Properties

Label 5586.2.a.ca.1.3
Level $5586$
Weight $2$
Character 5586.1
Self dual yes
Analytic conductor $44.604$
Analytic rank $1$
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] = [5586,2,Mod(1,5586)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5586, 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("5586.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5586 = 2 \cdot 3 \cdot 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5586.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: \(44.6044345691\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.202932.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 12x^{2} + 12x + 33 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 798)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.49111\) of defining polynomial
Character \(\chi\) \(=\) 5586.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.49111 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.49111 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +1.49111 q^{10} +0.794368 q^{11} -1.00000 q^{12} -2.00000 q^{13} -1.49111 q^{15} +1.00000 q^{16} -4.49111 q^{17} +1.00000 q^{18} -1.00000 q^{19} +1.49111 q^{20} +0.794368 q^{22} +3.15514 q^{23} -1.00000 q^{24} -2.77659 q^{25} -2.00000 q^{26} -1.00000 q^{27} -6.23498 q^{29} -1.49111 q^{30} -9.01157 q^{31} +1.00000 q^{32} -0.794368 q^{33} -4.49111 q^{34} +1.00000 q^{36} -2.49111 q^{37} -1.00000 q^{38} +2.00000 q^{39} +1.49111 q^{40} +4.18449 q^{41} -4.94951 q^{43} +0.794368 q^{44} +1.49111 q^{45} +3.15514 q^{46} -11.9106 q^{47} -1.00000 q^{48} -2.77659 q^{50} +4.49111 q^{51} -2.00000 q^{52} -5.42284 q^{53} -1.00000 q^{54} +1.18449 q^{55} +1.00000 q^{57} -6.23498 q^{58} -11.8093 q^{59} -1.49111 q^{60} +12.5825 q^{61} -9.01157 q^{62} +1.00000 q^{64} -2.98222 q^{65} -0.794368 q^{66} +16.0231 q^{67} -4.49111 q^{68} -3.15514 q^{69} +1.47333 q^{71} +1.00000 q^{72} +1.23498 q^{73} -2.49111 q^{74} +2.77659 q^{75} -1.00000 q^{76} +2.00000 q^{78} -2.33597 q^{79} +1.49111 q^{80} +1.00000 q^{81} +4.18449 q^{82} +1.57096 q^{83} -6.69674 q^{85} -4.94951 q^{86} +6.23498 q^{87} +0.794368 q^{88} -2.28548 q^{89} +1.49111 q^{90} +3.15514 q^{92} +9.01157 q^{93} -11.9106 q^{94} -1.49111 q^{95} -1.00000 q^{96} -17.2971 q^{97} +0.794368 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} - 4 q^{3} + 4 q^{4} - 2 q^{5} - 4 q^{6} + 4 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{2} - 4 q^{3} + 4 q^{4} - 2 q^{5} - 4 q^{6} + 4 q^{8} + 4 q^{9} - 2 q^{10} - 4 q^{12} - 8 q^{13} + 2 q^{15} + 4 q^{16} - 10 q^{17} + 4 q^{18} - 4 q^{19} - 2 q^{20} + 7 q^{23} - 4 q^{24} + 8 q^{25} - 8 q^{26} - 4 q^{27} - 5 q^{29} + 2 q^{30} + 3 q^{31} + 4 q^{32} - 10 q^{34} + 4 q^{36} - 2 q^{37} - 4 q^{38} + 8 q^{39} - 2 q^{40} - 12 q^{41} - 11 q^{43} - 2 q^{45} + 7 q^{46} + 9 q^{47} - 4 q^{48} + 8 q^{50} + 10 q^{51} - 8 q^{52} + 11 q^{53} - 4 q^{54} - 24 q^{55} + 4 q^{57} - 5 q^{58} - 21 q^{59} + 2 q^{60} - 11 q^{61} + 3 q^{62} + 4 q^{64} + 4 q^{65} - 14 q^{67} - 10 q^{68} - 7 q^{69} - 18 q^{71} + 4 q^{72} - 15 q^{73} - 2 q^{74} - 8 q^{75} - 4 q^{76} + 8 q^{78} - 7 q^{79} - 2 q^{80} + 4 q^{81} - 12 q^{82} - 16 q^{83} - 22 q^{85} - 11 q^{86} + 5 q^{87} + 2 q^{89} - 2 q^{90} + 7 q^{92} - 3 q^{93} + 9 q^{94} + 2 q^{95} - 4 q^{96} - 19 q^{97}+O(q^{100}) \) 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
\(5\) 1.49111 0.666845 0.333422 0.942778i \(-0.391796\pi\)
0.333422 + 0.942778i \(0.391796\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 1.49111 0.471531
\(11\) 0.794368 0.239511 0.119755 0.992803i \(-0.461789\pi\)
0.119755 + 0.992803i \(0.461789\pi\)
\(12\) −1.00000 −0.288675
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) −1.49111 −0.385003
\(16\) 1.00000 0.250000
\(17\) −4.49111 −1.08925 −0.544627 0.838678i \(-0.683329\pi\)
−0.544627 + 0.838678i \(0.683329\pi\)
\(18\) 1.00000 0.235702
\(19\) −1.00000 −0.229416
\(20\) 1.49111 0.333422
\(21\) 0 0
\(22\) 0.794368 0.169360
\(23\) 3.15514 0.657892 0.328946 0.944349i \(-0.393307\pi\)
0.328946 + 0.944349i \(0.393307\pi\)
\(24\) −1.00000 −0.204124
\(25\) −2.77659 −0.555318
\(26\) −2.00000 −0.392232
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −6.23498 −1.15781 −0.578904 0.815396i \(-0.696519\pi\)
−0.578904 + 0.815396i \(0.696519\pi\)
\(30\) −1.49111 −0.272238
\(31\) −9.01157 −1.61853 −0.809263 0.587446i \(-0.800133\pi\)
−0.809263 + 0.587446i \(0.800133\pi\)
\(32\) 1.00000 0.176777
\(33\) −0.794368 −0.138282
\(34\) −4.49111 −0.770219
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −2.49111 −0.409536 −0.204768 0.978811i \(-0.565644\pi\)
−0.204768 + 0.978811i \(0.565644\pi\)
\(38\) −1.00000 −0.162221
\(39\) 2.00000 0.320256
\(40\) 1.49111 0.235765
\(41\) 4.18449 0.653508 0.326754 0.945110i \(-0.394045\pi\)
0.326754 + 0.945110i \(0.394045\pi\)
\(42\) 0 0
\(43\) −4.94951 −0.754793 −0.377396 0.926052i \(-0.623181\pi\)
−0.377396 + 0.926052i \(0.623181\pi\)
\(44\) 0.794368 0.119755
\(45\) 1.49111 0.222282
\(46\) 3.15514 0.465200
\(47\) −11.9106 −1.73734 −0.868669 0.495394i \(-0.835024\pi\)
−0.868669 + 0.495394i \(0.835024\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) −2.77659 −0.392669
\(51\) 4.49111 0.628881
\(52\) −2.00000 −0.277350
\(53\) −5.42284 −0.744884 −0.372442 0.928055i \(-0.621479\pi\)
−0.372442 + 0.928055i \(0.621479\pi\)
\(54\) −1.00000 −0.136083
\(55\) 1.18449 0.159717
\(56\) 0 0
\(57\) 1.00000 0.132453
\(58\) −6.23498 −0.818694
\(59\) −11.8093 −1.53744 −0.768720 0.639585i \(-0.779106\pi\)
−0.768720 + 0.639585i \(0.779106\pi\)
\(60\) −1.49111 −0.192502
\(61\) 12.5825 1.61103 0.805514 0.592577i \(-0.201889\pi\)
0.805514 + 0.592577i \(0.201889\pi\)
\(62\) −9.01157 −1.14447
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −2.98222 −0.369899
\(66\) −0.794368 −0.0977799
\(67\) 16.0231 1.95754 0.978769 0.204964i \(-0.0657078\pi\)
0.978769 + 0.204964i \(0.0657078\pi\)
\(68\) −4.49111 −0.544627
\(69\) −3.15514 −0.379834
\(70\) 0 0
\(71\) 1.47333 0.174852 0.0874262 0.996171i \(-0.472136\pi\)
0.0874262 + 0.996171i \(0.472136\pi\)
\(72\) 1.00000 0.117851
\(73\) 1.23498 0.144544 0.0722720 0.997385i \(-0.476975\pi\)
0.0722720 + 0.997385i \(0.476975\pi\)
\(74\) −2.49111 −0.289586
\(75\) 2.77659 0.320613
\(76\) −1.00000 −0.114708
\(77\) 0 0
\(78\) 2.00000 0.226455
\(79\) −2.33597 −0.262817 −0.131409 0.991328i \(-0.541950\pi\)
−0.131409 + 0.991328i \(0.541950\pi\)
\(80\) 1.49111 0.166711
\(81\) 1.00000 0.111111
\(82\) 4.18449 0.462100
\(83\) 1.57096 0.172435 0.0862174 0.996276i \(-0.472522\pi\)
0.0862174 + 0.996276i \(0.472522\pi\)
\(84\) 0 0
\(85\) −6.69674 −0.726364
\(86\) −4.94951 −0.533719
\(87\) 6.23498 0.668460
\(88\) 0.794368 0.0846799
\(89\) −2.28548 −0.242260 −0.121130 0.992637i \(-0.538652\pi\)
−0.121130 + 0.992637i \(0.538652\pi\)
\(90\) 1.49111 0.157177
\(91\) 0 0
\(92\) 3.15514 0.328946
\(93\) 9.01157 0.934457
\(94\) −11.9106 −1.22848
\(95\) −1.49111 −0.152985
\(96\) −1.00000 −0.102062
\(97\) −17.2971 −1.75625 −0.878125 0.478432i \(-0.841205\pi\)
−0.878125 + 0.478432i \(0.841205\pi\)
\(98\) 0 0
\(99\) 0.794368 0.0798370
\(100\) −2.77659 −0.277659
\(101\) −3.85188 −0.383277 −0.191638 0.981466i \(-0.561380\pi\)
−0.191638 + 0.981466i \(0.561380\pi\)
\(102\) 4.49111 0.444686
\(103\) 1.79437 0.176804 0.0884021 0.996085i \(-0.471824\pi\)
0.0884021 + 0.996085i \(0.471824\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) −5.42284 −0.526713
\(107\) 11.5027 1.11201 0.556003 0.831180i \(-0.312334\pi\)
0.556003 + 0.831180i \(0.312334\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 3.47333 0.332685 0.166342 0.986068i \(-0.446804\pi\)
0.166342 + 0.986068i \(0.446804\pi\)
\(110\) 1.18449 0.112937
\(111\) 2.49111 0.236446
\(112\) 0 0
\(113\) −3.87421 −0.364455 −0.182228 0.983256i \(-0.558331\pi\)
−0.182228 + 0.983256i \(0.558331\pi\)
\(114\) 1.00000 0.0936586
\(115\) 4.70466 0.438712
\(116\) −6.23498 −0.578904
\(117\) −2.00000 −0.184900
\(118\) −11.8093 −1.08713
\(119\) 0 0
\(120\) −1.49111 −0.136119
\(121\) −10.3690 −0.942635
\(122\) 12.5825 1.13917
\(123\) −4.18449 −0.377303
\(124\) −9.01157 −0.809263
\(125\) −11.5958 −1.03716
\(126\) 0 0
\(127\) −1.55938 −0.138373 −0.0691865 0.997604i \(-0.522040\pi\)
−0.0691865 + 0.997604i \(0.522040\pi\)
\(128\) 1.00000 0.0883883
\(129\) 4.94951 0.435780
\(130\) −2.98222 −0.261558
\(131\) 8.91058 0.778521 0.389261 0.921128i \(-0.372731\pi\)
0.389261 + 0.921128i \(0.372731\pi\)
\(132\) −0.794368 −0.0691408
\(133\) 0 0
\(134\) 16.0231 1.38419
\(135\) −1.49111 −0.128334
\(136\) −4.49111 −0.385110
\(137\) 9.85188 0.841703 0.420852 0.907130i \(-0.361731\pi\)
0.420852 + 0.907130i \(0.361731\pi\)
\(138\) −3.15514 −0.268583
\(139\) −21.4195 −1.81678 −0.908388 0.418128i \(-0.862686\pi\)
−0.908388 + 0.418128i \(0.862686\pi\)
\(140\) 0 0
\(141\) 11.9106 1.00305
\(142\) 1.47333 0.123639
\(143\) −1.58874 −0.132857
\(144\) 1.00000 0.0833333
\(145\) −9.29705 −0.772078
\(146\) 1.23498 0.102208
\(147\) 0 0
\(148\) −2.49111 −0.204768
\(149\) 17.4051 1.42588 0.712939 0.701226i \(-0.247364\pi\)
0.712939 + 0.701226i \(0.247364\pi\)
\(150\) 2.77659 0.226708
\(151\) 2.88744 0.234976 0.117488 0.993074i \(-0.462516\pi\)
0.117488 + 0.993074i \(0.462516\pi\)
\(152\) −1.00000 −0.0811107
\(153\) −4.49111 −0.363085
\(154\) 0 0
\(155\) −13.4373 −1.07931
\(156\) 2.00000 0.160128
\(157\) 9.91058 0.790951 0.395475 0.918477i \(-0.370580\pi\)
0.395475 + 0.918477i \(0.370580\pi\)
\(158\) −2.33597 −0.185840
\(159\) 5.42284 0.430059
\(160\) 1.49111 0.117883
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) 9.02935 0.707233 0.353617 0.935390i \(-0.384952\pi\)
0.353617 + 0.935390i \(0.384952\pi\)
\(164\) 4.18449 0.326754
\(165\) −1.18449 −0.0922124
\(166\) 1.57096 0.121930
\(167\) 1.47333 0.114010 0.0570049 0.998374i \(-0.481845\pi\)
0.0570049 + 0.998374i \(0.481845\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) −6.69674 −0.513617
\(171\) −1.00000 −0.0764719
\(172\) −4.94951 −0.377396
\(173\) −16.0691 −1.22171 −0.610855 0.791742i \(-0.709174\pi\)
−0.610855 + 0.791742i \(0.709174\pi\)
\(174\) 6.23498 0.472673
\(175\) 0 0
\(176\) 0.794368 0.0598777
\(177\) 11.8093 0.887642
\(178\) −2.28548 −0.171304
\(179\) −0.505526 −0.0377848 −0.0188924 0.999822i \(-0.506014\pi\)
−0.0188924 + 0.999822i \(0.506014\pi\)
\(180\) 1.49111 0.111141
\(181\) −14.0000 −1.04061 −0.520306 0.853980i \(-0.674182\pi\)
−0.520306 + 0.853980i \(0.674182\pi\)
\(182\) 0 0
\(183\) −12.5825 −0.930127
\(184\) 3.15514 0.232600
\(185\) −3.71452 −0.273097
\(186\) 9.01157 0.660761
\(187\) −3.56759 −0.260888
\(188\) −11.9106 −0.868669
\(189\) 0 0
\(190\) −1.49111 −0.108177
\(191\) −26.5825 −1.92344 −0.961722 0.274026i \(-0.911644\pi\)
−0.961722 + 0.274026i \(0.911644\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −4.77204 −0.343499 −0.171749 0.985141i \(-0.554942\pi\)
−0.171749 + 0.985141i \(0.554942\pi\)
\(194\) −17.2971 −1.24186
\(195\) 2.98222 0.213561
\(196\) 0 0
\(197\) 16.0152 1.14104 0.570519 0.821284i \(-0.306742\pi\)
0.570519 + 0.821284i \(0.306742\pi\)
\(198\) 0.794368 0.0564532
\(199\) 10.8341 0.768009 0.384005 0.923331i \(-0.374545\pi\)
0.384005 + 0.923331i \(0.374545\pi\)
\(200\) −2.77659 −0.196334
\(201\) −16.0231 −1.13019
\(202\) −3.85188 −0.271017
\(203\) 0 0
\(204\) 4.49111 0.314441
\(205\) 6.23954 0.435788
\(206\) 1.79437 0.125020
\(207\) 3.15514 0.219297
\(208\) −2.00000 −0.138675
\(209\) −0.794368 −0.0549476
\(210\) 0 0
\(211\) 4.24992 0.292577 0.146288 0.989242i \(-0.453267\pi\)
0.146288 + 0.989242i \(0.453267\pi\)
\(212\) −5.42284 −0.372442
\(213\) −1.47333 −0.100951
\(214\) 11.5027 0.786307
\(215\) −7.38026 −0.503330
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) 3.47333 0.235244
\(219\) −1.23498 −0.0834525
\(220\) 1.18449 0.0798583
\(221\) 8.98222 0.604210
\(222\) 2.49111 0.167192
\(223\) 15.0807 1.00988 0.504938 0.863156i \(-0.331515\pi\)
0.504938 + 0.863156i \(0.331515\pi\)
\(224\) 0 0
\(225\) −2.77659 −0.185106
\(226\) −3.87421 −0.257709
\(227\) −5.19241 −0.344632 −0.172316 0.985042i \(-0.555125\pi\)
−0.172316 + 0.985042i \(0.555125\pi\)
\(228\) 1.00000 0.0662266
\(229\) 10.7802 0.712379 0.356189 0.934414i \(-0.384076\pi\)
0.356189 + 0.934414i \(0.384076\pi\)
\(230\) 4.70466 0.310216
\(231\) 0 0
\(232\) −6.23498 −0.409347
\(233\) 1.58874 0.104082 0.0520408 0.998645i \(-0.483427\pi\)
0.0520408 + 0.998645i \(0.483427\pi\)
\(234\) −2.00000 −0.130744
\(235\) −17.7600 −1.15853
\(236\) −11.8093 −0.768720
\(237\) 2.33597 0.151738
\(238\) 0 0
\(239\) −4.41126 −0.285341 −0.142670 0.989770i \(-0.545569\pi\)
−0.142670 + 0.989770i \(0.545569\pi\)
\(240\) −1.49111 −0.0962508
\(241\) 3.84486 0.247669 0.123835 0.992303i \(-0.460481\pi\)
0.123835 + 0.992303i \(0.460481\pi\)
\(242\) −10.3690 −0.666543
\(243\) −1.00000 −0.0641500
\(244\) 12.5825 0.805514
\(245\) 0 0
\(246\) −4.18449 −0.266793
\(247\) 2.00000 0.127257
\(248\) −9.01157 −0.572235
\(249\) −1.57096 −0.0995553
\(250\) −11.5958 −0.733380
\(251\) −18.9185 −1.19413 −0.597063 0.802195i \(-0.703666\pi\)
−0.597063 + 0.802195i \(0.703666\pi\)
\(252\) 0 0
\(253\) 2.50634 0.157572
\(254\) −1.55938 −0.0978444
\(255\) 6.69674 0.419366
\(256\) 1.00000 0.0625000
\(257\) 18.9399 1.18144 0.590720 0.806876i \(-0.298844\pi\)
0.590720 + 0.806876i \(0.298844\pi\)
\(258\) 4.94951 0.308143
\(259\) 0 0
\(260\) −2.98222 −0.184950
\(261\) −6.23498 −0.385936
\(262\) 8.91058 0.550498
\(263\) 30.9806 1.91034 0.955172 0.296052i \(-0.0956701\pi\)
0.955172 + 0.296052i \(0.0956701\pi\)
\(264\) −0.794368 −0.0488899
\(265\) −8.08605 −0.496722
\(266\) 0 0
\(267\) 2.28548 0.139869
\(268\) 16.0231 0.978769
\(269\) −5.92471 −0.361236 −0.180618 0.983553i \(-0.557810\pi\)
−0.180618 + 0.983553i \(0.557810\pi\)
\(270\) −1.49111 −0.0907461
\(271\) 13.0798 0.794544 0.397272 0.917701i \(-0.369957\pi\)
0.397272 + 0.917701i \(0.369957\pi\)
\(272\) −4.49111 −0.272314
\(273\) 0 0
\(274\) 9.85188 0.595174
\(275\) −2.20563 −0.133005
\(276\) −3.15514 −0.189917
\(277\) −12.2316 −0.734927 −0.367463 0.930038i \(-0.619774\pi\)
−0.367463 + 0.930038i \(0.619774\pi\)
\(278\) −21.4195 −1.28465
\(279\) −9.01157 −0.539509
\(280\) 0 0
\(281\) −0.150585 −0.00898313 −0.00449156 0.999990i \(-0.501430\pi\)
−0.00449156 + 0.999990i \(0.501430\pi\)
\(282\) 11.9106 0.709265
\(283\) −24.4017 −1.45053 −0.725265 0.688470i \(-0.758283\pi\)
−0.725265 + 0.688470i \(0.758283\pi\)
\(284\) 1.47333 0.0874262
\(285\) 1.49111 0.0883258
\(286\) −1.58874 −0.0939439
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) 3.17008 0.186475
\(290\) −9.29705 −0.545942
\(291\) 17.2971 1.01397
\(292\) 1.23498 0.0722720
\(293\) −12.6694 −0.740154 −0.370077 0.929001i \(-0.620669\pi\)
−0.370077 + 0.929001i \(0.620669\pi\)
\(294\) 0 0
\(295\) −17.6090 −1.02523
\(296\) −2.49111 −0.144793
\(297\) −0.794368 −0.0460939
\(298\) 17.4051 1.00825
\(299\) −6.31028 −0.364933
\(300\) 2.77659 0.160306
\(301\) 0 0
\(302\) 2.88744 0.166153
\(303\) 3.85188 0.221285
\(304\) −1.00000 −0.0573539
\(305\) 18.7619 1.07431
\(306\) −4.49111 −0.256740
\(307\) 19.8031 1.13022 0.565111 0.825015i \(-0.308833\pi\)
0.565111 + 0.825015i \(0.308833\pi\)
\(308\) 0 0
\(309\) −1.79437 −0.102078
\(310\) −13.4373 −0.763185
\(311\) 23.3918 1.32643 0.663215 0.748429i \(-0.269192\pi\)
0.663215 + 0.748429i \(0.269192\pi\)
\(312\) 2.00000 0.113228
\(313\) 27.6045 1.56030 0.780149 0.625594i \(-0.215143\pi\)
0.780149 + 0.625594i \(0.215143\pi\)
\(314\) 9.91058 0.559287
\(315\) 0 0
\(316\) −2.33597 −0.131409
\(317\) −21.4883 −1.20690 −0.603451 0.797400i \(-0.706208\pi\)
−0.603451 + 0.797400i \(0.706208\pi\)
\(318\) 5.42284 0.304098
\(319\) −4.95287 −0.277307
\(320\) 1.49111 0.0833556
\(321\) −11.5027 −0.642017
\(322\) 0 0
\(323\) 4.49111 0.249892
\(324\) 1.00000 0.0555556
\(325\) 5.55318 0.308035
\(326\) 9.02935 0.500090
\(327\) −3.47333 −0.192076
\(328\) 4.18449 0.231050
\(329\) 0 0
\(330\) −1.18449 −0.0652040
\(331\) −15.7377 −0.865020 −0.432510 0.901629i \(-0.642372\pi\)
−0.432510 + 0.901629i \(0.642372\pi\)
\(332\) 1.57096 0.0862174
\(333\) −2.49111 −0.136512
\(334\) 1.47333 0.0806171
\(335\) 23.8923 1.30537
\(336\) 0 0
\(337\) −31.7670 −1.73046 −0.865230 0.501375i \(-0.832828\pi\)
−0.865230 + 0.501375i \(0.832828\pi\)
\(338\) −9.00000 −0.489535
\(339\) 3.87421 0.210418
\(340\) −6.69674 −0.363182
\(341\) −7.15850 −0.387655
\(342\) −1.00000 −0.0540738
\(343\) 0 0
\(344\) −4.94951 −0.266859
\(345\) −4.70466 −0.253290
\(346\) −16.0691 −0.863879
\(347\) −12.3769 −0.664427 −0.332213 0.943204i \(-0.607795\pi\)
−0.332213 + 0.943204i \(0.607795\pi\)
\(348\) 6.23498 0.334230
\(349\) −29.2610 −1.56630 −0.783152 0.621830i \(-0.786389\pi\)
−0.783152 + 0.621830i \(0.786389\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) 0.794368 0.0423399
\(353\) −21.8423 −1.16255 −0.581274 0.813708i \(-0.697446\pi\)
−0.581274 + 0.813708i \(0.697446\pi\)
\(354\) 11.8093 0.627657
\(355\) 2.19690 0.116599
\(356\) −2.28548 −0.121130
\(357\) 0 0
\(358\) −0.505526 −0.0267179
\(359\) −24.2209 −1.27833 −0.639164 0.769070i \(-0.720720\pi\)
−0.639164 + 0.769070i \(0.720720\pi\)
\(360\) 1.49111 0.0785884
\(361\) 1.00000 0.0526316
\(362\) −14.0000 −0.735824
\(363\) 10.3690 0.544230
\(364\) 0 0
\(365\) 1.84150 0.0963884
\(366\) −12.5825 −0.657699
\(367\) 17.4373 0.910217 0.455109 0.890436i \(-0.349600\pi\)
0.455109 + 0.890436i \(0.349600\pi\)
\(368\) 3.15514 0.164473
\(369\) 4.18449 0.217836
\(370\) −3.71452 −0.193109
\(371\) 0 0
\(372\) 9.01157 0.467228
\(373\) −10.7374 −0.555960 −0.277980 0.960587i \(-0.589665\pi\)
−0.277980 + 0.960587i \(0.589665\pi\)
\(374\) −3.56759 −0.184476
\(375\) 11.5958 0.598802
\(376\) −11.9106 −0.614242
\(377\) 12.4700 0.642236
\(378\) 0 0
\(379\) −11.5887 −0.595273 −0.297637 0.954679i \(-0.596198\pi\)
−0.297637 + 0.954679i \(0.596198\pi\)
\(380\) −1.49111 −0.0764924
\(381\) 1.55938 0.0798896
\(382\) −26.5825 −1.36008
\(383\) −5.69704 −0.291105 −0.145552 0.989351i \(-0.546496\pi\)
−0.145552 + 0.989351i \(0.546496\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −4.77204 −0.242890
\(387\) −4.94951 −0.251598
\(388\) −17.2971 −0.878125
\(389\) −1.35793 −0.0688497 −0.0344249 0.999407i \(-0.510960\pi\)
−0.0344249 + 0.999407i \(0.510960\pi\)
\(390\) 2.98222 0.151011
\(391\) −14.1701 −0.716611
\(392\) 0 0
\(393\) −8.91058 −0.449480
\(394\) 16.0152 0.806836
\(395\) −3.48319 −0.175258
\(396\) 0.794368 0.0399185
\(397\) −28.3334 −1.42201 −0.711007 0.703185i \(-0.751761\pi\)
−0.711007 + 0.703185i \(0.751761\pi\)
\(398\) 10.8341 0.543064
\(399\) 0 0
\(400\) −2.77659 −0.138829
\(401\) −33.9577 −1.69577 −0.847884 0.530182i \(-0.822123\pi\)
−0.847884 + 0.530182i \(0.822123\pi\)
\(402\) −16.0231 −0.799162
\(403\) 18.0231 0.897797
\(404\) −3.85188 −0.191638
\(405\) 1.49111 0.0740939
\(406\) 0 0
\(407\) −1.97886 −0.0980883
\(408\) 4.49111 0.222343
\(409\) −14.6251 −0.723165 −0.361582 0.932340i \(-0.617763\pi\)
−0.361582 + 0.932340i \(0.617763\pi\)
\(410\) 6.23954 0.308149
\(411\) −9.85188 −0.485957
\(412\) 1.79437 0.0884021
\(413\) 0 0
\(414\) 3.15514 0.155067
\(415\) 2.34247 0.114987
\(416\) −2.00000 −0.0980581
\(417\) 21.4195 1.04892
\(418\) −0.794368 −0.0388538
\(419\) 1.52211 0.0743602 0.0371801 0.999309i \(-0.488162\pi\)
0.0371801 + 0.999309i \(0.488162\pi\)
\(420\) 0 0
\(421\) −35.7129 −1.74054 −0.870269 0.492576i \(-0.836055\pi\)
−0.870269 + 0.492576i \(0.836055\pi\)
\(422\) 4.24992 0.206883
\(423\) −11.9106 −0.579112
\(424\) −5.42284 −0.263356
\(425\) 12.4700 0.604882
\(426\) −1.47333 −0.0713832
\(427\) 0 0
\(428\) 11.5027 0.556003
\(429\) 1.58874 0.0767049
\(430\) −7.38026 −0.355908
\(431\) 39.9664 1.92512 0.962558 0.271076i \(-0.0873794\pi\)
0.962558 + 0.271076i \(0.0873794\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −15.5568 −0.747614 −0.373807 0.927507i \(-0.621948\pi\)
−0.373807 + 0.927507i \(0.621948\pi\)
\(434\) 0 0
\(435\) 9.29705 0.445759
\(436\) 3.47333 0.166342
\(437\) −3.15514 −0.150931
\(438\) −1.23498 −0.0590098
\(439\) 18.6395 0.889616 0.444808 0.895626i \(-0.353272\pi\)
0.444808 + 0.895626i \(0.353272\pi\)
\(440\) 1.18449 0.0564683
\(441\) 0 0
\(442\) 8.98222 0.427241
\(443\) 25.5929 1.21596 0.607978 0.793954i \(-0.291981\pi\)
0.607978 + 0.793954i \(0.291981\pi\)
\(444\) 2.49111 0.118223
\(445\) −3.40790 −0.161550
\(446\) 15.0807 0.714090
\(447\) −17.4051 −0.823231
\(448\) 0 0
\(449\) −4.49477 −0.212121 −0.106061 0.994360i \(-0.533824\pi\)
−0.106061 + 0.994360i \(0.533824\pi\)
\(450\) −2.77659 −0.130890
\(451\) 3.32402 0.156522
\(452\) −3.87421 −0.182228
\(453\) −2.88744 −0.135664
\(454\) −5.19241 −0.243692
\(455\) 0 0
\(456\) 1.00000 0.0468293
\(457\) −31.1633 −1.45776 −0.728880 0.684642i \(-0.759959\pi\)
−0.728880 + 0.684642i \(0.759959\pi\)
\(458\) 10.7802 0.503728
\(459\) 4.49111 0.209627
\(460\) 4.70466 0.219356
\(461\) 22.4815 1.04707 0.523535 0.852004i \(-0.324613\pi\)
0.523535 + 0.852004i \(0.324613\pi\)
\(462\) 0 0
\(463\) 1.16590 0.0541838 0.0270919 0.999633i \(-0.491375\pi\)
0.0270919 + 0.999633i \(0.491375\pi\)
\(464\) −6.23498 −0.289452
\(465\) 13.4373 0.623138
\(466\) 1.58874 0.0735967
\(467\) 23.0984 1.06887 0.534434 0.845210i \(-0.320525\pi\)
0.534434 + 0.845210i \(0.320525\pi\)
\(468\) −2.00000 −0.0924500
\(469\) 0 0
\(470\) −17.7600 −0.819208
\(471\) −9.91058 −0.456656
\(472\) −11.8093 −0.543567
\(473\) −3.93173 −0.180781
\(474\) 2.33597 0.107295
\(475\) 2.77659 0.127399
\(476\) 0 0
\(477\) −5.42284 −0.248295
\(478\) −4.41126 −0.201767
\(479\) −7.15349 −0.326851 −0.163426 0.986556i \(-0.552254\pi\)
−0.163426 + 0.986556i \(0.552254\pi\)
\(480\) −1.49111 −0.0680596
\(481\) 4.98222 0.227170
\(482\) 3.84486 0.175129
\(483\) 0 0
\(484\) −10.3690 −0.471317
\(485\) −25.7918 −1.17115
\(486\) −1.00000 −0.0453609
\(487\) −9.76867 −0.442661 −0.221330 0.975199i \(-0.571040\pi\)
−0.221330 + 0.975199i \(0.571040\pi\)
\(488\) 12.5825 0.569584
\(489\) −9.02935 −0.408321
\(490\) 0 0
\(491\) 9.03556 0.407769 0.203884 0.978995i \(-0.434643\pi\)
0.203884 + 0.978995i \(0.434643\pi\)
\(492\) −4.18449 −0.188651
\(493\) 28.0020 1.26115
\(494\) 2.00000 0.0899843
\(495\) 1.18449 0.0532389
\(496\) −9.01157 −0.404632
\(497\) 0 0
\(498\) −1.57096 −0.0703962
\(499\) 23.9405 1.07172 0.535861 0.844306i \(-0.319987\pi\)
0.535861 + 0.844306i \(0.319987\pi\)
\(500\) −11.5958 −0.518578
\(501\) −1.47333 −0.0658236
\(502\) −18.9185 −0.844374
\(503\) 37.5044 1.67224 0.836119 0.548548i \(-0.184819\pi\)
0.836119 + 0.548548i \(0.184819\pi\)
\(504\) 0 0
\(505\) −5.74358 −0.255586
\(506\) 2.50634 0.111420
\(507\) 9.00000 0.399704
\(508\) −1.55938 −0.0691865
\(509\) 4.10517 0.181958 0.0909791 0.995853i \(-0.471000\pi\)
0.0909791 + 0.995853i \(0.471000\pi\)
\(510\) 6.69674 0.296537
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 1.00000 0.0441511
\(514\) 18.9399 0.835405
\(515\) 2.67560 0.117901
\(516\) 4.94951 0.217890
\(517\) −9.46138 −0.416111
\(518\) 0 0
\(519\) 16.0691 0.705355
\(520\) −2.98222 −0.130779
\(521\) 9.14191 0.400514 0.200257 0.979743i \(-0.435822\pi\)
0.200257 + 0.979743i \(0.435822\pi\)
\(522\) −6.23498 −0.272898
\(523\) 34.5106 1.50904 0.754522 0.656275i \(-0.227869\pi\)
0.754522 + 0.656275i \(0.227869\pi\)
\(524\) 8.91058 0.389261
\(525\) 0 0
\(526\) 30.9806 1.35082
\(527\) 40.4720 1.76299
\(528\) −0.794368 −0.0345704
\(529\) −13.0451 −0.567178
\(530\) −8.08605 −0.351236
\(531\) −11.8093 −0.512480
\(532\) 0 0
\(533\) −8.36898 −0.362501
\(534\) 2.28548 0.0989023
\(535\) 17.1518 0.741536
\(536\) 16.0231 0.692095
\(537\) 0.505526 0.0218150
\(538\) −5.92471 −0.255432
\(539\) 0 0
\(540\) −1.49111 −0.0641672
\(541\) 19.6483 0.844744 0.422372 0.906422i \(-0.361198\pi\)
0.422372 + 0.906422i \(0.361198\pi\)
\(542\) 13.0798 0.561827
\(543\) 14.0000 0.600798
\(544\) −4.49111 −0.192555
\(545\) 5.17912 0.221849
\(546\) 0 0
\(547\) 26.3582 1.12700 0.563498 0.826117i \(-0.309455\pi\)
0.563498 + 0.826117i \(0.309455\pi\)
\(548\) 9.85188 0.420852
\(549\) 12.5825 0.537009
\(550\) −2.20563 −0.0940485
\(551\) 6.23498 0.265619
\(552\) −3.15514 −0.134292
\(553\) 0 0
\(554\) −12.2316 −0.519672
\(555\) 3.71452 0.157673
\(556\) −21.4195 −0.908388
\(557\) −25.7430 −1.09077 −0.545384 0.838186i \(-0.683616\pi\)
−0.545384 + 0.838186i \(0.683616\pi\)
\(558\) −9.01157 −0.381490
\(559\) 9.89901 0.418684
\(560\) 0 0
\(561\) 3.56759 0.150624
\(562\) −0.150585 −0.00635203
\(563\) −19.2383 −0.810800 −0.405400 0.914139i \(-0.632868\pi\)
−0.405400 + 0.914139i \(0.632868\pi\)
\(564\) 11.9106 0.501526
\(565\) −5.77688 −0.243035
\(566\) −24.4017 −1.02568
\(567\) 0 0
\(568\) 1.47333 0.0618196
\(569\) −2.20198 −0.0923117 −0.0461558 0.998934i \(-0.514697\pi\)
−0.0461558 + 0.998934i \(0.514697\pi\)
\(570\) 1.49111 0.0624558
\(571\) −11.7224 −0.490569 −0.245284 0.969451i \(-0.578881\pi\)
−0.245284 + 0.969451i \(0.578881\pi\)
\(572\) −1.58874 −0.0664284
\(573\) 26.5825 1.11050
\(574\) 0 0
\(575\) −8.76052 −0.365339
\(576\) 1.00000 0.0416667
\(577\) 14.5532 0.605857 0.302928 0.953013i \(-0.402036\pi\)
0.302928 + 0.953013i \(0.402036\pi\)
\(578\) 3.17008 0.131858
\(579\) 4.77204 0.198319
\(580\) −9.29705 −0.386039
\(581\) 0 0
\(582\) 17.2971 0.716986
\(583\) −4.30773 −0.178408
\(584\) 1.23498 0.0511040
\(585\) −2.98222 −0.123300
\(586\) −12.6694 −0.523368
\(587\) −0.509703 −0.0210377 −0.0105189 0.999945i \(-0.503348\pi\)
−0.0105189 + 0.999945i \(0.503348\pi\)
\(588\) 0 0
\(589\) 9.01157 0.371315
\(590\) −17.6090 −0.724950
\(591\) −16.0152 −0.658778
\(592\) −2.49111 −0.102384
\(593\) −29.2545 −1.20134 −0.600669 0.799498i \(-0.705099\pi\)
−0.600669 + 0.799498i \(0.705099\pi\)
\(594\) −0.794368 −0.0325933
\(595\) 0 0
\(596\) 17.4051 0.712939
\(597\) −10.8341 −0.443410
\(598\) −6.31028 −0.258046
\(599\) −10.6864 −0.436633 −0.218316 0.975878i \(-0.570056\pi\)
−0.218316 + 0.975878i \(0.570056\pi\)
\(600\) 2.77659 0.113354
\(601\) 24.0541 0.981189 0.490595 0.871388i \(-0.336780\pi\)
0.490595 + 0.871388i \(0.336780\pi\)
\(602\) 0 0
\(603\) 16.0231 0.652513
\(604\) 2.88744 0.117488
\(605\) −15.4613 −0.628591
\(606\) 3.85188 0.156472
\(607\) −5.56304 −0.225797 −0.112898 0.993607i \(-0.536013\pi\)
−0.112898 + 0.993607i \(0.536013\pi\)
\(608\) −1.00000 −0.0405554
\(609\) 0 0
\(610\) 18.7619 0.759649
\(611\) 23.8212 0.963701
\(612\) −4.49111 −0.181542
\(613\) −11.3975 −0.460341 −0.230171 0.973150i \(-0.573928\pi\)
−0.230171 + 0.973150i \(0.573928\pi\)
\(614\) 19.8031 0.799188
\(615\) −6.23954 −0.251602
\(616\) 0 0
\(617\) 23.5205 0.946898 0.473449 0.880821i \(-0.343009\pi\)
0.473449 + 0.880821i \(0.343009\pi\)
\(618\) −1.79437 −0.0721801
\(619\) 42.4393 1.70578 0.852889 0.522092i \(-0.174848\pi\)
0.852889 + 0.522092i \(0.174848\pi\)
\(620\) −13.4373 −0.539653
\(621\) −3.15514 −0.126611
\(622\) 23.3918 0.937927
\(623\) 0 0
\(624\) 2.00000 0.0800641
\(625\) −3.40761 −0.136304
\(626\) 27.6045 1.10330
\(627\) 0.794368 0.0317240
\(628\) 9.91058 0.395475
\(629\) 11.1879 0.446089
\(630\) 0 0
\(631\) −33.5730 −1.33652 −0.668259 0.743928i \(-0.732960\pi\)
−0.668259 + 0.743928i \(0.732960\pi\)
\(632\) −2.33597 −0.0929200
\(633\) −4.24992 −0.168919
\(634\) −21.4883 −0.853408
\(635\) −2.32521 −0.0922733
\(636\) 5.42284 0.215030
\(637\) 0 0
\(638\) −4.95287 −0.196086
\(639\) 1.47333 0.0582841
\(640\) 1.49111 0.0589413
\(641\) 2.35659 0.0930798 0.0465399 0.998916i \(-0.485181\pi\)
0.0465399 + 0.998916i \(0.485181\pi\)
\(642\) −11.5027 −0.453975
\(643\) 30.4488 1.20078 0.600392 0.799706i \(-0.295011\pi\)
0.600392 + 0.799706i \(0.295011\pi\)
\(644\) 0 0
\(645\) 7.38026 0.290597
\(646\) 4.49111 0.176700
\(647\) 28.4336 1.11784 0.558920 0.829222i \(-0.311216\pi\)
0.558920 + 0.829222i \(0.311216\pi\)
\(648\) 1.00000 0.0392837
\(649\) −9.38093 −0.368234
\(650\) 5.55318 0.217814
\(651\) 0 0
\(652\) 9.02935 0.353617
\(653\) 41.0008 1.60449 0.802243 0.596998i \(-0.203640\pi\)
0.802243 + 0.596998i \(0.203640\pi\)
\(654\) −3.47333 −0.135818
\(655\) 13.2867 0.519153
\(656\) 4.18449 0.163377
\(657\) 1.23498 0.0481813
\(658\) 0 0
\(659\) −9.41298 −0.366678 −0.183339 0.983050i \(-0.558691\pi\)
−0.183339 + 0.983050i \(0.558691\pi\)
\(660\) −1.18449 −0.0461062
\(661\) −24.4200 −0.949828 −0.474914 0.880032i \(-0.657521\pi\)
−0.474914 + 0.880032i \(0.657521\pi\)
\(662\) −15.7377 −0.611662
\(663\) −8.98222 −0.348841
\(664\) 1.57096 0.0609649
\(665\) 0 0
\(666\) −2.49111 −0.0965286
\(667\) −19.6722 −0.761712
\(668\) 1.47333 0.0570049
\(669\) −15.0807 −0.583052
\(670\) 23.8923 0.923039
\(671\) 9.99515 0.385859
\(672\) 0 0
\(673\) −3.30743 −0.127492 −0.0637461 0.997966i \(-0.520305\pi\)
−0.0637461 + 0.997966i \(0.520305\pi\)
\(674\) −31.7670 −1.22362
\(675\) 2.77659 0.106871
\(676\) −9.00000 −0.346154
\(677\) −6.27420 −0.241137 −0.120568 0.992705i \(-0.538472\pi\)
−0.120568 + 0.992705i \(0.538472\pi\)
\(678\) 3.87421 0.148788
\(679\) 0 0
\(680\) −6.69674 −0.256808
\(681\) 5.19241 0.198973
\(682\) −7.15850 −0.274113
\(683\) −34.2406 −1.31018 −0.655091 0.755550i \(-0.727370\pi\)
−0.655091 + 0.755550i \(0.727370\pi\)
\(684\) −1.00000 −0.0382360
\(685\) 14.6902 0.561285
\(686\) 0 0
\(687\) −10.7802 −0.411292
\(688\) −4.94951 −0.188698
\(689\) 10.8457 0.413187
\(690\) −4.70466 −0.179103
\(691\) 20.3834 0.775421 0.387710 0.921781i \(-0.373266\pi\)
0.387710 + 0.921781i \(0.373266\pi\)
\(692\) −16.0691 −0.610855
\(693\) 0 0
\(694\) −12.3769 −0.469821
\(695\) −31.9388 −1.21151
\(696\) 6.23498 0.236336
\(697\) −18.7930 −0.711836
\(698\) −29.2610 −1.10754
\(699\) −1.58874 −0.0600915
\(700\) 0 0
\(701\) 22.3620 0.844600 0.422300 0.906456i \(-0.361223\pi\)
0.422300 + 0.906456i \(0.361223\pi\)
\(702\) 2.00000 0.0754851
\(703\) 2.49111 0.0939540
\(704\) 0.794368 0.0299389
\(705\) 17.7600 0.668880
\(706\) −21.8423 −0.822046
\(707\) 0 0
\(708\) 11.8093 0.443821
\(709\) −18.3466 −0.689023 −0.344511 0.938782i \(-0.611955\pi\)
−0.344511 + 0.938782i \(0.611955\pi\)
\(710\) 2.19690 0.0824482
\(711\) −2.33597 −0.0876058
\(712\) −2.28548 −0.0856519
\(713\) −28.4328 −1.06482
\(714\) 0 0
\(715\) −2.36898 −0.0885948
\(716\) −0.505526 −0.0188924
\(717\) 4.41126 0.164742
\(718\) −24.2209 −0.903915
\(719\) 0.852403 0.0317893 0.0158946 0.999874i \(-0.494940\pi\)
0.0158946 + 0.999874i \(0.494940\pi\)
\(720\) 1.49111 0.0555704
\(721\) 0 0
\(722\) 1.00000 0.0372161
\(723\) −3.84486 −0.142992
\(724\) −14.0000 −0.520306
\(725\) 17.3120 0.642951
\(726\) 10.3690 0.384829
\(727\) −35.9487 −1.33326 −0.666631 0.745388i \(-0.732264\pi\)
−0.666631 + 0.745388i \(0.732264\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 1.84150 0.0681569
\(731\) 22.2288 0.822161
\(732\) −12.5825 −0.465064
\(733\) −35.0037 −1.29289 −0.646446 0.762960i \(-0.723745\pi\)
−0.646446 + 0.762960i \(0.723745\pi\)
\(734\) 17.4373 0.643621
\(735\) 0 0
\(736\) 3.15514 0.116300
\(737\) 12.7283 0.468852
\(738\) 4.18449 0.154033
\(739\) 5.49648 0.202191 0.101096 0.994877i \(-0.467765\pi\)
0.101096 + 0.994877i \(0.467765\pi\)
\(740\) −3.71452 −0.136549
\(741\) −2.00000 −0.0734718
\(742\) 0 0
\(743\) 10.2871 0.377398 0.188699 0.982035i \(-0.439573\pi\)
0.188699 + 0.982035i \(0.439573\pi\)
\(744\) 9.01157 0.330380
\(745\) 25.9529 0.950840
\(746\) −10.7374 −0.393123
\(747\) 1.57096 0.0574783
\(748\) −3.56759 −0.130444
\(749\) 0 0
\(750\) 11.5958 0.423417
\(751\) 19.4621 0.710180 0.355090 0.934832i \(-0.384450\pi\)
0.355090 + 0.934832i \(0.384450\pi\)
\(752\) −11.9106 −0.434334
\(753\) 18.9185 0.689429
\(754\) 12.4700 0.454129
\(755\) 4.30549 0.156693
\(756\) 0 0
\(757\) −39.9230 −1.45103 −0.725513 0.688209i \(-0.758397\pi\)
−0.725513 + 0.688209i \(0.758397\pi\)
\(758\) −11.5887 −0.420922
\(759\) −2.50634 −0.0909744
\(760\) −1.49111 −0.0540883
\(761\) 37.7029 1.36673 0.683365 0.730077i \(-0.260516\pi\)
0.683365 + 0.730077i \(0.260516\pi\)
\(762\) 1.55938 0.0564905
\(763\) 0 0
\(764\) −26.5825 −0.961722
\(765\) −6.69674 −0.242121
\(766\) −5.69704 −0.205842
\(767\) 23.6186 0.852819
\(768\) −1.00000 −0.0360844
\(769\) 8.86711 0.319756 0.159878 0.987137i \(-0.448890\pi\)
0.159878 + 0.987137i \(0.448890\pi\)
\(770\) 0 0
\(771\) −18.9399 −0.682105
\(772\) −4.77204 −0.171749
\(773\) 35.2043 1.26621 0.633105 0.774066i \(-0.281780\pi\)
0.633105 + 0.774066i \(0.281780\pi\)
\(774\) −4.94951 −0.177906
\(775\) 25.0214 0.898796
\(776\) −17.2971 −0.620928
\(777\) 0 0
\(778\) −1.35793 −0.0486841
\(779\) −4.18449 −0.149925
\(780\) 2.98222 0.106781
\(781\) 1.17037 0.0418790
\(782\) −14.1701 −0.506721
\(783\) 6.23498 0.222820
\(784\) 0 0
\(785\) 14.7778 0.527442
\(786\) −8.91058 −0.317830
\(787\) −17.1775 −0.612311 −0.306155 0.951982i \(-0.599043\pi\)
−0.306155 + 0.951982i \(0.599043\pi\)
\(788\) 16.0152 0.570519
\(789\) −30.9806 −1.10294
\(790\) −3.48319 −0.123926
\(791\) 0 0
\(792\) 0.794368 0.0282266
\(793\) −25.1651 −0.893637
\(794\) −28.3334 −1.00552
\(795\) 8.08605 0.286783
\(796\) 10.8341 0.384005
\(797\) 39.7758 1.40893 0.704465 0.709739i \(-0.251187\pi\)
0.704465 + 0.709739i \(0.251187\pi\)
\(798\) 0 0
\(799\) 53.4918 1.89240
\(800\) −2.77659 −0.0981672
\(801\) −2.28548 −0.0807534
\(802\) −33.9577 −1.19909
\(803\) 0.981031 0.0346198
\(804\) −16.0231 −0.565093
\(805\) 0 0
\(806\) 18.0231 0.634838
\(807\) 5.92471 0.208560
\(808\) −3.85188 −0.135509
\(809\) 33.0968 1.16362 0.581811 0.813324i \(-0.302345\pi\)
0.581811 + 0.813324i \(0.302345\pi\)
\(810\) 1.49111 0.0523923
\(811\) −40.7994 −1.43266 −0.716330 0.697762i \(-0.754179\pi\)
−0.716330 + 0.697762i \(0.754179\pi\)
\(812\) 0 0
\(813\) −13.0798 −0.458730
\(814\) −1.97886 −0.0693589
\(815\) 13.4638 0.471615
\(816\) 4.49111 0.157220
\(817\) 4.94951 0.173161
\(818\) −14.6251 −0.511355
\(819\) 0 0
\(820\) 6.23954 0.217894
\(821\) −22.7873 −0.795281 −0.397641 0.917541i \(-0.630171\pi\)
−0.397641 + 0.917541i \(0.630171\pi\)
\(822\) −9.85188 −0.343624
\(823\) −27.2189 −0.948792 −0.474396 0.880312i \(-0.657333\pi\)
−0.474396 + 0.880312i \(0.657333\pi\)
\(824\) 1.79437 0.0625098
\(825\) 2.20563 0.0767903
\(826\) 0 0
\(827\) 43.9995 1.53001 0.765005 0.644024i \(-0.222736\pi\)
0.765005 + 0.644024i \(0.222736\pi\)
\(828\) 3.15514 0.109649
\(829\) 10.9755 0.381195 0.190597 0.981668i \(-0.438958\pi\)
0.190597 + 0.981668i \(0.438958\pi\)
\(830\) 2.34247 0.0813083
\(831\) 12.2316 0.424310
\(832\) −2.00000 −0.0693375
\(833\) 0 0
\(834\) 21.4195 0.741696
\(835\) 2.19690 0.0760269
\(836\) −0.794368 −0.0274738
\(837\) 9.01157 0.311486
\(838\) 1.52211 0.0525806
\(839\) 39.4965 1.36357 0.681785 0.731553i \(-0.261204\pi\)
0.681785 + 0.731553i \(0.261204\pi\)
\(840\) 0 0
\(841\) 9.87503 0.340518
\(842\) −35.7129 −1.23075
\(843\) 0.150585 0.00518641
\(844\) 4.24992 0.146288
\(845\) −13.4200 −0.461662
\(846\) −11.9106 −0.409494
\(847\) 0 0
\(848\) −5.42284 −0.186221
\(849\) 24.4017 0.837464
\(850\) 12.4700 0.427716
\(851\) −7.85980 −0.269430
\(852\) −1.47333 −0.0504755
\(853\) 25.7492 0.881637 0.440819 0.897596i \(-0.354688\pi\)
0.440819 + 0.897596i \(0.354688\pi\)
\(854\) 0 0
\(855\) −1.49111 −0.0509949
\(856\) 11.5027 0.393154
\(857\) −33.1492 −1.13236 −0.566178 0.824283i \(-0.691579\pi\)
−0.566178 + 0.824283i \(0.691579\pi\)
\(858\) 1.58874 0.0542385
\(859\) −14.7630 −0.503707 −0.251853 0.967765i \(-0.581040\pi\)
−0.251853 + 0.967765i \(0.581040\pi\)
\(860\) −7.38026 −0.251665
\(861\) 0 0
\(862\) 39.9664 1.36126
\(863\) 49.1439 1.67288 0.836439 0.548060i \(-0.184633\pi\)
0.836439 + 0.548060i \(0.184633\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −23.9608 −0.814691
\(866\) −15.5568 −0.528643
\(867\) −3.17008 −0.107661
\(868\) 0 0
\(869\) −1.85562 −0.0629476
\(870\) 9.29705 0.315200
\(871\) −32.0463 −1.08585
\(872\) 3.47333 0.117622
\(873\) −17.2971 −0.585416
\(874\) −3.15514 −0.106724
\(875\) 0 0
\(876\) −1.23498 −0.0417262
\(877\) 21.4186 0.723256 0.361628 0.932322i \(-0.382221\pi\)
0.361628 + 0.932322i \(0.382221\pi\)
\(878\) 18.6395 0.629053
\(879\) 12.6694 0.427328
\(880\) 1.18449 0.0399292
\(881\) −38.2184 −1.28761 −0.643805 0.765189i \(-0.722645\pi\)
−0.643805 + 0.765189i \(0.722645\pi\)
\(882\) 0 0
\(883\) 21.3667 0.719045 0.359523 0.933136i \(-0.382940\pi\)
0.359523 + 0.933136i \(0.382940\pi\)
\(884\) 8.98222 0.302105
\(885\) 17.6090 0.591919
\(886\) 25.5929 0.859811
\(887\) −2.83126 −0.0950644 −0.0475322 0.998870i \(-0.515136\pi\)
−0.0475322 + 0.998870i \(0.515136\pi\)
\(888\) 2.49111 0.0835962
\(889\) 0 0
\(890\) −3.40790 −0.114233
\(891\) 0.794368 0.0266123
\(892\) 15.0807 0.504938
\(893\) 11.9106 0.398573
\(894\) −17.4051 −0.582112
\(895\) −0.753795 −0.0251966
\(896\) 0 0
\(897\) 6.31028 0.210694
\(898\) −4.49477 −0.149992
\(899\) 56.1870 1.87394
\(900\) −2.77659 −0.0925530
\(901\) 24.3546 0.811368
\(902\) 3.32402 0.110678
\(903\) 0 0
\(904\) −3.87421 −0.128854
\(905\) −20.8755 −0.693927
\(906\) −2.88744 −0.0959287
\(907\) −8.02480 −0.266459 −0.133230 0.991085i \(-0.542535\pi\)
−0.133230 + 0.991085i \(0.542535\pi\)
\(908\) −5.19241 −0.172316
\(909\) −3.85188 −0.127759
\(910\) 0 0
\(911\) −17.7253 −0.587265 −0.293632 0.955918i \(-0.594864\pi\)
−0.293632 + 0.955918i \(0.594864\pi\)
\(912\) 1.00000 0.0331133
\(913\) 1.24792 0.0413000
\(914\) −31.1633 −1.03079
\(915\) −18.7619 −0.620251
\(916\) 10.7802 0.356189
\(917\) 0 0
\(918\) 4.49111 0.148229
\(919\) −12.8445 −0.423700 −0.211850 0.977302i \(-0.567949\pi\)
−0.211850 + 0.977302i \(0.567949\pi\)
\(920\) 4.70466 0.155108
\(921\) −19.8031 −0.652534
\(922\) 22.4815 0.740390
\(923\) −2.94666 −0.0969906
\(924\) 0 0
\(925\) 6.91679 0.227423
\(926\) 1.16590 0.0383138
\(927\) 1.79437 0.0589348
\(928\) −6.23498 −0.204673
\(929\) −10.8552 −0.356149 −0.178075 0.984017i \(-0.556987\pi\)
−0.178075 + 0.984017i \(0.556987\pi\)
\(930\) 13.4373 0.440625
\(931\) 0 0
\(932\) 1.58874 0.0520408
\(933\) −23.3918 −0.765814
\(934\) 23.0984 0.755804
\(935\) −5.31968 −0.173972
\(936\) −2.00000 −0.0653720
\(937\) 20.9520 0.684473 0.342237 0.939614i \(-0.388816\pi\)
0.342237 + 0.939614i \(0.388816\pi\)
\(938\) 0 0
\(939\) −27.6045 −0.900838
\(940\) −17.7600 −0.579267
\(941\) 13.5893 0.442997 0.221499 0.975161i \(-0.428905\pi\)
0.221499 + 0.975161i \(0.428905\pi\)
\(942\) −9.91058 −0.322904
\(943\) 13.2026 0.429937
\(944\) −11.8093 −0.384360
\(945\) 0 0
\(946\) −3.93173 −0.127831
\(947\) 57.8118 1.87863 0.939316 0.343053i \(-0.111461\pi\)
0.939316 + 0.343053i \(0.111461\pi\)
\(948\) 2.33597 0.0758689
\(949\) −2.46997 −0.0801786
\(950\) 2.77659 0.0900844
\(951\) 21.4883 0.696805
\(952\) 0 0
\(953\) −13.7129 −0.444203 −0.222102 0.975024i \(-0.571292\pi\)
−0.222102 + 0.975024i \(0.571292\pi\)
\(954\) −5.42284 −0.175571
\(955\) −39.6375 −1.28264
\(956\) −4.41126 −0.142670
\(957\) 4.95287 0.160104
\(958\) −7.15349 −0.231119
\(959\) 0 0
\(960\) −1.49111 −0.0481254
\(961\) 50.2085 1.61963
\(962\) 4.98222 0.160633
\(963\) 11.5027 0.370669
\(964\) 3.84486 0.123835
\(965\) −7.11563 −0.229060
\(966\) 0 0
\(967\) −2.29900 −0.0739307 −0.0369654 0.999317i \(-0.511769\pi\)
−0.0369654 + 0.999317i \(0.511769\pi\)
\(968\) −10.3690 −0.333272
\(969\) −4.49111 −0.144275
\(970\) −25.7918 −0.828125
\(971\) 41.2755 1.32459 0.662297 0.749241i \(-0.269582\pi\)
0.662297 + 0.749241i \(0.269582\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) −9.76867 −0.313008
\(975\) −5.55318 −0.177844
\(976\) 12.5825 0.402757
\(977\) 41.2445 1.31953 0.659765 0.751472i \(-0.270656\pi\)
0.659765 + 0.751472i \(0.270656\pi\)
\(978\) −9.02935 −0.288727
\(979\) −1.81551 −0.0580239
\(980\) 0 0
\(981\) 3.47333 0.110895
\(982\) 9.03556 0.288336
\(983\) 18.3623 0.585665 0.292832 0.956164i \(-0.405402\pi\)
0.292832 + 0.956164i \(0.405402\pi\)
\(984\) −4.18449 −0.133397
\(985\) 23.8805 0.760895
\(986\) 28.0020 0.891765
\(987\) 0 0
\(988\) 2.00000 0.0636285
\(989\) −15.6164 −0.496572
\(990\) 1.18449 0.0376456
\(991\) −54.7673 −1.73974 −0.869870 0.493280i \(-0.835798\pi\)
−0.869870 + 0.493280i \(0.835798\pi\)
\(992\) −9.01157 −0.286118
\(993\) 15.7377 0.499420
\(994\) 0 0
\(995\) 16.1548 0.512143
\(996\) −1.57096 −0.0497777
\(997\) −32.5941 −1.03227 −0.516133 0.856509i \(-0.672629\pi\)
−0.516133 + 0.856509i \(0.672629\pi\)
\(998\) 23.9405 0.757822
\(999\) 2.49111 0.0788153
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 5586.2.a.ca.1.3 4
7.3 odd 6 798.2.j.k.457.3 8
7.5 odd 6 798.2.j.k.571.3 yes 8
7.6 odd 2 5586.2.a.cb.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
798.2.j.k.457.3 8 7.3 odd 6
798.2.j.k.571.3 yes 8 7.5 odd 6
5586.2.a.ca.1.3 4 1.1 even 1 trivial
5586.2.a.cb.1.2 4 7.6 odd 2