Properties

Label 8030.2.a.bd.1.9
Level $8030$
Weight $2$
Character 8030.1
Self dual yes
Analytic conductor $64.120$
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] = [8030,2,Mod(1,8030)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8030, 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("8030.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8030 = 2 \cdot 5 \cdot 11 \cdot 73 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8030.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(64.1198728231\)
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: \( x^{14} - 6 x^{13} - 15 x^{12} + 143 x^{11} - 13 x^{10} - 1176 x^{9} + 1018 x^{8} + 4076 x^{7} - 4865 x^{6} - 5483 x^{5} + 7607 x^{4} + 1210 x^{3} - 3153 x^{2} + 878 x - 54 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.9
Root \(1.13475\) of defining polynomial
Character \(\chi\) \(=\) 8030.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.13475 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.13475 q^{6} -2.24922 q^{7} +1.00000 q^{8} -1.71235 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.13475 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.13475 q^{6} -2.24922 q^{7} +1.00000 q^{8} -1.71235 q^{9} -1.00000 q^{10} +1.00000 q^{11} +1.13475 q^{12} +0.202024 q^{13} -2.24922 q^{14} -1.13475 q^{15} +1.00000 q^{16} +2.27259 q^{17} -1.71235 q^{18} +6.24810 q^{19} -1.00000 q^{20} -2.55229 q^{21} +1.00000 q^{22} -2.51752 q^{23} +1.13475 q^{24} +1.00000 q^{25} +0.202024 q^{26} -5.34733 q^{27} -2.24922 q^{28} +3.26404 q^{29} -1.13475 q^{30} +3.69060 q^{31} +1.00000 q^{32} +1.13475 q^{33} +2.27259 q^{34} +2.24922 q^{35} -1.71235 q^{36} -10.0765 q^{37} +6.24810 q^{38} +0.229246 q^{39} -1.00000 q^{40} +8.61893 q^{41} -2.55229 q^{42} -0.359773 q^{43} +1.00000 q^{44} +1.71235 q^{45} -2.51752 q^{46} -2.00249 q^{47} +1.13475 q^{48} -1.94102 q^{49} +1.00000 q^{50} +2.57882 q^{51} +0.202024 q^{52} +3.20981 q^{53} -5.34733 q^{54} -1.00000 q^{55} -2.24922 q^{56} +7.09001 q^{57} +3.26404 q^{58} -2.01046 q^{59} -1.13475 q^{60} +1.59340 q^{61} +3.69060 q^{62} +3.85144 q^{63} +1.00000 q^{64} -0.202024 q^{65} +1.13475 q^{66} +10.9959 q^{67} +2.27259 q^{68} -2.85675 q^{69} +2.24922 q^{70} +4.47296 q^{71} -1.71235 q^{72} +1.00000 q^{73} -10.0765 q^{74} +1.13475 q^{75} +6.24810 q^{76} -2.24922 q^{77} +0.229246 q^{78} +12.7687 q^{79} -1.00000 q^{80} -0.930819 q^{81} +8.61893 q^{82} -11.6129 q^{83} -2.55229 q^{84} -2.27259 q^{85} -0.359773 q^{86} +3.70386 q^{87} +1.00000 q^{88} +9.86267 q^{89} +1.71235 q^{90} -0.454396 q^{91} -2.51752 q^{92} +4.18790 q^{93} -2.00249 q^{94} -6.24810 q^{95} +1.13475 q^{96} -10.1623 q^{97} -1.94102 q^{98} -1.71235 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 14 q^{2} + 6 q^{3} + 14 q^{4} - 14 q^{5} + 6 q^{6} + 4 q^{7} + 14 q^{8} + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 14 q^{2} + 6 q^{3} + 14 q^{4} - 14 q^{5} + 6 q^{6} + 4 q^{7} + 14 q^{8} + 24 q^{9} - 14 q^{10} + 14 q^{11} + 6 q^{12} + 4 q^{13} + 4 q^{14} - 6 q^{15} + 14 q^{16} + 20 q^{17} + 24 q^{18} - 14 q^{20} + 25 q^{21} + 14 q^{22} - 4 q^{23} + 6 q^{24} + 14 q^{25} + 4 q^{26} + 21 q^{27} + 4 q^{28} + 7 q^{29} - 6 q^{30} + 8 q^{31} + 14 q^{32} + 6 q^{33} + 20 q^{34} - 4 q^{35} + 24 q^{36} + 17 q^{37} + 7 q^{39} - 14 q^{40} - 14 q^{41} + 25 q^{42} + 12 q^{43} + 14 q^{44} - 24 q^{45} - 4 q^{46} + 28 q^{47} + 6 q^{48} + 20 q^{49} + 14 q^{50} - 13 q^{51} + 4 q^{52} + 13 q^{53} + 21 q^{54} - 14 q^{55} + 4 q^{56} - 23 q^{57} + 7 q^{58} + 36 q^{59} - 6 q^{60} + 25 q^{61} + 8 q^{62} + 45 q^{63} + 14 q^{64} - 4 q^{65} + 6 q^{66} + 20 q^{68} - 7 q^{69} - 4 q^{70} + 17 q^{71} + 24 q^{72} + 14 q^{73} + 17 q^{74} + 6 q^{75} + 4 q^{77} + 7 q^{78} + 10 q^{79} - 14 q^{80} + 58 q^{81} - 14 q^{82} - 6 q^{83} + 25 q^{84} - 20 q^{85} + 12 q^{86} + 44 q^{87} + 14 q^{88} + 36 q^{89} - 24 q^{90} - 15 q^{91} - 4 q^{92} - 2 q^{93} + 28 q^{94} + 6 q^{96} - 19 q^{97} + 20 q^{98} + 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 1.13475 0.655147 0.327573 0.944826i \(-0.393769\pi\)
0.327573 + 0.944826i \(0.393769\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 1.13475 0.463259
\(7\) −2.24922 −0.850125 −0.425062 0.905164i \(-0.639748\pi\)
−0.425062 + 0.905164i \(0.639748\pi\)
\(8\) 1.00000 0.353553
\(9\) −1.71235 −0.570783
\(10\) −1.00000 −0.316228
\(11\) 1.00000 0.301511
\(12\) 1.13475 0.327573
\(13\) 0.202024 0.0560314 0.0280157 0.999607i \(-0.491081\pi\)
0.0280157 + 0.999607i \(0.491081\pi\)
\(14\) −2.24922 −0.601129
\(15\) −1.13475 −0.292991
\(16\) 1.00000 0.250000
\(17\) 2.27259 0.551185 0.275592 0.961275i \(-0.411126\pi\)
0.275592 + 0.961275i \(0.411126\pi\)
\(18\) −1.71235 −0.403604
\(19\) 6.24810 1.43341 0.716706 0.697376i \(-0.245649\pi\)
0.716706 + 0.697376i \(0.245649\pi\)
\(20\) −1.00000 −0.223607
\(21\) −2.55229 −0.556956
\(22\) 1.00000 0.213201
\(23\) −2.51752 −0.524939 −0.262470 0.964940i \(-0.584537\pi\)
−0.262470 + 0.964940i \(0.584537\pi\)
\(24\) 1.13475 0.231629
\(25\) 1.00000 0.200000
\(26\) 0.202024 0.0396201
\(27\) −5.34733 −1.02909
\(28\) −2.24922 −0.425062
\(29\) 3.26404 0.606116 0.303058 0.952972i \(-0.401992\pi\)
0.303058 + 0.952972i \(0.401992\pi\)
\(30\) −1.13475 −0.207176
\(31\) 3.69060 0.662851 0.331426 0.943481i \(-0.392470\pi\)
0.331426 + 0.943481i \(0.392470\pi\)
\(32\) 1.00000 0.176777
\(33\) 1.13475 0.197534
\(34\) 2.27259 0.389747
\(35\) 2.24922 0.380187
\(36\) −1.71235 −0.285391
\(37\) −10.0765 −1.65657 −0.828285 0.560307i \(-0.810683\pi\)
−0.828285 + 0.560307i \(0.810683\pi\)
\(38\) 6.24810 1.01357
\(39\) 0.229246 0.0367088
\(40\) −1.00000 −0.158114
\(41\) 8.61893 1.34605 0.673025 0.739620i \(-0.264995\pi\)
0.673025 + 0.739620i \(0.264995\pi\)
\(42\) −2.55229 −0.393828
\(43\) −0.359773 −0.0548649 −0.0274324 0.999624i \(-0.508733\pi\)
−0.0274324 + 0.999624i \(0.508733\pi\)
\(44\) 1.00000 0.150756
\(45\) 1.71235 0.255262
\(46\) −2.51752 −0.371188
\(47\) −2.00249 −0.292094 −0.146047 0.989278i \(-0.546655\pi\)
−0.146047 + 0.989278i \(0.546655\pi\)
\(48\) 1.13475 0.163787
\(49\) −1.94102 −0.277288
\(50\) 1.00000 0.141421
\(51\) 2.57882 0.361107
\(52\) 0.202024 0.0280157
\(53\) 3.20981 0.440902 0.220451 0.975398i \(-0.429247\pi\)
0.220451 + 0.975398i \(0.429247\pi\)
\(54\) −5.34733 −0.727679
\(55\) −1.00000 −0.134840
\(56\) −2.24922 −0.300564
\(57\) 7.09001 0.939095
\(58\) 3.26404 0.428589
\(59\) −2.01046 −0.261740 −0.130870 0.991400i \(-0.541777\pi\)
−0.130870 + 0.991400i \(0.541777\pi\)
\(60\) −1.13475 −0.146495
\(61\) 1.59340 0.204014 0.102007 0.994784i \(-0.467474\pi\)
0.102007 + 0.994784i \(0.467474\pi\)
\(62\) 3.69060 0.468707
\(63\) 3.85144 0.485236
\(64\) 1.00000 0.125000
\(65\) −0.202024 −0.0250580
\(66\) 1.13475 0.139678
\(67\) 10.9959 1.34336 0.671679 0.740842i \(-0.265573\pi\)
0.671679 + 0.740842i \(0.265573\pi\)
\(68\) 2.27259 0.275592
\(69\) −2.85675 −0.343912
\(70\) 2.24922 0.268833
\(71\) 4.47296 0.530843 0.265422 0.964132i \(-0.414489\pi\)
0.265422 + 0.964132i \(0.414489\pi\)
\(72\) −1.71235 −0.201802
\(73\) 1.00000 0.117041
\(74\) −10.0765 −1.17137
\(75\) 1.13475 0.131029
\(76\) 6.24810 0.716706
\(77\) −2.24922 −0.256322
\(78\) 0.229246 0.0259570
\(79\) 12.7687 1.43659 0.718295 0.695739i \(-0.244923\pi\)
0.718295 + 0.695739i \(0.244923\pi\)
\(80\) −1.00000 −0.111803
\(81\) −0.930819 −0.103424
\(82\) 8.61893 0.951801
\(83\) −11.6129 −1.27469 −0.637343 0.770580i \(-0.719967\pi\)
−0.637343 + 0.770580i \(0.719967\pi\)
\(84\) −2.55229 −0.278478
\(85\) −2.27259 −0.246497
\(86\) −0.359773 −0.0387953
\(87\) 3.70386 0.397095
\(88\) 1.00000 0.106600
\(89\) 9.86267 1.04544 0.522720 0.852504i \(-0.324917\pi\)
0.522720 + 0.852504i \(0.324917\pi\)
\(90\) 1.71235 0.180497
\(91\) −0.454396 −0.0476336
\(92\) −2.51752 −0.262470
\(93\) 4.18790 0.434265
\(94\) −2.00249 −0.206541
\(95\) −6.24810 −0.641041
\(96\) 1.13475 0.115815
\(97\) −10.1623 −1.03182 −0.515912 0.856641i \(-0.672547\pi\)
−0.515912 + 0.856641i \(0.672547\pi\)
\(98\) −1.94102 −0.196072
\(99\) −1.71235 −0.172097
\(100\) 1.00000 0.100000
\(101\) 11.1168 1.10616 0.553081 0.833127i \(-0.313452\pi\)
0.553081 + 0.833127i \(0.313452\pi\)
\(102\) 2.57882 0.255341
\(103\) 12.0478 1.18710 0.593550 0.804797i \(-0.297726\pi\)
0.593550 + 0.804797i \(0.297726\pi\)
\(104\) 0.202024 0.0198101
\(105\) 2.55229 0.249078
\(106\) 3.20981 0.311765
\(107\) −9.35618 −0.904496 −0.452248 0.891892i \(-0.649378\pi\)
−0.452248 + 0.891892i \(0.649378\pi\)
\(108\) −5.34733 −0.514547
\(109\) 15.2409 1.45981 0.729906 0.683548i \(-0.239564\pi\)
0.729906 + 0.683548i \(0.239564\pi\)
\(110\) −1.00000 −0.0953463
\(111\) −11.4343 −1.08530
\(112\) −2.24922 −0.212531
\(113\) 3.51943 0.331080 0.165540 0.986203i \(-0.447063\pi\)
0.165540 + 0.986203i \(0.447063\pi\)
\(114\) 7.09001 0.664040
\(115\) 2.51752 0.234760
\(116\) 3.26404 0.303058
\(117\) −0.345935 −0.0319817
\(118\) −2.01046 −0.185078
\(119\) −5.11156 −0.468576
\(120\) −1.13475 −0.103588
\(121\) 1.00000 0.0909091
\(122\) 1.59340 0.144260
\(123\) 9.78031 0.881860
\(124\) 3.69060 0.331426
\(125\) −1.00000 −0.0894427
\(126\) 3.85144 0.343114
\(127\) −0.831256 −0.0737621 −0.0368810 0.999320i \(-0.511742\pi\)
−0.0368810 + 0.999320i \(0.511742\pi\)
\(128\) 1.00000 0.0883883
\(129\) −0.408252 −0.0359445
\(130\) −0.202024 −0.0177187
\(131\) 5.24272 0.458058 0.229029 0.973420i \(-0.426445\pi\)
0.229029 + 0.973420i \(0.426445\pi\)
\(132\) 1.13475 0.0987671
\(133\) −14.0533 −1.21858
\(134\) 10.9959 0.949898
\(135\) 5.34733 0.460224
\(136\) 2.27259 0.194873
\(137\) −2.36028 −0.201652 −0.100826 0.994904i \(-0.532149\pi\)
−0.100826 + 0.994904i \(0.532149\pi\)
\(138\) −2.85675 −0.243183
\(139\) 3.95726 0.335650 0.167825 0.985817i \(-0.446326\pi\)
0.167825 + 0.985817i \(0.446326\pi\)
\(140\) 2.24922 0.190094
\(141\) −2.27232 −0.191364
\(142\) 4.47296 0.375363
\(143\) 0.202024 0.0168941
\(144\) −1.71235 −0.142696
\(145\) −3.26404 −0.271063
\(146\) 1.00000 0.0827606
\(147\) −2.20256 −0.181664
\(148\) −10.0765 −0.828285
\(149\) 9.40226 0.770263 0.385132 0.922862i \(-0.374156\pi\)
0.385132 + 0.922862i \(0.374156\pi\)
\(150\) 1.13475 0.0926517
\(151\) 10.4648 0.851615 0.425807 0.904814i \(-0.359990\pi\)
0.425807 + 0.904814i \(0.359990\pi\)
\(152\) 6.24810 0.506787
\(153\) −3.89147 −0.314607
\(154\) −2.24922 −0.181247
\(155\) −3.69060 −0.296436
\(156\) 0.229246 0.0183544
\(157\) 21.7044 1.73220 0.866099 0.499872i \(-0.166620\pi\)
0.866099 + 0.499872i \(0.166620\pi\)
\(158\) 12.7687 1.01582
\(159\) 3.64233 0.288855
\(160\) −1.00000 −0.0790569
\(161\) 5.66245 0.446264
\(162\) −0.930819 −0.0731320
\(163\) −2.25705 −0.176786 −0.0883930 0.996086i \(-0.528173\pi\)
−0.0883930 + 0.996086i \(0.528173\pi\)
\(164\) 8.61893 0.673025
\(165\) −1.13475 −0.0883400
\(166\) −11.6129 −0.901340
\(167\) 3.12464 0.241792 0.120896 0.992665i \(-0.461423\pi\)
0.120896 + 0.992665i \(0.461423\pi\)
\(168\) −2.55229 −0.196914
\(169\) −12.9592 −0.996860
\(170\) −2.27259 −0.174300
\(171\) −10.6989 −0.818166
\(172\) −0.359773 −0.0274324
\(173\) 19.7226 1.49948 0.749740 0.661733i \(-0.230179\pi\)
0.749740 + 0.661733i \(0.230179\pi\)
\(174\) 3.70386 0.280789
\(175\) −2.24922 −0.170025
\(176\) 1.00000 0.0753778
\(177\) −2.28136 −0.171478
\(178\) 9.86267 0.739238
\(179\) 16.1286 1.20551 0.602755 0.797926i \(-0.294069\pi\)
0.602755 + 0.797926i \(0.294069\pi\)
\(180\) 1.71235 0.127631
\(181\) 0.309823 0.0230290 0.0115145 0.999934i \(-0.496335\pi\)
0.0115145 + 0.999934i \(0.496335\pi\)
\(182\) −0.454396 −0.0336821
\(183\) 1.80811 0.133659
\(184\) −2.51752 −0.185594
\(185\) 10.0765 0.740840
\(186\) 4.18790 0.307072
\(187\) 2.27259 0.166188
\(188\) −2.00249 −0.146047
\(189\) 12.0273 0.874857
\(190\) −6.24810 −0.453284
\(191\) 21.0739 1.52485 0.762426 0.647075i \(-0.224008\pi\)
0.762426 + 0.647075i \(0.224008\pi\)
\(192\) 1.13475 0.0818933
\(193\) −5.84784 −0.420937 −0.210468 0.977601i \(-0.567499\pi\)
−0.210468 + 0.977601i \(0.567499\pi\)
\(194\) −10.1623 −0.729610
\(195\) −0.229246 −0.0164167
\(196\) −1.94102 −0.138644
\(197\) 4.39229 0.312938 0.156469 0.987683i \(-0.449989\pi\)
0.156469 + 0.987683i \(0.449989\pi\)
\(198\) −1.71235 −0.121691
\(199\) −18.1565 −1.28708 −0.643538 0.765414i \(-0.722534\pi\)
−0.643538 + 0.765414i \(0.722534\pi\)
\(200\) 1.00000 0.0707107
\(201\) 12.4775 0.880097
\(202\) 11.1168 0.782175
\(203\) −7.34153 −0.515274
\(204\) 2.57882 0.180553
\(205\) −8.61893 −0.601972
\(206\) 12.0478 0.839407
\(207\) 4.31087 0.299626
\(208\) 0.202024 0.0140078
\(209\) 6.24810 0.432190
\(210\) 2.55229 0.176125
\(211\) 13.2684 0.913435 0.456718 0.889612i \(-0.349025\pi\)
0.456718 + 0.889612i \(0.349025\pi\)
\(212\) 3.20981 0.220451
\(213\) 5.07569 0.347780
\(214\) −9.35618 −0.639575
\(215\) 0.359773 0.0245363
\(216\) −5.34733 −0.363839
\(217\) −8.30096 −0.563506
\(218\) 15.2409 1.03224
\(219\) 1.13475 0.0766791
\(220\) −1.00000 −0.0674200
\(221\) 0.459118 0.0308836
\(222\) −11.4343 −0.767420
\(223\) −10.7683 −0.721101 −0.360551 0.932740i \(-0.617411\pi\)
−0.360551 + 0.932740i \(0.617411\pi\)
\(224\) −2.24922 −0.150282
\(225\) −1.71235 −0.114157
\(226\) 3.51943 0.234109
\(227\) −7.05329 −0.468143 −0.234072 0.972219i \(-0.575205\pi\)
−0.234072 + 0.972219i \(0.575205\pi\)
\(228\) 7.09001 0.469547
\(229\) 3.70742 0.244993 0.122496 0.992469i \(-0.460910\pi\)
0.122496 + 0.992469i \(0.460910\pi\)
\(230\) 2.51752 0.166000
\(231\) −2.55229 −0.167929
\(232\) 3.26404 0.214294
\(233\) 7.71390 0.505354 0.252677 0.967551i \(-0.418689\pi\)
0.252677 + 0.967551i \(0.418689\pi\)
\(234\) −0.345935 −0.0226145
\(235\) 2.00249 0.130628
\(236\) −2.01046 −0.130870
\(237\) 14.4892 0.941177
\(238\) −5.11156 −0.331333
\(239\) 26.3200 1.70250 0.851249 0.524762i \(-0.175846\pi\)
0.851249 + 0.524762i \(0.175846\pi\)
\(240\) −1.13475 −0.0732476
\(241\) −9.57433 −0.616737 −0.308368 0.951267i \(-0.599783\pi\)
−0.308368 + 0.951267i \(0.599783\pi\)
\(242\) 1.00000 0.0642824
\(243\) 14.9857 0.961335
\(244\) 1.59340 0.102007
\(245\) 1.94102 0.124007
\(246\) 9.78031 0.623569
\(247\) 1.26226 0.0803160
\(248\) 3.69060 0.234353
\(249\) −13.1778 −0.835107
\(250\) −1.00000 −0.0632456
\(251\) −12.3728 −0.780966 −0.390483 0.920610i \(-0.627692\pi\)
−0.390483 + 0.920610i \(0.627692\pi\)
\(252\) 3.85144 0.242618
\(253\) −2.51752 −0.158275
\(254\) −0.831256 −0.0521577
\(255\) −2.57882 −0.161492
\(256\) 1.00000 0.0625000
\(257\) −17.8602 −1.11409 −0.557043 0.830483i \(-0.688064\pi\)
−0.557043 + 0.830483i \(0.688064\pi\)
\(258\) −0.408252 −0.0254166
\(259\) 22.6643 1.40829
\(260\) −0.202024 −0.0125290
\(261\) −5.58917 −0.345961
\(262\) 5.24272 0.323896
\(263\) 3.26368 0.201247 0.100624 0.994925i \(-0.467916\pi\)
0.100624 + 0.994925i \(0.467916\pi\)
\(264\) 1.13475 0.0698389
\(265\) −3.20981 −0.197177
\(266\) −14.0533 −0.861665
\(267\) 11.1916 0.684917
\(268\) 10.9959 0.671679
\(269\) 0.313678 0.0191253 0.00956264 0.999954i \(-0.496956\pi\)
0.00956264 + 0.999954i \(0.496956\pi\)
\(270\) 5.34733 0.325428
\(271\) −1.47405 −0.0895419 −0.0447709 0.998997i \(-0.514256\pi\)
−0.0447709 + 0.998997i \(0.514256\pi\)
\(272\) 2.27259 0.137796
\(273\) −0.515625 −0.0312070
\(274\) −2.36028 −0.142589
\(275\) 1.00000 0.0603023
\(276\) −2.85675 −0.171956
\(277\) 23.4699 1.41017 0.705085 0.709122i \(-0.250909\pi\)
0.705085 + 0.709122i \(0.250909\pi\)
\(278\) 3.95726 0.237341
\(279\) −6.31959 −0.378344
\(280\) 2.24922 0.134416
\(281\) −14.8057 −0.883231 −0.441616 0.897204i \(-0.645594\pi\)
−0.441616 + 0.897204i \(0.645594\pi\)
\(282\) −2.27232 −0.135315
\(283\) 18.1441 1.07855 0.539277 0.842129i \(-0.318698\pi\)
0.539277 + 0.842129i \(0.318698\pi\)
\(284\) 4.47296 0.265422
\(285\) −7.09001 −0.419976
\(286\) 0.202024 0.0119459
\(287\) −19.3858 −1.14431
\(288\) −1.71235 −0.100901
\(289\) −11.8353 −0.696195
\(290\) −3.26404 −0.191671
\(291\) −11.5316 −0.675996
\(292\) 1.00000 0.0585206
\(293\) 14.5058 0.847439 0.423719 0.905794i \(-0.360724\pi\)
0.423719 + 0.905794i \(0.360724\pi\)
\(294\) −2.20256 −0.128456
\(295\) 2.01046 0.117054
\(296\) −10.0765 −0.585686
\(297\) −5.34733 −0.310283
\(298\) 9.40226 0.544659
\(299\) −0.508599 −0.0294131
\(300\) 1.13475 0.0655147
\(301\) 0.809208 0.0466420
\(302\) 10.4648 0.602183
\(303\) 12.6148 0.724699
\(304\) 6.24810 0.358353
\(305\) −1.59340 −0.0912380
\(306\) −3.89147 −0.222461
\(307\) −4.43968 −0.253386 −0.126693 0.991942i \(-0.540436\pi\)
−0.126693 + 0.991942i \(0.540436\pi\)
\(308\) −2.24922 −0.128161
\(309\) 13.6712 0.777725
\(310\) −3.69060 −0.209612
\(311\) −17.4668 −0.990454 −0.495227 0.868764i \(-0.664915\pi\)
−0.495227 + 0.868764i \(0.664915\pi\)
\(312\) 0.229246 0.0129785
\(313\) 18.3946 1.03972 0.519862 0.854250i \(-0.325983\pi\)
0.519862 + 0.854250i \(0.325983\pi\)
\(314\) 21.7044 1.22485
\(315\) −3.85144 −0.217004
\(316\) 12.7687 0.718295
\(317\) −18.0011 −1.01104 −0.505521 0.862814i \(-0.668700\pi\)
−0.505521 + 0.862814i \(0.668700\pi\)
\(318\) 3.64233 0.204252
\(319\) 3.26404 0.182751
\(320\) −1.00000 −0.0559017
\(321\) −10.6169 −0.592577
\(322\) 5.66245 0.315556
\(323\) 14.1994 0.790075
\(324\) −0.930819 −0.0517121
\(325\) 0.202024 0.0112063
\(326\) −2.25705 −0.125007
\(327\) 17.2945 0.956390
\(328\) 8.61893 0.475901
\(329\) 4.50404 0.248316
\(330\) −1.13475 −0.0624658
\(331\) −5.15501 −0.283345 −0.141672 0.989914i \(-0.545248\pi\)
−0.141672 + 0.989914i \(0.545248\pi\)
\(332\) −11.6129 −0.637343
\(333\) 17.2545 0.945541
\(334\) 3.12464 0.170973
\(335\) −10.9959 −0.600768
\(336\) −2.55229 −0.139239
\(337\) 0.0479947 0.00261444 0.00130722 0.999999i \(-0.499584\pi\)
0.00130722 + 0.999999i \(0.499584\pi\)
\(338\) −12.9592 −0.704887
\(339\) 3.99366 0.216906
\(340\) −2.27259 −0.123249
\(341\) 3.69060 0.199857
\(342\) −10.6989 −0.578531
\(343\) 20.1103 1.08585
\(344\) −0.359773 −0.0193977
\(345\) 2.85675 0.153802
\(346\) 19.7226 1.06029
\(347\) −12.4765 −0.669771 −0.334886 0.942259i \(-0.608698\pi\)
−0.334886 + 0.942259i \(0.608698\pi\)
\(348\) 3.70386 0.198548
\(349\) −3.55815 −0.190463 −0.0952317 0.995455i \(-0.530359\pi\)
−0.0952317 + 0.995455i \(0.530359\pi\)
\(350\) −2.24922 −0.120226
\(351\) −1.08029 −0.0576615
\(352\) 1.00000 0.0533002
\(353\) 4.93425 0.262623 0.131312 0.991341i \(-0.458081\pi\)
0.131312 + 0.991341i \(0.458081\pi\)
\(354\) −2.28136 −0.121253
\(355\) −4.47296 −0.237400
\(356\) 9.86267 0.522720
\(357\) −5.80033 −0.306986
\(358\) 16.1286 0.852425
\(359\) 4.03340 0.212875 0.106437 0.994319i \(-0.466056\pi\)
0.106437 + 0.994319i \(0.466056\pi\)
\(360\) 1.71235 0.0902487
\(361\) 20.0387 1.05467
\(362\) 0.309823 0.0162839
\(363\) 1.13475 0.0595588
\(364\) −0.454396 −0.0238168
\(365\) −1.00000 −0.0523424
\(366\) 1.80811 0.0945114
\(367\) −15.6153 −0.815111 −0.407556 0.913180i \(-0.633619\pi\)
−0.407556 + 0.913180i \(0.633619\pi\)
\(368\) −2.51752 −0.131235
\(369\) −14.7586 −0.768302
\(370\) 10.0765 0.523853
\(371\) −7.21957 −0.374821
\(372\) 4.18790 0.217132
\(373\) −5.60101 −0.290009 −0.145005 0.989431i \(-0.546320\pi\)
−0.145005 + 0.989431i \(0.546320\pi\)
\(374\) 2.27259 0.117513
\(375\) −1.13475 −0.0585981
\(376\) −2.00249 −0.103271
\(377\) 0.659413 0.0339615
\(378\) 12.0273 0.618618
\(379\) −23.3380 −1.19879 −0.599396 0.800453i \(-0.704592\pi\)
−0.599396 + 0.800453i \(0.704592\pi\)
\(380\) −6.24810 −0.320521
\(381\) −0.943266 −0.0483250
\(382\) 21.0739 1.07823
\(383\) −17.4899 −0.893691 −0.446846 0.894611i \(-0.647453\pi\)
−0.446846 + 0.894611i \(0.647453\pi\)
\(384\) 1.13475 0.0579073
\(385\) 2.24922 0.114631
\(386\) −5.84784 −0.297647
\(387\) 0.616057 0.0313159
\(388\) −10.1623 −0.515912
\(389\) 19.0646 0.966614 0.483307 0.875451i \(-0.339435\pi\)
0.483307 + 0.875451i \(0.339435\pi\)
\(390\) −0.229246 −0.0116083
\(391\) −5.72130 −0.289339
\(392\) −1.94102 −0.0980362
\(393\) 5.94916 0.300095
\(394\) 4.39229 0.221280
\(395\) −12.7687 −0.642462
\(396\) −1.71235 −0.0860487
\(397\) 23.9517 1.20210 0.601051 0.799211i \(-0.294749\pi\)
0.601051 + 0.799211i \(0.294749\pi\)
\(398\) −18.1565 −0.910101
\(399\) −15.9470 −0.798348
\(400\) 1.00000 0.0500000
\(401\) 1.28203 0.0640214 0.0320107 0.999488i \(-0.489809\pi\)
0.0320107 + 0.999488i \(0.489809\pi\)
\(402\) 12.4775 0.622323
\(403\) 0.745589 0.0371404
\(404\) 11.1168 0.553081
\(405\) 0.930819 0.0462528
\(406\) −7.34153 −0.364354
\(407\) −10.0765 −0.499474
\(408\) 2.57882 0.127671
\(409\) −30.7455 −1.52027 −0.760133 0.649768i \(-0.774866\pi\)
−0.760133 + 0.649768i \(0.774866\pi\)
\(410\) −8.61893 −0.425658
\(411\) −2.67832 −0.132112
\(412\) 12.0478 0.593550
\(413\) 4.52196 0.222511
\(414\) 4.31087 0.211868
\(415\) 11.6129 0.570057
\(416\) 0.202024 0.00990504
\(417\) 4.49049 0.219900
\(418\) 6.24810 0.305604
\(419\) −26.6982 −1.30429 −0.652146 0.758094i \(-0.726131\pi\)
−0.652146 + 0.758094i \(0.726131\pi\)
\(420\) 2.55229 0.124539
\(421\) −13.4999 −0.657945 −0.328973 0.944339i \(-0.606702\pi\)
−0.328973 + 0.944339i \(0.606702\pi\)
\(422\) 13.2684 0.645896
\(423\) 3.42897 0.166722
\(424\) 3.20981 0.155882
\(425\) 2.27259 0.110237
\(426\) 5.07569 0.245918
\(427\) −3.58391 −0.173438
\(428\) −9.35618 −0.452248
\(429\) 0.229246 0.0110681
\(430\) 0.359773 0.0173498
\(431\) 16.9683 0.817333 0.408666 0.912684i \(-0.365994\pi\)
0.408666 + 0.912684i \(0.365994\pi\)
\(432\) −5.34733 −0.257273
\(433\) 14.5903 0.701165 0.350582 0.936532i \(-0.385984\pi\)
0.350582 + 0.936532i \(0.385984\pi\)
\(434\) −8.30096 −0.398459
\(435\) −3.70386 −0.177586
\(436\) 15.2409 0.729906
\(437\) −15.7297 −0.752454
\(438\) 1.13475 0.0542203
\(439\) 12.9426 0.617718 0.308859 0.951108i \(-0.400053\pi\)
0.308859 + 0.951108i \(0.400053\pi\)
\(440\) −1.00000 −0.0476731
\(441\) 3.32370 0.158271
\(442\) 0.459118 0.0218380
\(443\) −30.1339 −1.43170 −0.715852 0.698252i \(-0.753961\pi\)
−0.715852 + 0.698252i \(0.753961\pi\)
\(444\) −11.4343 −0.542648
\(445\) −9.86267 −0.467535
\(446\) −10.7683 −0.509896
\(447\) 10.6692 0.504636
\(448\) −2.24922 −0.106266
\(449\) −14.5155 −0.685028 −0.342514 0.939513i \(-0.611278\pi\)
−0.342514 + 0.939513i \(0.611278\pi\)
\(450\) −1.71235 −0.0807209
\(451\) 8.61893 0.405849
\(452\) 3.51943 0.165540
\(453\) 11.8749 0.557933
\(454\) −7.05329 −0.331027
\(455\) 0.454396 0.0213024
\(456\) 7.09001 0.332020
\(457\) −37.5708 −1.75749 −0.878745 0.477292i \(-0.841619\pi\)
−0.878745 + 0.477292i \(0.841619\pi\)
\(458\) 3.70742 0.173236
\(459\) −12.1523 −0.567221
\(460\) 2.51752 0.117380
\(461\) −33.7192 −1.57046 −0.785230 0.619204i \(-0.787455\pi\)
−0.785230 + 0.619204i \(0.787455\pi\)
\(462\) −2.55229 −0.118743
\(463\) 34.3995 1.59868 0.799341 0.600878i \(-0.205182\pi\)
0.799341 + 0.600878i \(0.205182\pi\)
\(464\) 3.26404 0.151529
\(465\) −4.18790 −0.194209
\(466\) 7.71390 0.357340
\(467\) −22.3369 −1.03363 −0.516814 0.856098i \(-0.672882\pi\)
−0.516814 + 0.856098i \(0.672882\pi\)
\(468\) −0.345935 −0.0159909
\(469\) −24.7321 −1.14202
\(470\) 2.00249 0.0923681
\(471\) 24.6290 1.13484
\(472\) −2.01046 −0.0925389
\(473\) −0.359773 −0.0165424
\(474\) 14.4892 0.665512
\(475\) 6.24810 0.286682
\(476\) −5.11156 −0.234288
\(477\) −5.49632 −0.251659
\(478\) 26.3200 1.20385
\(479\) 3.41665 0.156111 0.0780553 0.996949i \(-0.475129\pi\)
0.0780553 + 0.996949i \(0.475129\pi\)
\(480\) −1.13475 −0.0517939
\(481\) −2.03570 −0.0928198
\(482\) −9.57433 −0.436099
\(483\) 6.42545 0.292368
\(484\) 1.00000 0.0454545
\(485\) 10.1623 0.461446
\(486\) 14.9857 0.679767
\(487\) −23.2454 −1.05335 −0.526675 0.850067i \(-0.676562\pi\)
−0.526675 + 0.850067i \(0.676562\pi\)
\(488\) 1.59340 0.0721300
\(489\) −2.56118 −0.115821
\(490\) 1.94102 0.0876862
\(491\) −35.8534 −1.61804 −0.809021 0.587779i \(-0.800002\pi\)
−0.809021 + 0.587779i \(0.800002\pi\)
\(492\) 9.78031 0.440930
\(493\) 7.41783 0.334082
\(494\) 1.26226 0.0567920
\(495\) 1.71235 0.0769643
\(496\) 3.69060 0.165713
\(497\) −10.0607 −0.451283
\(498\) −13.1778 −0.590510
\(499\) 17.9926 0.805460 0.402730 0.915319i \(-0.368061\pi\)
0.402730 + 0.915319i \(0.368061\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 3.54568 0.158409
\(502\) −12.3728 −0.552226
\(503\) −20.1316 −0.897622 −0.448811 0.893627i \(-0.648152\pi\)
−0.448811 + 0.893627i \(0.648152\pi\)
\(504\) 3.85144 0.171557
\(505\) −11.1168 −0.494691
\(506\) −2.51752 −0.111917
\(507\) −14.7054 −0.653090
\(508\) −0.831256 −0.0368810
\(509\) −5.76386 −0.255479 −0.127739 0.991808i \(-0.540772\pi\)
−0.127739 + 0.991808i \(0.540772\pi\)
\(510\) −2.57882 −0.114192
\(511\) −2.24922 −0.0994996
\(512\) 1.00000 0.0441942
\(513\) −33.4106 −1.47511
\(514\) −17.8602 −0.787778
\(515\) −12.0478 −0.530887
\(516\) −0.408252 −0.0179723
\(517\) −2.00249 −0.0880695
\(518\) 22.6643 0.995812
\(519\) 22.3801 0.982379
\(520\) −0.202024 −0.00885933
\(521\) 27.0492 1.18505 0.592524 0.805553i \(-0.298131\pi\)
0.592524 + 0.805553i \(0.298131\pi\)
\(522\) −5.58917 −0.244631
\(523\) −38.9383 −1.70265 −0.851327 0.524635i \(-0.824202\pi\)
−0.851327 + 0.524635i \(0.824202\pi\)
\(524\) 5.24272 0.229029
\(525\) −2.55229 −0.111391
\(526\) 3.26368 0.142303
\(527\) 8.38723 0.365353
\(528\) 1.13475 0.0493835
\(529\) −16.6621 −0.724439
\(530\) −3.20981 −0.139425
\(531\) 3.44261 0.149396
\(532\) −14.0533 −0.609289
\(533\) 1.74123 0.0754210
\(534\) 11.1916 0.484309
\(535\) 9.35618 0.404503
\(536\) 10.9959 0.474949
\(537\) 18.3019 0.789787
\(538\) 0.313678 0.0135236
\(539\) −1.94102 −0.0836055
\(540\) 5.34733 0.230112
\(541\) 27.1146 1.16575 0.582875 0.812562i \(-0.301928\pi\)
0.582875 + 0.812562i \(0.301928\pi\)
\(542\) −1.47405 −0.0633157
\(543\) 0.351571 0.0150874
\(544\) 2.27259 0.0974366
\(545\) −15.2409 −0.652847
\(546\) −0.515625 −0.0220667
\(547\) −2.96129 −0.126616 −0.0633079 0.997994i \(-0.520165\pi\)
−0.0633079 + 0.997994i \(0.520165\pi\)
\(548\) −2.36028 −0.100826
\(549\) −2.72846 −0.116448
\(550\) 1.00000 0.0426401
\(551\) 20.3940 0.868814
\(552\) −2.85675 −0.121591
\(553\) −28.7196 −1.22128
\(554\) 23.4699 0.997141
\(555\) 11.4343 0.485359
\(556\) 3.95726 0.167825
\(557\) −23.7930 −1.00814 −0.504071 0.863662i \(-0.668165\pi\)
−0.504071 + 0.863662i \(0.668165\pi\)
\(558\) −6.31959 −0.267530
\(559\) −0.0726828 −0.00307415
\(560\) 2.24922 0.0950468
\(561\) 2.57882 0.108878
\(562\) −14.8057 −0.624539
\(563\) −5.10421 −0.215117 −0.107558 0.994199i \(-0.534303\pi\)
−0.107558 + 0.994199i \(0.534303\pi\)
\(564\) −2.27232 −0.0956821
\(565\) −3.51943 −0.148063
\(566\) 18.1441 0.762653
\(567\) 2.09361 0.0879235
\(568\) 4.47296 0.187681
\(569\) 16.6904 0.699697 0.349849 0.936806i \(-0.386233\pi\)
0.349849 + 0.936806i \(0.386233\pi\)
\(570\) −7.09001 −0.296968
\(571\) −5.38893 −0.225520 −0.112760 0.993622i \(-0.535969\pi\)
−0.112760 + 0.993622i \(0.535969\pi\)
\(572\) 0.202024 0.00844704
\(573\) 23.9135 0.999002
\(574\) −19.3858 −0.809150
\(575\) −2.51752 −0.104988
\(576\) −1.71235 −0.0713478
\(577\) −11.5172 −0.479469 −0.239734 0.970839i \(-0.577060\pi\)
−0.239734 + 0.970839i \(0.577060\pi\)
\(578\) −11.8353 −0.492284
\(579\) −6.63582 −0.275775
\(580\) −3.26404 −0.135532
\(581\) 26.1201 1.08364
\(582\) −11.5316 −0.478002
\(583\) 3.20981 0.132937
\(584\) 1.00000 0.0413803
\(585\) 0.345935 0.0143027
\(586\) 14.5058 0.599230
\(587\) −8.34988 −0.344636 −0.172318 0.985041i \(-0.555126\pi\)
−0.172318 + 0.985041i \(0.555126\pi\)
\(588\) −2.20256 −0.0908322
\(589\) 23.0592 0.950138
\(590\) 2.01046 0.0827693
\(591\) 4.98414 0.205020
\(592\) −10.0765 −0.414142
\(593\) 31.9432 1.31175 0.655875 0.754870i \(-0.272300\pi\)
0.655875 + 0.754870i \(0.272300\pi\)
\(594\) −5.34733 −0.219403
\(595\) 5.11156 0.209553
\(596\) 9.40226 0.385132
\(597\) −20.6030 −0.843224
\(598\) −0.508599 −0.0207982
\(599\) −0.695079 −0.0284001 −0.0142001 0.999899i \(-0.504520\pi\)
−0.0142001 + 0.999899i \(0.504520\pi\)
\(600\) 1.13475 0.0463259
\(601\) −24.4761 −0.998403 −0.499201 0.866486i \(-0.666373\pi\)
−0.499201 + 0.866486i \(0.666373\pi\)
\(602\) 0.809208 0.0329809
\(603\) −18.8287 −0.766766
\(604\) 10.4648 0.425807
\(605\) −1.00000 −0.0406558
\(606\) 12.6148 0.512440
\(607\) 2.61424 0.106109 0.0530543 0.998592i \(-0.483104\pi\)
0.0530543 + 0.998592i \(0.483104\pi\)
\(608\) 6.24810 0.253394
\(609\) −8.33078 −0.337580
\(610\) −1.59340 −0.0645150
\(611\) −0.404551 −0.0163664
\(612\) −3.89147 −0.157303
\(613\) −34.9165 −1.41027 −0.705133 0.709075i \(-0.749113\pi\)
−0.705133 + 0.709075i \(0.749113\pi\)
\(614\) −4.43968 −0.179171
\(615\) −9.78031 −0.394380
\(616\) −2.24922 −0.0906236
\(617\) 13.0864 0.526837 0.263419 0.964682i \(-0.415150\pi\)
0.263419 + 0.964682i \(0.415150\pi\)
\(618\) 13.6712 0.549935
\(619\) 2.66239 0.107010 0.0535052 0.998568i \(-0.482961\pi\)
0.0535052 + 0.998568i \(0.482961\pi\)
\(620\) −3.69060 −0.148218
\(621\) 13.4620 0.540212
\(622\) −17.4668 −0.700357
\(623\) −22.1833 −0.888755
\(624\) 0.229246 0.00917719
\(625\) 1.00000 0.0400000
\(626\) 18.3946 0.735196
\(627\) 7.09001 0.283148
\(628\) 21.7044 0.866099
\(629\) −22.8998 −0.913076
\(630\) −3.85144 −0.153445
\(631\) 26.9788 1.07401 0.537005 0.843579i \(-0.319556\pi\)
0.537005 + 0.843579i \(0.319556\pi\)
\(632\) 12.7687 0.507911
\(633\) 15.0563 0.598434
\(634\) −18.0011 −0.714915
\(635\) 0.831256 0.0329874
\(636\) 3.64233 0.144428
\(637\) −0.392132 −0.0155368
\(638\) 3.26404 0.129224
\(639\) −7.65927 −0.302996
\(640\) −1.00000 −0.0395285
\(641\) 9.91525 0.391629 0.195814 0.980641i \(-0.437265\pi\)
0.195814 + 0.980641i \(0.437265\pi\)
\(642\) −10.6169 −0.419016
\(643\) 15.6338 0.616535 0.308268 0.951300i \(-0.400251\pi\)
0.308268 + 0.951300i \(0.400251\pi\)
\(644\) 5.66245 0.223132
\(645\) 0.408252 0.0160749
\(646\) 14.1994 0.558667
\(647\) −11.9006 −0.467859 −0.233930 0.972254i \(-0.575159\pi\)
−0.233930 + 0.972254i \(0.575159\pi\)
\(648\) −0.930819 −0.0365660
\(649\) −2.01046 −0.0789175
\(650\) 0.202024 0.00792403
\(651\) −9.41950 −0.369179
\(652\) −2.25705 −0.0883930
\(653\) 8.13936 0.318518 0.159259 0.987237i \(-0.449090\pi\)
0.159259 + 0.987237i \(0.449090\pi\)
\(654\) 17.2945 0.676270
\(655\) −5.24272 −0.204850
\(656\) 8.61893 0.336513
\(657\) −1.71235 −0.0668051
\(658\) 4.50404 0.175586
\(659\) −23.4469 −0.913362 −0.456681 0.889631i \(-0.650962\pi\)
−0.456681 + 0.889631i \(0.650962\pi\)
\(660\) −1.13475 −0.0441700
\(661\) −43.2799 −1.68339 −0.841697 0.539950i \(-0.818443\pi\)
−0.841697 + 0.539950i \(0.818443\pi\)
\(662\) −5.15501 −0.200355
\(663\) 0.520983 0.0202333
\(664\) −11.6129 −0.450670
\(665\) 14.0533 0.544965
\(666\) 17.2545 0.668599
\(667\) −8.21728 −0.318174
\(668\) 3.12464 0.120896
\(669\) −12.2193 −0.472427
\(670\) −10.9959 −0.424807
\(671\) 1.59340 0.0615126
\(672\) −2.55229 −0.0984569
\(673\) 20.4588 0.788628 0.394314 0.918976i \(-0.370982\pi\)
0.394314 + 0.918976i \(0.370982\pi\)
\(674\) 0.0479947 0.00184869
\(675\) −5.34733 −0.205819
\(676\) −12.9592 −0.498430
\(677\) 4.22663 0.162443 0.0812213 0.996696i \(-0.474118\pi\)
0.0812213 + 0.996696i \(0.474118\pi\)
\(678\) 3.99366 0.153376
\(679\) 22.8572 0.877179
\(680\) −2.27259 −0.0871500
\(681\) −8.00370 −0.306702
\(682\) 3.69060 0.141320
\(683\) −9.43617 −0.361065 −0.180532 0.983569i \(-0.557782\pi\)
−0.180532 + 0.983569i \(0.557782\pi\)
\(684\) −10.6989 −0.409083
\(685\) 2.36028 0.0901815
\(686\) 20.1103 0.767815
\(687\) 4.20698 0.160506
\(688\) −0.359773 −0.0137162
\(689\) 0.648459 0.0247043
\(690\) 2.85675 0.108755
\(691\) 18.3790 0.699171 0.349585 0.936904i \(-0.386322\pi\)
0.349585 + 0.936904i \(0.386322\pi\)
\(692\) 19.7226 0.749740
\(693\) 3.85144 0.146304
\(694\) −12.4765 −0.473600
\(695\) −3.95726 −0.150107
\(696\) 3.70386 0.140394
\(697\) 19.5873 0.741922
\(698\) −3.55815 −0.134678
\(699\) 8.75333 0.331081
\(700\) −2.24922 −0.0850125
\(701\) −13.6547 −0.515731 −0.257866 0.966181i \(-0.583019\pi\)
−0.257866 + 0.966181i \(0.583019\pi\)
\(702\) −1.08029 −0.0407728
\(703\) −62.9591 −2.37455
\(704\) 1.00000 0.0376889
\(705\) 2.27232 0.0855807
\(706\) 4.93425 0.185703
\(707\) −25.0041 −0.940376
\(708\) −2.28136 −0.0857389
\(709\) −10.3750 −0.389640 −0.194820 0.980839i \(-0.562412\pi\)
−0.194820 + 0.980839i \(0.562412\pi\)
\(710\) −4.47296 −0.167867
\(711\) −21.8644 −0.819980
\(712\) 9.86267 0.369619
\(713\) −9.29116 −0.347957
\(714\) −5.80033 −0.217072
\(715\) −0.202024 −0.00755527
\(716\) 16.1286 0.602755
\(717\) 29.8665 1.11539
\(718\) 4.03340 0.150525
\(719\) 32.2858 1.20406 0.602028 0.798475i \(-0.294360\pi\)
0.602028 + 0.798475i \(0.294360\pi\)
\(720\) 1.71235 0.0638155
\(721\) −27.0980 −1.00918
\(722\) 20.0387 0.745763
\(723\) −10.8644 −0.404053
\(724\) 0.309823 0.0115145
\(725\) 3.26404 0.121223
\(726\) 1.13475 0.0421144
\(727\) 12.0134 0.445553 0.222777 0.974870i \(-0.428488\pi\)
0.222777 + 0.974870i \(0.428488\pi\)
\(728\) −0.454396 −0.0168410
\(729\) 19.7975 0.733240
\(730\) −1.00000 −0.0370117
\(731\) −0.817618 −0.0302407
\(732\) 1.80811 0.0668297
\(733\) −6.18587 −0.228480 −0.114240 0.993453i \(-0.536443\pi\)
−0.114240 + 0.993453i \(0.536443\pi\)
\(734\) −15.6153 −0.576371
\(735\) 2.20256 0.0812428
\(736\) −2.51752 −0.0927970
\(737\) 10.9959 0.405038
\(738\) −14.7586 −0.543272
\(739\) 3.59211 0.132138 0.0660690 0.997815i \(-0.478954\pi\)
0.0660690 + 0.997815i \(0.478954\pi\)
\(740\) 10.0765 0.370420
\(741\) 1.43235 0.0526187
\(742\) −7.21957 −0.265039
\(743\) −2.24031 −0.0821890 −0.0410945 0.999155i \(-0.513084\pi\)
−0.0410945 + 0.999155i \(0.513084\pi\)
\(744\) 4.18790 0.153536
\(745\) −9.40226 −0.344472
\(746\) −5.60101 −0.205068
\(747\) 19.8854 0.727569
\(748\) 2.27259 0.0830942
\(749\) 21.0441 0.768934
\(750\) −1.13475 −0.0414351
\(751\) −7.59820 −0.277262 −0.138631 0.990344i \(-0.544270\pi\)
−0.138631 + 0.990344i \(0.544270\pi\)
\(752\) −2.00249 −0.0730234
\(753\) −14.0400 −0.511647
\(754\) 0.659413 0.0240144
\(755\) −10.4648 −0.380854
\(756\) 12.0273 0.437429
\(757\) 16.7688 0.609473 0.304737 0.952437i \(-0.401432\pi\)
0.304737 + 0.952437i \(0.401432\pi\)
\(758\) −23.3380 −0.847673
\(759\) −2.85675 −0.103693
\(760\) −6.24810 −0.226642
\(761\) −36.1062 −1.30885 −0.654425 0.756127i \(-0.727089\pi\)
−0.654425 + 0.756127i \(0.727089\pi\)
\(762\) −0.943266 −0.0341709
\(763\) −34.2801 −1.24102
\(764\) 21.0739 0.762426
\(765\) 3.89147 0.140696
\(766\) −17.4899 −0.631935
\(767\) −0.406161 −0.0146656
\(768\) 1.13475 0.0409467
\(769\) 52.3082 1.88628 0.943140 0.332395i \(-0.107857\pi\)
0.943140 + 0.332395i \(0.107857\pi\)
\(770\) 2.24922 0.0810562
\(771\) −20.2668 −0.729890
\(772\) −5.84784 −0.210468
\(773\) −14.1422 −0.508659 −0.254330 0.967118i \(-0.581855\pi\)
−0.254330 + 0.967118i \(0.581855\pi\)
\(774\) 0.616057 0.0221437
\(775\) 3.69060 0.132570
\(776\) −10.1623 −0.364805
\(777\) 25.7182 0.922637
\(778\) 19.0646 0.683499
\(779\) 53.8519 1.92944
\(780\) −0.229246 −0.00820833
\(781\) 4.47296 0.160055
\(782\) −5.72130 −0.204593
\(783\) −17.4539 −0.623750
\(784\) −1.94102 −0.0693220
\(785\) −21.7044 −0.774663
\(786\) 5.94916 0.212200
\(787\) 24.2345 0.863867 0.431933 0.901906i \(-0.357832\pi\)
0.431933 + 0.901906i \(0.357832\pi\)
\(788\) 4.39229 0.156469
\(789\) 3.70345 0.131846
\(790\) −12.7687 −0.454289
\(791\) −7.91596 −0.281459
\(792\) −1.71235 −0.0608456
\(793\) 0.321905 0.0114312
\(794\) 23.9517 0.850014
\(795\) −3.64233 −0.129180
\(796\) −18.1565 −0.643538
\(797\) 2.00794 0.0711248 0.0355624 0.999367i \(-0.488678\pi\)
0.0355624 + 0.999367i \(0.488678\pi\)
\(798\) −15.9470 −0.564517
\(799\) −4.55085 −0.160998
\(800\) 1.00000 0.0353553
\(801\) −16.8883 −0.596719
\(802\) 1.28203 0.0452699
\(803\) 1.00000 0.0352892
\(804\) 12.4775 0.440049
\(805\) −5.66245 −0.199575
\(806\) 0.745589 0.0262623
\(807\) 0.355945 0.0125299
\(808\) 11.1168 0.391088
\(809\) −55.7732 −1.96088 −0.980441 0.196811i \(-0.936941\pi\)
−0.980441 + 0.196811i \(0.936941\pi\)
\(810\) 0.930819 0.0327056
\(811\) −12.3909 −0.435103 −0.217552 0.976049i \(-0.569807\pi\)
−0.217552 + 0.976049i \(0.569807\pi\)
\(812\) −7.34153 −0.257637
\(813\) −1.67267 −0.0586631
\(814\) −10.0765 −0.353182
\(815\) 2.25705 0.0790611
\(816\) 2.57882 0.0902767
\(817\) −2.24790 −0.0786439
\(818\) −30.7455 −1.07499
\(819\) 0.778084 0.0271885
\(820\) −8.61893 −0.300986
\(821\) −12.9323 −0.451340 −0.225670 0.974204i \(-0.572457\pi\)
−0.225670 + 0.974204i \(0.572457\pi\)
\(822\) −2.67832 −0.0934170
\(823\) 24.0588 0.838639 0.419319 0.907839i \(-0.362269\pi\)
0.419319 + 0.907839i \(0.362269\pi\)
\(824\) 12.0478 0.419703
\(825\) 1.13475 0.0395068
\(826\) 4.52196 0.157339
\(827\) −17.0786 −0.593880 −0.296940 0.954896i \(-0.595966\pi\)
−0.296940 + 0.954896i \(0.595966\pi\)
\(828\) 4.31087 0.149813
\(829\) −43.3214 −1.50461 −0.752307 0.658813i \(-0.771059\pi\)
−0.752307 + 0.658813i \(0.771059\pi\)
\(830\) 11.6129 0.403091
\(831\) 26.6324 0.923869
\(832\) 0.202024 0.00700392
\(833\) −4.41114 −0.152837
\(834\) 4.49049 0.155493
\(835\) −3.12464 −0.108133
\(836\) 6.24810 0.216095
\(837\) −19.7348 −0.682136
\(838\) −26.6982 −0.922273
\(839\) 25.5893 0.883439 0.441720 0.897153i \(-0.354369\pi\)
0.441720 + 0.897153i \(0.354369\pi\)
\(840\) 2.55229 0.0880625
\(841\) −18.3461 −0.632623
\(842\) −13.4999 −0.465238
\(843\) −16.8007 −0.578646
\(844\) 13.2684 0.456718
\(845\) 12.9592 0.445810
\(846\) 3.42897 0.117890
\(847\) −2.24922 −0.0772841
\(848\) 3.20981 0.110225
\(849\) 20.5890 0.706611
\(850\) 2.27259 0.0779493
\(851\) 25.3678 0.869599
\(852\) 5.07569 0.173890
\(853\) 24.7978 0.849060 0.424530 0.905414i \(-0.360439\pi\)
0.424530 + 0.905414i \(0.360439\pi\)
\(854\) −3.58391 −0.122639
\(855\) 10.6989 0.365895
\(856\) −9.35618 −0.319788
\(857\) −33.8755 −1.15717 −0.578583 0.815624i \(-0.696394\pi\)
−0.578583 + 0.815624i \(0.696394\pi\)
\(858\) 0.229246 0.00782633
\(859\) −19.8121 −0.675980 −0.337990 0.941150i \(-0.609747\pi\)
−0.337990 + 0.941150i \(0.609747\pi\)
\(860\) 0.359773 0.0122682
\(861\) −21.9980 −0.749691
\(862\) 16.9683 0.577942
\(863\) 16.9471 0.576886 0.288443 0.957497i \(-0.406862\pi\)
0.288443 + 0.957497i \(0.406862\pi\)
\(864\) −5.34733 −0.181920
\(865\) −19.7226 −0.670588
\(866\) 14.5903 0.495798
\(867\) −13.4301 −0.456110
\(868\) −8.30096 −0.281753
\(869\) 12.7687 0.433148
\(870\) −3.70386 −0.125572
\(871\) 2.22143 0.0752702
\(872\) 15.2409 0.516121
\(873\) 17.4014 0.588948
\(874\) −15.7297 −0.532065
\(875\) 2.24922 0.0760375
\(876\) 1.13475 0.0383396
\(877\) 7.51955 0.253917 0.126959 0.991908i \(-0.459478\pi\)
0.126959 + 0.991908i \(0.459478\pi\)
\(878\) 12.9426 0.436792
\(879\) 16.4604 0.555197
\(880\) −1.00000 −0.0337100
\(881\) −9.79564 −0.330023 −0.165012 0.986292i \(-0.552766\pi\)
−0.165012 + 0.986292i \(0.552766\pi\)
\(882\) 3.32370 0.111915
\(883\) 21.2176 0.714029 0.357014 0.934099i \(-0.383795\pi\)
0.357014 + 0.934099i \(0.383795\pi\)
\(884\) 0.459118 0.0154418
\(885\) 2.28136 0.0766872
\(886\) −30.1339 −1.01237
\(887\) 21.7273 0.729530 0.364765 0.931100i \(-0.381149\pi\)
0.364765 + 0.931100i \(0.381149\pi\)
\(888\) −11.4343 −0.383710
\(889\) 1.86968 0.0627070
\(890\) −9.86267 −0.330597
\(891\) −0.930819 −0.0311836
\(892\) −10.7683 −0.360551
\(893\) −12.5118 −0.418690
\(894\) 10.6692 0.356831
\(895\) −16.1286 −0.539121
\(896\) −2.24922 −0.0751411
\(897\) −0.577132 −0.0192699
\(898\) −14.5155 −0.484388
\(899\) 12.0462 0.401765
\(900\) −1.71235 −0.0570783
\(901\) 7.29460 0.243018
\(902\) 8.61893 0.286979
\(903\) 0.918247 0.0305573
\(904\) 3.51943 0.117054
\(905\) −0.309823 −0.0102989
\(906\) 11.8749 0.394518
\(907\) 26.2924 0.873026 0.436513 0.899698i \(-0.356213\pi\)
0.436513 + 0.899698i \(0.356213\pi\)
\(908\) −7.05329 −0.234072
\(909\) −19.0358 −0.631379
\(910\) 0.454396 0.0150631
\(911\) −0.148527 −0.00492093 −0.00246047 0.999997i \(-0.500783\pi\)
−0.00246047 + 0.999997i \(0.500783\pi\)
\(912\) 7.09001 0.234774
\(913\) −11.6129 −0.384332
\(914\) −37.5708 −1.24273
\(915\) −1.80811 −0.0597743
\(916\) 3.70742 0.122496
\(917\) −11.7920 −0.389407
\(918\) −12.1523 −0.401086
\(919\) 27.3087 0.900831 0.450416 0.892819i \(-0.351276\pi\)
0.450416 + 0.892819i \(0.351276\pi\)
\(920\) 2.51752 0.0830002
\(921\) −5.03792 −0.166005
\(922\) −33.7192 −1.11048
\(923\) 0.903646 0.0297439
\(924\) −2.55229 −0.0839643
\(925\) −10.0765 −0.331314
\(926\) 34.3995 1.13044
\(927\) −20.6300 −0.677577
\(928\) 3.26404 0.107147
\(929\) 43.6002 1.43048 0.715238 0.698881i \(-0.246318\pi\)
0.715238 + 0.698881i \(0.246318\pi\)
\(930\) −4.18790 −0.137327
\(931\) −12.1277 −0.397468
\(932\) 7.71390 0.252677
\(933\) −19.8205 −0.648893
\(934\) −22.3369 −0.730885
\(935\) −2.27259 −0.0743217
\(936\) −0.345935 −0.0113072
\(937\) −14.2353 −0.465047 −0.232523 0.972591i \(-0.574698\pi\)
−0.232523 + 0.972591i \(0.574698\pi\)
\(938\) −24.7321 −0.807532
\(939\) 20.8732 0.681172
\(940\) 2.00249 0.0653141
\(941\) −23.6631 −0.771395 −0.385697 0.922625i \(-0.626039\pi\)
−0.385697 + 0.922625i \(0.626039\pi\)
\(942\) 24.6290 0.802456
\(943\) −21.6983 −0.706595
\(944\) −2.01046 −0.0654349
\(945\) −12.0273 −0.391248
\(946\) −0.359773 −0.0116972
\(947\) 28.5512 0.927789 0.463894 0.885891i \(-0.346452\pi\)
0.463894 + 0.885891i \(0.346452\pi\)
\(948\) 14.4892 0.470588
\(949\) 0.202024 0.00655797
\(950\) 6.24810 0.202715
\(951\) −20.4267 −0.662381
\(952\) −5.11156 −0.165667
\(953\) 42.5823 1.37937 0.689687 0.724107i \(-0.257748\pi\)
0.689687 + 0.724107i \(0.257748\pi\)
\(954\) −5.49632 −0.177950
\(955\) −21.0739 −0.681935
\(956\) 26.3200 0.851249
\(957\) 3.70386 0.119729
\(958\) 3.41665 0.110387
\(959\) 5.30877 0.171429
\(960\) −1.13475 −0.0366238
\(961\) −17.3795 −0.560628
\(962\) −2.03570 −0.0656335
\(963\) 16.0210 0.516271
\(964\) −9.57433 −0.308368
\(965\) 5.84784 0.188249
\(966\) 6.42545 0.206736
\(967\) 34.8942 1.12212 0.561062 0.827774i \(-0.310393\pi\)
0.561062 + 0.827774i \(0.310393\pi\)
\(968\) 1.00000 0.0321412
\(969\) 16.1127 0.517615
\(970\) 10.1623 0.326292
\(971\) −3.98812 −0.127985 −0.0639924 0.997950i \(-0.520383\pi\)
−0.0639924 + 0.997950i \(0.520383\pi\)
\(972\) 14.9857 0.480668
\(973\) −8.90074 −0.285345
\(974\) −23.2454 −0.744831
\(975\) 0.229246 0.00734175
\(976\) 1.59340 0.0510036
\(977\) −25.6626 −0.821019 −0.410510 0.911856i \(-0.634649\pi\)
−0.410510 + 0.911856i \(0.634649\pi\)
\(978\) −2.56118 −0.0818976
\(979\) 9.86267 0.315212
\(980\) 1.94102 0.0620035
\(981\) −26.0977 −0.833235
\(982\) −35.8534 −1.14413
\(983\) 0.124549 0.00397248 0.00198624 0.999998i \(-0.499368\pi\)
0.00198624 + 0.999998i \(0.499368\pi\)
\(984\) 9.78031 0.311785
\(985\) −4.39229 −0.139950
\(986\) 7.41783 0.236232
\(987\) 5.11095 0.162683
\(988\) 1.26226 0.0401580
\(989\) 0.905736 0.0288007
\(990\) 1.71235 0.0544220
\(991\) −41.6751 −1.32385 −0.661926 0.749569i \(-0.730261\pi\)
−0.661926 + 0.749569i \(0.730261\pi\)
\(992\) 3.69060 0.117177
\(993\) −5.84963 −0.185632
\(994\) −10.0607 −0.319105
\(995\) 18.1565 0.575598
\(996\) −13.1778 −0.417553
\(997\) 9.64175 0.305357 0.152679 0.988276i \(-0.451210\pi\)
0.152679 + 0.988276i \(0.451210\pi\)
\(998\) 17.9926 0.569546
\(999\) 53.8824 1.70476
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 8030.2.a.bd.1.9 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8030.2.a.bd.1.9 14 1.1 even 1 trivial