Properties

Label 8018.2.a.i.1.18
Level $8018$
Weight $2$
Character 8018.1
Self dual yes
Analytic conductor $64.024$
Analytic rank $0$
Dimension $43$
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] = [8018,2,Mod(1,8018)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8018, 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("8018.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8018 = 2 \cdot 19 \cdot 211 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8018.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: \(64.0240523407\)
Analytic rank: \(0\)
Dimension: \(43\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.18
Character \(\chi\) \(=\) 8018.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -0.978925 q^{3} +1.00000 q^{4} -2.13476 q^{5} +0.978925 q^{6} +5.08930 q^{7} -1.00000 q^{8} -2.04171 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -0.978925 q^{3} +1.00000 q^{4} -2.13476 q^{5} +0.978925 q^{6} +5.08930 q^{7} -1.00000 q^{8} -2.04171 q^{9} +2.13476 q^{10} +5.28067 q^{11} -0.978925 q^{12} +1.87541 q^{13} -5.08930 q^{14} +2.08977 q^{15} +1.00000 q^{16} +5.22020 q^{17} +2.04171 q^{18} +1.00000 q^{19} -2.13476 q^{20} -4.98204 q^{21} -5.28067 q^{22} +6.98184 q^{23} +0.978925 q^{24} -0.442792 q^{25} -1.87541 q^{26} +4.93545 q^{27} +5.08930 q^{28} +1.78976 q^{29} -2.08977 q^{30} +5.33522 q^{31} -1.00000 q^{32} -5.16938 q^{33} -5.22020 q^{34} -10.8644 q^{35} -2.04171 q^{36} +2.19410 q^{37} -1.00000 q^{38} -1.83589 q^{39} +2.13476 q^{40} +3.33373 q^{41} +4.98204 q^{42} +5.18624 q^{43} +5.28067 q^{44} +4.35856 q^{45} -6.98184 q^{46} +9.99600 q^{47} -0.978925 q^{48} +18.9009 q^{49} +0.442792 q^{50} -5.11018 q^{51} +1.87541 q^{52} -8.83272 q^{53} -4.93545 q^{54} -11.2730 q^{55} -5.08930 q^{56} -0.978925 q^{57} -1.78976 q^{58} +6.70453 q^{59} +2.08977 q^{60} +9.23027 q^{61} -5.33522 q^{62} -10.3909 q^{63} +1.00000 q^{64} -4.00355 q^{65} +5.16938 q^{66} -8.67592 q^{67} +5.22020 q^{68} -6.83469 q^{69} +10.8644 q^{70} -10.4463 q^{71} +2.04171 q^{72} +13.2292 q^{73} -2.19410 q^{74} +0.433460 q^{75} +1.00000 q^{76} +26.8749 q^{77} +1.83589 q^{78} +7.35322 q^{79} -2.13476 q^{80} +1.29368 q^{81} -3.33373 q^{82} +3.29471 q^{83} -4.98204 q^{84} -11.1439 q^{85} -5.18624 q^{86} -1.75204 q^{87} -5.28067 q^{88} -5.14339 q^{89} -4.35856 q^{90} +9.54452 q^{91} +6.98184 q^{92} -5.22278 q^{93} -9.99600 q^{94} -2.13476 q^{95} +0.978925 q^{96} +3.82425 q^{97} -18.9009 q^{98} -10.7816 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 43 q - 43 q^{2} + 43 q^{4} + 19 q^{7} - 43 q^{8} + 53 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 43 q - 43 q^{2} + 43 q^{4} + 19 q^{7} - 43 q^{8} + 53 q^{9} + 14 q^{11} + 3 q^{13} - 19 q^{14} + q^{15} + 43 q^{16} + 8 q^{17} - 53 q^{18} + 43 q^{19} - 14 q^{22} + 43 q^{23} + 97 q^{25} - 3 q^{26} + 15 q^{27} + 19 q^{28} - 25 q^{29} - q^{30} + 22 q^{31} - 43 q^{32} + 22 q^{33} - 8 q^{34} - 6 q^{35} + 53 q^{36} + 42 q^{37} - 43 q^{38} + 9 q^{39} - 40 q^{41} + 72 q^{43} + 14 q^{44} - q^{45} - 43 q^{46} + 27 q^{47} + 86 q^{49} - 97 q^{50} - 3 q^{51} + 3 q^{52} - 5 q^{53} - 15 q^{54} + 86 q^{55} - 19 q^{56} + 25 q^{58} - 43 q^{59} + q^{60} + 31 q^{61} - 22 q^{62} + 38 q^{63} + 43 q^{64} - 32 q^{65} - 22 q^{66} + 15 q^{67} + 8 q^{68} - 7 q^{69} + 6 q^{70} + 14 q^{71} - 53 q^{72} + 93 q^{73} - 42 q^{74} - 13 q^{75} + 43 q^{76} + 38 q^{77} - 9 q^{78} + 15 q^{79} + 43 q^{81} + 40 q^{82} + 34 q^{83} + 16 q^{85} - 72 q^{86} + 60 q^{87} - 14 q^{88} - 37 q^{89} + q^{90} - 3 q^{91} + 43 q^{92} + 19 q^{93} - 27 q^{94} + 27 q^{97} - 86 q^{98} - 24 q^{99}+O(q^{100}) \) 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
\(3\) −0.978925 −0.565182 −0.282591 0.959240i \(-0.591194\pi\)
−0.282591 + 0.959240i \(0.591194\pi\)
\(4\) 1.00000 0.500000
\(5\) −2.13476 −0.954695 −0.477347 0.878715i \(-0.658402\pi\)
−0.477347 + 0.878715i \(0.658402\pi\)
\(6\) 0.978925 0.399644
\(7\) 5.08930 1.92357 0.961787 0.273799i \(-0.0882804\pi\)
0.961787 + 0.273799i \(0.0882804\pi\)
\(8\) −1.00000 −0.353553
\(9\) −2.04171 −0.680569
\(10\) 2.13476 0.675071
\(11\) 5.28067 1.59218 0.796091 0.605177i \(-0.206898\pi\)
0.796091 + 0.605177i \(0.206898\pi\)
\(12\) −0.978925 −0.282591
\(13\) 1.87541 0.520145 0.260073 0.965589i \(-0.416253\pi\)
0.260073 + 0.965589i \(0.416253\pi\)
\(14\) −5.08930 −1.36017
\(15\) 2.08977 0.539577
\(16\) 1.00000 0.250000
\(17\) 5.22020 1.26608 0.633042 0.774117i \(-0.281806\pi\)
0.633042 + 0.774117i \(0.281806\pi\)
\(18\) 2.04171 0.481235
\(19\) 1.00000 0.229416
\(20\) −2.13476 −0.477347
\(21\) −4.98204 −1.08717
\(22\) −5.28067 −1.12584
\(23\) 6.98184 1.45581 0.727907 0.685676i \(-0.240493\pi\)
0.727907 + 0.685676i \(0.240493\pi\)
\(24\) 0.978925 0.199822
\(25\) −0.442792 −0.0885584
\(26\) −1.87541 −0.367798
\(27\) 4.93545 0.949828
\(28\) 5.08930 0.961787
\(29\) 1.78976 0.332350 0.166175 0.986096i \(-0.446858\pi\)
0.166175 + 0.986096i \(0.446858\pi\)
\(30\) −2.08977 −0.381538
\(31\) 5.33522 0.958235 0.479117 0.877751i \(-0.340957\pi\)
0.479117 + 0.877751i \(0.340957\pi\)
\(32\) −1.00000 −0.176777
\(33\) −5.16938 −0.899873
\(34\) −5.22020 −0.895257
\(35\) −10.8644 −1.83643
\(36\) −2.04171 −0.340284
\(37\) 2.19410 0.360708 0.180354 0.983602i \(-0.442276\pi\)
0.180354 + 0.983602i \(0.442276\pi\)
\(38\) −1.00000 −0.162221
\(39\) −1.83589 −0.293977
\(40\) 2.13476 0.337535
\(41\) 3.33373 0.520642 0.260321 0.965522i \(-0.416172\pi\)
0.260321 + 0.965522i \(0.416172\pi\)
\(42\) 4.98204 0.768745
\(43\) 5.18624 0.790893 0.395447 0.918489i \(-0.370590\pi\)
0.395447 + 0.918489i \(0.370590\pi\)
\(44\) 5.28067 0.796091
\(45\) 4.35856 0.649735
\(46\) −6.98184 −1.02942
\(47\) 9.99600 1.45807 0.729033 0.684479i \(-0.239970\pi\)
0.729033 + 0.684479i \(0.239970\pi\)
\(48\) −0.978925 −0.141296
\(49\) 18.9009 2.70014
\(50\) 0.442792 0.0626202
\(51\) −5.11018 −0.715569
\(52\) 1.87541 0.260073
\(53\) −8.83272 −1.21327 −0.606633 0.794982i \(-0.707480\pi\)
−0.606633 + 0.794982i \(0.707480\pi\)
\(54\) −4.93545 −0.671630
\(55\) −11.2730 −1.52005
\(56\) −5.08930 −0.680086
\(57\) −0.978925 −0.129662
\(58\) −1.78976 −0.235007
\(59\) 6.70453 0.872856 0.436428 0.899739i \(-0.356243\pi\)
0.436428 + 0.899739i \(0.356243\pi\)
\(60\) 2.08977 0.269788
\(61\) 9.23027 1.18182 0.590908 0.806739i \(-0.298770\pi\)
0.590908 + 0.806739i \(0.298770\pi\)
\(62\) −5.33522 −0.677574
\(63\) −10.3909 −1.30912
\(64\) 1.00000 0.125000
\(65\) −4.00355 −0.496580
\(66\) 5.16938 0.636306
\(67\) −8.67592 −1.05993 −0.529966 0.848019i \(-0.677795\pi\)
−0.529966 + 0.848019i \(0.677795\pi\)
\(68\) 5.22020 0.633042
\(69\) −6.83469 −0.822800
\(70\) 10.8644 1.29855
\(71\) −10.4463 −1.23975 −0.619874 0.784701i \(-0.712816\pi\)
−0.619874 + 0.784701i \(0.712816\pi\)
\(72\) 2.04171 0.240617
\(73\) 13.2292 1.54836 0.774178 0.632968i \(-0.218164\pi\)
0.774178 + 0.632968i \(0.218164\pi\)
\(74\) −2.19410 −0.255059
\(75\) 0.433460 0.0500516
\(76\) 1.00000 0.114708
\(77\) 26.8749 3.06268
\(78\) 1.83589 0.207873
\(79\) 7.35322 0.827302 0.413651 0.910436i \(-0.364253\pi\)
0.413651 + 0.910436i \(0.364253\pi\)
\(80\) −2.13476 −0.238674
\(81\) 1.29368 0.143743
\(82\) −3.33373 −0.368149
\(83\) 3.29471 0.361641 0.180820 0.983516i \(-0.442125\pi\)
0.180820 + 0.983516i \(0.442125\pi\)
\(84\) −4.98204 −0.543585
\(85\) −11.1439 −1.20872
\(86\) −5.18624 −0.559246
\(87\) −1.75204 −0.187838
\(88\) −5.28067 −0.562921
\(89\) −5.14339 −0.545199 −0.272599 0.962128i \(-0.587883\pi\)
−0.272599 + 0.962128i \(0.587883\pi\)
\(90\) −4.35856 −0.459432
\(91\) 9.54452 1.00054
\(92\) 6.98184 0.727907
\(93\) −5.22278 −0.541577
\(94\) −9.99600 −1.03101
\(95\) −2.13476 −0.219022
\(96\) 0.978925 0.0999111
\(97\) 3.82425 0.388294 0.194147 0.980972i \(-0.437806\pi\)
0.194147 + 0.980972i \(0.437806\pi\)
\(98\) −18.9009 −1.90928
\(99\) −10.7816 −1.08359
\(100\) −0.442792 −0.0442792
\(101\) −19.0554 −1.89608 −0.948040 0.318151i \(-0.896938\pi\)
−0.948040 + 0.318151i \(0.896938\pi\)
\(102\) 5.11018 0.505984
\(103\) −2.82087 −0.277949 −0.138974 0.990296i \(-0.544381\pi\)
−0.138974 + 0.990296i \(0.544381\pi\)
\(104\) −1.87541 −0.183899
\(105\) 10.6355 1.03792
\(106\) 8.83272 0.857909
\(107\) −6.38920 −0.617667 −0.308834 0.951116i \(-0.599939\pi\)
−0.308834 + 0.951116i \(0.599939\pi\)
\(108\) 4.93545 0.474914
\(109\) 17.3683 1.66358 0.831790 0.555091i \(-0.187316\pi\)
0.831790 + 0.555091i \(0.187316\pi\)
\(110\) 11.2730 1.07484
\(111\) −2.14786 −0.203866
\(112\) 5.08930 0.480893
\(113\) −9.76251 −0.918379 −0.459190 0.888338i \(-0.651860\pi\)
−0.459190 + 0.888338i \(0.651860\pi\)
\(114\) 0.978925 0.0916847
\(115\) −14.9046 −1.38986
\(116\) 1.78976 0.166175
\(117\) −3.82904 −0.353995
\(118\) −6.70453 −0.617202
\(119\) 26.5672 2.43541
\(120\) −2.08977 −0.190769
\(121\) 16.8855 1.53504
\(122\) −9.23027 −0.835670
\(123\) −3.26347 −0.294258
\(124\) 5.33522 0.479117
\(125\) 11.6191 1.03924
\(126\) 10.3909 0.925691
\(127\) −9.09037 −0.806640 −0.403320 0.915059i \(-0.632144\pi\)
−0.403320 + 0.915059i \(0.632144\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −5.07693 −0.446999
\(130\) 4.00355 0.351135
\(131\) 0.917354 0.0801496 0.0400748 0.999197i \(-0.487240\pi\)
0.0400748 + 0.999197i \(0.487240\pi\)
\(132\) −5.16938 −0.449936
\(133\) 5.08930 0.441298
\(134\) 8.67592 0.749485
\(135\) −10.5360 −0.906796
\(136\) −5.22020 −0.447629
\(137\) −10.6395 −0.908990 −0.454495 0.890749i \(-0.650180\pi\)
−0.454495 + 0.890749i \(0.650180\pi\)
\(138\) 6.83469 0.581808
\(139\) −3.85031 −0.326579 −0.163289 0.986578i \(-0.552210\pi\)
−0.163289 + 0.986578i \(0.552210\pi\)
\(140\) −10.8644 −0.918213
\(141\) −9.78533 −0.824073
\(142\) 10.4463 0.876634
\(143\) 9.90342 0.828165
\(144\) −2.04171 −0.170142
\(145\) −3.82071 −0.317292
\(146\) −13.2292 −1.09485
\(147\) −18.5026 −1.52607
\(148\) 2.19410 0.180354
\(149\) −13.9757 −1.14493 −0.572467 0.819928i \(-0.694014\pi\)
−0.572467 + 0.819928i \(0.694014\pi\)
\(150\) −0.433460 −0.0353919
\(151\) 9.97240 0.811542 0.405771 0.913975i \(-0.367003\pi\)
0.405771 + 0.913975i \(0.367003\pi\)
\(152\) −1.00000 −0.0811107
\(153\) −10.6581 −0.861658
\(154\) −26.8749 −2.16564
\(155\) −11.3894 −0.914821
\(156\) −1.83589 −0.146988
\(157\) −17.5990 −1.40455 −0.702276 0.711904i \(-0.747833\pi\)
−0.702276 + 0.711904i \(0.747833\pi\)
\(158\) −7.35322 −0.584991
\(159\) 8.64656 0.685717
\(160\) 2.13476 0.168768
\(161\) 35.5326 2.80036
\(162\) −1.29368 −0.101641
\(163\) −9.78312 −0.766273 −0.383137 0.923692i \(-0.625156\pi\)
−0.383137 + 0.923692i \(0.625156\pi\)
\(164\) 3.33373 0.260321
\(165\) 11.0354 0.859104
\(166\) −3.29471 −0.255719
\(167\) 0.717227 0.0555007 0.0277504 0.999615i \(-0.491166\pi\)
0.0277504 + 0.999615i \(0.491166\pi\)
\(168\) 4.98204 0.384373
\(169\) −9.48284 −0.729449
\(170\) 11.1439 0.854697
\(171\) −2.04171 −0.156133
\(172\) 5.18624 0.395447
\(173\) −20.6011 −1.56627 −0.783136 0.621850i \(-0.786381\pi\)
−0.783136 + 0.621850i \(0.786381\pi\)
\(174\) 1.75204 0.132822
\(175\) −2.25350 −0.170349
\(176\) 5.28067 0.398045
\(177\) −6.56324 −0.493323
\(178\) 5.14339 0.385514
\(179\) −26.2816 −1.96438 −0.982191 0.187884i \(-0.939837\pi\)
−0.982191 + 0.187884i \(0.939837\pi\)
\(180\) 4.35856 0.324868
\(181\) 7.14860 0.531351 0.265676 0.964063i \(-0.414405\pi\)
0.265676 + 0.964063i \(0.414405\pi\)
\(182\) −9.54452 −0.707487
\(183\) −9.03574 −0.667941
\(184\) −6.98184 −0.514708
\(185\) −4.68388 −0.344366
\(186\) 5.22278 0.382953
\(187\) 27.5662 2.01584
\(188\) 9.99600 0.729033
\(189\) 25.1180 1.82706
\(190\) 2.13476 0.154872
\(191\) −2.34949 −0.170003 −0.0850015 0.996381i \(-0.527090\pi\)
−0.0850015 + 0.996381i \(0.527090\pi\)
\(192\) −0.978925 −0.0706478
\(193\) −3.49526 −0.251594 −0.125797 0.992056i \(-0.540149\pi\)
−0.125797 + 0.992056i \(0.540149\pi\)
\(194\) −3.82425 −0.274565
\(195\) 3.91918 0.280658
\(196\) 18.9009 1.35007
\(197\) 19.8215 1.41222 0.706112 0.708100i \(-0.250448\pi\)
0.706112 + 0.708100i \(0.250448\pi\)
\(198\) 10.7816 0.766213
\(199\) −17.8989 −1.26882 −0.634408 0.772998i \(-0.718756\pi\)
−0.634408 + 0.772998i \(0.718756\pi\)
\(200\) 0.442792 0.0313101
\(201\) 8.49307 0.599055
\(202\) 19.0554 1.34073
\(203\) 9.10861 0.639299
\(204\) −5.11018 −0.357785
\(205\) −7.11673 −0.497054
\(206\) 2.82087 0.196539
\(207\) −14.2549 −0.990781
\(208\) 1.87541 0.130036
\(209\) 5.28067 0.365271
\(210\) −10.6355 −0.733917
\(211\) −1.00000 −0.0688428
\(212\) −8.83272 −0.606633
\(213\) 10.2261 0.700684
\(214\) 6.38920 0.436757
\(215\) −11.0714 −0.755062
\(216\) −4.93545 −0.335815
\(217\) 27.1525 1.84323
\(218\) −17.3683 −1.17633
\(219\) −12.9503 −0.875103
\(220\) −11.2730 −0.760023
\(221\) 9.79002 0.658548
\(222\) 2.14786 0.144155
\(223\) −15.6798 −1.04999 −0.524997 0.851104i \(-0.675934\pi\)
−0.524997 + 0.851104i \(0.675934\pi\)
\(224\) −5.08930 −0.340043
\(225\) 0.904051 0.0602701
\(226\) 9.76251 0.649392
\(227\) 6.17225 0.409667 0.204833 0.978797i \(-0.434335\pi\)
0.204833 + 0.978797i \(0.434335\pi\)
\(228\) −0.978925 −0.0648309
\(229\) 21.8685 1.44511 0.722555 0.691314i \(-0.242968\pi\)
0.722555 + 0.691314i \(0.242968\pi\)
\(230\) 14.9046 0.982778
\(231\) −26.3085 −1.73097
\(232\) −1.78976 −0.117503
\(233\) 10.9588 0.717934 0.358967 0.933350i \(-0.383129\pi\)
0.358967 + 0.933350i \(0.383129\pi\)
\(234\) 3.82904 0.250312
\(235\) −21.3391 −1.39201
\(236\) 6.70453 0.436428
\(237\) −7.19825 −0.467576
\(238\) −26.5672 −1.72209
\(239\) 4.07796 0.263781 0.131891 0.991264i \(-0.457895\pi\)
0.131891 + 0.991264i \(0.457895\pi\)
\(240\) 2.08977 0.134894
\(241\) 5.48349 0.353222 0.176611 0.984281i \(-0.443486\pi\)
0.176611 + 0.984281i \(0.443486\pi\)
\(242\) −16.8855 −1.08544
\(243\) −16.0728 −1.03107
\(244\) 9.23027 0.590908
\(245\) −40.3490 −2.57780
\(246\) 3.26347 0.208072
\(247\) 1.87541 0.119329
\(248\) −5.33522 −0.338787
\(249\) −3.22527 −0.204393
\(250\) −11.6191 −0.734854
\(251\) −4.22467 −0.266659 −0.133329 0.991072i \(-0.542567\pi\)
−0.133329 + 0.991072i \(0.542567\pi\)
\(252\) −10.3909 −0.654562
\(253\) 36.8688 2.31792
\(254\) 9.09037 0.570380
\(255\) 10.9090 0.683150
\(256\) 1.00000 0.0625000
\(257\) 10.8613 0.677507 0.338753 0.940875i \(-0.389995\pi\)
0.338753 + 0.940875i \(0.389995\pi\)
\(258\) 5.07693 0.316076
\(259\) 11.1664 0.693848
\(260\) −4.00355 −0.248290
\(261\) −3.65416 −0.226187
\(262\) −0.917354 −0.0566743
\(263\) −19.4914 −1.20189 −0.600944 0.799291i \(-0.705209\pi\)
−0.600944 + 0.799291i \(0.705209\pi\)
\(264\) 5.16938 0.318153
\(265\) 18.8557 1.15830
\(266\) −5.08930 −0.312045
\(267\) 5.03500 0.308137
\(268\) −8.67592 −0.529966
\(269\) 16.9894 1.03586 0.517931 0.855423i \(-0.326703\pi\)
0.517931 + 0.855423i \(0.326703\pi\)
\(270\) 10.5360 0.641201
\(271\) −6.65000 −0.403959 −0.201979 0.979390i \(-0.564737\pi\)
−0.201979 + 0.979390i \(0.564737\pi\)
\(272\) 5.22020 0.316521
\(273\) −9.34337 −0.565486
\(274\) 10.6395 0.642753
\(275\) −2.33824 −0.141001
\(276\) −6.83469 −0.411400
\(277\) −7.22848 −0.434317 −0.217159 0.976136i \(-0.569679\pi\)
−0.217159 + 0.976136i \(0.569679\pi\)
\(278\) 3.85031 0.230926
\(279\) −10.8930 −0.652144
\(280\) 10.8644 0.649274
\(281\) −22.6400 −1.35059 −0.675294 0.737549i \(-0.735983\pi\)
−0.675294 + 0.737549i \(0.735983\pi\)
\(282\) 9.78533 0.582708
\(283\) −23.6356 −1.40499 −0.702496 0.711688i \(-0.747931\pi\)
−0.702496 + 0.711688i \(0.747931\pi\)
\(284\) −10.4463 −0.619874
\(285\) 2.08977 0.123787
\(286\) −9.90342 −0.585601
\(287\) 16.9664 1.00149
\(288\) 2.04171 0.120309
\(289\) 10.2505 0.602971
\(290\) 3.82071 0.224360
\(291\) −3.74365 −0.219457
\(292\) 13.2292 0.774178
\(293\) −18.1258 −1.05892 −0.529459 0.848335i \(-0.677605\pi\)
−0.529459 + 0.848335i \(0.677605\pi\)
\(294\) 18.5026 1.07909
\(295\) −14.3126 −0.833311
\(296\) −2.19410 −0.127529
\(297\) 26.0625 1.51230
\(298\) 13.9757 0.809591
\(299\) 13.0938 0.757234
\(300\) 0.433460 0.0250258
\(301\) 26.3943 1.52134
\(302\) −9.97240 −0.573847
\(303\) 18.6538 1.07163
\(304\) 1.00000 0.0573539
\(305\) −19.7044 −1.12827
\(306\) 10.6581 0.609284
\(307\) −30.3152 −1.73018 −0.865090 0.501617i \(-0.832739\pi\)
−0.865090 + 0.501617i \(0.832739\pi\)
\(308\) 26.8749 1.53134
\(309\) 2.76142 0.157092
\(310\) 11.3894 0.646876
\(311\) 8.45107 0.479216 0.239608 0.970870i \(-0.422981\pi\)
0.239608 + 0.970870i \(0.422981\pi\)
\(312\) 1.83589 0.103937
\(313\) −2.58724 −0.146240 −0.0731198 0.997323i \(-0.523296\pi\)
−0.0731198 + 0.997323i \(0.523296\pi\)
\(314\) 17.5990 0.993169
\(315\) 22.1820 1.24981
\(316\) 7.35322 0.413651
\(317\) −16.5315 −0.928499 −0.464250 0.885704i \(-0.653676\pi\)
−0.464250 + 0.885704i \(0.653676\pi\)
\(318\) −8.64656 −0.484875
\(319\) 9.45111 0.529161
\(320\) −2.13476 −0.119337
\(321\) 6.25455 0.349095
\(322\) −35.5326 −1.98016
\(323\) 5.22020 0.290460
\(324\) 1.29368 0.0718713
\(325\) −0.830416 −0.0460632
\(326\) 9.78312 0.541837
\(327\) −17.0022 −0.940226
\(328\) −3.33373 −0.184075
\(329\) 50.8726 2.80470
\(330\) −11.0354 −0.607478
\(331\) 3.66999 0.201721 0.100860 0.994901i \(-0.467840\pi\)
0.100860 + 0.994901i \(0.467840\pi\)
\(332\) 3.29471 0.180820
\(333\) −4.47971 −0.245486
\(334\) −0.717227 −0.0392449
\(335\) 18.5210 1.01191
\(336\) −4.98204 −0.271793
\(337\) 14.4640 0.787905 0.393953 0.919131i \(-0.371107\pi\)
0.393953 + 0.919131i \(0.371107\pi\)
\(338\) 9.48284 0.515798
\(339\) 9.55676 0.519052
\(340\) −11.1439 −0.604362
\(341\) 28.1735 1.52568
\(342\) 2.04171 0.110403
\(343\) 60.5675 3.27034
\(344\) −5.18624 −0.279623
\(345\) 14.5904 0.785523
\(346\) 20.6011 1.10752
\(347\) −14.7330 −0.790909 −0.395455 0.918486i \(-0.629413\pi\)
−0.395455 + 0.918486i \(0.629413\pi\)
\(348\) −1.75204 −0.0939191
\(349\) −28.8350 −1.54350 −0.771751 0.635925i \(-0.780619\pi\)
−0.771751 + 0.635925i \(0.780619\pi\)
\(350\) 2.25350 0.120455
\(351\) 9.25600 0.494048
\(352\) −5.28067 −0.281461
\(353\) −4.72643 −0.251563 −0.125781 0.992058i \(-0.540144\pi\)
−0.125781 + 0.992058i \(0.540144\pi\)
\(354\) 6.56324 0.348832
\(355\) 22.3004 1.18358
\(356\) −5.14339 −0.272599
\(357\) −26.0073 −1.37645
\(358\) 26.2816 1.38903
\(359\) −11.0980 −0.585730 −0.292865 0.956154i \(-0.594609\pi\)
−0.292865 + 0.956154i \(0.594609\pi\)
\(360\) −4.35856 −0.229716
\(361\) 1.00000 0.0526316
\(362\) −7.14860 −0.375722
\(363\) −16.5296 −0.867578
\(364\) 9.54452 0.500269
\(365\) −28.2411 −1.47821
\(366\) 9.03574 0.472306
\(367\) −37.1694 −1.94023 −0.970114 0.242648i \(-0.921984\pi\)
−0.970114 + 0.242648i \(0.921984\pi\)
\(368\) 6.98184 0.363953
\(369\) −6.80651 −0.354333
\(370\) 4.68388 0.243503
\(371\) −44.9523 −2.33381
\(372\) −5.22278 −0.270789
\(373\) 17.7787 0.920546 0.460273 0.887777i \(-0.347752\pi\)
0.460273 + 0.887777i \(0.347752\pi\)
\(374\) −27.5662 −1.42541
\(375\) −11.3742 −0.587361
\(376\) −9.99600 −0.515504
\(377\) 3.35653 0.172870
\(378\) −25.1180 −1.29193
\(379\) −1.89896 −0.0975428 −0.0487714 0.998810i \(-0.515531\pi\)
−0.0487714 + 0.998810i \(0.515531\pi\)
\(380\) −2.13476 −0.109511
\(381\) 8.89878 0.455899
\(382\) 2.34949 0.120210
\(383\) −7.13976 −0.364825 −0.182412 0.983222i \(-0.558391\pi\)
−0.182412 + 0.983222i \(0.558391\pi\)
\(384\) 0.978925 0.0499555
\(385\) −57.3715 −2.92392
\(386\) 3.49526 0.177904
\(387\) −10.5888 −0.538257
\(388\) 3.82425 0.194147
\(389\) 16.0209 0.812290 0.406145 0.913809i \(-0.366873\pi\)
0.406145 + 0.913809i \(0.366873\pi\)
\(390\) −3.91918 −0.198455
\(391\) 36.4466 1.84318
\(392\) −18.9009 −0.954642
\(393\) −0.898021 −0.0452992
\(394\) −19.8215 −0.998593
\(395\) −15.6974 −0.789820
\(396\) −10.7816 −0.541794
\(397\) 9.33819 0.468670 0.234335 0.972156i \(-0.424709\pi\)
0.234335 + 0.972156i \(0.424709\pi\)
\(398\) 17.8989 0.897189
\(399\) −4.98204 −0.249414
\(400\) −0.442792 −0.0221396
\(401\) −26.2078 −1.30876 −0.654378 0.756168i \(-0.727069\pi\)
−0.654378 + 0.756168i \(0.727069\pi\)
\(402\) −8.49307 −0.423596
\(403\) 10.0057 0.498421
\(404\) −19.0554 −0.948040
\(405\) −2.76171 −0.137230
\(406\) −9.10861 −0.452053
\(407\) 11.5863 0.574312
\(408\) 5.11018 0.252992
\(409\) 29.1369 1.44073 0.720364 0.693597i \(-0.243975\pi\)
0.720364 + 0.693597i \(0.243975\pi\)
\(410\) 7.11673 0.351470
\(411\) 10.4152 0.513746
\(412\) −2.82087 −0.138974
\(413\) 34.1214 1.67900
\(414\) 14.2549 0.700588
\(415\) −7.03341 −0.345257
\(416\) −1.87541 −0.0919496
\(417\) 3.76916 0.184577
\(418\) −5.28067 −0.258286
\(419\) −0.725090 −0.0354230 −0.0177115 0.999843i \(-0.505638\pi\)
−0.0177115 + 0.999843i \(0.505638\pi\)
\(420\) 10.6355 0.518958
\(421\) 5.90920 0.287997 0.143998 0.989578i \(-0.454004\pi\)
0.143998 + 0.989578i \(0.454004\pi\)
\(422\) 1.00000 0.0486792
\(423\) −20.4089 −0.992314
\(424\) 8.83272 0.428955
\(425\) −2.31146 −0.112122
\(426\) −10.2261 −0.495458
\(427\) 46.9756 2.27331
\(428\) −6.38920 −0.308834
\(429\) −9.69470 −0.468065
\(430\) 11.0714 0.533909
\(431\) 5.64753 0.272032 0.136016 0.990707i \(-0.456570\pi\)
0.136016 + 0.990707i \(0.456570\pi\)
\(432\) 4.93545 0.237457
\(433\) −10.5446 −0.506741 −0.253370 0.967369i \(-0.581539\pi\)
−0.253370 + 0.967369i \(0.581539\pi\)
\(434\) −27.1525 −1.30336
\(435\) 3.74018 0.179328
\(436\) 17.3683 0.831790
\(437\) 6.98184 0.333987
\(438\) 12.9503 0.618791
\(439\) −13.3575 −0.637517 −0.318758 0.947836i \(-0.603266\pi\)
−0.318758 + 0.947836i \(0.603266\pi\)
\(440\) 11.2730 0.537418
\(441\) −38.5902 −1.83763
\(442\) −9.79002 −0.465664
\(443\) −16.3288 −0.775803 −0.387902 0.921701i \(-0.626800\pi\)
−0.387902 + 0.921701i \(0.626800\pi\)
\(444\) −2.14786 −0.101933
\(445\) 10.9799 0.520498
\(446\) 15.6798 0.742458
\(447\) 13.6812 0.647097
\(448\) 5.08930 0.240447
\(449\) −1.40367 −0.0662433 −0.0331217 0.999451i \(-0.510545\pi\)
−0.0331217 + 0.999451i \(0.510545\pi\)
\(450\) −0.904051 −0.0426174
\(451\) 17.6043 0.828956
\(452\) −9.76251 −0.459190
\(453\) −9.76223 −0.458669
\(454\) −6.17225 −0.289678
\(455\) −20.3753 −0.955208
\(456\) 0.978925 0.0458424
\(457\) 26.7909 1.25323 0.626613 0.779331i \(-0.284441\pi\)
0.626613 + 0.779331i \(0.284441\pi\)
\(458\) −21.8685 −1.02185
\(459\) 25.7641 1.20256
\(460\) −14.9046 −0.694929
\(461\) −27.7961 −1.29459 −0.647296 0.762239i \(-0.724100\pi\)
−0.647296 + 0.762239i \(0.724100\pi\)
\(462\) 26.3085 1.22398
\(463\) 34.6332 1.60954 0.804770 0.593587i \(-0.202289\pi\)
0.804770 + 0.593587i \(0.202289\pi\)
\(464\) 1.78976 0.0830874
\(465\) 11.1494 0.517041
\(466\) −10.9588 −0.507656
\(467\) 37.5224 1.73633 0.868166 0.496275i \(-0.165299\pi\)
0.868166 + 0.496275i \(0.165299\pi\)
\(468\) −3.82904 −0.176997
\(469\) −44.1543 −2.03886
\(470\) 21.3391 0.984298
\(471\) 17.2281 0.793829
\(472\) −6.70453 −0.308601
\(473\) 27.3868 1.25925
\(474\) 7.19825 0.330626
\(475\) −0.442792 −0.0203167
\(476\) 26.5672 1.21770
\(477\) 18.0338 0.825711
\(478\) −4.07796 −0.186522
\(479\) −12.3795 −0.565633 −0.282817 0.959174i \(-0.591269\pi\)
−0.282817 + 0.959174i \(0.591269\pi\)
\(480\) −2.08977 −0.0953846
\(481\) 4.11484 0.187620
\(482\) −5.48349 −0.249766
\(483\) −34.7838 −1.58272
\(484\) 16.8855 0.767521
\(485\) −8.16386 −0.370702
\(486\) 16.0728 0.729076
\(487\) 24.0676 1.09061 0.545303 0.838239i \(-0.316414\pi\)
0.545303 + 0.838239i \(0.316414\pi\)
\(488\) −9.23027 −0.417835
\(489\) 9.57694 0.433084
\(490\) 40.3490 1.82278
\(491\) 2.95851 0.133515 0.0667577 0.997769i \(-0.478735\pi\)
0.0667577 + 0.997769i \(0.478735\pi\)
\(492\) −3.26347 −0.147129
\(493\) 9.34289 0.420783
\(494\) −1.87541 −0.0843787
\(495\) 23.0161 1.03450
\(496\) 5.33522 0.239559
\(497\) −53.1643 −2.38475
\(498\) 3.22527 0.144528
\(499\) −1.40518 −0.0629046 −0.0314523 0.999505i \(-0.510013\pi\)
−0.0314523 + 0.999505i \(0.510013\pi\)
\(500\) 11.6191 0.519620
\(501\) −0.702112 −0.0313680
\(502\) 4.22467 0.188556
\(503\) 20.9828 0.935577 0.467789 0.883840i \(-0.345051\pi\)
0.467789 + 0.883840i \(0.345051\pi\)
\(504\) 10.3909 0.462845
\(505\) 40.6787 1.81018
\(506\) −36.8688 −1.63902
\(507\) 9.28298 0.412272
\(508\) −9.09037 −0.403320
\(509\) 34.1506 1.51370 0.756849 0.653590i \(-0.226738\pi\)
0.756849 + 0.653590i \(0.226738\pi\)
\(510\) −10.9090 −0.483060
\(511\) 67.3271 2.97838
\(512\) −1.00000 −0.0441942
\(513\) 4.93545 0.217905
\(514\) −10.8613 −0.479070
\(515\) 6.02189 0.265356
\(516\) −5.07693 −0.223500
\(517\) 52.7855 2.32151
\(518\) −11.1664 −0.490624
\(519\) 20.1669 0.885230
\(520\) 4.00355 0.175567
\(521\) −12.1120 −0.530635 −0.265317 0.964161i \(-0.585477\pi\)
−0.265317 + 0.964161i \(0.585477\pi\)
\(522\) 3.65416 0.159938
\(523\) 4.35232 0.190314 0.0951569 0.995462i \(-0.469665\pi\)
0.0951569 + 0.995462i \(0.469665\pi\)
\(524\) 0.917354 0.0400748
\(525\) 2.20601 0.0962780
\(526\) 19.4914 0.849864
\(527\) 27.8509 1.21321
\(528\) −5.16938 −0.224968
\(529\) 25.7460 1.11939
\(530\) −18.8557 −0.819041
\(531\) −13.6887 −0.594039
\(532\) 5.08930 0.220649
\(533\) 6.25212 0.270809
\(534\) −5.03500 −0.217886
\(535\) 13.6394 0.589683
\(536\) 8.67592 0.374743
\(537\) 25.7278 1.11023
\(538\) −16.9894 −0.732465
\(539\) 99.8096 4.29911
\(540\) −10.5360 −0.453398
\(541\) −36.2102 −1.55680 −0.778399 0.627770i \(-0.783968\pi\)
−0.778399 + 0.627770i \(0.783968\pi\)
\(542\) 6.65000 0.285642
\(543\) −6.99794 −0.300310
\(544\) −5.22020 −0.223814
\(545\) −37.0771 −1.58821
\(546\) 9.34337 0.399859
\(547\) 40.9074 1.74907 0.874537 0.484958i \(-0.161165\pi\)
0.874537 + 0.484958i \(0.161165\pi\)
\(548\) −10.6395 −0.454495
\(549\) −18.8455 −0.804306
\(550\) 2.33824 0.0997028
\(551\) 1.78976 0.0762462
\(552\) 6.83469 0.290904
\(553\) 37.4227 1.59138
\(554\) 7.22848 0.307109
\(555\) 4.58517 0.194629
\(556\) −3.85031 −0.163289
\(557\) 2.10069 0.0890093 0.0445046 0.999009i \(-0.485829\pi\)
0.0445046 + 0.999009i \(0.485829\pi\)
\(558\) 10.8930 0.461136
\(559\) 9.72632 0.411379
\(560\) −10.8644 −0.459106
\(561\) −26.9852 −1.13932
\(562\) 22.6400 0.955010
\(563\) −41.5667 −1.75183 −0.875913 0.482469i \(-0.839740\pi\)
−0.875913 + 0.482469i \(0.839740\pi\)
\(564\) −9.78533 −0.412037
\(565\) 20.8406 0.876772
\(566\) 23.6356 0.993479
\(567\) 6.58394 0.276499
\(568\) 10.4463 0.438317
\(569\) 35.9343 1.50644 0.753222 0.657767i \(-0.228499\pi\)
0.753222 + 0.657767i \(0.228499\pi\)
\(570\) −2.08977 −0.0875309
\(571\) −28.7145 −1.20166 −0.600831 0.799376i \(-0.705164\pi\)
−0.600831 + 0.799376i \(0.705164\pi\)
\(572\) 9.90342 0.414083
\(573\) 2.29997 0.0960828
\(574\) −16.9664 −0.708162
\(575\) −3.09150 −0.128924
\(576\) −2.04171 −0.0850711
\(577\) 38.4150 1.59924 0.799618 0.600509i \(-0.205035\pi\)
0.799618 + 0.600509i \(0.205035\pi\)
\(578\) −10.2505 −0.426365
\(579\) 3.42160 0.142197
\(580\) −3.82071 −0.158646
\(581\) 16.7677 0.695643
\(582\) 3.74365 0.155179
\(583\) −46.6426 −1.93174
\(584\) −13.2292 −0.547426
\(585\) 8.17408 0.337957
\(586\) 18.1258 0.748769
\(587\) 5.49408 0.226765 0.113382 0.993551i \(-0.463831\pi\)
0.113382 + 0.993551i \(0.463831\pi\)
\(588\) −18.5026 −0.763035
\(589\) 5.33522 0.219834
\(590\) 14.3126 0.589240
\(591\) −19.4038 −0.798164
\(592\) 2.19410 0.0901769
\(593\) −13.6976 −0.562494 −0.281247 0.959635i \(-0.590748\pi\)
−0.281247 + 0.959635i \(0.590748\pi\)
\(594\) −26.0625 −1.06936
\(595\) −56.7146 −2.32507
\(596\) −13.9757 −0.572467
\(597\) 17.5216 0.717113
\(598\) −13.0938 −0.535446
\(599\) −14.6629 −0.599108 −0.299554 0.954079i \(-0.596838\pi\)
−0.299554 + 0.954079i \(0.596838\pi\)
\(600\) −0.433460 −0.0176959
\(601\) 16.9750 0.692425 0.346213 0.938156i \(-0.387468\pi\)
0.346213 + 0.938156i \(0.387468\pi\)
\(602\) −26.3943 −1.07575
\(603\) 17.7137 0.721356
\(604\) 9.97240 0.405771
\(605\) −36.0464 −1.46550
\(606\) −18.6538 −0.757758
\(607\) −43.5340 −1.76699 −0.883496 0.468438i \(-0.844817\pi\)
−0.883496 + 0.468438i \(0.844817\pi\)
\(608\) −1.00000 −0.0405554
\(609\) −8.91664 −0.361321
\(610\) 19.7044 0.797809
\(611\) 18.7466 0.758406
\(612\) −10.6581 −0.430829
\(613\) −36.8696 −1.48915 −0.744574 0.667540i \(-0.767347\pi\)
−0.744574 + 0.667540i \(0.767347\pi\)
\(614\) 30.3152 1.22342
\(615\) 6.96674 0.280926
\(616\) −26.8749 −1.08282
\(617\) 5.40290 0.217512 0.108756 0.994068i \(-0.465313\pi\)
0.108756 + 0.994068i \(0.465313\pi\)
\(618\) −2.76142 −0.111081
\(619\) −7.57463 −0.304450 −0.152225 0.988346i \(-0.548644\pi\)
−0.152225 + 0.988346i \(0.548644\pi\)
\(620\) −11.3894 −0.457411
\(621\) 34.4585 1.38277
\(622\) −8.45107 −0.338857
\(623\) −26.1763 −1.04873
\(624\) −1.83589 −0.0734942
\(625\) −22.5900 −0.903599
\(626\) 2.58724 0.103407
\(627\) −5.16938 −0.206445
\(628\) −17.5990 −0.702276
\(629\) 11.4536 0.456687
\(630\) −22.1820 −0.883752
\(631\) −4.57626 −0.182178 −0.0910891 0.995843i \(-0.529035\pi\)
−0.0910891 + 0.995843i \(0.529035\pi\)
\(632\) −7.35322 −0.292495
\(633\) 0.978925 0.0389088
\(634\) 16.5315 0.656548
\(635\) 19.4058 0.770094
\(636\) 8.64656 0.342859
\(637\) 35.4470 1.40446
\(638\) −9.45111 −0.374173
\(639\) 21.3283 0.843734
\(640\) 2.13476 0.0843839
\(641\) −38.7681 −1.53125 −0.765624 0.643289i \(-0.777570\pi\)
−0.765624 + 0.643289i \(0.777570\pi\)
\(642\) −6.25455 −0.246847
\(643\) 37.7450 1.48852 0.744259 0.667891i \(-0.232803\pi\)
0.744259 + 0.667891i \(0.232803\pi\)
\(644\) 35.5326 1.40018
\(645\) 10.8380 0.426748
\(646\) −5.22020 −0.205386
\(647\) 22.0464 0.866733 0.433367 0.901218i \(-0.357326\pi\)
0.433367 + 0.901218i \(0.357326\pi\)
\(648\) −1.29368 −0.0508207
\(649\) 35.4044 1.38975
\(650\) 0.830416 0.0325716
\(651\) −26.5803 −1.04176
\(652\) −9.78312 −0.383137
\(653\) −37.6120 −1.47187 −0.735935 0.677052i \(-0.763257\pi\)
−0.735935 + 0.677052i \(0.763257\pi\)
\(654\) 17.0022 0.664840
\(655\) −1.95833 −0.0765184
\(656\) 3.33373 0.130160
\(657\) −27.0100 −1.05376
\(658\) −50.8726 −1.98322
\(659\) 11.3226 0.441067 0.220534 0.975379i \(-0.429220\pi\)
0.220534 + 0.975379i \(0.429220\pi\)
\(660\) 11.0354 0.429552
\(661\) −37.3545 −1.45292 −0.726460 0.687209i \(-0.758836\pi\)
−0.726460 + 0.687209i \(0.758836\pi\)
\(662\) −3.66999 −0.142638
\(663\) −9.58369 −0.372200
\(664\) −3.29471 −0.127859
\(665\) −10.8644 −0.421305
\(666\) 4.47971 0.173585
\(667\) 12.4958 0.483839
\(668\) 0.717227 0.0277504
\(669\) 15.3493 0.593438
\(670\) −18.5210 −0.715529
\(671\) 48.7420 1.88166
\(672\) 4.98204 0.192186
\(673\) −8.71929 −0.336104 −0.168052 0.985778i \(-0.553748\pi\)
−0.168052 + 0.985778i \(0.553748\pi\)
\(674\) −14.4640 −0.557133
\(675\) −2.18538 −0.0841152
\(676\) −9.48284 −0.364724
\(677\) 22.7762 0.875362 0.437681 0.899130i \(-0.355800\pi\)
0.437681 + 0.899130i \(0.355800\pi\)
\(678\) −9.55676 −0.367025
\(679\) 19.4627 0.746912
\(680\) 11.1439 0.427349
\(681\) −6.04217 −0.231536
\(682\) −28.1735 −1.07882
\(683\) −5.11802 −0.195836 −0.0979178 0.995195i \(-0.531218\pi\)
−0.0979178 + 0.995195i \(0.531218\pi\)
\(684\) −2.04171 −0.0780666
\(685\) 22.7127 0.867808
\(686\) −60.5675 −2.31248
\(687\) −21.4076 −0.816750
\(688\) 5.18624 0.197723
\(689\) −16.5650 −0.631075
\(690\) −14.5904 −0.555449
\(691\) −3.05694 −0.116291 −0.0581457 0.998308i \(-0.518519\pi\)
−0.0581457 + 0.998308i \(0.518519\pi\)
\(692\) −20.6011 −0.783136
\(693\) −54.8706 −2.08436
\(694\) 14.7330 0.559257
\(695\) 8.21949 0.311783
\(696\) 1.75204 0.0664108
\(697\) 17.4028 0.659177
\(698\) 28.8350 1.09142
\(699\) −10.7278 −0.405764
\(700\) −2.25350 −0.0851743
\(701\) −1.23988 −0.0468295 −0.0234148 0.999726i \(-0.507454\pi\)
−0.0234148 + 0.999726i \(0.507454\pi\)
\(702\) −9.25600 −0.349345
\(703\) 2.19410 0.0827520
\(704\) 5.28067 0.199023
\(705\) 20.8893 0.786738
\(706\) 4.72643 0.177882
\(707\) −96.9784 −3.64725
\(708\) −6.56324 −0.246661
\(709\) 11.9659 0.449388 0.224694 0.974429i \(-0.427862\pi\)
0.224694 + 0.974429i \(0.427862\pi\)
\(710\) −22.3004 −0.836918
\(711\) −15.0131 −0.563036
\(712\) 5.14339 0.192757
\(713\) 37.2497 1.39501
\(714\) 26.0073 0.973297
\(715\) −21.1414 −0.790645
\(716\) −26.2816 −0.982191
\(717\) −3.99202 −0.149085
\(718\) 11.0980 0.414174
\(719\) 12.6319 0.471091 0.235546 0.971863i \(-0.424312\pi\)
0.235546 + 0.971863i \(0.424312\pi\)
\(720\) 4.35856 0.162434
\(721\) −14.3563 −0.534655
\(722\) −1.00000 −0.0372161
\(723\) −5.36792 −0.199635
\(724\) 7.14860 0.265676
\(725\) −0.792490 −0.0294323
\(726\) 16.5296 0.613471
\(727\) 11.8067 0.437887 0.218944 0.975738i \(-0.429739\pi\)
0.218944 + 0.975738i \(0.429739\pi\)
\(728\) −9.54452 −0.353743
\(729\) 11.8530 0.438999
\(730\) 28.2411 1.04525
\(731\) 27.0732 1.00134
\(732\) −9.03574 −0.333971
\(733\) 43.1923 1.59534 0.797671 0.603093i \(-0.206065\pi\)
0.797671 + 0.603093i \(0.206065\pi\)
\(734\) 37.1694 1.37195
\(735\) 39.4987 1.45693
\(736\) −6.98184 −0.257354
\(737\) −45.8146 −1.68760
\(738\) 6.80651 0.250551
\(739\) 36.1192 1.32867 0.664334 0.747436i \(-0.268715\pi\)
0.664334 + 0.747436i \(0.268715\pi\)
\(740\) −4.68388 −0.172183
\(741\) −1.83589 −0.0674429
\(742\) 44.9523 1.65025
\(743\) −9.80509 −0.359714 −0.179857 0.983693i \(-0.557563\pi\)
−0.179857 + 0.983693i \(0.557563\pi\)
\(744\) 5.22278 0.191477
\(745\) 29.8348 1.09306
\(746\) −17.7787 −0.650924
\(747\) −6.72682 −0.246122
\(748\) 27.5662 1.00792
\(749\) −32.5165 −1.18813
\(750\) 11.3742 0.415327
\(751\) −6.10985 −0.222952 −0.111476 0.993767i \(-0.535558\pi\)
−0.111476 + 0.993767i \(0.535558\pi\)
\(752\) 9.99600 0.364517
\(753\) 4.13563 0.150711
\(754\) −3.35653 −0.122238
\(755\) −21.2887 −0.774775
\(756\) 25.1180 0.913532
\(757\) −44.1144 −1.60337 −0.801683 0.597749i \(-0.796062\pi\)
−0.801683 + 0.597749i \(0.796062\pi\)
\(758\) 1.89896 0.0689732
\(759\) −36.0917 −1.31005
\(760\) 2.13476 0.0774360
\(761\) −40.7679 −1.47784 −0.738918 0.673796i \(-0.764663\pi\)
−0.738918 + 0.673796i \(0.764663\pi\)
\(762\) −8.89878 −0.322369
\(763\) 88.3924 3.20002
\(764\) −2.34949 −0.0850015
\(765\) 22.7525 0.822620
\(766\) 7.13976 0.257970
\(767\) 12.5738 0.454012
\(768\) −0.978925 −0.0353239
\(769\) −10.0368 −0.361936 −0.180968 0.983489i \(-0.557923\pi\)
−0.180968 + 0.983489i \(0.557923\pi\)
\(770\) 57.3715 2.06752
\(771\) −10.6324 −0.382915
\(772\) −3.49526 −0.125797
\(773\) 6.78116 0.243901 0.121951 0.992536i \(-0.461085\pi\)
0.121951 + 0.992536i \(0.461085\pi\)
\(774\) 10.5888 0.380605
\(775\) −2.36239 −0.0848597
\(776\) −3.82425 −0.137283
\(777\) −10.9311 −0.392151
\(778\) −16.0209 −0.574376
\(779\) 3.33373 0.119443
\(780\) 3.91918 0.140329
\(781\) −55.1635 −1.97390
\(782\) −36.4466 −1.30333
\(783\) 8.83326 0.315675
\(784\) 18.9009 0.675034
\(785\) 37.5697 1.34092
\(786\) 0.898021 0.0320313
\(787\) −24.7677 −0.882874 −0.441437 0.897292i \(-0.645531\pi\)
−0.441437 + 0.897292i \(0.645531\pi\)
\(788\) 19.8215 0.706112
\(789\) 19.0806 0.679286
\(790\) 15.6974 0.558487
\(791\) −49.6843 −1.76657
\(792\) 10.7816 0.383106
\(793\) 17.3105 0.614715
\(794\) −9.33819 −0.331400
\(795\) −18.4584 −0.654650
\(796\) −17.8989 −0.634408
\(797\) −32.5089 −1.15153 −0.575763 0.817617i \(-0.695295\pi\)
−0.575763 + 0.817617i \(0.695295\pi\)
\(798\) 4.98204 0.176362
\(799\) 52.1811 1.84604
\(800\) 0.442792 0.0156551
\(801\) 10.5013 0.371045
\(802\) 26.2078 0.925430
\(803\) 69.8588 2.46526
\(804\) 8.49307 0.299527
\(805\) −75.8537 −2.67349
\(806\) −10.0057 −0.352437
\(807\) −16.6313 −0.585451
\(808\) 19.0554 0.670365
\(809\) −2.33485 −0.0820888 −0.0410444 0.999157i \(-0.513069\pi\)
−0.0410444 + 0.999157i \(0.513069\pi\)
\(810\) 2.76171 0.0970364
\(811\) 18.8579 0.662192 0.331096 0.943597i \(-0.392582\pi\)
0.331096 + 0.943597i \(0.392582\pi\)
\(812\) 9.10861 0.319649
\(813\) 6.50985 0.228310
\(814\) −11.5863 −0.406100
\(815\) 20.8846 0.731557
\(816\) −5.11018 −0.178892
\(817\) 5.18624 0.181443
\(818\) −29.1369 −1.01875
\(819\) −19.4871 −0.680935
\(820\) −7.11673 −0.248527
\(821\) −43.5777 −1.52087 −0.760436 0.649413i \(-0.775014\pi\)
−0.760436 + 0.649413i \(0.775014\pi\)
\(822\) −10.4152 −0.363273
\(823\) 35.1308 1.22458 0.612291 0.790632i \(-0.290248\pi\)
0.612291 + 0.790632i \(0.290248\pi\)
\(824\) 2.82087 0.0982697
\(825\) 2.28896 0.0796913
\(826\) −34.1214 −1.18723
\(827\) 49.4794 1.72057 0.860284 0.509814i \(-0.170286\pi\)
0.860284 + 0.509814i \(0.170286\pi\)
\(828\) −14.2549 −0.495391
\(829\) −6.54886 −0.227451 −0.113726 0.993512i \(-0.536278\pi\)
−0.113726 + 0.993512i \(0.536278\pi\)
\(830\) 7.03341 0.244133
\(831\) 7.07614 0.245469
\(832\) 1.87541 0.0650182
\(833\) 98.6668 3.41860
\(834\) −3.76916 −0.130515
\(835\) −1.53111 −0.0529862
\(836\) 5.28067 0.182636
\(837\) 26.3317 0.910158
\(838\) 0.725090 0.0250478
\(839\) −3.93998 −0.136023 −0.0680116 0.997685i \(-0.521665\pi\)
−0.0680116 + 0.997685i \(0.521665\pi\)
\(840\) −10.6355 −0.366958
\(841\) −25.7968 −0.889544
\(842\) −5.90920 −0.203644
\(843\) 22.1628 0.763329
\(844\) −1.00000 −0.0344214
\(845\) 20.2436 0.696401
\(846\) 20.4089 0.701672
\(847\) 85.9351 2.95276
\(848\) −8.83272 −0.303317
\(849\) 23.1375 0.794077
\(850\) 2.31146 0.0792825
\(851\) 15.3188 0.525123
\(852\) 10.2261 0.350342
\(853\) −24.5391 −0.840204 −0.420102 0.907477i \(-0.638006\pi\)
−0.420102 + 0.907477i \(0.638006\pi\)
\(854\) −46.9756 −1.60747
\(855\) 4.35856 0.149059
\(856\) 6.38920 0.218378
\(857\) 11.0934 0.378942 0.189471 0.981886i \(-0.439323\pi\)
0.189471 + 0.981886i \(0.439323\pi\)
\(858\) 9.69470 0.330972
\(859\) 0.198137 0.00676035 0.00338017 0.999994i \(-0.498924\pi\)
0.00338017 + 0.999994i \(0.498924\pi\)
\(860\) −11.0714 −0.377531
\(861\) −16.6088 −0.566026
\(862\) −5.64753 −0.192356
\(863\) −51.2565 −1.74479 −0.872396 0.488799i \(-0.837435\pi\)
−0.872396 + 0.488799i \(0.837435\pi\)
\(864\) −4.93545 −0.167907
\(865\) 43.9784 1.49531
\(866\) 10.5446 0.358320
\(867\) −10.0345 −0.340789
\(868\) 27.1525 0.921617
\(869\) 38.8299 1.31721
\(870\) −3.74018 −0.126804
\(871\) −16.2709 −0.551318
\(872\) −17.3683 −0.588164
\(873\) −7.80800 −0.264261
\(874\) −6.98184 −0.236164
\(875\) 59.1329 1.99906
\(876\) −12.9503 −0.437552
\(877\) 45.6100 1.54014 0.770070 0.637960i \(-0.220221\pi\)
0.770070 + 0.637960i \(0.220221\pi\)
\(878\) 13.3575 0.450792
\(879\) 17.7438 0.598482
\(880\) −11.2730 −0.380012
\(881\) −18.2693 −0.615507 −0.307753 0.951466i \(-0.599577\pi\)
−0.307753 + 0.951466i \(0.599577\pi\)
\(882\) 38.5902 1.29940
\(883\) −27.1128 −0.912417 −0.456209 0.889873i \(-0.650793\pi\)
−0.456209 + 0.889873i \(0.650793\pi\)
\(884\) 9.79002 0.329274
\(885\) 14.0109 0.470973
\(886\) 16.3288 0.548576
\(887\) −13.0938 −0.439647 −0.219824 0.975540i \(-0.570548\pi\)
−0.219824 + 0.975540i \(0.570548\pi\)
\(888\) 2.14786 0.0720774
\(889\) −46.2636 −1.55163
\(890\) −10.9799 −0.368048
\(891\) 6.83151 0.228864
\(892\) −15.6798 −0.524997
\(893\) 9.99600 0.334503
\(894\) −13.6812 −0.457566
\(895\) 56.1051 1.87539
\(896\) −5.08930 −0.170021
\(897\) −12.8179 −0.427976
\(898\) 1.40367 0.0468411
\(899\) 9.54875 0.318469
\(900\) 0.904051 0.0301350
\(901\) −46.1086 −1.53610
\(902\) −17.6043 −0.586160
\(903\) −25.8380 −0.859836
\(904\) 9.76251 0.324696
\(905\) −15.2606 −0.507278
\(906\) 9.76223 0.324328
\(907\) 41.6152 1.38181 0.690905 0.722946i \(-0.257212\pi\)
0.690905 + 0.722946i \(0.257212\pi\)
\(908\) 6.17225 0.204833
\(909\) 38.9055 1.29041
\(910\) 20.3753 0.675434
\(911\) 21.8548 0.724083 0.362042 0.932162i \(-0.382080\pi\)
0.362042 + 0.932162i \(0.382080\pi\)
\(912\) −0.978925 −0.0324154
\(913\) 17.3982 0.575798
\(914\) −26.7909 −0.886164
\(915\) 19.2892 0.637680
\(916\) 21.8685 0.722555
\(917\) 4.66869 0.154174
\(918\) −25.7641 −0.850340
\(919\) 15.3017 0.504757 0.252378 0.967629i \(-0.418787\pi\)
0.252378 + 0.967629i \(0.418787\pi\)
\(920\) 14.9046 0.491389
\(921\) 29.6763 0.977867
\(922\) 27.7961 0.915415
\(923\) −19.5911 −0.644849
\(924\) −26.3085 −0.865486
\(925\) −0.971529 −0.0319437
\(926\) −34.6332 −1.13812
\(927\) 5.75939 0.189163
\(928\) −1.78976 −0.0587517
\(929\) 30.7509 1.00890 0.504452 0.863440i \(-0.331695\pi\)
0.504452 + 0.863440i \(0.331695\pi\)
\(930\) −11.1494 −0.365603
\(931\) 18.9009 0.619454
\(932\) 10.9588 0.358967
\(933\) −8.27296 −0.270845
\(934\) −37.5224 −1.22777
\(935\) −58.8472 −1.92451
\(936\) 3.82904 0.125156
\(937\) 21.7682 0.711135 0.355568 0.934651i \(-0.384288\pi\)
0.355568 + 0.934651i \(0.384288\pi\)
\(938\) 44.1543 1.44169
\(939\) 2.53272 0.0826521
\(940\) −21.3391 −0.696004
\(941\) 14.4802 0.472041 0.236020 0.971748i \(-0.424157\pi\)
0.236020 + 0.971748i \(0.424157\pi\)
\(942\) −17.2281 −0.561322
\(943\) 23.2756 0.757957
\(944\) 6.70453 0.218214
\(945\) −53.6209 −1.74429
\(946\) −27.3868 −0.890421
\(947\) −26.3763 −0.857116 −0.428558 0.903514i \(-0.640978\pi\)
−0.428558 + 0.903514i \(0.640978\pi\)
\(948\) −7.19825 −0.233788
\(949\) 24.8101 0.805370
\(950\) 0.442792 0.0143661
\(951\) 16.1831 0.524771
\(952\) −26.5672 −0.861047
\(953\) −10.6034 −0.343478 −0.171739 0.985142i \(-0.554939\pi\)
−0.171739 + 0.985142i \(0.554939\pi\)
\(954\) −18.0338 −0.583866
\(955\) 5.01560 0.162301
\(956\) 4.07796 0.131891
\(957\) −9.25193 −0.299072
\(958\) 12.3795 0.399963
\(959\) −54.1474 −1.74851
\(960\) 2.08977 0.0674471
\(961\) −2.53539 −0.0817866
\(962\) −4.11484 −0.132668
\(963\) 13.0449 0.420365
\(964\) 5.48349 0.176611
\(965\) 7.46155 0.240196
\(966\) 34.7838 1.11915
\(967\) −1.59404 −0.0512610 −0.0256305 0.999671i \(-0.508159\pi\)
−0.0256305 + 0.999671i \(0.508159\pi\)
\(968\) −16.8855 −0.542719
\(969\) −5.11018 −0.164163
\(970\) 8.16386 0.262126
\(971\) −60.1533 −1.93041 −0.965206 0.261490i \(-0.915786\pi\)
−0.965206 + 0.261490i \(0.915786\pi\)
\(972\) −16.0728 −0.515534
\(973\) −19.5954 −0.628198
\(974\) −24.0676 −0.771176
\(975\) 0.812915 0.0260341
\(976\) 9.23027 0.295454
\(977\) 1.46029 0.0467187 0.0233594 0.999727i \(-0.492564\pi\)
0.0233594 + 0.999727i \(0.492564\pi\)
\(978\) −9.57694 −0.306237
\(979\) −27.1606 −0.868055
\(980\) −40.3490 −1.28890
\(981\) −35.4609 −1.13218
\(982\) −2.95851 −0.0944097
\(983\) 14.0859 0.449271 0.224635 0.974443i \(-0.427881\pi\)
0.224635 + 0.974443i \(0.427881\pi\)
\(984\) 3.26347 0.104036
\(985\) −42.3142 −1.34824
\(986\) −9.34289 −0.297538
\(987\) −49.8005 −1.58517
\(988\) 1.87541 0.0596647
\(989\) 36.2094 1.15139
\(990\) −23.0161 −0.731499
\(991\) 39.3962 1.25146 0.625730 0.780040i \(-0.284801\pi\)
0.625730 + 0.780040i \(0.284801\pi\)
\(992\) −5.33522 −0.169394
\(993\) −3.59264 −0.114009
\(994\) 53.1643 1.68627
\(995\) 38.2098 1.21133
\(996\) −3.22527 −0.102197
\(997\) 19.9684 0.632406 0.316203 0.948692i \(-0.397592\pi\)
0.316203 + 0.948692i \(0.397592\pi\)
\(998\) 1.40518 0.0444803
\(999\) 10.8289 0.342610
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 8018.2.a.i.1.18 43
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8018.2.a.i.1.18 43 1.1 even 1 trivial