Properties

Label 4830.2.a.bw.1.3
Level $4830$
Weight $2$
Character 4830.1
Self dual yes
Analytic conductor $38.568$
Analytic rank $0$
Dimension $3$
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] = [4830,2,Mod(1,4830)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4830, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4830.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4830 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4830.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: \(38.5677441763\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.34292\) of defining polynomial
Character \(\chi\) \(=\) 4830.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +2.68585 q^{11} -1.00000 q^{12} +0.292731 q^{13} +1.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} -6.97858 q^{17} -1.00000 q^{18} +4.39312 q^{19} -1.00000 q^{20} +1.00000 q^{21} -2.68585 q^{22} +1.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} -0.292731 q^{26} -1.00000 q^{27} -1.00000 q^{28} +0.685846 q^{29} -1.00000 q^{30} +8.97858 q^{31} -1.00000 q^{32} -2.68585 q^{33} +6.97858 q^{34} +1.00000 q^{35} +1.00000 q^{36} -6.00000 q^{37} -4.39312 q^{38} -0.292731 q^{39} +1.00000 q^{40} -7.37169 q^{41} -1.00000 q^{42} -6.68585 q^{43} +2.68585 q^{44} -1.00000 q^{45} -1.00000 q^{46} +10.3503 q^{47} -1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} +6.97858 q^{51} +0.292731 q^{52} -6.00000 q^{53} +1.00000 q^{54} -2.68585 q^{55} +1.00000 q^{56} -4.39312 q^{57} -0.685846 q^{58} -4.00000 q^{59} +1.00000 q^{60} +7.37169 q^{61} -8.97858 q^{62} -1.00000 q^{63} +1.00000 q^{64} -0.292731 q^{65} +2.68585 q^{66} +1.31415 q^{67} -6.97858 q^{68} -1.00000 q^{69} -1.00000 q^{70} +7.27131 q^{71} -1.00000 q^{72} +11.9572 q^{73} +6.00000 q^{74} -1.00000 q^{75} +4.39312 q^{76} -2.68585 q^{77} +0.292731 q^{78} -15.3288 q^{79} -1.00000 q^{80} +1.00000 q^{81} +7.37169 q^{82} +0.393115 q^{83} +1.00000 q^{84} +6.97858 q^{85} +6.68585 q^{86} -0.685846 q^{87} -2.68585 q^{88} -9.66442 q^{89} +1.00000 q^{90} -0.292731 q^{91} +1.00000 q^{92} -8.97858 q^{93} -10.3503 q^{94} -4.39312 q^{95} +1.00000 q^{96} -10.2499 q^{97} -1.00000 q^{98} +2.68585 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{5} + 3 q^{6} - 3 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{5} + 3 q^{6} - 3 q^{7} - 3 q^{8} + 3 q^{9} + 3 q^{10} - 4 q^{11} - 3 q^{12} - 2 q^{13} + 3 q^{14} + 3 q^{15} + 3 q^{16} - 6 q^{17} - 3 q^{18} + 4 q^{19} - 3 q^{20} + 3 q^{21} + 4 q^{22} + 3 q^{23} + 3 q^{24} + 3 q^{25} + 2 q^{26} - 3 q^{27} - 3 q^{28} - 10 q^{29} - 3 q^{30} + 12 q^{31} - 3 q^{32} + 4 q^{33} + 6 q^{34} + 3 q^{35} + 3 q^{36} - 18 q^{37} - 4 q^{38} + 2 q^{39} + 3 q^{40} + 2 q^{41} - 3 q^{42} - 8 q^{43} - 4 q^{44} - 3 q^{45} - 3 q^{46} - 8 q^{47} - 3 q^{48} + 3 q^{49} - 3 q^{50} + 6 q^{51} - 2 q^{52} - 18 q^{53} + 3 q^{54} + 4 q^{55} + 3 q^{56} - 4 q^{57} + 10 q^{58} - 12 q^{59} + 3 q^{60} - 2 q^{61} - 12 q^{62} - 3 q^{63} + 3 q^{64} + 2 q^{65} - 4 q^{66} + 16 q^{67} - 6 q^{68} - 3 q^{69} - 3 q^{70} + 4 q^{71} - 3 q^{72} + 6 q^{73} + 18 q^{74} - 3 q^{75} + 4 q^{76} + 4 q^{77} - 2 q^{78} + 8 q^{79} - 3 q^{80} + 3 q^{81} - 2 q^{82} - 8 q^{83} + 3 q^{84} + 6 q^{85} + 8 q^{86} + 10 q^{87} + 4 q^{88} - 2 q^{89} + 3 q^{90} + 2 q^{91} + 3 q^{92} - 12 q^{93} + 8 q^{94} - 4 q^{95} + 3 q^{96} + 2 q^{97} - 3 q^{98} - 4 q^{99}+O(q^{100}) \) 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
\(5\) −1.00000 −0.447214
\(6\) 1.00000 0.408248
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) 2.68585 0.809813 0.404907 0.914358i \(-0.367304\pi\)
0.404907 + 0.914358i \(0.367304\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0.292731 0.0811890 0.0405945 0.999176i \(-0.487075\pi\)
0.0405945 + 0.999176i \(0.487075\pi\)
\(14\) 1.00000 0.267261
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) −6.97858 −1.69255 −0.846277 0.532743i \(-0.821161\pi\)
−0.846277 + 0.532743i \(0.821161\pi\)
\(18\) −1.00000 −0.235702
\(19\) 4.39312 1.00785 0.503925 0.863747i \(-0.331889\pi\)
0.503925 + 0.863747i \(0.331889\pi\)
\(20\) −1.00000 −0.223607
\(21\) 1.00000 0.218218
\(22\) −2.68585 −0.572624
\(23\) 1.00000 0.208514
\(24\) 1.00000 0.204124
\(25\) 1.00000 0.200000
\(26\) −0.292731 −0.0574093
\(27\) −1.00000 −0.192450
\(28\) −1.00000 −0.188982
\(29\) 0.685846 0.127358 0.0636792 0.997970i \(-0.479717\pi\)
0.0636792 + 0.997970i \(0.479717\pi\)
\(30\) −1.00000 −0.182574
\(31\) 8.97858 1.61260 0.806300 0.591507i \(-0.201467\pi\)
0.806300 + 0.591507i \(0.201467\pi\)
\(32\) −1.00000 −0.176777
\(33\) −2.68585 −0.467546
\(34\) 6.97858 1.19682
\(35\) 1.00000 0.169031
\(36\) 1.00000 0.166667
\(37\) −6.00000 −0.986394 −0.493197 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) −4.39312 −0.712657
\(39\) −0.292731 −0.0468745
\(40\) 1.00000 0.158114
\(41\) −7.37169 −1.15126 −0.575632 0.817709i \(-0.695244\pi\)
−0.575632 + 0.817709i \(0.695244\pi\)
\(42\) −1.00000 −0.154303
\(43\) −6.68585 −1.01958 −0.509791 0.860298i \(-0.670277\pi\)
−0.509791 + 0.860298i \(0.670277\pi\)
\(44\) 2.68585 0.404907
\(45\) −1.00000 −0.149071
\(46\) −1.00000 −0.147442
\(47\) 10.3503 1.50974 0.754871 0.655873i \(-0.227699\pi\)
0.754871 + 0.655873i \(0.227699\pi\)
\(48\) −1.00000 −0.144338
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) 6.97858 0.977196
\(52\) 0.292731 0.0405945
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 1.00000 0.136083
\(55\) −2.68585 −0.362159
\(56\) 1.00000 0.133631
\(57\) −4.39312 −0.581882
\(58\) −0.685846 −0.0900560
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 1.00000 0.129099
\(61\) 7.37169 0.943848 0.471924 0.881639i \(-0.343560\pi\)
0.471924 + 0.881639i \(0.343560\pi\)
\(62\) −8.97858 −1.14028
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) −0.292731 −0.0363088
\(66\) 2.68585 0.330605
\(67\) 1.31415 0.160549 0.0802747 0.996773i \(-0.474420\pi\)
0.0802747 + 0.996773i \(0.474420\pi\)
\(68\) −6.97858 −0.846277
\(69\) −1.00000 −0.120386
\(70\) −1.00000 −0.119523
\(71\) 7.27131 0.862946 0.431473 0.902126i \(-0.357994\pi\)
0.431473 + 0.902126i \(0.357994\pi\)
\(72\) −1.00000 −0.117851
\(73\) 11.9572 1.39948 0.699740 0.714398i \(-0.253299\pi\)
0.699740 + 0.714398i \(0.253299\pi\)
\(74\) 6.00000 0.697486
\(75\) −1.00000 −0.115470
\(76\) 4.39312 0.503925
\(77\) −2.68585 −0.306081
\(78\) 0.292731 0.0331453
\(79\) −15.3288 −1.72463 −0.862315 0.506372i \(-0.830986\pi\)
−0.862315 + 0.506372i \(0.830986\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) 7.37169 0.814067
\(83\) 0.393115 0.0431500 0.0215750 0.999767i \(-0.493132\pi\)
0.0215750 + 0.999767i \(0.493132\pi\)
\(84\) 1.00000 0.109109
\(85\) 6.97858 0.756933
\(86\) 6.68585 0.720953
\(87\) −0.685846 −0.0735304
\(88\) −2.68585 −0.286312
\(89\) −9.66442 −1.02443 −0.512213 0.858858i \(-0.671174\pi\)
−0.512213 + 0.858858i \(0.671174\pi\)
\(90\) 1.00000 0.105409
\(91\) −0.292731 −0.0306865
\(92\) 1.00000 0.104257
\(93\) −8.97858 −0.931035
\(94\) −10.3503 −1.06755
\(95\) −4.39312 −0.450724
\(96\) 1.00000 0.102062
\(97\) −10.2499 −1.04072 −0.520359 0.853948i \(-0.674202\pi\)
−0.520359 + 0.853948i \(0.674202\pi\)
\(98\) −1.00000 −0.101015
\(99\) 2.68585 0.269938
\(100\) 1.00000 0.100000
\(101\) 4.29273 0.427143 0.213571 0.976927i \(-0.431490\pi\)
0.213571 + 0.976927i \(0.431490\pi\)
\(102\) −6.97858 −0.690982
\(103\) −4.58546 −0.451819 −0.225909 0.974148i \(-0.572535\pi\)
−0.225909 + 0.974148i \(0.572535\pi\)
\(104\) −0.292731 −0.0287046
\(105\) −1.00000 −0.0975900
\(106\) 6.00000 0.582772
\(107\) 1.37169 0.132607 0.0663033 0.997800i \(-0.478880\pi\)
0.0663033 + 0.997800i \(0.478880\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 6.00000 0.574696 0.287348 0.957826i \(-0.407226\pi\)
0.287348 + 0.957826i \(0.407226\pi\)
\(110\) 2.68585 0.256085
\(111\) 6.00000 0.569495
\(112\) −1.00000 −0.0944911
\(113\) 15.9572 1.50112 0.750561 0.660801i \(-0.229783\pi\)
0.750561 + 0.660801i \(0.229783\pi\)
\(114\) 4.39312 0.411453
\(115\) −1.00000 −0.0932505
\(116\) 0.685846 0.0636792
\(117\) 0.292731 0.0270630
\(118\) 4.00000 0.368230
\(119\) 6.97858 0.639725
\(120\) −1.00000 −0.0912871
\(121\) −3.78623 −0.344203
\(122\) −7.37169 −0.667402
\(123\) 7.37169 0.664683
\(124\) 8.97858 0.806300
\(125\) −1.00000 −0.0894427
\(126\) 1.00000 0.0890871
\(127\) 12.0575 1.06993 0.534967 0.844873i \(-0.320324\pi\)
0.534967 + 0.844873i \(0.320324\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 6.68585 0.588656
\(130\) 0.292731 0.0256742
\(131\) −4.00000 −0.349482 −0.174741 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) −2.68585 −0.233773
\(133\) −4.39312 −0.380931
\(134\) −1.31415 −0.113526
\(135\) 1.00000 0.0860663
\(136\) 6.97858 0.598408
\(137\) −21.9143 −1.87227 −0.936133 0.351646i \(-0.885622\pi\)
−0.936133 + 0.351646i \(0.885622\pi\)
\(138\) 1.00000 0.0851257
\(139\) 18.7434 1.58979 0.794897 0.606745i \(-0.207525\pi\)
0.794897 + 0.606745i \(0.207525\pi\)
\(140\) 1.00000 0.0845154
\(141\) −10.3503 −0.871650
\(142\) −7.27131 −0.610195
\(143\) 0.786230 0.0657479
\(144\) 1.00000 0.0833333
\(145\) −0.685846 −0.0569564
\(146\) −11.9572 −0.989581
\(147\) −1.00000 −0.0824786
\(148\) −6.00000 −0.493197
\(149\) −16.7434 −1.37167 −0.685836 0.727756i \(-0.740563\pi\)
−0.685836 + 0.727756i \(0.740563\pi\)
\(150\) 1.00000 0.0816497
\(151\) −15.3288 −1.24744 −0.623722 0.781646i \(-0.714380\pi\)
−0.623722 + 0.781646i \(0.714380\pi\)
\(152\) −4.39312 −0.356329
\(153\) −6.97858 −0.564185
\(154\) 2.68585 0.216432
\(155\) −8.97858 −0.721177
\(156\) −0.292731 −0.0234372
\(157\) 10.5855 0.844812 0.422406 0.906407i \(-0.361186\pi\)
0.422406 + 0.906407i \(0.361186\pi\)
\(158\) 15.3288 1.21950
\(159\) 6.00000 0.475831
\(160\) 1.00000 0.0790569
\(161\) −1.00000 −0.0788110
\(162\) −1.00000 −0.0785674
\(163\) −8.00000 −0.626608 −0.313304 0.949653i \(-0.601436\pi\)
−0.313304 + 0.949653i \(0.601436\pi\)
\(164\) −7.37169 −0.575632
\(165\) 2.68585 0.209093
\(166\) −0.393115 −0.0305117
\(167\) 4.19235 0.324414 0.162207 0.986757i \(-0.448139\pi\)
0.162207 + 0.986757i \(0.448139\pi\)
\(168\) −1.00000 −0.0771517
\(169\) −12.9143 −0.993408
\(170\) −6.97858 −0.535232
\(171\) 4.39312 0.335950
\(172\) −6.68585 −0.509791
\(173\) 11.7648 0.894462 0.447231 0.894419i \(-0.352410\pi\)
0.447231 + 0.894419i \(0.352410\pi\)
\(174\) 0.685846 0.0519939
\(175\) −1.00000 −0.0755929
\(176\) 2.68585 0.202453
\(177\) 4.00000 0.300658
\(178\) 9.66442 0.724379
\(179\) 5.95715 0.445259 0.222629 0.974903i \(-0.428536\pi\)
0.222629 + 0.974903i \(0.428536\pi\)
\(180\) −1.00000 −0.0745356
\(181\) 24.5426 1.82424 0.912119 0.409925i \(-0.134445\pi\)
0.912119 + 0.409925i \(0.134445\pi\)
\(182\) 0.292731 0.0216987
\(183\) −7.37169 −0.544931
\(184\) −1.00000 −0.0737210
\(185\) 6.00000 0.441129
\(186\) 8.97858 0.658341
\(187\) −18.7434 −1.37065
\(188\) 10.3503 0.754871
\(189\) 1.00000 0.0727393
\(190\) 4.39312 0.318710
\(191\) −13.9572 −1.00990 −0.504952 0.863147i \(-0.668490\pi\)
−0.504952 + 0.863147i \(0.668490\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 25.3288 1.82321 0.911605 0.411067i \(-0.134844\pi\)
0.911605 + 0.411067i \(0.134844\pi\)
\(194\) 10.2499 0.735899
\(195\) 0.292731 0.0209629
\(196\) 1.00000 0.0714286
\(197\) 11.9572 0.851912 0.425956 0.904744i \(-0.359938\pi\)
0.425956 + 0.904744i \(0.359938\pi\)
\(198\) −2.68585 −0.190875
\(199\) 15.3288 1.08663 0.543317 0.839528i \(-0.317168\pi\)
0.543317 + 0.839528i \(0.317168\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −1.31415 −0.0926933
\(202\) −4.29273 −0.302035
\(203\) −0.685846 −0.0481370
\(204\) 6.97858 0.488598
\(205\) 7.37169 0.514861
\(206\) 4.58546 0.319484
\(207\) 1.00000 0.0695048
\(208\) 0.292731 0.0202972
\(209\) 11.7992 0.816170
\(210\) 1.00000 0.0690066
\(211\) 7.41454 0.510438 0.255219 0.966883i \(-0.417852\pi\)
0.255219 + 0.966883i \(0.417852\pi\)
\(212\) −6.00000 −0.412082
\(213\) −7.27131 −0.498222
\(214\) −1.37169 −0.0937670
\(215\) 6.68585 0.455971
\(216\) 1.00000 0.0680414
\(217\) −8.97858 −0.609506
\(218\) −6.00000 −0.406371
\(219\) −11.9572 −0.807990
\(220\) −2.68585 −0.181080
\(221\) −2.04285 −0.137417
\(222\) −6.00000 −0.402694
\(223\) 19.0790 1.27762 0.638811 0.769364i \(-0.279427\pi\)
0.638811 + 0.769364i \(0.279427\pi\)
\(224\) 1.00000 0.0668153
\(225\) 1.00000 0.0666667
\(226\) −15.9572 −1.06145
\(227\) 12.9786 0.861418 0.430709 0.902491i \(-0.358263\pi\)
0.430709 + 0.902491i \(0.358263\pi\)
\(228\) −4.39312 −0.290941
\(229\) 18.7862 1.24143 0.620715 0.784037i \(-0.286843\pi\)
0.620715 + 0.784037i \(0.286843\pi\)
\(230\) 1.00000 0.0659380
\(231\) 2.68585 0.176716
\(232\) −0.685846 −0.0450280
\(233\) 15.9572 1.04539 0.522694 0.852520i \(-0.324927\pi\)
0.522694 + 0.852520i \(0.324927\pi\)
\(234\) −0.292731 −0.0191364
\(235\) −10.3503 −0.675177
\(236\) −4.00000 −0.260378
\(237\) 15.3288 0.995716
\(238\) −6.97858 −0.452354
\(239\) 7.27131 0.470342 0.235171 0.971954i \(-0.424435\pi\)
0.235171 + 0.971954i \(0.424435\pi\)
\(240\) 1.00000 0.0645497
\(241\) 3.64973 0.235100 0.117550 0.993067i \(-0.462496\pi\)
0.117550 + 0.993067i \(0.462496\pi\)
\(242\) 3.78623 0.243388
\(243\) −1.00000 −0.0641500
\(244\) 7.37169 0.471924
\(245\) −1.00000 −0.0638877
\(246\) −7.37169 −0.470002
\(247\) 1.28600 0.0818263
\(248\) −8.97858 −0.570140
\(249\) −0.393115 −0.0249127
\(250\) 1.00000 0.0632456
\(251\) −13.6216 −0.859786 −0.429893 0.902880i \(-0.641449\pi\)
−0.429893 + 0.902880i \(0.641449\pi\)
\(252\) −1.00000 −0.0629941
\(253\) 2.68585 0.168858
\(254\) −12.0575 −0.756557
\(255\) −6.97858 −0.437015
\(256\) 1.00000 0.0625000
\(257\) 14.5855 0.909816 0.454908 0.890538i \(-0.349672\pi\)
0.454908 + 0.890538i \(0.349672\pi\)
\(258\) −6.68585 −0.416243
\(259\) 6.00000 0.372822
\(260\) −0.292731 −0.0181544
\(261\) 0.685846 0.0424528
\(262\) 4.00000 0.247121
\(263\) −31.3288 −1.93182 −0.965910 0.258879i \(-0.916647\pi\)
−0.965910 + 0.258879i \(0.916647\pi\)
\(264\) 2.68585 0.165302
\(265\) 6.00000 0.368577
\(266\) 4.39312 0.269359
\(267\) 9.66442 0.591453
\(268\) 1.31415 0.0802747
\(269\) −10.2499 −0.624947 −0.312473 0.949927i \(-0.601157\pi\)
−0.312473 + 0.949927i \(0.601157\pi\)
\(270\) −1.00000 −0.0608581
\(271\) −1.76481 −0.107204 −0.0536022 0.998562i \(-0.517070\pi\)
−0.0536022 + 0.998562i \(0.517070\pi\)
\(272\) −6.97858 −0.423138
\(273\) 0.292731 0.0177169
\(274\) 21.9143 1.32389
\(275\) 2.68585 0.161963
\(276\) −1.00000 −0.0601929
\(277\) 9.47208 0.569122 0.284561 0.958658i \(-0.408152\pi\)
0.284561 + 0.958658i \(0.408152\pi\)
\(278\) −18.7434 −1.12415
\(279\) 8.97858 0.537533
\(280\) −1.00000 −0.0597614
\(281\) 32.8009 1.95674 0.978370 0.206865i \(-0.0663261\pi\)
0.978370 + 0.206865i \(0.0663261\pi\)
\(282\) 10.3503 0.616350
\(283\) −9.03612 −0.537141 −0.268571 0.963260i \(-0.586551\pi\)
−0.268571 + 0.963260i \(0.586551\pi\)
\(284\) 7.27131 0.431473
\(285\) 4.39312 0.260226
\(286\) −0.786230 −0.0464908
\(287\) 7.37169 0.435137
\(288\) −1.00000 −0.0589256
\(289\) 31.7005 1.86474
\(290\) 0.685846 0.0402743
\(291\) 10.2499 0.600859
\(292\) 11.9572 0.699740
\(293\) −13.3288 −0.778680 −0.389340 0.921094i \(-0.627297\pi\)
−0.389340 + 0.921094i \(0.627297\pi\)
\(294\) 1.00000 0.0583212
\(295\) 4.00000 0.232889
\(296\) 6.00000 0.348743
\(297\) −2.68585 −0.155849
\(298\) 16.7434 0.969918
\(299\) 0.292731 0.0169291
\(300\) −1.00000 −0.0577350
\(301\) 6.68585 0.385366
\(302\) 15.3288 0.882076
\(303\) −4.29273 −0.246611
\(304\) 4.39312 0.251962
\(305\) −7.37169 −0.422102
\(306\) 6.97858 0.398939
\(307\) 10.1579 0.579743 0.289872 0.957066i \(-0.406387\pi\)
0.289872 + 0.957066i \(0.406387\pi\)
\(308\) −2.68585 −0.153040
\(309\) 4.58546 0.260858
\(310\) 8.97858 0.509949
\(311\) −5.03612 −0.285572 −0.142786 0.989754i \(-0.545606\pi\)
−0.142786 + 0.989754i \(0.545606\pi\)
\(312\) 0.292731 0.0165726
\(313\) 29.0790 1.64364 0.821820 0.569747i \(-0.192959\pi\)
0.821820 + 0.569747i \(0.192959\pi\)
\(314\) −10.5855 −0.597372
\(315\) 1.00000 0.0563436
\(316\) −15.3288 −0.862315
\(317\) −18.5855 −1.04386 −0.521932 0.852987i \(-0.674788\pi\)
−0.521932 + 0.852987i \(0.674788\pi\)
\(318\) −6.00000 −0.336463
\(319\) 1.84208 0.103137
\(320\) −1.00000 −0.0559017
\(321\) −1.37169 −0.0765604
\(322\) 1.00000 0.0557278
\(323\) −30.6577 −1.70584
\(324\) 1.00000 0.0555556
\(325\) 0.292731 0.0162378
\(326\) 8.00000 0.443079
\(327\) −6.00000 −0.331801
\(328\) 7.37169 0.407034
\(329\) −10.3503 −0.570629
\(330\) −2.68585 −0.147851
\(331\) −0.585462 −0.0321799 −0.0160899 0.999871i \(-0.505122\pi\)
−0.0160899 + 0.999871i \(0.505122\pi\)
\(332\) 0.393115 0.0215750
\(333\) −6.00000 −0.328798
\(334\) −4.19235 −0.229395
\(335\) −1.31415 −0.0717999
\(336\) 1.00000 0.0545545
\(337\) −5.85677 −0.319039 −0.159519 0.987195i \(-0.550994\pi\)
−0.159519 + 0.987195i \(0.550994\pi\)
\(338\) 12.9143 0.702446
\(339\) −15.9572 −0.866674
\(340\) 6.97858 0.378466
\(341\) 24.1151 1.30590
\(342\) −4.39312 −0.237552
\(343\) −1.00000 −0.0539949
\(344\) 6.68585 0.360477
\(345\) 1.00000 0.0538382
\(346\) −11.7648 −0.632480
\(347\) −22.7434 −1.22093 −0.610464 0.792044i \(-0.709017\pi\)
−0.610464 + 0.792044i \(0.709017\pi\)
\(348\) −0.685846 −0.0367652
\(349\) 22.3931 1.19868 0.599338 0.800496i \(-0.295431\pi\)
0.599338 + 0.800496i \(0.295431\pi\)
\(350\) 1.00000 0.0534522
\(351\) −0.292731 −0.0156248
\(352\) −2.68585 −0.143156
\(353\) 18.7862 0.999890 0.499945 0.866057i \(-0.333354\pi\)
0.499945 + 0.866057i \(0.333354\pi\)
\(354\) −4.00000 −0.212598
\(355\) −7.27131 −0.385921
\(356\) −9.66442 −0.512213
\(357\) −6.97858 −0.369345
\(358\) −5.95715 −0.314845
\(359\) −31.5296 −1.66407 −0.832035 0.554724i \(-0.812824\pi\)
−0.832035 + 0.554724i \(0.812824\pi\)
\(360\) 1.00000 0.0527046
\(361\) 0.299461 0.0157611
\(362\) −24.5426 −1.28993
\(363\) 3.78623 0.198726
\(364\) −0.292731 −0.0153433
\(365\) −11.9572 −0.625866
\(366\) 7.37169 0.385325
\(367\) 26.0722 1.36096 0.680480 0.732767i \(-0.261771\pi\)
0.680480 + 0.732767i \(0.261771\pi\)
\(368\) 1.00000 0.0521286
\(369\) −7.37169 −0.383755
\(370\) −6.00000 −0.311925
\(371\) 6.00000 0.311504
\(372\) −8.97858 −0.465518
\(373\) 19.1709 0.992633 0.496316 0.868142i \(-0.334686\pi\)
0.496316 + 0.868142i \(0.334686\pi\)
\(374\) 18.7434 0.969197
\(375\) 1.00000 0.0516398
\(376\) −10.3503 −0.533774
\(377\) 0.200768 0.0103401
\(378\) −1.00000 −0.0514344
\(379\) 6.62831 0.340473 0.170237 0.985403i \(-0.445547\pi\)
0.170237 + 0.985403i \(0.445547\pi\)
\(380\) −4.39312 −0.225362
\(381\) −12.0575 −0.617726
\(382\) 13.9572 0.714110
\(383\) −1.37169 −0.0700902 −0.0350451 0.999386i \(-0.511157\pi\)
−0.0350451 + 0.999386i \(0.511157\pi\)
\(384\) 1.00000 0.0510310
\(385\) 2.68585 0.136883
\(386\) −25.3288 −1.28920
\(387\) −6.68585 −0.339861
\(388\) −10.2499 −0.520359
\(389\) 18.7862 0.952500 0.476250 0.879310i \(-0.341996\pi\)
0.476250 + 0.879310i \(0.341996\pi\)
\(390\) −0.292731 −0.0148230
\(391\) −6.97858 −0.352922
\(392\) −1.00000 −0.0505076
\(393\) 4.00000 0.201773
\(394\) −11.9572 −0.602393
\(395\) 15.3288 0.771278
\(396\) 2.68585 0.134969
\(397\) −17.2797 −0.867245 −0.433622 0.901095i \(-0.642765\pi\)
−0.433622 + 0.901095i \(0.642765\pi\)
\(398\) −15.3288 −0.768366
\(399\) 4.39312 0.219931
\(400\) 1.00000 0.0500000
\(401\) −15.3142 −0.764752 −0.382376 0.924007i \(-0.624894\pi\)
−0.382376 + 0.924007i \(0.624894\pi\)
\(402\) 1.31415 0.0655440
\(403\) 2.62831 0.130925
\(404\) 4.29273 0.213571
\(405\) −1.00000 −0.0496904
\(406\) 0.685846 0.0340380
\(407\) −16.1151 −0.798795
\(408\) −6.97858 −0.345491
\(409\) −12.1579 −0.601171 −0.300585 0.953755i \(-0.597182\pi\)
−0.300585 + 0.953755i \(0.597182\pi\)
\(410\) −7.37169 −0.364062
\(411\) 21.9143 1.08095
\(412\) −4.58546 −0.225909
\(413\) 4.00000 0.196827
\(414\) −1.00000 −0.0491473
\(415\) −0.393115 −0.0192973
\(416\) −0.292731 −0.0143523
\(417\) −18.7434 −0.917867
\(418\) −11.7992 −0.577119
\(419\) 28.8353 1.40870 0.704349 0.709853i \(-0.251239\pi\)
0.704349 + 0.709853i \(0.251239\pi\)
\(420\) −1.00000 −0.0487950
\(421\) 12.0428 0.586932 0.293466 0.955969i \(-0.405191\pi\)
0.293466 + 0.955969i \(0.405191\pi\)
\(422\) −7.41454 −0.360934
\(423\) 10.3503 0.503247
\(424\) 6.00000 0.291386
\(425\) −6.97858 −0.338511
\(426\) 7.27131 0.352296
\(427\) −7.37169 −0.356741
\(428\) 1.37169 0.0663033
\(429\) −0.786230 −0.0379596
\(430\) −6.68585 −0.322420
\(431\) 19.3288 0.931038 0.465519 0.885038i \(-0.345868\pi\)
0.465519 + 0.885038i \(0.345868\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −4.99327 −0.239961 −0.119981 0.992776i \(-0.538283\pi\)
−0.119981 + 0.992776i \(0.538283\pi\)
\(434\) 8.97858 0.430985
\(435\) 0.685846 0.0328838
\(436\) 6.00000 0.287348
\(437\) 4.39312 0.210151
\(438\) 11.9572 0.571335
\(439\) 21.5640 1.02920 0.514598 0.857432i \(-0.327941\pi\)
0.514598 + 0.857432i \(0.327941\pi\)
\(440\) 2.68585 0.128043
\(441\) 1.00000 0.0476190
\(442\) 2.04285 0.0971683
\(443\) 34.6577 1.64664 0.823318 0.567580i \(-0.192120\pi\)
0.823318 + 0.567580i \(0.192120\pi\)
\(444\) 6.00000 0.284747
\(445\) 9.66442 0.458138
\(446\) −19.0790 −0.903415
\(447\) 16.7434 0.791935
\(448\) −1.00000 −0.0472456
\(449\) 18.7862 0.886577 0.443289 0.896379i \(-0.353812\pi\)
0.443289 + 0.896379i \(0.353812\pi\)
\(450\) −1.00000 −0.0471405
\(451\) −19.7992 −0.932309
\(452\) 15.9572 0.750561
\(453\) 15.3288 0.720212
\(454\) −12.9786 −0.609115
\(455\) 0.292731 0.0137234
\(456\) 4.39312 0.205726
\(457\) 11.8139 0.552632 0.276316 0.961067i \(-0.410886\pi\)
0.276316 + 0.961067i \(0.410886\pi\)
\(458\) −18.7862 −0.877823
\(459\) 6.97858 0.325732
\(460\) −1.00000 −0.0466252
\(461\) 21.4637 0.999662 0.499831 0.866123i \(-0.333395\pi\)
0.499831 + 0.866123i \(0.333395\pi\)
\(462\) −2.68585 −0.124957
\(463\) 40.6430 1.88884 0.944420 0.328741i \(-0.106624\pi\)
0.944420 + 0.328741i \(0.106624\pi\)
\(464\) 0.685846 0.0318396
\(465\) 8.97858 0.416372
\(466\) −15.9572 −0.739201
\(467\) −28.3074 −1.30991 −0.654956 0.755667i \(-0.727313\pi\)
−0.654956 + 0.755667i \(0.727313\pi\)
\(468\) 0.292731 0.0135315
\(469\) −1.31415 −0.0606820
\(470\) 10.3503 0.477422
\(471\) −10.5855 −0.487752
\(472\) 4.00000 0.184115
\(473\) −17.9572 −0.825671
\(474\) −15.3288 −0.704077
\(475\) 4.39312 0.201570
\(476\) 6.97858 0.319863
\(477\) −6.00000 −0.274721
\(478\) −7.27131 −0.332582
\(479\) 14.1579 0.646892 0.323446 0.946247i \(-0.395159\pi\)
0.323446 + 0.946247i \(0.395159\pi\)
\(480\) −1.00000 −0.0456435
\(481\) −1.75639 −0.0800843
\(482\) −3.64973 −0.166241
\(483\) 1.00000 0.0455016
\(484\) −3.78623 −0.172101
\(485\) 10.2499 0.465423
\(486\) 1.00000 0.0453609
\(487\) 15.4721 0.701107 0.350553 0.936543i \(-0.385994\pi\)
0.350553 + 0.936543i \(0.385994\pi\)
\(488\) −7.37169 −0.333701
\(489\) 8.00000 0.361773
\(490\) 1.00000 0.0451754
\(491\) 23.1281 1.04376 0.521878 0.853020i \(-0.325232\pi\)
0.521878 + 0.853020i \(0.325232\pi\)
\(492\) 7.37169 0.332342
\(493\) −4.78623 −0.215561
\(494\) −1.28600 −0.0578599
\(495\) −2.68585 −0.120720
\(496\) 8.97858 0.403150
\(497\) −7.27131 −0.326163
\(498\) 0.393115 0.0176159
\(499\) −24.5855 −1.10060 −0.550298 0.834968i \(-0.685486\pi\)
−0.550298 + 0.834968i \(0.685486\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −4.19235 −0.187300
\(502\) 13.6216 0.607961
\(503\) 23.4145 1.04400 0.522001 0.852945i \(-0.325186\pi\)
0.522001 + 0.852945i \(0.325186\pi\)
\(504\) 1.00000 0.0445435
\(505\) −4.29273 −0.191024
\(506\) −2.68585 −0.119400
\(507\) 12.9143 0.573545
\(508\) 12.0575 0.534967
\(509\) −14.4507 −0.640514 −0.320257 0.947331i \(-0.603769\pi\)
−0.320257 + 0.947331i \(0.603769\pi\)
\(510\) 6.97858 0.309017
\(511\) −11.9572 −0.528953
\(512\) −1.00000 −0.0441942
\(513\) −4.39312 −0.193961
\(514\) −14.5855 −0.643337
\(515\) 4.58546 0.202060
\(516\) 6.68585 0.294328
\(517\) 27.7992 1.22261
\(518\) −6.00000 −0.263625
\(519\) −11.7648 −0.516418
\(520\) 0.292731 0.0128371
\(521\) −8.09196 −0.354515 −0.177258 0.984164i \(-0.556723\pi\)
−0.177258 + 0.984164i \(0.556723\pi\)
\(522\) −0.685846 −0.0300187
\(523\) 24.3650 1.06541 0.532703 0.846302i \(-0.321176\pi\)
0.532703 + 0.846302i \(0.321176\pi\)
\(524\) −4.00000 −0.174741
\(525\) 1.00000 0.0436436
\(526\) 31.3288 1.36600
\(527\) −62.6577 −2.72941
\(528\) −2.68585 −0.116886
\(529\) 1.00000 0.0434783
\(530\) −6.00000 −0.260623
\(531\) −4.00000 −0.173585
\(532\) −4.39312 −0.190466
\(533\) −2.15792 −0.0934700
\(534\) −9.66442 −0.418220
\(535\) −1.37169 −0.0593034
\(536\) −1.31415 −0.0567628
\(537\) −5.95715 −0.257070
\(538\) 10.2499 0.441904
\(539\) 2.68585 0.115688
\(540\) 1.00000 0.0430331
\(541\) 32.4569 1.39543 0.697716 0.716374i \(-0.254200\pi\)
0.697716 + 0.716374i \(0.254200\pi\)
\(542\) 1.76481 0.0758050
\(543\) −24.5426 −1.05322
\(544\) 6.97858 0.299204
\(545\) −6.00000 −0.257012
\(546\) −0.292731 −0.0125277
\(547\) −14.8291 −0.634046 −0.317023 0.948418i \(-0.602683\pi\)
−0.317023 + 0.948418i \(0.602683\pi\)
\(548\) −21.9143 −0.936133
\(549\) 7.37169 0.314616
\(550\) −2.68585 −0.114525
\(551\) 3.01300 0.128358
\(552\) 1.00000 0.0425628
\(553\) 15.3288 0.651849
\(554\) −9.47208 −0.402430
\(555\) −6.00000 −0.254686
\(556\) 18.7434 0.794897
\(557\) 44.7434 1.89584 0.947919 0.318511i \(-0.103183\pi\)
0.947919 + 0.318511i \(0.103183\pi\)
\(558\) −8.97858 −0.380093
\(559\) −1.95715 −0.0827788
\(560\) 1.00000 0.0422577
\(561\) 18.7434 0.791346
\(562\) −32.8009 −1.38362
\(563\) 35.1365 1.48083 0.740413 0.672152i \(-0.234630\pi\)
0.740413 + 0.672152i \(0.234630\pi\)
\(564\) −10.3503 −0.435825
\(565\) −15.9572 −0.671323
\(566\) 9.03612 0.379816
\(567\) −1.00000 −0.0419961
\(568\) −7.27131 −0.305097
\(569\) 39.3435 1.64937 0.824683 0.565595i \(-0.191353\pi\)
0.824683 + 0.565595i \(0.191353\pi\)
\(570\) −4.39312 −0.184007
\(571\) 36.4998 1.52747 0.763734 0.645531i \(-0.223364\pi\)
0.763734 + 0.645531i \(0.223364\pi\)
\(572\) 0.786230 0.0328739
\(573\) 13.9572 0.583068
\(574\) −7.37169 −0.307688
\(575\) 1.00000 0.0417029
\(576\) 1.00000 0.0416667
\(577\) −18.5855 −0.773723 −0.386861 0.922138i \(-0.626441\pi\)
−0.386861 + 0.922138i \(0.626441\pi\)
\(578\) −31.7005 −1.31857
\(579\) −25.3288 −1.05263
\(580\) −0.685846 −0.0284782
\(581\) −0.393115 −0.0163092
\(582\) −10.2499 −0.424871
\(583\) −16.1151 −0.667418
\(584\) −11.9572 −0.494791
\(585\) −0.292731 −0.0121029
\(586\) 13.3288 0.550610
\(587\) −7.91431 −0.326658 −0.163329 0.986572i \(-0.552223\pi\)
−0.163329 + 0.986572i \(0.552223\pi\)
\(588\) −1.00000 −0.0412393
\(589\) 39.4439 1.62526
\(590\) −4.00000 −0.164677
\(591\) −11.9572 −0.491852
\(592\) −6.00000 −0.246598
\(593\) 2.67115 0.109691 0.0548456 0.998495i \(-0.482533\pi\)
0.0548456 + 0.998495i \(0.482533\pi\)
\(594\) 2.68585 0.110202
\(595\) −6.97858 −0.286094
\(596\) −16.7434 −0.685836
\(597\) −15.3288 −0.627368
\(598\) −0.292731 −0.0119707
\(599\) −4.14323 −0.169288 −0.0846439 0.996411i \(-0.526975\pi\)
−0.0846439 + 0.996411i \(0.526975\pi\)
\(600\) 1.00000 0.0408248
\(601\) 23.3717 0.953351 0.476676 0.879079i \(-0.341842\pi\)
0.476676 + 0.879079i \(0.341842\pi\)
\(602\) −6.68585 −0.272495
\(603\) 1.31415 0.0535165
\(604\) −15.3288 −0.623722
\(605\) 3.78623 0.153932
\(606\) 4.29273 0.174380
\(607\) −0.450654 −0.0182915 −0.00914573 0.999958i \(-0.502911\pi\)
−0.00914573 + 0.999958i \(0.502911\pi\)
\(608\) −4.39312 −0.178164
\(609\) 0.685846 0.0277919
\(610\) 7.37169 0.298471
\(611\) 3.02984 0.122574
\(612\) −6.97858 −0.282092
\(613\) −29.4439 −1.18923 −0.594614 0.804011i \(-0.702695\pi\)
−0.594614 + 0.804011i \(0.702695\pi\)
\(614\) −10.1579 −0.409940
\(615\) −7.37169 −0.297255
\(616\) 2.68585 0.108216
\(617\) −27.2860 −1.09849 −0.549247 0.835660i \(-0.685085\pi\)
−0.549247 + 0.835660i \(0.685085\pi\)
\(618\) −4.58546 −0.184454
\(619\) −23.1365 −0.929934 −0.464967 0.885328i \(-0.653934\pi\)
−0.464967 + 0.885328i \(0.653934\pi\)
\(620\) −8.97858 −0.360588
\(621\) −1.00000 −0.0401286
\(622\) 5.03612 0.201930
\(623\) 9.66442 0.387197
\(624\) −0.292731 −0.0117186
\(625\) 1.00000 0.0400000
\(626\) −29.0790 −1.16223
\(627\) −11.7992 −0.471216
\(628\) 10.5855 0.422406
\(629\) 41.8715 1.66952
\(630\) −1.00000 −0.0398410
\(631\) 15.3288 0.610232 0.305116 0.952315i \(-0.401305\pi\)
0.305116 + 0.952315i \(0.401305\pi\)
\(632\) 15.3288 0.609749
\(633\) −7.41454 −0.294701
\(634\) 18.5855 0.738123
\(635\) −12.0575 −0.478489
\(636\) 6.00000 0.237915
\(637\) 0.292731 0.0115984
\(638\) −1.84208 −0.0729285
\(639\) 7.27131 0.287649
\(640\) 1.00000 0.0395285
\(641\) 39.6300 1.56529 0.782645 0.622468i \(-0.213870\pi\)
0.782645 + 0.622468i \(0.213870\pi\)
\(642\) 1.37169 0.0541364
\(643\) −7.07896 −0.279167 −0.139583 0.990210i \(-0.544576\pi\)
−0.139583 + 0.990210i \(0.544576\pi\)
\(644\) −1.00000 −0.0394055
\(645\) −6.68585 −0.263255
\(646\) 30.6577 1.20621
\(647\) 2.73496 0.107522 0.0537612 0.998554i \(-0.482879\pi\)
0.0537612 + 0.998554i \(0.482879\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −10.7434 −0.421715
\(650\) −0.292731 −0.0114819
\(651\) 8.97858 0.351898
\(652\) −8.00000 −0.313304
\(653\) 8.54262 0.334298 0.167149 0.985932i \(-0.446544\pi\)
0.167149 + 0.985932i \(0.446544\pi\)
\(654\) 6.00000 0.234619
\(655\) 4.00000 0.156293
\(656\) −7.37169 −0.287816
\(657\) 11.9572 0.466493
\(658\) 10.3503 0.403496
\(659\) 30.2155 1.17703 0.588514 0.808487i \(-0.299713\pi\)
0.588514 + 0.808487i \(0.299713\pi\)
\(660\) 2.68585 0.104546
\(661\) −41.1281 −1.59970 −0.799848 0.600202i \(-0.795087\pi\)
−0.799848 + 0.600202i \(0.795087\pi\)
\(662\) 0.585462 0.0227546
\(663\) 2.04285 0.0793376
\(664\) −0.393115 −0.0152558
\(665\) 4.39312 0.170358
\(666\) 6.00000 0.232495
\(667\) 0.685846 0.0265561
\(668\) 4.19235 0.162207
\(669\) −19.0790 −0.737635
\(670\) 1.31415 0.0507702
\(671\) 19.7992 0.764341
\(672\) −1.00000 −0.0385758
\(673\) 34.7862 1.34091 0.670455 0.741950i \(-0.266099\pi\)
0.670455 + 0.741950i \(0.266099\pi\)
\(674\) 5.85677 0.225594
\(675\) −1.00000 −0.0384900
\(676\) −12.9143 −0.496704
\(677\) −16.7434 −0.643501 −0.321750 0.946825i \(-0.604271\pi\)
−0.321750 + 0.946825i \(0.604271\pi\)
\(678\) 15.9572 0.612831
\(679\) 10.2499 0.393354
\(680\) −6.97858 −0.267616
\(681\) −12.9786 −0.497340
\(682\) −24.1151 −0.923414
\(683\) 15.9143 0.608944 0.304472 0.952521i \(-0.401520\pi\)
0.304472 + 0.952521i \(0.401520\pi\)
\(684\) 4.39312 0.167975
\(685\) 21.9143 0.837303
\(686\) 1.00000 0.0381802
\(687\) −18.7862 −0.716739
\(688\) −6.68585 −0.254895
\(689\) −1.75639 −0.0669130
\(690\) −1.00000 −0.0380693
\(691\) −31.4439 −1.19618 −0.598092 0.801428i \(-0.704074\pi\)
−0.598092 + 0.801428i \(0.704074\pi\)
\(692\) 11.7648 0.447231
\(693\) −2.68585 −0.102027
\(694\) 22.7434 0.863327
\(695\) −18.7434 −0.710977
\(696\) 0.685846 0.0259969
\(697\) 51.4439 1.94858
\(698\) −22.3931 −0.847592
\(699\) −15.9572 −0.603555
\(700\) −1.00000 −0.0377964
\(701\) −8.62831 −0.325887 −0.162943 0.986635i \(-0.552099\pi\)
−0.162943 + 0.986635i \(0.552099\pi\)
\(702\) 0.292731 0.0110484
\(703\) −26.3587 −0.994137
\(704\) 2.68585 0.101227
\(705\) 10.3503 0.389814
\(706\) −18.7862 −0.707029
\(707\) −4.29273 −0.161445
\(708\) 4.00000 0.150329
\(709\) −5.12808 −0.192589 −0.0962945 0.995353i \(-0.530699\pi\)
−0.0962945 + 0.995353i \(0.530699\pi\)
\(710\) 7.27131 0.272887
\(711\) −15.3288 −0.574877
\(712\) 9.66442 0.362190
\(713\) 8.97858 0.336250
\(714\) 6.97858 0.261167
\(715\) −0.786230 −0.0294033
\(716\) 5.95715 0.222629
\(717\) −7.27131 −0.271552
\(718\) 31.5296 1.17667
\(719\) −37.5359 −1.39985 −0.699926 0.714215i \(-0.746784\pi\)
−0.699926 + 0.714215i \(0.746784\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 4.58546 0.170772
\(722\) −0.299461 −0.0111448
\(723\) −3.64973 −0.135735
\(724\) 24.5426 0.912119
\(725\) 0.685846 0.0254717
\(726\) −3.78623 −0.140520
\(727\) −31.3288 −1.16192 −0.580961 0.813931i \(-0.697323\pi\)
−0.580961 + 0.813931i \(0.697323\pi\)
\(728\) 0.292731 0.0108493
\(729\) 1.00000 0.0370370
\(730\) 11.9572 0.442554
\(731\) 46.6577 1.72570
\(732\) −7.37169 −0.272466
\(733\) 6.78623 0.250655 0.125328 0.992115i \(-0.460002\pi\)
0.125328 + 0.992115i \(0.460002\pi\)
\(734\) −26.0722 −0.962344
\(735\) 1.00000 0.0368856
\(736\) −1.00000 −0.0368605
\(737\) 3.52962 0.130015
\(738\) 7.37169 0.271356
\(739\) 22.7434 0.836629 0.418314 0.908302i \(-0.362621\pi\)
0.418314 + 0.908302i \(0.362621\pi\)
\(740\) 6.00000 0.220564
\(741\) −1.28600 −0.0472424
\(742\) −6.00000 −0.220267
\(743\) 24.9013 0.913540 0.456770 0.889585i \(-0.349006\pi\)
0.456770 + 0.889585i \(0.349006\pi\)
\(744\) 8.97858 0.329171
\(745\) 16.7434 0.613430
\(746\) −19.1709 −0.701897
\(747\) 0.393115 0.0143833
\(748\) −18.7434 −0.685326
\(749\) −1.37169 −0.0501205
\(750\) −1.00000 −0.0365148
\(751\) −37.0852 −1.35326 −0.676630 0.736323i \(-0.736560\pi\)
−0.676630 + 0.736323i \(0.736560\pi\)
\(752\) 10.3503 0.377435
\(753\) 13.6216 0.496398
\(754\) −0.200768 −0.00731155
\(755\) 15.3288 0.557874
\(756\) 1.00000 0.0363696
\(757\) −7.95715 −0.289208 −0.144604 0.989490i \(-0.546191\pi\)
−0.144604 + 0.989490i \(0.546191\pi\)
\(758\) −6.62831 −0.240751
\(759\) −2.68585 −0.0974900
\(760\) 4.39312 0.159355
\(761\) 0.243614 0.00883101 0.00441550 0.999990i \(-0.498594\pi\)
0.00441550 + 0.999990i \(0.498594\pi\)
\(762\) 12.0575 0.436799
\(763\) −6.00000 −0.217215
\(764\) −13.9572 −0.504952
\(765\) 6.97858 0.252311
\(766\) 1.37169 0.0495613
\(767\) −1.17092 −0.0422796
\(768\) −1.00000 −0.0360844
\(769\) 39.6791 1.43087 0.715433 0.698682i \(-0.246230\pi\)
0.715433 + 0.698682i \(0.246230\pi\)
\(770\) −2.68585 −0.0967912
\(771\) −14.5855 −0.525283
\(772\) 25.3288 0.911605
\(773\) −22.1151 −0.795424 −0.397712 0.917510i \(-0.630196\pi\)
−0.397712 + 0.917510i \(0.630196\pi\)
\(774\) 6.68585 0.240318
\(775\) 8.97858 0.322520
\(776\) 10.2499 0.367949
\(777\) −6.00000 −0.215249
\(778\) −18.7862 −0.673519
\(779\) −32.3847 −1.16030
\(780\) 0.292731 0.0104815
\(781\) 19.5296 0.698825
\(782\) 6.97858 0.249553
\(783\) −0.685846 −0.0245101
\(784\) 1.00000 0.0357143
\(785\) −10.5855 −0.377811
\(786\) −4.00000 −0.142675
\(787\) −6.67742 −0.238024 −0.119012 0.992893i \(-0.537973\pi\)
−0.119012 + 0.992893i \(0.537973\pi\)
\(788\) 11.9572 0.425956
\(789\) 31.3288 1.11534
\(790\) −15.3288 −0.545376
\(791\) −15.9572 −0.567371
\(792\) −2.68585 −0.0954374
\(793\) 2.15792 0.0766301
\(794\) 17.2797 0.613235
\(795\) −6.00000 −0.212798
\(796\) 15.3288 0.543317
\(797\) −21.2138 −0.751430 −0.375715 0.926735i \(-0.622603\pi\)
−0.375715 + 0.926735i \(0.622603\pi\)
\(798\) −4.39312 −0.155515
\(799\) −72.2302 −2.55532
\(800\) −1.00000 −0.0353553
\(801\) −9.66442 −0.341476
\(802\) 15.3142 0.540762
\(803\) 32.1151 1.13332
\(804\) −1.31415 −0.0463466
\(805\) 1.00000 0.0352454
\(806\) −2.62831 −0.0925782
\(807\) 10.2499 0.360813
\(808\) −4.29273 −0.151018
\(809\) −25.2432 −0.887502 −0.443751 0.896150i \(-0.646353\pi\)
−0.443751 + 0.896150i \(0.646353\pi\)
\(810\) 1.00000 0.0351364
\(811\) 33.1709 1.16479 0.582394 0.812906i \(-0.302116\pi\)
0.582394 + 0.812906i \(0.302116\pi\)
\(812\) −0.685846 −0.0240685
\(813\) 1.76481 0.0618945
\(814\) 16.1151 0.564833
\(815\) 8.00000 0.280228
\(816\) 6.97858 0.244299
\(817\) −29.3717 −1.02759
\(818\) 12.1579 0.425092
\(819\) −0.292731 −0.0102288
\(820\) 7.37169 0.257431
\(821\) −51.7282 −1.80533 −0.902664 0.430346i \(-0.858391\pi\)
−0.902664 + 0.430346i \(0.858391\pi\)
\(822\) −21.9143 −0.764349
\(823\) 6.80092 0.237065 0.118533 0.992950i \(-0.462181\pi\)
0.118533 + 0.992950i \(0.462181\pi\)
\(824\) 4.58546 0.159742
\(825\) −2.68585 −0.0935092
\(826\) −4.00000 −0.139178
\(827\) 29.2860 1.01837 0.509187 0.860656i \(-0.329946\pi\)
0.509187 + 0.860656i \(0.329946\pi\)
\(828\) 1.00000 0.0347524
\(829\) −39.8799 −1.38509 −0.692543 0.721377i \(-0.743510\pi\)
−0.692543 + 0.721377i \(0.743510\pi\)
\(830\) 0.393115 0.0136452
\(831\) −9.47208 −0.328583
\(832\) 0.292731 0.0101486
\(833\) −6.97858 −0.241793
\(834\) 18.7434 0.649030
\(835\) −4.19235 −0.145082
\(836\) 11.7992 0.408085
\(837\) −8.97858 −0.310345
\(838\) −28.8353 −0.996101
\(839\) 16.4015 0.566244 0.283122 0.959084i \(-0.408630\pi\)
0.283122 + 0.959084i \(0.408630\pi\)
\(840\) 1.00000 0.0345033
\(841\) −28.5296 −0.983780
\(842\) −12.0428 −0.415024
\(843\) −32.8009 −1.12972
\(844\) 7.41454 0.255219
\(845\) 12.9143 0.444266
\(846\) −10.3503 −0.355850
\(847\) 3.78623 0.130096
\(848\) −6.00000 −0.206041
\(849\) 9.03612 0.310119
\(850\) 6.97858 0.239363
\(851\) −6.00000 −0.205677
\(852\) −7.27131 −0.249111
\(853\) −18.4507 −0.631738 −0.315869 0.948803i \(-0.602296\pi\)
−0.315869 + 0.948803i \(0.602296\pi\)
\(854\) 7.37169 0.252254
\(855\) −4.39312 −0.150241
\(856\) −1.37169 −0.0468835
\(857\) 19.9572 0.681723 0.340862 0.940113i \(-0.389281\pi\)
0.340862 + 0.940113i \(0.389281\pi\)
\(858\) 0.786230 0.0268415
\(859\) −7.21377 −0.246131 −0.123065 0.992399i \(-0.539272\pi\)
−0.123065 + 0.992399i \(0.539272\pi\)
\(860\) 6.68585 0.227985
\(861\) −7.37169 −0.251227
\(862\) −19.3288 −0.658343
\(863\) −44.8156 −1.52554 −0.762771 0.646669i \(-0.776161\pi\)
−0.762771 + 0.646669i \(0.776161\pi\)
\(864\) 1.00000 0.0340207
\(865\) −11.7648 −0.400015
\(866\) 4.99327 0.169678
\(867\) −31.7005 −1.07661
\(868\) −8.97858 −0.304753
\(869\) −41.1709 −1.39663
\(870\) −0.685846 −0.0232524
\(871\) 0.384694 0.0130348
\(872\) −6.00000 −0.203186
\(873\) −10.2499 −0.346906
\(874\) −4.39312 −0.148599
\(875\) 1.00000 0.0338062
\(876\) −11.9572 −0.403995
\(877\) −47.1428 −1.59190 −0.795949 0.605364i \(-0.793028\pi\)
−0.795949 + 0.605364i \(0.793028\pi\)
\(878\) −21.5640 −0.727751
\(879\) 13.3288 0.449571
\(880\) −2.68585 −0.0905399
\(881\) 1.46365 0.0493118 0.0246559 0.999696i \(-0.492151\pi\)
0.0246559 + 0.999696i \(0.492151\pi\)
\(882\) −1.00000 −0.0336718
\(883\) −49.4011 −1.66248 −0.831239 0.555915i \(-0.812368\pi\)
−0.831239 + 0.555915i \(0.812368\pi\)
\(884\) −2.04285 −0.0687083
\(885\) −4.00000 −0.134459
\(886\) −34.6577 −1.16435
\(887\) 23.0508 0.773970 0.386985 0.922086i \(-0.373516\pi\)
0.386985 + 0.922086i \(0.373516\pi\)
\(888\) −6.00000 −0.201347
\(889\) −12.0575 −0.404397
\(890\) −9.66442 −0.323952
\(891\) 2.68585 0.0899792
\(892\) 19.0790 0.638811
\(893\) 45.4699 1.52159
\(894\) −16.7434 −0.559983
\(895\) −5.95715 −0.199126
\(896\) 1.00000 0.0334077
\(897\) −0.292731 −0.00977400
\(898\) −18.7862 −0.626905
\(899\) 6.15792 0.205378
\(900\) 1.00000 0.0333333
\(901\) 41.8715 1.39494
\(902\) 19.7992 0.659242
\(903\) −6.68585 −0.222491
\(904\) −15.9572 −0.530727
\(905\) −24.5426 −0.815824
\(906\) −15.3288 −0.509267
\(907\) −42.2155 −1.40174 −0.700871 0.713288i \(-0.747205\pi\)
−0.700871 + 0.713288i \(0.747205\pi\)
\(908\) 12.9786 0.430709
\(909\) 4.29273 0.142381
\(910\) −0.292731 −0.00970394
\(911\) −44.9013 −1.48765 −0.743823 0.668376i \(-0.766990\pi\)
−0.743823 + 0.668376i \(0.766990\pi\)
\(912\) −4.39312 −0.145471
\(913\) 1.05585 0.0349434
\(914\) −11.8139 −0.390770
\(915\) 7.37169 0.243701
\(916\) 18.7862 0.620715
\(917\) 4.00000 0.132092
\(918\) −6.97858 −0.230327
\(919\) −46.6577 −1.53910 −0.769548 0.638589i \(-0.779518\pi\)
−0.769548 + 0.638589i \(0.779518\pi\)
\(920\) 1.00000 0.0329690
\(921\) −10.1579 −0.334715
\(922\) −21.4637 −0.706868
\(923\) 2.12854 0.0700617
\(924\) 2.68585 0.0883579
\(925\) −6.00000 −0.197279
\(926\) −40.6430 −1.33561
\(927\) −4.58546 −0.150606
\(928\) −0.685846 −0.0225140
\(929\) 6.00000 0.196854 0.0984268 0.995144i \(-0.468619\pi\)
0.0984268 + 0.995144i \(0.468619\pi\)
\(930\) −8.97858 −0.294419
\(931\) 4.39312 0.143979
\(932\) 15.9572 0.522694
\(933\) 5.03612 0.164875
\(934\) 28.3074 0.926247
\(935\) 18.7434 0.612974
\(936\) −0.292731 −0.00956821
\(937\) −41.4637 −1.35456 −0.677279 0.735726i \(-0.736841\pi\)
−0.677279 + 0.735726i \(0.736841\pi\)
\(938\) 1.31415 0.0429086
\(939\) −29.0790 −0.948956
\(940\) −10.3503 −0.337589
\(941\) −4.82908 −0.157423 −0.0787117 0.996897i \(-0.525081\pi\)
−0.0787117 + 0.996897i \(0.525081\pi\)
\(942\) 10.5855 0.344893
\(943\) −7.37169 −0.240055
\(944\) −4.00000 −0.130189
\(945\) −1.00000 −0.0325300
\(946\) 17.9572 0.583837
\(947\) 1.75639 0.0570749 0.0285374 0.999593i \(-0.490915\pi\)
0.0285374 + 0.999593i \(0.490915\pi\)
\(948\) 15.3288 0.497858
\(949\) 3.50023 0.113622
\(950\) −4.39312 −0.142531
\(951\) 18.5855 0.602675
\(952\) −6.97858 −0.226177
\(953\) 1.12808 0.0365420 0.0182710 0.999833i \(-0.494184\pi\)
0.0182710 + 0.999833i \(0.494184\pi\)
\(954\) 6.00000 0.194257
\(955\) 13.9572 0.451643
\(956\) 7.27131 0.235171
\(957\) −1.84208 −0.0595459
\(958\) −14.1579 −0.457422
\(959\) 21.9143 0.707650
\(960\) 1.00000 0.0322749
\(961\) 49.6148 1.60048
\(962\) 1.75639 0.0566282
\(963\) 1.37169 0.0442022
\(964\) 3.64973 0.117550
\(965\) −25.3288 −0.815364
\(966\) −1.00000 −0.0321745
\(967\) −56.3565 −1.81230 −0.906152 0.422952i \(-0.860994\pi\)
−0.906152 + 0.422952i \(0.860994\pi\)
\(968\) 3.78623 0.121694
\(969\) 30.6577 0.984867
\(970\) −10.2499 −0.329104
\(971\) 26.2070 0.841024 0.420512 0.907287i \(-0.361850\pi\)
0.420512 + 0.907287i \(0.361850\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −18.7434 −0.600885
\(974\) −15.4721 −0.495757
\(975\) −0.292731 −0.00937489
\(976\) 7.37169 0.235962
\(977\) 23.4574 0.750468 0.375234 0.926930i \(-0.377562\pi\)
0.375234 + 0.926930i \(0.377562\pi\)
\(978\) −8.00000 −0.255812
\(979\) −25.9572 −0.829594
\(980\) −1.00000 −0.0319438
\(981\) 6.00000 0.191565
\(982\) −23.1281 −0.738047
\(983\) −0.200768 −0.00640352 −0.00320176 0.999995i \(-0.501019\pi\)
−0.00320176 + 0.999995i \(0.501019\pi\)
\(984\) −7.37169 −0.235001
\(985\) −11.9572 −0.380987
\(986\) 4.78623 0.152425
\(987\) 10.3503 0.329453
\(988\) 1.28600 0.0409131
\(989\) −6.68585 −0.212598
\(990\) 2.68585 0.0853618
\(991\) −9.84208 −0.312644 −0.156322 0.987706i \(-0.549964\pi\)
−0.156322 + 0.987706i \(0.549964\pi\)
\(992\) −8.97858 −0.285070
\(993\) 0.585462 0.0185791
\(994\) 7.27131 0.230632
\(995\) −15.3288 −0.485957
\(996\) −0.393115 −0.0124563
\(997\) −23.7073 −0.750817 −0.375408 0.926860i \(-0.622498\pi\)
−0.375408 + 0.926860i \(0.622498\pi\)
\(998\) 24.5855 0.778239
\(999\) 6.00000 0.189832
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 4830.2.a.bw.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4830.2.a.bw.1.3 3 1.1 even 1 trivial