Properties

Label 8022.2.a.p.1.8
Level $8022$
Weight $2$
Character 8022.1
Self dual yes
Analytic conductor $64.056$
Analytic rank $1$
Dimension $9$
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] = [8022,2,Mod(1,8022)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8022, 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("8022.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8022 = 2 \cdot 3 \cdot 7 \cdot 191 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8022.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: \(64.0559925015\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 17x^{7} - 12x^{6} + 72x^{5} + 81x^{4} - 67x^{3} - 105x^{2} - 17x + 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(2.17782\) of defining polynomial
Character \(\chi\) \(=\) 8022.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.12019 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.12019 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +2.12019 q^{10} -5.65813 q^{11} -1.00000 q^{12} -3.41673 q^{13} +1.00000 q^{14} -2.12019 q^{15} +1.00000 q^{16} +3.62524 q^{17} +1.00000 q^{18} -0.530988 q^{19} +2.12019 q^{20} -1.00000 q^{21} -5.65813 q^{22} -1.83642 q^{23} -1.00000 q^{24} -0.504804 q^{25} -3.41673 q^{26} -1.00000 q^{27} +1.00000 q^{28} +7.73144 q^{29} -2.12019 q^{30} -8.59340 q^{31} +1.00000 q^{32} +5.65813 q^{33} +3.62524 q^{34} +2.12019 q^{35} +1.00000 q^{36} -11.6878 q^{37} -0.530988 q^{38} +3.41673 q^{39} +2.12019 q^{40} +12.4762 q^{41} -1.00000 q^{42} -6.43569 q^{43} -5.65813 q^{44} +2.12019 q^{45} -1.83642 q^{46} -8.05558 q^{47} -1.00000 q^{48} +1.00000 q^{49} -0.504804 q^{50} -3.62524 q^{51} -3.41673 q^{52} +5.25315 q^{53} -1.00000 q^{54} -11.9963 q^{55} +1.00000 q^{56} +0.530988 q^{57} +7.73144 q^{58} +5.97094 q^{59} -2.12019 q^{60} +12.0771 q^{61} -8.59340 q^{62} +1.00000 q^{63} +1.00000 q^{64} -7.24411 q^{65} +5.65813 q^{66} -0.347898 q^{67} +3.62524 q^{68} +1.83642 q^{69} +2.12019 q^{70} -5.14641 q^{71} +1.00000 q^{72} -12.9220 q^{73} -11.6878 q^{74} +0.504804 q^{75} -0.530988 q^{76} -5.65813 q^{77} +3.41673 q^{78} -15.6065 q^{79} +2.12019 q^{80} +1.00000 q^{81} +12.4762 q^{82} -13.4142 q^{83} -1.00000 q^{84} +7.68619 q^{85} -6.43569 q^{86} -7.73144 q^{87} -5.65813 q^{88} +15.3853 q^{89} +2.12019 q^{90} -3.41673 q^{91} -1.83642 q^{92} +8.59340 q^{93} -8.05558 q^{94} -1.12579 q^{95} -1.00000 q^{96} -16.3823 q^{97} +1.00000 q^{98} -5.65813 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 9 q^{2} - 9 q^{3} + 9 q^{4} - 4 q^{5} - 9 q^{6} + 9 q^{7} + 9 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + 9 q^{2} - 9 q^{3} + 9 q^{4} - 4 q^{5} - 9 q^{6} + 9 q^{7} + 9 q^{8} + 9 q^{9} - 4 q^{10} - 3 q^{11} - 9 q^{12} - 9 q^{13} + 9 q^{14} + 4 q^{15} + 9 q^{16} - 12 q^{17} + 9 q^{18} - 18 q^{19} - 4 q^{20} - 9 q^{21} - 3 q^{22} + q^{23} - 9 q^{24} + 9 q^{25} - 9 q^{26} - 9 q^{27} + 9 q^{28} - 11 q^{29} + 4 q^{30} - 22 q^{31} + 9 q^{32} + 3 q^{33} - 12 q^{34} - 4 q^{35} + 9 q^{36} + 8 q^{37} - 18 q^{38} + 9 q^{39} - 4 q^{40} - 22 q^{41} - 9 q^{42} - 3 q^{43} - 3 q^{44} - 4 q^{45} + q^{46} - 49 q^{47} - 9 q^{48} + 9 q^{49} + 9 q^{50} + 12 q^{51} - 9 q^{52} + 8 q^{53} - 9 q^{54} - 19 q^{55} + 9 q^{56} + 18 q^{57} - 11 q^{58} - 18 q^{59} + 4 q^{60} + 3 q^{61} - 22 q^{62} + 9 q^{63} + 9 q^{64} - 32 q^{65} + 3 q^{66} - 11 q^{67} - 12 q^{68} - q^{69} - 4 q^{70} - 7 q^{71} + 9 q^{72} - 23 q^{73} + 8 q^{74} - 9 q^{75} - 18 q^{76} - 3 q^{77} + 9 q^{78} - 17 q^{79} - 4 q^{80} + 9 q^{81} - 22 q^{82} - 30 q^{83} - 9 q^{84} + 18 q^{85} - 3 q^{86} + 11 q^{87} - 3 q^{88} + 23 q^{89} - 4 q^{90} - 9 q^{91} + q^{92} + 22 q^{93} - 49 q^{94} - 30 q^{95} - 9 q^{96} - 46 q^{97} + 9 q^{98} - 3 q^{99}+O(q^{100}) \) 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
\(5\) 2.12019 0.948177 0.474088 0.880477i \(-0.342778\pi\)
0.474088 + 0.880477i \(0.342778\pi\)
\(6\) −1.00000 −0.408248
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 2.12019 0.670462
\(11\) −5.65813 −1.70599 −0.852995 0.521918i \(-0.825217\pi\)
−0.852995 + 0.521918i \(0.825217\pi\)
\(12\) −1.00000 −0.288675
\(13\) −3.41673 −0.947631 −0.473815 0.880624i \(-0.657124\pi\)
−0.473815 + 0.880624i \(0.657124\pi\)
\(14\) 1.00000 0.267261
\(15\) −2.12019 −0.547430
\(16\) 1.00000 0.250000
\(17\) 3.62524 0.879250 0.439625 0.898181i \(-0.355111\pi\)
0.439625 + 0.898181i \(0.355111\pi\)
\(18\) 1.00000 0.235702
\(19\) −0.530988 −0.121817 −0.0609085 0.998143i \(-0.519400\pi\)
−0.0609085 + 0.998143i \(0.519400\pi\)
\(20\) 2.12019 0.474088
\(21\) −1.00000 −0.218218
\(22\) −5.65813 −1.20632
\(23\) −1.83642 −0.382921 −0.191460 0.981500i \(-0.561322\pi\)
−0.191460 + 0.981500i \(0.561322\pi\)
\(24\) −1.00000 −0.204124
\(25\) −0.504804 −0.100961
\(26\) −3.41673 −0.670076
\(27\) −1.00000 −0.192450
\(28\) 1.00000 0.188982
\(29\) 7.73144 1.43569 0.717847 0.696201i \(-0.245128\pi\)
0.717847 + 0.696201i \(0.245128\pi\)
\(30\) −2.12019 −0.387092
\(31\) −8.59340 −1.54342 −0.771711 0.635974i \(-0.780599\pi\)
−0.771711 + 0.635974i \(0.780599\pi\)
\(32\) 1.00000 0.176777
\(33\) 5.65813 0.984954
\(34\) 3.62524 0.621724
\(35\) 2.12019 0.358377
\(36\) 1.00000 0.166667
\(37\) −11.6878 −1.92146 −0.960732 0.277479i \(-0.910501\pi\)
−0.960732 + 0.277479i \(0.910501\pi\)
\(38\) −0.530988 −0.0861377
\(39\) 3.41673 0.547115
\(40\) 2.12019 0.335231
\(41\) 12.4762 1.94846 0.974229 0.225560i \(-0.0724212\pi\)
0.974229 + 0.225560i \(0.0724212\pi\)
\(42\) −1.00000 −0.154303
\(43\) −6.43569 −0.981433 −0.490717 0.871319i \(-0.663265\pi\)
−0.490717 + 0.871319i \(0.663265\pi\)
\(44\) −5.65813 −0.852995
\(45\) 2.12019 0.316059
\(46\) −1.83642 −0.270766
\(47\) −8.05558 −1.17503 −0.587514 0.809214i \(-0.699893\pi\)
−0.587514 + 0.809214i \(0.699893\pi\)
\(48\) −1.00000 −0.144338
\(49\) 1.00000 0.142857
\(50\) −0.504804 −0.0713901
\(51\) −3.62524 −0.507635
\(52\) −3.41673 −0.473815
\(53\) 5.25315 0.721576 0.360788 0.932648i \(-0.382508\pi\)
0.360788 + 0.932648i \(0.382508\pi\)
\(54\) −1.00000 −0.136083
\(55\) −11.9963 −1.61758
\(56\) 1.00000 0.133631
\(57\) 0.530988 0.0703311
\(58\) 7.73144 1.01519
\(59\) 5.97094 0.777350 0.388675 0.921375i \(-0.372933\pi\)
0.388675 + 0.921375i \(0.372933\pi\)
\(60\) −2.12019 −0.273715
\(61\) 12.0771 1.54631 0.773157 0.634215i \(-0.218676\pi\)
0.773157 + 0.634215i \(0.218676\pi\)
\(62\) −8.59340 −1.09136
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) −7.24411 −0.898521
\(66\) 5.65813 0.696468
\(67\) −0.347898 −0.0425025 −0.0212513 0.999774i \(-0.506765\pi\)
−0.0212513 + 0.999774i \(0.506765\pi\)
\(68\) 3.62524 0.439625
\(69\) 1.83642 0.221079
\(70\) 2.12019 0.253411
\(71\) −5.14641 −0.610767 −0.305383 0.952229i \(-0.598785\pi\)
−0.305383 + 0.952229i \(0.598785\pi\)
\(72\) 1.00000 0.117851
\(73\) −12.9220 −1.51241 −0.756205 0.654334i \(-0.772949\pi\)
−0.756205 + 0.654334i \(0.772949\pi\)
\(74\) −11.6878 −1.35868
\(75\) 0.504804 0.0582898
\(76\) −0.530988 −0.0609085
\(77\) −5.65813 −0.644804
\(78\) 3.41673 0.386869
\(79\) −15.6065 −1.75587 −0.877934 0.478781i \(-0.841079\pi\)
−0.877934 + 0.478781i \(0.841079\pi\)
\(80\) 2.12019 0.237044
\(81\) 1.00000 0.111111
\(82\) 12.4762 1.37777
\(83\) −13.4142 −1.47240 −0.736200 0.676764i \(-0.763382\pi\)
−0.736200 + 0.676764i \(0.763382\pi\)
\(84\) −1.00000 −0.109109
\(85\) 7.68619 0.833685
\(86\) −6.43569 −0.693978
\(87\) −7.73144 −0.828898
\(88\) −5.65813 −0.603159
\(89\) 15.3853 1.63084 0.815420 0.578870i \(-0.196506\pi\)
0.815420 + 0.578870i \(0.196506\pi\)
\(90\) 2.12019 0.223487
\(91\) −3.41673 −0.358171
\(92\) −1.83642 −0.191460
\(93\) 8.59340 0.891095
\(94\) −8.05558 −0.830870
\(95\) −1.12579 −0.115504
\(96\) −1.00000 −0.102062
\(97\) −16.3823 −1.66337 −0.831685 0.555247i \(-0.812624\pi\)
−0.831685 + 0.555247i \(0.812624\pi\)
\(98\) 1.00000 0.101015
\(99\) −5.65813 −0.568664
\(100\) −0.504804 −0.0504804
\(101\) −10.9489 −1.08946 −0.544729 0.838612i \(-0.683368\pi\)
−0.544729 + 0.838612i \(0.683368\pi\)
\(102\) −3.62524 −0.358952
\(103\) −9.79451 −0.965082 −0.482541 0.875873i \(-0.660286\pi\)
−0.482541 + 0.875873i \(0.660286\pi\)
\(104\) −3.41673 −0.335038
\(105\) −2.12019 −0.206909
\(106\) 5.25315 0.510231
\(107\) −0.795378 −0.0768921 −0.0384461 0.999261i \(-0.512241\pi\)
−0.0384461 + 0.999261i \(0.512241\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 14.3889 1.37821 0.689104 0.724663i \(-0.258004\pi\)
0.689104 + 0.724663i \(0.258004\pi\)
\(110\) −11.9963 −1.14380
\(111\) 11.6878 1.10936
\(112\) 1.00000 0.0944911
\(113\) −8.17372 −0.768919 −0.384459 0.923142i \(-0.625612\pi\)
−0.384459 + 0.923142i \(0.625612\pi\)
\(114\) 0.530988 0.0497316
\(115\) −3.89356 −0.363076
\(116\) 7.73144 0.717847
\(117\) −3.41673 −0.315877
\(118\) 5.97094 0.549669
\(119\) 3.62524 0.332325
\(120\) −2.12019 −0.193546
\(121\) 21.0145 1.91041
\(122\) 12.0771 1.09341
\(123\) −12.4762 −1.12494
\(124\) −8.59340 −0.771711
\(125\) −11.6712 −1.04391
\(126\) 1.00000 0.0890871
\(127\) −15.4919 −1.37468 −0.687341 0.726335i \(-0.741222\pi\)
−0.687341 + 0.726335i \(0.741222\pi\)
\(128\) 1.00000 0.0883883
\(129\) 6.43569 0.566631
\(130\) −7.24411 −0.635351
\(131\) −17.5231 −1.53100 −0.765499 0.643437i \(-0.777508\pi\)
−0.765499 + 0.643437i \(0.777508\pi\)
\(132\) 5.65813 0.492477
\(133\) −0.530988 −0.0460425
\(134\) −0.347898 −0.0300538
\(135\) −2.12019 −0.182477
\(136\) 3.62524 0.310862
\(137\) −10.7313 −0.916836 −0.458418 0.888737i \(-0.651584\pi\)
−0.458418 + 0.888737i \(0.651584\pi\)
\(138\) 1.83642 0.156327
\(139\) 19.8459 1.68330 0.841652 0.540020i \(-0.181583\pi\)
0.841652 + 0.540020i \(0.181583\pi\)
\(140\) 2.12019 0.179189
\(141\) 8.05558 0.678402
\(142\) −5.14641 −0.431877
\(143\) 19.3323 1.61665
\(144\) 1.00000 0.0833333
\(145\) 16.3921 1.36129
\(146\) −12.9220 −1.06944
\(147\) −1.00000 −0.0824786
\(148\) −11.6878 −0.960732
\(149\) −12.2987 −1.00754 −0.503772 0.863836i \(-0.668055\pi\)
−0.503772 + 0.863836i \(0.668055\pi\)
\(150\) 0.504804 0.0412171
\(151\) 17.6647 1.43753 0.718766 0.695252i \(-0.244707\pi\)
0.718766 + 0.695252i \(0.244707\pi\)
\(152\) −0.530988 −0.0430688
\(153\) 3.62524 0.293083
\(154\) −5.65813 −0.455945
\(155\) −18.2196 −1.46344
\(156\) 3.41673 0.273557
\(157\) −18.2820 −1.45906 −0.729532 0.683947i \(-0.760262\pi\)
−0.729532 + 0.683947i \(0.760262\pi\)
\(158\) −15.6065 −1.24159
\(159\) −5.25315 −0.416602
\(160\) 2.12019 0.167616
\(161\) −1.83642 −0.144730
\(162\) 1.00000 0.0785674
\(163\) 8.89726 0.696888 0.348444 0.937330i \(-0.386710\pi\)
0.348444 + 0.937330i \(0.386710\pi\)
\(164\) 12.4762 0.974229
\(165\) 11.9963 0.933911
\(166\) −13.4142 −1.04114
\(167\) 19.7895 1.53136 0.765680 0.643222i \(-0.222403\pi\)
0.765680 + 0.643222i \(0.222403\pi\)
\(168\) −1.00000 −0.0771517
\(169\) −1.32595 −0.101996
\(170\) 7.68619 0.589504
\(171\) −0.530988 −0.0406057
\(172\) −6.43569 −0.490717
\(173\) −11.2793 −0.857546 −0.428773 0.903412i \(-0.641054\pi\)
−0.428773 + 0.903412i \(0.641054\pi\)
\(174\) −7.73144 −0.586119
\(175\) −0.504804 −0.0381596
\(176\) −5.65813 −0.426498
\(177\) −5.97094 −0.448803
\(178\) 15.3853 1.15318
\(179\) −3.64668 −0.272565 −0.136283 0.990670i \(-0.543516\pi\)
−0.136283 + 0.990670i \(0.543516\pi\)
\(180\) 2.12019 0.158029
\(181\) −1.91630 −0.142437 −0.0712186 0.997461i \(-0.522689\pi\)
−0.0712186 + 0.997461i \(0.522689\pi\)
\(182\) −3.41673 −0.253265
\(183\) −12.0771 −0.892765
\(184\) −1.83642 −0.135383
\(185\) −24.7803 −1.82189
\(186\) 8.59340 0.630099
\(187\) −20.5121 −1.49999
\(188\) −8.05558 −0.587514
\(189\) −1.00000 −0.0727393
\(190\) −1.12579 −0.0816737
\(191\) −1.00000 −0.0723575
\(192\) −1.00000 −0.0721688
\(193\) 0.0643161 0.00462957 0.00231479 0.999997i \(-0.499263\pi\)
0.00231479 + 0.999997i \(0.499263\pi\)
\(194\) −16.3823 −1.17618
\(195\) 7.24411 0.518762
\(196\) 1.00000 0.0714286
\(197\) 6.31491 0.449919 0.224959 0.974368i \(-0.427775\pi\)
0.224959 + 0.974368i \(0.427775\pi\)
\(198\) −5.65813 −0.402106
\(199\) 9.61618 0.681673 0.340836 0.940123i \(-0.389290\pi\)
0.340836 + 0.940123i \(0.389290\pi\)
\(200\) −0.504804 −0.0356951
\(201\) 0.347898 0.0245388
\(202\) −10.9489 −0.770364
\(203\) 7.73144 0.542641
\(204\) −3.62524 −0.253818
\(205\) 26.4519 1.84748
\(206\) −9.79451 −0.682416
\(207\) −1.83642 −0.127640
\(208\) −3.41673 −0.236908
\(209\) 3.00440 0.207819
\(210\) −2.12019 −0.146307
\(211\) 16.0386 1.10414 0.552072 0.833796i \(-0.313837\pi\)
0.552072 + 0.833796i \(0.313837\pi\)
\(212\) 5.25315 0.360788
\(213\) 5.14641 0.352626
\(214\) −0.795378 −0.0543709
\(215\) −13.6449 −0.930572
\(216\) −1.00000 −0.0680414
\(217\) −8.59340 −0.583358
\(218\) 14.3889 0.974540
\(219\) 12.9220 0.873191
\(220\) −11.9963 −0.808790
\(221\) −12.3865 −0.833205
\(222\) 11.6878 0.784434
\(223\) 19.9769 1.33775 0.668875 0.743375i \(-0.266776\pi\)
0.668875 + 0.743375i \(0.266776\pi\)
\(224\) 1.00000 0.0668153
\(225\) −0.504804 −0.0336536
\(226\) −8.17372 −0.543708
\(227\) 3.10798 0.206284 0.103142 0.994667i \(-0.467110\pi\)
0.103142 + 0.994667i \(0.467110\pi\)
\(228\) 0.530988 0.0351656
\(229\) 18.5609 1.22654 0.613268 0.789875i \(-0.289854\pi\)
0.613268 + 0.789875i \(0.289854\pi\)
\(230\) −3.89356 −0.256734
\(231\) 5.65813 0.372278
\(232\) 7.73144 0.507594
\(233\) 7.66569 0.502196 0.251098 0.967962i \(-0.419208\pi\)
0.251098 + 0.967962i \(0.419208\pi\)
\(234\) −3.41673 −0.223359
\(235\) −17.0793 −1.11413
\(236\) 5.97094 0.388675
\(237\) 15.6065 1.01375
\(238\) 3.62524 0.234990
\(239\) −4.31275 −0.278969 −0.139484 0.990224i \(-0.544545\pi\)
−0.139484 + 0.990224i \(0.544545\pi\)
\(240\) −2.12019 −0.136858
\(241\) −28.8232 −1.85667 −0.928334 0.371748i \(-0.878759\pi\)
−0.928334 + 0.371748i \(0.878759\pi\)
\(242\) 21.0145 1.35086
\(243\) −1.00000 −0.0641500
\(244\) 12.0771 0.773157
\(245\) 2.12019 0.135454
\(246\) −12.4762 −0.795455
\(247\) 1.81424 0.115438
\(248\) −8.59340 −0.545682
\(249\) 13.4142 0.850091
\(250\) −11.6712 −0.738153
\(251\) 8.56547 0.540648 0.270324 0.962769i \(-0.412869\pi\)
0.270324 + 0.962769i \(0.412869\pi\)
\(252\) 1.00000 0.0629941
\(253\) 10.3907 0.653259
\(254\) −15.4919 −0.972047
\(255\) −7.68619 −0.481328
\(256\) 1.00000 0.0625000
\(257\) −26.0282 −1.62359 −0.811796 0.583941i \(-0.801510\pi\)
−0.811796 + 0.583941i \(0.801510\pi\)
\(258\) 6.43569 0.400668
\(259\) −11.6878 −0.726245
\(260\) −7.24411 −0.449261
\(261\) 7.73144 0.478564
\(262\) −17.5231 −1.08258
\(263\) −6.37183 −0.392904 −0.196452 0.980513i \(-0.562942\pi\)
−0.196452 + 0.980513i \(0.562942\pi\)
\(264\) 5.65813 0.348234
\(265\) 11.1377 0.684182
\(266\) −0.530988 −0.0325570
\(267\) −15.3853 −0.941566
\(268\) −0.347898 −0.0212513
\(269\) −16.0331 −0.977555 −0.488777 0.872409i \(-0.662557\pi\)
−0.488777 + 0.872409i \(0.662557\pi\)
\(270\) −2.12019 −0.129031
\(271\) 22.3891 1.36004 0.680020 0.733194i \(-0.261971\pi\)
0.680020 + 0.733194i \(0.261971\pi\)
\(272\) 3.62524 0.219813
\(273\) 3.41673 0.206790
\(274\) −10.7313 −0.648301
\(275\) 2.85625 0.172238
\(276\) 1.83642 0.110540
\(277\) 1.87730 0.112796 0.0563980 0.998408i \(-0.482038\pi\)
0.0563980 + 0.998408i \(0.482038\pi\)
\(278\) 19.8459 1.19028
\(279\) −8.59340 −0.514474
\(280\) 2.12019 0.126705
\(281\) −1.54137 −0.0919502 −0.0459751 0.998943i \(-0.514639\pi\)
−0.0459751 + 0.998943i \(0.514639\pi\)
\(282\) 8.05558 0.479703
\(283\) −15.7354 −0.935373 −0.467687 0.883894i \(-0.654912\pi\)
−0.467687 + 0.883894i \(0.654912\pi\)
\(284\) −5.14641 −0.305383
\(285\) 1.12579 0.0666863
\(286\) 19.3323 1.14314
\(287\) 12.4762 0.736448
\(288\) 1.00000 0.0589256
\(289\) −3.85762 −0.226919
\(290\) 16.3921 0.962578
\(291\) 16.3823 0.960348
\(292\) −12.9220 −0.756205
\(293\) −3.70401 −0.216391 −0.108195 0.994130i \(-0.534507\pi\)
−0.108195 + 0.994130i \(0.534507\pi\)
\(294\) −1.00000 −0.0583212
\(295\) 12.6595 0.737065
\(296\) −11.6878 −0.679340
\(297\) 5.65813 0.328318
\(298\) −12.2987 −0.712442
\(299\) 6.27456 0.362867
\(300\) 0.504804 0.0291449
\(301\) −6.43569 −0.370947
\(302\) 17.6647 1.01649
\(303\) 10.9489 0.628999
\(304\) −0.530988 −0.0304543
\(305\) 25.6057 1.46618
\(306\) 3.62524 0.207241
\(307\) −18.3234 −1.04577 −0.522886 0.852403i \(-0.675145\pi\)
−0.522886 + 0.852403i \(0.675145\pi\)
\(308\) −5.65813 −0.322402
\(309\) 9.79451 0.557190
\(310\) −18.2196 −1.03481
\(311\) −19.4591 −1.10343 −0.551713 0.834034i \(-0.686026\pi\)
−0.551713 + 0.834034i \(0.686026\pi\)
\(312\) 3.41673 0.193434
\(313\) 7.82362 0.442217 0.221108 0.975249i \(-0.429033\pi\)
0.221108 + 0.975249i \(0.429033\pi\)
\(314\) −18.2820 −1.03171
\(315\) 2.12019 0.119459
\(316\) −15.6065 −0.877934
\(317\) −1.62203 −0.0911021 −0.0455510 0.998962i \(-0.514504\pi\)
−0.0455510 + 0.998962i \(0.514504\pi\)
\(318\) −5.25315 −0.294582
\(319\) −43.7455 −2.44928
\(320\) 2.12019 0.118522
\(321\) 0.795378 0.0443937
\(322\) −1.83642 −0.102340
\(323\) −1.92496 −0.107108
\(324\) 1.00000 0.0555556
\(325\) 1.72478 0.0956736
\(326\) 8.89726 0.492774
\(327\) −14.3889 −0.795709
\(328\) 12.4762 0.688884
\(329\) −8.05558 −0.444119
\(330\) 11.9963 0.660375
\(331\) 5.80972 0.319331 0.159665 0.987171i \(-0.448958\pi\)
0.159665 + 0.987171i \(0.448958\pi\)
\(332\) −13.4142 −0.736200
\(333\) −11.6878 −0.640488
\(334\) 19.7895 1.08283
\(335\) −0.737609 −0.0402999
\(336\) −1.00000 −0.0545545
\(337\) 34.2577 1.86613 0.933067 0.359702i \(-0.117122\pi\)
0.933067 + 0.359702i \(0.117122\pi\)
\(338\) −1.32595 −0.0721219
\(339\) 8.17372 0.443936
\(340\) 7.68619 0.416842
\(341\) 48.6226 2.63306
\(342\) −0.530988 −0.0287126
\(343\) 1.00000 0.0539949
\(344\) −6.43569 −0.346989
\(345\) 3.89356 0.209622
\(346\) −11.2793 −0.606376
\(347\) 0.654394 0.0351297 0.0175649 0.999846i \(-0.494409\pi\)
0.0175649 + 0.999846i \(0.494409\pi\)
\(348\) −7.73144 −0.414449
\(349\) 20.2461 1.08375 0.541874 0.840459i \(-0.317715\pi\)
0.541874 + 0.840459i \(0.317715\pi\)
\(350\) −0.504804 −0.0269829
\(351\) 3.41673 0.182372
\(352\) −5.65813 −0.301579
\(353\) −11.9714 −0.637172 −0.318586 0.947894i \(-0.603208\pi\)
−0.318586 + 0.947894i \(0.603208\pi\)
\(354\) −5.97094 −0.317352
\(355\) −10.9114 −0.579115
\(356\) 15.3853 0.815420
\(357\) −3.62524 −0.191868
\(358\) −3.64668 −0.192733
\(359\) 24.6520 1.30108 0.650542 0.759470i \(-0.274542\pi\)
0.650542 + 0.759470i \(0.274542\pi\)
\(360\) 2.12019 0.111744
\(361\) −18.7181 −0.985161
\(362\) −1.91630 −0.100718
\(363\) −21.0145 −1.10297
\(364\) −3.41673 −0.179085
\(365\) −27.3972 −1.43403
\(366\) −12.0771 −0.631280
\(367\) −25.7416 −1.34370 −0.671850 0.740687i \(-0.734500\pi\)
−0.671850 + 0.740687i \(0.734500\pi\)
\(368\) −1.83642 −0.0957302
\(369\) 12.4762 0.649486
\(370\) −24.7803 −1.28827
\(371\) 5.25315 0.272730
\(372\) 8.59340 0.445547
\(373\) 14.3450 0.742755 0.371378 0.928482i \(-0.378886\pi\)
0.371378 + 0.928482i \(0.378886\pi\)
\(374\) −20.5121 −1.06066
\(375\) 11.6712 0.602699
\(376\) −8.05558 −0.415435
\(377\) −26.4163 −1.36051
\(378\) −1.00000 −0.0514344
\(379\) −6.54180 −0.336030 −0.168015 0.985784i \(-0.553736\pi\)
−0.168015 + 0.985784i \(0.553736\pi\)
\(380\) −1.12579 −0.0577521
\(381\) 15.4919 0.793673
\(382\) −1.00000 −0.0511645
\(383\) −1.70210 −0.0869734 −0.0434867 0.999054i \(-0.513847\pi\)
−0.0434867 + 0.999054i \(0.513847\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −11.9963 −0.611388
\(386\) 0.0643161 0.00327360
\(387\) −6.43569 −0.327144
\(388\) −16.3823 −0.831685
\(389\) −13.5361 −0.686307 −0.343153 0.939279i \(-0.611495\pi\)
−0.343153 + 0.939279i \(0.611495\pi\)
\(390\) 7.24411 0.366820
\(391\) −6.65748 −0.336683
\(392\) 1.00000 0.0505076
\(393\) 17.5231 0.883922
\(394\) 6.31491 0.318141
\(395\) −33.0887 −1.66487
\(396\) −5.65813 −0.284332
\(397\) −8.64242 −0.433751 −0.216875 0.976199i \(-0.569586\pi\)
−0.216875 + 0.976199i \(0.569586\pi\)
\(398\) 9.61618 0.482015
\(399\) 0.530988 0.0265827
\(400\) −0.504804 −0.0252402
\(401\) 6.08530 0.303885 0.151943 0.988389i \(-0.451447\pi\)
0.151943 + 0.988389i \(0.451447\pi\)
\(402\) 0.347898 0.0173516
\(403\) 29.3614 1.46259
\(404\) −10.9489 −0.544729
\(405\) 2.12019 0.105353
\(406\) 7.73144 0.383705
\(407\) 66.1311 3.27800
\(408\) −3.62524 −0.179476
\(409\) −28.2163 −1.39520 −0.697602 0.716485i \(-0.745750\pi\)
−0.697602 + 0.716485i \(0.745750\pi\)
\(410\) 26.4519 1.30637
\(411\) 10.7313 0.529335
\(412\) −9.79451 −0.482541
\(413\) 5.97094 0.293811
\(414\) −1.83642 −0.0902553
\(415\) −28.4406 −1.39610
\(416\) −3.41673 −0.167519
\(417\) −19.8459 −0.971856
\(418\) 3.00440 0.146950
\(419\) 28.2638 1.38078 0.690390 0.723438i \(-0.257439\pi\)
0.690390 + 0.723438i \(0.257439\pi\)
\(420\) −2.12019 −0.103455
\(421\) 8.29174 0.404115 0.202057 0.979374i \(-0.435237\pi\)
0.202057 + 0.979374i \(0.435237\pi\)
\(422\) 16.0386 0.780748
\(423\) −8.05558 −0.391676
\(424\) 5.25315 0.255116
\(425\) −1.83004 −0.0887699
\(426\) 5.14641 0.249344
\(427\) 12.0771 0.584452
\(428\) −0.795378 −0.0384461
\(429\) −19.3323 −0.933373
\(430\) −13.6449 −0.658014
\(431\) −6.44130 −0.310266 −0.155133 0.987894i \(-0.549581\pi\)
−0.155133 + 0.987894i \(0.549581\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 25.3953 1.22042 0.610211 0.792239i \(-0.291085\pi\)
0.610211 + 0.792239i \(0.291085\pi\)
\(434\) −8.59340 −0.412497
\(435\) −16.3921 −0.785942
\(436\) 14.3889 0.689104
\(437\) 0.975119 0.0466463
\(438\) 12.9220 0.617439
\(439\) −14.9239 −0.712278 −0.356139 0.934433i \(-0.615907\pi\)
−0.356139 + 0.934433i \(0.615907\pi\)
\(440\) −11.9963 −0.571901
\(441\) 1.00000 0.0476190
\(442\) −12.3865 −0.589165
\(443\) 25.1056 1.19280 0.596402 0.802686i \(-0.296596\pi\)
0.596402 + 0.802686i \(0.296596\pi\)
\(444\) 11.6878 0.554679
\(445\) 32.6198 1.54632
\(446\) 19.9769 0.945932
\(447\) 12.2987 0.581706
\(448\) 1.00000 0.0472456
\(449\) −37.7012 −1.77923 −0.889615 0.456712i \(-0.849027\pi\)
−0.889615 + 0.456712i \(0.849027\pi\)
\(450\) −0.504804 −0.0237967
\(451\) −70.5921 −3.32405
\(452\) −8.17372 −0.384459
\(453\) −17.6647 −0.829959
\(454\) 3.10798 0.145865
\(455\) −7.24411 −0.339609
\(456\) 0.530988 0.0248658
\(457\) 3.99000 0.186644 0.0933221 0.995636i \(-0.470251\pi\)
0.0933221 + 0.995636i \(0.470251\pi\)
\(458\) 18.5609 0.867292
\(459\) −3.62524 −0.169212
\(460\) −3.89356 −0.181538
\(461\) 30.6319 1.42667 0.713336 0.700823i \(-0.247184\pi\)
0.713336 + 0.700823i \(0.247184\pi\)
\(462\) 5.65813 0.263240
\(463\) 17.6524 0.820377 0.410188 0.912001i \(-0.365463\pi\)
0.410188 + 0.912001i \(0.365463\pi\)
\(464\) 7.73144 0.358923
\(465\) 18.2196 0.844915
\(466\) 7.66569 0.355106
\(467\) −11.4024 −0.527641 −0.263820 0.964572i \(-0.584983\pi\)
−0.263820 + 0.964572i \(0.584983\pi\)
\(468\) −3.41673 −0.157938
\(469\) −0.347898 −0.0160644
\(470\) −17.0793 −0.787811
\(471\) 18.2820 0.842391
\(472\) 5.97094 0.274835
\(473\) 36.4140 1.67432
\(474\) 15.6065 0.716830
\(475\) 0.268045 0.0122988
\(476\) 3.62524 0.166163
\(477\) 5.25315 0.240525
\(478\) −4.31275 −0.197261
\(479\) −9.83500 −0.449373 −0.224686 0.974431i \(-0.572136\pi\)
−0.224686 + 0.974431i \(0.572136\pi\)
\(480\) −2.12019 −0.0967729
\(481\) 39.9341 1.82084
\(482\) −28.8232 −1.31286
\(483\) 1.83642 0.0835601
\(484\) 21.0145 0.955203
\(485\) −34.7336 −1.57717
\(486\) −1.00000 −0.0453609
\(487\) 17.9436 0.813102 0.406551 0.913628i \(-0.366731\pi\)
0.406551 + 0.913628i \(0.366731\pi\)
\(488\) 12.0771 0.546705
\(489\) −8.89726 −0.402348
\(490\) 2.12019 0.0957803
\(491\) 4.97688 0.224603 0.112302 0.993674i \(-0.464178\pi\)
0.112302 + 0.993674i \(0.464178\pi\)
\(492\) −12.4762 −0.562472
\(493\) 28.0284 1.26233
\(494\) 1.81424 0.0816267
\(495\) −11.9963 −0.539194
\(496\) −8.59340 −0.385855
\(497\) −5.14641 −0.230848
\(498\) 13.4142 0.601105
\(499\) −14.3583 −0.642765 −0.321382 0.946950i \(-0.604147\pi\)
−0.321382 + 0.946950i \(0.604147\pi\)
\(500\) −11.6712 −0.521953
\(501\) −19.7895 −0.884131
\(502\) 8.56547 0.382296
\(503\) 3.64411 0.162483 0.0812414 0.996694i \(-0.474112\pi\)
0.0812414 + 0.996694i \(0.474112\pi\)
\(504\) 1.00000 0.0445435
\(505\) −23.2138 −1.03300
\(506\) 10.3907 0.461924
\(507\) 1.32595 0.0588873
\(508\) −15.4919 −0.687341
\(509\) −20.6190 −0.913919 −0.456960 0.889487i \(-0.651062\pi\)
−0.456960 + 0.889487i \(0.651062\pi\)
\(510\) −7.68619 −0.340350
\(511\) −12.9220 −0.571638
\(512\) 1.00000 0.0441942
\(513\) 0.530988 0.0234437
\(514\) −26.0282 −1.14805
\(515\) −20.7662 −0.915068
\(516\) 6.43569 0.283315
\(517\) 45.5795 2.00459
\(518\) −11.6878 −0.513533
\(519\) 11.2793 0.495104
\(520\) −7.24411 −0.317675
\(521\) 16.5313 0.724248 0.362124 0.932130i \(-0.382052\pi\)
0.362124 + 0.932130i \(0.382052\pi\)
\(522\) 7.73144 0.338396
\(523\) 20.6771 0.904147 0.452073 0.891981i \(-0.350685\pi\)
0.452073 + 0.891981i \(0.350685\pi\)
\(524\) −17.5231 −0.765499
\(525\) 0.504804 0.0220315
\(526\) −6.37183 −0.277825
\(527\) −31.1532 −1.35705
\(528\) 5.65813 0.246239
\(529\) −19.6276 −0.853372
\(530\) 11.1377 0.483790
\(531\) 5.97094 0.259117
\(532\) −0.530988 −0.0230213
\(533\) −42.6279 −1.84642
\(534\) −15.3853 −0.665788
\(535\) −1.68635 −0.0729073
\(536\) −0.347898 −0.0150269
\(537\) 3.64668 0.157366
\(538\) −16.0331 −0.691235
\(539\) −5.65813 −0.243713
\(540\) −2.12019 −0.0912383
\(541\) −11.7578 −0.505507 −0.252753 0.967531i \(-0.581336\pi\)
−0.252753 + 0.967531i \(0.581336\pi\)
\(542\) 22.3891 0.961693
\(543\) 1.91630 0.0822362
\(544\) 3.62524 0.155431
\(545\) 30.5072 1.30678
\(546\) 3.41673 0.146223
\(547\) 15.4976 0.662631 0.331316 0.943520i \(-0.392507\pi\)
0.331316 + 0.943520i \(0.392507\pi\)
\(548\) −10.7313 −0.458418
\(549\) 12.0771 0.515438
\(550\) 2.85625 0.121791
\(551\) −4.10531 −0.174892
\(552\) 1.83642 0.0781634
\(553\) −15.6065 −0.663656
\(554\) 1.87730 0.0797588
\(555\) 24.7803 1.05187
\(556\) 19.8459 0.841652
\(557\) 5.34896 0.226643 0.113321 0.993558i \(-0.463851\pi\)
0.113321 + 0.993558i \(0.463851\pi\)
\(558\) −8.59340 −0.363788
\(559\) 21.9890 0.930036
\(560\) 2.12019 0.0895943
\(561\) 20.5121 0.866022
\(562\) −1.54137 −0.0650186
\(563\) 8.04695 0.339138 0.169569 0.985518i \(-0.445762\pi\)
0.169569 + 0.985518i \(0.445762\pi\)
\(564\) 8.05558 0.339201
\(565\) −17.3298 −0.729071
\(566\) −15.7354 −0.661409
\(567\) 1.00000 0.0419961
\(568\) −5.14641 −0.215939
\(569\) −13.5562 −0.568304 −0.284152 0.958779i \(-0.591712\pi\)
−0.284152 + 0.958779i \(0.591712\pi\)
\(570\) 1.12579 0.0471544
\(571\) 8.19030 0.342754 0.171377 0.985206i \(-0.445178\pi\)
0.171377 + 0.985206i \(0.445178\pi\)
\(572\) 19.3323 0.808325
\(573\) 1.00000 0.0417756
\(574\) 12.4762 0.520747
\(575\) 0.927035 0.0386600
\(576\) 1.00000 0.0416667
\(577\) −29.8019 −1.24067 −0.620335 0.784337i \(-0.713003\pi\)
−0.620335 + 0.784337i \(0.713003\pi\)
\(578\) −3.85762 −0.160456
\(579\) −0.0643161 −0.00267288
\(580\) 16.3921 0.680645
\(581\) −13.4142 −0.556515
\(582\) 16.3823 0.679068
\(583\) −29.7230 −1.23100
\(584\) −12.9220 −0.534718
\(585\) −7.24411 −0.299507
\(586\) −3.70401 −0.153011
\(587\) 2.15875 0.0891010 0.0445505 0.999007i \(-0.485814\pi\)
0.0445505 + 0.999007i \(0.485814\pi\)
\(588\) −1.00000 −0.0412393
\(589\) 4.56300 0.188015
\(590\) 12.6595 0.521184
\(591\) −6.31491 −0.259761
\(592\) −11.6878 −0.480366
\(593\) −14.4005 −0.591356 −0.295678 0.955288i \(-0.595546\pi\)
−0.295678 + 0.955288i \(0.595546\pi\)
\(594\) 5.65813 0.232156
\(595\) 7.68619 0.315103
\(596\) −12.2987 −0.503772
\(597\) −9.61618 −0.393564
\(598\) 6.27456 0.256586
\(599\) −8.37404 −0.342154 −0.171077 0.985258i \(-0.554725\pi\)
−0.171077 + 0.985258i \(0.554725\pi\)
\(600\) 0.504804 0.0206086
\(601\) 21.3849 0.872309 0.436155 0.899872i \(-0.356340\pi\)
0.436155 + 0.899872i \(0.356340\pi\)
\(602\) −6.43569 −0.262299
\(603\) −0.347898 −0.0141675
\(604\) 17.6647 0.718766
\(605\) 44.5546 1.81140
\(606\) 10.9489 0.444770
\(607\) −22.7477 −0.923301 −0.461651 0.887062i \(-0.652743\pi\)
−0.461651 + 0.887062i \(0.652743\pi\)
\(608\) −0.530988 −0.0215344
\(609\) −7.73144 −0.313294
\(610\) 25.6057 1.03674
\(611\) 27.5238 1.11349
\(612\) 3.62524 0.146542
\(613\) 8.38344 0.338604 0.169302 0.985564i \(-0.445849\pi\)
0.169302 + 0.985564i \(0.445849\pi\)
\(614\) −18.3234 −0.739472
\(615\) −26.4519 −1.06664
\(616\) −5.65813 −0.227973
\(617\) −30.0978 −1.21169 −0.605847 0.795581i \(-0.707166\pi\)
−0.605847 + 0.795581i \(0.707166\pi\)
\(618\) 9.79451 0.393993
\(619\) −18.8535 −0.757785 −0.378892 0.925441i \(-0.623695\pi\)
−0.378892 + 0.925441i \(0.623695\pi\)
\(620\) −18.2196 −0.731718
\(621\) 1.83642 0.0736931
\(622\) −19.4591 −0.780240
\(623\) 15.3853 0.616400
\(624\) 3.41673 0.136779
\(625\) −22.2212 −0.888846
\(626\) 7.82362 0.312695
\(627\) −3.00440 −0.119984
\(628\) −18.2820 −0.729532
\(629\) −42.3711 −1.68945
\(630\) 2.12019 0.0844703
\(631\) 4.80622 0.191332 0.0956662 0.995413i \(-0.469502\pi\)
0.0956662 + 0.995413i \(0.469502\pi\)
\(632\) −15.6065 −0.620793
\(633\) −16.0386 −0.637478
\(634\) −1.62203 −0.0644189
\(635\) −32.8457 −1.30344
\(636\) −5.25315 −0.208301
\(637\) −3.41673 −0.135376
\(638\) −43.7455 −1.73190
\(639\) −5.14641 −0.203589
\(640\) 2.12019 0.0838078
\(641\) −15.0629 −0.594950 −0.297475 0.954730i \(-0.596144\pi\)
−0.297475 + 0.954730i \(0.596144\pi\)
\(642\) 0.795378 0.0313911
\(643\) −48.3680 −1.90745 −0.953723 0.300686i \(-0.902784\pi\)
−0.953723 + 0.300686i \(0.902784\pi\)
\(644\) −1.83642 −0.0723652
\(645\) 13.6449 0.537266
\(646\) −1.92496 −0.0757366
\(647\) −8.55762 −0.336435 −0.168217 0.985750i \(-0.553801\pi\)
−0.168217 + 0.985750i \(0.553801\pi\)
\(648\) 1.00000 0.0392837
\(649\) −33.7843 −1.32615
\(650\) 1.72478 0.0676515
\(651\) 8.59340 0.336802
\(652\) 8.89726 0.348444
\(653\) 8.98741 0.351705 0.175852 0.984417i \(-0.443732\pi\)
0.175852 + 0.984417i \(0.443732\pi\)
\(654\) −14.3889 −0.562651
\(655\) −37.1522 −1.45166
\(656\) 12.4762 0.487115
\(657\) −12.9220 −0.504137
\(658\) −8.05558 −0.314039
\(659\) −21.4718 −0.836421 −0.418210 0.908350i \(-0.637343\pi\)
−0.418210 + 0.908350i \(0.637343\pi\)
\(660\) 11.9963 0.466955
\(661\) 31.8319 1.23812 0.619060 0.785344i \(-0.287514\pi\)
0.619060 + 0.785344i \(0.287514\pi\)
\(662\) 5.80972 0.225801
\(663\) 12.3865 0.481051
\(664\) −13.4142 −0.520572
\(665\) −1.12579 −0.0436565
\(666\) −11.6878 −0.452893
\(667\) −14.1982 −0.549757
\(668\) 19.7895 0.765680
\(669\) −19.9769 −0.772350
\(670\) −0.737609 −0.0284963
\(671\) −68.3338 −2.63800
\(672\) −1.00000 −0.0385758
\(673\) 9.52768 0.367265 0.183633 0.982995i \(-0.441214\pi\)
0.183633 + 0.982995i \(0.441214\pi\)
\(674\) 34.2577 1.31956
\(675\) 0.504804 0.0194299
\(676\) −1.32595 −0.0509979
\(677\) −1.47448 −0.0566690 −0.0283345 0.999598i \(-0.509020\pi\)
−0.0283345 + 0.999598i \(0.509020\pi\)
\(678\) 8.17372 0.313910
\(679\) −16.3823 −0.628695
\(680\) 7.68619 0.294752
\(681\) −3.10798 −0.119098
\(682\) 48.6226 1.86186
\(683\) −18.4990 −0.707843 −0.353921 0.935275i \(-0.615152\pi\)
−0.353921 + 0.935275i \(0.615152\pi\)
\(684\) −0.530988 −0.0203028
\(685\) −22.7523 −0.869322
\(686\) 1.00000 0.0381802
\(687\) −18.5609 −0.708141
\(688\) −6.43569 −0.245358
\(689\) −17.9486 −0.683788
\(690\) 3.89356 0.148225
\(691\) −40.3726 −1.53584 −0.767922 0.640543i \(-0.778709\pi\)
−0.767922 + 0.640543i \(0.778709\pi\)
\(692\) −11.2793 −0.428773
\(693\) −5.65813 −0.214935
\(694\) 0.654394 0.0248405
\(695\) 42.0770 1.59607
\(696\) −7.73144 −0.293060
\(697\) 45.2293 1.71318
\(698\) 20.2461 0.766326
\(699\) −7.66569 −0.289943
\(700\) −0.504804 −0.0190798
\(701\) 17.1461 0.647600 0.323800 0.946126i \(-0.395040\pi\)
0.323800 + 0.946126i \(0.395040\pi\)
\(702\) 3.41673 0.128956
\(703\) 6.20609 0.234067
\(704\) −5.65813 −0.213249
\(705\) 17.0793 0.643245
\(706\) −11.9714 −0.450549
\(707\) −10.9489 −0.411777
\(708\) −5.97094 −0.224402
\(709\) 28.5871 1.07361 0.536806 0.843706i \(-0.319631\pi\)
0.536806 + 0.843706i \(0.319631\pi\)
\(710\) −10.9114 −0.409496
\(711\) −15.6065 −0.585289
\(712\) 15.3853 0.576589
\(713\) 15.7811 0.591008
\(714\) −3.62524 −0.135671
\(715\) 40.9881 1.53287
\(716\) −3.64668 −0.136283
\(717\) 4.31275 0.161063
\(718\) 24.6520 0.920005
\(719\) 36.8258 1.37337 0.686685 0.726955i \(-0.259065\pi\)
0.686685 + 0.726955i \(0.259065\pi\)
\(720\) 2.12019 0.0790147
\(721\) −9.79451 −0.364767
\(722\) −18.7181 −0.696614
\(723\) 28.8232 1.07195
\(724\) −1.91630 −0.0712186
\(725\) −3.90287 −0.144949
\(726\) −21.0145 −0.779920
\(727\) −36.5052 −1.35390 −0.676951 0.736028i \(-0.736699\pi\)
−0.676951 + 0.736028i \(0.736699\pi\)
\(728\) −3.41673 −0.126632
\(729\) 1.00000 0.0370370
\(730\) −27.3972 −1.01401
\(731\) −23.3309 −0.862925
\(732\) −12.0771 −0.446382
\(733\) −4.78136 −0.176604 −0.0883019 0.996094i \(-0.528144\pi\)
−0.0883019 + 0.996094i \(0.528144\pi\)
\(734\) −25.7416 −0.950140
\(735\) −2.12019 −0.0782043
\(736\) −1.83642 −0.0676915
\(737\) 1.96845 0.0725089
\(738\) 12.4762 0.459256
\(739\) 2.78479 0.102440 0.0512201 0.998687i \(-0.483689\pi\)
0.0512201 + 0.998687i \(0.483689\pi\)
\(740\) −24.7803 −0.910943
\(741\) −1.81424 −0.0666479
\(742\) 5.25315 0.192849
\(743\) −19.0883 −0.700282 −0.350141 0.936697i \(-0.613866\pi\)
−0.350141 + 0.936697i \(0.613866\pi\)
\(744\) 8.59340 0.315049
\(745\) −26.0755 −0.955331
\(746\) 14.3450 0.525207
\(747\) −13.4142 −0.490800
\(748\) −20.5121 −0.749997
\(749\) −0.795378 −0.0290625
\(750\) 11.6712 0.426173
\(751\) −11.6462 −0.424976 −0.212488 0.977164i \(-0.568157\pi\)
−0.212488 + 0.977164i \(0.568157\pi\)
\(752\) −8.05558 −0.293757
\(753\) −8.56547 −0.312143
\(754\) −26.4163 −0.962024
\(755\) 37.4524 1.36303
\(756\) −1.00000 −0.0363696
\(757\) 4.55068 0.165397 0.0826986 0.996575i \(-0.473646\pi\)
0.0826986 + 0.996575i \(0.473646\pi\)
\(758\) −6.54180 −0.237609
\(759\) −10.3907 −0.377159
\(760\) −1.12579 −0.0408369
\(761\) 51.8367 1.87908 0.939539 0.342443i \(-0.111254\pi\)
0.939539 + 0.342443i \(0.111254\pi\)
\(762\) 15.4919 0.561212
\(763\) 14.3889 0.520914
\(764\) −1.00000 −0.0361787
\(765\) 7.68619 0.277895
\(766\) −1.70210 −0.0614995
\(767\) −20.4011 −0.736641
\(768\) −1.00000 −0.0360844
\(769\) −35.7537 −1.28931 −0.644655 0.764474i \(-0.722999\pi\)
−0.644655 + 0.764474i \(0.722999\pi\)
\(770\) −11.9963 −0.432317
\(771\) 26.0282 0.937381
\(772\) 0.0643161 0.00231479
\(773\) −2.01138 −0.0723442 −0.0361721 0.999346i \(-0.511516\pi\)
−0.0361721 + 0.999346i \(0.511516\pi\)
\(774\) −6.43569 −0.231326
\(775\) 4.33799 0.155825
\(776\) −16.3823 −0.588090
\(777\) 11.6878 0.419298
\(778\) −13.5361 −0.485292
\(779\) −6.62473 −0.237356
\(780\) 7.24411 0.259381
\(781\) 29.1191 1.04196
\(782\) −6.65748 −0.238071
\(783\) −7.73144 −0.276299
\(784\) 1.00000 0.0357143
\(785\) −38.7613 −1.38345
\(786\) 17.5231 0.625028
\(787\) 7.92134 0.282365 0.141183 0.989984i \(-0.454910\pi\)
0.141183 + 0.989984i \(0.454910\pi\)
\(788\) 6.31491 0.224959
\(789\) 6.37183 0.226843
\(790\) −33.0887 −1.17724
\(791\) −8.17372 −0.290624
\(792\) −5.65813 −0.201053
\(793\) −41.2642 −1.46533
\(794\) −8.64242 −0.306708
\(795\) −11.1377 −0.395013
\(796\) 9.61618 0.340836
\(797\) 10.9960 0.389499 0.194749 0.980853i \(-0.437611\pi\)
0.194749 + 0.980853i \(0.437611\pi\)
\(798\) 0.530988 0.0187968
\(799\) −29.2034 −1.03314
\(800\) −0.504804 −0.0178475
\(801\) 15.3853 0.543613
\(802\) 6.08530 0.214879
\(803\) 73.1146 2.58016
\(804\) 0.347898 0.0122694
\(805\) −3.89356 −0.137230
\(806\) 29.3614 1.03421
\(807\) 16.0331 0.564391
\(808\) −10.9489 −0.385182
\(809\) 49.8017 1.75093 0.875467 0.483278i \(-0.160554\pi\)
0.875467 + 0.483278i \(0.160554\pi\)
\(810\) 2.12019 0.0744958
\(811\) 10.8964 0.382626 0.191313 0.981529i \(-0.438725\pi\)
0.191313 + 0.981529i \(0.438725\pi\)
\(812\) 7.73144 0.271321
\(813\) −22.3891 −0.785219
\(814\) 66.1311 2.31790
\(815\) 18.8639 0.660773
\(816\) −3.62524 −0.126909
\(817\) 3.41727 0.119555
\(818\) −28.2163 −0.986558
\(819\) −3.41673 −0.119390
\(820\) 26.4519 0.923742
\(821\) 45.3251 1.58186 0.790929 0.611908i \(-0.209598\pi\)
0.790929 + 0.611908i \(0.209598\pi\)
\(822\) 10.7313 0.374297
\(823\) 18.2341 0.635599 0.317800 0.948158i \(-0.397056\pi\)
0.317800 + 0.948158i \(0.397056\pi\)
\(824\) −9.79451 −0.341208
\(825\) −2.85625 −0.0994419
\(826\) 5.97094 0.207755
\(827\) 57.3076 1.99278 0.996391 0.0848816i \(-0.0270512\pi\)
0.996391 + 0.0848816i \(0.0270512\pi\)
\(828\) −1.83642 −0.0638201
\(829\) −2.15392 −0.0748087 −0.0374043 0.999300i \(-0.511909\pi\)
−0.0374043 + 0.999300i \(0.511909\pi\)
\(830\) −28.4406 −0.987189
\(831\) −1.87730 −0.0651228
\(832\) −3.41673 −0.118454
\(833\) 3.62524 0.125607
\(834\) −19.8459 −0.687206
\(835\) 41.9575 1.45200
\(836\) 3.00440 0.103909
\(837\) 8.59340 0.297032
\(838\) 28.2638 0.976358
\(839\) 15.7268 0.542948 0.271474 0.962446i \(-0.412489\pi\)
0.271474 + 0.962446i \(0.412489\pi\)
\(840\) −2.12019 −0.0731534
\(841\) 30.7752 1.06121
\(842\) 8.29174 0.285752
\(843\) 1.54137 0.0530875
\(844\) 16.0386 0.552072
\(845\) −2.81125 −0.0967101
\(846\) −8.05558 −0.276957
\(847\) 21.0145 0.722065
\(848\) 5.25315 0.180394
\(849\) 15.7354 0.540038
\(850\) −1.83004 −0.0627698
\(851\) 21.4638 0.735768
\(852\) 5.14641 0.176313
\(853\) −1.58582 −0.0542976 −0.0271488 0.999631i \(-0.508643\pi\)
−0.0271488 + 0.999631i \(0.508643\pi\)
\(854\) 12.0771 0.413270
\(855\) −1.12579 −0.0385014
\(856\) −0.795378 −0.0271855
\(857\) −9.13838 −0.312161 −0.156081 0.987744i \(-0.549886\pi\)
−0.156081 + 0.987744i \(0.549886\pi\)
\(858\) −19.3323 −0.659994
\(859\) 12.4840 0.425948 0.212974 0.977058i \(-0.431685\pi\)
0.212974 + 0.977058i \(0.431685\pi\)
\(860\) −13.6449 −0.465286
\(861\) −12.4762 −0.425189
\(862\) −6.44130 −0.219391
\(863\) −30.9644 −1.05404 −0.527020 0.849853i \(-0.676691\pi\)
−0.527020 + 0.849853i \(0.676691\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −23.9141 −0.813105
\(866\) 25.3953 0.862969
\(867\) 3.85762 0.131012
\(868\) −8.59340 −0.291679
\(869\) 88.3036 2.99550
\(870\) −16.3921 −0.555745
\(871\) 1.18867 0.0402767
\(872\) 14.3889 0.487270
\(873\) −16.3823 −0.554457
\(874\) 0.975119 0.0329839
\(875\) −11.6712 −0.394559
\(876\) 12.9220 0.436595
\(877\) 13.4456 0.454025 0.227012 0.973892i \(-0.427104\pi\)
0.227012 + 0.973892i \(0.427104\pi\)
\(878\) −14.9239 −0.503657
\(879\) 3.70401 0.124933
\(880\) −11.9963 −0.404395
\(881\) −51.6111 −1.73882 −0.869412 0.494089i \(-0.835502\pi\)
−0.869412 + 0.494089i \(0.835502\pi\)
\(882\) 1.00000 0.0336718
\(883\) 20.3553 0.685009 0.342504 0.939516i \(-0.388725\pi\)
0.342504 + 0.939516i \(0.388725\pi\)
\(884\) −12.3865 −0.416602
\(885\) −12.6595 −0.425545
\(886\) 25.1056 0.843440
\(887\) −11.6889 −0.392475 −0.196238 0.980556i \(-0.562872\pi\)
−0.196238 + 0.980556i \(0.562872\pi\)
\(888\) 11.6878 0.392217
\(889\) −15.4919 −0.519581
\(890\) 32.6198 1.09342
\(891\) −5.65813 −0.189555
\(892\) 19.9769 0.668875
\(893\) 4.27742 0.143138
\(894\) 12.2987 0.411328
\(895\) −7.73164 −0.258440
\(896\) 1.00000 0.0334077
\(897\) −6.27456 −0.209502
\(898\) −37.7012 −1.25811
\(899\) −66.4394 −2.21588
\(900\) −0.504804 −0.0168268
\(901\) 19.0440 0.634446
\(902\) −70.5921 −2.35046
\(903\) 6.43569 0.214166
\(904\) −8.17372 −0.271854
\(905\) −4.06291 −0.135056
\(906\) −17.6647 −0.586870
\(907\) 40.3018 1.33820 0.669100 0.743172i \(-0.266680\pi\)
0.669100 + 0.743172i \(0.266680\pi\)
\(908\) 3.10798 0.103142
\(909\) −10.9489 −0.363153
\(910\) −7.24411 −0.240140
\(911\) −44.8871 −1.48718 −0.743588 0.668638i \(-0.766877\pi\)
−0.743588 + 0.668638i \(0.766877\pi\)
\(912\) 0.530988 0.0175828
\(913\) 75.8993 2.51190
\(914\) 3.99000 0.131977
\(915\) −25.6057 −0.846499
\(916\) 18.5609 0.613268
\(917\) −17.5231 −0.578663
\(918\) −3.62524 −0.119651
\(919\) −40.0817 −1.32217 −0.661087 0.750309i \(-0.729905\pi\)
−0.661087 + 0.750309i \(0.729905\pi\)
\(920\) −3.89356 −0.128367
\(921\) 18.3234 0.603776
\(922\) 30.6319 1.00881
\(923\) 17.5839 0.578781
\(924\) 5.65813 0.186139
\(925\) 5.90006 0.193993
\(926\) 17.6524 0.580094
\(927\) −9.79451 −0.321694
\(928\) 7.73144 0.253797
\(929\) 34.4310 1.12964 0.564822 0.825213i \(-0.308945\pi\)
0.564822 + 0.825213i \(0.308945\pi\)
\(930\) 18.2196 0.597445
\(931\) −0.530988 −0.0174024
\(932\) 7.66569 0.251098
\(933\) 19.4591 0.637063
\(934\) −11.4024 −0.373098
\(935\) −43.4895 −1.42226
\(936\) −3.41673 −0.111679
\(937\) −29.4960 −0.963593 −0.481797 0.876283i \(-0.660016\pi\)
−0.481797 + 0.876283i \(0.660016\pi\)
\(938\) −0.347898 −0.0113593
\(939\) −7.82362 −0.255314
\(940\) −17.0793 −0.557067
\(941\) 35.0316 1.14200 0.570998 0.820951i \(-0.306556\pi\)
0.570998 + 0.820951i \(0.306556\pi\)
\(942\) 18.2820 0.595661
\(943\) −22.9116 −0.746105
\(944\) 5.97094 0.194337
\(945\) −2.12019 −0.0689697
\(946\) 36.4140 1.18392
\(947\) 3.67687 0.119482 0.0597410 0.998214i \(-0.480973\pi\)
0.0597410 + 0.998214i \(0.480973\pi\)
\(948\) 15.6065 0.506876
\(949\) 44.1512 1.43321
\(950\) 0.268045 0.00869654
\(951\) 1.62203 0.0525978
\(952\) 3.62524 0.117495
\(953\) −49.4895 −1.60312 −0.801561 0.597913i \(-0.795997\pi\)
−0.801561 + 0.597913i \(0.795997\pi\)
\(954\) 5.25315 0.170077
\(955\) −2.12019 −0.0686077
\(956\) −4.31275 −0.139484
\(957\) 43.7455 1.41409
\(958\) −9.83500 −0.317755
\(959\) −10.7313 −0.346531
\(960\) −2.12019 −0.0684288
\(961\) 42.8466 1.38215
\(962\) 39.9341 1.28753
\(963\) −0.795378 −0.0256307
\(964\) −28.8232 −0.928334
\(965\) 0.136362 0.00438965
\(966\) 1.83642 0.0590859
\(967\) 45.2145 1.45400 0.727001 0.686637i \(-0.240914\pi\)
0.727001 + 0.686637i \(0.240914\pi\)
\(968\) 21.0145 0.675430
\(969\) 1.92496 0.0618387
\(970\) −34.7336 −1.11523
\(971\) 4.84691 0.155545 0.0777723 0.996971i \(-0.475219\pi\)
0.0777723 + 0.996971i \(0.475219\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 19.8459 0.636229
\(974\) 17.9436 0.574950
\(975\) −1.72478 −0.0552372
\(976\) 12.0771 0.386578
\(977\) 7.92023 0.253391 0.126695 0.991942i \(-0.459563\pi\)
0.126695 + 0.991942i \(0.459563\pi\)
\(978\) −8.89726 −0.284503
\(979\) −87.0521 −2.78220
\(980\) 2.12019 0.0677269
\(981\) 14.3889 0.459403
\(982\) 4.97688 0.158819
\(983\) −56.4428 −1.80025 −0.900123 0.435637i \(-0.856523\pi\)
−0.900123 + 0.435637i \(0.856523\pi\)
\(984\) −12.4762 −0.397727
\(985\) 13.3888 0.426603
\(986\) 28.0284 0.892605
\(987\) 8.05558 0.256412
\(988\) 1.81424 0.0577188
\(989\) 11.8186 0.375811
\(990\) −11.9963 −0.381267
\(991\) −19.0966 −0.606624 −0.303312 0.952891i \(-0.598093\pi\)
−0.303312 + 0.952891i \(0.598093\pi\)
\(992\) −8.59340 −0.272841
\(993\) −5.80972 −0.184366
\(994\) −5.14641 −0.163234
\(995\) 20.3881 0.646346
\(996\) 13.4142 0.425045
\(997\) −6.66164 −0.210976 −0.105488 0.994421i \(-0.533640\pi\)
−0.105488 + 0.994421i \(0.533640\pi\)
\(998\) −14.3583 −0.454503
\(999\) 11.6878 0.369786
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 8022.2.a.p.1.8 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8022.2.a.p.1.8 9 1.1 even 1 trivial