Properties

Label 4598.2.a.bk.1.3
Level $4598$
Weight $2$
Character 4598.1
Self dual yes
Analytic conductor $36.715$
Analytic rank $1$
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] = [4598,2,Mod(1,4598)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4598, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4598.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4598 = 2 \cdot 11^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4598.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: \(36.7152148494\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.568.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 6x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(3.12489\) of defining polynomial
Character \(\chi\) \(=\) 4598.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +2.12489 q^{3} +1.00000 q^{4} -3.12489 q^{5} -2.12489 q^{6} +0.515138 q^{7} -1.00000 q^{8} +1.51514 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +2.12489 q^{3} +1.00000 q^{4} -3.12489 q^{5} -2.12489 q^{6} +0.515138 q^{7} -1.00000 q^{8} +1.51514 q^{9} +3.12489 q^{10} +2.12489 q^{12} +0.484862 q^{13} -0.515138 q^{14} -6.64002 q^{15} +1.00000 q^{16} +1.51514 q^{17} -1.51514 q^{18} +1.00000 q^{19} -3.12489 q^{20} +1.09461 q^{21} +0.515138 q^{23} -2.12489 q^{24} +4.76491 q^{25} -0.484862 q^{26} -3.15516 q^{27} +0.515138 q^{28} +2.60975 q^{29} +6.64002 q^{30} -5.28005 q^{31} -1.00000 q^{32} -1.51514 q^{34} -1.60975 q^{35} +1.51514 q^{36} -10.6097 q^{37} -1.00000 q^{38} +1.03028 q^{39} +3.12489 q^{40} +6.09461 q^{41} -1.09461 q^{42} -1.03028 q^{43} -4.73463 q^{45} -0.515138 q^{46} +9.01468 q^{47} +2.12489 q^{48} -6.73463 q^{49} -4.76491 q^{50} +3.21949 q^{51} +0.484862 q^{52} -0.670300 q^{53} +3.15516 q^{54} -0.515138 q^{56} +2.12489 q^{57} -2.60975 q^{58} -2.90539 q^{59} -6.64002 q^{60} -9.92007 q^{61} +5.28005 q^{62} +0.780505 q^{63} +1.00000 q^{64} -1.51514 q^{65} -2.06055 q^{67} +1.51514 q^{68} +1.09461 q^{69} +1.60975 q^{70} -2.64002 q^{71} -1.51514 q^{72} +7.52982 q^{73} +10.6097 q^{74} +10.1249 q^{75} +1.00000 q^{76} -1.03028 q^{78} -1.60975 q^{79} -3.12489 q^{80} -11.2498 q^{81} -6.09461 q^{82} +1.60975 q^{83} +1.09461 q^{84} -4.73463 q^{85} +1.03028 q^{86} +5.54541 q^{87} -1.06433 q^{89} +4.73463 q^{90} +0.249771 q^{91} +0.515138 q^{92} -11.2195 q^{93} -9.01468 q^{94} -3.12489 q^{95} -2.12489 q^{96} +7.76491 q^{97} +6.73463 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} - 2 q^{3} + 3 q^{4} - q^{5} + 2 q^{6} + 2 q^{7} - 3 q^{8} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{2} - 2 q^{3} + 3 q^{4} - q^{5} + 2 q^{6} + 2 q^{7} - 3 q^{8} + 5 q^{9} + q^{10} - 2 q^{12} + q^{13} - 2 q^{14} - 12 q^{15} + 3 q^{16} + 5 q^{17} - 5 q^{18} + 3 q^{19} - q^{20} - 6 q^{21} + 2 q^{23} + 2 q^{24} - 2 q^{25} - q^{26} - 2 q^{27} + 2 q^{28} - q^{29} + 12 q^{30} - 3 q^{32} - 5 q^{34} + 4 q^{35} + 5 q^{36} - 23 q^{37} - 3 q^{38} + 4 q^{39} + q^{40} + 9 q^{41} + 6 q^{42} - 4 q^{43} + 3 q^{45} - 2 q^{46} - 6 q^{47} - 2 q^{48} - 3 q^{49} + 2 q^{50} - 8 q^{51} + q^{52} + 5 q^{53} + 2 q^{54} - 2 q^{56} - 2 q^{57} + q^{58} - 18 q^{59} - 12 q^{60} - 6 q^{61} + 20 q^{63} + 3 q^{64} - 5 q^{65} - 8 q^{67} + 5 q^{68} - 6 q^{69} - 4 q^{70} - 5 q^{72} - 10 q^{73} + 23 q^{74} + 22 q^{75} + 3 q^{76} - 4 q^{78} + 4 q^{79} - q^{80} - 17 q^{81} - 9 q^{82} - 4 q^{83} - 6 q^{84} + 3 q^{85} + 4 q^{86} + 18 q^{87} + 7 q^{89} - 3 q^{90} - 16 q^{91} + 2 q^{92} - 16 q^{93} + 6 q^{94} - q^{95} + 2 q^{96} + 7 q^{97} + 3 q^{98}+O(q^{100}) \) 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
\(3\) 2.12489 1.22680 0.613402 0.789771i \(-0.289801\pi\)
0.613402 + 0.789771i \(0.289801\pi\)
\(4\) 1.00000 0.500000
\(5\) −3.12489 −1.39749 −0.698746 0.715370i \(-0.746258\pi\)
−0.698746 + 0.715370i \(0.746258\pi\)
\(6\) −2.12489 −0.867481
\(7\) 0.515138 0.194704 0.0973519 0.995250i \(-0.468963\pi\)
0.0973519 + 0.995250i \(0.468963\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.51514 0.505046
\(10\) 3.12489 0.988176
\(11\) 0 0
\(12\) 2.12489 0.613402
\(13\) 0.484862 0.134477 0.0672383 0.997737i \(-0.478581\pi\)
0.0672383 + 0.997737i \(0.478581\pi\)
\(14\) −0.515138 −0.137676
\(15\) −6.64002 −1.71445
\(16\) 1.00000 0.250000
\(17\) 1.51514 0.367475 0.183737 0.982975i \(-0.441180\pi\)
0.183737 + 0.982975i \(0.441180\pi\)
\(18\) −1.51514 −0.357121
\(19\) 1.00000 0.229416
\(20\) −3.12489 −0.698746
\(21\) 1.09461 0.238863
\(22\) 0 0
\(23\) 0.515138 0.107414 0.0537069 0.998557i \(-0.482896\pi\)
0.0537069 + 0.998557i \(0.482896\pi\)
\(24\) −2.12489 −0.433740
\(25\) 4.76491 0.952982
\(26\) −0.484862 −0.0950893
\(27\) −3.15516 −0.607211
\(28\) 0.515138 0.0973519
\(29\) 2.60975 0.484618 0.242309 0.970199i \(-0.422095\pi\)
0.242309 + 0.970199i \(0.422095\pi\)
\(30\) 6.64002 1.21230
\(31\) −5.28005 −0.948324 −0.474162 0.880438i \(-0.657249\pi\)
−0.474162 + 0.880438i \(0.657249\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −1.51514 −0.259844
\(35\) −1.60975 −0.272097
\(36\) 1.51514 0.252523
\(37\) −10.6097 −1.74423 −0.872116 0.489299i \(-0.837253\pi\)
−0.872116 + 0.489299i \(0.837253\pi\)
\(38\) −1.00000 −0.162221
\(39\) 1.03028 0.164976
\(40\) 3.12489 0.494088
\(41\) 6.09461 0.951818 0.475909 0.879495i \(-0.342119\pi\)
0.475909 + 0.879495i \(0.342119\pi\)
\(42\) −1.09461 −0.168902
\(43\) −1.03028 −0.157116 −0.0785578 0.996910i \(-0.525032\pi\)
−0.0785578 + 0.996910i \(0.525032\pi\)
\(44\) 0 0
\(45\) −4.73463 −0.705797
\(46\) −0.515138 −0.0759530
\(47\) 9.01468 1.31493 0.657463 0.753487i \(-0.271630\pi\)
0.657463 + 0.753487i \(0.271630\pi\)
\(48\) 2.12489 0.306701
\(49\) −6.73463 −0.962090
\(50\) −4.76491 −0.673860
\(51\) 3.21949 0.450819
\(52\) 0.484862 0.0672383
\(53\) −0.670300 −0.0920727 −0.0460364 0.998940i \(-0.514659\pi\)
−0.0460364 + 0.998940i \(0.514659\pi\)
\(54\) 3.15516 0.429363
\(55\) 0 0
\(56\) −0.515138 −0.0688382
\(57\) 2.12489 0.281448
\(58\) −2.60975 −0.342677
\(59\) −2.90539 −0.378250 −0.189125 0.981953i \(-0.560565\pi\)
−0.189125 + 0.981953i \(0.560565\pi\)
\(60\) −6.64002 −0.857223
\(61\) −9.92007 −1.27013 −0.635067 0.772457i \(-0.719028\pi\)
−0.635067 + 0.772457i \(0.719028\pi\)
\(62\) 5.28005 0.670567
\(63\) 0.780505 0.0983344
\(64\) 1.00000 0.125000
\(65\) −1.51514 −0.187930
\(66\) 0 0
\(67\) −2.06055 −0.251737 −0.125868 0.992047i \(-0.540172\pi\)
−0.125868 + 0.992047i \(0.540172\pi\)
\(68\) 1.51514 0.183737
\(69\) 1.09461 0.131775
\(70\) 1.60975 0.192402
\(71\) −2.64002 −0.313313 −0.156657 0.987653i \(-0.550072\pi\)
−0.156657 + 0.987653i \(0.550072\pi\)
\(72\) −1.51514 −0.178561
\(73\) 7.52982 0.881299 0.440649 0.897679i \(-0.354748\pi\)
0.440649 + 0.897679i \(0.354748\pi\)
\(74\) 10.6097 1.23336
\(75\) 10.1249 1.16912
\(76\) 1.00000 0.114708
\(77\) 0 0
\(78\) −1.03028 −0.116656
\(79\) −1.60975 −0.181111 −0.0905554 0.995891i \(-0.528864\pi\)
−0.0905554 + 0.995891i \(0.528864\pi\)
\(80\) −3.12489 −0.349373
\(81\) −11.2498 −1.24997
\(82\) −6.09461 −0.673037
\(83\) 1.60975 0.176693 0.0883464 0.996090i \(-0.471842\pi\)
0.0883464 + 0.996090i \(0.471842\pi\)
\(84\) 1.09461 0.119432
\(85\) −4.73463 −0.513543
\(86\) 1.03028 0.111098
\(87\) 5.54541 0.594531
\(88\) 0 0
\(89\) −1.06433 −0.112819 −0.0564095 0.998408i \(-0.517965\pi\)
−0.0564095 + 0.998408i \(0.517965\pi\)
\(90\) 4.73463 0.499074
\(91\) 0.249771 0.0261831
\(92\) 0.515138 0.0537069
\(93\) −11.2195 −1.16341
\(94\) −9.01468 −0.929793
\(95\) −3.12489 −0.320606
\(96\) −2.12489 −0.216870
\(97\) 7.76491 0.788407 0.394204 0.919023i \(-0.371021\pi\)
0.394204 + 0.919023i \(0.371021\pi\)
\(98\) 6.73463 0.680301
\(99\) 0 0
\(100\) 4.76491 0.476491
\(101\) −6.70058 −0.666732 −0.333366 0.942797i \(-0.608185\pi\)
−0.333366 + 0.942797i \(0.608185\pi\)
\(102\) −3.21949 −0.318777
\(103\) −7.54920 −0.743844 −0.371922 0.928264i \(-0.621301\pi\)
−0.371922 + 0.928264i \(0.621301\pi\)
\(104\) −0.484862 −0.0475446
\(105\) −3.42053 −0.333809
\(106\) 0.670300 0.0651052
\(107\) 4.84484 0.468368 0.234184 0.972192i \(-0.424758\pi\)
0.234184 + 0.972192i \(0.424758\pi\)
\(108\) −3.15516 −0.303606
\(109\) −7.82924 −0.749905 −0.374953 0.927044i \(-0.622341\pi\)
−0.374953 + 0.927044i \(0.622341\pi\)
\(110\) 0 0
\(111\) −22.5445 −2.13983
\(112\) 0.515138 0.0486760
\(113\) −10.8751 −1.02304 −0.511522 0.859270i \(-0.670918\pi\)
−0.511522 + 0.859270i \(0.670918\pi\)
\(114\) −2.12489 −0.199014
\(115\) −1.60975 −0.150110
\(116\) 2.60975 0.242309
\(117\) 0.734633 0.0679168
\(118\) 2.90539 0.267463
\(119\) 0.780505 0.0715488
\(120\) 6.64002 0.606148
\(121\) 0 0
\(122\) 9.92007 0.898121
\(123\) 12.9503 1.16769
\(124\) −5.28005 −0.474162
\(125\) 0.734633 0.0657076
\(126\) −0.780505 −0.0695329
\(127\) 0.780505 0.0692586 0.0346293 0.999400i \(-0.488975\pi\)
0.0346293 + 0.999400i \(0.488975\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −2.18922 −0.192750
\(130\) 1.51514 0.132886
\(131\) 18.3103 1.59978 0.799890 0.600146i \(-0.204891\pi\)
0.799890 + 0.600146i \(0.204891\pi\)
\(132\) 0 0
\(133\) 0.515138 0.0446681
\(134\) 2.06055 0.178005
\(135\) 9.85952 0.848572
\(136\) −1.51514 −0.129922
\(137\) 0.969724 0.0828491 0.0414246 0.999142i \(-0.486810\pi\)
0.0414246 + 0.999142i \(0.486810\pi\)
\(138\) −1.09461 −0.0931793
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) −1.60975 −0.136048
\(141\) 19.1552 1.61316
\(142\) 2.64002 0.221546
\(143\) 0 0
\(144\) 1.51514 0.126262
\(145\) −8.15516 −0.677249
\(146\) −7.52982 −0.623172
\(147\) −14.3103 −1.18030
\(148\) −10.6097 −0.872116
\(149\) 16.0752 1.31693 0.658467 0.752609i \(-0.271205\pi\)
0.658467 + 0.752609i \(0.271205\pi\)
\(150\) −10.1249 −0.826693
\(151\) −22.2304 −1.80908 −0.904542 0.426385i \(-0.859787\pi\)
−0.904542 + 0.426385i \(0.859787\pi\)
\(152\) −1.00000 −0.0811107
\(153\) 2.29564 0.185592
\(154\) 0 0
\(155\) 16.4995 1.32528
\(156\) 1.03028 0.0824881
\(157\) −20.2304 −1.61456 −0.807281 0.590168i \(-0.799062\pi\)
−0.807281 + 0.590168i \(0.799062\pi\)
\(158\) 1.60975 0.128065
\(159\) −1.42431 −0.112955
\(160\) 3.12489 0.247044
\(161\) 0.265367 0.0209139
\(162\) 11.2498 0.883865
\(163\) −9.03028 −0.707306 −0.353653 0.935377i \(-0.615061\pi\)
−0.353653 + 0.935377i \(0.615061\pi\)
\(164\) 6.09461 0.475909
\(165\) 0 0
\(166\) −1.60975 −0.124941
\(167\) −12.2498 −0.947916 −0.473958 0.880548i \(-0.657175\pi\)
−0.473958 + 0.880548i \(0.657175\pi\)
\(168\) −1.09461 −0.0844509
\(169\) −12.7649 −0.981916
\(170\) 4.73463 0.363130
\(171\) 1.51514 0.115866
\(172\) −1.03028 −0.0785578
\(173\) −23.5942 −1.79383 −0.896915 0.442203i \(-0.854197\pi\)
−0.896915 + 0.442203i \(0.854197\pi\)
\(174\) −5.54541 −0.420397
\(175\) 2.45459 0.185549
\(176\) 0 0
\(177\) −6.17362 −0.464038
\(178\) 1.06433 0.0797751
\(179\) −23.6547 −1.76804 −0.884018 0.467453i \(-0.845172\pi\)
−0.884018 + 0.467453i \(0.845172\pi\)
\(180\) −4.73463 −0.352899
\(181\) −15.7044 −1.16730 −0.583648 0.812007i \(-0.698375\pi\)
−0.583648 + 0.812007i \(0.698375\pi\)
\(182\) −0.249771 −0.0185142
\(183\) −21.0790 −1.55821
\(184\) −0.515138 −0.0379765
\(185\) 33.1542 2.43755
\(186\) 11.2195 0.822653
\(187\) 0 0
\(188\) 9.01468 0.657463
\(189\) −1.62534 −0.118226
\(190\) 3.12489 0.226703
\(191\) 18.0450 1.30569 0.652844 0.757493i \(-0.273576\pi\)
0.652844 + 0.757493i \(0.273576\pi\)
\(192\) 2.12489 0.153350
\(193\) 18.2645 1.31470 0.657352 0.753584i \(-0.271677\pi\)
0.657352 + 0.753584i \(0.271677\pi\)
\(194\) −7.76491 −0.557488
\(195\) −3.21949 −0.230553
\(196\) −6.73463 −0.481045
\(197\) −21.5445 −1.53498 −0.767491 0.641060i \(-0.778495\pi\)
−0.767491 + 0.641060i \(0.778495\pi\)
\(198\) 0 0
\(199\) 2.18922 0.155190 0.0775948 0.996985i \(-0.475276\pi\)
0.0775948 + 0.996985i \(0.475276\pi\)
\(200\) −4.76491 −0.336930
\(201\) −4.37844 −0.308831
\(202\) 6.70058 0.471451
\(203\) 1.34438 0.0943570
\(204\) 3.21949 0.225410
\(205\) −19.0450 −1.33016
\(206\) 7.54920 0.525977
\(207\) 0.780505 0.0542489
\(208\) 0.484862 0.0336191
\(209\) 0 0
\(210\) 3.42053 0.236039
\(211\) −16.5639 −1.14030 −0.570152 0.821539i \(-0.693116\pi\)
−0.570152 + 0.821539i \(0.693116\pi\)
\(212\) −0.670300 −0.0460364
\(213\) −5.60975 −0.384374
\(214\) −4.84484 −0.331186
\(215\) 3.21949 0.219568
\(216\) 3.15516 0.214682
\(217\) −2.71995 −0.184642
\(218\) 7.82924 0.530263
\(219\) 16.0000 1.08118
\(220\) 0 0
\(221\) 0.734633 0.0494167
\(222\) 22.5445 1.51309
\(223\) −15.7990 −1.05798 −0.528989 0.848629i \(-0.677429\pi\)
−0.528989 + 0.848629i \(0.677429\pi\)
\(224\) −0.515138 −0.0344191
\(225\) 7.21949 0.481300
\(226\) 10.8751 0.723402
\(227\) −22.2791 −1.47872 −0.739359 0.673312i \(-0.764871\pi\)
−0.739359 + 0.673312i \(0.764871\pi\)
\(228\) 2.12489 0.140724
\(229\) 2.98440 0.197215 0.0986075 0.995126i \(-0.468561\pi\)
0.0986075 + 0.995126i \(0.468561\pi\)
\(230\) 1.60975 0.106144
\(231\) 0 0
\(232\) −2.60975 −0.171338
\(233\) −1.96972 −0.129041 −0.0645205 0.997916i \(-0.520552\pi\)
−0.0645205 + 0.997916i \(0.520552\pi\)
\(234\) −0.734633 −0.0480244
\(235\) −28.1698 −1.83760
\(236\) −2.90539 −0.189125
\(237\) −3.42053 −0.222187
\(238\) −0.780505 −0.0505926
\(239\) 28.3553 1.83415 0.917075 0.398714i \(-0.130543\pi\)
0.917075 + 0.398714i \(0.130543\pi\)
\(240\) −6.64002 −0.428612
\(241\) 3.07901 0.198337 0.0991683 0.995071i \(-0.468382\pi\)
0.0991683 + 0.995071i \(0.468382\pi\)
\(242\) 0 0
\(243\) −14.4390 −0.926262
\(244\) −9.92007 −0.635067
\(245\) 21.0450 1.34451
\(246\) −12.9503 −0.825684
\(247\) 0.484862 0.0308510
\(248\) 5.28005 0.335283
\(249\) 3.42053 0.216767
\(250\) −0.734633 −0.0464623
\(251\) −12.7493 −0.804729 −0.402365 0.915479i \(-0.631812\pi\)
−0.402365 + 0.915479i \(0.631812\pi\)
\(252\) 0.780505 0.0491672
\(253\) 0 0
\(254\) −0.780505 −0.0489733
\(255\) −10.0606 −0.630016
\(256\) 1.00000 0.0625000
\(257\) −6.95504 −0.433844 −0.216922 0.976189i \(-0.569602\pi\)
−0.216922 + 0.976189i \(0.569602\pi\)
\(258\) 2.18922 0.136295
\(259\) −5.46548 −0.339609
\(260\) −1.51514 −0.0939649
\(261\) 3.95413 0.244754
\(262\) −18.3103 −1.13122
\(263\) −10.6888 −0.659097 −0.329549 0.944139i \(-0.606897\pi\)
−0.329549 + 0.944139i \(0.606897\pi\)
\(264\) 0 0
\(265\) 2.09461 0.128671
\(266\) −0.515138 −0.0315851
\(267\) −2.26159 −0.138407
\(268\) −2.06055 −0.125868
\(269\) −4.92007 −0.299982 −0.149991 0.988687i \(-0.547924\pi\)
−0.149991 + 0.988687i \(0.547924\pi\)
\(270\) −9.85952 −0.600031
\(271\) −9.52982 −0.578895 −0.289448 0.957194i \(-0.593472\pi\)
−0.289448 + 0.957194i \(0.593472\pi\)
\(272\) 1.51514 0.0918687
\(273\) 0.530734 0.0321215
\(274\) −0.969724 −0.0585832
\(275\) 0 0
\(276\) 1.09461 0.0658877
\(277\) 13.5639 0.814974 0.407487 0.913211i \(-0.366405\pi\)
0.407487 + 0.913211i \(0.366405\pi\)
\(278\) 4.00000 0.239904
\(279\) −8.00000 −0.478947
\(280\) 1.60975 0.0962008
\(281\) 2.57947 0.153878 0.0769392 0.997036i \(-0.475485\pi\)
0.0769392 + 0.997036i \(0.475485\pi\)
\(282\) −19.1552 −1.14067
\(283\) −17.7796 −1.05689 −0.528443 0.848969i \(-0.677224\pi\)
−0.528443 + 0.848969i \(0.677224\pi\)
\(284\) −2.64002 −0.156657
\(285\) −6.64002 −0.393321
\(286\) 0 0
\(287\) 3.13957 0.185323
\(288\) −1.51514 −0.0892804
\(289\) −14.7044 −0.864962
\(290\) 8.15516 0.478888
\(291\) 16.4995 0.967220
\(292\) 7.52982 0.440649
\(293\) 1.95035 0.113940 0.0569702 0.998376i \(-0.481856\pi\)
0.0569702 + 0.998376i \(0.481856\pi\)
\(294\) 14.3103 0.834595
\(295\) 9.07901 0.528601
\(296\) 10.6097 0.616679
\(297\) 0 0
\(298\) −16.0752 −0.931213
\(299\) 0.249771 0.0144446
\(300\) 10.1249 0.584561
\(301\) −0.530734 −0.0305910
\(302\) 22.2304 1.27922
\(303\) −14.2380 −0.817949
\(304\) 1.00000 0.0573539
\(305\) 30.9991 1.77500
\(306\) −2.29564 −0.131233
\(307\) −20.1542 −1.15026 −0.575132 0.818061i \(-0.695049\pi\)
−0.575132 + 0.818061i \(0.695049\pi\)
\(308\) 0 0
\(309\) −16.0412 −0.912551
\(310\) −16.4995 −0.937111
\(311\) −4.90917 −0.278374 −0.139187 0.990266i \(-0.544449\pi\)
−0.139187 + 0.990266i \(0.544449\pi\)
\(312\) −1.03028 −0.0583279
\(313\) −27.5601 −1.55779 −0.778894 0.627155i \(-0.784219\pi\)
−0.778894 + 0.627155i \(0.784219\pi\)
\(314\) 20.2304 1.14167
\(315\) −2.43899 −0.137421
\(316\) −1.60975 −0.0905554
\(317\) 28.2791 1.58831 0.794157 0.607713i \(-0.207913\pi\)
0.794157 + 0.607713i \(0.207913\pi\)
\(318\) 1.42431 0.0798713
\(319\) 0 0
\(320\) −3.12489 −0.174686
\(321\) 10.2947 0.574596
\(322\) −0.265367 −0.0147883
\(323\) 1.51514 0.0843045
\(324\) −11.2498 −0.624987
\(325\) 2.31032 0.128154
\(326\) 9.03028 0.500141
\(327\) −16.6362 −0.919986
\(328\) −6.09461 −0.336519
\(329\) 4.64380 0.256021
\(330\) 0 0
\(331\) 26.0937 1.43424 0.717120 0.696950i \(-0.245460\pi\)
0.717120 + 0.696950i \(0.245460\pi\)
\(332\) 1.60975 0.0883464
\(333\) −16.0752 −0.880917
\(334\) 12.2498 0.670278
\(335\) 6.43899 0.351800
\(336\) 1.09461 0.0597158
\(337\) 19.2148 1.04670 0.523348 0.852119i \(-0.324683\pi\)
0.523348 + 0.852119i \(0.324683\pi\)
\(338\) 12.7649 0.694320
\(339\) −23.1084 −1.25507
\(340\) −4.73463 −0.256772
\(341\) 0 0
\(342\) −1.51514 −0.0819293
\(343\) −7.07523 −0.382027
\(344\) 1.03028 0.0555488
\(345\) −3.42053 −0.184155
\(346\) 23.5942 1.26843
\(347\) −3.17076 −0.170215 −0.0851076 0.996372i \(-0.527123\pi\)
−0.0851076 + 0.996372i \(0.527123\pi\)
\(348\) 5.54541 0.297265
\(349\) 11.5445 0.617963 0.308981 0.951068i \(-0.400012\pi\)
0.308981 + 0.951068i \(0.400012\pi\)
\(350\) −2.45459 −0.131203
\(351\) −1.52982 −0.0816556
\(352\) 0 0
\(353\) 5.43899 0.289488 0.144744 0.989469i \(-0.453764\pi\)
0.144744 + 0.989469i \(0.453764\pi\)
\(354\) 6.17362 0.328124
\(355\) 8.24977 0.437852
\(356\) −1.06433 −0.0564095
\(357\) 1.65848 0.0877763
\(358\) 23.6547 1.25019
\(359\) −37.0890 −1.95748 −0.978741 0.205099i \(-0.934248\pi\)
−0.978741 + 0.205099i \(0.934248\pi\)
\(360\) 4.73463 0.249537
\(361\) 1.00000 0.0526316
\(362\) 15.7044 0.825403
\(363\) 0 0
\(364\) 0.249771 0.0130915
\(365\) −23.5298 −1.23161
\(366\) 21.0790 1.10182
\(367\) 11.8245 0.617236 0.308618 0.951186i \(-0.400133\pi\)
0.308618 + 0.951186i \(0.400133\pi\)
\(368\) 0.515138 0.0268534
\(369\) 9.23417 0.480712
\(370\) −33.1542 −1.72361
\(371\) −0.345297 −0.0179269
\(372\) −11.2195 −0.581704
\(373\) 29.4948 1.52719 0.763593 0.645698i \(-0.223434\pi\)
0.763593 + 0.645698i \(0.223434\pi\)
\(374\) 0 0
\(375\) 1.56101 0.0806102
\(376\) −9.01468 −0.464897
\(377\) 1.26537 0.0651697
\(378\) 1.62534 0.0835987
\(379\) 29.4655 1.51354 0.756770 0.653681i \(-0.226776\pi\)
0.756770 + 0.653681i \(0.226776\pi\)
\(380\) −3.12489 −0.160303
\(381\) 1.65848 0.0849667
\(382\) −18.0450 −0.923260
\(383\) 11.8789 0.606983 0.303492 0.952834i \(-0.401848\pi\)
0.303492 + 0.952834i \(0.401848\pi\)
\(384\) −2.12489 −0.108435
\(385\) 0 0
\(386\) −18.2645 −0.929636
\(387\) −1.56101 −0.0793506
\(388\) 7.76491 0.394204
\(389\) 13.6244 0.690786 0.345393 0.938458i \(-0.387746\pi\)
0.345393 + 0.938458i \(0.387746\pi\)
\(390\) 3.21949 0.163025
\(391\) 0.780505 0.0394718
\(392\) 6.73463 0.340150
\(393\) 38.9073 1.96262
\(394\) 21.5445 1.08540
\(395\) 5.03028 0.253101
\(396\) 0 0
\(397\) 17.7943 0.893069 0.446534 0.894766i \(-0.352658\pi\)
0.446534 + 0.894766i \(0.352658\pi\)
\(398\) −2.18922 −0.109736
\(399\) 1.09461 0.0547990
\(400\) 4.76491 0.238245
\(401\) 19.0256 0.950092 0.475046 0.879961i \(-0.342431\pi\)
0.475046 + 0.879961i \(0.342431\pi\)
\(402\) 4.37844 0.218377
\(403\) −2.56009 −0.127527
\(404\) −6.70058 −0.333366
\(405\) 35.1542 1.74683
\(406\) −1.34438 −0.0667205
\(407\) 0 0
\(408\) −3.21949 −0.159389
\(409\) −5.58325 −0.276074 −0.138037 0.990427i \(-0.544079\pi\)
−0.138037 + 0.990427i \(0.544079\pi\)
\(410\) 19.0450 0.940563
\(411\) 2.06055 0.101640
\(412\) −7.54920 −0.371922
\(413\) −1.49668 −0.0736467
\(414\) −0.780505 −0.0383597
\(415\) −5.03028 −0.246927
\(416\) −0.484862 −0.0237723
\(417\) −8.49954 −0.416224
\(418\) 0 0
\(419\) 20.1698 0.985361 0.492681 0.870210i \(-0.336017\pi\)
0.492681 + 0.870210i \(0.336017\pi\)
\(420\) −3.42053 −0.166905
\(421\) −17.8292 −0.868944 −0.434472 0.900685i \(-0.643065\pi\)
−0.434472 + 0.900685i \(0.643065\pi\)
\(422\) 16.5639 0.806317
\(423\) 13.6585 0.664098
\(424\) 0.670300 0.0325526
\(425\) 7.21949 0.350197
\(426\) 5.60975 0.271793
\(427\) −5.11021 −0.247300
\(428\) 4.84484 0.234184
\(429\) 0 0
\(430\) −3.21949 −0.155258
\(431\) −10.7687 −0.518710 −0.259355 0.965782i \(-0.583510\pi\)
−0.259355 + 0.965782i \(0.583510\pi\)
\(432\) −3.15516 −0.151803
\(433\) 40.6353 1.95281 0.976405 0.215949i \(-0.0692846\pi\)
0.976405 + 0.215949i \(0.0692846\pi\)
\(434\) 2.71995 0.130562
\(435\) −17.3288 −0.830852
\(436\) −7.82924 −0.374953
\(437\) 0.515138 0.0246424
\(438\) −16.0000 −0.764510
\(439\) 15.0596 0.718757 0.359379 0.933192i \(-0.382989\pi\)
0.359379 + 0.933192i \(0.382989\pi\)
\(440\) 0 0
\(441\) −10.2039 −0.485900
\(442\) −0.734633 −0.0349429
\(443\) −26.1892 −1.24429 −0.622144 0.782903i \(-0.713738\pi\)
−0.622144 + 0.782903i \(0.713738\pi\)
\(444\) −22.5445 −1.06991
\(445\) 3.32592 0.157664
\(446\) 15.7990 0.748103
\(447\) 34.1580 1.61562
\(448\) 0.515138 0.0243380
\(449\) 30.5748 1.44291 0.721456 0.692460i \(-0.243473\pi\)
0.721456 + 0.692460i \(0.243473\pi\)
\(450\) −7.21949 −0.340330
\(451\) 0 0
\(452\) −10.8751 −0.511522
\(453\) −47.2370 −2.21939
\(454\) 22.2791 1.04561
\(455\) −0.780505 −0.0365907
\(456\) −2.12489 −0.0995069
\(457\) 16.4693 0.770400 0.385200 0.922833i \(-0.374133\pi\)
0.385200 + 0.922833i \(0.374133\pi\)
\(458\) −2.98440 −0.139452
\(459\) −4.78051 −0.223135
\(460\) −1.60975 −0.0750549
\(461\) −21.8742 −1.01878 −0.509391 0.860535i \(-0.670129\pi\)
−0.509391 + 0.860535i \(0.670129\pi\)
\(462\) 0 0
\(463\) 27.8089 1.29239 0.646196 0.763172i \(-0.276359\pi\)
0.646196 + 0.763172i \(0.276359\pi\)
\(464\) 2.60975 0.121154
\(465\) 35.0596 1.62585
\(466\) 1.96972 0.0912457
\(467\) 20.3709 0.942652 0.471326 0.881959i \(-0.343776\pi\)
0.471326 + 0.881959i \(0.343776\pi\)
\(468\) 0.734633 0.0339584
\(469\) −1.06147 −0.0490141
\(470\) 28.1698 1.29938
\(471\) −42.9873 −1.98075
\(472\) 2.90539 0.133731
\(473\) 0 0
\(474\) 3.42053 0.157110
\(475\) 4.76491 0.218629
\(476\) 0.780505 0.0357744
\(477\) −1.01560 −0.0465010
\(478\) −28.3553 −1.29694
\(479\) −32.2342 −1.47282 −0.736409 0.676537i \(-0.763480\pi\)
−0.736409 + 0.676537i \(0.763480\pi\)
\(480\) 6.64002 0.303074
\(481\) −5.14426 −0.234558
\(482\) −3.07901 −0.140245
\(483\) 0.563875 0.0256572
\(484\) 0 0
\(485\) −24.2645 −1.10179
\(486\) 14.4390 0.654966
\(487\) −1.81078 −0.0820543 −0.0410272 0.999158i \(-0.513063\pi\)
−0.0410272 + 0.999158i \(0.513063\pi\)
\(488\) 9.92007 0.449060
\(489\) −19.1883 −0.867725
\(490\) −21.0450 −0.950714
\(491\) −4.65940 −0.210276 −0.105138 0.994458i \(-0.533528\pi\)
−0.105138 + 0.994458i \(0.533528\pi\)
\(492\) 12.9503 0.583847
\(493\) 3.95413 0.178085
\(494\) −0.484862 −0.0218150
\(495\) 0 0
\(496\) −5.28005 −0.237081
\(497\) −1.35998 −0.0610033
\(498\) −3.42053 −0.153278
\(499\) −7.63911 −0.341973 −0.170987 0.985273i \(-0.554696\pi\)
−0.170987 + 0.985273i \(0.554696\pi\)
\(500\) 0.734633 0.0328538
\(501\) −26.0294 −1.16291
\(502\) 12.7493 0.569030
\(503\) −8.26537 −0.368535 −0.184267 0.982876i \(-0.558991\pi\)
−0.184267 + 0.982876i \(0.558991\pi\)
\(504\) −0.780505 −0.0347665
\(505\) 20.9385 0.931752
\(506\) 0 0
\(507\) −27.1240 −1.20462
\(508\) 0.780505 0.0346293
\(509\) 11.3756 0.504213 0.252107 0.967699i \(-0.418877\pi\)
0.252107 + 0.967699i \(0.418877\pi\)
\(510\) 10.0606 0.445489
\(511\) 3.87890 0.171592
\(512\) −1.00000 −0.0441942
\(513\) −3.15516 −0.139304
\(514\) 6.95504 0.306774
\(515\) 23.5904 1.03952
\(516\) −2.18922 −0.0963750
\(517\) 0 0
\(518\) 5.46548 0.240140
\(519\) −50.1349 −2.20068
\(520\) 1.51514 0.0664432
\(521\) −2.62065 −0.114813 −0.0574063 0.998351i \(-0.518283\pi\)
−0.0574063 + 0.998351i \(0.518283\pi\)
\(522\) −3.95413 −0.173067
\(523\) −15.3150 −0.669679 −0.334840 0.942275i \(-0.608682\pi\)
−0.334840 + 0.942275i \(0.608682\pi\)
\(524\) 18.3103 0.799890
\(525\) 5.21571 0.227632
\(526\) 10.6888 0.466052
\(527\) −8.00000 −0.348485
\(528\) 0 0
\(529\) −22.7346 −0.988462
\(530\) −2.09461 −0.0909840
\(531\) −4.40207 −0.191033
\(532\) 0.515138 0.0223341
\(533\) 2.95504 0.127997
\(534\) 2.26159 0.0978684
\(535\) −15.1396 −0.654540
\(536\) 2.06055 0.0890023
\(537\) −50.2635 −2.16903
\(538\) 4.92007 0.212119
\(539\) 0 0
\(540\) 9.85952 0.424286
\(541\) 9.09839 0.391170 0.195585 0.980687i \(-0.437339\pi\)
0.195585 + 0.980687i \(0.437339\pi\)
\(542\) 9.52982 0.409341
\(543\) −33.3700 −1.43204
\(544\) −1.51514 −0.0649610
\(545\) 24.4655 1.04799
\(546\) −0.530734 −0.0227133
\(547\) −19.6547 −0.840374 −0.420187 0.907437i \(-0.638036\pi\)
−0.420187 + 0.907437i \(0.638036\pi\)
\(548\) 0.969724 0.0414246
\(549\) −15.0303 −0.641477
\(550\) 0 0
\(551\) 2.60975 0.111179
\(552\) −1.09461 −0.0465897
\(553\) −0.829242 −0.0352630
\(554\) −13.5639 −0.576274
\(555\) 70.4490 2.99039
\(556\) −4.00000 −0.169638
\(557\) 16.2791 0.689769 0.344884 0.938645i \(-0.387918\pi\)
0.344884 + 0.938645i \(0.387918\pi\)
\(558\) 8.00000 0.338667
\(559\) −0.499542 −0.0211284
\(560\) −1.60975 −0.0680242
\(561\) 0 0
\(562\) −2.57947 −0.108808
\(563\) −18.0038 −0.758769 −0.379384 0.925239i \(-0.623864\pi\)
−0.379384 + 0.925239i \(0.623864\pi\)
\(564\) 19.1552 0.806578
\(565\) 33.9835 1.42970
\(566\) 17.7796 0.747332
\(567\) −5.79518 −0.243375
\(568\) 2.64002 0.110773
\(569\) 15.0303 0.630102 0.315051 0.949075i \(-0.397978\pi\)
0.315051 + 0.949075i \(0.397978\pi\)
\(570\) 6.64002 0.278120
\(571\) 22.5189 0.942387 0.471194 0.882030i \(-0.343823\pi\)
0.471194 + 0.882030i \(0.343823\pi\)
\(572\) 0 0
\(573\) 38.3435 1.60182
\(574\) −3.13957 −0.131043
\(575\) 2.45459 0.102363
\(576\) 1.51514 0.0631308
\(577\) −19.2947 −0.803250 −0.401625 0.915804i \(-0.631554\pi\)
−0.401625 + 0.915804i \(0.631554\pi\)
\(578\) 14.7044 0.611621
\(579\) 38.8099 1.61288
\(580\) −8.15516 −0.338625
\(581\) 0.829242 0.0344028
\(582\) −16.4995 −0.683928
\(583\) 0 0
\(584\) −7.52982 −0.311586
\(585\) −2.29564 −0.0949132
\(586\) −1.95035 −0.0805681
\(587\) −7.51044 −0.309989 −0.154995 0.987915i \(-0.549536\pi\)
−0.154995 + 0.987915i \(0.549536\pi\)
\(588\) −14.3103 −0.590148
\(589\) −5.28005 −0.217561
\(590\) −9.07901 −0.373777
\(591\) −45.7796 −1.88312
\(592\) −10.6097 −0.436058
\(593\) 17.2110 0.706772 0.353386 0.935478i \(-0.385030\pi\)
0.353386 + 0.935478i \(0.385030\pi\)
\(594\) 0 0
\(595\) −2.43899 −0.0999888
\(596\) 16.0752 0.658467
\(597\) 4.65184 0.190387
\(598\) −0.249771 −0.0102139
\(599\) −27.5592 −1.12604 −0.563019 0.826444i \(-0.690360\pi\)
−0.563019 + 0.826444i \(0.690360\pi\)
\(600\) −10.1249 −0.413347
\(601\) −21.5757 −0.880091 −0.440045 0.897976i \(-0.645038\pi\)
−0.440045 + 0.897976i \(0.645038\pi\)
\(602\) 0.530734 0.0216311
\(603\) −3.12202 −0.127139
\(604\) −22.2304 −0.904542
\(605\) 0 0
\(606\) 14.2380 0.578377
\(607\) 35.8889 1.45668 0.728342 0.685213i \(-0.240291\pi\)
0.728342 + 0.685213i \(0.240291\pi\)
\(608\) −1.00000 −0.0405554
\(609\) 2.85665 0.115757
\(610\) −30.9991 −1.25512
\(611\) 4.37088 0.176827
\(612\) 2.29564 0.0927959
\(613\) −5.12489 −0.206992 −0.103496 0.994630i \(-0.533003\pi\)
−0.103496 + 0.994630i \(0.533003\pi\)
\(614\) 20.1542 0.813359
\(615\) −40.4683 −1.63184
\(616\) 0 0
\(617\) −1.28096 −0.0515697 −0.0257848 0.999668i \(-0.508208\pi\)
−0.0257848 + 0.999668i \(0.508208\pi\)
\(618\) 16.0412 0.645271
\(619\) −12.9092 −0.518863 −0.259432 0.965761i \(-0.583535\pi\)
−0.259432 + 0.965761i \(0.583535\pi\)
\(620\) 16.4995 0.662638
\(621\) −1.62534 −0.0652228
\(622\) 4.90917 0.196840
\(623\) −0.548279 −0.0219663
\(624\) 1.03028 0.0412441
\(625\) −26.1202 −1.04481
\(626\) 27.5601 1.10152
\(627\) 0 0
\(628\) −20.2304 −0.807281
\(629\) −16.0752 −0.640961
\(630\) 2.43899 0.0971717
\(631\) −15.2039 −0.605258 −0.302629 0.953108i \(-0.597864\pi\)
−0.302629 + 0.953108i \(0.597864\pi\)
\(632\) 1.60975 0.0640323
\(633\) −35.1963 −1.39893
\(634\) −28.2791 −1.12311
\(635\) −2.43899 −0.0967883
\(636\) −1.42431 −0.0564776
\(637\) −3.26537 −0.129379
\(638\) 0 0
\(639\) −4.00000 −0.158238
\(640\) 3.12489 0.123522
\(641\) −0.775808 −0.0306426 −0.0153213 0.999883i \(-0.504877\pi\)
−0.0153213 + 0.999883i \(0.504877\pi\)
\(642\) −10.2947 −0.406300
\(643\) 39.5904 1.56129 0.780646 0.624974i \(-0.214890\pi\)
0.780646 + 0.624974i \(0.214890\pi\)
\(644\) 0.265367 0.0104569
\(645\) 6.84106 0.269366
\(646\) −1.51514 −0.0596123
\(647\) −13.0147 −0.511660 −0.255830 0.966722i \(-0.582349\pi\)
−0.255830 + 0.966722i \(0.582349\pi\)
\(648\) 11.2498 0.441933
\(649\) 0 0
\(650\) −2.31032 −0.0906183
\(651\) −5.77959 −0.226520
\(652\) −9.03028 −0.353653
\(653\) −49.2782 −1.92841 −0.964203 0.265166i \(-0.914573\pi\)
−0.964203 + 0.265166i \(0.914573\pi\)
\(654\) 16.6362 0.650529
\(655\) −57.2177 −2.23568
\(656\) 6.09461 0.237955
\(657\) 11.4087 0.445096
\(658\) −4.64380 −0.181034
\(659\) 25.0890 0.977328 0.488664 0.872472i \(-0.337484\pi\)
0.488664 + 0.872472i \(0.337484\pi\)
\(660\) 0 0
\(661\) −18.9882 −0.738555 −0.369277 0.929319i \(-0.620395\pi\)
−0.369277 + 0.929319i \(0.620395\pi\)
\(662\) −26.0937 −1.01416
\(663\) 1.56101 0.0606246
\(664\) −1.60975 −0.0624703
\(665\) −1.60975 −0.0624233
\(666\) 16.0752 0.622903
\(667\) 1.34438 0.0520546
\(668\) −12.2498 −0.473958
\(669\) −33.5710 −1.29793
\(670\) −6.43899 −0.248760
\(671\) 0 0
\(672\) −1.09461 −0.0422255
\(673\) 34.4196 1.32678 0.663389 0.748274i \(-0.269117\pi\)
0.663389 + 0.748274i \(0.269117\pi\)
\(674\) −19.2148 −0.740126
\(675\) −15.0341 −0.578661
\(676\) −12.7649 −0.490958
\(677\) −14.0185 −0.538773 −0.269387 0.963032i \(-0.586821\pi\)
−0.269387 + 0.963032i \(0.586821\pi\)
\(678\) 23.1084 0.887472
\(679\) 4.00000 0.153506
\(680\) 4.73463 0.181565
\(681\) −47.3406 −1.81410
\(682\) 0 0
\(683\) 20.8780 0.798874 0.399437 0.916761i \(-0.369206\pi\)
0.399437 + 0.916761i \(0.369206\pi\)
\(684\) 1.51514 0.0579328
\(685\) −3.03028 −0.115781
\(686\) 7.07523 0.270134
\(687\) 6.34152 0.241944
\(688\) −1.03028 −0.0392789
\(689\) −0.325003 −0.0123816
\(690\) 3.42053 0.130217
\(691\) −27.8889 −1.06094 −0.530471 0.847703i \(-0.677985\pi\)
−0.530471 + 0.847703i \(0.677985\pi\)
\(692\) −23.5942 −0.896915
\(693\) 0 0
\(694\) 3.17076 0.120360
\(695\) 12.4995 0.474135
\(696\) −5.54541 −0.210198
\(697\) 9.23417 0.349769
\(698\) −11.5445 −0.436966
\(699\) −4.18544 −0.158308
\(700\) 2.45459 0.0927746
\(701\) −38.5436 −1.45577 −0.727885 0.685699i \(-0.759497\pi\)
−0.727885 + 0.685699i \(0.759497\pi\)
\(702\) 1.52982 0.0577393
\(703\) −10.6097 −0.400154
\(704\) 0 0
\(705\) −59.8577 −2.25437
\(706\) −5.43899 −0.204699
\(707\) −3.45172 −0.129815
\(708\) −6.17362 −0.232019
\(709\) 45.9688 1.72639 0.863197 0.504867i \(-0.168458\pi\)
0.863197 + 0.504867i \(0.168458\pi\)
\(710\) −8.24977 −0.309608
\(711\) −2.43899 −0.0914693
\(712\) 1.06433 0.0398876
\(713\) −2.71995 −0.101863
\(714\) −1.65848 −0.0620672
\(715\) 0 0
\(716\) −23.6547 −0.884018
\(717\) 60.2517 2.25014
\(718\) 37.0890 1.38415
\(719\) 24.3784 0.909162 0.454581 0.890705i \(-0.349789\pi\)
0.454581 + 0.890705i \(0.349789\pi\)
\(720\) −4.73463 −0.176449
\(721\) −3.88888 −0.144829
\(722\) −1.00000 −0.0372161
\(723\) 6.54255 0.243320
\(724\) −15.7044 −0.583648
\(725\) 12.4352 0.461832
\(726\) 0 0
\(727\) 32.5739 1.20810 0.604049 0.796947i \(-0.293553\pi\)
0.604049 + 0.796947i \(0.293553\pi\)
\(728\) −0.249771 −0.00925712
\(729\) 3.06811 0.113634
\(730\) 23.5298 0.870878
\(731\) −1.56101 −0.0577361
\(732\) −21.0790 −0.779103
\(733\) 27.9154 1.03108 0.515539 0.856866i \(-0.327592\pi\)
0.515539 + 0.856866i \(0.327592\pi\)
\(734\) −11.8245 −0.436452
\(735\) 44.7181 1.64945
\(736\) −0.515138 −0.0189882
\(737\) 0 0
\(738\) −9.23417 −0.339915
\(739\) 50.8586 1.87086 0.935432 0.353507i \(-0.115011\pi\)
0.935432 + 0.353507i \(0.115011\pi\)
\(740\) 33.1542 1.21877
\(741\) 1.03028 0.0378481
\(742\) 0.345297 0.0126762
\(743\) −24.3709 −0.894081 −0.447040 0.894514i \(-0.647522\pi\)
−0.447040 + 0.894514i \(0.647522\pi\)
\(744\) 11.2195 0.411327
\(745\) −50.2333 −1.84040
\(746\) −29.4948 −1.07988
\(747\) 2.43899 0.0892380
\(748\) 0 0
\(749\) 2.49576 0.0911931
\(750\) −1.56101 −0.0570000
\(751\) −48.8392 −1.78217 −0.891084 0.453838i \(-0.850055\pi\)
−0.891084 + 0.453838i \(0.850055\pi\)
\(752\) 9.01468 0.328732
\(753\) −27.0908 −0.987245
\(754\) −1.26537 −0.0460820
\(755\) 69.4674 2.52818
\(756\) −1.62534 −0.0591132
\(757\) −4.27627 −0.155424 −0.0777118 0.996976i \(-0.524761\pi\)
−0.0777118 + 0.996976i \(0.524761\pi\)
\(758\) −29.4655 −1.07023
\(759\) 0 0
\(760\) 3.12489 0.113352
\(761\) 1.65940 0.0601532 0.0300766 0.999548i \(-0.490425\pi\)
0.0300766 + 0.999548i \(0.490425\pi\)
\(762\) −1.65848 −0.0600805
\(763\) −4.03314 −0.146009
\(764\) 18.0450 0.652844
\(765\) −7.17362 −0.259363
\(766\) −11.8789 −0.429202
\(767\) −1.40871 −0.0508657
\(768\) 2.12489 0.0766752
\(769\) 3.15046 0.113609 0.0568043 0.998385i \(-0.481909\pi\)
0.0568043 + 0.998385i \(0.481909\pi\)
\(770\) 0 0
\(771\) −14.7787 −0.532241
\(772\) 18.2645 0.657352
\(773\) −35.9338 −1.29245 −0.646225 0.763147i \(-0.723653\pi\)
−0.646225 + 0.763147i \(0.723653\pi\)
\(774\) 1.56101 0.0561094
\(775\) −25.1589 −0.903736
\(776\) −7.76491 −0.278744
\(777\) −11.6135 −0.416633
\(778\) −13.6244 −0.488459
\(779\) 6.09461 0.218362
\(780\) −3.21949 −0.115276
\(781\) 0 0
\(782\) −0.780505 −0.0279108
\(783\) −8.23417 −0.294265
\(784\) −6.73463 −0.240523
\(785\) 63.2177 2.25634
\(786\) −38.9073 −1.38778
\(787\) 8.44277 0.300952 0.150476 0.988614i \(-0.451919\pi\)
0.150476 + 0.988614i \(0.451919\pi\)
\(788\) −21.5445 −0.767491
\(789\) −22.7124 −0.808583
\(790\) −5.03028 −0.178969
\(791\) −5.60219 −0.199191
\(792\) 0 0
\(793\) −4.80986 −0.170803
\(794\) −17.7943 −0.631495
\(795\) 4.45080 0.157854
\(796\) 2.18922 0.0775948
\(797\) 34.3784 1.21775 0.608873 0.793267i \(-0.291622\pi\)
0.608873 + 0.793267i \(0.291622\pi\)
\(798\) −1.09461 −0.0387488
\(799\) 13.6585 0.483202
\(800\) −4.76491 −0.168465
\(801\) −1.61261 −0.0569788
\(802\) −19.0256 −0.671817
\(803\) 0 0
\(804\) −4.37844 −0.154416
\(805\) −0.829242 −0.0292269
\(806\) 2.56009 0.0901755
\(807\) −10.4546 −0.368019
\(808\) 6.70058 0.235725
\(809\) 20.4158 0.717782 0.358891 0.933379i \(-0.383155\pi\)
0.358891 + 0.933379i \(0.383155\pi\)
\(810\) −35.1542 −1.23519
\(811\) −8.84484 −0.310584 −0.155292 0.987869i \(-0.549632\pi\)
−0.155292 + 0.987869i \(0.549632\pi\)
\(812\) 1.34438 0.0471785
\(813\) −20.2498 −0.710190
\(814\) 0 0
\(815\) 28.2186 0.988454
\(816\) 3.21949 0.112705
\(817\) −1.03028 −0.0360448
\(818\) 5.58325 0.195214
\(819\) 0.378437 0.0132237
\(820\) −19.0450 −0.665079
\(821\) −7.69966 −0.268720 −0.134360 0.990933i \(-0.542898\pi\)
−0.134360 + 0.990933i \(0.542898\pi\)
\(822\) −2.06055 −0.0718700
\(823\) −41.1433 −1.43417 −0.717083 0.696987i \(-0.754523\pi\)
−0.717083 + 0.696987i \(0.754523\pi\)
\(824\) 7.54920 0.262989
\(825\) 0 0
\(826\) 1.49668 0.0520761
\(827\) 44.3472 1.54210 0.771052 0.636772i \(-0.219731\pi\)
0.771052 + 0.636772i \(0.219731\pi\)
\(828\) 0.780505 0.0271244
\(829\) −50.7252 −1.76176 −0.880880 0.473339i \(-0.843048\pi\)
−0.880880 + 0.473339i \(0.843048\pi\)
\(830\) 5.03028 0.174603
\(831\) 28.8217 0.999813
\(832\) 0.484862 0.0168096
\(833\) −10.2039 −0.353544
\(834\) 8.49954 0.294315
\(835\) 38.2791 1.32470
\(836\) 0 0
\(837\) 16.6594 0.575833
\(838\) −20.1698 −0.696756
\(839\) −7.92007 −0.273431 −0.136716 0.990610i \(-0.543655\pi\)
−0.136716 + 0.990610i \(0.543655\pi\)
\(840\) 3.42053 0.118019
\(841\) −22.1892 −0.765145
\(842\) 17.8292 0.614436
\(843\) 5.48108 0.188778
\(844\) −16.5639 −0.570152
\(845\) 39.8889 1.37222
\(846\) −13.6585 −0.469588
\(847\) 0 0
\(848\) −0.670300 −0.0230182
\(849\) −37.7796 −1.29659
\(850\) −7.21949 −0.247627
\(851\) −5.46548 −0.187354
\(852\) −5.60975 −0.192187
\(853\) 41.1055 1.40743 0.703713 0.710484i \(-0.251524\pi\)
0.703713 + 0.710484i \(0.251524\pi\)
\(854\) 5.11021 0.174868
\(855\) −4.73463 −0.161921
\(856\) −4.84484 −0.165593
\(857\) −21.2195 −0.724844 −0.362422 0.932014i \(-0.618050\pi\)
−0.362422 + 0.932014i \(0.618050\pi\)
\(858\) 0 0
\(859\) 20.3297 0.693640 0.346820 0.937932i \(-0.387261\pi\)
0.346820 + 0.937932i \(0.387261\pi\)
\(860\) 3.21949 0.109784
\(861\) 6.67122 0.227354
\(862\) 10.7687 0.366783
\(863\) −39.2782 −1.33705 −0.668523 0.743691i \(-0.733073\pi\)
−0.668523 + 0.743691i \(0.733073\pi\)
\(864\) 3.15516 0.107341
\(865\) 73.7290 2.50686
\(866\) −40.6353 −1.38084
\(867\) −31.2451 −1.06114
\(868\) −2.71995 −0.0923212
\(869\) 0 0
\(870\) 17.3288 0.587501
\(871\) −0.999083 −0.0338526
\(872\) 7.82924 0.265132
\(873\) 11.7649 0.398182
\(874\) −0.515138 −0.0174248
\(875\) 0.378437 0.0127935
\(876\) 16.0000 0.540590
\(877\) −45.6656 −1.54202 −0.771009 0.636824i \(-0.780248\pi\)
−0.771009 + 0.636824i \(0.780248\pi\)
\(878\) −15.0596 −0.508238
\(879\) 4.14426 0.139783
\(880\) 0 0
\(881\) 34.9301 1.17682 0.588412 0.808561i \(-0.299753\pi\)
0.588412 + 0.808561i \(0.299753\pi\)
\(882\) 10.2039 0.343583
\(883\) −7.58039 −0.255100 −0.127550 0.991832i \(-0.540711\pi\)
−0.127550 + 0.991832i \(0.540711\pi\)
\(884\) 0.734633 0.0247084
\(885\) 19.2919 0.648489
\(886\) 26.1892 0.879844
\(887\) 34.6987 1.16507 0.582535 0.812806i \(-0.302061\pi\)
0.582535 + 0.812806i \(0.302061\pi\)
\(888\) 22.5445 0.756544
\(889\) 0.402068 0.0134849
\(890\) −3.32592 −0.111485
\(891\) 0 0
\(892\) −15.7990 −0.528989
\(893\) 9.01468 0.301665
\(894\) −34.1580 −1.14242
\(895\) 73.9182 2.47081
\(896\) −0.515138 −0.0172096
\(897\) 0.530734 0.0177207
\(898\) −30.5748 −1.02029
\(899\) −13.7796 −0.459575
\(900\) 7.21949 0.240650
\(901\) −1.01560 −0.0338344
\(902\) 0 0
\(903\) −1.12775 −0.0375292
\(904\) 10.8751 0.361701
\(905\) 49.0743 1.63129
\(906\) 47.2370 1.56935
\(907\) −38.2148 −1.26890 −0.634451 0.772963i \(-0.718774\pi\)
−0.634451 + 0.772963i \(0.718774\pi\)
\(908\) −22.2791 −0.739359
\(909\) −10.1523 −0.336730
\(910\) 0.780505 0.0258735
\(911\) 6.51136 0.215731 0.107865 0.994166i \(-0.465598\pi\)
0.107865 + 0.994166i \(0.465598\pi\)
\(912\) 2.12489 0.0703620
\(913\) 0 0
\(914\) −16.4693 −0.544755
\(915\) 65.8695 2.17758
\(916\) 2.98440 0.0986075
\(917\) 9.43234 0.311483
\(918\) 4.78051 0.157780
\(919\) 38.4390 1.26799 0.633993 0.773339i \(-0.281415\pi\)
0.633993 + 0.773339i \(0.281415\pi\)
\(920\) 1.60975 0.0530718
\(921\) −42.8255 −1.41115
\(922\) 21.8742 0.720388
\(923\) −1.28005 −0.0421333
\(924\) 0 0
\(925\) −50.5545 −1.66222
\(926\) −27.8089 −0.913859
\(927\) −11.4381 −0.375676
\(928\) −2.60975 −0.0856692
\(929\) −40.6878 −1.33492 −0.667462 0.744643i \(-0.732620\pi\)
−0.667462 + 0.744643i \(0.732620\pi\)
\(930\) −35.0596 −1.14965
\(931\) −6.73463 −0.220719
\(932\) −1.96972 −0.0645205
\(933\) −10.4314 −0.341510
\(934\) −20.3709 −0.666555
\(935\) 0 0
\(936\) −0.734633 −0.0240122
\(937\) 41.6206 1.35969 0.679844 0.733357i \(-0.262048\pi\)
0.679844 + 0.733357i \(0.262048\pi\)
\(938\) 1.06147 0.0346582
\(939\) −58.5620 −1.91110
\(940\) −28.1698 −0.918799
\(941\) 44.5436 1.45208 0.726040 0.687653i \(-0.241359\pi\)
0.726040 + 0.687653i \(0.241359\pi\)
\(942\) 42.9873 1.40060
\(943\) 3.13957 0.102238
\(944\) −2.90539 −0.0945624
\(945\) 5.07901 0.165220
\(946\) 0 0
\(947\) 3.75779 0.122112 0.0610559 0.998134i \(-0.480553\pi\)
0.0610559 + 0.998134i \(0.480553\pi\)
\(948\) −3.42053 −0.111094
\(949\) 3.65092 0.118514
\(950\) −4.76491 −0.154594
\(951\) 60.0899 1.94855
\(952\) −0.780505 −0.0252963
\(953\) 18.1022 0.586387 0.293193 0.956053i \(-0.405282\pi\)
0.293193 + 0.956053i \(0.405282\pi\)
\(954\) 1.01560 0.0328811
\(955\) −56.3884 −1.82469
\(956\) 28.3553 0.917075
\(957\) 0 0
\(958\) 32.2342 1.04144
\(959\) 0.499542 0.0161310
\(960\) −6.64002 −0.214306
\(961\) −3.12110 −0.100681
\(962\) 5.14426 0.165858
\(963\) 7.34060 0.236548
\(964\) 3.07901 0.0991683
\(965\) −57.0743 −1.83729
\(966\) −0.563875 −0.0181424
\(967\) 6.45459 0.207565 0.103783 0.994600i \(-0.466905\pi\)
0.103783 + 0.994600i \(0.466905\pi\)
\(968\) 0 0
\(969\) 3.21949 0.103425
\(970\) 24.2645 0.779085
\(971\) 42.5033 1.36400 0.681998 0.731354i \(-0.261111\pi\)
0.681998 + 0.731354i \(0.261111\pi\)
\(972\) −14.4390 −0.463131
\(973\) −2.06055 −0.0660583
\(974\) 1.81078 0.0580212
\(975\) 4.90917 0.157219
\(976\) −9.92007 −0.317534
\(977\) 38.2351 1.22325 0.611624 0.791148i \(-0.290516\pi\)
0.611624 + 0.791148i \(0.290516\pi\)
\(978\) 19.1883 0.613574
\(979\) 0 0
\(980\) 21.0450 0.672256
\(981\) −11.8624 −0.378737
\(982\) 4.65940 0.148687
\(983\) 51.7602 1.65089 0.825447 0.564479i \(-0.190923\pi\)
0.825447 + 0.564479i \(0.190923\pi\)
\(984\) −12.9503 −0.412842
\(985\) 67.3241 2.14512
\(986\) −3.95413 −0.125925
\(987\) 9.86755 0.314088
\(988\) 0.484862 0.0154255
\(989\) −0.530734 −0.0168764
\(990\) 0 0
\(991\) 23.6391 0.750921 0.375460 0.926838i \(-0.377485\pi\)
0.375460 + 0.926838i \(0.377485\pi\)
\(992\) 5.28005 0.167642
\(993\) 55.4461 1.75953
\(994\) 1.35998 0.0431358
\(995\) −6.84106 −0.216876
\(996\) 3.42053 0.108384
\(997\) 27.6244 0.874874 0.437437 0.899249i \(-0.355886\pi\)
0.437437 + 0.899249i \(0.355886\pi\)
\(998\) 7.63911 0.241812
\(999\) 33.4755 1.05912
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 4598.2.a.bk.1.3 3
11.10 odd 2 4598.2.a.bn.1.3 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4598.2.a.bk.1.3 3 1.1 even 1 trivial
4598.2.a.bn.1.3 yes 3 11.10 odd 2