21} -3.41421 q^{22} -3.65685 q^{23} -4.41421 q^{24} +1.00000 q^{25} +6.24264 q^{26} -1.00000 q^{27} +13.0711 q^{28} +5.07107 q^{29} +2.41421 q^{30} -10.4853 q^{31} -1.58579 q^{32} +1.41421 q^{33} -16.4853 q^{34} -3.41421 q^{35} +3.82843 q^{36} -3.07107 q^{37} +2.41421 q^{38} -2.58579 q^{39} -4.41421 q^{40} -4.58579 q^{41} -8.24264 q^{42} +3.41421 q^{43} -5.41421 q^{44} -1.00000 q^{45} -8.82843 q^{46} +11.6569 q^{47} -3.00000 q^{48} +4.65685 q^{49} +2.41421 q^{50} +6.82843 q^{51} +9.89949 q^{52} +4.00000 q^{53} -2.41421 q^{54} +1.41421 q^{55} +15.0711 q^{56} -1.00000 q^{57} +12.2426 q^{58} -8.48528 q^{59} +3.82843 q^{60} -5.65685 q^{61} -25.3137 q^{62} +3.41421 q^{63} -9.82843 q^{64} -2.58579 q^{65} +3.41421 q^{66} +12.0000 q^{67} -26.1421 q^{68} +3.65685 q^{69} -8.24264 q^{70} +12.4853 q^{71} +4.41421 q^{72} -2.00000 q^{73} -7.41421 q^{74} -1.00000 q^{75} +3.82843 q^{76} -4.82843 q^{77} -6.24264 q^{78} +11.3137 q^{79} -3.00000 q^{80} +1.00000 q^{81} -11.0711 q^{82} +6.48528 q^{83} -13.0711 q^{84} +6.82843 q^{85} +8.24264 q^{86} -5.07107 q^{87} -6.24264 q^{88} -14.7279 q^{89} -2.41421 q^{90} +8.82843 q^{91} -14.0000 q^{92} +10.4853 q^{93} +28.1421 q^{94} -1.00000 q^{95} +1.58579 q^{96} +4.24264 q^{97} +11.2426 q^{98} -1.41421 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} + 4 q^{7} + 6 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} + 4 q^{7} + 6 q^{8} + 2 q^{9} - 2 q^{10} - 2 q^{12} + 8 q^{13} + 8 q^{14} + 2 q^{15} + 6 q^{16} - 8 q^{17} + 2 q^{18} + 2 q^{19} - 2 q^{20} - 4 q^{21} - 4 q^{22} + 4 q^{23} - 6 q^{24} + 2 q^{25} + 4 q^{26} - 2 q^{27} + 12 q^{28} - 4 q^{29} + 2 q^{30} - 4 q^{31} - 6 q^{32} - 16 q^{34} - 4 q^{35} + 2 q^{36} + 8 q^{37} + 2 q^{38} - 8 q^{39} - 6 q^{40} - 12 q^{41} - 8 q^{42} + 4 q^{43} - 8 q^{44} - 2 q^{45} - 12 q^{46} + 12 q^{47} - 6 q^{48} - 2 q^{49} + 2 q^{50} + 8 q^{51} + 8 q^{53} - 2 q^{54} + 16 q^{56} - 2 q^{57} + 16 q^{58} + 2 q^{60} - 28 q^{62} + 4 q^{63} - 14 q^{64} - 8 q^{65} + 4 q^{66} + 24 q^{67} - 24 q^{68} - 4 q^{69} - 8 q^{70} + 8 q^{71} + 6 q^{72} - 4 q^{73} - 12 q^{74} - 2 q^{75} + 2 q^{76} - 4 q^{77} - 4 q^{78} - 6 q^{80} + 2 q^{81} - 8 q^{82} - 4 q^{83} - 12 q^{84} + 8 q^{85} + 8 q^{86} + 4 q^{87} - 4 q^{88} - 4 q^{89} - 2 q^{90} + 12 q^{91} - 28 q^{92} + 4 q^{93} + 28 q^{94} - 2 q^{95} + 6 q^{96} + 14 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.41421 1.70711 0.853553 0.521005i \(-0.174443\pi\)
0.853553 + 0.521005i \(0.174443\pi\)
\(3\) −1.00000 −0.577350
\(4\) 3.82843 1.91421
\(5\) −1.00000 −0.447214
\(6\) −2.41421 −0.985599
\(7\) 3.41421 1.29045 0.645226 0.763992i \(-0.276763\pi\)
0.645226 + 0.763992i \(0.276763\pi\)
\(8\) 4.41421 1.56066
\(9\) 1.00000 0.333333
\(10\) −2.41421 −0.763441
\(11\) −1.41421 −0.426401 −0.213201 0.977008i \(-0.568389\pi\)
−0.213201 + 0.977008i \(0.568389\pi\)
\(12\) −3.82843 −1.10517
\(13\) 2.58579 0.717168 0.358584 0.933497i \(-0.383260\pi\)
0.358584 + 0.933497i \(0.383260\pi\)
\(14\) 8.24264 2.20294
\(15\) 1.00000 0.258199
\(16\) 3.00000 0.750000
\(17\) −6.82843 −1.65614 −0.828068 0.560627i \(-0.810560\pi\)
−0.828068 + 0.560627i \(0.810560\pi\)
\(18\) 2.41421 0.569036
\(19\) 1.00000 0.229416
\(20\) −3.82843 −0.856062
\(21\) −3.41421 −0.745042
\(22\) −3.41421 −0.727913
\(23\) −3.65685 −0.762507 −0.381253 0.924471i \(-0.624507\pi\)
−0.381253 + 0.924471i \(0.624507\pi\)
\(24\) −4.41421 −0.901048
\(25\) 1.00000 0.200000
\(26\) 6.24264 1.22428
\(27\) −1.00000 −0.192450
\(28\) 13.0711 2.47020
\(29\) 5.07107 0.941674 0.470837 0.882220i \(-0.343952\pi\)
0.470837 + 0.882220i \(0.343952\pi\)
\(30\) 2.41421 0.440773
\(31\) −10.4853 −1.88321 −0.941606 0.336717i \(-0.890684\pi\)
−0.941606 + 0.336717i \(0.890684\pi\)
\(32\) −1.58579 −0.280330
\(33\) 1.41421 0.246183
\(34\) −16.4853 −2.82720
\(35\) −3.41421 −0.577107
\(36\) 3.82843 0.638071
\(37\) −3.07107 −0.504880 −0.252440 0.967612i \(-0.581233\pi\)
−0.252440 + 0.967612i \(0.581233\pi\)
\(38\) 2.41421 0.391637
\(39\) −2.58579 −0.414057
\(40\) −4.41421 −0.697948
\(41\) −4.58579 −0.716180 −0.358090 0.933687i \(-0.616572\pi\)
−0.358090 + 0.933687i \(0.616572\pi\)
\(42\) −8.24264 −1.27187
\(43\) 3.41421 0.520663 0.260331 0.965519i \(-0.416168\pi\)
0.260331 + 0.965519i \(0.416168\pi\)
\(44\) −5.41421 −0.816223
\(45\) −1.00000 −0.149071
\(46\) −8.82843 −1.30168
\(47\) 11.6569 1.70033 0.850163 0.526519i \(-0.176503\pi\)
0.850163 + 0.526519i \(0.176503\pi\)
\(48\) −3.00000 −0.433013
\(49\) 4.65685 0.665265
\(50\) 2.41421 0.341421
\(51\) 6.82843 0.956171
\(52\) 9.89949 1.37281
\(53\) 4.00000 0.549442 0.274721 0.961524i \(-0.411414\pi\)
0.274721 + 0.961524i \(0.411414\pi\)
\(54\) −2.41421 −0.328533
\(55\) 1.41421 0.190693
\(56\) 15.0711 2.01396
\(57\) −1.00000 −0.132453
\(58\) 12.2426 1.60754
\(59\) −8.48528 −1.10469 −0.552345 0.833616i \(-0.686267\pi\)
−0.552345 + 0.833616i \(0.686267\pi\)
\(60\) 3.82843 0.494248
\(61\) −5.65685 −0.724286 −0.362143 0.932123i \(-0.617955\pi\)
−0.362143 + 0.932123i \(0.617955\pi\)
\(62\) −25.3137 −3.21484
\(63\) 3.41421 0.430150
\(64\) −9.82843 −1.22855
\(65\) −2.58579 −0.320727
\(66\) 3.41421 0.420261
\(67\) 12.0000 1.46603 0.733017 0.680211i \(-0.238112\pi\)
0.733017 + 0.680211i \(0.238112\pi\)
\(68\) −26.1421 −3.17020
\(69\) 3.65685 0.440234
\(70\) −8.24264 −0.985184
\(71\) 12.4853 1.48173 0.740865 0.671654i \(-0.234416\pi\)
0.740865 + 0.671654i \(0.234416\pi\)
\(72\) 4.41421 0.520220
\(73\) −2.00000 −0.234082 −0.117041 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) −7.41421 −0.861885
\(75\) −1.00000 −0.115470
\(76\) 3.82843 0.439151
\(77\) −4.82843 −0.550250
\(78\) −6.24264 −0.706840
\(79\) 11.3137 1.27289 0.636446 0.771321i \(-0.280404\pi\)
0.636446 + 0.771321i \(0.280404\pi\)
\(80\) −3.00000 −0.335410
\(81\) 1.00000 0.111111
\(82\) −11.0711 −1.22259
\(83\) 6.48528 0.711852 0.355926 0.934514i \(-0.384165\pi\)
0.355926 + 0.934514i \(0.384165\pi\)
\(84\) −13.0711 −1.42617
\(85\) 6.82843 0.740647
\(86\) 8.24264 0.888827
\(87\) −5.07107 −0.543676
\(88\) −6.24264 −0.665468
\(89\) −14.7279 −1.56116 −0.780578 0.625058i \(-0.785075\pi\)
−0.780578 + 0.625058i \(0.785075\pi\)
\(90\) −2.41421 −0.254480
\(91\) 8.82843 0.925471
\(92\) −14.0000 −1.45960
\(93\) 10.4853 1.08727
\(94\) 28.1421 2.90264
\(95\) −1.00000 −0.102598
\(96\) 1.58579 0.161849
\(97\) 4.24264 0.430775 0.215387 0.976529i \(-0.430899\pi\)
0.215387 + 0.976529i \(0.430899\pi\)
\(98\) 11.2426 1.13568
\(99\) −1.41421 −0.142134
\(100\) 3.82843 0.382843
\(101\) 0.828427 0.0824316 0.0412158 0.999150i \(-0.486877\pi\)
0.0412158 + 0.999150i \(0.486877\pi\)
\(102\) 16.4853 1.63229
\(103\) 9.65685 0.951518 0.475759 0.879576i \(-0.342173\pi\)
0.475759 + 0.879576i \(0.342173\pi\)
\(104\) 11.4142 1.11926
\(105\) 3.41421 0.333193
\(106\) 9.65685 0.937957
\(107\) 8.00000 0.773389 0.386695 0.922208i \(-0.373617\pi\)
0.386695 + 0.922208i \(0.373617\pi\)
\(108\) −3.82843 −0.368391
\(109\) 8.82843 0.845610 0.422805 0.906221i \(-0.361046\pi\)
0.422805 + 0.906221i \(0.361046\pi\)
\(110\) 3.41421 0.325532
\(111\) 3.07107 0.291493
\(112\) 10.2426 0.967839
\(113\) −4.48528 −0.421940 −0.210970 0.977493i \(-0.567662\pi\)
−0.210970 + 0.977493i \(0.567662\pi\)
\(114\) −2.41421 −0.226112
\(115\) 3.65685 0.341003
\(116\) 19.4142 1.80256
\(117\) 2.58579 0.239056
\(118\) −20.4853 −1.88582
\(119\) −23.3137 −2.13716
\(120\) 4.41421 0.402961
\(121\) −9.00000 −0.818182
\(122\) −13.6569 −1.23643
\(123\) 4.58579 0.413486
\(124\) −40.1421 −3.60487
\(125\) −1.00000 −0.0894427
\(126\) 8.24264 0.734313
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) −20.5563 −1.81694
\(129\) −3.41421 −0.300605
\(130\) −6.24264 −0.547516
\(131\) −8.72792 −0.762562 −0.381281 0.924459i \(-0.624517\pi\)
−0.381281 + 0.924459i \(0.624517\pi\)
\(132\) 5.41421 0.471247
\(133\) 3.41421 0.296050
\(134\) 28.9706 2.50268
\(135\) 1.00000 0.0860663
\(136\) −30.1421 −2.58467
\(137\) −14.0000 −1.19610 −0.598050 0.801459i \(-0.704058\pi\)
−0.598050 + 0.801459i \(0.704058\pi\)
\(138\) 8.82843 0.751526
\(139\) 6.82843 0.579180 0.289590 0.957151i \(-0.406481\pi\)
0.289590 + 0.957151i \(0.406481\pi\)
\(140\) −13.0711 −1.10471
\(141\) −11.6569 −0.981684
\(142\) 30.1421 2.52947
\(143\) −3.65685 −0.305802
\(144\) 3.00000 0.250000
\(145\) −5.07107 −0.421129
\(146\) −4.82843 −0.399603
\(147\) −4.65685 −0.384091
\(148\) −11.7574 −0.966449
\(149\) 7.65685 0.627274 0.313637 0.949543i \(-0.398453\pi\)
0.313637 + 0.949543i \(0.398453\pi\)
\(150\) −2.41421 −0.197120
\(151\) −21.7990 −1.77398 −0.886988 0.461792i \(-0.847207\pi\)
−0.886988 + 0.461792i \(0.847207\pi\)
\(152\) 4.41421 0.358040
\(153\) −6.82843 −0.552046
\(154\) −11.6569 −0.939336
\(155\) 10.4853 0.842198
\(156\) −9.89949 −0.792594
\(157\) 17.7990 1.42051 0.710257 0.703942i \(-0.248579\pi\)
0.710257 + 0.703942i \(0.248579\pi\)
\(158\) 27.3137 2.17296
\(159\) −4.00000 −0.317221
\(160\) 1.58579 0.125367
\(161\) −12.4853 −0.983978
\(162\) 2.41421 0.189679
\(163\) −11.8995 −0.932040 −0.466020 0.884774i \(-0.654313\pi\)
−0.466020 + 0.884774i \(0.654313\pi\)
\(164\) −17.5563 −1.37092
\(165\) −1.41421 −0.110096
\(166\) 15.6569 1.21521
\(167\) 10.0000 0.773823 0.386912 0.922117i \(-0.373542\pi\)
0.386912 + 0.922117i \(0.373542\pi\)
\(168\) −15.0711 −1.16276
\(169\) −6.31371 −0.485670
\(170\) 16.4853 1.26436
\(171\) 1.00000 0.0764719
\(172\) 13.0711 0.996660
\(173\) 22.1421 1.68344 0.841718 0.539918i \(-0.181545\pi\)
0.841718 + 0.539918i \(0.181545\pi\)
\(174\) −12.2426 −0.928112
\(175\) 3.41421 0.258090
\(176\) −4.24264 −0.319801
\(177\) 8.48528 0.637793
\(178\) −35.5563 −2.66506
\(179\) 22.8284 1.70628 0.853138 0.521685i \(-0.174696\pi\)
0.853138 + 0.521685i \(0.174696\pi\)
\(180\) −3.82843 −0.285354
\(181\) −24.8284 −1.84548 −0.922741 0.385420i \(-0.874057\pi\)
−0.922741 + 0.385420i \(0.874057\pi\)
\(182\) 21.3137 1.57988
\(183\) 5.65685 0.418167
\(184\) −16.1421 −1.19001
\(185\) 3.07107 0.225789
\(186\) 25.3137 1.85609
\(187\) 9.65685 0.706179
\(188\) 44.6274 3.25479
\(189\) −3.41421 −0.248347
\(190\) −2.41421 −0.175145
\(191\) −17.8995 −1.29516 −0.647581 0.761997i \(-0.724219\pi\)
−0.647581 + 0.761997i \(0.724219\pi\)
\(192\) 9.82843 0.709306
\(193\) −0.928932 −0.0668660 −0.0334330 0.999441i \(-0.510644\pi\)
−0.0334330 + 0.999441i \(0.510644\pi\)
\(194\) 10.2426 0.735379
\(195\) 2.58579 0.185172
\(196\) 17.8284 1.27346
\(197\) −9.17157 −0.653448 −0.326724 0.945120i \(-0.605945\pi\)
−0.326724 + 0.945120i \(0.605945\pi\)
\(198\) −3.41421 −0.242638
\(199\) 0.485281 0.0344007 0.0172003 0.999852i \(-0.494525\pi\)
0.0172003 + 0.999852i \(0.494525\pi\)
\(200\) 4.41421 0.312132
\(201\) −12.0000 −0.846415
\(202\) 2.00000 0.140720
\(203\) 17.3137 1.21518
\(204\) 26.1421 1.83032
\(205\) 4.58579 0.320285
\(206\) 23.3137 1.62434
\(207\) −3.65685 −0.254169
\(208\) 7.75736 0.537876
\(209\) −1.41421 −0.0978232
\(210\) 8.24264 0.568796
\(211\) 7.31371 0.503496 0.251748 0.967793i \(-0.418995\pi\)
0.251748 + 0.967793i \(0.418995\pi\)
\(212\) 15.3137 1.05175
\(213\) −12.4853 −0.855477
\(214\) 19.3137 1.32026
\(215\) −3.41421 −0.232847
\(216\) −4.41421 −0.300349
\(217\) −35.7990 −2.43019
\(218\) 21.3137 1.44355
\(219\) 2.00000 0.135147
\(220\) 5.41421 0.365026
\(221\) −17.6569 −1.18773
\(222\) 7.41421 0.497609
\(223\) 17.6569 1.18239 0.591195 0.806529i \(-0.298656\pi\)
0.591195 + 0.806529i \(0.298656\pi\)
\(224\) −5.41421 −0.361752
\(225\) 1.00000 0.0666667
\(226\) −10.8284 −0.720296
\(227\) 14.9706 0.993631 0.496816 0.867856i \(-0.334503\pi\)
0.496816 + 0.867856i \(0.334503\pi\)
\(228\) −3.82843 −0.253544
\(229\) 9.65685 0.638143 0.319071 0.947731i \(-0.396629\pi\)
0.319071 + 0.947731i \(0.396629\pi\)
\(230\) 8.82843 0.582129
\(231\) 4.82843 0.317687
\(232\) 22.3848 1.46963
\(233\) −19.6569 −1.28776 −0.643882 0.765125i \(-0.722677\pi\)
−0.643882 + 0.765125i \(0.722677\pi\)
\(234\) 6.24264 0.408094
\(235\) −11.6569 −0.760409
\(236\) −32.4853 −2.11461
\(237\) −11.3137 −0.734904
\(238\) −56.2843 −3.64837
\(239\) 5.41421 0.350216 0.175108 0.984549i \(-0.443972\pi\)
0.175108 + 0.984549i \(0.443972\pi\)
\(240\) 3.00000 0.193649
\(241\) 18.9706 1.22200 0.611001 0.791630i \(-0.290767\pi\)
0.611001 + 0.791630i \(0.290767\pi\)
\(242\) −21.7279 −1.39672
\(243\) −1.00000 −0.0641500
\(244\) −21.6569 −1.38644
\(245\) −4.65685 −0.297516
\(246\) 11.0711 0.705866
\(247\) 2.58579 0.164530
\(248\) −46.2843 −2.93905
\(249\) −6.48528 −0.410988
\(250\) −2.41421 −0.152688
\(251\) 27.0711 1.70871 0.854355 0.519689i \(-0.173952\pi\)
0.854355 + 0.519689i \(0.173952\pi\)
\(252\) 13.0711 0.823400
\(253\) 5.17157 0.325134
\(254\) 19.3137 1.21185
\(255\) −6.82843 −0.427613
\(256\) −29.9706 −1.87316
\(257\) 6.82843 0.425946 0.212973 0.977058i \(-0.431685\pi\)
0.212973 + 0.977058i \(0.431685\pi\)
\(258\) −8.24264 −0.513164
\(259\) −10.4853 −0.651524
\(260\) −9.89949 −0.613941
\(261\) 5.07107 0.313891
\(262\) −21.0711 −1.30177
\(263\) −3.85786 −0.237886 −0.118943 0.992901i \(-0.537951\pi\)
−0.118943 + 0.992901i \(0.537951\pi\)
\(264\) 6.24264 0.384208
\(265\) −4.00000 −0.245718
\(266\) 8.24264 0.505389
\(267\) 14.7279 0.901334
\(268\) 45.9411 2.80630
\(269\) −28.3848 −1.73065 −0.865325 0.501211i \(-0.832888\pi\)
−0.865325 + 0.501211i \(0.832888\pi\)
\(270\) 2.41421 0.146924
\(271\) −25.1716 −1.52906 −0.764532 0.644586i \(-0.777030\pi\)
−0.764532 + 0.644586i \(0.777030\pi\)
\(272\) −20.4853 −1.24210
\(273\) −8.82843 −0.534321
\(274\) −33.7990 −2.04187
\(275\) −1.41421 −0.0852803
\(276\) 14.0000 0.842701
\(277\) −22.9706 −1.38017 −0.690084 0.723730i \(-0.742426\pi\)
−0.690084 + 0.723730i \(0.742426\pi\)
\(278\) 16.4853 0.988721
\(279\) −10.4853 −0.627737
\(280\) −15.0711 −0.900669
\(281\) 10.7279 0.639974 0.319987 0.947422i \(-0.396321\pi\)
0.319987 + 0.947422i \(0.396321\pi\)
\(282\) −28.1421 −1.67584
\(283\) −2.24264 −0.133311 −0.0666556 0.997776i \(-0.521233\pi\)
−0.0666556 + 0.997776i \(0.521233\pi\)
\(284\) 47.7990 2.83635
\(285\) 1.00000 0.0592349
\(286\) −8.82843 −0.522036
\(287\) −15.6569 −0.924195
\(288\) −1.58579 −0.0934434
\(289\) 29.6274 1.74279
\(290\) −12.2426 −0.718913
\(291\) −4.24264 −0.248708
\(292\) −7.65685 −0.448084
\(293\) 7.79899 0.455622 0.227811 0.973705i \(-0.426843\pi\)
0.227811 + 0.973705i \(0.426843\pi\)
\(294\) −11.2426 −0.655684
\(295\) 8.48528 0.494032
\(296\) −13.5563 −0.787947
\(297\) 1.41421 0.0820610
\(298\) 18.4853 1.07082
\(299\) −9.45584 −0.546846
\(300\) −3.82843 −0.221034
\(301\) 11.6569 0.671890
\(302\) −52.6274 −3.02837
\(303\) −0.828427 −0.0475919
\(304\) 3.00000 0.172062
\(305\) 5.65685 0.323911
\(306\) −16.4853 −0.942401
\(307\) −31.7990 −1.81486 −0.907432 0.420199i \(-0.861960\pi\)
−0.907432 + 0.420199i \(0.861960\pi\)
\(308\) −18.4853 −1.05330
\(309\) −9.65685 −0.549359
\(310\) 25.3137 1.43772
\(311\) −23.7574 −1.34716 −0.673578 0.739116i \(-0.735244\pi\)
−0.673578 + 0.739116i \(0.735244\pi\)
\(312\) −11.4142 −0.646203
\(313\) −26.4853 −1.49704 −0.748518 0.663114i \(-0.769234\pi\)
−0.748518 + 0.663114i \(0.769234\pi\)
\(314\) 42.9706 2.42497
\(315\) −3.41421 −0.192369
\(316\) 43.3137 2.43659
\(317\) −11.3137 −0.635441 −0.317721 0.948184i \(-0.602917\pi\)
−0.317721 + 0.948184i \(0.602917\pi\)
\(318\) −9.65685 −0.541529
\(319\) −7.17157 −0.401531
\(320\) 9.82843 0.549426
\(321\) −8.00000 −0.446516
\(322\) −30.1421 −1.67976
\(323\) −6.82843 −0.379944
\(324\) 3.82843 0.212690
\(325\) 2.58579 0.143434
\(326\) −28.7279 −1.59109
\(327\) −8.82843 −0.488213
\(328\) −20.2426 −1.11771
\(329\) 39.7990 2.19419
\(330\) −3.41421 −0.187946
\(331\) 12.8284 0.705114 0.352557 0.935790i \(-0.385312\pi\)
0.352557 + 0.935790i \(0.385312\pi\)
\(332\) 24.8284 1.36264
\(333\) −3.07107 −0.168293
\(334\) 24.1421 1.32100
\(335\) −12.0000 −0.655630
\(336\) −10.2426 −0.558782
\(337\) 19.7574 1.07625 0.538126 0.842864i \(-0.319132\pi\)
0.538126 + 0.842864i \(0.319132\pi\)
\(338\) −15.2426 −0.829090
\(339\) 4.48528 0.243607
\(340\) 26.1421 1.41776
\(341\) 14.8284 0.803004
\(342\) 2.41421 0.130546
\(343\) −8.00000 −0.431959
\(344\) 15.0711 0.812578
\(345\) −3.65685 −0.196878
\(346\) 53.4558 2.87380
\(347\) −6.48528 −0.348148 −0.174074 0.984733i \(-0.555693\pi\)
−0.174074 + 0.984733i \(0.555693\pi\)
\(348\) −19.4142 −1.04071
\(349\) 6.68629 0.357909 0.178954 0.983857i \(-0.442729\pi\)
0.178954 + 0.983857i \(0.442729\pi\)
\(350\) 8.24264 0.440588
\(351\) −2.58579 −0.138019
\(352\) 2.24264 0.119533
\(353\) −7.65685 −0.407533 −0.203767 0.979019i \(-0.565318\pi\)
−0.203767 + 0.979019i \(0.565318\pi\)
\(354\) 20.4853 1.08878
\(355\) −12.4853 −0.662650
\(356\) −56.3848 −2.98839
\(357\) 23.3137 1.23389
\(358\) 55.1127 2.91280
\(359\) −9.89949 −0.522475 −0.261238 0.965275i \(-0.584131\pi\)
−0.261238 + 0.965275i \(0.584131\pi\)
\(360\) −4.41421 −0.232649
\(361\) 1.00000 0.0526316
\(362\) −59.9411 −3.15044
\(363\) 9.00000 0.472377
\(364\) 33.7990 1.77155
\(365\) 2.00000 0.104685
\(366\) 13.6569 0.713855
\(367\) −16.5858 −0.865771 −0.432886 0.901449i \(-0.642505\pi\)
−0.432886 + 0.901449i \(0.642505\pi\)
\(368\) −10.9706 −0.571880
\(369\) −4.58579 −0.238727
\(370\) 7.41421 0.385447
\(371\) 13.6569 0.709029
\(372\) 40.1421 2.08127
\(373\) 9.89949 0.512576 0.256288 0.966600i \(-0.417500\pi\)
0.256288 + 0.966600i \(0.417500\pi\)
\(374\) 23.3137 1.20552
\(375\) 1.00000 0.0516398
\(376\) 51.4558 2.65363
\(377\) 13.1127 0.675338
\(378\) −8.24264 −0.423956
\(379\) −4.14214 −0.212767 −0.106384 0.994325i \(-0.533927\pi\)
−0.106384 + 0.994325i \(0.533927\pi\)
\(380\) −3.82843 −0.196394
\(381\) −8.00000 −0.409852
\(382\) −43.2132 −2.21098
\(383\) 12.0000 0.613171 0.306586 0.951843i \(-0.400813\pi\)
0.306586 + 0.951843i \(0.400813\pi\)
\(384\) 20.5563 1.04901
\(385\) 4.82843 0.246079
\(386\) −2.24264 −0.114147
\(387\) 3.41421 0.173554
\(388\) 16.2426 0.824595
\(389\) −18.9706 −0.961846 −0.480923 0.876763i \(-0.659698\pi\)
−0.480923 + 0.876763i \(0.659698\pi\)
\(390\) 6.24264 0.316108
\(391\) 24.9706 1.26282
\(392\) 20.5563 1.03825
\(393\) 8.72792 0.440265
\(394\) −22.1421 −1.11550
\(395\) −11.3137 −0.569254
\(396\) −5.41421 −0.272074
\(397\) 24.3431 1.22175 0.610874 0.791728i \(-0.290818\pi\)
0.610874 + 0.791728i \(0.290818\pi\)
\(398\) 1.17157 0.0587256
\(399\) −3.41421 −0.170924
\(400\) 3.00000 0.150000
\(401\) −18.0416 −0.900956 −0.450478 0.892788i \(-0.648746\pi\)
−0.450478 + 0.892788i \(0.648746\pi\)
\(402\) −28.9706 −1.44492
\(403\) −27.1127 −1.35058
\(404\) 3.17157 0.157792
\(405\) −1.00000 −0.0496904
\(406\) 41.7990 2.07445
\(407\) 4.34315 0.215282
\(408\) 30.1421 1.49226
\(409\) 26.4853 1.30961 0.654806 0.755797i \(-0.272750\pi\)
0.654806 + 0.755797i \(0.272750\pi\)
\(410\) 11.0711 0.546761
\(411\) 14.0000 0.690569
\(412\) 36.9706 1.82141
\(413\) −28.9706 −1.42555
\(414\) −8.82843 −0.433894
\(415\) −6.48528 −0.318350
\(416\) −4.10051 −0.201044
\(417\) −6.82843 −0.334390
\(418\) −3.41421 −0.166995
\(419\) 18.8701 0.921863 0.460931 0.887436i \(-0.347515\pi\)
0.460931 + 0.887436i \(0.347515\pi\)
\(420\) 13.0711 0.637803
\(421\) −37.3137 −1.81856 −0.909279 0.416186i \(-0.863366\pi\)
−0.909279 + 0.416186i \(0.863366\pi\)
\(422\) 17.6569 0.859522
\(423\) 11.6569 0.566776
\(424\) 17.6569 0.857493
\(425\) −6.82843 −0.331227
\(426\) −30.1421 −1.46039
\(427\) −19.3137 −0.934656
\(428\) 30.6274 1.48043
\(429\) 3.65685 0.176555
\(430\) −8.24264 −0.397495
\(431\) 20.4853 0.986741 0.493371 0.869819i \(-0.335765\pi\)
0.493371 + 0.869819i \(0.335765\pi\)
\(432\) −3.00000 −0.144338
\(433\) 15.0711 0.724269 0.362135 0.932126i \(-0.382048\pi\)
0.362135 + 0.932126i \(0.382048\pi\)
\(434\) −86.4264 −4.14860
\(435\) 5.07107 0.243139
\(436\) 33.7990 1.61868
\(437\) −3.65685 −0.174931
\(438\) 4.82843 0.230711
\(439\) −32.9706 −1.57360 −0.786800 0.617209i \(-0.788263\pi\)
−0.786800 + 0.617209i \(0.788263\pi\)
\(440\) 6.24264 0.297606
\(441\) 4.65685 0.221755
\(442\) −42.6274 −2.02758
\(443\) 21.3137 1.01264 0.506322 0.862344i \(-0.331005\pi\)
0.506322 + 0.862344i \(0.331005\pi\)
\(444\) 11.7574 0.557980
\(445\) 14.7279 0.698170
\(446\) 42.6274 2.01847
\(447\) −7.65685 −0.362157
\(448\) −33.5563 −1.58539
\(449\) −11.8995 −0.561572 −0.280786 0.959770i \(-0.590595\pi\)
−0.280786 + 0.959770i \(0.590595\pi\)
\(450\) 2.41421 0.113807
\(451\) 6.48528 0.305380
\(452\) −17.1716 −0.807683
\(453\) 21.7990 1.02421
\(454\) 36.1421 1.69623
\(455\) −8.82843 −0.413883
\(456\) −4.41421 −0.206714
\(457\) 23.1716 1.08392 0.541960 0.840404i \(-0.317682\pi\)
0.541960 + 0.840404i \(0.317682\pi\)
\(458\) 23.3137 1.08938
\(459\) 6.82843 0.318724
\(460\) 14.0000 0.652753
\(461\) 33.3137 1.55157 0.775787 0.630995i \(-0.217353\pi\)
0.775787 + 0.630995i \(0.217353\pi\)
\(462\) 11.6569 0.542326
\(463\) −27.8995 −1.29660 −0.648300 0.761385i \(-0.724520\pi\)
−0.648300 + 0.761385i \(0.724520\pi\)
\(464\) 15.2132 0.706255
\(465\) −10.4853 −0.486243
\(466\) −47.4558 −2.19835
\(467\) 27.6569 1.27981 0.639903 0.768455i \(-0.278974\pi\)
0.639903 + 0.768455i \(0.278974\pi\)
\(468\) 9.89949 0.457604
\(469\) 40.9706 1.89184
\(470\) −28.1421 −1.29810
\(471\) −17.7990 −0.820134
\(472\) −37.4558 −1.72404
\(473\) −4.82843 −0.222011
\(474\) −27.3137 −1.25456
\(475\) 1.00000 0.0458831
\(476\) −89.2548 −4.09099
\(477\) 4.00000 0.183147
\(478\) 13.0711 0.597857
\(479\) 38.1838 1.74466 0.872330 0.488917i \(-0.162608\pi\)
0.872330 + 0.488917i \(0.162608\pi\)
\(480\) −1.58579 −0.0723809
\(481\) −7.94113 −0.362084
\(482\) 45.7990 2.08609
\(483\) 12.4853 0.568100
\(484\) −34.4558 −1.56617
\(485\) −4.24264 −0.192648
\(486\) −2.41421 −0.109511
\(487\) 2.82843 0.128168 0.0640841 0.997944i \(-0.479587\pi\)
0.0640841 + 0.997944i \(0.479587\pi\)
\(488\) −24.9706 −1.13036
\(489\) 11.8995 0.538114
\(490\) −11.2426 −0.507891
\(491\) 2.10051 0.0947945 0.0473972 0.998876i \(-0.484907\pi\)
0.0473972 + 0.998876i \(0.484907\pi\)
\(492\) 17.5563 0.791501
\(493\) −34.6274 −1.55954
\(494\) 6.24264 0.280870
\(495\) 1.41421 0.0635642
\(496\) −31.4558 −1.41241
\(497\) 42.6274 1.91210
\(498\) −15.6569 −0.701600
\(499\) 15.7990 0.707260 0.353630 0.935385i \(-0.384947\pi\)
0.353630 + 0.935385i \(0.384947\pi\)
\(500\) −3.82843 −0.171212
\(501\) −10.0000 −0.446767
\(502\) 65.3553 2.91695
\(503\) −4.82843 −0.215289 −0.107644 0.994189i \(-0.534331\pi\)
−0.107644 + 0.994189i \(0.534331\pi\)
\(504\) 15.0711 0.671319
\(505\) −0.828427 −0.0368645
\(506\) 12.4853 0.555038
\(507\) 6.31371 0.280402
\(508\) 30.6274 1.35887
\(509\) −4.38478 −0.194352 −0.0971759 0.995267i \(-0.530981\pi\)
−0.0971759 + 0.995267i \(0.530981\pi\)
\(510\) −16.4853 −0.729981
\(511\) −6.82843 −0.302072
\(512\) −31.2426 −1.38074
\(513\) −1.00000 −0.0441511
\(514\) 16.4853 0.727135
\(515\) −9.65685 −0.425532
\(516\) −13.0711 −0.575422
\(517\) −16.4853 −0.725022
\(518\) −25.3137 −1.11222
\(519\) −22.1421 −0.971932
\(520\) −11.4142 −0.500546
\(521\) 12.3848 0.542587 0.271293 0.962497i \(-0.412549\pi\)
0.271293 + 0.962497i \(0.412549\pi\)
\(522\) 12.2426 0.535846
\(523\) 23.7990 1.04066 0.520329 0.853966i \(-0.325809\pi\)
0.520329 + 0.853966i \(0.325809\pi\)
\(524\) −33.4142 −1.45971
\(525\) −3.41421 −0.149008
\(526\) −9.31371 −0.406097
\(527\) 71.5980 3.11886
\(528\) 4.24264 0.184637
\(529\) −9.62742 −0.418583
\(530\) −9.65685 −0.419467
\(531\) −8.48528 −0.368230
\(532\) 13.0711 0.566703
\(533\) −11.8579 −0.513621
\(534\) 35.5563 1.53867
\(535\) −8.00000 −0.345870
\(536\) 52.9706 2.28798
\(537\) −22.8284 −0.985119
\(538\) −68.5269 −2.95440
\(539\) −6.58579 −0.283670
\(540\) 3.82843 0.164749
\(541\) 12.6274 0.542895 0.271448 0.962453i \(-0.412498\pi\)
0.271448 + 0.962453i \(0.412498\pi\)
\(542\) −60.7696 −2.61028
\(543\) 24.8284 1.06549
\(544\) 10.8284 0.464265
\(545\) −8.82843 −0.378168
\(546\) −21.3137 −0.912143
\(547\) −5.85786 −0.250464 −0.125232 0.992127i \(-0.539968\pi\)
−0.125232 + 0.992127i \(0.539968\pi\)
\(548\) −53.5980 −2.28959
\(549\) −5.65685 −0.241429
\(550\) −3.41421 −0.145583
\(551\) 5.07107 0.216035
\(552\) 16.1421 0.687055
\(553\) 38.6274 1.64260
\(554\) −55.4558 −2.35609
\(555\) −3.07107 −0.130360
\(556\) 26.1421 1.10867
\(557\) 10.0000 0.423714 0.211857 0.977301i \(-0.432049\pi\)
0.211857 + 0.977301i \(0.432049\pi\)
\(558\) −25.3137 −1.07161
\(559\) 8.82843 0.373403
\(560\) −10.2426 −0.432831
\(561\) −9.65685 −0.407713
\(562\) 25.8995 1.09250
\(563\) −14.2843 −0.602010 −0.301005 0.953623i \(-0.597322\pi\)
−0.301005 + 0.953623i \(0.597322\pi\)
\(564\) −44.6274 −1.87915
\(565\) 4.48528 0.188697
\(566\) −5.41421 −0.227576
\(567\) 3.41421 0.143383
\(568\) 55.1127 2.31248
\(569\) 18.7279 0.785115 0.392558 0.919727i \(-0.371590\pi\)
0.392558 + 0.919727i \(0.371590\pi\)
\(570\) 2.41421 0.101120
\(571\) −19.7990 −0.828562 −0.414281 0.910149i \(-0.635967\pi\)
−0.414281 + 0.910149i \(0.635967\pi\)
\(572\) −14.0000 −0.585369
\(573\) 17.8995 0.747762
\(574\) −37.7990 −1.57770
\(575\) −3.65685 −0.152501
\(576\) −9.82843 −0.409518
\(577\) 1.79899 0.0748929 0.0374465 0.999299i \(-0.488078\pi\)
0.0374465 + 0.999299i \(0.488078\pi\)
\(578\) 71.5269 2.97513
\(579\) 0.928932 0.0386051
\(580\) −19.4142 −0.806131
\(581\) 22.1421 0.918611
\(582\) −10.2426 −0.424571
\(583\) −5.65685 −0.234283
\(584\) −8.82843 −0.365323
\(585\) −2.58579 −0.106909
\(586\) 18.8284 0.777795
\(587\) 0.343146 0.0141631 0.00708157 0.999975i \(-0.497746\pi\)
0.00708157 + 0.999975i \(0.497746\pi\)
\(588\) −17.8284 −0.735232
\(589\) −10.4853 −0.432038
\(590\) 20.4853 0.843366
\(591\) 9.17157 0.377268
\(592\) −9.21320 −0.378660
\(593\) −6.68629 −0.274573 −0.137287 0.990531i \(-0.543838\pi\)
−0.137287 + 0.990531i \(0.543838\pi\)
\(594\) 3.41421 0.140087
\(595\) 23.3137 0.955769
\(596\) 29.3137 1.20074
\(597\) −0.485281 −0.0198612
\(598\) −22.8284 −0.933524
\(599\) 45.9411 1.87710 0.938552 0.345139i \(-0.112168\pi\)
0.938552 + 0.345139i \(0.112168\pi\)
\(600\) −4.41421 −0.180210
\(601\) −23.1716 −0.945188 −0.472594 0.881280i \(-0.656682\pi\)
−0.472594 + 0.881280i \(0.656682\pi\)
\(602\) 28.1421 1.14699
\(603\) 12.0000 0.488678
\(604\) −83.4558 −3.39577
\(605\) 9.00000 0.365902
\(606\) −2.00000 −0.0812444
\(607\) 26.1421 1.06108 0.530538 0.847661i \(-0.321990\pi\)
0.530538 + 0.847661i \(0.321990\pi\)
\(608\) −1.58579 −0.0643121
\(609\) −17.3137 −0.701587
\(610\) 13.6569 0.552950
\(611\) 30.1421 1.21942
\(612\) −26.1421 −1.05673
\(613\) −37.1127 −1.49897 −0.749484 0.662023i \(-0.769698\pi\)
−0.749484 + 0.662023i \(0.769698\pi\)
\(614\) −76.7696 −3.09817
\(615\) −4.58579 −0.184917
\(616\) −21.3137 −0.858754
\(617\) −18.1421 −0.730375 −0.365187 0.930934i \(-0.618995\pi\)
−0.365187 + 0.930934i \(0.618995\pi\)
\(618\) −23.3137 −0.937815
\(619\) −4.20101 −0.168853 −0.0844264 0.996430i \(-0.526906\pi\)
−0.0844264 + 0.996430i \(0.526906\pi\)
\(620\) 40.1421 1.61215
\(621\) 3.65685 0.146745
\(622\) −57.3553 −2.29974
\(623\) −50.2843 −2.01460
\(624\) −7.75736 −0.310543
\(625\) 1.00000 0.0400000
\(626\) −63.9411 −2.55560
\(627\) 1.41421 0.0564782
\(628\) 68.1421 2.71917
\(629\) 20.9706 0.836151
\(630\) −8.24264 −0.328395
\(631\) −22.6274 −0.900783 −0.450392 0.892831i \(-0.648716\pi\)
−0.450392 + 0.892831i \(0.648716\pi\)
\(632\) 49.9411 1.98655
\(633\) −7.31371 −0.290694
\(634\) −27.3137 −1.08477
\(635\) −8.00000 −0.317470
\(636\) −15.3137 −0.607228
\(637\) 12.0416 0.477107
\(638\) −17.3137 −0.685456
\(639\) 12.4853 0.493910
\(640\) 20.5563 0.812561
\(641\) −11.4142 −0.450834 −0.225417 0.974262i \(-0.572375\pi\)
−0.225417 + 0.974262i \(0.572375\pi\)
\(642\) −19.3137 −0.762251
\(643\) −42.0416 −1.65796 −0.828980 0.559278i \(-0.811078\pi\)
−0.828980 + 0.559278i \(0.811078\pi\)
\(644\) −47.7990 −1.88354
\(645\) 3.41421 0.134435
\(646\) −16.4853 −0.648605
\(647\) −17.1127 −0.672770 −0.336385 0.941725i \(-0.609204\pi\)
−0.336385 + 0.941725i \(0.609204\pi\)
\(648\) 4.41421 0.173407
\(649\) 12.0000 0.471041
\(650\) 6.24264 0.244857
\(651\) 35.7990 1.40307
\(652\) −45.5563 −1.78412
\(653\) −42.4264 −1.66027 −0.830137 0.557560i \(-0.811738\pi\)
−0.830137 + 0.557560i \(0.811738\pi\)
\(654\) −21.3137 −0.833432
\(655\) 8.72792 0.341028
\(656\) −13.7574 −0.537135
\(657\) −2.00000 −0.0780274
\(658\) 96.0833 3.74572
\(659\) −20.0000 −0.779089 −0.389545 0.921008i \(-0.627368\pi\)
−0.389545 + 0.921008i \(0.627368\pi\)
\(660\) −5.41421 −0.210748
\(661\) 1.51472 0.0589157 0.0294579 0.999566i \(-0.490622\pi\)
0.0294579 + 0.999566i \(0.490622\pi\)
\(662\) 30.9706 1.20371
\(663\) 17.6569 0.685735
\(664\) 28.6274 1.11096
\(665\) −3.41421 −0.132398
\(666\) −7.41421 −0.287295
\(667\) −18.5442 −0.718033
\(668\) 38.2843 1.48126
\(669\) −17.6569 −0.682653
\(670\) −28.9706 −1.11923
\(671\) 8.00000 0.308837
\(672\) 5.41421 0.208858
\(673\) 2.10051 0.0809685 0.0404843 0.999180i \(-0.487110\pi\)
0.0404843 + 0.999180i \(0.487110\pi\)
\(674\) 47.6985 1.83728
\(675\) −1.00000 −0.0384900
\(676\) −24.1716 −0.929676
\(677\) −11.0294 −0.423896 −0.211948 0.977281i \(-0.567981\pi\)
−0.211948 + 0.977281i \(0.567981\pi\)
\(678\) 10.8284 0.415863
\(679\) 14.4853 0.555894
\(680\) 30.1421 1.15590
\(681\) −14.9706 −0.573673
\(682\) 35.7990 1.37081
\(683\) −5.65685 −0.216454 −0.108227 0.994126i \(-0.534517\pi\)
−0.108227 + 0.994126i \(0.534517\pi\)
\(684\) 3.82843 0.146384
\(685\) 14.0000 0.534913
\(686\) −19.3137 −0.737401
\(687\) −9.65685 −0.368432
\(688\) 10.2426 0.390497
\(689\) 10.3431 0.394042
\(690\) −8.82843 −0.336092
\(691\) 39.1127 1.48792 0.743959 0.668226i \(-0.232946\pi\)
0.743959 + 0.668226i \(0.232946\pi\)
\(692\) 84.7696 3.22245
\(693\) −4.82843 −0.183417
\(694\) −15.6569 −0.594326
\(695\) −6.82843 −0.259017
\(696\) −22.3848 −0.848493
\(697\) 31.3137 1.18609
\(698\) 16.1421 0.610989
\(699\) 19.6569 0.743491
\(700\) 13.0711 0.494040
\(701\) −11.6569 −0.440273 −0.220137 0.975469i \(-0.570650\pi\)
−0.220137 + 0.975469i \(0.570650\pi\)
\(702\) −6.24264 −0.235613
\(703\) −3.07107 −0.115828
\(704\) 13.8995 0.523857
\(705\) 11.6569 0.439023
\(706\) −18.4853 −0.695703
\(707\) 2.82843 0.106374
\(708\) 32.4853 1.22087
\(709\) −12.6863 −0.476444 −0.238222 0.971211i \(-0.576565\pi\)
−0.238222 + 0.971211i \(0.576565\pi\)
\(710\) −30.1421 −1.13121
\(711\) 11.3137 0.424297
\(712\) −65.0122 −2.43643
\(713\) 38.3431 1.43596
\(714\) 56.2843 2.10639
\(715\) 3.65685 0.136759
\(716\) 87.3970 3.26618
\(717\) −5.41421 −0.202198
\(718\) −23.8995 −0.891921
\(719\) −47.5563 −1.77355 −0.886776 0.462199i \(-0.847061\pi\)
−0.886776 + 0.462199i \(0.847061\pi\)
\(720\) −3.00000 −0.111803
\(721\) 32.9706 1.22789
\(722\) 2.41421 0.0898477
\(723\) −18.9706 −0.705523
\(724\) −95.0538 −3.53265
\(725\) 5.07107 0.188335
\(726\) 21.7279 0.806399
\(727\) −7.41421 −0.274978 −0.137489 0.990503i \(-0.543903\pi\)
−0.137489 + 0.990503i \(0.543903\pi\)
\(728\) 38.9706 1.44435
\(729\) 1.00000 0.0370370
\(730\) 4.82843 0.178708
\(731\) −23.3137 −0.862289
\(732\) 21.6569 0.800460
\(733\) 21.3137 0.787240 0.393620 0.919273i \(-0.371223\pi\)
0.393620 + 0.919273i \(0.371223\pi\)
\(734\) −40.0416 −1.47796
\(735\) 4.65685 0.171771
\(736\) 5.79899 0.213754
\(737\) −16.9706 −0.625119
\(738\) −11.0711 −0.407532
\(739\) 1.65685 0.0609484 0.0304742 0.999536i \(-0.490298\pi\)
0.0304742 + 0.999536i \(0.490298\pi\)
\(740\) 11.7574 0.432209
\(741\) −2.58579 −0.0949912
\(742\) 32.9706 1.21039
\(743\) 7.31371 0.268314 0.134157 0.990960i \(-0.457167\pi\)
0.134157 + 0.990960i \(0.457167\pi\)
\(744\) 46.2843 1.69686
\(745\) −7.65685 −0.280525
\(746\) 23.8995 0.875023
\(747\) 6.48528 0.237284
\(748\) 36.9706 1.35178
\(749\) 27.3137 0.998021
\(750\) 2.41421 0.0881546
\(751\) 25.1127 0.916375 0.458188 0.888855i \(-0.348499\pi\)
0.458188 + 0.888855i \(0.348499\pi\)
\(752\) 34.9706 1.27525
\(753\) −27.0711 −0.986525
\(754\) 31.6569 1.15287
\(755\) 21.7990 0.793346
\(756\) −13.0711 −0.475390
\(757\) 11.1716 0.406038 0.203019 0.979175i \(-0.434925\pi\)
0.203019 + 0.979175i \(0.434925\pi\)
\(758\) −10.0000 −0.363216
\(759\) −5.17157 −0.187716
\(760\) −4.41421 −0.160120
\(761\) 46.2843 1.67780 0.838902 0.544283i \(-0.183198\pi\)
0.838902 + 0.544283i \(0.183198\pi\)
\(762\) −19.3137 −0.699662
\(763\) 30.1421 1.09122
\(764\) −68.5269 −2.47922
\(765\) 6.82843 0.246882
\(766\) 28.9706 1.04675
\(767\) −21.9411 −0.792248
\(768\) 29.9706 1.08147
\(769\) −24.3431 −0.877836 −0.438918 0.898527i \(-0.644638\pi\)
−0.438918 + 0.898527i \(0.644638\pi\)
\(770\) 11.6569 0.420084
\(771\) −6.82843 −0.245920
\(772\) −3.55635 −0.127996
\(773\) 0.970563 0.0349087 0.0174544 0.999848i \(-0.494444\pi\)
0.0174544 + 0.999848i \(0.494444\pi\)
\(774\) 8.24264 0.296276
\(775\) −10.4853 −0.376642
\(776\) 18.7279 0.672293
\(777\) 10.4853 0.376157
\(778\) −45.7990 −1.64197
\(779\) −4.58579 −0.164303
\(780\) 9.89949 0.354459
\(781\) −17.6569 −0.631812
\(782\) 60.2843 2.15576
\(783\) −5.07107 −0.181225
\(784\) 13.9706 0.498949
\(785\) −17.7990 −0.635273
\(786\) 21.0711 0.751580
\(787\) −37.4558 −1.33516 −0.667578 0.744540i \(-0.732669\pi\)
−0.667578 + 0.744540i \(0.732669\pi\)
\(788\) −35.1127 −1.25084
\(789\) 3.85786 0.137344
\(790\) −27.3137 −0.971778
\(791\) −15.3137 −0.544493
\(792\) −6.24264 −0.221823
\(793\) −14.6274 −0.519435
\(794\) 58.7696 2.08565
\(795\) 4.00000 0.141865
\(796\) 1.85786 0.0658503
\(797\) 5.17157 0.183187 0.0915933 0.995797i \(-0.470804\pi\)
0.0915933 + 0.995797i \(0.470804\pi\)
\(798\) −8.24264 −0.291786
\(799\) −79.5980 −2.81597
\(800\) −1.58579 −0.0560660
\(801\) −14.7279 −0.520386
\(802\) −43.5563 −1.53803
\(803\) 2.82843 0.0998130
\(804\) −45.9411 −1.62022
\(805\) 12.4853 0.440048
\(806\) −65.4558 −2.30558
\(807\) 28.3848 0.999191
\(808\) 3.65685 0.128648
\(809\) 26.6863 0.938240 0.469120 0.883134i \(-0.344571\pi\)
0.469120 + 0.883134i \(0.344571\pi\)
\(810\) −2.41421 −0.0848268
\(811\) 7.31371 0.256819 0.128410 0.991721i \(-0.459013\pi\)
0.128410 + 0.991721i \(0.459013\pi\)
\(812\) 66.2843 2.32612
\(813\) 25.1716 0.882806
\(814\) 10.4853 0.367509
\(815\) 11.8995 0.416821
\(816\) 20.4853 0.717128
\(817\) 3.41421 0.119448
\(818\) 63.9411 2.23565
\(819\) 8.82843 0.308490
\(820\) 17.5563 0.613094
\(821\) 0.544156 0.0189912 0.00949559 0.999955i \(-0.496977\pi\)
0.00949559 + 0.999955i \(0.496977\pi\)
\(822\) 33.7990 1.17888
\(823\) 22.7279 0.792246 0.396123 0.918198i \(-0.370355\pi\)
0.396123 + 0.918198i \(0.370355\pi\)
\(824\) 42.6274 1.48500
\(825\) 1.41421 0.0492366
\(826\) −69.9411 −2.43356
\(827\) 3.37258 0.117276 0.0586381 0.998279i \(-0.481324\pi\)
0.0586381 + 0.998279i \(0.481324\pi\)
\(828\) −14.0000 −0.486534
\(829\) −21.5147 −0.747237 −0.373619 0.927582i \(-0.621883\pi\)
−0.373619 + 0.927582i \(0.621883\pi\)
\(830\) −15.6569 −0.543457
\(831\) 22.9706 0.796840
\(832\) −25.4142 −0.881079
\(833\) −31.7990 −1.10177
\(834\) −16.4853 −0.570839
\(835\) −10.0000 −0.346064
\(836\) −5.41421 −0.187254
\(837\) 10.4853 0.362424
\(838\) 45.5563 1.57372
\(839\) −35.1127 −1.21222 −0.606112 0.795379i \(-0.707272\pi\)
−0.606112 + 0.795379i \(0.707272\pi\)
\(840\) 15.0711 0.520001
\(841\) −3.28427 −0.113251
\(842\) −90.0833 −3.10447
\(843\) −10.7279 −0.369489
\(844\) 28.0000 0.963800
\(845\) 6.31371 0.217198
\(846\) 28.1421 0.967547
\(847\) −30.7279 −1.05582
\(848\) 12.0000 0.412082
\(849\) 2.24264 0.0769672
\(850\) −16.4853 −0.565440
\(851\) 11.2304 0.384975
\(852\) −47.7990 −1.63757
\(853\) −26.4853 −0.906839 −0.453419 0.891297i \(-0.649796\pi\)
−0.453419 + 0.891297i \(0.649796\pi\)
\(854\) −46.6274 −1.59556
\(855\) −1.00000 −0.0341993
\(856\) 35.3137 1.20700
\(857\) −29.9411 −1.02277 −0.511385 0.859352i \(-0.670867\pi\)
−0.511385 + 0.859352i \(0.670867\pi\)
\(858\) 8.82843 0.301398
\(859\) −41.9411 −1.43101 −0.715506 0.698606i \(-0.753804\pi\)
−0.715506 + 0.698606i \(0.753804\pi\)
\(860\) −13.0711 −0.445720
\(861\) 15.6569 0.533584
\(862\) 49.4558 1.68447
\(863\) −8.68629 −0.295685 −0.147842 0.989011i \(-0.547233\pi\)
−0.147842 + 0.989011i \(0.547233\pi\)
\(864\) 1.58579 0.0539496
\(865\) −22.1421 −0.752855
\(866\) 36.3848 1.23641
\(867\) −29.6274 −1.00620
\(868\) −137.054 −4.65191
\(869\) −16.0000 −0.542763
\(870\) 12.2426 0.415064
\(871\) 31.0294 1.05139
\(872\) 38.9706 1.31971
\(873\) 4.24264 0.143592
\(874\) −8.82843 −0.298626
\(875\) −3.41421 −0.115421
\(876\) 7.65685 0.258701
\(877\) −1.89949 −0.0641414 −0.0320707 0.999486i \(-0.510210\pi\)
−0.0320707 + 0.999486i \(0.510210\pi\)
\(878\) −79.5980 −2.68630
\(879\) −7.79899 −0.263053
\(880\) 4.24264 0.143019
\(881\) 3.45584 0.116430 0.0582152 0.998304i \(-0.481459\pi\)
0.0582152 + 0.998304i \(0.481459\pi\)
\(882\) 11.2426 0.378559
\(883\) −18.5269 −0.623480 −0.311740 0.950167i \(-0.600912\pi\)
−0.311740 + 0.950167i \(0.600912\pi\)
\(884\) −67.5980 −2.27357
\(885\) −8.48528 −0.285230
\(886\) 51.4558 1.72869
\(887\) −8.00000 −0.268614 −0.134307 0.990940i \(-0.542881\pi\)
−0.134307 + 0.990940i \(0.542881\pi\)
\(888\) 13.5563 0.454921
\(889\) 27.3137 0.916072
\(890\) 35.5563 1.19185
\(891\) −1.41421 −0.0473779
\(892\) 67.5980 2.26335
\(893\) 11.6569 0.390082
\(894\) −18.4853 −0.618240
\(895\) −22.8284 −0.763070
\(896\) −70.1838 −2.34468
\(897\) 9.45584 0.315721
\(898\) −28.7279 −0.958663
\(899\) −53.1716 −1.77337
\(900\) 3.82843 0.127614
\(901\) −27.3137 −0.909952
\(902\) 15.6569 0.521316
\(903\) −11.6569 −0.387916
\(904\) −19.7990 −0.658505
\(905\) 24.8284 0.825325
\(906\) 52.6274 1.74843
\(907\) 9.85786 0.327325 0.163663 0.986516i \(-0.447669\pi\)
0.163663 + 0.986516i \(0.447669\pi\)
\(908\) 57.3137 1.90202
\(909\) 0.828427 0.0274772
\(910\) −21.3137 −0.706543
\(911\) −0.686292 −0.0227379 −0.0113689 0.999935i \(-0.503619\pi\)
−0.0113689 + 0.999935i \(0.503619\pi\)
\(912\) −3.00000 −0.0993399
\(913\) −9.17157 −0.303535
\(914\) 55.9411 1.85037
\(915\) −5.65685 −0.187010
\(916\) 36.9706 1.22154
\(917\) −29.7990 −0.984049
\(918\) 16.4853 0.544095
\(919\) −12.0000 −0.395843 −0.197922 0.980218i \(-0.563419\pi\)
−0.197922 + 0.980218i \(0.563419\pi\)
\(920\) 16.1421 0.532190
\(921\) 31.7990 1.04781
\(922\) 80.4264 2.64870
\(923\) 32.2843 1.06265
\(924\) 18.4853 0.608121
\(925\) −3.07107 −0.100976
\(926\) −67.3553 −2.21343
\(927\) 9.65685 0.317173
\(928\) −8.04163 −0.263979
\(929\) −0.544156 −0.0178532 −0.00892659 0.999960i \(-0.502841\pi\)
−0.00892659 + 0.999960i \(0.502841\pi\)
\(930\) −25.3137 −0.830069
\(931\) 4.65685 0.152622
\(932\) −75.2548 −2.46505
\(933\) 23.7574 0.777781
\(934\) 66.7696 2.18477
\(935\) −9.65685 −0.315813
\(936\) 11.4142 0.373085
\(937\) 54.7696 1.78924 0.894622 0.446824i \(-0.147445\pi\)
0.894622 + 0.446824i \(0.147445\pi\)
\(938\) 98.9117 3.22958
\(939\) 26.4853 0.864314
\(940\) −44.6274 −1.45559
\(941\) 13.5563 0.441924 0.220962 0.975282i \(-0.429080\pi\)
0.220962 + 0.975282i \(0.429080\pi\)
\(942\) −42.9706 −1.40006
\(943\) 16.7696 0.546092
\(944\) −25.4558 −0.828517
\(945\) 3.41421 0.111064
\(946\) −11.6569 −0.378997
\(947\) 7.17157 0.233045 0.116522 0.993188i \(-0.462825\pi\)
0.116522 + 0.993188i \(0.462825\pi\)
\(948\) −43.3137 −1.40676
\(949\) −5.17157 −0.167876
\(950\) 2.41421 0.0783274
\(951\) 11.3137 0.366872
\(952\) −102.912 −3.33539
\(953\) −34.1421 −1.10597 −0.552986 0.833190i \(-0.686512\pi\)
−0.552986 + 0.833190i \(0.686512\pi\)
\(954\) 9.65685 0.312652
\(955\) 17.8995 0.579214
\(956\) 20.7279 0.670389
\(957\) 7.17157 0.231824
\(958\) 92.1838 2.97832
\(959\) −47.7990 −1.54351
\(960\) −9.82843 −0.317211
\(961\) 78.9411 2.54649
\(962\) −19.1716 −0.618116
\(963\) 8.00000 0.257796
\(964\) 72.6274 2.33917
\(965\) 0.928932 0.0299034
\(966\) 30.1421 0.969807
\(967\) −8.10051 −0.260495 −0.130247 0.991482i \(-0.541577\pi\)
−0.130247 + 0.991482i \(0.541577\pi\)
\(968\) −39.7279 −1.27690
\(969\) 6.82843 0.219361
\(970\) −10.2426 −0.328871
\(971\) 6.34315 0.203561 0.101781 0.994807i \(-0.467546\pi\)
0.101781 + 0.994807i \(0.467546\pi\)
\(972\) −3.82843 −0.122797
\(973\) 23.3137 0.747403
\(974\) 6.82843 0.218797
\(975\) −2.58579 −0.0828114
\(976\) −16.9706 −0.543214
\(977\) −39.5980 −1.26685 −0.633426 0.773803i \(-0.718352\pi\)
−0.633426 + 0.773803i \(0.718352\pi\)
\(978\) 28.7279 0.918618
\(979\) 20.8284 0.665679
\(980\) −17.8284 −0.569508
\(981\) 8.82843 0.281870
\(982\) 5.07107 0.161824
\(983\) −35.9411 −1.14634 −0.573172 0.819435i \(-0.694287\pi\)
−0.573172 + 0.819435i \(0.694287\pi\)
\(984\) 20.2426 0.645312
\(985\) 9.17157 0.292231
\(986\) −83.5980 −2.66230
\(987\) −39.7990 −1.26682
\(988\) 9.89949 0.314945
\(989\) −12.4853 −0.397009
\(990\) 3.41421 0.108511
\(991\) −50.3431 −1.59920 −0.799601 0.600531i \(-0.794956\pi\)
−0.799601 + 0.600531i \(0.794956\pi\)
\(992\) 16.6274 0.527921
\(993\) −12.8284 −0.407098
\(994\) 102.912 3.26416
\(995\) −0.485281 −0.0153845
\(996\) −24.8284 −0.786719
\(997\) −33.5147 −1.06142 −0.530711 0.847553i \(-0.678075\pi\)
−0.530711 + 0.847553i \(0.678075\pi\)
\(998\) 38.1421 1.20737
\(999\) 3.07107 0.0971643
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 285.2.a.f.1.2 2
3.2 odd 2 855.2.a.e.1.1 2
4.3 odd 2 4560.2.a.bj.1.1 2
5.2 odd 4 1425.2.c.j.799.4 4
5.3 odd 4 1425.2.c.j.799.1 4
5.4 even 2 1425.2.a.l.1.1 2
15.14 odd 2 4275.2.a.x.1.2 2
19.18 odd 2 5415.2.a.p.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
285.2.a.f.1.2 2 1.1 even 1 trivial
855.2.a.e.1.1 2 3.2 odd 2
1425.2.a.l.1.1 2 5.4 even 2
1425.2.c.j.799.1 4 5.3 odd 4
1425.2.c.j.799.4 4 5.2 odd 4
4275.2.a.x.1.2 2 15.14 odd 2
4560.2.a.bj.1.1 2 4.3 odd 2
5415.2.a.p.1.1 2 19.18 odd 2