Properties

Label 4830.2.a.bz.1.2
Level $4830$
Weight $2$
Character 4830.1
Self dual yes
Analytic conductor $38.568$
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] = [4830,2,Mod(1,4830)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4830, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("4830.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4830 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4830.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: \(38.5677441763\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.17009\) of defining polynomial
Character \(\chi\) \(=\) 4830.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +3.26180 q^{11} -1.00000 q^{12} -4.34017 q^{13} +1.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} +5.75872 q^{17} +1.00000 q^{18} -3.07838 q^{19} -1.00000 q^{20} -1.00000 q^{21} +3.26180 q^{22} +1.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} -4.34017 q^{26} -1.00000 q^{27} +1.00000 q^{28} +0.581449 q^{29} +1.00000 q^{30} +3.07838 q^{31} +1.00000 q^{32} -3.26180 q^{33} +5.75872 q^{34} -1.00000 q^{35} +1.00000 q^{36} +2.00000 q^{37} -3.07838 q^{38} +4.34017 q^{39} -1.00000 q^{40} -8.83710 q^{41} -1.00000 q^{42} +0.738205 q^{43} +3.26180 q^{44} -1.00000 q^{45} +1.00000 q^{46} +3.07838 q^{47} -1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} -5.75872 q^{51} -4.34017 q^{52} +8.83710 q^{53} -1.00000 q^{54} -3.26180 q^{55} +1.00000 q^{56} +3.07838 q^{57} +0.581449 q^{58} +6.15676 q^{59} +1.00000 q^{60} +8.15676 q^{61} +3.07838 q^{62} +1.00000 q^{63} +1.00000 q^{64} +4.34017 q^{65} -3.26180 q^{66} +0.738205 q^{67} +5.75872 q^{68} -1.00000 q^{69} -1.00000 q^{70} -5.41855 q^{71} +1.00000 q^{72} -6.31351 q^{73} +2.00000 q^{74} -1.00000 q^{75} -3.07838 q^{76} +3.26180 q^{77} +4.34017 q^{78} -2.15676 q^{79} -1.00000 q^{80} +1.00000 q^{81} -8.83710 q^{82} -9.60197 q^{83} -1.00000 q^{84} -5.75872 q^{85} +0.738205 q^{86} -0.581449 q^{87} +3.26180 q^{88} +10.8638 q^{89} -1.00000 q^{90} -4.34017 q^{91} +1.00000 q^{92} -3.07838 q^{93} +3.07838 q^{94} +3.07838 q^{95} -1.00000 q^{96} -4.34017 q^{97} +1.00000 q^{98} +3.26180 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{5} - 3 q^{6} + 3 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{5} - 3 q^{6} + 3 q^{7} + 3 q^{8} + 3 q^{9} - 3 q^{10} + 2 q^{11} - 3 q^{12} - 2 q^{13} + 3 q^{14} + 3 q^{15} + 3 q^{16} - 8 q^{17} + 3 q^{18} - 6 q^{19} - 3 q^{20} - 3 q^{21} + 2 q^{22} + 3 q^{23} - 3 q^{24} + 3 q^{25} - 2 q^{26} - 3 q^{27} + 3 q^{28} + 16 q^{29} + 3 q^{30} + 6 q^{31} + 3 q^{32} - 2 q^{33} - 8 q^{34} - 3 q^{35} + 3 q^{36} + 6 q^{37} - 6 q^{38} + 2 q^{39} - 3 q^{40} + 2 q^{41} - 3 q^{42} + 10 q^{43} + 2 q^{44} - 3 q^{45} + 3 q^{46} + 6 q^{47} - 3 q^{48} + 3 q^{49} + 3 q^{50} + 8 q^{51} - 2 q^{52} - 2 q^{53} - 3 q^{54} - 2 q^{55} + 3 q^{56} + 6 q^{57} + 16 q^{58} + 12 q^{59} + 3 q^{60} + 18 q^{61} + 6 q^{62} + 3 q^{63} + 3 q^{64} + 2 q^{65} - 2 q^{66} + 10 q^{67} - 8 q^{68} - 3 q^{69} - 3 q^{70} - 2 q^{71} + 3 q^{72} - 6 q^{73} + 6 q^{74} - 3 q^{75} - 6 q^{76} + 2 q^{77} + 2 q^{78} - 3 q^{80} + 3 q^{81} + 2 q^{82} - 10 q^{83} - 3 q^{84} + 8 q^{85} + 10 q^{86} - 16 q^{87} + 2 q^{88} + 6 q^{89} - 3 q^{90} - 2 q^{91} + 3 q^{92} - 6 q^{93} + 6 q^{94} + 6 q^{95} - 3 q^{96} - 2 q^{97} + 3 q^{98} + 2 q^{99}+O(q^{100}) \) 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
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) −1.00000 −0.408248
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 3.26180 0.983468 0.491734 0.870745i \(-0.336363\pi\)
0.491734 + 0.870745i \(0.336363\pi\)
\(12\) −1.00000 −0.288675
\(13\) −4.34017 −1.20375 −0.601874 0.798591i \(-0.705579\pi\)
−0.601874 + 0.798591i \(0.705579\pi\)
\(14\) 1.00000 0.267261
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) 5.75872 1.39670 0.698348 0.715759i \(-0.253919\pi\)
0.698348 + 0.715759i \(0.253919\pi\)
\(18\) 1.00000 0.235702
\(19\) −3.07838 −0.706228 −0.353114 0.935580i \(-0.614877\pi\)
−0.353114 + 0.935580i \(0.614877\pi\)
\(20\) −1.00000 −0.223607
\(21\) −1.00000 −0.218218
\(22\) 3.26180 0.695417
\(23\) 1.00000 0.208514
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) −4.34017 −0.851178
\(27\) −1.00000 −0.192450
\(28\) 1.00000 0.188982
\(29\) 0.581449 0.107972 0.0539862 0.998542i \(-0.482807\pi\)
0.0539862 + 0.998542i \(0.482807\pi\)
\(30\) 1.00000 0.182574
\(31\) 3.07838 0.552893 0.276446 0.961029i \(-0.410843\pi\)
0.276446 + 0.961029i \(0.410843\pi\)
\(32\) 1.00000 0.176777
\(33\) −3.26180 −0.567806
\(34\) 5.75872 0.987613
\(35\) −1.00000 −0.169031
\(36\) 1.00000 0.166667
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) −3.07838 −0.499379
\(39\) 4.34017 0.694984
\(40\) −1.00000 −0.158114
\(41\) −8.83710 −1.38012 −0.690062 0.723751i \(-0.742417\pi\)
−0.690062 + 0.723751i \(0.742417\pi\)
\(42\) −1.00000 −0.154303
\(43\) 0.738205 0.112575 0.0562876 0.998415i \(-0.482074\pi\)
0.0562876 + 0.998415i \(0.482074\pi\)
\(44\) 3.26180 0.491734
\(45\) −1.00000 −0.149071
\(46\) 1.00000 0.147442
\(47\) 3.07838 0.449028 0.224514 0.974471i \(-0.427921\pi\)
0.224514 + 0.974471i \(0.427921\pi\)
\(48\) −1.00000 −0.144338
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) −5.75872 −0.806383
\(52\) −4.34017 −0.601874
\(53\) 8.83710 1.21387 0.606935 0.794752i \(-0.292399\pi\)
0.606935 + 0.794752i \(0.292399\pi\)
\(54\) −1.00000 −0.136083
\(55\) −3.26180 −0.439820
\(56\) 1.00000 0.133631
\(57\) 3.07838 0.407741
\(58\) 0.581449 0.0763480
\(59\) 6.15676 0.801541 0.400771 0.916178i \(-0.368742\pi\)
0.400771 + 0.916178i \(0.368742\pi\)
\(60\) 1.00000 0.129099
\(61\) 8.15676 1.04437 0.522183 0.852834i \(-0.325118\pi\)
0.522183 + 0.852834i \(0.325118\pi\)
\(62\) 3.07838 0.390954
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) 4.34017 0.538332
\(66\) −3.26180 −0.401499
\(67\) 0.738205 0.0901861 0.0450930 0.998983i \(-0.485642\pi\)
0.0450930 + 0.998983i \(0.485642\pi\)
\(68\) 5.75872 0.698348
\(69\) −1.00000 −0.120386
\(70\) −1.00000 −0.119523
\(71\) −5.41855 −0.643064 −0.321532 0.946899i \(-0.604198\pi\)
−0.321532 + 0.946899i \(0.604198\pi\)
\(72\) 1.00000 0.117851
\(73\) −6.31351 −0.738941 −0.369470 0.929243i \(-0.620461\pi\)
−0.369470 + 0.929243i \(0.620461\pi\)
\(74\) 2.00000 0.232495
\(75\) −1.00000 −0.115470
\(76\) −3.07838 −0.353114
\(77\) 3.26180 0.371716
\(78\) 4.34017 0.491428
\(79\) −2.15676 −0.242654 −0.121327 0.992613i \(-0.538715\pi\)
−0.121327 + 0.992613i \(0.538715\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) −8.83710 −0.975895
\(83\) −9.60197 −1.05395 −0.526976 0.849880i \(-0.676674\pi\)
−0.526976 + 0.849880i \(0.676674\pi\)
\(84\) −1.00000 −0.109109
\(85\) −5.75872 −0.624621
\(86\) 0.738205 0.0796027
\(87\) −0.581449 −0.0623379
\(88\) 3.26180 0.347709
\(89\) 10.8638 1.15156 0.575778 0.817606i \(-0.304699\pi\)
0.575778 + 0.817606i \(0.304699\pi\)
\(90\) −1.00000 −0.105409
\(91\) −4.34017 −0.454974
\(92\) 1.00000 0.104257
\(93\) −3.07838 −0.319213
\(94\) 3.07838 0.317510
\(95\) 3.07838 0.315835
\(96\) −1.00000 −0.102062
\(97\) −4.34017 −0.440678 −0.220339 0.975423i \(-0.570716\pi\)
−0.220339 + 0.975423i \(0.570716\pi\)
\(98\) 1.00000 0.101015
\(99\) 3.26180 0.327823
\(100\) 1.00000 0.100000
\(101\) −1.81658 −0.180757 −0.0903784 0.995908i \(-0.528808\pi\)
−0.0903784 + 0.995908i \(0.528808\pi\)
\(102\) −5.75872 −0.570199
\(103\) −2.52359 −0.248657 −0.124328 0.992241i \(-0.539678\pi\)
−0.124328 + 0.992241i \(0.539678\pi\)
\(104\) −4.34017 −0.425589
\(105\) 1.00000 0.0975900
\(106\) 8.83710 0.858335
\(107\) 8.68035 0.839161 0.419580 0.907718i \(-0.362177\pi\)
0.419580 + 0.907718i \(0.362177\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 19.3607 1.85442 0.927209 0.374544i \(-0.122201\pi\)
0.927209 + 0.374544i \(0.122201\pi\)
\(110\) −3.26180 −0.311000
\(111\) −2.00000 −0.189832
\(112\) 1.00000 0.0944911
\(113\) 8.15676 0.767323 0.383662 0.923474i \(-0.374663\pi\)
0.383662 + 0.923474i \(0.374663\pi\)
\(114\) 3.07838 0.288316
\(115\) −1.00000 −0.0932505
\(116\) 0.581449 0.0539862
\(117\) −4.34017 −0.401249
\(118\) 6.15676 0.566775
\(119\) 5.75872 0.527901
\(120\) 1.00000 0.0912871
\(121\) −0.360692 −0.0327902
\(122\) 8.15676 0.738478
\(123\) 8.83710 0.796815
\(124\) 3.07838 0.276446
\(125\) −1.00000 −0.0894427
\(126\) 1.00000 0.0890871
\(127\) 4.73820 0.420448 0.210224 0.977653i \(-0.432581\pi\)
0.210224 + 0.977653i \(0.432581\pi\)
\(128\) 1.00000 0.0883883
\(129\) −0.738205 −0.0649953
\(130\) 4.34017 0.380658
\(131\) 17.6742 1.54420 0.772101 0.635500i \(-0.219206\pi\)
0.772101 + 0.635500i \(0.219206\pi\)
\(132\) −3.26180 −0.283903
\(133\) −3.07838 −0.266929
\(134\) 0.738205 0.0637712
\(135\) 1.00000 0.0860663
\(136\) 5.75872 0.493806
\(137\) 2.00000 0.170872 0.0854358 0.996344i \(-0.472772\pi\)
0.0854358 + 0.996344i \(0.472772\pi\)
\(138\) −1.00000 −0.0851257
\(139\) −0.680346 −0.0577062 −0.0288531 0.999584i \(-0.509186\pi\)
−0.0288531 + 0.999584i \(0.509186\pi\)
\(140\) −1.00000 −0.0845154
\(141\) −3.07838 −0.259246
\(142\) −5.41855 −0.454715
\(143\) −14.1568 −1.18385
\(144\) 1.00000 0.0833333
\(145\) −0.581449 −0.0482867
\(146\) −6.31351 −0.522510
\(147\) −1.00000 −0.0824786
\(148\) 2.00000 0.164399
\(149\) 10.0000 0.819232 0.409616 0.912258i \(-0.365663\pi\)
0.409616 + 0.912258i \(0.365663\pi\)
\(150\) −1.00000 −0.0816497
\(151\) −4.68035 −0.380881 −0.190441 0.981699i \(-0.560992\pi\)
−0.190441 + 0.981699i \(0.560992\pi\)
\(152\) −3.07838 −0.249689
\(153\) 5.75872 0.465565
\(154\) 3.26180 0.262843
\(155\) −3.07838 −0.247261
\(156\) 4.34017 0.347492
\(157\) 4.52359 0.361022 0.180511 0.983573i \(-0.442225\pi\)
0.180511 + 0.983573i \(0.442225\pi\)
\(158\) −2.15676 −0.171582
\(159\) −8.83710 −0.700828
\(160\) −1.00000 −0.0790569
\(161\) 1.00000 0.0788110
\(162\) 1.00000 0.0785674
\(163\) 9.84324 0.770982 0.385491 0.922712i \(-0.374032\pi\)
0.385491 + 0.922712i \(0.374032\pi\)
\(164\) −8.83710 −0.690062
\(165\) 3.26180 0.253930
\(166\) −9.60197 −0.745257
\(167\) −5.91548 −0.457754 −0.228877 0.973455i \(-0.573505\pi\)
−0.228877 + 0.973455i \(0.573505\pi\)
\(168\) −1.00000 −0.0771517
\(169\) 5.83710 0.449008
\(170\) −5.75872 −0.441674
\(171\) −3.07838 −0.235409
\(172\) 0.738205 0.0562876
\(173\) −11.2351 −0.854191 −0.427096 0.904206i \(-0.640463\pi\)
−0.427096 + 0.904206i \(0.640463\pi\)
\(174\) −0.581449 −0.0440796
\(175\) 1.00000 0.0755929
\(176\) 3.26180 0.245867
\(177\) −6.15676 −0.462770
\(178\) 10.8638 0.814273
\(179\) 12.3135 0.920355 0.460178 0.887827i \(-0.347786\pi\)
0.460178 + 0.887827i \(0.347786\pi\)
\(180\) −1.00000 −0.0745356
\(181\) 17.5174 1.30206 0.651031 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(182\) −4.34017 −0.321715
\(183\) −8.15676 −0.602965
\(184\) 1.00000 0.0737210
\(185\) −2.00000 −0.147043
\(186\) −3.07838 −0.225718
\(187\) 18.7838 1.37361
\(188\) 3.07838 0.224514
\(189\) −1.00000 −0.0727393
\(190\) 3.07838 0.223329
\(191\) −6.15676 −0.445487 −0.222744 0.974877i \(-0.571501\pi\)
−0.222744 + 0.974877i \(0.571501\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 4.15676 0.299210 0.149605 0.988746i \(-0.452200\pi\)
0.149605 + 0.988746i \(0.452200\pi\)
\(194\) −4.34017 −0.311606
\(195\) −4.34017 −0.310806
\(196\) 1.00000 0.0714286
\(197\) 12.1568 0.866133 0.433066 0.901362i \(-0.357432\pi\)
0.433066 + 0.901362i \(0.357432\pi\)
\(198\) 3.26180 0.231806
\(199\) 18.5236 1.31310 0.656551 0.754281i \(-0.272015\pi\)
0.656551 + 0.754281i \(0.272015\pi\)
\(200\) 1.00000 0.0707107
\(201\) −0.738205 −0.0520689
\(202\) −1.81658 −0.127814
\(203\) 0.581449 0.0408097
\(204\) −5.75872 −0.403191
\(205\) 8.83710 0.617210
\(206\) −2.52359 −0.175827
\(207\) 1.00000 0.0695048
\(208\) −4.34017 −0.300937
\(209\) −10.0410 −0.694553
\(210\) 1.00000 0.0690066
\(211\) 27.2039 1.87280 0.936398 0.350940i \(-0.114138\pi\)
0.936398 + 0.350940i \(0.114138\pi\)
\(212\) 8.83710 0.606935
\(213\) 5.41855 0.371273
\(214\) 8.68035 0.593376
\(215\) −0.738205 −0.0503451
\(216\) −1.00000 −0.0680414
\(217\) 3.07838 0.208974
\(218\) 19.3607 1.31127
\(219\) 6.31351 0.426628
\(220\) −3.26180 −0.219910
\(221\) −24.9939 −1.68127
\(222\) −2.00000 −0.134231
\(223\) −0.496928 −0.0332768 −0.0166384 0.999862i \(-0.505296\pi\)
−0.0166384 + 0.999862i \(0.505296\pi\)
\(224\) 1.00000 0.0668153
\(225\) 1.00000 0.0666667
\(226\) 8.15676 0.542579
\(227\) −10.5958 −0.703270 −0.351635 0.936137i \(-0.614374\pi\)
−0.351635 + 0.936137i \(0.614374\pi\)
\(228\) 3.07838 0.203871
\(229\) 7.16290 0.473338 0.236669 0.971590i \(-0.423944\pi\)
0.236669 + 0.971590i \(0.423944\pi\)
\(230\) −1.00000 −0.0659380
\(231\) −3.26180 −0.214610
\(232\) 0.581449 0.0381740
\(233\) −19.3607 −1.26836 −0.634181 0.773185i \(-0.718662\pi\)
−0.634181 + 0.773185i \(0.718662\pi\)
\(234\) −4.34017 −0.283726
\(235\) −3.07838 −0.200811
\(236\) 6.15676 0.400771
\(237\) 2.15676 0.140096
\(238\) 5.75872 0.373283
\(239\) −5.41855 −0.350497 −0.175248 0.984524i \(-0.556073\pi\)
−0.175248 + 0.984524i \(0.556073\pi\)
\(240\) 1.00000 0.0645497
\(241\) 5.75872 0.370952 0.185476 0.982649i \(-0.440617\pi\)
0.185476 + 0.982649i \(0.440617\pi\)
\(242\) −0.360692 −0.0231862
\(243\) −1.00000 −0.0641500
\(244\) 8.15676 0.522183
\(245\) −1.00000 −0.0638877
\(246\) 8.83710 0.563433
\(247\) 13.3607 0.850120
\(248\) 3.07838 0.195477
\(249\) 9.60197 0.608500
\(250\) −1.00000 −0.0632456
\(251\) 17.1773 1.08422 0.542110 0.840308i \(-0.317626\pi\)
0.542110 + 0.840308i \(0.317626\pi\)
\(252\) 1.00000 0.0629941
\(253\) 3.26180 0.205067
\(254\) 4.73820 0.297301
\(255\) 5.75872 0.360625
\(256\) 1.00000 0.0625000
\(257\) −11.6742 −0.728217 −0.364108 0.931357i \(-0.618626\pi\)
−0.364108 + 0.931357i \(0.618626\pi\)
\(258\) −0.738205 −0.0459586
\(259\) 2.00000 0.124274
\(260\) 4.34017 0.269166
\(261\) 0.581449 0.0359908
\(262\) 17.6742 1.09192
\(263\) −19.8843 −1.22612 −0.613059 0.790037i \(-0.710061\pi\)
−0.613059 + 0.790037i \(0.710061\pi\)
\(264\) −3.26180 −0.200750
\(265\) −8.83710 −0.542859
\(266\) −3.07838 −0.188747
\(267\) −10.8638 −0.664852
\(268\) 0.738205 0.0450930
\(269\) −9.70086 −0.591472 −0.295736 0.955270i \(-0.595565\pi\)
−0.295736 + 0.955270i \(0.595565\pi\)
\(270\) 1.00000 0.0608581
\(271\) 3.07838 0.186998 0.0934991 0.995619i \(-0.470195\pi\)
0.0934991 + 0.995619i \(0.470195\pi\)
\(272\) 5.75872 0.349174
\(273\) 4.34017 0.262679
\(274\) 2.00000 0.120824
\(275\) 3.26180 0.196694
\(276\) −1.00000 −0.0601929
\(277\) −5.26180 −0.316151 −0.158075 0.987427i \(-0.550529\pi\)
−0.158075 + 0.987427i \(0.550529\pi\)
\(278\) −0.680346 −0.0408044
\(279\) 3.07838 0.184298
\(280\) −1.00000 −0.0597614
\(281\) 5.57531 0.332595 0.166297 0.986076i \(-0.446819\pi\)
0.166297 + 0.986076i \(0.446819\pi\)
\(282\) −3.07838 −0.183315
\(283\) −6.65368 −0.395520 −0.197760 0.980250i \(-0.563367\pi\)
−0.197760 + 0.980250i \(0.563367\pi\)
\(284\) −5.41855 −0.321532
\(285\) −3.07838 −0.182347
\(286\) −14.1568 −0.837107
\(287\) −8.83710 −0.521638
\(288\) 1.00000 0.0589256
\(289\) 16.1629 0.950759
\(290\) −0.581449 −0.0341439
\(291\) 4.34017 0.254425
\(292\) −6.31351 −0.369470
\(293\) 6.68035 0.390270 0.195135 0.980776i \(-0.437486\pi\)
0.195135 + 0.980776i \(0.437486\pi\)
\(294\) −1.00000 −0.0583212
\(295\) −6.15676 −0.358460
\(296\) 2.00000 0.116248
\(297\) −3.26180 −0.189269
\(298\) 10.0000 0.579284
\(299\) −4.34017 −0.250999
\(300\) −1.00000 −0.0577350
\(301\) 0.738205 0.0425494
\(302\) −4.68035 −0.269324
\(303\) 1.81658 0.104360
\(304\) −3.07838 −0.176557
\(305\) −8.15676 −0.467054
\(306\) 5.75872 0.329204
\(307\) 31.8843 1.81973 0.909866 0.414902i \(-0.136184\pi\)
0.909866 + 0.414902i \(0.136184\pi\)
\(308\) 3.26180 0.185858
\(309\) 2.52359 0.143562
\(310\) −3.07838 −0.174840
\(311\) 9.17727 0.520395 0.260198 0.965555i \(-0.416212\pi\)
0.260198 + 0.965555i \(0.416212\pi\)
\(312\) 4.34017 0.245714
\(313\) −21.3340 −1.20587 −0.602935 0.797790i \(-0.706002\pi\)
−0.602935 + 0.797790i \(0.706002\pi\)
\(314\) 4.52359 0.255281
\(315\) −1.00000 −0.0563436
\(316\) −2.15676 −0.121327
\(317\) 6.99386 0.392814 0.196407 0.980522i \(-0.437073\pi\)
0.196407 + 0.980522i \(0.437073\pi\)
\(318\) −8.83710 −0.495560
\(319\) 1.89657 0.106187
\(320\) −1.00000 −0.0559017
\(321\) −8.68035 −0.484490
\(322\) 1.00000 0.0557278
\(323\) −17.7275 −0.986386
\(324\) 1.00000 0.0555556
\(325\) −4.34017 −0.240749
\(326\) 9.84324 0.545167
\(327\) −19.3607 −1.07065
\(328\) −8.83710 −0.487947
\(329\) 3.07838 0.169716
\(330\) 3.26180 0.179556
\(331\) 2.83710 0.155941 0.0779706 0.996956i \(-0.475156\pi\)
0.0779706 + 0.996956i \(0.475156\pi\)
\(332\) −9.60197 −0.526976
\(333\) 2.00000 0.109599
\(334\) −5.91548 −0.323681
\(335\) −0.738205 −0.0403324
\(336\) −1.00000 −0.0545545
\(337\) 8.94828 0.487444 0.243722 0.969845i \(-0.421632\pi\)
0.243722 + 0.969845i \(0.421632\pi\)
\(338\) 5.83710 0.317496
\(339\) −8.15676 −0.443014
\(340\) −5.75872 −0.312311
\(341\) 10.0410 0.543753
\(342\) −3.07838 −0.166460
\(343\) 1.00000 0.0539949
\(344\) 0.738205 0.0398013
\(345\) 1.00000 0.0538382
\(346\) −11.2351 −0.604005
\(347\) 8.68035 0.465985 0.232993 0.972478i \(-0.425148\pi\)
0.232993 + 0.972478i \(0.425148\pi\)
\(348\) −0.581449 −0.0311690
\(349\) 17.0784 0.914185 0.457092 0.889419i \(-0.348891\pi\)
0.457092 + 0.889419i \(0.348891\pi\)
\(350\) 1.00000 0.0534522
\(351\) 4.34017 0.231661
\(352\) 3.26180 0.173854
\(353\) −11.1629 −0.594141 −0.297071 0.954856i \(-0.596010\pi\)
−0.297071 + 0.954856i \(0.596010\pi\)
\(354\) −6.15676 −0.327228
\(355\) 5.41855 0.287587
\(356\) 10.8638 0.575778
\(357\) −5.75872 −0.304784
\(358\) 12.3135 0.650789
\(359\) 16.8781 0.890794 0.445397 0.895333i \(-0.353063\pi\)
0.445397 + 0.895333i \(0.353063\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −9.52359 −0.501242
\(362\) 17.5174 0.920697
\(363\) 0.360692 0.0189314
\(364\) −4.34017 −0.227487
\(365\) 6.31351 0.330464
\(366\) −8.15676 −0.426360
\(367\) 16.1978 0.845518 0.422759 0.906242i \(-0.361062\pi\)
0.422759 + 0.906242i \(0.361062\pi\)
\(368\) 1.00000 0.0521286
\(369\) −8.83710 −0.460041
\(370\) −2.00000 −0.103975
\(371\) 8.83710 0.458799
\(372\) −3.07838 −0.159606
\(373\) 7.67420 0.397355 0.198678 0.980065i \(-0.436335\pi\)
0.198678 + 0.980065i \(0.436335\pi\)
\(374\) 18.7838 0.971286
\(375\) 1.00000 0.0516398
\(376\) 3.07838 0.158755
\(377\) −2.52359 −0.129972
\(378\) −1.00000 −0.0514344
\(379\) 11.6332 0.597556 0.298778 0.954323i \(-0.403421\pi\)
0.298778 + 0.954323i \(0.403421\pi\)
\(380\) 3.07838 0.157917
\(381\) −4.73820 −0.242746
\(382\) −6.15676 −0.315007
\(383\) −5.36069 −0.273919 −0.136959 0.990577i \(-0.543733\pi\)
−0.136959 + 0.990577i \(0.543733\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −3.26180 −0.166236
\(386\) 4.15676 0.211573
\(387\) 0.738205 0.0375251
\(388\) −4.34017 −0.220339
\(389\) −18.9939 −0.963027 −0.481513 0.876439i \(-0.659913\pi\)
−0.481513 + 0.876439i \(0.659913\pi\)
\(390\) −4.34017 −0.219773
\(391\) 5.75872 0.291231
\(392\) 1.00000 0.0505076
\(393\) −17.6742 −0.891546
\(394\) 12.1568 0.612448
\(395\) 2.15676 0.108518
\(396\) 3.26180 0.163911
\(397\) 0.707008 0.0354837 0.0177419 0.999843i \(-0.494352\pi\)
0.0177419 + 0.999843i \(0.494352\pi\)
\(398\) 18.5236 0.928504
\(399\) 3.07838 0.154112
\(400\) 1.00000 0.0500000
\(401\) −22.6225 −1.12971 −0.564857 0.825189i \(-0.691069\pi\)
−0.564857 + 0.825189i \(0.691069\pi\)
\(402\) −0.738205 −0.0368183
\(403\) −13.3607 −0.665543
\(404\) −1.81658 −0.0903784
\(405\) −1.00000 −0.0496904
\(406\) 0.581449 0.0288568
\(407\) 6.52359 0.323362
\(408\) −5.75872 −0.285099
\(409\) −6.36683 −0.314820 −0.157410 0.987533i \(-0.550314\pi\)
−0.157410 + 0.987533i \(0.550314\pi\)
\(410\) 8.83710 0.436433
\(411\) −2.00000 −0.0986527
\(412\) −2.52359 −0.124328
\(413\) 6.15676 0.302954
\(414\) 1.00000 0.0491473
\(415\) 9.60197 0.471342
\(416\) −4.34017 −0.212794
\(417\) 0.680346 0.0333167
\(418\) −10.0410 −0.491123
\(419\) −19.0205 −0.929213 −0.464607 0.885517i \(-0.653804\pi\)
−0.464607 + 0.885517i \(0.653804\pi\)
\(420\) 1.00000 0.0487950
\(421\) 14.6270 0.712877 0.356439 0.934319i \(-0.383991\pi\)
0.356439 + 0.934319i \(0.383991\pi\)
\(422\) 27.2039 1.32427
\(423\) 3.07838 0.149676
\(424\) 8.83710 0.429168
\(425\) 5.75872 0.279339
\(426\) 5.41855 0.262530
\(427\) 8.15676 0.394733
\(428\) 8.68035 0.419580
\(429\) 14.1568 0.683495
\(430\) −0.738205 −0.0355994
\(431\) 15.8310 0.762550 0.381275 0.924462i \(-0.375485\pi\)
0.381275 + 0.924462i \(0.375485\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 10.6947 0.513956 0.256978 0.966417i \(-0.417273\pi\)
0.256978 + 0.966417i \(0.417273\pi\)
\(434\) 3.07838 0.147767
\(435\) 0.581449 0.0278784
\(436\) 19.3607 0.927209
\(437\) −3.07838 −0.147259
\(438\) 6.31351 0.301671
\(439\) 16.3857 0.782049 0.391024 0.920380i \(-0.372121\pi\)
0.391024 + 0.920380i \(0.372121\pi\)
\(440\) −3.26180 −0.155500
\(441\) 1.00000 0.0476190
\(442\) −24.9939 −1.18884
\(443\) 0.680346 0.0323242 0.0161621 0.999869i \(-0.494855\pi\)
0.0161621 + 0.999869i \(0.494855\pi\)
\(444\) −2.00000 −0.0949158
\(445\) −10.8638 −0.514992
\(446\) −0.496928 −0.0235302
\(447\) −10.0000 −0.472984
\(448\) 1.00000 0.0472456
\(449\) 33.7152 1.59112 0.795560 0.605874i \(-0.207177\pi\)
0.795560 + 0.605874i \(0.207177\pi\)
\(450\) 1.00000 0.0471405
\(451\) −28.8248 −1.35731
\(452\) 8.15676 0.383662
\(453\) 4.68035 0.219902
\(454\) −10.5958 −0.497287
\(455\) 4.34017 0.203470
\(456\) 3.07838 0.144158
\(457\) −29.0928 −1.36090 −0.680451 0.732794i \(-0.738216\pi\)
−0.680451 + 0.732794i \(0.738216\pi\)
\(458\) 7.16290 0.334700
\(459\) −5.75872 −0.268794
\(460\) −1.00000 −0.0466252
\(461\) −38.8638 −1.81007 −0.905033 0.425341i \(-0.860154\pi\)
−0.905033 + 0.425341i \(0.860154\pi\)
\(462\) −3.26180 −0.151752
\(463\) −20.2557 −0.941360 −0.470680 0.882304i \(-0.655991\pi\)
−0.470680 + 0.882304i \(0.655991\pi\)
\(464\) 0.581449 0.0269931
\(465\) 3.07838 0.142756
\(466\) −19.3607 −0.896867
\(467\) −12.0722 −0.558636 −0.279318 0.960199i \(-0.590108\pi\)
−0.279318 + 0.960199i \(0.590108\pi\)
\(468\) −4.34017 −0.200625
\(469\) 0.738205 0.0340871
\(470\) −3.07838 −0.141995
\(471\) −4.52359 −0.208436
\(472\) 6.15676 0.283388
\(473\) 2.40787 0.110714
\(474\) 2.15676 0.0990631
\(475\) −3.07838 −0.141246
\(476\) 5.75872 0.263951
\(477\) 8.83710 0.404623
\(478\) −5.41855 −0.247839
\(479\) −41.5585 −1.89886 −0.949428 0.313985i \(-0.898336\pi\)
−0.949428 + 0.313985i \(0.898336\pi\)
\(480\) 1.00000 0.0456435
\(481\) −8.68035 −0.395790
\(482\) 5.75872 0.262303
\(483\) −1.00000 −0.0455016
\(484\) −0.360692 −0.0163951
\(485\) 4.34017 0.197077
\(486\) −1.00000 −0.0453609
\(487\) −36.8827 −1.67131 −0.835657 0.549252i \(-0.814913\pi\)
−0.835657 + 0.549252i \(0.814913\pi\)
\(488\) 8.15676 0.369239
\(489\) −9.84324 −0.445127
\(490\) −1.00000 −0.0451754
\(491\) −23.5708 −1.06373 −0.531867 0.846828i \(-0.678509\pi\)
−0.531867 + 0.846828i \(0.678509\pi\)
\(492\) 8.83710 0.398407
\(493\) 3.34841 0.150805
\(494\) 13.3607 0.601126
\(495\) −3.26180 −0.146607
\(496\) 3.07838 0.138223
\(497\) −5.41855 −0.243055
\(498\) 9.60197 0.430274
\(499\) 4.79606 0.214701 0.107351 0.994221i \(-0.465763\pi\)
0.107351 + 0.994221i \(0.465763\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 5.91548 0.264284
\(502\) 17.1773 0.766659
\(503\) −14.4703 −0.645197 −0.322599 0.946536i \(-0.604556\pi\)
−0.322599 + 0.946536i \(0.604556\pi\)
\(504\) 1.00000 0.0445435
\(505\) 1.81658 0.0808369
\(506\) 3.26180 0.145004
\(507\) −5.83710 −0.259235
\(508\) 4.73820 0.210224
\(509\) 4.82273 0.213764 0.106882 0.994272i \(-0.465913\pi\)
0.106882 + 0.994272i \(0.465913\pi\)
\(510\) 5.75872 0.255001
\(511\) −6.31351 −0.279293
\(512\) 1.00000 0.0441942
\(513\) 3.07838 0.135914
\(514\) −11.6742 −0.514927
\(515\) 2.52359 0.111203
\(516\) −0.738205 −0.0324977
\(517\) 10.0410 0.441604
\(518\) 2.00000 0.0878750
\(519\) 11.2351 0.493168
\(520\) 4.34017 0.190329
\(521\) −31.8576 −1.39571 −0.697854 0.716240i \(-0.745862\pi\)
−0.697854 + 0.716240i \(0.745862\pi\)
\(522\) 0.581449 0.0254493
\(523\) −0.979481 −0.0428297 −0.0214149 0.999771i \(-0.506817\pi\)
−0.0214149 + 0.999771i \(0.506817\pi\)
\(524\) 17.6742 0.772101
\(525\) −1.00000 −0.0436436
\(526\) −19.8843 −0.866996
\(527\) 17.7275 0.772223
\(528\) −3.26180 −0.141951
\(529\) 1.00000 0.0434783
\(530\) −8.83710 −0.383859
\(531\) 6.15676 0.267180
\(532\) −3.07838 −0.133465
\(533\) 38.3545 1.66132
\(534\) −10.8638 −0.470121
\(535\) −8.68035 −0.375284
\(536\) 0.738205 0.0318856
\(537\) −12.3135 −0.531367
\(538\) −9.70086 −0.418234
\(539\) 3.26180 0.140495
\(540\) 1.00000 0.0430331
\(541\) −3.47641 −0.149463 −0.0747313 0.997204i \(-0.523810\pi\)
−0.0747313 + 0.997204i \(0.523810\pi\)
\(542\) 3.07838 0.132228
\(543\) −17.5174 −0.751746
\(544\) 5.75872 0.246903
\(545\) −19.3607 −0.829321
\(546\) 4.34017 0.185742
\(547\) −36.0288 −1.54048 −0.770239 0.637755i \(-0.779863\pi\)
−0.770239 + 0.637755i \(0.779863\pi\)
\(548\) 2.00000 0.0854358
\(549\) 8.15676 0.348122
\(550\) 3.26180 0.139083
\(551\) −1.78992 −0.0762532
\(552\) −1.00000 −0.0425628
\(553\) −2.15676 −0.0917146
\(554\) −5.26180 −0.223552
\(555\) 2.00000 0.0848953
\(556\) −0.680346 −0.0288531
\(557\) 2.62702 0.111310 0.0556552 0.998450i \(-0.482275\pi\)
0.0556552 + 0.998450i \(0.482275\pi\)
\(558\) 3.07838 0.130318
\(559\) −3.20394 −0.135512
\(560\) −1.00000 −0.0422577
\(561\) −18.7838 −0.793052
\(562\) 5.57531 0.235180
\(563\) 6.39803 0.269645 0.134822 0.990870i \(-0.456954\pi\)
0.134822 + 0.990870i \(0.456954\pi\)
\(564\) −3.07838 −0.129623
\(565\) −8.15676 −0.343157
\(566\) −6.65368 −0.279675
\(567\) 1.00000 0.0419961
\(568\) −5.41855 −0.227357
\(569\) −14.6225 −0.613007 −0.306503 0.951870i \(-0.599159\pi\)
−0.306503 + 0.951870i \(0.599159\pi\)
\(570\) −3.07838 −0.128939
\(571\) 6.10343 0.255421 0.127710 0.991812i \(-0.459237\pi\)
0.127710 + 0.991812i \(0.459237\pi\)
\(572\) −14.1568 −0.591924
\(573\) 6.15676 0.257202
\(574\) −8.83710 −0.368853
\(575\) 1.00000 0.0417029
\(576\) 1.00000 0.0416667
\(577\) −15.7275 −0.654746 −0.327373 0.944895i \(-0.606163\pi\)
−0.327373 + 0.944895i \(0.606163\pi\)
\(578\) 16.1629 0.672288
\(579\) −4.15676 −0.172749
\(580\) −0.581449 −0.0241434
\(581\) −9.60197 −0.398357
\(582\) 4.34017 0.179906
\(583\) 28.8248 1.19380
\(584\) −6.31351 −0.261255
\(585\) 4.34017 0.179444
\(586\) 6.68035 0.275963
\(587\) −6.32580 −0.261094 −0.130547 0.991442i \(-0.541673\pi\)
−0.130547 + 0.991442i \(0.541673\pi\)
\(588\) −1.00000 −0.0412393
\(589\) −9.47641 −0.390469
\(590\) −6.15676 −0.253470
\(591\) −12.1568 −0.500062
\(592\) 2.00000 0.0821995
\(593\) 3.36069 0.138007 0.0690035 0.997616i \(-0.478018\pi\)
0.0690035 + 0.997616i \(0.478018\pi\)
\(594\) −3.26180 −0.133833
\(595\) −5.75872 −0.236085
\(596\) 10.0000 0.409616
\(597\) −18.5236 −0.758120
\(598\) −4.34017 −0.177483
\(599\) −2.09890 −0.0857586 −0.0428793 0.999080i \(-0.513653\pi\)
−0.0428793 + 0.999080i \(0.513653\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −24.8371 −1.01313 −0.506563 0.862203i \(-0.669084\pi\)
−0.506563 + 0.862203i \(0.669084\pi\)
\(602\) 0.738205 0.0300870
\(603\) 0.738205 0.0300620
\(604\) −4.68035 −0.190441
\(605\) 0.360692 0.0146642
\(606\) 1.81658 0.0737936
\(607\) 8.38121 0.340183 0.170091 0.985428i \(-0.445594\pi\)
0.170091 + 0.985428i \(0.445594\pi\)
\(608\) −3.07838 −0.124845
\(609\) −0.581449 −0.0235615
\(610\) −8.15676 −0.330257
\(611\) −13.3607 −0.540516
\(612\) 5.75872 0.232783
\(613\) 14.9939 0.605596 0.302798 0.953055i \(-0.402079\pi\)
0.302798 + 0.953055i \(0.402079\pi\)
\(614\) 31.8843 1.28674
\(615\) −8.83710 −0.356346
\(616\) 3.26180 0.131421
\(617\) −30.5113 −1.22834 −0.614169 0.789174i \(-0.710509\pi\)
−0.614169 + 0.789174i \(0.710509\pi\)
\(618\) 2.52359 0.101514
\(619\) −9.23513 −0.371191 −0.185596 0.982626i \(-0.559421\pi\)
−0.185596 + 0.982626i \(0.559421\pi\)
\(620\) −3.07838 −0.123631
\(621\) −1.00000 −0.0401286
\(622\) 9.17727 0.367975
\(623\) 10.8638 0.435247
\(624\) 4.34017 0.173746
\(625\) 1.00000 0.0400000
\(626\) −21.3340 −0.852679
\(627\) 10.0410 0.401000
\(628\) 4.52359 0.180511
\(629\) 11.5174 0.459231
\(630\) −1.00000 −0.0398410
\(631\) 2.89043 0.115066 0.0575330 0.998344i \(-0.481677\pi\)
0.0575330 + 0.998344i \(0.481677\pi\)
\(632\) −2.15676 −0.0857911
\(633\) −27.2039 −1.08126
\(634\) 6.99386 0.277762
\(635\) −4.73820 −0.188030
\(636\) −8.83710 −0.350414
\(637\) −4.34017 −0.171964
\(638\) 1.89657 0.0750859
\(639\) −5.41855 −0.214355
\(640\) −1.00000 −0.0395285
\(641\) −37.7731 −1.49195 −0.745974 0.665975i \(-0.768016\pi\)
−0.745974 + 0.665975i \(0.768016\pi\)
\(642\) −8.68035 −0.342586
\(643\) 40.8638 1.61151 0.805755 0.592249i \(-0.201760\pi\)
0.805755 + 0.592249i \(0.201760\pi\)
\(644\) 1.00000 0.0394055
\(645\) 0.738205 0.0290668
\(646\) −17.7275 −0.697480
\(647\) −6.03120 −0.237111 −0.118555 0.992947i \(-0.537826\pi\)
−0.118555 + 0.992947i \(0.537826\pi\)
\(648\) 1.00000 0.0392837
\(649\) 20.0821 0.788290
\(650\) −4.34017 −0.170236
\(651\) −3.07838 −0.120651
\(652\) 9.84324 0.385491
\(653\) −38.8248 −1.51933 −0.759666 0.650313i \(-0.774638\pi\)
−0.759666 + 0.650313i \(0.774638\pi\)
\(654\) −19.3607 −0.757063
\(655\) −17.6742 −0.690588
\(656\) −8.83710 −0.345031
\(657\) −6.31351 −0.246314
\(658\) 3.07838 0.120008
\(659\) −3.62863 −0.141351 −0.0706757 0.997499i \(-0.522516\pi\)
−0.0706757 + 0.997499i \(0.522516\pi\)
\(660\) 3.26180 0.126965
\(661\) 4.10343 0.159605 0.0798025 0.996811i \(-0.474571\pi\)
0.0798025 + 0.996811i \(0.474571\pi\)
\(662\) 2.83710 0.110267
\(663\) 24.9939 0.970681
\(664\) −9.60197 −0.372629
\(665\) 3.07838 0.119374
\(666\) 2.00000 0.0774984
\(667\) 0.581449 0.0225138
\(668\) −5.91548 −0.228877
\(669\) 0.496928 0.0192124
\(670\) −0.738205 −0.0285193
\(671\) 26.6057 1.02710
\(672\) −1.00000 −0.0385758
\(673\) −9.00614 −0.347161 −0.173581 0.984820i \(-0.555534\pi\)
−0.173581 + 0.984820i \(0.555534\pi\)
\(674\) 8.94828 0.344675
\(675\) −1.00000 −0.0384900
\(676\) 5.83710 0.224504
\(677\) 32.3012 1.24144 0.620718 0.784034i \(-0.286841\pi\)
0.620718 + 0.784034i \(0.286841\pi\)
\(678\) −8.15676 −0.313258
\(679\) −4.34017 −0.166561
\(680\) −5.75872 −0.220837
\(681\) 10.5958 0.406033
\(682\) 10.0410 0.384491
\(683\) −29.7275 −1.13749 −0.568746 0.822513i \(-0.692571\pi\)
−0.568746 + 0.822513i \(0.692571\pi\)
\(684\) −3.07838 −0.117705
\(685\) −2.00000 −0.0764161
\(686\) 1.00000 0.0381802
\(687\) −7.16290 −0.273282
\(688\) 0.738205 0.0281438
\(689\) −38.3545 −1.46119
\(690\) 1.00000 0.0380693
\(691\) −20.5113 −0.780287 −0.390143 0.920754i \(-0.627575\pi\)
−0.390143 + 0.920754i \(0.627575\pi\)
\(692\) −11.2351 −0.427096
\(693\) 3.26180 0.123905
\(694\) 8.68035 0.329501
\(695\) 0.680346 0.0258070
\(696\) −0.581449 −0.0220398
\(697\) −50.8904 −1.92761
\(698\) 17.0784 0.646426
\(699\) 19.3607 0.732289
\(700\) 1.00000 0.0377964
\(701\) 7.16290 0.270539 0.135269 0.990809i \(-0.456810\pi\)
0.135269 + 0.990809i \(0.456810\pi\)
\(702\) 4.34017 0.163809
\(703\) −6.15676 −0.232206
\(704\) 3.26180 0.122934
\(705\) 3.07838 0.115938
\(706\) −11.1629 −0.420121
\(707\) −1.81658 −0.0683196
\(708\) −6.15676 −0.231385
\(709\) 10.8494 0.407457 0.203729 0.979027i \(-0.434694\pi\)
0.203729 + 0.979027i \(0.434694\pi\)
\(710\) 5.41855 0.203355
\(711\) −2.15676 −0.0808847
\(712\) 10.8638 0.407137
\(713\) 3.07838 0.115286
\(714\) −5.75872 −0.215515
\(715\) 14.1568 0.529433
\(716\) 12.3135 0.460178
\(717\) 5.41855 0.202359
\(718\) 16.8781 0.629887
\(719\) −8.44360 −0.314893 −0.157447 0.987527i \(-0.550326\pi\)
−0.157447 + 0.987527i \(0.550326\pi\)
\(720\) −1.00000 −0.0372678
\(721\) −2.52359 −0.0939834
\(722\) −9.52359 −0.354431
\(723\) −5.75872 −0.214169
\(724\) 17.5174 0.651031
\(725\) 0.581449 0.0215945
\(726\) 0.360692 0.0133865
\(727\) −9.72753 −0.360774 −0.180387 0.983596i \(-0.557735\pi\)
−0.180387 + 0.983596i \(0.557735\pi\)
\(728\) −4.34017 −0.160858
\(729\) 1.00000 0.0370370
\(730\) 6.31351 0.233674
\(731\) 4.25112 0.157233
\(732\) −8.15676 −0.301482
\(733\) −14.4826 −0.534925 −0.267463 0.963568i \(-0.586185\pi\)
−0.267463 + 0.963568i \(0.586185\pi\)
\(734\) 16.1978 0.597871
\(735\) 1.00000 0.0368856
\(736\) 1.00000 0.0368605
\(737\) 2.40787 0.0886951
\(738\) −8.83710 −0.325298
\(739\) 15.3197 0.563543 0.281771 0.959482i \(-0.409078\pi\)
0.281771 + 0.959482i \(0.409078\pi\)
\(740\) −2.00000 −0.0735215
\(741\) −13.3607 −0.490817
\(742\) 8.83710 0.324420
\(743\) 49.1917 1.80467 0.902333 0.431039i \(-0.141853\pi\)
0.902333 + 0.431039i \(0.141853\pi\)
\(744\) −3.07838 −0.112859
\(745\) −10.0000 −0.366372
\(746\) 7.67420 0.280973
\(747\) −9.60197 −0.351318
\(748\) 18.7838 0.686803
\(749\) 8.68035 0.317173
\(750\) 1.00000 0.0365148
\(751\) 8.73367 0.318696 0.159348 0.987222i \(-0.449061\pi\)
0.159348 + 0.987222i \(0.449061\pi\)
\(752\) 3.07838 0.112257
\(753\) −17.1773 −0.625975
\(754\) −2.52359 −0.0919037
\(755\) 4.68035 0.170335
\(756\) −1.00000 −0.0363696
\(757\) 7.72753 0.280862 0.140431 0.990090i \(-0.455151\pi\)
0.140431 + 0.990090i \(0.455151\pi\)
\(758\) 11.6332 0.422536
\(759\) −3.26180 −0.118396
\(760\) 3.07838 0.111664
\(761\) 11.7899 0.427384 0.213692 0.976901i \(-0.431451\pi\)
0.213692 + 0.976901i \(0.431451\pi\)
\(762\) −4.73820 −0.171647
\(763\) 19.3607 0.700904
\(764\) −6.15676 −0.222744
\(765\) −5.75872 −0.208207
\(766\) −5.36069 −0.193690
\(767\) −26.7214 −0.964853
\(768\) −1.00000 −0.0360844
\(769\) −12.4514 −0.449007 −0.224504 0.974473i \(-0.572076\pi\)
−0.224504 + 0.974473i \(0.572076\pi\)
\(770\) −3.26180 −0.117547
\(771\) 11.6742 0.420436
\(772\) 4.15676 0.149605
\(773\) 4.83710 0.173978 0.0869892 0.996209i \(-0.472275\pi\)
0.0869892 + 0.996209i \(0.472275\pi\)
\(774\) 0.738205 0.0265342
\(775\) 3.07838 0.110579
\(776\) −4.34017 −0.155803
\(777\) −2.00000 −0.0717496
\(778\) −18.9939 −0.680963
\(779\) 27.2039 0.974682
\(780\) −4.34017 −0.155403
\(781\) −17.6742 −0.632433
\(782\) 5.75872 0.205932
\(783\) −0.581449 −0.0207793
\(784\) 1.00000 0.0357143
\(785\) −4.52359 −0.161454
\(786\) −17.6742 −0.630418
\(787\) 21.8043 0.777239 0.388620 0.921398i \(-0.372952\pi\)
0.388620 + 0.921398i \(0.372952\pi\)
\(788\) 12.1568 0.433066
\(789\) 19.8843 0.707899
\(790\) 2.15676 0.0767339
\(791\) 8.15676 0.290021
\(792\) 3.26180 0.115903
\(793\) −35.4017 −1.25715
\(794\) 0.707008 0.0250908
\(795\) 8.83710 0.313420
\(796\) 18.5236 0.656551
\(797\) −51.1917 −1.81330 −0.906651 0.421882i \(-0.861370\pi\)
−0.906651 + 0.421882i \(0.861370\pi\)
\(798\) 3.07838 0.108973
\(799\) 17.7275 0.627155
\(800\) 1.00000 0.0353553
\(801\) 10.8638 0.383852
\(802\) −22.6225 −0.798828
\(803\) −20.5934 −0.726725
\(804\) −0.738205 −0.0260345
\(805\) −1.00000 −0.0352454
\(806\) −13.3607 −0.470610
\(807\) 9.70086 0.341487
\(808\) −1.81658 −0.0639071
\(809\) 33.8310 1.18943 0.594717 0.803935i \(-0.297264\pi\)
0.594717 + 0.803935i \(0.297264\pi\)
\(810\) −1.00000 −0.0351364
\(811\) 11.6332 0.408496 0.204248 0.978919i \(-0.434525\pi\)
0.204248 + 0.978919i \(0.434525\pi\)
\(812\) 0.581449 0.0204049
\(813\) −3.07838 −0.107963
\(814\) 6.52359 0.228652
\(815\) −9.84324 −0.344794
\(816\) −5.75872 −0.201596
\(817\) −2.27247 −0.0795038
\(818\) −6.36683 −0.222611
\(819\) −4.34017 −0.151658
\(820\) 8.83710 0.308605
\(821\) −22.4247 −0.782627 −0.391314 0.920257i \(-0.627979\pi\)
−0.391314 + 0.920257i \(0.627979\pi\)
\(822\) −2.00000 −0.0697580
\(823\) −19.2618 −0.671424 −0.335712 0.941965i \(-0.608977\pi\)
−0.335712 + 0.941965i \(0.608977\pi\)
\(824\) −2.52359 −0.0879134
\(825\) −3.26180 −0.113561
\(826\) 6.15676 0.214221
\(827\) 24.6803 0.858220 0.429110 0.903252i \(-0.358827\pi\)
0.429110 + 0.903252i \(0.358827\pi\)
\(828\) 1.00000 0.0347524
\(829\) 24.5958 0.854248 0.427124 0.904193i \(-0.359527\pi\)
0.427124 + 0.904193i \(0.359527\pi\)
\(830\) 9.60197 0.333289
\(831\) 5.26180 0.182530
\(832\) −4.34017 −0.150468
\(833\) 5.75872 0.199528
\(834\) 0.680346 0.0235585
\(835\) 5.91548 0.204714
\(836\) −10.0410 −0.347277
\(837\) −3.07838 −0.106404
\(838\) −19.0205 −0.657053
\(839\) 26.1568 0.903031 0.451516 0.892263i \(-0.350883\pi\)
0.451516 + 0.892263i \(0.350883\pi\)
\(840\) 1.00000 0.0345033
\(841\) −28.6619 −0.988342
\(842\) 14.6270 0.504080
\(843\) −5.57531 −0.192024
\(844\) 27.2039 0.936398
\(845\) −5.83710 −0.200802
\(846\) 3.07838 0.105837
\(847\) −0.360692 −0.0123935
\(848\) 8.83710 0.303467
\(849\) 6.65368 0.228354
\(850\) 5.75872 0.197523
\(851\) 2.00000 0.0685591
\(852\) 5.41855 0.185636
\(853\) 3.03281 0.103841 0.0519206 0.998651i \(-0.483466\pi\)
0.0519206 + 0.998651i \(0.483466\pi\)
\(854\) 8.15676 0.279118
\(855\) 3.07838 0.105278
\(856\) 8.68035 0.296688
\(857\) −27.1629 −0.927867 −0.463933 0.885870i \(-0.653562\pi\)
−0.463933 + 0.885870i \(0.653562\pi\)
\(858\) 14.1568 0.483304
\(859\) −20.5113 −0.699837 −0.349918 0.936780i \(-0.613791\pi\)
−0.349918 + 0.936780i \(0.613791\pi\)
\(860\) −0.738205 −0.0251726
\(861\) 8.83710 0.301168
\(862\) 15.8310 0.539205
\(863\) 14.0410 0.477962 0.238981 0.971024i \(-0.423187\pi\)
0.238981 + 0.971024i \(0.423187\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 11.2351 0.382006
\(866\) 10.6947 0.363421
\(867\) −16.1629 −0.548921
\(868\) 3.07838 0.104487
\(869\) −7.03489 −0.238642
\(870\) 0.581449 0.0197130
\(871\) −3.20394 −0.108561
\(872\) 19.3607 0.655636
\(873\) −4.34017 −0.146893
\(874\) −3.07838 −0.104128
\(875\) −1.00000 −0.0338062
\(876\) 6.31351 0.213314
\(877\) −35.8141 −1.20936 −0.604679 0.796469i \(-0.706698\pi\)
−0.604679 + 0.796469i \(0.706698\pi\)
\(878\) 16.3857 0.552992
\(879\) −6.68035 −0.225322
\(880\) −3.26180 −0.109955
\(881\) −42.3279 −1.42606 −0.713031 0.701132i \(-0.752678\pi\)
−0.713031 + 0.701132i \(0.752678\pi\)
\(882\) 1.00000 0.0336718
\(883\) −6.78378 −0.228292 −0.114146 0.993464i \(-0.536413\pi\)
−0.114146 + 0.993464i \(0.536413\pi\)
\(884\) −24.9939 −0.840634
\(885\) 6.15676 0.206957
\(886\) 0.680346 0.0228567
\(887\) 49.6307 1.66644 0.833218 0.552944i \(-0.186496\pi\)
0.833218 + 0.552944i \(0.186496\pi\)
\(888\) −2.00000 −0.0671156
\(889\) 4.73820 0.158914
\(890\) −10.8638 −0.364154
\(891\) 3.26180 0.109274
\(892\) −0.496928 −0.0166384
\(893\) −9.47641 −0.317116
\(894\) −10.0000 −0.334450
\(895\) −12.3135 −0.411595
\(896\) 1.00000 0.0334077
\(897\) 4.34017 0.144914
\(898\) 33.7152 1.12509
\(899\) 1.78992 0.0596972
\(900\) 1.00000 0.0333333
\(901\) 50.8904 1.69541
\(902\) −28.8248 −0.959761
\(903\) −0.738205 −0.0245659
\(904\) 8.15676 0.271290
\(905\) −17.5174 −0.582300
\(906\) 4.68035 0.155494
\(907\) −33.1050 −1.09923 −0.549617 0.835416i \(-0.685226\pi\)
−0.549617 + 0.835416i \(0.685226\pi\)
\(908\) −10.5958 −0.351635
\(909\) −1.81658 −0.0602522
\(910\) 4.34017 0.143875
\(911\) −26.3545 −0.873165 −0.436583 0.899664i \(-0.643811\pi\)
−0.436583 + 0.899664i \(0.643811\pi\)
\(912\) 3.07838 0.101935
\(913\) −31.3197 −1.03653
\(914\) −29.0928 −0.962303
\(915\) 8.15676 0.269654
\(916\) 7.16290 0.236669
\(917\) 17.6742 0.583654
\(918\) −5.75872 −0.190066
\(919\) 32.3668 1.06768 0.533842 0.845584i \(-0.320748\pi\)
0.533842 + 0.845584i \(0.320748\pi\)
\(920\) −1.00000 −0.0329690
\(921\) −31.8843 −1.05062
\(922\) −38.8638 −1.27991
\(923\) 23.5174 0.774086
\(924\) −3.26180 −0.107305
\(925\) 2.00000 0.0657596
\(926\) −20.2557 −0.665642
\(927\) −2.52359 −0.0828856
\(928\) 0.581449 0.0190870
\(929\) 43.1917 1.41707 0.708536 0.705675i \(-0.249356\pi\)
0.708536 + 0.705675i \(0.249356\pi\)
\(930\) 3.07838 0.100944
\(931\) −3.07838 −0.100890
\(932\) −19.3607 −0.634181
\(933\) −9.17727 −0.300450
\(934\) −12.0722 −0.395016
\(935\) −18.7838 −0.614295
\(936\) −4.34017 −0.141863
\(937\) −35.8576 −1.17142 −0.585709 0.810522i \(-0.699184\pi\)
−0.585709 + 0.810522i \(0.699184\pi\)
\(938\) 0.738205 0.0241032
\(939\) 21.3340 0.696210
\(940\) −3.07838 −0.100406
\(941\) 11.3607 0.370348 0.185174 0.982706i \(-0.440715\pi\)
0.185174 + 0.982706i \(0.440715\pi\)
\(942\) −4.52359 −0.147387
\(943\) −8.83710 −0.287776
\(944\) 6.15676 0.200385
\(945\) 1.00000 0.0325300
\(946\) 2.40787 0.0782867
\(947\) −27.9376 −0.907850 −0.453925 0.891040i \(-0.649977\pi\)
−0.453925 + 0.891040i \(0.649977\pi\)
\(948\) 2.15676 0.0700482
\(949\) 27.4017 0.889498
\(950\) −3.07838 −0.0998758
\(951\) −6.99386 −0.226791
\(952\) 5.75872 0.186641
\(953\) −38.8781 −1.25939 −0.629693 0.776844i \(-0.716819\pi\)
−0.629693 + 0.776844i \(0.716819\pi\)
\(954\) 8.83710 0.286112
\(955\) 6.15676 0.199228
\(956\) −5.41855 −0.175248
\(957\) −1.89657 −0.0613074
\(958\) −41.5585 −1.34269
\(959\) 2.00000 0.0645834
\(960\) 1.00000 0.0322749
\(961\) −21.5236 −0.694309
\(962\) −8.68035 −0.279866
\(963\) 8.68035 0.279720
\(964\) 5.75872 0.185476
\(965\) −4.15676 −0.133811
\(966\) −1.00000 −0.0321745
\(967\) 39.0928 1.25714 0.628569 0.777754i \(-0.283641\pi\)
0.628569 + 0.777754i \(0.283641\pi\)
\(968\) −0.360692 −0.0115931
\(969\) 17.7275 0.569490
\(970\) 4.34017 0.139355
\(971\) −2.50921 −0.0805245 −0.0402623 0.999189i \(-0.512819\pi\)
−0.0402623 + 0.999189i \(0.512819\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −0.680346 −0.0218109
\(974\) −36.8827 −1.18180
\(975\) 4.34017 0.138997
\(976\) 8.15676 0.261091
\(977\) −47.8720 −1.53156 −0.765780 0.643102i \(-0.777647\pi\)
−0.765780 + 0.643102i \(0.777647\pi\)
\(978\) −9.84324 −0.314752
\(979\) 35.4354 1.13252
\(980\) −1.00000 −0.0319438
\(981\) 19.3607 0.618139
\(982\) −23.5708 −0.752174
\(983\) 0.313511 0.00999945 0.00499972 0.999988i \(-0.498409\pi\)
0.00499972 + 0.999988i \(0.498409\pi\)
\(984\) 8.83710 0.281717
\(985\) −12.1568 −0.387346
\(986\) 3.34841 0.106635
\(987\) −3.07838 −0.0979858
\(988\) 13.3607 0.425060
\(989\) 0.738205 0.0234735
\(990\) −3.26180 −0.103667
\(991\) 43.7152 1.38866 0.694330 0.719657i \(-0.255701\pi\)
0.694330 + 0.719657i \(0.255701\pi\)
\(992\) 3.07838 0.0977386
\(993\) −2.83710 −0.0900327
\(994\) −5.41855 −0.171866
\(995\) −18.5236 −0.587237
\(996\) 9.60197 0.304250
\(997\) 6.38121 0.202095 0.101047 0.994882i \(-0.467781\pi\)
0.101047 + 0.994882i \(0.467781\pi\)
\(998\) 4.79606 0.151817
\(999\) −2.00000 −0.0632772
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 4830.2.a.bz.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4830.2.a.bz.1.2 3 1.1 even 1 trivial