Properties

Label 2013.2.a.h.1.5
Level $2013$
Weight $2$
Character 2013.1
Self dual yes
Analytic conductor $16.074$
Analytic rank $0$
Dimension $14$
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] = [2013,2,Mod(1,2013)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2013, 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("2013.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2013 = 3 \cdot 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2013.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: \(16.0738859269\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - x^{13} - 21 x^{12} + 20 x^{11} + 167 x^{10} - 148 x^{9} - 627 x^{8} + 497 x^{7} + 1123 x^{6} - 745 x^{5} - 802 x^{4} + 386 x^{3} + 74 x^{2} - 15 x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(1.61392\) of defining polynomial
Character \(\chi\) \(=\) 2013.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.61392 q^{2} -1.00000 q^{3} +0.604750 q^{4} -3.65776 q^{5} +1.61392 q^{6} +0.985739 q^{7} +2.25183 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.61392 q^{2} -1.00000 q^{3} +0.604750 q^{4} -3.65776 q^{5} +1.61392 q^{6} +0.985739 q^{7} +2.25183 q^{8} +1.00000 q^{9} +5.90335 q^{10} -1.00000 q^{11} -0.604750 q^{12} +2.65320 q^{13} -1.59091 q^{14} +3.65776 q^{15} -4.84378 q^{16} -6.25317 q^{17} -1.61392 q^{18} +1.35459 q^{19} -2.21203 q^{20} -0.985739 q^{21} +1.61392 q^{22} +2.24451 q^{23} -2.25183 q^{24} +8.37924 q^{25} -4.28207 q^{26} -1.00000 q^{27} +0.596126 q^{28} -4.46976 q^{29} -5.90335 q^{30} -4.00805 q^{31} +3.31383 q^{32} +1.00000 q^{33} +10.0921 q^{34} -3.60560 q^{35} +0.604750 q^{36} -1.24458 q^{37} -2.18620 q^{38} -2.65320 q^{39} -8.23665 q^{40} +6.27870 q^{41} +1.59091 q^{42} +0.305238 q^{43} -0.604750 q^{44} -3.65776 q^{45} -3.62247 q^{46} -10.8096 q^{47} +4.84378 q^{48} -6.02832 q^{49} -13.5235 q^{50} +6.25317 q^{51} +1.60452 q^{52} -10.0242 q^{53} +1.61392 q^{54} +3.65776 q^{55} +2.21971 q^{56} -1.35459 q^{57} +7.21386 q^{58} +8.28564 q^{59} +2.21203 q^{60} +1.00000 q^{61} +6.46869 q^{62} +0.985739 q^{63} +4.33928 q^{64} -9.70479 q^{65} -1.61392 q^{66} -15.5138 q^{67} -3.78160 q^{68} -2.24451 q^{69} +5.81917 q^{70} -8.28871 q^{71} +2.25183 q^{72} +8.53532 q^{73} +2.00866 q^{74} -8.37924 q^{75} +0.819186 q^{76} -0.985739 q^{77} +4.28207 q^{78} -13.4999 q^{79} +17.7174 q^{80} +1.00000 q^{81} -10.1333 q^{82} +4.84293 q^{83} -0.596126 q^{84} +22.8726 q^{85} -0.492631 q^{86} +4.46976 q^{87} -2.25183 q^{88} -15.4360 q^{89} +5.90335 q^{90} +2.61537 q^{91} +1.35737 q^{92} +4.00805 q^{93} +17.4459 q^{94} -4.95476 q^{95} -3.31383 q^{96} +10.8187 q^{97} +9.72925 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - q^{2} - 14 q^{3} + 15 q^{4} + q^{5} + q^{6} + 9 q^{7} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - q^{2} - 14 q^{3} + 15 q^{4} + q^{5} + q^{6} + 9 q^{7} + 14 q^{9} + 6 q^{10} - 14 q^{11} - 15 q^{12} + q^{13} - 7 q^{14} - q^{15} + 17 q^{16} - 9 q^{17} - q^{18} + 22 q^{19} + 23 q^{20} - 9 q^{21} + q^{22} + q^{23} + 25 q^{25} + 4 q^{26} - 14 q^{27} + 37 q^{28} - 6 q^{29} - 6 q^{30} + 9 q^{31} + 4 q^{32} + 14 q^{33} + 8 q^{34} + 18 q^{35} + 15 q^{36} + 18 q^{37} + 8 q^{38} - q^{39} + 16 q^{40} - 25 q^{41} + 7 q^{42} + 25 q^{43} - 15 q^{44} + q^{45} + 20 q^{46} + 36 q^{47} - 17 q^{48} + 25 q^{49} + 2 q^{50} + 9 q^{51} - 13 q^{52} + q^{54} - q^{55} - 40 q^{56} - 22 q^{57} + 33 q^{58} + 17 q^{59} - 23 q^{60} + 14 q^{61} - 13 q^{62} + 9 q^{63} - 6 q^{64} - 61 q^{65} - q^{66} + 22 q^{67} + 66 q^{68} - q^{69} + 44 q^{70} - 13 q^{71} + 20 q^{73} - 12 q^{74} - 25 q^{75} + 49 q^{76} - 9 q^{77} - 4 q^{78} + 31 q^{79} + 88 q^{80} + 14 q^{81} + 2 q^{82} + 32 q^{83} - 37 q^{84} + 2 q^{85} - 14 q^{86} + 6 q^{87} - 21 q^{89} + 6 q^{90} + 45 q^{91} - 14 q^{92} - 9 q^{93} - 31 q^{94} + 23 q^{95} - 4 q^{96} + 37 q^{97} - 38 q^{98} - 14 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.61392 −1.14122 −0.570608 0.821222i \(-0.693292\pi\)
−0.570608 + 0.821222i \(0.693292\pi\)
\(3\) −1.00000 −0.577350
\(4\) 0.604750 0.302375
\(5\) −3.65776 −1.63580 −0.817901 0.575359i \(-0.804862\pi\)
−0.817901 + 0.575359i \(0.804862\pi\)
\(6\) 1.61392 0.658882
\(7\) 0.985739 0.372574 0.186287 0.982495i \(-0.440355\pi\)
0.186287 + 0.982495i \(0.440355\pi\)
\(8\) 2.25183 0.796141
\(9\) 1.00000 0.333333
\(10\) 5.90335 1.86680
\(11\) −1.00000 −0.301511
\(12\) −0.604750 −0.174576
\(13\) 2.65320 0.735866 0.367933 0.929852i \(-0.380066\pi\)
0.367933 + 0.929852i \(0.380066\pi\)
\(14\) −1.59091 −0.425188
\(15\) 3.65776 0.944431
\(16\) −4.84378 −1.21094
\(17\) −6.25317 −1.51662 −0.758308 0.651896i \(-0.773974\pi\)
−0.758308 + 0.651896i \(0.773974\pi\)
\(18\) −1.61392 −0.380405
\(19\) 1.35459 0.310763 0.155382 0.987855i \(-0.450339\pi\)
0.155382 + 0.987855i \(0.450339\pi\)
\(20\) −2.21203 −0.494626
\(21\) −0.985739 −0.215106
\(22\) 1.61392 0.344090
\(23\) 2.24451 0.468012 0.234006 0.972235i \(-0.424816\pi\)
0.234006 + 0.972235i \(0.424816\pi\)
\(24\) −2.25183 −0.459652
\(25\) 8.37924 1.67585
\(26\) −4.28207 −0.839782
\(27\) −1.00000 −0.192450
\(28\) 0.596126 0.112657
\(29\) −4.46976 −0.830014 −0.415007 0.909818i \(-0.636221\pi\)
−0.415007 + 0.909818i \(0.636221\pi\)
\(30\) −5.90335 −1.07780
\(31\) −4.00805 −0.719867 −0.359934 0.932978i \(-0.617201\pi\)
−0.359934 + 0.932978i \(0.617201\pi\)
\(32\) 3.31383 0.585808
\(33\) 1.00000 0.174078
\(34\) 10.0921 1.73079
\(35\) −3.60560 −0.609458
\(36\) 0.604750 0.100792
\(37\) −1.24458 −0.204608 −0.102304 0.994753i \(-0.532621\pi\)
−0.102304 + 0.994753i \(0.532621\pi\)
\(38\) −2.18620 −0.354648
\(39\) −2.65320 −0.424852
\(40\) −8.23665 −1.30233
\(41\) 6.27870 0.980568 0.490284 0.871563i \(-0.336893\pi\)
0.490284 + 0.871563i \(0.336893\pi\)
\(42\) 1.59091 0.245482
\(43\) 0.305238 0.0465483 0.0232742 0.999729i \(-0.492591\pi\)
0.0232742 + 0.999729i \(0.492591\pi\)
\(44\) −0.604750 −0.0911695
\(45\) −3.65776 −0.545267
\(46\) −3.62247 −0.534103
\(47\) −10.8096 −1.57675 −0.788374 0.615197i \(-0.789077\pi\)
−0.788374 + 0.615197i \(0.789077\pi\)
\(48\) 4.84378 0.699139
\(49\) −6.02832 −0.861188
\(50\) −13.5235 −1.91251
\(51\) 6.25317 0.875619
\(52\) 1.60452 0.222507
\(53\) −10.0242 −1.37693 −0.688466 0.725269i \(-0.741715\pi\)
−0.688466 + 0.725269i \(0.741715\pi\)
\(54\) 1.61392 0.219627
\(55\) 3.65776 0.493213
\(56\) 2.21971 0.296622
\(57\) −1.35459 −0.179419
\(58\) 7.21386 0.947226
\(59\) 8.28564 1.07870 0.539350 0.842082i \(-0.318670\pi\)
0.539350 + 0.842082i \(0.318670\pi\)
\(60\) 2.21203 0.285572
\(61\) 1.00000 0.128037
\(62\) 6.46869 0.821524
\(63\) 0.985739 0.124191
\(64\) 4.33928 0.542410
\(65\) −9.70479 −1.20373
\(66\) −1.61392 −0.198660
\(67\) −15.5138 −1.89531 −0.947654 0.319298i \(-0.896553\pi\)
−0.947654 + 0.319298i \(0.896553\pi\)
\(68\) −3.78160 −0.458587
\(69\) −2.24451 −0.270207
\(70\) 5.81917 0.695524
\(71\) −8.28871 −0.983689 −0.491845 0.870683i \(-0.663677\pi\)
−0.491845 + 0.870683i \(0.663677\pi\)
\(72\) 2.25183 0.265380
\(73\) 8.53532 0.998984 0.499492 0.866319i \(-0.333520\pi\)
0.499492 + 0.866319i \(0.333520\pi\)
\(74\) 2.00866 0.233502
\(75\) −8.37924 −0.967552
\(76\) 0.819186 0.0939670
\(77\) −0.985739 −0.112335
\(78\) 4.28207 0.484848
\(79\) −13.4999 −1.51886 −0.759429 0.650590i \(-0.774522\pi\)
−0.759429 + 0.650590i \(0.774522\pi\)
\(80\) 17.7174 1.98087
\(81\) 1.00000 0.111111
\(82\) −10.1333 −1.11904
\(83\) 4.84293 0.531581 0.265790 0.964031i \(-0.414367\pi\)
0.265790 + 0.964031i \(0.414367\pi\)
\(84\) −0.596126 −0.0650427
\(85\) 22.8726 2.48088
\(86\) −0.492631 −0.0531217
\(87\) 4.46976 0.479209
\(88\) −2.25183 −0.240046
\(89\) −15.4360 −1.63621 −0.818105 0.575069i \(-0.804975\pi\)
−0.818105 + 0.575069i \(0.804975\pi\)
\(90\) 5.90335 0.622268
\(91\) 2.61537 0.274165
\(92\) 1.35737 0.141515
\(93\) 4.00805 0.415615
\(94\) 17.4459 1.79941
\(95\) −4.95476 −0.508347
\(96\) −3.31383 −0.338217
\(97\) 10.8187 1.09847 0.549235 0.835668i \(-0.314919\pi\)
0.549235 + 0.835668i \(0.314919\pi\)
\(98\) 9.72925 0.982802
\(99\) −1.00000 −0.100504
\(100\) 5.06735 0.506735
\(101\) 18.5401 1.84481 0.922406 0.386221i \(-0.126220\pi\)
0.922406 + 0.386221i \(0.126220\pi\)
\(102\) −10.0921 −0.999271
\(103\) 13.3353 1.31396 0.656982 0.753907i \(-0.271833\pi\)
0.656982 + 0.753907i \(0.271833\pi\)
\(104\) 5.97455 0.585853
\(105\) 3.60560 0.351871
\(106\) 16.1783 1.57138
\(107\) 12.9262 1.24962 0.624811 0.780776i \(-0.285176\pi\)
0.624811 + 0.780776i \(0.285176\pi\)
\(108\) −0.604750 −0.0581921
\(109\) 17.1301 1.64077 0.820384 0.571813i \(-0.193760\pi\)
0.820384 + 0.571813i \(0.193760\pi\)
\(110\) −5.90335 −0.562863
\(111\) 1.24458 0.118130
\(112\) −4.77470 −0.451167
\(113\) 12.6163 1.18685 0.593423 0.804891i \(-0.297776\pi\)
0.593423 + 0.804891i \(0.297776\pi\)
\(114\) 2.18620 0.204756
\(115\) −8.20988 −0.765576
\(116\) −2.70309 −0.250976
\(117\) 2.65320 0.245289
\(118\) −13.3724 −1.23103
\(119\) −6.16399 −0.565052
\(120\) 8.23665 0.751900
\(121\) 1.00000 0.0909091
\(122\) −1.61392 −0.146118
\(123\) −6.27870 −0.566131
\(124\) −2.42387 −0.217670
\(125\) −12.3605 −1.10555
\(126\) −1.59091 −0.141729
\(127\) 14.7816 1.31166 0.655828 0.754910i \(-0.272320\pi\)
0.655828 + 0.754910i \(0.272320\pi\)
\(128\) −13.6309 −1.20482
\(129\) −0.305238 −0.0268747
\(130\) 15.6628 1.37372
\(131\) 12.0226 1.05042 0.525210 0.850972i \(-0.323987\pi\)
0.525210 + 0.850972i \(0.323987\pi\)
\(132\) 0.604750 0.0526367
\(133\) 1.33527 0.115782
\(134\) 25.0380 2.16296
\(135\) 3.65776 0.314810
\(136\) −14.0811 −1.20744
\(137\) −12.6310 −1.07914 −0.539568 0.841942i \(-0.681413\pi\)
−0.539568 + 0.841942i \(0.681413\pi\)
\(138\) 3.62247 0.308365
\(139\) −16.6404 −1.41142 −0.705711 0.708500i \(-0.749372\pi\)
−0.705711 + 0.708500i \(0.749372\pi\)
\(140\) −2.18049 −0.184285
\(141\) 10.8096 0.910336
\(142\) 13.3774 1.12260
\(143\) −2.65320 −0.221872
\(144\) −4.84378 −0.403648
\(145\) 16.3493 1.35774
\(146\) −13.7754 −1.14006
\(147\) 6.02832 0.497207
\(148\) −0.752659 −0.0618682
\(149\) −3.29899 −0.270264 −0.135132 0.990828i \(-0.543146\pi\)
−0.135132 + 0.990828i \(0.543146\pi\)
\(150\) 13.5235 1.10419
\(151\) 13.8467 1.12683 0.563413 0.826176i \(-0.309488\pi\)
0.563413 + 0.826176i \(0.309488\pi\)
\(152\) 3.05029 0.247411
\(153\) −6.25317 −0.505539
\(154\) 1.59091 0.128199
\(155\) 14.6605 1.17756
\(156\) −1.60452 −0.128465
\(157\) −5.10610 −0.407512 −0.203756 0.979022i \(-0.565315\pi\)
−0.203756 + 0.979022i \(0.565315\pi\)
\(158\) 21.7878 1.73335
\(159\) 10.0242 0.794972
\(160\) −12.1212 −0.958267
\(161\) 2.21250 0.174369
\(162\) −1.61392 −0.126802
\(163\) 14.0106 1.09739 0.548696 0.836022i \(-0.315124\pi\)
0.548696 + 0.836022i \(0.315124\pi\)
\(164\) 3.79704 0.296499
\(165\) −3.65776 −0.284757
\(166\) −7.81612 −0.606648
\(167\) −19.1530 −1.48210 −0.741052 0.671448i \(-0.765673\pi\)
−0.741052 + 0.671448i \(0.765673\pi\)
\(168\) −2.21971 −0.171255
\(169\) −5.96052 −0.458502
\(170\) −36.9147 −2.83123
\(171\) 1.35459 0.103588
\(172\) 0.184593 0.0140751
\(173\) −22.4432 −1.70632 −0.853161 0.521647i \(-0.825318\pi\)
−0.853161 + 0.521647i \(0.825318\pi\)
\(174\) −7.21386 −0.546881
\(175\) 8.25975 0.624378
\(176\) 4.84378 0.365113
\(177\) −8.28564 −0.622787
\(178\) 24.9125 1.86727
\(179\) 13.1373 0.981927 0.490964 0.871180i \(-0.336645\pi\)
0.490964 + 0.871180i \(0.336645\pi\)
\(180\) −2.21203 −0.164875
\(181\) 7.41628 0.551247 0.275624 0.961266i \(-0.411116\pi\)
0.275624 + 0.961266i \(0.411116\pi\)
\(182\) −4.22100 −0.312881
\(183\) −1.00000 −0.0739221
\(184\) 5.05425 0.372604
\(185\) 4.55238 0.334698
\(186\) −6.46869 −0.474307
\(187\) 6.25317 0.457277
\(188\) −6.53713 −0.476769
\(189\) −0.985739 −0.0717020
\(190\) 7.99660 0.580134
\(191\) −6.22085 −0.450125 −0.225063 0.974344i \(-0.572259\pi\)
−0.225063 + 0.974344i \(0.572259\pi\)
\(192\) −4.33928 −0.313161
\(193\) −1.95576 −0.140779 −0.0703893 0.997520i \(-0.522424\pi\)
−0.0703893 + 0.997520i \(0.522424\pi\)
\(194\) −17.4605 −1.25359
\(195\) 9.70479 0.694974
\(196\) −3.64563 −0.260402
\(197\) 27.6283 1.96843 0.984217 0.176968i \(-0.0566288\pi\)
0.984217 + 0.176968i \(0.0566288\pi\)
\(198\) 1.61392 0.114697
\(199\) −3.84659 −0.272678 −0.136339 0.990662i \(-0.543534\pi\)
−0.136339 + 0.990662i \(0.543534\pi\)
\(200\) 18.8686 1.33421
\(201\) 15.5138 1.09426
\(202\) −29.9224 −2.10533
\(203\) −4.40602 −0.309242
\(204\) 3.78160 0.264765
\(205\) −22.9660 −1.60402
\(206\) −21.5221 −1.49952
\(207\) 2.24451 0.156004
\(208\) −12.8515 −0.891093
\(209\) −1.35459 −0.0936987
\(210\) −5.81917 −0.401561
\(211\) −11.4958 −0.791405 −0.395703 0.918379i \(-0.629499\pi\)
−0.395703 + 0.918379i \(0.629499\pi\)
\(212\) −6.06214 −0.416350
\(213\) 8.28871 0.567933
\(214\) −20.8619 −1.42609
\(215\) −1.11649 −0.0761439
\(216\) −2.25183 −0.153217
\(217\) −3.95089 −0.268204
\(218\) −27.6467 −1.87247
\(219\) −8.53532 −0.576764
\(220\) 2.21203 0.149135
\(221\) −16.5909 −1.11603
\(222\) −2.00866 −0.134812
\(223\) 9.82324 0.657813 0.328906 0.944363i \(-0.393320\pi\)
0.328906 + 0.944363i \(0.393320\pi\)
\(224\) 3.26658 0.218257
\(225\) 8.37924 0.558616
\(226\) −20.3618 −1.35445
\(227\) 12.7593 0.846867 0.423433 0.905927i \(-0.360825\pi\)
0.423433 + 0.905927i \(0.360825\pi\)
\(228\) −0.819186 −0.0542519
\(229\) 10.6052 0.700808 0.350404 0.936599i \(-0.386044\pi\)
0.350404 + 0.936599i \(0.386044\pi\)
\(230\) 13.2501 0.873688
\(231\) 0.985739 0.0648569
\(232\) −10.0651 −0.660809
\(233\) 23.6060 1.54648 0.773239 0.634115i \(-0.218635\pi\)
0.773239 + 0.634115i \(0.218635\pi\)
\(234\) −4.28207 −0.279927
\(235\) 39.5391 2.57925
\(236\) 5.01074 0.326172
\(237\) 13.4999 0.876913
\(238\) 9.94822 0.644847
\(239\) −7.79900 −0.504476 −0.252238 0.967665i \(-0.581167\pi\)
−0.252238 + 0.967665i \(0.581167\pi\)
\(240\) −17.7174 −1.14365
\(241\) −14.4302 −0.929531 −0.464766 0.885434i \(-0.653861\pi\)
−0.464766 + 0.885434i \(0.653861\pi\)
\(242\) −1.61392 −0.103747
\(243\) −1.00000 −0.0641500
\(244\) 0.604750 0.0387151
\(245\) 22.0502 1.40873
\(246\) 10.1333 0.646078
\(247\) 3.59399 0.228680
\(248\) −9.02544 −0.573116
\(249\) −4.84293 −0.306908
\(250\) 19.9489 1.26168
\(251\) 27.7902 1.75410 0.877050 0.480399i \(-0.159508\pi\)
0.877050 + 0.480399i \(0.159508\pi\)
\(252\) 0.596126 0.0375524
\(253\) −2.24451 −0.141111
\(254\) −23.8564 −1.49688
\(255\) −22.8726 −1.43234
\(256\) 13.3207 0.832545
\(257\) 1.48481 0.0926199 0.0463100 0.998927i \(-0.485254\pi\)
0.0463100 + 0.998927i \(0.485254\pi\)
\(258\) 0.492631 0.0306698
\(259\) −1.22683 −0.0762316
\(260\) −5.86897 −0.363978
\(261\) −4.46976 −0.276671
\(262\) −19.4036 −1.19876
\(263\) −1.16715 −0.0719693 −0.0359847 0.999352i \(-0.511457\pi\)
−0.0359847 + 0.999352i \(0.511457\pi\)
\(264\) 2.25183 0.138590
\(265\) 36.6662 2.25239
\(266\) −2.15502 −0.132133
\(267\) 15.4360 0.944666
\(268\) −9.38195 −0.573094
\(269\) 26.7060 1.62829 0.814147 0.580659i \(-0.197205\pi\)
0.814147 + 0.580659i \(0.197205\pi\)
\(270\) −5.90335 −0.359267
\(271\) 2.63024 0.159776 0.0798879 0.996804i \(-0.474544\pi\)
0.0798879 + 0.996804i \(0.474544\pi\)
\(272\) 30.2890 1.83654
\(273\) −2.61537 −0.158289
\(274\) 20.3854 1.23153
\(275\) −8.37924 −0.505287
\(276\) −1.35737 −0.0817039
\(277\) −7.34527 −0.441335 −0.220667 0.975349i \(-0.570824\pi\)
−0.220667 + 0.975349i \(0.570824\pi\)
\(278\) 26.8564 1.61074
\(279\) −4.00805 −0.239956
\(280\) −8.11919 −0.485215
\(281\) 6.09887 0.363828 0.181914 0.983314i \(-0.441771\pi\)
0.181914 + 0.983314i \(0.441771\pi\)
\(282\) −17.4459 −1.03889
\(283\) 22.2465 1.32242 0.661209 0.750202i \(-0.270044\pi\)
0.661209 + 0.750202i \(0.270044\pi\)
\(284\) −5.01260 −0.297443
\(285\) 4.95476 0.293494
\(286\) 4.28207 0.253204
\(287\) 6.18916 0.365335
\(288\) 3.31383 0.195269
\(289\) 22.1021 1.30013
\(290\) −26.3866 −1.54947
\(291\) −10.8187 −0.634202
\(292\) 5.16173 0.302068
\(293\) −7.69309 −0.449435 −0.224718 0.974424i \(-0.572146\pi\)
−0.224718 + 0.974424i \(0.572146\pi\)
\(294\) −9.72925 −0.567421
\(295\) −30.3069 −1.76454
\(296\) −2.80258 −0.162897
\(297\) 1.00000 0.0580259
\(298\) 5.32431 0.308429
\(299\) 5.95513 0.344394
\(300\) −5.06735 −0.292563
\(301\) 0.300885 0.0173427
\(302\) −22.3475 −1.28595
\(303\) −18.5401 −1.06510
\(304\) −6.56131 −0.376317
\(305\) −3.65776 −0.209443
\(306\) 10.0921 0.576929
\(307\) 10.6159 0.605879 0.302939 0.953010i \(-0.402032\pi\)
0.302939 + 0.953010i \(0.402032\pi\)
\(308\) −0.596126 −0.0339674
\(309\) −13.3353 −0.758617
\(310\) −23.6609 −1.34385
\(311\) −34.2582 −1.94261 −0.971303 0.237847i \(-0.923558\pi\)
−0.971303 + 0.237847i \(0.923558\pi\)
\(312\) −5.97455 −0.338242
\(313\) 6.50686 0.367789 0.183895 0.982946i \(-0.441129\pi\)
0.183895 + 0.982946i \(0.441129\pi\)
\(314\) 8.24086 0.465059
\(315\) −3.60560 −0.203153
\(316\) −8.16407 −0.459265
\(317\) 8.11911 0.456015 0.228007 0.973659i \(-0.426779\pi\)
0.228007 + 0.973659i \(0.426779\pi\)
\(318\) −16.1783 −0.907235
\(319\) 4.46976 0.250259
\(320\) −15.8721 −0.887276
\(321\) −12.9262 −0.721470
\(322\) −3.57081 −0.198993
\(323\) −8.47045 −0.471309
\(324\) 0.604750 0.0335972
\(325\) 22.2318 1.23320
\(326\) −22.6120 −1.25236
\(327\) −17.1301 −0.947298
\(328\) 14.1385 0.780671
\(329\) −10.6555 −0.587456
\(330\) 5.90335 0.324969
\(331\) −31.4323 −1.72768 −0.863839 0.503768i \(-0.831947\pi\)
−0.863839 + 0.503768i \(0.831947\pi\)
\(332\) 2.92876 0.160737
\(333\) −1.24458 −0.0682025
\(334\) 30.9115 1.69140
\(335\) 56.7457 3.10035
\(336\) 4.77470 0.260481
\(337\) −18.6087 −1.01368 −0.506841 0.862040i \(-0.669187\pi\)
−0.506841 + 0.862040i \(0.669187\pi\)
\(338\) 9.61982 0.523249
\(339\) −12.6163 −0.685225
\(340\) 13.8322 0.750157
\(341\) 4.00805 0.217048
\(342\) −2.18620 −0.118216
\(343\) −12.8425 −0.693431
\(344\) 0.687343 0.0370591
\(345\) 8.20988 0.442005
\(346\) 36.2216 1.94728
\(347\) −4.39085 −0.235713 −0.117856 0.993031i \(-0.537602\pi\)
−0.117856 + 0.993031i \(0.537602\pi\)
\(348\) 2.70309 0.144901
\(349\) 20.5921 1.10227 0.551134 0.834417i \(-0.314195\pi\)
0.551134 + 0.834417i \(0.314195\pi\)
\(350\) −13.3306 −0.712551
\(351\) −2.65320 −0.141617
\(352\) −3.31383 −0.176628
\(353\) −8.31752 −0.442697 −0.221348 0.975195i \(-0.571046\pi\)
−0.221348 + 0.975195i \(0.571046\pi\)
\(354\) 13.3724 0.710735
\(355\) 30.3182 1.60912
\(356\) −9.33490 −0.494749
\(357\) 6.16399 0.326233
\(358\) −21.2026 −1.12059
\(359\) 15.5845 0.822520 0.411260 0.911518i \(-0.365089\pi\)
0.411260 + 0.911518i \(0.365089\pi\)
\(360\) −8.23665 −0.434110
\(361\) −17.1651 −0.903426
\(362\) −11.9693 −0.629093
\(363\) −1.00000 −0.0524864
\(364\) 1.58164 0.0829006
\(365\) −31.2202 −1.63414
\(366\) 1.61392 0.0843611
\(367\) 10.5398 0.550175 0.275088 0.961419i \(-0.411293\pi\)
0.275088 + 0.961419i \(0.411293\pi\)
\(368\) −10.8719 −0.566737
\(369\) 6.27870 0.326856
\(370\) −7.34719 −0.381962
\(371\) −9.88126 −0.513009
\(372\) 2.42387 0.125672
\(373\) 19.3765 1.00328 0.501638 0.865078i \(-0.332731\pi\)
0.501638 + 0.865078i \(0.332731\pi\)
\(374\) −10.0921 −0.521852
\(375\) 12.3605 0.638292
\(376\) −24.3414 −1.25531
\(377\) −11.8592 −0.610779
\(378\) 1.59091 0.0818275
\(379\) −6.66214 −0.342211 −0.171106 0.985253i \(-0.554734\pi\)
−0.171106 + 0.985253i \(0.554734\pi\)
\(380\) −2.99639 −0.153711
\(381\) −14.7816 −0.757285
\(382\) 10.0400 0.513690
\(383\) 18.7882 0.960031 0.480016 0.877260i \(-0.340631\pi\)
0.480016 + 0.877260i \(0.340631\pi\)
\(384\) 13.6309 0.695601
\(385\) 3.60560 0.183759
\(386\) 3.15645 0.160659
\(387\) 0.305238 0.0155161
\(388\) 6.54259 0.332150
\(389\) −21.2173 −1.07576 −0.537880 0.843021i \(-0.680775\pi\)
−0.537880 + 0.843021i \(0.680775\pi\)
\(390\) −15.6628 −0.793116
\(391\) −14.0353 −0.709795
\(392\) −13.5747 −0.685627
\(393\) −12.0226 −0.606461
\(394\) −44.5899 −2.24641
\(395\) 49.3795 2.48455
\(396\) −0.604750 −0.0303898
\(397\) −1.82400 −0.0915440 −0.0457720 0.998952i \(-0.514575\pi\)
−0.0457720 + 0.998952i \(0.514575\pi\)
\(398\) 6.20810 0.311184
\(399\) −1.33527 −0.0668470
\(400\) −40.5872 −2.02936
\(401\) 18.6265 0.930164 0.465082 0.885268i \(-0.346025\pi\)
0.465082 + 0.885268i \(0.346025\pi\)
\(402\) −25.0380 −1.24878
\(403\) −10.6342 −0.529726
\(404\) 11.2121 0.557825
\(405\) −3.65776 −0.181756
\(406\) 7.11098 0.352912
\(407\) 1.24458 0.0616915
\(408\) 14.0811 0.697116
\(409\) −28.3454 −1.40159 −0.700795 0.713362i \(-0.747171\pi\)
−0.700795 + 0.713362i \(0.747171\pi\)
\(410\) 37.0654 1.83053
\(411\) 12.6310 0.623040
\(412\) 8.06450 0.397310
\(413\) 8.16749 0.401896
\(414\) −3.62247 −0.178034
\(415\) −17.7143 −0.869561
\(416\) 8.79227 0.431076
\(417\) 16.6404 0.814885
\(418\) 2.18620 0.106930
\(419\) 14.0074 0.684307 0.342154 0.939644i \(-0.388844\pi\)
0.342154 + 0.939644i \(0.388844\pi\)
\(420\) 2.18049 0.106397
\(421\) −0.242484 −0.0118180 −0.00590898 0.999983i \(-0.501881\pi\)
−0.00590898 + 0.999983i \(0.501881\pi\)
\(422\) 18.5534 0.903165
\(423\) −10.8096 −0.525583
\(424\) −22.5728 −1.09623
\(425\) −52.3968 −2.54162
\(426\) −13.3774 −0.648135
\(427\) 0.985739 0.0477033
\(428\) 7.81712 0.377855
\(429\) 2.65320 0.128098
\(430\) 1.80193 0.0868967
\(431\) −20.9192 −1.00764 −0.503822 0.863807i \(-0.668073\pi\)
−0.503822 + 0.863807i \(0.668073\pi\)
\(432\) 4.84378 0.233046
\(433\) 38.5719 1.85364 0.926822 0.375500i \(-0.122529\pi\)
0.926822 + 0.375500i \(0.122529\pi\)
\(434\) 6.37644 0.306079
\(435\) −16.3493 −0.783891
\(436\) 10.3594 0.496127
\(437\) 3.04038 0.145441
\(438\) 13.7754 0.658212
\(439\) −7.54989 −0.360337 −0.180168 0.983636i \(-0.557664\pi\)
−0.180168 + 0.983636i \(0.557664\pi\)
\(440\) 8.23665 0.392667
\(441\) −6.02832 −0.287063
\(442\) 26.7765 1.27363
\(443\) −7.89709 −0.375202 −0.187601 0.982245i \(-0.560071\pi\)
−0.187601 + 0.982245i \(0.560071\pi\)
\(444\) 0.752659 0.0357196
\(445\) 56.4611 2.67651
\(446\) −15.8540 −0.750707
\(447\) 3.29899 0.156037
\(448\) 4.27740 0.202088
\(449\) 21.7371 1.02584 0.512919 0.858437i \(-0.328564\pi\)
0.512919 + 0.858437i \(0.328564\pi\)
\(450\) −13.5235 −0.637502
\(451\) −6.27870 −0.295652
\(452\) 7.62973 0.358872
\(453\) −13.8467 −0.650573
\(454\) −20.5926 −0.966458
\(455\) −9.56639 −0.448479
\(456\) −3.05029 −0.142843
\(457\) −17.7422 −0.829943 −0.414972 0.909834i \(-0.636209\pi\)
−0.414972 + 0.909834i \(0.636209\pi\)
\(458\) −17.1159 −0.799774
\(459\) 6.25317 0.291873
\(460\) −4.96493 −0.231491
\(461\) −5.74408 −0.267529 −0.133764 0.991013i \(-0.542706\pi\)
−0.133764 + 0.991013i \(0.542706\pi\)
\(462\) −1.59091 −0.0740157
\(463\) 11.0072 0.511550 0.255775 0.966736i \(-0.417669\pi\)
0.255775 + 0.966736i \(0.417669\pi\)
\(464\) 21.6505 1.00510
\(465\) −14.6605 −0.679865
\(466\) −38.0982 −1.76487
\(467\) 20.3504 0.941704 0.470852 0.882212i \(-0.343947\pi\)
0.470852 + 0.882212i \(0.343947\pi\)
\(468\) 1.60452 0.0741691
\(469\) −15.2925 −0.706144
\(470\) −63.8131 −2.94348
\(471\) 5.10610 0.235277
\(472\) 18.6578 0.858797
\(473\) −0.305238 −0.0140349
\(474\) −21.7878 −1.00075
\(475\) 11.3504 0.520792
\(476\) −3.72768 −0.170858
\(477\) −10.0242 −0.458977
\(478\) 12.5870 0.575716
\(479\) 18.4382 0.842463 0.421232 0.906953i \(-0.361598\pi\)
0.421232 + 0.906953i \(0.361598\pi\)
\(480\) 12.1212 0.553256
\(481\) −3.30212 −0.150564
\(482\) 23.2892 1.06080
\(483\) −2.21250 −0.100672
\(484\) 0.604750 0.0274886
\(485\) −39.5721 −1.79688
\(486\) 1.61392 0.0732091
\(487\) 5.57853 0.252787 0.126394 0.991980i \(-0.459660\pi\)
0.126394 + 0.991980i \(0.459660\pi\)
\(488\) 2.25183 0.101935
\(489\) −14.0106 −0.633580
\(490\) −35.5873 −1.60767
\(491\) 18.1802 0.820461 0.410231 0.911982i \(-0.365448\pi\)
0.410231 + 0.911982i \(0.365448\pi\)
\(492\) −3.79704 −0.171184
\(493\) 27.9502 1.25881
\(494\) −5.80043 −0.260973
\(495\) 3.65776 0.164404
\(496\) 19.4141 0.871719
\(497\) −8.17051 −0.366498
\(498\) 7.81612 0.350249
\(499\) −10.6130 −0.475104 −0.237552 0.971375i \(-0.576345\pi\)
−0.237552 + 0.971375i \(0.576345\pi\)
\(500\) −7.47500 −0.334292
\(501\) 19.1530 0.855693
\(502\) −44.8512 −2.00181
\(503\) 34.9625 1.55890 0.779449 0.626465i \(-0.215499\pi\)
0.779449 + 0.626465i \(0.215499\pi\)
\(504\) 2.21971 0.0988739
\(505\) −67.8155 −3.01775
\(506\) 3.62247 0.161038
\(507\) 5.96052 0.264716
\(508\) 8.93918 0.396612
\(509\) 10.0360 0.444838 0.222419 0.974951i \(-0.428605\pi\)
0.222419 + 0.974951i \(0.428605\pi\)
\(510\) 36.9147 1.63461
\(511\) 8.41360 0.372196
\(512\) 5.76323 0.254701
\(513\) −1.35459 −0.0598064
\(514\) −2.39637 −0.105699
\(515\) −48.7773 −2.14938
\(516\) −0.184593 −0.00812624
\(517\) 10.8096 0.475407
\(518\) 1.98001 0.0869967
\(519\) 22.4432 0.985146
\(520\) −21.8535 −0.958340
\(521\) −34.0373 −1.49120 −0.745601 0.666393i \(-0.767837\pi\)
−0.745601 + 0.666393i \(0.767837\pi\)
\(522\) 7.21386 0.315742
\(523\) −0.293817 −0.0128477 −0.00642386 0.999979i \(-0.502045\pi\)
−0.00642386 + 0.999979i \(0.502045\pi\)
\(524\) 7.27067 0.317621
\(525\) −8.25975 −0.360485
\(526\) 1.88369 0.0821326
\(527\) 25.0630 1.09176
\(528\) −4.84378 −0.210798
\(529\) −17.9622 −0.780964
\(530\) −59.1765 −2.57046
\(531\) 8.28564 0.359566
\(532\) 0.807504 0.0350097
\(533\) 16.6587 0.721567
\(534\) −24.9125 −1.07807
\(535\) −47.2810 −2.04414
\(536\) −34.9343 −1.50893
\(537\) −13.1373 −0.566916
\(538\) −43.1014 −1.85824
\(539\) 6.02832 0.259658
\(540\) 2.21203 0.0951907
\(541\) −35.6219 −1.53151 −0.765753 0.643134i \(-0.777634\pi\)
−0.765753 + 0.643134i \(0.777634\pi\)
\(542\) −4.24501 −0.182339
\(543\) −7.41628 −0.318263
\(544\) −20.7220 −0.888447
\(545\) −62.6580 −2.68397
\(546\) 4.22100 0.180642
\(547\) 43.3050 1.85159 0.925794 0.378029i \(-0.123398\pi\)
0.925794 + 0.378029i \(0.123398\pi\)
\(548\) −7.63858 −0.326304
\(549\) 1.00000 0.0426790
\(550\) 13.5235 0.576642
\(551\) −6.05468 −0.257938
\(552\) −5.05425 −0.215123
\(553\) −13.3074 −0.565888
\(554\) 11.8547 0.503658
\(555\) −4.55238 −0.193238
\(556\) −10.0633 −0.426779
\(557\) 23.9291 1.01391 0.506954 0.861973i \(-0.330771\pi\)
0.506954 + 0.861973i \(0.330771\pi\)
\(558\) 6.46869 0.273841
\(559\) 0.809858 0.0342533
\(560\) 17.4647 0.738020
\(561\) −6.25317 −0.264009
\(562\) −9.84310 −0.415206
\(563\) 6.09496 0.256872 0.128436 0.991718i \(-0.459004\pi\)
0.128436 + 0.991718i \(0.459004\pi\)
\(564\) 6.53713 0.275263
\(565\) −46.1476 −1.94144
\(566\) −35.9042 −1.50917
\(567\) 0.985739 0.0413972
\(568\) −18.6648 −0.783156
\(569\) −11.4324 −0.479271 −0.239635 0.970863i \(-0.577028\pi\)
−0.239635 + 0.970863i \(0.577028\pi\)
\(570\) −7.99660 −0.334941
\(571\) 4.26007 0.178278 0.0891392 0.996019i \(-0.471588\pi\)
0.0891392 + 0.996019i \(0.471588\pi\)
\(572\) −1.60452 −0.0670885
\(573\) 6.22085 0.259880
\(574\) −9.98884 −0.416926
\(575\) 18.8073 0.784318
\(576\) 4.33928 0.180803
\(577\) 31.0517 1.29270 0.646351 0.763041i \(-0.276294\pi\)
0.646351 + 0.763041i \(0.276294\pi\)
\(578\) −35.6711 −1.48372
\(579\) 1.95576 0.0812786
\(580\) 9.88726 0.410546
\(581\) 4.77387 0.198053
\(582\) 17.4605 0.723761
\(583\) 10.0242 0.415160
\(584\) 19.2201 0.795332
\(585\) −9.70479 −0.401244
\(586\) 12.4161 0.512903
\(587\) 12.7689 0.527030 0.263515 0.964655i \(-0.415118\pi\)
0.263515 + 0.964655i \(0.415118\pi\)
\(588\) 3.64563 0.150343
\(589\) −5.42925 −0.223708
\(590\) 48.9131 2.01372
\(591\) −27.6283 −1.13648
\(592\) 6.02847 0.247768
\(593\) −20.4992 −0.841800 −0.420900 0.907107i \(-0.638286\pi\)
−0.420900 + 0.907107i \(0.638286\pi\)
\(594\) −1.61392 −0.0662201
\(595\) 22.5464 0.924314
\(596\) −1.99506 −0.0817209
\(597\) 3.84659 0.157430
\(598\) −9.61113 −0.393028
\(599\) 4.42860 0.180948 0.0904739 0.995899i \(-0.471162\pi\)
0.0904739 + 0.995899i \(0.471162\pi\)
\(600\) −18.8686 −0.770308
\(601\) 1.08793 0.0443774 0.0221887 0.999754i \(-0.492937\pi\)
0.0221887 + 0.999754i \(0.492937\pi\)
\(602\) −0.485605 −0.0197918
\(603\) −15.5138 −0.631770
\(604\) 8.37377 0.340724
\(605\) −3.65776 −0.148709
\(606\) 29.9224 1.21551
\(607\) 1.52522 0.0619067 0.0309534 0.999521i \(-0.490146\pi\)
0.0309534 + 0.999521i \(0.490146\pi\)
\(608\) 4.48887 0.182048
\(609\) 4.40602 0.178541
\(610\) 5.90335 0.239020
\(611\) −28.6801 −1.16027
\(612\) −3.78160 −0.152862
\(613\) 23.3564 0.943356 0.471678 0.881771i \(-0.343648\pi\)
0.471678 + 0.881771i \(0.343648\pi\)
\(614\) −17.1332 −0.691439
\(615\) 22.9660 0.926079
\(616\) −2.21971 −0.0894348
\(617\) 34.2890 1.38042 0.690211 0.723608i \(-0.257518\pi\)
0.690211 + 0.723608i \(0.257518\pi\)
\(618\) 21.5221 0.865746
\(619\) −9.90023 −0.397924 −0.198962 0.980007i \(-0.563757\pi\)
−0.198962 + 0.980007i \(0.563757\pi\)
\(620\) 8.86594 0.356065
\(621\) −2.24451 −0.0900690
\(622\) 55.2901 2.21693
\(623\) −15.2158 −0.609610
\(624\) 12.8515 0.514473
\(625\) 3.31550 0.132620
\(626\) −10.5016 −0.419727
\(627\) 1.35459 0.0540969
\(628\) −3.08792 −0.123221
\(629\) 7.78257 0.310311
\(630\) 5.81917 0.231841
\(631\) −12.6953 −0.505390 −0.252695 0.967546i \(-0.581317\pi\)
−0.252695 + 0.967546i \(0.581317\pi\)
\(632\) −30.3995 −1.20923
\(633\) 11.4958 0.456918
\(634\) −13.1036 −0.520412
\(635\) −54.0677 −2.14561
\(636\) 6.06214 0.240380
\(637\) −15.9943 −0.633719
\(638\) −7.21386 −0.285599
\(639\) −8.28871 −0.327896
\(640\) 49.8588 1.97084
\(641\) 14.9145 0.589087 0.294544 0.955638i \(-0.404832\pi\)
0.294544 + 0.955638i \(0.404832\pi\)
\(642\) 20.8619 0.823353
\(643\) −20.7823 −0.819576 −0.409788 0.912181i \(-0.634397\pi\)
−0.409788 + 0.912181i \(0.634397\pi\)
\(644\) 1.33801 0.0527250
\(645\) 1.11649 0.0439617
\(646\) 13.6707 0.537865
\(647\) 2.22546 0.0874919 0.0437460 0.999043i \(-0.486071\pi\)
0.0437460 + 0.999043i \(0.486071\pi\)
\(648\) 2.25183 0.0884601
\(649\) −8.28564 −0.325240
\(650\) −35.8805 −1.40735
\(651\) 3.95089 0.154848
\(652\) 8.47289 0.331824
\(653\) −11.3494 −0.444135 −0.222068 0.975031i \(-0.571281\pi\)
−0.222068 + 0.975031i \(0.571281\pi\)
\(654\) 27.6467 1.08107
\(655\) −43.9759 −1.71828
\(656\) −30.4126 −1.18741
\(657\) 8.53532 0.332995
\(658\) 17.1971 0.670414
\(659\) 1.23695 0.0481847 0.0240923 0.999710i \(-0.492330\pi\)
0.0240923 + 0.999710i \(0.492330\pi\)
\(660\) −2.21203 −0.0861033
\(661\) 34.0520 1.32447 0.662234 0.749297i \(-0.269608\pi\)
0.662234 + 0.749297i \(0.269608\pi\)
\(662\) 50.7294 1.97165
\(663\) 16.5909 0.644338
\(664\) 10.9054 0.423213
\(665\) −4.88410 −0.189397
\(666\) 2.00866 0.0778339
\(667\) −10.0324 −0.388457
\(668\) −11.5828 −0.448151
\(669\) −9.82324 −0.379788
\(670\) −91.5833 −3.53817
\(671\) −1.00000 −0.0386046
\(672\) −3.26658 −0.126011
\(673\) 17.1078 0.659456 0.329728 0.944076i \(-0.393043\pi\)
0.329728 + 0.944076i \(0.393043\pi\)
\(674\) 30.0331 1.15683
\(675\) −8.37924 −0.322517
\(676\) −3.60462 −0.138639
\(677\) 18.9338 0.727687 0.363843 0.931460i \(-0.381464\pi\)
0.363843 + 0.931460i \(0.381464\pi\)
\(678\) 20.3618 0.781990
\(679\) 10.6644 0.409262
\(680\) 51.5052 1.97513
\(681\) −12.7593 −0.488939
\(682\) −6.46869 −0.247699
\(683\) 34.1390 1.30629 0.653146 0.757232i \(-0.273449\pi\)
0.653146 + 0.757232i \(0.273449\pi\)
\(684\) 0.819186 0.0313223
\(685\) 46.2011 1.76525
\(686\) 20.7269 0.791355
\(687\) −10.6052 −0.404612
\(688\) −1.47850 −0.0563675
\(689\) −26.5963 −1.01324
\(690\) −13.2501 −0.504424
\(691\) 4.19429 0.159558 0.0797792 0.996813i \(-0.474578\pi\)
0.0797792 + 0.996813i \(0.474578\pi\)
\(692\) −13.5725 −0.515949
\(693\) −0.985739 −0.0374451
\(694\) 7.08649 0.268999
\(695\) 60.8668 2.30881
\(696\) 10.0651 0.381518
\(697\) −39.2618 −1.48715
\(698\) −33.2340 −1.25793
\(699\) −23.6060 −0.892859
\(700\) 4.99508 0.188796
\(701\) −44.7848 −1.69150 −0.845749 0.533581i \(-0.820846\pi\)
−0.845749 + 0.533581i \(0.820846\pi\)
\(702\) 4.28207 0.161616
\(703\) −1.68589 −0.0635845
\(704\) −4.33928 −0.163543
\(705\) −39.5391 −1.48913
\(706\) 13.4238 0.505213
\(707\) 18.2757 0.687330
\(708\) −5.01074 −0.188315
\(709\) −45.6124 −1.71301 −0.856504 0.516140i \(-0.827369\pi\)
−0.856504 + 0.516140i \(0.827369\pi\)
\(710\) −48.9312 −1.83636
\(711\) −13.4999 −0.506286
\(712\) −34.7591 −1.30265
\(713\) −8.99610 −0.336907
\(714\) −9.94822 −0.372303
\(715\) 9.70479 0.362939
\(716\) 7.94478 0.296910
\(717\) 7.79900 0.291259
\(718\) −25.1523 −0.938674
\(719\) 27.9564 1.04260 0.521299 0.853374i \(-0.325448\pi\)
0.521299 + 0.853374i \(0.325448\pi\)
\(720\) 17.7174 0.660288
\(721\) 13.1451 0.489549
\(722\) 27.7032 1.03100
\(723\) 14.4302 0.536665
\(724\) 4.48499 0.166683
\(725\) −37.4532 −1.39098
\(726\) 1.61392 0.0598983
\(727\) 34.9140 1.29489 0.647445 0.762112i \(-0.275837\pi\)
0.647445 + 0.762112i \(0.275837\pi\)
\(728\) 5.88935 0.218274
\(729\) 1.00000 0.0370370
\(730\) 50.3870 1.86491
\(731\) −1.90870 −0.0705960
\(732\) −0.604750 −0.0223522
\(733\) 27.5891 1.01903 0.509514 0.860462i \(-0.329825\pi\)
0.509514 + 0.860462i \(0.329825\pi\)
\(734\) −17.0105 −0.627869
\(735\) −22.0502 −0.813333
\(736\) 7.43793 0.274166
\(737\) 15.5138 0.571457
\(738\) −10.1333 −0.373014
\(739\) −35.9604 −1.32282 −0.661412 0.750023i \(-0.730042\pi\)
−0.661412 + 0.750023i \(0.730042\pi\)
\(740\) 2.75305 0.101204
\(741\) −3.59399 −0.132029
\(742\) 15.9476 0.585455
\(743\) 47.0587 1.72642 0.863208 0.504849i \(-0.168452\pi\)
0.863208 + 0.504849i \(0.168452\pi\)
\(744\) 9.02544 0.330889
\(745\) 12.0669 0.442098
\(746\) −31.2722 −1.14495
\(747\) 4.84293 0.177194
\(748\) 3.78160 0.138269
\(749\) 12.7419 0.465577
\(750\) −19.9489 −0.728430
\(751\) 3.48808 0.127282 0.0636410 0.997973i \(-0.479729\pi\)
0.0636410 + 0.997973i \(0.479729\pi\)
\(752\) 52.3595 1.90935
\(753\) −27.7902 −1.01273
\(754\) 19.1398 0.697031
\(755\) −50.6478 −1.84326
\(756\) −0.596126 −0.0216809
\(757\) −20.7003 −0.752364 −0.376182 0.926546i \(-0.622763\pi\)
−0.376182 + 0.926546i \(0.622763\pi\)
\(758\) 10.7522 0.390537
\(759\) 2.24451 0.0814705
\(760\) −11.1573 −0.404716
\(761\) 19.3864 0.702756 0.351378 0.936234i \(-0.385713\pi\)
0.351378 + 0.936234i \(0.385713\pi\)
\(762\) 23.8564 0.864226
\(763\) 16.8858 0.611308
\(764\) −3.76206 −0.136107
\(765\) 22.8726 0.826961
\(766\) −30.3227 −1.09560
\(767\) 21.9835 0.793778
\(768\) −13.3207 −0.480670
\(769\) 42.9108 1.54740 0.773702 0.633549i \(-0.218403\pi\)
0.773702 + 0.633549i \(0.218403\pi\)
\(770\) −5.81917 −0.209708
\(771\) −1.48481 −0.0534741
\(772\) −1.18275 −0.0425679
\(773\) −8.56024 −0.307890 −0.153945 0.988079i \(-0.549198\pi\)
−0.153945 + 0.988079i \(0.549198\pi\)
\(774\) −0.492631 −0.0177072
\(775\) −33.5844 −1.20639
\(776\) 24.3618 0.874537
\(777\) 1.22683 0.0440123
\(778\) 34.2431 1.22768
\(779\) 8.50504 0.304725
\(780\) 5.86897 0.210143
\(781\) 8.28871 0.296594
\(782\) 22.6519 0.810030
\(783\) 4.46976 0.159736
\(784\) 29.1998 1.04285
\(785\) 18.6769 0.666608
\(786\) 19.4036 0.692103
\(787\) 6.09934 0.217418 0.108709 0.994074i \(-0.465328\pi\)
0.108709 + 0.994074i \(0.465328\pi\)
\(788\) 16.7082 0.595205
\(789\) 1.16715 0.0415515
\(790\) −79.6947 −2.83541
\(791\) 12.4364 0.442188
\(792\) −2.25183 −0.0800152
\(793\) 2.65320 0.0942180
\(794\) 2.94380 0.104471
\(795\) −36.6662 −1.30042
\(796\) −2.32623 −0.0824509
\(797\) 17.6881 0.626546 0.313273 0.949663i \(-0.398575\pi\)
0.313273 + 0.949663i \(0.398575\pi\)
\(798\) 2.15502 0.0762869
\(799\) 67.5945 2.39132
\(800\) 27.7674 0.981726
\(801\) −15.4360 −0.545403
\(802\) −30.0618 −1.06152
\(803\) −8.53532 −0.301205
\(804\) 9.38195 0.330876
\(805\) −8.09281 −0.285234
\(806\) 17.1627 0.604532
\(807\) −26.7060 −0.940096
\(808\) 41.7492 1.46873
\(809\) −3.78295 −0.133001 −0.0665007 0.997786i \(-0.521183\pi\)
−0.0665007 + 0.997786i \(0.521183\pi\)
\(810\) 5.90335 0.207423
\(811\) 48.6839 1.70952 0.854761 0.519021i \(-0.173703\pi\)
0.854761 + 0.519021i \(0.173703\pi\)
\(812\) −2.66454 −0.0935071
\(813\) −2.63024 −0.0922466
\(814\) −2.00866 −0.0704034
\(815\) −51.2474 −1.79512
\(816\) −30.2890 −1.06033
\(817\) 0.413471 0.0144655
\(818\) 45.7473 1.59952
\(819\) 2.61537 0.0913883
\(820\) −13.8887 −0.485014
\(821\) −15.5709 −0.543429 −0.271714 0.962378i \(-0.587591\pi\)
−0.271714 + 0.962378i \(0.587591\pi\)
\(822\) −20.3854 −0.711023
\(823\) −23.6529 −0.824487 −0.412244 0.911074i \(-0.635255\pi\)
−0.412244 + 0.911074i \(0.635255\pi\)
\(824\) 30.0287 1.04610
\(825\) 8.37924 0.291728
\(826\) −13.1817 −0.458650
\(827\) −24.0287 −0.835560 −0.417780 0.908548i \(-0.637192\pi\)
−0.417780 + 0.908548i \(0.637192\pi\)
\(828\) 1.35737 0.0471717
\(829\) −40.8171 −1.41764 −0.708818 0.705391i \(-0.750771\pi\)
−0.708818 + 0.705391i \(0.750771\pi\)
\(830\) 28.5895 0.992357
\(831\) 7.34527 0.254805
\(832\) 11.5130 0.399141
\(833\) 37.6961 1.30609
\(834\) −26.8564 −0.929960
\(835\) 70.0572 2.42443
\(836\) −0.819186 −0.0283321
\(837\) 4.00805 0.138538
\(838\) −22.6069 −0.780943
\(839\) 15.1758 0.523927 0.261963 0.965078i \(-0.415630\pi\)
0.261963 + 0.965078i \(0.415630\pi\)
\(840\) 8.11919 0.280139
\(841\) −9.02121 −0.311076
\(842\) 0.391351 0.0134868
\(843\) −6.09887 −0.210056
\(844\) −6.95210 −0.239301
\(845\) 21.8022 0.750018
\(846\) 17.4459 0.599803
\(847\) 0.985739 0.0338704
\(848\) 48.5551 1.66739
\(849\) −22.2465 −0.763498
\(850\) 84.5645 2.90054
\(851\) −2.79347 −0.0957589
\(852\) 5.01260 0.171729
\(853\) 2.51030 0.0859511 0.0429756 0.999076i \(-0.486316\pi\)
0.0429756 + 0.999076i \(0.486316\pi\)
\(854\) −1.59091 −0.0544398
\(855\) −4.95476 −0.169449
\(856\) 29.1076 0.994876
\(857\) −30.5364 −1.04310 −0.521552 0.853219i \(-0.674647\pi\)
−0.521552 + 0.853219i \(0.674647\pi\)
\(858\) −4.28207 −0.146187
\(859\) 10.4340 0.356003 0.178001 0.984030i \(-0.443037\pi\)
0.178001 + 0.984030i \(0.443037\pi\)
\(860\) −0.675196 −0.0230240
\(861\) −6.18916 −0.210926
\(862\) 33.7621 1.14994
\(863\) 7.19925 0.245065 0.122533 0.992464i \(-0.460898\pi\)
0.122533 + 0.992464i \(0.460898\pi\)
\(864\) −3.31383 −0.112739
\(865\) 82.0918 2.79121
\(866\) −62.2520 −2.11541
\(867\) −22.1021 −0.750628
\(868\) −2.38930 −0.0810982
\(869\) 13.4999 0.457953
\(870\) 26.3866 0.894589
\(871\) −41.1612 −1.39469
\(872\) 38.5741 1.30628
\(873\) 10.8187 0.366156
\(874\) −4.90694 −0.165980
\(875\) −12.1842 −0.411901
\(876\) −5.16173 −0.174399
\(877\) −26.2304 −0.885738 −0.442869 0.896586i \(-0.646039\pi\)
−0.442869 + 0.896586i \(0.646039\pi\)
\(878\) 12.1849 0.411222
\(879\) 7.69309 0.259482
\(880\) −17.7174 −0.597253
\(881\) 16.5850 0.558764 0.279382 0.960180i \(-0.409870\pi\)
0.279382 + 0.960180i \(0.409870\pi\)
\(882\) 9.72925 0.327601
\(883\) −29.1642 −0.981452 −0.490726 0.871314i \(-0.663268\pi\)
−0.490726 + 0.871314i \(0.663268\pi\)
\(884\) −10.0334 −0.337458
\(885\) 30.3069 1.01876
\(886\) 12.7453 0.428187
\(887\) −20.2621 −0.680334 −0.340167 0.940365i \(-0.610484\pi\)
−0.340167 + 0.940365i \(0.610484\pi\)
\(888\) 2.80258 0.0940484
\(889\) 14.5708 0.488689
\(890\) −91.1240 −3.05448
\(891\) −1.00000 −0.0335013
\(892\) 5.94060 0.198906
\(893\) −14.6426 −0.489995
\(894\) −5.32431 −0.178072
\(895\) −48.0531 −1.60624
\(896\) −13.4365 −0.448884
\(897\) −5.95513 −0.198836
\(898\) −35.0820 −1.17070
\(899\) 17.9150 0.597500
\(900\) 5.06735 0.168912
\(901\) 62.6831 2.08828
\(902\) 10.1333 0.337403
\(903\) −0.300885 −0.0100128
\(904\) 28.4098 0.944896
\(905\) −27.1270 −0.901732
\(906\) 22.3475 0.742445
\(907\) 34.5315 1.14660 0.573300 0.819345i \(-0.305663\pi\)
0.573300 + 0.819345i \(0.305663\pi\)
\(908\) 7.71621 0.256071
\(909\) 18.5401 0.614937
\(910\) 15.4394 0.511812
\(911\) 5.96287 0.197559 0.0987794 0.995109i \(-0.468506\pi\)
0.0987794 + 0.995109i \(0.468506\pi\)
\(912\) 6.56131 0.217267
\(913\) −4.84293 −0.160278
\(914\) 28.6345 0.947145
\(915\) 3.65776 0.120922
\(916\) 6.41347 0.211907
\(917\) 11.8512 0.391360
\(918\) −10.0921 −0.333090
\(919\) 31.1321 1.02695 0.513476 0.858104i \(-0.328358\pi\)
0.513476 + 0.858104i \(0.328358\pi\)
\(920\) −18.4872 −0.609506
\(921\) −10.6159 −0.349804
\(922\) 9.27051 0.305308
\(923\) −21.9916 −0.723863
\(924\) 0.596126 0.0196111
\(925\) −10.4286 −0.342891
\(926\) −17.7648 −0.583789
\(927\) 13.3353 0.437988
\(928\) −14.8121 −0.486229
\(929\) 16.4949 0.541181 0.270591 0.962695i \(-0.412781\pi\)
0.270591 + 0.962695i \(0.412781\pi\)
\(930\) 23.6609 0.775873
\(931\) −8.16587 −0.267626
\(932\) 14.2757 0.467616
\(933\) 34.2582 1.12156
\(934\) −32.8440 −1.07469
\(935\) −22.8726 −0.748015
\(936\) 5.97455 0.195284
\(937\) −1.65979 −0.0542229 −0.0271114 0.999632i \(-0.508631\pi\)
−0.0271114 + 0.999632i \(0.508631\pi\)
\(938\) 24.6810 0.805863
\(939\) −6.50686 −0.212343
\(940\) 23.9113 0.779900
\(941\) 16.4332 0.535707 0.267853 0.963460i \(-0.413686\pi\)
0.267853 + 0.963460i \(0.413686\pi\)
\(942\) −8.24086 −0.268502
\(943\) 14.0926 0.458918
\(944\) −40.1338 −1.30624
\(945\) 3.60560 0.117290
\(946\) 0.492631 0.0160168
\(947\) 7.61673 0.247510 0.123755 0.992313i \(-0.460506\pi\)
0.123755 + 0.992313i \(0.460506\pi\)
\(948\) 8.16407 0.265157
\(949\) 22.6459 0.735118
\(950\) −18.3187 −0.594337
\(951\) −8.11911 −0.263280
\(952\) −13.8803 −0.449862
\(953\) −36.1956 −1.17249 −0.586246 0.810133i \(-0.699395\pi\)
−0.586246 + 0.810133i \(0.699395\pi\)
\(954\) 16.1783 0.523792
\(955\) 22.7544 0.736316
\(956\) −4.71645 −0.152541
\(957\) −4.46976 −0.144487
\(958\) −29.7579 −0.961433
\(959\) −12.4508 −0.402059
\(960\) 15.8721 0.512269
\(961\) −14.9355 −0.481791
\(962\) 5.32937 0.171826
\(963\) 12.9262 0.416541
\(964\) −8.72666 −0.281067
\(965\) 7.15371 0.230286
\(966\) 3.57081 0.114889
\(967\) −12.3745 −0.397936 −0.198968 0.980006i \(-0.563759\pi\)
−0.198968 + 0.980006i \(0.563759\pi\)
\(968\) 2.25183 0.0723765
\(969\) 8.47045 0.272110
\(970\) 63.8664 2.05063
\(971\) 46.6524 1.49715 0.748574 0.663052i \(-0.230739\pi\)
0.748574 + 0.663052i \(0.230739\pi\)
\(972\) −0.604750 −0.0193974
\(973\) −16.4031 −0.525860
\(974\) −9.00332 −0.288485
\(975\) −22.2318 −0.711988
\(976\) −4.84378 −0.155046
\(977\) 54.8844 1.75591 0.877953 0.478746i \(-0.158909\pi\)
0.877953 + 0.478746i \(0.158909\pi\)
\(978\) 22.6120 0.723052
\(979\) 15.4360 0.493336
\(980\) 13.3348 0.425966
\(981\) 17.1301 0.546923
\(982\) −29.3415 −0.936324
\(983\) 37.6301 1.20021 0.600107 0.799920i \(-0.295125\pi\)
0.600107 + 0.799920i \(0.295125\pi\)
\(984\) −14.1385 −0.450720
\(985\) −101.058 −3.21997
\(986\) −45.1095 −1.43658
\(987\) 10.6555 0.339168
\(988\) 2.17346 0.0691471
\(989\) 0.685109 0.0217852
\(990\) −5.90335 −0.187621
\(991\) −54.0536 −1.71707 −0.858535 0.512755i \(-0.828625\pi\)
−0.858535 + 0.512755i \(0.828625\pi\)
\(992\) −13.2820 −0.421704
\(993\) 31.4323 0.997475
\(994\) 13.1866 0.418253
\(995\) 14.0699 0.446047
\(996\) −2.92876 −0.0928014
\(997\) 9.87216 0.312655 0.156327 0.987705i \(-0.450035\pi\)
0.156327 + 0.987705i \(0.450035\pi\)
\(998\) 17.1286 0.542197
\(999\) 1.24458 0.0393768
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 2013.2.a.h.1.5 14
3.2 odd 2 6039.2.a.j.1.10 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2013.2.a.h.1.5 14 1.1 even 1 trivial
6039.2.a.j.1.10 14 3.2 odd 2