Properties

Label 6045.2.a.u.1.5
Level $6045$
Weight $2$
Character 6045.1
Self dual yes
Analytic conductor $48.270$
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] = [6045,2,Mod(1,6045)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6045, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("6045.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6045 = 3 \cdot 5 \cdot 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6045.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: \(48.2695680219\)
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: \( x^{9} - 3x^{8} - 7x^{7} + 20x^{6} + 20x^{5} - 38x^{4} - 27x^{3} + 13x^{2} + 6x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(0.139045\) of defining polynomial
Character \(\chi\) \(=\) 6045.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.139045 q^{2} +1.00000 q^{3} -1.98067 q^{4} +1.00000 q^{5} -0.139045 q^{6} -0.572827 q^{7} +0.553494 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-0.139045 q^{2} +1.00000 q^{3} -1.98067 q^{4} +1.00000 q^{5} -0.139045 q^{6} -0.572827 q^{7} +0.553494 q^{8} +1.00000 q^{9} -0.139045 q^{10} -2.21820 q^{11} -1.98067 q^{12} -1.00000 q^{13} +0.0796491 q^{14} +1.00000 q^{15} +3.88437 q^{16} +4.50920 q^{17} -0.139045 q^{18} -3.16026 q^{19} -1.98067 q^{20} -0.572827 q^{21} +0.308431 q^{22} -5.35633 q^{23} +0.553494 q^{24} +1.00000 q^{25} +0.139045 q^{26} +1.00000 q^{27} +1.13458 q^{28} -4.19191 q^{29} -0.139045 q^{30} -1.00000 q^{31} -1.64709 q^{32} -2.21820 q^{33} -0.626983 q^{34} -0.572827 q^{35} -1.98067 q^{36} +8.21655 q^{37} +0.439420 q^{38} -1.00000 q^{39} +0.553494 q^{40} +4.93542 q^{41} +0.0796491 q^{42} +8.80730 q^{43} +4.39352 q^{44} +1.00000 q^{45} +0.744774 q^{46} +3.29882 q^{47} +3.88437 q^{48} -6.67187 q^{49} -0.139045 q^{50} +4.50920 q^{51} +1.98067 q^{52} +4.15951 q^{53} -0.139045 q^{54} -2.21820 q^{55} -0.317056 q^{56} -3.16026 q^{57} +0.582866 q^{58} -7.12913 q^{59} -1.98067 q^{60} -7.14025 q^{61} +0.139045 q^{62} -0.572827 q^{63} -7.53972 q^{64} -1.00000 q^{65} +0.308431 q^{66} -6.43070 q^{67} -8.93121 q^{68} -5.35633 q^{69} +0.0796491 q^{70} -12.4404 q^{71} +0.553494 q^{72} -16.2281 q^{73} -1.14247 q^{74} +1.00000 q^{75} +6.25942 q^{76} +1.27065 q^{77} +0.139045 q^{78} +3.79218 q^{79} +3.88437 q^{80} +1.00000 q^{81} -0.686248 q^{82} -8.15054 q^{83} +1.13458 q^{84} +4.50920 q^{85} -1.22462 q^{86} -4.19191 q^{87} -1.22776 q^{88} +17.6272 q^{89} -0.139045 q^{90} +0.572827 q^{91} +10.6091 q^{92} -1.00000 q^{93} -0.458686 q^{94} -3.16026 q^{95} -1.64709 q^{96} +1.30778 q^{97} +0.927693 q^{98} -2.21820 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 3 q^{2} + 9 q^{3} + 5 q^{4} + 9 q^{5} - 3 q^{6} - 5 q^{7} - 18 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q - 3 q^{2} + 9 q^{3} + 5 q^{4} + 9 q^{5} - 3 q^{6} - 5 q^{7} - 18 q^{8} + 9 q^{9} - 3 q^{10} + 5 q^{12} - 9 q^{13} + 11 q^{14} + 9 q^{15} + 9 q^{16} - 5 q^{17} - 3 q^{18} - 12 q^{19} + 5 q^{20} - 5 q^{21} - 17 q^{22} - 21 q^{23} - 18 q^{24} + 9 q^{25} + 3 q^{26} + 9 q^{27} - 8 q^{28} - 15 q^{29} - 3 q^{30} - 9 q^{31} - 9 q^{32} + 9 q^{34} - 5 q^{35} + 5 q^{36} - 13 q^{37} - 24 q^{38} - 9 q^{39} - 18 q^{40} - 11 q^{41} + 11 q^{42} - 26 q^{43} + 5 q^{44} + 9 q^{45} + 14 q^{46} - 21 q^{47} + 9 q^{48} - 26 q^{49} - 3 q^{50} - 5 q^{51} - 5 q^{52} + 18 q^{53} - 3 q^{54} - 19 q^{56} - 12 q^{57} - 16 q^{58} - 24 q^{59} + 5 q^{60} - 4 q^{61} + 3 q^{62} - 5 q^{63} + 2 q^{64} - 9 q^{65} - 17 q^{66} - 20 q^{67} - 19 q^{68} - 21 q^{69} + 11 q^{70} - 13 q^{71} - 18 q^{72} + 15 q^{73} + 12 q^{74} + 9 q^{75} + 54 q^{76} - 28 q^{77} + 3 q^{78} - 15 q^{79} + 9 q^{80} + 9 q^{81} - 21 q^{82} - 5 q^{83} - 8 q^{84} - 5 q^{85} + 36 q^{86} - 15 q^{87} + 23 q^{88} + 31 q^{89} - 3 q^{90} + 5 q^{91} - 7 q^{92} - 9 q^{93} - 12 q^{95} - 9 q^{96} - 11 q^{97} - 23 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.139045 −0.0983200 −0.0491600 0.998791i \(-0.515654\pi\)
−0.0491600 + 0.998791i \(0.515654\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.98067 −0.990333
\(5\) 1.00000 0.447214
\(6\) −0.139045 −0.0567651
\(7\) −0.572827 −0.216508 −0.108254 0.994123i \(-0.534526\pi\)
−0.108254 + 0.994123i \(0.534526\pi\)
\(8\) 0.553494 0.195690
\(9\) 1.00000 0.333333
\(10\) −0.139045 −0.0439700
\(11\) −2.21820 −0.668813 −0.334406 0.942429i \(-0.608536\pi\)
−0.334406 + 0.942429i \(0.608536\pi\)
\(12\) −1.98067 −0.571769
\(13\) −1.00000 −0.277350
\(14\) 0.0796491 0.0212871
\(15\) 1.00000 0.258199
\(16\) 3.88437 0.971093
\(17\) 4.50920 1.09364 0.546820 0.837250i \(-0.315838\pi\)
0.546820 + 0.837250i \(0.315838\pi\)
\(18\) −0.139045 −0.0327733
\(19\) −3.16026 −0.725013 −0.362507 0.931981i \(-0.618079\pi\)
−0.362507 + 0.931981i \(0.618079\pi\)
\(20\) −1.98067 −0.442890
\(21\) −0.572827 −0.125001
\(22\) 0.308431 0.0657577
\(23\) −5.35633 −1.11687 −0.558436 0.829547i \(-0.688599\pi\)
−0.558436 + 0.829547i \(0.688599\pi\)
\(24\) 0.553494 0.112981
\(25\) 1.00000 0.200000
\(26\) 0.139045 0.0272691
\(27\) 1.00000 0.192450
\(28\) 1.13458 0.214415
\(29\) −4.19191 −0.778417 −0.389209 0.921150i \(-0.627252\pi\)
−0.389209 + 0.921150i \(0.627252\pi\)
\(30\) −0.139045 −0.0253861
\(31\) −1.00000 −0.179605
\(32\) −1.64709 −0.291167
\(33\) −2.21820 −0.386139
\(34\) −0.626983 −0.107527
\(35\) −0.572827 −0.0968255
\(36\) −1.98067 −0.330111
\(37\) 8.21655 1.35079 0.675396 0.737455i \(-0.263973\pi\)
0.675396 + 0.737455i \(0.263973\pi\)
\(38\) 0.439420 0.0712833
\(39\) −1.00000 −0.160128
\(40\) 0.553494 0.0875150
\(41\) 4.93542 0.770783 0.385392 0.922753i \(-0.374066\pi\)
0.385392 + 0.922753i \(0.374066\pi\)
\(42\) 0.0796491 0.0122901
\(43\) 8.80730 1.34310 0.671551 0.740959i \(-0.265629\pi\)
0.671551 + 0.740959i \(0.265629\pi\)
\(44\) 4.39352 0.662347
\(45\) 1.00000 0.149071
\(46\) 0.744774 0.109811
\(47\) 3.29882 0.481182 0.240591 0.970627i \(-0.422659\pi\)
0.240591 + 0.970627i \(0.422659\pi\)
\(48\) 3.88437 0.560661
\(49\) −6.67187 −0.953124
\(50\) −0.139045 −0.0196640
\(51\) 4.50920 0.631414
\(52\) 1.98067 0.274669
\(53\) 4.15951 0.571353 0.285676 0.958326i \(-0.407782\pi\)
0.285676 + 0.958326i \(0.407782\pi\)
\(54\) −0.139045 −0.0189217
\(55\) −2.21820 −0.299102
\(56\) −0.317056 −0.0423684
\(57\) −3.16026 −0.418587
\(58\) 0.582866 0.0765340
\(59\) −7.12913 −0.928134 −0.464067 0.885800i \(-0.653610\pi\)
−0.464067 + 0.885800i \(0.653610\pi\)
\(60\) −1.98067 −0.255703
\(61\) −7.14025 −0.914215 −0.457108 0.889411i \(-0.651115\pi\)
−0.457108 + 0.889411i \(0.651115\pi\)
\(62\) 0.139045 0.0176588
\(63\) −0.572827 −0.0721695
\(64\) −7.53972 −0.942465
\(65\) −1.00000 −0.124035
\(66\) 0.308431 0.0379652
\(67\) −6.43070 −0.785635 −0.392817 0.919617i \(-0.628500\pi\)
−0.392817 + 0.919617i \(0.628500\pi\)
\(68\) −8.93121 −1.08307
\(69\) −5.35633 −0.644827
\(70\) 0.0796491 0.00951988
\(71\) −12.4404 −1.47640 −0.738200 0.674582i \(-0.764324\pi\)
−0.738200 + 0.674582i \(0.764324\pi\)
\(72\) 0.553494 0.0652299
\(73\) −16.2281 −1.89936 −0.949680 0.313220i \(-0.898592\pi\)
−0.949680 + 0.313220i \(0.898592\pi\)
\(74\) −1.14247 −0.132810
\(75\) 1.00000 0.115470
\(76\) 6.25942 0.718005
\(77\) 1.27065 0.144804
\(78\) 0.139045 0.0157438
\(79\) 3.79218 0.426654 0.213327 0.976981i \(-0.431570\pi\)
0.213327 + 0.976981i \(0.431570\pi\)
\(80\) 3.88437 0.434286
\(81\) 1.00000 0.111111
\(82\) −0.686248 −0.0757834
\(83\) −8.15054 −0.894638 −0.447319 0.894375i \(-0.647621\pi\)
−0.447319 + 0.894375i \(0.647621\pi\)
\(84\) 1.13458 0.123793
\(85\) 4.50920 0.489091
\(86\) −1.22462 −0.132054
\(87\) −4.19191 −0.449420
\(88\) −1.22776 −0.130880
\(89\) 17.6272 1.86848 0.934238 0.356649i \(-0.116081\pi\)
0.934238 + 0.356649i \(0.116081\pi\)
\(90\) −0.139045 −0.0146567
\(91\) 0.572827 0.0600486
\(92\) 10.6091 1.10608
\(93\) −1.00000 −0.103695
\(94\) −0.458686 −0.0473099
\(95\) −3.16026 −0.324236
\(96\) −1.64709 −0.168106
\(97\) 1.30778 0.132785 0.0663924 0.997794i \(-0.478851\pi\)
0.0663924 + 0.997794i \(0.478851\pi\)
\(98\) 0.927693 0.0937112
\(99\) −2.21820 −0.222938
\(100\) −1.98067 −0.198067
\(101\) −15.0836 −1.50088 −0.750439 0.660939i \(-0.770158\pi\)
−0.750439 + 0.660939i \(0.770158\pi\)
\(102\) −0.626983 −0.0620806
\(103\) −13.1610 −1.29679 −0.648397 0.761303i \(-0.724560\pi\)
−0.648397 + 0.761303i \(0.724560\pi\)
\(104\) −0.553494 −0.0542745
\(105\) −0.572827 −0.0559022
\(106\) −0.578361 −0.0561754
\(107\) −4.09875 −0.396241 −0.198120 0.980178i \(-0.563484\pi\)
−0.198120 + 0.980178i \(0.563484\pi\)
\(108\) −1.98067 −0.190590
\(109\) 2.67095 0.255830 0.127915 0.991785i \(-0.459171\pi\)
0.127915 + 0.991785i \(0.459171\pi\)
\(110\) 0.308431 0.0294077
\(111\) 8.21655 0.779880
\(112\) −2.22507 −0.210250
\(113\) 13.5481 1.27450 0.637248 0.770659i \(-0.280073\pi\)
0.637248 + 0.770659i \(0.280073\pi\)
\(114\) 0.439420 0.0411554
\(115\) −5.35633 −0.499481
\(116\) 8.30277 0.770893
\(117\) −1.00000 −0.0924500
\(118\) 0.991273 0.0912541
\(119\) −2.58299 −0.236782
\(120\) 0.553494 0.0505268
\(121\) −6.07958 −0.552689
\(122\) 0.992820 0.0898857
\(123\) 4.93542 0.445012
\(124\) 1.98067 0.177869
\(125\) 1.00000 0.0894427
\(126\) 0.0796491 0.00709570
\(127\) 10.1759 0.902969 0.451484 0.892279i \(-0.350895\pi\)
0.451484 + 0.892279i \(0.350895\pi\)
\(128\) 4.34255 0.383831
\(129\) 8.80730 0.775440
\(130\) 0.139045 0.0121951
\(131\) 1.38145 0.120698 0.0603490 0.998177i \(-0.480779\pi\)
0.0603490 + 0.998177i \(0.480779\pi\)
\(132\) 4.39352 0.382406
\(133\) 1.81028 0.156971
\(134\) 0.894159 0.0772436
\(135\) 1.00000 0.0860663
\(136\) 2.49581 0.214014
\(137\) 0.0950928 0.00812433 0.00406216 0.999992i \(-0.498707\pi\)
0.00406216 + 0.999992i \(0.498707\pi\)
\(138\) 0.744774 0.0633994
\(139\) −13.9815 −1.18589 −0.592946 0.805242i \(-0.702035\pi\)
−0.592946 + 0.805242i \(0.702035\pi\)
\(140\) 1.13458 0.0958895
\(141\) 3.29882 0.277811
\(142\) 1.72978 0.145160
\(143\) 2.21820 0.185495
\(144\) 3.88437 0.323698
\(145\) −4.19191 −0.348119
\(146\) 2.25645 0.186745
\(147\) −6.67187 −0.550286
\(148\) −16.2742 −1.33773
\(149\) −1.12543 −0.0921987 −0.0460994 0.998937i \(-0.514679\pi\)
−0.0460994 + 0.998937i \(0.514679\pi\)
\(150\) −0.139045 −0.0113530
\(151\) 16.9006 1.37535 0.687674 0.726019i \(-0.258632\pi\)
0.687674 + 0.726019i \(0.258632\pi\)
\(152\) −1.74918 −0.141878
\(153\) 4.50920 0.364547
\(154\) −0.176678 −0.0142371
\(155\) −1.00000 −0.0803219
\(156\) 1.98067 0.158580
\(157\) −3.81444 −0.304426 −0.152213 0.988348i \(-0.548640\pi\)
−0.152213 + 0.988348i \(0.548640\pi\)
\(158\) −0.527286 −0.0419486
\(159\) 4.15951 0.329871
\(160\) −1.64709 −0.130214
\(161\) 3.06825 0.241812
\(162\) −0.139045 −0.0109244
\(163\) 14.5778 1.14182 0.570912 0.821011i \(-0.306590\pi\)
0.570912 + 0.821011i \(0.306590\pi\)
\(164\) −9.77542 −0.763332
\(165\) −2.21820 −0.172687
\(166\) 1.13330 0.0879608
\(167\) −0.499655 −0.0386645 −0.0193322 0.999813i \(-0.506154\pi\)
−0.0193322 + 0.999813i \(0.506154\pi\)
\(168\) −0.317056 −0.0244614
\(169\) 1.00000 0.0769231
\(170\) −0.626983 −0.0480874
\(171\) −3.16026 −0.241671
\(172\) −17.4443 −1.33012
\(173\) −3.60715 −0.274246 −0.137123 0.990554i \(-0.543786\pi\)
−0.137123 + 0.990554i \(0.543786\pi\)
\(174\) 0.582866 0.0441869
\(175\) −0.572827 −0.0433017
\(176\) −8.61632 −0.649479
\(177\) −7.12913 −0.535858
\(178\) −2.45098 −0.183709
\(179\) 2.39316 0.178873 0.0894364 0.995993i \(-0.471493\pi\)
0.0894364 + 0.995993i \(0.471493\pi\)
\(180\) −1.98067 −0.147630
\(181\) −12.1618 −0.903977 −0.451989 0.892024i \(-0.649285\pi\)
−0.451989 + 0.892024i \(0.649285\pi\)
\(182\) −0.0796491 −0.00590398
\(183\) −7.14025 −0.527822
\(184\) −2.96470 −0.218560
\(185\) 8.21655 0.604093
\(186\) 0.139045 0.0101953
\(187\) −10.0023 −0.731441
\(188\) −6.53386 −0.476531
\(189\) −0.572827 −0.0416671
\(190\) 0.439420 0.0318789
\(191\) −26.6319 −1.92702 −0.963509 0.267676i \(-0.913744\pi\)
−0.963509 + 0.267676i \(0.913744\pi\)
\(192\) −7.53972 −0.544133
\(193\) −19.3118 −1.39010 −0.695048 0.718963i \(-0.744617\pi\)
−0.695048 + 0.718963i \(0.744617\pi\)
\(194\) −0.181841 −0.0130554
\(195\) −1.00000 −0.0716115
\(196\) 13.2147 0.943910
\(197\) 10.7291 0.764420 0.382210 0.924076i \(-0.375163\pi\)
0.382210 + 0.924076i \(0.375163\pi\)
\(198\) 0.308431 0.0219192
\(199\) −1.32078 −0.0936278 −0.0468139 0.998904i \(-0.514907\pi\)
−0.0468139 + 0.998904i \(0.514907\pi\)
\(200\) 0.553494 0.0391379
\(201\) −6.43070 −0.453586
\(202\) 2.09731 0.147566
\(203\) 2.40124 0.168534
\(204\) −8.93121 −0.625310
\(205\) 4.93542 0.344705
\(206\) 1.82998 0.127501
\(207\) −5.35633 −0.372291
\(208\) −3.88437 −0.269333
\(209\) 7.01009 0.484898
\(210\) 0.0796491 0.00549631
\(211\) −25.0872 −1.72708 −0.863538 0.504284i \(-0.831757\pi\)
−0.863538 + 0.504284i \(0.831757\pi\)
\(212\) −8.23860 −0.565830
\(213\) −12.4404 −0.852400
\(214\) 0.569912 0.0389584
\(215\) 8.80730 0.600653
\(216\) 0.553494 0.0376605
\(217\) 0.572827 0.0388861
\(218\) −0.371383 −0.0251533
\(219\) −16.2281 −1.09660
\(220\) 4.39352 0.296211
\(221\) −4.50920 −0.303321
\(222\) −1.14247 −0.0766778
\(223\) −13.4404 −0.900039 −0.450019 0.893019i \(-0.648583\pi\)
−0.450019 + 0.893019i \(0.648583\pi\)
\(224\) 0.943499 0.0630402
\(225\) 1.00000 0.0666667
\(226\) −1.88380 −0.125308
\(227\) −26.9758 −1.79045 −0.895223 0.445618i \(-0.852984\pi\)
−0.895223 + 0.445618i \(0.852984\pi\)
\(228\) 6.25942 0.414540
\(229\) −2.87887 −0.190241 −0.0951206 0.995466i \(-0.530324\pi\)
−0.0951206 + 0.995466i \(0.530324\pi\)
\(230\) 0.744774 0.0491090
\(231\) 1.27065 0.0836024
\(232\) −2.32019 −0.152328
\(233\) −26.9954 −1.76853 −0.884264 0.466987i \(-0.845339\pi\)
−0.884264 + 0.466987i \(0.845339\pi\)
\(234\) 0.139045 0.00908969
\(235\) 3.29882 0.215191
\(236\) 14.1204 0.919162
\(237\) 3.79218 0.246329
\(238\) 0.359153 0.0232804
\(239\) 9.53702 0.616898 0.308449 0.951241i \(-0.400190\pi\)
0.308449 + 0.951241i \(0.400190\pi\)
\(240\) 3.88437 0.250735
\(241\) −21.8420 −1.40697 −0.703485 0.710711i \(-0.748374\pi\)
−0.703485 + 0.710711i \(0.748374\pi\)
\(242\) 0.845339 0.0543404
\(243\) 1.00000 0.0641500
\(244\) 14.1425 0.905378
\(245\) −6.67187 −0.426250
\(246\) −0.686248 −0.0437536
\(247\) 3.16026 0.201082
\(248\) −0.553494 −0.0351469
\(249\) −8.15054 −0.516519
\(250\) −0.139045 −0.00879401
\(251\) −28.5385 −1.80133 −0.900666 0.434512i \(-0.856921\pi\)
−0.900666 + 0.434512i \(0.856921\pi\)
\(252\) 1.13458 0.0714718
\(253\) 11.8814 0.746979
\(254\) −1.41492 −0.0887799
\(255\) 4.50920 0.282377
\(256\) 14.4756 0.904727
\(257\) 15.6846 0.978380 0.489190 0.872177i \(-0.337292\pi\)
0.489190 + 0.872177i \(0.337292\pi\)
\(258\) −1.22462 −0.0762412
\(259\) −4.70666 −0.292458
\(260\) 1.98067 0.122836
\(261\) −4.19191 −0.259472
\(262\) −0.192085 −0.0118670
\(263\) 7.54877 0.465477 0.232739 0.972539i \(-0.425231\pi\)
0.232739 + 0.972539i \(0.425231\pi\)
\(264\) −1.22776 −0.0755634
\(265\) 4.15951 0.255517
\(266\) −0.251712 −0.0154334
\(267\) 17.6272 1.07877
\(268\) 12.7371 0.778040
\(269\) 1.58459 0.0966139 0.0483069 0.998833i \(-0.484617\pi\)
0.0483069 + 0.998833i \(0.484617\pi\)
\(270\) −0.139045 −0.00846204
\(271\) 12.5812 0.764256 0.382128 0.924109i \(-0.375191\pi\)
0.382128 + 0.924109i \(0.375191\pi\)
\(272\) 17.5154 1.06203
\(273\) 0.572827 0.0346691
\(274\) −0.0132222 −0.000798784 0
\(275\) −2.21820 −0.133763
\(276\) 10.6091 0.638593
\(277\) 20.5670 1.23575 0.617875 0.786276i \(-0.287994\pi\)
0.617875 + 0.786276i \(0.287994\pi\)
\(278\) 1.94406 0.116597
\(279\) −1.00000 −0.0598684
\(280\) −0.317056 −0.0189477
\(281\) 22.9342 1.36814 0.684068 0.729418i \(-0.260209\pi\)
0.684068 + 0.729418i \(0.260209\pi\)
\(282\) −0.458686 −0.0273144
\(283\) −9.06800 −0.539037 −0.269518 0.962995i \(-0.586865\pi\)
−0.269518 + 0.962995i \(0.586865\pi\)
\(284\) 24.6402 1.46213
\(285\) −3.16026 −0.187198
\(286\) −0.308431 −0.0182379
\(287\) −2.82714 −0.166881
\(288\) −1.64709 −0.0970558
\(289\) 3.33284 0.196050
\(290\) 0.582866 0.0342270
\(291\) 1.30778 0.0766633
\(292\) 32.1425 1.88100
\(293\) −4.86408 −0.284163 −0.142081 0.989855i \(-0.545379\pi\)
−0.142081 + 0.989855i \(0.545379\pi\)
\(294\) 0.927693 0.0541042
\(295\) −7.12913 −0.415074
\(296\) 4.54781 0.264336
\(297\) −2.21820 −0.128713
\(298\) 0.156486 0.00906498
\(299\) 5.35633 0.309765
\(300\) −1.98067 −0.114354
\(301\) −5.04506 −0.290793
\(302\) −2.34995 −0.135224
\(303\) −15.0836 −0.866533
\(304\) −12.2756 −0.704055
\(305\) −7.14025 −0.408850
\(306\) −0.626983 −0.0358422
\(307\) 10.5924 0.604542 0.302271 0.953222i \(-0.402255\pi\)
0.302271 + 0.953222i \(0.402255\pi\)
\(308\) −2.51673 −0.143404
\(309\) −13.1610 −0.748704
\(310\) 0.139045 0.00789725
\(311\) −30.8772 −1.75089 −0.875443 0.483321i \(-0.839430\pi\)
−0.875443 + 0.483321i \(0.839430\pi\)
\(312\) −0.553494 −0.0313354
\(313\) −4.59814 −0.259902 −0.129951 0.991520i \(-0.541482\pi\)
−0.129951 + 0.991520i \(0.541482\pi\)
\(314\) 0.530381 0.0299311
\(315\) −0.572827 −0.0322752
\(316\) −7.51105 −0.422530
\(317\) 7.73196 0.434270 0.217135 0.976142i \(-0.430329\pi\)
0.217135 + 0.976142i \(0.430329\pi\)
\(318\) −0.578361 −0.0324329
\(319\) 9.29849 0.520616
\(320\) −7.53972 −0.421483
\(321\) −4.09875 −0.228770
\(322\) −0.426627 −0.0237750
\(323\) −14.2502 −0.792904
\(324\) −1.98067 −0.110037
\(325\) −1.00000 −0.0554700
\(326\) −2.02698 −0.112264
\(327\) 2.67095 0.147704
\(328\) 2.73172 0.150834
\(329\) −1.88965 −0.104180
\(330\) 0.308431 0.0169786
\(331\) 1.86476 0.102496 0.0512482 0.998686i \(-0.483680\pi\)
0.0512482 + 0.998686i \(0.483680\pi\)
\(332\) 16.1435 0.885989
\(333\) 8.21655 0.450264
\(334\) 0.0694748 0.00380149
\(335\) −6.43070 −0.351347
\(336\) −2.22507 −0.121388
\(337\) 21.4216 1.16691 0.583454 0.812146i \(-0.301701\pi\)
0.583454 + 0.812146i \(0.301701\pi\)
\(338\) −0.139045 −0.00756308
\(339\) 13.5481 0.735831
\(340\) −8.93121 −0.484363
\(341\) 2.21820 0.120122
\(342\) 0.439420 0.0237611
\(343\) 7.83162 0.422868
\(344\) 4.87479 0.262831
\(345\) −5.35633 −0.288375
\(346\) 0.501558 0.0269639
\(347\) 15.0132 0.805951 0.402975 0.915211i \(-0.367976\pi\)
0.402975 + 0.915211i \(0.367976\pi\)
\(348\) 8.30277 0.445075
\(349\) 13.2725 0.710461 0.355231 0.934779i \(-0.384402\pi\)
0.355231 + 0.934779i \(0.384402\pi\)
\(350\) 0.0796491 0.00425742
\(351\) −1.00000 −0.0533761
\(352\) 3.65358 0.194737
\(353\) −27.0581 −1.44016 −0.720079 0.693892i \(-0.755894\pi\)
−0.720079 + 0.693892i \(0.755894\pi\)
\(354\) 0.991273 0.0526856
\(355\) −12.4404 −0.660266
\(356\) −34.9136 −1.85041
\(357\) −2.58299 −0.136706
\(358\) −0.332757 −0.0175868
\(359\) −21.5595 −1.13787 −0.568934 0.822383i \(-0.692644\pi\)
−0.568934 + 0.822383i \(0.692644\pi\)
\(360\) 0.553494 0.0291717
\(361\) −9.01276 −0.474356
\(362\) 1.69104 0.0888791
\(363\) −6.07958 −0.319095
\(364\) −1.13458 −0.0594681
\(365\) −16.2281 −0.849420
\(366\) 0.992820 0.0518955
\(367\) 7.09284 0.370243 0.185121 0.982716i \(-0.440732\pi\)
0.185121 + 0.982716i \(0.440732\pi\)
\(368\) −20.8060 −1.08459
\(369\) 4.93542 0.256928
\(370\) −1.14247 −0.0593944
\(371\) −2.38268 −0.123703
\(372\) 1.98067 0.102693
\(373\) 12.3960 0.641839 0.320920 0.947106i \(-0.396008\pi\)
0.320920 + 0.947106i \(0.396008\pi\)
\(374\) 1.39077 0.0719153
\(375\) 1.00000 0.0516398
\(376\) 1.82588 0.0941624
\(377\) 4.19191 0.215894
\(378\) 0.0796491 0.00409671
\(379\) −29.2985 −1.50496 −0.752481 0.658614i \(-0.771143\pi\)
−0.752481 + 0.658614i \(0.771143\pi\)
\(380\) 6.25942 0.321101
\(381\) 10.1759 0.521329
\(382\) 3.70305 0.189464
\(383\) −22.5140 −1.15041 −0.575205 0.818009i \(-0.695078\pi\)
−0.575205 + 0.818009i \(0.695078\pi\)
\(384\) 4.34255 0.221605
\(385\) 1.27065 0.0647581
\(386\) 2.68522 0.136674
\(387\) 8.80730 0.447700
\(388\) −2.59027 −0.131501
\(389\) −19.1098 −0.968904 −0.484452 0.874818i \(-0.660981\pi\)
−0.484452 + 0.874818i \(0.660981\pi\)
\(390\) 0.139045 0.00704084
\(391\) −24.1528 −1.22146
\(392\) −3.69284 −0.186516
\(393\) 1.38145 0.0696851
\(394\) −1.49184 −0.0751578
\(395\) 3.79218 0.190806
\(396\) 4.39352 0.220782
\(397\) −32.3728 −1.62474 −0.812372 0.583140i \(-0.801824\pi\)
−0.812372 + 0.583140i \(0.801824\pi\)
\(398\) 0.183649 0.00920548
\(399\) 1.81028 0.0906275
\(400\) 3.88437 0.194219
\(401\) −37.5706 −1.87619 −0.938093 0.346383i \(-0.887410\pi\)
−0.938093 + 0.346383i \(0.887410\pi\)
\(402\) 0.894159 0.0445966
\(403\) 1.00000 0.0498135
\(404\) 29.8757 1.48637
\(405\) 1.00000 0.0496904
\(406\) −0.333881 −0.0165703
\(407\) −18.2260 −0.903427
\(408\) 2.49581 0.123561
\(409\) −14.2358 −0.703913 −0.351956 0.936016i \(-0.614483\pi\)
−0.351956 + 0.936016i \(0.614483\pi\)
\(410\) −0.686248 −0.0338914
\(411\) 0.0950928 0.00469058
\(412\) 26.0676 1.28426
\(413\) 4.08376 0.200949
\(414\) 0.744774 0.0366037
\(415\) −8.15054 −0.400094
\(416\) 1.64709 0.0807553
\(417\) −13.9815 −0.684675
\(418\) −0.974721 −0.0476752
\(419\) 4.18121 0.204266 0.102133 0.994771i \(-0.467433\pi\)
0.102133 + 0.994771i \(0.467433\pi\)
\(420\) 1.13458 0.0553618
\(421\) 36.2826 1.76830 0.884152 0.467199i \(-0.154737\pi\)
0.884152 + 0.467199i \(0.154737\pi\)
\(422\) 3.48827 0.169806
\(423\) 3.29882 0.160394
\(424\) 2.30226 0.111808
\(425\) 4.50920 0.218728
\(426\) 1.72978 0.0838080
\(427\) 4.09013 0.197935
\(428\) 8.11825 0.392410
\(429\) 2.21820 0.107096
\(430\) −1.22462 −0.0590562
\(431\) 26.4188 1.27255 0.636275 0.771462i \(-0.280474\pi\)
0.636275 + 0.771462i \(0.280474\pi\)
\(432\) 3.88437 0.186887
\(433\) −0.442744 −0.0212769 −0.0106385 0.999943i \(-0.503386\pi\)
−0.0106385 + 0.999943i \(0.503386\pi\)
\(434\) −0.0796491 −0.00382328
\(435\) −4.19191 −0.200987
\(436\) −5.29026 −0.253357
\(437\) 16.9274 0.809748
\(438\) 2.25645 0.107817
\(439\) −17.0399 −0.813270 −0.406635 0.913591i \(-0.633298\pi\)
−0.406635 + 0.913591i \(0.633298\pi\)
\(440\) −1.22776 −0.0585312
\(441\) −6.67187 −0.317708
\(442\) 0.626983 0.0298226
\(443\) 5.82115 0.276571 0.138285 0.990392i \(-0.455841\pi\)
0.138285 + 0.990392i \(0.455841\pi\)
\(444\) −16.2742 −0.772341
\(445\) 17.6272 0.835608
\(446\) 1.86883 0.0884918
\(447\) −1.12543 −0.0532310
\(448\) 4.31896 0.204052
\(449\) 25.9864 1.22637 0.613187 0.789938i \(-0.289887\pi\)
0.613187 + 0.789938i \(0.289887\pi\)
\(450\) −0.139045 −0.00655467
\(451\) −10.9478 −0.515510
\(452\) −26.8342 −1.26218
\(453\) 16.9006 0.794058
\(454\) 3.75086 0.176037
\(455\) 0.572827 0.0268546
\(456\) −1.74918 −0.0819130
\(457\) −1.19736 −0.0560100 −0.0280050 0.999608i \(-0.508915\pi\)
−0.0280050 + 0.999608i \(0.508915\pi\)
\(458\) 0.400294 0.0187045
\(459\) 4.50920 0.210471
\(460\) 10.6091 0.494652
\(461\) 13.2899 0.618974 0.309487 0.950904i \(-0.399843\pi\)
0.309487 + 0.950904i \(0.399843\pi\)
\(462\) −0.176678 −0.00821979
\(463\) 31.1387 1.44714 0.723569 0.690252i \(-0.242500\pi\)
0.723569 + 0.690252i \(0.242500\pi\)
\(464\) −16.2829 −0.755916
\(465\) −1.00000 −0.0463739
\(466\) 3.75359 0.173882
\(467\) 1.62106 0.0750136 0.0375068 0.999296i \(-0.488058\pi\)
0.0375068 + 0.999296i \(0.488058\pi\)
\(468\) 1.98067 0.0915563
\(469\) 3.68368 0.170097
\(470\) −0.458686 −0.0211576
\(471\) −3.81444 −0.175760
\(472\) −3.94593 −0.181626
\(473\) −19.5364 −0.898283
\(474\) −0.527286 −0.0242191
\(475\) −3.16026 −0.145003
\(476\) 5.11604 0.234493
\(477\) 4.15951 0.190451
\(478\) −1.32608 −0.0606534
\(479\) 17.2796 0.789526 0.394763 0.918783i \(-0.370827\pi\)
0.394763 + 0.918783i \(0.370827\pi\)
\(480\) −1.64709 −0.0751791
\(481\) −8.21655 −0.374642
\(482\) 3.03704 0.138333
\(483\) 3.06825 0.139610
\(484\) 12.0416 0.547347
\(485\) 1.30778 0.0593832
\(486\) −0.139045 −0.00630723
\(487\) −36.9963 −1.67646 −0.838230 0.545316i \(-0.816410\pi\)
−0.838230 + 0.545316i \(0.816410\pi\)
\(488\) −3.95208 −0.178902
\(489\) 14.5778 0.659232
\(490\) 0.927693 0.0419089
\(491\) −21.1470 −0.954351 −0.477176 0.878808i \(-0.658339\pi\)
−0.477176 + 0.878808i \(0.658339\pi\)
\(492\) −9.77542 −0.440710
\(493\) −18.9021 −0.851309
\(494\) −0.439420 −0.0197704
\(495\) −2.21820 −0.0997007
\(496\) −3.88437 −0.174413
\(497\) 7.12618 0.319653
\(498\) 1.13330 0.0507842
\(499\) −36.5425 −1.63587 −0.817935 0.575311i \(-0.804881\pi\)
−0.817935 + 0.575311i \(0.804881\pi\)
\(500\) −1.98067 −0.0885781
\(501\) −0.499655 −0.0223229
\(502\) 3.96814 0.177107
\(503\) −32.1191 −1.43212 −0.716059 0.698039i \(-0.754056\pi\)
−0.716059 + 0.698039i \(0.754056\pi\)
\(504\) −0.317056 −0.0141228
\(505\) −15.0836 −0.671213
\(506\) −1.65206 −0.0734430
\(507\) 1.00000 0.0444116
\(508\) −20.1551 −0.894240
\(509\) −9.09674 −0.403206 −0.201603 0.979467i \(-0.564615\pi\)
−0.201603 + 0.979467i \(0.564615\pi\)
\(510\) −0.626983 −0.0277633
\(511\) 9.29593 0.411228
\(512\) −10.6979 −0.472783
\(513\) −3.16026 −0.139529
\(514\) −2.18088 −0.0961943
\(515\) −13.1610 −0.579943
\(516\) −17.4443 −0.767944
\(517\) −7.31745 −0.321821
\(518\) 0.654440 0.0287545
\(519\) −3.60715 −0.158336
\(520\) −0.553494 −0.0242723
\(521\) 28.2263 1.23662 0.618309 0.785935i \(-0.287818\pi\)
0.618309 + 0.785935i \(0.287818\pi\)
\(522\) 0.582866 0.0255113
\(523\) 30.9402 1.35292 0.676461 0.736479i \(-0.263513\pi\)
0.676461 + 0.736479i \(0.263513\pi\)
\(524\) −2.73620 −0.119531
\(525\) −0.572827 −0.0250002
\(526\) −1.04962 −0.0457657
\(527\) −4.50920 −0.196424
\(528\) −8.61632 −0.374977
\(529\) 5.69032 0.247405
\(530\) −0.578361 −0.0251224
\(531\) −7.12913 −0.309378
\(532\) −3.58557 −0.155454
\(533\) −4.93542 −0.213777
\(534\) −2.45098 −0.106064
\(535\) −4.09875 −0.177204
\(536\) −3.55935 −0.153741
\(537\) 2.39316 0.103272
\(538\) −0.220329 −0.00949908
\(539\) 14.7995 0.637462
\(540\) −1.98067 −0.0852343
\(541\) 30.2412 1.30017 0.650085 0.759862i \(-0.274733\pi\)
0.650085 + 0.759862i \(0.274733\pi\)
\(542\) −1.74936 −0.0751416
\(543\) −12.1618 −0.521912
\(544\) −7.42706 −0.318433
\(545\) 2.67095 0.114411
\(546\) −0.0796491 −0.00340867
\(547\) 12.4204 0.531056 0.265528 0.964103i \(-0.414454\pi\)
0.265528 + 0.964103i \(0.414454\pi\)
\(548\) −0.188347 −0.00804579
\(549\) −7.14025 −0.304738
\(550\) 0.308431 0.0131515
\(551\) 13.2475 0.564363
\(552\) −2.96470 −0.126186
\(553\) −2.17227 −0.0923742
\(554\) −2.85974 −0.121499
\(555\) 8.21655 0.348773
\(556\) 27.6926 1.17443
\(557\) 10.6141 0.449735 0.224867 0.974389i \(-0.427805\pi\)
0.224867 + 0.974389i \(0.427805\pi\)
\(558\) 0.139045 0.00588626
\(559\) −8.80730 −0.372509
\(560\) −2.22507 −0.0940266
\(561\) −10.0023 −0.422298
\(562\) −3.18889 −0.134515
\(563\) −42.6666 −1.79818 −0.899091 0.437762i \(-0.855771\pi\)
−0.899091 + 0.437762i \(0.855771\pi\)
\(564\) −6.53386 −0.275125
\(565\) 13.5481 0.569972
\(566\) 1.26086 0.0529981
\(567\) −0.572827 −0.0240565
\(568\) −6.88567 −0.288916
\(569\) −21.1736 −0.887644 −0.443822 0.896115i \(-0.646378\pi\)
−0.443822 + 0.896115i \(0.646378\pi\)
\(570\) 0.439420 0.0184053
\(571\) 28.9731 1.21249 0.606244 0.795279i \(-0.292676\pi\)
0.606244 + 0.795279i \(0.292676\pi\)
\(572\) −4.39352 −0.183702
\(573\) −26.6319 −1.11256
\(574\) 0.393102 0.0164077
\(575\) −5.35633 −0.223375
\(576\) −7.53972 −0.314155
\(577\) 7.75154 0.322701 0.161350 0.986897i \(-0.448415\pi\)
0.161350 + 0.986897i \(0.448415\pi\)
\(578\) −0.463417 −0.0192756
\(579\) −19.3118 −0.802573
\(580\) 8.30277 0.344754
\(581\) 4.66885 0.193697
\(582\) −0.181841 −0.00753754
\(583\) −9.22663 −0.382128
\(584\) −8.98218 −0.371685
\(585\) −1.00000 −0.0413449
\(586\) 0.676328 0.0279389
\(587\) −8.81242 −0.363727 −0.181864 0.983324i \(-0.558213\pi\)
−0.181864 + 0.983324i \(0.558213\pi\)
\(588\) 13.2147 0.544967
\(589\) 3.16026 0.130216
\(590\) 0.991273 0.0408101
\(591\) 10.7291 0.441338
\(592\) 31.9161 1.31174
\(593\) 22.1476 0.909493 0.454746 0.890621i \(-0.349730\pi\)
0.454746 + 0.890621i \(0.349730\pi\)
\(594\) 0.308431 0.0126551
\(595\) −2.58299 −0.105892
\(596\) 2.22910 0.0913075
\(597\) −1.32078 −0.0540560
\(598\) −0.744774 −0.0304561
\(599\) 26.7739 1.09395 0.546977 0.837148i \(-0.315779\pi\)
0.546977 + 0.837148i \(0.315779\pi\)
\(600\) 0.553494 0.0225963
\(601\) 1.67854 0.0684689 0.0342344 0.999414i \(-0.489101\pi\)
0.0342344 + 0.999414i \(0.489101\pi\)
\(602\) 0.701493 0.0285907
\(603\) −6.43070 −0.261878
\(604\) −33.4744 −1.36205
\(605\) −6.07958 −0.247170
\(606\) 2.09731 0.0851975
\(607\) −15.2329 −0.618285 −0.309142 0.951016i \(-0.600042\pi\)
−0.309142 + 0.951016i \(0.600042\pi\)
\(608\) 5.20524 0.211100
\(609\) 2.40124 0.0973031
\(610\) 0.992820 0.0401981
\(611\) −3.29882 −0.133456
\(612\) −8.93121 −0.361023
\(613\) 35.5594 1.43623 0.718115 0.695925i \(-0.245005\pi\)
0.718115 + 0.695925i \(0.245005\pi\)
\(614\) −1.47283 −0.0594386
\(615\) 4.93542 0.199015
\(616\) 0.703295 0.0283366
\(617\) 19.1432 0.770675 0.385338 0.922776i \(-0.374085\pi\)
0.385338 + 0.922776i \(0.374085\pi\)
\(618\) 1.82998 0.0736126
\(619\) 11.7348 0.471660 0.235830 0.971794i \(-0.424219\pi\)
0.235830 + 0.971794i \(0.424219\pi\)
\(620\) 1.98067 0.0795455
\(621\) −5.35633 −0.214942
\(622\) 4.29334 0.172147
\(623\) −10.0973 −0.404541
\(624\) −3.88437 −0.155499
\(625\) 1.00000 0.0400000
\(626\) 0.639351 0.0255536
\(627\) 7.01009 0.279956
\(628\) 7.55514 0.301483
\(629\) 37.0500 1.47728
\(630\) 0.0796491 0.00317329
\(631\) 31.2055 1.24227 0.621135 0.783703i \(-0.286672\pi\)
0.621135 + 0.783703i \(0.286672\pi\)
\(632\) 2.09895 0.0834918
\(633\) −25.0872 −0.997128
\(634\) −1.07509 −0.0426974
\(635\) 10.1759 0.403820
\(636\) −8.23860 −0.326682
\(637\) 6.67187 0.264349
\(638\) −1.29291 −0.0511869
\(639\) −12.4404 −0.492133
\(640\) 4.34255 0.171654
\(641\) 7.15917 0.282770 0.141385 0.989955i \(-0.454844\pi\)
0.141385 + 0.989955i \(0.454844\pi\)
\(642\) 0.569912 0.0224926
\(643\) −30.5949 −1.20655 −0.603273 0.797535i \(-0.706137\pi\)
−0.603273 + 0.797535i \(0.706137\pi\)
\(644\) −6.07719 −0.239475
\(645\) 8.80730 0.346787
\(646\) 1.98143 0.0779583
\(647\) 20.6823 0.813107 0.406554 0.913627i \(-0.366730\pi\)
0.406554 + 0.913627i \(0.366730\pi\)
\(648\) 0.553494 0.0217433
\(649\) 15.8138 0.620748
\(650\) 0.139045 0.00545381
\(651\) 0.572827 0.0224509
\(652\) −28.8738 −1.13079
\(653\) 10.5815 0.414085 0.207043 0.978332i \(-0.433616\pi\)
0.207043 + 0.978332i \(0.433616\pi\)
\(654\) −0.371383 −0.0145222
\(655\) 1.38145 0.0539778
\(656\) 19.1710 0.748502
\(657\) −16.2281 −0.633120
\(658\) 0.262748 0.0102430
\(659\) 1.43861 0.0560403 0.0280202 0.999607i \(-0.491080\pi\)
0.0280202 + 0.999607i \(0.491080\pi\)
\(660\) 4.39352 0.171017
\(661\) 8.89638 0.346029 0.173015 0.984919i \(-0.444649\pi\)
0.173015 + 0.984919i \(0.444649\pi\)
\(662\) −0.259286 −0.0100775
\(663\) −4.50920 −0.175123
\(664\) −4.51127 −0.175071
\(665\) 1.81028 0.0701998
\(666\) −1.14247 −0.0442700
\(667\) 22.4533 0.869393
\(668\) 0.989650 0.0382907
\(669\) −13.4404 −0.519638
\(670\) 0.894159 0.0345444
\(671\) 15.8385 0.611439
\(672\) 0.943499 0.0363963
\(673\) −16.7089 −0.644080 −0.322040 0.946726i \(-0.604369\pi\)
−0.322040 + 0.946726i \(0.604369\pi\)
\(674\) −2.97857 −0.114730
\(675\) 1.00000 0.0384900
\(676\) −1.98067 −0.0761795
\(677\) −50.1028 −1.92561 −0.962803 0.270205i \(-0.912909\pi\)
−0.962803 + 0.270205i \(0.912909\pi\)
\(678\) −1.88380 −0.0723469
\(679\) −0.749131 −0.0287490
\(680\) 2.49581 0.0957100
\(681\) −26.9758 −1.03371
\(682\) −0.308431 −0.0118104
\(683\) −43.3776 −1.65980 −0.829899 0.557914i \(-0.811602\pi\)
−0.829899 + 0.557914i \(0.811602\pi\)
\(684\) 6.25942 0.239335
\(685\) 0.0950928 0.00363331
\(686\) −1.08895 −0.0415764
\(687\) −2.87887 −0.109836
\(688\) 34.2108 1.30428
\(689\) −4.15951 −0.158465
\(690\) 0.744774 0.0283531
\(691\) 26.5170 1.00875 0.504376 0.863484i \(-0.331722\pi\)
0.504376 + 0.863484i \(0.331722\pi\)
\(692\) 7.14455 0.271595
\(693\) 1.27065 0.0482679
\(694\) −2.08752 −0.0792411
\(695\) −13.9815 −0.530347
\(696\) −2.32019 −0.0879467
\(697\) 22.2548 0.842960
\(698\) −1.84548 −0.0698526
\(699\) −26.9954 −1.02106
\(700\) 1.13458 0.0428831
\(701\) −50.0706 −1.89114 −0.945570 0.325419i \(-0.894495\pi\)
−0.945570 + 0.325419i \(0.894495\pi\)
\(702\) 0.139045 0.00524793
\(703\) −25.9664 −0.979342
\(704\) 16.7246 0.630333
\(705\) 3.29882 0.124241
\(706\) 3.76231 0.141596
\(707\) 8.64032 0.324953
\(708\) 14.1204 0.530678
\(709\) −16.5320 −0.620874 −0.310437 0.950594i \(-0.600475\pi\)
−0.310437 + 0.950594i \(0.600475\pi\)
\(710\) 1.72978 0.0649174
\(711\) 3.79218 0.142218
\(712\) 9.75653 0.365641
\(713\) 5.35633 0.200596
\(714\) 0.359153 0.0134410
\(715\) 2.21820 0.0829560
\(716\) −4.74004 −0.177144
\(717\) 9.53702 0.356166
\(718\) 2.99775 0.111875
\(719\) −38.8188 −1.44770 −0.723849 0.689959i \(-0.757629\pi\)
−0.723849 + 0.689959i \(0.757629\pi\)
\(720\) 3.88437 0.144762
\(721\) 7.53899 0.280767
\(722\) 1.25318 0.0466387
\(723\) −21.8420 −0.812314
\(724\) 24.0884 0.895239
\(725\) −4.19191 −0.155683
\(726\) 0.845339 0.0313735
\(727\) 35.8064 1.32798 0.663992 0.747739i \(-0.268861\pi\)
0.663992 + 0.747739i \(0.268861\pi\)
\(728\) 0.317056 0.0117509
\(729\) 1.00000 0.0370370
\(730\) 2.25645 0.0835150
\(731\) 39.7138 1.46887
\(732\) 14.1425 0.522720
\(733\) 42.8390 1.58229 0.791147 0.611626i \(-0.209484\pi\)
0.791147 + 0.611626i \(0.209484\pi\)
\(734\) −0.986227 −0.0364023
\(735\) −6.67187 −0.246096
\(736\) 8.82237 0.325197
\(737\) 14.2646 0.525443
\(738\) −0.686248 −0.0252611
\(739\) −35.1972 −1.29475 −0.647374 0.762172i \(-0.724133\pi\)
−0.647374 + 0.762172i \(0.724133\pi\)
\(740\) −16.2742 −0.598253
\(741\) 3.16026 0.116095
\(742\) 0.331301 0.0121624
\(743\) −23.1843 −0.850551 −0.425275 0.905064i \(-0.639823\pi\)
−0.425275 + 0.905064i \(0.639823\pi\)
\(744\) −0.553494 −0.0202921
\(745\) −1.12543 −0.0412325
\(746\) −1.72360 −0.0631056
\(747\) −8.15054 −0.298213
\(748\) 19.8112 0.724370
\(749\) 2.34787 0.0857894
\(750\) −0.139045 −0.00507722
\(751\) 12.5973 0.459682 0.229841 0.973228i \(-0.426179\pi\)
0.229841 + 0.973228i \(0.426179\pi\)
\(752\) 12.8138 0.467273
\(753\) −28.5385 −1.04000
\(754\) −0.582866 −0.0212267
\(755\) 16.9006 0.615075
\(756\) 1.13458 0.0412643
\(757\) 5.43341 0.197481 0.0987403 0.995113i \(-0.468519\pi\)
0.0987403 + 0.995113i \(0.468519\pi\)
\(758\) 4.07382 0.147968
\(759\) 11.8814 0.431268
\(760\) −1.74918 −0.0634496
\(761\) −7.85036 −0.284575 −0.142288 0.989825i \(-0.545446\pi\)
−0.142288 + 0.989825i \(0.545446\pi\)
\(762\) −1.41492 −0.0512571
\(763\) −1.52999 −0.0553894
\(764\) 52.7489 1.90839
\(765\) 4.50920 0.163030
\(766\) 3.13046 0.113108
\(767\) 7.12913 0.257418
\(768\) 14.4756 0.522344
\(769\) −7.20271 −0.259736 −0.129868 0.991531i \(-0.541455\pi\)
−0.129868 + 0.991531i \(0.541455\pi\)
\(770\) −0.176678 −0.00636702
\(771\) 15.6846 0.564868
\(772\) 38.2503 1.37666
\(773\) −30.2615 −1.08843 −0.544215 0.838946i \(-0.683172\pi\)
−0.544215 + 0.838946i \(0.683172\pi\)
\(774\) −1.22462 −0.0440179
\(775\) −1.00000 −0.0359211
\(776\) 0.723847 0.0259846
\(777\) −4.70666 −0.168851
\(778\) 2.65713 0.0952626
\(779\) −15.5972 −0.558828
\(780\) 1.98067 0.0709192
\(781\) 27.5952 0.987435
\(782\) 3.35833 0.120094
\(783\) −4.19191 −0.149807
\(784\) −25.9160 −0.925572
\(785\) −3.81444 −0.136143
\(786\) −0.192085 −0.00685144
\(787\) 47.1870 1.68203 0.841017 0.541009i \(-0.181958\pi\)
0.841017 + 0.541009i \(0.181958\pi\)
\(788\) −21.2509 −0.757030
\(789\) 7.54877 0.268743
\(790\) −0.527286 −0.0187600
\(791\) −7.76071 −0.275939
\(792\) −1.22776 −0.0436266
\(793\) 7.14025 0.253558
\(794\) 4.50129 0.159745
\(795\) 4.15951 0.147523
\(796\) 2.61603 0.0927227
\(797\) −5.83478 −0.206679 −0.103339 0.994646i \(-0.532953\pi\)
−0.103339 + 0.994646i \(0.532953\pi\)
\(798\) −0.251712 −0.00891050
\(799\) 14.8750 0.526241
\(800\) −1.64709 −0.0582335
\(801\) 17.6272 0.622826
\(802\) 5.22402 0.184467
\(803\) 35.9973 1.27032
\(804\) 12.7371 0.449202
\(805\) 3.06825 0.108142
\(806\) −0.139045 −0.00489767
\(807\) 1.58459 0.0557801
\(808\) −8.34870 −0.293706
\(809\) 0.484590 0.0170373 0.00851864 0.999964i \(-0.497288\pi\)
0.00851864 + 0.999964i \(0.497288\pi\)
\(810\) −0.139045 −0.00488556
\(811\) −3.02746 −0.106309 −0.0531543 0.998586i \(-0.516928\pi\)
−0.0531543 + 0.998586i \(0.516928\pi\)
\(812\) −4.75605 −0.166905
\(813\) 12.5812 0.441243
\(814\) 2.53424 0.0888249
\(815\) 14.5778 0.510639
\(816\) 17.5154 0.613161
\(817\) −27.8334 −0.973766
\(818\) 1.97942 0.0692087
\(819\) 0.572827 0.0200162
\(820\) −9.77542 −0.341373
\(821\) −5.54554 −0.193541 −0.0967704 0.995307i \(-0.530851\pi\)
−0.0967704 + 0.995307i \(0.530851\pi\)
\(822\) −0.0132222 −0.000461178 0
\(823\) −10.9108 −0.380326 −0.190163 0.981753i \(-0.560902\pi\)
−0.190163 + 0.981753i \(0.560902\pi\)
\(824\) −7.28454 −0.253769
\(825\) −2.21820 −0.0772278
\(826\) −0.567829 −0.0197573
\(827\) −18.0861 −0.628917 −0.314458 0.949271i \(-0.601823\pi\)
−0.314458 + 0.949271i \(0.601823\pi\)
\(828\) 10.6091 0.368692
\(829\) 7.49750 0.260399 0.130199 0.991488i \(-0.458438\pi\)
0.130199 + 0.991488i \(0.458438\pi\)
\(830\) 1.13330 0.0393373
\(831\) 20.5670 0.713460
\(832\) 7.53972 0.261393
\(833\) −30.0848 −1.04238
\(834\) 1.94406 0.0673172
\(835\) −0.499655 −0.0172913
\(836\) −13.8846 −0.480211
\(837\) −1.00000 −0.0345651
\(838\) −0.581379 −0.0200834
\(839\) −13.1527 −0.454082 −0.227041 0.973885i \(-0.572905\pi\)
−0.227041 + 0.973885i \(0.572905\pi\)
\(840\) −0.317056 −0.0109395
\(841\) −11.4279 −0.394066
\(842\) −5.04493 −0.173860
\(843\) 22.9342 0.789894
\(844\) 49.6894 1.71038
\(845\) 1.00000 0.0344010
\(846\) −0.458686 −0.0157700
\(847\) 3.48255 0.119662
\(848\) 16.1571 0.554837
\(849\) −9.06800 −0.311213
\(850\) −0.626983 −0.0215053
\(851\) −44.0106 −1.50866
\(852\) 24.6402 0.844160
\(853\) −2.94671 −0.100893 −0.0504466 0.998727i \(-0.516064\pi\)
−0.0504466 + 0.998727i \(0.516064\pi\)
\(854\) −0.568714 −0.0194610
\(855\) −3.16026 −0.108079
\(856\) −2.26863 −0.0775402
\(857\) 48.8117 1.66738 0.833688 0.552235i \(-0.186225\pi\)
0.833688 + 0.552235i \(0.186225\pi\)
\(858\) −0.308431 −0.0105297
\(859\) 56.6764 1.93377 0.966887 0.255205i \(-0.0821429\pi\)
0.966887 + 0.255205i \(0.0821429\pi\)
\(860\) −17.4443 −0.594847
\(861\) −2.82714 −0.0963488
\(862\) −3.67342 −0.125117
\(863\) −14.2424 −0.484817 −0.242408 0.970174i \(-0.577937\pi\)
−0.242408 + 0.970174i \(0.577937\pi\)
\(864\) −1.64709 −0.0560352
\(865\) −3.60715 −0.122647
\(866\) 0.0615616 0.00209195
\(867\) 3.33284 0.113189
\(868\) −1.13458 −0.0385101
\(869\) −8.41183 −0.285352
\(870\) 0.582866 0.0197610
\(871\) 6.43070 0.217896
\(872\) 1.47835 0.0500633
\(873\) 1.30778 0.0442616
\(874\) −2.35368 −0.0796144
\(875\) −0.572827 −0.0193651
\(876\) 32.1425 1.08600
\(877\) 20.6287 0.696582 0.348291 0.937386i \(-0.386762\pi\)
0.348291 + 0.937386i \(0.386762\pi\)
\(878\) 2.36932 0.0799607
\(879\) −4.86408 −0.164061
\(880\) −8.61632 −0.290456
\(881\) 18.0664 0.608671 0.304336 0.952565i \(-0.401566\pi\)
0.304336 + 0.952565i \(0.401566\pi\)
\(882\) 0.927693 0.0312371
\(883\) 22.8825 0.770057 0.385029 0.922905i \(-0.374192\pi\)
0.385029 + 0.922905i \(0.374192\pi\)
\(884\) 8.93121 0.300389
\(885\) −7.12913 −0.239643
\(886\) −0.809404 −0.0271925
\(887\) 10.8095 0.362947 0.181474 0.983396i \(-0.441913\pi\)
0.181474 + 0.983396i \(0.441913\pi\)
\(888\) 4.54781 0.152614
\(889\) −5.82906 −0.195500
\(890\) −2.45098 −0.0821570
\(891\) −2.21820 −0.0743125
\(892\) 26.6210 0.891338
\(893\) −10.4251 −0.348864
\(894\) 0.156486 0.00523367
\(895\) 2.39316 0.0799944
\(896\) −2.48753 −0.0831026
\(897\) 5.35633 0.178843
\(898\) −3.61329 −0.120577
\(899\) 4.19191 0.139808
\(900\) −1.98067 −0.0660222
\(901\) 18.7560 0.624854
\(902\) 1.52224 0.0506849
\(903\) −5.04506 −0.167889
\(904\) 7.49878 0.249406
\(905\) −12.1618 −0.404271
\(906\) −2.34995 −0.0780718
\(907\) 31.4214 1.04333 0.521664 0.853151i \(-0.325311\pi\)
0.521664 + 0.853151i \(0.325311\pi\)
\(908\) 53.4301 1.77314
\(909\) −15.0836 −0.500293
\(910\) −0.0796491 −0.00264034
\(911\) 44.8119 1.48468 0.742342 0.670021i \(-0.233715\pi\)
0.742342 + 0.670021i \(0.233715\pi\)
\(912\) −12.2756 −0.406486
\(913\) 18.0795 0.598345
\(914\) 0.166487 0.00550691
\(915\) −7.14025 −0.236049
\(916\) 5.70208 0.188402
\(917\) −0.791334 −0.0261321
\(918\) −0.626983 −0.0206935
\(919\) 25.4826 0.840594 0.420297 0.907387i \(-0.361926\pi\)
0.420297 + 0.907387i \(0.361926\pi\)
\(920\) −2.96470 −0.0977432
\(921\) 10.5924 0.349033
\(922\) −1.84790 −0.0608575
\(923\) 12.4404 0.409480
\(924\) −2.51673 −0.0827942
\(925\) 8.21655 0.270158
\(926\) −4.32970 −0.142283
\(927\) −13.1610 −0.432264
\(928\) 6.90445 0.226650
\(929\) 42.5100 1.39471 0.697354 0.716727i \(-0.254361\pi\)
0.697354 + 0.716727i \(0.254361\pi\)
\(930\) 0.139045 0.00455948
\(931\) 21.0848 0.691028
\(932\) 53.4689 1.75143
\(933\) −30.8772 −1.01087
\(934\) −0.225401 −0.00737534
\(935\) −10.0023 −0.327110
\(936\) −0.553494 −0.0180915
\(937\) −46.2539 −1.51105 −0.755524 0.655121i \(-0.772618\pi\)
−0.755524 + 0.655121i \(0.772618\pi\)
\(938\) −0.512199 −0.0167239
\(939\) −4.59814 −0.150055
\(940\) −6.53386 −0.213111
\(941\) −43.3103 −1.41187 −0.705937 0.708274i \(-0.749474\pi\)
−0.705937 + 0.708274i \(0.749474\pi\)
\(942\) 0.530381 0.0172807
\(943\) −26.4358 −0.860867
\(944\) −27.6922 −0.901304
\(945\) −0.572827 −0.0186341
\(946\) 2.71644 0.0883192
\(947\) 8.06709 0.262145 0.131073 0.991373i \(-0.458158\pi\)
0.131073 + 0.991373i \(0.458158\pi\)
\(948\) −7.51105 −0.243948
\(949\) 16.2281 0.526788
\(950\) 0.439420 0.0142567
\(951\) 7.73196 0.250726
\(952\) −1.42967 −0.0463358
\(953\) −49.9390 −1.61768 −0.808842 0.588027i \(-0.799905\pi\)
−0.808842 + 0.588027i \(0.799905\pi\)
\(954\) −0.578361 −0.0187251
\(955\) −26.6319 −0.861789
\(956\) −18.8896 −0.610935
\(957\) 9.29849 0.300578
\(958\) −2.40265 −0.0776262
\(959\) −0.0544718 −0.00175899
\(960\) −7.53972 −0.243344
\(961\) 1.00000 0.0322581
\(962\) 1.14247 0.0368348
\(963\) −4.09875 −0.132080
\(964\) 43.2618 1.39337
\(965\) −19.3118 −0.621670
\(966\) −0.426627 −0.0137265
\(967\) −12.6369 −0.406377 −0.203188 0.979140i \(-0.565130\pi\)
−0.203188 + 0.979140i \(0.565130\pi\)
\(968\) −3.36501 −0.108156
\(969\) −14.2502 −0.457783
\(970\) −0.181841 −0.00583855
\(971\) −14.5601 −0.467255 −0.233628 0.972326i \(-0.575060\pi\)
−0.233628 + 0.972326i \(0.575060\pi\)
\(972\) −1.98067 −0.0635299
\(973\) 8.00896 0.256756
\(974\) 5.14416 0.164830
\(975\) −1.00000 −0.0320256
\(976\) −27.7354 −0.887788
\(977\) 14.8551 0.475257 0.237628 0.971356i \(-0.423630\pi\)
0.237628 + 0.971356i \(0.423630\pi\)
\(978\) −2.02698 −0.0648157
\(979\) −39.1006 −1.24966
\(980\) 13.2147 0.422130
\(981\) 2.67095 0.0852768
\(982\) 2.94040 0.0938318
\(983\) −0.0118985 −0.000379504 0 −0.000189752 1.00000i \(-0.500060\pi\)
−0.000189752 1.00000i \(0.500060\pi\)
\(984\) 2.73172 0.0870842
\(985\) 10.7291 0.341859
\(986\) 2.62826 0.0837007
\(987\) −1.88965 −0.0601484
\(988\) −6.25942 −0.199139
\(989\) −47.1749 −1.50007
\(990\) 0.308431 0.00980258
\(991\) 43.1316 1.37012 0.685060 0.728486i \(-0.259776\pi\)
0.685060 + 0.728486i \(0.259776\pi\)
\(992\) 1.64709 0.0522952
\(993\) 1.86476 0.0591764
\(994\) −0.990864 −0.0314283
\(995\) −1.32078 −0.0418716
\(996\) 16.1435 0.511526
\(997\) −34.5009 −1.09265 −0.546327 0.837572i \(-0.683974\pi\)
−0.546327 + 0.837572i \(0.683974\pi\)
\(998\) 5.08108 0.160839
\(999\) 8.21655 0.259960
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 6045.2.a.u.1.5 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6045.2.a.u.1.5 9 1.1 even 1 trivial