Properties

Label 1045.2.a.f.1.6
Level $1045$
Weight $2$
Character 1045.1
Self dual yes
Analytic conductor $8.344$
Analytic rank $1$
Dimension $6$
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] = [1045,2,Mod(1,1045)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1045, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1045.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1045 = 5 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1045.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: \(8.34436701122\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.7281497.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 5x^{4} + 7x^{3} + 6x^{2} - 2x - 1 \) 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 1.6
Root \(-0.326248\) of defining polynomial
Character \(\chi\) \(=\) 1045.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.73890 q^{2} -0.901323 q^{3} +1.02379 q^{4} -1.00000 q^{5} -1.56731 q^{6} +2.22757 q^{7} -1.69754 q^{8} -2.18762 q^{9} +O(q^{10})\) \(q+1.73890 q^{2} -0.901323 q^{3} +1.02379 q^{4} -1.00000 q^{5} -1.56731 q^{6} +2.22757 q^{7} -1.69754 q^{8} -2.18762 q^{9} -1.73890 q^{10} -1.00000 q^{11} -0.922763 q^{12} -3.64023 q^{13} +3.87353 q^{14} +0.901323 q^{15} -4.99943 q^{16} +0.644551 q^{17} -3.80406 q^{18} -1.00000 q^{19} -1.02379 q^{20} -2.00776 q^{21} -1.73890 q^{22} -6.15983 q^{23} +1.53003 q^{24} +1.00000 q^{25} -6.33001 q^{26} +4.67572 q^{27} +2.28056 q^{28} -1.88204 q^{29} +1.56731 q^{30} -0.183675 q^{31} -5.29846 q^{32} +0.901323 q^{33} +1.12081 q^{34} -2.22757 q^{35} -2.23966 q^{36} -4.11428 q^{37} -1.73890 q^{38} +3.28102 q^{39} +1.69754 q^{40} -1.53371 q^{41} -3.49130 q^{42} -1.53577 q^{43} -1.02379 q^{44} +2.18762 q^{45} -10.7113 q^{46} +1.75837 q^{47} +4.50610 q^{48} -2.03793 q^{49} +1.73890 q^{50} -0.580948 q^{51} -3.72682 q^{52} -2.81491 q^{53} +8.13063 q^{54} +1.00000 q^{55} -3.78139 q^{56} +0.901323 q^{57} -3.27269 q^{58} -3.90068 q^{59} +0.922763 q^{60} +0.734057 q^{61} -0.319392 q^{62} -4.87307 q^{63} +0.785358 q^{64} +3.64023 q^{65} +1.56731 q^{66} -1.30264 q^{67} +0.659883 q^{68} +5.55199 q^{69} -3.87353 q^{70} +10.5493 q^{71} +3.71357 q^{72} -5.17599 q^{73} -7.15433 q^{74} -0.901323 q^{75} -1.02379 q^{76} -2.22757 q^{77} +5.70538 q^{78} +2.89974 q^{79} +4.99943 q^{80} +2.34852 q^{81} -2.66698 q^{82} +13.8209 q^{83} -2.05552 q^{84} -0.644551 q^{85} -2.67055 q^{86} +1.69633 q^{87} +1.69754 q^{88} -6.39884 q^{89} +3.80406 q^{90} -8.10886 q^{91} -6.30635 q^{92} +0.165550 q^{93} +3.05763 q^{94} +1.00000 q^{95} +4.77562 q^{96} +6.23352 q^{97} -3.54376 q^{98} +2.18762 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{2} - q^{3} + 4 q^{4} - 6 q^{5} + 5 q^{7} - 12 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 2 q^{2} - q^{3} + 4 q^{4} - 6 q^{5} + 5 q^{7} - 12 q^{8} + q^{9} + 2 q^{10} - 6 q^{11} + q^{12} - 5 q^{13} - 8 q^{14} + q^{15} + 4 q^{16} + q^{17} + 6 q^{18} - 6 q^{19} - 4 q^{20} - 21 q^{21} + 2 q^{22} + 4 q^{23} - q^{24} + 6 q^{25} - 14 q^{26} - 16 q^{27} + 10 q^{28} - 9 q^{29} - 21 q^{31} - q^{32} + q^{33} - 5 q^{35} - 28 q^{36} - 3 q^{37} + 2 q^{38} + 20 q^{39} + 12 q^{40} - 23 q^{41} + q^{42} + 7 q^{43} - 4 q^{44} - q^{45} - 12 q^{46} - 18 q^{47} - 3 q^{49} - 2 q^{50} - 16 q^{51} + 13 q^{52} - 17 q^{53} + q^{54} + 6 q^{55} - 2 q^{56} + q^{57} + 23 q^{58} - 29 q^{59} - q^{60} + 17 q^{61} + 2 q^{62} + 6 q^{63} - 18 q^{64} + 5 q^{65} + 8 q^{67} - q^{68} - 38 q^{69} + 8 q^{70} - 12 q^{71} + 13 q^{72} + 2 q^{73} - 37 q^{74} - q^{75} - 4 q^{76} - 5 q^{77} + q^{78} + 3 q^{79} - 4 q^{80} - 2 q^{81} + 24 q^{82} - 11 q^{83} - 3 q^{84} - q^{85} - 12 q^{86} - 12 q^{87} + 12 q^{88} - 22 q^{89} - 6 q^{90} - 18 q^{91} - 15 q^{92} + 18 q^{93} + 22 q^{94} + 6 q^{95} - 17 q^{96} - 2 q^{97} - q^{98} - q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.73890 1.22959 0.614795 0.788687i \(-0.289239\pi\)
0.614795 + 0.788687i \(0.289239\pi\)
\(3\) −0.901323 −0.520379 −0.260190 0.965558i \(-0.583785\pi\)
−0.260190 + 0.965558i \(0.583785\pi\)
\(4\) 1.02379 0.511894
\(5\) −1.00000 −0.447214
\(6\) −1.56731 −0.639853
\(7\) 2.22757 0.841943 0.420971 0.907074i \(-0.361689\pi\)
0.420971 + 0.907074i \(0.361689\pi\)
\(8\) −1.69754 −0.600171
\(9\) −2.18762 −0.729206
\(10\) −1.73890 −0.549890
\(11\) −1.00000 −0.301511
\(12\) −0.922763 −0.266379
\(13\) −3.64023 −1.00962 −0.504809 0.863231i \(-0.668437\pi\)
−0.504809 + 0.863231i \(0.668437\pi\)
\(14\) 3.87353 1.03525
\(15\) 0.901323 0.232721
\(16\) −4.99943 −1.24986
\(17\) 0.644551 0.156327 0.0781633 0.996941i \(-0.475094\pi\)
0.0781633 + 0.996941i \(0.475094\pi\)
\(18\) −3.80406 −0.896625
\(19\) −1.00000 −0.229416
\(20\) −1.02379 −0.228926
\(21\) −2.00776 −0.438129
\(22\) −1.73890 −0.370736
\(23\) −6.15983 −1.28441 −0.642206 0.766532i \(-0.721981\pi\)
−0.642206 + 0.766532i \(0.721981\pi\)
\(24\) 1.53003 0.312316
\(25\) 1.00000 0.200000
\(26\) −6.33001 −1.24142
\(27\) 4.67572 0.899842
\(28\) 2.28056 0.430985
\(29\) −1.88204 −0.349487 −0.174743 0.984614i \(-0.555910\pi\)
−0.174743 + 0.984614i \(0.555910\pi\)
\(30\) 1.56731 0.286151
\(31\) −0.183675 −0.0329889 −0.0164945 0.999864i \(-0.505251\pi\)
−0.0164945 + 0.999864i \(0.505251\pi\)
\(32\) −5.29846 −0.936644
\(33\) 0.901323 0.156900
\(34\) 1.12081 0.192218
\(35\) −2.22757 −0.376528
\(36\) −2.23966 −0.373276
\(37\) −4.11428 −0.676383 −0.338192 0.941077i \(-0.609815\pi\)
−0.338192 + 0.941077i \(0.609815\pi\)
\(38\) −1.73890 −0.282088
\(39\) 3.28102 0.525384
\(40\) 1.69754 0.268405
\(41\) −1.53371 −0.239525 −0.119763 0.992803i \(-0.538213\pi\)
−0.119763 + 0.992803i \(0.538213\pi\)
\(42\) −3.49130 −0.538720
\(43\) −1.53577 −0.234202 −0.117101 0.993120i \(-0.537360\pi\)
−0.117101 + 0.993120i \(0.537360\pi\)
\(44\) −1.02379 −0.154342
\(45\) 2.18762 0.326111
\(46\) −10.7113 −1.57930
\(47\) 1.75837 0.256484 0.128242 0.991743i \(-0.459067\pi\)
0.128242 + 0.991743i \(0.459067\pi\)
\(48\) 4.50610 0.650400
\(49\) −2.03793 −0.291133
\(50\) 1.73890 0.245918
\(51\) −0.580948 −0.0813491
\(52\) −3.72682 −0.516817
\(53\) −2.81491 −0.386658 −0.193329 0.981134i \(-0.561929\pi\)
−0.193329 + 0.981134i \(0.561929\pi\)
\(54\) 8.13063 1.10644
\(55\) 1.00000 0.134840
\(56\) −3.78139 −0.505309
\(57\) 0.901323 0.119383
\(58\) −3.27269 −0.429726
\(59\) −3.90068 −0.507825 −0.253913 0.967227i \(-0.581718\pi\)
−0.253913 + 0.967227i \(0.581718\pi\)
\(60\) 0.922763 0.119128
\(61\) 0.734057 0.0939863 0.0469932 0.998895i \(-0.485036\pi\)
0.0469932 + 0.998895i \(0.485036\pi\)
\(62\) −0.319392 −0.0405629
\(63\) −4.87307 −0.613949
\(64\) 0.785358 0.0981698
\(65\) 3.64023 0.451515
\(66\) 1.56731 0.192923
\(67\) −1.30264 −0.159143 −0.0795717 0.996829i \(-0.525355\pi\)
−0.0795717 + 0.996829i \(0.525355\pi\)
\(68\) 0.659883 0.0800226
\(69\) 5.55199 0.668381
\(70\) −3.87353 −0.462976
\(71\) 10.5493 1.25197 0.625983 0.779837i \(-0.284698\pi\)
0.625983 + 0.779837i \(0.284698\pi\)
\(72\) 3.71357 0.437648
\(73\) −5.17599 −0.605804 −0.302902 0.953022i \(-0.597956\pi\)
−0.302902 + 0.953022i \(0.597956\pi\)
\(74\) −7.15433 −0.831674
\(75\) −0.901323 −0.104076
\(76\) −1.02379 −0.117437
\(77\) −2.22757 −0.253855
\(78\) 5.70538 0.646007
\(79\) 2.89974 0.326246 0.163123 0.986606i \(-0.447843\pi\)
0.163123 + 0.986606i \(0.447843\pi\)
\(80\) 4.99943 0.558954
\(81\) 2.34852 0.260947
\(82\) −2.66698 −0.294518
\(83\) 13.8209 1.51704 0.758522 0.651647i \(-0.225922\pi\)
0.758522 + 0.651647i \(0.225922\pi\)
\(84\) −2.05552 −0.224276
\(85\) −0.644551 −0.0699114
\(86\) −2.67055 −0.287973
\(87\) 1.69633 0.181866
\(88\) 1.69754 0.180958
\(89\) −6.39884 −0.678275 −0.339138 0.940737i \(-0.610135\pi\)
−0.339138 + 0.940737i \(0.610135\pi\)
\(90\) 3.80406 0.400983
\(91\) −8.10886 −0.850040
\(92\) −6.30635 −0.657483
\(93\) 0.165550 0.0171667
\(94\) 3.05763 0.315371
\(95\) 1.00000 0.102598
\(96\) 4.77562 0.487410
\(97\) 6.23352 0.632918 0.316459 0.948606i \(-0.397506\pi\)
0.316459 + 0.948606i \(0.397506\pi\)
\(98\) −3.54376 −0.357974
\(99\) 2.18762 0.219864
\(100\) 1.02379 0.102379
\(101\) −16.3563 −1.62751 −0.813756 0.581206i \(-0.802581\pi\)
−0.813756 + 0.581206i \(0.802581\pi\)
\(102\) −1.01021 −0.100026
\(103\) 0.645519 0.0636049 0.0318025 0.999494i \(-0.489875\pi\)
0.0318025 + 0.999494i \(0.489875\pi\)
\(104\) 6.17943 0.605943
\(105\) 2.00776 0.195937
\(106\) −4.89487 −0.475432
\(107\) −17.5900 −1.70049 −0.850243 0.526390i \(-0.823545\pi\)
−0.850243 + 0.526390i \(0.823545\pi\)
\(108\) 4.78694 0.460624
\(109\) 19.4359 1.86162 0.930808 0.365507i \(-0.119104\pi\)
0.930808 + 0.365507i \(0.119104\pi\)
\(110\) 1.73890 0.165798
\(111\) 3.70829 0.351976
\(112\) −11.1366 −1.05231
\(113\) −5.88383 −0.553504 −0.276752 0.960941i \(-0.589258\pi\)
−0.276752 + 0.960941i \(0.589258\pi\)
\(114\) 1.56731 0.146792
\(115\) 6.15983 0.574407
\(116\) −1.92681 −0.178900
\(117\) 7.96342 0.736219
\(118\) −6.78291 −0.624417
\(119\) 1.43578 0.131618
\(120\) −1.53003 −0.139672
\(121\) 1.00000 0.0909091
\(122\) 1.27645 0.115565
\(123\) 1.38237 0.124644
\(124\) −0.188044 −0.0168868
\(125\) −1.00000 −0.0894427
\(126\) −8.47381 −0.754907
\(127\) 15.9676 1.41690 0.708450 0.705762i \(-0.249395\pi\)
0.708450 + 0.705762i \(0.249395\pi\)
\(128\) 11.9626 1.05735
\(129\) 1.38422 0.121874
\(130\) 6.33001 0.555178
\(131\) −3.93173 −0.343517 −0.171759 0.985139i \(-0.554945\pi\)
−0.171759 + 0.985139i \(0.554945\pi\)
\(132\) 0.922763 0.0803162
\(133\) −2.22757 −0.193155
\(134\) −2.26517 −0.195681
\(135\) −4.67572 −0.402422
\(136\) −1.09415 −0.0938226
\(137\) 15.5422 1.32786 0.663929 0.747796i \(-0.268888\pi\)
0.663929 + 0.747796i \(0.268888\pi\)
\(138\) 9.65438 0.821836
\(139\) −6.69900 −0.568202 −0.284101 0.958794i \(-0.591695\pi\)
−0.284101 + 0.958794i \(0.591695\pi\)
\(140\) −2.28056 −0.192742
\(141\) −1.58486 −0.133469
\(142\) 18.3441 1.53941
\(143\) 3.64023 0.304411
\(144\) 10.9368 0.911404
\(145\) 1.88204 0.156295
\(146\) −9.00056 −0.744891
\(147\) 1.83683 0.151499
\(148\) −4.21215 −0.346236
\(149\) 0.783045 0.0641496 0.0320748 0.999485i \(-0.489789\pi\)
0.0320748 + 0.999485i \(0.489789\pi\)
\(150\) −1.56731 −0.127971
\(151\) 8.46749 0.689075 0.344537 0.938773i \(-0.388036\pi\)
0.344537 + 0.938773i \(0.388036\pi\)
\(152\) 1.69754 0.137689
\(153\) −1.41003 −0.113994
\(154\) −3.87353 −0.312138
\(155\) 0.183675 0.0147531
\(156\) 3.35907 0.268941
\(157\) 8.27024 0.660037 0.330019 0.943974i \(-0.392945\pi\)
0.330019 + 0.943974i \(0.392945\pi\)
\(158\) 5.04237 0.401150
\(159\) 2.53715 0.201209
\(160\) 5.29846 0.418880
\(161\) −13.7215 −1.08140
\(162\) 4.08385 0.320858
\(163\) 23.9030 1.87223 0.936113 0.351699i \(-0.114396\pi\)
0.936113 + 0.351699i \(0.114396\pi\)
\(164\) −1.57019 −0.122612
\(165\) −0.901323 −0.0701679
\(166\) 24.0333 1.86534
\(167\) −19.4405 −1.50435 −0.752176 0.658962i \(-0.770996\pi\)
−0.752176 + 0.658962i \(0.770996\pi\)
\(168\) 3.40825 0.262952
\(169\) 0.251253 0.0193272
\(170\) −1.12081 −0.0859624
\(171\) 2.18762 0.167291
\(172\) −1.57230 −0.119887
\(173\) −11.6888 −0.888682 −0.444341 0.895858i \(-0.646562\pi\)
−0.444341 + 0.895858i \(0.646562\pi\)
\(174\) 2.94975 0.223620
\(175\) 2.22757 0.168389
\(176\) 4.99943 0.376847
\(177\) 3.51577 0.264261
\(178\) −11.1270 −0.834001
\(179\) −6.77452 −0.506351 −0.253176 0.967420i \(-0.581475\pi\)
−0.253176 + 0.967420i \(0.581475\pi\)
\(180\) 2.23966 0.166934
\(181\) −4.34551 −0.322999 −0.161500 0.986873i \(-0.551633\pi\)
−0.161500 + 0.986873i \(0.551633\pi\)
\(182\) −14.1005 −1.04520
\(183\) −0.661622 −0.0489085
\(184\) 10.4565 0.770867
\(185\) 4.11428 0.302488
\(186\) 0.287876 0.0211081
\(187\) −0.644551 −0.0471342
\(188\) 1.80020 0.131293
\(189\) 10.4155 0.757616
\(190\) 1.73890 0.126153
\(191\) −1.37612 −0.0995728 −0.0497864 0.998760i \(-0.515854\pi\)
−0.0497864 + 0.998760i \(0.515854\pi\)
\(192\) −0.707861 −0.0510855
\(193\) 5.15711 0.371217 0.185609 0.982624i \(-0.440574\pi\)
0.185609 + 0.982624i \(0.440574\pi\)
\(194\) 10.8395 0.778231
\(195\) −3.28102 −0.234959
\(196\) −2.08641 −0.149029
\(197\) 12.0998 0.862072 0.431036 0.902335i \(-0.358148\pi\)
0.431036 + 0.902335i \(0.358148\pi\)
\(198\) 3.80406 0.270343
\(199\) −13.2948 −0.942445 −0.471223 0.882014i \(-0.656187\pi\)
−0.471223 + 0.882014i \(0.656187\pi\)
\(200\) −1.69754 −0.120034
\(201\) 1.17410 0.0828149
\(202\) −28.4420 −2.00117
\(203\) −4.19239 −0.294248
\(204\) −0.594768 −0.0416421
\(205\) 1.53371 0.107119
\(206\) 1.12250 0.0782080
\(207\) 13.4753 0.936601
\(208\) 18.1991 1.26188
\(209\) 1.00000 0.0691714
\(210\) 3.49130 0.240923
\(211\) −26.0866 −1.79588 −0.897939 0.440120i \(-0.854936\pi\)
−0.897939 + 0.440120i \(0.854936\pi\)
\(212\) −2.88188 −0.197928
\(213\) −9.50828 −0.651497
\(214\) −30.5873 −2.09090
\(215\) 1.53577 0.104738
\(216\) −7.93722 −0.540059
\(217\) −0.409148 −0.0277748
\(218\) 33.7971 2.28903
\(219\) 4.66524 0.315248
\(220\) 1.02379 0.0690238
\(221\) −2.34631 −0.157830
\(222\) 6.44837 0.432786
\(223\) 13.8606 0.928175 0.464087 0.885789i \(-0.346382\pi\)
0.464087 + 0.885789i \(0.346382\pi\)
\(224\) −11.8027 −0.788600
\(225\) −2.18762 −0.145841
\(226\) −10.2314 −0.680583
\(227\) −12.9104 −0.856893 −0.428446 0.903567i \(-0.640939\pi\)
−0.428446 + 0.903567i \(0.640939\pi\)
\(228\) 0.922763 0.0611115
\(229\) 7.93710 0.524499 0.262249 0.965000i \(-0.415536\pi\)
0.262249 + 0.965000i \(0.415536\pi\)
\(230\) 10.7113 0.706285
\(231\) 2.00776 0.132101
\(232\) 3.19484 0.209752
\(233\) 17.6265 1.15475 0.577374 0.816480i \(-0.304078\pi\)
0.577374 + 0.816480i \(0.304078\pi\)
\(234\) 13.8476 0.905248
\(235\) −1.75837 −0.114703
\(236\) −3.99347 −0.259953
\(237\) −2.61360 −0.169772
\(238\) 2.49669 0.161836
\(239\) −18.6909 −1.20901 −0.604507 0.796600i \(-0.706630\pi\)
−0.604507 + 0.796600i \(0.706630\pi\)
\(240\) −4.50610 −0.290868
\(241\) −0.877285 −0.0565109 −0.0282555 0.999601i \(-0.508995\pi\)
−0.0282555 + 0.999601i \(0.508995\pi\)
\(242\) 1.73890 0.111781
\(243\) −16.1439 −1.03563
\(244\) 0.751518 0.0481110
\(245\) 2.03793 0.130198
\(246\) 2.40381 0.153261
\(247\) 3.64023 0.231622
\(248\) 0.311795 0.0197990
\(249\) −12.4571 −0.789438
\(250\) −1.73890 −0.109978
\(251\) −0.448818 −0.0283291 −0.0141646 0.999900i \(-0.504509\pi\)
−0.0141646 + 0.999900i \(0.504509\pi\)
\(252\) −4.98899 −0.314277
\(253\) 6.15983 0.387265
\(254\) 27.7662 1.74221
\(255\) 0.580948 0.0363804
\(256\) 19.2311 1.20194
\(257\) −7.85809 −0.490174 −0.245087 0.969501i \(-0.578817\pi\)
−0.245087 + 0.969501i \(0.578817\pi\)
\(258\) 2.40703 0.149855
\(259\) −9.16484 −0.569476
\(260\) 3.72682 0.231128
\(261\) 4.11719 0.254848
\(262\) −6.83691 −0.422386
\(263\) 0.585215 0.0360859 0.0180429 0.999837i \(-0.494256\pi\)
0.0180429 + 0.999837i \(0.494256\pi\)
\(264\) −1.53003 −0.0941669
\(265\) 2.81491 0.172919
\(266\) −3.87353 −0.237502
\(267\) 5.76742 0.352960
\(268\) −1.33363 −0.0814645
\(269\) −3.59078 −0.218934 −0.109467 0.993990i \(-0.534914\pi\)
−0.109467 + 0.993990i \(0.534914\pi\)
\(270\) −8.13063 −0.494814
\(271\) −23.4466 −1.42428 −0.712139 0.702038i \(-0.752274\pi\)
−0.712139 + 0.702038i \(0.752274\pi\)
\(272\) −3.22239 −0.195386
\(273\) 7.30870 0.442343
\(274\) 27.0264 1.63272
\(275\) −1.00000 −0.0603023
\(276\) 5.68406 0.342140
\(277\) −19.2912 −1.15909 −0.579547 0.814939i \(-0.696771\pi\)
−0.579547 + 0.814939i \(0.696771\pi\)
\(278\) −11.6489 −0.698656
\(279\) 0.401809 0.0240557
\(280\) 3.78139 0.225981
\(281\) −5.82550 −0.347520 −0.173760 0.984788i \(-0.555592\pi\)
−0.173760 + 0.984788i \(0.555592\pi\)
\(282\) −2.75592 −0.164112
\(283\) −15.5166 −0.922368 −0.461184 0.887305i \(-0.652575\pi\)
−0.461184 + 0.887305i \(0.652575\pi\)
\(284\) 10.8002 0.640874
\(285\) −0.901323 −0.0533898
\(286\) 6.33001 0.374301
\(287\) −3.41645 −0.201667
\(288\) 11.5910 0.683006
\(289\) −16.5846 −0.975562
\(290\) 3.27269 0.192179
\(291\) −5.61842 −0.329357
\(292\) −5.29912 −0.310107
\(293\) −12.8089 −0.748305 −0.374152 0.927367i \(-0.622066\pi\)
−0.374152 + 0.927367i \(0.622066\pi\)
\(294\) 3.19407 0.186282
\(295\) 3.90068 0.227106
\(296\) 6.98415 0.405945
\(297\) −4.67572 −0.271313
\(298\) 1.36164 0.0788777
\(299\) 22.4232 1.29677
\(300\) −0.922763 −0.0532758
\(301\) −3.42103 −0.197185
\(302\) 14.7242 0.847280
\(303\) 14.7423 0.846923
\(304\) 4.99943 0.286737
\(305\) −0.734057 −0.0420320
\(306\) −2.45191 −0.140166
\(307\) −14.0052 −0.799321 −0.399661 0.916663i \(-0.630872\pi\)
−0.399661 + 0.916663i \(0.630872\pi\)
\(308\) −2.28056 −0.129947
\(309\) −0.581821 −0.0330987
\(310\) 0.319392 0.0181403
\(311\) 4.10325 0.232674 0.116337 0.993210i \(-0.462885\pi\)
0.116337 + 0.993210i \(0.462885\pi\)
\(312\) −5.56966 −0.315320
\(313\) 21.5394 1.21748 0.608740 0.793370i \(-0.291675\pi\)
0.608740 + 0.793370i \(0.291675\pi\)
\(314\) 14.3812 0.811576
\(315\) 4.87307 0.274567
\(316\) 2.96872 0.167004
\(317\) −8.36917 −0.470059 −0.235030 0.971988i \(-0.575519\pi\)
−0.235030 + 0.971988i \(0.575519\pi\)
\(318\) 4.41186 0.247405
\(319\) 1.88204 0.105374
\(320\) −0.785358 −0.0439029
\(321\) 15.8542 0.884898
\(322\) −23.8603 −1.32968
\(323\) −0.644551 −0.0358638
\(324\) 2.40439 0.133577
\(325\) −3.64023 −0.201923
\(326\) 41.5650 2.30207
\(327\) −17.5180 −0.968746
\(328\) 2.60353 0.143756
\(329\) 3.91689 0.215945
\(330\) −1.56731 −0.0862778
\(331\) −29.1194 −1.60055 −0.800274 0.599635i \(-0.795312\pi\)
−0.800274 + 0.599635i \(0.795312\pi\)
\(332\) 14.1497 0.776565
\(333\) 9.00046 0.493222
\(334\) −33.8052 −1.84974
\(335\) 1.30264 0.0711711
\(336\) 10.0377 0.547600
\(337\) 3.36844 0.183491 0.0917453 0.995783i \(-0.470755\pi\)
0.0917453 + 0.995783i \(0.470755\pi\)
\(338\) 0.436905 0.0237645
\(339\) 5.30323 0.288032
\(340\) −0.659883 −0.0357872
\(341\) 0.183675 0.00994653
\(342\) 3.80406 0.205700
\(343\) −20.1326 −1.08706
\(344\) 2.60702 0.140561
\(345\) −5.55199 −0.298909
\(346\) −20.3257 −1.09272
\(347\) 3.97272 0.213267 0.106633 0.994298i \(-0.465993\pi\)
0.106633 + 0.994298i \(0.465993\pi\)
\(348\) 1.73668 0.0930959
\(349\) 1.51616 0.0811581 0.0405790 0.999176i \(-0.487080\pi\)
0.0405790 + 0.999176i \(0.487080\pi\)
\(350\) 3.87353 0.207049
\(351\) −17.0207 −0.908496
\(352\) 5.29846 0.282409
\(353\) 4.76876 0.253816 0.126908 0.991915i \(-0.459495\pi\)
0.126908 + 0.991915i \(0.459495\pi\)
\(354\) 6.11359 0.324934
\(355\) −10.5493 −0.559896
\(356\) −6.55105 −0.347205
\(357\) −1.29410 −0.0684912
\(358\) −11.7802 −0.622605
\(359\) 0.542750 0.0286453 0.0143226 0.999897i \(-0.495441\pi\)
0.0143226 + 0.999897i \(0.495441\pi\)
\(360\) −3.71357 −0.195722
\(361\) 1.00000 0.0526316
\(362\) −7.55643 −0.397157
\(363\) −0.901323 −0.0473072
\(364\) −8.30176 −0.435130
\(365\) 5.17599 0.270924
\(366\) −1.15050 −0.0601375
\(367\) −26.1792 −1.36654 −0.683271 0.730165i \(-0.739443\pi\)
−0.683271 + 0.730165i \(0.739443\pi\)
\(368\) 30.7956 1.60533
\(369\) 3.35517 0.174663
\(370\) 7.15433 0.371936
\(371\) −6.27042 −0.325544
\(372\) 0.169488 0.00878755
\(373\) −12.7807 −0.661760 −0.330880 0.943673i \(-0.607345\pi\)
−0.330880 + 0.943673i \(0.607345\pi\)
\(374\) −1.12081 −0.0579558
\(375\) 0.901323 0.0465441
\(376\) −2.98490 −0.153934
\(377\) 6.85107 0.352848
\(378\) 18.1115 0.931557
\(379\) −26.0931 −1.34031 −0.670157 0.742219i \(-0.733773\pi\)
−0.670157 + 0.742219i \(0.733773\pi\)
\(380\) 1.02379 0.0525192
\(381\) −14.3920 −0.737325
\(382\) −2.39295 −0.122434
\(383\) −12.7812 −0.653091 −0.326545 0.945182i \(-0.605885\pi\)
−0.326545 + 0.945182i \(0.605885\pi\)
\(384\) −10.7821 −0.550224
\(385\) 2.22757 0.113528
\(386\) 8.96773 0.456445
\(387\) 3.35967 0.170781
\(388\) 6.38180 0.323987
\(389\) −8.97979 −0.455293 −0.227647 0.973744i \(-0.573103\pi\)
−0.227647 + 0.973744i \(0.573103\pi\)
\(390\) −5.70538 −0.288903
\(391\) −3.97032 −0.200788
\(392\) 3.45946 0.174729
\(393\) 3.54376 0.178759
\(394\) 21.0403 1.06000
\(395\) −2.89974 −0.145902
\(396\) 2.23966 0.112547
\(397\) 26.8417 1.34715 0.673573 0.739120i \(-0.264758\pi\)
0.673573 + 0.739120i \(0.264758\pi\)
\(398\) −23.1184 −1.15882
\(399\) 2.00776 0.100514
\(400\) −4.99943 −0.249972
\(401\) −9.25518 −0.462182 −0.231091 0.972932i \(-0.574229\pi\)
−0.231091 + 0.972932i \(0.574229\pi\)
\(402\) 2.04165 0.101828
\(403\) 0.668617 0.0333062
\(404\) −16.7454 −0.833114
\(405\) −2.34852 −0.116699
\(406\) −7.29016 −0.361804
\(407\) 4.11428 0.203937
\(408\) 0.986183 0.0488233
\(409\) 26.0835 1.28975 0.644874 0.764289i \(-0.276910\pi\)
0.644874 + 0.764289i \(0.276910\pi\)
\(410\) 2.66698 0.131713
\(411\) −14.0085 −0.690990
\(412\) 0.660875 0.0325590
\(413\) −8.68904 −0.427560
\(414\) 23.4323 1.15164
\(415\) −13.8209 −0.678443
\(416\) 19.2876 0.945652
\(417\) 6.03796 0.295680
\(418\) 1.73890 0.0850526
\(419\) −32.5181 −1.58861 −0.794307 0.607517i \(-0.792166\pi\)
−0.794307 + 0.607517i \(0.792166\pi\)
\(420\) 2.05552 0.100299
\(421\) 3.16934 0.154464 0.0772320 0.997013i \(-0.475392\pi\)
0.0772320 + 0.997013i \(0.475392\pi\)
\(422\) −45.3621 −2.20819
\(423\) −3.84664 −0.187030
\(424\) 4.77843 0.232061
\(425\) 0.644551 0.0312653
\(426\) −16.5340 −0.801074
\(427\) 1.63516 0.0791311
\(428\) −18.0084 −0.870469
\(429\) −3.28102 −0.158409
\(430\) 2.67055 0.128785
\(431\) −1.19700 −0.0576575 −0.0288288 0.999584i \(-0.509178\pi\)
−0.0288288 + 0.999584i \(0.509178\pi\)
\(432\) −23.3759 −1.12468
\(433\) 23.4218 1.12558 0.562789 0.826601i \(-0.309728\pi\)
0.562789 + 0.826601i \(0.309728\pi\)
\(434\) −0.711469 −0.0341516
\(435\) −1.69633 −0.0813327
\(436\) 19.8982 0.952950
\(437\) 6.15983 0.294664
\(438\) 8.11241 0.387626
\(439\) −21.4961 −1.02595 −0.512975 0.858403i \(-0.671457\pi\)
−0.512975 + 0.858403i \(0.671457\pi\)
\(440\) −1.69754 −0.0809270
\(441\) 4.45821 0.212295
\(442\) −4.08001 −0.194066
\(443\) 23.6133 1.12190 0.560950 0.827850i \(-0.310436\pi\)
0.560950 + 0.827850i \(0.310436\pi\)
\(444\) 3.79650 0.180174
\(445\) 6.39884 0.303334
\(446\) 24.1023 1.14128
\(447\) −0.705777 −0.0333821
\(448\) 1.74944 0.0826533
\(449\) −5.35004 −0.252484 −0.126242 0.991999i \(-0.540292\pi\)
−0.126242 + 0.991999i \(0.540292\pi\)
\(450\) −3.80406 −0.179325
\(451\) 1.53371 0.0722196
\(452\) −6.02379 −0.283335
\(453\) −7.63194 −0.358580
\(454\) −22.4499 −1.05363
\(455\) 8.10886 0.380149
\(456\) −1.53003 −0.0716503
\(457\) 39.7929 1.86143 0.930716 0.365742i \(-0.119185\pi\)
0.930716 + 0.365742i \(0.119185\pi\)
\(458\) 13.8019 0.644919
\(459\) 3.01374 0.140669
\(460\) 6.30635 0.294035
\(461\) −3.21566 −0.149768 −0.0748840 0.997192i \(-0.523859\pi\)
−0.0748840 + 0.997192i \(0.523859\pi\)
\(462\) 3.49130 0.162430
\(463\) 16.7065 0.776416 0.388208 0.921572i \(-0.373094\pi\)
0.388208 + 0.921572i \(0.373094\pi\)
\(464\) 9.40915 0.436809
\(465\) −0.165550 −0.00767720
\(466\) 30.6507 1.41987
\(467\) −37.8565 −1.75179 −0.875894 0.482503i \(-0.839728\pi\)
−0.875894 + 0.482503i \(0.839728\pi\)
\(468\) 8.15285 0.376866
\(469\) −2.90173 −0.133990
\(470\) −3.05763 −0.141038
\(471\) −7.45416 −0.343469
\(472\) 6.62156 0.304782
\(473\) 1.53577 0.0706146
\(474\) −4.54481 −0.208750
\(475\) −1.00000 −0.0458831
\(476\) 1.46994 0.0673744
\(477\) 6.15796 0.281953
\(478\) −32.5017 −1.48659
\(479\) −16.6101 −0.758936 −0.379468 0.925205i \(-0.623893\pi\)
−0.379468 + 0.925205i \(0.623893\pi\)
\(480\) −4.77562 −0.217976
\(481\) 14.9769 0.682888
\(482\) −1.52552 −0.0694853
\(483\) 12.3675 0.562739
\(484\) 1.02379 0.0465358
\(485\) −6.23352 −0.283050
\(486\) −28.0727 −1.27341
\(487\) 29.2975 1.32760 0.663798 0.747912i \(-0.268944\pi\)
0.663798 + 0.747912i \(0.268944\pi\)
\(488\) −1.24609 −0.0564079
\(489\) −21.5443 −0.974267
\(490\) 3.54376 0.160091
\(491\) −0.806717 −0.0364066 −0.0182033 0.999834i \(-0.505795\pi\)
−0.0182033 + 0.999834i \(0.505795\pi\)
\(492\) 1.41525 0.0638045
\(493\) −1.21307 −0.0546340
\(494\) 6.33001 0.284800
\(495\) −2.18762 −0.0983261
\(496\) 0.918269 0.0412315
\(497\) 23.4992 1.05408
\(498\) −21.6617 −0.970686
\(499\) −30.7212 −1.37527 −0.687634 0.726058i \(-0.741351\pi\)
−0.687634 + 0.726058i \(0.741351\pi\)
\(500\) −1.02379 −0.0457852
\(501\) 17.5222 0.782833
\(502\) −0.780451 −0.0348332
\(503\) −13.0527 −0.581991 −0.290995 0.956724i \(-0.593986\pi\)
−0.290995 + 0.956724i \(0.593986\pi\)
\(504\) 8.27223 0.368475
\(505\) 16.3563 0.727846
\(506\) 10.7113 0.476178
\(507\) −0.226460 −0.0100574
\(508\) 16.3475 0.725302
\(509\) −22.6615 −1.00445 −0.502226 0.864736i \(-0.667485\pi\)
−0.502226 + 0.864736i \(0.667485\pi\)
\(510\) 1.01021 0.0447330
\(511\) −11.5299 −0.510052
\(512\) 9.51581 0.420544
\(513\) −4.67572 −0.206438
\(514\) −13.6645 −0.602713
\(515\) −0.645519 −0.0284450
\(516\) 1.41715 0.0623865
\(517\) −1.75837 −0.0773329
\(518\) −15.9368 −0.700222
\(519\) 10.5354 0.462451
\(520\) −6.17943 −0.270986
\(521\) 19.0608 0.835070 0.417535 0.908661i \(-0.362894\pi\)
0.417535 + 0.908661i \(0.362894\pi\)
\(522\) 7.15940 0.313358
\(523\) 15.1864 0.664054 0.332027 0.943270i \(-0.392268\pi\)
0.332027 + 0.943270i \(0.392268\pi\)
\(524\) −4.02526 −0.175844
\(525\) −2.00776 −0.0876259
\(526\) 1.01763 0.0443709
\(527\) −0.118388 −0.00515704
\(528\) −4.50610 −0.196103
\(529\) 14.9435 0.649716
\(530\) 4.89487 0.212619
\(531\) 8.53319 0.370309
\(532\) −2.28056 −0.0988748
\(533\) 5.58305 0.241829
\(534\) 10.0290 0.433997
\(535\) 17.5900 0.760481
\(536\) 2.21129 0.0955132
\(537\) 6.10603 0.263495
\(538\) −6.24402 −0.269199
\(539\) 2.03793 0.0877798
\(540\) −4.78694 −0.205997
\(541\) 41.1432 1.76888 0.884442 0.466650i \(-0.154539\pi\)
0.884442 + 0.466650i \(0.154539\pi\)
\(542\) −40.7714 −1.75128
\(543\) 3.91671 0.168082
\(544\) −3.41513 −0.146422
\(545\) −19.4359 −0.832540
\(546\) 12.7091 0.543901
\(547\) 40.7714 1.74326 0.871630 0.490164i \(-0.163063\pi\)
0.871630 + 0.490164i \(0.163063\pi\)
\(548\) 15.9119 0.679722
\(549\) −1.60583 −0.0685354
\(550\) −1.73890 −0.0741471
\(551\) 1.88204 0.0801778
\(552\) −9.42473 −0.401143
\(553\) 6.45938 0.274681
\(554\) −33.5455 −1.42521
\(555\) −3.70829 −0.157408
\(556\) −6.85835 −0.290859
\(557\) −14.2315 −0.603009 −0.301504 0.953465i \(-0.597489\pi\)
−0.301504 + 0.953465i \(0.597489\pi\)
\(558\) 0.698708 0.0295787
\(559\) 5.59054 0.236454
\(560\) 11.1366 0.470607
\(561\) 0.580948 0.0245277
\(562\) −10.1300 −0.427308
\(563\) −11.1511 −0.469963 −0.234982 0.972000i \(-0.575503\pi\)
−0.234982 + 0.972000i \(0.575503\pi\)
\(564\) −1.62256 −0.0683220
\(565\) 5.88383 0.247534
\(566\) −26.9819 −1.13414
\(567\) 5.23149 0.219702
\(568\) −17.9078 −0.751393
\(569\) 11.4022 0.478005 0.239003 0.971019i \(-0.423180\pi\)
0.239003 + 0.971019i \(0.423180\pi\)
\(570\) −1.56731 −0.0656476
\(571\) 12.5111 0.523573 0.261787 0.965126i \(-0.415688\pi\)
0.261787 + 0.965126i \(0.415688\pi\)
\(572\) 3.72682 0.155826
\(573\) 1.24033 0.0518156
\(574\) −5.94088 −0.247967
\(575\) −6.15983 −0.256883
\(576\) −1.71806 −0.0715860
\(577\) 7.95520 0.331179 0.165590 0.986195i \(-0.447047\pi\)
0.165590 + 0.986195i \(0.447047\pi\)
\(578\) −28.8390 −1.19954
\(579\) −4.64822 −0.193174
\(580\) 1.92681 0.0800066
\(581\) 30.7871 1.27726
\(582\) −9.76989 −0.404975
\(583\) 2.81491 0.116582
\(584\) 8.78645 0.363586
\(585\) −7.96342 −0.329247
\(586\) −22.2735 −0.920109
\(587\) 45.7680 1.88905 0.944524 0.328443i \(-0.106524\pi\)
0.944524 + 0.328443i \(0.106524\pi\)
\(588\) 1.88052 0.0775515
\(589\) 0.183675 0.00756818
\(590\) 6.78291 0.279248
\(591\) −10.9058 −0.448604
\(592\) 20.5691 0.845383
\(593\) −26.3613 −1.08253 −0.541264 0.840853i \(-0.682054\pi\)
−0.541264 + 0.840853i \(0.682054\pi\)
\(594\) −8.13063 −0.333604
\(595\) −1.43578 −0.0588614
\(596\) 0.801672 0.0328378
\(597\) 11.9829 0.490429
\(598\) 38.9917 1.59449
\(599\) 17.3368 0.708361 0.354180 0.935177i \(-0.384760\pi\)
0.354180 + 0.935177i \(0.384760\pi\)
\(600\) 1.53003 0.0624633
\(601\) −15.7946 −0.644274 −0.322137 0.946693i \(-0.604401\pi\)
−0.322137 + 0.946693i \(0.604401\pi\)
\(602\) −5.94884 −0.242457
\(603\) 2.84969 0.116048
\(604\) 8.66891 0.352733
\(605\) −1.00000 −0.0406558
\(606\) 25.6355 1.04137
\(607\) −1.80754 −0.0733659 −0.0366829 0.999327i \(-0.511679\pi\)
−0.0366829 + 0.999327i \(0.511679\pi\)
\(608\) 5.29846 0.214881
\(609\) 3.77869 0.153120
\(610\) −1.27645 −0.0516821
\(611\) −6.40086 −0.258951
\(612\) −1.44357 −0.0583529
\(613\) 10.9246 0.441239 0.220619 0.975360i \(-0.429192\pi\)
0.220619 + 0.975360i \(0.429192\pi\)
\(614\) −24.3538 −0.982838
\(615\) −1.38237 −0.0557425
\(616\) 3.78139 0.152357
\(617\) −2.68742 −0.108191 −0.0540957 0.998536i \(-0.517228\pi\)
−0.0540957 + 0.998536i \(0.517228\pi\)
\(618\) −1.01173 −0.0406978
\(619\) 18.2439 0.733284 0.366642 0.930362i \(-0.380507\pi\)
0.366642 + 0.930362i \(0.380507\pi\)
\(620\) 0.188044 0.00755202
\(621\) −28.8016 −1.15577
\(622\) 7.13515 0.286094
\(623\) −14.2539 −0.571069
\(624\) −16.4032 −0.656655
\(625\) 1.00000 0.0400000
\(626\) 37.4550 1.49700
\(627\) −0.901323 −0.0359954
\(628\) 8.46697 0.337869
\(629\) −2.65186 −0.105737
\(630\) 8.47381 0.337604
\(631\) 9.12012 0.363066 0.181533 0.983385i \(-0.441894\pi\)
0.181533 + 0.983385i \(0.441894\pi\)
\(632\) −4.92243 −0.195804
\(633\) 23.5125 0.934537
\(634\) −14.5532 −0.577981
\(635\) −15.9676 −0.633657
\(636\) 2.59750 0.102998
\(637\) 7.41852 0.293932
\(638\) 3.27269 0.129567
\(639\) −23.0777 −0.912940
\(640\) −11.9626 −0.472862
\(641\) −2.90773 −0.114848 −0.0574241 0.998350i \(-0.518289\pi\)
−0.0574241 + 0.998350i \(0.518289\pi\)
\(642\) 27.5690 1.08806
\(643\) −3.35603 −0.132349 −0.0661745 0.997808i \(-0.521079\pi\)
−0.0661745 + 0.997808i \(0.521079\pi\)
\(644\) −14.0479 −0.553563
\(645\) −1.38422 −0.0545036
\(646\) −1.12081 −0.0440978
\(647\) 26.5428 1.04351 0.521753 0.853096i \(-0.325278\pi\)
0.521753 + 0.853096i \(0.325278\pi\)
\(648\) −3.98670 −0.156613
\(649\) 3.90068 0.153115
\(650\) −6.33001 −0.248283
\(651\) 0.368774 0.0144534
\(652\) 24.4716 0.958381
\(653\) 28.6844 1.12251 0.561253 0.827644i \(-0.310319\pi\)
0.561253 + 0.827644i \(0.310319\pi\)
\(654\) −30.4621 −1.19116
\(655\) 3.93173 0.153626
\(656\) 7.66768 0.299373
\(657\) 11.3231 0.441756
\(658\) 6.81110 0.265524
\(659\) −39.0719 −1.52203 −0.761013 0.648737i \(-0.775297\pi\)
−0.761013 + 0.648737i \(0.775297\pi\)
\(660\) −0.922763 −0.0359185
\(661\) −21.1901 −0.824199 −0.412099 0.911139i \(-0.635204\pi\)
−0.412099 + 0.911139i \(0.635204\pi\)
\(662\) −50.6358 −1.96802
\(663\) 2.11478 0.0821314
\(664\) −23.4616 −0.910486
\(665\) 2.22757 0.0863815
\(666\) 15.6509 0.606462
\(667\) 11.5931 0.448885
\(668\) −19.9030 −0.770069
\(669\) −12.4929 −0.483003
\(670\) 2.26517 0.0875113
\(671\) −0.734057 −0.0283379
\(672\) 10.6380 0.410371
\(673\) −24.6931 −0.951849 −0.475924 0.879486i \(-0.657886\pi\)
−0.475924 + 0.879486i \(0.657886\pi\)
\(674\) 5.85739 0.225618
\(675\) 4.67572 0.179968
\(676\) 0.257230 0.00989345
\(677\) 5.41516 0.208122 0.104061 0.994571i \(-0.466816\pi\)
0.104061 + 0.994571i \(0.466816\pi\)
\(678\) 9.22180 0.354161
\(679\) 13.8856 0.532881
\(680\) 1.09415 0.0419588
\(681\) 11.6364 0.445909
\(682\) 0.319392 0.0122302
\(683\) −13.7885 −0.527604 −0.263802 0.964577i \(-0.584977\pi\)
−0.263802 + 0.964577i \(0.584977\pi\)
\(684\) 2.23966 0.0856354
\(685\) −15.5422 −0.593836
\(686\) −35.0087 −1.33664
\(687\) −7.15389 −0.272938
\(688\) 7.67796 0.292719
\(689\) 10.2469 0.390377
\(690\) −9.65438 −0.367536
\(691\) −31.5750 −1.20117 −0.600586 0.799560i \(-0.705066\pi\)
−0.600586 + 0.799560i \(0.705066\pi\)
\(692\) −11.9668 −0.454911
\(693\) 4.87307 0.185113
\(694\) 6.90818 0.262231
\(695\) 6.69900 0.254108
\(696\) −2.87959 −0.109150
\(697\) −0.988554 −0.0374442
\(698\) 2.63645 0.0997912
\(699\) −15.8871 −0.600906
\(700\) 2.28056 0.0861971
\(701\) −17.4459 −0.658924 −0.329462 0.944169i \(-0.606867\pi\)
−0.329462 + 0.944169i \(0.606867\pi\)
\(702\) −29.5973 −1.11708
\(703\) 4.11428 0.155173
\(704\) −0.785358 −0.0295993
\(705\) 1.58486 0.0596892
\(706\) 8.29242 0.312089
\(707\) −36.4348 −1.37027
\(708\) 3.59940 0.135274
\(709\) −38.7266 −1.45441 −0.727205 0.686421i \(-0.759181\pi\)
−0.727205 + 0.686421i \(0.759181\pi\)
\(710\) −18.3441 −0.688443
\(711\) −6.34352 −0.237901
\(712\) 10.8623 0.407081
\(713\) 1.13140 0.0423714
\(714\) −2.25032 −0.0842162
\(715\) −3.64023 −0.136137
\(716\) −6.93567 −0.259198
\(717\) 16.8465 0.629145
\(718\) 0.943791 0.0352220
\(719\) −28.9402 −1.07929 −0.539644 0.841894i \(-0.681441\pi\)
−0.539644 + 0.841894i \(0.681441\pi\)
\(720\) −10.9368 −0.407592
\(721\) 1.43794 0.0535517
\(722\) 1.73890 0.0647153
\(723\) 0.790717 0.0294071
\(724\) −4.44888 −0.165341
\(725\) −1.88204 −0.0698973
\(726\) −1.56731 −0.0581685
\(727\) −0.331738 −0.0123035 −0.00615174 0.999981i \(-0.501958\pi\)
−0.00615174 + 0.999981i \(0.501958\pi\)
\(728\) 13.7651 0.510169
\(729\) 7.50533 0.277975
\(730\) 9.00056 0.333126
\(731\) −0.989879 −0.0366120
\(732\) −0.677361 −0.0250360
\(733\) −30.5241 −1.12743 −0.563716 0.825969i \(-0.690629\pi\)
−0.563716 + 0.825969i \(0.690629\pi\)
\(734\) −45.5231 −1.68029
\(735\) −1.83683 −0.0677525
\(736\) 32.6376 1.20304
\(737\) 1.30264 0.0479835
\(738\) 5.83432 0.214764
\(739\) −8.65314 −0.318311 −0.159155 0.987254i \(-0.550877\pi\)
−0.159155 + 0.987254i \(0.550877\pi\)
\(740\) 4.21215 0.154842
\(741\) −3.28102 −0.120531
\(742\) −10.9037 −0.400286
\(743\) 33.9919 1.24704 0.623521 0.781807i \(-0.285702\pi\)
0.623521 + 0.781807i \(0.285702\pi\)
\(744\) −0.281028 −0.0103030
\(745\) −0.783045 −0.0286886
\(746\) −22.2244 −0.813694
\(747\) −30.2349 −1.10624
\(748\) −0.659883 −0.0241277
\(749\) −39.1829 −1.43171
\(750\) 1.56731 0.0572302
\(751\) −50.2581 −1.83394 −0.916972 0.398951i \(-0.869374\pi\)
−0.916972 + 0.398951i \(0.869374\pi\)
\(752\) −8.79085 −0.320569
\(753\) 0.404530 0.0147419
\(754\) 11.9133 0.433859
\(755\) −8.46749 −0.308164
\(756\) 10.6633 0.387819
\(757\) 10.7948 0.392344 0.196172 0.980570i \(-0.437149\pi\)
0.196172 + 0.980570i \(0.437149\pi\)
\(758\) −45.3734 −1.64804
\(759\) −5.55199 −0.201525
\(760\) −1.69754 −0.0615762
\(761\) 15.5661 0.564272 0.282136 0.959374i \(-0.408957\pi\)
0.282136 + 0.959374i \(0.408957\pi\)
\(762\) −25.0263 −0.906608
\(763\) 43.2947 1.56737
\(764\) −1.40886 −0.0509707
\(765\) 1.41003 0.0509798
\(766\) −22.2254 −0.803035
\(767\) 14.1994 0.512709
\(768\) −17.3334 −0.625465
\(769\) 19.6520 0.708671 0.354335 0.935118i \(-0.384707\pi\)
0.354335 + 0.935118i \(0.384707\pi\)
\(770\) 3.87353 0.139592
\(771\) 7.08267 0.255076
\(772\) 5.27979 0.190024
\(773\) −41.0208 −1.47541 −0.737707 0.675121i \(-0.764092\pi\)
−0.737707 + 0.675121i \(0.764092\pi\)
\(774\) 5.84214 0.209991
\(775\) −0.183675 −0.00659778
\(776\) −10.5817 −0.379859
\(777\) 8.26048 0.296343
\(778\) −15.6150 −0.559825
\(779\) 1.53371 0.0549509
\(780\) −3.35907 −0.120274
\(781\) −10.5493 −0.377482
\(782\) −6.90401 −0.246887
\(783\) −8.79991 −0.314483
\(784\) 10.1885 0.363874
\(785\) −8.27024 −0.295178
\(786\) 6.16226 0.219801
\(787\) 25.2114 0.898688 0.449344 0.893359i \(-0.351658\pi\)
0.449344 + 0.893359i \(0.351658\pi\)
\(788\) 12.3876 0.441289
\(789\) −0.527467 −0.0187783
\(790\) −5.04237 −0.179400
\(791\) −13.1066 −0.466018
\(792\) −3.71357 −0.131956
\(793\) −2.67213 −0.0948902
\(794\) 46.6752 1.65644
\(795\) −2.53715 −0.0899833
\(796\) −13.6111 −0.482432
\(797\) −23.4053 −0.829057 −0.414528 0.910036i \(-0.636053\pi\)
−0.414528 + 0.910036i \(0.636053\pi\)
\(798\) 3.49130 0.123591
\(799\) 1.13336 0.0400953
\(800\) −5.29846 −0.187329
\(801\) 13.9982 0.494602
\(802\) −16.0939 −0.568294
\(803\) 5.17599 0.182657
\(804\) 1.20203 0.0423924
\(805\) 13.7215 0.483618
\(806\) 1.16266 0.0409530
\(807\) 3.23645 0.113929
\(808\) 27.7655 0.976786
\(809\) −25.3235 −0.890325 −0.445163 0.895450i \(-0.646854\pi\)
−0.445163 + 0.895450i \(0.646854\pi\)
\(810\) −4.08385 −0.143492
\(811\) −1.24610 −0.0437566 −0.0218783 0.999761i \(-0.506965\pi\)
−0.0218783 + 0.999761i \(0.506965\pi\)
\(812\) −4.29211 −0.150624
\(813\) 21.1329 0.741165
\(814\) 7.15433 0.250759
\(815\) −23.9030 −0.837285
\(816\) 2.90441 0.101675
\(817\) 1.53577 0.0537296
\(818\) 45.3567 1.58586
\(819\) 17.7391 0.619854
\(820\) 1.57019 0.0548335
\(821\) 20.4113 0.712359 0.356180 0.934418i \(-0.384079\pi\)
0.356180 + 0.934418i \(0.384079\pi\)
\(822\) −24.3595 −0.849634
\(823\) −15.4655 −0.539093 −0.269546 0.962987i \(-0.586874\pi\)
−0.269546 + 0.962987i \(0.586874\pi\)
\(824\) −1.09579 −0.0381738
\(825\) 0.901323 0.0313800
\(826\) −15.1094 −0.525723
\(827\) −46.2008 −1.60656 −0.803280 0.595602i \(-0.796913\pi\)
−0.803280 + 0.595602i \(0.796913\pi\)
\(828\) 13.7959 0.479440
\(829\) 2.91356 0.101192 0.0505961 0.998719i \(-0.483888\pi\)
0.0505961 + 0.998719i \(0.483888\pi\)
\(830\) −24.0333 −0.834207
\(831\) 17.3876 0.603168
\(832\) −2.85888 −0.0991139
\(833\) −1.31355 −0.0455117
\(834\) 10.4994 0.363566
\(835\) 19.4405 0.672767
\(836\) 1.02379 0.0354084
\(837\) −0.858810 −0.0296848
\(838\) −56.5459 −1.95334
\(839\) 28.2887 0.976633 0.488317 0.872667i \(-0.337611\pi\)
0.488317 + 0.872667i \(0.337611\pi\)
\(840\) −3.40825 −0.117596
\(841\) −25.4579 −0.877859
\(842\) 5.51117 0.189928
\(843\) 5.25066 0.180842
\(844\) −26.7072 −0.919299
\(845\) −0.251253 −0.00864337
\(846\) −6.68893 −0.229970
\(847\) 2.22757 0.0765402
\(848\) 14.0730 0.483268
\(849\) 13.9855 0.479981
\(850\) 1.12081 0.0384435
\(851\) 25.3432 0.868755
\(852\) −9.73446 −0.333497
\(853\) 41.9234 1.43543 0.717715 0.696337i \(-0.245188\pi\)
0.717715 + 0.696337i \(0.245188\pi\)
\(854\) 2.84339 0.0972989
\(855\) −2.18762 −0.0748149
\(856\) 29.8597 1.02058
\(857\) 52.4352 1.79115 0.895576 0.444909i \(-0.146764\pi\)
0.895576 + 0.444909i \(0.146764\pi\)
\(858\) −5.70538 −0.194778
\(859\) −21.2305 −0.724374 −0.362187 0.932105i \(-0.617970\pi\)
−0.362187 + 0.932105i \(0.617970\pi\)
\(860\) 1.57230 0.0536149
\(861\) 3.07932 0.104943
\(862\) −2.08147 −0.0708952
\(863\) 1.75582 0.0597690 0.0298845 0.999553i \(-0.490486\pi\)
0.0298845 + 0.999553i \(0.490486\pi\)
\(864\) −24.7741 −0.842832
\(865\) 11.6888 0.397431
\(866\) 40.7282 1.38400
\(867\) 14.9480 0.507662
\(868\) −0.418881 −0.0142177
\(869\) −2.89974 −0.0983670
\(870\) −2.94975 −0.100006
\(871\) 4.74192 0.160674
\(872\) −32.9931 −1.11729
\(873\) −13.6366 −0.461528
\(874\) 10.7113 0.362317
\(875\) −2.22757 −0.0753056
\(876\) 4.77622 0.161373
\(877\) 27.7873 0.938309 0.469155 0.883116i \(-0.344559\pi\)
0.469155 + 0.883116i \(0.344559\pi\)
\(878\) −37.3796 −1.26150
\(879\) 11.5450 0.389402
\(880\) −4.99943 −0.168531
\(881\) 38.8712 1.30960 0.654802 0.755800i \(-0.272752\pi\)
0.654802 + 0.755800i \(0.272752\pi\)
\(882\) 7.75239 0.261037
\(883\) −1.20467 −0.0405404 −0.0202702 0.999795i \(-0.506453\pi\)
−0.0202702 + 0.999795i \(0.506453\pi\)
\(884\) −2.40212 −0.0807922
\(885\) −3.51577 −0.118181
\(886\) 41.0612 1.37948
\(887\) −44.0947 −1.48056 −0.740278 0.672301i \(-0.765306\pi\)
−0.740278 + 0.672301i \(0.765306\pi\)
\(888\) −6.29497 −0.211245
\(889\) 35.5691 1.19295
\(890\) 11.1270 0.372977
\(891\) −2.34852 −0.0786784
\(892\) 14.1903 0.475127
\(893\) −1.75837 −0.0588415
\(894\) −1.22728 −0.0410463
\(895\) 6.77452 0.226447
\(896\) 26.6475 0.890230
\(897\) −20.2105 −0.674809
\(898\) −9.30320 −0.310452
\(899\) 0.345683 0.0115292
\(900\) −2.23966 −0.0746552
\(901\) −1.81436 −0.0604450
\(902\) 2.66698 0.0888006
\(903\) 3.08345 0.102611
\(904\) 9.98803 0.332197
\(905\) 4.34551 0.144450
\(906\) −13.2712 −0.440907
\(907\) 36.2297 1.20299 0.601493 0.798878i \(-0.294573\pi\)
0.601493 + 0.798878i \(0.294573\pi\)
\(908\) −13.2175 −0.438638
\(909\) 35.7813 1.18679
\(910\) 14.1005 0.467428
\(911\) −5.41417 −0.179380 −0.0896898 0.995970i \(-0.528588\pi\)
−0.0896898 + 0.995970i \(0.528588\pi\)
\(912\) −4.50610 −0.149212
\(913\) −13.8209 −0.457406
\(914\) 69.1960 2.28880
\(915\) 0.661622 0.0218726
\(916\) 8.12591 0.268488
\(917\) −8.75822 −0.289222
\(918\) 5.24060 0.172966
\(919\) 54.0931 1.78437 0.892183 0.451675i \(-0.149173\pi\)
0.892183 + 0.451675i \(0.149173\pi\)
\(920\) −10.4565 −0.344742
\(921\) 12.6232 0.415950
\(922\) −5.59172 −0.184153
\(923\) −38.4017 −1.26401
\(924\) 2.05552 0.0676217
\(925\) −4.11428 −0.135277
\(926\) 29.0510 0.954674
\(927\) −1.41215 −0.0463811
\(928\) 9.97193 0.327345
\(929\) 52.0785 1.70864 0.854320 0.519747i \(-0.173974\pi\)
0.854320 + 0.519747i \(0.173974\pi\)
\(930\) −0.287876 −0.00943981
\(931\) 2.03793 0.0667904
\(932\) 18.0457 0.591108
\(933\) −3.69835 −0.121079
\(934\) −65.8288 −2.15398
\(935\) 0.644551 0.0210791
\(936\) −13.5182 −0.441857
\(937\) 35.9861 1.17561 0.587807 0.809002i \(-0.299992\pi\)
0.587807 + 0.809002i \(0.299992\pi\)
\(938\) −5.04584 −0.164752
\(939\) −19.4140 −0.633551
\(940\) −1.80020 −0.0587159
\(941\) 11.9972 0.391098 0.195549 0.980694i \(-0.437351\pi\)
0.195549 + 0.980694i \(0.437351\pi\)
\(942\) −12.9621 −0.422327
\(943\) 9.44739 0.307649
\(944\) 19.5012 0.634709
\(945\) −10.4155 −0.338816
\(946\) 2.67055 0.0868271
\(947\) 37.1663 1.20774 0.603871 0.797082i \(-0.293624\pi\)
0.603871 + 0.797082i \(0.293624\pi\)
\(948\) −2.67577 −0.0869051
\(949\) 18.8418 0.611630
\(950\) −1.73890 −0.0564175
\(951\) 7.54332 0.244609
\(952\) −2.43730 −0.0789933
\(953\) 16.5500 0.536106 0.268053 0.963404i \(-0.413620\pi\)
0.268053 + 0.963404i \(0.413620\pi\)
\(954\) 10.7081 0.346687
\(955\) 1.37612 0.0445303
\(956\) −19.1355 −0.618886
\(957\) −1.69633 −0.0548345
\(958\) −28.8834 −0.933180
\(959\) 34.6213 1.11798
\(960\) 0.707861 0.0228461
\(961\) −30.9663 −0.998912
\(962\) 26.0434 0.839673
\(963\) 38.4801 1.24000
\(964\) −0.898154 −0.0289276
\(965\) −5.15711 −0.166013
\(966\) 21.5058 0.691939
\(967\) 37.7443 1.21377 0.606887 0.794788i \(-0.292418\pi\)
0.606887 + 0.794788i \(0.292418\pi\)
\(968\) −1.69754 −0.0545610
\(969\) 0.580948 0.0186628
\(970\) −10.8395 −0.348035
\(971\) −40.2912 −1.29301 −0.646504 0.762911i \(-0.723769\pi\)
−0.646504 + 0.762911i \(0.723769\pi\)
\(972\) −16.5280 −0.530134
\(973\) −14.9225 −0.478393
\(974\) 50.9455 1.63240
\(975\) 3.28102 0.105077
\(976\) −3.66987 −0.117470
\(977\) 1.24031 0.0396812 0.0198406 0.999803i \(-0.493684\pi\)
0.0198406 + 0.999803i \(0.493684\pi\)
\(978\) −37.4635 −1.19795
\(979\) 6.39884 0.204508
\(980\) 2.08641 0.0666478
\(981\) −42.5182 −1.35750
\(982\) −1.40280 −0.0447653
\(983\) 8.75471 0.279232 0.139616 0.990206i \(-0.455413\pi\)
0.139616 + 0.990206i \(0.455413\pi\)
\(984\) −2.34662 −0.0748077
\(985\) −12.0998 −0.385530
\(986\) −2.10942 −0.0671775
\(987\) −3.53038 −0.112373
\(988\) 3.72682 0.118566
\(989\) 9.46005 0.300812
\(990\) −3.80406 −0.120901
\(991\) 49.7066 1.57898 0.789491 0.613763i \(-0.210345\pi\)
0.789491 + 0.613763i \(0.210345\pi\)
\(992\) 0.973192 0.0308989
\(993\) 26.2460 0.832891
\(994\) 40.8629 1.29609
\(995\) 13.2948 0.421474
\(996\) −12.7534 −0.404108
\(997\) 46.3005 1.46635 0.733176 0.680039i \(-0.238037\pi\)
0.733176 + 0.680039i \(0.238037\pi\)
\(998\) −53.4211 −1.69102
\(999\) −19.2372 −0.608638
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 1045.2.a.f.1.6 6
3.2 odd 2 9405.2.a.z.1.1 6
5.4 even 2 5225.2.a.l.1.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1045.2.a.f.1.6 6 1.1 even 1 trivial
5225.2.a.l.1.1 6 5.4 even 2
9405.2.a.z.1.1 6 3.2 odd 2