Properties

Label 8022.2.a.q.1.6
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: \( x^{9} - 4x^{8} - 11x^{7} + 36x^{6} + 50x^{5} - 70x^{4} - 73x^{3} + 14x^{2} + 22x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(-0.411372\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.51454 q^{5} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.51454 q^{5} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.51454 q^{10} -1.68070 q^{11} -1.00000 q^{12} -1.35713 q^{13} -1.00000 q^{14} -1.51454 q^{15} +1.00000 q^{16} -4.50273 q^{17} +1.00000 q^{18} +3.92466 q^{19} +1.51454 q^{20} +1.00000 q^{21} -1.68070 q^{22} +3.51870 q^{23} -1.00000 q^{24} -2.70618 q^{25} -1.35713 q^{26} -1.00000 q^{27} -1.00000 q^{28} -8.90561 q^{29} -1.51454 q^{30} +4.29806 q^{31} +1.00000 q^{32} +1.68070 q^{33} -4.50273 q^{34} -1.51454 q^{35} +1.00000 q^{36} +8.83806 q^{37} +3.92466 q^{38} +1.35713 q^{39} +1.51454 q^{40} -4.69402 q^{41} +1.00000 q^{42} -2.40129 q^{43} -1.68070 q^{44} +1.51454 q^{45} +3.51870 q^{46} -2.64404 q^{47} -1.00000 q^{48} +1.00000 q^{49} -2.70618 q^{50} +4.50273 q^{51} -1.35713 q^{52} -5.94440 q^{53} -1.00000 q^{54} -2.54549 q^{55} -1.00000 q^{56} -3.92466 q^{57} -8.90561 q^{58} -9.59202 q^{59} -1.51454 q^{60} +4.89730 q^{61} +4.29806 q^{62} -1.00000 q^{63} +1.00000 q^{64} -2.05542 q^{65} +1.68070 q^{66} +8.20435 q^{67} -4.50273 q^{68} -3.51870 q^{69} -1.51454 q^{70} -6.75595 q^{71} +1.00000 q^{72} -4.24487 q^{73} +8.83806 q^{74} +2.70618 q^{75} +3.92466 q^{76} +1.68070 q^{77} +1.35713 q^{78} -12.4406 q^{79} +1.51454 q^{80} +1.00000 q^{81} -4.69402 q^{82} -4.27560 q^{83} +1.00000 q^{84} -6.81955 q^{85} -2.40129 q^{86} +8.90561 q^{87} -1.68070 q^{88} +6.68911 q^{89} +1.51454 q^{90} +1.35713 q^{91} +3.51870 q^{92} -4.29806 q^{93} -2.64404 q^{94} +5.94405 q^{95} -1.00000 q^{96} +3.02861 q^{97} +1.00000 q^{98} -1.68070 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 9 q^{2} - 9 q^{3} + 9 q^{4} + 6 q^{5} - 9 q^{6} - 9 q^{7} + 9 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + 9 q^{2} - 9 q^{3} + 9 q^{4} + 6 q^{5} - 9 q^{6} - 9 q^{7} + 9 q^{8} + 9 q^{9} + 6 q^{10} - 7 q^{11} - 9 q^{12} - q^{13} - 9 q^{14} - 6 q^{15} + 9 q^{16} - 10 q^{17} + 9 q^{18} + 6 q^{20} + 9 q^{21} - 7 q^{22} - 19 q^{23} - 9 q^{24} + 5 q^{25} - q^{26} - 9 q^{27} - 9 q^{28} - 21 q^{29} - 6 q^{30} + 6 q^{31} + 9 q^{32} + 7 q^{33} - 10 q^{34} - 6 q^{35} + 9 q^{36} - 14 q^{37} + q^{39} + 6 q^{40} + 12 q^{41} + 9 q^{42} - 17 q^{43} - 7 q^{44} + 6 q^{45} - 19 q^{46} - 11 q^{47} - 9 q^{48} + 9 q^{49} + 5 q^{50} + 10 q^{51} - q^{52} - 12 q^{53} - 9 q^{54} - 21 q^{55} - 9 q^{56} - 21 q^{58} - 8 q^{59} - 6 q^{60} + q^{61} + 6 q^{62} - 9 q^{63} + 9 q^{64} - 8 q^{65} + 7 q^{66} - 31 q^{67} - 10 q^{68} + 19 q^{69} - 6 q^{70} - 41 q^{71} + 9 q^{72} - q^{73} - 14 q^{74} - 5 q^{75} + 7 q^{77} + q^{78} - 17 q^{79} + 6 q^{80} + 9 q^{81} + 12 q^{82} - 36 q^{83} + 9 q^{84} - 30 q^{85} - 17 q^{86} + 21 q^{87} - 7 q^{88} - 17 q^{89} + 6 q^{90} + q^{91} - 19 q^{92} - 6 q^{93} - 11 q^{94} - 50 q^{95} - 9 q^{96} - 14 q^{97} + 9 q^{98} - 7 q^{99}+O(q^{100}) \) 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
\(5\) 1.51454 0.677322 0.338661 0.940909i \(-0.390026\pi\)
0.338661 + 0.940909i \(0.390026\pi\)
\(6\) −1.00000 −0.408248
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 1.51454 0.478939
\(11\) −1.68070 −0.506751 −0.253375 0.967368i \(-0.581541\pi\)
−0.253375 + 0.967368i \(0.581541\pi\)
\(12\) −1.00000 −0.288675
\(13\) −1.35713 −0.376399 −0.188200 0.982131i \(-0.560265\pi\)
−0.188200 + 0.982131i \(0.560265\pi\)
\(14\) −1.00000 −0.267261
\(15\) −1.51454 −0.391052
\(16\) 1.00000 0.250000
\(17\) −4.50273 −1.09207 −0.546036 0.837761i \(-0.683864\pi\)
−0.546036 + 0.837761i \(0.683864\pi\)
\(18\) 1.00000 0.235702
\(19\) 3.92466 0.900379 0.450190 0.892933i \(-0.351356\pi\)
0.450190 + 0.892933i \(0.351356\pi\)
\(20\) 1.51454 0.338661
\(21\) 1.00000 0.218218
\(22\) −1.68070 −0.358327
\(23\) 3.51870 0.733699 0.366850 0.930280i \(-0.380436\pi\)
0.366850 + 0.930280i \(0.380436\pi\)
\(24\) −1.00000 −0.204124
\(25\) −2.70618 −0.541235
\(26\) −1.35713 −0.266154
\(27\) −1.00000 −0.192450
\(28\) −1.00000 −0.188982
\(29\) −8.90561 −1.65373 −0.826865 0.562401i \(-0.809878\pi\)
−0.826865 + 0.562401i \(0.809878\pi\)
\(30\) −1.51454 −0.276515
\(31\) 4.29806 0.771955 0.385977 0.922508i \(-0.373864\pi\)
0.385977 + 0.922508i \(0.373864\pi\)
\(32\) 1.00000 0.176777
\(33\) 1.68070 0.292573
\(34\) −4.50273 −0.772212
\(35\) −1.51454 −0.256004
\(36\) 1.00000 0.166667
\(37\) 8.83806 1.45297 0.726484 0.687183i \(-0.241153\pi\)
0.726484 + 0.687183i \(0.241153\pi\)
\(38\) 3.92466 0.636664
\(39\) 1.35713 0.217314
\(40\) 1.51454 0.239469
\(41\) −4.69402 −0.733083 −0.366541 0.930402i \(-0.619458\pi\)
−0.366541 + 0.930402i \(0.619458\pi\)
\(42\) 1.00000 0.154303
\(43\) −2.40129 −0.366194 −0.183097 0.983095i \(-0.558612\pi\)
−0.183097 + 0.983095i \(0.558612\pi\)
\(44\) −1.68070 −0.253375
\(45\) 1.51454 0.225774
\(46\) 3.51870 0.518804
\(47\) −2.64404 −0.385672 −0.192836 0.981231i \(-0.561769\pi\)
−0.192836 + 0.981231i \(0.561769\pi\)
\(48\) −1.00000 −0.144338
\(49\) 1.00000 0.142857
\(50\) −2.70618 −0.382711
\(51\) 4.50273 0.630508
\(52\) −1.35713 −0.188200
\(53\) −5.94440 −0.816526 −0.408263 0.912864i \(-0.633865\pi\)
−0.408263 + 0.912864i \(0.633865\pi\)
\(54\) −1.00000 −0.136083
\(55\) −2.54549 −0.343233
\(56\) −1.00000 −0.133631
\(57\) −3.92466 −0.519834
\(58\) −8.90561 −1.16936
\(59\) −9.59202 −1.24877 −0.624387 0.781115i \(-0.714651\pi\)
−0.624387 + 0.781115i \(0.714651\pi\)
\(60\) −1.51454 −0.195526
\(61\) 4.89730 0.627035 0.313518 0.949582i \(-0.398492\pi\)
0.313518 + 0.949582i \(0.398492\pi\)
\(62\) 4.29806 0.545854
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) −2.05542 −0.254943
\(66\) 1.68070 0.206880
\(67\) 8.20435 1.00232 0.501160 0.865354i \(-0.332907\pi\)
0.501160 + 0.865354i \(0.332907\pi\)
\(68\) −4.50273 −0.546036
\(69\) −3.51870 −0.423602
\(70\) −1.51454 −0.181022
\(71\) −6.75595 −0.801784 −0.400892 0.916125i \(-0.631300\pi\)
−0.400892 + 0.916125i \(0.631300\pi\)
\(72\) 1.00000 0.117851
\(73\) −4.24487 −0.496825 −0.248412 0.968654i \(-0.579909\pi\)
−0.248412 + 0.968654i \(0.579909\pi\)
\(74\) 8.83806 1.02740
\(75\) 2.70618 0.312482
\(76\) 3.92466 0.450190
\(77\) 1.68070 0.191534
\(78\) 1.35713 0.153664
\(79\) −12.4406 −1.39968 −0.699841 0.714299i \(-0.746746\pi\)
−0.699841 + 0.714299i \(0.746746\pi\)
\(80\) 1.51454 0.169330
\(81\) 1.00000 0.111111
\(82\) −4.69402 −0.518368
\(83\) −4.27560 −0.469308 −0.234654 0.972079i \(-0.575396\pi\)
−0.234654 + 0.972079i \(0.575396\pi\)
\(84\) 1.00000 0.109109
\(85\) −6.81955 −0.739684
\(86\) −2.40129 −0.258938
\(87\) 8.90561 0.954781
\(88\) −1.68070 −0.179163
\(89\) 6.68911 0.709044 0.354522 0.935048i \(-0.384644\pi\)
0.354522 + 0.935048i \(0.384644\pi\)
\(90\) 1.51454 0.159646
\(91\) 1.35713 0.142265
\(92\) 3.51870 0.366850
\(93\) −4.29806 −0.445688
\(94\) −2.64404 −0.272712
\(95\) 5.94405 0.609846
\(96\) −1.00000 −0.102062
\(97\) 3.02861 0.307509 0.153754 0.988109i \(-0.450864\pi\)
0.153754 + 0.988109i \(0.450864\pi\)
\(98\) 1.00000 0.101015
\(99\) −1.68070 −0.168917
\(100\) −2.70618 −0.270618
\(101\) 12.0400 1.19803 0.599013 0.800739i \(-0.295560\pi\)
0.599013 + 0.800739i \(0.295560\pi\)
\(102\) 4.50273 0.445837
\(103\) −7.62868 −0.751676 −0.375838 0.926685i \(-0.622645\pi\)
−0.375838 + 0.926685i \(0.622645\pi\)
\(104\) −1.35713 −0.133077
\(105\) 1.51454 0.147804
\(106\) −5.94440 −0.577371
\(107\) −8.48436 −0.820214 −0.410107 0.912037i \(-0.634509\pi\)
−0.410107 + 0.912037i \(0.634509\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 0.166999 0.0159956 0.00799782 0.999968i \(-0.497454\pi\)
0.00799782 + 0.999968i \(0.497454\pi\)
\(110\) −2.54549 −0.242703
\(111\) −8.83806 −0.838871
\(112\) −1.00000 −0.0944911
\(113\) −10.7808 −1.01417 −0.507086 0.861895i \(-0.669277\pi\)
−0.507086 + 0.861895i \(0.669277\pi\)
\(114\) −3.92466 −0.367578
\(115\) 5.32920 0.496950
\(116\) −8.90561 −0.826865
\(117\) −1.35713 −0.125466
\(118\) −9.59202 −0.883017
\(119\) 4.50273 0.412765
\(120\) −1.51454 −0.138258
\(121\) −8.17524 −0.743204
\(122\) 4.89730 0.443381
\(123\) 4.69402 0.423245
\(124\) 4.29806 0.385977
\(125\) −11.6713 −1.04391
\(126\) −1.00000 −0.0890871
\(127\) −4.94951 −0.439198 −0.219599 0.975590i \(-0.570475\pi\)
−0.219599 + 0.975590i \(0.570475\pi\)
\(128\) 1.00000 0.0883883
\(129\) 2.40129 0.211422
\(130\) −2.05542 −0.180272
\(131\) −12.5335 −1.09505 −0.547527 0.836788i \(-0.684431\pi\)
−0.547527 + 0.836788i \(0.684431\pi\)
\(132\) 1.68070 0.146286
\(133\) −3.92466 −0.340311
\(134\) 8.20435 0.708748
\(135\) −1.51454 −0.130351
\(136\) −4.50273 −0.386106
\(137\) 19.1587 1.63684 0.818419 0.574622i \(-0.194851\pi\)
0.818419 + 0.574622i \(0.194851\pi\)
\(138\) −3.51870 −0.299531
\(139\) −10.0578 −0.853090 −0.426545 0.904466i \(-0.640269\pi\)
−0.426545 + 0.904466i \(0.640269\pi\)
\(140\) −1.51454 −0.128002
\(141\) 2.64404 0.222668
\(142\) −6.75595 −0.566947
\(143\) 2.28092 0.190740
\(144\) 1.00000 0.0833333
\(145\) −13.4879 −1.12011
\(146\) −4.24487 −0.351308
\(147\) −1.00000 −0.0824786
\(148\) 8.83806 0.726484
\(149\) −17.9407 −1.46976 −0.734881 0.678196i \(-0.762762\pi\)
−0.734881 + 0.678196i \(0.762762\pi\)
\(150\) 2.70618 0.220958
\(151\) −10.3551 −0.842687 −0.421344 0.906901i \(-0.638441\pi\)
−0.421344 + 0.906901i \(0.638441\pi\)
\(152\) 3.92466 0.318332
\(153\) −4.50273 −0.364024
\(154\) 1.68070 0.135435
\(155\) 6.50957 0.522862
\(156\) 1.35713 0.108657
\(157\) −0.575064 −0.0458951 −0.0229476 0.999737i \(-0.507305\pi\)
−0.0229476 + 0.999737i \(0.507305\pi\)
\(158\) −12.4406 −0.989725
\(159\) 5.94440 0.471421
\(160\) 1.51454 0.119735
\(161\) −3.51870 −0.277312
\(162\) 1.00000 0.0785674
\(163\) 5.54296 0.434158 0.217079 0.976154i \(-0.430347\pi\)
0.217079 + 0.976154i \(0.430347\pi\)
\(164\) −4.69402 −0.366541
\(165\) 2.54549 0.198166
\(166\) −4.27560 −0.331851
\(167\) 18.4617 1.42861 0.714306 0.699834i \(-0.246742\pi\)
0.714306 + 0.699834i \(0.246742\pi\)
\(168\) 1.00000 0.0771517
\(169\) −11.1582 −0.858324
\(170\) −6.81955 −0.523036
\(171\) 3.92466 0.300126
\(172\) −2.40129 −0.183097
\(173\) 0.979310 0.0744556 0.0372278 0.999307i \(-0.488147\pi\)
0.0372278 + 0.999307i \(0.488147\pi\)
\(174\) 8.90561 0.675132
\(175\) 2.70618 0.204568
\(176\) −1.68070 −0.126688
\(177\) 9.59202 0.720980
\(178\) 6.68911 0.501370
\(179\) −12.0842 −0.903212 −0.451606 0.892217i \(-0.649149\pi\)
−0.451606 + 0.892217i \(0.649149\pi\)
\(180\) 1.51454 0.112887
\(181\) −16.0985 −1.19659 −0.598297 0.801275i \(-0.704156\pi\)
−0.598297 + 0.801275i \(0.704156\pi\)
\(182\) 1.35713 0.100597
\(183\) −4.89730 −0.362019
\(184\) 3.51870 0.259402
\(185\) 13.3856 0.984126
\(186\) −4.29806 −0.315149
\(187\) 7.56775 0.553409
\(188\) −2.64404 −0.192836
\(189\) 1.00000 0.0727393
\(190\) 5.94405 0.431226
\(191\) 1.00000 0.0723575
\(192\) −1.00000 −0.0721688
\(193\) 8.73233 0.628567 0.314283 0.949329i \(-0.398236\pi\)
0.314283 + 0.949329i \(0.398236\pi\)
\(194\) 3.02861 0.217441
\(195\) 2.05542 0.147192
\(196\) 1.00000 0.0714286
\(197\) −9.75048 −0.694693 −0.347346 0.937737i \(-0.612917\pi\)
−0.347346 + 0.937737i \(0.612917\pi\)
\(198\) −1.68070 −0.119442
\(199\) 14.2985 1.01360 0.506798 0.862065i \(-0.330829\pi\)
0.506798 + 0.862065i \(0.330829\pi\)
\(200\) −2.70618 −0.191356
\(201\) −8.20435 −0.578690
\(202\) 12.0400 0.847132
\(203\) 8.90561 0.625051
\(204\) 4.50273 0.315254
\(205\) −7.10927 −0.496533
\(206\) −7.62868 −0.531515
\(207\) 3.51870 0.244566
\(208\) −1.35713 −0.0940998
\(209\) −6.59619 −0.456268
\(210\) 1.51454 0.104513
\(211\) 6.00269 0.413242 0.206621 0.978421i \(-0.433753\pi\)
0.206621 + 0.978421i \(0.433753\pi\)
\(212\) −5.94440 −0.408263
\(213\) 6.75595 0.462910
\(214\) −8.48436 −0.579979
\(215\) −3.63685 −0.248031
\(216\) −1.00000 −0.0680414
\(217\) −4.29806 −0.291771
\(218\) 0.166999 0.0113106
\(219\) 4.24487 0.286842
\(220\) −2.54549 −0.171617
\(221\) 6.11077 0.411055
\(222\) −8.83806 −0.593172
\(223\) −16.4263 −1.09999 −0.549993 0.835169i \(-0.685370\pi\)
−0.549993 + 0.835169i \(0.685370\pi\)
\(224\) −1.00000 −0.0668153
\(225\) −2.70618 −0.180412
\(226\) −10.7808 −0.717128
\(227\) −18.4802 −1.22657 −0.613287 0.789860i \(-0.710153\pi\)
−0.613287 + 0.789860i \(0.710153\pi\)
\(228\) −3.92466 −0.259917
\(229\) 17.9688 1.18741 0.593705 0.804683i \(-0.297665\pi\)
0.593705 + 0.804683i \(0.297665\pi\)
\(230\) 5.32920 0.351397
\(231\) −1.68070 −0.110582
\(232\) −8.90561 −0.584682
\(233\) 29.5440 1.93549 0.967746 0.251926i \(-0.0810640\pi\)
0.967746 + 0.251926i \(0.0810640\pi\)
\(234\) −1.35713 −0.0887181
\(235\) −4.00449 −0.261224
\(236\) −9.59202 −0.624387
\(237\) 12.4406 0.808107
\(238\) 4.50273 0.291869
\(239\) −8.96693 −0.580023 −0.290011 0.957023i \(-0.593659\pi\)
−0.290011 + 0.957023i \(0.593659\pi\)
\(240\) −1.51454 −0.0977630
\(241\) 1.20612 0.0776933 0.0388467 0.999245i \(-0.487632\pi\)
0.0388467 + 0.999245i \(0.487632\pi\)
\(242\) −8.17524 −0.525524
\(243\) −1.00000 −0.0641500
\(244\) 4.89730 0.313518
\(245\) 1.51454 0.0967602
\(246\) 4.69402 0.299280
\(247\) −5.32626 −0.338902
\(248\) 4.29806 0.272927
\(249\) 4.27560 0.270955
\(250\) −11.6713 −0.738157
\(251\) 13.6302 0.860334 0.430167 0.902749i \(-0.358455\pi\)
0.430167 + 0.902749i \(0.358455\pi\)
\(252\) −1.00000 −0.0629941
\(253\) −5.91388 −0.371803
\(254\) −4.94951 −0.310560
\(255\) 6.81955 0.427057
\(256\) 1.00000 0.0625000
\(257\) −5.82640 −0.363441 −0.181721 0.983350i \(-0.558167\pi\)
−0.181721 + 0.983350i \(0.558167\pi\)
\(258\) 2.40129 0.149498
\(259\) −8.83806 −0.549170
\(260\) −2.05542 −0.127472
\(261\) −8.90561 −0.551243
\(262\) −12.5335 −0.774321
\(263\) −31.0461 −1.91438 −0.957191 0.289458i \(-0.906525\pi\)
−0.957191 + 0.289458i \(0.906525\pi\)
\(264\) 1.68070 0.103440
\(265\) −9.00301 −0.553051
\(266\) −3.92466 −0.240636
\(267\) −6.68911 −0.409367
\(268\) 8.20435 0.501160
\(269\) 0.696142 0.0424445 0.0212223 0.999775i \(-0.493244\pi\)
0.0212223 + 0.999775i \(0.493244\pi\)
\(270\) −1.51454 −0.0921718
\(271\) 31.0053 1.88344 0.941720 0.336398i \(-0.109209\pi\)
0.941720 + 0.336398i \(0.109209\pi\)
\(272\) −4.50273 −0.273018
\(273\) −1.35713 −0.0821370
\(274\) 19.1587 1.15742
\(275\) 4.54828 0.274271
\(276\) −3.51870 −0.211801
\(277\) −0.447859 −0.0269093 −0.0134546 0.999909i \(-0.504283\pi\)
−0.0134546 + 0.999909i \(0.504283\pi\)
\(278\) −10.0578 −0.603226
\(279\) 4.29806 0.257318
\(280\) −1.51454 −0.0905109
\(281\) 11.2801 0.672915 0.336457 0.941699i \(-0.390771\pi\)
0.336457 + 0.941699i \(0.390771\pi\)
\(282\) 2.64404 0.157450
\(283\) −5.76688 −0.342806 −0.171403 0.985201i \(-0.554830\pi\)
−0.171403 + 0.985201i \(0.554830\pi\)
\(284\) −6.75595 −0.400892
\(285\) −5.94405 −0.352095
\(286\) 2.28092 0.134874
\(287\) 4.69402 0.277079
\(288\) 1.00000 0.0589256
\(289\) 3.27459 0.192623
\(290\) −13.4879 −0.792035
\(291\) −3.02861 −0.177540
\(292\) −4.24487 −0.248412
\(293\) −3.05098 −0.178240 −0.0891200 0.996021i \(-0.528405\pi\)
−0.0891200 + 0.996021i \(0.528405\pi\)
\(294\) −1.00000 −0.0583212
\(295\) −14.5275 −0.845822
\(296\) 8.83806 0.513702
\(297\) 1.68070 0.0975242
\(298\) −17.9407 −1.03928
\(299\) −4.77532 −0.276164
\(300\) 2.70618 0.156241
\(301\) 2.40129 0.138408
\(302\) −10.3551 −0.595870
\(303\) −12.0400 −0.691681
\(304\) 3.92466 0.225095
\(305\) 7.41715 0.424705
\(306\) −4.50273 −0.257404
\(307\) −19.5185 −1.11398 −0.556991 0.830518i \(-0.688044\pi\)
−0.556991 + 0.830518i \(0.688044\pi\)
\(308\) 1.68070 0.0957669
\(309\) 7.62868 0.433981
\(310\) 6.50957 0.369719
\(311\) 5.73920 0.325440 0.162720 0.986672i \(-0.447973\pi\)
0.162720 + 0.986672i \(0.447973\pi\)
\(312\) 1.35713 0.0768321
\(313\) −17.0827 −0.965574 −0.482787 0.875738i \(-0.660375\pi\)
−0.482787 + 0.875738i \(0.660375\pi\)
\(314\) −0.575064 −0.0324528
\(315\) −1.51454 −0.0853345
\(316\) −12.4406 −0.699841
\(317\) −13.1001 −0.735776 −0.367888 0.929870i \(-0.619919\pi\)
−0.367888 + 0.929870i \(0.619919\pi\)
\(318\) 5.94440 0.333345
\(319\) 14.9677 0.838029
\(320\) 1.51454 0.0846652
\(321\) 8.48436 0.473551
\(322\) −3.51870 −0.196089
\(323\) −17.6717 −0.983279
\(324\) 1.00000 0.0555556
\(325\) 3.67262 0.203721
\(326\) 5.54296 0.306996
\(327\) −0.166999 −0.00923509
\(328\) −4.69402 −0.259184
\(329\) 2.64404 0.145770
\(330\) 2.54549 0.140124
\(331\) −17.3251 −0.952271 −0.476135 0.879372i \(-0.657963\pi\)
−0.476135 + 0.879372i \(0.657963\pi\)
\(332\) −4.27560 −0.234654
\(333\) 8.83806 0.484323
\(334\) 18.4617 1.01018
\(335\) 12.4258 0.678893
\(336\) 1.00000 0.0545545
\(337\) 0.0407571 0.00222018 0.00111009 0.999999i \(-0.499647\pi\)
0.00111009 + 0.999999i \(0.499647\pi\)
\(338\) −11.1582 −0.606927
\(339\) 10.7808 0.585533
\(340\) −6.81955 −0.369842
\(341\) −7.22376 −0.391189
\(342\) 3.92466 0.212221
\(343\) −1.00000 −0.0539949
\(344\) −2.40129 −0.129469
\(345\) −5.32920 −0.286914
\(346\) 0.979310 0.0526480
\(347\) −11.3420 −0.608868 −0.304434 0.952533i \(-0.598467\pi\)
−0.304434 + 0.952533i \(0.598467\pi\)
\(348\) 8.90561 0.477391
\(349\) 14.3535 0.768324 0.384162 0.923266i \(-0.374490\pi\)
0.384162 + 0.923266i \(0.374490\pi\)
\(350\) 2.70618 0.144651
\(351\) 1.35713 0.0724380
\(352\) −1.68070 −0.0895817
\(353\) 7.44716 0.396372 0.198186 0.980164i \(-0.436495\pi\)
0.198186 + 0.980164i \(0.436495\pi\)
\(354\) 9.59202 0.509810
\(355\) −10.2321 −0.543065
\(356\) 6.68911 0.354522
\(357\) −4.50273 −0.238310
\(358\) −12.0842 −0.638667
\(359\) 15.3668 0.811028 0.405514 0.914089i \(-0.367092\pi\)
0.405514 + 0.914089i \(0.367092\pi\)
\(360\) 1.51454 0.0798231
\(361\) −3.59703 −0.189317
\(362\) −16.0985 −0.846119
\(363\) 8.17524 0.429089
\(364\) 1.35713 0.0711327
\(365\) −6.42902 −0.336510
\(366\) −4.89730 −0.255986
\(367\) −22.3719 −1.16780 −0.583902 0.811824i \(-0.698475\pi\)
−0.583902 + 0.811824i \(0.698475\pi\)
\(368\) 3.51870 0.183425
\(369\) −4.69402 −0.244361
\(370\) 13.3856 0.695883
\(371\) 5.94440 0.308618
\(372\) −4.29806 −0.222844
\(373\) 21.7486 1.12610 0.563049 0.826423i \(-0.309628\pi\)
0.563049 + 0.826423i \(0.309628\pi\)
\(374\) 7.56775 0.391319
\(375\) 11.6713 0.602703
\(376\) −2.64404 −0.136356
\(377\) 12.0860 0.622462
\(378\) 1.00000 0.0514344
\(379\) 20.1124 1.03311 0.516554 0.856255i \(-0.327215\pi\)
0.516554 + 0.856255i \(0.327215\pi\)
\(380\) 5.94405 0.304923
\(381\) 4.94951 0.253571
\(382\) 1.00000 0.0511645
\(383\) −24.0159 −1.22716 −0.613578 0.789634i \(-0.710270\pi\)
−0.613578 + 0.789634i \(0.710270\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 2.54549 0.129730
\(386\) 8.73233 0.444464
\(387\) −2.40129 −0.122065
\(388\) 3.02861 0.153754
\(389\) 1.60108 0.0811781 0.0405891 0.999176i \(-0.487077\pi\)
0.0405891 + 0.999176i \(0.487077\pi\)
\(390\) 2.05542 0.104080
\(391\) −15.8438 −0.801253
\(392\) 1.00000 0.0505076
\(393\) 12.5335 0.632230
\(394\) −9.75048 −0.491222
\(395\) −18.8418 −0.948035
\(396\) −1.68070 −0.0844584
\(397\) −4.99455 −0.250669 −0.125335 0.992115i \(-0.540000\pi\)
−0.125335 + 0.992115i \(0.540000\pi\)
\(398\) 14.2985 0.716721
\(399\) 3.92466 0.196479
\(400\) −2.70618 −0.135309
\(401\) 9.77927 0.488354 0.244177 0.969731i \(-0.421482\pi\)
0.244177 + 0.969731i \(0.421482\pi\)
\(402\) −8.20435 −0.409196
\(403\) −5.83301 −0.290563
\(404\) 12.0400 0.599013
\(405\) 1.51454 0.0752580
\(406\) 8.90561 0.441978
\(407\) −14.8541 −0.736292
\(408\) 4.50273 0.222918
\(409\) 14.4342 0.713723 0.356862 0.934157i \(-0.383847\pi\)
0.356862 + 0.934157i \(0.383847\pi\)
\(410\) −7.10927 −0.351102
\(411\) −19.1587 −0.945029
\(412\) −7.62868 −0.375838
\(413\) 9.59202 0.471992
\(414\) 3.51870 0.172935
\(415\) −6.47556 −0.317873
\(416\) −1.35713 −0.0665386
\(417\) 10.0578 0.492532
\(418\) −6.59619 −0.322630
\(419\) 3.44484 0.168292 0.0841458 0.996453i \(-0.473184\pi\)
0.0841458 + 0.996453i \(0.473184\pi\)
\(420\) 1.51454 0.0739018
\(421\) −7.85560 −0.382859 −0.191429 0.981506i \(-0.561312\pi\)
−0.191429 + 0.981506i \(0.561312\pi\)
\(422\) 6.00269 0.292206
\(423\) −2.64404 −0.128557
\(424\) −5.94440 −0.288685
\(425\) 12.1852 0.591068
\(426\) 6.75595 0.327327
\(427\) −4.89730 −0.236997
\(428\) −8.48436 −0.410107
\(429\) −2.28092 −0.110124
\(430\) −3.63685 −0.175384
\(431\) −32.8227 −1.58102 −0.790508 0.612452i \(-0.790183\pi\)
−0.790508 + 0.612452i \(0.790183\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 30.7560 1.47804 0.739021 0.673683i \(-0.235289\pi\)
0.739021 + 0.673683i \(0.235289\pi\)
\(434\) −4.29806 −0.206314
\(435\) 13.4879 0.646694
\(436\) 0.166999 0.00799782
\(437\) 13.8097 0.660608
\(438\) 4.24487 0.202828
\(439\) −10.8474 −0.517719 −0.258859 0.965915i \(-0.583347\pi\)
−0.258859 + 0.965915i \(0.583347\pi\)
\(440\) −2.54549 −0.121351
\(441\) 1.00000 0.0476190
\(442\) 6.11077 0.290660
\(443\) −15.0722 −0.716102 −0.358051 0.933702i \(-0.616559\pi\)
−0.358051 + 0.933702i \(0.616559\pi\)
\(444\) −8.83806 −0.419436
\(445\) 10.1309 0.480251
\(446\) −16.4263 −0.777808
\(447\) 17.9407 0.848567
\(448\) −1.00000 −0.0472456
\(449\) 33.9640 1.60286 0.801431 0.598088i \(-0.204073\pi\)
0.801431 + 0.598088i \(0.204073\pi\)
\(450\) −2.70618 −0.127570
\(451\) 7.88925 0.371490
\(452\) −10.7808 −0.507086
\(453\) 10.3551 0.486526
\(454\) −18.4802 −0.867319
\(455\) 2.05542 0.0963595
\(456\) −3.92466 −0.183789
\(457\) −39.9472 −1.86865 −0.934326 0.356419i \(-0.883997\pi\)
−0.934326 + 0.356419i \(0.883997\pi\)
\(458\) 17.9688 0.839626
\(459\) 4.50273 0.210169
\(460\) 5.32920 0.248475
\(461\) 4.14051 0.192843 0.0964213 0.995341i \(-0.469260\pi\)
0.0964213 + 0.995341i \(0.469260\pi\)
\(462\) −1.68070 −0.0781933
\(463\) −23.7143 −1.10209 −0.551047 0.834474i \(-0.685772\pi\)
−0.551047 + 0.834474i \(0.685772\pi\)
\(464\) −8.90561 −0.413432
\(465\) −6.50957 −0.301874
\(466\) 29.5440 1.36860
\(467\) 35.8186 1.65749 0.828743 0.559629i \(-0.189056\pi\)
0.828743 + 0.559629i \(0.189056\pi\)
\(468\) −1.35713 −0.0627332
\(469\) −8.20435 −0.378842
\(470\) −4.00449 −0.184713
\(471\) 0.575064 0.0264976
\(472\) −9.59202 −0.441508
\(473\) 4.03586 0.185569
\(474\) 12.4406 0.571418
\(475\) −10.6208 −0.487317
\(476\) 4.50273 0.206382
\(477\) −5.94440 −0.272175
\(478\) −8.96693 −0.410138
\(479\) −36.2129 −1.65461 −0.827305 0.561753i \(-0.810127\pi\)
−0.827305 + 0.561753i \(0.810127\pi\)
\(480\) −1.51454 −0.0691288
\(481\) −11.9944 −0.546896
\(482\) 1.20612 0.0549375
\(483\) 3.51870 0.160106
\(484\) −8.17524 −0.371602
\(485\) 4.58694 0.208282
\(486\) −1.00000 −0.0453609
\(487\) −22.0811 −1.00059 −0.500296 0.865855i \(-0.666775\pi\)
−0.500296 + 0.865855i \(0.666775\pi\)
\(488\) 4.89730 0.221691
\(489\) −5.54296 −0.250661
\(490\) 1.51454 0.0684198
\(491\) 3.79385 0.171214 0.0856070 0.996329i \(-0.472717\pi\)
0.0856070 + 0.996329i \(0.472717\pi\)
\(492\) 4.69402 0.211623
\(493\) 40.0996 1.80599
\(494\) −5.32626 −0.239640
\(495\) −2.54549 −0.114411
\(496\) 4.29806 0.192989
\(497\) 6.75595 0.303046
\(498\) 4.27560 0.191594
\(499\) 29.5409 1.32243 0.661216 0.750196i \(-0.270041\pi\)
0.661216 + 0.750196i \(0.270041\pi\)
\(500\) −11.6713 −0.521956
\(501\) −18.4617 −0.824810
\(502\) 13.6302 0.608348
\(503\) 30.0797 1.34119 0.670595 0.741824i \(-0.266039\pi\)
0.670595 + 0.741824i \(0.266039\pi\)
\(504\) −1.00000 −0.0445435
\(505\) 18.2350 0.811449
\(506\) −5.91388 −0.262904
\(507\) 11.1582 0.495553
\(508\) −4.94951 −0.219599
\(509\) −27.8390 −1.23394 −0.616971 0.786986i \(-0.711640\pi\)
−0.616971 + 0.786986i \(0.711640\pi\)
\(510\) 6.81955 0.301975
\(511\) 4.24487 0.187782
\(512\) 1.00000 0.0441942
\(513\) −3.92466 −0.173278
\(514\) −5.82640 −0.256992
\(515\) −11.5539 −0.509127
\(516\) 2.40129 0.105711
\(517\) 4.44384 0.195440
\(518\) −8.83806 −0.388322
\(519\) −0.979310 −0.0429869
\(520\) −2.05542 −0.0901360
\(521\) 19.4835 0.853586 0.426793 0.904349i \(-0.359643\pi\)
0.426793 + 0.904349i \(0.359643\pi\)
\(522\) −8.90561 −0.389788
\(523\) 10.6141 0.464124 0.232062 0.972701i \(-0.425453\pi\)
0.232062 + 0.972701i \(0.425453\pi\)
\(524\) −12.5335 −0.547527
\(525\) −2.70618 −0.118107
\(526\) −31.0461 −1.35367
\(527\) −19.3530 −0.843031
\(528\) 1.68070 0.0731432
\(529\) −10.6188 −0.461685
\(530\) −9.00301 −0.391066
\(531\) −9.59202 −0.416258
\(532\) −3.92466 −0.170156
\(533\) 6.37038 0.275932
\(534\) −6.68911 −0.289466
\(535\) −12.8499 −0.555549
\(536\) 8.20435 0.354374
\(537\) 12.0842 0.521470
\(538\) 0.696142 0.0300128
\(539\) −1.68070 −0.0723930
\(540\) −1.51454 −0.0651753
\(541\) −2.50986 −0.107907 −0.0539537 0.998543i \(-0.517182\pi\)
−0.0539537 + 0.998543i \(0.517182\pi\)
\(542\) 31.0053 1.33179
\(543\) 16.0985 0.690853
\(544\) −4.50273 −0.193053
\(545\) 0.252927 0.0108342
\(546\) −1.35713 −0.0580796
\(547\) −39.1666 −1.67464 −0.837321 0.546711i \(-0.815879\pi\)
−0.837321 + 0.546711i \(0.815879\pi\)
\(548\) 19.1587 0.818419
\(549\) 4.89730 0.209012
\(550\) 4.54828 0.193939
\(551\) −34.9515 −1.48898
\(552\) −3.51870 −0.149766
\(553\) 12.4406 0.529030
\(554\) −0.447859 −0.0190277
\(555\) −13.3856 −0.568186
\(556\) −10.0578 −0.426545
\(557\) −18.2800 −0.774547 −0.387273 0.921965i \(-0.626583\pi\)
−0.387273 + 0.921965i \(0.626583\pi\)
\(558\) 4.29806 0.181951
\(559\) 3.25886 0.137835
\(560\) −1.51454 −0.0640009
\(561\) −7.56775 −0.319511
\(562\) 11.2801 0.475822
\(563\) 13.3951 0.564537 0.282269 0.959335i \(-0.408913\pi\)
0.282269 + 0.959335i \(0.408913\pi\)
\(564\) 2.64404 0.111334
\(565\) −16.3279 −0.686921
\(566\) −5.76688 −0.242400
\(567\) −1.00000 −0.0419961
\(568\) −6.75595 −0.283473
\(569\) 1.34888 0.0565480 0.0282740 0.999600i \(-0.490999\pi\)
0.0282740 + 0.999600i \(0.490999\pi\)
\(570\) −5.94405 −0.248969
\(571\) −0.215473 −0.00901725 −0.00450863 0.999990i \(-0.501435\pi\)
−0.00450863 + 0.999990i \(0.501435\pi\)
\(572\) 2.28092 0.0953702
\(573\) −1.00000 −0.0417756
\(574\) 4.69402 0.195925
\(575\) −9.52222 −0.397104
\(576\) 1.00000 0.0416667
\(577\) 4.28061 0.178204 0.0891021 0.996022i \(-0.471600\pi\)
0.0891021 + 0.996022i \(0.471600\pi\)
\(578\) 3.27459 0.136205
\(579\) −8.73233 −0.362903
\(580\) −13.4879 −0.560054
\(581\) 4.27560 0.177382
\(582\) −3.02861 −0.125540
\(583\) 9.99076 0.413775
\(584\) −4.24487 −0.175654
\(585\) −2.05542 −0.0849811
\(586\) −3.05098 −0.126035
\(587\) −12.4431 −0.513584 −0.256792 0.966467i \(-0.582665\pi\)
−0.256792 + 0.966467i \(0.582665\pi\)
\(588\) −1.00000 −0.0412393
\(589\) 16.8684 0.695052
\(590\) −14.5275 −0.598086
\(591\) 9.75048 0.401081
\(592\) 8.83806 0.363242
\(593\) −40.5151 −1.66375 −0.831877 0.554960i \(-0.812733\pi\)
−0.831877 + 0.554960i \(0.812733\pi\)
\(594\) 1.68070 0.0689600
\(595\) 6.81955 0.279574
\(596\) −17.9407 −0.734881
\(597\) −14.2985 −0.585200
\(598\) −4.77532 −0.195277
\(599\) −45.8522 −1.87347 −0.936735 0.350040i \(-0.886168\pi\)
−0.936735 + 0.350040i \(0.886168\pi\)
\(600\) 2.70618 0.110479
\(601\) 5.92898 0.241848 0.120924 0.992662i \(-0.461414\pi\)
0.120924 + 0.992662i \(0.461414\pi\)
\(602\) 2.40129 0.0978694
\(603\) 8.20435 0.334107
\(604\) −10.3551 −0.421344
\(605\) −12.3817 −0.503388
\(606\) −12.0400 −0.489092
\(607\) 1.24505 0.0505351 0.0252676 0.999681i \(-0.491956\pi\)
0.0252676 + 0.999681i \(0.491956\pi\)
\(608\) 3.92466 0.159166
\(609\) −8.90561 −0.360873
\(610\) 7.41715 0.300312
\(611\) 3.58829 0.145167
\(612\) −4.50273 −0.182012
\(613\) −28.7860 −1.16265 −0.581327 0.813670i \(-0.697466\pi\)
−0.581327 + 0.813670i \(0.697466\pi\)
\(614\) −19.5185 −0.787704
\(615\) 7.10927 0.286673
\(616\) 1.68070 0.0677174
\(617\) 7.29625 0.293736 0.146868 0.989156i \(-0.453081\pi\)
0.146868 + 0.989156i \(0.453081\pi\)
\(618\) 7.62868 0.306871
\(619\) 19.3366 0.777203 0.388602 0.921406i \(-0.372958\pi\)
0.388602 + 0.921406i \(0.372958\pi\)
\(620\) 6.50957 0.261431
\(621\) −3.51870 −0.141201
\(622\) 5.73920 0.230121
\(623\) −6.68911 −0.267993
\(624\) 1.35713 0.0543285
\(625\) −4.14572 −0.165829
\(626\) −17.0827 −0.682764
\(627\) 6.59619 0.263426
\(628\) −0.575064 −0.0229476
\(629\) −39.7954 −1.58675
\(630\) −1.51454 −0.0603406
\(631\) −21.8594 −0.870210 −0.435105 0.900380i \(-0.643289\pi\)
−0.435105 + 0.900380i \(0.643289\pi\)
\(632\) −12.4406 −0.494862
\(633\) −6.00269 −0.238585
\(634\) −13.1001 −0.520272
\(635\) −7.49622 −0.297478
\(636\) 5.94440 0.235711
\(637\) −1.35713 −0.0537713
\(638\) 14.9677 0.592576
\(639\) −6.75595 −0.267261
\(640\) 1.51454 0.0598673
\(641\) −14.7589 −0.582942 −0.291471 0.956580i \(-0.594145\pi\)
−0.291471 + 0.956580i \(0.594145\pi\)
\(642\) 8.48436 0.334851
\(643\) 11.5866 0.456932 0.228466 0.973552i \(-0.426629\pi\)
0.228466 + 0.973552i \(0.426629\pi\)
\(644\) −3.51870 −0.138656
\(645\) 3.63685 0.143201
\(646\) −17.6717 −0.695284
\(647\) 25.9610 1.02063 0.510315 0.859987i \(-0.329529\pi\)
0.510315 + 0.859987i \(0.329529\pi\)
\(648\) 1.00000 0.0392837
\(649\) 16.1213 0.632817
\(650\) 3.67262 0.144052
\(651\) 4.29806 0.168454
\(652\) 5.54296 0.217079
\(653\) −13.9175 −0.544633 −0.272316 0.962208i \(-0.587790\pi\)
−0.272316 + 0.962208i \(0.587790\pi\)
\(654\) −0.166999 −0.00653019
\(655\) −18.9824 −0.741704
\(656\) −4.69402 −0.183271
\(657\) −4.24487 −0.165608
\(658\) 2.64404 0.103075
\(659\) −11.1201 −0.433177 −0.216589 0.976263i \(-0.569493\pi\)
−0.216589 + 0.976263i \(0.569493\pi\)
\(660\) 2.54549 0.0990829
\(661\) 40.6675 1.58178 0.790891 0.611957i \(-0.209617\pi\)
0.790891 + 0.611957i \(0.209617\pi\)
\(662\) −17.3251 −0.673357
\(663\) −6.11077 −0.237323
\(664\) −4.27560 −0.165926
\(665\) −5.94405 −0.230500
\(666\) 8.83806 0.342468
\(667\) −31.3361 −1.21334
\(668\) 18.4617 0.714306
\(669\) 16.4263 0.635077
\(670\) 12.4258 0.480050
\(671\) −8.23091 −0.317751
\(672\) 1.00000 0.0385758
\(673\) 3.41211 0.131527 0.0657635 0.997835i \(-0.479052\pi\)
0.0657635 + 0.997835i \(0.479052\pi\)
\(674\) 0.0407571 0.00156990
\(675\) 2.70618 0.104161
\(676\) −11.1582 −0.429162
\(677\) 28.5914 1.09886 0.549429 0.835540i \(-0.314845\pi\)
0.549429 + 0.835540i \(0.314845\pi\)
\(678\) 10.7808 0.414034
\(679\) −3.02861 −0.116227
\(680\) −6.81955 −0.261518
\(681\) 18.4802 0.708163
\(682\) −7.22376 −0.276612
\(683\) −20.6210 −0.789042 −0.394521 0.918887i \(-0.629090\pi\)
−0.394521 + 0.918887i \(0.629090\pi\)
\(684\) 3.92466 0.150063
\(685\) 29.0166 1.10867
\(686\) −1.00000 −0.0381802
\(687\) −17.9688 −0.685552
\(688\) −2.40129 −0.0915484
\(689\) 8.06730 0.307340
\(690\) −5.32920 −0.202879
\(691\) −9.53559 −0.362751 −0.181375 0.983414i \(-0.558055\pi\)
−0.181375 + 0.983414i \(0.558055\pi\)
\(692\) 0.979310 0.0372278
\(693\) 1.68070 0.0638446
\(694\) −11.3420 −0.430535
\(695\) −15.2329 −0.577816
\(696\) 8.90561 0.337566
\(697\) 21.1359 0.800580
\(698\) 14.3535 0.543287
\(699\) −29.5440 −1.11746
\(700\) 2.70618 0.102284
\(701\) −6.59066 −0.248926 −0.124463 0.992224i \(-0.539721\pi\)
−0.124463 + 0.992224i \(0.539721\pi\)
\(702\) 1.35713 0.0512214
\(703\) 34.6864 1.30822
\(704\) −1.68070 −0.0633438
\(705\) 4.00449 0.150818
\(706\) 7.44716 0.280278
\(707\) −12.0400 −0.452811
\(708\) 9.59202 0.360490
\(709\) 39.7191 1.49168 0.745841 0.666124i \(-0.232048\pi\)
0.745841 + 0.666124i \(0.232048\pi\)
\(710\) −10.2321 −0.384005
\(711\) −12.4406 −0.466561
\(712\) 6.68911 0.250685
\(713\) 15.1236 0.566383
\(714\) −4.50273 −0.168510
\(715\) 3.45455 0.129193
\(716\) −12.0842 −0.451606
\(717\) 8.96693 0.334876
\(718\) 15.3668 0.573484
\(719\) 23.0920 0.861188 0.430594 0.902546i \(-0.358304\pi\)
0.430594 + 0.902546i \(0.358304\pi\)
\(720\) 1.51454 0.0564435
\(721\) 7.62868 0.284107
\(722\) −3.59703 −0.133868
\(723\) −1.20612 −0.0448563
\(724\) −16.0985 −0.598297
\(725\) 24.1002 0.895057
\(726\) 8.17524 0.303412
\(727\) −4.50433 −0.167056 −0.0835281 0.996505i \(-0.526619\pi\)
−0.0835281 + 0.996505i \(0.526619\pi\)
\(728\) 1.35713 0.0502984
\(729\) 1.00000 0.0370370
\(730\) −6.42902 −0.237949
\(731\) 10.8124 0.399910
\(732\) −4.89730 −0.181010
\(733\) 20.4381 0.754897 0.377448 0.926031i \(-0.376802\pi\)
0.377448 + 0.926031i \(0.376802\pi\)
\(734\) −22.3719 −0.825763
\(735\) −1.51454 −0.0558645
\(736\) 3.51870 0.129701
\(737\) −13.7891 −0.507927
\(738\) −4.69402 −0.172789
\(739\) −26.0039 −0.956569 −0.478284 0.878205i \(-0.658741\pi\)
−0.478284 + 0.878205i \(0.658741\pi\)
\(740\) 13.3856 0.492063
\(741\) 5.32626 0.195665
\(742\) 5.94440 0.218226
\(743\) 13.4256 0.492536 0.246268 0.969202i \(-0.420796\pi\)
0.246268 + 0.969202i \(0.420796\pi\)
\(744\) −4.29806 −0.157575
\(745\) −27.1719 −0.995501
\(746\) 21.7486 0.796272
\(747\) −4.27560 −0.156436
\(748\) 7.56775 0.276704
\(749\) 8.48436 0.310012
\(750\) 11.6713 0.426175
\(751\) 43.2231 1.57723 0.788616 0.614886i \(-0.210798\pi\)
0.788616 + 0.614886i \(0.210798\pi\)
\(752\) −2.64404 −0.0964181
\(753\) −13.6302 −0.496714
\(754\) 12.0860 0.440147
\(755\) −15.6832 −0.570770
\(756\) 1.00000 0.0363696
\(757\) −11.5253 −0.418896 −0.209448 0.977820i \(-0.567167\pi\)
−0.209448 + 0.977820i \(0.567167\pi\)
\(758\) 20.1124 0.730517
\(759\) 5.91388 0.214660
\(760\) 5.94405 0.215613
\(761\) 41.6182 1.50866 0.754329 0.656497i \(-0.227962\pi\)
0.754329 + 0.656497i \(0.227962\pi\)
\(762\) 4.94951 0.179302
\(763\) −0.166999 −0.00604578
\(764\) 1.00000 0.0361787
\(765\) −6.81955 −0.246561
\(766\) −24.0159 −0.867730
\(767\) 13.0176 0.470038
\(768\) −1.00000 −0.0360844
\(769\) 4.01075 0.144631 0.0723157 0.997382i \(-0.476961\pi\)
0.0723157 + 0.997382i \(0.476961\pi\)
\(770\) 2.54549 0.0917329
\(771\) 5.82640 0.209833
\(772\) 8.73233 0.314283
\(773\) 47.5493 1.71023 0.855115 0.518438i \(-0.173486\pi\)
0.855115 + 0.518438i \(0.173486\pi\)
\(774\) −2.40129 −0.0863127
\(775\) −11.6313 −0.417809
\(776\) 3.02861 0.108721
\(777\) 8.83806 0.317064
\(778\) 1.60108 0.0574016
\(779\) −18.4224 −0.660052
\(780\) 2.05542 0.0735958
\(781\) 11.3547 0.406304
\(782\) −15.8438 −0.566571
\(783\) 8.90561 0.318260
\(784\) 1.00000 0.0357143
\(785\) −0.870956 −0.0310858
\(786\) 12.5335 0.447054
\(787\) −5.91975 −0.211016 −0.105508 0.994418i \(-0.533647\pi\)
−0.105508 + 0.994418i \(0.533647\pi\)
\(788\) −9.75048 −0.347346
\(789\) 31.0461 1.10527
\(790\) −18.8418 −0.670362
\(791\) 10.7808 0.383321
\(792\) −1.68070 −0.0597211
\(793\) −6.64626 −0.236016
\(794\) −4.99455 −0.177250
\(795\) 9.00301 0.319304
\(796\) 14.2985 0.506798
\(797\) −7.74622 −0.274385 −0.137193 0.990544i \(-0.543808\pi\)
−0.137193 + 0.990544i \(0.543808\pi\)
\(798\) 3.92466 0.138932
\(799\) 11.9054 0.421182
\(800\) −2.70618 −0.0956778
\(801\) 6.68911 0.236348
\(802\) 9.77927 0.345318
\(803\) 7.13436 0.251766
\(804\) −8.20435 −0.289345
\(805\) −5.32920 −0.187830
\(806\) −5.83301 −0.205459
\(807\) −0.696142 −0.0245053
\(808\) 12.0400 0.423566
\(809\) 15.2520 0.536230 0.268115 0.963387i \(-0.413599\pi\)
0.268115 + 0.963387i \(0.413599\pi\)
\(810\) 1.51454 0.0532154
\(811\) −11.3203 −0.397508 −0.198754 0.980049i \(-0.563689\pi\)
−0.198754 + 0.980049i \(0.563689\pi\)
\(812\) 8.90561 0.312526
\(813\) −31.0053 −1.08740
\(814\) −14.8541 −0.520637
\(815\) 8.39502 0.294065
\(816\) 4.50273 0.157627
\(817\) −9.42426 −0.329713
\(818\) 14.4342 0.504678
\(819\) 1.35713 0.0474218
\(820\) −7.10927 −0.248266
\(821\) 52.8540 1.84462 0.922308 0.386456i \(-0.126301\pi\)
0.922308 + 0.386456i \(0.126301\pi\)
\(822\) −19.1587 −0.668236
\(823\) −18.5931 −0.648115 −0.324057 0.946037i \(-0.605047\pi\)
−0.324057 + 0.946037i \(0.605047\pi\)
\(824\) −7.62868 −0.265758
\(825\) −4.54828 −0.158351
\(826\) 9.59202 0.333749
\(827\) −47.1275 −1.63878 −0.819391 0.573235i \(-0.805688\pi\)
−0.819391 + 0.573235i \(0.805688\pi\)
\(828\) 3.51870 0.122283
\(829\) 39.4965 1.37177 0.685886 0.727709i \(-0.259415\pi\)
0.685886 + 0.727709i \(0.259415\pi\)
\(830\) −6.47556 −0.224770
\(831\) 0.447859 0.0155361
\(832\) −1.35713 −0.0470499
\(833\) −4.50273 −0.156010
\(834\) 10.0578 0.348272
\(835\) 27.9610 0.967630
\(836\) −6.59619 −0.228134
\(837\) −4.29806 −0.148563
\(838\) 3.44484 0.119000
\(839\) −25.2536 −0.871850 −0.435925 0.899983i \(-0.643579\pi\)
−0.435925 + 0.899983i \(0.643579\pi\)
\(840\) 1.51454 0.0522565
\(841\) 50.3099 1.73482
\(842\) −7.85560 −0.270722
\(843\) −11.2801 −0.388507
\(844\) 6.00269 0.206621
\(845\) −16.8995 −0.581361
\(846\) −2.64404 −0.0909039
\(847\) 8.17524 0.280905
\(848\) −5.94440 −0.204131
\(849\) 5.76688 0.197919
\(850\) 12.1852 0.417948
\(851\) 31.0985 1.06604
\(852\) 6.75595 0.231455
\(853\) −27.8781 −0.954528 −0.477264 0.878760i \(-0.658372\pi\)
−0.477264 + 0.878760i \(0.658372\pi\)
\(854\) −4.89730 −0.167582
\(855\) 5.94405 0.203282
\(856\) −8.48436 −0.289989
\(857\) 5.58166 0.190666 0.0953329 0.995445i \(-0.469608\pi\)
0.0953329 + 0.995445i \(0.469608\pi\)
\(858\) −2.28092 −0.0778695
\(859\) −12.4408 −0.424475 −0.212238 0.977218i \(-0.568075\pi\)
−0.212238 + 0.977218i \(0.568075\pi\)
\(860\) −3.63685 −0.124015
\(861\) −4.69402 −0.159972
\(862\) −32.8227 −1.11795
\(863\) −2.57187 −0.0875473 −0.0437737 0.999041i \(-0.513938\pi\)
−0.0437737 + 0.999041i \(0.513938\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 1.48320 0.0504304
\(866\) 30.7560 1.04513
\(867\) −3.27459 −0.111211
\(868\) −4.29806 −0.145886
\(869\) 20.9090 0.709290
\(870\) 13.4879 0.457282
\(871\) −11.1343 −0.377273
\(872\) 0.166999 0.00565531
\(873\) 3.02861 0.102503
\(874\) 13.8097 0.467120
\(875\) 11.6713 0.394562
\(876\) 4.24487 0.143421
\(877\) 18.9735 0.640688 0.320344 0.947301i \(-0.396201\pi\)
0.320344 + 0.947301i \(0.396201\pi\)
\(878\) −10.8474 −0.366082
\(879\) 3.05098 0.102907
\(880\) −2.54549 −0.0858083
\(881\) −1.43436 −0.0483247 −0.0241623 0.999708i \(-0.507692\pi\)
−0.0241623 + 0.999708i \(0.507692\pi\)
\(882\) 1.00000 0.0336718
\(883\) 21.2692 0.715766 0.357883 0.933766i \(-0.383499\pi\)
0.357883 + 0.933766i \(0.383499\pi\)
\(884\) 6.11077 0.205528
\(885\) 14.5275 0.488335
\(886\) −15.0722 −0.506361
\(887\) −26.2972 −0.882974 −0.441487 0.897268i \(-0.645549\pi\)
−0.441487 + 0.897268i \(0.645549\pi\)
\(888\) −8.83806 −0.296586
\(889\) 4.94951 0.166001
\(890\) 10.1309 0.339589
\(891\) −1.68070 −0.0563056
\(892\) −16.4263 −0.549993
\(893\) −10.3769 −0.347251
\(894\) 17.9407 0.600028
\(895\) −18.3019 −0.611765
\(896\) −1.00000 −0.0334077
\(897\) 4.77532 0.159443
\(898\) 33.9640 1.13339
\(899\) −38.2769 −1.27660
\(900\) −2.70618 −0.0902059
\(901\) 26.7660 0.891706
\(902\) 7.88925 0.262683
\(903\) −2.40129 −0.0799100
\(904\) −10.7808 −0.358564
\(905\) −24.3818 −0.810478
\(906\) 10.3551 0.344026
\(907\) 13.4014 0.444986 0.222493 0.974934i \(-0.428580\pi\)
0.222493 + 0.974934i \(0.428580\pi\)
\(908\) −18.4802 −0.613287
\(909\) 12.0400 0.399342
\(910\) 2.05542 0.0681364
\(911\) −8.53780 −0.282870 −0.141435 0.989948i \(-0.545172\pi\)
−0.141435 + 0.989948i \(0.545172\pi\)
\(912\) −3.92466 −0.129959
\(913\) 7.18601 0.237822
\(914\) −39.9472 −1.32134
\(915\) −7.41715 −0.245203
\(916\) 17.9688 0.593705
\(917\) 12.5335 0.413892
\(918\) 4.50273 0.148612
\(919\) 15.3686 0.506964 0.253482 0.967340i \(-0.418424\pi\)
0.253482 + 0.967340i \(0.418424\pi\)
\(920\) 5.32920 0.175699
\(921\) 19.5185 0.643158
\(922\) 4.14051 0.136360
\(923\) 9.16868 0.301791
\(924\) −1.68070 −0.0552910
\(925\) −23.9174 −0.786398
\(926\) −23.7143 −0.779299
\(927\) −7.62868 −0.250559
\(928\) −8.90561 −0.292341
\(929\) −9.79727 −0.321438 −0.160719 0.987000i \(-0.551381\pi\)
−0.160719 + 0.987000i \(0.551381\pi\)
\(930\) −6.50957 −0.213457
\(931\) 3.92466 0.128626
\(932\) 29.5440 0.967746
\(933\) −5.73920 −0.187893
\(934\) 35.8186 1.17202
\(935\) 11.4616 0.374836
\(936\) −1.35713 −0.0443591
\(937\) 15.9801 0.522047 0.261024 0.965332i \(-0.415940\pi\)
0.261024 + 0.965332i \(0.415940\pi\)
\(938\) −8.20435 −0.267881
\(939\) 17.0827 0.557474
\(940\) −4.00449 −0.130612
\(941\) −25.1496 −0.819855 −0.409927 0.912118i \(-0.634446\pi\)
−0.409927 + 0.912118i \(0.634446\pi\)
\(942\) 0.575064 0.0187366
\(943\) −16.5168 −0.537862
\(944\) −9.59202 −0.312194
\(945\) 1.51454 0.0492679
\(946\) 4.03586 0.131217
\(947\) 21.8948 0.711487 0.355743 0.934584i \(-0.384228\pi\)
0.355743 + 0.934584i \(0.384228\pi\)
\(948\) 12.4406 0.404053
\(949\) 5.76083 0.187004
\(950\) −10.6208 −0.344585
\(951\) 13.1001 0.424800
\(952\) 4.50273 0.145934
\(953\) −13.7032 −0.443891 −0.221945 0.975059i \(-0.571241\pi\)
−0.221945 + 0.975059i \(0.571241\pi\)
\(954\) −5.94440 −0.192457
\(955\) 1.51454 0.0490093
\(956\) −8.96693 −0.290011
\(957\) −14.9677 −0.483836
\(958\) −36.2129 −1.16999
\(959\) −19.1587 −0.618667
\(960\) −1.51454 −0.0488815
\(961\) −12.5267 −0.404086
\(962\) −11.9944 −0.386714
\(963\) −8.48436 −0.273405
\(964\) 1.20612 0.0388467
\(965\) 13.2254 0.425742
\(966\) 3.51870 0.113212
\(967\) 41.4536 1.33306 0.666528 0.745480i \(-0.267779\pi\)
0.666528 + 0.745480i \(0.267779\pi\)
\(968\) −8.17524 −0.262762
\(969\) 17.6717 0.567697
\(970\) 4.58694 0.147278
\(971\) −1.22893 −0.0394382 −0.0197191 0.999806i \(-0.506277\pi\)
−0.0197191 + 0.999806i \(0.506277\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 10.0578 0.322438
\(974\) −22.0811 −0.707525
\(975\) −3.67262 −0.117618
\(976\) 4.89730 0.156759
\(977\) 21.7929 0.697215 0.348608 0.937269i \(-0.386655\pi\)
0.348608 + 0.937269i \(0.386655\pi\)
\(978\) −5.54296 −0.177244
\(979\) −11.2424 −0.359308
\(980\) 1.51454 0.0483801
\(981\) 0.166999 0.00533188
\(982\) 3.79385 0.121067
\(983\) 17.0163 0.542734 0.271367 0.962476i \(-0.412524\pi\)
0.271367 + 0.962476i \(0.412524\pi\)
\(984\) 4.69402 0.149640
\(985\) −14.7675 −0.470530
\(986\) 40.0996 1.27703
\(987\) −2.64404 −0.0841606
\(988\) −5.32626 −0.169451
\(989\) −8.44943 −0.268676
\(990\) −2.54549 −0.0809008
\(991\) 36.3089 1.15339 0.576696 0.816959i \(-0.304342\pi\)
0.576696 + 0.816959i \(0.304342\pi\)
\(992\) 4.29806 0.136464
\(993\) 17.3251 0.549794
\(994\) 6.75595 0.214286
\(995\) 21.6557 0.686531
\(996\) 4.27560 0.135478
\(997\) 46.0938 1.45980 0.729902 0.683551i \(-0.239566\pi\)
0.729902 + 0.683551i \(0.239566\pi\)
\(998\) 29.5409 0.935101
\(999\) −8.83806 −0.279624
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.q.1.6 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8022.2.a.q.1.6 9 1.1 even 1 trivial