Properties

Label 4830.2.a.cb.1.1
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: 3.3.229.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 4x - 1 \) 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 1.1
Root \(2.11491\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -2.94567 q^{11} -1.00000 q^{12} +1.28415 q^{13} +1.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} +4.22982 q^{17} +1.00000 q^{18} +6.22982 q^{19} +1.00000 q^{20} -1.00000 q^{21} -2.94567 q^{22} -1.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} +1.28415 q^{26} -1.00000 q^{27} +1.00000 q^{28} +4.94567 q^{29} -1.00000 q^{30} -2.22982 q^{31} +1.00000 q^{32} +2.94567 q^{33} +4.22982 q^{34} +1.00000 q^{35} +1.00000 q^{36} -6.45963 q^{37} +6.22982 q^{38} -1.28415 q^{39} +1.00000 q^{40} -10.4596 q^{41} -1.00000 q^{42} +10.9457 q^{43} -2.94567 q^{44} +1.00000 q^{45} -1.00000 q^{46} -2.22982 q^{47} -1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} -4.22982 q^{51} +1.28415 q^{52} -6.45963 q^{53} -1.00000 q^{54} -2.94567 q^{55} +1.00000 q^{56} -6.22982 q^{57} +4.94567 q^{58} +8.45963 q^{59} -1.00000 q^{60} -2.00000 q^{61} -2.22982 q^{62} +1.00000 q^{63} +1.00000 q^{64} +1.28415 q^{65} +2.94567 q^{66} +2.94567 q^{67} +4.22982 q^{68} +1.00000 q^{69} +1.00000 q^{70} +12.8370 q^{71} +1.00000 q^{72} +3.43171 q^{73} -6.45963 q^{74} -1.00000 q^{75} +6.22982 q^{76} -2.94567 q^{77} -1.28415 q^{78} +1.00000 q^{80} +1.00000 q^{81} -10.4596 q^{82} -4.79811 q^{83} -1.00000 q^{84} +4.22982 q^{85} +10.9457 q^{86} -4.94567 q^{87} -2.94567 q^{88} +3.17548 q^{89} +1.00000 q^{90} +1.28415 q^{91} -1.00000 q^{92} +2.22982 q^{93} -2.22982 q^{94} +6.22982 q^{95} -1.00000 q^{96} +15.6351 q^{97} +1.00000 q^{98} -2.94567 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} + 3 q^{5} - 3 q^{6} + 3 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} + 3 q^{5} - 3 q^{6} + 3 q^{7} + 3 q^{8} + 3 q^{9} + 3 q^{10} + 2 q^{11} - 3 q^{12} + 2 q^{13} + 3 q^{14} - 3 q^{15} + 3 q^{16} + 3 q^{18} + 6 q^{19} + 3 q^{20} - 3 q^{21} + 2 q^{22} - 3 q^{23} - 3 q^{24} + 3 q^{25} + 2 q^{26} - 3 q^{27} + 3 q^{28} + 4 q^{29} - 3 q^{30} + 6 q^{31} + 3 q^{32} - 2 q^{33} + 3 q^{35} + 3 q^{36} + 6 q^{37} + 6 q^{38} - 2 q^{39} + 3 q^{40} - 6 q^{41} - 3 q^{42} + 22 q^{43} + 2 q^{44} + 3 q^{45} - 3 q^{46} + 6 q^{47} - 3 q^{48} + 3 q^{49} + 3 q^{50} + 2 q^{52} + 6 q^{53} - 3 q^{54} + 2 q^{55} + 3 q^{56} - 6 q^{57} + 4 q^{58} - 3 q^{60} - 6 q^{61} + 6 q^{62} + 3 q^{63} + 3 q^{64} + 2 q^{65} - 2 q^{66} - 2 q^{67} + 3 q^{69} + 3 q^{70} + 6 q^{71} + 3 q^{72} + 14 q^{73} + 6 q^{74} - 3 q^{75} + 6 q^{76} + 2 q^{77} - 2 q^{78} + 3 q^{80} + 3 q^{81} - 6 q^{82} + 2 q^{83} - 3 q^{84} + 22 q^{86} - 4 q^{87} + 2 q^{88} - 14 q^{89} + 3 q^{90} + 2 q^{91} - 3 q^{92} - 6 q^{93} + 6 q^{94} + 6 q^{95} - 3 q^{96} - 2 q^{97} + 3 q^{98} + 2 q^{99}+O(q^{100}) \) 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
\(11\) −2.94567 −0.888152 −0.444076 0.895989i \(-0.646468\pi\)
−0.444076 + 0.895989i \(0.646468\pi\)
\(12\) −1.00000 −0.288675
\(13\) 1.28415 0.356158 0.178079 0.984016i \(-0.443012\pi\)
0.178079 + 0.984016i \(0.443012\pi\)
\(14\) 1.00000 0.267261
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) 4.22982 1.02588 0.512940 0.858424i \(-0.328556\pi\)
0.512940 + 0.858424i \(0.328556\pi\)
\(18\) 1.00000 0.235702
\(19\) 6.22982 1.42922 0.714609 0.699524i \(-0.246605\pi\)
0.714609 + 0.699524i \(0.246605\pi\)
\(20\) 1.00000 0.223607
\(21\) −1.00000 −0.218218
\(22\) −2.94567 −0.628018
\(23\) −1.00000 −0.208514
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) 1.28415 0.251842
\(27\) −1.00000 −0.192450
\(28\) 1.00000 0.188982
\(29\) 4.94567 0.918387 0.459194 0.888336i \(-0.348138\pi\)
0.459194 + 0.888336i \(0.348138\pi\)
\(30\) −1.00000 −0.182574
\(31\) −2.22982 −0.400487 −0.200243 0.979746i \(-0.564173\pi\)
−0.200243 + 0.979746i \(0.564173\pi\)
\(32\) 1.00000 0.176777
\(33\) 2.94567 0.512775
\(34\) 4.22982 0.725407
\(35\) 1.00000 0.169031
\(36\) 1.00000 0.166667
\(37\) −6.45963 −1.06196 −0.530978 0.847385i \(-0.678175\pi\)
−0.530978 + 0.847385i \(0.678175\pi\)
\(38\) 6.22982 1.01061
\(39\) −1.28415 −0.205628
\(40\) 1.00000 0.158114
\(41\) −10.4596 −1.63352 −0.816760 0.576978i \(-0.804232\pi\)
−0.816760 + 0.576978i \(0.804232\pi\)
\(42\) −1.00000 −0.154303
\(43\) 10.9457 1.66920 0.834599 0.550857i \(-0.185699\pi\)
0.834599 + 0.550857i \(0.185699\pi\)
\(44\) −2.94567 −0.444076
\(45\) 1.00000 0.149071
\(46\) −1.00000 −0.147442
\(47\) −2.22982 −0.325252 −0.162626 0.986688i \(-0.551996\pi\)
−0.162626 + 0.986688i \(0.551996\pi\)
\(48\) −1.00000 −0.144338
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) −4.22982 −0.592293
\(52\) 1.28415 0.178079
\(53\) −6.45963 −0.887298 −0.443649 0.896201i \(-0.646316\pi\)
−0.443649 + 0.896201i \(0.646316\pi\)
\(54\) −1.00000 −0.136083
\(55\) −2.94567 −0.397194
\(56\) 1.00000 0.133631
\(57\) −6.22982 −0.825159
\(58\) 4.94567 0.649398
\(59\) 8.45963 1.10135 0.550675 0.834720i \(-0.314370\pi\)
0.550675 + 0.834720i \(0.314370\pi\)
\(60\) −1.00000 −0.129099
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) −2.22982 −0.283187
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) 1.28415 0.159279
\(66\) 2.94567 0.362587
\(67\) 2.94567 0.359871 0.179935 0.983678i \(-0.442411\pi\)
0.179935 + 0.983678i \(0.442411\pi\)
\(68\) 4.22982 0.512940
\(69\) 1.00000 0.120386
\(70\) 1.00000 0.119523
\(71\) 12.8370 1.52347 0.761736 0.647888i \(-0.224347\pi\)
0.761736 + 0.647888i \(0.224347\pi\)
\(72\) 1.00000 0.117851
\(73\) 3.43171 0.401651 0.200825 0.979627i \(-0.435638\pi\)
0.200825 + 0.979627i \(0.435638\pi\)
\(74\) −6.45963 −0.750917
\(75\) −1.00000 −0.115470
\(76\) 6.22982 0.714609
\(77\) −2.94567 −0.335690
\(78\) −1.28415 −0.145401
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) −10.4596 −1.15507
\(83\) −4.79811 −0.526661 −0.263330 0.964706i \(-0.584821\pi\)
−0.263330 + 0.964706i \(0.584821\pi\)
\(84\) −1.00000 −0.109109
\(85\) 4.22982 0.458788
\(86\) 10.9457 1.18030
\(87\) −4.94567 −0.530231
\(88\) −2.94567 −0.314009
\(89\) 3.17548 0.336601 0.168300 0.985736i \(-0.446172\pi\)
0.168300 + 0.985736i \(0.446172\pi\)
\(90\) 1.00000 0.105409
\(91\) 1.28415 0.134615
\(92\) −1.00000 −0.104257
\(93\) 2.22982 0.231221
\(94\) −2.22982 −0.229988
\(95\) 6.22982 0.639166
\(96\) −1.00000 −0.102062
\(97\) 15.6351 1.58751 0.793753 0.608241i \(-0.208124\pi\)
0.793753 + 0.608241i \(0.208124\pi\)
\(98\) 1.00000 0.101015
\(99\) −2.94567 −0.296051
\(100\) 1.00000 0.100000
\(101\) 7.17548 0.713987 0.356994 0.934107i \(-0.383802\pi\)
0.356994 + 0.934107i \(0.383802\pi\)
\(102\) −4.22982 −0.418814
\(103\) 5.89134 0.580491 0.290245 0.956952i \(-0.406263\pi\)
0.290245 + 0.956952i \(0.406263\pi\)
\(104\) 1.28415 0.125921
\(105\) −1.00000 −0.0975900
\(106\) −6.45963 −0.627415
\(107\) 4.00000 0.386695 0.193347 0.981130i \(-0.438066\pi\)
0.193347 + 0.981130i \(0.438066\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 3.89134 0.372722 0.186361 0.982481i \(-0.440331\pi\)
0.186361 + 0.982481i \(0.440331\pi\)
\(110\) −2.94567 −0.280858
\(111\) 6.45963 0.613121
\(112\) 1.00000 0.0944911
\(113\) −2.45963 −0.231383 −0.115691 0.993285i \(-0.536908\pi\)
−0.115691 + 0.993285i \(0.536908\pi\)
\(114\) −6.22982 −0.583476
\(115\) −1.00000 −0.0932505
\(116\) 4.94567 0.459194
\(117\) 1.28415 0.118719
\(118\) 8.45963 0.778772
\(119\) 4.22982 0.387747
\(120\) −1.00000 −0.0912871
\(121\) −2.32304 −0.211186
\(122\) −2.00000 −0.181071
\(123\) 10.4596 0.943113
\(124\) −2.22982 −0.200243
\(125\) 1.00000 0.0894427
\(126\) 1.00000 0.0890871
\(127\) 13.5140 1.19917 0.599585 0.800311i \(-0.295332\pi\)
0.599585 + 0.800311i \(0.295332\pi\)
\(128\) 1.00000 0.0883883
\(129\) −10.9457 −0.963712
\(130\) 1.28415 0.112627
\(131\) −4.00000 −0.349482 −0.174741 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) 2.94567 0.256387
\(133\) 6.22982 0.540193
\(134\) 2.94567 0.254467
\(135\) −1.00000 −0.0860663
\(136\) 4.22982 0.362704
\(137\) −17.7827 −1.51928 −0.759638 0.650346i \(-0.774624\pi\)
−0.759638 + 0.650346i \(0.774624\pi\)
\(138\) 1.00000 0.0851257
\(139\) −2.56829 −0.217840 −0.108920 0.994051i \(-0.534739\pi\)
−0.108920 + 0.994051i \(0.534739\pi\)
\(140\) 1.00000 0.0845154
\(141\) 2.22982 0.187784
\(142\) 12.8370 1.07726
\(143\) −3.78267 −0.316323
\(144\) 1.00000 0.0833333
\(145\) 4.94567 0.410715
\(146\) 3.43171 0.284010
\(147\) −1.00000 −0.0824786
\(148\) −6.45963 −0.530978
\(149\) −9.32304 −0.763773 −0.381887 0.924209i \(-0.624726\pi\)
−0.381887 + 0.924209i \(0.624726\pi\)
\(150\) −1.00000 −0.0816497
\(151\) −1.43171 −0.116511 −0.0582553 0.998302i \(-0.518554\pi\)
−0.0582553 + 0.998302i \(0.518554\pi\)
\(152\) 6.22982 0.505305
\(153\) 4.22982 0.341960
\(154\) −2.94567 −0.237369
\(155\) −2.22982 −0.179103
\(156\) −1.28415 −0.102814
\(157\) −13.0279 −1.03974 −0.519871 0.854245i \(-0.674020\pi\)
−0.519871 + 0.854245i \(0.674020\pi\)
\(158\) 0 0
\(159\) 6.45963 0.512282
\(160\) 1.00000 0.0790569
\(161\) −1.00000 −0.0788110
\(162\) 1.00000 0.0785674
\(163\) 10.5683 0.827773 0.413886 0.910329i \(-0.364171\pi\)
0.413886 + 0.910329i \(0.364171\pi\)
\(164\) −10.4596 −0.816760
\(165\) 2.94567 0.229320
\(166\) −4.79811 −0.372406
\(167\) 21.2577 1.64497 0.822487 0.568784i \(-0.192586\pi\)
0.822487 + 0.568784i \(0.192586\pi\)
\(168\) −1.00000 −0.0771517
\(169\) −11.3510 −0.873151
\(170\) 4.22982 0.324412
\(171\) 6.22982 0.476406
\(172\) 10.9457 0.834599
\(173\) −16.6894 −1.26888 −0.634438 0.772974i \(-0.718768\pi\)
−0.634438 + 0.772974i \(0.718768\pi\)
\(174\) −4.94567 −0.374930
\(175\) 1.00000 0.0755929
\(176\) −2.94567 −0.222038
\(177\) −8.45963 −0.635865
\(178\) 3.17548 0.238013
\(179\) −7.02792 −0.525292 −0.262646 0.964892i \(-0.584595\pi\)
−0.262646 + 0.964892i \(0.584595\pi\)
\(180\) 1.00000 0.0745356
\(181\) 6.00000 0.445976 0.222988 0.974821i \(-0.428419\pi\)
0.222988 + 0.974821i \(0.428419\pi\)
\(182\) 1.28415 0.0951873
\(183\) 2.00000 0.147844
\(184\) −1.00000 −0.0737210
\(185\) −6.45963 −0.474921
\(186\) 2.22982 0.163498
\(187\) −12.4596 −0.911138
\(188\) −2.22982 −0.162626
\(189\) −1.00000 −0.0727393
\(190\) 6.22982 0.451958
\(191\) 20.4596 1.48041 0.740203 0.672383i \(-0.234729\pi\)
0.740203 + 0.672383i \(0.234729\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 22.4596 1.61668 0.808340 0.588716i \(-0.200366\pi\)
0.808340 + 0.588716i \(0.200366\pi\)
\(194\) 15.6351 1.12254
\(195\) −1.28415 −0.0919597
\(196\) 1.00000 0.0714286
\(197\) 22.2423 1.58470 0.792349 0.610068i \(-0.208858\pi\)
0.792349 + 0.610068i \(0.208858\pi\)
\(198\) −2.94567 −0.209339
\(199\) 12.4596 0.883240 0.441620 0.897202i \(-0.354404\pi\)
0.441620 + 0.897202i \(0.354404\pi\)
\(200\) 1.00000 0.0707107
\(201\) −2.94567 −0.207771
\(202\) 7.17548 0.504865
\(203\) 4.94567 0.347118
\(204\) −4.22982 −0.296146
\(205\) −10.4596 −0.730532
\(206\) 5.89134 0.410469
\(207\) −1.00000 −0.0695048
\(208\) 1.28415 0.0890396
\(209\) −18.3510 −1.26936
\(210\) −1.00000 −0.0690066
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) −6.45963 −0.443649
\(213\) −12.8370 −0.879577
\(214\) 4.00000 0.273434
\(215\) 10.9457 0.746488
\(216\) −1.00000 −0.0680414
\(217\) −2.22982 −0.151370
\(218\) 3.89134 0.263555
\(219\) −3.43171 −0.231893
\(220\) −2.94567 −0.198597
\(221\) 5.43171 0.365376
\(222\) 6.45963 0.433542
\(223\) −2.60719 −0.174590 −0.0872951 0.996182i \(-0.527822\pi\)
−0.0872951 + 0.996182i \(0.527822\pi\)
\(224\) 1.00000 0.0668153
\(225\) 1.00000 0.0666667
\(226\) −2.45963 −0.163612
\(227\) −14.2298 −0.944466 −0.472233 0.881474i \(-0.656552\pi\)
−0.472233 + 0.881474i \(0.656552\pi\)
\(228\) −6.22982 −0.412580
\(229\) 3.89134 0.257147 0.128573 0.991700i \(-0.458960\pi\)
0.128573 + 0.991700i \(0.458960\pi\)
\(230\) −1.00000 −0.0659380
\(231\) 2.94567 0.193811
\(232\) 4.94567 0.324699
\(233\) −10.4596 −0.685233 −0.342617 0.939475i \(-0.611313\pi\)
−0.342617 + 0.939475i \(0.611313\pi\)
\(234\) 1.28415 0.0839473
\(235\) −2.22982 −0.145457
\(236\) 8.45963 0.550675
\(237\) 0 0
\(238\) 4.22982 0.274178
\(239\) 20.8370 1.34783 0.673917 0.738807i \(-0.264611\pi\)
0.673917 + 0.738807i \(0.264611\pi\)
\(240\) −1.00000 −0.0645497
\(241\) −20.6894 −1.33272 −0.666362 0.745628i \(-0.732150\pi\)
−0.666362 + 0.745628i \(0.732150\pi\)
\(242\) −2.32304 −0.149331
\(243\) −1.00000 −0.0641500
\(244\) −2.00000 −0.128037
\(245\) 1.00000 0.0638877
\(246\) 10.4596 0.666882
\(247\) 8.00000 0.509028
\(248\) −2.22982 −0.141593
\(249\) 4.79811 0.304068
\(250\) 1.00000 0.0632456
\(251\) 5.17548 0.326674 0.163337 0.986570i \(-0.447774\pi\)
0.163337 + 0.986570i \(0.447774\pi\)
\(252\) 1.00000 0.0629941
\(253\) 2.94567 0.185193
\(254\) 13.5140 0.847941
\(255\) −4.22982 −0.264881
\(256\) 1.00000 0.0625000
\(257\) −21.3230 −1.33009 −0.665047 0.746801i \(-0.731589\pi\)
−0.665047 + 0.746801i \(0.731589\pi\)
\(258\) −10.9457 −0.681448
\(259\) −6.45963 −0.401382
\(260\) 1.28415 0.0796394
\(261\) 4.94567 0.306129
\(262\) −4.00000 −0.247121
\(263\) 23.4876 1.44830 0.724152 0.689640i \(-0.242231\pi\)
0.724152 + 0.689640i \(0.242231\pi\)
\(264\) 2.94567 0.181293
\(265\) −6.45963 −0.396812
\(266\) 6.22982 0.381974
\(267\) −3.17548 −0.194336
\(268\) 2.94567 0.179935
\(269\) 10.2034 0.622113 0.311056 0.950391i \(-0.399317\pi\)
0.311056 + 0.950391i \(0.399317\pi\)
\(270\) −1.00000 −0.0608581
\(271\) −14.0125 −0.851198 −0.425599 0.904912i \(-0.639937\pi\)
−0.425599 + 0.904912i \(0.639937\pi\)
\(272\) 4.22982 0.256470
\(273\) −1.28415 −0.0777201
\(274\) −17.7827 −1.07429
\(275\) −2.94567 −0.177630
\(276\) 1.00000 0.0601929
\(277\) −7.51396 −0.451470 −0.225735 0.974189i \(-0.572478\pi\)
−0.225735 + 0.974189i \(0.572478\pi\)
\(278\) −2.56829 −0.154036
\(279\) −2.22982 −0.133496
\(280\) 1.00000 0.0597614
\(281\) −1.40530 −0.0838330 −0.0419165 0.999121i \(-0.513346\pi\)
−0.0419165 + 0.999121i \(0.513346\pi\)
\(282\) 2.22982 0.132784
\(283\) −8.20341 −0.487642 −0.243821 0.969820i \(-0.578401\pi\)
−0.243821 + 0.969820i \(0.578401\pi\)
\(284\) 12.8370 0.761736
\(285\) −6.22982 −0.369022
\(286\) −3.78267 −0.223674
\(287\) −10.4596 −0.617412
\(288\) 1.00000 0.0589256
\(289\) 0.891336 0.0524315
\(290\) 4.94567 0.290420
\(291\) −15.6351 −0.916547
\(292\) 3.43171 0.200825
\(293\) −7.89134 −0.461017 −0.230508 0.973070i \(-0.574039\pi\)
−0.230508 + 0.973070i \(0.574039\pi\)
\(294\) −1.00000 −0.0583212
\(295\) 8.45963 0.492539
\(296\) −6.45963 −0.375458
\(297\) 2.94567 0.170925
\(298\) −9.32304 −0.540069
\(299\) −1.28415 −0.0742642
\(300\) −1.00000 −0.0577350
\(301\) 10.9457 0.630898
\(302\) −1.43171 −0.0823854
\(303\) −7.17548 −0.412221
\(304\) 6.22982 0.357304
\(305\) −2.00000 −0.114520
\(306\) 4.22982 0.241802
\(307\) −10.8106 −0.616993 −0.308497 0.951225i \(-0.599826\pi\)
−0.308497 + 0.951225i \(0.599826\pi\)
\(308\) −2.94567 −0.167845
\(309\) −5.89134 −0.335146
\(310\) −2.22982 −0.126645
\(311\) −7.74378 −0.439109 −0.219555 0.975600i \(-0.570460\pi\)
−0.219555 + 0.975600i \(0.570460\pi\)
\(312\) −1.28415 −0.0727005
\(313\) 17.0668 0.964674 0.482337 0.875986i \(-0.339788\pi\)
0.482337 + 0.875986i \(0.339788\pi\)
\(314\) −13.0279 −0.735208
\(315\) 1.00000 0.0563436
\(316\) 0 0
\(317\) −8.56829 −0.481243 −0.240622 0.970619i \(-0.577351\pi\)
−0.240622 + 0.970619i \(0.577351\pi\)
\(318\) 6.45963 0.362238
\(319\) −14.5683 −0.815668
\(320\) 1.00000 0.0559017
\(321\) −4.00000 −0.223258
\(322\) −1.00000 −0.0557278
\(323\) 26.3510 1.46621
\(324\) 1.00000 0.0555556
\(325\) 1.28415 0.0712317
\(326\) 10.5683 0.585324
\(327\) −3.89134 −0.215191
\(328\) −10.4596 −0.577536
\(329\) −2.22982 −0.122934
\(330\) 2.94567 0.162154
\(331\) 30.3510 1.66824 0.834120 0.551583i \(-0.185976\pi\)
0.834120 + 0.551583i \(0.185976\pi\)
\(332\) −4.79811 −0.263330
\(333\) −6.45963 −0.353986
\(334\) 21.2577 1.16317
\(335\) 2.94567 0.160939
\(336\) −1.00000 −0.0545545
\(337\) 27.2966 1.48694 0.743471 0.668768i \(-0.233178\pi\)
0.743471 + 0.668768i \(0.233178\pi\)
\(338\) −11.3510 −0.617411
\(339\) 2.45963 0.133589
\(340\) 4.22982 0.229394
\(341\) 6.56829 0.355693
\(342\) 6.22982 0.336870
\(343\) 1.00000 0.0539949
\(344\) 10.9457 0.590151
\(345\) 1.00000 0.0538382
\(346\) −16.6894 −0.897230
\(347\) 14.3510 0.770400 0.385200 0.922833i \(-0.374132\pi\)
0.385200 + 0.922833i \(0.374132\pi\)
\(348\) −4.94567 −0.265116
\(349\) 4.01249 0.214783 0.107392 0.994217i \(-0.465750\pi\)
0.107392 + 0.994217i \(0.465750\pi\)
\(350\) 1.00000 0.0534522
\(351\) −1.28415 −0.0685427
\(352\) −2.94567 −0.157005
\(353\) 24.8106 1.32053 0.660267 0.751031i \(-0.270443\pi\)
0.660267 + 0.751031i \(0.270443\pi\)
\(354\) −8.45963 −0.449624
\(355\) 12.8370 0.681317
\(356\) 3.17548 0.168300
\(357\) −4.22982 −0.223866
\(358\) −7.02792 −0.371437
\(359\) −0.676959 −0.0357285 −0.0178643 0.999840i \(-0.505687\pi\)
−0.0178643 + 0.999840i \(0.505687\pi\)
\(360\) 1.00000 0.0527046
\(361\) 19.8106 1.04266
\(362\) 6.00000 0.315353
\(363\) 2.32304 0.121928
\(364\) 1.28415 0.0673076
\(365\) 3.43171 0.179624
\(366\) 2.00000 0.104542
\(367\) −0.754747 −0.0393975 −0.0196987 0.999806i \(-0.506271\pi\)
−0.0196987 + 0.999806i \(0.506271\pi\)
\(368\) −1.00000 −0.0521286
\(369\) −10.4596 −0.544507
\(370\) −6.45963 −0.335820
\(371\) −6.45963 −0.335367
\(372\) 2.22982 0.115611
\(373\) 6.67696 0.345720 0.172860 0.984946i \(-0.444699\pi\)
0.172860 + 0.984946i \(0.444699\pi\)
\(374\) −12.4596 −0.644272
\(375\) −1.00000 −0.0516398
\(376\) −2.22982 −0.114994
\(377\) 6.35097 0.327091
\(378\) −1.00000 −0.0514344
\(379\) −7.78267 −0.399769 −0.199884 0.979819i \(-0.564057\pi\)
−0.199884 + 0.979819i \(0.564057\pi\)
\(380\) 6.22982 0.319583
\(381\) −13.5140 −0.692341
\(382\) 20.4596 1.04681
\(383\) −27.0279 −1.38106 −0.690531 0.723303i \(-0.742623\pi\)
−0.690531 + 0.723303i \(0.742623\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −2.94567 −0.150125
\(386\) 22.4596 1.14317
\(387\) 10.9457 0.556400
\(388\) 15.6351 0.793753
\(389\) 4.81060 0.243907 0.121953 0.992536i \(-0.461084\pi\)
0.121953 + 0.992536i \(0.461084\pi\)
\(390\) −1.28415 −0.0650253
\(391\) −4.22982 −0.213911
\(392\) 1.00000 0.0505076
\(393\) 4.00000 0.201773
\(394\) 22.2423 1.12055
\(395\) 0 0
\(396\) −2.94567 −0.148025
\(397\) 1.28415 0.0644495 0.0322248 0.999481i \(-0.489741\pi\)
0.0322248 + 0.999481i \(0.489741\pi\)
\(398\) 12.4596 0.624545
\(399\) −6.22982 −0.311881
\(400\) 1.00000 0.0500000
\(401\) 1.62263 0.0810301 0.0405151 0.999179i \(-0.487100\pi\)
0.0405151 + 0.999179i \(0.487100\pi\)
\(402\) −2.94567 −0.146917
\(403\) −2.86341 −0.142637
\(404\) 7.17548 0.356994
\(405\) 1.00000 0.0496904
\(406\) 4.94567 0.245449
\(407\) 19.0279 0.943179
\(408\) −4.22982 −0.209407
\(409\) −8.10866 −0.400948 −0.200474 0.979699i \(-0.564248\pi\)
−0.200474 + 0.979699i \(0.564248\pi\)
\(410\) −10.4596 −0.516564
\(411\) 17.7827 0.877154
\(412\) 5.89134 0.290245
\(413\) 8.45963 0.416271
\(414\) −1.00000 −0.0491473
\(415\) −4.79811 −0.235530
\(416\) 1.28415 0.0629605
\(417\) 2.56829 0.125770
\(418\) −18.3510 −0.897575
\(419\) −28.4985 −1.39224 −0.696122 0.717924i \(-0.745093\pi\)
−0.696122 + 0.717924i \(0.745093\pi\)
\(420\) −1.00000 −0.0487950
\(421\) 20.5683 1.00244 0.501219 0.865321i \(-0.332885\pi\)
0.501219 + 0.865321i \(0.332885\pi\)
\(422\) 4.00000 0.194717
\(423\) −2.22982 −0.108417
\(424\) −6.45963 −0.313707
\(425\) 4.22982 0.205176
\(426\) −12.8370 −0.621955
\(427\) −2.00000 −0.0967868
\(428\) 4.00000 0.193347
\(429\) 3.78267 0.182629
\(430\) 10.9457 0.527847
\(431\) −17.5962 −0.847580 −0.423790 0.905760i \(-0.639301\pi\)
−0.423790 + 0.905760i \(0.639301\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −9.28415 −0.446168 −0.223084 0.974799i \(-0.571612\pi\)
−0.223084 + 0.974799i \(0.571612\pi\)
\(434\) −2.22982 −0.107035
\(435\) −4.94567 −0.237127
\(436\) 3.89134 0.186361
\(437\) −6.22982 −0.298012
\(438\) −3.43171 −0.163973
\(439\) −19.6615 −0.938393 −0.469197 0.883094i \(-0.655456\pi\)
−0.469197 + 0.883094i \(0.655456\pi\)
\(440\) −2.94567 −0.140429
\(441\) 1.00000 0.0476190
\(442\) 5.43171 0.258360
\(443\) 39.2702 1.86578 0.932892 0.360155i \(-0.117276\pi\)
0.932892 + 0.360155i \(0.117276\pi\)
\(444\) 6.45963 0.306560
\(445\) 3.17548 0.150532
\(446\) −2.60719 −0.123454
\(447\) 9.32304 0.440965
\(448\) 1.00000 0.0472456
\(449\) −13.4876 −0.636517 −0.318259 0.948004i \(-0.603098\pi\)
−0.318259 + 0.948004i \(0.603098\pi\)
\(450\) 1.00000 0.0471405
\(451\) 30.8106 1.45081
\(452\) −2.45963 −0.115691
\(453\) 1.43171 0.0672674
\(454\) −14.2298 −0.667838
\(455\) 1.28415 0.0602018
\(456\) −6.22982 −0.291738
\(457\) 40.5110 1.89503 0.947513 0.319719i \(-0.103588\pi\)
0.947513 + 0.319719i \(0.103588\pi\)
\(458\) 3.89134 0.181830
\(459\) −4.22982 −0.197431
\(460\) −1.00000 −0.0466252
\(461\) −21.5264 −1.00259 −0.501293 0.865277i \(-0.667142\pi\)
−0.501293 + 0.865277i \(0.667142\pi\)
\(462\) 2.94567 0.137045
\(463\) −13.7563 −0.639308 −0.319654 0.947534i \(-0.603567\pi\)
−0.319654 + 0.947534i \(0.603567\pi\)
\(464\) 4.94567 0.229597
\(465\) 2.22982 0.103405
\(466\) −10.4596 −0.484533
\(467\) −0.338479 −0.0156630 −0.00783148 0.999969i \(-0.502493\pi\)
−0.00783148 + 0.999969i \(0.502493\pi\)
\(468\) 1.28415 0.0593597
\(469\) 2.94567 0.136018
\(470\) −2.22982 −0.102854
\(471\) 13.0279 0.600295
\(472\) 8.45963 0.389386
\(473\) −32.2423 −1.48250
\(474\) 0 0
\(475\) 6.22982 0.285844
\(476\) 4.22982 0.193873
\(477\) −6.45963 −0.295766
\(478\) 20.8370 0.953062
\(479\) −35.9472 −1.64247 −0.821234 0.570591i \(-0.806714\pi\)
−0.821234 + 0.570591i \(0.806714\pi\)
\(480\) −1.00000 −0.0456435
\(481\) −8.29512 −0.378225
\(482\) −20.6894 −0.942378
\(483\) 1.00000 0.0455016
\(484\) −2.32304 −0.105593
\(485\) 15.6351 0.709954
\(486\) −1.00000 −0.0453609
\(487\) −1.05433 −0.0477764 −0.0238882 0.999715i \(-0.507605\pi\)
−0.0238882 + 0.999715i \(0.507605\pi\)
\(488\) −2.00000 −0.0905357
\(489\) −10.5683 −0.477915
\(490\) 1.00000 0.0451754
\(491\) −12.9193 −0.583038 −0.291519 0.956565i \(-0.594161\pi\)
−0.291519 + 0.956565i \(0.594161\pi\)
\(492\) 10.4596 0.471557
\(493\) 20.9193 0.942156
\(494\) 8.00000 0.359937
\(495\) −2.94567 −0.132398
\(496\) −2.22982 −0.100122
\(497\) 12.8370 0.575818
\(498\) 4.79811 0.215008
\(499\) −29.8385 −1.33576 −0.667878 0.744271i \(-0.732797\pi\)
−0.667878 + 0.744271i \(0.732797\pi\)
\(500\) 1.00000 0.0447214
\(501\) −21.2577 −0.949726
\(502\) 5.17548 0.230993
\(503\) 14.8106 0.660372 0.330186 0.943916i \(-0.392889\pi\)
0.330186 + 0.943916i \(0.392889\pi\)
\(504\) 1.00000 0.0445435
\(505\) 7.17548 0.319305
\(506\) 2.94567 0.130951
\(507\) 11.3510 0.504114
\(508\) 13.5140 0.599585
\(509\) −28.6072 −1.26799 −0.633996 0.773337i \(-0.718586\pi\)
−0.633996 + 0.773337i \(0.718586\pi\)
\(510\) −4.22982 −0.187299
\(511\) 3.43171 0.151810
\(512\) 1.00000 0.0441942
\(513\) −6.22982 −0.275053
\(514\) −21.3230 −0.940519
\(515\) 5.89134 0.259603
\(516\) −10.9457 −0.481856
\(517\) 6.56829 0.288873
\(518\) −6.45963 −0.283820
\(519\) 16.6894 0.732585
\(520\) 1.28415 0.0563136
\(521\) 0.312072 0.0136721 0.00683606 0.999977i \(-0.497824\pi\)
0.00683606 + 0.999977i \(0.497824\pi\)
\(522\) 4.94567 0.216466
\(523\) −8.20341 −0.358710 −0.179355 0.983784i \(-0.557401\pi\)
−0.179355 + 0.983784i \(0.557401\pi\)
\(524\) −4.00000 −0.174741
\(525\) −1.00000 −0.0436436
\(526\) 23.4876 1.02411
\(527\) −9.43171 −0.410852
\(528\) 2.94567 0.128194
\(529\) 1.00000 0.0434783
\(530\) −6.45963 −0.280588
\(531\) 8.45963 0.367117
\(532\) 6.22982 0.270097
\(533\) −13.4317 −0.581792
\(534\) −3.17548 −0.137417
\(535\) 4.00000 0.172935
\(536\) 2.94567 0.127233
\(537\) 7.02792 0.303277
\(538\) 10.2034 0.439900
\(539\) −2.94567 −0.126879
\(540\) −1.00000 −0.0430331
\(541\) 0.108664 0.00467185 0.00233592 0.999997i \(-0.499256\pi\)
0.00233592 + 0.999997i \(0.499256\pi\)
\(542\) −14.0125 −0.601888
\(543\) −6.00000 −0.257485
\(544\) 4.22982 0.181352
\(545\) 3.89134 0.166687
\(546\) −1.28415 −0.0549564
\(547\) 5.59622 0.239277 0.119639 0.992818i \(-0.461826\pi\)
0.119639 + 0.992818i \(0.461826\pi\)
\(548\) −17.7827 −0.759638
\(549\) −2.00000 −0.0853579
\(550\) −2.94567 −0.125604
\(551\) 30.8106 1.31258
\(552\) 1.00000 0.0425628
\(553\) 0 0
\(554\) −7.51396 −0.319238
\(555\) 6.45963 0.274196
\(556\) −2.56829 −0.108920
\(557\) 40.5933 1.71999 0.859996 0.510301i \(-0.170466\pi\)
0.859996 + 0.510301i \(0.170466\pi\)
\(558\) −2.22982 −0.0943956
\(559\) 14.0558 0.594499
\(560\) 1.00000 0.0422577
\(561\) 12.4596 0.526046
\(562\) −1.40530 −0.0592789
\(563\) −4.79811 −0.202216 −0.101108 0.994875i \(-0.532239\pi\)
−0.101108 + 0.994875i \(0.532239\pi\)
\(564\) 2.22982 0.0938922
\(565\) −2.45963 −0.103477
\(566\) −8.20341 −0.344815
\(567\) 1.00000 0.0419961
\(568\) 12.8370 0.538629
\(569\) −22.3774 −0.938108 −0.469054 0.883169i \(-0.655405\pi\)
−0.469054 + 0.883169i \(0.655405\pi\)
\(570\) −6.22982 −0.260938
\(571\) −16.4596 −0.688814 −0.344407 0.938820i \(-0.611920\pi\)
−0.344407 + 0.938820i \(0.611920\pi\)
\(572\) −3.78267 −0.158161
\(573\) −20.4596 −0.854713
\(574\) −10.4596 −0.436577
\(575\) −1.00000 −0.0417029
\(576\) 1.00000 0.0416667
\(577\) −38.9193 −1.62023 −0.810115 0.586271i \(-0.800595\pi\)
−0.810115 + 0.586271i \(0.800595\pi\)
\(578\) 0.891336 0.0370747
\(579\) −22.4596 −0.933390
\(580\) 4.94567 0.205358
\(581\) −4.79811 −0.199059
\(582\) −15.6351 −0.648096
\(583\) 19.0279 0.788056
\(584\) 3.43171 0.142005
\(585\) 1.28415 0.0530930
\(586\) −7.89134 −0.325988
\(587\) −3.32304 −0.137157 −0.0685783 0.997646i \(-0.521846\pi\)
−0.0685783 + 0.997646i \(0.521846\pi\)
\(588\) −1.00000 −0.0412393
\(589\) −13.8913 −0.572383
\(590\) 8.45963 0.348277
\(591\) −22.2423 −0.914926
\(592\) −6.45963 −0.265489
\(593\) −46.2423 −1.89894 −0.949472 0.313852i \(-0.898380\pi\)
−0.949472 + 0.313852i \(0.898380\pi\)
\(594\) 2.94567 0.120862
\(595\) 4.22982 0.173406
\(596\) −9.32304 −0.381887
\(597\) −12.4596 −0.509939
\(598\) −1.28415 −0.0525127
\(599\) −33.2966 −1.36046 −0.680232 0.732997i \(-0.738121\pi\)
−0.680232 + 0.732997i \(0.738121\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −37.4876 −1.52915 −0.764575 0.644535i \(-0.777051\pi\)
−0.764575 + 0.644535i \(0.777051\pi\)
\(602\) 10.9457 0.446112
\(603\) 2.94567 0.119957
\(604\) −1.43171 −0.0582553
\(605\) −2.32304 −0.0944451
\(606\) −7.17548 −0.291484
\(607\) 7.57926 0.307633 0.153816 0.988099i \(-0.450844\pi\)
0.153816 + 0.988099i \(0.450844\pi\)
\(608\) 6.22982 0.252652
\(609\) −4.94567 −0.200409
\(610\) −2.00000 −0.0809776
\(611\) −2.86341 −0.115841
\(612\) 4.22982 0.170980
\(613\) −2.75475 −0.111263 −0.0556316 0.998451i \(-0.517717\pi\)
−0.0556316 + 0.998451i \(0.517717\pi\)
\(614\) −10.8106 −0.436280
\(615\) 10.4596 0.421773
\(616\) −2.94567 −0.118684
\(617\) −28.8106 −1.15987 −0.579935 0.814662i \(-0.696922\pi\)
−0.579935 + 0.814662i \(0.696922\pi\)
\(618\) −5.89134 −0.236984
\(619\) −44.2857 −1.77999 −0.889996 0.455969i \(-0.849293\pi\)
−0.889996 + 0.455969i \(0.849293\pi\)
\(620\) −2.22982 −0.0895515
\(621\) 1.00000 0.0401286
\(622\) −7.74378 −0.310497
\(623\) 3.17548 0.127223
\(624\) −1.28415 −0.0514070
\(625\) 1.00000 0.0400000
\(626\) 17.0668 0.682127
\(627\) 18.3510 0.732867
\(628\) −13.0279 −0.519871
\(629\) −27.3230 −1.08944
\(630\) 1.00000 0.0398410
\(631\) −3.78267 −0.150586 −0.0752929 0.997161i \(-0.523989\pi\)
−0.0752929 + 0.997161i \(0.523989\pi\)
\(632\) 0 0
\(633\) −4.00000 −0.158986
\(634\) −8.56829 −0.340290
\(635\) 13.5140 0.536285
\(636\) 6.45963 0.256141
\(637\) 1.28415 0.0508798
\(638\) −14.5683 −0.576764
\(639\) 12.8370 0.507824
\(640\) 1.00000 0.0395285
\(641\) −0.486038 −0.0191973 −0.00959867 0.999954i \(-0.503055\pi\)
−0.00959867 + 0.999954i \(0.503055\pi\)
\(642\) −4.00000 −0.157867
\(643\) 16.0389 0.632512 0.316256 0.948674i \(-0.397574\pi\)
0.316256 + 0.948674i \(0.397574\pi\)
\(644\) −1.00000 −0.0394055
\(645\) −10.9457 −0.430985
\(646\) 26.3510 1.03676
\(647\) −39.6087 −1.55718 −0.778589 0.627534i \(-0.784064\pi\)
−0.778589 + 0.627534i \(0.784064\pi\)
\(648\) 1.00000 0.0392837
\(649\) −24.9193 −0.978167
\(650\) 1.28415 0.0503684
\(651\) 2.22982 0.0873933
\(652\) 10.5683 0.413886
\(653\) 11.8913 0.465344 0.232672 0.972555i \(-0.425253\pi\)
0.232672 + 0.972555i \(0.425253\pi\)
\(654\) −3.89134 −0.152163
\(655\) −4.00000 −0.156293
\(656\) −10.4596 −0.408380
\(657\) 3.43171 0.133884
\(658\) −2.22982 −0.0869272
\(659\) −6.48604 −0.252660 −0.126330 0.991988i \(-0.540320\pi\)
−0.126330 + 0.991988i \(0.540320\pi\)
\(660\) 2.94567 0.114660
\(661\) −23.8913 −0.929265 −0.464633 0.885504i \(-0.653814\pi\)
−0.464633 + 0.885504i \(0.653814\pi\)
\(662\) 30.3510 1.17962
\(663\) −5.43171 −0.210950
\(664\) −4.79811 −0.186203
\(665\) 6.22982 0.241582
\(666\) −6.45963 −0.250306
\(667\) −4.94567 −0.191497
\(668\) 21.2577 0.822487
\(669\) 2.60719 0.100800
\(670\) 2.94567 0.113801
\(671\) 5.89134 0.227432
\(672\) −1.00000 −0.0385758
\(673\) 8.56829 0.330283 0.165142 0.986270i \(-0.447192\pi\)
0.165142 + 0.986270i \(0.447192\pi\)
\(674\) 27.2966 1.05143
\(675\) −1.00000 −0.0384900
\(676\) −11.3510 −0.436576
\(677\) −18.9193 −0.727126 −0.363563 0.931570i \(-0.618440\pi\)
−0.363563 + 0.931570i \(0.618440\pi\)
\(678\) 2.45963 0.0944615
\(679\) 15.6351 0.600021
\(680\) 4.22982 0.162206
\(681\) 14.2298 0.545288
\(682\) 6.56829 0.251513
\(683\) −43.0529 −1.64737 −0.823687 0.567045i \(-0.808086\pi\)
−0.823687 + 0.567045i \(0.808086\pi\)
\(684\) 6.22982 0.238203
\(685\) −17.7827 −0.679441
\(686\) 1.00000 0.0381802
\(687\) −3.89134 −0.148464
\(688\) 10.9457 0.417300
\(689\) −8.29512 −0.316019
\(690\) 1.00000 0.0380693
\(691\) 26.1336 0.994170 0.497085 0.867702i \(-0.334404\pi\)
0.497085 + 0.867702i \(0.334404\pi\)
\(692\) −16.6894 −0.634438
\(693\) −2.94567 −0.111897
\(694\) 14.3510 0.544755
\(695\) −2.56829 −0.0974210
\(696\) −4.94567 −0.187465
\(697\) −44.2423 −1.67580
\(698\) 4.01249 0.151875
\(699\) 10.4596 0.395620
\(700\) 1.00000 0.0377964
\(701\) 21.4876 0.811574 0.405787 0.913968i \(-0.366998\pi\)
0.405787 + 0.913968i \(0.366998\pi\)
\(702\) −1.28415 −0.0484670
\(703\) −40.2423 −1.51777
\(704\) −2.94567 −0.111019
\(705\) 2.22982 0.0839797
\(706\) 24.8106 0.933759
\(707\) 7.17548 0.269862
\(708\) −8.45963 −0.317932
\(709\) −2.00000 −0.0751116 −0.0375558 0.999295i \(-0.511957\pi\)
−0.0375558 + 0.999295i \(0.511957\pi\)
\(710\) 12.8370 0.481764
\(711\) 0 0
\(712\) 3.17548 0.119006
\(713\) 2.22982 0.0835072
\(714\) −4.22982 −0.158297
\(715\) −3.78267 −0.141464
\(716\) −7.02792 −0.262646
\(717\) −20.8370 −0.778172
\(718\) −0.676959 −0.0252639
\(719\) −31.2313 −1.16473 −0.582366 0.812927i \(-0.697873\pi\)
−0.582366 + 0.812927i \(0.697873\pi\)
\(720\) 1.00000 0.0372678
\(721\) 5.89134 0.219405
\(722\) 19.8106 0.737274
\(723\) 20.6894 0.769449
\(724\) 6.00000 0.222988
\(725\) 4.94567 0.183677
\(726\) 2.32304 0.0862161
\(727\) 39.4876 1.46451 0.732256 0.681029i \(-0.238467\pi\)
0.732256 + 0.681029i \(0.238467\pi\)
\(728\) 1.28415 0.0475937
\(729\) 1.00000 0.0370370
\(730\) 3.43171 0.127013
\(731\) 46.2982 1.71240
\(732\) 2.00000 0.0739221
\(733\) 9.54037 0.352382 0.176191 0.984356i \(-0.443622\pi\)
0.176191 + 0.984356i \(0.443622\pi\)
\(734\) −0.754747 −0.0278582
\(735\) −1.00000 −0.0368856
\(736\) −1.00000 −0.0368605
\(737\) −8.67696 −0.319620
\(738\) −10.4596 −0.385024
\(739\) −9.64903 −0.354945 −0.177473 0.984126i \(-0.556792\pi\)
−0.177473 + 0.984126i \(0.556792\pi\)
\(740\) −6.45963 −0.237461
\(741\) −8.00000 −0.293887
\(742\) −6.45963 −0.237140
\(743\) 38.9751 1.42986 0.714929 0.699197i \(-0.246459\pi\)
0.714929 + 0.699197i \(0.246459\pi\)
\(744\) 2.22982 0.0817490
\(745\) −9.32304 −0.341570
\(746\) 6.67696 0.244461
\(747\) −4.79811 −0.175554
\(748\) −12.4596 −0.455569
\(749\) 4.00000 0.146157
\(750\) −1.00000 −0.0365148
\(751\) −33.8385 −1.23479 −0.617393 0.786655i \(-0.711811\pi\)
−0.617393 + 0.786655i \(0.711811\pi\)
\(752\) −2.22982 −0.0813130
\(753\) −5.17548 −0.188605
\(754\) 6.35097 0.231289
\(755\) −1.43171 −0.0521051
\(756\) −1.00000 −0.0363696
\(757\) 14.7547 0.536270 0.268135 0.963381i \(-0.413593\pi\)
0.268135 + 0.963381i \(0.413593\pi\)
\(758\) −7.78267 −0.282679
\(759\) −2.94567 −0.106921
\(760\) 6.22982 0.225979
\(761\) −25.1057 −0.910081 −0.455041 0.890471i \(-0.650375\pi\)
−0.455041 + 0.890471i \(0.650375\pi\)
\(762\) −13.5140 −0.489559
\(763\) 3.89134 0.140876
\(764\) 20.4596 0.740203
\(765\) 4.22982 0.152929
\(766\) −27.0279 −0.976559
\(767\) 10.8634 0.392255
\(768\) −1.00000 −0.0360844
\(769\) −5.44419 −0.196323 −0.0981613 0.995171i \(-0.531296\pi\)
−0.0981613 + 0.995171i \(0.531296\pi\)
\(770\) −2.94567 −0.106154
\(771\) 21.3230 0.767931
\(772\) 22.4596 0.808340
\(773\) 36.1336 1.29964 0.649818 0.760090i \(-0.274845\pi\)
0.649818 + 0.760090i \(0.274845\pi\)
\(774\) 10.9457 0.393434
\(775\) −2.22982 −0.0800973
\(776\) 15.6351 0.561268
\(777\) 6.45963 0.231738
\(778\) 4.81060 0.172468
\(779\) −65.1616 −2.33466
\(780\) −1.28415 −0.0459798
\(781\) −37.8135 −1.35308
\(782\) −4.22982 −0.151258
\(783\) −4.94567 −0.176744
\(784\) 1.00000 0.0357143
\(785\) −13.0279 −0.464987
\(786\) 4.00000 0.142675
\(787\) 29.1755 1.03999 0.519997 0.854168i \(-0.325933\pi\)
0.519997 + 0.854168i \(0.325933\pi\)
\(788\) 22.2423 0.792349
\(789\) −23.4876 −0.836179
\(790\) 0 0
\(791\) −2.45963 −0.0874544
\(792\) −2.94567 −0.104670
\(793\) −2.56829 −0.0912028
\(794\) 1.28415 0.0455727
\(795\) 6.45963 0.229099
\(796\) 12.4596 0.441620
\(797\) −19.5962 −0.694134 −0.347067 0.937840i \(-0.612822\pi\)
−0.347067 + 0.937840i \(0.612822\pi\)
\(798\) −6.22982 −0.220533
\(799\) −9.43171 −0.333670
\(800\) 1.00000 0.0353553
\(801\) 3.17548 0.112200
\(802\) 1.62263 0.0572969
\(803\) −10.1087 −0.356727
\(804\) −2.94567 −0.103886
\(805\) −1.00000 −0.0352454
\(806\) −2.86341 −0.100859
\(807\) −10.2034 −0.359177
\(808\) 7.17548 0.252433
\(809\) −5.75770 −0.202430 −0.101215 0.994865i \(-0.532273\pi\)
−0.101215 + 0.994865i \(0.532273\pi\)
\(810\) 1.00000 0.0351364
\(811\) 45.6740 1.60383 0.801916 0.597437i \(-0.203814\pi\)
0.801916 + 0.597437i \(0.203814\pi\)
\(812\) 4.94567 0.173559
\(813\) 14.0125 0.491439
\(814\) 19.0279 0.666928
\(815\) 10.5683 0.370191
\(816\) −4.22982 −0.148073
\(817\) 68.1895 2.38565
\(818\) −8.10866 −0.283513
\(819\) 1.28415 0.0448717
\(820\) −10.4596 −0.365266
\(821\) 21.0235 0.733724 0.366862 0.930275i \(-0.380432\pi\)
0.366862 + 0.930275i \(0.380432\pi\)
\(822\) 17.7827 0.620242
\(823\) −44.3246 −1.54506 −0.772528 0.634980i \(-0.781008\pi\)
−0.772528 + 0.634980i \(0.781008\pi\)
\(824\) 5.89134 0.205234
\(825\) 2.94567 0.102555
\(826\) 8.45963 0.294348
\(827\) −20.9193 −0.727434 −0.363717 0.931510i \(-0.618492\pi\)
−0.363717 + 0.931510i \(0.618492\pi\)
\(828\) −1.00000 −0.0347524
\(829\) 33.3914 1.15973 0.579865 0.814712i \(-0.303105\pi\)
0.579865 + 0.814712i \(0.303105\pi\)
\(830\) −4.79811 −0.166545
\(831\) 7.51396 0.260656
\(832\) 1.28415 0.0445198
\(833\) 4.22982 0.146554
\(834\) 2.56829 0.0889328
\(835\) 21.2577 0.735654
\(836\) −18.3510 −0.634681
\(837\) 2.22982 0.0770737
\(838\) −28.4985 −0.984465
\(839\) −28.4596 −0.982536 −0.491268 0.871009i \(-0.663466\pi\)
−0.491268 + 0.871009i \(0.663466\pi\)
\(840\) −1.00000 −0.0345033
\(841\) −4.54037 −0.156564
\(842\) 20.5683 0.708830
\(843\) 1.40530 0.0484010
\(844\) 4.00000 0.137686
\(845\) −11.3510 −0.390485
\(846\) −2.22982 −0.0766626
\(847\) −2.32304 −0.0798206
\(848\) −6.45963 −0.221825
\(849\) 8.20341 0.281540
\(850\) 4.22982 0.145081
\(851\) 6.45963 0.221433
\(852\) −12.8370 −0.439788
\(853\) −27.4178 −0.938767 −0.469384 0.882994i \(-0.655524\pi\)
−0.469384 + 0.882994i \(0.655524\pi\)
\(854\) −2.00000 −0.0684386
\(855\) 6.22982 0.213055
\(856\) 4.00000 0.136717
\(857\) −30.1645 −1.03040 −0.515200 0.857070i \(-0.672282\pi\)
−0.515200 + 0.857070i \(0.672282\pi\)
\(858\) 3.78267 0.129138
\(859\) 23.7049 0.808800 0.404400 0.914582i \(-0.367480\pi\)
0.404400 + 0.914582i \(0.367480\pi\)
\(860\) 10.9457 0.373244
\(861\) 10.4596 0.356463
\(862\) −17.5962 −0.599330
\(863\) −40.4068 −1.37546 −0.687732 0.725965i \(-0.741394\pi\)
−0.687732 + 0.725965i \(0.741394\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −16.6894 −0.567458
\(866\) −9.28415 −0.315488
\(867\) −0.891336 −0.0302713
\(868\) −2.22982 −0.0756849
\(869\) 0 0
\(870\) −4.94567 −0.167674
\(871\) 3.78267 0.128171
\(872\) 3.89134 0.131777
\(873\) 15.6351 0.529168
\(874\) −6.22982 −0.210727
\(875\) 1.00000 0.0338062
\(876\) −3.43171 −0.115947
\(877\) −18.6197 −0.628742 −0.314371 0.949300i \(-0.601794\pi\)
−0.314371 + 0.949300i \(0.601794\pi\)
\(878\) −19.6615 −0.663544
\(879\) 7.89134 0.266168
\(880\) −2.94567 −0.0992984
\(881\) −45.9861 −1.54931 −0.774655 0.632384i \(-0.782077\pi\)
−0.774655 + 0.632384i \(0.782077\pi\)
\(882\) 1.00000 0.0336718
\(883\) 4.51245 0.151856 0.0759279 0.997113i \(-0.475808\pi\)
0.0759279 + 0.997113i \(0.475808\pi\)
\(884\) 5.43171 0.182688
\(885\) −8.45963 −0.284367
\(886\) 39.2702 1.31931
\(887\) −4.41627 −0.148284 −0.0741419 0.997248i \(-0.523622\pi\)
−0.0741419 + 0.997248i \(0.523622\pi\)
\(888\) 6.45963 0.216771
\(889\) 13.5140 0.453244
\(890\) 3.17548 0.106442
\(891\) −2.94567 −0.0986836
\(892\) −2.60719 −0.0872951
\(893\) −13.8913 −0.464856
\(894\) 9.32304 0.311809
\(895\) −7.02792 −0.234918
\(896\) 1.00000 0.0334077
\(897\) 1.28415 0.0428764
\(898\) −13.4876 −0.450086
\(899\) −11.0279 −0.367802
\(900\) 1.00000 0.0333333
\(901\) −27.3230 −0.910262
\(902\) 30.8106 1.02588
\(903\) −10.9457 −0.364249
\(904\) −2.45963 −0.0818061
\(905\) 6.00000 0.199447
\(906\) 1.43171 0.0475652
\(907\) 22.4860 0.746637 0.373318 0.927703i \(-0.378220\pi\)
0.373318 + 0.927703i \(0.378220\pi\)
\(908\) −14.2298 −0.472233
\(909\) 7.17548 0.237996
\(910\) 1.28415 0.0425691
\(911\) −15.4876 −0.513126 −0.256563 0.966528i \(-0.582590\pi\)
−0.256563 + 0.966528i \(0.582590\pi\)
\(912\) −6.22982 −0.206290
\(913\) 14.1336 0.467755
\(914\) 40.5110 1.33999
\(915\) 2.00000 0.0661180
\(916\) 3.89134 0.128573
\(917\) −4.00000 −0.132092
\(918\) −4.22982 −0.139605
\(919\) −17.4317 −0.575019 −0.287509 0.957778i \(-0.592827\pi\)
−0.287509 + 0.957778i \(0.592827\pi\)
\(920\) −1.00000 −0.0329690
\(921\) 10.8106 0.356221
\(922\) −21.5264 −0.708936
\(923\) 16.4846 0.542597
\(924\) 2.94567 0.0969054
\(925\) −6.45963 −0.212391
\(926\) −13.7563 −0.452059
\(927\) 5.89134 0.193497
\(928\) 4.94567 0.162349
\(929\) −2.97208 −0.0975106 −0.0487553 0.998811i \(-0.515525\pi\)
−0.0487553 + 0.998811i \(0.515525\pi\)
\(930\) 2.22982 0.0731185
\(931\) 6.22982 0.204174
\(932\) −10.4596 −0.342617
\(933\) 7.74378 0.253520
\(934\) −0.338479 −0.0110754
\(935\) −12.4596 −0.407473
\(936\) 1.28415 0.0419737
\(937\) 55.8774 1.82544 0.912718 0.408591i \(-0.133980\pi\)
0.912718 + 0.408591i \(0.133980\pi\)
\(938\) 2.94567 0.0961795
\(939\) −17.0668 −0.556955
\(940\) −2.22982 −0.0727285
\(941\) −22.2173 −0.724264 −0.362132 0.932127i \(-0.617951\pi\)
−0.362132 + 0.932127i \(0.617951\pi\)
\(942\) 13.0279 0.424473
\(943\) 10.4596 0.340612
\(944\) 8.45963 0.275338
\(945\) −1.00000 −0.0325300
\(946\) −32.2423 −1.04829
\(947\) 4.00000 0.129983 0.0649913 0.997886i \(-0.479298\pi\)
0.0649913 + 0.997886i \(0.479298\pi\)
\(948\) 0 0
\(949\) 4.40682 0.143051
\(950\) 6.22982 0.202122
\(951\) 8.56829 0.277846
\(952\) 4.22982 0.137089
\(953\) −47.1616 −1.52771 −0.763856 0.645386i \(-0.776696\pi\)
−0.763856 + 0.645386i \(0.776696\pi\)
\(954\) −6.45963 −0.209138
\(955\) 20.4596 0.662058
\(956\) 20.8370 0.673917
\(957\) 14.5683 0.470926
\(958\) −35.9472 −1.16140
\(959\) −17.7827 −0.574232
\(960\) −1.00000 −0.0322749
\(961\) −26.0279 −0.839610
\(962\) −8.29512 −0.267445
\(963\) 4.00000 0.128898
\(964\) −20.6894 −0.666362
\(965\) 22.4596 0.723001
\(966\) 1.00000 0.0321745
\(967\) 0.299586 0.00963402 0.00481701 0.999988i \(-0.498467\pi\)
0.00481701 + 0.999988i \(0.498467\pi\)
\(968\) −2.32304 −0.0746654
\(969\) −26.3510 −0.846515
\(970\) 15.6351 0.502013
\(971\) −15.5264 −0.498267 −0.249134 0.968469i \(-0.580146\pi\)
−0.249134 + 0.968469i \(0.580146\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −2.56829 −0.0823357
\(974\) −1.05433 −0.0337830
\(975\) −1.28415 −0.0411256
\(976\) −2.00000 −0.0640184
\(977\) 45.5125 1.45607 0.728037 0.685537i \(-0.240433\pi\)
0.728037 + 0.685537i \(0.240433\pi\)
\(978\) −10.5683 −0.337937
\(979\) −9.35392 −0.298953
\(980\) 1.00000 0.0319438
\(981\) 3.89134 0.124241
\(982\) −12.9193 −0.412270
\(983\) −11.6182 −0.370562 −0.185281 0.982686i \(-0.559319\pi\)
−0.185281 + 0.982686i \(0.559319\pi\)
\(984\) 10.4596 0.333441
\(985\) 22.2423 0.708699
\(986\) 20.9193 0.666205
\(987\) 2.22982 0.0709758
\(988\) 8.00000 0.254514
\(989\) −10.9457 −0.348052
\(990\) −2.94567 −0.0936195
\(991\) −27.2702 −0.866267 −0.433134 0.901330i \(-0.642592\pi\)
−0.433134 + 0.901330i \(0.642592\pi\)
\(992\) −2.22982 −0.0707967
\(993\) −30.3510 −0.963159
\(994\) 12.8370 0.407165
\(995\) 12.4596 0.394997
\(996\) 4.79811 0.152034
\(997\) −14.1256 −0.447363 −0.223681 0.974662i \(-0.571808\pi\)
−0.223681 + 0.974662i \(0.571808\pi\)
\(998\) −29.8385 −0.944522
\(999\) 6.45963 0.204374
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.cb.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4830.2.a.cb.1.1 3 1.1 even 1 trivial