Properties

Label 8023.2.a.b.1.5
Level $8023$
Weight $2$
Character 8023.1
Self dual yes
Analytic conductor $64.064$
Analytic rank $1$
Dimension $155$
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] = [8023,2,Mod(1,8023)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8023, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("8023.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8023 = 71 \cdot 113 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8023.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: \(64.0639775417\)
Analytic rank: \(1\)
Dimension: \(155\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Character \(\chi\) \(=\) 8023.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.76547 q^{2} +1.71486 q^{3} +5.64781 q^{4} -3.60382 q^{5} -4.74239 q^{6} -5.22895 q^{7} -10.0879 q^{8} -0.0592525 q^{9} +O(q^{10})\) \(q-2.76547 q^{2} +1.71486 q^{3} +5.64781 q^{4} -3.60382 q^{5} -4.74239 q^{6} -5.22895 q^{7} -10.0879 q^{8} -0.0592525 q^{9} +9.96624 q^{10} +4.65816 q^{11} +9.68521 q^{12} +3.15368 q^{13} +14.4605 q^{14} -6.18005 q^{15} +16.6021 q^{16} -4.13977 q^{17} +0.163861 q^{18} -3.95842 q^{19} -20.3537 q^{20} -8.96691 q^{21} -12.8820 q^{22} +3.55199 q^{23} -17.2993 q^{24} +7.98751 q^{25} -8.72141 q^{26} -5.24619 q^{27} -29.5321 q^{28} -2.64348 q^{29} +17.0907 q^{30} +4.89331 q^{31} -25.7368 q^{32} +7.98809 q^{33} +11.4484 q^{34} +18.8442 q^{35} -0.334647 q^{36} +0.163772 q^{37} +10.9469 q^{38} +5.40813 q^{39} +36.3550 q^{40} +4.13772 q^{41} +24.7977 q^{42} -4.26841 q^{43} +26.3084 q^{44} +0.213535 q^{45} -9.82290 q^{46} -10.6972 q^{47} +28.4703 q^{48} +20.3419 q^{49} -22.0892 q^{50} -7.09912 q^{51} +17.8114 q^{52} +9.86100 q^{53} +14.5082 q^{54} -16.7872 q^{55} +52.7491 q^{56} -6.78813 q^{57} +7.31047 q^{58} -3.58420 q^{59} -34.9037 q^{60} -7.46176 q^{61} -13.5323 q^{62} +0.309828 q^{63} +37.9702 q^{64} -11.3653 q^{65} -22.0908 q^{66} +4.54903 q^{67} -23.3806 q^{68} +6.09116 q^{69} -52.1129 q^{70} +1.00000 q^{71} +0.597733 q^{72} +12.6498 q^{73} -0.452906 q^{74} +13.6975 q^{75} -22.3564 q^{76} -24.3573 q^{77} -14.9560 q^{78} +12.5963 q^{79} -59.8311 q^{80} -8.81873 q^{81} -11.4427 q^{82} +3.80676 q^{83} -50.6434 q^{84} +14.9190 q^{85} +11.8042 q^{86} -4.53321 q^{87} -46.9910 q^{88} -8.70056 q^{89} -0.590525 q^{90} -16.4904 q^{91} +20.0609 q^{92} +8.39134 q^{93} +29.5829 q^{94} +14.2654 q^{95} -44.1351 q^{96} -14.2906 q^{97} -56.2548 q^{98} -0.276008 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 155 q - 21 q^{2} - 16 q^{3} + 151 q^{4} - 26 q^{5} - 10 q^{6} - 40 q^{7} - 57 q^{8} + 135 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 155 q - 21 q^{2} - 16 q^{3} + 151 q^{4} - 26 q^{5} - 10 q^{6} - 40 q^{7} - 57 q^{8} + 135 q^{9} - 2 q^{10} - 24 q^{11} - 32 q^{12} - 62 q^{13} - 18 q^{14} - 12 q^{15} + 155 q^{16} - 129 q^{17} - 42 q^{18} - 18 q^{19} - 59 q^{20} - 45 q^{21} - 17 q^{22} - 38 q^{23} - 27 q^{24} + 129 q^{25} - 44 q^{26} - 43 q^{27} - 100 q^{28} - 52 q^{29} - 39 q^{30} - 56 q^{31} - 145 q^{32} - 126 q^{33} - q^{34} - 49 q^{35} + 131 q^{36} - 30 q^{37} - 91 q^{38} - 29 q^{39} - 5 q^{40} - 163 q^{41} - 80 q^{42} - 15 q^{43} - 118 q^{44} - 66 q^{45} + 2 q^{46} - 111 q^{47} - 89 q^{48} + 101 q^{49} - 121 q^{50} + 5 q^{51} - 111 q^{52} - 93 q^{53} - 68 q^{54} - 60 q^{55} - 27 q^{56} - 106 q^{57} + 16 q^{58} - 79 q^{59} - 103 q^{60} - 74 q^{61} - 102 q^{62} - 118 q^{63} + 175 q^{64} - 109 q^{65} + 65 q^{66} - 18 q^{67} - 346 q^{68} - 39 q^{69} + 32 q^{70} + 155 q^{71} - 203 q^{72} - 108 q^{73} - 87 q^{74} - 22 q^{75} - 16 q^{76} - 121 q^{77} - 75 q^{78} - 6 q^{79} - 136 q^{80} + 107 q^{81} - 30 q^{82} - 116 q^{83} - 5 q^{84} - 53 q^{85} + 8 q^{86} - 100 q^{87} - 43 q^{88} - 189 q^{89} - 76 q^{90} + 14 q^{91} - 99 q^{92} - 72 q^{93} + 17 q^{94} - 18 q^{95} - 50 q^{96} - 184 q^{97} - 249 q^{98} - 114 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.76547 −1.95548 −0.977740 0.209819i \(-0.932713\pi\)
−0.977740 + 0.209819i \(0.932713\pi\)
\(3\) 1.71486 0.990075 0.495038 0.868872i \(-0.335154\pi\)
0.495038 + 0.868872i \(0.335154\pi\)
\(4\) 5.64781 2.82390
\(5\) −3.60382 −1.61168 −0.805839 0.592135i \(-0.798285\pi\)
−0.805839 + 0.592135i \(0.798285\pi\)
\(6\) −4.74239 −1.93607
\(7\) −5.22895 −1.97636 −0.988178 0.153312i \(-0.951006\pi\)
−0.988178 + 0.153312i \(0.951006\pi\)
\(8\) −10.0879 −3.56661
\(9\) −0.0592525 −0.0197508
\(10\) 9.96624 3.15160
\(11\) 4.65816 1.40449 0.702244 0.711937i \(-0.252182\pi\)
0.702244 + 0.711937i \(0.252182\pi\)
\(12\) 9.68521 2.79588
\(13\) 3.15368 0.874675 0.437337 0.899298i \(-0.355922\pi\)
0.437337 + 0.899298i \(0.355922\pi\)
\(14\) 14.4605 3.86473
\(15\) −6.18005 −1.59568
\(16\) 16.6021 4.15053
\(17\) −4.13977 −1.00404 −0.502020 0.864856i \(-0.667410\pi\)
−0.502020 + 0.864856i \(0.667410\pi\)
\(18\) 0.163861 0.0386224
\(19\) −3.95842 −0.908123 −0.454062 0.890970i \(-0.650025\pi\)
−0.454062 + 0.890970i \(0.650025\pi\)
\(20\) −20.3537 −4.55122
\(21\) −8.96691 −1.95674
\(22\) −12.8820 −2.74645
\(23\) 3.55199 0.740640 0.370320 0.928904i \(-0.379248\pi\)
0.370320 + 0.928904i \(0.379248\pi\)
\(24\) −17.2993 −3.53121
\(25\) 7.98751 1.59750
\(26\) −8.72141 −1.71041
\(27\) −5.24619 −1.00963
\(28\) −29.5321 −5.58104
\(29\) −2.64348 −0.490883 −0.245441 0.969411i \(-0.578933\pi\)
−0.245441 + 0.969411i \(0.578933\pi\)
\(30\) 17.0907 3.12032
\(31\) 4.89331 0.878864 0.439432 0.898276i \(-0.355180\pi\)
0.439432 + 0.898276i \(0.355180\pi\)
\(32\) −25.7368 −4.54967
\(33\) 7.98809 1.39055
\(34\) 11.4484 1.96338
\(35\) 18.8442 3.18525
\(36\) −0.334647 −0.0557745
\(37\) 0.163772 0.0269240 0.0134620 0.999909i \(-0.495715\pi\)
0.0134620 + 0.999909i \(0.495715\pi\)
\(38\) 10.9469 1.77582
\(39\) 5.40813 0.865994
\(40\) 36.3550 5.74822
\(41\) 4.13772 0.646203 0.323101 0.946364i \(-0.395275\pi\)
0.323101 + 0.946364i \(0.395275\pi\)
\(42\) 24.7977 3.82637
\(43\) −4.26841 −0.650927 −0.325463 0.945555i \(-0.605520\pi\)
−0.325463 + 0.945555i \(0.605520\pi\)
\(44\) 26.3084 3.96614
\(45\) 0.213535 0.0318320
\(46\) −9.82290 −1.44831
\(47\) −10.6972 −1.56035 −0.780177 0.625559i \(-0.784871\pi\)
−0.780177 + 0.625559i \(0.784871\pi\)
\(48\) 28.4703 4.10934
\(49\) 20.3419 2.90598
\(50\) −22.0892 −3.12389
\(51\) −7.09912 −0.994076
\(52\) 17.8114 2.47000
\(53\) 9.86100 1.35451 0.677256 0.735747i \(-0.263169\pi\)
0.677256 + 0.735747i \(0.263169\pi\)
\(54\) 14.5082 1.97431
\(55\) −16.7872 −2.26358
\(56\) 52.7491 7.04889
\(57\) −6.78813 −0.899110
\(58\) 7.31047 0.959912
\(59\) −3.58420 −0.466623 −0.233312 0.972402i \(-0.574956\pi\)
−0.233312 + 0.972402i \(0.574956\pi\)
\(60\) −34.9037 −4.50605
\(61\) −7.46176 −0.955380 −0.477690 0.878528i \(-0.658526\pi\)
−0.477690 + 0.878528i \(0.658526\pi\)
\(62\) −13.5323 −1.71860
\(63\) 0.309828 0.0390347
\(64\) 37.9702 4.74627
\(65\) −11.3653 −1.40969
\(66\) −22.0908 −2.71919
\(67\) 4.54903 0.555753 0.277876 0.960617i \(-0.410369\pi\)
0.277876 + 0.960617i \(0.410369\pi\)
\(68\) −23.3806 −2.83532
\(69\) 6.09116 0.733290
\(70\) −52.1129 −6.22869
\(71\) 1.00000 0.118678
\(72\) 0.597733 0.0704435
\(73\) 12.6498 1.48055 0.740275 0.672304i \(-0.234695\pi\)
0.740275 + 0.672304i \(0.234695\pi\)
\(74\) −0.452906 −0.0526493
\(75\) 13.6975 1.58165
\(76\) −22.3564 −2.56445
\(77\) −24.3573 −2.77577
\(78\) −14.9560 −1.69343
\(79\) 12.5963 1.41720 0.708598 0.705613i \(-0.249328\pi\)
0.708598 + 0.705613i \(0.249328\pi\)
\(80\) −59.8311 −6.68932
\(81\) −8.81873 −0.979859
\(82\) −11.4427 −1.26364
\(83\) 3.80676 0.417846 0.208923 0.977932i \(-0.433004\pi\)
0.208923 + 0.977932i \(0.433004\pi\)
\(84\) −50.6434 −5.52565
\(85\) 14.9190 1.61819
\(86\) 11.8042 1.27287
\(87\) −4.53321 −0.486011
\(88\) −46.9910 −5.00926
\(89\) −8.70056 −0.922258 −0.461129 0.887333i \(-0.652555\pi\)
−0.461129 + 0.887333i \(0.652555\pi\)
\(90\) −0.590525 −0.0622468
\(91\) −16.4904 −1.72867
\(92\) 20.0609 2.09150
\(93\) 8.39134 0.870141
\(94\) 29.5829 3.05124
\(95\) 14.2654 1.46360
\(96\) −44.1351 −4.50452
\(97\) −14.2906 −1.45099 −0.725493 0.688230i \(-0.758388\pi\)
−0.725493 + 0.688230i \(0.758388\pi\)
\(98\) −56.2548 −5.68259
\(99\) −0.276008 −0.0277398
\(100\) 45.1120 4.51120
\(101\) 9.94010 0.989077 0.494538 0.869156i \(-0.335337\pi\)
0.494538 + 0.869156i \(0.335337\pi\)
\(102\) 19.6324 1.94390
\(103\) 3.50256 0.345118 0.172559 0.984999i \(-0.444797\pi\)
0.172559 + 0.984999i \(0.444797\pi\)
\(104\) −31.8140 −3.11962
\(105\) 32.3151 3.15363
\(106\) −27.2703 −2.64872
\(107\) 2.81493 0.272130 0.136065 0.990700i \(-0.456554\pi\)
0.136065 + 0.990700i \(0.456554\pi\)
\(108\) −29.6295 −2.85110
\(109\) 5.93745 0.568705 0.284352 0.958720i \(-0.408221\pi\)
0.284352 + 0.958720i \(0.408221\pi\)
\(110\) 46.4243 4.42639
\(111\) 0.280846 0.0266567
\(112\) −86.8116 −8.20293
\(113\) 1.00000 0.0940721
\(114\) 18.7724 1.75819
\(115\) −12.8007 −1.19367
\(116\) −14.9299 −1.38621
\(117\) −0.186864 −0.0172756
\(118\) 9.91199 0.912473
\(119\) 21.6466 1.98434
\(120\) 62.3437 5.69117
\(121\) 10.6984 0.972585
\(122\) 20.6353 1.86823
\(123\) 7.09561 0.639789
\(124\) 27.6365 2.48183
\(125\) −10.7665 −0.962982
\(126\) −0.856819 −0.0763315
\(127\) 19.8850 1.76451 0.882254 0.470775i \(-0.156025\pi\)
0.882254 + 0.470775i \(0.156025\pi\)
\(128\) −53.5315 −4.73156
\(129\) −7.31973 −0.644466
\(130\) 31.4304 2.75663
\(131\) −6.35089 −0.554880 −0.277440 0.960743i \(-0.589486\pi\)
−0.277440 + 0.960743i \(0.589486\pi\)
\(132\) 45.1152 3.92678
\(133\) 20.6983 1.79477
\(134\) −12.5802 −1.08676
\(135\) 18.9063 1.62720
\(136\) 41.7615 3.58102
\(137\) −10.0216 −0.856205 −0.428103 0.903730i \(-0.640818\pi\)
−0.428103 + 0.903730i \(0.640818\pi\)
\(138\) −16.8449 −1.43393
\(139\) −12.4234 −1.05374 −0.526868 0.849947i \(-0.676634\pi\)
−0.526868 + 0.849947i \(0.676634\pi\)
\(140\) 106.428 8.99483
\(141\) −18.3443 −1.54487
\(142\) −2.76547 −0.232073
\(143\) 14.6904 1.22847
\(144\) −0.983718 −0.0819765
\(145\) 9.52664 0.791144
\(146\) −34.9827 −2.89519
\(147\) 34.8835 2.87714
\(148\) 0.924953 0.0760307
\(149\) 3.53692 0.289756 0.144878 0.989450i \(-0.453721\pi\)
0.144878 + 0.989450i \(0.453721\pi\)
\(150\) −37.8799 −3.09288
\(151\) 23.3839 1.90295 0.951476 0.307724i \(-0.0995672\pi\)
0.951476 + 0.307724i \(0.0995672\pi\)
\(152\) 39.9321 3.23892
\(153\) 0.245292 0.0198306
\(154\) 67.3592 5.42796
\(155\) −17.6346 −1.41644
\(156\) 30.5441 2.44548
\(157\) 16.2604 1.29772 0.648860 0.760907i \(-0.275246\pi\)
0.648860 + 0.760907i \(0.275246\pi\)
\(158\) −34.8347 −2.77130
\(159\) 16.9102 1.34107
\(160\) 92.7509 7.33261
\(161\) −18.5731 −1.46377
\(162\) 24.3879 1.91610
\(163\) 9.67616 0.757895 0.378948 0.925418i \(-0.376286\pi\)
0.378948 + 0.925418i \(0.376286\pi\)
\(164\) 23.3690 1.82481
\(165\) −28.7876 −2.24112
\(166\) −10.5275 −0.817089
\(167\) −12.8593 −0.995079 −0.497540 0.867441i \(-0.665763\pi\)
−0.497540 + 0.867441i \(0.665763\pi\)
\(168\) 90.4573 6.97893
\(169\) −3.05427 −0.234944
\(170\) −41.2579 −3.16434
\(171\) 0.234546 0.0179362
\(172\) −24.1072 −1.83815
\(173\) −21.5031 −1.63485 −0.817425 0.576035i \(-0.804599\pi\)
−0.817425 + 0.576035i \(0.804599\pi\)
\(174\) 12.5364 0.950385
\(175\) −41.7663 −3.15723
\(176\) 77.3353 5.82937
\(177\) −6.14640 −0.461992
\(178\) 24.0611 1.80346
\(179\) 11.4261 0.854030 0.427015 0.904244i \(-0.359565\pi\)
0.427015 + 0.904244i \(0.359565\pi\)
\(180\) 1.20601 0.0898904
\(181\) 23.4309 1.74161 0.870803 0.491633i \(-0.163600\pi\)
0.870803 + 0.491633i \(0.163600\pi\)
\(182\) 45.6038 3.38038
\(183\) −12.7959 −0.945899
\(184\) −35.8321 −2.64157
\(185\) −0.590205 −0.0433927
\(186\) −23.2060 −1.70154
\(187\) −19.2837 −1.41016
\(188\) −60.4160 −4.40629
\(189\) 27.4321 1.99539
\(190\) −39.4506 −2.86204
\(191\) 7.30726 0.528735 0.264367 0.964422i \(-0.414837\pi\)
0.264367 + 0.964422i \(0.414837\pi\)
\(192\) 65.1135 4.69916
\(193\) 1.72952 0.124493 0.0622467 0.998061i \(-0.480173\pi\)
0.0622467 + 0.998061i \(0.480173\pi\)
\(194\) 39.5201 2.83737
\(195\) −19.4899 −1.39570
\(196\) 114.887 8.20621
\(197\) −14.8370 −1.05709 −0.528545 0.848905i \(-0.677262\pi\)
−0.528545 + 0.848905i \(0.677262\pi\)
\(198\) 0.763290 0.0542446
\(199\) 23.6181 1.67424 0.837120 0.547020i \(-0.184238\pi\)
0.837120 + 0.547020i \(0.184238\pi\)
\(200\) −80.5772 −5.69767
\(201\) 7.80095 0.550237
\(202\) −27.4890 −1.93412
\(203\) 13.8226 0.970159
\(204\) −40.0945 −2.80718
\(205\) −14.9116 −1.04147
\(206\) −9.68622 −0.674871
\(207\) −0.210464 −0.0146283
\(208\) 52.3579 3.63037
\(209\) −18.4389 −1.27545
\(210\) −89.3665 −6.16687
\(211\) −18.8933 −1.30067 −0.650335 0.759648i \(-0.725371\pi\)
−0.650335 + 0.759648i \(0.725371\pi\)
\(212\) 55.6930 3.82501
\(213\) 1.71486 0.117500
\(214\) −7.78460 −0.532144
\(215\) 15.3826 1.04908
\(216\) 52.9230 3.60096
\(217\) −25.5868 −1.73695
\(218\) −16.4198 −1.11209
\(219\) 21.6927 1.46586
\(220\) −94.8107 −6.39213
\(221\) −13.0555 −0.878209
\(222\) −0.776671 −0.0521267
\(223\) −18.7263 −1.25401 −0.627004 0.779016i \(-0.715719\pi\)
−0.627004 + 0.779016i \(0.715719\pi\)
\(224\) 134.577 8.99178
\(225\) −0.473280 −0.0315520
\(226\) −2.76547 −0.183956
\(227\) 14.1276 0.937682 0.468841 0.883283i \(-0.344672\pi\)
0.468841 + 0.883283i \(0.344672\pi\)
\(228\) −38.3381 −2.53900
\(229\) 12.6243 0.834240 0.417120 0.908851i \(-0.363039\pi\)
0.417120 + 0.908851i \(0.363039\pi\)
\(230\) 35.4000 2.33420
\(231\) −41.7693 −2.74822
\(232\) 26.6672 1.75079
\(233\) −8.58335 −0.562314 −0.281157 0.959662i \(-0.590718\pi\)
−0.281157 + 0.959662i \(0.590718\pi\)
\(234\) 0.516765 0.0337820
\(235\) 38.5509 2.51479
\(236\) −20.2429 −1.31770
\(237\) 21.6009 1.40313
\(238\) −59.8630 −3.88034
\(239\) −24.1510 −1.56220 −0.781101 0.624405i \(-0.785341\pi\)
−0.781101 + 0.624405i \(0.785341\pi\)
\(240\) −102.602 −6.62293
\(241\) −6.53769 −0.421130 −0.210565 0.977580i \(-0.567530\pi\)
−0.210565 + 0.977580i \(0.567530\pi\)
\(242\) −29.5862 −1.90187
\(243\) 0.615679 0.0394958
\(244\) −42.1426 −2.69790
\(245\) −73.3084 −4.68350
\(246\) −19.6227 −1.25110
\(247\) −12.4836 −0.794312
\(248\) −49.3632 −3.13456
\(249\) 6.52806 0.413699
\(250\) 29.7743 1.88309
\(251\) 9.67000 0.610365 0.305182 0.952294i \(-0.401283\pi\)
0.305182 + 0.952294i \(0.401283\pi\)
\(252\) 1.74985 0.110230
\(253\) 16.5457 1.04022
\(254\) −54.9913 −3.45046
\(255\) 25.5840 1.60213
\(256\) 72.0993 4.50621
\(257\) −20.3720 −1.27077 −0.635385 0.772195i \(-0.719159\pi\)
−0.635385 + 0.772195i \(0.719159\pi\)
\(258\) 20.2425 1.26024
\(259\) −0.856355 −0.0532113
\(260\) −64.1891 −3.98084
\(261\) 0.156633 0.00969534
\(262\) 17.5632 1.08506
\(263\) 4.67398 0.288210 0.144105 0.989562i \(-0.453970\pi\)
0.144105 + 0.989562i \(0.453970\pi\)
\(264\) −80.5831 −4.95954
\(265\) −35.5373 −2.18304
\(266\) −57.2406 −3.50965
\(267\) −14.9203 −0.913105
\(268\) 25.6921 1.56939
\(269\) 21.5654 1.31486 0.657431 0.753514i \(-0.271643\pi\)
0.657431 + 0.753514i \(0.271643\pi\)
\(270\) −52.2848 −3.18195
\(271\) −2.48617 −0.151024 −0.0755122 0.997145i \(-0.524059\pi\)
−0.0755122 + 0.997145i \(0.524059\pi\)
\(272\) −68.7289 −4.16730
\(273\) −28.2788 −1.71151
\(274\) 27.7145 1.67429
\(275\) 37.2071 2.24367
\(276\) 34.4017 2.07074
\(277\) −24.5430 −1.47464 −0.737322 0.675541i \(-0.763910\pi\)
−0.737322 + 0.675541i \(0.763910\pi\)
\(278\) 34.3564 2.06056
\(279\) −0.289941 −0.0173583
\(280\) −190.098 −11.3605
\(281\) 4.81026 0.286956 0.143478 0.989654i \(-0.454171\pi\)
0.143478 + 0.989654i \(0.454171\pi\)
\(282\) 50.7305 3.02096
\(283\) −10.6271 −0.631718 −0.315859 0.948806i \(-0.602293\pi\)
−0.315859 + 0.948806i \(0.602293\pi\)
\(284\) 5.64781 0.335136
\(285\) 24.4632 1.44908
\(286\) −40.6257 −2.40225
\(287\) −21.6359 −1.27713
\(288\) 1.52497 0.0898599
\(289\) 0.137665 0.00809796
\(290\) −26.3456 −1.54707
\(291\) −24.5063 −1.43659
\(292\) 71.4438 4.18093
\(293\) −15.3014 −0.893916 −0.446958 0.894555i \(-0.647493\pi\)
−0.446958 + 0.894555i \(0.647493\pi\)
\(294\) −96.4691 −5.62619
\(295\) 12.9168 0.752046
\(296\) −1.65211 −0.0960272
\(297\) −24.4376 −1.41801
\(298\) −9.78124 −0.566612
\(299\) 11.2018 0.647819
\(300\) 77.3607 4.46642
\(301\) 22.3193 1.28646
\(302\) −64.6673 −3.72118
\(303\) 17.0459 0.979260
\(304\) −65.7181 −3.76919
\(305\) 26.8908 1.53976
\(306\) −0.678346 −0.0387784
\(307\) 4.56716 0.260662 0.130331 0.991471i \(-0.458396\pi\)
0.130331 + 0.991471i \(0.458396\pi\)
\(308\) −137.565 −7.83850
\(309\) 6.00641 0.341692
\(310\) 48.7679 2.76983
\(311\) −12.0484 −0.683204 −0.341602 0.939845i \(-0.610970\pi\)
−0.341602 + 0.939845i \(0.610970\pi\)
\(312\) −54.5567 −3.08866
\(313\) 13.7004 0.774390 0.387195 0.921998i \(-0.373444\pi\)
0.387195 + 0.921998i \(0.373444\pi\)
\(314\) −44.9676 −2.53767
\(315\) −1.11656 −0.0629113
\(316\) 71.1415 4.00202
\(317\) 22.2269 1.24839 0.624193 0.781271i \(-0.285428\pi\)
0.624193 + 0.781271i \(0.285428\pi\)
\(318\) −46.7647 −2.62243
\(319\) −12.3138 −0.689439
\(320\) −136.838 −7.64945
\(321\) 4.82722 0.269429
\(322\) 51.3634 2.86237
\(323\) 16.3869 0.911793
\(324\) −49.8065 −2.76703
\(325\) 25.1901 1.39730
\(326\) −26.7591 −1.48205
\(327\) 10.1819 0.563060
\(328\) −41.7409 −2.30475
\(329\) 55.9353 3.08381
\(330\) 79.6113 4.38246
\(331\) −25.2741 −1.38919 −0.694596 0.719400i \(-0.744417\pi\)
−0.694596 + 0.719400i \(0.744417\pi\)
\(332\) 21.4998 1.17996
\(333\) −0.00970390 −0.000531770 0
\(334\) 35.5619 1.94586
\(335\) −16.3939 −0.895694
\(336\) −148.870 −8.12152
\(337\) 21.3645 1.16380 0.581899 0.813261i \(-0.302310\pi\)
0.581899 + 0.813261i \(0.302310\pi\)
\(338\) 8.44649 0.459429
\(339\) 1.71486 0.0931385
\(340\) 84.2595 4.56961
\(341\) 22.7938 1.23435
\(342\) −0.648630 −0.0350739
\(343\) −69.7639 −3.76690
\(344\) 43.0593 2.32160
\(345\) −21.9514 −1.18183
\(346\) 59.4661 3.19692
\(347\) −30.3960 −1.63174 −0.815871 0.578234i \(-0.803742\pi\)
−0.815871 + 0.578234i \(0.803742\pi\)
\(348\) −25.6027 −1.37245
\(349\) −20.5264 −1.09875 −0.549377 0.835574i \(-0.685135\pi\)
−0.549377 + 0.835574i \(0.685135\pi\)
\(350\) 115.503 6.17391
\(351\) −16.5448 −0.883098
\(352\) −119.886 −6.38996
\(353\) −24.9856 −1.32985 −0.664925 0.746910i \(-0.731536\pi\)
−0.664925 + 0.746910i \(0.731536\pi\)
\(354\) 16.9977 0.903417
\(355\) −3.60382 −0.191271
\(356\) −49.1391 −2.60437
\(357\) 37.1209 1.96465
\(358\) −31.5986 −1.67004
\(359\) −7.40394 −0.390765 −0.195383 0.980727i \(-0.562595\pi\)
−0.195383 + 0.980727i \(0.562595\pi\)
\(360\) −2.15412 −0.113532
\(361\) −3.33093 −0.175312
\(362\) −64.7974 −3.40567
\(363\) 18.3463 0.962933
\(364\) −93.1349 −4.88159
\(365\) −45.5877 −2.38617
\(366\) 35.3866 1.84969
\(367\) −22.2691 −1.16244 −0.581219 0.813747i \(-0.697424\pi\)
−0.581219 + 0.813747i \(0.697424\pi\)
\(368\) 58.9705 3.07405
\(369\) −0.245170 −0.0127630
\(370\) 1.63219 0.0848536
\(371\) −51.5626 −2.67700
\(372\) 47.3927 2.45720
\(373\) 27.6798 1.43321 0.716603 0.697481i \(-0.245696\pi\)
0.716603 + 0.697481i \(0.245696\pi\)
\(374\) 53.3284 2.75755
\(375\) −18.4630 −0.953424
\(376\) 107.913 5.56517
\(377\) −8.33672 −0.429363
\(378\) −75.8624 −3.90194
\(379\) −2.61110 −0.134123 −0.0670616 0.997749i \(-0.521362\pi\)
−0.0670616 + 0.997749i \(0.521362\pi\)
\(380\) 80.5684 4.13307
\(381\) 34.1000 1.74699
\(382\) −20.2080 −1.03393
\(383\) 11.6605 0.595826 0.297913 0.954593i \(-0.403709\pi\)
0.297913 + 0.954593i \(0.403709\pi\)
\(384\) −91.7991 −4.68460
\(385\) 87.7792 4.47364
\(386\) −4.78293 −0.243445
\(387\) 0.252914 0.0128563
\(388\) −80.7103 −4.09745
\(389\) −0.429804 −0.0217919 −0.0108960 0.999941i \(-0.503468\pi\)
−0.0108960 + 0.999941i \(0.503468\pi\)
\(390\) 53.8987 2.72927
\(391\) −14.7044 −0.743633
\(392\) −205.207 −10.3645
\(393\) −10.8909 −0.549373
\(394\) 41.0311 2.06712
\(395\) −45.3948 −2.28406
\(396\) −1.55884 −0.0783345
\(397\) 4.47107 0.224396 0.112198 0.993686i \(-0.464211\pi\)
0.112198 + 0.993686i \(0.464211\pi\)
\(398\) −65.3150 −3.27394
\(399\) 35.4948 1.77696
\(400\) 132.610 6.63049
\(401\) −23.9135 −1.19418 −0.597092 0.802173i \(-0.703677\pi\)
−0.597092 + 0.802173i \(0.703677\pi\)
\(402\) −21.5733 −1.07598
\(403\) 15.4319 0.768720
\(404\) 56.1398 2.79306
\(405\) 31.7811 1.57922
\(406\) −38.2260 −1.89713
\(407\) 0.762876 0.0378144
\(408\) 71.6152 3.54548
\(409\) −30.6639 −1.51623 −0.758115 0.652121i \(-0.773880\pi\)
−0.758115 + 0.652121i \(0.773880\pi\)
\(410\) 41.2375 2.03657
\(411\) −17.1857 −0.847708
\(412\) 19.7818 0.974579
\(413\) 18.7416 0.922213
\(414\) 0.582031 0.0286053
\(415\) −13.7189 −0.673432
\(416\) −81.1659 −3.97949
\(417\) −21.3043 −1.04328
\(418\) 50.9923 2.49411
\(419\) −33.6937 −1.64605 −0.823023 0.568008i \(-0.807714\pi\)
−0.823023 + 0.568008i \(0.807714\pi\)
\(420\) 182.510 8.90556
\(421\) 30.0506 1.46458 0.732288 0.680995i \(-0.238453\pi\)
0.732288 + 0.680995i \(0.238453\pi\)
\(422\) 52.2488 2.54343
\(423\) 0.633838 0.0308183
\(424\) −99.4767 −4.83102
\(425\) −33.0664 −1.60396
\(426\) −4.74239 −0.229770
\(427\) 39.0171 1.88817
\(428\) 15.8982 0.768468
\(429\) 25.1919 1.21628
\(430\) −42.5400 −2.05146
\(431\) 19.1766 0.923704 0.461852 0.886957i \(-0.347185\pi\)
0.461852 + 0.886957i \(0.347185\pi\)
\(432\) −87.0979 −4.19050
\(433\) 1.40570 0.0675536 0.0337768 0.999429i \(-0.489246\pi\)
0.0337768 + 0.999429i \(0.489246\pi\)
\(434\) 70.7595 3.39657
\(435\) 16.3369 0.783293
\(436\) 33.5336 1.60597
\(437\) −14.0602 −0.672593
\(438\) −59.9904 −2.86645
\(439\) 2.27194 0.108434 0.0542170 0.998529i \(-0.482734\pi\)
0.0542170 + 0.998529i \(0.482734\pi\)
\(440\) 169.347 8.07331
\(441\) −1.20531 −0.0573956
\(442\) 36.1046 1.71732
\(443\) 11.2504 0.534522 0.267261 0.963624i \(-0.413881\pi\)
0.267261 + 0.963624i \(0.413881\pi\)
\(444\) 1.58617 0.0752761
\(445\) 31.3553 1.48638
\(446\) 51.7870 2.45219
\(447\) 6.06533 0.286880
\(448\) −198.544 −9.38032
\(449\) 10.1819 0.480511 0.240256 0.970710i \(-0.422769\pi\)
0.240256 + 0.970710i \(0.422769\pi\)
\(450\) 1.30884 0.0616993
\(451\) 19.2741 0.907584
\(452\) 5.64781 0.265651
\(453\) 40.1001 1.88407
\(454\) −39.0694 −1.83362
\(455\) 59.4286 2.78606
\(456\) 68.4780 3.20678
\(457\) −28.1358 −1.31614 −0.658068 0.752958i \(-0.728626\pi\)
−0.658068 + 0.752958i \(0.728626\pi\)
\(458\) −34.9122 −1.63134
\(459\) 21.7180 1.01371
\(460\) −72.2960 −3.37082
\(461\) −23.0105 −1.07170 −0.535852 0.844312i \(-0.680010\pi\)
−0.535852 + 0.844312i \(0.680010\pi\)
\(462\) 115.512 5.37409
\(463\) 16.5418 0.768761 0.384381 0.923175i \(-0.374415\pi\)
0.384381 + 0.923175i \(0.374415\pi\)
\(464\) −43.8875 −2.03742
\(465\) −30.2409 −1.40239
\(466\) 23.7370 1.09959
\(467\) 20.6829 0.957089 0.478544 0.878063i \(-0.341165\pi\)
0.478544 + 0.878063i \(0.341165\pi\)
\(468\) −1.05537 −0.0487845
\(469\) −23.7866 −1.09836
\(470\) −106.611 −4.91761
\(471\) 27.8843 1.28484
\(472\) 36.1570 1.66426
\(473\) −19.8829 −0.914218
\(474\) −59.7366 −2.74379
\(475\) −31.6179 −1.45073
\(476\) 122.256 5.60359
\(477\) −0.584289 −0.0267527
\(478\) 66.7889 3.05485
\(479\) −20.0057 −0.914083 −0.457041 0.889445i \(-0.651091\pi\)
−0.457041 + 0.889445i \(0.651091\pi\)
\(480\) 159.055 7.25983
\(481\) 0.516485 0.0235497
\(482\) 18.0798 0.823511
\(483\) −31.8503 −1.44924
\(484\) 60.4227 2.74649
\(485\) 51.5006 2.33852
\(486\) −1.70264 −0.0772334
\(487\) −17.8331 −0.808094 −0.404047 0.914738i \(-0.632397\pi\)
−0.404047 + 0.914738i \(0.632397\pi\)
\(488\) 75.2735 3.40747
\(489\) 16.5933 0.750374
\(490\) 202.732 9.15850
\(491\) 37.0479 1.67195 0.835975 0.548768i \(-0.184903\pi\)
0.835975 + 0.548768i \(0.184903\pi\)
\(492\) 40.0746 1.80670
\(493\) 10.9434 0.492866
\(494\) 34.5230 1.55326
\(495\) 0.994681 0.0447076
\(496\) 81.2393 3.64775
\(497\) −5.22895 −0.234550
\(498\) −18.0531 −0.808980
\(499\) 13.8040 0.617952 0.308976 0.951070i \(-0.400014\pi\)
0.308976 + 0.951070i \(0.400014\pi\)
\(500\) −60.8069 −2.71937
\(501\) −22.0518 −0.985204
\(502\) −26.7421 −1.19356
\(503\) 19.1957 0.855894 0.427947 0.903804i \(-0.359237\pi\)
0.427947 + 0.903804i \(0.359237\pi\)
\(504\) −3.12551 −0.139221
\(505\) −35.8223 −1.59407
\(506\) −45.7566 −2.03413
\(507\) −5.23765 −0.232612
\(508\) 112.307 4.98280
\(509\) −35.7007 −1.58241 −0.791203 0.611554i \(-0.790545\pi\)
−0.791203 + 0.611554i \(0.790545\pi\)
\(510\) −70.7516 −3.13293
\(511\) −66.1453 −2.92609
\(512\) −92.3253 −4.08024
\(513\) 20.7666 0.916869
\(514\) 56.3381 2.48497
\(515\) −12.6226 −0.556218
\(516\) −41.3404 −1.81991
\(517\) −49.8294 −2.19150
\(518\) 2.36822 0.104054
\(519\) −36.8748 −1.61863
\(520\) 114.652 5.02783
\(521\) 2.07701 0.0909956 0.0454978 0.998964i \(-0.485513\pi\)
0.0454978 + 0.998964i \(0.485513\pi\)
\(522\) −0.433164 −0.0189591
\(523\) −42.4656 −1.85689 −0.928447 0.371466i \(-0.878855\pi\)
−0.928447 + 0.371466i \(0.878855\pi\)
\(524\) −35.8686 −1.56693
\(525\) −71.6234 −3.12590
\(526\) −12.9257 −0.563589
\(527\) −20.2571 −0.882415
\(528\) 132.619 5.77152
\(529\) −10.3834 −0.451452
\(530\) 98.2771 4.26888
\(531\) 0.212373 0.00921620
\(532\) 116.900 5.06827
\(533\) 13.0491 0.565217
\(534\) 41.2615 1.78556
\(535\) −10.1445 −0.438585
\(536\) −45.8901 −1.98215
\(537\) 19.5943 0.845554
\(538\) −59.6383 −2.57119
\(539\) 94.7557 4.08142
\(540\) 106.779 4.59505
\(541\) 5.64948 0.242890 0.121445 0.992598i \(-0.461247\pi\)
0.121445 + 0.992598i \(0.461247\pi\)
\(542\) 6.87543 0.295325
\(543\) 40.1807 1.72432
\(544\) 106.545 4.56806
\(545\) −21.3975 −0.916568
\(546\) 78.2041 3.34683
\(547\) −15.4825 −0.661984 −0.330992 0.943634i \(-0.607383\pi\)
−0.330992 + 0.943634i \(0.607383\pi\)
\(548\) −56.6002 −2.41784
\(549\) 0.442128 0.0188696
\(550\) −102.895 −4.38746
\(551\) 10.4640 0.445782
\(552\) −61.4470 −2.61536
\(553\) −65.8654 −2.80088
\(554\) 67.8728 2.88364
\(555\) −1.01212 −0.0429621
\(556\) −70.1648 −2.97565
\(557\) −19.1853 −0.812908 −0.406454 0.913671i \(-0.633235\pi\)
−0.406454 + 0.913671i \(0.633235\pi\)
\(558\) 0.801821 0.0339438
\(559\) −13.4612 −0.569349
\(560\) 312.853 13.2205
\(561\) −33.0688 −1.39617
\(562\) −13.3026 −0.561137
\(563\) 1.14992 0.0484634 0.0242317 0.999706i \(-0.492286\pi\)
0.0242317 + 0.999706i \(0.492286\pi\)
\(564\) −103.605 −4.36256
\(565\) −3.60382 −0.151614
\(566\) 29.3890 1.23531
\(567\) 46.1127 1.93655
\(568\) −10.0879 −0.423279
\(569\) −21.1540 −0.886821 −0.443411 0.896319i \(-0.646232\pi\)
−0.443411 + 0.896319i \(0.646232\pi\)
\(570\) −67.6522 −2.83364
\(571\) −0.667874 −0.0279497 −0.0139748 0.999902i \(-0.504448\pi\)
−0.0139748 + 0.999902i \(0.504448\pi\)
\(572\) 82.9684 3.46908
\(573\) 12.5309 0.523487
\(574\) 59.8334 2.49740
\(575\) 28.3715 1.18317
\(576\) −2.24983 −0.0937428
\(577\) 38.1625 1.58873 0.794364 0.607443i \(-0.207805\pi\)
0.794364 + 0.607443i \(0.207805\pi\)
\(578\) −0.380709 −0.0158354
\(579\) 2.96588 0.123258
\(580\) 53.8046 2.23412
\(581\) −19.9053 −0.825812
\(582\) 67.7714 2.80921
\(583\) 45.9341 1.90240
\(584\) −127.610 −5.28055
\(585\) 0.673423 0.0278426
\(586\) 42.3154 1.74803
\(587\) 6.80014 0.280672 0.140336 0.990104i \(-0.455182\pi\)
0.140336 + 0.990104i \(0.455182\pi\)
\(588\) 197.015 8.12477
\(589\) −19.3697 −0.798116
\(590\) −35.7210 −1.47061
\(591\) −25.4433 −1.04660
\(592\) 2.71896 0.111749
\(593\) −8.46786 −0.347734 −0.173867 0.984769i \(-0.555626\pi\)
−0.173867 + 0.984769i \(0.555626\pi\)
\(594\) 67.5814 2.77290
\(595\) −78.0105 −3.19812
\(596\) 19.9758 0.818243
\(597\) 40.5017 1.65762
\(598\) −30.9783 −1.26680
\(599\) −34.2014 −1.39743 −0.698715 0.715400i \(-0.746244\pi\)
−0.698715 + 0.715400i \(0.746244\pi\)
\(600\) −138.179 −5.64112
\(601\) 26.2495 1.07074 0.535369 0.844618i \(-0.320173\pi\)
0.535369 + 0.844618i \(0.320173\pi\)
\(602\) −61.7233 −2.51565
\(603\) −0.269541 −0.0109766
\(604\) 132.068 5.37375
\(605\) −38.5552 −1.56749
\(606\) −47.1398 −1.91492
\(607\) −48.6074 −1.97291 −0.986456 0.164025i \(-0.947552\pi\)
−0.986456 + 0.164025i \(0.947552\pi\)
\(608\) 101.877 4.13167
\(609\) 23.7039 0.960530
\(610\) −74.3657 −3.01098
\(611\) −33.7357 −1.36480
\(612\) 1.38536 0.0559998
\(613\) −45.9740 −1.85687 −0.928435 0.371494i \(-0.878846\pi\)
−0.928435 + 0.371494i \(0.878846\pi\)
\(614\) −12.6303 −0.509719
\(615\) −25.5713 −1.03113
\(616\) 245.713 9.90008
\(617\) −25.6330 −1.03195 −0.515973 0.856605i \(-0.672570\pi\)
−0.515973 + 0.856605i \(0.672570\pi\)
\(618\) −16.6105 −0.668173
\(619\) 20.0863 0.807336 0.403668 0.914906i \(-0.367735\pi\)
0.403668 + 0.914906i \(0.367735\pi\)
\(620\) −99.5968 −3.99990
\(621\) −18.6344 −0.747773
\(622\) 33.3196 1.33599
\(623\) 45.4948 1.82271
\(624\) 89.7865 3.59434
\(625\) −1.13719 −0.0454874
\(626\) −37.8879 −1.51431
\(627\) −31.6202 −1.26279
\(628\) 91.8356 3.66464
\(629\) −0.677978 −0.0270327
\(630\) 3.08782 0.123022
\(631\) −17.5775 −0.699750 −0.349875 0.936796i \(-0.613776\pi\)
−0.349875 + 0.936796i \(0.613776\pi\)
\(632\) −127.070 −5.05458
\(633\) −32.3994 −1.28776
\(634\) −61.4677 −2.44119
\(635\) −71.6619 −2.84382
\(636\) 95.5058 3.78705
\(637\) 64.1518 2.54179
\(638\) 34.0533 1.34818
\(639\) −0.0592525 −0.00234399
\(640\) 192.918 7.62575
\(641\) −0.0789430 −0.00311806 −0.00155903 0.999999i \(-0.500496\pi\)
−0.00155903 + 0.999999i \(0.500496\pi\)
\(642\) −13.3495 −0.526863
\(643\) −35.1457 −1.38601 −0.693005 0.720933i \(-0.743713\pi\)
−0.693005 + 0.720933i \(0.743713\pi\)
\(644\) −104.898 −4.13354
\(645\) 26.3790 1.03867
\(646\) −45.3175 −1.78299
\(647\) 8.41572 0.330856 0.165428 0.986222i \(-0.447099\pi\)
0.165428 + 0.986222i \(0.447099\pi\)
\(648\) 88.9624 3.49477
\(649\) −16.6958 −0.655366
\(650\) −69.6624 −2.73238
\(651\) −43.8779 −1.71971
\(652\) 54.6491 2.14022
\(653\) 7.12071 0.278655 0.139327 0.990246i \(-0.455506\pi\)
0.139327 + 0.990246i \(0.455506\pi\)
\(654\) −28.1577 −1.10105
\(655\) 22.8875 0.894287
\(656\) 68.6949 2.68209
\(657\) −0.749534 −0.0292421
\(658\) −154.687 −6.03034
\(659\) −1.80686 −0.0703851 −0.0351925 0.999381i \(-0.511204\pi\)
−0.0351925 + 0.999381i \(0.511204\pi\)
\(660\) −162.587 −6.32869
\(661\) 14.1155 0.549028 0.274514 0.961583i \(-0.411483\pi\)
0.274514 + 0.961583i \(0.411483\pi\)
\(662\) 69.8948 2.71654
\(663\) −22.3884 −0.869493
\(664\) −38.4022 −1.49029
\(665\) −74.5931 −2.89260
\(666\) 0.0268358 0.00103987
\(667\) −9.38962 −0.363567
\(668\) −72.6266 −2.81001
\(669\) −32.1130 −1.24156
\(670\) 45.3367 1.75151
\(671\) −34.7581 −1.34182
\(672\) 230.780 8.90253
\(673\) −1.51990 −0.0585877 −0.0292939 0.999571i \(-0.509326\pi\)
−0.0292939 + 0.999571i \(0.509326\pi\)
\(674\) −59.0828 −2.27579
\(675\) −41.9040 −1.61289
\(676\) −17.2500 −0.663460
\(677\) −42.4817 −1.63271 −0.816353 0.577554i \(-0.804007\pi\)
−0.816353 + 0.577554i \(0.804007\pi\)
\(678\) −4.74239 −0.182130
\(679\) 74.7245 2.86766
\(680\) −150.501 −5.77145
\(681\) 24.2269 0.928376
\(682\) −63.0355 −2.41375
\(683\) 5.98137 0.228871 0.114435 0.993431i \(-0.463494\pi\)
0.114435 + 0.993431i \(0.463494\pi\)
\(684\) 1.32467 0.0506501
\(685\) 36.1161 1.37993
\(686\) 192.930 7.36610
\(687\) 21.6490 0.825961
\(688\) −70.8647 −2.70169
\(689\) 31.0985 1.18476
\(690\) 60.7060 2.31104
\(691\) −1.62066 −0.0616528 −0.0308264 0.999525i \(-0.509814\pi\)
−0.0308264 + 0.999525i \(0.509814\pi\)
\(692\) −121.445 −4.61666
\(693\) 1.44323 0.0548237
\(694\) 84.0591 3.19084
\(695\) 44.7716 1.69828
\(696\) 45.7305 1.73341
\(697\) −17.1292 −0.648814
\(698\) 56.7652 2.14859
\(699\) −14.7193 −0.556733
\(700\) −235.888 −8.91573
\(701\) 17.3195 0.654148 0.327074 0.944999i \(-0.393937\pi\)
0.327074 + 0.944999i \(0.393937\pi\)
\(702\) 45.7542 1.72688
\(703\) −0.648278 −0.0244503
\(704\) 176.871 6.66608
\(705\) 66.1095 2.48983
\(706\) 69.0969 2.60050
\(707\) −51.9762 −1.95477
\(708\) −34.7137 −1.30462
\(709\) 14.3440 0.538702 0.269351 0.963042i \(-0.413191\pi\)
0.269351 + 0.963042i \(0.413191\pi\)
\(710\) 9.96624 0.374026
\(711\) −0.746363 −0.0279908
\(712\) 87.7704 3.28933
\(713\) 17.3810 0.650922
\(714\) −102.657 −3.84183
\(715\) −52.9414 −1.97990
\(716\) 64.5327 2.41170
\(717\) −41.4157 −1.54670
\(718\) 20.4754 0.764134
\(719\) −27.3999 −1.02184 −0.510922 0.859627i \(-0.670696\pi\)
−0.510922 + 0.859627i \(0.670696\pi\)
\(720\) 3.54514 0.132120
\(721\) −18.3147 −0.682075
\(722\) 9.21158 0.342820
\(723\) −11.2112 −0.416950
\(724\) 132.333 4.91813
\(725\) −21.1149 −0.784187
\(726\) −50.7362 −1.88300
\(727\) −39.0786 −1.44935 −0.724673 0.689093i \(-0.758009\pi\)
−0.724673 + 0.689093i \(0.758009\pi\)
\(728\) 166.354 6.16549
\(729\) 27.5120 1.01896
\(730\) 126.071 4.66611
\(731\) 17.6702 0.653557
\(732\) −72.2687 −2.67113
\(733\) −24.8531 −0.917969 −0.458985 0.888444i \(-0.651787\pi\)
−0.458985 + 0.888444i \(0.651787\pi\)
\(734\) 61.5845 2.27313
\(735\) −125.714 −4.63702
\(736\) −91.4169 −3.36967
\(737\) 21.1901 0.780547
\(738\) 0.678010 0.0249579
\(739\) 3.31251 0.121853 0.0609263 0.998142i \(-0.480595\pi\)
0.0609263 + 0.998142i \(0.480595\pi\)
\(740\) −3.33336 −0.122537
\(741\) −21.4076 −0.786429
\(742\) 142.595 5.23482
\(743\) 36.0903 1.32402 0.662012 0.749493i \(-0.269703\pi\)
0.662012 + 0.749493i \(0.269703\pi\)
\(744\) −84.6509 −3.10345
\(745\) −12.7464 −0.466993
\(746\) −76.5476 −2.80261
\(747\) −0.225560 −0.00825280
\(748\) −108.911 −3.98217
\(749\) −14.7191 −0.537825
\(750\) 51.0588 1.86440
\(751\) −28.7424 −1.04883 −0.524413 0.851464i \(-0.675715\pi\)
−0.524413 + 0.851464i \(0.675715\pi\)
\(752\) −177.597 −6.47630
\(753\) 16.5827 0.604307
\(754\) 23.0549 0.839610
\(755\) −84.2712 −3.06694
\(756\) 154.931 5.63479
\(757\) −28.2415 −1.02645 −0.513227 0.858253i \(-0.671550\pi\)
−0.513227 + 0.858253i \(0.671550\pi\)
\(758\) 7.22091 0.262275
\(759\) 28.3736 1.02990
\(760\) −143.908 −5.22009
\(761\) 18.9696 0.687647 0.343824 0.939034i \(-0.388278\pi\)
0.343824 + 0.939034i \(0.388278\pi\)
\(762\) −94.3024 −3.41621
\(763\) −31.0466 −1.12396
\(764\) 41.2700 1.49310
\(765\) −0.883986 −0.0319606
\(766\) −32.2469 −1.16513
\(767\) −11.3034 −0.408143
\(768\) 123.640 4.46149
\(769\) 4.46954 0.161176 0.0805878 0.996748i \(-0.474320\pi\)
0.0805878 + 0.996748i \(0.474320\pi\)
\(770\) −242.750 −8.74812
\(771\) −34.9351 −1.25816
\(772\) 9.76799 0.351558
\(773\) 18.6811 0.671914 0.335957 0.941877i \(-0.390940\pi\)
0.335957 + 0.941877i \(0.390940\pi\)
\(774\) −0.699425 −0.0251403
\(775\) 39.0854 1.40399
\(776\) 144.162 5.17510
\(777\) −1.46853 −0.0526832
\(778\) 1.18861 0.0426137
\(779\) −16.3788 −0.586832
\(780\) −110.075 −3.94133
\(781\) 4.65816 0.166682
\(782\) 40.6645 1.45416
\(783\) 13.8682 0.495610
\(784\) 337.718 12.0614
\(785\) −58.5995 −2.09151
\(786\) 30.1184 1.07429
\(787\) −6.14304 −0.218976 −0.109488 0.993988i \(-0.534921\pi\)
−0.109488 + 0.993988i \(0.534921\pi\)
\(788\) −83.7964 −2.98512
\(789\) 8.01522 0.285349
\(790\) 125.538 4.46644
\(791\) −5.22895 −0.185920
\(792\) 2.78434 0.0989370
\(793\) −23.5320 −0.835647
\(794\) −12.3646 −0.438803
\(795\) −60.9414 −2.16137
\(796\) 133.390 4.72789
\(797\) 15.3512 0.543769 0.271885 0.962330i \(-0.412353\pi\)
0.271885 + 0.962330i \(0.412353\pi\)
\(798\) −98.1597 −3.47481
\(799\) 44.2841 1.56666
\(800\) −205.573 −7.26812
\(801\) 0.515530 0.0182154
\(802\) 66.1320 2.33520
\(803\) 58.9249 2.07941
\(804\) 44.0583 1.55382
\(805\) 66.9342 2.35912
\(806\) −42.6765 −1.50322
\(807\) 36.9816 1.30181
\(808\) −100.275 −3.52765
\(809\) 21.8813 0.769306 0.384653 0.923061i \(-0.374321\pi\)
0.384653 + 0.923061i \(0.374321\pi\)
\(810\) −87.8896 −3.08813
\(811\) −11.3827 −0.399699 −0.199850 0.979827i \(-0.564045\pi\)
−0.199850 + 0.979827i \(0.564045\pi\)
\(812\) 78.0676 2.73964
\(813\) −4.26344 −0.149525
\(814\) −2.10971 −0.0739452
\(815\) −34.8711 −1.22148
\(816\) −117.861 −4.12594
\(817\) 16.8962 0.591122
\(818\) 84.7999 2.96496
\(819\) 0.977100 0.0341426
\(820\) −84.2178 −2.94101
\(821\) −46.0663 −1.60773 −0.803863 0.594814i \(-0.797226\pi\)
−0.803863 + 0.594814i \(0.797226\pi\)
\(822\) 47.5265 1.65768
\(823\) 8.08494 0.281823 0.140912 0.990022i \(-0.454997\pi\)
0.140912 + 0.990022i \(0.454997\pi\)
\(824\) −35.3335 −1.23090
\(825\) 63.8050 2.22141
\(826\) −51.8293 −1.80337
\(827\) −11.6961 −0.406712 −0.203356 0.979105i \(-0.565185\pi\)
−0.203356 + 0.979105i \(0.565185\pi\)
\(828\) −1.18866 −0.0413088
\(829\) 13.8619 0.481445 0.240722 0.970594i \(-0.422616\pi\)
0.240722 + 0.970594i \(0.422616\pi\)
\(830\) 37.9391 1.31688
\(831\) −42.0878 −1.46001
\(832\) 119.746 4.15144
\(833\) −84.2106 −2.91772
\(834\) 58.9165 2.04011
\(835\) 46.3424 1.60375
\(836\) −104.140 −3.60174
\(837\) −25.6712 −0.887327
\(838\) 93.1789 3.21881
\(839\) 26.4799 0.914189 0.457094 0.889418i \(-0.348890\pi\)
0.457094 + 0.889418i \(0.348890\pi\)
\(840\) −325.992 −11.2478
\(841\) −22.0120 −0.759034
\(842\) −83.1039 −2.86395
\(843\) 8.24893 0.284108
\(844\) −106.706 −3.67297
\(845\) 11.0070 0.378654
\(846\) −1.75286 −0.0602645
\(847\) −55.9416 −1.92217
\(848\) 163.714 5.62195
\(849\) −18.2241 −0.625449
\(850\) 91.4442 3.13651
\(851\) 0.581716 0.0199410
\(852\) 9.68521 0.331810
\(853\) −11.0461 −0.378212 −0.189106 0.981957i \(-0.560559\pi\)
−0.189106 + 0.981957i \(0.560559\pi\)
\(854\) −107.901 −3.69228
\(855\) −0.845262 −0.0289073
\(856\) −28.3967 −0.970581
\(857\) 4.68870 0.160163 0.0800815 0.996788i \(-0.474482\pi\)
0.0800815 + 0.996788i \(0.474482\pi\)
\(858\) −69.6674 −2.37841
\(859\) −54.8407 −1.87114 −0.935570 0.353142i \(-0.885113\pi\)
−0.935570 + 0.353142i \(0.885113\pi\)
\(860\) 86.8779 2.96251
\(861\) −37.1025 −1.26445
\(862\) −53.0322 −1.80628
\(863\) 21.8944 0.745295 0.372647 0.927973i \(-0.378450\pi\)
0.372647 + 0.927973i \(0.378450\pi\)
\(864\) 135.020 4.59349
\(865\) 77.4933 2.63485
\(866\) −3.88741 −0.132100
\(867\) 0.236077 0.00801759
\(868\) −144.510 −4.90497
\(869\) 58.6756 1.99043
\(870\) −45.1791 −1.53171
\(871\) 14.3462 0.486103
\(872\) −59.8964 −2.02835
\(873\) 0.846751 0.0286582
\(874\) 38.8831 1.31524
\(875\) 56.2972 1.90319
\(876\) 122.516 4.13944
\(877\) 27.0585 0.913699 0.456850 0.889544i \(-0.348978\pi\)
0.456850 + 0.889544i \(0.348978\pi\)
\(878\) −6.28298 −0.212040
\(879\) −26.2397 −0.885044
\(880\) −278.703 −9.39506
\(881\) 41.4136 1.39526 0.697629 0.716459i \(-0.254238\pi\)
0.697629 + 0.716459i \(0.254238\pi\)
\(882\) 3.33324 0.112236
\(883\) 10.5273 0.354271 0.177135 0.984187i \(-0.443317\pi\)
0.177135 + 0.984187i \(0.443317\pi\)
\(884\) −73.7351 −2.47998
\(885\) 22.1505 0.744582
\(886\) −31.1126 −1.04525
\(887\) −47.5602 −1.59691 −0.798457 0.602052i \(-0.794350\pi\)
−0.798457 + 0.602052i \(0.794350\pi\)
\(888\) −2.83315 −0.0950742
\(889\) −103.978 −3.48729
\(890\) −86.7119 −2.90659
\(891\) −41.0790 −1.37620
\(892\) −105.763 −3.54120
\(893\) 42.3441 1.41699
\(894\) −16.7735 −0.560988
\(895\) −41.1778 −1.37642
\(896\) 279.913 9.35125
\(897\) 19.2096 0.641390
\(898\) −28.1576 −0.939631
\(899\) −12.9354 −0.431419
\(900\) −2.67300 −0.0890999
\(901\) −40.8222 −1.35999
\(902\) −53.3020 −1.77476
\(903\) 38.2745 1.27369
\(904\) −10.0879 −0.335518
\(905\) −84.4407 −2.80690
\(906\) −110.895 −3.68425
\(907\) −4.84484 −0.160870 −0.0804351 0.996760i \(-0.525631\pi\)
−0.0804351 + 0.996760i \(0.525631\pi\)
\(908\) 79.7900 2.64793
\(909\) −0.588976 −0.0195351
\(910\) −164.348 −5.44808
\(911\) 10.8648 0.359968 0.179984 0.983670i \(-0.442395\pi\)
0.179984 + 0.983670i \(0.442395\pi\)
\(912\) −112.697 −3.73179
\(913\) 17.7325 0.586859
\(914\) 77.8086 2.57368
\(915\) 46.1140 1.52448
\(916\) 71.2999 2.35581
\(917\) 33.2085 1.09664
\(918\) −60.0604 −1.98229
\(919\) 26.1625 0.863020 0.431510 0.902108i \(-0.357981\pi\)
0.431510 + 0.902108i \(0.357981\pi\)
\(920\) 129.132 4.25737
\(921\) 7.83205 0.258075
\(922\) 63.6347 2.09570
\(923\) 3.15368 0.103805
\(924\) −235.905 −7.76071
\(925\) 1.30813 0.0430111
\(926\) −45.7457 −1.50330
\(927\) −0.207536 −0.00681636
\(928\) 68.0350 2.23336
\(929\) 28.4562 0.933619 0.466810 0.884358i \(-0.345403\pi\)
0.466810 + 0.884358i \(0.345403\pi\)
\(930\) 83.6301 2.74234
\(931\) −80.5216 −2.63899
\(932\) −48.4771 −1.58792
\(933\) −20.6614 −0.676424
\(934\) −57.1978 −1.87157
\(935\) 69.4949 2.27273
\(936\) 1.88506 0.0616152
\(937\) 19.7018 0.643631 0.321815 0.946802i \(-0.395707\pi\)
0.321815 + 0.946802i \(0.395707\pi\)
\(938\) 65.7811 2.14783
\(939\) 23.4942 0.766705
\(940\) 217.728 7.10151
\(941\) −4.46392 −0.145519 −0.0727597 0.997349i \(-0.523181\pi\)
−0.0727597 + 0.997349i \(0.523181\pi\)
\(942\) −77.1132 −2.51248
\(943\) 14.6971 0.478604
\(944\) −59.5054 −1.93673
\(945\) −98.8602 −3.21592
\(946\) 54.9856 1.78774
\(947\) 16.3653 0.531800 0.265900 0.964001i \(-0.414331\pi\)
0.265900 + 0.964001i \(0.414331\pi\)
\(948\) 121.998 3.96231
\(949\) 39.8936 1.29500
\(950\) 87.4383 2.83687
\(951\) 38.1160 1.23600
\(952\) −218.369 −7.07737
\(953\) −25.9709 −0.841281 −0.420640 0.907227i \(-0.638195\pi\)
−0.420640 + 0.907227i \(0.638195\pi\)
\(954\) 1.61583 0.0523145
\(955\) −26.3340 −0.852150
\(956\) −136.400 −4.41151
\(957\) −21.1164 −0.682596
\(958\) 55.3251 1.78747
\(959\) 52.4025 1.69217
\(960\) −234.657 −7.57353
\(961\) −7.05556 −0.227599
\(962\) −1.42832 −0.0460510
\(963\) −0.166792 −0.00537479
\(964\) −36.9236 −1.18923
\(965\) −6.23287 −0.200643
\(966\) 88.0811 2.83396
\(967\) 50.3073 1.61777 0.808886 0.587965i \(-0.200071\pi\)
0.808886 + 0.587965i \(0.200071\pi\)
\(968\) −107.925 −3.46883
\(969\) 28.1013 0.902744
\(970\) −142.423 −4.57293
\(971\) 4.13076 0.132562 0.0662811 0.997801i \(-0.478887\pi\)
0.0662811 + 0.997801i \(0.478887\pi\)
\(972\) 3.47724 0.111533
\(973\) 64.9611 2.08256
\(974\) 49.3168 1.58021
\(975\) 43.1975 1.38343
\(976\) −123.881 −3.96534
\(977\) 8.97941 0.287277 0.143638 0.989630i \(-0.454120\pi\)
0.143638 + 0.989630i \(0.454120\pi\)
\(978\) −45.8881 −1.46734
\(979\) −40.5286 −1.29530
\(980\) −414.032 −13.2258
\(981\) −0.351809 −0.0112324
\(982\) −102.455 −3.26947
\(983\) 34.7871 1.10953 0.554767 0.832005i \(-0.312807\pi\)
0.554767 + 0.832005i \(0.312807\pi\)
\(984\) −71.5798 −2.28188
\(985\) 53.4698 1.70369
\(986\) −30.2636 −0.963790
\(987\) 95.9212 3.05321
\(988\) −70.5050 −2.24306
\(989\) −15.1613 −0.482102
\(990\) −2.75076 −0.0874248
\(991\) 35.6277 1.13175 0.565876 0.824490i \(-0.308538\pi\)
0.565876 + 0.824490i \(0.308538\pi\)
\(992\) −125.938 −3.99854
\(993\) −43.3416 −1.37540
\(994\) 14.4605 0.458658
\(995\) −85.1152 −2.69833
\(996\) 36.8692 1.16825
\(997\) 13.1334 0.415940 0.207970 0.978135i \(-0.433314\pi\)
0.207970 + 0.978135i \(0.433314\pi\)
\(998\) −38.1745 −1.20839
\(999\) −0.859179 −0.0271832
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 8023.2.a.b.1.5 155
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8023.2.a.b.1.5 155 1.1 even 1 trivial