Properties

Label 8034.2.a.q.1.6
Level $8034$
Weight $2$
Character 8034.1
Self dual yes
Analytic conductor $64.152$
Analytic rank $1$
Dimension $8$
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] = [8034,2,Mod(1,8034)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8034, 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("8034.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8034.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(64.1518129839\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 11x^{6} + 21x^{5} + 23x^{4} - 29x^{3} - 27x^{2} + x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(1.79755\) of defining polynomial
Character \(\chi\) \(=\) 8034.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +0.797548 q^{5} +1.00000 q^{6} -4.52438 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +0.797548 q^{5} +1.00000 q^{6} -4.52438 q^{7} +1.00000 q^{8} +1.00000 q^{9} +0.797548 q^{10} +0.726837 q^{11} +1.00000 q^{12} -1.00000 q^{13} -4.52438 q^{14} +0.797548 q^{15} +1.00000 q^{16} +0.987094 q^{17} +1.00000 q^{18} +1.78344 q^{19} +0.797548 q^{20} -4.52438 q^{21} +0.726837 q^{22} -5.45739 q^{23} +1.00000 q^{24} -4.36392 q^{25} -1.00000 q^{26} +1.00000 q^{27} -4.52438 q^{28} +2.41751 q^{29} +0.797548 q^{30} -0.500062 q^{31} +1.00000 q^{32} +0.726837 q^{33} +0.987094 q^{34} -3.60841 q^{35} +1.00000 q^{36} +0.242893 q^{37} +1.78344 q^{38} -1.00000 q^{39} +0.797548 q^{40} -11.0069 q^{41} -4.52438 q^{42} -6.15290 q^{43} +0.726837 q^{44} +0.797548 q^{45} -5.45739 q^{46} +6.55388 q^{47} +1.00000 q^{48} +13.4701 q^{49} -4.36392 q^{50} +0.987094 q^{51} -1.00000 q^{52} -0.739743 q^{53} +1.00000 q^{54} +0.579687 q^{55} -4.52438 q^{56} +1.78344 q^{57} +2.41751 q^{58} -2.94509 q^{59} +0.797548 q^{60} -11.9494 q^{61} -0.500062 q^{62} -4.52438 q^{63} +1.00000 q^{64} -0.797548 q^{65} +0.726837 q^{66} +0.738580 q^{67} +0.987094 q^{68} -5.45739 q^{69} -3.60841 q^{70} -0.256125 q^{71} +1.00000 q^{72} -8.07464 q^{73} +0.242893 q^{74} -4.36392 q^{75} +1.78344 q^{76} -3.28849 q^{77} -1.00000 q^{78} +1.16693 q^{79} +0.797548 q^{80} +1.00000 q^{81} -11.0069 q^{82} -8.73755 q^{83} -4.52438 q^{84} +0.787255 q^{85} -6.15290 q^{86} +2.41751 q^{87} +0.726837 q^{88} +2.51500 q^{89} +0.797548 q^{90} +4.52438 q^{91} -5.45739 q^{92} -0.500062 q^{93} +6.55388 q^{94} +1.42238 q^{95} +1.00000 q^{96} -10.1954 q^{97} +13.4701 q^{98} +0.726837 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{2} + 8 q^{3} + 8 q^{4} - 6 q^{5} + 8 q^{6} - 3 q^{7} + 8 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{2} + 8 q^{3} + 8 q^{4} - 6 q^{5} + 8 q^{6} - 3 q^{7} + 8 q^{8} + 8 q^{9} - 6 q^{10} - 15 q^{11} + 8 q^{12} - 8 q^{13} - 3 q^{14} - 6 q^{15} + 8 q^{16} - 11 q^{17} + 8 q^{18} - 15 q^{19} - 6 q^{20} - 3 q^{21} - 15 q^{22} + q^{23} + 8 q^{24} - 10 q^{25} - 8 q^{26} + 8 q^{27} - 3 q^{28} - 10 q^{29} - 6 q^{30} - 3 q^{31} + 8 q^{32} - 15 q^{33} - 11 q^{34} - 12 q^{35} + 8 q^{36} - 26 q^{37} - 15 q^{38} - 8 q^{39} - 6 q^{40} - 12 q^{41} - 3 q^{42} - 4 q^{43} - 15 q^{44} - 6 q^{45} + q^{46} - 6 q^{47} + 8 q^{48} - 5 q^{49} - 10 q^{50} - 11 q^{51} - 8 q^{52} - 4 q^{53} + 8 q^{54} - 3 q^{56} - 15 q^{57} - 10 q^{58} - 19 q^{59} - 6 q^{60} - 14 q^{61} - 3 q^{62} - 3 q^{63} + 8 q^{64} + 6 q^{65} - 15 q^{66} - 13 q^{67} - 11 q^{68} + q^{69} - 12 q^{70} - 31 q^{71} + 8 q^{72} - 27 q^{73} - 26 q^{74} - 10 q^{75} - 15 q^{76} - 30 q^{77} - 8 q^{78} - 13 q^{79} - 6 q^{80} + 8 q^{81} - 12 q^{82} - 28 q^{83} - 3 q^{84} + 15 q^{85} - 4 q^{86} - 10 q^{87} - 15 q^{88} - 2 q^{89} - 6 q^{90} + 3 q^{91} + q^{92} - 3 q^{93} - 6 q^{94} - 18 q^{95} + 8 q^{96} - 30 q^{97} - 5 q^{98} - 15 q^{99}+O(q^{100}) \) 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
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 0.797548 0.356674 0.178337 0.983969i \(-0.442928\pi\)
0.178337 + 0.983969i \(0.442928\pi\)
\(6\) 1.00000 0.408248
\(7\) −4.52438 −1.71006 −0.855028 0.518581i \(-0.826460\pi\)
−0.855028 + 0.518581i \(0.826460\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0.797548 0.252207
\(11\) 0.726837 0.219150 0.109575 0.993979i \(-0.465051\pi\)
0.109575 + 0.993979i \(0.465051\pi\)
\(12\) 1.00000 0.288675
\(13\) −1.00000 −0.277350
\(14\) −4.52438 −1.20919
\(15\) 0.797548 0.205926
\(16\) 1.00000 0.250000
\(17\) 0.987094 0.239406 0.119703 0.992810i \(-0.461806\pi\)
0.119703 + 0.992810i \(0.461806\pi\)
\(18\) 1.00000 0.235702
\(19\) 1.78344 0.409149 0.204574 0.978851i \(-0.434419\pi\)
0.204574 + 0.978851i \(0.434419\pi\)
\(20\) 0.797548 0.178337
\(21\) −4.52438 −0.987302
\(22\) 0.726837 0.154962
\(23\) −5.45739 −1.13794 −0.568972 0.822357i \(-0.692659\pi\)
−0.568972 + 0.822357i \(0.692659\pi\)
\(24\) 1.00000 0.204124
\(25\) −4.36392 −0.872784
\(26\) −1.00000 −0.196116
\(27\) 1.00000 0.192450
\(28\) −4.52438 −0.855028
\(29\) 2.41751 0.448920 0.224460 0.974483i \(-0.427938\pi\)
0.224460 + 0.974483i \(0.427938\pi\)
\(30\) 0.797548 0.145612
\(31\) −0.500062 −0.0898137 −0.0449069 0.998991i \(-0.514299\pi\)
−0.0449069 + 0.998991i \(0.514299\pi\)
\(32\) 1.00000 0.176777
\(33\) 0.726837 0.126526
\(34\) 0.987094 0.169285
\(35\) −3.60841 −0.609933
\(36\) 1.00000 0.166667
\(37\) 0.242893 0.0399314 0.0199657 0.999801i \(-0.493644\pi\)
0.0199657 + 0.999801i \(0.493644\pi\)
\(38\) 1.78344 0.289312
\(39\) −1.00000 −0.160128
\(40\) 0.797548 0.126103
\(41\) −11.0069 −1.71899 −0.859495 0.511144i \(-0.829222\pi\)
−0.859495 + 0.511144i \(0.829222\pi\)
\(42\) −4.52438 −0.698128
\(43\) −6.15290 −0.938309 −0.469154 0.883116i \(-0.655441\pi\)
−0.469154 + 0.883116i \(0.655441\pi\)
\(44\) 0.726837 0.109575
\(45\) 0.797548 0.118891
\(46\) −5.45739 −0.804648
\(47\) 6.55388 0.955981 0.477991 0.878365i \(-0.341365\pi\)
0.477991 + 0.878365i \(0.341365\pi\)
\(48\) 1.00000 0.144338
\(49\) 13.4701 1.92429
\(50\) −4.36392 −0.617151
\(51\) 0.987094 0.138221
\(52\) −1.00000 −0.138675
\(53\) −0.739743 −0.101611 −0.0508057 0.998709i \(-0.516179\pi\)
−0.0508057 + 0.998709i \(0.516179\pi\)
\(54\) 1.00000 0.136083
\(55\) 0.579687 0.0781650
\(56\) −4.52438 −0.604596
\(57\) 1.78344 0.236222
\(58\) 2.41751 0.317434
\(59\) −2.94509 −0.383417 −0.191709 0.981452i \(-0.561403\pi\)
−0.191709 + 0.981452i \(0.561403\pi\)
\(60\) 0.797548 0.102963
\(61\) −11.9494 −1.52997 −0.764985 0.644048i \(-0.777254\pi\)
−0.764985 + 0.644048i \(0.777254\pi\)
\(62\) −0.500062 −0.0635079
\(63\) −4.52438 −0.570019
\(64\) 1.00000 0.125000
\(65\) −0.797548 −0.0989236
\(66\) 0.726837 0.0894674
\(67\) 0.738580 0.0902319 0.0451160 0.998982i \(-0.485634\pi\)
0.0451160 + 0.998982i \(0.485634\pi\)
\(68\) 0.987094 0.119703
\(69\) −5.45739 −0.656993
\(70\) −3.60841 −0.431288
\(71\) −0.256125 −0.0303965 −0.0151982 0.999885i \(-0.504838\pi\)
−0.0151982 + 0.999885i \(0.504838\pi\)
\(72\) 1.00000 0.117851
\(73\) −8.07464 −0.945065 −0.472532 0.881313i \(-0.656660\pi\)
−0.472532 + 0.881313i \(0.656660\pi\)
\(74\) 0.242893 0.0282358
\(75\) −4.36392 −0.503902
\(76\) 1.78344 0.204574
\(77\) −3.28849 −0.374758
\(78\) −1.00000 −0.113228
\(79\) 1.16693 0.131290 0.0656452 0.997843i \(-0.479089\pi\)
0.0656452 + 0.997843i \(0.479089\pi\)
\(80\) 0.797548 0.0891686
\(81\) 1.00000 0.111111
\(82\) −11.0069 −1.21551
\(83\) −8.73755 −0.959071 −0.479535 0.877523i \(-0.659195\pi\)
−0.479535 + 0.877523i \(0.659195\pi\)
\(84\) −4.52438 −0.493651
\(85\) 0.787255 0.0853898
\(86\) −6.15290 −0.663485
\(87\) 2.41751 0.259184
\(88\) 0.726837 0.0774811
\(89\) 2.51500 0.266589 0.133295 0.991076i \(-0.457444\pi\)
0.133295 + 0.991076i \(0.457444\pi\)
\(90\) 0.797548 0.0840689
\(91\) 4.52438 0.474284
\(92\) −5.45739 −0.568972
\(93\) −0.500062 −0.0518540
\(94\) 6.55388 0.675981
\(95\) 1.42238 0.145933
\(96\) 1.00000 0.102062
\(97\) −10.1954 −1.03519 −0.517594 0.855626i \(-0.673172\pi\)
−0.517594 + 0.855626i \(0.673172\pi\)
\(98\) 13.4701 1.36068
\(99\) 0.726837 0.0730499
\(100\) −4.36392 −0.436392
\(101\) 9.95897 0.990954 0.495477 0.868621i \(-0.334993\pi\)
0.495477 + 0.868621i \(0.334993\pi\)
\(102\) 0.987094 0.0977369
\(103\) −1.00000 −0.0985329
\(104\) −1.00000 −0.0980581
\(105\) −3.60841 −0.352145
\(106\) −0.739743 −0.0718501
\(107\) −0.884089 −0.0854681 −0.0427340 0.999086i \(-0.513607\pi\)
−0.0427340 + 0.999086i \(0.513607\pi\)
\(108\) 1.00000 0.0962250
\(109\) −13.9459 −1.33578 −0.667888 0.744262i \(-0.732801\pi\)
−0.667888 + 0.744262i \(0.732801\pi\)
\(110\) 0.579687 0.0552710
\(111\) 0.242893 0.0230544
\(112\) −4.52438 −0.427514
\(113\) 8.78961 0.826857 0.413428 0.910537i \(-0.364331\pi\)
0.413428 + 0.910537i \(0.364331\pi\)
\(114\) 1.78344 0.167034
\(115\) −4.35253 −0.405876
\(116\) 2.41751 0.224460
\(117\) −1.00000 −0.0924500
\(118\) −2.94509 −0.271117
\(119\) −4.46599 −0.409397
\(120\) 0.797548 0.0728058
\(121\) −10.4717 −0.951973
\(122\) −11.9494 −1.08185
\(123\) −11.0069 −0.992459
\(124\) −0.500062 −0.0449069
\(125\) −7.46817 −0.667974
\(126\) −4.52438 −0.403064
\(127\) 14.9261 1.32448 0.662238 0.749294i \(-0.269607\pi\)
0.662238 + 0.749294i \(0.269607\pi\)
\(128\) 1.00000 0.0883883
\(129\) −6.15290 −0.541733
\(130\) −0.797548 −0.0699496
\(131\) 6.85518 0.598940 0.299470 0.954106i \(-0.403190\pi\)
0.299470 + 0.954106i \(0.403190\pi\)
\(132\) 0.726837 0.0632630
\(133\) −8.06896 −0.699668
\(134\) 0.738580 0.0638036
\(135\) 0.797548 0.0686420
\(136\) 0.987094 0.0846426
\(137\) 3.70447 0.316494 0.158247 0.987400i \(-0.449416\pi\)
0.158247 + 0.987400i \(0.449416\pi\)
\(138\) −5.45739 −0.464564
\(139\) −17.2252 −1.46102 −0.730511 0.682901i \(-0.760718\pi\)
−0.730511 + 0.682901i \(0.760718\pi\)
\(140\) −3.60841 −0.304967
\(141\) 6.55388 0.551936
\(142\) −0.256125 −0.0214936
\(143\) −0.726837 −0.0607812
\(144\) 1.00000 0.0833333
\(145\) 1.92808 0.160118
\(146\) −8.07464 −0.668262
\(147\) 13.4701 1.11099
\(148\) 0.242893 0.0199657
\(149\) −16.4849 −1.35049 −0.675247 0.737591i \(-0.735963\pi\)
−0.675247 + 0.737591i \(0.735963\pi\)
\(150\) −4.36392 −0.356312
\(151\) −5.22890 −0.425522 −0.212761 0.977104i \(-0.568246\pi\)
−0.212761 + 0.977104i \(0.568246\pi\)
\(152\) 1.78344 0.144656
\(153\) 0.987094 0.0798018
\(154\) −3.28849 −0.264994
\(155\) −0.398823 −0.0320342
\(156\) −1.00000 −0.0800641
\(157\) 13.1212 1.04719 0.523593 0.851968i \(-0.324591\pi\)
0.523593 + 0.851968i \(0.324591\pi\)
\(158\) 1.16693 0.0928363
\(159\) −0.739743 −0.0586654
\(160\) 0.797548 0.0630517
\(161\) 24.6913 1.94595
\(162\) 1.00000 0.0785674
\(163\) 2.05386 0.160871 0.0804354 0.996760i \(-0.474369\pi\)
0.0804354 + 0.996760i \(0.474369\pi\)
\(164\) −11.0069 −0.859495
\(165\) 0.579687 0.0451286
\(166\) −8.73755 −0.678165
\(167\) 6.47667 0.501180 0.250590 0.968093i \(-0.419375\pi\)
0.250590 + 0.968093i \(0.419375\pi\)
\(168\) −4.52438 −0.349064
\(169\) 1.00000 0.0769231
\(170\) 0.787255 0.0603797
\(171\) 1.78344 0.136383
\(172\) −6.15290 −0.469154
\(173\) −3.56976 −0.271404 −0.135702 0.990750i \(-0.543329\pi\)
−0.135702 + 0.990750i \(0.543329\pi\)
\(174\) 2.41751 0.183271
\(175\) 19.7440 1.49251
\(176\) 0.726837 0.0547874
\(177\) −2.94509 −0.221366
\(178\) 2.51500 0.188507
\(179\) −5.40943 −0.404319 −0.202160 0.979353i \(-0.564796\pi\)
−0.202160 + 0.979353i \(0.564796\pi\)
\(180\) 0.797548 0.0594457
\(181\) −6.80871 −0.506087 −0.253044 0.967455i \(-0.581432\pi\)
−0.253044 + 0.967455i \(0.581432\pi\)
\(182\) 4.52438 0.335370
\(183\) −11.9494 −0.883328
\(184\) −5.45739 −0.402324
\(185\) 0.193719 0.0142425
\(186\) −0.500062 −0.0366663
\(187\) 0.717457 0.0524656
\(188\) 6.55388 0.477991
\(189\) −4.52438 −0.329101
\(190\) 1.42238 0.103190
\(191\) 27.2893 1.97458 0.987291 0.158924i \(-0.0508024\pi\)
0.987291 + 0.158924i \(0.0508024\pi\)
\(192\) 1.00000 0.0721688
\(193\) 0.595016 0.0428302 0.0214151 0.999771i \(-0.493183\pi\)
0.0214151 + 0.999771i \(0.493183\pi\)
\(194\) −10.1954 −0.731988
\(195\) −0.797548 −0.0571136
\(196\) 13.4701 0.962147
\(197\) −15.9865 −1.13899 −0.569497 0.821994i \(-0.692862\pi\)
−0.569497 + 0.821994i \(0.692862\pi\)
\(198\) 0.726837 0.0516540
\(199\) −11.5511 −0.818836 −0.409418 0.912347i \(-0.634268\pi\)
−0.409418 + 0.912347i \(0.634268\pi\)
\(200\) −4.36392 −0.308576
\(201\) 0.738580 0.0520954
\(202\) 9.95897 0.700710
\(203\) −10.9377 −0.767678
\(204\) 0.987094 0.0691104
\(205\) −8.77853 −0.613119
\(206\) −1.00000 −0.0696733
\(207\) −5.45739 −0.379315
\(208\) −1.00000 −0.0693375
\(209\) 1.29627 0.0896648
\(210\) −3.60841 −0.249004
\(211\) −24.7761 −1.70565 −0.852827 0.522194i \(-0.825114\pi\)
−0.852827 + 0.522194i \(0.825114\pi\)
\(212\) −0.739743 −0.0508057
\(213\) −0.256125 −0.0175494
\(214\) −0.884089 −0.0604351
\(215\) −4.90723 −0.334671
\(216\) 1.00000 0.0680414
\(217\) 2.26247 0.153587
\(218\) −13.9459 −0.944536
\(219\) −8.07464 −0.545633
\(220\) 0.579687 0.0390825
\(221\) −0.987094 −0.0663991
\(222\) 0.242893 0.0163019
\(223\) 2.06150 0.138048 0.0690241 0.997615i \(-0.478011\pi\)
0.0690241 + 0.997615i \(0.478011\pi\)
\(224\) −4.52438 −0.302298
\(225\) −4.36392 −0.290928
\(226\) 8.78961 0.584676
\(227\) 19.0225 1.26257 0.631283 0.775552i \(-0.282528\pi\)
0.631283 + 0.775552i \(0.282528\pi\)
\(228\) 1.78344 0.118111
\(229\) 15.9093 1.05132 0.525658 0.850696i \(-0.323819\pi\)
0.525658 + 0.850696i \(0.323819\pi\)
\(230\) −4.35253 −0.286997
\(231\) −3.28849 −0.216367
\(232\) 2.41751 0.158717
\(233\) −15.4747 −1.01378 −0.506890 0.862011i \(-0.669205\pi\)
−0.506890 + 0.862011i \(0.669205\pi\)
\(234\) −1.00000 −0.0653720
\(235\) 5.22703 0.340974
\(236\) −2.94509 −0.191709
\(237\) 1.16693 0.0758005
\(238\) −4.46599 −0.289487
\(239\) −9.91541 −0.641375 −0.320687 0.947185i \(-0.603914\pi\)
−0.320687 + 0.947185i \(0.603914\pi\)
\(240\) 0.797548 0.0514815
\(241\) −24.1212 −1.55379 −0.776893 0.629633i \(-0.783205\pi\)
−0.776893 + 0.629633i \(0.783205\pi\)
\(242\) −10.4717 −0.673147
\(243\) 1.00000 0.0641500
\(244\) −11.9494 −0.764985
\(245\) 10.7430 0.686346
\(246\) −11.0069 −0.701775
\(247\) −1.78344 −0.113477
\(248\) −0.500062 −0.0317540
\(249\) −8.73755 −0.553720
\(250\) −7.46817 −0.472329
\(251\) 1.25196 0.0790232 0.0395116 0.999219i \(-0.487420\pi\)
0.0395116 + 0.999219i \(0.487420\pi\)
\(252\) −4.52438 −0.285009
\(253\) −3.96663 −0.249380
\(254\) 14.9261 0.936546
\(255\) 0.787255 0.0492998
\(256\) 1.00000 0.0625000
\(257\) −20.1530 −1.25711 −0.628553 0.777766i \(-0.716353\pi\)
−0.628553 + 0.777766i \(0.716353\pi\)
\(258\) −6.15290 −0.383063
\(259\) −1.09894 −0.0682850
\(260\) −0.797548 −0.0494618
\(261\) 2.41751 0.149640
\(262\) 6.85518 0.423514
\(263\) −13.0656 −0.805659 −0.402829 0.915275i \(-0.631973\pi\)
−0.402829 + 0.915275i \(0.631973\pi\)
\(264\) 0.726837 0.0447337
\(265\) −0.589980 −0.0362422
\(266\) −8.06896 −0.494740
\(267\) 2.51500 0.153915
\(268\) 0.738580 0.0451160
\(269\) 27.1939 1.65804 0.829020 0.559219i \(-0.188899\pi\)
0.829020 + 0.559219i \(0.188899\pi\)
\(270\) 0.797548 0.0485372
\(271\) 1.66544 0.101168 0.0505842 0.998720i \(-0.483892\pi\)
0.0505842 + 0.998720i \(0.483892\pi\)
\(272\) 0.987094 0.0598514
\(273\) 4.52438 0.273828
\(274\) 3.70447 0.223795
\(275\) −3.17186 −0.191270
\(276\) −5.45739 −0.328496
\(277\) −29.8720 −1.79483 −0.897417 0.441183i \(-0.854559\pi\)
−0.897417 + 0.441183i \(0.854559\pi\)
\(278\) −17.2252 −1.03310
\(279\) −0.500062 −0.0299379
\(280\) −3.60841 −0.215644
\(281\) −8.28440 −0.494206 −0.247103 0.968989i \(-0.579479\pi\)
−0.247103 + 0.968989i \(0.579479\pi\)
\(282\) 6.55388 0.390278
\(283\) 0.106605 0.00633701 0.00316850 0.999995i \(-0.498991\pi\)
0.00316850 + 0.999995i \(0.498991\pi\)
\(284\) −0.256125 −0.0151982
\(285\) 1.42238 0.0842544
\(286\) −0.726837 −0.0429788
\(287\) 49.7995 2.93957
\(288\) 1.00000 0.0589256
\(289\) −16.0256 −0.942685
\(290\) 1.92808 0.113221
\(291\) −10.1954 −0.597666
\(292\) −8.07464 −0.472532
\(293\) −2.52246 −0.147364 −0.0736819 0.997282i \(-0.523475\pi\)
−0.0736819 + 0.997282i \(0.523475\pi\)
\(294\) 13.4701 0.785590
\(295\) −2.34885 −0.136755
\(296\) 0.242893 0.0141179
\(297\) 0.726837 0.0421754
\(298\) −16.4849 −0.954944
\(299\) 5.45739 0.315609
\(300\) −4.36392 −0.251951
\(301\) 27.8381 1.60456
\(302\) −5.22890 −0.300889
\(303\) 9.95897 0.572128
\(304\) 1.78344 0.102287
\(305\) −9.53025 −0.545701
\(306\) 0.987094 0.0564284
\(307\) 22.0445 1.25815 0.629073 0.777347i \(-0.283435\pi\)
0.629073 + 0.777347i \(0.283435\pi\)
\(308\) −3.28849 −0.187379
\(309\) −1.00000 −0.0568880
\(310\) −0.398823 −0.0226516
\(311\) 0.982492 0.0557120 0.0278560 0.999612i \(-0.491132\pi\)
0.0278560 + 0.999612i \(0.491132\pi\)
\(312\) −1.00000 −0.0566139
\(313\) −3.79254 −0.214367 −0.107183 0.994239i \(-0.534183\pi\)
−0.107183 + 0.994239i \(0.534183\pi\)
\(314\) 13.1212 0.740472
\(315\) −3.60841 −0.203311
\(316\) 1.16693 0.0656452
\(317\) 14.9328 0.838708 0.419354 0.907823i \(-0.362257\pi\)
0.419354 + 0.907823i \(0.362257\pi\)
\(318\) −0.739743 −0.0414827
\(319\) 1.75713 0.0983806
\(320\) 0.797548 0.0445843
\(321\) −0.884089 −0.0493450
\(322\) 24.6913 1.37599
\(323\) 1.76042 0.0979525
\(324\) 1.00000 0.0555556
\(325\) 4.36392 0.242067
\(326\) 2.05386 0.113753
\(327\) −13.9459 −0.771210
\(328\) −11.0069 −0.607755
\(329\) −29.6523 −1.63478
\(330\) 0.579687 0.0319107
\(331\) −10.1335 −0.556990 −0.278495 0.960438i \(-0.589836\pi\)
−0.278495 + 0.960438i \(0.589836\pi\)
\(332\) −8.73755 −0.479535
\(333\) 0.242893 0.0133105
\(334\) 6.47667 0.354388
\(335\) 0.589053 0.0321834
\(336\) −4.52438 −0.246825
\(337\) 19.9561 1.08708 0.543538 0.839384i \(-0.317084\pi\)
0.543538 + 0.839384i \(0.317084\pi\)
\(338\) 1.00000 0.0543928
\(339\) 8.78961 0.477386
\(340\) 0.787255 0.0426949
\(341\) −0.363463 −0.0196826
\(342\) 1.78344 0.0964373
\(343\) −29.2730 −1.58059
\(344\) −6.15290 −0.331742
\(345\) −4.35253 −0.234332
\(346\) −3.56976 −0.191912
\(347\) 3.81997 0.205067 0.102533 0.994730i \(-0.467305\pi\)
0.102533 + 0.994730i \(0.467305\pi\)
\(348\) 2.41751 0.129592
\(349\) 11.6084 0.621381 0.310690 0.950511i \(-0.399440\pi\)
0.310690 + 0.950511i \(0.399440\pi\)
\(350\) 19.7440 1.05536
\(351\) −1.00000 −0.0533761
\(352\) 0.726837 0.0387405
\(353\) 10.4353 0.555416 0.277708 0.960666i \(-0.410425\pi\)
0.277708 + 0.960666i \(0.410425\pi\)
\(354\) −2.94509 −0.156530
\(355\) −0.204272 −0.0108416
\(356\) 2.51500 0.133295
\(357\) −4.46599 −0.236365
\(358\) −5.40943 −0.285897
\(359\) 5.73178 0.302512 0.151256 0.988495i \(-0.451668\pi\)
0.151256 + 0.988495i \(0.451668\pi\)
\(360\) 0.797548 0.0420345
\(361\) −15.8193 −0.832597
\(362\) −6.80871 −0.357858
\(363\) −10.4717 −0.549622
\(364\) 4.52438 0.237142
\(365\) −6.43991 −0.337080
\(366\) −11.9494 −0.624608
\(367\) 5.52511 0.288408 0.144204 0.989548i \(-0.453938\pi\)
0.144204 + 0.989548i \(0.453938\pi\)
\(368\) −5.45739 −0.284486
\(369\) −11.0069 −0.572997
\(370\) 0.193719 0.0100710
\(371\) 3.34688 0.173761
\(372\) −0.500062 −0.0259270
\(373\) 27.1285 1.40466 0.702330 0.711852i \(-0.252143\pi\)
0.702330 + 0.711852i \(0.252143\pi\)
\(374\) 0.717457 0.0370988
\(375\) −7.46817 −0.385655
\(376\) 6.55388 0.337990
\(377\) −2.41751 −0.124508
\(378\) −4.52438 −0.232709
\(379\) −31.7844 −1.63265 −0.816327 0.577590i \(-0.803993\pi\)
−0.816327 + 0.577590i \(0.803993\pi\)
\(380\) 1.42238 0.0729664
\(381\) 14.9261 0.764687
\(382\) 27.2893 1.39624
\(383\) −22.4750 −1.14842 −0.574210 0.818708i \(-0.694691\pi\)
−0.574210 + 0.818708i \(0.694691\pi\)
\(384\) 1.00000 0.0510310
\(385\) −2.62273 −0.133667
\(386\) 0.595016 0.0302855
\(387\) −6.15290 −0.312770
\(388\) −10.1954 −0.517594
\(389\) −1.87542 −0.0950874 −0.0475437 0.998869i \(-0.515139\pi\)
−0.0475437 + 0.998869i \(0.515139\pi\)
\(390\) −0.797548 −0.0403854
\(391\) −5.38696 −0.272430
\(392\) 13.4701 0.680341
\(393\) 6.85518 0.345798
\(394\) −15.9865 −0.805390
\(395\) 0.930686 0.0468279
\(396\) 0.726837 0.0365249
\(397\) 1.42324 0.0714303 0.0357152 0.999362i \(-0.488629\pi\)
0.0357152 + 0.999362i \(0.488629\pi\)
\(398\) −11.5511 −0.579005
\(399\) −8.06896 −0.403953
\(400\) −4.36392 −0.218196
\(401\) 16.9059 0.844241 0.422120 0.906540i \(-0.361286\pi\)
0.422120 + 0.906540i \(0.361286\pi\)
\(402\) 0.738580 0.0368370
\(403\) 0.500062 0.0249099
\(404\) 9.95897 0.495477
\(405\) 0.797548 0.0396305
\(406\) −10.9377 −0.542831
\(407\) 0.176544 0.00875095
\(408\) 0.987094 0.0488685
\(409\) −10.3569 −0.512113 −0.256057 0.966662i \(-0.582423\pi\)
−0.256057 + 0.966662i \(0.582423\pi\)
\(410\) −8.77853 −0.433541
\(411\) 3.70447 0.182728
\(412\) −1.00000 −0.0492665
\(413\) 13.3247 0.655666
\(414\) −5.45739 −0.268216
\(415\) −6.96861 −0.342076
\(416\) −1.00000 −0.0490290
\(417\) −17.2252 −0.843521
\(418\) 1.29627 0.0634026
\(419\) 17.3634 0.848255 0.424128 0.905602i \(-0.360581\pi\)
0.424128 + 0.905602i \(0.360581\pi\)
\(420\) −3.60841 −0.176073
\(421\) −26.7985 −1.30608 −0.653039 0.757324i \(-0.726506\pi\)
−0.653039 + 0.757324i \(0.726506\pi\)
\(422\) −24.7761 −1.20608
\(423\) 6.55388 0.318660
\(424\) −0.739743 −0.0359251
\(425\) −4.30760 −0.208949
\(426\) −0.256125 −0.0124093
\(427\) 54.0639 2.61633
\(428\) −0.884089 −0.0427340
\(429\) −0.726837 −0.0350920
\(430\) −4.90723 −0.236648
\(431\) 14.2900 0.688324 0.344162 0.938910i \(-0.388163\pi\)
0.344162 + 0.938910i \(0.388163\pi\)
\(432\) 1.00000 0.0481125
\(433\) 14.9149 0.716763 0.358382 0.933575i \(-0.383329\pi\)
0.358382 + 0.933575i \(0.383329\pi\)
\(434\) 2.26247 0.108602
\(435\) 1.92808 0.0924442
\(436\) −13.9459 −0.667888
\(437\) −9.73292 −0.465589
\(438\) −8.07464 −0.385821
\(439\) 1.51504 0.0723090 0.0361545 0.999346i \(-0.488489\pi\)
0.0361545 + 0.999346i \(0.488489\pi\)
\(440\) 0.579687 0.0276355
\(441\) 13.4701 0.641431
\(442\) −0.987094 −0.0469513
\(443\) −0.0224802 −0.00106807 −0.000534034 1.00000i \(-0.500170\pi\)
−0.000534034 1.00000i \(0.500170\pi\)
\(444\) 0.242893 0.0115272
\(445\) 2.00583 0.0950855
\(446\) 2.06150 0.0976149
\(447\) −16.4849 −0.779708
\(448\) −4.52438 −0.213757
\(449\) 4.30909 0.203358 0.101679 0.994817i \(-0.467578\pi\)
0.101679 + 0.994817i \(0.467578\pi\)
\(450\) −4.36392 −0.205717
\(451\) −8.00022 −0.376716
\(452\) 8.78961 0.413428
\(453\) −5.22890 −0.245675
\(454\) 19.0225 0.892770
\(455\) 3.60841 0.169165
\(456\) 1.78344 0.0835172
\(457\) 19.5132 0.912789 0.456394 0.889778i \(-0.349141\pi\)
0.456394 + 0.889778i \(0.349141\pi\)
\(458\) 15.9093 0.743393
\(459\) 0.987094 0.0460736
\(460\) −4.35253 −0.202938
\(461\) −37.7225 −1.75691 −0.878457 0.477822i \(-0.841426\pi\)
−0.878457 + 0.477822i \(0.841426\pi\)
\(462\) −3.28849 −0.152994
\(463\) 18.3765 0.854028 0.427014 0.904245i \(-0.359566\pi\)
0.427014 + 0.904245i \(0.359566\pi\)
\(464\) 2.41751 0.112230
\(465\) −0.398823 −0.0184950
\(466\) −15.4747 −0.716850
\(467\) 14.3930 0.666028 0.333014 0.942922i \(-0.391934\pi\)
0.333014 + 0.942922i \(0.391934\pi\)
\(468\) −1.00000 −0.0462250
\(469\) −3.34162 −0.154302
\(470\) 5.22703 0.241105
\(471\) 13.1212 0.604593
\(472\) −2.94509 −0.135559
\(473\) −4.47216 −0.205630
\(474\) 1.16693 0.0535991
\(475\) −7.78278 −0.357098
\(476\) −4.46599 −0.204699
\(477\) −0.739743 −0.0338705
\(478\) −9.91541 −0.453520
\(479\) 7.31044 0.334023 0.167011 0.985955i \(-0.446588\pi\)
0.167011 + 0.985955i \(0.446588\pi\)
\(480\) 0.797548 0.0364029
\(481\) −0.242893 −0.0110750
\(482\) −24.1212 −1.09869
\(483\) 24.6913 1.12349
\(484\) −10.4717 −0.475987
\(485\) −8.13133 −0.369225
\(486\) 1.00000 0.0453609
\(487\) 13.9513 0.632196 0.316098 0.948727i \(-0.397627\pi\)
0.316098 + 0.948727i \(0.397627\pi\)
\(488\) −11.9494 −0.540926
\(489\) 2.05386 0.0928788
\(490\) 10.7430 0.485320
\(491\) 31.1091 1.40393 0.701967 0.712209i \(-0.252305\pi\)
0.701967 + 0.712209i \(0.252305\pi\)
\(492\) −11.0069 −0.496230
\(493\) 2.38631 0.107474
\(494\) −1.78344 −0.0802407
\(495\) 0.579687 0.0260550
\(496\) −0.500062 −0.0224534
\(497\) 1.15881 0.0519797
\(498\) −8.73755 −0.391539
\(499\) 9.53910 0.427029 0.213514 0.976940i \(-0.431509\pi\)
0.213514 + 0.976940i \(0.431509\pi\)
\(500\) −7.46817 −0.333987
\(501\) 6.47667 0.289356
\(502\) 1.25196 0.0558778
\(503\) −17.6243 −0.785830 −0.392915 0.919575i \(-0.628533\pi\)
−0.392915 + 0.919575i \(0.628533\pi\)
\(504\) −4.52438 −0.201532
\(505\) 7.94275 0.353448
\(506\) −3.96663 −0.176338
\(507\) 1.00000 0.0444116
\(508\) 14.9261 0.662238
\(509\) 1.14732 0.0508542 0.0254271 0.999677i \(-0.491905\pi\)
0.0254271 + 0.999677i \(0.491905\pi\)
\(510\) 0.787255 0.0348602
\(511\) 36.5328 1.61611
\(512\) 1.00000 0.0441942
\(513\) 1.78344 0.0787407
\(514\) −20.1530 −0.888909
\(515\) −0.797548 −0.0351442
\(516\) −6.15290 −0.270866
\(517\) 4.76360 0.209503
\(518\) −1.09894 −0.0482848
\(519\) −3.56976 −0.156695
\(520\) −0.797548 −0.0349748
\(521\) 2.61627 0.114621 0.0573105 0.998356i \(-0.481748\pi\)
0.0573105 + 0.998356i \(0.481748\pi\)
\(522\) 2.41751 0.105811
\(523\) 36.4244 1.59273 0.796364 0.604817i \(-0.206754\pi\)
0.796364 + 0.604817i \(0.206754\pi\)
\(524\) 6.85518 0.299470
\(525\) 19.7440 0.861701
\(526\) −13.0656 −0.569687
\(527\) −0.493608 −0.0215019
\(528\) 0.726837 0.0316315
\(529\) 6.78312 0.294918
\(530\) −0.589980 −0.0256271
\(531\) −2.94509 −0.127806
\(532\) −8.06896 −0.349834
\(533\) 11.0069 0.476762
\(534\) 2.51500 0.108835
\(535\) −0.705103 −0.0304843
\(536\) 0.738580 0.0319018
\(537\) −5.40943 −0.233434
\(538\) 27.1939 1.17241
\(539\) 9.79053 0.421708
\(540\) 0.797548 0.0343210
\(541\) 35.3850 1.52132 0.760661 0.649150i \(-0.224875\pi\)
0.760661 + 0.649150i \(0.224875\pi\)
\(542\) 1.66544 0.0715368
\(543\) −6.80871 −0.292190
\(544\) 0.987094 0.0423213
\(545\) −11.1225 −0.476436
\(546\) 4.52438 0.193626
\(547\) 29.1934 1.24822 0.624109 0.781337i \(-0.285462\pi\)
0.624109 + 0.781337i \(0.285462\pi\)
\(548\) 3.70447 0.158247
\(549\) −11.9494 −0.509990
\(550\) −3.17186 −0.135248
\(551\) 4.31148 0.183675
\(552\) −5.45739 −0.232282
\(553\) −5.27966 −0.224514
\(554\) −29.8720 −1.26914
\(555\) 0.193719 0.00822292
\(556\) −17.2252 −0.730511
\(557\) −11.5853 −0.490887 −0.245443 0.969411i \(-0.578934\pi\)
−0.245443 + 0.969411i \(0.578934\pi\)
\(558\) −0.500062 −0.0211693
\(559\) 6.15290 0.260240
\(560\) −3.60841 −0.152483
\(561\) 0.717457 0.0302910
\(562\) −8.28440 −0.349456
\(563\) 40.2228 1.69519 0.847595 0.530644i \(-0.178050\pi\)
0.847595 + 0.530644i \(0.178050\pi\)
\(564\) 6.55388 0.275968
\(565\) 7.01013 0.294918
\(566\) 0.106605 0.00448094
\(567\) −4.52438 −0.190006
\(568\) −0.256125 −0.0107468
\(569\) −9.88422 −0.414368 −0.207184 0.978302i \(-0.566430\pi\)
−0.207184 + 0.978302i \(0.566430\pi\)
\(570\) 1.42238 0.0595768
\(571\) −9.62620 −0.402844 −0.201422 0.979505i \(-0.564556\pi\)
−0.201422 + 0.979505i \(0.564556\pi\)
\(572\) −0.726837 −0.0303906
\(573\) 27.2893 1.14003
\(574\) 49.7995 2.07859
\(575\) 23.8156 0.993179
\(576\) 1.00000 0.0416667
\(577\) −17.8517 −0.743177 −0.371589 0.928397i \(-0.621187\pi\)
−0.371589 + 0.928397i \(0.621187\pi\)
\(578\) −16.0256 −0.666579
\(579\) 0.595016 0.0247280
\(580\) 1.92808 0.0800591
\(581\) 39.5320 1.64007
\(582\) −10.1954 −0.422614
\(583\) −0.537672 −0.0222681
\(584\) −8.07464 −0.334131
\(585\) −0.797548 −0.0329745
\(586\) −2.52246 −0.104202
\(587\) −40.7854 −1.68339 −0.841697 0.539950i \(-0.818443\pi\)
−0.841697 + 0.539950i \(0.818443\pi\)
\(588\) 13.4701 0.555496
\(589\) −0.891829 −0.0367472
\(590\) −2.34885 −0.0967005
\(591\) −15.9865 −0.657598
\(592\) 0.242893 0.00998286
\(593\) −11.4018 −0.468215 −0.234107 0.972211i \(-0.575217\pi\)
−0.234107 + 0.972211i \(0.575217\pi\)
\(594\) 0.726837 0.0298225
\(595\) −3.56184 −0.146021
\(596\) −16.4849 −0.675247
\(597\) −11.5511 −0.472755
\(598\) 5.45739 0.223169
\(599\) 32.6152 1.33262 0.666311 0.745674i \(-0.267872\pi\)
0.666311 + 0.745674i \(0.267872\pi\)
\(600\) −4.36392 −0.178156
\(601\) −33.9546 −1.38504 −0.692518 0.721400i \(-0.743499\pi\)
−0.692518 + 0.721400i \(0.743499\pi\)
\(602\) 27.8381 1.13460
\(603\) 0.738580 0.0300773
\(604\) −5.22890 −0.212761
\(605\) −8.35169 −0.339544
\(606\) 9.95897 0.404555
\(607\) −20.1260 −0.816889 −0.408444 0.912783i \(-0.633929\pi\)
−0.408444 + 0.912783i \(0.633929\pi\)
\(608\) 1.78344 0.0723280
\(609\) −10.9377 −0.443219
\(610\) −9.53025 −0.385869
\(611\) −6.55388 −0.265142
\(612\) 0.987094 0.0399009
\(613\) 41.6300 1.68142 0.840711 0.541485i \(-0.182138\pi\)
0.840711 + 0.541485i \(0.182138\pi\)
\(614\) 22.0445 0.889643
\(615\) −8.77853 −0.353985
\(616\) −3.28849 −0.132497
\(617\) 3.23796 0.130355 0.0651776 0.997874i \(-0.479239\pi\)
0.0651776 + 0.997874i \(0.479239\pi\)
\(618\) −1.00000 −0.0402259
\(619\) 18.3610 0.737989 0.368995 0.929431i \(-0.379702\pi\)
0.368995 + 0.929431i \(0.379702\pi\)
\(620\) −0.398823 −0.0160171
\(621\) −5.45739 −0.218998
\(622\) 0.982492 0.0393943
\(623\) −11.3788 −0.455883
\(624\) −1.00000 −0.0400320
\(625\) 15.8634 0.634535
\(626\) −3.79254 −0.151580
\(627\) 1.29627 0.0517680
\(628\) 13.1212 0.523593
\(629\) 0.239759 0.00955980
\(630\) −3.60841 −0.143763
\(631\) 22.8885 0.911178 0.455589 0.890190i \(-0.349429\pi\)
0.455589 + 0.890190i \(0.349429\pi\)
\(632\) 1.16693 0.0464182
\(633\) −24.7761 −0.984760
\(634\) 14.9328 0.593056
\(635\) 11.9043 0.472406
\(636\) −0.739743 −0.0293327
\(637\) −13.4701 −0.533703
\(638\) 1.75713 0.0695656
\(639\) −0.256125 −0.0101322
\(640\) 0.797548 0.0315258
\(641\) −16.0348 −0.633336 −0.316668 0.948536i \(-0.602564\pi\)
−0.316668 + 0.948536i \(0.602564\pi\)
\(642\) −0.884089 −0.0348922
\(643\) −21.9170 −0.864322 −0.432161 0.901796i \(-0.642249\pi\)
−0.432161 + 0.901796i \(0.642249\pi\)
\(644\) 24.6913 0.972975
\(645\) −4.90723 −0.193222
\(646\) 1.76042 0.0692629
\(647\) −3.37893 −0.132839 −0.0664197 0.997792i \(-0.521158\pi\)
−0.0664197 + 0.997792i \(0.521158\pi\)
\(648\) 1.00000 0.0392837
\(649\) −2.14060 −0.0840258
\(650\) 4.36392 0.171167
\(651\) 2.26247 0.0886733
\(652\) 2.05386 0.0804354
\(653\) −1.86127 −0.0728372 −0.0364186 0.999337i \(-0.511595\pi\)
−0.0364186 + 0.999337i \(0.511595\pi\)
\(654\) −13.9459 −0.545328
\(655\) 5.46733 0.213626
\(656\) −11.0069 −0.429747
\(657\) −8.07464 −0.315022
\(658\) −29.6523 −1.15597
\(659\) 3.66110 0.142616 0.0713082 0.997454i \(-0.477283\pi\)
0.0713082 + 0.997454i \(0.477283\pi\)
\(660\) 0.579687 0.0225643
\(661\) 30.4869 1.18580 0.592902 0.805275i \(-0.297982\pi\)
0.592902 + 0.805275i \(0.297982\pi\)
\(662\) −10.1335 −0.393851
\(663\) −0.987094 −0.0383356
\(664\) −8.73755 −0.339083
\(665\) −6.43538 −0.249553
\(666\) 0.242893 0.00941193
\(667\) −13.1933 −0.510846
\(668\) 6.47667 0.250590
\(669\) 2.06150 0.0797022
\(670\) 0.589053 0.0227571
\(671\) −8.68530 −0.335292
\(672\) −4.52438 −0.174532
\(673\) 34.7610 1.33994 0.669970 0.742389i \(-0.266307\pi\)
0.669970 + 0.742389i \(0.266307\pi\)
\(674\) 19.9561 0.768679
\(675\) −4.36392 −0.167967
\(676\) 1.00000 0.0384615
\(677\) −21.6936 −0.833753 −0.416876 0.908963i \(-0.636875\pi\)
−0.416876 + 0.908963i \(0.636875\pi\)
\(678\) 8.78961 0.337563
\(679\) 46.1280 1.77023
\(680\) 0.787255 0.0301898
\(681\) 19.0225 0.728943
\(682\) −0.363463 −0.0139177
\(683\) 34.4730 1.31907 0.659536 0.751673i \(-0.270753\pi\)
0.659536 + 0.751673i \(0.270753\pi\)
\(684\) 1.78344 0.0681915
\(685\) 2.95449 0.112885
\(686\) −29.2730 −1.11765
\(687\) 15.9093 0.606978
\(688\) −6.15290 −0.234577
\(689\) 0.739743 0.0281819
\(690\) −4.35253 −0.165698
\(691\) −5.39903 −0.205389 −0.102694 0.994713i \(-0.532746\pi\)
−0.102694 + 0.994713i \(0.532746\pi\)
\(692\) −3.56976 −0.135702
\(693\) −3.28849 −0.124919
\(694\) 3.81997 0.145004
\(695\) −13.7379 −0.521109
\(696\) 2.41751 0.0916354
\(697\) −10.8649 −0.411536
\(698\) 11.6084 0.439383
\(699\) −15.4747 −0.585306
\(700\) 19.7440 0.746255
\(701\) 34.9808 1.32121 0.660604 0.750735i \(-0.270300\pi\)
0.660604 + 0.750735i \(0.270300\pi\)
\(702\) −1.00000 −0.0377426
\(703\) 0.433185 0.0163379
\(704\) 0.726837 0.0273937
\(705\) 5.22703 0.196861
\(706\) 10.4353 0.392739
\(707\) −45.0582 −1.69459
\(708\) −2.94509 −0.110683
\(709\) −25.2093 −0.946756 −0.473378 0.880859i \(-0.656966\pi\)
−0.473378 + 0.880859i \(0.656966\pi\)
\(710\) −0.204272 −0.00766620
\(711\) 1.16693 0.0437635
\(712\) 2.51500 0.0942535
\(713\) 2.72903 0.102203
\(714\) −4.46599 −0.167136
\(715\) −0.579687 −0.0216791
\(716\) −5.40943 −0.202160
\(717\) −9.91541 −0.370298
\(718\) 5.73178 0.213908
\(719\) −25.2591 −0.942006 −0.471003 0.882132i \(-0.656108\pi\)
−0.471003 + 0.882132i \(0.656108\pi\)
\(720\) 0.797548 0.0297229
\(721\) 4.52438 0.168497
\(722\) −15.8193 −0.588735
\(723\) −24.1212 −0.897079
\(724\) −6.80871 −0.253044
\(725\) −10.5498 −0.391810
\(726\) −10.4717 −0.388642
\(727\) 1.00734 0.0373601 0.0186800 0.999826i \(-0.494054\pi\)
0.0186800 + 0.999826i \(0.494054\pi\)
\(728\) 4.52438 0.167685
\(729\) 1.00000 0.0370370
\(730\) −6.43991 −0.238352
\(731\) −6.07349 −0.224636
\(732\) −11.9494 −0.441664
\(733\) 36.6669 1.35432 0.677161 0.735835i \(-0.263210\pi\)
0.677161 + 0.735835i \(0.263210\pi\)
\(734\) 5.52511 0.203935
\(735\) 10.7430 0.396262
\(736\) −5.45739 −0.201162
\(737\) 0.536827 0.0197743
\(738\) −11.0069 −0.405170
\(739\) 23.2457 0.855108 0.427554 0.903990i \(-0.359375\pi\)
0.427554 + 0.903990i \(0.359375\pi\)
\(740\) 0.193719 0.00712125
\(741\) −1.78344 −0.0655163
\(742\) 3.34688 0.122868
\(743\) 27.3497 1.00336 0.501681 0.865052i \(-0.332715\pi\)
0.501681 + 0.865052i \(0.332715\pi\)
\(744\) −0.500062 −0.0183332
\(745\) −13.1475 −0.481687
\(746\) 27.1285 0.993244
\(747\) −8.73755 −0.319690
\(748\) 0.717457 0.0262328
\(749\) 3.99996 0.146155
\(750\) −7.46817 −0.272699
\(751\) 28.4157 1.03690 0.518451 0.855107i \(-0.326509\pi\)
0.518451 + 0.855107i \(0.326509\pi\)
\(752\) 6.55388 0.238995
\(753\) 1.25196 0.0456240
\(754\) −2.41751 −0.0880404
\(755\) −4.17030 −0.151773
\(756\) −4.52438 −0.164550
\(757\) −17.9350 −0.651859 −0.325930 0.945394i \(-0.605677\pi\)
−0.325930 + 0.945394i \(0.605677\pi\)
\(758\) −31.7844 −1.15446
\(759\) −3.96663 −0.143980
\(760\) 1.42238 0.0515951
\(761\) −0.407740 −0.0147806 −0.00739028 0.999973i \(-0.502352\pi\)
−0.00739028 + 0.999973i \(0.502352\pi\)
\(762\) 14.9261 0.540715
\(763\) 63.0966 2.28425
\(764\) 27.2893 0.987291
\(765\) 0.787255 0.0284633
\(766\) −22.4750 −0.812055
\(767\) 2.94509 0.106341
\(768\) 1.00000 0.0360844
\(769\) −11.7330 −0.423104 −0.211552 0.977367i \(-0.567852\pi\)
−0.211552 + 0.977367i \(0.567852\pi\)
\(770\) −2.62273 −0.0945165
\(771\) −20.1530 −0.725791
\(772\) 0.595016 0.0214151
\(773\) −14.7850 −0.531779 −0.265890 0.964003i \(-0.585666\pi\)
−0.265890 + 0.964003i \(0.585666\pi\)
\(774\) −6.15290 −0.221162
\(775\) 2.18223 0.0783880
\(776\) −10.1954 −0.365994
\(777\) −1.09894 −0.0394244
\(778\) −1.87542 −0.0672370
\(779\) −19.6301 −0.703323
\(780\) −0.797548 −0.0285568
\(781\) −0.186161 −0.00666138
\(782\) −5.38696 −0.192637
\(783\) 2.41751 0.0863947
\(784\) 13.4701 0.481073
\(785\) 10.4648 0.373504
\(786\) 6.85518 0.244516
\(787\) −4.69636 −0.167407 −0.0837036 0.996491i \(-0.526675\pi\)
−0.0837036 + 0.996491i \(0.526675\pi\)
\(788\) −15.9865 −0.569497
\(789\) −13.0656 −0.465147
\(790\) 0.930686 0.0331123
\(791\) −39.7676 −1.41397
\(792\) 0.726837 0.0258270
\(793\) 11.9494 0.424337
\(794\) 1.42324 0.0505089
\(795\) −0.589980 −0.0209244
\(796\) −11.5511 −0.409418
\(797\) −34.0542 −1.20626 −0.603131 0.797642i \(-0.706080\pi\)
−0.603131 + 0.797642i \(0.706080\pi\)
\(798\) −8.06896 −0.285638
\(799\) 6.46930 0.228867
\(800\) −4.36392 −0.154288
\(801\) 2.51500 0.0888630
\(802\) 16.9059 0.596968
\(803\) −5.86894 −0.207111
\(804\) 0.738580 0.0260477
\(805\) 19.6925 0.694070
\(806\) 0.500062 0.0176139
\(807\) 27.1939 0.957270
\(808\) 9.95897 0.350355
\(809\) −45.7412 −1.60817 −0.804087 0.594512i \(-0.797345\pi\)
−0.804087 + 0.594512i \(0.797345\pi\)
\(810\) 0.797548 0.0280230
\(811\) −9.34669 −0.328207 −0.164103 0.986443i \(-0.552473\pi\)
−0.164103 + 0.986443i \(0.552473\pi\)
\(812\) −10.9377 −0.383839
\(813\) 1.66544 0.0584096
\(814\) 0.176544 0.00618786
\(815\) 1.63805 0.0573785
\(816\) 0.987094 0.0345552
\(817\) −10.9733 −0.383908
\(818\) −10.3569 −0.362119
\(819\) 4.52438 0.158095
\(820\) −8.77853 −0.306560
\(821\) −26.8650 −0.937595 −0.468797 0.883306i \(-0.655313\pi\)
−0.468797 + 0.883306i \(0.655313\pi\)
\(822\) 3.70447 0.129208
\(823\) 24.7424 0.862467 0.431234 0.902240i \(-0.358078\pi\)
0.431234 + 0.902240i \(0.358078\pi\)
\(824\) −1.00000 −0.0348367
\(825\) −3.17186 −0.110430
\(826\) 13.3247 0.463626
\(827\) −27.9021 −0.970250 −0.485125 0.874445i \(-0.661226\pi\)
−0.485125 + 0.874445i \(0.661226\pi\)
\(828\) −5.45739 −0.189657
\(829\) 38.3360 1.33146 0.665732 0.746191i \(-0.268119\pi\)
0.665732 + 0.746191i \(0.268119\pi\)
\(830\) −6.96861 −0.241884
\(831\) −29.8720 −1.03625
\(832\) −1.00000 −0.0346688
\(833\) 13.2962 0.460687
\(834\) −17.2252 −0.596459
\(835\) 5.16546 0.178758
\(836\) 1.29627 0.0448324
\(837\) −0.500062 −0.0172847
\(838\) 17.3634 0.599807
\(839\) 23.2516 0.802735 0.401367 0.915917i \(-0.368535\pi\)
0.401367 + 0.915917i \(0.368535\pi\)
\(840\) −3.60841 −0.124502
\(841\) −23.1557 −0.798471
\(842\) −26.7985 −0.923537
\(843\) −8.28440 −0.285330
\(844\) −24.7761 −0.852827
\(845\) 0.797548 0.0274365
\(846\) 6.55388 0.225327
\(847\) 47.3780 1.62793
\(848\) −0.739743 −0.0254029
\(849\) 0.106605 0.00365867
\(850\) −4.30760 −0.147749
\(851\) −1.32556 −0.0454398
\(852\) −0.256125 −0.00877471
\(853\) 7.32715 0.250877 0.125438 0.992101i \(-0.459966\pi\)
0.125438 + 0.992101i \(0.459966\pi\)
\(854\) 54.0639 1.85003
\(855\) 1.42238 0.0486443
\(856\) −0.884089 −0.0302175
\(857\) −14.7087 −0.502439 −0.251220 0.967930i \(-0.580832\pi\)
−0.251220 + 0.967930i \(0.580832\pi\)
\(858\) −0.726837 −0.0248138
\(859\) 24.5352 0.837130 0.418565 0.908187i \(-0.362533\pi\)
0.418565 + 0.908187i \(0.362533\pi\)
\(860\) −4.90723 −0.167335
\(861\) 49.7995 1.69716
\(862\) 14.2900 0.486718
\(863\) −7.54013 −0.256669 −0.128335 0.991731i \(-0.540963\pi\)
−0.128335 + 0.991731i \(0.540963\pi\)
\(864\) 1.00000 0.0340207
\(865\) −2.84706 −0.0968029
\(866\) 14.9149 0.506828
\(867\) −16.0256 −0.544259
\(868\) 2.26247 0.0767933
\(869\) 0.848171 0.0287722
\(870\) 1.92808 0.0653680
\(871\) −0.738580 −0.0250258
\(872\) −13.9459 −0.472268
\(873\) −10.1954 −0.345063
\(874\) −9.73292 −0.329221
\(875\) 33.7889 1.14227
\(876\) −8.07464 −0.272817
\(877\) 4.62436 0.156154 0.0780768 0.996947i \(-0.475122\pi\)
0.0780768 + 0.996947i \(0.475122\pi\)
\(878\) 1.51504 0.0511302
\(879\) −2.52246 −0.0850805
\(880\) 0.579687 0.0195412
\(881\) −42.9887 −1.44833 −0.724163 0.689629i \(-0.757774\pi\)
−0.724163 + 0.689629i \(0.757774\pi\)
\(882\) 13.4701 0.453560
\(883\) −7.23511 −0.243481 −0.121740 0.992562i \(-0.538848\pi\)
−0.121740 + 0.992562i \(0.538848\pi\)
\(884\) −0.987094 −0.0331996
\(885\) −2.34885 −0.0789556
\(886\) −0.0224802 −0.000755238 0
\(887\) 37.8766 1.27177 0.635886 0.771783i \(-0.280635\pi\)
0.635886 + 0.771783i \(0.280635\pi\)
\(888\) 0.242893 0.00815097
\(889\) −67.5314 −2.26493
\(890\) 2.00583 0.0672356
\(891\) 0.726837 0.0243500
\(892\) 2.06150 0.0690241
\(893\) 11.6884 0.391139
\(894\) −16.4849 −0.551337
\(895\) −4.31428 −0.144210
\(896\) −4.52438 −0.151149
\(897\) 5.45739 0.182217
\(898\) 4.30909 0.143796
\(899\) −1.20890 −0.0403192
\(900\) −4.36392 −0.145464
\(901\) −0.730196 −0.0243263
\(902\) −8.00022 −0.266378
\(903\) 27.8381 0.926394
\(904\) 8.78961 0.292338
\(905\) −5.43027 −0.180508
\(906\) −5.22890 −0.173719
\(907\) 7.01925 0.233071 0.116535 0.993187i \(-0.462821\pi\)
0.116535 + 0.993187i \(0.462821\pi\)
\(908\) 19.0225 0.631283
\(909\) 9.95897 0.330318
\(910\) 3.60841 0.119618
\(911\) 37.9417 1.25706 0.628532 0.777783i \(-0.283656\pi\)
0.628532 + 0.777783i \(0.283656\pi\)
\(912\) 1.78344 0.0590556
\(913\) −6.35077 −0.210180
\(914\) 19.5132 0.645439
\(915\) −9.53025 −0.315060
\(916\) 15.9093 0.525658
\(917\) −31.0155 −1.02422
\(918\) 0.987094 0.0325790
\(919\) 30.1113 0.993279 0.496639 0.867957i \(-0.334567\pi\)
0.496639 + 0.867957i \(0.334567\pi\)
\(920\) −4.35253 −0.143499
\(921\) 22.0445 0.726391
\(922\) −37.7225 −1.24233
\(923\) 0.256125 0.00843047
\(924\) −3.28849 −0.108183
\(925\) −1.05997 −0.0348515
\(926\) 18.3765 0.603889
\(927\) −1.00000 −0.0328443
\(928\) 2.41751 0.0793586
\(929\) −7.98142 −0.261862 −0.130931 0.991392i \(-0.541797\pi\)
−0.130931 + 0.991392i \(0.541797\pi\)
\(930\) −0.398823 −0.0130779
\(931\) 24.0230 0.787323
\(932\) −15.4747 −0.506890
\(933\) 0.982492 0.0321653
\(934\) 14.3930 0.470953
\(935\) 0.572206 0.0187131
\(936\) −1.00000 −0.0326860
\(937\) 42.9781 1.40403 0.702017 0.712160i \(-0.252283\pi\)
0.702017 + 0.712160i \(0.252283\pi\)
\(938\) −3.34162 −0.109108
\(939\) −3.79254 −0.123765
\(940\) 5.22703 0.170487
\(941\) −4.88484 −0.159241 −0.0796207 0.996825i \(-0.525371\pi\)
−0.0796207 + 0.996825i \(0.525371\pi\)
\(942\) 13.1212 0.427512
\(943\) 60.0690 1.95612
\(944\) −2.94509 −0.0958544
\(945\) −3.60841 −0.117382
\(946\) −4.47216 −0.145402
\(947\) 7.34327 0.238624 0.119312 0.992857i \(-0.461931\pi\)
0.119312 + 0.992857i \(0.461931\pi\)
\(948\) 1.16693 0.0379003
\(949\) 8.07464 0.262114
\(950\) −7.78278 −0.252507
\(951\) 14.9328 0.484228
\(952\) −4.46599 −0.144744
\(953\) 40.4781 1.31121 0.655607 0.755102i \(-0.272413\pi\)
0.655607 + 0.755102i \(0.272413\pi\)
\(954\) −0.739743 −0.0239500
\(955\) 21.7645 0.704282
\(956\) −9.91541 −0.320687
\(957\) 1.75713 0.0568001
\(958\) 7.31044 0.236190
\(959\) −16.7605 −0.541224
\(960\) 0.797548 0.0257407
\(961\) −30.7499 −0.991933
\(962\) −0.242893 −0.00783120
\(963\) −0.884089 −0.0284894
\(964\) −24.1212 −0.776893
\(965\) 0.474554 0.0152764
\(966\) 24.6913 0.794431
\(967\) −36.9299 −1.18759 −0.593793 0.804618i \(-0.702370\pi\)
−0.593793 + 0.804618i \(0.702370\pi\)
\(968\) −10.4717 −0.336573
\(969\) 1.76042 0.0565529
\(970\) −8.13133 −0.261081
\(971\) −7.92569 −0.254347 −0.127174 0.991880i \(-0.540591\pi\)
−0.127174 + 0.991880i \(0.540591\pi\)
\(972\) 1.00000 0.0320750
\(973\) 77.9334 2.49843
\(974\) 13.9513 0.447030
\(975\) 4.36392 0.139757
\(976\) −11.9494 −0.382492
\(977\) 2.74275 0.0877483 0.0438741 0.999037i \(-0.486030\pi\)
0.0438741 + 0.999037i \(0.486030\pi\)
\(978\) 2.05386 0.0656752
\(979\) 1.82799 0.0584229
\(980\) 10.7430 0.343173
\(981\) −13.9459 −0.445258
\(982\) 31.1091 0.992731
\(983\) −25.6816 −0.819115 −0.409557 0.912284i \(-0.634317\pi\)
−0.409557 + 0.912284i \(0.634317\pi\)
\(984\) −11.0069 −0.350887
\(985\) −12.7500 −0.406250
\(986\) 2.38631 0.0759955
\(987\) −29.6523 −0.943842
\(988\) −1.78344 −0.0567387
\(989\) 33.5788 1.06774
\(990\) 0.579687 0.0184237
\(991\) 31.4969 1.00053 0.500267 0.865871i \(-0.333235\pi\)
0.500267 + 0.865871i \(0.333235\pi\)
\(992\) −0.500062 −0.0158770
\(993\) −10.1335 −0.321578
\(994\) 1.15881 0.0367552
\(995\) −9.21256 −0.292058
\(996\) −8.73755 −0.276860
\(997\) 34.4563 1.09124 0.545620 0.838032i \(-0.316294\pi\)
0.545620 + 0.838032i \(0.316294\pi\)
\(998\) 9.53910 0.301955
\(999\) 0.242893 0.00768481
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 8034.2.a.q.1.6 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8034.2.a.q.1.6 8 1.1 even 1 trivial