Properties

Label 5054.2.a.x.1.3
Level $5054$
Weight $2$
Character 5054.1
Self dual yes
Analytic conductor $40.356$
Analytic rank $0$
Dimension $4$
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] = [5054,2,Mod(1,5054)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5054, 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("5054.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5054 = 2 \cdot 7 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5054.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: \(40.3563931816\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.151572.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 10x^{2} + 8x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 266)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(1.16566\) of defining polynomial
Character \(\chi\) \(=\) 5054.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.16566 q^{3} +1.00000 q^{4} -1.55010 q^{5} +1.16566 q^{6} +1.00000 q^{7} +1.00000 q^{8} -1.64123 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.16566 q^{3} +1.00000 q^{4} -1.55010 q^{5} +1.16566 q^{6} +1.00000 q^{7} +1.00000 q^{8} -1.64123 q^{9} -1.55010 q^{10} +0.834338 q^{11} +1.16566 q^{12} -5.35699 q^{13} +1.00000 q^{14} -1.80689 q^{15} +1.00000 q^{16} +3.47557 q^{17} -1.64123 q^{18} -1.55010 q^{20} +1.16566 q^{21} +0.834338 q^{22} +8.74143 q^{23} +1.16566 q^{24} -2.59719 q^{25} -5.35699 q^{26} -5.41011 q^{27} +1.00000 q^{28} +9.23842 q^{29} -1.80689 q^{30} -6.90710 q^{31} +1.00000 q^{32} +0.972556 q^{33} +3.47557 q^{34} -1.55010 q^{35} -1.64123 q^{36} +1.10020 q^{37} -6.24445 q^{39} -1.55010 q^{40} +5.16566 q^{41} +1.16566 q^{42} +12.9071 q^{43} +0.834338 q^{44} +2.54407 q^{45} +8.74143 q^{46} +9.23842 q^{47} +1.16566 q^{48} +1.00000 q^{49} -2.59719 q^{50} +4.05134 q^{51} -5.35699 q^{52} +10.9071 q^{53} -5.41011 q^{54} -1.29331 q^{55} +1.00000 q^{56} +9.23842 q^{58} +7.09113 q^{59} -1.80689 q^{60} -9.16389 q^{61} -6.90710 q^{62} -1.64123 q^{63} +1.00000 q^{64} +8.30388 q^{65} +0.972556 q^{66} -0.309907 q^{67} +3.47557 q^{68} +10.1896 q^{69} -1.55010 q^{70} -0.734136 q^{71} -1.64123 q^{72} +6.79029 q^{73} +1.10020 q^{74} -3.02744 q^{75} +0.834338 q^{77} -6.24445 q^{78} +7.61379 q^{79} -1.55010 q^{80} -1.38266 q^{81} +5.16566 q^{82} -6.34039 q^{83} +1.16566 q^{84} -5.38748 q^{85} +12.9071 q^{86} +10.7689 q^{87} +0.834338 q^{88} -10.3313 q^{89} +2.54407 q^{90} -5.35699 q^{91} +8.74143 q^{92} -8.05134 q^{93} +9.23842 q^{94} +1.16566 q^{96} +1.49699 q^{97} +1.00000 q^{98} -1.36934 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} + q^{3} + 4 q^{4} + q^{5} + q^{6} + 4 q^{7} + 4 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{2} + q^{3} + 4 q^{4} + q^{5} + q^{6} + 4 q^{7} + 4 q^{8} + 9 q^{9} + q^{10} + 7 q^{11} + q^{12} + 5 q^{13} + 4 q^{14} + 12 q^{15} + 4 q^{16} + 2 q^{17} + 9 q^{18} + q^{20} + q^{21} + 7 q^{22} + 5 q^{23} + q^{24} + 15 q^{25} + 5 q^{26} + q^{27} + 4 q^{28} - 4 q^{29} + 12 q^{30} + 6 q^{31} + 4 q^{32} - 19 q^{33} + 2 q^{34} + q^{35} + 9 q^{36} - 10 q^{37} - 6 q^{39} + q^{40} + 17 q^{41} + q^{42} + 18 q^{43} + 7 q^{44} - 19 q^{45} + 5 q^{46} - 4 q^{47} + q^{48} + 4 q^{49} + 15 q^{50} - 22 q^{51} + 5 q^{52} + 10 q^{53} + q^{54} - 10 q^{55} + 4 q^{56} - 4 q^{58} + 20 q^{59} + 12 q^{60} + 9 q^{61} + 6 q^{62} + 9 q^{63} + 4 q^{64} + 3 q^{65} - 19 q^{66} + 7 q^{67} + 2 q^{68} - 24 q^{69} + q^{70} - 21 q^{71} + 9 q^{72} + 21 q^{73} - 10 q^{74} - 35 q^{75} + 7 q^{77} - 6 q^{78} - 8 q^{79} + q^{80} + 40 q^{81} + 17 q^{82} - 12 q^{83} + q^{84} + 10 q^{85} + 18 q^{86} + 36 q^{87} + 7 q^{88} - 34 q^{89} - 19 q^{90} + 5 q^{91} + 5 q^{92} + 6 q^{93} - 4 q^{94} + q^{96} - 5 q^{97} + 4 q^{98} + 14 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 1.16566 0.672995 0.336498 0.941684i \(-0.390758\pi\)
0.336498 + 0.941684i \(0.390758\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.55010 −0.693226 −0.346613 0.938008i \(-0.612668\pi\)
−0.346613 + 0.938008i \(0.612668\pi\)
\(6\) 1.16566 0.475880
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) −1.64123 −0.547077
\(10\) −1.55010 −0.490185
\(11\) 0.834338 0.251562 0.125781 0.992058i \(-0.459856\pi\)
0.125781 + 0.992058i \(0.459856\pi\)
\(12\) 1.16566 0.336498
\(13\) −5.35699 −1.48576 −0.742881 0.669423i \(-0.766541\pi\)
−0.742881 + 0.669423i \(0.766541\pi\)
\(14\) 1.00000 0.267261
\(15\) −1.80689 −0.466538
\(16\) 1.00000 0.250000
\(17\) 3.47557 0.842949 0.421475 0.906840i \(-0.361513\pi\)
0.421475 + 0.906840i \(0.361513\pi\)
\(18\) −1.64123 −0.386842
\(19\) 0 0
\(20\) −1.55010 −0.346613
\(21\) 1.16566 0.254368
\(22\) 0.834338 0.177881
\(23\) 8.74143 1.82271 0.911357 0.411616i \(-0.135036\pi\)
0.911357 + 0.411616i \(0.135036\pi\)
\(24\) 1.16566 0.237940
\(25\) −2.59719 −0.519438
\(26\) −5.35699 −1.05059
\(27\) −5.41011 −1.04118
\(28\) 1.00000 0.188982
\(29\) 9.23842 1.71553 0.857766 0.514041i \(-0.171852\pi\)
0.857766 + 0.514041i \(0.171852\pi\)
\(30\) −1.80689 −0.329892
\(31\) −6.90710 −1.24055 −0.620275 0.784384i \(-0.712979\pi\)
−0.620275 + 0.784384i \(0.712979\pi\)
\(32\) 1.00000 0.176777
\(33\) 0.972556 0.169300
\(34\) 3.47557 0.596055
\(35\) −1.55010 −0.262015
\(36\) −1.64123 −0.273539
\(37\) 1.10020 0.180872 0.0904360 0.995902i \(-0.471174\pi\)
0.0904360 + 0.995902i \(0.471174\pi\)
\(38\) 0 0
\(39\) −6.24445 −0.999912
\(40\) −1.55010 −0.245092
\(41\) 5.16566 0.806741 0.403370 0.915037i \(-0.367839\pi\)
0.403370 + 0.915037i \(0.367839\pi\)
\(42\) 1.16566 0.179866
\(43\) 12.9071 1.96831 0.984157 0.177300i \(-0.0567364\pi\)
0.984157 + 0.177300i \(0.0567364\pi\)
\(44\) 0.834338 0.125781
\(45\) 2.54407 0.379248
\(46\) 8.74143 1.28885
\(47\) 9.23842 1.34756 0.673781 0.738931i \(-0.264669\pi\)
0.673781 + 0.738931i \(0.264669\pi\)
\(48\) 1.16566 0.168249
\(49\) 1.00000 0.142857
\(50\) −2.59719 −0.367298
\(51\) 4.05134 0.567301
\(52\) −5.35699 −0.742881
\(53\) 10.9071 1.49820 0.749102 0.662454i \(-0.230485\pi\)
0.749102 + 0.662454i \(0.230485\pi\)
\(54\) −5.41011 −0.736222
\(55\) −1.29331 −0.174390
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) 9.23842 1.21306
\(59\) 7.09113 0.923187 0.461593 0.887092i \(-0.347278\pi\)
0.461593 + 0.887092i \(0.347278\pi\)
\(60\) −1.80689 −0.233269
\(61\) −9.16389 −1.17332 −0.586658 0.809835i \(-0.699557\pi\)
−0.586658 + 0.809835i \(0.699557\pi\)
\(62\) −6.90710 −0.877202
\(63\) −1.64123 −0.206776
\(64\) 1.00000 0.125000
\(65\) 8.30388 1.02997
\(66\) 0.972556 0.119713
\(67\) −0.309907 −0.0378612 −0.0189306 0.999821i \(-0.506026\pi\)
−0.0189306 + 0.999821i \(0.506026\pi\)
\(68\) 3.47557 0.421475
\(69\) 10.1896 1.22668
\(70\) −1.55010 −0.185272
\(71\) −0.734136 −0.0871260 −0.0435630 0.999051i \(-0.513871\pi\)
−0.0435630 + 0.999051i \(0.513871\pi\)
\(72\) −1.64123 −0.193421
\(73\) 6.79029 0.794744 0.397372 0.917658i \(-0.369922\pi\)
0.397372 + 0.917658i \(0.369922\pi\)
\(74\) 1.10020 0.127896
\(75\) −3.02744 −0.349579
\(76\) 0 0
\(77\) 0.834338 0.0950816
\(78\) −6.24445 −0.707044
\(79\) 7.61379 0.856618 0.428309 0.903632i \(-0.359109\pi\)
0.428309 + 0.903632i \(0.359109\pi\)
\(80\) −1.55010 −0.173307
\(81\) −1.38266 −0.153629
\(82\) 5.16566 0.570452
\(83\) −6.34039 −0.695949 −0.347974 0.937504i \(-0.613130\pi\)
−0.347974 + 0.937504i \(0.613130\pi\)
\(84\) 1.16566 0.127184
\(85\) −5.38748 −0.584354
\(86\) 12.9071 1.39181
\(87\) 10.7689 1.15454
\(88\) 0.834338 0.0889407
\(89\) −10.3313 −1.09512 −0.547559 0.836767i \(-0.684443\pi\)
−0.547559 + 0.836767i \(0.684443\pi\)
\(90\) 2.54407 0.268169
\(91\) −5.35699 −0.561566
\(92\) 8.74143 0.911357
\(93\) −8.05134 −0.834885
\(94\) 9.23842 0.952870
\(95\) 0 0
\(96\) 1.16566 0.118970
\(97\) 1.49699 0.151996 0.0759980 0.997108i \(-0.475786\pi\)
0.0759980 + 0.997108i \(0.475786\pi\)
\(98\) 1.00000 0.101015
\(99\) −1.36934 −0.137624
\(100\) −2.59719 −0.259719
\(101\) 1.80689 0.179793 0.0898963 0.995951i \(-0.471346\pi\)
0.0898963 + 0.995951i \(0.471346\pi\)
\(102\) 4.05134 0.401142
\(103\) 11.8069 1.16337 0.581684 0.813415i \(-0.302394\pi\)
0.581684 + 0.813415i \(0.302394\pi\)
\(104\) −5.35699 −0.525297
\(105\) −1.80689 −0.176335
\(106\) 10.9071 1.05939
\(107\) 10.5317 1.01814 0.509070 0.860725i \(-0.329989\pi\)
0.509070 + 0.860725i \(0.329989\pi\)
\(108\) −5.41011 −0.520588
\(109\) −13.9011 −1.33148 −0.665740 0.746183i \(-0.731884\pi\)
−0.665740 + 0.746183i \(0.731884\pi\)
\(110\) −1.29331 −0.123312
\(111\) 1.28246 0.121726
\(112\) 1.00000 0.0944911
\(113\) −1.19311 −0.112238 −0.0561190 0.998424i \(-0.517873\pi\)
−0.0561190 + 0.998424i \(0.517873\pi\)
\(114\) 0 0
\(115\) −13.5501 −1.26355
\(116\) 9.23842 0.857766
\(117\) 8.79207 0.812827
\(118\) 7.09113 0.652792
\(119\) 3.47557 0.318605
\(120\) −1.80689 −0.164946
\(121\) −10.3039 −0.936716
\(122\) −9.16389 −0.829659
\(123\) 6.02142 0.542933
\(124\) −6.90710 −0.620275
\(125\) 11.7764 1.05331
\(126\) −1.64123 −0.146213
\(127\) −9.79029 −0.868748 −0.434374 0.900733i \(-0.643030\pi\)
−0.434374 + 0.900733i \(0.643030\pi\)
\(128\) 1.00000 0.0883883
\(129\) 15.0453 1.32467
\(130\) 8.30388 0.728298
\(131\) −14.6609 −1.28093 −0.640463 0.767989i \(-0.721258\pi\)
−0.640463 + 0.767989i \(0.721258\pi\)
\(132\) 0.972556 0.0846501
\(133\) 0 0
\(134\) −0.309907 −0.0267719
\(135\) 8.38621 0.721770
\(136\) 3.47557 0.298028
\(137\) 15.7140 1.34254 0.671268 0.741214i \(-0.265750\pi\)
0.671268 + 0.741214i \(0.265750\pi\)
\(138\) 10.1896 0.867393
\(139\) 14.3172 1.21437 0.607185 0.794561i \(-0.292299\pi\)
0.607185 + 0.794561i \(0.292299\pi\)
\(140\) −1.55010 −0.131007
\(141\) 10.7689 0.906903
\(142\) −0.734136 −0.0616074
\(143\) −4.46954 −0.373762
\(144\) −1.64123 −0.136769
\(145\) −14.3205 −1.18925
\(146\) 6.79029 0.561969
\(147\) 1.16566 0.0961422
\(148\) 1.10020 0.0904360
\(149\) 2.89980 0.237561 0.118780 0.992921i \(-0.462102\pi\)
0.118780 + 0.992921i \(0.462102\pi\)
\(150\) −3.02744 −0.247190
\(151\) −10.3039 −0.838518 −0.419259 0.907867i \(-0.637710\pi\)
−0.419259 + 0.907867i \(0.637710\pi\)
\(152\) 0 0
\(153\) −5.70421 −0.461158
\(154\) 0.834338 0.0672329
\(155\) 10.7067 0.859982
\(156\) −6.24445 −0.499956
\(157\) 14.4572 1.15381 0.576905 0.816811i \(-0.304260\pi\)
0.576905 + 0.816811i \(0.304260\pi\)
\(158\) 7.61379 0.605720
\(159\) 12.7140 1.00828
\(160\) −1.55010 −0.122546
\(161\) 8.74143 0.688921
\(162\) −1.38266 −0.108632
\(163\) −8.21098 −0.643133 −0.321567 0.946887i \(-0.604209\pi\)
−0.321567 + 0.946887i \(0.604209\pi\)
\(164\) 5.16566 0.403370
\(165\) −1.50756 −0.117363
\(166\) −6.34039 −0.492110
\(167\) −11.7629 −0.910237 −0.455118 0.890431i \(-0.650403\pi\)
−0.455118 + 0.890431i \(0.650403\pi\)
\(168\) 1.16566 0.0899328
\(169\) 15.6974 1.20749
\(170\) −5.38748 −0.413201
\(171\) 0 0
\(172\) 12.9071 0.984157
\(173\) 13.7505 1.04543 0.522716 0.852507i \(-0.324919\pi\)
0.522716 + 0.852507i \(0.324919\pi\)
\(174\) 10.7689 0.816386
\(175\) −2.59719 −0.196329
\(176\) 0.834338 0.0628906
\(177\) 8.26586 0.621300
\(178\) −10.3313 −0.774366
\(179\) −10.4481 −0.780930 −0.390465 0.920618i \(-0.627686\pi\)
−0.390465 + 0.920618i \(0.627686\pi\)
\(180\) 2.54407 0.189624
\(181\) 3.16389 0.235170 0.117585 0.993063i \(-0.462485\pi\)
0.117585 + 0.993063i \(0.462485\pi\)
\(182\) −5.35699 −0.397087
\(183\) −10.6820 −0.789636
\(184\) 8.74143 0.644427
\(185\) −1.70542 −0.125385
\(186\) −8.05134 −0.590353
\(187\) 2.89980 0.212054
\(188\) 9.23842 0.673781
\(189\) −5.41011 −0.393527
\(190\) 0 0
\(191\) −10.8283 −0.783509 −0.391755 0.920070i \(-0.628132\pi\)
−0.391755 + 0.920070i \(0.628132\pi\)
\(192\) 1.16566 0.0841244
\(193\) −7.50301 −0.540079 −0.270039 0.962849i \(-0.587037\pi\)
−0.270039 + 0.962849i \(0.587037\pi\)
\(194\) 1.49699 0.107477
\(195\) 9.67952 0.693165
\(196\) 1.00000 0.0714286
\(197\) 11.3875 0.811325 0.405662 0.914023i \(-0.367041\pi\)
0.405662 + 0.914023i \(0.367041\pi\)
\(198\) −1.36934 −0.0973149
\(199\) −2.90710 −0.206079 −0.103039 0.994677i \(-0.532857\pi\)
−0.103039 + 0.994677i \(0.532857\pi\)
\(200\) −2.59719 −0.183649
\(201\) −0.361247 −0.0254804
\(202\) 1.80689 0.127133
\(203\) 9.23842 0.648410
\(204\) 4.05134 0.283651
\(205\) −8.00730 −0.559254
\(206\) 11.8069 0.822625
\(207\) −14.3467 −0.997166
\(208\) −5.35699 −0.371441
\(209\) 0 0
\(210\) −1.80689 −0.124688
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) 10.9071 0.749102
\(213\) −0.855755 −0.0586354
\(214\) 10.5317 0.719934
\(215\) −20.0073 −1.36449
\(216\) −5.41011 −0.368111
\(217\) −6.90710 −0.468884
\(218\) −13.9011 −0.941499
\(219\) 7.91519 0.534859
\(220\) −1.29331 −0.0871948
\(221\) −18.6186 −1.25242
\(222\) 1.28246 0.0860733
\(223\) −11.8142 −0.791137 −0.395568 0.918437i \(-0.629452\pi\)
−0.395568 + 0.918437i \(0.629452\pi\)
\(224\) 1.00000 0.0668153
\(225\) 4.26259 0.284172
\(226\) −1.19311 −0.0793642
\(227\) −17.1057 −1.13535 −0.567673 0.823254i \(-0.692156\pi\)
−0.567673 + 0.823254i \(0.692156\pi\)
\(228\) 0 0
\(229\) −23.7837 −1.57167 −0.785836 0.618435i \(-0.787767\pi\)
−0.785836 + 0.618435i \(0.787767\pi\)
\(230\) −13.5501 −0.893467
\(231\) 0.972556 0.0639895
\(232\) 9.23842 0.606532
\(233\) 9.71399 0.636385 0.318192 0.948026i \(-0.396924\pi\)
0.318192 + 0.948026i \(0.396924\pi\)
\(234\) 8.79207 0.574755
\(235\) −14.3205 −0.934165
\(236\) 7.09113 0.461593
\(237\) 8.87510 0.576500
\(238\) 3.47557 0.225288
\(239\) −7.82349 −0.506060 −0.253030 0.967458i \(-0.581427\pi\)
−0.253030 + 0.967458i \(0.581427\pi\)
\(240\) −1.80689 −0.116634
\(241\) 13.4970 0.869417 0.434709 0.900571i \(-0.356851\pi\)
0.434709 + 0.900571i \(0.356851\pi\)
\(242\) −10.3039 −0.662359
\(243\) 14.6186 0.937784
\(244\) −9.16389 −0.586658
\(245\) −1.55010 −0.0990323
\(246\) 6.02142 0.383912
\(247\) 0 0
\(248\) −6.90710 −0.438601
\(249\) −7.39076 −0.468370
\(250\) 11.7764 0.744805
\(251\) −9.57881 −0.604609 −0.302305 0.953211i \(-0.597756\pi\)
−0.302305 + 0.953211i \(0.597756\pi\)
\(252\) −1.64123 −0.103388
\(253\) 7.29331 0.458526
\(254\) −9.79029 −0.614298
\(255\) −6.27998 −0.393268
\(256\) 1.00000 0.0625000
\(257\) 7.98588 0.498145 0.249073 0.968485i \(-0.419874\pi\)
0.249073 + 0.968485i \(0.419874\pi\)
\(258\) 15.0453 0.936680
\(259\) 1.10020 0.0683632
\(260\) 8.30388 0.514985
\(261\) −15.1624 −0.938528
\(262\) −14.6609 −0.905752
\(263\) −16.5483 −1.02041 −0.510207 0.860052i \(-0.670431\pi\)
−0.510207 + 0.860052i \(0.670431\pi\)
\(264\) 0.972556 0.0598567
\(265\) −16.9071 −1.03859
\(266\) 0 0
\(267\) −12.0428 −0.737010
\(268\) −0.309907 −0.0189306
\(269\) −0.575771 −0.0351054 −0.0175527 0.999846i \(-0.505587\pi\)
−0.0175527 + 0.999846i \(0.505587\pi\)
\(270\) 8.38621 0.510369
\(271\) −3.62108 −0.219965 −0.109983 0.993934i \(-0.535080\pi\)
−0.109983 + 0.993934i \(0.535080\pi\)
\(272\) 3.47557 0.210737
\(273\) −6.24445 −0.377931
\(274\) 15.7140 0.949317
\(275\) −2.16693 −0.130671
\(276\) 10.1896 0.613339
\(277\) 30.3278 1.82222 0.911110 0.412164i \(-0.135227\pi\)
0.911110 + 0.412164i \(0.135227\pi\)
\(278\) 14.3172 0.858689
\(279\) 11.3361 0.678677
\(280\) −1.55010 −0.0926362
\(281\) −3.35877 −0.200367 −0.100184 0.994969i \(-0.531943\pi\)
−0.100184 + 0.994969i \(0.531943\pi\)
\(282\) 10.7689 0.641277
\(283\) 3.33437 0.198208 0.0991038 0.995077i \(-0.468402\pi\)
0.0991038 + 0.995077i \(0.468402\pi\)
\(284\) −0.734136 −0.0435630
\(285\) 0 0
\(286\) −4.46954 −0.264290
\(287\) 5.16566 0.304919
\(288\) −1.64123 −0.0967105
\(289\) −4.92042 −0.289436
\(290\) −14.3205 −0.840928
\(291\) 1.74498 0.102293
\(292\) 6.79029 0.397372
\(293\) 22.8839 1.33689 0.668446 0.743761i \(-0.266960\pi\)
0.668446 + 0.743761i \(0.266960\pi\)
\(294\) 1.16566 0.0679828
\(295\) −10.9920 −0.639977
\(296\) 1.10020 0.0639479
\(297\) −4.51386 −0.261921
\(298\) 2.89980 0.167981
\(299\) −46.8278 −2.70812
\(300\) −3.02744 −0.174790
\(301\) 12.9071 0.743953
\(302\) −10.3039 −0.592922
\(303\) 2.10623 0.121000
\(304\) 0 0
\(305\) 14.2049 0.813373
\(306\) −5.70421 −0.326088
\(307\) −18.2867 −1.04368 −0.521839 0.853044i \(-0.674754\pi\)
−0.521839 + 0.853044i \(0.674754\pi\)
\(308\) 0.834338 0.0475408
\(309\) 13.7629 0.782941
\(310\) 10.7067 0.608099
\(311\) 7.23112 0.410039 0.205020 0.978758i \(-0.434274\pi\)
0.205020 + 0.978758i \(0.434274\pi\)
\(312\) −6.24445 −0.353522
\(313\) 6.51031 0.367984 0.183992 0.982928i \(-0.441098\pi\)
0.183992 + 0.982928i \(0.441098\pi\)
\(314\) 14.4572 0.815867
\(315\) 2.54407 0.143342
\(316\) 7.61379 0.428309
\(317\) −29.4461 −1.65386 −0.826929 0.562306i \(-0.809914\pi\)
−0.826929 + 0.562306i \(0.809914\pi\)
\(318\) 12.7140 0.712965
\(319\) 7.70796 0.431563
\(320\) −1.55010 −0.0866533
\(321\) 12.2764 0.685204
\(322\) 8.74143 0.487141
\(323\) 0 0
\(324\) −1.38266 −0.0768147
\(325\) 13.9131 0.771761
\(326\) −8.21098 −0.454764
\(327\) −16.2040 −0.896081
\(328\) 5.16566 0.285226
\(329\) 9.23842 0.509331
\(330\) −1.50756 −0.0829884
\(331\) −20.7976 −1.14314 −0.571569 0.820554i \(-0.693665\pi\)
−0.571569 + 0.820554i \(0.693665\pi\)
\(332\) −6.34039 −0.347974
\(333\) −1.80569 −0.0989509
\(334\) −11.7629 −0.643635
\(335\) 0.480387 0.0262464
\(336\) 1.16566 0.0635921
\(337\) 28.2017 1.53624 0.768121 0.640304i \(-0.221192\pi\)
0.768121 + 0.640304i \(0.221192\pi\)
\(338\) 15.6974 0.853825
\(339\) −1.39076 −0.0755357
\(340\) −5.38748 −0.292177
\(341\) −5.76285 −0.312076
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 12.9071 0.695904
\(345\) −15.7948 −0.850366
\(346\) 13.7505 0.739232
\(347\) 28.3064 1.51956 0.759782 0.650177i \(-0.225305\pi\)
0.759782 + 0.650177i \(0.225305\pi\)
\(348\) 10.7689 0.577272
\(349\) 19.6211 1.05029 0.525146 0.851012i \(-0.324011\pi\)
0.525146 + 0.851012i \(0.324011\pi\)
\(350\) −2.59719 −0.138826
\(351\) 28.9819 1.54694
\(352\) 0.834338 0.0444704
\(353\) 1.82831 0.0973112 0.0486556 0.998816i \(-0.484506\pi\)
0.0486556 + 0.998816i \(0.484506\pi\)
\(354\) 8.26586 0.439326
\(355\) 1.13799 0.0603980
\(356\) −10.3313 −0.547559
\(357\) 4.05134 0.214420
\(358\) −10.4481 −0.552201
\(359\) 15.3006 0.807535 0.403767 0.914862i \(-0.367700\pi\)
0.403767 + 0.914862i \(0.367700\pi\)
\(360\) 2.54407 0.134084
\(361\) 0 0
\(362\) 3.16389 0.166290
\(363\) −12.0108 −0.630406
\(364\) −5.35699 −0.280783
\(365\) −10.5256 −0.550937
\(366\) −10.6820 −0.558357
\(367\) 19.9011 1.03883 0.519414 0.854523i \(-0.326150\pi\)
0.519414 + 0.854523i \(0.326150\pi\)
\(368\) 8.74143 0.455679
\(369\) −8.47805 −0.441350
\(370\) −1.70542 −0.0886607
\(371\) 10.9071 0.566268
\(372\) −8.05134 −0.417443
\(373\) −36.9071 −1.91098 −0.955488 0.295028i \(-0.904671\pi\)
−0.955488 + 0.295028i \(0.904671\pi\)
\(374\) 2.89980 0.149945
\(375\) 13.7273 0.708875
\(376\) 9.23842 0.476435
\(377\) −49.4902 −2.54887
\(378\) −5.41011 −0.278266
\(379\) 3.10750 0.159621 0.0798107 0.996810i \(-0.474568\pi\)
0.0798107 + 0.996810i \(0.474568\pi\)
\(380\) 0 0
\(381\) −11.4122 −0.584664
\(382\) −10.8283 −0.554025
\(383\) 17.7200 0.905450 0.452725 0.891650i \(-0.350452\pi\)
0.452725 + 0.891650i \(0.350452\pi\)
\(384\) 1.16566 0.0594850
\(385\) −1.29331 −0.0659131
\(386\) −7.50301 −0.381893
\(387\) −21.1835 −1.07682
\(388\) 1.49699 0.0759980
\(389\) 25.8069 1.30846 0.654231 0.756295i \(-0.272993\pi\)
0.654231 + 0.756295i \(0.272993\pi\)
\(390\) 9.67952 0.490142
\(391\) 30.3815 1.53646
\(392\) 1.00000 0.0505076
\(393\) −17.0896 −0.862058
\(394\) 11.3875 0.573693
\(395\) −11.8021 −0.593830
\(396\) −1.36934 −0.0688120
\(397\) 2.81292 0.141176 0.0705882 0.997506i \(-0.477512\pi\)
0.0705882 + 0.997506i \(0.477512\pi\)
\(398\) −2.90710 −0.145720
\(399\) 0 0
\(400\) −2.59719 −0.129859
\(401\) 13.7067 0.684480 0.342240 0.939613i \(-0.388814\pi\)
0.342240 + 0.939613i \(0.388814\pi\)
\(402\) −0.361247 −0.0180174
\(403\) 37.0013 1.84316
\(404\) 1.80689 0.0898963
\(405\) 2.14327 0.106500
\(406\) 9.23842 0.458495
\(407\) 0.917939 0.0455006
\(408\) 4.05134 0.200571
\(409\) −27.9237 −1.38074 −0.690369 0.723458i \(-0.742552\pi\)
−0.690369 + 0.723458i \(0.742552\pi\)
\(410\) −8.00730 −0.395452
\(411\) 18.3172 0.903521
\(412\) 11.8069 0.581684
\(413\) 7.09113 0.348932
\(414\) −14.3467 −0.705103
\(415\) 9.82825 0.482450
\(416\) −5.35699 −0.262648
\(417\) 16.6890 0.817265
\(418\) 0 0
\(419\) −32.5209 −1.58875 −0.794375 0.607428i \(-0.792201\pi\)
−0.794375 + 0.607428i \(0.792201\pi\)
\(420\) −1.80689 −0.0881674
\(421\) 21.1075 1.02872 0.514358 0.857575i \(-0.328030\pi\)
0.514358 + 0.857575i \(0.328030\pi\)
\(422\) 4.00000 0.194717
\(423\) −15.1624 −0.737220
\(424\) 10.9071 0.529695
\(425\) −9.02671 −0.437860
\(426\) −0.855755 −0.0414615
\(427\) −9.16389 −0.443472
\(428\) 10.5317 0.509070
\(429\) −5.20998 −0.251540
\(430\) −20.0073 −0.964838
\(431\) 2.89980 0.139678 0.0698392 0.997558i \(-0.477751\pi\)
0.0698392 + 0.997558i \(0.477751\pi\)
\(432\) −5.41011 −0.260294
\(433\) −23.9952 −1.15314 −0.576569 0.817049i \(-0.695609\pi\)
−0.576569 + 0.817049i \(0.695609\pi\)
\(434\) −6.90710 −0.331551
\(435\) −16.6928 −0.800361
\(436\) −13.9011 −0.665740
\(437\) 0 0
\(438\) 7.91519 0.378202
\(439\) −38.5355 −1.83920 −0.919599 0.392858i \(-0.871486\pi\)
−0.919599 + 0.392858i \(0.871486\pi\)
\(440\) −1.29331 −0.0616560
\(441\) −1.64123 −0.0781539
\(442\) −18.6186 −0.885597
\(443\) −26.6805 −1.26763 −0.633815 0.773485i \(-0.718512\pi\)
−0.633815 + 0.773485i \(0.718512\pi\)
\(444\) 1.28246 0.0608630
\(445\) 16.0146 0.759165
\(446\) −11.8142 −0.559418
\(447\) 3.38019 0.159877
\(448\) 1.00000 0.0472456
\(449\) −28.6284 −1.35106 −0.675528 0.737334i \(-0.736084\pi\)
−0.675528 + 0.737334i \(0.736084\pi\)
\(450\) 4.26259 0.200940
\(451\) 4.30991 0.202946
\(452\) −1.19311 −0.0561190
\(453\) −12.0108 −0.564319
\(454\) −17.1057 −0.802811
\(455\) 8.30388 0.389292
\(456\) 0 0
\(457\) 7.28728 0.340885 0.170442 0.985368i \(-0.445480\pi\)
0.170442 + 0.985368i \(0.445480\pi\)
\(458\) −23.7837 −1.11134
\(459\) −18.8032 −0.877659
\(460\) −13.5501 −0.631777
\(461\) 14.2019 0.661449 0.330724 0.943727i \(-0.392707\pi\)
0.330724 + 0.943727i \(0.392707\pi\)
\(462\) 0.972556 0.0452474
\(463\) 29.8308 1.38635 0.693177 0.720767i \(-0.256210\pi\)
0.693177 + 0.720767i \(0.256210\pi\)
\(464\) 9.23842 0.428883
\(465\) 12.4804 0.578764
\(466\) 9.71399 0.449992
\(467\) 5.87965 0.272078 0.136039 0.990703i \(-0.456563\pi\)
0.136039 + 0.990703i \(0.456563\pi\)
\(468\) 8.79207 0.406413
\(469\) −0.309907 −0.0143102
\(470\) −14.3205 −0.660554
\(471\) 16.8522 0.776509
\(472\) 7.09113 0.326396
\(473\) 10.7689 0.495153
\(474\) 8.87510 0.407647
\(475\) 0 0
\(476\) 3.47557 0.159302
\(477\) −17.9011 −0.819634
\(478\) −7.82349 −0.357838
\(479\) 16.8739 0.770988 0.385494 0.922710i \(-0.374031\pi\)
0.385494 + 0.922710i \(0.374031\pi\)
\(480\) −1.80689 −0.0824730
\(481\) −5.89377 −0.268733
\(482\) 13.4970 0.614771
\(483\) 10.1896 0.463641
\(484\) −10.3039 −0.468358
\(485\) −2.32048 −0.105368
\(486\) 14.6186 0.663113
\(487\) −17.1002 −0.774884 −0.387442 0.921894i \(-0.626641\pi\)
−0.387442 + 0.921894i \(0.626641\pi\)
\(488\) −9.16389 −0.414830
\(489\) −9.57122 −0.432826
\(490\) −1.55010 −0.0700264
\(491\) −5.04886 −0.227852 −0.113926 0.993489i \(-0.536343\pi\)
−0.113926 + 0.993489i \(0.536343\pi\)
\(492\) 6.02142 0.271466
\(493\) 32.1088 1.44611
\(494\) 0 0
\(495\) 2.12262 0.0954045
\(496\) −6.90710 −0.310138
\(497\) −0.734136 −0.0329305
\(498\) −7.39076 −0.331188
\(499\) −11.1108 −0.497387 −0.248693 0.968582i \(-0.580001\pi\)
−0.248693 + 0.968582i \(0.580001\pi\)
\(500\) 11.7764 0.526657
\(501\) −13.7115 −0.612585
\(502\) −9.57881 −0.427523
\(503\) −18.8190 −0.839098 −0.419549 0.907733i \(-0.637812\pi\)
−0.419549 + 0.907733i \(0.637812\pi\)
\(504\) −1.64123 −0.0731063
\(505\) −2.80087 −0.124637
\(506\) 7.29331 0.324227
\(507\) 18.2979 0.812636
\(508\) −9.79029 −0.434374
\(509\) −32.6359 −1.44656 −0.723281 0.690554i \(-0.757367\pi\)
−0.723281 + 0.690554i \(0.757367\pi\)
\(510\) −6.27998 −0.278082
\(511\) 6.79029 0.300385
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 7.98588 0.352242
\(515\) −18.3019 −0.806477
\(516\) 15.0453 0.662333
\(517\) 7.70796 0.338996
\(518\) 1.10020 0.0483401
\(519\) 16.0284 0.703571
\(520\) 8.30388 0.364149
\(521\) 1.17296 0.0513882 0.0256941 0.999670i \(-0.491820\pi\)
0.0256941 + 0.999670i \(0.491820\pi\)
\(522\) −15.1624 −0.663640
\(523\) 24.5209 1.07222 0.536112 0.844147i \(-0.319893\pi\)
0.536112 + 0.844147i \(0.319893\pi\)
\(524\) −14.6609 −0.640463
\(525\) −3.02744 −0.132128
\(526\) −16.5483 −0.721541
\(527\) −24.0061 −1.04572
\(528\) 0.972556 0.0423251
\(529\) 53.4126 2.32229
\(530\) −16.9071 −0.734397
\(531\) −11.6382 −0.505054
\(532\) 0 0
\(533\) −27.6724 −1.19863
\(534\) −12.0428 −0.521144
\(535\) −16.3252 −0.705802
\(536\) −0.309907 −0.0133859
\(537\) −12.1790 −0.525562
\(538\) −0.575771 −0.0248232
\(539\) 0.834338 0.0359375
\(540\) 8.38621 0.360885
\(541\) 37.0418 1.59255 0.796275 0.604935i \(-0.206801\pi\)
0.796275 + 0.604935i \(0.206801\pi\)
\(542\) −3.62108 −0.155539
\(543\) 3.68803 0.158268
\(544\) 3.47557 0.149014
\(545\) 21.5481 0.923017
\(546\) −6.24445 −0.267238
\(547\) −0.506290 −0.0216474 −0.0108237 0.999941i \(-0.503445\pi\)
−0.0108237 + 0.999941i \(0.503445\pi\)
\(548\) 15.7140 0.671268
\(549\) 15.0401 0.641894
\(550\) −2.16693 −0.0923983
\(551\) 0 0
\(552\) 10.1896 0.433696
\(553\) 7.61379 0.323771
\(554\) 30.3278 1.28850
\(555\) −1.98795 −0.0843836
\(556\) 14.3172 0.607185
\(557\) 0.513586 0.0217613 0.0108807 0.999941i \(-0.496537\pi\)
0.0108807 + 0.999941i \(0.496537\pi\)
\(558\) 11.3361 0.479897
\(559\) −69.1432 −2.92445
\(560\) −1.55010 −0.0655037
\(561\) 3.38019 0.142712
\(562\) −3.35877 −0.141681
\(563\) 29.3235 1.23584 0.617920 0.786241i \(-0.287976\pi\)
0.617920 + 0.786241i \(0.287976\pi\)
\(564\) 10.7689 0.453452
\(565\) 1.84943 0.0778063
\(566\) 3.33437 0.140154
\(567\) −1.38266 −0.0580664
\(568\) −0.734136 −0.0308037
\(569\) 1.21452 0.0509155 0.0254577 0.999676i \(-0.491896\pi\)
0.0254577 + 0.999676i \(0.491896\pi\)
\(570\) 0 0
\(571\) 21.2097 0.887599 0.443799 0.896126i \(-0.353630\pi\)
0.443799 + 0.896126i \(0.353630\pi\)
\(572\) −4.46954 −0.186881
\(573\) −12.6222 −0.527298
\(574\) 5.16566 0.215611
\(575\) −22.7031 −0.946787
\(576\) −1.64123 −0.0683846
\(577\) 7.78427 0.324063 0.162032 0.986786i \(-0.448195\pi\)
0.162032 + 0.986786i \(0.448195\pi\)
\(578\) −4.92042 −0.204662
\(579\) −8.74598 −0.363470
\(580\) −14.3205 −0.594626
\(581\) −6.34039 −0.263044
\(582\) 1.74498 0.0723318
\(583\) 9.10020 0.376892
\(584\) 6.79029 0.280984
\(585\) −13.6286 −0.563473
\(586\) 22.8839 0.945325
\(587\) 10.7128 0.442164 0.221082 0.975255i \(-0.429041\pi\)
0.221082 + 0.975255i \(0.429041\pi\)
\(588\) 1.16566 0.0480711
\(589\) 0 0
\(590\) −10.9920 −0.452532
\(591\) 13.2740 0.546018
\(592\) 1.10020 0.0452180
\(593\) −31.9859 −1.31350 −0.656751 0.754107i \(-0.728070\pi\)
−0.656751 + 0.754107i \(0.728070\pi\)
\(594\) −4.51386 −0.185206
\(595\) −5.38748 −0.220865
\(596\) 2.89980 0.118780
\(597\) −3.38869 −0.138690
\(598\) −46.8278 −1.91493
\(599\) 10.7916 0.440931 0.220466 0.975395i \(-0.429242\pi\)
0.220466 + 0.975395i \(0.429242\pi\)
\(600\) −3.02744 −0.123595
\(601\) −15.3908 −0.627802 −0.313901 0.949456i \(-0.601636\pi\)
−0.313901 + 0.949456i \(0.601636\pi\)
\(602\) 12.9071 0.526054
\(603\) 0.508629 0.0207130
\(604\) −10.3039 −0.419259
\(605\) 15.9721 0.649356
\(606\) 2.10623 0.0855597
\(607\) −17.8142 −0.723056 −0.361528 0.932361i \(-0.617745\pi\)
−0.361528 + 0.932361i \(0.617745\pi\)
\(608\) 0 0
\(609\) 10.7689 0.436377
\(610\) 14.2049 0.575142
\(611\) −49.4902 −2.00216
\(612\) −5.70421 −0.230579
\(613\) −45.5415 −1.83940 −0.919702 0.392617i \(-0.871570\pi\)
−0.919702 + 0.392617i \(0.871570\pi\)
\(614\) −18.2867 −0.737992
\(615\) −9.33380 −0.376375
\(616\) 0.834338 0.0336164
\(617\) −27.5390 −1.10868 −0.554340 0.832291i \(-0.687029\pi\)
−0.554340 + 0.832291i \(0.687029\pi\)
\(618\) 13.7629 0.553623
\(619\) −13.6370 −0.548116 −0.274058 0.961713i \(-0.588366\pi\)
−0.274058 + 0.961713i \(0.588366\pi\)
\(620\) 10.7067 0.429991
\(621\) −47.2921 −1.89777
\(622\) 7.23112 0.289942
\(623\) −10.3313 −0.413916
\(624\) −6.24445 −0.249978
\(625\) −5.26867 −0.210747
\(626\) 6.51031 0.260204
\(627\) 0 0
\(628\) 14.4572 0.576905
\(629\) 3.82383 0.152466
\(630\) 2.54407 0.101358
\(631\) 27.6284 1.09987 0.549934 0.835208i \(-0.314653\pi\)
0.549934 + 0.835208i \(0.314653\pi\)
\(632\) 7.61379 0.302860
\(633\) 4.66265 0.185324
\(634\) −29.4461 −1.16945
\(635\) 15.1759 0.602239
\(636\) 12.7140 0.504142
\(637\) −5.35699 −0.212252
\(638\) 7.70796 0.305161
\(639\) 1.20489 0.0476646
\(640\) −1.55010 −0.0612731
\(641\) −14.2631 −0.563359 −0.281680 0.959509i \(-0.590892\pi\)
−0.281680 + 0.959509i \(0.590892\pi\)
\(642\) 12.2764 0.484512
\(643\) −1.75378 −0.0691623 −0.0345812 0.999402i \(-0.511010\pi\)
−0.0345812 + 0.999402i \(0.511010\pi\)
\(644\) 8.74143 0.344461
\(645\) −23.3217 −0.918293
\(646\) 0 0
\(647\) −5.58180 −0.219443 −0.109722 0.993962i \(-0.534996\pi\)
−0.109722 + 0.993962i \(0.534996\pi\)
\(648\) −1.38266 −0.0543162
\(649\) 5.91640 0.232239
\(650\) 13.9131 0.545718
\(651\) −8.05134 −0.315557
\(652\) −8.21098 −0.321567
\(653\) 25.0098 0.978708 0.489354 0.872085i \(-0.337233\pi\)
0.489354 + 0.872085i \(0.337233\pi\)
\(654\) −16.2040 −0.633625
\(655\) 22.7258 0.887972
\(656\) 5.16566 0.201685
\(657\) −11.1444 −0.434786
\(658\) 9.23842 0.360151
\(659\) −27.2971 −1.06334 −0.531671 0.846951i \(-0.678436\pi\)
−0.531671 + 0.846951i \(0.678436\pi\)
\(660\) −1.50756 −0.0586817
\(661\) 33.5647 1.30552 0.652758 0.757567i \(-0.273612\pi\)
0.652758 + 0.757567i \(0.273612\pi\)
\(662\) −20.7976 −0.808321
\(663\) −21.7030 −0.842875
\(664\) −6.34039 −0.246055
\(665\) 0 0
\(666\) −1.80569 −0.0699689
\(667\) 80.7570 3.12692
\(668\) −11.7629 −0.455118
\(669\) −13.7714 −0.532431
\(670\) 0.480387 0.0185590
\(671\) −7.64578 −0.295162
\(672\) 1.16566 0.0449664
\(673\) −4.50428 −0.173627 −0.0868137 0.996225i \(-0.527668\pi\)
−0.0868137 + 0.996225i \(0.527668\pi\)
\(674\) 28.2017 1.08629
\(675\) 14.0511 0.540826
\(676\) 15.6974 0.603746
\(677\) −17.3839 −0.668119 −0.334059 0.942552i \(-0.608419\pi\)
−0.334059 + 0.942552i \(0.608419\pi\)
\(678\) −1.39076 −0.0534118
\(679\) 1.49699 0.0574491
\(680\) −5.38748 −0.206601
\(681\) −19.9395 −0.764083
\(682\) −5.76285 −0.220671
\(683\) −17.4268 −0.666817 −0.333408 0.942783i \(-0.608199\pi\)
−0.333408 + 0.942783i \(0.608199\pi\)
\(684\) 0 0
\(685\) −24.3583 −0.930681
\(686\) 1.00000 0.0381802
\(687\) −27.7238 −1.05773
\(688\) 12.9071 0.492078
\(689\) −58.4292 −2.22598
\(690\) −15.7948 −0.601299
\(691\) 28.0830 1.06833 0.534165 0.845381i \(-0.320626\pi\)
0.534165 + 0.845381i \(0.320626\pi\)
\(692\) 13.7505 0.522716
\(693\) −1.36934 −0.0520170
\(694\) 28.3064 1.07449
\(695\) −22.1931 −0.841833
\(696\) 10.7689 0.408193
\(697\) 17.9536 0.680042
\(698\) 19.6211 0.742669
\(699\) 11.3232 0.428284
\(700\) −2.59719 −0.0981645
\(701\) 30.7201 1.16028 0.580141 0.814516i \(-0.302998\pi\)
0.580141 + 0.814516i \(0.302998\pi\)
\(702\) 28.9819 1.09385
\(703\) 0 0
\(704\) 0.834338 0.0314453
\(705\) −16.6928 −0.628689
\(706\) 1.82831 0.0688094
\(707\) 1.80689 0.0679552
\(708\) 8.26586 0.310650
\(709\) −24.7213 −0.928427 −0.464214 0.885723i \(-0.653663\pi\)
−0.464214 + 0.885723i \(0.653663\pi\)
\(710\) 1.13799 0.0427078
\(711\) −12.4960 −0.468636
\(712\) −10.3313 −0.387183
\(713\) −60.3779 −2.26117
\(714\) 4.05134 0.151618
\(715\) 6.92824 0.259102
\(716\) −10.4481 −0.390465
\(717\) −9.11955 −0.340576
\(718\) 15.3006 0.571013
\(719\) −23.2396 −0.866692 −0.433346 0.901228i \(-0.642667\pi\)
−0.433346 + 0.901228i \(0.642667\pi\)
\(720\) 2.54407 0.0948120
\(721\) 11.8069 0.439712
\(722\) 0 0
\(723\) 15.7329 0.585114
\(724\) 3.16389 0.117585
\(725\) −23.9939 −0.891112
\(726\) −12.0108 −0.445764
\(727\) 5.33380 0.197820 0.0989099 0.995096i \(-0.468464\pi\)
0.0989099 + 0.995096i \(0.468464\pi\)
\(728\) −5.35699 −0.198543
\(729\) 21.1883 0.784754
\(730\) −10.5256 −0.389571
\(731\) 44.8595 1.65919
\(732\) −10.6820 −0.394818
\(733\) −26.8646 −0.992265 −0.496132 0.868247i \(-0.665247\pi\)
−0.496132 + 0.868247i \(0.665247\pi\)
\(734\) 19.9011 0.734562
\(735\) −1.80689 −0.0666483
\(736\) 8.74143 0.322213
\(737\) −0.258567 −0.00952444
\(738\) −8.47805 −0.312081
\(739\) 3.98943 0.146753 0.0733767 0.997304i \(-0.476622\pi\)
0.0733767 + 0.997304i \(0.476622\pi\)
\(740\) −1.70542 −0.0626926
\(741\) 0 0
\(742\) 10.9071 0.400412
\(743\) 14.6986 0.539239 0.269620 0.962967i \(-0.413102\pi\)
0.269620 + 0.962967i \(0.413102\pi\)
\(744\) −8.05134 −0.295176
\(745\) −4.49498 −0.164683
\(746\) −36.9071 −1.35126
\(747\) 10.4061 0.380738
\(748\) 2.89980 0.106027
\(749\) 10.5317 0.384821
\(750\) 13.7273 0.501251
\(751\) −7.56854 −0.276180 −0.138090 0.990420i \(-0.544096\pi\)
−0.138090 + 0.990420i \(0.544096\pi\)
\(752\) 9.23842 0.336890
\(753\) −11.1657 −0.406899
\(754\) −49.4902 −1.80233
\(755\) 15.9721 0.581282
\(756\) −5.41011 −0.196764
\(757\) 17.6934 0.643076 0.321538 0.946897i \(-0.395800\pi\)
0.321538 + 0.946897i \(0.395800\pi\)
\(758\) 3.10750 0.112869
\(759\) 8.50153 0.308586
\(760\) 0 0
\(761\) 0.277917 0.0100745 0.00503724 0.999987i \(-0.498397\pi\)
0.00503724 + 0.999987i \(0.498397\pi\)
\(762\) −11.4122 −0.413420
\(763\) −13.9011 −0.503252
\(764\) −10.8283 −0.391755
\(765\) 8.84211 0.319687
\(766\) 17.7200 0.640250
\(767\) −37.9871 −1.37164
\(768\) 1.16566 0.0420622
\(769\) −46.1347 −1.66366 −0.831829 0.555031i \(-0.812706\pi\)
−0.831829 + 0.555031i \(0.812706\pi\)
\(770\) −1.29331 −0.0466076
\(771\) 9.30884 0.335250
\(772\) −7.50301 −0.270039
\(773\) 30.0451 1.08065 0.540323 0.841458i \(-0.318302\pi\)
0.540323 + 0.841458i \(0.318302\pi\)
\(774\) −21.1835 −0.761426
\(775\) 17.9390 0.644389
\(776\) 1.49699 0.0537387
\(777\) 1.28246 0.0460081
\(778\) 25.8069 0.925222
\(779\) 0 0
\(780\) 9.67952 0.346582
\(781\) −0.612518 −0.0219176
\(782\) 30.3815 1.08644
\(783\) −49.9809 −1.78617
\(784\) 1.00000 0.0357143
\(785\) −22.4101 −0.799851
\(786\) −17.0896 −0.609567
\(787\) 9.54682 0.340308 0.170154 0.985418i \(-0.445574\pi\)
0.170154 + 0.985418i \(0.445574\pi\)
\(788\) 11.3875 0.405662
\(789\) −19.2898 −0.686734
\(790\) −11.8021 −0.419901
\(791\) −1.19311 −0.0424220
\(792\) −1.36934 −0.0486574
\(793\) 49.0909 1.74327
\(794\) 2.81292 0.0998268
\(795\) −19.7080 −0.698969
\(796\) −2.90710 −0.103039
\(797\) 32.1652 1.13935 0.569674 0.821871i \(-0.307069\pi\)
0.569674 + 0.821871i \(0.307069\pi\)
\(798\) 0 0
\(799\) 32.1088 1.13593
\(800\) −2.59719 −0.0918245
\(801\) 16.9561 0.599114
\(802\) 13.7067 0.484000
\(803\) 5.66540 0.199928
\(804\) −0.361247 −0.0127402
\(805\) −13.5501 −0.477578
\(806\) 37.0013 1.30331
\(807\) −0.671154 −0.0236257
\(808\) 1.80689 0.0635663
\(809\) −32.6175 −1.14677 −0.573386 0.819286i \(-0.694370\pi\)
−0.573386 + 0.819286i \(0.694370\pi\)
\(810\) 2.14327 0.0753068
\(811\) −10.8009 −0.379270 −0.189635 0.981855i \(-0.560730\pi\)
−0.189635 + 0.981855i \(0.560730\pi\)
\(812\) 9.23842 0.324205
\(813\) −4.22096 −0.148036
\(814\) 0.917939 0.0321738
\(815\) 12.7278 0.445837
\(816\) 4.05134 0.141825
\(817\) 0 0
\(818\) −27.9237 −0.976329
\(819\) 8.79207 0.307220
\(820\) −8.00730 −0.279627
\(821\) 17.9597 0.626798 0.313399 0.949622i \(-0.398532\pi\)
0.313399 + 0.949622i \(0.398532\pi\)
\(822\) 18.3172 0.638886
\(823\) 36.4784 1.27156 0.635778 0.771872i \(-0.280679\pi\)
0.635778 + 0.771872i \(0.280679\pi\)
\(824\) 11.8069 0.411313
\(825\) −2.52591 −0.0879409
\(826\) 7.09113 0.246732
\(827\) 24.6365 0.856694 0.428347 0.903614i \(-0.359096\pi\)
0.428347 + 0.903614i \(0.359096\pi\)
\(828\) −14.3467 −0.498583
\(829\) 5.95091 0.206684 0.103342 0.994646i \(-0.467046\pi\)
0.103342 + 0.994646i \(0.467046\pi\)
\(830\) 9.82825 0.341144
\(831\) 35.3519 1.22635
\(832\) −5.35699 −0.185720
\(833\) 3.47557 0.120421
\(834\) 16.6890 0.577894
\(835\) 18.2336 0.631000
\(836\) 0 0
\(837\) 37.3681 1.29163
\(838\) −32.5209 −1.12342
\(839\) 1.94866 0.0672752 0.0336376 0.999434i \(-0.489291\pi\)
0.0336376 + 0.999434i \(0.489291\pi\)
\(840\) −1.80689 −0.0623438
\(841\) 56.3484 1.94305
\(842\) 21.1075 0.727412
\(843\) −3.91519 −0.134846
\(844\) 4.00000 0.137686
\(845\) −24.3325 −0.837065
\(846\) −15.1624 −0.521294
\(847\) −10.3039 −0.354046
\(848\) 10.9071 0.374551
\(849\) 3.88675 0.133393
\(850\) −9.02671 −0.309613
\(851\) 9.61734 0.329678
\(852\) −0.855755 −0.0293177
\(853\) −9.65428 −0.330556 −0.165278 0.986247i \(-0.552852\pi\)
−0.165278 + 0.986247i \(0.552852\pi\)
\(854\) −9.16389 −0.313582
\(855\) 0 0
\(856\) 10.5317 0.359967
\(857\) −4.07276 −0.139123 −0.0695614 0.997578i \(-0.522160\pi\)
−0.0695614 + 0.997578i \(0.522160\pi\)
\(858\) −5.20998 −0.177866
\(859\) 1.60321 0.0547010 0.0273505 0.999626i \(-0.491293\pi\)
0.0273505 + 0.999626i \(0.491293\pi\)
\(860\) −20.0073 −0.682243
\(861\) 6.02142 0.205209
\(862\) 2.89980 0.0987675
\(863\) −16.0574 −0.546601 −0.273301 0.961929i \(-0.588115\pi\)
−0.273301 + 0.961929i \(0.588115\pi\)
\(864\) −5.41011 −0.184056
\(865\) −21.3147 −0.724720
\(866\) −23.9952 −0.815392
\(867\) −5.73555 −0.194789
\(868\) −6.90710 −0.234442
\(869\) 6.35247 0.215493
\(870\) −16.6928 −0.565940
\(871\) 1.66017 0.0562527
\(872\) −13.9011 −0.470750
\(873\) −2.45690 −0.0831535
\(874\) 0 0
\(875\) 11.7764 0.398115
\(876\) 7.91519 0.267429
\(877\) 0.567266 0.0191552 0.00957760 0.999954i \(-0.496951\pi\)
0.00957760 + 0.999954i \(0.496951\pi\)
\(878\) −38.5355 −1.30051
\(879\) 26.6749 0.899722
\(880\) −1.29331 −0.0435974
\(881\) −5.63641 −0.189896 −0.0949478 0.995482i \(-0.530268\pi\)
−0.0949478 + 0.995482i \(0.530268\pi\)
\(882\) −1.64123 −0.0552631
\(883\) 0.276708 0.00931196 0.00465598 0.999989i \(-0.498518\pi\)
0.00465598 + 0.999989i \(0.498518\pi\)
\(884\) −18.6186 −0.626211
\(885\) −12.8129 −0.430702
\(886\) −26.6805 −0.896350
\(887\) −5.35850 −0.179921 −0.0899604 0.995945i \(-0.528674\pi\)
−0.0899604 + 0.995945i \(0.528674\pi\)
\(888\) 1.28246 0.0430366
\(889\) −9.79029 −0.328356
\(890\) 16.0146 0.536810
\(891\) −1.15361 −0.0386474
\(892\) −11.8142 −0.395568
\(893\) 0 0
\(894\) 3.38019 0.113050
\(895\) 16.1956 0.541361
\(896\) 1.00000 0.0334077
\(897\) −54.5854 −1.82255
\(898\) −28.6284 −0.955341
\(899\) −63.8106 −2.12820
\(900\) 4.26259 0.142086
\(901\) 37.9084 1.26291
\(902\) 4.30991 0.143504
\(903\) 15.0453 0.500677
\(904\) −1.19311 −0.0396821
\(905\) −4.90435 −0.163026
\(906\) −12.0108 −0.399034
\(907\) 35.6437 1.18353 0.591765 0.806111i \(-0.298431\pi\)
0.591765 + 0.806111i \(0.298431\pi\)
\(908\) −17.1057 −0.567673
\(909\) −2.96553 −0.0983605
\(910\) 8.30388 0.275271
\(911\) −1.71071 −0.0566784 −0.0283392 0.999598i \(-0.509022\pi\)
−0.0283392 + 0.999598i \(0.509022\pi\)
\(912\) 0 0
\(913\) −5.29003 −0.175074
\(914\) 7.28728 0.241042
\(915\) 16.5582 0.547396
\(916\) −23.7837 −0.785836
\(917\) −14.6609 −0.484145
\(918\) −18.8032 −0.620598
\(919\) 7.46981 0.246406 0.123203 0.992381i \(-0.460683\pi\)
0.123203 + 0.992381i \(0.460683\pi\)
\(920\) −13.5501 −0.446734
\(921\) −21.3161 −0.702390
\(922\) 14.2019 0.467715
\(923\) 3.93277 0.129449
\(924\) 0.972556 0.0319947
\(925\) −2.85743 −0.0939517
\(926\) 29.8308 0.980301
\(927\) −19.3778 −0.636452
\(928\) 9.23842 0.303266
\(929\) −25.3305 −0.831068 −0.415534 0.909578i \(-0.636405\pi\)
−0.415534 + 0.909578i \(0.636405\pi\)
\(930\) 12.4804 0.409248
\(931\) 0 0
\(932\) 9.71399 0.318192
\(933\) 8.42905 0.275955
\(934\) 5.87965 0.192388
\(935\) −4.49498 −0.147002
\(936\) 8.79207 0.287378
\(937\) −22.9297 −0.749081 −0.374541 0.927210i \(-0.622200\pi\)
−0.374541 + 0.927210i \(0.622200\pi\)
\(938\) −0.309907 −0.0101188
\(939\) 7.58882 0.247652
\(940\) −14.3205 −0.467083
\(941\) −14.1626 −0.461688 −0.230844 0.972991i \(-0.574149\pi\)
−0.230844 + 0.972991i \(0.574149\pi\)
\(942\) 16.8522 0.549075
\(943\) 45.1553 1.47046
\(944\) 7.09113 0.230797
\(945\) 8.38621 0.272804
\(946\) 10.7689 0.350126
\(947\) 36.7165 1.19313 0.596563 0.802566i \(-0.296532\pi\)
0.596563 + 0.802566i \(0.296532\pi\)
\(948\) 8.87510 0.288250
\(949\) −36.3756 −1.18080
\(950\) 0 0
\(951\) −34.3242 −1.11304
\(952\) 3.47557 0.112644
\(953\) −22.6679 −0.734285 −0.367142 0.930165i \(-0.619664\pi\)
−0.367142 + 0.930165i \(0.619664\pi\)
\(954\) −17.9011 −0.579568
\(955\) 16.7850 0.543149
\(956\) −7.82349 −0.253030
\(957\) 8.98488 0.290440
\(958\) 16.8739 0.545171
\(959\) 15.7140 0.507431
\(960\) −1.80689 −0.0583172
\(961\) 16.7080 0.538967
\(962\) −5.89377 −0.190023
\(963\) −17.2850 −0.557001
\(964\) 13.4970 0.434709
\(965\) 11.6304 0.374397
\(966\) 10.1896 0.327844
\(967\) −18.7379 −0.602570 −0.301285 0.953534i \(-0.597416\pi\)
−0.301285 + 0.953534i \(0.597416\pi\)
\(968\) −10.3039 −0.331179
\(969\) 0 0
\(970\) −2.32048 −0.0745061
\(971\) 41.0496 1.31735 0.658673 0.752430i \(-0.271118\pi\)
0.658673 + 0.752430i \(0.271118\pi\)
\(972\) 14.6186 0.468892
\(973\) 14.3172 0.458988
\(974\) −17.1002 −0.547926
\(975\) 16.2180 0.519392
\(976\) −9.16389 −0.293329
\(977\) −58.8681 −1.88336 −0.941678 0.336515i \(-0.890752\pi\)
−0.941678 + 0.336515i \(0.890752\pi\)
\(978\) −9.57122 −0.306054
\(979\) −8.61981 −0.275490
\(980\) −1.55010 −0.0495161
\(981\) 22.8149 0.728423
\(982\) −5.04886 −0.161116
\(983\) 4.33132 0.138148 0.0690739 0.997612i \(-0.477996\pi\)
0.0690739 + 0.997612i \(0.477996\pi\)
\(984\) 6.02142 0.191956
\(985\) −17.6517 −0.562431
\(986\) 32.1088 1.02255
\(987\) 10.7689 0.342777
\(988\) 0 0
\(989\) 112.827 3.58767
\(990\) 2.12262 0.0674612
\(991\) −43.5388 −1.38305 −0.691527 0.722351i \(-0.743062\pi\)
−0.691527 + 0.722351i \(0.743062\pi\)
\(992\) −6.90710 −0.219300
\(993\) −24.2430 −0.769327
\(994\) −0.734136 −0.0232854
\(995\) 4.50629 0.142859
\(996\) −7.39076 −0.234185
\(997\) −34.3389 −1.08752 −0.543762 0.839239i \(-0.683001\pi\)
−0.543762 + 0.839239i \(0.683001\pi\)
\(998\) −11.1108 −0.351705
\(999\) −5.95221 −0.188320
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 5054.2.a.x.1.3 4
19.8 odd 6 266.2.f.d.197.3 8
19.12 odd 6 266.2.f.d.239.3 yes 8
19.18 odd 2 5054.2.a.w.1.2 4
57.8 even 6 2394.2.o.v.1261.2 8
57.50 even 6 2394.2.o.v.505.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
266.2.f.d.197.3 8 19.8 odd 6
266.2.f.d.239.3 yes 8 19.12 odd 6
2394.2.o.v.505.2 8 57.50 even 6
2394.2.o.v.1261.2 8 57.8 even 6
5054.2.a.w.1.2 4 19.18 odd 2
5054.2.a.x.1.3 4 1.1 even 1 trivial