Properties

Label 4002.2.a.bk.1.2
Level $4002$
Weight $2$
Character 4002.1
Self dual yes
Analytic conductor $31.956$
Analytic rank $0$
Dimension $8$
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] = [4002,2,Mod(1,4002)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4002, 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("4002.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4002 = 2 \cdot 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4002.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: \(31.9561308889\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 30x^{6} + 52x^{5} + 267x^{4} - 352x^{3} - 632x^{2} + 240x + 288 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-3.42323\) of defining polynomial
Character \(\chi\) \(=\) 4002.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.42323 q^{5} -1.00000 q^{6} -1.25167 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.42323 q^{5} -1.00000 q^{6} -1.25167 q^{7} +1.00000 q^{8} +1.00000 q^{9} -3.42323 q^{10} -2.48043 q^{11} -1.00000 q^{12} -5.96144 q^{13} -1.25167 q^{14} +3.42323 q^{15} +1.00000 q^{16} -5.90366 q^{17} +1.00000 q^{18} +0.0494308 q^{19} -3.42323 q^{20} +1.25167 q^{21} -2.48043 q^{22} -1.00000 q^{23} -1.00000 q^{24} +6.71852 q^{25} -5.96144 q^{26} -1.00000 q^{27} -1.25167 q^{28} -1.00000 q^{29} +3.42323 q^{30} +1.65549 q^{31} +1.00000 q^{32} +2.48043 q^{33} -5.90366 q^{34} +4.28477 q^{35} +1.00000 q^{36} +11.1995 q^{37} +0.0494308 q^{38} +5.96144 q^{39} -3.42323 q^{40} -6.37459 q^{41} +1.25167 q^{42} +9.44108 q^{43} -2.48043 q^{44} -3.42323 q^{45} -1.00000 q^{46} -0.286731 q^{47} -1.00000 q^{48} -5.43332 q^{49} +6.71852 q^{50} +5.90366 q^{51} -5.96144 q^{52} +1.42752 q^{53} -1.00000 q^{54} +8.49109 q^{55} -1.25167 q^{56} -0.0494308 q^{57} -1.00000 q^{58} +1.26100 q^{59} +3.42323 q^{60} +4.48043 q^{61} +1.65549 q^{62} -1.25167 q^{63} +1.00000 q^{64} +20.4074 q^{65} +2.48043 q^{66} -8.37070 q^{67} -5.90366 q^{68} +1.00000 q^{69} +4.28477 q^{70} -5.41935 q^{71} +1.00000 q^{72} -4.43236 q^{73} +11.1995 q^{74} -6.71852 q^{75} +0.0494308 q^{76} +3.10469 q^{77} +5.96144 q^{78} +12.1466 q^{79} -3.42323 q^{80} +1.00000 q^{81} -6.37459 q^{82} +11.8542 q^{83} +1.25167 q^{84} +20.2096 q^{85} +9.44108 q^{86} +1.00000 q^{87} -2.48043 q^{88} -0.796776 q^{89} -3.42323 q^{90} +7.46177 q^{91} -1.00000 q^{92} -1.65549 q^{93} -0.286731 q^{94} -0.169213 q^{95} -1.00000 q^{96} -0.811353 q^{97} -5.43332 q^{98} -2.48043 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{2} - 8 q^{3} + 8 q^{4} + 2 q^{5} - 8 q^{6} + 5 q^{7} + 8 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{2} - 8 q^{3} + 8 q^{4} + 2 q^{5} - 8 q^{6} + 5 q^{7} + 8 q^{8} + 8 q^{9} + 2 q^{10} + 3 q^{11} - 8 q^{12} + 13 q^{13} + 5 q^{14} - 2 q^{15} + 8 q^{16} + 5 q^{17} + 8 q^{18} + 7 q^{19} + 2 q^{20} - 5 q^{21} + 3 q^{22} - 8 q^{23} - 8 q^{24} + 24 q^{25} + 13 q^{26} - 8 q^{27} + 5 q^{28} - 8 q^{29} - 2 q^{30} + 15 q^{31} + 8 q^{32} - 3 q^{33} + 5 q^{34} + 9 q^{35} + 8 q^{36} + 22 q^{37} + 7 q^{38} - 13 q^{39} + 2 q^{40} - 8 q^{41} - 5 q^{42} - q^{43} + 3 q^{44} + 2 q^{45} - 8 q^{46} - 9 q^{47} - 8 q^{48} + 33 q^{49} + 24 q^{50} - 5 q^{51} + 13 q^{52} + 14 q^{53} - 8 q^{54} - 17 q^{55} + 5 q^{56} - 7 q^{57} - 8 q^{58} - 4 q^{59} - 2 q^{60} + 13 q^{61} + 15 q^{62} + 5 q^{63} + 8 q^{64} + 21 q^{65} - 3 q^{66} - 3 q^{67} + 5 q^{68} + 8 q^{69} + 9 q^{70} + 7 q^{71} + 8 q^{72} + 16 q^{73} + 22 q^{74} - 24 q^{75} + 7 q^{76} - 13 q^{78} + 14 q^{79} + 2 q^{80} + 8 q^{81} - 8 q^{82} + 36 q^{83} - 5 q^{84} + 47 q^{85} - q^{86} + 8 q^{87} + 3 q^{88} - 12 q^{89} + 2 q^{90} + 20 q^{91} - 8 q^{92} - 15 q^{93} - 9 q^{94} + 7 q^{95} - 8 q^{96} + 10 q^{97} + 33 q^{98} + 3 q^{99}+O(q^{100}) \) 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
\(5\) −3.42323 −1.53092 −0.765458 0.643486i \(-0.777488\pi\)
−0.765458 + 0.643486i \(0.777488\pi\)
\(6\) −1.00000 −0.408248
\(7\) −1.25167 −0.473088 −0.236544 0.971621i \(-0.576015\pi\)
−0.236544 + 0.971621i \(0.576015\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −3.42323 −1.08252
\(11\) −2.48043 −0.747878 −0.373939 0.927453i \(-0.621993\pi\)
−0.373939 + 0.927453i \(0.621993\pi\)
\(12\) −1.00000 −0.288675
\(13\) −5.96144 −1.65341 −0.826703 0.562639i \(-0.809786\pi\)
−0.826703 + 0.562639i \(0.809786\pi\)
\(14\) −1.25167 −0.334524
\(15\) 3.42323 0.883875
\(16\) 1.00000 0.250000
\(17\) −5.90366 −1.43185 −0.715924 0.698178i \(-0.753994\pi\)
−0.715924 + 0.698178i \(0.753994\pi\)
\(18\) 1.00000 0.235702
\(19\) 0.0494308 0.0113402 0.00567011 0.999984i \(-0.498195\pi\)
0.00567011 + 0.999984i \(0.498195\pi\)
\(20\) −3.42323 −0.765458
\(21\) 1.25167 0.273137
\(22\) −2.48043 −0.528830
\(23\) −1.00000 −0.208514
\(24\) −1.00000 −0.204124
\(25\) 6.71852 1.34370
\(26\) −5.96144 −1.16913
\(27\) −1.00000 −0.192450
\(28\) −1.25167 −0.236544
\(29\) −1.00000 −0.185695
\(30\) 3.42323 0.624994
\(31\) 1.65549 0.297335 0.148668 0.988887i \(-0.452502\pi\)
0.148668 + 0.988887i \(0.452502\pi\)
\(32\) 1.00000 0.176777
\(33\) 2.48043 0.431788
\(34\) −5.90366 −1.01247
\(35\) 4.28477 0.724258
\(36\) 1.00000 0.166667
\(37\) 11.1995 1.84119 0.920595 0.390518i \(-0.127704\pi\)
0.920595 + 0.390518i \(0.127704\pi\)
\(38\) 0.0494308 0.00801874
\(39\) 5.96144 0.954594
\(40\) −3.42323 −0.541261
\(41\) −6.37459 −0.995543 −0.497772 0.867308i \(-0.665848\pi\)
−0.497772 + 0.867308i \(0.665848\pi\)
\(42\) 1.25167 0.193137
\(43\) 9.44108 1.43975 0.719876 0.694103i \(-0.244199\pi\)
0.719876 + 0.694103i \(0.244199\pi\)
\(44\) −2.48043 −0.373939
\(45\) −3.42323 −0.510305
\(46\) −1.00000 −0.147442
\(47\) −0.286731 −0.0418240 −0.0209120 0.999781i \(-0.506657\pi\)
−0.0209120 + 0.999781i \(0.506657\pi\)
\(48\) −1.00000 −0.144338
\(49\) −5.43332 −0.776188
\(50\) 6.71852 0.950142
\(51\) 5.90366 0.826678
\(52\) −5.96144 −0.826703
\(53\) 1.42752 0.196085 0.0980425 0.995182i \(-0.468742\pi\)
0.0980425 + 0.995182i \(0.468742\pi\)
\(54\) −1.00000 −0.136083
\(55\) 8.49109 1.14494
\(56\) −1.25167 −0.167262
\(57\) −0.0494308 −0.00654727
\(58\) −1.00000 −0.131306
\(59\) 1.26100 0.164168 0.0820841 0.996625i \(-0.473842\pi\)
0.0820841 + 0.996625i \(0.473842\pi\)
\(60\) 3.42323 0.441937
\(61\) 4.48043 0.573660 0.286830 0.957981i \(-0.407398\pi\)
0.286830 + 0.957981i \(0.407398\pi\)
\(62\) 1.65549 0.210248
\(63\) −1.25167 −0.157696
\(64\) 1.00000 0.125000
\(65\) 20.4074 2.53123
\(66\) 2.48043 0.305320
\(67\) −8.37070 −1.02264 −0.511322 0.859389i \(-0.670844\pi\)
−0.511322 + 0.859389i \(0.670844\pi\)
\(68\) −5.90366 −0.715924
\(69\) 1.00000 0.120386
\(70\) 4.28477 0.512127
\(71\) −5.41935 −0.643158 −0.321579 0.946883i \(-0.604214\pi\)
−0.321579 + 0.946883i \(0.604214\pi\)
\(72\) 1.00000 0.117851
\(73\) −4.43236 −0.518769 −0.259384 0.965774i \(-0.583520\pi\)
−0.259384 + 0.965774i \(0.583520\pi\)
\(74\) 11.1995 1.30192
\(75\) −6.71852 −0.775788
\(76\) 0.0494308 0.00567011
\(77\) 3.10469 0.353812
\(78\) 5.96144 0.675000
\(79\) 12.1466 1.36660 0.683299 0.730139i \(-0.260545\pi\)
0.683299 + 0.730139i \(0.260545\pi\)
\(80\) −3.42323 −0.382729
\(81\) 1.00000 0.111111
\(82\) −6.37459 −0.703955
\(83\) 11.8542 1.30117 0.650586 0.759433i \(-0.274524\pi\)
0.650586 + 0.759433i \(0.274524\pi\)
\(84\) 1.25167 0.136569
\(85\) 20.2096 2.19204
\(86\) 9.44108 1.01806
\(87\) 1.00000 0.107211
\(88\) −2.48043 −0.264415
\(89\) −0.796776 −0.0844580 −0.0422290 0.999108i \(-0.513446\pi\)
−0.0422290 + 0.999108i \(0.513446\pi\)
\(90\) −3.42323 −0.360840
\(91\) 7.46177 0.782206
\(92\) −1.00000 −0.104257
\(93\) −1.65549 −0.171667
\(94\) −0.286731 −0.0295740
\(95\) −0.169213 −0.0173609
\(96\) −1.00000 −0.102062
\(97\) −0.811353 −0.0823804 −0.0411902 0.999151i \(-0.513115\pi\)
−0.0411902 + 0.999151i \(0.513115\pi\)
\(98\) −5.43332 −0.548848
\(99\) −2.48043 −0.249293
\(100\) 6.71852 0.671852
\(101\) 0.311563 0.0310016 0.0155008 0.999880i \(-0.495066\pi\)
0.0155008 + 0.999880i \(0.495066\pi\)
\(102\) 5.90366 0.584550
\(103\) −16.0371 −1.58018 −0.790090 0.612991i \(-0.789966\pi\)
−0.790090 + 0.612991i \(0.789966\pi\)
\(104\) −5.96144 −0.584567
\(105\) −4.28477 −0.418150
\(106\) 1.42752 0.138653
\(107\) 7.79784 0.753846 0.376923 0.926245i \(-0.376982\pi\)
0.376923 + 0.926245i \(0.376982\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −9.71695 −0.930715 −0.465358 0.885123i \(-0.654074\pi\)
−0.465358 + 0.885123i \(0.654074\pi\)
\(110\) 8.49109 0.809594
\(111\) −11.1995 −1.06301
\(112\) −1.25167 −0.118272
\(113\) −2.63452 −0.247835 −0.123917 0.992293i \(-0.539546\pi\)
−0.123917 + 0.992293i \(0.539546\pi\)
\(114\) −0.0494308 −0.00462962
\(115\) 3.42323 0.319218
\(116\) −1.00000 −0.0928477
\(117\) −5.96144 −0.551135
\(118\) 1.26100 0.116085
\(119\) 7.38945 0.677390
\(120\) 3.42323 0.312497
\(121\) −4.84746 −0.440678
\(122\) 4.48043 0.405639
\(123\) 6.37459 0.574777
\(124\) 1.65549 0.148668
\(125\) −5.88289 −0.526181
\(126\) −1.25167 −0.111508
\(127\) 16.8037 1.49109 0.745543 0.666458i \(-0.232190\pi\)
0.745543 + 0.666458i \(0.232190\pi\)
\(128\) 1.00000 0.0883883
\(129\) −9.44108 −0.831241
\(130\) 20.4074 1.78985
\(131\) −3.20126 −0.279695 −0.139848 0.990173i \(-0.544661\pi\)
−0.139848 + 0.990173i \(0.544661\pi\)
\(132\) 2.48043 0.215894
\(133\) −0.0618712 −0.00536491
\(134\) −8.37070 −0.723119
\(135\) 3.42323 0.294625
\(136\) −5.90366 −0.506235
\(137\) 14.1466 1.20862 0.604312 0.796748i \(-0.293448\pi\)
0.604312 + 0.796748i \(0.293448\pi\)
\(138\) 1.00000 0.0851257
\(139\) 6.46531 0.548380 0.274190 0.961675i \(-0.411590\pi\)
0.274190 + 0.961675i \(0.411590\pi\)
\(140\) 4.28477 0.362129
\(141\) 0.286731 0.0241471
\(142\) −5.41935 −0.454782
\(143\) 14.7869 1.23655
\(144\) 1.00000 0.0833333
\(145\) 3.42323 0.284284
\(146\) −4.43236 −0.366825
\(147\) 5.43332 0.448132
\(148\) 11.1995 0.920595
\(149\) 0.0280400 0.00229713 0.00114856 0.999999i \(-0.499634\pi\)
0.00114856 + 0.999999i \(0.499634\pi\)
\(150\) −6.71852 −0.548565
\(151\) −2.83192 −0.230458 −0.115229 0.993339i \(-0.536760\pi\)
−0.115229 + 0.993339i \(0.536760\pi\)
\(152\) 0.0494308 0.00400937
\(153\) −5.90366 −0.477283
\(154\) 3.10469 0.250183
\(155\) −5.66714 −0.455195
\(156\) 5.96144 0.477297
\(157\) −12.1009 −0.965754 −0.482877 0.875688i \(-0.660408\pi\)
−0.482877 + 0.875688i \(0.660408\pi\)
\(158\) 12.1466 0.966330
\(159\) −1.42752 −0.113210
\(160\) −3.42323 −0.270630
\(161\) 1.25167 0.0986456
\(162\) 1.00000 0.0785674
\(163\) 5.14348 0.402869 0.201434 0.979502i \(-0.435440\pi\)
0.201434 + 0.979502i \(0.435440\pi\)
\(164\) −6.37459 −0.497772
\(165\) −8.49109 −0.661031
\(166\) 11.8542 0.920067
\(167\) 17.0100 1.31627 0.658136 0.752899i \(-0.271345\pi\)
0.658136 + 0.752899i \(0.271345\pi\)
\(168\) 1.25167 0.0965686
\(169\) 22.5388 1.73375
\(170\) 20.2096 1.55001
\(171\) 0.0494308 0.00378007
\(172\) 9.44108 0.719876
\(173\) −15.5687 −1.18366 −0.591832 0.806062i \(-0.701595\pi\)
−0.591832 + 0.806062i \(0.701595\pi\)
\(174\) 1.00000 0.0758098
\(175\) −8.40938 −0.635690
\(176\) −2.48043 −0.186970
\(177\) −1.26100 −0.0947826
\(178\) −0.796776 −0.0597209
\(179\) −15.5798 −1.16449 −0.582243 0.813015i \(-0.697825\pi\)
−0.582243 + 0.813015i \(0.697825\pi\)
\(180\) −3.42323 −0.255153
\(181\) 14.8151 1.10120 0.550599 0.834770i \(-0.314399\pi\)
0.550599 + 0.834770i \(0.314399\pi\)
\(182\) 7.46177 0.553103
\(183\) −4.48043 −0.331203
\(184\) −1.00000 −0.0737210
\(185\) −38.3386 −2.81871
\(186\) −1.65549 −0.121387
\(187\) 14.6436 1.07085
\(188\) −0.286731 −0.0209120
\(189\) 1.25167 0.0910458
\(190\) −0.169213 −0.0122760
\(191\) 3.44769 0.249466 0.124733 0.992190i \(-0.460193\pi\)
0.124733 + 0.992190i \(0.460193\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 24.1237 1.73646 0.868232 0.496159i \(-0.165257\pi\)
0.868232 + 0.496159i \(0.165257\pi\)
\(194\) −0.811353 −0.0582517
\(195\) −20.4074 −1.46140
\(196\) −5.43332 −0.388094
\(197\) 19.0298 1.35581 0.677907 0.735147i \(-0.262887\pi\)
0.677907 + 0.735147i \(0.262887\pi\)
\(198\) −2.48043 −0.176277
\(199\) −2.13847 −0.151592 −0.0757960 0.997123i \(-0.524150\pi\)
−0.0757960 + 0.997123i \(0.524150\pi\)
\(200\) 6.71852 0.475071
\(201\) 8.37070 0.590424
\(202\) 0.311563 0.0219215
\(203\) 1.25167 0.0878502
\(204\) 5.90366 0.413339
\(205\) 21.8217 1.52409
\(206\) −16.0371 −1.11736
\(207\) −1.00000 −0.0695048
\(208\) −5.96144 −0.413351
\(209\) −0.122610 −0.00848110
\(210\) −4.28477 −0.295677
\(211\) −13.7573 −0.947094 −0.473547 0.880768i \(-0.657027\pi\)
−0.473547 + 0.880768i \(0.657027\pi\)
\(212\) 1.42752 0.0980425
\(213\) 5.41935 0.371328
\(214\) 7.79784 0.533050
\(215\) −32.3190 −2.20414
\(216\) −1.00000 −0.0680414
\(217\) −2.07213 −0.140666
\(218\) −9.71695 −0.658115
\(219\) 4.43236 0.299511
\(220\) 8.49109 0.572469
\(221\) 35.1943 2.36743
\(222\) −11.1995 −0.751663
\(223\) −1.26560 −0.0847510 −0.0423755 0.999102i \(-0.513493\pi\)
−0.0423755 + 0.999102i \(0.513493\pi\)
\(224\) −1.25167 −0.0836309
\(225\) 6.71852 0.447901
\(226\) −2.63452 −0.175246
\(227\) 15.7506 1.04540 0.522701 0.852516i \(-0.324924\pi\)
0.522701 + 0.852516i \(0.324924\pi\)
\(228\) −0.0494308 −0.00327364
\(229\) 28.2236 1.86506 0.932532 0.361086i \(-0.117594\pi\)
0.932532 + 0.361086i \(0.117594\pi\)
\(230\) 3.42323 0.225721
\(231\) −3.10469 −0.204273
\(232\) −1.00000 −0.0656532
\(233\) 5.63781 0.369345 0.184672 0.982800i \(-0.440878\pi\)
0.184672 + 0.982800i \(0.440878\pi\)
\(234\) −5.96144 −0.389711
\(235\) 0.981547 0.0640290
\(236\) 1.26100 0.0820841
\(237\) −12.1466 −0.789005
\(238\) 7.38945 0.478987
\(239\) −22.9285 −1.48312 −0.741561 0.670885i \(-0.765914\pi\)
−0.741561 + 0.670885i \(0.765914\pi\)
\(240\) 3.42323 0.220969
\(241\) 15.0573 0.969924 0.484962 0.874535i \(-0.338833\pi\)
0.484962 + 0.874535i \(0.338833\pi\)
\(242\) −4.84746 −0.311607
\(243\) −1.00000 −0.0641500
\(244\) 4.48043 0.286830
\(245\) 18.5995 1.18828
\(246\) 6.37459 0.406429
\(247\) −0.294679 −0.0187500
\(248\) 1.65549 0.105124
\(249\) −11.8542 −0.751231
\(250\) −5.88289 −0.372066
\(251\) 11.7565 0.742066 0.371033 0.928620i \(-0.379004\pi\)
0.371033 + 0.928620i \(0.379004\pi\)
\(252\) −1.25167 −0.0788480
\(253\) 2.48043 0.155943
\(254\) 16.8037 1.05436
\(255\) −20.2096 −1.26557
\(256\) 1.00000 0.0625000
\(257\) −9.62315 −0.600276 −0.300138 0.953896i \(-0.597033\pi\)
−0.300138 + 0.953896i \(0.597033\pi\)
\(258\) −9.44108 −0.587776
\(259\) −14.0181 −0.871045
\(260\) 20.4074 1.26561
\(261\) −1.00000 −0.0618984
\(262\) −3.20126 −0.197774
\(263\) −1.50326 −0.0926949 −0.0463475 0.998925i \(-0.514758\pi\)
−0.0463475 + 0.998925i \(0.514758\pi\)
\(264\) 2.48043 0.152660
\(265\) −4.88673 −0.300190
\(266\) −0.0618712 −0.00379357
\(267\) 0.796776 0.0487619
\(268\) −8.37070 −0.511322
\(269\) −12.9944 −0.792285 −0.396142 0.918189i \(-0.629651\pi\)
−0.396142 + 0.918189i \(0.629651\pi\)
\(270\) 3.42323 0.208331
\(271\) −15.7996 −0.959760 −0.479880 0.877334i \(-0.659320\pi\)
−0.479880 + 0.877334i \(0.659320\pi\)
\(272\) −5.90366 −0.357962
\(273\) −7.46177 −0.451607
\(274\) 14.1466 0.854626
\(275\) −16.6648 −1.00493
\(276\) 1.00000 0.0601929
\(277\) −18.5873 −1.11680 −0.558401 0.829571i \(-0.688585\pi\)
−0.558401 + 0.829571i \(0.688585\pi\)
\(278\) 6.46531 0.387763
\(279\) 1.65549 0.0991118
\(280\) 4.28477 0.256064
\(281\) −29.8134 −1.77852 −0.889259 0.457405i \(-0.848779\pi\)
−0.889259 + 0.457405i \(0.848779\pi\)
\(282\) 0.286731 0.0170746
\(283\) 25.8779 1.53828 0.769141 0.639079i \(-0.220684\pi\)
0.769141 + 0.639079i \(0.220684\pi\)
\(284\) −5.41935 −0.321579
\(285\) 0.169213 0.0100233
\(286\) 14.7869 0.874370
\(287\) 7.97890 0.470979
\(288\) 1.00000 0.0589256
\(289\) 17.8532 1.05019
\(290\) 3.42323 0.201019
\(291\) 0.811353 0.0475623
\(292\) −4.43236 −0.259384
\(293\) −21.9314 −1.28125 −0.640623 0.767856i \(-0.721324\pi\)
−0.640623 + 0.767856i \(0.721324\pi\)
\(294\) 5.43332 0.316877
\(295\) −4.31670 −0.251328
\(296\) 11.1995 0.650959
\(297\) 2.48043 0.143929
\(298\) 0.0280400 0.00162431
\(299\) 5.96144 0.344759
\(300\) −6.71852 −0.387894
\(301\) −11.8171 −0.681129
\(302\) −2.83192 −0.162959
\(303\) −0.311563 −0.0178988
\(304\) 0.0494308 0.00283505
\(305\) −15.3376 −0.878226
\(306\) −5.90366 −0.337490
\(307\) 19.0530 1.08741 0.543705 0.839276i \(-0.317021\pi\)
0.543705 + 0.839276i \(0.317021\pi\)
\(308\) 3.10469 0.176906
\(309\) 16.0371 0.912317
\(310\) −5.66714 −0.321872
\(311\) 29.8045 1.69006 0.845029 0.534721i \(-0.179583\pi\)
0.845029 + 0.534721i \(0.179583\pi\)
\(312\) 5.96144 0.337500
\(313\) 29.0561 1.64235 0.821173 0.570679i \(-0.193320\pi\)
0.821173 + 0.570679i \(0.193320\pi\)
\(314\) −12.1009 −0.682891
\(315\) 4.28477 0.241419
\(316\) 12.1466 0.683299
\(317\) −24.7475 −1.38996 −0.694980 0.719029i \(-0.744587\pi\)
−0.694980 + 0.719029i \(0.744587\pi\)
\(318\) −1.42752 −0.0800514
\(319\) 2.48043 0.138877
\(320\) −3.42323 −0.191364
\(321\) −7.79784 −0.435233
\(322\) 1.25167 0.0697530
\(323\) −0.291823 −0.0162375
\(324\) 1.00000 0.0555556
\(325\) −40.0520 −2.22169
\(326\) 5.14348 0.284871
\(327\) 9.71695 0.537349
\(328\) −6.37459 −0.351978
\(329\) 0.358893 0.0197864
\(330\) −8.49109 −0.467419
\(331\) −2.82184 −0.155102 −0.0775512 0.996988i \(-0.524710\pi\)
−0.0775512 + 0.996988i \(0.524710\pi\)
\(332\) 11.8542 0.650586
\(333\) 11.1995 0.613730
\(334\) 17.0100 0.930744
\(335\) 28.6549 1.56558
\(336\) 1.25167 0.0682843
\(337\) 9.01515 0.491086 0.245543 0.969386i \(-0.421034\pi\)
0.245543 + 0.969386i \(0.421034\pi\)
\(338\) 22.5388 1.22595
\(339\) 2.63452 0.143088
\(340\) 20.2096 1.09602
\(341\) −4.10634 −0.222371
\(342\) 0.0494308 0.00267291
\(343\) 15.5624 0.840293
\(344\) 9.44108 0.509029
\(345\) −3.42323 −0.184301
\(346\) −15.5687 −0.836976
\(347\) −2.56587 −0.137743 −0.0688716 0.997626i \(-0.521940\pi\)
−0.0688716 + 0.997626i \(0.521940\pi\)
\(348\) 1.00000 0.0536056
\(349\) 9.48440 0.507688 0.253844 0.967245i \(-0.418305\pi\)
0.253844 + 0.967245i \(0.418305\pi\)
\(350\) −8.40938 −0.449501
\(351\) 5.96144 0.318198
\(352\) −2.48043 −0.132207
\(353\) −18.1416 −0.965582 −0.482791 0.875736i \(-0.660377\pi\)
−0.482791 + 0.875736i \(0.660377\pi\)
\(354\) −1.26100 −0.0670214
\(355\) 18.5517 0.984621
\(356\) −0.796776 −0.0422290
\(357\) −7.38945 −0.391091
\(358\) −15.5798 −0.823416
\(359\) −17.1102 −0.903042 −0.451521 0.892260i \(-0.649118\pi\)
−0.451521 + 0.892260i \(0.649118\pi\)
\(360\) −3.42323 −0.180420
\(361\) −18.9976 −0.999871
\(362\) 14.8151 0.778664
\(363\) 4.84746 0.254426
\(364\) 7.46177 0.391103
\(365\) 15.1730 0.794192
\(366\) −4.48043 −0.234196
\(367\) −12.4083 −0.647707 −0.323853 0.946107i \(-0.604978\pi\)
−0.323853 + 0.946107i \(0.604978\pi\)
\(368\) −1.00000 −0.0521286
\(369\) −6.37459 −0.331848
\(370\) −38.3386 −1.99313
\(371\) −1.78679 −0.0927654
\(372\) −1.65549 −0.0858333
\(373\) 27.2148 1.40913 0.704565 0.709640i \(-0.251142\pi\)
0.704565 + 0.709640i \(0.251142\pi\)
\(374\) 14.6436 0.757204
\(375\) 5.88289 0.303791
\(376\) −0.286731 −0.0147870
\(377\) 5.96144 0.307030
\(378\) 1.25167 0.0643791
\(379\) −11.3839 −0.584751 −0.292375 0.956304i \(-0.594446\pi\)
−0.292375 + 0.956304i \(0.594446\pi\)
\(380\) −0.169213 −0.00868045
\(381\) −16.8037 −0.860878
\(382\) 3.44769 0.176399
\(383\) −35.7156 −1.82498 −0.912490 0.409100i \(-0.865843\pi\)
−0.912490 + 0.409100i \(0.865843\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −10.6281 −0.541656
\(386\) 24.1237 1.22786
\(387\) 9.44108 0.479917
\(388\) −0.811353 −0.0411902
\(389\) 6.85337 0.347480 0.173740 0.984792i \(-0.444415\pi\)
0.173740 + 0.984792i \(0.444415\pi\)
\(390\) −20.4074 −1.03337
\(391\) 5.90366 0.298561
\(392\) −5.43332 −0.274424
\(393\) 3.20126 0.161482
\(394\) 19.0298 0.958706
\(395\) −41.5806 −2.09215
\(396\) −2.48043 −0.124646
\(397\) −5.16414 −0.259181 −0.129590 0.991568i \(-0.541366\pi\)
−0.129590 + 0.991568i \(0.541366\pi\)
\(398\) −2.13847 −0.107192
\(399\) 0.0618712 0.00309743
\(400\) 6.71852 0.335926
\(401\) 3.17994 0.158799 0.0793994 0.996843i \(-0.474700\pi\)
0.0793994 + 0.996843i \(0.474700\pi\)
\(402\) 8.37070 0.417493
\(403\) −9.86912 −0.491616
\(404\) 0.311563 0.0155008
\(405\) −3.42323 −0.170102
\(406\) 1.25167 0.0621195
\(407\) −27.7797 −1.37699
\(408\) 5.90366 0.292275
\(409\) 7.97642 0.394409 0.197204 0.980362i \(-0.436814\pi\)
0.197204 + 0.980362i \(0.436814\pi\)
\(410\) 21.8217 1.07770
\(411\) −14.1466 −0.697800
\(412\) −16.0371 −0.790090
\(413\) −1.57836 −0.0776660
\(414\) −1.00000 −0.0491473
\(415\) −40.5798 −1.99198
\(416\) −5.96144 −0.292284
\(417\) −6.46531 −0.316608
\(418\) −0.122610 −0.00599704
\(419\) −10.4857 −0.512260 −0.256130 0.966642i \(-0.582448\pi\)
−0.256130 + 0.966642i \(0.582448\pi\)
\(420\) −4.28477 −0.209075
\(421\) 15.7041 0.765373 0.382686 0.923878i \(-0.374999\pi\)
0.382686 + 0.923878i \(0.374999\pi\)
\(422\) −13.7573 −0.669697
\(423\) −0.286731 −0.0139413
\(424\) 1.42752 0.0693265
\(425\) −39.6639 −1.92398
\(426\) 5.41935 0.262568
\(427\) −5.60803 −0.271392
\(428\) 7.79784 0.376923
\(429\) −14.7869 −0.713920
\(430\) −32.3190 −1.55856
\(431\) −12.8101 −0.617040 −0.308520 0.951218i \(-0.599834\pi\)
−0.308520 + 0.951218i \(0.599834\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 21.3978 1.02831 0.514156 0.857696i \(-0.328105\pi\)
0.514156 + 0.857696i \(0.328105\pi\)
\(434\) −2.07213 −0.0994657
\(435\) −3.42323 −0.164131
\(436\) −9.71695 −0.465358
\(437\) −0.0494308 −0.00236460
\(438\) 4.43236 0.211787
\(439\) −28.2476 −1.34818 −0.674091 0.738648i \(-0.735465\pi\)
−0.674091 + 0.738648i \(0.735465\pi\)
\(440\) 8.49109 0.404797
\(441\) −5.43332 −0.258729
\(442\) 35.1943 1.67402
\(443\) −3.60019 −0.171050 −0.0855251 0.996336i \(-0.527257\pi\)
−0.0855251 + 0.996336i \(0.527257\pi\)
\(444\) −11.1995 −0.531506
\(445\) 2.72755 0.129298
\(446\) −1.26560 −0.0599280
\(447\) −0.0280400 −0.00132625
\(448\) −1.25167 −0.0591360
\(449\) 26.4204 1.24685 0.623427 0.781881i \(-0.285740\pi\)
0.623427 + 0.781881i \(0.285740\pi\)
\(450\) 6.71852 0.316714
\(451\) 15.8117 0.744545
\(452\) −2.63452 −0.123917
\(453\) 2.83192 0.133055
\(454\) 15.7506 0.739211
\(455\) −25.5434 −1.19749
\(456\) −0.0494308 −0.00231481
\(457\) −7.19372 −0.336508 −0.168254 0.985744i \(-0.553813\pi\)
−0.168254 + 0.985744i \(0.553813\pi\)
\(458\) 28.2236 1.31880
\(459\) 5.90366 0.275559
\(460\) 3.42323 0.159609
\(461\) −29.3838 −1.36854 −0.684270 0.729229i \(-0.739879\pi\)
−0.684270 + 0.729229i \(0.739879\pi\)
\(462\) −3.10469 −0.144443
\(463\) −20.4254 −0.949247 −0.474623 0.880189i \(-0.657416\pi\)
−0.474623 + 0.880189i \(0.657416\pi\)
\(464\) −1.00000 −0.0464238
\(465\) 5.66714 0.262807
\(466\) 5.63781 0.261166
\(467\) 41.6099 1.92548 0.962739 0.270431i \(-0.0871663\pi\)
0.962739 + 0.270431i \(0.0871663\pi\)
\(468\) −5.96144 −0.275568
\(469\) 10.4774 0.483801
\(470\) 0.981547 0.0452754
\(471\) 12.1009 0.557578
\(472\) 1.26100 0.0580423
\(473\) −23.4180 −1.07676
\(474\) −12.1466 −0.557911
\(475\) 0.332102 0.0152379
\(476\) 7.38945 0.338695
\(477\) 1.42752 0.0653617
\(478\) −22.9285 −1.04873
\(479\) −25.8977 −1.18330 −0.591649 0.806196i \(-0.701523\pi\)
−0.591649 + 0.806196i \(0.701523\pi\)
\(480\) 3.42323 0.156248
\(481\) −66.7653 −3.04424
\(482\) 15.0573 0.685840
\(483\) −1.25167 −0.0569531
\(484\) −4.84746 −0.220339
\(485\) 2.77745 0.126117
\(486\) −1.00000 −0.0453609
\(487\) −16.7191 −0.757617 −0.378808 0.925475i \(-0.623666\pi\)
−0.378808 + 0.925475i \(0.623666\pi\)
\(488\) 4.48043 0.202820
\(489\) −5.14348 −0.232596
\(490\) 18.5995 0.840240
\(491\) 2.14894 0.0969804 0.0484902 0.998824i \(-0.484559\pi\)
0.0484902 + 0.998824i \(0.484559\pi\)
\(492\) 6.37459 0.287389
\(493\) 5.90366 0.265888
\(494\) −0.294679 −0.0132582
\(495\) 8.49109 0.381646
\(496\) 1.65549 0.0743338
\(497\) 6.78325 0.304270
\(498\) −11.8542 −0.531201
\(499\) −29.2331 −1.30865 −0.654327 0.756211i \(-0.727048\pi\)
−0.654327 + 0.756211i \(0.727048\pi\)
\(500\) −5.88289 −0.263091
\(501\) −17.0100 −0.759950
\(502\) 11.7565 0.524720
\(503\) −21.1768 −0.944226 −0.472113 0.881538i \(-0.656509\pi\)
−0.472113 + 0.881538i \(0.656509\pi\)
\(504\) −1.25167 −0.0557539
\(505\) −1.06655 −0.0474609
\(506\) 2.48043 0.110269
\(507\) −22.5388 −1.00098
\(508\) 16.8037 0.745543
\(509\) 32.7617 1.45214 0.726069 0.687622i \(-0.241345\pi\)
0.726069 + 0.687622i \(0.241345\pi\)
\(510\) −20.2096 −0.894897
\(511\) 5.54787 0.245423
\(512\) 1.00000 0.0441942
\(513\) −0.0494308 −0.00218242
\(514\) −9.62315 −0.424459
\(515\) 54.8986 2.41912
\(516\) −9.44108 −0.415621
\(517\) 0.711216 0.0312793
\(518\) −14.0181 −0.615922
\(519\) 15.5687 0.683388
\(520\) 20.4074 0.894923
\(521\) −23.1467 −1.01408 −0.507038 0.861923i \(-0.669260\pi\)
−0.507038 + 0.861923i \(0.669260\pi\)
\(522\) −1.00000 −0.0437688
\(523\) 14.7123 0.643322 0.321661 0.946855i \(-0.395759\pi\)
0.321661 + 0.946855i \(0.395759\pi\)
\(524\) −3.20126 −0.139848
\(525\) 8.40938 0.367016
\(526\) −1.50326 −0.0655452
\(527\) −9.77347 −0.425739
\(528\) 2.48043 0.107947
\(529\) 1.00000 0.0434783
\(530\) −4.88673 −0.212266
\(531\) 1.26100 0.0547228
\(532\) −0.0618712 −0.00268246
\(533\) 38.0017 1.64604
\(534\) 0.796776 0.0344799
\(535\) −26.6938 −1.15407
\(536\) −8.37070 −0.361559
\(537\) 15.5798 0.672316
\(538\) −12.9944 −0.560230
\(539\) 13.4770 0.580494
\(540\) 3.42323 0.147312
\(541\) 5.69481 0.244839 0.122419 0.992478i \(-0.460935\pi\)
0.122419 + 0.992478i \(0.460935\pi\)
\(542\) −15.7996 −0.678653
\(543\) −14.8151 −0.635777
\(544\) −5.90366 −0.253117
\(545\) 33.2634 1.42485
\(546\) −7.46177 −0.319334
\(547\) 14.6491 0.626349 0.313174 0.949696i \(-0.398608\pi\)
0.313174 + 0.949696i \(0.398608\pi\)
\(548\) 14.1466 0.604312
\(549\) 4.48043 0.191220
\(550\) −16.6648 −0.710590
\(551\) −0.0494308 −0.00210582
\(552\) 1.00000 0.0425628
\(553\) −15.2035 −0.646521
\(554\) −18.5873 −0.789699
\(555\) 38.3386 1.62738
\(556\) 6.46531 0.274190
\(557\) −1.69393 −0.0717740 −0.0358870 0.999356i \(-0.511426\pi\)
−0.0358870 + 0.999356i \(0.511426\pi\)
\(558\) 1.65549 0.0700826
\(559\) −56.2825 −2.38049
\(560\) 4.28477 0.181064
\(561\) −14.6436 −0.618255
\(562\) −29.8134 −1.25760
\(563\) −17.9267 −0.755520 −0.377760 0.925904i \(-0.623306\pi\)
−0.377760 + 0.925904i \(0.623306\pi\)
\(564\) 0.286731 0.0120736
\(565\) 9.01858 0.379414
\(566\) 25.8779 1.08773
\(567\) −1.25167 −0.0525653
\(568\) −5.41935 −0.227391
\(569\) −30.0759 −1.26085 −0.630424 0.776251i \(-0.717119\pi\)
−0.630424 + 0.776251i \(0.717119\pi\)
\(570\) 0.169213 0.00708756
\(571\) −26.1559 −1.09459 −0.547295 0.836940i \(-0.684342\pi\)
−0.547295 + 0.836940i \(0.684342\pi\)
\(572\) 14.7869 0.618273
\(573\) −3.44769 −0.144029
\(574\) 7.97890 0.333033
\(575\) −6.71852 −0.280182
\(576\) 1.00000 0.0416667
\(577\) −1.35438 −0.0563834 −0.0281917 0.999603i \(-0.508975\pi\)
−0.0281917 + 0.999603i \(0.508975\pi\)
\(578\) 17.8532 0.742597
\(579\) −24.1237 −1.00255
\(580\) 3.42323 0.142142
\(581\) −14.8376 −0.615568
\(582\) 0.811353 0.0336317
\(583\) −3.54087 −0.146648
\(584\) −4.43236 −0.183413
\(585\) 20.4074 0.843742
\(586\) −21.9314 −0.905977
\(587\) −18.3243 −0.756327 −0.378163 0.925739i \(-0.623444\pi\)
−0.378163 + 0.925739i \(0.623444\pi\)
\(588\) 5.43332 0.224066
\(589\) 0.0818324 0.00337184
\(590\) −4.31670 −0.177716
\(591\) −19.0298 −0.782780
\(592\) 11.1995 0.460298
\(593\) 45.0673 1.85069 0.925346 0.379125i \(-0.123775\pi\)
0.925346 + 0.379125i \(0.123775\pi\)
\(594\) 2.48043 0.101773
\(595\) −25.2958 −1.03703
\(596\) 0.0280400 0.00114856
\(597\) 2.13847 0.0875217
\(598\) 5.96144 0.243781
\(599\) −12.4545 −0.508877 −0.254438 0.967089i \(-0.581891\pi\)
−0.254438 + 0.967089i \(0.581891\pi\)
\(600\) −6.71852 −0.274282
\(601\) −15.2181 −0.620757 −0.310379 0.950613i \(-0.600456\pi\)
−0.310379 + 0.950613i \(0.600456\pi\)
\(602\) −11.8171 −0.481631
\(603\) −8.37070 −0.340881
\(604\) −2.83192 −0.115229
\(605\) 16.5940 0.674641
\(606\) −0.311563 −0.0126564
\(607\) −31.9059 −1.29502 −0.647510 0.762057i \(-0.724189\pi\)
−0.647510 + 0.762057i \(0.724189\pi\)
\(608\) 0.0494308 0.00200468
\(609\) −1.25167 −0.0507203
\(610\) −15.3376 −0.621000
\(611\) 1.70933 0.0691521
\(612\) −5.90366 −0.238641
\(613\) 13.3171 0.537873 0.268936 0.963158i \(-0.413328\pi\)
0.268936 + 0.963158i \(0.413328\pi\)
\(614\) 19.0530 0.768915
\(615\) −21.8217 −0.879936
\(616\) 3.10469 0.125091
\(617\) 45.9054 1.84808 0.924041 0.382293i \(-0.124865\pi\)
0.924041 + 0.382293i \(0.124865\pi\)
\(618\) 16.0371 0.645105
\(619\) 37.2120 1.49568 0.747839 0.663880i \(-0.231092\pi\)
0.747839 + 0.663880i \(0.231092\pi\)
\(620\) −5.66714 −0.227598
\(621\) 1.00000 0.0401286
\(622\) 29.8045 1.19505
\(623\) 0.997302 0.0399561
\(624\) 5.96144 0.238649
\(625\) −13.4541 −0.538164
\(626\) 29.0561 1.16131
\(627\) 0.122610 0.00489656
\(628\) −12.1009 −0.482877
\(629\) −66.1182 −2.63631
\(630\) 4.28477 0.170709
\(631\) −12.2235 −0.486612 −0.243306 0.969950i \(-0.578232\pi\)
−0.243306 + 0.969950i \(0.578232\pi\)
\(632\) 12.1466 0.483165
\(633\) 13.7573 0.546805
\(634\) −24.7475 −0.982850
\(635\) −57.5229 −2.28273
\(636\) −1.42752 −0.0566049
\(637\) 32.3904 1.28335
\(638\) 2.48043 0.0982012
\(639\) −5.41935 −0.214386
\(640\) −3.42323 −0.135315
\(641\) −5.18516 −0.204802 −0.102401 0.994743i \(-0.532652\pi\)
−0.102401 + 0.994743i \(0.532652\pi\)
\(642\) −7.79784 −0.307756
\(643\) 25.7268 1.01457 0.507284 0.861779i \(-0.330650\pi\)
0.507284 + 0.861779i \(0.330650\pi\)
\(644\) 1.25167 0.0493228
\(645\) 32.3190 1.27256
\(646\) −0.291823 −0.0114816
\(647\) 31.0162 1.21937 0.609687 0.792642i \(-0.291295\pi\)
0.609687 + 0.792642i \(0.291295\pi\)
\(648\) 1.00000 0.0392837
\(649\) −3.12783 −0.122778
\(650\) −40.0520 −1.57097
\(651\) 2.07213 0.0812134
\(652\) 5.14348 0.201434
\(653\) 36.3135 1.42106 0.710529 0.703668i \(-0.248456\pi\)
0.710529 + 0.703668i \(0.248456\pi\)
\(654\) 9.71695 0.379963
\(655\) 10.9587 0.428190
\(656\) −6.37459 −0.248886
\(657\) −4.43236 −0.172923
\(658\) 0.358893 0.0139911
\(659\) 18.4974 0.720557 0.360279 0.932845i \(-0.382682\pi\)
0.360279 + 0.932845i \(0.382682\pi\)
\(660\) −8.49109 −0.330515
\(661\) −32.0191 −1.24540 −0.622699 0.782461i \(-0.713964\pi\)
−0.622699 + 0.782461i \(0.713964\pi\)
\(662\) −2.82184 −0.109674
\(663\) −35.1943 −1.36683
\(664\) 11.8542 0.460033
\(665\) 0.211800 0.00821323
\(666\) 11.1995 0.433973
\(667\) 1.00000 0.0387202
\(668\) 17.0100 0.658136
\(669\) 1.26560 0.0489310
\(670\) 28.6549 1.10703
\(671\) −11.1134 −0.429028
\(672\) 1.25167 0.0482843
\(673\) 24.3732 0.939518 0.469759 0.882795i \(-0.344341\pi\)
0.469759 + 0.882795i \(0.344341\pi\)
\(674\) 9.01515 0.347250
\(675\) −6.71852 −0.258596
\(676\) 22.5388 0.866875
\(677\) 23.2758 0.894562 0.447281 0.894393i \(-0.352392\pi\)
0.447281 + 0.894393i \(0.352392\pi\)
\(678\) 2.63452 0.101178
\(679\) 1.01555 0.0389732
\(680\) 20.2096 0.775003
\(681\) −15.7506 −0.603563
\(682\) −4.10634 −0.157240
\(683\) 28.5651 1.09301 0.546506 0.837455i \(-0.315957\pi\)
0.546506 + 0.837455i \(0.315957\pi\)
\(684\) 0.0494308 0.00189004
\(685\) −48.4270 −1.85030
\(686\) 15.5624 0.594177
\(687\) −28.2236 −1.07680
\(688\) 9.44108 0.359938
\(689\) −8.51008 −0.324208
\(690\) −3.42323 −0.130320
\(691\) 38.4394 1.46230 0.731152 0.682215i \(-0.238983\pi\)
0.731152 + 0.682215i \(0.238983\pi\)
\(692\) −15.5687 −0.591832
\(693\) 3.10469 0.117937
\(694\) −2.56587 −0.0973992
\(695\) −22.1323 −0.839524
\(696\) 1.00000 0.0379049
\(697\) 37.6334 1.42547
\(698\) 9.48440 0.358990
\(699\) −5.63781 −0.213241
\(700\) −8.40938 −0.317845
\(701\) −40.0888 −1.51413 −0.757066 0.653338i \(-0.773368\pi\)
−0.757066 + 0.653338i \(0.773368\pi\)
\(702\) 5.96144 0.225000
\(703\) 0.553602 0.0208795
\(704\) −2.48043 −0.0934848
\(705\) −0.981547 −0.0369672
\(706\) −18.1416 −0.682770
\(707\) −0.389974 −0.0146665
\(708\) −1.26100 −0.0473913
\(709\) 46.7043 1.75402 0.877008 0.480475i \(-0.159536\pi\)
0.877008 + 0.480475i \(0.159536\pi\)
\(710\) 18.5517 0.696232
\(711\) 12.1466 0.455533
\(712\) −0.796776 −0.0298604
\(713\) −1.65549 −0.0619987
\(714\) −7.38945 −0.276543
\(715\) −50.6191 −1.89305
\(716\) −15.5798 −0.582243
\(717\) 22.9285 0.856281
\(718\) −17.1102 −0.638547
\(719\) 7.71868 0.287858 0.143929 0.989588i \(-0.454026\pi\)
0.143929 + 0.989588i \(0.454026\pi\)
\(720\) −3.42323 −0.127576
\(721\) 20.0732 0.747563
\(722\) −18.9976 −0.707016
\(723\) −15.0573 −0.559986
\(724\) 14.8151 0.550599
\(725\) −6.71852 −0.249519
\(726\) 4.84746 0.179906
\(727\) −9.09828 −0.337437 −0.168718 0.985664i \(-0.553963\pi\)
−0.168718 + 0.985664i \(0.553963\pi\)
\(728\) 7.46177 0.276552
\(729\) 1.00000 0.0370370
\(730\) 15.1730 0.561578
\(731\) −55.7370 −2.06151
\(732\) −4.48043 −0.165602
\(733\) 20.7658 0.767002 0.383501 0.923541i \(-0.374718\pi\)
0.383501 + 0.923541i \(0.374718\pi\)
\(734\) −12.4083 −0.457998
\(735\) −18.5995 −0.686053
\(736\) −1.00000 −0.0368605
\(737\) 20.7630 0.764813
\(738\) −6.37459 −0.234652
\(739\) 7.72286 0.284090 0.142045 0.989860i \(-0.454632\pi\)
0.142045 + 0.989860i \(0.454632\pi\)
\(740\) −38.3386 −1.40935
\(741\) 0.294679 0.0108253
\(742\) −1.78679 −0.0655951
\(743\) 18.3274 0.672368 0.336184 0.941796i \(-0.390864\pi\)
0.336184 + 0.941796i \(0.390864\pi\)
\(744\) −1.65549 −0.0606933
\(745\) −0.0959875 −0.00351671
\(746\) 27.2148 0.996405
\(747\) 11.8542 0.433724
\(748\) 14.6436 0.535424
\(749\) −9.76035 −0.356635
\(750\) 5.88289 0.214813
\(751\) 8.13997 0.297032 0.148516 0.988910i \(-0.452550\pi\)
0.148516 + 0.988910i \(0.452550\pi\)
\(752\) −0.286731 −0.0104560
\(753\) −11.7565 −0.428432
\(754\) 5.96144 0.217103
\(755\) 9.69432 0.352812
\(756\) 1.25167 0.0455229
\(757\) 8.06059 0.292967 0.146484 0.989213i \(-0.453204\pi\)
0.146484 + 0.989213i \(0.453204\pi\)
\(758\) −11.3839 −0.413481
\(759\) −2.48043 −0.0900340
\(760\) −0.169213 −0.00613801
\(761\) 20.6510 0.748599 0.374299 0.927308i \(-0.377883\pi\)
0.374299 + 0.927308i \(0.377883\pi\)
\(762\) −16.8037 −0.608733
\(763\) 12.1624 0.440310
\(764\) 3.44769 0.124733
\(765\) 20.2096 0.730680
\(766\) −35.7156 −1.29046
\(767\) −7.51738 −0.271437
\(768\) −1.00000 −0.0360844
\(769\) −3.88096 −0.139951 −0.0699755 0.997549i \(-0.522292\pi\)
−0.0699755 + 0.997549i \(0.522292\pi\)
\(770\) −10.6281 −0.383009
\(771\) 9.62315 0.346569
\(772\) 24.1237 0.868232
\(773\) −16.9976 −0.611359 −0.305680 0.952134i \(-0.598884\pi\)
−0.305680 + 0.952134i \(0.598884\pi\)
\(774\) 9.44108 0.339353
\(775\) 11.1225 0.399531
\(776\) −0.811353 −0.0291259
\(777\) 14.0181 0.502898
\(778\) 6.85337 0.245705
\(779\) −0.315101 −0.0112897
\(780\) −20.4074 −0.730702
\(781\) 13.4423 0.481004
\(782\) 5.90366 0.211115
\(783\) 1.00000 0.0357371
\(784\) −5.43332 −0.194047
\(785\) 41.4241 1.47849
\(786\) 3.20126 0.114185
\(787\) 22.4425 0.799989 0.399994 0.916518i \(-0.369012\pi\)
0.399994 + 0.916518i \(0.369012\pi\)
\(788\) 19.0298 0.677907
\(789\) 1.50326 0.0535174
\(790\) −41.5806 −1.47937
\(791\) 3.29756 0.117248
\(792\) −2.48043 −0.0881383
\(793\) −26.7098 −0.948494
\(794\) −5.16414 −0.183268
\(795\) 4.88673 0.173315
\(796\) −2.13847 −0.0757960
\(797\) 28.7644 1.01889 0.509443 0.860504i \(-0.329851\pi\)
0.509443 + 0.860504i \(0.329851\pi\)
\(798\) 0.0618712 0.00219022
\(799\) 1.69276 0.0598857
\(800\) 6.71852 0.237535
\(801\) −0.796776 −0.0281527
\(802\) 3.17994 0.112288
\(803\) 10.9942 0.387976
\(804\) 8.37070 0.295212
\(805\) −4.28477 −0.151018
\(806\) −9.86912 −0.347625
\(807\) 12.9944 0.457426
\(808\) 0.311563 0.0109607
\(809\) 30.8036 1.08300 0.541499 0.840701i \(-0.317857\pi\)
0.541499 + 0.840701i \(0.317857\pi\)
\(810\) −3.42323 −0.120280
\(811\) 0.602248 0.0211478 0.0105739 0.999944i \(-0.496634\pi\)
0.0105739 + 0.999944i \(0.496634\pi\)
\(812\) 1.25167 0.0439251
\(813\) 15.7996 0.554118
\(814\) −27.7797 −0.973676
\(815\) −17.6073 −0.616758
\(816\) 5.90366 0.206670
\(817\) 0.466681 0.0163271
\(818\) 7.97642 0.278889
\(819\) 7.46177 0.260735
\(820\) 21.8217 0.762047
\(821\) −2.10468 −0.0734537 −0.0367269 0.999325i \(-0.511693\pi\)
−0.0367269 + 0.999325i \(0.511693\pi\)
\(822\) −14.1466 −0.493419
\(823\) 5.93359 0.206832 0.103416 0.994638i \(-0.467023\pi\)
0.103416 + 0.994638i \(0.467023\pi\)
\(824\) −16.0371 −0.558678
\(825\) 16.6648 0.580195
\(826\) −1.57836 −0.0549182
\(827\) 26.0644 0.906349 0.453174 0.891422i \(-0.350291\pi\)
0.453174 + 0.891422i \(0.350291\pi\)
\(828\) −1.00000 −0.0347524
\(829\) 15.2371 0.529205 0.264602 0.964358i \(-0.414759\pi\)
0.264602 + 0.964358i \(0.414759\pi\)
\(830\) −40.5798 −1.40855
\(831\) 18.5873 0.644786
\(832\) −5.96144 −0.206676
\(833\) 32.0765 1.11138
\(834\) −6.46531 −0.223875
\(835\) −58.2291 −2.01510
\(836\) −0.122610 −0.00424055
\(837\) −1.65549 −0.0572222
\(838\) −10.4857 −0.362223
\(839\) 7.48575 0.258437 0.129218 0.991616i \(-0.458753\pi\)
0.129218 + 0.991616i \(0.458753\pi\)
\(840\) −4.28477 −0.147838
\(841\) 1.00000 0.0344828
\(842\) 15.7041 0.541200
\(843\) 29.8134 1.02683
\(844\) −13.7573 −0.473547
\(845\) −77.1554 −2.65423
\(846\) −0.286731 −0.00985801
\(847\) 6.06743 0.208479
\(848\) 1.42752 0.0490213
\(849\) −25.8779 −0.888128
\(850\) −39.6639 −1.36046
\(851\) −11.1995 −0.383915
\(852\) 5.41935 0.185664
\(853\) 28.8071 0.986336 0.493168 0.869934i \(-0.335839\pi\)
0.493168 + 0.869934i \(0.335839\pi\)
\(854\) −5.60803 −0.191903
\(855\) −0.169213 −0.00578697
\(856\) 7.79784 0.266525
\(857\) 3.15892 0.107907 0.0539533 0.998543i \(-0.482818\pi\)
0.0539533 + 0.998543i \(0.482818\pi\)
\(858\) −14.7869 −0.504818
\(859\) 25.7180 0.877486 0.438743 0.898613i \(-0.355424\pi\)
0.438743 + 0.898613i \(0.355424\pi\)
\(860\) −32.3190 −1.10207
\(861\) −7.97890 −0.271920
\(862\) −12.8101 −0.436313
\(863\) −7.72221 −0.262867 −0.131434 0.991325i \(-0.541958\pi\)
−0.131434 + 0.991325i \(0.541958\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 53.2951 1.81209
\(866\) 21.3978 0.727127
\(867\) −17.8532 −0.606328
\(868\) −2.07213 −0.0703328
\(869\) −30.1288 −1.02205
\(870\) −3.42323 −0.116058
\(871\) 49.9015 1.69085
\(872\) −9.71695 −0.329057
\(873\) −0.811353 −0.0274601
\(874\) −0.0494308 −0.00167202
\(875\) 7.36345 0.248930
\(876\) 4.43236 0.149756
\(877\) −25.7601 −0.869857 −0.434929 0.900465i \(-0.643226\pi\)
−0.434929 + 0.900465i \(0.643226\pi\)
\(878\) −28.2476 −0.953309
\(879\) 21.9314 0.739727
\(880\) 8.49109 0.286235
\(881\) 27.5117 0.926893 0.463447 0.886125i \(-0.346613\pi\)
0.463447 + 0.886125i \(0.346613\pi\)
\(882\) −5.43332 −0.182949
\(883\) −36.2104 −1.21858 −0.609288 0.792949i \(-0.708545\pi\)
−0.609288 + 0.792949i \(0.708545\pi\)
\(884\) 35.1943 1.18371
\(885\) 4.31670 0.145104
\(886\) −3.60019 −0.120951
\(887\) −37.2658 −1.25126 −0.625632 0.780118i \(-0.715159\pi\)
−0.625632 + 0.780118i \(0.715159\pi\)
\(888\) −11.1995 −0.375831
\(889\) −21.0327 −0.705414
\(890\) 2.72755 0.0914276
\(891\) −2.48043 −0.0830976
\(892\) −1.26560 −0.0423755
\(893\) −0.0141733 −0.000474293 0
\(894\) −0.0280400 −0.000937798 0
\(895\) 53.3331 1.78273
\(896\) −1.25167 −0.0418154
\(897\) −5.96144 −0.199047
\(898\) 26.4204 0.881659
\(899\) −1.65549 −0.0552138
\(900\) 6.71852 0.223951
\(901\) −8.42760 −0.280764
\(902\) 15.8117 0.526473
\(903\) 11.8171 0.393250
\(904\) −2.63452 −0.0876229
\(905\) −50.7155 −1.68584
\(906\) 2.83192 0.0940842
\(907\) 10.5076 0.348899 0.174449 0.984666i \(-0.444185\pi\)
0.174449 + 0.984666i \(0.444185\pi\)
\(908\) 15.7506 0.522701
\(909\) 0.311563 0.0103339
\(910\) −25.5434 −0.846755
\(911\) −11.1668 −0.369971 −0.184986 0.982741i \(-0.559224\pi\)
−0.184986 + 0.982741i \(0.559224\pi\)
\(912\) −0.0494308 −0.00163682
\(913\) −29.4036 −0.973117
\(914\) −7.19372 −0.237947
\(915\) 15.3376 0.507044
\(916\) 28.2236 0.932532
\(917\) 4.00693 0.132320
\(918\) 5.90366 0.194850
\(919\) 41.0557 1.35430 0.677151 0.735844i \(-0.263215\pi\)
0.677151 + 0.735844i \(0.263215\pi\)
\(920\) 3.42323 0.112861
\(921\) −19.0530 −0.627817
\(922\) −29.3838 −0.967704
\(923\) 32.3071 1.06340
\(924\) −3.10469 −0.102137
\(925\) 75.2442 2.47401
\(926\) −20.4254 −0.671219
\(927\) −16.0371 −0.526726
\(928\) −1.00000 −0.0328266
\(929\) 0.615142 0.0201822 0.0100911 0.999949i \(-0.496788\pi\)
0.0100911 + 0.999949i \(0.496788\pi\)
\(930\) 5.66714 0.185833
\(931\) −0.268573 −0.00880214
\(932\) 5.63781 0.184672
\(933\) −29.8045 −0.975755
\(934\) 41.6099 1.36152
\(935\) −50.1285 −1.63938
\(936\) −5.96144 −0.194856
\(937\) 20.0072 0.653608 0.326804 0.945092i \(-0.394028\pi\)
0.326804 + 0.945092i \(0.394028\pi\)
\(938\) 10.4774 0.342099
\(939\) −29.0561 −0.948209
\(940\) 0.981547 0.0320145
\(941\) −37.6403 −1.22704 −0.613520 0.789679i \(-0.710247\pi\)
−0.613520 + 0.789679i \(0.710247\pi\)
\(942\) 12.1009 0.394268
\(943\) 6.37459 0.207585
\(944\) 1.26100 0.0410421
\(945\) −4.28477 −0.139383
\(946\) −23.4180 −0.761384
\(947\) 12.1147 0.393675 0.196837 0.980436i \(-0.436933\pi\)
0.196837 + 0.980436i \(0.436933\pi\)
\(948\) −12.1466 −0.394503
\(949\) 26.4233 0.857736
\(950\) 0.332102 0.0107748
\(951\) 24.7475 0.802493
\(952\) 7.38945 0.239494
\(953\) −41.1265 −1.33222 −0.666110 0.745854i \(-0.732042\pi\)
−0.666110 + 0.745854i \(0.732042\pi\)
\(954\) 1.42752 0.0462177
\(955\) −11.8022 −0.381912
\(956\) −22.9285 −0.741561
\(957\) −2.48043 −0.0801810
\(958\) −25.8977 −0.836718
\(959\) −17.7069 −0.571785
\(960\) 3.42323 0.110484
\(961\) −28.2593 −0.911592
\(962\) −66.7653 −2.15260
\(963\) 7.79784 0.251282
\(964\) 15.0573 0.484962
\(965\) −82.5811 −2.65838
\(966\) −1.25167 −0.0402719
\(967\) −36.3071 −1.16756 −0.583779 0.811912i \(-0.698427\pi\)
−0.583779 + 0.811912i \(0.698427\pi\)
\(968\) −4.84746 −0.155803
\(969\) 0.291823 0.00937471
\(970\) 2.77745 0.0891785
\(971\) 5.75572 0.184710 0.0923549 0.995726i \(-0.470561\pi\)
0.0923549 + 0.995726i \(0.470561\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −8.09245 −0.259432
\(974\) −16.7191 −0.535716
\(975\) 40.0520 1.28269
\(976\) 4.48043 0.143415
\(977\) 9.58165 0.306544 0.153272 0.988184i \(-0.451019\pi\)
0.153272 + 0.988184i \(0.451019\pi\)
\(978\) −5.14348 −0.164470
\(979\) 1.97635 0.0631643
\(980\) 18.5995 0.594139
\(981\) −9.71695 −0.310238
\(982\) 2.14894 0.0685755
\(983\) −50.2321 −1.60215 −0.801077 0.598561i \(-0.795739\pi\)
−0.801077 + 0.598561i \(0.795739\pi\)
\(984\) 6.37459 0.203214
\(985\) −65.1433 −2.07564
\(986\) 5.90366 0.188011
\(987\) −0.358893 −0.0114237
\(988\) −0.294679 −0.00937499
\(989\) −9.44108 −0.300209
\(990\) 8.49109 0.269865
\(991\) −27.3240 −0.867975 −0.433987 0.900919i \(-0.642894\pi\)
−0.433987 + 0.900919i \(0.642894\pi\)
\(992\) 1.65549 0.0525620
\(993\) 2.82184 0.0895484
\(994\) 6.78325 0.215152
\(995\) 7.32047 0.232075
\(996\) −11.8542 −0.375616
\(997\) 17.4923 0.553987 0.276994 0.960872i \(-0.410662\pi\)
0.276994 + 0.960872i \(0.410662\pi\)
\(998\) −29.2331 −0.925359
\(999\) −11.1995 −0.354337
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 4002.2.a.bk.1.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4002.2.a.bk.1.2 8 1.1 even 1 trivial